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+{"text":"\\section{Introduction}\nTwo important concepts in modern condensed matter physics are\ntopological and dynamical phase transitions. These two ideas differ\nfrom the traditional phenomenology of phase transitions which can be\ntraced back to Landau.\\cite{Landau1980} Key to the Landau\nclassification of phase transitions is the concept of order parameters\nindicative of symmetries broken across phase boundaries. Contrarily,\ntopological phase transitions separate regions with the same\nsymmetries but with different topological properties of the ground\nstate. Dynamical phase transitions---rather than focusing on the\nnon-analyticities which occur in the derivatives of the free\nenergy---concern non-analyticities which occur in dynamical\nquantities after a perturbation of the system.\n\nThe definition of a dynamical phase transition which we want to follow\nhere is based on the Loschmidt echo\\cite{Heyl2013} \n\\begin{equation}\n\\label{LE1}\nL(t)= \\langle \\Psi_0 | \\e^{-iH_1t} | \\Psi_0\\rangle.\n\\end{equation}\nHere $|\\Psi_0\\rangle$ is the initial state of the system before the\nquench and $H_1$ the time-independent Hamiltonian which induces the\nunitary time evolution of the system. The Loschmidt echo can be viewed\nas a `partition function' with fixed boundaries. Similar to the\ncanonical partition function\\cite{Fisher1965,Bena2005} there are\n`Fisher zeros' for complex times $t$ with the Loschmidt echo vanishing\nif the Fisher zeros are real or approach the real axis in the\nthermodynamic limit. For the Ising model in a transverse field the\nauthors of Ref.~\\onlinecite{Heyl2013} have shown that the Loschmidt\necho becomes zero at real critical times $t_c$ only for quenches\nacross the critical point, i.e. in cases where the initial state is\nthe ground state of the Hamiltonian on one side of the transition\nwhile the time-evolving Hamiltonian belongs to the other phase. In\nthis case there is therefore a direct connection between the\nequilibrium and the dynamical phase transition. In recent years,\ndynamical phase transitions have also been studied in a number of\nother\nmodels.\\cite{Mitra2012a,Karrasch2013,Heyl2014,Andraschko2014,Sharma2016,Karrasch2017,Gomez-Leon2017,Sedlmayr2017a}\nContrary to the transverse Ising model it has been found that, in\ngeneral, there is no connection between equilibrium and dynamical\nphase transitions: crossing an equilibrium phase transition does not\nnecessarily lead to zeros at real times in the Loschmidt echo while\nsuch zeros can also occur for quenches within the same\nphase.\\cite{Andraschko2014,Vajna2014} A special case are quenches in\nGaussian models with topological order. Here it has been shown that,\nunder certain conditions, a quench across a topological phase\ntransition is guaranteed to lead to dynamical phase transitions while\nthe opposite is not true.\\cite{Vajna2015} The phase of the Loschmidt\necho can then be used to define a dynamical topological order\nparameter which changes at critical times $t_c$.\\cite{Budich2016}\n\nA natural question to ask which we want to address in this manuscript\nis then: is there a dynamical analogue to the bulk boundary\ncorrespondence of equilibrium topological phase transitions? Our paper is organized as follows. In\nSec.~\\ref{Models} we introduce the class of models we will discuss. In\nSec.~\\ref{LE} we review known results for the Loschmidt echo and the\nreturn rate in the periodic case. We then present results of\nnumerical calculations of the return rate for open systems in\nSec.~\\ref{Numerics}. To understand the origins of the observed sudden\nchanges of the boundary contribution to the return rate at dynamical\nphase transitions we investigate the dynamical entanglement structure\nin Sec.~\\ref{Entangle}. Our results are summarized in\nSec.~\\ref{Concl}.\n\n\\section{Models}\n\\label{Models}\nWe focus here on one-dimensional (1D) models with symmetry protected\ntopological (SPT) phases. Following the ten-fold way symmetry\nclassification\\cite{Schnyder2008} we have three symmetry classes with\nground states labeled by a $\\mathbb{Z}$ topological invariant AIII,\nBDI, and CII; and two labeled by a $\\mathbb{Z}_2$ topological\ninvariant D, and DIII. The unitary particle-hole operator $\\C$ for\nBDI, D, and DIII obeys $\\{\\C,H\\}=0$ with $H$ the Hamiltonian of the\nsystem and $\\C^2=1$. The other two symmetries we require are the time\nreversal symmetries $\\mathcal{T}_\\pm$ satisfying\n$\\mathcal{T}_\\pm^2=\\pm1$. They must also anticommute with $\\C$. BDI\nhas additionally $[\\mathcal{T}_+,H]=0$ and a $\\mathbb{Z}$ invariant in\n1D. DIII has additionally $[\\mathcal{T}_-,H]=0$ and a $\\mathbb{Z}_2$\ninvariant in 1D. As $\\mathcal{T}_-$ is the time reversal symmetry of\nthe electrons, DIII has Kramer's pairs. D, with a $\\mathbb{Z}_2$\ninvariant in 1D, has no additional symmetry beyond particle-hole. If\nboth $\\mathcal{T}_\\pm$ symmetries are present the system is best\nthought of as in BDI with an additional TR symmetry protecting the\nKramer's pairs and this will have a $\\mathbb{Z}$ invariant in\n1D.\\cite{Schnyder2008,Sedlmayr2015a}\n\nWe will use examples principally in the BDI class, which possesses\nboth a unitary `time reversal symmetry', and a particle hole\nsymmetry. It therefore also possesses chiral or sublattice\nsymmetry. The topological superconductors in which Majorana bound\nstates are sought all belong to either BDI, D, or\nDIII.\\cite{Kitaev2001,Lutchyn2010,Oreg2010,Alicea2011,Mourik2012,Wong2012}\n\nFor concreteness we consider 1D Hamiltonians, which after a Fourier\ntransform on a periodic lattice, are of the form\n\\begin{equation}\n\\label{1dham}\nH=\\sum_{ k}\\Psi^\\dagger_{ k}\\mathcal{H}( k)\\Psi_{ k} \\textrm{ with } \\mathcal{H}(k)=\\mathbf{d}_k \\cdot {\\bm\\tau}\\,,\n\\end{equation}\nwhere ${\\bm\\tau}$ is the vector of Pauli matrices acting in some\nsubspace and $\\Psi_{ k}$ are the appropriate operators for that\nsubspace. This will be particle-hole space for examples such as the\nKitaev chain, which is a topological superconductor, or a unit-cell\nsubspace for the Su-Schrieffer-Heeger (SSH) chain. These two examples\nwill be those we focus on and we will introduce them in more detail\nbelow. In general $\\mathbf{d}_k=(d^x_k,d^y_k,d^z_k)$, and\ndiagonalizing $\\mathbf{d}_k\n\\cdot {\\bm\\tau}$ one can find $\\mathbf{\\tilde d}_k \\cdot\n{\\bm\\tilde\\tau}$ with $\\mathbf{\\tilde d}_k=(0,0,\\epsilon_k)$ and such\nHamiltonians have pairs of eigenenergies $\\pm \\epsilon_k$, a result of\nthe particle-hole symmetry of the Hamiltonians we consider.\n\n\\subsection{SSH model}\n\\label{SSH}\nThe first example we will consider is the SSH model with open or\nperiodic boundary conditions,\n\\begin{equation}\n\\label{ssh_model}\nH=-J\\sum_{j}\\left[(1+\\delta\\e^{i\\pi j})c^\\dagger_jc_{j+1}+\\textrm{H.c.}\\right]\\,.\n\\end{equation}\nHere $J$ is the nearest-neighbor hopping amplitude, $\\delta$ is the\ndimerization, and $c^\\dagger_j$ is the creation operator on site\n$j$. For periodic boundary conditions the sum is taken up to site $N$\nand $c_{N+1}=c_N$ while the sum is taken up to $N-1$ for open boundary\nconditions. The main reason to consider this specific model first, is\nthat an exact solution for open boundary conditions\nexists\\cite{Shin1997,Sirker2014a} which depends on a set of parameters\ndetermined by non-linear equations. As a result, numerically accurate\ndata for very large open systems can be easily obtained. The system is\ntopologically non-trivial for $\\delta>0$. The particle-hole symmetry\nis then $\\psi_j\\to\\textrm{i}\\psi_j^\\dagger$ and $T_+$ is\n$\\psi_j\\to(-1)^j\\psi_j^\\dagger$. Note that the phase of $\\C$ must be\nfixed such that $\\{\\C,\\T_+\\}=0$.\n\nFor periodic boundary conditions the Hamiltonian can be easily diagonalized. Firstly after a Fourier transform and a convenient rotation,\n\\begin{equation}\n\\Psi^\\dagger_k=\\sqrt{\\frac{2}{N}}\\sum_{j=1}^{N\/2}\\e^{\\textrm{i} 2kj}\\begin{pmatrix}\n1&0\\\\0&\\e^{\\textrm{i} 2k}\n\\end{pmatrix}\\underbrace{\\begin{pmatrix}\nc_{2j-1}\\\\\nc_{2j}\n\\end{pmatrix}}_{=\\Psi_j}\\,,\n\\end{equation}\nthe Hamiltonian takes the form \\eqref{1dham} with\n\\begin{equation}\n\\label{d_SSH}\n\\mathbf{d}_k=\\begin{pmatrix}\n-2J\\cos[k],2J\\delta\\sin[k],0\n\\end{pmatrix}\\,,\n\\end{equation}\nwhich can be readily diagonalized. The momenta are $k=2\\pi n\/N$ with\n$n=1,2,\\ldots N\/2$. The particle-hole symmetry is now\n$\\C=\\e^{\\textrm{i}\\pi\/2}{\\bm \\tau}^x\\hat K$ and $\\T_+=\\hat K$, where $\\hat K$ is the complex conjugation operator. \n\nAlthough this model has a $\\mathbb{Z}$ winding number, the values are\nnevertheless confined to be either 0 or 1. Extensions of this model\nin the same symmetry class but with a higher winding number are however\npossible.\\cite{Rice1982}\n\n\\subsection{Long-range Kitaev chain}\n\\label{Kitaev}\nAs our second example we will consider the Kitaev chain of $M$\nsites with long-range hopping terms,\\cite{Kitaev2001} \n\\begin{eqnarray}\\label{khlr}\nH&=&\\sum_{i,j}\\Psi^\\dagger_{i}\\left(\n\\Delta_{|i-j|}\\textrm{i}{\\bm\\tau}^y-J_{|i-j|}{\\bm\\tau}^z\\right)\\Psi_{j+1}+\\textrm{H.c.}\\nonumber\\\\&&-\\mu\\sum_{j}\\Psi^\\dagger_{j}{\\bm\\tau}^z\\Psi_{j}\\,,\n\\end{eqnarray}\nwith open or periodic boundary conditions. For the periodic case we\nhave $\\Psi_{M+1}=\\Psi_{1}$.  The operators in particle-hole space are\ngiven by $\\Psi^\\dagger_{j}=(c^\\dagger_{j},c_{j})$, and $c_{\nj}^{(\\dagger)}$ annihilates (creates) a spinless fermionic particle at\na site $j$. In this case we have again $\\C=\\e^{\\textrm{i}\\pi\/2}{\\bm\n\\tau}^x\\hat K$, $\\T_+=\\hat K$, and a Fourier transform brings the\nHamiltonian into the form of Eq.~\\eqref{1dham} where\n$\\Psi^\\dagger_k=(c^\\dagger_k,c_{-k})$ and\n\\begin{equation}\n\\label{d_Kit}\n\\mathbf{d}_k=\\sum_{m=1}^3 \\begin{pmatrix}\n-2J_m\\cos[mk]-\\mu\/3,2\\Delta_m\\sin[mk],0\n\\end{pmatrix}\\,.\n\\end{equation}\nThe long-range hopping has been truncated here at a distance of 3\nsites and we define $\\vec J=(J_1,J_2,J_3)$ and\n$\\vec\\Delta=(\\Delta_1,\\Delta_2,\\Delta_3)$ . Note that contrary to the\nSSH model phases with higher winding numbers in $\\mathbb{Z}$ exist,\nallowing for a more general investigation of quenches between\ntopological phases with different invariants, and therefore also\ndifferent numbers of boundary states.  The momenta are $k=2\\pi n\/M$\nwith $n=1,2,\\ldots M$ and the total system size is $N=2M$.\n\n\\section{The Loschmidt echo and return rate}\n\\label{LE}\nIn this section we will define the quantities studied throughout the following,\nand review results for periodic boundary conditions. \nThe initial state $|\\Psi_0\\rangle$ in Eq.~\\eqref{LE1} is the many-body\nground state of the initial Hamiltonian $H_0$ before the quench. The\nunitary time evolution is then determined by the Hamiltonian\n$H_1$. The Loschmidt echo in a translationally invariant system of the\nform of Eq.~\\eqref{1dham} can be easily calculated\\cite{Quan2006} as\nthe momentum $k$ remains a good quantum number during the quench. One\nfinds\n\\begin{equation}\\label{pbcloschmidt}\nL(t)=\\prod_k\\left[\\cos(\\epsilon^1_kt)+\\textrm{i}\\hat{\\mathbf{d}}^0_k\\cdot\\hat{\\mathbf{d}}^1_k\\sin(\\epsilon^1_kt)\\right]\\,,\n\\end{equation}\nwith\n$\\hat{\\mathbf{d}}^{0,1}_k=\\mathbf{d}^{0,1}_k\/\\sqrt{\\mathbf{d}^{0,1}_k\\cdot\\mathbf{d}^{0,1}_k}$\nand $\\mathbf{d}^{0,1}_k$ being the parameter vector in the\nHamiltonian \\eqref{1dham} before and after the quench respectively. The\nproduct in $k$ is over all filled states of the lower band.\n\nMore generally, for any free fermion system the Loschmidt echo can\nalways be obtained from the single-particle correlation matrix defined\nby\n\\begin{equation}\n\\label{Cij}\n\\C_{ij}=\\langle\\Psi_0|\\Psi^\\dagger_i\\Psi_j|\\Psi_0\\rangle \\, .\n\\end{equation}\nHere $i$ and $j$ run over all lattice sites. The Loschmidt echo in\nterms of the correlation matrix is given\nby\\cite{Levitov1996,Klich2003,Rossini2007}\n\\begin{equation}\n\\label{rle}\nL(t)=\\det\\mathbf{M}\\equiv\\det\\left[1-\\mathbf{\\C}+\\mathbf{\\C}\\e^{\\textrm{i} {\\bm H}_1t}\\right]\\,.\n\\end{equation}\nHere ${\\bm H}_1$ is the Hamiltonian matrix written in the same basis\nas $\\C$. We will use \\eqref{rle} to calculate $L(t)$ in the open\nboundary case and call $\\mathbf{M}$ the {\\it Loschmidt matrix} in the\nfollowing. Eq.~\\eqref{pbcloschmidt} is easily recovered for periodic\nboundary conditions by transforming to momentum space and by using the\neigenbasis of the initial (momentum resolved) Hamiltonian.\n\nIn a many-body system we expect, in general, an orthogonality\ncatastrophe: the overlap between the initial state and the states in\nthe time evolution will become exponentially small in system size\n$N$. It is therefore useful to define the return rate as\n\\begin{equation}\n\\label{return}\nl(t)=-\\frac{1}{N}\\ln|L(t)|\\,.\n\\end{equation}\nThe non-analytic points of the return rate are determined by the zeros\nof the Loschmidt echo.\\cite{Heyl2013} Fisher zeros in the complex\nplane occur in the translationally invariant case at\ntimes\\cite{Vajna2015}\n\\begin{equation}\n\\label{tn1}\nt_n(k)=\\frac{\\pi}{\\epsilon^1_k}\\left(n+\\frac{1}{2}\\right)+\\frac{i}{\\epsilon^1_k}\\tanh^{-1}\\left(\\hat{\\mathbf{d}}^0_k\\cdot\\hat{\\mathbf{d}}^1_k\\right)\\,\n\\end{equation}\nwit $n$ being an integer. These zeros lie on the real axis and\ntherefore give rise to non-analytic behavior for the return rate at critical\ntimes\n\\begin{equation}\n\\label{tn2}\nt_n=\\frac{\\pi}{2\\epsilon^1_{k^*}}\\left(2n-1\\right)\\,, \\textrm{ where  }n\\in\\mathbb{Z}\\,,\n\\end{equation}\nif a critical momentum $k^*$ exists with\n\\begin{equation}\n\\label{tn3}\n\\hat{\\mathbf{d}}^0_{k^*}\\cdot\\hat{\\mathbf{d}}^1_{k^*}=0 \\, .\n\\end{equation}\nThis is the condition for the vanishing of the imaginary part in\nEq.~\\eqref{tn1}. We introduce the critical time scale\n$t_c=\\pi\/(2\\epsilon^1_{k^*})$. Where multiple critical times exist, we\ntake the smallest critical time to be the timescale $t_c$.\n\n\\section{Boundary contributions to the return rate}\n\\label{Numerics}\nIn this study we are interested in the boundary contributions to the\nreturn rate \\eqref{return} for systems with open boundaries. In the\nlarge $N$ limit we can expand the return rate as\n\\begin{equation}\n\\label{bbreturn}\nl(t)\\sim l_0(t)+\\frac{l_{B}(t)}{N}\\,.\n\\end{equation}\nHere $l_0$ is the bulk contribution which is equivalent to the return\nrate in the thermodynamic limit for periodic boundary\nconditions. $l_B$ is the boundary contribution which contains information about the topologically protected edge states, as we will demonstrate  in the following.\n\n\\subsection{Finite-size scaling}\n\\label{scaling}\nThe most straightforward approach to find $l_B(t)$ is a numerical\ncalculation of the correlation matrix \\eqref{Cij} followed by a finite\nsize scaling analysis of the return rate, Eq.~\\eqref{bbreturn}. We\nwill discuss the results of such an approach here for both our\nexamples, the SSH and the long-range Kitaev chain.\n\n\\subsubsection{SSH chain}\nWe start with the SSH chain where the semi-analytical solution for\nopen boundary conditions allows one to obtain highly accurate results\nfor very large systems. We consider quenches between the topologically\nordered phase with edge states ($\\delta>0$) and the topologically\ntrivial phase without edge states ($\\delta<0$) in the half-filled\ncase. If we perform a symmetric quench $\\delta\\to -\\delta$ then the\ndirection of the quench does not matter for the bulk contribution\n$l_0(t)$ shown in Fig.~\\ref{Fig1} as is obvious from\nEq.~\\eqref{pbcloschmidt}.\n\\begin{figure}[!ht]\n\\includegraphics*[width=0.99\\linewidth]{Figure01.pdf}\n\\caption{Bulk contributions for symmetric quenches $\\delta\\to -\\delta$ in the SSH model for $\\delta=0.3$ and $\\delta=0.95$.\n}\n\\label{Fig1}\n\\end{figure}\nIn both examples cusps in the bulk return rate are present at the\ncritical times determined by Eqs.~(\\ref{tn2}, \\ref{tn3}).\n\nThe direction of the quench does, however, strongly affect the\nboundary contribution $l_B(t)$. For a quench from the trivial into the\ntopological phase we find a boundary contribution $l_B(t)$ which shows\nlarge jumps at the critical times, see Fig.~\\ref{Fig2}.\n\\begin{figure}[!ht]\n\\includegraphics*[width=0.99\\linewidth]{Figure02.pdf}\n\\caption{Boundary contribution to the return rate $l_B(t)$ for symmetric quenches from \nthe trivial into the topological phase. $l_B(t)$ is extracted from a\n$1\/N$ scaling analysis of chains of up to $N=2200$ sites.}\n\\label{Fig2}\n\\end{figure}\nFor a quench from the topological into the trivial phase we also\nobserve jumps in $l_B(t)$ at $t_c$ as shown in Fig.~\\ref{Fig3},\nhowever, the jumps are more than two orders of magnitude smaller than\nfor the quench in the opposite direction.\n\\begin{figure}[!ht]\n\\includegraphics*[width=0.99\\linewidth]{Figure03.pdf}\n\\caption{Boundary contribution to the return rate $l_B(t)$ for symmetric quenches from \nthe topological into the trivial phase. $l_B(t)$ is extracted from a\n$1\/N$ scaling analysis of chains of up to $N=2200$ sites. Note the\ndifferent scale of $l_B(t)$ compared to the quenches in\nFig.~\\ref{Fig2}. The top panel shows a close-up of $l_B(t)$ for the quench from $\\delta=0.95$ to $\\delta=-0.95$.}\n\\label{Fig3}\n\\end{figure}\nThe data in the two figures clearly point to a bulk-boundary\ncorrespondence: At the same critical times where the bulk contribution\nshows cusps and the bulk dynamical winding number\nchanges,\\cite{Budich2016} the boundary contribution also shows\ndiscontinuities. The dependence on the direction of the quench\nfurthermore suggests, that the boundary contribution $l_B(t)$ is\nstrongly affected by the presence or absence of symmetry protected\nedge states in the final Hamiltonian. That the boundary contribution\nis directly related to the edge states can be seen from the\ntime-dependent occupation of these states, see Fig.~\\ref{Fig4}.\n\\begin{figure}[!ht]\n\\includegraphics*[width=0.99\\linewidth]{Figure04.pdf}\n\\caption{Occupation of the edge modes for a symmetric quench at $|\\delta|=0.95$ from the topological to the trivial phase for a system of size $N=80$.}\n\\label{Fig4}\n\\end{figure}\nFor a system with $N\/2$ spinless fermions in the topological phase,\none of the edge modes is filled at time $t=0$ while the other one is\nempty. At the critical times $t_c$, where the cusps in $l_0(t)$ occur,\nboth edge modes are approximately half-filled. In the remainder of the\npaper we will investigate the relation between the edge modes and the\nsingularities in the Loschmidt echo in more detail.\n\n\\subsubsection{Long-range Kitaev chain}\nBefore doing so, we will first present numerical results for $l_B(t)$\nfor the other model system we study here, the long-range Kitaev\nchain. Contrary to the SSH model, this chain has different topological\nphases characterized by an integer winding number\n$\\nu_{\\textrm{w}}$. Dynamical phase transitions are expected for any\nquench between phases with different winding numbers. In\nFig.~\\ref{Fig5} the bulk return rate is shown for two examples. \n\\begin{figure}[!ht]\n\\includegraphics*[width=0.99\\linewidth]{Figure05.pdf}\n\\caption{Bulk return rate for quenches in the \nlong-range Kitaev model from (a) $\\nu_{\\textrm{w}}=1$ to $\\nu_{\\textrm{w}}=3$ and (b) $\\nu_{\\textrm{w}}=1$ to $\\nu_{\\textrm{w}}=-1$. The gray and dashed red lines show the positions of the critical times.}\n\\label{Fig5}\n\\end{figure}\nIn the case shown in Fig.~\\ref{Fig5}(a) the quench is from\n$\\nu_{\\textrm{w}}=1$ with $\\vec J=(1,-2,2)$, $\\mu=2$, and\n$\\vec\\Delta=(1.3,-0.6,0.6)$ to $\\nu_{\\textrm{w}}=3$ with $\\vec\nJ=(1,-2,2)$, $\\mu=0.1$, and $\\vec\\Delta=(0.45,-0.9,1.35)$. Two\ncritical momenta exist leading to two distinct critical times at which\nDPT's occur. In Fig.~\\ref{Fig5}(b) a quench from $\\nu_{\\textrm{w}}=1$\nwith $\\vec J=(1,-2,-2)$, $\\mu=3$, and $\\vec\\Delta=(1.3,-0.6,0.6)$ to\n$\\nu_{\\textrm{w}}=-1$ with $\\vec J=(1,-2,-2)$, $\\mu=3$, and\n$\\vec\\Delta=-(1.3,-0.6,0.6)$ is considered. Cusps in the return rate\nare again clearly visible, demonstrating that DPT's also occur for\nquenches where only the sign of the winding number changes.\n\nBy a finite size scaling analysis of chains up to $N=1600$, we\nhave also extracted the boundary contribution. As an example we show\n$l_B(t)$ for the same quench as in Fig.~\\ref{Fig5}(a) in\nFig.~\\ref{Fig6}(a), and for the reverse quench in Fig.~\\ref{Fig6}(b). Note that contrary to the SSH model an analytic\nsolution for the eigensystem is not known so that we cannot increase\nthe system size till a clear scaling also for times very close to\n$t_c$ emerges. Finite-size corrections are still present near\nDPT's. Furthermore, the edge states have energies which are\nexponentially close to zero so that an exact diagonalization in\nmulti-precision is required.\n\\begin{figure}[!ht]\n\\includegraphics*[width=0.99\\linewidth]{Figure06.pdf}\n\\caption{Boundary contribution $l_B(t)$ in the long-range Kitaev model for (a)\nthe same quench as in Fig.~\\ref{Fig5}(a) from $\\nu_{\\textrm{w}}=1$ to\n$\\nu_{\\textrm{w}}=3$, and (b) the reverse quench from\n$\\nu_{\\textrm{w}}=3$ to $\\nu_{\\textrm{w}}=1$. $l_B(t)$ is extracted from a\n$1\/N$ scaling analysis of chains of up to $N=1600$. The gray and dashed red\nlines show the positions of the critical times.}\n\\label{Fig6}\n\\end{figure}\nWhile the obtained data are therefore not as accurate as for the SSH\nchain, we nevertheless observe a behavior which is qualitatively\nsimilar. At the critical times $t_c$ the boundary contribution shows\njumps. However, here the jumps are of similar magnitude for both\nquench directions.\n\n\\subsection{Loschmidt eigenvalues}\nIn order to understand the origin of the discontinuous boundary\ncontribution we next investigate the spectrum of the dynamical\nLoschmidt matrix $\\mathbf{M}$ defined in Eq.~\\eqref{rle}.\n\nWe concentrate first on symmetric quenches in the SSH model for large\ndimerizations $\\delta$ where the structure of the spectrum is\nparticularly simple. In Fig.~\\ref{Fig7} the spectrum of the matrix\n$\\mathbf{M}$ for a quench from the trivial into the topological phase\nis shown.\n\\begin{figure}[!ht]\n\\includegraphics*[width=0.99\\linewidth]{Figure07.pdf}\n\\caption{Absolute value [black circles] and argument [green diamonds] of the eigenvalues of the matrix $\\mathbf{M}$ for a symmetric quench at $|\\delta|=0.95$ from the trivial into the topological phase for a system of size $N=80$.}\n\\label{Fig7}\n\\end{figure}\nAt the first critical time $t_c$ there are two eigenvalues\n$\\lambda_{1,2}$ whose absolute values become exponentially small in\nsystem size. Between $t_c$ and $3t_c$ these eigenvalues stay close to\nzero and this structure repeats itself in time. At $t_c$, the argument\nof the eigenvalues also changes abruptly from zero to $\\pm\\pi$. These\ntwo eigenvalues are related to the edges of the system as a comparison\nto the case with periodic boundary shows where they are absent.\n\nThe return rate in terms of the eigenvalues $\\lambda_i$ of the matrix\n$\\mathbf{M}$ is given by\n\\begin{equation}\n\\label{EVs}\nl(t)=-\\frac{1}{N}\\sum_{j=1}^N \\ln |\\lambda_j| \\,.\n\\end{equation}\nWe can isolate the contribution to $l(t)$ from $\\lambda_{1,2}$, the two eigenvalues which periodically approach\nzero:\n\\begin{equation}\n\\label{2EVs}\n\\Lambda\\equiv-\\frac{1}{N}\\ln |\\lambda_1\\lambda_2|\\,.\n\\end{equation}\nAt a particular system size $N$, $\\Lambda$ reproduces the same  behavior as the finite size boundary contribution which we define as $l(t)-l_0(t)$, see\nFig.~\\ref{Fig8}. It is the principle \n source of the large boundary contribution for $t\\in[t_c,3t_c]$.\n\\begin{figure}[!ht]\n\\includegraphics*[width=0.99\\linewidth]{Figure08.pdf}\n\\caption{The boundary contribution to the return rate $l(t)-l_0(t)$ [red circles] compared to $\\Lambda$ [green triangles], the contribution to the return rate from\nthe two eigenvalues which are close to zero for $t\\in [t_c,3t_c]$, see Fig.~\\ref{Fig7}. Shown for a system of\nsize $N=500$.}\n\\label{Fig8}\n\\end{figure}\n\nAt a DPT the system thus not only changes back and forth between the\ntrivial phase with dynamical winding number $\\nu=0$ and the\ntopological one with $\\nu=1$\\cite{Budich2016} there is also---at the\nsame time---a transition in the edge degrees of freedom. The presence\nor absence of a pair of zero eigenvalues of the dynamical matrix\n$\\mathbf{M}$ can serve as an equivalent order parameter to the bulk\nwinding number, establishing a concrete bulk-boundary correspondence\nin this case.\n\nFor the quench from the topological to the trivial phase we obtain a\nslightly different picture. While at the critical times there is still\na pair of eigenvalues which approach zero, the eigenvalues no longer\nremain close to zero for $t\\in [t_c,3t_c]$, see Fig.~\\ref{Fig9}.\n\\begin{figure}[!ht]\n\\includegraphics*[width=0.99\\linewidth]{Figure09.pdf}\n\\caption{Absolute value [black circles] and argument [green diamonds] of the eigenvalues of the matrix $\\mathbf{M}$ for a symmetric quench at $|\\delta|=0.95$ from the topological into the trivial phase for a system of size $N=80$.}\n\\label{Fig9}\n\\end{figure}\nThe direction of the quench can thus clearly be distinguished from the\nspectrum of $\\mathbf{M}$. From the almost symmetric spectrum around\n$t_c$ for the topological to trivial quench it is in particular clear\nthat the boundary contribution is of similar magnitude on both sides\nof the transition consistent with the scaling results shown in\nFig.~\\ref{Fig3}.\n\nSo far we have investigated quenches for large dimerizations\n$|\\delta|\\lesssim 1$ where the system is almost perfectly dimerized\nand the Loschmidt spectrum is particularly simple. A natural question\nto ask is whether or not the spectrum is still useful to detect if\nedge states are present in the final Hamiltonian for smaller\ndimerizations. To this end, we present in Fig.~\\ref{Fig10} data for\nsymmetric quenches at $|\\delta|=0.3$.\n\\begin{figure}[!ht]\n\\includegraphics*[width=0.99\\linewidth]{Figure10.pdf}\n\\caption{Absolute value [black circles] of the smallest 20\neigenvalues of the matrix $\\mathbf{M}$ for a symmetric quench at\n$|\\delta|=0.3$ from (a) the trivial to the topological phase, and (b)\nvice versa, for a system of size $N=160$.}\n\\label{Fig10}\n\\end{figure}\nWhile the spectrum becomes more complex, the direction of the quench\nis still obvious. In particular, a pair of eigenvalues close to zero\nfor $t\\in [t_c,3t_c]$ persists for the trivial to topological quench.\n\nIn the long range Kitaev model where we are able to consider quenches\nbetween more general winding numbers a similar structure is seen in\nthe spectrum of $\\mathbf{M}$, see Fig.~\\ref{Fig11}. Although finite\nsize effects are still present, it is already obvious that the results\nare again very different for the two quench directions. Note that for\nthe quench from $\\nu_{\\textrm{w}}=1$ to $\\nu_{\\textrm{w}}=3$ the time\nevolving Hamiltonian, $H_1$, has two additional pairs of edge states,\ncompared to the initial Hamiltonian, $H_0$.\n\\begin{figure}[!ht]\n\\includegraphics*[width=0.99\\linewidth]{Figure11.pdf}\n\\caption{Absolute value of the smallest four eigenvalues of the matrix $\\mathbf{M}$ for the quenches in Fig.~\\ref{Fig6}, for (a) $\\nu_{\\textrm{w}}=1$ to $\\nu_{\\textrm{w}}=3$, and (b) vice versa, for a system of size $N=600$. Eigenvalues are colored as an aid to the eye.}\n\\label{Fig11}\n\\end{figure}\nFor the quench where the number of edge states increases\n[Fig.~\\ref{Fig11}(a)] we find, in particular, an extended time\ninterval between critical times where $2$ or $4$ eigenvalues are\nzero. In fact two eigenvalues are close to zero for $t\\in[t_c,3t_c]$\nand two different eigenvalues are close to zero for\n$t\\in[t'_c,3t'_c]$, where $t'_c$ is the larger of the two critical\ntimes for this quench. For the opposite direction, on the other hand,\nwe find a roughly symmetric structure around the first two critical\ntimes very similar to the SSH case. Note that finite size effects\nstrongly influence the results for $t\/t_c\\gtrsim 4$. Compatible with\nthe data for the SSH and the Kitaev chain is thus the idea that for\nthe boundary contribution $l_B(t)$ and for the Loschmidt spectrum,\n$\\text{spec}(\\mathbf{M})$, it is important how many more or fewer edge\nstates are present for $H_1$ as compared to $H_0$.\n\n\\section{Long-range entanglement}\n\\label{Entangle}\nWe have established numerically a bulk-boundary correspondence and\ncould show that the spectrum of $\\mathbf{M}$ contains information\nabout the edge states. However, the spectrum of this matrix is not an\neasily measurable quantity and also does not provide a physical\npicture about what happens to the edge states during the time\nevolution. In this section we therefore want to connect the changes on\nthe Loschmidt echo with the dynamical entanglement properties of the\nsystem. To this end, we consider the entanglement between the two\nhalves of an open chain with an even number of sites. The entanglement\nentropy is defined as the von-Neumann entropy of a reduced density\nmatrix\n\\begin{equation}\n S_{\\textrm{ent}}(t) =-\\Tr \\{\\rho_A(t) \\ln \\rho_A(t)\\}\\,\n\\end{equation}\nwith $\\rho_A(t)=\\Tr_B |\\Psi(t)\\rangle\\langle\\Psi(t)|$ and\n$|\\Psi(t)\\rangle = \\text{e}^{-iH_1t}|\\Psi_0\\rangle$ being the\ntime-evolved state. The system has been divided up into two blocks $A$\nand $B$ of equal size. Here we will focus on the SSH model.\n\nFor a Gaussian model the entanglement between a subsystem and the rest\ncan be calculated from the correlation matrix $\\mathbf{\\C}(t)$ defined\nin Eq.~\\eqref{Cij} with the now time-dependent two-point correlations\nrestricted to lattice sites within the subsystem.\\cite{Peschel2003}\nThe entanglement entropy is then given by\n\\begin{equation}\n\\label{Sent}\nS_{\\textrm{ent}} = -\\sum_j \\left[ \\eta_j\\ln\\eta_j +(1-\\eta_j)\\ln(1-\\eta_j)\\right] \\, \n\\end{equation}\nwith $\\eta_j$ being the eigenvalues of $\\mathbf{\\C}(t)$.\n\nIn Fig.~\\ref{Fig12} the entanglement entropy for a symmetric quench\nfrom the trivial into the topological phase at large $|\\delta|$ is\nshown. \n\\begin{figure}[!ht]\n\\includegraphics*[width=0.99\\linewidth]{Figure12.pdf}\n\\caption{$S_{\\textrm{ent}}(t)$ between two halves of the system for a symmetric quench at $|\\delta|=0.95$ from the trivial to the topological phase with $N=32$.}\n\\label{Fig12}\n\\end{figure}\n$S_{\\textrm{ent}}(t)$ is showing oscillations with local maxima\nlocated exactly at the DPT's. For short times, in particular,\n$S_{\\textrm{ent}}(t_c)\\approx 2\\ln 2$ while at longer times the\nentanglement entropy on average starts to increase linearly as is\nexpected for a global quench. These observatios can be explained as\nfollows: In the strongly dimerized case, each dimer bond between the\ntwo subsystems is in a fully entangled state, $(|10\\rangle \\pm\n|01\\rangle)\/\\sqrt{2}$, and contributes $\\ln 2$ to the entanglement\nentropy. The data in Fig.~\\ref{Fig12} therefore indicate that there\nare two dimer bonds between the subsystems at the critical times\n$(2n+1)t_c$. At times $2nt_c$, on the other hand, the two subsystems\nare almost completely disentangled. This picture is confirmed by\ndirectly considering the two-point correlations in the system as a\nfunction of time, see Fig.~\\ref{Fig13}.\n\\begin{figure}[!ht]\n\\includegraphics*[width=0.99\\linewidth]{Figure13.pdf}\n\\caption{Time evolution of two-point correlations for the same quench as in Fig.~\\ref{Fig12} for a system with $N=16$ sites and times $t=0,t_c,2t_c,3t_c$. The opacity indicates the strength of the correlation and the dashed gray line is the cut between the two subsystems.}\n\\label{Fig13}\n\\end{figure}\nThe system starts in a topologically trivial, strongly dimerized state\n$|\\Psi_0\\rangle=|\\Psi(t=0)\\rangle$ and thus\n$S_{\\textrm{ent}}(t=0)\\approx 0$. At the critical time $t_c$, two\ndimer bonds have formed which cross the cut between the subsystems\nleading to a $S_{\\textrm{ent}}(t_c)\\approx 2\\ln 2$. For short times,\nthe system oscillates between these two configurations with other\ncorrelations slowly starting to build up and finally leading to a\nlinearly increasing entanglement entropy. Note that for a finite\nsystem this linear increase will be cut off at\n$S^{\\textrm{max}}_{\\textrm{ent}}=\\frac{N}{2}\\ln 2$.\n\nFor the quench in the opposite direction, shown in Fig.~\\ref{Fig14},\n$S_{\\textrm{ent}}$ also shows oscillations with a frequency set by the\ncritical time $t_c$.\n\\begin{figure}[!ht]\n\\includegraphics*[width=0.99\\linewidth]{Figure14.pdf}\n\\caption{$S_{\\textrm{ent}}(t)$ between two halves of the system for a symmetric quench at $|\\delta|=0.95$ from the topological to the trivial phase with $N=32$.}\n\\label{Fig14}\n\\end{figure}\nIn this case $S_{\\textrm{ent}}(t=0)\\approx 2\\ln 2$ and for long times\nthe entanglement entropy shows again the expected linear increase. To\nunderstand this behavior it is again instructive to consider the time\nevolution of the two-point correlations, see Fig.~\\ref{Fig15}.\n\\begin{figure}[!ht]\n\\includegraphics*[width=0.99\\linewidth]{Figure15.pdf}\n\\caption{Time evolution of two-point correlations for the same quench as in Fig.~\\ref{Fig14} for a system with $N=16$ sites and times $t=0,t_c,2t_c,3t_c$. The opacity indicates the strength of the correlation and the dashed gray line is the cut between the two subsystems. Long-range entanglement between the edges of the system is present at all times.}\n\\label{Fig15}\n\\end{figure}\nAt time $t=0$ a nearest-neighbor dimer and a dimer between the two\nedge sites is cut. Such long-range entanglement between the boundaries\nof an open SSH chain has been discussed previously in\nRef.~\\onlinecite{CamposVenuti2007a}. Interestingly, the long-range\nentanglement persists at all times during the unitary time evolution\ndespite the strong quench perturbing the system. While at times\n$2nt_c$ the two edge sites are strongly entangled, the long-range\nentanglement moves to the sites one removed from the edge for times\n$(2n+1)t_c$, see Fig.~\\ref{Fig15}. The different entanglement\nstructure in the two cases explains the oscillations seen in the\nentanglement entropy shown in Fig.~\\ref{Fig14} while the slow build-up\nof additional correlations explains the linear increase of\n$S_{\\textrm{ent}}(t)$ at longer times.\n\n\\section{Conclusions}\n\\label{Concl}\nIn this paper we have studied dynamical phase transitions in open\nchains with symmetry protected topological phases. Specifically, we\nhave concentrated on two examples: the SSH chain and a long-range\nKitaev model. In both cases we have shown that for a quench between\ndifferent topological phases there is not only a cusp in the bulk\nreturn rate but also a jump in the boundary ($1\/N$) contribution. In\ncontrast to the bulk part, the boundary return rate $l_B(t)$ is\nsensitive to the direction of the quench.  For the SSH model, in\nparticular, we found that the jump in $l_B(t)$ at a DPT is orders of\nmagnitude larger for a quench from the trivial to the topological\nphase than in the other direction.  A clear qualitative difference\nbetween the quench directions can also be seen in the Loschmidt\neigenvalue spectrum: While for the quench into the topological phase\ntwo eigenvalues which behave very differently on both sites of the DPT\nare largely responsible for the boundary contribution, the spectrum\nnear a DPT is almost symmetric for a quench in the opposite direction.\n\nThe critical times $t_c$ at which DPT's occur are also clearly visible\nas oscillations in the entanglement entropy between two halves of the\nSSH chain. We found that these oscillations can be explained by the\ntime-dependent structure of two-point correlations. At short times and\nlarge dimerizations the system oscillates between two configurations\nof correlations. Starting from the topological phase we discussed, in\nparticular, the long-range entanglement which is transferred from the\nboundary sites onto the sites one removed from the boundary going from\ntimes $2nt_c$ to $(2n+1)t_c$. Quite surprisingly, the long-range\nentanglement in this case remains stable despite the fact that the\nquench introduces a large perturbation to the system.\n\n\\acknowledgments\nJS acknowledges support by the Natural Sciences and Engineering\nResearch Council (NSERC, Canada) and by the Deutsche\nForschungsgemeinschaft (DFG) via Research Unit FOR 2316. Support for this research at Michigan State University (NS) was provided by the Institute for Mathematical and Theoretical Physics with funding from the office of the Vice President for Research and Graduate Studies.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn the last few years, it has become increasingly clear that SN~1987A\nin the Large Magellanic Cloud (LMC) was a remarkable, but highly\nunusual event. One of the major surprises of SN~1987A was the fact\nthat the star that exploded was a blue supergiant rather than a red\nsupergiant as had been predicted. While there have been many attempts\nin the early years to explain the blue-supergiant progenitor within\nthe framework of single stellar evolution theory (e.g., Woosley,\nPinto \\& Ensman 1988; for a review see Podsiadlowski 1992), these\nmodels are no longer viable with the best, up-to-date input physics\n(Woosley 1997). In particular, the recent large increase in stellar\nopacities (Rogers \\& Iglesias 1992) had as an immediate consequence\nthat there are no longer any plausible parameters for which a massive\nstar first becomes a red supergiant and then experiences a late blue\nloop after helium core burning (Woosley 1997). Of course, even if such\nmodels could be constructed, they still would not be promising models\nfor the progenitor of SN~1987A, since they still could not explain any\nof the other major anomalies of this event: the complex triple-ring\nnebula surrounding the progenitor and the various chemical anomalies\n(see Podsiadlowski 1992 and section~3). It is now almost certain\nthat these anomalies are somehow connected to binary evolution. In\nsection~2, I therefore first review some of the main, relevant\nprinciples of binary stellar evolution theory and, in section~3,\nsummarize the observational constraints any model for the progenitor\nhas to fulfill.  In section~4, I present in detail an updated version\nof a merger scenario that at present provides the only framework in\nwhich not only some, but all of these anomalies can be\nunderstood. Throughout this review, I emphasize the still substantial\ntheoretical uncertainties in this model and indicate how observations\nmay help to conclusively verify it in all its details.\n\n\\section{Binary interactions}\n\nAs is not very widely known in the supernova community, most stars in the\nsky are actually members of binary (or multiple) systems. To zeroth order,\n{\\em all} stars are members of binaries (see, e.g., the references in\nPodsiadlowski, Joss \\& Hsu 1991 [PJH] and Ghez 1996). Of course to have its\npresupernova structure altered, a star has to be in a close,\ninteracting binary where at least one star fills its Roche lobe during\nits evolution. The fraction of massive stars in interacting binaries\ncan be estimated to be in the range of 30$\\,$--$\\,50\\,$\\%. For\nexample, Garmany, Conti \\& Massey (1980) found that $36\\,$\\% of massive stars\nare spectroscopic binaries with massive companions with periods less\nthan $\\sim 1\\,$yr. This estimate would imply a {\\em true}\ninteracting-binary frequency of around $50\\,$\\%. It is worth noting\nthat Roche-lobe overflow occurs more frequently in evolved phases\nsimply because the radius of a star expands only by a factor of $\\sim\n2$ during the main sequence, while it expands by a factor of $\\sim 100$\nsubsequently. This means that, for any plausible orbital-period\ndistribution, a star is much more likely to encounter mass transfer after\nits main-sequence phase. Since stars spend the largest fraction of their\nlives on the main sequence, most stars observed in the sky have not\n(yet) experienced a binary interaction. On the other hand, supernovae\nprobe the very final stage in the evolution of a star. Therefore a large\nfraction of all supernova progenitors are affected by a previous binary\ninteraction. This is, at least in part, responsible for the \nlarge variety of observed supernova types.\n\n\nIn general, two qualitatively very different modes of mass transfer\ncan be distinguished: more-or-less conservative mass transfer and\ndynamically unstable mass transfer.\n\n{\\em (Quasi-)conservative mass transfer} usually occurs when the\nmass donor has a radiative envelope and the mass ratio of the\nmass-accreting to the mass-losing component is not too small (e.g.,\nde Loore \\& de Gr\\`eve 1992). Then, a large fraction\n($\\ga 0.5$) of the mass lost from the primary is accreted by\nthe secondary. Thus in this case, both components of the binary are\naffected, one by losing mass, the other by accreting it. During mainly\nconservative mass transfer, the orbital period tends to increase.\n\n\n\n{\\em Dynamical mass transfer} usually takes place when the secondary\nis a giant star with a deep convective envelope. In this case, mass\ntransfer is dynamically unstable and leads to the formation of a\ncommon envelope surrounding the core of the giant and the secondary\n(Paczy\\'nski 1976).  Due to friction between this immersed binary with\nthe common envelope, the orbit of the binary starts to shrink.\nDependent on how much energy is released in the orbital decay of the\nbinary and is deposited in the envelope, two different outcomes are\npossible. If the deposited energy exceeds the binding energy of the\nenvelope, the common envelope can be ejected, leaving a very close\nbinary consisting of a helium star (or Wolf-Rayet star) and a normal\ncompanion star (which is hardly affected by the common-envelope\nphase).\nIf the energy is not sufficient to unbind\nthe envelope, the less dense component of the immersed binary will\nultimately be tidally destroyed and the two components merge completely\nto form a single, but rapidly rotating giant (see section~4.2).\t\n\n\nIn a typical binary scenario, a binary system may \nexperience several different phases of\nmass transfer, which is the reason for the large variety of possible\nbinary scenarios. In the following, I will concentrate on the main\nconsequences of the various mass-transfer types for the structure\nof the immediate supernova progenitors (for a more detailed discussion\nsee PJH and Hsu et al.\\ 1996).\n\n\\subsection{Mass loss}\n\nSome 30$\\,$--$\\,50\\,$\\% of all massive stars experience Roche-lobe\noverflow and mass loss at some point during their evolution. \nIn most cases, they lose all of their\nhydrogen-rich envelopes and become helium stars or Wolf-Rayet stars,\nwhich are excellent candidates for type Ib\/Ic supernovae. In some\ncases, however, it is possible for the primary to retain part of\nits hydrogen-rich envelope, in particular when mass transfer occurs\nvery late during the evolution of the primary and when the initial mass\nratio is very close to one. However, even in this case, at most a few\nsolar masses can be retained in the envelope. The immediate supernova\nprogenitor will be a {\\em stripped supergiant} with a small envelope\nmass (Joss et al.\\ 1988). \nIt is interesting to note that the outer appearance of this\nstar will be very similar to that of a star that has lost no mass at\nall, since the radius and the luminosity of the star are almost\nindependent of the envelope mass (PJH). The\nresulting supernova will, however, be quite different. The light curve\nwill show no extended plateau phase but resemble a type II-L\nor, in the most extreme case, a type IIb supernova (Nomoto et al.\\ 1993;\nPodsiadlowski et al.\\ 1993; Woosley et al.\\ 1994; Hsu et al.\\ 1996). \n\n\\subsection{Mass accretion}\n\n\nJust as the structure and appearance of the mass-losing star can be\nstrongly affected, the structure and further evolution \nof the accreting companion can also be dramatically altered by mass transfer.\nThis depends, however, strongly on the evolutionary stage of the\naccreting secondary at the beginning of the mass-transfer phase.\n\\par\\medskip\n\\noindent {\\em Secondary on the main sequence}\\par\\noindent\nIf the secondary is still on the main sequence at the beginning of\nthe mass-transfer phase, the secondary is usually {\\em\nrejuvenated} and will behave subsequently (after the mass-transfer \nphase) like a single,\nbut now more massive star (Hellings 1983; PJH).\nHowever, as was shown by Braun \\& Langer (1995), this need\nnot be the case if accretion occurs very late on the main sequence\nand if semi-convection is very slow (combined with the Ledoux\ncriterion for convective instability), since the convective core will\nthen not grow significantly as a result of accretion. The subsequent\nevolution will  resemble the evolution of a star that accreted after the\nmain-sequence phase (see below); in particular, the star may never \nbecome a red supergiant and spend the whole post-main-sequence phase\nin the blue-supergiant region of the Hertzsprung-Russell (H-R)\ndiagram.\n\\par\\medskip\\noindent\n{\\em Secondary has completed hydrogen core burning}\\par\\noindent\nIf the secondary has already left the main sequence before the\nbeginning of the mass-transfer phase, the subsequent evolution will\ngenerally differ quite substantially \nfrom the evolution of a normal single star. \nSince the mass of the helium core will not grow as a result\nof mass accretion, the main effect of post-main-sequence accretion is\nto increase the envelope mass relative to the core mass. As was first\nshown by Podsiadlowski \\& Joss (1989) (see also De Loore \\&\nVanbeveren 1992; PJH), this has the consequence that\nthe star will now not become a red supergiant or, if it was a red\nsupergiant at the time of the accretion phase, leave the\nred-supergiant region and spend its remaining lifetime as a blue\nsupergiant (see also Barkat \\& Wheeler 1989 for a related scenario).\nThe final location of the star in the H-R diagram\nat the time of the supernova depends on how much mass has been\naccreted in the accretion phase. The star will be the bluer, the more mass\nit has accreted. The final supernova will be of the SN~1987A variety \n(type II[blue]). Since a star has to accrete only a few percent of its\nown mass from a binary companion to be spun up to critical rotation,\nthe progenitor is expected to pass through a phase where it is \nrapidly rotating. Rapid rotation may also induce large-scale mixing in the\naccreting star (even across chemical gradients). \nHowever, whether this happens depends critically on the angular-momentum\ntransport inside and the structure of the accreting star.\n\n\\subsection{Binary mergers}\n\n\nThe most dramatic type of binary interaction is dynamical mass\ntransfer leading to a common-envelope and spiral-in phase. If the\norbital energy released during the spiral-in is sufficient to eject\nthe common envelope, the end product is a short-period binary\nconsisting of a Wolf-Rayet primary and a normal stellar companion.\nThe primary will eventually explode as a type Ib\/Ic supernova. If the\nsystem remains bound, the secondary is also likely to experience\nmass loss in the future leading to a second mass-transfer phase, just\nas in the mass-accretion scenarios discussed in the previous\nsubsection.\n\nIf the common envelope is not ejected, the two components will merge\ncompletely to produce a more massive and rapidly rotating\nsingle star. This end product resembles in many ways the outcome\nof the post-main-sequence accretion models in section~2.2. In\nparticular, it may end its evolution as a blue supergiant rather than\nas a red supergiant either because of the added mass in the envelope\n(Podsiadlowski, Joss \\& Rappaport 1990) or because of the dredge-up\nof helium (Hillebrandt \\& Meyer 1989) or both.\nThe resulting supernova may again be of the SN~1987A variety.\nPodsiadlowski et al.\\ (1991) have estimated that the combined frequency\nfor a blue-supergiant progenitor in either an accretion or a merger\nscenario is about 5$\\,$\\% of all core-collapse supernovae with an uncertainty\nof about a factor of two.\n\\par\\medskip\n\n\n\n\\noindent{\\em Complications.} \nIn this section, I only discussed the main types of interactions.\nHowever, many binaries experience more than one phase of mass transfer.\nFor example, low-mass helium stars, produced in a first mass-transfer phase,\nwill expand again after core helium burning to become helium giants \nand may fill their Roche lobes for a second time (so-called case BB \nmass transfer; De Gr\\`eve \\& De Loore 1977; Delgado \\& Thomas 1981).\nThis still does not exhaust the whole variety of multiple \nmass-transfer scenarios. Nomoto {et al.}\\ (1994) \nsummarized various scenarios to produce bare CO cores as progenitors\nof type Ic supernovae, which involve up to {\\it three} mass-transfer phases.\n\\par\n\n\n\\section{Observational Constraints on the Progenitor of SN~1987A}\n\n\\subsection{The blue-supergiant progenitor}\n\nOne of the major early surprises of SN~1987A was the fact that its\nknown progenitor, \\mbox{SK~$-69^{\\circ}$202}, was a blue supergiant\nrather than a red supergiant, the normally expected supernova\nprogenitor. Moreover, from the dynamical age of the surrounding\nlow-velocity nebula ($\\sim 30000\\,$yr) one can infer that it was a red\nsupergiant just a few $10^4\\,$yr ago. Any model of the progenitor has\nto explain this recent transition {\\em and} be consistent with the\ngeneral behaviour of massive stars in the LMC in the H-R diagram.\nMany of the early models already failed this obvious test (see\nPodsiadlowski 1992).\n\n\\subsection{The triple-ring nebula} \n\nOne of the most spectacular features of SN~1987A is \nthe complex, but very axisymmetric nebula surrounding\nthe supernova, which was formed out of material that was ejected from\nthe progenitor in the not-too-distant past. The main geometry of the\nnebula consists of three rings, an inner ring centered on the\nsupernova (Wampler et al.\\ 1990; Jacobson et al.\\ 1991) and two rings\ndisplaced to the South and the North, but in approximate alignment\nwith the symmetry axis of the inner ring (Wampler et al.\\ 1990;\nBurrows et al.\\ 1995).\nThis geometry implies an axisymmetric, but highly non-spherical\nstructure of the envelope of the progenitor and\/or its winds.\nThe origin of this non-sphericity provides a severe constraint\nfor models of the progenitor. A plausible mechanism to produce\nthe required asymmetry is the flattening of the progenitor's envelope\ncaused by rapid rotation (Chevalier \\& Soker 1989). However,\nstraightforward angular-momentum considerations show that a single\nstar which was rapidly rotating on the main sequence would be a slow\nrotator in any subsequent supergiant phase and could not possibly \nbe significantly flattened at the time of the supernova explosion.\nRecent HST observations (Pun 1997) which show that the supernova ejecta\nthemselves are elongated along the symmetry axis of the nebula\nprovide further evidence for a rapidly rotating, flattened (oblate) \nprogenitor, since in this case the supernova shock will propagate faster\nin the polar directions, thereby generating a prolate structure\nof the ejecta.\n\n\\subsection{The chemical anomalies}\n\nThe third major surprise of the supernova involves a number of\nchemical anomalies in the progenitor's hydrogen-rich envelope\nand in the presupernova ejecta.\\par\\medskip\n\n{\\noindent\\em The inner ring: the helium anomaly}\\par\\nobreak\\medskip\\noindent\nAs is now firmly established, the composition of the inner ring shows\nthat significant amounts of CNO-processed material have been dredged up to\nthe surface (with N\/C$\\,\\sim\\,$5, N\/O$\\,\\sim\\,$1 [all ratios are by number]) \nand that, most importantly, the helium abundance in the inner ring is \nabout twice solar (He\/H$\\,\\sim\\,$0.25; H\\\"oflich 1988; Allen, Meikle \\& \nSpyromilio 1989; Fransson et al.\\ 1989; Wang 1991; Lundqvist \\& Fransson 1996;\nSonneborn et al.\\ 1997).\n\n\\medskip\n{\\noindent\\em The outer rings: two dredge-up phases?}\n\\par\\medskip\\noindent\nRecently, Panagia et al.\\ (1996) found that the chemical composition\nof the outer rings is significantly different from the composition of\nthe inner ring, indicating a smaller amount of dredge-up of\nCNO-processed material (N\/C$\\,\\sim\\,$2; N\/O$\\,\\sim\\,$0.6). This, if confirmed,\nmight suggest that there were two dredge-up phases, the first\nassociated with the normal convective dredge-up when a star becomes a\nred-supergiant and develops a deep convective envelope and the second,\na few $10^4\\,$yr ago, connected with the event that produced all the\nother anomalies as well. Determination of the helium abundance of the\nouter rings could shed more light on the details of this process.\n\\par\\medskip\n{\\noindent\\em The barium anomaly}\n\\par\\medskip\\nobreak\\noindent\nThe third chemical anomaly, most people had hoped would go away, is the\nenhancement of barium (by a factor of 5$\\,$--$\\,$10) and other\ns-process elements in the progenitor's envelope\n(Williams 1987; H\\\"oflich 1988; \nMazzali, Lucy \\& Butler 1992; Mazzali \\& Chugai 1995). This suggests\nthe simultaneous occurrence of hydrogen and some helium-burning\nreactions (in particular, C$^{13} + \\alpha$) in the outer\nparts of the core, similar to the process that produces these elements\nin S stars on the asymptotic-giant branch (Sanders 1967).\n\n\nAll of these anomalies together suggest that there was a single,\ndramatic event a few $10^4\\,$yr ago that is responsible for them.\nSeveral of these show clear fingerprints of binary interactions.\nJust as in any good mystery story, there are plenty of traces and clues,\nand it is just up to us to decipher their meanings and to reconstruct what\n{\\em really} happened to the progenitor of SN~1987A.\n\n\\section{The Progenitor of SN~1987A: a Binary Merger}\n\\begin{figure}[p]\n\\plotone{podsiadlowski1.eps}\n\\end{figure}\n\nOver the last ten years, numerous binary models have been suggested\n(for a review, see Podsiadlowski 1992; see also Rathnasree 1993;\nBraun \\& Langer 1995). Both accretion and merger\nmodels (see section~2) can explain a rapidly rotating blue-supergiant\nprogenitor.  However, since in the former class of models it is the\nsecondary that accretes from a more evolved star and since all of this\nshould have occurred in the very recent past, this would require the\nmasses of the stars to have been extremely close initially. This\nimplies enormous fine-tuning of the binary parameters and therefore is\nno longer a favoured model.  It now appears most likely that the\nprogenitor was a binary that merged with its companion in the recent\npast (as originally proposed independently by Hillebrandt \\& Meyer\n[1989] and Podsiadlowski {et al.}\\ [1990]; also see Chevalier \\& Soker\n[1988] for an earlier suggestion).\n\nIn this section, I will present an updated version of the merger\nscenario outlined in Podsiadlowski (1992) and schematically\nillustrated in Figure~1 (details will be published in Podsiadlowski\n1997). The calculations use the chemical abundances\nfor massive LMC stars determined by Russell \\& Bessell (1989) and\nRussell \\& Dopita (1990) (with $Z=0.01$ but an uncertain carbon\nabundance), updated opacities by Rogers \\& Iglesias (1992) and\nAlexander (1994), as provided by Eggleton (1997).\n\nInitially, the progenitor was a member of a very typical binary,\nconsisting of a primary of $\\sim 15\\,M_{\\odot}$ and a significantly\nless massive secondary (5$\\,$--$\\,10\\,M_{\\odot}$) in a fairly wide\norbit (with an orbital period of $\\sim 10\\,$yr), so that the primary\nstarted to fill its Roche lobe only on the asymptotic-giant branch\n(i.e., on its second ascent of the red-supergiant branch after helium\ncore burning). The mass of the companion is not well determined by the\nmodel at the present time. Indeed, a relatively low-mass companion\ncould be sufficient.  The system then experienced dynamical mass\ntransfer, leading to the complete merger of the two stars. The end\nproduct of this evolution is a single, but very rapidly rotating red\nsupergiant, which has been thoroughly stirred up during the merging\nprocess (explaining the main chemical anomalies, provided that this\nenvironment allows for s-processing). The star will now want to shrink\nto become a blue supergiant, producing a rotationally forced disk-like\noutflow in the process. In the subsequent blue-supergiant phase, the\nenergetic blue-supergiant wind sweeps up all the structures\ngenerated previously and produces the triple-ring nebula (for a model\nof the outer rings in a merger scenario, see Podsiadlowski, Fabian \\&\nStevens 1991; Lloyd, O'Brien \\& Kahn 1995; Podsiadlowski \\& Cumming\n1995).\\par\n\n\\begin{figure}[t]\n\\special{psfile='podsiadlowski2.eps' vscale=35 hscale=35 angle=-90 hoffset=30}\n\\vspace{2.6truein}\n\\noindent{{\\bf Figure 2.} \nRepresentative merger calculations for a star with an initial mass of\n18\\hbox{$\\,M_\\odot $}\\ merging with a 10\\hbox{$\\,M_\\odot $}\\ star after helium core burning for different\namounts of helium dredge-up (solid and dashed curves) and final\nmasses as indicated.}\n\\vspace{-0.1in}\n\\end{figure}\n\nFigure~2 shows several representative merger calculations based on\nan analytic theory for the merger process \n(Podsiadlowski 1997; and Podsiadlowski\n\\& Spruit 1997), for a primary with an initial mass of 18\\hbox{$\\,M_\\odot $}\\ merging\nwith a 10\\hbox{$\\,M_\\odot $}\\ companion. In the calculations represented by solid\ncurves, 1.2\\hbox{$\\,M_\\odot $}\\ of the helium core is dredged-up during the merger and\nthe final masses of the merged stars are 20 and 25\\hbox{$\\,M_\\odot $}, respectively (as\nindicated). Both stars have surface abundances of helium and the CNO elements \nthat are in excellent agreement with the observational constraints\n(section~3.3). In the model shown as a dashed curve, the merger is even\nmore dramatic and the final object would chemically be classified as a\nbarium star, though this particular model is somewhat too hot and too\nluminous for the progenitor of SN~1987A.\n\n\\subsection{Formation of the inner ring}\n\nThe production of the inner ring requires a disk-like outflow in the\nred-supergiant phase. It has been suggested that gravitational\nfocusing by a non-interacting companion (similar to what may happen in\nsome planetary nebulae, e.g., Morris 1981) could provide such a\nfocusing mechanism. However, the constraints on any distant companion\nafter the supernova are quite stringent, and its mass could not be\nlarger than $\\sim 2\\hbox{$\\,M_\\odot $}$ (e.g., Plait et al.\\ 1995). It is easy to\nshow, using a Bondi-Hoyle-type wind theory, that the fraction of the\nred-supergiant wind that is gravitationally affected by the companion\nis of order ($v_{\\rm orb}\/v_{\\rm wind})\\,[M_2\/(M_1+M_2)]^2$, where\n$v_{\\rm orb}$ and $v_{\\rm wind}$ are the orbital velocity of the\nsecondary and the wind velocity, respectively, while $M_1$ and $M_2$\nare the masses of the primary and the secondary, respectively. For\nplausible parameters for the SN~1987A progenitor, this fraction cannot\nexceed $\\sim 5\\,$\\%, which is completely insufficient to explain the\nlarge observed asymmetry.\n\nOn the other hand in a merger scenario, the problem is not a lack of\nrotation but an excess thereof (Podsiadlowski 1992; Chen \\& Colgate 1995),\nsince all the orbital angular momentum in the pre-merger binary will be\ndeposited in the envelope of the progenitor, spinning it up in the process.\nIndeed, for the parameters of the model shown in Figure~2, the total orbital\nangular momentum ($\\sim 9\\times 10^{54}\\,$erg s) substantially exceeds \nthe maximum angular momentum of a dynamically stable blue supergiant \n($\\sim 4\\times 10^{54}\\,$erg s). This immediately implies that the\nmerged system will pass through a phase of critical surface rotation in the\nfinal red-blue transition, leading to rotationally enforced, equatorial\nmass loss. One can obtain a rough estimate for the minimum amount of mass that\nneeds to be shed in this process by dividing the excess angular momentum\nby the specific orbital angular momentum of the initial binary. This\nyields\n$$\\Delta M \\sim {\\Delta L\\over\\sqrt{G M D}}\\sim 4\\hbox{$\\,M_\\odot $}\\,\\left({D\\over 10 \n{\\rm AU}}\\right)^{-1\/2},$$\nwhere $D$ is the characteristic co-rotation radius  \nassociated with the mass loss (assumed here to be of order the initial\nbinary separation). \n\nThis estimate implies that one expects at least\nseveral solar masses to be lost after the merger. How this mass loss took\nplace in detail is not so clear at the moment. It may involve\na rotationally focused wind, as in the standard model for Be-star disks\nby Bjorkman \\& Cassinelli \n(1993), or dynamical instabilities in a \ncritically rotating object (e.g., Durisen et al.\\ 1986;\nLivio \\& Soker 1988; Taam \\& Bodenheimer\n1991; Chen \\& Colgate 1995). In both cases, one would\nexpect the formation of a disk-like, equatorial structure with a small\nradial velocity (as required to explain the low expansion velocity of \n$10\\,$km s$^{-1}$ of the inner ring).\n\nIn this context it is worth noting that there is an observed system\nthat seems to have merged in the very recent past and that shows some\nsimilarities to the SN~1987A progenitor: V Hydrae is a carbon-star\ngiant rotating near break-up (Barnbaum, Morris \\& Kahane 1995).  The\nsystem also shows evidence for a disk-like equatorial and a biconical\npolar outflow. Barnbaum et al.\\ (1995) even suggested that the\ncarbon-star characteristics of V Hyd may by a direct consequence of\nthe dredge-up of carbon {\\em caused} by the merging process. While\nthis system is not entirely comparable to the progenitor of SN~1987A,\none may hope to learn more about this relatively unexplored, but not\nat all uncommon ($\\sim\\,$5\\%--$\\,10\\,$\\%) phase of binary evolution from V Hyd\nand, of course, SN~1987A.\n\n\\subsection{The merger phase}\n\nOne of the least studied aspects of the merger scenario involves the details\nof the final merging of the two binary components, the secondary and\nthe helium core of the red supergiant, inside the common envelope.\nWe (Podsiadlowski \\& Spruit 1997) have started to look at this process in\nsome detail (also see Meyer \\& Meyer-Hofmeister 1979; Taam \\& Bodenheimer\n1989, 1991; Terman, Taam \\& Hernquist 1994).\n\nThe spiral-in phase ends when the embedded secondary starts to fill\nits own Roche lobe (at a separation of $\\sim 10\\hbox{$\\,R_\\odot $}$). It then begins\nto transfer mass to the helium core. This mass-transfer process\nresembles in many respects the ``normal'' mass transfer in interacting\nbinaries, except that it occurs inside a low-density, opaque\nenvelope. The mass-transfer rate is determined by the frictional drag\nthe orbiting binary experiences with the common envelope.  The\ntimescale for the final destruction of the secondary is uncertain.\nRough guesses range from a few days (assuming that the envelope does\nnot expand as a result of the spiral-in) to hundreds of years\n(assuming that the frictional luminosity is self-regulated at the\nEddington limit; e.g., Meyer \\& Meyer-Hofmeister [1979]). In the\ncalculations presented earlier, I adopted, somewhat arbitrarily, a\ntimescale of $1\\,$yr.  Because of the relatively large size of the\nhelium core, the stream emanating from the Roche-lobe filling\nsecondary does not self-intersect and form an accretion disk but\nrather impacts with the core and penetrates it. In other words, it\ndrills a hole into the core and starts to erode it, causing\nthe dredge-up of helium-rich material. The\npenetration depth can be estimated from the condition that the ram\npressure in the stream must be of the order of the ambient pressure in\nthe helium core. One major uncertainty in this estimate is how much\nenergy is dissipated inside the stream by internal shocks (which must\noccur because of the pressure focusing of the stream and\nangular-momentum constraints). The characteristic temperature of the\nshock-heated material in the stream-impact region is $T\\sim 2\\times\n10^8\\,$K. Since this material is hydrogen-rich, but the temperature\nmore typical of helium burning, one can expect some unusual\nnucleosynthesis in this region, possibly responsible for the more\nexotic chemical anomalies of SN~1987A like the barium anomaly.\n\nPreliminary calculations using extended nuclear-reaction networks\nshow that significant s-processing (with neutron exposures\nof $\\sim 10^{27}\\,$cm$^{-2}$ with C$^{13}+\\alpha$ as neutron source) \nis possible in this environment. This requires\nthat the dredge-up region of the core is continually mixed with\nsome of the hydrogen-rich envelope that serves as a proton reservoir\nfor an extended period of time (of order the merger timescale). \nUnfortunately, some of the key reaction rates are very temperature sensitive,\nwhile the burning conditions are not sufficiently well determined by\nthe present analytic model to allow very firm conclusions.\nIn addition, these calculations suggest that the abundances of the\nrare nitrogen and oxygen isotopes N$^{15}$ and O$^{17}$ may\nbe overabundant by a large factor. This could provide a potentially\nsensitive tracer for the burning conditions in this environment.\n\n\n\n\n\\section{Concluding Remarks}\n\nAs I have shown in this review, binary interactions are almost certainly\nresponsible for some of the large variety of observed supernova types\nin general and therefore cannot be ignored. As far as SN~1987A is concerned,\na merger scenario provides at present the only model for the progenitor \nthat can explain all the major features of this remarkable event. \nThere are still many theoretical uncertainties,\nin particular involving the details of the merging process and the\nassociated nucleosynthesis. Observations of abundances in different\nparts of the nebula may be particularly useful in helping to reconstruct the\ndetailed merger history (for example, the helium abundance in the\nouter rings). In addition, there may be other chemical anomalies \n(overabundances of N$^{15}$ and O$^{17}$) that may be used as direct\ntracers of the unusual burning conditions that occur during the final\nmerging process. \n\nWhile a lot has been learned about the progenitor of SN~1987A in the last\nten years, there still remains much more to be done, both theoretically and\nobservationally. Ultimately, when the supernova blast wave reaches the\ninner ring, at least the structure of the whole nebula will be finally\nrevealed and confirm or refute some of the aspects of the merger scenario.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\subsection{Proof of Lemma~\\ref{corollary:hellinger_projection_lb_product}}\n\\label{proof:lemma:hellinger_projection_lb_bayes_net}\n\nWe start by defining the \\emph{projection} of a distribution unto a DAG structure, a notion which will be used in the next subsection. \n\\begin{definition}\n  The \\emph{projection} of a distribution $P$ on $\\{ 0, 1 \\}^n$ onto a DAG $G :=\n  \\{ (i, \\Pi_i) \\mid i = \\{ 1, \\ldots, n \\} \\}$ (where, without loss of generality,  $\\{ X_i \\}_{i \\in [n]}$ is assumed in topological order) is the distribution $Q$, Markov w.r.t. $G$, defined as\n  follows. Let $Q_{X_i, X_{\\Pi_i}} := P_{X_i, X_{\\Pi_i}}$ and $Q_{X_{\\Pi_i}} :=\n  P_{X_{\\Pi_i}}$ for $i \\in [n]$, and, for $x\\in\\{0,1\\}^n$, define \n  \\[ Q (x) = Q_{X_1} (x_1)  \\prod_{i = 2}^n Q_{X_i \\mid X_{\\Pi_i}} (x_i \\mid x_{\\Pi_i})\n   . \\]\n\\end{definition}\n\nSince squared Hellinger distance is an $f$-divergence, by the data processing inequality, we have that\n\\begin{equation}\\label{eq:hellinger:projection}\nd_{\\rm{}H} (P, Q) \\geqslant d_{\\rm{}H} (\\pi P, \\pi Q)\n\\end{equation}\nfor any $P$ and $Q$ on $\\{ 0, 1 \\}^n$ and any projection $\\pi$ on a subset of random variables. We will establish the more general statement below, which readily implies Lemma~\\ref{corollary:hellinger_projection_lb_product} by taking $k=0$:\n\\begin{lemma}\n  \\label{lemma:hellinger_projection_lb_bayes_net}Let $G_k$ be a degree-$k$ DAG, $k \\leqslant n$. Let $P'$\n  a projection of $P$ onto $G_k$ with a support of $\\{0, 1\\}^n$ and let\n  $\\mathcal{Q}$ be the set of all distributions Markov w.r.t. $G_k$ on $\\{0,\n  1\\}^n$. Then, we have\n  \\[\n    \\min_{Q \\in \\mathcal{Q}} d_{\\rm{}H} (P, Q) \\geqslant \\frac{1}{1 + \\sqrt{n}} d_{\\rm{}H}\n     (P, P') .\n    \\]\n\\end{lemma}\n  \\begin{proof}\n    For $\\mathcal{Q}$ be the set of\n    all distributions Markov w.r.t. $G_k$, by the subadditivity of Hellinger\n    %\n    for Bayes nets from Corollary~\\ref{corollary:squared_hellinger_subadditivity} along with~\\eqref{eq:hellinger:projection},\n    \\begin{eqnarray*}\n      d_{\\rm{}H} (P, P') & \\leqslant & d_{\\rm{}H} (P, Q) + d_{\\rm{}H} (Q, P')\\\\\n      & \\leqslant & d_{\\rm{}H} (P, Q) + \\Big(\\sum_{i = 1}^n d_{\\rm{}H}^2 (P'_{X_i,\n      X_{\\Pi_i}}, Q_{X_i, \\Pi_i}) \\Big)^{1\/2}\\\\\n      & = & d_{\\rm{}H} (P, Q) + \\Big(\\sum_{i = 1}^n d_{\\rm{}H}^2 (P_{X_i,\n      X_{\\Pi_i}}, Q_{X_i, \\Pi_i})\\Big)^{1\/2}\\\\\n      & \\leqslant & d_{\\rm{}H} (P, Q) + \\sqrt{nd_{\\rm{}H}^2 (P, Q)}\\\\\n      & = & \\left( 1 + \\sqrt{n} \\right) d_{\\rm{}H} (P, Q).\n    \\end{eqnarray*}\n  \\end{proof}\n\\subsection{Proof of Lemma~\\ref{lemma:technical_distance_lb_2xbinomials_conditional_tree}}\n\\begin{lemma}[Lemma~\\ref{lemma:technical_distance_lb_2xbinomials_conditional_tree}, restated] \n  There exist $C_1,C_2>0$ such that the following holds. Let $\\varepsilon\\in(0,1]$ and $n \\geq C_1$, and let $a,b$ be two integers such that $a+b=n$  and $b \\geqslant\n  \\frac{1}{4} n$. Then, for $\\delta \\coloneqq \\frac{\\varepsilon}{\\sqrt{n}}$, we have\n  \\[ \n    \\frac{(1 - \\delta)^n}{2^n}  \\sum_{k_1 = 0}^{a}\\sum_{k_2 = 0}^{b} \\binom{a}{k_1} \\binom{b}{k_2} \\mleft|\n      \\mleft( \\mleft( \\frac{1 + \\delta}{1 - \\delta} \\mright)^{k_1 + k_2} - \\mleft(\n      \\frac{1 + \\delta}{1 - \\delta} \\mright)^{k_1 + b - k_2} \\mright) \\mright|\n      \\geqslant C_2\\varepsilon . \\]\n\\end{lemma}\n\\begin{proof}\n  \\label{proof:lemma:technical_distance_lb_2xbinomials_conditional_tree}\n  By concentration of Binomials,\n  \\begin{eqnarray*}\n    &  & \\left( \\frac{1}{2} \\right)^n (1 - \\delta)^n \\sum_{k_1 =\n    0}^{a} \\sum_{k_2 = 0}^{b} \\binom{a}{k_1}\n    \\binom{b}{k_2} \\left| \\left( \\left( \\frac{1 + \\delta}{1 - \\delta}\n    \\right)^{k_1 + k_2} - \\left( \\frac{1 + \\delta}{1 - \\delta} \\right)^{k_1\n    + b - k_2} \\right) \\right|\\\\\n    & \\geqslant & \\left( \\frac{1}{2} \\right)^n (1 - \\delta)^n \\sum_{k_1 =\n    \\frac{a}{2} + \\sqrt{a}}^{\\frac{a}{2} + 2 \\sqrt{a}}\n    \\sum_{k_2 = \\frac{b}{2} + \\sqrt{b}}^{\\frac{b}{2} + 2\n    \\sqrt{b}} \\binom{a}{k_1} \\binom{b}{k_2} \\left| \\left(\n    \\left( \\frac{1 + \\delta}{1 - \\delta} \\right)^{k_1 + k_2} - \\left(\n    \\frac{1 + \\delta}{1 - \\delta} \\right)^{k_1 + b - k_2} \\right)\n    \\right|\\\\\n    & \\geqslant & \\left( \\frac{1}{2} \\right)^n \\frac{C}{\\sqrt{a\n    b}} \\cdot 2^{| a | + | b |} \\sum_{k_1 = \\frac{a}{2} +\n    \\sqrt{a}}^{\\frac{a}{2} + 2 \\sqrt{a}} \\sum_{k_2 =\n    \\frac{b}{2} + \\sqrt{b}}^{\\frac{b}{2} + 2 \\sqrt{b}} (1 -\n    \\delta)^n \\left| \\left( \\left( \\frac{1 + \\delta}{1 - \\delta}\n    \\right)^{k_1 + k_2} - \\left( \\frac{1 + \\delta}{1 - \\delta} \\right)^{k_1\n    + b - k_2} \\right) \\right|\\\\\n    & \\geqslant & \\frac{C}{\\sqrt{a b}} \\sum_{k_1 =\n    \\frac{a}{2} + \\sqrt{a}}^{\\frac{a}{2} + 2 \\sqrt{a}}\n    \\sum_{k_2 = \\frac{b}{2} + \\sqrt{b}}^{\\frac{b}{2} + 2\n    \\sqrt{b}} (1 - \\delta)^n \\left| \\left( \\left( \\frac{1 + \\delta}{1 -\n    \\delta} \\right)^{k_1 + k_2} - \\left( \\frac{1 + \\delta}{1 - \\delta}\n    \\right)^{k_1 + k_2 + b - 2 k_2} \\right) \\right|\n  \\end{eqnarray*}\n  where $C$ is some constant larger than 0. For $k_1 + k_2 - \\frac{n}{2} = l \\in\n  \\left[ \\sqrt{a} + \\sqrt{b}, 2 \\left( \\sqrt{a} + \\sqrt{b}\n  \\right) \\right]$, $l \\geqslant \\sqrt{a + b} = \\sqrt{n}$.\n  \\begin{eqnarray}\n    &  & (1 - \\delta)^n \\left| \\left( \\left( \\frac{1 + \\delta}{1 - \\delta}\n    \\right)^{k_1 + k_2} - \\left( \\frac{1 + \\delta}{1 - \\delta} \\right)^{k_1 +\n    k_2 + b - 2 k_2} \\right) \\right| \\nonumber\\\\\n    & = & \\left| \\left( (1 - \\delta)^n \\left( \\frac{1 + \\delta}{1 - \\delta}\n    \\right)^{n \/ 2 + l} - (1 - \\delta)^n \\left( \\frac{1 + \\delta}{1 - \\delta}\n    \\right)^{n \/ 2 + l + b - 2 k_2} \\right) \\right| \\nonumber\\\\\n    & = & (1 - \\delta^2)^{n \/ 2} \\left( \\frac{1 + \\delta}{1 - \\delta} \\right)^l\n    \\left| 1 - \\left( \\frac{1 + \\delta}{1 - \\delta} \\right)^{b - 2 k_2}\n    \\right| \\nonumber\\\\\n    & \\geqslant & e^{- \\delta^2 n} e^{2 \\delta \\cdot l}  \\left| 1 - \\left(\n    \\frac{1 + \\delta}{1 - \\delta} \\right)^{b - 2 k_2} \\right| \\geqslant e^{2\n    \\varepsilon - \\varepsilon^2}  \\left| 1 - \\left( \\frac{1 + \\delta}{1 -\n    \\delta} \\right)^{b - 2 k_2} \\right|  \\label{eq:bin_asym_1}\\\\\n    & \\geqslant & e^{\\varepsilon} \\left( 1 - \\left( \\frac{1 + \\delta}{1 -\n    \\delta} \\right)^{b - 2 k_2} \\right) \\geqslant e^{\\varepsilon} (1 - e^{4\n    \\delta (b - 2 k_2)}) .  \\label{eq:bin_asym_2}\n  \\end{eqnarray}\n  where (\\ref{eq:bin_asym_1}) and (\\ref{eq:bin_asym_2}) follows from $1 - x\n  \\geqslant e^{- 2 x}$, $0 < x < 0.79$ and $e^{4 x} \\geqslant \\left( \\frac{1 +\n  x}{1 - x} \\right) \\geqslant e^{2 x}$, $0 < x < 0.95$. All these inequalities\n  hold for $n$ larger some constant and every $\\varepsilon \\in (0, 1]$. Since\n  $b \\geqslant \\frac{1}{4} n$, and by the summation above $b - 2 k_2 \\in\n  \\left[ - 4 \\sqrt{b}, - 2 \\sqrt{b} \\right]$,\n  \\[ e^{\\varepsilon} (1 - e^{4 \\delta (b - 2 k_2)}) \\geqslant\n     e^{\\varepsilon} \\left( 1 - e^{- 8 \\frac{\\varepsilon}{\\sqrt{n}} \n     \\sqrt{b}} \\right) \\geqslant e^{\\varepsilon} (1 - e^{- 4 \\varepsilon})\n     \\geqslant \\varepsilon . \\]\n  and therefore, summing up every term, we have our lower bound\n  \\[\n  \\frac{C}{\\sqrt{a b}} \\sum_{k_1 = \\frac{a}{2} +\n    \\sqrt{a}}^{\\frac{a}{2} + 2 \\sqrt{a}} \\sum_{k_2 =\n    \\frac{b}{2} + \\sqrt{b}}^{\\frac{b}{2} + 2 \\sqrt{b}} (1 -\n    \\delta)^n \\left| \\left( \\left( \\frac{1 + \\delta}{1 - \\delta}\n    \\right)^{k_1 + k_2} - \\left( \\frac{1 + \\delta}{1 - \\delta} \\right)^{k_1\n    + k_2 + b - 2 k_2} \\right) \\right| \\geqslant C \\varepsilon\n  \\]\n  concluding the proof.\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n\\end{proof}\n\n\\subsection{Proof of~\\eqref{fact:intermedia_upper_bound_on_multigraph_cycle_term:restated}}\n\\label{app:fact:intermedia_upper_bound_on_multigraph_cycle_term:proof}\n\n\\begin{fact}\n  \\label{fact:intermedia_upper_bound_on_multigraph_cycle_term}\n  For any set of cycles such that $\\sum_i\n  | \\sigma_i | \\leqslant n$, we have\n  \\[\n    \\prod_{\\substack{\\sigma_i : \\tmop{even}\\\\ | \\sigma_i | \\geqslant 4}} (1 + (- 4\n    \\delta)^{| \\sigma_i |}) \\prod_{\\substack{\\sigma_i : \\tmop{odd}\\\\ | \\sigma_i | \\geqslant 4}} (1 - (- 4 \\delta)^{| \\sigma_i |})\n    \\leqslant e^{O \\left( \\varepsilon^5 \/ n^{\\frac{3}{2}} \\right)}\n    \\prod_{\\substack{\\sigma_i : \\tmop{even}\\\\ | \\sigma_i | = 4}} (1 + (4 \\delta)^4)\n    \\prod_{\\substack{\\sigma_i : \\tmop{odd}\\\\ | \\sigma_i | = 4}} (1 - (4 \\delta)^4)\n  \\]\n  \\begin{proof}\n  We have\n    \\begin{eqnarray*}\n      \\prod_{\\sigma_i : \\tmop{even}} &  & (1 + (- 4 \\delta)^{| \\sigma_i |})\n      \\prod_{\\sigma_i : \\tmop{odd}} (1 - (- 4 \\delta)^{| \\sigma_i |})\\\\\n      & = & \\prod_{\\substack{\\sigma_i : \\tmop{even}\\\\ | \\sigma_i | \\geq 5}} (1 + (-\n      4 \\delta)^{| \\sigma_i |}) \\prod_{\\substack{\\sigma_i : \\tmop{odd}\\\\ | \\sigma_i | \\geq 5}} (1 - (- 4 \\delta)^{| \\sigma_i |})\n      \\prod_{\\substack{\\sigma_i : \\tmop{even}\\\\ | \\sigma_i | = 4}} (1 + (- 4\n      \\delta)^{| \\sigma_i |}) \\prod_{\\substack{\\sigma_i : \\tmop{odd}\\\\ | \\sigma_i | = 4}}\n      (1 - (- 4 \\delta)^{| \\sigma_i |})\\\\\n      & \\leqslant & \\prod_{\\sigma_i : | \\sigma_i | > 5} (1 + (4 \\delta)^{|\n      \\sigma_i |}) \\prod_{\\sigma_i : \\tmop{even}, | \\sigma_i | = 4} (1 + (- 4\n      \\delta)^{| \\sigma_i |}) \\prod_{\\sigma_i : \\tmop{odd}, | \\sigma_i | = 4}\n      (1 - (- 4 \\delta)^{| \\sigma_i |})\\\\\n      & \\leqslant & (1 + (4 \\delta)^5)^{\\frac{n}{4}} \\prod_{\\sigma_i :\n      \\tmop{even}, | \\sigma_i | = 4} (1 + (- 4 \\delta)^{| \\sigma_i |})\n      \\prod_{\\sigma_i : \\tmop{odd}, | \\sigma_i | = 4} (1 - (- 4 \\delta)^{|\n      \\sigma_i |})\\\\\n      & \\leqslant & e^{(4 \\delta)^5 n \/ 4} \\prod_{\\sigma_i : \\tmop{even}, |\n      \\sigma_i | = 4} (1 + (- 4 \\delta)^{| \\sigma_i |}) \\prod_{\\sigma_i :\n      \\tmop{odd}, | \\sigma_i | = 4} (1 - (- 4 \\delta)^{| \\sigma_i |})\\\\\n    %\n    %\n    %\n    %\n      & \\leq & e^{256\\varepsilon^5 \/ n^{\\frac{3}{2}}}\n      \\prod_{\\sigma_i : \\tmop{even}, | \\sigma_i | = 4} (1 + (- 4 \\delta)^4)\n      \\prod_{\\sigma_i : \\tmop{odd}, | \\sigma_i | = 4} (1 - (- 4 \\delta)^4)\n    \\end{eqnarray*}\n    \n  \\end{proof}\n\\end{fact}\n\n\n\n\n\\section{Structured Testing Lower Bound}\\label{sec:gen}\n\nLetting $D = 2^n$, we will rely on the construction from the ``standard'' lower bound of~{\\cite{paninski2008coincidence}} by picking a uniformly\nrandom subset $S$ of $\\{0, 1\\}^n$ of size $\\frac{D}{2}$. Denote\n$\\mathcal{S}$ the set of all such combinations of $S$, and define\n$\\mathcal{P}_{\\tmop{no}}$ to be $\\mathcal{P}_{\\tmop{no}} := \\left\\{ P =\n\\frac{1 + C \\varepsilon}{2} U_S + \\frac{1 - C \\varepsilon}{2} U_{S^c} \\mid S \\in \\mathcal{S}\n\\right\\}$, where $C>0$ is a suitable normalizing constant. As before, $U_S$ denotes the uniform distribution on the set of variable $S$\nand $P \\in \\mathcal{P}_{\\tmop{no}}$ is a mixture of two uniform\ndistributions on disjoint parts, with different weights.\n\nIt is known that $\\Omega (2^{n \/ 2} \/ \\varepsilon^2)$\nsamples are required to distinguish between such a randomly chosen $P$ and\nthe uniform distribution $U$; further, assume we know that the uniform distribution $U$\nis in $\\mathcal{C}$. What remains to show is the \\emph{distance}, that is, ``most'' choices of $P \\in\n\\mathcal{P}_{\\tmop{no}}$ are $\\varepsilon$-far from $\\mathcal{C}$. To argue that\nlast part, we will use our assumption that $\\mathcal{C}$ can be learned with\n$m$ samples to conclude by a counting argument: i.e., we will show that there can be at most $2^{mn}$\nor so ``relevant'' elements of $\\mathcal{C}$, while there are at least $2^{2^{\\Omega (n)}}$ $\\mathcal{P}_{\\tmop{no}}$ that are $\\varepsilon$-far from each\nother. Suitably combining the two will establish the theorem below:\n\\begin{theorem}\n  \\label{theorem:testing_hard_for_easy_to_learn_dist}Let $\\mathcal{C}$ be a\n  class of probability distributions over $\\{0, 1\\}^n$ such that the following\n  holds: (1) the uniform distribution belongs to $\\mathcal{C}$ (2) there\n  exists a learning algorithm for $\\mathcal{C}$ with sample complexity $m = m\n  (n, \\varepsilon)$. Then, as long as $mn \\ll 2^{O (n)}$, testing whether an\n  arbitrary distribution over $\\{0, 1\\}^n$ belongs to $\\mathcal{C}$ or is $\\varepsilon$-far\n  from every distribution in $\\mathcal{C}$ in total variation distance requires\n  $\\Omega (2^{n \/ 2} \/ \\varepsilon^2)$ samples.\n\\end{theorem}\n\n\\begin{proof}\n  As discussed above, indistinguishability follows from the literature~{\\citep{paninski2008coincidence}}, and thus all we need to show now is that $\\mathcal{P}_{\\tmop{no}}$ is far from every distribution in $\\mathcal{C}$.\n  By assumption (2), there exists an algorithm $H\\colon \\{0, 1\\}^{mn} \\to\n  \\mathcal{P}$ (without loss of generality, we assume $H$ deterministic) that can output an estimated\n  distribution given $m = m (n, \\varepsilon)$ samples from $P \\in \\mathcal{C}$. Thus, for every $P\n  \\in \\mathcal{C}$ given $m$ $i.i.d$. samples $X \\in \\{0, 1\\}^{mn}$, $\\Pr_{X \\sim P^{\\otimes m}} (d_{\\tmop{TV}} (H (X), P) < \\varepsilon)\n  \\geqslant 2 \/ 3$. \n  \n  In particular, this implies the weaker statement that, for every $P \\in \\mathcal{C}$, there\n  exists \\emph{some} $x$ in $\\{0, 1\\}^{mn}$ s.t. $P \\in B(H (x), \\varepsilon)$ (where $B(x,r)$ denotes the TV distance ball of radius $r$ centered at $x$).\n  By enumerating all possible values in $\\{0, 1\\}^{mn}$, we then can obtain an\n  $\\varepsilon$-cover $\\{H (x_1), \\ldots, H (x_{2^{mn}})\\}$ of $\\mathcal{C}$, that is, such that\n  $\\mathcal{C} \\subseteq \\bigcup_{i = 1}^{2^{mn}} B (H (x_i), \\varepsilon)$. The $\\varepsilon$-covering number of $\\mathcal{C}$ is thus upper bounded by $2^{O (mn)}$.\n  \n  Next, we lower bound the size of $\\mathcal{P}_{\\rm{}no}$ by constructing an $\\varepsilon$-packing $P_{\\varepsilon}$, where $P_{\\varepsilon} = \\{P_i \\in \\mathcal{P}_{\\tmop{no}}, i\n  \\in \\mathbb{N}: d_{\\tmop{TV}} (P_i, P_j) > \\varepsilon, i \\neq j\\}$.\n  For $P,Q \\in \\mathcal{P}_{\\tmop{no}}$ corresponding to two sets $S,S'$, each of size $\\frac{D}{2}=2^{n-1}$, we have\n  \\begin{eqnarray*}\n  d_{\\tmop{TV}} (P,Q) & = & \\frac{1}{2} | S \\triangle S' | \\cdot \\left|\n  \\frac{1 + C \\varepsilon}{2} - \\frac{1 - C \\varepsilon}{2} \\right| \\cdot \n  \\frac{2}{D} = C\\varepsilon\\cdot \\frac{| S \\triangle S' |}{D} > \\varepsilon\n  \\end{eqnarray*}\n  For this to be at least $\\varepsilon$, the pairrwise\n  symmetric difference of (the sets corresponding to the) distributions in $P_{\\varepsilon}$ should be at least $\\frac{D}{C} = \\Omega(2^n)$. We know, by, e.g., \\citet[Proposition~3.3]{Blais_2019} that there exist such families of balanced subsets of $\\{0,1\\}^n$ of cardinality at least $\\Omega(2^{2^{\\rho n}})$, where \n  $\\rho>0$ is a constant that only depends on $C$.\n  \n  Thus, the size of $\\mathcal{P}_{\\tmop{no}}$ is itself $\\Omega(2^{2^{\\rho n}})$; combining this lower bound with the upper bound on the covering number of $\\mathcal{C}$ concludes the proof.\n\\end{proof}\n\nAs a corollary, instantiating the above to the class $\\mathcal{C}$ of degree-$d$ Bayes nets over $n$ nodes readily yields the following:\n\\begin{corollary}\n  \\label{corollary:lb_general}\n  For large enough $n$, testing whether an arbitrary distribution over\n  $\\{0, 1\\}^n$ is a degree-$d$ Bayes net or is $\\varepsilon$-far from every such\n  Bayes net requires $\\Omega (2^{n \/ 2} \/ \\varepsilon^2)$ samples, for any $d = o\n  (n)$ and $\\varepsilon \\geq 2^{- O (n)}$.\n\\end{corollary}  \n  \\begin{proof}\n    We can obtain a learning upper bound of $m = O (2^d n \\log (2^{d + 1} n)\n    \\log (n^{d n}) \/ \\varepsilon^2)$ for degree-$d$ Bayes nets by combining\n    the known-structure case (proven in~{\\citet{BhattacharyyaGMV20}}) with the\n    reduction from known-structure to unknown-structure (via hypothesis\n    selection\/tournament~{\\citep{CanonneDKS20}}). We have $m n \\leqslant O\n    (2^d n^6 \/ \\varepsilon^2)$. To have $2^{mn} \\ll 2^{2^{\\rho n}}$, where\n    $\\rho$ is some constant, we need $m n < 2^{O (n)}$, which requires $d = o\n    (n)$ and $\\varepsilon \\geq 2^{- O (n)}$ for large enough $n$.\n  \\end{proof}\n\n\\subsection{Related Work}\nDistribution testing has been an active and rapidly progressing research program for the last 20+ years; see \\cite{rubinfeld2012taming} and \\cite{canonne2020survey} for surveys. One of the earliest works in this history was that of \\cite{batu2001testing} who studied testing independence of two random variables. There followed a series of papers \\citep{alon2007testing, levi2013testing, AcharyaDK15}, strengthening and generalizing bounds for this problem, culminating in the work of \\cite{diakonikolas2016new} who gave tight bounds for testing independence of distributions over $[n_1]\\times \\cdots \\times [n_d]$. \\citet{HaoL20} recently considered the (harder) problem of estimating the distance to the closest product distribution (i.e., \\emph{tolerant} testing), showing this task could, too, be performed with a sublinear sample complexity.\n\nThough most of the focus has been on testing properties of arbitrary input distributions, it has long been recognized that distributional restrictions are needed to obtain sample complexity improvements. For example, \\cite{rubinfeld2009testing, adamaszek2010testing} studied testing uniformity of monotone distributions on the hypercube. Similarly, \\cite{daskalakis2012learning} considered the problem of testing monotonicity of $k$-modal distributions. More recently, \\cite{CanonneDKS20, daskalakis2016square, BhattacharyyaGMV20, bhattacharyya2021testing} have studied identity testing and closeness testing for distributions that are structured as degree-$d$ Bayes nets. Our work here continues this research direction in the context of independence testing.\n\n\\subsection{Our techniques}\n\\noindent\\textit{Upper bound.} The starting point of our upper bound is the (standard) observation that a distribution $P$ over $\\{0,1\\}^n$ is far from being a product if, and only if, it is far from the product of its marginals. By itself, this would not lead to any savings over the trivial exponential sample complexity. However, we can combine this with a localization result due to~\\cite{daskalakis2016square}, which then guarantees that if the degree-$d$ Bayes net $P$ is at \\emph{Hellinger} distance $\\varepsilon$ from the product of its marginals $P'$, then there exists \\emph{some} vertex $i\\in[n]$ such that $P_{i,\\Pi_i}$ (the marginalization of $P$ onto the set of nodes consisting of $i$ and its $d$ parents) is at Hellinger distance at least $\\frac{\\varepsilon}{\\sqrt{n}}$ from $P'_{i,\\Pi_i}$. \nThese two facts, combined, seem to provide exactly what is needed: indeed, given access to samples from $P$ and any fixed set of $d+1$ vertices $S$, one can simulate easily samples from both $P_S$ and $P'_S$ (for the second, using $d+1$ samples from $P$ to generate one from $P'_S$, as $P'_S$ is the product of marginals of $P_S$). A natural idea is then to iterate over all $\\binom{n}{d+1}$ possible subsets $S$ of $d+1$ variables and check whether $P_S=P'_S$ for each of them using a closeness testing algorithm for arbitrary distributions over $\\{0,1\\}^{d+1}$: the overhead due to a union bound and the sampling process for $P_S,P'_S$ adds a factor $O(d\\cdot \\log\\binom{n}{d+1}) = O(d^2\\log n)$ to the closeness testing procedure. However, since testing closeness over $\\{0,1\\}^{d+1}$ to total variation distance $\\varepsilon'$ has sample complexity $O(2^{2d\/3}\/\\varepsilon'^2)$ and, by the quadratic relation between Hellinger and total variation distances, we need to take $\\varepsilon' = \\frac{\\varepsilon^2}{n}$, we would then expect the overall test to result in a $\\tilde{O}(2^{2d\/3}n^2\/\\varepsilon^4)$ sample complexity~--~much more than what we set out for.\n\nA first natural idea to improve upon this is to avoid the back and forth between Hellinger and total variation distance, and save on this quadratic blowup. Namely, by using an optimal closeness testing algorithm directly for Hellinger distance~\\citep{diakonikolas2016new}, we could in the last step keep $\\varepsilon' =\\frac{\\varepsilon}{\\sqrt{n}}$ (saving on this quadratic blowup), and pay only overall $\\tilde{O}(2^{3d\/4}\/\\varepsilon'^2)=\\tilde{O}(2^{3d\/4}n\/\\varepsilon^2)$. This is better, but still falls short of our original goal.\n\nThe second idea is to forego closeness testing in the last step entirely, and instead use directly an \\emph{independence} testing algorithm for arbitrary distributions over $\\{0,1\\}^{d+1}$, to test if $P_S$ is indeed a product distribution for every choice of $S$ considered. Unfortunately, while promising, this idea suffers from a similar drawback as our very first attempt: namely, the known independence testing algorithms are all designed for testing in total variation distance (not Hellinger)! Thus, even using an optimal TV testing algorithm for independence~\\citep{AcharyaDK15,diakonikolas2016new} would still lead to this quadratic loss in the distance parameter $\\varepsilon'$, and a resulting $\\tilde{O}(2^{d\/2}n^2\/\\varepsilon^4)$ sample complexity.\n\nTo combine the best of our last two approaches and achieve the claimed $\\tilde{O}(2^{d\/2}n\/\\varepsilon^2)$, we combine the two insights and perform, for each set $S$ of $d+1$ variables, independence testing on $P_S$ in Hellinger distance. In order to do so, however, we first need to design a testing algorithm for this task, as none was previously available in the literature. Fortunately for us, we are able to design such a testing algorithm (Lemma~\\ref{lemma:hellinger_independence_tester}) achieving the desired~--~and optimal~--~sample complexity. Combining this Hellinger independence testing algorithm over $\\{0,1\\}^{d+1}$ with the above outline finally leads to the $\\tilde{O}(2^{d\/2}n\/\\varepsilon^2)$ upper bound of Theorem~\\ref{theo:main:informal}.\\medskip\n\n\\noindent\\textit{Lower bound.} To obtain our $\\Omega(2^{d\/2}n\/\\varepsilon^2)$ lower bound on testing independence of a degree-$d$ Bayes net, we start with the construction introduced by \\cite{CanonneDKS20} to prove an $\\Omega(n\/\\varepsilon^2)$ sample complexity lower bound on testing \\emph{uniformity} of degree-1 Bayes nets. At a high level, this construction relies on picking uniformly at random a perfect matching $M$ of the $n$ vertices, which defines the structure of the Bayes net; and, for each of the $n\/2$ resulting edges, picking either a positive or negative correlation (with value $\\pm \\varepsilon\/\\sqrt{n}$) between the two vertices, again uniformly at random. One can check relatively easily that every Bayes net $P_{\\lambda}$ obtained this way, where $\\lambda$ encodes the matching $M$ and the $2^{n\/2}$ signs, is (1)~a degree-$1$ Bayes net, (2)~at total variation $\\varepsilon$ from the uniform distribution $U$. The bulk of their analysis then lies in showing that (3)~$\\Omega(n\/\\varepsilon^2)$ samples are necessary to distinguish between $U$ and such a randomly chosen $P_\\lambda$. Generalizing this lower bound construction and analysis to independence testing (not just uniformity), and to degree-$d$ (and not just degree-$1$) Bayes nets turns out to be highly non-trivial, and is our main technical contribution.\n\nIndeed, in view of the simpler and different $\\Omega(2^{d\/2}\\sqrt{n}\/\\varepsilon^2)$ sample complexity lower bound for uniformity testing degree-$d$ Bayes nets obtained in~\\cite{CanonneDKS20} \\emph{when the structure of the Bayes net is known},\\footnote{We note that generalizing this (weaker, in view of the dependence on $n$) $\\Omega(2^{d\/2} \\sqrt{n}\/\\varepsilon^2)$ sample complexity lower bound to our setting is relatively simple, and we do so in Appendix~\\ref{section:lb_2} (specifically, Theorem~\\ref{theorem:lb_2}).} one is tempted to adapt the same idea to the matching construction: that is, reserve $d-1$ out of the $n$ vertices to ``encode'' a pointer towards one of $2^{d-1}$ independently chosen $P_\\lambda$'s as above (i.e., the hard instances are now uniform mixtures over $2^{d-1}$ independently generated degree-$1$ hard instances).\nThe degree-$1$ hard instances lead in previous work to a tighter dependence on $n$ (linear instead of $\\sqrt{n}$) because their Bayesian structure is \\emph{unknown}.\nThus, by looking at the mixture of these degree-$1$ Bayes nets, one could hope to extend the analysis of that second lower bound from~\\cite{CanonneDKS20} and get the desired  $\\Omega(2^{d\/2}n\/\\varepsilon^2)$ lower bound.\nUnfortunately, there is a major issue with this idea: namely, if the $2^{d-1}$ matchings are chosen independently, then the resulting overall distribution is unlikely to be a degree-$d$ Bayes net~--~instead, each vertex will have expected degree $\\Omega(2^d)$! (This was not an issue in the corresponding lower bound of~\\cite{CanonneDKS20}, since for them each of the $2^d$ components of the mixture was a degree-$0$ Bayes net, i.e., a product distribution; so degrees could not ``add up'' across the components).\n\nTo circumvent this, we instead choose the matching $M$ to be common to all $2^{d-1}$ components of the mixture and only pick the sign of their $n\/2$ correlations independently; thus ensuring that every node in the resulting distribution $P$ has degree $d$. This comes at a price, however: the analysis of the $\\Omega(2^{d\/2}\\sqrt{n}\/\\varepsilon^2)$ lower bound from~\\cite{CanonneDKS20} crucially relied on independence across those components, and thus can no longer be extended to our case (where the $2^{d-1}$ distributions $P_\\lambda$ share the same matching $M$, and thus we only have independence across components conditioned on $M$). Handling this requires entirely new ideas, and constitutes the core of our lower bound. In particular, from a technical point of view this requires us to handle the moment-generating-function of squares of Binomials (Lemma~\\ref{lemma:MGF_Binomial_square_bound}), as well as that of (squares of) truncated Binomials (Lemma~\\ref{lemma:MGF_Binomial_square_bound_with_min}). To do so, we develop in Section~\\ref{ssec:tools:mgf} a range of results on Binomials and Multinomial distributions which we believe are of independent interest.\n\nFinally, after establishing that $\\Omega(2^{d\/2}{n}\/\\varepsilon^2)$ samples are necessary to distinguish the resulting ``mixture of trees'' $P$ from the uniform distribution $U$ (Lemma~\\ref{theorem:lower_bound_uniformity_testing_on_Bayes_Net}), it remains to show that this implies our stronger statement on testing \\emph{independence} (not just uniformity). To do so, we need to show that $P$ is not only far from $U$, but from \\emph{every} product distribution: doing so is itself far from immediate, and is established in Lemma~\\ref{lemma;distance:mixture:of:trees} by relating the distance from the mixture $P$ to every product distribution to the distance between distinct components of the mixture (Lemma~\\ref{lemma:technical:TV_lower_bound_of_its_marginals}), and lower bounding those directly by analyzing the concentration properties of each component $P_\\lambda$ of our construction (Lemma~\\ref{lemma:technical_distance_lb_2xbinomials_conditional_tree}).\n\n\\subsection{Sample Complexity to distinguish from Uniform (Lemma~\\ref{theorem:lower_bound_uniformity_testing_on_Bayes_Net})}\n    \\label{ssec:distinguish:uniform}\n\\iffalse %\nWe will also require the two following technical lemmas.\n\\begin{lemma}\n\\label{lemma:proof:mgf:twonorm}\nLet $\\mathbf{p}$ be an arbitrary distribution over $[k]$, $m\\geq 0$, and let $X_1,\\dots,X_k$ denote the vector of counts obtained by drawing $m$ i.i.d.\\ samples from $\\mathbf{p}$ (i.e., $X\\coloneqq (X_1,\\dots,X_k)$ follows a multinomial distribution with parameters $m$ and $(\\mathbf{p}_1,\\dots,\\mathbf{p}_k)$). Then,\n\\[\n\\mathbb{E}[e^{t\\norm{X}_2^2}] \\leq e^{16 m^2 t \\norm{\\mathbf{p}}_2^2 + 2mt}\n\\]\nfor every $0 < t < \\frac{1}{16m}$.\n\\end{lemma}\n\\begin{proof}\nBy negative association~\\cite[Lemma 26 and Theorem~31]{Dubhashi96ballsand}, we have\n  \\begin{align*}\n    \\mathbb{E}[e^{t\\norm{X}_2^2}] \n    &= \\mathbb{E}\\mleft[ \\prod_{j = 1}^k e^{t X_j^2} \\mright]\n    \\leq \\prod_{j = 1}^k \\mathbb{E}_{X_j \\sim \\tmop{Bin} \\mleft( m, \\mathbf{p}_j \\mright)} [e^{t X^2}]\n  \\end{align*}\n  From Lemma~\\ref{lemma:MGF_Binomial_square_bound:restated}, we then get that, for every $0< t \\leq 1\/(16m)$, \n  \\begin{align*}\n    \\mathbb{E}[e^{t\\norm{X}_2^2}] \n    &\\leq \\prod_{j = 1}^k e^{16 t m^2 \\mathbf{p}_j^2 + 2 t m \\mathbf{p}_j}\n    = e^{16 t m^2 \\norm{\\mathbf{p}}_2^2 + 2tm}\n  \\end{align*}\nconcluding the proof.\n\\end{proof}\nThis result will itself be useful in deriving the next lemma, whose proof is deferred to Appendix~\\ref{proof:lemma:average_of_product_with_binomial_raise_by_m}.\n\\begin{lemma}\n  \\label{lemma:average_of_product_with_binomial_raise_by_m:restated}\n  There exists an absolute constant $\\delta_0\\approx 0.96$ such that the following holds. Let $K\\geq 1$ and $R_1,\\dots, R_K$ be integers, and $\\delta_1,\\dots, \\delta_K\\in(0,\\delta_0]$. \n  Suppose that $\\kappa_{j, 1},\n  \\ldots, \\kappa_{j, D} \\sim \\tmop{Bin} \\mleft( R_j, \\frac{1}{2} \\mright)$,\n  are i.i.d., and mutually independent across $1\\leq j\\leq K$, and $z_j \\coloneqq \\frac{1 +\n  \\delta_j}{1 - \\delta_j}$. Then\n  \\[\n      \\mathbb{E} \\mleft[ \\mleft( \\frac{1}{D} \\sum_{i = 1}^D \\prod_{j=1}^K\n      z_j^{\\kappa_{j, i}} \\mright)^m \\mright]\n    \\leq \\mleft( \\prod_{j=1}^K z_j^{\\frac{m}{2}R_j} \\mright) \\mleft(\n    \\mathbb{E}_{X \\sim \\tmop{Bin} \\mleft( m, \\frac{1}{D} \\mright)} [e^{t\n    X^2}] \\mright)^D,\n    \\]\n  where $t \\coloneqq \\sum_j 2 \\delta_j^2 R_j \\geq 0$. Moreover, if $tm \\leq \\frac{1}{16}$, then the factor $\\mleft(\n    \\mathbb{E}_{X \\sim \\tmop{Bin} \\mleft( m, \\frac{1}{D} \\mright)} [e^{t\n    X^2}] \\mright)^D$ in the RHS can be replaced by $e^{16 t \\frac{m^2}{D} + 2mt}.\n  $ \n\\end{lemma}\n\\fi %\n\nWe here proceed with the proof of Lemma~\\ref{theorem:lower_bound_uniformity_testing_on_Bayes_Net}, starting with some convenient notations; some of the technical lemmas and facts used here are stated and proven in Section~\\ref{ssec:tools:mgf}. \nLet $\\theta = (\\lambda, \\mu_1, \\ldots, \\mu_{2^{d-1}})$, where each $\\mu_k\\in\\{\\pm 1\\}^{N\/2}$, and let $P_{\\theta}$ be\nthe distribution for the mixture of trees construction from Definition~\\ref{def:mixture:trees}. The\nfollowing denotes the \\emph{matching count} between $(\\lambda, \\mu)$ and $x$ as the quantity\n\\[ \nc (\\lambda, \\mu, x) \\coloneqq  \\mleft| \\mleft\\{ (i, j) \\in \\{d, \\dots, n\\}^2\n    : \\exists 1\\leq k \\leq \\tfrac{N}{2}, \\lambda_k = (i, j) \\text{ and } (- 1)^{x_i + x_j}\n    = (- 1)^{\\mu_{\\iota(x),k}} \\mright\\} \\mright| . \n\\]\nWe will also introduce an analogous quantity with an ``offset'', for $x_{\\tmop{ch}} = (x_{d},\n\\ldots, x_n)$, referring exclusively to the child nodes of $x$ (i.e., the last $N$ nodes, which are the ``children'' of the first $d-1$ ``pointer nodes'' in our construction),\n\\begin{align}\n&c_{\\tmop{ch}} (\\lambda, \\mu, x_{\\tmop{ch}}) \\coloneqq\\label{def:ch}\\\\ \n  &\\mleft| \\mleft\\{ (i, j) \\in \\{d, \\dots, n\\}^2 : \\exists 1\\leq k \\leq \\tfrac{N}{2},\n  \\lambda_k = (i - d + 1, j - d + 1) \\text{ and } (- 1)^{x_i + x_j} = (- 1)^{\\mu_{\\iota(x),k}}\n  \\mright\\} \\mright| \\,. \\notag\n\\end{align}\nTo denote the parameters of the ``mixture of trees,'' we write $\\theta_i \\coloneqq (\\lambda, \\mu_i)$ (for $i\\in[D]$), recalling that the matching parameter $\\lambda$ is common to all $D$ tree components. Since each $\\mu_i$\ncorresponds to one of the values of $(x_1, \\ldots, x_{d-1}) \\in \\{0, 1\\}^{d-1}$, we as before use \n$\\iota\\colon \\{0,1\\}^{d-1} \\to [D]$ to denote the indexing function (so that, for instance, $\\iota(x_1 =\n\\cdots = x_{d-1} = 0) = 0$). We finally introduce three more quantities, related to the matching and orientations parameters across the $D$ components of the mixture:\n\\begin{align*}\nA_{\\theta_i, \\theta_i'} &\\coloneqq  \\{(s, t) \\in \\{1,\\dots,N \/ 2\\}^2 :\n    \\lambda [s] = \\lambda' [t], \\mu_i [s] = \\mu_i' [t] \\} \\tag{common\n    pairs, same orientation} \\\\\nB_{\\theta_i, \\theta_i'} &\\coloneqq  \\{(s, t) \\in \\{1,\\dots,N \/ 2\\}^2 :\n    \\lambda [s] = \\lambda' [t], \\mu_i [s] \\neq \\mu_i' [t] \\}\n    \\tag{common pairs, different orientation} \\\\\nC_{\\theta_i, \\theta_i'} & \\coloneqq \n    (\\lambda \\cup \\lambda') \\setminus (A \\cup B)  \\tag{pairs unique to\n    $\\theta$ or $\\theta'$} \n\\end{align*}\nFor ease of notation, we define $A_i \\coloneqq A_{\\theta_i, \\theta_i'}, B_i \\coloneqq\nB_{\\theta_i, \\theta_i'} \\text{ and } C_i \\coloneqq C_{\\theta_i, \\theta_i'}$; and note that $C_i=C_1$, as it only depends on $\\lambda$ (not on the orientation $\\mu_i$).\\label{def:AB:lb}\n\n\n  To prove the indistinguishability, we will bound the squared total variation distance (or equivalently,\nsquared $\\ell_1$) distance between the distributions of $m$ samples from (the uniform mixture of) $P_{\\theta}$, or $U$ by a small constant; that is, between $Q \\coloneqq \\mathbb{E}_{\\theta}\n[P_{\\theta}^{\\otimes m}]$ and $U^{\\otimes m}$. From Ingster's method (see, e.g., \\cite[Lemma~III.8.]{acharya2018inference}), by using chi-square divergence as an intermediate step we get\n\\begin{align}\n  \\norm{Q - U^{\\otimes m} }_1^2 & \\leq d_{\\chi^2} (Q, U^{\\otimes m})  \n  %\n  = \\mathbb{E}_{\\theta, \\theta'} [(1 + \\tau (\\theta, \\theta'))^m] - 1 \\label{eq:Ingster:restated}\n  \\text{,}\n\\end{align}\nwhere $\\tau (\\theta, \\theta') \\coloneqq \\mathbb{E}_{x \\sim U} \\mleft[ \\mleft(\n\\frac{P_{\\theta} (x) - U (x)}{U (x)} \\mright) \\mleft( \\frac{P_{\\theta'} (x) - U\n(x)}{U (x)} \\mright) \\mright]$. \nIn order to get a handle on this quantity $\\tau (\\theta, \\theta')$, we start by writing the expression for the density $P_{\\theta}$ (for a given parameter $\\theta$ of the mixture of trees). For any  $x\\in\\{0,1\\}^n$, recalling Item~\\ref{def:mixture:trees:item4} of Definition~\\ref{def:mixture:trees},\n\\begin{eqnarray*}\n  P_{\\theta} (x) & = & P_{\\theta} (x_{d}, \\ldots, x_n \\mid x_1, \\ldots, x_{d-1}) U_{d-1}\n  (x_1, \\ldots, x_{d-1})\\\\\n  & = & \\frac{1}{2^{d-1}} P_{\\lambda, \\mu_{\\iota(x)}} (x_{d}, \\ldots, x_n) \\\\\n  & = & \\frac{1}{2^{d-1}} \\cdot \\frac{1}{2^N}  (1 + 4 \\delta)^{c (\\lambda, \\mu_{\\iota(x)}, x)}  (1 -\n  4 \\delta)^{\\frac{N}{2} - c (\\lambda, \\mu_{\\iota(x)}, x)}\\\\\n  & = & \\frac{1}{2^n}  (1 + 4 \\delta)^{c (\\lambda, \\mu_{\\iota(x)}, x)}  (1 - 4\n  \\delta)^{\\frac{N}{2} - c (\\lambda, \\mu_{\\iota(x)}, x)} .\n\\end{eqnarray*}\nSubstituting this in the definition of $\\tau$, we get\n\\begin{align}\n  \\tau (\\theta,\\theta')\n  & = \\mathbb{E}_{x \\sim U}  \\mleft[ \\mleft( \\frac{P_{\\theta} (x)}{U (x)} - 1\n  \\mright)  \\mleft( \\frac{P_{\\theta'} (x)}{U (x)} - 1 \\mright) \\mright]\\notag\\\\\n  & = \\mathbb{E}_{x \\sim U}  \\mleft[ \\mleft( (1 - 4 \\delta)^{\\frac{N}{2}}\n  \\mleft( \\frac{1 + 4 \\delta}{1 - 4 \\delta} \\mright)^{c (\\lambda, \\mu_{\\iota(x)},\n  x)} - 1 \\mright)  \\mleft( (1 - 4 \\delta)^{\\frac{N}{2}} \\mleft( \\frac{1 + 4\n  \\delta}{1 - 4 \\delta} \\mright)^{c (\\lambda', \\mu_{\\iota(x)}', x)} - 1 \\mright)\n  \\mright] \\notag\\\\\n  %\n  %\n  &= 1 +(1-4\\delta)^N\\mathbb{E}_{x \\sim U} \\mleft[ z_0^{c (\\lambda, \\mu_{\\iota(x)}, x) + c (\\lambda', \\mu_{\\iota(x)}', x)} \\mright] \\notag\\\\\n  &\\quad- (1 - 4 \\delta)^{\\frac{N}{2}}\\mathbb{E}_{x \\sim U} \\mleft[ z_0^{c (\\lambda, \\mu_{\\iota(x)}, x)}\n  \\mright] -(1 - 4 \\delta)^{\\frac{N}{2}} \\mathbb{E}_{x \\sim U} \\mleft[ z_0^{c (\\lambda', \\mu_{\\iota(x)}', x)}\n  \\mright] \\,, \\label{eq:calculation:tau}\n\\end{align}\nwhere $z_0 \\coloneqq \\frac{1 + 4\n\\delta}{1 - 4 \\delta}$. \nAs $x \\sim U_N$, for fixed $\\lambda,\\mu$ we have that $c_{\\tmop{ch}} (\\lambda, \\mu, x)$ follows a $\\tmop{Bin}\n\\mleft( \\frac{N}{2}, \\frac{1}{2} \\mright)$ distribution; recalling the expression of the Binomial distribution's probability\ngenerating function, we then have\n\\begin{eqnarray*}\n  (1 - 4 \\delta)^{\\frac{N}{2}} \\mathbb{E}_{x \\sim U} [z_0^{c (\\lambda, \\mu_{\\iota\n  (x)}, x)}] & = & (1 - 4 \\delta)^{\\frac{N}{2}} \\mathbb{E}_{\\orgvec{x}_{1} \\sim U_{d-1}}\n  \\mleft[ \\mathbb{E}_{\\orgvec{x}_2 \\sim U_N} \\mleft[ z_0^{c_{\\tmop{ch}} (\\lambda, \\mu_{\\iota\n  (\\orgvec{x}_1)}, \\orgvec{x}_2)} \\mright] \\mright]\\\\\n  & = & (1 - 4 \\delta)^{\\frac{N}{2}}  \\frac{1}{D} \\Bigg\\{ \\sum_{i = 1}^D\n  \\mathbb{E}_{m \\sim \\tmop{Bin} \\mleft( \\frac{N}{2}, \\frac{1}{2} \\mright)}\n  [z_0^m] \\Bigg\\} \\\\\n  & = & (1 - 4 \\delta)^{\\frac{N}{2}}  \\mleft(\n  \\frac{1 + z_0}{2} \\mright)^{\\frac{N}{2}} = 1 \\text{.}\n\\end{eqnarray*}\nUsing this to simplify the last two terms of~\\eqref{eq:calculation:tau}, we obtain\n\\begin{align}\n  1 + \\tau (\\theta, \\theta') \n  & = (1 - 4 \\delta)^N \\mathbb{E}_{x \\sim U}  [z_0^{c (\\lambda, \\mu_{\\iota(x)}, x) + c (\\lambda', \\mu_{\\iota(x)}', x)}] \\notag\\\\\n  & = \\frac{1}{D}  \\Bigg\\{ \\sum_{i = 1}^D (1 - 4 \\delta)^N \\mathbb{E}_{x \\sim U_N}  [ z_0^{c_{\\tmop{ch}} (\\lambda, \\mu_i, x) + c_{\\tmop{ch}}\n  (\\lambda', \\mu_i', x)}] \\Bigg\\} \\notag\\\\\n  & = \\frac{1}{D}  \\Bigg\\{ \\sum_{i = 1}^D (1 - 4 \\delta)^N \\cdot z_0^{|B_i |}\n  \\mathbb{E}_{\\alpha \\sim \\tmop{Bin} \\mleft( |A_i |, \\frac{1}{2} \\mright)}\n  [z_0^{2 \\alpha}]  \\prod_{\\sigma_i : | \\sigma_i | \\geq 4}\n  \\mathbb{E}_{\\alpha \\sim \\mathcal{B} (\\sigma_i)}\n  [z_0^{\\alpha}] \\Bigg\\} \\notag\\\\\n  & = \\frac{1}{D}  \\Bigg\\{ \\sum_{i = 1}^D (1 - 4 \\delta)^N \\cdot z_0^{|B_i |}\n  \\mleft( \\frac{1 + z_0^2}{2} \\mright)^{|A_i |}  \\prod_{\\sigma_i : | \\sigma_i |\n  \\geq 4} \\mathbb{E}_{\\alpha \\sim \\mathcal{B} (\\sigma_i)}\n  [z_0^{\\alpha}] \\Bigg\\} , \\label{eq:1+tau:restated}\n\\end{align}\nwhere the product is taken over all cycles in the multigraph $G_{\\theta,\\theta'}$ induced by the two matchings; and, given a cycle $\\sigma$, $\\mathcal{B} (\\sigma)$ is the probability distribution defined as follows. Say that a cycle $\\sigma$ is \\emph{even} (resp., \\emph{odd}) if the number of edges with weight $1$ along $\\sigma$ is even (resp., odd); that is, a cycle is even or odd depending on whether the number of negatively correlated pairs along the cycle is an even or odd number. \nIf $\\sigma$ is an \\emph{even cycle}, then $\\mathcal{B} (\\sigma)$ is a Binomial with parameters $|\\sigma|$ and $1\/2$, conditioned on taking even values. Similarly, if $\\sigma$ is an  \\emph{odd cycle}, $\\mathcal{B} (\\sigma)$ is a Binomial with parameters $|\\sigma|$ and $1\/2$, conditioned on taking odd values. \nIt follows that $\\mathbb{E}_{\\alpha \\sim\n  \\mathcal{B} (\\sigma)} [z_0^{\\alpha}]$ is given by the following expression.\n  \\[ \\mathbb{E}_{\\alpha \\sim \\mathcal{B} (\\sigma)}\n      [z_0^{\\alpha}] = \\mleft\\{\\begin{array}{ll}\n        \\mathbb{E}_{\\alpha \\sim \\tmop{Bin} \\mleft( | \\sigma |, \\frac{1}{2}\n        \\mright)} \\mleft[ z^{\\alpha} \\mid \\alpha \\text{ even} \\mright] = \\frac{(1 +\n        z_0)^{| \\sigma |} + (1 - z_0)^{| \\sigma |}}{2^{| \\sigma |}} & \\text{,\n        if } \\sigma \\text{ is even}\\\\\n        \\mathbb{E}_{\\alpha \\sim \\tmop{Bin} \\mleft( | \\sigma |, \\frac{1}{2}\n        \\mright)} \\mleft[ z^{\\alpha} \\mid \\alpha \\text{ odd} \\mright] = \\frac{(1 +\n        z_0)^{| \\sigma |} - (1 - z_0)^{| \\sigma |}}{2^{| \\sigma |}} & \\text{,\n        if } \\sigma \\text{ is odd}\n      \\end{array}\\mright. \\text{.} \\]\n  Denote\n  \\begin{align*}\n      \\mathcal{S}_{e, i} &\\coloneqq \\mathcal{S}_e (\\theta_i, \\theta_i') = \\mleft\\{ \\sigma\n      \\in \\tmop{cycle} (\\theta_i, \\theta_i') : | \\sigma | \\geq 4, \\sigma\n      \\text{ is even} \\mright\\}, \\\\\n      \\mathcal{S}_{o, i} &\\coloneqq\\mathcal{S}_o (\\theta_i, \\theta_i') = \\mleft\\{ \\sigma\n      \\in \\tmop{cycle} (\\theta_i, \\theta_i') : | \\sigma | \\geq 4, \\sigma\n      \\text{ is odd} \\mright\\} .\n  \\end{align*} \n  %\n  We will often drop $\\lambda, \\mu$ or $i$, when clear from context. We expand $\\mathbb{E}_{\\alpha \\sim \\mathcal{B} (\\sigma)}\n[z_0^{\\alpha}]$ as follows: \n\\begin{eqnarray*}\n  \\prod_{\\sigma : | \\sigma | \\geq 4} \\mathbb{E}_{\\alpha \\sim\n  \\mathcal{B} (\\sigma)} [z_0^{\\alpha}]\n  & = & \\prod_{\\sigma : | \\sigma | \\geq 4, \\tmop{even}} \\frac{(1 +\n  z_0)^{| \\sigma |} + (1 - z_0)^{| \\sigma |}}{2^{| \\sigma |}}  \\prod_{\\sigma :\n  | \\sigma | \\geq 4, \\tmop{odd}} \\frac{(1 + z_0)^{| \\sigma |} - (1 -\n  z_0)^{| \\sigma |}}{2^{| \\sigma |}}\\\\\n  & = & \\prod_{\\substack{\\sigma : | \\sigma | \\geq 4\\\\ \\tmop{even}}} \\frac{(1 +\n  z_0)^{| \\sigma |}}{2^{| \\sigma |}} \\!\\! \\mleft( 1 + \\mleft( \\frac{1 - z_0}{1 +\n  z_0} \\mright)^{| \\sigma |} \\mright)\n    \\prod_{\\substack{\\sigma : | \\sigma | \\geq 4 \\\\\\tmop{odd}}} \\!\\!\\frac{(1 + z_0)^{|\n  \\sigma |}}{2^{| \\sigma |}}  \\mleft( 1 - \\mleft( \\frac{1 - z_0}{1 + z_0}\n  \\mright)^{| \\sigma |} \\mright)\\\\\n  & = & \\prod_{\\sigma : | \\sigma | \\geq 4} \\frac{(1 + z_0)^{| \\sigma\n  |}}{2^{| \\sigma |}}  \\prod_{\\sigma \\in \\mathcal{S}_e} (1 + (- 4 \\delta)^{|\n  \\sigma |})  \\prod_{\\sigma \\in \\mathcal{S}_o} (1 - (- 4 \\delta)^{| \\sigma\n  |})\\\\\n  & = & \\frac{(1 + z_0)^{\\sum_{\\sigma: |\\sigma|\\geq 4} |\\sigma|}}{2^{\\sum_{\\sigma: |\\sigma|\\geq 4} |\\sigma|}}  \\prod_{\\sigma \\in \\mathcal{S}_e} (1 + (- 4 \\delta)^{|\n  \\sigma |})  \\prod_{\\sigma \\in \\mathcal{S}_o} (1 - (- 4 \\delta)^{| \\sigma\n  |})\\\\\n  & = & \\frac{(1 + z_0)^{|C|}}{2^{|C|}}  \\prod_{\\sigma \\in \\mathcal{S}_e} (1\n  + (- 4 \\delta)^{| \\sigma |})  \\prod_{\\sigma \\in \\mathcal{S}_o} (1 - (- 4\n  \\delta)^{| \\sigma |})\n\\end{eqnarray*}\nwhere for the last equality we used that $\\sum_{\\sigma: |\\sigma|\\geq 4} |\\sigma| = |C|$. We now improve upon the analogous analysis from~\\cite[Claim~12]{CanonneDKS20} to\nobtain a better upper bound for the remaining terms; indeed, the bound they derived is $e^{O(\\varepsilon^4\/n)}$, which was enough for their purposes but not ours (since it does not feature any dependence on $d$). Let\n$z_1 \\coloneqq \\frac{1 + (4 \\delta)^2}{1 - (4 \\delta)^2}$. In view of using the above expression to bound~\\eqref{eq:1+tau:restated}, we first simplify (part of) the summands of~\\eqref{eq:1+tau:restated} by using the fact that $2|A_i|+2|B_i|+|C_i|=N$ for all $i$, and following the same computations as in~\\citet{CanonneDKS20}:\n\\begin{align*}\n  (1 - 4 \\delta)^N  z_0^{|B_i |} &\n  \\mleft( \\frac{1 + z_0^2}{2} \\mright)^{|A_i |}  \\frac{(1 + z_0)^{|C_i|}}{2^{|C_i|}} \\\\\n  &= (1 - 4 \\delta)^N  z_0^{|B_i |} \\mleft( \\frac{1 + z_0^2}{2}\n  \\mright)^{|A_i |} \\frac{(1 + z_0)^{|C_i |}}{2^{|C_i |}}\\\\\n  & = ((1 - 4 \\delta)^2 z_0)^{|B_i |}  \\mleft( (1 -\n  4 \\delta)^2  \\frac{1 + z_0^2}{2} \\mright)^{|A_i |}\n  \\underbrace{\\mleft( (1 - 4 \\delta) \\frac{1 + z_0}{2} \\mright)^{|C_i |}}_{=\n  1}\\\\\n  & =  (1 - (4 \\delta)^2)^{|B_i |}  (1 + (4 \\delta)^2)^{|A_i |}\\\\\n  & =  (1 - (4 \\delta)^2)^{|A_i | + |B_i |} z_1^{|A_i |}\n  =  (1 - (4 \\delta)^2)^{|A_1 | + |B_1 |} z_1^{|A_i |} \\,,\n\\end{align*}\nwhere the last equality uses the fact that the sum $|A_i | + |B_i |$ only depends on the matchings $\\lambda,\\lambda'$ (not the orientations $\\mu_i, \\mu_i'$), and thus is independent of $i$. \nPlugging this simplification into~\\eqref{eq:1+tau:restated}, and letting $R \\coloneqq |A_1|+|B_1| \\leq N\/2$ for convenience, we get\n\\begin{align*}\n  1 + \\tau (\\theta, \\theta')\n  & = \\frac{1}{D} \\Bigg\\{ \\sum_{i = 1}^D (1 - 4 \\delta)^N \\cdot z_0^{|B_i |}\n  \\mleft( \\frac{1 + z_0^2}{2} \\mright)^{|A_i |}  \\prod_{\\sigma_i}\n  \\mathbb{E}_{\\alpha \\sim \\mathcal{B}(\\sigma_i)} [z_0^{\\alpha}] \\Bigg\\}\\\\\n  & = (1 - (4 \\delta)^2)^{R} \\cdot  \\frac{1}{D}  \\Bigg\\{ \\sum_{i = 1}^D \n  z_1^{|A_i |}  \\prod_{\\sigma \\in \\mathcal{S}_{e, i}} (1 + (- 4\n  \\delta)^{| \\sigma |}) \\prod_{\\sigma \\in \\mathcal{S}_{o, i}} (1 - (- 4\n  \\delta)^{| \\sigma |}) \\Bigg\\} \\text{.}\n\\end{align*}\nNext, we compute the expectation after raising the above to the power $m$.\n\\begin{align}\n  \\mathbb{E}_{\\theta, \\theta'} &[(1 + \\tau (\\theta, \\theta'))^m] \\notag\\\\\n  & = \\mathbb{E}_{\\theta, \\theta'} \\mleft[ \\mleft( (1 - (4 \\delta)^2)^{R} \n  \\frac{1}{D} \\Bigg\\{ \\sum_{i = 1}^D  z_1^{|A_i |}  \\prod_{\\sigma \\in\n  \\mathcal{S}_{e, i}} (1 + (- 4 \\delta)^{| \\sigma |}) \\prod_{\\sigma \\in\n  \\mathcal{S}_{o, i}} (1 - (- 4 \\delta)^{| \\sigma |}) \\Bigg\\} \\mright)^m\n  \\mright] \\notag\\\\\n  & = \\mathbb{E}_{\\lambda, \\lambda'}  \\mleft[ (1 - (4 \\delta)^2)^{mR}\n  \\mathbb{E}_{\\orgvec{\\mu}, \\orgvec{\\mu}'} \\mleft[ \\mleft( \\frac{1}{D} \\Bigg\\{ \\sum_{i = 1}^D \n  z_1^{|A_i |}  \\prod_{\\sigma \\in \\mathcal{S}_{e, i}} (1 + (- 4 \\delta)^{|\n  \\sigma |}) \\prod_{\\sigma \\in \\mathcal{S}_{o, i}} (1 - (- 4 \\delta)^{| \\sigma\n  |}) \\Bigg\\} \\mright)^m \\mright] \\mright]. \\label{eq:boundexpect:partial}\n\\end{align}\nThe quantity inside the inner expectation is quite unwieldy; to proceed, we will rely on the following identity, which lets us bound the two product terms:\n\\begin{equation}\n\\label{fact:intermedia_upper_bound_on_multigraph_cycle_term:restated}\n  \\prod_{\\sigma \\in \\mathcal{S}_e} (1 + (- 4 \\delta)^{| \\sigma |}) \n  \\prod_{\\sigma \\in \\mathcal{S}_o} (1 - (- 4 \\delta)^{| \\sigma |})\n  \\leq e^{c'\\frac{\\varepsilon^5}{N^{3 \/ 2}}}\\!\\!\\!\\!\\!\\!  \\prod_{\\sigma \\in\n  \\mathcal{S}_e : | \\sigma | = 4} \\!\\!\\!\\!\\!\\!(1 + (- 4 \\delta)^{| \\sigma |}) \n  \\!\\!\\!\\!\\!\\!\\prod_{\\sigma \\in \\mathcal{S}_o : | \\sigma | = 4} \\!\\!\\!\\!\\!\\!(1 - (- 4 \\delta)^{|\n  \\sigma |})\\,,\n\\end{equation}\nfor some absolute constant $c'>0$. We defer the proof of this inequality to Appendix~\\ref{app:fact:intermedia_upper_bound_on_multigraph_cycle_term:proof}, and proceed assuming it. Note that as long as $D = \\new{O}(n\/\\varepsilon^6)$, we will have $e^{c'\\cdot  m\\varepsilon^5 \/ N^{3 \/ 2}} \\leq e^{128 m \\varepsilon^2 \/ (\\sqrt{D} n)}$,\\footnote{\\new{As per the condition set in Lemma \\ref{lemma:multinomial_truncated_MGF}, we will from now on assume that $n \/ \\varepsilon^2 \\geqslant n \\geqslant 40 D$, which gives us $N^{3 \/ 2} \\geqslant \\left( n\/2 \\right)^{3 \/ 2} \\geqslant \\frac{n}{2}\n\\cdot (20 D)^{1 \/ 2} \\geqslant 2 n \\sqrt{D}$; and some more calculations give us $c'\\frac{m \\varepsilon^2}{N^{3 \/ 2}} \\leqslant \\frac{128\nm \\varepsilon^2}{\\sqrt{D} n}$.}} and this restriction on $D$ is satisfied for the regime of parameters considered in our lower bound, $d = O(\\log n)$.\n\nFix a pair $\\lambda, \\lambda'$; we have that the $|A_i |$'s are i.i.d.\\ $\\tmop{Bin}\n(R, 1 \/ 2)$ random variables. We now introduce \\[\nR' \\coloneqq |\\{\\sigma_1 : | \\sigma_1 | = 4\\}| =\n|\\{\\sigma_i : | \\sigma_i | = 4\\}|,\n\\] which is the random variable denoting how many cycles have length exactly 4. In particular, we have $R'\n\\leq \\frac{N}{4}$, since $\\sum_{\\sigma: |\\sigma|\\geq 4} \\sigma = |C | \\leq\nN$; more specifically, we have $R' \\leq \\frac{N - 2 R}{4} \\leq \\frac{N}{4}$.     \nFurther, define $\\kappa_i$ as the number of cycles of length 4 which have an even total number of negative correlations; that is, the number of cycles $\\sigma$ such that  $\\mu_i, \\mu_i'$ impose either 0, 2, or 4 negatively correlated pairs along that cycle. \n\n\nSince $\\mu, \\mu'$ are uniformly distributed, being odd or even each has probability $1 \/ 2$, and thus $\\kappa_i \\sim \\tmop{Bin} \\mleft(\nR', \\frac{1}{2} \\mright)$. Moreover, while $\\kappa_i$ and $A_i$ both depend on $\\mu_i, \\mu_i'$, they by definition depend on disjoint subsets of those two random variables: thus, because each correlation parameter is chosen independently, we have that \n$\\kappa_i$ and $A_i$ are independent conditioned on $(R, R')$. Now, recalling our setting of $z_2 = \\frac{1 + (4 \\delta)^4}{1\n- (4 \\delta)^4}$ and fixing a realization of $R,R'$, we have\n\n\n\\iffalse\n\\begin{align}\n  \\mathbb{E}_{\\orgvec{\\mu}, \\orgvec{\\mu}'} &\\mleft[ \\mleft( \\frac{1}{D}  \\sum_{i = 1}^D\n    z_1^{|A_i |}  \\prod_{\\sigma \\in \\mathcal{S}_e (i) : | \\sigma | =\n  4} (1 + (4 \\delta)^4) \\prod_{\\sigma_i \\in \\mathcal{S}_o (i) : | \\sigma | =\n  4} (1 - (4 \\delta)^4) \\mright)^m \\mright]\\notag\\\\\n  & = \\mathbb{E}_{\\orgvec{\\mu}, \\orgvec{\\mu}'} \\mleft[ \\mleft( \\frac{1}{D}  \\sum_{i = 1}^D\n  z_1^{|A_i |} (1 + (4 \\delta)^4)^{\\kappa_i} (1 - (4 \\delta)^4)^{R' -\n  \\kappa_i} \\mright)^m \\mright]\\notag\\\\\n  & = (1 - (4 \\delta)^4)^{m R'} \\mathbb{E}_{\\orgvec{\\mu}, \\orgvec{\\mu}'} \\mleft[ \\mleft(\n  \\frac{1}{D}  \\sum_{i = 1}^D z_1^{|A_i |} z_2^{\\kappa_i} \\mright)^m\n  \\mright]\\notag\\\\\n  & \\leq (1 - (4 \\delta)^4)^{m R'} z_1^{\\frac{mR}{2}}\n  z_2^{\\frac{mR'}{2}}  \\mleft( \\mathbb{E}_{X \\sim \\tmop{Bin}\n  \\mleft( m, \\frac{1}{D} \\mright)} [e^{t X^2}] \\mright)^D, \\label{eq:boundinnerexpect:partial}\n\\end{align}\nwhere the last step follows from the first part of Lemma~\\ref{lemma:average_of_product_with_binomial_raise_by_m:restated}, setting $t \\coloneqq 2 (4\n\\delta)^4 R + 2 (4 \\delta)^8 R'$. \n\nTo continue, we would like to invoke the second part of Lemma~\\ref{lemma:average_of_product_with_binomial_raise_by_m:restated} to bound the RHS further. However, this requires $mt \\leq 1\/16$, which does not always hold: indeed, for $m = O(\\sqrt{D}n\/\\varepsilon^2)$,\\footnote{This is our regime of interest, as we want to prove a lower bound of $m = \\Omega(\\sqrt{D}n\/\\varepsilon^2)$, and thus assume that $m \\leq C'\\cdot \\sqrt{D}n\/\\varepsilon^2$ for some absolute constant $C'>0$ in order to derive a contradiction. Similarly, as for $m = o(n\/\\varepsilon^2)$ or $m = o(2^{d\/2} \\sqrt{n}\/\\varepsilon^2)$ the uniformity testing lower bounds of~\\citet[Theorems 13 and 14]{CanonneDKS20} (which can be adapted, after establishing the corresponding analogue of Lemma~\\ref{lemma;distance:mixture:of:trees:restated}, to generalize to independence testing) apply, we can assume that $m \\geq c' n\/\\varepsilon^2$ and $m \\geq c' \\sqrt{D n}\/\\varepsilon^2$ (for some  small absolute constant $c'>0$), which lets us later use that $m^2\/D > m$ and $m^2\/D > Dn\/m^2$.} $mt \\asymp m \\delta^4 R \\asymp m \\frac{\\varepsilon^4}{n^2} R$ could be as large as $\\sqrt{D}\\varepsilon^2$, since all we can say with probability one is that $R\\leq N\/2$. Yet, as we will see, we have that $mt = O(1)$ with overwhelming probability, which will suffice.\n\nFirst, note that since $R' \\leq N\/4\\leq n\/4$, \n\\begin{equation}\n  \\label{eq:inequality:t}\nmt \\leq 256\\delta^4 mR + 2^{15} m\\frac{\\varepsilon^8}{n^3} \\leq 256\\delta^4 mR + \\frac{\\varepsilon^4}{32n},\n\\end{equation}\nthe last inequality since $m = O(\\sqrt{D}n\/\\varepsilon^2)$, and $d \\ll \\log n$. In particular, given the bound on $m$ and the choice of $\\delta$, there exists an absolute constant $c>0$ such that $mt \\leq \\frac{1}{16}$ whenever $R \\leq r^\\ast \\coloneqq c\\cdot n\/(\\sqrt{D}\\varepsilon^2)$. To show that this happens with high enough probability, we will use the fact that, for every $k\\geq 0$,\n\\begin{equation}\n  \\label{eq:tail:bound:R}\n  \\Pr [R > k] \\leq \\frac{1}{k!},\n\\end{equation}\nwhich was established in~\\citet[p.46]{CanonneDKS20}. First, combining~\\eqref{eq:boundexpect:partial}, Claim~\\ref{fact:intermedia_upper_bound_on_multigraph_cycle_term:restated} and~\\eqref{eq:boundinnerexpect:partial}, we have\n\\begin{align}\n  \\mathbb{E}_{\\theta, \\theta'} [(1 + \\tau (\\theta, \\theta'))^m]\n  &\\leq e^{\\bigO{\\frac{m \\varepsilon^2}{\\sqrt{D} n}}}\\mathbb{E}_{\\lambda, \\lambda'}  \\mleft[ (1 - (4 \\delta)^2)^{mR} (1 - (4\\delta)^4)^{m R'} z_1^{\\frac{mR}{2}} z_2^{\\frac{mR'}{2}} (\\mathbb{E}_X[e^{tX^2}])^D \\mright]   \\notag\\\\\n  &= e^{\\bigO{\\frac{m \\varepsilon^2}{\\sqrt{D} n}}}\\mathbb{E}_{\\lambda, \\lambda'}  \\mleft[ (1 - (4 \\delta)^4)^{\\frac{mR}{2}} (1 - (4\\delta)^8)^{\\frac{m R'}{2}}(\\mathbb{E}_X[e^{tX^2}])^D \\mright]   \\notag\\\\\n  &\\leq e^{\\bigO{\\frac{m \\varepsilon^2}{\\sqrt{D} n}}}\\mathbb{E}_{\\lambda, \\lambda'}  \\mleft[ (\\mathbb{E}_X[e^{tX^2}])^D \\mright],   \\label{eq:almost:there}\n\\end{align}\nwhere we recall that $X$ is Binomial with parameters $m$ and $1\/D$ and that, in the inner expectation, $t$ is a random variable (function of $R$, and thus of $\\lambda,\\lambda'$). To handle this, we use the above tail bound on $R$ to write, recalling that $t \\leq 256\\delta^4 R + \\frac{\\varepsilon^4}{32mn} \\leq 128\\delta^4 n + \\frac{\\varepsilon^4}{32mn}\n= \\frac{128 \\varepsilon^4}{n} + \\frac{\\varepsilon^4}{32mn} \\leq \\frac{129Dn}{m^2}$, \n\\begin{align*}\n\\mathbb{E}_{R}  \\mleft[ (\\mathbb{E}_X[e^{tX^2}])^D \\mright]\n&\\leq \\mathbb{E}_{R}  \\mleft[ (\\mathbb{E}_X[e^{tX^2}])^D  \\mathbbm{1}[R\\leq r^\\ast] \\mright]\n+ \\mathbb{E}_{R}  \\mleft[ (\\mathbb{E}_X[e^{t X^2}])^D  \\mathbbm{1}[R > r^\\ast] \\mright]\\\\\n&\\leq \\mathbb{E}_{R}  \\mleft[ (\\mathbb{E}_X[e^{tX^2}])^D  \\mathbbm{1}[R\\leq r^\\ast] \\mright] + \\mathbb{E}_{R}  \\mleft[ (\\mathbb{E}_X[e^{\\frac{129 Dn}{m^2} X^2}])^D  \\mathbbm{1}[R > r^\\ast] \\mright]\\\\\n&\\leq \\mathbb{E}_{R}  \\mleft[ e^{16 t \\frac{m^2}{D} + 2mt}  \\mathbbm{1}[R\\leq r^\\ast] \\mright] + e^{ 129 D^2 n }  \\Pr[R > r^\\ast] \\tag{Lemma~\\ref{lemma:average_of_product_with_binomial_raise_by_m:restated}}\\\\\n&\\leq \\mathbb{E}_{R}  \\mleft[ e^{18 t \\frac{m^2}{D}}  \\mright] + e^{ 129 D^2 n } \\cdot \\frac{1}{r^*!} \\tag{as $\\frac{m^2}{D} > m$, $\\frac{m^2}{D} > \\frac{Dn}{m^2}$} \\\\\n&\\leq \\mathbb{E}_{R}  \\mleft[ e^{C'' \\mleft(\\delta^4\\frac{m^2}{D} R + \\frac{\\varepsilon^8m^2}{n^3 D} \\mright) }  \\mright] + e^{ 129 D^2 n  - C'''\\frac{n}{\\sqrt{D}\\varepsilon^2} \\log \\frac{n}{\\sqrt{D}\\varepsilon^2} },\n\\end{align*}    \nwhen for the first term we used the setting of $t$ and the bound from (the first part of)~\\eqref{eq:inequality:t}; and, for the second, we used the value of $r^\\ast$, along with Stirling's inequality ($C'',C'''>0$ are again absolute constants). This means that the second term can be made arbitrary small as long as \n\\[\nD^{5\/2} \\ll \\log \\frac{n}{\\sqrt{D}\\varepsilon^2},\n\\]\nfor which $D \\ll \\log^{2\/5} n$ (i.e., $d \\ll \\log\\log n$) suffices. For the first term, we have, since\n$\\frac{\\varepsilon^8m^2}{n^3 D} = O(\\varepsilon^4\/n)$ and \n$\\alpha \\coloneqq C''\\delta^4\\frac{m^2}{D} = O(1)$,\n\\[\n\\mathbb{E}_{R}  \\mleft[ e^{C'' \\mleft(\\delta^4\\frac{m^2}{D} R + \\frac{\\varepsilon^8m^2}{n^3 D} \\mright) }  \\mright]\n= e^{O(\\varepsilon^4\/n)}\\mathbb{E}_{R}  \\mleft[ e^{\\alpha R}  \\mright]\n\\leq  e^{O(\\varepsilon^4\/n)} (1+O(\\alpha)),\n\\]\nusing again the tail bound on $R$ from~\\eqref{eq:tail:bound:R} along with a summation by parts. \n\\iffalse  %\n\\begin{align*}\n\\mathbb{E}_{R}  \\mleft[ e^{\\alpha R}  \\mright]\n&= \\sum_{k=0}^\\infty e^{\\alpha k} \\Pr[ R = k]\n= \\sum_{k=0}^\\infty e^{\\alpha k} (\\Pr[ R \\geq k]-\\Pr[ R \\geq k+1])\\\\\n&= \\sum_{k=0}^\\infty e^{\\alpha k} \\Pr[ R \\geq k]-\n  \\sum_{k=1}^\\infty e^{\\alpha (k-1)} \\Pr[ R \\geq k] \\\\\n  &= 1+\\sum_{k=1}^\\infty (e^{\\alpha k} - e^{\\alpha (k-1)}) \\Pr[ R \\geq k] \\\\\n  &= 1+(1-e^{-\\alpha})\\sum_{k=1}^\\infty e^{\\alpha k} \\Pr[ R \\geq k] \\\\\n  &\\leq 1+(1-e^{-\\alpha})\\sum_{k=1}^\\infty  \\frac{e^{\\alpha k}}{k!} \\\\\n  &= 1+(1-e^{-\\alpha}) (e^{e^\\alpha}-1)\n  \\xrightarrow[\\alpha\\to 0]{} 1+ \\alpha(e-1)\n\\end{align*}\n\\fi\nIn particular, as $\\alpha$ can be made arbitrarily close to $0$ by choosing a smaller value for the constant $C'$ (in the bound for $m$), this term can made made arbitrarily close to $1$.\n\nPlugging this back in~\\eqref{eq:almost:there}, we finally obtain\n\\begin{align}\n  \\mathbb{E}_{\\theta, \\theta'} [(1 + \\tau (\\theta, \\theta'))^m]\n  &\\leq e^{\\bigO{\\frac{m \\varepsilon^2}{\\sqrt{D}n}}} (1+O(\\alpha) + o(1))\n  = 1+o(1),\n\\end{align}\nshowing, in view of~\\eqref{eq:Ingster:restated}, that our family of ``mixture-of-trees'' distributions cannot be distinguished from uniform given $m \\leq C' \\sqrt{D} n\/\\varepsilon^2$ samples. This concludes the proof of Lemma~\\ref{theorem:lower_bound_uniformity_testing_on_Bayes_Net:restated}.\n\\fi\n \n \n \\begin{align}\n  \\mathbb{E}_{\\orgvec{\\mu}, \\orgvec{\\mu}'} & \\mleft[ \\mleft( \\frac{1}{D} \n  \\sum_{i = 1}^D z_1^{|A_i |}  \\prod_{\\sigma \\in \\mathcal{S}_e (i) : | \\sigma\n  | = 4} (1 + (4 \\delta)^4) \\prod_{\\sigma_i \\in \\mathcal{S}_o (i) : | \\sigma |\n  = 4} (1 - (4 \\delta)^4) \\mright)^m \\mright] \\nonumber\\\\\n  & =\\mathbb{E}_{\\orgvec{\\mu}, \\orgvec{\\mu}'} \\mleft[ \\mleft( \\frac{1}{D} \n  \\sum_{i = 1}^D z_1^{|A_i |} (1 + (4 \\delta)^4)^{\\kappa_i} (1 - (4\n  \\delta)^4)^{R' - \\kappa_i} \\mright)^m \\mright] \\nonumber\\\\\n  & = (1 - (4 \\delta)^4)^{mR'} \\mathbb{E}_{\\orgvec{\\mu}, \\orgvec{\\mu}'}\n  \\mleft[ \\mleft( \\frac{1}{D}  \\sum_{i = 1}^D z_1^{|A_i |} z_2^{\\kappa_i}\n  \\mright)^m \\mright] \\nonumber\\\\\n  & \\leq (1 - (4 \\delta)^4)^{mR'} z_1^{\\frac{mR}{2}} z_2^{\\frac{mR'}{2}}\n  \\mathbb{E}_{\\orgvec{\\alpha}} \\mleft[ \\prod_{i = 1}^D (\\cosh (2 \\alpha_i\n  \\delta^2))^R  (\\cosh (2 \\alpha_i \\delta^4))^{R'} \\mright], \n  \\label{eq:boundinnerexpect:partial_improve_attempt}\n\\end{align}\nwhere \\eqref{eq:boundinnerexpect:partial_improve_attempt} follows from the following lemma, whose proof we defer to the end of the section:\n\\begin{restatable*}[]{lemma}{scaryExpectation}\n  \\label{lemma:average_of_product_with_binomial_raise_by_m:restated_improve_attempt}\n  There\n  exists an absolute constant $\\delta_0 \\approx 0.96$ such that the following\n  holds. Let $K \\geq 1$ and $R_1, \\ldots, R_K$ be integers, and $\\delta_1,\n  \\ldots, \\delta_K \\in (0, \\delta_0]$. Suppose that $\\kappa_{j, 1}, \\ldots,\n  \\kappa_{j, D} \\sim \\tmop{Bin} \\mleft( R_j, \\frac{1}{2} \\mright)$, are i.i.d.,\n  and mutually independent across $1 \\leq j \\leq K$, and $z_j \\coloneqq \\frac{1 +\n  \\delta_j}{1 - \\delta_j}$. Then\n  \\[ \\mathbb{E} \\mleft[ \\mleft( \\frac{1}{D}  \\sum_{i = 1}^D \\prod_{j = 1}^K\n     z_j^{\\kappa_{j, i}} \\mright)^m \\mright] \\leq \\mleft( \\prod_{j = 1}^K\n     z_j^{\\frac{m}{2} R_j} \\mright) \\mathbb{E}_{\\orgvec{\\alpha}} \\mleft[\n     \\prod_{i = 1}^D \\prod^K_{j = 1} \\cosh (2 \\alpha_i \\delta_j) \\mright], \\]\n  where $(\\alpha_1, \\ldots, \\alpha_D)$ follows a multinomial distribution with\n  parameters $m$ and $(1\/D,\\dots,1\/D)$.\n\\end{restatable*}\nWe now focus on the expectation on the right (last factor of the RHS of~\\eqref{eq:boundinnerexpect:partial_improve_attempt}): using that $\\cosh u \\leq \\min(e^{u^2\/2}, e^u)$ for $u\\geq 0$, we have, setting $\\Delta\\coloneqq 1\/\\delta^2 = n\/\\varepsilon^2$,\n\\begin{align}\n  \\mathbb{E}_{\\orgvec{\\alpha}} &\\mleft[ \\prod_{i = 1}^D (\\cosh (2 \\alpha_i\n  \\delta^2))^R  (\\cosh (2 \\alpha_i \\delta^4))^{R'} \\mright] \\nonumber\\\\\n  & \\leq \\mathbb{E}_{\\orgvec{\\alpha}} \\mleft[ \\prod_{i = 1}^D \\min(e^{2 \\alpha_i \\delta^2 R}, e^{2\\alpha_i^2 \\delta^4 R}) e^{2 \\alpha_i^2 \\delta^8 R'} \\mright] \\nonumber\\\\\n  & \\leq \\mathbb{E}_{\\orgvec{\\alpha}} \\mleft[ \\prod_{i = 1}^D e^{2\\alpha_i \\delta^2 R \\mathbbm{1} [\\alpha_i > \\Delta]} e^{2 \\alpha_i^2 \\delta^4 R  \\mathbbm{1} [\\alpha_i \\leq \\Delta]} e^{2 \\alpha_i^2 \\delta^8 R'}\n  \\mright] \\nonumber\\\\\n  %\n  & \\leq \\mathbb{E} \\mleft[ \\prod_{i = 1}^D e^{8 \\alpha_i\n  \\delta^2 R \\mathbbm{1} [\\alpha_i > \\Delta]} \\mright]^{1\/4} \\mathbb{E} \\mleft[ \\prod_{i =  1}^D e^{8 \\alpha_i^2 \\delta^4 R \\mathbbm{1} [\\alpha_i \\leq \\Delta]} \\mright]^{1 \/ 4}  \\mathbb{E} \\mleft[ \\prod_{i = 1}^D e^{4  \\alpha_i^2 \\delta^8 R'} \\mright]^{1 \/ 2} \\label{eq:three_scary_expactations_multinomial}\n\\end{align}\nwhere the last step comes from the generalized H\\\"older inequality (or, equivalently, two applications of the Cauchy--Schwarz\ninequality), and the threshold $\\Delta$ was chosen as the value for which the term realizing the minimum changes. We first bound the product of the last two expectations:\n\\begin{align}\n  \\mathbb{E} \\mleft[ \\prod_{i = 1}^D e^{8 \\alpha_i^2 \\delta^4 R\n  \\mathbbm{1} [\\alpha_i\\leq \\Delta]} \\mright]^{1 \/ 4} &\\mathbb{E} \\mleft[ \\prod_{i = 1}^D e^{4 \\alpha_i^2 \\delta^8 R'}\n  \\mright]^{1 \/ 2} \\notag\\\\\n  & \\leq \\mathbb{E} \\mleft[ \\prod_{i = 1}^D e^{8 \\delta^4 R \\min(\\alpha_i^2, \\Delta^2)} \\mright]^{1 \/ 4} \\mathbb{E} \\mleft[\n  \\prod_{i = 1}^D e^{4 \\alpha_i^2 \\delta^8 R'} \\mright]^{1 \/\n  2} \\nonumber\\\\\n  & \\leq  (\\mathbb{E} [e^{8 \\min (\\alpha_j^2, \\Delta^2) \\delta^4 R}])^{D \/\n  4} (\\mathbb{E} [e^{4 \\alpha_1^2 \\delta^8 R'}])^{D \/ 2} \n  \\label{equation:step_negative_association}\\\\\n  & \\leq  \\exp \\mleft( 32 \\delta^4  \\frac{m^2}{D} R \\mright) \\exp \\mleft( 32\\delta^8 \\frac{m^2}{D} R' \\mright) .  \n  \\label{equation:apply_MGF_min_square_bin}\n\\end{align}\nwhere we applied negative\nassociation on both expectations for~\\eqref{equation:step_negative_association};\nand then got~\\eqref{equation:apply_MGF_min_square_bin} by Lemmas~\\ref{lemma:MGF_Binomial_square_bound_with_min} and~\\ref{lemma:MGF_Binomial_square_bound} (for the latter, noting that $t m = 2 \\delta^8 m R' \\leq 1 \/ 16$; and, for the former, assuming with little loss of generality that $\\varepsilon \\leq 1\/(4\\sqrt{2})$). \nApplying Lemma~\\ref{lemma:bound_on_truncated_multinomial_MGF} to the first\n(remaining) factor of the LHS above as $8\\delta^2 R\\leq 4$ and $D \\geq \\Omega(1)$, we get\n\\begin{align*}\n  &\\mathbb{E} \\mleft[ \\prod_{i = 1}^D e^{8 \\alpha_i \\delta^2 R\n  \\mathbbm{1} [\\alpha_i > \\Delta]} \\mright]^{1\/4} \\mathbb{E} \\mleft[ \\prod_{i = 1}^D e^{8 \\alpha_i^2 \\delta^4 R \\mathbbm{1} [\\alpha_i \\leq \\Delta]} \\mright]^{1\/ 4}\\mathbb{E} \\mleft[ \\prod_{i = 1}^D e^{4 \\alpha_i^2\n  \\delta^8 R'} \\mright]^{1 \/ 2}\\\\\n  &\\qquad \\leq (1 + o (1)) \\cdot \\exp \\mleft( 32 \\delta^4  \\frac{m^2}{D} R\n  \\mright) \\exp \\mleft( 32 \\frac{m^2}{D} \\delta^8 R' \\mright)\\\\\n  &\\qquad = (1 + o(1))\\exp (32C'^2 R),\n\\end{align*}\nrecalling that $R' \\leq N\/4\\leq n\/4$, and our assumption that $m \\leq C'\\sqrt{D}n\/\\varepsilon^2$. \\new{Combining~\\eqref{eq:boundexpect:partial}, \\eqref{fact:intermedia_upper_bound_on_multigraph_cycle_term:restated} and~\\eqref{eq:boundinnerexpect:partial_improve_attempt}, what we showed is}\n \\begin{align*}\n  \\mathbb{E}_{\\theta, \\theta'} [(1 + \\tau (\\theta, \\theta'))^m]\n  &\\leq (1 + \\new{e^{128 \\frac{m \\varepsilon^2}{\\sqrt{D} n}}})\\mathbb{E}_{\\lambda, \\lambda'}  \\mleft[ (1 - (4 \\delta)^2)^{mR}(1 - (4 \\delta)^4)^{mR'} z_1^{\\frac{mR}{2}} z_2^{\\frac{mR'}{2}}e^{32C'^2 \\cdot R} \\mright]\\\\\n  &= (1 + \\new{e^{128 \\cdot C'}})\\mathbb{E}_{\\lambda, \\lambda'}  \\mleft[ (1 - (4 \\delta)^4)^{\\frac{mR}{2}}(1 - (4 \\delta)^8)^{\\frac{mR'}{2}}e^{32C'^2 \\cdot R} \\mright]\\\\\n  &\\leq (1 + \\new{e^{128 \\cdot C'}})\\mathbb{E}_{\\lambda, \\lambda'}  \\mleft[e^{32C'^2 \\cdot R} \\mright]\\,,\n\\end{align*}\nwhere the equality follows from the definition of $z_1,z_2$. To conclude, we will use the fact that, for every $k\\geq 0$,\n\\begin{equation}\n  \\label{eq:tail:bound:R}\n  \\Pr [R > k] \\leq \\frac{1}{k!},\n\\end{equation}\nwhich was established in~\\citet[p.46]{CanonneDKS20}. By summation by parts, one can show that this implies\n\\[\n    \\mathbb{E}_{R}  \\mleft[ e^{\\alpha R}  \\mright] \\leq 1+(1-e^{-\\alpha}) (e^{e^\\alpha}-1)\n  \\xrightarrow[\\alpha\\to 0]{} 1+ \\alpha(e-1)\n\\]\nfor any $\\alpha>0$, and so, in our case,\n\\iffalse %\n\\begin{align*}\n\\mathbb{E}_{R}  \\mleft[ e^{\\alpha R}  \\mright]\n&= \\sum_{k=0}^\\infty e^{\\alpha k} \\Pr[ R = k]\n= \\sum_{k=0}^\\infty e^{\\alpha k} (\\Pr[ R \\geq k]-\\Pr[ R \\geq k+1])\\\\\n&= \\sum_{k=0}^\\infty e^{\\alpha k} \\Pr[ R \\geq k]-\n  \\sum_{k=1}^\\infty e^{\\alpha (k-1)} \\Pr[ R \\geq k] \\\\\n  &= 1+\\sum_{k=1}^\\infty (e^{\\alpha k} - e^{\\alpha (k-1)}) \\Pr[ R \\geq k] \\\\\n  &= 1+(1-e^{-\\alpha})\\sum_{k=1}^\\infty e^{\\alpha k} \\Pr[ R \\geq k] \\\\\n  &\\leq 1+(1-e^{-\\alpha})\\sum_{k=1}^\\infty  \\frac{e^{\\alpha k}}{k!} \\\\\n  &= 1+(1-e^{-\\alpha}) (e^{e^\\alpha}-1)\n  \\xrightarrow[\\alpha\\to 0]{} 1+ \\alpha(e-1)\n\\end{align*}\n\\fi %\n \\begin{align}\n  \\mathbb{E}_{\\theta, \\theta'} [(1 + \\tau (\\theta, \\theta'))^m]\n  &\\leq (1 + e^{128 \\cdot C'})\\mleft( 1+(1-e^{-32C'^2}) (e^{e^{32C'^2}}-1) \\mright)\n\\end{align}\nIn particular, the RHS can be made arbitrarily close to $1$ by choosing a small enough value for the constant $C'$ (in the bound for $m$). By~\\eqref{eq:Ingster:restated}, this implies the desired bound on $\\norm{Q - U^{\\otimes m} }_1^2$, and thus establishes Lemma~\\ref{theorem:lower_bound_uniformity_testing_on_Bayes_Net}. \\qed\n\n\n\\paragraph{The remaining technical lemma.} It only remains to establish Lemma~\\ref{lemma:average_of_product_with_binomial_raise_by_m:restated_improve_attempt}, which we do now.\n\\scaryExpectation\n\\begin{proof}[Proof\nof~Lemma~\\ref{lemma:average_of_product_with_binomial_raise_by_m:restated_improve_attempt}]\nWe will require the following simple fact, which follows from the multinomial theorem and the definition of the multinomial distribution:\n\\begin{fact}\\label{fact:power_of_average_equates_multinomial_expect}Let $D$ be a\npositive integer and $m$ be a non-negative integer. For any $x_1, \\ldots, x_D\n\\in \\mathbb{R}$, we have\n\\[ \\mleft( \\frac{1}{D}  \\sum_{i = 1}^D x_i \\mright)^m =\\mathbb{E}_{\\alpha_1,\n   \\ldots, \\alpha_D} \\mleft[ \\prod_{i = 1}^D x_i^{\\alpha_i} \\mright], \\]\nwhere $(\\alpha_1, \\ldots, \\alpha_D)$ follows a multinomial distribution with\nparameters $m$ and $(1\/D, \\ldots, 1\/D)$.\n\\end{fact}\n\\iffalse %\n\\begin{proof}\n  By the multinomial theorem and then the definition of multinomial\n  distribution,\n  \\[ \\mleft( \\frac{1}{D}  \\sum_{i = 1}^D x_i \\mright)^m = \\sum_{\\alpha_1 +\n     \\cdots + \\alpha_m = D} \\binom{m}{\\alpha_1, \\ldots, \\alpha_D} \\mleft(\n     \\prod_{i = 1}^D \\frac{1}{D^{\\alpha_i}} \\mright) \\cdot \\mleft( \\prod_{i =\n     1}^D x_i^{\\alpha_i} \\mright) =\\mathbb{E}_{\\alpha_1, \\ldots, \\alpha_D}\n     \\mleft[ \\prod_{i = 1}^D x_i^{\\alpha_i} \\mright] . \\]\n\\end{proof}\n\\fi\n\nWe now apply Fact~\\ref{fact:power_of_average_equates_multinomial_expect} inside\nthe expectation of the LHS of the statement. Note that the sets of random variables $\\orgvec{\\alpha} =\n\\{\\alpha_1, \\ldots, \\alpha_D \\}$, $\\orgvec{\\kappa}_1 = \\{\\kappa_{1, 1},\n\\ldots, \\kappa_{1, D} \\}, \\ldots, \\orgvec{\\kappa}_K = \\{\\kappa_{K, 1}, \\ldots,\n\\kappa_{K, D} \\}$ are mutually independent, since $\\orgvec{\\alpha}$ are a\nset of auxiliary random variables derived from an averaging operation and by\nthe assumption on $\\orgvec{\\kappa}_j$; and we have that $\\kappa_{j, 1},\n\\ldots, \\kappa_{j, D}$ are i.i.d.,\n\\begin{align*}\n  \\mathbb{E} \\mleft[ \\mleft( \\frac{1}{D}  \\sum_{i = 1}^D \\prod^K_{j = 1}\n  z_j^{\\kappa_{j, i}} \\mright)^m \\mright] & =  \\mathbb{E} \\mleft[\n  \\mathbb{E}_{\\alpha_1, \\ldots, \\alpha_D} \\mleft[ \\prod_{i = 1}^D \\mleft(\n  \\prod^K_{j = 1} z_j^{\\kappa_{j, i}} \\mright)^{\\alpha_i} \\mright] \\mright]\\\\\n  & = \\mathbb{E}_{\\orgvec{\\alpha}, \\orgvec{\\kappa}_j} \\mleft[ \\prod_{i =\n  1}^D \\prod^K_{j = 1} z_j^{\\alpha_i \\kappa_{j, i}} \\mright]\\\\\n  & = \\mathbb{E}_{\\orgvec{\\alpha}} \\mleft[ \\mathbb{E}_{\\orgvec{\\kappa}_j}\n  \\mleft[ \\prod_{i = 1}^D \\prod^K_{j = 1} z_j^{\\alpha_i \\kappa_{j, i}} \\mright]\n  \\mright]\\\\\n  & = \\mathbb{E}_{\\orgvec{\\alpha}} \\mleft[ \\mleft[ \\prod_{i = 1}^D \\prod^K_{j\n  = 1} \\mathbb{E}_{\\orgvec{\\kappa}_j} [z_j^{\\alpha_i \\kappa_{j, i}}] \\mright]\n  \\mright]\\\\\n  & = \\mathbb{E}_{\\orgvec{\\alpha}} \\mleft[ \\mleft[ \\prod_{i = 1}^D \\prod^K_{j\n  = 1} \\mleft( \\frac{1 + z_j^{\\alpha_i}}{2} \\mright)^{R_j} \\mright] \\mright] \\tag{Probability-Generating Function of a Binomial}\\\\\n  & = \\mathbb{E}_{\\orgvec{\\alpha}} \\mleft[ \\prod_{i = 1}^D \\prod^K_{j = 1}\n  z_j^{\\frac{\\alpha_i R_j}{2}} \\mleft( \\frac{z_j^{- \\alpha_i \/ 2} +\n  z_j^{\\alpha_i \/ 2}}{2} \\mright)^{R_j} \\mright]\\\\\n  & = \\mleft( \\prod^K_{j = 1} z_j^{\\frac{mR_j}{2}} \\mright)\n  \\mathbb{E}_{\\orgvec{\\alpha}} \\mleft[ \\prod_{i = 1}^D \\prod^K_{j = 1} \\mleft(\n  \\frac{z_j^{- \\alpha_i \/ 2} + z_j^{\\alpha_i \/ 2}}{2} \\mright)^{R_j} \\mright] .\n\\end{align*}\nNext, we will simplify the\nexpression left inside by upper bounding it, using the fact that, given our\nassumption on $\\delta_j$ being bounded above by $\\delta_0$, we have $z_j =\n\\frac{1 + \\delta_j}{1 - \\delta_j} \\leq e^{4 \\delta_j}$. Thus,\n\\begin{align*}\n  \\mathbb{E}_{\\orgvec{\\alpha}} \\mleft[ \\prod_{i = 1}^D \\prod^K_{j = 1} \\mleft(\n  \\frac{z_j^{- \\alpha_i \/ 2} + z_j^{\\alpha_i \/ 2}}{2} \\mright)^{R_j} \\mright] &\n  \\leq \\mathbb{E}_{\\orgvec{\\alpha}} \\mleft[ \\prod_{i = 1}^D \\prod^K_{j =\n  1} \\mleft( \\frac{e^{- 2 \\alpha_i \\delta_j} + e^{2 \\alpha_i \\delta_j}}{2}\n  \\mright)^{R_j} \\mright]\\\\\n  & = \\mathbb{E}_{\\orgvec{\\alpha}} \\mleft[ \\prod_{i = 1}^D \\prod^K_{j = 1}\n  (\\cosh (2 \\alpha_i \\delta_j))^{R_j} \\mright]\n\\end{align*}\nas claimed.\n\\end{proof}\n\n\n %\n %\n\\subsection{Product Distributions Are Far from Mixture of Trees (Lemma~\\ref{lemma;distance:mixture:of:trees})}\n\\label{ssec:mixturetree:farness}\nIn this subsection, we outline the proof of Lemma \\ref{lemma;distance:mixture:of:trees}.\nOur argument starts with Lemma~\\ref{lemma:technical:TV_lower_bound_of_its_marginals}, which allows us to relate the total variation distance between the mixture and the product of its marginals to a simpler quantity, the difference between two components of this mixture. \n\\begin{lemma}\n\\label{lemma:technical:TV_lower_bound_of_its_marginals}\nLet $p$ be a distribution on $\\{0, 1\\}^N \\times \\{0, 1\\}^M$ (with $N,M\\geq 2$), and denote its\nmarginals on $\\{0, 1\\}^N$, $\\{0, 1\\}^M$ by $p_1, p_2$ respectively. Then, if $p_1$ is uniform, %\n\\[ \\begin{array}{l}\n    d_{\\tmop{TV}} (p, p_1 \\otimes p_2)\n  \\end{array} \\geq d_{\\tmop{TV}} (p (\\cdot \\mid x_1 = 0), p (\\cdot  \\mid x_1\n  = 1))\\,. \\]\n\\end{lemma}   \nThis in turn will be much more manageable, as the parameters of these two mixture components are independent, and thus analyzing this distance can be done by analyzing Binomial-like expressions. This second step is reminiscent of \\cite[Lemma 8]{CanonneDKS20}, which can be seen as a simpler version involving only one Binomial instead of two:\n\\begin{lemma}\n  \\label{lemma:technical_distance_lb_2xbinomials_conditional_tree} \n  There exist $C_1,C_2>0$ such that the following holds. Let $\\varepsilon\\in(0,1]$ and $n \\geq C_1$, and let $a,b$ be two integers such that $a+b=n$  and $b \\geq\n  \\frac{1}{4} n$. Then, for $\\delta \\coloneqq \\frac{\\varepsilon}{\\sqrt{n}}$, we have\n  \\[ \n    \\frac{(1 - \\delta)^n}{2^n}  \\sum_{k_1 = 0}^{a}\\sum_{k_2 = 0}^{b} \\binom{a}{k_1} \\binom{b}{k_2} \\mleft|\n      \\mleft( \\mleft( \\frac{1 + \\delta}{1 - \\delta} \\mright)^{k_1 + k_2} - \\mleft(\n      \\frac{1 + \\delta}{1 - \\delta} \\mright)^{k_1 + b - k_2} \\mright) \\mright|\n      \\geq C_2\\varepsilon . \\]\n\\end{lemma}\nThis parameter $b$ corresponds to the\ndifference between the orientations parameters \n$\\mu, \\mu'$ being large, which happens with high constant probability as long\nas $n$ is large enough. The proof of Lemma\n\\ref{lemma:technical_distance_lb_2xbinomials_conditional_tree} is deferred to Appendix\n\\ref{proof:lemma:technical_distance_lb_2xbinomials_conditional_tree}, and we hereafter proceed with the rest of the argument.\nFor fixed $\\theta$ and $x_2, \\ldots, x_d$, $z := \\frac{1 + 4 \\delta}{1 -\n4 \\delta}$.  We will denote by $\\mu, \\mu'$ the two (randomly chosen) orientation parameters corresponding to the mixture components indexed by $(0,x_2,\\dots, x_d)$ and $(1,x_2,\\dots, x_d)$.     \nBy\nLemma \\ref{lemma:technical:TV_lower_bound_of_its_marginals} and Lemma\n\\ref{lemma:technical_approximated_lb_product_distributions}, for any product\ndistribution $q$,\n\\begin{align}\n  2 d_{\\tmop{TV}} (p, q) & \\geq \\frac{1}{3\\cdot 2^N}  (1 - 4 \\delta)^{\\frac{N}{2}}  \\sum_{x_{d + 1}, \\ldots,\n  x_n} |z^{c (\\lambda, \\mu, x)} - z^{c (\\lambda, \\mu', x)} | \\label{eq:expression:tv}\n\\end{align}\nLet $S_1$ denote the set of pairs in the child nodes with common parameters\nbetween $\\mu$ and $\\mu'$, and $S_2$ the set of pairs with different\nparameters (that is, the definition of $S_1, S_2$ is essentially that of $A$ and $B$ from the previous section (p.\\pageref{def:AB:lb}), but for equal matching parameters $\\lambda = \\lambda'$). In particular, we have that $|S_2 | = \\tmop{Hamming} (\\mu, \\mu') \\sim\n\\tmop{Binomial} \\left( \\frac{N}{2}, \\frac{1}{2} \\right)$ and $|S_1 \\cup S_2 |\n= \\frac{N}{2}$. Let $\\tilde{c}(S, \\mu, x)$ be the analogue of $c_{\\tmop{ch}}\n(\\lambda, \\mu, x)$ from~\\eqref{def:ch}, but only on a subset of pairs $S$ instead of $\\{d,\\dots,n\\}^2$; i.e.,\n\\[ \\tilde{c}(S, \\mu, x) \\coloneqq \\left| \\left\\{ (i, j) \\in S : \\exists k \\in\n    \\mathbb{N}, \\lambda_k = (i - d + 1, j - d + 1) \\text{ and } (- 1)^{x_i + x_j} = (-\n    1)^{\\mu_k} \\right\\} \\right| . \\]\nGiven any $x, \\mu \\text{ and } \\mu'$, the following holds from the definitions of $\\tilde{c}$ and $c_{\\tmop{ch}}$:\n\\begin{itemize}\n  \\item Since $S_1 \\cup S_2$ contains all the pairs, $c_{\\tmop{ch}} (\\lambda, \\mu, x)\n  = \\tilde{c}(S_1, \\mu, x) + \\tilde{c}(S_2, \\mu, x)$ (similarly for $\\mu'$).\n  \n  \\item Since $S_1$ (resp., $S_2$) contains exactly the pairs whose orientation is the same (resp., differs) between $\\mu$ and $\\mu'$, we have\n  $\\tilde{c}(S_1, \\mu', x) = \\tilde{c}(S_1, \\mu, x)$\n  and\n  $\\tilde{c}(S_2, \\mu', x) + \\tilde{c}(S_2, \\mu, x) = |S_2 |$\n  \n  \\item For a fixed matching and a partition $S_1,S_2$ of its $N\/2$ pairs, given an orientation vector $\\mu\\in\\{0,1\\}^{N\/2}$, and fixed values $0\\leq k_1\\leq |S_1|$, $0\\leq k_2\\leq |S_2|$, there are $2^{N\/2}\\binom{|S_1|}{k_1}\\binom{|S_2|}{k_2}$ different vectors $x\\in\\{0,1\\}^N$ such that $\\tilde{c}(S_1, \\mu, x)=k_1$ and $\\tilde{c}(S_2, \\mu, x)=k_2$.\n\\end{itemize}\nUsing these properties, we have, assuming $|S_2 | \\geqslant \\frac{1}{4} \\cdot \\frac{N}{2}$ and $N$ bigger than some constant,\n\\begin{align*}\n    \\sum_{x \\in \\{0, 1\\}^N}  |z^{c_{\\tmop{ch}} (\\lambda,\n  \\mu, x)} - z^{c_{\\tmop{ch}} (\\lambda, \\mu', x)} |\n  &= \\sum_{x \\in \\{0, 1\\}^N} |z^{\\tilde{c}(S_1, \\mu, x) + \\tilde{c}(S_2, \\mu, x)} - z^{\\tilde{c}(S_1, \\mu', x) + \\tilde{c}(S_2, \\mu', x)} |\\\\\n  &= \\sum_{x \\in \\{0, 1\\}^N} |z^{\\tilde{c}(S_1, \\mu, x)+\\tilde{c}(S_2, \\mu, x)} - z^{\\tilde{c}(S_1, \\mu, x)+|S_2|-\\tilde{c}(S_2, \\mu, x)} |\\\\\n  &= 2^{\\frac{N}{2}}\\sum_{k_1=0}^{|S_1|} \\binom{|S_1|}{k_1}\\sum_{k_2=0}^{|S_2|} \\binom{|S_2|}{k_2} |z^{k_1+k_2} - z^{k_1+|S_2|-k_2} |\\\\&\\geq C\\cdot \\frac{2^N}{(1-4\\delta)^{N\/2}}\\cdot \\varepsilon,\n\\end{align*}\nwhere $C>0$ is an absolute constant, and for the last inequality we invoked Lemma~\\ref{lemma:technical_distance_lb_2xbinomials_conditional_tree}.\n Recalling now that $|S_2 | \\sim \\tmop{Bin}\\mleft( \\frac{N}{2},\n    \\frac{1}{2} \\mright)$, for $N$ large enough we also have\n\\[ \\Pr \\mleft[ |S_2| \\geq \\frac{N}{8} \\mright] \\geq 1 - e^{\n    - \\frac{N}{16}} > 9\/10. \\]\nThus, combining the two along with~\\eqref{eq:expression:tv}, we conclude that\n\\[\n\\Pr [d_{\\tmop{TV}} (P_{\\theta} (\\cdot \\mid x_1 = 0, x_2, \\ldots, x_d),\n    P_{\\theta} (\\cdot \\mid x_1 = 1, x_2, \\ldots, x_d)) \\geq \\Omega\n    (\\varepsilon)] \\geq 9\/10,\n\\]\nestablishing Lemma~\\ref{lemma;distance:mixture:of:trees}. \\hfill$\\blacksquare$\n\n\\subsection{Bounds on moment-generating functions}\nWe start with some relatively simple statements:\n\\begin{fact}\n\\label{fact:bound_on_MGF_Binomial_not_general}If $X \\sim \\tmop{Bin}\n(m, p)$, then, for any $0 \\leq t \\leq 1$,\n$\\mathbb{E} [e^{t X}] \\leq \\exp (2 t mp).$\n\\end{fact}\n\\begin{proof}\n  This follows from computing explicitly \n  $\n  \\mathbb{E} [e^{t X}] = (1 + p (e^{t} - 1))^m \\leq (1 + 2\n    t p)^m \\leq e^{2 t mp},\n    $\n    where the first inequality uses that $t\\leq 1$. \n\\end{proof}\n\\noindent We will also require the following decoupling inequality:\n\\begin{lemma}\n  \\label{lemma:decoupling}Let $F\\colon \\mathbb{R} \\to \\mathbb{R}$ be a convex,\n  {\\tmem{non-decreasing}} function, and $X = (X_1, \\ldots, X_n)$ be a vector\n  of independent {\\tmem{non-negative}} random variables. Then\n  \\[ \\mathbb{E} \\mleft[ F \\mleft( \\sum_{i \\neq j} X_i X_j \\mright) \\mright] \\leq \\mathbb{E} \\mleft[\n    F \\mleft( 4 \\sum_{i, j} X_i Y_j \\mright) \\mright] \\]\n  where $Y$ is an independent copy of $X$.\n\\end{lemma}\n\\begin{proof}\n  \\label{proof:lemma:decoupling}\n  Introduce a vector of independent\n  (and independent of $X$) $\\tmop{Bern} (1 \/ 2)$ random variables $\\delta =\n  (\\delta_1, \\ldots, \\delta_n)$; so that $\\mathbb{E} [\\delta_i (1 - \\delta_j)]\n  = \\frac{1}{4} \\mathbbm{1}_{i \\neq j}$. For any realization of $X$, we can write\n  \\[ \\sum_{i \\neq j} X_i X_j = 4 \\mathbb{E}_{\\delta} \\mleft[ \\sum_{i \\neq j} \\delta_i\n     (1 - \\delta_j) X_i X_j \\mright] = 4 \\mathbb{E}_{\\delta} \\mleft[ \\sum_{i, j}\n     \\delta_i (1 - \\delta_j) X_i X_j \\mright], \\]\n  and so, by Jensen's inequality and Fubini, as well as independence of $X$ and $\\delta$,  \n  \\begin{align*}\n    \\mathbb{E}_X \\mleft[ F \\mleft( \\sum_{i \\neq j} X_i X_j \\mright) \\mright] &\n    =\\mathbb{E}_X \\mleft[ F \\mleft( 4 \\mathbb{E}_{\\delta} \\mleft[ \\sum_{i, j} \\delta_i (1\n    - \\delta_j) X_i X_j \\mright] \\mright) \\mright]\\\\\n    & \\leq \\mathbb{E}_{\\delta} \\mleft[ \\mathbb{E}_X \\mleft[ F \\mleft( 4\n    \\sum_{i, j} \\delta_i (1 - \\delta_j) X_i X_j \\mright) \\mright] \\mright]\n  \\end{align*}\n  \n  This implies that there exists some realization $\\delta^{\\ast} \\in \\{0,\n  1\\}^n$ such that\n  \\[ \\mathbb{E}_X \\mleft[ F \\mleft( \\sum_{i \\neq j} X_i X_j \\mright) \\mright]\n     \\leq \\mathbb{E}_X \\mleft[ F \\mleft( 4 \\sum_{i, j} \\delta^{\\ast}_i (1 -\n     \\delta^{\\ast}_j) X_i X_j \\mright) \\mright] \\hspace{0.17em} . \\]\n  Let $I := \\{i \\in [n] : \\delta^{\\ast}_i = 1\\}$. Then $\\sum_{i, j}\n  \\delta^{\\ast}_i  (1 - \\delta^{\\ast}_j) X_i X_j = \\sum_{(i, j) \\in I \\times\n  I^c} X_i X_j$, and we get\n  \n  \\begin{align}\n    \\mathbb{E}_X \\mleft[ F \\mleft( \\sum_{i \\neq j} X_i X_j \\mright) \\mright] &\n    \\leq \\mathbb{E}_X \\mleft[ F \\mleft( 4 \\sum_{(i, j) \\in I \\times I^c}\n    X_i X_j \\mright) \\mright] \\label{eq:decoupling:intermediate}\\\\\n    & =\\mathbb{E}_X \\mleft[ F \\mleft( 4 \\sum_{(i, j) \\in I \\times I^c} X_i Y_j \\mright) \\mright] \\notag\\\\\n    & \\leq \\mathbb{E}_X \\mleft[ F \\mleft( 4 \\sum_{(i, j) \\in I \\times I^c} X_i Y_j + 4 \\sum_{(i, j) \\not\\in I \\times I^c} X_i Y_j \\mright) \\mright] \\notag\\\\\n    & =\\mathbb{E}_X \\mleft[ F \\mleft( 4 \\sum_{i, j} X_i Y_j \\mright) \\mright] \\notag\n\\;,\n  \\end{align}\n  where the equality uses the fact that $(X_i)_{i \\in I}$ and $(X_j)_{j \\in\n  I^c}$ are independent (as $I, I^c$ are disjoint), and so replacing $\\sum_{j\n  \\in I^c} X_j$ by the identically distributed $\\sum_{j \\in I^c} Y_j$ does not\n  change the expectation; and the second inequality uses monotonicity of $F$\n  and non-negativity of $X, Y$, as $4 \\sum_{(i, j) \\not\\in I \\times I^c} X_i Y_j\n  \\geq 0$. (Note that up to (and including)~\\eqref{eq:decoupling:intermediate}, the assumption that the $X_i$'s are independent is not necessary; we will use this fact later on.)\n\\end{proof}\nNote that compared to the usual version of the inequality, we do not require\nthat the $X_i$'s have mean zero; but instead require that they be\nnon-negative, and that $F$ be monotone. We will, in the next lemma, apply\nLemma~\\ref{lemma:decoupling} to the function $F (x) = e^{2 tx}$, for some\nfixed {\\tmem{positive}} parameter $t > 0$ (so that $F$ is indeed\nnon-decreasing), and to $X_1, \\ldots, X_n$ independent Bernoulli r.v.'s. Specifically, we obtain the following bound on the MGF of the square of a Binomial:\n\n\\MGFSquaredBinomial\n\\begin{proof}\n  Write $X=\\sum_{i = 1}^m X_i$, where the $X_i$ are i.i.d.\\ $\\tmop{Bern} (p)$ (in particular, $X_i = X_i^2$).         \n  Then, by the Cauchy--Schwarz inequality and the decoupling inequality from Lemma~\\ref{lemma:decoupling}, we have, for $t>0$,\n  \\begin{eqnarray}\n    \\mathbb{E} [e^{tX^2}]  \n    & = & \\mathbb{E} \\mleft[ e^{t \\sum_i X_i} e^{t \\sum_{i \\neq j} X_i\n    X_j} \\mright] \\nonumber\\\\\n    & \\leq & \\sqrt{\\mathbb{E} \\mleft[ e^{2 t \\sum_i X_i} \\mright]} \n    \\sqrt{\\mathbb{E} \\mleft[ e^{2 t \\sum_{i \\neq j} X_i X_j} \\mright]} \\nonumber\\\\\n    \\mleft( \\text{decoupling} \\mright) & \\leq & \\sqrt{\\mathbb{E} \\mleft[ e^{2\n    t \\sum_i X_i} \\mright]}  \\sqrt{\\mathbb{E} \\mleft[ e^{8 t \\sum_{i, j} X_i Y_j}\n    \\mright]} \\text{.}  \\label{eq:decoupling_step}\n  \\end{eqnarray}\n  where $Y_j \\sim \\tmop{Bern} (p)$ are i.i.d., and independent of the $X_i$'s. Let $Y = \\sum_{i = 1}^m Y_i\\sim \\tmop{Bin}(m, p)$.         \n  From Fact~\\ref{fact:bound_on_MGF_Binomial_not_general}, as long as $2 t\n  \\leq 1$, $8 t m \\leq 1$, and $16 tmp \\leq 1$\n  (all conditions satisfied in view of our assumption),\n  \\[ \\mathbb{E}_{X, Y} [e^{8 tXY}] =\\mathbb{E}_X [\\mathbb{E}_Y [e^{8 tXY}]]\n    \\leq \\mathbb{E}_X [e^{16 tXmp}] \\leq e^{32 tm^2 p^2},\n  \\]\n  and $\\mathbb{E} \\mleft[ e^{2t X} \\mright] \\leq e^{4tmp}$. Going back to~\\eqref{eq:decoupling_step}, this implies\n  \\[\n    \\mathbb{E} [e^{tX^2}] \n    \\leq \\sqrt{\\exp \\left(4tmp\\right)}\\sqrt{\\exp \\left(32 tm^2 p^2 \\right)}\n    = \\exp \\left(2 tmp + 16 tm^2 p^2 \\right)\\,,\n  \\]\n  concluding the proof.\n\\end{proof}\nWe will prove an MGF bound on the truncated Multinomial in Lemma \\ref{lemma:bound_on_truncated_multinomial_MGF} (noting that using MGF bound of Multinomial distribution is not nearly enough), as required by our analysis on the independence testing lower bound; prior to that, we will need two important lemmas: Lemma \\ref{lemma:factorial_fraction_inequality} and Lemma \\ref{lemma:multinomial_truncated_MGF}. These two lemmas both try to bound the expression with a uniform and more manageable term.\n\n\\iffalse\nThe intuition for this particular analysis starts with the the following\narguments (using the notation in the proof):\n\\begin{itemize}\n  \\item We can obtain a ``trivial'' bound via Gibbs' inequality (which is\n  based on Jensen's inequality) in\n  (\\ref{equation:everything-grouped-together-log-sum}), $L_1 \\log\n  \\mleft( \\frac{L_1 \\cdot D}{m k} + L_2 \\log \\mleft( \\frac{L_2 \\cdot\n  D}{m (D - k)} \\mright) \\mright) \\geq m \\log (1) = 0$; but this is\n  not enough for us, as we need to sum over a huge collection of different\n  values of $\\alpha_i$, which could be up to $\\binom{D}{k}  (m - n)^k n^{D -\n  k}$.\n  \n  \\item We notice that in order for the ``trivial'' bound to be tight, we need\n  the following to hold:\n  \\[ \\frac{L_1}{m k \/ D} = \\frac{L_2}{m (D - k) \/ D} \\leftrightarrow\n     \\frac{L_1}{k} = \\frac{m - L_1}{D - k} \\leftrightarrow L_1 D = m k. \\]\n  However, we know that $L_1 \\gtrsim O (k n) = m k \/ \\sqrt{D}$, and there $L_1\n  D \\gg m k$, leaving a potential gap for us. As such we suspect that it is\n  closely related to an application of the more general Jensen's gap.\n  \n  \\item One key step of our analysis in\n  (\\ref{equation:swapping-D-with-lower-order-D}) replaces $D$ with $D^{3 \/\n  4}$, as it is more likely to make Jensen's gap smaller ($D^{1 \/ 2}$ is where\n  the Jensen's inequality above becomes tight, but it makes the analysis\n  messy; and thus we choose a $3 \/ 4$ in place of $1 \/ 2$).\n\\end{itemize}\n\\fi\n\\begin{lemma}\n  \\label{lemma:factorial_fraction_inequality}\n  Fix $m,\\Delta,D$ such that $\\frac{m}{\\Delta} \\new{\\leq}c\\sqrt{D}$ for some $c>0$ (and $D> \\max(16c^4, e^{100})$). Fix any integer $k > 0$ and a tuple of non-negative integers $(a_1,\\dots,a_D)$ summing to $m$ such that $L \\coloneqq \\sum_{i = 1}^k a_i > k\\Delta$ (in particular, $k \\leq c\\sqrt{D}$). \n  Suppose $(\\alpha_1, \\ldots, \\alpha_D)$ follows\n  a multinomial distribution with\n  parameters $m$ and $(1\/D, \\ldots, 1\/D)$. Then,\n  \\[\n    e^{4 L} \\Pr \\mleft[ \\orgvec{\\alpha} = (a_1, \\ldots, a_D)\n     \\mright] \\leq m\\cdot  \\exp (- \\frac{1}{5} L \\log D) .\n    \\]\n\\end{lemma}\n\n\\begin{proof}\n  Via a multinomial distribution grouping argument, the probability can be\n  bounded by considering a grouping of two random variables, $L_1 = \\sum_{i = 1}^k\n  \\alpha_i$ and $L_2 = \\sum_{i = k + 1}^D \\alpha_i$, where $(L_1,L_2)$ follows\n  a multinomial distribution with parameters $m$ and $\\mleft( \\frac{k}{D},\n  \\frac{D - k}{D} \\mright)$, namely, recalling $L = \\sum_{i = 1}^k a_i$ and setting $T \\coloneqq \\sum_{i = k + 1}^D a_i$, \n  \\[ \\Pr \\mleft[ \\orgvec{\\alpha} = (a_1, \\ldots, a_D) \\mright] \\leq \n  \\Pr\\mleft[ L_1 = L, L_2 = T    \\mright] = \\frac{m!}{L! T!}  \\mleft( \\frac{k}{D}\n     \\mright)^{L} \\mleft( \\frac{D - k}{D} \\mright)^{T} \\]\n  Moreover, note that $m=L+T$. Via Stirling's approximation, we have\n  \\begin{equation}\n    \\label{equation:stirling_gibbs_application} \n    \\frac{m!}{L ! T!}\n      \\leq \\exp(m \\log m + \\log m - L \\log L - T \\log T)\n  \\end{equation}\n from which we can write, taking the logarithm for convenience,\n  \\begin{align}\n    \\log\\mleft( \\frac{m!}{L!T!}  \\mleft( \\frac{k}{D} \\mright)^{L}\n    \\mleft( \\frac{D - k}{D} \\mright)^{T} \\mright) \n    & \\leq \\log m - \\mleft( L \\log \\frac{L D}{m k} + T \\log \\frac{T  D}{m (D - k)}\n    \\mright)\n    \\label{equation:everything-grouped-together-log-sum}\\\\\n    & = \\log m - \\mleft( L \\log \\frac{LD}{m\n    k }   + T \\log \\mleft( \\frac{TD}{m ({D^{3 \/ 4}} -\n    k)}  \\frac{{D^{3 \/ 4}} - k}{D - k} \\mright) \\mright) \\label{equation:swapping-D-with-lower-order-D}\\\\\n    & \\leq \\log m - m \\log (D^{1 \/ 4}) + (m - L) \\log \\mleft(\n    \\frac{D - k}{D^{3 \/ 4} - k} \\mright)\n    \\label{equation:Gibbs-ineqaulity-again}\\\\\n    %\n    & = \\log m + m \\log \\mleft( 1 + k\\frac{D^{1 \/ 4} - 1}{D\n    - k D^{1 \/ 4}} \\mright) - L \\log \\mleft( 1 + \\frac{D^{1 \/ 4} - 1}{1 - k \/\n    D^{3 \/ 4}} \\mright)  \\nonumber\\\\\n    & \\leq \\log m + m k \\frac{D^{1 \/ 4} - 1}{D - kD^{1 \/ 4}} - L\n    \\log (D^{1 \/ 4})\n    \\label{equation:setting-k-to-extreme}\\\\\n    & \\leq  \\log m +\n    \\frac{m k}{D^{1 \/ 2}}  \\frac{1 - 1 \/ D^{1 \/ 4}}{D^{1 \/ 4} - c} - \\frac{1}{4} L \\log D  \\nonumber\\\\\n    & \\leq  \\log m +  \n    \\frac{c L}{D^{1 \/ 4} - c} - \\frac{1}{4} L \\log D \n    \\label{equation:upperbound-for-mk-over-D-square}\\\\\n    & \\leq  \\log m + L - \\frac{1}{4} L \\log D  \\tag{as $c\/D^{1\/4}\\leq 1\/2$} \n  \\end{align}\n  where we used Gibbs' inequality for \\eqref{equation:Gibbs-ineqaulity-again};\n  we then have (\\ref{equation:setting-k-to-extreme}) by $\\log (1 + x)\n  \\leq x$ for the first term. \\eqref{equation:upperbound-for-mk-over-D-square} then follows from $k \\leq c\\sqrt{D}$ and $km \\new{\\leq} k\\Delta \\cdot c\\sqrt{D} \\leq L\\cdot  c\\sqrt{D} $. Finally,\n  \\[\n    e^{4 L}  \\frac{m!}{L!T!}  \\mleft( \\frac{k}{D}\n    \\mright)^{L} \\mleft( \\frac{D - k}{D} \\mright)^{T} \n    \\leq   \\exp (5L- \\frac{1}{4}L \\log D)\n     \\leq  m \\exp \\mleft( - \\frac{1}{5} L \\log D \\mright) \\,,\n  \\]\n  the last inequality as long as $\\log D > 100$.\n\\end{proof}\n\n\n\\begin{lemma}\n  \\label{lemma:multinomial_truncated_MGF}\n  Suppose $\\orgvec{\\alpha} =\n  (\\alpha_1, \\ldots, \\alpha_D)^T$ follows a multinomial distribution with\n  parameters $m$ and $(1\/D, \\ldots, 1\/D)$, and that $\\frac{m}{\\Delta} \\new{\\leq} c\\sqrt{D}$ for some $c>0,\\Delta\\geq 1$ with $\\Delta \\geq 40D \\geq \\Omega(c^4)$ and $D > \\Omega(1)$. For any integer $c \\sqrt{D} \\geq k \\geq 1$ and any $t\\leq 4$,\n    \\[ \n  \\mathbb{E} \\mleft[ \\prod_{i : \\alpha_i > \\Delta} e^{t \\alpha_i}\n     \\cdot \\mathbbm{1} [\\nu(\\orgvec{\\alpha})= k] \\mright] \\leq \\exp \\mleft( -\n     \\frac{1}{80} \\Delta \\log (D) \\mright) . \n    \\]\n    where $\\nu(\\orgvec{\\alpha}) \\coloneqq |\\{ i: \\alpha_i > \\Delta \\}|$ denotes the number of coordinates of $\\orgvec{\\alpha}$ greater than $\\Delta$.\n\\end{lemma}\n\\begin{proof}\n  Without loss of generality, (as later, we will sum over all combinations)\n  assume that $\\alpha_1, \\ldots, \\alpha_k$ are the coordinates larger than $\\Delta$, for some integer $k$; and denote their sum by $L$. Note that we then have $k\\Delta < L \\leq m \\new{\\leq} c\\Delta\\sqrt{D}$, and thus $0 \\leq k \\leq c\\sqrt{D}$.\n  \\begin{equation}\n    \\mathbb{E} \\mleft[ \\prod_{i : \\alpha_i > \\Delta} e^{4\\alpha_i}\n    \\cdot \\mathbbm{1} [\\nu(\\orgvec{\\alpha})= k] \\mright]\n    = \\binom{D}{k} \\!\\!\\sum_{\\alpha_1, \\ldots, \\alpha_k > \\Delta} \\sum_{\\alpha_{k\n    + 1}, \\ldots, \\alpha_D \\leq \\Delta} \\!\\!\\!e^{4 \\sum_{i = 1}^k \\alpha_i} \\Pr\n    [\\orgvec{\\alpha} = (\\alpha_1, \\ldots, \\alpha_D)] \n    \\label{equation:MGF_bound_of_truncated_multinomial}\n  \\end{equation}\n  A uniform bound on any $\\alpha_1, \\ldots, \\alpha_D$ as specified can be\n  obtained from Lemma \\ref{lemma:factorial_fraction_inequality}; and, combining it\n  with (\\ref{equation:MGF_bound_of_truncated_multinomial}), we have an\n  expression that does not depend on the value of $\\orgvec{\\alpha}$; from which\\footnote{We have the number of terms in the summation upper bounded by the following\n  analysis: $(m - \\Delta)^k$ is an upper bound of combinations of $\\alpha_1,\n  \\ldots, \\alpha_k$ with values larger than $\\Delta$; and similarly, $(\\Delta + 1)^{D -\n  k}$ will be the upper bound for the combinations of $\\alpha_{k + 1}, \\ldots,\n  \\alpha_D$ with values up to $\\Delta$.}\n  \\begin{eqnarray}\n    \\mathbb{E} \\mleft[ e^{4\\sum_{i = 1}^k \\alpha_i} \n    \\mathbbm{1} [\\nu(\\orgvec{\\alpha})= k] \\mright] \n    & \\leq & \\binom{D}{k} \\sum_{\\alpha_1, \\ldots, \\alpha_k > \\Delta, \\alpha_{k + 1},\n    \\ldots, \\alpha_D \\leq \\Delta} m e^{- \\frac{1}{5} \\Delta \\log D} \\nonumber\\\\\n    & \\leq & \\binom{D}{k} (m - \\Delta)^k \\Delta^{D - k} \\exp \\mleft( \\log m - \\frac{1}{5}\n    \\Delta \\log D \\mright) \\nonumber\\\\\n    & \\leq & \\exp \\mleft( k \\log D + k \\log (m - \\Delta) + (D - k) \\log \\Delta + \\log m\n    - \\frac{\\Delta}{5} \\log (D) \\mright) \\nonumber\\\\\n    & = & \\exp \\mleft( k \\log (D \\cdot \\frac{m - \\Delta}{\\Delta}) + D \\log \\Delta + \\log m -\n    \\frac{\\Delta}{5} \\log (D) \\mright) \\nonumber\\\\\n    & \\leq & \\exp \\mleft( - \\frac{1}{5} \\Delta \\log D+ (D+1) \\log \\Delta + \\log (c\\sqrt{D})+\n    \\frac{3}{2} k \\log (c D) \\mright) \\nonumber\\\\\n    & \\leq & \\exp \\mleft( - \\frac{1}{10} \\Delta \\log D + 2c \\sqrt{D}\n    \\log (c D)  \\mright)  \\label{weight_log_swap}\\\\\n    & \\leq & \\exp \\mleft( - \\frac{1}{80} \\Delta \\log D \\mright) . \\nonumber\n  \\end{eqnarray}\n  where (\\ref{weight_log_swap}) follows from $20\\frac{D}{\\log D} \\leq \\frac{\\Delta}{\\log \\Delta}$, which holds for $\\Delta \\geq 40D$ and $D$ large enough (larger than some absolute constant); and the last inequality holds, given the above constraints, for $D \\geq 16 c^4$.\n\\end{proof}\n\n\\MGFCappedMultinomial\n\\begin{proof}\n  Let $\\nu(\\orgvec{\\alpha}) \\coloneqq |\\{ i: \\alpha_i > \\Delta \\}|$ denote the number of coordinates of $\\orgvec{\\alpha}$ greater than $\\Delta$. Note that $\\nu(\\orgvec{\\alpha}) < L\\coloneqq \\frac{m}{\\Delta}$, and that $L = c \\sqrt{D}$s by assumption. We break down the expectation by enumerating over the possible values for $\\nu(\\orgvec{\\alpha})$,\n  from $0\\leq k\\leq L$:\n  \\begin{eqnarray}\n    \\mathbb{E} \\left[ \\prod_{i = 1}^D e^{t \\alpha_i \\mathbbm{1} [\\alpha_i > \\Delta]} \\right] \n    & = & \\mathbb{E} \\mleft[ \\sum_{k = 1}^L \\prod_{i : \\alpha_i > \\Delta} e^{t\n    \\alpha_i} \\cdot \\mathbbm{1} [\\nu(\\orgvec{\\alpha}) = k] +\n    \\mathbbm{1} [\\nu(\\orgvec{\\alpha}) = 0] \\mright] \\nonumber\\\\\n    & = & \\sum_{k = 1}^L \\mathbb{E} \\mleft[ \\prod_{i : \\alpha_i > \\Delta} e^{t\n    \\alpha_i} \\cdot \\mathbbm{1} [\\nu(\\orgvec{\\alpha}) = k] \\mright]\n    + 1 \\cdot \\Pr [\\nu(\\orgvec{\\alpha}) = 0] \\nonumber\\\\\n    & \\leq & L \\exp \\mleft( - \\frac{1}{80} \\Delta \\log D \\mright) + \\Pr [\\nu(\\orgvec{\\alpha}) = 0]  \n    \\label{equation:application_multinomial_truncated_MGF}\\\\\n    & \\leq & c D^{1\/2} \\exp \\mleft( - \\frac{1}{80} \\Delta \\log D \\mright) + 1\\,, \\nonumber\n  \\end{eqnarray}\n  where (\\ref{equation:application_multinomial_truncated_MGF}) follows from\n  Lemma \\ref{lemma:multinomial_truncated_MGF}.\n\\end{proof}\n\n\n\nWe now state and prove our last lemma, Lemma~\\ref{lemma:MGF_Binomial_square_bound_with_min}, on the MGF of the square of a truncated Binomial:\n\n\\MGFSquaredCappedBinomial\n\\begin{proof}\n  We will analyze the sampling process in Definition \\ref{definition:coupling-decoupling-sampling-procedure-attempt-2}:\n  %\n  \\begin{definition}\n    \\label{definition:coupling-decoupling-sampling-procedure-attempt-2}\n    Fix integers $m \\geq \\Delta \\geq 1$, and let $X'_1,\\dots, X'_m$ be i.i.d.\\ $\\tmop{Bern} (p)$ random variables. Define the distribution of $X_1,\\dots,X_m$ through the following sampling process:\n    \\begin{enumerate}\n      \\item Initialize $X_i = 0$ for all $i \\in [m]$; sample $\\{ X_i' \\}_{1\\leq i\\leq m}$ as $m$\n      i.i.d.\\ $\\tmop{Bern} (p)$;\n      \\item If $\\sum_{i \\in [m]} X_i' < \\Delta$, let $X_i = X_i'$ for all $i \\in\n      [m]$;\n      \\item If $\\sum_{i \\in [m]} X_i' \\geq \\Delta$, let $\\mathcal{S}' = \\{ i \\in\n      [m] : X_i' = 1 \\}$ and let $\\mathcal{S}$ be a uniformly random subset of\n      $\\mathcal{S}'$ of size $\\Delta$; set $X_i = X_i'$ for $i \\in \\mathcal{S}$.\n    \\end{enumerate}\n  \\end{definition}\n  %\n\n  \n  %\n  %\n  %\n  \\noindent Consider a sequence of random variable $X_1,\\dots,X_m$ as defined in\n  Definition~\\ref{definition:coupling-decoupling-sampling-procedure-attempt-2}; each\n  $X_i$ (for $1\\leq i\\leq m$) is supported on $\\{ 0, 1 \\}$ (so that, in particular, $X_i^2=X_i$); and $X = \\sum_{i \\in [m]}\n  X_i$. By the Cauchy--Schwarz inequality,\n  \\begin{eqnarray}\n    \\mathbb{E} [e^{t X^2}] & = & \\mathbb{E} \\left[ e^{t \\sum_{i = 1}^m X_i + t\n    \\sum_{i \\neq j} X_i X_j} \\right] \\nonumber\\\\\n    & \\leqslant & \\sqrt{\\mathbb{E} \\left[ e^{2 t \\sum_{i = 1}^m X_i} \\right]}\n    \\sqrt{\\mathbb{E} \\left[ e^{2 t \\sum_{i \\neq j} X_i X_j} \\right]}\n    \\nonumber\\\\\n    & \\leqslant & \\sqrt{\\mathbb{E} \\left[ e^{2 t \\sum_{i = 1}^m X_i} \\right]}\n    \\sqrt{\\mathbb{E} \\left[ e^{8 t \\sum_{(i, j) \\in I \\times I^c} X_i X_j}\n    \\right]} \n    \\label{equation:coupling-decoupling-application-square-truncated-binomials}\\\\\n    & \\leqslant & \\sqrt{\\mathbb{E} \\left[ e^{2 t X} \\right]}\n    \\sqrt{\\mathbb{E} [e^{8 t Y_1 Y_2}]} \n    \\label{equation:dominance_application}\n  \\end{eqnarray}\n  where $Y_1 \\sim \\min (\\tmop{Bin} (| I |, p), \\Delta)$, $Y_2 \\sim \\min (\\tmop{Bin}\n  (| I^c |, p), \\Delta)$ and $Y_1$ is independent of $Y_2$ (and $(I, I^c)$ is some fixed, but   unknown partition of $[m]$).\n  \\eqref{equation:coupling-decoupling-application-square-truncated-binomials} follows from the intermediate step~\\eqref{eq:decoupling:intermediate} in the proof of Lemma~\\ref{lemma:decoupling} (observing that $x\\mapsto e^{t x}$ is convex, and\n  non-decreasing as $t>0$; and using the remark from that proof about the independence of $X_i$'s not being required up to that step) and (\\ref{equation:dominance_application}) follows from Lemma\n  \\ref{lemma:coupling-for-decoupling-attempt-2}. We will implicitly use Facts~\\ref{fact:useful_dominance_order},\n  \\ref{fact:min_preserves_dominance_order}, and~\\ref{fact:dominance_between_binomials} for the remaining calculations, eventually replacing most expressions with $X' \\sim \\tmop{Bin} (m, p)$.\n  \n  Recalling that $X \\leq X'$ by definition, the first term \n  of~\\eqref{equation:dominance_application} can be bounded as $\\mathbb{E} [e^{2 t X}] \\leqslant e^{4 t m p}$. Moreover, from our assumption, $t Y_1 \\leq t \\Delta \\leq 1 \/ 8$ and $t m p\n  \\leq 1 \/ 16$. Combined with the fact that $Y_1, Y_2$ is dominated by $X\n  \\sim \\min (\\tmop{Bin} (m, p), \\Delta)$ and thus by $X' \\sim \\tmop{Bin} (m, p)$,\n  we have\n    \\[\n        \\mathbb{E} [e^{8 t Y_1 Y_2}] = \\mathbb{E}_{Y_1} [\\mathbb{E}_{Y_2}\n        [e^{8 t Y_1 Y_2}]]\n        \\leqslant \\mathbb{E}_{Y_1} [e^{16 t Y_1 m p}]\n        \\leqslant e^{32 t m^2 p^2} .\n    \\]\n  Going back to~\\eqref{equation:dominance_application},\n  this implies\n    \\[\n        \\mathbb{E} [\\exp \\left(t X^2 \\right)] \\leq \\sqrt{\\exp \\left( 4 t m p \\right)}  \\sqrt{\\exp \\left( 32 t m^2 p^2 \\right)} = \\exp \\left(2 t m p + 16 t m^2 p^2 \\right),\n    \\]\n  concluding the proof.\n\\end{proof}\n\n\\subsection{Stochastic dominance results between truncated Binomials} %\n\\begin{fact}\\label{fact:useful_dominance_order}\nLet $X \\sim \\tmop{Bin} (m, p)$, and $0 < n \\leq m$. Defining $Y \\coloneqq \\min (X, n)$ and $Z \\coloneqq X \\mid X\\leq\nn$, we have, for every $k\\geq 0$,\n\\[ \\Pr [X \\geq k] \\geq \\Pr [Y \\geq k] \\geq\n   \\Pr [Z \\geq k], \\]\ni.e., $X \\succeq Y \\succeq Z$, where $\\succeq$ denotes first-order stochastic dominance.\n\\end{fact}\n\n\\begin{proof}\n  We can write the PMF of $Z$ and $Y$, for all $0\\leq k\\leq n$,\n  \\[ \\Pr [Y = k] = \\left\\{\\begin{array}{ll}\n       \\Pr [X = k], & k < n\\\\\n       \\Pr [X \\geq n], & k = n\n     \\end{array}\\right., \\qquad \\Pr [Z = k] = \\frac{\\Pr [X = k]}{\\Pr [X \\leq n]} . \\]\n  It follows that $\\Pr [Y \\geq k] = \\Pr [X \\geq k]\\mathbbm{1}\\{k\\leq n\\}$, which gives the first part of the statement.\n  \n  The second part follows from a direct comparison between the two CDF of $Z, Y$: indeed, for $0\\leq k \\leq n$,\n  \\begin{eqnarray*}\n   \\Pr [Y \\geq k] \\geq \\Pr [Z \\geq k]\n    & \\Leftrightarrow & \\Pr [X \\geq k] \\geq \\frac{\\Pr [n \\geq\n    X \\geq k]}{\\Pr [X \\leq n]}\\\\\n    & \\Leftrightarrow & \\Pr [X \\geq k] (1 - \\Pr [X > n])\n    \\geq \\Pr [X \\geq k] - \\Pr [X > n]\\\\\n    & \\Leftrightarrow & \\Pr [X \\geq k] \\Pr [X > n] \\leq \\Pr [X > n]\\\\\n    & \\Leftrightarrow & \\Pr [X \\geq k] \\leq 1\\,,\n  \\end{eqnarray*}\n  and this last inequality clearly holds.\n\\end{proof}\n\nWe also record the facts below, which follow respectively from the more general result that first-order stochastic dominance is preserved by non-decreasing mappings, and from a coupling argument.\n\\begin{fact}\n\\label{fact:min_preserves_dominance_order}\nConsider two real-valued random variables $X,Y$, \nand $n \\geq 0$. If $X \\succeq Y$, then $\\min(X,n) \\succeq \\min(Y,n)$: for all $k$,\n\\[ \\Pr [\\min (X, n) \\geq k] \\geq \\Pr [\\min (Y, n) \\geq k]\\,; \\]\ni.e., the $\\min$ operator preserves first-order stochastic dominance relation.\n\\end{fact}\n\\begin{fact}\n\\label{fact:dominance_between_binomials}\nLet $X \\sim \\tmop{Bin} (n, p)$ and $Y \\sim \\tmop{Bin} (m, p)$, where $m \\geq n$. Then $X \\preceq Y$.\n\\end{fact}\n\\begin{lemma}\n  \\label{lemma:coupling-for-decoupling-attempt-2}\n  Let\n  $X_1,\\dots,X_m$ be sampled from the sampling process in Definition~\\ref{definition:coupling-decoupling-sampling-procedure-attempt-2}, and $I,\n  I^c$ be any partition of $[m]$. \n  Define $Z_I \\coloneqq \\sum_{i \\in I} X_i$, $Z_{I^c} \\coloneqq \\sum_{i \\in I^c} X_i$, and $Y_I\n  \\sim \\min (\\tmop{Bin} (| I |, p), n)$, $Y_{I^c} \\sim \\min (\\tmop{Bin} (| I^c\n  |, p), n)$. Then \n  \\[\n    Z_I \\cdot Z_{I^c} \\preceq Y_I \\cdot Y_{I^c}\\,.\n  \\]\n\\end{lemma}\n\n\\begin{proof}\n  We prove the lemma by defining a coupling $Z_I, Z_{I^c}, Y_I, Y_{I^c}$ such that $Z_I \\cdot Z_{I^c} \\leq Y_I \\cdot Y_{I^c}$ with probability one. The sampling process below will\n    generate samples $(X_i)_{1\\leq i\\leq m},  Z_I, Z_{I^c}, Y_I, Y_{I^c}$ for all possible\n    realizations of $I$ and $I^c$. In other words, from a given sequence $\\{\n    X_i' \\}_{i \\in [m]}$, we will generate $\\{ X_i \\}_{i \\in [m]}, Y_{I_1},\n    Y_{I^c_1}, Y_{I_2}, Y_{I_2^c}, \\ldots, Y_{I_{2^m}}, Y_{I_{2^m}^c}, Z_{I_1},\n    Z_{I^c_1}, Z_{I_2}, Z_{I_2^c}, \\ldots, Z_{I_{2^m}}, Z_{I_{2^m}^c}$, where the $(I_i, I^c_i)$ enumerate all partitions of $[m]$ in two sets.\n    \\begin{enumerate}\n      \\item Initialize $X_i = 0$ for all $i \\in [m]$; sample $( X_i' )_{1\\leq i\\leq m}$ as\n      $m$ i.i.d.\\ $\\tmop{Bern} (p)$;\n      \n      \\item If $\\sum_{i \\in [m]} X_i' < n$, let $X_i = X_i'$ for all $i \\in\n      [m]$;\n      \n      \\item If $\\sum_{i \\in [m]} X_i' \\geq n$, let $\\mathcal{S}' = \\{ i\n      \\in [m] : X_i' = 1 \\}$ and let $\\mathcal{S}$ be a uniformly random subset\n      of $\\mathcal{S}'$ with size $n$; set $X_i =\n      X_i'$ for $i \\in \\mathcal{S}$.\n      \n      \\item \\label{enumerate:the-coupling-step}For each $I \\in \\{ I_1, \\ldots,\n      I_{2^m} \\}$, denote $\\mathcal{S}_I' =\\mathcal{S}' \\cap I$. Select a\n      uniformly random subset of $\\mathcal{S}_I'$ with at most $n$ indices which is a\n      superset of $\\mathcal{S} \\cap I$. In more detail, if $| \\mathcal{S} \\cap\n      I | < n$, select $\\min (| \\mathcal{S}_I' |, n) - | \\mathcal{S} \\cap I |$\n      elements uniformly at random from $\\mathcal{S}_I' \\setminus (\\mathcal{S} \\cap\n      I)$ to add to $\\mathcal{S} \\cap I$, which becomes $\\mathcal{S}_I$;\n      else, let $\\mathcal{S}_I =\\mathcal{S} \\cap I$. Repeat a similar process for $I^c$ to obtain\n      $\\mathcal{S}_{I^c}$.\n      \n      \\item For each $I \\in \\{ I_1, \\ldots, I_{2^m} \\}$, set $Y_I = \\sum_{i \\in\n      \\mathcal{S}_I} X_i'$ and $Y_{I^c} = \\sum_{i \\in S_{I^c}} X_i'$.\n    \\end{enumerate}\n \n  From the above definition, we can\n  readily see that for any $I$, $Y_I \\geq Z_I$ and $Y_{I^c} \\geq\n  Z_{I^c}$. What is left is to argue that the $Y_I \\sim \\min (\\tmop{Bin} (| I\n  |, p), n)$ and $Y_{I^c} \\sim \\min (\\tmop{Bin} (| I^c |, p), n)$. We start by\n  noting that for any $k < n$, $\\{ Y_I = k \\} = \\{ | \\mathcal{S}_I | = k \\} =\n  \\{ | \\mathcal{S}_I' | = k \\}$. The last equality comes from the fact that $|\n  \\mathcal{S}_I | < n$ can only mean that $| \\mathcal{S}_I' | < n$, and the\n  selection process in step \\ref{enumerate:the-coupling-step} will thus add all elements from\n  $\\mathcal{S}_I'$ to $\\mathcal{S}_I$. From here, we have $\\Pr [Y_I = k] = \\Pr\n  [| \\mathcal{S}_I' | = k] = \\Pr [\\tmop{Bin} (m, p) = k]$, for $k < n$; and we\n  have $\\Pr [Y_I = n] = 1 - \\Pr [Y_I < n] = \\Pr [\\tmop{Bin} (m, p) \\geq\n  n]$. As a result, $Y_I \\sim \\min (\\tmop{Bin} (| I |, p), n)$. Similarly, we\n  can argue that $Y_{I^c} \\sim \\min (\\tmop{Bin} (| I^c |, p), n)$.\n\\end{proof}\n\n\n\n\\section{Introduction}\n\\input{sec-intro}\n\n\\section{Preliminaries}\n\\input{sec-preliminaries}\n\n\\section{Upper Bound for Independence Testing}\n\\input{sec-upperbound}\n\n\\section{The $\\Omega(2^{d\/2}{n} \/ \\varepsilon^2)$ Lower Bound}\n    \\label{sec:lb:outline}\n\\input{sec-lowerbound}\n\n\\iffalse\n\\section{Extension to Testing Substructure}\n    \\input{..\/testing_substructure}\n\\fi\n\n\\section{Useful results on the MGFs of Binomials and Multinomials}\n    \\label{ssec:tools:mgf}\n\\input{sec-mgfs}\n\n\\section*{Acknowledgments}\nYang would like to thank Vipul Arora and Philips George John for the helpful discussions on the lower bound analysis; and Vipul, specifically, for providing valuable feedback on the manuscript.\n\n\\bibliographystyle{plainnat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn this paper we derive stability estimates in several norms for the $L^{2}$ projection and a family\nof quasiinterpolants in the space of smooth, periodic, polynomial splines on a uniform mesh in a \nfinite interval. Such estimates are of course useful in deriving approximation properties for the\n$L^{2}$ projection and the quasiinterpolant. They have also been used in deriving optimal-order-of\n-accuracy error estimates for Galerkin finite-element approximations to the solution of e.g. first-\norder hyperbolic problems and nonlinear dispersive wave pde's, cf. e.g. \\cite{bdk},\n\\cite{dk}, \\cite{bdkmc}, \\cite{adm}, \\cite{ad1}, \\cite{adk}. Stability estimates for the $L^{2}$ \nprojection onto finite element spaces have been derived in several works. For example, see \n\\cite{ddw} for estimates in $L^{\\infty}$ in the case of a quasiuniform mesh in one dimension, \n\\cite{ct} for estimates in $L^{p}$ and the Sobolev spaces $W_{\\infty}^{p}$, $1\\leq p\\leq \\infty$, in one and two \ndimensions for more general than quasiuniform types of meshes, and also the references of these \npapers. Here, we take advantage of the uniform mesh, and the periodicity of the underlying space of \nsmooth splines, and use matrix methods to derive suitable decay estimates of the elements of the\ninverse of a Gram matrix associated with the standard basis of the space of splines. To this end, we\nuse properties of cyclic (circulant) matrices listed in \\cite{se}, and relevant results of Demko\n\\emph{et al.}, \\cite{dms}, and Bini and Capovani, \\cite{bc}. These decay properties enable us to\nprove stability estimates for the $L^{2}$ projection in the $L^{2}$-based Sobolev spaces \n$H_{per}^{l}$ of periodic functions, for $l=0,\\dots,r-1$, where $r$ is the order of the spline space,\nand for continuous, periodic functions in $W_{l}^{\\infty}$, for $l=0,\\dots,r-1$. \\par\nQuasiinterpolants of continuous, periodic functions in the space of smooth, periodic splines have\nbeen studied, among other, in \\cite{s}, \\cite{tw}, \\cite{t}, \\cite{ds}, and in  references of \nthese works. All these quasiinterpolants achieve optimal-order accuracy in $L^{2}$ provided the\nfunctions they approximate are smooth enough. Of particular interest is the Thom{\\'e}e-Wendroff\nquasiinterpolant, \\cite{tw}, for which $L^{2}$ inner products of truncation errors with elements of\na special basis of the spline space are superaccurate due to cancellations. This enables one to prove\noptimal-order $L^{2}$-error estimates for Galerkin approximations of the solutions of the periodic\ninitial-value-problems for the types of pde's previously mentioned.\\par\nIn section 2 of the paper at hand we introduce the spline spaces and their standard basis. In \nsection 3 we list a series of properties of cyclic matrices that will be needed in the sequel, \nmostly following \\cite{se}. In section 4, based on results from \\cite{dms} and \\cite{bc}, we \nestablish the required decay estimates for the elements of the inverse of the Gram matrix. These are\nneed in section 5 in order to establish stability results for the $L^{2}$ projection onto the spline\nspaces in the Sobolev spaces $H_{per}^{l}$ and $W_{\\infty}^{l}\\cap H_{per}^{l}$ for \n$l=0,1,\\dots r-1$. Finally, in section 6, we consider the quasiinterpolants and prove stability \nestimates for them in $H_{per}^{l}$ and $W_{\\infty}^{l}\\cap H_{per}^{l}$, for $l=1,\\dots,r-1$, and in\n$C_{per}$. These estimates do not depend of course on the decay results of section 4.\\par\nWe use the following notation: For integer $k\\geq 0$, $C_{per}^{k}$ ($C_{per}\\equiv C_{per}^{0}$)\nwill denote the space of continuous, 1-periodic functions that are $k$ times continuously\ndifferentiable. (By $C_{per}^{k}([0,1])$ we mean the restriction on $[0,1]$ of such functions).\nAnalogously, $L_{per}^{2}$ will denote the space of 1-periodic functions that are \nsquare-integrable over one period. The $L^{2}$ inner product, resp. norm, on $[0,1]$ will be denoted by\n$(\\cdot,\\cdot)$, resp. $\\|\\cdot\\|$. For integer $l\\geq 0$, the norm on $[0,1]$ of the \n$L^{2}$-based Sobolev spaces of 1-periodic functions will be denoted by $\\|\\cdot\\|_{l}$, while\nthe analogous norm of $W_{\\infty}^{l}$ by $\\|\\cdot\\|_{l,\\infty}$. \nThe Euclidean inner product, resp. norm, on $\\mathbb{R}^{N}$  will be denoted by \n$\\eup{\\cdot,\\cdot}$, resp. $\\abs{\\cdot}$, while\n$\\mathbb{P}_{r}$ will stand for the polynomials of degree at most $r$.\nFinally, for $a>0$ we will  denote by $\\floor{a}$ the largest integer that is less or equal to \n$a$.\n\\section{Spline spaces, basis} Let $r$ and $N$ be integers such that $r\\geq 2$, $N\\geq 4r$. Let\n$h=1\/N$ and $x_{i}=ih$, $i=0,1,\\dots,N$, be a uniform partition of $[0,1]$. We consider the \n$N$-dimensional space of the 1-periodic, smooth, piecewise polynomial splines\n\\[\n\\mathcal{S}_{h}^{r}=\\{v\\in C_{per}^{r-2}[0,1] : v_{|(x_{i-1},x_{i})}\\in\n\\mathbb{P}_{r-1}, \\,\\, i=1,2,\\dots,N\\},\n\\]\nand the space\n\\[\n\\mathcal{S}_{h}^{1} = \\{v\\in L_{per}^{2}(0,1) : v_{|(x_{i-1},x_{i})}=\\text{constant},\n\\,\\, i=1,2,\\dots,N\\}.\n\\]\nFollowing e.g. \\cite{tw}, we define a standard basis of $\\mathcal{S}_{h}^{r}$ as follows. Let \n$v_{1}$ be the characteristic function of $[-\\tfrac{1}{2},\\tfrac{1}{2}]$ and $v_{r}$ the convolution\nof $v_{1}$ with itself $r-1$ times. Thus, $v_{r}$, with support $[-\\tfrac{1}{2}r,\\tfrac{1}{2}r]$, is\nthe $B$-spline of order $r$. If\n\\[\n\\phi_{l}(x)=v_{r}(h^{-1}x - l - \\tfrac{r-2}{2}),\\,\\, l\\in \\mathbb{Z},\\quad\n\\text{and}\\quad\n\\Phi_{j}(x) =\\sum_{l\\in \\mathbb{Z}}\\phi_{j+lN}(x),\\,\\, j=1,2,\\dots, N,\n\\]\nthen, the restrictions of $\\{\\Phi_{j}\\}_{j=1}^{N}$ on $[0,1]$ are a basis of $\\mathcal{S}_{h}^{r}$. \\par\n{\\bf{Remarks:}} 1. Here we define $\\Phi_{j}$ in terms of \n$\\phi_{l}(x)=v_{r}(h^{-1}x - l - \\tfrac{r-2}{2})$ and not in the standard way via \n$\\phi_{l}(x)=v_{r}(h^{-1}x - l)$, since we wish that the support of $\\Phi_{j}$ in $[0,1]$ to be\nan interval or union of intervals with endpoints that are integer multiples of $h$. Thus, in \n$[0,1]$, supp$\\Phi_{j}=[x_{j-1},x_{j+r-1}]$, for $j=1,2,\\dots,N-(r-1)$, and \nsupp$\\Phi_{j}=[x_{j-1},1]\\cup [0,x_{j-(N-(r-1))}]$ for $j=N-(r-2),N-(r-3),\\dots,N$. \\\\\n2. If $v\\in\\mathcal{S}_{h}^{r}$, then $v$ may be written in the form \n\\[\nv(x) = \\sum_{j=1}^{N}V_{j}\\Phi_{j}(x), \\quad \\text{or}\\quad \nv(x)=\\sum_{l\\in\\mathbb{Z}}V_{l}\\phi_{l}(x),\n\\]\nwhere the coefficients $V_{l}$ are periodic with period $N$, i.e. $V_{l}=V_{l+N}$, $l\\in\\mathbb{Z}$.\nIn addition it holds that $\\Phi_{l}(x)=\\Phi_{l+N}(x)$ for any integer $l$.\\\\\n3. A basis of $\\mathcal{S}_{h}^{1}$ is, obviously, the set $\\{\\Phi_{j}\\}_{j=1}^{N}$, where\n$\\Phi_{j}(x)=v_{1}(h^{-1}x-j+\\tfrac{1}{2})$.\\\\\n4. If  $\\{\\Phi_{j}\\}_{j=1}^{N}$, $\\{\\Psi_{j}\\}_{j=1}^{N}$ are the bases of $\\mathcal{S}_{h}^{r}$,\n$\\mathcal{S}_{h}^{r-1}$, respectively, then $\\Phi_{j}'(x)=h^{-1}(\\Psi_{j}(x) - \\Psi_{j+1}(x))$.\nIndeed, it follows by the definition of $v_{r}$ that\n\\[\nv_{r}(x)=\\int_{x-\\frac{1}{2}}^{x+\\frac{1}{2}}v_{r-1}(y)dy,\n\\]\nfrom which $v_{r}'(x) = v_{r-1}(x+\\frac{1}{2}) - v_{r-1}(x - \\frac{1}{2})$, and since\n$\\Phi_{j}(x)=\\sum_{l\\in\\mathbb{Z}}v_{r}(h^{-1}x - lN - j - \\frac{r-2}{2})$, it follows that\n\\begin{align*}\n\\Phi_{j}'(x) & = h^{-1}\\sum_{l\\in\\mathbb{Z}}\n\\Bigl[v_{r-1}\\Bigl(h^{-1}x - lN-j-\\frac{r-2}{2} +\n\\frac{1}{2}\\Bigr) - v_{r-1}\\Bigl(h^{-1}x -\nlN-j-\\frac{r-2}{2} -\\frac{1}{2}\\Bigr)\\Bigr]\\\\\n& = h^{-1}\\sum_{l\\in\\mathbb{Z}}\\Bigl[v_{r-1}\\Bigl(h^{-1}x - lN-j - \\frac{r-3}{2}\\Bigr)\n- v_{r-1}\\Bigl(h^{-1}x - lN - j-1 - \\frac{r-3}{2}\\Bigr)\\Bigr]\\\\\n& = h^{-1}\\bigl(\\Psi_{j}(x) - \\Psi_{j+1}(x)\\bigr).\n\\end{align*}\n5. If $G$  is the Gram matrix with elements $h^{-1}(\\Phi_{k},\\Phi_{j})$, then, cf. \n\\cite[Lemma 2.1]{tw}, $G$ is a symmetric, positive definite, cyclic $N\\times N$ matrix with\neigenvalues $g(2\\pi l\/N)$, $l=1,2,\\dots,N$, where\n\\[\ng(x) = \\sum_{l\\in\\mathbb{Z}}\\hat{v}_{r}(x + 2\\pi l)^{2}, \\qquad\n\\hat{v}_{r}(x)=\\Biggl(\\frac{2\\sin\\frac{1}{2}x}{x}\\Biggr)^{r}.\n\\]\nMoreover,\n\\[\n\\underline{g}\\abs{V}^{2} \\leq \\eup{GV,V} \\leq \\overline{g}\\abs{V}^{2},\n\\]\nprovided $\\underline{g}\\leq g(x)\\leq \\overline{g}$. It follows that: \\\\\n(i)\\,\\, The elements of $G$ are numbers independent of $h$.\\\\\n(ii)\\, The function $g$ is periodic with period $2\\pi$, $g(0)=g(2\\pi)=1$, and\n$g(\\theta)\\in [\\underline{g},\\overline{g}]$ for all $\\theta\\in\\mathbb{R}$, \n\\indent\\,\\,\\,\\, where $\\overline{g}=1$\nand $\\underline{g} >0$. In addition, the maximum eigenvalue of $G$ is $\\lambda_{\\max}(G)=1$.\\\\\n(iii)\\, The eigenvalues of $G$ are included in the interval $[\\underline{g},1]$ and \n$\\underline{g}$ does not depend on $N$ but only \\indent\\,\\,\\,\\,\\, on $r$.\\\\\n(iv)\\,\\, For each element of the inverse of $G$ there holds \n$\\abs{(G^{-1})_{ij}}\\leq 1\/\\lambda_{\\min}(G)\\leq 1\/\\underline{g}$, where $\\lambda_{\\min}(G)$ \n\\indent \\,\\,\\,\\,\\,\nis the smallest eigenvalue of $G$. In particular, $\\underline{g}\\leq (G^{-1})_{11}\\leq 1$.\\\\\n(v)\\,\\, If $v_{h}=\\sum_{j=1}^{N}V_{j}\\Phi_{j}$ is an element of $\\mathcal{S}_{h}^{r}$, then\n\\begin{equation}\n\\underline{g}h\\abs{V}^{2} \\leq \\|v_{h}\\|^{2} \\leq \\overline{g}h\\abs{V}^{2}.\n\\label{eq21}\n\\end{equation} \nIt is also well known that the following inverse inequality holds in $\\mathcal{S}_{h}^{r}$: There\nexists a constant $C$ independent of $h$, such that\n\\begin{equation}\n\\|v_{hx}\\|\\leq Ch^{-1}\\|v_{h}\\|,\n\\label{eq22}\n\\end{equation}\nfor all $v_{h}\\in \\mathcal{S}_{h}^{r}$.\n\\section{Cyclic matrices} \nIn what follows we state some facts about cyclic (or circulant) matrices that will be useful in the\nsequel. A $N\\times N$ matric $C$ is called cyclic when\n\\begin{equation*}\nc_{jk}=\n\\begin{cases}\nc_{k-j+1}\\,, &  \\text{if} \\,\\,\\, k\\geq j,\\\\\nc_{k-j+N+1}\\,,& \\text{if} \\,\\,\\, k<j,\n\\end{cases}\n\\end{equation*}\nwhere $c_{j}=c_{1j}$ are the elements of the first row of $C$. A cyclic matrix may be written in the\nform \n\\[\nC = \\sum_{j=1}^{N}c_{j}P^{j-1},\n\\]\nwhere $P$ is the $N\\times N$ permutation matrix whose columns are the vectors $e_{j}$ of the\nstandard basis of $\\mathbb{R}^{N}$ written in the sequence\n\\begin{equation}\nP=[e_{N},e_{1},e_{2},\\dots,e_{N-1}].\n\\label{eq31}\n\\end{equation}\nThe following are properties of cyclic matrices, cf. e.g. \\cite{se}:\\\\\n(i)\\,\\,\\, The matrix $P$ is cyclic.\\\\\n(ii)\\,\\, $P^{N}=I_{N}$, where $I_{N}$ is the $N\\times N$ identity matrix.\\\\\n(iii)\\, $PP^{T} = P^{T}P = I_{N}$. This follows from the fact that \n$PP^{T} = [Pe_{2},Pe_{3},\\dots,Pe_{N},Pe_{1}]$, and \\\\ \\indent\\,\\,\\,\\,\\, then \n$PP^{T}=[e_{1},e_{2},\\dots,e_{N-1},e_{N}]=I_{N}$. \\\\\n(iv)\\, The product of cyclic $N\\times N$ matrices is cyclic.\\\\\n(v)\\,\\,\\, If $A$, $B$ are $N\\times N$ cyclic matrices, then $AB=BA$.\\\\\n(vi)\\, The inverse of a cyclic matrix, if it exists, is cyclic.\\\\\n(vii)\\, If $C$ is a symmetric cyclic $N\\times N$ matrix, then $c_{N+1-j}=c_{j+1}$, for \n$j=1,2,\\dots,\\floor{\\frac{N-1}{2}}$. Such \\indent \\,\\,\\,\\,\\, a matrix may be written in the form\n\\begin{equation}\nC = c_{1}I_{N} + \\sum_{j=2}^{\\frac{N}{2}}c_{j}\\Bigl(P^{j-1} + P^{1-j}\\Bigr) +\nc_{\\frac{N}{2} + 1}P^{\\frac{N}{2}}, \\quad \\text{if}\\,\\,\\, N\\,\\,\\text{is even},\n\\label{eq32}\n\\end{equation}\\indent\\,\\,\\,\\,\nor\n\\begin{equation}\nC = c_{1}I_{N} + \\sum_{j=2}^{\\frac{N+1}{2}}c_{j}\\Bigl(P^{j-1} + P^{1-j}\\Bigr),\n\\quad \\text{if}\\,\\,\\, N\\,\\,\\text{is odd}.\n\\label{eq33}\n\\end{equation}\\indent\\,\\,\\,\\,\nTo check this note that if $N$ is even, then \n\\[\nC = \\sum_{j=1}^{N}c_{j}P^{j-1}=c_{1}I_{N} + \\sum_{j=2}^{\\frac{N}{2}}c_{j}P^{j-1}\n+c_{\\frac{N}{2}+1}P^{\\frac{N}{2}} + \\sum_{j=\\frac{N}{2}+2}^{N}c_{j}P^{j-1}.\n\\]\\indent\\,\\,\\,\\,\nBut $c_{j} = c_{N+2-j}$, $j=N\/2+2, N\/2+3,\\dots,N$, from which we obtain \n\\[\n\\sum_{j=\\frac{N}{2}+2}^{N}c_{j}P^{j-1}=\\sum_{j=\\frac{N}{2}+2}^{N}c_{N+2-j}P^{j-1}\n=\\sum_{j=2}^{\\frac{N}{2}}c_{j}P^{N-(j-1)}=\\sum_{j=2}^{\\frac{N}{2}}c_{j}P^{1-j},\n\\]\\indent\\,\\,\\,\\,\nand \\eqref{eq32} follows. If $N$ is odd, then\n\\[\nC = \\sum_{j=1}^{N}c_{j}P^{j-1}=c_{1}I_{N} + \\sum_{j=2}^{\\frac{N+1}{2}}c_{j}P^{j-1}\n+ \\sum_{j=\\frac{N+1}{2}+1}^{N}c_{j}P^{j-1}.\n\\]\\indent\\,\\,\\,\\,\nHere $c_{j}=c_{N+2-j}$, $j=(N+1)\/2+1, (N+1)\/2 +2,\\dots,N$, giving \n\\[\n\\sum_{j=\\frac{N+1}{2}+1}^{N}c_{j}P^{j-1}= \\sum_{j=\\frac{N+1}{2}+1}^{N}c_{N+2-j}P^{j-1}\n=  \\sum_{j=2}^{\\frac{N+1}{2}}c_{j}P^{N+2-j-1}= \\sum_{j=2}^{\\frac{N+1}{2}}c_{j}P^{1-j},\n\\]\\indent\\,\\,\\,\\,\nfrom which \\eqref{eq33} follows.\\\\\n(viii)\\, If $C$ is an antisymmetric cyclic $N\\times N$ matrix, then $c_{N+1-j}=-c_{j+1}$,\nfor $j=1,2,\\dots,\\floor{\\frac{N-1}{2}}$, \\indent\\,\\,\\,\\,\\,\\,\\, with $c_{\\frac{N}{2}+1}=0$, if $N$ \nis even. The matrix $C$ is written in the form \n\\begin{equation}\nC = \\sum_{j=2}^{\\floor{\\frac{N+1}{2}}}c_{j}\\Bigl(P^{j-1} - P^{1-j}\\Bigr).\n\\label{eq34}\n\\end{equation}\n(ix)\\,\\, If $V\\in \\mathbb{R}^{N}$ with $V=(V_{1},V_{2},\\dots,V_{N})^{T}$ and $V_{l}=V_{l+N}$, \n$l\\in\\mathbb{Z}$, then $(P^{k}V)_{i}=V_{i+k}$, if \\\\ \\indent\\,\\,\\,\\, $i=1,2\\dots,N$, and $k$ \nany integer.\n\\section{An estimate of the elements of the inverse of $G$}\nFor the purposes of proving stability properties for the $L^{2}$ projection operator onto\n$\\mathcal{S}_{h}^{r}$, we shall prove some estimates for the elements of the inverse of the Gram\nmatrix $G$, introduced in Remark 5 of section 2.\\par\nLet $\\{\\Phi_{j}\\}_{j=1}^{N}$ be the basis of $\\mathcal{S}_{h}^{r}$ defined in section 2, let $G$ be\nthe matrix with elements $G_{ij}=h^{-1}(\\Phi_{j},\\Phi_{i})$, and $\\Gamma$ be the inverse of $G$. The \nmatrices $G$, $\\Gamma$ are cyclic, symmetric, and positive definite. If \n$g_{j}=h^{-1}(\\Phi_{j},\\Phi_{1})$, then the first row of $G$ is the row vector\n\\[ \n(g_{1},g_{2},\\dots,g_{r},0,\\dots,0,g_{r},g_{r-1},\\dots,g_{2}),\n\\]\nand according to \\eqref{eq32}, and\n\\eqref{eq33}, $G$ may be written as \n\\begin{equation}\nG = g_{1}I_{N} + \\sum_{j=2}^{r}g_{j}\\Bigl(P^{j-1} + P^{1-j}\\Bigr).\n\\label{eq41}\n\\end{equation}\n$\\Gamma$ is a full matrix and its first row is\n\\begin{align*}\n\\gamma & =(\\gamma_{1},\\gamma_{2},\\dots,\\gamma_{\\frac{N}{2}},\\gamma_{\\frac{N}{2}+1},\n\\gamma_{\\frac{N}{2}}, \\gamma_{\\frac{N}{2}-1},\\dots,\\gamma_{2}), \\quad \\text{if}\n\\quad N \\,\\, \\text{is even, or}\\\\\n\\gamma & =(\\gamma_{1},\\gamma_{2},\\dots,\\gamma_{\\frac{N-1}{2}},\\gamma_{\\frac{N+1}{2}},\n\\gamma_{\\frac{N+1}{2}}, \\gamma_{\\frac{N-1}{2}},\\dots,\\gamma_{2}), \\quad \\text{if}\n\\quad N \\,\\, \\text{is odd}.\n\\end{align*}\nTherefore, in view again of \\eqref{eq32}, \\eqref{eq33}, $\\Gamma$ may be written in the form\n\\begin{equation}\n\\begin{aligned}\n\\Gamma & =\\gamma_{1}I_{N} + \\sum_{j=2}^{\\frac{N}{2}}\\gamma_{j}\\Bigl(P^{j-1}+P^{1-j}\\Bigr)\n+ \\gamma_{\\frac{N}{2}+1}P^{\\frac{N}{2}}, \\quad \\text{if}\\quad N \\,\\, \\text{is even},\\\\\n\\Gamma & =\\gamma_{1}I_{N} + \\sum_{j=2}^{\\frac{N+1}{2}}\\gamma_{j}\\Bigl(P^{j-1}+P^{1-j}\\Bigr),\n\\quad \\text{if}\\quad N \\,\\, \\text{is odd}.\n\\end{aligned}\n\\label{eq42}\n\\end{equation}\n\\par\nDemko \\emph{et al.}, \\cite[Proposition 2.2]{dms}, proved that if a banded matrix is positive\ndefinite, then the elements of its inverse decay exponentially as they move away from the diagonal.\nThis result may be stated as follows for a symmetric, positive definite, banded matrix.\n\\begin{proposition} Let $B$ be a $N\\times N$ positive definite, symmetric, banded matrix, for which\n$B_{ij}=0$, if $\\abs{i-j} > k$ for some positive integer $k$. If $\\lambda_{\\min}(B)$, \n$\\lambda_{\\max}(B)$ are the minimum and the maximum eigenvalue, respectively, of $B$ and \n$\\lambda_{\\mu}=(\\lambda_{\\min}(B))^{1\/2}$, $\\lambda_{m}=(\\lambda_{\\max}(B))^{1\/2}$, then\n\\begin{equation}\n\\abs{(B^{-1})_{ij}} \\leq C_{B} q_{B}^{-\\abs{i-j}},\n\\label{eq43}\n\\end{equation}\nwhere\n\\begin{equation}\nC_{B} = \\frac{1}{\\lambda_{\\mu}^{2}}\\max \\Bigl(1, \\frac{(\\lambda_{\\mu} \n+ \\lambda_{m})^{2}}{2\\lambda_{m}^{2}}\\Bigr),\\quad\nq_{B} = \\Bigl(\\frac{\\lambda_{m}+\\lambda_{\\mu}}{\\lambda_{m} - \\lambda_{\\mu}}\\Bigr)^{1\/k}.\n\\label{eq44}\n\\end{equation}\n\\end{proposition} \nBased on this result we will prove the following lemma for the inverse of $G$.\n\\begin{lemma} Let $\\{\\Phi_{j}\\}_{j=1}^{N}$ be the basis of $\\mathcal{S}_{h}^{r}$ defined in section 2,\n$G$ be the $N\\times N$ matrix defined by $G_{ij} = h^{-1}(\\Phi_{j},\\Phi_{i})$, and let $\\Gamma$ be\nthe inverse of $G$. If $\\gamma$ is the first row of $\\Gamma$, there exist positive constants\n$C_{1}$, $C_{2}$, $q$, independent of $N$, such that \n\\begin{equation}\n\\abs{\\gamma_{i}} \\leq C_{1}q^{-(i-1)} + C_{2}q^{-(N-i)},\n\\label{eq45}\n\\end{equation}\nfor $i=r,r+1,\\dots, \\floor{N\/2}+1$. Moreover, there exists a constant $C_{3}$, independent of $N$,\nsuch that \n\\begin{equation}\n\\sum_{i=r}^{\\floor{\\frac{N}{2}}+1}(1 + i)\\abs{\\gamma_{i}} \\leq C_{3}.\n\\label{eq46}\n\\end{equation}\n\\end{lemma}\n\\begin{proof} Let $\\wt{G}$ be the $N\\times N$ symmetric, banded matrix with elements\n$\\wt{G}_{ij} = G_{ij}$, if $\\abs{i-j}\\leq r-1$, and $\\wt{G}_{ij}=0$, if $\\abs{i-j}>r-1$. Let\n$\\mathcal{G}$ be the $(N+2r-2)\\times (N+2r-2)$ cyclic matrix with elements \n$\\mathcal{G}_{ij}=\\mathfrak{h}^{-1}(\\Upphi_{j},\\Upphi_{i})$, $1\\leq i,j\\leq N+2r-2$,\nwhere $\\mathfrak{h}=1\/(N+2r-2)$, and $\\{ \\Upphi_{j} \\}_{j=1}^{N+2r-2}$ is the basis of \n$\\mathcal{S}_{\\mathfrak{h}}^{r}$. Hence $\\mathcal{G}$ is a\n`cyclic extension' of $\\wt{G}$, in the sense that $\\wt{G}$ is obtained from $\\mathcal{G}$ if we omit\nthe first $r-1$ and the last $r-1$ columns of $\\mathcal{G}$, and also the first $r-1$ and the last\n$r-1$ rows of $\\mathcal{G}$. Following Bini and Capovani, cf. \\cite[Proposition 4.2]{bc},\nwe obtain that $\\lambda_{\\min}(\\mathcal{G})\\leq \\lambda_{\\min}(\\wt{G})$. Hence \n$\\lambda_{\\min}(\\wt{G}) \\geq \\underline{g} >0$, i.e. $\\wt{G}$ is positive definite, and by\nProposition 4.1, we have \n\\begin{equation}\n\\abs{(\\wt{G}^{-1})_{ij}} \\leq C_{\\wt{G}}q_{\\wt{G}}^{-\\abs{i-j}},\n\\label{eq47}\n\\end{equation}\nwhere $C_{\\wt{G}}$, $q_{\\wt{G}}$ are defined as in Proposition 4.1 for $B=\\wt{G}$ and $k=r-1$. Since\n$\\lambda_{min}(\\wt{G})\\geq\\underline{g}>0$, it follows that $C_{\\wt{G}}$ is bounded above by a\nconstant independent of $N$ (see Remark 5 (iii) in section 2.) In addition, using Proposition 4.2 of\n\\cite{bc}, we also conclude that $\\lambda_{min}(\\wt{G}) < \\lambda_{max}(\\wt{G})$ since the\neigenvalues of $\\mathcal{G}$ have the same property. We conclude that $q_{\\wt{G}}$ is also\nindependent of $N$. (Of course $C_{\\wt{G}}$, $q_{\\wt{G}}$ depend on $r$.) Now, the cyclic matrix\n$G$ is written in the form $G=\\wt{G}+\\wt{W}$, where\n\\begin{equation*}\n\\wt{W} = \n\\begin{pNiceMatrix}\n0 &  & W \\\\\n\\hline\n & 0 & \\\\\n\\hline\nW^{T} & & 0 \n\\end{pNiceMatrix}\n, \\qquad\nW =\n\\begin{pmatrix}\ng_{r} & g_{r-1} & \\cdots & g_{2} \\\\\n0 & g_{r} & \\cdots & g_{3} \\\\\n\\vdots  & \\ddots  & \\ddots & \\vdots  \\\\\n 0 & \\cdots & 0 & g_{r}\n\\end{pmatrix},\n\\end{equation*}\nand therefore  $\\wt{W}\\gamma=(W\\ul{\\gamma},0,\\dots,0,W^{T}\\ol{\\gamma})^{T}$, where\n$\\ul{\\gamma} = (\\gamma_{r},\\gamma_{r-1},\\dots,\\gamma_{2})^{T}$,\n$\\ol{\\gamma}=(\\gamma_{1},\\gamma_{2},\\dots,\\gamma_{r-1})^{T}$. Since\n$G\\gamma=e_{1}$ we have $\\gamma = (\\wt{G})^{-1}e_{1} - (\\wt{G}^{-1})\\wt{W}\\gamma$, from which, for\n$r\\leq i\\leq \\floor{N\/2}+1$, we see that\n\\begin{equation}\n\\begin{aligned}\n\\gamma_{i} & =(\\wt{G}^{-1})_{i1} - \\sum_{j=1}^{r-1}(W\\ul{\\gamma})_{j}(\\wt{G}^{-1})_{ij}\n-\\sum_{j=1}^{r-1}(W^{T}\\ol{\\gamma})_{j}(\\wt{G}^{-1})_{i,N-(r-1-j)} \n= (1 - (W\\ul{\\gamma})_{1})(\\wt{G}^{-1})_{i1} \\\\\n& \\hspace{53pt} \n- (W^{T}\\ol{\\gamma})_{1}(\\wt{G}^{-1})_{i,N-(r-2)} \n-\\sum_{j=2}^{r-1}\\Bigl((W\\ul{\\gamma})_{j}(\\wt{G}^{-1})_{ij}\n+ (W^{T}\\ol{\\gamma})_{j}(\\wt{G}^{-1})_{i,N-(r-1-j)}\\Bigr).\n\\end{aligned}\n\\label{eq48}\n\\end{equation}\nBut $g_{1}\\gamma_{1} + 2(g_{2}\\gamma_{2} + \\dots g_{r}\\gamma_{r})=1$, which yields\n\\[\n(W\\ul{\\gamma})_{1} = \\sum_{j=2}^{r}g_{j}\\gamma_{j}=\\frac{1}{2}(1-g_{1}\\gamma_{1}).\n\\]\nIn addition, $(W^{T}\\ol{\\gamma})_{1}=g_{r}\\gamma_{1}$, and therefore \\eqref{eq48} may be written as\n\\begin{equation}\n\\gamma_{i}=\\frac{1+g_{1}\\gamma_{1}}{2}(\\wt{G}^{-1})_{i1}\n- g_{r}\\gamma_{1}(\\wt{G}^{-1})_{i,N-(r-2)}\n-\\sum_{j=2}^{r-1}\\Bigl((W\\ul{\\gamma})_{j}(\\wt{G}^{-1})_{ij}\n+(W^{T}\\ol{\\gamma})_{j}(\\wt{G}^{-1})_{i,N-(r-1-j)}\\Bigr).\n\\label{eq49}\n\\end{equation}\nNow, since $g_{1}+2(g_{2}+g_{3}+\\dots+g_{r})=1$, $g_{j}>0$, for\n$j=1,2,\\dots,r$, $0<\\gamma_{1}\\leq 1$ and $\\abs{\\gamma_{j}}\\leq 1\/\\ul{g}$,\nit follows that \n$\\abs{(W\\ul{\\gamma})_{j}} \\leq 1\/\\ul{g}$ and $\\abs{(W^{T}\\ol{\\gamma})_{j}} \\leq 1\/\\ul{g}$,\nfor $j=2,3,\\dots,r$. Hence, \\eqref{eq48} and \\eqref{eq43} give, for\n$r\\leq i\\leq \\floor{N\/2}+1$,\n\\begin{align*}\n\\abs{\\gamma_{i}} & \\leq Cq^{-(i-1)}+  Cq^{-(N-r+2-i)}\n+\\frac{C}{\\ul{g}}\\Bigl(\\sum_{j=2}^{r-1}q^{-(i-j)}\n+ \\sum_{j=2}^{r-1}q^{-(N-r+1+j-i)}\\Bigr)\\\\\n& = Cq^{-(i-1)} + Cq^{r-2}\\cdot q^{-(N-i)} + \\frac{C}{\\ul{g}}\\Bigl(\nq^{-(i-1)}\\cdot\\frac{q^{r-1}-q}{q-1} + q^{-(N-i)}\\cdot\\frac{q^{r-2}-1}{q-1}\\Bigr),\n\\end{align*}\nwhere $C$ is a multiple of ${C}_{\\wt{G}}$ by a constant and where, for simplicity, we have put \n$q = q_{\\wt{G}}$. This finally yields \\eqref{eq45} with\n\\[\nC_{1} = \\Bigl(1 + \\frac{q^{r-1}-q}{\\ul{g}(q-1)}\\Bigr)C,\\quad\nC_{2} = \\Bigl(q^{r-2} + \\frac{q^{r-2}-1}{\\ul{g}(q-1)}\\Bigr)C.\n\\]\nIn order to prove \\eqref{eq46} we consider the polynomial \n$p(x)=x^{r} + x^{r+1} + \\dots +x^{M-1}=(x^{M}-x^{r})\/(x-1)$, for $0<x\\ne 1$, and\n$M=\\floor{N\/2}+2$, and put $f(x)=xp(x)$. Then, it follows from \\eqref{eq45} that\n\\[\n\\sum_{i=r}^{M-1}(1+i)\\abs{\\gamma_{i}}\\leq C_{1}qf'(q^{-1})\n+ C_{2}q^{-N}f'(q).\n\\]\nSince \n\\begin{align*}\nf'(x)=\\Bigl(\\frac{x^{M+1}-x^{r+1}}{x-1}\\Bigr)' & =\n\\frac{\\bigl((M+1)x^{M}-(r+1)x^{r}\\bigr)(x-1)-(x^{M+1}-x^{r+1})}{(x-1)^{2}}\\\\\n& = \\frac{Mx^{M+1}-rx^{r+1}-(M+1)x^{M}+(r+1)x^{r}}{(x-1)^{2}}\\\\\n& < \\frac{Mx^{M+1} + (r+1)x^{r}}{(x-1)^{2}},\n\\end{align*}\nwe get the estimate\n\\[\n\\sum_{i=r}^{M-1}(1+i)\\abs{\\gamma_{i}} \\leq\nC_{1}\\frac{Mq^{-M} + (r+1)q^{-(r-1)}}{(q^{-1}-1)^{2}}\n+ C_{2}\\frac{Mq^{-N+M+1} + (r+1)q^{-N+r}}{(q-1)^{2}}.\n\\]\nFrom this, the fact that \n\\[\nMq^{-N+M+1} \\leq \\Bigl(\\frac{N}{2}+2\\Bigr)q^{-(\\frac{N}{2}+2)} \\cdot q^{5},\n\\]\nand taking into account that the function $\\omega(x)=xq^{-x}$ is bounded for $x>0$, we obtain\n\\eqref{eq46}.\n\\end{proof}\n{\\bf{Remarks:}} 1. It Follows from \\cite[Theorem 1]{se}, that the elements of  $\\Gamma$ for $r=2$\nmay be computed exactly.\\\\\n2. From \\eqref{eq46} and the fact that $\\abs{\\gamma_{i}}\\leq 1\/\\ul{g}$, $1\\leq i<r$, it follows that\nthere exists a constant $C_{4}$, independent of $N$, such that \n\\begin{equation}\n\\sum_{i=1}^{\\floor{\\frac{N}{2}}+1}(1+i)\\abs{\\gamma_{i}}\\leq C_{4}.\n\\label{eq410}\n\\end{equation} \n\\section{Stability of the $L^{2}$ projection onto $\\mathcal{S}_{h}^{r}$}\nIn this section we shall show the $L^{2}$-projection operator onto $\\mathcal{S}_{h}^{r}$ is stable\nin $H_{per}^{l}$ and $W_{\\infty}^{l}$ for $l=0,1,\\dots,r-1$.\n\\begin{proposition} Let $P_{h}$ be the $L^{2}$-projection operator onto $\\mathcal{S}_{h}^{r}$. Then,\nfor $l=0,1,\\dots,r-1$, there exist constants $C_{1}$, $C_{2}$, depending only on $l$ and $r$, such\nthat \\\\\n(i)\\,\\, $\\|\\partial_{x}^{l}(P_{h}u)\\|\\leq C_{1} \\|\\partial_{x}^{l}u\\|$, \\quad\nif \\quad $u \\in H^{l}_{per}$, and \\\\\n(ii)\\,\\, $\\|\\partial_{x}^{l}(P_{h}u)\\|_{\\infty}\\leq C_{2}\\|\\partial_{x}^{l}u\\|_{\\infty}$,\n\\quad if \\quad $u\\in W_{\\infty}^{l}\\cap H^{l}_{per}$.\n\\end{proposition}\n\\begin{proof} (i)\\,\\, If $l=0$ the estimate is obvious with $C_{1}=1$. Let now $1\\leq l\\leq r-1$ and\nlet $\\{\\Phi_{j}\\}_{j=1}^{N}$ be the basis of $\\mathcal{S}_{h}^{r}$, defined in section 2. Let\n$P_{h}u=\\sum_{j=1}^{N}c_{j}\\Phi_{j}$. Then, applying Remark 4 of section 2, we get by induction that\n\\begin{equation}\n\\partial_{x}^{l}(P_{h}u) = h^{-l}\\sum_{j=1}^{N}\n\\bigl((I_{N}-P^{-1})^{l}c\\bigr)_{j}\\Psi_{j},\n\\label{eq51}\n\\end{equation}\nwhere $P$ is the `translation' matrix defined in \\eqref{eq31}, $\\{\\Psi_{j}\\}_{j=1}^{N}$ is the basis\nof $\\mathcal{S}_{h}^{r-l}$ and \n\\[\nc=(c_{1},c_{2},\\dots,c_{N})^{T}\n\\]\nis the solution of the linear system\n$hGc=b$, with $G_{i,j}=h^{-1}(\\Phi_{j},\\Phi_{i})$, $1\\leq i,j\\leq N$ as usual, and\n$b=(b_{1},b_{2},\\dots,b_{N})^{T}$, $b_{i}=(u,\\Phi_{i})$, $1\\leq i\\leq N$. We easily see by \ninduction that \n\\[\n(I_{N} - P^{-1})^{l} = \\sum_{m=0}^{l}\\binom{l}{m}(-1)^{m}P^{-m}.\n\\]\nTherefore, we get $hG\\bigl((I_{N}-P^{-1})^{l}\\bigr)c=(I_{N}-P^{-1})^{l}b=:\\beta$, where\n\\begin{equation}\n\\beta_{i} = \\sum_{m=0}^{l}\\binom{l}{m}(-1)^{m}b_{i-m}.\n\\label{eq52}\n\\end{equation}\nNow \n\\begin{align*}\nb_{i-m}=(u,\\Phi_{i-m}) & = \\int_{0}^{1}u(x)\\Phi_{i}(x+mh)dx =\n\\int_{0}^{1}u(x-mh)\\Phi_{i}(x)dx \\\\\n& = \\int_{0}^{1}u(x-mh)\\Phi_{i-\\wt{l}}(x-\\wt{l}h)dx\\\\\n& = \\int_{0}^{1}u(x+(\\wt{l}-m)h)\\Phi_{i-\\wt{l}}(x)dx\n=:(\\wt{u}_{l,m},\\Phi_{i-\\wt{l}}),\n\\end{align*}\nwhere $\\wt{l} = \\floor{\\frac{l}{2}}$ and $\\wt{u}_{l,m}(x)=u(x+(\\wt{l}-m)h)$. Hence\n\\begin{align*}\n\\wt{u}_{l,m}(x) & =\\sum_{k=0}^{l-1}\\frac{(\\wt{l}-m)^{k}h^{k}}{k!}\n(\\partial_{x}^{k}u)(x) + \\frac{1}{(l-1)!}\\int_{\\delta_{x}}(\\partial_{x}^{l}u)(y)\n\\bigl(x+(\\wt{l}-m)h - y\\bigr)^{l-1}dy \\\\\n& =: U_{l,m}(x) + \\frac{1}{(l-1)!}J_{l,m}(x),\n\\end{align*}\nwhere $\\delta_{x}$ is the interval with endpoints $x$, $x+(\\wt{l}-m)h$. It follows from \n\\eqref{eq52} that\n\\begin{align*}\n\\beta_{i} & = \\sum_{m=0}^{l}\\binom{l}{m}(-1)^{m}(U_{l,m},\\Phi_{i-\\wt{l}}) +\n\\frac{1}{(l-1)!}\\sum_{m=0}^{l}\\binom{l}{m}(-1)^{m}(J_{l,m}, \\Phi_{i-\\wt{l}})\\\\\n& = \\sum_{k=0}^{l-1}\\frac{h^{k}}{k!}(\\partial_{x}^{k}u,\\Phi_{i-\\wt{l}})\n\\sum_{m=0}^{l}\\binom{l}{m}(\\wt{l}-m)^{k}(-1)^{m}\n+ \\frac{1}{(l-1)!}\\sum_{m=0}^{l}\\binom{l}{m}(-1)^{m}(J_{l,m},\\Phi_{i-\\wt{l}}).\n\\end{align*}\nAccording to Lemma 5.2 that we prove at the end of the section, the sum \n$\\sum_{m=0}^{l}\\binom{l}{m}(\\wt{l}-m)^{k}(-1)^{m}$, for $0\\leq k\\leq l$, is equal to zero. This\nyields \n\\begin{equation}\n\\beta_{i} = \\frac{1}{(l-1)!}\n\\sum_{m=0}^{l}\\binom{l}{m}(-1)^{m}(J_{l,m},\\Phi_{i-\\wt{l}}).\n\\label{eq53}\n\\end{equation}\nSince\n\\begin{equation}\n\\abs{J_{l,m}(x)}^{2} \\leq\n\\bigabs{\\int_{\\delta_{x}}(\\partial_{x}^{l}u)^{2}(y)dy}\\cdot\n\\bigabs{\\int_{\\delta_{x}}(x+(\\wt{l}-m)h-y)^{2l-2}dy},\n\\label{eq54}\n\\end{equation}\nand\n\\begin{align*}\n\\bigabs{\\int_{\\delta_{x}}(x+(\\wt{l}-m)h-y)^{2l-2}dy}\n& = \\int_{0}^{\\abs{\\wt{l}-m}h}y^{2l-2}dy\n=\\frac{1}{2l-1}\\abs{\\wt{l}-m}^{2l-1}h^{2l-1} \\\\\n& \\leq \\frac{1}{2l-1}\\floor{\\tfrac{l+1}{2}}^{2l-1}h^{2l-1}=:\\wt{C}_{1,l}h^{2l-1},\n\\end{align*}\nit follows that\n\\begin{align*}\n\\|J_{l,m}\\|^{2} & \\leq \\wt{C}_{1,l}h^{2l-1}\\sum_{j=1}^{N}\n\\int_{x_{j-1}}^{x_{j}}\\bigabs{\\int_{\\delta_{x}}(\\partial_{x}^{l}u)^{2}(y)dy}dx\\\\\n& \\leq \\wt{C}_{1,l}h^{2l-1}\\sum_{j=1}^{N}\\int_{x_{j-1}}^{x_{j}}\\int_{I_{l,m}}\n(\\partial_{x}^{l}u)^{2}(y)dydx,\n\\end{align*}\nwhere $I_{l,m}$ is an interval of length $(\\abs{\\wt{l}-m}+1)h$. Therefore\n\\[\n\\|J_{l,m}\\|^{2}\\leq \\wt{C}_{1,l}h^{2l}(\\abs{\\wt{l}-m}+1)\\|\\partial_{x}^{l}u\\|^{2},\n\\]\nfor some constant $\\wt{C}_{1,l}$. Finally\n\\begin{equation}\n\\|J_{l,m}\\| \\leq C_{1,l}h^{l}\\|\\partial_{x}^{l}u\\|, \\quad \\text{if}\\quad\nu\\in H^{l}_{per},\n\\label{eq55}\n\\end{equation}\nfor some constant $C_{1,l}$. Moreover\n\\[\n\\abs{J_{l,m}(x)} = \\bigabs{\\int_{\\delta_{x}}(\\partial_{x}^{l}u)(y)\n\\bigl(x+(\\wt{l}-m)h - y\\bigr)^{l-1}dy},\n\\]\nand therefore \n\\[\n\\abs{J_{l,m}(x)}\\leq \\|\\partial_{x}^{l}u\\|_{\\infty}\\int_{0}^{\\abs{\\wt{l}-m}h}y^{l-1}dy,\n\\]\ngiving\n\\begin{equation}\n\\|J_{l,m}\\|_{\\infty} \\leq C_{2,l} h^{l}\\|\\partial_{x}^{l}u\\|_{\\infty}, \\quad\n\\text{if}\\quad u\\in H^{l}_{per}\\cap W_{l}^{\\infty},\n\\label{eq56}\n\\end{equation}\nfor some constant $C_{2,l}$. From \\eqref{eq53}, taking into account \\eqref{eq55} and \\eqref{eq21} we\nsee that \n\\begin{align*}\n\\abs{\\beta}^{2} & = \\frac{1}{(l-1)!}\n\\sum_{m=0}^{l}\\binom{l}{m}(-1)^{m}\n\\bigl(J_{l,m},\\sum_{i=1}^{N}\\beta_{i}\\Phi_{i-\\wt{l}}\\bigr)\\\\\n& \\leq C_{l}\\|J_{l,m}\\|h^{0.5}\\abs{\\beta}\\leq C_{3,l}h^{l+0.5}\n\\|\\partial_{x}^{l}u\\|\\abs{\\beta},\n\\end{align*}\nwhere $C_{l}$ , $C_{3,l}$ are constants depending on $l$. Hence\n\\[\n\\abs{\\beta} \\leq C_{3,l}h^{l+0.5}\\|\\partial_{x}^{l}u\\|.\n\\]\nFrom this estimate, the fact that $hG(I_{N}-P^{-1})^{l}c=\\beta$, \\eqref{eq21} and \\eqref{eq51}, it\nfollows that there exists a constant $C=C(l,r)$, such that\n\\[\n\\|\\partial_{x}^{l}(P_{h}u)\\|\\leq h^{-l+0.5}\n\\abs{(I_{N}-P^{-1})^{l}c} \\leq Ch^{-l+0.5} \\cdot h^{l-0.5}\n\\|\\partial_{x}^{l}u\\|,\n\\]\nfrom which (i) follows with $C_{1}=C$. \\par\n(ii)\\,\\, Let $1\\leq l\\leq r-1$. If now $u\\in H^{l}_{per}\\cap W_{l}^{\\infty}$, then from \\eqref{eq53}\nand \\eqref{eq56}, we obtain\n\\[\n\\max_{1\\leq i\\leq N}\\abs{\\beta_{i}} \\leq C_{4,l}h^{l+1}\\|\\partial_{x}^{l}u\\|_{\\infty},\n\\]\nfor some constant $C_{4,l}$. This estimate, the definition of $\\beta$, and \\eqref{eq410}, imply that\nthere is a constant $C=C(l,r)$, such that\n\\[\n\\max_{1\\leq i\\leq N}\\abs{((I_{N}-P^{-1})c)_{i}}\n\\leq Ch^{l}\\|\\partial_{x}^{l}u\\|_{\\infty}.\n\\]\nTherefore, from this inequality, \\eqref{eq51}, and the fact that $\\sum_{j=1}^{N}\\Psi_{j}=1$, we see\nthat (ii) follows. \\par\nIf $l=0$ and $\\Gamma$ is the (cyclic, symmetric) inverse of $G$, then \\eqref{eq410} implies that\n$\\|\\Gamma\\|_{\\infty}\\leq C_{4}$, and therefore \n$\\abs{c}_{\\infty}=\\abs{\\Gamma b}_{\\infty}\\leq C_{4}\\abs{b}$. But $b_{i}=(u,\\Phi_{i})$, so that\n\\[\n\\abs{b_{i}}\\leq \\|u\\|_{\\infty}\\int_{I_{i}}\\abs{\\Phi_{i}(x)}dx, \\quad 1\\leq i\\leq N,\n\\]\nwhere $I_{i}$ is an interval (or union of intervals) of length $rh$. Therefore, there exists a\nconstant $C$, independent of $h$, such that \n\\[\n\\abs{b}_{\\infty}:=\\max_{1\\leq i\\leq N}\\abs{b_{i}}\\leq Ch\\|u\\|_{\\infty},\n\\]\ni.e. that $\\abs{c}_{\\infty}\\leq C_{4}\\cdot Ch\\|u\\|_{\\infty}$. Given that $0\\leq \\Phi_{j}(x)\\leq 1$\nfor $j=1,2,\\dots,N$ and all $x\\in [0,1]$, it follows that\n\\[\n\\abs{(P_{h}u)(x)} \\leq \\sum_{j=1}^{N}\\abs{c_{j}} \\Phi_{j}(x)\\le \\sum_{j=1}^{N}\\abs{c_{j}}\n\\leq C\\|u\\|_{\\infty},\n\\]\ni.e. $\\|P_{h}u\\|_{\\infty}\\leq C\\|u\\|_{\\infty}$. The proof of proposition 5.1. is now complete.\n\\end{proof}\n\\begin{lemma} Let $l\\geq 0$ be an integer, and for a nonnegative integer $k$ let\n\\[\na_{k} = \\sum_{m=0}^{l}\\binom{l}{m}m^{k}(-1)^{m}.\n\\]\nThen, if $l=0$ and $k\\geq 0$, or $l\\geq 1$ and $0\\leq k\\leq l-1$, we have that $a_{k}=0$.\n\\end{lemma}\n\\begin{proof} For $l=0$, $k\\geq 0$ we immediately get the conclusion. If $l\\geq 1$, \n$0\\leq k\\leq l-1$, consider the polynomial \n\\[\np(x) = (1+x)^{l} = \\sum_{m=0}^{l}\\binom{l}{m}x^{m}.\n\\]\nThen\n\\[\np^{(k)}(x) = \\frac{l!}{(l-k)!}(1+x)^{l-k} =\n\\sum_{m=k}^{l}\\binom{l}{m}\\frac{m!}{(m-k)!}x^{m-k}:=b_{k}(x),\n\\]\nand therefore $p^{(k)}(-1) = 0 = b_{k}(-1)$, for $k=0,1,\\dots,l-1$, and also $a_{0}=b_{0}(-1)=0$,\n$a_{1}= - b_{1}(-1) = 0$. Assuming $a_{j}=0$ for $j=0,1,\\dots,k-1$, we will show that $a_{k}=0$. We\nhave\n\\begin{align*}\nb_{k}(-1) & = (-1)^{k}\\sum_{m=k}^{l}\\binom{l}{m}\\frac{m!}{(m-k)!}(-1)^{m}\\\\\n& =(-1)^{k}\n\\sum_{m=k}^{l}\\binom{l}{m}m(m-1)\\cdot\\cdot\\cdot\\bigl(m-(k-1)\\bigr)(-1)^{m}\\\\\n& = (-1)^{k}\\sum_{m=1}^{l}\\binom{l}{m}\\bigl(m^{k}+ q_{k-1}(m)\\bigr)(-1)^{m},\n\\end{align*}\nwhere $q_{k-1}(m)$ is a polynomial of $m$ of degree $k-1$. Since $b_{k}(-1)=0$ it follows that\n\\begin{equation}\n\\sum_{m=1}^{l}\\binom{l}{m}\\bigl(m^{k} + q_{k-1}(m)\\bigr)(-1)^{m}=0.\n\\label{eq57}\n\\end{equation}\nFrom our hypothesis that $a_{j}=0$ for $j=0,1,2,\\dots,k-1$, we obtain\n\\[\n\\sum_{m=1}^{l}\\binom{l}{m}q_{k-1}(m)(-1)^{m}=0,\n\\]\nand, hence, using \\eqref{eq57}, we get $a_{k}=0$, and the inductive step is complete.\n\\end{proof}\n\\section{Stability of a family of quasiinterpolants in $\\mathcal{S}_{h}^{r}$}\nWe shall prove now that a family of quasiinterpolants in $\\mathcal{S}_{h}^{r}$, cf. e.g.\n\\cite{s}, \\cite{tw}, \\cite{t}, \\cite{ds}, is stable in $H_{per}^{l}$ for $1\\leq l\\leq r-1$, in\n$(C_{per}, \\|\\cdot\\|_{\\infty})$, and in $W_{l}^{\\infty}$ for $1\\leq l\\leq r-1$. We recall from the\nabove references that if $u$ is a continuous, 1-periodic function, then a \\emph{quasiinterpolant}\nin $\\mathcal{S}_{h}^{r}$, is defined as \n\\[\nQ_{h}u= \\sum_{j=1}^{N}u(jh)\\wt{\\Phi}_{j}\\,,\n\\]\nwhere $\\{\\wt{\\Phi}_{j}\\}_{j=1}^{N}$ is a modified basis of $\\mathcal{S}_{h}^{r}$ defined as\n$\\wt{\\Phi}_{j}=q_{0}\\Phi_{j}+\\sum_{k=1}^{r-1}q_{k}(\\Phi_{j+k} + \\Phi_{j-k})$, where\n$\\{\\Phi_{j}\\}_{j=1}^{N}$ is the standard basis of $\\mathcal{S}_{h}^{r}$ as introduced in section 2.\nTherefore, if $v\\in \\mathcal{S}_{h}^{r}$, then,\n\\[\nv = \\sum_{j=1}^{N}V_{j}\\Phi_{j}=\\sum_{j=1}^{N}\\wt{V}_{j}\\wt{\\Phi}_{j},\n\\] \nwith $V=Q\\wt{V}$, where $Q$ is a symmetric $N\\times N$ cyclic matrix with first row\n\\[\n(q_{0},q_{1},\\dots,q_{r-1},0,\\dots,0,q_{r-1},\\dots,q_{1}).\n\\]\nActually, see Remark 1 in section 2, the quasiinterpolant that we define \nhere satisfies $(Q_h u)(x) = (Q^{TW}_{h} u)(x - h(r - 2)\/2)$, but due to \nperiodicity the results of this section are valid for $Q^{TW}_{h}$  as well.\n Note that for the Thom{\\'e}e-Wendroff quasiinterpolant, \\cite{tw}, the $q_{j}$'s are chosen so that the trigonometric polynomial \n\\[\nq(\\xi)=\\sum_{j=-(r-1)}^{r-1}q_{j}e^{-ij\\xi}=\n\\sum_{j=0}^{r-1}\\delta_{j}\\Bigl(\\sin\\frac{\\xi}{2}\\Bigr)^{2j},\n\\]\ncorresponds to coefficients obtained from the series \n\\[\n\\sum_{j=0}^{\\infty}\\delta_{j}\\tau^{2j}=\\Bigl(\\frac{\\arcsin\\tau}{\\tau}\\Bigr)^{r}.\n\\]\nAs a result of this choice, this quasiinterpolant has several superconvergent approximation\nproperties which will not concern us here.\n\\begin{proposition} Let $Q_{h}$ be a quasiinterpolant in $\\mathcal{S}_{h}^{r}$ defined as above.\nThen: \\\\\n(i)\\,\\, There exists a constant $C_{0}$ independent of $h$, such that \n$\\|Q_{h}u\\|_{\\infty}\\leq C_{0}\\|u\\|_{\\infty}$, for all $u\\in C_{per}$.\\\\\nMoreover, for $l=1,2,\\dots,r-1$, there exist constants $C_{1}$, $C_{2}$ depending only on $l$ and\n$r$ such that \\\\\n(ii)\\,\\, $\\|\\partial_{x}^{l}(Q_{h}u)\\|_{\\infty} \\leq C_{1} \\|\\partial_{x}^{l}u\\|_{\\infty}$, \\,\\, if \n\\,\\, $u\\in W_{\\infty}^{l}\\cap H^{l}_{per}$, and \\\\\n(iii)\\, $\\|\\partial_{x}^{l}(Q_{h}u)\\| \\leq C_{2}\\|\\partial_{x}^{l}u\\|$,\n\\,\\,if \\,\\, $u\\in H_{per}^{l}$.\n\\end{proposition}\n\\begin{proof} (i)\\,\\, We have\n\\begin{equation}\nQ_{h}u=\\sum_{j=1}^{N}u(jh)\\wt{\\Phi}_{j} = \\sum_{j=1}^{N}(Q\\wt{u})_{j}\\Phi_{j},\n\\label{eq61}\n\\end{equation}\nwhere $Q = q_{0}I_{N} + \\sum_{m=1}^{r-1}q_{m}(P^{m} + P^{-m})$ and \n$\\wt{u}=(u_{1},u_{2},\\dots,u_{N})^{T}$, $u_{j}=u(jh)$, $1\\leq j\\leq N$. Since\n\\begin{equation}\n(Q\\wt{u})_{j}=q_{0}u_{j} + \\sum_{m=1}^{r-1}q_{m}(u_{j+m} + u_{j-m}),\n\\label{eq62}\n\\end{equation}\nit follows that\n\\[\n\\max_{1\\leq j\\leq N}\\abs{(Q\\wt{u})_{j}} \\leq \n(\\abs{q_{0}}+2\\sum_{m=1}^{r-1}\\abs{q_{m}})\\|u\\|_{\\infty} =: C_{0}\\|u\\|_{\\infty},\n\\]\ni.e. the result (i) of the proposition. \\par\nFrom \\eqref{eq61} and Remark 4 of section 2,\n\\begin{equation}\n\\partial_{x}^{l}(Q_{h}u) =h^{-l}\\sum_{j=1}^{N}\\omega_{j}^{l}\\Psi_{j},\n\\label{eq63}\n\\end{equation}\nwhere $\\{\\Psi\\}_{j=1}^{N}$ the basis of $\\mathcal{S}_{h}^{r-l}$ and   \n$\\omega_{j}^{l} = \\bigl[(I_{N} - P^{-1})^{l}(Q\\wt{u})\\bigr]_{j}$. Hence, from the relation\n\\[\n(I_{N}-P^{-1})^{l}=\\sum_{k=0}^{l}\\binom{l}{k}(-1)^{k}P^{-k},\n\\]\nand \\eqref{eq62}, we obtain\n\\begin{equation}\n\\omega_{j}^{l} = q_{0}\\sum_{k=0}^{l}\\binom{l}{k}(-1)^{k}u_{j-k}\n+ \\sum_{m=1}^{r-1}q_{m}\\sum_{k=0}^{l}\\binom{l}{k}(-1)^{k}(u_{j-k+m} + u_{j-k-m}).\n\\label{eq64}\n\\end{equation}\nNow, for integer $k$ it holds that \n\\[\nu_{j-k} = \\sum_{\\gamma=0}^{l-1}\\frac{(-1)^{\\gamma}k^{\\gamma}h^{\\gamma}}{\\gamma !}\n(\\partial_{x}^{\\gamma}u)(jh) + J(l,j,k),\n\\]\nwhere\n\\[\nJ(l,j,k) = \\frac{-1}{(l-1)!}\\int_{(j-k)h}^{jh}\n(\\partial_{x}^{l}u)(x)\\bigl((j-k)h - x\\bigr)^{l-1}dx.\n\\] \nTherefore, from \\eqref{eq64} and Lemma 5.2, we get that\n\\begin{equation}\n\\omega_{j}^{l}=q_{0}\\sum_{k=0}^{l}{l}\\binom{l}{k}(-1)^{k}J(l,j,k)\n+ \\sum_{m=1}^{r-1}q_{m}\\sum_{k=0}^{l}\\binom{l}{k}(-1)^{k}\n\\bigl(J(l,j,k-m) + J(l,j,k+m)\\bigr).\n\\label{eq65}\n\\end{equation}\nSince for integer $k\\in \\{-2(r-2),\\dots,r-1\\}$, it holds that\n\\[\n\\abs{J(l,j,k)} \\leq \\frac{\\abs{k}^{l}h^{l}}{l!}\\|\\partial_{x}^{l}u\\|_{\\infty},\n\\]\nand \n\\[\n\\abs{J(l,j,k)}^{2}  \n\\leq \\frac{\\abs{k}^{2l-1}h^{2l-1}}{(2l-1)((l-1)!)^{2}}\n\\bigabs{\\int_{(j-k)h}^{jh}(\\partial_{x}^{l}u)^{2}(x)dx}\n\\leq \\frac{\\abs{k}^{2l-1}h^{2l-1}}{(2l-1)((l-1)!)^{2}}\n\\int_{(j-2r+4)h}^{(j+r-1)h}(\\partial_{x}^{l}u)^{2}(x)dx,\n\\]\nwe obtain from \\eqref{eq65} that\n$\\abs{\\omega_{j}^{l}} \\leq \\wt{C}_{1}h^{l}\\|\\partial_{x}^{l}u\\|_{\\infty}$,\nand\n\\[\n\\abs{\\omega_{j}^{l}}^{2}\\leq \\wt{C}_{2}h^{2l-1}\n\\int_{(j-2r+4)h}^{(j+r-1)h}(\\partial_{x}^{l}u)^{2}(x)dx.\n\\]\nFrom these two inequalities, \\eqref{eq63}, and \\eqref{eq21}, we get the estimates (ii) and (iii)\nof the Proposition.\n\\end{proof}\n\\bibliographystyle{amsalpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \n\nFor $m \\geq 3$, we define the $m$th order pyramidal number by \\[ \\mathrm{Pyr}_m(x) = \\frac{1}{6} x(x+1)((m-2)x+5-m). \\] Pyramidal numbers are a special type of figurate number with many interesting properties and a rich history. The properties of figurate numbers, and in particular their relationship with perfect powers, have received much attention in the literature, see \\citep{part1, poly, decagonal} for example, and the references therein. \\freefootnote{\\emph{Date}: \\date{\\today}.}\n\\freefootnote{\\emph{Keywords}: Pyramidal, perfect power, binomial Thue equations, modular method, Frey curve, level-lowering.}\n\\freefootnote{\\emph{MSC2010}: 11D41, 11D59, 11D61, 14G99.}\n\\freefootnote{A. D. is supported by the Croatian Science Foundation under the project no. IP-2018-011313\n and the QuantiXLie Center of Excellence, a project co-financed by the Croatian Government and \n European Union through the European Regional Development Fund - the Competitiveness and Cohesion \n Operational Programme (Grant KK.01.1.1.01.0004). }\n\\freefootnote{K. Gy. is supported in part by Grants K115479 and K128088 from \n the Hungarian National Foundation for Scientific Research (OTKA) and from the Austrian-Hungarian \njoint project ANN 130909 (FWF-NKFIH).}\n\\freefootnote{P. M-J. is supported by an EPSRC studentship and has previously used the name Philippe Michaud-Rodgers.}\n\\freefootnote{\\'A. P. is supported by Grants K115479 and K128088 from \n the Hungarian National Foundation for Scientific Research (OTKA) and from the Austrian-Hungarian \njoint project ANN 130909 (FWF-NKFIH) and E\u00f6tv\u00f6s Lor\u00e1nd Research Network (ELKH).}\n\nThis paper is the second in a series of two papers. In the first paper \\citep{part1}, written by the first-, second-, and fourth-named authors in 2012, the question of when $\\mathrm{Pyr}_m(x)$ is a square is considered, and all solutions are found with $3 \\leq m \\leq 100$ and $m \\ne 5$ (the cases $m = 3$ and $m=4$ are classical). In this paper, we consider the case of higher powers and obtain the following result.\n\n\\begingroup\n\\renewcommand\\thetheorem{1}\n\\begin{theorem}\\label{Mainthm} All the solutions of the equation\n\\begin{equation}\\label{maineq} \\mathrm{Pyr}_m(x)=\\frac{1}{6}x(x+1)((m-2)x+5-m)=y^n \\end{equation}\nin positive integers $m,x,y,n$  with $y>1, \\, 3\\leq m\\leq 50$ and $n \\geq 3$ are \n\\begin{align*} (m,x,y,n)= &  (5,57121,3107,4), \\; (7,2,2,3), \\;  (15,2,2,4), \\; (17,8,6,4), \\\\ & (26,2,3,3), \\; (31,2,2,5) \\; \\text{ and } \\; (50,15,30,3). \\end{align*}\n\\end{theorem}\n\\endgroup\n\nWhen $m = 3$ or $m = 4$, these are known results (see \\citep{m=3} and \\citep{m=4}), and so we will consider the cases $5 \\leq m \\leq 50$. We note that there are no apparent obstructions to extending Theorem \\ref{Mainthm} to larger values of $m$, say $m \\leq 100$ for example, although we do not pursue this here. \n\nThe techniques we use to prove Theorem \\ref{Mainthm} when $n $ is odd will be of a very different flavour to those used in \\citep{part1}, where the main ideas are centred around the study of integral points on elliptic curves. Starting from equation (\\ref{maineq}) we will form various systems of binomial Thue equations. We will then use a combination of local arguments  and the modular method via Frey curves, along with bounds arising from linear forms in logarithms to prove Theorem \\ref{Mainthm}.\n\nWe now outline the rest of the paper. In Section 2, we use the results of \\citep{part1} to find all solutions to equation (\\ref{maineq}) in the case $n = 4$. We also treat the case $m = 5$. For the remainder of the paper, we will consider the case $n$ an odd prime and $ 6 \\leq m \\leq 50$. In Section 3, we reduce the problem to the study of finitely many systems of binomial Thue equations, and obtain a bound on $n$ using results on linear forms in logarithms. In Section 4, we solve the majority of these Thue equations using local arguments, and finally we use the modular method in various guises in Section 5 to deal with the remaining cases. \n\n\\bigskip\n\nThe \\texttt{Magma} \\citep{magma} code used to support the computations in this paper can be found at: \n\n\\vspace{3pt}\n\n \\url{https:\/\/warwick.ac.uk\/fac\/sci\/maths\/people\/staff\/michaud\/c\/}\n\n\\bigskip\n\nThe third-named author would like to thank Samir Siksek and Damiano Testa for many useful discussions.\n\n\\section{The case \\texorpdfstring{$n = 4$}{} and the case \\texorpdfstring{$m = 5$}{}}\n\nIn this section we will treat certain cases that are not amenable to the methods of the later parts of the paper.\n\n\\begin{lemma}\\label{n=4Lem} Let $(m,x,y)$ be a solution to equation (\\ref{maineq}) with $n = 4$, $x > 0$, $y >1 $, and $6 \\leq m \\leq 50$. Then \\[(m,x,y) = (15,2,2) \\; \\text{ or } \\; (17,8,6). \\]\n\\end{lemma}\n\n\\begin{proof} Let $(m,x,y)$ be such a solution to equation (\\ref{maineq}). Then $(m,x,y^2)$ is a solution to equation (\\ref{maineq}) with $n = 2$, and so we can apply the results of \\citep[pp.~218--219]{part1} to obtain all solutions for $6 \\leq m \\leq 50$. The two solutions we find are those stated in the lemma.\n\\end{proof}\n \n\\begin{lemma}\\label{m=5Lem} Let $m = 5$. Let $(x,y,n)$ be a solution to equation (\\ref{maineq}) with $x > 0$, $y >1 $, and $n \\geq 3$. Then \\[(x,y,n) = (57121, 3107, 4). \\]\n\\end{lemma} \n \n\\begin{proof} Suppose $(x,y,n)$ is such a solution to equation (\\ref{maineq}). We have \\begin{equation}\\label{m=5} x^2(x+1) = 2y^n. \\end{equation} Suppose first that $n = 4$. Then $2\\operatorname{ord}_2(x)+ \\operatorname{ord}_2(x+1) = 1 + 4 \\operatorname{ord}_2(y)$, so $x$ is odd and $x + 1$ is even. We have \\[ x^2 \\left (\\frac{x+1}{2} \\right) = y^4, \\] and $\\gcd(x^2, (x+1)\/2) = 1$. It follows that $x^2 = y_1^4$ and $x+ 1 = 2y_2^4$ for some coprime integers $y_1$ and $y_2$ with $y = y_1y_2$. Then $x = y_1^2$, since $x$ is positive, so \\begin{equation} y_1^2 + 1 = 2y_2^4. \\end{equation} This equation is known as Ljunggren's equation, since Ljunggren proved that the only positive integer solutions to this equation are given by $(y_1,y_2) = (1,1)$ and $(y_1,y_2) = (239,13)$ (see \\citep{ljunggren} for the original proof, which is somewhat involved, or \\citep{simpler} for an example of a simpler proof). Since $y > 1$, we obtain the solution $(x,y) = (y_1^2, y_1y_2) = (57121,3107)$ to equation (\\ref{maineq}).\n\nSuppose instead that $n$ is odd. In this case $x$ may be odd or even. If $x$ is odd, then arguing similarly to above, there exist coprime integers $y_1$ and $y_2$ satisfying \\begin{equation*} y_1^n + 1 = 2y^n. \\end{equation*} This equation has no solutions with $y_1y_2 > 1$  and $n \\geq 3$ by \\citep[Main~Theorem]{ll2}. If $x$ is even, then we find that there exist coprime integers $y_1$ and $y_2$ satisfying \\[ x^2 = 2y_1^n \\; \\text{ and } \\; x+1 = y_2^n. \\] From $x^2 = 2y_1^n$, we see that $\\operatorname{ord}_2(y) = k \\geq 1$, with $k$ odd, and we may write $y_1 = 2^k z_1^2$ for some positive integer $z_1$ coprime to $y_2$. Then  \\[ x =  2^{\\frac{kn+1}{2}} z_1^n, \\] and so \\[2^{\\frac{kn+1}{2}} z_1^n - y_2^n = 1. \\] By \\citep[Theorem~1.2]{consecutive}, this equation has no solutions in positive coprime integers. This completes the case $m = 5$.\n\\end{proof}\n\n\\section{Systems of binomial Thue equations}\n\nThanks to Lemmas \\ref{n=4Lem} and \\ref{m=5Lem}, we may suppose that $n \\geq 3$ is odd and that $6 \\leq m \\leq 50$. Moreover, we will assume that $n = p$ is an odd prime. We write equation (\\ref{maineq}) as \\begin{equation}\\label{eq2} x(x+1)(a_mx-b_m) = c_m y^p, \\end{equation} \nwhere \\begin{align*} (a_m,b_m,c_m) = \\begin{cases} \\left(\\frac{m-2}{3}, \\frac{m-5}{3}, 2 \\right) &  \\text{ if } \\; m \\equiv 2 \\pmod{3}, \\\\ (m-2, m-5, 6) & \\text{ if } \\;  m \\not\\equiv 2 \\pmod{3}. \\end{cases} \\end{align*}\nWe introduce the notation \\begin{equation*} d_1 = \\gcd(x, a_mx-b_m) \\quad \\text{and} \\quad d_2 = \\gcd(x+1,a_mx-b_m). \\end{equation*} By writing $a_mx-b_m = a_m(x+1) - (a_m+b_m)$, it is straightforward to see that \\[ \\gcd(x,x+1) = 1, \\; \\; d_1 \\mid b_m, \\; \\; \\text{ and} \\; \\; d_2 \\mid a_m+b_m. \\] Even though $d_1$ and  $d_2$ are unknown, we know a finite list of possibilities for each one; namely the divisors of $b_m$ and $a_m+b_m$ respectively. We also note that $\\gcd(a_m,b_m) = 1$, since $m \\ne 3$.\n\nWe may now divide both sides of equation (\\ref{eq2}) by $c_m$ and the appropriate $p$th powers of $d_1$ and $d_2$ to obtain  \\begin{equation} \\label{denom} \\left( \\frac{x}{A} \\right) \\left( \\frac{x+1}{B} \\right) \\left( \\frac{a_mx-b_m}{C} \\right) = Y^p.  \\end{equation} Here, $Y \\mid y$ is a positive integer,  $A, B,$ and $C$ are positive integers satisfying \\[ A \\mid c_m (b_m)^p, \\quad B \\mid c_m(a_m+b_m)^p, \\quad C \\mid c_m (b_m)^p (a_m+b_m)^p, \\] and the three factors on the left-hand side of equation (\\ref{denom}) are integral and pairwise coprime. Moreover, we may assume that $A$, $B$, and $C$ are $p$th power free. Later in this section we will provide, for each $m$, the precise list of possibilities for the triple $(A,B,C)$. We emphasise that the triple $(A,B,C)$ depends on $m$, and will usually also depend on $p$. It follows that there exist positive pairwise coprime integers $y_1, y_2,$ and $y_3$ satisfying\n\\begin{equation*} x = Ay_1^p, \\qquad x+1 = By_2^p, \\qquad a_mx-b_m = Cy_3^p. \\end{equation*} This leads us to consider the following system of binomial Thue equations \\begin{align*} \nB \\, y_2^p - A \\, y_1^p & = 1 \\\\\na_m A \\, y_1^p - C \\, y_3^p & = b_m \\\\\na_m B \\, y_2^p - C \\,  y_3^p & = a_m+b_m. \n\\end{align*} We observe that the third equation is dependent on the first two, so we in fact have the following system of two Thue equations:\n\\begin{align} \\label{system}\n\\begin{split}\nB \\, y_2^p - A \\, y_1^p & = 1 \\\\\na_m A \\, y_1^p - C \\, y_3^p & = b_m. \n\\end{split}\n\\end{align}\nFor each fixed value of $m$, we aim to solve this system of equations for each possible triple $(A,B,C)$. We start by providing a bound for $p$ using results on linear forms in logarithms.\n\n\\begin{proposition}\\label{LogBound} Let $(x,y,m,p)$ be a solution to equation (\\ref{eq2}) with $y > 1$. Then \\[ p < 10676 \\cdot \\log \\left( {c_m}^2 \\cdot b_m \\cdot (a_m+b_m) \\right). \\] \n\\end{proposition}\n\n\\begin{proof} We aim to obtain a binomial Thue equation with coefficients independent of $p$. We divide both sides of equation (\\ref{eq2}) by $c_m \\cdot {d_1}^p \\cdot {d_2}^p$. to obtain  \\[ \\left(\\frac{x}{e_1 \\cdot {d_1}^{r_1}} \\right) \\left(\\frac{x+1}{e_2 \\cdot {d_2}^{r_2}} \\right) \\left(\\frac{a_mx-b_m}{D} \\right) = \\left(\\frac{y}{d_1d_2}\\right)^p, \\] where $r_1, r_2 \\in \\{1,p-1\\}$, $e_1, e_2 \\mid c_m$, $D$ is some integer, $Y \\mid y$, and the three factors on the left-hand side are integral and pairwise coprime. Now, if $r_i = p-1$ then we rewrite $1\/{d_i}^{r_i}$ as $d_i \/ {d_i}^p$. It follows that \\[ x = \\frac{u_1}{v_1} z_1^p \\; \\text{ and } \\; x+1 = \\frac{u_2}{v_2} z_2^p, \\] for some positive integers $u_i, v_i,$ and $z_i$. Moreover, we observe that \\[ u_1v_1 \\mid c_m b_m \\; \\text{ and } \\; u_2v_2 \\mid c_m (a_m+b_m). \\] \nWe then obtain the binomial Thue equation \\begin{equation} \\label{logthue} u_2v_1z_2^p - u_1v_2 z_1^p = v_1v_2, \\end{equation} with $\\max \\{u_2v_1, u_1v_2, v_1v_2 \\} \\leq {c_m}^2 \\cdot b_m \\cdot (a_m+b_m)$. The proposition now follows by applying \\citep[Theorem~2]{mignotte}, a result due to Mignotte obtained using linear forms in logarithms, where we have used $\\lambda \\geq \\log(2)$ in this result, so that $7400 \/ \\lambda < 10676$. \n\\end{proof}\n\nTo give some indication of the magnitude of this bound, when $m = 6$ the bound we obtain is $55440$, and  when $m = 50$ the bound is $80372$. We note that for each triple $(A,B,C)$ we could provide a distinct bound on $p$ using the system (\\ref{system}), and this bound will usually be smaller than the one obtained in Proposition \\ref{LogBound}, but for simplicity we use a single bound for each value of $m$.\n\nWe will now list the possibilities for the triple $(A,B,C)$ for each value of $m$. In general, we write \\[b_m = 2^t \\cdot {p_1}^{r_1} \\cdot {q_1}^{s_1} \\quad a_m+b_m = {p_2}^{r_2} \\cdot {q_2}^{s_2}, \\] where $p_i$ and $q_i$ are primes that do not divide $c_m$, and $t, r_i, s_i$ are non-negative integers. We split into six cases dependent on $t = \\operatorname{ord}_2(b_m)$.\n\n\n\n\\subsection*{Case 1: \\texorpdfstring{$\\operatorname{ord}_2(b_m) = 0$}{}}\n\nHere,\n\\begin{align*} m \\in \\{ & 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46,  \\\\ & 48, 50 \\}.\\end{align*} We have\n\\begin{itemize}\n\\item $b_m = {p_1}^{r_1} \\cdot {q_1}^{s_1}$ for some $r_1 \\in \\{ 0,1,2 \\}$, $s_1 \\in \\{ 0,1 \\}$, and $p_1, q_1 \\nmid c_m$ are prime;\n\\item $a_m+b_m = {p_2}^{r_2} \\cdot {q_2}^{s_2}$ for some $r_2 \\in \\{ 1,2,3 \\}$, $s_2 \\in \\{0,1\\}$, and $p_2, q_2 \\nmid c_m$ are prime.\n\\end{itemize}\nThen \\begin{align*} A & = 2^{\\alpha_1} \\cdot 3^{\\beta_1} \\cdot {p_1}^{\\gamma_1} \\cdot {q_1}^{\\delta_1}, \\\\\nB & = 2^{\\alpha_2} \\cdot 3^{\\beta_2} \\cdot {p_2}^{\\gamma_2} \\cdot {q_2}^{\\delta_2},\\\\ \nC & = 3^{\\beta_3} \\cdot {p_1}^{p-\\gamma_1} \\cdot {q_1}^{p-\\delta_1} \\cdot {p_2}^{p-\\gamma_2} \\cdot {q_2}^{p-\\delta_2}, \\end{align*}\nwhere \\begin{align*} (\\alpha_1,\\alpha_2) & \\in \\{(0,1),(1,0) \\}, & \\\\\n(\\beta_1, \\beta_2, \\beta_3) & \\in \\begin{cases} \\{ (1,0,0),(0,1,0),(0,0,1) \\} & \\text{ if } c_m = 6, \\\\\n \\{ (0,0,0) \\} & \\text{ if } c_m = 2, \\end{cases} \\\\\n\\gamma_i & \\in \\{0,r_i,p-r_i \\}, & \\\\\n\\delta_i & \\in \\{0,s_i,p-s_i\\}.  & \\\\\n\\end{align*}\nWe note that if, say, $\\gamma_1 = 0$, then we use the convention of removing the perfect $p$th power $p_1^p$ from $C$. We do this (in each case) to avoid introducing too many variables.\n\n\\begin{proof}[Proof of Case 1] As discussed earlier in this section, the basic idea is to divide both sides of (\\ref{eq2}) by $p$th powers of $d_1$ and $d_2$ in order to obtain three pairwise coprime factors. We will work one prime at a time.\n\nWe start by dividing both sides of equation (\\ref{eq2}) by $2$. If $x$ is even then $A$ will have a factor of $2$ and both $B$ and $C$ will be odd. Otherwise, both $x+1$ and $a_mx-b_m$ are even. In this case we choose to divide $x+1$ by $2$ so that $B$ is even and $A$ and $C$ are odd.\n\nNext, if $c_m = 6$, we divide both sides by $3$, and precisely one of $x$, $x+1$, and $a_mx-b_m$ will be divisible by $3$, so precisely one of $A$, $B$, and $C$ will have a factor of $3$.\n\nWe now consider the prime $p_2$ and split into three cases depending on the value of $r_2$. The other primes ($p_1$, $q_1$, and $q_2$) are dealt with in the same manner.\n\n\\begin{enumerate}[(i)]\n\\item Case $r_2 = 1$. If $p_2 \\nmid d_2$, then our three factors will be pairwise coprime at $p_2$ and there is nothing more to do, so we assume that $p_2 \\mid d_2$. Since $p_2 \\parallel a_m+b_m$, we have that  $p_2 \\parallel d_2$.\n\nWe then divide both sides of equation (\\ref{eq2}) by ${p_2}^p$. If $p_2 \\parallel x$, then ${p_2}^{p-1} \\mid a_mx-b_m$, so $B$ will have a factor of $p_2$ and $C$ will have a factor of ${p_2}^{p-1}$. Otherwise, $p_2 \\parallel a_mx-b_m$, so $C$ will have a factor of $p_2$ and $B$ will have a factor of ${p_2}^{p-1}$.\n\\item Case $r_2 = 2$. If $p_2 \\nmid d_2$ then we already have coprimality at $p_2$. If ${p_2}^2 \\parallel d_2$, then after dividing by ${p_2}^p$, one of $B$ and $C$ will have a factor of ${p_2}^2$, and the other will have a factor of ${p_2}^{p-2}$. \n\nNext, if $p_2 \\parallel d_2$, then one of $x+1$ and $a_mx-b_m$ will be divisible by ${p_2}^{p-1}$, and in particular by ${p_2}^2$. If ${p_2}^2 \\mid x+1$, then since ${p_2}^2 \\mid a_m+b_m$, we obtain ${p_2}^2 \\mid a_mx-b_m$, contradicting $p_2 \\parallel d_2$. We obtain a similar contradiction in the case ${p_2}^2 \\mid a_mx-b_m$ and therefore conclude that $\\operatorname{ord}_{p_2}(d_2) \\ne 1$.\n\n\\item Case $r_2 = 3$. If $p_2 \\nmid d_2$ then we argue as in the first two cases. Arguing as in Case (ii), we see that $\\operatorname{ord}_{p_2}(d_2) \\ne 1$ or $2$. Suppose ${p_2}^3 \\parallel d_2$. We then divide both sides of equation (\\ref{eq2}) by ${p_2}^p$. One of $B$ and $C$ will have a factor of ${p_2}^3$ and the other a factor of ${p_2}^{p-3}$.\n\\end{enumerate}\n\nWe repeat this process with the primes $p_1$, $q_1$, and $q_2$, until the factors are pairwise coprime. This gives the possibilities listed for the triple $(A,B,C)$. \\end{proof}\n\n\\subsection*{Case 2: \\texorpdfstring{$\\operatorname{ord}_2(b_m) = 1$}{}}\n\nHere, \\[ m \\in \\{ 7, 11, 15, 19, 23, 27, 31, 35, 39, 47 \\}. \\] We have\n\\begin{itemize}\n\\item $b_m = 2 \\cdot {p_1}^{r_1}$ for some $r_1 \\in \\{ 0,1 \\}$, and $p_1 \\nmid c_m$ is prime;\n\\item $a_m+b_m = p_2 \\cdot {q_2}^{s_2}$, for some $s_2 \\in \\{ 0,1 \\}$, and $p_2, q_2 \\nmid c_m$ are prime.\n\\end{itemize}\nThen \\begin{align*} A & = 2^{\\alpha_1} \\cdot 3{^\\beta_1} \\cdot {p_1}^{\\gamma_1}, \\\\\nB & = 2^{\\alpha_2} \\cdot 3^{\\beta_2} \\cdot {p_2}^{\\gamma_2} \\cdot {q_2}^{\\delta_2},\\\\ \nC & = 2^{\\alpha_3} \\cdot 3^{\\beta_3} \\cdot {p_1}^{p-\\gamma_1} \\cdot {p_2}^{p-\\gamma_2} \\cdot {q_2}^{p-\\delta_2},\\end{align*}\nwhere \\begin{align*} (\\alpha_1,\\alpha_2, \\alpha_3) & \\in \\{(1,0,0),(0,1,0),(0,0,1) \\}, & \\\\\n(\\beta_1, \\beta_2, \\beta_3) & \\in \\begin{cases} \\{ (1,0,0),(0,1,0),(0,0,1) \\} & \\text{ if } c_m = 6, \\\\\n \\{ (0,0,0) \\} & \\text{ if } c_m = 2, \\end{cases} \\\\\n\\gamma_1 & \\in \\{0,r_1,p-r_1\\}, &  \\\\\n\\gamma_2 & \\in \\{0,1,p-1\\}, & \\\\\n\\delta_2 & \\in \\{0,s_2,p-s_2 \\}. &\n\\end{align*}\n\n\\begin{proof}[Proof of Case 2] For primes away from $2$, we argue in exactly the same way as in Case 1. We only need to consider what happens at the prime $2$. If $x$ is odd, then $a_mx-b_m$ is also odd, and we simply divide equation (\\ref{eq2}) by $2$ so that $B$ has a factor of $2$. Since $a_m+b_m$ is odd, there is nothing more to do.\n\nNow we suppose $x$ is even. Since $\\operatorname{ord}_2{b_m} = 1$, we must have $2 \\parallel  d_1$. We divide both sides of equation (\\ref{eq2}) by $2^{p+1}$. Either $2 \\parallel x$ and $2^p \\mid a_mx-b_m$, or $2^p \\mid x$ and $ 2 \\parallel a_m x- b_m$. Since we may absorb any $p$th powers, we have  $(\\alpha_1,\\alpha_3) = (0,1)$ or $(1,0)$.\n\\end{proof}\n\n\\subsection*{Case 3: \\texorpdfstring{$\\operatorname{ord}_2(b_m) = 2$}{}}\n\nHere, \\[m \\in \\{ 9,17,25,33,41,49  \\}. \\] We have\n\\begin{itemize}\n\\item $b_m = 4 \\cdot {p_1}^{r_1}$ for some $r_1 \\in \\{ 0,1 \\}$, and $p_1 \\nmid c_m$ is prime;\n\\item $a_m+b_m = {p_2}^{r_2} \\cdot {q_2}^{s_2}$, for some $r_2 \\in \\{1,2\\}$, $s_2 \\in \\{ 0,1 \\}$, and $p_2, q_2 \\nmid c_m$ are prime.\n\\end{itemize}\nThen \\begin{align*} A & = 2^{\\alpha_1} \\cdot 3^{\\beta_1} \\cdot {p_1}^{\\gamma_1}, \\\\\nB & = 2^{\\alpha_2} \\cdot 3^{\\beta_2} \\cdot {p_2}^{\\gamma_2} \\cdot {q_2}^{\\delta_2},\\\\ \nC & = 2^{\\alpha_3} \\cdot 3^{\\beta_3} \\cdot {p_1}^{p-\\gamma_1} \\cdot {p_2}^{p-\\delta_2} \\cdot {q_2}^{p-\\delta_2},\\end{align*}\nwhere \\begin{align*} (\\alpha_1,\\alpha_2, \\alpha_3) & \\in \\{(0,1,0),(2,0,p-1),(p-1,0,2) \\}, & \\\\\n(\\beta_1, \\beta_2, \\beta_3) & \\in \\begin{cases} \\{ (1,0,0),(0,1,0),(0,0,1) \\} & \\text{ if } c_m = 6, \\\\\n \\{ (0,0,0) \\} & \\text{ if } c_m = 2, \\end{cases} \\\\\n\\gamma_1 & \\in \\{0,r_1,p-r_1\\}, &\\\\\n\\gamma_2 & \\in \\{0,r_2,p-r_2\\}, &\\\\\n\\delta_2 & \\in \\{0,s_2,p-s_2 \\}. &\n\\end{align*}\n\n\\begin{proof}[Proof of Case 3]\nWe will only consider what happens at the prime $2$ in the case $x$ even. The other primes and the case when $x$ is odd can be dealt with as in the proofs of cases 1 and 2.\n\nWe first claim that $2^2 \\parallel d_1$. If not, then we must have $2 \\parallel d_1$, and then either $2^2 \\mid x$ or $2^2 \\mid (a_mx-b_m)$. Since $2^2 \\mid b_m$, we will have that $2^2 \\mid x$ and $2^2 \\mid a_mx-b_m$, a contradiction, proving the claim.\n\nWe now divide both sides of equation (\\ref{eq2}) by $2^{p+1}$. One of $x$ and $a_mx-b_m$ will be exactly divisible by $2^2$ and the other will be divisible by $2^{p-1}$, and so $(\\alpha_1, \\alpha_3) = (2,p-1)$ or $(p-1,2)$. \n\\end{proof}\n\n\\subsection*{Case 4: \\texorpdfstring{$\\operatorname{ord}_2(b_m) = 3$}{}}\n\nHere, \\[m \\in \\{ 13,29,45 \\}. \\] We have\n\\begin{itemize}\n\\item $b_m = 8 \\cdot {p_1}^{r_1}$ for some $r_1 \\in \\{ 0,1 \\}$, and $p_1 \\nmid c_m$ is prime;\n\\item $a_m+b_m = p_2$, and $p_2 \\nmid c_m$ is prime.\n\\end{itemize}\nThen \\begin{align*} A & = 2^{\\alpha_1} \\cdot 3^{\\beta_1} \\cdot {p_1}^{\\gamma_1}, \\\\\nB & = 2^{\\alpha_2} \\cdot 3^{\\beta_2} \\cdot {p_2}^{\\gamma_2},\\\\ \nC & = 2^{\\alpha_3} \\cdot 3^{\\beta_3} \\cdot {p_1}^{p-\\gamma_1} \\cdot {p_2}^{p-\\gamma_2}, \\end{align*}\nwhere \\begin{align*} (\\alpha_1,\\alpha_2, \\alpha_3) & \\in \\{(0,1,0),(3,0,p-2),(p-2,0,3) \\}, & \\\\\n(\\beta_1, \\beta_2, \\beta_3) & \\in \\begin{cases} \\{ (1,0,0),(0,1,0),(0,0,1) \\} & \\text{ if } c_m = 6, \\\\\n \\{ (0,0,0) \\} & \\text{ if } c_m = 2, \\end{cases} \\\\\n\\gamma_1 & \\in \\{0,r_1,p-r_1\\}, & \\\\\n\\gamma_2 & \\in \\{0,1,p-1\\}. &\n\\end{align*}\nWhen $p = 3$, we must also consider the cases $(\\alpha_1, \\alpha_2, \\alpha_3) = (2,0,2)$, with $\\beta_i$ and $\\gamma_i$ varying as above.\n\n\\begin{proof}[Proof of Case 4] As in Case 3, we will only consider $x$ even and the prime $2$. If $2 \\parallel d_1$ then we obtain a contradiction as in Case 3. Suppose $2^2 \\parallel d_1$. Then if $2^3 \\mid x$ or $a_mx-b_m$ we obtain a contradiction as before, so we must have $2^2 \\parallel x, a_mx-b_m$. The valuation at $2$ of the left-hand side of equation (\\ref{eq2}) is thus $4$, and it is $p\\operatorname{ord}_2(y)+1$ for the right-hand side of equation (\\ref{eq2}). This forces $p = 3$ and $(\\alpha_1,\\alpha_3)  =(2,2)$.\n\nFinally, if $2^3 \\parallel \\gcd(x,a_mx-b_m)$ then we divide by $2^{p+1}$ and one of $A$ and $C$ will have a factor of $2^3$, and the other a factor of $2^{p-2}$.\n\\end{proof}\n\n\\subsection*{Case 5: \\texorpdfstring{$\\operatorname{ord}_2(b_m) = 4$}{}}\n\nHere, \\[m = 21.\\] We have\n\\begin{itemize}\n\\item $b_m = 16$ and $c_m = 6$;\n\\item $a_m+b_m = p_2 \\cdot q_2$, and $p_2, q_2 \\nmid c_m$ are prime.\n\\end{itemize}\nThen \\begin{align*} A & = 2^{\\alpha_1} \\cdot 3^{\\beta_1}, \\\\\nB & = 2^{\\alpha_2} \\cdot 3^{\\beta_2} \\cdot {p_2}^{\\gamma_2} \\cdot {q_2}^{\\delta_2},\\\\ \nC & = 2^{\\alpha_3} \\cdot 3^{\\beta_3} \\cdot {p_2}^{p-\\gamma_2} \\cdot {q_2}^{p-\\delta_2}, \\end{align*}\nwhere \\begin{align*} (\\alpha_1,\\alpha_2, \\alpha_3) & \\in \\{(0,1,0),(4,0,p-3),(p-3,0,4) \\}, \\\\\n(\\beta_1, \\beta_2, \\beta_3) & \\in \\{ (1,0,0),(0,1,0),(0,0,1) \\}, \\\\\n\\gamma_2, \\delta_2 & \\in \\{0,1,p-1\\}.\n\\end{align*}\nWhen $p = 3$ or $p = 5$ we must also consider the cases \\[(\\alpha_1, \\alpha_2, \\alpha_3) = \\left((p+1)\/2, 0, (p+1)\/2 \\right), \\] with $\\beta_i, \\gamma_2,$ and $\\delta_2$ varying as above.\n\n\\begin{proof}[Proof of Case 5]\nWe are in a very similar set-up to Case 4. If $2^2 \\parallel d_1$ then we must have $p = 3$ and $(\\alpha_1,\\alpha_3) = (2,2)$. If $2^3 \\parallel d_1$, then by comparing valuations at $2$ on each side of equation (\\ref{eq2}), we have $6 = p \\operatorname{ord}_2(y) + 1$, which forces $p = 5$ and $(\\alpha_1,\\alpha_3) = (3,3)$.\nNext, if $2^4 \\parallel d_1$, then we divide through by $2^{p+1}$ and argue as in previous cases.\n\\end{proof}\n\n\\subsection*{Case 6: \\texorpdfstring{$\\operatorname{ord}_2(b_m) = 5$}{}}\n\nHere, \\[ m = 37. \\] We have\n\\begin{itemize}\n\\item $b_m = 32$ and $c_m = 6$;\n\\item $a_m+b_m = p_2$, and $p_2 \\nmid c_m$ is prime.\n\\end{itemize}\nThen \\begin{align*} A & = 2^{\\alpha_1} \\cdot 3{^\\beta_1}, \\\\\nB & = 2^{\\alpha_2} \\cdot 3^{\\beta_2} \\cdot {p_2}^{\\gamma_2},\\\\ \nC & = 2^{\\alpha_3} \\cdot 3^{\\beta_3} \\cdot {p_2}^{p-\\gamma_2},\n\\end{align*}\nwhere \\begin{align*} (\\alpha_1,\\alpha_2, \\alpha_3) & \\in \\begin{cases} \\{(0,1,0),(5,0,p-4),(p-4,0,5) \\} & \\text{ if } p > 3, \\\\ \\{(0,1,0),(2,0,2) \\} & \\text{ if } p = 3,\n\\end{cases}\n \\\\\n(\\beta_1, \\beta_2, \\beta_3) & \\in \\{ (1,0,0),(0,1,0),(0,0,1) \\}, & \\\\\n\\gamma_2 & \\in \\{0,1,p-1\\}. &\n\\end{align*}\nWhen $p = 3$, $p = 5$, or $p = 7$ we must also consider the cases \\[(\\alpha_1, \\alpha_2, \\alpha_3) = \\left((p+1)\/2, 0, (p+1)\/2 \\right), \\] with $\\beta_i$ and  $\\gamma_2$ varying as above.\n\n\\begin{proof}[Proof of Case 6] This is almost identical to the argument given in Case 5, apart when $2^5 \\parallel d_1$ and $p = 3$. In this case, dividing by $2^{p+1} = 2^4$ is not enough to make the factors coprime. Instead, we divide through by $2^{2p+1} = 2^7$. One of $A$ and $C$ will have a factor of $2^5$, and the other a factor of $2^2$. After removing the perfect cube $2^3$ from $2^5$, we have $(\\alpha_1, \\alpha_3) = (2,2)$.\n\\end{proof}\n\n\\section{The Local Method}\n\nWe consider the system (\\ref{system}) of Thue equations for a fixed value of $m$ and triple $(A,B,C)$. We will start by considering this system mod $\\ell$, for many auxilliary primes $\\ell$ to try and obtain a contradiction; since if the system of equations has no local solution then it will certainly not have a global solution. When the system of equations does not have a (global) solution, we found this method to be extremely effective (as we see below). The strategy we present here is used with a single binomial Thue equation in \\citep[p.~492]{local}. \n\nFix a prime $p > 2$. We search for a prime $\\ell$ such that $\\ell = 2kp+1$ for some $k \\geq 1$ (i.e. $\\ell \\equiv 1 \\pmod{p}$), such that $\\ell \\nmid ABC$, and for which the system of equations has no solution mod $\\ell$. If we can find such an $\\ell$, then we have obtained a contradiction. The reason for choosing $\\ell$ of this form is that we have, for each $i \\in \\{1,2,3\\}$, either $\\ell \\mid y_i$, or \\[ ({{y_i}^p})^{2k} = {y_i}^{\\ell-1} \\equiv 1 \\pmod{\\ell}. \\] In particular, ${y_i}^p \\in \\mu_{2k}(\\mathbb{F}_\\ell) \\cup \\{0\\},$ where $ \\mu_{2k}(\\mathbb{F}_\\ell)  = \\{ \\alpha \\in \\mathbb{F}_\\ell : \\alpha^{2k}  = 1 \\}$. We therefore only have $2k+1$ possibilities for ${y_i}^p \\pmod{\\ell}$, and moreover the set $\\mu_{2k}(\\mathbb{F}_\\ell)$ can be computed extremely quickly using a primitive root modulo $\\ell$. Indeed, if $g$ is a primitive root modulo $\\ell$, then  \\[ \\mu_{2k}(\\mathbb{F}_\\ell) = \\{({g^p})^r : 0 \\leq r \\leq 2k-1 \\}.\\]\n\nFor each triple $(A,B,C)$, we searched for a prime $\\ell$ by testing with $1 \\leq k \\leq 150$. For $p>5$, with $p$ less than the prime bound for $m$ obtained in Proposition \\ref{LogBound}, apart from the cases where we have a global solution, and a single case when $p = 7$, we succeeded in obtaining a contradiction.\n\nWhen $p=3$ or $p = 5$, the method sometimes fails even when there is no global solution. In these cases, as $p$ is small we can simply solve the two Thue equations using \\texttt{Magma} and verify whether we have a solution $(y_1,y_2,y_3)$ with $y_1,y_2 > 0$ (since $x > 0$). As mentioned above, the local method also fails for $p = 7$ in a single case. This is for the case $m = 21$ and $(A,B) = (2^4 \\cdot 3, 1)$. Here we also simply solve the corresponding Thue equations directly to conclude there are no non-zero solutions.\n\nFor certain triples, the local method will fail \\emph{for all values of $p$} as we have a global solution for all $p$. There are three cases when this happens.\n\n\\begin{enumerate}[(I)]\n\\item $A = 1, B = 2$, and $a_m-C = b_m$ . Here we have a global solution $(y_1,y_2,y_3) = (1,1,1)$ for all $p$, which comes from the solution $x= y = 1$ to our original equation. However, in this case, our first Thue equation is \\[ 2{y_2}^p - {y_1}^p = 1.\\] Applying a well-known result of Bennett \\citep[Theorem~1.1]{approx}, we see that $y_1 = y_2 = 1$ for all $p$, so $x = 1$.\n\n\\item $A = 1$ and $C = a_m + b_m$. This admits the solution $(y_1,y_2,y_3) = (-1,0,-1)$.\n\n\\item $B = 1$ and $C = b_m$. This admits the solution $(y_1,y_2,y_3) = (0,1,-1)$. \n\\end{enumerate}\n\nIn cases (II) and (III) we must use a different strategy. We use Theorem \\ref{ThmD} (stated below) together with the modular method.\n\n\\section{The Modular Method}\n\nIt remains to deal with cases (II) and (III), outlined in Section 4, for each $6 \\leq m \\leq 50$. In each case, we have $A = 1$ or $B = 1$, and this leads to an equation of the form \\begin{equation}\\label{eqD} z_1^p - Dz_2^p = 1 \\end{equation} for integers $z_1$ and  $z_2$. The following result of Bartolom\\'e and Mih{\\u{a}}ilescu will be extremely helpful. \n\n\\begin{theorem}[{\\citep[Theorem~1.3]{bartmihai}}]\\label{ThmD} Let $D>1$ and and let $p$ be an odd prime satisfying \\[ \\gcd(\\mathrm{Rad}(\\varphi(D),p) = 1. \\] Suppose $z_1$ and $z_2$ are integers satisfying equation (\\ref{eqD}) with $ \\abs{z_2} > 1$. Then either $(z_1,z_2,D,p) = (18,7,17,3)$ or $p > 163 \\cdot 10^{12}$.\n\\end{theorem}\n\nHere, $\\varphi$ denotes Euler's totient function, and $\\mathrm{Rad}(\\varphi(D))$ denotes the product of all primes dividing $\\varphi(D)$. We note that the theorem stated in \\citep{bartmihai} goes on to give further constraints in the case $p > 163 \\cdot 10^{12}$.\n\nSince the prime bound obtained in Proposition \\ref{LogBound} is smaller than $10^6$ in each case, Theorem \\ref{ThmD} reduces our problem to only needing to consider finitely many small primes in each case. When $ p = 3$, $5$, or $7$, we can solve the relevant Thue equations directly with \\texttt{Magma}. For $p \\geq 11$ and for each value of $m$ we are then left with at most one triple $(A,B,C)$ and at most one value of $p$ that we are unable to eliminate. Table \\ref{TabRat} records these remaining values of $p$ and corresponding triples. We note that one of $A$ and $B$ is equal to $1$, and the other is exactly divisible by $2$ in each case.\n\n\\begingroup \n\\renewcommand*{\\arraystretch}{1.5}\n\\begin{table}[ht!]\n\\begin{center} \\small\n\\begin{tabular}{ |c|c|c|c|c|c| } \n \\hline\n $m$ & $p$ & $A$ & $B$ & $C$  \\\\\n \\hline\n $15$  & $11$ & $1$ & $2 \\cdot 3 \\cdot 23^{10}$ & $23$               \\\\ \\hline\n $27$ & $23$ & $1$ & $2 \\cdot 3 \\cdot 47^{22}$ & $47$               \\\\ \\hline\n $28$ & $11$ & $2 \\cdot 3 \\cdot 23^{10}$ & $1$ & $23$ \\\\ \\hline\n $30$ & $13$ &  $1$ & $2 \\cdot 3 \\cdot 53^{12}$ & $53$               \\\\ \\hline\n $33$ & $29$ &  $1$ & $2 \\cdot 3 \\cdot 59^{28}$ & $59$               \\\\ \\hline\n $37$  & $11$ & $1$ & $2 \\cdot 3 \\cdot 67^{10}$ & $67$               \\\\ \\hline\n $38$  & $11$ & $1$ & $2 \\cdot 3 \\cdot 23^{10}$ & $23$               \\\\ \\hline\n $43$ & $13$ &  $1$ & $2 \\cdot 3 \\cdot 79^{12}$ & $79$               \\\\ \\hline\n $45$ & $41$ &  $1$ & $2 \\cdot 3 \\cdot 83^{40}$ & $83$               \\\\ \\hline\n $48$  & $11$ & $1$ & $2 \\cdot 3 \\cdot 89^{10}$ & $89$               \\\\ \\hline\n \n\\end{tabular}\n\\caption{\\label{TabRat}\\normalsize Remaining cases after applying Theorem \\ref{ThmD} and solving Thue equations for $p \\leq 7$.}\n\\end{center}\n\\end{table} \n\nIn order to eliminate the cases remaining in Table \\ref{TabRat}, we will use some of the local arguments of Section 4 in combination with the modular method. We start by seeing how one may associate, following standard recipes (see \\citep{ppp} for example), a Frey curve to equation (\\ref{eqD}). We rewrite equation (\\ref{eqD}) as \\begin{equation}\\label{ppp} -1 - Dz_2^p +z_1^p = 0,  \\end{equation} and we assume that $p \\geq 11$ and $\\operatorname{ord}_2(D) = 1$, since this will be our set-up. The Frey curve we associate to this equation is \\[E: \\; Y^2 = X(X+1)(X-Dz_2^p). \\] The conductor, $N$, of $E$ is then given by \\[ N = \\begin{cases} 2 \\cdot  \\mathrm{Rad}_2(Dz_1z_2) & \\text{ if } \\; \\; 2 \\mid z_2, \\\\ 2^5 \\cdot  \\mathrm{Rad}_2(Dz_1z_2) & \\text{ if }  \\; \\;2 \\nmid z_2.  \\end{cases} \\] Here, $\\mathrm{Rad}_2(Dz_1z_2)$ denotes the product of all \\emph{odd} primes dividing $Dz_1z_2$. We write $\\overline{\\rho}_{E,p}$ for the mod $p$ Galois representation of $E$. Applying standard level-lowering results, we obtain that \\[ \\overline{\\rho}_{E,p} \\sim \\overline{\\rho}_{f,\\mathfrak{p}}, \\] for $f$ a newform at level $N_p$, where \\[N_p = \\begin{cases} 2 \\cdot \\mathrm{Rad}_2(D) & \\text{ if } \\; \\; 2 \\mid z_2, \\\\ 2^5 \\cdot  \\mathrm{Rad}_2(D) & \\text{ if }  \\; \\;2 \\nmid z_2,  \\end{cases} \\] and $\\mathfrak{p}$ a prime above $p$ in the coefficient field of $f$.\n\nWe are now in a position to complete the proof of Theorem \\ref{Mainthm}.\n\n\\begin{proof}[Proof of Theorem \\ref{Mainthm}] It remains to deal with the cases appearing in Table \\ref{TabRat}. Suppose we are in one of these cases, and let $(y_1,y_2,y_3)$ be a non-zero solution to the system (\\ref{system}) of Thue equations. By rewriting ${y_i}^p$ as $-(-y_i)^p$ if necessary, we obtain an equation of the same form as (\\ref{ppp}). As described above, we attach a Frey curve $E$ to this equation, and level lower so that $\\overline{\\rho}_{E,p} \\sim \\overline{\\rho}_{f,\\mathfrak{p}}$, for $f$ a newform at level $2 \\cdot \\mathrm{Rad}_2(D)$ or $2^5 \\cdot \\mathrm{Rad}_2(D)$.\n\n Now, if $\\ell \\mid y_1y_2$ is a prime, then it must be a prime of multiplictive reduction for $E$, and by comparing traces of Frobenius, we have \\[\\ell+1 \\equiv \\pm c_\\ell(f) \\pmod{\\mathfrak{p}},\\] where $c_\\ell(f)$ denotes the $\\ell$th Fourier coefficient of the newform $f$. It follows that\\begin{equation}\\label{traces} p \\mid \\mathrm{Norm}((\\ell+1)^2 - c_\\ell(f)^2) \\end{equation}\n\nWe now search for a prime $\\ell \\nmid D$ with $\\ell \\equiv 1 \\pmod{p}$, for which the system of Thue equations (\\ref{system}) has a unique solution  mod ${\\ell}$, \\emph{and} for which (\\ref{traces}) does not hold. If the system has a unique solution mod $\\ell$, then this solution must be the reduction mod $\\ell$ of the known global solution, for which $y_1y_2 = 0$, so either $y_1 \\equiv 0 \\pmod{\\ell}$ or   $y_2 \\equiv 0 \\pmod{\\ell}$. So $\\ell \\mid y_1y_2$, and we have therefore obtained a contradiction if (\\ref{traces}) does not hold. For each newform $f$ in each case we were able to find such a prime $\\ell$, apart from the cases listed in Table \\ref{TabSturm}.\n\nFor the remaining newforms in Table \\ref{TabSturm}, we find that for any prime $q \\nmid 2D$ that we test, \\begin{equation}\\label{strm} p \\mid \\mathrm{Norm}(q+1 - c_q(f)). \\end{equation} This suggests that the representation $\\overline{\\rho}_{f,\\mathfrak{p}}$ is reducible, which would be a contradiction. We proceed by applying \\citep[Proposition~2.2]{tripexp} to the newform $f$. We obtain that $p \\mid \\# E(\\mathbb{F}_q)$ for any prime $q \\nmid D$, and so $E$ must have a rational subgroup of order $p$, a contradiction since $p \\geq 11$.\n\\begingroup \n\\renewcommand*{\\arraystretch}{1.5}\n\\begin{table}[ht!]\n\\begin{center} \\small\n\\begin{tabular}{ |c|c|c| } \n \\hline\n $m$ & $p$ & $f$  \\\\\n \\hline\n $15$ & $11$ & 138.2.a.d \\\\ \\hline\n $27$ & $23$ & 282.2.a.e \\\\ \\hline\n $28$ & $11$ & 138.2.a.d \\\\ \\hline\n $30$ & $13$ & 318.2.a.g \\\\ \\hline\n $33$ & $29$ & 354.2.a.h \\\\ \\hline\n $37$ & $11$ & 402.2.a.g \\\\ \\hline\n $38$ & $11$ & ---      \\\\ \\hline\n $43$ & $13$ & 474.2.a.e \\\\ \\hline\n $45$ & $41$ & 498.2.a.g \\\\ \\hline\n $48$ & $11$ & 534.2.a.f \\\\ \\hline \n\\end{tabular}\n\\caption{\\label{TabSturm}\\normalsize Remaining newforms. We use the notation of the LMFDB \\citep{lmfdb}.}\n\\end{center}\n\\end{table} \n\\endgroup\n\\end{proof}\n\n\\bibliographystyle{plainnat}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nWe perform a resurgence analysis of the $SU(2)$ Chern-Simons partition function on a Bireksorn homology sphere, following \\cite{GMP}. Consider the Chern-Simons action with a gauge group $G$ on a 3-manifold $M_{3}$:\n$$CS(A) = \\frac{1}{8\\pi^{2}} \\int_{M_{3}} A \\wedge dA + \\frac{2}{3}A \\wedge A \\wedge A,$$\nwhere $A$ is a Lie algebra (ad$G$) valued 1-form on $M_{3}$. Classical solutions of this action are the flat connections, satisfying $F_{A} = dA + A \\wedge A = 0$. The Chern-Simons partition function at level $k$ can be expanded with a perturbation parameter $1\/k$, around the flat connections:\n\n\\begin{equation}\nZ_{CS}(M_{3}) = \\sum_{\\alpha \\in \\mathcal{M}_{\\text{flat}}(M_{3}, G)} e^{2 \\pi i k CS(\\alpha)}Z^{\\text{pert}}_{\\alpha}.\n\\label{eqn:pert}\n\\end{equation}\nAbove, $\\mathcal{M}_{\\text{flat}}(M_{3}, G)$ is the moduli space of flat $G$-connections on $M_{3}$, and we have assumed a discrete moduli space. When $k$ is an integer, $CS(A)$ is only defined modulo 1. \n\nThe exact partition function $Z_{CS}(M_{3}) = \\int \\mathcal{D}A e^{2 \\pi i k CS(A)}$ can be recovered from its perturbative expansion by a resurgence analysis of Jean \\'{E}calle \\cite{Ecalle}. We first analytically continue $k$ to compex values and apply the method of steepest descent. Then, perform a Borel transformation and resummation of the perturbative partition function, to recover the exact partition function. Surprisingly, the exact partition function is now written as a linear sum of the ``homological blocks'' \\cite{GMP}:\n\\begin{equation}\nZ_{CS}(M_{3}) = \\sum_{a \\, \\text{abelian}} e^{2 \\pi i k CS(\\alpha)}Z_{a}.\n\\label{eqn:abelianDecomposition}\n\\end{equation}\nAbove, $Z_{a}$ gets contributions from \\textit{both} the abelian flat connection $a$ and the irreducible flat connections. In \\cite{GPV}, it was proposed that the partition function in this form allows a ``categorification,'' in a sense that it is a ``S-transform'' of a vector whose entries are integer-coefficient Laurent series in $q = e^{2\\pi i \/k}$.\n\nIn this paper, we provide a supporting example of \\cite{GMP}. First, we perform a resurgence analysis of $SU(2)$ Chern-Simons partition function on  a Brieskorn homology sphere, $M_{3} = \\Sigma(2,5,7)$. We start with the exact partition function $Z_{CS}(\\Sigma(2,5,7))$, which is written as a linear sum of ``mock modular forms'' \\cite{HikamiBrieskorn}. Then, we consider its perturbative expansion and perform a Borel resummation. The Borel resummation in effect recovers the full partition function $Z_{CS}(\\Sigma(2,5,7))$, and we observe a Stokes phenomenon which encodes the non-perturbative contributons to the partition function. \n\n\\section{Setups for the Borel resummation in Chern-Simons theory}\nIn this section, we provide necessary notations and setups for the Borel resummation in Chern-Simons theory. A complete and concise review can be found in section 2 of \\cite{GMP}. \n\nLet us start with the exact Chern-Simons partition function $Z_{CS}(M_{3}) = \\int \\mathcal{D}A e^{2 \\pi i k CS(A)}$, integrated over $G=SU(2)$ connections. Next, analytically continue $k$ to complex values and apply the method of steepest descent on the Feynman path integral \\cite{Marino, Garoufalidis, GLM, ArgyresUnsal, Kashani-Poor, CostinGaroufalidis, WittenAnalytic, KontsevichPerimeter,KontsevichSCGP,KontsevichTFC}. Then, the integration domain is altered to a middle-dimensional cycle $\\Gamma$ in the moduli space of $G_{\\mathbb{C}} = SL(2,\\mathbb{C})$ connections, which is the union of the steepest descent flows from the saddle points. To elaborate, the moduli space is the universal cover of the space of $SL(2,\\mathbb{C})$ connections modulo ``based'' gauge transformations, in which the gauge transformations are held to be $1$ at the designated points. In sum, the partition function becomes:\n\\begin{equation}\nZ_{CS}(M_{3}) = \\int_{\\Gamma} \\mathcal{D}A e^{2 \\pi i k CS(A)}, \\quad k \\in \\mathbb{C}.\n\\label{eqn:ZoverGamma}\n\\end{equation}\n\n\n\\subsection{Borel resummation basics}\nPartition function of form Equation \\ref{eqn:ZoverGamma} is interesting, for its perturbative expansion can be regarded as a \\textit{trans-series} expansion, which can be Borel resummed. Let us provide here the basics of Borel resummation, following \\cite{Marino}. The simplest example of a trans-series is a formal power series solution of Euler's equation:\n$$\\frac{d \\varphi}{dz} + A \\varphi(z) = \\frac{A}{z}, \\quad \\varphi_{0}(z) = \\sum_{n \\geq 0} \\frac{A^{-n}n!}{z^{n+1}}.$$\nOne may view the above trans-series as a perturbative (in $1\/z$) solution to the differential equation, but the solution has zero radius of convergence. By the Borel resummation, however, one can recover a convergent solution. When a trans-series is of form $\\varphi(z) = \\sum_{n \\geq 0} a_{n}\/z^{n}$ with $a_{n} \\sim n!$, its Borel transformation is defined as:\n$$\\hat{\\varphi}(\\zeta) = \\sum_{n \\geq 1} a_{n} \\frac{\\zeta^{n-1}}{(n-1)!}.$$\nThe Borel transformation $\\hat{\\varphi}(\\zeta)$ is analytic near the origin of $\\zeta$-plane. If we can analytically continue $\\hat{\\varphi}(\\zeta)$ to a neighborhood of the positive real axis, we can perform the Laplace transform:\n$$S_{0}\\varphi(z) = a_{0} + \\int_{0}^{\\infty} e^{- z \\zeta}\\hat{\\varphi}(\\zeta) d \\zeta,$$\nwhere the subscript ``0'' indicates that the integration contour is along the positive real axis, $\\{ \\arg(z) = 0 \\}$. It can be easily checked that the asymptotics of the above integral coincides with that of $\\varphi(z)$. When $S_{0}\\varphi(z)$ converges in some region in the $z$-plane, $\\varphi(z)$ is said to be Borel summable, and $S_{0}\\varphi(z)$ is called the Borel sum of $\\varphi(z)$. \n\n\\subsection{Chern-Simons partition function as a trans-series}\n\nSaddle points of the Chern-Simons action form the moduli space of flat connections $\\tilde{M}$,\nwhose connected components $\\tilde{M}_{\\tilde{\\alpha}}$ are indexed by their ``instanton numbers,''\n$$\\tilde{\\alpha} = (\\alpha, CS(\\tilde{\\alpha})) \\in \\mathcal{M}_{\\text{flat}}(M_{3},SL(2, \\mathbb{C})) \\times \\mathbb{Z}.$$ \nHere, $CS(\\tilde{\\alpha})$ denotes the value of Chern-Simons action at $\\alpha$, without moding out by 1. Following \\cite{GMP}, we will call a flat connection \\textit{abelian} (\\textit{irreducible}, resp.), if the stabilizer is $SU(2)$ or $U(1)$ ($\\{ \\pm 1\\}$, resp.) action on  $Hom(\\pi_{1}(M_{3}),SU(2))$. \n\nNow, let $\\Gamma_{\\tilde{\\alpha}}$ be the union of steepest descent flows in $\\tilde{M}$, starting from $\\tilde{\\alpha}$. The integration cycle $\\Gamma$ is then given by a linear sum of these ``Lefshetz thimbles.''\n\n\\begin{equation}\n\\Gamma = \\sum_{\\tilde{\\alpha}} n_{\\tilde{\\alpha},\\theta}\\Gamma_{\\tilde{\\alpha},\\theta},\n\\label{eqn:GammaDecomposition}\n\\end{equation}\nwhere $\\theta = \\arg(k)$, and $n_{\\tilde{\\alpha},\\theta} \\in \\mathbb{Z}$ are the \\textit{trans-series} parameters, given by the pairing between the submanifolds of steepest descent and ascent. The value of $\\theta$ is adjusted so that there is no steepest descent flow between the saddle points. Let $I_{\\tilde{\\alpha},\\theta}$ be the contribution from a Lefshetz thimble $\\Gamma_{\\tilde{\\alpha},\\theta}$ to $Z_{CS}(M_{3})$ in Equation \\ref{eqn:ZoverGamma}:\n\n$$I_{\\tilde{\\alpha},\\theta} = \\int_{\\Gamma_{\\tilde{\\alpha},\\theta}} \\mathcal{D}A e^{2 \\pi i k CS(A)},$$\nwhich can be expanded in $1\/k$ near $\\tilde{\\alpha}$ as:\n$$I_{\\tilde{\\alpha},\\theta} \\sim e^{2 \\pi i k CS(\\tilde{\\alpha})}Z^{\\text{pert}}_{\\alpha}, \\quad \\text{where} \\quad Z^{\\text{pert}}_{\\alpha} = \\sum_{n=0}^{\\infty} a_{n}^{\\alpha}k^{-n+(d_{\\alpha}-3)\/2}, \\quad d_{\\alpha} = dim_{\\mathbb{C}}\\tilde{\\mathcal{M}}_{\\tilde{\\alpha}}.$$\nIn sum, we can write the Chern-Simons partition function in the form:\n\n\\begin{equation}\nZ_{CS}(M_{3};k) = \\sum_{\\tilde{\\alpha}} n_{\\tilde{\\alpha},\\theta}I_{\\tilde{\\alpha},\\theta} \\sim \\sum_{\\tilde{\\alpha}} n_{\\tilde{\\alpha},\\theta}e^{2 \\pi i k CS(\\tilde{\\alpha})}Z^{\\text{pert}}_{\\alpha}(k),\n\\label{eqn:trans-series}\n\\end{equation}\nwhich is a trans-series expansion of the Chern-Simons partition function. From the asymptotics given by this trans-series, we can apply Borel resummation and recover the full Chern-Simons partition function. Note that Equation \\ref{eqn:trans-series} depends on the choice of $\\theta = \\arg(k)$. In fact, as we vary $\\theta$, the value of $I_{\\tilde{\\alpha},\\theta}$ jumps to keep the whole expression continuous in $\\theta$ as follows:\n\\begin{equation}\nI_{\\tilde{\\alpha},\\theta_{\\tilde{\\alpha}\\tilde{\\beta}}+\\epsilon} = I_{\\tilde{\\alpha},\\theta_{\\tilde{\\alpha}\\tilde{\\beta}}-\\epsilon} + m_{\\tilde{\\alpha}}^{\\tilde{\\beta}}I_{\\tilde{\\beta},\\theta_{\\tilde{\\alpha}\\tilde{\\beta}}-\\epsilon}.\n\\label{eqn:Stokes}\n\\end{equation}\nThis is called the Stokes phenomenon, and it happens near the Stokes rays $\\theta = \\theta_{\\tilde{\\alpha}\\tilde{\\beta}} \\equiv \\frac{1}{i}\\arg(S_{\\tilde{\\alpha}}-S_{\\tilde{\\beta}})$. The trans-series parameters $n_{\\tilde{\\alpha},\\theta}$ jump accordingly to keep $Z_{CS}(M_{3};k)$ continuous in $\\theta$. The coefficients $m_{\\tilde{\\alpha}}^{\\tilde{\\beta}}$ are called Stokes monodromy coefficients.\n\n\\section{Exact partition function $Z_{CS}(\\Sigma(2,5,7))$}\n\\label{sec:exact}\nBefore going into the resurgence analysis of $Z_{CS}(\\Sigma(2,5,7))$, let us provide here the exact partition function $Z_{CS}(\\Sigma(2,5,7))$. We first compute the Witten-Reshetikhin-Turaev (WRT) invariant $\\tau_{k}(\\Sigma(p_{1},p_{2},p_{3}))$ and then write the exact $SU(2)$ Chern-Simons partition function in terms of WRT invariants as follows:\n\\begin{equation}\nZ_{CS}(\\Sigma(p_{1},p_{2},p_{3})) = \\frac{\\tau_{k}(\\Sigma(p_{1},p_{2},p_{3}))}{\\tau_{k}(S^{2} \\times S^{1})}.\n\\label{eqn:WRTandCSpartition}\n\\end{equation}\nHere, $k$ is the level of Chern-Simons theory.\\footnote{To be more precise, $k$ must be replaced by $k+2$. However, our interest in this paper is to recover the full partition function from a perturbative expansion in $1\/k$. Therefore, we will assume $k$ to be large, and replace $k+2$ with $k$ here.}\n\nWRT invariants for Seifert homology spheres can be computed from their surgery presentations \\cite{LawrenceRozansky}. In this paper, we focus on a specific type of Seifert homology spheres, the so-called Bireskorn homology spheres. A Brieskorn manifold $\\Sigma(p_{1},p_{2},p_{3})$ is defined as an intersection of a complex unit sphere $|z_{1}|^{2}+|z_{2}|^{2}+|z_{3}|^{2}=1$ and a hypersurface $z_{1}^{p_{1}}+z_{2}^{p_{2}}+z_{3}^{p_{3}}=0$. When $p_{1},p_{2},p_{3}$ are coprime integers, $\\Sigma(p_{1},p_{2},p_{3})$ is a homology sphere with three singular fibers. From the surgery presentation of $\\Sigma(p_{1},p_{2},p_{3})$, we can write its WRT invariant, which can be written a linear sum of mock modular forms \\cite{HikamiBrieskorn, LawrenceZagier}. In particular, when $1\/p_{1}+1\/p_{2}+1\/p_{3} < 1$, we can write:\n\\begin{equation}\ne^{\\frac{2 \\pi i}{k}(\\frac{\\phi(p_{1},p_{2},p_{3})}{4}-\\frac{1}{2})}(e^{\\frac{2 \\pi i}{k}} -1) \\tau_{k}(\\Sigma(p_{1},p_{2},p_{3})) = \\frac{1}{2}\\tilde{\\Psi}_{p_{1}p_{2}p_{3}}^{(1,1,1)}(1\/k).\n\\label{eqn:WRTmodular}\n\\end{equation}\nLet us decode Equation \\ref{eqn:WRTmodular}. First of all, $\\tau_{k}(\\Sigma(p_{1},p_{2},p_{3}))$ is the desired WRT invariant, normalized such that $\\tau_{k}(S^{3}) = 1$ and $\\tau_{k}(S^{2} \\times S^{1}) = \\sqrt{\\frac{k}{2}}\\frac{1}{\\sin(\\pi \/ k)}.$ Next, the number $\\phi(p_{1},p_{2},p_{3})$ is defined as:\n\\begin{gather}\n\\phi(p_{1},p_{2},p_{3}) = 3 - \\frac{1}{p_{1}p_{2}p_{3}} + 12 (s(p_{1}p_{2},p_{3})+s(p_{2}p_{3},p_{1})+s(p_{3}p_{1},p_{2})), \\nonumber \\\\\n\\text{where} \\quad s(a,b) = \\frac{1}{4b}\\sum_{n=1}^{b-1}\\cot(\\frac{n \\pi}{b})\\cot(\\frac{n a \\pi}{b}). \\nonumber\n\\end{gather}\nFinally, $\\tilde{\\Psi}_{p_{1}p_{2}p_{3}}^{(1,1,1)}$ is a linear sum of mock modular forms $\\tilde{\\Psi}_{p_{1}p_{2}p_{3}}^{a}$, namely:\n\\begin{gather}\n\\tilde{\\Psi}_{P}^{a}(1\/k) = \\sum_{n \\geq 0} \\psi_{2P}^{a}(n)q^{n^{2}\/4P}, \\quad \\text{where} \\quad \\psi_{2P}^{a}(n) = \\begin{cases}\n\\pm 1 & \\text{$n \\equiv \\pm a$ mod $2P$} \\\\   0 & \\text{otherwise}\n\\end{cases} \\label{eqn:modular} \\\\\n\\tilde{\\Psi}_{p_{1}p_{2}p_{3}}^{(1,1,1)}(1\/k) = -\\frac{1}{2}\\sum_{\\epsilon_{1},\\epsilon_{2},\\epsilon_{3} = \\pm 1} \\epsilon_{1}\\epsilon_{2}\\epsilon_{3}\\tilde{\\Psi}_{p_{1}p_{2}p_{3}}^{p_{1}p_{2}p_{3}(1+\\sum_{j}\\epsilon_{j}\/p_{j})}(1\/k), \\label{eqn:linSumModular}\n\\end{gather}\nwhere $q$ in Equation \\ref{eqn:modular} is given by $e^{2 \\pi i \/k }$.\n\nNow, let us restrict ourselves to $(p_{1},p_{2},p_{3}) = (2,5,7)$. First of all, $p_{1}=2,p_{2}=5,p_{3}=7$ are relatively prime, so $\\Sigma(2,5,7)$ is a homology sphere. Next, $1\/p_{1}+1\/p_{2}+1\/p_{3} < 1$, so we can write the WRT invariant as a linear sum of mock modular forms:\n\n\\begin{align}\ne^{\\frac{2 \\pi i}{k}(\\frac{\\phi(2,5,7)}{4}-\\frac{1}{2})}(e^{\\frac{2 \\pi i}{k}} -1) \\tau_{k}(\\Sigma(2,5,7)) &= \\frac{1}{2}\\tilde{\\Psi}_{70}^{(1,1,1)}(1\/k) \\nonumber \\\\\n&= \\frac{1}{2}(\\tilde{\\Psi}_{70}^{11} - \\tilde{\\Psi}_{70}^{31}  - \\tilde{\\Psi}_{70}^{39} + \\tilde{\\Psi}_{70}^{59})(1\/k),\n\\label{eqn:WRTinvariantModularDecomposition}\n\\end{align}\nwhere $\\phi(2,5,7) = -\\frac{19}{70}$. From Equation \\ref{eqn:WRTandCSpartition} and \\ref{eqn:WRTinvariantModularDecomposition}, we can explicitly write the exact Chern-Simons partition function $Z_{CS}(\\Sigma(2,5,7))$ as follows:\n\n\\begin{equation}\nZ_{CS}(\\Sigma(2,5,7)) = \\frac{1}{i q^{\\phi(2,5,7)\/4}\\sqrt{8k}}(\\tilde{\\Psi}_{70}^{11} - \\tilde{\\Psi}_{70}^{31}  - \\tilde{\\Psi}_{70}^{39} + \\tilde{\\Psi}_{70}^{59})(1\/k).\n\\label{eqn:CSpartition}\n\\end{equation} \n\n\\section{Asymptotics of $Z_{CS}(\\Sigma(2,5,7))$}\n\\label{sec:asymptotics}\nBefore proceeding to the Borel transform and resummation of the exact partition function, let us briefly consider the its asymptotics in the large $k$ limit. This can be most easily done by considering the ``mock modular'' property of mock modular forms:\n\\begin{gather}\n\\tilde{\\Psi}_{p}^{a}(q) = -\\sqrt{\\frac{k}{i}} \\sum_{b=1}^{p-1}\\sqrt{\\frac{2}{p}}\\sin{\\frac{\\pi a b}{p}}\\tilde{\\Psi}_{p}^{b}(e^{-2 \\pi i k}) + \\sum_{n \\geq 0} \\frac{L(-2n,\\psi_{2p}^{a})}{n!}\\bigg(\\frac{\\pi i}{2 p k}\\bigg)^{n}, \\label{eqn:mockmodular} \\\\\n\\text{where} \\quad L(-n,\\psi_{2p}^{a}) = -\\frac{(2p)^{n}}{n+1}\\sum_{m=1}^{2p}\\psi_{2p}^{a}(m)B_{n+1}\\bigg(\\frac{m}{2p}\\bigg),\n\\end{gather}\nand $B_{n+1}$ stands for the $(n+1)$-th Bernoulli polynomial. For integer values of $k$, \n$$\\tilde{\\Psi}^{b}_{p}(e^{-2 \\pi i k}) = (1-\\tfrac{b}{p})e^{-\\frac{2 \\pi i k b^{2}}{2p}},$$ and in large $k$ limit, we may consider the second summation in Equation \\ref{eqn:mockmodular} as ``perturbative'' contributions, while the first summation standing for ``non-perturbative'' contributions. Therefore, the asymptotics of $Z_{CS}(\\Sigma(2,5,7))$ can be written as ($p=70$, below):\n\\begin{multline}\ni q^{-19\/280}\\sqrt{8k}Z_{CS}(\\Sigma(2,5,7)) = \\\\\n -\\sqrt{\\frac{k}{i}} \\sum_{b=1}^{70-1} \\sqrt{\\frac{2}{70}}\\bigg( \\sin\\frac{11b \\pi}{70} - \\sin\\frac{31b \\pi}{70} - \\sin\\frac{39b \\pi}{70} + \\sin\\frac{59b \\pi}{70}\\bigg)(1-\\tfrac{b}{p})e^{-\\frac{2 \\pi i k b^{2}}{4p}} \\\\\n+ i q^{-19\/280}\\sqrt{8k}Z_{\\text{pert}}(1\/k),\n\\label{eqn:asymptotics}\n\\end{multline}\nwhere the perturbative contributions $i q^{-19\/280}\\sqrt{8k}Z_{\\text{pert}}(1\/k)$ can be explicitly written as:\n\\begin{gather}\nZ_{\\text{pert}}(1\/k) = Z^{11}_{\\text{pert}}(1\/k)-Z^{31}_{\\text{pert}}(1\/k)-Z^{39}_{\\text{pert}}(1\/k)+Z^{59}_{\\text{pert}}(1\/k), \\nonumber \\\\\n\\text{where} \\quad i \\sqrt{8} q^{-19\/280} Z^{a}_{\\text{pert}}(1\/k) = \\sum_{n \\geq 0} \\frac{b_{n}^{a}}{k^{n+1\/2}} \\quad \\text{for} \\quad a = 11, 31, 39, 59 \\nonumber \\\\\n\\text{and} \\quad b_{n}^{a} = \\frac{L(-2n,\\psi_{2p}^{a})}{n!}\\bigg(\\frac{\\pi i}{2p}\\bigg)^{n}.\n\\end{gather}\n\nOne can easily see that the sum $\\big( \\sin\\frac{11b \\pi}{70} - \\sin\\frac{31b \\pi}{70} - \\sin\\frac{39b \\pi}{70} + \\sin\\frac{59b \\pi}{70}\\big)$ in Equation \\ref{eqn:asymptotics} is nonzero if and only if $b$ is not divisible by $2,5$ or $7$. We will later see that these $b$'s correspond to the positions of the poles in the Borel plane. \n\n\n\\section{Resurgence analysis of $Z_{CS}(\\Sigma(2,5,7))$}\nIn this section, we perform a resurgence analysis of the partition function and decompose $Z_{CS}(M(2,5,7))$ into the homological blocks:\n$$Z_{CS}(\\Sigma(2,5,7)) = \\sum_{\\alpha} n_{\\alpha}e^{2 \\pi i k CS(\\alpha)} Z_{\\alpha},$$\nwhere $\\alpha$ runs over the abelian\/reducible flat connections. Since $Z_{\\alpha}$ gets contributions from both the abelian\/reducble flat connection $\\alpha$ and the irreducible flat connections, it is necessary to study how the contributions from the irreducible flat connections regroup themselves into the homological blocks. We accomplish the goal in three steps. First, we study the Borel transform and resummation of the partition function and identify the contributions from the irreducible flat connections. Then, the contributions from the irreducible flat connections are shown to enter in the homological blocks via Stokes monodromy coefficients.\n\n\\subsection{Borel transform and resummation of $Z_{CS}(M(2,5,7))$}\nRecall that the perturbative contributions $Z^{a}_{\\text{pert}}(1\/k)$ have the following asymptotics:\n\\begin{equation}\ni\\sqrt{8}q^{\\phi(2,5,7)\/4}Z^{a}_{\\text{pert}}(1\/k) = \\sum_{n \\geq 0} \\frac{b^{a}_{n}}{k^{(n+1\/2)}}.\n\\end{equation}\nNow, consider its Borel transform:\n\\begin{align}\nBZ^{a}_{\\text{pert}}(\\zeta) &= \\sum_{n \\geq 1} \\frac{b^{a}_{n}}{\\Gamma(n+1\/2)} \\zeta^{n-1\/2} \\\\\n&= \\frac{1}{\\sqrt{\\zeta}} \\sum_{n \\geq 0} b^{a}_{n}\\frac{4^{n}}{\\sqrt{\\pi}} \\frac{n!}{(2n)!}\\zeta^{n} \\quad \\bigg(\\because \\Gamma(n+1\/2) = \\frac{\\sqrt{\\pi}}{4^{n}}\\frac{(2n)!}{n!} \\bigg) \\\\\n&= \\frac{1}{\\sqrt{\\pi \\zeta}} \\sum_{n \\geq 0} c^{a}_{n} \\frac{n!}{(2n)!} z^{2n}, \\quad \\text{where} \\quad z = \\sqrt{\\frac{2 \\pi i}{p}\\zeta}.\n\\end{align}\nIn the last equality, we have simply changed the variable from $\\zeta$ to $z$ and absorbed all other factors into the coefficients $c_{n}^{a}$.\n\nAlthough the coefficients $c^{a}_{n}$ only appear in the perturbative piece of the partition function, we can recover the exact partition function from them. Let us first consider generating functions which package the coefficients $c^{a}_{n}$:\n$$\\frac{\\sinh((p-a)z)}{\\sinh(pz)} = \\sum_{n \\geq 0} c^{a}_{n} \\frac{n!}{(2n)!} z^{2n} = \\sum_{n \\geq 0} \\psi_{2p}^{a} e^{-nz}.$$ \nNow we can write the mock modular forms in an integral from, using these generating functions:\n\\begin{gather}\n\\frac{\\sinh(p-a)\\eta}{\\sinh p \\eta} = \\sum_{n \\geq 0} \\psi_{2p}^{a}(n)e^{-n \\eta} \\\\\n\\Rightarrow \\quad \\int_{i \\mathbb{R} + \\epsilon} d \\eta \\frac{\\sinh(p-a)\\zeta}{\\sinh p \\eta} e^{-\\frac{k p \\eta^{2}}{2 \\pi i}} = \\int_{i \\mathbb{R}+\\epsilon} d \\eta \\sum_{n \\geq 0} \\psi_{2p}^{a}(n)e^{-n \\eta} e^{-\\frac{k p \\eta^{2}}{2 \\pi i}} \\label{eqn:Borel1} \\\\\n\\Rightarrow  \\quad \\int_{i \\mathbb{R} + \\epsilon} d \\eta \\frac{\\sinh(p-a)\\eta}{\\sinh p \\eta} e^{-\\frac{k p \\eta^{2}}{2 \\pi i}} = \\sqrt{\\frac{2 \\pi^{2} i}{p}} \\frac{1}{\\sqrt{k}} \\tilde{\\Psi}_{p}^{a}(q). \\label{eqn:Borel2}\n\\end{gather}\nIn the second line, the integral is taken along a line $Re[\\eta] = \\epsilon > 0$, where the integral converges, and the third line is simply a Gaussian integral. The change of variables\n$$ \\zeta = \\frac{p \\eta^{2}}{2 \\pi i}$$\nalters the integration contour from a single line to the union of two rays from the origin, $i e^{i \\delta} \\mathbb{R}_{+}$ and $i e^{-i \\delta} \\mathbb{R}_{+}$. In sum, \n\\begin{equation}\n\\frac{1}{\\sqrt{k}} \\tilde{\\Psi}_{p}^{a}(q) = \\frac{1}{2}\\bigg(\\int_{i e^{i \\delta} \\mathbb{R}_{+}} + \\int_{i e^{-i \\delta} \\mathbb{R}_{+}} \\bigg) \\frac{d \\zeta}{\\sqrt{\\pi \\zeta}} \\frac{\\sinh \\bigg( (p-a)\\sqrt{\\frac{2 \\pi i \\zeta}{p}}\\bigg)}{\\sinh \\bigg(p \\sqrt{\\frac{2 \\pi i \\zeta}{p}}\\bigg)} e^{-k\\zeta}. \\label{eqn:BorelZeta}\n\\end{equation}\n\nThus we have recovered the entire mock modular form from its perturbative expansion. Since the partition function is a linear sum of mock modular forms, this implies that the Borel resummation of $BZ_{\\text{pert}}$ will return the exact partition function. Furthermore, the poles of generating functions $\\sinh((p-a)z)\/\\sinh(pz)$ encodes the information of the non-perturbative contributions, as we exhibit below.\n\nFirst of all, since $Z_{CS}(\\Sigma(2,5,7)) \\sim (\\tilde{\\Psi}_{70}^{11} - \\tilde{\\Psi}_{70}^{31}  - \\tilde{\\Psi}_{70}^{39} + \\tilde{\\Psi}_{70}^{59})(q)$, the Borel transform of $Z_{\\text{pert}}$ is given by:\n\\begin{equation}\n\\frac{\\sinh(59\\eta)-\\sinh(39\\eta)-\\sinh(31\\eta)+\\sinh(11\\eta)}{\\sinh(70\\eta)} = \\frac{4\\sinh(35\\eta)\\sinh(14\\eta)\\sinh(10\\eta)}{\\sinh(70\\eta)},\n\\label{eqn:psiBorel}\n\\end{equation}\nNote that the RHS of Equation \\ref{eqn:psiBorel} has only simple poles at $\\eta = n \\pi i\/ 70$ for $n$ non-divisible by 2, 5, or 7. In particular, the poles are aligned on the imaginary axis, so we choose the same integration contours as in Equation \\ref{eqn:Borel1} - \\ref{eqn:BorelZeta}. The Borel resummation of Equation \\ref{eqn:psiBorel} is then the average of Borel sums along the two rays depicted in Figure \\ref{fig:integrationContour}(a):\n\\begin{equation}\nZ_{CS}(\\Sigma(2,5,7)) = \\frac{1}{2} \\bigg[ S_{\\frac{\\pi}{2} - \\delta}Z_{\\text{pert}}(1\/k) + S_{\\frac{\\pi}{2} + \\delta}Z_{\\text{pert}}(1\/k) \\bigg].\n\\label{eqn:Borelsums}\n\\end{equation}\n\n\\begin{figure} [htb]\n\\centering\n\\includegraphics{integrationContour}\n\\caption{(a) An integration contour in the $\\zeta$-plane, made of two rays from the origin. Dots represent the poles. (b) An equivalent integration contour. The contribution from the integration along the real axis must be doubled.}\n\\label{fig:integrationContour}\n\\end{figure}\n\nTo evaluate the RHS of Equation \\ref{eqn:Borelsums}, we integrate along an equivalent contour in Figure \\ref{fig:integrationContour}(b). Note that as we change to the contour in Figure \\ref{fig:integrationContour}(b), a Stokes ray $i e^{-i \\delta}\\mathbb{R}_{+}$ has crossed the poles on the imaginary axis, towards the positive real axis. As a reult, the poles contribute to the Borel sums with residues, which is precisely a Stokes phenomenon. Since each pole is located at $\\eta = n \\pi i \/ 70$, its residue includes a factor of $e^{-k \\zeta} = e^{-k \\frac{70 \\eta^{2}}{2 \\pi i}} = e^{2 \\pi i k (- \\frac{n^{2}}{280})}$. Shortly, we will exhibit that these factors precisely correspond to the Chern-Simons instanton actions, so let us regroup the poles ($n$ modulo 140) by their instanton actions:\n\n\\begin{itemize}\n\\item $n = 9, 19, 51, 61, 79, 89, 121, 131$, for which $CS = -\\frac{9^{2}}{280}$ and residues $\\{1,1,1,1,-1,-1,-1,-1\\}$ with overall factor $\\frac{i}{35}(\\cos \\frac{3 \\pi}{35} - \\sin \\frac{\\pi}{70})$.\n\\item $n = 3, 17, 53, 67, 73, 87, 123, 137$, for which $CS = -\\frac{3^{2}}{280}$ and residues $\\{ -1, -1, -1,-1,1,1,1,1\\}$ with overall factor $\\frac{i}{35}(\\cos \\frac{\\pi}{35} + \\cos \\frac{6 \\pi}{35})$.\n\\item $n = 23, 33, 37, 47, 93, 103, 107, 117$, for which $CS = -\\frac{23^{2}}{280}$ and residues $\\{1, 1, 1, 1,-1,-1,-1,-1\\}$ with overall factor $\\frac{i}{35}(\\cos \\frac{4 \\pi}{35} + \\sin \\frac{13 \\pi}{70})$.\n\\item $n = 13, 27, 43, 57, 83, 97, 113, 127$, for which $CS = -\\frac{13^{2}}{280}$ and residues $\\{ -1, -1, -1, -1,1,1,1,1\\}$ with overall factor $\\frac{i}{35}(\\sin \\frac{3 \\pi}{70} + \\sin \\frac{17 \\pi}{70})$.\n\\item $n = 11, 31, 39, 59, 81, 101, 109, 129$, for which $CS = -\\frac{11^{2}}{280}$ and residues $\\{1, -1, -1, 1, -1,1,1,-1\\}$ with overall factor $\\frac{i}{35}(\\cos \\frac{8 \\pi}{35} + \\sin \\frac{9 \\pi}{70})$.\n\\item $n = 1, 29, 41, 69, 71, 99, 111, 139$, for which $CS = -\\frac{1^{2}}{280}$ and residues $\\{1,-1,-1,1,-1,1,1,-1\\}$ with overall factor $\\frac{i}{35}(\\cos \\frac{2 \\pi}{35} - \\sin \\frac{11 \\pi}{70})$.\n\\end{itemize}\n\nThe top four groups of poles correspond to the four irreducible $SU(2)$ flat connections, while the remaining two correspond to the complex flat connections. To see this, first consider the moduli space of flat connections $\\mathcal{M}_{\\text{flat}}(\\Sigma(2,5,7), SL(2,\\mathbb{C}))$. Since $\\Sigma(2,5,7)$ is a homology 3-sphere, it has only one abelian flat connection $\\alpha_{0}$, which is trivial. Next, there are total $\\frac{(2-1)(5-1)(7-1)}{4} = 6$ irreducible $SL(2,\\mathbb{C})$ flat connections, four of which are conjugate to $SU(2)$ and the remaining two are ``complex'' (conjugate to $SL(2,\\mathbb{R})$) \\cite{KitanoYamaguchi,BodenCurtis,FintushelStern}. To compute their Chern-Simons instanton actions, we characterize all six flat connections by their ``rotation angles,'' which we will briefly explain here. Consider the following presentation of the fundamental group of $\\Sigma(2,5,7)$.\n\\begin{equation}\n\\pi_{1}(\\Sigma(2,5,7)) = \\langle x_{1},x_{2},x_{3},h \\, | \\, h \\, \\text{central}, x_{1}^{2} = h^{-1}, x_{2}^{5} = h^{-9}, x_{3}^{7} = h^{-5}, x_{1}x_{2}x_{3} = h^{-3} \\rangle.\n\\label{eqn:fundPresentation}\n\\end{equation}\nWhen a representation $\\alpha: \\pi_{1}(\\Sigma(2,5,7)) \\rightarrow SL(2,\\mathbb{C})$ is conjugate in $SU(2)$, $\\alpha(h)$ is equal to $\\pm 1$, and the conjugacy classes of $\\alpha(x_{j})$ can be represented in the form $\\bigl(\\begin{smallmatrix}\n\\lambda_{j} & 0 \\\\ 0 & \\lambda_{j}^{-1}\n\\end{smallmatrix} \\bigr)$ for some $| \\lambda_{j} | = 1$. There are four triples $(\\lambda_{1}, \\lambda_{2}, \\lambda_{3})$ satisfying the relations in Equation \\ref{eqn:fundPresentation}:\n\\begin{equation}\n(l_{1},l_{2},l_{3}) = (1,1,3), \\, (1,3,1), \\, (1,3,3), \\, (1,3,5) \\quad \\text{where} \\quad \\lambda_{j} = e^{\\pi i l_{j} \/ p_{j}}. \\label{eqn:flatCon}\n\\end{equation} \nEach triple corresponds to one of the four irreducible $SU(2)$ flat connections, which we will call $\\alpha_{1}, \\alpha_{2}, \\alpha_{3}$ and $\\alpha_{4}$. From the rotation angles of an irreducible flat connection $A$, we can read off its Chern-Simons instanton action:\n\\begin{gather}\nCS(A) = -\\frac{p_{1}p_{2}p_{3}}{4}(1+\\sum_{i}l_{j}\/p_{j})^{2} \\nonumber \\\\\n\\Rightarrow \\quad CS(\\alpha_{1}) = -\\frac{9^{2}}{280}, \\quad CS(\\alpha_{2}) = -\\frac{3^{2}}{280}, \\quad CS(\\alpha_{3}) = -\\frac{23^{2}}{280}, \\quad CS(\\alpha_{4}) = -\\frac{13^{2}}{280}, \\label{eqn:CSinstanton}\n\\end{gather}\nwhich is in agreement with the instanton actions of the poles in the Borel plane. Likewise, one can compute the Chern-Simons instanton actions of the two complex flat connections $\\alpha_{5}$ and $\\alpha_{6}$, \n$$CS(\\alpha_{5}) = -\\frac{11^{2}}{280}, \\quad CS(\\alpha_{6}) = -\\frac{1^{2}}{280}.$$\n\nNow, let us sum the residues to reproduce the non-perturbative contributions in Equation \\ref{eqn:asymptotics}. When $k$ is an integer, the residues from the poles $SU(2)$ connection $\\alpha_{1}$ are summed into:\n\\begin{multline}\n\\frac{i}{35}(\\cos \\frac{3 \\pi}{35} - \\sin \\frac{\\pi}{70}) e^{-2 \\pi i k \\frac{9^{2}}{280}} \\bigg[ \\sum_{n \\equiv \\pm 9 \\, (mod \\, 140)} \\pm 1 \\, + \\sum_{n \\equiv \\pm 19 \\, (mod \\, 140)} \\pm 1 \\\\\n+ \\sum_{n \\equiv \\pm 51\\, (mod \\, 140)} \\pm 1 \\, + \\sum_{n \\equiv \\pm 61\\, (mod \\, 140)} \\pm 1 \\bigg].\n\\label{eqn:alpha1Contribution}\n\\end{multline}\nVia zeta-function regularization $\\sum_{n \\equiv \\pm a \\, (mod \\, 2p)} \\pm 1 = 1 - \\frac{a}{p}$, we can rewrite Equation \\ref{eqn:alpha1Contribution} as follows:\n\\begin{gather*}\n\\frac{i}{35}(\\cos \\frac{3 \\pi}{35} - \\sin \\frac{\\pi}{70}) e^{-2 \\pi i k \\frac{9^{2}}{280}}\\bigg( (1-\\frac{9}{70}) + (1-\\frac{19}{70}) + (1-\\frac{51}{70}) + (1-\\frac{61}{70}) \\bigg) \\\\\n= \\frac{2i}{35}(\\cos \\frac{3 \\pi}{35} - \\sin \\frac{\\pi}{70}) e^{-2 \\pi i k \\frac{9^{2}}{240}} = n_{\\alpha_{1}}Z_{\\text{pert}}^{\\alpha_{1}}e^{2 \\pi i k CS(\\alpha_{1})},\n\\end{gather*}\nwhere $n_{\\alpha_{1}}$ is the trans-series parameter. Similarly for connections $\\alpha_{2}, \\alpha_{3}$ and $\\alpha_{4}$, \n\\begin{itemize}\n\\item $n_{\\alpha_{2}}Z_{\\text{pert}}^{\\alpha_{2}}e^{2 \\pi i k CS(\\alpha_{2})} = -\\frac{2i}{35}(\\cos \\frac{\\pi}{35} + \\cos \\frac{6 \\pi}{35})e^{-2 \\pi i k \\frac{3^{2}}{280}}$.\n\\item $n_{\\alpha_{3}}Z_{\\text{pert}}^{\\alpha_{3}}e^{2 \\pi i k CS(\\alpha_{3})} = \\frac{2i}{35}(\\cos \\frac{4 \\pi}{35} + \\sin \\frac{13 \\pi}{70})e^{-2 \\pi i k \\frac{23^{2}}{280}}$.\n\\item$n_{\\alpha_{4}}Z_{\\text{pert}}^{\\alpha_{4}}e^{2 \\pi i k CS(\\alpha_{4})} = -\\frac{2i}{35}(\\sin \\frac{3 \\pi}{70} + \\sin \\frac{17 \\pi}{70})e^{-2 \\pi i k \\frac{13^{2}}{280}}$.\n\\end{itemize}\nAnd the contributions from the two complex connections vanish. Notice that the poles grouped by their instanton actions correspond to the $b$'s with non-vanishing contributions in Equation \\ref{eqn:asymptotics}. Furthermore, the sum of residues is proportional to the sum $\\big( \\sin\\frac{11b \\pi}{70} - \\sin\\frac{31b \\pi}{70} - \\sin\\frac{39b \\pi}{70} + \\sin\\frac{59b \\pi}{70}\\big)$ at each $b$, so the Borel sum correctly captures the non-perturbative contributions to the exact partition function. \n\n\\section{Homological block decomposition of $Z_{CS}(\\Sigma(2,5,7))$ and the modular transform}\nWe conclude this paper by writing the partition function in a categorification-friendly form, as was advertised in Equation \\ref{eqn:abelianDecomposition}. To summarize, we started with the exact partition function $Z_{CS}(\\Sigma(2,5,7))$, considered its perturbative expansion and performed a Borel resummation. Although our example is a homology 3-sphere and has only one abelian flat connection, more generally the Borel sum results in a decomposition into homological blocks \\cite{GMP}:\n$$Z_{CS}(\\Sigma(2,5,7)) = \\sum_{a} e^{2 \\pi i CS_{a}}Z_{a},$$\nwhere the summation runs over abelian flat connections. Each ``homological block'' $Z_{a}$ gets contributions from both the abelian flat connection $a$ and the irreducible $SU(2)$ flat connections. How the irreducible flat connections regroup themselves into each homological block is encoded in the Stokes monodromy coefficients as follows:\n\\begin{gather}\nZ_{CS}(\\Sigma(2,5,7)) = \\frac{1}{2}\\bigg[S_{\\frac{\\pi}{2}-\\epsilon}Z_{\\text{pert}}(k) + S_{\\frac{\\pi}{2}+\\epsilon}Z_{\\text{pert}}(k) \\bigg] = Z^{\\alpha_{0}}_{\\text{pert}} + \\frac{1}{2} \\sum_{\\tilde{\\beta}}m_{\\tilde{\\beta}}^{(\\alpha_{0},0)}e^{2 \\pi i k S_{\\tilde{\\beta}}}Z^{\\beta}_{\\text{pert}} \\nonumber \\\\\n=\\sum_{\\tilde{\\beta}} n_{\\tilde{\\beta},0}e^{2 \\pi i k S_{\\tilde{\\beta}}}Z^{\\beta}_{\\text{pert}},  \\quad \\text{where} \\quad n_{\\tilde{\\beta}} = \\begin{cases} 1 & \\tilde{\\beta}=(\\alpha_{0},0) \\\\ \\frac{1}{2}m_{\\tilde{\\beta}}^{(\\alpha_{0},0)} & \\text{otherwise.} \\end{cases} \\label{eqn:trans-seriesCoeff}\n\\end{gather}\n\\begin{equation}\nm_{\\tilde{\\beta}}^{(\\alpha_{0},0)} = \\begin{cases}\n1 & \\tilde{\\beta} = (\\alpha_{1}, -n^{2}\/280), \\quad \\text{for} \\quad n = 9, 19, 51, 61 \\quad (\\text{mod} \\, 140) \\\\\n-1 & \\tilde{\\beta} = (\\alpha_{1}, -n^{2}\/280), \\quad \\text{for} \\quad n = 79, 89, 121, 131 \\quad (\\text{mod} \\, 140) \\\\\n1 & \\tilde{\\beta} = (\\alpha_{2}, -n^{2}\/280), \\quad \\text{for} \\quad n = 73, 87, 123, 137 \\quad (\\text{mod} \\, 140) \\\\\n-1 & \\tilde{\\beta} = (\\alpha_{2}, -n^{2}\/280), \\quad \\text{for} \\quad n = 3, 17, 53, 67 \\quad (\\text{mod} \\, 140) \\\\\n1 & \\tilde{\\beta} = (\\alpha_{3}, -n^{2}\/280), \\quad \\text{for} \\quad n = 23, 33, 37, 47 \\quad (\\text{mod} \\, 140) \\\\\n-1 & \\tilde{\\beta} = (\\alpha_{3}, -n^{2}\/280), \\quad \\text{for} \\quad n = 93, 103, 107, 117 \\quad (\\text{mod} \\, 140) \\\\\n1 & \\tilde{\\beta} = (\\alpha_{4}, -n^{2}\/280), \\quad \\text{for} \\quad n = 83, 97, 113, 127 \\quad (\\text{mod} \\, 140) \\\\\n-1 & \\tilde{\\beta} = (\\alpha_{4}, -n^{2}\/280), \\quad \\text{for} \\quad n = 13, 27, 43, 57 \\quad (\\text{mod} \\, 140) \\\\\n1 & \\tilde{\\beta} = (\\alpha_{5}, -n^{2}\/280), \\quad \\text{for} \\quad n = 11, 59, 101, 109 \\quad (\\text{mod} \\, 140) \\\\\n-1 & \\tilde{\\beta} = (\\alpha_{5}, -n^{2}\/280), \\quad \\text{for} \\quad n = 31, 39, 81, 129 \\quad (\\text{mod} \\, 140) \\\\\n1 & \\tilde{\\beta} = (\\alpha_{6}, -n^{2}\/280), \\quad \\text{for} \\quad n = 1, 69, 99, 111 \\quad (\\text{mod} \\, 140) \\\\\n-1 & \\tilde{\\beta} = (\\alpha_{6}, -n^{2}\/280), \\quad \\text{for} \\quad n = 29, 41, 71, 139 \\quad (\\text{mod} \\, 140) \\end{cases}\n\\end{equation}\nThe formula \\ref{eqn:trans-seriesCoeff} holds for any Seifert manifold with three singular fibers (which includes our example $\\Sigma(2,5,7)$) \\cite{CostinGaroufalidis}. In \\cite{GPV}, it was conjectured that there is a ``modular transform'' of the homological blocks $Z_{a}$, which turns it into a ``categorification-friendly'' form. Namely,\n$$Z_{a} = \\frac{1}{i \\sqrt{2k}} \\sum_{b} S_{ab} \\hat{Z}_{b},$$\nfor some $k$-independent $S_{ab}$. Above, $b$ runs over the abelian flat connections, and each $\\hat{Z}_{b}$ is an element of $q^{\\Delta_{b}}\\mathbb{Z}[[q]]$ for some $\\Delta_{b} \\in \\mathbb{Q}$. Suppose the exact partition function is a linear sum of mock modular forms, and there are multiple abelian flat connections. Then, a homological block decomposition regroups the mock modular forms (see \\cite{GPV} for examples.) In our example, however, there is only one abelian flat connection $\\alpha_{0}$, because $\\Sigma(2,5,7)$ is a homology sphere. Therefore, it suffices to find $\\hat{Z}_{\\alpha_{0}}$ which is an element of $q^{\\Delta_{\\alpha_{0}}} \\mathbb{Z}[[q]]$. From the exact partition function\n$$i q^{\\phi(2,5,7)\/4}\\sqrt{2k}Z_{CS}(\\Sigma(2,5,7)) = \\frac{1}{2}(\\tilde{\\Psi}_{70}^{11} - \\tilde{\\Psi}_{70}^{31}  - \\tilde{\\Psi}_{70}^{39} + \\tilde{\\Psi}_{70}^{59})(q),$$\nand the definition of the mock modular forms\n$$\\tilde{\\Psi}_{p}^{a}(q) = \\sum_{n \\geq 0} \\psi_{2p}^{a}q^{n^{2}\/4p},$$\nwe can easily see that the partition function is an element of $q^{121\/280}\\mathbb{Z}[[q]]$. Thus,\n$$\\hat{Z}_{\\alpha_{0}} = q^{1\/2}(1 - q^{3} - q^{5} + q^{12} + \\cdots) \\quad \\text{and} \\quad S_{\\alpha_{0}\\alpha_{0}} = \\frac{1}{2}.$$\n\n\n\\acknowledgments{The author is deeply indebted to Sergei Gukov for his suggestions and invaluable discussions.\n\nThe work is funded in part by the DOE Grant DE-SC0011632 and the Walter Burke Institute for Theoretical Physics, and also by the Samsung Scholarship.}\n\n\n\n\\newpage\n\n\\bibliographystyle{JHEP_TD}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nWhereas many upper bounds for the probability that a sum of independent\nreal-valued (integrable) random variables exceeds its expectation by a\nspecified threshold value $t\\in\\mathbb{R}$ are documented in the literature\n(see \\textit{e.g.} \\cite{BLM13} and the references therein), very few results\nare available when the random variables involved in the summation are sampled\nfrom a finite population according to a given survey scheme and next\nappropriately normalized (using the related survey weights as originally\nproposed in \\cite{HT51} for approximating a total). The sole situation where\nresults in the independent setting straightforwardly carry over to survey\nsamples (without replacement) corresponds to the case where the variables are\nsampled independently with possibly unequal weights, \\textit{i.e.} Poisson\nsampling. For more complex sampling plans, the dependence structure between\nthe sampled variables makes the study of the fluctuations of the resulting\nweighted sum approximating the total (referred to as the\n\\textit{Horvitz-Thompson total estimate}) very challenging. The case of basic\n\\textit{sampling without replacement} (\\textsc{SWOR} in abbreviated form) has\nbeen first considered in \\cite{Hoeffding63}, and refined in \\cite{Serfling74}\nand \\cite{BardenetMaillard}. In contrast, the asymptotic behavior of the\nHorvitz-Thompson estimator as $N$ tends to infinity is well-documented in the\nlitterature. Following in the footsteps of the seminal contribution\n\\cite{Hajek64}, a variety of limit results (\\textit{e.g.} consistency,\nasymptotic normality) have been established for Poisson sampling and next\nextended to rejective sampling viewed as conditional Poisson sampling given\nthe sample size and to sampling schemes that are closed to the latter in a\n\\textit{coupling} sense in \\cite{Rob82} and \\cite{Ber98}. Although the nature\nof the results established in this paper are nonasymptotic, these arguments\n(conditioning upon the sampling size and coupling) are involved in their proofs.\n\nIt is indeed the major purpose of this article to extend tail bounds proved\nfor \\textsc{SWOR} to the case of rejective sampling, a fixed size sampling\nscheme generalizing it. The approach we develop is thus based on viewing\nrejective sampling as conditional Poisson sampling given the sample size and\nwriting then the deviation probability as a ratio of two quantities: the joint\nprobability that a Poisson sampling-based total estimate exceeds the threshold\n$t$ and the size of the cardinality of the Poisson sample equals the\n(deterministic) size $n$ of the rejective plan considered in the numerator and\nthe probability that the Poisson sample size is equal to $n$ in the\ndenominator. Whereas a sharp lower bound for the denominator can be\nstraightforwardly derived from a local Berry-Esseen bound proved in\n\\cite{Deheuvels} for sums of independent, possibly non indentically\ndistributed, Bernoulli variables, an accurate upper bound for the numerator can be\nestablished by means of an appropriate exponential change of measure\n(\\textit{i.e.} Escher transformation), following in the footsteps of the\nmethod proposed in \\cite{Talagrand95}, a refinement of the classical argument\nof Bahadur-Rao's theorem in order to improve exponential bounds in the\nindependent setting. The tail bounds (of Bennett\/Bernstein type) established\nby means of this method are shown to be sharp in the sense that they\nexplicitely involve the 'small' asymptotic variance of the Horvitz-Thompson total\nestimate based on rejective sampling, in contrast to those proved by using the\n\\textit{negative association} property of the sampling scheme.\n\nThe article is organized as follows. A few key concepts pertaining to survey\ntheory are recalled in section \\ref{sec:background}, as well as specific\nproperties of Poisson and rejective sampling schemes. For comparison purpose,\npreliminary tail bounds in the (conditional) Poisson case are stated in\nsection \\ref{sec:Poisson}. The main results of the paper, sharper exponential\nbounds for conditional Poisson sampling namely, are proved in section\n\\ref{sec:main}, while section \\ref{sec:extension} explains how they can be\nextended to other sampling schemes, sufficiently close to rejective sampling in the sense of the total variation norm. A few remarks are finally collected in section \\ref{sec:concl} and some\ntechnical details are deferred to the Appendix section.\n\n\\section{Background and Preliminaries}\n\n\\label{sec:background} As a first go, we start with briefly recalling basic\nnotions in survey theory, together with key properties of (conditional)\nPoisson sampling schemes. Here and throughout, the indicator function of any\nevent $\\mathcal{E}$ is denoted by $\\mathbb{I}\\{\\mathcal{E}\\}$, the power set\nof any set $E$ by $\\mathcal{P}(E)$, the variance of any square integrable r.v.\n$Y$ by $Var(Y)$, the cardinality of any finite set $E$ by $\\#E$ and the Dirac\nmass at any point $a$ by $\\delta_{a}$. For any real number $x$, we set\n$x^{+}=\\max\\{x,\\; 0 \\}$, $x^{-}=\\max\\{ -x,\\; 0 \\}$, $\\lceil x\\rceil=\\inf\\{\nk\\in\\mathbb{Z}:\\; x\\leq k \\}$ and $\\lfloor x \\rfloor=\\sup\\{ k\\in\\mathbb{Z}:\\;\nk\\leq x \\}$.\n\n\\subsection{Sampling schemes and Horvitz-Thompson estimation}\n\nConsider a finite population of $N\\geq1$ distinct units, $\\mathcal{I\n_{N}=\\{1,\\ \\ldots,\\; N \\}$ say, a survey sample of (possibly random) size\n$n\\leq N$ is any subset $s=\\{i_{1},\\; \\ldots,\\; i_{n(s)} \\}\\in\\mathcal{P\n(\\mathcal{I}_{N})$ of size $n(s)=n$. A sampling design without replacement is\ndefined as a probability distribution $R_{N}$ on the set of all possible\nsamples $s\\in\\mathcal{P}(\\mathcal{I}_{N})$. For all $i\\in\\mathcal{I}_{N}$, the\nprobability that the unit $i$ belongs to a random sample $S$ defined on a\nprobability space $(\\Omega,\\; \\mathcal{F},\\; \\mathcal{P})$ and drawn from\ndistribution $R_{N}$ is denoted by $\\pi_{i}=\\mathbb{P}\\{i\\in S \\}=R_{N\n(\\{i\\})$. The $\\pi_{i}$'s are referred to as \\textit{first order inclusion\nprobabilities}. The \\textit{second order inclusion probability} related to any\npair $(i,j)\\in\\mathcal{I}_{N}^{2}$ is denoted by $\\pi_{i,j}=\\mathbb{P\n\\{(i,j)\\in S^{2} \\}=R_{N}(\\{i,\\; j \\})$ (observe that $\\pi_{i,i}=\\pi_{i}$).\nHere and throughout, we denote by $\\mathbb{E}[.]$ the $\\mathbb{P}$-expectation\nand by $Var(Z)$ the conditional variance of any $\\mathbb{P}$-square integrable\nr.v. $Z:\\Omega\\rightarrow\\mathbb{R}$.\n\nThe random vector $\\boldsymbol{\\epsilon} _{N}=(\\epsilon_{1},\\; \\ldots,\\;\n\\epsilon_{N})$ defined on $(\\Omega,\\; \\mathcal{F},\\; \\mathcal{P})$, where\n$\\epsilon_{i}=\\mathbb{I}\\{ i\\in S \\}$ fully characterizes the random sample\n$S\\in\\mathcal{P}(\\mathcal{I}_{N})$. In particular, the sample size $n(S)$ is\ngiven by $n=\\sum_{i=1}^{N}\\epsilon_{i}$, its expectation and variance by\n$\\mathbb{E}[n(S)]=\\sum_{i=1}^{N}\\pi_{i}$ and $Var(n(S))=\\sum_{1\\leq i,\\; j\\leq\nN}\\{\\pi_{i,j}-\\pi_{i}\\pi_{j}\\}$ respectively. The $1$-dimensional marginal\ndistributions of the random vector $\\boldsymbol{\\epsilon} _{N}$ are the\nBernoulli distributions $Ber(\\pi_{i})=\\pi_{i}\\delta_{1}+(1-\\pi_{i})\\delta_{0\n$, $1\\leq i \\leq N$ and its covariance matrix is $\\Gamma_{N}=(\\pi_{i,j\n-\\pi_{i}\\pi_{j})_{1\\leq i,\\;j \\leq N}$. \\medskip\n\nWe place ourselves here in the \\textit{fixed-population} or\n\\textit{design-based} sampling framework, meaning that we suppose that a fixed\n(unknown) real value $x_{i}$ is assigned to each unit $i\\in\\mathcal{I}_{N}$.\nAs originally proposed in the seminal contribution \\cite{HT51}, the\nHorvitz-Thompson estimate of the population total $S_{N}=\\sum_{i=1}^{N} x_{i}$\nis given by\n\\begin{equation}\n\\label{eq:HT}\\widehat{S}_{\\boldsymbol{\\pi} _{N}}^{\\boldsymbol{\\epsilon} _{N\n}=\\sum_{i=1}^{N} \\frac{\\epsilon_{i}}{\\pi_{i}}x_{i}=\\sum_{i\\in S}\\frac{1\n{\\pi_{i}}x_{i},\n\\end{equation}\nwith $0\/0=0$ by convention. Throughout the article, we assume that the\n$\\pi_{i}$'s are all strictly positive. Hence, the conditional expectation of\n\\eqref{eq:HT}  is $\\mathbb{E}[\\widehat{S}_{\\boldsymbol{\\pi} _{N\n}^{\\boldsymbol{\\epsilon} _{N}}]=S_{N}$ and, in the case where the size of the\nrandom sample is deterministic, its variance is given by\n\\begin{equation}\nVar(\\widehat{S}_{\\boldsymbol{\\pi} }^{\\boldsymbol{\\epsilon} _{N}})=\\sum_{i<\nj}\\left(  \\frac{x_{i}}{\\pi_{i}}-\\frac{x_{j}}{\\pi_{j}} \\right)  ^{2\n\\times\\left(  \\pi_{i}\\pi_{j}-\\pi_{i,j} \\right)  .\n\\end{equation}\n\nThe goal of this paper is to establish accurate bounds for tail probabilities\n\\begin{equation}\n\\label{eq:tailprob}\\mathbb{P}\\{\\widehat{S}_{\\boldsymbol{\\pi} _{N\n}^{\\boldsymbol{\\epsilon} _{N}}-S_{N} >t \\},\n\\end{equation}\nwhere $t\\in\\mathbb{R}$, when the sampling scheme $\\boldsymbol{\\epsilon} _{N}$\nis \\textit{rejective}, a very popular sampling plan that generalizes \\textit{random\nsampling without replacement} and can be expressed as a conditional Poisson\nscheme, as recalled in the following subsection for clarity. One may refer to\n\\cite{Dev87} for instance for an excellent account of survey theory, including\nmany more examples of sampling designs.\n\n\\subsection{Poisson and conditional Poisson sampling}\n\n\\label{subsec:Poisson} Undoubtedly, one of the simplest sampling plan is the\n\\textit{Poisson survey scheme} (without replacement), a generalization of\n\\textit{Bernoulli sampling} originally proposed in \\cite{Goodman} for the case\nof unequal weights: the $\\epsilon_{i}$'s are independent and the sampling\ndistribution $P_{N}$ is thus entirely determined by the first order inclusion\nprobabilities $\\mathbf{p}_{N}=(p_{1},\\;\\ldots,\\;p_{N})\\in]0,1[^{N}$:\n\\begin{equation}\n\\forall s\\in\\mathcal{P}(\\mathcal{I}_{N}),\\;\\;P_{N}(s)=\\prod_{i\\in S}p_{i\n\\prod_{i\\notin S}(1-p_{i}). \\label{eq:Poisson\n\\end{equation}\nObserve in addition that the behavior of the quantity \\eqref{eq:HT}  can be\ninvestigated by means of results established for sums of independent random\nvariables. However, the major drawback of this sampling plan lies in the\nrandom nature of the corresponding sample size, impacting significantly the\nvariability of \\eqref{eq:HT}. The variance of the Poisson sample size is\ngiven by $d_{N}=\\sum_{i=1}^{N}p_{i}(1-p_{i})$, while the variance of\n\\eqref{eq:HT}  is in this case:\n\\[\nVar\\left(  \\widehat{S}_{\\boldsymbol{\\pi} _{N}}^{\\boldsymbol{\\epsilon} _{N\n}\\right)  =\\sum_{i=1}^{N}\\frac{1-p_{i}}{p_{i}}x_{i}^{2}.\n\\]\nFor this reason, \\textit{rejective sampling}, a sampling design $R_{N}$ of\nfixed size $n\\leq N$, is often preferred in practice. It generalizes the\n\\textit{simple random sampling without replacement} (where all samples with\ncardinality $n$ are equally likely to be chosen, with probability $(N-n)!\/n!$,\nall the corresponding first and second order probabilities being thus equal to\n$n\/N$ and $n(n-1)\/(N(N-1))$ respectively). Denoting by $\\boldsymbol{\\pi}\n_{N}^{R}=(\\pi_{1}^{R},\\;\\ldots,\\;\\pi_{N})$ its first order inclusion\nprobabilities and by $\\mathcal{S}_{n}=\\{s\\in\\mathcal{P}(\\mathcal{I\n_{N}):\\;\\#s=n\\}$ the subset of all possible samples of size $n$, it is defined\nby:\n\\begin{equation}\n\\forall s\\in\\mathcal{S}_{n},\\;\\;R_{N}(s)=C\\prod_{i\\in s}p_{i}^{R\n\\prod_{i\\notin s}(1-p_{i}^{R}), \\label{eq:Rejective\n\\end{equation}\nwhere $C=1\/\\sum_{s\\in\\mathcal{S}_{n}}\\prod_{i\\in s}p_{i}^{R}\\prod_{i\\notin\ns}(1-p_{i}^{R})$ and the vector of parameters $\\mathbf{p}_{N}^{R}=(p_{1}^{R},\\;\\ldots\n,\\;p_{N}^{R})\\in]0,1[^{N}$ yields first order inclusion probabilities equal to\nthe $\\pi_{i}^{R}$'s and is such that $\\sum_{i=1}^{N}p_{i}^{R}=n$. Under this\nlatter additional condition, such a vector $\\mathbf{p}_{N}^{R}$ exists and is\nunique (see \\cite{Dupacova}) and the related representation\n\\eqref{eq:Rejective}  is then said to be \\textit{canonical}. Notice\nincidentally that any vector $\\mathbf{p}_{N}^{\\prime}\\in]0,1[^{N}$ such that\n$p_{i}^{R}\/(1-p_{i}^{R})=cp_{i}^{\\prime}\/(1-p_{i}^{\\prime})$ for all\n$i\\in\\{1,\\;\\ldots,\\;n\\}$ for some constant $c>0$ can be used to write a\nrepresentation of $R_{N}$ of the same type as \\eqref{eq:Rejective}. Comparing\n\\eqref{eq:Rejective}  and \\eqref{eq:Poisson}  reveals that rejective $R_{N}$\nsampling of fixed size $n$ can be viewed as Poisson sampling given that the\nsample size is equal to $n$. It is for this reason that rejective sampling is\nusually referred to as \\textit{conditional Poisson sampling}. For simplicity's\nsake, the superscrit $R$ is omitted in the sequel. One must pay attention not\nto get the $\\pi_{i}$'s and the $p_{i}$'s mixed up (except in the SWOR case, where these quantities are all equal to $n\/N$): the latter are the first\norder inclusion probabilities of $P_{N}$, whereas the former are those of its\nconditional version $R_{N}$. However they can be related by means of the\nresults stated in \\cite{Hajek64} (see Theorem 5.1 therein, as well as Lemma \\ref{lem:bias} in section \\ref{sec:main} and \\cite{BLRG12}): $\\forall\ni\\in\\{1,\\;\\ldots,\\;N\\}$,\n\\begin{align}\n\\pi_{i}(1-p_{i})  &  =p_{i}(1-\\pi_{i})\\times\\left(  1-\\left(  \\tilde{\\pi\n-\\pi_{i}\\right)  \/d_{N}^{\\ast}+o(1\/d_{N}^{\\ast})\\right)  ,\\label{eq:rel1}\\\\\np_{i}(1-\\pi_{i})  &  =\\pi_{i}(1-p_{i})\\times\\left(  1-\\left(  \\tilde{p\n-p_{i}\\right)  \/d_{N}+o(1\/d_{N})\\right)  , \\label{eq:rel2\n\\end{align}\nwhere $d_{N}^{\\ast}=\\sum_{i=1}^{N}\\pi_{i}(1-\\pi_{i})$, $d_{N}=\\sum_{i=1\n^{N}p_{i}(1-p_{i})$, $\\tilde{\\pi}=(1\/d_{N}^{\\ast})\\sum_{i=1}^{N}\\pi_{i\n^{2}(1-\\pi_{i})$ and $\\tilde{p}=(1\/d_{N})\\sum_{i=1}^{N}(p_{i})^{2}(1-p_{i})$. \\medskip\n\nSince the major advantage of conditional Poisson sampling lies in its reduced\nvariance property (compared to Poisson sampling in particular, see the discussion in section \\ref{sec:main}), focus is next\non exponential inequalities involving a variance term, of Bennett\/Bernstein\ntype namely.\n\n\\section{Preliminary Results}\n\n\\label{sec:Poisson}\n\nAs a first go, we establish tail bounds for the Horvitz-Thompson estimator in\nthe case where the variables are sampled according to a Poisson scheme. We\nnext show how to exploit the \\textit{negative association} property satisfied\nby rejective sampling in order to extend the latter to conditional Poisson\nsampling. Of course, this approach do not account for the reduced variance\nproperty of Horvitz-Thompson estimates based on rejective sampling, it is the\npurpose of the next section to improve these first exponential bounds.\n\n\\subsection{Tails bounds for Poisson sampling}\n\nAs previously observed, bounding the tail probability \\eqref{eq:tailprob}  is\neasy in the Poisson situation insofar as the variables summed up in\n\\eqref{eq:HT}  are independent though possibly non identically distributed\n(since the inclusion probabilities are not assumed to be all equal). The\nfollowing theorem thus directly follows from well-known results related to\ntail bounds for sums of independent random variables.\n\n\\begin{theorem}\n\\label{thm:poisson}\\textsc{(Poisson sampling)} Assume that the survey scheme\n$\\boldsymbol{\\epsilon} _{N}$ defines a Poisson sampling plan with first order\ninclusion probabilities $p_{i}>0$, with $1\\leq i \\leq N$. Then, we\nalmost-surely have: $\\forall t>0$, $\\forall N\\geq1$,\n\\begin{align}\n\\mathbb{P}\\left\\{  \\widehat{S}_{\\boldsymbol{p} _{N}}^{\\boldsymbol{\\epsilon}\n_{N}}-S_{N} >t \\right\\}   &  \\leq\\exp\\left(  -\\frac{\\sum_{i=1}^{N\n\\frac{1-p_{i}}{p_{i}}x_{i}^{2}}{\\left(  \\max_{1\\leq i \\leq N}\\frac{x_{i\n}{p_{i}} \\right)  ^{2}} H\\left(  \\frac{\\max_{1\\leq i \\leq N}\\frac{\\vert\nx_{i}\\vert}{p_{i}}t}{\\sum_{i=1}^{N}\\frac{1-p_{i}}{p_{i}}x_{i}^{2}} \\right)\n\\right) \\label{eq:Bennett}\\\\\n&  \\leq\\exp\\left(  \\frac{-t^{2}}{\\frac{2}{3}\\max_{1\\leq i\\leq N}\\frac{\\vert\nx_{i}\\vert}{p_{i}}+ 2\\sum_{i=1}^{N}\\frac{1-p_{i}}{p_{i}}x_{i}^{2}} \\right)  ,\n\\label{eq:Bern\n\\end{align}\nwhere $H(x)=(1+x)\\log(1+x)-x$ for $x\\geq0$.\n\\end{theorem}\n\nBounds \\eqref{eq:Bennett}  and \\eqref{eq:Bern}  straightforwardly result from\nBennett inequality \\cite{Bennett} and Bernstein exponential inequality\n\\cite{Bernstein} respectively, when applied to the independent random\nvariables $(\\epsilon_{i}\/p_{i})x_{i}$, $1\\leq i \\leq N$. By applying these\nresults to the variables $-(\\epsilon_{i}\/p_{i})x_{i}$'s, the same bounds\nnaturally hold for the deviation probability $\\mathbb{P}\\{\\widehat\n{S}_{\\boldsymbol{p} _{N}}^{\\boldsymbol{\\epsilon} _{N}}-S_{N} <-t \\}$ (and,\nincidentally, for $\\mathbb{P}\\{\\vert\\widehat{S}_{\\boldsymbol{p} _{N\n}^{\\boldsymbol{\\epsilon} _{N}}-S_{N} \\vert>t \\}$ up to a factor $2$).\nDetails, as well as extensions to other deviation inequalities (see\n\\textit{e.g.} \\cite{FukNagaev}), are left to the reader.\n\n\n\\subsection{Exponential inequalities for sums of negatively associated random variables}\n\nFor clarity, we first recall the definition of \\textit{negatively associated\nrandom variables}, see \\cite{JDP83}.\n\n\\begin{definition}\n\\label{def:negassoc} Let $Z_{1},\\; \\ldots,\\; Z_{n}$ be random variables\ndefined on the same probability space, valued in a measurable space\n$(E,\\mathcal{E})$. They are said to be negatively associated iff for any pair\nof disjoint subsets $A_{1}$ and $A_{2}$ of the index set $\\{1,\\; \\ldots,\\; n\n\\}$\n\\begin{equation}\n\\label{eq:neg}Cov \\left(  f((Z_{i})_{i\\in A_{1}}),\\; g((Z_{j})_{j\\in A_{2}})\n\\right)  \\leq0,\n\\end{equation}\nfor any real valued measurable functions $f:E^{\\#A_{1}}\\rightarrow\\mathbb{R}$\nand $g:E^{\\#A_{2}}\\rightarrow\\mathbb{R}$ that are both increasing in each variable.\n\\end{definition}\n\nThe following result provides tail bounds for sums of negatively associated\nrandom variables, which extends the usual Bennett\/Bernstein inequalities in the\ni.i.d. setting, see \\cite{Bennett} and \\cite{Bernstein}.\n\n\\begin{theorem}\n\\label{thm:BernNeg} Let $Z_{1},\\;\\ldots,\\;Z_{N}$ be square integrable\nnegatively associated real valued random variables such that $|Z_{i}|\\leq c$\na.s. and $\\mathbb{E}[Z_{i}]=0$ for $1\\leq i\\leq N$. Let $a_{1},\\; \\ldots,\\; a_N$ be\nnon negative constants and set $\\sigma^{2}=\\frac{1}{N}\\sum_{i=1}^{N}a_{i\n^{2}Var(Z_{i})$. Then, for all $t>0$, we have: $\\forall N\\geq1$,\n\\begin{align}\n\\mathbb{P}\\left\\{  \\sum_{i=1}^{N}a_{i}Z_{i}\\geq t\\right\\}   &  \\leq\\exp\\left(\n-\\frac{N\\sigma^{2}}{c^{2}}H\\left(  \\frac{ct}{N\\sigma^{2}}\\right)  \\right) \\\\\n&  \\leq\\exp\\left(  -\\frac{t^{2}}{2N\\sigma^{2}+\\frac{2ct}{3}}\\right)  .\n\\end{align}\n\\end{theorem}\n\nBefore detailing the proof, observe that the same bounds hold true for the\ntail probability $\\mathbb{P}\\left\\{ \\sum_{i=1}^{N}a_{i}Z_{i}\\leq-t\\right\\}  $\n(and for $\\mathbb{P}\\left\\{  |\\sum_{i=1}^{N}a_{i}Z_{i}|\\geq t\\right\\}  $ as\nwell, up to a multiplicative factor $2$). Refer also to Theorem 4 in\n\\cite{Janson} for a similar result in a more restrictive setting\n(\\textit{i.e.} for tail bounds related to sums of \\textit{negatively related}\nr.v.'s) and to \\cite{Shao00} as well. \\begin{proof\n\nThe proof starts off with the usual Chernoff method: for all $\\lambda>0$,\n\\begin{equation\n\\label{eq:Chernoff}\n\\mathbb{P}\\left\\{\\sum_{i=1}\n^N a_i Z_i  \\geq t\\right\\}\\leq \\exp\\left( -t\\lambda +\\log \\mathbb{E\n\\left[e^{t\\sum_{i=1}^N a_i Z_i} \\right] \\right).\n\\end{equation}\n\nNext, observe that, for all $t>0$, we have\n\\begin{eqnarray}\\label{eq:neg2}\n\\mathbb{E}\\left[\\exp\\left(t\\sum_{i=1}^n a_i Z_i\\right)\\right]&=&\\mathbb{E}\n\\left[\\exp(t a_n Z_n )\\exp\\left(t\\sum_{i=1}^{n-1}\na_i Z_i\\right)\\right]\\nonumber\\\\\n&\\leq &\\mathbb{E}\n\\left[ \\exp(ta_n Z_n) \\right]\\mathbb{E}\\left[\\exp\\left(t\\sum_{i=1}^{n-1}\na_i Z_i\\right)  \\right]\\nonumber\\\\\n&\\leq & \\prod_{i=1}^n\\mathbb{E}\n\\left[ \\exp(ta_iZ_i) \\right],\\label{eq:neg2}\n\\end{eqnarray}\n\nusing the property \\eqref{eq:neg}\ncombined with a descending recurrence on $i$. The proof is finished by plugging \\eqref{eq:neg2}\ninto \\eqref{eq:Chernoff}\nand optimizing finally the resulting bound w.r.t. $\\lambda>0$, just like in the proof of the classic Bennett\/Bernstein inequalities, see \\cite{Bennett}\nand \\cite{Bernstein}. $\\square$\n\\end{proof} \\medskip\n\nThe first assertion of the theorem stated below reveals that any rejective\nscheme $\\boldsymbol{\\epsilon} ^{*}_{N}$ forms a collection of negatively\nrelated r.v.'s, the second one appearing then as a direct consequence of Theorem \\ref{thm:BernNeg}.\nWe underline that many sampling schemes (\\textit{e.g.} Rao-Sampford sampling, Pareto sampling, Srinivasan sampling) of fixed size are actually described by random vectors $\\boldsymbol{\\epsilon}_N$ with negatively associated components, see \\cite{BJ12} or \\cite{KCR11}, so that exponential bounds similar to that stated below can be proved for such sampling plans.\n\n\\begin{theorem}\n\\label{thm:neg} Let $N\\geq1$ and $\\boldsymbol{\\epsilon} ^{*}_{N}=(\\epsilon\n^{*}_{1},\\; \\ldots,\\; \\epsilon^{*}_{N})$ be the vector of indicator variables\nrelated to a rejective plan on $\\mathcal{I}_{N}$ with first order inclusion probabilities $(\\pi_1,\\; \\ldots,\\; \\pi_N)\\in ]0,1]^N$. Then, the following\nassertions hold true.\n\n\\begin{itemize}\n\\item[(i)] The binary random variables $\\epsilon^{*}_{1},\\; \\ldots,\\;\n\\epsilon^{*}_{N}$ are negatively related.\n\n\\item[(ii)] For any $t\\geq0$ and $N\\geq1$, we have:\n\\begin{align*}\n\\mathbb{P}\\left\\{  \\widehat{S}_{\\boldsymbol{\\pi} }^{\\boldsymbol{\\epsilon}\n^{*}_{N}}-S_{N} \\geq t \\right\\}   &  \\leq2 \\exp\\left(  -\\frac{\\sum_{i=1\n^{N}\\frac{1-\\pi_{i}}{\\pi_{i}}x_{i}^{2}}{\\left(  \\max_{1\\leq i \\leq\nN}\\frac{x_{i}}{\\pi_{i}} \\right)  ^{2}} H\\left(  \\frac{\\max_{1\\leq i \\leq\nN}\\frac{\\vert x_{i}\\vert}{\\pi_{i}}t\/2}{\\sum_{i=1}^{N}\\frac{1-\\pi_{i}}{\\pi_{i\n}x_{i}^{2}} \\right)  \\right) \\\\\n&  \\leq2 \\exp\\left(  \\frac{-t^{2}\/4}{\\frac{2}{3}\\max_{1\\leq i\\leq N}\\frac{\\vert\nx_{i}\\vert}{\\pi_{i}}t+ 2\\sum_{i=1}^{N}\\frac{1-\\pi_{i}}{\\pi_{i}}x_{i}^{2}}\n\\right)  .\n\\end{align*}\n\\end{itemize}\n\\end{theorem}\n\n\\begin{proof\n\nConsidering the usual representation of the distribution of $(\\epsilon_1,\\; \\ldots,\\; \\epsilon_N)$ as the conditional distribution of a sample of independent Bernoulli variables $(\\epsilon^*_1,\\; \\ldots,\\; \\epsilon^*_N)$ conditioned upon the event $\\sum_{i=1\n^N\\epsilon^*_i=n$ (see subsection \\ref{subsec:Poisson\n), Assertion $(i)$ is a straightforward consequence from Theorem 2.8 in \\cite{JDP83}\n(see also \\cite{Barbour\n).\n\\medskip\nAssertion $(i)$ shows in particular that Theorem \\ref{thm:BernNeg}\ncan be applied to the random variables $\\{ (\\epsilon_i^*\/\\pi_i-1)x_i^+:\\; 1\\leq i \\leq N \\}$ and to the random variables $\\{ (\\epsilon_i^*\/\\pi_i-1)x_i^-:\\; 1\\leq i \\leq N \\}$ as well. Using the union bound, we obtain that\n\\begin{multline*}\n\\mathbb{P}\\left\\{ \\widehat{S}_{\\boldsymbol{\\pi}}^{\\boldsymbol{\\epsilon\n^*_N}-S_N \\geq t \\right\\}\\leq \\mathbb{P}\\left\\{ \\sum_{i=1\n^N\\left( \\frac{\\epsilon^*_i}{\\pi_i\n-1 \\right)x^+_i\\geq t\/2 \\right\\} \\\\+ \\mathbb{P}\\left\\{ \\sum_{i=1\n^N\\left( \\frac{\\epsilon^*_i}{\\pi_i\n-1 \\right)x^-_i\\leq -t\/2 \\right\\},\n\\end{multline*}\nand a direct application of Theorem \\ref{thm:BernNeg} to each of the terms involved in this bound straightforwardly proves Assertion $(ii)$. $\\square$\n\\end{proof}\n \\medskip\n\nThe negative association property permits to handle the dependence of the\nterms involved in the summation. However, it may lead to rather loose\nprobability bounds. Indeed, except the factor $2$, the bounds of Assertion\n$(ii)$ exactly correspond to those stated in Theorem \\ref{thm:poisson}, as if\nthe $\\epsilon_{i}^{*}$'s were independent, whereas the asymptotic variance $\\sigma^2_N$ of\n$\\widehat{S}_{\\boldsymbol{\\pi} }^{\\boldsymbol{\\epsilon} _{N}^{*}}$ can be much smaller than $\\sum_{i=1}^{N}(1-\\pi_{i})x_{i}^{2}\/\\pi_{i}$.\nIt is the goal of the subsequent analysis to improve these preliminary results and establish exponential bounds involving the asymptotic variance $\\sigma^2_N$.\n\\begin{remark}{\\sc (SWOR)} We point out that in the specific case of sampling without replacement, \\textit{i.e.} when $\\pi_i=n\/N$ for all $i\\in \\{1,\\; \\ldots,\\; N \\}$, the inequality stated in Assertion $(ii)$ is quite comparable (except the factor $2$) to that which can be derived from the Chernoff bound given in \\cite{Hoeffding63}, see Proposition 2 in \\cite{BardenetMaillard}.\n\n\\end{remark}\n\n\n\\section{Main Results - Exponential Inequalities for Rejective Sampling}\n\n\\label{sec:main} The main results of the paper are stated and discussed in the\npresent section. More accurate deviation probabilities related to the total estimate\n\\eqref{eq:HT}  based on a rejective sampling scheme $\\boldsymbol{\\epsilon}\n_{N}^{\\ast}$ of (fixed) sample size $n\\leq N$ with first order inclusion\nprobabilities $\\boldsymbol{\\pi} _{N}=(\\pi_{1},\\;\\ldots,\\;\\pi_{N})$ and\ncanonical representation $\\mathbf{p}_{N}=(p_{1},\\;\\ldots,\\;p_{N})$ are now\ninvestigated. Consider $\\boldsymbol{\\epsilon} _{N}$ a Poisson scheme with\n$\\mathbf{p}_{N}$ as vector of first order inclusion probabilities. As\npreviously recalled, the distribution of $\\boldsymbol{\\epsilon} _{N}^{\\ast}$\nis equal to the conditional distribution of $\\boldsymbol{\\epsilon} _{N}$ given\n$\\sum_{i=1}^{N}\\varepsilon_{i}=n$:\n\\begin{equation}\n(\\varepsilon_{1}^{\\ast},\\varepsilon_{2}^{\\ast},....,\\varepsilon_{N}^{\\ast\n})\\overset{d}{=}(\\varepsilon_{1},....,\\varepsilon_{N})\\ |\\sum_{i=1\n^{N}\\varepsilon_{i}=n.\\label{eq:distribution\n\\end{equation}\nHence, we almost-surely have: $\\forall t>0$, $\\forall N\\geq1$,\n\\begin{equation}\n\\mathbb{P}\\left\\{  \\widehat{S}_{\\boldsymbol{\\pi} _{N}\n^{\\boldsymbol{\\epsilon} _{N}^{\\ast}}-S_{N}>t\\right\\}  =\\mathbb{P}\\left\\{  \\sum_{i=1}^{N}\\frac{\\epsilon_{i}}{\\pi_{i}}x_{i}-S_{N}>t\\mid\n\\sum_{i=1}^{N}\\epsilon_{i}=n\\right\\}  .\\label{eq:cond_tail\n\\end{equation}\nAs a first go, we shall prove tail bounds for the quantity\n\\begin{equation}\n\\widehat{S}_{\\boldsymbol{p} _{N}}^{\\boldsymbol{\\epsilon} _{N}^{\\ast}\n\\overset{def}{=}\\sum_{i=1}^{N}\\frac{\\epsilon_{i}^{\\ast}}{p_{i}}x_{i\n.\\label{eq:HT2\n\\end{equation}\nObserve that this corresponds to the HT estimate of the total $\\sum_{i=1\n^{N}\\frac{p_{i}}{\\pi_{i}}x_{i}$. Refinements of relationships \\eqref{eq:rel1}  and\n\\eqref{eq:rel2}  between the $p_{i}$'s and the $\\pi_{i}$'s shall next allow us\nto obtain an upper bound for \\eqref{eq:cond_tail}. Notice incidentally that,\nthough slightly biased (see Assertion $(i)$ of Theorem \\ref{thm:final}), the statistic\n\\eqref{eq:HT2}  is commonly used as an estimator of $S_{N}$, insofar as the\nparameters $p_{i}$'s are readily available from the canonical representation\nof $\\boldsymbol{\\epsilon} _{N}^{\\ast}$, whereas the computation of the\n$\\pi_{i}$'s is much more complicated. One may refer to \\cite{CDL94} for\npractical algorithms dedicated to this task. Hence, Theorem \\ref{thm:rejective} is of practical interest to build non asymptotic confidence intervals for the total $S_N$.\n\\medskip\n\n\\noindent {\\bf Asymptotic variance.} Recall that $d_{N}=\\sum_{i=1}^{N}p_{i}(1-p_{i})$ is the variance\n$Var(\\sum_{i=1}^{N}\\epsilon_{i})$ of the size of the Poisson plan\n$\\boldsymbol{\\epsilon} _{N}$ and set\n\\[\n\\theta_{N}=\\frac{\\sum_{i=1}^{N}x_{i}(1-p_{i})}{d_{N}}.\n\\]\nAs explained in \\cite{BCC2013}, the quantity $\\theta_{N}$ is the coefficient of the linear regression relating $\\sum_{i=1}^{N}\\frac{\\epsilon_{i}}{p_{i}}x_{i}-S_{N}$ to the sample\nsize $\\sum_{i=1}^{N}\\epsilon_{i}$. We may thus write\n\\[\n\\sum_{i=1}^{N}\\frac{\\epsilon_{i}}{p_{i}}x_{i}-S_{N}=\\theta_{N}\\times \\sum_{i=1\n^{N}\\epsilon_{i}+r_{N},\n\\]\nwhere the residual $r_{N}$\\ is orthogonal to $\\sum\n_{i=1}^{N}\\epsilon_{i}$. Hence, we have the following decomposition\n\\begin{equation}\\label{eq:Poisson_var}\nVar\\left(  \\sum_{i=1}^{N}\\frac{\\epsilon_{i}}{p_{i}}x_{i}\\right)  =\\sigma^2_{N}+\\theta_{N}^{2}d_{N},\n\\end{equation}\nwhere \n\\begin{equation}\\label{eq:Asympt_Var}\n\\sigma^2_{N}=Var\\left(  \\sum_{i=1}^{N}(\\epsilon_{i}-p_{i})\\left(\n\\frac{x_{i}}{p_{i}}-\\theta_{N}\\right)  \\right)  \n\\end{equation}\n is the asymptotic variance\nof the statistic $\\widehat{S}_{\\boldsymbol{p} _{N}}^{\\boldsymbol{\\epsilon} _{N}^{\\ast}}$, see \\cite{Hajek64}. In other words,\nthe variance reduction resulting from the use of a rejective sampling plan instead of a Poisson plan is\nequal to $\\theta_{N}^{2}d_{N}$, and can be very large in practice. A sharp Bernstein type probability inequality for $\\widehat{S}_{\\boldsymbol{p} _{N}}^{\\boldsymbol{\\epsilon} _{N}^{\\ast}}$ should thus involve $\\sigma^2_N$ rather than the Poisson variance $Var(  \\sum_{i=1}^{N}(\\epsilon_{i}\/p_{i})x_{i})$.\nUsing the fact that $\\sum_{i=1}^{N}(\\epsilon_{i}-p_{i})=0$ on the event\n$\\{\\sum_{i=1}^{N}\\epsilon_{i}=n\\}$, we may now write:\n\\begin{align}\n\\mathbb{P}\\left\\{  \\widehat{S}_{\\boldsymbol{p} _{N}}^{\\boldsymbol{\\epsilon}\n_{N}^{\\ast}}-S_{N}>t\\right\\}   &  =\\mathbb{P}\\left\\{  \\sum_{i=1\n^{N}\\frac{\\epsilon_{i}}{p_{i}}x_{i}-S_{N}>t\\mid\\sum_{i=1}^{N}\\epsilon\n_{i}=n\\right\\}  \\nonumber\\label{eq:ratio}\\\\\n&  =\\frac{\\mathbb{P}\\left\\{  \\sum_{i=1}^{N}(\\epsilon_{i}-p_{i})\\frac{x_{i\n}{p_{i}}>t,\\;\\sum_{i=1}^{N}\\epsilon_{i}=n\\right\\}  }{\\mathbb{P}\\left\\{\n\\sum_{i=1}^{N}\\epsilon_{i}=n\\right\\}  }\\nonumber\\\\\n&  =\\frac{\\mathbb{P}\\left\\{  \\sum_{i=1}^{N}(\\epsilon_{i}-p_{i})\\left(\n\\frac{x_{i}}{p_{i}}-\\theta_{N}\\right)  >t,\\sum_{i=1}^{N}\\epsilon\n_{i}=n\\right\\}  }{\\mathbb{P}\\left\\{  \\sum_{i=1}^{N}\\epsilon_{i}=n\\right\\}  }.\n\\end{align}\nBased on the\nobservation that the random variables $\\sum_{i=1}^{N}(\\epsilon_{i\n-p_{i})(x_{i}\/p_{i}-\\theta_{N})$ and $\\sum_{i=1}^{N}(\\epsilon_{i}-p_{i})$ are\nuncorrelated, Eq. \\eqref{eq:ratio} thus permits to establish directly the CLT $\\sigma_N^{-1}(\\widehat{S}_{\\boldsymbol{p} _{N}}^{\\boldsymbol{\\epsilon} _{N}^{\\ast}}-S_{N})\\Rightarrow \\mathcal{N}(0,1)$,\nprovided that $d_{N}\\rightarrow+\\infty$, as $N\\rightarrow+\\infty$, symplifying\nasymptotically the ratio, see \\cite{Hajek64}. Hence, the asymptotic variance of $\\widehat{S}_{\\boldsymbol{p} _{N}}^{\\boldsymbol{\\epsilon} _{N}^{\\ast}}-S_{N}$ is the variance $\\sigma^2_N$ of the quantity $\\sum_{i=1}^{N}(\\epsilon_{i\n-p_{i})(x_{i}\/p_{i}-\\theta_{N})$, which is less than that of the Poisson HT estimate $\\eqref{eq:Poisson_var}$, since it eliminates the variability due to the sample size. We also point out that Lemma \\ref{lem:bias} proved in the Appendix section straightforwardly shows that the \"variance term\" $\\sum_{i=1}^Nx_i^2(1-\\pi_i)\/\\pi_i$ involved in the bound stated in Theorem \\ref{thm:BernNeg} is always larger than $(1+6\/d_N)^{-1}\\sum_{i=1}^Nx_i^2(1-p_i)\/p_i$.\n\\medskip\n\nThe desired result here is non\nasymptotic and accurate exponential bounds are required for both the numerator\nand the denominator of \\eqref{eq:ratio}. It is proved in \\cite{Hajek64} (see\nLemma 3.1 therein) that, as $N\\rightarrow+\\infty$:\n\\begin{equation}\n\\mathbb{P}\\left\\{  \\sum_{i=1}^{N}\\epsilon_{i}=n\\right\\}  =(2\\,\\pi\n\\,d_{N})^{-1\/2}\\,(1+o(1)).\\label{local\n\\end{equation}\nAs shall be seen in the proof of the theorem stated below, the approximation\n\\eqref{local}  can be refined by using a local Berry-Essen bound or the\nresults in \\cite{Deheuvels} and we thus essentially need to establish an\nexponential bound for the numerator with a constant of order $d_{N}^{-1\/2}$,\nsharp enough so as to simplify the resulting ratio bound and cancel off the\ndenominator. We shall prove that this can be achieved by using a similar\nargument as that considered in \\cite{BERCLEM2010} for establishing an accurate\nexponential bound for i.i.d. $1$-lattice random vectors, based on a device\nintroduced in \\cite{Talagrand95} for refining Hoeffding's inequality.\n\n\\begin{theorem}\n\\label{thm:rejective}Let $N\\geq1$. Suppose that $\\boldsymbol{\\epsilon}\n_{N}^{\\ast}$ is a rejective scheme of size $n\\leq N$ with canonical parameter\n$\\boldsymbol{p} _{N}=(p_{1},\\;\\ldots,\\;p_{N})\\in]0,\\;1[^{N}$. Then, there\nexist universal constants $C$ and $D$ such that we have for all $t>0$ and\nfor all $N\\geq1$,\n\\begin{align*}\n\\mathbb{P}\\left\\{  \\widehat{S}_{\\boldsymbol{p} _{N}}^{\\boldsymbol{\\epsilon}\n_{N}^{\\ast}}-S_{N}>t\\right\\}   &  \\leq C\\exp\\left(  -\\frac{\\sigma^2_{N\n}{\\left(  \\max_{1\\leq j\\leq N}\\frac{|x_{j}|}{p_{j}}\\right)  ^{2}}H\\left(\n\\frac{t\\max_{1\\leq j\\leq N}\\frac{|x_{j}|}{p_{j}}\n{\\sigma^2_{N}}\\right)  \\right)  \\\\\n&  \\leq C\\exp\\left(  -\\frac{t^{2}}{2\\left(  \\sigma^2_{N}+\\frac{1\n{3}t\\max_{1\\leq j\\leq N}\\frac{|x_{j}|}{p_{j}}\\right)\n}\\right)  ,\n\\end{align*}\nas soon as $\\min\\{d_{N},\\;d_{N}^{\\ast}\\}\\geq1$ and $d_N\\geq D$.\n\\end{theorem}\n\nAn overestimated value of the constant $C$ can be deduced by a careful\nexamination of the proof given below. Before we detail it, we point out that\nthe exponential bound in Theorem \\ref{thm:rejective} involves the asymptotic variance of \\eqref{eq:HT2}, in contrast to bounds\nobtained by exploiting the \\textit{negative association} property of the\n$\\epsilon_{i}^{\\ast}$'s.\n\n\\begin{remark}{\\sc (SWOR (bis))} We underline that, in the particular case of sampling without replacement (\\textit{i.e.} when $p_i=\\pi_i=n\/N$ for $1\\leq i\\leq N$),  the Bernstein type exponential inequality stated above provides a control of the tail similar to that obtained in \\cite{BardenetMaillard}, see Theorem 2 therein, with $k=n$. In this specific situation, we have $d_N=n(1-n\/N)$ and $\\theta_N=S_N\/n$, so that the formula \\eqref{eq:Asympt_Var} then becomes $$\n\\sigma_N^2=\\left(1-\\frac{n}{N}\\right)\\frac{N^2}{n}\\left\\{ \\frac{1}{N}\\sum_{i=1}^Nx_i^2 -\\left(\\frac{1}{N}\\sum_{i=1}^N x_i\\right)^2 \\right\\}.\n$$\nThe control induced by Theorem \\ref{thm:rejective} is actually slightly better than that given by Theorem 2 in \\cite{BardenetMaillard}, insofar as the factor $(1-n\/N)$ is involved in the variance term, rather than $(1-(n-1)\/N)$, that is crucial when considering situations where $n$ gets close to $N$ (see the discussion preceded by Proposition 2 in \\cite{BardenetMaillard}).\n\\end{remark}\n\n\\begin{proof}\nWe first introduce additional notations.\nSet $Z_{i\n=(\\epsilon _{i}-p_{i})(x_{i}\/p_{i}-\\theta _{N})$ and $m_{i}=\\epsilon _{i\n-p_{i\n$ for $1\\leq i\\leq N$ and, for convenience, consider the standardized variables given by\n\\begin{equation*}\n\\mathcal{Z}_{N}=n^{1\/2}\\frac{1}{N}\\sum_{1\\leq i\\leq N}Z_{i} \\text{ and }\n\\mathcal{M}_{N}=d_N^{-1\/2}\\sum_{1\\leq i\\leq N}m_{i}.\n\\end{equation*\nAs previously announced, the proof is based on Eq. \\eqref{eq:ratio}. The lemma below first provides a sharp lower bound for the denominator, $ \\mathbb{P\n^*\\left\\{ \\mathcal{M}_{N\n=0\\right\\}$ with the notations above. As shown in the proof given in the Appendix section, it can be obtained by applying the local Berry-Esseen bound established in \\cite{Deheuvels}\nfor sums of independent (and possibly non identically) Bernoulli random variables.\n\\begin{lemma}\n\\label{lem:denominator}\nSuppose that Theorem \\ref{thm:rejective\n's assumptions are fulfilled. Then, there exist universal constants $C_1$ and $D$ such that: $\\forall N\\geq 1$,\n\\begin{equation}\n\\mathbb{P}\\{ \\mathcal{M}_N=0 \\}\\geq C_1\\frac{1}{\\sqrt{d_N}},\n\\end{equation}\nprovided that $d_N\\geq D$.\n\\end{lemma}\n\nThe second lemma gives an accurate upper bound for the numerator. Its proof can be found in the Appendix section.\n\\begin{lemma\n\\label{lem:numerator}\nSuppose that Theorem \\ref{thm:rejective\n's assumptions are fulfilled. Then, we have for all $x\\geq 0$, and for all $N\\geq 1$ such that $\\min\\{d_N,\\; d_N^*\\}\\geq 1$:\n\\begin{multline*}\n\\mathbb{P}\\left\\{\\mathcal{Z}_{N}\\geq  x,\\mathcal{M}_{N\n=0\\right\\}\\leq C \\frac{1}{\\sqrt{d_N}\n\\times \\\\\\exp\\left( -\\frac{Var\\left( \\sum_{i=1}^NZ_i \\right)\n{\\left(\\max_{1\\leq j\\leq N}\\frac{\\vert x_j\\vert}{p_j} \\right)^2\nh\\left( \\frac{N}{\\sqrt{n}}\\frac{x\\max_{1\\leq j\\leq N}\\frac{\\vert x_j\\vert\n{p_j}}{Var\\left( \\sum_{i=1}^NZ_i \\right)}\n\\right) \\right)\\\\\n\\leq  C_2 \\frac{1}{\\sqrt{d_N}}\\exp\\left( -\\frac{N^2x^2\/n\n{2\\left(Var\\left( \\sum_{i=1}^NZ_i \\right)+\\frac{1}{3}\\frac{N}{\\sqrt{n\n}x\\max_{1\\leq j\\leq N}\\frac{\\vert x_j\\vert}{p_j} \\right)}\n\\right),\n\\end{multline*\nwhere $C_2<+\\infty$ is a universal constant.\n\\end{lemma}\n\nThe bound stated in Theorem \\ref{thm:rejective}\nnow directly results from Eq. \\eqref{eq:ratio}\ncombined with Lemmas \\ref{lem:denominator} and \\ref{lem:numerator}, with $x=t\\frac{\\sqrt{n}}{N}$. $\\square$\n\\end{proof} \n\n\\bigskip\n\nEven if the computation of the biased statistic \\eqref{eq:HT2} is much more tractable from a practical perspective, we now come back to the study of the HT total estimate \\eqref{eq:HT}. The first part of the result stated below provides an estimation of the bias that replacement of \\eqref{eq:HT} by \\eqref{eq:HT2} induces, whereas its second part finally gives a tail bound for $\\eqref{eq:HT}$.\n\n\\begin{theorem}\\label{thm:final} Suppose that the assumptions of Theorem \\ref{thm:rejective} are fulfilled and set $M_N=(6\/d_N)\\sum_{i=1}^N\\vert x_i\\vert \/\\pi_i$. The following assertions hold true.\n\\begin{itemize}\n\\item[(i)] For all $N\\geq 1$, we almost-surely have:\n\\begin{equation*}\n\\left\\vert \\widehat{S}^{\\boldsymbol{\\epsilon}_N^*}_{\\boldsymbol{\\pi}_N } - \\widehat{S}^{\\boldsymbol{\\epsilon}_N^*}_{\\mathbf{p}_N } \\right\\vert \\leq M_N .\n\\end{equation*}\n\\item[(ii)] There exist universal\nconstants $C$ and $D$ such that, for all $t>M_{N}$ and for all $N\\geq1$, we have:\n\\begin{multline*}\n\\mathbb{P}\\left\\{  \\widehat{S}^{\\boldsymbol{\\epsilon}^*_N}_{\\boldsymbol{\\pi}_N}-S_{N}>t \\right\\}   \n\\leq \\\\ C\\exp\\left(  -\\frac{\\sigma_{N}^2}{\\left(  \\max_{1\\leq j\\leq\nN}\\frac{|x_{j}|}{p_{j}}\\right)  ^{2}}H\\left(  \\frac{N}{\\sqrt{n}\n\\frac{(t-M_{N})\\max_{1\\leq j\\leq N}\\frac{|x_{j}|}{p_{j}}}{\\sigma^2_{N\n}\\right)  \\right)  \\\\\n  \\leq  C\\exp\\left(  -\\frac{N^{2}(t-M_{N})^{2}\/n}{2\\left(  \\sigma^2_{N}+\\frac{1}{3}\\frac{N}{\\sqrt{n}}(t-M_{N})\\max_{1\\leq j\\leq N}\\frac{|x_{j\n|}{p_{j}}\\right)  }\\right)  ,\n\\end{multline*}\nas soon as $\\min\\{d_{N},\\;d_{N}^{\\ast}\\}\\geq1$ and $d_N\\geq D$.\n\\end{itemize}\n\\end{theorem}\n\nThe proof is given in the Appendix section. We point out that, for nearly uniform weights, \\textit{i.e.} when $c_1n\/N\\leq\\pi_i\\leq c_2n\/N$ for all $i\\in\\{1,\\; \\ldots,\\; N  \\}$ with $0<c_1\\leq c_2<+\\infty$, if there exists $K<+\\infty$ such that $\\max_{1\\leq i\\leq N}\\vert x_i\\vert \\leq K$ for all $N\\geq 1$, then the bias term $M_N$ is of order $o(N)$, provided that $\\sqrt{N}\/n\\rightarrow 0$ as $N\\rightarrow +\\infty$.\n\n\n\\section{Extensions to more general sampling schemes}\\label{sec:extension}\n\nWe finally explain how the results established in the previous section for rejective sampling may permit to control tail probabilities for more general sampling plans. A similar argument is used in \\cite{Ber98} to derive CLT's for HT estimators based on complex sampling schemes that can be approximated by more simple sampling plans, see also \\cite{BCC2013}. \nLet $\\widetilde{R}_{N}$ and $R_{N}$ be two sampling plans on the population $\\mathcal{I}_N$ and consider the\n\\emph{total variation metric}\n\\[\n\\Vert\\widetilde{R}_{N}-R_{N}\\Vert_{1}\\overset{def}{=}\\sum_{s\\in \\mathcal{P}(\\mathcal{I}_{N})}\\left|\n\\widetilde{R}_{N}(s)-R_{N}(s)\\right|  ,\n\\]\nas well as the \\emph{Kullback-Leibler divergence}\n\\[\nD_{KL}(R_{N}\\vert\\vert \\widetilde{R}_{N})\\overset{def}{=}\\sum_{s\\in \\mathcal{P}(\\mathcal{I}_{N})}R_{N\n(s)\\,\\log\\left(  \\frac{R_{N}(s)}{\\widetilde{R}_{N}(s)}\\right)  .\n\\]\nEquipped with these notations, we can state the following result.\n\\begin{lemma} \\label{lem:ext} Let $\\boldsymbol{\\epsilon}_N$ and $\\widetilde{\\boldsymbol{\\epsilon}}_N$ be two schemes defined on the same probability space and drawn from plans $R_N$ and $\\widetilde{R}_N$ respectively and let $\\mathbf{p}_N\\in ]0,1]^N$. Then, we have: $\\forall N\\geq 1$, $\\forall t\\in \\mathbb{R}$,\n\\begin{eqnarray*}\n \\left|  \\mathbb{P}\\left\\{  \\widehat{S}_{\\mathbf{p} _{N\n}^{\\boldsymbol{\\epsilon} _{N}}-S_{N}>t\\right\\}  -\\mathbb{P}\\left\\{  \\widehat{S}_{\\mathbf{p} _{N}\n^{\\widetilde{\\boldsymbol{\\epsilon}}_{N}}-S_{N}>t\\right\\}  \\right| \n& \\leq &\\Vert\\widetilde{R}_{N}-R_{N}\\Vert_{1} \\\\ &\\leq&\\sqrt{2 D_{KL}(R_{N}\\vert\\vert\\widetilde\n{R}_{N})}.\n\\end{eqnarray*}\n\\end{lemma}\n\\begin{proof} The first bound immediately results from the following elementary observation:\n\\begin{multline*}\n \\mathbb{P}\\left\\{  \\widehat{S}_{\\mathbf{p} _{N\n}^{\\boldsymbol{\\epsilon} _{N}}-S_{N}>t\\right\\}  -\\mathbb{P}\\left\\{  \\widehat{S}_{\\mathbf{p} _{N}\n^{\\widetilde{\\boldsymbol{\\epsilon}}_{N}}-S_{N}>t\\right\\} =\\\\\n\\sum_{s\\in \\mathcal{P}(\\mathcal{I}_N)}\\mathbb{I}\\{\\sum_{i\\in s}x_i\/p_i-S_N  >t\\}\\times \\left(R_N(s) -\\widetilde{R}_N(s) \\right),\n\\end{multline*}\nwhile the second bound is the classical Pinsker inequality.\n$\\square$\n\\end{proof}\n\\medskip\n\nIn practice, $R_{N}$ is typically the rejective sampling plan\ninvestigated in the previous subsection (or eventually the Poisson sampling scheme) and $\\widetilde{R}_N$ a sampling plan from which the Kullback-Leibler divergence to $R_N$ asymptotically vanishes, \\textit{e.g.} the rate at which $D_{KL}(R_{N}\\vert\\vert\\widetilde{R}_{N})$ decays to zero has been investigated in \\cite{Ber98} when $\\widetilde{R}_N$ corresponds to Rao-Sampford, successive sampling or Pareto sampling\nunder appropriate regular conditions (see also \\cite{BTL06}). Lemma \\ref{lem:ext} combined with Theorem \\ref{thm:rejective} or Theorem \\ref{thm:final} permits then to obtain upper bounds for the tail probabilities $\\mathbb{P}\\{  \\widehat{S}_{\\mathbf{p} _{N}\n^{\\widetilde{\\boldsymbol{\\epsilon}}_{N}}-S_{N}>t\\} $.\n\n\\section{Conclusion}\\label{sec:concl}\nIn this article, we proved Bernstein-type tail bounds to quantify the deviation between a total and its Horvitz-Thompson estimator when based on conditional Poisson sampling, extending (and even slightly improving) results proved in the case of basic sampling without replacement. The original proof technique used to establish these inequalities relies on expressing the deviation probablities related to a conditional Poisson scheme as conditional probabilities related to a Poisson plan. This permits to recover tight exponential bounds, involving the asymptotic variance of the Horvitz-Thompson estimator. Beyond the fact that rejective sampling is of prime importance in the practice of survey sampling, extension of these tail bounds to sampling schemes that can be accurately approximated by rejective sampling in the total variation sense is also discussed. \n\n\n\\section*{Appendix - Technical Proofs}\n\n\\subsection*{Proof of Lemma \\ref{lem:denominator}}\n\nFor clarity, we first recall the following result.\n\n\\begin{theorem}\n\\label{thm:denominator}(\\cite{Deheuvels}, Theorem 1.3) Let $(Y_{j,n})_{1\\leq\nj\\leq n}$ be a triangular array of independent Bernoulli random variables with\nmeans $q_{1,n},\\; \\ldots,\\; q_{n,n}$ in $(0,1)$ respectively. Denote by\n$\\sigma^{2}_{n}=\\sum_{i=1}^{n}q_{i,n}(1-q_{i,n})$ the variance of the sum\n$\\Sigma_{n}=\\sum_{i=1}^{n}Y_{i,n}$ and by $\\nu_{n}=\\sum_{i=1}^{n}q_{i,n}$ its\nmean. Considering the cumulative distribution function (cdf) $F_{n\n(x)=\\mathbb{P}\\{ \\sigma_{n}^{-1}(\\Sigma_{n}-\\nu_{n} )\\leq x \\}$, we have:\n$\\forall n\\geq1$,\n\\[\n\\sup_{k\\in\\mathbb{Z}}\\left\\vert F_{n}(x_{n,k})-\\Phi(x_{n,k})-\\frac{1-x_{n,k\n^{2}}{6\\sigma_{n}}\\phi(x_{n,k})\\left\\{  1-\\frac{2\\sum_{i=1}^{n}q^{2\n_{i,n}(1-q_{i,n})}{\\sigma_{n}^{2}} \\right\\}  \\right\\vert \\leq\\frac{C\n{\\sigma^{2}_{n}},\n\\]\nwhere $x_{n,k}=\\sigma_{n}^{-1}(k-\\nu_{n}+1\/2)$ for any $k\\in\\mathbb{Z}$,\n$\\Phi(x)=(2\\pi)^{-1\/2}\\int_{-\\infty}^{x}\\exp(-z^{2}\/2)dz$ is the cdf of the\nstandard normal distribution $\\mathcal{N}(0,1)$, $\\phi(x)=\\Phi^{\\prime}(x)$\nand $C<+\\infty$ is a universal constant.\n\\end{theorem}\n\nObserve first that we can write:\n\\begin{multline*}\n\\mathbb{P}\\left\\{  \\mathcal{M}_{N}=0\\right\\}  =\\mathbb{P}\\left\\{\\sum_{i=1}^{N}(\\epsilon\n_{i}-p_{i})\\in]-1\/2,1\/2]\\right\\}\\\\\n=\\mathbb{P}\\left\\{  d_{N}^{-1\/2}\\sum_{i=1}^{N\nm_{i}\\leq \\frac1 2 d_{N}^{-1\/2}\\right\\}  -\\mathbb{P}\\left\\{  d_{N}^{-1\/2\n\\sum_{i=1}^{N}m_{i}\\leq-\\frac1 2 d_{N}^{-1\/2}\\right\\}  .\n\\end{multline*}\nApplying Theorem \\ref{thm:denominator} to bound the first term of this\ndecomposition (with $k=\\nu_{n}$ and $x_{n,k}=1\/(2\\sqrt{d_{N}})$) directly yields\nthat\n\\begin{multline*}\n \\mathbb{P}\\left\\{  \\frac{\\sum_{i=1}^{N}m_{i}}{\\sqrt{d_N}}\\leq\n\\frac{1}{2\\sqrt{d_N}}\\right\\}  \\geq \\Phi\\left(\\frac{1}{2\\sqrt{d_{N}}}\\right)\\\\+\\frac{1-\\frac{1}{4d_{N}}\n{6\\sqrt{d_{N}}}\\phi\\left(\\frac{1}{2\\sqrt{d_{N}}}\\right)\\left\\{  1-\\frac{2\\sum_{i=1}^{n}p_{i\n^{2}(1-p_{i})}{d_{N}}\\right\\}  - \\frac{C}{d_{N}}.\n\\end{multline*}\nFor the second term, its application with $k=\\nu_{n}-1$ entails that:\n\\begin{multline*}\n-\\mathbb{P}\\left\\{  \\frac{1}{2\\sqrt{d_N}}\\sum_{i=1}^{N}m_{i}\\geq\n-\\frac{1}{2\\sqrt{d_N}} \\right\\}  \\leq -\\Phi\\left(-\\frac{1}{2\\sqrt{d_{N}}}\\right)\\\\-\\frac{1-\\frac{1}{4d_{N}}}{6\\sqrt{d_{N}}}\\phi\\left(-\\frac{1}{2\\sqrt{d_{N}}}\\right) \n\\left\\{  1-\\frac{2\\sum_{i=1}^{n}p_{i\n^{2}(1-p_{i})}{d_{N}}\\right\\} -\\frac{C}{d_{N}}.\n\\end{multline*}\nIf $d_N\\geq 1$, it follows that\n\\begin{eqnarray*}\n\\mathbb{P}\\left\\{  \\mathcal{M}_{N}=0\\right\\}  &\\geq &\\Phi\\left(\\frac{1}{2\\sqrt{d_{N}}\n\\right)-\\Phi\\left(-\\frac{1}{2\\sqrt{d_{N}}}\\right)- \\frac{2C}{d_{N}}\\\\\n&= & 2\\int_{0}^{\\frac{1}{2\\sqrt{d_N}}}\\phi(t)dt- \\frac{2C}{d_{N}}\n\\geq \\left( \\phi(1\/2)-\\frac{2C}{\\sqrt{d_N}}\\right)\\frac{1}{\\sqrt{d_N}}.\n\\end{eqnarray*}\n We\nthus obtain the desired result for $d_{N}\\geq D$, where $D>0$ is any constant strictly larger than $4C^2\\phi^2(1\/2)$.\n\n\\subsection{Proof of Lemma \\ref{lem:numerator}}\n\nObserve that\n\\begin{equation}\nVar \\left(  \\sum_{i=1}^{N} Z_{i}\\right)  =\\sum_{i=1}^{N}Var\\left(\nZ_{i}\\right)  =Var \\left(  \\sum_{i=1}^{N} \\epsilon_{i}^{*} \\frac{x_{i}}{p_{i\n}\\right)  =Var \\left(  \\widehat{S}^{\\boldsymbol{\\epsilon} ^{*}_{N\n}_{\\boldsymbol{p} _{N}} \\right)  .\n\\end{equation}\nLet $\\psi_{N}(u)=\\log\\mathbb{E}^{*}[\\exp(\\langle u,(\\mathcal{Z}_{N\n,\\;\\mathcal{M}_{N})\\rangle)]$, $u=(u_{1},u_{2})\\in\\mathbb{R}^{+\n\\times\\mathbb{R}$, be the log-Laplace of the $1$-lattice random vector\n$(\\mathcal{Z}_{N},\\;\\mathcal{M}_{N})$, where $\\langle.,\\; .\\rangle$ is the\nusual scalar product on $\\mathbb{R}^{2}$. Denote by $\\psi_{N}^{(1)}(u)$ and\n$\\psi_{N}^{(2)}(u)$ its gradient and its Hessian matrix respectively. Consider\nnow the conditional probability measure $\\mathbb{P}^{*}_{u,N}$ given\n$\\mathcal{D}_{N}$ defined by the Esscher transform\n\\begin{equation}\nd\\mathbb{P}_{u,N}=\\exp\\left(  \\left\\langle u,(\\mathcal{Z}_{N},\\mathcal{M\n_{N})\\right\\rangle -\\psi_{N}(u)\\right)  d\\mathbb{P}.\n\\end{equation}\nThe $\\mathbb{P}_{u,N}$-expectation is denoted by $\\mathbb{E}_{u,N}[.]$, the\ncovariance matrix of a $\\mathbb{P}_{u,N}$-square integrable random vector $Y$\nunder $\\mathbb{P}_{u,N}$ by $Var_{u^{*},N}(Y)$. With $x=t\\sqrt{n}\/N$, by\nexponential change of probability measure, we can rewrite the numerator of\n\\eqref{eq:ratio}  as\n\\begin{multline}\n\\mathbb{P}\\left\\{  \\mathcal{Z}_{N}\\geq x,\\mathcal{M}_{N}=0\\right\\}\n=\\mathbb{E}_{u,N}\\left[  e^{\\psi_{N}(u)-\\left\\langle u,(\\mathcal{Z\n_{N},\\mathcal{M}_{N})\\right\\rangle }\\mathbb{I}\\{\\mathcal{Z}_{N}\\geq\nx,\\mathcal{M}_{N}=0\\}\\right] \\nonumber\\\\\n=H(u)\\mathbb{E}_{u,N}\\left[  e^{-\\left\\langle u,(\\mathcal{Z}_{N\n-x,\\mathcal{M}_{N})\\right\\rangle }\\mathbb{I}\\{\\mathcal{Z}_{N}\\geq\nx,\\mathcal{M}_{N}=0\\}\\right]  ,\n\\end{multline}\nwhere we set $H(u)=\\exp(-\\left\\langle u,(x,0)\\right\\rangle +\\psi_{N}(u))$.\nNow, as $\\psi_{N}$ is convex, the point defined by\n\\[\nu^{\\ast}=(u_{1}^{*},0)=\\arg\\sup_{u\\in\\mathbb{R}_{+}\\times\\{0\\}}\\{\\langle\nu,(x,0)\\rangle-\\psi_{N}(u)\\}\n\\]\nis such that $\\psi_{N}^{(1)}(u^{\\ast})=(x,0)$. Since $\\mathbb{E\n[\\exp(<u,(\\mathcal{Z}_{N},\\mathcal{M}_{N})>)]=\\exp(\\psi_{N}(u))$, by\ndifferentiating one gets\n\\[\n\\mathbb{E}[e^{<u^{\\ast},(\\mathcal{S}_{N},\\;\\mathcal{M}_{N})>}(\\mathcal{S\n_{N},\\;\\mathcal{M}_{N})]=\\psi_{N}^{(1)}(u^{\\ast})e^{\\psi_{N}(u^{\\ast\n)}=(x,0)e^{\\psi_{N}(u^{\\ast})},\n\\]\nso that $\\mathbb{E}_{u^{\\ast},N}[(\\mathcal{Z}_{N},\\mathcal{M}_{N})]=(x,0)$ and\n$Var_{u^{\\ast},N}[(\\mathcal{Z}_{N},\\mathcal{M}_{N})]=\\psi_{N}^{(2)}(u^{\\ast\n)$. Choosing $u=u^{*}$, integration by parts combined with straightforward\nchanges of variables yields\n\\[\n\\mathbb{E}_{u^{*},N}[e^{-\\left\\langle u^{*},(\\mathcal{Z}_{N}-x,\\mathcal{M\n_{N})\\right\\rangle }\\mathbb{I}\\{\\mathcal{Z}_{N}\\geq x,\\mathcal{M\n_{N}=0\\}]\\newline \\leq\\mathbb{P}^{*}_{u^{*},N}\\left\\{  \\mathcal{M\n_{N}=0\\right\\}  .\n\\]\nHence, we have the bound:\n\\begin{equation}\n\\label{eq:prod}\\mathbb{P}\\left\\{  \\mathcal{Z}_{N}\\geq x,\\mathcal{M\n_{N}=0\\right\\}  \\leq H(u^{*})\\times\\mathbb{P}_{u^{*},N}\\left\\{  \\mathcal{M\n_{N}=0\\right\\}  .\n\\end{equation}\nWe shall bound each factor involved in \\eqref{eq:prod}  separately. We start\nwith bounding $H(u^{*})$, which essentially boils down to bounding\n$\\mathbb{E}[e^{\\langle u^{*},(\\mathcal{Z}_{N},\\mathcal{M}_{N})\\rangle}]$.\n\n\\begin{lemma}\n\\label{lem:factor1} Under Theorem \\ref{thm:rejective}'s assumptions, we have:\n\\begin{align}\nH(u^{*})  &  \\leq\\exp\\left(  -\\frac{Var\\left(  \\sum_{i=1}^{N}Z_{i} \\right)\n}{\\left(  \\max_{1\\leq j\\leq N}\\vert x_{j}\\vert\/p_{j} \\right)  ^{2}}h\\left(\n\\frac{N}{\\sqrt{n}}\\frac{x\\max_{1\\leq j\\leq N}\\vert x_{j}\\vert\/p_{j\n}{Var\\left(  \\sum_{i=1}^{N}Z_{i} \\right)  } \\right)  \\right) \\\\\n&  \\leq\\exp\\left(  -\\frac{N^{2}x^{2}\/n}{2\\left(  Var\\left(  \\sum_{i=1\n^{N}Z_{i} \\right)  +\\frac{1}{3}\\frac{N}{\\sqrt{n}}x \\max_{1\\leq j\\leq N}\\vert\nx_{j}\\vert\/p_{j} \\right)  } \\right)  ,\n\\end{align}\nwhere $h(x)=(1+x)\\log(1+x)-x$ for $x\\geq0$.\n\\end{lemma}\n\n\\begin{proof}\nUsing the standard argument leading to the Bennett-Bernstein bound, observe that: $\\forall i\\in \\{1,\\; \\ldots,\\; N  \\}$, $\\forall u_1> 0$,\n\\begin{equation*}\n\\mathbb{E}[e^{u_1Z_i\n]\\leq \\exp\\left( Var(Z_i)\\frac{\\exp\\left(u_1\\max_{1\\leq j\\leq N\n\\frac{\\vert x_j\\vert}{p_j} \\right) - 1- u_1\\max_{1\\leq j\\leq N\n\\frac{\\vert x_j\\vert}{p_j}}{\\left(\\max_{1\\leq j\\leq N}\\frac{\\vert x_j\\vert\n{p_j}\\right)^2} \\right).\n\\end{equation*}\nsince we $\\mathbb{P\n$-almost surely have  $\\vert Z_{i}\\vert \\leq \\max_{1\\leq j\\leq N\n\\vert x_j\\vert\/p_j $ for all $i\\in \\{1,\\; \\ldots,\\; N  \\}$. Using the independence of the $Z_i$'s, we obtain that: $\\forall u_1> 0$,\n\\begin{multline*}\n\\mathbb{E}[e^{u_1 \\mathcal{Z}_N}]\\leq \\\\\n\\exp\\left( Var\\left(  \\sum_{i=1\n^NZ_i\\right)\\frac{\\exp\\left(\\frac{\\sqrt{n}}{N}u_1\\max_{1\\leq j\\leq N\n\\frac{\\vert x_j\\vert}{p_j} \\right) - 1- \\frac{\\sqrt{n}}{N\nu_1\\max_{1\\leq j\\leq N}\\frac{\\vert x_j\\vert}{p_j}}{\\left(\\max_{1\\leq j\\leq N\n\\frac{\\vert x_j\\vert}{p_j}\\right)^2} \\right).\n\\end{multline*\n\nThe resulting upper bound for $H((u_1,0))$ being minimum for\n$$\nu_1=\\frac{N\n{\\sqrt{n}}\\frac{\\log \\left( 1+\\frac{N}{\\sqrt{n}}\\frac{x\\max_{1\\leq j\\leq N\n\\vert x_j\\vert\/p_j}{Var(\\sum_{i=1}^NZ_i)} \\right)}{\\max_{1\\leq j\\leq N\n\\vert x_j\\vert\/p_j},\n$$\nthis yields\n\\begin{equation}\nH(u^*)\\leq \\exp\\left( -\\frac{Var\\left( \\sum_{i=1}^NZ_i \\right)\n{\\left(\\max_{1\\leq j\\leq N}\\vert x_j\\vert\/p_j \\right)^2}h\\left( \\frac{N\n{\\sqrt{n}}\\frac{x\\max_{1\\leq j\\leq N}\\vert x_j\\vert\/p_j}{Var\\left( \\sum_{i=1\n^NZ_i \\right)} \\right) \\right).\n\\end{equation\n\nUsing the classical\ninequality\n\\begin{equation*}\nh(x)\\geq \\frac{x^{2\n}{2(1+x\/3)},\\text{ for }x\\geq 0,\n\\end{equation*\n\nwe also get that\n\\begin{equation}\\label{eq:prod}\nH(u^*)\\leq \\exp\\left( -\\frac{N^2x^2\/n}{2\\left(Var\\left( \\sum_{i=1\n^NZ_i \\right)+\\frac{1}{3}\\frac{N}{\\sqrt{n}}x \\max_{1\\leq j\\leq N\n\\vert x_j\\vert\/p_j \\right)} \\right).\n\\end{equation}\n$\\square$\n\\end{proof}\n \n\n\nWe now prove the lemma stated below, which provides an upper bound for\n$\\mathbb{P}^{*}_{u^{*},N}\\{ \\mathcal{M}_{N}=0\\}$.\n\n\\begin{lemma}\n\\label{lem:factor2} Under Theorem \\ref{thm:rejective}'s assumptions, there\nexists a universal constant $C^{\\prime}$ such that: $\\forall N\\geq1$,\n\\begin{equation}\n\\mathbb{P}_{u^{*},N}\\left\\{  \\mathcal{M}_{N}=0\\right\\}  \\leq C^{\\prime\n}\\frac{1}{\\sqrt{d_{N}}}.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nUnder the probability measure $\\mathbb{P}_{u^*,N\n$, the $\\varepsilon _{i\n$'s are still\nindependent Bernoulli variables, with means now given by\n\\begin{equation*}\n\\pi^* _{i}\\overset{def}{=}\\sum_{s\\in\n\\mathcal{P}(\\mathcal{I}_{N\n)}e^{\\left\\langle u^*,(\\mathcal{Z}_{N}(s)\n\\mathcal{M}_{N\n(s))\\right\\rangle -\\psi _{N}(u^*)}\\mathbb{I}\\left\\{ i\\in\ns\\right\\} R_{N\n(s)>0,\n\\end{equation*}\nfor $i\\in \\{1,\\; \\ldots,\\; N  \\}$.\nSince $\\mathbb{E\n_{u^{\\ast },N}[\\mathcal{M}_{N}]=0$, we have $\\sum_{i=1}^{N}\\pi^*_{i\n=n$ and thus\n\\begin{equation*}\nd_{N,u^{\\ast\n}}\\overset{def}{=\nVar_{u^{\\ast },N}\\left(\\sum_{i=1}^{N}\\varepsilon _{i}\\right)=\\sum_{i=1\n^{N}\\pi^* _{i}(1-\\pi ^*_{i})\\leq n.\n\\end{equation*\n\nWe can thus apply the local Berry-Esseen bound established in \\cite{Deheuvels}\nfor sums of independent (and possibly non identically) Bernoulli random variables, recalled in Theorem \\ref{thm:numerator\n.\n\\begin{theorem}\\label{thm:numerator}(\\cite{Deheuvels\n, Theorem 1.2) Let $(Y_{j,n})_{1\\leq j\\leq n\n$ be a triangular array of  independent Bernoulli random variables with means $q_{1,n\n,\\; \\ldots,\\; q_{n,n\n$ in $(0,1)$ respectively. Denote by $\\sigma^2_n=\\sum_{i=1}^nq_{i,n\n(1-q_{i,n})$ the variance of the sum $\\Sigma_n=\\sum_{i=1}^nY_{i,n\n$ and by $\\nu_n=\\sum_{i=1}^nq_{i,n\n$ its mean. Considering the cumulative distribution function (cdf) $F_n(x)=\\mathbb{P\n\\{ \\sigma_n^{-1\n(\\Sigma_n-\\nu_n )\\leq x \\}$, we have: $\\forall n\\geq 1$,\n\\begin{equation}\n\\sup_x\\left(1+\\vert x\\vert^3  \\right)\\left\\vert  F_n(x)-\\Phi(x)\\right\\vert\\leq \\frac{C\n{\\sigma_n},\n\\end{equation}\nwhere $\\Phi(x)=(2\\pi)^{-1\/2}\\int_{-\\infty\n^x\\exp(-z^2\/2)dz$ is the cdf of the standard normal distribution $\\mathcal{N\n(0,1)$ and $C<+\\infty$ is a universal constant.\n\\end{theorem\n\nApplying twice a pointwise version of the bound recalled above (for $x=0$ and $x=1\/\\sqrt{d_{N,u^*\n}$), we obtain that\n\\begin{multline*}\n\\mathbb{P}_{u^*,N}\\left\\{ \\mathcal{M\n_{N}=0\\right\\}= \\mathbb{P}_{u^*,N}\\left\\{d_{N,u^{\\ast\n}}^{-1\/2}\n\\sum_{i=1}^Nm_i\\leq 0\\right\\}\\\\-  \\mathbb{P}_{u^*,N}\\left\\{d_{N,u^{\\ast\n\n}^{-1\/2} \\sum_{i=1}^Nm_i\\leq -d_{N,u^{\\ast\n}}^{-1\/2\n\\right\\}\\\\\n\\leq \\frac{2C}{\\sqrt{d_{N,u^*}}}+\\Phi(0)-\\Phi(-d_{N,u^*\n^{-1\/2})\\leq  \\left(\\frac{1}{\\sqrt{2\\pi}}+2C\\right)\\frac{1}{\\sqrt{d_{N,u^*}\n},\n\\end{multline*\n\nby means of the finite increment theorem.\nFinally, observe that:\n\\begin{multline*}\nd_{N,u^*\n=\\mathbb{E}_{u^*, N}\\left[\\left(\\sum_{i=1}^Nm_i\\right)^2 \\right]=\\mathbb{E\n\\left[ \\left(\\sum_{i=1}^Nm_i\\right)^2\/H(u^*) \\right]\\\\\n\\geq \\mathbb{E\n\\left[ \\left(\\sum_{i=1}^Nm_i\\right)^2 \\right]=d_N,\n\\end{multline*\n\nsince we proved that $H(u^*)\\leq 1$. Combined with the previous bound, this yields the desired result.\n$\\square$\n\\end{proof\n\n \n\nLemmas \\ref{lem:factor1} and \\ref{lem:factor2} combined with Eq.\n\\eqref{eq:prod}  leads to the bound stated in Lemma \\ref{lem:numerator}.\n\n\n\n\\subsection*{Proof of Theorem \\ref{thm:final}}\nWe start with proving the preliminary result below.\n\n\\begin{lemma}\n\\label{lem:bias}Let $\\pi_{1},\\; \\ldots, \\pi_N$ be the first order\ninclusion probabilities of a rejective sampling of size $n$ with canonical representation characterized by the Poisson weights $p_1,\\; \\ldots,\\; p_N$.Provided that $d_{N}=\\sum_{i=1}^{N}p_{i}(1-p_{i\n)\\geq1$, we have: $\\forall i\\in\\{1,\\; \\ldots,\\; N  \\}$,\n\\[\n\\left\\vert \\frac{1}{\\pi_{i}}-\\frac{1}{p_{i}}\\right\\vert\\leq\\frac{6}{d_{N}}\\times \\frac{1-\\pi_{i}\n{\\pi_{i}}.\n\\]\n\\end{lemma}\n\\begin{proof}\nThe proof follows the representation (5.14) on page 1509 of \\cite{Hajek64}.\nWe have\n For all $i\\in \\{1,\\; \\ldots,\\; N\\}$, we have:\n\\begin{eqnarray*}\n\\frac{\\pi_{i}}{p_{i}}\\frac{1-p_{i}}{1-\\pi_{i}}  &=&\\frac{\\sum_{s\\in \\mathcal{P}(\\mathcal{I}_N):\\;  i\\in \\mathcal{I}_N\\setminus\\{s \\}}P(s)\\sum_{h\\in s}\\frac{1-p_{h}\n{\\sum_{j\\in s}(1-p_{j})+(p_{h}-p_{i})}}{ \\sum_{s\\in \\mathcal{P}(\\mathcal{I}_N):\\; i\\in\n\\mathcal{I}_N\\setminus\\{s \\}}P(s)}\\\\\n&=&\\frac{\\sum_{s:\\ i\\in \\mathcal{I}_N\\setminus\\{s \\}\nP_N(s)\\sum_{h\\in s}\\frac{1-p_{h}}{\\sum_{j\\in s}(1-p_{j})\\left(  1+\\frac\n{(p_{h}-p_{i})}{\\sum_{j\\in s}(1-p_{j})}\\right)  }}{\\sum_{s:\\ i\\in \\mathcal{I}_N\\setminus\\{s \\}}P_N(s)}.\n\\end{eqnarray*}\nNow recall that for any $x\\in]-1,1[ $, we have:\n\\[\n1-x\\leq\\frac{1}{1+x}\\leq1-x+x^{2}.\n\\]\nIt follows that\n\\begin{align*}\n\\frac{\\pi_{i}}{p_{i}}\\frac{1-p_{i}}{1-\\pi_{i}}  & \\leq1-\\left(  \\sum_{s:\\ i\\in\n\\mathcal{I}_N\\setminus\\{s \\}}P(s)\\right)  ^{-1}\\sum_{s:\\ i\\in \\mathcal{I}_N\\setminus\\{s \\}}P(s)\\sum_{h\\in s}\\frac{(1-p_{h\n)(p_{h}-p_{i})}{\\left(  \\sum_{j\\in s}(1-p_{j})\\right)  ^{2}}\\\\\n& +\\left(  \\sum_{s:\\ i\\in \\mathcal{I}_N\\setminus\\{s \\}}P(s)\\right)  ^{-1}\\sum_{s:\\ i\\in \\mathcal{I}_N\\setminus\\{s \\}\nP(s)\\sum_{h\\in s}\\frac{(1-p_{h})(p_{h}-p_{i})^{2}}{\\left(  \\sum_{j\\in\ns}(1-p_{j})\\right)  ^{3}\n\\end{align*}\nFollowing now line by line the proof on p. 1510 in \\cite{Hajek64} and noticing that $\\sum_{j\\in s\n(1-p_{j})\\geq1\/2d_{N}$ (see Lemma 2.2 in \\cite{Hajek64}), we have\n\\begin{align*}\n\\left\\vert \\sum_{h\\in s}\\frac{(1-p_{h})(p_{h}-p_{i})}{\\left(  \\sum_{j\\in\ns}(1-p_{j})\\right)  ^{2}}\\right\\vert  & \\leq\\frac{1}{\\left(  \\sum_{j\\in\ns}(1-p_{j})\\right)  } \\leq\\frac{2}{d_{N}}\n\\end{align*}\nand similarl\n\\begin{align*}\n\\sum_{h\\in s}\\frac{(1-p_{h})(p_{h}-p_{i})^{2}}{\\left(  \\sum_{j\\in s\n(1-p_{j})\\right)  ^{3}}  & \\leq\\frac{1}{\\left(  \\sum_{j\\in s}(1-p_{j})\\right)\n^{2}} \\leq\\frac{4}{d_{N}^{2}}.\n\\end{align*}\nThis yieds: $\\forall i\\in\\{1,\\; \\ldots,\\; N  \\}$,\n\\[\n1-\\frac{2}{d_{N}}\\leq\\frac{\\pi_{i}}{p_{i}}\\frac{1-p_{i}}{1-\\pi_{i}}\\leq\n1+\\frac{2}{d_{N}}+\\frac{4}{d_{N}^{2}\n\\]\nand\n\\[\np_{i}(1-\\pi_{i})(1-\\frac{2}{d_{N}})\\leq\\pi_{i}(1-p_{i})\\leq p_{i}(1-\\pi\n_{i})\\left(1+\\frac{2}{d_{N}}+\\frac{4}{d_{N}^{2}}\\right),\n\\]\nleading then to\n\\[\n-\\frac{2}{d_{N}}(1-\\pi_{i})p_{i}\\leq\\pi_{i}-p_{i}\\leq p_{i}\\left(1-\\pi_{i\n\\right)\\left(\\frac{2}{d_{N}}+\\frac{4}{d_{N}^{2}}\\right)\n\\]\nand finally to\n\\[\n-\\frac{(1-\\pi_{i})}{\\pi_{i}}\\frac{2}{d_{N}}\\leq\\frac{1}{p_{i}}-\\frac{1\n{\\pi_{i}}\\leq\\frac{(1-\\pi_{i})}{\\pi_{i}}\\left(\\frac{2}{d_{N}}+\\frac{4}{d_{N}^{2}}\\right).\n\\]\nSince $1\/d_{N}^{2}\\leq 1\/d_{N}$ as soon as $d_{N}\\geq1$, the lemma is proved.  $\\square$\n\\end{proof}\n\\medskip\n\nBy virtue of lemma \\ref{lem:bias}, we obtain that:\n\\begin{equation*}\n\\left\\vert \\widehat{S}_{\\boldsymbol{\\pi}_N}^{\\boldsymbol{\\epsilon}_N^*}-\\widehat{S}_{\\boldsymbol{p} _{N\n}^{\\boldsymbol{\\epsilon} _{N}^{\\ast}}\\right\\vert\n \\leq\\frac{6}{d_{N}}\\sum_{i=1}^{N}\\frac{1}{\\pi_{i}}|x_{i}|=M_{N\n\\end{equation*}\nIt follows that\n\\begin{align*}\n\\mathbb{P}\\left\\{  \\widehat{S}_{\\boldsymbol{\\pi}_N}^{\\boldsymbol{\\epsilon}_N^*}-S_N>x\\right\\}    &\n\\leq\\mathbb{P}\\left\\{  |\\widehat{S}_{\\boldsymbol{\\pi}_N}^{\\boldsymbol{\\epsilon}_N^*}-\\widehat{S}_{\\boldsymbol{p} _{N\n}^{\\boldsymbol{\\epsilon} _{N}^{\\ast}}|+\\widehat\n{S}_{\\boldsymbol{p} _{N}}^{\\boldsymbol{\\epsilon} _{N}^{\\ast}}-S_{N}>x\\right\\}\n\\\\\n& \\leq\\mathbb{P}\\left\\{  M_{N}+\\widehat{S}_{\\boldsymbol{p} _{N}\n^{\\boldsymbol{\\epsilon} _{N}^{\\ast}}-S_{N}>x\\right\\}\n\\end{align*}\nand a direct application of Theorem \\ref{thm:rejective} finally gives the desired result. \n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{s:Intro}\n\nIn recent years, there has been increasing interest in the study of structures \nthat can be presented by automata. The underlying idea \nis to apply techniques of automata theory to decision \nproblems that arise in logic and applications such as databases and verification. A typical decision problem \nis the model checking problem: for a \nstructure $\\mathcal A$ (e.g.\\ a graph), design an algorithm that, given a formula $\\phi(\\bar{x})$ in a formal system and a tuple $\\bar{a}$ from the structure, decides if \n$\\phi(\\bar{a})$ is true in $\\mathcal A$. In particular, when the formal system is the first order predicate logic or the monadic second order logic, we would like to know if the \ntheory of the structure  is decidable.  Fundamental early results in this direction by B\\\"uchi (\\cite{Buc60}, \\cite{Buc62}) and Rabin (\\cite{Rab69}) proved the decidability of the monadic second order theories of the successor on the natural numbers and of the binary tree.\nThere have  been numerous applications and extensions of these results in logic, algebra \\cite{EPCHLT92}, verification and model checking \\cite{VW84} \\cite{Var96}, and databases \\cite{Var05}.    Moreover, automatic structures provide a theoretical framework for constraint databases over discrete  domains such as strings and trees \\cite{BL02}.  Using simple closure properties and the decidability of the emptiness problem for automata, one can prove that  the first order (and monadic second order) theories of some well-known structures are decidable. Examples of such structures are Presburger arithmetic and some of its extensions, the term algebra, the\nreal numbers under addition, finitely generated abelian groups, and the atomless Boolean algebra. Direct proofs of  these results, without the use of automata, require non-trivial  technical work.\n\n\nA structure $\\mathcal A=(A; R_0, \\ldots, R_m)$ is {\\bf automatic} if the domain $A$ and all the relations $R_0, \\ldots, R_m$ of the structure are recognised by finite automata (precise definitions are in the next section). \nIndependently, Hodgson \\cite{H82} and later Khoussainov and Nerode \\cite{KhN95} proved that for any given automatic structure there is an algorithm that solves the model checking problem for the first order logic. In particular, the first order theory of the structure is decidable.  Blumensath and Gr\\\"adel proved a logical characterization theorem stating that automatic structures are exactly those definable in the fragment of arithmetic $(\\omega; +, |_2, \\leq, 0)$, where $+$ and $\\leq$ have their usual meanings and $|_2$ is a weak divisibility predicate for which $x|_2 y$ if and only if $x$ is a power of $2$ and divides $y$ (see \\cite{BG00}). In addition, for some classes of automatic structures there are characterization theorems that have direct algorithmic implications. For example, in \\cite{Del04}, Delhomm\\'e proved that automatic well-ordered sets are all strictly less than $\\omega^\\omega$. Using this characterization, \\cite{KhRS03} gives an algorithm which decides the isomorphism problem for automatic well-ordered sets. The algorithm is based on extracting the Cantor normal form for the ordinal isomorphic to the given automatic well-ordered set.  Another characterization theorem of this ilk gives that automatic Boolean algebras are exactly those that are finite products of the Boolean algebra of finite and co-finite subsets of $\\omega$ \\cite{KhNRS04}. Again, this result can be used to show that the isomorphism problem for automatic Boolean algebras is decidable.  \n\nAnother body of work is devoted to the study of resource-bounded complexity of the model checking problem for automatic structures. On the one hand, Gr\\\"adel and Blumensath (\\cite{BG00}) constructed examples of automatic structures whose first order theories are non-elementary. On the other hand, Lohrey in \\cite{Loh03} proved that the first order theory of any automatic graph of bounded degree is elementary.  It is noteworthy that when both a first order formula $\\phi$ and an automatic structure $\\mathcal A$ are fixed, determining if a tuple $\\bar{a}$ from $\\mathcal A$ satisfies \n$\\phi(\\bar{x})$ can be done in linear time. There are also feasible time bounds on deciding the first order theories of automatic structures over the unary alphabet (\\cite{Blu99}, \\cite{KhLM}). \n\nMost current results demonstrate that automatic structures are not complex in various concrete senses.\nHowever, in this paper we use well-established concepts from both logic and model theory to prove results in the opposite direction.  We now briefly describe the measures of complexity we use (ordinal heights of well-founded relations, Scott ranks of \nstructures, and Cantor-Bendixson ranks of trees) and connect them with the results of this paper.\n\n\n\nA relation $R$ is called {\\bf well-founded} if there is no infinite sequence $x_1,x_2,x_3, \\ldots$ such that $(x_{i+1}, x_{i})\\in R$ for $i \\in \\omega$. In computer science, well-founded relations are of interest due to a natural connection between well-founded sets and terminating programs. \nWe say that a program is {\\bf terminating} if every computation from an initial state is finite.\nThis is equivalent to well-foundedness of the collection of states reachable from the initial state, under the reachability relation \\cite{BG06}.  The {\\bf ordinal height} is a measure of the depth of well-founded relations.  Since all automatic structures are also computable structures, the obvious bound for ordinal heights of automatic well-founded relations is $\\omega_1^{CK}$ (the first non-computable ordinal). Sections \\ref{s:RanksofOrders} and \\ref{s:ranksWF} study the sharpness of this bound.\nTheorem \\ref{thm:OrderRank} characterizes automatic well-founded partial orders in terms of their (relatively low) ordinal heights, whereas Theorem \\ref{thm:HeightRank} shows that $\\omega_1^{CK}$ is the sharp bound \nin the general case.\n\n\\begin{theorem}\\label{thm:OrderRank} For each ordinal $\\alpha$, $\\alpha$ is the ordinal height of an automatic well-founded partial order if and only if $\\alpha< \\omega^\\omega$. \n\\end{theorem}\n\n\\begin{theorem}\\label{thm:HeightRank}\nFor each (computable) ordinal $\\alpha < \\omega_{1}^{CK}$, there is an automatic well-founded relation $\\mathcal A$ with ordinal height greater than $\\alpha$.\n\\end{theorem}\n\n\nSection \\ref{s:SR} is devoted to building automatic structures with high Scott ranks. The concept of Scott rank comes from a well-known theorem of Scott stating that for every countable structure $\\mathcal A$ there exists a sentence $\\phi$ in $L_{\\omega_1,\\omega}$-logic which characterizes $\\mathcal A$ up to isomorphism \\cite{Sco65}. The minimal quantifier rank of such a  formula is called the Scott rank of $\\mathcal A$.  A known upper bound on the Scott rank of computable structures implies that the Scott rank of automatic structures is at most $\\omega_1^{CK}+1$.\nBut, until now,  all the known examples of automatic structures had low Scott ranks. Results in \\cite{Loh03}, \\cite{Del04}, \\cite{KhRS05} \nsuggest that the Scott ranks of automatic structures could be bounded by small ordinals. This intuition is falsified in Section \\ref{s:SR} with the theorem:\n\n\n\\begin{theorem}\\label{thm:ScottRank}\nFor each computable ordinal $\\alpha$\nthere is an automatic structure of Scott rank at least $\\alpha$.\n\\end{theorem}\n\nIn particular, this theorem gives a new proof that the isomorphism problem for automatic structures is $\\Sigma_1^1$-complete (another proof may be found in \\cite{KhNRS04}).\n\nIn the last section, we investigate the Cantor-Bendixson ranks of automatic trees. A\n{\\bf partial order tree} is  a partially ordered  set $(T, \\leq)$ such that  there is a $\\leq$-minimal element of $T$, and each subset $\\{x \\in T : x \\leq y\\}$ is finite and is linearly ordered\nunder $\\leq$.  A {\\bf successor tree} is a pair $(T, S)$ such that the reflexive and transitive closure  $\\leq_S$ of $S$ produces a partial order tree \n$(T, \\leq_{S})$. The {\\bf derivative} of a tree $\\mathcal T$ is obtained by removing all the nonbranching paths of the tree. One applies the derivative operation to $\\mathcal T$ successively until a fixed point is reached. The minimal ordinal that is needed to reach the fixed point is called the {\\bf Cantor-Bendixson (CB) rank} of the tree. The CB rank plays an important role in logic, algebra, and topology. Informally, the CB rank tells us how far the structure is from algorithmically (or algebraically) simple structures. Again, the obvious bound on $CB$ ranks of automatic successor trees is $\\omega_1^{CK}$.  \nIn \\cite{KhRS03}, it is proved that the CB rank of any automatic partial order tree is finite and can be computed from the automaton for the $\\leq$ relation on the tree. It has been an open question whether\nthe CB ranks of automatic successor trees can also be bounded by small ordinals. We answer this question in the following theorem.\n\n\n\n\\begin{theorem}\\label{thm:RecTrees}\nFor $\\alpha < \\omega_1^{CK}$ there is an automatic successor tree of CB rank $\\alpha$.\n\\end{theorem}\n\nThe main tool we use to prove results about high ranks is the configuration spaces of Turing machines, considered as automatic graphs.\nIt is important to note that graphs which arise as configuration spaces have very low model-theoretic complexity: their Scott ranks are at most $3$, and if they are well-founded then their ordinal heights are at most $\\omega$ (see Propositions \\ref{ppn:ConfigWF} and \\ref{ppn:ConfigScott}). Hence, the configuration spaces serve merely as building blocks in the construction of automatic structures with high complexity, rather than contributing materially to the high complexity themselves.\n\n\n\\section*{Acknowledgement}\nWe thank Moshe Vardi who posed the question about ranks of automatic well-founded relations. We also thank Anil Nerode and Frank Stephan with whom \nwe discussed Scott and Cantor-Bendixson ranks of automatic structures.\n\n\n\n\\section{Preliminaries}\\label{s:Prelim}\n\nA (relational) {\\bf vocabulary} is a finite sequence $(P_1^{m_1}, \\ldots, P_t^{m_t}, c_1, \\ldots, c_s)$, where each $P_j^{m_j}$ is a predicate symbol of arity $m_j>0$, and each $c_k$ is a constant symbol.  A {\\bf structure} with this vocabulary is a tuple $\\mathcal A=(A;P_1^{\\mathcal A}, \\ldots, P_t^{\\mathcal A}, c_1^{\\mathcal A}, \\ldots, c_s^{\\mathcal A})$, where $P_j^{\\mathcal A}$ and $c_k^{\\mathcal A}$ are interpretations of the symbols of the vocabulary.  When convenient, we may omit the superscripts $\\mathcal A$. We only consider infinite structures, that is, those whose universe is an infinite set.\n\n\nTo establish notation, we briefly recall some definitions associated with finite automata.  A {\\bf finite automaton} $\\mathcal M$ over an alphabet $\\Sigma$ is a tuple\n$(S,\\iota,\\Delta,F)$, where $S$ is a finite set of {\\bf states}, $\\iota \\in S$\nis the {\\bf initial state}, $\\Delta \\subset S \\times \\Sigma \\times S$ is the\n{\\bf transition table}, and $F \\subset S$ is the set of {\\bf final states}.\nA {\\bf computation} of $\\mathcal A$ on a word $\\sigma_1 \\sigma_2 \\dots \\sigma_n$\n($\\sigma_i \\in \\Sigma$) is a sequence of states, say $q_0,q_1,\\dots,q_n$, such\nthat $q_0 = \\iota$ and $(q_i,\\sigma_{i+1},q_{i+1}) \\in \\Delta$ for all $i \\in\n\\{0,\\ldots,n-1\\}$.  If $q_n \\in F$, then the computation is {\\bf successful}\nand we say that the automaton $\\mathcal M$ {\\bf accepts} the word $\\sigma_1 \\sigma_2 \\dots \\sigma_n$. The {\\bf language}\naccepted by the automaton $\\mathcal M$ is the set of all words accepted by $\\mathcal M$. In\ngeneral, $D \\subset \\Sigma^{\\star}$ is {\\bf finite automaton recognisable},\nor {\\bf regular}, if $D$ is the language accepted by some finite automaton~$\\mathcal M$.\n\n\n\nTo define  automaton recognisable  relations, we use $n$-variable (or $n$-tape) automata.\nAn {\\bf $n$--tape automaton} can be thought of as a one-way \nTuring machine with $n$ input tapes \\cite{Eil69}.  Each tape is regarded as \nsemi-infinite, having written on it a word over the alphabet $\\Sigma$ followed \nby an infinite succession of blanks (denoted by $\\diamond$ symbols).  The automaton \nstarts in the initial state, reads simultaneously the first symbol of each tape, \nchanges state, reads simultaneously the second symbol of each tape, \nchanges state, etc., until it reads a blank on each tape. The automaton then \nstops and accepts the $n$--tuple of words if it is in a final state. The set of all \n$n$--tuples accepted by the automaton is the relation recognised by the automaton.  \nFormally,  an $n$--tape  automaton on $\\Sigma$ is a finite automaton over the alphabet $(\\Sigma_{\\diamond})^n$, where $\\Sigma_{\\diamond}=\\Sigma \\cup \\{\\diamond\\}$ and\n$\\diamond \\not \\in \\Sigma$.\nThe {\\bf convolution of a tuple} $(w_1,\\cdots,w_n) \\in\n\\Sigma^{\\star n}$ is the string $ c(w_1,\\cdots,w_n)$ of length $\\max_i|w_i|$\nover the alphabet $(\\Sigma_{\\diamond})^n$ which is defined as follows.\nIts $k$'th symbol is $(\\sigma_1,\\ldots,\\sigma_n)$ where $\\sigma_i$ is the\n$k$'th symbol of $w_i$ if $k \\leq |w_i|$ and $\\diamond$ otherwise.\nThe {\\bf convolution of a relation} $R \\subset \\Sigma^{\\star n}$ is the language\n$c(R) \\subset (\\Sigma_{\\diamond})^{n\\star}$ formed as the set of convolutions\nof all the tuples in $R$. \n An $n$--ary relation $R \\subset \\Sigma^{{\\star}n}$ is {\\bf finite automaton recognisable},\n  or {\\bf regular}, if its convolution $c(R)$ is recognisable by an $n$--tape automaton.\n\n\\begin{definition} \\label{dfn:automatic} A structure $\\mathcal A=(A; R_0, R_1, \\ldots, R_m)$ is {\\bf automatic} over $\\Sigma$ if its domain $A$ and all relations \n$R_0$, $R_1$, $\\ldots$, $R_m$  are regular over $\\Sigma$.  If $\\mathcal B$ is isomorphic to an automatic structure $\\mathcal A$\nthen we call $\\mathcal A$ an {\\bf automatic presentation} of $\\mathcal B$ and say that\n$\\mathcal B$ is {\\bf automatically presentable}.\n\\end{definition}\n\nThe configuration graph of any Turing machine\nis an example of an automatic structure. The graph is defined by letting the \nconfigurations of the Turing machine be the vertices, \nand putting an edge from configuration $c_1$ to configuration $c_2$ if the machine can make an instantaneous move from $c_1$ to $c_2$. Examples of automatically\npresentable structures are $(\\mathbb N, +)$, $(\\mathbb N, \\leq)$, $(\\mathbb N, S)$,\n$(\\mathbb{Z},+)$, the order on the rationals $(Q, \\leq)$, and the Boolean algebra\nof finite and co-finite subsets of $\\mathbb N$. In the following, we abuse terminology and identify the notions of \\textquotedblleft automatic\\textquotedblright~ and \\textquotedblleft automatically presentable\\textquotedblright~.\nMany examples of automatic structures can be formed using the $\\omega$-fold disjoint union of a structure $\\mathcal A$ (the disjoint union of $\\omega$ many  \ncopies of $\\mathcal A$). \n\n\\begin{lemma}\\cite{Rub04}\\label{lm:omega-fold} If $\\mathcal A$ is automatic then its $\\omega$-fold disjoint union is isomorphic to an automatic structure.\n\\end{lemma}\n\\begin{proof}\nSuppose that $\\mathcal A = (A; R_{1}, R_2, \\ldots)$ is automatic.  Define $\\mathcal A' = (A \\times 1^{\\star}; R'_{1}, R_2', \\ldots)$ by\n\\[\n\\langle (x,i), (y, j) \\rangle \\in R'_{m} \\qquad \\iff \\qquad i = j~ \\& ~\\langle x, y \\rangle \\in R_{m},  \\   \\  m=1,2,\\ldots.\n\\]\nIt is clear that $\\mathcal A'$ is automatic and is isomorphic to the $\\omega$-fold disjoint union of $\\mathcal A$. \n\\end{proof}\n\nThe class of automatic structures is a proper subclass of the computable structures.  \nWe therefore mention some crucial definitions and facts about computable structures.\nGood references for the theory of computable structures include \\cite{Hariz98}, \\cite{KhSh99}.  \n\n\\begin{definition}\nA {\\bf computable structure} is $\\mathcal A = (A; R_{1}, \\ldots, R_m)$ whose domain and relations are all computable.  \n\\end{definition}\n\n The domains of computable structures can always be identified with the set $\\omega$ of natural numbers. Under this assumption, we introduce  new constant symbols \n $c_n$ for each $n\\in \\omega$ and interpret $c_n$ as $n$. We expand the vocabulary of each structure to include these new constants $c_{n}$.   In this context, $\\mathcal A$  is computable if and only if \n  the {\\bf atomic diagram} of $\\mathcal A$ (the set of G\\\"odel numbers of all quantifier-free sentences in the extended vocabulary that are true in $\\mathcal A$) is a computable set.\n  If $\\mathcal A$ is computable and $\\mathcal B$ is isomorphic to $\\mathcal A$ then we say that $\\mathcal A$ is a {\\bf computable presentation}\nof $\\mathcal B$. Note that if $\\mathcal B$ has a computable presentation then $\\mathcal B$ has  $\\omega$ many computable presentations.  In this paper, we will be coding computable structures into automatic ones.\n\n\nThe ranks that we use to measure the complexity of automatic structures take values in the ordinals.  In particular, we will see that only a subset of the countable ordinals will play an important role.  An ordinal is called {\\bf computable} if it is the order-type of a computable well-ordering of the natural numbers.  The least ordinal which is not computable is denoted $\\omega_1^{CK}$ (after Church and Kleene). \n\n\\section{Ranks of automatic well-founded partial orders} \\label{s:RanksofOrders}\n\nIn this section we consider structures $\\mathcal A = (A; R)$ with a single binary relation.  \nAn element $x$ is said to be {\\bf $R$-minimal for a set $X$} if for each \n$y \\in X$, $(y,x) \\notin R$.  The relation $R$ is said to be {\\bf well-founded} \nif every non-empty  subset of $A$ has an $R$-minimal element.  \nThis is equivalent to saying that $(A; R)$ has no infinite chains \n$x_1, x_2, x_3, \\ldots$ where $(x_{i+1}, x_i) \\in R$ for all $i$. \n\n\n A {\\bf ranking function} for $\\mathcal A$ is an ordinal-valued function $f$ such that $f(y)< f(x)$ whenever $(y,x)\\in R$. If $f$ is a ranking function on $\\mathcal A$, let $ord(f)= \\sup\\{ f(x) : x \\in A \\}$.  The structure $\\mathcal A$ is well-founded if and only if  $\\mathcal A$ admits a ranking function.  The {\\bf ordinal height} of $\\mathcal A$, denoted $r(\\mathcal A)$, is the least ordinal $\\alpha$ which is $ord(g)$ for some ranking function $g$ on $\\mathcal A$.  An equivalent definition for the rank of $\\mathcal A$ is the following.  We define the function $r_{\\mathcal A}$ by induction: for the $R$-minimal elements $x$,\nset $r_{\\mathcal A}(x)=0$; for $z$ not $R$-minimal, put $r_{\\mathcal A}(z)=\\sup\\{ r(y)+1 : (y,z) \\in R\\}$. Then $r_{\\mathcal A}$ is a ranking function admitted by $\\mathcal A$ and $r(\\mathcal A) = \\sup\\{ r_{\\mathcal A}(x) : x \\in A\\}$.\nFor $B \\subseteq A$, we write $r(B)$ for the ordinal height of the structure obtained by restricting the relation $R$ to the subset $B$. \n\n\\begin{lemma}\\label{lm:compRank}\nIf $\\alpha<\\omega_1^{CK}$, there is a computable well-founded relation of \nordinal height $\\alpha$.\n\\end{lemma}\n\\begin{proof}\nThis lemma is trivial: the ordinal height of an ordinal $\\alpha$ is $\\alpha$ itself.  Since all computable ordinals are computable and well-founded relations, we are done.\n\\end{proof}\n  \nThe next lemma follows easily from the well-foundedness of ordinals and of $R$.  The proof is left to the reader.\n\n\\begin{lemma}\\label{lm:witnessRank}\nFor a structure $\\mathcal A = (A; R)$ where $R$ is well-founded, if $r(\\mathcal A) = \\alpha$ and $\\beta < \\alpha$ then there is an $x \\in A$ such that $r_{\\mathcal A}(x) = \\beta$.\n\\end{lemma}\n\nFor the remainder of this section, we assume further that $R$ is a partial order. For\nconvenience, we write $\\leq$ instead of $R$. Thus,  we consider automatic well-founded partial orders $\\mathcal A=(A,\\leq)$.  We will use the notion of {\\bf natural sum of ordinals}. The natural sum of ordinals $\\alpha, \\beta$ (denoted $\\alpha +' \\beta$) is defined recursively: $\\alpha +' 0 = \\alpha$, $0 +' \\beta = \\beta$, and $\\alpha +' \\beta$ is the least ordinal strictly greater than $\\gamma +' \\beta$ for all $\\gamma < \\alpha$ and strictly greater than $\\alpha +' \\gamma$ for all $\\gamma < \\beta$.\n\n\\begin{lemma}\nLet $A_1$ and $A_2$ be disjoint subsets of $A$ such that $A=A_1\\cup A_2$.  \nConsider the partially ordered sets $\\mathcal A_1=(A_1,\\leq_1)$ and $\\mathcal A_2=(A_2,\\leq_2)$ obtained by restricting $\\leq$ to $A_1$ and $A_2$ respectively. Then, $r(\\mathcal A)\\leq \\alpha_1 +' \\alpha_2$, where $\\alpha_i=r(\\mathcal A_i)$. \n\\end{lemma}\n\\begin{proof}\nWe will show that there is a ranking function on $A$ whose range is contained in the ordinal\n$\\alpha_1 +' \\alpha_2$.   \nFor each $x\\in A$\nconsider the partially ordered sets $\\mathcal A_{1,x}$ and $\\mathcal A_{2,x}$ obtained by restricting\n$\\leq$ to $\\{z\\in A_1 \\mid z < x\\}$ and $\\{z\\in A_2 \\mid z < x\\}$, respectively. \nDefine $f(x)=r(\\mathcal A_{1,x}) +' r(\\mathcal A_{2,x})$.\nWe claim that $f$ is a ranking function. Indeed, assume that $x<y$. Then, since $\\leq$ is transitive, it must be the case that $\\mathcal A_{1,x} \\subseteq A_{1,y}$ and $\\mathcal A_{2,x} \\subseteq A_{2,y}$. Therefore, $r(\\mathcal A_{1,x})\\leq r(\\mathcal A_{1,y})$ and $r(\\mathcal A_{2,x})\\leq r(\\mathcal A_{2,y})$. At least one of these inequalities must be strict. To see this, assume that $x\\in A_1$ (the case $x\\in A_2$ is similar). Then since $x\\in A_{1,y}$, \nit is the case that $r(\\mathcal A_{1,x})+1 \\leq  r(\\mathcal A_{1,y})$ by the definition of ranks. Therefore, we have that $f(x) < f(y)$. \nMoreover, the image of $f(x)$ is contained in $\\alpha_1 +' \\alpha_2$.\n \\end{proof}\n \n \n\\begin{corollary} \\label{cr:PartitionRank}\nIf $r(\\mathcal A)=\\omega^n$ and $A=A_1\\cup A_2$, where $A_1\\cap A_2=\\emptyset$,\nthen either $r(\\mathcal A_1)=\\omega^n$ or $r(\\mathcal A_2)=\\omega^n$.\n\\end{corollary}\n\nKhoussainov and Nerode \\cite{KhN95} show that, for each $n$, there is an automatic presentation of the ordinal $\\omega^n$.  It is clear that such a presentation has ordinal height $\\omega^n$. The next theorem proves that $\\omega^\\omega$ is the sharp bound on ranks of all automatic well-founded partial orders.  Once Corollary \\ref{cr:PartitionRank} has been established, the proof of Theorem \\ref{thm:OrderRank} follows Delhomm\\'e \\cite{Del04} and Rubin \\cite{Rub04}.\\\\\n\n\\noindent{\\bf Theorem \\ref{thm:OrderRank}.~}{\\em\nFor each ordinal $\\alpha$, $\\alpha$ is the ordinal height of an automatic well-founded partial order if and only if $\\alpha< \\omega^\\omega$.}\n\n\n\\begin{proof}\nOne direction of the proof is clear. For the other, assume for a contradiction that there is an automatic well-founded partial order $\\mathcal A = (A, \\leq)$ with $r(\\mathcal A) = \\alpha\\geq\\omega^\\omega$.  Let $(S_{A}, \\iota_{A}, \\Delta_{A}, F_{A})$ and $(S_{\\leq}, \\iota_{\\leq}, \\Delta_{\\leq}, F_{\\leq})$ be finite automata over $\\Sigma$ recognizing $A$ and $\\leq$ (respectively).  \n By Lemma \\ref{lm:witnessRank}, for each $n>0$ there is\n$u_n\\in A$ such that $r_{\\mathcal A}(u_n)=\\omega^n$.   For each $u \\in A$ we define the set\n\\[\nu \\downarrow = \\{ x \\in A : x < u \\}.\n\\]\nNote that if $r_{\\mathcal A}(u)$ is a limit ordinal then $r_{\\mathcal A}(u) = r(u\\downarrow)$. We define a finite partition of $u \\downarrow$ in order to apply Corollary \\ref{cr:PartitionRank}.  To do so, \nfor $u, v \\in \\Sigma^{\\star}$, define\n$X_{v}^{u} = \\{ vw \\in A : w \\in \\Sigma^{\\star} \\ \\& \\ vw < u \\}$.  \nEach set of the form $u \\downarrow$ can then be partitioned based on the prefixes of words\nas follows:\n\\[\nu \\downarrow = \\{ x \\in A : |x| < |u | \\ \\& \\ x < u \\} \\cup \\bigcup_{v \\in \\Sigma^{\\star} : |v| = |u|} X_{v}^{u}.\n\\]\n(All the unions above are finite and disjoint.)  Hence, applying Corollary \\ref{cr:PartitionRank}, for each $u_n$ there exists  a $v_n$ such that $|u_n|=|v_n|$ and $r(X_{v_n}^{u_n})=r(u_n \\downarrow)=\\omega^n$.\n\n\nOn the other hand, we use the automata to define the following equivalence relation on pairs of words of equal lengths:\n\\begin{align*}\n(u,v) \\sim (u', v') \\ \\iff \\ &\\Delta_{A}(\\iota_{A}, v) = \\Delta_{A}(\\iota_{A}, v') \\ \\& \\\\ &\\Delta_{\\leq}(\\iota_{\\leq}, \\binom{v}{u}) = \\Delta_{\\leq}(\\iota_{\\leq}, \\binom{v'}{u'})\n\\end{align*}\nThere are at most  $|S_{A}|\\times |S_{\\leq}|$ equivalence classes. Thus, the infinite sequence $(u_1, v_1)$, $(u_2, v_2)$, $\\ldots$ contains $m$, $n$ such that $m \\neq n$ and $(u_{m}, v_{m}) \\sim (u_{n}, v_{n})$.  \n\n\\begin{lemma}\\label{lm:IsoXvu}\nFor any $u,v,u',v' \\in \\Sigma^{\\star}$, if $(u,v) \\sim (u', v')$ then $r(X_{v}^{u}) = r(X_{v'}^{u'})$.\n\\end{lemma}\n\nTo prove the lemma, consider $g: X_{v}^{u} \\to X_{v'}^{u'}$ defined as $g(vw) = v'w$.  From the equivalence relation, we see that $g$ is well-defined, bijective, and order preserving.  Hence $X_v^u \\cong X_{v'}^{u'}$ (as partial orders).  Therefore, $r(X_{v}^{u}) = r(X_{v'}^{u'})$.\n\n\n\nBy Lemma \\ref{lm:IsoXvu}, $\\omega^{m} = r(X_{v_{m}}^{u_{m}}) = r(X_{v_{n}}^{u_{n}}) = \\omega^{n}$, a contradiction with the assumption that $m \\neq n$.  Therefore, there is no automatic well-founded partial order of ordinal height greater than or equal to $\\omega^{\\omega}$.\n\\end{proof}\n\n\n\\section{Ranks of automatic well-founded relations}\\label{s:ranksWF}\n\n\\subsection{Configuration spaces of Turing machines}\\label{s:Config}\nIn the forthcoming constructions, we embed computable structures into \nautomatic ones via configuration spaces of  Turing machines.\nThis subsection provides terminology and background for these constructions. \nLet $\\mathcal M$ be an $n$-tape deterministic Turing machine.  \nThe {\\bf configuration space}  of $\\mathcal M$, denoted by $Conf(\\mathcal M)$, is a directed graph whose nodes are configurations of $\\mathcal M$. The nodes are $n$-tuples, each of whose coordinates \nrepresents the contents of a tape.  Each tape is encoded as $(w ~q ~ w')$, \nwhere $w, w' \\in \\Sigma^{\\star}$ are the symbols on the tape before and \nafter the location of the read\/write head, and $q$ is one of the  states \nof $\\mathcal M$.  The edges of the graph are all the pairs of the form $(c_1,c_2)$ such that \nthere is an instruction of $\\mathcal M$ that  transforms  \n$c_{1}$ to $c_{2}$.  The configuration space is an automatic graph.  The out-degree of every vertex in $Conf(\\mathcal M)$ is $1$; the in-degree need not be $1$. \n\n\\begin{definition}\nA deterministic Turing machine $\\mathcal M$ is {\\bf reversible} if $Conf(\\mathcal M)$ consists only of finite chains and chains of type $\\omega$.\n\\end{definition}\n\n\\begin{lemma} \\cite{Ben73} \\label{lm:reverse}\nFor any deterministic $1$-tape Turing machine there is a reversible $3$-tape Turing machine which accepts the same language.\n\\end{lemma}\n\n\\begin{proof}(Sketch)\nGiven a deterministic Turing machine, define a $3$-tape Turing machine\nwith a modified set of instructions. \nThe modified instructions have the property that neither the domains nor the ranges overlap.  The first tape performs the computation exactly as the original machine would have done.  As the new machine executes each instruction, it stores the index of the instruction on the second tape, forming a history.  Once the machine enters a state which would have been halting for the original machine, the output of the computation is copied onto the third tape.  Then, the machine runs the computation backwards  and erases the history tape.  The halting configuration contains the input on the first tape, blanks on the second tape, and the output on the third tape\n\\end{proof}\n\nWe establish the following notation for a $3$-tape reversible Turing machine $\\mathcal M$ given by the construction in this lemma.  A {\\bf valid initial configuration} of $\\mathcal M$ is of the form $(\\lambda~ \\iota ~ x , \\lambda, \\lambda )$, where $x$ in the domain,  $\\lambda$ is the empty string,  and $\\iota$ is the initial state of $\\mathcal M$. From the  proof of Lemma \\ref{lm:reverse}, observe that a {\\bf final (halting)} configuration is of the form $(x, \\lambda, \\lambda ~q_{f} ~y)$, \nwith $q_{f}$ a halting state of $\\mathcal M$.   Also, because of the reversibility assumption, all the chains in $Conf(\\mathcal M)$ \nare either finite or $\\omega$-chains (the order type of the natural numbers).  In particular, this means that $Conf(\\mathcal M)$ is well-founded.  We call an element of in-degree $0$ a {\\bf base} (of a chain).  The set of valid initial or final configurations is regular. We classify the components (chains) of $Conf(\\mathcal M)$ as follows:\n\\begin{itemize}\n\\item {\\bf Terminating computation chains}: finite chains whose base is a valid initial configuration; that is, one of the form $(\\lambda ~\\iota~ x, \\lambda, \\lambda )$, for $x \\in \\Sigma^{\\star}$.\n\\item {\\bf Non-terminating computation chains}: infinite chains whose base is a valid initial configuration. \n\\item {\\bf Unproductive chains}: chains whose base is not a valid initial configuration.\n\\end{itemize} \n\nConfiguration spaces of reversible Turing machines are locally finite graphs (graphs of finite degree) and well-founded.  Hence, the following proposition guarantees that their ordinal heights are small.\n\n\\begin{proposition} \\label{ppn:ConfigWF} \nIf $G = (A,E)$ is a locally finite graph then $E$ is well-founded and the ordinal height of $E$ is not above $\\omega$, or $E$ has an infinite chain.\n\\end{proposition}\n\n\\begin{proof}\nSuppose $G$ is a locally finite graph and $E$ is well-founded.  For a contradiction, suppose $r(G) > \\omega$.  Then there is $v \\in A$ with $r(v) = \\omega$. By definition, $r(v) = \\sup\\{ r(u) : u E v \\}$.  But, this implies that there are infinitely many elements $E$-below $v$, a contradiction with local finiteness of $G$\n\\end{proof}\n\n\\subsection{Automatic well-founded relations of high rank}\n\nWe are now ready to prove that $\\omega_1^{CK}$ is the sharp bound for ordinal heights of automatic well-founded relations.\n\n\\vspace{5pt}\n\n\\noindent{\\bf Theorem \\ref{thm:HeightRank}.~}{\\em\nFor each computable ordinal $\\alpha < \\omega_{1}^{CK}$, there is an automatic well-founded relation $\\mathcal A$ with ordinal height greater than $\\alpha$}\n\n\n\\begin{proof} \nThe proof of the theorem uses properties of Turing machines and their configuration spaces.  We take a computable well-founded relation whose ordinal height is $\\alpha$, and ``embed\" it into an automatic well-founded relation with similar ordinal height.  \n\n\nBy Lemma \\ref{lm:compRank}, let $\\mathcal C=(C, L_\\alpha)$ be a computable well-founded relation of ordinal height $\\alpha$.  \nWe assume without loss of generality that $C = \\Sigma^{\\star}$ for some finite alphabet $\\Sigma$. Let $\\mathcal M$ be the Turing machine computing the relation $L_{\\alpha}$.  On each pair $(x,y)$ from the domain, $\\mathcal M$ halts and outputs \\textquotedblleft yes\\textquotedblright~ or \\textquotedblleft no\\textquotedblright~. By Lemma \\ref{lm:reverse}, we can assume that $\\mathcal M$ is reversible.  Recall that $Conf(\\mathcal M) = (D, E)$ is an automatic graph. \nWe define the domain of our automatic structure to be $A = \\Sigma^{\\star} \\cup D$.  The binary relation of the automatic structure is:\n\\begin{align*}\nR = E ~\\cup~ &\\{ (x, (\\lambda ~ \\iota ~ (x, y), \\lambda, \\lambda) ) : x,y \\in \\Sigma^{\\star}\\}  ~\\cup \\\\\n&\\{ (( (x,y), \\lambda, \\lambda~q_{f}~\\text{\\textquotedblleft yes\\textquotedblright~}), y) : x,y \\in \\Sigma^{\\star}\\}.\n\\end{align*}\nIntuitively, the structure $(A; R)$ is a stretched out version of $(C, L_\\alpha)$ with infinitely many finite pieces extending from elements of $C$, and with disjoint pieces which are either finite chains or chains of type $\\omega$.  The structure $(A; R)$ is automatic because its domain is a regular set of words\nand the relation $R$ is recognisable by a $2$-tape automaton.  We should verify, however, that $R$ is well-founded.  Let $Y \\subset A$.  If $Y \\cap C \\neq \\emptyset$ then since $(C, L_{\\alpha})$ is well-founded, there is $x \\in Y \\cap C$ which is $L_{\\alpha}$-minimal. The only possible elements $u$ in $Y$ for which $(u,x) \\in R$ are those which lie on computation chains connecting some $z \\in C$ with $x$.  Since each such computation chain is finite, there is an $R$-minimal $u$ below $x$ on each chain.  Any such $u$ is $R$-minimal for $Y$.  On the other hand, if $Y \\cap C = \\emptyset$, then $Y$ consists of disjoint finite chains and chains of type $\\omega$. Any such chain has a minimal element, and any of these elements are $R$-minimal for $Y$. \nTherefore, $(A; R)$ is an automatic well-founded structure. \n\n\nWe now consider the ordinal height of $(A; R)$.  \nFor each element $x \\in C$, an easy induction on $r_{C}(x)$, shows that \n$r_{\\mathcal C} (x) \\leq r_{\\mathcal A}(x) \\leq \\omega+r_{\\mathcal C} (x)$. \nWe denote by $\\ell(a,b)$ the (finite) length of the computation chain of $\\mathcal M$ with input $(a,b)$.  For any element $a_{x,y}$ in the computation chain which represents the computation of $\\mathcal M$ determining whether $(x,y) \\in R$, we have \\ \n$r_{\\mathcal A}(x) \\leq r_{\\mathcal A}(a_{x,y}) \\leq r_{\\mathcal A}(x) + \\ell(x,y)$. \\ \nFor any element $u$ in an unproductive chain of the configuration space,  $0\\leq r_{\\mathcal A}(u)<\\omega$.  Therefore, since $C \\subset A$,  \\ \n$r(\\mathcal C) \\leq r(\\mathcal A) \\leq \\omega + r(C)$.\n\\end{proof}\n\n\n\n\\section{Automatic Structures and Scott Rank}\\label{s:SR}\n  \n The Scott rank of a structure is introduced in the proof of Scott's Isomorphism Theorem \\cite{Sco65}. Since then, variants of the Scott rank have been used in the computable model theory literature. Here we follow the definition of Scott rank from \\cite{CGKn05}.\n\n\\begin{definition}\nFor structure $\\mathcal A$ and tuples $\\bar{a}, \\bar{b} \\in A^{n}$ (of equal length), define\n\\begin{itemize}\n\\item $\\bar{a} \\equiv^{0} \\bar{b}$ if $\\bar{a}, \\bar{b}$ satisfy the same quantifier-free formulas in the language of $\\mathcal A$; \n\\item For $\\alpha > 0$, $\\bar{a} \\equiv^{\\alpha} \\bar{b}$ if for all $\\beta < \\alpha$, for each $\\bar{c}$ (of arbitrary length) there is $\\bar{d}$ such that \n$\\bar{a}, \\bar{c} \\equiv^{\\beta} \\bar{b}, \\bar{d}$; and for each $\\bar{d}$ (of arbitrary length) there is $\\bar{c}$  such that \n$\\bar{a}, \\bar{c} \\equiv^{\\beta} \\bar{b}, \\bar{d}$.\n\\end{itemize}\nThen, the {\\bf Scott rank} of the tuple $\\bar{a}$, denoted by $\\mathcal{SR}(\\bar{a})$, is  the least \n$\\beta$ such that for all $\\bar{b} \\in A^{n}$, $\\bar{a}\\equiv^{\\beta} \\bar{b}$ implies that $(\\mathcal A, \\bar{a}) \\cong (\\mathcal A, \\bar{b})$.  Finally, the Scott rank of $\\mathcal A$, denoted by $\\mathcal{SR}(\\mathcal A)$, is the\nleast $\\alpha$ greater than the Scott ranks of all tuples of $\\mathcal A$.\n\\end{definition}\n\n\\begin{example} $\\mathcal{SR}(\\mathbb Q, \\leq) = 1$, $\\mathcal{SR}(\\omega, \\leq) = 2$, and $\\mathcal{SR}( n \\cdot \\omega, \\leq) = n+1$.\n\\end{example}\n\nConfiguration spaces of reversible Turing machines are locally finite graphs.  By the proposition below, they all have low Scott Rank.\n\\begin{proposition}\\label{ppn:ConfigScott}\nIf $G = (V,E)$ is a locally finite graph, $SR(G) \\leq 3$.\n\\end{proposition}\n\\begin{proof}\nThe neighbourhood of diameter $n$ of a subset $U$, denoted $B_{n}(U)$, is defined as follows: $B_0(U) = U$ and $B_n(U)$ is the set of $v \\in V$ which can be reached from $U$ by $n$ or fewer edges.  The proof of the proposition relies on two lemmas.\n\n\\begin{lemma}\\label{lm:ConfigScott_1}\nLet $\\bar{a}, \\bar{b} \\in V$ be such that $\\bar{a} \\equiv^2 \\bar{b}$.  Then for all $n$, there is a bijection of the $n$-neighbourhoods around $\\bar{a}, \\bar{b}$ which sends $\\bar{a}$ to $\\bar{b}$ and which respects $E$.\n\\end{lemma}\n\\begin{proof}\nFor a given $n$, let $\\bar{c} = B_{n}(\\bar{a})\\setminus \\bar{a}$. Note that $\\bar{c}$ is a finite tuple because of the local finiteness condition.  Since $\\bar{a} \\equiv^2 \\bar{b}$, there is $\\bar{d}$ such that $\\bar{a} \\bar{c} \\equiv^1 \\bar{b} \\bar{d}$.  If $B_{n}(\\bar{b}) = \\bar{b} \\bar{d}$, we are done.  Two set inclusions are needed.  First, we show that $d_{i} \\in B_{n}(\\bar{b})$.  By definition, we have that $c_{i} \\in B_{n}(\\bar{a})$, and let $a_{j}, u_{1}, \\ldots, u_{n-1}$ witness this.  Then since $\\bar{a} \\bar{c} \\equiv^1 \\bar{b} \\bar{d}$, there are $v_{1}, \\ldots, v_{n-1}$ such that $\\bar{a}\\bar{c} \\bar{u}\\equiv^0 \\bar{b}\\bar{d}\\bar{v}$.  In particular, we have that if $c_{i} E u_{i} E \\cdots E u_{n-1} E a_{j}$, then also $d_{i} E v_{i} E \\cdots E v_{n-1} E b_{j}$ (and likewise if the $E$ relation is in the other direction).  Hence, $d_{i} \\in B_{n}(\\bar{b})$.  Conversely, suppose $v \\in B_{n}(\\bar{b}) \\setminus \\bar{d}$.  Let $v_{1}, \\ldots, v_{n}$ be witnesses and this will let us find a new element of $B_{n}(\\bar{a})$ which is not in $\\bar{c}$, a contradiction.\n\\end{proof}\n\n\n\\begin{lemma}\\label{lm:ConfigScott_2}\nLet $G=(V,E)$ be a graph.  Suppose $\\bar{a}, \\bar{b} \\in V$ are such that for all $n$, $(B_{n}(\\bar{a}), E, \\bar{a}) \\cong (B_{n}(\\bar{b}), E, \\bar{b})$.  Then there is an isomorphism between the component of $G$ containing $\\bar{a}$ and that containing $\\bar{b}$ which sends $\\bar{a}$ to $\\bar{b}$.\n\\end{lemma}\n\\begin{proof}\nWe consider a tree of partial isomorphisms of $G$.  The nodes of the tree are bijections from $B_{n}(\\bar{a})$ to $B_{n}(\\bar{b})$ which respect the relation $E$ and map $\\bar{a}$ to $\\bar{b}$.  Node $f$ is the child of node $g$ in the tree if $\\text{dom}(f) = B_{n}(\\bar{a})$, $\\text{dom}(g)  = B_{n+1}(\\bar{a})$ and $f \\supset g$.  Note that the root of this tree is the map which sends $\\bar{a}$ to $\\bar{b}$.  Moreover, the tree is finitely branching and is infinite by Lemma \\ref{lm:ConfigScott_1}.  Therefore, K\\\"onig's Lemma gives an infinite path through this tree.  The union of all partial isomorphisms along this path is the required isomorphism.\n\\end{proof}\n\nTo prove the proposition, we note that for any $\\bar{a}, \\bar{b}$ in $V$ such that $\\bar{a} \\equiv^2 \\bar{b}$, Lemmas \\ref{lm:ConfigScott_1} and \\ref{lm:ConfigScott_2} yield an isomorphism from the component of $\\bar{a}$ to the component of $\\bar{b}$ that maps $\\bar{a}$ to $\\bar{b}$.  Hence, if $\\bar{a} \\equiv^2 \\bar{b}$, there is an automorphism of $G$ that maps $\\bar{a}$ to $\\bar{b}$.  Therefore, for each $\\bar{a} \\in V$, $SR(\\bar{a}) \\leq 2$, so $SR(G) \\leq 3$.\n\\end{proof}\n\nLet $\\mathcal C=(C; R_{1}, \\ldots, R_{m})$ be a computable structure.  Recall that since $C$ is a computable set, we may assume it is $\\Sigma^\\star$ for some finite alphabet $\\Sigma$.  We construct an \nautomatic structure $\\mathcal A$ whose Scott rank is (close to) the Scott rank of $\\mathcal C$.  Since the domain of $\\mathcal C$ is computable, we assume that $C=\\Sigma^{\\star}$ for some finite \n$\\Sigma$.  The construction of $\\mathcal A$ involves connecting the configuration spaces of Turing machines computing relations $R_{1}, \\ldots, R_{m}$. Note that Proposition \\ref{ppn:ConfigScott} suggests that the high Scott rank of the resulting automatic structure is the main part of the construction because it is not provided by the configuration spaces themselves.  The construction in some sense expands $\\mathcal C$ into an automatic structure.  We comment that expansions do not  necessarily preserve the Scott rank.  For example, any computable structure, $\\mathcal C$, has an expansion with Scott rank $2$.  The expansion is obtained by adding the successor relation into the signature.\n\n\n We detail the construction for  $R_{i}$.  Let $\\mathcal M_{i}$ be a Turing machine for $R_{i}$. \nBy a simple modification of the machine we assume that $\\mathcal M_{i}$ halts if and only if its output is \\textquotedblleft yes\\textquotedblright~.  By Lemma \\ref{lm:reverse}, we can also assume that $\\mathcal M_{i}$ is reversible.  We now modify the configuration space $Conf(\\mathcal M_{i})$ so as to respect the isomorphism type of $\\mathcal C$. \nThis will ensure that the construction (almost) preserves the Scott rank of $\\mathcal C$. We use the terminology from Subsection \\ref{s:Config}.\n\n\n\n\n{\\bf Smoothing out unproductive parts}. The length and number of unproductive chains is determined by the machine $\\mathcal M_{i}$ and hence may differ even for Turing machines computing the same set.  In this stage, we standardize the format of this unproductive part of the configuration space.  We wish to add enough redundant information in the unproductive section of the structure so that if two given computable structures are isomorphic, the unproductive parts of the automatic representations will also be isomorphic .\nWe add $\\omega$-many chains of length $n$ (for each $n$) and $\\omega$-many copies of $\\omega$.  This ensures that the (smoothed) unproductive section of the configuration space of any Turing machine will be isomorphic and preserves automaticity.   We comment that adding this redundancy preserves automaticity since the operation is a disjoint union of automatic structures.\n\n\n\n\n{\\bf Smoothing out lengths of computation chains}. We turn our attention to the chains  which have valid initial configurations at their base.  The length of each finite chain denotes the length of computation required to return a \\textquotedblleft yes\\textquotedblright~ answer.  We will smooth out these chains by adding \\textquotedblleft fans\\textquotedblright~ to each base.  For this, we connect to each base of a computation chain a structure which consists of $\\omega$ many chains of each finite length.  To do so we follow Rubin \\cite{Rub04}:  consider the structure whose domain is $0^{\\star} 0 1^{\\star}$ and whose relation is given by $x E y$ if and only if $|x| = |y|$ and $y$ is the least lexicographic successor of $x$. This structure has a finite chain of every finite length.  As in Lemma \\ref{lm:omega-fold}, we take the $\\omega$-fold disjoint union of the structure and identify the bases of all the finite chains.  We get a \\textquotedblleft fan\\textquotedblright~ with infinitely many chains of each finite size whose base can be identified with a valid initial computation state.  Also, the fan has an infinite component if and only if $R_{i}$ does not hold of the input tuple corresponding to the base. The result is an automatic graph, $Smooth(R_{i}) = ( D_{i}, E_{i})$, which extends $Conf(\\mathcal M_{i})$.\n\n\n\n\n{\\bf Connecting domain symbols to the computations of the relation}. \nWe apply the construction above to each  $R_{i}$ in the signature of $\\mathcal C$. \nTaking the union of the resulting automatic graphs and adding vertices for the domain, we have the structure $(\\Sigma^{\\star} \\cup \\cup_i D_{i}, E_{1}, \\ldots, E_{n})$ (where we assume that the $D_i$ are disjoint).\nWe assume without loss of generality that each $\\mathcal M_{i}$ has a different initial state, and denote it by $\\iota_{i}$.   We add $n$ predicates $F_i$ to the signature of the automatic structure connecting the elements of the domain of $\\mathcal C$ with the computations of the relations $R_{i}$:\n$$F_i =  \\{ (x_{0}, \\ldots, x_{m_{i}-1}, (\\lambda~\\iota_{i}~(x_{0},\\ldots, x_{m_{i}-1}), \\lambda, \\lambda)) \\mid  x_{0}, \\ldots, x_{m_{i}-1} \\in \\Sigma^{\\star} \\}.$$\nNote that for $\\bar{x} \\in \\Sigma^{\\star}$,  $R_{i}(\\bar{x})$ if and only if \n$F_{i}  (\\bar{x}, (\\lambda ~\\iota_{i}~\\bar{x}, \\lambda, \\lambda))$ holds and all $E_{i}$ chains emanating from $(\\lambda~\\iota_{i}~\\bar{x}, \\lambda, \\lambda)$ are finite. \nWe have built  the automatic structure $$\\mathcal A= (\\Sigma^{\\star} \\cup \\cup_i D_{i},  E_{1}, \\ldots, E_{n}, F_{1}, \\ldots, F_{n}).$$ Two technical lemmas are used to show that the Scott rank of $\\mathcal A$ is close to $\\alpha$:\n\n\\begin{lemma}\\label{lm:EquivTransfer}\nFor $\\bar{x}, \\bar{y}$ in the domain of $\\mathcal C$ and for ordinal $\\alpha$, if $\\bar{x} \\equiv_{\\mathcal C}^{\\alpha} \\bar{y}$ then $\\bar{x} \\equiv_{\\mathcal A}^{\\alpha} \\bar{y}$.\n\\end{lemma}\n\n\\begin{proof}\nLet $X = \\text{dom}{\\mathcal A} \\setminus \\Sigma^{\\star}$.  We prove the stronger result that for any ordinal $\\alpha$, and for all $\\bar{x}, \\bar{y} \\in \\Sigma^{\\star}$ and $\\bar{x}', \\bar{y}' \\in X$, if the following assumptions hold\n\\begin{enumerate}\n\\item \\label{asmp:C}$\\bar{x} \\equiv_{\\mathcal C}^{\\alpha} \\bar{y}$;\n\\item \\label{asmp:I}$\\langle \\bar{x}', E_{i} : i =1 \\ldots n\\rangle_{\\mathcal A} \\cong_{f}   \\langle \\bar{y}', E_{i}, : i =1 \\ldots n\\rangle_{\\mathcal A}$ (hence the substructures in $A$ are isomorphic) with $f(\\bar{x}') = \\bar{y}'$; and\n\\item \\label{asmp:E}for each $x'_{k} \\in \\bar{x}'$, each $i=1, \\ldots, n$ and each subsequence of indices of length $m_{i}$, \n\\[\nx'_{k} = (\\lambda ~\\iota_{i}~\\bar{x}_{j}, \\lambda, \\lambda) ~~ \\iff ~~ y'_{k} = (\\lambda ~\\iota_{i}~\\bar{y}_{j}, \\lambda, \\lambda)\n\\]\n\\end{enumerate} \nthen $\\bar{x} \\bar{x}' \\equiv_{\\mathcal A}^{\\alpha} \\bar{y} \\bar{y}'$.  The lemma follows if we take $\\bar{x}' = \\bar{y}' = \\lambda$ (the empty string).\n\nWe show the stronger result by induction on $\\alpha$.  If $\\alpha = 0$, we need to show that for each $i,k, k', k_{0}, \\ldots, k_{m_{i}-1}$, \n\\[\nE_{i}(x'_{k}, x'_{k'}) \\iff E_{i}(y'_{k}, y'_{k'}), \n\\]\nand that \n\\[\nF_{i}(x_{k_{0}}, \\ldots, x_{k_{m_{i}-1}}, x'_{k'}) \\iff F_{i}(y_{k_{0}}, \\ldots, y_{k_{m_{i}-1}}, y'_{k'}).  \n\\]\nThe first statement follows by assumption \\ref{asmp:I}, since the isomorphism must preserve the $E_{i}$ relations and maps $\\bar{x}'$ to $\\bar{y}'$.  The second statement follows by assumption \\ref{asmp:E}.  \n\nAssume now that $\\alpha >0$ and that the result holds for all $\\beta < \\alpha$.  Let $\\bar{x}, \\bar{y} \\in \\Sigma^{\\star}$ and $\\bar{x}', \\bar{y}' \\in A$ be such that the assumptions of the lemma hold.  We will show that $\\bar{x} \\bar{x}' \\equiv_{\\mathcal A}^{\\alpha} \\bar{y} \\bar{y}'$.  Let $\\beta < \\alpha$ and suppose $\\bar{u} \\in \\Sigma^{\\star}, \\bar{u}' \\in A$.  By assumption \\ref{asmp:C}, there is $\\bar{v} \\in \\Sigma^{\\star}$ such that $\\bar{x}\\bar{u} \\equiv_{\\mathcal C}^{\\beta} \\bar{y} \\bar{v}$.  By the construction (in particular, the smoothing steps), we can find a corresponding $\\bar{v}' \\in A$ such that assumptions \\ref{asmp:I}, \\ref{asmp:E} hold.  Applying the inductive hypothesis, we get that $\\bar{x} \\bar{u} \\bar{x}'\\bar{u}' \\equiv_{\\mathcal A}^{\\beta} \\bar{y} \\bar{v}\\bar{y}' \\bar{v}'$.  Analogously, given $\\bar{v}, \\bar{v}'$ we can find the necessary $\\bar{u}, \\bar{u}'$.  Therefore, $\\bar{x}\\bar{x}' \\equiv_{\\mathcal A}^{\\alpha}\\bar{y} \\bar{y}'$.\n\\end{proof}\n\n\\begin{lemma}\\label{lm:CRankHigher}\nIf $\\bar{x} \\in \\Sigma^{\\star} \\cup \\cup_i D_i$, there is $\\bar{y} \\in \\Sigma^{\\star}$ with $\\mathcal{SR}_{\\mathcal A} (\\bar{x} \\bar{x}' \\bar{u} ) \\leq 2 + \\mathcal{SR}_{\\mathcal C} (\\bar{y})$.\n\\end{lemma}\n\\begin{proof}\nWe use the notation $X_{P}$ to mean the subset of $X = A \\setminus \\Sigma^{\\star}$ which corresponds to elements on fans associated with productive chains of the configuration space.  We write $X_{U}$ to mean the subset of $X$ which corresponds to the unproductive chains of the configuration space.  Therefore, $A = \\Sigma^{\\star} \\cup X_{P} \\cup X_{U}$, a disjoint union.  Thus, we will show that for each $\\bar{x} \\in \\Sigma^{\\star}$, $\\bar{x}' \\in X_{P}$, $\\bar{u} \\in X_{U}$ there is $\\bar{y} \\in \\Sigma^{\\star}$ such that $\\mathcal{SR}_{\\mathcal A} (\\bar{x} \\bar{x}' \\bar{u} ) \\leq 2 + \\mathcal{SR}_{\\mathcal C} (\\bar{y})$.\n\nGiven $\\bar{x}, \\bar{x}', \\bar{u}$, let $\\bar{y} \\in \\Sigma^{\\star}$ be a minimal element satisfying that $\\bar{x} \\subset \\bar{y}$ and that $\\bar{x}' \\subset \\langle \\bar{y}, E_{i}, F_{i} : i =1 \\ldots n\\rangle_{\\mathcal A} $.  Then we will show that $\\bar{y}$ is the desired witness.  First, we observe that since the unproductive part of the structure is disconnected from the productive elements we can consider the two independently.  Moreover, because the structure of the unproductive part is predetermined and simple, for $\\bar{u}, \\bar{v} \\in X_{U}$, if $\\bar{u} \\equiv_{\\mathcal A}^{1} \\bar{v}$ then $(\\mathcal A, \\bar{u}) \\cong (\\mathcal A, \\bar{v})$. It remains to consider the productive part of the structure.  \n\n\nConsider any $\\bar{z} \\in \\Sigma^{\\star}$, $\\bar{z}' \\in X_{P}$ satisfying $\\bar{z}' \\subset \\langle \\bar{z}, E_{i}, F_{i} : i =1 \\ldots n\\rangle_{\\mathcal A} $.  We claim that $SR_{\\mathcal A}(\\bar{z} \\bar{z}') \\leq 2 + \\mathcal{SR}_{\\mathcal C}(\\bar{z})$.  It suffices to show that for all $\\alpha$, for all $\\bar{w} \\in \\Sigma^{\\star}, \\bar{w}' \\in X_{P}$, \n\\[\n\\bar{z}\\bar{z}'~\\equiv_{\\mathcal A}^{2+\\alpha}~\\bar{w}\\bar{w}' \\qquad \\implies \\qquad \\bar{z}~\\equiv_{\\mathcal C}^{\\alpha}~\\bar{w}.\n\\]\nThis is sufficient for the following reason. If $\\bar{z} \\bar{z}' \\equiv_{\\mathcal A}^{2 + \\mathcal{SR}_{\\mathcal C}(\\bar{z})} \\bar{w} \\bar{w}'$ then $\\bar{z} \\equiv_{\\mathcal C}^{\\mathcal{SR}_{\\mathcal C}(\\bar{z})} \\bar{w}$ and hence $(\\mathcal C, \\bar{z}) \\cong (\\mathcal C, \\bar{w})$.  From this automorphism, we can define an automorphism of $\\mathcal A$ mapping $\\bar{z} \\bar{z}'$ to $\\bar{w} \\bar{w}'$ because $\\bar{z} \\bar{z}' \\equiv_{\\mathcal A}^{2} \\bar{w} \\bar{w}'$  and hence for each $i$, the relative positions of $\\bar{z}'$ and $\\bar{w}'$ in the fans above $\\bar{z}$ and $\\bar{w}$ are isomorphic.  Therefore, $2 + \\mathcal{SR}_{\\mathcal C}(\\bar{z}) \\geq \\mathcal{SR}_{\\mathcal A}(\\bar{z}\\bar{z}')$.\n\n\nSo, we now show that for all $\\alpha$, for all $\\bar{w} \\in \\Sigma^{\\star}, \\bar{w}' \\in X_{P}$, $\\bar{z}\\bar{z}' \\equiv_{\\mathcal A}^{2+\\alpha} \\bar{w} \\bar{w}'$ implies that $\\bar{z} \\equiv_{\\mathcal C}^{\\alpha} \\bar{w}$.   We proceed by induction on $\\alpha$.  For $\\alpha = 0$, suppose that $\\bar{z}\\bar{z}' \\equiv_{\\mathcal A}^{2} \\bar{w} \\bar{w}'$.  This implies that for each $i$ and for each subsequence of length $m_{i}$ of the indices, the $E_{i}$-fan above $\\bar{z}_{j}$ has an infinite chain if and only if the $E_{i}$-fan above $\\bar{w}_{j}$ does. Therefore, $R_{i}(\\bar{z}_{j})$ if and only if $R_{i}(\\bar{w}_{j})$.  Hence, $\\bar{z} \\equiv_{\\mathcal C}^{0} \\bar{w}$, as required. For the inductive step, we assume the result holds for all $\\beta < \\alpha$.  Suppose that $\\bar{z}\\bar{z}' \\equiv_{\\mathcal A}^{2+\\alpha} \\bar{w} \\bar{w}'$.  Let $\\beta < \\alpha$ and $\\bar{c} \\in \\Sigma^{\\star}$.  Then $2 + \\beta < 2 +\\alpha$ so by definition there is $\\bar{d} \\in \\Sigma^{\\star}, \\bar{d}' \\in X_{P}$ such that $\\bar{z}\\bar{z}' \\bar{c}\\equiv_{\\mathcal A}^{2+\\beta} \\bar{w} \\bar{w}' \\bar{d} \\bar{d}'$.  However, since $2 + \\beta > 1$, $\\bar{d}'$ must be empty (elements in $\\Sigma^{\\star}$ cannot be $1$-equivalent to elements in $X_{P}$).   Then by the induction hypothesis, $\\bar{z} \\bar{c} \\equiv_{\\mathcal C}^{\\beta} \\bar{w} \\bar{d}$.  The argument works symmetrically if we are given $\\bar{d}$ and want to find $\\bar{c}$.   Thus, $\\bar{z} \\equiv_{\\mathcal C}^{\\alpha} \\bar{w}$, as required.\n\\end{proof}\n\n\nPutting Lemmas \\ref{lm:EquivTransfer} and \\ref{lm:CRankHigher} together, we can prove the main result about our construction.\n\n\\begin{theorem}\\label{thm:SR}  Let $\\mathcal C$ be a computable structure and construct the automatic structure $\\mathcal A$ from it as above.  \nThen $\\mathcal{SR}(\\mathcal C) \\leq \\mathcal{SR}(\\mathcal A) \\leq 2 + \\mathcal{SR}(\\mathcal C)$.\n\\end{theorem}\n\n\\begin{proof}\nLet $\\bar{x}$ be a tuple in the domain of $\\mathcal C$.  Then, by the definition of Scott rank, $\\mathcal{SR}_{A}(\\bar{x})$ is the least ordinal $\\alpha$ such that for all $\\bar{y} \\in \\text{dom}(\\mathcal A)$, $\\bar{x} \\equiv_{A}^{\\alpha} \\bar{y}$ implies that $(\\mathcal A, \\bar{x}) \\cong (\\mathcal A, \\bar{y})$; and similarly for $\\mathcal{SR}_{C}(\\bar{x})$.  We first show that $\\mathcal{SR}_{\\mathcal A}(\\bar{x}) \\geq \\mathcal{SR}_{\\mathcal C}(\\bar{x})$.  Suppose $\\mathcal{SR}_{C}(\\bar{x}) = \\beta$.  We assume for a contradiction that $\\mathcal{SR}_{A}(\\bar{x})= \\gamma < \\beta$.  Consider an arbitrary $\\bar{z} \\in \\Sigma^{\\star}$ (the domain of $\\mathcal C$) such that $\\bar{x} \\equiv_{\\mathcal C}^{\\gamma} \\bar{z}$.  By Lemma \\ref{lm:EquivTransfer}, $\\bar{x} \\equiv_{\\mathcal A}^{\\gamma} \\bar{z}$.  But, the definition of $\\gamma$ as the Scott rank of $\\bar{x}$ in $\\mathcal A$ implies that $(\\mathcal A, \\bar{x}) \\cong (\\mathcal A, \\bar{z})$.  Now, $\\mathcal C$ is $L_{\\omega_{1}, \\omega}$ definable in $\\mathcal A$ and therefore inherits the isomorphism.  Hence, $(\\mathcal C, \\bar{x}) \\cong (\\mathcal C, \\bar{z})$.  But, this implies that $\\mathcal{SR}_{\\mathcal C}(\\bar{x}) \\leq \\gamma < \\beta = \\mathcal{SR}_{\\mathcal C}(\\bar{x})$, a contradiction.\n\nSo far, we have that for each $\\bar{x} \\in \\Sigma^{\\star}$, $\\mathcal{SR}_{\\mathcal A}(\\bar{x}) \\geq \\mathcal{SR}_{\\mathcal C}(\\bar{x})$.  Hence, since $\\text{dom}(\\mathcal C) \\subset \\text{dom}(\\mathcal A)$, \n\\begin{align*}\n\\mathcal{SR} (\\mathcal A) &= \\sup\\{\\mathcal{SR}_{\\mathcal A}(\\bar{x}) +1: \\bar{x} \\in \\text{dom}(\\mathcal A)\\} \\\\\n&\\geq \\sup\\{\\mathcal{SR}_{\\mathcal A}(\\bar{x}) +1: \\bar{x} \\in \\text{dom}(\\mathcal C)\\} \\\\\n&\\geq \\sup\\{\\mathcal{SR}_{\\mathcal C}(\\bar{x}) +1: \\bar{x} \\in \\text{dom}(\\mathcal C)\\} = \\mathcal{SR}(C).\n\\end{align*}\n\nIn the other direction, we wish to show that $ \\mathcal{SR}(\\mathcal A) \\leq 2+ \\mathcal{SR}(\\mathcal C)$.  Suppose this is not the case.  Then there is $\\bar{x} \\bar{x}' \\bar{u} \\in \\mathcal A$ such that $\\mathcal{SR}_{\\mathcal A}(\\bar{x} \\bar{x}' \\bar{u} ) \\geq 2 + \\mathcal{SR}(\\mathcal C)$.  By Lemma \\ref{lm:CRankHigher}, there is $\\bar{y} \\in \\Sigma^{\\star}$ such that $2 + \\mathcal{SR}_{\\mathcal C} (\\bar{y}) \\geq 2 + \\mathcal{SR}(\\mathcal C)$, a contradiction.\n\\end{proof}\n\nRecent work in the theory of computable structures has focussed on finding computable structures of high Scott rank.  Nadel \\cite{Nad85} proved that any computable structure has Scott rank at most $\\omega_{1}^{CK} + 1$. Early on, Harrison \\cite{Harr68} showed that there is a computable ordering of type $\\omega_{1}^{CK}( 1 + \\eta)$ (where $\\eta$ is the order type of the rational numbers). This ordering has Scott rank $\\omega_{1}^{CK}+1$, as witnessed by any element outside the initial $\\omega_{1}^{CK}$ set.   However, it was not until much more recently that a computable structure of Scott rank $\\omega_{1}^{CK}$ was produced (see Knight and Millar \\cite{KnM}).  A recent result of Cholak, Downey, and Harrington gives the first natural example of a structure with Scott rank $\\omega_1^{CK}$: the computable enumerable sets under inclusion \\cite{CDH}.\n\n\\begin{corollary}\nThere is an automatic structure with Scott rank $\\omega_1^{CK}$.  There is an automatic structure with Scott rank $\\omega_1^{CK}+1$.\n\\end{corollary}\n\nWe also apply the construction to \\cite{GonKn02}, where it is proved that there are computable structures with Scott ranks above each computable ordinal.  In this case, we get the following theorem.\\\\\n\n\\noindent{\\bf Theorem \\ref{thm:ScottRank}.~}{\\em For each computable ordinal $\\alpha$, there is an automatic structure of Scott rank at least $\\alpha$.}\n\n\n\n\\section{Cantor-Bendixson Rank of Automatic Successor Trees}\\label{s:CBdefs}\n\nIn this section we show that there are automatic successor trees of high Cantor-Bendixson (CB) \nrank. Recall the definitions of partial order trees and successor trees from Section \\ref{s:Intro}.\nNote that if $(T,\\leq)$ is an automatic partial order tree then the successor tree $(T,S)$, where the relation $S$ is defined by $S(x,y) \\iff (x <  y) \\  \\& \\ \\neg \\exists z (x < z < y)$, is automatic. \n\n\\begin{definition}\nThe {\\bf derivative} of a (partial order or successor) tree $T$, $d(T)$, is the subtree of $T$ whose domain is \n\\[\n\\{ x \\in T: \\text{ $x$ lies on at least two infinite paths in $T$}\\}.\n\\]\nBy induction, $d^{0}(T) = T$, $d^{\\alpha+1} (T) = d(d^{\\alpha}(T))$, and for $\\gamma$ a limit ordinal, $d^{\\gamma}(T) = \\cap_{\\beta < \\gamma} d^{\\beta}(T)$.  The {\\bf CB rank} of the tree, $CB(T)$, is the least $\\alpha$ such that $d^{\\alpha}(T) = d^{\\alpha+1}(T)$.\n\\end{definition}\n\nThe CB ranks of automatic partial order trees are finite \\cite{KhRS03}.  This is not true of automatic successor trees.\nThe main theorem of this section provides a general technique for building trees of given CB ranks. Before we get to it, we give some examples of automatic successor ordinals whose CB ranks are low.\n\n\\begin{example} \nThere is an automatic partial order tree (hence an automatic successor tree) whose CB rank is $n$ for each $n \\in \\omega$.\n\\end{example}\n\n\\begin{proof}\nThe tree $T_n$ is defined over the $n$ letter alphabet $\\{a_1,\\ldots, a_n\\}$ as follows. The domain of the tree is  $a_{1}^{\\star} \\cdots a_{n}^{\\star}$. The order $\\leq_n$ is the prefix partial order. Therefore, the successor relation is given as follows:\n\\begin{equation*}\nS(a_{1}^{\\ell_{1}} \\cdots a_{i}^{\\ell_{i}}) = \\begin{cases}\n\t\\{ a_{1}^{\\ell_{1}} \\cdots a_{i}^{\\ell_{i}+1}, a_{1}^{\\ell_{1}} \\cdots a_{i}^{\\ell_{i}}a_{i+1} \\} \\qquad \\text{if $1 \\leq i <n$} \\\\\n\t\\{ a_{1}^{\\ell_{0}} \\cdots a_{i}^{\\ell_{i}+1} \\} \\qquad \\qquad \\text{if $i=n$}\n\\end{cases}\n\\end{equation*}\n\nNote that if $n=0$ then the tree is empty, which is consistent with it having CB rank $0$. \nIt is obvious that $T_n$ is an automatic partial order tree. The rank of $T_{n}$ can be shown, by induction, to be equal to $n$.\n \\end{proof}\n\nThe following examples code the finite rank successor trees uniformly into one automaton in order to push the rank higher.  We begin by building an automatic successor tree $T_{\\omega+1}$ of rank $\\omega+1$. We note that the CB ranks of all trees with at most countably many paths are successor ordinals. Thus, \n$T_{\\omega+1}$ will have countably many paths. Later, we construct a tree of rank $\\omega$ which must embed the perfect\ntree because its CB rank is a limit ordinal.\n\n\\begin{example} \nThere is an automatic successor tree $T_{\\omega+1}$ whose CB rank is $\\omega+1$.\n\\end{example}\n\n\\begin{proof}\nInformally, this tree is a chain of trees of increasing finite CB ranks.  Let $T_{\\omega+1} = (\\{0,1\\}^{\\star}, S) $ with $S$ defined as follows:\n\\begin{align*}\nS(1^{n}) = \\{ 1^{n}0, 1^{n+1} \\} \\qquad &\\text{for all $n$}\\\\ \nS(0u) = 0u0 \\qquad &\\text{for all $u \\in \\{0,1\\}^{\\star}$}\\\\\nS(1^{n}0 u) = \\{ 1^{n}0u0, 1^{n-1} 0 u 1 \\} \\qquad &\\text{for $n \\geq 1$ and $u \\in \\{0,1\\}^{\\star}$}\n\\end{align*}\n\nIntuitively, the subtree of rank $n$ is coded by the set $X_n$ of nodes which contain exactly $n$  $1$s.\nBy induction on the length of strings, we can show that $\\text{range}(S) = \\{0,1\\}^{\\star}$ and hence the domain of the tree is also $\\{0,1\\}^{\\star}$.  It is also not hard to show that the transitive closure of the relation $S$ satisfies the conditions of being a tree. It is also easy to see that $T_{\\omega+1}$ is automatic.\nFinally, we compute the rank of $T_{\\omega + 1}$.  We note that in successive derivatives, each of the finite rank sub-trees $X_n$ is reduced in rank by $1$.  Therefore \n\\[\nd^{\\omega}(T) = 1^{\\star}.\n\\]\nBut, since each point in $1^{\\star}$ is on exactly one infinite path, $d^{\\omega+1}(T) = \\emptyset$, and this is a fixed-point.  Thus, $CB(T_{\\omega+1}) = \\omega+1$, as required.\n\\end{proof}\n\nThe following example gives a tree $T_{\\omega}$ of rank $\\omega$. The idea is to code the trees $T_n$ provided above into the leftmost path of the full binary tree.\n\n\\begin{example}\\label{ex:LimitOrd}\nThere is an automatic successor tree $T_{\\omega}$ whose CB rank is $\\omega$.\n\\end{example}\n\n\\begin{proof}\nThe tree is the full binary tree, where at each node on the leftmost branch we append trees of increasing finite CB rank. Thus, \ndefine $T_{\\omega} = ( \\{0,1\\}^{\\star} \\cup \\{0,a\\}^{\\star}, S)$ where $S$ is given as follows:\n\\begin{align*}\nS(u1v) = \\{ u1v0, u1v1 \\} \\qquad &\\text{for all $u,v \\in \\{0,1\\}^{\\star}$}\\\\ \nS(0^{n}) = \\{0^{n+1}, 0^{n} 1, 0^{n} a\\} \\qquad &\\text{for all $n$} \\\\\nS(au) = aua \\qquad &\\text{for all $u \\in \\{0,a\\}^{\\star}$}\\\\\nS(0^{n}a u) = \\{ 0^{n}aua, 0^{n-1} a u 0 \\} \\qquad &\\text{for $n \\geq 1$ and $u \\in \\{0,a\\}^{\\star}$}\n\\end{align*}\n\nProving that $T_{\\omega}$ is an automatic successor tree is a routine check.   So, we need only compute its rank.  \nEach derivative leaves the right part of the tree (the full binary tree) fixed.  However, the trees appended to the leftmost path of the tree are affected by taking derivatives. Successive derivatives decrease the rank of the protruding finite rank trees by $1$.  Therefore, \n$d^{\\omega}(T_{\\omega}) = \\{0,1\\}^{\\star}$, a fixed point.  Thus, \n$CB(T_{\\omega}) = \\omega$.\n\\end{proof}\n\nTo extend these examples to higher ordinals, we consider the {\\bf product} operation on \ntrees defined as follows. Let $(T_1, S_1)$ and $(T_2,S_2)$ be successor trees. The product of these trees is the tree $(T,S)$ with domain $T=T_1\\times T_2$ and successor relations given by:\n\\begin{align*}\nS((x,y), (u,v))\\iff\\begin{cases}\ny ~\\text{is root of $T_{2}$~ and $\\left( u=x, S_{2}(y,v)\\right)$ or $\\left( S_{1}(x,u),  y=v\\right)$} \\\\\ny ~\\text{is not the root of $T_{2}$ and $u=x$, $S_{2}(y,v)$}.\n\\end{cases}\n\\end{align*}\n\nThe following is an easy proposition.\n\\begin{proposition}\nAssume that $T_1$ and $T_2$ are successor trees of CB ranks $\\alpha$ and $\\beta$, respectively, each having at most countably many paths. Then $T_1\\times T_2$ has \nCB rank $\\alpha +\\beta$. Moreover, if $T_1$ and $T_2$ are automatic successor trees then so is the product.\n\\end{proposition}\n\nThe examples and the proposition above yield tools for building automatic successor trees of CB ranks up to $\\omega^2$. However, it is not clear that these methods can be applied to obtain automatic successor trees of higher CB ranks.  We will see that a different approach to building automatic successor trees will yield all possible CB ranks.\n\n\nWe are now ready for the main theorem of this section. As before, we will transfer results from computable trees to automatic trees. We note that every computable successor \ntree $(T,S)$ is also a computable partial order tree. Indeed, in order to compute\nif $x\\prec_S y$, we effectively find the distances of $y$ and $x$ from the root. If $y$ is closer to the root or is at the same distance as $x$ then $\\neg (x\\prec_S y)$; otherwise,\nwe start computing the trees above all $z$ at the same distance from the root as $x$ is.\nThen $y$ must appear in one of these trees. Thus, we have computed whether $x\\prec_S y$. Hence, every computable successor tree is a computable partial order tree.  However, not every computable partial order tree is a computable successor tree.  We have the following inclusions:\n\\[\n\\text{Aut.\\ PO trees} \\subset \\text{Aut.\\ Succ.\\ trees} \\subset \\text{Comp.\\ Succ.\\ trees} \\subset \\text{Comp.\\ PO trees}\n\\]\nWe use the fact that for each $\\alpha < \\omega_1^{CK}$  there is a computable successor tree of CB rank $\\alpha$.  This fact can be proven by recursively coding up computable trees of increasing CB rank.\n\n\\vspace{5pt}\n\n\\noindent{\\bf Theorem \\ref{thm:RecTrees}.~}{\\em\nFor $\\alpha < \\omega_1^{CK}$ there is an automatic successor tree of CB rank $\\alpha$.}\n\n\\begin{proof} \nSuppose we are given $\\alpha < \\omega_{1}^{CK}$.  Take a computable tree $R_{\\alpha}$ of CB rank $\\alpha$.  We use the same construction as in the case of well-founded relations (see the proof of Theorem \\ref{thm:HeightRank}).\nThe result is\na stretched out version of the tree $R_{\\alpha}$, where between each two elements of the original tree we have a coding of their computation.  In addition, extending from each $x \\in \\Sigma^{\\star}$ we have infinitely many finite computation chains.  Those chains which correspond to output \\textquotedblleft no\\textquotedblright~ are not connected to any other part of the automatic structure.  Finally, there is a disjoint part of the structure consisting of chains whose bases are not valid initial configurations.  By the reversibility assumption, each unproductive component of the configuration space is isomorphic either to a finite chain or to an $\\omega$-chain.  Moreover,  the set of invalid initial configurations which are the base of such an unproductive chain is regular.  We connect all such bases of unproductive chains to the root and get an automatic successor tree, $T_{\\alpha}$.\n\n\nWe now consider the CB rank of $T_{\\alpha}$.  Note that the first derivative removes \nall the subtrees whose roots are at distance $1$ from the root  and are invalid initial computations.  This occurs because each of the invalid computation chains has no branching and is not connected to any other element of the tree.  Next, if we consider the subtree of $T_{\\alpha}$ rooted at some $x \\in \\Sigma^{\\star}$, we see that all the paths which correspond to computations whose output is \\textquotedblleft no\\textquotedblright~ vanish after the first derivative.  Moreover, $x \\in d(T_{\\alpha})$ if and only if $x \\in d(R_{\\alpha})$ because the construction did not add any new infinite paths.  Therefore, after one derivative, the structure is exactly a stretched out version of $d(R_{\\alpha})$.  Likewise, for all $\\beta < \\alpha$, $d^{\\beta}(T_{\\alpha})$ is a stretched out version of $d^{\\beta}(R_{\\alpha})$. Hence, $CB(T_{\\alpha}) = CB(R_{\\alpha}) = \\alpha$\n\\end{proof}\n\nAutomatic successor trees have also been recently studied by Kuske and Lohrey in \\cite{KuLoh}.  In that paper, techniques similar to those above are used to show that the existence of an infinite path in an automatic successor tree is $\\Sigma_1^1$-complete.  In addition, Kuske and Lohrey look at graph questions for automatic graphs and show that the existence of a Hamiltonian path is $\\Sigma_1^1$-complete whereas the set cover problem is decidable.\n\n\\section{Conclusion}\nThis paper studies the complexity of automatic structures.  In particular, we seek to understand the difference in complexity between automatic and computable structures.  We show that automatic well-founded partial orders are considerably simpler than their computable counterparts, because the ordinal heights of automatic partial orders are bounded below $\\omega^\\omega$.  On the other hand, computable well-founded relations, computable successor trees, and computable structures in general can be transformed into automatic objects in a way which (almost) preserves the ordinal height, Cantor-Bendixson rank, or Scott ranks (respectively).  Therefore, the corresponding classes of automatic structures are as complicated as possible.  \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nAfter the successful discovery of a Higgs Boson consistent with the predictions of the Standard Model (SM), the focus of the current and upcoming runs of the Large Hadron Collider (LHC) at 13 TeV will be to discover evidence for physics beyond the SM. Among the prime targets of this search is dark matter (DM), which has so far only been observed via its gravitational interactions at astrophysical and cosmological scales. Since no particle within the SM has the required properties to explain these observations, DM searches at the LHC are necessarily searches for new particles. \n\nIn fact, LHC DM searches are also likely to be searches for new interactions. Given the severe experimental constraints on the interactions between DM and SM particles, it is a plausible and intriguing possibility that the DM particle is part of a (potentially rich) hidden sector, which does not couple directly to SM particles or participate in the known gauge interactions. In this setup, the visible sector interacts with the hidden sector only via one or several new mediators, which have couplings to both sectors.\n\nIn the simplest case the mass of these mediators is large enough that they can be integrated out and interactions between DM particles and the SM can be described by higher-dimensional contact interactions~\\cite{Beltran:2008xg,Beltran:2010ww}. This effective field theory (EFT) approach has been very popular for the analysis and interpretation of DM searches at the LHC~\\cite{Goodman:2010ku,Fox:2011pm,Rajaraman:2011wf}. Nevertheless, as any effective theory it suffers from the problem that unitarity breaks down if the relevant energy scales become comparable to the cut-off scale of the theory~\\cite{Shoemaker:2011vi,Fox:2012ee,Busoni:2013lha,Busoni:2014sya,Xiang:2015lfa} (for other examples of applying unitarity arguments in the DM context see refs.~\\cite{Griest:1989wd, Walker:2013hka, Endo:2014mja, Hedri:2014mua}).\n\nThe easiest way to avoid this problem appears to be to explicitly retain the (lightest) mediator in the theory. The resulting models are referred to as simplified DM models, in which couplings are only specified after electroweak symmetry breaking (EWSB) and no ultraviolet (UV) completion is provided~\\cite{Abdallah:2015ter}. Compared to the EFT approach, simplified models have a richer phenomenology~\\cite{Busoni:2013lha,Buchmueller:2013dya,Buchmueller:2014yoa,Harris:2014hga,Garny:2014waa,Buckley:2014fba,Jacques:2015zha,Alves:2015dya,Choudhury:2015lha}, including explicit searches for the mediator itself~\\cite{Frandsen:2012rk,Fairbairn:2014aqa,Chala:2015ama}. Moreover, it is possible to achieve the DM relic abundance in large regions of parameter space~\\cite{Busoni:2014gta,Chala:2015ama,Blennow:2015gta}. Constraining the parameter space of simplified DM models is therefore a central objective of experimental collaborations~\\cite{Khachatryan:2014rra, Aad:2015zva,Abercrombie:2015wmb}.\n\nIn the present work we focus on the case of a spin-1 $s$-channel\nmediator~\\cite{Dudas:2009uq,Fox:2011qd,Frandsen:2012rk,Alves:2013tqa,Arcadi:2013qia,Jackson:2013pjq,Jackson:2013rqp,Duerr:2013lka,Duerr:2014wra,Lebedev:2014bba,Hooper:2014fda,Martin-Lozano:2015vva,Alves:2015pea,Alves:2015mua,Blennow:2015gta,Duerr:2015wfa,Heisig:2015ira}. Our central observation is that the simplified model approach is not generally\nsufficient to avoid the problem of unitarity violation at high\nenergies and that further amendments are required if the model\n  is to be both simple and realistic. In particular, a spin-1\nmediator with axial couplings violates perturbative unitarity at large\nenergies, pointing towards the presence of additional new physics to\nrestore unitarity. \n\nIndeed, the simplest way to restore unitarity is to assume that the\nspin-1 mediator is the gauge boson of an additional $U(1)'$ gauge\nsymmetry~\\cite{Holdom:1985ag,Babu:1997st} and that its mass as well as\nthe DM mass are generated by a new Higgs field in the hidden\nsector. The famous Lee-Quigg-Thacker bound~\\cite{Lee:1977eg} implies\nthat the additional Higgs boson cannot be arbitrarily heavy and may\ntherefore play an important role for LHC and DM phenomenology. In particular,\nit can mix with the SM-like Higgs boson and mediate interactions\nbetween DM particles and quarks.\n\n\nFurthermore, we require for a consistent simplified DM model that the coupling structure respects gauge invariance of the full SM gauge group before EWSB (see~\\cite{Bell:2015sza} for a similar discussion in the EFT context). If the mediator has axial couplings to quarks, this requirement implies that the new mediator will also have couplings to leptons and mixing with the SM $Z$ boson, both of which are tightly constrained by experiments. Much weaker constraints are obtained for the simplified DM model containing a spin-1 mediator with vectorial couplings to quarks.  Constraints from direct detection can be evaded if the mediator has only axial couplings to DM, which naturally arises in the case that the DM particle is a Majorana fermion. We discuss the importance of loop-induced mixing effects in this context, which can play a crucial role for both direct detection experiments and LHC phenomenology.\n\nThe outline of the paper is as follows. Starting from a simplified model for a \\mbox{spin-1} \\mbox{$s$-channel} mediator, we explore in section~\\ref{sec:unitarity} the implications of perturbative unitarity, deriving a number of constraints on the model parameters and in particular an upper bound on the scale of additional new physics. In section~\\ref{sec:higgs} we then consider the case where this additional new physics is a Higgs field in the hidden sector and derive an upper bound on the mass of the extra Higgs boson. We then discuss additional constraints on the SM couplings implied by gauge invariance. Section~\\ref{sec:axial} focuses on the case of non-zero axial couplings between SM fermions and the mediator, whereas in section~\\ref{sec:vector} we assume that the SM couplings of the mediator are purely vectorial. \nFinally, we discuss the experimental implications of a possible mixing between the SM Higgs and the hidden sector Higgs in section~\\ref{sec:higgsmixing}. A discussion of our results and our conclusions are presented in section~\\ref{sec:discussion}.\n\n\n\\section{Unitarity constraints on simplified models}\n\\label{sec:unitarity}\n\n\\subsection{Brief review of $S$ matrix unitarity constraints}\n\nConsider the scattering matrix element $\\mathcal{M}_{if}(s, \\cos\n\\theta)$ between 2-particle initial and final states ($i,f$), with $\\sqrt{s}$\nand $\\theta$ being the centre of mass energy and scattering angle,\nrespectively. We define the helicity matrix element for the $J$th\npartial wave by\n\\begin{equation}\\label{eq:Jexpansion}\n  \\mathcal{M}_{if}^J(s) = \\frac{1}{32\\pi} \\beta_{if} \n  \\int_{-1}^1 \\mathrm{d}\\cos \\theta \\, d^J_{\\mu \\mu'}(\\theta) \\, \\mathcal{M}_{if}(s, \\cos \\theta) \\,,\n\\end{equation}\nwhere $d^J_{\\mu \\mu'}$ is the $J$th Wigner d-function, $\\mu$ and $\\mu'$ denote the total spin of the initial and the final state (see e.g.~\\cite{Chanowitz:1978mv}), and $\\beta_{if}$ is a\nkinematical factor. In the high-energy limit $s\n\\to \\infty$, which we are going to consider below, $\\beta_{if} \\to 1$. The right-hand side of eq.~\\eqref{eq:Jexpansion} is to be multiplied with a factor of $1\/\\sqrt{2}$ each if the initial or final state particles are identical~\\cite{Schuessler:2007av}.\nUnitarity of the $S$ matrix implies\n\\begin{align}\n  {\\rm Im}(\\mathcal{M}_{ii}^J) & = \\sum_f | \\mathcal{M}_{if}^J|^2 \\nonumber\\\\\n  &=\n  | \\mathcal{M}_{ii}^J|^2 + \\sum_{f \\neq i} | \\mathcal{M}_{if}^J|^2 \\ge | \\mathcal{M}_{ii}^J|^2\n  \\label{eq:unity}\n\\end{align}\nfor all $J$ and all $s$. The sum over $f$ in the first line runs over\nall possible final states. Restricting these to be all possible\n2-particle states leads to a conservative bound. \nIf the relation \\eqref{eq:unity} is strongly violated\nfor matrix elements calculated at leading order in perturbation theory\none can conclude that either higher-order terms in perturbation theory\nrestore unitarity (i.e.\\ break-down of\nperturbativity) or that the theory is not complete and additional\ncontributions to the matrix element are needed. \n\nFrom eq.~\\eqref{eq:unity} one obtains the necessary conditions\n\\begin{equation}\\label{eq:unitary-bound}\n  0 \\le {\\rm Im}({\\mathcal{M}}_{ii}^J) \\le 1\\,, \\quad\n  \\left| \\text{Re}({\\mathcal{M}}_{ii}^J) \\right| \\le \\frac{1}{2} \\,.\n\\end{equation}\nIn the following we will apply these inequalities to leading-order matrix elements in order to identify regions in parameter space where perturbative unitarity is violated. Since these matrix elements are always real in the present context, only the second constraint will be relevant.\n\nIf the matrix $\\mathcal{M}_{if}^J$ is diagonalized the\ninequality in eq.~\\eqref{eq:unity} becomes an equality. Hence,\nstronger constraints can be obtained by considering the full\ntransition matrix connecting all possible 2-particle states with each\nother (or some submatrix thereof) and calculating the eigenvalues of\nthat matrix. Then the bounds from eq.~\\eqref{eq:unitary-bound} have to\nhold for each of the eigenvalues~\\cite{Schuessler:2007av}.\n\nWe note that $d^J_{00}(\\theta) = P_J(\\cos \\theta)$, where $P_J$ are the Legendre polynomials. If initial and final state both have zero total spin, eq.~\\eqref{eq:Jexpansion} therefore becomes identical to the familiar partial wave expansion of the matrix element. In the following we will focus on the $J=0$ partial wave, which typically provides the strongest constraint. Since $d^0_{\\mu\\mu'}$ is non-zero only for $\\mu = \\mu' = 0$, we then obtain from eq.~\\eqref{eq:Jexpansion}\n\\begin{equation}\n  \\mathcal{M}_{if}^0(s) = \\frac{1}{64\\pi} \\beta_{if} \\, \\delta_{\\mu0} \\delta_{\\mu'0} \n  \\int_{-1}^1 \\mathrm{d}\\cos \\theta \\, \\mathcal{M}_{if}(s, \\cos \\theta) \\,.\n\\end{equation}\n\n\\subsection{Application to a simplified model with a $Z'$ mediator}\n\nLet us consider a simplified model for a spin-1 mediator $Z'^\\mu$ with mass $m_{Z'}$ and a Dirac DM particle $\\psi$ with mass $m_\\text{DM}$.\\footnote{In the case of Majorana DM the vector current vanishes and hence there can only be an axial coupling on the DM side. We will come back to this case shortly but will consider Dirac DM here to allow for both vectorial and axial couplings.} The most general coupling structure is captured by the following Lagrangian:\n\\begin{equation}\\label{eq:L_VA}\n\\mathcal{L} =  - \\sum_{f = q,l,\\nu} Z'^\\mu \\, \\bar{f} \\left[ g_{f}^V \\gamma_\\mu + g_f^A \\gamma_\\mu \\gamma^5 \\right] f - Z'^\\mu \\, \\bar{\\psi} \\left[ g_\\text{DM}^V \\gamma_\\mu + g_\\text{DM}^A \\gamma_\\mu \\gamma^5 \\right] \\psi \\; .\\\\\n\\end{equation}\nAlthough these interactions appear renormalisable, the presence of a massive vector boson implies that perturbative unitarity may be violated at large energies. In the following, we will study this issue in detail and derive constraints on the parameter space of the model.\n\nLet us first consider diagrams between 2-fermion states with the $Z'$\nas mediator. The appropriate propagator for the mediator is\n\\begin{equation}\n\\langle Z'^\\mu(k)Z'^\\nu(-k)\\rangle =  \\frac{1}{k^2 - m_{Z'}^2} \\left(g^{\\mu\\nu} - \\frac{k^\\mu k^\\nu}{m_{Z'}^2}\\right) \\;,\n\\end{equation}\nwhere $k^\\mu$ is the momentum of the mediator.  For the case of a\ngauge boson this corresponds to unitary gauge in which the Goldstone\nboson has been absorbed. Since we are interested in the high-energy\nbehaviour of the theory we concentrate on the second term, which does\nnot vanish in the limit $k \\rightarrow \\infty$. This corresponds to\nrestricting to the longitudinal component of the mediator, $Z'_L$,\nwhich dominates at high energy \\cite{Chanowitz:1978mv}.\\footnote{It\n  turns out that for certain processes the transversal part of the\n  propagator leads to a logarithmic divergence for $m_{Z'}^2 \\ll s$.\n  This divergence is not related to the UV completeness of the theory,\n  but signals breakdown of perturbativity in the IR, see also\n  \\cite{Hedri:2014mua}. By restricting to the longitudinal components\n  of the $Z'$ \\cite{Chanowitz:1978mv} we can avoid the occurence of\n  those IR divergences.}  For\ninstance, considering DM annihilations, we can contract the longitudinal\npart of the propagator with the DM current.  Making use of $k = p_1 +\np_2$, where $p_1$ and $p_2$ are the momenta of the two DM particles in\nthe initial state, leads to a factor\n\\begin{align}\n k^\\mu \\bar{v}(p_2) \\left( g^V_\\text{DM} \\gamma_\\mu + g^A_\\text{DM} \\gamma_\\mu \\gamma^5 \\right) u(p_1)  \n & = \\bar{v}(p_2) \\left[ g^V_\\text{DM} (\\slashed{p}_2 + \\slashed{p}_1) + g^A_\\text{DM} (\\slashed{p}_2 \\gamma^5 - \\gamma^5 \\slashed{p}_1) \\right] u(p_1) \\nonumber \\\\\n & = - 2 \\, g^A_\\text{DM} \\, m_\\text{DM} \\, \\bar{v}(p_2) \\gamma^5 u(p_1) \\;.\n\\end{align}\nHence, the second term in the propagator behaves exactly like\na pseudoscalar with mass $m_{Z'}$ and couplings to DM equal to $2 \\,\ng^A_\\text{DM} \\, m_\\text{DM} \/ m_{Z'}$, just like the Goldstone boson\npresent in Feynman gauge. Note that the term is independent of the vector couplings. The same argument holds for the quark\ncouplings, which are found to be given by $2 \\, g^A_f \\, m_f \/\nm_{Z'}$.\nThis consideration suggests that perturbative unitarity will not only lead to bounds on $g^{V,A}$, but also on the combination $g^A_f \\, m_f \/ m_{Z'}$.\n\n\nWe can make this statement more precise by applying the methods outlined in the previous subsection to the self-scattering of two DM particles or two SM fermions.\nWe obtain for any fermion $f$ with axial couplings $g^A_f \\neq 0$ that the fermion mass must satisfy the bound\n\\begin{equation}\n m_f \\lesssim \\sqrt{\\frac{\\pi}{2}} \\frac{m_{Z'}}{g^A_f} \\; .\n \\label{eq:DMmass}\n\\end{equation}\nHere $f$ can be any fermion, including SM fermions and the DM\nparticle. As suggested by the above discussion we do not obtain any\nbound on the masses of fermions with purely vectorial couplings, nor\non the scale of new physics.\n\nLet us now turn to the discussion of processes involving $Z'$ in the\nexternal state, in particular $Z'$ with longitudinal polarisation. For\nconcreteness, we study the process $\\psi \\bar{\\psi} \\rightarrow Z'_L\nZ'_L$.\\footnote{Note that this process corresponds to an off-diagonal\n  element of $\\mathcal{M}_{if}$, with $i \\neq f$, whereas the bounds\n  from eq.~\\eqref{eq:unitary-bound} apply for diagonal elements. In\n  order to apply the unitarity constraint we consider the $2\\times 2$\n  submatrix of $\\mathcal{M}_{if}$ spanned by the states $\\psi\n  \\bar{\\psi}$ and $Z'_L Z'_L$. For $s\\to \\infty$ only the off-diagonal\n  element survives, and hence the eigenvalues of the matrix become\n  equal to the off-diagonal element, and we can apply\n  eq.~\\eqref{eq:unitary-bound}.} At large momenta, $k^2 \\gg\nm_{Z'}^2$, the polarisation vectors of the gauge bosons can be\nreplaced by $\\epsilon_L^\\mu(k) = k^\\mu \/ m_{Z'}$. One might therefore\nexpect the matrix element for this process to grow proportional to $s\n\/ m_{Z'}^2$. However, such a term is absent due to a cancellation\nbetween the $t$- and $u$-channel diagram. To obtain a non-zero\ncontribution, one needs to include a mass insertion along the fermion\nline~\\cite{Shu:2007wg}. It turns out that the contribution\nproportional to $g^V_\\text{DM}$ still cancels in this case and that\nthe leading contribution at high energies becomes proportional to\n$(g^A_\\text{DM})^2 \\sqrt{s} \\, m_\\text{DM} \/ m_{Z'}^2$. As a result,\nperturbative unitarity is violated\nunless~\\cite{Hosch:1996wu,Shu:2007wg,Babu:2011sd}\\footnote{Our result differs from the one in~\\cite{Shu:2007wg} by a factor $1\/\\sqrt{2}$.}\n\\begin{equation}\\label{eq:s}\n\\sqrt{s} < \\frac{\\pi \\, m_{Z'}^2}{(g^A_\\text{DM})^2 \\, m_\\text{DM}} \\; .\n\\end{equation}\nFor larger energies new physics must appear to restore unitarity.  This can be accomplished by including an additional diagram with an $s$-channel Higgs boson, since both contributions have the same high-energy behaviour. The consideration above implies an upper bound on the mass of the Higgs that breaks the $U(1)'$ and gives mass to the $Z'$:\n\\begin{equation}\nm_s < \\frac{\\pi \\, m_{Z'}^2}{(g^A_\\text{DM})^2 \\, m_\\text{DM}} \\; .\n\\label{eq:higgsmass}\n\\end{equation}\nWe will discus the consequences  of such an extension of the minimal model in section~\\ref{sec:higgs}.\n \n\n\\begin{figure}[tb]\n\\centering\n\\includegraphics[height=0.3\\textheight]{uni.pdf}\n\\caption{Parameter space forbidden by the requirement of perturbative unitarity in the $\\sqrt{s}-m_{Z'}$ plane for $m_\\text{DM}=500\\:\\text{GeV}$. The constraint resulting from DM scattering is shown in grey (solid), the constraint resulting from DM annihilation into $Z'$s is shown in blue (dashed). Thick (thin) lines correspond to $g^A_\\text{DM}=1$ ($g^A_\\text{DM}=0.1$). In these cases, the $Z'$ can never be lighter than about $400\\:\\text{GeV}$ ($40\\:\\text{GeV}$) irrespective of the UV completion.}\n\\label{fig:ubound}\n\\end{figure}\n\nIn summary we have found that there are two different types of\nconstraints on the parameters of this simplified model, even for\nperturbative couplings. For non-vanishing axial couplings there is an\nenergy scale for which the theory violates perturbative unitarity and\nneeds to be UV completed, see eq.~\\eqref{eq:s}.  In addition, imposing that the coupling between the longitudinal component of the vector mediator and the DM particle remain perturbative, we find that the\nvector mediator cannot be much lighter than the DM, see eq.~\\eqref{eq:DMmass}. This constraint is not related to missing degrees of freedom and is therefore completely\nindependent of the UV completion. We illustrate both constraints in figure~\\ref{fig:ubound} for different axial couplings\nand a DM mass $m_\\text{DM} = 500\\:\\text{GeV}$.\n\nTo conclude this section, we emphasise that for pure vector couplings of the $Z'$ ($g^A_\\text{DM} = g^A_f = 0$) the simplified model considered in this section is well-behaved in the UV in the sense that there is no problem with perturbative unitarity.\\footnote{As discussed below there can be anomalies which require additional fermions.} Indeed in this specific case a bare mass term for the dark matter is allowed such that it is sufficient to generate the vector boson mass via a Stueckelberg mechanism without the need for additional degrees of freedom~\\cite{Stueckelberg:1900zz,Kors:2005uz}. However, this specific coupling configuration is highly constrained, since it is very difficult to evade bounds from direct detection experiments and still reproduce the observed DM relic abundance. This is illustrated in figure~\\ref{fig:vector} where we show the parameter region excluded by the bound on the spin-independent DM-nucleon scattering cross section from LUX~\\cite{Akerib:2013tjd} and the parameter region where the DM annihilation cross section becomes so small that DM is overproduced in the early Universe. One can clearly see that only a finely-tuned region of parameter space close to the resonance $m_\\text{DM} = m_{Z'}\/2$ is still allowed. For the rest of the paper, we will therefore not consider this case further and always assume that at least one of the vector couplings vanishes such that direct detection constraints can be weakened.\n\n\n\\begin{figure}[tb]\n\\centering\n\\includegraphics[height=0.3\\textheight]{vectorvector.pdf}\n\\caption{Vector(SM)--Vector(DM): Parameter space excluded by the bound on the spin-independent DM-nucleon scattering cross section from LUX (green, dashed) and the parameter region where the DM annihilation cross section becomes so small that DM is overproduced in the early Universe (red, solid).}\n\\label{fig:vector}\n\\end{figure}\n\n\\section{Including an additional Higgs field}\n\\label{sec:higgs}\n\nAs we have seen in the previous section, for non-zero axial couplings the simplified model violates perturbative unitarity at high energies, implying that additional new physics must appear below these scales. This observation motivates a detailed discussion of how to generate the vector boson mass from an additional Higgs mechanism. To restore unitarity let us therefore now consider the case that the $Z'$ is the gauge boson of a new $U(1)'$ gauge group. To break this gauge group and give a mass to the $Z'$, we introduce a dark Higgs singlet $S$, which needs to be complex in order to allow for a $U(1)'$ charge. We then obtain the following Lagrangian\n\\begin{equation}\n \\mathcal{L} = \\mathcal{L}_\\text{SM}  + \\mathcal{L}_\\text{DM} + \\mathcal{L}'_\\text{SM} + \\mathcal{L}_\\text{S} \\; ,\n\\end{equation}\nwhere the first term is the usual SM Lagrangian and the second term describes the interactions of DM. The third term contains the interactions between SM states and the new $Z'$ gauge boson while the fourth term contains the extended Higgs sector. \n\n\\subsection{Implications for the dark sector}\n\nAs mentioned above, it is well-motivated from a phenomenological perspective to consider the case that vector couplings to the $Z'$ mediator vanish in at least one of the two sectors, so that direct detection is suppressed. On the DM side this is naturally achieved for a Majorana fermion, which we will focus on from now. We therefore write\n\\begin{equation}\n\\psi =\n\\left(\n    \\begin{array}{c}\n      \\chi \\\\\n      \\epsilon \\chi^\\ast\n    \\end{array}\n  \\right) \\; ,\n\\end{equation}\nwhere $\\chi$ is a Weyl spinor. We assume that $\\chi$ carries a charge $q_\\text{DM}$ under the new $U(1)'$ gauge group, such that under a gauge transformation\n\\begin{equation}\n\\psi \\rightarrow \\exp\\left[i \\, g' q_\\text{DM} \\, \\alpha(x) \\, \\gamma^5\\right] \\psi \\; ,\n\\end{equation}\nwhere $g'$ is the gauge coupling of the new $U(1)'$.\nThe kinetic term for $\\psi$ can hence be written as\n\\begin{equation}\n\\mathcal{L}_\\text{kin} = \\frac{1}{2} \\bar{\\psi} (i \\slashed{\\partial} - g' \\, q_\\text{DM} \\, \\gamma^5 \\slashed{Z}') \\psi = \\frac{i}{2} \\bar{\\psi} \\slashed{\\partial} \\psi - \\frac{1}{2} g_\\text{DM}^A Z'^\\mu \\bar{\\psi} \\gamma^5 \\gamma_\\mu \\psi \\; ,\n\\end{equation}\nwith $g_\\text{DM}^A \\equiv g' q_\\text{DM}$.\nThe $U(1)'$ charge forbids a Majorana mass term. Nevertheless, if the Higgs field $S$ carries charge $q_S = - 2 q_\\text{DM}$, we can write down the gauge-invariant combination\n\\begin{equation}\n\\mathcal{L}_\\text{mass} = -\\frac{1}{2} y_\\text{DM} \\bar{\\psi} (P_L S + P_R S^\\ast) \\psi \\; .\n\\end{equation}\nIncluding the kinetic and potential terms for the Higgs singlet, the\nfull dark Lagrangian therefore reads\n\\begin{align}\n \\mathcal{L}_\\text{DM} = & \\frac{i}{2} \\bar{\\psi} \\slashed{\\partial} \\psi - \\frac{1}{2} g_\\text{DM}^A Z'^\\mu \\bar{\\psi} \\gamma^5 \\gamma_\\mu \\psi - \\frac{1}{2} y_\\text{DM} \\bar{\\psi} (P_L S + P_R S^\\ast) \\psi \\,, \\nonumber \\\\\n\\mathcal{L}_S = &  \\left[ (\\partial^\\mu + i \\, g_S \\, Z'^\\mu) S \\right]^\\dagger \\left[ (\\partial_\\mu + i \\, g_S \\, Z'_\\mu) S \\right] + \\mu_s^2 \\, S^\\dagger S - \\lambda_s \\left(S^\\dagger S  \\right)^2 \\,.\n\\end{align}\n\nOnce the Higgs singlet aquires a vacuum expectation value (vev), it will spontaneously break the $U(1)'$ symmetry, thus giving mass to the $Z'$ gauge boson and the DM particle. After symmetry breaking, we obtain the following Lagrangian (defining $S = 1\/\\sqrt{2} (s + w)$ and using $g_S \\equiv g'q_S = -2g_\\text{DM}^A$)\n\\begin{align}\n \\mathcal{L} = & \\frac{i}{2} \\bar{\\psi} \\slashed{\\partial} \\psi - \\frac{1}{4} F'^{\\mu\\nu}F'_{\\mu\\nu} - \\frac{1}{2} g_\\text{DM}^A Z'^\\mu \\bar{\\psi} \\gamma^5 \\gamma_\\mu \\psi - \\frac{m_\\text{DM}}{2} \\bar{\\psi} \\psi - \\frac{y_\\text{DM}}{2\\sqrt{2}} s \\bar{\\psi} \\psi \\nonumber \\\\\n & + \\frac{1}{2} m_{Z'}^2 \\, Z'^\\mu Z'_\\mu + \\frac{1}{2} \\partial^\\mu s \\partial_\\mu s + 2 (g_\\text{DM}^A)^2 \\, Z'^\\mu Z'_\\mu (s^2 + 2\\,s\\,w) \n+ \\frac{\\mu_s^2}{2} (s+w)^2 - \\frac{\\lambda_s}{4} (s+w)^4 \\; ,\n\\end{align}\nwith $F'^{\\mu\\nu} = \\partial^\\mu Z'^\\nu - \\partial^\\nu Z'^\\mu$ and\n\\begin{equation} \\label{eq:masses}\n  m_\\text{DM} = \\frac{1}{\\sqrt{2}} \\, y_\\text{DM} \\, w\\,,\\quad\n  m_{Z'} \\approx 2 g_\\text{DM}^A \\, w \\,.  \n\\end{equation}\nIf the SM Higgs is charged under the $U(1)'$ the $Z'$ mass will receive an\nadditional contribution from the SM Higgs vev, see eq.~\\eqref{eq:mZpr}\nbelow. Electroweak precisison data requires that this\n  contribution is small, and therefore we neglect this term in\n  eq.~\\eqref{eq:masses} and for the rest of this subsection.\nNote that without loss of\ngenerality we can choose $w$ and $y_\\text{DM}$ to be real (ensuring\nreal masses) by absorbing complex phases in the field definitions for\n$S$ and $\\psi$.\\footnote{This will no longer be true if we allow for\n  an explicit mass term for $\\psi$. In this case the relative phase\n  between $y_\\text{DM}$ and the mass term is physical (see\n  e.g.~\\cite{LopezHonorez:2012kv}). Here we do not allow for an\n  explicit mass term and we assume that the vev of the singlet is the\n  only source of $U(1)'$ symmetry breaking.}\n\nAs discussed above, the mass of the additional Higgs particle must satisfy\n\\begin{equation}\nm_s < \\frac{\\pi \\, m_{Z'}^2}{(g^A_\\text{DM})^2 \\, m_\\text{DM}}\n\\end{equation}\nin order for perturbative unitarity to be satisfied, which when substituting the masses of the $Z'$ and DM becomes\n\\begin{equation}\nm_s < \\frac{4\\sqrt{2} \\pi w}{y_\\text{DM}} \\;.\n\\end{equation}\nOnce we include such a new particle coupling to the $Z'$, however, there are additional scattering processes such as $s s \\rightarrow s s$ that need to be taken into account when checking perturbative unitarity~\\cite{Basso:2011na}. \nHere we consider the scattering of the states $ss\/\\sqrt{2}$ and $Z'_L Z'_L\/\\sqrt{2}$.\nIn the limit $\\sqrt{s} \\gg m_s \\gg m_{Z'}$, the $J=0$ partial wave of the scattering matrix takes the form~\\cite{Lee:1977eg} \\begin{equation}\n  \\lim_{\\sqrt{s} \\rightarrow \\infty}  \\mathcal{M}^0_{if}\n  = - \\frac{(g^A_\\text{DM})^2 m_s^2}{8 \\pi m_{Z'}^2} \n\\begin{pmatrix}\n3 & 1\\\\\n1 & 3\n\\end{pmatrix} \\; .\n\\end{equation}\nPartial wave unitarity requires the real part of the largest eigenvalue, which corresponds to the eigenvector $(ss + Z'_L Z'_L)\/2$, to be smaller than $1\/2$. We hence obtain the inequality\n\\begin{equation}\n m_s \\leq \\frac{\\sqrt{\\pi} \\, m_{Z'}}{g_\\text{DM}^A} = \\sqrt{4 \\pi} w \\; .\n \\label{eq:perturb}\n\\end{equation}\nThis inequality together with eq.~(\\ref{eq:DMmass}) gives a stronger bound on the Higgs mass than the one obtained in eq.~(\\ref{eq:higgsmass}). In other words, the bound in~(\\ref{eq:higgsmass}) can never actually be saturated in this UV completion. We note that eqs.~\\eqref{eq:DMmass} and \\eqref{eq:perturb} can be unified to\n\\begin{equation}\\label{eq:bound_w}\n  \\sqrt{\\pi} \\, \\frac{m_{Z'}}{g_\\text{DM}^A} \\ge \\text{max}\\left[ m_s ,\n    \\sqrt{2} m_\\text{DM}\\right] \\,.\n\\end{equation}\n\n\\subsection{Implications for the visible sector}\n\nFor the discussion above we only needed to consider the DM part of the Lagrangian. Let us now also look at the coupling to the SM, see e.g.~\\cite{Carena:2004xs}. The interactions between SM states and the new $Z'$ gauge boson can be written as\n\\begin{align}\n  \\mathcal{L}'_\\text{SM} = &\n  \\left[ (D^\\mu H)^\\dagger (-i \\, g' \\, q_H \\, Z'_\\mu \\, H) + \\text{h.c.} \\right] +\n  g'^2 \\, q_H^2 \\, Z'^\\mu Z'_\\mu \\, H^\\dagger H \\nonumber \\\\\n  & - \\sum_{f = q,\\ell,\\nu} g' \\, Z'^\\mu \\, \\left[ q_{f_L} \\, \\bar{f}_L \\gamma_\\mu f_L  + q_{f_R} \\, \\bar{f}_R \\gamma_\\mu f_R \\right] \\; ,\n\\end{align}\nwhere $D^\\mu$ denotes the SM covariant derivative.  We can now\nimmediately write down a list of relations between the different\ncharges $q$ required by gauge invariance of the SM Yukawa\nterms:\\footnote{If right-handed neutrinos exist their charge\n  $q_{\\nu_R}$ would be constrained by $q_H = q_{\\ell_L} -\n  q_{\\nu_R}$ to allow for a Yukawa term with the lepton doublet. In the following\n  we assume that if right-handed neutrinos exist they are heavy enough\n  to decouple from all relevant phenomenology.}\n\\begin{align} \\label{eq:charges}\nq_H = q_{q_L} - q_{u_R} = q_{d_R} - q_{q_L} = q_{e_R} - q_{\\ell_L} \\; .\n\\end{align}\n\nAfter electroweak symmetry breaking, we obtain\n\\begin{align}\n  \\mathcal{L}'_\\text{SM} & =\n  \\frac{1}{2} \\frac{e \\, g' \\, q_{H}}{ s_\\mathrm{W} \\, c_\\mathrm{W}} (h+v)^2 \\, Z^\\mu Z'_\\mu\n  + \\frac{1}{2} g'^2 \\, q_{H}^2 \\, (h+v)^2 \\, Z'^\\mu Z'_\\mu \\nonumber \\\\\n  & \\quad -\n  \\sum_{f = q,l,\\nu} \\frac{1}{2} g' Z'^\\mu \\, \\bar{f}\n  \\left[ (q_{f_R} + q_{f_L}) \\gamma_\\mu + (q_{f_R} - q_{f_L}) \\gamma_\\mu \\gamma^5 \\right] f \\; .\n  \\label{eq:LprSM}\n\\end{align}\nComparing the second line of eq.~\\eqref{eq:LprSM} with\neq.~\\eqref{eq:L_VA} we can read off the vector and axial vector\ncouplings of the fermions:\n\\begin{equation}\\label{eq:g_f_VA}\n  g_f^V = \\frac{1}{2}g'(q_{f_R} + q_{f_L}) \\,,\\quad\n  g_f^A = \\frac{1}{2}g'(q_{f_R} - q_{f_L}) \\,.\n\\end{equation}\n\nIt is well known that a $U(1)'$ under which only SM fields are charged is in general anomalous,\nunless the SM fields have very specific charges (e.g.\\ $U(1)_{B-L}$ is anomaly free). The relevant anomaly coefficients can e.g.\\ be found in \\cite{Carena:2004xs}. The presence of these anomalies implies that the theory has to include new fermions to cancel the anomalies. While these fermions can be vectorlike with respect to the SM, they will then need to be chiral with respect to the $U(1)'$. The mass of the additional fermions is therefore constrained by the breaking scale of the $U(1)'$. In particular, the bound from eq.~\\eqref{eq:DMmass} applies to these fermions as well and therefore they cannot be decoupled from the low-energy theory.\n\nIt is however interesting to note that the anomaly involving two gluons and a $Z'$ is proportional to\n\\begin{equation}\nA_{ggZ'}  =   3 \\left( 2q_{q_L} - q_{u_R} - q_{d_R} \\right) \\; ,\n\\end{equation}\nwhich always vanishes if we restrict the charges based on gauge invariance of the Yukawa couplings (see eq.~\\eqref{eq:charges}). This implies that no new coloured states are needed to cancel the anomalies, greatly reducing the \nsensitivity of colliders to these new states.\\footnote{This conclusion is in disagreement with the observations made in~\\cite{Hooper:2014fda}.} In any case, there are many different possibilities for cancelling the anomalies via new fermions. While the existence of additional fermions will lead to new signatures, a detailed investigation of these is beyond the scope of this work.\n\nIf the SM Higgs is charged under $U(1)'$ ($q_H \\neq 0$) the mass of\nthe $Z'$ receives a contribution from both Higgses:\n\\begin{equation}\\label{eq:mZpr}\n  m_{Z'}^2 = (g'q_H v)^2 + 4(g_\\text{DM}^A w)^2  \\,,\n\\end{equation}\nand we obtain a mass mixing term of the form $\\delta m^2 \\, Z^\\mu Z'_\\mu$ with\n\\begin{align}\n\\delta m^2 =  \\frac{1}{2} \\frac{e \\, g' \\, q_{H}}{ s_\\mathrm{W} \\, c_\\mathrm{W}} v^2 \\; ,\n\\end{align}\nwhere $s_\\mathrm{W} \\,(c_\\mathrm{W})$ is the sine (cosine) of the Weinberg angle.\n\nAs we are going to discuss below, electroweak precision data requires\n$|\\delta m^2| \\ll |m_Z-m_{Z'}|$ (see also\nApp.~\\ref{app:Z}). Using $m_Z = e v \/ (2 s_\\mathrm{W} c_\\mathrm{W})$, $g'q_s =\n-2g_\\text{DM}^A$, and neglecting order one factors this requirement\nimplies either $g'q_H \\ll e$ or $q_s w \\gg v$. In the parameter regions of interest it follows from those conditions\n  that the first term in eq.~\\eqref{eq:mZpr} is small and hence the mass of\n  the $Z'$ is dominated by the vev of the dark Higgs. Taking into account\neqs.~\\eqref{eq:charges} and \\eqref{eq:g_f_VA}, the condition $|\\delta m^2| \\ll |m_Z - m_{Z'}|$ then implies either small axial couplings ($g_f^A \\ll 1$) or $m_{Z'} \\gg m_Z$. We are going to present more quantitative results in the next section and discuss a number of interesting experimental signatures resulting from the new interactions due to eq.~\\eqref{eq:LprSM}.\n\n\nTo conclude this section, it should be noted, that  the Lagrangian introduced above is UV-complete (up to anomalies) and  gauge invariant but does not correspond to the most general realization of this model. In particular the term \n\\begin{equation}\n  \\mathcal{L} \\supset - \\lambda_{hs} (S^*S)(H^\\dagger H) \\,,\n\\end{equation}\nwhich will lead to mixing between the SM Higgs $h$ and the dark Higgs $s$ \ncan be expected to be present at tree level. Furthermore, the term\n\\begin{equation}\\label{eq:kin-mix}\n\\mathcal{L} \\supset -\\frac{1}{2} \\sin \\epsilon F'^{\\mu\\nu} B_{\\mu\\nu} \\; ,\n\\end{equation}\nwhich generates kinetic mixing between the $Z'$ and the $Z$-boson, respects all symmetries of the Lagrangian. It can be argued that $\\epsilon$\nmight vanish at high scales in certain UV-completions, but even in this case\nkinetic mixing is necessarily generated at the one-loop level and can have a substantial impact on EWPT. We will return to these issues and the resulting phenomenology of the model in sections~\\ref{sec:vector} and \\ref{sec:higgsmixing}. For the moment, however, we are going to neglect these additional effects and focus on the impact of $\\delta m \\neq 0$, which necessarily leads to mass mixing between the neutral gauge bosons in the case of non-vanishing axial couplings.\n\n\n\\section{Non-zero axial couplings to SM fermions}\n\\label{sec:axial}\n\nLet us start with the case that axial couplings on the SM side are\nnon-vanishing. An immediate consequence is that the SM Higgs is\ncharged under the $U(1)'$, which follows from eqs.~\\eqref{eq:charges}\nand \\eqref{eq:g_f_VA} for $g_f^A \\neq 0$. Note that these equations also imply that it is inconsistent to set the vectorial couplings for all quarks equal to zero. For example, if we impose that the vectorial couplings of up quarks vanish, i.e. $g_u^V = 0$, eq.~\\eqref{eq:g_f_VA} implies $q_{u_R} = - q_{q_L}$, which using eq.~\\eqref{eq:charges} leads to $g_d^V = 2 g' q_{q_L}$. In the following, whenever $g^A_f \\neq 0$, we always fix $g^V_f = g^A_f$, which corresponds to setting $q_{q_L} = q_{\\ell_L} = 0$.\n\nFurthermore, eq.~\\eqref{eq:charges} requires that $Z'$ couplings are\nflavour universal and leptons couple with the same strength to the\n$Z'$ as quarks. This conclusion could potentially be modified by considering an extended Higgs sector, e.g.\\ a two-Higgs-doublet  model. Here we focus on the simplest case where a single Higgs doublet generates all SM fermion masses. This implies that the leading search channel at the\nLHC will be dilepton resonances, which give severe constraints.  In principle\nalso electron-positron colliders can constrain this scenario\nefficiently.  Limits on a $Z'$ lighter than 209~GeV derived from LEP\ndata imply $g \\lesssim 10^{-2}$ \\cite{Agashe:2014kda} (see also\n\\cite{Appelquist:2002mw,LEP:2003aa}). We do not include LEP constraints here since other constraints will turn out to be at least equally strong.\n\nFor general couplings, the partial decay width of the mediator into SM fermions is given by\n\\begin{equation}\n\\Gamma(Z'\\rightarrow f\\bar{f}) = \\frac{m_{Z'} N_c}{12\\pi} \n \\sqrt{1-\\frac{4 m_f^2}{m_{Z'}^2}} \\, \\left[(g^{V}_{f})^2+(g^{A}_{f})^2 + \\frac{m_f^2}{m_{Z'}^2}\\left(2 (g^{V}_{f})^2 - 4 (g^{A}_{f})^2\\right)  \\right]  \\; ,\n\\end{equation}\nwhere $N_c = 3 \\;(1)$ for quarks (leptons). The decay width into DM pairs is\n\\begin{equation}\n\\Gamma(Z'\\rightarrow \\psi\\psi) = \\frac{m_{Z'}}{24\\pi} \n (g^{A}_\\text{DM})^2 \\left(1-\\frac{4 m_\\text{DM}^2}{m_{Z'}^2}\\right)^{3\/2}  \\; .\n\\end{equation}\nConsequently, for $m_\\text{DM} \\ll m_{Z'}$ and $g^A_{\\ell} = g^A_{q} \\ll g^A_\\text{DM}$ the branching ratio into $\\ell = e,\\,\\mu$ is given by $\\text{BR}(R\\rightarrow \\ell \\ell) \\approx 8 (g^A_{\\ell})^2 \/ (g^A_\\text{DM})^2$. For $m_\\text{DM} > m_{Z'} \/ 2$, on the other hand, the branching ratio is given by $\\text{BR}(R\\rightarrow \\ell \\ell) \\approx 0.08\\text{--}0.10$ depending on the ratio $m_{Z'} \/ m_t$.\n\nWe implement the latest ATLAS dilepton search \\cite{Aad:2014cka},\ncomplemented by a Tevatron dilepton search \\cite{Jaffre:2009dg}\n  for the low mass region, and show the resulting bounds in\nfigure~\\ref{fig:axialaxial}. One can see that the bounds strongly\ndepend on the assumed branching ratio of the $Z'$. As a conservative\nlimiting case we show $g_\\text{DM}^A=1$ and $m_\\text{DM}=100$ GeV,\nwhich leads to a rather large branching fraction into DM and hence\nsuppressed bounds. The second benchmark, $m_\\text{DM}=500$~GeV,\nallows for $Z'$ decays to DM only for rather heavy $Z'$s, leading to correspondingly\nmore restrictive dilepton constraints. Overall the bounds turn out to be very stringent\nand the $Z'$ coupling to leptons and quarks needs to be significantly\nsmaller than unity for $100 \\; \\text{GeV} \\lesssim m_{Z'} \\lesssim 4 \\;\n\\text{TeV}$, so that dijet constraints are basically irrelevant in\nthis case given that $g_q=g_l$.\n\n\\begin{figure}[tb]\n\\centering\n\\includegraphics[height=0.3\\textheight]{axialaxial_100_label.pdf}\\qquad\n\\includegraphics[height=0.3\\textheight]{axialaxial_500.pdf}\n\\caption{Axial+Vector(SM)--Axial(DM): \n  Parameter space forbidden by constraints from dilepton resonance searches from ATLAS (light green, dashed) and Tevatron (dark green, dashed), electroweak precision observables (blue, dotted) and DM overproduction (red, solid) in the $m_{Z'}-g_{q,l}^A$ parameter plane for two exemplary DM masses 100 GeV (left) and 500~GeV (right). In the shaded region to the left of the vertical grey line the $Z'$-mass violates the bound from perturbative unitarity from eq.~\\eqref{eq:DMmass}.}\n\\label{fig:axialaxial}\n\\end{figure}\n\nThe fact that the SM Higgs is charged also implies potentially large corrections to electroweak precision observables.\nIn particular we obtain the non-diagonal mass term $\\delta m^2 \\,\nZ^\\mu Z'_\\mu$ leading to mass mixing between the SM $Z$ and the new\n$Z'$.  The diagonalisation required to obtain mass eigenstates is discussed in the appendix. \nIn the absence of kinetic mixing between the $U(1)'$ and the SM $U(1)$ gauge bosons ($\\epsilon = 0$), the resulting effects can be expressed in\nterms of the mixing parameter $\\xi = \\delta m^2 \/(m_Z^2-m_{Z'}^2)$ (see eq.~\\eqref{eq:xi_def} in the appendix with $\\epsilon=0$). In particular, we can calculate the constraints from electroweak\nprecision measurements, which are encoded in the $S$ and $T$\nparameters. To quadratic order in $\\xi$ we find~\\cite{Frandsen:2011cg}\n\\begin{align}\n  \\alpha S = & - 4 c_\\mathrm{W}^2 s_\\mathrm{W}^2 \\xi^2 \n  \\; , \\nonumber\\\\\n  \\alpha T = & \\xi^2\\left(\\frac{m_{Z'}^2}{m_{Z}^2}-2\\right) \\; , \\label{eq:ST}\n\\end{align}\nwhere $\\alpha=e^2\/4\\pi$. The resulting bounds are shown in figure~\\ref{fig:axialaxial}.  To infer\nour bounds we use the $90\\%$ CL limit on the $S$ and $T$ parameters as\ngiven in \\cite{Agashe:2014kda}.\n\nNote that the bound from electroweak precision data is\n  completely independent of the $Z'$ couplings to the DM as well as\n  the DM mass. Hence, the same bound would apply also in the\n  case of Dirac DM with vector couplings to the $Z'$. Note however\n  that since vectorial couplings to quarks are necessarily non-zero if $g_q^A \\neq 0$ (see\n    eqs.~\\eqref{eq:charges}, \\eqref{eq:g_f_VA}), there will be very\n  stringent bounds from direct detection experiments on any model with\n  $g^V_\\text{DM} \\neq 0$ due to unsuppressed spin-independent\n    scattering. For Majorana DM, the vectorial coupling always\n  vanishes and the constraints from direct detection are much weaker.\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[height=0.3\\textheight]{aamm1_monojet.pdf}\\qquad\n\\includegraphics[height=0.3\\textheight]{aamm025_label_monojet.pdf}\n\\includegraphics[height=0.3\\textheight]{aamm01_monojet.pdf}\\qquad\n\\includegraphics[height=0.3\\textheight]{aamm0025_monojet.pdf}\n\\caption{Axial+Vector(SM)--Axial(DM): Parameter space forbidden by constraints from dilepton resonance searches (green, dashed) and electroweak precision observables (blue, dotted) in the $m_\\text{DM}-m_{Z'}$ plane for four different sets of couplings. We also show the regions excluded by DM overproduction (red), direct detection bounds (purple, dot-dashed) and the parameter space where perturbative unitarity is violated (grey). For the relic density calculation we have assumed that the mass of the hidden sector Higgs saturates the unitarity bound.}\n\\label{fig:axialaxial2}\n\\end{figure}\n\nIn figure~\\ref{fig:axialaxial2} we show the constraints from\nelectroweak precision data as well as LHC dilepton searches in the\n$m_\\text{DM}-m_{Z'}$ plane for different values of the axial vector\ncoupling to fermions. In the lower right corner of the plots (grey area) the\nperturbative unitarity condition from eq.~\\eqref{eq:DMmass} is\nviolated. We also show the region excluded by direct detection\nsearches (dark region in the lower left corners). For the axial-axial\ncouplings DM-nucleus scattering proceeds through spin-dependent\ninteractions, with a scattering cross section given by\n\\begin{equation}\n\\sigma^\\text{SD}_N = \\frac{3 \\, a_N^2 \\, (g^A_\\text{DM})^2 \\, (g^A_q)^2}{\\pi} \\frac{\\mu^2}{m_{Z'}^4} \\; ,\n\\end{equation}\nwhere $\\mu$ is the DM-nucleon reduced mass, $N = p,\\,n$ and $a_p = -a_n = 1.18$ is the effective nucleon coupling~\\cite{Agashe:2014kda}. \nThis is the dominant contribution in this case as the vector-axial coupling combination is even further suppressed.\nIn the plots we show the bound on the spin-dependent scattering cross section that can be calculated from the published LUX results~\\cite{Akerib:2013tjd}, following the method described in~\\cite{Feldstein:2014ufa}. We observe\nthat in this case direct detection is never competitive with other\nconstraints.\n\nThe red solid curves in Figs.~\\ref{fig:axialaxial} and\n  \\ref{fig:axialaxial2} show the parameter values that lead to the correct relic abundance.  In\norder to calculate the relic abundance we have implemented the model\nin micrOMEGAs\\_v4~\\cite{Belanger:2014vza}, assuming that the mass of the Higgs\nsinglet saturates the unitarity bound and setting the mixing with the\nSM Higgs to zero.\\footnote{Note that since $\\xi$ can be large in some regions of parameter space, it is not a good approximation to expand the annihilation cross section in $\\xi$. We therefore use the exact expression for the mixing between the neutral gauge bosons in terms of $\\epsilon$ and $\\delta m^2$ as derived in the appendix.}\nIn the regions shaded in red (to the right\/above the solid curve) there is overproduction of DM. In\nthis region additional annihilation channels are required to avoid\noverclosure of the Universe, since the interactions provided by the\n$Z'$ are insufficient to keep DM in thermal equilibrium long\nenough. Such additional interactions could be obtained for instance from\nthe scalar mixing discussed in section~\\ref{sec:higgsmixing}. Conversely, to the left\/below the red\nsolid curve the model does not provide all of the DM matter in the\nUniverse, since the annihilation rate is too high. \n\nLet us briefly discuss the various features that can be observed in the relic abundance curve. First there is a significant decrease of the predicted abundance as the DM mass crosses the top-quark threshold, $m_\\chi > m_t$, resulting from the fact that the $s$-wave contribution to the annihilation cross section is helicity suppressed and hence annihilation into top-quarks becomes the dominant annihilation channel as soon as it is kinematically allowed. The second feature occurs at $m_\\chi \\sim m_{Z'}$ and reflects the resonant enhancement of the annihilation process $\\chi \\chi \\rightarrow q \\bar{q}$ as the mediator can be produced on-shell. A third visible feature is a very narrow resonance at $2 \\, m_\\chi \\sim m_s = \\sqrt{\\pi} \\,  m_{Z'} \/ g^A_\\text{DM}$ due to a resonant enhancement of the process $\\chi \\chi \\rightarrow s \\rightarrow Z' Z$. The position and magnitude of this effect depends on the mass of the dark Higgs, which has been (arbitrarily) fixed to saturate the unitarity bound. However, even for this extreme choice, it turns out to give a non-negligible contribution to the relic abundance. For $m_\\chi > m_{Z'}$ direct annihilation into two mediators becomes possible, leading to a significant decrease of the predicted relic abundance. Finally, the fact that the relic abundance curve in figure~\\ref{fig:axialaxial2} touches the unitarity bound for high DM masses reflects the well-known unitarity bound on the mass of a thermally produced DM particle \\cite{Griest:1989wd}.\n\nAll in all we find the case with non-vanishing axial couplings on the\nSM side to be strongly constrained by dilepton searches as well as\nelectroweak precision observables, implying that in a UV complete\nmodel this is where a signal should first be seen. For comparison, we show recent bounds from LUX as well as from the CMS monojet search~\\cite{Khachatryan:2014rra}.\\footnote{To interpret the CMS results in the context of our model, we implement our model in \\texttt{Feynrules~v2}~\\cite{Alloul:2013bka} and simulate the monojet signal with \\texttt{MadGraph~v5}~\\cite{Alwall:2014hca} and \\texttt{Pythia~v6}~\\cite{Sjostrand:2006za}. Imposing a cut on the missing transverse energy of $\\slashed{E}_T > 450\\:\\text{GeV}$, we exclude all parameter points that predict a contribution to the monojet cross section larger than $7.8\\:\\text{fb}$. We find good agreement between this procedure and an analogous implementation using \\texttt{CalcHEP v3}~\\cite{Belyaev:2012qa} and \\texttt{DELPHES v3}~\\cite{deFavereau:2013fsa}.}\nWe find that these searches, as well as searches for dijet resonances, are not competitive. Note that in figure~\\ref{fig:axialaxial2} we assume $g_\\text{DM}^A = 1$. We comment on smaller couplings on the DM side later in the context of figure~\\ref{fig:smallgA}. Let us now look at the case where axial couplings to\nquarks are taken to be zero, which will turn out to be somewhat less constrained.\n\n\n\\section{Purely vectorial couplings to SM fermions}\n\\label{sec:vector}\n\nLet us now consider the case with purely vectorial couplings on the SM\nside, i.e.\\ $g^A_q = g^A_\\ell = g^A_\\nu = 0$. In this case the SM\nHiggs does not carry a $U(1)'$ charge and therefore the charges of\nquarks and leptons are independent. In particular, it is conceivable\nthat $g^V_q \\gg g^V_\\ell$, so that constraints from dilepton resonance\nsearches can be evaded. Also there can in principle be a flavour dependence\n  of the $Z'$ couplings to quarks. Nevertheless, to avoid large flavour-changing neutral currents, we will always assume the\n  same coupling for all quark families in what follows~\\cite{Abdallah:2015ter}. Finally, in contrast to the case discussed above,\ntree-level $Z-Z'$ mass mixing is absent. It therefore seems plausible that\n the $Z'$ is the only state coupling to both the visible\n and the dark sector.\n Nevertheless, as mentioned above, potentially important effects in this scenario can be kinetic mixing of the $U(1)$ gauge bosons as well as effects induced by the dark Higgs, which we are going to discuss below. Let us just mention that all these effects will also be present in the scenario discussed in the previous section.  They are, however, typically less important than the effects of tree-level $Z$-$Z'$ mixing.\n\nWe first consider the effects of kinetic mixing between the $Z'$ and the SM hypercharge gauge boson $B$:\n\\begin{equation}\n\\mathcal{L} \\supset -\\frac{1}{2} \\sin \\epsilon \\, F'^{\\mu\\nu} B_{\\mu\\nu} \\; ,\n\\end{equation}\nwhere $F'^{\\mu\\nu} = \\partial^\\mu Z'^\\nu - \\partial^\\nu Z'^\\mu$ and $B^{\\mu\\nu} = \\partial^\\mu B^\\nu - \\partial^\\nu B^\\mu$. A non-zero value of $\\epsilon$ leads to mixing between the $Z'$ and the neutral gauge bosons of the SM (see App.~\\ref{app:Z}). As in the case of mass mixing discussed above, there are strong constraints on kinetic mixing from searches for dilepton resonances and electroweak precision observables. \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[height=0.3\\textheight]{vectoraxial_100.pdf}\\qquad\n\\includegraphics[height=0.3\\textheight]{vectoraxialmm.pdf}\n\\caption{Vector(SM)--Axial(DM): Parameter space forbidden by\n  constraints from ATLAS and Tevatron dileptons (green, dashed),\n  electroweak precision observables (blue, dotted) and relic DM\n  overproduction (red, solid) in the $m_{Z'}$-$\\epsilon$ parameter\n  plane (left) and the $m_\\text{DM}$-$m_{Z'}$ parameter plane\n  (right). In both panels we\n    show the parameter space where perturbative unitarity is violated\n    (grey). For the relic density calculation we have assumed that\n  the mass of the hidden sector Higgs saturates the unitarity bound.}\n\\label{fig:va}\n\\end{figure}\n\nThe dilepton couplings induced via the kinetic mixing parameter $\\epsilon$ can be inferred from the mixing matrices and are given in the appendix, cf.~eq.~\\eqref{eq:dilepton}. The $S$ and $T$ parameters are given by\n\\begin{align}\n  \\alpha S = & 4 c_\\mathrm{W}^2 s_\\mathrm{W} \\xi (\\epsilon - s_\\mathrm{W} \\xi) \n  \\; , \\nonumber\\\\\n  \\alpha T = & \\xi^2\\left(\\frac{m_{Z'}^2}{m_{Z}^2}-2\\right)+2\n  s_\\mathrm{W} \\xi \\epsilon \\; , \\label{eq:ST2}\n\\end{align}\nwhere for $\\delta m^2 = 0$ the mixing parameter $\\xi$ is given by $\\xi = m_{Z}^2  s_\\mathrm{W} \\epsilon \/ (m_{Z}^2 - m_{Z'}^2)$ at leading order. If $\\epsilon$ is sizeable, i.e.\\ if mixing is present at tree level, the resulting bounds can be quite strong. This expectation is confirmed in figure~\\ref{fig:va}. Note that the relic density curves shown in figure~\\ref{fig:va} are basically independent of $\\epsilon$, because freeze-out is dominated by direct $Z'$ exchange for the adopted choice of couplings.\n\nWhile tree-level mixing is tightly constrained, it is reasonable to expect that $\\epsilon$ vanishes at high scales, for example if both $U(1)$s originate from the same underlying non-Abelian gauge group, as in Grand Unified Theories. Since quarks carry charge under both $U(1)'$ and $U(1)_Y$, quark loops will still induce kinetic mixing at lower scales~\\cite{Holdom:1985ag}, but the magnitude of $\\epsilon$ can be much smaller than what we considered above. The precise magnitude of the kinetic mixing depends on the underlying theory, but if we assume that $\\epsilon(\\Lambda) = 0$ at some scale $\\Lambda \\gg 1\\:\\text{TeV}$, the kinetic mixing at a lower scale $\\mu > m_t$ will be given by~\\cite{Carone:1995pu}\n\\begin{equation}\n\\epsilon(\\mu) = \\frac{e \\, g^V_q}{2 \\pi^2 \\, \\cos \\theta_\\text{W}} \\log \\frac{\\Lambda}{\\mu} \\simeq   0.02 \\, g^V_q \\, \\log \\frac{\\Lambda}{\\mu} \\; .\n\\end{equation}\nWe can use this equation (setting $\\mu = m_{Z'}$) to translate the\nbounds from figure~\\ref{fig:va} into constraints on\n$g^V_q$. The results of such an analysis are shown in\n  figure~\\ref{fig:va_loop} assuming $\\Lambda = 10$~TeV.  As can be seen\nin figure~\\ref{fig:va_loop} (left), searches for dilepton\nresonances give again stringent constraints, implying $g_q^V <\n0.1$ for $m_{Z'} = 200\\:\\text{GeV}$ and $g_q^V < 1$ for $m_{Z'} = 1\\:\\text{TeV}$.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[height=0.3\\textheight]{vectoraxial_loop_100.pdf}\\qquad\n\\includegraphics[height=0.3\\textheight]{vectoraxialmm_loop.pdf}\n\\caption{Vector(SM)--Axial(DM): Parameter space forbidden by\n  constraints from ATLAS and Tevatron dileptons (green, dashed) and\n  electroweak precision observables (blue, dotted) in the\n  $m_{Z'}$-$g^V_q$ parameter plane (left) and the\n  $m_\\text{DM}$-$m_{Z'}$ parameter plane (right), assuming that\n  $\\epsilon=0$ at $\\Lambda = 10\\:\\text{TeV}$ so that kinetic mixing is\n  only induced at the one-loop level.  In the right panel we\n    show also the region excluded by LHC monojet (orange, dashed) and\n    dijet (violet, dot-dashed) searches due to tree-level $Z'$\n    exchange for the adopted coupling choice. In both panels we show\n    the parameter space where perturbative unitarity is violated\n    (grey). For the relic density calculation we have assumed that the\n  mass of the hidden sector Higgs saturates the unitarity bound.  }\n\\label{fig:va_loop}\n\\end{figure}\n\nIn the right panel of figure~\\ref{fig:va_loop} we also show the constraints coming from LHC searches for monojets (i.e.\\ jets in association with large amounts of missing transverse energy) and for dijet resonances, adopted from ref.~\\cite{Chala:2015ama}.\\footnote{Note that as long as the mediator is produced on-shell, the production cross section is proportional to $(g^V_q)^2 + (g^A_q)^2$ and hence it is a good approximation to apply the bounds obtained for $g^V_q = 0$ and $g^A_q \\neq 0$ also to the case $g^A_q = 0$ and $g^V_q \\neq 0$. However, ref.~\\cite{Chala:2015ama} assumes a Dirac DM particle, while we focus on Majorana DM. As a result, the invisible branching fraction will be somewhat smaller in our case and bounds from dijet resonance searches will be strengthened. The dijet bounds we show are therefore conservative.}\nThese limits are independent of the kinetic mixing $\\epsilon$ since they originate from the tree-level\n  $Z'$ exchange and probe larger values of $m_{Z'}$ and smaller values of $m_\\text{DM}$. \n  Nevertheless, dilepton resonance searches and EWPT give relevant constraints for small $m_{Z'}$ and large $m_\\text{DM}$, which are difficult to probe with monojet and dijet searches.\n\nWe conclude from figure~\\ref{fig:va_loop} that the combination of constraints due to loop-induced kinetic mixing and bounds from LHC DM searches leave only a small region in parameter space (a small strip close to the $Z'$ resonance), where DM overabundance is avoided. While this result depends somewhat on our choice $\\Lambda = 10$~TeV and $\\mu = m_{Z'}$, it is only logarithmically sensitive to these choices.\n\nIt is worth emphasising that the unitarity constraints shown in figures~\\ref{fig:axialaxial2}--~\\ref{fig:va_loop} depend sensitively on the choice of $g^A_\\text{DM}$ (cf.~figure~\\ref{fig:ubound}). We therefore show in figure~\\ref{fig:smallgA} how these constraints change if we take $g^A_\\text{DM} = 0.1$ rather than $g^A_\\text{DM} = 1$. In this case both $m_\\text{DM}$ and $m_s$ can be much larger than $m_{Z'}$. At the same time, however, relic density constraints become significantly more severe, excluding almost the entire parameter space with $m_\\text{DM} < m_{Z'}$ (apart from the resonance region $m_\\text{DM} \\sim m_{Z'}$). Even for $m_\\text{DM} > m_{Z'}$ is it difficult to reproduce the observed relic abundance, because the annihilation channel $\\chi \\chi \\rightarrow Z' Z'$ is significantly suppressed due to the smallness of $g^A_\\text{DM}$. It only becomes relevant close to the unitarity bound, where also the process $\\chi \\chi \\rightarrow Z Z'$ mediated by the dark Higgs gives a sizeable contribution. While in the case with axial couplings on the SM side the region compatible with thermal freeze-out becomes fully excluded by dilepton resonance searches, the case with vector couplings on the SM side is very difficult to probe at colliders and direct detection, leading to a small allowed parameter region close to the bound from perturbative unitarity.\n\n\\begin{figure}[tb]\n\\centering\n\\includegraphics[height=0.3\\textheight]{aamm_gAx01_gAq01_label.pdf}\\qquad\n\\includegraphics[height=0.3\\textheight]{vectoraxialmm_loop_gax01_gvq01.pdf}\n\\caption{Constraints for small DM couplings ($g^A_\\text{DM} = 0.1$). The left panel considers the case Axial+Vector(SM)--Axial(DM) and should be compared to figure~\\ref{fig:axialaxial2}. The right panel considers the case Vector(SM)--Axial(DM) with loop-induced kinetic mixing, assuming that $\\epsilon=0$ at $\\Lambda = 10\\:\\text{TeV}$ (cf.\\ figure~\\ref{fig:va_loop}).}\n\\label{fig:smallgA}\n\\end{figure}\n\n\\bigskip\n\nIn addition to the effects of kinetic mixing, we have shown above that for $g^A_\\text{DM} \\neq 0$ the dark sector necessarily contains a new Higgs particle. The presence of this additional Higgs can change the phenomenology of the model in two important ways. First, loop-induced couplings of the dark Higgs to SM states may give an important contribution to direct detection signals. And second, there may be mixing between the SM Higgs and the dark Higgs, leading to pertinent modifications of the properties of the SM Higgs as well as opening another portal for DM-SM interactions. We will discuss loop-induced couplings in this section and then return to a detailed study of the Higgs potential in the next section.\n\nFor $g^A_q = g^V_\\text{DM} = 0$, scattering in direct detection experiments is momentum-suppressed in the non-relativistic limit and the corresponding event rates are very small. This conclusion may change if loop corrections induce unsuppressed scattering~\\cite{Haisch:2013uaa}. Indeed, at the one-loop level the dark Higgs can couple to quarks and can therefore mediate unsuppressed spin-independent interactions. The resulting interaction can be written as $\\mathcal{L} \\propto \\sum_q \\, m_q \\, s \\, \\bar{q} q$. After integrating out heavy-quark loops as well as the dark Higgs this interaction leads to an effective coupling between DM and nucleons of the form $\\mathcal{L} \\propto f_N \\, m_N \\, m_\\text{DM} \\, \\bar{N}N \\, \\bar{\\psi}\\psi$, where $m_N$ is the nucleon mass, $N = p, n$ and $f_N \\approx 0.3$ is the effective nucleon coupling.\n\nIn the non-relativistic limit, the diagram in the left of figure~\\ref{fig:SI} induces the effective interaction\n\\begin{equation}\n\\mathcal{L}_\\text{eff} \\supset \\frac{(g^A_\\text{DM})^2 \\, (g^V_q)^2}{\\pi^2} \\frac{1}{m_s^2 \\, m_{Z'}^2} \\times m_\\text{DM} \\, f_N \\, m_N \\, \\bar{N} N \\, \\bar{\\psi} \\psi \\; .\n\\end{equation}\nThe corresponding spin-independent scattering cross section is given by\n\\begin{equation}\n\\sigma_N^\\text{SI} = \\frac{m_\\text{DM}^2 \\, f_N^2 \\, m_N^2 \\, \\mu^2}{\\pi} \\frac{(g^A_\\text{DM})^4 \\, (g^V_q)^4}{\\pi^4} \\frac{1}{m_s^4 \\, m_{Z'}^4} \\; ,\n\\end{equation}\nwhere $\\mu$ is the DM-nucleon reduced mass. For masses of order $300\\:\\text{GeV}$ and couplings of order unity this expression yields $\\sigma_N^\\text{SI} \\sim 10^{-46} \\:\\text{cm}^2$, which is below the current bounds from LUX but well within the potential sensitivity of XENON1T.\n\n\\begin{figure}[tb]\n\\centering\n\\includegraphics[width=0.75\\textwidth,clip,trim=25 670 100 15]{diagrams.pdf}\n\\caption{Contributions to spin-independent scattering.}\n\\label{fig:SI}\n\\end{figure}\n\nWe note that there are two additional diagrams (shown in the second and third panel of figure~\\ref{fig:SI}) that also lead to unsuppressed spin-independent scattering of DM particles~\\cite{Haisch:2013uaa}. For $m_\\text{DM} \\gg m_N$, the resulting contribution is given by\n\\begin{equation}\n\\mathcal{L}_\\text{eff} \\supset \\frac{(g^A_\\text{DM})^2 \\, (g^V_q)^2}{4 \\pi^2} \\frac{m_\\text{DM}^2 - m_{Z'}^2 + m_{Z'}^2 \\log (m_{Z'}^2 \/ m_\\text{DM}^2)}{m_{Z'}^2 (m_{Z'}^2 - m_\\text{DM}^2)^2} \\times m_\\text{DM} \\, f_N \\, m_N \\, \\bar{N} N \\, \\bar{\\psi} \\psi \\; .\n\\end{equation}\nIf there is no large hierarchy between $m_\\text{DM}$, $m_{Z'}$ and $m_s$, this contribution is of comparable magnitude to the one from dark Higgs exchange and interference effects can be important. Moreover, there may be a relevant contribution from loop-induced spin-dependent scattering. We leave a detailed study of these effects to future work. \n\n\n\\section{Mixing between the two Higgs bosons}\n\\label{sec:higgsmixing}\n\nIn addition to the loop-induced couplings of the dark Higgs to SM fermions discussed in the previous\nsection, such couplings can also arise at tree-level from mixing. In fact, an important\nimplication of the presence of a second Higgs field is that the two\nHiggs fields will in general mix, thus modifying the properties of the\nmostly SM-like Higgs. Furthermore, the mixing opens up the so-called Higgs portal between the\nDM and SM particles, leading to a much richer DM phenomenology than in\nthe case of DM-SM interactions only via the vector mediator. \n\n\nThe mixing between the scalars is due to an additional term in the scalar potential:\n\\begin{equation}\n  V(S, H) \\supset \\lambda_{hs} (S^*S)(H^\\dagger H) \\,.\n\\end{equation}\nThe coupling $\\lambda_{hs}$ is a free parameter, independent of the vector mediator. For non-zero $\\lambda_{hs}$, the scalar mass eigenstates $H_{1,2}$ are given by\n\\begin{align}\n H_1 = s \\sin \\theta + h \\cos \\theta \\nonumber \\\\\n H_2 = s \\cos \\theta - h \\sin \\theta \n\\end{align}\nwhere, as shown in App.~\\ref{app:scalar},\n\\begin{equation}\n \\theta \\approx - \\frac{\\lambda_{hs} \\, v \\, w}{m_s^2 - m_h^2} + \\mathcal{O}(\\lambda_{hs}^3) \\; .\n\\end{equation}\nWe emphasise  that perturbative unitarity implies that $m_s$ cannot be arbitrarily large (for given $m_{Z'}$ and $g^A_\\text{DM}$) and hence it is impossible to completely decouple the dark Higgs.\n\nThe resulting Higgs mixing leads to three important consequences. First, the (mostly) dark Higgs obtains couplings to SM particles, enabling us to produce it at hadron colliders and to search for its decay products (or monojet signals). Second, the properties of the (mostly) SM-like Higgs, in particular its total production cross section and potentially also its branching ratios, are modified. And finally, both Higgs particles can mediate interactions between DM and nuclei, leading to potentially observable signals at direct detection experiments.\n\nHiggs portal DM has been extensively studied, see for instance\n\\cite{Djouadi:2011aa, Lebedev:2012zw, LopezHonorez:2012kv,\n  Baek:2012uj,  Walker:2013hka, Esch:2013rta, Freitas:2015hsa} for an incomplete\nselection of references.  A full analysis of Higgs mixing effects is beyond\nthe scope of the present paper. Nevertheless, to illustrate the\nmagnitude of potential effects, let us consider the induced coupling\nof the SM-like Higgs $H_1 \\approx h$ to DM particles\n\\begin{equation}\n\\mathcal{L} \\supset - \\frac{m_\\text{DM} \\, \\sin \\theta}{2 \\, w} h \\, \\bar{\\psi} \\psi \\simeq \\frac{m_\\text{DM} \\, \\lambda_{hs} \\, v}{2 (m_s^2 - m_h^2)} h \\, \\bar{\\psi} \\psi \\; . \n\\end{equation}\nFor small $\\lambda_{hs}$, the resulting direct detection cross section is given by~\\cite{Djouadi:2011aa}\n\\begin{equation}\n \\sigma_N^\\text{SI} \\simeq \\frac{\\mu^2}{\\pi \\, m_h^4} \\frac{f_N^2 \\, m_N^2 \\, m_\\text{DM}^2\\,\\lambda_{hs}^2}{(m_s^2 - m_h^2)^2} \\; ,\n\\end{equation}\nwhere we can neglect an additional contribution from the exchange of a dark Higgs provided $m_s^4 \\gg m_h^4$.\nThe parameter regions excluded by the LUX results~\\cite{Akerib:2013tjd} are shown in figure~\\ref{fig:higgsmixing} (green regions).\n\n\\begin{figure}[tb]\n\\centering\n\\includegraphics[height=0.3\\textheight]{HiggsPlot1.pdf}\\qquad\\includegraphics[height=0.3\\textheight]{HiggsPlot2.pdf}\n\\caption{Constraints on $m_s$ and $\\lambda_{hs}$ from bounds on the Higgs invisible branching ratio (blue, dotted) and from bounds on the spin-independent DM-nucleon scattering cross section (green, dashed). In the grey parameter region unitarity constraints are in conflict with the stability of the potential.}\n\\label{fig:higgsmixing}\n\\end{figure}\n\nWe note that (in the linear approximation) the direct detection cross section is independent of $w$ and does therefore not depend on $m_{Z'}$ or $g^A_\\text{DM}$. Nevertheless $w$ is not arbitrary, because unitarity gives a lower bound $\\sqrt{4\\pi} w > \\text{max}\\left[\\sqrt{2} m_\\text{DM}, m_s \\right]$. At the same time, stability of the Higgs potential requires $4 \\lambda_s \\, \\lambda_h > \\lambda_{hs}$. These two inequalities can only be satisfied at the same time if\n\\begin{equation}\nm_\\text{DM} < \\frac{\\sqrt{2 \\pi} \\, m_s \\, m_h}{v} \\; .\n\\end{equation}\nIn figure~\\ref{fig:higgsmixing}, we show the parameter region where unitarity and stability are in conflict in grey.\n\nFinally, if the DM mass is sufficiently small, the SM-like Higgs can decay into pairs of DM particles, with a partial width given by~\\cite{Djouadi:2011aa}\n\\begin{equation}\n \\Gamma^\\text{inv} = \\frac{1}{8 \\pi} \\frac{m_\\text{DM}^2 \\, \\lambda_{hs}^2}{(m_s^2 - m_h^2)^2} v^2 m_h \\left(1-\\frac{4 \\, m_\\text{DM}^2}{m_h^2}\\right)^{3\/2} \\; .\n\\end{equation}\nThe invisible branching fraction is tightly constrained by LHC measurements: $\\text{BR}(h\\rightarrow \\text{inv}) < 0.27$~\\cite{Khachatryan:2014jba}. Furthermore, a combined fit from ATLAS and CMS yields $\\mu = 1.09^{+0.11}_{-0.10}$ for the total Higgs signal strength~\\cite{ATLAS-CONF-2015-044}, which can be used to deduce $\\text{BR}(h\\rightarrow \\text{inv}) < 0.11$ at 95\\% CL. The resulting constraints, compared to the ones on $\\sigma_N^\\text{SI}$ from LUX, are shown in figure~\\ref{fig:higgsmixing} (blue regions).\n\nThe crucial observation is that the necessary presence of a dark Higgs will in general induce additional signatures and therefore lead to new ways to constrain models with a $Z'$ mediator using both direct detection experiments and Higgs measurements. However, since $\\lambda_{hs}$ and $m_s$ are effectively free parameters, it is difficult to directly compare the constraints shown in figure~\\ref{fig:higgsmixing} to the ones obtained from monojet and dijet searches at the LHC. Nevertheless, we can conservatively estimate the relevance of these effects by fixing the dark Higgs mass $m_s$ to the largest value consistent with perturbative unitarity. \n\n\\begin{figure}[tb]\n\\centering\n\\includegraphics[height=0.3\\textheight]{HiggsPlot_01_label.pdf}\\qquad\\includegraphics[height=0.3\\textheight]{HiggsPlot_02.pdf}\n\\caption{Constraints in the $m_{Z'}$-$m_\\text{DM}$ plane for different values of $\\lambda_{hs}$, taking the mass of the hidden sector Higgs to saturate the unitarity bound. The blue (dotted) region is excluded by bounds on the Higgs invisible branching ratio and the green (dashed) region is in conflict with bounds on the spin-independent DM-nucleon scattering cross section. The orange (dashed) region shows constraints from the CMS monojet search, the purple (dot-dashed) region is excluded by a combination of dijet searches from the LHC, Tevatron and UA2 (adopted from ref.~\\cite{Chala:2015ama}). In the grey parameter region unitarity constraints are in conflict with the stability of the potential, the red region corresponds to DM overproduction. Note the change of scale in these figures.}\n\\label{fig:higgsmixing2}\n\\end{figure}\n\nThe resulting constraints in the conventional $m_{Z'}$-$m_\\text{DM}$ parameter plane with fixed couplings are shown in figure~\\ref{fig:higgsmixing2}. For comparison we show the constraints from the CMS monojet search~\\cite{Khachatryan:2014rra} and a combination of dijet searches from the LHC, Tevatron and UA2 (adopted from ref.~\\cite{Chala:2015ama}). We find that the additional constraints due to Higgs mixing provide valuable complementary information in the parameter region with small $m_{Z'}$ and large $m_\\text{DM}$, which is difficult to probe with monojet or dijet searches. Note that for $m_{Z'} > 2\\:\\text{TeV}$ (not shown in figure~\\ref{fig:higgsmixing2}) there is still an allowed parameter region if either $m_\\text{DM} \\approx m_{Z'} \/ 2$ or $m_\\text{DM} > m_{Z'}$ (cf.\\ figure 6). Furthermore, it is worth emphasising that for smaller values of $m_s$ significantly stronger constraints are expected from Higgs mixing. Moreover, these constraints are independent of the SM couplings of the $Z'$ and will therefore become increasingly important in the case of small $g_q$.\n\n\n\\section{Discussion and Outlook}\n\\label{sec:discussion}\n\nIn this paper we have studied the so-called simplified model approach to DM used to parametrise the interactions of a DM\n  particle with the SM via one or several new mediators. It should be clear that simplified models are\n  considered merely as an effective description, used as a tool to\n  combine different DM search strategies. Nevertheless, it is\n  important that such models fulfil basic requirements, such as gauge invariance and that\n  perturbative unitarity is guaranteed in the regions of the parameter\n  space where the model is used to describe data. To ensure gauge\n  invariance, one needs to impose certain relations between the\n  different couplings, whereas it is necessary to introduce additional\n  states in order to restore perturbative unitarity.\n  \nWe have illustrated these issues by considering a simplified model consisting of a fermionic DM particle and a vector mediator, which may for example be the $Z'$ gauge boson of a new $U(1)'$ gauge symmetry in the hidden sector. The phenomenology of this model depends decisively on whether the couplings of the mediator are purely vectorial or whether there are non-zero axial couplings (implying that left- and right-handed fields are charged differently under the new $U(1)'$). Since the coupling structure on the SM side may be different from the one of the DM side, there are four different cases of interest: purely vectorial couplings on both sides, non-zero axial couplings on either the SM side or the DM side, and non-zero axial couplings in both sectors. Our results can be summarized as follows:\n\n\\begin{enumerate}\n\\item \\label{it:VV} {\\bf Vector(SM)--Vector(DM):} In this case no additional new physics is needed to guarantee perturbative unitarity and the mass of the $Z'$ can be generated via the Stueckelberg mechanism. This model is however highly constrained phenomenologically and a thermal DM is excluded for large parts of the parameter space due to strong limits on the spin-independent DM-nucleon scattering cross section.\n\\end{enumerate}\n\n\\noindent Generally, if at least one of the axial couplings is non-zero one needs new physics to unitarize the longitudinal component of the $Z'$. As a simple example we consider a SM-singlet Higgs breaking the dark $U(1)'$. Unitarity then requires the mass of the new Higgs to be comparable to the $Z'$ mass. Models with non-zero axial couplings are therefore expected to have a rich phenomenology with promising experimental signatures in DM direct detection experiments and invisible Higgs decays as well as additional DM annihilation channels.\n\n\\begin{enumerate}\n\\setcounter{enumi}{1}\n\\item \\label{it:AA} {\\bf Axial(SM)--Axial(DM):} The crucial observation in this case is that gauge invariance of the SM Yukawa terms requires that the SM Higgs has to be charged under the $U(1)'$. This requirement has important implications for phenomenology:\n  \\begin{itemize}\n    \\item[(a)] Electroweak symmetry breaking leads to mass mixing between the $Z'$ and the SM $Z$-boson, which is strongly constrained by EWPT.\n    \\item[(b)] The axial couplings of SM fermions to the $Z'$ are necessarily flavour universal and equal for quarks and leptons. Hence, it is not possible to couple the DM particle to quarks without also inducing couplings of the $Z'$ to leptons. Since the LHC is very sensitive to dilepton resonances, the resulting bounds severely constrain the model (dominating over constraints from monojet and dijet searches).\n  \\end{itemize}\n\n\\item \\label{it:AV} {\\bf Axial(SM)--Vector(DM):} The constraints from EWPT and dilepton resonance searches are largely independent of the coupling between the $Z'$ and DM and therefore also apply in the case of purely vectorial couplings on the DM side. However, gauge invariance of the Yukawa couplings implies that it is impossible for the $Z'$ to have purely axial couplings to quarks. Consequently, as soon as there is a vectorial coupling on the DM side, one necessarily obtains a vector-vector component inducing unsuppressed spin-independent DM-nucleus scattering, which is strongly constrained by direct detection (see item~\\ref{it:VV} above).\n\n\\item \\label{it:VA} {\\bf Vector(SM)--Axial(DM):} In contrast to the couplings between the $Z'$ and quarks it is possible for the DM-$Z'$ coupling to be purely axial. Indeed, this situation arises naturally in the case that the DM particle is a Majorana fermion such that the vector current vanishes. If the couplings on the SM side are purely vectorial (i.e.\\ left- and right-handed SM fields have the same charge), the SM Higgs is uncharged under the $U(1)'$ and consequently the constraints discussed in item~\\ref{it:AA} do not apply. Furthermore, the tree-level direct detection cross section is velocity suppressed, leading to much weaker constraints on this particular scenario.\n\nNevertheless, sizeable spin-independent DM-nucleus scattering can be induced at loop level. In addition, kinetic mixing between the $Z'$ and SM gauge bosons (at tree level or loop-induced) can be potentially important for EWPT and dilepton signatures. Assuming $\\epsilon = 0$ at $\\Lambda=10$~TeV, we find that bounds from searches for dilepton resonances due to loop-induced kinetic mixing can still be relevant and give constraints that are complementary to the ones obtained from monojet and dijet searches.\n\n\n\\end{enumerate}\n\nAll in all we find that imposing gauge invariance and conservation of perturbative unitarity has important implications for the phenomenology of DM interacting via a vector mediator and that relevant experimental signatures are not captured by considering only the interactions of the vector mediator with DM and quarks. This observation is relevant for the interpretation of various recent analyses of $Z'$-based simplified models,  e.g.~\\cite{Lebedev:2014bba,Hooper:2014fda,Chala:2015ama,Alves:2015dya,Blennow:2015gta,Heisig:2015ira}. Indeed, the $Z'$ model considered here is severely constrained by EWPT and dilepton resonance searches, either due to tree-level effects or loop-induced kinetic mixing. Moreover, the general expectation is that the mixing between the dark Higgs and the SM Higgs is sizeable and that as a result Higgs portal interactions are present in addition to the interactions mediated by the $Z'$. \n\nThe weakest constraints are obtained in the case of purely vectorial couplings on the SM side and purely axial couplings on the DM side. Indeed, this is the only case where LHC monojet and dijet searches are potentially competitive with other kinds of constraints.   In all cases that we have considered we find the hypothesis of thermal DM production to be under significant pressure. In large regions of the  parameter space which is still allowed by experiments additional annihilation channels (beyond the $Z'$-mediated interactions) are necessary to avoid DM\noverabundance. A more systematic parameter scan of the model will be performed in a forthcoming publication~\\cite{upcoming}. \n\nTwo final comments are in order. First, in this work we have not taken into account gauge anomalies. In general new fermions are needed to cancel the anomalous triangle diagrams, potentially leading to additional signatures and further constraints on the model.\nHowever, due to the constraints implied by gauge invariance of the SM Yukawa terms, the gluon-gluon-$Z'$ anomaly vanishes automatically, so that the new fermions need not be charged under colour, making them difficult to probe at the LHC.\n\nSecond, the requirement of universality of all axial fermion charges (including leptons) follows from the gauge invariance of the SM Yukawa term. It relies on the fact that in the SM all fermion masses are generated by the same Higgs doublet. If the Higgs sector is more complicated, for example in a two-Higgs-doublet model, this condition is relaxed and it is possible to have different axial couplings to up- and down-type quarks or different axial couplings to quarks and to leptons. In any case, such extensions of the SM go significantly beyond the simplified model approach and would most likely have a number of implications for Higgs physics, EWPT and other searches for new physics.\n\n\\bigskip\n\nIn conclusion we would like to emphasise that one of the most intriguing implications of our study is that a model with a vector mediator should generically also contain a scalar mediator, corresponding to the dark Higgs that generates the vector mass. In the limit that the mass of the vector mediator is much larger than the mass of the scalar, our $Z'$ model can also be used to study simplified models with a scalar mediator, where gauge invariance and perturbative unitarity can be similarly problematic. Indeed, our findings suggest that a strict distinction between simplified models with scalar and vector mediators is unnatural and many of the issues with these models may be best addressed in a more realistic set-up combining the two. Future direct detection experiments together with the upcoming runs of the LHC will be able to thoroughly explore the parameter space of such a realistic simplified model and test the hypothesis of DM as a thermal relic.\n\n\n\n\\subsection*{Acknowledgements}\n\nWe thank Sonia El Hedri, Juan Herrero-Garcia, Matthew McCullough and Oscar St\\aa l for helpful discussions, and Michael Duerr and Jure Zupan for valuable comments on the manuscript. FK would like to thank the Oskar Klein Center and Stockholm University for hospitality. This work is supported by the German Science Foundation (DFG) under the Collaborative Research Center (SFB) 676 Particles, Strings and the Early Universe as well as the ERC Starting Grant `NewAve' (638528).\n\n\n\\begin{appendix}\n\\section{Coupling structure from mixing}\n\\label{ap:mixing}\n\n\\subsection{Gauge boson mixing}\n\\label{app:Z}\n\nIn this appendix we discuss the mixing of a gauge boson $\\hat{Z}'$ of a new $U(1)'$ gauge group with the SM $U(1)_Y$ gauge field $\\hat B$ and the neutral component $\\hat W^3$ of the $SU(2)_\\mathrm{L}$ weak fields, where we use hats to denote the interaction eigenstates in the original basis. The mixing will then lead to the mass eigenstates $Z'$, $Z$ and $A$. Following the discussion in~\\cite{Frandsen:2011cg}, we consider an effective Lagrangian including both kinetic mixing and mass mixing (see also~\\cite{Babu:1997st})\n\\begin{align}\n  {\\cal L} =& \\; {\\cal L}_{SM}\n  -\\frac{1}{4}\\hat{X}^{\\mu\\nu}\\hat{X}_{\\mu\\nu} + {\\frac{1}{2}} m_{\\hat\n    Z'}^2 \\hat{Z'}_\\mu \\hat{Z'}^\\mu - {\\frac{1}{2}} \\sin \\epsilon\\, \\hat{B}_{\\mu\\nu} \\hat{X}^{\\mu\\nu} +\\delta\n  m^2 \\hat{Z}_\\mu \\hat{Z'}^\\mu \\;,\n\\label{eq:Lappendix}\n\\end{align}\nwhere $\\hat{X}^{\\mu \\nu} \\equiv \\partial^\\mu \\hat{Z'}^\\nu - \\partial^\\nu \\hat{Z'}^\\mu$. Furthermore, we have defined $\\hat{Z}\\equiv \\hat{c}_\\mathrm{W} \\hat{W}^3- \\hat{s}_\\mathrm{W} \\hat{B}$, where $\\hat{s}_\\mathrm{W} \\, (\\hat{c}_\\mathrm{W})$ is the sine (cosine) of the Weinberg angle and $\\hat g', \\, \\hat g$ are the corresponding gauge couplings.\n\nThe field strengths are diagonalised and canonically normalised via the following two consecutive transformations~\\cite{Babu:1997st,Chun:2010ve,Frandsen:2011cg}\n\\begin{align}\n\\label{eq:Zpmixing}\n\\left(\\begin{array}{c} \\hat B_\\mu \\\\ \\hat W_\\mu^3 \\\\ \\hat\n    Z'_\\mu \\end{array}\\right) & = \\left(\\begin{array}{ccc} 1 & 0 &\n    -t_\\epsilon \\\\ 0 & 1 & 0 \\\\ 0 & 0 &\n    1\/c_\\epsilon \\end{array}\\right)\n\\left(\\begin{array}{c} B_\\mu \\\\ W_\\mu^3 \\\\ Z'_\\mu \\end{array}\\right) \\ , \\\\\n\\left(\\begin{array}{c} B_\\mu \\\\ W_\\mu^3 \\\\ Z'_\\mu \\end{array}\\right) &\n= \\left(\\begin{array}{ccc}\n    \\hat c_\\mathrm{W} & -\\hat s_\\mathrm{W} c_\\xi &  \\hat s_\\mathrm{W} s_\\xi \\\\\n    \\hat s_\\mathrm{W} & \\hat c_\\mathrm{W} c_\\xi & - \\hat c_\\mathrm{W} s_\\xi \\\\\n    0 & s_\\xi & c_\\xi\n\\end{array} \\right) \n\\left(\\begin{array}{c} A_\\mu \\\\ Z_\\mu \\\\ R_\\mu \\end{array}\\right)\n \\; ,\n\\end{align}\nwhere\n\\begin{align}\n  t_{2\\xi}=\\frac{-2c_\\epsilon(\\delta m^2+m_{\\hat Z}^2 \\hat\n    s_\\mathrm{W} s_\\epsilon)} {m_{\\hat Z'}^2-m_{\\hat\n      Z}^2 c_\\epsilon^2 +m_{\\hat Z}^2\\hat s_\\mathrm{W}^2 s_\\epsilon^2\n    +2\\,\\delta m^2\\,\\hat s_\\mathrm{W} s_\\epsilon} \\; .\n\\label{eq:xi}\n\\end{align}\nFor $\\epsilon \\ll 1$ and $\\delta m^2 \\ll m_{\\hat Z}^2, m_{\\hat Z'}^2$, this equation can be approximated by\n\\begin{equation}\\label{eq:xi_def}\n \\xi = \\frac{\\delta m^2 + m_{\\hat Z}^2 \\hat s_\\mathrm{W} \\epsilon}{m_{\\hat Z}^2 - m_{\\hat Z'}^2} \\; .\n\\end{equation}\n\nThe mass eigenvalues $m_Z$ and $m_{Z'}$ are given by\n\\begin{align} \n m_Z^2 & = m_{\\hat Z}^2 (1+{\\hat s}_\\mathrm{W} \\, t_\\xi \\, t_\\epsilon)+\\frac{\\delta m^2 \\, t_\\xi}{c_\\epsilon} \\nonumber \\\\ & \\approx m_{\\hat Z}^2 + (m_{\\hat Z}^2 - m_{\\hat Z'}^2) \\xi^2 \\; ,  \\\\\n m_{Z'}^2 & = \\frac{m_{\\hat Z'}^2 + \\delta m^2 ({\\hat s}_\\mathrm{W} \\, s_\\epsilon-c_\\epsilon \\, t_\\xi)}{c_\\epsilon^2 \\, (1+{\\hat s}_\\mathrm{W} \\, t_\\xi \\, t_\\epsilon)} \\nonumber \\\\ &  \\approx m_{\\hat Z'}^2 + m_{\\hat Z'}^2 \\xi (\\xi - {\\hat s_\\mathrm{W}} \\epsilon) - m_{\\hat Z}^2 (\\xi - {\\hat s_\\mathrm{W}} \\epsilon)^2\\; .\n\\end{align}\nWe define the `physical' weak angle via\n\\begin{align}\n  s_\\mathrm{W}^2 \\, c_\\mathrm{W}^2=\\frac{\\pi \\, \\alpha(m_{Z})}{\\sqrt{2} \\, G_\\mathrm{F} \\, m_{Z}^2} \\; ,\n\\label{eq:swcw}\n\\end{align}\nwhere $\\alpha = e^2 \/ (4\\pi)$. Eq.~(\\ref{eq:swcw}) also holds with the replacements $s_\\mathrm{W}\\to\\hat s_\\mathrm{W}$, $c_\\mathrm{W}\\to\\hat c_\\mathrm{W}$ and $m_{Z}\\to m_{\\hat Z}$, leading to the identity $s_\\mathrm{W} \\, c_\\mathrm{W} \\, m_{Z}=\\hat s_\\mathrm{W} \\, \\hat c_\\mathrm{W} \\, m_{\\hat Z}$.  This equation implies\n\\begin{equation}\n s_\\mathrm{W}^2 = \\hat s_\\mathrm{W}^2 - \\frac{\\hat s_\\mathrm{W}^2 \\, \\hat c_\\mathrm{W}^2}{\\hat c_\\mathrm{W}^2 - \\hat s_\\mathrm{W}^2} \\left(1 - \\frac{m_{\\hat Z'}^2}{m_{\\hat Z}^2}\\right) \\xi^2 \\; .\n\\end{equation}\nThese equations allow us to fix $\\hat s_\\mathrm{W}$ and $m_{\\hat Z}$ in such a way that we reproduce the experimentally well-measured quantities $s_\\mathrm{W}$ and $m_{Z}$.\n\nThe couplings of the $Z'$ to SM fermions induced via mixing can e.g.\\ be found in \\cite{Frandsen:2012rk}.\nOf particular interest to our current analysis are the couplings to leptons which are strongly constrained.\nIn terms of the mixing parameters they can be written as\n\\begin{align}\n  g_{\\ell}^\\mathrm{V} &= \\frac{1}{4}(3 {\\hat g'} (\\hat s_\\mathrm{W} s_\\xi-c_\\xi t_\\epsilon)- {\\hat\n    g}\\hat c_\\mathrm{W} s_\\xi) \\ , & g_{\\ell}^\\mathrm{A} &= -\\frac{1}{4}\n  ({\\hat g'} (\\hat s_\\mathrm{W} s_\\xi-c_\\xi t_\\epsilon) + {\\hat g} \\hat c_\\mathrm{W} s_\\xi) \\; ,\n\\label{eq:dilepton}\n\\end{align}\nwith $\\hat{g}$ and $\\hat{g}'$ the fundamental gauge couplings of \n$SU(2)_\\text{L}$ and $U(1)_Y$.\n\n\\subsection{Scalar mixing}\n\\label{app:scalar}\n\nConsidering the SM Higgs $h$ plus the dark Higgs $s$, the most general\nscalar potential after electroweak and dark symmetry breaking can be\nwritten as\n\\begin{equation}\n V(s, h) = - \\frac{\\mu_s^2}{2} (s+w)^2  - \\frac{\\mu_h^2}{2} (h+v)^2 + \\frac{\\lambda_h}{4} (h+v)^4 + \\frac{\\lambda_s}{4} (s+w)^4 + \\frac{\\lambda_{hs}}{4} (h+v)^2(s+w)^2 \\; .\n \\end{equation}\nFor $\\lambda_{hs} = 0$, we obtain the usual formulas\n\\begin{align}\n v^2 & = \\frac{\\mu_h^2}{\\lambda_h} \\; , \\quad m_h^2 = 2 \\, \\lambda_h \\, v^2 \\; ,\\\\\n w^2 & = \\frac{\\mu_s^2}{\\lambda_s} \\; , \\quad m_s^2 = 2 \\, \\lambda_s \\, w^2 \\; .\n\\end{align}\nIn this case, there is no mixing between the two Higgs fields even at one-loop level. Nevertheless, there is no reason why $\\lambda_{hs}$ should be negligible and therefore the two fields will in general mix. One then obtains for the minimum (assuming $4 \\, \\lambda_h \\, \\lambda_s > \\lambda_{hs}^2$)\n\\begin{align}\n v^2 & = 2 \\frac{2 \\, \\lambda_s \\, \\mu_h^2 - \\lambda_{hs} \\, \\mu_s^2}{4 \\, \\lambda_s \\, \\lambda_h - \\lambda_{hs}^2} \\; ,\\\\\n w^2 & = 2 \\frac{2 \\, \\lambda_h \\, \\mu_s^2 - \\lambda_{hs} \\, \\mu_h^2}{4 \\, \\lambda_s \\, \\lambda_h - \\lambda_{hs}^2} \\; ,\\;\n\\end{align}\nand for the mass squared eigenvalues\n\\begin{equation}\n m_{1,2}^2 = \\lambda_h \\, v^2 + \\lambda_s w^2 \\mp \\sqrt{(\\lambda_s w^2 - \\lambda_h v^2)^2 + \\lambda_{hs}^2 w^2 v^2} \\; .\n\\end{equation}\nThe corresponding mass eigenstates are\n\\begin{align}\n H_1 = s \\sin \\theta + h \\cos \\theta \\nonumber \\\\\n H_2 = s \\cos \\theta - h \\sin \\theta \n\\end{align}\nwith\n\\begin{equation}\n \\tan 2\\theta = \\frac{\\lambda_{hs} \\, v \\, w}{\\lambda_h \\, v^2 - \\lambda_s \\, w^2} \\; .\n\\end{equation}\nFor small $\\lambda_{hs}$ we find $m_1^2 \\approx 2 \\, \\lambda_h \\, v^2 \\equiv m_h^2$ and $m_2^2 \\approx 2 \\, \\lambda_s \\, w^2 \\equiv m_s^2$. This yields\n\\begin{equation}\n \\theta \\approx - \\frac{\\lambda_{hs} \\, v \\, w}{m_s^2 - m_h^2} + \\mathcal{O}(\\lambda_{hs}^3) \\; .\n\\end{equation}\n\n\\end{appendix}\n\n\\providecommand{\\href}[2]{#2}\\begingroup\\raggedright","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\subsection*{Author information}\nThe authors declare no competing financial interest.\n\n\\begin{acknowledgement}\nThis work was financially supported by the Deutsche Forschungsgemeinschaft (DFG) via Sonderforschungsbereich SFB 631 (Projects B1 and B5) and the Emmy Noether Program (KR3790\/2-1) and by the European Union via SOLID and the FP7 Marie-Curie Reintegration Grant.\n\\end{acknowledgement}\n\n\\begin{suppinfo}\n(i) Emission spectra from typical NWs from this growth run excited above the \\ce{Al_{0.3}Ga_{0.7}As} bandgap. (ii) Cross-sectional HRTEM of a reference sample\\cite{Funk2013a}. (iii) PL suppression by SAW of the GaAs core and QW emissions. 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A.~C.; Farrer,~I.; Ritchie,~D.~A. {On-demand Single-electron\n  Transfer Between Distant Quantum Dots.} \\emph{Nature} \\textbf{2011},\n  \\emph{477}, 439--442\\relax\n\\mciteBstWouldAddEndPuncttrue\n\\mciteSetBstMidEndSepPunct{\\mcitedefaultmidpunct}\n{\\mcitedefaultendpunct}{\\mcitedefaultseppunct}\\relax\n\\unskip.}\n\\end{mcitethebibliography}\n\n\n\n\n\\end{document}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nHigh-resolution and high-cadence  observations have shown the ubiquity of  the magnetohydrodynamic (MHD) waves throughout the solar atmosphere \\citep[see, e.g.,][among many others]{Aschwanden99,Nakariakov99,Tomczyk07,Depontieu07,Mcintosh11,Morton15,Jafarzadeh17,Srivastava17}. MHD wave  dissipation could play an important role in the heating of the solar corona \\citep[see, e.g.,][]{Hollweg78,Cranmer05,Cargill11,Mathioudakis13,Soler19,Tom20,Nakariakov20}, and the acceleration of the solar wind \\citep[see, e.g.,][]{Charbonneau95,Cranmer09,Matsumoto12,Shoda18}.\n\nTorsional Alfv\\'{e}n waves are a sub-type of axisymmetric Alfv\\'{e}n waves in cylindrical flux tubes. They are nearly incompressible, the restoring force is the magnetic tension, and their direction of polarization is perpendicular to the magnetic field lines. This kind of waves  produce  axisymmetric perturbations in the perpendicular components of velocity and magnetic field. Furthermore, unlike Alfv\\'{e}n waves with other azimuthal symmetries,  torsional Alfv\\'{e}n waves do not couple with magneto-acoustic waves when the magnetic field is straight \\citep{Goossens11}. Torsional Alfv\\'{e}n waves have been reported in bright points \\citep{Jess09} and in a coronal active region structure \\citep{Kohutova20}. Torsional motions found in the chromosphere and transition region by \\citet{DePontieu12,DePontieu14} can also be interpreted as torsional Alfv\\'en waves.  \\citet{Aschwanden20} interpreted  oscillations in the magnetic energy during solar flares \nas torsional Alfv\\' {e}n waves. This kind of waves has also been reported in the solar wind during the interaction between coronal mass ejections \\citep{Raghav18}.\n\nIn coronal loops, closed magnetic structures with  two footpoints anchored at the solar photosphere, there is no report of such waves. Nonetheless, torsional Alfv\\'{e}n waves can be excited at their footpoints through vortex\/twisting plasma motions in the solar photosphere \\citep[see, e.g.,][]{Fedun11,Shelyag11,Shelyag12,Wedmeyer12,Mumford15,Srivastava17b}. Recently, \\citet{Soler21} have investigated the propagation of torsional Alfv\\'{e}n waves from the photosphere to a coronal loop. As the waves reach the coronal loop, they can resonate with standing modes of the loop and drive global torsional oscillations \\citep[see also][]{Hollweg84,Hollweg84b}. \\citet{Soler21} found that large amounts of energy can be transmitted  at coronal heights despite the chromospheric filtering. Particularly, they find that the energy flux is channeled mostly through the fundamental standing torsional mode of the loop.\n\nInspired by these results, in \\citeauthor{Diazsoler21} (\\citeyear{Diazsoler21}; hereafter Paper \\citetalias{Diazsoler21}), we investigated the nonlinear evolution of torsional Alfv\\'{e}n waves in a straight coronal flux tube with a constant axial magnetic field. In Paper \\citetalias{Diazsoler21}, see also the torsional model in \\citet{Guo19}, we showed that the phase mixing of the torsional Alfv\\'{e}n waves generates azimuthal shear flows. These flows eventually excite the Kelvin-Helmholtz instability (KHi), as \\citet{HeyvaertPriest83} and \\citet{Browning84} analytically predicted. Phase mixing is a linear phenomenon that occurs when waves in adjacent radial positions oscillate with different frequencies. The cause of phase-mixing is the presence of a frequency continuum, whose origin is in the inhomogeneities in density and\/or magnetic field \\citep[see, e.g.,][]{HeyvaertPriest83,Nocera84,Moortel00,Smith07,Prokopyszyn2019}. Phase-mixing increases the values of vorticity and current density and generates small scales, although at a relatively slow pace.\n\nThe KHi triggered by phase mixing can evolve nonlinearly forming large eddies that break into smaller eddies. This process initiates and drives turbulence. Turbulence accelerates the energy cascade to small scales, which might enhance the efficiency of wave energy dissipation \\citep{Hillier20}. There is evidence that coronal loops may be in a turbulent state \\citep{Demoortel14,Hahn14,Liu14,Xie17}. The generation of turbulence is also a result obtained by numerous studies of numerical simulations of kink oscillations of coronal loops \\citep[see, e.g.,][among others]{Terradas08,Terradas18,Antolin15,Magyar16,Howson17a,Karampelas19,Antolin19}. \n\n\n\nIn the present paper, we extend the investigation of Paper~\\citetalias{Diazsoler21} by replacing the constant axial magnetic field by a twisted magnetic field. The existence of twisted coronal loops has been confirmed by observations \\citep[see, e.g.,][]{Kwon08,Aschwanden12,Thalmann14,Aschwanden19}. The behavior of linear MHD waves in twisted flux tubes has extensively been  investigated analytically or semi-analytically \\citep[see, e.g.,][among many others]{Bennet99,Erdelyi06,Erdelyi07,Erdelyi10,Rudermann07,Karami09,Karami10,Terradas12,Ebrahimi17,Ebrahimi22}. The nonlinear evolution of kink MHD waves in twisted flux tubes has  also been investigated numerically \\citep{Howson17a,Terradas18}. Here, we explore how the presence of  magnetic twist affects the nonlinear evolution of torsional Alfv\\'en waves.\n\n\\section{Numerical setup}\n\\label{Sect_Num}\n\\subsection{Initial Configuration} \\label{Subsect_model}\n\nAs in Paper~\\citetalias{Diazsoler21}, we use the standard coronal loop model \\citep[see, e.g.,][]{Edwinroberts83}. The model is made of an overdense cylindrical flux tube of radius, $ R $, and length, $ L, $ embedded in a low-$\\beta$ uniform coronal environment. We set $L\/R = 10$. We used a shorter loop length than what is reported from observations, typically $L\/R \\sim 100$. The main reason for doing so is to speed up the simulation times. The periods of the standing torsional oscillations are proportional to the loop length. Thus, if a longer loop were considered, the periods and simulation times would be longer, but the dynamics would be essentially the same. The effect of considering larger loop lengths was explored in Paper~\\citetalias{Diazsoler21}.\n\n\nThe loop footpoints are fixed at two rigid walls representing the solar photosphere.  We neglect the curvature of the coronal loop and the thin chromospheric layer at the loop feet.  In Paper~\\citetalias{Diazsoler21} we assumed a uniform magnetic field along the cylinder axis. Here, we improve the model by considering a twisted magnetic field, which is a more realistic representation of the magnetic field in coronal loops, namely\n\\begin{equation}\n    {\\bf B} = B_{\\varphi}(r) \\hat{e}_\\varphi + B_{z}(r) \\hat{e}_z, \n\\end{equation}\nwhere  $B_{\\varphi}$ and $B_{z}$ are the  azimuthal and longitudinal components of the background magnetic field in a cylindrical coordinate system denoted by $r$, $\\varphi$, and $z$. In our model, the $z$-axis coincides with the flux tube axis. We consider the same force-free twist model as that used in \\citet{Terradas18}, namely\n\\begin{eqnarray}\nB_{\\varphi}(r)&=&\\left\\lbrace \\begin{matrix} cr\/R, & \\mbox{if}\\;r < R,\\label{bphi} \\\\\n cR\/r, & \\mbox{if}\\;r\\geq R,\\end{matrix}\\right. \\\\\nB_{z}(r)&=&\\left\\lbrace \\begin{matrix} \\sqrt{B^{2}_{0}+2c^{2}\\left(1-r^{2}\/R^{2}\\right)}, & \\mbox{if}\\;r < R, \\label{bz}\\\\\n B_{0}, & \\mbox{if}\\;r\\geq R,\\end{matrix}\\right.  \n\\end{eqnarray}\nwhere  $ B_{0} $ is a constant that corresponds to the magnetic field strength at the tube axis and $c$ is a dimensionless parameter that controls the amount of magnetic twist. \n\nThe largest value of the magnetic twist parameter used here is $ c = 0.4$, which is chosen according to observational constraints. \\citet{Aschwanden13p3} computed the twist angle, $ \\theta $, in coronal loops as $ \\theta=\\arctan \\sqrt{q_{\\mathrm{free}}}$, where $ q_{\\mathrm{free}}$ is the free energy ratio. The calculation of $q_{\\mathrm{free}}$ can be achieved from observations through line-of-sight magnetograms, the three-dimensional (3D) position of the coronal loop, and a nonlinear force-free field code. Tables 3 in \\citet{Aschwanden14} and \\citet{Aschwanden16} show that the free energy ratio ranges from 0 to $\\sim 0.25 $. Therefore, twist angles lower than  $ \\sim 25^{\\circ}$ are expected. The twist angle of our model can be  calculated as\n\\begin{equation}\n\\theta=\\arcsin\\left(\\frac{B_{\\varphi}(r)}{B(r)}\\right),\n\\label{angles}\n\\end{equation}\nwhere $ B(r)=\\sqrt{B^{2}_{\\varphi}\\;(r)+B^{2}_{z}\\;(r)} $ is the modulus of the magnetic field.\nFor $c=0.4$, the twist angle is $ \\theta =21.8^{\\circ}$ at $r=R$, which is slightly lower than the largest angle inferred by  \\citet{Aschwanden14} and \\citet{Aschwanden16}. On the other hand, the number of turns of the magnetic field lines over the cylinder length can be computed as,\n\\begin{equation}\nN\\left(r\\right)=\\frac{1}{2\\pi} \\frac{L}{r} \\frac{B_{\\varphi}\\left(r\\right)}{B_{z}\\left(r\\right)},\n\\label{ntw}\n\\end{equation}\nwhere the factor $L B_{\\varphi}\\left(r\\right)\/rB_{z}\\left(r\\right) \\equiv  \\Phi(r) $ is the absolute twist. \\citet{Kwon08} considered a sample of 14 loops from  Transition Region And Coronal Explorer (TRACE; \\citeauthor{TRACE} \\citeyear{TRACE}) observations in extreme ultraviolet and determined that their absolute twist values ranged  from $0.22\\pi$ to $1.73\\pi$. This result implies that $0.11 < N < 0.865$.  At $ r=R $ in our model, where $B_{\\varphi} $ is maximum, we obtain $ N= 0.64$ for $c=0.4$ and $ L\/R= 10$, so that the maximum twist considered here is below the upper limit of the  interval inferred by \\citet{Kwon08}.\n\n\n\nA twisted magnetic field is prone to develop the kink instability. In the case of uniform twist,  \\citet{HoodPriest79} found that the kink instability can develop when the absolute twist is larger than $ 3.3\\pi $. With an improved analysis, \\citet{HoodPriest81} reduced the threshold value to $ 2.49\\pi $. Using the definition of $N$ in Eq.~(\\ref{ntw}), we find that the kink instability may appear in our model if $ N \\geq 1.65$ according to the critical twist of  \\citet{HoodPriest79} and if $ N \\geq 1.245 $ according to the critical twist of  \\citet{HoodPriest81}. Clearly, we are sufficiently far away from the critical twist for the kink instability because our maximum twist ($ N= 0.64$) is still relatively weak.\n\nWe note that we used a pre-existing twisted magnetic field in our model. A different way to proceed would be to create twist in the model by dynamically rotating the magnetic field lines anchored in the photosphere, as done in \\cite{Ofman09}. This alternative approach imposes a more complex scenario and it is not considered here for simplicity.\n\nSince the magnetic field is force-free, we consider in the model a background uniform gas pressure, $p_0$. In turn, the equilibrium density, $\\rho_0$, is the same as that used in Paper \\citetalias{Diazsoler21}. Thus, we used\n\\begin{equation}\n\\rho_{0}(r)=\\left\\{\n\\begin{array}{lll}\n\\rho_{\\mathrm{i}}, & \\mbox{if} & r \\leq R-\\frac{l}{2},\\\\\n\\rho_{\\mathrm{tr}}(r), & \\mbox{if} & R-\\frac{l}{2}<r< R+\\frac{l}{2}, \\\\\n\\rho_{\\mathrm{e}},  & \\mbox{if} & r \\geq R+\\frac{l}{2}.\n\\end{array}\n\\right.\n\\label{rhogen}\n\\end{equation}\nIn Eq.~(\\ref{rhogen}), $ \\rho_{\\rm i} $ is the density in the uniform inner core, $ \\rho_{\\rm e} $ is the uniform external density, and $ \\rho_{\\rm tr}(r) $ is the density in a nonuniform transitional layer of thickness $ l $ that continuously connects both uniform regions as\n\\begin{equation}\n\\rho_{\\mathrm{tr}}\\left(r\\right)=\\frac{\\rho_{\\mathrm{i}}}{2}\\left\\lbrace \\left[1+\\frac{\\rho_{\\mathrm{e}}}{\\rho_{\\mathrm{i}}}\\right]- \\left[1-\\frac{\\rho_{\\mathrm{e}}}{\\rho_{\\mathrm{i}}}\\right]\\sin\\left[\\frac{\\pi}{l}\\left(r-R\\right)\\right]\\right\\rbrace,\n\\label{rhotr}\n\\end{equation}\nwhere $ l $ may range from 0 (abrupt transition) to $2R$ (fully nonuniform loop). We set $\\rho_{\\rm i}\/\\rho_{\\rm e}=2$ and $ l=0.4R $. A scheme of our model is depicted in Fig. \\ref{Fig_sketch}.\n\n\\begin{figure}[htbp!]\n\\resizebox{\\hsize}{!}{\\includegraphics{sketch.pdf}} \n\\centering\n\\caption{Sketch of the coronal flux tube model. The isosurface of density with value equal to $(\\rho_{\\mathrm{i}} + \\rho_{\\mathrm{e}})\/2$ is shown in orange. Magnetic field lines located at $r=2R$, $r=R$ and $r=0$ are drawn in green, blue, and red colors, respectively, for the model with $c=0.4$. The bottom and top gray planes represent the solar photosphere. }\n\\label{Fig_sketch}\n\\end{figure}\n\nThe equilibrium Alfv\\'{e}n, $v_{A}(r)$, and sound, $ v_{s}(r) $,  speeds are,\n\\begin{eqnarray}\nv_{\\rm A}(r) &=&\\frac{B (r)}{\\sqrt{\\mu \\rho_0(r)}}, \\\\\n v_{s}(r) &=&\\sqrt{\\frac{\\gamma p_{0}}{\\rho_{0} (r)}},\n\\end{eqnarray}\nwhere  $\\gamma$ is the adiabatic constant. Figure~\\ref{Fig_soundalfven} shows the radial profiles of both speeds for a particular set of parameters. Both  Alfv\\'{e}n and sound speeds vary inside the transition region  because the density is nonuniform. However, when $c \\neq 0$ the Alfv\\'{e}n speed is also position-dependent in the inner core and in the external medium, where the density is uniform. This happens because the magnetic field is nonuniform everywhere when twist is present. \n\n\n\\begin{figure}[htbp!]\n\\resizebox{\\hsize}{!}{\\includegraphics{valfcsl04.pdf}} \n\\centering\n\\caption{The equilibrium radial  profiles of the Alfv\\'{e}n, $ v_{\\rm A}(r)$, and sound speeds, $v_{s}(r) $, for the twist parameter ranging from $c =0$ to $c=0.4$. The sound speed is the same in all cases. The region between the  vertical dashed lines denotes the inhomogeneous region in density. The values are normalized with respect to the  Alfv\\'{e}n speed at $r \\to \\infty$. We used $\\rho_{\\rm i}\/\\rho_{\\rm e}=2$ and $ l\/R=0.4 $.} \n\\label{Fig_soundalfven}\n\\end{figure}\n\nAccording to \\cite{Halberstadt93}, the line-tied Alfv\\'{e}n frequency continuum is \n\\begin{equation}\n\\omega_{\\rm A}(r)=\\frac{n\\pi}{L}\\frac{B_{z}(r)}{\\sqrt{\\mu_{0}\\rho_{0}(r)}},\n\\label{conteq}\n\\end{equation}\nwhere $n$ is the harmonic number. A continuous spectrum is present because of the radial variation of the $z$-component of the magnetic field, $B_z(r)$, and the density, $\\rho_0(r)$. The density is only nonuniform in the transition region. However, $B_z(r)$ is also nonuniform in the inner core of the loop when twist is present. Because of twist, the Alfv\\'{e}n continuum  also extends into the uniform core of the loop, as it can be  seen in Fig.~\\ref{Fig_conti}.\n\n\\begin{figure}[htbp!]\n\\resizebox{\\hsize}{!}{\\includegraphics{continuum.pdf}} \n\\centering\n\\caption{The radial profile of the line-tied  Alfv\\'{e}n frequency continuum for the twist parameter ranging from $c =0$ to $c=0.4$. The region between the green vertical dashed lines corresponds to the nonuniform region in density. The values are normalized to the external Alfv\\'{e}n frequency. We used $\\rho_{\\rm i}\/\\rho_{\\rm e}=2$ and $ l\/R=0.4 $.}\n\\label{Fig_conti}\n\\end{figure}\n\nThroughout the paper, length and density are normalized with respect to the radius of the flux tube, $ R $, and the external density, $ \\rho_{e} $, respectively. Velocities are normalized with respect to the external Alfv\\'{e}n speed at $ r \\to \\infty $, namely $v_{A,\\infty} = v_{A}\\left( r \\to \\infty \\right)$. The normalized time is $ \\overline{t} = t\/\\tau_{\\rm A}$, with $\\tau_{\\rm A} =  R\/v_{A,\\infty} $. \n\n\\subsection{Numerical code}\n\nAs in Paper~\\citetalias{Diazsoler21}, the numerical simulations are performed with the  PLUTO code \\citep{Mignone07}. We  solve the 3D ideal MHD equations with a finite-volume, shock capturing spatial discretization on a structured mesh. The equations are as follows:\n\\begin{eqnarray}\n\\frac{\\partial\\rho}{\\partial t} &=& -\\nabla \\cdot\\left(\\rho \\mathbf{v}\\right), \\label{cont} \\\\\n\\rho \\frac{\\mathrm{D}\\mathbf{v}}{\\mathrm{D} t} &=&  -\\nabla p + \\frac{1}{\\mu_0} \\left(\\nabla \\times \\mathbf{B}\\right) \\times \\mathbf{B}, \\label{mome} \\\\ \n\\frac{\\partial \\mathbf{B}}{\\partial t} &=& \\nabla \\times \\left(\\mathbf{v}\\times \\mathbf{B}\\right), \\label{indu} \\\\\n\\frac{\\mathrm{D} p}{\\mathrm{D} t} &=& \\frac{\\gamma p}{\\rho}\\frac{\\mathrm{D} \\rho}{\\mathrm{D} t}. \\label{ener} \n\\end{eqnarray}\nIn Eqs. (\\ref{cont})-(\\ref{ener}),  $ \\frac{\\mathrm{D}}{\\mathrm{D}t}  \\equiv \\frac{\\partial}{\\partial t}+ \\mathbf{v} \\cdot\\nabla$ denotes the total derivative, $ \\rho $ is the mass density, $ \\mathbf{v} $ is the velocity,  $ \\mathbf{B} $ is the magnetic field, $ p$ is the gas pressure, and  $\\mu_{0}$ is the vacuum magnetic permeability. Gravity and nonideal terms are neglected.\n\nThe PLUTO code solves Eqs. (\\ref{cont})-(\\ref{ener}) in Cartesian coordinates. As in Paper \\citetalias{Diazsoler21}, the code uses a Roe-Riemann solver \\citep{Roe81} to compute the numerical fluxes, a piecewise parabolic scheme for spatial reconstruction, and a second-order Runge-Kutta algorithm for temporal evolution. In order to maintain the solenoidal constraint, the Generalized Lagrange Multiplier (GLM) method \\citep{Dedner02} is used. The background magnetic field  is not current-free because of magnetic twist, so the background splitting technique cannot be used in the present simulations, contrary to Paper~\\citetalias{Diazsoler21}. For comparison purposes, the background splitting technique is not used for the straight magnetic field case either. So, here the code advances the full magnetic field vector.\n\nWe used the same base computational box as in Paper \\citetalias{Diazsoler21}, namely a numerical resolution of 100x100x100 cells distributed uniformly from $-3R$ to $3R$ in the $x$-  and $y$-directions, and from $-L\/2$ to $L\/2$ in the $z$-direction. The code implements the adaptive mesh refinement (AMR) strategy \\citep{Mignone12} by which the cells split in two, and so the spatial resolution doubles, if a certain criterion \\textbf{is} satisfied. The refinement criterion is based on the second derivative error norm \\citep{Lohner87} of a parameter closely related with the kinetic energy density, since the goal of the AMR strategy in the present simulations is to conserve wave energy as much as possible. The use of AMR decreases the computational cost by keeping low resolution where the dynamics is not relevant and generating high resolution in the regions of interest. We included four levels of refinement,  so that the maximum effective resolution is 1600x1600x1600. For typical values of a coronal loop radius, the effective transverse resolution is $\\sim$ 10 km.\n\nAlthough the AMR scheme allows us to deal with extremely fine scales,  the dynamics developed in the flux tube eventually cascades the energy to scales below the maximum effective resolution. As in Paper \\citetalias{Diazsoler21}, to determine the simulation time at which this unphysical energy loss starts to happen, we monitored the kinetic, magnetic, and internal energies. In such a way, we could determine the time at which numerical dissipation is beginning to play an important role. This time also coincides with the time when the total integrated vorticity saturates and decreases (see later). \n\nRegarding the boundary conditions, we set the same set of boundary conditions as in Paper \\citetalias{Diazsoler21}. We considered outflow conditions, that is, zero gradient for pressure, density, and the $x$- and $y$-components of the magnetic field at all boundaries. We also set the outflow condition for the $z$-component of magnetic field, $ B_{z} $, at the lateral boundaries. To mimic the line-tying of the magnetic field lines at the solar photosphere, $ B_{z} $ is fixed to the equilibrium value through Eq. (\\ref{bz}) at the top and bottom boundaries. The three components of velocity vanish at all boundaries to avoid energy injection from outside the domain.\n\n\n\n\n\n\\subsection{Initial perturbation}\n\n\n\nIn the solar atmosphere, torsional Alfv\\'{e}n waves can be driven by photospheric plasma motions \\citep[see, e.g.,][]{Fedun11,Shelyag11,Shelyag12,Wedmeyer12,Mumford15,Srivastava17b}. The waves propagate  to the corona through the chrosmosphere, where reflection and dissipation heavily reduces the upward wave transmission \\citep[see][]{Soler19}. When a broadband spectrum of  waves driven at the photosphere are transmitted into a closed coronal structure, like a coronal loop, \\citet{Soler21} showed that most of the energy is deposited in the longitudinally fundamental mode. In view of this result and as in Paper~\\citetalias{Diazsoler21},  we excited at $t=0$ the longitudinally fundamental mode of standing torsional Alfv\\'{e}n waves. We did so by perturbing the components of velocity. To avoid that the perturbation also excites slow MHD waves initially, we impose an initial velocity field in the same manner as in \\citet{DiazSuarezsolerlet2021}, so that the velocity is purely perpendicular to the magnetic field lines, namely\n\\begin{eqnarray}\nv_{r}(r,z) &=&0, \\label{vvr} \\\\\nv_{\\varphi}(r,z) &=& v_{\\perp} (r,z) \\frac{B_{z}(r)}{B(r)} ,\\label{vvphi}\\\\\nv_{z}(r,z) &=&  -v_{\\perp} (r,z) \\frac{B_{\\varphi}(r)}{B(r)},\\label{vvz}\n\\end{eqnarray}\nwhere $ v_{\\perp}$ is the perpendicular velocity to the magnetic field lines.  We consider the same form as that used in Paper~\\citetalias{Diazsoler21}, namely\n\\begin{equation}\nv_{\\perp} (r,z) = v_0 A\\left(r\\right) \\cos\\left(\\frac{\\pi}{L} z\\right),\n\\label{vperp}\n\\end{equation} \nwhere $v_0$ is the maximum velocity amplitude and  $A\\left(r\\right)$ contains the radial profile (see Paper~\\citetalias{Diazsoler21}). We set $v_0 = 0.1 v_{A,\\infty} $. The radial profile of $ v_{\\varphi} $ and $ v_{z} $ at $ z=0 $ is shown in Fig. \\ref{Fig_ampli} for different values of the twist parameter, $c$.\n\n\n\\begin{figure}[htbp!]\n\\resizebox{\\hsize}{!}{\\includegraphics{veloc.pdf}} \n\\centering\n\\caption{Radial profiles of the azimuthal (top) and longitudinal (bottom) components of velocity at $t=0$ for the twist parameter ranging from $c =0$ to $c=0.4$. The inset in the top panel shows a close-up view of the azimuthal velocity around its maximum. Both velocity components are normalized to the maximum amplitude  of the perpendicular velocity perturbation, $v_{0}$.}\n\\label{Fig_ampli}\n\\end{figure}\n\nWe recover the same initial condition as in Paper~\\citetalias{Diazsoler21} for the untwisted case, $ c=0 $. When we increase the magnetic twist, the maximum of $v_{\\varphi} $  slightly decreases,  but its profile remains mostly unaltered. The longitudinal component of velocity, $v_{z} $, displays a significant increase when the twist grows. This is a direct effect of the relative variation of the azimuthal and longitudinal components of the magnetic field. \n\nAlthough the initial perturbation will mostly excite torsional Alfv\\'{e}n waves, other types of MHD can also appear in the flux tube. The inclusion of a twisted magnetic field introduces linear coupling between torsional Alfv\\'{e}n waves and fast magneto-acoustic sausage waves \\citep{Sakurai91,Goossens92,Goossens11,Giagkiozis15}. The waves excited by the initial perturbation when $c\\neq 0$ will develop mixed properties of both torsional Alfv\\'{e}n waves and fast magneto-acoustic sausage waves. Nevertheless, since the twist is weak, the torsional Alfv\\'{e}n wave character will be dominant. In addition, later in the evolution the ponderomotive force will nonlinearly  drive  slow MHD waves \\citep{hollweg1971,Rankin1994,Tikhonchuk1995,Terradas2004}.\n\n\n\\section{Revisiting the straight magnetic field case}\n\\label{Subsect_stra}\n\nThe main goal of this paper is to illustrate the effects of a twisted magnetic field in the nonlinear evolution of the standing torsional Alfv\\'{e}n waves. In particular, we investigate its influence on the triggering of the KHi and the associated turbulence generation. However, before tackling the investigation of twist, we seize the opportunity to revisit the  straight  magnetic field case of Paper \\citetalias{Diazsoler21} and to perform an extended study.\n\n\n\nWe rerun the simulation corresponding to the thin-layer case of  Paper~\\citetalias{Diazsoler21} but for larger times and deeper into the turbulent phase. So, the simulation is continued beyond the capacity of the AMR scheme to correctly describe the smallest spatial scales developed in the computational box. Although the dynamics of the smallest scales will then be affected by numerical dissipation, we can assume that the evolution at spatial scales larger than the maximum effective resolution will remain valid. Having this in mind, we can study how turbulence evolves globally at larger times despite the non-reliable information at the smallest scales.\n\nWe refer readers to Paper~\\citetalias{Diazsoler21} for detailed explanations of the simulation evolution. We give a brief summary. Standing torsional Alfv\\'{e}n waves excited in the flux tube develop the process of phase mixing \\citep[see, e.g.,][]{HeyvaertPriest83,Nocera84,Moortel00,Smith07,Prokopyszyn2019,Ebrahimi2020}. In the transition region, where density is nonuniform, waves in adjacent magnetic surfaces oscillate with a slightly different frequency (see Eq.~(\\ref{conteq})) and become out of phase as time increases. The effect of phase mixing is to generate an azimuthal velocity shear, which eventually triggers the KHi \\cite[see, e.g.,][]{HeyvaertPriest83,Soler10,Zaqarashvili15,Barbulescu19}. Initially, the KHi develops locally in the transition region, so it does not globally perturb the flux tube. Nonetheless, the KHi evolves nonlinearly as time increases. The nonlinear evolution of the KHi breaks large eddies into smaller eddies. The instability naturally drives turbulence in the flux tube that develops perpendicularly to the background magnetic field.  Turbulence extends beyond the boundaries of the initial nonuniform transition, and further accelerates the energy cascade towards small scales. Because of turbulence, mixing of the internal and external plasmas occurs. Thus, the dynamics of torsional Alfv\\'{e}n oscillations is similar to that of linearly polarized kink \\citep[see, e.g.,][]{Terradas08,Antolin15,Magyar16,Karampelas19,Antolin19,Pascoe20,Martinez22} or circularly polarized kink oscillations \\citep{Magyar22}. Nonetheless, the flux tube is not displaced laterally in the case of torsional oscillations, as occurs with the excitation of kink oscillations, and the resonant absorption process is absent for torsional Alfv\\'{e}n waves \\citep[see, e.g., ][]{Goossens11}.\n\n \n As shown in Paper~\\citetalias{Diazsoler21}, a  parameter that helps us understand the different phases of the evolution is the vorticity. We investigated the vorticity squared, $  \\omega^{2} (\\mathbf{r},t) = \\left| \\nabla \\times \\mathbf{v}(\\mathbf{r},t)\\right|^2$, and its value integrated over the whole computational domain, $ \\Omega^{2} $, namely\n\\begin{equation}\n\\Omega^{2} \\left(t\\right)=\\int \\omega^{2}\\left(\\mathbf{r},t \\right) \\mathrm{d}^{3}\\mathbf{r}.\n\\label{vortyintegrate}\n\\end{equation}\nFigure \\ref{Fig_vortynotwist} shows a cross-sectional cut at the tube center, $z=0 $, of $\\omega^{2}\\left(\\mathbf{r},t \\right) $ in  logarithmic scale. Only a subdomain of the complete numerical domain is shown, where the relevant dynamics occurs. The inset to the right shows the  integrated vorticity squared, $ \\Omega^{2} \\left(t\\right)$, as a function of the computational time. The blue dot tracks the position on the curve of $ \\Omega^{2} $ as time varies. The complete temporal evolution is available as a movie. \n\nIn the small panel of Fig.~\\ref{Fig_vortynotwist}, we see that the temporal evolution of $ \\Omega^{2} $ undergoes three phases (see Paper~\\citetalias{Diazsoler21}). In the first phase for $\\bar{t} \\lesssim 65$, $ \\Omega^{2} $ oscillates and slightly increases owing to the linear development of phase-mixing. In this phase, the spatial distribution of $\\omega^{2}\\left(\\mathbf{r},t \\right) $ is mostly in the form of concentric rings.  The concentric rings deform after the KHi is locally excited. In the second phase for $65 \\lesssim \\bar{t} \\lesssim 87$, the KHi has grown enough to behave nonlinearly and induces turbulence. The KHi vortices break into smaller and smaller vortices. As a result, there is a dramatic increase in vorticity. The spatial distribution of $\\omega^{2}\\left(\\mathbf{r},t \\right) $ clearly reveals the formation of very fine scales owing to turbulence. Finally, in the third phase for $\\bar{t} \\gtrsim 87$, the AMR scheme can no longer fully describe the  smallest scales that appear in the evolution. Thus, vorticity decreases unphysically because of numerical dissipation. As a reference, the periods of the internal and external Alfv\\'en oscillations are $20 \\sqrt{2}$ and $20$ normalized time units, respectively.\n\nThe particular snapshot displayed  in the large panel of Fig.~\\ref{Fig_vortynotwist} shows the spatial distribution of $\\omega^{2}\\left(\\mathbf{r},t \\right) $ at the time for which  $ \\Omega^{2} $ is maximum. We can clearly see  that the loop boundary is fully turbulent and  extremely fine structures in vorticity have already been generated. Several authors have studied the evolution of vorticity in numerical simulations where the kink mode is excited \\citep[see, e.g.,][]{Terradas08,Antolin15,Howson17res,Karampelas17,Karampelas18,Guo19,Howson2020,Howson2021} and some of their results are comparable with the present ones. Similar vorticity structures as those displayed in Fig.~\\ref{Fig_vortynotwist} can also be seen in previous works of nonlinear kink oscillations \\citep[see, e.g.,][]{Antolin14,Howson17a,Antolin19}. Nonetheless, the spatial distribution is  different due to the different azimuthal symmetry of torsional and kink oscillations.\n\n\n\\begin{figure*}[htbp!]\n\\resizebox{0.75\\hsize}{!}{\\includegraphics{vortyc0.pdf}} \n\\centering\n\\caption{Cross-sectional cut at $z=0 $ of the logarithm with base 10 of the vorticity squared, $\\omega^{2}\\left(\\mathbf{r},t \\right) $, in arbitrary units when $ \\overline{t}=87 $.  The small panel to the right shows the temporal evolution of $\\omega^{2}\\left(\\mathbf{r},t \\right) $ integrated over the whole computational domain, $\\Omega^{2}(t)$,  normalized with respect to the value at $\\overline{t}=0$. The blue dot marks the position in the curve for the considered simulation time. The complete temporal evolution is available as a movie.} \n\\label{Fig_vortynotwist}\n\\end{figure*}\n\nThe turbulent evolution of the flux tube is clearly highlighted by the formation of extremely fine vorticity structures captured by the high resolution of the AMR scheme. A consequence of turbulence is the mixing of the internal and external plasmas, which heavily modifies  the  transverse density profile adopted initially. To illustrate this effect, Fig.~\\ref{Fig_averagedens} displays an azimuthal average of the transverse density profile at $z=0$ for various times. Figure~\\ref{Fig_averagedens} is similar to density panels from Fig.~5 of \\citet{Karampelas18} for  nonlinear kink waves. On average, we find that the width of the nonuniform region increases with time. For large times in our simulation, part of the initially uniform core becomes turbulent, and so nonuniform. Consequently, at sufficiently large times beyond those simulated here, the whole flux tube should become turbulent. It is likely that for kink waves, which are global oscillations of the whole tube, the fully turbulent regime is reached faster than for torsional Alfv\\'en waves, which are local waves in nature. This would require further analysis. We note that, because of the azimuthal average, the density variations are smoothed. Along specific directions, we can find density variations that are much more important than those displayed in Fig.~\\ref{Fig_averagedens}.In addition, the full temporal evolution of the results displayed in Fig.~\\ref{Fig_averagedens} also shows periodic density increases in the loop core caused by the ponderomotive force \\citep[see, e.g.,][]{hollweg1971}.\n\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{average1ddens.pdf}} \n\\centering\n\\caption{Azimuthally averaged density at the tube center, $z=0$, for different simulation times. The vertical dashed purple lines show the limits of the nonuniform region at $\\overline{t}=0$. Density is normalized to the external density.}\n\\label{Fig_averagedens}\n\\end{figure}\n\n\\section{Exploring the effect of  magnetic twist}\n\nHere we turn to investigate the influence of  magnetic twist. Besides the simulation without magnetic twist ($c=0$) discussed before, we performed four additional simulations including magnetic twist with $c=$~0.1, 0.2, 0.3, and 0.4. \n\nBefore the onset  of the KHi, when the dynamics is governed by the development of phase-mixing, the results of all simulations are quite similar independently of the strength of magnetic twist. The reason is that the shape of the Alfv\\'en frequency continuum in the nonuniform layer is very similar in all cases (see Fig.~\\ref{Fig_conti}). Therefore, we focus on later times. In Fig. \\ref{Fig_Densitytwist} we show cross-sectional cuts of density at the tube center, $z=0 $,  at three different simulation times corresponding to already advanced stages of the simulations. The results for $c=0$ are shown in the first row. The twist parameter increases from $c=0.1$ to $c=0.4$ as we move from top to bottom in the remaining rows. The results for $c=0.3$ are not shown as they are visually indistinguishable from those for $c=0.4$. Figure~\\ref{Fig_Densitytwist} suggests that the inclusion of the magnetic twist in our model delays the onset  of KHi. For the same time, the KHi vortices appear less and less developed when the twist increases. \n\n\\begin{figure}[htb]\n\\includegraphics[width=\\columnwidth]{densitymultin.pdf}\n\\centering\n\\caption{Cross-sectional cuts of density at the tube centre, $z=0$, for different simulation times: $ \\overline{t}=76 $ (left column), $ \\overline{t}=82 $ (mid column), and  $ \\overline{t}=88 $ (right column). The first row corresponds to the simulation with no magnetic twist, $c=0$. In descending order, the remaining rows correspond to the simulations with magnetic twist parameter of $ c =$~0.1, 0.2, and 0.4. Density is normalized to the external density.} \n\\label{Fig_Densitytwist}\n\\end{figure}\n\nOn the other hand, when magnetic twist is included the KHi vortices are not strictly perpendicular to the flux tube axis and can also develop in the $z$-direction \\citep{Terradas18}. To illustrate this fact, we show in Fig.~\\ref{Fig_rholong}  a longitudinal cut of density of the simulation with $c=0.2$. This cut is done for the same time as that in the third column of Fig.~\\ref{Fig_Densitytwist}.\n\n\n\n\\begin{figure}[htbp!]\n\\includegraphics[width=0.6\\columnwidth]{rholong.pdf}\n\\centering\n\\caption{Longitudinal cut of density at $y=0$ for the simulation with magnetic twist parameter, $c = 0.2$ when $ \\overline{t}=88 $, i.e., the same time as in the right column of Fig.~\\ref{Fig_Densitytwist}.  Density is normalized to the external density.}\n\\label{Fig_rholong}\n\\end{figure}\n\n\n\\subsection{Delay or suppression of the KHi}\n\nWe aim to quantify how magnetic twist postpone the KHi onset time.  Theoretically, a component of the magnetic field along the direction of the shear flow has a stabilizing effect on the KHi \\citep[see, e.g.,][]{Chandrasekhar61}. Significant efforts have been made to understand this problem for the KHi driven by transverse MHD waves. However, the analytical obtainment of the onset time of the KHi in a cylindrical tube with time-varying flows under the presence of a twisted magnetic field remains a challenge. \\citet{Browning84} analytically obtained an expression for the onset time of the KHi for time-varying flows in Cartesian geometry and with a straight field. \\citet{Soler10} considered the KHi at the boundary of a cylindrical tube, but with constant azimuthal flows. \\citet{Soler10} studied the effect of magnetic twist by considering a Cartesian analog with an inclined magnetic field and found that twist can decresase the growth rates and even suppress the instability. \\citet{Zaqarashvili15} studied the instability of a cylindrical, twisted, and rotating jet. They found that jets are unstable to the KHi only when the kinetic energy of rotation is larger than the magnetic energy of the twist. \\citet{Hillier2019MNRAS} studied the linear stability of a discontinuous oscillatory shear flow  in Cartesian geometry in the presence of a uniform magnetic field. They show that parametric instabilities can also grow in addition to the KHi. \\citet{Barbulescu19} also considered a Cartesian model of time-varying flows, but including an inclined magnetic field on one side of the interface to mimic the effect of twist, as was previously done by \\citet{Soler10} for constant flows. \\citet{Barbulescu19} conclude that the magnetic shear may reduce the instability growth rate, but it cannot   completely stabilize the interface, contrary to the constant flow case of \\citet{Soler10}.\n\n\nTo understand the effect of twist on the triggering of the KHi, here we follow the method introduced by \\citet{Terradas18}. Their method is based on studying the excitation of different azimuthal wave numbers. This approach was also used in \\citet{Antolin19} and in  Paper~\\citetalias{Diazsoler21}. We proceed as follows. In a cross-sectional cut at the tube center, $z=0$, we calculated the perpendicular component of velocity to the magnetic field lines, $v_\\perp$, at the middle of the transition region, $ r=R $, as a function of the azimuthal angle, $\\varphi$, from $\\varphi = 0$ to $\\varphi =  2\\pi$. After that, we computed the discrete Fourier transform to the data using the fast Fourier transform (FFT) algorithm with the Scipy module \\citep{Virtanen20}. Following the notation of \\citet{Terradas18}, the discrete Fourier transform is\n\\begin{equation}\nG\\left(p\\right)=\\sum^{N-1}_{k=0} g\\left(k\\right)\\exp{\\left(\\frac{-2\\pi i p k}{N}\\right)},\n\\label{Gdep}    \n\\end{equation}\nwhere $ N $ is the number of sample points, $ g\\left(k\\right) $ is the angular sampling of $v_\\perp$, and $ p $ is an integer that plays the role of the azimuthal wavenumber. Unlike in Paper \\citetalias{Diazsoler21}, here, we consider positive and negative values of $p$, namely $ p=0,\\pm 1, \\pm 2, ... , \\pm (N-1) $, since  twist breaks the degeneracy in the sign of the azimuthal wavenumber.   $p=0$ is the torsional plus the sausage mode. $ p=\\pm 1 $ are the kink modes, and $ |p| \\geq 2 $ are the fluting modes \\citep[see, e.g.,][]{roberts19}. In the presence of magnetic twist, the torsional mode and the fast sausage mode are coupled linearly \\citep[see, e.g.,][]{Goossens11}. However, the contribution of the fast sausage mode is weak because the magnetic twist considered here is relatively weak, even for $ c=0.4 $. Thus, for $p=0$, the torsional mode dominates over the fast sausage mode.\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{modosperp.pdf}\n\\centering\n\\caption{\\textit{Top}: temporal evolution of the Fourier coefficient associated with the torsional mode, $ p=0 $, at $z=0$ and  $r=R$ for the twist parameter ranging from $c=0$ to $c=0.4$. \\textit{Bottom}: same as the top panel but for the sum of the first twenty positive and the first twenty negative Fourier coefficients excluding $p=0$. Arbitrary units are used.}\n\\label{Fig_Modos}\n\\end{figure}\n\nFigure~\\ref{Fig_Modos} shows the results of the azimuthal Fourier analysis. To better visualize  the results, in the top panel, we plot the temporal evolution of the $ p=0 $ mode alone. In the bottom panel, we plot the temporal evolution of the sum of the first twenty positive modes and the first twenty negative modes, excluding the $p=0$ mode. We checked that adding more modes to the sum does not affect the results.\n\nAs expected, we obtain that the $p=0$ mode is dominant during the linear evolution of phase-mixing independently of the magnetic twist. The dominance of the $p=0$ mode lasts until the onset  of KHi, which excites higher azimuthal wave numbers. At that moment, the sum of the $p\\neq 0$ Fourier coefficients begins to display significant variations. These results are consistent with those of \\citet{Terradas18} and \\citet{Antolin19} for the standing kink mode and also agree with those of Paper~\\citetalias{Diazsoler21}. During the development of KHi and the subsequent turbulence the initially low-amplitudes of the sum of the $p\\neq 0$ Fourier coefficients become comparable with or even larger than the amplitude of the $p=0$ mode. Moreover, we find that the $p=0$ mode loses its periodicity.\n\nSimilarly to the results of Paper~\\citetalias{Diazsoler21}, independently on the presence or absence of magnetic twist, we verified that the excitation of high-order azimuthal wave numbers initially occurs in the nonuniform transition region. This analysis is not shown here for simplicity. We also verified the persistence of the $p=0$ mode in the core where the KHi cannot develop. However, once the turbulence expands into the core, high-order azimuthal wave numbers are also excited in the core.  This is consistent with the results from Fig. \\ref{Fig_averagedens} for the untwisted case.  \n\nTherefore, the azimuthal Fourier analysis confirms what was already visualized in Fig.~\\ref{Fig_Densitytwist}: the inclusion of the magnetic twist  delays the development of the KHi. Indeed, if magnetic twist is sufficiently weak, the KHi is just delayed but the dynamics is similar to that in the absence of twist, although evolving at a slower pace. In a sense, this conclusion is not completely new, although this is the first time that the role of twist  in the nonlinear evolution of standing torsional Alfv\\'{e}n waves has been investigated. Similar results can be found in simulations of nonlinear kink waves in twisted flux tubes \\citep[see, e.g.,][]{Howson17a,Terradas18}. For the different twist profiles considered by \\citet{Howson17a}, they find that the KHi vortices grow less as the magnetic twist is increased. \\citet{Howson17a} also find that the location of the maximum twist may affect at the development of KHi vortices. \\citet{Terradas18} also studied nonlinear kink  waves considering the same twist profile used here. Similarly to \\citet{Howson17a}, \\citet{Terradas18} find that the KHi vortices grow less when the magnetic twist increases. \n\nWhen the  magnetic twist is strong enough, however, the physical scenario can be different. The simulations with strong magnetic twist suggest that the KHi can be suppressed altogether even if the phase-mixing generated shear flows should remain unstable according to linear theory \\citep[see][]{Barbulescu19}. Although the KHi itself can linearly be triggered in a local fashion, its perturbations are not allowed to fully grow into the nonlinear regime when twist provides a strong enough tension force \\citep{DiazSuarezsolerlet2021}. The important role of magnetic tension in  nonlinearly preventing the growth of the KHi vortices was already discussed by, e.g., \\citet{Galinsky94}, \\citet{Ryu2000}, and \\citet{Hillier19}. In our model, the nonlinear suppression of the KHi seems to occur for $c\\gtrsim 0.3$. \n\nSince the KHi suppression for strong twist is a nonlinear phenomenon, we cannot exclude that the initial amplitude of the excited torsional wave may affect the subsequent stability. Indeed, one should expect that for sufficiently large amplitudes, the KHi may break the stabilizing effect of magnetic tension even if twist is strong. So, we speculate that the ability of the KHi to grow might depend on the relation between the kinetic energy of the shear flow that drives the KHi and the magnetic energy of the twisted magnetic field, in a similar fashion as in the model by \\citet{Zaqarashvili15}. The investigation of the effect of the initial amplitude is not done here and is left for forthcoming works.\n\n\n\\subsection{Turbulence and vorticity generation}\n\\label{Sub_vortic_twist}\n\nThe delay or suppression of the KHi has important consequences in the dynamics of the flux tube.  In our model, the generation of turbulence, which results from the KHi evolution, can just be delayed or be absent depending on the strength of magnetic twist. In addition, magnetic twist may affect the subsequent evolution of turbulence. To explore that, we resort again to the vorticity to study how magnetic twist affects the evolution of turbulence.\n\n\nFigure \\ref{Fig_Vorty_local_twist} shows cross-sectional cuts of $\\omega^{2}\\left(\\mathbf{r},t \\right) $ at the tube center, $z=0 $, for the  same three times as in Fig.~\\ref{Fig_Densitytwist}. In each row, the strength of magnetic twist is different and sorted in increasing order from  top to bottom. For comparison purposes, we also included in the first row the results from the simulation without magnetic twist. Unlike in Fig.~\\ref{Fig_Densitytwist}, we find slightly different vorticity patterns for the cases with a magnetic twist of $c=0.3$ and $c=0.4$. This will be highlighted later. Figure~\\ref{Fig_Vorty_local_twist} evidences that magnetic twist not only plays a role in delaying or suppressing the KHi, but also it heavily affects the value and spatial distribution of vorticity when turbulence is present. Another plot that helps us in the present discussion is Fig.~\\ref{Fig_Vorty_twist}. There, we show the temporal evolution of the vorticity squared integrated in the whole computational domain, $ \\Omega^{2} $, as a function of the computational time for all simulations. The curve for $c=0$ plotted in Fig.~\\ref{Fig_Vorty_twist} is the same as that already shown in the small panel of Fig.~\\ref{Fig_vortynotwist}.\n\n\\begin{figure*}[htbp!]\n\\includegraphics[width=1.7\\columnwidth]{vortymulti5.pdf}\n\\centering\n\\caption{Cross-sectional cuts at $z=0$ of the logarithm with base 10 of the vorticity squared, $\\omega^{2}\\left(\\mathbf{r},t \\right) $, for different simulation times: $ \\overline{t}=76 $ (left column), $ \\overline{t}=82 $ (mid column), and  $ \\overline{t}=88 $ (right column). The first row corresponds to the simulation with no magnetic twist, $c=0$. In descending order, the remaining rows correspond to the simulations with magnetic twist parameter of $ c =$~0.1, 0.2, 0,3, and 0.4. Arbitrary units are used.}\n\\label{Fig_Vorty_local_twist}\n\\end{figure*}\n\n\n\\begin{figure}[htbp!]\n\\resizebox{\\hsize}{!}{\\includegraphics{evolutionvorticity.pdf}} \n\\centering\n\\caption{Temporal evolution of the vorticity squared integrated in the whole computational domain, $ \\Omega^{2} $, for the twist parameter ranging from $c =0$ to $c=0.4$. The curve for $c=0$ (red dot-dashed line) is the same as in the small panel of Figure~\\ref{Fig_vortynotwist}. All curves are normalized to their  values at $t=0$.}\n\\label{Fig_Vorty_twist}\n\\end{figure}\n\nTo start with, we focus on comparing the simulations with $c=0$ (no twist) and $c=0.1$ (weakest magnetic twist considered). Figure~\\ref{Fig_Vorty_twist} indicates that the onset  of the KHi occurs  practically at the same time in both simulations, since the sharp increase in vorticity in the two cases is almost simultaneous. Thus,  the delaying effect of twist when $c=0.1$ is still not very relevant. However, slightly smaller values of integrated vorticity are found when $c=0.1$ compared with those of $c=0$.\nIndeed,  the first two rows in Fig.~\\ref{Fig_Vorty_local_twist} show that the spatial distribution of vorticity once turbulence sets in is very similar in the two cases. Nevertheless, smaller vorticity structures appear when magnetic twist is absent. This effect is clearly visible when one compares the two simulations when $ \\overline{t}=88 $, a time for which turbulence is already well established. \n\nThe effect of magnetic twist becomes more pronounced when we consider the simulation with $c=0.2$.  Figure~\\ref{Fig_Vorty_twist} shows that the KHi onset is significantly delayed with respect to the cases with $c=0$ and $c=0.1$ discussed before.  Figure~\\ref{Fig_Vorty_local_twist} plainly evidences that the dynamics  advances at a slower pace when $c=0.2$. For instance, the spatial distribution of $\\omega^{2}\\left(\\mathbf{r},t \\right) $ at $ \\overline{t}=88 $ is  similar to that found at $ \\overline{t}=76 $ in the simulation with $c=0.1$. The case with $c=0.2$ corresponds to an intermediate situation in the sense that twist already has an important impact in delaying the KHi onset  and in the appearance and evolution of turbulence. However, magnetic twist is still not strong enough to avoid the KHi growth and so prevent the generation of turbulence.\n\nThe simulations with stronger magnetic twist, $c=0.3$ and $c=0.4$, represent a completely different scenario in which turbulence is not driven because the KHi is not allowed to grow. Even for the largest considered simulation times, the shape of $\\omega^{2}\\left(\\mathbf{r},t \\right) $ in the two simulations is essentially in the form of  concentric rings, which is the typical structure caused by phase-mixing. However, when $ \\overline{t}=88 $ a close look at the shape of $\\omega^{2}\\left(\\mathbf{r},t \\right) $ in the case with $c=0.3$  reveals that some of the phase-mixing-driven concentric rings are slightly deformed in a wavy fashion. Such a deformation is not seen in the simulation with $c=0.4$ at that time. This deformation in the vorticity rings  when $c=0.3$ is caused by the very local triggering of the KHi within the nonuniform transition layer. Such a local inception of the KHi is not appreciable as discernible deformations in density. Nonetheless, vorticity, which is computed from the velocity field, is much more sensitive to the very initial appearance of the KHi.\n\nWe have run the simulation with $c=0.3$ up to larger times than those displayed in  Fig.~\\ref{Fig_Vorty_local_twist} and have confirmed that the KHi does not grow and vortices do not form. As explained before, the reason is the nonlinear suppresion of the KHi by magnetic tension \\citep[see, e.g.,][]{Galinsky94,Ryu2000,Hillier19,DiazSuarezsolerlet2021}. Thus, turbulence is not driven. In the simulation with $c=0.4$, the local inception of the KHi is eventually observed at later times, but the perturbations are neither allowed to grow  and turbulence is equally absent. However, Fig.~\\ref{Fig_Vorty_twist} shows that vorticity keeps increasing for both $c=0.3$ and $c=0.4$ at large times. The reason is that the linear phase mixing remains at work for large times and it is responsible for the increase of vorticity. Nevertheless, the increase of vorticity owing to phase mixing alone occurs at a much slower pace than when the KHi is present.\n\nAgain, our results about the effect of magnetic twist on the vorticity are comparable with those obtained from nonlinear kink wave simulations \\citep{Howson17a,Terradas18}. \\citet{Howson17a} showed cross-sectional cuts of the magnitude of vorticity at the loop apex for a twisted and an untwisted case at two different simulation times. In both cases, they find big and small vortices due to KHi in their untwisted case, but their twisted case does not show small vortices. Thus, the magnetic twist is suppressing the small vortices associated with KHi. \\citet{Howson17a} also find that the increase in vorticity due to KHi happens earlier when twist is absent.\n\n\n\\subsection{Kinetic energy cascade to small scales}\n\nIn Paper~\\citetalias{Diazsoler21} we showed that the onset of turbulence in the flux tube accelerates the cascade of energy from large to small spatial scales. Such a process is slower before the appearance of turbulence, when phase mixing is the only working mechanism. The delay or even suppression of the turbulent regime when magnetic twist is included should affect the rate at which small scales are generated.\n\nWe investigate how kinetic energy is transported to small scales depending on the strength of twist. For this purpose, we only consider the kinetic energy associated with motions perpendicular to the magnetic field lines, namely $ E =\\frac{1}{2}\\rho v^{2}_{\\perp}$. We exclude the contribution from the other components of velocity because we are mainly interested in the Alfv\\'enic part of the energy. The presence of  twist linearly couples  torsional Alfv\\'{e}n waves and  fast magneto-acoustic sausage waves \\citep[see, e.g.,][]{Sakurai91}. Fast magneto-acoustic sausage waves would predominantly perturb the radial component of velocity. In turn, Alfv\\'en waves and slow magneto-acoustic waves are nonlinearly coupled through the ponderomotive force \\citep[see, e.g.,][]{hollweg1971}. Slow waves would mainly affect the longitudinal component of velocity.\n\nWe  consider  a cross-sectional cut at the tube centre, $z=0$, and average the Alfv\\'{e}nic kinetic energy in the azimuthal direction using 512 radial cuts. Thus, the angular resolution is approximately $ \\sim 0.7^{\\circ}$. We denote the averaged energy by $ \\overline{E}$, which is a function of $r$ and $t$. Then, we apply the continuous Fourier transform in the radial direction, discretized due to the limited numerical resolution. Following Paper~\\citetalias{Diazsoler21}, it can be defined as\n\\begin{align}\n\\overline{E}_F(k,t)\\approx\\frac{\\Delta r}{\\sqrt{2\\pi}}\\exp{\\left(-i k r_{0}\\right)} \\sum^{N-1}_{m=0} \\overline{E}(r,t)\\exp{\\left(-\\frac{2\\pi i m k}{N}\\right)},\n\\label{fftcont}\n\\end{align}\nwhere $N$ is the number of samples, $ k $ is the radial wave number, $ \\Delta r $ is the spatial resolution, and $ r_{0} $ is the upper limit of the radial domain. The summatory in Eq.~(\\ref{fftcont}) is exactly the 1D discrete Fourier transform, which we compute using the fft module of Scipy \\citep{Virtanen20} based on the FFT algorithm \\citep{Cooley65}. Finally, we calculate the modulus of $ \\overline{E}_F(k,t) $ and normalize it with respect to the maximum value in the spectrum at each time.\n\nFigure~\\ref{Fig_Scales} displays the results of the Fourier transform for all simulations at the last common simulation time for which the vorticity still increases in all simulations, $ \\overline{t}=87 $. The reason for choosing this time is that, for larger times, the simulations with $c=0$ and $c=0.1$ are already significantly affected by numerical dissipation (see the decrease of the integrated vorticity in Fig.~\\ref{Fig_Vorty_twist}), which may  also affect the results of the Fourier transform at the small scales.\n\n\\begin{figure}[htb!]\n\\resizebox{\\hsize}{!}{\\includegraphics{nokmaxfft2v87.pdf}} \n\\centering\n\\caption{Azimuthally-averaged amplitude spectrum of the Alfv\\'enic (perpendicular to the magnetic field) part of the kinetic energy for the twist parameter ranging from $c =0$ to $c=0.4$. We considered the last common  time for which vorticity increases in all simulations, $\\bar{t}=87$. Arbitrary units are used.}\n\\label{Fig_Scales}\n\\end{figure}\n\nAt large scales, $kR < 10^0 $, the amplitude spectrum is independent on the wave number. For intermediate scales, $ 10^0 < kR < 10^2 $, we do not find that the values of the amplitude spectrum are ordered in any particular way regarding the twist parameter, $c$. The temporal evolution of the amplitude spectrum (not shown here) shows oscillations by which the various curves with different $c$ exchange their relative positions depending on the considered time. The reason for such a behavior is unclear to us. The understanding of the behavior of the amplitude spectrum in these intermediate scales would require additional research.\n\nWe are more interested in the spectrum at small scales, $ 10^2 < kR < 10^3 $, where the amplitude spectrum decreases as the wavenumber increases. In the absence of  twist, the values of the amplitude spectrum are larger than those obtained when  twist is considered. In addition, we find a decreasing trend with respect to the twist parameter, $c$, so that the stronger the magnetic twist, the lower the values of the amplitude spectrum. These results just confirm again the relevant role of magnetic twist in suppressing the generation of small scales and in the rate at which Alfv\\'enic energy is deposited into those small scales. We intentionally avoided $ kR >  10^3 $ in our analysis owing to the possible role of numerical dissipation at the smallest scales.\n\n\\section{Concluding remarks}\n\\label{Sect_con}\n\nWe conclude the investigation started in Paper~\\citetalias{Diazsoler21} about the ability of torsional Alfv\\'en waves to drive turbulence in coronal loops. Because of plasma and\/or magnetic nonuniformity, standing torsional Alfv\\'{e}n waves undergo the process of phase mixing, generating shear flows perpendicular to the magnetic field direction in adjacent radial positions as time increases.  Eventually, the shear flows trigger the KHi, as \\citet{HeyvaertPriest83} and \\citet{Browning84} first predicted.\n\nIn the absence of magnetic twist (the case studied in Paper~\\citetalias{Diazsoler21}), the KHi can grow nonlinearly without opposition. The nonlinear evolution of the KHi naturally induces turbulence. First, large KHi eddies are formed, which later break into smaller eddies in a cascade-like dynamics. Mixing of plasma takes place.  An important increase in vorticity is found and the generation of small scales speeds up compared with the initial phase dominated by phase mixing alone. Turbulence evolves perpendicularly to the magnetic field, so that for a straight magnetic field turbulence is pseudo-2D.\n\nIn the present paper, we have explored the role of magnetic twist. While the linear evolution of phase mixing is similar in the presence or in the absence of twist, the dynamics of the subsequent KHi growth and turbulence generation depends strongly on the strength of the twist.\n\nIf magnetic twist is sufficiently weak, the onset  and growth of the KHi is just delayed, but the dynamics is similar to that in the case of a straight magnetic field. In this regime, the stronger the magnetic twist, the longer the delay.  Although the vorticity still increases dramatically during the development of  turbulence, the increase is smaller compared with  that in absence of magnetic twist. Small scales are still quickly generated by turbulence, although at a slower pace than when twist is absent. Turbulence still evolves perpendicularly to the magnetic field lines, but since the field is twisted, turbulence is no longer confined to perpendicular planes to the tube axis. \n\nConversely, under the presence of a strong enough magnetic twist, the scenario is completely different. If the strength of  magnetic twist surpasses a critical value, or a critical twist angle, the KHi vortices cannot grow. The KHi itself is still locally excited by the phase-mixing generated shear flows, but the tension of the twisted magnetic field prevents now the nonlinear development of the instability \\citep[see, e.g.,][]{Galinsky94,Ryu2000,Hillier19,DiazSuarezsolerlet2021}.  As a consequence, there is no enhancement of vorticity, neither turbulence is driven.  Thus, the generation of small scales is not accelerated and continues slowly evolving at the rate dictated by phase-mixing. \n\nAlthough the effect of magnetic twist may oppose the process, the nonlinear evolution of torsional Alfv\\'en waves remains as a viable mechanism to induce turbulence as long as coronal loops are weakly twisted. There are other effects that should be explored in the future. For instance, the tension of the background field in curved coronal loops may also affect the triggering and evolution of the KHi and associated turbulence. It could be also interesting to investigate turbulence generation in multithreaded loops \\citep[see, e.g.,][]{Ofman09}.\n\n\n\\begin{acknowledgement}\nThis publication is part of the R+D+i project PID2020-112791GB-I00, financed by MCIN\/AEI\/10.13039\/501100011033. S.D. acknowledges the financial support from MCIN\/AEI\/10.13039\/501100011033 and European Social Funds for the predoctoral FPI fellowship PRE2018-084223. We also thank the UIB for the use of the Foner cluster. For the simulation data analysis we have used VisIT \\citep{VisIt} and Python 3.6. In particular, we have used Matplotlib \\citep{Hunter07}, Scipy \\citep{Virtanen20}, and Numpy \\citep{Harris20}. We are thankful to B. Vaidya and his contributors for the tool pyPLUTO.\n\\end{acknowledgement}\n\n\n\\bibpunct{(}{)}{;}{a}{}{,} \n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\nX-ray Binaries (hereafter XrBs) display cycles of strong activity, where their luminosity increases by several orders of magnitude and their spectral shape changes drastically on long timescales, before decreasing back to quiescence. We call this entire phase an outburst. At the beginning of such an outburst, they are in the so-called hard state where their X-ray spectra are dominated by a power-law component with a hard photon index $\\Gamma<2$ \\citep{Remillard06}. During this state, they also show flat or slightly inverted radio spectra \\citep[e.g.,][]{CorbelFender02}, interpreted as self-absorbed synchrotron emission from collimated, mildly relativistic jets \\citep{BK79}. At some point, the X-ray spectrum of these objects undergoes a smooth transition from this power-law dominated spectral shape, to a dominant blackbody of temperature $\\sim~1$~keV only. In addition, steady radio emission disappears, suggesting a quenching of the jets \\citep{Corbel04, Fender04, Fender09}. X-ray binaries remain in this soft state until a decline in luminosity makes them transit back to a hard state at the end of the outburst, along with reappearance of the jets. This surprising behavior has been observed multiple times in the past decades, and in dozens of different objects, where some have even undergone multiple outbursts \\citep[see][for a global overview]{Dunn10}. What is even more striking in these outbursts, is that they seem to be very similar in different objects, while being different from one outburst to another in the same object! \\\\\n\n\nA general scenario has been proposed by \\citet{Esin97}, where changes in the accretion flow geometry provokes the spectral variations. In this view, the interplay between two different accretion flows is responsible for the spectral changes in the disk: in the outer parts, a cold standard accretion disk \\citep[SAD, ][]{SS73} extends down to a given truncation (or transition) radius where an advection dominated hot flow\\footnote{The inner hot flow is often referred to as a \"hot corona\". However, this designation remains ambiguous and we choose not to utilize it.} takes place \\citep{Ichimaru77, Rees82}. The inner hot flow is expected to be responsible for the power-law component, while the outer cold flow produces the blackbody radiation. While the presence of a SAD in the outer regions of the disk is highly accepted to date \\citep{Done07}, the physical properties of the advection dominated inner flow remain an open question.\n\nBetween Slim disks \\citep{Abramowicz88}, ADAFs \\citep{Narayan94}, ADIOS \\citep{BB99}, LHAFs \\citep{Yuan01} or more peculiar models \\citep[e.g.,][]{Meyer00, Lasota01}, no satisfactory explanation has been provided so far \\citep{Yuan14}. A discussion about the major models and their current state can be found in \\citet{paperII}. Many questions remain open but in this article we focus on: (1) reproducing the X-ray spectral shape of all the generic spectral states, (2) explaining the correlated accretion-ejection processes through their observables, i.e. radio and X-rays fluxes. \\\\\n\n\n\nIn this work we consider an accretion flow extended down to the inner-most stable circular orbit, and thread by a large scale vertical magnetic field $B_z$.\nIt is well known that matter can only accrete by transferring away its angular momentum. This can be achieved by few physical mechanisms, namely internal turbulent (\"viscous\") torques and magnetic torques from an outflow:\n\\begin{itemize}\n\\item When accretion is mostly due to internal (turbulent) viscosity, angular momentum is transported radially. This produces an optically thick and geometrically thin accretion disk, a SAD, which is observed as a cold multicolor-disk blackbody. The production of winds by SAD is still a debated question, but few simulations and observations have shown the possible existence of winds from standard disks (see discussion in section~\\ref{sec:innerSAD}). However, these winds cannot explain the powerful jets associated with XrBs so that the SAD suits perfectly for the jet-less thermal states in XrBs, i.e. soft states.\n\\item Alternatively, self-confined super-Alfv\\'enic jets can also provide a feedback torque on the disk, carrying away vertically both energy and angular momentum. This accretion mode, referred to as jet-emitting disks \\citep[JED, ][and subsequent work]{Ferreira95}, presents a supersonic accretion speed. Therefore, for the same accretion rate, it has a much smaller density than the SAD, leading to optically thin and geometrically thick disks. Disks accreting under this JED mode are thus good candidates to explain power-law dominated and jetted states in XrBs, i.e. hard states.\n\\end{itemize}\n\n\\noindent The magnetic field strength is characterized by the mid-plane magnetization $\\mu(r)= B_z^2\/\\mu_o P_{tot}$, where $P_{tot}$ is the total pressure, the sum of the kinetic plasma pressure and the radiation pressure. At large magnetization, the SAD can no longer be maintained as magneto-centrifugally driven jets are launched: a JED arises \\citep{Ferreira95,Ferreira97}. Full MHD calculations of JEDs have shown that the transition occurs around $\\mu \\sim 0.1$ \\citep{Casse00a,Casse00b,Lesur13}. \\\\\n\n\n\n\nA global scenario based on these possible dynamical transitions in accretion modes has then been proposed to explain XrB cycles (\\citealt{Ferreira06}, hereafter paper I; \\citealt{Petrucci08}). What would be the causes of the evolution of the disk magnetization distribution $\\mu(r)$ is still a highly debated question. The main uncertainty comes from the interplay between the magnetic field advection and diffusion in turbulent accretion disks, geometrically thin or thick and with or without jets. Modern global 3D MHD simulations do show that large scale magnetic fields are indeed advected \\citep{Avara16, ZhuStone18}, but these simulations are always done on quite short time scales, up to few seconds, and it is hard to scale them to the duration of XrB cycles, typically lasting over several month. In this paper, we assume that cycles result from transitions in accretion modes and focus on their observational consequences. \\\\\n\n\n\\citet{Petrucci10} computed the thermal states of a pure JED solutions and successfully reproduced the spectral emission, jet power and jet velocity during hard states of Cygnus X-1. However, their calculations were done assuming a one temperature (1T) plasma, but the necessity of a (2T) plasma seem inevitable to cover the large variation of accretion rate expected during an entire outburst \\citep{Yuan14}.\n\n\\citet{paperII}, hereafter paper II, extended this work by developing a two-temperature (2T) plasma code that computes the disk local thermal equilibrium, including advection of energy and addressing optically thin-to-thick transitions in both radiation and gas supported regimes.\nFor a range in radius and accretion rates, they showed that JEDs exhibit three thermal equilibria, one thermally unstable and two stable ones. Only the stable equilibria are of physical importance \\citep{Frank92}. One solution consists of a cold plasma, leading to an optically thick and geometrically thin disk, whereas the second solution describes a hot plasma, leading to an optically thin and geometrically thick disk. Due to the existence of these two thermally stable solutions a hysteresis cycle is naturally obtained. But large outbursting cycles, such as those exhibited by GX 339-4, cannot be reproduced (paper II). Nevertheless, JEDs have the striking property of being able to reproduce very well hard states spectral shape, all the way up to very luminous hard states $L > 30 \\% L_{Edd}$.\\\\\n\n\nHowever, SAD-JED local transitions are expected to occur locally on dynamical time scales, typically $\\sim 1$~ms Kepler orbital time at $10~R_g$, whereas hard-to-soft transitions involve time scales of days or even weeks. This implies that, at any given time, the disk must be in some hybrid configuration with some regions emitting jets, while others do not. It is expected that jets, namely magnetocentrifugally-driven flows \\citep{BP82}, are only launched from the disk innermost regions. This translates into a hybrid configuration where an inner JED is established from the last stable orbit $R_{in}$ until an unknown transition radius $R_J$, and then surrounded by an outer SAD until $R_{out}$. The exact location of the transition $R_J$ depends on the global response of the magnetic field $B_z$ to accretion rate evolution at the outer edge $\\dot{M}_{out}$, two unknowns. Hence $R_J$ is treated here as a free parameter of the model. Such radial transition between two flows has already been studied in the context of non-magnetized accretion flows, advocating for mechanism such as evaporation or turbulent diffusion at the origin of the transition \\citep[see, e.g.,][]{Meyer94, Honma96}. We study however configurations where the magnetization $\\mu$ is large (near equipartition) and uniform in the JED region, and drops at the transition radius $R_J$, organizing the two-flow structure. Although the main properties of isolated JED and SAD are well understood, hybrid configurations imply mutual interactions that need to be described. For instance, part of cold radiation emitted from the SAD region must be intercepted by the geometrically thick JED and provides an additional cooling term that might change its general properties. \\\\\n\nIn this paper, we explore the observational signatures of disk configurations with an inner JED and an outer SAD. Section~\\ref{sec2} describes this hybrid configuration, including interactions between the two regions, and explores some of its dynamical consequences. Section~\\ref{sec3} presents the procedure followed to simulate and fit synthetic X-ray data from our theoretical spectra as well as to estimate the jet radio emission. Section~\\ref{sec4} is devoted to the exploration of the parameter space, by varying the disk accretion rate $\\dot{M}_{in}$ and transition radius $R_J$. Playing with these two parameters allows to completely cover the disk fraction luminosity diagram \\citep[hereafter DFLD, ][]{Kording06}. As an illustrative example, we apply our model and reproduce canonical states of GX 339-4, both in X-ray spectral shape and radio fluxes. We end with concluding remarks in section~\\ref{sec:Ccl}.\n\n\n\\section{Hybrid disk configuration: internal JED and external SAD} \\label{sec2}\n\n\\subsection{General properties}  \n\\label{sec:GeneralPpts}\n\nAs introduced in paper II, we consider an axisymmetric accretion disk orbiting a black hole of mass $M$. For simplicity, the disk is assumed to be in global steady-state so that any radial variation of the disk accretion rate $\\dot{M}(R)$ is only due to mass loss in outflows. We define $H(R)$ the half-height of the disk, $\\varepsilon(R) = H\/R$ its aspect ratio, $\\dot{M}(R) = - 4 \\pi R u_R \\Sigma$ the local disk accretion rate, $u_R$ the radial (accretion) velocity and $\\Sigma = \\rho_0 H$ the vertical column density with $\\rho_0$ the mid plane density. Throughout the paper, calculations are done within the Newtonian approximation. Moreover, and for the sake of simplicity, the disk is assumed to be always quasi-Keplerian with a local angular velocity $\\Omega \\simeq \\Omega_K = \\sqrt[]{G M R^{-3}}$, where $G$ is the gravitational constant. \\\\\n\nThe disk is assumed to be thread by a large scale vertical magnetic field $B_z(R)$. We assume that such a field is the result of field advection and diffusion and we neglect thereby any field amplification by dynamo.\nClearly, the existence of cycles shows that some evolution is ongoing within the disk. However, the timescales involved (days to months) are always much longer than accretion timescales inferred from the X-ray emitting regions. Thus, as for any other disk quantity, the local magnetic field is assumed to be stationary on dynamical time scales (Keplerian orbital time). \\\\\n  \n\n\\begin{figure}[h!\n   \\centering\n   \\includegraphics[width=\\columnwidth]{config.pdf}  \n      \\caption{Example of hybrid disk configuration in the JED-SAD paradigm. The inner disk regions are in a jet-emitting disk (JED) mode, up to a transition radius $r_J$, beyond which a standard accretion disk (SAD) is settled. The disk scale height $H(R)$ is accurately displayed, while colors corresponds to the central electronic temperature $T_e$ in Kelvin. The disk switches from an outer optically thick, geometrically thin jet-less disk to an inner optically thin, geometrically thick disk launching self-confined jets (not shown here). This solution has been computed for a transition radius $r_J=15$ and a disk accretion rate $\\dot{m}_{in}=0.1$ at the disk inner radius $r_{in}=2$ (see section~\\ref{sec:GeneralPpts} for more details). Other similar examples are shown in Fig.~\\ref{fig:JoliesImages} for different pairs $(\\dot{m}_{in}, r_J)$.}\n         \\label{fig:config}\n\\end{figure\n\nThe hybrid disk configuration is composed of a black hole of mass $M$, an inner jet-emitting disk from the last stable orbit $R_{in}$ to the transition radius $R_{J}$ and an outer standard accretion disk from $R_{J}$ to $R_{out}$. The system is assumed to be at a distance $D$ from the observer. In the following, we adopt the dimensionless scalings: $r = R\/R_g$, $h = H\/R_g= \\varepsilon r$, where $R_g = GM\/c^2$ is the gravitational radius, $m = M\/M_{\\odot}$, and $\\dot{m} = \\dot{M}\/\\dot{M}_{Edd}$, where $\\dot{M}_{Edd} = L_{Edd} \/ c^2$ is the Eddington accretion rate\\footnote{Note that this definition does not include the accretion efficiency, usually of the order $\\sim 10\\%$ for a Schwarzschild black hole. This means that reaching Eddington luminosities would require $\\dot{m}_{in} \\gtrsim 10$ (see Fig.~\\ref{fig:Lxmdot}).} and $L_{Edd}$ is the Eddington luminosity. Since GX 339-4 appears to be an archetypal object, we decided to concentrate only on this object. We thus use a black hole mass $m = 5.8$, a spin $a = 0.93$ corresponding to $r_{in} = 2.1$ and a distance $D = 8$~kpc \\citep{Miller04}\\footnote{See also \\citet{Munoz08}, \\citet{Parker16} or \\citet{Heida17} for more recent estimations.}. All luminosities and powers are expressed in terms of the Eddington luminosity $L_{Edd}$. An example of disk configuration is shown in Fig.~\\ref{fig:config} for $\\dot{m}_{in} = \\dot{M}_{in}\/\\dot{M}_{Edd} = 0.1$ and $r_J = R_J \/ R_g = 15$.\\\\\n\nOur goal is then to compute, as accurately as possible, the radial disk thermal equilibrium from $r_{out}$ to $r_{in}$ by taking into account the known dynamical properties associated with each accretion mode. The inflow-outflow sctructure is described by the following midplane quantities:\n\\begin{eqnarray}\n\\mu &= &\\frac{B_z^2\/\\mu_0}{P_{tot}} = \\frac{B_z^2 \/ \\mu_0}{P_{gas} + P_{rad}}   \\nonumber \\\\\n\\xi &=& \\frac{d \\ln \\dot{m}}{d \\ln r} \\nonumber \\\\\nm_s &=& \\frac{-u_R}{c_s} = \\frac{-u_R}{\\Omega_K H} =m_{s,turb} + m_{s, jet} = \\alpha_{\\nu} \\varepsilon + 2 q \\mu \\\\\n b &= & \\frac{2 P_{jet}}{P_{acc, JED}} \\nonumber\n\\end{eqnarray}\nwhere $\\mu$ is the disk magnetization, $\\xi$ the local ejection index, $m_s$ the sonic accretion Mach number and $b$ the fraction of the JED accretion power $\\displaystyle P_{acc, JED} = \\left [ \\frac{G M \\dot{M}}{2 R} \\right ]^{R_{in}}_{R_{J}}$ that is carried away by the jets in the JED. The parameter $q \\simeq -B_\\phi^+\/B_z $ is the magnetic shear of the magnetic configuration and $B_\\phi^+$ is the toroidal magnetic field at the disk surface \\citep[see][for more details]{Ferreira97}. From the above radial distributions, we can deduce the expressions of the vertical magnetic field $B_z$, the accretion speed $u_R$ and the disk surface density $\\Sigma$ as a function of $\\dot{m}_{in}$ and the disk aspect ratio $\\varepsilon=H\/R$. The latter is obtained by solving the coupled energy equations for the ions and electrons in order to compute both electronic and ion temperatures (see paper II for a full description of the method). \n\n\n\\subsubsection{Inner JED}\n\nJet-emitting disks solutions from a large radial disk extent are known to exist in a restricted region of the parameter space \\citep[][Fig. 2]{Ferreira97}. For simplicity, we assume that any given configuration is stationary and that parameters are constant. An extensive study of the thermal structure and associated spectra of JED can be found in paper II, where it is shown that the following set of parameters best reproduces XrB hard states, from low to very high luminosities: \n\\begin{itemize}\n\\item $\\mu= 0.5$: the disk magnetization $\\mu$ has very little influence on X-ray spectra, because the synchrotron emission does not contribute much to the equilibrium (paper II). This value has thus been chosen to lie between the two extreme values allowed for JED solutions, namely $\\mu_{min}=0.1$ and $\\mu_{max}=0.8$ \\citep{Ferreira97}.\n\\item $\\xi=0.01$: the smaller the ejection index, the less mass is being ejected and the larger the asymptotic jet velocity. A value $\\xi=0.01$ is consistent with mildly relativistic speeds \\citep[e.g., the case study for Cyg X-1 in][]{Petrucci10}.\n\\item $m_s=1.5$: within the JED accretion mode, the jets torque is dominant and imposes $m_s= m_{s, jet} =2 q \\mu $. The precise value of $m_{s,jet}$ depends on the trans-Alfv\\'enic constraint, but accretion in a JED is always at least sonic and usually supersonic $m_{s,jet}>1$ \\citep{Ferreira97}. In paper II we showed that a supersonic accretion with $m_s=1.5$ allows to reproduce luminous hard states. \n\\item $b=0.3$: the fraction of the released accretion energy $P_{acc}$ transferred to the jets has been computed within self-similar models and goes from almost $1$ to roughly $0.2$ \\citep{Ferreira97, Petrucci10}. The chosen value also appears as a good compromise and facilitates the reproduction of luminous hard states (paper II).\n\\end{itemize}\n\n\n\n\\noindent The fact that the disk accretion rate necessarily varies with the radius has been first introduced in accretion-ejection models by \\citet{Ferreira93a}, in the context of magnetically driven jets, and later by \\citet{BB99} in the context of thermally-driven outflows. In both cases, the disk ejection efficiency is characterized by the radial exponent $\\xi$ in $\\dot m (r) \\propto r^\\xi$. While it has been shown that magnetically-driven jets require $\\xi < 0.1$ \\citep{Ferreira97}, the values measured in many simulations is usually higher, lying between $0.5$ and $1$ \\citep[][and references therein]{CasseKeppens04, YBW12, Yuan15}. It is somewhat troublesome that different simulations lead to a comparable value regardless of the strength of the magnetic field. Moreover, they were mostly done in the context of non-radiating hot accretion flows. But on the other hand, \\citet{ZhuStone18} obtained $\\xi \\sim 0.003$ with an isothermal equation of state. Our guess is that this issue is not settled yet, especially given the extreme sensitivity of the disk ejection efficiency to the local thermodynamics \\citep{Casse00b}. As discussed above, we thus simply assume a small value for $\\xi$ that is compatible with the existence of relativistic jets.\n\n\n\n\n\n\n\n\\subsubsection{Outer SAD} \\label{sec:innerSAD}\n\nAs argued in introduction, the outer disk regions are assumed to accrete under the SAD mode. This implies that the relevant torque is turbulent, probably due to the magneto-rotational instability (MRI hereafter). In this case, $m_{s,turb} = \\alpha_{\\nu} \\varepsilon $. The Shakura-Sunyaev viscosity parameter $\\alpha_{\\nu}$ needs to be specified and we use $\\alpha_{\\nu}=0.1$ throughout this paper \\citep{Hawley02, King04, Penna13}. The magnetization must be small enough to allow for the development of the MRI, we choose $\\mu=10^{-3}$. As long as the SAD remains optically thick the value of $\\mu$ does not affect our calculations of the SAD thermal equilibria. \\\\\n\n\nBy definition, no jets are present in a SAD. This translates into $\\xi_{SAD}=0$ (no mass loss) and $b_{SAD}=0$ so that all released energy is either radiated or advected by the flow. Doing so, we neglect the potential presence of winds usually observed in XrB, especially at high luminosities \\citep{Ponti12, Tetarenko16}. This assumption sounds reasonable for two reasons. First, although mass loss from turbulent disks is indeed possible and actually observed in MHD simulations \\citep[e.g.,][]{Proga00, BaiStone13, Suzuki14, Bethune17, ZhuStone18}, these magneto-thermally driven flows carry away a negligible fraction of the disk angular momentum and released accretion energy, introducing thereby no significant change in the disk structure. Second, the mass rate feeding the inner JED is simply $\\dot{m}(r_J)= \\dot{m}_{in} (r_J\/r_{in})^{\\xi}$ and is independent from $\\xi_{SAD}$. Increasing $\\xi_{SAD}$ up to say, $0.5$, would imply a strong increase of the disk accretion rate in the regions beyong $r_J$ up to $r_{out}$. This would of course lead to a significant change of the emitted spectrum from these outer regions, but with no detectable counterpart \\citep[see however][and references therein]{Susmita16} as long as the disk remains optically thick, which is the case here.\n\n\n\n\n\n\\subsubsection{JED-SAD radial transition}\n\\label{sec:inter}\n\nWe examine here some properties of the transition, assuming that it occurs over a radial extent of the order of a few local disk scale heights. \\\\\n\nThe first striking property is the existence of a trans-sonic critical point near $R_J$. Indeed, while the accretion flow is subsonic in the SAD with a Mach number $m_{s,SAD} = m_{s,turb} = \\alpha_{\\nu} \\varepsilon_{SAD} \\ll 1$, it is supersonic in the inner JED with $m_{s,JED} = m_s = 1.5$. This property is a natural consequence of the transition from a turbulent \"viscous\" torque acting within the outer SAD to a dominant jet torque in the JED. Since the disk is assumed to be in a steady-state, the continuity of the mass flux $\\dot{M}_{JED}=\\dot{M}_{SAD}$ must be fulfilled at the transition radius. Given the difference between the JED and SAD sonic Mach number, this implies a drastic density decrease $\\displaystyle \\Sigma_{JED}\/\\Sigma_{SAD} \\sim \\alpha_{\\nu} \\varepsilon_{SAD}^2\/\\varepsilon_{JED} \\ll 1$ between the SAD and the JED. The Thomson optical depth being defined by $\\tau_T \\propto \\Sigma$, this density drop therefore implies a huge drop in the disk optical depth. Thus, a dynamical JED-SAD radial transition naturally goes with an optically thin - optically thick transition (see paper II). \n \nThe second striking property is the possible existence of a thin super-Keplerian layer between the JED and the SAD. In the outer SAD, the disk is slightly sub-Keplerian with a deviation due to the radial pressure gradient and of the order of $\\varepsilon_{SAD}^2$. Within the JED, the much larger magnetic radial tension leads to a larger deviation of the order of $\\mu \\varepsilon$ \\citep{Ferreira95}. This requires that the radial profile $\\Omega(R)$ has two extrema (with $d\\Omega\/dR \\simeq 0$). Since all the disk angular momentum is carried away vertically in JEDs, there is no outward angular momentum flux into the SAD at $R_J$. This translates into a \"no-torque\" condition for the SAD. Such a situation has already been discussed in the context of a radial transition between an outer cold (SAD) and inner hot (ADAF) accretion flows, leading to a super-Keplerian layer \\citep{Honma96, Abramowicz98}. It is, however, not clear whether such a thin layer would still be present in our context given the existence of magnetic forces. But these two extrema of the angular velocity clearly define the radial end points of each dynamical (SAD or JED) solution, and the trans-sonic transition occurring in-between. \\\\\n\n\nConstructing a dynamical solution describing the radial transition between a SAD and a JED is beyond the scope of the present paper and will be studied elsewhere.\nFrom now on, we assume that the two accretion modes can always be matched at a transition radius $R_J$. The calculation of the global disk equilibrium can then be undergone using $\\dot{m}_{in} = \\dot{M}_{in} \/ \\dot{M}_{Edd}$ and $r_J = R_J \/ R_g$ as independent variables.\n\n\n\n\n\n\n\\subsection{Thermal structure of hybrid JED-SAD configurations}\n\\label{sec:discussionJED}\n\nAs described in paper II, our accretion flow is locally described by a two-temperatures ($T_e$, $T_i$), fully ionized plasma of densities $n_e = n_i$, embedded in a magnetic field $B_z$. We recall here the main equations used to compute the thermal equilibrium (see paper II for full explanations). The electron and proton temperatures are computed at each radius using the coupled steady-state local energy balance equations\n  \\begin{eqnarray}\n     \\left( 1-\\delta \\right) \\cdot q_{turb} &=&  q_{adv,i} + q_{ie} ~~~~~~~~~~~~~~~~~~~~ \\text{ions} \\label{eq:IONS} \\\\\n      \\delta \\cdot q_{turb} &=& q_{adv,e} - q_{ie} + q_{rad} ~~~~~~~~~ \\text{electrons} \\label{eq:ELECTRONS}\n   \\end{eqnarray}\nwhere the local heating term, of turbulent origin, varies according to the radial zone considered. Within the JED in $R<R_J$, it gives\n\\begin{equation}\nq_{turb,JED} =   (1-b) (1-\\xi) \\frac{G M \\dot{M}(R)}{8 \\pi H R^3} \n\\end{equation}\nwhereas within the SAD in $R>R_J$ it is \n\\begin{equation}\nq_{turb,SAD}  =   \\frac{3 G M \\dot{M}(R)}{8 \\pi H R^3} \\left( 1 - \\sqrt[]{R_{J}\/R} \\right) \\label{eq:qturbSAD}\n\\end{equation}\nmaking use of the no-torque condition imposed at $R_{J}$ and a constant $\\dot{M}$ at the transition (section~\\ref{sec:inter}). In principle, the released turbulent energy could be unevenly shared between ions and electrons by a factor $\\delta$. Throughout this paper, we use $\\delta=0.5$ \\citep[see paper II,][section 2.3 and references therein]{Yuan14}. The other terms appearing in Eq.~(\\ref{eq:IONS}) and (\\ref{eq:ELECTRONS}) are the ion (electron) advection of internal energy $q_{adv,i}$ ($q_{adv,e}$), the Coulomb collisional interaction between ions and electrons $q_{ie}$ and the radiative cooling term due to the electrons $q_{rad}$. This term, as well as the radiation pressure $P_{rad}$ term, is computed using a bridge function allowing to accurately deal with both the optically thin and the optically thick regimes (paper II). The optically thin cooling regime $q_{thin}$ is computed with the BELM code \\citep{Belmont08}, which includes Compton scattering, emission and absorption through bremsstrahlung and synchrotron processes.\n\n\nThe thermal equilibrium of the SAD region is well known. For large accretion rates, required for outbursting XrBs, the disk is mostly in the optically thick, geometrically thin cold regime ($T_e \\sim 10^5-10^7$~K). \nHeating of the SAD surface layers by the hard X-rays emitted by the inner JED might produce some disk evaporation \\citep[][]{Meyer94, Meyer00, Liu05, Meyer05}. This would require to solve the 2D (vertical and radial) stratification of the disk, which is beyond our vertical one-zone approach. However, this is not expected to be crucial for the scenario depicted here. We therefore neglect the feedback of the inner JED over the outer SAD structure. In this approximation and since the resolution is computed outside-in, the temperature of the outer SAD does not depend on any assumption made on the inner JED.\\\\\n\n\nOn the contrary, the effects of the outer SAD on the inner JED are twofold and cannot be neglected. The first effect is the cooling due to advection of the outer cold material into the JED. Indeed the local advection term $q_{adv}$ can be either a cooling or a heating term, depending on the sign of the radial derivatives. This is self-consistently taken into account in our code (see Eq. (12) and Appendix A.2 in paper II).\nThe second effect results from the Compton scattering of the SAD photons on the JED electrons. This effect occurs whenever the JED is in the optically thin, geometrically thick thermal solution. This effect was not taken into account in paper II. \\\\ \n\n\nIllumination is estimated from the SAD properties, using the following geometrical prescriptions. Outside of the transition radius $R_J$, the disk luminosity is $\\displaystyle L_{SAD} = G M \\dot{M} \/ 2 R_J \\simeq 4 \\pi R_*^2 \\sigma T_s^4$, where $T_s= T_{eff}(R_*)$ is the effective temperature at the radius $R_*$ where $T_{eff}$ reaches its maximum \\citep{Frank02}. JED solutions being optically thin (or slim at worse), we assume that the radiation field in the region below $R_J$ is well described by the average energy density\n\\begin{equation} \nU_{rad} = \\omega \\frac{L_{SAD}}{4\\pi R_J^2 c} \\label{eq:Urad}\n\\end{equation}\nwhere $0 < \\omega < 1$ is a geometrical dilution factor that describes the fraction of the SAD power that irradiates the region below $R_J$ (see section~\\ref{sec:EffectOfOmega}). This applies to the bolometric luminosity, but also to the luminosity in any given energy band.\n\nThis prescription allows to compute all properties of the illumination field. This radiation is then provided to the BELM code as an external source of seed photons, and the associated cooling and reprocessed spectrum are computed. More precisely, the JED is divided in many spheres of radius $H$ in which radiation processes are computed (see paper II for more details). Here, each sphere of radius $H$ receives the power\n\\begin{equation} \nL_{s} = U_{rad} 4\\pi H^2 c = \\omega (H\/R_J)^2L_{SAD} \\label{eq:Ls}\n\\end{equation}\n\\noindent where the value of $\\omega$ is discussed in section~\\ref{sec:EffectOfOmega}.\n\n\n\n\\begin{figure*}[t]\n  \\centering\n  \\includegraphics[width=\\textwidth]{EffectOfOmega.pdf}\n   \\caption{Effect of the external Comptonisation of the SAD photon field on the inner JED for $\\dot{m}_{in} = \\dot{M}_{in} \/ \\dot{M}_{Edd} = 1$ and $r_J = 15$ (green vertical line on the top figures), with different dilution factors, from left to right $\\omega = 0,~1,~10,~50 \\%$. Top panels show the disk aspect as well as its Thomson optical depth in colors. Middle panels show the electron temperature as function of radius. Bottom panel displays the local spectra emitted by each radius (dashed lines) and the corresponding total disk spectrum (black solid line). For comparison, in each panel we have overplotted in gray solid lines the total spectra obtained in the other three panels. The spectra are given in Eddington fluxes $F_{Edd} = L_{Edd} \/ 4 \\pi D^2$ units for GX 339-4 (see section~\\ref{sec:GeneralPpts}). The blue lines and dots correspond to the SAD zone, while the red lines and triangles represent the JED. The white part of the spectra shows the $3-200$~keV energy range. Approximate values of the photon index $\\Gamma$ and energy cutoff $E_{cut}$, derived by comparison with a simple cutoff power-law model in this energy range, are indicated on each plot.}\n     \\label{fig:EffectOfOmega}\n\\end{figure*}\n\n\\subsection{Effect of an external illumination on the JED thermal structure} \\label{sec:EffectOfOmega}\n\nAdding an outer standard accretion disk may have a colossal impact on the inner hot JED, depending on the transition radius $r_J = R_J \/ R_g$. For large values of $r_J$ (say larger than $50-100$), the power emitted by the SAD is too low to affect the global disk spectrum. But this is no longer the case when the transition radius becomes smaller. \nBesides, the geometrical dilution factor $\\omega$ used in Eq.~(\\ref{eq:Urad}) plays an important role. It is however quite tricky to obtain an accurate estimate of its value within our framework. It depends on the solid angle under which the SAD photosphere is seen by the JED and corresponds thereby to the fraction of the SAD photons that are intercepted by the JED. \n\nConsidering a spherical hot corona of radius $r_J$, centered on the black hole and embedded in an infinitely thin disk, former studies \\citep[e.g.,][]{Zdz99, Ibragimov05} led to $\\omega \\sim 2 - 25 \\%$ depending on the dynamical (\"no-torque\", \"torque\") hypothesis made at $r_J$. The inner JED accretion flow is clearly different from a sphere of radius $r_J$ (see Fig.~\\ref{fig:config}), which might naively suggest a value smaller than the above estimates. Moreover, Compton cooling also should be a function of the radius within the JED, and in the case of the geometry shown in Fig.~\\ref{fig:config}, we could expect $\\omega$ to decrease with decreasing radius. \\\\\n\n\nThere are however numerous effects that should magnify $\\omega$. \nFirst, the SAD is clearly flared, thus the infinitely thin disk approximation generally used is rather crude and tends to decrease the value of $\\omega$ \\citep[see, e.g.,][]{Meyer05, Mayer07}.\nSecond, although not considered in the litterature, the photons emitted radially by the innermost region of the SAD are also expected to radiate directly towards the JED (see Fig.~\\ref{fig:EffectOfOmega} top panel). Indeed, the photosphere $\\tau_T = 1$ necessarily crosses the disk midplane near $r_J$, allowing cold SAD photons to enter directly into the outer parts of the JED (and not only from the SAD surfaces). This should be responsible for another radiative contribution. The estimation of the corresponding photon flux emitted by the SAD and entering the JED in such a way is quite complex. It would require to solve the full (radial + vertical) radiative transfer problem to determine the photosphere properties of the SAD close to $r_J$. While this is far beyond the goal of the present paper, this effect should be similar to estimations\\footnote{Again, in the case of an infinitely thin disk, and for a disk penetrating the hot corona in $r_J$ \\citep[see again][]{Zdz99}, resulting in $\\omega$ of the order of tens of percent depending on how far the disk penetrates inside the corona.}. \nThird, the reprocessing of the X-ray emission from the JED inside the disk will also naturally increase the SAD emission \\citep{Poutanen17}. In our model, we only take into account the intrinsic disk emission, but X-ray reprocessing can be mimicked by increasing $\\omega$.\nFourth, another important effect that should be taken into account is the gravitational light bending close to the black hole. This should strongly magnify the flux of disk photons impinging the JED in comparison to the Newtonian situation where they would mainly escape away from the disk. This effect should depend on the transition radius $r_J$ as well as the radial position inside the JED. Ray tracing simulations are required here for a rigorous computation, and here again this is out of the scope of the present paper. But this effect could be the dominant one especially for small radii inside the JED or for small $r_J$, since the closer to the black hole the stronger the light bending effect. This could result in a factor of a few to be applied to the number of SAD photons entering the JED in comparison to the absence of light bending \\citep[see, e.g.,][]{Miniutti03}. All included, we believe that $\\omega$ of the order of a few tens of percent seems rather reasonable. \\\\\n\n\n\nWe report in Fig.~\\ref{fig:EffectOfOmega} the JED radial temperature distribution, as well as the corresponding spectral energy distribution (SED), for a constant $\\omega$ varying from $0$ to $50 \\%$, the two physical extremes values for this parameter. Increasing $\\omega$ obviously decreases the JED temperature and softens the SED. The variation is quite important between $\\omega=0\\%$ and $50 \\%$. Using a simple power-law model we find a spectral softening of $\\Delta \\Gamma \\simeq 0.4$ of the resulting power-law, along with a modification of its energy cutoff, from $E_{cut} \\simeq 500$~keV to $E_{cut} \\simeq 200$~keV.\n\nClearly, this effect can not be neglected, as the value of $\\omega$ has an important impact on the spectra. In this article, unless otherwise specified, we use $\\omega = 0.2$. This value appears to be close to the upper limit for previous estimations, but, considering the number of assumptions diminishing this value, we thought this was a good compromise. We note however that this does not mean that the inner JED captures $20\\%$ of the SAD luminosity, as there is still a factor $\\left( H\/R_J \\right)^2$ in Eq.~(\\ref{eq:Ls}).\n\n\n\n\n\n\\section{Synthetic observations: X-ray disk spectra and radio jet emission} \\label{sec3}\n\\label{sec:Syntheticspectra}\n\n\\subsection{From theoretical SEDs to simulated data}\n\\label{sec:Py2Xspec}\n\nThis work aims at providing synthetic hardness-intensity diagrams, or more precisely, disk fraction luminosity diagrams \\citep[DFLD, see, e.g.,][]{Kording06, Dunn10}. To that purpose, our synthetic spectra must be processed in a way similar to observational data to derive the disk and power-law components from the fits, and place the corresponding points in a DFLD. This procedure, too rarely performed, is mandatory as we intend to compare our synthetic data to observations.\n\n\nFrom our theoretical SED, an \\textsc{XSPEC} table model was first built using the \\textsc{flx2tab}\\footnote{https:\/\/heasarc.gsfc.nasa.gov\/ftools\/caldb\/help\/flx2tab.html} command of \\textsc{ftools}\\footnote{https:\/\/heasarc.gsfc.nasa.gov\/ftools\/}. Disk inclination was ignored for simplification but we add background and galactic absorption (\\textsc{wabs} model in \\textsc{XSPEC} with $N_H = 4 \\times 10^{21}$~cm$^{-2}$, see \\citealt{Dickey90, Dunn10, Clavel16}). Then we simulated RXTE\/PCA and HEXTE spectrum with the \\textsc{XSPEC} simulation command \\textsc{fakeit}. We use exposure times $t_{exp}$ between $1$ and $10$~ks depending on the model flux in the $3-200$~keV band in order to have a reasonably good signal-to-noise ratio. In this article, we use $t_{exp} = 10$~ks for the quiescent state (section~\\ref{sec:XrB5states}) and $t_{exp} = 1$~ks elsewhere. In Fig. \\ref{fig:Xspec}, we plot for example the simulated spectrum from the theoretical SED produced by the JED-SAD configuration with $\\omega = 0.1$ (Fig.~\\ref{fig:EffectOfOmega}, 3rd panel). \\\\\n\nEach simulated spectrum is then fitted with three different models (a): \\textsc{wabs} $\\times$ (\\textsc{cutoffpl} + \\textsc{ezdiskbb}), (b): \\textsc{wabs} $\\times$ \\textsc{cutoffpl} or (c): \\textsc{wabs} $\\times$ \\textsc{ezdiskbb}. We keep the best one according to a \\textsc{ftest} procedure \\cite[see, e.g.,][section 3.2]{Clavel16}. In the example shown in Fig.~\\ref{fig:Xspec}, model (a) gives the best fit with a reduced $\\chi^2_{red} = 248.4 \/ 213 = 1.17$. The best fit parameters are a disk blackbody temperature $T_{in} = 1.0 \\pm 0.3$~keV, a photon index $\\Gamma = 1.59 ^{+0.03}_{-0.06}$ and a lower limit in cutoff $E_{cut} > 400$~keV. These values are consistent with simple estimations performed on the theoretical data $\\Gamma = 1.64$ and $E_{cut} \\simeq 500$~keV. The corresponding best fit disk and power-law components are plotted in dashed lines in Fig.~\\ref{fig:Xspec}.\n\nThe total unabsorbed disk luminosity $L_{Disk}$ and unabsorbed power law luminosity $L_{PL}$ are computed in the $3-200$~keV range. In the example shown in Fig.~\\ref{fig:Xspec}, the powerlaw fraction, defined by $PL_f = L_{PL} \/ (L_{PL} + L_{Disk})$ is equal to $0.99 \\pm 0.01$ and the total flux in the $3-200$~keV band is $L_{tot} = L_{PL} + L_{Disk} = 2.9 \\pm 0.1 ~ \\% L_{Edd}$.\n\n\n\\begin{figure}[h!]\n  \\includegraphics[width=1\\linewidth]{Py2Xspec.png}\n   \\caption{Example of simulated \\textsc{RXTE\/PCA} ($3-25$~keV, in black) and \\textsc{RXTE\/Hexte} ($20-200$~keV, in red) data sets from the theoretical SED produced by the JED-SAD configuration shown in Fig.~\\ref{fig:EffectOfOmega}, third panel. The dotted lines are the power law and disk components corresponding to the best fit model. See section~\\ref{sec:Py2Xspec} for more details.}\n     \\label{fig:Xspec}\n\\end{figure}\n\n\n\n\\subsection{Hard tail}\n\\label{sec:HardTail}\n\n\\begin{figure}[h!]\n  \\includegraphics[width=1\\linewidth]{HardTail_Python.pdf}\n  \\includegraphics[width=1\\linewidth, trim = 0.5cm 1.3cm 2.5cm 2.5cm, clip]{HardTail_Xspec.png\n   \\caption{Electron temperature (top-left) and theoretical spectrum (top-right) of the configuration $\\dot{m}_{in} = 1$, $\\mu_{SAD} \\ll 1$, $\\alpha_{\\nu} = 0.1$, $r_J = r_{in}$. Each annulus is displayed as a blue dot, its associated spectrum in blue dashed lines and the total disk spectrum in black solid line. In the bottom panel, final faked and fitted data after the addition of the hard power-law tail, black for \\textsc{PCA} and red for \\textsc{\\textit{HEXTE}}. Dashed lines show the best fit obtained with \\textsc{XSPEC}, see section~\\ref{sec:HardTail} for details.}\n     \\label{fig:HardTail}\n\\end{figure}\n\nIt is well know that soft states show non thermal tails generally observed above few keV \\citep{McConnell02,Remillard06}. Although uncertain, these tails are thought to be produced by a population of non-thermal electrons \\citep[see][for few investigations]{Galeev79, Gierlinski99} that are not taken into account in our model. In order to reproduce such soft states, we added a power-law component to the synthetic spectra each time the fitting procedure favors a pure blackbody emission (model (c)) and the new data were re-fitted with a modified model (c) \\textsc{wabs} $\\times$ (\\textsc{pl} + \\textsc{ezdiskbb}) model. The photon index of this power-law component is set to $\\Gamma=2.5$ and it is normalized in order to contribute to a fixed fraction of the $3-20$~keV energy range \\citep[typically between $1\\%$ and $10\\%$, ][]{Remillard06}. \\\\\n\nAn example is shown in Fig.~\\ref{fig:HardTail}. The theoretical model plotted on top of this figure corresponds to a SAD extending down to $r_J = r_{in}$ (no JED), with $\\dot{m}_{in} = 1$. A fit with an absorbed disk component (model (c)) gives a disk temperature $T_{in} = 0.97 \\pm 0.01$~keV and a total flux $L_{tot} = 4.1 \\pm 0.1 \\% L_{Edd}$ in the $3-200$~keV band. The data simulated in XSPEC include the additional power-law tail (Fig.~\\ref{fig:HardTail}, bottom panel). The best fit with new model (c) then gives $T_{in} = 0.97 \\pm 0.01$~keV, $\\Gamma = 2.5 \\pm 0.2$ and $L_{tot} = 4.5 \\pm 0.1 \\% L_{Edd}$ with a reduced $\\chi^2_{red} = 214.5 \/ 213 = 1.01$. We note that this procedure slightly increases the total flux detected in X-rays due to the addition of the hard power-law tail.\n\n\n\n\n\n\\begin{figure*}[h!\n  \\centering\n  \\includegraphics[width=1.\\textwidth]{Pavage_DFLD.pdf} \n   \\caption{Total, disk + power-law, luminosity $L_{tot}=L_{disk} + L_{PL}$ in the $3-200$~keV energy range (in Eddington luminosity unit) is shown in function of the power-law fraction $L_{PL}\/L_{tot}$. Each point within this plot corresponds to a fully computed and then XSPEC processed hybrid JED-SAD configuration. Contours (black solid lines) are for a constant disk accretion rate $\\dot{m}_{in}$ while the color background display the disk transition radius $r_J$. Dashed black line shows the 2010-2011 cycle of GX 339-4. XSPEC fits were done with a hard tail level of $1\\%$ (left) and $10\\%$ (right). See section \\ref{sec:EffectOfRjMdot} for a description of the figure.}\n     \\label{fig:EffectOfRj}\n\\end{figure*\n\n\n\n\\subsection{Jet power and radio luminosity}\n\\label{sec:Pjet}\n\nIn a JED, the jets power available is a given fraction of the accretion power\n\\begin{eqnarray}\n  P_{jets} &=& b  P_{acc, JED} = b  \\left [ \\frac{G M \\dot{M}}{2 R} \\right ]^{R_{in}}_{R_{out}} \\nonumber \\\\\n  \t\t       &=& \\frac{b}{2} \\frac{\\dot{m}_{in}}{r_{in}} \\left( 1 - \\left( \\frac{r_J}{r_{in}} \\right)^{\\xi-1} \\right) L_{Edd} \\label{eq:Pjet}\n\\end{eqnarray}\nAssuming that $b$ is roughly a constant throughout an entire evolution, the jets power depends on both $\\dot{m}_{in}$ and $r_J$. We follow the computations of \\citet{Heinz03} to deduce the expected radio luminosity emitted by one jet component. It assumes that the jet emission is explained by self-absorbed synchrotron emission of non-thermal particles along the jet. We need however to modify their equations in order to account for the finite radial extent of the JED, imposing a finite radial extent of each jet. This leads to the following expression (see Appendix~\\ref{sec:JetRadio} for more details):\n\\begin{equation}\n\\frac{\\nu_R L_{R}}{L_{Edd}} \\simeq f_{R}\\,   m^{\\beta-1} \\,  r_{in}^{- \\frac{6p+49}{4p+16}} \\, \\dot{m}_{in}^\\beta \\,  r_J^{\\frac{p+9}{p+4}}\\,  \\left (1- \\frac{r_{in}}{r_J} \\right )^{\\frac{5}{p+4}}\n\\label{eq:Fr}\n\\end{equation}\nwhere $L_R= 4\\pi D^2 F_R$ is the monochromatic power emitted at the radio frequency $\\nu_R$ from an object at a distance $D$. The parameter $f_R$ is a normalization constant, $\\beta= \\frac{2p+13}{2p+8}$ and $p$ is the usual exponent of the non-thermal particle energy distribution. In the standard case with $p=2$ and $r_J$ constant, one gets the \\citet{Heinz03} dependencies, namely a radio power $\\nu_R L_{R} \\propto \\dot{m}_{in}^{17\/12}$.\\\\\n\nOur model then provides naturally both $L_R$ and $L_X$. Indeed, for any given set of parameters $(\\dot{m}_{in}, r_J)$, we can compute $L_X$ from our simulated SED, whereas an estimate of the radio luminosity can be obtained using Eq.~(\\ref{eq:Fr}). This is discussed in section~\\ref{sec4}.\n\n\n\n\n\n\\section{Reproducing typical XrB behavior: DFLD and canonical spectral states} \\label{sec4}\n\nIn this section, we show that hybrid JED-SAD configurations can, in principle, reproduce the outbursting cycles of XrBs by varying only two parameters, the disk accretion rate $\\dot{m}_{in}$ and the transition radius $r_J$. This is done in two steps. First, we need to find which ranges in $\\dot{m}_{in}$ and $r_J$ allow to cover the full DFLD. However, this diagram includes only information on X-ray emission while a cycle also deals with jet production and quenching. We thus require that the same framework reproduces radio emission at the correct level. Then, as a second step, we define five canonical spectral states characteristic of an XrB spectral evolution during an outburst and show more precisely how well our framework is able to reproduce them.\n\n\\subsection{Disk fraction luminosity diagram}\n\\label{sec:EffectOfRjMdot}\n\n\nWe perform a large parameter survey for $\\dot{m}_{in} \\in [0.01, ~ 10]$ and $r_J \\in [r_{in},~ \\sim 50 r_{in}] = [2, ~ 100]$. We compute the whole thermal structure and corresponding theoretical SED for each pair $(\\dot{m}_{in}, r_J)$, and we fit as described in section~\\ref{sec:Syntheticspectra} to get the corresponding position in the DFLD. The fits\\footnote{Only fits with $\\chi^2_{red} < 3$ have been displayed here. Few fits $(\\dot{m}_{in} \\gtrsim 5,~ r_J = r_{in})$ require the addition of a second blackbody component to better describe their spectral shape, but their position (top-left) beyond the extension of usual DFLDs makes them meaningless in the current study.} are shown in Fig.~\\ref{fig:EffectOfRj}. The smoothness of our DFLD is indicative of the absence of spectral degeneracy in our modeling (Fig.~\\ref{fig:EffectOfRj}).\nIn this figure, the mean transition radius is color-coded and the accretion rate is shown in contours of constant values. For comparison, the 2010-2011 outburst of GX 339-4 is overplotted in black dashed line\\footnote{The spectral analysis of the 2010-2011 outburst was done in the $3-25$~keV energy range (\\textsc{RXTE\/PCA}), but the models were integrated in the $3-200$~keV range and not the usual ranges \\citep[e.g.,][]{Dunn10}\\label{fnt:ranges}}.\n\n\nFig.~\\ref{fig:EffectOfRj} shows that we can cover the whole domain usually followed by XrBs within our framework. Concerning the hard states, we are able to reproduce their evolution up to high luminosities. Concerning the soft states, their position in the DFLD depends on the amplitude of the additional hard tail. With a hard tail representing $10\\%$ of the flux in the $3-20$~keV energy range throughout the cycle, we can only reproduce soft states with $L_{PL}\/L_{tot} > 0.1$ (Fig.~\\ref{fig:EffectOfRj}, right). Softer states, populating the very left part of the DFLD, require a hard tail flux fraction lower than $1\\%$ (Fig.~\\ref{fig:EffectOfRj}, left).\n\nAs expected, the accretion rate is mainly responsible for the global X-ray luminosity of the system, leading to almost horizontal isocontours for $\\dot{m}_{in}$ in the DFLD. Indeed, the higher the accretion rate, the higher the total available accretion energy $P_{acc} \\propto \\dot{m}_{in}$. However, part of the energy can be advected, which explains the sharp variations of the isocontours at high luminosities (see discussion below). The effect of the transition radius $r_J$ follows the predictions of paper I. At large transition radii, most of the emission originates from the JED, as the outer SAD has no detectable influence. These solutions display power-law spectra for all accretion rates (see paper II). This is the reason why they appear on the right-side of the DFLD. At small transition radii, two effects appear in the RXTE energy range of our simulated data. First, as its temperature and flux increase with decreasing $r_J$, the SAD blackbody emission starts appearing in the SED around $3$~keV. Second, the closer the SAD, the stronger its illumination becomes, cooling down the inner JED and producing softer spectra. Combining these two effects leads to a disk dominated spectrum, with a power-law fraction becoming entirely dominated by the high energy tail when $r_J \\rightarrow r_{in}$. \\\\\n\n\n\\subsection{$L_X$ dependencies on $\\dot{m}_{in}$}\n\n\n\\begin{figure}[h\n  \\centering\n  \\includegraphics[width=\\columnwidth]{LbolLx39_vs_mdot.pdf}\n   \\caption{Bolometric (top) and $3-9$~keV (bottom) luminosities in function of the mass accretion rate $\\dot{m}_{in}$ onto the black hole. This plot is extracted from Fig.~\\ref{fig:EffectOfRj}, done with a $10\\%$ hard tail (right). The colors are for different values of the transition radius $r_J$. Four different $L \\propto \\dot{m}_{in}^{\\alpha}$ regimes are shown. Also, the $r_J = r_{in}$ has been drawn in dashed black to be visible at low accretion rate in the bottom panel.}\n     \\label{fig:Lxmdot}\n\\end{figure\n\n\n\nFigure~\\ref{fig:Lxmdot} shows the bolometric ($L_{bol}$, top) and $3-9$~keV ($L_{3-9}$, bottom) luminosity deduced from our synthetic SED as a function of the accretion rate at the inner radius $\\dot{m}_{in}$. The colors correspond to different transition radius $r_J$.\nThis figure illustrates different concerns about the radiative efficiency of accretion flows. \\\\\n\n\n\n\nOn the top panel and in the SAD mode ($r_J = r_{in} = 2$), the bolometric luminosity follows the radiatively efficient regime $L_{bol} \\propto \\dot{m}_{in}$ as long as $\\dot{m}_{in} < 5$. Below this accretion rates, the SAD is indeed radiating all of its available energy. At $\\dot{m}_{in} > 5$, the SAD enters the slim domain, where more and more energy becomes advected instead of being radiated away. As a consequence, the global luminosity has a steeper slope $L_{bol} \\propto \\dot{m}_{in}^{0.5}$.\n\n\nAs $r_J$ increases, the JED mode starts to have a bigger extent, and at any given accretion rate $\\dot{m}_{in}$ the luminosity decreases as $r_J$ increases. This is the result of two different effects. First, as $r_J$ increases, more and more energy is controlled by the JED and transferred to the jets $b = P_{jets} \/ P_{acc, JED} = 0.3$ (papers I and II) instead of being radiated. Second, the JED thermal equilibrium is often strongly affected by advection, as $f = P_{adv} \/ P_{acc, JED} \\propto \\varepsilon^2$ is no more negligible (paper II). Combining these two effects, a more representative formulation is $L_{bol} \\propto (1-b-f) ~ \\dot{m}_{in}$, where $f \\propto \\varepsilon^2 = \\varepsilon^2 (\\dot{m}_{in})$ also is a function of accretion rate.\n\nAt low accretion rates $\\dot{m}_{in} < 0.5$, the JED is optically thin and geometrically thick with $\\varepsilon \\simeq 0.2-0.3$ (termed thick disk solution in paper II). In the thick disk branch, the low density of the plasma allows $q_{ie} \\propto n_e^{2}$ to be negligible. Ions are neither cooled down by radiation nor by Coulomb interactions: $T_i \\gg T_e$. Contrarily to usual one-temperature plasmas, the disk thickness is then only linked to the ion pressure, $P_{gas,i} \\gg P_{gas,e} + P_{rad}$, leading to $q_{adv} \\simeq q_{adv,i} \\gg q_{adv,e}$ (see Eq. (13) in paper II). In the ion thermal equilibrium from Eq. (\\ref{eq:IONS}), advection is directly determined by ion heating, $q_{adv} \\simeq q_{adv,i} \\simeq (1-\\delta) q_{turb}$, leading to $f = q_{adv} \/ q_{turb} \\simeq 0.5$. Since $q_{adv,e} \\ll q_{adv,i} \\simeq (1-\\delta) q_{turb}$, we obtain $q_{adv,e} \\ll \\delta q_{turb}$. In the electron equilibrium Eq. (\\ref{eq:ELECTRONS}), radiation is then determined by $q_{rad} \\sim \\delta q_{turb} \\propto \\dot{m}_{in}$, leading to the trend $L_{bol} \\propto \\dot{m}_{in}$, unexpected in a thick disk\\footnote{This in only true if $(1-\\delta) \\sim \\delta$, which is the case here. In ADAFs, where $\\delta = 1\/2000$, even if $q_{adv,e} \\ll (1-\\delta) q_{turb}$ the factor $\\delta \\ll 1$ would not ensure that $q_{adv,e} \\ll \\delta q_{turb}$, and this reasoning would not stand.}. In the end, combining the loss of power through jets ($b$) and in advection ($f$) reduces the JED luminosity. For instance, for $r_J = 50$ (yellow color) it is reduced by a factor $1 \/ (1-b-f) \\sim 5$ compared to the SAD mode power.\n\nAt high accretion rates $\\dot{m}_{in} > 2$, the JED mode is optically thick and geometrically slim with $\\varepsilon \\simeq 0.1$ (termed slim disk solution in paper II). The bolometric luminosity now has a slope $L_{bol} \\propto \\dot{m}_{in}^{0.5}$ (see the previous discussion on the slim SAD mode), with a factor $\\sim 2$ times lower in luminosity. Indeed, in this case, $f \\simeq 0.15$, due to the lower temperature (and then the lower $\\varepsilon$) compared to the thick disk solution.\n\nBetween these two solutions, from the thick to the slim disks, $f$ decreases from $0.45$ to $0.15$. The disk radiates more and more energy and the luminosity increases until it reaches the $L_{bol} \\propto \\dot{m}_{in}^{0.5}$ slope. During this transition, the slope is more abrupt and fits well with $L_{bol} \\propto \\dot{m}_{in}^{1.5}$. \\\\\n\n\nThe bottom panel of Fig.~\\ref{fig:Lxmdot} shows that the $L_{3-9}$ variation with the accretion rate is different from $L_{bol}$. In the SAD mode ($r_J = r_{in} = 2$), $L_{3-9} \\propto \\dot{m}_{in}^{\\sim 2}$ for $\\dot{m}_{in} < 1$ and it slowly drops down to $L_{3-9} \\propto \\dot{m}_{in}^{0.5}$ while approaching the slim region. The JED mode with $r_J = 50$ remains closer to the bolometric behavior, with still $L_{3-9} \\propto \\dot{m}_{in}$ when $\\dot{m}_{in} < 0.5$ and roughly $L_{3-9} \\propto \\dot{m}_{in}^{1.5}$ during the transition and in the slim region. At very low accretion rates, the JED mode radiates more energy in the $3-9$~keV band up to a factor $\\sim 4$, while at higher accretion rates the SAD can radiate as high as $\\sim 17$ times more energy in this range. \\\\\n\n\nTwo interesting comments can be done from Fig.~\\ref{fig:Lxmdot}. First a \"radiatively efficient accretion flow\" does not necessarily mean $L \\propto \\dot{m}_{in}$. Indeed, as shown before in the JED dominated mode ($r_J = 50$) we find $L \\propto \\dot{m}_{in}$, while the JED radiates only few tens of its total available energy. The term \"radiatively efficient\" seems therefore inappropriate. Second, the evolution of the luminosity with the accretion rate strongly depends on the energy range used. In the SAD mode, while the disk is indeed radiatively efficient and $L_{bol} \\propto \\dot{m}_{in}$, the $3-9$~keV luminosity follows a $L_{3-9} \\propto \\dot{m}_{in}^{1.5-2}$ regime, which could be considered as the signature of a radiatively inefficient flow. This clearly means that the interpreation of the luminosity variation with the accretion rate in terms of radiative efficiency can be strongly misleading and should be done with caution.\\\\\n\n\n\n\n\n\nFinally, Fig.~\\ref{fig:Lxmdot} also illustrates that the functional dependence $L_{3-9} (\\dot{m}_{in})$ can be much more complex than a single power law. This is quite promising as it is known that $L_{3-9} (\\dot{m}_{in})$ may need to vary from one object to another \\citep[][section~4.3.3 and Fig.~7]{Coriat11}. However, we cannot go further without considering a proper outbursting cycle. For instance, Fig.~\\ref{fig:EffectOfRj} clearly shows that, in order to successfully reproduce the 2010-2011 cycle of GX 339-4 (black lines in Fig~\\ref{fig:DFLD_complet}), one would need to (1) rise up, namely increase $\\dot{m}_{in}$ from the quiescent state until the highest hard state; then (2) transit left by decreasing the transition radius $r_J$ until the full disappearance of the JED; (3) drop down in the soft realm by decreasing $\\dot{m}_{in}$ and finally (4) transit right back to the hard zone by increasing $r_J$. It can thus be inferred from Fig.~\\ref{fig:Lxmdot} that the evolution in time of the X-ray luminosity is sharper than $L_{3-9} \\propto \\dot{m}_{in}$ at high luminosities, because of the necessary decrease of $r_J$. A detailed modelling of the actual track followed by GX 339-4 during a full cycle will be presented in a forthcoming paper.\n\n\n\n\\subsection{Radio fluxes} \\label{sec:RadioFluxes}\n\n\n\\begin{figure}[h!\n  \\centering\n  \\includegraphics[width=\\columnwidth]{LradioLx.pdf}\n   \\caption{Power in the 8.6 GHz radio band in function of the 3-9 keV X-ray power (both in Eddington luminosity). Black solid lines are for constant transition radius $r_J$, while the color background shows the accretion rate $\\dot{m}_{in}$. This figure has been done using a similar procedure than in Fig.~\\ref{fig:EffectOfRj}. The black points are the observed values for GX 339-4 during its cycles between 2003 and 2011. A possible theoretical jet line has been drawn in red (see section~\\ref{sec:RadioFluxes}).}\n     \\label{fig:PJET}\n\\end{figure\n\n\n\n\n\nFor any given hybrid JED-SAD disk configuration, computed with a pair of parameters ($\\dot{m}_{in}, r_J$), one can also derive an estimation of the one-sided jet radiative power $P_R$ emitted at the radio frequency $\\nu_R = 8.6$~GHz. Assuming an electron distribution with $p=2$, Eq.~(\\ref{eq:Fr}) leads to\n\\begin{equation}\nP_R \\equiv \\nu_R L_R = \\tilde f_R \\, \\dot{m}_{in}^{17\/12} r_J \\left ( r_J - r_{in} \\right )^{5\/6} \\, L_{Edd}\n\\label{eq:PR}\n\\end{equation}\nwhere the dimensionless factor $\\tilde f_R$ incorporates factors such as $m$ or $r_{in}$, see Appendix \\ref{sec:JetRadio}. For the sake of simplicity, we assume that this factor is a constant throughout the whole cycle. This is far from being obvious, but our goal here is simply to show that hybrid configurations have the potential to reproduce simultaneously both X-rays and jet radio emission. \nTo have an estimate of $\\tilde f_R$, we require that two radio observations at $\\nu_R= 8.6$~GHz, one during the quiescent state and the other during the high-luminosity hard state (see their definition below), are qualitatively reproduced. We obtain $\\tilde f_R = 3 \\times 10^{-11}$. This allows us to compute the radio power $P_R$ as a function of ($\\dot{m}_{in}, r_J$) and finally relate the radio power to the observed $3-9$~keV power.\nIn Fig.~\\ref{fig:PJET}, the same procedure than in Fig.~\\ref{fig:EffectOfRj} is used, but with binning of the true integrated luminosity $L_{3-9~\\text{keV}}$ and radio power $P_R$. In addition, radio and X-ray observations of GX 339-4 were overplotted in black dots. It can be seen that most of them correspond to $r_J \\sim 10-50$ while the accretion rate spans $\\dot{m}_{in} \\sim 0.01$ to almost $5$. Thus, reproducing the observed radio\/X-ray diagram requires variations in mass accretion rate, from $\\dot{m}_{in} < 0.1$ to almost $5$ here. However, in order to describe the disappearance or reappearance of the steady radio emission, i.e. the crossing of the jet line \\citep{Fender04}, one needs to invoke variations in transition radius: a decrease in $r_J$ when the jet is quenched (bottom-right), and an increase when the jet re-appears (top-left). This is very promising for the model to reproduce observations, but further investigation needs to be done.\n\n\n\\begin{table*}[h!]\n\\centering\n{\\renewcommand{\\arraystretch}{1.3}\n \\begin{tabular}{c | c c c c | c c | c c c c c}\n & \\multicolumn{4}{c|}{Typical observed states} & \\multicolumn{2}{c|}{Parameters} & \\multicolumn{5}{c}{Results of fits} \\\\ \\hline\nSpectral & $\\mathbf{PLf}$ & $\\textbf{X-rays}$ & $\\mathbf{\\Gamma}$ & $\\textbf{Radio}$ & $\\mathbf{\\dot{m}_{in}}$ & $\\mathbf{r_J} $  & $\\mathbf{PLf}$ & $\\textbf{X-rays}$ & $\\mathbf{\\Gamma}$ & $\\textbf{Radio}$ & $ \\chi^2_{red}$\\\\\nstate & & $(\\% L_{Edd})$ & & (mJy) & $(L_{Edd}\/c^2)$ & $(R_g)$ & & $(\\% L_{Edd})$ & & (mJy) & \\\\ \\hline\n\\textbf{Q} & $ 1 $ & $ < 0.1 $ & $ 1.5-2.1 $ & $ 1 $& $ 0.06 $ & $ 100 $ & $ 1 $ & $ 0.11 \\pm 0.01 $ & $ 1.9^{+0.3}_{-0.3}$ & $ 2.8 $ & $0.96$ \\\\\n\\textbf{LH} & $ 1 $ & $ 1 $ & $ 1.5-1.6 $ & $ 5 $ & $ 0.4 $ & $ 50 $ & $ 1 $ & $ 0.92 \\pm 0.05 $ & $ 1.64^{+0.04}_{-0.04} $ & $ 11.5 $ & $1.10$ \\\\\n\\textbf{HH} & $ 1 $ & $ >10 $ & $ 1.6-1.8 $ & $ 25 $& $ 2 $ & $ 15 $ & $ 0.98 \\pm 0.02 $ & $ 13.1 \\pm 0.1 $ & $ 1.68^{+0.03}_{-0.02} $ & $ 20.2 $ & $1.60$ \\\\ \n\\textbf{HS} & $ < 0.3 $ & $ 5 $ & $ 2-3 $ & $ < 0.01 $& $ 0.75 $ & $ 2 $ & $ 0.13 \\pm 0.05$ & $ 4.6 \\pm 0.1 $ & $ 2.4^{+0.3}_{-0.7} $ & $ 0 $ & $0.90$ \\\\\n\\textbf{LS} & $ < 0.3 $ & $ 1 $ & $ 2-3 $ & $ < 0.01 $& $ 0.45 $ & $ 2 $ & $ 0.2 \\pm 0.05 $ & $ 1.40 \\pm 0.04 $ & $ 2.5^{+0.3}_{-0.6} $ & $ 0 $ & $1.04$ \\\\ \\hline\n\\end{tabular}}\n \\caption{Typical observed properties of the five canonical states (left), pairs of parameters ($\\dot{m}_{in}$ and $r_J$, center) and XSPEC fits (right) results associated to the five chosen canonical states Q, LH, HH, HS and LS. Fits are performed using the simple model described in section~\\ref{sec:Py2Xspec}, and the definition of the five states is detailed in section~\\ref{sec:XrB5states}.}\n \\label{table:States}\n\\end{table*}\n\n\n\\subsection{XrB canonical spectral states} \\label{sec:XrB5states}\n\n\nIn section \\ref{sec:EffectOfRjMdot}, it is shown that hybrid JED-SAD configurations can cover the observed DFLDs by varying independently $\\dot{m}_{in}$ and $ r_J$. \nWe focus here on the five typical spectral states that any given XrB needs to cross (or get close to) when making a full cycle, and detail their characteristics in our JED-SAD framework. These five states are shown in Fig.~\\ref{fig:DFLD_complet} and are named quiescent state (hereafter Q), low-luminosity hard state (LH) and low-luminosity soft state (LS), both at the soft-to-hard lower transition branch, and high-luminosity hard state (HH) and high-luminosity soft state (HS) both at the hard-to-soft upper transition branch.\n\n\\begin{figure}[h!]\n\\centering\n\\sidecaption\n  \\includegraphics[width=1.\\columnwidth]{b1_4.pdf}\n   \\caption{DFLDs for the past outbursts of GX 339-4 between MJD$50290$ and MJD$55650$ extrapolated in the $3-200$~keV energy range\\footref{fnt:ranges}. A typical cycle goes from Q (black), and crosses LH (orange) up to HH (red). It then transits to the HS (blue), decreases to LS (cyan) until it transits back to LH, before decreasing down to Q. The stars mark the positions of the five canonical spectral states Q, LH, HH, HS and LS defined in section~\\ref{sec:XrB5states}.}\n     \\label{fig:DFLD_complet}\n\\end{figure}\n\n\n\n\\begin{figure*}[h!\n \\centering\n \\includegraphics[width=1.\\textwidth]{TypicalStates.pdf}\n  \\caption{Computed radial structures of hybrid JED-SAD disk configurations associated to the canonical states defined in Table~\\ref{table:States}. From left to right: \\textbf{Q} for quiescent state, \\textbf{LH} for low-hard, \\textbf{HH} for high-hard, \\textbf{HS} for high-soft and \\textbf{LS} for low-soft. Top: electron temperature $T_e$ at the disk mid plane. Bottom: Thomson optical depth $\\tau_T$. Red triangles show the JED zone, and blue dots describe the SAD zone. The vertical green line marks their separation at the transition radius $r_J$. The vertical yellow line marks the transition from a gas to radiation pressure supported regime within the SAD. In addition, the other two possible thermal solutions are shown in grey when present, the unstable in circles and the thin disk in triangles, see paper II.}\n  \\label{fig:TypicalStates}\n\\end{figure*\n\n\nThe quiescent state chosen is clearly not the most quiescent state that can be reached by an XrB. It is at position in the DFLD that any object needs to cross while going up and down. Our choice of the two soft states in the DFLD is somewhat arbitrary as it depends on the chosen level of the hard tail, see section~\\ref{sec:HardTail}. Our computed spectra are also shown from $0.5$ to $500$~keV but remember that the observational PCA and HEXTE data are available only from $3$ to $200$~keV. \\\\\n\n\n\n\nTable~\\ref{table:States} middle panel shows the values of the parameters $\\dot{m}_{in}$ and $r_J$ that better characterize these five canonical states. We also have reported in this table the values (and their associated $3 \\sigma$ errors) of the power-law fraction $PLf$, X-ray luminosity and spectral index $\\Gamma$ derived from XSPEC fits, as well as the expected radio flux density at $8.6$~GHz, $F_{8.6~\\text{GHz}}$. These fluxes have been computed in mJy using Eq.~(\\ref{eq:PR}), namely $F_{8.6~\\text{GHz}}= 10^{26} \\times P_R\/(4\\pi D^2 \\nu_R)$, with $\\nu_R= 8.6$~GHz and $\\tilde{f}_R = 3 \\times 10^{-11}$. Figs.~\\ref{fig:TypicalStates} and \\ref{fig:SpectralStates} illustrate the thermal state radial distribution (temperature, optical depth) and the theoretical and faked spectra. An accurate representation of the physical structure (size) and temperature (color) of the disk in those five states is shown in Fig.~\\ref{fig:JoliesImages}.\n \n\\begin{figure*}[h!\n \\centering\n \\includegraphics[width=1.\\textwidth]{SpectralStates.pdf}\n  \\caption{Theoretical SED (top) and XSPEC spectral fits (bottom) for the five canonical spectral states computed in Fig.~\\ref{fig:TypicalStates}. It can be seen that hard states spectra are always dominated by the comptonization of soft photons, mostly due to local Bremsstrahlung and cold photons from the outer SAD. The white area in the theoretical SED corresponds to the observationally relevant $3-200$~keV energy band. The value of the spectral index $\\Gamma$ is shown for each state, with its errors.}\n  \\label{fig:SpectralStates}\n\\end{figure*\nIn the following, we discuss each of the canonical states with more details. \\\\% A summary of the canonical states can be found in Table~\\ref{table:TypicalStates}, and the results of the resolution and fits shown as examples are presented in Fig.~\\ref{fig:TypicalStates} (temperature and optical depth), Fig.~\\ref{fig:SpectralStates} (spectra) and Table~\\ref{table:States} (fits).\n\n\n\\noindent {\\bf - Q state:} the quiescent state has an X-ray luminosity lower than $0.1 \\%$ Eddington, with a typical power-law spectrum of index $\\Gamma \\simeq 1.5-2.1$ in the observed $3-200$~keV band \\citep{Remillard06}. It exhibits faint but steady radio and IR luminosities fluxes \\citep{Fender01, Corbel13}, probing weak but detectable jets. \nIt is located at the bottom-right of the DFLD. In our JED-SAD framework it is characterized by a very low accretion rate $\\dot{m}_{in} \\lesssim 1$ and a relatively high transition radius $r_J \\gg r_{in}$. In the example shown in this section we choose $\\dot{m}_{in} = 0.06$ and $r_J = 100$. The innermost region of the disk (from $r \\simeq 30$ down to $r_{in}$) is optically thin with electron temperature as high as $T_e \\simeq 10^{10}$~K (Fig.~\\ref{fig:TypicalStates}, left panel). This results in a global spectrum that is the sum of multiple power-law spectra with roughly the same shape $\\Gamma \\sim 1.6-2$ and $E_{cut} \\gg 200$~keV (Fig.~\\ref{fig:SpectralStates}, left panel), in good agreement with the observations. In addition, the power available in the jets is relatively small due to a very low accretion rate (Table \\ref{table:States}).\\\\\n\n\n\n\n\\noindent {\\bf - LH state:} the low-luminosity hard state is characterized by a power-law dominated spectrum, with spectral index $\\Gamma \\simeq 1.5-1.6$, no cutoff detected $E_{cut} > 200$~keV \\citep[][and references therein]{Grove98, Zdz04} and a typical luminosity $L_{tot} \\simeq 1 \\% L_{Edd}$. This state is also associated to steady and high radio and IR luminosities suggesting powerful jets. In this article we choose $\\dot{m}_{in} = 0.4$ and $r_J =50$. As shown in Fig.~\\ref{fig:TypicalStates} top panel, the temperature of this flow increases quickly from $T_e \\simeq 10^{6}$~K in the outer parts of the JED to $T_e \\gtrsim 5 \\times 10^{9}$~K in the inner parts for most of the JED extension (from $r = 20$ down to $r_{in}$). This state does not strongly depend on $r_J$ as long as it is larger than a few tens of $r_{in}$, as the global spectrum is the sum of similar spectra with $\\Gamma \\sim 1.2-1.8$ (Fig.~\\ref{fig:SpectralStates}, LH-panel). This configuration is also accompanied by more powerful jets, due to larger accretion rate compared to Q-states, in agreement with stronger observed radio emission. \\\\\n\n\\noindent {\\bf - HH state:} the high-luminosity hard state is also characterized by a power-law dominated spectrum, with a spectral index $\\Gamma \\simeq 1.6-1.8$, but with a high energy cutoff generally detected $E_{cut} \\simeq 50-200$~keV \\citep{Motta09} and luminosities as high as $30 \\% L_{Edd}$ \\citep{Dunn10}. Actually, as the luminosity increases, $E_{cut}$ is observed to decrease from $>200$~keV to $\\sim 50$~keV, while the spectral index slightly changes from $\\Gamma = 1.6$ to $\\Gamma = 1.8$ before transiting to the soft state \\citep[see Figure 6 in][]{Motta09}. Those states also show the highest radio and IR fluxes \\citep{Coriat09}, suggesting the most powerful jets of the cycle. In our JED-SAD framework this state is characterized by a larger accretion rate $\\dot{m}_{in} > 1$, we choose $\\dot{m}_{in} = 3$. As shown in paper II, at such high accretion rate the hot geometrically thick disk solution switches to a denser and cooler solution, the so-called slim disk. The disk is optically slim $\\tau \\sim 1-10$ and rather warm $T_e \\sim 10^{8-9}$~K (Fig.~\\ref{fig:TypicalStates}, middle panel). The spectra associated to those slim disks solutions are closer to a very hot multi-temperature disk blackbody emission (paper II). The combination between electron temperature and optical thickness distribution with radius produces a spectral shape in agreement with observations (i.e. $\\Gamma \\simeq 1.6-1.8$ and $E_{cut} \\in [50, ~ 200]$~keV). Once we choose $\\dot{m}_{in} = 3$, the range of appropriate values for transition radius is rather narrow to reproduce the value of $\\Gamma$, we adopt $r_J = 15$. The large $\\dot{m}_{in}$ and $r_J$ result in a large jet power consistent with observations (see Table~\\ref{table:States}).\n\n\\noindent {\\bf - HS state:} the high-luminosity soft state is defined by a dominant multi-temperature blackbody with maximum effective temperature $T_{in} \\lesssim 1$~keV and total flux $L_{tot} \\sim 5 \\% L_{Edd}$. In addition, many of the soft states display a minor component, the hard tail (see section \\ref{sec:HardTail}). These states are also characterized by the absence of steady radio emission, interpreted as the jet disappearance \\citep[][]{Fender99, Corbel00}. In our JED-SAD framework this translates to $r_J = r_{in}$, i.e. our accretion flow is entirely in a SAD mode. In the example shown in Fig.~\\ref{fig:TypicalStates}, we choose $\\dot{m}_{in} = 0.75$ with a $10\\%$ hard tail. \\\\\n\n\\noindent {\\bf - LS state:} the spectral shape of the low-luminosity soft state is similar to the HS, with a typical maximum effective disk temperature $T_{in} \\gtrsim 0.6-0.7$~keV, a $\\Gamma \\simeq 2-3$ hard tail, and total luminosity $\\sim 5$ times lower. \nWe define our canonical LS state with the same level of hard tail. In our JED-SAD framework, this correspond to lower $\\dot{m}_{in}$ but still $r_J = r_{in}$ (no JED). In the example shown in this section, we choose $\\dot{m}_{in} = 0.45$. The absence of JED means there are no jets, i.e. no radio emitted. \\\\\n\n\n\n\n\\begin{figure}[h!]\n\\centering\n  \\includegraphics[width=1.\\columnwidth]{b2.pdf}\n   \\caption{Computed geometrical shape of the hybrid disk, consistent with the dynamical resolution (Fig.~\\ref{fig:TypicalStates}) and SED (Fig.~\\ref{fig:SpectralStates}) for each of the five canonical states. The color background is the central electron temperature. Note that the X-scale is logarithmic and the Y-scale is linear.}\n     \\label{fig:JoliesImages}\n\\end{figure}\n\n\n\n\n\n\n\n\\section{Concluding remarks} \n\\label{sec:Ccl}\n\nIn paper II, we studied the thermal equilibrium of jet-emitting disks (JED). JEDs are assumed to be thread by a large scale vertical magnetic fields, building two jets that produce a torque responsible for supersonic accretion.\n\nIn this article, we extend the code to compute thermal equilibria of hybrid disk configurations. This configuration assumes an inner JED and an outer standard accretion disk (SAD), characterized by a highly subsonic accretion speed. The transition between those two flows is assumed to be abrupt ($\\Delta R \/ R \\ll 1$) at some transition radius $r_J$. As argued in section~\\ref{sec:inter}, such a transition requires a discontinuity in the disk magnetization $\\mu$ that can be obtained if the transition radius $r_J$ is a steep density front. The transition radius $r_J$ would thus correspond to a density front advancing or receding within the disk during an outburst, as also found in the context of ADAF-SAD transitions \\citep{Honma96, Manmoto00}. Why such a density front would be present is an open question, possibly answered by how matter is initially brought in towards the disk inner regions. In any case, if such a front is indeed produced, it is not clear how it would be maintained over the long duration of the outburst.\n\nNow, such a density front is known to be favorable to the Rossby wave instability \\citep[][and references therein]{Tagger99, Lovelace99, Li00, Tagger04, Meheut10}, which leads to the formation of non-axisymmetric vortices within the disk. Whether or not the density front is smeared out and destroyed or simply perturbed (leading possibly to quasi-periodic oscillations) remains to be investigated. We refer the interested reader to the discussion on timing properties in paper I, section 4.\n\nOn the other hand, one might argue as well that such a discontinuity in the disk magnetization is unrealistic and that, instead, there is a continuous increase in $\\mu$ towards the disk inner regions \\citep{Petrucci08}. Assuming that such a situation would be indeed realized, the transition radius $r_J$ required in our spectral calculations would then be interpreted as the transition from the outer optically thick disk to the inner optically thin disk. Correspondingly, one could argue that the outer low-magnetized disk regions would give rise to winds whereas jets would be launched from the inner highly magnetized disk regions (JED). The difficulty with this scenario is that it relies on the disk mass loss and the radial distribution of the large scale vertical magnetic field, both unknown to date. Our simple approach, that assumes a sharp JED-SAD radial transition, can be seen as a first step towards addressing this difficult topic in XrB accretion disks. \\\\\n\n\n\nThe outside-in radial transition in accretion speed translates thermally, from an outer optically thick and cold accretion flow to an inner optically thin and hot flow. The soft photons emitted by the outer disk also provide a non-local cooling term which, added to advection of internal energy, allow a smooth thermal transition between these two regions. For a given JED-SAD dynamical solution the corresponding spectrum depends only on the mass accretion rate onto the black hole $\\dot{m}_{in}$ and the transition radius $r_J$ between the two flows. We explore in this article a large range in $\\dot{m}_{in}$ and $r_J$. Using XSPEC, we build synthetic spectra and fit them using a standard observers procedure (section~\\ref{sec:Py2Xspec}), allowing us to easily compare the resulting fits to observations.\n\nWe show that this framework is able to cover the whole domain explored by typical cycles in a disk fraction luminosity diagram (Fig.~\\ref{fig:EffectOfRj}). Furthermore, five canonical X-ray spectral states representative of a standard outburst are quantitatively reproduced with a reasonable set of parameters (Figs.~\\ref{fig:TypicalStates} to \\ref{fig:JoliesImages} and Table~\\ref{table:States}). A very interesting and important aspect of this framework is its ability to simultaneously explain both X-ray and radio emissions (Fig~\\ref{fig:PJET} and Table~\\ref{table:States}). In a forthcoming paper we will show the required time sequences $\\dot{m}_{in}(t)$ and $r_J(t)$ needed to reproduce a full cycle within the JED-SAD paradigm. \\\\\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{acknowledgements}\n   We are grateful to the anonymous referee for his\/her careful reading of the manuscript. The authors acknowledge funding support from French Research National Agency (CHAOS project ANR-12-BS05-0009 http:\/\/www.chaos-project.fr), Centre National de l'Enseignement Superieur (CNES) and Programme National des Hautes Energies (PNHE) in France. SC is supported by the SERB National Postdoctoral Fellowship (File No. PDF\/2017\/000841).\n\\end{acknowledgements}\n\n\n\\newpage \n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\IEEEPARstart{F}{or} a given $q{\\in}\\mathbb{N}$, define a set of complex exponential sequences as $H_q = \\{S_{q,k}(n) = e^{\\frac{j2{\\pi}kn}{q}}|0{\\leq}k{\\leq}q-1, (k,q)=1\\}$, where $0{\\leq}n{\\leq}q-1$ and $(k,q)$ denotes the greatest common divisor (gcd) between $k$ and $q$. \nBy adding all the elements of $H_q$, mathematician Srinivasa Ramanujan introduced a trigonometric summation called as Ramanujan Sum (RS) in $1918$, denoted as $c_q(n)$ \\cite{Ramanujan}.\nLater in 2014, P. P. Vaidyanathan introduced a finite length signal representation using $c_q(n)$ and its circular shifts \\cite{6839014}, \\cite{6839030}.\n\n\n\nMotivated by this, we introduced a trigonometric summation known as Complex Conjugate Pair Sum (CCPS) in one of our previous works \\cite{Shah}. \nAs $(k,q)$ $=$ $(q-k,q)$, for every $S_{q,k}(n){\\in}H_q$, there exists a complex conjugate sequence $S_{q,q-k}(n){\\in}H_q$, both together form a complex conjugate pair. \nCCPS $(c_{q,k}(n))$ is defined by adding each complex conjugate pair, i.e.,\n\\begin{equation}\nc_{q,k}(n) = 2Mcos\\left(\\frac{2{\\pi}kn}{q}\\right),\n\\end{equation}\nwhere \\footnotesize$M=\\begin{cases}\n\\frac{1}{2}&\\ \\text{if } q=1\\text{ (or) }2\\\\\n1&\\ \\text{if } q{\\geq}3 \n\\end{cases}$\\normalsize, $k{\\in}{\\hat{U}_q}$ if $q>1$ and $k=1$ if $q = 1$, refer \\textit{Notations} for ${\\hat{U}_q}$. \nRecently, S. W. Deng et al., introduced a two-dimensional subspace spanned by a complex conjugate pair $\\{S_{q,k}(n),S_{q,q-k}(n)\\}$ known as Complex Conjugate Subspace (CCS) \\cite{7544641}, denoted as $v_{q,k}$. \nIn \\cite{Shah}, we provided a new basis for CCS using CCPS.\nFurther, a finite length signal is represented as a linear combination of signals which belong to CCSs known as Complex Conjugate Periodic Transform (CCPT) \\cite{Shah}.\n\nInspired from \\cite{6839014,6839030,7964706}, in this letter we discuss several properties of CCPSs and CCSs, which may find  applications in signal processing.\nContributions of this letter can be divided into two parts. \nIn the first part, we show that\nthe operation of linear convolution between a given signal $x(n)$ and $\\bar{c}_{q,k}(n)$ is equivalent to computing the first derivative of $x(n)$, where $\\bar{c}_{q,k}(n)$ denotes one period data of ${c}_{q,k}(n)$. \nThis operation is also equivalent to the second derivative if we consider an odd number $q$ and a circular shift of $\\frac{q-1}{2}$ for $\\bar{c}_{q,k}(n)$. \nThen, the problem of edge detection in an image is addressed using this derivative equivalent operation. \nMoreover, we compare these results with the results obtained by using RSs.\n\nIn the second part, we prove the following properties:\n\\begin{itemize}\n\\item CCS is a shift-invariant subspace.\n\\item Since CCPT is a non-orthogonal transform \\cite{Shah}, we\ncompute the projections onto CCSs. \nFurther, we show that the projection matrix is a circulant matrix, which reduces the computational complexity of projections.\n\\item CCS is closed under circular cross-correlation. Moreover, the circular auto-correlation of any finite length sequence is equal to the weighted linear combination of its projections auto-correlation.\n\\end{itemize}\n\n\nThe structure of this letter is as follows: Using CCPSs as derivatives and the problem of detecting edges in an image are studied in Section \\RNum{2}.\nProperties of CCSs are discussed in Section \\RNum{3}. Finally, conclusions are drawn in Section \\RNum{4}.\n \n\\textit{Notations:}\n$(a,b)$ indicates the gcd between $a$ and $b$. A least common multiple is denoted as $lcm$.\nRounding the value $a$ to the greatest integer less than or equal to $a$ is denoted as $\\floor*{a}$.\nFor a given $n\\in\\mathbb{N}$, Euler's totient function $\\varphi(n)$ is defined as $\\varphi(n) = \\sum\\limits_{i=1}^{n}\\floor*{\\frac{1}{(i,n)}}$. As $(k,n) = (n-k,n)$, $\\varphi(n)$ is even for $n\\geq 3$. \nSymbol $d|N$ denotes that $d$ is a divisor of $N$. \nDefine a set $\\hat{U}_n = \\{a{\\in}\\mathbb{N}\\ |\\ 1{\\leq}a{\\leq}\\floor*{\\frac{n}{2}}, (a,n)=1\\}$, hence $\\#\\hat{U}_n = \\frac{\\varphi(n)}{2}$.\n$M_{m,n}(\\mathbb{C})$ indicates set of all $m\\times n$ matrices with entries from complex numbers. If $m=n$, $M_{m,n}(\\mathbb{C})=M_n(\\mathbb{C}).$\n\\section{CCPSs as Derivatives and Application}\nHere we perform an operation using CCPSs, which is equivalent to the derivative. In particular, we prove the following:\n\\begin{theorem}\nConsider an LTI system whose impulse response is $\\bar{c}_{q,k}(n),\\ q>1$, then for a given input $x(n)$  the output $y(n)$ of the system, i.e., $x(n)*\\bar{c}_{q,k}(n)$ is equivalent to computing the first order derivative of $x(n)$.\n\\label{Th1}\n\\end{theorem}\n\\textit{Proof:} Let $x(n) = C$, where $C$ is a constant value, then\n\\begin{equation}\ny(n) = C\\sum\\limits_{l=0}^{q-1}\\bar{c}_{q,k}(l) = 0,\\ \\text{as }\\sum\\limits_{l=0}^{q-1}\\bar{c}_{q,k}(l) = 0,\\ \\text{for }q>1.\n\\end{equation}\nIf $x(n) = u(n-n_0)$, where $u(n)$ is an unit step sequence, then \n\\begin{equation}\ny(n) = \\sum\\limits_{l=0}^{q-1}\\bar{c}_{q,k}(l)u(n-n_0-l),\n\\end{equation}\nhere $y(n){\\neq}0,\\ \\forall\\ n_0{\\leq}n{\\leq}{n_0+q-2}$. If $x(n) = n$, then\n\\begin{equation}\n\\footnotesize\n\\begin{aligned}\ny(n) &= n\\underbrace{\\sum\\limits_{l=0}^{q-1}\\bar{c}_{q,k}(l)}_{=0}-\\underbrace{\\sum\\limits_{l=0}^{q-1}l\\bar{c}_{q,k}(l)}_{\\mathbf{P}}\\\\\n&= -M\\left[\\sum\\limits_{l=0}^{q-1}le^{\\frac{j2{\\pi}kl}{q}}+\\sum\\limits_{l=0}^{q-1}l e^{\\frac{-j2{\\pi}kl}{q}}\\right]\\\\\n&= M\\left[\\frac{q}{1-e^{\\frac{j2{\\pi}k}{q}}}+\\frac{q}{1-e^{\\frac{-j2{\\pi}k}{q}}}\\right] = Mq.\n\\end{aligned}\n\\label{Der1}\n\\end{equation}\n\\normalsize\nFrom the above analysis, we draw the following conclusions regarding the system output. That is, the system output is:\n\\begin{enumerate}\n\\item Zero for constant input.\n\\item Non-zero at the on transient of the unit step sequence.\n\\item Non-zero constant along the ramps.\n\\end{enumerate}\nIn the \\textit{context of image processing}, any function\/operation satisfying the above three properties is equivalent to first order derivative \\cite{Jain,Gonzalez,7964706}.\nTherefore, $x(n)*\\bar{c}_{q,k}(n)$ is equivalent to computing the first order derivative of $x(n)$.\n\nFurther, with some modifications in the impulse response, the above operation is  equivalent to the second order derivative.\nTo be a second order derivative, \nthe system output should satisfy the first two conclusions mentioned in  \\textbf{Theorem} \\ref{Th1} and \nit should be zero for $x(n)=n$ \\cite{7964706}.\nThat is, the term $\\mathbf{P}$ in \\eqref{Der1} should be equal to zero, but the assumption of $\\bar{c}_{q,k}(n)$ as impulse response leads to $\\mathbf{P}=Mq$. So, instead of $\\bar{c}_{q,k}(n)$, we try by considering its circular shifts as an impulse response. Therefore,\n\\begin{equation}\n\\footnotesize\n\\sum\\limits_{l=0}^{q-1}l\\bar{c}_{q,k}(l-m) = \\frac{q}{1-cos\\left(v\\right)}\\left[cos\\left(u+v\\right)-cos\\left(u\\right)\\right],\\ 1{\\leq}m{\\leq}q-1,\n\\end{equation}\n\\normalsize\nwhere $u = \\frac{2{\\pi}km}{q}$ and $v = \\frac{2{\\pi}k}{q}$. Now for what value of $m$, \\footnotesize$ cos\\left(u+v\\right)=cos\\left(u\\right)$\\normalsize? Figure \\ref{f1} depicts \\footnotesize$cos\\left(u+v\\right)$ vs $cos\\left(u\\right)\\ $\\normalsize  for different $q$ and $k$ values. It is observed that independent of $k$ both the values are equal whenever $q$ is an odd number and $m = \\frac{q-1}{2}$.\n\\begin{figure}[!h]\n\\centering\n \\includegraphics[width=4.3in,height=1.8in]{Images\/secondDerivative.eps} \n\\caption[\\small{(a)-(d) $cos\\left(u+v\\right)$ vs $cos\\left(u\\right)$ for different $q$ and $k$ values}]{\\footnotesize{(a)-(d) $cos\\left(u+v\\right)$ vs $cos\\left(u\\right)$ for different $q$ and $k$ values.}}\n\\label{f1}\n\\end{figure}\nHence, we can summarize the above discussion as:\n\\begin{theorem}\nFor a given odd number $q$, the operation of linear convolution between a given signal $x(n)$ and $\\bar{c}_{q,k}\\left(n-\\frac{q-1}{2}\\right)$ is equivalent to computing the second order derivative of $x(n)$.\n\\end{theorem}\n\\subsection{Application}\nEdge detection of an image is crucial in many applications, where the derivative functions are used \\cite{Jain,Gonzalez}. \nIn this work, we address this problem using CCPSs and compare the results with the results obtained using RSs \\cite{7964706}.\nConsider the Lena image and convert it into two one-dimensional signals, namely $x_1(n)$ and $x_2(n)$, by column-wise appending and row-wise appending respectively.\nNow compute $x_1(n)*\\bar{c}_{5,1}(n),\\ x_2(n)*\\bar{c}_{5,1}(n),\\ x_1(n)*\\bar{c}_{5,2}(n)$ and $x_2(n)*\\bar{c}_{5,2}(n)$, the results are depicted in figure \\ref{f2} (a)-(d) respectively.\nFrom the results, it is clear that we can find the edges using CCPSs.\nNote that, performing convolution on $x_1(n)$ and $x_2(n)$ gives better detection of horizontal (Figure \\ref{f2} (a) and (c)) and vertical (Figure \\ref{f2} (b) and (d)) edges respectively.\nLet $\\hat{U}_q = \\{k_1,k_2,\\dots,k_{\\frac{\\varphi(q)}{2}}\\}$, $q{\\in}\\mathbb{N}$, then we can write the relationship between RSs and CCPSs as\n\\begin{equation}\n\\bar{c}_q(n) = \\bar{c}_{q,k_1}(n)+\\bar{c}_{q,k_2}(n)+\\dots+\\bar{c}_{q,k_{\\frac{\\varphi(q)}{2}}}(n).\n\\end{equation}\nUsing this and linearity property of convolution sum, we can write $x_1(n)*\\bar{c}_5(n) = x_1(n)*\\bar{c}_{5,1}(n)+x_1(n)*\\bar{c}_{5,2}(n)$. Figure \\ref{f2} (e)-(f) validates the same.\nSo, fine edge detection can be achieved using CCPSs over RSs, where RSs are integer-valued sequences and CCPSs are real-valued sequences.\nA similar kind of analysis can be done using CCPSs as the second derivative.\n\\begin{figure}[!h]\n\\centerline{\n \\includegraphics[width=3.3in,height=1.8in]{Images\/edgeDetectionUsingCCPS.PNG} }\n\\caption[\\small{(a)-(d) Applying convolution between Lena image and $\\bar{c}_{q,k}(n)$ in both vertical (column-wise appending) and horizontal (row-wise appending) directions respectively: (a)-(b) With $q=5$ and $k=1$. (c)-(d) With $q=5$ and $k=2$. (e) Adding (a) and (c). (f) Convolving Lena image with $\\bar{c}_{5}(n)$ in a vertical direction.}]{\\footnotesize{(a)-(d) Applying convolution between Lena image and $\\bar{c}_{q,k}(n)$ in both vertical (column-wise appending) and horizontal (row-wise appending) directions respectively: (a)-(b) With $q=5$ and $k=1$. (c)-(d) With $q=5$ and $k=2$. (e) Convolving Lena image with $\\bar{c}_{5}(n)$ in a vertical direction. (f) Adding (a) and (c) then subtract the result from (e).}}\n\\label{f2}\n\\end{figure}\n\n\\section{Properties of CCS}\nConstruct a circulant matrix $\\mathbf{D_{q,k}}{\\in}M_{q}(\\mathbb{R})$ as given below\n\\begin{equation}\n\\mathbf{D_{q,k}} = \\begin{bmatrix}\n\\mathbf{\\bar{c}}^0_\\mathbf{{q,k}} & \\dots & \\mathbf{\\bar{c}}^l_\\mathbf{{q,k}}\\dots & \\mathbf{\\bar{c}}^{q-1}_\\mathbf{{q,k}}\n\\end{bmatrix},\n\\end{equation}\nwhere $\\mathbf{\\bar{c}}^l_\\mathbf{{q,k}}$ indicates $l$ times circular downshift of the sequence $\\mathbf{\\bar{c}_{q,k}}$.\nUsing the \\textit{factorization} property of $\\mathbf{D_{q,k}}$, we can write $\\mathbf{D_{q,k}} = \\mathbf{B_{q,k}}\\mathbf{B_{q,k}^H}$, where\n\\begin{equation}\n\\mathbf{B_{q,k}^H} = \\begin{bmatrix}\nS_{q,{q}-k}(0)&S_{q,{q}-k}(1)&\\dots&S_{q,{q}-k}(q-1) \\\\\nS_{q,k}(0)&S_{q,k}(1)&\\dots&S_{q,k}({q}-1)\\\\\n\\end{bmatrix}_{2{\\times}q}.\n\\label{Factorization}\n\\end{equation}\nFrom \\eqref{Factorization}, the column space of $\\mathbf{D_{q,k}}$ is equal to $v_{q,k}$. Moreover, the first two columns of $\\mathbf{D_{q,k}}$ are linearly independent \\cite{Shah}.\nSo, any signal $\\mathbf{x}{\\in}v_{q,k}$ can be written as,\n\\begin{equation}\n\\left[\\mathbf{x}\\right]_{q{\\times}1} =\\begin{bmatrix}\n\\mathbf{\\bar{c}}^0_\\mathbf{{q,k}}& \\mathbf{\\bar{c}}^1_\\mathbf{{q,k}}\n\\end{bmatrix}\\left[\\mathbf{\\hat{\\beta}_{q,k}}\\right] = \\left[\\mathbf{F_{q,k}}\\right]_{q{\\times}2}\\left[\\mathbf{\\hat{\\beta}_{q,k}}\\right]_{2{\\times}1}.\n\\label{ccsBasis}\n\\end{equation}\nLet $q_1{\\in}\\mathbb{N}$, $q_2{\\in}\\mathbb{N}$ and $N = lcm(q_1,q_2)$, then the orthogonality property of CCPSs is \\footnotesize$\\left[\\mathbf{\\hat{c}_{q_1,k_1}}^{l_1}\\right]_{N{\\times}1}^T\\left[\\mathbf{\\hat{c}_{q_2,k_2}}^{l_2}\\right]_{N{\\times}1}= 2NMcos\\left(\\frac{2{\\pi}{k_1}(l_1-l_2)}{q_1}\\right)\\delta(q_1-q_2)\\delta(k_1-k_2)$ \\normalsize\\cite{Shah},\nwhere $\\mathbf{\\hat{c}_{q_i,k_i}}^{l_i}$ is obtained by repeating $\\mathbf{\\bar{c}}^{l_i}_\\mathbf{{q_i,k_i}}$ periodically $\\frac{N}{q_i}$ times.\nUsing this we can conclude the following: The vectors in the basis of $v_{q,k}$ are not orthogonal; The subspaces $v_{q_1,k_1}$ and $v_{q_2,k_2}$ are orthogonal to each other.\nWith this basic introduction, now we discuss a few of CCS properties in the following subsections.\n\n\n\\subsection{shift-invariant Subspace}\nA natural question for a signal belongs to CCS is:\nWhat is the effect if we consider a shifted version of the input signal?\nWe answer this question by proving the following theorem:\n\\begin{theorem}\nCCS is a circular shift-invariant subspace, i.e., if $\\mathbf{x}{\\in}v_{q,k}$, then $\\mathbf{x}^l{\\in}v_{q,k}$, here the shift is interpreted as circular shift $(\\text{modulo }q)$.\n\\end{theorem}\n\\textit{Proof:} From \\eqref{ccsBasis}, circular shifting $\\mathbf{x}$ by an amount $l$ gives \n\\begin{equation}\n\\mathbf{x}^l = \\begin{bmatrix}\n\\mathbf{\\bar{c}}^{l}_\\mathbf{{q,k}} & \\mathbf{\\bar{c}}^{l+1}_\\mathbf{{q,k}}\n\\end{bmatrix}\\left[\\mathbf{\\hat{\\beta}_{q,k}}\\right].\n\\end{equation}\n$\\mathbf{\\bar{c}}^{l}_\\mathbf{{q,k}}$ for any $l{\\in}\\mathbb{Z}$ is still a column in $\\mathbf{D_{q,k}}$. \nIt implies both $\\mathbf{\\bar{c}}^{l}_\\mathbf{{q,k}}{\\in}v_{q,k}$ and $\\mathbf{\\bar{c}}^{l+1}_\\mathbf{{q,k}}{\\in}v_{q,k}$.\nThis results in $\\mathbf{x}^l{\\in}v_{q,k}$.\nIn fact, using the row-Vandermonde structure of $\\mathbf{B_{q,k}^H}$ one can prove that any two consecutive columns of $\\mathbf{D_{q,k}}$ act as the basis for $v_{q,k}$,\nhence $\\mathbf{x}^l{\\in}v_{q,k}$.\n\nSince CCS is Shift-invariant, it may be useful in applications like wireless communication \\cite{Forsythe},  subspace tracking:\nwhich play a vital role in video surveillance, source localization in radar and sonar, etc., \\cite{8683025}.\n \n\\subsection{Computing the Projections}\nIn CCPT, a signal $\\mathbf{x}{\\in}M_{N,1}(\\mathbb{C})$ is represented as a linear combination of signals belongs to CCSs \\cite{Shah},\n\\begin{equation}\n\\mathbf{x} = \\sum\\limits_{q_i|N}^{}\\sum\\limits_{\\substack{k=1\\\\(k,q_i)=1}}^{\\floor*{\\frac{q_i}{2}}}\\underbrace{\\mathbf{E_{q_i,k}}\\boldsymbol{\\beta}_\\mathbf{{q_i,k}}}_{\\mathbf{x_{q_i,k}}{\\in}v_{q_i,k}} = \\left[\\mathbf{T_N}\\right]_{N{\\times}N}\\left[\\boldsymbol{\\beta}\\right]_{N{\\times}1}.\n\\label{Synthesis}\n\\end{equation}\nHere, $\\mathbf{T_N}$ is the transformation matrix, $\\boldsymbol{\\beta}$ is the transform coefficient vector,\n$\\mathbf{x_{q_i,k}}$ denotes the projection of $\\mathbf{x}$ onto $v_{q_i,k}$,\n$\\mathbf{[E_{q_i,k}]_{N{\\times}2}}= [\\mathbf{F_{q_i,k}},\\dots,\\mathbf{F_{q_i,k}}]^T$ \nand $\\boldsymbol{\\beta}_\\mathbf{{q_i,k}}{\\in}M_{2,1}(\\mathbb{C})$ is the transform coefficients vector corresponds to $v_{q_i,k}$. \nAs given in \\cite{Shah}, a major application of CCPT is \nestimating the period and frequency information of a signal using $\\boldsymbol{\\beta}$.\nWhile the non-orthogonal basis of $v_{q_i,k}$ makes CCPT a non-orthogonal transform, i.e., it requires computation of $\\mathbf{T_N^{-1}}$ to find $\\boldsymbol{\\beta}$. \nAccording to Strassen's algorithm, computing $\\mathbf{T}^{-1}_\\mathbf{N}$ requires a computational complexity of $\\mathcal{O}(N^{2.81})$ \\cite{Cormen}.\nTo use CCPT in an efficient way, one has to overcome this limitation, for this\nwe compute $\\mathbf{x_{q_i,k}}$ instead of $\\boldsymbol{\\beta}$, which can serve for the same purpose.\nSince $\\left[\\mathbf{E_{q_i,k}}\\right]^H\\left[\\mathbf{E_{q_x,k_y}}\\right] = 0,\\ \\forall\\ i{\\neq}x$ (where $q_i|N \\text{ and } q_x|N$), the projection of $\\mathbf{x}$ onto $v_{q_i,k}$ can be computed as follows:\n\\begin{equation}\n\\mathbf{x_{q_i,k}} = \\underbrace{\\mathbf{E_{q_i,k}}(\\mathbf{E_{q_i,k}^H}\\mathbf{E_{q_i,k}})^{-1}\\mathbf{E_{q_i,k}^H}}_{\\mathbf{P_{q_i,k}}{\\in}M_{N}:\\text{ Projection matrix}}\\mathbf{x}.\n\\label{projMatrix}\n\\end{equation}\nHere $\\mathbf{P_{q_i,k}}$ can be further reduced as given below:\n\\begin{equation}\n\\mathbf{P_{q_i,k}} = \\frac{q_i}{N}\\begin{bmatrix}\n\\mathbf{\\hat{P}_{q_i,k}}&\\dots&\\mathbf{\\hat{P}_{q_i,k}}\\\\\n\\vdots&\\ddots&\\vdots\\\\\n\\mathbf{\\hat{P}_{q_i,k}}&\\dots&\\mathbf{\\hat{P}_{q_i,k}}\n\\end{bmatrix}, \\text{where}\n\\label{redProj}\n\\end{equation} \n\\begin{equation}\n\\mathbf{[\\hat{P}_{q_i,k}]_{q_i{\\times}q_i}} = \\mathbf{F_{q_i,k}}(\\mathbf{F_{q_i,k}^H}\\mathbf{F_{q_i,k}})^{-1}\\mathbf{F_{q_i,k}^H}.\n\\end{equation}\nLet us divide the input signal $\\mathbf{x}$ into $\\frac{N}{q_i}$ blocks, that is $\\mathbf{x} = \\left[\\mathbf{x^{(1)}},\\mathbf{x^{(2)}},\\dots,\\mathbf{x^{(\\frac{N}{q_i})}}\\right]^T$, where $i^{th}$ block $\\mathbf{x^{(i)}}{\\in}M_{{q_i}{\\times}1}$.\nThen, from \\eqref{projMatrix} and \\eqref{redProj} we can write\n\\begin{equation}\n\\mathbf{x_{q_i,k}} = \\frac{q_i}{N}\\begin{bmatrix}\n\\mathbf{\\hat{P}_{q_i,k}}&\\dots&\\mathbf{\\hat{P}_{q_i,k}}\\\\\n\\vdots&\\ddots&\\vdots\\\\\n\\mathbf{\\hat{P}_{q_i,k}}&\\dots&\\mathbf{\\hat{P}_{q_i,k}}\n\\end{bmatrix}\\begin{bmatrix}\n\\mathbf{x^{(1)}}\\\\\n\\vdots\\\\\n\\mathbf{x^{(\\frac{N}{q_i})}}\n\\end{bmatrix} = \\begin{bmatrix}\n\\mathbf{y_{q_i,k}}\\\\\n\\vdots\\\\\n\\mathbf{y_{q_i,k}}\n\\end{bmatrix},\n\\label{Proj}\n\\end{equation}\nwhere\n\\begin{equation}\n\\mathbf{y_{q_i,k}} =\\frac{q_i}{N}\\mathbf{\\hat{P}_{q_i,k}}\\sum\\limits_{i=1}^{\\frac{N}{q_i}}\\mathbf{x^{(i)}}.\n\\label{Proj1}\n\\end{equation}\nNow consider a matrix $\\mathbf{\\tilde{P}_{q_i,k}} = \\frac{\\mathbf{D_{q_i,k}}}{q_i}$, then $\\mathbf{\\tilde{P}_{q_i,k}}^2 = \\frac{\\mathbf{B_{q_i,k}}\\mathbf{B}^H_\\mathbf{{q_i,k}}\\mathbf{B_{q_i,k}}\\mathbf{B}^H_\\mathbf{{q_i,k}}}{q_i^2} = \\mathbf{\\tilde{P}_{q_i,k}}$, since $\\mathbf{B}^H_\\mathbf{{q_i,k}}\\mathbf{B_{q_i,k}} = q_i\\mathbf{I}$.\nMoreover, the even symmetry property of CCPSs makes $\\mathbf{\\tilde{P}_{q_i,k}}^H = \\mathbf{\\tilde{P}_{q_i,k}}$.\nSo, we can say that $\\mathbf{\\tilde{P}_{q_i,k}}$ is an orthogonal projection matrix. \nSince $v_{q_i,k}$ is the column space of $\\mathbf{D_{q_i,k}}$, \nwe can write $\\mathbf{\\hat{P}_{q_i,k}} = \\mathbf{\\tilde{P}_{q_i,k}}= \\frac{\\mathbf{D_{q_i,k}}}{q_i}$.\nWith this, \\eqref{Proj1} can be modified as,\n\\begin{equation}\n\\mathbf{y_{q_i,k}} =\\frac{1}{N}\\mathbf{D_{q_i,k}}\\sum\\limits_{i=1}^{\\frac{N}{q_i}}\\mathbf{x^{(i)}} =\\frac{1}{N}\\mathbf{D_{q_i,k}}\\mathbf{\\hat{x}} .\n\\label{Proj2}\n\\end{equation}\nSo, the orthogonal projection $\\mathbf{x_{q_i,k}}$ is computed as follows: First, multiply $\\mathbf{\\hat{x}}$ with the circulant matrix $\\mathbf{D_{q_i,k}}$, then multiply the result with a scale factor $\\frac{1}{N}$ to get $\\mathbf{y_{q_i,k}}$. Now repeat $\\mathbf{y_{q_i,k}}$ periodically $\\frac{N}{q_i}$ times to obtain $\\mathbf{x_{q_i,k}}$.\nThe number of multiplications (computational complexity) involved in computing $\\mathbf{x_{q_i,k}}$ are ${q_i}^2+q_i$.\nFrom \\eqref{Synthesis}, there are total of $\\frac{1}{2}\\sum\\limits_{q_i|N}^{}\\varphi(q_i) = \\frac{N}{2}$ number of CCSs are involved in representing $\\mathbf{x}$. \nSo, the total number of multiplications ($M_{total}$) required for computing projections for all $\\frac{N}{2}$ CCSs are \n\\begin{equation}\nM_{total} = \\begin{cases}\n2+\\frac{1}{2}\\sum\\limits_{\\substack{q_i|N\\\\ q_i{\\geq}3}}^{}\\varphi(q_i)(q_i^2+q_i),\\ &\\text{if }2\\nmid N \\\\\n8+\\frac{1}{2}\\sum\\limits_{\\substack{q_i|N\\\\ q_i{\\geq}3}}^{}\\varphi(q_i)(q_i^2+q_i),\\ &\\text{if }2\\mid N \n\\end{cases}.\n\\end{equation}\n\\begin{table}[h]\n\\centering\n\\caption{C\\scriptsize{OMPARISON OF $M_{total}\\ vs\\ \\floor*{N^{2.81}}$ FOR FEW $N$ VALUES}}\n\\label{tab:Comparison}\n\\begin{adjustbox}{max width=\\textwidth}\n\\scalebox{0.8}{\n\\begin{tabular}{|M{1.1cm}|M{1cm}|M{1.2cm}|M{1.4cm}|M{1.4cm}|M{1.4cm}|} \\hline\nN &\t$3$\t&\t$6$ & $8$&\t$32$\t& $82$\t\\\\\t\\hline\n\\small{$M_{total}$} &\t$14$\t&\t$62$ &\t$172$& $9708$\t&  $170568$  \\\\\t\\hline\n\\small{$\\floor*{N^{2.81}}$} & $21$\t&\t$153$ &$344$&\t$16961$\t& $238680$  \\\\    \\hline\n\\end{tabular}}\n\\end{adjustbox}\n\\end{table}\nFor few $N$ values both $M_{total}$ and $\\floor*{N^{2.81}}$ are tabulated in Table \\ref{tab:Comparison}. \nFrom the table, we can conclude that $M_{total} << \\mathcal{O}(N^{2.81})$ and there is an approximate of $40\\%$ reduction in the computational complexity for the values given in the table.\nTherefore, estimating the period and frequency information through\nprojection computation requires less computational complexity over the direct inverse computation method.\nApart from this computational advantage, in some applications, it is required to know the existence of certain periods in an observed signal.\nIn such scenarios, computing projections for those CCSs are sufficient.\n\\subsection{Correlation of Sequences in CCS}\n\\textbf{DFT of CCPS:} For a given $q{\\in}\\mathbb{N}$ and $k{\\in}\\hat{U}_q$,\n\\begin{equation}\nDFT[\\bar{c}_{q,k}(n)] = \\bar{C}_{q,k}(K) = \\begin{cases}\nq,\\ \\text{if }K=k\\ \\text{(or)}\\ q-k,\\\\\n0,\\ \\text{Otherwise}.\n\\end{cases}\n\\end{equation}\nLet $r_{c}(l)$ denotes the circular autocorrelation of $\\bar{c}_{q,k}(l)$, then\n\\begin{equation}\n\\nonumber\nDFT[r_{c}(l)] = R_{c}(K) = \\bar{C}_{q,k}(K)\\bar{C}_{q,k}(K) = q\\bar{C}_{q,k}(K).\n\\end{equation}\nTaking IDFT of above equation leads to $r_{c}(l) = q\\bar{c}_{q,k}(l)$.\nUsing this relation, we prove the following Theorem.\n\\begin{theorem} CCS is closed under circular cross-correlation operation, i.e., if $x(n){\\in}v_{q,k}$ and $y(n){\\in}v_{q,k}$ then the circular cross-correlation $r_{xy}(l)$ also belongs to $v_{q,k}$.\n\\end{theorem}\n\\textit{Proof:} Given $x(n){\\in}v_{q,k}$ and $y(n){\\in}v_{q,k}$ then\n\\begin{equation}\n\\nonumber\n\\footnotesize\n\\begin{aligned}\nr_{xy}(l)& = \\sum\\limits_{n=0}^{q-1}\\underbrace{\\left[\\sum\\limits_{m_1=0}^{1}{\\beta_{m_1}}\\bar{c}_{q,k}(n-m_1)\\right]}_{x(n)}\\underbrace{\\left[\\sum\\limits_{m_2=0}^{1}{\\gamma^*_{m_2}}\\bar{c}_{q,k}(n-l-m_2)\\right]}_{y^*(n-l)}\\\\\n& = \\sum\\limits_{m_1=0}^{1}\\sum\\limits_{m_2=0}^{1}{\\beta_{m_1}}{\\gamma^*_{m_2}}\\left[\\sum\\limits_{n=0}^{q-1}\\bar{c}_{q,k}(n-m_1)\\bar{c}_{q,k}(n-l-m_2)\\right]\\\\\n& = \\sum\\limits_{m_1=0}^{1}\\sum\\limits_{m_2=0}^{1}{\\beta_{m_1}}{\\gamma^*_{m_2}}q\\underbrace{\\bar{c}_{q,k}(l+m_2-m_1)}_{{\\in}v_{q,k}}.\n\\end{aligned}\n\\normalsize\n\\end{equation}\nFrom above, $r_{xy}(l)$ is a weighted linear combination of signals belongs to $v_{q,k}$, hence $r_{xy}(l){\\in}v_{q,k}$.\n\nFurther, the autocorrelation of a $N$-length signal $x(n)$ is\n\\begin{equation}\nr_x(l) = \\sum\\limits_{n=0}^{N-1}x(n)x^*(n-l)\n\\end{equation}\nUsing \\eqref{Synthesis}, the above equation can be modified as\n\\footnotesize\n\\begin{equation}\nr_x(l) = \\sum\\limits_{q_i|N}^{}\\sum\\limits_{\\substack{k_1=1\\\\(k_1,q_i)=1}}^{\\floor*{\\frac{q_i}{2}}}\\sum\\limits_{q_j|N}^{}\\sum\\limits_{\\substack{k_2=1\\\\(k_2,q_j)=1}}^{\\floor*{\\frac{q_j}{2}}}\\underbrace{\\left[\\sum\\limits_{n=0}^{N-1}x_{q_i,k_1}(n)x^*_{q_j,k_2}(n-l)\\right]}_{\\mathbf{Q}}.\n\\label{Corr}\n\\end{equation}\n\\normalsize\nwhere $\\mathbf{Q} = \\begin{cases}\n\\frac{N}{q_i}r_{x_{q_i,k_1}}(l),&\\ \\text{if } q_i=q_j\\ \\&\\ k_1=k_2\\\\\n0,&\\ \\text{if }{k_1}{\\neq}{k_2}\n\\end{cases},\n$\nsince CCS is shift-invariant and both $v_{q_i,k_1}$, $v_{q_j,k_2}$ are orthogonal to each other  for $k_1{\\neq}k_2$.\nNow substituting $\\mathbf{Q}$ in \\eqref{Corr} leads to\n\\begin{equation}\n\\footnotesize\n\\frac{1}{N}r_x(l) = \\sum\\limits_{q_i|N}^{}\\sum\\limits_{\\substack{k_1=1\\\\(k_1,q_i)=1}}^{\\floor*{\\frac{q_i}{2}}}\\frac{1}{q_i}r_{x_{q_i,k_1}}(l).\n\\normalsize\n\\end{equation}\nThe above result is summarized as follows:\n\\begin{theorem}\nThe circular autocorrelation of a signal is\nequal to the linear combination of \nits  projections (onto CCSs) autocorrelation, with a proper normalization.\n\\end{theorem}\nThis property is useful if the given signal is contaminated with additive noise. In such scenarios projecting $r_x(l)$ onto CCSs gives an accurate period and frequency estimation over projecting $x(n)$.\n\n\n\\section{Conclusion}\nIn this work, we have shown how to use CCPSs as first and second order derivatives. Here, the first order derivative is used to find the edges in an image.\nIt is shown that fine edge detection can be achieved using CCPSs over RSs.\nLater we discussed few properties of CCSs, which may find their applications in signal processing.\nIn particular, the signal information can be estimated with lesser computational complexity using CCS projections.\n\\section*{Acknowledgement}\nThe authors would like to thank Mr. Shiv Nadar, the founder and chairman of HCL and Shiv Nadar Foundation.\n\n\n\\ifCLASSOPTIONcaptionsoff\n  \\newpage\n\\fi\n\n\n\n\n\n\\bibliographystyle{ieeetr}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Abstract}\nPhylogenomic analyses are subject to bias from convergence in\nmacromolecular compositions and noise from horizontal gene transfer\n(HGT). Accordingly, compositional convergence leads to contradictory\nresults on the phylogeny of taxa such as the ecologically dominant\nSAR11 group of Alphaproteobacteria, which have extremely streamlined,\nA+T-biased genomes. While careful modeling can reduce bias artifacts\ncaused by convergence, the most consistent and robust phylogenetic\nsignal in genomes may lie distributed among encoded functional\nfeatures that govern macromolecular interactions.  Here we develop a\nnovel phyloclassification method based on signatures derived from\nbioinformatically defined tRNA Class-Informative Features (CIFs). tRNA\nCIFs are enriched for features that underlie tRNA-protein\ninteractions. Using a simple tRNA-CIF-based phyloclassifier, we\nobtained results consistent with bias-corrected whole proteome\nphylogenomic studies, rejecting monophyly of SAR11 and affiliating\nmost strains with Rhizobiales with strong statistical support. Yet, as\nexpected by their elevated genomic A+T contents, SAR11 and\nRickettsiales tRNA genes are also similarly and distinctly A+T-rich\nwithin Alphaproteobacteria. Using conventional supermatrix methods on\ntotal tRNA sequence data, we could recover the artifactual result of a\nmonophyletic SAR11 grouping with Rickettsiales. Thus tRNA CIF-based\nphyloclassification is more robust to base content convergence than\nsupermatrix phylogenomics with whole tRNA sequences. Also, given the\nnotoriously promiscuous HGT rates of aminoacyl-tRNA synthetase genes,\ntRNA CIF-based phyloclassification may be at least partly robust to\nHGT of network components. We describe how unique features of the\ntRNA-protein interaction network facilitate mining of traits governing\nmacromolecular interactions from genomic data, and discuss why\ninteraction-governing traits may be especially useful to solve\ndifficult problems in microbial classification and phylogeny.\n\n\n\\section*{Author Summary}\nIn this study, we describe a new way to classify living things using\ninformation from whole genomes. First, for a group of related\norganisms, we bioinformatically predict features by which specific\nclasses of tRNAs are recognized by certain proteins or\ncomplexes. Second, we train an artificial neural network to recognize\nwhich code a new, unknown genome belongs to. We apply our method to\nSAR11, one of the most abundant bacteria in the world's oceans. We\nfind that different strains of SAR11 are more distantly related, both\nto each other and to mitochondria, than previously thought. However,\nwith more traditional treatments of whole tRNA sequence data, we\nobtain different results, best explained as artifacts of base content\nconvergence. Our tRNA features are therefore more robust to genomic\nbase content convergence than the tRNAs in which they are embedded;\nthis is additional evidence of their functional importance. The tRNA\nfeatures we study form a clade-specific and slowly diverging ``feature\nnetwork'' that underlies a universally conserved macromolecular\ninteraction network. We discuss on theoretical grounds why traits\ngoverning macromolecular interactions may be especially well-suited to\nresolve deep relationships in the Tree of Life.\n\n\\section*{Introduction}\n\nWhat parts of genomes are most robust to compositional convergence?\nWhat information is most faithfully inherited vertically? The key\nassumptions of compositional stationarity and consistency in gene\nhistories underpin most current approaches in phylogenomics and are\nfrequently violated (reviewed in {\\it e.g.}\\cite{Gribaldo:2002}). HGT\nis so widespread that the very existence of a ``Tree of Life'' has\nbeen questioned~\\cite{Gogarten:2002us,Bapteste:2009ci}. Better\nunderstanding of ancient phylogenetic relationships requires discovery\nof new universal, slowly-evolving phylogenetic markers that are robust\nto compositional convergence and HGT.\n\nThe controversial phylogeny of {\\it Ca.}Pelagibacter ubique (SAR11) is\na case in point. SAR11 make up between a fifth and a third of the\nbacterial biomass in marine and freshwater\necosystems~\\cite{Morris:2002bn}. Adaptations to extreme environmental\nnutrient limitation may explain why SAR11 have very small cell and\ngenome sizes and small fractions of intergenic\nDNA~\\cite{Giovannoni:2005}. While some recent phylogenomic studies\ndefine a clade among SAR11, the largely endoparasitic Rickettsiales,\nand the alphaproteobacterial ancestor of\nmitochondria~\\cite{Williams:2007p5327,Georgiades:2011gx,Thrash:2011kl},\nothers argue that this placement of SAR11 is an artifact of\nindependent convergence towards increased genomic A+T content, and\nthat SAR11 belongs closer to other free-living Alphaproteobacteria\nsuch as the Rhizobiales and\nRhodobacteraceae~\\cite{Brindefalk:2011ei,RodriguezEzpeleta:2012fw,Viklund:2012jr}.\nMonophyly of SAR11 was also recently\nrejected~\\cite{RodriguezEzpeleta:2012fw}.\n\nNonstationary macromolecular compositions are a known source of bias\nin phylogenomics~\\cite{Foster01062004,Losos14122012}. Widespread variation in\nmacromolecular compositions may be associated with loss of DNA repair\npathways in reduced genomes~\\cite{Dale:2003hc,Viklund:2012jr},\nunveiling an inherent A+T-bias of mutation in\nbacteria~\\cite{Hershberg:2010ig} and elevating genomic A+T\ncontent~\\cite{Moran:2002p5737,Lind:2008cs}. A process such as this has\nlikely altered protein and RNA compositions genome-wide in SAR11, and\nif such effects are accounted for, the placement of SAR11 with\nRickettsiales drops away as an apparent\nartifact~\\cite{RodriguezEzpeleta:2012fw,Viklund:2012jr}. Consistent\nwith this interpretation, SAR11 strain HTTC1062 shares a surprising\nand unique codivergence of \\tRNAaa{His} and histidyl-tRNA synthetase\n(HisRS) with a clade of free-living\nAlphaproteobacteria~\\cite{Wang:2007p1040,Ardell:2010du} that likely\narose only once in bacteria~\\cite{Ardell:2006ko}. This synapomorphy\ncontradicts the placement of SAR11 with Rickettsiales.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=3.2in]{AmrineEtAl_Fig1.pdf\n\\caption{\\figcapone}\n\\label{fig:network}\n\\end{figure*}\n\nThis work was motivated to determine whether the entire system of\ntRNA-protein interactions could be exploited to address phylogeny of\nbacteria, particularly SAR11. The highly conserved tRNA-protein\ninteraction network (Fig.~\\ref{fig:network}) has special advantages\nfor comparative systems biological study from genomic data. First, the\ncomponents and interactions of this network are highly\nconserved. Second, bioinformatic mining of interaction-determining\ntraits from genomic tRNA data is favorable because tRNA structures are\nhighly conserved not just across extant taxa but also across different\nfunctional classes of tRNAs\n(``conformity''~\\cite{WOLFSON01012001}). Yet each functional class of\ntRNA must maintain a hierarchy of increasingly specific interactions\nwith various proteins and other factors\n(``identity''~\\cite{Giege:2008}). The conflicting requirements of\nconformity and identity allow structural comparison and contrast to\npredict class-informative traits of tRNAs from sequence data by\nrelatively simple bioinformatic methods~\\cite{Ardell:2010du}. The\nfeatures that govern tRNA-protein interactions diverge across the\nthree domains of life (reviewed in~\\cite{Giege:1998um}) and also\nwithin the domain of bacteria~\\cite{Ardell:2006ko}.\n\nIn prior work, we developed ``function logos'' to predict, at the\nlevel of individual nucleotides before post-transcriptional\nmodification, what genetically templated information in tRNA gene\nsequences is associated to specific functional identity\nclasses~\\cite{Freyhult:2006dr}. We now call these function-logo-based\npredictions Class-Informative Features (CIFs). A tRNA CIF answers a\nquestion like: ``if a tRNA gene from a group of related genomes\ncarries a specific nucleotide at a specific structural position, how\nmuch informaiton do we gain about that tRNAs specific function?''\nSuch information estimates are corrected for biased sampling of\nfunctional classes and sample size effects~\\cite{Freyhult:2006dr}, and\ntheir statistical significance may be\ncalculated~\\cite{Ardell:2010du}. Although an individual bacterial\ngenome does not present enough data to generate a function logo,\nrelated genome data may be lumped, weakly assuming homogeneity of tRNA\nidentity rules (although heterogeneity generally reduces\nsignal). Function logos recover known tRNA identity elements ({\\it\n  i.e.} features that govern the specificity of interactions between\ntRNAs and proteins)~\\cite{Giege:1998um}, and more generally, predict\nfeatures governing interactions with class-specific network partners\nsuch as amidotransferases~\\cite{Bailly:2006cs}. A recent molecular\ndynamics study on a \\tRNAaa{Glu}-GluRS (Glutaminal tRNA-synthetase)\ncomplex identified tRNA functional sites involved in intra- and\ninter-molecular allosteric signalling within GluRS that couples\nsubstrate recognition to reaction catalysis~\\cite{Sethi:2009}. The\npredicted sites are correlated with those from proteobacterial\nfunction logos~\\cite{Freyhult:2007jj}.\n\nIn this work, we show that tRNA CIFs have diverged among\nAlphaproteobacteria in a phylogenetically informative manner. Second,\nas phylogenetic markers, tRNA CIFs are more robust to compositional\nconvergence than the tRNA bodies in which they are embedded. Using our\ntRNA-CIF-based phyloclassification approach, we confirm that SAR11 are\npolyphyletic with the majority of strains clustering with the\nfree-living Alphaproteobacteria. Our results have implications for how\nto best mine genomic data for phylogenetic signals.\n\n\\section*{Results}\n\nWe reannotated Alphaproteobacterial tDNA data from tRNAdb-CE\n2011~\\cite{Abe:2011js} and other prepublication genomic data, and\nsplit them into two groups according to whether or not their source\ngenome contained the uniquely derived synapomorphic traits previously\ndescribed~\\cite{Ardell:2006ko}: a gene for \\tRNAaa{His} containing A73\n(using ``Sprinzl coordinates'',~\\cite{Sprinzl:1998vz}) and lacking\ntemplated $-1G$. We could thereby partition the data into an RRCH\nclade (Rhodobacteraceae, Rhizobiales, Caulobacterales,\nHyphomonadaceae), which present the uniquely derived \\tRNAaa{His}, and\nthe RSR grade (Rhodospirillales, Sphingomonadales, and Rickettsiales,\nexcluding SAR11), which present ``normal'' bacterial \\tRNAaa{His} with\nC73 and genomically templated $-1G$.  In all, data from 214\nAlphaproteobacterial genomes represented 11644 predicted tRNA\nsequences (8773 sequences unique within genomes and 3064 total unique\nsequences). Our final dataset contained 147 genomes (8597 tRNAs) for\nthe RRCH clade, 59 genomes (2792 tRNAs) for the RSR grade, and 8\ngenomes (255 tRNAs) of SAR11 strains.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=5in]{AmrineEtAl_Fig2.pdf\n\\caption{\\figcaptwo}\n\\label{fig:over}\n\\end{figure*}\n\nThe unique traits of the RRCH \\tRNAaa{His} are perfectly associated to\nsubstitutions of key residues in the motif IIb tRNA-binding loops of\nHisRS involved in tRNA recognition~\\cite{Ardell:2006ko}. Seven of\neight SAR11 strains exhibited the unique \\tRNAaa{His}\/HisRS\ncodivergence traits in common with RRCH genomes. In contrast, strain\nHIMB59 presented ancestral bacterial characters in both \\tRNAaa{His}\nand HisRS (Fig. S1). These results immediately suggest that\nHIMB59 is not monophyletic with the other SAR11 strains, consistent\nwith~\\cite{RodriguezEzpeleta:2012fw}.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=5in]{AmrineEtAl_Fig3.pdf}\n\\caption{\\figcapthree}\n\\label{fig:binclass}\n\\end{figure*}\n\nWe computed function logos~\\cite{Freyhult:2006dr} of the RRCH clade\nand RSR grade to form the basis of a tRNA-CIF-based binary\nphyloclassifier as shown schematically in Fig.~\\ref{fig:over}. To\nreduce bias, we used a Leave-One-Out Cross-Validation (LOOCV)\napproach. For comparison, we also performed LOOCV phyloclassification\nusing sequence profiles of entire tRNAs, with typical results shown in\nFig.~\\ref{fig:binclass}B. Although the tRNA-CIF-based phyloclassifier\n(Fig.~\\ref{fig:binclass}A) was biased positively by the much larger\nRRCH sample size, it achieved better phylogenetic separation of\ngenomes than the total-tRNA-sequence-based phyloclassifier\n(Fig.~\\ref{fig:binclass}B). The Sphingomonadales and Rhodospirillales\nseparated in scores from the Rickettsiales in both classifiers. Most\nimportantly, the tRNA-CIF-based phyloclassifier placed all eight SAR11\ngenomes closer to the RRCH clade and far away from the Rickettsiales\nwith HIMB59 overlapping the Rhodospirillales, while the\ntotal-tRNA-sequence-based phyloclassifier placed all eight SAR11\ngenomes closer to the Rickettsiales. Fig.~S2 shows the effects of\ndifferent treatments of missing data in the total-tRNA-sequence-based\nclassifier. Method ``zero,'' shown in Fig.~\\ref{fig:binclass}B, is\nmost analogous to the method used to generate\nFig.~\\ref{fig:binclass}A. Method ``skip'' (Fig. S2B) shows that\nSAR11 tRNAs share sequence characters in common with the RSR grade\nthat are not seen in the RRCH clade. Methods ``small'' and ``pseudo''\n(Figs. S2C and S2D) show that SAR11 have sequence traits not\nobserved in either RSR or RRCH.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=5in]{AmrineEtAl_Fig4.pdf\n\\caption{\\figcapfour}\n\\label{fig:breakout}\n\\end{figure*}\n\nMany other tRNA classes besides \\tRNAaa{His} contribute to the\ndifferentiated classification of RRCH and RSR genomes by the CIF-based\nbinary classifier (Fig.~\\ref{fig:breakout}). Other tRNA classes are\nalso differentiated between these two groups, including \\tRNAaa{Cys},\n\\tRNAaa{Asp}, \\tRNAaa{Glu}, \\tRNA{Ile}{LAU} (symbolized ``J''),\n\\tRNAaa{Lys}, \\tRNAaa{Tyr}. These results extend the observations of\n\\cite{Wang:2007p1040} who discovered unusual base-pair features of\n\\tRNAaa{Glu} in the RRCH clade. In classes for which the RRCH and RSR\ngroups are well-differentiated, HIMB59 uniquely groups with RSR while\nother strains group with RRCH, while for other tRNA classes, all\nputative SAR11 strains lie outside the RRCH and RSR\ndistributions. This implies that more diverse Alphaproteobacterial\ngenomic data are necessary to completely resolve the phylogenetic\naffiliation of SAR11 strains, but strongly contradict a monophyletic\naffiliation of SAR11 with Rickettsiales. \n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=5in]{AmrineEtAl_Fig5.pdf\n\\caption{\\figcapfive}\n\\label{fig:comp}\n\\end{figure*}\n\nThe increases in genomic A+T contents in SAR11 and Rickettsiales have\nalso driven elevated A+T contents of their tRNA genes\n(Fig.~\\ref{fig:comp}A).  Rickettsiales and SAR11 tRNA genes are both\nnotably elevated in both A and T, and share an overall similarity in\ncomposition distinct from other Alphaproteobacteria. Hierarchical\nclustering of Alphaproteobacterial taxa based on tRNA gene base\ncontents closely group SAR11 and Rickettsiales together\n(Fig.~\\ref{fig:comp}B).\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=5in]{AmrineEtAl_Fig6.pdf\n\\caption{\\figcapsix}\n\\label{fig:unifrac}\n\\end{figure*}\n\nNonstationary tRNA base content --- convergence to greater A+T content\n--- causes all eight SAR11 strains in our dataset to group with\nRickettsiales using phylogenomic approaches based on total tRNA\nsequence evidence. In a ``supermatrix'' phylogenomic approach,\nconcatenating genes for 28 isoacceptor classes from 169 species (2156\ntotal sites) and using the GTR+Gamma model in RAxML, we estimated a\nMaximum Likelihood tree in which all eight putative SAR11 strains\nbranch together with Rickettsiales (Fig.~S3). For this analysis,\nin 31\\% of instances when isoacceptor genes were picked from a genome,\nwe randomly picked one gene from a set of isoacceptor\nparalogs. However, our results did not depend on which paralog we\npicked. Using a distance-based approach with FastTree, we computed a\nconsensus cladogram over 100 replicate alignments each representing\ndifferent randomized picks over paralogs. As the consensus cladogram\nshows (Fig. S4) each replicate distance tree placed all\neight putative SAR11 strains together with Rickettsiales. The recently\nintroduced tRNA-specific FastUniFrac-based method for microbial\nclassification~\\cite{Widmann:2010ea} also places all SAR11 strains\ntogether with Rickettsiales (Fig.~\\ref{fig:unifrac}).\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=6in]{AmrineEtAl_Fig7.pdf\n\\caption{\\figcapsix}\n\\label{fig:multiway}\n\\end{figure*}\n\nHowever, as shown in Fig.~\\ref{fig:multiway}, a multiway classifier\nbased on tRNA CIFs bins all SAR11 strains with the Rhizobiales except\nfor HIMB59, which bins with the Rhodospirillales, consistent with the\nresults of~\\cite{RodriguezEzpeleta:2012fw}. These results use a\nmultilayer perceptron (MLP) classifier implemented in\nWEKA~\\cite{Hall:2009ud} and only seven taxon-specific CIF-based\nsummary scores. The MLP is the simplest non-linear classifier able to\nhandle the interdependent signals in the CIF-based scores for\ntree-like data~\\cite{duda2012pattern}. In a Leave-One-Out\ncross-classification, all other genomes scored consistently with NCBI\nTaxonomy except three placed in Rhodobacteraceae based on 16S\nribosomal RNA evidence: {\\it Stappia aggregata}, {\\it Labrenzia\n  alexandrii} and the denitrifying {\\it Pseudovibrio sp.} JE062. None\nof these genomes scored strongly against Rhodobacteraceae except {\\it\n  Pseudovibrio}, which scored four times greater against the\nRhizobiales. \n\nTo assess robustness of our results we performed two controls: we\nbootstrapped sites of tRNA data in each genome to be classified, and\nwe filtered away small CIFs with Gorodkin heights $< 0.5$ bits from\nour models, retrained the classifier and bootstrapped sites\nagain. Generally bootstrap support values correspond to original\nclassification probabilities. All SAR11 strains have support values\n$>80\\%$ as Rhizobiales, majority bootstrap values as Rhizobiales\n(HIMB114 at $70\\%$ with Rickettsiales at $15\\%$ and HTCC7211 at $54\\%$\nwith Rickettsiales at $13\\%$), or plurality bootstrap value as\nRickettsiales (HIM5 at $48\\%$ with Rickettsiales at $18\\%$) except\nHIM59 which had a bootstrap support value of $87\\%$ to be in the\nRhodospirillales. Full bootstrap statistics with these model are\nprovided in Table S1.\n\n\\section*{Discussion}\n\nOur results provide strong, albeit unconventional, evidence that most\nSAR11 strains are affiliated with Rhizobiales, while strain HIMB59 is\naffiliated with Rhodospirillales.  These results are entirely\nconsistent with comprehensive phylogenomic studies that control for\nnonstationary macromolecular compositions in\nAlphaproteobacteria~\\cite{Brindefalk:2011ei,RodriguezEzpeleta:2012fw,Viklund:2012jr}\nor a site-rate-filtered analysis~\\cite{Gupta:2007}. Our CIF-based\nmethod works even though SAR11 and Rickettsiales tRNAs have converged\nin base content, so that total tRNA sequence-based phylogenomics gives\nopposite results.  tRNA CIFs must be at least partly robust to\ncompositional convergence of the tRNA bodies in which they are\nembedded.\n\nIt is well known that aminoacyl-tRNA synthetases (aaRS) are highly\nprone to\nHGT~\\cite{Doolittle:1998bn,Brown:1999ur,Wolf:1999vo,Woese:2000uy,Andam:2011gh}\nincluding in\nAlphaproteobacteria~\\cite{Ardell:2006ko,Dohm:2006ts,Brindefalk:2006jda}. We\nhypothesize that our tRNA-CIF-based phyloclassifiers are also robust\nto HGT of components of the tRNA-protein interaction network,\nconsistent with \\cite{Shiba:1997te}, who argued that a horizontally\ntransferred aaRS is more likely to functionally ameliorate to a\ntRNA-protein network into which it has been transferred rather than\nremodel that network to accomodate itself. HGT of aaRSs may also\nperturb a network so as to cause a distinct pattern of divergence\n(\\cite{Ardell:2006ko} and this work). Wang {\\it et\n  al.}~\\cite{Wang:2007p1040} discuss the possibility that RRCH\n\\tRNAaa{His} and HisRS were co-transferred into an ancestral SAR11\ngenome. However, this fails to explain the correlations of many other\ntRNA traits of SAR11 genomes with the RRCH clade reported\nhere. Further study is needed to address the robustness of our method\nto component HGT.\n\nA more distant relationship between most SAR11 strains and\nRickettsiales actually strengthens the genome streamlining\nhypothesis~\\cite{Giovannoni:2005}. If SAR11 were a true branch within\nRickettsiales, it becomes more difficult to claim that genome\nreduction in SAR11 occurred by a selection-driven evolutionary process\ndistinct from the drift-dominated erosion of genomes in the\nRickettsiales~\\cite{Andersson:1998p7319,Moran:2002p5737,Itoh:2002dr}. By\nthe same token, polyphyly of nominal SAR11 strains implies that the\nextensive similarity in genome structure and other traits between\nHIMB59 and SAR11 reported by~\\cite{Grote:2012ge} may have originated\nindependently. Perhaps convergence in some traits is consistent with\nstreamlining, which could also explain trait-sharing between SAR11 and\n{\\it Prochlorococcus}, marine cyanobacteria also argued to have\nundergone streamlining~\\cite{Dufresne:2005gn}. Clear signs of\ndata-limitation in our study should be taken to mean that better\ntaxonomic sampling will improve our results and could ultimately\nresolve more than two origins of SAR11-type genomes among\nAlphaproteobacteria.\n\nWe extracted accurate and robust phylogenetic signals from tRNA gene\nsequences by first integrating within genomes to identify features\nlikely to govern functional interactions with other\nmacromolecules. Unlike small molecule interactions, macromolecular\ninteractions are mediated by genetically determined structural and\ndynamic complementarities. These are intrinsically relative; a large\n{\\it neutral network}~\\cite{Schuster:1994gf} of\ninteraction-determining features should be compatible with the same\ninteraction network. Coevolutionary divergence --- turnover---of\nfeatures that mediate macromolecular interactions, while conserving\nnetwork architecture, has been described in the transcriptional\nnetworks of yeast~\\cite{Kuo:2010fk,Baker:2011fz} and\nworms~\\cite{Barriere:2012dp} and in post-translational modifications\nunderlying protein-protein interactions~\\cite{Krogan:2012cs}.  This\nwork demonstrates that divergence of interaction-governing features is\nphylogenetically informative.\n\nIt remains open how such features diverge, with possibilities\nincluding compensatory nearly neutral mutations~\\cite{Hartl:1996dy},\nfluctuating selection~\\cite{He:2011jg}, adaptive\nreversals~\\cite{Bullaughey:2012dg}, and functionalization of\npre-existent variation~\\cite{Haag:2005ty}. Major changes to\ninteraction interfaces may be sufficient to induce genetic isolation\nbetween related lineages, as discussed for the 16S rRNA- and 23S\nrRNA-based standard model of the ``Tree of Life,'' in which many\nimportant and deep branches associate with large, rare macromolecular\nchanges (``signatures'') in ribosome structure and\nfunction~\\cite{Winker:1991jd,Roberts:2008di,Chen:2010et}.\n\nInteraction-mediating features of macromolecules may be systems\nbiology's answer to the phylogeny problem. Perhaps no other traits of\ngenomes are vertically inherited more consistently than those that\nmediate functional interactions with other macromolecules in the same\nlineage. In fact, the structural and dynamic basis of interaction\namong macromolecular components --- essential to their collaborative\nfunction in a system --- may define a lineage better than any of those\ncomponents can themselves, either alone or in ensemble.\n\n\\section*{Materials and Methods}\n\nSupplementary data packages are provided to reproduce all figures from\nraw data and enable third-party classification of alphaproteobacterial\ngenomes.\n\n\\subsection*{Data}\nThe 2011 release of the tRNAdb-CE\ndatabase~\\cite{Abe:2011js} was downloaded on August 24,\n2011.  From this master database, we selected Alphaproteobacteria data\nas specified by NCBI Taxonomy data (downloaded September 24, 2010,\n\\cite{Sayers:2010p4006}). Also using NCBI Taxonomy, we further\ntripartitioned Alphaproteobacterial tRNAdb-CE data into those from the\nRRCH clade, the RSR grade (excluding SAR11), and three SAR11\ngenomes, as documented in Supplementary data for figure 2. Five\nadditional SAR11 genomes (for strains HIMB59, HIMB5, HIMB114, IMCC9063\nand HTCC9565) were obtained from J. Cameron Thrash courtesy of the lab\nof S. Giovannoni. We custom annotated tRNA genes in these genomes as\nthe union of predictions from tRNAscan-SE version 1.3.1 (with {\\tt -B}\noption, \\cite{LoweEddy97}) and Aragorn version\n1.2.34~\\cite{Laslett:2004ih}. We classified initiator tRNAs and\n\\tRNA{Ile}{CAU} using TFAM version 1.4~\\cite{Taquist:2007jl} using a\nmodel previously created to do this based on identifications\nin~\\cite{Silva:2006jc} provided as supplementary data.  We aligned\ntRNAs with covea version 2.4.4~\\cite{Eddy:1994ul} and the prokaryotic\ntRNA covariance model~\\cite{LoweEddy97}, removed sites with more than\n97\\% gaps with a bioperl-based utility~\\cite{Stajich:2002}, and edited\nthe alignment manually in Seaview 4.1~\\cite{gouy2010seaview} to remove\nCCA tails and remove sequences with unusual secondary structures.  We\nmapped sites to Sprinzl coordinates manually~\\cite{Sprinzl:1998vz} and\nverified by spot-checks against tRNAdb~\\cite{juhling2009trnadb}.  We\nadded a gap in the -1 position for all sequences and -1G for\n\\tRNAaa{His} in the RSR group~\\cite{Wang:2007p1040}.\n\n\\subsection*{tRNA CIF Estimation and Binary Classifiers}\nOur tRNA-CIF-based binary phyloclassifier with Leave-One-Out\nCross-Validation (LOO CV) is computed directly from function logos,\nestimated from tDNA alignments as described in~\\cite{Freyhult:2006dr}.\nHere, we define a {\\it feature} $f \\in F$ as a nucleotide $n \\in N$ at\na position $l \\in L$ in a structurally aligned tDNA, where $N =\n\\{A,C,G,T\\}$ and $L$ is the set of all Sprinzl\ncoordinates~\\cite{Sprinzl:1998vz}. The set $F$ of all possible\nfeatures is the Cartesian product $F = N \\times L$.  A {\\it functional\n  class} or {\\it class} of a tDNA is denoted $c \\in \\mathcal{C}$ where\n$\\mathcal{C} = \\{A,C,D,E,F,G,H,I,J,K,L,M,N,P,Q,R,S,T,V,W,X,Y\\}$ is the\nuniverse of functions we here consider, symbolized by IUPAC one-letter\namino acid codes (for aminoacylation classes), $X$ for initiator\ntRNAs, and $J$ for \\tDNA{Ile}{LAU}. A {\\it taxon set of genomes} or\njust {\\it taxon set} $S \\in \\mathcal{P}(G)$ is a set of genomes, where\n$G$ is the set of all genomes, and $\\mathcal{P}(G)$ is the power set\nof $G$. In this work a genome $G$ is represented by the multiset of\ntDNA sequences it contains, denoted $T_G$. The functional information of\nfeatures is computed with a map $h:(F \\times C \\times \\mathcal{P}(G))\n\\longrightarrow \\mathbb{R}_{\\geq 0}$ from the Cartesian product of\nfeatures, classes and taxon sets to non-negative real numbers. For a\nfeature $f \\in F$, class $c \\in \\mathcal{C}$ and taxon set $S \\in\n\\mathcal{P}(G)$, $h(f,c,S)$ is the fraction of functional information\nor ``Gorodkin height''~\\cite{Gorodkin:1997tv}, measured in bits,\nassociated to that feature, class and taxon set. In this work, for a\ngiven taxon set $S$, a function logo $H(S)$ is the tuple:\n\n\\begin{equation}\nH(S) = \\{(\\alpha,\\beta) \\mid  \\beta = h(\\alpha, S), \\forall \\alpha \\in (F \\times \\mathcal{C})\\}.\n\\end{equation}\n\nFurthermore the set $I(S) \\subset (F \\times \\mathcal{C})$ of {\\it tRNA\n  Class-Informative Features} for taxon set $S$ is defined:\n\n\\begin{equation}\nI(S) = \\{\\alpha \\in (F \\times \\mathcal{C}) \\mid h(\\alpha, S) > 0\\}.\n\\end{equation}\n\nBriefly, a tRNA Class-Informative Feature is a tRNA structural feature\nthat is informative about the functional classes it associates with,\ngiven the context of tRNA structural features that actually co-occur\namong a taxon set of related cells, and corrected for biased sampling\nof classes and finite sampling of\nsequences~\\cite{Freyhult:2006dr}. Let $A$ denote a set of\nAlphaproteobacterial genomes partitioned into three disjoint subsets\n$X$, $Y$ and $Z$ with $X \\cup Y \\cup Z = A$, representing genomes from\nthe RRCH clade, the RSR grade, and the eight nominal {\\it Ca.}\nPelagibacter strains respectively. To execute Leave-One-Out\nCross-Validation of a tRNA CIF-based binary phyloclassifier for a\ngenome $G \\in A$, we compute a score $S_C(G, S_1, S_2)$, averaging\ncontributions from the multiset $T_G$ of tDNAs in $G$ scored against\ntwo function logos $H(S_1)$ and $H(S_2)$ computed respectively from\ntwo disjoint taxon sets $S_1 \\subset A$ and $S_2 \\subset A$, with $G\n\\notin S_1 \\cup S_2$. In this study, those sets are $X \\setminus G$\nand $Y \\setminus G$, denoted $X_G$ and $Y_G$ respectively. Each tDNA\n$t \\in T_G$ presents a set of features $F_t \\subset F$ and has a\nfunctional class $c_t \\in \\mathcal{C}$ associated to it. The score\n$S_C(G,X_G,Y_G)$ is then defined:\n\n\\begin{equation}\nS_C(G,X_G, Y_G) \\equiv \\frac{1}{| T_G |}\\sum_{t \\in T_G}\\sum_{f \\in\n  F_t}h(f,c_t,X_G) - h(f,c_t,Y_G).\n\\label{eqn:cif}\n\\end{equation}\n\nAs controls, we implemented four total-tDNA-sequence based binary\nphyloclassifiers to score a genome $G$. All are slight variations in\nwhich a tRNA $t \\in T_G$ of class $c(t)$ contributes a score that is a\ndifference in log relative frequencies of the features it shares in\nclass-specific profile models generated from $X_G$ and $Y_G$. The\ndefault ``zero'' scoring scheme method $S^Z_T(G,X_G, Y_G)$ shown in\nFig.~\\ref{fig:binclass}B is defined as:\n\n\\begin{equation}\nS^Z_T(G,X_G, Y_G) \\equiv \\frac{1}{| T_G |} \\sum_{t \\in T_G}\\sum_{f \\in\n  F_t}\\log_2 \\frac{p^*(f | c_t,  X_G)}{p^*(f  | c_t, Y_G)}, \n\\label{weaksol}\n\\end{equation}\n\n\\noindent where \n\n\\begin{eqnarray}\np^*(f | c, S) \\equiv \n\\begin{cases} \n\\#\\{f , c, S\\} \/ \\#\\{c,S\\}  & \\#\\{f , c, S \\} > 0 \\\\ \n1 & \\#\\{f , c, S \\} = 0  \n\\end{cases},\n\\end{eqnarray}\n\n\\noindent $\\#\\{f, c, S \\}$ is the observed frequency of feature $f$ in\ntDNAs of class $c$ in set $S$, and $\\#\\{c, S \\}$ is the frequency of\ntDNAs of class $c$ in set $S$. \n\nMethod ``skip'' corresponds to scoring scheme $S^K_T(G,X_G,\nY_G)$ defined as:\n\n \\begin{equation}\nS^K_T(G,X_G, Y_G) \\equiv \\frac{1}{| T_G |} \\sum_{t \\in T_G}\\sum_{f \\in F_t} s^k(f,c_t,  X_G, Y_G),\n\\end{equation}\n\n\\noindent where \n\n\\begin{eqnarray}\ns^k(f, c,  S, T) \\equiv \n\\begin{cases} \n\\log_2 \\frac{p(f | c, S)}{p(f | c, T)} & \\#\\{f , c, S \\} > 0 \\wedge \\#\\{f , c, T \\} > 0 \\\\\n 0 & \\#\\{f , c, S \\} = 0 \\vee \\#\\{f , c, T \\} = 0  \\end{cases},\n\\end{eqnarray}\n\n\\noindent and $p(f | c, R) \\equiv \\#\\{f , c, R \\}\/ \\#\\{c,R\\}$ for $R\n\\in \\{S,T\\}$ as before.\n\n\n\nMethods ``pseudo'' and ``small'' correspond to scoring schemes $S^I_T(G,X_G,\nY_G)$:\n\n \\begin{equation}\nS^I_T(G,X_G, Y_G) \\equiv \n \\frac{1}{| T_G |} \\sum_{t \\in T_G}\\sum_{f \\in F_t} \\log_2\n \\frac{p^I(f | c_t, X_G)}{p^I(f  | c_t, Y_G)},\n\\end{equation}\n\n\\noindent where \n\n\\begin{eqnarray}\np^I(f | c, S) \\equiv \n\\begin{cases} o \/ t  &  \\forall n \\in N: \\#\\{(n,l) , c, S \\} > 0  \\\\  \n\\frac{o + I}{t + 4I} & \\exists n \\in N: \\#\\{(n,l) , c, S \\} = 0\n\\end{cases},\n\\end{eqnarray}\n\n\\noindent where $f = (n,l)$, $o \\equiv \\#\\{f, c, S \\}$, $t \\equiv \\#\\{c,S\\}$,\n$I = 1$ for method ``pseudo,'' and, for method ``small,'' $I = 1 \/\nT_A$, where $T_A = \\sum_{G \\in A} T_G$.\n\n\\subsection*{Analysis of tRNA Base Composition}\nWe computed the base composition of tRNAs aggregated by clades using\nbioperl-based~\\cite{Stajich:2002} scripts, and transformed them by the\ncentered log ratio transformation~\\cite{aitchison1986statistical} with\na custom script provided as supplementary data. We then computed\nEuclidean distances on the transformed composition data, and then\nperformed hierarchical clustering by UPGMA on those distances as implemented in\nthe program NEIGHBOR from Phylip 3.6b~\\cite{PHYLIP} and visualized in\nFigTree v.1.4.\n\n\\subsection*{Supermatrix and FastUniFrac Analysis}\nFor supermatrix approaches, we created concatenated tRNA alignments\nfrom 169 Alphaproteobacteria genomes (117 RRCH, 44 RSR, 8 PEL) that\nall shared the same 28 isoacceptors with 77 sites per gene (2156 total\nsites).  In cases where a species contained more than a single\nisoacceptor, one was chosen at random.  Using a GTR+Gamma model, we\nran RAxML by means of The iPlant\nCollaborative project RAxML server\n(\\url{http:\/\/www.iplantcollaborative.org}, \\cite{Stamatakis01102008})\non January 23, 2013 with their installment of RAxML version\n7.2.8-Alpha (executable raxmlHPC-SSE3, a sequential version of RAxML\noptimized for parallelization).  We tested the robustness of our\nresult to random picking of isoacceptors by creating 100 replicate\nconcatenated alignments and running them through\nFastTree~\\cite{Price:2010vy}. For the FastUniFrac\nanalysis we used the FastUniFrac~\\cite{Hamady:2010dr}\nweb-server at \\url{http:\/\/bmf2.colorado.edu\/fastunifrac\/} to\naccomodate our large dataset. We removed two genomes from our dataset\nfor containing fewer than 20 tRNAs, and following~\\cite{Widmann:2010ea}\nremoved anticodon sites.  Following~\\cite{Widmann:2010ea}\ndeliberately, we computed an approximate ML tree based on Jukes-Cantor\ndistances using FastTree~\\cite{Price:2010vy}. We then\nqueried the FastUniFrac webserver with this tree, defining\nenvironments as genomes. We then computed a UPGMA tree based on the\nserver's output FastUniFrac distance matrix in NEIGHBOR from Phylip\n3.6b~\\cite{PHYLIP}.\n\n\\subsection*{Multiway Classifier}\nAll tDNA data from the RSR and RRCH clades were partitioned into one\nof seven monophyletic clades: orders Rickettsiales (N = 40 genomes),\nRhodospirillales (N = 10), Sphingomonadales (N = 9), Rhizobiales (N =\n91), and Caulobacterales (N = 6), or families Rhodobacteraceae (N =\n43) or Hyphomonadaceae (N = 4) as specified by NCBI taxonomy\n(downloaded September 24, 2010,~\\cite{Sayers:2010p4006}) and\ndocumented in supplementary data for figure 7. We withheld data from\nthe eight nominal SAR11 strains, as well as from three genera {\\it\n  Stappia}, {\\it Pseudovibrio}, and {\\it Labrenzia}, based on\npreliminary analysis of tDNA and CIF sequence variation. Following a\nrelated strategy as with the binary classifier, we computed, for each\ngenome, seven tRNA-CIF-based scores, one for each of the seven\nAlphaproteobacterial clades as represented by their function logos,\nusing the principle of Leave-One-Out Cross-Validation (LOO CV), that\nis, excluding data from the genome to be scored. Function logos were\ncomputed for each clade as described in~\\cite{Freyhult:2006dr}.  For\neach taxon set $X_G$ (with genome $G$ left out if it occurs), genome\n$G$ obtains a score $S^M(G,X_G)$ defined by:\n\n\\begin{equation}\nS_M(G,X_G) \\equiv \\frac{1}{| T_G |}\\sum_{t \\in T_G}  \\sum_{f \\in\n  F_t}h(f,c_t,X_G).\n\\label{eqn:multi}\n\\end{equation}\n\nEach genome $G$ is then represented by a vector of seven scores, one\nfor each taxon set modeled. These labeled vectors were then used to\ntrain a multilayer perceptron classifier in WEKA 3.7.7~(downloaded\nJanuary 24, 2012,~\\cite{Hall:2009ud}) by their defaults through the\ncommand-line interface, which include a ten-fold cross-validation\nprocedure. We bootstrap resampled sites in genomic tRNA alignment data\n(100 replicates) and also bootstrap resampled a reduced (and\nretrained) model including only CIFs with a Gorodkin height~\\cite{Freyhult:2006dr} $\\ge\n0.5$ bits.\n\n\n\\section*{Acknowledgments}\n  We thank J. Cameron Thrash and Stephen Giovannoni for sharing data\n  in advance of publication, Harish Bhat, Torgeir Hvidsten, Carolin\n  Frank and Suzanne Sindi for helpful suggestions.\n\n\n\\bibliographystyle{plos2009}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction and motivation}\n\n\n\n\n\n\nIn this paper, we aim to classify Grothendieck topologies on a poset. This is motivated by the research project of describing quantum theory in terms of topos theory (see e.g., \\cite{BI}, \\cite{DI}, \\cite{HLS}). Here the central object of research are the topoi $\\Sets^{\\CCC(\\A)}$ and $\\Sets^{\\CCC(\\A)^\\op}$, where $\\CCC(\\A)$ is the poset of the commutative C*-subalgebras of some C*-algebra $\\A$ ordered by inclusion. $\\A$ describes some quantum system, whereas the elements of $\\CCC(\\A)$ are interpreted as ``classical snapshots of reality''. The motivation behind this approach is Niels Bohr's doctrine of classical concepts, which, roughly speaking, states that a measurement provides a classical snapshot of quantum reality, and knowledge of all classical snapshots should provide a picture of quantum reality that is as complete as (humanly) possible. In terms of C*-algebras, this means we should be able to reconstruct the algebra $\\A$ in some way from the posetal structure of $\\CCC(\\A)$. Partial results in this direction can be found in \\cite{DH} and \\cite{Hamhalter}, where, at least for certain classes of C*-algebras and von Neumann algebras, it is proven that the Jordan structure of $\\A$ may be recovered from the poset $\\CCC(\\A)$ of commutative subalgebras of $\\A$. Since there is a C*-algebra $\\A$ that is not isomorphic to its opposite C*-algebra $\\A^0$, although $\\CCC(\\A)\\cong\\CCC(\\A^0)$ as posets (see \\cite{Connes}), it is known that the C*-algebraic structure of a C*-algebra cannot always be completely recovered from the posetal structure of $\\CCC(\\A)$. Therefore, extra information is needed, which might be found in sheaf theory, for instance in the form of a Grothendieck topology on $\\CCC(\\A)$.\n\nThe simplest C*-algebras are the finite-dimensional algebras $\\B(\\C^n)$ of $n\\times n$-matrices, where $n$ is a finite positive integer. However, even the posets $\\CCC_n=\\CCC(\\B(\\C^n))$ are already far from trivial (see for instance \\cite[Example 5.3.5]{Heunen}), hence interesting. Familiar with the posetal structure of $\\CCC_n$, one easily sees that these posets are Artinian, i.e., posets for which every non-empty downwards directed subset contains a least element. Moreover, we show in Appendix A that $\\CC(\\A)$ is Artinian if and only if $\\A$ is finite dimensional. The Artinian property is extremely powerful, since it allows one to use a generalization of the principle of induction called \\emph{Artinian induction}\\index{Artinian induction}.\n\nTwo main sources of this paper are \\cite{J&T} and \\cite{EGP}. In the first, several order-theoretic notions are related to the notion of Grothendieck topologies, whilst in the second, the relation is between these order-theoretic notions and the Artinian property is investigated. This article aims to combine both articles. We define a family of Grothendieck topologies on a poset $\\P$ generated by a subset of $\\P$ and show that these Grothendieck topologies in fact exhaust the possible Grothendieck topologies on the given poset if and only $\\P$ is Artinian, where we use Artinian induction for the 'if' direction.\n\nA poset equipped with a Grothendieck topology is called a \\emph{site}. If the extra structure on a $\\CCC(\\A)$ is indeed given by a Grothendieck topology, we also need a notion of 'morphisms of sites'. Finally, we calculate the sheaves corresponding to all classes of Grothendieck topologies we have found. For this calculation, we make use of a posetal version of a sheaf-theoretic result known as the Comparison Lemma, going back to Grothendieck et al \\cite{SGA4} (see also \\cite{Elephant2}), which relates sheaves on a given category to sheaves on some subcategory.\n\n\nIn the appendix, we show that the notion of a Grothendieck topology on $\\P$ is equivalent with the notion of a frame quotient of the frame $\\D(\\P)$ of down-sets of $\\P$, which in turn is equivalent with the notion of a nuclei on $\\D(\\P)$, a congruence on $\\D(\\P)$, a sublocale of $\\D(\\P)$ if we consider $\\D(\\P)$ as a locale. Since all these notions are equivalent, this gives a description of nuclei, congruences, and sublocales of $\\D(\\P)$ that correspond to Grothendieck topologies corresponding to subsets of $\\P$.\n\n\n\n\\section{Preliminaries on order theory}\\label{Order Theory}\\label{Noetherian induction}\nThis section can be skipped if one is familiar with the basic concepts of order theory. We refer to \\cite{DP} for a detailed exposition of order theory.\n\n\\begin{definition}\n A \\emph{poset} $(\\P,\\leq)$ is a set $\\P$ equipped with a \\emph{(partial) order}\\index{order!partial} $\\leq$. That is, $\\leq$, is binary relation, which is reflexive, antisymmetric and transitive. We often write $\\P$ instead of $(\\P,\\leq)$ if it is clear which order is used. If either $p\\leq q$ or $q\\leq p$ for each $p,q\\in\\P$, we say that $\\leq$ is a \\emph{linear order}\\index{order!linear}, and we call $\\P$ a \\emph{linearly ordered set}\\index{linearly ordered set}. Given a poset $\\P$ with order $\\leq$, we define the opposite poset $\\P^\\op$ as the poset with the same underlying set $\\P$, but where $p\\leq q$ if and only if $q\\leq p$ in the original order. .\n\\end{definition}\n\n\n\\begin{definition}\n Let $(\\P,\\leq)$ be a poset and $M\\subseteq \\P$ a subset. We say that $M$ is an \\emph{up-set}\\index{up-set} if for each $x\\in M$ and $y\\in \\P$ we have $x\\leq y$ implies $y\\in M$. Similarly, $M$ is called a \\emph{down-set}\\index{down-set} if for each $x\\in M$ and $y\\in \\P$ we have $x\\geq y$ implies $y\\in M$. Given an element $x\\in \\P$, we define the up-set and down-set generated by $x$ by $\\up x=\\{y\\in \\P:y\\geq x\\}$ and $\\down x=\\{y\\in \\P:y\\leq x\\}$, respectively. We can also define the up-set generated by a subset $M$ of $\\P$ by $\\up M=\\{x\\in \\P:x\\geq m$ for some $m\\in M\\}=\\bigcup_{m\\in M}\\up m$, and similarly, we define the down-set generated by $M$ by $\\down M=\\bigcup_{m\\in M}\\down m$. We denote the collection of all up-sets of a poset $\\P$ by $\\U(\\P)$ and the set of all down-sets by $\\D(\\P)$. If we want to emphasize that we use the order $\\leq$, we write $\\U(\\P,\\leq)$ instead of $\\U(\\P)$.\n\\end{definition}\n\n\\begin{lemma}\\label{cosievegen}\nLet $\\P$ be a poset and $M\\subseteq \\P$. Then $\\down M$ is the smallest down-set containing $M$, and $M=\\down M$ if and only if $M$ is an down-set.\n\\end{lemma}\n\n\n\\begin{definition}\n Let $M$ be a non-empty subset of a poset $\\P$.\n\\begin{enumerate}\n \\item An element $x\\in M$ such that $\\up x\\cap M=\\{x\\}$ is called a \\emph{maximal element}\\index{maximal element}. The set of maximal elements of $M$ is denoted by $\\max M$. The set of all \\emph{minimal elements}\\index{minimal element} $\\min M$ is defined dually.\n \\item If there is an element $x\\in M$ such that $x\\geq y$ for each $y\\in M$, we call $x$ the \\emph{greatest element}\\index{greatest element} of $M$, which is necesarrily unique. The \\emph{least element}\\index{least element} of $M$ is defined dually. If $\\P$ itself has a greatest element, it is denoted by $1$. Dually, the least element of $\\P$ is denoted by $0$.\n \\item An element $x\\in \\P$ is called an \\emph{upper bound}\\index{upper bound} of $M$ if $y\\leq x$ for each $y\\in M$. If the set of upper bounds of $M$ has a least element, it is called the \\emph{join}\\index{join} of $M$, which is necesarrily unique and is denoted by $\\bigvee M$. Dually, we can define \\emph{lower bounds}\\index{lower bound} and the \\emph{meet}\\index{meet} of $M$, which is denoted by $\\bigwedge M$. The (binary) join and the (binary) meet of $\\{x,y\\}$ are denoted by $x\\vee y$ and $x\\wedge y$, respectively.\n \\item $M$ is called upwards (downwards) \\emph{directed}\\index{directed subset of a poset} if for every $x,y\\in M$ there is an upper (lower) bound $z\\in M$.\n\\item $M$ is called a \\emph{filter}\\index{filter of a poset} if $M\\in\\U(\\P)$ and $M$ is downwards directed. We denote the set of filters of $\\P$ by $\\F(\\P)$. A filter is called \\emph{principal}\\index{principal filter} if it is equal to $\\up p$ for some $p\\in \\P$.\n\\item $M$ is called an \\emph{ideal}\\index{ideal of a poset} if $M\\in\\D(\\P)$ and $M$ is upwards directed. We denote the set of ideals of $\\P$ by $\\Idl(\\P)$. An ideal is called \\emph{principal}\\index{principal ideal} if it is equal to $\\down p$ for some $p\\in \\P$.\n\\end{enumerate}\n\\end{definition}\nIf we consider $\\P$ as a category, meets are exactly products, joins are coproducts, and the greatest (least) element of $\\P$ is exactly the terminal (initial) object. These concepts are unique in posets, whereas in arbitrary categories they are only unique up to isomorphism. This follows from the fact that there is at most one morphism between two elements in a poset, hence isomorphic elements in a poset are automatically equal.\n\n\n\\begin{definition}\nLet $\\P$ be a poset. Then $\\P$ is called\n\\begin{enumerate}\n \\item \\emph{Artinian}\\index{Artinian poset} (\\emph{Noetherian})\\index{Noetherian poset} if every non-empty subset contains a minimal (maximal) element;\n \\item a \\emph{meet-semilattice}\\index{meet-semilattice} (\\emph{join-semilattice})\\index{join-semilattice} if all binary meets (joins) exist;\n \\item a \\emph{lattice}\\index{lattice} if all binary meets and binary joins exist;\n \\item a \\emph{complete lattice}\\index{complete lattice} if all meets and joins exist.\n\\item a \\emph{distributive lattice}\\index{distributive lattice} if it is a lattice such that the \\emph{distributive law}\\index{distributive law}\n \\begin{equation}\n  x\\wedge(y\\vee z)=(x\\wedge y)\\vee(x\\wedge z)\n \\end{equation}\n holds for each $x,y,z\\in \\P$\n \\item a \\emph{frame} if all joins and all finite meets exists and the \\emph{infinite distributive law}\\index{infinite distributive law}\n \\begin{equation}\\label{eq:infdist}\n  x\\wedge\\bigvee_i y_i=\\bigvee_i(x\\wedge y_i)\n \\end{equation}\n holds for each element $x$ and each family $y_i$ in $\\P$.\n\\end{enumerate}\n\\end{definition}\n\nRemark that in a meet-semilattice (and therefore also in a lattice and in a frame), we have $x\\wedge y=x$ if and only if $x\\leq y$.\n\n\n\\begin{lemma}\\label{lem:completelattice}\n Let $\\P$ be a poset. If $\\P$ has all meets, then $\\P$ becomes a complete lattice with join operation defined by\n\\begin{equation*}\n \\bigvee S=\\bigwedge\\{p\\in\\P:s\\leq p\\ \\forall s\\in S\\}\n\\end{equation*}\nfor each $S\\subseteq\\P$. Dually, if $\\P$ has all joins, then $\\P$ becomes a complete lattice with meet operation defined by\n\\begin{equation*}\n\\bigwedge S=\\bigvee\\{p\\in\\P:p\\leq s\\ \\forall s\\in S\\}\n\\end{equation*}\nfor each $S\\subseteq\\P$. In particular a frame is a complete lattice.\n\\end{lemma}\nAs a consequence, if $\\P$ is a complete lattice, $\\P^\\op$ is a complete lattice as well.\n\n\n\n\nFinally we define morphisms between posets as follows.\n\\begin{definition}\n Let $\\P,\\mathbf{Q}$ be posets and $f:\\P\\to \\mathbf{Q}$ a map. Then $f$ is called\n\\begin{enumerate}\n \\item an \\emph{order morphism} if $x\\leq y$ implies $f(x)\\leq f(y)$ for each $x,y\\in \\P$;\n \\item a \\emph{embedding of posets} if $f(x)\\leq f(y)$ if and only if $x\\leq y$ for each $x,y\\in\\P$;\n \\item a \\emph{meet-semilattice morphism} if $\\P$ and $\\mathbf{Q}$ are both meet-semilattices and\\\\ $f(x\\wedge y)=f(x)\\wedge f(y)$ for each $x,y\\in \\P$;\n \\item a \\emph{lattice morphism} if $\\P$ and $\\mathbf{Q}$ are both lattices and $f$ is a meet-semilattice morphism such that $f(x\\vee y)=f(x)\\vee f(y)$ for each $x,y\\in \\P$;\n \\item a \\emph{frame morphism} if $\\P$ and $\\mathbf{Q}$ are both frames and $f\\left(\\bigvee M\\right)=\\bigvee f(M)$,\\\\ $f\\left(\\bigwedge N\\right)=\\bigwedge f(N)$ for each $M\\subseteq \\P$ and each finite $N\\subseteq \\P$.\n\\end{enumerate}\nAn order morphism, meet-semilattice morphism, lattice morphism, frame morphism is called an \\emph{order isomorphism}, \\emph{meet-semilattice isomorphism}, \\emph{lattice isormphism}, \\emph{frame isomorphism}\\index{isomorphism}, respectively, if there is a morphism $g:\\mathbf{Q}\\to\\P$ of the same type such that $f\\circ g=1_{\\mathbf{Q}}$ and $g\\circ f=1_\\P$.\n\\end{definition}\nClearly an embedding of posets $f$ is injective. If $f(x)=f(y)$, then $f(x)\\leq f(y)$, so $x\\leq y$, and in a similar way, we find $y\\leq x$, so $x=y$.\nThe converse does not always hold. Consider for instance the poset $\\P=\\{p_1,p_2,p_3\\}$ with $p_1,p_2<p_3$ and $\\mathbf{Q}=\\{q_1,q_2,q_3\\}$ with $q_1<q_2<q_3$. Then $f:\\P\\to\\mathbf{Q}$ defined by $f(p_i)=q_i$ for $q=1,2,3$ is clearly an injective order morphism (it is even bijective), but $p_1\\nleq p_2$, whereas $f(p_1)\\leq  f(p_2)$.\n\n\\begin{lemma}\n Let $f:\\P\\to \\mathbf{Q}$ be an order isomorphism. Then $f$ respects meets and joins. Moreover, if $\\P$ and $\\mathbf{Q}$ are both meet-semilattices, lattices or frames, then $f$ is a meet-semilattice morphism, lattice morphism, or frame morphism, respectively, if and only if $f$ is an order isomorphism.\n\\end{lemma}\n\n\n\n\n\n\n\\begin{definition}\n Let $\\P$, $\\mathbf Q$ be posets and $f :\\P \\to \\QQ$, $g : \\QQ \\to \\P$ order morphism. If $f(p) \\leq q$ if and only if $p \\leq g(q)$ holds for each $p \\in\\P$ and $q \\in \\QQ$, then $g$ is called the \\emph{upper adjoint}\\index{adjoint} of $f$ and $f$ is called the \\emph{lower adjoint} of $g$. If the upper adjoint of $f$ exists, it is sometimes denoted by $f_*$. Similarly, the lower adjoint of $g$, if it exists, is sometimes denoted by $g^*$.\n\\end{definition}\nIf we consider posets as categories, an upper adjoint is exactly a right adjoint.\n\nThe following lemma can be found in \\cite[Chapter 7]{DP}. Categorical generalizations of these facts are also well-known, but we will not need them here.\n\\begin{lemma}\\label{lem:adjointequivalent}\nLet $\\P$, $\\mathbf Q$ be posets and $f :\\P \\to \\QQ$, $g : \\QQ \\to \\P$ order morphisms. The following two statements are equivalent:\n\\begin{enumerate}\n\\item[(a)] $g$ is the upper adjoint of $f$.\n\\item[(b)] Both $1_{\\P} \\leq g \\circ f$ and $f \\circ g \\leq 1_{\\QQ}$.\n\\end{enumerate}\nIf these statements hold, then we also have the following:\n\\begin{enumerate}\n\\item[(i)] Both $f \\circ g \\circ f = f$ and $g \\circ f \\circ g = g$.\n\\item[(ii)] $f$ is surjective if and only if $g$ is injective if and only if $f \\circ g = 1_{\\QQ}$.\n\\item[(iii)] $f$ is injective if and only if $g$ is surjective if and only if $g\\circ f=1_{\\P}$.\n\\item[(iv)] If $g' : \\QQ \\to \\P$ is order-preserving and (a) or (b) holds when $g$ is replaced by $g'$, then $g' = g$.\n\\item[(v)] If $f' : \\P \\to \\QQ$ is order-preserving and (a) or (b) holds when $f$ is replaced by $f'$, then $f' = f$.\n\\end{enumerate}\n\\end{lemma}\n\n\n\\begin{lemma}\\label{lem:reflection}\n Let $g:\\QQ\\to\\P$ be an embedding of posets. If $g$ has a lower adjoint $f:\\P\\to\\QQ$, then $f$ is the left-inverse of $g$.\n\\end{lemma}\n\n\n\\begin{lemma}\\label{lem:adjoint}\n Let $f:\\P\\to\\QQ$ be an order morphism with an upper adjoint $g:\\QQ\\to\\P$. Then $f$ preserves all existing joins, whereas $g$ preserves all existing meets. If $\\P$ and $\\QQ$ are frames, then an order morphism (not necessarily a frame morphism) $f:\\P\\to\\QQ$ has an upper (lower) adjoint $g:\\QQ\\to\\P$ if it preserves all joins (meets).\n\\end{lemma}\n\n\n\n\n\\begin{lemma}\\label{lem:isomorphismisadjoint}\n Let $f:\\P\\to\\mathbf{R}$ and $g:\\mathbf{R}\\to\\QQ$ be order morphism with upper adjoints $f_*$ and $g_*$, respectively. Then $g\\circ f$ has an upper adjoint $(g\\circ f)_*$ equal to $f_*\\circ g_*$. Moreover, if $f$ is an order isomorphism, then its upper adjoint $f_*$ is its inverse $f^{-1}$.\n\\end{lemma}\n\n\n\n\\begin{lemma}\\label{lem:frameadjoint}\n Let $f:F\\to G$ be a frame morphism. Then $f$ has an adjoint $g:G\\to F$ given by $g(y)=\\bigvee\\{x\\in F:f(x)\\leq y\\}$.\n\\end{lemma}\n\n\\begin{example}\\label{ex:defheyimp}\nLet $F$ be a frame. Then for fixed $x\\in F$, we define $f_x:F\\to F$ by $f_x(z)=x\\wedge z$. Then $f_x\\left(\\bigvee_{i}z_i\\right)=x\\wedge\\left(\\bigvee_iz_i\\right)=\\bigvee_i(x\\wedge z_i)=\\bigvee_i f_x(z_i)$ by the infinite distributivity law, so $f$ preserves joins. So $f$ has an upper adjoint given by $g_x(y)=\\bigvee\\{z:x\\wedge z\\leq y\\}$.\n\\end{example}\n\n\\begin{definition}\\label{def:heyimplicationinframes}\n The element $g_x(y)$ as defined in the previous example is denoted by $x\\to y$ and is called the \\emph{relative pseudo-complement}\\index{relative pseudo-complement} of $x$ with respect to $y$. We refer to the map $F\\times F\\to F$ given by $(x,y)\\mapsto(x\\to y)$ as the \\emph{Heyting implication}\\index{Heyting implication} or \\emph{Heyting operator}\\index{Heyting operator}. So we have\n\\begin{equation}\\label{eq:HeyImp}\n x\\wedge z\\leq y \\ \\mathrm{if\\ and\\ only\\ if\\ }z\\leq(x\\to y).\n\\end{equation}\nFor each $x\\in F$ we denote the element $x\\to 0$ by $\\neg x$, which we call the \\emph{negation} of $x$.\n\\end{definition}\n\n\nIn the next lemma we state some useful identities for the Heyting implication and especially the negation.\n\n\\begin{lemma}\\cite[Chapter I.8]{M&M}\\label{lem:negationidentities}\n Let $F$ be a frame. Then for each $x,y\\in F$ we have\n\\begin{enumerate}\n \\item[(i)] $x\\wedge(x\\to y)=x\\wedge y$;\n \\item[(ii)] $x\\wedge\\neg x=0$;\n \\item[(iii)] $y\\leq\\neg x$ if and only if $x\\wedge y=0$;\n \\item[(iv)] $x\\leq y$ implies $\\neg y\\leq\\neg x$;\n \\item[(v)] $x\\leq\\neg\\neg x$;\n \\item[(vi)] $\\neg x=\\neg\\neg\\neg x$;\n \\item[(vii)] $\\neg\\neg(x\\wedge y)  =  (\\neg\\neg x)\\wedge(\\neg\\neg y).$\n\\end{enumerate}\n\\end{lemma}\n\n\n\n\nThere are several categories which have frames as objects, but different classes of morphisms.\n\\begin{definition}\n The category $\\mathbf{Frm}$ of frames is the category with frames as objects and frame morphisms as morphisms. The category $\\mathbf{cHA}$ of \\emph{complete Heyting algebras}\\index{complete Heyting algebra} is defined as the category with frames as objects and as morphisms exactly the frame morphisms $f:H\\to G$ which also satisfy $f(x\\to y)=f(x)\\to f(y)$ for each $x,y\\in H$. Finally, the category $\\mathbf{Loc}$ of \\emph{locales}\\index{locale} is defined as the opposite category of $\\mathbf{Frm}$. That is, the objects of $\\mathbf{Loc}$ are frames, and $f:L\\to K$ is a locale morphism if it is the opposite of a frame morphism $K\\to L$. If we want to stress that we work in the $\\mathbf{Loc}$, we call the objects locales rather than frames.\n\\end{definition}\n\n\n\n\n\\begin{lemma}\\label{lem:equivalentdefinitionsArtinian}\nLet $\\P$ be a poset. Then the following statements are equivalent:\n\\begin{enumerate}\n\\item $\\P$ is Artinian;\n\\item All non-empty downwards directed subsets of $\\P$ have a least element.\n\\item $\\P$ satisfies the \\emph{descending chain condition}\\index{descending chain condition}. That is, if we have a sequence of\nelements $x_1\\geq x_2\\geq \\ldots$ in $\\P$ (a descending chain), then the sequence stabilizes. That is, there is an\n$n\\in\\N$ such that $x_k=x_{n}$ for all $k>n$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nAssume $\\P$ is Artinian and let $M$ a non-empty downwards directed subset of $\\P$. Then $M$ must have a minimal element $x$. Now, if $y\\in M$, then there must be an $z\\in M$ such that $z\\leq x,y$. Since $x$ is minimal, it follows that $z=x$, so $x\\leq y$, whence $x$ is the least element of $M$.\n\nNow, assume that every non-empty downwards directed subset of $\\P$ has a least element. If $x_1\\geq x_2\\geq x_3\\geq \\ldots$ is a descending chain, then $M=\\{x_i\\}_{i\\in\\N}$ is clearly a directed subset, so it has a least element, say $x_n$. So we must have $x_k=x_n$ for all $k>n$, hence $\\P$ satisfies the descending chain condition.\n\nFinally, we show by contraposition that it follows from the descending chain condition that $\\P$ must be Artinian.\nSo assume that $\\P$ does not satisfy the descending chain\ncondition. Using the Axiom of Dependent Choice, we can construct a sequence $x_1\\geq x_2\\geq \\ldots$ that\ndoes not terminate. The set \\mbox{$M=\\{x_n:n\\in\\N\\}$} is then a non-empty\nsubset of $\\P$ without a minimal element. Thus $\\P$ is not\nArtinian.\n\\end{proof}\n\nSince $\\P$ is Noetherian if and only if $\\P^\\op$ is Artinian, we obtain an equivalent characterization of Noetherian posets.\n\\begin{lemma}\\label{lem:equivalentdefinitionsNoetherian}\nLet $\\P$ be a poset. Then the following statements are equivalent:\n\\begin{enumerate}\n\\item $\\P$ is Noetherian;\n\\item All non-empty upwards directed subsets of $\\P$ have a greatest element.\n\\item $\\P$ satisfies the \\emph{ascending chain condition}\\index{ascending chain condition}. That is, if we have a sequence of\nelements $x_1\\leq x_2\\leq \\ldots$ in $\\P$ (an ascending chain), then the sequence stabilizes. That is, there is an\n$n\\in\\N$ such that $x_k=x_{n}$ for all $k>n$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proposition}[Principle of Artinian induction]\\index{Artinian induction}\\label{prop:ArtinianInduction}\nLet $\\P$ be an Artinian poset and $\\PP$ a property such that:\n\\begin{enumerate}\n\\item Induction basis: $\\PP(x)$ is true for each minimal $x\\in \\P$;\n\\item Induction step: $\\PP(y)$ is true for all $y<x$ implies that $\\PP(x)$ is true.\n\\end{enumerate}\nThen $\\PP(x)$ is true for each $x\\in \\P$.\n\\end{proposition}\n\\begin{proof}\nAssume that $M=\\{x\\in \\P:\\PP(x)$ is not true$\\}$ is non-empty. Since\n$\\P$ is Artinian, this means that $M$ has a minimal element $x$.\nHence $\\PP(y)$ is true for all elements $y<x$, so $\\PP(x)$ is true by\nthe induction step, contradicting the definition of $M$.\n\\end{proof}\n\n\n\\section{Grothendieck topologies on posets}\n\\begin{definition}\n Let $\\P$ be a poset. Given an element $p\\in\\P$, a subset $S\\subseteq\\P$ is called a \\emph{sieve} on $p$ if $S\\in\\D(\\down p)$, where $\\down p=\\{q\\in\\P: q\\leq p\\}$. Equivalently, $S$ is a sieve on $p$ if $q\\leq p$ for each $q\\in S$ and $r\\in S$ if $r\\leq q$ for some $q\\in S$. Then a \\emph{Grothendieck topology}\\index{Grothendieck topology} $J$ on $\\P$ is a map $p\\mapsto J(p)$ that assigns to each element $p\\in\\P$ a collection $J(p)$ of sieves on $p$ such that\n\\begin{enumerate}\n \\item the maximal sieve $\\down p$ is an element of $J(p)$;\n \\item if $S\\in J(p)$ and $q\\leq p$, then $S\\cap\\down q\\in J(q)$;\n \\item if $S\\in J(p)$ and $R$ is any sieve on $p$ such that $R\\cap\\down q\\in J(q)$ for each $q\\in S$, then $R\\in J(p)$.\n\\end{enumerate}\nThe second and third axioms are called the \\emph{stability axiom}\\index{stability axiom} and the \\emph{transitivity axiom}\\index{transitivity axiom}, respectively. Given $p\\in\\P$, we refer to elements of $J(p)$ as $J$-\\emph{covers}\\index{$J$-cover} of $p$. A pair $(\\P,J)$ consisting of a poset $\\P$ and a Grothendieck topology $J$ on $\\P$ is called a \\emph{site}\\index{site}. We denote the set of Grothendieck topologies on $\\P$ by $\\G(\\P)$. This set can be ordered pointwisely: for any two Grothendieck topologies $J$ and $K$ on $\\P$ we define $J\\leq K$ if $J(p)\\subseteq K(p)$ for each $p\\in\\P$.\n\\end{definition}\n\nNotice that $J$ becomes a functor $\\P^\\op\\to\\Sets$ if we define $J(q\\leq p)S=S\\cap\\down q$ for each $S\\in J(p)$. Indeed, if $r\\leq q\\leq p$ and $S\\in J(p)$, we have\n\\begin{equation*}\nJ(r\\leq p)S=S\\cap\\down r=(S\\cap\\down q)\\cap\\down r=J(r\\leq q)J(q\\leq p)S.\n\\end{equation*}\nThus the stability axiom exactly expresses that fact that $J\\in\\Sets^{\\P\\op}$. In fact, it is enough to require that $S\\in J(p)$ for some $p\\in\\P$ implies $S\\cap\\down q\\in J(q)$ for each $q<p$. Indeed, if $q=p$, then $S\\cap\\down q=S$, which was already assumed to be in $J(p)$.\n\n\\begin{example}\n If $\\P=\\O(X)$, the set of open subsets of some topological space $X$ ordered by inclusion, the Grothendieck topology corresponding to the usual notion of covering is given by $J(U)=\\{S\\in\\D(\\down U):\\bigcup S=U\\}$.\n\\end{example}\n\n\n\\begin{example}\\label{ex:examplesGT}\nLet $\\P$ be a poset. Then\n\\begin{itemize}\n \\item The \\emph{indiscrete}\\index{Grothendieck topology!indiscrete} Grothendieck topology on $\\P$ is given by $J_{\\mathrm{ind}}(p)=\\{\\down p\\}$.\n \\item The \\emph{discrete}\\index{Grothendieck topology!discrete} Grothendieck topology on $\\P$ is given by $J_{\\mathrm{dis}}(p)=\\D(\\down p)$.\n\\item The \\emph{atomic}\\index{Grothendieck topology!atomic} Grothendieck topology on $\\P$ can only be defined if $\\P$ is downwards directed, and is given by $J_{\\mathrm{atom}}(p)=\\D(\\down p)\\setminus\\{\\emptyset\\}$.\n\\end{itemize}\nNotice that the existence of (finite) meets is sufficient for a poset to be downwards directed. An example of a poset that is not downwards directed and where the stability axiom for $J_{\\mathrm{atom}}$ fails is as follows. Let $\\P_3=\\{x,y,z\\}$ with $y\\leq x$ and $z\\leq x$. Then $\\down y\\in J_{\\mathrm{atom}}(x)$ and since $z\\leq x$, we should have $\\down y\\cap\\down z\\in J(x)$. But this means that $\\emptyset\\in J_{\\mathrm{atom}}(x)$, a contradiction. On the other hand, if $\\P$ is downwards directed, the stability axiom always holds. Indeed, let $S\\in J_{\\mathrm{atom}}(x)$ and $z\\leq x$. Even with $z\\notin S$, we have $S\\cap\\down z\\neq\\emptyset$, so $S\\cap\\down z\\in J_{\\mathrm{atom}}(z)$, since if we choose an arbitrary $y\\in S$, there must be a $w\\in\\P$ such that $w\\leq z$ and $w\\leq y$. The latter inequality implies $w\\in S$, so $w\\in S\\cap\\down z$.\n\\end{example}\n\n\\begin{convention}\nIf we want to emphasize the poset on which a Grothendieck topology is defined, we add the poset as a superscript. So $J_\\atom^\\P$ is the atomic topology defined on the poset $\\P$.\n\\end{convention}\n\n\nThe following lemma will be very useful; for arbitrary categories it can be found in \\cite[pp. 110-111]{M&M}. We give a direct proof for posets.\n\\begin{lemma}\\label{lem:filter}\n Let $J$ be a Grothendieck topology on $\\P$. Then $J(p)$ is a filter of sieves on $p$ in the sense that:\n \\begin{itemize}\n  \\item $S\\in J(p)$ implies $R\\in J(p)$ for each sieve $R$ on $p$ containing\n  $S$;\n  \\item $S,R\\in J(p)$ implies $S\\cap R\\in J(p)$.\n  \\end{itemize}\n\\end{lemma}\n\\begin{proof}\nLet $S\\in J(p)$ and $R\\in\\D(\\down p)$ such that $S\\subseteq R$. Then if $q\\in S$, we have $q\\in R$, so $R\\cap\\down q=\\down q\\in J(q)$. It follows now from the transitivity axiom that $R\\in J(p)$.\n\nIf $S,R\\in J(p)$, and let $q\\in R$. Then $$(S\\cap\\down R)\\cap\\down q=S\\cap(R\\cap\\down q)=S\\cap\\down q\\in J(q)$$ by the stability axiom. So $(S\\cap R)\\cap\\down q\\in J(q)$ for each $q\\in R$, hence by the transitivity axiom, it follows that $S\\cap R\\in J(p)$.\n\\end{proof}\n\nWe see that a Grothendieck topology is pointwise closed under finite intersections. In general, a Grothendieck topology is not closed under arbitrary intersections, which leads to the following definition.\n\n\\begin{definition}\n Let $J$ be a Grothendieck topology on a poset $\\P$. We say that $J$ is \\emph{complete}\\index{complete Grothendieck topology} if for each $p\\in\\P$ and each family $\\{S_i\\}_{i\\in I}$ in $J(p)$, for some index set $I$, we have $\\bigcap_{i\\in I}S_i\\in J(p)$.\n\\end{definition}\n\n\n\\begin{lemma}\n For each $p\\in\\P$, denote $\\bigcap\\{S\\in J(p)\\}$ by $S_p$. Then $J$ is complete if and only if $S_p\\in J(p)$ for each $p\\in\\P$.\n\\end{lemma}\n\\begin{proof}\nIf $J$ is complete, it follows directly from the definitions of $S_p$ and of completeness that $S_p\\in J(p)$ for each $p\\in\\P$.\n Conversely, assume that $S_p\\in J(p)$ for each $p\\in\\P$. For an arbitrary $p\\in\\P$ let $\\{S_i\\}_{i\\in I}$ a family of sieves in $J(p)$. Then $$S_p=\\bigcap\\{S\\in J(p)\\}\\subseteq\\bigcap_{i\\in I}S_i,$$ so by Lemma \\ref{lem:filter}, we find that $\\bigcap_{i\\in I}S_i\\in J(p)$.\n\\end{proof}\n\n\\begin{proposition}\\label{prop:GTisCompleteLattice}\n Let $\\P$ be a poset. Then $\\G(\\P)$ is a complete lattice where the meet $\\bigwedge_{i\\in I}J_i$ of a each collection $\\{J_i\\}_{i\\in I}$ of Grothendieck topologies on $\\P$ is defined by pointwise intersection: $\\left(\\bigwedge_{i\\in I} J_i\\right)(p)=\\bigcap_{i\\in I}J_i(p)$ for any collection $\\{J_i\\}_{i\\in I}$ of Grothendieck topologies on $\\P$.\n\\end{proposition}\n\\begin{proof}\n Let $\\{J_{i}\\}_{i\\in I}$ be a collection of Grothendieck topologies on $\\P$. We shall prove that $\\bigwedge_{i\\in I}J_i$ is a Grothendieck topology on $\\P$. Let $p\\in \\P$, then $\\down p\\in J_i(p)$ for each $i\\in I$, so $\\down p\\in\\bigcap_{i\\in I}J_i(p)$. For stability, assume $S\\in\\bigcap_{i\\in I}J_i(p)$ and let $q\\leq p$. Thus $S\\in J_i(p)$ for each $i\\in I$, so by the stability axiom for each $J_i$, we find $S\\cap\\down q\\in J_i(q)$ for each $i\\in I$. So $S\\cap\\down q\\in\\bigcap_{i\\in I}J_i(q)$. Finally, let $S\\in\\bigcap_{i\\in I}J_i(p)$ and $R\\in\\D(\\down p)$ be such that $R\\cap\\down q\\in \\bigcap_{i\\in I}J_i(q)$ for each $q\\in S$. Then for each $i\\in I$ we have $S\\in J_i(p)$ and $R\\cap\\down q\\in J_i(q)$ for each $q\\in S$. So for each $i\\in I$, by the transitivity axiom for $J_i$ we find that $R\\in J_i(p)$. Thus $R\\in\\bigcap_{i\\in I}J_i(p)$. We conclude that $\\bigwedge J_i$ is indeed a Grothendieck topology on $\\P$. Now, for each $k\\in I$, we have $\\bigwedge_{i\\in I}J_i\\leq J_k$, since $\\bigcap_{i\\in I}J_i(p)\\subseteq J_k(p)$ for each $p\\in\\P$. If $K$ is another Grothendieck topology on $\\P$ such that $J_k\\leq K$ for each $k\\in I$, then for each $p\\in\\P$ $$\\bigcap_{i\\in I}J_i(p)\\subseteq J_k(p)\\subseteq K(p).$$ Hence $\\bigwedge_{i\\in I}J_i\\leq K$. This shows that pointwise intersection indeed defines a meet operation on $\\G(\\P)$, and by Lemma \\ref{lem:completelattice}, it follows that $\\G(\\P)$ is a complete lattice.\n\\end{proof}\n\n\n\n\nWe will now describe a special class of Grothendieck topologies on a poset that are generated by subsets of the poset.\n\\begin{proposition}\\label{prop:defJX}\nLet $\\P$ be a poset and $X$ a subset of $\\P$. Then\n\\begin{equation}\\label{eq:inducedtopology1}\nJ_X(p)  = \\{S\\in\\D(\\down p): X\\cap\\down p\\subseteq S\\}\n\\end{equation}\nis a complete Grothendieck topology on $\\P$. Moreover, if $X\\subset Y\\subseteq\\P$, then $J_Y\\leq J_X$.\n\\end{proposition}\n\\begin{proof}\n We have $\\down p\\in J_X(p)$, for $\\down p$ contains $X\\cap\\down p$. If\n\\mbox{$S\\in J_X(p)$}, i.e. $X\\cap\\down p\\subseteq S$, and if $q<p$, then\n$\\down q\\subset \\down p$, so\n\\begin{equation*}\nX\\cap \\down q=X\\cap \\down p\\cap \\down q \\subseteq S\\cap\n\\down q.\n\\end{equation*}\nHence we find that $S\\cap\\down q\\in J_X(q)$, so\nstability holds.\n\nFor transitivity, let $S\\in J_X(p)$, so $X\\cap\\down p\\subseteq S$. Let\n$R$ be a sieve on $p$ such that for each $q\\in S$ we have $R\\cap \\down q\\in J_X(q)$, so $X\\cap \\down q\\subseteq R\\cap\\down q$. Since $S$ is a sieve, we have $S=\\down S=\\bigcup_{q\\in S}\\down q$, whence we\nfind\n\\begin{equation*}\nX\\cap S  = X\\cap\\bigcup_{q\\in S}\\down q=\\bigcup_{q\\in S}(X\\cap \\down q)\n\\subseteq  \\bigcup_{q\\in S}(R\\cap \\down q)=R\\cap\\bigcup_{q\\in\nS}\\down q=R\\cap S,\n\\end{equation*}\nfrom which $$\nX\\cap\\down p=X\\cap (X\\cap \\down p)\\subseteq X\\cap S\\subseteq\nR\\cap S\\subseteq R$$ follows. Thus $R\\in J_X(p)$ and the transitivity\naxiom holds.\n\nWe next have to show that $J_X$ is complete. So let $p\\in\\P$ and let $\\{S_i\\}_{i\\in I}\\subseteq J_X(p)$ be a collection of covers with index set $I$. This means that $X\\cap\\down p\\subseteq S_i$ for each $i\\in I$. But this implies that $X\\cap\\down p\\subseteq\\bigcap_{i\\in I}S_i$, hence $\\bigcap_{i\\in I}S_i\\in J_X(p)$.\n\nFinally, let $Y\\subseteq\\P$ such that $X\\subseteq Y$. Let $p\\in\\P$ and $S\\in J_Y(p)$. Then $Y\\cap\\down p\\subseteq S$, and since $X\\subseteq Y$ this implies $X\\cap\\down p\\subseteq S$. So $S\\in J_X(p)$. We conclude that $J_Y\\leq J_X$.\n\\end{proof}\n\nWe call $J_X$ the \\emph{subset Grothendieck topology}\\index{subset Grothendieck topology} generated by the subset $X$ of $\\P$. It is easy to see that the indiscrete Grothendieck topology is exactly $J_\\P$, whereas the discrete Grothendieck topology is $J_\\emptyset$.\n\n\n\n\n\\begin{lemma}\\label{lem:defXJ}\n Let $J$ be a Grothendieck topology on a poset $\\P$ and define\n$X_J\\subseteq \\P$ by\n\\begin{equation}\\label{mJ}\nX_J=\\big\\{p\\in \\P:J(p)=\\{\\down p\\}\\big\\}.\n\\end{equation}\nIf $K$ is another Grothendieck topology on $\\P$ such that $K\\leq J$, then $X_J\\subseteq X_K$.\n\\end{lemma}\n\\begin{proof}\n Let $p\\in X_J$. Then $K(p)\\subseteq J(p)=\\{\\down p\\}$, and since $\\down p\\in K(p)$ by definition of a Grothendieck topology, this implies $K(p)=\\{\\down p\\}$. So $p\\in X_K$.\n\\end{proof}\n\n\n\n\\begin{lemma}\\label{lem:sublemma}\n Let $\\P$ be a poset and $J$ a Grothendieck topology on $\\P$. If $p\\in \\P$ such that $J(p)=\\{\\down p\\}$, then for each sieve $S$ on $p$ such that $X_J\\cap\\down p\\subseteq S$, we have $S\\in J(p)$.\n\\end{lemma}\n\\begin{proof}\n By definition of $X_J$ we have $p\\in X_J$, so each sieve $S$ on $p$ containing $X_J\\cap\n\\down p$ contains $p$. Since the only sieve on $p$ containing $p$ must be equal to $\\down p$, we find that\n$X_J\\cap\\down p\\subseteq S$ implies $S\\in J(p)$.\n\\end{proof}\n\n\\begin{proposition}\\label{prop:correspsubsetsandtopologies}\nLet $\\P$ be a poset. Then\n\\begin{enumerate}\n\\item $Y=X_{J_Y}$ for each $Y\\subseteq\\P$;\n\\item $J_Y\\leq J_Z$ if and only if $Z\\subseteq Y$ for each $Y,Z\\subseteq\\P$;\n\\item $K\\leq J_{X_K}$ for each Grothendieck topology $K$ on $\\P$;\n\\item If $\\P$ is Artinian, then the map $\\PP(\\P)^\\op\\to\\G(\\P)$ given by $X\\mapsto J_X$ is an order isomorphism with inverse $J\\mapsto X_J$;\n\\item If $\\P$ is Artinian, then every Grothendieck topology on $\\P$ is complete.\n\\end{enumerate}\n\\end{proposition}\n\\begin{proof}\\\n\\begin{enumerate}\n\\item Let $x\\in Y$. Then $x\\in Y\\cap \\down x$, so if $S\\in\nJ_Y(x)$, that is $S$ is a sieve on $x$ containing $Y\\cap\\down x$, we\nmust have $x\\in S$. The only sieve on $x$ containing $x$ is $\\down x$,\nso we find that $J_Y(x)=\\{\\down x\\}$. By definition of $X_{J_Y}$ we find\nthat $x\\in X_{J_Y}$.\n\nConversely, let $x\\in X_{J_Y}$. This means that $J_Y(x)=\\{\\down x\\}$, or\nequivalently, that the only sieve on $x$ containing $Y\\cap \\down x$ is\n$\\down x$. Now, assume that $x\\notin Y$. Then $x\\notin Y\\cap\\down x$, so\n$\\down x\\setminus\\{x\\}$ (possibly empty) is a sieve on $x$ which\nclearly contains $X\\cap \\down x$, but which is clearly not equal to\n$\\down x$ in any case. So we must have $x\\in Y$.\n\n\\item Let $Y,Z\\subseteq\\P$. We already found in Proposition \\ref{prop:defJX} that $Z\\subseteq Y$ implies $J_Y\\leq J_Z$. So assume that $J_Y\\leq J_Z$. By Lemma \\ref{lem:defXJ}, this implies that $X_{J_Z}\\subseteq X_{J_Y}$. But by the first statement of this proposition, this is exactly $Z\\subseteq Y$.\n\\item Let $S\\in K(p)$ and $q\\in X_K\\cap\\down p$. So $q\\leq p$ and $K(q)=\\{\\down q\\}$.\nSince $S\\in K(p)$ and $q\\leq p$, we have $\\down q\\cap S\\in K(q)$ by\nstability. In other words $S\\cap\\down q=\\down q$. Thus $\\down q\\subseteq S$ and\nso certainly $q\\in S$. We see that $X_K\\cap\\down p\\subseteq S$, hence\n$S\\in J_{X_K}$.\n\\item It follows from Proposition \\ref{prop:defJX} and Lemma \\ref{lem:defXJ} that $X\\mapsto J_X$ and $J\\mapsto X_J$ are order morphisms.\nSince the first statement of this proposition is equivalent with saying that $J\\mapsto X_J$ is a left inverse of $X\\mapsto J_X$, we only have to show that it is also a right inverse. In other words, we have to show that each\nGrothendieck topology $K$ on $\\P$ equals $J_{X_K}$. So\nlet $K$ be a Grothendieck topology on $\\P$. We shall\nshow using Artinian induction (see Lemma \\ref{prop:ArtinianInduction}) that for any $p\\in \\P$, each sieve\n$S$ on $p$ containing $X_K\\cap\\down p$ is an element of $K(p)$.\n\nLet $p$ be a minimal element. Then the only possible sieves on $p$\nare $\\down p=\\{p\\}$ and $\\emptyset$. Hence there are only two options\nfor $K(p)$, namely either $K(p)=\\{\\down p\\}$ or\n$K(p)=\\{\\emptyset,\\down p\\}$. In the first case, we find by Lemma \\ref{lem:sublemma} that\n$X_K\\cap\\down p\\subseteq S$ implies $S\\in K(p)$. If\n$K(p)=\\{\\emptyset,\\down p\\}$, then $K(p)$ contains all possible sieves on\n$p$, so we automatically have that $S\\in K(p)$ for any sieve $S$ on $p$ such that\n$X_K\\cap \\down p\\subseteq S$.\n\nFor the induction step, assume that $p\\in \\P$ is not minimal and assume that for each $q<p$ the condition $X_K\\cap\n\\down q\\subseteq R$ implies $R\\in K(q)$ for each sieve $R$ on $q$. If\n$K(p)=\\{\\down p\\}$, we again apply Lemma \\ref{lem:sublemma} to conclude that\n$S\\in K(p)$ for all sieves $S$ on $p$ such that $X_K\\cap\\down p\\subseteq S$. If\n$K(p)\\neq\\{\\down p\\}$, we must have $\\down p\\setminus\\{p\\}\\in K(p)$, which is non-empty, since $p$ is not minimal. If $S$ is a sieve on $p$ such that $X_K\\cap\n\\down p\\subseteq S$, we have for each $q\\in \\down p\\setminus\\{p\\}$, that is, for each $q<p$, that\n\\begin{equation}\\label{eq:identityXJ}\nX_K\\cap \\down q=X_K\\cap \\down p\\cap \\down q\\subseteq S\\cap\\down q.\n\\end{equation}\nOur induction assumption on $q<p$ implies now that $S\\cap\\down q\\in K(q)$ for each $q\\in \\down p\\setminus\\{p\\}$, hence by the transitivity axiom we find\n$S\\in K(p)$.\n\nSo for all sieves $S$ on $p$ such that $X_K\\cap\\down p\\subseteq S$, we found\nthat $S\\in K(p)$, hence we have $J_{X_K}\\subseteq K$. The other inclusion follows from the third statement of this proposition.\n\\item Since all subset Grothendieck topologies are complete, this follows from the fourth statement of this proposition.\n\\end{enumerate}\n\\end{proof}\n\n\n\n\n\\begin{theorem}\\label{thm:equivalenceArtinianandSubsettopologies}\nLet $\\P$ be a poset. Then $\\P$ is Artinian if and only if all Grothendieck topologies on $\\P$ are subset Grothendieck topologies.\n\\end{theorem}\n\\begin{proof}\n The previous proposition states that all Grothendieck topologies on $\\P$ are subset Grothendieck topologies if $\\P$ is Artinian. For the other direction, we first introduce another Grothendieck topology. Let $\\P$ be a poset and $X\\subseteq\\P$, and define $L_X$ for each $p\\in\\P$ by $$L_X(p)=\\{S\\in\\D(\\down p): x\\in X\\cap\\down p\\implies S\\cap\\down x\\cap X\\neq\\emptyset\\}.$$ To see that this is a Grothendieck topology on $\\P$, assume that $x\\in X\\cap\\down p$, then clearly $\\down p\\cap\\down x\\cap X\\neq\\emptyset$, so $\\down p\\in L_X(p)$. If $S\\in L_X(p)$ and $q\\leq p$, assume that $x\\in X\\cap\\down q$. Then $x\\in X\\cap\\down p$, so $S\\cap\\down x\\cap X\\neq\\emptyset$. Since $x\\leq q$, we have $\\down x=\\down q\\cap\\down x$, hence $S\\cap\\down q\\cap\\down x\\cap X\\neq\\emptyset$. We conclude that $S\\cap\\down q\\in L_X(q)$. Finally, let $S\\in L_X(p)$ and $R\\in\\D(\\down p)$ such that $R\\cap\\down q\\in L_X(q)$ for each $q\\in S$. Let $x\\in X\\cap\\down p$. Then $S\\cap\\down x\\cap X\\neq\\emptyset$, so there is some $q\\in S\\cap\\down x\\cap X$. Since $q\\in S$, we find $R\\cap\\down q\\in L_X(q)$. Since $q\\in X$, we find $q\\in X\\cap\\down q$, so $(R\\cap\\down q)\\cap\\down q\\cap X\\neq\\emptyset$. Since $q\\leq x$, we find $\\down q\\subseteq\\down x$, whence $R\\cap\\down x\\cap X\\neq\\emptyset$. So $R\\in L_X(p)$.\n\nNow assume that $\\P$ is non-Artinian. Then $\\P$ contains a non-empty subset $X$ without a minimal element. We show that $L_X\\neq J_Y$ for each $Y\\subseteq\\P$. First take $Y=\\emptyset$. Then $\\emptyset\\in J_Y(p)$ for each $p\\in\\P$. However, since $X$ is assumed to be non-empty, there is some $p\\in X$. Then $p\\in X\\cap\\down p$, but $\\emptyset\\cap\\down p\\cap X=\\emptyset$, so $\\emptyset\\notin L_X(p)$. We conclude that $L_X\\neq J_Y$ if $Y=\\emptyset$.\n\nAssume that $Y$ is non-empty. Then there is some $p\\in Y$, and $J_Y(p)=\\{\\down p\\}$. Assume that $X\\cap\\down p=\\emptyset$. Then $x\\in X\\cap\\down p\\implies \\emptyset\\cap\\down x\\cap X\\neq\\emptyset$ holds, so $\\emptyset\\in L_X(p)$. Thus $L_X\\neq J_Y$ in this case. Assume that $X\\cap\\down p\\neq\\emptyset$. Hence there is some $x\\in X\\cap\\down p$. Even if $p\\in X$, we can assume that $x$ is strictly smaller than $p$, since $X$ does not contain a minimal element, so $X\\cap\\down p\\setminus\\{p\\}\\neq\\emptyset$. Let $S=\\down x$, then $S\\cap\\down x\\cap X\\neq\\emptyset$, so $\\down x\\in L_X(p)$. We conclude that $L_X(p)\\neq\\{\\down p\\}$, so $L_X\\neq J_Y$ in all cases.\n\\end{proof}\n\nThe Grothendieck topology $L_X$ in the proof somewhat falls out of the sky. In the next section we explore techniques, which helped us to find this Grothendieck topology.\n\n\n\\section{Non-Artinian and downwards directed posets}\n\nIf a poset is non-Artinian, it is much harder to find all Grothendieck topologies. This section is devoted to the question how to find a Grothendieck topology $J$ on a non-Artinian poset such that $J$ is not a subset Grothendieck topology. It turns out that this gives a wide class of new Grothendieck topologies.\n\n\\begin{proposition}\\label{prop:densetopologydef}\n Let $\\P$ be a poset. Then $J_\\dense$ defined by $$J_\\dense(p)=\\{S\\in\\D(\\down p):\\down p\\subseteq\\up S\\},$$ for each $p\\in\\P$ is a Grothendieck topology on $\\P$, called the \\emph{dense topology}\\index{dense topology} \\cite[p. 115]{M&M}.\n\nIf $\\P$ is downwards directed, the dense topology is exactly the atomic topology on $\\P$, and is not complete if $\\P$ does not contain a least element.\n\\end{proposition}\n\\begin{proof}\n Clearly $\\down p\\subseteq\\up(\\down p)$, so $\\down p\\in J_\\dense(p)$. Let $S\\in J_\\dense(p)$ and $q\\leq p$. Then $\\down p\\subseteq\\up S$, so if $r\\in\\down q$, then $r\\in\\down p$, so there must be an $s\\in S$ such that $r\\leq s$. Then $s\\in S\\cap\\down q$, since $q\\geq r$. We conclude that $\\down q\\subseteq\\up(S\\cap\\down q)$, so $S\\cap\\down q\\in J_\\dense(q)$. Finally, let $S\\in J_\\dense(p)$ and $R\\in\\D(\\down p)$ such that $R\\cap\\down q\\in J_\\dense(q)$ for each $q\\in S$. So $\\down p\\subseteq\\up S$ and $\\down q\\subseteq\\up(R\\cap\\down q)$ for each $q\\in S$. Now, let $r\\in\\down p$. We aim to prove that $r\\in\\up R$. Since $\\down p\\subseteq \\up S$, there is a $q\\in S$ such that $q\\leq r$. Moreover, $\\down q\\subseteq(R\\cap\\down q)$ holds, so there is a $t\\in R\\cap\\down q)$ such that $t\\leq q$. Then $r\\in\\up R$, for $r\\geq q\\geq t$. We conlude that $R\\in J_\\dense(p)$.\n\n\nIf we assume that $\\P$ is downwards directed, let $S$ be a non-empty sieve on $p\\in\\P$. So $S$ contains an $s\\leq p$, since $S\\in\\D(\\down p)$. Let $q\\in\\down p$. Since $\\P$ is downwards directed, there is an $r\\leq q,s$, which implies that $r\\in S$, whence $q\\in\\up S$. So $S\\in J_\\dense(p)$. The other way round, if $S\\in J_\\dense(p)$, we should have $p\\in\\down p\\subseteq\\up S$, which is only possible if $S$ is non-empty. So $S\\in J_\\atom(p)$.\n\nIf $\\P$ does not contain a least element, $\\min(\\P)=\\emptyset$, since if $x\\in\\min(\\P)$, and $y\\in\\P$, there is some $z\\in\\P$ such that $z\\leq x,y$ for $\\P$ is downwards directed. But $x\\in\\min(\\P)$, so $z\\leq x$ implies $z=x$, hence $x\\leq y$ contradicting that $\\P$ does not contain a least element. Now, let $p\\in\\P$. Then $\\down q\\in J_\\atom(q)$ for each $q\\leq p$. Moreover, if $x\\leq p$, then there is a $q<x$, since otherwise $x$ would be minimal. Hence $x\\notin\\bigcap_{q\\leq p}\\down q$, so $\\bigcap_{q\\leq p}\\down q=\\emptyset\\notin J_\\atom(p)$. Hence $J_\\atom$ is not complete.\n\\end{proof}\n\n\n\\begin{proposition}\\label{prop:conditiondensetopologynotsubsettopology}\n Let $\\P$ be a poset. Then $J_\\dense=J_{\\min(\\P)}$ if and only if $\\up\\min(\\P)=\\P$. In particular, $J_\\dense=J_{\\min(\\P)}$ if $\\P$ is Artinian. If $\\up\\min(\\P)\\neq\\P$, then $J_\\dense$ is not equal to any subset Grothendieck topology.\n\\end{proposition}\n\\begin{proof}\n Assume that $\\up\\min(\\P)=\\P$ and let $p\\in\\P$. If $S\\in J_{\\min(\\P)}(p)$, we have by definition $\\min(\\P)\\cap\\down p\\subseteq S$. Let $q\\leq p$. Since $\\up\\min(\\P)=\\P$, there is an $m\\in\\min(\\P)$ such that $m\\leq q$. Then $$m\\in\\min(\\P)\\cap\\down q\\subseteq\\min(\\P)\\cap\\down p.$$ Thus $m\\in S$, whence $q\\in\\up S$. So we find that $\\down p\\subseteq\\up S$, hence $S\\in J_\\dense(p)$.\n\nConversely, if $S\\in J_\\dense(p)$, we have $\\down p\\subseteq\\up S$. Now, $\\min(\\P)\\cap\\down p$ is non-empty, since $\\up\\min(\\P)=\\P$, so let $m\\in\\min(\\P)\\cap\\down p$. Then $m\\in\\up S$, so there is an $s\\in S$ such that $s\\leq m$. But $m\\in\\min(\\P)$, so $s=m$, and we conclude that $\\min(\\P)\\cap\\down p\\subseteq S$, hence $S\\in J_{\\min(\\P)}(p)$.\n\nIf $\\P$ is Artinian, there are two ways to show that $J_\\dense=J_{\\min(\\P)}$. Firstly, if $p\\in\\P$, the set $\\down p$ is non-empty, so by the Artinian property, it contains a minimal element $m$. Now, if $q\\leq m$, we have $q\\in\\down p$, so $m\\leq q$ by the minimality of $m$. So $q=m$, which implies that $m\\in\\min(\\P)$. Thus $p\\in\\up\\min(\\P)$, so $\\up\\min(\\P)=\\P$ and we conclude that $J_\\dense=J_{\\min(\\P)}$.\n\n Another path we could take is using Proposition \\ref{prop:correspsubsetsandtopologies}, which says that $J_\\dense$ is equal to $J_Y$, with $$Y=X_{J_{\\dense}}=\\{p\\in\\P:J_\\dense(p)=\\{\\down p\\}\\},$$ if $J_\\dense$ is a subset topology, which is always the case if $\\P$ is Artinian. Then we find $J_{\\dense(p)}=\\{\\down p\\}$ if and only if $\\down p\\setminus\\{p\\}\\notin J_\\dense(p)$ (using Lemma \\ref{lem:filter}). Thus $p\\in Y$ if and only if $\\down p\\nsubseteq\\up(\\down p\\setminus\\{p\\})$ if and only if $\\down p\\setminus\\{p\\}=\\emptyset$ if and only if $p\\in\\min(\\P)$, so $Y=\\min(\\P)$. Proposition \\ref{prop:correspsubsetsandtopologies} implies $X_{J_Z}=Z$ for each $Z\\subseteq\\P$. So if $J_\\dense=J_Z$ for some $Z$, this implies that $Z=\\min(\\P)$. Thus $J_\\dense\\neq J_Z$ for each $Z\\subseteq\\P$ such that $Z\\neq\\min(\\P)$, hence $J_\\dense$ is not equal to any subset topology if $\\up\\min(\\P)\\neq\\P$.\n\\end{proof}\n\nThe next example shows that there exists a non-Artinian poset $\\P$ with $\\up\\min(\\P)=\\P$, so with $J_\\dense=J_{\\min(\\P)}$.\n\\begin{example}\nLet $\\P=X\\cup Y$ with $X=\\{x_n\\}_{n\\in\\N}$ and $Y=\\{y_n\\}_{n\\in\\N}$ with the ordering $x_n\\geq y_n,x_{n+1}$ for each $n\\in\\N$. We can visualize this in the following diagram\n\n\\begin{equation*}\n\\xymatrix{x_1\\\\\nx_2\\ar[u] & y_1\\ar[ul]\\\\\nx_3\\ar[u] & y_2\\ar[ul]\\\\\n\\vdots\\ar[u]& y_3\\ar[ul]\\\\\n}\n\\end{equation*}\nwhere $a\\to b$ represents the inequality $a\\leq b$ for each $a,b\\in\\P$. Then $\\P$ is clearly non-Artinian, since it contains the chain $x_1\\geq x_2\\geq x_3\\geq\\ldots$, which does not terminate. However, we have $\\min(\\P)=Y$, whence $\\up\\min(\\P)=\\P$.\n\\end{example}\n\n\n\n\n\n\n\n\n\nSo the dense topology fails to be a universal counterexample of a Grothendieck topology that is not a subset Grothendieck topology on non-Artinian posets. Given a non-Artinian poset with non-empty subset $X$ without minimal elements, we found in the proof of Theorem \\ref{thm:equivalenceArtinianandSubsettopologies}\na Grothendieck topology $L_X$ on $\\P$ that is never a subset Grothendieck topology. This Grothendieck topology was found by ``extending'' the dense Grothendieck topology on $X$ (which is on $X$ not a subset Grothendieck topology by Proposition \\ref{prop:conditiondensetopologynotsubsettopology}) to a Grothendieck topology on $\\P$.\n\nIn order to explain in detail how $L_X$ was found, we have to introduce a notion of an extension of a Grothendieck topology to a larger poset. Since a structure on some object is called an extension of a structure on a smaller object if its restriction equals the structure on the smaller object, we also have to define how to restrict Grothendieck topologies to subsets. First we have to introduce some notation in order to discriminate between down-sets of $X$ and down-sets of $\\P$.\n\\begin{definition}\\label{def:subposetdownset}\n Let $\\P$ be a poset and $X\\subseteq\\P$ a subset. For any other subset $A\\subseteq\\P$, we shall use the notation $\\bar\\down A=\\down A\\cap X$. In the same way, $\\bar\\up A$ is defined as the set $\\up A\\cap X$.\n\\end{definition}\n Given this definition, if $x\\in X$, the set $\\down x$ is the set of elements in $\\P$ below $x$, whereas $\\bar\\down x$ is the set of elements of $X$ below $x$. Hence a sieve on $x\\in X$ with respect to the subposet $X$ of $\\P$ is precisely an element of $\\D(\\bar\\down x)$.\n\n\n\\begin{lemma}\\label{lem:inducedtopology}\n Let $J$ be a Grothendieck topology on a poset $\\P$ and let $X\\subseteq\\P$ be a subset such that $J_X\\leq J$. Then $\\bar J$ defined by $$\\bar J(x)=\\{S\\cap X:S\\in J(x)\\}$$ for each $x\\in X$ is a Grothendieck topology on $X$ called the \\emph{induced Grothendieck topology}\\index{induced Grothendieck topology}. Furthermore, let $x\\in X$. Then for each $\\bar S\\in\\D(\\bar\\down x)$ we have $\\bar S=\\down \\bar S\\cap X$. Moreover, we have $\\bar S\\in \\bar J(x)$ if and only if $\\down\\bar S\\in J(x)$, i.e.,\n     $$\\bar J(x)=\\{\\bar S\\in\\D(\\bar\\down x):\\down\\bar S\\in J(x)\\}$$\n     for each $x\\in X$.\n\\end{lemma}\n\\begin{proof}\nLet $x\\in X$ and $\\bar S\\in\\D(\\bar \\down x)$. Then $\\bar S\\subseteq X$, so $\\bar S\\subseteq(\\down\\bar S)\\cap X$. On the other hand, if $y\\in(\\down\\bar S)\\cap X$, we have $y\\in X$ and $y\\leq s$ for some $s\\in\\bar S$. Since $\\bar S\\in\\D(X)$, we obtain $y\\in\\bar S$. So $(\\down\\bar S)\\cap X=\\bar S$. Then if $\\down\\bar S\\in J(x)$, it follows immediately that $\\bar S\\in \\bar J(x)$. Now assume $\\bar S\\in\\bar J(x)$. Then there is an $S\\in J(x)$ such that $\\bar S=S\\cap X$. Let  $y\\in S$, so $\\down y\\subseteq S$, then we have $$\\down(X\\cap\\down y)=\\down(X\\cap\\down y)\\cap\\down y\\subseteq\\down(X\\cap S)\\cap\\down y=\\down\\bar S\\cap \\down y.$$ Since $J_X\\leq J$ and so $\\down(X\\cap\\down y)\\in J(y)$, we find by Lemma \\ref{lem:filter} that $\\down\\bar S\\cap \\down y\\in J(y)$ for each $y\\in S$. Hence by the transitivity axiom for $J$ it follows that $\\down\\bar S\\in J(x)$.\n\nIn order to show that $\\bar J$ is a Grothendieck topology on $X$, let $x\\in X$. Then $\\bar\\down x=\\down x\\cap X$, and since $\\down x\\in J(x)$, we have $\\bar\\down x\\in \\bar J(x)$. If $\\bar S\\in \\bar J(x)$ and $y\\leq x$ in $X$, then $\\bar S=S\\cap X$ for some $S\\in J(x)$. Since $J$ is a Grothendieck topology, we have $S\\cap\\down y\\in J(y)$. Now, $\\bar S\\cap\\bar\\down y=S\\cap X\\cap\\down y$, so $\\bar S\\cap\\bar\\down y\\in \\bar J(y)$. Finally, let $\\bar S\\in \\bar J(x)$ and $\\bar R\\in\\D(\\bar\\down x)$ such that $\\bar R\\cap\\bar \\down y\\in \\bar J(y)$ for each $y\\in \\bar S$. This means that $\\down\\bar S\\in J(x)$ and $\\down(\\bar R\\cap\\bar\\down y)\\in J(y)$ for each $y\\in \\bar S$. Now, let $R=\\down\\bar R$. If $y\\in\\bar S$, we have $\\bar R\\cap\\bar\\down y\\subseteq\\down R\\cap\\down y$, so $\\down(\\bar R\\cap\\bar\\down y)\\subseteq\\down\\bar R\\cap\\down y$. Since $\\down(\\bar R\\cap\\bar\\down y)\\in J(y)$, Lemma \\ref{lem:filter} assures that $\\down\\bar R\\cap\\down y\\in J(y)$ for each $y\\in\\bar S$. Now assume $z\\in\\down\\bar S$. Then $z\\leq y$ for some $y\\in\\bar S$, and since $\\down\\bar R\\cap\\down y\\in J(y)$, it follows that $$\\down\\bar R\\cap\\down z=(\\down\\bar R\\cap\\down y)\\cap\\down z\\in J(z).$$ Again by the transitivity axiom for $J$ it follows that $\\down\\bar R\\in J(x)$, so $\\bar R\\in\\bar J(x)$.\n\\end{proof}\n\n\nIf we regard induced Grothendieck topologies as restrictings of Grothendieck topologies to subsets, we aim to find a Grothendieck topology $J_K$ on $\\P$ such that $\\bar J_K=K$ given a Grothendieck topology $K$ on $X$. First we recall Definition \\ref{def:subposetdownset}, where, for a fixed subset $X$ of a poset $\\P$, for each subset $A$ of $\\P$, we denoted the set $X\\cap\\down A$ by $\\bar\\down A$. So given $x\\in X$, this means that $\\down x$ is the set of elements in $\\P$ below $x$, whereas $\\bar\\down x$ means the set of elements of $X$ below $x$. Hence a sieve on $x\\in X$ with respect to the subposet $X$ of $\\P$ is precisely an element of $\\D(\\bar\\down x)$. Using this notation, we can formulate the following definition.\n\\begin{definition}\\label{def:extensionGrothTop}\nLet $K$ be a Grothendieck topology of a subset $X$ of a poset $\\P$. Then we define the \\emph{extension}\\index{extension of a Grothendieck topology} $J_K$ of $K$ on $\\P$ by\n\\begin{equation*}\nJ_K(p)=\\{S\\in\\D(\\down p): S\\cap\\bar\\down x\\in K(x)\\ \\forall x\\in\\bar\\down p\\}.\n\\end{equation*}\n\\end{definition}\n\nWe perform a sanity check that this is a Grothendieck topology. Let $x\\in\\bar\\down p$, so $x\\in X$ and $x\\leq p$. Then $$\\down p\\cap\\bar\\down x=\\down p\\cap X\\cap\\down x=X\\cap\\down x=\\bar\\down x\\in K(x).$$ So $\\down p\\in J_K(p)$. For stability, let $S\\in J_K(p)$, so $S\\cap \\bar\\down x\\in K(x)$ for each $x\\in\\bar\\down p$. Let $q\\leq p$. If $x\\in\\bar\\down q$, then $x\\in\\bar\\down p$ and $\\down x\\cap\\down q=\\down x$, hence $$S\\cap\\down q\\cap\\bar\\down x=S\\cap\\down q\\cap X\\cap\\down x=S\\cap X\\cap\\down x=S\\cap\\bar\\down x\\in K(x),$$ so $S\\cap\\down q\\in J_K(q)$. Finally, for transitivity, let $S\\in J_K(p)$ and $R\\in\\D(\\down p)$ be such that $R\\cap\\down q\\in J_K(q)$ for each $q\\in S$. Let $x\\in\\bar\\down p$. Since $S\\in J_K$ by definition of $J_K$ we find that $S\\cap\\bar\\down x\\in K(x)$. Then for each $y\\in S\\cap\\bar\\down x$, we have $y\\in S$, so $R\\cap\\down y\\in J_K(y)$. This means that $R\\cap\\down y\\cap\\bar\\down z\\in K(z)$ for each $z\\in\\bar\\down y$. Since also $y\\in\\bar\\down x\\subseteq X$, we have $y\\in\\bar\\down y$, so we are allowed to choose $z=y$. Hence we find for each $y\\in\\bar\\down x$ that $$R\\cap\\bar\\down x\\cap\\bar\\down y=R\\cap\\bar\\down y=R\\cap\\down y\\cap\\bar\\down y \\in K(y).$$ Since $\\bar\\down x\\in K(x)$, the transitivity axiom for $K$ implies that $R\\cap\\bar\\down x\\in K(x)$. We conclude that $R\\in J_K(p)$.\n\nThe next proposition assures that $J_K$ is indeed an extension of $K$.\n\n\\begin{proposition}\\label{prop:extensionofGrothTop}\n Let $X$ be a subset of a poset $\\P$ and $K$ a Grothendieck topology on $X$. Then:\n \\begin{enumerate}\n \\item[(i)] $J_X\\leq J_K$, where $J_K$ is the extension of $K$ on $\nP$;\n \\item[(ii)] $\\bar J_K$, defined in Lemma \\ref{lem:inducedtopology} for each $x\\in X$ by\n \\begin{eqnarray*}\n \\bar J_K(x) & = &  \\{S\\cap X:S\\in J_K(x)\\}\\\\\n & = & \\{\\bar S\\in\\D(\\bar\\down x):\\down\\bar S\\in J_K(x)\\},\n \\end{eqnarray*}\nis a well-defined Grothendieck topology on $X$;\n\\item[(iii)] $\\bar J_K=K$;\n\\item[(iv)] The assignment $\\G(X)\\to\\G(\\P)$ given by $K\\mapsto J_K$ is injective;\n\\item[(v)] For $Y\\subseteq X$, where the subset Grothendieck topology on $X$ induced by $Y\\subseteq X$ is denoted by $J_Y^X$, we have $J_{J_Y^X}=J_Y$ and $J^X_Y=\\bar J_Y$;\n\\item[(vi)] If $J_K$ is a subset Grothendieck topology on $\\P$, then $K$ is a subset Grothendieck topology on $X$;\n\\item[(vii)] If $J_K$ is a complete Grothendieck topology on $\\P$, then $K$ is a complete Grothendieck topology on $X$;\n\\item[(viii)] Denote the dense topology on $X$ by $J_\\dense^X$, then $L_X$ defined by\n     \\begin{equation*}\n    L_X(p)=\\{S\\in\\D(\\down p): x\\in X\\cap\\down p\\implies S\\cap\\down x\\cap X\\neq\\emptyset\\}\n    \\end{equation*}\n    for each $p\\in\\P$ is a well-defined Grothendieck topology on $\\P$ such that $J_X\\leq L_X$, $J_{J_\\dense^X}=L_X$ and $\\bar L_X=J^X_\\dense$.\n\\end{enumerate}\n\\end{proposition}\n\\begin{proof}\\\n\\begin{enumerate}\n\\item[(i)]\n Let $p\\in\\P$ and $S\\in J_X(p)$. So $X\\cap\\down p\\subseteq S$. If $x\\in X$ and $x\\leq p$, then $$\\bar\\down x=X\\cap\\down x=(X\\cap\\down p)\\cap (X\\cap\\down x)\\subseteq S\\cap\\bar\\down x,$$ and since $\\bar\\down x\\in K(x)$, it follows that $S\\cap \\bar\\down x\\in K(x)$. We conclude that $S\\in J_K(p)$, so $J_X\\leq J_K$.\n\\item[(ii)] Since the inequality in (i) holds, Lemma \\ref{lem:inducedtopology} garantuees that $\\bar J_K$ is a well-defined Grothendieck topology.\n\\item[(iii)] Let $x\\in X$. Since $\\down\\bar S\\in J_K(x)$ if and only $\\down\\bar S\\cap\\down y\\in K(y)$ for each $y\\in\\bar\\down x$, and $$\\down\\bar S\\cap\\bar\\down y=\\down\\bar S\\cap X\\cap\\down y=\\bar S\\cap\\bar\\down y,$$ where we again used Lemma \\ref{lem:inducedtopology}, we find $$\\bar J_K(x)=\\{\\bar S\\in\\D(\\bar\\down x): \\bar S\\cap\\bar\\down y\\in K(y)\\ \\forall y\\in\\bar\\down x\\}$$ for each $x\\in X$. Let $x\\in X$. Since $\\bar\\down x\\in K(x)$, it follows immediately by the transitivity axiom for $K$ that $\\bar S\\in K(x)$ if $\\bar S\\in\\bar J_K(x)$. Conversely, if $\\bar S\\in K(x)$, it follows immediately by the stability axiom for $K$ that $\\bar S\\in\\bar J_K(x)$. Hence $\\bar J_K=K$.\n\\item[(iv)] Let $K_1,K_2$ be Grothendieck topologies on $X$ such that $J_{K_1}=J_{K_2}$. Then by (iii), $$K_1=\\bar J_{K_1}=\\bar J_{K_2}=K_2,$$ hence $K\\mapsto J_K$ is an injective map.\n\\item[(v)] By a direct calculation, we show that $J_K=J_Y$, where we take $K=J_Y^X$, which is more explicitly given by\n $$J_Y^X(x)=\\{\\bar S\\in\\D(\\bar\\down x):Y\\cap\\bar\\down x\\subseteq\\bar S\\}$$ for each $x\\in X$. Then for each $p\\in\\P$,\n\\begin{eqnarray*}\nJ_K(p) & = & \\{S\\in\\D(\\down p): Y\\cap\\bar\\down x\\subseteq S\\cap\\bar\\down x\\ \\forall x\\in\\bar\\down p\\}\\\\\n& = & \\{S\\in\\D(\\down p): Y\\cap\\down x\\subseteq S\\cap X\\cap\\down x\\ \\forall x\\in X\\cap\\down p\\},\n\\end{eqnarray*}\nwhere we used $Y\\subseteq X$, so $Y\\cap\\bar\\down x=Y\\cap\\down x$. Let $p\\in\\P$ and $S\\in J_Y(p)$. If $x\\in X\\cap\\down p$, then the stability axiom for $J_Y$ implies that $Y\\cap\\down x\\subseteq S\\cap\\down x$, and since $Y\\subseteq X$, it follows that $Y\\cap\\down x\\subseteq S\\cap X\\cap\\down x$. So $S\\in J_K(p)$. Now assume that $S\\in J_K(p)$ and let $x\\in Y\\cap\\down p$. Since $Y\\subseteq X$, this implies $x\\in X\\cap\\down p$, so $Y\\cap\\down x\\subseteq S\\cap X\\cap\\down x$ for $S\\in J_K(p)$. But $x\\in Y$, so this implies that $x\\in S$. Thus $Y\\cap\\down p\\subseteq S$, whence $S\\in J_Y(p)$.\n\nBy (iii), $J^X_Y=\\bar J_Y$ follows, but we shall also give a direct proof. We have\n    \\begin{eqnarray*}\n    \\bar J_Y(x) &=&\\{\\bar S\\in\\D(\\bar\\down x):\\down\\bar S\\in J_Y(x)\\}\\\\\n    & = & \\{\\bar S\\in\\D(\\bar\\down x):Y\\cap\\down x\\subseteq\\down\\bar S\\}\n    \\end{eqnarray*}\n    for each $x\\in X$. Now let $x\\in X$ and assume that $Y\\cap\\down x\\subseteq \\down\\bar S$. Then we have $$Y\\cap\\bar\\down x=Y\\cap\\down x\\cap X\\subseteq\\down \\bar S\\cap X=\\bar S,$$\n    where we used Lemma \\ref{lem:inducedtopology} in the last equality. Hence $\\bar J_Y(x)\\subseteq J_Y^X(x)$. Conversely, let $\\bar S\\in J_Y^X(x)$. Since $Y\\subseteq X$, we find that $$Y\\cap\\down x=Y\\cap X\\cap\\down x=Y\\cap \\bar\\down x\\subseteq\\bar S\\subseteq\\down\\bar S,$$ so $\\down\\bar S\\in J_Y(x)$. Thus $\\bar S\\in\\bar J_Y(x)$.\n\\item[(vi)]\nAssume that $J_K$ is a subset Grothendieck topology on $\\P$, so there is a subset $Y$ of $\\P$ such that $J_K=J_Y$. By (i), we have $J_X\\leq J_Y$. By Proposition \\ref{prop:correspsubsetsandtopologies}(ii) this implies $Y\\subseteq X$. Thus by (iii) and (v), we find $$K=\\bar J_K=\\bar J_Y=J_Y^X.$$\nSo $K$ is a subset Grothendieck topology on $X$.\n\\item[(vii)] Let $J$ be a Grothendieck topology on $\\P$ such that $J_X\\leq J$. Then $\\bar J$ is well defined. Now, let $x\\in X$ and let $\\{\\bar S_i\\}_{i\\in I}\\subseteq\\bar J(x)$ be a collection of covers of $x$. Then for each $i\\in I$, there is some $S_i\\in J(x)$ such that $\\bar S_i=S_i\\cap X$. Since $J$ is complete, we have $\\bigcap_{i\\in I}S_i\\in J(x)$, hence $$\\bigcap_{i\\in I}\\bar S_i=\\bigcap_{i\\in I}(S_i\\cap X)=\\left(\\bigcap_{i\\in I}S_i\\right)\\cap X\\in\\bar J(x).$$ Now, let $K$ be a Grothendieck topology on $X$ such that $J_K$ is complete. By (i), we have $J_X\\leq J_K$, so by (iii) it follows that $K$ is complete.\n\\item[(viii)]Let $X$ be a subset of $\\P$. It was already shown in Theorem \\ref{thm:equivalenceArtinianandSubsettopologies} that $L_X$ is a well-defined Grothendieck topology on $\\P$.\n\nAnother method is to take $K=J^X_\\dense$ and to show that $J_K=L_X$. It follows then by (ii) that $L_X$ is well-defined. Moreover, by (i) and (iii) it follows that $J_X\\leq L_X$ and $\\bar L_X=J^X_\\dense$, respectively.\nFirst notice that $S\\in J_\\dense(p)$ if and only if $S\\cap\\down y\\neq\\emptyset$ for each $y\\in\\down p$. Similarly, we have $\\bar S\\in J_\\dense^X(x)$ if and only if $\\bar S\\cap\\bar\\down y\\neq\\emptyset$ for each $y\\in\\bar\\down x$. Now take $K=J_\\dense^X$. Then we find for each $p\\in\\P$:\n\\begin{eqnarray*}\n J_K(p) & = & \\{S\\in\\D(\\down p):\\forall x\\in\\bar\\down p(S\\cap\\bar\\down x\\in K(x))\\}\\\\\n& = & \\{S\\in\\D(\\down p):\\forall x\\in X\\cap\\down p(S\\cap X\\cap\\down x\\in J^X_\\dense(x))\\}\\\\\n& = & \\{S\\in\\D(\\down p):\\forall x\\in X\\cap\\down p(\\forall y\\in\\bar\\down x(S\\cap X\\cap\\down x\\cap\\bar\\down y\\neq\\emptyset))\\}\\\\\n& = & \\{S\\in\\D(\\down p):\\forall x\\in X\\cap\\down p(\\forall y\\in X\\cap\\down x(S\\cap X\\cap\\down x\\cap X\\cap\\down y\\neq\\emptyset))\\}\\\\\n& = & \\{S\\in\\D(\\down p):\\forall x\\in X\\cap\\down p(\\forall y\\in X\\cap\\down x(S\\cap X\\cap\\down y\\neq\\emptyset))\\}\\\\\n& = & \\{S\\in\\D(\\down p):\\forall x\\in X\\cap\\down p(S\\cap X\\cap\\down x\\neq\\emptyset)\\}\\\\\n& = & L_X(p).\n\\end{eqnarray*}\n\\end{enumerate}\n\\end{proof}\n\nSo we see that $L_X$ is indeed the extension of the dense Grothendieck topology on $X$. Moreover, (vi) of the Proposition shows that $L_X$ is not a subset Grothendieck topology if $J_dense^X$ is not a subset Grothendieck topology.\n\n\n\n\n\\section{Sheaves and morphisms of sites}\nIn this section we want to explore several notions of the equivalence of sites. A possibility is to say that $(\\P,J)$ and $(\\QQ,K)$ are equivalent if there exists an equivalence between their categories of sites. However, it might be more natural to explore notions of morphisms of sites, so we postpone the discussion of equivalence of sites in terms of sheaves.\n\n\\begin{definition}\\cite[Section VII.10]{M&M}\n Let $(\\P,J)$ and $(\\QQ,K)$ be sites. An order morphism $\\phi:\\P\\to\\QQ$ \\emph{preserves covers}\\index{cover preserving order morphism}\\index{clp} if $\\down\\phi[S]\\in K(\\phi(p))$ for each $p\\in\\P$ and each $S\\in J(p)$. An order morphism $\\pi:\\QQ\\to\\P$ has the \\emph{covering lifting property}\\index{covering lifting property} (abbreviated by ``clp'') if for each $q\\in\\QQ$ and each $S\\in J(\\pi(q))$ there is an $R\\in K(q)$ such that $\\pi[R]\\subseteq S$.\n\\end{definition}\n\n\n\\begin{proposition}\n Let $\\phi:(\\P_1,J_1)\\to(\\P_2,J_2)$ and $\\pi:(\\P_2,J_2)\\to(\\P_3,J_3)$ be order morphisms. Then $\\pi\\circ\\phi$ has the clp if $\\pi$ and $\\phi$ have the clp. Moreover, if $\\pi$ and $\\phi$ preserve covers, then $\\pi\\circ\\phi$ preserves covers.\n\\end{proposition}\n\\begin{proof}\n Let $p\\in\\P_1$ and $S_3\\in J_3(\\pi\\circ\\phi(p))$. Since $\\pi$ has the clp, there must be an $S_2\\in J_2(\\phi(p))$ such that $\\pi[S_2]\\subseteq S_3$. Since $\\phi$ has the clp, there must be an $S_1\\in J_1(p)$ such that $\\phi[S_1]\\subseteq S_2$, hence $\\pi\\circ\\phi[S_1]\\subseteq\\pi[S_2]$. Combining both inclusions, we obtain $\\pi\\circ\\phi[S_1]\\subseteq S_3$, hence $\\pi\\circ\\phi$ must have the clp.\n\nNow assume that $\\pi$ and $\\phi$ preserve covers and let $p\\in\\P$ and $S\\in J_1(p)$. Then $\\down\\phi[S]\\in J_2(\\phi(p))$, since $\\phi$ preserves covers, hence $\\down\\pi[\\down\\phi[S]]\\in J_3(\\pi\\circ\\phi(p))$ for $\\pi$ preserves covers. However, in order to show that $\\pi\\circ\\phi$ preserves covers, we have to show that $\\down\\pi\\circ\\phi[S]\\in J_3(\\pi\\circ\\phi(p))$. Let $x\\in\\down\\pi[\\down\\phi[S]]$. Then there is a $y\\in\\down\\phi[S]$ such that $x\\leq\\pi(y)$. Moreover, there must be an $s\\in S$ such that $y\\leq\\phi(s)$. Since $\\pi$ is an order morphism, we find $\\pi(y)\\leq\\pi\\circ\\phi(s)$, so $x\\leq\\pi\\circ\\phi(s)$. We conclude that $\\down\\pi[\\down\\phi[S]]\\subseteq\\down\\pi\\circ\\phi[S]$, hence by Lemma \\ref{lem:filter} it follows that $\\down\\pi\\circ\\phi[S]\\in J_3(\\pi\\circ\\phi(p))$.\n\\end{proof}\n\n\n\nIn order to define a correct and suitable notion of a morphism of sites, it seems like we have to choose between the cover preserving property and the clp. However, both notions are related to each other as the following lemma shows.\n\\begin{lemma}\\cite[Lemma VII.10.3]{M&M}\n Let $(\\P,J)$ and $(\\QQ,K)$ be sites and $\\phi:\\P\\to\\QQ$ the upper adjoint of $\\pi:\\QQ\\to\\P$. Then $\\phi$ preserves covers if and only if $\\pi$ has the clp.\n\\end{lemma}\n\\begin{proof}\n Assume that $\\phi$ preserves covers. Let $q\\in\\QQ$ and $S\\in J(\\pi(q))$. Let $p=\\pi(q)$, so $S\\in J(p)$. Then $\\pi(q)\\leq p$, so using the adjunction, we obtain $q\\leq\\phi(p)$. Since $\\phi$ preserves covers, we have $\\down\\phi[S]\\in K(\\phi(p)$, and since $q\\leq\\phi(p)$, it follows by stability of $K$ that $R=\\down\\phi[S]\\cap\\down q\\in K(q)$. Let $x\\in R$, then $\\pi(x)\\in\\pi[R]$. Moreover, we have $R\\subseteq\\down\\phi[S]$, so there is a $y\\in S$ such that $x\\leq\\phi(y)$. By the adjunction, we obtain $\\pi(x)\\leq y$ and since $S$ is a down-set, we find $\\pi(x)\\in S$. Thus $\\pi[R]\\subseteq S$, which shows that $\\pi$ has the clp.\n\nConversely, assume that $\\pi$ has the clp and let $p\\in\\P$ and $S\\in J(p)$. Let $q=\\phi(p)$, then $q\\leq\\phi(p)$, so using the adjunction, we find $\\pi(q)\\leq p$. By stability of $J$, it follows that $S\\cap\\down\\pi(q)\\in J(\\pi(q)$, and since $\\pi$ has the clp, we obtain an $R\\in K(q)$ such that $\\pi[R]\\subseteq S\\cap\\down\\pi(q)$. Let $x\\in R$. Then $s=\\pi(x)\\in S$, so $\\pi(x)\\leq s$. Using the adjunction we obtain $x\\leq\\phi(s)$, whence $x\\in\\down\\phi[S]$, so $R\\subseteq\\down\\phi[S]$. By Lemma \\ref{lem:filter} it follows that $\\down\\phi[S]\\in J(q)$ if we can show that $\\down\\phi[S]\\in\\D(\\down q)$. Thus we have to show that $q$ is an upper bound of $\\down\\phi[S]$. Let $y\\in\\down\\psi[S]$. Then there is an $s\\in S$ such that $y\\leq\\phi(s)$. Using the adjunction, we find $\\pi(y)\\leq s\\leq p$, so using the adjunction again yields $y\\leq\\phi(p)=q$. We conclude that $\\down\\phi[S]\\in K(\\phi(p))$, hence $\\phi$ preserves covers.\n\\end{proof}\n\nIf we assume that we have a definition of site morphism, $\\phi:(\\P,J)\\to(\\QQ,K)$ is a site isomorphism if it has an inverse $\\pi:(\\QQ,K)\\to(\\P,J)$. On the poset part of the site, this means that $\\phi$ and $\\pi$ are order isomorphisms, which are each others inverses. Since this implies that $\\phi$ is both the upper adjoint and lower adjoint of $\\pi$, the preceding lemma implies that it does not matter whether we define an order morphism $\\phi:(\\P,J)\\to(\\QQ,K)$ to be a site morphism if it preserves covers of if it has the clp. Independent of which choice we make, we obtain the same notion of site isomorphisms, which is in the end the notion in which we are interested.\n\n\\begin{definition}\n Let $(\\P,J)$ and $(\\QQ,K)$ be sites. Then an order isomorphism $\\phi:\\P\\to\\QQ$ is called an \\emph{isomorphism of sites}\\index{isomorphism} if it satisfies one of the following equivalent conditions:\n\\begin{enumerate}\n \\item $\\phi$ preserves covers and has the clp;\n \\item $\\phi$ and $\\phi^{-1}$ both preserve covers;\n \\item $\\phi$ and $\\phi^{-1}$ both have the clp.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{lemma}\\label{lem:orderisopreservessieves}\n Let $\\phi:\\P\\to\\QQ$ be an order isomorphism. Let $p\\in\\P$. Then $S\\in\\D(\\down p)$ implies $\\phi[S]\\in\\D(\\down\\phi(p))$. Moreover, $\\down\\phi(p)=\\phi[\\down p]$.\n\\end{lemma}\n\\begin{proof}\n For each $s\\in S\\in\\D(\\down p)$ we have $s\\leq p$, so $\\phi(s)\\leq\\phi(p)$. If $x\\in\\phi[S]$ and $y\\leq x$, then $\\phi^{-1}(x)\\in S$ and $\\phi^{-1}(y)\\leq\\phi^{-1}(x)$, so $\\phi^{-1}(y)\\in S$. Hence $y\\in\\phi[S]$, and we conclude that $\\phi[S]\\in\\D(\\down\\phi(p))$.\n\nSince $\\down p\\in\\D(\\down p)$, we immediately obtain $\\phi[\\down p]\\in\\D(\\down\\phi(p))$, hence $\\phi[\\down p]=\\down\\phi[\\down p]$. Now $q\\in\\down\\phi[\\down p]$ if and only if $q\\leq\\phi(p')$ for some $p'\\leq p$ if and only if $q\\leq\\phi(p)$, so $\\down\\phi[\\down p]=\\down\\phi(p)$.\n\\end{proof}\n\n\\begin{proposition}\n Let $(\\P,J)$ and $(\\QQ,K)$ be sites. Then an order isomorphism $\\phi:\\P\\to\\QQ$ is an isomorphism of sites $(\\P,J)\\to(\\QQ,K)$ if and only if for each $p\\in\\P$ and $S\\in\\PP(\\P)$, we have $S\\in J(p)$ if and only if $\\phi[S]\\in K(\\phi(p))$.\n\\end{proposition}\n\\begin{proof}\n Assume that $\\phi$ is an isomorphism of sites and let $p\\in\\P$ and $S\\in J(p)$. Since $\\phi$ preserves covers, we have $\\down\\phi[S]\\in K(\\phi(p))$. By the preceding lemma, we have $\\phi[S]\\in\\D(\\down\\phi(p))$, so $\\down\\phi[S]=\\phi[S]$. Hence $\\phi[S]\\in K(\\phi(p))$. Conversely, if $S\\in\\PP(\\P)$ such that $\\phi[S]\\in K(\\phi(p))$, then the same argument for $\\phi^{-1}$ instead of $\\phi$ yields $\\phi^{-1}[\\phi[S]]\\in J(\\phi^{-1}\\circ\\phi(p))$, so $S\\in J(p)$.\n\nNow assume that for each $p\\in\\P$, we have $S\\in J(p)$ if and only if $\\phi[S]\\in K(\\phi(p))$. Let $p\\in\\P$ and $S\\in J(p)$. Then $\\phi[S]\\in K(\\phi(p))$, so certainly $\\down\\phi[S]\\in K(\\phi(p))$. Let $q\\in\\QQ$ and $R\\in K(q)$. By Lemma \\ref{lem:orderisopreservessieves} we find $\\phi^{-1}[R]\\in\\D(\\down\\phi^{-1}(q))$. Since $\\phi[\\phi^{-1}[R]]=R$, we find that $\\phi[\\phi^{-1}[R]]\\in K(\\phi\\circ\\phi^{-1}(q))$, so $\\phi^{-1}[R]\\in J(\\phi^{-1}(q))$. We conclude that both $\\phi$ and $\\phi^{-1}$ preserve covers, so $\\phi$ is an isomorphism of sites.\n\\end{proof}\n\n\n\\begin{corollary}\n Let $\\P$ and $\\QQ$ posets, and $X\\subseteq\\P$ and $Y\\subseteq\\QQ$ subsets. Then an order isomorphism $\\phi:\\P\\to\\QQ$ is an isomorphism of sites $(\\P,J_X)\\to(\\QQ,J_Y)$ if and only if $\\phi[X]=Y$.\n\\end{corollary}\n\\begin{proof}\n Assume that $\\phi:(\\P,J_X)\\to(\\QQ,J_Y)$ is an isomorphism of sites and let $x\\in X$. Then $J(x)=\\{\\down x\\}$, so $J_Y(\\phi(x))=\\{\\phi[\\down x]\\}$ for $\\phi$ is an isomorphism of sites. Now $J_Y(\\phi(x))$ contains only one element if and only if $\\phi(x)\\in Y$, hence $\\phi[X]\\subseteq Y$. Replacing $\\phi$ by $\\phi^{-1}$ gives $\\phi^{-1}[Y]\\subseteq X$, hence $\\phi[X]=Y$.\n\nConversely assume that $\\phi[X]=Y$. Let $p\\in\\P$ and $S\\in J_X(p)$. Then $X\\cap\\down p\\subseteq S$, hence $\\phi[X]\\cap\\phi[\\down p]\\subseteq\\phi[S]$. By Lemma \\ref{lem:orderisopreservessieves}, $\\phi[S]\\in\\D(\\down\\phi(p))$ and $\\phi[\\down p]=\\down\\phi(p)$. Moreover, $\\phi[X]=Y$, hence $Y\\cap\\down\\phi(p)\\subseteq\\phi[S]$, and we conclude that $\\phi[S]\\in J_Y(\\phi(p))$. Since $\\phi$ is an order isomorphism, we have $\\phi^{-1}[Y]=X$. Hence applying the same arguments to $\\phi^{-1}$ gives the implication $\\phi[S]\\in J_Y(\\phi(p))$ implies $S\\in J_X(p)$. We conclude that $\\phi$ is an isomorphism of sites.\n\\end{proof}\n\nIf $(\\P,J_X)$ and $(\\QQ,J_Y)$ are sites, it might be interesting as well to examine how to express the cover preserving property and the clp of an order morphism $\\phi:\\P\\to\\QQ$ in terms of $\\phi$, $X$ and $Y$.\n\n\\begin{proposition}\n Let $\\P$ and $\\QQ$ posets and $X$ and $Y$ subsets of $\\P$ and $\\QQ$, respectively. Let $\\phi:\\P\\to\\QQ$ an order morphism. Then $\\phi:(\\P,J_X)\\to(\\QQ,J_Y)$ has the clp if and only if $\\phi[X]\\subseteq Y$.\n\\end{proposition}\n\\begin{proof}\nAssume that $\\phi$ has the clp and let $x\\in X$. Since $\\phi$ has the clp and $\\down(Y\\cap\\down\\phi(x))\\in J_Y(\\phi(x))$, there must be an $R\\in J_X(x)$ such that $\\phi[R]\\subseteq\\down(Y\\cap\\down\\phi(x))$. Since $x\\in X$, it follows that $J_X(x)$ contains only $\\down x$, hence $\\phi[\\down x]\\subseteq\\down(Y\\cap\\down\\phi(x))$. In particular, we must have $\\phi(x)\\in\\down (Y\\cap\\down\\phi(x))$. Now assume that $\\phi(x)\\notin Y$. Then for each $y\\in Y\\cap\\down\\phi(x)$ we have $y<\\phi(x)$. If $z\\in\\down(Y\\cap\\down\\phi(x))$, we must have $z\\leq y$ for some $y\\in Y\\cap\\down\\phi(x)$, so $z<\\phi(x)$ for each $z\\in \\down(Y\\cap\\down\\phi(x)$. Thus the choice $z=\\phi(x)$ gives a contradiction, so we must have $\\phi(x)\\in Y$. We conclude that $\\phi[X]\\subseteq Y$.\n\nAssume that $\\phi[X]\\subseteq Y$ and let $p\\in\\P$ and $S\\in J_Y(\\phi(p))$. Then $Y\\cap\\down\\phi(p)\\subseteq S$, so $\\phi[X]\\cap\\down \\phi(p)\\subseteq S$. Moreover, since $S$ is a down-set, we obtain $\\down(\\phi[X]\\cap\\down\\phi(p))\\subseteq S$. Let $R=\\down (X\\cap\\down p)$, then $R\\in J_X(p)$. Let $y\\in R$. Then there is an $x\\in X$ such that $y\\leq x\\leq p$. Since $\\phi$ is an order morphism, we find $\\phi(y)\\leq\\phi(x)\\leq\\phi(p)$. So $\\phi(x)\\in \\phi[X]\\cap\\down\\phi(p)$, whence $\\phi(y)\\in\\down(\\phi[X]\\cap\\down\\phi(p))$. Thus we find that $\\phi(y)\\in S$, hence $\\phi[R]\\subseteq S$, and we conclude that $\\phi$ has the clp.\n\\end{proof}\n\nFor $\\phi:(\\P,J_X)\\to(\\QQ,J_Y)$ preserving covers, we can state a similar statement, although we have add bijectivity of $\\phi$ as extra condition.\n\n\\begin{proposition}\nLet $\\P$ and $\\QQ$ posets and $X$ and $Y$ subsets of $\\P$ and $\\QQ$, respectively. Let $\\phi:\\P\\to\\QQ$ an order isomorphism. Then $\\phi:(\\P,J_X)\\to(\\QQ,J_Y)$ preserves covers if and only if $Y\\subseteq\\phi[X]$.\n\\end{proposition}\n\\begin{proof}\n Assume that $\\phi$ preserves covers. Then $\\down\\phi[S]\\in J_Y(\\phi(p))$ for each $p\\in\\P$ and $S\\in J_X(p)$. Notice that Lemma \\ref{lem:orderisopreservessieves} assures that $\\down\\phi[S]=\\phi[S]$ for $\\phi$ is assumed to be an order isomorphism. Since $\\down(X\\cap\\down p)$ is the least element of $J_X(p)$, Lemma \\ref{lem:filter} assures that $\\phi[S]\\in J_Y(\\phi(p))$ for each $S\\in J_X(p)$ if and only if $\\phi[\\down(X\\cap\\down p))]\\in J_Y(\\phi(p))$, we find that $\\phi$ preserves covers if and only if for\n\\begin{equation}\\label{eq:coveringpreservingforsubsettopologies}\nY\\cap\\down\\phi(p)\\subseteq\\phi[\\down(X\\cap\\down p)]\n\\end{equation} for each $p\\in\\P$.\n\nLet $y\\in Y$. By the surjectivity of $\\phi$ there is some $p\\in\\P$ such that $\\phi(p)=y$. Since $\\phi$ preserves covers, we find that $y=\\phi(p)\\in\\phi[\\down(X\\cap\\down p)]$. Assume that $p\\notin X$. Then for each $x\\in X\\cap\\down p$, we must have $x<p$. Then if $p'\\in\\down(X\\cap\\down p)$, there must be an $x\\in X\\cap\\down p$ such that $p'\\leq x$, hence $p'<p$. Then $\\phi(p')<\\phi(p)$ for $\\phi$ is an order isomorphism, so $z<\\phi(p)$ for each $z\\in\\phi[\\down(X\\cap\\down p)$. Since the choice $z=y$ gives a contradiction, we must have $\\phi(p)\\in X$. We conclude that $Y\\subseteq\\phi[X]$.\n\nConversely, assume that $Y\\subseteq\\phi[X]$ and let $p\\in\\P$. We have to show that (\\ref{eq:coveringpreservingforsubsettopologies}) holds, so let $y\\in Y\\cap\\down\\phi(p)$. Since $Y\\subseteq\\phi[X]$, there is some $x\\in X$ such that $\\phi(x)=y$. Hence $\\phi(x)\\leq\\phi(p)$, whence $x\\leq p$ for $\\phi$ is an order isomorphism. Hence $y=\\phi(x)$ with $x\\in X\\cap\\down p$, which shows that (\\ref{eq:coveringpreservingforsubsettopologies}) indeed holds.\n\\end{proof}\n\n\n\n\n\n\n\n\n\nIf $\\P$ is a poset and $X\\subseteq\\P$ a subset, it might be interesting to find a description of all $J_X$-sheaves.\n\\begin{definition}\\cite[Chapter III.4]{M&M}\nLet $\\P$ be a poset and $J$ a Grothendieck topology on $\\P$.\nFurthermore, let $F:\\P^\\op\\to\\Sets$ be a functor (a contravariant functor to $\\Sets$ is also called a \\emph{presheaf}\\index{presheaf}). Let $p\\in\\P$ and $S\\in J(p)$. Then a family $\\langle a_x\\rangle_{x\\in S}\\in\\prod_{x\\in S}F(x)$ is called a \\emph{matching family}\\index{matching family} for the cover $S$ with elements of $F$ if \\begin{equation}\\label{matchfam}\n F(y\\leq x)a_x=a_y\n\\end{equation}\nfor each $x,y\\in S$ such that $y\\leq x$. An element $a\\in F(p)$ such that $F(x\\leq p)a=a_x$ for each $x\\in S$ is called an \\emph{amalgamation}\\index{amalgamation}. We say that $F$ is a $J$-\\emph{sheaf}\\index{sheaf} if for each $p\\in\\P$ for each $S\\in J(p)$ and for each matching family $\\langle a_x\\rangle_{x\\in S}$ there is a unique\n\\emph{amalgamation}\\index{amalgamation} $a\\in F(p)$.\n\\end{definition}\n\n\n\n\n\\begin{example}\\label{ex:indiscretetopologysheaves}\n Let $J$ be a topology on a poset $\\P$. If we consider the indiscrete topology given by $J_{\\mathrm{ind}}(p)=\\{\\down p\\}$ on $\\P$, we have $\\Sh(\\P,J_{\\mathrm{ind}})=\\Sets^{\\P^\\op}$. Indeed, let $F\\in\\Sets^{\\P^\\op}$ and $p\\in\\P$. Then there is only one cover of $p$, namely $\\down p$. Hence if $\\langle a_q\\rangle_{q\\in\\down p}$ is a matching family for $\\down p$, we have $p\\in\\down p$, so $a_q=F(q\\leq p)a_p$. Thus $a_p$ is an amalgamation of the matching family. It is also unique, since if $a\\in F(p)$ is another amalgamation, we have $a_p=F(p\\leq p)a=a$.\n\\end{example}\n\n\n\\begin{example}\\label{ex:discretetopologysheaves}\n Let $J$ be a topology on a poset $\\P$. Given a $J$-sheaf $F$ and a point $p\\in\\P$ such that $\\emptyset\\in J(p)$. Then a matching family for $\\emptyset$ must be a function on the empty set. There exists only one function with the empty set as domain, and it is clearly a matching family. An amalgamation must be a point in $F(p)$, but since there are no restrictions on this point except that it must be unique, it follows that $F(p)$ is a singleton set $1$. In particular, if $J$ is the discrete topology, then $\\Sh(\\P,J)=1$, since $\\emptyset\\in J(p)$ for each $p\\in\\P$.\n\\end{example}\n\n\n\\begin{example}\\label{ex:atomsheaves}\n Let $\\P$ be a poset and let $J$ be a topology on $\\P$. Let $F$ be a $J$-sheaf on $\\P$ and let $q\\leq p$ be an element such that $\\down q\\in J(p)$. Notice that $F(q)$ is non-empty if $F(p)$ is non-empty, since we have a map $F(q\\leq p):F(p)\\to F(q)$. Then given any $b\\in F(q)$, we can define $a_r:=F(r\\leq q)b$ for each $r\\in\\down q$, which yields a matching family for $\\down q\\in J(p)$. Since $F$ is a sheaf, we find that there is a unique $a\\in F(p)$ such that $F(r\\leq p)a=a_r$. This shows not only that $F(p)$ must be non-empty if $F(q)$ is non-empty, but also that $a$ is the unique element such that $F(q\\leq p)a=b$, so $F(q\\leq p)$ is a bijection.\n\nNow assume that $\\P$ is downwards directed consider the atomic topology $J_\\atom$. Then $\\down q\\in J_\\atom(p)$ for each $q\\leq p$. Let $p_1,p_2\\in\\P$. Since $\\P$ is directed, we find that there is a $q\\in\\P$ such that $p_1,p_2\\geq q$. From the preceding, we find that if $F$ is a $J_\\atom$-sheaf, $F(q\\leq p_1)$ and $F(q\\leq p_2)$ are bijections, so $F(q\\leq p_2)^{-1}\\circ F(q\\leq p_1)$ is a bijection from $F(p_1)$ to $F(p_2)$. Thus we find that all sheaves on $\\P$ are constant up to isomorphism. Hence, $\\Sh(\\P,J_\\atom)\\cong\\Sets$.\n\\end{example}\n\nThe indiscrete topology is the coarsest topology, but from the first example, we see that all presheaves are sheaves with respect to this topology. So $\\Sh(\\P,J_{\\mathrm{ind}})=\\Sets^{\\P^\\op}$. On the other hand, from the second example we see that only one presheaf is a sheaf with respect to the discrete topology, which is the finest topology possible. So $\\Sh(\\P,J_{\\mathrm{dis}})\\cong 1\\cong\\Sets^{\\emptyset^\\op}$ if we consider the empty set as a subposet of $\\P$.\nIn the third example, we consider the atomic topology, which lies between the indiscrete and the discrete topology if we order Grothendieck topologies from coarse to fine. Recall that the atomic topology is only defined for downwards directed posets, but if $\\P$ is Artinian, this is certainly the case. Then the atomic topology is induced by the least element $0$, and we see that $\\Sh(\\P,J_\\atom)\\cong\\Sets\\cong\\Sets^{\\{0\\}^\\op}$. In all these cases, the topology in question is induced by some subset $X$ of $\\P$, and the category of sheaves with respect to this topology is equivalent to $\\Sets^{X^\\op}$.\n\nIt turns out to be a direct consequence of the so-called \\emph{Comparison Lemma}\\index{Comparison Lemma} that this is true for each subset $X$ of $\\P$. The idea behind the Comparison Lemma is the following. Given a subset $X$ of a poset $\\P$, the inclusion map $I:X\\to\\P$ induces a map $I^*:\\Sets^{\\P^\\op}\\to\\Sets^{X^\\op}$ given by $F\\mapsto F\\circ I$. If $J$ is a Grothendieck topology on $\\P$, one could ask about the image of $\\Sh(\\P,J)$ under $I^*$. The Comparison Lemma gives a sufficient condition on $X$ for the existence of a Grothendieck topology $\\bar J$ on $X$ such that $I^*$ provides an equivalence $\\Sh(X,\\bar J)\\cong\\Sh(\\P,J)$, where we consider $X$ a subposet of $\\P$, so if $x,y\\in X$ and $x\\leq y$ in $\\P$, then $x\\leq y$ in $X$. In categorical terms, this says that $X$ is a full subcategory of $\\P$. We shall state the Comparison Lemma restricted to the posetal case as Theorem \\ref{thm:ComparisonLemma} below. The proof we present is based on the proof of \\cite[Theorem C.2.2.3]{Elephant2}, where the Comparison Lemma is stated for arbitrary categories, although we do not explicitely make use of Kan extensions. Another proof of the Comparison Lemma for arbitrary categories can be found in \\cite{K&M}. In these references, the Lemma is also stated in a sharper version, i.e., the fullness condition is dropped. The condition that we will put on $X$ is called $J$-\\emph{denseness}\\index{$J$-dense}, which says that for each $p\\in\\P$ one should have $\\down(X\\cap\\down p)\\in J(p)$. Since $\\down(X\\cap\\down p)$ is the smallest cover in $J_X(p)$, we find by Lemma \\ref{lem:filter} that $X$ is $J$-dense if and only if $J_X\\subseteq J$.\n\nLet $X\\subseteq\\P$ and $I:X\\embeds\\P$ be the inclusion. Then we write $\\bar F=I^*(F)=F\\circ I$, if we want to stress when we restrict the domain of $F$ to $X$.\n\\begin{lemma}\n Let $J$ be a Grothendieck topology on a poset $\\P$ and let $X\\subseteq\\P$ be a subset such that $J_X\\leq J$. For each $x\\in X$, let $\\bar J$ be the Grothendieck topology on $X$ defined in Lemma \\ref{lem:inducedtopology}. Then the functor $I^*:\\Sets^{\\P^\\op}\\to\\Sets^{X^\\op}$ restricts to a functor $\\Sh(\\P,J)\\to\\Sh(X,\\bar J)$, which we will denote by $I^*$ as well.\n\\end{lemma}\n\\begin{proof}\n  Let $F\\in\\Sh(\\P,J)$ and $\\bar F=F\\circ I$ with $I:X\\embeds\\P$ the inclusion. Let $x\\in X$ and $\\bar S\\in\\bar J(x)$. Let $\\langle a_y\\rangle_{y\\in\\bar S}$ be a matching family for $\\bar S$ of elements of $\\bar F$. By Lemma \\ref{lem:inducedtopology}, $S=\\down\\bar S\\in J(x)$. Thus we are going to extend the matching family $\\langle a_y\\rangle_{y\\in\\bar S}$ to a matching family for $S$ of elements of $F$, as follows. If $z\\in S$, there must be at least one $y\\in\\bar S$ such that $z\\leq y$, hence we define $a_z=F(z\\leq y)a_y$. Now, if there is another $y'\\in\\bar S$ such that $z\\leq y'$, we should have $a_z=F(z\\leq y')a_{y'}$. Call the right-hand side $b_z$. Since $$R=\\down(X\\cap\\down z)\\in J(z),$$ we have matching families $\\langle F(w\\leq z)a_z\\rangle_{w\\in R}$ and $\\langle F(w\\leq z)b_z\\rangle_{w\\in R}$ for $R$ of elements of $F$. Now, the matching families must be equal, since if $u\\in\\down(X\\cap\\down z)$, there is a $v\\in X\\cap\\down z$ such that $u\\leq v\\leq z$. Notice that also $v\\in\\bar S$, since $v\\in X\\cap z$, and $z\\leq y$ with $y\\in\\bar S\\in\\D(X)$. Then $F(v\\leq y)a_y=F(v\\leq y')a_{y'}$, since $v,y,y'\\in\\bar S$. So indeed\n\\begin{eqnarray*}\n F(u\\leq z)a_z & = & F(u\\leq v)F(v\\leq z)a_z=F(u\\leq v)F(v\\leq z)F(z\\leq y)a_y\\\\\n & = & F(u\\leq v)F(v\\leq y)a_y=F(u\\leq v)F(v\\leq y')a_{y'}\\\\\n& = & F(u\\leq v)F(v\\leq z)F(z\\leq y')a_{y'}=F(u\\leq z)b_z.\n\\end{eqnarray*}\nHence both matching families must have the same amalgamation, for $F$ is a $J$-sheaf, but since both $a_z$ and $b_z$ are amalgamations, we find $a_z=b_z$. Hence $\\langle a_z\\rangle_{z\\in\\down\\bar S}$ is a matching family for $\\down\\bar S\\in J(x)$, so again since $F$ is a $J$-sheaf, it has a unique amalgamation $a\\in F(x)$ such that $F(z\\leq x)a=a_z$ for each $z\\in\\down\\bar S$, so in particular for $z\\in\\bar S$. We conclude that $a$ is an amalgamation of $\\langle a_y\\rangle_{y\\in\\bar S}$. Now assume $b$ is another amalgamation. Then $a_y=F(y\\leq x)b$ for each $y\\in\\bar S$, and if $z\\in\\down\\bar S$, we have $$F(z\\leq x)b=F(z\\leq y)F(y\\leq x)b=a_y$$ for some $y\\in\\bar S$ such that $z\\leq y$. So $b$ is also an amalgamation for $\\langle a_z\\rangle_{z\\in\\down\\bar S}$, whence $b=a$.\n\\end{proof}\n\n\n\nThe functor $E:\\Sets^{X^\\op}\\to\\Sets^{\\P^\\op}$ in the opposite direction is defined on objects by sending $F\\mapsto\\hat F$, where for each $p\\in\\P$, $\\hat F(p)$ is defined as the collection of all functions $f:X\\cap\\down p\\to\\bigcup_{x\\in X}F(x)$ such that $f(x)\\in F(x)$ for each $x\\in X\\cap\\down p$, and\n\\begin{equation}\\label{eq:elFhat}\nF(y\\leq x)f(x)=f(y)\n\\end{equation}\n for each $x,y\\in X\\cap\\down p$ such that $y\\leq x$. The action of $\\hat F$ on morphisms $q\\leq p$ in\n$\\P$ is defined by\n\\begin{equation}\\label{eq:hatFmor}\n\\hat F(q\\leq p)(f)=\\left.f\\right|_{X\\cap\\down q},\n\\end{equation}\nwhere $f\\in \\hat F(p)$. It is immediate that $\\hat F(q\\leq\np)f\\in\\hat F(q)$ and that $\\hat F$ is a functor.\n\nIf $\\alpha:F\\to G$ is a natural transformation between\n$F,G\\in\\Sets^{X^\\op}$, we define the action of $E$ on $\\alpha$, denoted by\n$\\hat\\alpha:\\hat F\\to\\hat G$, by\n\\begin{equation}\\label{eq:hatalph}\n\\hat\\alpha_p(f)(x)=\\alpha_x\\big(f(x)\\big),\n\\end{equation}\nwhere $p\\in\\P$, $f\\in\\hat F(p)$ and $x\\in X\\cap\\down p$.\n\n\nThis action on morphisms is well defined. For this, we need the naturality of $\\alpha$:\n\\begin{equation*}\n\\xymatrix{F(x)\\ar[rr]^{\\alpha_x}\\ar[dd]_{F(y\\leq x)} && G(x)\\ar[dd]^{G(y\\leq x)}\\\\\n\\\\\nF(y)\\ar[rr]_{\\alpha_y} && G(y)}\n\\end{equation*}\nWe have to show that $\\hat\\alpha_p(f)\\in\\hat G(p)$. So let $y\\leq x$ in $X\\cap\\down p$. Then\n\\begin{align*}\n G(y\\leq x)\\hat\\alpha_p(f)(x) &  =G(y\\leq x)\\alpha_x\\big(f(x)\\big) =\\alpha_y\\big(F(y\\leq x)f(x)\\big)\\\\\n & =\\alpha_y\\big(f(y)\\big) =\\hat\\alpha_p(f)(y).\n\\end{align*}\nFurthermore, we have to show that $\\hat\\alpha$ is a natural transformation between $\\hat F$ and $\\hat G$. That is,\n\\begin{equation*}\n\\xymatrix{\\hat F(p)\\ar[rr]^{\\hat \\alpha_p}\\ar[dd]_{\\hat F(q\\leq p)} && \\hat G(p)\\ar[dd]^{\\hat G(q\\leq p)}\\\\\n\\\\\n\\hat F(q)\\ar[rr]_{\\hat \\alpha_q} && \\hat G(q)}\n\\end{equation*}\ncommutes. Notice that given a function $f$ with domain $X\\cap\\down p$, both $\\hat G(q\\leq p)\\circ\\hat\\alpha_p(f)$ and $\\hat\\alpha_q\\circ\\hat F(q\\leq p)(f)$ are functions with domain $X\\cap\\down q$. Hence for arbitrary $x\\in X\\cap\\down q$ we find\n\\begin{align*}\n\\hat G(q\\leq p)\\circ\\hat\\alpha_p(f)(x) & = \\left.\\hat\\alpha_p(f)\\right|_{X\\cap\\down q}(x) = \\hat\\alpha_p(f)(x) = \\alpha_x(f(x))\\\\\n& = \\alpha_x\\left(\\left.f\\right|_{X\\cap\\down q}(x)\\right) =  \\hat\\alpha_q\\left(\\left.f\\right|_{X\\cap\\down q}\\right)(x)\\\\\n&  = \\hat\\alpha_q\\circ\\hat F(q\\leq p)(f)(x),\n\\end{align*}\nwhere we used (\\ref{eq:hatFmor}) in the first and the last equality, and used (\\ref{eq:hatalph}) in the third and fifth equality.\nSo the\ndiagram indeed commutes.\n\n\n\\begin{lemma}\nLet $J$ be a Grothendieck topology on $\\P$ such that $J_X\\leq J$. Then $E$ restricts to a functor $\\Sh(X,\\bar J)\\to \\Sh(\\P,J)$.\n\\end{lemma}\n\\begin{proof}\nWe have to show that $\\hat F\\in\\Sh(\\P,J)$ for each $F\\in\\Sh(X,\\bar J)$.\n So let $\\langle\nf_q\\rangle_{q\\in S}$ be a matching family for $S\\in J(p)$. Then by definition of $f_q\\in\\hat F(q)$, $f_q$ is a function $X\\cap\\down\nq\\to\\bigcup_{x\\in X}F(x)$ with $f_q(x)\\in F(x)$ for each $x\\in X\\cap\\down q$, and such that (\\ref{eq:elFhat}) holds if we substitute $f_q$ for $f$ and $q$ for $p$.\nFurthermore, the condition that $\\langle f_q\\rangle_{q\\in S}$ is a matching family for\n$S$ of elements of $\\hat F$ translates to $\\hat F(r\\leq q)f_q=f_r$. By (\\ref{eq:hatFmor}), it follows that for each $q\\in S$, $r\\leq q$ and $x\\in X\\cap\\down r$, we have\n\\begin{align}\\label{eq:mfforhatF}\nf_q(x)=f_r(x).\n\\end{align}\n\nWe have to find a unique amalgamation $f$ for $\\langle f_q\\rangle_{q\\in S}$. This means that we have to construct a function $f\\in\\hat F(p)$ such that $\\hat F(q\\leq p)f=f_q$ for each $q\\in S$, and if there is another function $g\\in\\hat F(p)$ such that\n\\begin{equation}\\label{eq:uniquenessf}\n\\hat F(q\\leq p)g=f_q,\n\\end{equation}\nfor each $q\\in S$, we must have $f=g$.\n\nLet $x\\in X\\cap\\down p$. We would like to define $f(x)=f_x(x)$, but since it might be possible that $x\\notin S$ and $f_x$ is only given for $x\\in S$, we cannot do so. However, it is still possible to construct $f$. First notice that if $x\\in S$, $f_x\\in \\hat F(x)$ is defined such that $f_x(y)\\in F(y)$ for each $y\\in X\\cap\\down x$, so $f_x(x)\\in F(x)$ for each $x\\in S$.\n\nFor each $x\\in X\\cap\\down p$, we have $x\\leq p$, and since $S\\in J(p)$, we have $S\\cap\\down x\\in J(x)$ by the stability axiom for Grothendieck topologies. It follows that $S\\cap\\down x\\cap X\\in\\bar J(x)$. Now, $f_y$ exists for each $y\\in S\\cap\\down x\\cap X$, and $\\langle f_y(y)\\rangle_{y\\in S\\cap\\down x\\cap X}$ is a matching family for $S\\cap\\down x\\cap X$ of elements of $F$. Indeed, if $z\\leq y$ in $S\\cap\\down x\\cap X\\subseteq X\\cap\\down x$, we have\n\\begin{equation*}\nF(z\\leq y)f_y(y)=f_y(z)=f_z(z),\n\\end{equation*}\nwhere, in the first equality, we used $f_y\\in\\hat F(y)$, and in the second equality, we used (\\ref{eq:mfforhatF}).\nSince $F$ is a $\\bar J$-sheaf, we find that $\\langle f_y(y)\\rangle_{y\\in S\\cap\\down x\\cap X}$ has a unique amalgamation $f(x)\\in F(x)$. Notice that if $x\\in S$, then $f(x)=f_x(x)$.\n\n If $y\\leq x$ in $X\\cap\\down p$, then $\\langle f_z(z)\\rangle_{z\\in S\\cap\\down y\\cap X}\\subseteq\\langle f_z(z)\\rangle_{z\\in S\\cap\\down x\\cap X}$. The unique amalgamation of the left-hand side of the inclusion is $f(y)$. Since $f(x)$ is the amalgamation of the right-hand side of the inclusion, we have $$f_z(z)=F(z\\leq x)f(x)=F(z\\leq y)F(y\\leq x)f(x)$$ for each $z\\in S\\cap\\down y\\cap X$. Thus $F(y\\leq x)f(x)$ is another amalgamation of the left-hand side of the inclusion, whence $F(y\\leq x)f(x)=f(y)$. So $f:X\\cap\\down p\\to\\bigcup_{x\\in X}F(x)$, defined by $x\\mapsto f(x)$, is an element of $\\hat F(p)$.\n\nLet $q\\in S$. Then for each $x\\in X\\cap\\down q$, we have $x\\in S$, so $f(x)=f_x(x)$ as we have noted before. By (\\ref{eq:mfforhatF}) we have $f_q(x)=f_x(x)$, so $\\hat F(q\\leq p)f=f_q$, and we see that $f$ is indeed an amalgamation for the matching family\n$f_q$.\n\nIf $g\\in\\hat F(p)$ satisfies (\\ref{eq:uniquenessf}) for each $q\\in S$, then $$g(q)=\\left.g\\right|_{X\\cap\\down q}=\\left(\\hat F(q\\leq p)g\\right)(q)=f_q(q).$$\nSince $g\\in\\hat F(p)$, we have $F(y\\leq x)g(x)=g(y)$ for each $y\\leq x$ in $X\\cap\\down p$ and since we just noticed that $f_y(y)=g(y)$ for each $y\\in S\\cap\\down x\\cap X$, which is a subset of $X\\cap\\down p$, it follows that also $g(x)$ is an amalgamation of $\\langle f_y(y)\\rangle_{y\\in S\\cap\\down x\\cap X}$. Hence $g(x)=f(x)$ for each $x\\in X\\cap\\down p$.\nIt follows that $f=g$, so $f$ is the unique amalgamation of $\\langle f_q\\rangle_{q\\in S}$.\n\n\\end{proof}\n\nIf $J_X\\leq J$, we have $\\down(X\\cap\\down p)\\in J(p)$ for each $p\\in\\P$. It turns out that if $F\\in\\Sh(\\P,J)$, then $F$ satisfies a sheaflike condition for the generating set $X\\cap\\down p$:\n\\begin{lemma}\nLet $X$ be a subset of a poset $\\P$ and $J$ a Grothendieck topology on $\\P$ such that $J_X\\leq J$. Let $F\\in\\Sh(\\P,J)$. Then for each $p\\in\\P$, if $\\langle a_x\\rangle_{x\\in X\\cap\\down p}$ such that $a_x\\in F(x)$ for each $x\\in X\\cap\\down p$ and $F(y\\leq x)a_x=a_y$ for each $y\\leq x$ in $X\\cap\\down p$, then there is a unique $a\\in F(p)$ such that $F(x\\leq p)a=a_x$.\n\\end{lemma}\nAlthough $X\\cap\\down p$ generates a sieve on $p$, but is not necessarily a sieve itself, we shall also refer to $a$ as the amalgamation of $\\langle a_x\\rangle_{x\\in X\\cap\\down p}$.\n\\begin{proof}\nFirst we consider the case that $X\\cap\\down p=\\emptyset$. Then there\nis only one family $\\langle a_x\\rangle_{x\\in X\\cap\\down p}$ such that the conditions of the lemma hold, namely the empty family, and clearly any point in $F(p)$ is an amalgamation of this empty family. So we have to show that $F(p)\\cong 1$. Since $X\\cap\\down\np=\\emptyset$, it follows that $\\emptyset=\\down(X\\cap\\down p)\\in J(p)$. Now, there\nis only one matching family for this sieve, namely the empty\nmatching family, which must have a unique amalgamation $a$, so indeed\n$F(p)=\\{a\\}$. Notice that the same conclusion could be drawn from\nExample \\ref{ex:discretetopologysheaves}.\n\nIf $X\\cap\\down p\\neq\\emptyset$, for each\n$z\\in\\down(X\\cap\\down p)$ we have an $x\\in X\\cap\\down p$ such that $z\\leq\nx$. So given a family $\\langle a_x\\rangle_{x\\in X\\cap\\down p}$ satisfying the conditions stated in the lemma, we define $a_z:=F(z\\leq x)a_x$. We\nhave to show that $a_z$ does not depend on the point $x\\geq z$. As we have seen above, we have $F(z)\\cong 1$ if\n$X\\cap\\down z=\\emptyset$. Since $a_z\\in F(z)$, we find that\n$F(z)=\\{a_z\\}$, so if $y\\in X\\cap\\down p$ such that $z\\leq y$, we\nalso must have $a_z=F(z\\leq y)a_y$.\n\nIf $X\\cap\\down z\\neq\\emptyset$, then also $\\down(X\\cap\\down z)\\in\nJ(z)$ is non-empty. So for each $v\\in\\down(X\\cap\\down z)$ there is\na $w\\in X\\cap\\down z$ such that $v\\leq w$. Hence\n\\begin{eqnarray*}\nF(v\\leq z)F(z\\leq x)a_x & = & F(v\\leq w)F(w\\leq z)F(z\\leq\ny)a_y=F(v\\leq w)F(w\\leq y)a_y\\\\\n& =& F(v\\leq w)a_w,\n\\end{eqnarray*}\nsince $w,x\\in X\\cap\\down p$. In a similar way, we find $$F(v\\leq\nz)F(z\\leq y)a_y=F(v\\leq w)a_w.$$ In other words, for each\n$v\\in\\down(X\\cap\\down z)\\in J(z)$, the images under $F(v\\leq z)$\nof $F(z\\leq x)a_x$ and $F(z\\leq y)a_y$ are the same, and since $F$\nis a $J$-sheaf, it follows that $$F(z\\leq x)a_x=F(z\\leq y)a_y.$$\nSo $a_z$ is uniquely determined and it is clear from the way it is\ndefined that $\\langle a_z\\rangle_{z\\in\\down(X\\cap\\down p)}$ is a\nmatching family. It follows that there is a unique amalgamation $a\\in F(p)$ for $\\langle a_x\\rangle_{x\\in\\down(X\\cap\\down p)}$, so $a$ is also an amalgamation for $\\langle a_x\\rangle_{x\\in X\\cap\\down p}$. In order to show that $a$ is also the unique amalgamation of $\\langle a_x\\rangle_{x\\in X\\cap\\down p}$, let $b\\in F(p)$ such that $a_x=F(x\\leq p)b$ for each $x\\in X\\cap\\down p$. Then for each $z\\in\\down(X\\cap\\down p)$, there must be an $x\\in X\\cap\\down p$ such that $z\\leq x$, whence\n\\begin{equation*}\nF(z\\leq p)b=F(z\\leq x)F(x\\leq p)b=F(z\\leq x)a_x=a_z.\n\\end{equation*}\nSo we see that $b$ is an amalgamation for $\\langle a_z\\rangle_{z\\in\\down(X\\cap\\down p)}$, whence $b=a$.\n\\end{proof}\n\n\n\\begin{lemma}\n Let $J$ be a Grothendieck topology such that $J_X\\leq J$. Then for each $F\\in\\Sh(\\P,J)$, there is a natural bijection $\\phi_F:E\\circ I^*(F)\\to F$. Moreover, the family $\\{\\phi_F\\}_F$ constitutes a natural isomorphism $\\phi:E\\circ I^*\\to 1_{\\Sh(\\P,J)}$.\n\\end{lemma}\n\\begin{proof}\nLet $F\\in\\Sh(\\P,J)$ and write $\\hat F$ instead of $E\\circ I^*(F)=\\widehat{F\\circ I}$. Then $\\hat F(p)$ is the set of functions $f:X\\cap\\down p\\to\\bigcup_{x\\in X}F(x)$ such that $f(x)\\in F(x)$ for each $x\\in X$ and $F(y\\leq x)f(x)=f(y)$ for each $x,y\\in X$ such that $y\\leq x$.\nWe define the natural bijection $\\phi_F: \\hat F\\to F$ as follows. Since the family $\\langle f(x)\\rangle_{x\\in X\\cap\\down p}$ satisfies the conditions of the previous lemma, we define $(\\phi_F)_p(f)$ to be the unique amalgamation of this family, whose existence is assured by the same lemma. Clearly $(\\phi_F)_p$ is a bijection, but we also have to show that it is natural, i.e., we have to show that for each $q\\leq p$, in $\\P$\nthe following diagram commutes:\n\\begin{equation*}\n\\xymatrix{\\hat F(p)\\ar[rr]^{(\\phi_F)_p}\\ar[dd]_{\\hat F(q\\leq p)} && F(p)\\ar[dd]^{F(q\\leq p)}\\\\\n\\\\\n\\hat F(p)\\ar[rr]_{(\\phi_F)_q} && F(q)}\n\\end{equation*}\nLet $f\\in\\hat F(p)$, so that $(\\phi_F)_p(f)$ is the amalgamation of the family $\\langle f(x)\\rangle_{x\\in X\\cap\\down p}$. Let $$g=\\hat\nF(q\\leq p)f=\\left.f\\right|_{X\\cap\\down q}.$$ Then for each $x\\in X\\cap\\down q$, we find $$g(x)=f(x)=F(x\\leq p)(\\phi_F)_p(f)=F(x\\leq q)F(q\\leq p)(\\phi_F)_p(f).$$ In other words, $F(q\\leq p)(\\phi_F)_p(f)$ is the amalgamation of the family $\\langle g(x)\\rangle_{x\\in X\\cap\\down q}$. But by definition of $\\phi_F$, this is exactly $(\\phi_F)_q(g)$. Hence $$F(q\\leq p)(\\phi_F)_p(f)=(\\phi_F)_q\\circ F(q\\leq p)(f).$$\n\nFinally, we have to show that $\\phi$ is a natural isomorphism. Since each component $\\phi_F$ is an isomorphism in $\\Sets$, we only have to show that $\\phi$ is natural in $F$. In other words, let $\\alpha:F\\to G$ be a natural transformation in $\\Sh(\\P,J)$. Then for each $p\\in\\P$,\n\\begin{equation*}\n\\xymatrix{\\hat F(p)\\ar[rr]^{(\\phi_F)_p}\\ar[dd]_{\\hat\\alpha_p} && F(p)\\ar[dd]^{\\alpha_p}\\\\\n\\\\\n\\hat G(p)\\ar[rr]_{(\\phi_G)_p} && G(p)}\n\\end{equation*}\nshould commute. So let $f\\in\\hat F(p)$, let $a=(\\phi_F)_p(f)$ be the amalgamation of $\\langle f(x)\\rangle_{x\\in X\\cap\\down p}$, and let $g\\in\\hat G(p)$ be given by $\\hat\\alpha(f)$, which means that $g(x)=\\alpha_x(f(x))$ for each $x\\in X\\cap\\down p$. Then $(\\phi_G)_p(g)$ is the unique amalgamation of $\\langle g(x)\\rangle_{x\\in X\\cap\\down p}$, but since $\\alpha:F\\to G$ is natural, we find for each $x\\in X\\cap\\down p$ that\n\\begin{equation*}\n G(x\\leq p)\\alpha_p(a)=\\alpha_xF(x\\leq p)a=\\alpha_x(f(x))=g(x),\n\\end{equation*}\nwhence $\\alpha_p(a)$ is an amalgamation of $\\langle g(x)\\rangle_{x\\in X\\cap\\down p}$. In other words, we must have $(\\phi_G)_p(g)=\\alpha_p(a)$.\n\\end{proof}\n\n\n\\begin{lemma}\n  Let $J$ be a Grothendieck topology such that $J_X\\leq J$. Then for each $F\\in\\Sh(X,\\bar J)$, there is a natural bijection $\\psi_F:I^*\\circ E(F)\\to F$. Moreover, the $\\psi_F$ constitute a natural isomorphism $\\psi:I^*\\circ E\\to 1_{\\Sh(X,\\bar J)}$.\n\\end{lemma}\n\\begin{proof}\n Let $F\\in\\Sh(X,\\bar J)$. Then for each $x\\in X$ we have $$I^*\\circ E(F)(x)=\\hat F\\circ I(x)=\\hat F(x).$$ Then define $(\\psi_F)_x:\\hat F\\circ I(x)\\to F(x)$ by $f\\mapsto f(x)$. This is well defined, for $x\\in X$ and $f$ is a function $X\\cap\\down x\\to\\bigcup_{x\\in X}$, so $x$ lies in the domain of $f$. Now, $\\down(X\\cap\\down x)\\in J(x)$, so $X\\cap\\down x\\in\\bar J(x)$, so $\\langle f(y)\\rangle_{y\\in X\\cap\\down x}$ is a matching family for the cover $X\\cap\\down x$ of elements of $F\\in\\Sh(X,\\bar J)$, hence there is a unique amalgamation $a\\in F(x)$ such that $F(y\\leq x)a=f(y)$. But since clearly $f(x)$ is also an amalgamation, we see that $a=f(x)$, whence the assignment $(\\psi_F)_x:f\\mapsto f(x)$ is a bijection. We have to show that $\\psi_F$ is natural, so for each $x,y\\in X$ such that $y\\leq x$,\n\\begin{equation*}\n\\xymatrix{\\hat F(x)\\ar[rr]^{(\\psi_F)_x}\\ar[dd]_{\\hat F(y\\leq x)} && F(x)\\ar[dd]^{F(y\\leq x)}\\\\\n\\\\\n\\hat F(x)\\ar[rr]_{(\\psi_F)_y} && F(y)}\n\\end{equation*}\nshould commute. Now, if $f\\in\\hat F(x)$, let $$g=\\hat F(y\\leq x)f=\\left.f\\right|_{X\\cap\\down q}.$$ Then $$(\\psi_F)_y(g)=g(y)=f(y),$$ whilst $F(y\\leq x)f(x)=f(y)$ by definition of $f\\in\\hat F(x)$. So the diagram clearly commutes.\n\nFinally, we show that $\\psi$ is a natural isomorphism. Again, since each component $\\psi_F$ is a bijection, we only have to show that $\\psi$ is natural in $F$, i.e., if $\\alpha:F\\to G$ in $\\Sh(X,\\bar J)$, then the following diagram\n\\begin{equation*}\n\\xymatrix{\\hat F(x)\\ar[rr]^{(\\psi_F)_x}\\ar[dd]_{\\hat\\alpha_x} && F(x)\\ar[dd]^{\\alpha_x}\\\\\n\\\\\n\\hat G(x)\\ar[rr]_{(\\psi_G)_x} && G(x)}\n\\end{equation*}\nshould commute for each $x\\in X$. Let $g=\\hat\\alpha(f)\\in\\hat G(x)$. Then $g(y)=\\alpha_y(f(y))$, so $(\\psi_G)_x(g)=g(x)=\\alpha_x(f(x))$. But this is exactly $\\alpha_x\\circ(\\psi_F)_x(f)$.\n\\end{proof}\n\n\n\n\n\\begin{theorem}[Comparison Lemma, posetal case]\\label{thm:ComparisonLemma}\n Let $\\P$ be a poset and $X\\subseteq\\P$ a subset. If $I:X\\to\\P$ is the inclusion, and $J$ is a Grothendieck topology on $\\P$ such that $J_X\\leq J$, then $I^*:\\Sh(\\P,J)\\to\\Sh(X,\\bar J)$ and $E:\\Sh(X,\\bar J)\\to\\Sh(\\P,J)$ form an equivalence of categories.\n\\end{theorem}\n\\begin{proof}\nThe previous two lemmas show that $I^*\\circ E\\cong 1$ and $E\\circ I^*\\cong 1$, so $\\Sh(\\P,J)\\cong\\Sh(X,\\bar J)$.\n\\end{proof}\n\n\n\\begin{corollary}\\label{cor:JXsheaves}\n Let $\\P$ be a poset and $X\\subseteq\\P$ a subset. Then $\\Sh(\\P,J_X)\\cong\\Sets^{X^\\op}$.\n\\end{corollary}\n\\begin{proof}\n For each $x\\in $, we have $\\bar J_X(x)=\\{S\\cap X:S\\in J_X(x)\\}$. Since $x\\in X$, it follows that $$J_X(x)=\\{S\\in\\D(\\down x):X\\cap\\down x\\subseteq S\\}=\\{\\down x\\},$$ so $\\bar{J}_X(x)=\\{\\bar\\down x\\}$, the indiscrete topology on $X$. By Example \\ref{ex:indiscretetopologysheaves}, we find $\\Sh(X,\\bar{J}_X)=\\Sets^{X^\\op}$.\n\\end{proof}\n\n\nWe can put an equivalence relation on the set of Grothendieck topologies of a poset by declaring two Grothendieck topologies $J_1$ and $J_2$ to be equivalent if and only if $\\Sh(\\P,J_1)\\cong\\Sh(\\P,J_2)$. If $X,Y\\subseteq\\P$ are subsets, using the previous corollary we find that $J_X\\simeq J_Y$ if and only if there is an order isomorphism between $X$ and $Y$ if we equip both sets with the order inherited from $\\P$.\n\n\n\nA question which might be interesting is which subset topologies are subcanonical. In the case of an Artinian poset, we are also able to determine the condition of canonicity. We start by introducing the notion of canonical and subcanonical topologies.\n\\begin{definition}\n Let $J$ be a Grothendieck topology on a category $\\CC$. Then $J$ is called \\emph{subcanonical}\\index{subcanonical Grothendieck topology}\nif all representable presheaves are $J$-sheaves. That is, if\n$\\y(C)=\\Hom(-,C)$ is a $J$-sheaf for each $C\\in\\CC$. The\n\\emph{canonical}\\index{canonical Grothendieck topology} topology on\n$\\CC$ is the finest subcanonical topology.\n\\end{definition}\n\nIt turns out to be quite easy to find a condition for subcanonicity of an arbitrary Grothendieck topology on an arbitrary poset, since every homset of each poset $\\P$ contains at most one element. Therefore, given a matching family for a cover of elements of $\\y(p)$ for $p\\in\\P$, the existence of an amalgamation automatically implies uniqueness. Another way of expressing this is saying that $\\y(p)$ is a \\emph{separated presheaf}. The existence of an amalgamation for a matching family for a cover of an element $q\\in\\P$ is assured if $q\\leq p$. So if $q\\nleq p$, the sheaf condition only holds if there is no matching family for every cover of $q$. The next proposition assures that this condition is necessary and sufficient.\n\n\n\\begin{proposition}\\label{prop:representablesheaf}\n Let $J$ be a Grothendieck topology on a poset $\\P$ and let $p\\in\\P$. Then $\\y(p)$ is an $J$-sheaf if and only if $J(q)\\cap\\D(\\down p)=\\emptyset$ for each $q\\nleq p$.\n\\end{proposition}\n\\begin{proof}\nWe first show that for $q\\leq p$, the sheaf condition is always satisfied. Let $q\\leq p$ and $S\\in J(q)$. Then if $\\langle a_x\\rangle_{x\\in S}$ is a matching family for $S$ of elements of $\\y(p)$, we have $x\\leq p$, so $a_x\\in\\y(p)(x)=\\Hom(x,p)\\cong 1$. Thus $a_x$ is fixed for each $x\\in S$, so there is only one matching family for $S$.  As a consequence, if $R\\in J(q)$ such that $S\\subseteq R$ and $\\langle b_x\\rangle_{x\\in R}$ is a matching family for $R$, we must have $b_x=a_x$ for each $x\\in S$. In particular, we can take $R=\\down q$. Now, $\\langle\\y(p)(x\\leq q)a\\rangle_{x\\leq q}$ is a matching family for $R$, where $a$ is the unique point in $\\y(p)(q)=\\Hom(q,p)$, which exists since $q\\leq p$. So $a_x=\\y(p)(x\\leq q)a$ for each $x\\in S$, which proves that $a$ is the unique amalgamation of $\\langle a_x\\rangle_{x\\in S}$.\n\nNow assume that $q\\nleq p$ and $J(q)\\cap\\D(\\down p)=\\emptyset$. Let $S\\in J(q)$, then $S\\notin\\D(\\down p)$, so $S\\nsubseteq\\down p$. Then there must be an $x\\in S$ such that $x\\notin\\down p$, or equivalently, there is an $x\\in S$ such that $x\\nleq p$. Hence $\\y(p)(x)=\\Hom(x,p)=\\emptyset$, so there is no matching family $\\langle a_y\\rangle_{y\\in S}$ for $S$, since this requires the existence of an element $a_x\\in\\y(p)(x)$. We conclude that $\\y(p)\\in\\Sh(\\P,J)$.\n\nThe other direction follows by contraposition. So assume that $q\\nleq p$ and furthermore, assume that $J(q)\\cap\\D(\\down p)\\neq\\emptyset$. So there is an $S\\in J(q)$ such that $S\\subseteq\\down p$. Then for each $x\\in S$ we have $x\\leq p$, so there is a unique element $a_x\\in\\y(p)(x)=\\Hom(x,p)\\cong 1$, and $\\langle a_x\\rangle_{x\\in S}$ define a matching family for $S$. However, since $q\\nleq p$, we have $\\y(p)(q)=\\Hom(q,p)=\\emptyset$, so there is no amalgamation of the matching family. We conclude that $\\y(p)\\notin\\Sh(\\P,J)$.\n\\end{proof}\n\n\n\\begin{proposition}\\label{prop:subcanonicalJX}\n Let $X$ be a subset of a poset $\\P$, and let $p\\in\\P$. Then $\\y(p)\\in\\Sh(\\P,J_X)$ for each $p\\in\\P$ if and only if $X\\to\\down p=\\down p$. As a consequence, $J_X$ is subcanonical if and only if $X\\to\\down p=\\down p$ for each $p\\in \\P$.\n\\end{proposition}\n\\begin{proof}\nAssume that $X\\to\\down p=\\down p$ and let $q\\nleq p$. Then $\\down q\\nsubseteq X\\to\\down p$, so by definition of the Heyting implication, we find that $X\\cap\\down q\\nsubseteq \\down p$. Let $S\\in J_X(q)$. Then $S$ contains $X\\cap\\down q$, hence $S\\nsubseteq\\down p$. Thus $J_X(q)\\cap\\D(\\down p)=\\emptyset$.\n\nNow let $X\\to\\down p\\neq\\down p$. Since $X\\cap\\down p\\subseteq\\down p$, we have $\\down p\\subseteq X\\to\\down p$. Hence, there must be a $q\\notin\\down p$, i.e. $q\\nleq p$, such that $q\\in X\\cap\\down p$. But then $\\down q\\subseteq X\\to\\down p$, so $X\\cap\\down q\\subseteq\\down p$. Hence $\\down(X\\cap\\down q)\\subseteq\\down p$, and since $\\down(X\\cap\\down q)\\in J_X(q)$, we find that $J_X(q)\\cap\\D(\\down p)\\neq\\emptyset$. By contraposition, we find that $J_X(q)\\cap\\D(\\down p)=\\emptyset$ for each $q\\nleq p$ implies $X\\to\\down p=\\down p$. The claim now follows directly from Proposition \\ref{prop:representablesheaf}.\n\\end{proof}\n\n\n\n\\begin{example}\n Let $\\P=\\{x,y,z\\}$ with the order specified by $x<y$ and $x<z$. Then $J_X$ with $X=\\{y,z\\}$ is subcanonical, since $X\\to\\down p=\\down p$ for $p=x,y,z$. On the other hand, $J_Y$ with $Y=\\{y\\}$ is not subcanonical, since $Y\\to\\down x=\\down z$.\n\\end{example}\n\nLet $\\P$ be an Artinian poset with $X,Y\\subseteq\\P$ such that $X\\subseteq Y$. Then $S\\in J_Y(p)$ if and only if $Y\\cap\\down p\\subseteq S$, which implies that $X\\cap\\down p\\subseteq S$, or equivalently, $S\\in J_X(p)$. So $X\\subseteq Y$ implies $J_Y\\subseteq J_X$. The canonical topology $J_{\\mathrm{canonical}}$ on $\\P$ is defined as the largest subcanonical topology on $\\P$. Since on Artinian posets all topologies are of the form $J_X$ for some subset $X$ of $\\P$, we find that $J_{\\mathrm{canonical}}=J_X$, where $X$ is the smallest subset of $\\P$ satisfying $X\\to\\down p=\\down p$ for all $p\\in\\P$.\n\n\\section{Other Grothendieck topologies and its sheaves}\n\nWe found that if $\\P$ is downwards-directed, then the dense Grothendieck topology is actually the atomic Grothendieck topology on $\\P$.\nIt turns out that we can find a whole new class of Grothendieck topologies on $X$ related to the atomic Grothendieck topology that are not always subset Grothendieck topologies. Since we can extend these Grothendieck topologies to Grothendieck topologies on $\\P$ that are not subset Grothendieck topologies, they might be interesting to investigate. Another reason for investigating this class of Grothendieck topologies is that, together with the subset Grothendieck topologies, they exhaust all Grothendieck topologies on $\\Z$, which can be found in the Appendix as Proposition \\ref{prop:classificationGTofZ}.\n\n\\begin{proposition}\n Let $\\P$ be a downwards directed poset and $X\\subseteq\\P$. Then $K_X$ defined pointwise by\n\\begin{equation}\n K_X(p)=J_X(p)\\setminus\\{\\emptyset\\}\n\\end{equation}\nis a Grothendieck topology on $\\P$, which we call the \\emph{derived Grothendieck\ntopology}\\index{derived Grothendieck topology} with respect to the subset $X$.\n\\end{proposition}\n\\begin{proof}\n Since $\\down p\\in J_X(p)$ for each $p\\in\\P$, we have $\\down p\\in K_X(p)$ for each $p\\in\\P$. If $S\\in K_X(p)$, then $S\\in J_X(p)$ so $S\\cap\\down q\\in J_X(q)$ for each $q\\leq p$. Moreover, since $S\\in K_X(p)$, we must have $S\\neq\\emptyset$. Since $\\P$ is downwards directed, this implies that $S\\cap\\down q\\neq\\emptyset$, so $S\\cap\\down q\\in K_X(q)$ for each $q\\leq p$. Finally, let $S\\in K_X(p)$ and $R\\in\\D(\\down p)$ be such that $R\\cap\\down q\\in K_X(q)$ for each $q\\in S$. So $R\\neq\\emptyset$, since $R\\cap\\down q\\neq\\emptyset$ if there exists a $q\\in S$, which is the case since $S\\in K_X(p)$ implies that $S$ is non-empty. Moreover, $S\\in K_X(p)$ implies $S\\in J_X(p)$, and $R\\cap\\down q\\in K_X(q)$ for each $q\\in S$ implies $R\\cap\\down q\\in J_X(q)$ for each $q\\in S$. So $R\\in J_X(p)$, and since $R\\neq\\emptyset$, we find that $R\\in K_X(p)$.\n\\end{proof}\n\n In the definition it might not be directly visible that the derived Grothendieck topologies are related to the atomic Grothendieck topology, but this follows from the next lemma.\n\n\\begin{lemma}\\label{lem:derivedtopologies}\n Let $\\P$ be a downward directed poset and $X\\subseteq\\P$ a subset of $\\P$. Then\n \\begin{equation}\n K_X(p)=\\left\\{\n       \\begin{array}{ll}\n         J_X(p), & p\\in\\up X; \\\\\n         J_{\\mathrm{atom}}(p), & p\\notin \\up X.\n       \\end{array}\n     \\right.\n\\end{equation}\nIn particular, if $\\up X=\\P$, we have $K_X=J_X$.\n\\end{lemma}\n\\begin{proof}\nLet $p\\in\\up X$. Then $X\\cap\\down p\\neq\\emptyset$, since otherwise\nwe must have $x\\nleq p$ for each $x\\in X$, contradicting $p\\in\\up X$.\nIt follows that $X\\cap\\down p\\nsubseteq \\emptyset$, so\n$\\emptyset\\notin J_X(p)$, whence $J_X(p)=K_X(p)$. If $p\\notin\\up X$,\nwe have $X\\cap\\down p=\\emptyset$, so $J_X(p)=\\D(\\down p)$. It\nfollows that $K_X(p)=\\D(\\down\np)\\setminus\\{\\emptyset\\}=J_{\\mathrm{atom}}(p)$.\n\\end{proof}\nSince the atomic topology is only defined on downwards directed\nsubsets, $(\\up X)^c$ should be downwards directed. This is indeed\nthe case: if $p_1,p_2\\in(\\up X)^c$, then there is a $q\\in\\P$ such\nthat $q\\leq p_1,p_2$, since $\\P$ is downwards directed. However, we\ncannot have $q\\in\\up X$, since this would imply that $p_1,p_2\\in\\up\nX$. Hence $q\\in(\\up X)^c$.\n\nFurthermore, we see that $K_\\emptyset=J_{\\mathrm{atom}}$. Even if\n$X$ is not dense, there might be another subset $Y\\subseteq\\P$ such\nthat $K_X=J_Y$.\n\n\n\\begin{lemma}\\label{lem:derivedtopologiesforposetswithleastelement}\n Let $\\P$ be a downwards directed poset with a least element $0$. Then for each subset $X\\subseteq\\P$, we have $K_X=J_{X_0}$, where $X_0=X\\cup\\{0\\}$.\n\\end{lemma}\n\\begin{proof}\nLet $p\\in\\P$. Then if $S\\in\\D(\\down p)$, we clearly have\n$S\\neq\\emptyset$ if and only if $0\\in S$, since $S$ is downwards\nclosed and $0$ lies below each element of $S$. Assume $p\\in\\up X$\nand $S\\in\\D(\\down p)$. Then $X_0\\cap\\down p\\subseteq S$ implies\n$X\\cap\\down p\\subseteq S$, so $J_{X_0}(p)\\subseteq J_X(p)$. On the\nother hand, since $p\\in\\up X$, we have $X\\cap\\down p\\neq\\emptyset$,\nso if $X\\cap\\down p\\subseteq S$, then $S\\neq\\emptyset$, so $0\\in S$.\nHence we find that $X_0\\cap\\down p\\subseteq S$, whence\n$J_X(p)\\subseteq J_{X_0}(p)$. Thus $J_{X_0}(p)=J_X(p)$.\n\nNow assume $p\\notin\\up X$. Then $X\\cap\\down p=\\emptyset$, so\n$X_0\\cap\\down p=\\{0\\}$. We find that $J_{X_0}(p)$ is exactly the set\nof all $S\\in\\D(\\down p)$ such that $0\\in S$, which are exactly all\nnon-empty sieves on $p$. Thus $J_{X_0}(p)=J_\\atom(p)$. By the\nprevious lemma, we conclude that $J_{X_0}(p)=K_X(p)$ for each\n$p\\in\\P$.\n\\end{proof}\nNotice that if $\\P$ is downwards directed and Artinian, we\nautomatically have a least element $0$. An example of a downwards\ndirected poset without a least element is $\\Z$. Then for any\n$p\\in\\Z$ we have\n\\begin{equation*}\n\\bigcap K_\\emptyset(p)=\\bigcap\\left(\\D(\\down\np)\\setminus\\{\\emptyset\\}\\right)=\\bigcap_{q\\leq p}\\down q=\\emptyset,\n\\end{equation*}\nso $K_\\emptyset$ is not complete. Since $J_X$ is complete for each\n$X\\subseteq\\Z$, we find that $K_\\emptyset\\neq J_X$ for each\n$X\\subseteq\\P$. So we see that if $\\P$ is non-Artinian, all\ntopologies are neither necessarily a subset Grothendieck topology, nor are they necessarily complete.\n\nIn Lemma \\ref{lem:derivedtopologies} we found that $K_X$ is a\nGrothendieck topology, which is constructed from two other\nGrothendieck topologies, but since one of these topologies is the\natomic topology, we have to consider posets which are downwards\ndirected. Since the dense topology can be seen as a generalisation of the atomic topology for posets that are not necessarily downwards directed, one could try to generalize $K_X$ by defining\n \\begin{equation*}\n K'_X(p)=\\left\\{\n       \\begin{array}{ll}\n         J_X(p), & p\\in\\up X; \\\\\n         J_{\\mathrm{dense}}(p), & p\\notin \\up X.\n       \\end{array}\n     \\right.\n\\end{equation*}\nHowever, this is not always a Grothendieck topology. For instance,\nconsider the poset $\\P=\\{x,y_1,y_2\\}$ with $y_i\\leq x$. Let\n$X=\\{y_1\\}$. Then $K'_X(x)=J_X(x)=\\{\\down x,\\down y_1\\}$,\n$K'_X(y_1)=J_X(y_1)=\\{\\down y_1\\}$, whereas\n$K'_X(y_2)=J_{\\mathrm{dense}}(y_2)=\\{\\down y_2\\}$. Now, since\n$y_2\\leq x$ and $\\down y_1\\in K'_X(x)$, we should have\n$\\emptyset=\\down y_1\\cap\\down y_2\\in K'_X(y_2)$ by the stability\naxiom. Since this is not the case, $K'_X$ cannot be a Grothendieck\ntopology.\n\n\n\nWe shall calculate $K_X$-sheaves on a downwards directed poset $\\P$. Moreover, if $\\P$ is a poset, $X\\subseteq\\P$ is downwards directed and $Y\\subseteq X$, we can consider the Grothendieck topology $L_{X,Y}$ on $\\P$ obtained by extending the Grothendieck topology $K_Y^X$ on $X$ to $\\P$. Since $L_{X,Y}$ is defined by the extension of a Grothendieck topology, whose sheaves are known, we can easily calculate the $L_{X,Y}$-sheaves on $\\P$ as well. For the calculation of the $K_X$-sheaves we need the following definition.\n\n\\begin{definition}\nLet $\\P$ be a downwards directed poset. Even if $\\P$ already has a\nleast element, we can add a new least element, which we denote by\n$0$. We define $\\P_0=\\P\\cup\\{0\\}$, where $0<p$ for each $p\\in\\P$. If\n$A\\subseteq\\P$, then we write $A_0=A\\cup\\{0\\}$. If $p\\in\\P$, then\n$\\down p$ with respect to $\\P_0$ is exactly $(\\down p)_0$, where\n$\\down p$ is defined with respect to $\\P$. If $X\\subseteq\\P$, then $X\\cap\\down\np=X\\cap(\\down p)_0$, in which case the notation does not matter. If\n$J$ is a topology on $\\P$ and we want to define a similar topology\non $\\P_0$, we denote this topology by $J^0$. For instance, $K^0_X$\nis the derived topology on $\\P_0$ with respect to $X$. By the\nprevious remark about $X\\cap\\down p=X\\cap(\\down p)_0$, we have $S\\in\nJ_X^0(p)$ if and only if $X\\cap\\down p\\subseteq S$.\n\\end{definition}\n\n\\begin{lemma}\\label{lem:KX0intermsofKX}\n Let $\\P$ be a downwards-directed poset. Then we have a bijection $\\D(\\P)\\to\\D(\\P_0)\\setminus\\{\\emptyset\\}$ given by $A\\mapsto A_0$, with inverse $B\\mapsto B\\setminus\\{0\\}$, which, for each $X\\subseteq\\P$ and $p\\in\\P$, restricts to a bijection $K_X(p)\\to K^0_X(p)\\setminus\\{\\{0\\}\\}$. Moreover, $\\{0\\}\\in K_X^0(p)$ if and only if $p\\in(\\up X)^c$, where the complement is taken with respect to $\\P_0$.\n\\end{lemma}\n\\begin{proof}\nClearly, $A_0\\in\\D(\\P_0)\\setminus\\{\\emptyset\\}$ if $A\\in\\D(\\P)$ and $B\\setminus\\{0\\}\\in\\D(\\P)$ if $B\\in\\D(\\P_0)\\setminus\\{\\emptyset\\}$. It is also clear that $A_0\\setminus\\{0\\}=A$ for $A\\in\\D(\\P)$. It follows from $0\\in B$ for each $B\\in\\D(\\P_0)\\setminus\\{\\emptyset\\}$ that $(B\\setminus\\{0\\}_0=B$.\n\nLet $S\\in K_X(p)$. If $p\\in\\up X$, this means that $\\emptyset\\neq X\\cap\\down p\\subseteq S$. Then $X\\cap\\down p\\subseteq S_0$, so $S_0\\in K^0_X(p)$. If $p\\notin\\up X$, then $S\\in J_\\atom(p)$, so $S$ is non-empty. Then also $S_0\\neq\\emptyset$, so $S_0\\in J^0_\\atom(p)=K^0_X(p)$. In both cases, since $S\\neq\\emptyset$, we have $S_0\\neq\\{0\\}$.\n\nLet $R\\in K^0(p)$ and assume that $R\\neq\\{0\\}$. If $p\\in\\up X$, this means that $R\\in J^0_X(p)$, so $X\\cap\\down p\\subseteq R$. Since $0\\notin X$, we find that  $X\\cap\\down p\\subseteq R\\setminus\\{0\\}$, hence $R\\setminus\\{0\\}\\in J_X(p)=K_X(p)$. If $p\\notin\\up X$, we have $R\\in J^0_\\atom(p)$, so $R$ is non-empty. Since also $R\\neq\\{0\\}$, we must have $R\\setminus\\{0\\}\\neq\\emptyset$, so $R\\setminus\\{0\\}\\in J_\\atom(p)=K_X(p)$.\n\nIf $p\\in(\\up X)^c$, then for each $x\\in X$ we have $x\\nleq p$, so $X\\cap\\down p=\\emptyset\\subseteq\\{0\\}$, whence $\\{0\\}\\in K_X^0(p)=J_X^0(p)\\setminus\\{\\emptyset\\}$. If $p\\in\\up X$, we have $x\\leq p$ for some $x\\in X$ and since $X\\subseteq\\P$, we have $x\\neq 0$. Hence $x\\in X\\cap\\down p$, so $X\\cap\\down p\\nsubseteq\\{0\\}$. Thus $\\{0\\}\\notin K_X^0(p)$.\n\n\\end{proof}\n\n\n\\begin{lemma}\n Let $\\P$ be a downwards directed poset. Let $X\\subseteq\\P$ be a subset. If $I_0:\\P\\embeds\\P_0$ denotes the map, then $I_0^*:\\Sets^{\\P_0^\\op}\\to\\Sets^{\\P^\\op}$ given by $F\\mapsto F\\circ I_0$ restricts to a functor $\\Sh(\\P_0,K^0_X)\\to\\Sh(\\P,K_X)$.\n\\end{lemma}\n\\begin{proof}\n Let $F\\in\\Sh(\\P_0,K^0_X)$ and let $p\\in\\P$. Let $S\\in K_X(p)$ and let $\\langle a_x\\rangle_{x\\in S}$ be a matching family for $S$ of elements of $I_0^*(F)=F\\circ I_0$. So $a_x\\in F(x)$ for each $x\\in S$. Since $K_X$-covers are always non-empty, we can find a $q\\in S$ that allows us to define $a_0\\in F(0)$ by $a_0=F(0\\leq q)a_q$. This is independent of the chosen $q\\in S$; if $q'\\in S$, there exists an $r\\leq q,q'$, for $\\P$ is downwards directed, hence\n\\begin{eqnarray*}\nF(0\\leq q')a_{q'} & = & F(0\\leq r)F(r\\leq q')a_{q'}=F(0\\leq r)a_r\\\\\n & = & F(0\\leq r)F(r\\leq q)a_q=F(0\\leq q)a_q=a_0.\n\\end{eqnarray*}\nBy the previous lemma, $S_0\\in K^0_X(p)$, and $a_0$ is constructed exactly in such a way that $\\langle a_x\\rangle_{x\\in S_0}$ is a matching family for $S_0$ of elements of $F$. Since $F\\in\\Sh(\\P_0,K^0_X)$, we find that there is a unique amalgamation $a\\in F(p)$ such that $a_x=F(x\\leq p)$ for each $x\\in S_0$. To see that this is also a unique amalgamation for $S$, let $b\\in F(p)$ such that $a_x=F(x\\leq p)b$ for each $x\\in S$. Then $$F(0\\leq p)b=F(0\\leq q)F(q\\leq p)b=F(0\\leq q)a_q=a_0,$$ so $b$ is also an amalgamation of $\\langle a_x\\rangle_{x\\in S_0}$, whence $b=a$.\n\\end{proof}\n\n\\begin{definition}\\label{def:E0}\n Let $\\P$ be a downwards directed poset and take $X\\subseteq\\P$ such that $\\up X\\neq\\P$. Given a fixed $p_0\\in(\\up X)^c$, define $E_0:\\Sh(\\P,K_X)\\to\\Sh(\\P_0,K_X^0)$ as follows. $E_0$ acts on $F\\in\\Sh(\\P,K_X)$ by $F\\mapsto\\bar F$, where $\\bar F$ acts on objects of $\\P_0$ by\n \\begin{equation*}\n \\bar F(p)=\\left\\{\n       \\begin{array}{ll}\n         F(p), & p\\neq 0; \\\\\n         F(p_0), & p=0,\n       \\end{array}\n     \\right.\n\\end{equation*}\nand acts on morphisms by\n \\begin{equation*}\n \\bar F(q\\leq p)=\\left\\{\n       \\begin{array}{ll}\n         F(q\\leq p), & q\\neq 0; \\\\\n         F(r\\leq p_0)^{-1}F(r\\leq p), & q=0\\mathrm{\\ for\\ some\\ }r\\leq p,p_0\\mathrm{\\ in\\ }\\P.\n       \\end{array}\n     \\right.\n\\end{equation*}\nFurthermore, if $\\alpha:F\\to G$ in $\\Sh(\\P,K_X)$, then $E_0$ acts on $\\alpha$ by $\\alpha\\mapsto\\bar\\alpha$, where\n \\begin{equation*}\n \\bar\\alpha_p=\\left\\{\n       \\begin{array}{ll}\n         \\alpha_p, & p\\neq 0; \\\\\n         \\alpha_{p_0}, & p=0.\n       \\end{array}\n     \\right.\n\\end{equation*}\n\\end{definition}\n\\begin{lemma}\n$E_0$ is well defined.\n\\end{lemma}\n\\begin{proof}\nSince $\\up X\\neq\\P$, we can indeed find a $p_0\\in(\\up X)^c$. Let $F\\in\\Sh(\\P,K_X)$. We have to show that $\\bar F(0\\leq p)$ is well defined. Since $\\P$ is downwards directed, there is indeed an $r\\in\\P$ such that $r\\leq p,p_0$. Now, $r\\in\\up X$ implies $p_0\\in\\up X$, which is not the case, so we have $r\\notin\\up X$. Since $p_0\\notin\\up X$, we have $K_X(p_0)=J_\\atom(p_0)$. So $\\down r\\in K_X(p_0)$, and hence by Example \\ref{ex:atomsheaves}, $F(r\\leq p_0)$ is a bijection, so its inverse is defined. We have to show that the definition of $\\bar F(0\\leq p)$ is independent of the choice of $r\\leq p,p_0$. That is, if $r'$ is another element of $\\P$ such that $r'\\leq p,p_0$, then we have to show that\n\\begin{equation}\\label{eq:independentofr}\n F(r'\\leq p_0)^{-1}F(r'\\leq p)=F(r\\leq p_0)^{-1}F(r\\leq p).\n\\end{equation}\nSince $\\P$ is downwards directed, there is an $r''\\leq r,r'$, which is necesarrily in $(\\up X)^c$, so $F(r''\\leq r)$ and $F(r''\\leq r')$ are bijections.\nThen\n\\begin{eqnarray*}\n F(r'\\leq p_0)^{-1}F(r'\\leq p) &  =  &  F(r'\\leq p_0)^{-1}F(r''\\leq r')^{-1}F(r''\\leq r')F(r'\\leq p)\\\\\n&  = &  F(r''\\leq p_0)^{-1}F(r''\\leq p),\n\\end{eqnarray*}\nand in the same way, we find $$F(r\\leq p_0)^{-1}F(r\\leq p)=F(r''\\leq p_0)^{-1}F(r''\\leq p),$$ which shows that (\\ref{eq:independentofr}) indeed holds. To prove that $\\bar F$ is a functor, we must show that $$\\bar F(0\\leq p)=\\bar F(0\\leq p')\\bar F(p'\\leq p)$$ if $p'\\leq p$ in $\\P$. Therefore, take an $r\\leq p',p_0$, then also $r\\leq p$, so\n\\begin{equation*}\n \\bar F(0\\leq p)=F(r\\leq p_0)^{-1}F(r\\leq p)=F(r\\leq p_0)F(r\\leq p')F(p'\\leq p)=\\bar F(0\\leq p')\\bar F(p'\\leq p).\n\\end{equation*}\n\n\nWe have to show that $\\bar F$ is a $K_X^0$-sheaf if $F\\in\\Sh(\\P,K_X)$. Let $p\\in\\P_0$, $R\\in K_X^0(p)$, and let $\\langle a_x\\rangle_{x\\in R}$ be a matching family for $R$ of elements of $\\bar F$. If $p=0$, then $R=\\{0\\}$, since there are no other covers. Then the matching family consists only of the single element $a_0\\in\\bar F(0)$, which is also automatically the unique amalgamation, so the sheaf condition is satisfied. Let $p\\in\\P$ and assume that $R\\neq\\{0\\}$. Then by Lemma \\ref{lem:KX0intermsofKX},  we have $R\\setminus\\{0\\}\\in K_X(p)$. Since $a_x\\in\\bar F(x)=F(x)$ for $x\\in R\\setminus\\{0\\}$, we find that $\\langle a_x\\rangle_{x\\in R\\setminus\\{0\\}}$ is a matching family for $R\\setminus\\{0\\}$ of elements of $F$, which is a $K_X$-sheaf, so there is a unique amalgamation $a\\in F(p)$ for this matching family. Since $R\\neq\\{0\\}$, there is an $r_0\\in\\P$ such that $r_0\\in R$. Now, take some $r\\leq r_0,p_0$, then $r\\in R$, so we find that\n\\begin{eqnarray*}\n\\bar F(0\\leq p)a & = & F(r\\leq p_0)^{-1}F(r\\leq p)a=F(r\\leq p_0)^{-1}F(r\\leq r_0)F(r_0\\leq p)a\\\\\n& = & \\bar F(r\\leq r_0)a_{r_0}=a_0,\n\\end{eqnarray*}\nwhere $F(r_0\\leq p)a=a_{r_0}$ in the third equality, since $a$ is the amalgamation for $\\langle a_x\\rangle_{x\\in R\\setminus\\{0\\}}$ of elements of $F$. The last equality holds since $r\\leq r_0$ in $R$ and $\\langle a_x\\rangle_{x\\in R}$ is a matching family of elements of $\\bar F$. Thus $a$ is also an amalgamation of the matching family for $R$ of elements of $\\bar F$. If $b\\in F(p)$ is another amalgamation of the family for $R$ of elements of $\\bar F$, then $b$ is also an amalgamation of the family for $R\\setminus\\{\\emptyset\\}$ of elements of $F$, so $b=a$. This shows that $a$ is the unique amalgamation of $\\langle a_x\\rangle_{x\\in R}$.\n\nNow assume $R=\\{0\\}$. By Lemma \\ref{lem:KX0intermsofKX}, this is only possible if $p\\notin\\up X$. So $K_X(p)=J_\\atom(p)$, hence $\\down r\\in K_X(p)$ for each $r\\leq p$ in $\\P$. By Example \\ref{ex:atomsheaves}, we find that $F(r\\leq p)$ is a bijection for each $r\\leq p$ in $\\P$. Then if we choose $r\\leq p,p_0$, we find that $\\bar F(0\\leq p)=F(r\\leq p_0)^{-1}F(r\\leq p)$ is also a bijection. So if $\\langle a_x\\rangle_{x\\in R}$ is the matching family for $R=\\{0\\}$ with elements of $\\bar F$, we see that this matching family consists only of a single element $a_0\\in\\bar F(0)$. Since $\\bar F(0\\leq p)$ is a bijection, it is clear that the matching family has an amalgamation $\\bar F(0\\leq p)^{-1}a_0$, which is necesarrily unique.\n\nFinally, we have to show that $\\bar\\alpha:\\bar F\\to\\bar G$ is natural if $\\alpha:F\\to G$ is a natural transformation in $\\Sh(\\P,K_X)$.\nNow, $\\bar\\alpha_p=\\alpha_p$, $\\bar F(p)=F(p)$ and $\\bar G(p)=G(p)$ if $p\\in\\P$, so we clearly have $\\bar G(q\\leq p)\\bar\\alpha_p=\\bar\\alpha_q\\bar F(q\\leq p)$ if $q,p\\in\\P$, since this is exactly the naturality of $\\alpha$. If $q=0$, then\n\\begin{eqnarray*}\n\\bar G(0\\leq p)\\bar\\alpha_p &\n= & G(r\\leq p_0)^{-1}G(r\\leq p)\\alpha_p=G(r\\leq p_0)^{-1}\\alpha_rF(r\\leq p)\\\\\n& = & G(r\\leq p_0)^{-1}F(r\\leq p_0)^{-1}F(r\\leq p_0)\\alpha_rF(r\\leq p)\\\\\n & = & G(r\\leq p_0)^{-1}\\alpha_rF(r\\leq p_0)F(r\\leq p_0)^{-1}F(r\\leq p)\\\\\n& = & G(r\\leq p_0)^{-1}G(r\\leq p_0)\\alpha_{p_0}F(r\\leq p_0)^{-1}F(r\\leq p) = \\bar\\alpha_{0}\\bar F(0\\leq p).\n\\end{eqnarray*}\nSo $\\bar\\alpha$ is indeed a natural transformation.\n\\end{proof}\n\n\\begin{theorem}\n Let $\\P$ be a poset and take $X\\subseteq\\P$ such that $\\up X\\neq\\P$. Then\n\\begin{eqnarray*}\nI_0^*&:&\\Sh(\\P_0,K_X^0)\\to\\Sh(\\P,K_X);\\\\\nE_0&:&\\Sh(\\P,K_X)\\to\\Sh(\\P_0,K_X^0)\n\\end{eqnarray*}\n form an equivalence of categories.\n\\end{theorem}\n\\begin{proof}\n Clearly, we have $I^*_0\\circ E_0(F)=\\bar F\\circ I_0=F$ for each $F\\in\\Sh(\\P,K_X)$, so $I^*_0\\circ E_0=1_{\\Sh(\\P,K_X)}$. We aim to construct a natural isomorphism $\\beta: E_0\\circ I_0^*\\to 1_{\\Sh(\\P_0,K_X^0)}$. First, given $F\\in\\Sh(\\P_0,K_X^0)$, we have $E_0\\circ I_0^*(F)=\\widehat{F\\circ I_0}$. If we abbreviate the right-hand side by $\\bar F$, notice that $\\bar F(p)=F(p)$ for each $p\\in\\P$ and $\\bar F(q\\leq p)=F(q\\leq p)$ if $q\\leq p$ in $\\P$. Let $p_0$ the fixed point we chose in the previous lemma in order to construct $E_0$. Since $p_0\\notin\\up X$, we have $\\{0\\}=\\down 0\\in K_X^0(p_0)$, where we used Lemma \\ref{lem:KX0intermsofKX}. Since $F\\in\\Sh(\\P_0,K^0_X)$, we find by Example \\ref{ex:atomsheaves} that $F(0\\leq p_0)$ is a bijection. This allows us to define the component $\\beta_F$ by\n \\begin{equation*}\n (\\beta_F)_p=\\left\\{\n       \\begin{array}{ll}\n         1_{F(p)}, & p\\neq 0; \\\\\n         F(0\\leq p_0), & p=0.\n       \\end{array}\n     \\right.\n\\end{equation*}\nNotice that indeed $(\\beta_F)_0$ maps $\\bar F(0)$ into $F(0)$, since $\\bar F(0)=F(p_0)$ by definition of $\\bar F$. Furthermore, $(\\beta_F)_p$ is clearly a bijection for each $p\\in\\P_0$. We have to check that it is natural, that is,\n\\begin{equation*}\n\\xymatrix{\\bar F(p)\\ar[rr]^{(\\beta_F)_p}\\ar[dd]_{\\bar F(q\\leq p)} &&  F(p)\\ar[dd]^{ F(q\\leq p)}\\\\\n\\\\\n\\bar F(q)\\ar[rr]_{(\\beta_F)_q} && F(q)}\n\\end{equation*}\ncommutes for each $q\\leq p$ in $\\P_0$. If $q\\in\\P$ (so also $p\\in\\P$), this is exactly $$F(q\\leq p)1_{F(p)}=1_{F(q)}F(q\\leq p),$$ so assume $q=0$.\nChoose an $r\\leq p,p_0$. Then\n\\begin{eqnarray*}\n (\\beta_F)_0\\bar F(0\\leq p) & = & F(0\\leq p_0)F(r\\leq p_0)^{-1}F(r\\leq p)\\\\\n& =& F(0\\leq r)F(r\\leq p_0)F(r\\leq p_0)^{-1}F(r\\leq p)\\\\\n & = & F(0\\leq p)1_{F(p)} = \\bar F(0\\leq p)(\\beta_F)_p.\n\\end{eqnarray*}\nSo we find that $\\beta_F$ is a natural bijection for each $F$.\nFinally, we have to show that $\\beta$ is natural in $F$. So let $\\alpha:F\\to G$ in $\\Sh(\\P_0,K_X^0)$ and abbreviate $E_0\\circ I_0^*(\\alpha)=\\widehat{\\alpha I_0}$ by $\\bar\\alpha$. Then $\\bar\\alpha_p=\\alpha_p$ for $p\\in\\P$ and $\\bar\\alpha_0=\\alpha_{p_0}$. We have to show that for each $p\\in\\P_0$, the diagram\n\\begin{equation*}\n\\xymatrix{\\bar F(p)\\ar[rr]^{(\\beta_F)_p}\\ar[dd]_{\\bar \\alpha_p} && F(p)\\ar[dd]^{ \\alpha_p}\\\\\n\\\\\n\\bar G(p)\\ar[rr]_{(\\beta_G)_p} && G(p)}\n\\end{equation*}\ncommutes. If $p\\in\\P$, this says that $1_{G(p)}\\alpha_p=\\alpha_p1_{F(p)}$, so assume $p=0$. Then by the naturality of $\\alpha$ we obtain\n\\begin{equation*}\n (\\beta_G)_0\\bar\\alpha_0=G(0\\leq p_0)\\alpha_{p_0}=\\alpha_0F(0\\leq p_0)=\\alpha_0(\\beta_F)_0.\n\\end{equation*}\nSo $\\beta$ is a natural transformation such that each component $\\beta_F$ is a natural bijection, hence $\\beta:E_0\\circ I_0^*\\to 1$ is a natural isomorphism. We conclude that $E_0$ and $I_0^*$ constitute an equivalence of categories.\n\\end{proof}\n\n\n\n\n\\begin{corollary}\\label{cor:KXsheaves}\n Let $\\P$ be a downwards directed poset and $X\\subseteq\\P$. Then $$\\Sh(\\P,K_X)\\cong\\Sets^{Y^\\op},$$ where $Y=X$ if $\\up X=\\P$, and $Y=X_0$ (considered as subset of $\\P_0$) otherwise.\n\\end{corollary}\n\\begin{proof}\nAssume that $\\up X=\\P$. Then by Lemma \\ref{lem:derivedtopologies}, we have $K_X=J_X$, so $$\\Sh(\\P,K_X)=\\Sh(\\P,J_X)\\cong\\Sets^{X^\\op},$$ by Corollary \\ref{cor:JXsheaves}. Now assume that $\\up X\\neq\\P$. Then $\\up (X_0)=\\P_0$, so $K^0_{X}=J^0_{X_0}$ by Lemma \\ref{lem:derivedtopologiesforposetswithleastelement}. Using the previous theorem and Corollary \\ref{cor:JXsheaves}, we find $$\\Sh(\\P,K_X)\\cong\\Sh(\\P_0,K^0_X)=\\Sh(\\P_0,J^0_{X_0})\\cong\\Sets^{X_0^\\op}.$$\n\\end{proof}\n\n\nWe are also interested in the subcanonicity of the derived Grothendieck topologies. It turns out that the condition for subcanonicity is exactly the same as for subset topologies.\n\\begin{proposition}\n Let $\\P$ be a downwards directed poset and $X\\subseteq\\P$. Then for each $p\\in\\P$, $\\y(p)$ is a $K_X$-sheaf if and only if $X\\to\\down p=\\down p$. As a consequence, $K_X$ is subcanonical if and only if $X\\to\\down p=\\down p$ for each $p\\in\\P$.\n\\end{proposition}\n\\begin{proof}\nLet $X\\to\\down p=\\down p$. Proposition \\ref{prop:subcanonicalJX} assures that $\\y(p)\\in\\Sh(\\P,J_X)$, so by Proposition \\ref{prop:representablesheaf}, we obtain $J_X(q)\\cap\\D(\\down p)=\\emptyset$ for each $q\\nleq p$. Now $K_X\\subseteq J_X$, so certainly $K_X(q)\\cap\\D(\\down p)=\\emptyset$ for each $q\\nleq p$. Again by Proposition \\ref{prop:representablesheaf}, we obtain $\\y(p)\\in\\Sh(\\P,K_X)$.\n\nNow assume that $X\\to\\down p\\neq\\down p$. In the proof of Proposition \\ref{prop:subcanonicalJX}, we showed that this implies that $X\\cap\\down q\\subseteq\\down p$ for some $q\\nleq p$. If $q\\in\\up X$, we have $X\\cap\\down q\\neq\\emptyset$, so $\\down(X\\cap\\down q)\\in K_X(q)$, whereas also $\\down(X\\cap\\down q)\\subseteq\\down p$. If $q\\notin\\up X$, it follows from Lemma \\ref{lem:derivedtopologies} that $K_X(q)=J_\\atom(q)$. Now, $\\P$ is downwards directed, so there is an $r\\leq p,q$, which implies that $\\down r\\subseteq\\down p$ and $\\down r\\in K_X(q)$. So in both cases we find that $K_X(q)\\cap\\D(\\down p)\\neq\\emptyset$, whence by Proposition \\ref{prop:representablesheaf}, we obtain that $\\y(p)\\notin\\Sh(\\P,K_X)$.\n\\end{proof}\n\nFinally, if $X$ is a downwards directed subset of a poset $\\P$, we once again consider the map $\\G(X)\\to\\G(\\P)$ that assigns to each Grothendieck topology on $X$ its extension on $\\P$ in Definition \\ref{def:extensionGrothTop}.\n\\begin{definition}\n Let $\\P$ be a poset and $X$ a downwards directed subset of $\\P$. For each subset $Y$ of $X$, we define $L_{X,Y}$ to be the extension on $\\P$ of the derived Grothendieck topology induced by $Y\\subseteq X$ on $X$, which we denote by $K_Y^X$.\n\\end{definition}\nNotice that $K_Y^X=J^X_\\atom$, the atomic Grothendieck topology on $X$ if $Y=\\emptyset$. So $L_{X,\\emptyset}=L_X$.\n Recall the notion $\\bar\\up Z=\\up Z\\cap X$ in Definition \\ref{def:subposetdownset}.\n\\begin{proposition}\n Let $\\P$ be a poset and $X$ a downwards directed subset of $\\P$. For each subset $Y$ of $X$ we have $$\\Sh(\\P,L_{X,Y})\\cong\\Sets^{Z^\\op},$$ where $Z=Y$ if $\\bar\\up Z=X$, whereas $Z=Y_0$ if $\\bar\\up Y\\neq X$.\n\\end{proposition}\n\\begin{proof}\n By Proposition \\ref{prop:extensionofGrothTop}, $J_X\\leq L_{X,Y}$, so $\\bar L_{X,Y}$ is well defined and is equal to $K_Y^X$. Moreover, we are allowed to use Theorem \\ref{thm:ComparisonLemma}, i.e., the Comparison Lemma. Hence $$\\Sh(\\P,L_{X,Y})\\cong\\Sh(X,K_X^Y)\\cong\\Sets^{Z^\\op},$$ where we used Corollary \\ref{cor:KXsheaves} in the last equivalence.\n\\end{proof}\n\n\n\n\\section*{Conclusion}\nWe have found that a poset $\\P$ is Artinian if and only if every Grothendieck topology on $\\P$ equals $J_X$ for some subset $X$ of $\\P$.\nIf $\\P$ is not Artinian, we used the extension of the atomic Grothendieck topology on a certain subset of $\\P$ in order to show the existence of Grothendieck topologies that are not generated by some subset of $\\P$. Now, one could ask the following: if we completely identify all Grothendieck topologies on downwards directed subsets of $\\P$, do the extensions of these Grothendieck topologies together with the subset Grothendieck topologies exhaust all Grothendieck topologies on $\\P$? And if so, how do we characterize all Grothendieck topologies on a downwards directed poset $\\P$? A partial answer to the last question is given by the Grothendieck topologies of the form $K_X$ for some subset $X$ of $\\P$. In case of $\\P=\\Z$, these Grothendieck topologies together with the $J_X$-topologies exhaust all Grothendieck topologies on $\\Z$.\n\nIf there exist more Grothendieck topologies on a downwards directed poset $\\P$ not equal to $\\Z$, it might be possible to find the corresponding sheaf topoi using \\emph{Artin glueing}. Uniqueness of sites yielding a sheaf topos is not assured, see for instance the remark below Corollary \\ref{cor:JXsheaves}, hence this might not give a method of finding new Grothendieck topologies, but at least it provides some information about the corresponding Grothendieck topologies. A technique of obtaining new sheaf topoi is provided by \\emph{Artin glueing}\\index{Artin glueing}, which is described in \\cite{CJ}. We illustrate why this technique might be useful by considering $K_X$-sheaves again.\n\nBy Lemma \\ref{lem:derivedtopologies}, we know that $K_X$ is a Grothendieck topology on $\\P$, which is equal to $J_X$ on $\\up X$, whereas it is equal to $J_\\atom$ on its complement in $\\P$. By Corollary \\ref{cor:JXsheaves}, $\\Sets^{X^\\op}$ is equivalent to $\\Sh(\\up X,J_X)$, with $J_X$ the subset Grothendieck topology on $\\up X$ generated by $X$. Moreover, $\\Sh((\\up X)^c,J_\\atom)\\cong\\Sets$ by Example \\ref{ex:atomsheaves}, since each sheaf on $(\\up X)^c$ is constant. Moreover, by Lemma \\ref{lem:derivedtopologies}, we know that $K_X$ is a Grothendieck topology on $\\P$, which is equal to $J_X$ on $\\up X$, whereas it is equal to $J_\\atom$ on its complement in $\\P$. Therefore, in some way the $K_X$-sheaves are obtained by glueing $J_X$-sheaves on $\\up X$ and $J_\\atom$-sheaves on $(\\up X)^c$, or equivalently, by glueing $\\Sets^{X^\\op}$ together with $\\Sets$.\n\nArtin glueing can be described as follows. Given two categories $\\CC$ and $\\mathcal{D}$ and a functor $F:\\CC\\to\\mathcal{D}$, the category obtained by Artin glueing along $F$ is the comma category $(\\mathcal{D}\\down F)$ with objects $(C,D,f)$, where $C$ is an object of $\\CC$, $D$ an object of $\\mathcal{D}$ and $f$ a morphism $f:D\\to FC$. A morphism $(C,D,f)\\to (C',D',f')$ between objects of $(\\mathcal{D}\\down F)$ is a pair $(\\alpha,\\beta)$, where $\\alpha:C\\to C'$ in $\\CC$ and $\\beta:D\\to D'$ in $\\mathcal{D}$ such that\n\\begin{equation*}\n\\xymatrix{D\\ar[rr]^{\\beta}\\ar[dd]_{f} &&  D'\\ar[dd]^{f'}\\\\\n\\\\\n FC\\ar[rr]_{F\\alpha} && FC'}\n\\end{equation*}\n\nLet $\\P$ be a downwards directed poset and take $X\\subseteq\\P$ such that $\\up X\\neq\\P$. Let $\\CC=\\Sets$, $\\mathcal{D}=\\Sets^{X^\\op}$, and $F=\\Delta_{X^\\op}$. Let the functor $\\Sets\\to\\Sets^{X^\\op}$ be defined as $\\Delta_{X^\\op}(A)(x)=A$ for each $x\\in X$ and $A\\in\\Sets$. Then the category obtained by Artin glueing along $\\Delta_{X^\\op}$ consists of objects $(A,G,f)$, where $G\\in\\Sets^{X^\\op}$, $A\\in\\Sets$ and $f$ a natural transformation $G\\to\\Delta_{X^\\op}(A)$. This corresponds exactly to a presheaf $G_0\\in\\Sets^{X_0^\\op}$, where $X_0$ is the set obtained by adding a least element to $X$. The correspondence is given by defining $G_0(x)=G(x)$ if $x\\in X$ and $G_0(0)=A$, $G_0(x\\leq y)=G(x\\leq y)$ if $x\\in X$ and $G_0(0\\leq y)=f_y$ for each $y\\in X$. Moreover, if $(\\alpha,\\beta):(A,G,f)\\to(A',G',f')$ is a morphism, this corresponds to a natural transformation $\\gamma:G_0\\to G_0'$ defined by $\\gamma_x=\\beta_x$ if $x\\in X$ and $\\gamma_0=\\alpha$. Since $f'\\circ\\beta=\\Delta_{X^\\op}(A)\\circ f$, this is indeed a natural transformation. Conversely, given an element $H\\in\\Sets^{X_0^\\op}$, we obtain an element $(H(0),H|_{X},f)$ in $(\\Sets\\down\\Delta_{X^\\op})$, the category obtained by Artin glueing along $\\Delta_{X^\\op}$, where $f:H|_X\\to\\Delta_{X^\\op}(H(0))$ is defined as $f_x=H(0\\leq x)$ for each $x\\in X$. It turns out that these correspondences constitute an equivalence between $(\\Sets\\down\\Delta_{X^\\op})$ and $\\Sets^{X_0^\\op}$. In Corollary \\ref{cor:KXsheaves}, we already found that the latter is equivalent to $\\Sh(\\P,K_X)$, hence Artin glueing yields exactly the $K_X$-sheaves.\n\nIt might be interesting to see what other topoi we obtain by Artin glueing along other functors, by glueing different categories together. For instance, the $K_X$ sheaves are obtained by glueing $\\Sets^{X^\\op}$ and $\\Sets$, which corresponds to adding an element $0$ below $X$. What happens if we replace $\\Sets$ by $\\Sets^{Y^\\op}$ for some poset $Y$? Can we glue this presheaf category together with $\\Sets^{X^\\op}$. Does this correspond to a topos equivalent to $\\Sh(\\P,J)$ for some Grothendieck topology $J$ on $\\P$? If so, how can we guess what form $J$ should have? Is $J$ a mixture of two already known Grothendieck topologies $J_1$ and $J_2$ just like $K_X$ is a mixture of $J_X$ and $J_\\atom$? In any case, Artin glueing might be an interesting technique for further research.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nComputer Vision (CV) has evolved as a key technology in the last decade for numerous applications replacing human supervision. CV has the ability to gain a high-level understanding and deriving information by analyzing and processing digital images or videos. These systems are also designed to automate various tasks that the human visual system can do. There are numerous interdisciplinary fields where CV is used; Automatic Inspection, Modelling Objects, Controlling Processes, Navigation, Video Surveillance, etc.\n \n'Video Surveillance' is a key application of CV which is used in most public and private places for observation and monitoring. Nowadays intelligent video surveillance systems are used which detect, track and gain a high-level understanding of objects without human supervision. Such intelligent video surveillance systems are used in homes, offices, hospitals, malls, parking areas depending upon the preference of the user. \n \nThere are several computer vision-based studies that primarily discuss on aspects such as scene understanding and analysis \\cite{wu2019deep} \\cite{shirahama2015weakly}, video analysis \\cite{long2017edge} \\cite{suresha2020study}, anomaly\/abnormality detection methods \\cite{xu2018anomaly}, human-object detection and tracking \\cite{guo2019detecting}, activity recognition \\cite{singh2017graph}, recognition of facial expressions \\cite{elgarrai2016robust}, urban traffic monitoring \\cite{zhou2017fast}, human behavior monitoring \\cite{popoola2012video}, detection of unusual events in surveillance scenes \\cite{muhammad2019efficient}, etc. Out of these different aspects anomaly detection in video surveillance scenes has been discussed further in our review.\n \nAnomaly Detection is a subset of behavior classification. Anomalies are unusual behavior or events that deviate from the normal. Anomaly detection in video scenes is the cutting-edge technology that monitors unusual activities using artificial intelligence. Examples of an anomaly in video surveillance scenes are shown in Figure 1; a person walking in a restricted area, vehicles moving in the wrong direction, a cyclist riding on a footpath; a sudden crowd of people; a person carrying a suspicious bag, a person climbing over the fence, etc.\n\n\\begin{figure}\n    \\centering\n   \\includegraphics[width=7cm]{\"Fig_2\"}\\\\\n  \\caption{Anomaly Detection in video surveillance scenes. Reference: UCSD Dataset. (a) Cyclist Riding on Footpath. (b)(c) Vehicles moving on footpath (d) Pedestrian walking on lawn.} \n\\end{figure}\n\n\nIntelligent video surveillance systems track unusual suspicious behavior and raise alarms without human intervention. Various machine learning tools are used for the detection and tracking of human objects in video scenes and to classify the events as normal and abnormal. The general overview of the anomaly detection is shown in Figure 2. In this process, visual sensors in the surveillance environment collect the data. This raw visual data is then subjected to pre-processing and feature extraction \\cite{georgiou2020survey}. The resulting data is provided to a modeling algorithm, in which a learning method is applied to model the behavior of surveillance targets and to determine whether the behavior is abnormal or not. \nFor the purpose of anomaly detection, various machine learning tools use cloud computing for data processing and storage \\cite{ghosh2020edge}. Cloud computing requires large bandwidth and longer response time \\cite{srivastava2016survey} \\cite{yang2019edge}. Anomaly detection in video surveillance is a delay sensitive application and requires low latency. So considering these aspects, cloud computing in combination with edge computing provides a better solution for real-time intelligent video surveillance \\cite{shi2016edge}. \n\nThe research efforts in anomaly detection for video surveillance are not only scattered in the learning methods but also approaches. Initially, the researchers broadly focused on the use of different handcrafted spatio-temporal features, conventional image processing methods. Recently, more advanced methods like object-level information and machine learning methods for tracking, classification, and clustering have been used to detect anomalies in video scenes. In this survey, we aim to bring together all these methods and approaches to provide a better view of different anomaly detection schemes.\n\nFurther, the choice of surveillance target varies according to the application of the system. The reviews done so far have a disparity in the surveillance targets. We have categorized the surveillance targets primarily focusing on four types; automobile, individual, crowd, object, or event.  \n\n\n\nMoreover, the evolution of cloud and edge devices and their employment in the automated surveillance and anomaly detection is important. Traditionally, massive surveillance data is sent to cloud servers where the data is analyzed and modelled to detect abnormal behaviour or events. With the advancement in the cloud technology, the physical servers is replaced by cloud servers to perform computationally heavy task of computer vision. However, inevitable network latency and operational delays make cloud computing inefficient for the time sensitive applications such as anomaly detection. Thus, this survey discuss the application of Edge Computing (EC) with cloud computing which enhance the response time for anomaly detection. This survey also presents recent techniques in anomaly detection using edge computing in video surveillance.\\\\\n\nNone of the previous surveys address the confluence anomaly detection in video surveillance and edge computing. In this study, we provide a detailed review of latest publications on anomaly detection in video surveillance using edge computing. This review will also address the challenges and opportunities involved in anomaly detection using edge computing.\\par  \n\nThe research contributions of this review article are as follows:\n\\begin{enumerate}\n    \\item Presented review attempts to connect the disparity in the evaluation of the problem formulations and suggested solutions for the anomaly detection. \n    \\item The suitability of anomaly detection techniques in the context of application area, surveillance targets, learning methods, and modeling techniques.\n    \\item  We explore anomaly detection techniques used in vehicle parking, automobile traffic, public places, industrial and home surveillance. The emphasis in these surveillance scenarios is on humans (individual\/crowd), objects, automobiles, events and their interactions. \n    \\item  The review will also focus on modern-age edge computing technology employed to detect anomalies in video surveillance applications and further discuss the challenges and opportunities involved. \n    \n\\end{enumerate}\n\n\n\n\n\\begin{figure}\n    \\centering\n   \\includegraphics[width=10cm]{\"Basicblockdiagram\"}\\\\\n  \\caption{General Block Diagram of Anomaly Detection} \n\\end{figure}\n\nFurther, to the best of our knowledge anomaly detection using edge computing paradigm in video surveillance systems is less explored and not surveyed.\\\\\n\nWe present this survey from the aforementioned perspectives and organize it into seven sections; Section-II presents the prior published surveys, Section-III presents different surveillance targets in corresponding application areas. Section-IV article explores methodologies employed in anomaly detection. Section-V talks about the adoption of edge computing, its challenges, and opportunities in video surveillance and anomaly detection. Section-VI presents critical analysis followed by conclusion in Section-VII\n\n\\section{Related Work}\nThe context of anomaly is different in various scenarios, but can broadly be categorized into anomalies in road traffic and anomalies in human or crowd behavior. Apart from anomalies in human\/crowd behavior, on-road vehicle tracking and surveillance are also extensively studied and reviewed over the past decade. The advances in vehicle detection using monocular, stereo vision, and active sensor\u2013vision fusion for an on-road vehicle are surveyed in \\cite{sivaraman2013looking}. Automobile detection and tracking are surveyed in \\cite{tian2014hierarchical}. The performance dependency of a vehicle surveillance system on traffic conditions is also discussed and a general architecture for the hierarchical and networked vehicle surveillance is presented. The techniques for recognizing vehicles based on attributes such as color, logos, license plates are discussed in \\cite{shobha2018review}. The anomaly detection methodologies in road traffic are surveyed in \\cite{kumaran2019anomaly}. As the anomaly detection schemes cannot be applied universally across all traffic scenarios, the paper categorizes the methods according to features, object representation, approaches, and models. \n\nUnlike anomaly detection in-vehicle surveillance, anomalies in human or crowd behavior are much more complex. Approaches to understanding human behavior are surveyed in \\cite{popoola2012video} based on human tracking,human-computer interactions, activity tracking, and rehabilitation. In \\cite{sodemann2012review}, the learning methods and classification algorithms are discussed considering crowd and individuals as separate surveillance targets to detect the anomaly. However, the occlusions and visual disparity in the crowded scenes reduce the accuracy in detecting the anomalies. A review \\cite{li2014crowded} focuses on the aforementioned aspects and learns the motion pattern of the crowd to detect abnormal behaviour. The recognition of complex human behaviour and various anomaly detection techniques are discussed in \\cite{li2016anomaly}. Further, the use of moving object trajectory-clustering \\cite{yuan2017review}, and trajectory-based surveillance \\cite{ahmed2018trajectory} to detect abnormal events are observed in the literature. The recent surveys on anomaly detection and automated video surveillance are listed in Table 1.\\\\ \n\n\\begin{table}[]\n    \\centering\n    \\caption{Recent Surveys on Video Surveillance}\n    \\label{table_1}\n    \\begin{tabular}{l|l|l}\n        Year & Existing Work & Broad Topics \\\\\n        \\hline\n        2019 & Ahmed et al. \\cite{ahmed2018trajectory} & Trajectory-Based Surveillance \\\\\n         2018 & Shobha et al.\\cite{shobha2018review} & Vehicle Detection, Recognition and \\\\\n     & & Tracking \\\\\n     2016 & Yuan et al. \\cite{yuan2017review} & Moving object trajectory clustering\\\\\n     2016 & Xiaoli Li et al. \\cite{li2016anomaly} & Anomaly Detection Techniques \\\\\n    \n     \n     2015 & Li et al. \\cite{li2014crowded} & Crowded Scene Analysis \\\\\n     2014 & Tian et al. \\cite{tian2014hierarchical}& Vehicle Surveillance \\\\\n     2013 & Sivaraman et al. \\cite{sivaraman2013looking}& Vehicle Detection, Tracking, \\\\\n     2012 & Popoola et al. \\cite{popoola2012video} & Abnormal Human Behaviour Recognition \\\\\n     2012 & Angela A. Sodemann \\cite{sodemann2012review} & Human Behaviour Detection \\\\\n    \\end{tabular}\n    \n\\end{table}\n\n\n\\section{Surveillance Targets}\nThe entities upon which the surveillance operates are called surveillance targets. Surveillance targets are those entities among which the anomaly detection method aims to detect anomalies. In the context of surveillance areas, the surveillance targets can be categorized as; the individual, crowd, automobile traffic, object or event, the interaction between humans and objects, etc. A Venn diagram showcasing the relationship between video surveillance, anomaly detection, and the surveillance targets (individual, crowd, automobile traffic, object) is illustrated in Figure 3. As shown in Figure 3, there is a large domain of research emphasizing on automated surveillance of targets that are included in the process of anomaly detection and scene understanding.\n\n\\subsection{Individual}\nAnomaly detection is deployed to ensure safety of individuals in public places like hospitals, offices, public places or at home. It recognizes patterns of human behavior based on sequential actions and detect abnormalities \\cite{torres2018detection} \\cite{lee2009nonsupervised} . Several approaches have been proposed to detect anomalies in behavior involving breach of security \\cite{puvvadi2015cost}, running \\cite{zhou2019anomalynet}, lawbreaking actions like robbery \\cite{lao2009automatic}, fall of elderly people \\cite{liu2021privacy}.\n\n\\subsection{Crowd}\nThis review distinguishes between individuals and crowds as shown in Figure 3. Both of these targets consist of a single entity, the methods used to identify abnormalities are distinct for individuals and crowds \\cite{li2013anomaly} \\cite{tripathi2019convolutional}. Any change in motion vector or density or kinetic energy indicates an anomalous crowd motion \\cite{yuan2014online, kaltsa2015swarm, colque2016histograms, sabokrou2017deep, sabokrou2018deep}. In \\cite{li2016anomaly}, behavior such as people suddenly running in different directions or the same direction is considered anomalous. A crowd cannot only be a crowd of individuals but a fleet of taxis as well; \\cite{bock2019smart} allows the scene understanding and monitoring on a fleet of taxis. Crowd counting in high density becomes difficult. \\cite{khan2020scale} detects only the head of people as it is the only visible part as the body is occluded in public. \n\n\n\\subsection{Automobiles and Traffic}\nThe automobile and traffic surveillance intends to monitor and understand automobile traffic, traffic density, traffic law violations, safety issues like; accident or parking occupancy. In smart cities, automobiles become important surveillance targets and extensively surveyed for traffic monitoring, lane congestion, and behaviour understanding \\cite{zhou2017fast} \\cite{kumaran2019anomaly} \\cite{bock2019smart} \\cite{wang2018offloading} \\cite{ke2016real} \\cite{ke2018real} \\cite{chen2019distributed}. In metro cities, finding a vacant parking spot for vehicles is a tedious job for drivers; \\cite{nieto2018automatic} allows drivers to find a vacant parking area. For better accessibility, security, and comfort of the citizens, studies also focus on traffic law violations which include vehicles parked in an incorrect place \\cite{ke2020smart}, predicting anomalous driving behavior, abnormal license plate detection \\cite{liu2018hybrid}, detection of road accidents \\cite{singh2018deep} and detection of collision-prone behavior of vehicles \\cite{roy2020detection}.\n\n\n\\subsection{Inanimate objects or events}\nThe target in this category is divided into events and inanimate objects. Some of the examples of abnormal events are; an outbreak of fire, which is a common calamity in industries \\cite{muhammad2019efficient} and needs automatic detection and quick response. Similarly, it is challenging to detect smoke in the foggy environment; \\cite{muhammad2019edge} presents smoke detection in such an environment which plays a key role in disaster management. \nSometimes there are defects in the manufacturing system and it is tedious for humans to examine small details; \\cite{li2018deep} proposes a scheme for detecting manufacturing defects in industries.\n\n\\subsection{Interaction between humans and objects}\nIn this category, anomaly detection schemes are associated with the interaction between humans and objects. Both individuals and objects together give the potential benefits of detecting interaction between them such as an individual carrying a suspicious baggage \\cite{nawaratne2019spatiotemporal}, individual throwing a chair \\cite{angelini2019privacy}. Some studies attempt to account for both pedestrians and vehicles in the same scene such as cyclists driving on a footpath, pedestrians walking on the road \\cite{nawaratne2019spatiotemporal} \\cite{xu2018real} \\cite{wang2019saliencygan} \\cite{shojaei2019semi}. In \\cite{sabokrou2017deep} abnormal behavior is identified by objects like a skateboarder, a vehicle or a wheelchair on footpath.\n\n\\begin{figure}\n    \\centering\n   \\includegraphics[width=6cm]{\"Surveillance_Target\"}\n    \\caption{Venn Diagram of Surveillance Targets} \n\\end{figure}\n\n\n\\section{Anomaly Detection Methodologies in Video Surveillance}\nTo improve the safety and well-being of individuals and surrounding, the surveillance has become imperative nowadays. However, it is not possible to view video surveillance scenarios for twenty-four hours and track anomalous events, there is a need for an intelligent surveillance system. \n\nAnomalies can be contextual, point, or collective. Contextual anomalies are data instances that are considered anomalous when viewed against a certain context associated with the data instance. Point anomalies are single data instances which are different with respect to others \\cite{bozcan2021context}. Finally, collective anomalies are data instances which are considered anomalous when viewed with other data instances, concerning entire dataset \\cite{cheng2015gaussian}. To detect anomalies in automated surveillance, advanced detection schemes have been developed over a decade. In this survey, we categorize them broadly into; learning-based and modeling-based approaches and further sub-categorize for clear understanding. The anomaly classification based on learning and based on approaches is shown in Figure 4.\n\n\n\\begin{comment}\n\\subsection{Types of Anomaly}\nIn the context of video surveillance, the anomalies are commonly classified as global anomalies and local anomalies. Global anomalies are present in a video segment or a frame of the video without specifying the exact location where the discrepancy occurred. Whereas, the local anomalies occur within a specific area of the scene. Such, global and local anomalies are detected using a hierarchical framework employing Gaussian Process Regression (GPR) and hierarchical feature representation \\cite{cheng2015gaussian}. \n\\end{comment}\n\n\\subsection{Learning}\nAny event or behaviour that deviates from the normal is called an anomaly. The learning algorithms learn anomalies or normal situations based on the training data which can be labeled or unlabeled. Depending upon the methodologies used the various learning methods for anomaly detection can be classified as;\n\n\\begin{enumerate}\n    \\item Supervised Learning,\n    \\item Unsupervised Learning and\n    \\item Semi-supervised Learning.\n\\end{enumerate}\n\nThe anomaly detection classification on the basis of learning is shown in Table 2.\n\\subsubsection{Supervised learning}\n\nSupervised learning uses labelled dataset to train the algorithm to predict or classify the outcomes. It gives a categorical output or probabilistic output for the different categories. The training data is processed to form different class formulations; single class, two-class or multi-class.\\\\ \nWhen the training data contains data samples either of normal situations or anomalous situations only then it is called single class formulation \\cite{sodemann2012review}. Since the training data consists of a single class, the labeling is difficult. In a single class approach, if the detector is trained on normal events then the events that fall outside the learned class are classified as anomalous \\cite{asad2021multi}. Various approaches to classify and model anomalies with such training data use a 3D Convolutional Neural Network Model \\cite{ji20123d}, Stacked Sparse Coding (SSC) \\cite{xu2018anomaly}, adaptive iterative hard-thresholding algorithm \\cite{zhou2019anomalynet}.\\\\\nApart from single and two class formulation, an approach where multiple classes of events are learned is called multi-class formulation. In this approach before anomaly detection, certain rules are defined regarding behavior classification. Anomaly detection is then performed using these set of rules \\cite{lao2009automatic} \\cite{saligrama2012video}. However, this approach has a drawback that, the events that are learned can only be reliably recognized and the events that do not span the learned domain are incorrectly classified. Thus, the multi-class approach may not provide optimum results outside a scripted environment.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=7cm]{\"AnomalyClassification\"}\n    \\caption{Anomaly Classification} \n\\end{figure}\n\n \n\n\\begin{table}\n    \\centering\n\\caption{Anomaly Detection Classification}\n\\begin{tabular}{l|l}\n    Type of Learning & Reference \\\\\n    \\hline\n    Supervised &    \\cite{xu2018anomaly} \\cite{zhou2019anomalynet} \\cite{lao2009automatic}  \\cite{asad2021multi} \\cite{ji20123d} \\cite{saligrama2012video} \\\\\n    Unsupervised &  \\cite{wu2019deep} \\cite{nawaratne2019spatiotemporal}  \\cite{song2019learning} \\cite{kumar2017visual} \\cite{lamba2019segmentation} \\cite{ren2015unsupervised} \\cite{chu2018sparse} \\cite{sun2020adversarial} \\cite{cocsar2016toward}  \\cite{roy2018unsupervised} \\cite{perveen2018spontaneous} \\cite{chowdhury2020unsupervised} \\cite{chriki2020uav} \\cite{chriki2021deep} \\cite{bansod2020crowd} \\\\\n    Semi-supervised & \\cite{tripathi2019convolutional} \\cite{wang2019saliencygan} \\cite{liu2016semi} \\\\\n\\end{tabular}\n\\end{table}\n\n\\subsubsection{Unsupervised Learning}\n\nIn unsupervised learning, given a set of unlabeled data, we discover patterns in data by cohesive grouping, association, or frequent occurrence in the data. In this approach, both normal and abnormal training data samples which do not have any label. An algorithm discovers patterns and groups them together with an assumption that the training data consist of mostly normal events and occurs frequently while rare events are termed as anomalous \\cite{sodemann2012review}\\cite{wu2019deep}. However, because the non-deterministic nature of anomalous events and insufficient training data it is challenging to automatically detect anomalies in surveillance videos. To address these issues, \\cite{song2019learning} presented an adversarial attention-based auto-encoder network. It uses a reconstruction error produced by the auto-encoder to diverge abnormal events\/anomalies from normal events. \n\n\\cite{kumar2017visual} introduces a trajectory-based hierarchical clustering algorithm framework named Visual Assessment of Tendency (VAT) to detect abnormal trajectories. For anomaly detection in a crowd, \\cite{lamba2019segmentation} estimates contour-based trajectory clustering method where the crowd flow is segmented using density-based DBSCAN clustering. For experimentation high density-based UCF, Collective motion, Violent Flows datasets are used to provide optimum results. \\\\ \n\nRecently, deep 3-dimensional convolutional network (C3D) \\cite{chu2018sparse} is widely used for video surveillance applications. The local spatiotemporal patterns captured by C3D are suitable for video data however, it is tedious to incorporate this supervised model to detect anomalies as there are no categorical labels involved and hence, all the events will be viewed as normal events. To cope with this, self-supervised signals are developed by extracting spatiotemporal patterns in videos and agglomerative clustering is employed to obtain a similarity relationship between the inputs to train C3D. Similarly by utilizing the self-similarity property of the training data, Unsupervised Kernel Learning with Clustering Constraint (CCUKL) is deployed to learn abnormal events where the feature space is learned using Non-negative matrix factorization (NMF) and the clustering degree is measured using support vector data description (SVDD) \\cite{ren2015unsupervised}.\n\nSome studies have also proposed to used C3D and adversarial auto-encoder for detecting anomalous events in videos \\cite{sun2020adversarial}. The 3D convolution auto-encoder model aims to learn the spatiotemporal patterns and train the auto-encoder by using the de-noising reconstruction error and adversarial learning strategy to detect anomalies without supervision \\cite{cocsar2016toward}. To distinguish between new anomalies and normality that evolve, Incremental Spatio-Temporal Learner (ISTL) remain updated about the changing nature of anomalies by utilizing active learning with fuzzy aggregation \\cite{nawaratne2019spatiotemporal}. ISTL approach is estimated on a spatiotemporal auto-encoder model which consists of convolution layers and ConvLSTM (Convolution Long-Short Term Memory)layers that learn spatial-temporal regularities. Any anomalies in the scene are periodically monitored by a human observer so that the system dynamically evolves normal behavior using fuzzy aggregation.\n\nFor the purpose of action recognition in surveillance scenes \\cite{roy2018unsupervised} proposes a Gaussian mixture model called Universal Attribute Modelling (UAM) using unsupervised learning approach. The UAM is also been used for facial expression recognition where it captures the attributes of all expressions \\cite{perveen2018spontaneous}\n\nFurther, for autonomous vehicles like cars or UAVs (Unmanned Aerial Vehicles) it is very essential to distinguish between normal and anomalous states. Chowdhury et al. estimates the degree of abnormality using an unsupervised heterogeneous system from real-time images and IMU (Inertial Measurement Unit) sensor data in a UAV \\cite{chowdhury2020unsupervised}. They also demonstrated a CNN architecture to estimate an angle between a normal image and query image, to provide a measure of anomaly. \nRecently, one-class classifiers \\cite{chriki2020uav} \\cite{chriki2021deep} are evolved as state-of-the-art for anomaly detection. They use a CNN with One-Class Support Vector Machine (OCSVM) to detect anomalies or abnormalities in the data. \n\n\\cite{bansod2020crowd} uses a Physics theory of momentum which is used to establish a relationship between the foreground region and object motion. By utilizing the statistical and positional features, a histogram of magnitude and momentum (HoMM) is derived, and further by K-means clustering and distance calculation anomalous and non-anomalous events are recognized.\n\n\n\\subsubsection{Semi-supervised Learning}\nSemi-supervised learning falls between supervised learning and unsupervised learning. It combines a small amount of labeled data with a large amount of unlabeled data during training. Semi-supervised learning is used where less variety of labeled training dataset is available such as in extracting suspicious events in smart security camera \\cite{wang2019saliencygan}. In such situations, Salient Object Detection (SOD) is a commonly used fundamental pre-processing tool for deep learning models like SaliencyGAN (Saliency Generative Adversarial Network). Different combinations of labelled and unlabelled data are used in training of SaliencyGAN to obtain misclassified events. In some applications, Laplacian Support Vector Machine (LapSVM) utilizes unlabeled samples to learn a more accurate classifier \\cite{liu2016semi}. It is observed that, there is a considerable improvement in learning accuracy when unlabeled data is used in conjunction with small amount of labelled data. \\cite{tripathi2019convolutional} proposes an object tracking method using structural sparse representation where edge detection is deployed to avoid tracking error accumulation in case of occlusion in complex scenes.\n\n\n\n\n\\begin{table*}[]\n    \\centering\n    \\caption{Categorization of Anomaly Detection Techniques}\n    \\begin{tabular}{l|l|l|l}\n       \n       Approach  & Ref & Technique & Highlights \\\\\n        \\hline\n         \n        Statistical & \\cite{cheng2015gaussian} & Gaussian Process  & Hierarchical Feature Representation, Gaussian Process  \\\\\n         Based & & Regression &  Regression. STIP ( spatio-temporal interest points) is used   \\\\\n        & & & to detect local and global video anomaly detection;  \\\\\n        &&& Dataset: Subway, UCSD, Behave, QMUL Junction;\\\\\n        & & & Parameters: AUC, EER\\\\\n        &&& (AUC: Area Under Curve), EER: Equal Error Rate)\\\\\n        \n        & \\cite{kaltsa2015swarm} & Histogram Based & HOG, Histograms of Oriented Swarms (HOS) KLT interest \\\\\n     & & Model  & point tracking are used to detect anomalous event in crowd;  \\\\ \n     & & & Aims at achieving high accuracy and low computational cost; \\\\\n     &&& Dataset used: UCSD, UMN; Parameters: ROI, EER, DR; \\\\\n     &&& (ROI: Region of Interest, DR: Detection Rate)\\\\\n     \n     & \\cite{bansod2020crowd} & Histogram Based   & Crowd anomaly detection using HoMM (histogram of \\\\\n     & &   &  magnitude and momentum); background removal,\\\\\n     & &  & feature extraction (optical flow), anomaly detection \\\\\n     & & & (using K-means clustering); Dataset: UCSD, UMN;\\\\\n     & & &  Parameters: AUC, EER, RD (Rate  Reduction) \\\\\n& & &  True Positive Rate (TPR), False Positive Rate (FPR)\\\\\n     \n      & \\cite{nguyen2015bayesian} & Bayesian Model & Bayesian Non-Parametric (BNP) approach,  Hidden Markov\\\\\n     & & & Model (HMM) and Bayesian non-parametric factor  \\\\\n     & & & analysis is employed for data segmentation and pattern\\\\\n     & & &  discovery of abnormal events. Dataset: MIT\\\\\n     & & & Parameters: Energy Distribution \\\\\n     \n     \n     \n     & \\cite{isupova2016anomaly} & Bayesian Model & Bayesian non-parametric dynamic topic model is used. \\\\ \n     & & &Hierarchical Dirichlet process (HDP) is used to detect anomaly.\\\\ \n     & & & Dataset: QMUL-junction; Paremeters: ROC, AUC\\\\\n     &&& (ROC: Receiver Operating Characteristics)\\\\\n     \n        \n       & \\cite{sabokrou2018deep} & Gaussian Classifier & Fully convolutional neural network, Gaussian classifier is \\\\\n    &  &  &  used. It extracts distinctive features of video regions to  \\\\\n     &  &  & detect anomaly. Dataset: UCSD, Subway; \\\\\n     &&&  Parameters: ROC, EER, AUC; \\\\\n     & & & Aims to increase the speed and accuracy.\\\\\n    \n        \n     \n     & \\cite{yuan2014online} & Histogram Based & Structural Context Descriptor (SCD), SHOF, 3D DCT  \\\\\n     & & Model & Selective Histogram of Optical Flow (SHOF),\\\\\n     & & & Discrete Cosine Transform (DCT)\\\\\n     & & &  object tracker, Spatiotemporal analysis for abnormal   \\\\\n     & & & detection is crowd using Energy Function  \\\\\n     & & & Dataset: UMN and UCSD; \\\\\n     &&& Parameters: ROC, AUC, TPR, FPR\\\\\n     \n     \n    \n\n     \\hline\n     Proximity  & \\cite{colque2016histograms} & Histogram  &  Histogram of optical flow and motion entropy (HOFME) is \\\\\n     Based & & &  used to detect the pixel level features diverse anomalous  \\\\\n     & & & events in crowd  anomaly scenarios as compared with  \\\\\n     & & &  conventional features. Nearest Neighbor threshold is used  \\\\\n     &&&  by HOFME.\\\\\n     &&& Dataset: UCSD, Subway, Badminton;  \\\\\n     &&& Parameters: AUC, EER \\\\\n     \n     &  \\cite{liu2018accumulated} & Accumulated & ARD method is used for large-scale traffic data and detect\\\\\n     && Relative &   outliers;  Dataset: Self-deployed  \\\\\n     & & Density (ARD) & Parameter: Detection Success Rate (DSR) \\\\\n     &\\cite{hu2018anomaly} & Density Based & A weighted neighborhood density estimation is used to detect  \\\\\n     & & & anomalies. Hierarchical context-based local kernel regression. \\\\\n     & & & Dataset: KDD, Shuttle; Parameters: Precision, recall \\\\\n     \n    \n    \n\n\n\n\\hline\n  \\end{tabular}\n    \n    \\label{table_2}\n\\end{table*}\n\n\\begin{table*}[]\n    \\centering\n  \n    \\begin{tabular}{l|l|l|l}\n       \n       Approach  & Ref & Technique & Highlights \\\\\n       \\hline\n       \n       Proximity & \\cite{lamba2019segmentation} & Trajectory Extraction & Foreground segmentation by using active contouring \\\\\n     Based & & Flow analysis & Density based DBSCAN Clustering.\\\\\n     && &Dataset: UCF Web, Collective Motion, and Violent Flows \\\\\n     & & & Parameters: MAE (Mean Absolute Error) and F-score \\\\\n     \n     \\hline\n       \n      Classification & \\cite{mo2013adaptive} & Adaptive Sparse  & Trajectory-based video anomaly detection using joint  \\\\\n Based & & Representation & sparsity model. Dataset: CAVIAR. Parameters: ROC\\\\\n \n  & \\cite{wang2014detection}  & One-Class  & OCC based anomaly detection techniques using SVM. \\\\\n   & & classification & Pixel level features are extracted. \\\\\n     &&&  Dataset: PETS2009 and UMN; Parameters: ROC \\\\\n     \n      & \\cite{nikouei2018smart} & Harr-Cascade,  & Smart Surveillance as an Edge Network using Harr-\\\\\n     & & HOG feature  & Cascade, SVM, L-CNN. Provides fast object detection  \\\\\n     & & extraction & and tracking. \\\\\n     & & & Dataset: VOC07, ImageNet; Parameters: Accuracy\\\\\n     \n \n\n & \\cite{xu2018real} & Stacked Sparse & Intraframe classification strategy; Dataset: UCSD, Avenue, \\\\\n  & & Coding  & Subway; Parameters: EER, AUC, Accuracy\\\\\n\n\n     \n & \\cite{shi2017sequential} & Sequential Deep   & Dense trajectories are projected into 2D plane, \\\\\n &  & Trajectory &  long term motion CNN-RNN network is employed.\\\\\n& &  Descriptor & Dataset: KTH, HMDB51, UCF101; Parameters: Accuracy\\\\\n\n&\\cite{ghosh2020edge} & Autoencoders,  & Feature learning with deep learning, autoencoder is placed \\\\\n & &Data Reduction & on the edge, decoder part is placed on the cloud\\\\\n& & & Dataset: HAR, MHEALTH; Parameters: Accuracy \\\\\n\n& \\cite{ke2016real} & Kanade\u2013Lucas\u2013& k-means clustering, connected graphs and traffic flow\\\\\n&  & Tomasi  (KLT)& theory are used estimate real-time parameters from \\\\\n& &  tracker  & aerial videos; Dataset: Manual;  \\\\\n&&& Parameters: speed, density, volume of vehicles\\\\\n\n\n \\hline\n \n       \n       Reconstruction & \\cite{lei2020discriminative} &  Hyperspectral Image  & Discriminative reconstruction method for HSI anomaly  \\\\\n& & (HSI) Analysis &  detection with spectral learning (SLDR). Loss Function  \\\\\n& & & of SLDR Model is generated. \\\\\n& & & Dataset: ABU,  San Diego\\\\\n& & & Parameters: ROC, AUC\\\\\n\n& \\cite{xu2019low} & Hyperspectral  & Low-rank sparse matrix decomposition (LRaSMD) \\\\\n& & Imagery (HSI) & based dictionary reconstruction is used for anomaly \\\\\n& & & detection. Dataset: STONE, AVIRIS\\\\\n& & & Parameters: ROC \\\\\n\n& \\cite{chu2018sparse} &  3D convolutional  & C3D network is used to perform feature extraction \\\\\n& & network  &  and detect anomaly using sparse coding, DL. \\\\\n& & & Dataset: Avenue, Subway, UCSD ;\\\\\n&&&  Parameters: AUC, EER\\\\\n\n& \\cite{wu2019deep} & Deep OC Neural  & One stage\nmodel is used to learn compact features   \\\\\n& & Network & and train a DeepOC (Deep One Class)classifier. \\\\\n& & & Dataset: UCSD, Avenue, Live Video; Parameters: ROC\\\\ \n\n& \\cite{jiang2020discriminative}& Generative  & Anomaly detection using Generative Adversarial\n\\\\\n& & Adversarial & Network for hyperspectral images (HADGAN).\\\\\n& &  Network (GAN) & Dataset: ABU, San Diego, HYDICE\\\\\n& & & Parameters: ROC, Computing time\\\\\n\n\n\n& \\cite{song2019learning} & Adversarial  & Normal patterns are learnt through adversarial \\\\\n& &  attention based , & attention based auto-encoder and anomaly is detected. \\\\\n& &  auto-encoder GAN & Dataset: ShanghaiTech, Avenue, UCSD, Subway;\\\\\n&&&  Parameters: AUC EER \\\\\n\n\n& \\cite{sun2020adversarial} & Adversarial 3D Conv,  & spatio temporal patterns are learnt using\\\\\n& & Autoencoder & adversarial 3D Conv, Autoencoder to detect abnormal \\\\\n& & & events in videos; Dataset: Subway, UCSD, \\\\\n&&&  Avenue, ShanghaiTech; Parameters: AUC\/EER\\\\\n\n\n\n\n\n \\hline\n\n  \\end{tabular}\n    \n    \\label{table_3}\n\\end{table*}\n\n\\begin{table*}[]\n    \\centering\n   \n    \\begin{tabular}{l|l|l|l}\n       \n       Approach  & Ref & Technique & Highlights \\\\\n        \\hline\n  Reconstruction   & \\cite{yu2016content} & Sparse  & Sparsity based reconstruction  method is used with low  \\\\\nBased& & Reconstruction  & rank property to  determine abnormal events.   \\\\ \n& & & Datasets: UCSD, Avenue; Parameters: ROC, AUC, EER\\\\\n\n \n & \\cite{zhang2015abnormal} & Sparsity based  & Abnormal event detection in traffic surveillance using  \\\\\n & & method & low-rank sparse representation (CLSR).\\\\\n & & & Dataset: UCSD, Subway, Avenue; \\\\\n &&&  Parameters: AUC, EER\\\\\n\n\\hline   \n Prediction & \\cite{liu2018future} & Video prediction & Spatial\/motion constraints are used for future  \\\\\n\n Based & & framework & frame prediction for normal events and \\\\\n & & &identifies abnormal events; Dataset: CUHK, UCSD, \\\\\n & & & ShanghaiTech; Parameters: AUC \\\\\n \n & \\cite{nawaratne2019spatiotemporal} & Incremental Spatio- & ISTL, an unsupervised deep learning approach with  \\\\\n && Temporal Learner &  fuzzy aggregation is used to distinguish between \\\\\n&&&  anomalies that evolve over time in assistance with  \\\\\n&&& spatiotemporal autoencoder, ConvLSTM.  \\\\\n&&& Dataset: CUHK, Avenue, UCSD;  Parameters: AUC, EER\\\\\n \n         & \\cite{lai2019lstm} & LSTM, Cross  & Recognizing industrial\nequipment in manufacturing  \\\\\n & & Entropy & system using edge computing; Big Data, smart meter\\\\\n\n& & & Dataset: Manual ; Parameter: Accuracy\\\\\n\n& \\cite{cheng2020securead} & Optical flow, GMM, & Detection accuracy, less computational time \\\\ \n\n& & HoF  & Dataset: UMN, UCSD, Subway, LV; \\\\\n&&& Parameter: AUC, EER;\\\\\n\n       & \\cite{zhou2019anomalynet} & spatiotemporal feature  & CNN, Adaptive\nISTA and SLSTM \\\\\n & & extraction & Dataset: CUHK Avenue, UCSD, UMN ;  \\\\ \n&&& Parameter:AUC EER\\\\\n\n\\hline\n\n Other  & \\cite{li2018road} & Fuzzy theory & Anomaly detection in road traffic using Fuzzy  \\\\\n \nApproaches & & & theory. The Gaussian distribution model is trained.\\\\\n & & & Dataset: SNA2014-Nomal\\\\\n & & & Parameter: Accuracy, False Detection Rate \\\\\n \n      &  \\cite{mo2013adaptive} & Sparse Reconstruction &  Joint sparsity model for abnormality detection, \\\\\n      & & & multi-object anomaly detection for real world \\\\\n      & & & scenarios. \\\\\n      & & & Dataset: CAVIAR \\\\\n      & & & Parameters:ROC\\\\\n\n     & \\cite{li2020adaptive} & Sparsity Based & Video surveillance of traffic. Background subtraction.\\\\\n     & & &  Non-convex optimization; Generalized Shrinkage\\\\\n     & & &  Thresholding Operator (GSTO), Joint estimation\\\\\n     & & & Dataset: I2R, CDnet2014 \\\\\n     & & & Parameters: F-measure \\\\\n     \n     &  \\cite{guo2019detecting} & Frequency Domain & Detection of Vehicle Anomaly using Edge Computing\\\\\n     & & (Power Spectral & high-frequency correlation, sensors, reduces \\\\\n     & &  Density; Correlation) &  computation overhead, privacy\\\\\n     & & & Dataset: Open Source Platform \\\\\n     & & & Parameters: FPR, TPR, ROC\\\\\n\n\\hline\n\n  \\end{tabular}\n    \n    \\label{table_4}\n\\end{table*}\n\n\\subsection{Modeling Algorithms for Anomaly Detection}\n\\subsubsection{Statistical Based}\n\nIn statistical based approach the parameters of the model are learnt to estimate anomalous activity. The aim is to model the distribution of normal-activity data. The expected outcome under the probabilistic model will have higher likelihood for normal-activities and lower likelihood for abnormal activities \\cite{bergman2020classification}. Statistical approaches can further be classified as parametric method and non-parametric method. Parametric methods assume that the normal-activity data can be represented by some kind of probability density function \\cite{kumaran2019anomaly}. Some methods use Gaussian Mixture Model (GMM) which works only if the data satisfies the probabilistic assumptions implicated by the model \\cite{cheng2015gaussian}. Non-parametric statistical model is determined dynamically from the data. Examples of non-parametric models are histogram-based \\cite{kaltsa2015swarm} models, HoMM (histogram of magnitude and momentum) based models \\cite{bansod2020crowd}, Bayesian \\cite{nguyen2015bayesian} \\cite{isupova2016anomaly} models. Recently, efficient way to detect and localize anomalies in surveillance videos is to use Fully Convolutional Networks (FCNs) \\cite{sabokrou2018deep} and Structural Context Descriptor (SCD) \\cite{yuan2014online}.\n\nThe detailed categorization of anomaly detection techniques are shown in Table 3.\n\n\\subsubsection{Proximity Based}\nWhen the video frame is sparsely crowded it is easier to detect anomalies, but it becomes a tedious job to find irregularities in densely crowded frame. Proximity based technique utilizes the distance between the object and its surrounding to detect anomalies. In \\cite{colque2016histograms}, a distance-based approach is used that assumes normal data has dense neighborhood and anomalies are identified by their proximity to their neighbours. Further, density-based approaches identify distinctive groups or clusters depending upon the density and sparsity and the anomaly is detected \\cite{liu2018accumulated} \\cite{hu2018anomaly} \\cite{lamba2019segmentation}. \n\n\n\\subsubsection{Classification Based} \nAnother commonly used methods of anomaly detection are classification based which aims to distinguish between events by determining the margin of separation. In \\cite{mo2013adaptive}, Support Vector Machine (SVM) uses classic kernel to learn a feature space with to detect anomaly. Further, a non-linear one-class SVM trained with histogram of optical flow orientation to encode the moving information of each video frame \\cite{wang2014detection}. Aiming at intelligent human object surveillance scheme, Harr-cascade and HOG+SVM is applied together to enable a real-time human-objects identification \\cite{nikouei2018smart}. Some approaches utilize object trajectories to understand the nature of object in the scene and detect anomalies, various tracking algorithms \\cite{xu2018real}\\cite{parate2017structurally}\\cite{parate2018global} are used to estimate trajectories of an object. Trajectory based descriptors are also widely used to capture long term motion information and to estimate the dynamic information of foreground objects for action recognition \\cite{shi2017sequential}. Some approaches use PCA \\cite{ghosh2020edge}, K-means \\cite{ke2016real} for detection of anomaly.\n\n\n\\subsubsection{Reconstruction Based}\nIn reconstruction-based techniques, the anomalies are estimated based on reconstruction error. In this technique every normal sample is reconstructed accurately using a limited set of basis functions whereas abnormal data is observed to have larger reconstruction loss \\cite{kumaran2019anomaly}. Depending on the model type, different loss functions and basis functions are used. Some of the methods use Hyperspectral Image (HSI) \\cite{lei2020discriminative}, \\cite{xu2019low}, 3D convolution network \\cite{chu2018sparse}.\n\nRecently, a deep neural network DeepOC in \\cite{wu2019deep} can simultaneously train a classifier and learn compact feature representations. This framework uses the reconstruction error between the ground truth and predicted future frame to detect anomalous events. Another set of methods use Generative Adversarial Network (GAN) \\cite{kumar2020generative} to learn the reconstruction loss function \\cite{jiang2020discriminative}. GAN based auto-encoder proposed in \\cite{song2019learning} produce reconstruction error and detect anomalous events by distinguishing them from the normal events. Further, an adversarial learning strategy and denoising reconstruction error are used to train a 3D convolutional  auto-encoder to discriminate abnormal events \\cite{sun2020adversarial}.\n\nAnother paradigm to detect anomalous events is by exploiting the low-rank property of video sequences. In the phase of learning, lower construction costs are assigned to instances that describe important characteristics of the normal behavior. Depending on low-rank approximation, a weighted sparse reconstruction method is estimated to describe the abnormality of testing samples \\cite{yu2016content} \\cite{zhang2015abnormal}. \n\n\n\\subsubsection{Prediction based}\nPrediction-based approach use known results to train a model. Such a model predicts the probability of the target variable based on the estimated significance from the set of input variables. In prediction-based approach, the difference between the actual and predicted spatio-temporal characteristics of the feature descriptor is calculated to detect the anomaly \\cite{liu2018future}. Also, Incremental Spatio-Temporal Learning (ISTL) approach with fuzzy aggregation is used to distinguish anomalies that evolve over time \\cite{nawaratne2019spatiotemporal}.\nFurther, in sequence prediction, Long Short Term Memory (LSTMs) are very powerful as they store past information to estimate future predictions . LSTM networks are used to learn the temporal representation to remember the history of the motion information to achieve better predictions \\cite{lai2019lstm}. To enhance the approach, \\cite{cheng2020securead}  integrates autoencoder and LSTM in a convolutional framework to detect video anomaly. Another technique of learning spatiotemporal characteristics is estimating an adaptive iterative hard-thresholding algorithm (ISTA) where a recurrent neural network is used to learn sparse representation and dictionary to detect anomalies \\cite{zhou2019anomalynet}. \n\n\\subsubsection{Other Approaches}\n\nTo handle complex issues in traffic surveillance, \\cite{li2018road} estimates a fuzzy theory and propose a traffic anomaly detection algorithm. To perform the state evaluation, virtual detection lines are used to design the fuzzy traffic flow, pixel statistics are used to design fuzzy traffic density, and vehicle trajectory is used to design the fuzzy motion of the target. To recognize abnormal events in traffic such as accidents, unsafe driving behavior, on-street crime, traffic violations, \\cite{mo2013adaptive} proposes adaptive sparsity model to detect such anomalous events. Similarly, \\cite{li2020adaptive} estimates sparsity based background subtraction method and shrinkage operators. Other approaches also include \\cite{guo2019detecting}, which uses high-frequency correlation sensors to detect vehicle anomaly.\n\n\n\n\\section{Edge Computing}\n\n In traditional video surveillance systems, raw video data gathered from all visual sensors is sent to the centralized servers for storage and further processing. Some approaches use cloud-based computing methodologies for the same. However, the inevitable network latency and bandwidth requirements are not suitable for real-time applications, especially in time-critical applications such as anomaly detection. An edge computing \\cite{shi2016edge} is proposed as a potential solution to this problem as it requires low bandwidth and is applicable where network latency and privacy are concerns \\cite{varghese2016challenges}.\n \n Edge computing is a distributed, decentralized computing method that provides on-board computation and storage of data. Most of the data produced at the device are processed at the device itself \\cite{schneible2017anomaly}. With the advancement in the terminal or edge devices, few contributions are observed in detecting anomalies at the edge or terminal devices. Schneible et al. present a federated learning approach in which autoencoders are deployed on edge devices to identify anomalies. Utilizing a centralized server as a back-end processing system, the local models are updated and redistributed to the edge devices \\cite{8170817}. Despite the rapid development of learning methods, CNNs, and edge devices for computer vision purposes particularly, the gap between software and hardware implementations is already considerable \\cite{8114708}.\n \n \\begin{figure}\n    \\centering\n   \\includegraphics[width=7cm]{\"Fig_1\"}\\\\\n    \\caption{Architectural Overview of Edge Computing}\n\\end{figure}\n\nThe general architectural overview of the edge computing paradigm is shown in Figure 5. The top-level entities are cloud storage and computing devices which comprise data centers and servers. The middle level represents fog computing. Any device with compute capability, memory, and network connectivity is called a fog node. Examples of fog devices are switches, routers, servers, controllers. The bottom-most part of the pyramid includes Edge Devices like sensors, actuators, smartphones, mobile phones. These terminal devices participate in processing a particular task using a user access encryption \\cite{kang2017privacy}. \n\n\\section{Anomaly Detection using Edge Devices}\n\nThe terminal processing at edge devices for automated surveillance is considered to be the alternative for cloud and network-based processing when response time is a relatively important parameter. Over the decade, some approaches have been crafted for automated surveillance\/object tracking using edge devices and a few of them talk about anomaly detection in video surveillance as summarized in Table 4.\n\nTraditional computer vision methods such as feature-based classification approaches are noticeable candidates for edge application; for example, Harr-Cascaded and HOG+SVM algorithms are widely used for human detection in surveillance. Nikouei et al. \\cite{nikouei2018smart} deployed an SVM classifier trained on Harr-Cascade and HOG feature at the edge and introduces a Lightweight Convolutional Neural Network (L-CNN) for smart surveillance. The model is trained using VOC07 and ImageNet datasets with the MXNet platform for neural networks. Again, Histogram of Oriented Gradients (HOG) and Support Vector Machine (SVM) along with a Kernelized Correlation Filters (KCF) is implemented to be deployed on Raspberry pi 3 which is an only CPU edge device \\cite{xu2018real}. On similar lines, a Kerman algorithm \\cite{nikouei2019toward}, which is a combination of Kernelized Kalman filter, Kalman Filter (KF), and Background subtraction (BS) is proposed to achieve enhanced performance on edge. Kernelized Kalman filter is based on decision trees and is suitable for human object tracking. Kerman outperforms Haar-cascade, HOG+SVM, SSD Google Net in terms of FPR (False Positive Rate), FPS (Frames Per Second), and speed and can track\/re-find the human objects in real-time.\\\\\n\nThe traditional methods, though performing well in some scenarios are not as accurate as of the modern learning approaches. Wang et al. propose SaliencyGAN  \\cite{wang2019saliencygan}, that uses a semi-supervised method for Salient Object Detection in the fog-IoT network. Salient Object Detection (SOD) is a useful tool for human-object detection and tracking. However, due to diverse fog devices, implementing SOD on fog nodes is a cumbersome task.  The proposed SaliencyGAN is trained with 10\\% to 100\\% labeled training data. SaliencyGAN showcased comparable performance to the supervised approaches when the labeled data reached 30\\% and outperformed the unsupervised and weakly supervised approaches. Unlike, SaliencyGAN, Zhao et al. opt lightweight deep learning-based model to reduce network occupancy and reduce the system response delay by using edge and cloud computing together \\cite{zhao2020lightweight}. Altogether the system has benefits of low time response and resource management.\nRecently, to ensure passenger safety in public transportation, Ajay et al. propose a Binary Neural Network (BNN) based on real-time emotion detection of passengers using edge computing devices \\cite{ajay2021binary}. Here the facial expressions are classified into six categories namely: fear, happy, sad, angry, surprise, and disgust. The LBP-BNN based improved and faster emotion recognition model keeps the track of facial expressions of people and is suitable for is used for applications including tracking of facial expressions for physically disordered people in hospitals.\n\n\n\n\\begin{table*}\n\\centering\n \\caption{Anomaly Detection at the Edge}\n \\begin{tabular}{l l l l l l}\nRef & Features & Learning & Anomaly Criteria & Dataset & Parameters\\\\\n\\hline\n\n\\cite{ajay2021binary}  & Accuracy & Binary Neural & Recognition & JAFFE dataset & Accuracy\\\\\n&&Network; FPGA &of facial &&\\\\\n&&system design&emotions&&\\\\\n\\hline\n\n \\cite{deng2021air} & Reliability, & Air-Ground & Target tracking  & Self deployed & Probability of   \\\\\n &timeliness, &  Surveillance & in visually & &target retrieval,\\\\\n & opportunistic & Sensor Network &  obstacle blocking & &delay, workload \\\\\n &&&scenarios.\\\\\n \n\n \n\\hline\n \\cite{zhao2020lightweight} & Reduce network   & Intelligent Edge  &  Cloud,  & Self deployed& Accuracy \\\\\n\n         & occupancy and &      Surveillance   &  DL, Edge     &  & Model loss  \\\\\n        &  system response  & & & \\\\\n        & delay &&&\\\\\n        \n        \\hline\n       \\cite{ke2020smart} & Smart parking & CNN, HOG  & Parking & Self deployed & Accuracy\\\\\n       &  &      &    Surveillance    &   \\\\\n     \n     \n       \\hline\n   \\cite{nikouei2019toward}  & Real-time, good & LCNN, Kerman & Tracking  &  VOC07, VOC12 & Accuracy \\\\\n         & accuracy, with  & (KCF, KF, BS) & Human Objects        &  & Precision \\\\\n         & limited resources & & & \\\\\n         \\hline\n   \\cite{wang2019saliencygan}  & Increases & SaliencyGAN & Object & \n  & MAE    \\\\\n        & computing & Deep SOD CNN &   Detection &  PASCALS  & F-Measure \\\\\n         & performance & Adversarial  & \/Tracking  & & Precision  \\\\\n          & & Learning  & &  &Recall \\\\\n          && Semi-supervised && & \\\\\n         \n         \\hline\n         \n          \\cite{muhammad2019edge}  & Processing   & LCNN & Smoke   & ImageNet & FP, FN, \n \\\\\n       & requires &  &  Detection    &  &  Accuracy\\\\\n      & less memory & &in Foggy&& precision, \\\\\n      &&&Surveillance &&  recall,  \\\\\n      &&&&& F-measure\\\\\n      \\hline\n      \n       \\cite{lai2019lstm} & Parallel & LSTM, & Industrial   & Self deployed & Accuracy\\\\\n      & Computing to  &     Cross Entropy &  Electrical  &     &    \\\\\n       &improve&  &Equipment&& \\\\\n       & efficiency &  &&& \\\\\n    \\hline \n    \n    \\cite{guo2019detecting} & Reduced & high-frequency  & Vehicle  & Open Source & FPR, TPR, \\\\\n       &  computation , &    correlation, &    Anomaly    & Platform & ROC   \\\\\n       & overhead & sensors&&& \\\\\n       & privacy  &&&& \\\\\n       \n       \\hline\n       \n       \\cite{chen2019distributed} & Balances  & DIVS & Vehicle  & Self deployed & Efficiency  \\\\\n       & computational &       &   classification   &   &  \\\\\n      & power and &&& \\\\\n      & workload &&&& \\\\\n      \\hline\n      \n      \n   \\cite{xu2018real} &  Low   & HOG,SVM, & Human Object  & Self deployed   & Speed,\\\\\n      & computational  &   KCF &  Tracking  &  & Performance   \\\\\n     & cost with & & &   & \\\\\n     &  high accuracy  & & &  & \\\\\n     & and performance & &  \\\\\n     \\hline\n     \n     \n      \\cite{li2018deep}  & Improved  & CNN, DL & Industry  & Self deployed & FPR, TPR,\\\\\n       & efficiency  &              & Manufacture  &   &  ROC   \\\\\n       & & &  Inspection  &\\\\\n       \\hline\n     \n     \n   \\cite{wang2018offloading} & Resource  & FORT & Traffic & Self deployed & average  \\\\\n      & management,   &     (Real-Time Traffic  & Management   &      & response    \\\\\n    & reduction in& management    &System& & time \\\\\n    & response time  &in Fog)&&& \\\\\n   \n        \n       \\hline\n       \n\\end{tabular}\\\\\n\\end{table*}\n      \n       \n\\begin{table*}\n\\centering\n\n \\begin{tabular}{l l l l l l}       \n       \n        \\hline\n     \n      \\cite{nikouei2018smart} & Processing   & Harr-Cascade,  & Smart  & ImageNet & FP, FN, Accuracy\\\\\n        & requires &  SVM, L-CNN  &   Surveillance         &  &   precision, recall,  \\\\\n       &less memory&&&& F-measure (F) \\\\\n       \n       \\hline\n       \n       \\cite{kang2017privacy}  & Reduce  & Privacy  & Smart IoV & Self & Entropy comparison  \\\\\n        &  computational  &   -Preserved    &  (Internet of Vehicles)        & deployed&  Nash Equilibrium, \\\\\n       &  overheads and   & Pseudonym  & address privacy &&   \\\\\n       & enhance location & Algorithm  &location issues &&   \\\\\n       & privacy &&&&\\\\\n      \n       \n       \\hline\n       \n\\end{tabular}\\\\\n\\end{table*}\n\n\nFurther, UAV (Unmanned Aerial Vehicle) proves to be beneficial as it offers good performance in sight-blocking scenarios \\cite{chriki2021deep}. On one hand there is a cloud-based tracking system where \\cite{zhai2020smart} proposes a cloud-enabled autopilot drone system in video surveillance that uses a Deep Neural Network (DNN) for abnormal event detection. The problem with such a kind of implementation is that the time response is large as the amount of the data to be transferred is large. As this process needs to be done in real-time it becomes tedious to locate the target. To address this challenge, Air-Ground Surveillance Sensor Network (AGSSN) tracking system \\cite{deng2021air} is proposed which is based on edge computing. To reduce latency and huge network communication overhead, an option of dividing the computationally expensive tasks among the edge nodes is always open and termed as task offloading \\cite{wang2018offloading}.\\\\\n\nTo detect anomalies in traffic surveillance, Chen et al. deployed a Distributed Intelligent Video Surveillance (DIVS) system \\cite{chen2019distributed} on an edge computing environment. It includes multi-layer edge computing architecture and a distributed Deep Learning (DL) training model. To reduce network overhead and gain workload balance a multi-layer edge computing architecture is employed for vehicle classification and traffic flow prediction. The experimental setup includes 200 monitoring terminals and 35 EC servers to monitor traffic for 7 days. The results show that the execution time is less even if we increase the number of nodes or number of tasks. An Edge Computing-based Vehicle Anomaly Detection (EVAD) scheme is proposed in \\cite{guo2019detecting} to prevent attacks on vehicles. It detects anomalies in the intra-vehicle system by sensors using edge computing. To reduce the computation overhead and improve security, the correlations are organized in the form of a ring architecture. Further, an attempt to identify a real-time parking occupancy is made in \\cite{ke2020smart}. This Angle Lake parking garage experiment employed edge devices empowered with a single shot multi-box detector  (SSD-Mobilenet) and is implemented using Tensorflow Lite. This experiment aims to track multiple objects for vehicle parking and occupancy judgments under different environmental conditions such as; rain, fog, sunlight, snow, weekend, weekday, day, and night. The data transmission volume is kept small to be handled by the limited network bandwidth. Results show the network latency will always exist and increases with an increase in the number of cameras employed for the surveillance.\\\\\n\nMoreover, the edge is also employed to detect industrial anomalies, for example, Muhammad et al. proposed a real-time CNN-based smoke detection system surveillance system for foggy environment \\cite{muhammad2019edge}. Owing to the problems related to foggy environmental conditions the author focuses on building a lightweight CNN model on MobileNet V2 and tested different conditions like; smoke, without smoke, smoke with fog, and smoke without fog. In industries, to detect manufacturing anomalies a \"DeepIn\" model is proposed in \\cite{li2018deep}. It is composed of three modules (fog side,  back-end communication module, server-side) designed using CNN layers. Fog side computing module is used for computational offloading, back-end communication module is used for data exchanges and command traffic, and finally, the server-side is used for defect classification and degree regression.\\\\\n\nAlthough many intelligent surveillance methods based on machine learning algorithms are available, it is still challenging to efficiently migrate those smart algorithms to the edge due to the very restricted constraints on resources. However, attempts are made to combine the edge computing architecture with the parallel computing of an artificial neural network \\cite{lai2019lstm}. Also, an Edge Artificial Intelligence seems to be a promising technology that combines edge computing, artificial intelligence, and Internet-of-Things (IoT) that migrates computation workloads from central cloud to the edge of the network but has its challenges; security and accessibility \\cite{kang2017privacy}, the balance of workload among the edge nodes under the complicated scenarios \\cite{wang2018offloading}, synchronization of distributed models in an edge computed environment \\cite{chen2019distributed}, and reduced network occupancy and reduced system response \\cite{zhao2020lightweight}.\n\n\\begin{comment}\n\\subsection{Parameters Observed}\n\\textcolor{red}{After analysis of the videos using different detection techniques, the parameters that are observed are Area Under the Curve (AUC) or Receiver Operating Characteristics (ROC), Equal Error Rate (EER), Accuracy, Precision and Recall (True Positive Rate, False Positive Rate) pixel-wise mean absolute error (MAE) of saliency maps, maximum F-measure scores (maxF), Precision Recall curves (PR-curves), F-measure curves. It is observed that there is an impact on these parameters due to the variation in environmental conditions, occlusions, increase in traffic, camera angle.\n}\n\\end{comment}\n\n\\subsection{Datasets}\nThere are many publicly available data-sets for validating the surveillance and anomaly detection algorithms. UCSD \\cite{mahadevan2010anomaly}, CUHK \\cite{nawaratne2019spatiotemporal}, Avenue \\cite{lu2013abnormal}, UMN \\cite{sabokrou2017deep} and Subway \\cite{adam2008robust} are some of the popularly used data-sets for anomaly detection consisting of individuals, crowd, objects, vehicles and human object interaction. A detailed list of datasets corresponding to their surveillance targets is given in Table 5.\n\nOther datasets that are often found in the literature are Shanghai \\cite{luo2017revisit}, Badminton \\cite{colque2016histograms} Behave and QMUL Junction \\cite{cheng2015gaussian}, Mind's Eye and Vanaheim dataset \\cite{cocsar2016toward}. These datasets include normal videos and abnormal videos for training and testing purposes depending upon the application. For example, the normal events in the CUHK dataset include pedestrians walking on the footpath, group of pedestrians congregating on the footpath whereas abnormal events include individuals littering, walking on the grass, walking towards the camera, and carrying suspicious objects \\cite{nawaratne2019spatiotemporal}. \n\n\nEvents in the UCSD dataset include events captured from various crowd scenes that range from sparse to dense. The data-set represents different situations like; walking on the road, walking on the grass, vehicular movement on the footpath, unexpected behavior like skateboarding, etc. \\cite{nawaratne2019spatiotemporal}.\n\nAvenue anomalous dataset includes a random person running, any abandoned object, person walking with an object \\cite{zhou2019anomalynet}. UVSD dataset includes individuals and vehicles while DAVIS dataset is composed of various objects (human, vehicles, animals) to obtain the class diversity \\cite{zhang2020industrial}. \nAnomalous situations in the Subway data-set include walking in the wrong way (people entering the exit gate) and jumping over the ticket gate \\cite{colque2016histograms}. Uturn dataset is a video of a road crossing with trams, vehicles, and pedestrians in the scene. The abnormal activity videos cover illegal U-turns and trams \\cite{saligrama2012video}.\nVanaheim Dataset consists of videos containing people passing turnstiles while entering\/exiting stations recorded in metro stations \\cite{cocsar2016toward}. The abnormal events encountered were a person loitering, a group of people suddenly stopping, a person jumping over turnstiles.\n\nSome authors have also used live videos for the implementation of their respective methods \\cite{wu2019deep}. Anomalous events from live videos like an accident, kidnapping, robbery, crime (a man being murdered) are seen in the literature. \n\nTo evaluate the anomaly detection model, Mini-Drone Video Dataset (MDVD) has been shot in a car parking area\\cite{chriki2020uav}\\cite{chriki2021deep}. Anomalous behavior in MDVD includes people fighting, wrongly parked vehicles, or people stealing items or vehicles.\n\nVarious algorithms have been developed to tackle challenges in video surveillance in different datasets.\n\n\\begin{table}\n\\centering\n \\caption{List of Datasets}\n\\begin{tabular}{l|l|l}\n\n   Dataset  & References & Surveillance Target \\\\\n   \\hline\n  \n   UCSD &  \\cite{wu2019deep} \\cite{xu2018anomaly} \\cite{singh2017graph} \\cite{zhou2019anomalynet} \\cite{yuan2014online} &  Automobile, Individual, Crowd (Public Places)\\\\\n   & \\cite{colque2016histograms} \\cite{sabokrou2017deep}  \\cite{sabokrou2018deep} \\cite{nawaratne2019spatiotemporal}  \\cite{cheng2015gaussian} \\\\\n   &  \\cite{saligrama2012video}  \\cite{song2019learning}  \\cite{chu2018sparse}  \\cite{sun2020adversarial}  \\cite{yu2016content} \\\\\n   &  \\cite{liu2018future} \\\\\n   \n   \n   Avenue, CUHK & \\cite{wu2019deep} \\cite{xu2018anomaly}  \\cite{zhou2019anomalynet} \\cite{nawaratne2019spatiotemporal}  \\cite{song2019learning} & Individual, Crowd, Object  (Public Places)  \\\\\n   & \\cite{chu2018sparse}  \\cite{sun2020adversarial} \\cite{cheng2020securead}  \\cite{liu2018future}   \\\\\n   \n   \n   \n   UMN & \\cite{singh2017graph} \\cite{zhou2019anomalynet}  \\cite{sabokrou2017deep}  \\cite{saligrama2012video} \\cite{yu2016content} & Individual, Crowd (Public Places) \\\\\n   \n   \n   Subway & \\cite{wu2019deep} \\cite{xu2018anomaly} \\cite{kaltsa2015swarm} \\cite{colque2016histograms} \\cite{sabokrou2018deep}    & Individual, Crowd (Entrance and Exit of Subway \\\\\n   & \\cite{cheng2015gaussian} \\cite{saligrama2012video}  \\cite{song2019learning} \\cite{chu2018sparse}  \\cite{sun2020adversarial}   & Stations)\\\\ \n   & \\cite{cocsar2016toward} \\\\\n   \n   \n   Uturn & \\cite{saligrama2012video} & Objects (Vehicles), Individuals (Pedestrians)\\\\\n   Vanaheim & \\cite{cocsar2016toward} & Individual, Crowd (Metro Stations)\\\\\n   Mind's Eye & \\cite{cocsar2016toward} & Individual, Crowd, Objects (Vehicles) (Parking Area)\\\\\n   UVSD, DAVIS  & \\cite{zhang2020industrial} & Individuals, Objects (Vehicles)\\\\\n    Shanghai & \\cite{song2019learning} \\cite{sun2020adversarial} \\cite{liu2018future} \\cite{luo2017revisit}  & Individual, Crowd, Objects (Vehicles) (Public Places)\\\\\n    Badminton & \\cite{colque2016histograms} & Individual, Crowd (Badminton Game)\\\\\n    Behave &  \\cite{cheng2015gaussian} & Individual, Crowd Public Places)\\\\\n    ABU Dataset & \\cite{lei2020discriminative} \\cite{jiang2020discriminative} & Objects (Aerial View)\\\\\n    MDVD & \\cite{chriki2020uav}\\cite{chriki2021deep} & Objects (Aerial View)\\\\\n    IEEE SP Cup-2020 & \\cite{chowdhury2020unsupervised}  \\cite{chowdhury2020anomaly} & Objects (Lobby, Parking Area)\\\\\n    AU-AIR & \\cite{bozcan2021context} & Objects (Aerial View)\\\\\n    Facial Expression  & \\cite{ajay2021binary} & Individuals (Facial Expressions) \\\\\n    2013 (FER-2013) & & \\\\\n    ISLD-A & \\cite{angelini2019privacy} & Individuals, Objects (Human Action Recognition)\\\\\n    \n    \n    Self-deployed &  \\cite{lao2009automatic}  \\cite{ke2016real} \\cite{nieto2018automatic}  \\cite{ke2020smart} \\cite{liu2018hybrid}& Individuals, Objects, Crowd, Automobiles \\\\\n    & \\cite{singh2018deep}  \\cite{li2018deep}  \\cite{ji20123d}  \\cite{liu2016semi}   \\cite{lai2019lstm} &  \\\\\n    & \\cite{li2018road}  \\cite{liu2018accumulated}  \\cite{zhao2020lightweight} \\cite{deng2021air} \\cite{zhai2020smart}\\\\\n    & \\cite{ma2019cost} \\cite{xu2020trust} \\\\\n    \n    \n    \n    STONE & \\cite{xu2019low} & Objects (Grassy Scene)\\\\\n    AVIRIS & \\cite{xu2019low} & Objects (Aerial View)\\\\\n   SkyEye & \\cite{roy2020detection} & Objects (Aerial View)\\\\\n   UCF101 & \\cite{roy2018unsupervised} & Individual (Action Videos) \\\\\n   HMDB51 & \\cite{roy2018unsupervised} & Individual (Action Videos)\\\\\n   THUMOS14 & \\cite{roy2018unsupervised} \\cite{guo2018fully}  & Individual (Action Videos) \\\\\n   BP4D & \\cite{perveen2018spontaneous} & Individual (Facial Expressions)\\\\\n   AFEW & \\cite{perveen2018spontaneous} &  Individual (Facial Expressions)\\\\\n   CAVIAR  & \\cite{mo2013adaptive} & Individuals, Objects (Vehicles)\\\\\n   PASCAL VOC& \\cite{zhou2017fast} \\cite{wang2019saliencygan} \\cite{nikouei2018smart} \\cite{nikouei2019toward}   & Objects (Vehicles)\\\\\n   DS1, DS2 & \\cite{muhammad2019efficient} & Events (Fire\/Smoke\/Fog)\\\\\n   SFpark & \\cite{bock2019smart} & Object (Taxis) \\\\\n   LISA 2010 & \\cite{zhou2017fast} & Objects (Vehicles)\\\\\n   ImageNet & \\cite{muhammad2019edge} & Events (Fire\/Smoke\/Fog)\\\\\n   QMUL Junction & \\cite{isupova2016anomaly} \\cite{cheng2015gaussian} & Objects (Vehicles)\\\\\n   HockeyFight & \\cite{asad2021multi} & Individuals \\\\\n   ViolentFlow & \\cite{asad2021multi} & Individuals \\\\\n   \n   \\hline\n  \n \n\\end{tabular}\n\\end{table}\n\n\\subsection{Edge Computing in Anomaly Detection: Challenges and Opportunities}\n\n\\subsubsection{Challenge 1: Locating Edge Nodes}\nLocating edge nodes in a distributed computing paradigm is well explored through a variety of techniques \\cite{de2011toward} \\cite{povedano2013dargos} \\cite{montes2013gmone}. Bench-marking techniques are used for mapping tasks onto the most suitable resources and thereby discovering edge nodes. However, a proper mechanism is required to explore the edge of the network as several devices from different generations will be available at this layer. For example, machine learning tasks were rarely used previously but nowadays, they are used as a first-hand option for anomaly detection setups. Bench-marking methods should be efficient in finding the capability and availability of resources.\nA diagram illustrating the challenges and opportunities is given in Figure 6.\n\n\\subsubsection{Challenge 2: Security and accessibility}\nIn edge computing, a significantly large number of nodes (edge devices) participate in processing tasks and each device requires a user access encryption \\cite{kang2017privacy}. Also, the data that is processed needs to be secured as it is handled by many devices during the process of offloading \\cite{guo2019detecting}. \n\n\\subsubsection{Challenge 3: Quality of Service}\nThe quality of service delivered by the edge nodes is determined by the throughput where the aim is to ensure that the nodes achieve high throughput while delivering workloads. The overall framework should ensure that the nodes are not overloaded with work however if they are overloaded in the peak hours, the tasks should be partitioned and scheduled accordingly \\cite{rodrigues2016hybrid} \\cite{suto2015qoe}. \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=7cm]{\"ChallengesOpportunities\"}\\\\\n    \\caption{Edge Computing in Anomaly Detection: Challenges and Opportunities}\n\\end{figure}\n\n\n\\subsubsection{Opportunity 1: Distributed computing}\nEdge computing uses the technique of dividing computationally expensive tasks to other nodes available in the network thereby reducing response time. The transfer of these intensive tasks to a separate processor such as a cluster, cloud-let, or grid is called computation offloading. It is used to accelerate applications by dividing the tasks between the nodes such as mobile devices. Mobile devices have physical limitations and are restricted in memory, battery, and processing. This is the reason that many computationally heavy applications do not run on such devices. To cope with this problem, the anomaly detection task is migrated to various edge devices according to the computing capabilities of respective devices. Xu et al. tried to optimize running performance, responsive time, and privacy by deploying task offloading for video surveillance in edge computing enabled Internet of Vehicles \\cite{xu2020trust}. Similarly, a Fog-enabled real-time traffic management system uses a resource management offloading system to minimize the average response time of the traffic management server \\cite{wang2018offloading} \\cite{dinh2018learning} \\cite{yang2015cost}. The resources are efficiently managed with the help of distributed computing or task offloading \\cite{wang2018offloading}  \\cite{zhao2020lightweight} \\cite{wang2018traffic}.\n\n\n\\subsubsection{Opportunity 2: Low Latency Computing and Decentralized Cloud}\nAs far as anomaly detection using the cloud, the data is captured on the device and is processed away from the device leading to a delay. Moreover, if the cloud centers are geographically distant the time response is hampered further. Edge computing has the capability of processing the data where it is produced thereby reducing the latency \\cite{nikouei2018smart} \\cite{yang2019edge}.\nOther conventional methods focused on improving either transmission delay or processing delay, but not both. Service delay puts forth a solution that reduces both \\cite{rodrigues2016hybrid}. \n\n\n\\subsubsection{Opportunity 3: Smart Computation}\nTo perform meaningful analytics, the data generated at the user end needs to be transported to the cloud. There are inevitable delays and energy implications in this process. Computations can be performed smartly by hierarchically distributing the computations \\cite{li2018deep}. Smart computation involves accuracy \\cite{nikouei2019toward}, efficiency  \\cite{xu2019data}, lower computation \\cite{xu2018real}, latency \\cite{wang2019saliencygan} which is essential for abnormal event detection. If there are limited resources on edge nodes, data centers offload the task to volunteer nodes to enhance computational capabilities of the front-end devices \\cite{lai2019lstm}.\n\n\n\\section{Observations}\nAfter studying different paradigms of anomaly detection in video surveillance systems, we observe that only benchmark data-set-based comparison may not be relevant for all real-life situations, as they are not enough to consider all real-life scenarios. Further, the performance depends on the density of the crowd, as the crowd increases the performance of the anomaly detection model decreases and it works best when the crowd is sparse. Some approaches intend to neglect background and focus only on foreground features for anomaly detection. We think that background information would be useful to model environmental conditions like rainy, sunny, or snowy weather that can cause anomalies. Further, for delay-sensitive applications like intelligence surveillance and anomaly detection, edge computing is a promising approach. It offers more privacy and security as the data is processed on the device itself. With continuous improvement in edge devices and task offloading the workload is divided thereby improving the overall efficiency. \n  \n\n\\section{Conclusion}\nIn this paper, we survey anomaly detection in video surveillance. We explored various anomaly detection techniques applied for different surveillance scenarios including vehicular pedestrian, crowd, traffic, industries, and public places. We emphasized the learning techniques, models, approaches, and different scenarios for anomaly detection. The survey intended to provide detailed inside and diversities in anomaly detection techniques. In context to anomaly detection using edge computing, the area is less explored and needs attention.  A lot of work can be done in this field to achieve state-of-the-art anomaly detection and intelligence surveillance on edge devices.\n\n\n\\begin{comment}\n\\section{Non-aligned Material}\n The reviews done so far focus on anomaly detection in video systems using cloud. None of the surveys satisfactorily explain the learning methods and approaches. This survey emphasizes on different techniques of anomaly detection in video surveillance systems. The review later focuses on using edge computing devices in video surveillance systems. \n\n\n\n\n\nClassification based approaches can be further divided into one class \\cite{wang2014detection} \\cite{nikouei2019toward} and multi-class categories. One class approach assumes that all the training data consists of one label and multi-class consists of more than one labels as mentioned in section IV.\\\\\n\n\nAs far as deep learning is concerned cascaded convolutional neural networks are used for detecting complex visual surveillance scenes \\cite{liu2018hybrid}. A deep one class neural network can simultaneously learn compact feature representations and train a one-class classifier \\cite{wu2019deep}. It is observed that for the purpose of training of model, the public data-sets available are insufficient. To address this problem \\cite{xu2018anomaly} proposed stacked sparse coding (SSC) with intra-frame classification strategy to evaluate abnormality in various scenes.\\\\\n\nClassification technique is mostly used in case of supervised learning. One of the main challenges in semi-supervised learning is to provide an objective to learn the features that are relevant to the task. \\cite{liu2016semi} has provided a self-supervised approach using Laplacian support vector machine to study its influence on visual location recognition. In this study the time and weather factors are also taken into account thereby providing economical and convenient visual location recognition system.\\\\\n\nAmong previous methods, trajectory based descriptor are widely used to capture long term motion information effectively for action recognition. The trajectories are projected to CNN-RNN to learn effective representation for long-term motion \\cite{shi2017sequential}. Trajectory based descriptor have an ability to describe the dynamic information of foreground objects. This is done by learning spatiotemporal patterns based on hand-crafted features such as optical flow, texture and color. The problem with this method is that most hand-crafted features are designed under some a prior knowledge that variation is caused by factors like deformation, scale or illumination. When the video surveillance scenes are crowded these features do not generalize well.\\\\\n\n\nDeep neural networks based approaches have achieved remarkable progress in various computer vision applications. Although there are a number of deep learning methods for dealing with video data, most of the methods need large data-sets to learn visual representations. \n\nOne possible solution is unsupervised deep learning approach based on auto-encoder. As 3 dimensional convolution (C3D) is suitable for video data \\cite{chu2018sparse} estimates a sparse coding method to obtain self-supervised signals for training this C3D network.\\\\\n\n\nIn recent times, neural networks have been widely used to detect anomalous events as they have stronger ability of feature representation than traditional hand-crafted methods \\cite{lai2019lstm}. However, due to the high cost of training a neural network, it is challenging to achieve satisfactory detection results especially for resource-limited user.\\\\\n\n\\subsubsubsection{Opportunity 2: Network Traffic}\nThe network traffic reduces as the majority of the data is processed on the device itself and only essential data is sent to the cloud \\cite{muhammad2019edge}. The number of user devices is growing at an enormous rate and consequently the volume of data generated will also increase and thereby placing a need for expanding data centers. With the increase in data generation, the volume of network traffic with also increase. There are attempts to mitigate this challenge by using edge devices. Edge devices provide a potential to cope with the growth of data by on-board processing and distributing traffic (task offloading) in a network thereby reducing network traffic.\n\nIn the next decade, the cloud centers  are likely to consume three times (http:\/\/www.independent.co.uk\/environment\/global-warming-datacentres- to-consume-three-times-as-much-energy-in-next-decade-expertswarn- a6830086.html) as much energy consumed today.\n\\subsubsection{Opportunity 3: Energy Consumption}\nIn the next decade, the cloud centers  are likely to consume three times as much energy consumed today. There is a need for adopting strategies that can minimise the energy usage. This challenge can be alleviated by incorporating strategies that facilitate edge computing \\cite{peng2014energy}.\\\\\n\n\n\n\n\nEdge computing is important because it creates new and improved ways for industrial and enterprise-level businesses to maximize operational efficiency, improve performance and safety, automate all core business processes, and ensure \"always on\" availability. At its basic level, edge computing brings computation and data storage closer to the devices where it's being gathered, rather than relying on a central location that can be thousands of miles away. This is done so that data, especially real-time data, does not suffer latency issues that can affect an application's performance. This on device processing improves response time and saves bandwidth. The merits of Edge Computing are listed below:\n\n\n\n\\begin{enumerate}\n    \\item Response Time\n     Edge computing brings computation and data storage closer to the devices where it's being gathered, rather than relying on a central location that can be at geographically distant location. The benefit of on-device is that the data, especially real-time data, does not suffer latency issues and affect application's performance.\n     \\item Bandwidth\n     With the ever increasing demand of edge devices, the bandwidth requirements have come to a standstill. With the growing quantity of data generated at the edge, speed of data transportation has come to a bottleneck. If the data is processed at the edge and only the data that is important is sent to the cloud will reduce the bandwidth requirements of the system.\n\\end{enumerate}\n    \n    \n    \n    \n    \nWith the advances in science and technology, the electrical equipment in industries upgrade every year. Thus, identification complexity of power equipment increases gradually and a simple machine learning method cannot have good recognition ability.\nFor the purpose of accurately recognizing industrial power equipment in manufacturing system, multiple power features of the industrial equipment like current harmonics, steady state, radio frequency have to be studied for identification. This statistical analysis of the data is required to achieve a high degree of accuracy.\nIn recent years, artificial neural network became a popular solution for deep learning in the machine learning analytics \\cite{manning2015computational}. \nThus, in order to enhance the accuracy for industrial power equipment identification,\n\\end{comment}\n\n\n\n\\bibliographystyle{ieeetr}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:Intro}\n\n\nHolographic renormalization was first understood as an important ingredient in gauge\/gravity dualities \\cite{Henningson:1998gx,Balasubramanian:1999re,Emparan:1999pm,Bianchi:2001kw}. In that context, the goal is to address a class of large volume (IR) divergences in the gravitational action. This is accomplished by adding a number of surface terms that render the action and other quantities of interest finite. Soon afterward, the technique was extended to theories with qualitatively different spacetime asymptotics, like de Sitter space \\cite{Balasubramanian:2001nb,Balasubramanian:2002zh,GM2002}, and inflating FRW spacetimes \\cite{Larsen:2003pf, Larsen:2004kf}. This marked a new application of the technique, as such theories are not known to possess a holographic dual. It also makes ``holographic renormalization\" something of a misnomer, but the term has stuck and so we will use it.\n\n\nBefore the development of holographic renormalization, the problem of IR divergences in calculations involving gravitational actions was usually addressed via a technique known as `background subtraction' \\cite{Gibbons:1976ue}. Given a spacetime $(\\MM,g)$, one attempts to isometrically embed a regulating boundary $(\\dM,h)$ in some suitable reference spacetime $(\\MM_{\\textrm{\\tiny ref}}, g_{\\textrm{\\tiny ref}})$ \\cite{Brown:1992br, Brown:2000dz}. The action is then obtained from the difference of the regulated actions for $(\\MM,g)$ and $(\\MM_{\\textrm{\\tiny ref}}, g_{\\textrm{\\tiny ref}})$, in the limit that the regulating boundary is removed to infinity. In many calculations this technique produces a physically sensible result. Indeed, several important results in Euclidean quantum gravity are based on these sorts of calculations. However, background subtraction also suffers from a number of problems \\cite{Emparan:1999pm, Mann:2005yr}. For example, there may be more than one candidate for the reference spacetime, and these may lead to inequivalent results. Furthermore, in spacetime dimension $d+1 \\geq 4$, Weyl's embedding theorem states that an isometric embedding of the regulating boundary in the reference spacetime may not exist. In other words, there is no guarantee that background subtraction can be implemented for a particular spacetime. These problems are not restricted to exotic or pathological spacetimes; even well-known solutions like the Kerr spacetime suffer from ambiguities \\cite{Martinez:1994ja}. Holographic renormalization, on the other hand, is a completely intrinsic technique that does not rely on reference spacetimes or particular embeddings.\n\n\nThere is another motivation for holographic renormalization that is entirely separate from gauge\/gravity dualities. In the 1970s, Regge and Teitelboim \\cite{Regge:1974zd} showed that gravity with asymptotically flat boundary conditions requires additional surface terms in order for the Hamiltonian to have the appropriate variational properties. The same is true for the Lagrangian approach: in non-compact spacetimes, physically interesting boundary conditions for the metric \\emph{require} additional surface terms in the action. In fact, the AdS boundary counterterms serve precisely this purpose \\cite{Papadimitriou:2005ii}. The IR divergences that plague the standard form of the action are symptomatic of a deeper issue: the lack of a well-defined variational principle. This is a fundamental problem that background subtraction cannot address. Even if the regulating boundary for a spacetime can be embedded in an appropriate reference background, such embeddings cannot be constructed for an open family around that spacetime \\cite{Mann:2005yr}. This means that background subtraction cannot be used -- even in principle -- to construct a well-defined variational principle. Holographic renormalization offers a complete and unambiguous solution to this problem.\n\n\nWhether we are discussing gauge\/gravity dualities or Euclidean quantum gravity, it is essential to have a well-defined variational principle. This is the key physical insight that guides the introduction of surface terms in the action, as well as the derivation and interpretation of conserved charges. The variational approach to holographic renormalization is especially relevant for gravity with asymptotically flat boundary conditions. Early results for AAdS spacetimes make use of the constant-curvature asymptotics, and hence do not generalize to asymptotically flat spacetimes (though some progress was made in \\cite{Kraus:1999di}). A different approach, which makes use of the Gauss-Codazzi relations, was recently proposed for gravity with asymptotically flat boundary conditions \\cite{Mann:2005yr}. The new surface term is proportional to the trace of a tensor $\\hK_{ab}$, which is constructed from the intrinsic curvature of the induced metric on the (regulating) boundary. The resulting action\n \\begin{equation}\n \\label{covaction} I = \\frac{1}{2\\,\\kappa^{\\,2}}\\int_{\\MM} \\nts d^{\\,d+1}x \\,\\sqrt{-g}\\,R +\n \\frac{1}{\\kappa^{\\,2}}\\int_{\\dM} \\bns d^{\\,d}x \\, \\sqrt{-h} \\,\\big(K - \\hat K\\big),\n \\end{equation}\nis finite on-shell, and its first variation vanishes for all asymptotically flat variations of the metric. Note that ``asymptotically flat variation\" has a precise meaning here: it refers to a specific set of fall-off conditions on components of the metric, as well as certain kinematic constraints on the metric at spatial infinity \\cite{Ashtekar:1990gc, Ashtekar:1991vb, Beig:1980be, Beig:1982xx}. The boundary stress tensor obtained from the variation of this action is proportional to the electric part of the Weyl tensor, and it can be shown \\cite{Mann:2008ay} that currents built from the stress tensor can be integrated over a cut of spatial infinity to yield conserved charges that are equivalent to both the ADM construction \\cite{ADM1} and the covariant charges defined by Ashtekar and Hansen \\cite{AH}.\n\n\nFor gravity with asymptotically flat boundary conditions the ``$\\hK$\" counterterm prescription \\cite{Mann:2005yr, Mann:2006bd, Astefanesei:2006zd, Mann:2008ay} has been shown to be quite robust. It has also been applied to a broad class of spacetimes obeying these boundary conditions \\cite{Ross2008}. This was also the case with the original AdS\/CFT construction for boundary counterterms, which applies to all spacetimes with AAdS boundary conditions \\cite{Emparan:1999pm, Papadimitriou:2005ii}. So it is a bit odd that the two constructions seem to be incompatible. That is, the $\\hK$ approach does not work for AAdS boundary conditions, and the AdS\/CFT constructions do not work for AF boundary conditions. To investigate this situation, we turn to a recent application of the $\\hK$ construction by Marolf and Virmani \\cite{Marolf:2007ys}, who consider NS 5-brane backgrounds in ten dimensional type II supergravity. They develop a hybrid approach which depends on a $\\hK$ surface term, as well as additional surface terms reminiscent of those used in AdS\/CFT. Motivated by their example, we consider a broader class of theories in which gravity is coupled to an asymptotically linear dilaton field and magnetic $p$-form flux. Because of the relation to the NS 5-brane background, we refer to this class of theories as asymptotically linear dilaton (ALD) gravity. In some respects, these theories are similar to the more familiar AF and AAdS cases. But obtaining the proper surface terms reveals new aspects of holographic renormalization. For instance, we find that certain ALD theories allow for a family of actions that admit a well-defined variational principle, have the same value on-shell, and yield the same conserved charges. These results add to existing work on non-conformal brane backgrounds \\cite{Cai:1999xg, Kanitscheider:2008kd, Wiseman:2008qa}.\n\n\nThe outline of our paper is as follows. We begin in section \\ref{sec:Prelim} by defining ALD gravity and showing that the usual form of the action does not give a well-defined variational principle. In section \\ref{sec:KHat} we discuss how the $\\hK$ prescription can be applied to ALD gravity, and then examine a broader range of possible boundary counterterms for this class of theories. We continue along these lines in section \\ref{sec:OC}, where we offer an alternative prescription for the holographic renormalization of ALD gravity.  In section \\ref{sec:LRF} we compute the boundary stress tensor and the associated conserved charges.  In section \\ref{sec:Examples} we apply our results to two examples in ten dimensional, type II supergravity: spacetimes asymptotic to a stack of NS 5-branes, and spacetimes asymptotic to a stack of D5-branes.  We summarize our results in section \\ref{sec:Discussion}, and present technical details of our calculations in a collection of Appendices. \n\nThroughout most of the paper we set $2\\,\\kappa^{\\,2} = 16 \\pi G = 1$, but these factors are restored for the examples in section \\ref{sec:Examples}. Furthermore, the dimension of spacetime is taken to be $d+1 > 4$. This assumption is meant to expedite our calculations, which would be complicated in lower dimensions by the mixing of terms at different orders in the asymptotic expansions. Our expectation, based on a similar situation in \\cite{Mann:2008ay}, is that the results of sections \\ref{sec:KHat}, \\ref{sec:OC}, and \\ref{sec:LRF} readily generalize to the case $d+1=4$.\n\n\n\n\\section{Asymptotically Linear Dilaton Gravity}\n\\label{sec:Prelim}\n\nThis section is divided into four parts. In the first part we give the action for gravity coupled to a dilaton and field strength on a manifold with boundary. In the second part we consider the same theory on a spacetime with non-compact spatial slices, and define a set of asymptotically linear dilaton (ALD) boundary conditions that characterize the behavior of the fields near spatial infinity. The third part gives a prescription for working with functionals that are ill-defined because of the asymptotic behavior of the fields. Finally, in the last part we show that solutions of the equations of motion with ALD boundary conditions are not stationary points of the original action.\n\n\\subsection{Equations of Motion}\n\\label{sec:EOM1}\n\nWe begin by considering gravity coupled to a scalar field and a $p-1$ form ${\\bf A}$, on a $d+1$ dimensional spacetime $(\\MM,g)$ with boundary $(\\dM,h)$. This theory is described by the Einstein frame action\n\\begin{equation}\\label{bulkaction}\n  I = \\int_{\\MM} \\nts \\nts d^{\\,d+1}x \\sqrt{g} \\, \\left( R - \\frac{4}{d-1} \\, \\nabla^{\\mu} \\phi\n    \\nabla_{\\mu} \\phi - \\frac{1}{2 \\cdot p!} \\, e^{2 \\alpha \\phi} \\, F^{a_1 \\ldots a_p}\n    F_{a_1 \\ldots a_p}\\right) +  2 \\int_{\\dM} \\bns \\nts d^{\\,d} x \\,\\sqrt{h} \\, K\n\\end{equation}\nwhere ${\\bf F} = {\\rm d} {\\bf A}$, and $K$ is the trace of the extrinsic curvature of the boundary. Under a small variation of the fields $(\\delta g_{ab}, \\delta \\phi, \\delta {\\bf A})$, with the restriction to $\\dM$ denoted by $(\\delta h_{ab}, \\delta \\varphi, \\delta {\\bf \\abig})$, the change in the action is given by\n\\begin{align}\\label{ActionVariation1}\n \\delta I = & \\int_{\\MM} \\nts \\nts d^{\\,d+1}x \\sqrt{g} \\, \\left( \\EE^{ab} \\, \\delta g_{ab}\n + \\EE_{\\phi} \\, \\delta \\phi + \\EE^{a_1 \\ldots a_{p-1}} \\, \\delta A_{a_1 \\ldots a_{p-1}}  \\right) \\\\ \\nonumber\n &  +  \\int_{\\dM} \\bns \\nts d^{\\,d} x \\, \\sqrt{h} \\, \\left( \\pi^{ab} \\, \\delta h_{ab}\n + \\pi_{\\phi} \\, \\delta \\varphi + \\pi^{a_1 \\ldots a_{p-1}} \\, \\delta \\abig_{a_1 \\ldots a_{p-1}} \\right) ~.\n\\end{align}\nThe coefficients of the field variations in the bulk integral, when set to zero, yield the equations of motion of the theory\n\\begin{gather}\\label{EinsteinEOM}\n R_{ab} - \\frac{1}{2}\\,g_{ab}\\, R = \\frac{4}{d-1}\\,\\left( \\nabla_{a} \\phi \\nabla_{b}\n    \\phi -\\frac{1}{2}\\,g_{ab}\\,(\\nabla \\phi)^2 \\right)   + \\frac{1}{2 \\, (p-1)!}\\,e^{2\\alpha \\phi} \\, \\left(F_{a}{}^{\\ldots}\\,F_{b \\ldots } - \\frac{1}{2\\,p}\\,g_{ab}\\,F^{\\,2}\\right)\\\\\\label{DilatonEOM}\n \\nabla^{2} \\phi - \\frac{\\alpha \\, (d-1)}{8\\,p!}\\, e^{2 \\alpha \\phi} \\, F^{\\,2} = 0 \\\\ \\label{MaxwellEOM}\n \\nabla_{a_1} \\left(e^{2\\alpha \\phi} \\, F^{a_1 \\ldots a_p} \\right) = 0~.\n\\end{gather}\nThe boundary terms that remain in \\eqref{ActionVariation1} have the form $p\\,\\delta q$, where $\\delta q$ is a field variation on $\\dM$ and $p$ is the momentum conjugate to $q$. As long as these momenta are finite at $\\dM$, the boundary terms in $\\delta I$ will vanish if the fields satisfy Dirichlet boundary conditions.\n\nThis derivation of the equations of motion assumes that spacetime has a boundary where the field variations vanish. But we are often interested in spacetimes with non-compact spatial slices, where instead of a boundary one is interested in the behavior of the fields at spatial infinity. In that case these equations do not give stationary points of the action \\eqref{bulkaction}. This is not a surprising result: gravity with asymptotically flat boundary conditions suffers from the same problem \\cite{Regge:1974zd, Mann:2005yr, Mann:2006bd}. The main goal of this paper is to construct an action with a well-defined variational principle that describes asymptotically linear dilaton gravity on spacetimes with non-compact spatial slices. Our first assumption regarding this action is that its stationary points should satisfy the equations \\eqref{EinsteinEOM}-\\eqref{MaxwellEOM}.\n\n\n\n\\subsection{ALD Boundary Conditions}\n\\label{sec:BndyCond}\n\nFor spacetimes with non-compact spatial slices, the variational principle must distinguish between different field configurations that exhibit the same behavior near spatial infinity. This requires that we set fall-off conditions for the fields along some spacelike coordinate $\\rho$, with spatial infinity at $\\rho \\to \\infty$, and determine appropriate kinematic constraints on the asymptotic values of the fields. In a slight abuse of terminology we will refer to these specifications as ``boundary conditions\".\n\n\n\\vskip1em\n\\noindent{\\bf Fall-off Conditions}\n\\vskip.5em\n\n\\noindent The fall-off conditions are expressed in terms of asymptotic expansions of the fields. For $\\rho \\gg 1$ the line element takes the form\n\\begin{gather}\\label{metricBC}\n  ds^{2} = d\\rho^{2} + \\rho^{2} \\, \\left(h_{ab}^\\ts{(0)} + \\rho^{2-d} \\,\n    h_{ab}^\\ts{(1)} + \\rho^{1-d} \\, h_{ab}^\\ts{(2)} + \\ldots \\right)\\,dx^{a} dx^{b} ~.\n\\end{gather}\nLet $\\Sigma$ denote a $d$-dimensional surface of constant $\\rho$\\,; then the leading term $h_{ab}^\\ts{(0)}$ is a metric on $\\Sigma$, and the $h_{ab}^\\ts{(n)}$ are smooth, symmetric tensor fields on $\\Sigma$. The properties of constant $\\rho$ surfaces are reviewed in appendix \\ref{sec:Hyper}. We assume that $\\Sigma$ consists of a timelike component $\\Sigma_\\ts{A}$ with dimension $d-p$ and a spacelike component $\\Sigma_\\ts{B}$ with dimension $p$\n\\begin{gather}\n  \\Sigma \\sim \\Sigma_\\ts{A} \\times \\Sigma_\\ts{B} ~.\n\\end{gather}\nThe component $\\Sigma_\\ts{A}$ has coordinates $x^{i}$ and metric $h_{ij}^\\ts{(0)}$, and the component $\\Sigma_\\ts{B}$ has coordinates $x^{\\alpha}$ and metric $h_{\\alpha\\beta}^\\ts{(0)}$.\n\nThe asymptotic expansion of the dilaton is similar to \\eqref{metricBC}, except the leading term is logarithmic\n\\begin{align}\\label{dilatonBC}\n    \\phi = & \\,\\,\\bar{\\phi} \\,\\log(\\rho) + \\phi^\\ts{(0)} + \\rho^{\\,2-d}\\,\\phi^\\ts{(1)} + \\rho^{\\,1-d}\\,\\phi^\\ts{(2)} + \\ldots ~.\n\\end{align}\nThe quantities $\\bar{\\phi}$ and $\\phi^\\ts{(0)}$ are constants, and the remaining $\\phi^\\ts{(n)}$ are smooth functions on $\\Sigma$. Instead of specifying fall-off conditions for the $p-1$ form ${\\bf A}$, it is more convenient to work directly with the field strength ${\\bf F}$. The expansion for the field strength is\n\\begin{align} \\label{fluxBC}\n    F_{a_1 \\ldots a_p} = & F_{a_1 \\ldots a_p}^\\ts{(0)} + \\rho^{\\,2-d} \\, F_{a_1 \\ldots a_p}^\\ts{(1)} + \\rho^{\\,1-d} \\, F_{a_1 \\ldots a_p}^\\ts{(2)} \\\\ \\nonumber\n    & \\, \\, + \\rho^{\\,1-d} \\, n_{{\\textrm{\\tiny [}}\\,a_1} \\, t^\\ts{(1)}_{a_2 \\ldots a_p \\, \\textrm{\\tiny ]}}\n    + \\rho^{-d} \\, n_{{\\textrm{\\tiny [}}\\,a_1} \\, t^\\ts{(2)}_{a_2 \\ldots a_p \\, \\textrm{\\tiny ]}} + \\ldots ~.\n\\end{align}\nThe leading term in this expansion is assumed to have support only on the $p$-dimensional subspace $\\Sigma_{B}$, i.e.\n\\begin{gather}\nF_{\\alpha_1 \\ldots \\alpha_p}^\\ts{(0)} = Q \\, \\eps_{\\alpha_1 \\ldots \\alpha_p}^\\ms{(0)}~,\n\\end{gather}\nwith all other components at this order equal to zero. The remaining terms are smooth, anti-symmetric tensors on $\\Sigma$, with $n^{a}$ the spacelike unit vector normal to $\\Sigma$.\n\n\\vskip1em\n\\noindent{\\bf Boundary Data}\n\\vskip.5em\n\n\\noindent The leading terms in the asymptotic expansions \\eqref{metricBC}, \\eqref{dilatonBC}, and \\eqref{fluxBC} constitute the boundary data for the fields. In principle one may specify any boundary data and consider the set of all field configurations that satisfy the resulting boundary conditions. However, this set will only contain solutions of the equations of motion if the boundary data satisfies the equations obtained from the leading terms in the asymptotic expansions of \\eqref{EinsteinEOM}-\\eqref{MaxwellEOM}. These kinematic constraints must therefore be imposed as part of our boundary conditions.\n\nThe leading term in the dilaton equation \\eqref{DilatonEOM} fixes $\\bar{\\phi}$ and $\\phi^{_{(0)}}$ in terms of other constants in the theory. If we define the parameter $\\beta$ as\n\\begin{equation}\n  \\beta^{\\,2\\,(p-1)} \\defeq \\frac{8\\,(p-1)}{\\alpha^{2}\\,Q^{2}}~,\n\\end{equation}\nthen the first two terms in \\eqref{dilatonBC} are required to take the form\n\\begin{gather}\\label{DilatonConstraint}\n \\bar{\\phi} \\,\\log(\\rho) + \\phi^\\ts{(0)} = \\frac{p-1}{\\alpha}\\,\\log(\\beta\\,\\rho) ~.\n\\end{gather}\nThe leading term in the Einstein equation \\eqref{EinsteinEOM} fixes the Ricci tensor of the $d$-dimensional metric $h_{ab}^\\ts{(0)}$. The components of $\\RR_{ab}^\\ts{(0)}$ on $\\Sigma_\\ts{A}$ and $\\Sigma_\\ts{B}$ are\n\\begin{gather} \\label{Ricci2}\n  \\RR_{ij}^\\ts{(0)} = (d-1)\\,\\left(1-\\frac{4\\,(p-1)^2}{\\alpha^{2}\\,(d-1)^2} \\right)\\,\n    h_{ij}^\\ts{(0)} \\\\ \\label{Ricci1}\n  \\RR_{\\alpha\\beta}^\\ts{(0)} = (d-1)\\,\\left( 1+ \\frac{4\\,(p-1)\\,(d-p)}{\\alpha^{2}\\,(d-1)^{2}} \\right)\\,h_{\\alpha\\beta}^\\ts{(0)} ~.\n\\end{gather}\nFor convenience we denote the numerical factors in these expressions by $A$ and $B$, respectively. Notice that $B$, the factor that appears in the Ricci tensor on $\\Sigma_\\ts{B}$, is strictly positive. The factor $A$, on the other hand, may be positive, negative, or zero, depending on the details of the theory.\n\nTaken together, the fall-off conditions \\eqref{metricBC}-\\eqref{fluxBC} and the constraints \\eqref{DilatonConstraint}-\\eqref{Ricci1} on the boundary data are the ALD boundary conditions. There are two different motivations for this particular choice of boundary conditions. First, they are relevant for various supergravity theories. For example, the near horizon description of Neveu-Schwarz and Dirichlet 5-branes in type II supergravity take this form \\cite{Narayan:2001dr, Parnachev:2005hh}. Second, they represent an interesting generalization of gravity with asymptotically flat (AF) boundary conditions \\cite{Beig:1980be, Beig:1982xx, Ashtekar:1990gc, Ashtekar:1991vb, Mann:2005yr, Mann:2008ay}. This can be seen in the fall-off conditions for the metric, which are the same in both cases. Notice, though, that the constraints on the metric are very different in the two theories. For AF gravity the constraint on the metric $h_{ab}^\\ms{(0)}$ at spatial infinity is\n\\begin{gather}\\label{AFgravity}\n\\RR_{ab}^\\ms{(0)} = (d-1) \\, h_{ab}^{(0)} ~. \n\\end{gather}\nThe analogous conditions \\eqref{Ricci2} and \\eqref{Ricci1} for ALD gravity are deformed due to the non-vanishing field strength at spatial infinity. \n\n\n\n\\subsection{Spacetime Cut-Offs and Regulated Integrals}\n\\label{sec:BndyDef}\n\nThe integrals that appear in the action \\eqref{bulkaction} are not defined for fields that satisfy the fall-off conditions \\eqref{metricBC}-\\eqref{fluxBC}. For example, the bulk integral contains terms that diverge when the upper limit of the integral over $\\rho$ is taken to infinity. Before we can verify the claim that this action is not suitable for spacetimes with non-compact spatial slices, we need to establish a systematic method for regulating these integrals.\n\nWe follow the same approach that was used in \\cite{Mann:2005yr}. Let $\\MM_\\ms{\\Omega} \\subset \\MM$ be a compact region with boundary $\\dM_\\ms{\\Omega}$. The metric $g$ on $\\MM$ induces a metric $h_\\ms{\\Omega} = g|_{\\dM_\\ms{\\Omega}}$ on $\\dM_\\ms{\\Omega}$. Now consider a one-parameter family of these regions that converges to $\\MM$ as $\\Omega \\to \\infty$; i.e., $\\MM_\\ms{\\Omega'}\\subset \\MM_\\ms{\\Omega}$ for $\\Omega > \\Omega'$ and $\\cup_\\ms{\\Omega} \\MM_\\ms{\\Omega} = \\MM$. For finite values of $\\Omega$ this defines a method of cutting-off the spacetime. Integrals like the ones appearing in \\eqref{bulkaction} are regulated by evaluating them on the compact region $(\\MM_\\ms{\\Omega},g)$ or its boundary $(\\dM_\\ms{\\Omega}, h_\\ms{\\Omega})$. The resulting functionals are perfectly well-defined for any finite value of the cut-off $\\Omega$. Of course, the regulated integrals contain terms that diverge as $\\Omega \\to \\infty$, and the cut-off cannot be removed until those terms have been addressed.\n\nThere are different families of cut-offs that can be used to regulate the action, with a family defined as a collection of cut-off prescriptions that all give the same results when the cut-off is removed. Therefore, the family of cut-offs used to regulate the action represents an additional structure that enters into the definition of the theory. Given the ALD boundary conditions for the theory, it is natural to focus on cut-offs that take the form $\\Omega = \\rho + \\OO(\\rho^{\\,0})$\\,\\footnote{This is the hyperbolic cut-off for AF gravity discussed, along with other families of cut-offs, in \\cite{Mann:2005yr}.}. With this choice the boundary of the compact region $\\MM_\\ms{\\rho}$ is the constant $\\rho$ surface $\\Sigma$. Throughout the rest of this paper it is always assumed that integrals and functionals are regulated in this manner. When a functional is written explicitly in terms of integrals over $\\MM$ and\/or $\\dM$, this means that the process of regulating the functional, evaluating it, and then removing the cut-off gives a finite result. If this is not the case, the functional will be expressed in terms of integrals over $\\MM_\\ms{\\rho}$ and $\\Sigma$.\n\n\n\n\\subsection{Variation of the Action}\n\\label{sec:ActionVariation}\n\n\nUsing the regulator defined in the last section, we can show that the action \\eqref{bulkaction} is not stationary with respect to variations of the fields that preserve the ALD boundary conditions. The restriction of these variations to the regulating boundary $\\Sigma$ take the form\n\\begin{align} \\label{hvariation}\n\\delta h_{ab} = & \\,\\,\\,\\, \\rho^{\\,4-d}\\,\\delta h_{ab}^\\ms{(1)} + \\rho^{\\,3-d}\\,\\delta h_{ab}^\\ms{(2)} \\\\ \\label{phivariation}\n\\delta \\varphi = & \\,\\,\\,\\, \\rho^{\\,2-d}\\,\\delta \\phi^\\ms{(1)} + \\rho^{\\,1-d}\\,\\delta \\phi^\\ms{(2)} \\\\ \\label{Avariation}\n\\delta \\abig_{a_1 \\ldots a_{p-1}} = & \\,\\,\\,\\, \\rho^{\\,2-d}\\,\\delta \\abig_{a_1 \\ldots a_{p-1}}^\\ms{(1)} +\n    \\rho^{\\,1-d}\\,\\delta \\abig_{a_1 \\ldots a_{p-1}}^\\ms{(2)} ~.\n\\end{align}\nThe scalars and tensors on the right-hand-side of these equations are arbitrary smooth fields on $\\Sigma$ with the appropriate symmetries. The change in the action, evaluated on a solution of the equations of motion, is\n\\begin{align}\\label{OnShellDeltaI}\n \\delta I \\big|_\\ts{EOM} = &   \\int_{\\Sigma} \\nts d^{\\,d} x \\, \\sqrt{h} \\, \\left( \\pi^{ab} \\, \\delta h_{ab}\n + \\pi_{\\phi} \\, \\delta \\varphi + \\pi^{a_1 \\ldots a_{p-1}} \\, \\delta \\abig_{a_1 \\ldots a_{p-1}} \\right) ~.\n\\end{align}\nFor the action to be stationary these surface terms must vanish as $\\rho \\to \\infty$ for arbitrary field variations of the form \\eqref{hvariation}-\\eqref{Avariation}. To evaluate the surface terms we need the following expressions for the momenta\n\\begin{align}\\label{hmomentum}\n\\pi^{ab} = & \\,\\,\\,\\, h^{ab}\\,K - K^{ab} \\\\ \\label{phimomentum}\n\\pi_{\\phi} = & \\,\\,\\,\\, - \\frac{8}{d-1}\\,n^{a}\\,\\partial_{a} \\phi \\\\ \\label{Amomentum}\n\\pi^{a_{1}\\ldots a_{p-1}} = & \\,\\,\\,\\, - \\frac{1}{2\\,(p-1)!}\\,n_{b}\\,F^{b\\,a_1 \\ldots b_{p-1}} ~.\n\\end{align}\nThe boundary conditions on the $(p-1)$-form insure that the momentum \\eqref{Amomentum} falls off very rapidly, so that terms in \\eqref{OnShellDeltaI} proportional to $\\delta \\abig$ vanish as $\\rho \\to \\infty$. This is not the case for the metric and dilaton. Their variation changes the action by\n\\begin{gather}\\label{ActionVariationNoCT}\n \\delta I \\big|_\\ts{EOM} = \\int_{\\Sigma} \\nts d^{\\,d}x \\sqrt{h^\\ms{(0)}} \\, \\left[\n    (d-1)\\,h^{ab}_\\ms{(0)}\\,\\left( \\rho \\, \\delta h_{ab}^\\ms{(1)} + \\delta h_{ab}^\\ms{(2)} \\right)\n    - \\frac{8\\,(p-1)}{\\alpha\\,(d-1)}\\,\\left( \\rho \\, \\delta \\phi^\\ms{(1)} + \\delta \\phi^\\ms{(2)} \\right)\\right] ~.\n\\end{gather}\nThis does not vanish, despite the fact that the variations $\\delta h_{ab}$ and $\\delta \\varphi$ `vanish at the boundary'. We conclude that the desired equations of motion are not stationary points of the action \\eqref{bulkaction}. \n\nThe result \\eqref{ActionVariationNoCT} contains terms linear in $\\rho$, so that the cut-off cannot be removed in this calculation. One might try to modify the boundary conditions so that these terms do not appear~\\footnote{For example, the boundary conditions used in \\cite{Marolf:2007ys} do not permit such terms.}. But even in that case \\eqref{ActionVariationNoCT} still contains finite terms that do not vanish. Additional `pruning' of the boundary conditions results in a trivial variational principle, with the allowed field variations falling off too rapidly to distinguish between interesting solutions of the equations of motion. Instead, our only option is to modify the action.\n\n\n\\section{$\\hK$ Counterterms}\n\\label{sec:KHat}\n\nThe fall-off conditions on the metric \\eqref{metricBC} are the same as for an asymptotically flat spacetime in hyperbolic coordinates. This suggests that the $\\hK$ counterterm constructed in \\cite{Mann:2005yr} may also be relevant for the theories we are considering. The authors of \\cite{Marolf:2007ys} used this approach -- with modifications -- to define an action for fields in a background of NS 5-branes in 10 dimensional type II supergravity. In this section we generalize the constructions of \\cite{Mann:2005yr, Marolf:2007ys, Mann:2008ay} to the ALD theories described in the last section.\n\n\n\\subsection{$\\hK$ for AF Gravity}\n\\label{sec:PureGravityKhat}\n\nLet us briefly review the $\\hK$ counterterm for gravity with AF boundary conditions. It was shown in \\cite{Mann:2005yr, Mann:2006bd, Mann:2008ay} that solutions of the vacuum Einstein equations are stationary points of the action\n\\begin{gather}\\label{puregravaction}\nI = \\int_{\\MM}\\bns d^{\\,d+1}x \\,\\sqrt{g}\\, R \\, +\\, 2 \\int_{\\dM} \\bns \\nts d^{\\,d}x \\,\\sqrt{h}\\,\\left(K - \\hK \\right) ~.\n\\end{gather}\nThe term $\\hK$ in the boundary integral is the trace of a symmetric tensor $\\hK_{ab}$ defined (implicitly) by the equation\n\\begin{equation}\\label{flatgravKhat}\n \\hK \\, \\hK_{ab} - \\hK_{a}{}^{c}\\,\\hK_{b c} = \\RR_{ab} ~,\n\\end{equation}\nwith $\\RR_{ab}$ the intrinsic Ricci tensor on $(\\dM,h)$. The tensor $\\hK_{ab}$ can be thought of as the part of the extrinsic curvature that is fixed by the kinematics of AF gravity. Furthermore, $\\hK_{ab}$ is defined for any AF space time, which ensures that \\eqref{puregravaction} has a well-defined variational problem.\n\n\nIn order to adapt this construction to ALD gravity, we first need to understand the origin of the defining equation \\eqref{flatgravKhat}. The starting point is the expression \\eqref{ElectricWeyl1} for the electric part of the bulk Weyl tensor evaluated on $\\Sigma$\n\\begin{equation}\\label{EWeyl1}\n  E_{ab} = \\perp(R_{a c b d} \\, n^{c} n^{d}) - \\perp(S_{a b}) -\n    h_{ab} \\, S_{c d} \\, n^{c} n^{d} ~,\n\\end{equation}\nwhere $n^{a}$ is the outward-pointing unit normal vector, $\\perp$ represents projection of indices along $\\Sigma$ via the induced metric $h_{ab}$, and $S_{ab}$ is the Schouten tensor defined in \\eqref{Schouten}. Applying both the Gauss-Codazzi equations and the Einstein equations gives\n\\begin{equation}\\label{EWeyl2}\nK\\,K_{ab} - K_{a}{}^{c}\\,K_{b c} = \\RR_{ab} + E_{ab} ~.\n\\end{equation}\nThis result does not hold for an arbitrary AF spacetime -- it only applies to solutions of the Einstein equations. However, all AF spacetimes must satisfy the kinematic constraints obtained from the leading term in an asymptotic expansion of the Einstein equations near spatial infinity. Therefore \\emph{all} AF spacetimes satisfy the leading term in the asymptotic expansion of \\eqref{EWeyl2}\n\\begin{equation}\\label{preDefiningEqn}\nK^\\ms{(0)}\\,K_{ab}^\\ms{(0)} - K_{a}^\\ms{(0)}{}^{c}\\,K_{bc}^\\ms{(0)} = \\RR_{ab}^\\ms{(0)} ~,\n\\end{equation}\nwhich depends only on geometric identities and the kinematic constraints of the theory. \nThe relation between  \\eqref{preDefiningEqn} and \\eqref{flatgravKhat} is apparent, but it will be useful to introduce one additional bit of notation before proceeding. Let $\\widehat{=}$ denote equality between quantities on $\\Sigma$ up to terms that are sub-leading at large $\\rho$. Using this notation, the equation \\eqref{preDefiningEqn} is equivalent to\n\\begin{equation}\\label{preDefiningEqn2}\nK\\,K_{ab} - K_{a}{}^{c}\\,K_{bc} \\,\\,\\,\\, \\widehat{=} \\,\\,\\,\\, \\RR_{ab} ~.\n\\end{equation}\nThe symmetric tensor $\\hK_{ab}$ is defined as the tensor that satisfies \\eqref{preDefiningEqn2} when `$\\widehat{=}$' is replaced by `$=$'. This is the origin of the definition \\eqref{flatgravKhat}.\n\n\\subsection{$\\hK$ for ALD Gravity}\n\\label{sec:DefiningKhat}\n\nThis procedure outlined in the previous section can be generalized to ALD gravity. Starting from the definition \\eqref{EWeyl1} and applying the equations of motion, we obtain\n\\begin{equation}\\label{EWeyl3}\nK\\,K_{ab} - K_{a}{}^{c} \\, K_{bc} = E_{ab} + \\RR_{ab} -\\left(\\frac{d-2}{d-1}\\right)\\,\\perp(T_{ab}) - \\frac{1}{d\\,(d-1)}\\,h_{ab}\\,T^{c}{}_{c} + \\frac{1}{d-1}\\,h_{ab}\\,n^{c}\\,n^{d}\\,T_{cd}~,\n\\end{equation}\nwhere $T_{ab}$ is the right-hand side of equation \\eqref{EinsteinEOM}. The various projections of $T_{ab}$ can be worked out using the results of appendix \\ref{sec:Hyper}. As before, we begin by expressing the leading term in the asymptotic expansion of this equation in terms of the boundary data. Using the `$\\widehat{=}$' notation, the terms in \\eqref{EWeyl3} that are relevant at leading order are\n\\begin{align}\\label{preDefiningEqn3} \\nonumber\nK\\,K_{ab} - K_{a}{}^{c} \\, K_{bc} \\,\\,\\,\\, \\widehat{=} \\,\\, & \\,\\, E_{ab} + \\RR_{ab} + \\frac{4}{d\\,(d-1)}\\,h_{ab}\\,\\left(n^{c} \\partial_{c} \\phi \\right)^{2} - \\frac{(d-2)}{2\\,(d-1)\\,(p-1)!}\\,e^{2\\,\\alpha\\,\\phi}\\perp\\left(F_{a}{}^{\\ldots}\\,F_{b \\ldots} \\right) \\\\ & \\,\\,\\,\\, \\,\\,\\,\\, + \\frac{\\big(p\\,(d-2)-(d-1)\\big)}{2\\,d\\,(d-1)\\,p!}\\,h_{ab}\\,e^{2\\,\\alpha\\,\\phi}\\,F^{\\,2} ~.\n\\end{align}\nNow the right-hand side of this equation must be expressed in terms of intrinsic quantities on $\\Sigma$. This is straightforward for the field strength terms -- the boundary conditions allow us to replace $F_{a_1 \\ldots a_p}$ with $\\FF_{a_1 \\ldots a_p}$, its projection along $\\Sigma$. The normal derivative of $\\phi$ can be expressed, up to sub-leading terms, as a simple function of $\\phi$ on $\\Sigma$\n\\begin{equation}\nn^{c} \\partial_{c} \\phi \\,\\,\\,\\, \\widehat{=} \\,\\,\\,\\, \\frac{(p-1)}{\\alpha}\\,\\beta\\,e^{-\\frac{\\alpha}{p-1}\\,\\phi} ~.\n\\end{equation}\nLikewise, the result \\eqref{E0} gives the leading behavior of the electric part of the Weyl tensor as \n\\begin{equation}\nE_{ab} \\,\\,\\,\\, \\widehat{=} \\,\\,\\,\\, - \\frac{1}{d-1}\\,\\left(\\RR_{ab} - \\frac{1}{d}\\,h_{ab}\\,\\RR \\right) ~.\n\\end{equation}\nNotice that this vanishes for the AF boundary conditions \\eqref{AFgravity}, but not in the case of ALD gravity.\n\nAt this point \\eqref{preDefiningEqn3} could be rewritten to furnish a defining equation for $\\hK_{ab}$. However, in order to make a direct connection with the defining equation \\eqref{flatgravKhat} for AF gravity, it is useful to express the leading part of $E_{ab}$ as a function of the dilaton and field strength on $\\Sigma$. Using the various kinematic constraints on the fields, we obtain\n\\begin{equation}\nE_{ab} \\,\\,\\,\\, \\widehat{=} \\,\\,\\,\\, - \\frac{1}{2\\,(d-1)\\,(p-1)!}\\,e^{2\\,\\alpha\\,\\phi}\\,\\left( \\FF_{a}{}^{\\ldots}\\,\\FF_{b \\ldots} -\\frac{1}{d}\\,h_{ab}\\,\\FF^{\\,2} \\right)~.\n\\end{equation}\nSubstituting this result in \\eqref{preDefiningEqn3} now gives a defining equation for $\\hK_{ab}$ that resembles the AF definition, with additional terms involving the dilaton and field strength\n\\begin{align}\\label{DefiningEquation}\n\\hK\\,\\hK_{ab} - \\hK_{a}{}^{c}\\,\\hK_{bc} = & \\,\\,\\,\\, \\RR_{ab} + \\frac{4\\,(p-1)^{2}\\,\\beta^{\\,2}}{d\\,(d-1)\\,\\alpha^{2}}\\,h_{ab}\\,e^{-\\frac{2\\,\\alpha}{p-1}\\,\\phi} \\\\ \\nonumber & - \\frac{1}{2\\,(p-1)!}\\,e^{2\\,\\alpha\\,\\phi}\\,\\FF_{a}{}^{\\ldots}\\,\\FF_{b \\ldots} + \\frac{(p-1)}{2\\,d\\,p!}\\,h_{ab}\\,e^{2\\,\\alpha\\,\\phi}\\,\\FF^{2} ~.\n\\end{align}\nThis equation can be solved order-by-order in the asymptotic expansion for any set of fields satisfying the ALD boundary conditions\\,\\footnote{In $3+1$ dimensions with AF boundary conditions it is possible to obtain $\\hK_{ab}$ explicitly \\cite{Visser:2008gx}.}. The result is an asymptotic expansion for $\\hK_{ab}$ of the form\n\\begin{equation}\\label{KhatExpansion}\n  \\hK_{ab} = \\rho \\,\\hK_{ab}^\\ms{(0)} + \\rho^{\\,3-d}\\,\\hK_{ab}^\\ms{(1)} + \\rho^{\\,2-d}\\,\\hK_{ab}^\\ms{(2)} + \\ldots ~.\n\\end{equation}\nThe equation for the leading term is straightforward and gives\n\\begin{equation}\\label{LeadingKhat}\n  \\hK_{ab}^\\ms{(0)} = h_{ab}^\\ms{(0)} ~,\n\\end{equation}\nwhich shows that $\\hK_{ab}$ and $K_{ab}$ agree at leading order\\,\\footnote{The equation \\eqref{DefiningEquation} admits two solutions with opposite signs. We have chosen the solution whose sign agrees with that of the leading term in the extrinsic curvature.}. Expressions for the sub-leading terms in \\eqref{KhatExpansion} are more complicated, and will be given as needed later in the paper.\n\nAn important caveat regarding the procedure described above is that it does not give a unique definition for $\\hK_{ab}$\\,! The boundary data for the different fields are related by the kinematic constraints in the boundary conditions, and these relations can be used to rewrite the right-hand side of \\eqref{DefiningEquation}. For example, the boundary conditions imply the following asymptotic relations\n\\begin{align}\\label{FManipulations}\n  \\frac{1}{2\\,p!}\\,e^{2\\,\\alpha\\,\\phi}\\,\\FF^{\\,2} \\,\\,\\,\\, \\widehat{=} & \\,\\,\\,\\, \\frac{4\\,(p-1)\\,\\beta^{\\,2}}{\\alpha^{2}}\\,e^{-\\frac{2\\,\\alpha}{p-1}\\,\\phi} \\\\ \\label{RManipulations}\n  \\RR \\,\\,\\,\\, \\widehat{=} & \\,\\,\\,\\, \\RR^\\ms{(0)} \\, \\beta^{\\,2}\\,e^{-\\frac{2\\,\\alpha}{p-1}\\,\\phi}~.\n\\end{align}\nThese sorts of replacements will not change the value of $\\hK_{ab}$ at leading order -- that is fixed by the boundary conditions -- but they \\emph{will} change the functional dependence of $\\hK_{ab}$ on the different fields. In other words, one obtains inequivalent definitions of $\\hK_{ab}$. The obvious criteria for choosing one of these definitions is whether the corresponding counterterm, when added to the action \\eqref{bulkaction}, leads to a well-defined variational principle. As we show in the next section, there is no definition that manages to do this.\n\n\\subsection{Variation of the Action}\n\nWe now investigate the variational properties of the action \\eqref{bulkaction} with the additional surface term\n\\begin{equation}\\label{hKterm}\nI_{\\hK} = -2\\int_{\\Sigma}\\nts d^{\\,d}x \\,\\sqrt{h}\\,\\hK ~.\n\\end{equation}\nSpecifically, we want to determine whether or not the equations of motion \\eqref{EinsteinEOM}-\\eqref{MaxwellEOM} are stationary points of the new action with respect to small changes in the fields that preserve the boundary conditions. The result \\eqref{ActionVariationNoCT} showed that this was not the case for the action \\eqref{bulkaction} on its own.\n\nThe change in \\eqref{hKterm} due to a small variation of the fields is\n\\begin{equation}\\label{deltaLKhat}\n\\delta I_{\\hK} = -2\\int_{\\Sigma}\\nts d^{\\,d}x \\,\\sqrt{h}\\,\\left[\\left(\\frac{1}{2}\\,\\hK\\,h^{ab} - \\hK^{ab} \\right)\\,\\delta h_{ab} + h^{ab}\\,\\delta \\hK_{ab}\\right] ~.\n\\end{equation}\nThe response $\\delta\\hK_{ab}$ is obtained by varying the defining equation \\eqref{DefiningEquation}, and solving order-by-order in the asymptotic expansion. In fact, since it is the trace of $\\delta\\hK_{ab}$ that appears in \\eqref{deltaLKhat}, it is sufficient to work with the trace of the defining equation\n\\begin{equation} \\label{TraceDefEqn}\n\\hK^{2} - \\hK^{ab}\\,\\hK_{ab} = \\RR + \\frac{4\\,(p-1)^{2}}{\\alpha^{\\,2}\\,(d-1)}\\,\\beta^{\\,2} \\, e^{-\\frac{2\\,\\alpha}{p-1}\\,\\phi} - \\frac{1}{2\\,p!}\\,e^{2\\,\\alpha\\,\\phi}\\,\\FF^{\\,2} ~.\n\\end{equation}\nVarying both sides of this equation and rearranging terms, we arrive at\n\\begin{align}\\label{VarTraceDefEqn} \\nonumber\n2\\,\\left(\\hK\\,h^{ab} - \\hK^{ab} \\right)\\,\\delta \\hK_{ab} = & \\,\\, \\left(2\\,\\hK\\,\\hK^{ab} - 2\\,\\hK^{a}{}_{c}\\,\\hK^{bc} - \\RR^{ab} + \\frac{1}{2\\,(p-1)!}\\,e^{2\\,\\alpha\\,\\phi}\\,\\FF^{a}{}_{\\ldots} \\, \\FF^{b \\ldots} \\right)\\,\\delta h_{ab} \\\\\n & \\,\\,\\,\\, - \\left(\\frac{8\\,(p-1)\\,\\beta^{\\,2}}{\\alpha\\,(d-1)}\\,e^{-\\frac{2\\,\\alpha}{p-1}\\,\\phi} + \\frac{\\alpha}{p!}\\,e^{2\\,\\alpha\\,\\phi}\\,\\FF^{\\,2} \\right) \\,\\delta \\varphi + h^{ab}\\,\\delta \\RR_{ab} \\\\ \\nonumber\n & \\,\\,\\,\\, - \\frac{1}{(p-1)!}\\,e^{2\\,\\alpha\\,\\phi}\\,\\FF^{a_{1} \\ldots a_{p}}\\,\\DD_{a_{1}} \\delta \\abig_{a_{2}\\ldots a_{p}} ~.\n\\end{align}\nOnly the leading term in this equation is relevant for determining \\eqref{deltaLKhat}; the sub-leading terms fall off too rapidly to contribute. Furthermore, total derivatives in \\eqref{VarTraceDefEqn} can be ignored because the term we are calculating is integrated over $\\Sigma$\\,\\footnote{Terms in \\eqref{VarTraceDefEqn} that contribute total derivatives at leading order also produce sub-leading terms that are \\emph{not} total derivatives. These terms must be accounted for when solving for $\\delta \\hK_{ab}$ beyond leading order.}. Using the field variations \\eqref{hvariation} - \\eqref{Avariation}, the leading terms in \\eqref{VarTraceDefEqn} are\n\\begin{align}\\label{TraceKhatVariation}\nh^{ab}_\\ms{(0)} \\, \\delta \\hK_{ab} = & \\,\\, \\left( h^{ab}_{(0)} - \\frac{1}{2\\,(d-1)}\\,\\RR^{ab}_\\ms{(0)} + \\frac{2\\,(p-1)}{\\alpha^{\\,2}\\,(d-1)}\\,\\delta^{a}{}_{\\alpha}\\,\\delta^{b}{}_{\\beta}\\,h^{\\alpha\\beta}_\\ms{(0)} \\right)\\,\\left(\\rho^{\\,3-d}\\,\\delta h_{ab}^\\ms{(1)} + \\rho^{\\,2-d}\\,\\delta h_{ab}^\\ms{(2)} \\right) \\\\ \\nonumber\n& \\,\\,\\,\\, - \\frac{4\\,d\\,(p-1)}{\\alpha\\,(d-1)^{2}}\\,\\left( \\rho^{\\,3-d}\\,\\delta \\phi^\\ms{(1)} + \\rho^{\\,2-d}\\,\\delta \\phi^\\ms{(2)}\\right) + \\ldots ~.\n\\end{align}\nThe `$\\ldots$' refer to terms that are either sub-leading or total derivatives. In particular, the terms involving the variation of the $p-1$ form are total derivatives. The result \\eqref{TraceKhatVariation} applies for any field configuration that obeys the ALD boundary conditions, because it depends only on the defining equation and the constraints on the boundary data.\n\nThe new contribution to the variation of the action can be evaluated using \\eqref{LeadingKhat} and \\eqref{TraceKhatVariation}. Combining these with \\eqref{ActionVariationNoCT} gives\n\\begin{align}\\label{NewActionVariation}\n\\delta ( I + I_{\\hK} ) = \\int_{\\Sigma} \\nts d^{\\,d}x \\,\\sqrt{h^\\ms{(0)}}\\,& \\left[-\\frac{4\\,(p-1)^{2}}{\\alpha^{\\,2}\\,(d-1)^{2}}\\,h^{ab}_\\ms{(0)}\\, \\left(\\rho\\,\\delta h_{ab}^\\ms{(1)} + \\delta h_{ab}^\\ms{(2)} \\right) \\right. \\\\ \\nonumber\n & \\left. \\,\\,\\,\\, \\,\\,\\,\\, + \\frac{8\\,(p-1)}{\\alpha\\,(d-1)^{2}}\\,\\left(\\rho\\,\\delta \\phi^\\ms{(1)} + \\delta \\phi^\\ms{(2)} \\right) \\right] ~.\n\\end{align}\nThis is qualitatively no different than the result we were trying to fix! The presence of the $\\hK$ boundary term in the action \\eqref{puregravaction} ensures a well-defined variational principle for gravity with AF boundary conditions, but the analogous construction does not work for theories with ALD boundary conditions. Of course, in AF gravity a unique definition of $\\hK_{ab}$ follows from a combination of geometric identities and boundary conditions. But in the present case the definition is somewhat arbitrary -- terms on the right-hand side of \\eqref{DefiningEquation} can be rewritten using relations between the boundary data. Does the result \\eqref{NewActionVariation} mean that the $\\hK$ construction does not work for ALD gravity, or does it tell us that we have chosen the wrong defining equation for $\\hK_{ab}$?\n\nWe can answer this question by parameterizing the possible defining equations for $\\hK_{ab}$, and showing that there is no definition such that \\eqref{NewActionVariation} vanishes. First, note that the result \\eqref{NewActionVariation} depends only on the trace of the defining equation. The trace belongs to a two-parameter family of equations, related to \\eqref{TraceDefEqn} via \\eqref{FManipulations} and \\eqref{RManipulations}\n\\begin{align}\n\\hK^{2} - \\hK^{ab}\\,\\hK_{ab} = & \\,\\,\\,\\, (1+x)\\,\\RR - \\frac{(1+y)}{2\\,p!}\\,e^{2\\,\\alpha\\,\\phi}\\,\\FF^{\\,2} \\\\ \\nonumber\n& \\,\\,\\,\\, + \\left(\\frac{4\\,(p-1)^{2}}{\\alpha^{\\,2}\\,(d-1)} - x\\,\\RR^\\ms{(0)} + y\\,\\frac{4\\,(p-1)}{\\alpha^{\\,2}}\\right)\\,\\beta^{\\,2}\\,e^{-\\frac{2\\,\\alpha}{p-1}\\,\\phi}\n\\end{align}\nwhere $x$ and $y$ are arbitrary real numbers. Repeating the analysis of this section shows that there are no values of $x$ and $y$ for which $\\delta I$ vanishes on-shell. This seems odd at first; there are two terms in \\eqref{NewActionVariation} that need to be canceled, and two free parameters ($x$ and $y$) that can be adjusted. The problem becomes apparent when one chooses $x$ so that the $\\delta \\phi$ terms in \\eqref{NewActionVariation} vanish. The variation of the action now contains terms proportional to $\\delta h_{ab}$, with a coefficient that depends on the remaining free parameter $y$. But it also contains new terms proportional to $\\delta h_{\\alpha\\beta}$, which were not present when \\eqref{TraceDefEqn} was taken as the defining equation. There is no choice of $y$ that cancels the coefficients of both sets of terms. Therefore, it is not possible to obtain $\\delta I = 0$ using only the $\\hK$ counterterm. This same problem was encountered in \\cite{Marolf:2007ys}, where it was addressed by adding additional surface terms to the action.\n\n\\subsection{Additional Boundary Terms}\n\nFollowing \\cite{Marolf:2007ys} we introduce three new surface terms, in addition to \\eqref{hKterm}. These terms are functionals of quantities intrinsic to the boundary and take the form\n\\begin{align}\\label{NewBoundaryTerms}\nI_\\ts{CT} = \\int_{\\Sigma} \\nts d^{\\,d}x\\,\\sqrt{h}\\,\\left(c_{_0} \\, e^{-\\frac{\\alpha}{p-1}\\,\\phi} + c_{_1}\\,e^{\\frac{\\alpha}{p-1}\\,\\phi}\\,\\RR + c_{_2}\\,e^{\\frac{2p-1}{p-1}\\,\\alpha\\,\\phi}\\,\\FF^{\\,2} \\right) ~.\n\\end{align}\nThe constants $c_{i}$ are determined by requiring that the full action satisfy $\\delta I = 0$ on-shell. We will discuss a similar calculation in much more detail in the next section, so for now we simply note that for most values of $\\alpha$ the condition $\\delta I = 0$ requires\n\\begin{equation}\\label{deltaIconditions}\nc_{0} = \\frac{8\\,(p-1)^{2}\\,\\beta}{\\alpha^{\\,2}\\,(d-1)^{2}} \\quad \\quad c_{1} = c_{2} = 0 ~.\n\\end{equation}\nThere is also a special case, $\\alpha = \\pm 2(p-1)\/(d-1)$, where the equations that determine the $c_i$ become degenerate and admit a one-parameter family of solutions. In that case the $c_i$ are given by\n\\begin{equation}\\label{deltaIconditions2}\nc_{0} = 2\\,\\beta - (d-1)^{2}\\,\\beta^{\\,2}\\,c_{1} \\quad \\quad  c_{2} = -\\frac{1}{2\\,p!}\\,c_{1} \\end{equation}\nwith $c_1$ arbitrary. The special values of $\\alpha$ corresponds to the coupling between the dilaton and the NS-NS\\,(-) or R-R\\,(+) 3-form field strengths in ten dimensional type II supergravity.\n\n\nThe authors of \\cite{Marolf:2007ys} considered the NS-NS sector of type II supergravity in 10 dimensions.  However, instead of adding the third boundary term in \\eqref{NewBoundaryTerms} they included a term proportional to  $\\RR\\,\\FF^{\\,2}$. This raises an important question: what if we did not limit ourselves to the three boundary terms in \\eqref{NewBoundaryTerms}? For instance, one might add the following boundary terms to the action\n\\begin{align}\\label{NewBoundaryTerms2}\nI_\\ts{CT} = \\int_{\\Sigma} \\nts d^{\\,d}x\\,\\sqrt{h}\\,\\left(c_{_0} \\, e^{-\\frac{\\alpha}{p-1}\\,\\phi} + c_{_1}\\,e^{\\frac{\\alpha}{p-1}\\,\\phi}\\,\\RR + c_{_2}\\,e^{\\frac{2p-1}{p-1}\\,\\alpha\\,\\phi}\\,\\FF^{\\,2} + c_{4}\\,e^{\\frac{3\\,\\alpha}{p-1}\\,\\phi}\\,\\RR^{2} + \\ldots  \\right)\n\\end{align}\nFor generic values of $\\alpha$ this leads to a one-parameter family of actions such that $\\delta I = 0$, and for the special value of $\\alpha$ mentioned above a two-parameter family is obtained. This is qualitatively different than what one finds in asymptotically anti-de Sitter spacetimes, where a higher derivative boundary term like $\\RR^{2}$ always falls off faster than a term like $\\RR$. In a theory with ALD boundary conditions, dressing the higher derivative boundary terms with suitable functions of the dilaton ensures that all of the counterterms contribute at the same orders in the asymptotic expansion. This means that one may add an arbitrary number of boundary terms to obtain a family of actions, all of which satisfy $\\delta I = 0$, with a corresponding number of free parameters! We will return to this issue later. For now, we focus on the boundary terms \\eqref{NewBoundaryTerms}.\n\n\\subsection{Finiteness of the On-Shell Action}\n\nThe conditions \\eqref{deltaIconditions} or \\eqref{deltaIconditions2} also imply that the on-shell action is finite in the $\\rho \\to \\infty$ limit. The equations of motion can be used to rewrite the bulk terms in the action as a total derivative, so that the on-shell action is given by\n\\begin{align}\\label{OnShellAction}\nI \\big|_\\ts{EOM} = \\int_{\\dM} \\bns d^{\\,d}x \\, \\sqrt{h}\\,& \\bigg[  \\raisebox{16pt}{} - \\frac{8\\, (p-1)}{\\alpha\\,(d-1)^{2}}\\,n^{a}\\,\\partial_{a}\\phi + 2\\,K\n - 2\\,\\hK + c_{_0} \\, e^{-\\frac{2\\,\\alpha}{p-1}\\,\\phi} \\\\ \\nonumber\n &  + c_{_1}\\,e^{-\\frac{2\\,\\alpha}{p-1}\\,\\phi}\\,\\RR + c_{_2}\\,e^{-\\frac{2p-1}{p-1}\\,\\alpha\\,\\phi}\\,\\FF^{\\,2} \\raisebox{16pt}{} \\bigg] + \\ldots ~,\n\\end{align}\nwhere `$\\ldots$' represents a contribution from the total derivative evaluated at the lower limit of the bulk integral. These terms can be ignored, since they do not depend on the cut-off $\\rho$ and we are only interested in establishing finiteness at this point. Using the asymptotic expansions of the various boundary terms, we find that the on-shell action contains a leading term of order $\\rho^{d-1}$ and a sub-leading term proportional to $\\rho$. The remaining terms are finite, or vanish as the cut-off is removed. It is straightforward to show that the leading term in the expansion vanishes as a result of the boundary conditions and \\eqref{LeadingKhat}. The order $\\rho$ term needs to cancel as well, and to show this we need the first sub-leading term in the asymptotic expansion of $\\hK$. The asymptotic expansion of $\\hK$ is\n\\begin{equation}\n \\hK= d\\,\\rho^{-1} + \\rho^{\\,1-d}\\,\\left(h^{ab}_\\ms{(0)}\\hK_{ab}^\\ms{(1)}- h^\\ms{(1)} \\right) + \\OO(\\rho^{-d}) ~,\n\\end{equation}\nwhere we have used the result \\eqref{LeadingKhat} for the leading term, and terms of $\\OO(\\rho^{-d})$ give finite contributions to the action. Solving \\eqref{TraceDefEqn}, we find that the order $\\rho^{1-d}$ term in $\\hK$ is\n\\begin{align}\n\\hK^{(1)} -  h^{(1)} = \\frac{1}{2\\,(d-1)}\\,& \\left(\\RR^\\ms{(1)} - h^{ab}_\\ms{(1)}\\,\\RR_{ab}^\\ms{(0)}\n-\\frac{8\\,(p-1)}{\\alpha\\,(d-1)}\\,\\phi^\\ms{(1)} \\right. \\\\ \\nonumber\n & \\left. \\quad - \\frac{4\\,(p-1)}{\\alpha^{\\,2}}\\,\\left(Y^\\ms{(1)} - h^\\ms{(1)}_{s} + 2\\,\\alpha\\,\\phi^\\ms{(1)} \\right) \\right) ~\n\\end{align}\nwhere $h^\\ms{(1)}_{s} = h^{\\alpha\\beta}_\\ms{(0)}\\,h_{\\alpha\\beta}^\\ms{(1)}$, and $Y^\\ms{(1)}$ is defined in equation \\eqref{YDef}. Using this result in \\eqref{OnShellAction} shows that the order $\\rho$ term in the on-shell action vanishes. For generic values of $\\alpha$, the conditions \\eqref{deltaIconditions} and the equations of motion in appendix \\ref{sec:EOMandBianchi} imply that this term vanishes identically. In the special cases $\\alpha = \\pm2(p-1)\/(d-1)$, terms at this order vanish up to total derivatives. In either case, the lesson is that finiteness of the on-shell action follows from the requirement that the action admits a well-defined variational principle.\n\n\n\\section{Explicit Counterterms}\n\\label{sec:OC}\n\n\n\nThe $\\hK$ construction is of interest because it makes contact with a procedure that works for gravity with AF boundary conditions. But this approach is not sufficient if the theory contains additional fields with support at spatial infinity. In that case new surface terms must be included in the action. These terms contribute to the action and its variation at the same orders in the asymptotic expansion as the $\\hK$ term; it stands to reason that one might be able to define a viable action using \\emph{only} terms of the form \\eqref{NewBoundaryTerms}.\n\nIn this section we drop the $\\hK$ counterterm and construct actions using boundary counterterms that are defined explicitly as local functionals of the fields on the boundary. The results are reminiscent of the counterterms used in the AdS\/CFT correspondence \\cite{Henningson:1998gx, Balasubramanian:1999re, Emparan:1999pm, Bianchi:2001kw, Martelli:2002sp, Batrachenko:2004fd, Papadimitriou:2005ii}, but there are also significant differences. In AdS\/CFT a well-defined variational principle identifies a finite set of counterterms \\cite{Papadimitriou:2005ii}. For ALD theories we encounter families of actions that admit a well-defined variational problem but depend on an arbitrary number of free parameters. Surprisingly, we find that the on-shell action and the conserved charges do not depend on the choice of action. This is reassuring, because even in cases where there are compelling reasons to limit ourselves to a small number of possible counterterms, we still find interesting examples that admit a one-parameter family of actions.\n\n\n\n\\subsection{The Action}\n\\label{sec:ECAction}\n\n\nConsider a counterterm action that takes the following form\n\\begin{equation}\\label{CT}\n  I_\\ts{CT} = \\int_{\\Sigma} \\nts d^{\\,d}x \\, \\sqrt{h}\\,\\bigg(U(\\phi) + C(\\phi) \\, \\RR + Z(\\phi)\\,\\FF^{a_1 \\ldots a_p} \\FF_{a_1 \\ldots a_p} + \\ldots \\bigg) ~.\n\\end{equation}\nThis can be thought of as a derivative expansion on $\\Sigma$, neglecting terms that vanish due to the boundary conditions. For example, a term of the form $M(\\phi) (D \\phi)^2$ will not contribute to the action or its first variation. At this point we restrict our attention to terms containing no more than two (boundary) derivatives of the fields on $\\Sigma$. This is motivated by our interest in various supergravity theories obtained as the low-energy limit of string theory: we do not wish to include surface terms corresponding to higher orders in the $\\alpha'$ expansion than the terms that appear in the bulk part of the action.\n\nIncluding the counterterms \\eqref{CT}, the on-shell variation of the action is\n\\begin{align}\\label{1stVariation}\n \\delta I \\big|_\\ts{EOM} = \\int_{\\dM} \\bns d^{\\,d}x \\,\\sqrt{h}\n    \\, & \\bigg[ \\left( \\pi^{ab} + P^{ab} \\right)\\,\\delta h_{ab} + \\left( \\pi_{\\phi} + P_{\\phi} \\right) \\delta \\phi \\\\ \\nonumber\n    & \\quad + \\left(\\pi^{a_1 \\ldots a_{p-1}} + P^{a_1 \\ldots a_{p-1}} \\right)\n    \\,\\delta\\abig_{a_1 \\ldots a_{p-1}} \\bigg]\n\\end{align}\nwhere $\\pi^{ab}$, $\\pi_{\\phi}$, and $\\pi^{a_1 \\ldots a_{p-1}}$ are the momenta given in \\eqref{hmomentum}-\\eqref{Amomentum}. The terms $P^{ab}$, $P_{\\phi}$, and $P^{a_1 \\ldots a_{p-1}}$ are functional derivatives of \\eqref{CT}\n\\begin{align} \\label{CTMomentum1}\nP^{ab}  = \\,\\,\\,\\, & \\frac{1}{2}\\,h^{ab}\\,U - C \\left(\\RR^{ab} - \\frac{1}{2}\\,h^{ab}\\,\\RR \\right) + D^{a} D^{b} C - h^{ab}\\,D^{2}C\\\\  \\nonumber\n& \\,\\,\\,\\, + Z \\, \\left( \\frac{1}{2}\\,h^{ab}\\,\\FF^{\\,2} - p\\,\\FF^{a}{}_{\\ldots}\\,\\FF^{b \\ldots} \\right) \\\\\n \\label{CTMomentum2}\nP_{\\phi} = \\,\\,\\,\\, & \\frac{\\partial U}{\\partial \\phi} + \\frac{\\partial C}{\\partial \\phi} \\, \\RR + \\frac{\\partial Z}{\\partial \\phi} \\, \\FF^{\\,2}  \\raisebox{12pt}{}\\\\ \\label{CTMomentum3}\nP^{a_1 \\ldots a_{p-1}} = \\,\\,\\,\\, & -2\\,p\\,D_{b}\\big(Z \\,\\FF^{b a_{1} \\ldots a_{p-1}} \\big)\\raisebox{12pt}{}\n\\end{align}\nThe functions $U$, $C$, and $Z$ are determined by the requirement that \\eqref{1stVariation} vanishes for field variations that preserve the boundary conditions. Based on the results of the previous section, these functions are expected to take the form\n\\begin{equation}\n  U(\\phi) = c_0 \\, e^{- \\frac{\\alpha}{p-1} \\, \\phi} \\quad\\quad C(\\phi) = c_1 \\, e^{\n    \\frac{\\alpha}{p-1} \\, \\phi}\n    \\quad\\quad Z(\\phi) = c_2 \\, e^{\\frac{2p-1}{p-1} \\, \\alpha\\, \\phi} ~.\n\\end{equation}\nThe constants $c_i$ are determined by canceling the appropriate terms in \\eqref{1stVariation}. It is interesting to note that the dilaton dependence in these expressions means that the terms in \\eqref{CT} correspond to an expansion in powers of the inverse metric in string frame.\n\n\nAs before, the boundary conditions on the gauge field imply that the terms proportional to $\\delta \\abig$ fall off very rapidly and do not contribute to \\eqref{1stVariation}. This leaves the terms proportional to $\\delta h_{ab}$ and $\\delta \\phi$. The leading behavior of the coefficients of these terms are\n\\begin{align}\\label{metriccancel}\n\\pi^{ab} + P^{ab} = \\rho^{-3} & \\left(d-1 + \\frac{c_0}{2\\,\\beta} - c_1\\,\\beta\\,A + \\frac{1}{2}\\,c_1\\,\\beta\\,\\big(A \\,(d-p) + B\\,p\\big) +c_2\\,\\beta\\,p!\\,(B-A) \\right)\\,h^{ab}_\\ms{(0)} \\\\ \\nonumber\n & \\quad + \\beta \\, \\rho^{-3}\\,\\big((B-A)\\,c_1 + c_2\\,p!\\,Q^{2}\\,\\beta^{\\,2(p-1)} \\big)\\,\\delta^{a}{}_{\\alpha} \\,\\delta^{b}{}_{\\beta}\\,h^{\\alpha\\beta}_\\ms{(0)} \\\\ \\label{dilatoncancel}\n\\pi_{\\phi} + P_{\\phi} = \\rho^{-1} & \\frac{\\alpha}{p-1}\\,\\bigg(-\\frac{8\\,(p-1)^2}{\\alpha^{\\,2}\\,(d-1)} - \\frac{c_0}{\\beta} + c_1 \\, \\beta\\, \\big(A\\,(d-p) + B\\,p\\big) \\\\ \\nonumber \n  & \\quad \\quad \\quad \\,\\,\\,\\,  + 2\\,c_2\\,\\beta\\,p!\\,(2p-1)\\,(B-A)\\bigg)\n\\end{align}\nwhere $A$ and $B$ are the numerical factors appearing in the expressions \\eqref{Ricci2} and \\eqref{Ricci1} for the Ricci tensor on $\\Sigma$.  The $\\rho \\to \\infty$ limit of \\eqref{1stVariation} vanishes if these expressions are set to zero. This yields three  equations, because the terms in \\eqref{metriccancel} with support along $\\Sigma_\\ts{B}$ must vanish independently. These equations uniquely determine the $c_i$ for most values of $\\alpha$\n\\begin{gather}\\label{uniqueci}\n c_0 = \\beta\\,\\left(-d + \\frac{4\\,(p-1)^2}{\\alpha^{\\,2}\\,(d-1)^{2}} \\right) \\quad \\quad c_1 = - \\frac{1}{\\beta\\,(d-1)}\n \\quad \\quad c_2 = \\frac{1}{2\\,\\beta\\,(d-1)\\,p!} ~.\n\\end{gather}\nHowever, two of the conditions on the $c_i$ become linearly dependent when $\\alpha$ takes the special values \n\\begin{equation}\\label{SpecialAlpha}\n  \\alpha = \\pm \\,2 \\, \\left(\\frac{p-1}{d-1}\\right) ~.\n\\end{equation}\nIn that case there is a one-parameter family of solutions \n\\begin{gather}\\label{1ParamFamily}\nc_{0} = -2\\,(d-1)\\,\\beta - c_1\\,(d-1)^{2}\\,\\beta^{\\,2} \\quad \\quad c_2 = - \\frac{1}{2\\,p!}\\,c_1 ~.\n\\end{gather}\nThis result is relevant for type II supergravity in ten dimensions, as it corresponds to the coupling between the dilaton and the 3-form field strengths in the NS-NS and R-R sectors.\n\n\nTo summarize, we have found that adding the surface terms\n\\begin{align}\\label{FinalCTs}\n I_\\ts{CT} = \\int_{\\Sigma} \\nts d^{\\,d}x \\,\\sqrt{h}\\,& \\left[ \\left(\\frac{4}{\\alpha^{2}}\\,\n   \\left(\\frac{p-1}{d-1} \\right)^2 - d \\right) \\beta\\,e^{-\\frac{\\alpha}{p-1}\\,\\phi}\n   - \\frac{1}{\\beta\\,(d-1)}\\,e^{\\frac{\\alpha}{p-1}\\,\\phi} \\left(\\RR\n   - \\, \\frac{1}{2\\,p!} \\, e^{2\\,\\alpha\\,\\phi}\n  \\, \\FF^{\\,2}\\right)\n  \\right]\n\\end{align}\nto the action \\eqref{bulkaction} leads to a well-defined variational problem for ALD\nboundary conditions. Notice that the $\\RR$ and $\\FF^{\\,2}$ terms appear in \\eqref{FinalCTs} with the same relative coefficients as the $R$ and $F^{2}$ term in the bulk part of the action, including the factor of $e^{2\\,\\alpha\\,\\phi}$. It is straightforward to show that the full action is finite on-shell. Using the same arguments as in the previous section, we can write the on-shell bulk term as a total derivative. The surface terms in the on-shell action are then given by\n\\begin{align}\\label{OnShellSurface}\n  I \\big|_\\ts{EOM} = \\int_{\\dM} \\bns d^{\\,d} x \\,\\sqrt{h} \\, & \\left[\n    - \\frac{8\\,(p-1)}{\\alpha\\,(d-1)^2}\\,n^{a}\\, \\partial_{a} \\phi + 2\\,K + \\left(\\frac{4}{\\alpha^{2}}\\,\n   \\left(\\frac{p-1}{d-1} \\right)^2 - d \\right)\\,\\beta\\,e^{-\\frac{\\alpha}{p-1}\\,\\phi}\n   \\right. \\\\ \\nonumber\n   & \\,\\,\\,\\, \\left. - \\frac{1}{\\beta\\,(d-1)}\\,e^{\\frac{\\alpha}{p-1}\\,\\phi}\\,\\RR + \\frac{1}{2\\,\\beta\\,p!\\,(d-1)}\n   \\, e^{\\frac{2p-1}{p-1}\\,\\alpha\\,\\phi} \\, \\FF^{\\,2} \\raisebox{20pt}{~}\\right]  ~.\n\\end{align}\nOf course, the on-shell action may also receive contributions from the total derivative evaluated at the lower limit of the bulk integral, but those terms are manifestly independent of the cut-off. Using the asymptotic expansions of the various fields in \\eqref{OnShellSurface} gives\n\\begin{align}\nI \\big|_\\ts{EOM} = \\int_{\\dM} \\bns d^{\\,d}x \\, \\sqrt{h^{_{(0)}}}\\, \\rho^{\\,d-1} & \\left[\n    - \\frac{8\\,(p-1)^2}{\\alpha^{2}\\,(d-1)^{2}} + 2\\,d + \\frac{1}{\\beta}\\,c_0\n    + \\beta\\,\\RR^\\ms{(0)}\\,c_1 \\right. \\\\ \\nonumber\n    & \\,\\,\\,\\, \\left. + \\beta\\,p!\\,\\frac{8\\,(p-1)}{\\alpha^{2}}\\,c_2 +\n    \\OO(\\rho^{1-d}) \\,\\right] ~.\n\\end{align}\nThe leading term vanishes when evaluated using the expressions for $\\RR^\\ms{(0)}$ and the\n$c_i$, so that the action is finite as $\\rho \\to \\infty$. Once again, finiteness of the on-shell action follows from the requirement that the action possess the proper variational properties.\n\n\n\n\\subsection{Higher Derivative Counterterms}\n\\label{sec:HDC}\n\n The counterterms used in the previous section contain no more than two derivatives of the fields. This seems appropriate for theories obtained as the low-energy limit of some quantum theory of gravity: in that case the boundary terms in the action should contain no more derivatives than the bulk terms. But relaxing this assumption reveals that there are many possible counterterms that lead to a well-defined variational principle for ALD boundary conditions. This is closely related to the ambiguity in the action when $\\alpha$ takes the special value \\eqref{SpecialAlpha}, so it is worth examining the issue more closely. \n\n\nTo illustrate how families of actions can be constructed, suppose that we remove the restriction on the number of derivatives that can appear in the counterterms and make the following ansatz for $I_\\ts{CT}$\n\\begin{equation}\\label{HDCT}\n  I_\\ts{CT} = \\int_{\\dM} \\bns d^{\\,d}x \\, \\sqrt{h}\\,\\left( b_0 \\, e^{- \\frac{\\alpha}{p-1} \\, \\phi} + b_1 \\, e^{\\frac{\\alpha}{p-1} \\, \\phi}\\,\\RR + b_2 \\, e^{\\frac{2p-1}{p-1} \\, \\alpha\\,\\phi}\\,\\FF^{a_1 \\ldots a_p} \\FF_{a_1 \\ldots a_p} + b_3 \\, e^{\\frac{3\\,\\alpha}{p-1}\\,\\phi}\n \\, \\RR^{2} \\right) ~.\n\\end{equation}\nThis differs from \\eqref{CT} by an $\\RR^{2}$ term, with a dilaton prefactor chosen to give the appropriate $\\rho$ dependence. The coefficients are labeled $b_i$ to emphasize that this set of counterterms is distinct from the previous result \\eqref{uniqueci}. The momenta $P_{\\phi}$ and $P_{ab}$ obtained from this action are similar to \\eqref{CTMomentum1} and \\eqref{CTMomentum2}, with the $c_i$ replaced by $b_i$ and the following additional terms\n\\begin{align}\n \\Delta P_{\\phi} & = b_3 \\, \\frac{3\\,\\alpha}{p-1} \\, e^{\\frac{3\\,\\alpha}{p-1}\\,\\phi}\n \\, \\RR^{2} \\\\\n \\Delta P^{ab}  & = b_3 \\, e^{\\frac{3\\,\\alpha}{p-1}\\,\\phi} \\, \\left( \\frac{1}{2} \\, h^{ab} \\, \\RR^{2} -\n    2 \\, \\RR \\, \\RR^{ab} \\right) + 2\\,b_{3}\\,D^{a}D^{b}\\left(e^{\\frac{3\\,\\alpha}{p-1}\\,\\phi}\\,\\RR\\right)\n    -2\\,b_{3}\\,h^{ab}\\,D^{\\,2} \\left( e^{\\frac{3\\,\\alpha}{p-1}\\,\\phi}\\,\\RR \\right)~.\n\\end{align}\nRequiring $\\delta I|_\\ts{EOM} = 0$ gives three equations for the four constants $b_i$. The one-parameter family of solutions can be expressed in terms of the previous solution \\eqref{uniqueci} as\n\\begin{equation}\\label{HDCTCoefficients}\nb_0 = c_0 + b_3 \\, \\beta^{\\,4}\\,\\RR_\\ms{(0)}^{\\,2} \\quad \\quad b_{1} = c_{1} - 2\\,b_3\\,\\beta^{\\,2}\\,\\RR_\\ms{(0)} \\quad \\quad b_{2} = c_{2} ~,\n\\end{equation}\nwhere $b_3$ an arbitrary real number. In the special case $\\alpha = \\pm 2(p-1)\/(d-1)$ this procedure gives a two-parameter family of counterterms. \n\n\n\nThe full action with the new set of counterterms exhibits two important properties. First, despite the presence of the new surface term, the on-shell action remains finite for all values of the coefficient $b_3$. The relationships \\eqref{HDCTCoefficients} ensure that a divergent contribution from the $\\RR^2$ counterterm is canceled by the other surface terms. In other words, a well-defined variational principle may not determine a unique action, but it is sufficient to render the on-shell action finite. Second, the value of the on-shell action remains the same for all values of $b_3$. The shifts in the coefficients \\eqref{HDCTCoefficients} and the new counterterm give additional finite contributions to the on-shell action, but these terms combine to form total derivatives or otherwise vanish by the equations of motion. Both of these results generalize as more counterterms are added to the action.  Thus, allowing counterterms with more than two derivatives leads to families of actions with a well-defined variational principle that depend on an arbitrary number of free parameters. However, all of these actions are finite and have the same on-shell value. \n\nWhen the parameter $\\alpha$ takes the special value \\eqref{SpecialAlpha}, restricting to two-derivative counterterms still results in a one-parameter family of actions. The results described above apply to this case as well: the action is finite, and its on-shell value is independent of the free parameter $c_1$ in \\eqref{1ParamFamily}. In that case it is still possible that other considerations might identify a unique action. For instance, requiring the full action to exhibit a particular symmetry or duality might constrain the sorts of surface terms that can be introduced. This is known to be the case in other examples of holographic renormalization, such as dilaton gravity in two dimensions. A construction similar to the one described above \\cite{Davis:2004xi, Grumiller:2007ju} yields a one-parameter family of counterterms, but only a particular value of the free parameter preserves Buscher duality \\cite{Buscher:1987qj} in the full action.\n\n\n\n\\section{Linear Response Functions and Conserved Charges}\n\\label{sec:LRF}\n\n\nIn previous sections we were concerned with the properties of the action under field variations that preserve the ALD boundary conditions. Now we focus our attention on general variations of the fields. The resulting change in the action is characterized by a set of linear response functions related to the coefficients of the field variations in \\eqref{1stVariation}.\n\n\n\n\\subsection{The Stress Tensor}\n\\label{sec:ST}\n\nThe linear response functions include a boundary stress tensor, obtained by varying the on-shell action with respect to the boundary metric\n\\begin{gather}\\label{BST}\nT^{ab} \\defeq \\left. \\frac{2}{\\sqrt{h}}\\,\\frac{\\delta \\, I}{\\delta \\, h_{ab}}\\right|_\\ts{EOM} = 2\\,\\left( \\pi^{ab} + P^{ab} \\right) ~.\n\\end{gather}\nThis differs from the definition used by Brown and York \\cite{Brown:1992br, Brown:2000dz}, who obtain a quasi-local stress tensor via a Hamilton-Jacobi variation of the action\\,\\footnote{A variation within the space of classical field configurations due to a small change in the boundary data}. Their approach is not possible in the present case, because the boundary data for the metric is restricted by the kinematic constraints \\eqref{Ricci2} and \\eqref{Ricci1} in the ALD boundary conditions. Instead we adopt the construction \\eqref{BST}, which was used in \\cite{Mann:2005yr, Mann:2006bd, Mann:2008ay} and gives a sensible definition of conserved quantities. \n\nThe asymptotic expansion of the boundary stress tensor is given by\n\\begin{gather}\\label{StressTensorExpansion}\nT_{ab} = \\rho^{\\,3-d}\\,T_{ab}^\\ms{(1)} + \\rho^{\\,2-d}\\,T_{ab}^\\ms{(2)} + \\OO(\\rho^{1-d}) ~.\n\\end{gather}\nThe various terms are evaluated using the asymptotic expansions of the tensors $\\pi^{ab}$ and $P^{ab}$, and the equations of motion. For generic values of $\\alpha$ we obtain\n\\begin{gather}\\label{genericSTpot}\nT_{ab}^\\ms{(1)} =  \\frac{\\,(d-2)^2}{(d-1)}\\,\\gamma_{ab}^\\ms{(1)} - \\frac{2\\,\\alpha}{(p-1)(d-1)}\\,\\delta_{ab}^\\ms{(1)}\\\\\nT_{ab}^\\ms{(2)} =  (d-1)\\,\\gamma_{ab}^\\ms{(2)} - \\frac{2\\,\\alpha}{(p-1)(d-1)} \\, \\delta_{ab}^\\ms{(2)} ~.\n\\end{gather}\nThe symmetric tensors $\\gamma_{ab}^\\ms{(n)}$ and $\\delta_{ab}^\\ms{(n)}$ are defined as\n\\begin{gather} \\label{gammadef}\n \\gamma_{ab}^\\ms{(n)} = h_{ab}^\\ms{(n)} - h_{ab}^\\ms{(0)}\\,h^\\ms{(n)} +\n   \\frac{8\\,(p-1)}{\\alpha\\,(d-1)\\,(d+n-3)}\\,h_{ab}^\\ms{(0)}\\,\\phi^\\ms{(n)} \\\\ \\label{deltadef}\n \\delta_{ab}^\\ms{(n)} = \\DD_{a} \\DD_{b} \\phi^\\ms{(n)} - h_{ab}^\\ms{(0)}\\,\\DD^{2} \\phi^\\ms{(n)} -\n   A\\,h_{ab}^\\ms{(0)}\\,\\phi^\\ms{(n)} ~,\n\\end{gather}\nwhere $\\DD$ is the boundary covariant derivative compatible with the metric $h_{ab}^\\ts{(0)}$, and $A$ is the numerical coefficient in the result \\eqref{Ricci2} for the Ricci tensor on $\\Sigma_\\ts{A}$. For the special case $\\alpha = \\pm 2 \\, (p-1)\/(d-1)$, the components of the stress tensor are instead given by\n\\begin{align}\\label{specialST1}\nT_{ab}^\\ms{(1)} = & \\,\\, (d-2)\\,\\gamma_{ab}^\\ms{(1)} + c_1\\,\\beta\\, (d-2)\\,\\gamma_{ab}^\\ms{(1)} - \\frac{4\\,c_1 \\, \\beta}{(d-1)}\\, \\delta_{ab}^\\ms{(1)} \\\\ \\label{specialST2}\nT_{ab}^\\ms{(2)} = & \\,\\,(d-1)\\,\\gamma_{ab}^\\ms{(2)} - \\frac{4\\,c_1 \\, \\beta}{(d-1)}\\,\\delta_{ab}^\\ms{(2)} ~.\n\\end{align}\nDespite the appearance of the free parameter $c_1$ in these expressions, we show in the next section that the conserved charges are independent of the choice of counterterms.\n\n\n\n\\subsection{Conserved Charges}\n\\label{sec:CC}\n\nConsider a $d-1$ dimensional, spacelike surface $\\CC \\subset \\Sigma$, with timelike unit normal vector $u^{a}$. We will assume that $u^{a}$ lies in $\\Sigma_\\ts{A}$, so that $u^{a}\\,v_{a} = 0$ for any $v_{a}$ in $\\Sigma_\\ts{B}$. If $\\xi^{a}$ is an asymptotic Killing vector that preserves the ALD boundary conditions, then the conserved charge associated with $\\xi^{a}$ is given by the flux of the current $T_{ab}\\,\\xi^{b}$ across $\\CC$\n\\begin{equation}\\label{Charges}\n  Q[\\xi] = \\int_{\\,\\CC} d^{\\,d-1}x \\,\\sqrt{h_\\ms{\\CC}}\\,u^{a}\\,T_{ab}\\,\\xi^{b} ~.\n\\end{equation}\nConserved charges constructed in this manner necessarily generate the asymptotic symmetries of the theory \\cite{HIM2}. It is straightforward to show that the $\\xi^{a}$ that preserve the ALD boundary conditions are given by the Killing vectors of $h_{ab}^\\ms{(0)}$, up to sub-leading terms that generate gauge transformations. The asymptotic expansion of these vectors is\\,\\footnote{In AF gravity, asymptotic boosts are associated with terms of order $\\rho^{-1}$ in $\\xi^{a}$. These terms are not present for ALD boundary conditions because they do not preserve the asymptotic form of the dilaton.}\n\\begin{gather}\n\\xi^{a}(\\rho,x) = \\xi^{a}_\\ms{(0)}(x) + \\OO(\\rho^{2-d}) ~.\n\\end{gather}\nGiven this result, and the expansion \\eqref{StressTensorExpansion} for the boundary stress tensor, only the leading terms in the asymptotic expansions of $h_\\ms{\\CC}$ and $u^{a}$ contribute to the conserved charges\n\\begin{equation}\\label{QExpression}\n  Q[\\xi] = \\int_{\\CC} \\nts d^{\\,d-1}x \\,\\sqrt{h_\\ms{\\CC}^\\ms{(0)}}\\,\\left( \\rho\\,u^{a}_\\ms{(0)}\\,T_{ab}^{(1)}\\,\\xi^{b}_\\ms{(0)}  + u^{a}_\\ms{(0)}\\,T_{ab}^{(2)}\\,\\xi^{b}_\\ms{(0)} + \\OO(\\rho^{-1}) \\right) ~,\n\\end{equation}\nwhere $h_{ab}^\\ms{(0)}\\,u^{a}_\\ms{(0)}\\,u^{b}_\\ms{(0)} = -1$. Obviously, this is only well-defined if the first term in the integrand has a vanishing integral over $\\CC$. Then the remaining term, which is finite as $\\rho \\to \\infty$, gives the conserved charge associated with $\\xi^{a}$. We will now show that this is indeed the case.\n\n\n\\vskip1em\n\\noindent{\\bf Scalar and Tensor Potentials}\n\\vskip.5em\n\n\\noindent In order to prove that the conserved charges are well-defined and independent of any free parameters in the action, it is useful to first establish several results concerning potentials for symmetric tensors on $\\Sigma$. Terms in the boundary stress tensor that admit such potentials do not contribute to the conserved charges. See \\cite{Mann:2008ay} for a recent discussion in the context of AF gravity.\n\nA tensor $\\theta_{ab}$ is said to admit a scalar potential $\\psi$ if it can be expressed in the form\n\\begin{gather}\\label{ScalarPotential}\n  \\theta_{ab}[\\psi] = \\DD_{a} \\DD_{b} \\psi - h_{ab}^\\ms{(0)}\\,\\DD^{2} \\psi - A\\,h_{ab}^\\ms{(0)}\\,\\psi ~ ,\n\\end{gather}\nTaking the covariant divergence we find that $\\theta_{ab}$ is not, strictly speaking, conserved. However, its divergence is restricted to lie in the $p$-dimensional subspace $\\Sigma_\\ts{B} \\subset \\Sigma$\n\\begin{gather}\n\\DD^{b} \\theta_{ab} = ( B - A ) \\,\\delta_{a}{}^{\\alpha}\\,\\DD_{\\alpha} \\psi ~.\n\\end{gather}\nAs a result the tensor $\\theta_{ab}$ is `effectively conserved' in the following sense: the current $\\theta_{ab} \\, \\xi_\\ms{(0)}^{b}$, where $\\xi_\\ms{(0)}^{b}$ is a Killing vector of the metric $h_{ab}^\\ms{(0)}$, does not contribute to conserved charges. Such currents take the form\n\\begin{gather}\\label{thetacurrent}\n\\theta_{ab} \\, \\xi_\\ms{(0)}^{b} = \\DD^{b} \\left(2\\,\\xi^\\ms{(0)}_{[b} \\DD^{\\vphantom{\\ms{(0)}}}_{a]}\\psi + \\psi \\, \\DD^{\\vphantom{\\ms{(0)}}}_{[b} \\,\\xi_{a]}^\\ms{(0)} \\right) + (B-A)\\,\\delta_{a}{}^{\\alpha}\\,\\xi_{\\alpha}^\\ms{(0)}\\,\\psi ~.\n\\end{gather}\nThis expression is obtained using properties of the Killing vector, which imply $\\DD^{2}\\xi_\\ms{(0)}^{b} = -\\RR^{b c}_\\ms{(0)}\\,\\xi_{c}^\\ms{(0)}$. Now consider the flux of the current across $\\CC$, which is given by the integral of $u_\\ms{(0)}^{a} \\theta_{ab} \\xi_\\ms{(0)}^{b}$. The last term in \\eqref{thetacurrent} does not contribute because $u_\\ms{(0)}^{a}\\,\\delta_{a}{}^{\\alpha} = 0$. The remaining term takes the form\n\\begin{gather}\nu_\\ms{(0)}^{a}\\,\\theta_{ab}\\,\\xi_\\ms{(0)}^{b} = \\DD^{b}_\\ms{\\CC} \\left(2\\,u_\\ms{(0)}^{a}\\,\\xi^\\ms{(0)}_{[b} \\DD^{\\vphantom{\\ms{(0)}}}_{a]}\\psi + u_\\ms{(0)}^{a}\\,\\psi \\, \\DD^{\\vphantom{\\ms{(0)}}}_{[b} \\, \\xi^\\ms{(0)}_{a]} \\right)\n\\end{gather}\nwhere $\\DD_\\ms{\\CC}^{a}$ is the covariant derivative compatible with the induced metric on $\\CC$. The integral vanishes, so a current of the form \\eqref{thetacurrent} has vanishing flux across $\\CC$. The tensors $\\delta_{ab}^\\ms{(n)}$ appearing in the boundary stress tensor have the form \\eqref{ScalarPotential}, and therefore do not contribute to the conserved charges. \n\nSimilarly, a tensor $\\chi_{ab}$ admits a symmetric, transverse tensor potential if\n\\begin{equation}\\label{TensorPotential}\n\\chi_{ab} = \\DD^{2} \\gamma_{ab} + 2\\,\\RR^\\ms{(0)}_{acbd}\\,\\gamma^{cd} +\n  (B - A)\\,\\left(\\delta_{a}{}^{\\alpha}\\,\\gamma_{\\alpha b}+\\delta_{b}{}^{\\alpha}\\,\\gamma_{\\alpha a}\\right)\n\\end{equation}\nfor some symmetric $\\gamma_{ab}$ with $\\DD^{a}\\gamma_{ab} = 0$. As in the case of the scalar potential, the divergence of the tensor $\\chi_{ab}$ does not vanish. However, after commuting covariant derivatives and applying the contracted Bianchi identities, the divergence can be written in the form\n\\begin{gather}\n \\DD^{b} \\chi_{ab} = 2\\,(B-A)\\,\\DD^{\\alpha} \\gamma_{\\alpha a} ~.\n\\end{gather}\nThe current formed from $\\chi_{ab}$ and the Killing vector $\\xi_\\ms{(0)}^{b}$ takes a form similar to \\eqref{thetacurrent}\n\\begin{gather}\n\\chi_{ab} \\, \\xi_\\ms{(0)}^{b} = 2\\,\\DD^{c}\\left( \\xi_\\ms{(0)}^{b}\\,\\DD_{[c}\\,\\gamma_{a]b} + \\gamma_{b[c} \\DD_{a]}\\,\\xi_\\ms{(0)}^{b} \\right) + 2\\,(B-A)\\,\\delta_{a}{}^{\\alpha}\\,\\xi_\\ms{(0)}^{b}\\,\\gamma_{\\alpha b} ~.\n\\end{gather}\nThis current also has zero flux across $\\CC$, and so it does not contribute to a conserved charge. Notice that an arbitrary tensor along $\\Sigma_\\ts{B}$ can be added to the right-hand side of \\eqref{TensorPotential} (or \\eqref{ScalarPotential}, for that matter) without changing this conclusion.\n\n\nThe equations of motion in appendix \\ref{sec:EOMandBianchi} can be used to show that certain terms in the boundary stress tensor admit scalar and tensor potentials. Consider the tensor $\\gamma_{ab}^\\ms{(1)}$, defined in \\eqref{gammadef}. The divergence of $\\gamma_{ab}^\\ms{(1)}$ vanishes by \\eqref{AsympMom1}. Rewriting the equations of motion in terms of $\\gamma_{ab}^\\ms{(1)}$ gives\n\\begin{align}\\nonumber\n(d-2)\\,\\gamma_{ab}^\\ms{(1)} + \\frac{8\\,(p-1)}{\\alpha^{2}}\\,\\left(\\delta_{a}{}^{\\alpha} \\, \\gamma_{\\alpha b}^\\ms{(1)} + \\delta_{b}{}^{\\alpha} \\, \\gamma_{a \\alpha}^\\ms{(1)}\\right)\n = & \\,\\,\\,\\, \\chi_{ab}[\\gamma_{ab}^\\ms{(1)}] - \\theta_{ab}[h^\\ms{(1)}] + \\frac{16\\,(p-1)}{\\alpha\\,(d-1)\\,(d-2)}\\,\\theta_{ab}[\\phi^\\ms{(1)}] \\\\  \\label{gamma1}\n    & \\,\\,\\,\\, + \\frac{8\\,(p-1)}{\\alpha^{2}}\\,\\left(\\delta_{a}{}^{\\alpha} \\, W_{\\alpha b}^\\ms{(1)}\n + \\delta_{b}{}^{\\alpha} \\, W_{\\alpha a}^\\ms{(1)}\\right) \\\\ \\nonumber\n  & \\,\\,\\,\\, + \\frac{8\\,(p-1)}{\\alpha^{2}}\\,\\delta_{a}{}^{\\alpha}\\,\\delta_{b}{}^{\\beta}\\,\n  \\left(h_{\\alpha\\beta}^\\ms{(1)} - h_{\\alpha\\beta}^\\ms{(0)}\\,h_{s}^\\ms{(1)} + 2\\,\\alpha\\,h_{\\alpha\\beta}^\\ms{(0)}\\,\\phi^\\ms{(1)} \\right) ~.\n\\end{align}\nMost of the terms on the right-hand side of this expression do not contribute to conserved charges, either because they admit potentials or because they have both indices along $\\Sigma_\\ts{B}$. The only exception occurs for asymptotic Killing fields $\\xi^{\\alpha}$ with a component along $\\Sigma_\\ts{B}$. In that case a term proportional to $u_\\ms{(0)}^{a}\\,W_{\\alpha a}^\\ms{(1)}\\,\\xi_\\ms{(0)}^{\\alpha}$ appears in the expression for the flux of the current $\\gamma_{a\\alpha}^\\ms{(1)}\\,\\xi_\\ms{(0)}^{\\alpha}$ across ${\\cal C}$\\,\\footnote{This term indicates the presence of an electric field along $\\Sigma_\\ts{B}$ at the first sub-leading order in the asymptotic expansion.}. We require that such a term is not present: $u_\\ms{(0)}^{a}\\,W_{\\alpha a}^\\ms{(1)} = 0$ for any timelike $u^{a}$. This condition ensures that terms proportional to $\\gamma_{ab}^\\ms{(1)}$ do not contribute to the conserved charges. Furthermore, this result and the analogous result for $\\delta_{ab}^\\ms{(1)}$ ensure that the integral of the first term in \\eqref{QExpression} vanishes, which proves that the conserved charges are well-defined. \n\nNow consider the tensor $\\gamma_{ab}^\\ms{(2)}$\n\\begin{gather}\n \\gamma_{ab}^\\ms{(2)} = h_{ab}^\\ms{(2)} - h_{ab}^\\ms{(0)}\\,h^\\ms{(2)} + \\frac{8\\,(p-1)}{\\alpha\\,(d-1)^{2}}\\,h_{ab}^\\ms{(0)}\\,\\phi^\\ms{(2)} ~.\n\\end{gather}\nIn this case the equations of motion only constrain the components of $\\gamma_{ab}^\\ms{(2)}$ with at least one index along $\\Sigma_\\ts{B}$\n\\begin{align}\\label{gamma2}\n\\frac{8\\,(p-1)}{\\alpha^{2}}\\,\\left(\\delta_{a}{}^{\\alpha} \\, \\gamma_{\\alpha b}^\\ms{(2)}\n + \\delta_{b}{}^{\\alpha} \\, \\gamma_{\\alpha a}^\\ms{(2)}\\right) =\n & \\,\\,\\,\\, \\chi_{ab}[\\gamma_{ab}^\\ms{(2)}] - \\theta_{ab}[h^\\ms{(2)}] + \\frac{16\\,(p-1)}{\\alpha\\,(d-1)^{2}}\\,\\theta_{ab}[\\phi^\\ms{(2)}] \\\\ \\nonumber\n    & \\,\\,\\,\\, + \\frac{8\\,(p-1)}{\\alpha^{2}}\\,\\left(\\delta_{a}{}^{\\alpha} \\, W_{\\alpha b}^\\ms{(2)}\n + \\delta_{b}{}^{\\alpha} \\, W_{\\alpha a}^\\ms{(2)}\\right) \\\\ \\nonumber\n  & \\,\\,\\,\\, + \\frac{8\\,(p-1)}{\\alpha^{2}}\\,\\delta_{a}{}^{\\alpha}\\,\\delta_{b}{}^{\\beta}\\,\n  \\left(h_{\\alpha\\beta}^\\ms{(2)} - h_{\\alpha\\beta}^\\ms{(0)}\\,h_{s}^\\ms{(2)} + 2\\,\\alpha\\,h_{\\alpha\\beta}^\\ms{(0)}\\,\\phi^\\ms{(2)} \\right) ~.\n\\end{align}\nThus, the term in $T_{ab}^\\ms{(2)}$ proportional to $\\gamma_{ab}^\\ms{(2)}$ can contribute to the conserved charges. In fact, we have shown that this is the \\emph{only} term that contributes to the conserved charges, which can now be expressed as\n\\begin{equation}\\label{FinalQExpression}\n  Q[\\xi] = (d-1)\\,\\int_{\\CC} \\nts d^{\\,d-1}x \\,\\sqrt{h_\\ms{\\CC}^\\ms{(0)}}\\,u^{a}_\\ms{(0)}\\,\\gamma_{ab}^{(2)}\\,\\xi^{b}_\\ms{(0)} ~.\n\\end{equation}\n\n\n\n\\vskip1em\n\\noindent{\\bf Conserved Charges are Independent of Free Parameters in the Action}\n\\vskip.5em\n\n\\noindent In the special case $\\alpha = \\pm 2\\,(p-1)\/(d-1)$ the boundary stress tensor contains terms proportional to the free parameter $c_1$. However, all of these terms are of the sort described in the previous section -- they admit scalar or tensor potentials and therefore do not contribute to the conserved charges. So, for theories that admit a one-parameter family of two-derivative counterterms, the conserved charges are independent of the choice of action. This is also true for actions that include higher-derivative counterterms, as we will now show. \n\nConsider the counterterm action discussed in section \\ref{sec:HDC}, which includes a four derivative term proportional to $\\RR^{2}$. This leads to a one-parameter family of actions for generic values of $\\alpha$, and a corresponding family of boundary stress tensors. We will now show that these stress tensors differ only by terms that admit potentials, so that the conserved charges are independent of the choice of action. The difference between $P_{ab}$ obtained from \\eqref{HDCT} and $P_{ab}$ obtained from \\eqref{CT} is\n\\begin{align}\n\\Delta P_{ab} = & \\,\\, \\frac{1}{2}\\,b_3\\,\\beta^{4}\\,\\RR_{(0)}^{\\,2}\\,h_{ab}\\,e^{-\\frac{\\alpha}{p-1}\\,\\phi} + 2\\,b_3\\,\\beta^{2}\\,\\RR^{(0)}\\,e^{\\frac{\\alpha}{p-1}\\,\\phi}\\,\\left(\\RR_{ab} - \\frac{1}{2}\\,h_{ab}\\,\\RR \\right) \\\\ \\nonumber\n& \\,\\, -2\\,b_3\\,\\beta^{2}\\,\\RR^{(0)}\\,\\left(D_{a}D_{b}\\left(e^{\\frac{\\alpha}{p-1}\\,\\phi}\\right)  - h_{ab}\\,D^{2}\\left(e^{\\frac{\\alpha}{p-1}\\,\\phi}\\right)\\right) \\\\ \\nonumber\n& \\,\\, + b_3 \\, e^{\\frac{3\\,\\alpha}{p-1}\\,\\phi} \\left(\\frac{1}{2}\\,h_{ab}\\,\\RR^{2} - 2\\,\\RR\\,\\RR_{ab} \\right) + 2\\,b_3\\,D_{a}D_{b}\\left(e^{\\frac{3\\,\\alpha}{p-1}\\,\\phi}\\,\\RR \\right) \\\\ \\nonumber\n& \\,\\, - 2\\,b_3\\,h_{ab}\\,D^{2}\\left(e^{\\frac{3\\,\\alpha}{p-1}\\,\\phi}\\,\\RR \\right) ~,\n\\end{align}\nwith $b_3$ an arbitrary real number. An extended calculation involving the asymptotic expansions of all the terms gives\n\\begin{align}\n\\Delta P_{ab} = & \\,\\, b_3 \\, \\beta^{\\,3}\\,\\rho^{3-d}\\,\\left(\\frac{4\\,\\alpha}{p-1}\\,\\RR^{(0)}\\,\\theta_{ab}[\\phi^{(1)}] +\\theta_{ab}[\\RR^{(1)}-\\RR_{cd}^{(0)}\\,h^{cd}_{(1)}]\\right) \\\\ \\nonumber\n& \\, \\, + b_3 \\, \\beta^{\\,3}\\,\\rho^{2-d}\\,\\left(\\frac{4\\,\\alpha}{p-1}\\,\\RR^{(0)}\\,\\theta_{ab}[\\phi^{(2)}] +\\theta_{ab}[\\RR^{(2)}-\\RR_{cd}^{(0)}\\,h^{cd}_{(2)}]\\right) + \\OO(\\rho^{1-d}) ~.\n\\end{align}\nThus, when the higher derivative counterterm proportional to $\\RR^{2}$ is included, the only affect on the boundary stress tensor is to generate terms admitting scalar potentials. Such terms do not change the conserved charges of the theory. Similar results were also obtained for higher derivative counterterms proportional to $\\RR^{ab}\\,\\RR_{ab}$, $\\RR\\,\\FF^{2}$, $\\RR^{ab}\\,\\FF_{a}{}^{c}\\,\\FF_{bc}$, and $\\RR^{k}$ for $k> 2$. We conclude that this result holds in general: the conserved charges are always independent of free parameters that appear in the action. This is true for any value of $\\alpha$, and for actions containing boundary terms with any number of derivatives.\n\n\n\n\\section{Examples}\n\\label{sec:Examples}\n\nA nice application of the techniques developed in the preceding sections is the holographic renormalization of asymptotically linear dilaton spacetimes that arise in type II supergravity theories. The field content in the NS-NS sector is a metric $g_{ab}$, a dilaton $\\phi$, and a two-form potential ${\\bf B}$ with field strength ${\\bf H} = \\textrm{d} {\\bf B}$. In addition, there is another two-form potential ${\\bf C}$ in the R-R sector, with field strength ${\\bf F} = \\textrm{d} {\\bf C}$. The bulk part of the two-derivative action for these fields is\n\\begin{equation}\n L = \\frac{1}{2\\,\\kappa^{2}}\\,\\sqrt{g}\\,\\left[R - \\frac{1}{2}\\,\\left(\\nabla \\phi \\right)^{2} - \\frac{1}{12}\\,e^{-\\phi}\\,H^{abc}\\,H_{abc} -\\frac{1}{12}\\,g_{s}^{\\,2}\\,e^{\\,\\phi}\\,F^{abc}\\,F_{abc}\\right]\n\\end{equation}\nwith $2\\,\\kappa^{2}$ = $2\\,\\kappa_{0}^{\\,2}\\,g_{s}^{\\,2}$ and $2\\,\\kappa_{0}^{\\,2} = (2\\pi)^{7}\\,(\\alpha')^{4}$. This is the same form as \\eqref{bulkaction}, with $\\alpha = -1\/2$ for the NS-NS coupling and $\\alpha = 1\/2$ for the R-R coupling. The R-R field strength appears in the Einstein frame action with an additional factor of $g_{s}^{\\,2}$, since it does not couple to the dilaton in string frame.\n\n\\subsection{Thermally Excited NS5-Branes}\n\\label{sec:NS5}\n\n\nA counterterm action based on the $\\hK$ construction was used in \\cite{Marolf:2007ys} to holographically renormalize the NS-NS sector of the action, for spacetimes asymptotic to a stack of non-extremal NS 5-branes. As a first application of our results we repeat that analysis using the explicit boundary counterterms of section \\ref{sec:OC}. Notice that $\\alpha=-1\/2$ corresponds to the special case $\\alpha = - 2 (p-1)\/(d-1)$, since $p=3$ and $d=9$. This means that there is a one parameter family of holographically renormalized, two-derivative actions to consider. In the last section we showed that all such actions yield the same conserved charges. For the present example, it turns out that the stress tensor itself is also independent of the choice of action. Thus, the results given below hold for any value of the free parameter $c_1$.  \n\n\nConsider a stack of $N$ coincident non-extremal NS 5-branes. The field strength is\n\\begin{equation}\\label{NS5flux}\n  H_{\\alpha\\beta\\gamma} = 2\\,N\\,\\alpha'\\,\\eps^\\ms{\\,\\Omega}_{\\alpha\\beta\\gamma}~,\n\\end{equation}\nwith $\\eps^\\ms{\\Omega}_{\\alpha\\beta\\gamma}$ the volume form on a unit three sphere in the directions transverse to the NS 5-branes. The near-horizon solution describing thermal fluctuations of the fields is \\cite{Narayan:2001dr}\n\\begin{gather}\\label{NearHorizonNS5}\nds^{2} = \\left(\\frac{N\\,\\alpha'}{r^{2}}\\right)^{-\\tfrac{1}{4}}\\left( -\\left( 1 - \\frac{r_{h}^{\\,2}}{r^{2}}\\right)\\,dt^{2} + d \\vec{x}_{5}^{\\,2} \\right) + \\left(\\frac{N\\,\\alpha'}{r^{2}}\\right)^{\\tfrac{3}{4}} \\left( \\left(1-\\frac{r_{h}^{\\,2}}{r^{2}}\\right)^{-1} \\nts dr^{2} + r^{2}\\,d\\Omega_{3}^{\\,2} \\right) \\\\\n\\phi  = \\frac{1}{2}\\,\\log\\left(\\frac{N\\,\\alpha'}{r^{2}}\\right)\n\\end{gather}\nwhere $r_{h}$ is the non-extremality parameter. To express this solution in the form \\eqref{metricBC} and \\eqref{dilatonBC}, we make two changes of coordinates. First, define $r = r_h \\, \\cosh \\sigma$ and $\\ell^{\\,2} = N\\,\\alpha'$, so that the solution becomes\n\\begin{gather}\\label{coord1}\nds^{2}  = \\left(\\frac{r_h}{\\ell}\\,\\cosh \\sigma \\right)^{1\/2} \\bigg[\\, \\ell^{\\,2}\\,\\left( d\\sigma^{2} + d\\Omega_{3}^{\\,2} \\right)  - \\tanh^{2} \\sigma \\, d t^{2} + d \\vec{x}_{5}^{\\,2} \\bigg] \\\\\n\\phi = - \\log \\left( \\frac{r_h}{\\ell}\\,\\cosh \\sigma  \\right) ~.\n\\end{gather}\nNow convert to asymptotic coordinates via\n\\begin{equation}\\label{coord2}\n \\sigma = \\log \\left( \\frac{\\rho^{\\,4}}{2^{7}\\,r_h\\,\\ell^{\\,3}} \\right) + \\frac{4^{7}\\,r_{h}^{\\,2}\\,\\ell^{\\,6}}{7\\,\\rho^{\\,8}} + \\OO(\\rho^{\\,-9})~,\n\\end{equation}\nand rescale the worldvolume coordinates as\n\\begin{equation}\nt = 4\\,\\ell\\,\\tau \\quad\\quad\\quad \\vec{x}_{5} = 4\\,\\ell\\,\\vec{y}_{5} ~.\n\\end{equation}\nIn these coordinates the solution becomes\n\\begin{gather}\\label{metricNS5branes}\nds^{2} = d \\rho^{\\,2} + \\rho^{\\,2}\\,\\bigg[ -\\left(1 - \\frac{6}{7}\\,\\frac{\\mu}{\\rho^{8}} + \\ldots\\right)\\,d\\tau^{2} + \\left(1+\\frac{1}{7}\\,\\frac{\\mu}{\\rho^{8}} + \\ldots \\right)\\,\\left(d\\vec{y}_{5}{}^{2} + \\frac{1}{16}\\,d\\Omega_{3}{}^{2}\\right)\\bigg] \\\\ \\label{dilatonNS5branes}\n\\phi = - 4\\,\\log \\left(\\frac{\\rho}{4\\,\\ell} \\right) - \\frac{2}{7}\\,\\frac{\\mu}{\\rho^{8}} + \\ldots \\\\ \\label{FSNS5branes}\nH_{\\alpha\\beta\\gamma} = 2\\,\\ell^{\\,2}\\,\\eps^\\ms{\\,\\Omega}_{\\alpha\\beta\\gamma}\n\\end{gather}\nwhere $\\mu$ is related to the non-extremality parameter $r_h$ and the length scale $\\ell$ by\n\\begin{gather}\\label{MassParam}\n\\mu = 4^{8}\\,r_{h}^{\\,2}\\,\\ell^{\\,6}\n\\end{gather}\nAt this point, the components of the stress tensor can be directly obtained from the asymptotic expansions of the fields.\n\n\nThe first thing to notice is that $h_{ab}^\\ts{(1)} = 0$ and $\\phi^\\ts{(1)}=0$, and there are no sub-leading terms in the field strength. The boundary stress tensor can be read off from the solution \\eqref{metricNS5branes}. The only non-vanishing component is\n\\begin{equation}\nT_{\\tau\\tau}^\\ts{(2)} = \\frac{4\\,\\mu}{\\kappa^{\\,2}} ~.\n\\end{equation}\nThe conserved charge associated with the asymptotic Killing vector $\\partial_{\\tau}$ is then given by\n\\begin{align}\nQ[\\partial_{\\tau}] = \\int_{\\CC}\\nts d^{\\,8}x\\,\\sqrt{h_{\\CC}^{_{(0)}}}\\,u_\\ms{(0)}^{\\tau}\\,T_{\\tau\\tau}^{(2)}\\,u_\\ms{(0)}^{\\tau} ~.\n\\end{align}\nTaking $\\CC$ to be a surface of constant $\\tau$, with $\\omega_{\\alpha\\beta}$\nthe metric on the unit 3-sphere, we have from \\eqref{metricNS5branes}\n\\begin{equation}\n\\sqrt{h_{\\CC}^{_{(0)}}} = \\frac{1}{4^{3}}\\,\\sqrt{\\omega} \\quad \\quad u^{t} = 1 ~.\n\\end{equation}\nThis gives\n\\begin{equation}\\label{PreMassNS}\nQ[\\partial_{\\tau}] = \\frac{\\mu}{4^{2}\\kappa^{2}}\\,{\\textrm {A}}(\\Omega_{3})\\,{\\textrm {Vol}_{y}}(\\mathbb{R}^{5})~,\n\\end{equation}\nwhere $\\textrm{A}(\\Omega_{3})$ is the area of the unit 3-sphere, and the volume factor represents the integral over the rescaled coordinates $\\vec{y}_5$ on the worldvolume of the branes. Using the definition \\eqref{MassParam} and transforming back to the original worldvolume coordinates $t = 4\\,\\ell\\,\\tau$ and $\\vec{x}_5 = 4\\,\\ell\\,\\vec{y}_5$, we then have\n\\begin{gather}\\label{Mass1}\nQ[\\partial_{t}] = \\frac{r_{h}^{\\,2}}{\\kappa^{2}}\\,{\\textrm {A}}(\\Omega_{3})\\,{\\textrm {Vol}_{x}}(\\mathbb{R}^{5})~,\n\\end{gather}\nwhich we interpret as the mass (internal energy) $M$ of the spacetime. \n\nAs a test of this result we can check it against the first law. Continuing the solution \\eqref{NearHorizonNS5} to Euclidean time, regularity at the horizon implies that the Hawking temperature is\n\\begin{equation}\nT_{H} = \\frac{1}{2\\pi\\sqrt{N\\,\\alpha'}} ~.\n\\end{equation}\nThis result is independent of the non-extremality parameter, which leads to a vanishing free energy. The entropy of the system is one-quarter of the area of the horizon in Planck units\n\\begin{equation}\n  S = \\frac{A_{h}}{4\\,G} = \\frac{1}{\\kappa^{2}}\\,2\\pi\\sqrt{N\\,\\alpha'}\\,r_{h}^{\\,2}\\,{\\textrm {A}}(\\Omega_{3})\\,{\\textrm {Vol}_{x}}(\\mathbb{R}^{5})\n\\end{equation}\nComparing with the result \\eqref{Mass1}, we have\n\\begin{equation}\n dM = T_{H}\\,dS ~,\n\\end{equation}\nin agreement with the first law of thermodynamics.\n\n\n\\subsection{D5-Branes}\n\\label{sec:D5}\n\nThe situation is similar for a stack of near-extremal D5-branes, except that the solution contains additional factors of the string coupling $g_s$. The R-R field strength is\n\\begin{equation}\\label{D5flux}\n  F_{\\alpha\\beta\\gamma} = 2\\,N\\,\\alpha'\\,\\eps_{\\alpha\\beta\\gamma} ~,\n\\end{equation}\nand the near-horizon solution is\n\\begin{gather}\\label{NearHorizonD5}\nds^{2} = \\left(\\frac{g_s N \\alpha'}{r^{2}}\\right)^{-\\frac{1}{4}}\\left( -\\left( 1 - \\frac{r_{h}^{\\,2}}{r^{2}}\\right)\\,dt^{2} + d \\vec{x}_{5}^{\\,2} \\right) + \\left(\\frac{g_s N \\alpha'}{r^{2}}\\right)^{\\tfrac{3}{4}} \\left( \\left(1-\\frac{r_{h}^{\\,2}}{r^{2}}\\right)^{-1} \\nts dr^{2} + r^{2}\\,d\\Omega_{3}^{\\,2} \\right) \\\\\n\\phi  = \\frac{1}{2}\\,\\log\\left(\\frac{g_s N \\alpha'}{r^{2}}\\right) ~.\n\\end{gather}\nUsing the same coordinate transformations as in the previous section, the solution for a stack of D5-branes can be put in the form\n\\begin{gather}\\label{metricD5branes}\nds^{2} = d \\rho^{\\,2} + \\rho^{\\,2}\\,\\bigg[ -\\left(1 - \\frac{6}{7}\\,\\frac{\\bar{\\mu}}{\\rho^{8}} + \\ldots\\right)\\,d\\tau^{2} + \\left(1+\\frac{1}{7}\\,\\frac{\\bar{\\mu}}{\\rho^{8}} + \\ldots \\right)\\,\\left(d\\vec{y}_{5}{}^{2} + \\frac{1}{16}\\,d\\Omega_{3}{}^{2}\\right)\\bigg] \\\\ \\label{dilatonD5branes}\n\\phi = - 4\\,\\log \\left(\\frac{\\rho}{4\\,\\bar{\\ell}} \\right) - \\frac{2}{7}\\,\\frac{\\bar{\\mu}}{\\rho^{8}} + \\ldots \\\\ \\label{FSD5branes}\nF_{\\alpha\\beta\\gamma} = 2\\,\\frac{\\,\\bar{\\ell}^{\\,2}}{g_{s}}\\,\\eps^\\ms{\\,\\Omega}_{\\alpha\\beta\\gamma} ~,\n\\end{gather}\nwhere $\\bar{\\ell}$ and $\\bar{\\mu}$ are now defined as\n\\begin{gather}\n\\bar{\\ell} = \\sqrt{g_s \\, N\\, \\alpha'} \\quad \\quad \\quad \\bar{\\mu} = 4^{8}\\,r_{h}^{\\,2}\\,\\bar{\\ell}^{\\,6} ~.\n\\end{gather}\nThe $\\tau$-$\\tau$ component of the boundary stress tensor is now\n\\begin{equation}\nT_{\\tau\\tau}^{(2)} = \\frac{4\\,\\bar{\\mu}}{\\kappa^{\\,2}} ~,\n\\end{equation}\nso the conserved charge associated with the Killing vector $\\partial_{\\tau}$ is\n\\begin{equation}\\label{PreMassD}\nQ[\\partial_{\\tau}] = \\frac{\\bar{\\mu}}{4^{2}\\kappa^{2}}\\,{\\textrm {A}}(\\Omega_{3})\\,{\\textrm {Vol}_{y}}(\\mathbb{R}^{5})~.\n\\end{equation}\nThis differs from the result \\eqref{PreMassNS} for the NS 5-branes by a factor of $g_{s}^{\\,3}$. However, in terms of the coordinates $t = 4\\,\\bar{\\ell}\\,\\tau$ and $\\vec{x}_{5} = 4\\,\\bar{\\ell}\\,\\vec{x}_{5}$ appearing in \\eqref{NearHorizonD5}, we obtain the same result as before\\,\\footnote{The results \\eqref{Mass1} and \\eqref{Mass2} are the same, but the coordinates used in these two examples are not.}\n\\begin{equation}\\label{Mass2}\nM = \\frac{r_{h}^{\\,2}}{\\kappa^{\\,2}}\\,{\\textrm{A}(\\Omega_3)}\\,{\\textrm{Vol}_{x}}(\\mathbb{R}^{5})~.\n\\end{equation}\nIt is straightforward to show, using the same arguments as the previous section, that the first law of thermodynamics is upheld. Both the Hawking temperature and the entropy depend on $g_s$ in this case, but that this dependence cancels out when they are multiplied.\n\n\n\\section{Discussion}\n\\label{sec:Discussion}\n\n\nThe main lesson of this paper is that a well-defined variational problem is the key physical principle guiding the introduction of surface terms in the action. In the case of asymptotically flat or asymptotically AdS spacetimes the potential boundary counterterms are quite restrictive. But for ALD gravity the boundary terms are not necessarily unique -- it is possible to construct families of actions, all of which have a well-defined variational principle. However, these actions all have the same value on-shell, and they all yield the same conserved charges. \n\n\nFor ALD gravity one can introduce a set of counterterms based on the $\\hK$ approach. This makes contact with similar constructions in asymptotically flat gravity. Specifically, the $\\alpha \\to \\infty$ limit in \\eqref{DefiningEquation} recovers the definition of $\\hK_{ab}$ used previously \\cite{Mann:2005yr, Mann:2006bd}. However the definition of $\\hK$ for ALD gravity is inherently ambiguous because of relations between the boundary data that are not present in AF gravity. Regardless of which definition is used, the $\\hK$ construction does not lead to an action with a well-defined variational principle. Additional surface terms must be included in the action, as was first pointed out in \\cite{Marolf:2007ys}. This suggests an alternate approach in which $\\hK$ -- whose implicit definition makes certain calculations cumbersome -- is eliminated from the boundary counterterm action. Assuming a set of surface terms with no more than two derivatives, in section \\ref{sec:OC} we arrived at an action of the form\n\\begin{align}\\label{FinalAction}\nI = & \\,\\,\\,\\, \\frac{1}{2\\,\\kappa^{\\,2}}\\,\\int_{\\MM} \\nts \\nts d^{\\,d+1}x \\sqrt{g} \\, \\left( R - \n  \\frac{4}{d-1} \\, \\nabla^{\\mu} \\phi \\nabla_{\\mu} \\phi - \\frac{1}{2 \\cdot p!} \\, e^{2 \\alpha \n  \\phi} \\, F^{\\,a_1 \\ldots a_p} F_{a_1 \\ldots a_p}\\right) \\\\ \\nonumber\n    & \\,\\,\\,\\, \\,\\,\\,\\, +  \\frac{1}{\\kappa^{\\,2}}\\,\\int_{\\dM} \\bns \\nts d^{\\,d} x \\,\\sqrt{h} \\, \n  \\bigg( K  +c_{0}\\,e^{-\\frac{\\alpha}{p-1}\\,\\phi} + e^{\\frac{\\alpha}{p-1}\\,\\phi} \\Big(\n  c_{1}\\,\\RR + c_{2} \\,e^{2\\,\\alpha\\,\\phi}\\,\\FF^{\\,a_\\ms{1}\\ldots a_\\ms{p}} \\,\\FF_{a_\\ms{1}\\ldots a_\\ms{p}}\\Big) \\bigg) ~.\n\\end{align}\nFor most values of $\\alpha$, requiring a well-defined variational principle determines the coefficients of the surface terms uniquely \\eqref{uniqueci}. In the special case $\\alpha = \\pm 2(p-1)\/(d-1)$ the conditions that determine the $c_i$ become linearly dependent, and there is a one-parameter family of actions \\eqref{SpecialAlpha} with a well-defined variational principle. \n\nAllowing boundary counterterms with more than two derivatives reveals the existence of families of actions that contain an arbitrary number of free parameters. This is quite unlike the situation in AAdS gravity, where counterterms with more derivatives are less relevant in the asymptotic expansion. Higher derivative counterterms in ALD gravity may be dressed with factors of the dilaton so that they are just as relevant as their lower-derivative counterparts. However, we were able to show that both the on-shell action and the conserved charges are independent of the choice of counterterms. Quantities like the boundary stress tensor do, in principle, depend on the choice of action, though in the examples we considered in section \\ref{sec:Examples} this was not the case. It would be interesting to find an example where this scheme dependence could be examined in the context of a dual holographic theory. \n\n\nOf course, there are various reasons for restricting our attention to a finite number of counterterms. For example, a particular supergravity theory may be of interest as the low energy limit of string theory. At lowest order in $\\alpha'$ one would expect terms with no more than two derivatives in either the bulk or boundary part of the action. Even in that case there are still examples that admit a one-parameter family of counterterms. It is worth pointing out a similar situation in two dimensions: the low energy limit of type 0 supergravity also seems to admit a one parameter family of boundary counterterms, but requiring the full action to respect Buscher duality \\cite{Buscher:1987qj} identifies a unique action \\cite{Davis:2004xi}. Perhaps T-duality or another symmetry can be used to identify a unique action for the ten dimensional type II supergravity examples we considered in section \\ref{sec:Examples}.\n\n\nThe constructions presented in this paper suggest a number of generalizations. The two most obvious are additional field content and a broader class of boundary conditions. For instance, one might consider additional scalar fields or field strengths with support at spatial infinity. At the same time, the boundary conditions could be weakened by allowing terms that fall off more slowly as $\\rho \\to \\infty$. In particular, it would be interesting to investigate boundary conditions associated with more complicated (and less symmetric) brane configurations than the simple stacks of branes analyzed in section \\ref{sec:Examples}.\n\n\nFinally, there is a puzzling aspect to our results that we have not found a satisfactory explanation for. The fall-off conditions in section \\ref{sec:Prelim} allow for terms like $h_{ab}^\\ms{(1)}$ and $\\phi^\\ms{(1)}$ at relative order $\\OO(\\rho^{2-d})$ in the asymptotic expansion. As shown in section \\ref{sec:LRF}, these terms do not contribute to the conserved charges. Yet they seem to have non-trivial equations of motion, and so we would expect that they are dynamical. This is certainly the case in AF gravity, where the analogous terms encode useful information about the spacetime and contribute to the conserved charges associated with asymptotic translations. It is possible that we have simply missed some consequence of the equations of motion or Bianchi identities that force these terms to vanish for ALD gravity, as is the case for the two examples in section \\ref{sec:Examples}. On the other hand, it may be that these terms are relevant for some aspect of the theory that is simply not apparent when linearizing the equations of motion.\n\n\\vskip1em\n\\noindent{\\bf Acknowledgments}\n\\vskip.5em\nThe authors would like to thank Don Marolf, Amitabh Virmani, and Oleg Lunin for useful correspondence and conversations. Research at Perimeter Institute is supported by the Government of Canada\nthrough Industry Canada and by the Province of Ontario through the Ministry of Research\n\\& Innovation. R. Mann would also like to thank the Kavli Institute for Theoretical Physics, where part of this work was carried out, and the Fulbright Foundation and the Natural Sciences and Engineering Research Council of Canada for financial support.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn \\cite{Cra}, T.A. Crawford showed that if $E \\leq 5 |d|+10$, the\nsubset $\\mbox{\\rm Harm}_{d,E} (\\CC P^2)$ of the space of harmonic maps from\n$S^2$ to $\\CC P^2$ consisting of those maps of degree $d$ and\nenergy $4\\pi E$ can be given the structure of a complex manifold\nand that this manifold is connected.\n\nThis he did by showing that the space $\\mbox{\\rm Hol}_{k,r}^*(\\CC P^2)$ of\nfull holomorphic maps from $S^2$ to $\\CC P^2$ of degree $k$ and\nramification index $r$ is\na complex manifold if $r \\leq (k+1)\/2$, and that the ``Gauss\ntransform'' $G'_{k,r}$ which maps $\\mbox{\\rm Hol}_{k,r}^*(\\CC P^2)$ to\n$\\mbox{\\rm Harm}_{k-r-2,3k-r-2}(\\CC P^2)$ bijectively is a homeomorphism,\nso that the manifold structure of $\\mbox{\\rm Hol}_{k,r}^*(\\CC P^2)$ can be\ntransported to the topological space $\\mbox{\\rm Harm}_{d,E}(\\CC P^2)$. This\ndoes not prove that the transported structure is the one induced\nby the natural inclusion of $\\mbox{\\rm Harm}_{d,E}(\\CC P^2)$ in the space\nof maps from $S^2$ to $\\CC P^2$.\n\nIn this paper, after giving a treatment of Crawford's result\nadapted to our needs, we show that $G'_{k,r}$ is a {\\bf smooth}\nmap from $\\mbox{\\rm Hol}_{k,r}^*(\\CC P^2)$ to $C^j(S^2,\\CC P^2)$ \\ (for $j\n\\geq 2$) and has {\\bf injective differential}. From this we\nobtain: \\begin{theorem} \\label{th:1.1}\nFor $0 \\leq r \\leq \\displaystyle{\\frac{k+1}{2}}$\nand $\\displaystyle{\\frac{4k-11}{3} \\leq r \\leq\n\\frac{3}{2}k-3}$ \\ the map\n$$\nG'_{k,r} : \\mbox{\\rm Hol}_{k,r}^* (\\CC P^2) \\to C^j(S^2,\\CC P^2)\n$$\nis a smooth embedding onto $\\mbox{\\rm Harm}_{k-r-2,3k-r-2}(\\CC P^2)$\nfor any $j \\geq 2$.\n\nEach component $\\mbox{\\rm Harm}_{d,E}(\\CC P^2)$ of $\\mbox{\\rm Harm}(\\CC P^2)$ with $E\n\\leq 5 |d| + 10$ is a closed  smooth submanifold of $C^j(S^2,\\CC\nP^2)$ of dimension $6E+4$ if $E=|d|$ (in which case it consists\nof holomorphic or antiholomorphic maps) and of dimension $2E+8$\n(otherwise). \\end{theorem}\n\nAdded in proof: In a revised version of [4], Crawford has shown that\n$\\mbox{\\rm Hol}_{k,r}^* (\\CC P^2)$ is a manifold for\nall $k,r$.  It follows that all our results (Theorems \\ref{th:1.1},\n\\ref{th:1.3}, Proposition \\ref{prop:3.1}, \\ref{prop:4.1}, \\ref{prop:5.2} and\nLemma \\ref{lem:5.1}) are valid for this range and the restriction $E \\leq 5 |d|\n+ 10$ can be removed from Theorem \\ref{th:1.1}.  Proofs are\nunchanged except for those in Sec. 3, see below.\n\n\n\\begin{remark} \\label{rem:1.2} The space $\\mbox{\\rm Hol}_{k,r}^*(\\CC P^2)$\nis non empty precisely for $k \\geq 2$, \\\\ $0 \\leq r \\leq\n\\frac{3}{2} k-3$ (see Proposition \\ref{prop:2.7} below).\n\\end{remark}\n\nWe shall first of all prove that $\\mbox{\\rm Hol}^*_{k,r}(\\CC P^2)$ is a\ncomplex manifold for the range $k \\geq 2$, \\ $0 \\leq r \\leq\n(k+1)\/2$\\ . The passage from this range to the second $k \\geq 3$,\n\\ $(4k-11)\/3 \\leq r \\leq 3k\/2 -3$ is achieved by the conjugate\npolar (see Definition \\ref{def:2.3} below).\n\nIn fact, we have:\n\n\\begin{theorem} \\label{th:1.3} For $0 \\leq r \\leq\n\\displaystyle{\\frac{k+1}{2}}$ and for\n$\\displaystyle{\\frac{4k-11}{3} \\leq r \\leq \\frac{3}{2}k-3}$,\n$\\mbox{\\rm Hol}_{k,r}^*(\\CC P^2)$ is a complex submanifold of the complex\nmanifold $\\mbox{\\rm Hol}_{k}^*(\\CC P^2)$ of dimension $3k-r+2$.\n\\end{theorem}\n\nAdded in proof: Note that  Lemma \\ref{lem:3.3} in the proof is\n  false outside the above range; we give a counter example for $k=6$, $r=4$.\nHence the description of the manifold structure on $\\mbox{\\rm Hol}_{k,r}^*(\\CC P^2)$\nin Sec. 3 is special\nto the range $r \\leq (k+1)\/2$ --- see the revised version\nof \\cite{Cra} for a description valid for any $k,r$.\n\n\\bigskip\n\nThe contents of the subsequent sections are as follows.\n\nIn Section 2, we recall the construction of J. Eells and the\nsecond author of the harmonic maps of $S^2$ to $\\CC P^2$,\nstressing the limits of the values of the parameters involved,\nand we give some examples illustrating the behaviour of this\nconstruction.\n\nIn Section 3, we present a proof of the result of T.A. Crawford\non the structure of $\\mbox{\\rm Hol}_{k,r}^*(\\CC P^2)$, adapted to our\nneeds. We use ideas from the paper of Crawford, and a\nconstruction suggested to us by M. Guest.\n\nIn Sections 4 and 5, we show successively that the Gauss\ntransform is smooth and that it is an embedding. The main\ningredient is a property of smooth dependence of the common\nfactor of smooth families of polynomials (Lemma \\ref{lem:4.5}).\nWe note at this point that the methods of the present paper will\nnot generalize easily to ${\\CC} P^n$ for $n > 2$, however M.\nGuest and Y. Ohnita \\cite{G-O} show that some topological\nquestions on $\\CC P^n$ reduce to $\\CC P^2$.\n\nRecall finally that the harmonic maps from $S^2$ to any manifold\nare precisely the minimal branched immersions of $S^2$, in\nparticular, they are conformal (see e.g. \\cite{E-L} (5.15))\n\n\\noindent\n{\\bf Acknowledgements}\\\\\nBoth authors would like to thank the organizers of the first MSJ\nInternational Research Institute (Sendai, 1993) for invitations\nto the conference where this work was begun, F. Burstall, J.\nEells, M. Guest and A. West for helpful discussions, and L. Simon\nfor posing questions at the above conference which led to this\nresearch.\n\nThe second author would like to thank the Belgian Contact Group\nfor an invitation to their 1993 meeting which allowed this work\nto continue.\n\nThis research was partially supported by the EU Human Capital and\nMobility Programme (Contract CHRX-CT92-0050), the Belgian\nF.N.R.S. and the British Council.\n\n\\section{Gauss transform and polars}\n\nFollowing work of others \\cite{G-S,D-Z}, see also \\cite{Bur}, J.\nEells and the second author classified the harmonic maps from the\n2-sphere $S^2$ to a complex projective space ${\\CC} P^n$, as\nfollows:\n\n\\begin{theorem} \\label{th:2.1} \\cite{E-W}. There is a bijective\ncorrespondence between the set of pairs $(f,s)$ where $f : S^2\n\\to C P^n$ is a full holomorphic map and $s$ an integer, $0 \\leq\ns \\leq n$, and the set of full harmonic maps $\\varphi : S^2 \\to\n\\CC P^n$.\n\\end{theorem}\n\nHere ``full'' means ``not having image in any proper projective\nsubspace of $\\CC P^n$\\ ''. For $s=0$, $\\varphi=f$ is holomorphic,\nfor $s \\neq 0,n$, $\\varphi$ is neither holomorphic nor\nantiholomorphic, and for $s=n$, $\\varphi$ is antiholomorphic and\nis called the {\\bf polar} of $f$.\n\nWe now restrict to the case of $\\CC P^2$, where the description\nof this construction is simpler.\n\nLet $f : S^2 \\to \\CC P^2$ be a holomorphic map. Identifying $S^2$\nwith $\\CC \\cup \\{ \\infty \\}$ by stereographic projection, $f$ can\nbe represented on $\\CC$ by a map $p : \\CC \\to \\CC^3 \\setminus\n\\{0\\}$ where $p(z) = (p_0(z),p_1(z),p_2(z))$ is a triple of\ncoprime polynomials with max (degree $p_0$, degree $p_1$, degree\n$p_2$) = degree of $f$. We shall write $f=[p_0,p_1,p_2]$.\n\nA map $S^2 \\to \\CC P^2$ is called {\\bf full} if its image lies in\nno complex projective line. Note that if a harmonic map is not\nfull, its image lies in a $\\CC P^1$and it is then\n$\\pm$-holomorphic, since it is a conformal map between surfaces.\nThus all harmonic non $\\pm$-holomorphic maps $f : S^2 \\to \\CC\nP^2$ are full.\n\nWe denote by $\\mbox{\\rm Hol}_k^*(\\CC P^2)$ the space of full holomorphic\nmaps of degree $k$. All values of $k \\geq 2$ occur and $\\mbox{\\rm Hol}_k^*\n(\\CC P^2)$ is a complex manifold of dimension $3k+2$ with\ncoordinate charts given by the coefficients.\n\nRecall that a holomorphic map $f : S^2 \\to \\CC P^2$ is said to be\n{\\bf ramified} at a point $z \\in S^2$ if $df(z)=0$. The\nramification index of $f$ at $z$ is the order of the zero of\n$df(z)$ and the {\\bf total ramification index} $r$ of $f$ is the\nsum of ramification indices.\n\nConsider a full holomorphic map $f$. The harmonic map $\\varphi$\nassociated to the pair $(f,1)$ in Theorem \\ref{th:2.1} is\nobtained by means of the $\\partial'$-{\\bf Gauss transform} (in\nthe terminology of \\cite{B-W} - in \\cite{Wol} it is called the\n$\\partial$-{\\bf transform}) which is defined as follows\n\\cite{E-W}:\n\nLet $\\pi : \\CC^3 \\setminus \\{0\\} \\to \\CC P^2$ be the canonical\nprojection sending $(z_0,z_1,z_2)$ to the point of $\\CC P^2$ with\nhomogeneous coordinates $[z_0,z_1,z_2]$. For a map $f : S^2 \\to\n\\CC P^2$, say that $F : U \\to \\CC^3 \\setminus \\{0\\}$ represents\n$f$ on the open set $U$ if $f |_U = \\pi \\circ F$, in which case\nwe write $f=[F]$.\n\nFor $f$ holomorphic and full, the {\\bf first associated curve}\n$f_{(1)} : S^2 \\to G_2(\\CC^3)$ is the holomorphic map defined as\nfollows. Let $F : U \\to \\CC^3 \\setminus \\{0\\}$  represent $f$ on\na domain $(U,z)$ of $\\CC$. Consider the map $F \\wedge F' : U \\to\n\\wedge^2 \\CC^3$ where $\\displaystyle{F'=\\frac{dF}{dz}}$. At a\npoint $x$ where $f$ is not ramified, $F \\wedge F'$ is non zero\nand so defines a complex two-dimensional subspace $f_{(1)}(x)$.\nIf, on the other hand, $f$ is ramified at $x$ with ramification\nindex $k$, then $F \\wedge F' = (z-x)^k \\cdot \\Psi$ for some\nsmooth nonzero map $\\Psi : U' \\to \\wedge^2 \\CC^3$ on an open\nneighbourhood of $x$. Since $\\Psi(y)$ is decomposable for all $y\n\\in U'$, $y \\neq x$, it remains decomposable for $y=x$ and we can\ndefine $f_{(1)}(x)$ as the complex two-dimensional subspace\ndefined by $\\Psi(x)$. The resulting map $f_{(1)} : S^2 \\to\nG_2(\\CC^3)$ is well-defined and smooth. This leads us to two maps\nassociated to $f$~:\n\n\\begin{definition} \\label{def:2.2} The {\\bf $\\partial'$-Gauss\ntransform} $\\varphi=G'(f) : S^2 \\to \\CC P^2$ is defined by the\nformula \\begin{equation}\\label{equ:2.1} \\varphi(x) = f(x)^\\perp\n\\cap f_{(1)}(x). \\end{equation} \\end{definition}\n\n\\noindent By Theorem 2.1, it is a smooth and full harmonic map.\n\n\\begin{definition} \\label{def:2.3} The {\\bf polar} of the holomorphic\nmap $f$ is the antiholomorphic map $$ g(x) =f_{(1)}(x)^\\perp \\ .\n$$\n\\end{definition}\n\nNote that for any $x \\in S^2$, $f(x), \\varphi(x)$ and $g(x)$ are\nHermitian orthogonal complex lines.\n\nFor convenience, we consider the {\\bf conjugate polar} $h$ of $f$\ndefined by taking in $\\CC^3$ the complex conjugate of the values\nof $g(x) : h(x) = \\overline{f_{(1)}(x)^\\perp}$.\n\n\nMore explicitly, represent $f \\in \\mbox{\\rm Hol}_k^*(\\CC P^2)$ by\n$[p_0,p_1,p_2]$ as above.  Identifying $\\wedge^2 \\CC^3$ with\n$\\CC^3$, the first associated curve, or equivalently the\nconjugate polar $h$ of $f$, is represented by the polynomials\n\\begin{eqnarray*} h &=& [p_{12},p_{20},p_{01}]\\\\ &=&\n[p_1p'_2-p'_1p_2,p_2p'_0-p'_2p_0,p_0p'_1-p'_0p_1] \\end{eqnarray*}\nonce they have been divided by their common factor. Explicitly,\nif $f$ has finite ramification points $z_I$ with multiplicities\n$r_I$ $(I=1,\\ldots,R)$, then the {\\bf ramification divisor}\n$R(f)$ is the monic polynomial $R(f)=\\prod (z-z_I)^{r_I}$.\n\nIf $f$ is not ramified at $\\infty$ this has degree equal to the\nramification index, otherwise it has lower degree.\n\nIn either case, $R(f)$ is the highest common factor of the\npolynomials $p_{ij}$, and $$ h = \\left[ \\frac{p_{12}}{R(f)},\n\\frac{p_{20}}{R(f)}, \\frac{p_{01}}{R(f)} \\right]. $$ One checks\neasily that $h$ has degree $2k-2-r$ (indeed, the terms of degree\n$2k-1$ in $p_{ij}$ cancel).\n\nNote that formula (\\ref{equ:2.1}) has a counterpart:\n$$\n\\overline{\\varphi(x)} = h(x)^\\perp \\cap h_{(1)} (x).\n$$\nTheorem \\ref{th:2.1} can be rephrased in $\\CC P^2$ by saying that\n{\\em the Gauss transform defines a bijection between the space of\nfull holomorphic maps and the space of harmonic non\n$\\pm$-holomorphic maps}, and that {\\em the passage to the polar\ndefines a bijection between full holomorphic and antiholomorphic\nmaps}.\n\nHowever, this does not immediately provide a simple description\nof the space of harmonic maps. Indeed, the Gauss transform $G':\n\\mbox{\\rm Hol}_k^*(\\CC P^2) \\to \\mbox{\\rm Harm}(\\CC P^2)$ is not a continuous map,\nwhen the spaces are equipped with their $C^0$-topology. This\nappears for instance in the following example (brought to our\nattention by F. Burstall).\n\n\\begin{example} \\label{ex:2.4} Let $f_t : S^2 \\to \\CC P^2$ be\ndefined by $f_t(z) = [F_t(z)]$, where $$ F_t(z) = (1,tz+z^3,z^2)\n\\quad (z \\in \\CC, t\\in {\\bf R}) $$ (so that\n$f_t(\\infty)=[0,1,0]$). Note that $f_t(0)=[1,0,0]$ for all $t$\nand that $f_t$ is a smooth family of full holomorphic maps. Then\n$F'_t(z)=(0,t+3z^2,2z)$ and $F_t \\wedge\nF'_t(z)=(tz^2-z^4,-2z,t+3z^2)$. If $t \\neq 0$, at $z=0$ this\nequals $(0,0,t)=t(1,0,0) \\wedge (0,1,0)$ so that the first\nassociated curve has the value $f_{t(1)}(0) =\n\\mbox{span}\\{(1,0,0),(0,1,0)\\}$ and $G'(f_t)(0)=[0,1,0]$.\nHowever, if $t=0$, then $F_t \\wedge\nF'_t(z)=(-z^4,-2z,3z^2)=z\\psi(z)$ where $\\psi(z)=(-z^3,-2,3z)$.\nIn particular $\\psi(0)=(0,-2,0)=2(1,0,0) \\wedge(0,0,1)$ so that\n$f_{0(1)}(0) = \\mbox{span}\\{(1,0,0),(0,0,1)\\}$ and\n$G'(f_0)(0)=[0,0,1]$. This shows that $f_{t(1)}$ and $G'(f_t)$ do\nnot vary continuously with $t$. The reason for this is that $f_t$\nis unramified when $t \\neq 0$ but ramified with ramification\nindex 1 at $z=0$ when $t=0$, and that in the presence of\nramification, both $G'$ and $g$ involve division of polynomials\nby their common factor, a discontinuous process when the degree\nof the factor changes.\n\\end{example}\n\nTo proceed, define $\\mbox{\\rm Hol}_{k,r}^*(\\CC P^2)$ as the space of full\nholomorphic maps of degree $k$ and total ramification index $r$,\nand $\\mbox{\\rm Harm}_{d,E}(\\CC P^2)$ the space of all harmonic maps of\ndegree $d$ and energy $4 \\pi E$. By results of \\cite{E-W},\nTheorem \\ref{th:2.1} specializes to\n\n\\begin{proposition} \\label{prop:2.5} For each pair of integers $k\n\\geq 2$, $0 \\leq r \\leq \\frac{3}{2}k-3$, there is a bijective\ncorrespondence $$ G'_{k,r} : \\mbox{\\rm Hol}_{k,r}^*(\\CC P^2) \\to\n\\mbox{\\rm Harm}_{d,E}(\\CC P^2) $$ given by the restriction of the Gauss\ntransform, where $d = k-r-2$ and $E=3k-r-2$.\n\\end{proposition}\n\n\\begin{proposition} \\label{prop:2.6} For each pair of integers $k\n\\geq 2$, $0\\leq r \\leq \\frac{3}{2}k-3$, the map $f \\mapsto$\nconjugate polar of $f$ restricts to a bijection $$\n\\mbox{\\rm Hol}_{k,r}^*(\\CC P^2) \\to \\mbox{\\rm Hol}_{k',r'}^*(\\CC P^2) $$ where\n$k'=2k-r-2$, $r'=3k-2r-6$.\n\\end{proposition}\n\nThis allows us to specify the values of $k$ and $r$ as follows:\n\n\\begin{proposition} \\label{prop:2.7} The space $\\mbox{\\rm Hol}_{k,r}^*(\\CC\nP^2)$ is non-empty precisely for the range $k \\geq 2$, $0 \\leq r\n\\leq \\frac{3}{2}k-3$. \\end{proposition}\n\n\\noindent\n{\\bf Proof.} The well known examples (see \\cite{C-M-R}) $$ f(z) =\n[1,(z+1)^{k-r+1},z^k] $$ provide maps $f$ in $\\mbox{\\rm Hol}_{k,r}^*(\\CC\nP^2)$ for all $k \\geq 2$, $0 \\leq r \\leq k-2$.\n\nThe Pl\\\"ucker formulae (see e.g. \\cite{E-W}) show that the\ninvolutive map $f \\mapsto$ conjugate polar of $f$ restricts to\nbijections $$ \\mbox{\\rm Hol}_{k,r}^*(\\CC P^2) \\to \\mbox{\\rm Hol}_{k',r'}^*(\\CC P^2)\n$$ with $k',r'$ as in Proposition 2.6. Thus to get\n$\\mbox{\\rm Hol}_{k,r}^*(\\CC P^2)$ (and $\\mbox{\\rm Hol}_{k',r'}^*(\\CC P^2)$) non\nempty, we need $r' \\geq 0$, i.e. $r \\leq \\frac{3}{2}k-3$.\n\nOn the other hand, the conjugate polars of the above examples\nprovide maps in $\\mbox{\\rm Hol}_{k,r}^*(\\CC P^2)$ for all $$ k-2 \\leq r\n\\leq \\frac{3}{2} k-3 \\ . $$ \\medskip\n\nAs in \\cite{C-M-R}, we note that the above shows that\n$\\mbox{\\rm Harm}_{d,E} (\\CC P^2)$ is non-empty precisely for pairs $(d,E)$\nof integers with either $E=|d|$ (in which case it consists of\n$\\pm$-holomorphic maps) or $E=3|d|+4+2r$ for some $r \\geq 0$\n(otherwise).\n\nIndeed, all such values of $E$ with $d \\geq 0$ are achieved with\n$0 \\leq r \\leq k-2$, the range $k-2 < r \\leq 3k\/2-3$ giving $d <\n0$.\n\nThe main contribution of the present paper is to show that $$\nG'_{k,r} : \\mbox{\\rm Hol}_{k,r}^*(\\CC P^2) \\to \\mbox{\\rm Harm}_{d,E}(\\CC P^2) \\subset\nC^j(S^2,\\CC P^2) $$ is a smooth embedding and that the map $f\n\\mapsto$ conjugate polar of $f$ is a complex analytic equivalence\nfrom $\\mbox{\\rm Hol}_{k,r}^*(\\CC P^2)$ to $\\mbox{\\rm Hol}_{k',r'}^*(\\CC P^2)$.\n\nWe conclude this section by an example showing that in\n$\\mbox{\\rm Hol}_{k,r}^*(\\CC P^2)$, the ramification divisor of a smooth\nfamily of polynomials can vary smoothly, even when individual\ncommon roots of the $p_{ij}$'s vary only continuously.\n\n\\begin{example} \\label{ex:2.8} Identifying $S^2$ with $\\CC \\cup\n\\{0\\}$ by stereographic projection, let $f_t : S^2 \\to \\CC P^2$\nbe defined by $$ F_t(z)=\\big\n(z^4+1,(1-3t^2)z^3+(-3t+t^3)z,2tz^2+(1-t^2) \\big) \\quad (z \\in\n\\CC, t \\in {{\\Bbb R}}) $$ (so that $f_t(\\infty)=[1,0,0]$).\n\nIdentifying $\\Lambda^2 \\CC^3$ with $\\CC^3$ we have $(F_t \\wedge\nF'_t)(z)=(z^2-t)\\psi(z)$ where $$ \\psi(z) = \\big(\n(-2t+6t^3)z^2+(-3+t^2)(1-t^2),4z(tz^2+1),(-1+3t^2)z^4+8tz^2+3-t^2\n\\big) \\ . $$ which shows that, if $t \\neq 0$, $f_t$ is ramified at\n$z=\\pm \\sqrt{t}$ with index 1, but if $t=0$, these ramification\npoints coalesce into a ramification point at $z=0$ of index 2.\nFurther $f_{t(1)}(z)=[\\psi(z)]$. We see from this that $f_t \\in\n\\mbox{\\rm Hol}_{4,2}^*(\\CC P^2)$ for all $t$ and that $f_{t(1)}$ and so\n$G'(f_t)$ vary smoothly with $t$, even though each root does not.\n\\end{example}\n\n\\section{Spaces of holomorphic maps} In this section, we give a\nproof of the\n\n\\begin{proposition} \\label{prop:3.1} \\cite{Cra} For any $k \\geq\n2$ and $0 \\leq r \\leq \\displaystyle{\\frac{k+1}{2}}$,\n$\\mbox{\\rm Hol}_{k,r}^* (\\CC P^2)$ is a complex submanifold of\n$\\mbox{\\rm Hol}_k^*(\\CC P^2)$ of dimension $3k-r+2$.\n\\end{proposition}\n\n\\noindent {\\bf Proof.} Let\n\\begin{eqnarray*}\n\\mbox{\\rm Hol}'_k(\\CC P^2) & = & \\{ f \\in \\mbox{\\rm Hol}_k^*(\\CC P^2) : f =\n[p_0,p_1,p_2],\\ p_0 \\mbox{ is monic of}\\\\ & & \\mbox{degree } k\n\\mbox{ with distinct roots}, f \\mbox{ is not ramified at } \\infty\n\\} \\ .\n\\end{eqnarray*}\nHere, $p_0,p_1$ and $p_2$ are always assumed to be coprime.\n$\\mbox{\\rm Hol}'_k(\\CC P^2)$ is an open subset of $\\mbox{\\rm Hol}_k^*(\\CC P^2)$ and\nso a complex manifold. It can be embedded as an open subset in\n$\\CC^{3k+2}$ by sending the polynomials $(p_0,p_1,p_2)$ to their\ncoefficients (omitting the leading coefficient of $p_0$, which is\nequal to 1).\n\nThe group $G=PGL_2(\\CC) \\times PGL_3(\\CC)$ acts on $\\mbox{\\rm Hol}_k^*(\\CC\nP^2)$ in a natural way preserving the subsets $\\mbox{\\rm Hol}_{k,r}^*(\\CC\nP^2)$. Given $f \\in \\mbox{\\rm Hol}_k^*(\\CC P^2)$, a variation of the proof\nof Lemma \\ref{lem:4.5} below shows that there exist $g \\in G$ and\na neighbourhood $U$ of $f$ in $\\mbox{\\rm Hol}_k^*({\\bf C } P^2)$ such that\n$g(U) \\subset \\mbox{\\rm Hol}'_k(\\CC P^2)$. Setting $\\mbox{\\rm Hol}'_{k,r}(\\CC\nP^2)=\\mbox{\\rm Hol}_{k,r}^*(\\CC P^2) \\cap \\mbox{\\rm Hol}'_k(\\CC P^2)$, it suffices\ntherefore to show that $\\mbox{\\rm Hol}'_{k,r}(\\CC P^2)$ is a complex\nsubmanifold of $\\mbox{\\rm Hol}'_k(\\CC P^2) \\subset {\\CC}^{3k+2}$.\n\nTo do this, we use a construction suggested by M. Guest following\nideas of T.A. Crawford. Set \\begin{eqnarray*}\n   X''_{k,r}  & = & \\{ (a,f)=(a, (p_0,p_1,p_2)) \\in \\CC^r \\times\n\\CC^{3k+2}~:\\ f \\in \\mbox{\\rm Hol}'_k(\\CC P^2) \\ , \\\\\n               & &  a \\mbox{ is a monic polynomial of degree } r ,\n        (a,p_0)\\mbox{ are coprime and } a  \\mbox{ divides } R(f) \\} \\ .\n\\end{eqnarray*}\nNote that $X''_{k,r}$ is an algebraic subvariety of $\\CC^r \\times {\\bf\nC}^{3k+2}$.\nWe shall prove that $X''_{k,r}$ is a complex submanifold provided\n$r \\leq (k+1)\/2$.\n\nThere is an injective map\n$$\ni : \\mbox{\\rm Hol}'_{k,r}(\\CC P^2) \\to X''_{k,r}\n$$\ngiven by $i(f)=(R(f),(p_0,p_1,p_2))$ where $f=[p_0,p_1,p_2]$ as\nabove, with image $X'_{k,r}=\\{(a,f) \\in X''_{k,r} : \\deg f = k,\n\\mbox{ ramification index } (f)=r \\}$.\n\n\nTo check this, we need only show that in\n$(R(f),f)=(a,(p_0,p_1,p_2))$, $a$ and $p_0$ are coprime.\n\nSuppose, to the contrary, that there is an $x$ with\n$a(x)=p_0(x)=0$. Then $p_0(z)=(z-x)p(z)$, and since $a$ divides\n$p_0p'_i-p_ip'_0$, it follows that $p_i(x)p(x)=0$ for $i=1,2$.\nSince $p_0$ has distinct roots, $p(x) \\neq 0$ so that $p_i(x)=0$\nfor $i=1,2$, contradicting the fact that $p_0,p_1$ and $p_2$ are\ncoprime.\n\nBy Lemma \\ref{lem:4.5} below, the injective map $i$ is complex\nanalytic, since the map $f \\mapsto R(f)$ is.\n\nNote that since $f \\in \\mbox{\\rm Hol}'_{k,r}(\\CC P^2)$ is not ramified at\ninfinity, $R(f)$ is a polynomial of degree $r$. The complement of\n$X'_{k,r}$ in $X''_{k,r}$ is a proper subvariety of $X''_{k,r}$,\nso that if $X''_{k,r}$ is a complex submanifold of $\\CC^r \\times\n\\CC^{3k+2}$, so is $X'_{k,r}$.\n\nTo study $X''_{k,r}$, we embed it in the trivial holomorphic\nvector bundle $\\pi : E \\to A$, where $A$ is the open set in\n$\\CC^{r+k}$ given by \\begin{eqnarray*} A & = & \\{(a,p_0) \\in\n\\CC^r \\times \\CC^k : a \\mbox{ and } p_0 \\mbox{ are monic coprime\npolynomials } \\\\ & & \\mbox{of degrees } r \\mbox{ and } k \\mbox{\nrespectively and } p_0 \\mbox{ has no repeated root} \\} \\ , \\\\ E &\n= & \\{(a,p_0,p_1,p_2):(a,p_0)\\in A, (p_1,p_2) \\in \\CC^{k+1}\n\\times \\CC^{k+1} \\} \\end{eqnarray*} and $\\pi$ is the natural\nprojection.\n\nFor $(a,p_0) \\in A$, let $T_{(a,p_0)} : \\CC^{k+1} \\to \\CC^r$ be\nthe linear map which sends a polynomial $p$ of degree $\\leq k$\n(represented by its coefficients) to the remainder of the\ndivision of $p_0p'-p'_0p$ by $a$. We have:\n\n\\begin{lemma} \\label{lem:3.2} Let $(a,p_0,p_1,p_2) \\in E$ and\n$f=[p_0,p_1,p_2]$. Then $a | R(f)$ if and only if $p_1$ and $p_2$\nlie in $\\ker T_{(a,p_0)}$.\n\\end{lemma}\n\n\\noindent\n{\\bf Proof.} With the notation $p_{ij}=p_ip'_j-p_jp'_i$, we have\nimmediately $$ p_1  p_{02} - p_2 p_{01}=p_0p_{12}. $$\n\nIf $p_1,p_2 \\in \\ker T_{(a,p_0)}$, then $a$ divides the left hand\nside, and since $a$ and $p_0$ are coprime, $a$ must divide\n$p_{12}$. Therefore $a$ divides $R(f)$.\n\nThe converse is immediate.\n\n\\medskip\n\nNow, note that $X''_{k,r}$ is the kernel of the morphism of\nholomorphic vector bundles\n$$\nE=A \\times (\\CC^{k+1})^2 \\to A \\times (\\CC^r)^2\n$$\ndefined by\n$$\n((a,p_0),(p_1,p_2)) \\to ((a,p_0),T_{(a,p_0)}(p_1),T_{(a,p_0)}(p_2)).\n$$\nHence $X''_{k,r}$ is a complex submanifold of $\\CC^r \\times\n{\\CC}^{k+2}$ if $\\dim \\ker T_{(a,p_0)}$ is independent of\n$(a,p_0) \\in A$.\n\n\\begin{lemma} \\label{lem:3.3} If $\\displaystyle{r \\leq\n\\frac{k+1}{2}}$, $\\dim \\ker T_{(a,p_0)}=k+1-r$ \\ $\\forall (a,p_0)\n\\in A$. \\end{lemma}\n\n\\noindent\n{\\bf Proof.} Let the zeros of $a$ be $\\alpha_1, \\ldots, \\alpha_R$ with\nmultiplicities $m_1,\\ldots,m_R$, so that $\\displaystyle{\\sum_{J=1}^R\nm_J=r}$. For any $p$, set $h(p)=p_0p'-p'_0p$. Then $p \\in \\ker\nT_{(a,p_0)}$ iff\n\\begin{equation}\\label{equ:3.1}\n(h(p))^{(I)}(\\alpha_J)=0 \\quad \\forall J = 1,\\ldots,R,\\;\nI=0,\\ldots,m_J-1\n\\end{equation}\nwhere $(h(p))^{(I)}$ denotes the $I^{\\mbox{th}}$ derivative of the\npolynomial $h(p)$.\n\nNow (\\ref{equ:3.1}) is a system of $r$ linear equations in $k+1$\nunknows. Indeed, we can replace $T_{(a,p_0)}$ by the linear map\n$\\CC^{k+1} \\to \\CC^r$ which sends $p \\in \\CC^{k+1}$ to the vector\n$$\n\\left( (h(p))^{(I)}(\\alpha_J), \\, J=1,\\ldots,R, \\, I=0,\\ldots,m_J-1 \\right)\n\\in \\CC^r.\n$$\n\nWe shall show that this map has rank $r$ by finding $r$\npolynomials $P_{K,L} \\in \\CC^{k+1}$ $(L=1,\\ldots,R$, $K=1,\n\\ldots,m_L)$ whose images are linearly independent.\n\nTo do this, choose for $P_{K,L}$ a polynomial of degree $\\leq k$\nwith roots $\\alpha_L$ of multiplicity $K$ and $\\alpha_J$ (for $J\n\\neq L$) of multiplicity $m_J+1$.\n\nThis is possible since \\begin{eqnarray*}\nm_L + \\sum_{J \\neq L}(m_J+1) &=& r + R -1 \\\\\n&\\leq& 2r-1 \\leq k\n\\end{eqnarray*}\nby the hypothesis $r \\leq (k+1)\/2$.\n\n\\noindent\nThen\n$$\n(h(P_{K,L}))^{(I)}(\\alpha_J) = \\left\\{\n\\begin{array}{lll}\n0 & &\\mbox{if } J \\neq L \\\\\n&\\mbox{or} &J=L \\mbox{ and } I < K-1 \\ . \\\\\n\\mbox{non zero } & &\\mbox{if } J=L,\\ I=K-1\n\\end{array}\n\\right.\n$$\nIf we order the components of the vector $(h(p))^{(I)}(\\alpha_J)$\nin lexicographical order, viz.\n$$\n(J,I)=(1,0),(1,1),\\ldots,(1,m_1-1),(2,0),\\ldots,(R,m_R-1),\n$$\nwe observe that the matrix\n$$\n(h(P_{K,L}))^{(I)}(\\alpha_J)\n$$\nis in echelon form.\n\n\\noindent\nThus, the images of the $P_{K,L}$ are linearly independent, which\nshows that the rank of $T_{(a,p_0)}$ is $r$, and so its kernel\nhas dimension $k+1-r$.\n\n\\begin{remark} \\label{rem:3.4} An example, obtained in conjunction\nwith M. Guest, shows this lemma to be false for $k=6,r=4$.\nNamely, when\n$$\np_0=4z^6-12z^5+10z^4+2z^2-4z+4\n$$\nand\n$$\na = z(z-1)(z+1)(z-2),\n$$\nthe kernel of $T_{(a,p_0)}$ is of dimension 4, instead of 3.\n\\end{remark}\n\nWe deduce from the lemma that  $X''_{k,r}$, and so\n$X'_{k,r}$, is a complex submanifold of dimension $3k-r+2$.\n\nNow the restriction of the projection $X'_{k,r} \\to \\mbox{\\rm Hol}'_k(\\CC\nP^2)$ which forgets $a$ is complex analytic and has image\n$\\mbox{\\rm Hol}'_{k,r}(\\CC P^2)$ (indeed, it is the inverse of the map $i :\n\\mbox{\\rm Hol}'_{k,r}(\\CC P^2) \\to X''_{k,r}, \\, f \\mapsto (R(f),f))$.\n\nThus $\\mbox{\\rm Hol}'_{k,r}(\\CC P^2)$ is a complex analytic submanifold of\n$\\CC^{3k+r+2}$, which concludes the proof of Proposition\n\\ref{prop:3.1}.\n\n\\section{The smooth nature of $G'_{k,r}$} In this section we\nshall prove, with notation as in \\S 1,\n\n\\begin{proposition} \\label{prop:4.1} For $0 \\leq r \\leq\n\\displaystyle{\\frac{k+1}{2}}$ and for\n$\\displaystyle{\\frac{4k-11}{3} \\leq r \\leq \\frac{3}{2}k-3}$, \\\n$G'_{k,r} : \\mbox{\\rm Hol}_{k,r}^*(\\CC P^2) \\to C^j (S^2,\\CC P^2)$ is a\n$C^\\infty$ map between $C^\\infty$ manifolds, for any $2 \\leq j <\n\\infty$. \\end{proposition}\n\n\\begin{lemma} \\label{lem:4.2} Let $g_t$ and $h_t$ be two families\nof polynomials in a single (complex) variable which depend\nsmoothly (resp. complex analytically) on a parameter $t \\in U\n\\subset {\\Bbb R}^N$ (resp. $\\CC^N$), where $U$ is open. Suppose that\nthe degrees of $g_t$, $h_t$ and of their highest common factor\n$l_t$ are all constant, i.e. do not vary with $t$. Then the\npolynomial $l_t$ depends smoothly (resp. complex analytically) on\n$t$.\n\\end{lemma}\n\n\\begin{remark} \\label{rem:4.3} We can take the polynomial $l_t$ to\nbe monic; then the statement of the lemma means that the\nremaining coefficients depend smoothly (or complex analytically)\non $t$. \\end{remark}\n\n\\begin{remark} \\label{rem:4.4} In the situation of Lemma\n\\ref{lem:4.2}, each root of the polynomials depends continuously\non $t$, but not smoothly, in general, when the multiplicity\nchanges. So the linear factors of the polynomials do not always\nvary smoothly. \\end{remark}\n\n\\noindent\n{\\bf Proof of Lemma \\ref{lem:4.2}.} It is sufficient to prove the\nlemma for $t$ close to a fixed $t_0 \\in U$. We can suppose deg\n$l_t > 0$, or there is nothing to prove.\n\n\\noindent\n{\\bf First case:} Suppose that for one value of $t$, $g_t$ divides\n$h_t$ or $h_t$ divides $g_t$. If for instance $g_t$ divides\n$h_t$, we have $l_t=a(t)g_t$, where $a(t)$ is a scalar function.\nSince all degrees are constant, we see that, for all $s$, deg\n$l_s=$ deg $g_s$, so that again $l_s=a(s)g_s$, and $l_s$ is\nsmooth.\\\\\n{\\bf Second case:} For each $t,\\ g_t$ (resp. $h_t$) does not divide\n$h_t$ (resp. $g_t$). In particular, $g_t$ and $h_t$ are not\nproportional.\n\nFor brevity of notation we shall now omit the parameter $t$ from\nthe notation.\n\n{\\bf Claim 1.} There exist unique polynomials $\\lambda$ and $\\mu$\nwith \\begin{equation}\\label{equ:4.1}\n\\mbox{deg } \\lambda < \\mbox{ deg } (h\/l) \\quad \\mbox{ and } \\quad\n\\mbox{ deg } \\mu < \\mbox{ deg } (g\/l)\n\\end{equation}\nsuch that $\\lambda g + \\mu h=l$.\n\nThe Euclidean algorithm ensures the existence of $\\lambda$ and\n$\\mu$ such that $\\lambda g + \\mu h = l$. Suppose deg $\\lambda\n\\geq$ deg $(h\/l)$ and let $\\tilde{\\lambda}$ be the unique\npolynomial such that\n$$\n\\lambda = q \\cdot \\frac{h}{l} + \\tilde{\\lambda},\n$$\nwith $\\tilde{\\lambda}=0$ or deg $\\tilde{\\lambda} <$ deg $(h\/l)$.\n\nIf $\\tilde{\\lambda} = 0$, we have $\\lambda = q \\cdot h\/l$ and\n$\\lambda g + \\mu h = 1$ becomes $(qg\/l+\\mu)h=l$, which is\nimpossible with $l \\not \\equiv 0$ and deg $l <$ deg $h$.\n\nIf, instead, $\\tilde{\\lambda} \\neq 0$, and deg $\\tilde{\\lambda}\n<$ deg $(h\/l)$, we have $\\tilde{\\lambda}=\\lambda -q \\cdot h\/l$.\n\\\\ Setting $\\tilde{\\mu} = \\mu + q \\cdot g\/l$, we see that $$\n\\tilde{\\lambda} g + \\tilde{\\mu}h = l\n$$\nand\n$$\n\\mbox{deg } \\tilde{\\mu} + \\mbox{ deg } h = \\mbox{ deg }\n\\tilde{\\lambda} + \\mbox{ deg } g.\n$$\nThis implies\n\\begin{eqnarray*}\n\\mbox{deg } \\tilde{\\mu} &<& \\mbox{ deg } h - \\mbox{ deg } l +\n\\mbox{ deg } g - \\mbox{ deg } h \\\\ &=& \\mbox{ deg } (g\/l) \\ .\n\\end{eqnarray*}\n\nThus we have existence of $\\tilde{\\lambda}$ and $\\tilde{\\mu}$\nsatisfying (\\ref{equ:4.1}).\n\nUnicity is easily checked.\n\n{\\bf Claim 2}. Let $\\lambda$ and $\\mu$ be polynomials satisfying\n(\\ref{equ:4.1}) and such that deg$(\\lambda g + \\mu h)  \\leq$ deg\n$l$. Then $\\lambda g + \\mu h=a \\cdot l$, with $a(t)$ a scalar\nfunction.\n\nSince $l$ divides $g$ and $h$, we have deg$(\\lambda g + \\mu h)\n\\geq$ deg $l$ or $\\lambda g + \\mu h=0$.\n\nIn the second case, we have $\\lambda g\/l=- \\mu h\/l$, and by\n(\\ref{equ:4.1}) $g\/l$ must have a common factor with $h\/l$, a\ncontradiction.\n\nTherefore, deg $(\\lambda g + \\mu h) \\geq$ deg $l$. With the\nhypotheses of the claim, we have deg $(\\lambda g + \\mu h)=$ deg\n$l$ and $l$ divides $\\lambda g + \\mu h$ so that $\\lambda g + \\mu\nh = a \\cdot l$. \\medskip\n\nWe conclude from the two claims that for $\\lambda$ and $\\mu$\nsatisfying (\\ref{equ:4.1}), $l$ is characterized up to a non-zero\nscalar factor as the unique polynomial of the form $\\lambda g +\n\\mu h$ such that deg $(\\lambda g + \\mu h) \\leq$ deg $l \\equiv\nL$.\n\nWriting the parameter $t$ back in, this is equivalent to\n$\\displaystyle{\\frac{d^{L+1}}{dz^{L+1}}(\\lambda_tg_t+\\mu_th_t)=0}$,\na system of homogeneous linear equations in the unknown\ncoefficients of $\\lambda_t$ and $\\mu_t$, with coefficients smooth\nin $t$.\n\nAt $t=t_0$, consider any non zero coefficient of $\\lambda_t$ or\n$\\mu_t$ and scale the solution by setting the coefficient equal\nto 1 for $t$ close to $t_0$. The system becomes inhomogeneous,\nand can be solved by Cramer's rule, so that the solutions are\nsmooth, which proves Lemma \\ref{lem:4.2}.\n\n\\begin{lemma} \\label{lem:4.5} Let $g_t, h_t$ and $k_t$ be three\nfamilies of polynomials in a single complex variable which depend\nsmoothly (resp. complex analytically) on a parameter $t \\in U\n\\subset {\\Bbb R}^N$ (resp. $\\CC^N$). Suppose that the degrees of $g_t$\nand of the highest common factor $l_t$ of $g_t, h_t, k_t$ are\nconstant and that deg $h_t \\leq$ deg $ g_t$ and deg $k_t \\leq$\ndeg $g_t$ for all $t$. Then $l_t$ depends smoothly (resp. complex\nanalytically) on $t$.\n\\end{lemma}\n\n\\noindent\n{\\bf Proof.} The idea of the proof is to replace $g_t, h_t$ and\n$k_t$ by linear combinations $\\tilde{g}_t, \\tilde{h}_t$ and\n$\\tilde{k}_t$, so that the common factor remains the same, but\nany two of the three polynomials have no further common factor.\n\nFirst, replace $h_t$ by $h_t + a \\cdot g_t$ and $k_t$ by $k_t + b\n\\cdot g_t$ so that the three polynomials (still denoted by $g_t,\nh_t$ and $k_t$) all have the same constant degree.\n\nConsider now a fixed value of the parameter -- say $t=0$. We\nshall show that there exists $\\epsilon > 0$ such that $l_t$ is\nsmooth for $\\| t \\| < \\epsilon$.\n\nLet $A$ be a 3 by 3 matrix, which we shall choose close to the\nidentity matrix $I$, and in particular invertible. Set \\begin{equation}\n\\left( \\begin{array}{c}\n\\tilde{g}_t \\\\\n\\tilde{h}_t \\\\\n\\tilde{k}_t\n\\end{array}\n\\right)\n = A\n\\left( \\begin{array}{c}\n{g}_t \\\\\n{h}_t \\\\\n{k}_t\n\\end{array}\n\\right) \\ . \\label{equ:4.2}\n\\end{equation}\n\nFor $\\| t \\| < \\epsilon_1$, all roots of the three polynomials move in a\ncompact set of $\\CC$, since the degrees are constant.\n\nIf $(a_t^1,\\ldots,a_t^r)$ are the roots common to $g_t,h_t$ and\n$k_t$, they are also the roots common to $\\tilde{g}_t,\n\\tilde{h}_t$ and $\\tilde{k}_t$. We shall now study the common\nroots of two (but not three) of these polynomials.\n\nSuppose that $\\alpha$ is a root of $g_0$ and $\\beta$ a root of\n$h_0$, with $\\alpha \\neq \\beta$. By continuity of the roots of a\nfamily of polynomials, for $A-I$ small enough, the corresponding\nroots of $\\tilde{g}_0$ and $\\tilde{h}_0$ remain distinct.\nApplying this remark a finite number of times, we see that any\npair of distinct roots of the polynomials remain distinct after\ntranformation by $A$, when $A-I$ is small enough.\n\nConsider now a complex number $b$ which is a root of $g_t$ and\n$h_t$, but not $k_t$, so that $g_t(b)=h_t(b)=0$ and $k_t(b)=B\n\\neq 0$.\n\nThen for $\\theta \\in \\CC$ small enough, replace $g_0$ by $g_0 +\n\\theta k_0$. We see that $g_0+\\theta k_0$ and $h_0$ do not any\nmore have the common root $b$. By the preceeding remark no new\ncommon root has been created.\n\nNote that the same applies if $b$ is one of the $a_0^s$, by which\nwe mean that $b$ is a root of the three polynomials, of order\n$m+n$ for $g_0$ and $h_0$ and of order $m$ for $k_0$, with $m \\ge\n1$, $n \\geq 1$. Indeed, in this case,\n\\begin{eqnarray*}\ng_0(z) &=& (z-b)^m(z-b)^n\\tilde{g}(z), \\\\\nk_0(z) &=& (z-b)^m\\tilde{k}(z)\n\\end{eqnarray*}\nand\n$$\n(g_0+\\theta k_0)(z)= (z-b)^m((z-b)^n\\tilde{g}(z)+\\tilde{k}(z)),\n$$\nthe last factor being non zero at $b$.\n\nThus $b$ is not any more a common root of $g_0+\\theta k_0$ and $h_0$,\nexcept for the multiplicity of the root in all three polynomials.\n\nRepeating this argument a finite number of times, we can replace $g_0,h_0$\nand $k_0$ by three new polynomials given by\n$$\n\\left( \\begin{array}{c}\n\\tilde{g}_0 \\\\\n\\tilde{h}_0 \\\\\n\\tilde{k}_0\n\\end{array}\n\\right)\n = A\n\\left( \\begin{array}{c}\n{g}_0 \\\\\n{h}_0 \\\\\n{k}_0\n\\end{array}\n\\right)\n$$\nin such a way that the only roots common to two of them are in\nfact common to all three.\n\nDefining $\\tilde{g}_t, \\tilde{h}_t$ and $\\tilde{k}_t$ by\n(\\ref{equ:4.2}) for the same matrix $A$, we see that for $\\|t\\|$\nsmall enough the roots have the same property.\n\nThus, the common factor $l_t$ of $g_t,h_t$ and $k_t$ is also the\ncommon factor of, for example, $g_t$ and $h_t$. By Lemma 4.2, it\nvaries smoothly (resp. complex analytically) with $t$.\n\n\\medskip\n\n\\noindent {\\bf Proof of Proposition \\ref{prop:4.1}.} Let $f \\in\n\\mbox{\\rm Hol}_{k,r}^* (\\CC P^2)$.  Then identifying $S^2$ with $\\CC \\cup\n\\{ \\infty \\}$ by stereographic projection, $f$ can be represented\n(at least on $\\CC \\subset S^2)$ by a map $p : \\CC \\to \\CC^3\n\\setminus \\{0\\}$ with $p(z) =(p_0(z),p_1(z),p_2(z))$, \\ $(z \\in\n\\CC)$, a triple  of polynomials with no common zeros and with max\n(degree $p_0$, degree $p_1$, degree $p_2)=k$. Now since $f$ has a\nfinite number of ramification points (in fact no more than $r$)\nwe can choose the pole of the stereographic projection such that\nnone of them is at $\\infty$\\ , then let the ramification points\nbe $z_1,\\ldots,z_t \\in \\CC$ with ramification indices\n$k_1,\\ldots,k_t$. Note that $\\sum_{i=1}^t k_i = r$.\n\nNext, the first associated curve (see \\S 2) $f_{(1)}$  is\nrepresented by $q=p \\wedge p' : \\CC \\to \\Lambda^2 \\CC^3 \\equiv\n\\CC^3$. This is a triple of polynomials $q=(g,h,k)$ and it is\neasily seen that, since $f$ is not ramified at infinity, the\nmaximum of the degrees of $f,g,k$ is equal to $2k-2$. Further\n$(g,h,k)$ have a common zero at $z_i$ of order $k_i$ if and only\nif $z_i$ is a ramification point of $f$ of ramification index\n$k_i$ so that the highest common factor of $(g,h,k)$ is the {\\em\nramification divisor} of $f$ given by\n$R(z)=\\displaystyle{\\prod_{i=1}^t(z-z_i)^{k_i}}$, a polynomial of\ndegree $r$. Then $q\/R$ is a triple of polynomials with no common\nroots and, for $z \\in \\CC$, $f_{(1)}(z)$ is the 2-plane spanned\nby $q(z)\/R(z)$.\n\nNow suppose that $f_t \\in \\mbox{\\rm Hol}_{k,r}^*(\\CC P^2)$ is a family of\nholomorphic maps depending smoothly on a parameter $t \\in U\n\\subset {\\Bbb R}^N$. Then we can choose a family of polynomial maps\n$p_t : \\CC \\to \\CC^3 \\setminus \\{0\\}$ as above representing $f_t$\nand depending smoothly on $t$ with no ramification point at\ninfinity.\n\nHere we use the fact that the ramification points, being roots of\npolynomials, vary continuously with $t$.\n\nSince the total ramification stays constant ($=r$), the\nramification divisor $R_t$ of $f_t$ has constant degree and we\ncan apply Lemma 4.5 to see that $R_t$ depends smoothly on $t$.\nHence the corresponding $q_t\/R_t$ depends smoothly on $t$ and so\ndoes $(f_t)_{(1)}$. Since $\\varphi_t=f_t^\\perp \\cap f_{t(1)}$ it\nis clear that this too varies smoothly with $t$ and the\nproposition is proven.\n\n\\noindent {\\bf Proof of Theorem \\ref{th:1.3}.} Similarly, Lemma\n\\ref{lem:4.5} allows us to conclude that the passage from $f$ to\nits conjugate polar is a complex analytic map from\n$\\mbox{\\rm Hol}_{k,r}^*(\\CC P^2)$ to $\\mbox{\\rm Hol}_{k',r'}^*(\\CC P^2)$, and that\nthe same applies to its inverse.\n\n\\section{The diffeomorphic nature of $G'_{k,r}$}\nIn this section, we complete the proof of Theorem 1.1. First we\nshow\n\n\\begin{lemma} \\label{lem:5.1}  For $k \\geq 2$, \\ $0 \\leq r \\leq\n\\displaystyle{\\frac{k+1}{2}}$ and $k \\geq 3$,\\\n$\\displaystyle{\\frac{4k-11}{3} \\leq r \\leq \\frac{3}{2}k-3}$ \\\n$G'_{k,r} : \\mbox{\\rm Hol}_{k,r}^* (\\CC P^2) \\to C^j(S^2,{\\bf C}P^2)$ has\ninjective differential at all points $f_0 \\in \\mbox{\\rm Hol}_{k,r}^* (\\CC\nP^2)$, for any $j \\geq 2$. \\end{lemma}\n\n\\noindent\n{\\bf Proof.} Let $f_t \\in \\mbox{\\rm Hol}_{k,r}^* (\\CC P^2)$ be a family of\nholomorphic maps depending smoothly on a real parameter $t \\in (-\n\\epsilon, \\epsilon)$, $\\epsilon > 0$, i.e. a smooth curve in\n$\\mbox{\\rm Hol}_{k,r}^* (\\CC P^2)$. Then, by Proposition \\ref{prop:4.1} and\nthe proof of Theorem \\ref{th:1.3}, $\\varphi_t = G'_{k,r}(f_t)$\nand $g_t$ (the polar of $f_t$) are smooth curves in $C^j(S^2,\\CC\nP^2)$. Working on a coordinate domain $(U,z)$, let $F_t,\n\\Phi_t,G_t : U \\to \\CC^3 \\setminus \\{0\\}$ be families of smooth\nmaps representing $f_t, \\phi_t, g_t$ respectively with $F_t$\nholomorphic and $G_t$ antiholomorphic. Then for each $z \\in U, \\\nt \\in (-\\epsilon, \\epsilon)$, $\\{F_t(z), \\Phi_t(z),G_t(z)\\}$ is a\nHermitian orthogonal basis of $\\CC^3$. Now $\\displaystyle{\\frac{d\n\\varphi_t}{dt} = d \\pi \\left( \\frac{d \\Phi_t}{dt} \\right)}$\n(where, as before, $\\pi : \\CC^3 \\setminus \\{0\\} \\to \\CC P^2$ is\nthe canonical projection). Suppose that $\\displaystyle{\\frac{d\n\\varphi_t}{dt}=0}$ for some value of $t$. Then\n$\\displaystyle{\\frac{d \\Phi_t}{dt}}$ must be in direction\n$\\Phi_t$, so that, in particular,\n$$ <\\frac{d \\Phi_t}{dt},F_t>=0.\n$$\n(Here $<,>$ denotes the standard Hermitian inner product on\n${\\bf C}^3$). This last equation is equivalent to\n$$\n<\\frac{d F_t}{dt},\\Phi_t>=0,\n$$\nhence\n\\begin{equation}\\label{equ:5.1}\n\\frac{dF_t}{dt}=\\alpha F_t + \\beta G_t\n\\end{equation}\nfor some smooth functions $\\alpha, \\beta$ on $U \\times\n(-\\epsilon,\\epsilon)$. Differentiating with respect to\n$\\overline{z}$, since $F_t$ is holomorphic we obtain\n$$\n0=\\frac{\\partial \\alpha}{\\partial \\overline{z}}F_t +\n\\frac{\\partial \\beta}{\\partial \\overline{z}}G_t +\n\\beta\\frac{\\partial G_t}{\\partial \\overline{z}}.\n$$\nNow the triple $\\big\\{ F_t,\\displaystyle{\\frac{\\partial\nG_t}{\\partial \\overline{z}},G_t} \\big\\}$ is linearly independent\nexcept at the isolated points where $h_t=\\bar{g}_t$ is ramified.\nHence $\\beta \\equiv 0$ and so from (\\ref{equ:5.1})\n$$\n\\frac{d F_t}{dt} = \\alpha F_t\n$$\nwhich implies that $df_t\/dt=0$. The lemma follows.\n\nThe proof of Theorem \\ref{th:1.1} is completed by\n\n\\begin{proposition} \\label{prop:5.2} For $k \\geq 2$, \\ $0 \\leq r\n\\leq \\displaystyle{\\frac{k+1}{2}}$ and $k \\geq 3$,\\\n$\\displaystyle{\\frac{4k-11}{3} \\leq r \\leq \\frac{3}{2}k-3}$ \\\n$G'_{k,r} : \\mbox{\\rm Hol}_{k,r}^*(\\CC P^2) \\to C^j(S^2,\\CC P^2)$ is an\nembedding with image the closed submanifold\n$\\mbox{\\rm Harm}_{k-2-r,3k-2-r}(\\CC P^2)$ of $C^j(S^2,\\CC P^2)$ for any $j\n\\geq 2$.\n\\end{proposition}\n\n\\noindent\n{\\bf Proof.} Since $\\mbox{\\rm Hol}_{k,r}^*(\\CC P^2)$ is finite dimensional,\nthe differential of $G'_{k,r}$ splits at each point and so by\n\\cite{Lan} $G'_{k,r}$ is an immersion. By Proposition\n\\ref{prop:2.5} it is injective and has image\n$\\mbox{\\rm Harm}_{k-2-r,3k-2-r}(\\CC P^2)$, and we show below that it is a\nhomeomorphism onto its image. Thus it is an embedding and its image\nis a closed submanifold of $C^j(S^2,\\CC P^2)$. The dimension of\nthe space $\\mbox{\\rm Harm}_{k-2-r,3k-2-r}(\\CC P^2)$ is thus equal to the\n(real) dimension of $\\mbox{\\rm Hol}_{k,r}^* (\\CC P^2)$ which is $6k-2r+4$;\nthe stated dimensions follow easily.\n\nTo show that $G'_{k,r}$ is a homeomorphism it suffices to show that\nit is proper. To do this, following \\cite{Cra},\\cite[Lemma\n3.3]{F-G-K-O} consider a sequence $(\\phi_n)$ which converges to\n$\\phi$ in $G'_{k,r}(\\mbox{\\rm Hol}^*_{k,r}(\\CC P^2)) = \\mbox{\\rm Harm}_{k-2-r,\n3k-2-r}(\\CC P^2)$.  Let $f_n = (G'_{k,r})^{-1}(\\phi_n) \\in\n\\mbox{\\rm Hol}^*_{k,r}(\\CC P^2)$.  It suffices to prove that a subsequence\nconverges in $\\mbox{\\rm Hol}^*_{k,r}(\\CC P^2)$.  Note first that\n$\\mbox{\\rm Hol}^*_{k,r}(\\CC P^2)$ can be injected into the projective space\n$P(\\CC^{3k+3})$ by the map $i:f \\mapsto$ projective class of the\ncoefficients of the polynomials describing $f$.  Since that space\nis compact, a subsequence of $(i(f_n))$ converges to $[p] =\n[p_0,p_1,p_2]$. (Note that $(p_0, p_1, p_2)$ need not be coprime\nand that their maximal degree need not be $k$.)  We retain the\nnotation $i(f_n)$ for the subsequence.  We can then write $[p] =\n[b q_0,b q_1, b q_2]$ where the $q_i$'s are coprime and $[q] =\n[q_0, q_1, q_2]$  lies in $\\mbox{\\rm Hol}_{k-m}(\\CC P^2)$, the space of not\nnecessarily full holomorphic maps from $S^2$ to $\\CC P^2$ of\ndegree $k-m$, for some $m \\geq 0$ . For each $n$, let $h_n$ be the\nconjugate polar of $f_n$ belonging to $\\mbox{\\rm Hol}_{2k-2-r}^*(\\CC P^2)$,\nand consider its image by $i$ in the appropriate projective space\nof coefficients $P(\\CC^{6k-3-3r})$. Again, a subsequence\nconverges to $[t] = [t_0,t_1,t_2]$, and we have $[t_0,t_1,t_2] =\n[a s_0, a s_1, a s_2]$ with the $s_i$'s coprime and of possibly\nlower degree than $2k-2-r$.  Now $f_n \\perp \\overline{g_n}$ for\nall $n$ \\ so that $q \\perp \\bar{s}$ and $\\psi = (q \\oplus\n\\bar{s})^{\\perp}$ is well-defined. Further on $S^2 \\setminus\n\\{\\{\\mbox{zeros of } a \\} \\cup \\{ \\mbox{zeros of } b \\} \\cup\n\\{\\infty\\}\\}$, \\ $G'(f_n)(x)$ converges to $\\psi(x)$ so that\n$\\phi$ coincides with $\\psi$ on a dense set, and therefore\neverywhere. Then $3k-2-r = E(\\phi) = E(\\psi) = \\deg q + \\deg s$\nwith $\\deg q \\leq k$ and $\\deg s \\leq 2k-2-r$ so that both these\ninequalities must be equalities and there can be no loss of\ndegree above. Thus $q$ must have degree $k$ and ramification\nindex $r$.  Further, since $r \\leq 3k\/2 -3$, \\ $q$ is full,\notherwise by the Riemann-Hurwitz formula we would have $r = 2k-2\n> 3k\/2 -3$.  Hence $q \\in \\mbox{\\rm Hol}_{k,r}^*(\\CC P^2)$ as required.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nConditional image-to-image translation aims to generate images from certain given conditional inputs such as semantic segmentation \\cite{park2019spade,wang2018pix2pixhd}, layout \\cite{li2020bachgan}, and key points \\cite{tang2019cycle}.\nWith the advance of Generative Adversarial Networks (GANs), it has made rapid progress and achieved quite promising translation performance in recent years.\nHowever, most existing methods have very loose control over the translation process which often affects the translation quality greatly and so the wide application of image translation in various tasks. Optimal style control is still an open challenge in high-fidelity realistic image translation.\n\nSeveral prior works attempted to tackle the style control challenge by using a latent code that is encoded by either Variational Auto-Encoder (VAE) \\cite{park2019spade} or style encoder \\cite{choi2020starganv2}.\nHowever, latent codes often impair style control accuracy as they do not have sufficient capacity to capture detailed style information. \nA different approach is to inject specific style codes for different semantic regions \\cite{zhu2020sean}, but it is specifically designed for conditional input of semantic segmentation and cannot well generalize to other conditional inputs. Recently, Zhang et al.  \\cite{zhang2020cocosnet} explore to establish dense semantic correspondence between conditioned input and a given style exemplar so as to offer dense style guidance in translation.\nHowever, it constructs the semantic correspondence based on cosine similarity that often leads to many-to-one matching (i.e. multiple conditional input features match to the same exemplar feature as illustrated in Fig. \\ref{im_intro}) and missing of details in image translation.\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=1.0\\linewidth]{im_intro.pdf}\n\\caption{\nDifferent feature matching in image translation: \\textit{Cosine Similarity} tends to match each feature separately which often leads to many-to-one matching. Classical optimal transport (\\textit{Classical OT}) suppresses the many-to-one matching problem but it matches all feature points including undesired outliers (existing between deviational feature distributions). Our designed unbalanced optimal transport (\\textit{Unbalanced OT}) mitigates many-to-one matching and avoid outlier matching effectively.\n}\n\\label{im_intro}\n\\end{figure}\n\n\nWe designed \\textit{UNITE}, UNbalanced optImal feature Transport for Exemplar-based image translation that achieves high-fidelity image generation with faithful style to given exemplars. UNITE consists of a \\textit{feature transport network} and a \\textit{translation network} that are inter-connected and can be jointly optimized in training. The \\textit{feature transport network} introduces optimal transport \\cite{peyre2019computational} which matches two sets of features as a whole and effectively overcomes many-to-one matching as in the widely adopted cosine similarity \\cite{zhang2020cocosnet} that matches individual features separately. To tackle the distribution deviations between conditional inputs and exemplars, we design an unbalanced optimal transport technique that adaptively learns the mass (or weight) of each individual feature for effective transport between distributions of different masses. In the \\textit{translation network}, we design a semantic-activation normalization scheme that injects the aligned features into the translation process, where the exemplar features are transported in a multi-stage manner for preserving rich and complicated textural details. Extensive experiments show that UNITE translates images with superior realism and fidelity. \n\nThe contributions of this work can be summarized in three aspects. First, we propose a conditional image translation framework that introduces optimal transport for proper feature alignment and faithful style control in image translation. Second, we design an unbalanced optimal transport technique with adaptive mass learning scheme that is capable of aligning features with deviational distributions, and a multi-stage transport strategy that can preserve complex textures at different scales. Third, we design a novel semantic-activation normalization that is capable of injecting the aligned style features into the image translation process effectively.\n\n\n\n\n\\section{Related Work}\n\n\\subsection{Image-to-Image Translation}\n\nGAN-based image-to-image translation has been investigated extensively due to its wide applications in different tasks such as domain adaptation \\cite{shrivastava2017simgan,zhan2019gadan,zhan2019esir,zhan2020sagan},\ndata augmentation \\cite{zhang2021defect,zhan2018verisimilar,zhan2019scene,xue2018accurate},\nimage editing \\cite{yu2018inpainting,wu2020cascade,wu2020leed,koksal2020rf,yu2021diverse}, image composition \\cite{zhan2019sfgan,zhan2019acgan,zhan2020aicnet,zhan2020emlight,zhan2021gmlight,zhan2021rabit,zhan2020towards}, etc. Existing works explored different conditional inputs such as semantic segmentation \\cite{isola2017pix2pix,wang2018pix2pixhd,park2019spade}, scene layouts \\cite{sun2019lostgan,zhao2019layout2im,li2020bachgan}, key points \\cite{ma2017pose,men2020adgan,zhang2021deep}, edge maps \\cite{isola2017pix2pix,zhu2017toward,lee2018diverse}, etc. for photo-realistic image translation. On the other hand, optimal style control remains a critical yet challenging task that has attracted increasing attention in recent years. For example, \\cite{huang2018multimodal} and \\cite{ma2018exemplar} transfer style codes from exemplars to source images via adaptive instance normalization (AdaIN) \\cite{huang2017adain}. \\cite{park2019spade} uses variational autoencoder (VAE) \\cite{kingma2013vae} to encode exemplars for image translation. \\cite{choi2020starganv2} employs a style encoder for style consistency between exemplars and the translated images. \n\nDifferent from the aforementioned methods that adopt latent vectors for style control, \\cite{zhang2020cocosnet} learns dense semantic correspondences between conditional inputs and exemplars for image translation. Similar ideas have been explored in other translation tasks such as image colorization \\cite{he2018colorization,zhang2019colorization} that also employs exemplars to build up semantic correspondences. On the other hand, most existing works use cosine similarity to build up semantic correspondences which often suffer from many-to-one matching and resultant feature missing. We introduce optimal transport for feature matching that treats the whole feature set as a whole and overcomes the many-to-one matching effectively.\n\n\n\\subsection{Optimal Transport}\n\nOptimal transport (OT) \\cite{villani2008ot} provides a principal way of comparing distributions and offers optimal plans for matching distributions. As a linear programming problem, classic OT is computationally intensive and \\cite{cuturi2013sinkhorn} presents entropy regularized optimal transport that is differentiable and can be solved by the Sinkhorn-Knopp algorithm \\cite{sinkhorn1967concerning,knight2008sinkhorn} efficiently. On the other hand, the classical optimal transport has a typical constraint that they can only handle distributions with equal mass and thus become inapplicable while facing unbalanced distributions with different masses and deviations as widely existed in various tasks. Different approaches have been reported to address this new challenge. For example, \\cite{chizat2016uot} presents a unified treatment of unbalanced optimal transport that allows for both static and dynamic formulations. \\cite{liero2015uot} introduce an entropic version of unbalanced optimal transport.\n\nIn recent years, optimal transport has been widely explored in various computer vision tasks such as domain adaptation \\cite{courty2016optimal}, semantic matching \\cite{liu2020scot}, style transfer \\cite{kolkin2019style}, etc. In this work, we adapt unbalanced feature transport for aligning deviational features between conditional inputs and exemplars for high-fidelity image translation.\n\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=1.0\\linewidth]{im_stru.pdf}\n\\caption{\nThe framework of our proposed network: The \\textit{Conditional Input} and \\textit{Exemplar} are fed to feature extractors $F_{X}$ and $F_{Z}$ to extract feature vectors $X$ and $Z$. The mass (or weight) of the feature vectors ($\\alpha$ and $\\beta$ masses) are then determined collectively by $X$ and $Z$. \nThe weights and the feature vectors form two sets of Dirac masses $\\bm{\\alpha}$ and $\\bm{\\beta}$, which are further aligned through \\textit{Unbalanced Optimal Transport}. With an obtained \\textit{Transport Plan}, the feature of the \\textit{Exemplar} is transported in a multi-stage manner to be aligned with that of the \\textit{Conditional Input}. The aligned features will be injected into the translation network through a proposed SEmantic-ACtivation (dE)normalization (SEACE) to synthesize the final output image. (Blue and orange parts for feature transport network, green part for translation network)}\n\\label{im_stru}\n\\end{figure*}\n\n\n\n\\section{Proposed Method}\nOur UNITE consists of a feature transport network (in blue and orange) and a translation network (in green) which are inter-connected as shown in Fig. \\ref{im_stru}. The feature transport network aligns the features of conditional inputs and exemplars and the translation network produces the final synthesis, more details to be described in the following subsections.\n\n\n\\subsection{Feature Transport Network}\n\nThe feature transport network aims to transport the feature of exemplars to be aligned with that of conditional inputs, thus providing accurate style guidance for the image translation. As shown in Fig. \\ref{im_stru}, both conditional input and exemplar are fed to two feature extractors $F_{X}$ and $F_{Z}$ to extract two sets of feature vectors $X = (x_1, \\cdots, x_{n}) \\in \\mathbb{R}^{d}$ and $Z = (z_1,\\cdots,z_{n}) \\in \\mathbb{R}^d$, where $n$ denotes the number of feature vectors and $d$ denotes the feature dimension.\n\nTo align feature sets $X$ and $Z$, most existing methods \\cite{zhang2020cocosnet,he2018colorization,zhang2019colorization} build a dense correspondence matrix between $X$ and $Z$ by measuring the Cosine similarity between any two feature vectors. As each feature vector $x_{i}$ is matched to the feature vector $z_{j}$ with the maximum Cosine similarity separately, multiple feature vectors in $X$ may correspond to the same feature vector in $Z$ (i.e. many-to-one matching), which leads to blurry translation as illustrated in Fig. \\ref{im_align}. To avoid many-to-one matching between sets of feature vectors, we introduce optimal transport method to align the features of conditional inputs and exemplars.\n\n\\textbf{Classical Optimal Transport.}\nThe classical Optimal transport aims to determine the best transport plan (namely the minimum amount of total work required) to transform one measure into another with the same mass. Here the `work' is evaluated by the product of the cost and the amount of mass to be transported. With constraints on the total masses in transport, optimal transport penalizes the many-to-one matching effectively.\n\nTo formulate the feature alignment as an optimal transport problem and derive the constraints of total masses, we encode the conditional input feature $X$ and exemplar feature $Z$ as Dirac masses: $\\alpha = \\sum_{i=1}^{n} \\alpha_{i} \\delta_{x_{i}}$ and  $\\beta = \\sum_{i=1}^{n} \\beta_{i} \\delta_{z_{j}}$, where the masses $\\alpha_{i}, \\beta_{i} \\geq 0$ and feature vectors $x_{i},z_{i}$ denote the locations of $\\alpha_{i},\\beta_{i}$. Then we define a distance matrix $C$, where each entry $C_{ij}$ in $C$ gives the cost of moving mass $\\alpha_{i}$ to mass $\\beta_{j}$ which can be defined by:\n$\nC_{ij} = 1 - \\frac{x_{i}^{\\top} \\cdot z_{j}} {||x_{i}|| \\, ||z_{j}||}\n$\nA transport plan $T$ can be defined, where each entry $T_{ij}$ is the amount of masses transported between $\\alpha_{i}$ and $\\beta_{j}$. Then the classical optimal transport problem can be formed as:\n\\begin{equation}\n\\begin{split}\n& OT(\\alpha,\\beta) = \\mathop{\\min}\\limits_{T} (\\sum_{i,j=1}^{n} C_{ij} T_{ij}) = \\mathop{\\min}\\limits_{T} \\langle C, T \\rangle \\\\\n&{\\rm subject \\ to} \\quad (T \\vec{1}) = \\alpha, \\quad (T^\\top \\vec{1}) = \\beta \\\\ \n\\end{split}\n\\label{formula_ot}\n\\end{equation}\nThe constraints of the total masses $\\quad (T \\vec{1}) = \\alpha$ and $(T^\\top \\vec{1}) = \\beta$ naturally penalize the many-to-one matching in optimal transport as illustrated in Fig. \\ref{im_align}.\n\n\n\n\\textbf{Unbalanced Optimal Transport.}\nFor classical optimal transport, the total masses of the two measures should be the same, namely $\\sum_{i=1}^{n} \\alpha_{i} = \\sum_{j=1}^{n} \\beta_{j}$. But for conditional inputs and exemplars, their features are usually not perfectly matched so have different total masses.\nFor example, the conditional input (key-point map) in Fig. \\ref{im_stru} does not contain feet which exist in the exemplar, so the feature of feet region in the exemplar is treated as outliers in optimal transport and should not be matched to any feature of the conditional input.\nHowever, classical optimal transport inevitably matches all features, leading to inaccurate or false matching as illustrated in Fig. \\ref{im_align}.\nWe handle it by introducing a relaxed version of classical optimal transport, namely unbalanced optimal transport (UOT or unbalanced OT) \\cite{chizat2016uot} that aims to determine an optimal transport plan between measures of different total masses. We formulate unbalanced OT by replacing the `Hard' conservation of masses in (\\ref{formula_ot}) by a `Soft' penalty with a divergence metric. An unbalanced OT problem can thus be formulated as follows:\n\\begin{equation}\n\\begin{split}\n & \\mathop{\\min}\\limits_{T} \\left [ \\langle C, T \\rangle + \\tau {\\rm KL} (T \\vec{1} || \\alpha) + \\tau {\\rm KL} (T^{\\top} \\vec{1}|| \\beta) \\right ]  \\\\\n\\end{split}\n\\end{equation}\nwhere $\\tau$ is regularization parameter, ${\\rm KL}$ is the Kullback-Leibler divergence which is defined as ${\\rm KL} (a||b) = \\sum_{i = 1}^{n} a_{i} \\log (\\frac{a_{i}}{b_{i}}) - a_{i} + b_{i}$.\n\nWe employ cross-inner product to generate the masses $\\alpha_{i}, \\beta_{j} (i,j \\in [1,n])$ associated with each feature vector. The masses are highly correlated with specific conditional inputs and exemplars, thus it should be determined collectively by both of them. Intuitively, the feature vector that is more related with another feature set should have higher mass. We therefore determine the mass of a feature vector by computing its relevance with another feature set:\n\\begin{equation}\n    \\alpha_{i} = x_{i}  \\cdot \\frac{\\sum_{i=1}^{n}(z_{i})}{n}, \\;\n    \\beta_{j} = z_{j}  \\cdot \\frac{\\sum_{j=1}^{n}(x_{j})}{n}\n\\end{equation}\nThe mass parameters are adaptively updated in training. They capture the mass of each single feature vector accurately and mitigate the false matching problem effectively.\n\nTo implement UOT in a differentiable manner, an entropic regularization term $H(T) = - \\sum_{i,j=1}^{n} T_{ij} \\log T_{ij}$ is introduced. An entropic UOT problem can be defined by:\n\\begin{equation*}\n\\begin{split}\n & \\mathop{\\min}\\limits_{T} \\left [ \\langle C, T \\rangle + \\tau {\\rm KL} (T \\vec{1} || \\alpha) + \\tau {\\rm KL} (T^{\\top} \\vec{1}|| \\beta) - \\eta H(T)  \\right ]  \\\\\n\\end{split}\n\\end{equation*}\nwhere $\\eta$ is the regularization coefficients that denotes the smoothness of the transport plan $T$. In our network, $\\eta$ is fixed at 0.0001 empirically.\n\nTo obtain $T$, we consider the Fenchel-Legendre dual form of the entropic UOT that is defined by:\n\\begin{footnotesize}\n\\begin{equation}\n\\begin{split}\n& \\mathop{\\max}\\limits_{u, v} \\left[ -F^*(-u) - G^*(-v) - \\eta \\sum_{i,j}\\exp(\\frac{u_i + v_j - C_{ij}}{\\eta})\n\\right] \\\\\n\\label{formula_dual}\n\\end{split}\n\\end{equation}\n\\end{footnotesize}\nwhere $F^*$ and $G^*$ are the Legendre conjugate of ${\\rm KL}$ divergence which can be computed by:\n\\begin{equation*}\n\\begin{split}\n& F^*(u) = \\mathop{\\max}\\limits_{z} z^{\\top}u - \\tau {\\rm KL}(z||\\alpha) = \\tau \\langle e^{u\/\\tau}, \\alpha \\rangle - \\alpha^{\\top} \\vec{1} \\\\\n& G^*(v) = \\mathop{\\max}\\limits_{x} x^{\\top}v - \\tau {\\rm KL} (x||\\beta) = \\tau \\langle e^{v\/\\tau}, \\beta \\rangle - \\beta^{\\top} \\vec{1} \\\\\n\\end{split}\n\\end{equation*}\nThen the Sinkhorn algorithm \\cite{cuturi2013sinkhorn} can be applied to (\\ref{formula_dual}) for approximating UOT solution, with a desired transport plan $T$ encoded by optimal dual vectors $u$ and $v$ as below:\n\\begin{equation}\n    T_{ij} = \\alpha_{i} \\beta_{j} \\, \\exp \\frac{1}{\\eta} \\left [ u_{i} + v_{j} - C_{ij} \\right]\n\\end{equation}\n\n\\textbf{Multi-Stage Feature Transport}.\nWith the transport plan, the exemplar features can be transported to be aligned with conditional input features for translation. Different from CoCosNet \\cite{zhang2020cocosnet} that warps exemplar images directly, we adopt a multi-stage manner to transport exemplar features as shown in Fig. \\ref{im_stru}. This multi-stage transport helps to preserve detailed exemplar features especially for textures with complicated patterns as illustrated in Fig. \\ref{im_com}.\n\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=1.0\\linewidth]{im_align.pdf}\n\\caption{\nThe comparison of different feature alignment methods: \nFor visual comparison, we directly apply the feature alignment result to warp the exemplar.\nThe \\textit{Cosine Matching} using cosine similarity often leads to many-to-one matching that introduces blurry feature alignment as highlighted by red box, which further leads to blurry synthesis result as shown in \\textit{Cosine Synthesis}. \\textit{OT Matching} using classical optimal transport suppresses the many-to-one matching but tends to introduce false matching as highlighted by orange box. Our proposed \\textit{UOT Matching} using unbalanced optimal transport mitigates both many-to-one matching and false matching effectively, which achieve the best feature alignment and synthesis fidelity as illustrated in \\textit{UOT Synthesis}.\n}\n\\label{im_align}\n\\end{figure}\n\n\n\n\n\n\\subsection{Translation Network}\n\nThe translation network aims to synthesize images under the semantic guidance of conditional inputs and style guidance of aligned exemplar features. The overall architecture of the translation network is similar to SPADE \\cite{park2019spade} as illustrated in Fig. \\ref{im_stru} (green part). More details of the network structure are available in the supplementary material.\n\nIn translation network, the aligned exemplar features are injected into the generation process at multiple stages to control the style of output image. Although style feature injection can be handled by several different approaches such as SPADE \\cite{park2019spade}, all prevalent approaches fail to consider the semantic correlation between style features in feature injection. We designed an innovative semantic-aware injection method to be described in the following subsection.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=1.0\\linewidth]{im_norm.pdf}\n\\caption{\nThe structure of the proposed SEmantic-ACtivation (dE)normalization (SEACE): To build the long range dependency between style features, a semantic-activation matrix is obtained by computing the self-attention of the condition input features $X$ that are extracted in the feature transport network. With the semantic-activation matrix, $\\gamma_{\\scriptscriptstyle X}$ is determined collectively by the entire region in $\\gamma_{\\scriptscriptstyle Z}$ with the same semantic as shown at the bottom.\n}\n\\label{im_norm}\n\\end{figure}\n\n\n\\textbf{Semantic-Activation Denormalization.}\nIdeally, the style of a spatial position should be determined by all the style feature with the same semantic instead of only relying on the local feature in the exemplar. In addition, building long-range dependencies between style features is usually beneficial to image generation \\cite{zhang2019self} as it allows to leverage the complementary style features of distant image regions. Based on these observations, we propose a novel SEmantic-ACtivation (dE)normalization (SEACE) to model the long-range dependencies across style features in style injection.\n\nAs shown in Fig. \\ref{im_norm}, two sets of modulation parameters $\\gamma_{\\scriptscriptstyle Z}$ and $\\mu_{\\scriptscriptstyle Z}$ are generated from the \\textit{Aligned Feature}.\nTo aggregate the style within each semantic region and build their long-range correlation, we introduce a semantic-activation matrix $M$, which can be obtained from the extracted feature of conditional input $X = (x_1,\\cdots,x_n)$ by computing its self-attention $M_{ij} = x_{i} \\cdot x_{j}$.\nAs there is only semantic feature in conditional input feature, the semantic-activation matrix accurately measures the self-semantic correlation.\nThen the semantic-activation matrix is employed to aggregate the modulation parameters by $\\gamma_{\\scriptscriptstyle X} = M \\cdot \\gamma_{\\scriptscriptstyle Z}$ and $\\mu_{\\scriptscriptstyle X} = M \\cdot \\mu_{\\scriptscriptstyle Z}$.\nThus the feature in each position of $\\gamma_{\\scriptscriptstyle X}$ is determined collectively by a region with the same semantic in $\\gamma_{\\scriptscriptstyle Z}$ as shown at the bottom of Fig. \\ref{im_norm}.\nMeanwhile, the long range correlation between modulation parameters with the same semantic is established.\n\nSpecially, instead of modulating the generation network directly with these modulation parameters, we first apply the modulation parameters $\\gamma_{\\scriptscriptstyle X}$ and $\\mu_{\\scriptscriptstyle X}$ to modulate the activation $X_{act}$ of the conditional input as follows:\n\\begin{equation}\n    X_{act}' = \\gamma_{\\scriptscriptstyle X} \\cdot X_{act} +  \\mu_{\\scriptscriptstyle X}\n\\end{equation}\nThe intuition is that some features cannot be correctly matched if the conditional input contains some parts that do not exist in the exemplar. Thus before injecting the aligned style feature into the generation process, the unmatched feature of conditional input can be effectively corrected according to the accurate semantic information of the conditional input.\nThen two sets of modulation parameters $\\gamma$ and $\\mu$ are further generated from the modulated conditional input $X_{act}'$.\n\nA positional normalization \\cite{li2019positional} with variance $\\gamma_p$ and mean $\\mu_p$ is applied to the activation of the translation network $L_{act}$ to preserve the structure information synthesized in prior layers, followed by a denormalization with $\\gamma$ and $\\mu$ as follows:\n\\begin{equation}\n    L_{act}' = \\gamma  \\frac{L_{act} - \\mu_p}{\\gamma_p} + \\mu\n\\end{equation}\n\n\n\n\\renewcommand\\arraystretch{1.0}\n\\begin{table*}[t]\n\\small \n\\caption{\nComparing UNITE with state-of-the-art image translation methods: The comparisons were performed over four public datasets with 3 widely used evaluation metrics FID, SWD and LPIPS.\n}\n\\renewcommand\\tabcolsep{3.75pt}\n\\centering \n\\begin{tabular}{l||ccc||ccc||ccc||ccc} \\hline\n\\multirow{2}{*}{\\textbf{Methods}}\n& \n\\multicolumn{3}{c||}{\\textbf{ADE20K}} & \n\\multicolumn{3}{c||}{\\textbf{COCO-Stuff}} &\n\\multicolumn{3}{c||}{\\textbf{DeepFashion}} &\n\\multicolumn{3}{c}{\\textbf{CelebA-HQ}}\n\\\\\n\\cline{2-13}\n& FID $\\downarrow$ & SWD $\\downarrow$ & LPIPS $\\uparrow$ & FID $\\downarrow$ & SWD  $\\downarrow$ & LPIPS  $\\uparrow$ & FID  $\\downarrow$ & SWD  $\\downarrow$ & LPIPS  $\\uparrow$ & FID  $\\downarrow$ & SWD  $\\downarrow$ & LPIPS  $\\uparrow$  \\\\\\hline\n\n\\textbf{Pix2pixHD\\cite{wang2018pix2pixhd}} & 81.80 & 35.70 & N\/A    & 121.2 & 44.82 & N\/A    & 25.20  & 16.40  & N\/A       & 42.70 & 33.30 & N\/A     \\\\\n\n\\textbf{Pix2pixSC\\cite{wang2019pix2pixsc}} & 56.23 & 24.52 & 0.378    &  77.63  & 26.34 &  0.307    & 28.49  & 21.13  & 0.172    & 49.39 & 33.20 & 0.193      \\\\\n\n\\textbf{StarGAN v2\\cite{choi2020starganv2}} & 98.72 & 65.47 & 0.451    &  153.2  & 61.87 &  0.394      & 43.29  & 30.87  & \\textbf{0.296}     & 48.63 & 41.96 & 0.214    \\\\\n\n\\textbf{SPADE\\cite{park2019spade}} & 33.90 & 19.70 & 0.344     & 49.27 & 19.78 & 0.254   & 36.20  & 27.80  & 0.231  & 31.50 & 26.90 & 0.187       \\\\\n\n\\textbf{SelectionGAN\\cite{tang2019selectiongan}}     &  35.10 & 21.82 & 0.382     & 52.41 & 20.32 & 0.277       & 38.31  & 28.21  & 0.223     & 34.67 & 27.34 &  0.191   \\\\\n\n\\textbf{SMIS\\cite{zhu2020smis}} & 42.17 &  22.67 &  0.416     & 58.21 &22.65 & 0.311      & 22.23  & 23.73  & 0.240       &  23.71  &  22.23  &  0.201   \\\\\n\n\\textbf{SEAN\\cite{zhu2020sean}}     & 24.84 &  10.42  & 0.499       & 37.74 & 16.31 & 0.355      & 16.28  &  17.52  & 0.251       &  18.88    & 19.94 & 0.203      \\\\\n\n\\textbf{CoCosNet\\cite{zhang2020cocosnet}}     & 26.40 & 10.50 & 0.560     & 35.23 & 14.54  &  0.391     & 14.40  & 17.20 & 0.272        & 14.30 & 15.30 & 0.208      \\\\\n\n\\hline\n\\textbf{UNITE}\n& \\textbf{25.15} & \\textbf{10.13} & \\textbf{0.571}\n& \\textbf{33.65} & \\textbf{12.18} & \\textbf{0.401}\n& \\textbf{13.08} & \\textbf{16.65} & 0.278\n& \\textbf{13.15} & \\textbf{14.91} & \\textbf{0.213}\n  \\\\\\hline\n\\end{tabular}\n\\label{tab_com}\n\\end{table*}\n\n\n\n\n\n\n\n\\renewcommand\\arraystretch{1.25}\n\\begin{table}[t]\n\\footnotesize\n\\caption{\nComparing UNITE with state-of-the-art image translation methods over evaluation metrics semantic consistency and style consistency (on dataset ADE20k \\cite{zhou2017ade20k}). \n}\n\\renewcommand\\tabcolsep{2pt}\n\\centering \n\\begin{tabular}{l|p{1.3cm}<{\\centering}p{1.3cm}<{\\centering}|p{1.3cm}<{\\centering}p{1.3cm}<{\\centering}} \\hline\n\\multirow{2}{*}{\\textbf{Methods}} & \n\\multicolumn{2}{c|}{\\textbf{Semantic Consistency}} & \n\\multicolumn{2}{c}{\\textbf{Style Consistency}}  \n\\\\\n\\cline{2-5}\n & ${\\rm VGG_{42}}$ $\\uparrow$ & ${\\rm VGG_{52}}$ $\\uparrow$ & ${\\rm VGG_{M}}$  $\\uparrow$  & ${\\rm VGG_{V}}$ $\\uparrow$   \\\\\\hline\n\n\\textbf{Pix2PixSC \\cite{wang2019pix2pixsc}}  &  0.840    & 0.751 &  0.941   &  0.932    \\\\\n\n\\textbf{SPADE \\cite{park2019spade}}  &  0.861   & 0.772   & 0.934 &  0.884      \\\\\n\n\\textbf{StarGAN v2 \\cite{choi2020starganv2}}  &  0.741  & 0.718   & 0.919 &  0.907      \\\\\n\n\\textbf{SelectionGAN \\cite{tang2019selectiongan}}  &  0.843    & 0.785 &  0.951 &  0.912     \\\\\n\n\\textbf{SMIS \\cite{zhu2020smis}}  & 0.862     & 0.787 &  0.951  &  0.933     \\\\\n\n\\textbf{SEAN \\cite{zhu2020sean}}  &  0.868     & 0.791 &  0.962 &  0.942      \\\\ \n\n\\textbf{CoCosNet \\cite{zhang2020cocosnet}} & 0.878 & 0.790 &  0.986 &  0.965      \\\\\n\n\\hline\n\\textbf{UNITE}  & \\textbf{0.883} & \\textbf{0.795} & \\textbf{0.990} & \\textbf{0.969}     \\\\\\hline\n\\end{tabular}\n\\label{tab_consistency}\n\\end{table}\n\n\n\n\n\n\n\n\n\\subsection{Loss Functions}\n\nThe feature transport network and translation network are trained jointly, and will drive each other to achieve better translation.\nFor clarity purpose, we denote the conditional input and exemplar as $X$ and $Z$, the ground truth as $X'$, the generated image as $Y$,\nthe feature extractor network for conditional input and exemplar as $F_{X}$ and $F_{Z}$, the translation network as $G$, the discriminator as $D$.\n\n\\textbf{Feature Transport Network.}\nFirst, the transported features should be cycle consistent, i.e. the original features should be able to be recovered from the transported features. We thus employ a cycle-consistency loss as follows:\n\\begin{equation}\n    \\mathcal{L}_{cyc} = || T^{\\top} \\cdot T\\cdot Z - Z ||_{1}\n\\end{equation}\nwhere $T$ is the transport plan.\nAs the two feature extractor networks $F_{X}$ and $F_{Z}$ aim to extract semantic information, the extracted features from the conditional input $X$ and the corresponding ground truth $X'$ should be consistent. A feature consistency loss can thus be defined as follows:\n\\begin{equation}\n    \\mathcal{L}_{cst}= || F_{X}(X) - F_{Z}(X') ||_{1}\n\\end{equation}\n\n\n\\textbf{Translation Network.}\nSeveral losses are employed in the translation network to drive the generation of high-fidelity images.\nAs the semantic of the generated image should be consistent with the conditional input $X$ or the ground truth $X'$, we employ a perceptual loss $\\mathcal{L}_{perc}$ \\cite{johnson2016perceptual} to penalize the semantic discrepancy as below:\n\\begin{equation}\n    \\mathcal{L}_{perc} = || \\phi_{l}(Y) - \\phi(X') ||_{1}\n\\end{equation}\nwhere $\\phi_{l}$ represent the activation of layer $l$ in pre-trained VGG-19 \\cite{simonyan2014vgg} model.\nTo ensure the consistency of statistics between the generated image $Y$ and the exemplar $Z$, a contextual loss in \\cite{mechrez2018contextual} is adopted as follows:\n\\begin{equation}\n    \\mathcal{L}_{cxt} = - \\log( \\sum_{i} \\mathop{\\max}\\limits_{j} CX_{ij} ( \\phi_{l}^{i} (Z), \\phi_{l}^{j} (Y) ) ) \n\\end{equation}\nwhere $i$ and $j$ are the indexes of the feature map in layer $\\phi_{l}$.\nBesides, a pseudo pairs loss $\\mathcal{L}_{pse}$ as described in \\cite{zhang2020cocosnet} is included in training.\n\nThe discriminator adopts the same architecture with Patch-GAN  \\cite{isola2017pix2pix}. With the adversarial loss $\\mathcal{L}_{adv}$, the model can be optimized with the following objective:\n\\begin{equation}\n\\begin{split}\n    \\mathcal{L} = & \\mathop{\\min}\\limits_{F_{X},F_{Z},G} \\mathop{\\max}\\limits_{D} (\\lambda_1 \\mathcal{L}_{cyc} + \\lambda_2 \\mathcal{L}_{cst} + \\lambda_3 \\mathcal{L}_{perc} \\\\\n    &  + \\lambda_4 \\mathcal{L}_{cxt} + \\lambda_5  \\mathcal{L}_{pse} + \\lambda_6 \\mathcal{L}_{adv}) \\\\\n\\end{split}\n\\end{equation}\nwhere the weights $\\lambda$ balance the losses in objective.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Experiments}\n\\subsection{Experimental Settings}\n\n\\textbf{Datasets:}\nWe experiment over multiple public datasets that handle different conditional image translation tasks.\n\n\\noindent\n$\\bullet$ ADE20k \\cite{zhou2017ade20k} consists of 20k training images and each image is associated with a 150-class segmentation mask. This is a challenging dataset to most existing methods due to its rich data diversity. We conduct image generation by using its semantic segmentation as conditional inputs.\n\n\\noindent\n$\\bullet$ COCO-Stuff \\cite{caesar2018cocostuff} augments COCO \\cite{lin2014coco} with pixel-level stuff annotations including 80 thing classes and 91 stuff classes.\nWe use its layout as conditional inputs. Following \\cite{li2019positional}, objects covering less than 2\\% of the image are ignored and images with 3 to 8 objects are used in experiments.\n\n\\noindent\n$\\bullet$ CelebA-HQ \\cite{liu2015celebahq} consists of 30,000 high quality face images. We use its edge maps as conditional inputs.\nThe face landmarks are connected as face edges, and the edges in the background are detected by Canny edge detector.\n\n\\noindent\n$\\bullet$ Deepfashion \\cite{liu2016deepfashion} contains 52,712 person images with various appearances and poses. 29,000 images are selected as training set and the rest as validation set.\nWe use its key points as conditional inputs in experiments.\n\n\n\n\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=1.0\\linewidth]{im_com.pdf}\n\\caption{\nQualitative illustration of UNITE and state-of-the-art image translation methods over four different types of conditional inputs.\n}\n\\label{im_com}\n\\end{figure*}\n\n\n\n\n\n\\textbf{Evaluation Metrics:} We adopt several evaluation metrics to assess image translation performance.\n\\textit{Fr{\\'e}chet Inception Score (FID)} \\cite{fid} is adopted to measures the distance between the distribution of generated images and real images.\nWe also adopt \\textit{Sliced Wasserstein distance (SWD)} \\cite{swd} to measure statistical distance of low level patch distributions.\nBesides, \\textit{Learned Perceptual Image Patch Similarity (LPIPS)} \\cite{zhang2018lpips} is adopted to evaluate the diversity of the translated images with different exemplars, which computes the perceptual distance between image features extracted by AlexNet \\cite{alexnet}. \n\nWe also adopt and extend the metrics in \\cite{zhang2020cocosnet} to evaluate semantic consistency and style consistency. Specifically, a pre-trained VGG model \\cite{simonyan2014vgg} is used to extract high-level features ($relu4\\_2$ and $relu5\\_2$) of the ground truth and generated images that capture semantic features. The semantic consistency ($\\rm VGG_{42}$ and $\\rm VGG_{52}$) is defined by the distance between the extracted high-level features as computed by cosine similarity. Similarly, the pre-trained VGG model is applied to extract the low-level feature ($relu1\\_2$) of the generated images and exemplars that capture style features. The style consistency ($\\rm VGG_{M}$ and $\\rm VGG_{V}$) is defined by the distance of channel-wise mean and standard deviation as computed by cosine similarity. \n\nBesides, we conduct user study (US) to evaluate the images generated under different ablation settings. 100 pairs of generated images were shown to 20 users who select the image with the best visual quality.\n\n\n\\textbf{Implementation Details:}\nThe learning rate for translation network and discriminator is 1$e$-4 and 4e-4 (the feature transport network is optimized jointly with the translation network).\nWe use Adam solver with $\\beta_{1}=0$ and $\\beta_{2}=0.999$. The experiments are conducted on 4 32GB Tesla V100 GPUs with synchronized BatchNorm applied.\nThe feature size for optimal transport is $64 \\times 64$ with feature dimension of 128. The image size is set at $256 \\times 256$ for generation tasks using semantic map, edge map, keypoints, and $128\\times 128$ for generation task using layout which is consistent with \\cite{sun2019lostgan}.\n\n\n\n\\subsection{Experimental Results} \nWe compare UNITE with several state-of-the-art translation methods including \n1) Pix2pixHD \\cite{wang2018pix2pixhd}, a supervised image translation method ; \n2) Pix2PixSC \\cite{wang2019pix2pixsc}, an example-guided image synthesis model based on Pix2PixHD \\cite{wang2018pix2pixhd};\n3) StarGAN v2\\cite{choi2020starganv2}, a model for multi-modal translation with support for style encoding from reference images;\n4) SPADE \\cite{park2019spade}, a supervised translation method that supports style injection from an exemplar image;\n5) SelectionGAN \\cite{tang2019selectiongan}, a guided translation framework with cascaded semantic guidance;\n6) SMIS \\cite{zhu2020smis}, a network for semantically multi-modal synthesis task with all group convolutions;\n7) SEAN \\cite{zhu2020sean}, a conditional translation network that can control the style of each individual semantic region;\n8) CoCosNet \\cite{zhang2020cocosnet}, a leading exemplar-based translation framework that works by building cross-domain correspondences.\n\n\\textbf{Quantitative Results:}\nIn quantitative experiments, all methods synthesize images with the same exemplars except Pix2PixHD \\cite{wang2018pix2pixhd} which synthesizes images directly without exemplar guidance (it doesn't support style injection from exemplars). As shown in Table \\ref{tab_com}, we compare UNITE with state-of-the-art methods in image quality as measured by FID and SWD and image diversity as measured by LPIPS. We can observe that UNITE outperforms all compared methods over all metrics and tasks consistently. Specifically, UNITE achieves the best FID and SWD which is largely attributed to our designed unbalance optimal transport in accurate feature alignments and semantic-activation normalization in effective style feature injection.\nBesides generation quality, UNITE achieves the best generation diversity in LPIPS, thanks to the multi-stage feature transport that aligns features in different scales to faithfully preserve rich textures in exemplars.\n\nExcept for high quality and rich diversity, the generated image should preserve consistent semantics with conditional inputs and present consistent styles with exemplars.\nTable \\ref{tab_consistency} shows the semantic consistency and style consistency evaluated by the metrics described in \\textit{Evaluation Metrics}. With our UOT for accurate semantic feature matching and SEACE for effective style injection, UNITE achieves the best semantic consistency and style consistency.\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=1.0\\linewidth]{im_diverse.pdf}\n\\caption{\nQualitative illustration of our proposed UNITE with different types of conditional inputs and exemplars.\n}\n\\label{im_diverse}\n\\end{figure*}\n\n\n\\textbf{Qualitative Evaluation:}\nWe compare images as generated by different translation methods as shown in Fig. \\ref{im_com}. It can be seen that UNITE achieves faithful styles to the exemplars. SPADE \\cite{park2019spade}, SMIS \\cite{zhu2020smis} and StarGAN v2 \\cite{choi2020starganv2} adopt single latent code to encode image styles, which tend to capture global exemplar styles but miss local details. Although SEAN \\cite{zhu2020sean} employs multiple latent codes for feature injection, it still struggles to preserve faithful and detailed exemplar style. CoCosNet \\cite{zhang2020cocosnet} can preserve certain details, but it adopts cosine similarity to align features which often lead to many-to-one matching and missing details as demonstrated by blurry textures in CoCosNet synthesized images. \nOur UNITE instead adopts UOT to achieve accurate feature alignment and a multi-stage transport to preserve the detailed texture.\nBesides, most existing methods tend to produce various artefacts as they do not build long-range dependency between style features. Our UNITE designs SEACE to explicitly build long-range dependency between style features which leads to superior synthesis fidelity as illustrated.\n\nThe proposed UNITE also demonstrates superior diversity in image translation as illustrated in Fig. \\ref{im_diverse}. We can observe that UNITE is capable of synthesizing various realistic images with faithful style to the given exemplars.\n\n\n\\renewcommand\\arraystretch{1.2}\n\\begin{table}[t]\n\\small\n\\caption\n{\nAblation studies of our UNITE designs over CelebA-HQ \\cite{liu2015celebahq}: The baseline is SPADE that uses spatial denormalization \\cite{park2019spade}. COS, OT and UOT mean to include cosine similarity, classical optimal transport and unbalanced optimal transport in feature alignment. SEACE means to use the proposed semantic-activation denormalization to inject style features. MS denotes the multi-stage feature transportation. Model in the last row is the standard UNITE. US denotes the user study metric.\n}\n\\renewcommand\\tabcolsep{4.5pt}\n\\centering \n\\begin{tabular}{p{2.5cm}|\np{1cm}<{\\centering} \np{1cm}<{\\centering}\np{1.2cm}<{\\centering}\np{1cm}<{\\centering}} \\hline\n\\textbf{Models} & \\textbf{FID}  $\\downarrow$ & \\textbf{SWD}  $\\downarrow$ & \\textbf{LPIPS}  $\\uparrow$ & \\textbf{US}  $\\uparrow$\n\\\\\n\\cline{2-3}\n\\hline\\hline\n\\textbf{SPADE}      & 31.50  &  26.90  &  0.187   & 0\\% \\\\\n\\textbf{SPADE+COS}     & 16.32  & 16.10  &  0.201  & 13\\% \\\\\n\\textbf{SPADE+OT}      & 17.87  & 17.24  &  0.202  & 10\\% \\\\\n\\textbf{SPADE+UOT}     & 14.02  & 15.41  &  0.206  & 22 \\% \\\\\n\\textbf{SEACE+UOT}      & 13.46  & 15.12  & 0.208 &  25 \\% \\\\\n\\hline\n\\textbf{SEACE+UOT+MS}   & \\textbf{13.15} & \\textbf{14.91} &  \\textbf{0.213} & \\textbf{30} \\% \\\\\n\\hline\n\\end{tabular}\n\\label{tab_ablation}\n\\end{table}\n\n\n\n\n\n\n\n\\subsection{Ablation Study}\n\nWe conduct extensive ablation studies over CelebA-HQ \\cite{liu2015celebahq} to validate the effectiveness of our designs. As Table \\ref{tab_ablation} shows, SPADE \\cite{park2019spade} is the baseline which achieves image translation directly without feature alignment. When cosine similarity is included to align features, the translation is improved significantly. While replacing cosine similarity with classical optimal transport, the performance does is clearly aggravated as classical optimal transport introduce many false matchings. However, the translation performance improves clearly when our UOT is included, largely attributed to that UOT adaptively learns the feature masses and suppresses false and many-to-one matching effectively. When replacing SPADE with our proposed SEACE, the FID score is improved clearly by 0.73. Additionally, the SWD and LPIP scores are improved clearly when our proposed multi-stage feature transport is included. We also performed qualitative ablation studies on DeepFashion \\cite{liu2016deepfashion} by removing each of our designs from the complete UNITE model.\nAs Fig. \\ref{im_ablation} shows, our designed UOT, MS and SEACE all contribute to the high-fidelity realistic image translation clearly.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=1.0\\linewidth]{im_ablation.pdf}\n\\caption{\nThe ablation study of each different design in UNITE as evaluated over a sample from dataset DeepFashion \\cite{liu2016deepfashion}. Specially, `w\/o OT' denotes image translation without feature alignment, `w\/o UOT' denotes using classical OT (without learnt unbalanced weights) to align features.\n}\n\\label{im_ablation}\n\\end{figure}\n\n\n\n\\section{Conclusions}\nThis paper presents UNITE, an exemplar-based image translation framework that adopts unbalanced optimal transport to align the feature between conditional input and exemplar, which effectively transport the style of the exemplar to the conditional input. A multi-stage feature transport manner is applied to preserved more detailed deep features. \nTo inject aligned the style feature into the generation process, we propose a novel semantic-activation normalization which builds the semantic coherence between style features with the same semantic in style injection.\nQuantitative and qualitative experiments show that UNITE is capable of generating high-fidelity images with consistent semantic with the conditional input and faithful style to the exemplar.\n\n\n\n{\\small\n\\bibliographystyle{ieee_fullname}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\nThe implementation of photo-association techniques in ultracold atomic traps \n\\cite{ThoHodWie05} opened a new route for quantitative  \ndetermination of universal properties in few-body systems \\cite{BraHam06}.\nIn these experiments, radio-frequency (rf) induced trimer formation leads \nto enhanced atom loss rates from the traps.\nScanning the rf field frequency, the change in the measured atom loss rate \nindicates various molecular thresholds and structures.\n\nRecently, trimer formation through rf association was discovered in both \nfermionic $^6$Li \\cite{LomOttSer10, NakHorMuk11}\nand bosonic $^7$Li \\cite{MacShoGro12} systems.\nThe three body case attracts special attention as the simplest non-trivial \nsystem. Moreover, in the 70's Efimov predicted that in the limit of a \nresonant 2-body interaction, the system reveals universal properties \n\\cite{Efi70}.\nA peculiar prediction is the existence of a series of giant three body \nmolecules, known as Efimov trimers, that was verified experimentally \nfew years ago \\cite{ZacDeiDer09,PolDriHul09,GroShoKok09,GroShoMac11}.\n\nIn a previous work \\cite{BazLivBar12}, \nwe have presented the multipole analysis of an rf association process \nbinding a molecule of $N$ identical bosons.\nWe have shown that the spin-flip and frozen-spin processes differ by \ntheir operator structure and by the de-excitation modes that contribute \nto the photoassociation rate. \nPrevious analysis of these rf experiments \n\\cite{ChiJul05, BerJesMol06, HanKohBur07, KleHenTop08,TscRit11},\nwhich relied on the Franck-Condon factor, \nis appropriate for describing spin-flip reactions. \nFor frozen-spin reactions we have applied our results to study \nthe dimer formation \\cite{BazLivBar12}, and to study numerically \nthe quadrupole response of a bound bosonic trimer  \n\\cite{LivBazBar12,BazLivBar13}. \n\nHere, we study trimer photoassociation using the hyperspherical adiabatic approximation. \nA zero-range potential is used to derive analytic results \nfor the transition rates at the unitary point. \nSimilarly to the dimer case \\cite{BazLivBar12}, the $s$-mode and the $d$-mode \nare found to be the leading order contributions.  A new fingerprint of\nEfimov physics is studied, which is a log-periodic oscillation in the response.\n\n\\section{Multipole expansion}\n\nThe molecular photoassosiaction rate is given by Fermi's golden rule,\n\\begin{equation} \\label{golden_rule}\nr_{i\\rightarrow f}=\\frac{2\\pi}{\\hbar} \\bar{\\sum_i} \\sum_f\n|\\langle f, \\bs k \\zeta | \\hat{H}_I | i\\rangle|^2 \n\\delta(E_i-E_f-\\hbar\\omega_k) \\;,\n\\end{equation} \nwhere three particles in an initial continuum state with energy $E_i$ \nform a bound state with energy $E_f$\nby emitting a photon with momentum $\\bs k$, polarization $\\zeta$ and energy \n$\\hbar \\omega_k=\\hbar c k$.\n$\\bar\\sum_i$ is an average on the appropriate initial continuum states \nand $\\sum_f$ is a sum on the final bound states.\nThe coupling between the neutral atoms and the radiation field takes the form \n$\\hat{H}_I=-e \\int d\\bs x \\; \\bs \\mu(\\bs x) \\cdot \\nabla \\times \\bs A(\\bs x)$,\nwhere $\\bs A$ is the electromagnetic (EM) photon field, and $\\mu(\\bs x)$ is\nthe magnetization current. Here we consider \nonly the one-body current\n$\\mu(\\bs x)=\\mu_0 \\sum_j\\bs S_j \\delta(\\bs x-\\bs r_j)$,\nwhere $\\mu_0$ is the magnetic moment of a single particle, and \n$\\bs S_j$ and $\\bs r_j$ are the spin and position of particle $j$.\n\nWe assume that the initial and final atomic wave functions can be written as \na product of symmetric spin $|\\chi\\rangle$ and configuration space $|\\psi\\rangle$ terms,\nand that the photon does not induce change in the spin structure of the system. \nIn this case the transition matrix element can be written as\n\\begin{equation} \\label{frozen-spin}\n\\langle f,\\bs k \\zeta | \\hat H_I | i\\rangle = -i \\mu_0 \\sqrt{\\frac{\\hbar c^2}{2\n    {\\cal V}\\omega_k}} k \\langle S_{0}  \\rangle\n\\langle \\psi^f_{L' M'} | \\sum_{j=1}^3 e^{i\\bs k \\cdot \\bs r_j} | \\psi^i_{L M} \\rangle \\;,\n\\end{equation}\nwhere \n$\\langle S_{0} \\rangle = \\frac{1}{3} \\sum_j \\langle \\chi^i_{M_F} |S_{j,0}| \\chi^i_{M_F}\\rangle $ \nis the average single particle magnetic moment, which plays the role \nof an effective charge. We normalize the EM field in a box of volume ${\\cal V}$.\n\nThe photon wavelength of rf radiation is much larger than the typical \ndimension of the system $R$, therefore $kR\\ll1$ and the lowest order \nin $kR$ dominates the interaction. \nTherefore the exponent can be expanded to get \n\\begin{equation} \\label{expand}\n\\sum_{j=1}^3 e^{i\\bs k \\cdot \\bs r_j} \\approx  3\n+i\\sum_{j=1}^3 \\bs k \\cdot \\bs r_j\n-\\frac{1}{6} \\sum_{j=1}^3 k^2 r_j^2\n-\\frac{4\\pi}{15} \\sum_{j=1}^3 k^2 r_j^2 \\sum_m Y_{2 -m}(\\hat{k}) Y_{2 m}(\\hat{r}_j), \n\\end{equation}\nwhere $Y_{lm}$ are the spherical harmonics.\nEach order in this expansion has clear physical meaning.\nThe zeroth order operator stands for elastic interaction.\nThe first order operator is the dipole, which for identical particles is\nproportional to the center of mass and hence does not affect the relative \nmotion of the atoms. \nAt second order two operators appear: the $r^2$ operator, corresponding \nto $s$-mode reaction, and the quadrupole terms, corresponding \nto $d$-mode reaction.\nFor identical particles this is the leading term in low energy frozen spin reactions, \nand the transition probability scales as $k^5$.   \nSumming over the initial and final magnetic numbers $M$, $M'$, \nthe transition matrix element reads \n\\begin{equation} \\label{t-frozen}\n\\sum_{M,M'}\n| \\langle f,\\bs k \\zeta | \\hat H_I | i \\rangle |^2  = \n\\frac{4\\pi \\hbar c k^5 \\mu_0^2}{2\\Omega} \\ev{S_0}\n\\left( \\frac{1}{6^2} \n| \\langle \\psi^f_{L'} \\Vert \\sum_{j=1}^3 r_j^2 Y_0\\Vert \\psi^i_{L} \\rangle |^2\n+\\frac{1}{15^2} \n| \\langle \\psi^f_{L'} \\Vert \\sum_{j=1}^3 r_j^2 Y_2(\\hat r_j)  \\Vert \\psi^i_{L} \\rangle |^2\n\\right).\n\\end{equation}\n\n\\section{The Three Body Problem}\n\nThe dynamics of a quantum 3 particle system is governed \nby the Schroedinger equation \n\\begin{equation} \\label{Hamiltonian}\n\\left(T+W\\right)\\psi=E\\psi\n\\end{equation}\nwhere $T$ is the center of mass kinetic energy operator and $W$ is the \npotential. In this study we shall limit our attention to\nshort range 2-body forces, thus $W=\\sum_{i<j}V(|\\bs{r}_i-\\bs{r}_j|)$.\nTo eliminate the center of mass motion, we define the Jacobi coordinates,\n$\\bs x=\\sqrt \\frac {1}{2} (\\bs r_2-\\bs r_1)$, and $\\bs y = \n\\sqrt \\frac {2}{3}\\left(-\\bs r_3 + \\frac{\\bs r_1+\\bs r_2}{2}\\right)$,\nwhich we transform into the hyperspherical coordinates\n$\\rho^2=x^2+y^2$, and $\\Omega=(\\alpha,\\hat x, \\hat y)$, where \n$\\tan \\alpha=x\/y$.\n\nIn the limit of infinite scattering length, $|a|\\rightarrow\\infty$,\nthe spatial wave function can be written as \\cite{Mac68} \n$ \\psi(\\rho,\\Omega)=\\rho^{-5\/2}\\mathcal R(\\rho)\\Phi(\\Omega) $.\nThe hyperspherical functions $\\Phi(\\Omega)$ and the corresponding \neigenvalue $\\nu^2$, are the solutions of the hyperangular equation,\n\\begin{equation} \\label{hae}\n \\left ( \\hat K ^2 + \\frac{2m\\rho^2}{\\hbar^2}\\sum_i \nV(\\sqrt 2 \\rho \\sin \\alpha_i) +4 \\right)\\Phi=\n \\nu^2 \\Phi, \\end{equation}\nwhere\n$\n\\hat K^2=-\\frac{1}{\\sin 2\\alpha}\\frac{\\partial^2}{\\partial \\alpha^2}\\sin2\\alpha\n+\\frac{\\hat l^2_x}{\\sin^2\\alpha}+\\frac{\\hat l^2_y}{\\cos^2\\alpha}-4.\n$\nFor low energy physics, when the extension of the wave function is\nmuch larger than the range of the potential, one can utilize the zero range\napproximation. In this approximation the lateral extension of the potential is\nneglected all together, and the action of the potential is represented through\nthe appropriate boundary conditions.\nFor a two-particle system the low energy interaction is dominated by the\n$s$-wave scattering length $a$ and the wave function fulfills the boundary\ncondition $[u'\/u]_{r=0}=-1\/a$. The corresponding 3-body condition is\n\\begin{equation} \\label{bc}\n\\left[\\frac{1}{2\\alpha \\Phi}\\frac{\\partial}{\\partial \\alpha}2 \\alpha \\Phi\n\\right]_{\\alpha=0}=-\\sqrt 2 \\frac{\\rho}{a}\\;.\n\\end{equation}\n\nPlugging the solution of Eq. (\\ref{hae}) into Eq. (\\ref{bc}), one gets \ntranscendental equations for $\\nu$.\nFor $L=0$ the resulting equation is \\cite{FedJen01}\n\\begin{equation}\\label{l0}\n  \\nu\\cos(\\nu\\pi\/2)-\\frac{8}{\\sqrt 3}\\sin (\\nu\\pi\/6)=\\frac{\\sqrt 2 \\rho}{a}\n  \\sin(\\nu\\pi\/2).\n\\end{equation}\nFor $|a|=\\infty$ the solution with lowest $\\nu^2$ is $\\nu_0\\approx\n1.00624i$, corresponding to the Efimov trimer. \n\nFor $L=2$ the corresponding equation is,\n\\begin{equation} \\label{l2}\n  \\nu(4-\\nu^2)\\cos(\\nu\\pi\/2)+\n  24 \\nu \\cos(\\nu \\pi\/6) +\n  \\frac{8}{\\sqrt3} (\\nu^2-10) \\sin (\\nu \\pi\/6)=\n  -\\frac{\\rho}{a}(\\nu^2-1)\\sin(\\nu\\pi\/2)\n\\end{equation}\nFor $|a|=\\infty$ the lowest non-trivial solution is\n$\\nu_2\\approx2.82334$. \n \n\nIn the limit of $|a|\\rightarrow\\infty$, $\\nu_L(\\rho)=\\nu_L$ and \nthe hyperradial equation for $\\mathcal R(\\rho)$ is similar to the Bessel equation, \n\\begin{equation}\\label{Bessel} \n-\\mathcal R''(\\rho)+\\frac{\\nu_L^2-1\/4}{\\rho^2} \\mathcal R(\\rho)\n=\\epsilon \\mathcal R(\\rho).\n\\end{equation}\nwhere $\\epsilon=2mE\/\\hbar^2$.\n\nWe seek the solution for the bound ($\\epsilon<0$) and continuum ($\\epsilon>0$) cases:\n\nI. A \\emph{bound state} exists only for $L=0$, and $\\nu_0\\approx1.00624i$.\nIn this case the relevant solution is\n$\\sqrt \\rho K_{\\nu_0}(\\kappa \\rho)$, where $\\kappa=\\sqrt {-\\epsilon}$. \nAt the origin, this solution behaves like \n$\\sin (\\nu \\ln (\\kappa \\rho\/2)+0.301)$, \ntherefore regularization is needed to avoid collapse, \ne.g. setting $\\mathcal R(\\rho \\le \\rho_0)=0$ for some finite $\\rho_0$.\nThe result is the discrete Efimov spectrum,\n$\\epsilon_n\/\\epsilon_0=e^{-2\\pi n\/|\\nu_0|}\\approx 515^{-n}.$\nThe normalized wave functions are \n\\begin{equation}\n\\mathcal R^{(n)}_B(\\rho)=\\sqrt{2 \\sin \\nu_0 \\pi\/\\nu_0 \\pi} \\kappa_n \\sqrt \\rho K_{\\nu_0}(\\kappa_n \\rho)\n\\end{equation}.\n\nII. For \\emph{continuum state}, the solution is \n\\begin{equation}\n\\mathcal R_L(\\rho)=\\sqrt{\\frac{q \\rho N_s}{2 R}} \\left[ \\sin \\delta \\mathrm{Re}[J_{\\nu_L}(q \\rho)] +\n\\cos \\delta \\mathrm{Re}[Y_{\\nu_L}(q \\rho)] \\right]\n\\end{equation}\nwhere $q=\\sqrt {\\epsilon}$, $N_s=1\/2(\\pi)$ for imaginary (real) $\\nu$, \nand we assume normalization in a sphere of radius $R$.\nThe phase shift $\\delta$ is to be found from the boundary condition, \n$\\mathcal R_L(\\rho_0)=0$.\n\n\\section{The Transition Matrix Elements}\n\nNow that we have obtained the initial and final wave functions\nwe are in position to evaluate the transition matrix elements, \nEq. (\\ref{t-frozen}), \n$I_\\lambda=\\langle \\psi^f_{L'} \\Vert \\sum_{j=1}^N r_j^2 Y_\\lambda \\Vert \\psi^i_L \\rangle$,\nfor $\\lambda=s (d)$ corresponding to the $r^2$ (quadrupole) operator, respectively.\n\n\\emph{The $r^2$ matrix element} ---\nThe $r^2$ operator connects the $L=0$ bound state to an $L=0$ scattering \nstate. We note that $\\sum_j r_j^2 = \\rho^2 + 3R_{CM}^2$\nwhere $R_{CM}$ is the center of mass radius. As we neglect the center of mass \nexcitations, the matrix element is reduced into the hyperradial integral \n$ I_s(\\kappa,q)=\\int_0^\\infty d\\rho \\mathcal R_B^*(\\rho) \\rho^2 \\mathcal R_s(\\rho)$.\nEvaluating this integral, \nthe resulting response function reveals log periodic oscillations \\cite{BazBar13b}.\nAt threshold, the matrix element gets a particularly simple form which can be well approximated by\n\\begin{equation}\nI_s(\\kappa,q) \\approx C \\kappa^{-3}\\sqrt{q}\\left(1+\\frac{B_3}{2}\\cos(2s_0\\ln\\frac{q}{\\kappa})\\right)\n\\end{equation}\nwhere $C$ is a constant that contains the normalization factors, and $B_3 \\approx\n8.475\\%$ is the normalized amplitude of the oscillations.  \nThe oscillations modulate the matrix element all the way to the high energy tail. \nIn Fig. \\ref{fig:Osc} we present the \nlog periodic oscillations of the $r^2$ matrix element.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[trim=0cm 0cm 0cm 0cm, clip=true, height=5.4cm]{TrOs.pdf}\n  \\includegraphics[trim=0cm 0cm 0cm 0cm, clip=true, height=5.4cm]{TaOs.pdf}\n  \\caption{The log periodic oscillation in the $r^2$ transition matrix element.\n\tLeft: near the threshold. Right: at the high frequency tail.}\n  \\label{fig:Osc} \n\\end{figure}\n\n\\emph{The Quadrupole matrix element} ---\nThe quadrupole operator $\\sum_j r_j^2 Y_2^M(\\hat r_j)$ connects the $L=0$ \nbound state with $L=2$ scattering states. \nIn this case the reduced matrix element takes the form\n\\begin{equation}\nI_d(\\kappa,q) =\n\\frac{3}{2}\\sqrt{5}\n\\int_0^\\infty d\\rho \\mathcal R_B^*(\\rho) \\rho^2 \\mathcal R_d(\\rho)\n\\int d\\Omega \\Phi_f^*(\\Omega) \\sum_j \\cos^2 \\alpha_j Y_{20}(\\hat y_j)\\Phi_i(\\Omega).\n\\end{equation}\nIn this case we find no log-periodic oscillations near threshold.\nSuch oscillations, however,  modulate \nthe high energy tail of the response function, \nattenuated by $q^{-4}$ and masked by the linear phase shift variation.\nThese high energy oscillations appear not only in these cases but in\nall partial waves \\cite{BazBar13b}. \n\nThe relative contribution of the $s,d$ modes to the trimer formation is \ndisplayed in  Fig. \\ref{fig:ME}, where the last term in parenthesis on \nthe rhs of Eq. (\\ref{t-frozen}) is presented normalized, along with the \n$s$ and $d$ components. \nSimilarly to the dimer formation case \\cite{BazLivBar12}, \nthe $s$-wave association is peaked around $q=\\kappa \/2$, \nwhile the $d$-wave association is peaked at $q=\\kappa$. \n\n\\begin{figure}\n\\centering\n\\includegraphics[height=5cm]{ME}\n\\caption{\\label{fig:ME}  \nThe normalized three-body transition matrix element, Eq. (\\ref{t-frozen}), \nas a function of the relative momentum $q\/\\kappa$. \nThe sum (black), $r^2$ (red, peaked at $q = \\kappa\/2$), and \nquadrupole (blue, peaked at $q = \\kappa$) terms are given for the unitary point,\n$|a|=\\infty$.}\n\\end{figure}\n\n\\section{Conclusion}\nWe have applied the multipole expansion to study\ntrimer photoassociation in ultracold atomic gases.\nThe two dominant modes, at order $k^5$, are studied and their relative \ncontribution is shown.\nLog periodic oscillations are shown in two cases: (i) for the leading s-wave\nmode, near the threshold, and (ii) for all partial waves at the high frequency \ntail.\n\n\\begin{acknowledgements}\nThis work was supported by the ISRAEL SCIENCE FOUNDATION (Grant No.~954\/09).\nWe would like to thank L. Khaykovich and N. Nevo-Dinur \nfor their useful comments and suggestions during the preparation \nof this work.\n\\end{acknowledgements}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n Pour nous, une $C$-relation est une relation ternaire satisfaisant les axiomes suivants:\n\\begin{enumerate}\n\\item $\\forall x,y,z \\;\\:  { C}(x,y,z)\\rightarrow { C}(x,z,y) $,\n\\item $\\forall x,y,z \\;\\: { C}(x,y,z) \\rightarrow \\neg { C}(y,x,z)$,\n\\item $\\forall x,y,z,w \\;\\:  { C}(x,y,z)\\rightarrow ({ C}(w,y,z)\\vee { C}(x,w,z)),$\n\n\\item $\\forall x,y \\;\\: x\\neq y \\rightarrow { C}(x,y,y)$.\n\\end{enumerate}\n\n\n\nUne $C$-relation interpr\u00e8te toujours un arbre (cf. \\cite{Adeleke1998}): pour des cha\u00eenes maximales $x,y$ et $z$,  ${ C}(x,y,z)$, exprime que $y$ et $z$  branchent au-dessus de l\u00e0 o\u00f9 $x$ et $y$ (ou $z$) branchent.\n\nOn rappelle qu'une\ndistance ultram\u00e9trique, est un triplet $(M,\\Gamma, d)$,  o\\`u $M$ est un ensemble, $\\Gamma$ est un ordre linaire ayant un minimum $0$, et $d:M^2\\to \\Gamma$, une application surjective, telle que, pour tout $x,y,z \\in M$, \n\\begin{enumerate}\n\t\\item $d(x,y)=0$ si et seulement si $x=y$,\n\t\\item $d(x,z)\\leq \\min\\{d(x,y),d(y,z)\\}$.\n\\end{enumerate}  \n\nUne distance ultram\\'etrique induit la $C$-relation \n$${ C}(x,y,z) \\Leftrightarrow d(x,y)=d(x,z)>d(y,z).$$\nDonc un groupe ab\u00e9lien ou un corps valu\u00e9 porte en particulier la $C$-relation canonique en inversant l'in\\'egalit\\'e ci-dessus:\n$${ C}(x,y,z) \\Leftrightarrow v(x-y)=v(x-z)<v(y-z).$$ \nUne valuation triviale, c'est-\u00e0-dire  prenant au plus deux valeurs, induit la\n$C$-relation triviale: ${ C}(x,y,z) \\Leftrightarrow x\\neq y=z$.\n\n\n\n\\begin{defn}On dit qu'une expansion $\\mathcal{U}=(U,{ C}, \\dots)$ d'une $C$-relation est $C$-minimale si, pour tout $\\mathcal{U}'=(U',{ C}, \\dots)$, \u00e9l\u00e9mentairement \u00e9quivalente \u00e0 $\\mathcal{U}$, tout sous-ensemble d\u00e9finissable (avec param\u00e8tres) de $U'$, est d\u00e9finissable sans quantificateur en n'utilisant que la relation ternaire $C$.\n\\end{defn}\n \n\\begin{remarque} En pr\u00e9sence d'une distance ultram\u00e9trique, la $C$-minimalit\u00e9 se traduit par le fait que tout sous-ensemble d\u00e9finissable  (de la structure elle-m\u00eame) est une combinaison bool\u00e9enne de boules \u00abouvertes\u00bb ou \u00abferm\u00e9es\u00bb.\n\\end{remarque}\n\n\n\n\n\n \n  \n   La $C$-minimalit\u00e9 appara\u00eet comme une g\u00e9n\u00e9ralisation commune de la minimalit\u00e9 forte et de l'$o$-minimalit\u00e9. Par exemple, une structure munie de la $C$-relation triviale est $C$-minimale si et seulement si elle est \nfortement minimale. Plus g\u00e9n\u00e9ralement ces trois notions sont \u00e9troitement li\u00e9es. Notons \u00e9galement que les th\u00e9ories $C$-minimales sont d\u00e9pendantes, i.e, elles ont la propri\u00e9t\u00e9  NIP, comme les th\u00e9ories fortement minimales, $o$-minimales et $P$-minimales (on pourra consulter \\cite{Haskell1997} pour les d\u00e9finitions des th\u00e9ories $o$-minimales, $C$-minimales et $P$-minimales, dont  le th\u00e9or\u00e8me 4.2.6 et la discussion qui le pr\u00e9c\u00e8de pour le r\u00e9sultat concernant NIP). Noter que si une structure est  $o$-minimale, alors il va de m\u00eame pour sa th\u00e9orie compl\u00e8te, i.e. toute structure qui lui est \u00e9l\u00e9mentairement \u00e9quivalente est  $o$-minimale. Cette derni\u00e8re propri\u00e9t\u00e9 est fausse pour les structures fortement minimales et par  voie de cons\u00e9quence pour les structures $C$-minimales.\n\nLe groupe additif de $\\mathbb{Q}_{ p}$, muni de sa valuation  $p$-adique,  est $C$-minimal mais le corps $\\mathbb{Q}_{p}$ ne l'est pas, par exemple\nl'ensemble des carr\u00e9s n'y est pas d\u00e9finissable sans quantificateur dans le langage des corps valu\u00e9s, donc  dans le langage des corps \u00e9quip\u00e9s\nde la $C$-relation non plus. De fait, un corps valu\u00e9 est $C$-minimal si et seulement s'il est alg\u00e9briquement clos (cf. \\cite{Haskell1994}).\n\nDans cette \\'etude, on reprend les notations et les d\\'efinitions (cf. Section 2 pour un rappel) figurant dans nos articles \\cite{Onay2017} et \\cite{Onay2018a}. Soit $K$ un corps et $\\vfi$ un endomorphisme de $K$. On pose $R:=K[t;\\vfi]$, $K$-alg\\`ebre engendr\\'e par $t$ selon la loi de commutativit\\'e \n$$at=ta^{\\vfi}$$ pour tout $a\\in K$. \nDans cette article, on  caract\u00e9rise les $R$-modules valu\u00e9s  $C$-minimaux.\n\n\n\n\nNos r\u00e9sultats principaux  se r\u00e9sument dans les trois th\u00e9or\u00e8mes \nci-dessous. \n\n\\begin{thm}\\label{thmfrtmin}\nUn $R$-module trivialement valu\u00e9 infini $M$ est $C$-minimal (i.e. fortement minimal dans ce cas, justifiant la notation $M$, \u00e0 la place de $(M,v)$) si et seulement si l'une des conditions ci-dessous est v\u00e9rifi\u00e9e:\n\\begin{enumerate}[$1.$]\n\\item $M$ est divisible et, pour tout $r \\in R\\setminus\\{0\\}$, $\\text{ann}_M(r)$ est fini\n\\item l'id\u00e9al annulateur de $M$, $\\{q \\in R \\;\\: | \\:\\: x.q=0, \\;\\: \\forall x \\in M\\}$, est engendr\u00e9 par un polyn\u00f4me irr\\'eductible $r$, $R\/rR$ est un corps et \n$M$ est un $(R\/rR)$-espace vectoriel. \n\\end{enumerate}\n\\end{thm}\n\n\n\\begin{thm}\\label{thmCmintrivi}\nSoit $(M,v)$ un module $K$-trivialement valu\u00e9 infini et non-trivialement valu\u00e9.  Alors\n$(M,v)$ est $C$-minimal si et seulement si $v(M)$ est $o$-minimal dans le langage $L'_V$, pour tout $r \\in R\\setminus\\{0\\}$, $\\text{ann}_M(r)$ est fini, $(M,v)$ est henselien et si l'une des conditions ci-dessous est v\u00e9rifi\u00e9e;\n\\begin{enumerate}[$1.$]\n\\item $(M,v)$ est divisible,\n\\item  $M_{>\\theta}.t$ contient une boule d'indice fini et $M=M_{tor}\\oplus M_{>\\theta}$, o\u00f9 $M_{tor}$ est divisible et fortement minimal s'il n'est pas fini.  \n\\end{enumerate}\n\\end{thm}\n\n\\begin{thm}\\label{thmKnontrivi}\nSoit $(M,v)$ un module  valu\u00e9  non $K$-trivialement. Alors\nil est $C$-minimal si et seulement s'il est affinement maximal, r\u00e9siduellement divisible et tel que $v(M)$ soit une $L_V$-structure $o$-minimale. \n\\end{thm}\n\n\n\nCet article est organiz\\'ee comme suit. La section 2 rappelle les d\\'efinitions introduites dans \\cite{Onay2017} et \\cite{Onay2018a}. La section 3,5,6 tra\\^itent respectivement les cas d'une valuation triviale, d'une valuation non-triviale et $K$-triviale et d'une valuation non $K$-triviale. La section 4\ns'agit d'une interlude des r\\'esultats g\\'en\\'eraux sur la $C$-minimialit\\'e utilis\\'es dans les sections 5 et 6.\n\\section{Pr\\'eliminaires}\n\nSoit $K$ un corps infini, $\\vfi$ un endomorphisme\nde $K$, et $R:=K[t,\\vfi]$. On renvoi le lecteur \\`a la section 2 de \\cite{Onay2017} pour les g\\'en\\'eralit\\'es sur l'anneau $R$ et les $R$-modules.\n\nRappelons les faits suivants.\n\nOn appelle polyn\\^ome un \\'el\\'ement de  $R$ et donc un mon\\^ome  un term de la forme $t^na$, o\\`u $a\\in K$ et  $n\\in \\mathbb{N}$.  Cet anneau est euclidien \\`a droite, en cons\u00e9quence tout $r \\in R$ s'\\'ecrit comme \n$$r=t^ns_1\\ldots s_1,$$ o\\`u les $s_i$ sont irr\\'educibles et s\\'eparables (i.e., non divible par $t$). Il s'en suit l'existence du $\\text{ppcm}(r,q)$ pour $r,q \\in R$. Il vient aussi qu'un $R$-module est divisible si et seulement \ns'il l'est par $t$ et par tout $s$ s\\'eparable. \nPar un $R$-module, on entendra un $R$-module \\`a droite.\nPar $ann_M(r)$ $(r\\in R)$ on d\\'esigne l'ensemble des \\'el\\'ements annul\\'es par $r$ dans un $R$-module $M$, i.e., l'ensemble \n$$\\{x\\in M \\;\\: | \\:\\: x.r=0\\}.$$\n La cl\\^oture divisible d'un sous module $A$ dans un $R$-module divisible $M$, est un sous-module $A' \\supseteq A$ tel que pour tout $a \\in A'$ il existe un $r\\in R$, non nul avec $a.r \\in A$. Les cl\\^otures divisibles de $A$ sont isomorphes au-dessus de $A$ en tant que $R$-modules (cf. Lemma 2.6 (4) \\cite{Onay2017} et la discussion suivant la preuve).\n\n  Une $K$-cha\\^ine est un ordre lin\\'eaire $\\Delta$ avec un maximum $\\infty$, muni d'une action de $K$ satisfaisant, pour tout $a,b \\in K^{\\times}$ et $\\delta, \\gamma \\in \\Delta\\setminus\\{\\infty\\}$, les axiomes suivantes:\n\n  \\begin{enumerate}\n  \t\\item $\\gamma > \\delta \\to \\gamma \\cdot a> \\delta \\cdot a$,\n  \t\\item $\\gamma \\cdot ab = (\\gamma \\cdot a) \\cdot b$,\n  \t\\item $\\gamma \\cdot a > \\gamma \\to \\delta \n  \t\\cdot a > \\delta$,\n  \t\\item $\\gamma \\cdot (a \\pm b)\\geq \n  \t\\min\\{\\gamma\\cdot a, \\gamma \\cdot b\\}$,\n  \t\\item $\\gamma \\cdot 0= \\infty \\; ;\n  \t\\gamma \\cdot 1 =\\gamma\\; ; \\infty\\cdot b=\\infty$.\n  \\end{enumerate}\n  \n Cet action induit une valuation $v_K$ sur $K$ (cf. \\cite{Onay2018a} Proposition 2.4). \n \\begin{defn}\n \tOn dit que cette action est $K$-triviale si $v_K$ est la valuation triviale, sinon on dit qu'elle est $K$-non-triviale. \n \\end{defn}\n \n \\begin{defn}[$R$-chains]\\label{rchains}\n \tUne $R$-cha\\^ine  $(\\Delta, <, \\infty, \\cdot r_{r \\in R})$ est une  $K$-cha\\^ine telle que\n \t\\begin{enumerate}\n \t\t\\item $\\cdot t$ is strictement croissant $\\Delta \\setminus \\{\\infty\\}$, et $\\infty \\cdot t=\\infty$,\n \t\t\\item $\\gamma\\cdot ta=(\\gamma \\cdot t)\\cdot a$ et $\\gamma\\cdot t^na = (\\gamma \\cdot t^{n-1})\\cdot ta$ pour tout $\\gamma \\in \\Delta$, et $a\\in K$,\n \t\t\\item $\\gamma \\cdot r = \\min_{i} \\{\\gamma \\cdot \\mathbf{m}_i\\}$, pour tout $r \\in R$ non nul, o\\`u\n \t\tles $\\mathbf{m}_i$ sont les mon\\^omes de $r$ et pour tout $\\gamma \\in \\Delta$,\n \t\t\\item $\\gamma \\cdot \\mathbf{m}_1 \\leq \\gamma \\cdot \\mathbf{m}_2 \\to \\left( \n \t\t\\forall \\delta \\;\\: (\\delta <\\gamma \\to \\delta \\cdot \n \t\t\\mathbf{m}_1 < \\delta \\cdot \\mathbf{m}_2)\\right)$ pout tout mon\\^omes \n \t\t$\\mathbf{m}_1, \\mathbf{m}_2 \\in R$ tels que $0\\leq\\deg(\\mathbf{m}_2)<\\deg(\\mathbf{m}_1)$,\n \t\tet pour tout $\\gamma\n \t\t\\neq \\infty$.\n \t\\end{enumerate}\n \\end{defn}\n \n\n \\begin{defn} Un $R$-module valu\\'e est un $R$-module, qui est de plus un groupe ab\\'elien valu\\'e \n \t$(M,\\Delta,v)$ (en particulier $v:M\\to \\Delta$ est surjective),  o\\`u $\\Delta$ est une $R$-cha\\^ine\n \ttelle que pour tout $x\\in M$, l'on ait\n \t\\begin{enumerate}\n \t\t\\item $v(x.a)=v(x)\\cdot a$, for all $a \\in K$,\n \t\t\\item  $v(x.t)=v(x)\\cdot t.$\n \t\\end{enumerate}\nUn $R$-module valu\\'e est  dit $K$-trivialement  valu\\'e (respectivement $K$-non-trivialement valu\\'e) si l'action de $K$ sur $\\Delta$ est $K$-triviale (respectivement $K$-non-triviale), i.e., $v(x.a)=v(x)$ pour tout $x\\in M$ et $a\\in K$. Enfin, $(M,\\Delta,v)$ est dit trivialement valu\\'e si $\\vert v(M) \\vert \\leq 2$.  \n \\end{defn}\n \nIl est clair que la d\\'efinition ci-dessus d'un $R$-module $K$-trivialement valu\\'e co\\\"incide bien avec la celle donn\\'ee dans \\cite{Onay2017} (cf. la d\\'efinition 3.4). En particulier l'application $x \\mapsto x.t$ est injective dans tout $R$-module valu\\'e.\n \nSoit $(M,\\Delta,v)$ un $R$-module valu\\'e. \n\\begin{nota}\n\tOn rappelle que l'on note  $\\theta$ la coupure de $\\Delta$, d\\'efinie par l'axiome (4) en prenant $\\mathbf{m}_1=t$ et $\\mathbf{m}_2=1$. Par cons\\'equent, on note, dans un $R$-module valu\\'e $(M,\\ldots)$,  ind\\'ependemment du fait que cette coupure soit realis\\'ee dans $\\Delta$;\n\t\\begin{itemize}\n\t\t\\item $M_{>\\theta}:=\\{x\\in M \\;\\: | \\:\\: v(x.t)>v(x)\\}$\n\t\t\\item $M_{\\geq \\theta}:=\\{x\\in M \\;\\: | \\:\\: v(x.t)\\geq v(x)\\}$.\n\t\\end{itemize}\n\\end{nota} \n\nPar ailleurs, tout $\\gamma \\in v(M)\\setminus\\{\\theta, \\infty\\}$, on note $M_{\\geq \\gamma}$ (respectivement $M_{>\\gamma}$) la boule ferm\\'ee (respectiement la boule ouverte) centr\\'ee  $0$ et de rayon $\\gamma$.\nNotez  que si la sous-module de torsion $M_{tor}$  est non triviale alors $v(M_{tor})=\\{\\theta, \\infty\\}$ est un sous $R$-module trivialement valu\\'e (cf. Lemma 3.10 (3) \\cite{Onay2017}). \n\n\\begin{nota}\n\tComme dans une $R$-cha\\^ine munie d'une action $K$-triviale,\n\tl'application $\\gamma \\mapsto  \\gamma \\cdot r$  est \\'egale \\`a $\\gamma \\mapsto \\gamma \\cdot t^k$, pour un certain $k \\in \\mathbb{N}$, suivant \\cite{Onay2017}, on notera parfois $\\tau$,  l'application $\\gamma \\mapsto   \\gamma \\cdot t$.  \n\\end{nota}\n \n \n\\begin{defn}[cf. Definition 3.23 \\cite{Onay2017}]\n\tSoit $(M,\\Delta,v)$ un $R$-module $K$-trivialement valu\\'e. On dit qu'il est hens\\'elien si pour tout $s$ s\\'eparable, $x\\mapsto x.s$ est une bijection de $M_{>\\theta}$ (autrement dit, le sous-module $M_{>\\theta}$ est divisible par les polyn\\^omes s\\'eparables).\n\\end{defn} \n\nOn peut prendre l'\\'enonc\\'e ci-dessous comme la d\\'efition d'un $R$-module {\\it r\\'esiduellement divisible et affinement maximal}. \n \n\\begin{fait}[reformulation du  Th\\'eor\\`eme 3.35, \\cite{Onay2018a}] $(M,\\Delta,v)$\n\test r\\'esiduelement divisible et affinement maximal si et seulement si pour tout non z\\'ero $r\\in R$ pour tout $z\\in M$, il y a un $y \\in M$ tel que \n\t\\begin{itemize}\n\t\t\\item $y.r=z$,\n\t\t\\item $v(y.r)=v(y)\\cdot r$.\n\t\\end{itemize}\t\n\\end{fait}\n\n\\begin{defn}\n\tUn \\'el\\'ement $y \\in M$, tel que $v(y.r)=v(y)\\cdot r$, est appell\\'e {\\it r\\'egulier} pour $r$.\n\\end{defn}\n Rappelons que $L_V=\\{<,(\\cdot r)_{r \\in R},\n \\infty\\}$, dit, le langage des $R$-cha\\^ines et $L_V'=L_V\\cup\\{(R_n)_{n \\in \\mathbb{N}} \\}$, o\\`u  les $R_n$ sont des pr\\'edicats unaires telles que,  $M \\models R_n(\\gamma)$ si et seulement si, $\\vert M_{\\geq \\gamma}\/M_{>\\gamma} \\vert\\geq n$.\nPar la suite, on notera $(M,v)$ pour d\\'esigner un $R$-module valu\\'e.\n\\section{Cas fortement minimal}\n\nOn consid\\`ere les $R$-modules trivialement valu\\'es. Notez qu'un $R$-module trivialement valu\\'e est $C$-minimale si et seulement s'il est fortement minimal.\n\\begin{lem}\\label{frtmin} \nSi $M$ est un $R$-module fortement minimal infini alors, pour tout $r \\in R \\setminus \\{0\\}$,\n$${ann}_M(r) \\;\\: \\text{fini} \\;\\: \\Leftrightarrow (M.r \\neq 0) \\Leftrightarrow M.r=M.$$\n\\begin{proof}\nPour tout $r \\in R\\setminus\\{0\\}$, ${ann}_M(r)$ et $M.r$, \u00e9tant d\u00e9finissables, sont finis ou cofinis; de plus, \\'{e}tant des sous-groupes, ils ne peuvent \u00eatre cofinis propres. Si tous les deux sont finis, en consid\u00e9rant l'application\n$M \\to M, x \\mapsto x.r$, qui a comme noyau $ann_M(r)$, on devrait avoir $M$ fini. Donc la seule possibilit\u00e9 restante est celle qui est \u00e9nonc\u00e9e.\n\\end{proof}\n\\end{lem}\n\n\n\\begin{lem}\\label{generateurI}\nSoit $I$ un id\u00e9al bilat\u00e8re non nul de $R$. Si $\\vfi^n=1$ pour un certain $n \\in \\mathbb{N}$, alors il existe $q\\in R$ \u00e0 coefficients dans $Fix(\\vfi)$  tel que $I=t^nqR$. Si $\\vfi$ n'est pas\nd'ordre fini, alors $I$ est de la forme $t^kR$ pour un certain $k \\in \\mathbb{N}$.\n \n\\begin{proof}\nVoir la proposition 2.2.8 dans \\cite{cohn}.\n\\end{proof}\n\\end{lem}\n\n\n\n\n\n\\begin{corr}\nSoit $(M,v)$ un $R$-module   trivialement valu\u00e9, infini et fortement minimal. Soit $I=\\{r \\in R \\;\\: | \\:\\: M.r=0\\}$. Alors, ou bien  $I=\\{0\\}$ et dans ce cas $M$ est divisible et les annulateurs \n$\\text{ann}_M(r)$ sont finis, ou bien $I\\neq \\{0\\}$ et dans ce cas, $I=rR$ pour un $r\\in R$ irr\\'eductible et $M$ est un $R\/rR$-espace vectoriel.\n\\begin{proof}\nLe cas o\u00f9 $I=\\{0\\}$ d\u00e9coule du lemme \\ref{frtmin}. \nSupposons donc $I\\neq \\{0\\}$. Remarquons que $I$ est un id\u00e9al bilat\u00e8re et que,\npar le lemme \\ref{generateurI},  il existe\n$r \\in K_0[t;\\vfi]$, o\\`u $K_0=Fix(\\vfi)$), tel que $I=rR$. Soit $d$ le degr\u00e9 de $r$. Alors pour tout $q$ de degr\u00e9 $<d$ v\u00e9rifiant $r=qs$ on a $M.q \\neq 0$ d'o\u00f9 par \\ref{frtmin}, $M.q=M$. Par cons\u00e9quent $M.r=M.qs=M.s=0$, d'o\u00f9 $s \\in I$. Donc\n$q \\in K$. C'est-\u00e0-dire que $r$ est irr\\'eductible. Par l'existence du ppcm, et par le th\u00e9or\u00e8me de B\u00e9zout, le quotient $R\/rR$ est un corps.\n\\end{proof}\n\\end{corr}\n\\begin{lem}\\label{excorr4.6}\n Si $M$ est un $R$-module fortement minimal, si l'action de $t$ est injective et si $\\vfi$ n'est pas d'ordre fini, alors l'id\u00e9al $I$ ci-dessus est nul.\n\\begin{proof}\nSi $\\vfi$ n'est pas d'ordre fini et si $I\\neq \\{0\\}$, par le lemme \\ref{generateurI}, $I$ est engendr\u00e9 par $t^k$, pour un certain $k \\in \\mathbb{N}_{>0}$. Puisque la multiplication par $t$ est injective ceci n'est pas possible. \n\\end{proof}\n\\end{lem}\n\n\n\n\n{\\bf Preuve du th\u00e9or\u00e8me \\ref{thmfrtmin}}.\nPar ce qui pr\u00e9c\u00e8de il suffit de montrer qu'un $R$-module satisfaisant $1$ ou \n$2$ est fortement minimal. Si $M$  est un $R$-module satisfaisant $1$,  alors,  par la proposition 2.15 \\cite{Onay2017}, toute partie d\u00e9finissable de $M$ est finie ou cofinie. Si $M$ satisfait $2$, alors \nla structure de $R$-module  sur $M$ co\u00efncide avec sa structure de $R\/rR$-espace vectoriel. Or tout espace vectoriel est fortement minimal.\n\n\n\n\\section{Interlude: cons\u00e9quences de la $C$-minimalit\u00e9}\n\nRappelons que dans un groupe (ou module )valu\\'e $G$, on note $G_{\\geq \\gamma}$ (respectivement $G_{>\\gamma}$ la boule ferm\\'ee de centre $0$ et de rayon $\\gamma$ (respectivement la boule ouverte de centre $0$ et de rayon $\\gamma$). On utilisera tr\u00e8s fr\u00e9quemment la proposition ci-dessous.\n\\begin{prop}\\label{c-min_d'indice_fini}\nSi $G$ est un groupe valu\u00e9  ${C}$-minimal alors, pour tout sous-groupe  d\u00e9finissable infini propre $F$ de G, il existe $\\gamma \\in v(G)$, tel que $F$ contient $G_{\\geq \\gamma}$ (ou $G_{> \\gamma}$), et $G_{\\geq \\gamma}$ (ou $G_{> \\gamma}$) est d'indice fini dans $F$.\n\\begin{proof}\nVoir  \\cite{Macpherson1996} ou \\cite{Simonetta2001}  1.6(ii).\n\\end{proof}\n\\end{prop}\n\n\\begin{prop}\nSoit $\\mathbb{G}=(G,v,\\dots)$  un groupe valu\u00e9, porteur de structure additionnelle, et $C$-minimal. Alors \\begin{enumerate}[$1.$]\n\\item pour tout $\\gamma \\in v(G)$, le groupe $G_{\\geq \\gamma}\/G_{>\\gamma}$ \nmuni de la structure induite par $\\mathbb{G}$ est fortement minimal,\n\\item la cha\u00eene $v(G)$ munie de la structure induite par $\\mathbb{G}$ est\n$o$-minimale.\n\\end{enumerate} \n\\begin{proof}\nVoir  \\cite{Macpherson1996}.\n\\end{proof}\n\\end{prop}\n\n\n\\begin{corr} Soit $(M,v)$ un $R$-module  valu\u00e9  $C$-minimal.\n\tNotons $K\/v_K$ le corps r\\'esiduel associ\\'e \\`a la valuation $v_K$ induit par l'action $K$ sur $v(M)$. Alors,\n\\begin{enumerate}\n\\item pour tout  $\\gamma \\in v(M)$, $M_{\\geq \\gamma}\/M_{>\\gamma}$ est un $(K\/v_K)$-espace vectoriel fortement minimal et,\n\\item la $R$-cha\u00eene $v(M)$ est $o$-minimale.\n\\end{enumerate}\n\\end{corr}\n\n\n\\begin{remarque}\nDans le point $1$ ci-dessus, si $(M,v)$ est $K$-trivialement valu\u00e9, i.e. si $v_K$ est triviale, alors $K\/v_K=K$ et donc tout   $M_{\\geq \\gamma}\/M_{>\\gamma}$ est un $K$ espace-vectoriel. \\end{remarque}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Cas d'une valuation non triviale et $K$-triviale}\nUne des implications du th\u00e9or\u00e8me \\ref{thmCmintrivi}  est contenue  dans la proposition suivante.\n\n\\begin{prop}\\label{divhens->c-min}\nSi $(M,v)$  est un module $K$-trivialement valu\u00e9 divisible henselien, si pour tout $q \\in R\\setminus\\{0\\}$, $\\text{ann}_M(q)$ est fini  et si\n$v(M)$ est ${o}$-minimal dans le langage $L'_V$, alors $(M,v)$ est ${ C}$-minimal.\n\\begin{proof}\nNoter qu'un tel $(M,v)$ est r\\'esiduellement divisible et affinement maximal par le corollaire 3.29, \\cite{Onay2017}.  Nous renvoyons donc \u00e0 la preuve de la proposition \\ref{cmin}  \ndont la pr\u00e9sente proposition n'est qu'un cas particulier. \n\\end{proof}\n\\end{prop}\n\n\n\n\n\n\\begin{lem}\\label{puiseux-stayla-c_min}\nSoit $(M,v)$ un module $K$-trivialement valu\u00e9, tel que \n\\begin{enumerate}\n\\item $M_{tor}$ est fini et $M=M_{tor} \\oplus M_{>\\theta}$, et\n\\item $M_{>\\theta}$ est divisible et $v(M)$ est $o$-minimal.\n\\end{enumerate}\nAlors $(M,v)$ est $C$-minimal.\n\\begin{proof} Si $M_{tor}=0$ alors, $M=M_{>\\theta}$ est  divisible henselien donc $C$-minimal par la proposition pr\u00e9c\u00e9dante. Sinon, soit $N'$ une cl\u00f4ture divisible de $M_{tor}$. Posons $N:=N'\\oplus M_{>\\theta}$ et on \u00e9tend $v$ \u00e0 $N$ en posant $v(x_{tor} + x_{>\\theta})=\\theta$ si $x_{tor}$ est non nul. D'o\u00f9 $(N,v)$ est divisible henselien et $v(N)=v(M)$, donc $(N,v)$ est $C$-minimal par la proposition pr\u00e9c\u00e9dente. Alors $M=\\bigcup_{a \\in M_{tor}} M_{>\\theta} + a$, est une union finie de boules de $N$, par cons\u00e9quent il est $C$-minimal.\n\\end{proof}\n\\end{lem}\n\n\nUne cons\u00e9quence imm\u00e9diate des deux r\u00e9sultats pr\u00e9c\u00e9dents est:\n\\begin{corr}\\label{puiseuxCmin}\nL'anneau de valuation d'un corps de caract\u00e9ristique $p>0$, parfait, muni d'une valuation henselienne,  de corps r\u00e9siduel fini ou p-clos, et de groupe de valuation divisible, est {C}-minimal en tant que $\\mathbb{F}_p[t;x\\to x^p]$-module valu\u00e9.\n\\begin{proof}\nUn groupe ab\u00e9lien ordonn\u00e9 divisible est $o$-minimal, donc a fortiori sa restriction \u00e0 la structure de  $\\mathbb{F}_p[t;x\\to x^p]$-cha\u00eene. \nLe corollaire d\u00e9coule donc du \n lemme 3.24 \\cite{Onay2017}, avec la proposition \\ref{divhens->c-min} si le corps r\u00e9siduel est $p$-clos et avec le lemme  \\ref{puiseux-stayla-c_min}\nsi le corps r\u00e9siduel est fini.\n\\end{proof}\n\\end{corr}\n\n\\begin{exemple}\nSoit $R=\\mathbb{F}_p[t;x\\mapsto x^p]$.\nL'anneau des s\u00e9ries de Puiseux sur $\\mathbb{F}_p$ ou sur un corps $p$-clos est $C$-minimal en tant que $R$-module valu\u00e9, mais le corps des s\u00e9ries de Puiseux ne l'est pas (en tant que $R$-module). \n\\end{exemple}\n\n\nLe lemme ci-dessous, avec le lemme pr\u00e9c\u00e9dent, montreront que la condition $2.$ cit\u00e9 dans le th\u00e9or\u00e8me \\ref{thmCmintrivi}, implique la $C$-minimalit\u00e9. \n\n\n\\begin{lem}\nSoit $(M,v)$  un module $K$-trivialement valu\u00e9 henselien et tel que\n\\begin{enumerate}\n\\item $M_{tor}$ est fini  et $M=M_{tor} \\oplus M_{>\\theta}$,\n\\item $M_{>\\theta}.t$ contient une boule  de la forme $M_{\\geq \\gamma}$ ou $M_{>\\gamma}$,  qui est d'indice fini dans $M_{>\\theta}.t$   et $v(M_{>\\theta})$ est $o$-minimal.\n\\end{enumerate}\nAlors $(M,v)$ est $C$-minimal.\n\\begin{proof}\nSi $M_{>\\theta}$ est divisible, alors l'assertion suit de lemme ci-dessus. \n\nMontrons d'abord que la valuation sur $M_{>\\theta}$  d\u00e9termine une et une seule valuation sur chacune de ses cl\u00f4tures divisibles (toutes sont d\\'ej\\`a isomorphes en tant que $R$-modules). \nSoit $N$ une cl\u00f4ture divisible   de $M_{>\\theta}$. Alors, puisque $M_{>\\theta}$ est divisible par tous les polyn\u00f4mes s\u00e9parables, \n $x \\in N$ si et seulement si  $x.t^n \\in M_{>\\theta}$ pour un certain $n \\in N$.  Soit alors $x \\in N\\setminus M_{>\\theta}$ et $n$ minimal tel que $x.t^n \\in M_{>\\theta}$. Soit  $\\gamma=v(x.t^n)$. On d\u00e9finit $\\bar{v}(x)$ comme le couple $(\\gamma, n)$. Si $y \\in N\\setminus M_{>\\theta}$ est tel que $\\bar{v}(y)=(\\delta, k)$, on d\u00e9cr\u00e8te que \n$\\bar{v}(x) < \\bar{v}(y)$ si et seulement si $\\tau^k(\\gamma) < \\tau^n(\\delta)$. Il est imm\u00e9diat de v\u00e9rifier que $(N,\\bar{v})$ est un module $K$-trivialement valu\u00e9, et que $\\bar{v}$ prolonge $v$. \n\nMontrons maintenant que $M_{>\\theta}$ s'identifie \u00e0 une union finie de boules de $(N,\\bar{v})$. Soit $B$ la boule de $M$,  d'indice fini dans $M_{>\\theta}.t$, dont l'existence est donn\u00e9e par l'hypoth\u00e8se du corollaire. Montrons que  $B$ est en fait une boule de $N$.  Soit  $\\gamma$ le rayon de $B$, i.e. $B=M_{>\\gamma}$ ou $B=M_{\\geq \\gamma}$. Soit $x \\in N \\setminus M_{>\\theta}$ , avec $v(x) > \\gamma$, et soit $k\\geq 1$ minimal tel que $\\overline{x}.t^k \\in  M_{>\\theta}$. On a, $v(x.t^k) > \n\\tau^{k}(\\gamma)> \\gamma$. Or $M_{>\\theta}.t \\supset B$ implique $x.t^{k-1} \\in M_{>\\theta}$. Contradiction. D'o\u00f9 $B=N_{\\geq \\gamma}$ ou\n$B=N_{>\\gamma}$. En \u00e9crivant $M_{>\\theta}.t= \\bigcup_i B +a_i$, on a $M_{>\\theta}=B.t^{-1} + a_.t^{-1}$, o\u00f9 $B.t^{-1}$ est n\u00e9cessairement la boule ouverte ou ferm\u00e9e de rayon $\\tau^{-1}(\\gamma)$ et $a_i.t^{-1}$ est l'unique $b_i \\in N$ tel que $b_i.t=a_i$. Par cons\u00e9quent, $M_{>\\theta}$   est une union finie de boules de $N$ donc $M$ est une union finie de boules de $N\\oplus M_{tor}$ et il suffit de montrer que $N\\oplus M_{tor}$ est $C$-minimal.  \nPuisque $(N, \\overline{v})$ est divisible henselien sans-torsion, le lemme \\ref{puiseux-stayla-c_min} dit  qu'il suffit de montrer que \n$\\overline{v}(N)$ est $o$-minimal.\n \nMontrons  d'abord que  $\\overline{v}(N)$\nest la cl\u00f4ture de $v(B)$ par $\\tau^{-1}$.\n Soit  $x \\in M_{>\\theta}$, tel que $x.t \\notin B$. On a $v(x.t-a_i)>v(x.t)=v(a_i)$ pour un certain $i \\in \\{1,\\dots, n\\}$. Dans ce cas , $v(x.t^2)>v(x.t)$ et donc pour tout $i$, $v(x.t^2-a_i)=v(a_i)$. Alors $x.t^2 \\in B$ n\u00e9cessairement. Par ailleurs, si $x.t \\in B$ de m\u00eame pour  $x.t^2$. Donc $N$ est aussi la cl\u00f4ture divisible de $B$. \nD'o\u00f9 $\\overline{v}(N)$\nest la cl\u00f4ture de $v(B)$ par $\\tau^{-1}$,  $\\overline{v}(N)=\\bigcup_{i \\in \\mathbb{N} } \\tau^{-i}v(B)$, et chaque   $\\tau^{-i}v(B)$ est isomorphe \u00e0 $v(B)$ donc ils sont  tous $o$-minimaux. \n\nPuisque $v(M)$ est $o$-minimal $v(M)\\setminus\\{\\theta,\\infty\\}$ est dense ou discret: Soit  $DE$ l'ensemble des points  ayant un voisinage dense et $DI$ l'ensemble des points ayant un pr\u00e9d\u00e9cesseur et un successeur; par $o$-minimalit\u00e9, $v(M)$ se partionne comme $DE\\cup DI\\cup F$, avec $F$ fini, plus pr\u00e9cisement $DE$ et $DI$ sont des unions finis d'intervalles s\u00e9par\u00e9s par au moins un point de $F$; par ce que $\\tau$ est strictement croissant, il pr\u00e9serve chacun des trois ensembles, $DE$, $DI$ et $F$, et parce qu'il n'admet aucun point fixe sur $v(M)\\setminus\\{\\theta,\\infty\\}$,  $F\\setminus\\{\\theta,\\infty\\}=\\emptyset$, par cons\u00e9quent $v(M)\\setminus\\{\\theta,\\infty\\}=DE$ ou $v(M)\\setminus\\{\\theta,\\infty\\}=DI$.\n\nMaintenant, par \\cite{Pillay1987}, si $v(M)$ est discret alors $\\tau$ est une translation sur $v(M)$ et donc $\\overline{v}(N)$ est discret et $\\tau$ reste une translation. Sinon si $v(M)$ est dense, par construction $\\overline{v}(N)$ est dense. Dans le \ndeux cas, la $\\overline{v}(N)$ est $o$-minimal par le corollaire 1.16 de \\cite{Maalouf2010a}.\n\\end{proof}\n\\end{lem}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nLes lemmes qui suivent   \u00e9tablissent des r\u00e9ciproques  aux r\u00e9sultat ci-dessus et ainsi compl\u00e8tent la preuve du th\u00e9or\u00e8me \\ref{thmCmintrivi}.\n\\begin{lem}\\label{c-min2div}\n Soit $(M,v)$  un module $K$-trivialement valu\u00e9, ${ C}$-minimal, infini.\nS'il existe  $a \\in M$ tel que $v(a)< \\theta$   alors M est divisible.\n\\begin{proof}\nSoit $q\\in R$, non nul. S'il existe un $a \\in M$ comme ci-dessus, alors $v(M.q)$\nn'a pas de premier \u00e9l\u00e9ment. Si $M.q \\neq M$ alors par la proposition \\ref{c-min_d'indice_fini} il contient un sous-groupe $H$ de la forme\n$M_{\\geq \\gamma}$ ou de la forme $M_{>\\gamma}$ qui est d'indice fini dans $M.q$. Alors $M.q$ est la r\u00e9union disjointe finie des $H + a_i$ et donc $v(M.q)$ a un  premier \u00e9l\u00e9ment. Contradiction.  \n\\end{proof}\n\\end{lem}\n\n\\begin{lem}\\label{anninfini_implique_Cmin}\nSoit $(M,v)$ un module non trivialement valu\u00e9 qui est $K$-trivialement valu\u00e9 et $C$-minimal. Alors, pour tout $r \\in R\\setminus\\{0\\}$, $\\text{ann}_M(r)$   est fini.\n\\begin{proof}\nSi $\\text{ann}_M(r)$,  est une partie propre infini de $M$, alors par la proposition \n\\ref{c-min_d'indice_fini} il existe $\\gamma \\in \\Delta$ tel que $\\text{ann}_M(r)$  contient une boule de la forme $M_{\\geq \\gamma}$ ou $M_{>\\gamma}$ qui est d'indice fini dans $\\text{ann}_M(r)$. Comme $v(\\text{ann}_M(r)) = \\{\\theta\\}$, on obtient  $\\gamma=\\theta$; dans ce cas l\u00e0, on a n\u00e9cessairement $M_{>\\theta}={0}$ et donc $\\text{ann}_M(r)=M_{\\geq\\theta}=M_{tor}$. Cela implique en particulier $\\infty$ est l'unique \u00e9l\u00e9ment qui est strictement$>\\theta$. Puisque $(M,v)$ est non trivialement valu\u00e9, il existe donc des \u00e9l\u00e9ments de valuation $<\\theta$. Par le lemme ci-dessus $M$ est alors divisible; d'o\u00f9 $M_{tor}$ est divisible.  Or ceci contredit le fait que $M_{tor}$ est annul\u00e9 par un seul \u00e9l\u00e9ment.\n\\end{proof}\n\\end{lem}\n\n\\begin{corr}\nSoit $(M,v)$ non trivialement valu\u00e9, $K$-trivialement valu\u00e9 et $C$-minimal. Si $M_{tor}$ est infini alors $M_{tor}$ est divisible et fortement minimal.\n\\end{corr}\n\\begin{proof} $M_{tor}$ se plonge dans  le $R$-module $M_{\\geq \\theta}\/M_{>\\theta}$ qui est fortement minimal, on a la divisibilit\u00e9 de $M_{\\geq \\theta}\/M_{>\\theta}$ par le lemme ci-dessus et par le lemme \\ref{frtmin}, d'o\u00f9 la divisibilit\u00e9 de $M_{tor}$. Alors par la proposition 2.15 dans \\cite{Onay2017}, tout sous-ensemble d\u00e9finissable de $M_{tor}$ est une combinaison bool\u00e9enne  d'ensembles du type $\\text{ann}_M(r)$, avec $r \\in R\\setminus\\{0\\}$. D'o\u00f9, par le th\u00e9or\u00e8me \\ref{frtmin} et par le fait\n\tque chaque annulateur est fini, la forte minimalit\u00e9 de $M_{tor}.$ \n\\end{proof}\n\n\\begin{lem}\nSoit $(M,v)$  un module $K$-trivialement valu\u00e9 ${C}$-minimal. Alors $(M,v)$ est henselien.\n\\begin{proof}\nOn va consid\u00e9rer $2$ cas:\\\\\n{\\bf a.} {\\it $v(M_{>\\theta})$ n'a pas de premier \u00e9l\u00e9ment.} \nPar la proposition\n\\ref{c-min_d'indice_fini}, si  $M_{>\\theta}.r$ ($r \\in R\\setminus 0$) n'est pas \u00e9gal \u00e0 $M_{>\\theta}$, alors il contiendrait \nune boule ouverte ou ferm\u00e9e d'indice fini. \nEn particulier, ou bien $v(M_{>\\theta }.r)$ aurait un plus petit \u00e9l\u00e9ment ou bien pour un certain $\\gamma>\\theta$ on aurait \n$M_{>\\theta }.r=M_{>\\gamma}$. Ces deux cas sont impossibles car si $r$ est s\u00e9parable et $x \\in M_{>\\theta}$, on a  $v(x.r)=v(x)$.\n\n\\noindent\n{\\bf b.} {\\it $v(M_{>\\theta})$ a un plus petit \u00e9l\u00e9ment.} \nSoit $\\gamma_{0}$ ce plus petit \u00e9l\u00e9ment $> \\theta$. Soit $s \\in R_{sep}$. \n\u00c9crivons $s$ comme $tq+ a$ avec $a \\in K$. Encore par le lemme \n\\ref{c-min_d'indice_fini}, si  $M_{>\\theta}.s \\neq M_{>\\theta}$ alors il contient\nun certain $M_{>\\gamma}$ ou $M_{\\geq \\gamma}$ qui est d'indice fini dans $M_{>\\theta}.s$. Supposons que c'est $M_{> \\gamma}$, l'autre cas se traite de m\u00eame. Puisque $M_{>\\gamma}$ est d'indice fini dans $M_{>\\theta}.s$, il existe  $\\gamma_1, \\dots \\gamma_k$ tel que $\\gamma_0 < \\gamma_1 < \\dots < \\gamma_k= \\gamma$ avec $\\gamma_{i+1}$  le successeur de $\\gamma_{i}$ dans $v(M)$. Soit $x$ de valuation $\\gamma_0$, alors \n$x_0:=(x - x.a^{-1}(tq + a))=(x - x.a^{-1}s)$ est de valuation $ \\geq \\gamma_1 >\\gamma_{0}$. Ainsi\non d\u00e9finit $x_i:=(x_{i-1} - x_{i-1}.a^{-1}s)$, avec $v(x_i) > \\gamma_i$, pour $1\\leq i \\leq k$. Donc $v(x_{k})>\\gamma$. Par cons\u00e9quent $x_k$ est divisible\npar $s$, et donc $x$ aussi. On a montr\u00e9 que $M_{>\\theta}$ est divisible\npar les polyn\u00f4mes s\u00e9parables, i.e. $(M,v)$ est henselien.\n\\end{proof}\n\\end{lem}\n\n\n\n\n\n\n\n\n\n\nPar ce qui pr\u00e9c\u00e8de, on peut supposer que $M=M_{\\geq \\theta}$ et que $M_{tor}$ est fini. Il nous  reste alors \u00e0 montrer le r\u00e9sultat suivant, avec lequel s'ach\u00e8ve la preuve du th\u00e9or\u00e8me \\ref{thmCmintrivi}.\n\n\\begin{lem}\nSoit $(M,v)$ un module $K$-trivialement valu\u00e9 $C$-minimal non divisible et tel que \n$M_{\\geq \\theta}=M$ et $M_{tor}$ est fini. Alors $M=M_{tor} \\oplus M_{>\\theta}$.\n\\begin{proof}\nPuisque $M_{tor}$ est fini la multiplication par $t$ induit une bijection\nde $M_{tor}$. Donc $M_{tor}=\\text{ann}_M(t^k-1)$ pour un certain $k \\in \\mathbb{N}\\setminus \\{0\\}$. Par cons\u00e9quent, $M_{tor}$ n'est pas divisible donc $M$ non plus. Donc $M.(t^k-1)$ contient une boule (ouverte ou ferm\u00e9e) qui y est d'indice fini, d'apr\u00e8s le lemme \\ref{c-min_d'indice_fini}. En particulier le quotient \n$M.(t^k-1)\/(M_{>\\theta} \\cap M.(t^k - \n1))$ est fini; de plus il est sans torsion. Or, puisque $R$ est infini, le \nseul module sans torsion fini est le module $0$. Cela implique  que \n$M.(t^k-1) \\subset M_{>\\theta}$. \n\nMontrons  l'inclusion r\u00e9ciproque $M_{>\\theta}\\subset M.(t^k-1)$: $(M,v)$ est henselien par le lemme ci-dessus; en particulier $M_{>\\theta}$ est divisible par le polyn\u00f4me $t^k-1$; cela implique  que la suite  ci-dessous est exacte (et clairement scind\u00e9e):\n$$0 \\longrightarrow M_{tor} \\longrightarrow M \\xrightarrow{.(t^k-1)} (M_{>\\theta})  \\longrightarrow 0.$$\nD'o\u00f9 $M_{>\\theta}$ est facteur direct dans $M$.\n\\end{proof}\n\\end{lem}\n\n\n\n\\section{Cas d'une valuation non $K$-triviale}\nLes propositions ci-dessous \u00e9tablissent la preuve du th\u00e9or\u00e8me \\ref{thmKnontrivi}. Rappelons que $(M,v)$ est dit r\\'esiduellement divisible, si pour tout non zero $r\\in R$, non zero $z\\in M$ il existe $y\\in M$, tel que \n$$v(z-y.r)>v(y.r)=v(y)\\cdot r.$$\n\nNotons que l'\\'egalit\\'e dans l'expression ci-dessus  peut ne peut \\^etre toujours v\\'erifi\\'ee (on peut penser au cas o\\`u $y$ est non zero et annul\\'e par $r$.)\n\n\n\\begin{prop}\\label{cmin}\nUn module  valu\u00e9 $(M,v)$ affinement maximal, r\u00e9siduellement divisible, ayant sa cha\u00eene $v(M)$ $o$-minimale et tel que, pour chaque $r \\in R\\setminus\\{0\\}$, $\\text{ann}_M(r)$ est fini, est $C$-minimal.\n\\begin{proof}\nPar \\cite{Onay2018a}, Th\\'eor\\`eme 4.6, on sait que toute formule \u00e0 une seule variable libre $x$, avec param\u00e8tres, est \u00e9quivalente \u00e0 une formule de la forme $$\\phi(x) \\wedge \\psi(v(t_1(x)),\\dots,v(t_k(x)), \\overline{\\gamma}),$$ o\u00f9 \n$\\phi$ est une formule sans quantificateur avec param\u00e8tres dans le langage des $R$-modules, $\\psi$ est\nune formule sans param\u00e8tres de $L_V'$,  les $t_i$ sont des termes avec param\u00e8tres dans le langage des $R$-modules  et $\\overline{\\gamma}$ est un $n$-uplet d'\u00e9l\u00e9ments de $v(M)$.\nEn particulier, $\\phi$ est une combinaison bool\u00e9enne de formules de la forme $x.r=b$. Puisque $\\text{ann}_M(r)$ est fini, elle d\u00e9finit un ensemble fini ou cofini. On se ram\u00e8ne donc \u00e0 montrer que $\\psi(v(t_i(x))_{i})$ d\u00e9crit une combinaison bool\u00e9enne de boules ouvertes ou ferm\u00e9es.\nOn se contentera de montrer que l'on peut remplacer les termes $v(t_i(x))$ par des termes\nde la forme $\\tau^k(v(x-c_i))$ o\u00f9 $k \\in \\mathbb{N}$ et $c_i \\in M$. Ainsi $\\psi$ sera de la forme $\\psi'(v(x-c_1),\\dots,v(x-c_n))$. \nLa $C$-minimalit\u00e9 de $(M,v)$ suit alors de l'$o$-minimalit\u00e9 de la $R$-cha\u00eene $v(M)$   par des consid\u00e9rations ultram\u00e9trique g\u00e9n\u00e9rales (comme il a \u00e9t\u00e9 explicit\u00e9 dans la preuve de la proposition 36 de \\cite{Maalouf2010a}). \n\n\\noindent\nLes termes $t_i(x)$ sont de la forme $x.r_i - a_i$ ($r_i \\in R$, $a_i \\in M$). \nPar divisiblit\u00e9 de $M$, il existe $b_i \\in M$ tel que \n$b_i.r_i= a_i$. Ainsi chaque terme $t_i(x)$ est \u00e9gal \u00e0 un terme de la forme $(x-b_i).r_i$. La preuve sera donc achev\u00e9e d\u00e8s que nous aurons prouv\u00e9 l'assertion suivante:\n\n\\noindent\nEtant donn\u00e9 $r \\in R\\setminus\\{0\\}$, il existe un recouvrement fini de $M$ par des ensembles dont chacun, disons $E$, est une combinaison bool\u00e9enne de boules de $M$ et tel que pour un certain param\u00e8tre $d$,  \n$(M,v) \\models \\forall x \\in E, \n           v(x  \\cdot r ) = v(x-d) \\cdot r$. \n\n\\noindent\nEn effet, d'une part par le r\u00e9sultat 3.36 dans \\cite{Onay2018a}, on a \n$$\\forall x \\bigvee_{c \\in ann_M(r)} v(x.r)=v((x-c).r)=v(x-c)\\cdot r.$$\nEt d'autre  part, pour tout $c_0 \\in \\text{ann}_M(r)$, l'ensemble \n$\\{x \\in M \\;\\: | \\:\\: v((x-c_0).r)=v(x-c_0)\\cdot r\\}$ est exactement l'ensemble des \u00e9l\u00e9ments $x$ tel que $x-c_0$ est r\u00e9gulier pour $r$,  et est \u00e9gal, par le m\\^eme r\\'esultat, l'ensemble $\\{x \\;\\: | \\:\\: v(x-c_0)=\\max\\{v(x-c_0-c); c\\in \\text{ann}_M(r) \\}\\}$. Ce dernier ensemble est clairement une combinaison bool\u00e9enne de boules.\n\n\\end{proof}\n\\end{prop}\n\nPour la d\\'emonstration suivante on rappelle que l'ensemble $Saut_M(r)$ (not\\'e $\\jumpm{M}{r}$ dans \\cite{Onay2018a}) est l'ensemble des valeurs $\\gamma$ tel qu'il existe un $x\\in M$ de valuation $\\gamma$ satisfaisant \n$$v(x.r)>v(x)\\cdot r.$$\n\n\\begin{prop}\nSoit $(M,v)$ un module valu\u00e9 non $K$-trivialement  et $C$-minimal. Alors\n$(M,v)$ est affinement maximal et r\u00e9siduellement divisible, et pour tout \n$r\\in R\\setminus \\{0\\}$, \n$\\text{ann}_M(r)$ est fini.\n\\begin{proof}\nPuisque $v_K$ est non triviale sur $K$,  $v(M)$ n'est pas minor\u00e9. Or \n$Saut_M(r)$ est fini pour tout $r \\in R \\setminus \\{0\\}$ donc $\\text{ann}_M(r)$\nne peut pas contenir de boule propre d'indice fini et doit \u00eatre fini. \nDe plus, pour les m\u00eames raisons, $v(M.r)$ n'est pas minor\u00e9, donc $M.r$ \nne peut contenir de boule propre d'indice fini et $M.r=M$. D'o\u00f9 $M$ est divisible. \n\nSupposons que $(M,v)$ n'est pas affinement maximal. Comme $M$ est d\\'ej\\`a divisible, il est r\\'esiduellement divisible. Dans ce cas, par le th\u00e9or\u00e8me 3.35 \\cite{Onay2018a},\n  il existe $r \\in R \\setminus\\{0\\}$ et $y \\in M\\setminus\\{0\\}$ tels que pour tout $x$ v\u00e9rifiant $x.r=y$, $x$ est irr\u00e9gulier pour $r$. Puisque \n$M$ est divisible, l'ensemble $A:=\\{x \\in M \\;\\: | \\:\\: x.r=y\\}$ est non-vide et puisque \n$\\text{ann}_M(r)$ est fini, il s'\u00e9crit comme $A=\\{x_1,\\dots, x_k\\}$. Posons\n$\\gamma=\\max_i\\{v(x_i)\\}$ et prenons $x\\in A$ de valuation $\\gamma$. Alors\n$y \\in M_{>\\gamma \\cdot r} \\setminus \\left(M_{>\\gamma}\\right).r$. On va montrer que ceci conduit \u00e0 une contradiction. Puisque $\\gamma$ est un saut, il est limite inf\u00e9rieure dans $v(M)$  des \u00e9l\u00e9ments $\\delta \\in v(M)\\setminus Saut_M(r)$ par le lemme 2.11 \\cite{Onay2018a} et par la finitude de $Saut_M(r)$; donc il est limite des \u00e9l\u00e9ments $\\delta$ tel que  $v(M_{>\\delta}.r)$ et $v(M_{>\\delta \\cdot r})$ d\u00e9finissent le m\u00eame segment final de $v(M)$. Cela impose que \n $M_{>\\gamma \\cdot r}=\\left(M_{>\\gamma}\\right).r$. Contradiction.\n \\end{proof}\n\\end{prop}\n\nEnfin, on donne une cons\u00e9quence du r\u00e9sultat ci-dessus dans le cas des corps valu\u00e9s: par  le lemme 2.11 \\cite{Onay2018a}, il suit qu'une $R$-cha\u00eene $o$-minimale est dense et pleine et il suit par le corollaire \\cite{Onay2018a} 2.26, qu'une $R$-cha\\^ine pleine et dense est $o$-minimale. De plus,\n  par la remarque 2.29 \\cite{Onay2018a},  un groupe ab\u00e9lien ordonn\u00e9 plein comme $R$-cha\u00eene est divisible. Ensuite, par l'exemple 2.28(2) \\cite{Onay2018a},  la $R$-cha\\^ine $v(\\mathcal{K})$, d'un corps aux difference valu\\'e $(\\mathcal{K},\\sigma, v)$, o\\`u $R=\\mathcal{K}[t;\\sigma]$ est pleine et dense que $v(\\mathcal{K})$ est un $\\mathbb{Z}[\\sigma]$-module (de nouveau par \\cite{Onay2018a}, 2.28). Enfin par le corollaire \\cite{Onay2018a}, un corps   valu\u00e9 hens\\'elien $(K,v)$ de caract\u00e9ristique $p>0$ est alg\u00e9briquement maximal si et seulement s'il est affinement maximal comme $K[t;x\\mapsto x^p]$-module valu\u00e9. \n\n\\begin{corr} On a:\n\\begin{itemize}\n\t\n\\item Soit $(K,v)$  un corps   valu\u00e9 infini hens\\'elien et de caract\u00e9ristique $p>0$. \n Alors $(K,v)$  est $C$-minimal comme $K[t;x\\mapsto x^p]$-module valu\u00e9 si et seulement s'il est alg\u00e9briquement maximal avec  un corps r\u00e9siduel $p$-clos\net un groupe de valuation divisible.\n\n\\item Soit $\\mathcal{K}$ l'ultraproduit  des corps alg\\`ebriquement valu\\'es $\\mathcal{K}_{p^n}$,  de caract\\'eristique  $p>0$, \\'equipp\\'e chacun du morphisme $x\\mapsto x^{p^n}$, selon un ultrafiltre $U$ sur $\\{p^n \\;\\: | \\:\\: n\\in \\mathbb{N}, \\; \\text{et} \\; p \\; \\text{premier}\\}$. $\\mathcal{K}_{p^n}$   muni de l'automorphisme  limite, {\\it Frobenius non-standard}, i.e., $\\sigma_{U}:=\\lim_{U} x \\mapsto x^{p^n}$, est $C$-minimal comme $\\mathcal{K}[t;\\sigma]$-module valu\\'e.\n\\end{itemize}\n\\end{corr}\n\\begin{remarque}\nNoter que en prenant un ultrafiltre trivial, le second point ci-dessus implique le premir.\n\\end{remarque}\n\\bibliographystyle{apalike}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{s:intro}\n\\textcolor{black}{Geometric median-of-means (GMOM) has been widely used for robust estimation of multivariate means, and it has been broadly adopted in machine learning \\citep{Minsker2015,Hsu2016,Prasad2020}. The idea of GMOM is to first divide the data into disjoint subsamples and calculate the empirical means of each of the subsamples. Then the GMOM estimator is computed as the spatial median (also called geometric median) of the obtained empirical means. The previous studies on the GMOM focused on establishing its non-asymptotic error bounds under certain heavy-tailed assumptions. Its distributional properties, which are essential for statistical inference, remain unknown.}\n\nHigh-dimensional data with the dimension increases to infinity as the number of observations goes to infinity have been encountered in many scientific disciplines.\nThere is a growing evidence of the multivariate normal distribution is problematic to model high-dimensional data due to the presents of heavy-tailedness and inadequate to accommodate tail dependence.\nFor example, the distributions of the microarray expression are observed to be non-normal and have heavy tails even after log transformation in many gene expression data \\citep{Purdom2005,Wang2015}.\nAs another example, elliptical distributions, in particular the multivariate $t$-distribution and symmetric multivariate normal inverse Gaussian distribution, provided far superior models to the multivariate normal for daily and weekly US stock-return data \\citep{McNeil2005}.\nIn such cases, the sample spatial median is favored against the sample mean for estimating the location parameter. \n\\textcolor{black}{The above discussions strongly motivate studying the spatial median under high-dimensionality, especially its distributional properties and the implementation in statistical inference for high-dimensional location parameter.}\n\nLet $X_1,\\ldots,X_n$ be a sequence of independent and identically distributed (i.i.d.) $p$-dimensional random vectors from a population $X$ with cumulative distribution function $F_X$ in $\\mathbb{R}^p$. \nIn this paper, we work on a general multivariate model where $X$ admits the following stochastic representation:\n\\begin{eqnarray}\\label{eq:Model_X}\n\tX = {\\boldsymbol \\theta} + \\nu\\Gamma U\\,,\n\\end{eqnarray}\nwhere ${\\boldsymbol \\theta}$ is the location parameter, $\\nu$ is a nonnegative univariate random variable and $U$ is a $p$-dimensional random vector with independent components. Model \\eqref{eq:Model_X} covers many commonly used multivariate models and distribution families, including the independent components model \\citep{Yao2015} and the elliptical distribution family \\citep{Fang1990}. We refer to Section \\ref{sec:note} for more detailed discussions.\n\nSpatial median, an extension of the univariate median to multivariate distributions, was proposed for robust inference of the location parameter \\citep{Haldane1948,Weber1929}. \nThe sample spatial median $\\hat{{\\boldsymbol \\theta}}_{n}\\in\\mathbb{R}^{p}$ minimizes the empirical criteria function $L_n({\\boldsymbol \\beta}) =\\sum_{i=1}^{n}(\\|X_i-{\\boldsymbol \\beta}\\|-\\|X_i\\|)$, where $\\|\\cdot\\|$ is the Euclidean norm. Equivalently,\n\\begin{eqnarray}\\label{eq:btheta_criteria}\n\t\\hat{{\\boldsymbol \\theta}}_{n} = \\argmin_{{\\boldsymbol \\beta}\\in {\\mathbb R}^p}L_{n}({\\boldsymbol \\beta}) = \\argmin_{{\\boldsymbol \\beta}\\in {\\mathbb R}^p} \\sum_{i=1}^{n}(\\|X_i-{\\boldsymbol \\beta}\\|-\\|X_i\\|)\\,. \n\\end{eqnarray}\nThe function $L_{n}({\\boldsymbol \\beta})$ is convex, and $\\hat{{\\boldsymbol \\theta}}_{n}$ is unique if the observations $\\{X_i\\}_{i=1}^{n}$ are not concentrated on a line in $\\mathbb{R}^{p}$ when $p> 2$ \\citep{Milasevic1987}.\nWhen the dimension $p$ is fixed, the spatial median has been well studied in the literature. \nWe refer to Chapter 6.2 of \\citet{Oja2010} for a nice review. \n\nIn the high-dimensional setting, where the dimension $p$ diverges to infinity as the number of observations $n\\to\\infty$, there are several existing works that study the asymptotic properties of the sample spatial median.\n\\citet{Zou2014} offered an expansion of $\\hat{{\\boldsymbol \\theta}}_{n}$ under elliptical distributions with identical shape matrix, and \\citet{Cheng2019} extended the result to a general shape matrix. \nAs a recent work, \\citet{LiXu2022} improved the expansion in \\citet{Cheng2019} with a smaller order remainder term under stronger conditions, and established a central limit theorem for the squared Euclidean distance $\\|\\hat{{\\boldsymbol \\theta}}_{n}-{\\boldsymbol \\theta}\\|^2$.\nIn \\citet{Zou2014} and \\citet{Cheng2019}, they both require that $p=O(n^2)$.\nIn addition, it is required in \\citet{LiXu2022} that $p$ diverges at the same rate as $n$.\nHowever, in modern areas such as genomics and proteomics, the dimension of the data may grow exponentially with the sample size, which lies in the ``ultrahigh dimensional'' region \\citep{FanLv2008}.\nThe previous works with restrictions on the polynomial dimensionality limit the usage of the spatial median under ultrahigh-dimensionality.\nMoreover, the previous results are all under elliptical distributions.\nThus, it is of great importance to establish asymptotic properties of the spatial median and investigate its applications under ultrahigh dimensionality and beyond elliptical distributions.\n\nIn this paper, we first establish Gaussian and bootstrap approximations hit hyperrectangles for the sample spatial median under the general model \\eqref{eq:Model_X} beyond elliptical distributions, which are valid when the dimension diverges exponential with the sample size.\nThey serve as the theoretical foundations of statistical inference for the location parameter based on the sample spatial median under ultrahigh dimensionality. \nConsistent simultaneous confidence intervals (SCIs) and global tests for the location parameters are established.\nWe also study multiple testing for every component of ${\\boldsymbol \\theta}$ based on $\\hat{{\\boldsymbol \\theta}}_n$.\nMotivated by simultaneous inference of ${\\boldsymbol \\theta}$, we define a high-dimensional asymptotic relative efficiency of the sample spatial median relative to the sample mean.\nMost importantly, our theoretical results guarantee the validity of the proposed inferential methods for exponentially divergent $p$.\nThe advantages of our proposed approaches have been justified by simulations and a real data analysis.\n\nThe main contributions of this paper are summarized as follow. \nFirstly, we establish SCIs for the location parameter ${\\boldsymbol \\theta}$ based on the sample spatial median $\\hat{{\\boldsymbol \\theta}}_n$, which is new in the literature. \nThe consistency of bootstrap approximation guarantees that the probability that the SCIs cover all components of the location parameter approaches the nominal confidence level under ultrahigh dimensionality.\nWe also propose a novel test for ultrahigh dimensional location parameter based on the maximum-norm of the sample spatial median. \nThe proposed test not only maintains nominal significance level asymptotically for exponentially divergent $p$, but also is more powerful under sparse alternatives compared to those based on $L_2$-norms \\citep{LiXu2022,Wang2015}.\nAs another major inference, we study multiple testing for every component of the location parameter, and the false discovery rate (FDR) can be well controlled combined with the Benjamini-Hochberg procedure based on the sample spatial median, which extends the existing methods based on the sample mean \\citep{Liu2014}. \nIn all inferential methods, the procedures based on the sample spatial median advances those based on the sample mean for heavy-tailed distributions.\n\nSecondly, this paper serves as the first work that provides Gaussian and bootstrap approximations for the sample spatial median under ultrahigh dimensionality. \nGaussian and bootstrap approximations for high-dimensional sample mean have received extensive attraction in the last decade. \n\\citet{Cher2013} and \\citet{Cher2017} established Gaussian and bootstrap approximations for the maxima of a sum of centered independent random vectors under Kolmogorov distance and on hyperrectangles, respectively. \nSee also \\citet{Chen2018}, \\citet{Cher2019} and \\citet{Cher2020} for related works.\nCompared to the sample mean, which has a simple linear form, \nthe theoretical difficulty for the sample spatial median lies in that it does not enjoy an explicit form.\nThis issue is addressed by deriving a novel Bahadur representation of the sample spatial median with a maximum-norm bound on the remainder term, which extends the results of \\citet{Zou2014}, \\citet{Cheng2019} and \\citet{LiXu2022} under elliptical distributions and polynomial dimensionality.\n\\textcolor{black}{Moreover, our results can be applied to the GMOM under reasonable conditions, and thus enhance the practice usage of GMOM.}\n\nThirdly, we propose a novel multiplier bootstrap method for the sample spatial median.\nInstead of multiplying on the loss function, which is generally the case for M-estimator \\citep{Imaizumi2021}, the multiplier is applied on the centralized $X_i$. Specifically, the bootstrap version of $\\hat{{\\boldsymbol \\theta}}_n$ is defined as $\\tilde{{\\boldsymbol \\theta}}_{n}= \\argmin_{{\\boldsymbol \\beta}\\in {\\mathbb R}^d}\\sum_{i=1}^n\\|Z_i(X_i-\\hat{{\\boldsymbol \\theta}}_n)-{\\boldsymbol \\beta}\\|$, where $Z_1,\\ldots,Z_n$ are the multipliers.\nThe multiplier bootstrap is consistent under ultrahigh dimensionality thanks to this novel formulation.\nThis is, however, different from the multiplier bootstrap method for the sample mean, which again has an explicit form \\citep{Cher2013,Cher2017}.\n\n\nThe rest of the paper is organized as follows. \nSection \\ref{sec:note} introduces model and assumptions. \nSection \\ref{sec:main} establishes Gaussian and bootstrap approximations to the distribution of the sample spatial median. \nStatistical inference for the location parameter based on the sample spatial median is presented in Section \\ref{sec:applications}.\nSection \\ref{sec:numerical} reports numerical results including simulations and a real data analysis. \nPreliminary lemmas and proofs of main results are presented in Appendix A of the supplementary material.\nProofs of preliminary lemmas and additional simulations are given in Appendices B and C of the supplementary material.\n\n\n\\textbf{Notation:}\nDenote\n$|x|_{\\infty}=\\max(|x_1|,\\ldots,|x_d|)$ as the maximum-norm of $x=(x_1,\\ldots,x_d)^{\\top}$.\nDenote $a_n\\lesssim b_n$ if $a_n\\leq Cb_n$ for a positive constant $C$, and $a_n\\asymp b_n$ means $a_n\\lesssim b_n$ and $b_n\\lesssim a_n$. \nFor $\\alpha>0$, let $\\psi_{\\alpha}(x)=\\exp(x^{\\alpha})-1$ be a function defined on $[0,\\infty)$. Then the Orlicz norm $\\|\\cdot\\|_{\\psi_{\\alpha}}$ of a random variable $X$ is defined as\n$\n\\|X\\|_{\\psi_{\\alpha}}=\\inf\\left\\{t>0, {\\mathbb{E}}\\{\\psi_{\\alpha}\\left(|X|\/t\\right)\\}\\leq 1\\right\\}.\n$\nWe use ${\\rm tr}(\\cdot)$ to denote the trace operator for square matrices.\nMoreover, we denote $I_p$ as the $p\\times p$ identity matrix.\nFor $a,b\\in\\mathbb{R}$, we write $a \\wedge b=\\min(a,b)$.\n\n\n\n\\section{Model and assumptions}\\label{sec:note}\n\nIn this paper, we consider a general multivariate model for the distribution $F_X$ such that $X_i$ admits the following stochastic representation:\n\\begin{eqnarray}\\label{eq:model}\n\tX_i = {\\boldsymbol \\theta} +\n\t\\nu_i\\Gamma U_i\\,,\n\\end{eqnarray}\nwhere ${\\boldsymbol \\theta}$ is the location parameter, $\\Gamma$ is a nonrandom and invertible $p\\times p$ matrix, \n$U_i$ is a $p$-dimensional random vector with independent~standardized components, and $\\nu_i$ is a nonnegative univariate random variable independent with the spatial sign of $U_i$.\nThe distribution of $X_i$ depends on $\\Gamma$ through the shape matrix $\\Omega=\\Gamma\\Gamma^{\\top}$.\n\n\\begin{remark}\n\tModel \\eqref{eq:model} covers many commonly used multivariate models and distribution families. \n\tFirst, the independent components model \\citep{Yao2015} follows \\eqref{eq:model} with $\\nu_i$ being a nonnegative constant.\n\tSecond, model \\eqref{eq:model} also includes elliptical distributions by choosing $U_i\\sim N(0, I_{p})$ and $\\nu_i=\\xi_i\/\\|U_{i}\\|$ for some nonnegative random variable $\\xi_i$ independent of $U_{i}$. In this case, $\\nu_{i}$ is independent of the spatial sign of $U_{i}$, but not $U_{i}$.\n\tThe independent components model has received great extension in high-dimensional data analysis as well as signal processing and machine learning \\citep{Oja2001}.\n\tIn addition, the elliptical distribution family covers many non-Gaussian distributions such as multivariate $t$-distribution, multivariate logistic distribution, and so on. It is commonly adopted in the literature on studying the sample spatial median\n\t\\citep{Cheng2019,LiXu2022,Zou2014}. \n\t\\textcolor{black}{In terms of the GMOM, if the data are from the independent components model, the subsample means satisfy model \\eqref{eq:model} clearly. In addition, some subfamilies of elliptical distributions are closed under convolution, and thus the subsample means also follow model \\eqref{eq:model}. Our results can be applied to the GMOM estimator directly in those cases.}\n\n\n\\end{remark}\n\nFor $i=1,\\ldots,n$, and  $k=1,2,3,4$, denote\n\\begin{eqnarray}\\label{eq:nota}\n\tW_i = S(X_i-{\\boldsymbol \\theta}) \\text{~~and~~} R_i=\\|X_i-{\\boldsymbol \\theta}\\|\\,\n\\end{eqnarray}\nas the spatial-sign and radius of $X_i-{\\boldsymbol \\theta}$, where $S(X)=\\|X\\|^{-1}X{\\mathbb{I}}(X\\neq 0)$ is the multivariate sign function with ${\\mathbb{I}}(\\cdot)$ being the indicator function. \nThus,\n$\\hat{{\\boldsymbol \\theta}}_n$ satisfies\n$\\sum_{i=1}^{n}S(X_i-\\hat{{\\boldsymbol \\theta}}_{n}) = 0\\,. $\n\nDenote $U_{i}=(U_{i,1},\\ldots,U_{i,p})^{\\top}$,\nwe impose the following three conditions.\n\n\n\\begin{condition}\\label{c1}\n\t$U_{i,1},\\ldots,U_{i,p}$ are i.i.d.~symmetric random variables with ${\\mathbb{E}}(U_{i,j})=0$, ${\\mathbb{E}}(U_{i,j}^2)=1$, and\n\t$\\|U_{i,j}\\|_{\\psi_{\\alpha}}\\leq c_0$ with some constant $c_0>0$ and $1\\leq \\alpha\\leq 2$.\n\\end{condition}\n\n\t\\begin{condition}\\label{c2}\n\t\n\t\n\t\n\t\n\t\tThe moments $\\zeta_{k}={\\mathbb{E}}(R_i^{-k})$ for $k=1,2,3,4$ exist for large enough $p$. In addition, there exist two positive constants $\\underline{b}$ and ${\\bar{B}}$ such that\n\t\t$\n\t\t{\\underline b}\\leq  \\limsup_{p}{\\mathbb{E}}(R_i\/\\sqrt{p})^{-k}\\leq {\\bar{B}}\n\t\t$ for $k=1,2,3,4$.\n\t\\end{condition}\n\t\n\t\\begin{condition}\\label{c3}\n\t\tThe shape matrix $\\Omega=(\\omega_{j\\ell})_{p\\times p}$ satisfies ${\\rm tr}(\\Omega)=p$ and it belongs to the following class:\n\t\t\\begin{eqnarray}\n\t\t\t\\notag & & {\\mathcal U}(a_0(p), \\underline{m}, \\bar{M})  = \\left\\{\\Omega: \\underline{m} \\leq \\omega_{jj}\\leq \\bar{M},~  \\sum_{\\ell=1}^{p}|\\omega_{j\\ell}|\\leq a_0(p), \\text{~~for all~} j=1,\\ldots,p\\right\\}\\,,\n\t\t\\end{eqnarray}\n\t\twhere $\\underline{m}\\leq \\bar{M}$ are bounded positive constants.\n\t\\end{condition}\n\t\n\t\\begin{remark}\n\t\tIn Condition \\ref{c1}, the symmetric assumption is to ensure that ${\\boldsymbol \\theta}$ in model \\eqref{eq:model} coincides with the population spatial median, which minimizes $L({\\boldsymbol \\beta})={\\mathbb{E}}(\\|X-{\\boldsymbol \\beta}\\|-\\|X\\|)$.\n\t\tIt is obvious that Condition \\ref{c1} is satisfied by elliptical distributions with $U_i\\sim N(0,I_p)$.\n\t\tThe condition $\\|U_{i,j}\\|_{\\psi_{\\alpha}}\\leq c_0$ implies that $U_{i,j}$ has a sub-exponential distribution.\n\t\tIt is worth highlighting that with slight modification of the proofs of main theorems, the i.i.d.~condition on $U_{i,1},\\ldots,U_{i,p}$ can be weaken by replacing Condition \\ref{c1} with the following assumption:\n\t\t$U_{i,1},\\ldots,U_{i,p}$ are independent symmetric random variables with ${\\mathbb{E}}(U_{i,j})=0$, ${\\mathbb{E}}(U_{i,j}^2)=1$ for all $j=1,\\ldots,p$, and\n\t\t$\\sup_{1\\leq j\\leq p}\\|U_{i,j}\\|_{\\psi_{\\alpha}}\\leq c_0$ with some constant $c_0>0$ and $1\\leq \\alpha\\leq 2$.\n\t\\end{remark}\n\t\n\t\\begin{remark}\n\t\tThe condition ${\\underline b}\\leq  \\limsup_{p}{\\mathbb{E}}(R_i\/\\sqrt{p})^{-k}\\leq {\\bar{B}}$ indicates that $\\zeta_{k}\\asymp p^{-k\/2}$ for $k=1,2,3,4$. It is introduced to avoid $X_i$ from concentrating too much near ${\\boldsymbol \\theta}$. For elliptical distributions, it is a generalization of Assumption 1 of \\citet{Zou2014}, which is satisfied by many common distributions.\n\t\tFor the independent components model, Condition \\ref{c2} is equivalent to that\n\t\t$\n\t\t{\\underline b}\\leq  \\limsup_{p}{\\mathbb{E}}(\\|\\Gamma U_{i}\\|\/\\sqrt{p})^{-k}\\leq {\\bar{B}}\\,.\n\t\t$\n\t\tAccording to Lemma \\ref{lemma:SM_moments} in Appendix A, ${\\mathbb{E}}(\\|\\Gamma U_{i}\\|^{k}) = p^{k\/2}\\{1+o(1)\\}$ for $k=1,2,3,4$. Then the Cauchy-Schwarz inequality implies that\n\t\t${\\mathbb{E}}(\\|\\Gamma U_{i}\\|^{-k}) \\geq \\{{\\mathbb{E}}(\\|\\Gamma U_{i}\\|^{k})\\}^{-1} = p^{-k\/2}\\{1+o(1)\\}\\,,$\n\t\tfrom which we know ${\\mathbb{E}}(\\|\\Gamma U_{i}\\|^{-k})\\gtrsim p^{-k\/2}$. Furthermore, denote $\\Gamma_{j}$ as the $j$th row of $\\Gamma$, then by the inequality of harmonic and quadratic means,\n\t\t\\begin{eqnarray}\n\t\t\t\\notag \\notag p^{2}\\|\\Gamma U_{i}\\|^{-4} = \\left\\{\\frac{p}{(\\Gamma_{1}U_{i})^2+\\cdots+(\\Gamma_{p}U_{i})^2}\\right\\} \\leq \\frac{(\\Gamma_{1}U_{i})^{-4}+\\cdots+(\\Gamma_{p}U_{i})^{-4}}{p}\\,.\n\t\t\\end{eqnarray}\n\t\tIt follows that ${\\mathbb{E}}(\\|\\Gamma U_{i}\\|^{-4})\\lesssim p^{-2}$ if ${\\mathbb{E}}\\{(\\Gamma_{1}U_{i})^{-4}\\},\\ldots,{\\mathbb{E}}\\{(\\Gamma_{p}U_{i})^{-4}\\}$ are uniformly bounded, and from which ${\\mathbb{E}}(\\|\\Gamma U_{i}\\|^{-k})\\lesssim p^{-k\/2}$ by Jensen's inequality. Thus, Condition \\ref{c2} is satisfied by the independent components models as long as $\\Gamma_{1}U_{i},\\ldots,\\Gamma_{p}U_{i}$ are not concentrating too much near $0$.\n\t\tSee also discussions in\n\t\t\\citet{Cardot2013} on similar conditions.\n\t\\end{remark}\n\t\n\t\\begin{remark}\n\t\tIt is noticed that the shape matrix $\\Omega$ is only well defined up to a scalar multiple, the condition ${\\rm tr}(\\Omega)=p$ is used to regularize $\\Omega$ to make model \\eqref{eq:model} identifiable.\n\t\tThe class ${\\mathcal U}(a_0(p), \\underline{m}, \\bar{M})$\n\t\tcovers a wide range of symmetric square matrices, and it is commonly adopted in the literature on high-dimensional analysis. \n\t\tFor example, a similar matrix class is introduced in \\citet{Bic2008}. The condition $\\underline{m} \\leq \\omega_{jj}\\leq \\bar{M}$ requires bounded diagonal elements. \n\t\tThe order of $a_0(p)$, which will be specified later, controls the orders of the off-diagonal elements of $\\Omega$.\n\t\\end{remark}\n\t\n\t\\section{Gaussian and bootstrap approximations}\\label{sec:main}\n\t\n\t\\subsection{Bahadur representation and Gaussian approximation}\\label{sec:02}\n\t\n\n\tIn this section, we establish Gaussian approximation for $\\hat{{\\boldsymbol \\theta}}_n$, which is valid when $p$ diverges exponentially over $n$.\n\tThe following lemma offers a Bahadur representation of $\\hat{{\\boldsymbol \\theta}}_n$, and it severs as the foundation of the Gaussian approximation result in Theorem \\ref{theo1}.\n\t\n\t\\begin{lemma}\\label{lem:Br}\n\t\t(Bahadur representation) Assume Conditions \\ref{c1}, \\ref{c2} and \\ref{c3} with $a_0(p)\\asymp p^{1-\\delta}$ for some positive constant\n\t\t$\\delta\\leq 1\/2$ hold. If\n\t\t$\\log p=o(n^{1\/3})$ and $\\log n=o(p^{1\/3 \\wedge \\delta})$, then\n\t\t\\[\n\t\tn^{1\/2}(\\hat{{\\boldsymbol \\theta}}_{n}-{\\boldsymbol \\theta})=n^{-1\/2}\\zeta_{1}^{-1}\\sum_{i=1}^nW_i+C_n\\,,\n\t\t\\]\n\t\twhere $|C_n|_{\\infty}=O_p\\{n^{-1\/4}\\log^{1\/2}(np)+ p^{-(1\/6 \\wedge \\delta\/2)}\\log^{1\/2}(np)\\}$.\n\t\\end{lemma}\n\t\n\t\\begin{remark}\n\t\tTo the best of our knowledge, Lemma \\ref{lem:Br} serves as the first result that offers the Bahadur representation of the sample spatial median with a maximum-norm bound on the remainder term.\n\t\tIn \\citet{Zou2014} and \\citet{Cheng2019}, the same expansion with the remainder term $C_n$ satisfies $\\|C_n\\|=o_{p}(\\zeta_{1}^{-1})$ was obtained, and their result was improved to $\\|C_n\\|=o_p(1)$ in \\citet{LiXu2022}, by replacing $\\zeta_{1}$ with $n^{-1}\\sum_{i=1}^{n}R_i^{-1}$ in the linear term, but under a more restricted condition that $p$ and $n$ are of the same order.\n\t\tIt is worth noticing that the previous results \\citep{Cheng2019,LiXu2022,Zou2014} are all derived under elliptical distributions.\n\t\n\t\\end{remark}\n\t\n\tLet ${\\mathcal A}^{\\mathrm{re}}=\\{\\prod_{j=1}^{p}[a_j, b_j]: -\\infty\\leq a_j\\leq b_j\\leq \\infty, j=1,\\ldots,p\\}$ be the class of rectangles in $\\mathbb{R}^{p}$.\n\tWith the Bahadur representation in Lemma \\ref{lem:Br} on hand, we establish the following Gaussian approximation result for $\\hat{{\\boldsymbol \\theta}}_n$ over hyperrectangles.\n\n\t\\begin{theorem}\\label{theo1}\n\t\t\\textit{(Gaussian approximation)}\n\t\tAssume Conditions \\ref{c1}, \\ref{c2} and \\ref{c3} with $a_0(p)\\asymp p^{1-\\delta}$ for some positive constant\n\t\t$\\delta\\leq 1\/2$ hold. If\n\t\t$\\log p=o(n^{1\/5})$ and $\\log n=o(p^{1\/3 \\wedge \\delta})$,\n\t\n\t\n\t\n\t\tthen\n\t\t\\[\n\t\t\\rho_n({\\mathcal A}^{\\mathrm{re}}) = \\sup_{A \\in {\\mathcal A}^{{\\rm re}}}\\left|{\\mathbb{P}}\\{{n}^{1\/2}(\\hat{{\\boldsymbol \\theta}}_{n}-{\\boldsymbol \\theta}) \\in A \\}-{\\mathbb{P}}\\left(G \\in A\\right)\\right|\\rightarrow 0\\,\n\t\t\\]\n\t\tas $n\\to\\infty$, where $G\\sim N(0, \\zeta_{1}^{-2}{\\mathbb{B}})$ with ${\\mathbb{B}}={\\mathbb{E}}(W_1W_1^{\\top})$.\n\t\n\\end{theorem}\n\nThe Gaussian approximation for $\\hat{{\\boldsymbol \\theta}}_n$\nindicates that the probabilities ${\\mathbb{P}}\\{n^{1\/2}(\\hat{{\\boldsymbol \\theta}}_{n}-{\\boldsymbol \\theta}) \\in A\\}$ can be approximated by that of a centered Gaussian random vector with covariance matrix $\\zeta_{1}^{-2}{\\mathbb{B}}$ for hyperrectangles $A\\in{\\mathcal A}^{\\mathrm{re}}$.\nTheorem \\ref{theo1} allows for an exponentially divergent $p$, which fits the ultrahigh dimensional setting. Compared to the asymptotic normality of\n$\\|\\hat{{\\boldsymbol \\theta}}_{n}-{\\boldsymbol \\theta}\\|^2$ in \\citet{LiXu2022}, in which $p$ is assumed to have the same order as $n$, the Gaussian approximation result in Theorem \\ref{theo1} requires much weaker conditions on the rates of $n$ and $p$.\n\n\\begin{remark}\\label{remark:B}\n\tLet ${\\mathbb{B}}_{j\\ell}$ be the $(j,\\ell)$th element of ${\\mathbb{B}}$. According to Lemma \\ref{lemma:05} (iii) in Appendix A, $\\zeta_{1}^{-2}{\\mathbb{B}}_{j\\ell} = \\zeta_{1}^{-2}p^{-1}\\omega_{j,\\ell} + O(p^{-\\delta\/2})$ for all $1\\leq j,\\ell\\leq p$. Thus, the covariance matrix of $G$ in Theorem \\ref{theo1} is asymptotically proportional to the shape matrix $\\Omega$.\n\\end{remark}\n\n\\begin{remark}\\label{remark:M-estimator}\n\tAs the sample spatial median is a special M-estimator, \n\tGaussian approximation for M-estimator in \\citet{Imaizumi2021} is potentially applicable to the spatial median under high-dimensionality.\n\tHowever, it is worth highlighting that the results in \\citet{Imaizumi2021} cannot be applied to our framework. \n\tTo be precise, Assumption 1 (A3) in \\citet{Imaizumi2021} assumes that there exist constants $C>0$ and $\\alpha\\in(0,2)$ such that $\\log \\mathcal{N}(\\varepsilon, \\Theta, \\|\\cdot\\|)\\leq C\\varepsilon^{-\\alpha}$ holds for all $\\varepsilon\\in(0,1)$, where $\\Theta$ is the parameter space, and $\\mathcal{N}(\\varepsilon, \\Theta, \\|\\cdot\\|)$ is the $\\varepsilon$-covering number of $\\Theta$ under the Euclidean norm $\\|\\cdot\\|$ \\citep{VanderVaart1996}. \n\tWhen $\\Theta$ is a compact subset of $\\mathbb{R}^{p}$, $\\mathcal{N}(\\varepsilon, \\Theta, \\|\\cdot\\|)$ is of order $O(\\varepsilon^{-p})$. \n\tIn this case, $\\log \\mathcal{N}(\\varepsilon, \\Theta, \\|\\cdot\\|)\\leq C\\varepsilon^{-\\alpha}$ cannot be satisfied when $p\\to\\infty$. \n\tThus, our theoretical findings are independent of those in \\citet{Imaizumi2021}.\n\\end{remark}\n\n\\iffalse\n\\begin{remark}\\label{remark:04}\n\tWhen $p$ is fixed, the classical CLT for $\\hat{{\\boldsymbol \\theta}}_n$ \\citep{Oja2010} is\n\t$ n^{1\/2}(\\hat{{\\boldsymbol \\theta}}_{n}-{\\boldsymbol \\theta}) \\to N(0, {\\mathbb{C}}^{-1}{\\mathbb{B}} {\\mathbb{C}}^{-1}) $\n\tin distribution, where ${\\mathbb{C}}={\\mathbb{E}}\\{R_i^{-1}(I_p-W_iW_i^{\\top} )\\}$. Compared to Theorem \\ref{theo1}, the asymptotic covariance matrix changes from ${\\mathbb{C}}^{-1}{\\mathbb{B}} {\\mathbb{C}}^{-1}$ to $\\zeta_{1}^{-2}{\\mathbb{B}}$ under high-dimensionality. \n\tIn fact, \n\n\t$\n\t\\left\\|{\\mathbb{C}}-\\zeta_{1}I_p \\right\\|_2= \\left\\|{\\mathbb{E}}(R_i^{-1}W_iW_i^{\\top})\\right\\|_2 \\lesssim p^{-1}{\\mathbb{E}}(R_i^{-1})=p^{-1}\\zeta_{1}\\,\n\t$\n\tin probability, where the last inequality can be shown similar to Lemmas A.5 and A.6 of \\citet{Wang2015}.\n\tNote that $p^{-1}\\zeta_{1}\/ \\|\\zeta_{1}I_p\\|_2\\to0$ under Condition \\ref{c2}, the distance between ${\\mathbb{C}}$ and $\\zeta_{1}I_p$, so as ${\\mathbb{C}}^{-1}{\\mathbb{B}} {\\mathbb{C}}^{-1}$ and $\\zeta_{1}^{-2}{\\mathbb{B}}$, is negligible under the matrix 2-norm as $p\\to\\infty$.\n\\end{remark}\n\\fi\n\nTheorem \\ref{theo1} immediately implies the following corollary\nsince the Kolmogorov distance of sup-norm is a subset of $\\mathcal{A}^{\\mathrm{re}}$ corresponding to max-hyperrectangles in $\\mathbb{R}^{p}$. \n\\begin{corollary}\\label{co1}\n\tUnder the conditions assumed in Theorem \\ref{theo1}, as $n\\to\\infty$,\n\t\\[\n\t\\rho_n=\\sup_{t\\in{\\mathbb{R}}}\\left|{\\mathbb{P}}(n^{1\/2}|\\hat{{\\boldsymbol \\theta}}_{n}-{\\boldsymbol \\theta}|_{\\infty}\\leq t)- {\\mathbb{P}}(|G|_{\\infty}\\leq t)\\right|\\rightarrow  0.\n\t\\]\n\\end{corollary}\n\\subsection{Multiplier bootstrap approximation}\\label{sec:boot}\n\nTheorem \\ref{theo1} allows us to approximate the distribution of $n^{1\/2}(\\hat{{\\boldsymbol \\theta}}_{n}-{\\boldsymbol \\theta})$ by that of $G$ hit hyperrectangles, where $G\\sim N(0, \\zeta_{1}^{-2}{\\mathbb{B}})$. \nHowever, it cannot be used directly in statistical inference for ${\\boldsymbol \\theta}$ as\nthe quantity $\\zeta_{1}$ and the matrix ${\\mathbb{B}}$ depend on the underlying distribution $F_X$ and are thus unknown. \nTo solve this issue, we propose an easy-to-implement bootstrap method to approximate the distribution of $n^{1\/2}(\\hat{{\\boldsymbol \\theta}}_{n}-{\\boldsymbol \\theta})$.\n\nLet $Z_1,\\ldots,Z_n$ be a sequence of i.i.d.~random variables with mean zero and unit variance. Define the bootstrap version of the sample spatial median as\n\\begin{align}\\label{btheta}\n\t\\tilde{{\\boldsymbol \\theta}}_{n}= \\argmin_{{\\boldsymbol \\beta}\\in {\\mathbb R}^d}\\sum_{i=1}^n\\|Z_i(X_i-\\hat{{\\boldsymbol \\theta}}_n)-{\\boldsymbol \\beta}\\|\\,.\n\\end{align}\nThen, the distribution of $n^{1\/2}\\tilde{{\\boldsymbol \\theta}}_{n}$ conditional on $X_1,\\ldots,X_n$ is used to approximate that of $n^{1\/2}(\\hat{{\\boldsymbol \\theta}}_{n}-{\\boldsymbol \\theta})$.\nThis algorithm is called the multiplier bootstrap, and $Z_1,\\ldots,Z_n$ are the multiplier weights.\n\nRegarding the proof of Lemma \\ref{lem:SM_boot_br} in Appendix B, it is preferred that the multiplier weights $Z_1,\\ldots,Z_n$ are bounded and satisfy ${\\mathbb{E}}(Z_{i}^{-2})<\\infty$. \nThus, we choose the Rademacher variables as the multipliers \\citep{Cher2019}, that is, ${\\mathbb{P}}(Z_i=1)={\\mathbb{P}}(Z_i=-1)=1\/2$.\n\\begin{theorem}\\label{theo:boot}\n\t\\textit{(Bootstrap approximation)}\n\tUnder the conditions assumed in Theorem \\ref{theo1},\n\t\\[\n\t\\rho_n^{\\mathrm{MB}}({\\mathcal A}^{\\mathrm{re}})=\\sup_{A \\in {\\mathcal A}^{\\mathrm{re}}}\\left|{\\mathbb{P}}\\{n^{1\/2}(\\hat{{\\boldsymbol \\theta}}_{n}-{\\boldsymbol \\theta})\\in A\\}- {\\mathbb{P}}^{*}(n^{1\/2}\\tilde{{\\boldsymbol \\theta}}_{n} \\in A)\\right|\\rightarrow 0\n\t\\]\n\tin probability as $n\\to\\infty$,\n\twhere ${\\mathbb{P}}^{*}$ denotes the conditional probability given $X_1,\\ldots,X_n$.\n\\end{theorem}\n\nUnder the same conditions on the divergence rates of $n$ and $p$ as in Theorem \\ref{theo1}, Theorem \\ref{theo:boot} validates that conditional on $X_1,\\ldots,X_n$, the distribution of the bootstrap sample spatial median $\\tilde{{\\boldsymbol \\theta}}_n$ approximates that of $\\hat{{\\boldsymbol \\theta}}_n$ consistently over hyperrectangles.\n\n\\begin{remark}\n\tThe proof of Theorem \\ref{theo:boot} is nontrivial and does not follow directly from existing results since $\\tilde{{\\boldsymbol \\theta}}_n$ has no explicit form, which is different from the multiplier bootstrap methods for high-dimensional sample mean that have been analysed in the literature.\n\tThe key step in the proof is to obtain a Bahadur representation of $\\tilde{{\\boldsymbol \\theta}}_n$ similar as $\\hat{{\\boldsymbol \\theta}}_n$ in Lemma \\ref{lem:Br}. Specifically, we show that $n^{1\/2}\\tilde{{\\boldsymbol \\theta}}_{n}=n^{-1\/2}\\zeta_{1}^{-1}\\sum_{i=1}^n Z_i W_i+{\\tilde C}_n$ with $|{\\tilde C}_n|_{\\infty}=O_p\\{n^{-1\/4}\\log^{1\/2} (np)+p^{-(1\/6\\wedge\\delta\/2)}\\log^{1\/2} (np)\\}$ in Lemma \\ref{lem:SM_boot_br} in Appendix A.\n\n\\end{remark}\n\nThe next corollary is an immediate consequence of Theorem \\ref{theo:boot}.\n\\begin{corollary}\\label{co2}\n\tUnder the conditions assumed in Theorem \\ref{theo:boot}, as $n\\to\\infty$,\n\t\\[\n\t\\rho_n^{\\mathrm{MB}}=\\sup_{t\\in{\\mathbb{R}}}\\left|{\\mathbb{P}}\\{n^{1\/2}|\\hat{{\\boldsymbol \\theta}}_{n}-{\\boldsymbol \\theta}|_{\\infty}\\leq t\\}- {\\mathbb{P}}^{*}(n^{1\/2}|\\tilde{{\\boldsymbol \\theta}}_{n}|_{\\infty}\\leq t)\\right|\\rightarrow  0 \\text{~~in probability.}\n\t\\]\n\t\n\\end{corollary}\n\n\n\\section{Statistical inference}\\label{sec:applications}\n\n\nThe Gaussian and multiplier bootstrap approximations for the sample spatial median\nenable many statistical inferential methods for ultrahigh dimensional  population location parameter.\nIn this section, we present the following statistical inferences: simultaneous confidence intervals (SCIs) and global tests for the population location parameter, multiple testing for every component of ${\\boldsymbol \\theta}$, and high-dimensional asymptotic relatively efficient of the sample spatial median compared to the sample mean.\n\n\n\\subsection{Simultaneous confidence intervals}\\label{sec:SCIs}\n\nWe are interested in building SCIs for all components of ${\\boldsymbol \\theta}=(\\theta_1,\\ldots,\\theta_p)^{\\top}$.\nCorollary \\ref{co2} motivates the following way of constructing SCIs for ${\\boldsymbol \\theta}$.\nGiven a nominal confidence level $1-\\tau$,\ndefine the set ${\\mathcal C}_{\\tau}$ as\n\\[\n{\\mathcal C}_{\\tau}=\\left\\{{\\boldsymbol \\theta}\\in {\\mathbb{R}}^p, n^{1\/2}|\\hat{{\\boldsymbol \\theta}}_{n}-{\\boldsymbol \\theta}|_{\\infty}<q^{B}_{1-\\tau}\\right\\},\n\\]\nwhere $q^{B}_{1-\\tau}$ is the $(1-\\tau)$th quantile of $n^{1\/2}|\\tilde{{\\boldsymbol \\theta}}_{n}|_{\\infty}$ given $X_1,\\ldots,X_n$.\nDenote $\\hat{{\\boldsymbol \\theta}}_n=(\\hat{\\theta}_{n,1},\\ldots,\\hat{\\theta}_{n,p})^{\\top}$, the confidence intervals are $[\\theta_{n,j}^{-},\\theta_{n,j}^{+}]$ for $j=1,\\ldots,p$, where\n$$\n\\theta_{n,j}^{-}=\\hat{\\theta}_{n,j}-n^{-1\/2}q^{B}_{1-\\tau} \\text{~~and~~} \\theta_{n,j}^{+}=\\hat{\\theta}_{n,j}+n^{-1\/2}q^{B}_{1-\\tau}.\n$$\nThe next theorem shows that $\\mathcal{C}_{\\tau}$ preserves the nominal simultaneous confidence level $1-\\tau$ asymptotically under ultrahigh dimensionality.\n\n\\begin{theorem}\\label{theo:SCIs}\n\tSuppose the conditions of Theorem \\ref{theo:boot} hold, then ${\\mathbb{P}}({\\boldsymbol \\theta}\\in {\\mathcal C}_{\\tau})\\rightarrow 1-\\tau$ as $n\\to\\infty$. Equivalently,\n\t${\\mathbb{P}}(\\theta_j\\in [\\theta_{n,j}^{-},\\theta_{n,j}^{+}] \\text{~~for all~} 1\\leq j\\leq p)\\rightarrow 1-\\tau$ as $n\\to\\infty$.\n\\end{theorem}\n\n\\begin{remark}\n\tUnlike the fixed dimensional setting, $n^{1\/2}|\\tilde{{\\boldsymbol \\theta}}_{n}|_{\\infty}$ and $n^{1\/2}|\\hat{{\\boldsymbol \\theta}}_{n}-{\\boldsymbol \\theta}|_{\\infty}$ are maxima of divergent numbers of variables, and their quantiles are generally divergent as $p\\to\\infty$.\n\tThus, Theorem \\ref{theo:SCIs} is not a direct consequence of Corollary \\ref{co2}.\n\tTo ascertain the consistency of $\\mathcal{C}_{\\tau}$ theoretically, we show that, with probability approaching one, $q_{1-\\tau}^{B}$ is bounded by two quantiles of $n^{1\/2}|\\hat{{\\boldsymbol \\theta}}_{n}-{\\boldsymbol \\theta}|_{\\infty}$ with quantile levels close enough to $1-\\tau$ using an anti-concentration inequality for divergent random sequences.\n\\end{remark}\n\n\\iffalse\nWe summarise the approach of building SCIs for ${\\boldsymbol \\theta}$ in Algorithm \\ref{al1}.\n\\RestyleAlgo{ruled}\n\\begin{algorithm}\\label{al1}\n\t\\caption{Construction of SCIs for ${\\boldsymbol \\theta}$}\n\t\\SetAlgoLined\n\t\\KwData{Data $X_1, \\ldots, X_n$; $\\alpha$, confidence level; $B$, number of bootstrap iterations}\n\t\\KwResult{Simultaneous confidence intervals: $[\\theta_{n,j}^{-},\\theta_{n,j}^{+}]$ for $j=1,\\ldots,p$}\n\t\\BlankLine\n\t\n\tCompute $\\hat{{\\boldsymbol \\theta}}_n=(\\hat{\\theta}_{n,1},\\ldots,\\hat{\\theta}_{n,p})^{\\top}$ by minimizing $L_n({\\boldsymbol \\beta})$ defined in \\eqref{eq:btheta_criteria};\n\t\n\t\n\t\\For{{$b\\leftarrow 1$ \\KwTo $B$}}{\n\t\tgenerate independent Rademacher random variables $Z_1^{(b)}, \\ldots, Z_n^{(b)}$;\n\t\t\n\t\tcompute $\\tilde{{\\boldsymbol \\theta}}_{n}^{(b)}$ by minimizing \\eqref{btheta} using $Z_1^{(b)}, \\ldots, Z_n^{(b)}$ as the multiplier weights;\n\t\t\n\t\tcompute $n^{1\/2}|\\tilde{{\\boldsymbol \\theta}}_{n}^{(b)}|_{\\infty},$;\n\t}\n\t\n\tCompute $q^{B}_{1-\\alpha}$ as the $(1-\\alpha)$-th quantiles of $n^{1\/2}|\\tilde{{\\boldsymbol \\theta}}_{n}^{(1)}|_{\\infty}, \\ldots, |n^{1\/2}\\tilde{{\\boldsymbol \\theta}}_{n}^{(B)}|_{\\infty}$;\n\t\n\t\\For{{$j\\leftarrow 1$ \\KwTo $p$}}{\n\t\tcompute $\\theta_{n,j}^{-}=\\hat{\\theta}_{n,j}-n^{-1\/2}q^{B}_{1-\\alpha}$;\n\t\t\n\t\tcompute $\\theta_{n,j}^{+}=\\hat{\\theta}_{n,j}+n^{-1\/2}q^{B}_{1-\\alpha}$;\n\t}\n\\end{algorithm}\n\\fi\n\n\n\\begin{remark}\\label{remark:SCIs_mean}\n\tThe Gaussian approximation for  the sample mean $\\bar{X}_n=n^{-1}\\sum_{i=1}^{n}X_i$ \\citep{Cher2013,Cher2017,Cher2019} indicate that if $\\log p = o(n^{1\/5})$,\n\t\\begin{eqnarray}\\label{eq:gaussian_approx_mean}\n\t\n\t\t\\sup_{t\\in{\\mathbb{R}}}\\left|{\\mathbb{P}}(n^{1\/2}|\\bar{X}_{n}-{\\boldsymbol \\theta}|_{\\infty}\\leq t)- {\\mathbb{P}}(|G_0|_{\\infty}\\leq t)\\right|\\rightarrow  0\n\t\\end{eqnarray}\n\tas $n\\to\\infty$ under some moderate conditions, where $G_0\\sim N(0, \\Sigma)$ with $\\Sigma={\\mathbb{E}}(XX^{\\top})$.\n\tDefine $X_{i}^{*} = Z_i(X_i-\\bar{X}_n)$ for $i=1,\\ldots,n$, where $Z_1,\\ldots,Z_n$ are the Rademacher weights. Denote $\\bar{X}_{n}^{*}=n^{-1}\\sum_{i=1}^{n}X_{i}^{*}$,\n\tit has been shown in \\citet{Cher2019} that\n\t\\begin{eqnarray}\\label{eq:boot_mean}\n\t\t& \\sup_{t\\in{\\mathbb{R}}}\\left|{\\mathbb{P}}(n^{1\/2}|\\bar{X}_{n}-{\\boldsymbol \\theta}|_{\\infty}\\leq t)- {\\mathbb{P}}^{*}(n^{1\/2}|\\bar{X}_{n}^{*}|_{\\infty}\\leq t)\\right|\\rightarrow  0 &\n\t\\end{eqnarray}\n\tin probability as $n\\to\\infty$ when $\\log p = o(n^{1\/5})$.\n\tBased on \\eqref{eq:boot_mean}, define\n\t$$\n\t{\\mathcal C}_{\\tau}^{\\prime}=\\left\\{{\\boldsymbol \\theta}\\in {\\mathbb{R}}^p, n^{1\/2}|\\bar{X}_{n}-{\\boldsymbol \\theta}|_{\\infty}<q^{B\\prime}_{1-\\tau}\\right\\},\n\t$$\n\twhere $q^{B\\prime}_{1-\\tau}$ is the $(1-\\tau)$th quantile of $n^{1\/2}|\\bar{X}_{n}^{*}|_{\\infty}$ conditional on $X_1,\\ldots,X_n$.\n\tThen $\\mathcal{C}_{\\tau}^{\\prime}$ is also an asymptotic $1-\\tau$ SCIs for ${\\boldsymbol \\theta}$.\n\tBased on the discussion in Section \\ref{sec:ARE}, $\\mathcal{C}_{\\tau}$ has advantage (relative shorter intervals) over $\\mathcal{C}_{\\tau}^{\\prime}$ under heavy-tailed distributions.\n\tWe refer to Section \\ref{sec:simulations} for finite-sample justifications on this.\n\\end{remark}\n\n\n\\subsection{Global tests for high-dimensional location parameters}\\label{sec:global_tests}\n\nIn this section, we propose a novel approach for global tests on high-dimensional location parameters.\nLet ${\\boldsymbol \\theta}_0$ be a known $p$-dimensional vector, we are interested in testing\n\\begin{eqnarray}\\label{eq:one_sample}\n\tH_0: {\\boldsymbol \\theta} = {\\boldsymbol \\theta}_0 \\text{~~versus~~} H_1:{\\boldsymbol \\theta}\\neq{\\boldsymbol \\theta}_0.\n\\end{eqnarray}\n\nTheorems \\ref{theo1} and \\ref{theo:boot} motivate us proposing a maximum-norm type test statistic.\nDefine\n\\begin{align}\\label{test}\n\tT_{n} = n^{1\/2}|\\hat{{\\boldsymbol \\theta}}_{n}-{\\boldsymbol \\theta}_0|_{\\infty}\n\\end{align}\nas the test statistic, and $H_0$ is rejected when $T_{n}$ is larger than a critical value.\nWe can use the multiplier bootstrap to approximate the distribution of $T_n$ under $H_0$.\nSpecifically, with a nominal significance level $\\tau$, the null hypothesis is rejected if $T_n>q_{1-\\tau}^{B}$.\nTheorem \\ref{theo:SCIs} guarantees that the test based on $T_n$ maintains nominal significance level asymptotically under ultrahigh dimensionality, that is, ${\\mathbb{P}}(T_n>q^B_{1-\\tau}\\mid H_0)\\rightarrow \\tau$ as $n\\to\\infty$ when $\\log p = o(n^{1\/5})$.\n\n\\begin{remark}\\label{remark:06}\n\n\tAn alternative test for \\eqref{eq:one_sample} can be constructed based on $\\bar{X}_n$ by defining the test statistic as\n\n\t$ T_{\\mathrm{Mean}} = n^{1\/2}|\\bar{X}_{n}-{\\boldsymbol \\theta}_0|_{\\infty}$.\n\n\tThen, the null hypothesis is rejected if $T_{\\mathrm{Mean}}>q_{1-\\tau}^{B\\prime}$.\n\tThe test based on $T_n$ can be deemed as a nonparametric extension of the test based on $T_{\\mathrm{Mean}}$ .\n\tAs $\\hat{{\\boldsymbol \\theta}}_n$ is more efficient than $\\bar{X}_n$ for simultaneous inference of ${\\boldsymbol \\theta}$ under heavy-tailed distributions as discussed in Section \\ref{sec:ARE}, we expect that the proposed test based on $T_n$ is more powerful than that based on $T_{\\mathrm{Mean}}$ in those cases.\n\tThis has been reflected by the simulation results in Appendix C of the supplementary material.\n\\end{remark}\n\nThe next theorem summarises the asymptotic power of the proposed test based on $T_n$.\n\\begin{theorem}\\label{theo:power}\n\tSuppose the conditions of Theorem \\ref{theo:boot} hold.\n\tFor any given $0<\\tau<1$, if\n\t$|{\\boldsymbol \\theta}-{\\boldsymbol \\theta}_0|_{\\infty}\\geq C\\log^{1\/2}(\\tau^{-1}) n^{-1\/2}\\log^{1\/2}(np)$ for some large enough constant $C>0$,\n\tthen ${\\mathbb{P}}(T_n>q^B_{1-\\tau}\\mid H_1)\\rightarrow 1$ as $n\\rightarrow \\infty$.\n\\end{theorem}\n\nTheorem \\ref{theo:power} indicates that the test based on $T_n$ achieves consistency when the maximum element of $n^{1\/2}|{\\boldsymbol \\theta}-{\\boldsymbol \\theta}_0|$ has a magnitude much large than $\\log^{1\/2}(\\tau^{-1})\\log^{1\/2}(np)$ for a fixed significant level $\\tau$.\n\n\\begin{remark}\\label{remark:07}\n\t\\citet{Wang2015} proposed a $L_2$-norm type test (WPL test) for \\eqref{eq:one_sample} with ${\\boldsymbol \\theta}_0=0$ based on\n\n\t$ T_{\\mathrm{WPL}} = \\sum_{i=1}^{n}\\sum_{j=1}^{i-1}W_{i}^{\\top}W_i$.\n\n\tIt has been argued in \\citet{Wang2015} and \\citet{LiXu2022} that the\n\n\n\n\n\tsignal of the WPL test is determined by the magnitude of $\\|{\\boldsymbol \\theta}\\|$, which is the $L_2$-norm of ${\\boldsymbol \\theta}$.\n\tAs a contrast, the power of the test based on $T_n$  depends on $|{\\boldsymbol \\theta}|_{\\infty}$.\n\tThus, the proposed test based on $T_n$ is expected to be more powerful under sparse alternatives, when ${\\boldsymbol \\theta}$ contains only a limited number of non-zero components and its maximum element has certain order of magnitude.\n\tIn such cases, $\\|{\\boldsymbol \\theta}\\|$ is not big enough for the rejection of the WPL test.\n\tSee Appendix C in the supplementary material and Section \\ref{sec:real_data} for numerical justifications.\n\\end{remark}\n\\subsection{Multiple testing with FDR control in large-scale tests}\\label{sec:FDR}\n\nMultiple testing with false discovery rate (FDR) control has been applied to many real problems, such as detecting differentially expressed genes in genomic study.\nIn this section, we study multiple testing for every component of ${\\boldsymbol \\theta}$ based on the spatial median with the Benjamini and Hochberg (B-H) method for FDR control.\nFor $j=1,\\ldots,p$, we are interested in testing\n\\begin{eqnarray}\n\t\\notag H_{0j}:  \\theta_j=\\theta_{0,j}  \\text{~~versus~~} H_{1j}:\\theta_{j} \\neq\\theta_{0,j}\n\\end{eqnarray}\nsimultaneously, where $\\theta_{0,1},\\ldots,\\theta_{0,p}$ are given values.\n\nDefine the test statistics as\n\\begin{eqnarray}\n\t\\notag T_{n,j} = n^{1\/2}(\\hat{\\theta}_{n,j}-\\theta_{0,j})\/s_{n,j}\n\\end{eqnarray}\nfor $j=1,\\ldots,p$, where $s_{n,j}^2=\\hat{\\zeta}_1^{-2}\\hat{{\\mathbb{B}}}_{jj}$ with $\\hat{\\zeta}_1=n^{-1}\\sum_{i=1}^n\\|X_i-\\hat{{\\boldsymbol \\theta}}_n\\|^{-1}$, and $\\hat{{\\mathbb{B}}}_{jj}$ is the $j$th diagonal element of ${\\hat {\\mathbb{B}}}=n^{-1}\\sum_{i=1}^n\\|X_i-\\hat{{\\boldsymbol \\theta}}_n\\|^{-2}(X_i-\\hat{{\\boldsymbol \\theta}}_n)(X_i-\\hat{{\\boldsymbol \\theta}}_n)^{\\top}$.\n\nAccording to the proof of Theorem \\ref{theo:FDR} in Appendix A, $T_{n,j}$ converges in distribution to a standard normal under $H_{0j}$ for $j=1,\\ldots,p$.\nThus, we utilise the standard normal distribution to estimate the marginal $p$-values.\nFor $j=1,\\ldots,p$, define the $p$-value for $H_{0j}$ as\n$P_{j} = 2-2\\Phi(|T_{n,j}|)$.\nDenote $P_{(1)}\\leq \\cdots\\leq P_{(p)}$ be the ordered $p$-values, and define\n\\begin{eqnarray}\n\t\\notag \\hat{k} = \\max\\left\\{j=0,\\ldots,p: P_{(j)}\\leq \\tau j\/p\\right\\}\n\\end{eqnarray}\nfor a pre-specific significance level $\\tau$.\nThen, the B-H procedure rejects the null hypotheses for which $P_{j}\\leq P_{(\\hat{k})}$. Denote $\\mathcal{H}_{R}=\\{j:P_{j}\\leq P_{(\\hat{k})}\\}$ as the set of indices $j$ such that $H_{0j}$ is rejected by the B-H method, and let $|\\mathcal{H}_{R}|$ be the cardinality of $\\mathcal{H}_{R}$ that equals the total number of rejected null hypotheses.\n\nLet $\\mathcal{H}_0\\subset\\{1,\\ldots,p\\}$ be the set of indices $j$ corresponding to the true null hypotheses $H_{0j}$.\nThe false discovery proportion (FDP) and false discovery rate (FDR) of the B-H method are defined as\n\\begin{eqnarray}\n\t\\notag \\mathrm{FDP}_{M}=\\frac{|\\mathcal{H}_{0}\\cap\\mathcal{H}_{R}|}{|\\mathcal{H}_{R}|\\vee 1} \\text{~~and~~} \\mathrm{FDR}_{M} = \\mathbb{E}(\\mathrm{FDP}_{M}).\n\\end{eqnarray}\n\nRegarding that $T_{n,1},\\ldots,T_{n,p}$ are dependent, we impose the following condition on the weak dependence between any two components of $W_i$. Define $(r_{j\\ell})_{p\\times p}=\\{\\mathrm{diag}({\\mathbb{B}})\\}^{-1\/2}{\\mathbb{B}}\\{\\mathrm{diag}({\\mathbb{B}})\\}^{-1\/2}$ as the correlation matrix, where $\\mathrm{diag}({\\mathbb{B}})$ is the diagonal matrix of ${\\mathbb{B}}$.\n\n\\begin{condition}\\label{c5}\n\tSuppose $\\max_{1\\leq j , \\ell \\leq p} |r_{j\\ell}|\\leq r$ with some constant $0< r< 1$. In addition, $\\sum_{j=1}^p{\\mathbb{I}}(r_{j\\ell}=0)=O(p^{\\eta})$ for some constant $0<\\eta<(1-r)\/(1+r)$.\n\\end{condition}\n\nSimilar conditions are assumed in \\citet{Liu2014} and \\citet{Bell2018}.\nLet $p_0=|\\mathcal{H}_0|$ be the number of true null hypotheses and ${\\mathbb{B}}_{jj}$ be the $j$th diagonal element of ${\\mathbb{B}}$.\n\n\\begin{theorem}\\label{theo:FDR}\n\tSuppose Condition \\ref{c5} and the conditions of Theorem \\ref{theo1} hold. In addition, there exists $\\mathcal{H}\\subset\\{1,\\ldots,p\\}$ such that\n\t${\\mathcal H}=\\big\\{j: \\zeta_{1}{\\mathbb{B}}_{jj}^{-1\/2}n^{1\/2}|\\theta_{j}-\\theta_{0,j}|\\geq 2\\log^{1\/2} (p)\\big\\}$\n\tand $|\\mathcal H|\\geq \\log \\log p \\rightarrow \\infty$ as $p\\rightarrow \\infty$.  Assume that the number of false null hypotheses $p_1\\leq p^{\\varpi}$ for some $0<\\varpi<1$. Then, $\\mathrm{FDR}_{M}\/(\\tau p_0\/p)\\to1$ as $n \\rightarrow \\infty$.\n\\end{theorem}\n\nTheorem \\ref{theo:FDR}  shows the B-H procedure based on $P_1, \\ldots, P_p$ controls the FDR asymptotically, and it extends Theorem 4.1 in \\citet{Liu2014} to spatial median-based test statistic.\n\n\n\n\n\\subsection{High-dimensional asymptotic relative efficiency}\\label{sec:ARE}\n\n\nAs two candidate estimators of the location parameter ${\\boldsymbol \\theta}$, it is of interest to study the asymptotic relative efficiency (ARE) of the sample spatial median $\\hat{{\\boldsymbol \\theta}}_n$  relative to the sample mean $\\bar{X}_n$.\nWhen $p$ is fixed, for spherical multivariate normal distribution, \\citet{Brown1983} showed that the asymptotic efficiency of $\\hat{{\\boldsymbol \\theta}}_n$ relative $\\bar{X}_n$, denoted as $\\mathrm{ARE}(\\hat{{\\boldsymbol \\theta}}_{n},\\bar{X}_n)$, exceeds the usual univariate case $2\/\\pi$. \nIn addition, $\\mathrm{ARE}(\\hat{{\\boldsymbol \\theta}}_{n},\\bar{X}_n)$ increases as the dimension increases, and it approaches to $1$ as $p$ tends to be sufficient large \\citep{Magyar2011}. \nHowever, when $p\\to\\infty$, the ARE is not straightforward to quantify as there are no obvious ``final'' limit distributions for $\\hat{{\\boldsymbol \\theta}}_{n}$ and $\\bar{X}_n$.\nMotivated by the discussions in Sections \\ref{sec:SCIs} and \\ref{sec:global_tests}, we compare $\\hat{{\\boldsymbol \\theta}}_n$ and $\\bar{X}_n$ in terms of their efficiencies in simultaneous inference for ${\\boldsymbol \\theta}$, which are determined by the variations of $|\\hat{{\\boldsymbol \\theta}}-{\\boldsymbol \\theta}|_{\\infty}$ and $|\\bar{X}_n-{\\boldsymbol \\theta}|_{\\infty}$.\nAccording to Corollary \\ref{co1} and \\eqref{eq:gaussian_approx_mean}, we define the high-dimensional ARE of $\\hat{{\\boldsymbol \\theta}}_n$ compared to $\\bar{X}_n$ in simultaneous inference for ${\\boldsymbol \\theta}$ as\n\\begin{align}\\label{AE}\n\t\\mathrm{ARE}(\\hat{{\\boldsymbol \\theta}}_{n}, \\bar{X}_n)=\\mathrm{Var}(|G_0|_{\\infty})\/\\mathrm{Var}(|G|_{\\infty})\\,,\n\\end{align}\nwhich approximates $\\mathrm{Var}(|\\bar{X}_n-{\\boldsymbol \\theta}|_{\\infty})\/\\mathrm{Var}(|\\hat{{\\boldsymbol \\theta}}_{n}-{\\boldsymbol \\theta}|_{\\infty})$. If $\\lim_{p\\to\\infty}\\mathrm{ARE}(\\hat{{\\boldsymbol \\theta}}_{n}, \\bar{X}_n)>1$, we say that $\\hat{{\\boldsymbol \\theta}}_n$ is more efficient than $\\bar{X}_n$ in simultaneous inference for ${\\boldsymbol \\theta}$ under high-dimensionality.\n\nAs discussed in Remark \\ref{remark:B}, $G\\sim N(0, \\zeta_{1}^{-2}{\\mathbb{B}})$ with $\\zeta_{1}^{-2}{\\mathbb{B}}_{j\\ell}=\\zeta_{1}^{-2}p^{-1}\\omega_{i\\ell}$ for all $1\\leq j,\\ell\\leq p$.\nMoreover, we can show that $\\Sigma_{j\\ell} = {\\mathbb{E}}(\\nu_{i}^{2})\\omega_{j\\ell} + O(p^{-1\/2})$ similar to the proof of Lemma \\ref{lemma:SM_Q} in Appendix B of the supplementary material, where $\\Sigma_{j\\ell}$ is the $(j,\\ell)$th element of $\\Sigma$.\nThus, both the covariance matrix $\\Sigma$ and $\\zeta_{1}^{-2}{\\mathbb{B}}$ are proportional to $\\Omega$ asymptotically, and $\\mathrm{ARE}(\\hat{{\\boldsymbol \\theta}}_{n}, \\bar{X}_n)$ is approximately ${\\mathbb{E}}(\\nu_{i}^{2})\\zeta_{1}^{2}p$.\n\nAs $\\Sigma$ and $\\zeta_{1}^{-2}{\\mathbb{B}}$ are rarely known in practice, we use bootstrap approximation to estimate the value of $\\mathrm{Var}(|G_0|_{\\infty})\/\\mathrm{Var}(|G|_{\\infty})$.\nCombining Corollary \\ref{co2} and \\eqref{eq:boot_mean}, we propose using\n$$\n\\mathrm{Var}^{*}(|\\bar{X}_{n}^{*}|_{\\infty})\/ \\mathrm{Var}^{*}(|\\tilde{{\\boldsymbol \\theta}}_{n}|_{\\infty}),\n$$\nto estimate $\\mathrm{ARE}(\\hat{{\\boldsymbol \\theta}}_{n}, \\bar{X}_n)$.\n\n\\begin{example}\n\tSuppose $X_1, \\ldots X_n$ are i.i.d.~from $N({\\boldsymbol \\theta}, I_p)$, then $\\nu_i^2$ follows a chi-square distribution with $p$ degrees of freedom. \n\tIt follows that ${\\mathbb{E}}(\\nu_i^2)=p$ and ${\\mathbb{E}}(\\nu_i^{-1})=\\Gamma(p\/2-1\/2)\/\\{2^{1\/2}\\Gamma(p\/2)\\}$, where $\\Gamma(\\cdot)$ is the gamma function.. So the ARE is\n\t$\n\t\\mathrm{ARE}(\\hat{{\\boldsymbol \\theta}}_{n}, \\bar{X}_n)=p\\{\\Gamma(p\/2-1\/2)\\}^2\/\\{2^{1\/2}\\Gamma(p\/2)\\}^2.\n\t$\n\tUsing Stirling's formula,\n\t$ \\lim_{p\\to\\infty} \\mathrm{ARE}(\\hat{{\\boldsymbol \\theta}}_{n}, \\bar{X}_n) = 1. $\n\tThus, for high-dimensional Gaussian data, the sample spatial median has the same asymptotically efficiency as the sample mean in simultaneous inference for ${\\boldsymbol \\theta}$.\n\\end{example}\n\n\n\\begin{example}\n\tWhen the data are from the multivariate $t$-distribution with degrees of freedom $v>2$ and shape matrix $\\Omega=I_{p}$, $\\nu_i^2\/p\\sim F_{p, v}$, where $F_{p,v}$ is the $F$ distribution with parameters $p$ and $v$.\n\tThen, ${\\mathbb{E}}(\\nu_i^2)=pv\/(v-2)$ and ${\\mathbb{E}}(\\nu_i^{-1})=\\Gamma(v\/2+1\/2)\\Gamma(p\/2-1\/2)\/\\{v^{1\/2}\\Gamma(v\/2)\\Gamma(p\/2)\\}$. Thus, the ARE is\n\t$\\mathrm{ARE}(\\hat{{\\boldsymbol \\theta}}_{n}, \\bar{X}_n)=(v-2)^{-1}p\\{\\Gamma(v\/2+1\/2)\\Gamma(p\/2-1\/2)\\}^2\/\\{\\Gamma(v\/2)\\Gamma(p\/2)\\}^2. $\n\tIt is clear that $\\mathrm{ARE}(\\hat{{\\boldsymbol \\theta}}_{n},~ \\bar{X})>1$ for large enough $p$.~ In addition, \t\n\t$\n\t\\lim_{p\\to\\infty} \\mathrm{ARE}(\\hat{{\\boldsymbol \\theta}}_{n}, \\bar{X}_n) = 2(v-2)^{-1}\\{\\Gamma(v\/2+1\/2)\\}^2\/\\{\\Gamma(v\/2)\\}^2>1\\,.\n\t$\n\tThus, for high-dimensional $t$-distribution, the sample spatial median is asymptotically more efficient than the sample mean in simultaneous inference for ${\\boldsymbol \\theta}$.\n\\end{example}\n\nFigure \\ref{Fig1} plots the simulated values of $\\mathrm{Var}(|\\bar{X}_n|_{\\infty})\/\\mathrm{Var}(|\\hat{{\\boldsymbol \\theta}}_n|_{\\infty})$ with a range of dimensions and sample sizes under different models.\nFor Gaussian data, the relative efficiency kept increasing in $p$, and it approached $1$ as $p$ getting larger.\nFor the data simulated from multivariate $t$-distribution, the relative efficiency was greater than $1$ for all combinations of $n$ and $p$. This indicates that the sample spatial median is more efficiency than the sample mean for $t$-distribution.\nThe results were consistent under different covariance structure considered in the simulation.\n\n\\begin{figure}[hbt]\n\t\\centerline{\n\t\t\\includegraphics[width=14cm]{p2.pdf}\n\t}\n\t\\caption{Finite sample relative efficiency of $|\\hat{{\\boldsymbol \\theta}}_{n}|_{\\infty}$ compared to $|\n\t\t\\bar{X}_n|_{\\infty}$ based on $5000$ replications,  the data are generated from multivariate normal distribution (Gaussian) and $t$-distribution with $5$ degrees of freedom ($t_{5}$). The shape matrix $\\Omega=(\\rho^{|j-\\ell|})_{p\\times p}$ with $\\rho=0$ and $0.8$.}\n\t\\label{Fig1}\n\\end{figure}\n\n\\iffalse\n\\section{Extension to two-sample Gaussian and bootstrap approximations}\\label{sec:two-sample}\n\nIn this section, we extend the one-sample Gaussian and bootstrap approximations for the sample spatial median to the two-sample case. \nLet $X_1, \\ldots, X_{n_1}$ and $Y_1, \\ldots, Y_{n_2}$ be two independent sequences of i.i.d.~$p$-dimensional random vectors from two populations with cumulative distribution functions $F_X$ and $F_Y$, respectively. Assume that $X_{i_1}$ and $Y_{i_2}$ admit the following representations:\n\\begin{eqnarray}\n\t\\notag X_{i_1} = {\\boldsymbol \\theta}^X+ \\nu_{X,i_1}\\Gamma_XU_{X,i_1}, \\text{~~and~~} Y_{i_2} = {\\boldsymbol \\theta}^Y+ \\nu_{Y,i_2}\\Gamma_YU_{Y,i_2}\\,,\n\\end{eqnarray}\nwhere $\\nu_{X,i_1}$ is independent of the spatial sign of $U_{X,i_1}$, and $\\nu_{Y,i_2}$ is independent of the spatial sign of $U_{Y,i_2}$.\nThe shape  matrices are $\\Omega_X=\\Gamma_X\\Gamma_X^{\\top}$ and $\\Omega_Y=\\Gamma_Y\\Gamma_Y^{\\top}$.\nSimilarly, denote $W_{X,i_1}, R_{X,i_1}$ and $W_{Y,i_2}, R_{Y,i_2}$ as in \\eqref{eq:nota} with additional subscripts $X$ or $Y$ to distinguish populations.\nLet $U_{X,i_1}=(U_{X,i_1,1},\\ldots,U_{X,i_1,p})^{\\top}$ and $U_{Y,i_2}=(U_{X,i_2,1},\\ldots,U_{X,i_2,p})^{\\top}$.\nThe following condition is parallel to Conditions \\ref{c1} and \\ref{c2}.\n\\begin{condition}\\label{c4}\n\t$U_{X,i_1,1},\\ldots,U_{X,i_1,p}$ are i.i.d.~symmetric random variables satisfies ${\\mathbb{E}}(U_{X,i_1,j_1})=0$, ${\\mathbb{E}}(U_{X,i_1,j_1}^2)=1$ for $j_1=1,\\ldots,p$, and\n\t$U_{Y,i_2,1},\\ldots,U_{Y,i_2,p}$ are i.i.d.~symmetric with ${\\mathbb{E}}(U_{Y,i_2,j_2})=0$, ${\\mathbb{E}}(U_{Y,i_2,j_2}^2)=1$ for $j_2=1,\\ldots,p$,\n\tIn addition, $\\|U_{X,i_1,j_1}\\|_{\\psi_{\\alpha}}\\leq c_0$ and $\\|U_{Y,i_2,j_2}\\|_{\\psi_{\\alpha}}\\leq c_0$ with some constant $c_0>0$ and $1\\leq \\alpha\\leq 2$.\n\tThe moments $\\zeta_{X,k}={\\mathbb{E}}(R_{X,i_1}^{-k})$ and $\\zeta_{Y,k}={\\mathbb{E}}(R_{Y,i_2}^{-k})$ for $k=1,2,3,4$ exist for large enough $p$. In addition, there exist two positive constants $\\underline{b}$ and ${\\bar{B}}$ such that\n\t$\n\t{\\underline b}\\leq \\limsup_{p}{\\mathbb{E}}(R_{X,i_1}\/\\sqrt{p})^{-k} \\leq {\\bar{B}}\n\t$ and $\n\t{\\underline b}\\leq \\limsup_{p}{\\mathbb{E}}(R_{Y,i_2}\/\\sqrt{p})^{-k} \\leq {\\bar{B}}\n\t$ for $k=1,2,3,4$.\n\\end{condition}\n\nLet $\\hat{{\\boldsymbol \\theta}}_{n_1}^{X}$ and $\\hat{{\\boldsymbol \\theta}}_{n_2}^{Y}$ be the spatial medians\nbased on $X_1, \\ldots, X_{n_1}$ and $Y_1, \\ldots, Y_{n_2}$, respectively.\n\\begin{theorem}\\label{theo:two}\n\tUnder Condition \\ref{c4}, assume that $\\Omega_X$ and $\\Omega_Y$ belong to $\\mathcal{U}(a_0(p), \\underline{m}, \\bar{M})$ with $a_0(p)\\asymp p^{1-\\delta}$ for some positive constant $\\delta \\leq 1\/2$. Let $N=n_1+n_2$, assume that $b_1\\leq n_1\/N\\leq b_2$ for some universal constants $0<b_1<b_2<1$. In addition, if\n\t$\\log p=o(N^{1\/5})$ and $\\log N=o(p^{1\/3\\wedge\\delta})$,\n\n\n\n\tthen\n\t\\begin{eqnarray\n\t\t\\notag & & \\sup_{A\\in {\\mathcal A}^{{\\rm re}}}\\left|{\\mathbb{P}}\\left\\{n_1^{1\/2}(\\hat{{\\boldsymbol \\theta}}_{n_1}^{X}-{\\boldsymbol \\theta}^{X})-\\gamma_{n_1,n_2}n_2^{1\/2}(\\hat{{\\boldsymbol \\theta}}_{n_2}^{Y}-{\\boldsymbol \\theta}^{Y})\\in A\\right\\}  - {\\mathbb{P}}(G_X+\\gamma_{n_1,n_{2}}G_Y\\in A)\\right|\n\t\t\\rightarrow 0\n\t\\end{eqnarray}\n\tas $N\\to\\infty$, where $\\gamma_{n_1,n_2}=n_1^{1\/2}n_2^{-1\/2}$, $G_X\\sim N(0, \\zeta_{X,1}^{-2}{\\mathbb{B}}_{X})$ with ${\\mathbb{B}}_X={\\mathbb{E}}(W_{X,1}W_{X,1}^{\\top})$ and \n\t$G_Y\\sim N(0, \\zeta_{Y,1}^{-2}{\\mathbb{B}}_{Y})$ with ${\\mathbb{B}}_Y={\\mathbb{E}}(W_{Y,1}W_{Y,1}^{\\top})$ are independent.\n\\end{theorem}\n\nThe condition $b_1\\leq n_1\/N\\leq b_2$ assumes that the sample sizes $n_1$ and $n_2$ are comparable.\nNote that \n$$n_1^{1\/2}(\\hat{{\\boldsymbol \\theta}}_{n_1}^{X}-{\\boldsymbol \\theta}^{X})-\\gamma_{n_1,n_2}n_2^{1\/2}(\\hat{{\\boldsymbol \\theta}}_{n_2}^{Y}-{\\boldsymbol \\theta}^{Y})=n_1^{1\/2}\\big\\{(\\hat{{\\boldsymbol \\theta}}_{n_1}^{X}-\\hat{{\\boldsymbol \\theta}}^{Y}_{n_2})-({\\boldsymbol \\theta}^{X}-{\\boldsymbol \\theta}^{Y})\\big\\},$$\nthe distribution of $G_X+\\gamma_{n_1,n_{2}}G_Y$ can be used to approximation that of $n_1^{1\/2}\\big\\{(\\hat{{\\boldsymbol \\theta}}_{n_1}^{X}-\\hat{{\\boldsymbol \\theta}}^{Y}_{n_2})-({\\boldsymbol \\theta}^{X}-{\\boldsymbol \\theta}^{Y})\\big\\}$ over hyperrectangles. As the covariance matrices of $G_X$ and $G_Y$ are unknown and inaccessible in practice, we consider the following bootstrap approximation.\n\nLet $Z_{X,1},\\ldots,Z_{X,n_1}; Z_{Y,1},\\ldots,Z_{Y,n_2}$ be i.i.d.~Rademacher random variables. \nDefine the bootstrap version of the $\\hat{{\\boldsymbol \\theta}}_{n_1}^{X}$ and $\\hat{{\\boldsymbol \\theta}}_{n_2}^{Y}$ as\n\\begin{align}\n\t\\notag \\tilde{{\\boldsymbol \\theta}}_{n_1}^{X}= \\argmin_{{\\boldsymbol \\beta}\\in {\\mathbb R}^p}\\sum_{i=1}^{n_1}\\|Z_{X,i}(X_i-\\hat{{\\boldsymbol \\theta}}_{n_1}^{X})-{\\boldsymbol \\beta}\\|, {\\rm ~and~}\n\t\\tilde{{\\boldsymbol \\theta}}_{n_2}^{Y}= \\argmin_{{\\boldsymbol \\beta}\\in {\\mathbb R}^p}\\sum_{j=1}^{n_2}\\|Z_{Y,j}(Y_j-\\hat{{\\boldsymbol \\theta}}_{n_2}^{Y})-{\\boldsymbol \\beta}\\|\\,.\n\\end{align}\nThen, the distribution of $n_1^{1\/2}(\\tilde{{\\boldsymbol \\theta}}_{n_1}^{X}-\\tilde{{\\boldsymbol \\theta}}_{n_2}^{Y})$ conditional on $X_1, \\ldots, X_{n_1}$ and $Y_1, \\ldots, Y_{n_2}$ is used to approximate that of $n_1^{1\/2}\\big\\{(\\hat{{\\boldsymbol \\theta}}_{n_1}^{X}-\\hat{{\\boldsymbol \\theta}}_{n_2}^{Y})-({\\boldsymbol \\theta}^{X}-{\\boldsymbol \\theta}^{Y})\\big\\}$. \n\n\\begin{theorem}\\label{theo:boottwo}\n\tUnder the conditions assumed in Theorem \\ref{theo:two}, as $N\\to\\infty$,\n\t\\begin{eqnarray}\n\t\t\\notag & & \\sup_{A\\in {\\mathcal A}^{{\\rm re}}}\\left|{\\mathbb{P}}\\left(n_1^{1\/2}\\big\\{(\\hat{{\\boldsymbol \\theta}}_{n_1}^{X}-\\hat{{\\boldsymbol \\theta}}_{n_2}^{Y})-({\\boldsymbol \\theta}^{X}-{\\boldsymbol \\theta}^{Y})\\big\\}\\in A\\right)  - {\\mathbb{P}}^{*}\\left\\{n_1^{1\/2}(\\tilde{{\\boldsymbol \\theta}}_{n_1}^{X}-\\tilde{{\\boldsymbol \\theta}}_{n_2}^{Y})\\in A\\right\\}\\right| \\to 0\\, \n\t\\end{eqnarray}\n\tin probability.\n\\end{theorem}\n\nSame to the one-sample case, the multiplier bootstrap is valid under an exponentially divergent $p$.\nLet $q^{\\mathrm{TS},B}_{1-\\alpha}$ be the $(1-\\alpha)$th quantile of $\\sqrt{n_1}|\\tilde{{\\boldsymbol \\theta}}_{n_1}^{X}-\\tilde{{\\boldsymbol \\theta}}_{n_2}^{Y}|_{\\infty}$ given $X_1,\\ldots,X_{n_1}$ and $Y_1,\\ldots,Y_{n_2}$.  Similar to the one-sample case, an asymptotic $1-\\alpha$ simultaneous confidence intervals for ${\\boldsymbol \\theta}^{X}-{\\boldsymbol \\theta}^{Y}$ is\n\\begin{eqnarray\n\t\\notag {\\mathcal C}_{\\alpha}^{\\mathrm{TS}}=\\left\\{{\\boldsymbol \\theta}^{X}-{\\boldsymbol \\theta}^{Y}\\in {\\mathbb{R}}^p, n_1^{1\/2}|(\\tilde{{\\boldsymbol \\theta}}_{n_1}^{X}-\\tilde{{\\boldsymbol \\theta}}_{n_2}^{Y})-({\\boldsymbol \\theta}^{X}-{\\boldsymbol \\theta}^{Y})|_{\\infty}<q^{\\mathrm{TS},B}_{1-\\alpha}\\right\\}.\n\\end{eqnarray}\n\nFor the two-sample global testing problem, suppose we are interested in testing\n\\begin{eqnarray\n\t\\notag H_0^{\\mathrm{TS}}: {\\boldsymbol \\theta}^{X} = {\\boldsymbol \\theta}^{Y} \\text{~~versus~~} H_1^{\\mathrm{TS}}:{\\boldsymbol \\theta}^{X}\\neq{\\boldsymbol \\theta}^{Y}.\n\\end{eqnarray}\nA direct test statistic is\n$T_{N}^{\\mathrm{TS}} = n^{1\/2}\\left|\\hat{{\\boldsymbol \\theta}}_{n_1}^{X}-\\hat{{\\boldsymbol \\theta}}_{n_2}^{Y}\\right|_{\\infty}$,\nand the null hypothesis $H_{0}^{\\mathrm{TS}}$ is rejected when $T_{N}^{\\mathrm{TS}}>q_{1-\\alpha}^{\\mathrm{TS},B}$.\nThe size and power analysis of the two-sample test based on $T_{N}^{\\mathrm{TS}}$ is similar to the one-sample case and is thus straightforward.\n\\fi\n\n\\section{Numerical studies}\\label{sec:numerical}\n\nIn this section, we report Monte Carlo simulations on simultaneous confidence intervals and multiple testing with FDR control, along with a real data analysis, to demonstrate the performance of the proposed approaches. Additional simulations on global tests can be found in Appendix C of the supplementary material.\nIn the simulations, all results were based on $2500$ replications. In the bootstrap implementation, the number of bootstrap iterations was set to $B=400$. \n\n\\subsection{Simulations on simultaneous confidence intervals}\\label{sec:simulations}\n\nWe first examine the performance of the SCIs based on $\\hat{{\\boldsymbol \\theta}}_n$, and compare it with the SCIs based on $\\bar{X}_n$.\nThe sample size $n$ is taken to be $100$ or $200$, and the dimensions $p= 100$ and $1000$ are considered for each sample size. \nTwo types of commonly used elliptical distributions are considered: (I) the multivariate normal distribution $N({\\boldsymbol \\theta},\\Sigma)$; (II) the multivariate $t$-distribution with $3$ degrees of freedom, mean vector ${\\boldsymbol \\theta}$, and covariance matrix $\\Sigma$.\nIn addition, we include the following independent components model: (III) $X_i={\\boldsymbol \\theta} + \\Sigma^{1\/2}Z_i$, where each component of $Z_i$ are i.i.d.~from the standard Laplace distribution.\nWe set $\\Sigma=(\\rho^{|j-\\ell|})$ with $\\rho=0, 0.2, 0.5$ and $0.8$.\nTo save space, we present the results for $\\rho=0$ and $0.8$ here. \nThe results for $\\rho\\in\\{0.2,0.5\\}$ are similar and are reported in the supplementary material. \nWe consider both sparse and dense case scenarios for ${\\boldsymbol \\theta}$: (i) ${\\boldsymbol \\theta}_{1}=(2,-2,3,0,\\ldots, 0)$; (ii) ${\\boldsymbol \\theta}_{2}=(0.2,\\ldots,0.2_{\\lfloor p\/4\\rfloor},0,\\ldots,0)$. Here $\\lfloor\\cdot\\rfloor$ is the floor function.\n\nTable \\ref{tab1} reports the coverage probability and median length of the SCIs based on $\\hat{{\\boldsymbol \\theta}}_n$, the results of the SCIs based on $\\bar{X}_n$ are presented in parentheses. \nFor Models I and II from elliptical distributions, we observe that the SCIs based on $\\hat{{\\boldsymbol \\theta}}_n$ and $\\bar{X}_n$ both achieve satisfying coverage probability for different choices for $\\rho$, ${\\boldsymbol \\theta}$, $n$ and $p$. \nFor the data simulated from the multivariate normal distribution, the median length of the SCIs based on $\\hat{{\\boldsymbol \\theta}}_n$ is very close to that of the the SCIs based on $\\bar{X}_n$. \nThese results indicate that $\\hat{{\\boldsymbol \\theta}}_n$ has similar asymptotic efficiency as $\\bar{X}_n$ in simultaneous inference for ${\\boldsymbol \\theta}$ under high-dimensional Gaussian model as discussed in Section \\ref{sec:ARE}.\nFor the multivariate $t$-distribution, the SCIs based on $\\hat{{\\boldsymbol \\theta}}_n$ is much narrower than the SCIs based on $\\bar{X}_n$.\nThese results suggest that the SCIs based on $\\hat{{\\boldsymbol \\theta}}_n$ is more efficient than the SCIs based on $\\bar{X}_n$ for multivariate $t$-distribution, which is heavy-tailed.\nThis is consistent with the asymptotic analysis in Section \\ref{sec:ARE}.\nMoreover, the results for Model III, which does not belong to the elliptical distribution family, shows the robustness of the SCIs based on the spatial median, and it performs similar to the SCIs based on the sample mean.\nWe also note that the median length of the SCIs decreases when $n$ increases or $p$ decreases for each model.\n\n\\begin{table}[ht]\n\t\\footnotesize\n\t\\centering\n\t{\n\t\t\\caption{Coverage probability (in $\\%$) and median length of the SCIs based on $\\hat{{\\boldsymbol \\theta}}_n$, the results of the SCIs based on $\\bar{X}_n$ are in parentheses.}\n\t\t\\label{tab1}\n\t\t\\setlength\\tabcolsep{3pt}\n\t\t\\resizebox{\\textwidth}{!}{\n\t\t\t\\begin{tabular}{@{}cccccccccccccc@{}}\n\t\t\t\t\n\t\t\t\t\\hline\n\t\t\t\t&&&&\\multicolumn{4}{c}{${\\boldsymbol \\theta}={\\boldsymbol \\theta}_1$}& &\\multicolumn{4}{c}{${\\boldsymbol \\theta}={\\boldsymbol \\theta}_2$}\\cr\n\t\t\t\t&&&&\\multicolumn{2}{c}{Coverage probability}&\\multicolumn{2}{c}{Median length}&&\\multicolumn{2}{c}{Coverage probability}&\\multicolumn{2}{c}{Median length}\\cr\n\t\t\t\t\n\t\t\t\tModel & $\\rho$ & $n$&$p$& 90\\% & 95\\% & 90\\% & 95\\%& &90\\% & 95\\% & 90\\% & 95\\%\\cr\n\t\t\t\t\n\t\t\t\t\\hline\n\t\t\t\tI & 0 &100&~$100$ & 89.6 (89.9) & 94.4 (94.4) & 0.65 (0.65) & 0.69 (0.69)\n\t\t\t\t&& 88.9 (88.8) & 94.1 (93.9) & 0.65 (0.65) & 0.69 (0.69) \\cr\n\t\t\t\t\n\t\t\t\t&&&$1000$& 89.5 (89.6) & 94.7 (94.4) & 0.77 (0.77) & 0.80 (0.80)\n\t\t\t\t&& 89.5 (89.5) & 94.0 (94.0) & 0.77 (0.77) & 0.81 (0.80) \\cr\n\t\t\t\t\n\t\t\t\t&&200&~$100$& 89.8 (89.8) & 95.1 (95.1) & 0.46 (0.46) & 0.49 (0.49)\n\t\t\t\t&& 88.6 (88.8) & 94.4 (94.7) & 0.46 (0.46) & 0.49 (0.49) \\cr\n\t\t\t\t\n\t\t\t\t&&&$1000$& 89.7 (89.7) & 94.4 (94.6) & 0.55 (0.55) & 0.57 (0.57)\n\t\t\t\t&& 89.1 (89.2) & 94.7 (94.6) & 0.55 (0.55) & 0.57 (0.57) \\cr\n\t\t\t\t\n\t\t\t\n\t\t\t\t\n\t\t\t\t\\cline{2-13}\n\t\t\t\t\n\t\t\t\t& 0.8 &100&~$100$ & 89.1 (88.7) & 94.6 (94.6) & 0.64 (0.63) & 0.68 (0.67)\n\t\t\t\t&& 88.4 (88.6) & 93.7 (94.1) & 0.64 (0.63) & 0.68 (0.67) \\cr\n\t\t\t\t\n\t\t\t\t&&&$1000$& 88.4 (88.4) & 93.8 (93.7) & 0.76 (0.76) & 0.80 (0.79)\n\t\t\t\t&& 89.0 (89.2) & 94.6 (94.6) & 0.76 (0.76) & 0.80 (0.79) \\cr\n\t\t\t\t\n\t\t\t\t&&200&~$100$& 90.5 (90.1) & 95.2 (94.9) & 0.45 (0.45) & 0.48 (0.48)\n\t\t\t\t&& 89.6 (89.6) & 94.0 (94.1) & 0.45 (0.45) & 0.48 (0.48) \\cr\n\t\t\t\t\n\t\t\t\t&&&$1000$& 90.4 (90.4) & 94.5 (94.4) & 0.54 (0.54) & 0.56 (0.56)\n\t\t\t\t&& 88.4 (88.5) & 93.6 (93.8) & 0.54 (0.54) & 0.56 (0.56) \\cr\n\t\t\t\t\n\t\t\t\t\\hline\n\t\t\t\t\n\t\t\t\tII & 0 &100&~$100$ & 89.7 (88.6) & 94.7 (93.7) & 0.71 (1.05) & 0.75 (1.11)\n\t\t\t\t&& 88.8 (88.8) & 94.5 (94.2) & 0.71 (1.05) & 0.75 (1.11) \\cr\n\t\t\t\t\n\t\t\t\t&&&$1000$& 89.4 (91.0) & 95.8 (95.0) & 0.84 (1.25) & 0.88 (1.30) \n\t\t\t\t&& 89.1 (89.0) & 94.4 (94.5) & 0.84 (1.25) & 0.88 (1.31) \\cr\n\t\t\t\t\n\t\t\t\t&&200&~$100$& 88.6 (89.1) & 94.2 (95.1) & 0.50 (0.76) & 0.53 (0.81)\n\t\t\t\t&& 89.5 (89.7) & 94.4 (94.8) & 0.50 (0.76) & 0.53 (0.80) \\cr\n\t\t\t\t\n\t\t\t\t&&&$1000$& 89.6 (88.7) & 94.8 (94.6) & 0.59 (0.90) & 0.62 (0.94)\n\t\t\t\t&& 90.1 (89.5) & 94.8 (93.9) & 0.59 (0.90) & 0.62 (0.94) \\cr\n\t\t\t\t\n\t\t\t\t\\cline{2-13}\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t& 0.8 &100&~$100$ & 89.1 (90.7) & 94.4 (94.9) & 0.69 (1.02) & 0.74 (1.09)\n\t\t\t\t&& 89.4 (89.7) & 94.2 (94.4) & 0.69 (1.02) & 0.74 (1.09) \\cr\n\t\t\t\t\n\t\t\t\t&&&$1000$& 89.3 (89.1) & 94.6 (94.4) & 0.83 (1.23) & 0.87 (1.29)\n\t\t\t\t&& 89.8 (88.8) & 94.7 (94.4) & 0.83 (1.23) & 0.87 (1.29) \\cr\n\t\t\t\t\n\t\t\t\t&&200&~$100$& 87.6 (87.7) & 93.4 (93.6) & 0.49 (0.73) & 0.52 (0.78)\n\t\t\t\t&& 90.3 (90.1) & 94.9 (95.2) & 0.49 (0.73) & 0.52 (0.78) \\cr\n\t\t\t\t\n\t\t\t\t&&&$1000$& 88.7 (89.7) & 94.7 (94.6) & 0.59 (0.88) & 0.61 (0.92)\n\t\t\t\t&& 90.2 (90.8) & 94.7 (95.7) & 0.59 (0.89) & 0.61 (0.93) \\cr\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\\hline\n\t\t\t\t\n\t\t\t\tIII&0&100&$100$& 89.8 (89.4) & 94.6 (94.5) & 0.65 (0.66) & 0.69 (0.70)\n\t\t\t\t&& 89.1 (89.0) & 94.4 (94.4) & 0.65 (0.66) & 0.69 (0.70) \\cr\n\t\t\t\t\n\t\t\t\t&&&$1000$ & 88.3 (88.2) & 93.6 (93.7) & 0.78 (0.78) & 0.82 (0.82)\n\t\t\t\t&& 89.1 (89.0) & 94.2 (93.8) & 0.78 (0.78) & 0.82 (0.82) \\cr\n\t\t\t\t\n\t\t\t\n\t\t\t\t\n\t\t\t\t&&200&~$100$ & 90.6 (91.1) & 95.0 (95.0) & 0.46 (0.46) & 0.49 (0.49)\n\t\t\t\t&& 90.6 (90.1) & 95.2 (95.2) & 0.46 (0.46) & 0.49 (0.49) \\cr\n\t\t\t\t\n\t\t\t\t&&&$1000$ & 90.1 (90.4) & 95.0 (94.6) & 0.55 (0.55) & 0.57 (0.58)\n\t\t\t\t&& 88.7 (89) & 93.6 (93.8) & 0.55 (0.55) & 0.57 (0.58)\\cr\n\t\t\t\t\n\t\t\t\t\\cline{2-13}\n\t\t\t\t\n\t\t\t\t&0.8&100&$100$& 90.4 (89.7) & 95.0 (94.8) & 0.63 (0.63) & 0.68 (0.68)\n\t\t\t\t&& 89.0 (88.9) & 95.0 (94.9) & 0.63 (0.63) & 0.67 (0.68) \\cr\n\t\t\t\t\n\t\t\t\t&&&$1000$ & 88.7 (88.9) & 93.8 (94.0) & 0.77 (0.77) & 0.80 (0.80)\n\t\t\t\t&& 89.0 (89.0) & 94.6 (94.3) & 0.76 (0.76) & 0.80 (0.80) \\cr\n\t\t\t\t\n\t\t\t\n\t\t\t\t\n\t\t\t\t&&200&~$100$ & 88.8 (89.1) & 94.2 (94.0) & 0.45 (0.45) & 0.48 (0.48)\n\t\t\t\t&& 90.2 (89.7) & 94.8 (95.0) & 0.45 (0.45) & 0.48 (0.48) \\cr\n\t\t\t\t\n\t\t\t\t&&&$1000$ & 90.0 (90.3) & 95.0 (95.0) & 0.54 (0.54) & 0.57 (0.57)\n\t\t\t\t&& 88.8 (89.1) & 94.2 (94.1) & 0.54 (0.54) & 0.57 (0.57) \\cr\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\\hline\n\t\t\t\\end{tabular}\n\t\t}\n\t}\n\\end{table}\n\n\\subsection{Simulations on multiple testing with FDR control}\n\nIn this section, we examine the performance of the sample spatial median-based B-H method introduced in Section \\ref{sec:FDR}, and compare it to the B-H procedure based on the sample mean with p-values calculated from $N(0,1)$ in \\citet{Liu2014}. We set $\\theta_{0,j}=0$ for all $j=1,\\ldots,p$. The data are generated from Models I and II with $p=1000$. For ${\\boldsymbol \\theta}=(\\theta_1,\\ldots,\\theta_{p})^{\\top}$, let $\\theta_j=2(\\log p\/n)^{1\/2}$ for $1\\leq j\\leq p_1$ and $\\theta_j=0$ for $(p_1+1)\\leq j\\leq p$, where\n$p_1=0.1p$.\n\nTable \\ref{tab2} reports the empirical FDR and power for the sample spatial median-based ($\\mathrm{FDR}_{M}$ and $\\mathrm{power}_{M}$) and the sample mean-based ($\\mathrm{FDR}_{A}$ and $\\mathrm{power}_{A}$) B-H procedures \\citep{Liu2014} with nominal level $\\alpha=0.1$ and $0.2$. The results indicate that the FDR are well controlled by both methods. For the multivariate normal distribution, the B-H procedures based on the spatial median and the sample mean have similar performance. However, the sample spatial median-based B-H method outperforms the sample mean-based B-H procedure in terms of empirical power under multivariate $t$-distribution, which is heavy-tailed.\n\n\\begin{table}[ht]\n\t\\centering\n\t{\n\t\t\\caption{Empirical FDR and power for the spatial median-based ($\\mathrm{FDR}_{M}$ and $\\mathrm{power}_{M}$) and the sample mean-based ($\\mathrm{FDR}_{A}$ and $\\mathrm{power}_{A}$) in \\citet{Liu2014} via B-H procedures.}\n\t\t\\label{tab2}\n\t\t\\setlength\\tabcolsep{3pt}\n\t\n\t\t\t\\begin{tabular}{@{}ccccccccccccc@{}}\n\t\t\t\t\\hline\n\t\t\t\t\n\t\t\t\n\t\t\t\t&&&\\multicolumn{4}{c}{$\\alpha=0.1$}& &\\multicolumn{4}{c}{$\\alpha=0.2$}\\cr\n\t\t\t\n\t\t\t\t\n\t\t\t\tModel & $\\rho$ & $n$& $\\mathrm{FDR}_{M}$ & $\\mathrm{FDR}_{A}$ & $\\mathrm{power}_{M}$ & $\\mathrm{power}_{A}$ & & $\\mathrm{FDR}_{M}$ & $\\mathrm{FDR}_{A}$ & $\\mathrm{power}_{M}$ & $\\mathrm{power}_{A}$ \\cr\n\t\t\t\t\\hline\n\t\t\t\t\n\t\t\t\n\t\t\t\tI & 0 &50& 0.124 & 0.124 & 0.996 & 0.996\n\t\t\t\t&& 0.224 & 0.222 & 0.999 & 0.999 \\cr\n\t\t\t\t\n\t\t\t\t&&100& 0.107 & 0.106 & 0.997 & 0.997\n\t\t\t\t&& 0.202 & 0.201 & 0.999 & 0.999 \\cr\n\t\t\t\t\n\t\t\t\n\t\t\t\t\n\t\t\t\t& 0.2 &50& 0.125 & 0.124 & 0.996 & 0.996\n\t\t\t\t&& 0.224 & 0.223 & 0.999 & 0.999 \\cr\n\t\t\t\t\n\t\t\t\t&&100& 0.107 & 0.106 & 0.997 & 0.997\n\t\t\t\t&& 0.202 & 0.201 & 0.999 & 0.999 \\cr\n\t\t\t\t\n\t\t\t\n\t\t\t\t\n\t\t\t\t& 0.5 &50& 0.125 & 0.124 & 0.996 & 0.996\n\t\t\t\t&& 0.225 & 0.223 & 0.999 & 0.999 \\cr\n\t\t\t\t\n\t\t\t\t&&100& 0.107 & 0.105 & 0.997 & 0.997\n\t\t\t\t&& 0.202 & 0.201 & 0.999 & 0.999 \\cr\n\t\t\t\t\n\t\t\t\n\t\t\t\t\n\t\t\t\t& 0.8 &50& 0.127 & 0.124 & 0.996 & 0.996\n\t\t\t\t&& 0.227 & 0.223 & 0.999 & 0.999 \\cr\n\t\t\t\t\n\t\t\t\t&&100& 0.108 & 0.105 & 0.997 & 0.997\n\t\t\t\t&& 0.204 & 0.199 & 0.999 & 0.999 \\cr\n\t\t\t\t\n\t\t\t\n\t\t\t\t\n\t\t\t\t\\hline\n\t\t\t\t\n\t\t\t\tII & 0 &50& 0.117 & 0.099 & 0.984 & 0.728\n\t\t\t\t&& 0.215 & 0.193 & 0.992 & 0.805 \\cr\n\t\t\t\t\n\t\t\t\t&&100& 0.103 & 0.088 & 0.987 & 0.710\n\t\t\t\t&& 0.197 & 0.179 & 0.994 & 0.795 \\cr\n\t\t\t\t\n\t\t\t\n\t\t\t\t\n\t\t\t\t& 0.2 &50& 0.117 & 0.098 & 0.984 & 0.727\n\t\t\t\t&& 0.215 & 0.194 & 0.992 & 0.805 \\cr\n\t\t\t\t\n\t\t\t\t&&100& 0.103 & 0.087 & 0.987 & 0.709\n\t\t\t\t&& 0.198 & 0.179 & 0.994 & 0.795 \\cr\n\t\t\t\t\n\t\t\t\n\t\t\t\t\n\t\t\t\t& 0.5 &50& 0.118 & 0.099 & 0.984 & 0.727\n\t\t\t\t&& 0.216 & 0.194 & 0.992 & 0.803 \\cr\n\t\t\t\t\n\t\t\t\t&&100& 0.103 & 0.087 & 0.987 & 0.708\n\t\t\t\t&& 0.198 & 0.178 & 0.994 & 0.794 \\cr\n\t\t\t\t\n\t\t\t\n\t\t\t\t\n\t\t\t\t& 0.8 &50& 0.120 & 0.098 & 0.984 & 0.724 \n\t\t\t\t&& 0.218 & 0.192 & 0.992 & 0.800 \\cr\n\t\t\t\t\n\t\t\t\t&&100& 0.104 & 0.087 & 0.987 & 0.705\n\t\t\t\t&& 0.199 & 0.177 & 0.994 & 0.791 \\cr\n\t\t\t\t\n\t\t\t\t\\hline \n\t\t\t\t\n\t\t\t\\end{tabular}\n\t\t\n\t}\n\\end{table}\n\n\n\n\\subsection{Real data analysis}\\label{sec:real_data}\n\nType 2 diabete\nis a disease in which the body becomes resistant to normal effects of insulin and gradually loses the capacity to produce enough insulin.\nBecause skeletal muscle is the main tissue for insulin-stimulated glucose disposal, skeletal muscle insulin resistance is commonly viewed as the critical component of whole-body insulin resistance, and thus is critical to the pathogenesis of Type 2 diabetes.\nTo investigate the effects of insulin on gene expression in skeletal muscle, a microarray study was performed in 15 diabetic patients using the Affymetrix Hu95A chip of muscle biopsies both before and after insulin treatment \\citep{Wu2007}. \nIn this paper, we are interested in the gene expression alteration, that is, the change of the gene expression level, due to the treatment. \nThe data are available at \\texttt{https:\/\/www.ncbi.nlm.nih.gov\/geo\/query\/acc.cgi?acc=GSE22309}.\nThe data were normalized by quantile normalization by the \\texttt{normalizeQuantiles} function in the \\texttt{limma} R package.\nFollow \\citet{Wang2015}, we focused on $2547$ curated gene sets with at least 15 genes, which are from the C2 collection of the GSEA online pathway databases. \nThe gene expression values are consolidated by taking the average when multiple probes are associated with the same gene.\n\nWe implemented the Median test based on $T_n$ on the $2519$ gene sets.\nThis is equivalent to testing whether the median change vector of gene expression levels is equal to 0.\nThe number of bootstrap iterations is $B=10^5$.\nWith the Bonferroni correction, there are $1242$ gene sets identified as significant at $5\\%$ level.\nFor comparison, we applied the WPL test \\citep{Wang2015} and the CQ test \\citep{Chen2010} on the same gene sets. \nFor the WPL test, $1060$ gene sets are selected as significant; and for the CQ test, $630$ gene sets are identified as significant. \nOut of the $630$ gene sets selected by the CQ test, $605$ of them are also identified by our proposed method, and $629$ of them are identified by the WPL test.\nIt has been argued in \\citet{Wang2015} that some gene expression levels have heavy tails as their kurtosises are much larger than the kurtosis of a normal distribution, 3. \nThus, the methods based on the spatial median (Median test and the WPL test) are expected to be more robust and efficient than those based on moments (CQ test).\nIn addition, out of the $1060$ gene sets identified by the WPL test, $958$ of them are significant based on our proposed approach.\n\n\nAs argued in Remark \\ref{remark:07}, the Median test based on $T_n$ is more powerful in detecting strong sparse signal compared to the WPL test.\nTo see this, we look into the following three gene sets: \\\\\n(1) ZHAN\\_MULTIPLE\\_MYELOMA\\_UP; \\\\\n(2) MIKKELSEN\\_MEF\\_HCP\\_WITH\\_H3K27ME3; \\\\\n(3) JAZAG\\_TGFB1\\_SIGNALING\\_VIA\\_SMAD4\\_UP. \\\\\nThe p-values of the WPL test for these three gene sets are $0.41$, $0.31$, $0.27$, respectively.\nHowever, the p-values of the Median test are all less than $1.0\\times 10^{-5}$ with $B=10^5$ bootstrap iterations for these three gene sets.\nFigure \\ref{Fig_real_data} \nplots the SCIs for the spatial median vectors of the change of gene expression levels for these three gene sets.\nThe confidence intervals that do not cover $0$ are colored in red.\nIt is very clear that the only one or two big values in the spatial median results in a rejection of the Median test, while the signals from other dimensions are not strong enough to land a rejection by the the WPL test.\n\n\\begin{figure}[htp]\n\t\\centerline{\n\t\t\\includegraphics[width=14cm,height=16cm]{genePlot.pdf}\n\t}\n\t\\caption{Simultaneous Confidence intervals (SCIs) for spatial medians of three gene sets. \n\t\n\t}\n\t\\label{Fig_real_data}\n\\end{figure}\n\n\nFinally, we use the spatial median-based B-H procedure to perform multiple testing with FDR control on the three gene sets to detect differentially expressed genes (DEG), which is one of the most important targets in genomic analysis.\nTable \\ref{tab3} reports the detected differentially expressed genes (DEG) in each gene set with nominal level $\\alpha=0.1$, along with the corresponding marginal p-value $P_{j}=2-2\\Phi(|T_{n,j}|)$ and the confidence interval in the SCIs for the selected genes.\nIt can be seen that for all the selected genes, the marginal p-values are very small, and the corresponding confidence intervals do not cover $0$.\n\n\n\\begin{table}[htp]\n\t\\centering\n\t{\n\t\t\\caption{Detected differentially expressed genes (DEG) by the spatial median-based B-H procedure for three gene sets with $\\alpha=0.1$; ``$p$-value'' refers to the marginal $p$-value $P_{j}=2-2\\Phi(|T_{n,j}|)$, and ``CI'' refers to the confidence interval in the SCIs for the selected genes.}\n\t\t\\label{tab3}\n\t\t\\setlength\\tabcolsep{3pt}\n\t\t\\resizebox{\\textwidth}{!}{\n\t\t\t\\begin{tabular}{@{}lccc@{}}\n\t\t\t\t\\hline\n\t\t\t\t\n\t\t\t\tGene set & DEG & p-value & CI \\\\\n\t\t\t\t\n\t\t\t\t\\hline \n\t\t\t\t\n\t\t\t\tZHAN\\_MULTIPLE\\_MYELOMA\\_UP & CDKN1A & 0.00082 & (0.234, 0.550) \\\\\n\t\t\t\t\n\t\t\t\tMIKKELSEN\\_MEF\\_HCP\\_WITH\\_H3K27ME3 & MYOD1 & $<0.00001$ & (0.433, 0.791) \\\\\n\t\t\t\t\n\t\t\t\tJAZAG\\_TGFB1\\_SIGNALING\\_VIA\\_SMAD4\\_UP & HDAC4 & 0.00058 & (0.254, 0.644) \\\\\n\t\t\t\t\n\t\t\t\t\\hline \n\t\t\t\t\n\t\t\t\\end{tabular}\n\t\t}\n\t}\n\\end{table}\n\n\\section{Discussion}\\label{sec:conclusion}\n\nIn this paper, we established one-sample and two-sample Gaussian and bootstrap approximations for ultrahigh dimensional sample spatial median under a general model beyond elliptical distributions.\nIt is of interest to study whether our results are potentially extendable to some other distribution families. \nWe leave this to a future work.\nIn addition, the proposed test based on the maxima of the sample spatial median is more powerful under sparse alternatives compared to those based on $L_2$-norms.\nIt is well known that the $L_2$-norm type tests are more powerful under dense alternatives. Thus, it is of interest to consider combining the test based on the maximum-norm and $L_2$-norm, which could be potentially powerful under both sparse and dense alternatives. We also leave this to a future study.\n\n\n\n\n\\section*{Supplementary material}\nThe supplementary material\nincludes all the technical proofs and  some additional numerical results.\n\n\n\\bibliographystyle{agsm}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec-intro}\n\\nobreak\nThe Gemini NICI Planet-Finding Campaign was a direct-imaging survey of\n\\new{about} 250 nearby stars for substellar and planetary-mass companions\nconducted at Gemini South Observatory between 2008 and 2012\n\\citep{2010SPIE.7736E..53L}.  The Campaign used the Near-Infrared\nCoronagraphic Imager \\citep[NICI,][]{2008SPIE.7015E..49C}, which\ncombines adaptive optics, coronagraphy, angular differential imaging\n\\citep{2004Sci...305.1442L, 2006ApJ...641..556M}, and dual-channel\nmethane-band infrared imaging to achieve an H-band contrast\ndetection limit of 14.4~magnitudes at 1$''$ radius\n\\citep{2013ApJ...779...80W}.  The principal scientific results have\nbeen published by \n\\citet{2010ApJ...720L..82B}, \n\\citet{2011ApJ...729..139W},\n\\citet{2012ApJ...750...53N},\n\\citet{2013ApJ...773..179W},\n\\citet{2013ApJ...776....4N}, and \n\\citet{2013ApJ...777..160B}; \nthe pipeline processing algorithm is described by\n\\citet{2013ApJ...779...80W}.\n\nThe multi-epoch imaging data acquired for the Campaign require\naccurate astrometric calibration in order to detect common proper\nmotion and parallax between the host star and any candidate companions and\ntherefore distinguish true companions from background stars.  In this\npaper, we describe the process used to establish the alignment of the NICI science\ndetectors relative to the celestial coordinate system and calibrate the World\nCoordinate System (WCS) contained in the data headers.\n\n\n\\section{Instrument Design and Data Format}\n\\label{sec-instdesign}\n\\nobreak\nNICI consists of two parts: an adaptive optics (AO) system and a\ndual-channel science camera \\citep{2008SPIE.7015E..49C}.  The\nAO-corrected wavefront entering the science camera first passes\nthrough a focal plane coronagraph mask and a pupil\/spider mask, then is divided\nbetween the two science channels by the\n{\\it Dichroic Wheel} (DW).  Light transmitted through the DW enters the\n{\\it Red} channel while reflected light enters the {\\it Blue}. \nThe DW can be set to one of the following elements: {\\it H-50\/50}\nbeamsplitter, which directs 50\\% of H-band (1.65~\\micron) light to\neach science channel; {\\it H\/K Dichroic}, which divides light between the H and\nK (2.2~\\micron) bands; {\\it Mirror}, which reflects all light to the\nBlue channel; or {\\it Open}, which passes all light to the Red channel.\n\nAfter the DW, each science channel contains a filter wheel, reimaging\noptics, and an ALADDIN~II 1024$\\times$1024 InSb array detector.  The\nCampaign most frequently used 4\\% bandpass filters centered in and out\nof the $\\lambda = 1.63~\\mu$m methane absorption feature (named\nCH4-H4\\%L and CH4-H4\\%S, respectively) to search for methane-bearing\nplanets very close to the primary star, and a broadband H filter for\ndeeper searches at larger separations.  The on-sky pixel size is\n$\\approx 18$~milli-arcseconds (mas) and the field size is $\\approx\n18\\farcs4 \\times 18\\farcs4$.  The mapping of the sky onto each of the\ntwo science detectors is different and changes with the DW element and\nfilter.\n\nNICI data are processed by the Gemini Data-Handling System (DHS) and\nare written to Multi-Extension FITS (MEF) files\n\\citep{1981A&AS...44..363W, 2010A&A...524A..42P} which contain a\nPrimary Header followed by two extensions, one for each of the two\nscience channels.  The header for each extension contains its own WCS\nspecifying the astrometric parameters for that channel\n\\citep{2002A&A...395.1061G}.  The calibration of the WCS for each channel is a\nprincipal subject of this work.\n\nDetails on the data format, WCS, and other header data are specified\nin the Appendix, and more information on NICI is available at\n\\url{http:\/\/www.gemini.edu\/sciops\/instruments\/nici}.\n\n\n\\section{Astrometric Requirements}\n\\label{sec-astreq}\n\\nobreak\nThe astrometric calibration requirements for the NICI Planet-Finding\nCampaign arise from three aspects of the observing strategy.\n\nFirst, most Campaign data were taken in a mode called \n{\\it Angular Differential Imaging} \n\\citep[ADI,][]{2004Sci...305.1442L, 2006ApJ...641..556M}, \nin which the Cassegrain rotator is held fixed and the sky \nallowed to rotate on the science detectors so that astronomical\nobjects surrounding the primary star under observation move relative\nto the fixed speckles in the NICI point-spread function (PSF).  \nThe individual $t \\approx 1$ minute exposures taken over a range\nof field angles (in some cases $> 90^\\circ$) were \nderotated and registered before being co-added by the \npipeline-processing software.  To achieve accurate alignment over the \nentire field, image distortion must be corrected before derotation.\n\nSecond, targets were imaged simultaneously in the two science channels\nin and out of the $\\lambda = 1.63~\\mu$m methane absorption feature, in a mode\nknown as {\\it Angular Spectral Differential Imaging} (ASDI), to\nsuppress the stellar continuum further and enhance potential\nmethane-bearing planets.  The distortion, rotation and scale\ndifference between the two channels must be corrected before the\npipeline can accurately register and subtract the two images,\nespecially when they are added in order to find non-methane-bearing\nplanets.  In practice, sufficient accuracy can be achieved because\nthese differences are static properties of the instrument that can be\nmeasured once and applied to all data.\n\nThird, the Campaign strategy required measuring the relative proper\nmotion (PM) between a target star and any candidate companions\ndetected in the surrounding field in order to distinguish true\ncompanions (with common PM) from background stars.  Because only the\nrelative PM is required, we are not concerned with absolute astrometry\n(i.e. the absolute celestial coordinate zero points), but only with\nthe image distortion, field rotation, and pixel scale.  \\new{In\norder to assess proper motion and parallax robustly, this calibration\nhad to be maintained over multi-year timescales}, during which time NICI\nwas dismounted and remounted on the telescope, sometimes on different\ninstrument ports with different numbers of reflections in the optical\npath to the instrument.  Therefore, the astrometric calibration\nrequired observations of astrometric fields during each instrument\nmounting.  In addition, because a field observed at multiple epochs was\nnot typically observed at the same hour angle each time, the field\nrotation was often significantly different, again requiring correction of the\nimage distortion before the two datasets could be compared.\n\nThe required astrometric accuracy is dictated by the PM of the target\nstars.  In the Campaign's target list of nearby stars, the lowest PMs\nare \\new{ $\\approx 20$~mas yr$^{-1}$, or $\\approx 1$~NICI pixel yr$^{-1}$};\nmost are at least a few times larger.  \\new{An accuracy of 1~pixel\n  from the center of the NICI field (the usual location of a primary\n  star) to the edge} therefore permits most candidate companions to be\nchecked for common PM within one year, especially since accuracy near\nthe field center will be considerably better.\n\nFor the pixel scale, 1~pixel out of \\new{512 is 0.2\\%}, while a\nrotation error of \\new{$0\\fdg112$ corresponds to 1~pixel across half the\nfield.}  These constraints will serve as guidelines throughout this\npaper.\n\n\n\\section{Astrometric Calibration}\n\\label{sec-astromcalib}\n\\nobreak\nOur astrometric calibration process has three principal components:\n\n\\begin{enumerate}\n\\item correcting the image distortion introduced by the instrument optics;\n\\item measuring the field rotation and setting the instrument alignment angle;\n\\item constructing the WCS.\n\\end{enumerate}\n\n\\subsection{Image Distortion}\n\\label{sec-distortion}\n\\nobreak\nThe two NICI channels suffer from image distortion arising from the\noff-axis reflective optics in both the AO system and science camera.\nTo measure the distortion, we imaged a pinhole grid mask mounted in\nthe {\\em Fiber Optic Calibration Source} (FOCS), which can be deployed\ninto the telescope focal plane at the entrance to the AO\nsystem.  The FOCS mask distortion at the science detectors represents\nthe combined distortions of the AO system and science camera optics.\n\\new{We preferred to use the mask rather than an astronomical field due\nto the grid spots' very high signal-to-noise, their uniform size and\ndistribution over almost the entire field, and the freedom from seeing\nand anisoplanatism effects.}\n\nFOCS Grid Mask images are shown in Figure~\\ref{fig-gridmask}.  The\ngrid is rotated by $22\\fdg3$ relative to the detector.  The\nBlue channel image is reflected left-right relative to the Red due to\nthe extra reflection at the DW optic in the path to the Blue channel.\n\n\n\\subsubsection{Fitting Procedure}\n\\label{sec-fitproc}\n\\nobreak\nTo determine the distortion, a grid mask image is first\nbackground-subtracted and the centroid of each spot\nmeasured.  A synthetic rectilinear grid is then fitted to the spot\npositions within 256 pixels of the image center, the region\nof lowest distortion, by minimizing the $\\chi^2$ statistic\n\\begin{equation}\n\\label{eq-chisq}\n\\chi^2 = \\sum_{i=1}^{N} \\frac{|{\\bf r}_i - {\\bf r}[x_i(x_0,y_0,d,\\theta),\\, y_i(x_0,y_0,d,\\theta)]|^2}{\\sigma_i^2},\n\\end{equation}\nwhere ${\\bf r}_i$ is the measured centroid of spot $i$,\n${\\bf r}[x_i(x_0,y_0,d,\\theta),\\, y_i(x_0,y_0,d,\\theta)]$ are\nthe spot positions in the synthetic grid, and \n$\\sigma_i$ is the uncertainty of the centroid measurement.  \nThere are four free parameters: $x_0$ and $y_0$ represent the overall \ngrid position, $d$ is the spot separation in pixels (assumed to be \nthe same in X and Y), and $\\theta$ is the grid rotation.  \nThe $\\chi^2$ value is minimized using the Nelder-Mead simplex method\n\\citep{1965TCJ..7..308P,Press:1992:NRC:148286} as implemented in the\nIDL built-in routine {\\tt AMOEBA} (Exelis Visual Information Solutions, \nBoulder, Colorado).  \\new{The simplex algorithm begins by computing $\\chi^2$ for a set of\ntrial parameters, then finds the trajectory through parameter space\nthat steadily reduces $\\chi^2$ until a minimum is reached.}  The uncertainty $\\sigma$ is \nset to 1 for all points because we are only interested in the best-fit\nrectangular grid at this first stage, not the positional uncertainties.\n\n\\new{We used the IDL routine {\\tt POLY\\_2D} to transform the images\n  geometrically to correct the distortion.  The transformation is\n  defined by polynomials of degree $N$:\n\\begin{equation}\n\\label{eq-distrans-x}\nx' = \\sum_{i=0}^{N}\\sum_{j=0}^{N} \\, P_{i,j} \\, x^j \\, y^i\n\\end{equation}\n\\begin{equation}\n\\label{eq-distrans-y}\ny' = \\sum_{i=0}^{N}\\sum_{j=0}^{N} \\, Q_{i,j} \\, x^j \\, y^i\n\\end{equation}\nwhere $x$ and $y$ are the initial pixel coordinates and $x'$ and $y'$\nare the transformed coordinates.  $P_{i,j}$ and $Q_{i,j}$ are\ncoefficients determined from a least-squares fit performed by the\ncompanion routine {\\tt POLY\\_WARP}, based on the measured grid mask\n($x, y$) positions and the rotated rectilinear grid ($x', y'$)\npositions.  After computing ($x', y'$) for all the pixels, \n{\\tt POLY\\_2D} generates a transformed image using cubic convolution.}\nThe residual errors of the distortion correction were computed by applying\nthe correction to the grid mask images, then measuring the difference\nbetween the rectilinear points and the mask spots.\n\n\\subsubsection{Distortion Corrections}\n\\label{sec-distcorr}\n\\nobreak\nWe derived distortion corrections for six pairs of images taken in\nfive different instrument configurations as shown in\nTable~\\ref{tab-dcparams}.  By experimentation, we determined that a\nfifth-order ($N = 5$) fit had a residual error of 0.5 -- 0.6 pixels\nroot-mean-square (rms) for both channels, and that higher-order\npolynomials did not reduce the error.  The initial distortion is as\nhigh as 12~pixels in the array corners.\n\nGiven the high signal-to-noise images of the spots, the uncertainty of\nthe individual centroid measurements is much smaller than 0.5 pixels,\nso the quality of the fits is limited by some source of systematic\nerror.  (In statistical terms, if $\\sigma$ is set to 0.1 pixel, the\nresultant $\\chi^2$ per degree of freedom of the perfect grid relative\nto the distortion-corrected images is $\\approx 100$, much higher than\nthe value of 1 expected when random errors dominate.)  The main source\nof systematic error is most likely the use of a polynomial to\napproximate the distortion; an optical raytrace model may be required\nto improve the fit.  The accuracy of the polynomial fit is within our\n1~pixel goal, so we accept it as satisfactory for the Campaign.\n\nWe fitted six different images to evaluate the consistency of the\ndistortion across multiple instrument configurations.  The consistency\nis a concern because the grid position on the Blue channel array\nvaries by up to a few dozen pixels (Table~\\ref{tab-spotpos}) due to\nthe slightly different mounting angles of the optical elements in the\nDichroic Wheel, and in some cases the failure of DW to seat\nconsistently in its detent.  The Blue channel variations are much\nlarger than the Red because the reflected beam from DW is deviated\nmore by optical element tilts than the transmitted beam.  For the\nH-50\/50 beamsplitter, the Blue channel ($X, Y$) positions vary by up to\n25 pixels between different positionings (two extremes ``min''\nand ``max'' are listed in Table~\\ref{tab-spotpos}); evidently the DW\ndoes not fully settle into its detent at the H-50\/50 position.  Such\nlarge variations are not detected for the other DW elements.\n\nAcross the six datasets, there are only small variations in each of\nthe parameters for scale and rotation (see Table~\\ref{tab-dcparams}).\nThe spot separations vary by 0.05\\% for the Red channel and 0.10\\% for\nthe Blue, while the rotations are consistent to $0\\fdg096$ for the Red\nchannel and $0\\fdg088$ for the Blue.  If the Open position is omitted,\na reasonable action given that DW substrates probably introduce\naberrations into the transmitted beam that would\nbe absent for the Open position, the Red channel rotation range is\nonly $0\\fdg027$.\n\nThe Blue channel exhibits DW-dependent rotations which appear to be\ncorrelated with the largest position offsets listed in\nTable~\\ref{tab-spotpos}, indicating that the irreproducible positions\nof the DW are causing both effects.  The rotations are smaller than\nour $0\\fdg1$ precision goal, and not much larger than the rms error of\nthe fit, but nevertheless appear to be systematic.\n\nThe fitting results do reveal significant differences in scale and\nrotation between the Red and Blue channels.  How these differences are\nmanaged is described in the WCS section below.\n\nTo evaluate distortion variations between the instrument\nconfigurations, we applied the correction for the H-50\/50 + CH4-H4\\%\nimages to the other four datasets.  Table~\\ref{tab-dch5050} shows that\nin all cases the errors are $< 1$~pixel rms, or $\\leq 0.1\\%$.\n\n\\new{\nGiven the acceptable repeatability of the H-50\/50 + CH4-H4\\%\ndistortion correction across all the instrument configurations, we\nadopted it as the standard correction for all Campaign data.  The\ndistortion correction coefficients for this mode are listed\nin Table~\\ref{tab-dccoeffs}, and maps of the distortion correction\nvectors and the post-correction residual errors are displayed in\nFigure~\\ref{fig-distmap}.}  More than half of\nthe Campaign data were taken in the dual-channel ASDI mode with this\nconfiguration.  Applications which require the highest possible\naccuracy, however, may wish to use more detailed calibration datasets\ntaken in a particular configuration.\n\n\n\\subsection{Instrument Alignment Angle}\n\\label{sec-iaa}\n\\nobreak\nThe Telescope Control System (TCS) uses a simple formula to control\nthe Cassegrain Rotator (CR) so that the desired position angle on sky,\n{$\\theta_{\\rm IPA}$}, the {\\em Instrument Position Angle}, is oriented\nparallel to the detector columns toward the top of the image.  A \nquantity {$\\theta_{\\rm IAA}$}, the {\\em Instrument Alignment\n  Angle}, is defined to represent the rotation between\nthe CR and science detector reference frames.  The CR angle\n$\\theta_{\\rm CR}$\\ is related to the other angles by\n\\begin{equation}\n   \\theta_{\\rm CR} = \\theta_{\\rm IPA} - \\theta_{\\rm par} - \\theta_{\\rm IAA} + 180^\\circ,\n\\end{equation}\nwhere $\\theta_{\\rm par}$\\ is the parallactic angle of the target under\nobservation.\n\nFor example, if a detector is mounted exactly aligned with the CR\nframe, $\\theta_{\\rm IAA} = 0$.  To achieve $\\theta_{\\rm IPA} = 0$\n(North up on detector) on the meridian where $\\theta_{\\rm par} = 0$,\n$\\theta_{\\rm CR}$\\ would be 180$^\\circ$.  Note that $\\theta_{\\rm IAA}$\\ is dependent on the\nmechanical orientation of the instrument on the telescope; small\nrotations or other changes in the mounting may in turn require a\ndifferent $\\theta_{\\rm IAA}$\\ setting in order to achieve an accurate $\\theta_{\\rm IPA}$.\n\nObservations may be defined with CR in two modes: {\\em Follow\\\/} mode, in\nwhich case the TCS sets and continuously updates the CR to keep the\nspecified $\\theta_{\\rm IPA}$\\ vertical on the detector, and {\\em Fixed\\\/} mode where the TCS\nsets the CR to a fixed value, allowing the sky (and therefore $\\theta_{\\rm IPA}$) to\nrotate on the detector during the observation.  Follow mode is used\nfor most observations with other science instruments, but Fixed is\nused for NICI ADI and ASDI observations.\n\nFor NICI, $\\theta_{\\rm IAA}$\\ can represent only one of the two science\nchannels.  The natural choice is the Red channel, due to its smaller\nDW-dependent systematic variations in scale and rotation than the\nBlue's.  With this calibration, at $\\theta_{\\rm IPA} = 0$, north is exactly\nvertical on the Red channel, while it is offset by $-1\\fdg1$ on the\nBlue channel.\n\nSetting $\\theta_{\\rm IAA}$\\ requires observing an astrometric standard\ntarget or field, an exercise which must be performed each time NICI is\nmounted on the telescope.  Table~\\ref{tab-iaa} lists the individual\nNICI mountings between 2008 and 2012, the astrometric standard\nobserved, and the $\\theta_{\\rm IAA}$\\ and pixel scale results.  Over the course of the\nCampaign, our calibration technique grew more sophisticated as we\nadded distortion corrections and switched from binary stars to a more\nprecise astrometric field.\n\n\n\\subsubsection{Binary Stars}\n\\label{sec-binstars}\n\nFor the Campaign's first two years, from 2008 August to 2010 October, we\nused two binary stars for astrometric calibration.  An ideal binary\nwould have a separation between 5$''$ and 10$''$ (small enough to fit\ninto NICI's 18$''$ field yet large enough to provide a sufficient lever\narm for an accurate measurement), a separation precision $<$ a few mas,\nand a position angle precision $< 0\\fdg1$.  Such high precision requires\nmodern speckle measurements; orbits based on older historical data\nare not sufficiently precise.  Several well-known systems with\nprecise orbits were rejected because their stars were too bright,\nsaturating on NICI even with the shortest possible exposure times.\n\nWe adopted \\object{70 Oph} (STF~2272AB, WDS~18055+0230), which has an\naccurate speckle-based orbit \\citep{2000A&AS..145..215P,\n  2008A&A...482..631E} and 5\\farcs5 separation in mid-2008, as our\ninitial astrometric standard.  The $V = 4.2$ primary component was\nbright enough to provide sufficient S\/N when placed under the\ncoronagraph mask.  The off-mask image of the secondary component, with\n$V = 6.2$, saturated slightly in the PSF core, but its position could\nstill be measured to $\\pm 0.5$~pixel based on the unsaturated halo.\nIn this way, $\\theta_{\\rm IAA}$\\ for mounting 1 was measured (see\nTable~\\ref{tab-iaa}), with an estimated uncertainty of $\\pm 0\\fdg09$.\nWhen NICI changed ISS ports (from side-looking to up-looking) in late\n2009, \\object{70 Oph} was not accessible, so we observed the binary \n\\objectname[CCDM~J22398]{HDO~171~B-C}\n(CCDM~J22398-1942~B, WDS 22398-1942) instead.  \\object{70 Oph} was\nobserved again for mounting 3 in 2010 March.  At this time we had not\ndeveloped the distortion-correction algorithm, so these data were\nreduced and $\\theta_{\\rm IAA}$\\ set with no distortion correction.\n\nFollow-up checks and corrections of these early binary star\ncalibrations are described in \\S\\ref{sec-earlycheck}.\n\n\n\\subsubsection{Trapezium Cluster}\n\\label{sec-trapezium}\n\\nobreak At the start of mounting 4 in 2010 October, we observed the\nTrapezium Cluster in Orion as an astrometric field.  We used the {\\em\n  HST}-derived coordinates from \\citet{2008AJ....136.2136R}, applied\nthe grid mask distortion correction procedure described in\n\\S\\ref{sec-distortion}, and fitted the measured position centroids to\nthe celestial coordinates as described for the LMC field in\n\\S\\ref{sec-lmc}.  Although the technique appeared to be superior to\nthe binary star calibrations, the fitting errors were unacceptably\nlarge: $\\approx 2$~pixels rms, \\new{considerably larger than\n  what we eventually achieved for the LMC field using the same\n  observing and reduction techniques.  The large errors\n  may be caused by systematic errors in the\n  {\\em HST} optical astrometry or the underlying Two Micron All Sky\n  Survey (2MASS) catalog \\citep{2003tmc..book.....C} absolute astrometry against\n  which the {\\em HST} data were registered.}  The magnitude of these\nerrors led us to develop the LMC-based calibration described in the\nnext section.\n\nNote that during this period in 2010 October-December, while new\nsoftware was being developed to apply the distortion correction,\nmeasure the alignment, and compute $\\theta_{\\rm IAA}$\\ changes, a number of\nerrors were made computing the direction of the $\\theta_{\\rm IAA}$\\ \ncorrections and the WCS parameters in the headers, as indicated in\nTable~\\ref{tab-iaa}.  Methods for correcting these errors are\ndescribed in \\S\\ref{sec-wcs}.\n\n\n\\subsubsection{LMC Astrometric Field}\n\\label{sec-lmc}\n\\nobreak\nIn 2010 November we began to observe a field in the Large Magellanic\nCloud (LMC) that was the subject of a detailed {\\em HST}-based\nastrometric calibration in support of {\\em JWST}\\\/\n\\citep{2007ASPC..364...81D}.  For {\\em JWST}, the field was chosen to\nbe relatively free of bright stars, but fortunately it contains three\n$R = 11-12$~mag stars bright enough to serve as natural guide stars\nfor NICI's AO system.  The data file containing high-precision\ncoordinates was supplied to us by J.\\,Anderson \\citetext{private\ncommunication}.\n\nThe subfield with the brightest guide star, named ``f606w\\_11.341'' in\nthe STScI documentation but which we refer to as ``LMC-11mag,''\ncontains 12 stars with $H < 17$ which can be detected by NICI\nin about 5~minutes observing time.  For consistency, the field was\nobserved in Cass Rotator Follow mode at {$\\theta_{\\rm IPA}$}~=~0, always with\nthe Clear focal-plane mask (no coronagraph), the H-50\/50 beamsplitter,\nand the {CH4-H4\\%L} and S filters.  Image pairs taken with a 4$''$\ndither were subtracted to cancel the background emission\n(Fig.~\\ref{fig-lmc11mag}), then the standard distortion correction\nwas applied and the stars' centroid positions in the Red channel\nmeasured.\n\nThe centroid position probable errors were estimated by synthesizing\nimages with Gaussian PSF's over a range of S\/N combined with\nbackgrounds with random variations.  After the centroid for a given\nS\/N PSF was measured for each of the random backgrounds, the standard\ndeviation $\\sigma$ could be computed directly, and a lookup table of\nS\/N vs. $\\sigma$ constructed.  The probable error of each star in the\nreal image was then estimated by measuring its S\/N and finding the\ncorresponding $\\sigma$ in the lookup table.\n\nThe centroid positions were fitted with a $\\chi^2$ minimization\ntechnique similar to the grid mask fitting described by\nEqn.\\ref{eq-chisq} to determine the pixel scale and the $\\theta_{\\rm IAA}$\\ \ncorrection, using the probable errors $\\sigma$ estimated for each\nstar.\n\nTable~\\ref{tab-lmc} displays the results of an LMC-11mag astrometric\nfit taken for mounting~10 on 2012 December 22 UTC.  The $R = 11$~mag guide star\nconsistently shows a large 5 - 6 pixel error in all images, so it is\nnot included in the fit.  (The exact reason for the error is unknown;\nthe star is certainly a foreground star, but the error appears to be\ntoo large to be explained by proper motion, and no proper motion was\ndetected between the different NICI astrometric images over a period\nof 3 years.)  As with the pinhole-grid images, the value\nof $\\chi^2$ per degree of freedom $> 30$ indicates that the errors are\ndominated by systematic effects.  In addition to the distortion\ncorrection errors described in \\S\\ref{sec-fitproc}, additional sources\nof error on a star field are errors in the provided celestial\ncoordinates, proper motions, atmospheric refraction, and\nanisoplanicity effects which cause stars far from the center of the\nfield to become elongated.\n\nThe residual error between the expected and measured positions \\new{of\n  all the LMC field stars is 0.68~pixels rms, with no errors $\\gtrsim\n  1$~pixel, better than our goal of $< 1$ pixel across half the field.\n  This 2012 December dataset was taken in good seeing and is one of\n  our best results, but in all our calibrations the residual errors\n  are $< 0.8$~pixels across the field, or $\\approx 0.08$\\% in scale\n  and 0\\fdg046 in rotation.  This corresponds to a center-to-edge\n  accuracy of $< 0.4$~pixels or 7~mas, better than our 20~mas\n  requirement.  The errors are similar for both up-looking and \n  side-looking port mountings, even with the extra reflection \n  on the side port.}\n\nAfter 2010 the LMC-11mag field became the primary NICI astrometric\nreference, observed regularly after most instrument mountings and port\nchanges and periodically during observing semesters.  We find that\n$\\theta_{\\rm IAA}$\\ varies by a few tenths degree between mountings, but\nperiodic checks within a mounting are consistent to $<0\\fdg03$,\nexceeding our calibration requirements.  We calibrated a secondary\nfield, \\object{HIP 62403}, relative to LMC-11mag for use during the southern\nwinter months when the LMC is inaccessible; our derived astrometric\ndata for this field are listed in Table~\\ref{tab-HIP62403}.\n\nThe LMC fits also yield the image pixel scale which, unlike {$\\theta_{\\rm IAA}$},\nshould not vary between mountings.  Over 12 separate LMC\nmeasurements, we find a Red channel scale of\n17.958$\\pm$0.022~mas\/pixel.  The relative probable error is 0.12\\%, or\nabout 1 pixel across the detector, again within our requirements.\n\n\\subsubsection{Checks and Corrections of Early Calibrations}\n\\label{sec-earlycheck}\n\\nobreak\nAfter developing the distortion correction and LMC-based calibration\nin late 2010, we checked the earlier binary star calibrations\nin two ways.  First, the binary star centroids were remeasured\nafter applying the grid mask distortion correction.  We found that the\nmounting~1 and 2 calibrations were unaffected by the distortion\ncorrection to a level of $\\approx 0\\fdg1$, because both binary components\nwere placed within the low-distortion central region.\nFor mounting~3, however, 70~Oph~B was placed in the lower left corner\nof the field and suffered a distortion of $\\approx 2.5$~pixels, causing a\nrotation error of $0\\fdg7$.\n\nAs a second check method, we utilized several Campaign target fields\nwith multiple background stars that had been observed both with binary\nstar and later LMC-based astrometric calibration.  The results are\nsummarized in Table~\\ref{tab-astromcorr}.  This method verified the\naccuracy of the mounting 1 and 2 calibrations and the error with\nmounting 3.  Two good comparison fields for mounting 3 indicated an\nerror of $0\\fdg5$ and a true {$\\theta_{\\rm IAA}$}~=~112.2, which we judge to be\nmore reliable than the 70~Oph derived value due to saturation of\n70~Oph~B and its position in the corner of the field.  A detailed\nprocedure for correcting the $\\theta_{\\rm IAA}$\\ error is presented in\n\\S\\ref{sec-wcs}; these corrections were applied to the Campaign data\nreductions, but {\\em not\\\/} to the raw data stored in the Gemini\nScience Archive.\n\nThese checks revealed one range of dates, 2009 April 26-27 UTC, for\nwhich the field rotation is in error by $0\\fdg5$, an unusually large\namount.  \\object{Proxima Cen} and \\object{HD 196544} data taken on\nthese dates both exhibit the anomaly.  We suspect that an error in the\nCass Rotator position or datum sometime between April 8 and 26 caused\nthis error.  After April 27, NICI operations were suspended for the\nwinter, so we do not have futher data until 2009 August, by which time\nthe Cass Rotator position is correct.\n\n\n\\subsection{WCS}\n\\label{sec-wcs}\n\\nobreak\nThe WCS in each NICI MEF extension specifies the mapping between sky\nand detector for that channel, including the field rotation, pixel\nscale, and reflection between sky and detector.  It has\nlimitations, however, in that it describes the absolute astrometric position\nonly to low accuracy and does not represent the image distortion.  In\naddition, during NICI's operations there were many instances when $\\theta_{\\rm IAA}$\\ \nwas set incorrectly and therefore the WCS does not\nrepresent the true astrometry.  Therefore, in this section we will\nreview how the WCS is constructed, its limitations and errors, and how\nthe errors can be corrected.\n\nThe Gemini algorithm that builds the WCS for each MEF file begins with\na mapping file that contains a series of $X, Y$ positions in detector\ncoordinates along with their corresponding on-sky angular offsets from\nthe field center in the telescope coordinate system.  The algorithm\nfits a transformation to these points, then applies the current\ntelescope pointing and Cassegrain rotator angles to compute the on-sky\nWCS.  The original intent was to populate the mapping file with points\nmeasured by offsetting a star to different positions on the detector\nand recording both the detector and celestial positions.  For NICI's\nvery narrow field of view and small pixels, however, this technique is\ninsufficiently precise. (It relies on the mapping accuracy of the\nPeripheral Wavefront Sensor probe arm, which could contain errors of\nseveral tenths of an arcsec.)  Therefore, we developed a simple script\nto generate synthetic mapping files based on the pixel scale and $\\theta_{\\rm IAA}$\\ \nrotation angle determined from the calibrations.  This\ntechnique produces a WCS limited only by the quality of the\nastrometric field measurements as described in preceeding sections.\n\nThe Red channel mapping file is generated directly from the measured\nRed channel scale and {$\\theta_{\\rm IAA}$}.  The Blue channel file is generated\nusing the more precise scale and rotation angle relative to the Red\nchannel determined from the grid mask images.  For the LMC\ncalibrations after 2010 November, the Blue WCS rotation angle is set\n$-1\\fdg1$ from the Red angle, and the scale is 1\/0.9980 = 1.0020 $\\times$\nthe Red scale (with the scale, in mas\/pixel, being inversely\nproportional to the magnification).\n\nGemini's WCS standard does not include distortion parameters such as\nthe SIP system, which defines FITS header distortion coefficients to\nsupplement the WCS \\citep{2005ASPC..347..491S}.  Therefore, for NICI\nfiles after 2010 November, the WCS represents the coordinate system\n{\\em after}\\\/ the standard distortion correction (for H-50\/50 +\nCH4-H4\\%) described in \\S\\ref{sec-distortion} has been applied.\n\nIn addition, note that the WCS represents the orientation at the {\\em\n  beginning}\\\/ of the exposure.  If the CR mode is Follow, then this\norientation is correct throughout the exposure.  However, if the CR\nmode is Fixed, as is the case for ADI observations, a rotation must be\napplied to correct the change in field angle between the start and\nmid-point of the exposure.\n\n\\subsubsection{WCS  Corrections}\n\\label{sec-wcs-corr}\n\\nobreak\nAs we previously described in \\S\\ref{sec-binstars} and\n\\ref{sec-trapezium}, the $\\theta_{\\rm IAA}$\\ and pixel scale values measured\nduring several calibrations contain errors.  These\nerrors are also manifested in the WCS.  Known issues are:\n\\begin{enumerate}\n\\item{The WCS values for each mounting are based on the pixel scale derived from\n    a particular calibration dataset.  For highest\n    accuracy and consistency, we recommend using the mean LMC-based scales of\n    17.958~mas\/pixel for the Red channel and 17.994 for Blue \n    for all epochs.}\n\\item{The values of $\\theta_{\\rm IAA}$\\ and WCS may be incorrect for data taken\n    immediately after each mounting, until astrometric calibrations\n    could be taken and analyzed.  The affected dates are listed in Table~\\ref{tab-iaa}.\n    Mounting 4 is the most severe example: data from 2010 October\n    and November contain several errors due to problems with the\n    distortion-correction and astrometric fitting software used at\n    that time for the Trapezium and LMC fields.}\n\\item{Before 2010-12-14, the rotation of the Blue WCS relative to the\n    Red did not use the correct value of $-1\\fdg1$.}\n\\item{For mounting 3 from 2010 March to October, the $\\theta_{\\rm IAA}$\\ \n    value of 111.70 was too low by $0\\fdg5$.}\n\\item{Data taken on 2009 April 26-27 have a rotation error of\n    $0\\fdg5$, apparently caused by a short-term error in the Cass Rotator.  (See last\n    paragraph of \\S\\ref{sec-earlycheck}}\n\\end{enumerate}\n\nThe WCS rotation correction for the Red channel is listed in\nTable~\\ref{tab-iaa}.  In addition, we have developed an IDL script\nthat automatically corrects a MEF file's WCS according to the date\nof observation (see Appendix).  Note that the script does not modify\nthe image data; only the WCS is updated to represent the true\nastrometry of the data.  The updated file with the corrected WCS can\nthen be pipeline-processed as usual.\n\n\n\\section{Campaign Astrometry Statistics}\n\\label{sec-campstats}\n\\nobreak\nSeveral dozen of the Campaign's target fields contained candidate\ncompanions which required follow-up checks for common proper motion\nand parallax.  The vast majority of these candidates were determined\nto be background stars.  We can use these multi-epoch observations to\nevaluate the consistency of the astrometric calibration over the\ncourse of the Campaign.\n\nOur technique is simply to compare the relative positions of the\nprimary target and background stars at the available epochs, after\ncorrecting the primary's relative parallax and proper motion to the\nfirst epoch.  Differences in the corrected positions between epochs\nindicate errors in the position measurements themselves, the proper\nmotion correction, the astrometric calibrations, or a combination of\nall three effects.  Some dense fields contained multiple background\nstars, permitting more detailed checks independent of the proper\nmotion correction, but because many fields just contained the primary\nand one background star we will discuss only the primary-to-background\nposition differences here.\n\nFigure~\\ref{fig-offsetdistrib} shows the offset distributions for a\ntotal of 218 pairs of observations of 205 background stars in 113\nfields.  The offsets are indicated parallel and perpendicular to the\nradius vector from the primary to the background star, which is\nappropriate given the large derotations applied to the ADI frames\nbefore stacking and helps to identify systematic errors in scale and\nrotation.\n\nThe left panel of Figure~\\ref{fig-offsetdistrib} shows the separation\n$r$ and perpendicular $d = r\\, \\tan(\\theta)$ offsets vs. $r$ itself.\nThe distribution of the $r$ offsets shows a possible increase with\nradius, suggesting scale errors on the order of 0.1\\%.  The $d$\noffsets, however, maintain a consistent width from small to large $r$,\nindicating that the primary source of error is in the individual\nposition measurements of the stars.  In other words, if the dominant\nsource of errors were in the rotation calibration, the distribution\nwould be expected to increase proportional to $r$, which is not\nobserved.\n\nThe right panel shows the distribution histograms in the same pixel\nunits.  The distributions have a central core with $\\sigma =\n0.85$~pixels in $r$ and 1.20 in $d$; values that are only slightly\nlarger than the rms errors of the distortion correction and the LMC\nfield astrometric fits.  Above 3.5~pixels total offset lie 9 {\\it\n  outlier} points, all of which have $r > 7''$.  The large errors in\nthese cases are caused by inaccurate distortion correction near the\nedges of the field and very different positions of the stars on the detector\nbetween the two epochs -- due to either different field angles, or to\nthe two observations being taken on different telescope ports with an\nodd and even number of reflections, which reversed the image and\ncaused the same object to appear on opposite sides of the detector.\n\nThat the primary source of the $d$ offset errors appears to be due to\nindividual position errors does not have an obvious explanation.\nBecause the PSF core FWHM is usually 3-4~pixels, centroid measurement\nerrors as large as 3~pixels would typically be several $\\sigma$.  It\nshould be noted that the astrometry is measured from final images\nwhich are coadded from many derotated ADI frames, which tends to\nbroaden the individual stars.  Possible causes of systematic error are\nsmall-scale irregularities in the distortion correction, shifts in the\nposition of the primary star which is dimmed by more than 6 magnitudes\nby the partially transparent occulting mask, or other irregularities\nin the mask.\n\n\n\n\\section{Conclusions}\n\\label{sec-conclusions}\n\\nobreak \\new{We have developed an astrometric calibration for the\n  NICI Planet-Finding Campaign based on a grid mask distortion\n  correction and an LMC astrometric field.  The accuracy\n  achieved by the calibration is $\\lesssim0.08$\\% in scale and\n  $\\lesssim0\\fdg046$ in rotation, corresponding to $\\lesssim7$~mas\n  center-to-edge.}  The calibration for each channel is represented by\nthe MEF header WCS.  Before 2010 November the calibrations and WCS\ncontain known errors; we provide tables and an IDL routine to correct\narchival NICI data to the final calibration.\n\n\\acknowledgements\n\nThis work was supported in part by NSF grants AST-0713881 and\nAST-0709484.  The authors thank the NICI team members J.~Hinds and\nC.~Lockhart, and Gemini engineers R.~Galvez, G.~Gausachs, J.~Luhrs,\nG.~Perez, R.~Rogers, R.~Rojas, L.~Solis, G.~Trancho, and C.~Urrutia\nfor their expert support of NICI during commissioning and operations.\nF.~Rantakyro, E.~Christensen, and E.~Artigau provided valuable\noperational and software support.  We thank J. Anderson for supplying\nthe {\\em HST\\\/} data for the LMC fields, and the anonymous referee\nfor providing valuable suggestions.  This research has made use of\nthe SIMBAD database, operated at CDS, Strasbourg, France; the\nWashington Double Star Catalog maintained at the U.S. Naval\nObservatory; and NASA's Astrophysics Data System.\n\n{\\it Facilities:} \\facility{Gemini (NICI)}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\nThe anchored $k$-core problem can be explained by the following illustrative example. We want\nto organize a workshop on Theory of Social Networks. We send invitations to most  distinguished researchers in the area and\nreceived many  replies of the following nature: ``Yes, in theory, I would be happy to come but my final decision  depends on\nhow many   people I know will be there.\"  Thus  we have a list of  tentative participants, but some of them can cancel their\nparticipation and we are afraid that the cancellation process\nmay escalate. On the other hand, we also have  limited funds to reimburse travel expenses for\na small number of participants, which we believe, will guarantee their participation. Thus we want to ``anchor\" a small subset\nof  participants whose guaranteed participation would prevent the unraveling process, and by\nfixing a small group   we hope to minimize the number of cancellations, or equivalently,\nmaximize the number of participants, or the core.\n\nUnraveling processes are common for social networks where the behavior of an individual is often influenced by the actions of\nher\/his friends. New events occur quite often in social networks: some examples are usage of a particular cell phone brand,\nadoption of a new drug within the medical profession, or the rise of a political movement in an unstable society. To estimate\nwhether these events or ideas spread extensively or die out soon, one has  to model and study the dynamics of \\emph{influence\npropagation} in social networks. Social networks are generally represented by making use of  undirected or directed graphs,\nwhere the edge set represents  the relationship between individuals in the network. Undirected graph  model works fine for\nsome networks, say Facebook, but the nature of interaction on some social networks such as Twitter is asymmetrical:  the fact\nthat  user $A$ follows  user $B$ does not imply that that user $B$ also follows $A$.\\footnote{The first author follows LeBron\nJames on Twitter (and so do 8,017,911 other people), but he only follows 302 people with the first author not being one of\nthem.}  In this case, it is more appropriate to model interactions in  the network by \\textbf{directed} graphs. We add a\ndirected edge $(u,v)$ if $v$ follows $u$.\n\n\nIn this work we are interested in the  model of \\emph{user engagement}, where each individual with less than $k$ people to\nfollow (or equivalently whose in-degree is less than $k$) drops out of the network. This\nprocess can be contagious, and may affect even those individuals who initially were linked to more than $k$ people, say follow\non Twitter. An extreme example of this was given by Schelling (see page 17 of ~\\cite{schelling2006micromotives}): consider a\ndirected path on $n$ vertices and let $k=1$. The left-endpoint has in-degree zero, it drops out and now the in-degree of its\nonly out-neighbor in the path becomes zero and it drops out as well. It is not hard to see that this way the whole network\neventually drops out as the result of a \\emph{cascade of iterated withdrawals}. In general at the end of all the iterated\nwithdrawals the remaining engaged individuals form a unique maximal induced subgraph whose minimum in-degree is at least $k$.\nThis is called as the \\emph{$k$-core} and is a well-known concept in the theory of social networks. It was introduced by\nSeidman~\\cite{seidman-k-core} and also been studied in various social sciences\nliterature~\\cite{chwe1999structure,chwe2000communication}.\n\n\\medskip\n\\noindent\\textbf{Preventing Unraveling:} The unraveling process described above in Schelling's example of a directed path can\nbe  highly undesirable in many scenarios.  How can one attempt to prevent this unraveling? In Schelling's example it is easy\nto see: if we ``buy\" the left end-point person into being engaged then the whole path becomes engaged. In general we overcome\nthe issue of unraveling by allowing some ``anchors\":  these are the vertices that remain engaged irrespective of their\npayoffs. This can be achieved by giving them extra incentives or discounts. The hope is that with a few \\emph{anchors} we can\nnow ensure a large subgraph remains engaged. This subgraph is  called as the \\emph{anchored $k$-core}: each non-anchor vertex\nin this induced subgraph must have in-degree at least $k$ while the anchored vertices can have arbitrary in-degrees. The\nproblem of identifying $k$-cores in a network also has the following  game-theoretical interpretation   introduced by\nBhawalkar et al.~\\cite{BhawalkarKLRS12}: each user in the social network pays a cost of $k$ to remain engaged. On the other\nhand, he\/she receives a profit of one from every neighbor who is engaged. The ``network effects\" come into play, and an\nindividual decides to remain engaged if has non-negative payoff, i.e., it has at least $k$ in-neighbors who are engaged. The\n$k$-core can be viewed as the unique maximal equilibrium in this model.\n\n\nBhawalkar et al.~\\cite{BhawalkarKLRS12} introduced  the \\textsc{Anchored $k$-Core} problem for (undirected) graphs. In the\n\\textsc{Anchored $k$-Core} problem the input is an undirected graph $G=(V,E)$ and integers $b, k$, and the task is to find  an\ninduced subgraph $H$ of maximum size with all vertices but at most $b$ (which are anchored)\nto be of degree at least $k$. In this work we extend the\nnotion of {anchored $k$-core} to directed graphs.  We are interested in the case, when  in-degrees of all but $b$ vertices of\n$H$ are at least  $k$.  More formally, we study the following parameterized version of the problem.\n\\begin{center}\n\\noindent\\framebox{\\begin{minipage}{4.50in} { \\textsc{Directed Anchored $k$-Core}  (\\textsc{Dir-AKC}\\xspace)}\\\\\n\\emph{Input}: A directed graph $G=(V,E)$ and integers $b, k,p$. \\\\\n\\emph{Parameter~1}: $b$.\\\\\n\\emph{Parameter~2}: $k$.\\\\\n\\emph{Parameter~3}: $p$.\\\\\n\\emph{Question}:  Do there exist sets of vertices $A\\subseteq H\\subseteq V(G)$ such that $|A|\\leq b$, $|H|\\geq p$, and every\n$v\\in H\\setminus a$ satisfies $d^{-}_{G[H]}(v)\\geq k$?\n\\end{minipage}}\n\\end{center}\nWe will call the set $A$ as the \\emph{anchors}, the graph $H$ as the \\emph{anchored $k$-core}. Note that the undirected\nversion of \\textsc{Anchored $k$-Core} problem can be modeled by the directed version: simply replace each edge $\\{u,v\\}$ by\narcs $(u,v)$ and $(v,u)$. Keeping the parameters $b,k,p$ unchanged it is now easy to see that the two instances are\nequivalent.\n\n\n\n\\medskip\n\\noindent\\textbf{Parameterized Complexity:} We are mainly interested in the parameterized complexity of  \\textsc{Anchored\n$k$-Core}. For the general background, we refer to the books by Downey and Fellows~\\cite{downey-fellows-book},  Flum and\nGrohe~\\cite{flum-grohe-book} and Niedermeier~\\cite{niedermeier-book}. Parameterized complexity is basically a two dimensional\nframework for studying the computational complexity of a problem. One dimension is the input size $n$ and another one is a\nparameter $k$. A problem is said to be \\emph{fixed parameter tractable} (or \\ensuremath{\\operatorClassFPT}) if it can be solved in time $f(k)\\cdot\nn^{O(1)}$ for some function $f$. A problem is said to be in \\ensuremath{\\operatorClassXP}, if it can be solved in time $O(n^{f(k)})$ for some\nfunction $f$. The $\\operatorClassW$-hierarchy is a collection of computational complexity classes: we omit the technical\ndefinitions here. The following relation is known amongst the classes in the $\\operatorClassW$-hierarchy:\n$\\ensuremath{\\operatorClassFPT}=\\classW{0}\\subseteq W[1]\\subseteq W[2]\\subseteq \\ldots$. It is widely believed that $FPT\\neq W[1]$, and hence if a\nproblem is hard for the class $W[i]$ (for any $i\\geq 1$) then it is considered to be fixed-parameter intractable.\n\n\\medskip\n\\noindent\\textbf{Previous Results:} Bhawalkar et al.~\\cite{BhawalkarKLRS12} initiated the algorithmic  study of\n\\textsc{Anchored $k$-Core}  on undirected graphs and obtained an interesting dichotomy result:  the decision version of the\nproblem is solvable in polynomial time for $k\\leq 2$ and is  \\ensuremath{\\operatorClassNP}-complete for all $k\\geq 3$. For $k\\geq 3$, they also\nstudied the problem from the viewpoint of parameterized complexity and approximation algorithms.\nThe current set of authors~\\cite{ChitnisFG13} improved and generalized these results by showing that for $k\\geq 3$ the problem\nremains \\ensuremath{\\operatorClassNP}-complete even on planar graphs.\n\n\n\\medskip\\noindent\\textbf{Our Results:} In this paper we provide a number of new results on the algorithmic complexity of\n\\textsc{Directed Anchored $k$-Core}  (\\textsc{Dir-AKC}\\xspace).\nWe start (Section~\\ref{sec:defs}) by showing that that the decision version of \\textsc{Dir-AKC}\\xspace{} is \\ensuremath{\\operatorClassNP}-complete for every $k\\geq\n1$  even if the input graph is restricted to be a planar directed acyclic graph (DAG) of maximum degree at most $k+2$. Note\nthat this shows that the directed version is in some sense strictly harder than the undirected version since it is known be in\n\\ensuremath{\\operatorClassP}\\ if $k\\leq 2$, and \\ensuremath{\\operatorClassNP}-complete if $k\\geq 3$~\\cite{BhawalkarKLRS12}. The \\ensuremath{\\operatorClassNP}-hardness result for \\textsc{Dir-AKC}\\xspace\nmotivates us to make a more refined analysis of the \\textsc{Dir-AKC}\\xspace problem via the paradigm of parameterized complexity. In\nSection~\\ref{sec:saved}, we  obtain the following dichotomy result: \\textsc{Dir-AKC}\\xspace is \\ensuremath{\\operatorClassFPT} \\, parameterized by $p$ if $k=1$, and\n\\classW1-hard if $k\\geq 2$. This fixed-parameter intractability result parameterized by $p$ forces us to consider the\ncomplexity on special classes of graphs such as bounded-degree directed graphs or directed acyclic graphs. In\nSection~\\ref{sec:bound-deg}, for graphs of degree upper bounded by $\\Delta$, we show that the \\textsc{Dir-AKC}\\xspace problem is FPT\nparameterized by $p+\\Delta$ if $k\\geq \\frac{\\Delta}{2}$. In particular, it implies that \\textsc{Dir-AKC}\\xspace is FPT parameterized by $p$ for\ndirected graphs of maximum degree at most four. We complement these results by showing in Section~\\ref{sec:concl} that  if $k<\n\\frac{\\Delta}{2}$ and $\\Delta\\geq 3$, then \\textsc{Dir-AKC}\\xspace is \\classW2-hard when parameterized by the number of anchors $b$ even for\nDAGs, but the problem is \\ensuremath{\\operatorClassFPT}\\ when parameterized by $\\Delta+p$ for DAGs of maximum degree at most $\\Delta$. Note that we\ncan always assume that $b\\leq p$, and hence any \\ensuremath{\\operatorClassFPT}\\ result with parameter $b$ implies \\ensuremath{\\operatorClassFPT}\\ result with parameter\n$p$ as well. On the other side, any hardness result with respect to $p$ implies the same hardness with respect to $b$.\n\n\\section{Preliminaries}\\label{sec:defs}\nWe consider finite directed and undirected graphs without loops or multiple arcs. The vertex set of a (directed) graph $G$ is\ndenoted by $V(G)$ and its edge set (arc set for a directed graph) by $E(G)$. The subgraph of $G$ induced by a subset\n$U\\subseteq V(G)$ is denoted by $G[U]$. For $U\\subset V(G)$ by $G-U$ we denote the graph $G[V(G)\\setminus U]$. For a directed\ngraph $G$, we denote by $G^*$ the undirected graph with the same set of vertices such that $\\{u,v\\}\\in E(G^*)$ if and only if\n$(u,v)\\in E(G)$. We say that $G^*$ is the \\emph{underlying} graph of $G$.\n\nLet $G$ be a directed graph. For a vertex $v\\in V(G)$, we say that $u$ is an \\emph{in-neighbor} of $v$ if $(u,v)\\in E(G)$. The\nset of all in-neighbors of $v$ is denoted by $N_G^-(v)$. The \\emph{in-degree} $d_G^-(v)=|N_G^-(v)|$. Respectively, $u$ is an\n\\emph{out-neighbor} of $v$ if $(v,u)\\in E(G)$, the set of all out-neighbors of $v$ is denoted by $N_G^+(v)$, and the\n\\emph{out-degree} $d_G^+(v)=|N_G^+(v)|$. The \\emph{degree} $d_G(v)$ of a vertex $v$ is the sum $d_G^-(v)+d_G^+$, and the\n\\emph{maximum degree} of $G$ is $\\Delta(G)=\\max_{v\\in V(G)}d_G(v)$. A vertex $v$ of $d_G^-(v)=0$ is called a \\emph{source},\nand if  $d_G^+(v)=0$, then $v$ is a \\emph{sink}. Observe that isolated vertices are sources and sinks simultaneously.\n\n\n\nLet $G$ be a directed graph. For $u,v\\in V(G)$, it is said that $v$ can be \\emph{reached} (or \\emph{reachable}) from $u$ if\nthere is a directed $u\\rightarrow v$ path in $G$. Respectively, a vertex $v$ can be reached from a set $U\\subseteq V(G)$ if\n$v$ can be reached from some vertex $u\\in U$. Notice that each vertex is reachable from itself. We denote by $R_G^+(u)$\n($R_G^+(U)$ respectively) the set of vertices that can be reached from a vertex $u$ (a set $U\\subseteq V(G)$ respectively).\nLet $R_G^-(u)$ denote the set of all vertices $v$ such that $u$ can be reached from $v$.\n\nFor two non-adjacent vertices $s,t$ of a directed graph $G$, a set $S\\subseteq V(G)\\setminus\\{s,t\\}$ is said to be a\n\\emph{$s-t$ separator} if $t\\notin R_{G-S}^+(s)$. An $s-t$ separator $S$ is \\emph{minimal} if no proper subset $S'\\subset S$\nis a $s-t$ separator.\n\nThe notion of important separators was introduced by Marx~\\cite{Marx06} and generalized for directed graphs\nin~\\cite{ChitnisHM12}. We need a special variant of this notion. Let $G$ be a directed graph, and let $s,t$ be non-adjacent\nvertices of $G$. An minimal $s-t$ separator is an \\emph{important $s-t$ separator} if there is no $s-t$ separator $S'$ with\n$|S'|\\leq |S|$ and $R_{G-S}^-(t)\\subset R_{G-S'}^-(t)$. The following lemma is a variant of Lemma~4.1 of~\\cite{ChitnisHM12}.\nNotice that to obtain it, we should replace the directed graph in Lemma~4.1 of~\\cite{ChitnisHM12} by the graph obtained from\nit by reversing direction of all arcs.\n\n\n\\begin{lemma}[\\cite{ChitnisHM12}]\\label{lem:imp-sep}\nLet $G$ be a directed graph with $n$ vertices, and let $s,t$ be non-adjacent vertices of $G$. Then for every $h\\geq 0$, there\nare at most $4^h$ important $s-t$ separators of size at most $h$. Furthermore, all these separators can be enumerated in time\n$O(4^{h}\\cdot n^{O(1)})$.\n\\end{lemma}\n\n\nAs further we are interested in the parameterized complexity of \\textsc{Dir-AKC}\\xspace, we show first \\ensuremath{\\operatorClassNP}-hardness of the problem. \n\n\\begin{theorem}\\label{thm:NPc}\nFor any $k\\geq 1$, \\textsc{Dir-AKC}\\xspace is \\ensuremath{\\operatorClassNP}-complete, even for planar DAGs of maximum degree at\nmost $k+2$.\n\\end{theorem}\n\n\n\\begin{proof}\nWe reduce {\\sc Satisfiability}:\n\\begin{center}\n\\noindent\\framebox{\\begin{minipage}{4.50in}{Satisfiability}\\\\\n\\emph{Input}: Sets of Boolean variables $x_1,\\ldots,x_n$ and clauses $C_1,\\ldots,C_m$. \\\\\n\\emph{Question}: Can the formula $\\phi=C_1\\vee\\ldots\\vee C_m$ be satisfied?\n\\end{minipage}}\n\\end{center}\nIt is known (see e.g.~\n\\cite{DahlhausJPSY94}) that this problem remains \\ensuremath{\\operatorClassNP}-hard even if each clause contains at most 3 literals (notice that clauses of size one or two are allowed), each variable\nis used in at most 3 clauses: at least once in positive and at least once in negation, and the graph that correspond to a\nboolean formula is planar. Consider an instance of {\\sc Satisfiability} with $n$ variables $x_1,\\ldots,x_n$ and $m$ clauses\n$C_1,\\ldots,C_m$ that satisfies these restrictions on planarity and the number of occurrences of the variables. We construct the graph $G$\nas follows.\n\n\n\\begin{figure}[ht]\n\\centering\\scalebox{0.7}{\\input{fig1.pdf_t}}\n\\caption{Construction of $G$ for $k=3$.\n\\label{fig:NPh}}\n\\end{figure}\n\n\n\n\\begin{itemize}\n\\item For each $i\\in\\{1,\\ldots,n\\}$,\n\\begin{itemize}\n\\item add vertices $x_i,\\overline{x}_i,r_i$ and add arcs $(x_i,r_i),(\\overline{x}_i,r_i)$;\n\\item add a set of $k-1$ vertices $Y_i$ and draw an arc from each of them to $r_i$;\n\\item for each vertex $y\\in Y_i$, add $k$ vertices and draw an arc from each of them to $y$, denote the set of these\n    $k(k-1)$ vertices $Z_i$.\n\\end{itemize}\n\\item For each $j\\in\\{1,\\ldots,j\\}$,\n\\begin{itemize}\n\\item add a vertex $v_j$, and for each literal $x_i$ ($\\overline{x}_i$ respectively) in the clause $C_j$, join the\n    vertex $x_i$ ($\\overline{x}_i$ respectively) with $v_j$ by an arc;\n\\item add a set of $k-1$ vertices $U_j$ and draw an arc from each of them to $v_j$;\n\\item for each vertex $u\\in U_j$, add $k$ vertices and draw an arc from each of them to $u$, denote the set of these\n    $k(k-1)$ vertices $W_j$.\n\\end{itemize}\n\\end{itemize}\nNotice that if $k=1$, then $Y_i=Z_i=U_j=W_j=\\emptyset$. The construction of $G$ is shown in Fig.~\\ref{fig:NPh}. We set\n$b=n(k(k-1)+1)+mk(k-1)$ and $p=n((k+1)(k-1)+2)+m((k+1)(k-1)+1)$. It is straightforward to see that $G$ is acyclic. Because\neach variable $x_i$ is used at most 2 times in positive and at most 2 times in negations, $d_G(x_i), d_G(\\overline{x}_i)\\leq\n3$ for all $i\\in\\{1,\\ldots,n\\}$, and $\\Delta(G)\\leq k+2$. \nBecause the graph of the boolean formula is a subcubic planar graph,  \n$G$ is planar.\n\nWe claim that all clauses $C_1,\\ldots,C_m$ can be satisfied if and only if there are a set $A\\subseteq V(G)$\nand an induced subgraph $H$ of $G$ such\nthat $A\\subseteq V(H)$, $|A|\\leq b, |V(H)|\\geq p$, and for every $v\\in V(H)\\setminus A$, we have $d^{-}_{H}(v)\\geq k$.\n\nSuppose that we have a YES-instance of {\\sc Satisfiability} and consider a truth assignment of $x_1,\\ldots,x_n$ such that\nall clauses are satisfied. We construct $A$ by including all the vertices $Z_1\\cup\\ldots\\cup Z_n\\cup W_1\\cup\\ldots\\cup W_m$ in\nthis set, and for each $i\\in\\{1,\\ldots,n\\}$, if $x_i=\\text{true}$, then $x_i$ is included in $A$ and $\\overline{x}_i$ is\nincluded otherwise. Clearly, $|A|=|Z_1|+\\ldots+|Z_n|+|W_1|+\\ldots+|W_m|+n=n(k(k-1)+1)+mk(k-1)=b$. Let $H=G[A\\cup\nY_1\\cup\\ldots\\cup Y_n\\cup U_1\\cup\\ldots U_m\\cup\\{r_1,\\ldots,r_n\\}\\cup\\{v_1,\\ldots,v_m\\}]$. Consider $w\\in V(H)\\setminus A$. If\n$w\\in Y_i$ for $i\\in\\{1,\\ldots,n\\}$, then $w$ has $k$ in-neighbors in $Z_i\\subseteq A$. If $w=r_i$ for $i\\in\\{1,\\ldots,n\\}$,\nthen $w$ has $k-1$ in-neighbors in $Y_i$ and either $x_i$ or $\\overline{x}_i$ is an in-neighbor of $w$ as well. If $w\\in U_j$\nfor $j\\in\\{1,\\ldots,m\\}$, then $w$ has $k$ in-neighbors in $W_j\\subseteq A$. Finally, if $w=v_j$ for some\n$j\\in\\{1,\\ldots,m\\}$, then $w$ has $k-1$ in-neighbors in $U_j$. As the clause $C_j$ is satisfied, it contains a literal $x_i$\nor $\\overline{x_i}$ that has the value $\\text{true}$. Then by the construction of $A$, the corresponding vertex $x_i$ or\n$\\overline{x}_i$ respectively is in $A$, and $w$ has one in-neighbor in $A$. It remains to observe that\n$|V(H)|=|A|+|Y_1|+\\ldots+|Y_n|+|U_1|+\\ldots+|U_m|=n(k(k-1)+1)+mk(k-1)+k(n+m)=p$.\n\nAssume now there are a set $A\\subseteq V(G)$\nand an induced subgraph $H$ of $G$ such\nthat $A\\subseteq V(H)$, $|A|\\leq b, |V(H)|\\geq p$ and for every $v\\in V(H)\\setminus A$ we have $d^{-}_{H}(v)\\geq k$.\n\nLet $S=\\{w\\in V(G)\\ |\\ d_G^-(w)=0\\}=(\\cup_{i=1}^n\\{x_i,\\overline{x_i}\\})\\cup (\\cup_{i=1}^nZ_i)\\cup(\\cup_{j=1}^mW_j)$ and\n$T=V(G)\\setminus S=\\{r_1,\\ldots,r_n\\}\\cup (\\cup_{i=1}^nY_i)\\cup(\\cup_{j=1}^mU_j)$. \n We claim that\n$A\\subseteq S$ and $T\\subseteq V(H)$. To show it, observe that any vertex $w\\in S$ is in $H$ if and only if $w\\in A$ as\n$d_G^-(w)=0$. Because $|V(G)|-|V(H)|\\leq n$, at least $|S|-n$ vertices of $S$ are in $A$. Since $|S|=b+n$, we conclude that\nexactly $b=|S|-n$ vertices of $S$ are in $A$ and $A\\subseteq S$. Moreover, $V(H)=T\\cup A$.\n\nLet $z\\in Z_i$ for some $i\\in\\{1,\\ldots,n\\}$ and assume that $z$ is adjacent to $y\\in Y_i$. If $z\\notin A$, then $y\\in T$ has\nat most $k-1$ in-neighbors in $H$, a contradiction. Hence, $Z_1\\cup\\ldots\\cup Z_n\\subseteq A$. By the same arguments we\nconclude that $W_1\\cup\\ldots\\cup W_m\\subseteq A$. Then we have exactly $n$ elements of $A$ in\n$\\cup_{i=1}^n\\{x_i,\\overline{x}_i\\}$. Consider a pair of vertices $x_i,\\overline{x}_i$ for $i\\in\\{1,\\ldots,n\\}$. If\n$x_i,\\overline{x}_i\\notin A$, then $r_i\\in T$ has at most $k-1$ in-neighbors in $H$, a contradiction. Therefore, for each\n$i\\in\\{1,\\ldots,n\\}$, exactly one vertex from the pair  $x_i, \\overline{x}_i$ is in $A$. For $i\\in\\{1,\\ldots,n\\}$, we set the\nvariable $x_i=\\text{true}$ if the vertex $x_i\\in A$, and $x_i=\\text{false}$ otherwise.\n\nIt remains to prove that we have a satisfying truth assignment. Consider a clause $C_j$ for $j\\in\\{1,\\ldots,m\\}$. The vertex\n$v_j\\in T$ has $k-1$ in-neighbors in $H$ that are vertices of $T$. Hence, it has at least one in-neighbor in $A$. It can be\neither a vertex $x_i$ or $\\overline{x}_i$ that correspond to a literal in $C_j$. It is sufficient to observe that if $x_i\\in\nA$, then the literal $x_i=\\text{true}$, and if  $\\overline{x}_i\\in A$, then the literal $\\overline{x}_i=\\text{true}$ by our\nassignment.\n\\end{proof}\n\n\nWe conclude this section by the simple observation that \\textsc{Dir-AKC}\\xspace is in \\ensuremath{\\operatorClassXP}\\ when parameterized by the number of anchors $b$.\nFor a directed graph $G$ with $n$ vertices, we can consider all the at most $n^b$ possibilities to\nchoose the anchors, and then recursively delete non-anchor vertices that have the in-degree at most $k-1$. Trivially, if we\nobtain a directed graph with at least $p$ vertices for some selection of the anchors, we have a solution and otherwise we can\nanswer NO.\n\n\\section{\\textsc{Dir-AKC}\\xspace parameterized by the size of the core}\n\\label{sec:saved}\n\\vskip-2mm\nIn this section we consider the \\textsc{Dir-AKC}\\xspace problem for fixed $k$ when $p$ is a parameter and obtain the following dichotomy: If\n$k=1$ then the \\textsc{Dir-AKC}\\xspace problem is FPT parameterized by $p$, otherwise for $k\\geq 2$ it is \\classW1-hard parameterized by $p$.\n\n\\begin{theorem}\\label{thm:fpt-d-1}\nFor $k=1$, the \\textsc{Dir-AKC}\\xspace problem is solvable in time $2^{O(p)}\\cdot n^2\\log n$ on digraphs with $n$ vertices.\n\\end{theorem}\n\n\\begin{proof}\nThe proof is constructive, and we describe an \\ensuremath{\\operatorClassFPT}~ algorithm for the problem. Without loss of generality, we assume that\n$b< p\\leq n$.\n\nWe apply the following preprocessing rule reducing the  instance to  an acyclic graph. Let  $C_1,\\ldots,C_r$ be strongly\nconnected components of $G$. By making use of Tarjan's algorithm~\\cite{Tarjan72}, the sets  $C_1,\\ldots,C_r$ can be found  in\nlinear time.  Let $R=R_G^{+}\\Big(\\bigcup_{i=1}^rV(C_i)\\Big)$ be the set of vertices reachable from strongly connected\ncomponents. Then every $v\\in R$ satisfies $d_{G[R]}^-(v)\\geq 1$. If $b\\geq p-|R|$, then we select in $V(G)\\setminus R$ any\narbitrary $b'=p-|R|$ vertices $a_1,\\ldots,a_{b'}$. In this case we  output the set of anchors $A=\\{a_1,\\ldots,a_{b'}\\}$ and\ngraph $H=G[A\\cup R]$. Otherwise, if $b< p-|R|$,  we set $G'=G-R$ and $p'=p-|R|$ and consider a new instance of\n\\textsc{Dir-AKC}\\xspace  with the graph $G'$ and the parameter $p'$.\n\nTo see that the rule is safe, it is sufficient to observe that a set of anchors $A$ and a subgraph $H'$ of size at least $p'$\nis a solution of the obtained instance if and only if $(A,H=G[V(H')\\cup R])$ is a solution for the original problem. Let us\nremark that the preprocessing rule can be easily  performed in  time $O(n^2)$.\n\nFrom now we can assume that $G$ has no strongly connected components, i.e., $G$ is a directed acyclic graph. Denote by\n$S=\\{s_1,\\ldots,s_h\\}$ the set of sources of $G$. If $|S|\\leq b$, then set $A=S$. In this case, we output the pair $(A,H=G)$.\nThe pair $(A,H)$ is a solution because every vertex $v\\in V(G)\\setminus S$ satisfies $d_G^-(v)\\geq 1$. It remains to consider\nthe case when $|S|>b$. For $i\\in\\{1,\\ldots,h\\}$, let $R_i=R_G^+(s_i)$. Then $V(G)=R_G^+(S)=\\bigcup_{i=1}^hR_i$. Without loss\nof generality, we can assume that every anchored vertex is from $S$. Indeed, if $s_i$ is an anchor, then each vertex of $R_i$\ncan be included in a solution. Hence for every anchor $a\\in R_j\\setminus \\{s_j\\}$, we can delete anchor from $a$ and anchor\n$s_j$, if it is not yet anchored.\nSince we  can choose anchors only from $S$, we are able to reduce the problem to {\\sc Partial Set Cover}.\n\n\\begin{center}\n\\noindent\\framebox{\\begin{minipage}{4.50in}{\\textsc{Partial Set Cover}\\xspace }\\\\\n\\emph{Input }: A collection $X=\\{X_1,\\ldots,X_r\\}$ of subsets of a finite $n$-element set $U$ and positive integers $p, b$. \\\\\n\\emph{Parameter}: $p$.\\\\\n\\emph{Question}: Are there at most $b$ subsets $X_{i_1},\\ldots,X_{i_{b}}$, $1\\leq i_1<\\ldots<i_{b}\\leq r$, covering at least\n$p$ elements of $U$, i.e., $|\\bigcup_{j=1}^{b} X_{i_j}|\\geq p$?\n\\end{minipage}}\n\\end{center}\n\nBl\\\"{a}ser~\\cite{Blaser03} showed that \\textsc{Partial Set Cover}\\xspace if \\ensuremath{\\operatorClassFPT} parameterized by $p$ and can be solved in time $O(2^{O(p)}\\cdot r\nn\\log n)$.\nFor \\textsc{Dir-AKC}\\xspace, \nwe consider the collection of subsets $\\{R_1,\\ldots,R_r\\}$ of $V(G)$. If we can select at most $b$\nsubsets $R_{i_1},\\ldots,R_{i_{b}}$ such that  $|\\cup_{j=1}^{b}R_{i_j}|\\geq p$, we return the solution with anchors\n$A=\\{s_{i_1},\\ldots,s_{i_{b}}\\}$ and $H=G[\\bigcup_{j=1}^{b}R_{i_j}]$. Otherwise, we return a NO-answer.\n\nBecause our preprocessing can be done in time $O(n^2)$ and  {\\sc Partial Set Cover} is solvable in time $2^{O(p)}\\cdot n^2\\log\nn$, we conclude that the total running time is $2^{O(p)}\\cdot n^2\\log n$.\n\\end{proof}\n\nNow we complement Theorem~\\ref{thm:fpt-d-1} by showing that for $k\\geq 2$,  \\textsc{Dir-AKC}\\xspace becomes hard\nparameterized by the core size.\n\n\\begin{theorem}\\label{thm:w-saved}\nFor any fixed $k\\geq 2$, the \\textsc{Dir-AKC}\\xspace problem is \\classW1-hard parameterized by $p$, even for DAGs.\n\\end{theorem}\n\n\\begin{proof}\nWe reduce from the {\\sc $b$-Clique} problem which is known to be \\classW1-hard~\\cite{downey-fellows-book}:\n\n\\begin{center}\n\\noindent\\framebox{\\begin{minipage}{4.50in}{$b$-Clique}\\\\\n\\emph{Input}: A undirected graph  $G$ and a positive integer $b$. \\\\\n\\emph{Parameter}: $b$\\\\\n\\emph{Question}: Is there a clique of size $b$ in $G$?\n\\end{minipage}}\n\\end{center}\n\n\nFrom a given graph $G$ we construct a directed graph $G'$ as follows.\n\\begin{itemize}\n\\item Construct a copy of $V(G)$.\n\\item For each edge $\\{u,v\\}\\in E(G)$, construct a new vertex $w_{uv}$ and join $u,v$  with $w_{uv}$  the copy of $V(G)$\n    by arcs $(u,w_{uv})$ and $(v,w_{uv})$.\n\\item Construct $k-2$ vertices $z_1,\\ldots,z_{k-2}$, and for each $e\\in E(G)$, join $z_1,\\ldots,z_{k-2}$ with $w_e$ by\n    arcs.\n\\end{itemize}\nIt is straightforward to see that $G'$ is a directed acyclic graph. We say that the vertex $w_{uv}$ for $\\{u,v\\}\\in E(G)$ is a\n\\emph{subdivision} vertex, and we say that $v\\in V(G)$  is a \\emph{branch} vertex. Let $b'=b+k-2$ and\n$p=\\frac{b(b+1)}{2}+k-2$. Let $Z=\\{z_1\\ldots,z_{k-1}\\}$. We claim that $G$ has a clique of size $b$ if and only if there is a\nset of at most $b'$ vertices $A\\subseteq V(G')$ such that there exists an an induced subgraph $H$ of $G'$ with at least $p$\nvertices, $A\\subseteq V(H)$ and for any $v\\in V(H)\\setminus A$ we have  $d_H^-(v)\\geq k$.\n\nSuppose that $K$ is a clique in $G$ of size $b$. We let $A=K\\cup Z$ and define $U=\\{w_{uv}|u,v\\in K\\}$. Notice that\n$|U|=\\frac{b(b-1)}{2}$ and each vertex of $U$ has two in-neighbors in $A\\cap K$ and $k-2$ in-neighbors in $Z$. We conclude\nthat $H=G'[A\\cup U]$ has $p$ vertices and  for any $v\\in V(H)\\setminus A$ satisfies $d_H^-(v)\\geq k$.\n\nAssume now that there is a set of at most $b'$ vertices $A\\subseteq V[G']$ such that there exists an induced subgraph $H$ of\n$G'$ with at least $p$ vertices, $A\\subseteq V(H)$ and for any $v\\in V(H)\\setminus A$ we have  $d_H^-(v)\\geq k$. Since\nsubdivision vertices of $G'$ are sinks, we can assume that $A$ contains only \\emph{branch} vertices and vertices from $Z$, as\notherwise we can replace an anchor $a\\in A$ that is a subdivision vertex of $G'$ by an arbitrary branch vertex or a vertex of\n$Z$. Because branch vertices of $G'$ and the vertices of $Z$ are sources, any such vertex $v$ is in $H$ if and only if $v\\in\nA$. Hence, $H$ has at most $b'$ sources of $G'$ and at least $\\frac{b(b-1)}{2}$ subdivision vertices. If there is a vertex\n$z_i\\in Z$ such that $z_i\\notin A$, then each subdivision vertex $w_e$ has at most $k-1$ in-neighbors and $H$ cannot contain\nsubdivision vertices. Therefore $Z\\subseteq A$ and $A$ has at most $b'-(k-2)=b$ branch vertices. It remains to observe that a\nsubdivision vertex $w_{uv}$ has $k$ in-neighbors in $H$ if and only if $u,v\\in A$. Then the claim follows.\n\\end{proof}\n\n\\section{\\textsc{Dir-AKC}\\xspace on graphs of bounded degree}\n\\label{sec:bound-deg} \nIn this section we show that \\textsc{Dir-AKC}\\xspace problem is \\ensuremath{\\operatorClassFPT}\\ parameterized by $\\Delta+p$ if\n$k\\geq\\frac{\\Delta}{2}$.\n\nIn our algorithms we need to check the existence of solutions for \\textsc{Dir-AKC}\\xspace that have bounded size. It can be observed that if we\nare interested in solutions $(A,H)$ such that $p\\leq |V(H)|\\leq q$, then for every positive $q$, we can express this problem\nin the first order logic. It was proved by Seese~\\cite{Seese96} that any graph problem expressible in the first-order logic\ncan be solved in linear time on (directed) graphs of bounded degree. Later this result was extended for much more rich graph\nclasses (see~\\cite{DvorakKT10} ). These meta theorems are very general, but do not provide\ngood upper bounds for running time for particular problems. Hence, we give the following lemma.\nOur algorithms use the random separation technique due to Cai et al.~\\cite{CaiCC06} (which is a variant of the color coding\nmethod introduced by Alon et al.~\\cite{AlonYZ95}) .\n\n\n\\begin{lemma}\\label{lem:bounded}\nThere is a randomized algorithm with running time $2^{O(\\Delta q)}\\cdot n$ that for an instance of \\textsc{Dir-AKC}\\xspace with an $n$-vertex\ndirected graph of maximum degree at most $\\Delta$ and a positive integer $q\\geq p$, either returns a solution $(A,H)$ with $V(H)\\geq p$ or gives the answer\nthat there is no solution with $|V(H)|\\leq q$. Furthermore, the algorithm can be derandomized, and the deterministic variant\nruns in time  $2^{O(\\Delta q)}\\cdot n\\log n$.\n\\end{lemma}\n\n\\begin{proof}\nConsider an instance of \\textsc{Dir-AKC}\\xspace with an $n$-vertex directed graph $G$ of maximum degree at most $\\Delta$. We assume that $b\\leq\np\\leq n$. For given $q\\geq p$, to decide if $G$ contains a solution of size at  most $q$, we do the following.\n\nWe color each vertex of $G$ uniformly at random with probability $\\frac{1}{2}$ by one of  two colors, say red or\nblue. Let  $R$ be the set of vertices  colored red. Observe that if there is a solution $(A,H)$ with $|V(H)|\\leq q$, then with\nprobability at least $\\frac{1}{2^q}$ all vertices of $H$ are colored red and with probability at least $\\frac{1}{2^{\\Delta q}}$\nall \nin- and out-neighbors of the vertices of $H$ that are outside of $H$ are colored blue. Using this observation, we assume that\n$H$ is the union of some weakly connected components of  the graph $G[R]$ induced by red vertices.\n\nIn time $O(\\Delta n)$ we find all weakly connected components of $G[R]$. If there is a component $C$ with at least $b+1$\nvertices of in-degree at most $k-1$ (in $C$), then we discard this component as it cannot be a part of any solution. Denote by\n$C_1,\\ldots,C_r$ the remaining components. For $i\\in\\{1,\\ldots,r\\}$, let $A_i=\\{v\\in V(C_i)|d_{C_i}^-(v)<k\\}$, $b_i=|A_i|$ and\n$p_i=|V(C_i)|$.\n\nThus everything boils down to the problem of finding a set $I\\subseteq \\{1,\\ldots,r\\}$ such that $\\sum_{i\\in I} b_i\\leq b$ and\n$\\sum_{i\\in I}p_i \\geq p$.  But this is  the well known {\\sc Knapsack} problem, which  is solvable in time $O(bn)$ by dynamic\nprogramming. If we obtain a solution $I$, then we output $(A,H)$, where $A=\\cup_{i\\in I} A_i$ and $H=G[\\cup_{i\\in I}V(C_i)]$.\nOtherwise, we return a NO-answer. Notice that this algorithm can also find a solution $(A,H)$  with $|V(H)|> q \\geq p$.\n\n\nIt remains to observe that for any positive number $\\alpha<1$, there is a constant $c_{\\alpha}$ such that after running our\nrandomized algorithm $c_{\\alpha}\\cdot 2^{\\Delta q}$ times, we either find a solution $(A,H)$ or can claim that with\nprobability $\\alpha$ that it does not exist.\n\n\nThis algorithm can be derandomized by the  technique proposed by Alon et al.~\\cite{AlonYZ95}: replace the random colorings by\na family of at most $2^{O(\\Delta q)}\\cdot \\log n$ hash functions which are known to be constructible  in time $2^{O(\\Delta\nq)}\\cdot n\\log n$.\n\\end{proof}\n\n\nOur next aim is to prove that for $k>\\Delta\/2$ the \\textsc{Dir-AKC}\\xspace problem is \\ensuremath{\\operatorClassFPT}\\ when parameterized by the number of anchors\n$b$.\n\n\\begin{lemma}\\label{lem:bound-deg-anchors}\nLet $\\Delta$ be a positive integer. If $k>\\Delta\/2$, then the \\textsc{Dir-AKC}\\xspace problem can be solved in time $2^{O(\\Delta^2 b)}\\cdot\nn\\log n$ for $n$-vertex directed graphs of maximum degree at most $\\Delta$.\n\\end{lemma}\n\n\\begin{proof}\nSuppose $(A,H)$ is a solution for the \\textsc{Dir-AKC}\\xspace problem. Let us observe that because $k>\\Delta\/2$, for every vertex $v\\in\nV(H)\\setminus A$,  we have $d_H^-(v)>d_H^+(v)$. Recall that for any directed graph, the sum of in-degrees equals the sum of\nout-degrees. Then\n$$  \\sum_{v\\in V(H)\\setminus A}(d_H^-(v)-d_H^+(v))=\\sum_{v\\in A}(d_H^+(v)-d_H^-(v)).$$\nSince for every vertex $v\\in V(H)\\setminus A$,  $d_H^-(v)-d_H^+(v)\\geq 1$, we have that\n$$|V(H)\\setminus A|\\leq \\sum_{v\\in V(H)\\setminus A}(d_H^-(v)-d_H^+(v)).$$\nOn the other hand,  $ d_H^+(v)-d_H^-(v)\\leq \\Delta$, and we arrive at\n$$|V(H)\\setminus A|\\leq \\sum_{v\\in V(H)\\setminus A}(d_H^-(v)-d_H^+(v))=\\sum_{v\\in A}(d_H^+(v)-d_H^-(v))\\leq \\Delta|A|.$$\nHence, $|V(H)|\\leq (\\Delta+1)|A|\\leq (\\Delta+1)b$. Using this observation, we can solve the \\textsc{Dir-AKC}\\xspace problem as follows. If\n$p>(\\Delta+1)b$, then we return a NO-answer. If $p\\leq(\\Delta+1)b$, we apply Lemma~\\ref{lem:bounded} for $q=(\\Delta+1)b$, and\nsolve that problem in time  $2^{O(\\Delta^2 b)}\\cdot n\\log n$.\n\\end{proof}\n\n\nNow we show that if $k=\\frac{\\Delta}{2}$ then the \\textsc{Dir-AKC}\\xspace problem is FPT parameterized by $\\Delta+p$. Due the space restrictions we only sketch the proof of the following lemma.\n\n\\begin{lemma}\\label{lem:bound-deg-saved}\nLet $\\Delta$ be a positive integer. If $k=\\Delta\/2$, then the \\textsc{Dir-AKC}\\xspace problem can be solved in time $2^{O(\\Delta^3 b+ \\Delta^2 b\np)}\\cdot n^{O(1)}$ for $n$-vertex directed graphs of maximum degree at most $\\Delta$.\n\\end{lemma}\n\n\\begin{proof}\nWe describe an \\ensuremath{\\operatorClassFPT}~ algorithm. Consider an instance of the \\textsc{Dir-AKC}\\xspace problem. Without loss of generality we assume that $b<p\\leq n$.\n\nWe apply the following preprocessing rule. Suppose that $G$ has a (weakly) connected component $C$ such that for any $v\\in\nV(C)$, $d_C^-(v)=d_C^+(v)=k$. If $b\\geq p-|V(C)|$, then we choose a set $A$ of $b'=p-|V(C)|$ vertices arbitrary in\n$V(G)\\setminus V(C)$. Then we return a YES-answer, as the anchors $A$ and $H=G[A\\cup V(C)]$ is a solution. Otherwise, if $b<\np-|V(C)|$, we let $G'=G-V(C)$ and $p'=p-|V(C)|$. Now we consider a new instance of the problem with the graph $G'$ and the\nparameter $p'$. To see that the rule is safe, it is sufficient to observe that a set of anchors $A$ and a subgraph $H'$ of\nsize at least $p'$ is a solution of the obtained instance if and only if $A$ and $H=G[V(H')\\cup V(C)]$ is a solution for the\noriginal problem. From now we assume that $G$ has no such components.\n\n\\medskip\n\nWe need the following claim.\n\n\\medskip\n\\noindent\n{\\bf Claim A.} {\\it\nIf an instance of the \\textsc{Dir-AKC}\\xspace problem has a core with at least $(\\Delta p+1)b+1$ vertices, then it has a solution\n$(A,H)$ with the following property: there is a vertex $t\\in V(H)\\setminus A$  reachable in $H$ from any vertex of $H$.\nMoreover, for each vertex $v$ of $H$, there is a path from $v$ to $t$ with all vertices except $v$ in  $ V(H)\\setminus A$.\n}\n\n\n\n\\begin{proof}[Proof of Claim~A]\nLet  $(A,H')$ be a solution with the set of anchors $A$ and such that $V(H')> (\\Delta p+1)b$.\n\nWe show that $V(H')=R_{H'}^+(A)$, i.e., all vertices of $H'$ are reachable from the anchors. To obtain a contradiction,\nsuppose that there is a vertex $u\\in V(H')$ such that $u\\notin R_{H'}^+(A)$. Let $U=R_{H'}^-(u)$, i.e., $U$ is the set of\nvertices from which we can reach $u$. Clearly, $A\\cap U=\\emptyset$. Therefore, $d_{H'}^-(v)\\geq k=\\Delta\/2$ for $v\\in U$.\nNotice that for a vertex $v\\in U$, $N_{H'}^{-}(v)\\subseteq U$ by the definition. Hence, $d_{G[U]}^-(v)\\geq k=\\Delta\/2$ for $v\\in\nU$. Because the sum of in-degrees equals the sum of out-degrees, for  every vertex $v\\in U$, we have that\n$d_{G[U]}^-(v)=d_{G[U]}^+(v)=k=\\Delta\/2$. Then $C=G[U]$ is a component of $G$ such that for every $v\\in V(C)$,\n$d_C^-(v)=d_C^+(v)=k$, but such components are excluded by the preprocessing; a contradiction.\n\nObserve now that if $d_{H'}^-(v)<d_{H'}^+(v)$, then $d_{H'}^-(v)<k$ and thus $v\\in A$. Hence, by adding at most $\\Delta b$\n(maybe multiple) arcs from $V(H')\\setminus A$ to $A$, joining the vertices $v\\in V(H')$ of degrees $d_{H'}^-(v)>d_{H'}^+(v)$ with\nvertices of degrees  $d_{H'}^-(v)<d_{H'}^+(v)$, we  can  transform $H'$ into a disjoint union of  directed Eulerian graphs.\nSince $V(H')=R_{H'}^+(A)$, each of these directed Eulerian graphs contains at least one vertex of $A$. Thus  the set of arcs\nof $H'$ can be covered by at most $\\Delta b$ arc-disjoint directed walks,   each walk starting from a vertex of $A$ and never\ncoming back to $A$. Because $d_{H'}^-(v)\\geq k$ for $v\\in V(H')\\setminus A$, we have that $|E(G')|\\geq k(|V(H')|-b)> \\Delta\nkbp$. Then there is a walk $W$ with at least $kp+1$ arcs.  Let $a\\in A$ be the first vertex of $W$ and let $t$ be the last\nvertex of the walk. The walk $W$ visits $a$ only once, $t$  and all other vertices of $W$ are visited at most $k$ times. We\nconclude  that $W$ has at least $p$ vertices.\n\nLet $R=R_{H'-A}^-(t)$ and let $A'=\\{a\\in A\\ |\\ N_{H'}^+(a)\\cap R\\neq\\emptyset\\}$. Consider $H=G[R\\cup A']$. Since\n$V(W)\\subseteq V(H)$, $|V(H)|\\geq p$.  For any $v\\in V(H)\\setminus A$, the in-neighbors of $v$ in $H'$ are in $H$ by the\nconstruction and, therefore, $d_H^-(v)\\geq k$. It remains to observe that to select at most $b$ anchors, we take $A'\\subseteq\nV(H)$.\n\\end{proof}\n\nUsing  Claim~A, we proceed with our algorithm. We try to find a solution such that $H$ has at most $q=(\\Delta\np+1)b$ vertices by applying  Lemma~\\ref{lem:bounded}. It takes time $O(2^{O(\\Delta^2 bp)}\\cdot n\\log n)$. If we obtain a\nsolution, then we return it and stop. Otherwise, we conclude that every core  contains at least $(\\Delta p+1)b+1$\nvertices. By Claim~A, we can search for a solution   $H$ with a  non-anchor vertex $t$ which is reachable from\nall other vertices of $H$ by directed paths  avoiding $A$. Notice that since $t$ is a non-anchor vertex, we have that\n$d_G^-(t)\\geq k$. We try  at most $n$ possibilities for all possible choices of  $t$, and solve our problem for each choice.\nClearly, if we get a YES-answer for one of the choices, we return it and stop. Otherwise, if we fail,  we return a NO-answer.\n\nFrom now we assume that we already selected $t$. We denote by $G'$ the graph obtained from $G$ by adding  an artificial source\nvertex $s$ joined by arcs with all the vertices $v\\in V(G)$ with $d_G^-(v)<k$. Observe that $(s,t)\\notin E(G')$.\n\n\\medskip\n\nSuppose that $(A,H)$ is a solution with the set of anchors $A$ such that $t\\in V(H)\\setminus A$ is reachable in $H$ from any\nvertex of $H$ by a path with all inner vertices in $V(H)\\setminus A$. Denote by $\\delta_{G'}(H)$ the set $\\{v\\in V(H)\\ |\\\nN_{G'}^-(v)\\setminus V(H)\\neq \\emptyset\\}$, i.e., $\\delta_{G'}(H)$ contains vertices that have in-neighbors outside $H$. We\nneed a chain of claims about the structure of $H$ in $G'$.\n\n\n\\medskip\n\\noindent\n{\\bf Claim~B.} {\\it\n$|\\delta_{G'}(H)\\setminus A|\\leq \\Delta b$.\n}\n\n\n\\begin{proof}[Proof of Claim~B]\nLet $X=\\{v\\in V(H)\\ |\\ d_H^-(v)\\geq k\\text{ and }d_H^+(v)<k\\}$, $Y=\\{v\\in V(H)\\ |\\ d_H^-(v)=d_H^+(v)=k\\}$ and $Z=\\{v\\in V(H)\\\n|\\ d_H^-(v)<k\\}$. Clearly, $$\\sum_{v\\in X}(d_H^-(v)-d_H^+(v))+ \\sum_{v\\in Y}(d_H^-(v)-d_H^+(v))=\\sum_{v\\in\nZ}(d_H^+(v)-d_H^-(v))$$ Observe that $d_H^-(v)-d_H^+(v)\\geq 1$ for $v\\in X$,  $d_H^-(v)-d_H^+(v)=0$ for $v\\in Y$ and\n$d_H^+(v)-d_H^-(v)\\leq \\Delta$ for $v\\in Z$. Hence, $|X|\\leq \\Delta|Z|$. If $d_H^-(v)<k$ for $v\\in V(H)$, then $v\\in A$. It\nfollows that $Z\\subseteq A$ and $|Z|\\leq b$. We have $|X|\\leq \\Delta b$. Consider a vertex $v\\in \\delta_{G'}(H)\\setminus A$.\nIt has at least one in-neighbor outside $H$ in $G$ and $d_H^-(v)\\geq k$. Then $d_H^+(v)<k$ and  $v\\in X$. We conclude that\n$\\delta_{G'}(H)\\setminus A\\subseteq X$ and $|\\delta_{G'}(H)\\setminus A|\\leq \\Delta b$.\n\\end{proof}\n\n\n\\medskip\n\\noindent\n{\\bf Claim~C.} {\\it\nThere is  an $s-t$ separator $S$ in $G'$ of size at most $(\\Delta(k-1)+1)b$ such that\n $V(H)\\setminus A\\subseteq R_{G'-S}^-(t)$.\n}\n\n\\begin{proof}[Proof of Claim~C]\nLet $S=\\Big(\\delta_{G'}(H)\\cap A\\Big)\\cup\\Big(\\bigcup_{v\\in \\delta_{G'}(H)\\setminus A}(N_G^-(v)\\setminus V(H)\\Big)$, i.e., the\nset containing all anchors that are in $\\delta_{G'}$, and for each non-anchor vertex of $\\delta_{G'}$ containing all its\nin-neighbors outside of $H$. Consider a directed $(s,t)$-path $P$ in $G'$. Let $v$ be the first vertex in $P$ that is in\n$V(H)$ and let $u$ be its predecessor in $P$. If $v\\in A$, then $v\\in S$. If $v\\notin A$, then $u\\neq s$ as $H$ has no\nnon-anchor vertices with in-degree at most $k-1$ in $G$. Then $u\\in S$. We conclude that each $(s,t)$-path contains a vertex\nof $S$, i.e., this set is an $s-t$ separator.\n\nObserve that $V(H)\\setminus A\\subseteq R_{G'-S}^-(t)$ by the definition of $S$ and the fact that $t$ can be reached from any\nvertex of $H$ in this graph by a path with all inner vertices in $V(H)\\setminus A$.\n\nIt remains to show that $|S|\\leq (\\Delta(k-1)+1)b$.  By Claim~B, $|\\delta_{G'}(H)\\setminus A|\\leq \\Delta b$. A\nvertex $v\\in \\delta_{G'}(H)\\setminus A$ has at least one out-neighbor in $H$ because $t$ is reachable from $v$. Then $v$ has\nat most $k-1$ in-neighbors outside $H$. Hence $|S|\\leq |A|+(k-1)(\\delta_{G'}(H)\\setminus A)\\leq (\\Delta(k-1)+1)b$.\n\\end{proof}\n\nNow we can prove the following claim about important $s-t$ separators in $G'$.\n\n\\medskip\n\\noindent\n{\\bf Claim~D.} {\\it\nThere is an important $s-t$ separator $S^*$ of size at most $(\\Delta(k-1)+1)b$ in $G'$ such that $V(H)\\subseteq\nR_{G'-S^*}^-(t)\\cup S^*$.\n}\n\n\n\n\\begin{proof}[Proof of Claim~D]\nBy Claim~C, there is an $s-t$ separator $S'$ in $G'$ of size at most $(\\Delta(k-1)+1)b$ such that $V(H)\\setminus\nA\\subseteq R_{G'-S'}^-(t)$. Notice that $S'$ not necessary a minimal separator, but there is a minimal $s-t$ separator\n$S\\subseteq S'$. Clearly, $|S|\\leq (\\Delta(k-1)+1)b$.\n\nWe show that  $V(H)\\subseteq R_{G'-S}^-(t)\\cup S$.  \nBecause $R_{G'-S'}^-(t)\\subseteq  R_{G'-S}^-(t)$, we have that $V(H)\\setminus A\\subseteq R_{G'-S}^-(t)$. Also if an anchor $a$\nis in  $R_{G'-S'}^-(t)$, then $a\\in R_{G'-S}^-(t)$. Let $a\\in A\\cap S'$. If $a\\in A\\cap S$, then $a\\in R_{G'-S}^-(t)\\cup S$.\nIf  $a\\notin S$, then by Claim~C, $a$ has an out-neighbor $v\\in R_{G'-S'}^-(t)$ and in this case we have $a\\in\nR_{G'-S}^-(t)$.\n\nIt remains to observe that there is an important $s-t$ separator $S^*$ such that $|S^*|\\leq |S|\\leq (\\Delta(k-1)+1)b$ and\n$R_{G'-S}^-(t)\\subseteq   R_{G'-S^*}^-(t)$. Therefore, $V(H)\\subseteq R_{G'-S}^-(t)\\cup S\\subseteq R_{G'-S^*}^-(t)\\cup S^*$.\n\\end{proof}\n\nThe next step of our algorithm is to check all important $s-t$ separators in $G'$ of size at most  $(\\Delta(k-1)+1)$. By\nLemma~\\ref{lem:imp-sep}, there are at most $4^{(\\Delta(k-1)+1)b}$ important $s-t$ separators and they can be listed in time\n$2^{O(\\Delta^2b)}\\cdot n^c$. For each important $s-t$ separator $S^*$, we consider the set of vertices $U= R_{G'-S^*}^-(t)\\cup\nS^*$ and decide whether there is a solution such that $V(H)\\subseteq U$. If we have a solution for some $S^*$, then we return\na YES-answer and stop. Otherwise, if we fail  to find such a solution for all important separators, we use\nClaim~D  to deduce that there is no solution.\n\n\\medskip\nFrom now on, we assume that an important $s-t$ separator $S^*$ is given and that  $U= R_{G'-S^*}^-(t)\\cup S^*$.\nIn what follows, we describe a procedure of finding  a solution with  $V(H)\\subseteq U$.\n\n\n\\medskip\nDenote by $D$ the set $\\{v\\in U\\ |\\ d_G^-(v)>0\\}$. We need the following observation.\n\n\n\n\\medskip\n\\noindent\n{\\bf Claim~E.} {\\it\nSet $D$ contains at most $(\\Delta+1) (\\Delta(k-1)+1)b$ vertices.\n}\n\n\n\\begin{proof}[Proof of Claim~E]\nLet $Q=G[U]$. Let $X=\\{v\\in V(Q)\\ |\\ d_Q^-(v)\\geq k\\text{ and }d_Q^+(v)<k\\}$, $Y=\\{v\\in V(Q)\\ |\\ d_Q^-(v)=d_Q^+(v)=k\\}$ and\n$Z=\\{v\\in V(Q)\\ |\\ d_Q^-(v)<k\\}$. Clearly,\n$$\\sum_{v\\in X}(d_Q^-(v)-d_Q^+(v))+ \\sum_{v\\in Y}(d_Q^-(v)-d_Q^+(v))=\\sum_{v\\in Z}(d_Q^+(v)-d_Q^-(v))$$\nObserve that $d_Q^-(v)-d_Q^+(v)\\geq 1$ for $v\\in X$,  $d_Q^-(v)-d_Q^+(v)= 0$ for $v\\in Y$ and $d_Q^+(v)-d_Q^-(v)\\leq \\Delta$\nfor $v\\in Z$. Hence, $|X|\\leq \\Delta|Z|$.\n\nRecall that $G'$ is obtained from $G$ by joining $s$ with all vertices of in-degree at most $k-1$. Since $S^*$ is an $s-t$\nseparator, if for $v\\in U$, $d_Q^-(v)<k$, then $v\\in S^*$. Hence, $Z\\subseteq S^*$ and $|Z|\\leq |S^*|\\leq (\\Delta(k-1)+1)b$.\nIf for for $v\\in U$, $d_G^-(v)>k$, then $v\\in X\\cup Z$. We conclude that $|D|\\leq|X|+|Z|\\leq (\\Delta+1)|Z|\\leq (\\Delta+1)\n(\\Delta(k-1)+1)b$.\n\\end{proof}\n\nRecall   that  set $\\delta_{G'}(H)$ contains vertices of $H$ that have in-neighbors outside of $H$. If $v\\in\n\\delta_{G'}(H)\\setminus A$, then it has at least $k$ in-neighbors in $H$ and at least one in-neighbor outside $H$. Notice that\n$s\\notin N_{G'}^-(v)$ because $d_G^-(v)\\geq d_H^-(v)\\geq k$. Hence, $d_G^-(v)>k$. Because $V(H)\\subseteq U$,\n$\\delta_{G'}(H)\\setminus A\\subseteq D$. By Claim~C, $|\\delta_{G'}(H)\\setminus A|\\leq \\Delta b$, and by\nClaim~E, $|D|\\leq (\\Delta+1) (\\Delta(k-1)+1)b$.\nWe consider  all at most\n$  2^{(\\Delta+1) (\\Delta(k-1)+1)b}$ possibilities to select  $\\delta_{G'}(H)\\setminus A$. For each choice of\n$\\delta_{G'}(H)\\setminus A$, we guess the arcs that join the vertices that are outside $H$ with the vertices of\n$\\delta_{G'}(H)\\setminus A$ and delete them. Denote the graph obtained from $G$ by $F$. Recall that from each vertex $v$ of\n$\\delta_{G'}(H)\\setminus A$, there is a directed path to $t$ that avoids $A$. Hence, $v$ has at least one out-neighbor in $H$\nand at most $\\Delta-1$ in-neighbors in $G$. Also $v$ has at least $k$ in-neighbors in $H$, and we delete at most $d_G^-(v)-k$\narcs. Therefore, for $v$ we choose at most $k-1$ arcs out of at most $\\Delta-1$ arcs. We can upper bound the number of\npossibilities for $v$ by $2^{\\Delta-1}$, and the total number of possibilities for $\\delta_{G'}(H)\\setminus A$ is\n$2^{(\\Delta-1)\\Delta b}$.\n\nObserve that $(A,H)$ is a solution for the new instance of \\textsc{Dir-AKC}\\xspace, where $G$ is replaced by $F$ for a correct guess of the\ndeleted arcs. Also each solution for the new instance provides a solution for the graph $G$, because if we put deleted arcs\nback, then we can only increase in-degrees. Hence, we can check for each possible choice of the set of deleted arcs, whether\nthe new instance has a solution. If for some choice we obtain a solution, then we return a YES-answer. Otherwise, if we fail\nfor all choices, then we return a NO-answer. Further we assume that $F$ is given.\n\n\nDenote by $F'$ the graph obtained from $F$ by the addition of a vertex $s$ joined by arcs with all the vertices $N_{G'}^+(s)$.\nNow $\\delta_{F'}(H)=\\{v\\in V(H)\\ |\\ N_{F'}^-(v)\\setminus V(H)\\neq \\emptyset\\}$. By the choice of $F$,\n$\\delta_{F'}(H)=\\delta_{G'}(H)\\cap A$ and, therefore, $|\\delta_{F'}(H)|\\leq b$. Also $\\delta_{F'}(H)$ is an $s-t$ separator in\n$F'$ by Claim~C.\n\nNow we can prove the following.\n\n\\medskip\n\\noindent\n{\\bf Claim~F.} {\\it\nThere is an important $s-t$ separator $\\hat{S}$ of size at most $b$ in $F'$ such that $(\\hat{S},G[R_{F'-\\hat{S}}^-(t)\\cup\n\\hat{S}])$ is a solution for the instance of the \\textsc{Dir-AKC}\\xspace problem for the graph $G$.\n}\n\n\n\\begin{proof}[Proof of Claim~F]\nLet $U=R_{F'-\\hat{S}}^-(t)\\cup \\hat{S}$. It was already observed that $\\delta_{G'}^*(H)$ is an $s-t$ separator in $F'$ of size\nat most $b$. Then there is a minimal $s-t$ separator $S\\subseteq \\delta_{G'}^*(H)$. Clearly, $|S|\\leq b$.\n\nAs before in the proof of Claim~D, we show that  $V(H)\\subseteq R_{F'-S}^-(t)\\cup S$. Because for any vertex\n$v$ of $H$, there is a directed $(v,t)$ path with all inner vertices in $V(H)\\setminus A$, $V(H)\\setminus A\\subseteq\nR_{F'-\\delta_{F'}(H)}^-(t)$. Because $R_{F'-\\delta_{F'}(H)}^-(t)\\subseteq  R_{F'-S}^-(t)$ we have $V(H)\\setminus A\\subseteq\nR_{F'-S}^-(t)$. Also if  $a\\in A$ is in  $R_{F'-\\delta_{F'}(H)}^-(t)$, then $a\\in R_{F'-S}^-(t)$. Let $a\\in A\\cap\n\\delta_{F'}(H)$. Trivially, if $a\\in A\\cap S$, then $a\\in R_{F'-S}^-(t)\\cup S$. If  $a\\notin S$, then $a$ has an out-neighbor\n$v\\in R_{F'-\\delta_{F'}(H)}^-(t)$ and $a\\in R_{F'-S}^-(t)$. Then there is an important $s-t$ separator $\\hat{S}$ such that\n$|\\hat{S}|\\leq |S|\\leq b$ and $R_{F'-S}^-(t)\\subseteq R_{F'-\\hat{S}}^-(t)$. Therefore, $V(H)\\subseteq R_{F'-S}^-(t)\\cup\nS\\subseteq R_{F'-S^*}^-(t)\\cup S^*$, and $|U|\\geq p$.\n\nIt remains to observe that $s$ is adjacent to all vertices of $G$ with in-degrees at most $k-1$ and $S^*$ is an $s-t$\nseparator. It immediately follows that for any  vertex $v\\in R_{F'-S^*}^-(t)$, $d_{F(U)}^-(v)\\geq k$. Then\n$(\\hat{S},G[R_{F'-\\hat{S}}^-(t)\\cup \\hat{S}])$ is a solution.\n\\end{proof}\n\nThe final step of our algorithm is to enumerate all important $s-t$ separators $\\hat{S}$ of size at most $b$ in $F'$, which\nnumber by Lemma~\\ref{lem:imp-sep} is at most $4^b$, and for each $\\hat{S}$, check whether $(\\hat{S},G[R_{F'-\\hat{S}}^-(t)\\cup\n\\hat{S}])$ is a solution. Recall that all these separators can be listed in time $2^{O(b)}\\cdot n^c$. We return a YES-answer\nif we obtain a solution for some important separator, and a NO-answer otherwise.\n\nTo complete the proof, let us observe that each step of the algorithm runs either in polynomial or \\ensuremath{\\operatorClassFPT}~ time.\nParticularly, the preprocessing is done in time $O(\\Delta n)$. Then we check the existence of a solution of a bounded size in\ntime $2^{O(\\Delta^2 bp)}\\cdot n\\log n$. Further we consider at most $n$ possibilities to choose $t$. For each $t$, we consider\nat most $4^{(\\Delta(k-1)+1)b}$ important $s-t$ separators $S^*$. Recall, that they can be listed in time\n$2^{O(\\Delta^2b)}\\cdot n^c$ for some constant $c$. Then for each $S^*$, we have  at most  $2^{(\\Delta+1)\n(\\Delta(k-1)+1)b+(\\Delta-1)}$ possibilities to construct $F$, and it can be done in time $2^{O(\\Delta^3b)}+O(\\Delta n)$.\nFinally, there are at most $4^b$ important $s-t$ separators $\\hat{S}$ and they can be listed in time $2^{O(b)}\\cdot n$ for\nsome $c$. We conclude that the total running time is $2^{O(\\Delta^3b+\\Delta^2bp)}\\cdot n^c$ for some constant $c$.\n\\end{proof}\n\nCombining Lemmas~\\ref{lem:bound-deg-anchors} and \\ref{lem:bound-deg-saved}, we obtain the following theorem.\n\n\\begin{theorem}\\label{thm:bound-deg}\nLet $\\Delta$ be a positive integer. If $k\\geq \\frac{\\Delta}{2}$, then the \\textsc{Dir-AKC}\\xspace problem can be solved in time $2^{O(\\Delta^3\nb+ \\Delta^2 b p)}\\cdot n^{O(1)}$ for $n$-vertex directed graphs of maximum degree at most $\\Delta$.\n\\end{theorem}\n\nTheorems~\\ref{thm:fpt-d-1} and \\ref{thm:bound-deg} give the next corollary.\n\n\\begin{corollary}\\label{cor:fpt-delta}\nThe \\textsc{Dir-AKC}\\xspace problem can be solved in time $2^{O(bp)}\\cdot n^{O(1)}$  for $n$-vertex directed graphs of maximum degree at most\n$4$.\n\\end{corollary}\n\n\n\\section{Conclusions}\\label{sec:concl}\nWe proved that \\textsc{Dir-AKC}\\xspace is \\ensuremath{\\operatorClassNP}-complete even for planar DAGs of maximum degree at most $k+2$. It was also shown  that \\textsc{Dir-AKC}\\xspace\nis \\ensuremath{\\operatorClassFPT}~ when parameterized by $p+\\Delta$ for directed graphs of maximum degree at most\n$\\Delta$ whenever $k\\geq\\Delta\/2$. It is natural to ask whether the problem is \\ensuremath{\\operatorClassFPT}~ for other values $k$. This question\nis interesting even for the special case $\\Delta=5$ and $k=2$.\n\nFor the special case of  directed acyclic graphs (DAGs) we understand  the complexity of the problem much better.\nTheorem~\\ref{thm:w-saved} showed that \\textsc{Dir-AKC}\\xspace on DAGs is \\classW{1}-hard parameterized by $p$ for every fixed $k\\geq 2$, when\nthe degree of the graph is not bounded. We now show the following theorem  that gives\n\\classW{2}-hardness of \\textsc{Dir-AKC}\\xspace when parameterized  by the number of anchors $b$ (recall that we can always assume that $b\\leq\np$).\n\n\\begin{theorem}\\label{thm:w-hardness-dags-bounded-degree}\nFor any $\\Delta\\geq 3$ and any positive $k<\\frac{\\Delta}{2}$,  \\textsc{Dir-AKC}\\xspace  is \\classW{2}-hard (even on DAGs) when parameterized by\nthe number of anchors $b$ on graphs of maximum degree at most $\\Delta$.\n\\end{theorem}\n\n\n\\begin{proof}\nFirst, we prove the claim for $k=1$ and $\\Delta=3$. We reduce from the {\\sc $b$-Set Cover} problem which is known to be\n\\classW2-hard~\\cite{downey-fellows-book}:\n\n\\begin{center}\n\\noindent\\framebox{\\begin{minipage}{4.50in}{\\textsc{$b$-Set Cover} }\\\\\n\\emph{Input }: A collection $X=\\{X_1,\\ldots,X_r\\}$ of subsets of a finite $n$-element set $U$ and a positive integer $b$. \\\\\n\\emph{Parameter}: $b$\\\\\n\\emph{Question}: Are there at most $b$ subsets $X_{i_1},\\ldots,X_{i_{b}}$ such that these sets cover $U$, i.e., $U=\n    \\bigcup_{j=1}^{b}X_{i_j}$?\n\\end{minipage}}\n\\end{center}\n\n\\begin{figure}[ht]\n\\centering\\scalebox{0.7}{\\input{fig2.pdf_t}}\n\\caption{Construction of $G$ for $U=\\{u_1,u_2,u_3\\}$ and $X_1=\\{u_1,u_2\\},X_2=\\{u_2,u_3\\} $.\n\\label{fig:W2h-1}}\n\\end{figure}\n\n\nLet $U=\\{u_1,\\ldots,u_n\\}$. We construct the directed graph $G$ as follows (see Fig.~\\ref{fig:W2h-1}).\n\\begin{itemize}\n\\item For $i\\in\\{1,\\dots,r\\}$, assume that $X_i=\\{u_{j_1},\\ldots,u_{j_s}\\}$ and\n\\begin{itemize}\n\\item construct a vertex $v_i$ and $s$ vertices $x_{ij_1},\\ldots,x_{ij_s}$;\n\\item construct arcs $(v_i,x_{ij_1}),(x_{ij_1},x_{ij_2}),\\ldots,(x_{ij_{s-1}},x_{ij_s})$.\n\\end{itemize}\n\\item For $j\\in\\{1,\\ldots,n\\}$, assume that $u_j$ is included in the sets $X_{i_1},\\ldots,X_{i_t}$ and\n\\begin{itemize}\n\\item construct a vertex $w_j$ and $t$ vertices $y_{ji_1},\\ldots,y_{ji_t}$;\n\\item construct arcs $(y_{ji_1},y_{ji_2}),\\ldots,(y_{ji_{t-1}},y_{ji_t})$;\n\\item join $y_{ji_t}$ with $w_j$ by a directed path $P_j$ of length $\\ell=2rn+r$.\n\\end{itemize}\n\\item For $i\\in\\{1,\\dots,r\\}$ and $j\\in\\{1,\\ldots,n\\}$, if $u_j\\in X_i$, then construct an arc $(x_{ij},y_{ji})$.\n\\end{itemize}\nIt is straightforward to see that $G$ is a directed acyclic graph of maximum degree at most 3. We set $p=n \\ell$. We claim\nthat $U$ can be covered by at most $b$ sets if and only if there is a set of at most $b$ vertices $A$ such that there exists\nan induced subgraph $H$ of $G$ with at least $p$ vertices, $A\\subseteq V(H)$ and for any $v\\in V(H)\\setminus A$, $d_H^-(v)\\geq\n1$.\n\nNotice that $v_1,\\ldots,v_r$ are the sources of $G$, $w_1,\\ldots,w_n$ are the sinks, and $V(G)=\\bigcup_{i=1}^rR_{G}^+(v_i)$.\nObserve also that $w_j$ can be reached from $v_i$ if and only if $u_j\\in X_i$.\n\nSuppose that $U$ can be covered by at most $b$ sets say $X_{i_1},\\ldots,X_{i_{b}}$. Let $A=\\{v_{i_1},\\ldots,v_{i_{b}}\\}$ and\n$H=G[R_{G}^+(A)]$. It is straightforward to see that for any vertex $z\\in V[H]$, $d_{H}^-(z)\\geq 1$. Because $U$ is covered,\nall vertices $w_1,\\ldots,w_n$ are in $H$ and, therefore, $V(P_1)\\cup\\ldots\\cup V(P_n)\\subseteq V(H)$. It remains to observe\nthat $|V(P_1)\\cup\\ldots\\cup V(P_n)|=n(\\ell+1)\\geq p$ and we conclude that $(A,H)$ is a solution of our instance of \\textsc{Dir-AKC}\\xspace.\n\nAssume now that $(A,H)$ is a solution of the \\textsc{Dir-AKC}\\xspace problem. Without loss of generality we can assume that that each $a\\in A$\nis a source of $G$. Otherwise, $a\\in R_G^+(v_i)$ for some source $v_i$, and we can replace $a$ by $v_i$ in $A$ (or delete it\nif $v_i\\in A$ already). Let $\\{i\\ |\\ 1\\leq i\\leq n,~v_i\\in A\\}=\\{i_1,\\ldots,i_{b}\\}$. We show that $X_1,\\ldots,X_{i_{b}}$\ncover $U$. To obtain a contradiction, assume that there is an element $u_j\\in U$ such that $u_j\\notin X_{i_1}\\cup \\ldots\\cup\nX_{i_{b}}$. Then the vertex $w_j$ is not reachable from $A$. Hence, the vertices of $P_j$ are not reachable from $A$. It\nfollows that $V(P_j)\\cap V(H)=\\emptyset$. We have that $|V(H)|\\leq |V(G)|-|V(P_j)|$. Because $|X_i|\\leq n$ for\n$i\\in\\{1,\\ldots,r\\}$ and each $u_h$ is included in at most $r$ sets for $h\\in\\{1,\\ldots,n\\}$, $|V(G)|\\leq\nr(n+1)+n(r+\\ell)=2rn+r+n \\ell=2rn+r+p$. Therefore, $|V(H)|\\leq p+(2rn+r-(\\ell+1))<p$ because $P_j$ has $\\ell+1$ vertices; a\ncontradiction.\n\nNow we prove \\classW{2}-hardness for $k\\geq 2$ and $\\Delta>2k$. We reduce from an instance of the \\textsc{Dir-AKC}\\xspace problem with $k=1$\nand $\\Delta=3$. Consider an instance of this problem with a directed acyclic graph $G$ and positive integers $b, p$. Assume\nthat $b\\leq p\\leq |V(G)|$ and $|V(G)|\\geq 3$. We construct the graph $G'$ as follows (see Fig.~\\ref{fig:W2h-2}).\n\n\\begin{figure}[ht]\n\\centering\\scalebox{0.7}{\\input{fig3.pdf_t}}\n\\caption{Construction of $G'$ for $k=4$.\n\\label{fig:W2h-2}}\n\\end{figure}\n\n\\begin{itemize}\n\\item Construct a copy of $G$ and denote its vertices by $v_1,\\ldots,v_n$.\n\\item For each $i\\in\\{1,\\ldots,n\\}$, construct a set of $k$ vertices $D_i$ and join $k-1$ vertices of this set with $v_i$\n    by arcs.\n\\item For each $i\\in \\{2,\\ldots,n\\}$, join each vertex of $D_{i-1}$ with all vertices of $D_i$ by arcs.\n\\end{itemize}\nClearly, $G'$ is a directed acyclic graph. We let $b'=b+k$ and $p'=p+nk$. Let also $D=D_1\\cup\\ldots\\cup D_n$. Notice that for\neach $v\\in V(G)$, $d_{G'}(v)=d_G(v)+k-1\\leq k+2\\leq\\Delta$ as maximum degree of $G$ is 3. For $v\\in D$, $d_{G'}(v)\\leq\n2k+1\\leq\\Delta$. Hence maximum degree of $G'$ is at most $\\Delta$. We now claim that there is a set of at most $b$ vertices\n$A\\subseteq V(G)$ such that there exists an an induced subgraph $H$ of $G$ with at least $p$ vertices, $A\\subseteq V(H)$ and\nfor any $v\\in V(H')\\setminus A$,  $d_{H}^-(v)\\geq 1$ if and only if there is a set of at most $b'$ vertices $A'\\subseteq\nV(G')$ such that there exists an an induced subgraph $H'$ of $G'$ with at least $p'$ vertices, $A'\\subseteq V(H')$ and for any\n$v\\in V(H)\\setminus A$,  $d_{H'}^-(v)\\geq k$.\n\nSuppose that our original instance of \\textsc{Dir-AKC}\\xspace has a solution $(A,H)$. We let $A'=A\\cup D_1$ and $H'=G'[V(H)\\cup D]$. Then each\nvertex $v\\in D\\setminus A'$ has $k$ in-neighbors in $D$. It remains to observe that each vertex $v$ of $G'$ from\n$V(G)\\setminus A'$ has at least one in-neighbor in $V(G)$ and $k-1$ in-neighbors in $D$. Therefore, $d_{G'}^-(v)\\geq k$.\n\nAssume now that $(A',H')$ is a solution for the constructed instance of \\textsc{Dir-AKC}\\xspace with $|A'|\\leq b'$ and $|V(H)|\\geq p'$. If\n$|D\\cap A'|<k$, then we claim that $D\\cap V(H')\\subseteq A'$. To prove it, suppose that $(V(H')\\cap D)\\setminus\nA\\neq\\emptyset$ and consider the smallest index $i$ such that there is $v\\in (V(H')\\cap D_i)\\setminus A$. Clearly, $i\\geq 2$.\nThe vertex $v$ has in-neighbors only in $D_{i-1}$. By the choice of $i$, $D_{i-1}$ has at most $k-1$ vertices of $H'$, because\nthey can be only anchors and $|D\\cap A'|<k$. Then $d_{H'}^-(v)<k$, a contradiction.\n\nThen if $|D\\cap A'|<k$, $V(H')\\subseteq V(G)\\cup A'$ and $|V(H')|\\leq n+b+k\\leq n+p+k<p'$ as $n\\geq 3$ and $k\\geq 2$. This\ncontradicts our assumption about size of $H'$. Hence, at least $k$ anchors are in $D$ and $|A'\\setminus D|\\leq b$. Let\n$A=A'\\setminus D$ and $H=H'-D$. If $v\\in V(H)\\setminus A$, then $d_{H'}^-(v)\\geq k$ and $v$ has at most $k-1$ in-neighbors\nfrom $D$ in $H'$. Then $v$ has at least one in-neighbor in $V(H)$ and $d_H^-(v)\\geq 1$.\n\\end{proof}\n\n\nThe case of $k\\geq \\frac{\\Delta}{2}$ the complexity of parameterization by $b$ on DAGs is left open. However we can show that\n\\textsc{Dir-AKC}\\xspace is \\ensuremath{\\operatorClassFPT} on DAGs of maximum degree $\\Delta$, when parameterized by $\\Delta+p$.\n\n\\begin{theorem}\\label{thm:dags}\nFor any positive integers $p$ and $\\Delta$, \\textsc{Dir-AKC}\\xspace can be solved in time $2^{O(\\Delta p)}\\cdot n^2\\log n$ for\n$n$-vertex DAGs of maximum degree at most $\\Delta$.\n\\end{theorem}\n\n\\begin{proof}\nConsider an instance of \\textsc{Dir-AKC}\\xspace with an $n$-vertex directed acyclic graph $G$. Without loss of generality we can assume that\n$b\\leq p\\leq n$.\n\nWe apply Lemma~\\ref{lem:bounded} for $q=p$. In time $2^{O(\\Delta p)}\\cdot n\\log n$ we either obtain a solution or conclude\nthat for any solution $(A,H)$, $H$ has size at least $p+1$. If we obtain a solution, we return it. Suppose that we got a\nNO-answer. If $p=n$, then we return a NO-answer. Otherwise,  we select a sink $t\\in V(G)$ using the fact that any directed\nacyclic graph has at least one such vertex. Observe that we can assume that $t$ is not an anchor in any solution. Also if $t$\nis included in a solution $H$ of size at least $p+1$, then $H-t$ is a solution of size at least $p$, because $t$ is not joined\nby arcs with other vertices of $H$. Then we solve the instance $G-t$ of \\textsc{Dir-AKC}\\xspace recursively.\n\nAs each step is done in time $2^{O(\\Delta p)}\\cdot n\\log n$ and the number of steps is at most $n$, the claim follows.\n\\end{proof}\n\nLet us remark that this result can be easily extended for any class of directed acyclic graphs $\\mathcal{G}$ such that the\ncorresponding class of underlaying graphs $\\{G^*|G\\in \\mathcal{G}\\}$ has (locally) bounded expansion by making use of  the\nresults by Dvorak et al.~\\cite{DvorakKT10}.\nFinally, what happens when the input graph is planar? We know that the problem is \\ensuremath{\\operatorClassNP}-complete on planar graphs for fixed\n$k\\geq 1$ and maximum degree $k+2$. Is the problem \\ensuremath{\\operatorClassFPT}\\ on planar directed graphs when parameterized by the size of the\ncore $p$?\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nKnee osteoarthritis (OA) is a leading cause of disability in the United States, affecting over 10\\% percent of the adult population, and accounting for over 600,000 knee replacements costing in excess of \\$10 billion per year \\cite{Zhang2010, Murphy2012, Cram2012, Kremers2015}. Although commonly characterized by joint pain and functional disability, OA is a heterogeneous disease with multiple risk factors and pathways contributing to variable disease symptoms and progression. The pathogenesis of OA is complex, with biomechanical joint stressors, inflammatory mediators, and structural changes to bone, ligaments, joint synovium, and menisci leading to articular cartilage degradation and loss \\cite{Robinson2016, Lane2011}. No FDA-approved drugs are available which can forestall cartilage degeneration. Imaging biomarkers that are sensitive to structural changes are needed for early detection of OA in an effort to provide useful insights to clinical trials investigating disease modifying drugs. Magnetic resonance imaging (MRI) of the knee provides a complete 3D view of the knee anatomy unlike traditional 2D x-rays. Moreover, changes in cartilage morphology are detected earlier in MRIs than x-rays \\cite{Sharma2008}.  In a longitudinal study by Raynauld et al.\\ \\cite{Raynauld2005}, MRI was used to reliably measure structural changes in arthritic knees and detect cartilage volume loss as early as 12 months from baseline. Balamoody et al. \\cite{Balamoody2010} demonstrated that similar MRI protocols across different 3T scanners gave comparable morphologic results, further suggesting that MR imaging is a dependable tool for quantitative cartilage morphology analysis.\n\nFor analysis of the cartilage structures, accurate segmentation is a crucial first step. In clinical research there is a need for reproducible fully or highly automated segmentation as it offers consistent accuracy and speed over manual segmentation efforts. However, automated knee segmentation is challenging due to the thinning cartilage, appearance of osteophytes, bone marrow and cartilage lesions and surface fibrillation in MRI data. Several of these disease artifacts appear similar to the cartilage further increasing the segmentation complexity (Fig.~\\ref{fig:4d}a).\n\nAutomated knee segmentation approaches previously reported in the literature were surveyed based on the employed methodology. Active shape and appearance models (ASM\/AAM) were widely used in a variety of combinations such as deformable active shape models \\cite{Fripp2010, Fripp2007}, volumetric and surface based appearance models \\cite{Williams2010}, semi-automated initialization of active shape models \\cite{Lynch2001}, and minimum descriptor length based group-wise image registration followed by active appearance model  segmentation \\cite{Vincent2010}. In the presence of pathology, active shape\/active appearance models need to be flexible enough to properly represent many local changes of shape\/appearance, thus requiring large training-set sizes and yielding high-dimensional models. The associated model-training and computational complexity is already substantial in 3D and may become impractical if 4D longitudinal analysis is considered. Multi-atlas based methods were used in several algorithms such as \\cite{Shan2014} with a non-local patch based fusion, Dam et al.\\ \\cite{Dam2015}  extended their multi-atlas based rigid registration by k-nearest neighbor voxel based classification in \\cite{Folkesson2007}. Multi-atlas registration followed by locally weighted voting was reported in \\cite{Lee2014},  multi-atlas based registration followed by outlier detection to segment the tibio-femoral joint in \\cite{Tamez-Pena2012}. Since multi-atlas methods optimize a match between the overall template labeling as well as the individual labeled templates in the set, they may be sensitive to imperfect label templates in the presence of pathology. Both multi-atlas and ASM\/AAM approaches may also suffer from converging to a locally rather than globally optimal solutions. Markov random fields were used to construct and optimize local image patches for the region and boundary probabilities from local shape and appearance information in \\cite{Lee2011}. Graph cuts optimization was applied to the output of a hierarchical two-stage classifier, which was trained to identify the cartilage and  bone voxels \\cite{Wang2014}. As the underlying multi-label graph cuts jointly consider independent classifier outputs for cartilage,  bone, and background, label-conflict-resolution  may be challenging in regions with multiple labels. In Zhang et al.\\ \\cite{Zhang2013}, four different MR acquisitions of the same patients were used to leverage the benefits of different sequence contrasts to model the spatial contextual information using support vector machines in a discrete random field framework. While a conceptually interesting approach, the need for multiple MR acquisitions limits the method's applicability.\n\nUsing cost functions designed as gradient-based features with heuristically determined weighting combination based on human expert knowledge (hand-tuned cost functions, Section II) limits the segmentation performance especially in the pathologic cases. Machine learning based methods can be used to identify the boundary properties of the anatomy of interest. Such learning-based cost function design has shown promise previously \\cite{Brejl-TMI-00}. \n\nA major limitation of learning-based approaches is the time-consuming task of manually segmenting a large number of accurate examples used for training. Several post-processing interactive techniques were reported to ease the corrections, e.g., thin plate splines \\cite{ross2009lung}, live wire techniques \\cite{schenk2000efficient} and active shape models \\cite{schwarz2008interactive} with the major issue being that they correct for the surfaces directly to match the object boundaries and are sometimes infeasible to be computed in close-to-real time.\n\nWe present a fully automated LOGISMOS segmentation algorithm that guarantees global optimality with respect to the cost functions provided. A novel hierarchical random forest (RF) classifiers was designed to provide the learned costs. An interactive correction method called just-enough interaction (JEI) \\cite{Sun2013,Kashyap-IMIC2016} was used to substantially reduce the interaction time needed to prepare the training data in comparison with voxel-by-voxel editing approaches. In our method, two variations of the RF classifier were used --- a neighborhood approximation forests (NAF) followed by an RF classifier collected on a geometric graph. The novelty of this approach is that the NAF classifier gathers contextual and textural information from a global neighborhood of 3D image patches while the RF classifier collects local feature information along the geometric graph search columns. This combination enables training on contextual and textural features  both locally and globally thus improving segmentation accuracy.\n \nOA as a disease progresses slowly in the early stages with the rate of cartilage loss accelerating with advancing disease \\cite{Eckstein2014a}. Every patient in the OAI was enrolled in the longitudinal study with follow-up MRI scans done at regular clinical visits. The availability of multi-time-point information can be leveraged to provide additional information proving beneficial in patients with progressively worsening OA. Using the spatial and temporal contextual information from the neighboring time-points of the same patient helps reducing the inter-time-point variability ensuring that the cartilage losses are within physiologically feasible ranges. The LOGISMOS framework was extended to 4D to incorporate this information and simultaneously segment the multiple follow-up scans of the same patient. To the best of our knowledge, this is the first attempt at a 4D knee MRI segmentation algorithm incorporating the additional available information in a LOGISMOS framework.\n\nThe work reported here improves several aspects of the LOGISMOS algorithm for cartilage segmentation previously presented in \\cite{Yin2010}. Our novel hierarchical random forest classifier outperforms the previous method, in which a single RF classifier did not account for the regionally-specific appearance of the surrounding menisci, muscle, bone and other anatomies. As a result, certain intensity profiles appearing in normal-cartilage regions of the knee also occurred in other local regions in a pathological case. This ambiguity resulted in the imperfect training of the classifier and contributed to segmentation inaccuracy.  Further, the LOGISMOS algorithm was extended to handle simultaneous multi-3D (4D) segmentation.\n\nThe remainder of the paper is organized as follows: Section \\ref{LOGISMOS} introduces the LOGISMOS algorithm framework and the 3D graph construction. In Section \\ref{cost_function} our proposed hierarchical RF classifier design is described in detail followed by the 4D LOGISMOS extension in Section \\ref{4D}. The experimental design, validation, and discussion follow in Sections \\ref{expMethods}, \\ref{results} and \\ref{conclusions}.\n\n\\subsection{LOGISMOS Segmentation}\n\\label{LOGISMOS}\n\nLOGISMOS algorithm \\cite{Yin2010} solves the simultaneous segmentation of multiple objects and surfaces in a graph-based framework with guarantee of global optimality. To ensure global optimality, the LOGISMOS algorithm has to satisfy two important criteria: a) The geometric graph must have non-intersecting column construction and b) the graph edge construction must satisfy the self-closure property (see \\cite{Wu2005, Li2006PAMI} for proof). The focus of this paper is reporting a new method for cost function design and extending the 3D LOGISMOS to 4D. The workflow originally proposed in \\cite{Yin2010} is maintained.\n\nThe algorithm is initialized by identifying the volume of interest (VOI) which uses an AdaBoost classifier \\cite{Freund1997} trained on manually identified VOI's. The purpose of VOI detection was to localize smaller regions to reduce computation time. Further the VOI bounds are used for affine fitting of the mean femur and tibia bone shape mesh $S_0$. A patient specific bone mesh $S$ is computed by a single surface single object LOGISMOS segmentation using $S_0$. This step is necessary since the final segmentation of the bone and the cartilage surfaces is defined by this prior.\n\nA geometric graph with non-intersecting columns representing each surface (bone and cartilage of femur and tibia) is built by mimicking the behavior of charged particles resulting in electric lines of force (ELF) based columns. The cost functions designed are assigned to the graph nodes of every column. The graph-based multiple surfaces and objects segmentation edge construction and optimization are described in \\cite{Yin2010}. \n\n\\section{Cost function design}\n\\label{cost_function}\nLOGISMOS guarantees global optimality of its solution with respect to the cost functions provided. \n\nSimple hand-tuned cost functions were first designed to capture the bone to cartilage interface, which appeared as a strong dark to bright edge when traversing the image along the graph column from inside of the bone surface outwards to the bright cartilage. Directional 1D derivative operators gave costs based on the edges encountered along the search columns. For the cartilage, a weighted combination of the first and second derivative operators was used. This helps prevent interpreting cartilage inhomogeneities as edges. Although human expert designed cost functions are very effective at capturing the desired features there are limitations. 1) Choosing the correct weighting combination is challenging. 2) Same costs may not work for all parts of the anatomy. 3) The same anatomical objects (e.g., bone, cartilage) appear differently in pathological cases due to the loss of structure and\/or the presence of lesions. \n\nRF classifiers \\cite{Breiman2001} were used to provide better and location-specific cost functions. RF classifiers use the concept of bagging, where for each decision tree a random subset of features is chosen thereby reducing inter-decision-tree correlation,  which improves accuracy. A single RF classifier was used in \\cite{Yin2010} where the limitation was that all the information used was localized along the graph columns. Because of this locality, features were unable to capture information of the neighborhood that appeared larger than a few nodes along the column. There are several anatomical features that appear locally like a cartilage boundary but when examined in a global neighborhood may be combined with pathology (e.g., synovial fluid, Fig.~\\ref{fig:4d}). Further, the single RF classifier did not account for the regionally-specific appearance of the surrounding menisci, muscle, bone and other anatomies. As a  result, certain intensity profiles appearing in a specific normal-cartilage region of the knee also occurred in another local region in a pathological case. This discrepancy resulted in the improper training of the classifier and contributed to segmentation inaccuracy.\n\nIn this work, a combination of global and local contextual features was explored. Two RF based classifiers working in hierarchy were used to learn image appearance properties in the cartilage regions (Fig.\\ \\ref{fig:workflow_train}). The first stage used a NAF \\footnote{Based on code available at: \\url{http:\/\/www.nmr.mgh.harvard.edu\/~enderk\/software.html}} \\cite{konukoglu2013} trained on example image patches. The output probability maps of the NAF were used with other image-based features for training each of the regionally specific second-stage RF classifiers. The second RF classifier collected features along the geometric graph columns. For the second RF classifier, the local regions of the knee were spatially clustered using $k$-means clustering to account for variable local anatomy appearances surrounding the cartilage (Fig.~\\ref{fig:clusterRF}). Each of the spatially clustered regions was trained using a different RF classifier in the second stage. Disjoint training sets were used to help build a more realistic RF model based on actual NAF performance on unseen images.\n\n\\subsection{Neighborhood Approximation Forests}\nNAF uses a random forest framework consisting of a several binary decision trees where each tree independently learns to predict the closest neighborhood. NAF uses the training set to learn which clusters of image patches have the most similar neighborhood structures based on a similarity criterion. The definition of similarity is application specific with the advantage of using a wide variety of distance-based metrics without modifying the core underlying approach. Upon training, NAF approximates the neighbors of an unseen test image based on this similarity criterion. \n\nFor each tree in the forest, the training phase starts with the root node and continues to add new branches. Each node branch is trained to learn a set of binary tests which progressively partition the image into subsets with respect to the user-defined distance metric. The node split is optimized such that the user-defined distance function yields the most compact partitioning. \nFor every tree in the forest, a subset of the entire feature set is chosen for training thereby improving generalization and producing an independent prediction. \n\nIn this work, the NAF was trained on image patches where the pairwise distance function $\\rho(I,J)$ captured the similarity of the image patches according to their segmentation labels. The distance function between training image patches were defined as $\\rho(I,J)~=~\\parallel seg(I) - seg(J) \\parallel_{l_0}$ where $seg(.)$ is the segmentation label map for the corresponding image patch. The algorithm learned to group image patches that appeared similar to each other based on the segmentation similarity. \n\nThe test image was partitioned into several smaller patches. Each patch passed through a trained NAF classifier. The output probability of all patches were combined to produce the final probability map (Fig.\\ \\ref{fig:NAF}). This was used as one of the inputs for the second clustered RF classifier. \n\n\\begin{figure}[htb]\n\t\\centering\n\t{\\includegraphics*[height=0.5\\columnwidth,width=0.6\\columnwidth]{NAP_probablity_Ver2_9496443.png}}\n\t\\vspace{-.3cm}\n\t\\caption{The output probability map of the NAF for an image overlaid on the image volume. The color map indicates the probability output values for each 3D voxel belonging to the femur (top) and tibia cartilage. The brighter yellow denotes higher probability while the red  indicates regions of lower probability of the voxel being a cartilage.}\n\t\\label{fig:NAF}\n\t\t\\vspace{-.4cm}\n\\end{figure}\n\n\\subsection{Clustered Random Forest Classifier}\nThe second RF classifier was trained on features collected at every node along the search columns of the geometric graph. Bone mesh surfaces that were corrected using just-enough interaction (see Appendix \\ref{jei}) were used for geometric graph construction during training. Positive example labels corresponded to the nearest cartilage mesh intersection along each graph column. The different features collected at each graph node are shown in Table \\ref{featureLabel} with volumes of every feature computed individually. The value at each node point along the geometric ELF graph column was linearly interpolated from the closest point in the feature volume.\n\nTo handle the variability of cartilage image intensities and the regionally specific surrounding anatomy, a $k$-means clustering algorithm was applied on the coordinates of femur and tibia mean shape $S_0$ mesh points. The spatial parcellation of the pre-segmented mesh in 40 clusters each was empirically designed (total 80, Fig.~\\ref{fig:clusterRF}). Having too many clusters resulted in a reduced set of graph columns yielding insufficient sets of training examples. Having only a small number of clusters resulted in larger more inhomogeneous sets  while negatively affecting the quality of  classification. Training used a separate RF classifier for each of the clusters to learn variable cartilage appearance and consider surrounding anatomies. \n\nThe probability response to the features along the search nodes in the testing datasets provided the node costs for LOGISMOS segmentation. \n\n\\begin{figure}[htb]\n\\vspace{-.2cm}\n\t\\centering\n\t\\includegraphics*[height=1in]{ClusteredRF_FEMUR.png} \n  \\includegraphics*[height=1in]{ClusteredRF_TIBIA.png}\n  \\vspace{-.2cm}\n\t\\caption{Parcellation of the $S_0$ mesh using $k$-means clustering of the femur and tibia into 40 clusters each. Each colored region represents a different cluster that is trained using a separate RF classifier to account for the regionally-specific appearance of surrounding menisci, muscle bone and other anatomies.\n}\n\t\\label{fig:clusterRF}\n\t\\vspace{-.4cm}\n\\end{figure}\n\n\\begin{table}[htb]\n\t\\large\n\t\\caption{List of features for the second RF classifier.}\n\t\\label{featureLabel}\n\t\\vspace{-.6cm}\n\t\\begin{center}\n\t  \\resizebox{\\columnwidth}{!}{\n\t\t\\begin{tabular}{l|l}\n\t\t\t\\hline\n\t\t\t{\\bf Index} & {\\bf Description} \\\\ \\hline\n\t\t\t1--9           & 3 eigenvalues of Hessian matrices on \\\\ & intensity image at $\\sigma=0.5,1.0,2.0$ mm \\cite{frangi1998multiscale}\\\\\n\t\t\t10--15         & Gaussian gradient on intensity and NAF \\\\&probability volumes at $\\sigma=0.36,0.7,1.4$ mm \\cite{zhao2009congenital}\\\\\n\t\t\t16--18         & Intensity, Gaussian smoothed intensity,\\\\ &and NAF probability volumes \\\\\n\t\t\t19--20         & Laplacian derivative of intensity volume\\\\ &at $\\sigma=0.36,0.7$ mm \\cite{zhao2009congenital} \\\\\n\t\t\t21             & Gabor texture feature \\cite{daugman1985uncertainty} \\\\\n\t\t\t22--25         & Intensity statistics: mean, variance, skewness\\\\& and kurtosis of a 2 mm$^3$ region\\\\& centered around each graph node \\cite{antonie2001application}\\\\\n\t\t\t26--28         & Haar features (1.5mm kernel) along \\\\& horizontal, vertical \\& diagonal directions \\cite{viola2004robust} \\\\\n\t\t\t29--30         & 1D directional gradient along the search column \\\\& direction on NAF probability and intensity volume\\\\\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t  }\n\t\t\\vspace{-.6cm}\n  \\end{center}\n\\end{table}\n\n\\section{4D Longitudinal Segmentation of Knee MRI}\n\\label{4D}\nAlong with the proposed learning-based cost functions, it is beneficial to leverage the contextual information from multiple time-points to improve the overall segmentation accuracy. Sequential yearly follow-up MRIs of the same knee provide spatial and temporal contextual information which helps to reduce the inter-time-point variability while ensuring physiologically feasible losses of cartilage during osteoarthritis progression. The crucial first step in 4D segmentation is to register the pre-segmented mesh surfaces and the respective images across time-points to establish correspondences temporally between the similar regions of the knee in 4D. A rigid registration was used given that only slight changes in bone shape occur with disease progression. The same configuration is maintained across all time-points in terms of the geometric graph parameters and topology. Enforcing contextual information temporally is done by linking adjacent time-points using  inter-time-point edges in the underlying 4D LOGISMOS graph.\n\n\\subsection{Establishing Temporal Correspondences}\n\nAfter pre-segmentation of each of the time-points, iterative closest points (ICP) algorithm was used to register the pre-segmented mesh surfaces. With a large translational or rotational movement between the two-time-points, there is a tendency to mismatch the surfaces, i.e., femur matched to the tibia or vice versa if they were the closest in terms of the least squares optimization. To prevent the mismatch, a two step registration was employed. The first step used the femur mesh only for ICP registration. This transform matrix was applied on both the femur and tibia meshes. After the first transform ensured that femoral and tibial surfaces were reasonably aligned, the ICP registration was run again using both the femur and tibia points together as a unified point cloud to further refine the registration. This ensured that vertex to vertex correspondences and thereby column correspondences were established. Fig.\\ \\ref{fig:icp} illustrates the establishment of correspondence between two time-points. The same two-step transformation matrices were applied to the entire images. \n\n\\begin{figure}[htb]\n\t\\includegraphics*[width=0.49\\columnwidth]{BeforeICP_with_Borders.png} \n\t\\includegraphics*[width=0.49\\columnwidth,height=3.7cm]{AfterICP_with_Borders.png}\n\t\\vspace{-.3cm}\n\t\\caption{Establishing point correspondences between time-points after pre-segmentation. The meshes are registered using ICP to establish column correspondences. The same transformation is applied to the corresponding volumes. (a)~Two time-points before registration. The meshes are colored differently and the volume border is highlighted in yellow to indicate its positioning. (b)~Meshes registered after ICP.}\n\t\\vspace{-.3cm}\n\t\\label{fig:icp}\n\\end{figure}\n\n\\subsection{Incorporating Inter-Time Point Context}\nFor the 4D segmentation, the datasets at each time-point have their respective 3D multi-surface multi-object constraints enforced as described earlier in \\cite{Yin2010}. In order to enforce the contextual constraints across adjacent time-points, inter-time-point edges are constructed connecting nodes in every column with the corresponding columns on the registered surfaces. Using the prior information we can set the different minimum and maximum permissible limits for the bones and cartilages respectively so that cartilage thickness changes are within a physiologically feasible range. The edges introduced define inter-time-point min and max ($\\delta_{tmin}, \\delta_{tmax}$) where the inter-time-point maximum allowed change ($\\delta_{tmax}$) was set to 0.6 mm\/year based on available clinical literature \\cite{Eckstein2014a}. The minimum $\\delta_{tmin}$ was set to zero.\n\nThe inter-time-point edges $E^{t}$ between neighboring time-points $t_1$, $t_2$ are constructed for all nodes of the graph:\n\\begin{align*}\nE^{t} = \\{\\langle v_i(t_1,n_{cart},k), v_i(t_2,n_{cart},k-\\delta_{tmin})\\rangle\\}\\\\\n\\cup \\{\\langle v_i(t_1,n_{cart},k-\\delta_{tmax}), v_i(t_2,n_{cart},k)\\rangle\\}\\\\\n\\cup \\{\\langle v_i(t_1,n_{bone},k), v_i(t_2,n_{bone},k - \\delta_{tmin})\\rangle\\}\\\\\n\\cup  \\{\\langle v_i(t_1,n_{bone},k-\\delta_{tmax}), v_i(t_2,n_{bone},k)\\rangle\\},\n\\end{align*}\nwhere $v_i(t,n,k)$ represents the $k$th node on the $i$th column, the inter-time-point edges are constructed between nodes $v_i(t_1,n,k), v_i(t_2,n,k)$ of longitudinally corresponding columns $t_1$, $t_2$ for every $n \\in \\{cart,bone\\}$. After the graph edges and nodes have been constructed, the 4D LOGISMOS segmentation problem is solved by computing a minimum $\\emph{s-t}$ cut in a derived edge weighted digraph \\cite{Boykov2004}. \n\n\\section{Experimental Methods} \n\\label{expMethods}\nKnee MRI data used for evaluation originated from the Osteoarthritis Initiative (OAI) database available for public access\\footnote{\\url{https:\/\/oai.epi-ucsf.org\/datarelease\/}}. All MR acquisitions used double echo steady state (DESS) pulse sequence with in-plane resolution of 0.36$\\times$0.36mm and a slice spacing of 0.7mm, resulting in image volumes with 384 $\\times$ 384 $\\times$ 160 voxels. 88 datasets at baseline (BL) and 12-months follow-up (12M) scans (176 3D MRIs in total) were used for which manually defined independent standards are available via OAI. BL images of 34 patients were used for training the hierarchical RF classifiers. The 12M data of the same patients were excluded from the testing set given the similarity of appearance resulting in a testing set of 54 MRIs at BL and 12M respectively (total of 108 MRIs).\n\nFig.\\ \\ref{fig:workflow_train} shows the training workflow for the hierarchical RF classifiers. For bone surface segmentation, the initially employed gradient-based costs were very robust and remained unchanged. For learning the cartilage-surface costs, the 34 patients were divided into two training sets with 15 and 19 patients each that were used to train the NAF and the clustered RF classifier respectively. The OAI provided independent standards had the cartilage and bone boundaries only in the region of articular cartilage (Fig.\\ \\ref{fig:qual}a). However the LOGISMOS segmentation and further analysis required the entire bone structure. The datasets used for training the clustered RF classifier were first inspected and JEI edited (see Appendix \\ref{jei} for details). \n\n\\begin{figure}[htb]\n\t\\centering\n\t\\includegraphics[width=\\columnwidth]{Workflow_Training2.png}\n\t\\vspace{-.3cm}\n\t\\caption{Learning-based workflow for training the hierarchical RF classifiers.}\n\t\\label{fig:workflow_train}\n\\end{figure}\n\n\\begin{figure}[htb]\n\t\\centering\n\t\\vspace{-.3cm}\n\t\\includegraphics[width=0.8\\columnwidth]{Workflow_Testing_Ver4.png}\n  \\vspace{-.3cm}\n\t\\caption{Testing workflow to compare the clustered random forest classifiers with the existing methods on a set of 108 patients.}\n\t\\vspace{-.3cm}\n\t\\label{fig:workflow_test}\n\\end{figure}\n\n\\begin{table}[htb]\n\t\\caption{Parameters used for graph construction.}\n\t\\centering\n\t\\label{graph-parameters}\n\t\\begin{tabular}{lccccc}\n\t\t\\hline\n\t\t& Inter-surface & Inter-object & Smoothness & Column size \\\\\n\t\t& max (mm)   & max (mm)  & (mm)    & (mm)     \\\\\n\t\t\\hline\n\t\tLearned cost  & 6            & 18          & 0.6          & 18.15 \\\\\n\t\tGradient cost & 4            & 12           & 0.4         & 12.2   \\\\\n\t\t\\hline\n\t\t& & & & \\\\\n\t\t\\multicolumn{5}{l}{*Minimum inter-surface, inter-object and inter-time-point separations are zero.}\n\t\\end{tabular}\n\\end{table}\n\nAll image volumes were first LOGISMOS-segmented using gradient costs. The geometric graphs had 8006 and 8002 graph columns for the femur and tibia objects, respectively. The graph parameters are listed in Table \\ref{graph-parameters} with the additional inter-time-point constraint set at 0.6 mm for the 4D LOGISMOS. \n\nThe NAF features were sampled over 15 datasets with 1521 samples per image patch. Due to the imbalance in the negative to positive labels ratio, a neighborhood around the cartilage was marked as negative examples. A set of 200 trees with 40,000 image patches were used as inputs to the training. The second RF classifier was trained on 19 JEI corrected datasets with every node along the ELF search column collecting 30 features (Table~\\ref{featureLabel}). A total of 80 RF classifiers (one per cluster) were trained with 800 trees per each forest. \n\nOA is characterized by thickness losses in subregions of the surface. Analyzing the entire cartilage structure may not be sensitive to these localized and regionally smaller changes. Therefore a consensus based nomenclature was proposed by clinicians identifying regions of the cartilage most likely affected by OA \\cite{Eckstein2006a}. The existing methodologies in the literature require manual initialization for the sub-plate analysis \\cite{Wirth2008,Wirth2010}. We developed a fully automated nomenclature-compliant sub-plate analysis algorithm. The algorithm subdivided the segmented surface by automated identification of the trochlear notch anatomy and implicit cutting plane geometry (Appendix \\ref{sub-plate}). All of our experimental results were subplate analyzed and reported for the entire femur and tibia cartilage regions, central load-bearing lateral femur\/tibia (cLF\/cLT), and central load-bearing medial femur\/tibia (cMF\/cMT).\n\nThe experimental results are reported as signed and unsigned errors. The signed errors indicate the measurement bias with the positive errors denoting underestimation and the negative errors denoting overestimation with respect to the independent standard. The unsigned errors indicate the variability of the measurement. The error measurements were collected in regions for which the independent standard was available. The first experiment was to validate the accuracy of the different cost functions in 3D LOGISMOS segmentation. The costs by the proposed hierarchical RF classifier were compared against the gradient costs. To evaluate the benefits of adding the NAF stage, a single RF classifier was trained on the same 19 patient datasets excluding the NAF features while maintaining the same graph parameters as the hierarchical classifier system. The testing workflow is shown in Fig.~\\ref{fig:workflow_test}.\nThe second experiment simultaneously 4D segmented the 54 knees using the proposed hierarchical costs and compared against the 108 individually 3D LOGISMOS segmented results using the same segmentation cost functions. It is designed to reveal the benefits of adding contextual information between the adjacent time-points. Given that OA is a slowly progressing disease, the expectation was that there would not be a significant appearance of artifacts or thickness losses over a one year study period. Fig.\\ \\ref{fig:3Dvs4D-flowchart} shows the flowchart highlighting the key differences between the 3D and 4D LOGISMOS pipelines. \n\n\\begin{figure}[htb]\n\t\\centering\n\t\\includegraphics[width=0.9\\columnwidth]{3D_vs_4D_flowchart.png}\n\t\\caption{Comparison of 3D and 4D LOGISMOS pipelines. The black and red arrows highlight the flow of 3D and 4D LOGISMOS algorithms respectively.}\n\t\\label{fig:3Dvs4D-flowchart}\n\\end{figure}\n\nTo demonstrate the benefits of 4D analysis over a longer study period, the same 54 patient datasets with BL, 12M, 24M, 36M and 48M follow-up scans were used resulting in a total of 237 3D MRIs analyzed for the third experiment (out of 270 due to subjects missing some follow-up visits). Since OAI-provided  independent standards were available only for BL and 12 M, a new independent standard was defined in house by an expert analyst using JEI approach (Appendix A).  To highlight the improvements, we compared the maximum cartilage thickness errors for 3D and 4D across all time-points and determined statistical significance of their differences.\n\n\n\\section{Results}\n\\label{results}\nTable \\ref{bone_error} shows surface positioning errors for the bone segmentation with the signed errors being sub-voxel accurate. Since high bone-segmentation accuracy was already achieved using the gradient-based costs, a learning based cost function design was not necessary. Although we saw presence of bone lesions with progression of OA, the gradient-based cost was sufficiently robust. Thus all the different methodologies in both 3D and 4D LOGISMOS used the same gradient-based cost functions for the bone.\n\n\\begin{table}[htb]\n\t\\centering\n\t\\caption{Bone segmentation errors (in mm) using the gradient-based cost functions.}\n\t\\label{bone_error}\n\t\\begin{tabular}{l|cc}\n\t\t\\hline\n\t\t& Signed & Unsigned \\\\\n\t\t\\hline\n\t\tFemur & -0.19$\\pm$0.07 & 0.40$\\pm$0.06 \\\\\n\t\tTibia & -0.13$\\pm$0.09 &  0.39$\\pm$0.09 \\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n\nCatilage surface positioning errors (compared against independent standard) using the hierarchical classifier, gradient cost, and single-stage RF classifier are reported. Each resulting surface from the above methods were sub-plate analyzed. Table \\ref{error1} shows a significant reduction in signed and unsigned errors for all the femoral sub-plates ($p\\ll0.001$). Significant reduction in unsigned errors over the segmentation using gradient costs was seen for all the tibial sub-plates while the signed errors showed a significant reduction of errors over the gradient-based costs on all sub-plates except the central medial tibia (cMT). Comparison with the single-stage RF classifier saw significant reduction of unsigned errors in the whole tibia and central medial tibia (cMT) plates. \n\nFig.\\ \\ref{fig:qual} qualitatively compares the segmentation accuracies between the gradient-based costs and the hierarchical classifier with respect to the independent standard. Both the femur and tibia are shown with their respective bone and cartilage segmentations showing good agreement between learning-based segmentation and the independent standard.  \n\n\\begin{table*}[htb]\n\t\\centering\n\t\\caption{Cartilage surface positioning errors (in mm) achieved by hierarchical classifier, gradient cost and single-stage RF classifier using 3D LOGISMOS algorithm. Paired $t$-test comparisons are reported for (NAF+RF vs. Gradient) and (NAF+RF vs. RF only)}\n\t\\label{error1}\n\t\\begin{tabular}{lr|rr|rr}\n\t\t\\hline\n\t\t\\bf n=108 & NAF+RF           & Gradient          & $p$-value & RF only & $p$-value       \\\\ \n\t\t\\hline                                           \n\t\tFemur signed   & \\bf -0.01$\\pm$0.18 &-0.31$\\pm$0.24     & $\\ll0.001$  & -0.10$\\pm$0.17& $\\ll0.001$  \\\\ \n\t\tFemur unsigned & \\bf 0.55$\\pm$0.11 & 0.69$\\pm$0.13     & $\\ll0.001$  &  0.56$\\pm$0.10& $\\ll0.001$  \\\\ \n\t\t\\hline     \n\t\tcMF signed   & \\bf -0.04$\\pm$0.29 &-0.38$\\pm$0.58     & $\\ll0.001$  & -0.11$\\pm$0.27& $\\ll0.001$  \\\\ \n\t\tcMF unsigned & \\bf 0.52$\\pm$0.16 & 0.78$\\pm$0.35     & $\\ll0.001$  &  0.55$\\pm$0.17& $\\ll0.001$  \\\\ \n\t\t\\hline\n\t\tcLF signed   & \\bf -0.26$\\pm$0.24 &-0.52$\\pm$0.35     & $\\ll0.001$  & -0.36$\\pm$0.20& $\\ll0.001$  \\\\ \n\t\tcLF unsigned & \\bf 0.42$\\pm$0.12 & 0.65$\\pm$0.20     & $\\ll0.001$  &  0.47$\\pm$0.11& $\\ll0.001$  \\\\ \n\t\t\\hline\n\t\tTibia signed   &    \\bf 0.06$\\pm$0.17 & -0.11$\\pm$0.35 & $\\ll0.001$      & 0.11$\\pm$0.22& $\\ll0.001$  \\\\\n\t\tTibia unsigned & \\bf 0.60$\\pm$0.14 &     0.79$\\pm$0.20 & $\\ll0.001$  & 0.62$\\pm$0.18& $\\ll0.001$  \\\\\n\t\t\\hline\n\t\tcMT signed   & -0.15$\\pm$0.31 &-0.25$\\pm$0.77     & $0.193$  & \\bf -0.09$\\pm$0.34& $0.003$ \\\\ \n\t\tcMT unsigned & \\bf 0.52$\\pm$0.20 & 0.92$\\pm$0.41     & $\\ll0.001$  &  0.58$\\pm$0.22& $\\ll0.001$ \\\\ \n\t\t\\hline\n\t\tcLT signed   & \\bf -0.03$\\pm$0.32 &0.36$\\pm$1.12     & $\\ll0.001$  & -0.01$\\pm$0.32& $0.16$ \\\\ \n\t\tcLT unsigned & \\bf 0.46$\\pm$0.17 & 0.79$\\pm$0.99     & $\\ll0.001$  &  0.47$\\pm$0.18& $0.31$ \\\\ \n\t\t\\hline \n\t\t\\multicolumn{6}{l}{*Bold indicates statistically significant reduction in positioning errors.}\n\t\\end{tabular}\n\\end{table*}\n\n\\begin{figure}[htb]\n  \\centering\n\t\\begin{tabular}{cc}\n\t\t\\multicolumn{2}{c}{\\includegraphics[width=0.4\\columnwidth]{IndependentStandard.png}}\\\\\n\t\t\\includegraphics[width=0.4\\columnwidth]{Gradient.png} &\n\t\t\\includegraphics[width=0.4\\columnwidth]{Learned.png} \\\\\n\t\\end{tabular}\n\t\\caption{Segmentation accuracy in a representative subject.\n\t(a)~Independent standard.\n\t(b)~Gradient-costs LOGISMOS segmentation.\n\t(c)~Learned-costs LOGISMOS segmentation. Region marked by the arrow shows clear improvement in the segmentation quality when using the learned costs.}\n\t\\label{fig:qual}\n\\end{figure}\n\nTable \\ref{error-4Dvs3D} compares signed and unsigned border positioning errors with respect to the independent standard for 3D and 4D LOGISMOS segmentation at BL and 12M follow-up respectively. As expected, the overall segmentation accuracy is almost the same between the two methods except for a few of the plate regions highlighted in bold showing statistically significant differences over 3D ($p<0.05$).\n\nTable VI shows the performance improvement obtained by the  4D LOGISMOS over 3D for a longer study period of five years (BL\u2013--48 M, 5 time-points). To demonstrate improvements in local accuracy, Table VI gives the unsigned  errors of cartilage thickness averaged for three quantiles within 90-100\\% of the largest errors  per patient combined for all available time-points for each sub-plate. These unsigned maximum thickness errors  were statistically significantly  smaller in each of cMF, cMT and cLT  for 4D LOGISMOS compared with separate 3D analyses.\n\nFig.\\ \\ref{fig:4d} qualitatively shows the improvement of 4D LOGISMOS over 3D. Note the lack of an obvious edge distinguishing the tibia and femur cartilage. Using the spatial and temporal contextual information from all the time-points,  the 4D method is able to correctly position the cartilage for the tibia and femur despite the lack of image information locally. \n\n\\begin{figure}[htb]\n  \\centering\n\t\\begin{tabular}{cc}\n\t\t\\multicolumn{2}{c}{\\includegraphics[width=0.4\\columnwidth]{9279291L_IS.png}}\\\\\n\t\t\\includegraphics[width=0.4\\columnwidth]{9279291L_3D.png} &\n\t\t\\includegraphics[width=0.4\\columnwidth]{9279291L_4D.png} \\\\\n\t\\end{tabular}\n\t\\caption{Qualitative improvement of 4D segmentation versus 3D for a patient with severe osteoarthritis. Parts of the region between the femur and tibia exhibit synovial fluid leakage, the appearance and texture of which mimic that of a cartilage. (a) Independent standard, (b) 3D segmentation, (c) 4D segmentation. }\n\t\\label{fig:4d}\n\\end{figure}\n\n\\begin{table*}[htb]\n  \\caption{Cartilage surface positioning errors (in mm) of 4D versus 3D LOGISMOS.}\n  \\label{error-4Dvs3D}\n  \\centering\n  \\begin{tabular}{l|rrr|rrr}\n  \\hline\n                  & \\multicolumn{3}{c|}{\\bf n=54 Baseline} & \\multicolumn{3}{c}{\\bf n=54 12 months} \\\\\n                  & \\multicolumn{1}{c}{4D} & \\multicolumn{1}{c}{3D} & $p$-value & \\multicolumn{1}{c}{4D} & \\multicolumn{1}{c}{3D} & $p$-value \\\\ \\hline\n  Femur signed    & \\bf  0.01$\\pm$0.18  & 0.01$\\pm$0.19     & 0.027      & -0.02$\\pm$0.17    &-0.04$\\pm$0.17 & 0.062 \\\\ \n  Femur unsigned  & \\bf  0.53$\\pm$0.11  & 0.54$\\pm$0.11     & $\\ll$0.001 &  0.55$\\pm$0.11    & 0.55$\\pm$0.11 & 0.601 \\\\ \\hline\n  cMF signed      & \\bf -0.01$\\pm$0.25  &-0.03$\\pm$0.27     & 0.015      & \\bf 0.02$\\pm$0.27 &-0.04$\\pm$0.30 & 0.016 \\\\\n  cMF unsigned    &  0.51$\\pm$0.16      & 0.52$\\pm$0.17     & 0.092      &  0.52$\\pm$0.15    & 0.53$\\pm$0.16 & 0.614 \\\\ \\hline\n  cLF signed      & -0.26$\\pm$0.22      &-0.25$\\pm$0.23     & 0.108      & -0.30$\\pm$0.21    &-0.30$\\pm$0.23 & 0.853 \\\\\n  cLF unsigned    &  0.42$\\pm$0.10      & 0.42$\\pm$0.10     & 0.790      &  0.43$\\pm$0.12    & 0.44$\\pm$0.13 & 0.318 \\\\ \\hline\n  Tibia signed    &  0.08$\\pm$0.17      &\\bf 0.07$\\pm$0.16  & 0.039      &  0.07$\\pm$0.19    & 0.06$\\pm$0.18 & 0.464 \\\\\n  Tibia unsigned  &  0.59$\\pm$0.14      & 0.60$\\pm$0.14     & 0.140      &  0.60$\\pm$0.16    & 0.60$\\pm$0.14 & 0.310 \\\\ \\hline\n  cMT signed      & -0.13$\\pm$0.28      &-0.14$\\pm$0.29     & 0.125      & -0.14$\\pm$0.29    &-0.15$\\pm$0.31 & 0.619 \\\\\n  cMT unsigned    & \\bf 0.50$\\pm$0.18   & 0.51$\\pm$0.20     & 0.039      &  0.52$\\pm$0.17    & 0.53$\\pm$0.17 & 0.341 \\\\ \\hline\n  cLT signed      &  0.00$\\pm$0.30      & 0.00$\\pm$0.31     & 0.338      & -0.06$\\pm$0.31    &-0.06$\\pm$0.30 & 0.629 \\\\\n  cLT unsigned    & \\bf 0.45$\\pm$0.17   & 0.46$\\pm$0.18     & $\\ll$0.001 &  0.45$\\pm$0.16    & 0.44$\\pm$0.16 & 0.256 \\\\ \n  \\hline\n\t\\multicolumn{7}{l}{*Bold indicates statistically significant reduction in positioning errors.}\n  \\end{tabular}\n\\end{table*}\n\n\\begin{table*}[htb]\n\t\\caption{Longitudinal max thickness error values of 4D versus 3D LOGISMOS.}\n\t\\label{max_thicknessError-4Dvs3D}\n\t\\centering\n\t\\begin{tabular}{l|rrr|rrr|rrr}\n\t\t\\hline\n    & \\multicolumn{3}{c|}{\\bf 2\\% Max Average Errors (98-100\\%)} & \\multicolumn{3}{c|}{\\bf 5\\% Max Average Errors (95-100\\%)} & \\multicolumn{3}{c}{\\bf 10\\% Max Average Errors (90-100\\%)} \\\\\n\t\t& \\multicolumn{1}{c}{4D} & \\multicolumn{1}{c}{3D} & $p$-value & \\multicolumn{1}{c}{4D} & \\multicolumn{1}{c}{3D} & $p$-value & \\multicolumn{1}{c}{4D} & \\multicolumn{1}{c}{3D} & $p$-value \\\\ \\hline\n    cMF unsigned    &  \\bf 0.76$\\pm$0.60 & 1.11$\\pm$0.63     & $\\ll$0.001 & \\bf 0.57$\\pm$0.43    & 0.87$\\pm$0.41 & $\\ll$0.001  &  \\bf 0.43$\\pm$0.32    & 0.68$\\pm$0.29 & $\\ll$0.001 \\\\ \\hline\n    cLF unsigned    &  1.46$\\pm$2.36     & 1.67$\\pm$1.82     & 0.610      &  1.07$\\pm$1.73    & 1.16$\\pm$1.02    & 0.74        &  0.74$\\pm$0.95    & 0.84$\\pm$0.55 & 0.528 \\\\ \\hline\n    cMT unsigned    & \\bf 1.28$\\pm$0.47  & 1.78$\\pm$1.36     & 0.008      &  \\bf 1.09$\\pm$0.42    & 1.48$\\pm$1.00 & 0.004      &  \\bf 0.92$\\pm$0.38    & 1.19$\\pm$0.63 & $\\ll$0.001 \\\\ \\hline\n    cLT unsigned    & \\bf 1.29$\\pm$0.47  & 1.40$\\pm$0.49     & $\\ll$0.001 &  \\bf 1.09$\\pm$0.40    & 1.17$\\pm$0.41 & $\\ll$0.001 &  \\bf 0.90$\\pm$0.33    & 0.98$\\pm$0.34 & $\\ll$0.001 \\\\ \n    \\hline\n\t\t\\multicolumn{7}{l}{*Bold indicates statistically significant reduction in positioning errors.}\n\t\\end{tabular}\n\\end{table*}\n\n\\section{Discussion and Conclusion}\n\\label{conclusions}\n\nA novel method for learning segmentation cost functions using  hierarchical RF classifiers  was reported and its performance was demonstrated in 3D and 4D LOGISMOS segmentation of knee MR images. The preparation of the training data was done using our just-enough interaction (JEI) approach, which provided fast and accurate post-processing correction. To minimize the impact of the underlying automated LOGISMOS segmentation on the independent standard used for training and evaluation, a great level of flexibility was embedded in the graph construction so that the expert analyst could freely modify the surfaces as needed and thus create unbiased corrections of the knee surfaces.\n\nUsing the single RF classifier improved the segmentation accuracy over the initial gradient-based costs. In addition, using a hierarchy of NAF and RF stages showed further statistically significant improvement in segmentation performance. The previous LOGISMOS-based method \\cite{Yin2010} used a single-stage RF with the graph constructed in the image-voxel space and a subset of current features contributing to the cost function (Table~\\ref{featureLabel}) and it placed third on the publicly available SKI10 challenge \\cite{heimann2010segmentation}. The method reported here shows a significant  improvement. Further comparison with the various other methods would inevitably be imperfect due to the different testing datasets utilized. In the OAI cohort, our method achieved high accuracy in comparison with the publicly available independent standard. Training of the hierarchical RF classifier and the subsequent 3D\/4D LOGISMOS analyses ran on a Linux system with CentOS-7.3, 128 GB RAM. Once trained, the average segmentation times were 34 and 22~minutes per time-point for individual 3D and simultaneous 4D LOGISMOS, respectively.\n\nLeveraging the benefits of the available multi-time-point MRI datasets for the OA patients, we designed a 4D extension of the LOGISMOS segmentation framework. The added temporal inter-time-point contextual constraints helped constrain the changes of cartilage thickness to  what was physiologically feasible. This benefit of using information from the previous time-points along with the new cost function design helped notably improve the segmentation accuracy. Further, the advantages of the 4D LOGISMOS were demonstrated on an OAI longitudinal patient dataset with 5 imaging sessions from BL to 48 month. Our sub-plate detection algorithm was used to study the accuracy of segmentation on the load-bearing regions of the knee and provided good performance insight. The robustness of the automated sub-plate division was demonstrated showing consistent measurements between different surfaces used to initialize the automated extraction of the sub-plate load-bearing regions. \n\nThe combination of LOGISMOS and learning-based methods has been  very effective. Other learning-based techniques such as deep learning may further improve the performance once more ground-truth data become available to determine further benefits of such approaches. 4D LOGISMOS also offers a platform for future clinically-oriented work, such as investigating the rate of cartilage degradation and its impact on OA progression, and identifying cartilage thickness that correlates to disease severity and\/or predicts the need for joint replacement.\nAssessing reproducibility of the reported method is left for future work and the OAI scan-rescan dataset is well-suited for this task. Patellar segmentation was not considered in the work reported here due to the limited validation material available for patellar segmentation development. Although patellar cartilage was not segmented, our early LOGISMOS approach \\cite{Yin2010} was designed to segment all three bones and cartilages of the knee joint simultaneously.\n\n\\appendices\n\n\\section{Just Enough Interaction} \n\\label{jei}\n\nIn this work, JEI was used to prepare data for training the proposed hierarchical RF classifier (Fig.~\\ref{fig:workflow_train}) and to assess the benefits of 4D LOGISMOS over a longitudinal study period. JEI is a novel interaction technique that uses the graph based LOGISMOS framework. The user inspects the automated segmentation results and provides segmentation correction points that interact directly with the underlying graph framework. This strategy differs from traditional voxel-by-voxel editing by only requiring limited (i.e., just-enough) interactions to correct the automated LOGISMOS segmentation. The JEI interaction method still guarantees global optimality even with the cost-changes resulting from the JEI corrections.\n\nFollowing automated LOGISMOS segmentation, the JEI editing was performed on the resulting surface as needed. A custom graphical user interface (GUI) was designed onto which the image volume, residual graph and the ELF geometric graph were loaded. The JEI workflow is shown in Fig. \\ref{fig:jei_workflow}. The details of the JEI architecture and GUI are given in \\cite{Kashyap-IMIC2016}.\n \n\\begin{figure}[htb]\n \t{\\includegraphics[width=\\columnwidth]{JEI_Workflow.png}}\n \t\\caption{A graphical depiction of the JEI workflow showing the interactive correction steps. A single slice that is identified with improper segmentation of the tibia cartilage (yellow) is shown. The user provides the correcting set of boundary points (cyan) which identifies the 3D neighborhood of graph columns on which the costs are modified locally. The max-flow was recomputed in 3D resulting in the corrected surfaces within milliseconds. The resulting correction from JEI on a single 2D slice can be appreciated in the corresponding 3D model representation of the surface.}\n \t\\label{fig:jei_workflow}\n\\end{figure}\n \n\n\\section{Automated Extraction of Load-Bearing Sub-Regions}\n\\label{sub-plate}\nAnalyzing the whole cartilage structure may not be sensitive to cartilage losses that occur locally. Cartilage sub-regions have been identified as areas which bear the maximum stresses during motion with a consensus based nomenclature by the clinicians on the sub-regions of the femur and tibia that need to be analyzed separately \\cite{Eckstein2006a}. Several techniques \\cite{Wirth2008,Wirth2010} for identifying the sub-plates exist all of which require human interaction to provide an initialization for the sub-region analysis. We have developed a fully automated sub-plate detection algorithm that uses LOGISMOS surface meshes to define individual sub-plates.\n\n\\subsection{Trochlear Notch Identification}\n\nThe first step in the sub-plate extraction is the detection of the trochlear notch. The notch is at the base of a groove along which the patella (knee cap) slides over the femur (Fig.\\ \\ref{fig:notch_eg}). The main anatomic feature exploited is anterior to posterior (AP) curvature of the groove of the femoral bone (Fig.\\ \\ref{fig:notch_eg}b). The notch is at the base of the curvature before the bone ridge rises sharply. An implicit cutting plane was used to isolate the analysis region represented by $\\hat{n}.p$ where $ \\hat{n}$ is the normal direction from the plane and $p$ is a point on the plane.  When isolating the groove regions using simple implicit plane cutting, a family of contours were drawn along this surface. The sharp rise of the bone structure along the contour near the base of the ridge gives a large change in contour positioning value traversing in the AP direction. To increase robustness we find the positions of largest change on all of the closely positioned groove contour lines and average them to find the desired trochlear notch (Fig.\\ \\ref{fig:notch}). \n\n\\begin{figure}[htb]\n\t\t{\\includegraphics*[width=0.49\\columnwidth,height=4.1cm]{Trochlear_Groove.png}}\n\t\t{\\includegraphics*[width=0.40\\columnwidth]{notch_Identification.png}}\\\\\t  \n\t\\caption{Trochlear notch.  (a)~LOGISMOS-segmented bone mesh with the trochlear groove region highlighted. \n\t(b)~Bone mesh highlighting the trochlear groove (overlayed in yellow) in the anterior to posterior direction. The blue arrow indicates the trochlear notch on the groove at the base of the groove curvature before the bone ridge structure rises sharply.\n\t}\n\t\\label{fig:notch_eg}\n\\end{figure}\n\n\\begin{figure}[htb]\n\t\\includegraphics[width=0.46\\columnwidth]{Trochlear_Notch.png}\n\t\\includegraphics[width=0.46\\columnwidth]{Central_Femur.png}\n\t\\caption{ Trochlear notch.  (a)~Use of AP curvature of the groove on the femoral bone. After isolating the groove region, the trochlear notch is identified by taking an average gradient on a family of closely positioned contour lines along this surface. (b)~The load-bearing regions of the femur identified by isolating 60 \\% of the distance from the trochlear notch to  posterior most in the AP direction on each condyle respectively.}\n\t\\label{fig:notch}\n\\end{figure}\n\\subsection{Sub-Plate Detection using Implicit Cutting Plane Geometry}\n\nUsing implicit cutting planes at the trochlear notch separates the posterior region from the anterior and further splits them into the medial and lateral condyle. For each femoral condyle, the load-bearing regions is defined as 60\\% of the distance in the AP direction from the trochlear notch to the posterior-most point of the respective condyles. Isolating them using cutting planes normal to the AP plane at the 60\\% region isolates the load-bearing regions of the central medial and lateral femur (Fig.\\ \\ref{fig:notch}b).\n\n\\begin{figure}[htb]\n\t\\centering\n\t\\includegraphics[height=0.31\\columnwidth]{Tibia_separation.png}\n\t\\includegraphics[height=0.31\\columnwidth]{Tibia_Subplate.png}\n\t\\caption{(a)~Isolated medial and lateral tibia, (b)~All the tibial sub-plates.}\n\t\\label{fig:tibia}\n\\end{figure}\n\nThe tibia is divided into the medial and lateral compartments using cutting planes positioned at the trochlear notch with its normal perpendicular to the AP plane (Fig.\\ \\ref{fig:tibia}). After subdividing the plates into the medial and lateral regions, we isolate the central tibia and the peripheral sub-regions. The central 20\\% elliptical area of the medial and the lateral plate is computed around the center of mass of each respective plate. The radius of the major and the minor axis is computed as a ratio of bounds of the respective medial\/lateral compartments. The major axis radius is computed from the compartment bounds along the AP direction and similarly, the minor axis radius is computed from the ratio of the compartment bounds perpendicular to the AP direction. Furthermore, the remaining regions are isolated using a 45$\\degree$ and 135$\\degree$ cutting planes around the center of mass of their respective compartments to give all four peripheral sub-regions as shown in Fig.\\ \\ref{fig:tibia}. The final list of all the extracted sub-plates from the automated algorithm is visually shown in Fig.\\ \\ref{fig:all_plates}. \n\n\\begin{figure}[htb]\n\t\\includegraphics[width=\\columnwidth]{Sub-platesCropped.png}\n\t\\includegraphics[width=\\columnwidth,height=0.6\\columnwidth]{Screenshot_2.png}\n\t\\caption{ Automated sub-plate division of the cartilage. The 60\\% central lateral (cLF) and central medial femur (cMF) are shown. Each Medial (MT) and lateral tibia (LT) regions are subdivided as: central (cLT\/cMT), interior (iLT\/iMT), exterior (eLT\/eMT), anterior (aLT\/aMT) and posterior (pLT\/pMT) regions respectively.}\n\t\\label{fig:all_plates}\n\\end{figure}\n\n\\subsection{Robustness of Sub-plate Detection}\nAutomated identification of sub-plate regions uses multi-surface knee joint segmentation as input. Consequently, the sensitivity of such surface segmentation on sub-plate region identification and thus on cartilage thickness measurement must be established. Load-bearing sub-plates were derived from two surface segmentation results (4D LOGISMOS, JEI-defined surfaces) and used to determine sub-plate-specific cartilage thickness from the manually segmented independent standard.\nIn 108 patients, the two sets of sub-plate-specific cartilage thickness measurements were compared using regression analysis, which showed high robustness (low sensitivity) of the sub-plate detection algorithm on the identification of the load-bearing regions (using manually segmented independent standard surface) with respect to the underlying knee-joint segmentation. The sub-plate-specific correlation  of the two thickness measurements ranged from $R^2 = 0.98$ to $0.99$.\n\n\\ifCLASSOPTIONcaptionsoff\n  \\newpage\n\\fi\n\n\\bibliographystyle{IEEEtranbib}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nICRU Report 36 on microdosimetry~[\\onlinecite{ICRU36}] provides general formulations for the mean chord length in terms of the geometrical dimension of the sensitive volumes of interest (SVOI).\nAccordingly, $\\langle l \\rangle \\equiv \\overline{l}$ is the mean length of randomly oriented chords in that volume and can be calculated based on the geometrical structure of the target of interest.\nThe mean chord length that is an intrinsic geometrical character of SVOI, results from the random interception of the SVOI by a straight geometrical line, superseded for a (single) physical track of a charged particle.\nThe chord length calculated as such is the mean of shortest Euclidean distances between pairs of end points, crossing closed surface of SVOI.\n\nAs originally introduced by Rossi, Kellerer and colleagues (e.g., see for example Refs.~[\\onlinecite{Kellerer1971:RR,Kellerer_Rossi1972:CTRR,Kellerer1984:RR,Kellerer1985:Book,Kellerer1975:REB}]), the lineal energy, $y = \\varepsilon \/ \\langle l \\rangle$,\nis the energy imparted in a single event, $\\varepsilon$, divided by the mean chord length, $\\langle l \\rangle$.\n$y$ is a random variable analogous to particle linear energy transfer (LET). Accordingly, the randomness stems from $\\varepsilon$ as the chord-length has been averaged out and it is a predetermined number.\nFor a convex volume, $V$, such as spherical geometries with area $A$, a pure geometrical calculation, first performed by Cauchy~[\\onlinecite{Cauchy1908}], yields $\\langle l \\rangle = 4V\/A$.\nFor a rectangular slab with the transversal thickness $d$ perpendicular to the unidirectional beam central axis, $\\langle l \\rangle = d$~[\\onlinecite{Magrin2018:PMB}].\nThe mean chord length, as such, which is based on straight tracks in a micro-meter size in a SVOI, has been vastly used in microdosimetry literature as well as in modelings in the radio-biological effects of any field of radiation.\n\nOne of the first studies based on MC technique to calculate chord-length distributions was reported by Birkhoff {\\em et al.}~[\\onlinecite{Birkhoff1970:HP}].\nIn this type of MC calculations, particles are not scored via a collision-by-collision approach using coarse-grained quantum mechanical scattering cross-sections, as we perform in now-a-days MC models such as Geant4~[\\onlinecite{Agostinelli2003:NIMA,Allison2006:TNS,Allison2016:NIMPRA}] and Geant4-DNA~[\\onlinecite{Incerti2010:IJMSSC}] where the tracks, including primary and secondary particles, exhibit the inchoate distribution of energy transfers~[\\onlinecite{Conte2012:NJP,Abolfath2011:JPC,Abolfath2013:PMB,Abolfath2016:MP,Abolfath2017:SR,Abolfath2019:EPJD}].\nInstead, Birkhoff {\\em et al.}~[\\onlinecite{Birkhoff1970:HP}] considered straight geometrical lines intercepting a sensitive volume of energy-proportional devices such as spherical \/ cylindrical gas proportional counters.\nThe rationale behind these simplifications is based on uniform and continuous energy transfer to matter along the tracks of charged particles.\nMoreover, the tracks considered in simulations performed by Birkhoff {\\em et al.}, were assumed to be part of a uniform source beam of radiation.\n\nThe MC techniques as such,\nare therefore analogous to the classical Metropolis MC algorithm proposed for calculation of the numerical value of $\\pi$, or numerical multi-variable integrations~[\\onlinecite{NumericalRecipes}].\nBy taking a large random sample, the resulting chord-length distribution approaches the correct geometrical distribution in a statistical manner.\n\n\nThis is seemingly a plausible approach, applicable to cavities of the instruments designed for measuring pulse height distributions in a field of radiation such as energy-proportional devices, as details in localities of the collisions, their spatial distribution and compactness within the cavity sensitive volume are irrelevant to overall observable electrical responses. The mathematical model including the calculation of the chord length,\nas presented in Birkhoff {\\em et al.}~[\\onlinecite{Birkhoff1970:HP}] and recent modifications and refinements in Refs.~[\\onlinecite{Bolst2017:PMB,Anderson2017:MP,Bolst2018:PMB}], attempted to introduce an alternative approach to the Cauchy path length, is consistent with the occurrence of the physical processes in micro-dosimetry devices, considering Rossi's formulation of lineal energy $y$.\n\nThe responses from biological systems, however, seems slightly different from the physical responses we expect to observe from the design of micro-dosimetry proportional counters.\nFor example, the nano-meter scale linear density of DNA double strand breaks (DSBs) induced by traversing of a track of charged particle in a cell nucleus that determines complexities in lethal pathways of the lesions, cell death and tissue late effects, is expected to be more sensitive to the compactness and linear distribution of the individual collisions, rather than the geometrical dimensions of the cell nuclei~[\\onlinecite{Abolfath2019:EPJD}].\nThe former roots in nano-meter scale localities in physio-chemical phenomenon and effects occurring in DNA material in cell nuclei whereas the latter is global characters associated with the outer structure of the cell nuclei boundaries and their membranes, prone to macroscopic-scale volume effects.\nMore precisely, such model calculations, as also pointed out, e.g., in Ref.~[\\onlinecite{Carlson2008:RR}], only provide an estimate of overall imparted energy within cell nucleus volume without offering an appropriate analysis in resolving nano-scale biologically relevant events, as the geometric chord length of a cell nucleus is in $\\mu m$-scale, three orders of magnitude larger than nano-meter scale, the microscopic resolution of DNA-damage.\n\nMoreover, as particles penetrate in tissue, they lose their kinetic energy, hence the compactness of collisions and the complexities in DNA damage are expected to rise. The geometrical chord-length as calculated by Cauchy and implemented in microdosimetry, is a fixed number, e.g., a factor proportional to the dimension of cell nucleus and does not vary as a function of depth.\n\nLet us demonstrate the subtleties and lack of sufficiency in the standard microdosimetry formalism to fully describe and capturing the microscopic \/ nano-meter responses and critical variabilities in biological complexities,\nby considering passage of a single charged particle in a $\\mu$m-scale SVOI, a typical representation of a cell nucleus, as schematically shown in Fig.~\\ref{fig0}.\nHence occurrence of various damages in DNA-materials and chromosomes such as double strand breaks (DSBs) are expected to be seen in these volumes~[\\onlinecite{Abolfath2011:JPC,Abolfath2013:PMB,Bianco2015:RRD}].\n\nIn Fig.~\\ref{fig0}, the black cross-lines ($\\times$) represent locations of the site of damages in DNA materials connected diagrammatically to the charged particle by the wiggly lines. The lines are Feynman diagram representation of the photon field propagators in quantum electrodynamics (QED) that describe interaction of charged particles in scattering processes with random interaction sites on DNA and\/or the environment of DNA in a cell nucleus.\nA process that describes release of OH-free radicals and\/or reactive oxygen species (ROS) in indirect DNA damage (the latter), or shell electrons localized initially in DNA in direct damage processes (the former).\nFor further details in simulating such microscopic events at nano-scales, we refer the readers to our previous studies and publications, for example Ref.~[\\onlinecite{Abolfath2013:PMB}].\n\nFor the present discussion, it is critical to recognize that the locations of the sites of interaction are random, hence the path lengths and distances between two sequential interaction sites along the beam central axis, $l$, are random variable. This is in addition to randomness in energy imparted at the location of interaction, $\\varepsilon$.\n\nAlthough it should be obvious that both $l$ and $\\varepsilon$ are two random independent variables, following two independent distribution functions, in the standard microdosimetry formalism, only $\\varepsilon$ was treated directly a random variable associated with the beam quality.\nThe sub-micrometer randomness in steps in $l$ was neglected, because such information was not necessary to be recorded from the measurement theory standpoints.\nInstead a constant chord-length that is a characteristic \/ geometrical length of microdosimetric sites was superseded.\n\nMore rigourously, the statistical fluctuations in $l$ with regards to all collisions scored at nano-scale, crucial to DNA-damage statistical analysis, cannot be captured by micrometer-scale chord-length calculated by averaging $l$ over the track-distribution functions.\nIn fact, in the microdosimetry distribution functions, the location of the collisions over a single track were already traced out \/ integrated over thus any information regarding to individual collision was wiped out.\nIn our approach, presented in this work, we substituted the standard microdosimetry distribution functions by collision distribution functions.\nBecause a single track consists of several collisions in a microdosimetry site, the collision-based distribution functions of $\\varepsilon$ and $l$ are more informative and appropriate for modeling biological responses.\n\n\nIn addition to aforementioned shortcomings, substantial number of studies and publications, including the microdosimetric kinetic model (MKM)~[\\onlinecite{Hawkins1998:MP,Hawkins2003:RR,Kase2006}], implemented in the Particle and Heavy Ion Transport Code System\n(PHITS)~[\\onlinecite{Takada2018:JRR}], formulated based on an alternative approach by introducing a concept of virtual spherical domains in cell nuclei with a size that varies within nano- to $\\mu$-meter.\nIn these models, it is not clear the rational for a choice of domain size but considering it spuriously an additional phenomenological parameter.\nTherefore the application of current micro-dosimetry formulation in radio-biology does not seem appropriate and is prone to mixing the above discernible aspects.\n\nWe note that the Cauchy formula for the mean chord length is valid only for isotropic radiation (for non-spherical detectors)~[\\onlinecite{Magrin2018:PMB}].\nHadron therapy beams are, at the contrary, non-isotropic and the primary particles have essentially a unique direction.\nThus several microdosimetry publications were developed based on non-Cauchy algorithms for the mean chord length.\nHowever, the extension of Cauchy to non-Cauchy calculation of chord length does not seem to overcome the shortcomings of application of chord length in biological models, because the chord-length, by definition, is a character of a surface boundary of microdosimetry sites, whether it is a detector, cell nucleus or MKM domains.\nThe chord-length as such does not contain detailed information on occurrence of the biological events and complexities inside the bulk of the site.\n\nThus it is appealing to revisit and refine these well established micro-dosimetry models and generalize them for modelings in biological response theories, radio-biological applications, and nano-dosimetry.\nThis is the main goal of the present study to propose necessary modifications essential in tweaking micro-dosimetry and suit it for nano-dosimetry formulations.\nAlong this line of thoughts, we propose substitution of the cellular geometrical chord-length (regardless if it was computed using Cauchy or non-Cauchy techniques) by the tracks {\\it diffusive mean free path}, $\\langle\\langle l \\rangle\\rangle$.\n\n\n\n\n\n\n\n\n\n\n\n\n\nWe demonstrate that to simulate biologically relevant passage of a charge particle, it is necessary to collect the events, collision-by-collision, and combine them together to shape a track-structure, a bottom-up approach.\nThis led us to perform a double averaging over collisions and tracks to calculate the track mean-free path, denoted by $\\langle\\langle l \\rangle\\rangle$.\nSimilar to a gas phase of matter, the molecules collide with one another, hence the diffusive mean free path is the average distance a particle travels between collisions. The larger the particles or the denser the gas, the more frequent the collisions are and the shorter the mean free path.\nTurning back to original problem, the shorter the mean free path of the charged particle tracks manifests in higher complexities in DNA damage, hence the higher biological impact.\nWe thus supersede the track-averaged chord length, $\\langle l \\rangle$, by the mean-free path, $\\langle\\langle l \\rangle\\rangle$.\nIn calculation of lineal energy and LET, to be consistent with changes in calculating $\\langle\\langle l \\rangle\\rangle$, we also supersede the single event energy imparted, $\\varepsilon$, by energy imparted in a single collision in the cellular SVOI. We then average over individual collisions and tracks in SVOI, wherever is necessary.\n\nWe substantiate our proposal by presenting numerical illustrations on recently reported experimental data based on in-{\\it vitro} clonogenic cell survival assay of non-small cell lung cancer (NSCLC) cells, i.e., an observation which was obtained by performing a high-throughput and high accuracy clonogenic cell-survival data acquired under exposure of the therapeutic scanned proton beams~[\\onlinecite{Guan2015:SR}].\nWe compare two RBE models using (1) ICRU 36 geometrical chord length of typical spherical cell nuclei with radius of $\\langle l \\rangle = 5 \\mu m$ and (2) charge particle diffusive mean-free path, $\\langle\\langle l \\rangle\\rangle$ that is a variable and a function of depth.\nWe show that a monotonic decrease in $\\langle\\langle l \\rangle\\rangle$ as a function of depth leads to a monotonic increase up to a factor of 4 in RBE.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1.0\\linewidth]{Fig1_epsilon_length_New.pdf}\\\\\n\\noindent\n\\caption{\nSchematically shown passage of a charged particle (red bold arrow) through a $\\mu$m-scale SVOI.\nCollision-by-collision following by event-by-event length and imparted energy scales, $\\langle\\langle l \\rangle\\rangle$ and $\\langle\\langle \\varepsilon \\rangle\\rangle$, respectively, describe typical distance between sequential interaction sites and imparted energy between charged particle and biological materials.\n}\n\\label{fig0}\n\\end{center}\\vspace{-0.5cm}\n\\end{figure}\n\n\n\\section{Method and Materials}\n\\subsection{Mathematical formalism}\nIn our recent study~[\\onlinecite{Abolfath2019:EPJD}], we demonstrated that the linear density of DNA DSBs, $\\Delta_l$,\na fundamental biological response function in a radiation field, is a variable of linear density of the collisions.\nTo this ends, we recall the following relation derived in our first principle multi-scale study~[\\onlinecite{Abolfath2019:EPJD}]\n\\begin{eqnarray}\n\\Delta_l = \\frac{\\mu}{m}\n\\frac{\\langle\\langle \\varepsilon^2 \\rangle\\rangle}{\\langle\\langle \\varepsilon \\rangle\\rangle}\n\\frac{1}{\\langle\\langle l \\rangle\\rangle},\n\\label{eq1}\n\\end{eqnarray}\nwhere $\\varepsilon$ and $l$ are single collision energy imparted and stepping-length, calculated by Geant4 MC toolkit~[\\onlinecite{Agostinelli2003:NIMA,Allison2006:TNS,Allison2016:NIMPRA}].\nHere $\\mu$ and $m$ are the average number of DSBs per deposition of 1 Gy of ionizing dose and cellular \/ DNA mass, respectively.\n\nHence, the relevant quantity in unit of lineal-energy (energy per length), consistent with Eq.(\\ref{eq1}), was defined\n\\begin{eqnarray}\ny_{1D} = \\frac{\\langle\\langle \\varepsilon^2 \\rangle\\rangle}{\\langle\\langle \\varepsilon \\rangle\\rangle}\n\\frac{1}{\\langle\\langle l \\rangle\\rangle},\n\\label{eq2}\n\\end{eqnarray}\nsuch that\n\\begin{eqnarray}\n\\Delta_l = \\frac{\\mu}{m} y_{1D}.\n\\label{eq3}\n\\end{eqnarray}\n\nIn terms of these quantities, calculation of cell survival was performed\n\\begin{eqnarray}\n-\\ln(SF) = \\sum_{i=1}^N \\sum_{j=1}^i b_{i,j} \\Delta_l^{i-j} D^j.\n\\label{eq4}\n\\end{eqnarray}\nExpanding Eq.(\\ref{eq4}) and keeping the series up to quadratic term in dose, $D$, we found radio-biological $\\alpha$, $\\beta$ indices, as following\n\\begin{eqnarray}\n\\alpha = \\sum_{i=1}^N b_{i,1} \\Delta_l^{i-1},\n\\label{eq5_1}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\beta = \\sum_{i=1}^{N-1} b_{i+1,2} \\Delta_l^{i-1},\n\\label{eq5_2}\n\\end{eqnarray}\nThe dependence of $\\alpha$ and $\\beta$ on $y_{1D}$ can be derived from the dependence of $\\Delta_l$ on $y_{1D}$.\nThe coefficients in Eq.(\\ref{eq4}) are identified by fitting to the experimental cell-survival data.\n\n\\subsection{Geant4 Monte Carlo simulations}\nIn Geant4 each pencil proton beams were simulated by irradiating a cylinder water phantom with 20 cm radius and 20 cm length. The mean deposited energies $\\langle\\langle\\varepsilon\\rangle\\rangle$ and $\\langle\\langle\\varepsilon^2\\rangle\\rangle$, the mean track length $\\langle\\langle l \\rangle\\rangle$ were scored within a linear array of voxels with 0.5 $mm$ down to 5 $\\mu m$ thickness.\nIn our notations, averaging over all single point collisions, following by averaging over all single primary and secondary particle tracks are denoted by double averaging, $\\langle\\langle\\cdots\\rangle\\rangle$.\nIn MC we scored $y_{1D}$, $y_{D}$ and LET$_d$ from collision-by-collision following by event-by-event averaging and energy deposition $\\varepsilon_j$ and stepping length $l_j$, using the following identities\n$y_{1D} = (\\sum_j \\varepsilon_j^2\/\\sum_j \\varepsilon_j)\/(\\sum_j l_j\/\\sum_j 1_j)$, $y_D = \\sum_j (\\varepsilon_j\/l_j)^2\/\\sum_j (\\varepsilon_j\/l_j)$, and\n${\\rm LET}_d = \\sum_j (\\varepsilon_j^2\/l_j) \/ \\sum_j \\varepsilon_j$\nwhere sum over $j$ includes all energy deposition events from primary and secondary processes in all steps and tracks in a specific voxel, hence $\\sum_j 1_j$ represents total number of single-collisions, scored in all energy imparted events.\n\nTherefore $y_{1D}=(\\langle\\langle\\varepsilon^2\\rangle\\rangle\/\\langle\\langle\\varepsilon\\rangle\\rangle)\/\\langle\\langle l \\rangle\\rangle$ as well as other types of LET's were calculated.\nThe number of primary protons and the number of interactions per track were saved, in the same volume.  Then, the energy deposition, the track length, the number of primary proton and the number of interactions were accumulated, in each cell. All simulations used $10^6$ protons with series of energy cut-off, corresponding to particle range that vary within 1 $mm$ and 1 $\\mu m$.\nIn Geant4, any particle with energy below the cut-off value is assumed to not produce secondary particles (i.e. production threshold).\nBelow these cuts, the particle is transported further according to the CSDA approximation which will still imply a varying energy loss, i.e., no tracking cuts ~[\\onlinecite{Agostinelli2003:NIMA}].\nAll simulation results presented used the QGSP-BIC-EMY physics list.\nWe used Gaussian proton energy spectrums with very small FWHM (0.18 MeV). Because of small divergence the simulated beam is mono-energetic.\n\n\n\nIn Geant4, when primary particle (in our case proton) collide with other particle, secondary particles are generated.\nThis includes photon, electron, proton, neutron, He and heavier ions. The primary particle will not be processed until all secondaries are dealt with.\nThe importance of the energy cut-off, requires the program to stop producing more secondaries when the energy of the secondaries become lower than energy cut-off, defined by the user as a cut-off in length-scale.\nOtherwise the simulation will not be practical as will take considerably long time to simulate the events.\nHowever, the very low energy secondaries will not be killed, but it will follow the continuous slowing down approximation, CSDA.\nThe CSDA works fine with low energy particles as they tend to not travel far in the material.\nIn other words, the cut-off represents the accuracy of the stopping position, and any particle will always be tracked down to zero kinetic energy.\nIn our previous studies, we did validate the Geant4 simulation using the cut-off values (as we used in current study) and experimentally measured Bragg peak via production of Cerenkov light~[\\onlinecite{Helo2014:PMB}].\nThe difference was found to be less than 1\\% for cut-off of 0.01 mm.\n\nIn the current manuscript, we used the terms cut-off, and cut-all interchangeably.\nThe term cut-all used in the legend of figures to emphasize a specific cut-off value applied uniformly for ``all\" primary and secondary particles.\nNote that the corresponding cut-off values in energy depends on the type of secondary particles.\nFor example the length-scale cut-off of 0.01 mm, approximately corresponds to 0.025 MeV cut-off for electrons in water.\nThus this value of energy for proton and the rest of particles is different than the corresponding value of cut-off in energy for electron.\n\nFinally we note that the mean free path calculated by Geant4 methodology, as described above, should not be confused with the distance between individual ionizations, as this would be a true track structure approach. But from the Geant4 MC simulations, it must be clear what we calculated is a condensed history simulation where a large number of ionizations are grouped in a single energy loss step so it is rather the mean free path between individual energy loss events above a certain production cut.\n\n\n\n\n\n\\section{Results}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1.0\\linewidth]{Fig2_stepping_length_v0.pdf}\\\\\n\\noindent\n\\caption{\nShown $\\langle\\langle l \\rangle\\rangle$ calculated for various cutoff values as a function depth for\na pencil beam of proton with nominal energy 80 MeV, traversing a cylindrical water phantom with slice thickness equal to 0.5 $mm$.\n}\n\\label{fig1}\n\\end{center}\\vspace{-0.5cm}\n\\end{figure}\n\nIn Fig.~\\ref{fig1}, we illustrate the effect of particle cutoff length on $\\langle\\langle l \\rangle\\rangle$.\nThe calculation was performed for a pencil beam of proton with nominal energy 80 MeV, hitting a cylindrical water with radius and length of 20 cm, and slice thickness of 0.5 $mm$.\nAs shown in Fig.~\\ref{fig1}, the depth dependence of $\\langle\\langle l \\rangle\\rangle$ is strongly influenced by the user defined cutoff choice.\nThe cutoff larger or equal to size of slice thickness, as shown by the black circles and red triangles respectively, yield the collision averaged chord-length equal to the slice thickness, a result that coincides with the geometrical chord-length of the water slice-thickness.\nIn this limit of large cut-off's, our numerical results recover the CSDA limits and the geometrical chord length calculated by Cauchy, Kellerer {\\em et al.} and ICRU 36.\nWith lowering the cutoff down to 10th of slice thickness (green triangles) or even further (blue squares), the track mean-free path becomes significantly smaller than the geometrical slice thickness.\nBy lowering cutoff, below 0.01 mm, we observe negligible change in $\\langle\\langle l \\rangle\\rangle$,\ni.e., if we lower the cut-off below 0.01 mm, the generated curve exhibits slight difference relative to the curve corresponding to 0.01 mm cut-off, shown by squares in Fig.~\\ref{fig1}.\nNumerically, we reached to a domain that resulted in convergence of the output data with respect to variations in cutoff values.\nA continuous decrease in $\\langle\\langle l \\rangle\\rangle$ as a function of depth reveals an increase in linear compactness of collisions.\n\nNote that, for slice thickness of 0.5 mm, as in Fig.~\\ref{fig1}, and large cut-off values, no secondary particles are generated.\nThe primary particle, i.e., proton, loss energy according to CSDA and in results the simulated chord length turns out to be identical to the slice thickness, e.g., 0.5 mm as in Fig.~\\ref{fig1}, which fits very well with Rossi-Kellerer's theory of chord-length.\nHowever, if we lower the cut-off value the simulated chord length will become considerably less (0.15 mm as seen in Fig.~\\ref{fig1}).\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1.0\\linewidth]{Fig3_stepping_length_PDD_v2.pdf}\\\\\n\\noindent\n\\caption{\nShown $\\langle\\langle l \\rangle\\rangle$ (black lines) and relative energy deposition (red lines) calculated for 80 MeV pencil beam of proton, traversing a cylindrical water phantom with 5.0 $\\mu m$ slice thickness, using a cutoff values for all particles equal to 1 $\\mu m$. The vertical dash line indicates the position of Bragg peak.\n}\n\\label{fig2}\n\\end{center}\\vspace{-0.5cm}\n\\end{figure}\n\nIn Fig.~\\ref{fig2}, we show $\\langle\\langle l \\rangle\\rangle$ (black lines) and relative energy deposition, $\\langle\\langle \\varepsilon \\rangle\\rangle$, (red lines) calculated for a pencil beam of $10^6$ protons with nominal energy 80 MeV, traversing a cylindrical water phantom with slice thickness equal to 5.0 $\\mu m$.\nTo score such fine slices, we divided the water phantom thickness, equal to 20 cm, into 40,000 divisions and scored collision-by-collision energy deposition, $\\varepsilon$, and stepping length, $l$, in each slice, using a cutoff values equal to 1 $\\mu m$.\nThe vertical dash line indicates the position of Bragg peak at approximate 5.2 cm depth corresponding with $LET_d \\approx 10 keV\/\\mu m$.\n\nNote that because of 5 $\\mu$m slice thickness and use of 1 $\\mu$m cut-off, the effect of delta-rays and secondary particles, imparting their energies outside of the slices they were generated, effectively alter the shape of PDD. However, the location of the Bragg peak does not change as the slice thickness reduces from 1 mm to 5 $\\mu$m.\nWe should also remark that if we lower the thickness of the SVOI to nano-scales, we must lower the cut-off values to below 1 nm to obtain sensible results.\n\n\nIt is intriguing to note that from beam entrance to the end of the proton range, $\\langle\\langle l \\rangle\\rangle$ decreases by a factor of eight.\nCoincidentally, a factor of 4 has been reported experimentally in increase in proton RBE distal and proximal to Bragg peak (for more discussion see Fig.~\\ref{fig4}, below).\nThe wiggling lines in the vicinity of proton range stem from statistical noises due to particle energy straggling and lack of enough statistics because of small number of particle fluence.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1.0\\linewidth]{Fig4_LETd_y1_v2.pdf}\\\\\n\\noindent\n\\caption{\nShown $y_{1D}$ vs. $LET_d$ scored for a MC setup identical to Fig.~\\ref{fig2}.\n}\n\\label{fig3}\n\\end{center}\\vspace{-0.5cm}\n\\end{figure}\n\nIn Fig.~\\ref{fig3}, $y_{1D} = \\left[\\langle\\langle \\varepsilon^2 \\rangle\\rangle \/ \\langle\\langle \\varepsilon \\rangle\\rangle \\right] \/ \\langle\\langle l \\rangle\\rangle$ vs. $LET_d = \\langle\\langle \\varepsilon^2\/l \\rangle\\rangle \/ \\langle\\langle \\varepsilon \\rangle\\rangle$ is shown.\nThe trace shown by black dots obtained from variable $\\langle\\langle l \\rangle\\rangle$, as shown in Fig.~\\ref{fig2} whereas the trace by the red dots uses an ICRU 36 type of chord-length estimate, i.e., $\\langle\\langle l \\rangle\\rangle = 5 \\mu m$ everywhere, regardless of the depth.\nNote that $\\langle\\langle l \\rangle\\rangle = 5 \\mu m$ is a geometrical chord-length for a slab with thickness equal to $5 \\mu m$.\nThe green solid line is a CSDA line, implying $LET_d = y_{1D}$ under assumption $\\langle\\langle 1\/l \\rangle\\rangle \\approx 1 \/ \\langle\\langle l \\rangle\\rangle$.\nA similar non-linear dependencies between different types of LET's were previously presented in~[\\onlinecite{Abolfath2019:EPJD}].\nA non-linear dependence between LET$_d$ and $y_{1D}$ suggests advantageous in using $y_{1D}$ over LET$_d$ for RBE studies because of linear dependence between $y_{1D}$ and $\\Delta_l$, as is given by Eq.(\\ref{eq1}).\n\nWe note that the scattered dots in Fig.~\\ref{fig3} are due to the statistical fluctuations in a MC simulation performed under limited number of primary protons. By increasing number of protons, the scattered dots merge together and form a single curve for each value of cutoff as shown in Fig.~\\ref{fig3}.\nIn this calculation we used 10$^6$ protons. Because of small thickness of slices and small cut-off value, the MC calculation is very slow. To generate similar curves with lower statistical variations, a simulation based on larger number of protons, e.g., 10$^7$ protons is required. However, because in Fig.~\\ref{fig3}, we only intend to show a non-linear dependence between LET$_d$ and y$_{1D}$, the envelope dependence obtained from scoring a million of protons is adequate for our current qualitative analysis. We note that, the red and black envelop-curves in Fig.~\\ref{fig3} were collected through merging the scattered points. We also note that we did not perform any analytical fitting procedure to derive a specific curve, to investigate the analytical dependencies of LET$_d$ on $y_{1D}$.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1.0\\linewidth]{Fig5_RBE_PDD_AAPM2019.pdf}\\\\\n\\noindent\n\\caption{\nShown RBE calculated using variable $\\langle\\langle l \\rangle\\rangle$ (red dots) and constant chord length equal to $5\\mu m$ (black dots), as recommended by ICRU 36.\nRBE calculated after a surface fitting in 3D space to dose, LET and SF data for NSCLC H1437 cell lines.\nA significant non-linear increase in RBE by a factor of $\\approx 4$ in domains distal to the Bragg peak can be captured if $\\langle\\langle l \\rangle\\rangle$ is considered a variable of depth.\nThe biological endpoint used for calculation of RBE, corresponds to 10\\% cell-survival fraction, as reported by Guan {\\em et al.}~[\\onlinecite{Guan2015:SR}].\n}\n\\label{fig4}\n\\end{center}\\vspace{-0.5cm}\n\\end{figure}\n\nWe now turn to present RBE of in-{\\it vitro} clonogenic cell survival assay of H460 cell lines, a type of non-small cell lung cancer (NSCLC).\nThe dependence of SF on dose and LET were measured experimentally in our group~[\\onlinecite{Guan2015:SR}]\nand fitted by 3D global fitting method by the present authors~[\\onlinecite{Abolfath2013:PMB,Abolfath2016:MP,Abolfath2017:SR,Abolfath2019:EPJD}].\nFig.~\\ref{fig4} shows the result of RBE calculated using variable $\\langle\\langle l \\rangle\\rangle$ (red dots) and constant $\\langle\\langle l \\rangle\\rangle = 5\\mu m$ (black dots),\ncorresponding a biological endpoint with 10\\% cell-survival fraction, as reported by Guan {\\em et al.}~[\\onlinecite{Guan2015:SR}].\nThe numerical values of the coefficients in the polynomial expansion of $\\alpha$ and $\\beta$ on $\\Delta_l$ and LET are given by Eqs.(\\ref{eq5_1}-\\ref{eq5_2}).\nFor $LET \\leq 10 keV\/\\mu m$ we obtained the optimal fitting to the following polynomials with\n$N=2$, $b_{1,1} = 0.05237$, $b_{2,1} = 0.0151$ and $b_{2,2} = 0.03265$, corresponding to reduced $\\chi^2 = 0.00541$, $R^2$(COD)$=0.98233$ and adjusted $R^2 = 0.98233$.\nThis is a limit that dependence of $\\alpha$ on $\\Delta_l$ and LET is linear whereas $\\beta$ shows no dependence on $\\Delta_l$ (and LET),\nNote that this is a limit frequently used in literature for all range of LET\n(see e.g., Eq.(II.28) in Ref.~[\\onlinecite{Hawkins1998:MP}] or Eq. (8) in Ref.~[\\onlinecite{Kase2006}]).\nFor $LET > 10 keV\/\\mu m$, we obtained $N=6$ with\n$b_{1,1} = 0.12288$, $b_{2,1} = 0.00624$,\n$b_{3,1} = 1.322\\times 10^{-6}$, $b_{4,1} = 9.449\\times 10^{-7}$,\n$b_{5,1} = 8.812\\times 10^{-7}$,\n$b_{6,1} = 7.725\\times 10^{-8}$, and\n$b_{2,2} = 0.00382$, $b_{3,2} = 0.0007821$, $b_{4,2} = 9.275\\times 10^{-5}$, $b_{5,2} = 9.511\\times 10^{-7}$,\n$b_{6,2} = 1.68\\times 10^{-7}$\ncorresponding to\nreduced $\\chi^2 = 0.05472$, $R^2$(COD)$=0.96777$ and adjusted $R^2 = 0.93911$.\n\n\n\nThe latter is a method of calculation, recommended by ICRU 36.\nA comparison between two methods (a) considering variability in $\\langle\\langle l \\rangle\\rangle$ as a function of depth by double averaging over collisions and tracks and (b) considering geometrical chord length equal to the cell line thickness, $5\\mu m$, evidently exhibits significant difference in predicting RBE by a factor of 4 in domains distal and proximal to the Bragg peak.\n\nSimilar variabilities in RBE as a function of proton range were reported in other experimental works using conventional microdosimetry approach and a methodology based on an empirical or phenomenological ``biological waiting functions\" in calculation of RBE [\\onlinecite{DeNardo2004:RPD}].\nAccordingly, the biological waiting functions fitted to the spectrum of a spread-out Bragg peaks (SOBP) beam of proton collected by a 2.3 mm a microdosimetric prob, using a tissue-equivalent proportional counter (TEPC), resulted in a monotonic increase in RBE up to 2.5, a value close to the RBE reported in this work.\nWe note that the proposed modifications presented in this work, is free from such phenomenological convolution between the beam spectrum and RBE.\n\n\\section{Discussion and conclusion}\nThere are substantial evidence that proton (similar to heavier charged particles) RBE increases as a function of depth.\nThis is partly due to increase in spatial density of collisions as protons pass through tissue and lose energy at an increasing rate.\nTo make connection between experimental radiobiological data and mechanistic models, it is customary approach in literature to use microdosimetry models, developed by Rossi, Kellerer and colleagues~[\\onlinecite{Kellerer1971:RR,Kellerer_Rossi1972:CTRR,Kellerer1984:RR,Kellerer1985:Book,Kellerer1975:REB}].\nEmbedded in these models, geometrical chord length of a SVOI plays a crucial role.\nWe discussed that although this quantity is a good parameter to describe microdosimetry processes and charge collections in tissue proportional chambers, but because it is intrinsically character of the surface and geometrical boundaries of cell nuclei, it cannot directly describe nano-meter scale localities in stochastic microscopic physio-chemical processes in DNA materials and DNA-damage.\nThus we proposed the diffusive mean-free path length of the particle tracks as a new metric that describes appropriately variabilities in collisions compactness as a function of depth in tissue to substitute the geometrical chord-lengths considered in micro-dosimetry modelings.\nThis variation of micro-dosimetry is potentially more appropriate for radio-biological studies.\n\n{\\bf Acknowledgement:}\nThe authors would like to acknowledge useful discussion and scientific exchanges with Drs. Alejandro Carabe-Fernandez and Alejandro Bertolet Reina.\nThe work at the University of Texas, MD Anderson Cancer Center was supported by the NIH \/ NCI under Grant No. U19 CA021239.\n\n\\noindent{\\bf Authors contributions:}\nRA: wrote the main manuscript, prepared figures, performed mathematical derivations and computational steps including Geant4 and Geant4-DNA Monte Carlo simulations and three dimensional surface fitting to the experimental data.\nYH: contributed to Geant4 Monte Carlo simulations and writing the manuscript.\nDC, RS, DG and RM: wrote the main manuscript, contributed to scientific problem and co-supervised the project.\n\n\\noindent{\\bf Corresponding Authors:}\\\\\n$^\\dagger$ ramin1.abolfath@gmail.com \/ Ramin.Abolfath@pennmedicine.upenn.edu \\\\\n$^*$ rmohan@mdanderson.org\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn recent years experiments with strongly-interacting cold atomic gases\nhave attracted much attention.\\cite{ReviewBloch2008}  A particular\nadvantage of these systems is that their parameters can be controlled to a\nhigh degree either directly or via oscillating forces that lead to\nsynthetic gauge fields.\\cite{Creffield2011a, Dalibard2011a}  This allows a flexible\nengineering and simulation of many-body Hamiltonians.  For a theoretical\ndescription, one frequently employs the Hubbard model.  Despite its seeming\nsimplicity, it captures a great variety of condensed-matter phenomena\nranging from metallic behavior to insulators, magnetism, and\nsuperconductivity.\n\nIn the strongly interacting limit of the Hubbard model, particles occupying\nthe same lattice site can bind together, even for repulsive interactions.\nThis occurs when the onsite interaction is much larger than the tunneling \nsuch that energy conservation inhibits the decay into a\nstate with two distant particles.  In principle, both bosons\n\\cite{Valiente2008, Compagno2017} and fermions \\cite{BookEssler2005} can\nform such $N$-particle states.  While the former allow any occupation\nnumber, for fermions with spin $s$, the occupation of one site is\nrestricted to at most $2s+1$ particles. \nIn particular, two spin-1\/2 fermions may reside in a singlet spin \nconfiguration on one lattice site and, thus, form a doublon. Over the last \nyears, they have been investigated both theoretically \n\\cite{Hofmann2012, Bello2016, Bello2017a} and experimentally \n\\cite{Winkler2006, Folling2007, Strohmaier2010, Preiss2015} \nwith cold atoms in optical lattices.\n\nIn the context of solid-state based quantum information and quantum\ntechnologies, arrays of tunnel coupled quantum dots represent a recent platform\nfor similar experiments with electrons.\\cite{Puddy2015a, Zajak2016a}  In\ncomparison to optical lattices, however, these systems are way more\nsensitive to decoherence and dissipation stemming from the interaction with\nenvironmental degrees of freedom such as phonons or charge and current\nnoise.  Since environments may absorb energy, the separation of two\nelectrons in a doublon state is no longer energetically forbidden.  In this\npaper we cast some light on this issue by studying the life times of\ndoublons in a one-dimensional lattice in the presence of charge and current\nnoise, as is sketched in Fig.~\\ref{fig:setup}.  For the environment we\nemploy a Caldeira-Leggett model, \\cite{Leggett1987a, Hanggi1990a} where\ndepending on the type of noise, the bath couples locally to the onsite\nenergies or to the tunnel matrix elements.\n\n\\begin{figure}[t]\n    \\centering\\includegraphics{figure1.pdf}\n    \\caption{Tight-binding lattice occupied by two electrons.  The\n    initial state with a doubly occupied site (doublon) may decay dissipatively\n    into a single-occupancy state with lower energy.  The released energy\n    is of the order of the onsite interaction $U$ and will be absorbed by\n    heat baths representing environmental charge and current noise.\n\\label{fig:setup}\n}\n\\end{figure}\n\nIn Sec.~\\ref{sec:model}, we specify our model and sketch the derivation of\na Bloch-Redfield master equation for the dissipative dynamics.  Section\n\\ref{sec:charge} is devoted to the influence of charge noise, while the\nresults for current noise are worked out in Sec.~\\ref{sec:current}.\nBoundary effects and experimental consequences are discussed in\nSec.~\\ref{sec:discussion}, while the appendix contains details of the\nmaster equation and the averaging of decay rates.\n\n\\section{Model and master equation}\n\\label{sec:model}\n\nThe Fermi-Hubbard model considers particles on a lattice with nearest\nneighbor tunneling and onsite interaction.  For electrons, its\nHamiltonian reads\n\\begin{align}\n    H_S & = -J\\sum\\limits_{j=1}^{N-1}\\sum\\limits_{\\sigma=\\uparrow, \\downarrow}\n    \\left( c^\\dagger_{j+1\\sigma}c_{j\\sigma} + \\mathrm{H.c.}\\right)  \n+ U\\sum\\limits_{j=1}^{N} n_{j\\uparrow}n_{j\\downarrow} \\nonumber \\\\\n& \\equiv -J T + UD \\,,\n\\label{eq:hubbard}\n\\end{align}\nwith the hopping matrix element $J$ and the interaction strength $U$.  The\nfermionic operator $c^\\dagger_{j\\sigma}$ creates an electron with spin\n$\\sigma$ on site $j$, while $n_{j\\sigma}$ is the corresponding number\noperator.  For convenience, we define the hopping operator between \nsites $j$ and $j+1$, as $T_j = \\sum_\\sigma c^\\dagger_{j+1\\sigma}c_{j\\sigma} +\n\\mathrm{H.c.}$\nWhile the Hamiltonian~\\eqref{eq:hubbard} has open boundary conditions,\nwe will also study the case of periodic boundary conditions (ring\nconfiguration) by adding the corresponding term for the hopping between\nthe first and the last site.\n\nHenceforth, we focus on the case of two fermions forming a spin singlet.\nThen we work in a Hilbert space that contains two types of states,\n\\textit{single-occupancy states}\n\\begin{equation}\n\\label{single}\n \\frac{1}{\\sqrt{2}} (c^\\dagger_{i\\uparrow}c^\\dagger_{j\\downarrow}-\n c^\\dagger_{i\\downarrow}c^\\dagger_{j\\uparrow})\\ket{0} \\ , \n \\quad 1\\leq i < j \\leq N \\ , \n\\end{equation}\nand the \\textit{double-occupancy states}, known as \\textit{doublons},\n\\begin{equation}\n\\label{double}\n c^\\dagger_{j\\uparrow}c^\\dagger_{j\\downarrow}\\ket{0} \\ , \\quad j=1,\\cdots,N \\,. \n\\end{equation}\nBoth kinds of states are eigenstates of the operator $D$, which in the\nHilbert space considered is equal to the projector onto the doublon\nstates~\\eqref{double}, in the following denoted as $P_D$.\n\nWhile being different from the states in Eqs.~\\eqref{single} and\n\\eqref{double}, for sufficiently large values of $U$, the eigenstates of \n$H_S$ also discern into two groups, namely $N(N-1)\/2$ states with energies \n$|\\epsilon_n|\\lesssim 4J$ and $N$ states, with energies $|\\epsilon_n|\\approx U$.\nWe will refer to the two groups as the \\textit{low-energy subspace} \n$\\mathcal{H}_0$, and the span of the latter as the \\textit{high-energy \nsubspace} $\\mathcal{H}_1$. In the strongly-interacting regime with $U\\gg J$,\ntreating the tunneling term as a perturbation, it is possible to express the \nprojector onto the high-energy subspace $P_1$ as a power series in $J\/U$,\nsee Ref.~\\onlinecite{MacDonald1988},\n\\begin{equation}\n    P_1 = P_D - \\frac{J}{U}(T^+ + T^-) + \n    \\mathcal{O}\\left( \\frac{J^2}{U^2} \\right) \\ , \\label{eq:projector}\n\\end{equation}\nwhere ${T^+=P_DT(\\mathbb{I}-P_D)}$ and ${T^-=(\\mathbb{I}-P_D)TP_D}$\ncomprise the hopping processes that increase and decrease the double\noccupancy respectively. $\\mathbb{I}$ is the identity operator.\n\nA key ingredient to our model is the coupling to environmental degrees of\nfreedom described as $N$ independent baths of harmonic oscillators,\n\\cite{Leggett1987a,Hanggi1990a}\n${H_B=\\sum_{j,n}\\omega_n a^\\dagger_{jn} a_{jn}}$.  They couple to the\nFermi-Hubbard chain via the Hamiltonian ${H_{SB}=\\sum_j X_j \\xi_j}$,\nwhere the $X_j$ are system operators that will be specified below.\nFor ease of notation, we introduce the collective bath coordinates\n${\\xi_j=\\sum_n g_n (a^\\dagger_{jn}+a_{jn})}$.  Moreover, we assume that all\nbaths are equal and statistically independent, such that\n${\\mean{\\xi_i(t),\\xi_j(t')}= 2S(t-t')\\delta_{ij}}$.\n\nAssuming weak coupling and Markovianity, the time evolution of the \nsystem's density matrix $\\rho$, can be suitably described by a master equation \nof the form \\cite{Redfield1957a,Breuer2007}\n\\begin{align}\n\\label{BlochRedfield}\n    \\dot{\\rho} & = -i[H_S,\\rho]-\\sum_j [X_j,[Q_j,\\rho]]- \n    \\sum_j [X_j,\\{R_j,\\rho\\}] \\\\\n    & \\equiv -i[H_S,\\rho] + \\mathcal{L}[\\rho] \\ .\n    \\nonumber\n\\end{align}\nwith the operators\n\\begin{align}\n  Q_j & = \\frac{1}{\\pi}\\int_0^\\infty d\\tau\\int_0^\\infty d\\omega \n  \\mathcal{S}(\\omega)\\tilde{X_j}(-\\tau) \\cos \\omega\\tau \\ , \\label{eq:R}\\\\\n  R_j &= \\frac{-i}{\\pi}\\int_0^\\infty d\\tau\\int_0^\\infty d\\omega \n  \\mathcal{J}(\\omega)\\tilde{X_j}(-\\tau) \\sin \\omega\\tau \\ . \\label{eq:Q}\n\\end{align}\nThe tilde denotes the interaction picture with respect to the system\nHamiltonian, ${\\tilde{X}_j(-\\tau)=e^{-i H_S\\tau}X_je^{i H_S\\tau}}$, while\n${\\mathcal{J}(\\omega)=\\pi\\sum_n |g_n|^2 \\delta(\\omega- \\omega_n)}$ is the\nspectral density of the baths and\n${\\mathcal{S}(\\omega)=\\mathcal{J}(\\omega)\\coth(\\beta\\omega\/2)}$ is the\nFourier transformed of the symmetrically-ordered equilibrium\nautocorrelation function $\\mean{\\{\\xi_j(\\tau),\\xi_j(0)\\}}\/2$.\n$\\mathcal{J}(\\omega)$ and $\\mathcal{S}(\\omega)$ are independent of $j$\nsince all baths are identical. We will assume an ohmic spectral density\n${\\mathcal{J}(\\omega)=\\pi\\alpha\\omega\/2}$, where the dimensionless parameter\n$\\alpha$ characterizes the dissipation strength.\n \n\n\\section{Charge noise}\n\\label{sec:charge}\n\nFluctuations of the background charges in the substrate essentially act\nupon the charge distribution of the chain. Therefore, we model it by\ncoupling the occupation of each site to a heat bath, such that\n\\begin{equation}\nH_{SB}^Q = \\sum_{j,\\sigma} n_{j,\\sigma} \\xi_j \\,,\n\\end{equation}\nwhich means $X_j = n_j$. This fully specifies the master equation~\\eqref{BlochRedfield}.\n\nTo get a qualitative impression of the decay dynamics of a doublon, let us\nstart by discussing the time evolution of a doublon state in the strongly\ninteracting regime shown in Fig.~\\ref{fig:dynamics}.  For $\\alpha=0$,\ni.e., in the absence of dissipation, the two electrons will essentially\nremain together throughout time evolution. This is due to energy\nconservation and the fact that kinetic energy in a lattice is bounded, it\ncan be at most $2|J|$ per particle. Thus, particles forming a doublon\ncannot split, as they would not have enough kinetic energy on their own to\ncompensate for the large $U$.  However, since the doublon states are not\neigenstates of the system Hamiltonian, we observe some slight oscillations\nof the double occupancy $\\langle D\\rangle$.  Still the time average of\nthis quantity stays close to unity, see Fig.~\\ref{fig:dynamics}(a). \n\nOn the contrary, if the system is coupled to a bath, doublons will be able to \nsplit releasing energy into the environment.  Then the density operator\neventually becomes the thermal state $\\rho_\\infty\\propto e^{-\\beta H_S}$.\nDepending on the temperature and the interaction strength, the\ncorresponding asymptotic doublon occupancy $\\mean{D}_\\infty$ may still\nassume an appreciable value.\n\n\\begin{figure}\n    \\centering\\includegraphics{figure2.pdf}\n    \\caption{Time evolution of the double occupancy in a system with charge\n        noise. The initial state consists of a doublon localized in a \n        particular site of a chain with periodic boundary conditions.  \n        Parameters: $N=5$, $U=10J$ and $\\alpha=0.04$. (a) Comparison between \n        free dynamics ($\\alpha=0$) and dissipative dynamics ($\\alpha\\neq 0$). \n        Temperature is set to $k_B T=0.01J$. The green line corresponds to the \n        occupancy of the high-energy subspace for the case with $\\alpha\\neq 0$ \n        and illustrates the bound given in \\eqref{eq:bound}. (b) Decay of the \n        high-energy subspace occupancy for different temperatures ranging from  \n        $0.01J$ to $1000J$. The slope of the curves at time $t=0$ is the same \n        in all cases and coincides with the value given by \\eqref{eq:avergamma} \n        (red dashed line).} \n        \\label{fig:dynamics} \n\\end{figure} \n\n\\subsection{Numerical analysis \\label{sec:numerics}}\n\nTo gain quantitative insight, we decompose our master equation\n\\eqref{BlochRedfield} into the system eigenbasis and obtain a form\nconvenient for numerical treatment (for details, see\nAppendix~\\ref{app:masterequation}). A typical time evolution of the occupancy\n$\\langle D\\rangle$ is shown in Fig.~\\ref{fig:dynamics}(a).  It exhibits an\nalmost mono-exponential decay, such that the doublon life time $T_1$ can be\ndefined as the $1\/e$ time of the difference between initial and final\nvalue of $\\langle D\\rangle$,\n\\begin{equation}\n    \\frac{\\mean{D}_{T_1}-\\mean{D}_\\infty}{1-\\mean{D}_\\infty}=\\frac{1}{e} \\,.\n    \\label{eq:numericgamma}\n\\end{equation}\n\nThe corresponding decay rate $\\Gamma = 1\/T_1$ is shown in\nFig.~\\ref{fig:numerics_occup} as a function of the temperature for different\nvalues of the dissipation strength $\\alpha$. For small $\\alpha$ and\nintermediate temperatures, $\\Gamma$ increases with the temperature,\nreaching a maximum after which the tendency inverts.  For sufficiently\nlarge temperatures, $\\Gamma \\propto (\\alpha k_B T)^{-1}$.\n\n\\begin{figure}[tb] \n    \\centering\\includegraphics{figure3.pdf} \n    \\caption{Temperature dependence of the numerically obtained decay rate \n        for a chain with $N=5$ and periodic boundary conditions in the presence\n        of charge noise. The interaction energy is set to $U=10J$. For values \n        of the coupling strength $\\alpha\\lesssim 0.04$ we obtain approximately \n        the same curve (continuous line). Inset: Values for $\\Gamma$ on a \n        logarithmic scale demonstrating the proportionality \n        $\\propto 1\/\\alpha T$.} \n\\label{fig:numerics_occup}\n\\end{figure}\n\n\\subsection{Analytical estimate for the decay rate}\n\nAn analytical estimate for the decay rates can often be gained from the\nbehavior at the initial time $t=0$, i.e.\\ from $\\dot\\rho(0) =\n-i[H_S,\\rho_0]+ \\mathcal{L}\\rho_0$ with $\\rho_0=\\rho(0)$ being the pure\ninitial state.  In the present case, however, the calculation is hindered\nby the fast initial oscillations witnessed in Fig.~\\ref{fig:dynamics}(a).\nThese oscillations stem from the mixing of the doublon states with the\nsingle-occupancy states.  To circumvent this problem, we focus for the\npresent purpose on the occupancy of the high-energy subspace, $\\langle\nP_1\\rangle$ shown in Fig.~\\ref{fig:dynamics}(b).  It turns out that this\nquantity evolves more smoothly while it decays also on the time scale\n$T_1$.  The reason for its lack of fast oscillations is that the projector\n$P_1$ commutes with the system Hamiltonian, so that it expectation value is\ndetermined solely by dissipation.  Notice that the initial decay is\ntemperature independent, while at a later stage, the decay is strongest for\nintermediate temperatures.\n\nA formal way of understanding the similarity of the long time dynamics of\n$\\langle D\\rangle$ and $\\langle P_1\\rangle$ is provided by the estimate\n\\begin{align}\n    |\\tr{P_1\\rho}-\\tr{D\\rho}|\n    \\leq {}& \\sqrt{2}\\|\\rho\\|\\sqrt{N-\\tr{P_1P_D}} \\\\\n    \\simeq {}& 2\\sqrt{2N}J\/U \\ ,\n    \\label{eq:bound}\n\\end{align}\nwhere the first lines follows from the Cauchy-Schwarz inequality for the\ninner product of operators, $(A,B)=\\tr{A^\\dagger B}$,\nwhile the second line stems from the\nperturbative expansion of $P_1$ given by Eq.~\\eqref{eq:projector}.  The\nresult implies that when neglecting corrections of the order of $J\/U$, we may\ndetermine $T_1$ and $\\Gamma$ from either quantity.  Nevertheless it is\ninstructive to analytically evaluate $\\Gamma$ for the decay of both\n$\\langle D\\rangle$ and $\\langle P_1\\rangle$.\n\nFollowing our hypothesis of a mono-exponential decay, we expect\n\\begin{equation}\n    \\label{eq:P1decay}\n    \\mean{P_1}\\simeq \\Delta e^{-\\Gamma t} + \\mean{P_1}_\\infty \\ ,\n\\end{equation} \ntherefore,\n\\begin{equation}\n    \\Gamma\\simeq - \\frac{1}{\\Delta}\n    \\left. \\frac{d \\mean{P_1}}{d t}\\right|_{t=0}=\n    -\\frac{\\tr{P_1\\mathcal{L}[\\rho_0]}}{\\mean{P_1}_0-\\mean{P_1}_\\infty} \\ .\n    \\label{eq:gamma}\n\\end{equation}\nThis expression still depends slightly on the specific choice of the\ninitial doublon state, in particular for open boundary conditions (see\nSec.~\\ref{sec:boundary}, below).  To obtain a more global picture, we\nconsider an average over all doublon states, which can be performed\nanalytically.\\cite{Storcz2005a}  From Eq.~\\eqref{eq:gamma}, we find the\naverage decay rate\n\\begin{multline}\n    \\overline{\\Gamma}=\\frac{1}{N\\Delta}\\sum_j \\tr{P_D[Q_j,[X_j,P_1]]} \\\\\n    -\\tr{P_D\\{R_j,[X_j,P_1]\\}} \\ .\n    \\label{eq:avergamma}\n\\end{multline}\nFor details of the averaging procedure, see Appendix~\\ref{app:average}.\n\nFor a further simplification, we have to evaluate the expressions\n\\eqref{eq:R} and \\eqref{eq:Q} which is possible by approximating the\ninteraction picture coupling operator as $\\tilde{X}_j(-\\tau)\\simeq X_j\n-i\\tau[H_S,X_j]$.  This is justified as long as the decay of the\nenvironmental excitations is much faster than the typical system evolution,\ni.e., in the high-temperature regime (HT).  Inserting our approximation for\n$\\tilde{X}_j$ and neglecting the imaginary part of the integrals, we arrive at\n\\begin{align} \n    Q_j & \\simeq \\frac{1}{2}\\lim_{\\omega\\rightarrow 0^+} \n    \\mathcal{S}(\\omega) X_j = \\frac{\\pi}{2}\\alpha k_B T X_j\\ , \\\\ R_j & \\simeq -\n    \\frac{1}{2}\\lim_{\\omega\\rightarrow 0^+} \\mathcal{J}'(\\omega) [H_S,X_j] = \n    \\frac{\\pi}{4}\\alpha [H_S,X_j] \\ .  \n\\end{align} \nWith these expressions, Eq.~\\eqref{eq:avergamma} results in a temperature\nindependent decay rate.  Notice that any temperature dependence stems from\nthe $Q_j$ in the first term of Eq.~\\eqref{eq:avergamma} which vanishes in\nthe present case.  While this observation agrees with the numerical findings in\nFig.~\\ref{fig:dynamics} for very short times, it does not reflect the\ntemperature dependent decay of $\\langle P_1\\rangle$ at the more relevant\nintermediate stage.\n\nThis particular behavior hints at the mechanism of the bath-induced doublon\ndecay.  Let us notice that the coupling to charge noise, $X_j=n_j$,\ncommutes with $D$.  Therefore, the initial state is robust against the\ninfluence of the bath.  Only after mixing with the single-occupancy\nstates due to the coherent dynamics, the system is no longer in an\neigenstate of the $n_j$, such that decoherence and dissipation become\nactive.  Thus, it is the combined action of the system's unitary evolution\nand the effect of the environment which leads to the doublon decay.\n\nAn improved estimate of the decay rate, can be calculated by averaging the\ntransition rate of states from the high-energy subspace to the low-energy\nsubspace.  Let us first focus on regime $k_BT\\gtrsim U$ in which we can\nevaluate the operators $Q_j$ in the high-temperature limit.  Then the\naverage rate can be computed using expression \\eqref{eq:avergamma} and\nreplacing $P_D$ by $P_1$, see Appendix \\ref{app:masterequation}. With the\nperturbative expansion of $P_1$ in Eq.~\\eqref{eq:projector} we obtain to\nleading order in $J\/U$ the averaged rate\n\\begin{equation} \n    \\overline{\\Gamma}_\\mathrm{HT} \\simeq \n    \\frac{4\\pi\\alpha J^2}{U^2 \\Delta}\\left(2k_B T + U \\right) \\,,\n    \\label{eq:GOHT}\n\\end{equation}\nvalid for periodic boundary conditions.  For open boundary conditions, the\nrate acquires an additional factor $(N-1)\/N$.  Notice that we have\nneglected back transitions via thermal excitations from singly occupied\nstates to doublon states.  We will see that this leads to some smaller\ndeviations when the temperature becomes extremely large.  Nevertheless, we\nrefer to this case as the high-temperature limit.\n\nIn the opposite limit, for temperatures ${k_B T < U}$, the decay rate\nsaturates at a constant value. To evaluate $\\overline{\\Gamma}$ in this limit, it\nwould be necessary to find an expression for $\\tilde{X}_j(-\\tau)$ dealing\nproperly with the $\\tau$-dependence for evaluating the noise kernel, a\nformidable task that may lead to rather involved expressions.\nNevertheless, one can make some progress by considering the transition of\none initial doublon to one particular single-occupancy state.  This\ncorresponds to approximating our two-particle lattice model by the\ndissipative two-level system for which the decay rates in the Ohmic case\ncan be taken from the literature,\\cite{Weiss1989a, Makhlin2001b} see\nAppendix \\ref{app:TLS}.  Relating $J$ to the tunnel matrix element of the\ntwo-level system and $U$ to the detuning, we obtain from\nEq.~\\eqref{app:Gammaii} the temperature-independent expression\n\\begin{equation}\n    \\overline{\\Gamma}_\\text{LT}  \\simeq \\frac{8\\pi\\alpha J^2}{U \\Delta} \\ ,\n\\label{eq:GOLT}\n\\end{equation}\nwhich formally corresponds to Eq.~\\eqref{eq:GOHT} with the temperature set\nto $k_B T=U\/2$.\n\nFigure~\\ref{fig:analyticsOccup} provides a comparison of these analytical\nfindings with numerical results.  The data in panel (a) reveal that the\ntransition between the low-temperature regime and the high-temperature\nregime is rather sharp and occurs at $U\\approx k_BT$.  Panel (b) shows\n$\\Gamma$ as a function of the temperature.  For low temperatures, the\nnumerical values saturate at $\\overline{\\Gamma}_\\text{LT}$ obtained from the\napproximate mapping to a two-level system.  For high temperatures, the\nanalytical prediction $\\overline{\\Gamma}_\\text{HT}$ seems slightly too \nlarge.  The discrepancy stems from neglecting thermal excitations, as \nmentioned above.\n\n\\begin{figure}[tb]\n    \\centering\\includegraphics{figure4.pdf}\n    \\caption{Comparison between the numerically computed decay rate and the \n        analytic formulas \\eqref{eq:GOHT} and \\eqref{eq:GOLT} for a chain\n        with $N=5$ sites and periodic boundary conditions in the case of \n        charge noise.  The dissipation strength is $\\alpha=0.02$.\n        (a) Dependence on the interaction strength for a fixed temperature $k_B \n        T=20J$. (b) Dependence on the temperature for a fixed interaction \n        strength $U=20J$.}\n    \\label{fig:analyticsOccup}\n\\end{figure}\n\n\\section{Current noise}\n\\label{sec:current}\n\nFluctuating background currents mainly couple to the tunnel matrix elements\nof the system.  Then the system-bath interaction is given by setting \n$X_j=T_j$ and reads\n\\begin{equation}\nH_{SB}^I = \\sum_{j,\\sigma} (c^\\dagger_{j+1\\sigma}c_{j\\sigma} \n+ c^\\dagger_{j\\sigma} c_{j+1\\sigma} ) \\xi_j \\,.\n\\end{equation}\nDepending on the boundary conditions, the sum may include the term with\n$j=N$.  The main qualitative difference of this choice is that in contrast\nto charge noise, $H_{SB}^I$ does not commute with the projector to the\ndoublon subspace and, thus, generally ${\\tr{D\\mathcal{L}[\\rho]}\\neq 0}$.\nThis enables a direct dissipative decay without the detour via an admixture\nof single-occupancy states to the doublon states.  As a consequence,\nfor the same value of the dimensionless dissipation parameter $\\alpha$, the\ndecay may be much faster.  Also the temperature dependence of the decay\nchanges significantly, as can bee seen in Fig.~\\ref{fig:numerics_hopp}.\nWhile $\\overline{\\Gamma}$ is still proportional to $\\alpha$, it now grows\nmonotonically with the temperature.\n\n\\begin{figure}\n    \\centering\\includegraphics{figure5.pdf}\n    \\caption{Average decay rate of the doublon states under the influence\n    of current noise for various dissipation strengths as a function of\n    the temperature.  The chain consists of $N=5$ sites with periodic boundary\n    conditions, while the interaction is $U=10J$.}\n    \\label{fig:numerics_hopp}\n\\end{figure}\n\nAs in the last section, we proceed by calculating analytical estimates for\nthe decay rates.  However, since the time evolution is no longer\nmono-exponential (not shown), we no longer start from the\nansatz~\\eqref{eq:avergamma}, but estimate the rate from the slope of the\noccupancy $\\langle P_1\\rangle$ at initial time,\n\\begin{equation}\n    \\Gamma \\simeq - \\left. \\frac{d \\mean{P_1}}{d t} \\right|_{t=0} =\n        - \\tr{P_1\\mathcal{L}[\\rho_0]} \\ .\n    \\label{eq:decay_hopp}\n\\end{equation}\nWe again perform the average over all doublon states for $\\rho_0$\nin the limits of high and low temperatures.  For periodic boundary\nconditions, we obtain to lowest order in $J\/U$ the high and low temperature\nrates \n\\begin{align}\n    \\overline{\\Gamma}_\\mathrm{HT} & = 2\\pi\\alpha \\left(2k_B T+U \\right) \\,,\n    \\label{eq:GTHT}  \\\\\n    \\overline{\\Gamma}_\\text{LT} & = 4\\pi\\alpha U \\,,\n    \\label{eq:GTLT}\n\\end{align}\nrespectively,\nwhile open boundary conditions lead to the same expressions but with a\ncorrection factor $(N-1)\/N$.\nIn Fig.~\\ref{fig:analyricsHopp}, we compare these results with the\nnumerically evaluated ones as a function of the interaction\n[Fig.~\\ref{fig:analyricsHopp}(a)] and\nthe temperature [Fig.~\\ref{fig:analyricsHopp}(b)].  Both show that the\nanalytical approach correctly predicts the (almost) linear behavior at\nlarge values of $U$ and $k_BT$, as well as the saturation for small values.\nHowever, the approximation slightly overestimates the influence of the\nbath.\n\n\\begin{figure}\n    \\centering\\includegraphics{figure6.pdf}\n    \\caption{Numerically obtained decay rate in comparison with the\n        approximations~\\eqref{eq:decay_hopp}, \\eqref{eq:GTHT} and \n        \\eqref{eq:GTLT} for a chain with $N=5$ sites and periodic boundary \n        conditions in the case of current noise with strength $\\alpha=0.02$.  \n        The results are plotted as a function of (a) the interaction and the \n        temperature $k_B T=20J$ and (b) for a fixed interaction $U=20J$ as a \n        function of the temperature.}\n    \\label{fig:analyricsHopp}\n\\end{figure}\n\nWhile the rates reflect the decay at short times, it is worthwhile to\ncomment on the long time behavior under the influence of current\nnoise.  For open chains as well as for closed chains with an even\nnumber of sites, it is not ergodic as the long-time solution is not unique.\nThe reason for this is the existence of a doublon state\n${\\ket{\\Phi}=\\frac{1}{\\sqrt{N}}\\sum_{j=1}^N (-1)^j\nc^\\dagger_{j\\uparrow}c^\\dagger_{j\\downarrow}\\ket{0}}$ which is an\neigenstate of $H_S$ without any admixture of single-occupancy states.\nSince $T_j\\ket{\\Phi}=0$ for all sites $j$, current noise may affect\nthe phase of ${\\ket{\\Phi}}$, but cannot induce its dissipative decay.\nFor a closed chain with an odd number of sites, by contrast, the alternating\nphase of the coefficients of $\\ket{\\Phi}$ is incompatible with periodic\nboundary conditions, unless a flux threatens the ring.  As a consequence,\nthe chain eventually resides in the thermal state $\\propto\\exp(-\\beta\nH_S)$.  The difference is manifest in the final value of the doublon\noccupancy at low temperatures.  For closed chains with an odd number of\nsites, it will fully decay, while in the other cases, the population of\n$\\ket{\\Phi}$ will survive.\n\n\\section{Discussion}\n\\label{sec:discussion}\n\\subsection{Dimension and boundary effects}\n\\label{sec:boundary}\n\n\\begin{figure}[tb]\n    \\centering\\includegraphics{figure7.pdf}\n    \\caption{Decay rates of the double occupancy for a chain with $N=5$ sites \n        with open boundary as a function of the initial location of the doublon.\n\tThe values for $\\Gamma$ are taken as the inverse of the $T_1$ time\n\tobtained from a numerical propagation of the master equation.\n        The red dashed line marks the value for closed boundary conditions.\n\tThe other parameters are $U=20J$, $\\alpha=0.01$, $k_B T=5J$.}\n    \\label{fig:site_dependence}\n\\end{figure}\n\nSo far, we have considered decay rates as the averages of all possible\ninitial doublon or high-energy states.  While this is sufficient for a\ngeneric estimate of the life times, it ignores the fact that the behavior\nof individual states may differ significantly, in particular when the\ninitial state is located at a boundary, which reduces the number of\naccessible decay channels.  In Fig.~\\ref{fig:site_dependence} we present\nthe decay rates for doublons as a function of the initial site.  It reveals\nthat in comparison to states at the center, an initial localization at the\nfirst or last site, may double the life time for charge noise and enhance by\nit by a factor three for current noise.  The dashed lines in these plots\nmarks the value for periodic boundary conditions, for which the value is\npractically the same as for a states in the center.\n\nThis knowledge about the role of boundaries and nearest neighbors provides\nsome hint on the doublon life time in higher-dimensional lattices.  Let us\nnotice that Let the decay rates \\eqref{eq:gamma} and \\eqref{eq:decay_hopp} \ncontain one term for each single-occupancy state that is directly tunnel \ncoupled to the initial site.  Assuming that all terms are of the same order, \nwe expect that $\\overline{\\Gamma}$ is by and large proportional to the \ncoordination number of the lattice sites.  Therefore the life time should \ndecrease only moderately with the dimension, roughly as \n$T_1 = \\Gamma^{-1} \\sim 2^{-D}$.  From the data in \nFig.~\\ref{fig:site_dependence}(b), we can appreciate that for current\nnoise, the difference between center and border is even larger.  Thus,\nincreasing dimensionality should have a slightly larger impact on the\ndoublon life times.\n\n\n\\subsection{Experimental implications}\n\\label{sec:experiment}\n\nA current experimental trend is the fabrication of larger arrays of quantum\ndots,\\cite{Puddy2015a, Zajak2016a} which triggered our question on the\nfeasibility of doublon experiments in solid-state systems.  While the size\nof these arrays would be sufficient for this purpose, their dissipative\nparameters are not yet fully known.  For an estimate we therefore consider\nthe values for GaAs\/InGaAs quantum dots which have been determined recently\nvia Landau-Zener interference.\\cite{Forster2014, onalpha}\nNotice, that for the strength of the current noise, only an upper\nbound has been reported.  We nevertheless use this value, but keep in mind\nthat it leads to a conservative estimate.  In contrast to the former\nsections, we now compute the decay for the simultaneous action of charge\nnoise and current noise.\n\n\\begin{figure}\n    \\centering\\includegraphics{figure8.pdf}\n    \\caption{(a) Spatially resolved doublon dynamics in a chain with $N=5$ \n    sites and open boundary conditions for the dissipative parameters\n    determined in Ref.~\\onlinecite{Forster2014}, i.e., for the dissipation\n    strengths\\cite{onalpha} $\\alpha_Q=3\\times 10^{-4}$ and $\\alpha_I=5\\times 10^{-6}$, \n    the tunnel coupling $J=13\\,\\mu\\mathrm{eV}$, interaction $U=1.3$\\,meV, and \n    temperature $T=10$\\,mK.  (b) Corresponding decay of the double occupancy \n    (solid line) and state purity (dashed).}\n    \\label{fig:realistic_dynamics}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\\includegraphics{figure9.pdf}\n    \\caption{Doublon life time as a function of the temperature for\n\tdifferent interaction strengths.  The other parameters are as in\n\tFig.~\\ref{fig:realistic_dynamics}. Vertical dashed lines mark the \n    temperature corresponding to the tunneling energy and the Hubbard \n    interaction energy.  \n    Inset: $T_1$ time for the optimized value of the interaction,\n\t$U =10J =130\\,\\mu$eV and a current noise with\n\t$\\alpha_I = 2\\times10^{-6}$.  The latter is smaller than the value in\n\tFig.~\\ref{fig:realistic_dynamics}, but still realistic.\n\t}\n    \\label{fig:realistic_experiment}\n\\end{figure}\n\nFigure~\\ref{fig:realistic_dynamics}(a) shows the dissipative time evolution\nfor a doublon initially localized at the center of a chain with 5 sites.\nThe dynamics exhibits a few coherent oscillations in which the doublon\nevolves into a superposition of the kind $|2,0,0\\rangle + |0,0,2\\rangle$,\nwhich represents an example of a NOON state.\\cite{Lee2002a}\nEach component propagates to one end of the chain, where is it reflected\nsuch that subsequently the initial states revives.  In\nFig.~\\ref{fig:realistic_dynamics}(b), we depict the evolution of the\ncorresponding doublon occupancy and the purity.  Both quantities decay\nrather smoothly.  This agrees to the finding found in\nSec.~\\ref{sec:current} for pure current noise which obviously dominates.\nIt is also consistent with the values for the respective analytical decay\nrates in the low-temperature limit.  Figure~\\ref{fig:realistic_experiment}\nshows the $T_1$ times for two different interaction strengths.  It reveals\nthat for low temperatures $T\\lesssim J\/k_BT$, the life time is essentially\nconstant, while for larger temperatures, it decreases moderately until\n$k_BT$ comes close to the interaction $U$.  For higher temperatures,\n$\\Gamma$ starts to grow linearly.  On a quantitative level, we expect life\ntimes of the order $T_1\\sim 5$\\,ns already for a moderately low\ntemperatures $T\\lesssim 100$\\,mK.  Since we employed the value of the upper\nbound for the current noise, the life time might be even larger.\n\nConsidering the analytical estimates for the decay rates at low\ntemperatures, Eqs.~\\eqref{eq:GOLT} and \\eqref{eq:GTLT}, separately,\nlets us conclude that for smaller values of $U$, current noise becomes\nless important, while the impact of charge noise grows.  Therefore, a\nstrategy for reaching larger $T_1$ times is to\ndesign quantum dots arrays with smaller onsite interaction, such\nthat the ratio $U\/J$ becomes more favorable.  The largest $T_1$ is expected\nin the case in which both low-temperature decay rates are equal,\n$\\overline{\\Gamma}_\\text{LT,charge} = \\overline{\\Gamma}_\\text{LT,current}$, \nwhich for the present experimental parameters is found at $U\\sim 10J$ (while \nour data is for $U\\sim 100 J$).  This implies that in an optimized device, \nthe doublon life times could be larger by one order of magnitude to reach \nvalues of $T_1\\sim 50$\\,ns, which is corroborated by the data in the inset \nof Fig.~\\ref{fig:realistic_experiment}.\n\n\n\\section{Conclusions}\n\nWe have investigated the life times of double-occupancy states or doublons\nin a one-dimensional Hubbard model under the influence of dissipating\nenvironments.  While in optical lattices, the resulting dissipative decay\nmay be of minor influence, for quantum dot arrays, it will be a limiting\nfactor.\n\nWe have considered two different couplings between the system and its\nenvironment, which physically correspond to the impact of charge noise and\ncurrent noise, respectively.  Within a Bloch-Redfield formalism, this model\ncan be treated with a master equation, which allows one to\nnumerically determine the life times from the time evolution of the reduced\ndensity operator.  Moreover, it provides analytical estimates for the\ninitial decay rates.  It turned out that the striking difference between\nthe two couplings is that the impact of charge noise decreases with the\ninteraction, while current noise becomes increasingly relevant.\n\nFor present quantum dots, the doublon life time is expected to be of the\norder 5\\,ns, which would limit the coherent dynamics to only a few\nperiods.  However, our analytical estimates suggest that for quantum dot\narrays with smaller onsite interaction, an extension by one order of\nmagnitude should be feasible.  Thus, the recent trend towards arrays with\never more coherently coupled quantum dots will allow the experimental\nrealization of effects that so far have been measured only in optical\nlattices.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nShort-pulse, high-intensity laser and beam plasma interaction is an active and robust research area. It involves relativistic, nonlinear and ultrafast plasma physics. It is also a critical topic to the field of plasma based acceleration (PBA). When an intense laser or particle beam propagates through a plasma, it excites a relativistic plasma wave (wakefield). These wakefields support extremely high and coherent accelerating fields which can be more than three orders of magnitude in excess of those in conventional accelerators. The field of PBA has seen rapid experimental progress with many milestones being achieved, including electron acceleration driven by an electron \\cite{blumenfeld2007,litos2014}, laser \\cite{faure2004,geddes2004,mangles2004,gonsalves2011,wang2013,steinke2016,guenot2017} or proton beam \\cite{adli2018}, positron acceleration \\cite{corde2015} and PBA-based radiation generation \\cite{kneip2010,cipiccia2011,nie2018}.\n\nThe rapid progress in experiments has been greatly facilitated by start-to-end simulations using high fidelity particle based methods. The nonlinear aspects of the physics requires the use of fully kinetic tools and the particle-in-cell (PIC) method has proven indispensable. The fully explicit relativistic electromagnetic (EM) PIC method has been used very successfully \\cite{dawson1983,birdsall2005,hockney1988}. In this method, individual macro-particles described by Lagrangian coordinates are tracked in continuous phase space as finite size particles (positions and momentum can have continuous values), and then moments of the distribution (\\emph{e.g.} charge and current density) are deposited onto  stationary mesh\/grid points. The electromagnetic fields are advanced forward in time on the grid points using a discretized version of Maxwell's equations. The new fields are then interpolated to the particles positions to push the particles to new momenta and positions using the relativistic equations of motion. This sequence is repeated for a desired number of time steps.\n\nIn most fully explicit PIC codes, a finite-difference time-domain (FDTD) method is used to advance the time-dependent Maxwell's equations and the differential operators are approximated through a finite difference representation (usually accurate to second order of the cell size).  However, to prevent a numerical instability, the time step is constrained by the Courant-Friedrichs-Lewy (CFL) condition which fundamentally couples the spatial and temporal resolution. Roughly speaking, the time step size needs to be less than the smallest cell size which in turn is determined by the smallest physical scale of interest. A second order representation of the time derivative is then used to push the particles. When modeling short-pulse laser and beam-plasma interactions, the moving window technique \\cite{decker1994} is always used. In this technique, only a finite window that keeps up with the laser is simulated. New cells and fresh plasma are added to the front, while cells and plasma are dropped off the back. This works because no information and physics that has been dropped can effect the plasma in front of it during the simulation.\n\nToday's supercomputers are capable of providing $\\sim10^{16}$ to $\\sim10^{17}$ floating point operations per second \\cite{top500}. To utilize such computers the algorithm needs data structures that permit many thousands of cores to simultaneously push particles. Effective utilization of such computers has enabled full-scale 3D modeling of intense laser or relativistic charged particles interaction with plasma in some cases. However, even with today's computers and PIC software, it is still not possible to carry out start-to-end simulations of every experiment or proposed concept in full 3D using standard PIC codes. In addition, explicit EM PIC codes can be susceptible to numerical issues including the numerical Cerenkov instability (NCI) \\cite{xu2013} and errors to the fields that surround relativistically moving charges \\cite{xu2019}. Furthermore, beam loading studies can require very fine resolution in the transverse direction when ion collapse within a particle beam arises \\cite{rosenzweig2005,gholizadeh2010,an2017}.\n\nVarious methods have been developed to more efficiently model the short-pulse laser and beam-plasma interactions in PBA. These include the boosted frame technique \\cite{vay2007}, the quasi-static approximation \\cite{mora1997,whittum1997,lotov2003,huang2006,an2013,mehrling2014}, and an azimuthal mode expansion method \\cite{lifschitz2009,davidson2015,lehe2016}. The first two are based on the assumption that all relevant waves move forward with velocities near the speed of light, \\emph{e.g.}, no radiation propagates backwards. Some of these methods can be combined \\cite{yu2016}.\n\nThe quasi-static approximation (QSA) was first presented as an analytical tool for studying short-pulse laser interactions \\cite{sprangle1990a,sprangle1990b}. The applicability of QSA originates from the disparity in time\/length scales between how the laser or particle beam evolves and the period\/wavelength of the plasma wake (the plasma response). In the QSA the plasma response is calculated by assuming that the shape of the laser or particle beam (envelope and energy or frequency) is static and the resulting fields from the plasma response are then used to advance the laser or beam forward using a very large time step. It was not until the work of Antonsen and Mora that a PIC algorithm was developed to utilized the QSA. They showed how to push a slice of plasma through a static laser (or move a static laser past a slice of plasma). Their code WAKE \\cite{mora1997} is two dimensional (2D) using r-z coordinates and it can model both lasers and particle beams. Whittam also independently developed a QSA PIC code for modeling particle beam-plasma interaction \\cite{whittum1997}. In this implementation, it was assumed that plasma particles motion is approximated to be non-relativistic so plasma particles do not move in the beam propagation direction. LCODE \\cite{lotov2003} is another 2D r-z PIC code based on the QSA that only models particle beam drivers. QuickPIC \\cite{huang2006,an2013} was the first fully 3D QSA based code and it is fully parallelized including a pipelining parallel algorithm \\cite{feng2009,an2013}. HiPACE \\cite{mehrling2014} is a more recent 3D PIC code based on the QSA. QuickPIC can efficiently simulate both laser pulses and particle beams. It can achieve $10^2$ to $10^4$ speedup without loss of accuracy when compared against fully explicit PIC codes (\\emph{e.g.} OSIRIS \\cite{fonseca2002}) for relevant problems.\n\nAnother recently developed method to enhance the computational efficiency applies the azimuthal Fourier decomposition \\cite{lifschitz2009}. In this method, all the field components and current (and charge) density are expanded into a Fourier series in $\\phi$ in the azimuthal direction (into azimuthal harmonics denoted by $m$); and the series can be truncated at a value of $m$ determined by the degree of asymmetry for the problem of interest. This algorithm can be viewed as a hybrid method where the PIC algorithm is used in r-z grid and a gridless method is used in $\\phi$ and it is sometimes referred to as quasi-3D. By using this algorithm, the problem reduces to solving the complex amplitude (coefficients for Fourier series) for each harmonic on a 2D grid. The complex amplitude, as a function of $r$ and $z$, is updated only at a cost similar to an r-z 2D code. Therefore, if only a few harmonics are kept the algorithm is very efficient. For example, a linearly polarized laser with a symmetric spot size can be described by only the first harmonic. In addition to the much lower cost for advancing fields, much fewer macro-particles are needed for high fidelity. It has been found that speedups of more than two orders of magnitude over a full 3D code are possible.\n\nThe quasi-3D method has been implemented into some fully explicit 3D PIC codes \\cite{lifschitz2009,davidson2015,lehe2016} and used to study laser \\cite{nie2018,ferri2018} and beam \\cite{corde2015} plasma interactions. It also been successfully combined with the boosted frame method \\cite{yu2016}. However, the azimuthal mode expansion has not been combined with the QSA method or implemented into a quasi-static PIC code. If the quasi-3D technique can be successfully combined with the QSA then dramatic speedups will be possible for problems which are nearly azimuthally symmetric. Such a code will greatly extend the scope of PBA research problems that can be studied numerically.\n\nIn this paper, we describe a new code that combines a QSA 3D PIC code with an azimuthal Fourier decomposition, called QPAD (QuickPIC with azimuthal decomposition). The code contains similar procedures and workflow as the 3D quasi-static PIC code QuickPIC, but with the entirely new framework to utilize the azimuthal decomposition. While QuickPIC uses FFT based Poisson solvers to update the fields in each 2D slice of plasma, QPAD computes the fields by means of finite-difference (FD) solvers using the cyclic reduction method \\cite{press2010}.\nWithout loss of accuracy, the code achieves dramatic speedup over fully 3D QuickPIC for a wide range of beam-driven plasma acceleration problems. QPAD currently only supports particle beam drivers.\n\nThe paper is organized as follows. In Section \\ref{sec:theory}, we derive the governing equations for the complex amplitudes for each harmonic of the relevant fields under the QSA. Section \\ref{sec:algorithm} provides details of how the algorithm is implemented. First, the entire numerical workflow that utilizes the three-layer nested loop is described. Next, we introduce the FD implementation of Poisson solvers for each harmonic amplitude and the boundary conditions associated with them. This is followed by a description of the deposition schemes for the source terms for each harmonic needed for the field equations. In Section \\ref{sec:simulation}, we compare simulation results between QPAD, QuickPIC and OSIRIS for the beam-driven wakefields and for the hosing instability. A qualitative discussion on the computational speedup is presented in Section \\ref{sec:performance}. Lastly, we give a conclusion and a discussion for future work.\n\n\\section{Azimuthal decomposition of electromagnetic fields under QSA}\n\\label{sec:theory}\n\nIn this section, we describe the physics arguments behind QPAD including a detailed description of the field equations. As mentioned above, the fundamental differences between a fully explicit 3D PIC code and QPAD are twofold. First, QPAD is a code based on the QSA which separates the time scale of the plasma evolution from that of a drive laser pulse or high-energy particle beam that moves at the speed of light $c$. The assumptions behind the QSA are based on the fact that the characteristic evolution time for a laser driver or a particle beam driver\nis several orders of magnitude larger than the plasma oscillation period, $2\\pi\/\\omega_p$ where $\\omega_p$ is the plasma frequency. In a quasi-static code, a Galilean spatial transformation is made from $(x,y,z,t)$ (where the laser or beam moves in the $\\hat{z}$ direction) to a co-moving frame described by coordinates $(x,y,\\xi=ct-z,s=z)$. All the Lagrangian quantities associated with the plasma particles evolve on the fast-varying time-like variable, $\\xi$, while those of the beam particles moving close to $c$ evolve on the slow-varying \"time\" scale, $s$. The transformations $\\partial_t=c\\partial_\\xi,\\ \\partial_z=\\partial_s-\\partial_\\xi$ are applied for all the Eulerian quantities, \\emph{i.e.}, fields, charge density and current density. The QSA assumes that $s$ is the slow-varying time-like scale, \\emph{i.e.} $\\partial_s\\ll\\partial_\\xi$, so that all the terms associated with $\\partial_s$ are small and can thus be neglected.\n\nFor remainder of the paper, we use normalized units for all the physical quantities; time, length and mass are normalized to $\\omega_p^{-1}$, $c\/\\omega_p$ and the electron rest mass $m_e$. The normalized Maxwell's equations under the QSA can thus be written as\n\\begin{align}\n\\label{eq:maxwell_1}\n\\nabla_\\perp\\times \\bm{E}_\\perp &=-\\PP{B_z}{\\xi}\\bm{e}_z, \\\\\n\\label{eq:maxwell_2}\n\\nabla_\\perp\\times E_z\\bm{e}_z &= -\\PP{}{\\xi}(\\bm{B}_\\perp-\\bm{e}_z\\times\\bm{E}_\\perp), \\\\\n\\label{eq:maxwell_3}\n\\nabla_\\perp\\times \\bm{B}_\\perp - J_z\\bm{e}_z &= \\PP{E_z}{\\xi}\\bm{e}_z, \\\\\n\\label{eq:maxwell_4}\n\\nabla_\\perp\\times B_z\\bm{e}_z - \\bm{J}_\\perp &= \\PP{}{\\xi}(\\bm{E}_\\perp+\\bm{e}_z\\times\\bm{B}_\\perp), \\\\\n\\label{eq:maxwell_5}\n\\nabla_\\perp\\cdot\\bm{E}_\\perp-\\rho &= \\PP{E_z}{\\xi}, \\\\\n\\label{eq:maxwell_6}\n\\nabla_\\perp\\cdot\\bm{B}_\\perp &= \\PP{B_z}{\\xi},\n\\end{align}\nwhere $\\nabla_\\perp=\\bm{e}_x\\partial_x+\\bm{e}_y\\partial_y$. For convenience, the equations for the transverse and longitudinal fields are written separately. In this context, transverse and longitudinal are defined with respect to the direction of laser or particle beam propagation and not to the direction of the wavenumber of the fields. Taking linear combinations of Eqs. (\\ref{eq:maxwell_1}), (\\ref{eq:maxwell_3}), (\\ref{eq:maxwell_5}) and (\\ref{eq:maxwell_6}) leads to equations for the divergence and curl of the transverse force, $\\bm{E}_\\perp+\\bm{e}_z\\times\\bm{B}_\\perp$, on a particle moving at the speed of light along $\\hat{z}$,\n\\begin{align}\n\\nabla_\\perp\\times(\\bm{E}_\\perp+\\bm{e}_z\\times\\bm{B}_\\perp)&=\\bm{0}, \\nonumber \\\\\n\\nabla_\\perp\\cdot(\\bm{E}_\\perp+\\bm{e}_z\\times\\bm{B}_\\perp)&=\\rho-J_z. \\nonumber\n\\end{align}\nWe can infer from the first of these equations  that the transverse force can be described by the transverse gradient of a scalar potential which we call $\\psi$,\n\\begin{equation}\n\\label{eq:def_psi}\n\\bm{E}_\\perp+\\bm{e}_z\\times\\bm{B}_\\perp=-\\nabla_\\perp\\psi.\n\\end{equation}\nSubstituting this relationship into the second equation, leads to a Poisson equation for the pseudo potential $\\psi$,\n\\begin{equation}\n\\label{eq:poisson_psi}\n-\\nabla_\\perp^2\\psi=(\\rho-J_z).\n\\end{equation}\nBy taking $\\bm{e}_z \\times$ on both sides of Eq. (\\ref{eq:maxwell_2}) and using the relation (\\ref{eq:def_psi}), it can be inferred that $E_z=\\frac{\\partial\\psi}{\\partial\\xi}$. This relationship also follows directly from the definition, $E_z=-\\frac{\\partial \\varphi}{\\partial z}-\\frac{\\partial A_z}{\\partial t}$, and the QSA, where $\\varphi$ and $A_z$ are the scalar potential and the $\\hat{z}$-component of the vector potential.\n\nThe transverse force $\\bm{E}_\\perp+\\bm{e}_z\\times\\bm{B}_\\perp$ in Eq. (\\ref{eq:maxwell_4}) and the quantity $\\bm{B}_\\perp-\\bm{e}_z\\times\\bm{E}_\\perp$ in Eq. (\\ref{eq:maxwell_2}) are not independent. Therefore, the quasi-static form of Maxwell's equations given above cannot be used to advance the fields forward in time, \\emph{i.e.}, $\\xi$, using the FDTD methods as is done in fully explicit PIC codes. Therefore, in QuickPIC, a set of Poisson-like equations are employed to directly solve the fields,\n\\begin{align}\n\\label{eq:poisson_bxy}\n\\nabla_\\perp^2\\bm{B}_\\perp&=\\bm{e}_z\\times\\left(\\PP{\\bm{J}_\\perp}{\\xi}+\\nabla_\\perp J_z\\right), \\\\\n\\label{eq:poisson_bz}\n\\nabla_\\perp^2 B_z&=-\\bm{e}_z\\cdot(\\nabla_\\perp\\times\\bm{J}_\\perp), \\\\\n\\label{eq:poisson_ez}\n\\nabla_\\perp^2 E_z&=\\nabla_\\perp\\cdot\\bm{J}_\\perp.\n\\end{align}\nwhich can be derived by applying the QSA to the wave equations for $\\bm{E}$ and $\\bm{B}$. After obtaining $\\bm{B}_\\perp$ from Eq. (\\ref{eq:poisson_bxy}) and $\\psi$ from Eq. (\\ref{eq:poisson_psi}), we can calculate $\\bm{E}_\\perp$ by subtracting $\\bm{e}_z\\times\\bm{B}_\\perp$ from $-\\nabla_\\perp\\psi$. Although it is not directly used in QuickPIC, for completeness we write out the Poisson-like equation for $\\bm{E}_\\perp$,\n\\begin{equation}\n\\label{eq:poisson_exy}\n\\nabla_\\perp^2\\bm{E}_\\perp=\\nabla_\\perp\\rho+\\PP{\\bm{J}_\\perp}{\\xi}.\n\\end{equation}\n\nWe next expand the electromagnetic fields, charge density and current density in cylindrical coordinates with each quantity being decomposed into a Fourier series in the azimuthal direction. To obtain a set of equations for the Fourier amplitude of each azimuthal harmonic, we first write the field equations, Eqs. (\\ref{eq:def_psi})-(\\ref{eq:poisson_exy}), in cylindrical coordinates,\n\\begin{align}\n\\label{eq:fperp}\n\\bm{e}_r\\PP{\\psi}{r}+\\bm{e}_\\phi\\frac{1}{r}\\PP{\\psi}{\\phi}&=(-E_r+B_\\phi)\\bm{e}_r-(E_\\phi+B_r)\\bm{e}_\\phi, \\\\\n\\nabla_\\perp^2\\psi&=-(\\rho-J_z), \\\\\n\\label{eq:pois_br}\n\\nabla_\\perp^2B_r-\\frac{B_r}{r^2}-\\frac{2}{r^2}\\PP{B_\\phi}{\\phi}&=-\\PP{J_\\phi}{\\xi}-\\frac{1}{r}\\PP{J_z}{\\phi}, \\\\\n\\nabla_\\perp^2B_\\phi-\\frac{B_\\phi}{r^2}+\\frac{2}{r^2}\\PP{B_r}{\\phi}&=\\PP{J_r}{\\xi}+\\PP{J_z}{r}, \\\\\n\\nabla_\\perp^2B_z&=-\\frac{1}{r}\\PP{}{r}(rJ_\\phi)+\\frac{1}{r}\\PP{J_r}{\\phi}, \\\\\n\\nabla_\\perp^2E_r-\\frac{E_r}{r^2}-\\frac{2}{r^2}\\PP{E_\\phi}{\\phi}&=\\PP{\\rho}{r}+\\PP{J_r}{\\xi}, \\\\\n\\nabla_\\perp^2E_\\phi-\\frac{E_\\phi}{r^2}+\\frac{2}{r^2}\\PP{E_r}{\\phi}&=\\frac{1}{r}\\PP{\\rho}{\\phi}+\\PP{J_\\phi}{\\xi}, \\\\\n\\label{eq:pois_ez}\n\\nabla_\\perp^2E_z&=\\frac{1}{r}\\PP{}{r}(rJ_r)+\\frac{1}{r}\\PP{J_\\phi}{\\phi}.\n\\end{align}\nwhere the 2D (transverse) Laplacian is defined as $\\nabla_\\perp^2\\equiv\\frac{1}{r}\\PP{}{r}\\left(r\\PP{}{r}\\right)+\\frac{1}{r}\\PP{^2}{\\phi^2}$. Expanding the electromagnetic fields, charge and current density into a Fourier series in the azimuthal direction, gives\n\\begin{equation}\n\\label{eq:fourier_series}\n\\begin{aligned}\nU(r,\\phi)&=\\sum_{m=-\\infty}^{+\\infty}U^m(r)e^{\\mathrm{i} m\\phi}\\\\\n&=U^0(r)+2\\sum_{m=1}\\mathfrak{Re}\\{U^m\\}\\cos(m\\phi)-2\\sum_{m=1}\\mathfrak{Im}\\{U^m\\}\\sin(m\\phi)\n\\end{aligned}\n\\end{equation}\nwhere $U$ represents an arbitrary scalar field or components of a vector field, and note that the amplitude of each harmonic $U^m$ is complex. It follows that $U^{-m}=(U^m)^*$ because $U(r,\\phi)$ is real, which indicates that only the evolution of $m\\ge0$ modes need to be considered. Substituting the expansion into Eqs. (\\ref{eq:fperp})-(\\ref{eq:pois_ez}) yields the following governing equations for each mode\n\\begin{align}\n\\label{eq:fperp_m}\n\\bm{e}_r\\PP{\\psi^m}{r}+\\bm{e}_\\phi\\frac{\\mathrm{i} m}{r}\\psi^m&=(-E_r^m+B_\\phi^m)\\bm{e}_r-(E_\\phi^m+B_r^m)\\bm{e}_\\phi,\\\\\n\\label{eq:pois_psi_m}\n\\mathop{}\\!\\mathbin\\bigtriangleup_m \\psi^m&=-(\\rho^m-J_z^m), \\\\\n\\label{eq:pois_br_m}\n\\mathop{}\\!\\mathbin\\bigtriangleup_m B_r^m-\\frac{B_r^m}{r^2}-\\frac{2\\mathrm{i} m}{r^2}B_\\phi^m&=-\\PP{J_\\phi^m}{\\xi}-\\frac{\\mathrm{i} m}{r}J_z^m, \\\\\n\\label{eq:pois_bphi_m}\n\\mathop{}\\!\\mathbin\\bigtriangleup_m B_\\phi^m-\\frac{B_\\phi^m}{r^2}+\\frac{2\\mathrm{i} m}{r^2}B_r^m&=\\PP{J_r^m}{\\xi}+\\PP{J_z^m}{r}, \\\\\n\\label{eq:pois_bz_m}\n\\mathop{}\\!\\mathbin\\bigtriangleup_m B_z^m&=-\\frac{1}{r}\\PP{}{r}(rJ_\\phi^m)+\\frac{\\mathrm{i} m}{r}J_r^m, \\\\\n\\mathop{}\\!\\mathbin\\bigtriangleup_m E_r^m-\\frac{E_r^m}{r^2}-\\frac{2\\mathrm{i} m}{r^2}E_\\phi^m&=\\PP{\\rho^m}{r}+\\PP{J_r^m}{\\xi}, \\\\\n\\mathop{}\\!\\mathbin\\bigtriangleup_m E_\\phi^m-\\frac{E_\\phi^m}{r^2}+\\frac{2\\mathrm{i} m}{r^2}E_r^m&=\\frac{\\mathrm{i} m}{r}\\rho^m+\\PP{J_\\phi^m}{\\xi}, \\\\\n\\label{eq:pois_ez_m}\n\\mathop{}\\!\\mathbin\\bigtriangleup_m E_z^m&=\\frac{1}{r}\\PP{}{r}(rJ_r^m)+\\frac{\\mathrm{i} m}{r}J_\\phi^m\n\\end{align}\nwhere $\\mathop{}\\!\\mathbin\\bigtriangleup_m\\equiv\\frac{1}{r}\\PP{}{r}\\left(r\\PP{}{r}\\right)-\\frac{m^2}{r^2}$. This set of equations is overdetermined and therefore, similarly to what is currently used in the 3D QuickPIC algorithm \\cite{an2013}, we select Eqs. (\\ref{eq:fperp_m})-(\\ref{eq:pois_bz_m}) and (\\ref{eq:pois_ez_m}) to solve for the electromagnetic fields.\n\nSimilar to other QSA codes and Darwin model codes \\cite{nielson1976}, it is not straightforward to solve the Poisson-like equations and therefore a predictor-corrector iteration is necessary to implicitly determine part of field components. The difficulty in our code arises because the source terms in Eqs. (\\ref{eq:pois_br_m}) and (\\ref{eq:pois_bphi_m}) are not known at the appropriate time step.  We use the same time indexing as in QuickPIC \\cite{huang2006,an2013}. The momentum $\\bm{p}$ and Lorentz factor $\\gamma$ for the plasma particles are defined on integer time steps, $\\xi=n_\\xi\\Delta_\\xi$, while the transverse position $\\bm{x}_\\perp$ and all the Eulerian quantities including $\\psi^m$, $\\bm{E}^m$, $\\bm{B}^m$, $(\\rho-J_z)^m$, $\\bm{J}^m$ and $\\pp{\\bm{J}_\\perp^m}{\\xi}$ are defined on half-integer time steps, $\\xi=(n_\\xi+\\frac{1}{2})\\Delta_\\xi$. In order to deposit $\\pp{\\bm{J}_\\perp^m}{\\xi}$ and $\\bm{J}^m$, the momentum $\\bm{p}^{n_\\xi+\\frac{1}{2}}$ (the superscript denotes the index of $\\xi$) needs to be known. These could be obtained by averaging  $\\bm{p}^{n_\\xi+1}$ and $\\bm{p}^{n_\\xi}$ but $\\bm{p}^{n_\\xi+1}$ is not known because the fields at $\\xi=(n_\\xi+\\frac{1}{2})\\Delta_\\xi$  are not known.  Therefore, an iteration procedure is needed.\nThe $\\bm{B}^m$ and $\\bm{E}^m$ solved at $\\xi=(n_\\xi-\\frac{1}{2})\\Delta_\\xi$ are used as an appropriate initial guess at $\\xi=(n_\\xi+\\frac{1}{2})\\Delta_\\xi$.\nThese are then used to predict $\\bm{p}^{n_\\xi+1}$ in a leapfrog manner and the $\\bm{p}^{n_\\xi+\\frac{1}{2}}$ is simply evaluated by the average $(\\bm{p}^{n_\\xi}+\\bm{p}^{n_\\xi+1})\/2$, which we call the predictor procedure. We note that as described in ref. \\cite{an2013}, $\\pp{\\bm{J}_\\perp^m}{\\xi}$ is obtained by analytically evaluating the derivative of the shape function and not through a finite difference operation of $\\bm{J}_\\perp^m$. Using this method the particle positions do not need be updated within the predictor procedure. The predicted $\\bm{p}^{n_\\xi+\\frac{1}{2}}$ are then used to deposit the source terms $\\pp{\\bm{J}_\\perp^m}{\\xi}$ and $\\bm{J}^m$  which are used to  improve  the values of $\\bm{B}^m$ and $\\bm{E}^m$ from the initial guesses\/predictions. This operation is called corrector procedure. To guarantee the procedure for correcting $\\bm{B}^m_\\perp$ is stable and that it converges, an iterative form of the Poisson equation is used\n\\begin{align}\n\\mathop{}\\!\\mathbin\\bigtriangleup_m B_r^{m,l+1}-\\left(1+\\frac{1}{r^2}\\right)B_r^{m,l+1}-\\frac{2\\mathrm{i} m}{r^2}B_\\phi^{m,l+1}&=-\\left(\\PP{J_\\phi^m}{\\xi}\\right)^l-\\frac{\\mathrm{i} m}{r}J_z^{m,l}-B_r^{m,l}, \\nonumber \\\\\n\\mathop{}\\!\\mathbin\\bigtriangleup_m B_\\phi^{m,l+1}-\\left(1+\\frac{1}{r^2}\\right)B_\\phi^{m,l+1}+\\frac{2\\mathrm{i} m}{r^2}B_r^{m,l+1}&=\\left(\\PP{J_r^m}{\\xi}\\right)^l+\\PP{J_z^{m,l}}{r}-B_\\phi^{m,l}, \\nonumber\n\\end{align}\nwhere the superscript $l$ denotes the iteration step. The other components of the fields, $\\bm{E}_\\perp^m$, $B_z^m$, $E_z^m$ can then be obtained once $\\bm{B}_\\perp$ is known via Eqs. (\\ref{eq:fperp_m}), (\\ref{eq:pois_bz_m}) and ($\\ref{eq:pois_ez_m}$) respectively (note that $\\psi^m$ is already known before the predictor-corrector iteration). This predictor-corrector iteration can be conducted for an arbitrary number of times until the answers are convergence to a desired accuracy.\n\nUnlike in 3D QuickPIC where the equations for the two components of the $\\bm{B}_\\perp$ are decoupled, Eqs. (\\ref{eq:pois_br_m}) and (\\ref{eq:pois_bphi_m}) are coupled. For numerical reasons, we instead seek solutions to a set of decoupled equations by introducing new variables $B_+^m=B_r^m+\\mathrm{i} B_\\phi^m$ and $B_-^m=B_r^m-\\mathrm{i} B_\\phi^m$ in QPAD. With these new field variables, the decoupled equations can be written as\n\\begin{equation}\n\\label{eq:pois_decouple_m_iter}\n\\left(\\PP{^2}{r^2}+\\frac{1}{r}\\PP{}{r}-\\frac{(m\\pm1)^2}{r^2}-1\\right)B_\\pm^{m,l+1}=S_\\pm^{m,l}-B_\\pm^{m,l}\n\\end{equation}\nwhere\n\\begin{equation}\nS_\\pm^m=-\\PP{J_\\phi^m}{\\xi}-\\frac{\\mathrm{i} m}{r}J_z^m\\pm\\mathrm{i}\\left(\\PP{J_r^m}{\\xi}+\\PP{J_z^m}{r}\\right). \\nonumber\n\\end{equation}\nAs we will see in the next section, after discretization, the decoupled equations become tri-diagonal linear systems for which the efficient cyclic reduction algorithm \\cite{press2010} can be applied. On the other hand, the original coupled equations would be solved using classic iterative methods or sparse matrix techniques which typically are computationally less efficient.\n\nFor computationally simplicity, in the azimuthal mode expansion method, we treat the fields from the beam separately. Due to the approximation that the transverse current $\\bm{J}_\\perp$ is negligible for beam particles and these particles travel at a speed very closed to the speed of light, $c$ it follows that $\\rho^m_\\text{beam}\\simeq J_{z,\\text{beam}}^m$. There is thus no transverse current from the beam which implies that longitudinal fields $B_z^m$ and $E_z^m$ from the beam vanish, and that Eqs. (\\ref{eq:pois_br_m}) and (\\ref{eq:pois_bphi_m}) reduce to an electrostatics problem,\n\\begin{equation}\n\\bm{B}^m_{\\perp,\\text{beam}}=\\bm{e}_r\\frac{\\mathrm{i} m}{r}A_z^m+\\bm{e}_\\phi\\PP{A_z^m}{r} \\nonumber\n\\end{equation}\nand $A_z^m$ satisfies\n\\begin{equation}\n\\label{eq:pois_phi_m}\n-\\mathop{}\\!\\mathbin\\bigtriangleup_m A^m_z=J^m_{z,\\text{beam}}=\\rho^m_\\text{beam}.\n\\end{equation}\nOnce  $\\bm{B}^m_{\\perp,\\text{beam}}$ is known then the electric fields can be obtained through $E^m_{r,\\text{beam}}=B^m_{\\phi,\\text{beam}}$ and $E^m_{\\phi,\\text{beam}}=-B^m_{r,\\text{beam}}$.\n\n\\section{Algorithm implementation}\n\\label{sec:algorithm}\n\n\\subsection{Numerical workflow in QPAD}\n\\label{subsec:workflow}\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=\\textwidth]{workflow.pdf}\n\\caption{The numerical workflow of QPAD}\n\\label{fig:workflow}\n\\end{figure}\n\nIn this section, we briefly introduce the numerical workflow in QPAD.  QPAD consists of three loops (see figure \\ref{fig:workflow}). The outermost level is the the quasi-3D loop in which the charge (current) of the beam particles are deposited onto the $r$-$\\xi$ plane for multiple Fourier harmonics, and the beam particles are pushed in $s$ in the full 3D space described by $(x,y,\\xi)$ coordinates. The particles are pushed using the  leapfrog method with second-order accuracy.\n\nThe quasi-2D loop is embedded into the quasi-3D loop to solve the harmonic amplitudes for all the fields with the plasma and beam charges and currents as sources. The motion of plasma particles is in the 2D space described by $(x, y)$ and particles are pushed in the coordinate $\\xi$. In this loop, the evolution of fields and the motion of plasma particles are updated slice by slice along the negative $\\xi$-direction. The transverse fields from the particle beam are first calculated at a given slice. This together with the self-consistent fields from the plasma particles are used to advance the particles to new position and momenta at the next slice. In the quasi-static algorithm the particle's charge depends on its speed in the $\\hat z$ direction and there are well defined relationships between $\\bm p_z$, $\\bm p_\\perp$ and $\\psi$. Therefore, the pseudo-potential $\\psi$ must also be interpolated to each particle's position and stored for the subsequent particle push. The equation of motion for a plasma particle is,\n\\begin{equation}\n\\DD{\\bm{p}_\\perp}{\\xi}=\\frac{q\\gamma}{1+\\frac{q}{m}\\psi}\\left[\\bm{E}_\\perp+\\left(\\frac{\\bm{p}}{\\gamma}\\times\\bm{B}\\right)_\\perp\\right] \\nonumber\n\\end{equation}\nand\n\\begin{equation}\np_z=\\frac{1+p_\\perp^2-(1-\\frac{q}{m}\\psi)^2}{2(1-\\frac{q}{m}\\psi)}. \\nonumber\n\\end{equation}\nOnce $\\psi$ is known, then the transverse fields $E_r-B_\\phi$ and $E_\\phi+B_r$ can be obtained by taking a transverse gradient of $\\psi$ according to Eq. (\\ref{eq:fperp_m}). The next step is to call the predictor-corrector iteration to implicitly solve the fields induced by plasma as described earlier. The iteration loop starts with updating the particle momenta by using an initial guess for $\\bm{E}$ and $\\bm{B}$. The predicted momenta are then used to deposit the source terms $\\bm{J}$ and $\\pp{\\bm{J}_\\perp}{\\xi}$ needed to solve for $\\bm{B}_\\perp$. The updated $\\bm{E}_\\perp$ is evaluated by subtracting $\\bm{B}_\\perp$ from $-\\nabla_\\perp\\psi$ according to Eq. (\\ref{eq:fperp_m}). With the updated $\\bm{J}$, the longitudinal field components $E_z$ and $B_z$ can be straightforwardly solved using Eqs. (\\ref{eq:pois_ez_m}) and (\\ref{eq:pois_bz_m}). This iteration is terminated when a maximum iterative step is reached or the updated fields meet a specified criterion for convergence\n\\begin{equation}\n\\frac{\\max|\\bm{B}^{l+1}-\\bm{B}^l|}{\\max|\\bm{B}^l|}<tol. \\nonumber\n\\end{equation}\nwhere the chosen tolerance is typically $10^{-3}$ or smaller. The last step in the quasi-2D loop is pushing the plasma particles with the converged electromagnetic fields (interpolating the fields onto the particle position) and $\\psi$ previously stored on each particle.\n\nIn QPAD, the position and momenta of plasma particles are advanced using the classic Boris integrator \\cite{boris1972,qin2013} in a leapfrog manner. All the fields in QPAD are solved using  second-order accurate finite difference methods in conjunction with the multi-grid method. A finite difference version of free boundary conditions for different types of field components are implemented into QPAD as well. New current and charge deposition schemes needed for cylindrical geometry and azimuthal decomposition are also developed and implemented. In the next two sections, we describe the numerical implementation of the field solver and deposition in detail.\n\n\n\\subsection{Finite difference Poisson field solver}\n\nIn 3D quasi-static PIC codes based on Cartesian coordinates, e.g., QuickPIC and HiPACE \\cite{mehrling2014} the fields are solved using FFTs as they are fast and accurate. High parallel scalability is obtained through careful considerations on minimizing data transfer and the use of a pipelining algorithm \\cite{feng2009,an2013}. In QPAD, we adopt finite difference (FD) methods to solve Poisson equations because FFTs can no longer be directly used in cylindrical geometry. We define all the fields on the integer grid points $r_i$, \\emph{i.e.} $r_i=i\\Delta_r$ for $i=1,\\dots,N$ where $N$ is the total number of grid points in $\\hat{r}$-direction and $\\Delta_r$ is the radial cell size. Using a 3-point discretization, the $\\mathop{}\\!\\mathbin\\bigtriangleup_m$ operator with second-order precision can be written as\n\\begin{equation}\n\\mathop{}\\!\\mathbin\\bigtriangleup_m U\\rightarrow\n\\beta_i^- U_{i-1}-\\alpha_i U_i+\\beta_i^+ U_{i+1} \\nonumber\n\\end{equation}\nwhere\n\\begin{equation}\n\\beta_i^\\pm=\\frac{1}{\\Delta_r^2}\\pm\\frac{1}{2r_i\\Delta_r},\\quad\n\\alpha_i=\\frac{2}{\\Delta_r^2}+\\frac{m^2}{r_i^2} \\nonumber\n\\end{equation}\nand $U$ is an arbitrary scalar field. The operator $\\pp{}{r}$ is approximated with the a central difference indexing with second-order precision. Similarly, the operator in Eq. (\\ref{eq:pois_decouple_m_iter}) is discretized as\n\\begin{equation}\n\\left(\\PP{^2}{r^2}+\\frac{1}{r}\\PP{}{r}-\\frac{(m\\pm1)^2}{r^2}-1\\right)U\\rightarrow\n\\beta_i^- U_{i-1}- \\mu_i U_i+\\beta_i^+ U_{i+1} \\nonumber\n\\end{equation}\nwith\n\\begin{equation}\n\\mu_i=\\frac{2}{\\Delta_r^2}+\\frac{(m\\pm1)^2}{r_i^2}+1. \\nonumber\n\\end{equation}\n\nIn QPAD, the governing Eqs. (\\ref{eq:fperp_m}), (\\ref{eq:pois_psi_m}), (\\ref{eq:pois_bz_m}), (\\ref{eq:pois_ez_m}), (\\ref{eq:pois_decouple_m_iter}) and (\\ref{eq:pois_phi_m}) are all discretized. These Poisson equations are converted into tri-diagonal linear systems which can benefit from fast solvers using the cyclic reduction method. These solvers are implemented with the library Hypre \\cite{falgout2002} developed and maintained by LLNL.\n\n\n\\subsection{Boundary conditions}\n\nBoth conducting and free (open) boundary conditions have been implemented in QPAD. Conducting boundary conditions are more standard and are not described further. The basic idea for free or open boundaries is to assume that the space outside the computational domain is vacuum and that it extends to infinity. Therefore, solutions can be obtained by solving a series of Laplace equations. The boundary values can be determined by utilizing the fact that the fields are continuous at the boundary. See \\ref{sec:app:free_bnd} for the details of the derivation.\n\nWhen implementing the field solvers in cylindrical geometry, issues with respect to singularities on the axis are inevitable issue because of the presence of the $1\/r$ term. As we discussed above, all the field components in QPAD are defined on integer grid points. Therefore,  all of the Poisson equations can have a singularity at $r=0$. These singularities can be properly treated by considering the properties of different field components at $r=0$. As pointed out by Constantinescu and Lele \\cite{constantinescu2002}, for any scalar of component of a field in a Cartesian coordinate directions, \\emph{i.e.}, $(\\psi, \\varphi, E_z, B_z, \\rho-J_z, J_z)$, the $m=0$ mode is non-zero at $r=0$ while other modes are zero at $r=0$.\nOn the other hand, for the field components in cylindrical coordinate directions $(E_r, E_\\phi, B_r, B_\\phi, J_r, J_\\phi)$ the $m=1$ mode is non-zero at $r=0$ and the other modes are zero at $r=0$.\n\nThe field components whose boundary values at $r=0$ need to be determined therefore only include $\\psi^0$, $\\varphi^0$, $B_z^0$, $E_z^0$ and $B_\\pm^1$; all other field components vanish at $r=0$. The singularity of the  $\\frac{1}{r}\\PP{U}{r}$ term on the LHS of each Poisson equation (where $U$ denotes any of fields mentioned above) can be eliminated by applying L'Hospital's rule, so that $\\frac{1}{r}\\PP{U}{r}\\rightarrow\\PP{^2U}{r^2}$. The terms having $1\/r$ on the RHS of Poisson equation can be treated in the same manner. There is another important property for components in the cylindrical coordinate directions \\cite{constantinescu2002}. The combinations $U_r+imU_\\phi$ and $U_\\phi-imU_r$ vanish at $r=0$ for arbitrary $m$, which implies $B^1_+$ (recalling the definition is $B_+^m\\equiv B_r^m+iB_\\phi^m$) vanishes at $r=0$ even though both $B_r^1$ and $B_\\phi^1$ are non-zero on the axis. For $m\\neq1$ modes $B_r^m$ and $B_\\phi^m$ are naturally zero at $r=0$ according to previous discussion, therefore, $B_+^m$ vanishes on the axis for arbitrary $m$. Considering the symmetry of different fields around the axis, the discrete version of boundary conditions at $r=0$ can therefore be summarized as follows,\n\\begin{align}\n4(\\psi_1^{m=0}-\\psi_0^{m=0}) &= -(\\rho-J_z)_0^{m=0} \\Delta_r^2 \\\\\n2(E_{z,1}^{m=0}-E_{z,0}^{m=0}) &= J_{r,1}^{m=0} \\Delta_r \\\\\n2(B_{z,1}^{m=0}-B_{z,0}^{m=0}) &= -J_{\\phi,1}^{m=0} \\Delta_r \\\\\n4(B_{-,1}^{m=1}-B_{-,0}^{m=1}) &= -\\left[\\left(\\PP{J_\\phi}{\\xi}\\right)_0^{m=1} + i\\left(\\PP{J_r}{\\xi}\\right)_0^{m=1} + 2i\\frac{J_{z,1}^{m=1}}{\\Delta_r} \\right] \\Delta_r^2\n\\end{align}\n\n\\subsection{Deposition of source terms}\n\nIn order to solve the 1D Poisson equations for each harmonic amplitude, the source terms on the RHS of the governing Eqs. (\\ref{eq:fperp_m})-(\\ref{eq:pois_bz_m}) and (\\ref{eq:pois_ez_m}) must be deposited from the particle information (charge, position and momentum) onto the grid points. The source terms to be deposited include $\\rho^m-J_z^m$, $\\bm{J}^m$ and $\\pp{\\bm{J}_{\\perp}^m}{\\xi}$. Since these source terms are defined on the grid in the $\\hat{r}$-direction while the particles are described by Cartesian coordinates, we need to transform the particle positions and momenta from the cylindrical to Cartesian coordinates in QPAD. The following equation is used to deposit the current as in QuickPIC,\n\\begin{equation}\n\\label{eq:dep_jay}\n\\bm{J}=\\frac{1}{\\text{Vol.}}\\sum_i\\frac{q_i\\bm{v}_i}{1-v_{iz}}S(\\bm{x}_\\perp-\\bm{x}_{i\\perp})=\\frac{1}{\\text{Vol.}}\\sum_i\\frac{q_i\\bm{p}_i}{1-\\frac{q_i}{m_i}\\psi_i}S(\\bm{x}_\\perp-\\bm{x}_{i\\perp})\n\\end{equation}\nwhere $S(\\bm{x}_\\perp-\\bm{x}_{i\\perp})$ is the particle shape function to interpolate the particle quantities at $i$th particle's transverse position $\\bm{x}_{i\\perp}$ onto the grid position $\\bm{x}_\\perp$. The pseudo-potential felt by an individual particle $\\psi_i$ is obtained by interpolating the $\\psi$ solved on the grid to the position of the particle. The second expression for $\\bm J$ can be obtained by multiplying the numerator and denominator of the first expression by the Lorentz factor $\\gamma_i$, and using the constant of motion under the QSA, $\\gamma-p_z=1-(q\/m)\\psi$. In order to derive the deposition scheme in QPAD in which the azimuthal direction is gridless, we  expand $S(\\bm{x}_\\perp-\\bm{x}_{i\\perp})$ into a Fourier series as well. In cylindrical geometry, the interpolation function is defined as\n\\begin{equation}\nS(\\bm{x}_\\perp-\\bm{x}_{i\\perp})\\equiv\n\\frac{1}{r}S_r(r-r_i)S_\\phi(\\phi-\\phi_i), \\nonumber\n\\end{equation}\nwhich is subject to the normalization condition $\\int dr d\\phi\\ S_rS_\\phi=1$. Next, we expand $S_\\phi$ into azimuthal harmonics\n\\begin{equation}\nS_\\phi(\\phi-\\phi_i)=\\sum_m S_\\phi^m(\\phi_i)e^{\\mathrm{i} m\\phi} \\nonumber\n\\end{equation}\nwhere\n\\begin{equation}\nS_\\phi^m(\\phi_i)=\\frac{1}{2\\pi}\\int_0^{2\\pi}d\\phi'\\ S_\\phi(\\phi'-\\phi_i)e^{-\\mathrm{i} m\\phi'}, \\nonumber\n\\end{equation}\nand require both $S_r$ and $S_\\phi$ to satisfy the normalization condition $\\int drS_r=1$ and $\\int d\\phi S_\\phi=1$. If we take $S_\\phi$ to be a Dirac delta function(which we do in QPAD), then $S_\\phi^m=\\frac{1}{2\\pi}e^{-\\mathrm{i} m\\phi_i}$. In addition,  the current $\\bm{J}$ defined on the {\\it r-z} grid can be expanded as\n\\begin{equation}\n\\bm{J}(r,\\phi)=\\sum_m \\bm{J}^m(r) e^{\\mathrm{i} m\\phi}, \\nonumber\n\\end{equation}\nthe deposition for $\\bm{J}^m$ is found to be\n\\begin{equation}\n\\bm{J}^m=\\frac{1}{\\text{Vol.}}\\sum_i\\frac{q_i\\bm{p}_i}{1-\\frac{q_i}{m_i}\\psi_i}\\frac{1}{r}S_r(r-r_i)S^m_\\phi(\\phi_i). \\nonumber\n\\end{equation}\nTherefore, it is actually not necessary to calculate each {\\it m} mode but only the $m=0$ mode from each particle. Any $m>0$ mode for an individual particle can be obtained from the $m=0$ contribution by simply multiplying by a phase factor through the relation $\\bm{J}^m=\\bm{J}^0e^{-\\mathrm{i} m\\phi_i}$ or recursively through $\\bm{J}^m=\\bm{J}^{m-1}e^{-\\mathrm{i}\\phi_i}$ if $S(\\phi-\\phi_i)=\\delta(\\phi-\\phi_i)$ is used.\n\nLikewise, according to ref. \\cite{an2013}, the deposition for $(\\rho-J_z)^m$ can be written as\n\\begin{equation}\n(\\rho-J_z)^m=\\frac{1}{\\text{Vol.}}\\sum_i\\frac{q_i}{r}S_r(r-r_i)S_\\phi^m(\\phi_i), \\nonumber\n\\end{equation}\nwhere $(\\rho-J_z)^m=(\\rho-J_z)^{m-1}e^{-\\mathrm{i}\\phi_i}$ for each particle.\n\nIn section \\ref{subsec:workflow}, we showed that in the predictor-corrector iteration the source term $\\pp{\\bm{J}^m_\\perp}{\\xi}$ at the half-integer time step $\\xi=(n+1\/2)\\Delta_\\xi$ needs to be calculated. This can be done in two ways. The first method, which was adopted in the original version of QuickPIC \\cite{huang2006}, is to predict $J^m_r$ and $J_\\phi^m$ at the next integer time step $\\xi=(n_\\xi+1)\\Delta_\\xi$ and approximate the derivative using the centered difference $\\pp{J_{r,\\phi}^m}{\\xi}\\vert^{n_\\xi+\\frac{1}{2}}=(J_{r,\\phi}^m\\vert^{n_\\xi+1}-J_{r,\\phi}^m\\vert^{n_\\xi})\/\\Delta_\\xi$. However, this approach requires repartitioning the particles within a single pass through the iteration loop when using domain decomposition as it requires updating the particle positions and storing previous and predicted values. In the current version of QuickPIC \\cite{an2013}, this approach is replaced by analytically calculating the derivative of the current in terms of $\\bm{x}_\\perp$, $\\bm{p}_\\perp$ and $\\psi$ using their particle shapes, which allows direct deposition without the computationally expensive particle repartitioning procedure. In QPAD, we use the approach in the current version of  QuickPIC to deposit $\\pp{\\bm{J}_\\perp^m}{\\xi}$. By definition, we have\n\\begin{equation}\n\\begin{aligned}\n\\label{eq:dep_djrdxi}\n\\PP{J_r^m}{\\xi}&=\\frac{1}{\\text{Vol.}}\\sum_i \\PP{}{\\xi}\\left(\\frac{q_ip_{r,i}}{1-\\frac{q_i}{m_i}\\psi_i}\\frac{1}{r}S_rS_\\phi^m\\right) \\\\\n&=\\frac{1}{\\text{Vol.}}\\sum_i \\frac{q_i}{r}\\left(\\frac{\\dd{p_{r,i}}{\\xi}}{1-\\frac{q_i}{m_i}\\psi_i}S_rS_\\phi^m+\\frac{p_{r,i}\\dd{(\\frac{q}{m_i}\\psi_i)}{\\xi}}{(1-\\frac{q_i}{m_i}\\psi_i)^2}S_rS_\\phi^m+\\frac{p_{r,i}}{1-\\frac{q_i}{m_i}\\psi_i}\\PP{(S_rS_\\phi^m)}{\\xi}\\right).\n\\end{aligned}\n\\end{equation}\nIt should be pointed out that the $\\psi_i$ in the denominator is the total value which is obtained by summing all the harmonics. The derivative of $\\psi_i$ with respect to $\\xi$ is calculated by\n\\begin{equation}\n\\label{eq:dpsidxi}\n\\DD{\\psi_i}{\\xi}=E_{z,i}+\\PP{\\psi_i}{r_i}\\DD{r_i}{\\xi}+\\PP{\\psi_i}{\\phi_i}\\DD{\\phi_i}{\\xi}\n\\end{equation}\nwhere the terms $E_{z,i}$, $\\PP{\\psi_i}{r_i}$ and $\\PP{\\psi_i}{\\phi_i}$ are regarded as the interpolated value of $E_z$, $\\PP{\\psi}{r}$ and $\\PP{\\psi}{\\phi}$ at the particle's position $(r_i,\\phi_i)$. The terms $\\DD{r_i}{\\xi}$ and $\\DD{\\phi_i}{\\xi}$ are evaluated by\n\\begin{equation}\n\\DD{r_i}{\\xi}=\\frac{p_{r,i}}{1-\\frac{q_i}{m_i}\\psi_i},\\quad\n\\DD{\\phi_i}{\\xi}=\\frac{1}{r_i}\\frac{p_{\\phi,i}}{1-\\frac{q_i}{m_i}\\psi_i}. \\nonumber\n\\end{equation}\nFor the last term in the bracket of Eq. (\\ref{eq:dep_djrdxi}), $\\pp{(S_rS_\\phi^m)}{\\xi}$ is calculated by\n\\begin{equation}\n\\begin{aligned}\n\\PP{}{\\xi}\\left(S_r(r-r_i)S_\\phi^m(\\phi_i)\\right)&=-\\DD{r_i}{\\xi}\\PP{S_r}{r}S_\\phi^m+\\DD{\\phi_i}{\\xi}\\PP{S_\\phi^m}{\\phi_i}S_r \\\\\n&=-\\frac{p_{r,i}}{1-\\frac{q_i}{m_i}\\psi_i}\\PP{S_r}{r}S_\\phi^m+\\frac{p_{\\phi,i}}{1-\\frac{q_i}{m_i}\\psi_i}\\frac{1}{r_i}\\PP{S_\\phi^m}{\\phi_i}S_r \\\\\n&=-\\frac{e^{-\\mathrm{i} m\\phi_i}}{2\\pi}\\left(\\frac{p_{r,i}}{1-\\frac{q_i}{m_i}\\psi_i}\\PP{S_r}{r}+\\frac{p_{\\phi,i}}{1-\\frac{q_i}{m_i}\\psi_i}\\frac{\\mathrm{i} mS_r}{r_i}\\right). \\nonumber\n\\end{aligned}\n\\end{equation}\nwhere we have applied $S_\\phi=\\delta(\\phi-\\phi_i)$ again. Substituting these expressions into Eq. (\\ref{eq:dep_djrdxi}), we finally obtain the deposition for $\\pp{J_r^m}{\\xi}$\n\\begin{equation}\n\\begin{aligned}\n\\PP{J_r^m}{\\xi}&=\\frac{1}{2\\pi\\text{Vol.}}\\left\\{\\sum_i q_ie^{-\\mathrm{i} m\\phi_i}\\left(\\frac{\\dd{p_{r,i}}{\\xi}}{1-\\frac{q_i}{m_i}\\psi_i}+\\frac{p_{r,i}\\dd{(\\frac{q}{m}\\psi_i)}{\\xi}}{(1-\\frac{q_i}{m_i}\\psi_i)^2}-\\frac{p_{r,i}p_{\\phi,i}}{(1-\\frac{q_i}{m_i}\\psi_i)^2}\\frac{\\mathrm{i} m}{r_i}\\right.\\right.\\\\\n&\\left.\\left.-\\frac{p_{r,i}^2}{(1-\\frac{q_i}{m_i}\\psi_i)^2}\\frac{1}{r}\\right)\\frac{S_r}{r}-\\PP{}{r}\\left(\\sum_i q_i e^{-\\mathrm{i} m\\phi_i}\\frac{p_{r,i}^2}{(1-\\frac{q_i}{m_i}\\psi_i)^2}\\frac{S_r}{r}\\right)\\right\\}\n\\end{aligned}, \\nonumber\n\\end{equation}\nand likewise we can derive the deposition formula for $\\pp{J_\\phi^m}{\\xi}$\n\\begin{equation}\n\\begin{aligned}\n\\PP{J_\\phi^m}{\\xi}&=\\frac{1}{2\\pi\\text{Vol.}}\\left\\{\\sum_i q_ie^{-\\mathrm{i} m\\phi_i}\\left(\\frac{\\dd{p_{\\phi,i}}{\\xi}}{1-\\frac{q_i}{m_i}\\psi_i}+\\frac{p_{\\phi,i}\\dd{(\\frac{q}{m}\\psi_i)}{\\xi}}{(1-\\frac{q_i}{m_i}\\psi_i)^2}-\\frac{p_{\\phi,i}^2}{(1-\\frac{q_i}{m_i}\\psi_i)^2}\\frac{\\mathrm{i} m}{r_i}\\right.\\right.\\\\\n&\\left.\\left.-\\frac{p_{r,i}p_{\\phi,i}}{(1-\\frac{q_i}{m_i}\\psi_i)^2}\\frac{1}{r}\\right)\\frac{S_r}{r}-\\PP{}{r}\\left(\\sum_i q_i e^{-\\mathrm{i} m\\phi_i}\\frac{p_{r,i}p_{\\phi,i}}{(1-\\frac{q_i}{m_i}\\psi_i)^2}\\frac{S_r}{r}\\right)\\right\\}.\n\\end{aligned}\n\\end{equation}\n\n\\section{Simulation results}\n\\label{sec:simulation}\n\nIn this section, we present a small sample of benchmark tests for QPAD compared against results from QuickPIC and 3D OSIRIS. These benchmarks are related to the plasma wakefield accelerator (PWFA) concept which uses high-energy particle beams to excite a plasma wave wake. The plasma wake provides very large accelerating and focusing forces as compared with conventional accelerator structures. These fields can be used to accelerate and\/or focus a trailing beam riding on an appropriate phase inside the wake. We present benchmarks for driving wakefields in both the linear and nonlinear regimes with only a single mode (only $m=0$ mode). We also present a benchmark for a case where a second witness beam is placed inside a nonlinear wakefield \\cite{lu2006,rosenzweig1991} with an offset in one direction with respect to the drive beam. This leads to a hosing instability \\cite{whittum1991,huang2007} and requires keeping at least the $m=1$ mode.\n\n\\subsection{Plasma wakefield excitation}\n\nWe start by simulating linear wakefield excitation. The linear regime refers to the case that the peak density of the drive beam $n_b$ is much smaller than the background plasma density $n_p$, so that the drive beam only introduces a weak perturbation to the plasma and the background electrons oscillate in a nearly sinusoidal fashion. In this case, the drive beam has a bi-Gaussian density profile with a spot size $k_p\\sigma_r=2.0$, bunch length $k_p\\sigma_z=0.5$, and peak density $n_b\/n_p=0.1$, where $k_p^{-1}$ is the plasma skin depth where $n_b=\\frac{N}{(2\\pi)^{3\/2}} \\exp[-(\\frac{r^2}{2\\sigma^2_r}+\\frac{z^2}{2\\sigma^2_z})]$ and $N$ is the number of particles in the bunch. Since this scenario possesses azimuthal symmetry, we only include the $m=0$ mode in QPAD which is equivalent to a 2D r-z simulation using codes such as WAKE or LCODE. In the QuickPIC and OSIRIS simulations, the cell size is $\\Delta_x=\\Delta_y=0.0234\\ k_p^{-1}, \\Delta_z=0.0195\\ k_p^{-1}$. In the QPAD simulation, $\\Delta_r=0.0234\\ k_p^{-1}, \\Delta_z=0.0195\\ k_p^{-1}$. The drive beams are initialized with $128\\times128\\times256$ particles in $x, y$ and $z$ for the QuickPIC simulation and with $128\\times32\\times256$ particles in $r,\\phi$ and $z$ for the QPAD simulation. For the plasma, we use $2\\times2$ particles per 2D cell in QuickPIC and uniformly distribute $2\\times32$ particles within a ring of width $\\Delta_r$ in QPAD. In OSIRIS, $2\\times2\\times2$ particles per 3D cell are used to initialize both the plasma and beam.\n\nThe simulation results are shown in Fig. \\ref{fig:linear_wake}. In figure \\ref{fig:linear_wake}(a) and (b), we compare the plasma electron density and $E_z$ field between QuickPIC and QPAD runs. The drive beams, whose centers reside at $\\xi=2$, move downward and are not displayed in these figures. Figure \\ref{fig:linear_wake}(c) compares the lineouts of $E_z$ on the $r=0$  axis between QPAD, QuickPIC and OSIRIS. Here, only one predictor-corrector iteration is conducted in QPAD and this already gives excellent agreement with QuickPIC and OSIRIS. We also conducted convergence tests for the predictor-corrector loop by iterating 1, 3 and 5 times. We found in this scenario, the predictor-corrector loop converges so rapidly that only one iteration is sufficient to reach the desired simulation accuracy.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=\\textwidth]{linear_wake.pdf}\n\\caption{Comparison of beam-driven wakefield in linear regime between OSIRIS, QuickPIC and QPAD. (a) Background electron density. (b) $E_z$ field. (c) On-axis lineouts of $E_z$ fields from OSIRIS, QuickPIC and QPAD.}\n\\label{fig:linear_wake}\n\\end{figure}\n\nNext, we simulate drive beam parameters for which a nonlinear plasma wakefield is excited. In this case the peak density of the beam is much larger than the plasma density, \\emph{i.e.}, $n_b\\gg n_p$.\nHere, we show an example for which $n_b\/n_p=4$, $k_p\\sigma_r=0.25$, $\\Lambda\\equiv(n_b\/n_p)(k_p\\sigma_r)^2=0.25$ and keep other numerical parameters the same as those in the linear regime case. In the nonlinear regime, the $E_z$ on axis now looks similar to a sawtooth wave as shown in fig. \\ref{fig:nonlinear_wake}(c). In the region where the background plasma electrons are fully evacuated by the drive beam (from $\\xi=3$ to $\\xi=7$), the $E_z$ field almost drops linearly to its minimum at the rear of the first ion bubble. From fig. \\ref{fig:nonlinear_wake}, we can see that QPAD with only one predictor-corrector iteration still gives results in almost perfect agreement with OSIRIS and QuickPIC. Similarly to the convergence test for the linear regime, the predictor-corrector iteration is found to converge rapidly. Running the iteration more than once does not make an observable difference to the simulation results.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=\\textwidth]{nonlinear_wake.pdf}\n\\caption{Comparison of beam-driven wakefield in nonlinear regime between OSIRIS, QuickPIC and QPAD. (a) Background electron density. (b) $E_z$ field. (c) On-axis lineouts of $E_z$ fields from OSIRIS, QuickPIC and QPAD.}\n\\label{fig:nonlinear_wake}\n\\end{figure}\n\nBesides an electron beam, a very short positron or proton beam can also excite a bubble-like plasma wake. Due to the attractive force from the positron bunch, the background electrons are ``sucked in'' first by the drive beam rather than ``blown out'' as is the case for an electron beam driver. This leads to the background electrons forming a density peak at the front of the first bucket, and the $E_z$ field being negative in that region. After the plasma electrons collapse to the axis, they then overshoot and eventually form a blowout type wake in the second wavelength. In figure \\ref{fig:positron}(a) and (b), a bi-Gaussian positron beam with $n_b\/n_p=2.5,\\ k_p\\sigma_r=0.8,\\ k_p\\sigma_\\xi=0.46$ and the center resides at $\\xi=3$ moves downward. Again, we use only one predictor-corrector iteration to achieve good agreements with the results of QuickPIC and OSIRIS.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=\\textwidth]{positron.pdf}\n\\caption{Comparison of positron-beam-driven wakefield between OSIRIS, QuickPIC and QPAD. (a) Background electron density. (b) $E_z$ field. (c) On-axis lineouts of $E_z$ fields from OSIRIS, QuickPIC and QPAD.}\n\\label{fig:positron}\n\\end{figure}\n\n\\subsection{Hosing instability}\n\nIn this section, we present a simulation of what is called the hosing instability in PWFA \\cite{huang2007}. The hosing instability is one of the major impediments for PWFA and can lead to beam breakup. Although an azimuthally symmetric r-z code such as WAKE and LCODE is very efficient to model PWFA, it cannot be used to investigate the physics involving asymmetries such as the hosing instability. For hosing we only compare QPAD against QuickPIC. The drive beam has a bi-Gaussian profile with a peak density $n_b\/n_p=93.5$, an rms spot-size $k_p\\sigma_r=0.14$ and an rms bunch length $k_p\\sigma_z=0.48$ which corresponds to $\\Lambda\\simeq1.8$. The trailing beam parameters are $n_b\/n_p=56$, $k_p\\sigma_r=0.14$ and $k_p\\sigma_z=0.24$.  For both the plasma and the beams there are 16 macro-particles distributed in $\\phi$ while for the plasma there are 2 macro-particles per r-z cell. Within the region $[-5\\sigma_r,+5\\sigma_r]\\times[-5\\sigma_z,+5\\sigma_z]$ the drive beam and trailing beam have $128\\times512$ and $128\\times256$ particles respectively, and have 16 particles azimuthally. The drive beam is initialized axisymmetrically while the trailing beam has a small centroid offset of $0.038\\ k_p^{-1}$ in $x$-direction. For the full 3D QuickPIC simulation, the plasma has $2\\times2\\times2$ particles per cell and the drive beam and trailing beam have $128\\times128\\times512$ and $128\\times128\\times256$ particles within the $5\\sigma$ rectangular block. The initial longitudinal proper velocity corresponds to $\\gamma \\beta_z=20000$ for both the drive and trailing beams.  In the QPAD simulation, modes $m=0, 1, 2$, and 3 are included.  Figure \\ref{fig:hosing_density} shows the density distribution with the background plasma electrons and beams colored blue and red respectively. The snapshots were taken at $\\omega_p t=20000$. It can be seen that there is excellent agreement between QPAD and QuickPIC for the motion of the trailing beam even for this nonlinear problem.\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{hosing_density.pdf}\n\\caption{Density distribution of plasma electrons and beams in (a) full 3D QuickPIC and (b) QPAD simulations.}\n\\label{fig:hosing_density}\n\\end{figure}\n\nA more careful comparison between the hosing results is obtained by investigating the beam centroid oscillation during the entire acceleration distance for different beam slices. Figure \\ref{fig:hosing_centroid}(a)-(c) plots the centroid oscillation for three slices, residing at $+\\sigma_z,\\ 0$ and $-\\sigma_z$ with respect to the beam center $\\xi_0$. The centroid is defined as $\\frac{1}{N}\\sum x_i$ where the sum is taken over all particles within a slice at $z \\pm 0.1k_p^{-1}$ and $N$ is the number of particles. The amplitude of the centroid oscillation for the slice closer to the beam head [figure \\ref{fig:hosing_centroid}(c)] remains nearly constant in $s$, the amplitude grows in $s$ with a larger growth rate the farther the slice is behind the center of the beam [figure \\ref{fig:hosing_centroid}(a) and (b)]. This qualitatively agrees well with the theoretical prediction on the instability growth. Except for a slight phase difference that is evident for larger values of $s$, there is excellent agreement between QPAD and full 3D QuickPIC simulations. These differences may be due to the truncation of the azimuthal mode expansion at $m=3$. We emphasize that a code such as QPAD is also a powerful too for carrying out large parameter scans even if the results are not quantitatively correct.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{hosing_centroid.pdf}\n\\caption{Beam centroid oscillation of slice residing at (a) $+\\sigma_z$, (b) 0 and (c) $-\\sigma_z$ with respect to the beam center $\\xi_0$.}\n\\label{fig:hosing_centroid}\n\\end{figure}\n\n\\section{Algorithm complexity}\n\\label{sec:performance}\n\n\nThe azimuthal-decomposition-based algorithm has the potential to greatly reduce the computational requirements without much loss in accuracy when modeling 3D physics when the problem only has low order azimuthal asymmetry. This is because it requires fewer grid points and hence few particles. We can make a straightforward estimation of the speedup over a full 3D quasi-static code.\n\nIn QuickPIC, the fields are solved on a 2D slab (usually a square) with $n_\\text{mesh}=N^2$ grid points, so the cost of the Poisson solver is $O(N^2\\log(N))$ assuming the fast FFT method is used. In QPAD, we solve fields on a 1D mesh with $n_\\text{mesh}=N\/2$ grid points for $2m_\\text{max}+1$ components ($m=0$ mode and real\/imaginary parts for $m>0$ modes) where $m_\\text{max}$ is the index of the highest azimuthal mode that is kept.  Therefore, the cost of the Poisson solver is $(2m_\\text{max}+1)O[(N\/2)\\log(N\/2)]$ using the cyclic reduction method. The speedup for the field solve will therefore scale as $\\sim O(N)\/(m_\\text{max}+\\frac{1}{2})$ compared with the FFT method used in QuickPIC. In QuickPIC, a total number of $N^2N_{\\text{ppc},x}N_{\\text{ppc},y}$ macro-particles for plasma species are used where $N_{\\text{ppc},\\eta},\\ (\\eta=x,y)$ denotes the particle number per cell in the $\\eta$-direction. In QPAD, there are only $NN_{\\text{ppc},r}N_{p,\\phi}\/2$ macro-particles for each plasma species, where $N_{\\text{ppc},r}$ is number of particles per $r$-$z$ cell  and $N_{p,\\phi}$ is the number of particles distributed over $0<\\phi<2\\pi$. Assuming the computational cost of pushing particles is proportional to the total macro-particle number, the speedup therefore scales as $2NN_{\\text{ppc},x}N_{\\text{ppc},y}\/(N_{\\text{ppc},r}N_{p,\\phi})\\sim O(N)$. For a majority of PWFA problems, the configuration with $m_\\text{max}\\leq2$ and particle number $N_{p,\\phi}\\sim10,\\ N_{\\text{ppc},r}\\sim N_{\\text{ppc},x}$ or $N_{\\text{ppc},y}$ are enough to capture the dominant azimuthal asymmetry to effectively simulate the physics with nearly round drive beams, so that considerable speedup can be achieved for typical numerical parameters. The goal of this paper is to describe how to implement an azimuthal mode expansion into a quasi-static PIC code. Issues with respect to optimization will be addressed in future publications. The parallelization in QPAD is also similar to that in QuickPIC. The code is parallelized using MPI to run on distributed memory clusters, which is implemented by means of spatial decomposition in r and z dimensions. However, owing to the basic numerical scheme of a quasi-static code, the parallelization in $r$ direction differs essentially from the that in $\\xi$ direction. The parallelization in $r$ is similar to that in full explicit PIC codes with the macro-particles exchange between neighboring processors. The interprocess exchange of field values at the domain boundaries is handled by the built-in routines of Hypre library. In the $\\xi$ direction, we use pipelining algorithm to allow the transverse process slabs to run asynchronously, which can significantly inhibit the idle time.\n\n\n\n\\section{Conclusion}\n\nWe have describe QPAD, a new quasi-static PIC code that uses the azimuthal Fourier decomposition for the fields. The new code utilizes the workflow and routines of QuickPIC in which a 2D code for evolving the plasma particles in a time like variable $\\xi$ is embedded into a 3D code that advances beam particles in a time like variable $s$. In QPAD, all the field components are decomposed into a few Fourier harmonics in $\\phi$. In the 2D part of the code each amplitude depends on $r$ and evolves in $\\xi$. Therefore, in this part of the code the fields are only defined on a 1D grid in $r$. The quasi-static version of  Maxwell's equations for each harmonic amplitude are therefore one-dimensional, making the new code much faster. A full set of Poisson-like equations that exactly correspond to those used in the full 3D QuickPIC are written in cylindrical geometry. A full set of 1D Poisson equations in $r$ are solved for the Fourier amplitudes in $\\phi$ for the relevant fields. To simplify the calculation, we introduced linear combinations of the complex amplitudes, $B^m_\\pm$, to decouple the equations for $B^m_r$ and $B^m_\\phi$. Open (free) boundary conditions are implemented for all the fields. For the particle module, the macro-particles are distributed and advanced in $\\xi$ in a 2D space $(r,\\phi)$.  A predictor-corrector routine is described. A novel deposition method for $\\PP{\\bm{J}}{\\xi}$, $\\bm{J}$ and $\\rho-J_z$ for each harmonics is described and implemented. This scheme does not require updating the particle positions to obtain $\\PP{\\bm{J}}{\\xi}$ which reduces the complexity of the predictor corrector routine.  The new code was benchmarked and compared against results from 3D OSIRIS and QuickPIC for a few sample cases. Excellent agreement was found for both wake excitation of plasma wave wakes from particle beam drivers (electrons and positrons) and for the electron hosing instability. Directions for future work include optimizing the field solver to reduce the across-node data communication, adding multi-threading features (OpenMP), and implementing more physics including field-ionization, radiation reaction, and the ponderomotive guiding center model for a laser.\n\n\\section*{Acknowledgments}\n\n\nWork supported by the U.S. Department of Energy under SciDAC FNAL subcontract 644405, DE-SC0010064, and contract number DE-AC02-76SF00515, and by the U.S. National Science Foundation under NSF 1806046. The simulations were performed on the UCLA Hoffman 2 and Dawson 2 Clusters, the resources of the National Energy Research Scientific Computing Center, and the Super Computing Center of Beijing Normal University. \n\n\\begin{appendix}\n\\section{Implementation of free boundary conditions for electromagnetic fields}\n\\label{sec:app:free_bnd}\n\nIn this appendix, we describe the implementation of the open (free) boundary conditions used to solve Eqs. (\\ref{eq:pois_psi_m})-(\\ref{eq:pois_bz_m}) and (\\ref{eq:pois_ez_m}). Figure \\ref{fig:boundary} shows the grid setup for solving the fields with total $N$ grid points within the solution region. The dashed line defines the boundary and the the physical domain. It is assume that outside of this region there is vacuum out to infinity.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.6\\textwidth]{boundary.pdf}\n\\caption{Grid points layout in $r$-direction.}\n\\label{fig:boundary}\n\\end{figure}\n\nThe basic idea is to obtain the analytic solution in vacuum by solving Laplace equations and then applying solutions at the boundary. We first consider the scalar Laplace equation,\n\\begin{equation}\n\\label{eq:laplace_scalar}\n\\mathop{}\\!\\mathbin\\bigtriangleup_m U^m=0,\\quad\\text{for}\\ r>R,\n\\end{equation}\nwhere $U^m$ represents the $m$th mode of $\\psi$, $A_z$, $B_z$ and $E_z$. It has the solution\n\\begin{equation}\n\\label{eq:laplace_sol_0}\nU^0=C_{U,0}+D_{U,0}\\ln(r)\n\\end{equation}\nand\n\\begin{equation}\nU^{m>0}=C_{U,m} r^{-m}+D_{U,m} r^m.\n\\end{equation}\nThe determination of the constants $C_{U,m}$ and $D_{U,m}$ differs depending on the types of fields. For $\\psi$, it can be shown that $D_{\\psi,0}=0$. By applying Gauss's theorem to Eq. (\\ref{eq:poisson_psi}) and considering a circular  region $S$ of integration with a radius greater than $R$, leads to\n\\begin{equation}\n\\label{eq:gauss_theorem}\n\\oint_{\\partial S} \\nabla_\\perp\\psi\\ dl=2\\pi r\\PP{\\psi^0}{r}=-\\int_S (\\rho-J_z) dS.\n\\end{equation}\nNote that the $m>0$ modes of $\\psi$ do not contribute to the integral on the left because of the presence of the term $e^{\\mathrm{i} m\\phi}$. From the continuity equation under the QSA\n\\begin{equation}\n\\PP{}{\\xi}(\\rho-J_z)+\\nabla_\\perp\\cdot\\bm{J}_\\perp=0.\n\\end{equation}\nand using the fact that $\\bm{J}_\\perp$ vanishes at the boundary of the surface integral $\\partial S$, we have\n\\begin{equation}\n\\PP{}{\\xi}\\int_S (\\rho-J_z)dS=0.\n\\end{equation}\nwhich indicates this integral is zero for any $\\xi$ because it is initially zero (neutral plasma). Therefore, according to Eq. (\\ref{eq:gauss_theorem}), we have $2\\pi r\\PP{\\psi^0}{r}|_{r>R}=0$ which gives $D_{\\psi,0}=0$ by inserting Eq. (\\ref{eq:laplace_sol_0}). Requiring $\\psi\\rightarrow 0$ while $r\\rightarrow 0$, we can determine that $C_{\\psi,0}=0$ and $D_{\\psi,m}=0\\ (m>0)$, and thus the solution in the vacuum has the form\n\\begin{equation}\n\\psi^0=0,\\quad\n\\psi^m=\\frac{C_{\\psi,m}}{r^m}.\n\\end{equation}\n\nFor the longitudinal component of beam's vector potential $A_z$, $D_{A_z,0}\\neq0$ because the charge of the beam is apparently non-neutralized. Applying the natural boundary condition $B_{\\phi,\\text{beam}}=\\PP{A_z}{r}\\rightarrow0$ when $r\\rightarrow0$ and ignoring the arbitrary constant, we have\n\\begin{equation}\nA_z^0=D_{A_z,0}\\ln(r),\\quad\nA_z^m=\\frac{C_{A_z,m}}{r^m}.\n\\end{equation}\n\nFor $B_z$ and $E_z$, the only constraint is $B_z,E_z\\rightarrow 0$ when $r\\rightarrow 0$, so that\n\\begin{equation}\nE_z^0=0,\\quad\nE_z^m=\\frac{C_{E_z,m}}{r^m}\n\\end{equation}\nand\n\\begin{equation}\nB_z^0=0,\\quad\nB_z^m=\\frac{C_{B_z,m}}{r^m}.\n\\end{equation}\n\nThe transverse magnetic fields induced by the plasma satisfy the coupled Laplace equations in the vacuum,\n\\begin{align}\n\\mathop{}\\!\\mathbin\\bigtriangleup_m B_r^m-\\frac{B_r^m}{r^2}-\\frac{2\\mathrm{i} m}{r^2}B_\\phi^m&=0, \\\\\n\\mathop{}\\!\\mathbin\\bigtriangleup_m B_\\phi^m-\\frac{B_\\phi^m}{r^2}+\\frac{2\\mathrm{i} m}{r^2}B_r^m&=0.\n\\end{align}\nIt can be verified that the general solution can be written as\n\\begin{equation}\nB_r^0=\\frac{C_{B_r,0}}{r},\\ B_r^m=\\frac{C_{B_r,m}}{r^{m+1}}+D_{B_r,m}r^{m-1},\n\\end{equation}\nand\n\\begin{equation}\nB_\\phi^0=\\frac{C_{B_\\phi,0}}{r},\\ B_\\phi^m=\\frac{C_{B_\\phi,m}}{r^{m+1}}+D_{B_\\phi,m}r^{m-1}.\n\\end{equation}\nHere, $D_{B_r,m}=D_{B_\\phi,m}=0$ because of the natural boundary conditions that $B_r,B_\\phi\\rightarrow 0$ when $r\\rightarrow 0$.\n\nAfter obtaining the analytical solution for each field (components) in the vacuum, we derive the finite difference form of the boundary conditions used for solving the discrete Poisson-like equations. For an arbitrary field $U^m$, the value on the ghost cell can be evaluated through Taylor expansion (central difference)\n\\begin{equation}\nU^m_{N+1}=U^m_N+\\left.\\PP{U^m}{r}\\right|_N\\Delta_r+O(\\Delta_r^2).\n\\end{equation}\nThe derivative at $r_{N+\\frac{1}{2}}$ (note that $r_{N+\\frac{1}{2}}=R$) is evaluated using the analytical formula. For $A_z^0$,\n\\begin{equation}\nA^0_{z,N+1}\\simeq A^0_{z,N}+\\frac{D_{A_z,0}}{R}\\Delta_r\\simeq A_{z,N}^0+\\frac{\\Delta_r}{R\\ln R}A^0_{z,N+1}\n\\end{equation}\ntherefore\n\\begin{equation}\nA^0_{z,N+1}\\simeq\\left(1+\\frac{\\Delta_r}{R\\ln(R)}\\right)A^0_{z,N}.\n\\end{equation}\nSimilarly, for $m>0$ modes of $\\psi,A_z,B_z$ and $E_z$, we can obtain\n\\begin{equation}\n\\begin{pmatrix}\n\\psi \\\\ A_z \\\\ B_z \\\\ E_z\n\\end{pmatrix}^m_{N+1}\\simeq\\left(1-\\frac{m\\Delta_r}{R}\\right)\n\\begin{pmatrix}\n\\psi \\\\ A_z \\\\ B_z \\\\ E_z\n\\end{pmatrix}^m_N\n\\end{equation}\nand for all the modes of $B_r$ and $B_\\phi$ associated with plasma\n\\begin{equation}\n\\begin{pmatrix}\nB_r \\\\ B_\\phi\n\\end{pmatrix}^m_{N+1}\\simeq\\left(1-\\frac{(m+1)\\Delta_r}{R}\\right)\n\\begin{pmatrix}\nB_r \\\\ B_\\phi\n\\end{pmatrix}^m_N.\n\\end{equation}\nAs $B_\\pm^m$ rather that $B_r^m$ and $B_\\phi^m$ are directly solved in QPAD, we need to perform the linear transformation  $B_\\pm^m=B_r^m\\pm\\mathrm{i} B_\\phi^m$ on both sides of the above equation to obtain  the boundary condition for $B_\\pm^m$\n\\begin{equation}\nB_{\\pm,N}^m\\simeq\\left(1-\\frac{(m+1)\\Delta_r}{R}\\right)B_{\\pm,N+1}^m.\n\\end{equation}\n\n\\end{appendix}\n\n\\bibliographystyle{elsarticle-num}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\subsection{}}\n\n\n\\newcommand{\\subsection}{\\subsection}\n\n\n\n\\newcommand{\\bC^\\times}{{\\mathbb C}^\\times}\n\n\n\n\\newcommand{\\operatorname{Ker}}{\\operatorname{Ker}}\n\n\\newcommand{\\operatorname{Exp}}{\\operatorname{Exp}}\n\n\n\n\n\n\n\n\n\n\n\\title[Centralizers of generic elements of Newton strata]{Centralizers\nof generic elements of Newton strata in the adjoint quotients of reductive\ngroups}\n\\author[Mitya Boyarchenko and Maria Sabitova]{Mitya\nBoyarchenko\\address{Mitya Boyarchenko: University of Chicago, \\hfill\\newline\nDepartment of Mathematics, Chicago, IL 60637} and Maria Sabitova\\address{Maria\nSabitova: University of Illinois at Urbana-Champaign, \\hfill\\newline Department\nof Mathematics, Urbana, IL 61801}}\n\n\\thanks{The research of M.B. was partially supported by NSF grant\nDMS-0401164.  \\hfill\\newline {\\em Email addresses:}\\ \\ {\\tt\nmitya@math.uchicago.edu} (M.B.), \\ \\ {\\tt sabitova@math.upenn.edu} (M.S.)}\n\n\n\n\n\n\n\n\\begin{document}\n\n\n\n\n\n\n\n\\begin{abstract}\nWe study the Newton stratification of the adjoint quotient of a connected split\nreductive group $G$ with simply connected derived group over the field $F={\\mathbb C}((\\epsilon))$ of formal Laurent series. Our main result describes the\ncentralizer of a regular semisimple element in $G(F)$ whose image in the\nadjoint quotient lies in a certain generic subset of a given Newton stratum.\nOther noteworthy results include analogues of some results of Springer on\nregular elements of finite reflection groups, as well as a geometric\nconstruction of a well known homomorphism $\\psi:P(R^\\vee)\/Q(R^\\vee)\\to W(R)$ defined\nfor every reduced and irreducible root system $R$.\n\\end{abstract}\n\n\\maketitle\n\n\\setcounter{tocdepth}{1}\n\n\\tableofcontents\n\n\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*{Introduction}\n\nIf $F={\\mathbb C}((\\epsilon))$ is the field of formal Laurent series and $G=GL_n(\\overline{F})$, it\nis well known that a semisimple element of $G$ is conjugate to an element lying\nin $GL_n(F)$ if and only if the coefficients of its characteristic polynomial\nlie in $F$, and, conversely, every monic polynomial of degree $n$ over $F$\nwhose constant term in nonzero is the characteristic polynomial of a semisimple\nelement of $G$ lying in $GL_n(F)$. Hence the $F$-points of the adjoint quotient\n${\\mathbb A}$ of $G$ are identified with the set  of such\npolynomials: ${\\mathbb A}(F)\\cong F^{n-1}\\times F^\\times$. We can partition ${\\mathbb A}(F)$ into a disjoint union of subsets\n${\\mathbb A}(F)_{\\nu}$ consisting of polynomials of ${\\mathbb A}(F)$ having the same Newton\npolygon $\\nu$, which gives the {\\em Newton stratification} of ${\\mathbb A}(F)$.\n\n\\medbreak\n\nIn his paper \\cite{k} R.~Kottwitz generalizes the notion of Newton\nstratification to a general connected split reductive group $G$ over $F$ whose\nderived group is simply connected. One of the main goals of this paper is to\nanswer a question (also formulated by Kottwitz) which arises naturally in this\ncontext. Namely, we prove that for every Newton stratum\n${\\mathbb A}(F)_\\nu\\subset{\\mathbb A}(F)$ and for every regular semisimple element $\\gamma\\in\nG(F)$ whose image in ${\\mathbb A}(F)$ lies within a certain open dense subset of\n${\\mathbb A}(F)_\\nu$, the centralizer $G_\\gamma$ is isomorphic to the twist of a split\nmaximal torus $A\\subset G$ by an element $w=w(\\nu)$ of the Weyl group of $G$\nwith respect to $A$. The element $w$ is independent of the choice of $\\gamma$ and admits a\nrather simple description in terms of $\\nu$ (see Theorem \\ref{t:1}). The\ndefinition of the Newton stratification appears in Section \\ref{s:notation}\ntogether with some other preliminaries, while the statement and the proof of\nour main result occupy Sections \\ref{s:main} and \\ref{s:systems}.\n\n\\medbreak\n\nIn the course of the proof we encountered several questions about semisimple\ngroups over ${\\mathbb C}$ that, to the best of our knowledge, have not been studied\nbefore. In particular, if $G$ is a simple, connected and simply connected\nalgebraic group over ${\\mathbb C}$ with maximal torus $A\\subset G$, we establish in\nSection \\ref{s:Springer} natural analogues of some of the results of\nT.~Springer \\cite{s} in the nonlinear context of the Weyl group $W=W(G,A)$\nacting on $A$. More precisely, we prove that, given an element $x$ of the center $Z(G)$ of $G$,\nthere exist $g\\in W$ and a regular element $u\\in A$ satisfying $g(u)=xu$; in\naddition, this condition determines the element $g$ uniquely up to conjugation,\nand the eigenvalues of the action of $g$ on the Lie algebra $\\operatorname{Lie}(A)$ can be\ncomputed easily in terms of $x$. Aside from the applications appearing in this\npaper, the usefulness of these results is demonstrated by Remark\n\\ref{r:kottwitz}.\n\n\\medbreak\n\nSection \\ref{s:geometric} is devoted to the study of a certain important\nhomomorphism $\\psi:Z(G)\\to W$ (which already enters the formulation of our\nfirst main result). This homomorphism is well defined up to conjugation by\nelements of $W$. So far the only known (to us) description of this homomorphism\nwas purely algebraic. Our last main result is Theorem \\ref{t:monodromy}, which\nprovides a geometric (or, if the reader wishes, topological) description of\n$\\psi$ as the monodromy of a certain $W$-torsor over a manifold whose\nfundamental group is identified with $Z(G)$, constructed by restricting the\nSpringer covering $\\widetilde{G}^{rs}\\to G^{rs}$ to the set of regular elements\nof a maximal compact subgroup of $G$.\n\n\n\n\n\\subsection*{Acknowledgements} We would like to express our deepest\ngratitude to Robert Kottwitz for introducing us to this subject and suggesting\nthe statement of one of our main results, as well as carefully reading our article. We would also like to thank him for\nsharing the preprint \\cite{k} before it was made available to the general\naudience, for his time, for numerous stimulating discussions, and in particular\nfor pointing out a way of greatly simplifying the proof of the first part of\nTheorem \\ref{t:monodromy}. We are grateful to Misha Finkelberg for useful and\ninspiring conversations, for referring us to the paper \\cite{ss}, and for\nexplaining to us Zakharevich's construction that appears in\n\\S\\ref{ss:monodromy}. We are indebted to Jim Humphreys, David Kazhdan, George\nLusztig, Jean-Pierre Serre, and Tonny Springer for very helpful e-mail\ncorrespondence. Part of the research was conducted when the second author was a\npostdoc at MSRI in the Spring of 2006, and she would like to thank the\nInstitute for its hospitality.\n\n\n\n\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{General facts and notation}\\label{s:notation}\n\n\nSince the main reference for the first part of this paper is \\cite{k}, we will\nkeep the same notation as in that article. In this section we review this\nnotation and recall some results that will be used in the remainder of the\npaper. The reader familiar with \\cite{k} may wish to skip directly to Section\n\\ref{s:main} and refer back to this one as the need arises.\n\n\n\n\\subsection{The setup}\\label{ss:setup} Let $G$ be a connected reductive group\nover ${\\mathbb C}$ of rank $n$. We assume that the derived group $G_{sc}=[G,G]$ of $G$\nis simply connected, and let $l$ be its rank (so that $l\\leq n$). The principal\nexample to keep in mind is $G=GL_n$. We fix a maximal torus $A\\subset G$ and a\nBorel subgroup $B\\subset G$ containing $A$, and write $A_{sc}=A\\cap G_{sc}$,\nwhich is a maximal torus of $G_{sc}$. We let $R\\subset X^*(A)$ denote the root\nsystem of $G$ with respect to $A$, and $\\Delta=\\{\\alpha_i\\}_{i\\in I}\\subset R$ the\nbasis of simple roots determined by $B$, where $I=\\{1,2,\\dotsc,l\\}$. Further,\nwe let $Q(R)$ and $Q(R^\\vee)$ denote the root lattice and the coroot lattice of\n$R$, and we let $P(R)$ and $P(R^\\vee)$ denote the dual lattices to $Q(R^\\vee)$\nand $Q(R)$, respectively. It is known that $P(R)$ can be canonically identified\nwith $X^*(A_{sc})$, and $Q(R^\\vee)$ can be canonically identified with\n$X_*(A_{sc})$. We will view these identifications as equalities. Finally, let\n$\\{\\varpi_i\\}_{i\\in I}$ denote the basis of fundamental weights of\n$P(R)=X^*(A_{sc})$ which is dual to the basis of $Q(R^\\vee)$ consisting of the\nsimple coroots $\\{\\alpha_i^\\vee\\}_{i\\in I}$, and let $\\omega_1,\\ldots,\\omega_n\\in\nX^*(A)$ be a ${\\mathbb Z}$-basis chosen as in \\cite{k} (\\S 1.3, p.~3). Namely,\n$\\omega_1,\\ldots,\\omega_l$ are preimages of the fundamental weights\n$\\varpi_1,\\ldots,\\varpi_l\\in X^*(A_{sc})$ under the natural map\n$\\operatorname{Res}:X^*(A)\\twoheadrightarrow X^*(A_{sc})$ and $\\omega_{l+1},\\ldots,\\omega_n$ is a\n${\\mathbb Z}$-basis of the kernel of $\\operatorname{Res}$.\n\n\n\n\n\n\\subsection{The adjoint quotient of $G$}\\label{ss:adj-quotient}\nLet $W=W(G,A)=N_G(A)\/A$ be the Weyl group of $G$ with respect to $A$. For each\n$\\lambda\\in X^*(A)$ we write $e^{\\lambda}$ for $\\lambda$ viewed as an element of the group\nalgebra ${\\mathbb C}[X^*(A)]$ of $X^*(A)$ over ${\\mathbb C}$; note that\n${\\mathbb C}[X^*(A)]\\cong{\\mathbb C}[A]$, the coordinate ring of $A$. Put\n$$\nc_i:=\\sum_{\\lambda\\in W\\omega_i}e^{\\lambda}\\in {\\mathbb C}[X^*(A)]^W,\\quad i\\in\\{1,\\ldots,n \\},\n$$\nwhere $W\\omega_i$ denotes the $W$-orbit of $\\omega_i$ in $X^*(A)$. Let\n${\\mathbb A}:={\\mathbb G}_a^l\\times{\\mathbb G}_m^{n-l}$. It is well-known that the map $c:A\\longrightarrow{\\mathbb A}$ defined\nby $$a\\mapsto (c_1(a),\\ldots,c_n(a))$$ induces an isomorphism of algebraic\nvarieties between $A\/W$ and ${\\mathbb A}$ (see \\cite{k}, \\S1.5, p. 5, for example). We will often\nidentify $A\/W$ with ${\\mathbb A}$ via $c$. Note that, in particular, we obtain an\nidentification $A(\\overline{F})\/W\\cong{\\mathbb A}(\\overline{F})$ for any algebraically\nclosed field $\\overline{F}$ containing ${\\mathbb C}$.\n\n\\medbreak\n\nThe variety ${\\mathbb A}$ is called the {\\em adjoint quotient} of $G$, because if we\nlet $G$ act on itself by conjugation, Chevalley's restriction theorem implies\nthat the restriction map ${\\mathbb C}[G]\\to{\\mathbb C}[A]$ induces an isomorphism of algebras\n${\\mathbb C}[G]^G\\cong{\\mathbb C}[A]^W$, and hence we obtain an isomorphism of algebraic\nvarieties $A\/W\\cong G\/\\!\/(\\operatorname{Ad} G)$.\n\n\n\n\n\n\n\n\n\\subsection{The base field $F$}\\label{ss:base-field}\nLet $F={\\mathbb C}((\\epsilon))$ be the field of formal Laurent power series\nover ${\\mathbb C}$ in an indeterminate $\\epsilon$.  For each $n\\in{\\mathbb N}$ we let $\\epsilon^{1\/n}\\in\\overline{F}^{\\times}$\nbe a fixed $n$-th root of $\\epsilon$\nsuch that\n$$\n\\left(\\epsilon^{\\frac{1}{mn}}\\right)^{m}=\\epsilon^{1\/n},\\quad\n\\forall m,n\\in{\\mathbb N}.\n$$\nIt is known that $\\overline{F}=\\bigcup_{n\\in{\\mathbb N}}{\\mathbb C}((\\epsilon^{1\/n}))$. Thus, the element\n$\\sigma\\in\\Gamma=\\operatorname{Gal}(\\overline{F}\/F)$ given by\n$$\n\\sigma(\\epsilon^{1\/n})=\\exp\\left(\\frac{2\\pi\n\\sqrt{-1}}{n}\\right)\\epsilon^{1\/n},\\quad\\forall n\\in{\\mathbb N},\n$$\nis a topological generator of $\\Gamma$, i.e., it determines an\nisomorphism $\\widehat{{\\mathbb Z}}\\rarab{\\simeq}\\Gamma$. We also fix the valuation $\\operatorname{val}$ on $\\overline{F}$ such that\n$$\n\\operatorname{val}(\\epsilon^{1\/n})=-1\/n,\\quad\\forall n\\in{\\mathbb N}, \\quad\\text{ and }\\,\\,\\operatorname{val}(0)=-\\infty.\n$$\n\n\\medbreak\n\nWe may view $G$ and $A$ as algebraic groups over $F$ by extending scalars. This\nshould cause no confusion, since in the remainder of this section, as well as\nin Sections \\ref{s:main} and \\ref{s:systems} we always think of $G$ and $A$ as\nalgebraic groups over $F$, whereas in Sections \\ref{s:Springer} and\n\\ref{s:geometric} we will only work with algebraic groups over ${\\mathbb C}$. We will\nconsider ${\\mathbb A}(F)=F^l\\times (F^\\times)^{n-l}$ as a topological space with\nrespect to the topology induced from $F^n$, where $F$ is identified with a\nproduct of countably many copies of ${\\mathbb C}$ and the closed sets in $F^n$ are\ndefined by polynomial equations over ${\\mathbb C}$ in finitely many coordinates.\n\n\n\n\n\\subsection{The map $f:G^{rs}(F)\\longrightarrow{\\mathbb A}(F)$}\\label{subsec:f}\nLet $G^{rs}(F)$ denote the set of regular semisimple elements of $G$ defined\nover $F$. We will need the map $f:G^{rs}(F)\\longrightarrow{\\mathbb A}(F)$ defined in the following\nway. Let $\\gamma\\in G^{rs}(F)$ be a regular semisimple element of $G$, i.e.,\n$\\gamma\\in G(F)$ and the centralizer $G_{\\gamma}$ of $\\gamma$ in $G$ is a maximal torus\n(defined over $F$). There is $g\\in G(\\overline{F})$ which conjugates $\\gamma$ to an element\n$\\gamma_0\\in A(\\overline{F})$. Note that $\\gamma_0$ is only determined up to the $W$-action.\nNamely, if $g_1\\gamma g_1^{-1}\\in A(\\overline{F})$ and $g_2\\gamma g_2^{-1}\\in A(\\overline{F})$, then\nthe element $g_1g_2^{-1}\\in G(\\overline{F})$ conjugates $g_2\\gamma g_2^{-1}$ into $g_1\\gamma\ng_1^{-1}$. Since $\\gamma$ is regular and semisimple,  we have\n$$\nA(\\overline{F})=G_{g_1\\gamma g_1^{-1}}(\\overline{F})=G_{g_2\\gamma g_2^{-1}}(\\overline{F}),\n$$\nhence $g_1g_2^{-1}\\in N_G(A)(\\overline{F})$, where $N_G(A)$ denotes the normalizer of\n$A$ in $G$. Thus, $g_1\\gamma g_1^{-1}$ and $g_2\\gamma g_2^{-1}$ are in the same\n$W$-orbit. So we get a well-defined map $f:G^{rs}(F)\\longrightarrow{\\mathbb A}(\\overline{F})$ given by\n$$\nf(\\gamma)=c(\\gamma_0).\n$$\nWe claim that, in fact, $f(\\gamma)\\in{\\mathbb A} (F)$. To check the claim we need to show\nthat the $W$-orbit of $\\gamma_0$ in $A(\\overline{F})$ is stable under the action of $\\Gamma$\non $A(\\overline{F})$. Let $\\lambda\\in\\Gamma$, then\n$$\n\\lambda(\\gamma_0)=\\lambda(g\\gamma g^{-1})=\\lambda(g)\\gamma\\lambda(g)^{-1},\n$$\nbecause $\\gamma\\in G(F)$. This computation shows that $\\lambda(\\gamma_0)$ is conjugate to\n$\\gamma_0$ by an element of $G(\\overline{F})$ and hence, by the argument above,\n$\\lambda(\\gamma_0)$ and $\\gamma_0$ lie in the same $W$-orbit, hence the image of $f$ lies\nin ${\\mathbb A}(F)$.\n\n\n\n\\subsection{The homomorphism $\\psi_G:\\Lambda_G\\longrightarrow W$}\\label{ss:psi-G}\nLet $\\Lambda_G=X_*(A)\/X_*(A_{sc})$, and let $A_G\\subseteq A$ be the identity\ncomponent of the center of $G$. We have the following natural maps:\n\\begin{equation}\\label{eq:l} \\Lambda_G\\twoheadrightarrow\\Lambda_G\/X_*(A_G)\\hookrightarrow P(R^{\\vee})\/Q(R^{\\vee}), \\end{equation}\nwhere $X_*(A_G)$ is identified with its image under the embedding\n$X_*(A_G)\\hookrightarrow X_*(A)\\twoheadrightarrow \\Lambda_G$ and $R^{\\vee}$ denotes the coroot system of\n$G$. Let $\\psi:P(R^{\\vee})\/Q(R^{\\vee})\\longrightarrow W$ be the map defined as in \\cite{b}\n(Chapter VI, \\S2, no.~3). (See also \\S\\ref{ss:psi} below.) We denote by\n$\\psi_G$ the composition\n$$\n\\Lambda_G\\longrightarrow P(R^{\\vee})\/Q(R^{\\vee})\\rarab{\\psi} W,\n$$\nwhere the first map is given by \\eqref{eq:l}.\n\n\\begin{rem} We note that $\\psi_G$ is the map $ps$ introduced\nin \\cite{k} (\\S 1.9, p. 9).\n\\end{rem}\n\n\n\n\\subsection{Newton stratification of ${\\mathbb A}(F)$} Let\n${\\mathfrak a}=X_*(A)\\otimes_{{\\mathbb Z}}{\\mathbb R}$ and let $P=MN$ be a parabolic subgroup of $G$\ncontaining $B$ (i.e., $P$ is a {\\em standard} parabolic subgroup) with  Levi\nsubgroup $M$ and  unipotent radical $N$. Let $W_M$ be the Weyl group of $M$\nidentified with a subgroup of $W$. Put ${\\mathfrak a}_M:=X_*(A_M)\\otimes_{{\\mathbb Z}}{\\mathbb R}$, where\nas above $A_M\\subseteq A$ denotes the identity component of the center of $M$.\nThen ${\\mathfrak a}_M$ can be identified with the set of fixed points of ${\\mathfrak a}$ under the\naction of $W_M$, and we have a surjection\n$$\np_M:{\\mathfrak a}\\twoheadrightarrow {\\mathfrak a}_M,\\quad x\\mapsto\\frac{1}{W_M}\\cdot\\sum_{w\\in W_M}w(x).\n$$\nSince $G_{sc}$ is simply connected, the derived group of $M$ is also simply\nconnected, which implies that $\\Lambda_M$ can be identified with the image of\n$X_*(A)$ under $p_M$. In what follows we write $\\Lambda_M\\hookrightarrow{\\mathfrak a}_M$ to indicate\nthat $\\Lambda_M$ is considered as  $p_M(X_*(A))\\subseteq{\\mathfrak a}_M$.\n\n\\medbreak\n\nLet ${\\mathcal N}_G$ be the subset of ${\\mathfrak a}$ defined as in \\cite{k} (\\S 1.3, p. 4). For the\nsake of completeness we repeat the facts about ${\\mathcal N}_G$ which will be used\nin this paper. First,\n$$\n{\\mathcal N}_G=\\coprod_{P}\\Lambda_P^+ \\subseteq{\\mathfrak a},\n$$\nwhere the union is taken over all standard parabolic subgroups of $G$ and for\neach such $P$ (in the above notation) $\\Lambda_P^+$ is a subset of\n$\\Lambda_M\\hookrightarrow{\\mathfrak a}_M$. Moreover, $\\Lambda_G^+=\\Lambda_G$, where $\\Lambda_G\\hookrightarrow{\\mathfrak a}_G$. Thus,\nfor each $\\nu\\in{\\mathcal N}_G$ there is a unique parabolic subgroup $P=MN$ such that\n$\\nu\\in\\Lambda_P^+$, hence the element $\\psi_M(\\nu)\\in W_M$ is defined and we put\n\\begin{equation}\\label{eq:we} w(\\nu):=\\psi_M(\\nu)\\in W.\\end{equation}\n\nAlso, for each $\\nu\\in{\\mathcal N}_G$ there is a certain non-empty irreducible subset\n${\\mathbb A}(F)_{\\nu}$ of ${\\mathbb A}(F)$ such that\n$$\n{\\mathbb A}(F)=\\coprod_{\\nu\\in{\\mathcal N}_G}{\\mathbb A}(F)_{\\nu}\n$$\n(\\cite{k}, Thm. 1.5.2, p. 6).\n\n\\medbreak\n\nIn what follows we will need one result about the sets ${\\mathbb A}(F)_{\\nu}$, which we\nstate as a lemma for future reference.\n\n\\begin{lem}\\label{l:kot}\nLet $\\nu\\in{\\mathcal N}_G$. Then $c\\in{\\mathbb A}(\\overline{F})$ belongs to ${\\mathbb A}(F)_{\\nu}$ if and only if\nthere is $a\\in A(\\overline{F})$ such that\n\\begin{eqnarray}\n&& c=c(a),\\label{eq:surj1}\\\\\n&&\\sigma(a)=g(a)\\quad\\text{for some }g\\in W,\\text{ and}\\label{eq:surj2}\\\\\n&& \\pair{\\lambda,\\nu}=\\operatorname{val}\\lambda(a)\\quad\\text{for any }\\lambda\\in X^*(A).\\label{eq:surj3}\n\\end{eqnarray}\n\\end{lem}\n\n\\begin{proof}\nSee Theorem 1.5.2(5) of \\cite{k}.\n\\end{proof}\n\n\\medbreak\n\nWe consider ${\\mathbb A}(F)_{\\nu}$ endowed with the topology induced from the topology\non ${\\mathbb A}(F)$ described in \\S\\ref{ss:base-field}. By a {\\em generic subset} of\n${\\mathbb A}(F)_{\\nu}$ we mean a subset containing an open dense subset of\n${\\mathbb A}(F)_{\\nu}$. Also, we say that a certain property {\\em holds for a generic\nelement of ${\\mathbb A}(F)_{\\nu}$} if it holds for all elements of some generic subset\nof ${\\mathbb A}(F)_{\\nu}$.\n\n\n\n\\subsection{Twists of a torus}\\label{ss:tor} Let $w\\in W$. In what follows we\nwill denote by $A^w$ the torus of $G$ obtained from $A$ by twisting the Galois\nstructure on $A$ by $w$. Then $X^*(A^w)=X^*(A)$ as abelian groups, and if\n$\\lambda\\in X^*(A^w)$, then the Galois action on $\\lambda$ is given by\n$$\n\\sigma\\lambda(a)=\\lambda(w^{-1}(a)), \\quad a\\in A.\n$$\nWe say that a torus $T$ of $G$ defined over $F$ is {\\em obtained from $A$ by\ntwisting by $w$} if and only if $X^*(T)\\cong X^*(A^w)$ as $\\Gamma$-modules (with\nthe natural action of $\\Gamma$ on $X^*(T)$).\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Statement of the main theorem}\\label{s:main}\n\nWe keep all the notation introduced above. In this section we view $G$ and $A$\nas algebraic groups over $F$. Our goal is to state the main result mentioned in\nthe introduction (Theorem \\ref{t:1}), give a more precise reformulation of this\nresult (Theorem \\ref{t:2}), and reduce its proof to a special case (Theorem\n\\ref{t:3}).\n\n\n\\subsection{Main result} The next two sections, as well as \\S\\ref{ss:Conclusion},\nare devoted to the proof of\n\n\\begin{thm}\\label{t:1}\nLet $\\nu\\in{\\mathcal N}_G.$ Then for generic $c\\in{\\mathbb A}(F)_{\\nu}$  and a regular\nsemisimple element $\\gamma\\in G(F)$ such that $f(\\gamma)=c$ we have\n$$\nG_{\\gamma}\\cong A^{w(\\nu)},\n$$\nwhere $w(\\nu)$ is given by \\eqref{eq:we}.\n\\end{thm}\n\nLet us slightly reformulate Theorem \\ref{t:1}. As we explained in\n\\S\\ref{subsec:f}, there exists $g\\in G(\\overline{F})$ such that\n\\begin{eqnarray*}\ng\\gamma g^{-1}&=&\\gamma_0\\in A(\\overline{F})\\quad\\text{and}\\\\\ng\\sigma(g)^{-1}&\\in &N_G(A)(\\overline{F}).\n\\end{eqnarray*}\nLet $h=h(\\gamma)$ denote the element of $W$ corresponding to $g\\sigma(g)^{-1}$. Note\nthat $h$ is defined up to conjugation in $W$.\n\n\\begin{lem}\nWe have $G_{\\gamma}\\cong A^h.$\n\\end{lem}\n\\begin{proof} Since $gG_{\\gamma}g^{-1}=A$, conjugation by $g$ induces\nthe isomorphism $\\phi:X^*(A)\\longrightarrow X^*(G_{\\gamma})$. By \\S\\ref{ss:tor} it is enough\nto show that $\\phi$ commutes with the action of $\\sigma$. Equivalently,\n\\begin{equation}\\label{eq:gx} (\\sigma\\lambda)(gxg^{-1})=\\sigma\\big(\\phi(\\lambda)\\big)(x),\\quad\\text{for\nany }\\lambda\\in X^*(A),\\,x\\in G_{\\gamma}.\\end{equation} Since $A$ splits, the right side of\n\\eqref{eq:gx} is equal to $\\lambda\\big(\\sigma(g)x\\sigma(g)^{-1}\\big)$, and by the\ndefinition of the Galois action on $X^*(A^h)$, the left hand side of \\eqref{eq:gx} is equal to\n$$\\lambda\\big(h^{-1}(gxg^{-1})\\big)=\\lambda\\big(\\sigma(g)x\\sigma(g)^{-1}\\big).$$\n\\end{proof}\n\nAccording to this lemma to prove Theorem \\ref{t:1} it is enough to prove the\nfollowing\n\n\\begin{thm}\\label{t:2'}\nLet $\\nu\\in{\\mathcal N}_G$. There exists a generic subset $X_\\nu\\subset{\\mathbb A}(F)_\\nu$ such\nthat given $a\\in A(\\overline{F})$ for which $G_a$ is a torus and $c(a)\\in X_{\\nu}$, the\nelement $h\\in W$ given by \\begin{equation} \\sigma(a)=h(a)\\end{equation} is conjugate to $w(\\nu)$.\n\\end{thm}\n\n\n\n\n\n\n\\subsection{Construction of $X_\\nu$}\nRecall that $R\\subset X^*(A)$ denotes the root system of $G$, let $R^+\\subset\nR$ be the set of positive roots with respect to $B$, and let\n$$\n\\Omega:=\\prod_{\\alpha\\in R}\\big(1-e^{\\alpha}\\big)\\in {\\mathbb C}[X^*(A)].\n$$\nFor $\\nu\\in{\\mathcal N}_G$ denote\n$$\nm_{\\nu}:=\\sum_{\\alpha\\in R}\\max\\left\\{0,\\pair{\\alpha,\\nu}\\right\\}.\n$$\nObserve that $m_{\\nu}=\\pair{2\\rho,\\nu}$, where $$\\rho=\\frac{1}{2}\\cdot\\sum_\n{\\alpha\\in R^{+}}\\alpha\\in X^*(A).$$ (This follows from the facts mentioned in the\nfirst section of the proof of Lemma \\ref{l:en} below.) Also, if\n$\\nu\\in\\Lambda_G\\hookrightarrow{\\mathfrak a}_G$ then $m_{\\nu}=0$. Note that for any $a\\in A$ such that\n$c(a)\\in{\\mathbb A}(F)_{\\nu}$, we have $\\operatorname{val}\\Omega(a)\\leq m_{\\nu}$. Indeed, by Lemma\n\\ref{l:kot} there exists $b\\in A$ such that $c(a)=c(b)$ and\n$\\operatorname{val}\\lambda(b)=\\pair{\\lambda,\\nu}$ for any $\\lambda\\in X^*(A)$. Hence $a=g(b)$ for some\n$g\\in W$, and consequently\n$$\n\\operatorname{val}\\Omega(a)=\\operatorname{val}\\Omega(b)=\\sum_{\\alpha\\in R}\\operatorname{val}(1-\\alpha(b))\\leq\\sum_{\\alpha\\in\nR}\\max\\left\\{0,\\pair{\\alpha,\\nu}\\right\\}=m_{\\nu}.\n$$\n(Recall that with our conventions, $\\operatorname{val}$ is the negative of the usual valuation, so that $$\\operatorname{val}(a+b)\\leq\\max\\{\\operatorname{val} a,\\operatorname{val} b\\},\\quad\\text{for any }a,b\\in\\overline{F}.)$$\nOn the other hand, $\\Omega\\in{\\mathbb C}[X^*(A)]$ belongs to the subring ${\\mathbb C}[X^*(A)]^W$\nof $W$-invariant elements in ${\\mathbb C}[X^*(A)]$. Since $G$ comes from an algebraic\ngroup defined over ${\\mathbb C}$,\n$$\n{\\mathbb C}[X^*(A)]^W={\\mathbb C}[c_1,\\ldots,c_n,c_{l+1}^{-1},\\ldots,c_{n}^{-1}].\n$$\nTogether with the note above, this implies that\n$$\nX_{\\nu}:=\\big\\{c\\in{\\mathbb A}(F)_{\\nu}\\,\\big\\vert\\,\\exists\\, a\\in A\\text{ s.t. }\nc=c(a)\\text{ and }\\operatorname{val}\\Omega(a)=m_{\\nu}\\big\\}\n$$\nis an open subset of ${\\mathbb A}(F)_{\\nu}$.\n\n\\medbreak\n\nSince ${\\mathbb A}(F)_{\\nu}$ is irreducible, Theorem \\ref{t:2'} is a consequence of the\nfollowing result:\n\\begin{thm}\\label{t:2}\nFor each $\\nu\\in{\\mathcal N}_G$ the set $X_{\\nu}$ is non-empty and for any $a\\in A$ such\nthat $c(a)\\in X_{\\nu}$, the element $h\\in W$ given by \\begin{equation} \\sigma(a)=h(a) \\end{equation} is\nconjugate to $w(\\nu)$.\n\\end{thm}\n\n\\begin{rem}\\label{r:chert}\nObserve that the condition $c(a)\\in X_{\\nu}$ implies automatically that $G_a$\nis torus, or equivalently, $G_a=A$. Indeed, write $a=a_1a_2$, where $a_1\\in\nA_{sc}$ and $a_2\\in A_G$. Then $G_a=(G_{sc})_{a_1}\\cdot A_G$. Since $a_1\\in\nA_{sc}$, it is in particular a semisimple element of $G_{sc}$, and hence the\ncentralizer $(G_{sc})_{a_1}$ of $a_1$ in $G_{sc}$ is {\\em connected} by a deep\nresult of Springer and Steinberg (\\cite{ss}, Theorem II.3.9), which is false\nwithout the assumption that $G_{sc}$ is simply connected. Thus, $G_a$ is\nconnected, as a product of two connected algebraic groups. Now, assume that\n$G_a$ does not coincide with $A$. Then the Lie algebra $\\operatorname{Lie}(A)$ of $A$ is a\nproper subspace of the Lie algebra $\\operatorname{Lie}(G_a)$ of $G_a$. Moreover, $\\operatorname{Lie}(G_a)$\nis invariant under the adjoint action of $A$, since $A\\subset G_a$. This\nimplies that $\\operatorname{Lie}(G_a)$ contains a root subspace ${\\mathfrak g}_{\\alpha}$ for some $\\alpha\\in R$. Thus\n$\\alpha(a)=1$, and we get a contradiction with $c(a)\\in X_{\\nu}$.\n\\end{rem}\n\n\n\n\\subsection{Reduction from ${\\mathcal N}_G$ to $\\Lambda_G$}\nTo end this section, we observe that in Theorem \\ref{t:2} it is enough to\nassume that $\\nu\\in\\Lambda_G\\hookrightarrow{\\mathfrak a}_G$, which means that we only have to prove\n\n\\begin{thm}\\label{t:3}\nFor each $\\nu\\in\\Lambda_G\\hookrightarrow{\\mathfrak a}_G$ the set $X_{\\nu}$ is non-empty and for any\n$a\\in A$ such that $c(a)\\in X_{\\nu}$, the element $h\\in W$ given by \\begin{equation}\n\\sigma(a)=h(a) \\end{equation} is conjugate to $w(\\nu)=\\psi_G(\\nu)$.\n\\end{thm}\n\nIndeed, we have the following result.\n\n\\begin{lem}\\label{l:en}\nTheorem $\\ref{t:3}$ implies Theorem $\\ref{t:2}.$\n\\end{lem}\n\n\\begin{proof}\nLet $\\nu\\in\\Lambda_P^+$ for some standard parabolic subgroup $P$ of $G$ with the\nLevi subgroup $M$ containing $A$, so that $\\Lambda_P^{+}\\subseteq\\Lambda_M\\hookrightarrow{\\mathfrak a}_M$.\nDenote by $\\Delta_M\\subseteq\\Delta$ the set of simple roots of $M$. Also, let\n$R_M\\subseteq R$ denote the root system of $M$ and let $$\\Omega_M:=\\prod_{\\alpha\\in\nR_M}(1-e^{\\alpha})\\in{\\mathbb C}[X^*(A)].$$ Recall (\\cite{k}, p. 4) that $\\Lambda_P^+$ is contained in\nthe set of elements\n $x\\in{\\mathfrak a}_M$ such that $\\pair{\\alpha,x}>0$ for any $\\alpha\\in\\Delta\\backslash\\Delta_M$.\nSince\n$${\\mathfrak a}_M=\\big\\{x\\in{\\mathfrak a}\\,\\big\\vert\\,\\pair{\\alpha,x}=0,\\,\\forall\\alpha\\in \\Delta_M\\big\\},\n$$\nwe conclude that $\\pair{\\alpha,\\nu}=0$ for any $\\alpha\\in R_M$ and $\\pair{\\alpha,\\nu}\\ne\n0$ for any $\\alpha\\in R\\backslash R_M$. Thus, if $a\\in A$ satisfies\n\\eqref{eq:surj3} and $c(a)\\in{\\mathbb A}(F)_{\\nu}$, then\n\\begin{equation}\\label{eq:op}\\operatorname{val}\\Omega(a)=m_{\\nu}\\Longleftrightarrow\\operatorname{val}\\Omega_M(a)=0.\\end{equation}\n\n\\medbreak\n\nLet ${\\mathbb A}_M$ and $c_M:A\/W_M\\rarab{\\sim}{\\mathbb A}_M$ denote respectively the set ${\\mathbb A}$\nand the map $c$ corresponding to $M$. Consider the map from ${\\mathbb A}_M$ to ${\\mathbb A}$\ndefined by the composition of $c_M^{-1}$ with the natural map $A\/W_M\\longrightarrow A\/W$\n(induced by the embedding $W_M\\hookrightarrow W$) followed by $c$. It is easy\nto check that this map induces a map $\\pi_M:{\\mathbb A}_M(F)_{\\nu}\\longrightarrow{\\mathbb A}(F)_{\\nu}$. We\nare now ready to prove the lemma. Let us show first that $X_{\\nu}$ is\nnon-empty. Let $X_{\\nu}(M)$ denote the set $X_{\\nu}$ defined for $M$. By\nTheorem \\ref{t:3} and Lemma \\ref{l:kot}, there exists $a\\in A$ satisfying\n\\eqref{eq:surj3} such that $c_M(a)\\in{\\mathbb A}_M(F)_{\\nu}$ and $\\operatorname{val}\\Omega_M(a)=0$. Then\n$c(a)=\\pi_M(c_M(a))\\in{\\mathbb A}(F)_{\\nu}$ and $\\operatorname{val}\\Omega(a)=m_{\\nu}$ by \\eqref{eq:op}.\nThus, $c(a)\\in X_{\\nu}$.\n\n\\medbreak\n\nNow we will show that Theorem \\ref{t:2} holds for $X_{\\nu}$. Let $a\\in A$\nsatisfy $c(a)\\in X_{\\nu}$ and $\\sigma(a)=h(a)$. Then by Lemma \\ref{l:kot} there\nexists $b\\in A$ satisfying  \\eqref{eq:surj3} such that $c(a)=c(b)$, hence\n$b=s(a)$ for some $s\\in W$. Thus, $\\sigma(b)=g(b)$ for some $g\\in W$. Since $a$ (and hence $b$) is regular semisimple, as it follows\nfrom Remark \\ref{r:chert}, we see that $g$ is conjugate to $h$. Hence it is enough to\nshow that $g$ is conjugate to $w(\\nu)=\\psi_M(\\nu)$. For any $\\lambda\\in X^*(A)$ we\nhave\n$$\n\\pair{\\lambda,\\nu}=\\operatorname{val}\\lambda(b)=\\operatorname{val}\\sigma(\\lambda(b))=\\operatorname{val} g^{-1}\\lambda(b)=\\pair{\\lambda,g\\nu},\n$$\nwhich implies $\\nu=g\\nu$. Thus, $g\\in W_M$ by standard facts about reflection\ngroups, cf. \\cite{b}. By Lemma \\ref{l:kot} this gives $c_M(b)\\in{\\mathbb A}_M(F)_{\\nu}$\nand also $\\operatorname{val}\\Omega_M(b)=0$ by \\eqref{eq:op}. Hence, $c_M(b)\\in X_{\\nu}(M)$ and\n$g$ is conjugate to $\\psi_M(\\nu)$ by Theorem \\ref{t:3}.\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{A result about root systems}\\label{s:systems}\n\n\n\nThe goal of this section is to show that Theorem \\ref{t:3} follows from a\ncertain statement about root systems (Proposition \\ref{p:sys} below). This\nprovides a link between our main result (which is specific to algebraic groups\nover $F$) and the theory of semisimple algebraic groups over ${\\mathbb C}$. Note that\nthe torus $T$ appearing in \\S\\ref{ss:generalities} corresponds to the torus\n$A_{sc}$ defined in \\S\\ref{ss:setup}; otherwise our notation remains the same\nas before.\n\n\n\n\\subsection{Generalities}\\label{ss:generalities}\nLet $R$ be a (reduced) root system in a real vector space $V$ and let $T$ be\nthe complex torus with character lattice $P(R)$, i.e.,\n$$\nT=\\operatorname{Spec}{\\mathbb C}[P(R)].\n$$\nFor each $\\mu\\in P(R^{\\vee})$ we let $x_{\\mu}$ denote the composition\n$$\nP(R)\\rarab{\\pair{\\cdot,\\mu}}{\\mathbb Q}\/{\\mathbb Z}\\hrarab{\\exp(-2\\pi\n\\sqrt{-1}\\cdot)}{\\mathbb C}^{\\times}.\n$$\nThus $x_{\\mu}$ is a group homomorphism from $P(R)$ to ${\\mathbb C}^{\\times}$, and we\nidentify it with the (complex) point of $T$ it defines.\n\n\\medbreak\n\nLet $W$ denote the Weyl group of $R$, acting on $P(R)$ and hence on $T$ in  the\nusual way, and let\n$$\n\\psi:P(R^{\\vee})\/Q(R^{\\vee})\\longrightarrow W\n$$\ndenote the homomorphism mentioned in \\S\\ref{ss:psi-G} above (see also \\S\\ref{ss:psi}).\n\n\\medbreak\n\nRecall that an element $u\\in T$ is said to be {\\em regular} if the stabilizer\nof $u$ in $W$ is trivial, or, equivalently, if\n$$\n\\alpha(u)\\ne 1,\\quad\\forall\\alpha\\in R.\n$$\n(The equivalence of the two properties is not an obvious statement, and follows\nfrom the same argument as in Remark \\ref{r:chert}.)\n\n\n\n\n\\subsection{Reduction of Theorem \\ref{t:3} to Proposition \\ref{p:sys}}\\label{ss:reduction}\nIn this section we assume\n\n\\begin{prop}\\label{p:sys}\nLet $\\mu\\in P(R^{\\vee})$ and let $h\\in W$. There exists a regular element\n$u\\in T$ with $h(u)=x_{\\mu} u$ if and only if $h$ is conjugate to\n$\\psi(\\bar{\\mu})$.\n\\end{prop}\n\nThe proof is given in \\S\\ref{ss:Conclusion}.\n\n\\begin{lem}\\label{l:enough}\nProposition $\\ref{p:sys}$ implies Theorem $\\ref{t:3}.$\n\\end{lem}\n\n\\begin{proof}\nIn the notation of Theorem \\ref{t:3}, let $R$ be the root system of $G$ with\nrespect to $A$, and note that the torus $T=\\operatorname{Spec}{\\mathbb C}[P(R)]$ is identified with the maximal\ntorus $A_{sc}$ of $G_{sc}$. Fix $\\nu\\in \\Lambda_G$. We first show that $X_{\\nu}$ is\nnon-empty. Put\n$$\nm_i:=\\pair{\\omega_i,\\nu}\\in{\\mathbb Q},\\quad i\\in\\{1,\\ldots,n\\},\n$$\nwhere as usual we consider $\\nu$ as an element of ${\\mathfrak a}_G$ under the embedding\n$\\Lambda_G\\hookrightarrow{\\mathfrak a}_G$. Let $\\mu\\in P(R^{\\vee})$ be such that\n$\\nu\\mapsto\\bar{\\mu}$ under the map\n$$\n\\Lambda_G\\twoheadrightarrow\\Lambda_G\/X_*(A_G)\\hookrightarrow P(R^{\\vee})\/Q(R^{\\vee}).\n$$\nThen \\begin{equation}\\label{eq:equiv}\n\\pair{\\omega_i,\\nu}\\equiv-\\pair{\\varpi_i,\\mu}\\mod\\,{\\mathbb Z}\\quad(1\\leq i\\leq l) \\end{equation}\n(\\cite{k}, p. 14). Furthermore, by definition $\\psi_G(\\nu)=\\psi(\\bar{\\mu})$. By\nProposition \\ref{p:sys} applied to $R$ and $\\mu$ there exists $g\\in W$ and a\nregular element $u\\in T$ such that $g(u)=x_{\\mu} u$. Note that in view of\n\\eqref{eq:equiv} we have $g(u)=x_{\\mu} u$ if and only if \\begin{equation}\\label{eq:starr}\ng^{-1}\\varpi_i(u)=\\exp\\left(2\\pi\\sqrt{-1}m_i\\right)\\cdot\\varpi_i(u),\\quad\n\\forall i\\in\\{1,\\ldots,l\\}. \\end{equation} Let $a\\in A$ be given by\n\\begin{eqnarray*}\n\\omega_i(a)&=&\\epsilon^{m_i}\\cdot\\varpi_i(u),\\quad i\\in\\{1,\\ldots,l\\},\\\\\n\\omega_i(a)&=&\\epsilon^{m_i},\\,\\,\\,\\,\\,\\,\\,\\quad\\quad\\quad i\\in\\{l+1,\\ldots,n\\}.\n\\end{eqnarray*}\nWe claim that $c(a)\\in X_{\\nu}$. Recall that $c\\in{\\mathbb A}(\\overline{F})$ belongs to\n$X_{\\nu}$ if and only if there is $b\\in A$ such that\n\\begin{eqnarray}\n c & = & c(b),\\label{eq:surj10}\\\\\n \\pair{\\omega_i,\\nu} & = & \\operatorname{val}\\omega_i(b),\\quad\\forall i\\in \\{1,\\ldots,n\\},\\label{eq:surj20}\\\\\n\\sigma(b) & = & s(b)\\,\\text{ for some }s\\in W,\\label{eq:surj30}\\text{\nand}\\\\\n\\operatorname{val}\\Omega(b) & = & 0\\label{eq:surj40}.\n\\end{eqnarray}\nClearly, \\eqref{eq:surj20} holds for $b=a$. For \\eqref{eq:surj40}, note that\n$$\n\\alpha(a)=\\epsilon^{\\pair{\\alpha,\\nu}}\\cdot\\alpha(u)=\\alpha(u),\\quad\\forall\\alpha\\in R,\n$$\nwhere on the right we consider $\\alpha$ as a character of $A_{sc}$. Since $u$ is\nregular, this gives \\eqref{eq:surj40} for $b=a$. Thus, it is enough to show\nthat $\\sigma(a)=g(a)$. Since $m_{l+1},\\ldots,m_n\\in{\\mathbb Z}$ and $W$ acts trivially on\n$\\omega_{l+1},\\ldots,\\omega_n$, this is equivalent to \\begin{equation}\\label{eq:**}\n\\sigma(\\omega_i(a))=g^{-1}\\omega_i(a),\\quad\\forall i\\in\\{1,\\ldots,l\\}.\\end{equation} For each $i$\nusing \\eqref{eq:starr} we have\n$$\n\\sigma(\\omega_i(a))=\\epsilon^{m_i}\\cdot\\exp\\left(2\\pi\\sqrt{-1}m_i\\right)\\cdot\\varpi_i(u)=\\epsilon^{m_i}\\cdot\ng^{-1}\\varpi_i(u)=g^{-1}\\omega_i(a),\n$$\nwhich proves \\eqref{eq:**}.\n\n\\medbreak\n\nLet us show now that Theorem \\ref{t:3} holds for $X_{\\nu}$. Let $a\\in A$ and\n$h\\in W$ be such that $c(a)\\in X_{\\nu}$ and $\\sigma(a)=h(a)$. As was explained in\nthe last paragraph of the proof of Lemma \\ref{l:en}, without loss of generality we\ncan assume that $\\operatorname{val}\\lambda(a)=\\pair{\\lambda,\\nu}$ for any $\\lambda\\in X^*(A)$. We need to\nshow that $h$ is conjugate to $w(\\nu)=\\psi_G(\\nu)$. Note first that there\nexists an element $z$ of the center $Z$ of $G$ such that\n\\begin{eqnarray*}\n&& \\omega_i(za)=\\epsilon^{m_i}\\cdot(v_i+\\dotsb),\\quad v_i\\in{\\mathbb C}^{\\times},\\,\\, i\\in\\{1,\\ldots,n\\},\\\\\n&& v_{l+1}=\\dotsb=v_n=1.\n\\end{eqnarray*}\nHere the dots in the formula for $\\omega_i(za)$ denote an element in the maximal ideal of the valuation ring in $\\overline{F}$. Let $u\\in T=\\operatorname{Spec}{\\mathbb C}[P(R)]$ be given by\n$$\n\\varpi_i(u)=v_i,\\quad 1\\leq i\\leq l.\n$$\nWe claim that $u$ is regular and \\begin{equation}\\label{eq:***} h(u)=x_{\\mu} u,\\end{equation}\nwhich together with Proposition \\ref{p:sys} and the fact that $\\psi_G(\\nu)=\\psi\n(\\bar{\\mu})$ implies Theorem \\ref{t:3}.\n\n\\medbreak\n\nSince $z\\in Z$, we have\n$$\n\\alpha(a)=\\alpha(za)=\\alpha(u)+\\dotsb,\\quad \\forall\\alpha\\in R,\n$$\nwhere the dots denote elements in the maximal ideal of the valuation ring in $\\overline{F}$. Since\n$\\operatorname{val}\\Omega(a)=0$, we conclude that $\\alpha(u)\\ne 1$ for any $\\alpha\\in R$, i.e., $u$ is\nregular. Furthermore,\nfor each $i\\in\\{1,\\ldots,l\\}$ we get\n\\begin{eqnarray*}\nh^{-1}\\omega_i(za)&=&\\epsilon^{m_i}\\cdot(h^{-1}\\varpi_i(u)+\\dotsb),\\text{ and}\\\\\n\\sigma(\\omega_i(za))&=&\\epsilon^{m_i}\\cdot\\exp(2\\pi\\sqrt{-1}m_i)\\cdot(\\varpi_i(u)+\\dotsb),\n\\end{eqnarray*}\nwhere as usual the dots denote elements in the maximal ideal of the valuation ring in $\\overline{F}$. Since\n$$\n\\sigma(za)=h(za)\\Longleftrightarrow \\sigma(a)=h(a),\n$$ we obtain\n$$\nh^{-1}\\varpi_i(u)=\\exp(2\\pi\\sqrt{-1}m_i)\\cdot\\varpi_i(u),\\quad\\forall\ni\\in\\{1,\\ldots,l\\},\n$$\nwhich is equivalent to \\eqref{eq:***} by \\eqref{eq:starr}.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Springer's theory for the torus $A$}\\label{s:Springer}\n\n\nThis section (which contains analogues of some results of Springer \\cite{s})\nand the next one can be read independently of the rest of the paper, and are\ninteresting in their own right. The main results are Theorem \\ref{t:abst} and\nTheorem \\ref{t:s}. The reader who is only interested in Newton stratifications\nmay decide to take the statement of Theorem \\ref{t:s} on faith and skip\ndirectly to \\S\\ref{ss:Conclusion}, which contains a proof of Proposition\n\\ref{p:sys}.\n\n\n\\subsection{General conjugacy theorem}\\label{ss:abstract}\nLet $X$ be a separated algebraic variety over ${\\mathbb C}$, and let $Z$ and $W$ be finite\ngroups acting on $X$ by morphisms. For $z\\in Z$ (resp., $w\\in W$) and $x\\in X$\nwe denote by $z\\cdot x$ (resp., $w(x)$) the action of $z$ (resp., of $w$) on\n$x$. Assume that the actions of $Z$ and $W$ commute, i.e.,\n$$\nw(z\\cdot x)=z\\cdot w(x),\\quad\\forall w\\in W,\\,z\\in Z,\\,x\\in X.\n$$\nDenote $X^\\circ:=\\left\\{x\\in X\\vert\\,W_x=\\{1\\}\\right\\}$. Let $Y=X\/W$ be the\nquotient considered as a topological space with the quotient topology of the Zariski topology on $X$. Since\nthe actions of $Z$ and $W$ on $X$ commute, we have the induced action of $Z$ on\n$Y$.\n\n\\begin{thm}\\label{t:abst}\nLet $z\\in Z$ be such that $$Y^z:=\\left\\{y\\in Y\\vert\\,z\\cdot y=y\\right\\}$$ is\nirreducible. If there exist $w_1,w_2\\in W$ and $x_1,x_2\\in X^\\circ$ such that\n$$\nw_i(x_i)=z\\cdot x_i,\\quad i=1,2,\n$$\nthen $w_1$ is conjugate to $w_2$.\n\\end{thm}\n\\begin{proof}\nLet $f:X\\twoheadrightarrow Y$ denote the quotient map. First, note that without loss of\ngenerality we can assume that $X=X^\\circ$. Indeed, $X^\\circ$ is an open, $W$- and\n$Z$-invariant subset of $X$. In particular, $X^\\circ$ is separated. Also, by\nassumption, $f(X^\\circ)^z$ is nonempty and it is an open subset of $Y^z$, because\n$f$ is open and $f(X^\\circ)^z=f(X^\\circ)\\cap Y^z$. Thus, $f(X^\\circ)^z$ is irreducible.\n\n\\smallbreak\n\nThus assume that $X=X^\\circ$. For $w\\in W$ denote $$X(w,z):=\\left\\{x\\in X\\vert\\,w(x)=z\\cdot\nx\\right\\}.$$ Since $X$ is separated, each $X(w,z)$ is closed. Also, for $w\\ne\nw'\\in W$ we have $X(w,z)\\cap X(w',z)=\\emptyset$. Therefore\n$$\nf^{-1}(Y^z)=\\coprod_{w\\in W}X(w,z),\n$$\nand each $X(w,z)$ is open in $f^{-1}(Y^z)$. Since the restriction of $f$ to\n$f^{-1}(Y^z)$ is also an open map, we see that $f\\left(X(w_1,z)\\right)$ and\n$f\\left(X(w_2,z)\\right)$ intersect, being nonempty open subsets of the irreducible space\n$Y^z$. Hence there exist $u,v\\in X$ and $g\\in W$ such that\n$u=g(v)$, $w_1(u)=z\\cdot u$, and $w_2(v)=z\\cdot v$. Combining these three\nequalities and taking into account that $X=X^\\circ$, we get $w_1=gw_2g^{-1}$.\n\\end{proof}\n\n\n\\begin{rem}\nThis argument was obtained by a careful analysis of the proof of Theorem 4.2(iv) in \\cite{s}.\n\\end{rem}\n\n\n\\begin{cor}\\label{c:dim}\nSuppose $X$ is a quasi-projective variety $($so that $Y$ has the structure of an\nalgebraic variety where the underlying topology coincides with the quotient topology$)$, and\n$z\\in Z$ is such that $Y^z$ is irreducible. If $X^\\circ\\cap\nX(w,z)\\ne\\emptyset$ for some $w\\in W$, then $\\operatorname{dim} X(w,z)=\\operatorname{dim} Y^z$. Conversely,\nassume that $Y^z\\cap f(X^\\circ)\\ne\\emptyset$, and let $w\\in W$. If $C$ is an irreducible\ncomponent of $X(w,z)$ with $\\operatorname{dim} C=\\operatorname{dim} Y^z$, then $X^\\circ\\cap C\\ne\\emptyset$.\n\\end{cor}\n\n\\begin{proof}\nLet  $X^\\circ\\cap X(w,z)\\ne\\emptyset$. Since $f$ is open and finite, we have\n\\[\n\\operatorname{dim} X(w,z) \\geq \\operatorname{dim}\n\\left(X^\\circ\\cap X(w,z)\\right)=\\operatorname{dim} f(X^\\circ\\cap X(w,z))=\\operatorname{dim} Y^z \\geq \\operatorname{dim} X(w,z),\n\\]\nwhere the second equality holds because $f(X^\\circ\\cap X(w,z))$ is nonempty and open in $Y^z$.\n\n\\smallbreak\n\nLet $\\operatorname{dim} C=\\operatorname{dim} Y^z$. Then $f(C)$ is dense in $Y^z$, hence intersects $f(X^\\circ)$\nnontrivially, which implies $X^\\circ\\cap C\\ne\\emptyset$, because $X^\\circ$ is\n$W$-invariant.\n\\end{proof}\n\n\n\\subsection{New conventions}\\label{ss:Conventions} In the rest of the paper we will\nbe working in a special case of the setup described in \\S\\ref{ss:setup}. Let\n$G$ be a connected and simply connected simple algebraic group over ${\\mathbb C}$; thus\n$G_{sc}=G$ and $A_{sc}=A$. All our algebraic varieties (resp., groups) will be\ndefined over ${\\mathbb C}$, and will be implicitly identified with their sets (resp.,\ngroups) of complex points; in particular, $G$, $A$, $B$ will stand for\n$G({\\mathbb C})$, $A({\\mathbb C})$, $B({\\mathbb C})$, respectively. The tangent space to an algebraic\nvariety $Y$ at a point $y\\in Y$ will be denoted by $T_y Y$, and if $f:Y\\longrightarrow Y'$\nis a morphism to another algebraic variety $Y'$, its differential at $y$ will\nbe denoted by $D_y f : T_y Y \\longrightarrow T_{f(y)} Y'$.\n\n\n\n\n\\subsection{A variation on Springer's theory of regular elements}\\label{ss:springer}\nThe Weyl group $W=W(G,A)$ acts naturally on $A$ and on the Lie algebra\n$\\operatorname{Lie}(A)$ of $A$. It is well known that $W$ is a finite complex reflection\ngroup in $\\operatorname{Lie}(A)$; thus Springer's results \\cite{s} apply to it. In\nparticular, we recall that Springer introduces the notion of a {\\em regular\nelement} of $W$, namely, it is an element $g\\in W$ which has a regular\neigenvector $v\\in\\operatorname{Lie}(A)$ (the order of the corresponding eigenvalue is then\nnecessarily equal to the order of $g$). Further, he proves a number of useful\nand nontrivial results about regular elements; this includes the fact that if\nthere exists a regular element $g\\in W$ of order $d\\in{\\mathbb N}$, then all such\nelements form a single conjugacy class ({\\em op.~cit.}, Theorem 4.2(iv)), as\nwell as an explicit determination of the eigenvalues of a regular element ({\\em\nop.~cit.}, Theorem 4.2(v)).\n\n\\medbreak\n\nIf one tries to find a version of Springer's theory for the action of $W$ on\nthe torus $A$ itself, rather than on $\\operatorname{Lie}(A)$, the first problem one\nencounters is how to define the analogue of an eigenvector. Indeed, if $\\zeta$ is\na root of unity in ${\\mathbb C}$, the equation $g(v)=\\zeta v$ makes no sense in $A$. Thus\none is led to trying to replace the group of roots of unity by an abelian group\nacting on $A$ in a way which commutes with the Weyl group action. We choose the\nmost naive answer (in some sense), yet one which leads to nontrivial results\nthat have several interesting applications, as the present paper already\ndemonstrates. Namely, the group of roots of unity will be replaced by $Z$, the\ncenter of $G$, acting on $A$ by multiplication. We recall that $Z$ is a finite\nabelian group which is non-canonically isomorphic to $P(R^\\vee)\/Q(R^\\vee)$, and\nwe recall the homomorphism $\\psi:P(R^\\vee)\/Q(R^\\vee)\\longrightarrow W$ used earlier in the\npaper (its definition is reviewed in \\S\\ref{ss:psi} below).\n\n\\medbreak\n\nOur main result in this setup is the following\n\n\\begin{thm}\\label{t:s}\n\\begin{enumerate}[$(a)$]\n\\item For every $x\\in Z$, there exists $g\\in W$ for which there is\na regular element $u\\in A$ satisfying $g(u)=x u$. Moreover, if we choose an\nembedding of groups ${\\mathbb Q}\/{\\mathbb Z}\\hookrightarrow\\bC^\\times$, which induces an isomorphism $Z\\cong\nP(R^\\vee)\/Q(R^\\vee)$, and denote by $\\mu_x\\in P(R^\\vee)\/Q(R^\\vee)$ the element\ncorresponding to $x$ under this isomorphism, then one can take $g=\\psi(\\mu_x)$.\n\\item If $x\\in Z$ is fixed, the element $g$ satisfying the property above is\ndetermined uniquely up to conjugation.\n\\item The eigenvalues of such an element $g$ acting on\n$\\operatorname{Lie}(A)$, counting the multiplicities, are precisely the complex numbers\n$\\{\\varpi_i(x)^{-1}\\}_{i\\in I}$.\n\\end{enumerate}\n\\end{thm}\n\nPart (a) of this theorem follows from a stronger result (Theorem\n\\ref{t:monodromy}) proved in Section \\ref{s:geometric}. Namely, it is possible\nto choose {\\em one} regular element $u\\in A$ that works for all $x\\in Z$\nsimultaneously. The rest of the section is devoted to the proofs of parts (b)\nand (c).\n\n\n\n\\subsection{Proof of Theorem \\ref{t:s}(b)}\\label{ss:Springer} Let $l$ denote the\nrank of $G$ and for $x\\in Z$ let $a(x)$ denote the number of elements in the\nset $\\{ 1\\leq i\\leq l \\,\\bigl\\lvert\\, \\varpi_i(x)=1 \\}$. Also, as in\n\\S\\ref{ss:abstract}, we introduce, for each $g\\in W$ and $x\\in Z$, the subset\n\\[\nA(g,x) = \\{ u\\in A \\,\\bigl\\lvert\\, g(u)=x u \\}.\n\\]\nClearly, Theorem \\ref{t:s}(b) follows from the following more\nprecise proposition, which is an analogue of Theorem 4.2(iv) in \\cite{s}:\n\\begin{prop}\nIf $g\\in W$, $x\\in Z$, the following conditions are equivalent:\n\\begin{enumerate}[$(i)$]\n\\item $\\operatorname{dim} A(g,x)=a(x)$;\n\\item there exists a regular element $u\\in A(g,x)$;\n\\item there exists a regular element in every connected component of $A(g,x)$.\n\\end{enumerate}\nMoreover, for fixed $x$, the elements of $W$ satisfying these properties form a\nsingle conjugacy class.\n\\end{prop}\n\n\\begin{proof}\nNote that although $A(g,x)$ may be disconnected, it is clearly a torsor for the\nsubgroup $A^g$ consisting of the elements of $A$ that are fixed by $g$. In\nparticular, $A(g,x)$ is smooth, its connected components ($=$irreducible\ncomponents) correspond to the connected components of $A^g$, and consequently\nhave the same dimension.\n\n\\smallbreak\n\nWe recall from \\S\\ref{ss:adj-quotient} that we have a $W$-invariant polynomial\nmorphism $c:A\\longrightarrow{\\mathbb A}={\\mathbb A}^l$ whose coordinates $c_i$ are given by\n\\[\nc_i(u) = \\sum_{\\lambda\\in W\\varpi_i} e^\\lambda(u),\\quad u\\in A,\n\\]\nand that $c$ identifies ${\\mathbb A}$ with the quotient $A\/W$. We apply Theorem\n\\ref{t:abst} and Corollary \\ref{c:dim} to $A$ with the action of $Z$ by\nmultiplication and the natural action of $W$. We only need to show that ${\\mathbb A}^x$\nis irreducible. We have\n$$\n{\\mathbb A}^x=\\left\\{(c_i)\\in{\\mathbb A} \\,\\,\\bigl\\lvert\\,\\, c_i=\\varpi_i(x)\\cdot c_i,\\,\\,\n1\\leq i\\leq l \\right\\},\n$$\nwhich implies that ${\\mathbb A}^x$ is an affine space of dimension $a(x)$. Thus, the\nassumptions of Theorem \\ref{t:abst} are satisfied. Using Theorem \\ref{t:s}(a),\nwe see that Corollary \\ref{c:dim} can be applied to prove the equivalence of (i) -- (iii), and\nthe proposition follows from these two results.\n\\end{proof}\n\n\n\n\n\n\\begin{rem}\nIf $G$ is a finite reflection group in a complex vector space $V$, the quotient\n$V\/G$ admits a description similar to the description of $A\/W$ used above.\nUsing this description one can prove Theorem 4.2(iv) of \\cite{s} in a way which\nis completely analogous to the last proof. In the notation of\n\\S\\ref{ss:abstract}, the role of $X$ is played by $V$, the role of $W$ is\nplayed by $G$, and the role of $Z$ is played by the group of roots of unity of\na fixed order $d\\in{\\mathbb N}$. In fact, this argument is essentially identical to the\none used by Springer, modulo some elementary simplifications.\n\\end{rem}\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Proof of Theorem \\ref{t:s}(c)}\\label{ss:Eigenvalues} This subsection is\nan imitation of the proof of Theorem 4.2(v) in \\cite{s}. However, we give a\ncoordinate-free version of Springer's argument.\n\n\\smallbreak\n\nNote that the map $c:A\\longrightarrow{\\mathbb A}$ has the property that the differential $D_u\nc:T_u A \\longrightarrow T_{c(u)}{\\mathbb A}$ is an isomorphism if and only if $u\\in A$ is regular.\nFor each $x\\in Z$, we define $m_x:A\\longrightarrow A$ to be the map $u\\mapsto xu$; since\n$x$ is $W$-invariant, it is clear that we have a commutative diagram\n\\begin{equation}\\label{e:Dilation}\n\\xymatrix{\n  A \\ar[dd]_c \\ar[rr]^{m_x}   &    &   A \\ar[dd]^c \\\\\n    &     &     \\\\\n  {\\mathbb A}  \\ar[rr]^{\\overline{m}_x} &    &   {\\mathbb A}\n   }\n\\end{equation}\nwhere $\\overline{m}_x:{\\mathbb A}\\longrightarrow{\\mathbb A}$ is defined by $(a_i)_{i\\in I}\\mapsto (\\varpi_i(x)\\cdot\na_i)_{i\\in I}$. In particular, given an element $u\\in A$, we have (imitating\n\\cite{s}, \\S3):\n\\[\nxu=g(u)\\quad\\text{for some  } g\\in W \\iff c(xu)=c(u) \\iff \\overline{m}_x(c(u))=c(u)\n\\]\n\\[\n\\iff \\text{for each } i\\in I, \\text{ either } c_i(u)=0, \\text{ or } \\varpi_i(x)=1.\n\\]\n\n\\smallbreak\n\nLet $g\\in G$ and $x\\in Z$ be fixed, and let us also fix a regular element $u\\in\nA(g,x)$ (assuming it exists). The diagram \\eqref{e:Dilation} and the fact that\n$c\\circ g=c$ yield two commutative diagrams (note that $c(xu)=c(g(u))=c(u)$):\n\\[\n\\xymatrix{\n  T_u A \\ar[ddr]_{D_u c} \\ar[rr]^{D_u g}   &    &   T_{xu}A \\ar[ddl]^{D_{xu}c} \\\\\n    &     &     \\\\\n   &     T_{c(u)}{\\mathbb A}     &\n   }\n  \\qquad\\text{and}\\qquad\n\\xymatrix{\n  T_u A \\ar[dd]_{D_u c} \\ar[rr]^{D_u m_x}   &    &   T_{xu}A \\ar[dd]^{D_{xu}c} \\\\\n    &     &     \\\\\n  T_{c(u)} {\\mathbb A}  \\ar[rr]^{D_{c(u)}\\overline{m}_x} &    &  T_{c(u)} {\\mathbb A}\n   }\n\\]\nSince $u$, and hence $xu$, are regular, the maps $D_u c$ and $D_{xu} c$ are\nisomorphisms. Therefore the automorphism\n\\[\n(D_u g)^{-1}\\circ D_u m_x : T_u A \\rarab{\\simeq} T_u A\n\\]\nis equal to\n\\begin{eqnarray*}\n(D_u g)^{-1}\\circ D_u m_x &=& (D_u c)^{-1} \\circ (D_{xu} c) \\circ (D_{xu} c)^{-1} \\circ (D_{c(u)}\\overline{m}_x) \\circ D_u c \\\\\n &=& (D_u c)^{-1} \\circ (D_{c(u)}\\overline{m}_x) \\circ (D_u c).\n\\end{eqnarray*}\nIn particular, $(D_u g)^{-1}\\circ D_u m_x$ has the same eigenvalues as\n$D_{c(u)} \\overline{m}_x$. But $D_{c(u)} \\overline{m}_x$ is obviously given by a diagonal matrix\nin the standard basis, whose eigenvalues are the numbers $\\varpi_i(x)$. On the\nother hand, the chain rule yields\n\\[\n(D_u g)^{-1}\\circ D_u m_x = D_u (g^{-1}\\circ m_x) = D_u(m_x\\circ g^{-1}) =\n(D_{g^{-1}(u)} m_x)\\circ D_u(g^{-1}),\n\\]\nand $(D_{g^{-1}(u)} m_x)\\circ D_u(g^{-1})$ has the same eigenvalues as\n\\[\n(D_1 m_u)^{-1} \\circ (D_{g^{-1}(u)} m_x) \\circ D_u(g^{-1}) \\circ D_1 m_u = D_1(\nm_u^{-1}\\circ m_x \\circ g^{-1} \\circ m_u) = D_1(g^{-1}).\n\\]\nBut $D_1(g^{-1})$, the differential of $g^{-1}$ at the identity element $1\\in\nA$, is nothing but the automorphism by which $g^{-1}\\in W$ acts on $\\operatorname{Lie}(A)$.\nHence we have shown that the eigenvalues of $g^{-1}$ are the numbers\n$\\varpi_i(x)$, completing the proof of part (c) of Theorem \\ref{t:s}.\n\n\n\n\n\n\n\n\n\n\n\\subsection{Proof of Proposition \\ref{p:sys}}\\label{ss:Conclusion} It is clear that\nif Proposition \\ref{p:sys} holds for two root systems, then it also holds for\ntheir direct sum; thus it suffices to prove it for a reduced and irreducible\nroot system. It is then obvious that parts (a) and (b) of Theorem \\ref{t:s} imply\nthe desired result, which finally completes the proof of Theorem \\ref{t:3}, and hence of Theorem \\ref{t:1}.\n\n\\begin{rem}\\label{r:kottwitz}\nAs an additional bonus, we note that Theorem \\ref{t:s} yields an alternate\nproof of Lemma 4.1.1 of \\cite{k}, namely, a geometric one which does not use\nthe classification of simple Lie algebras over ${\\mathbb C}$. Indeed, as explained on\np.~14 of {\\em op.~cit.}, the lemma reduces to the statement that, with our\nnotation, if we choose the embedding $\\iota:{\\mathbb Q}\/{\\mathbb Z}\\hookrightarrow\\bC^\\times$ defined by\n$\\iota(q)=\\exp(-2\\pi\\sqrt{-1}\\cdot q)$ (this is the same choice as the one that\nwas made in Section \\ref{s:systems}, but is complex conjugate to the one used\nby Kottwitz in \\cite{k}), and the corresponding isomorphism $Z\\cong\nP(R^\\vee)\/Q(R^\\vee)$, then the representation of $Z$ on $\\operatorname{Lie}(A)^*$ obtained\nfrom the natural representation of $W$ via the homomorphism $Z\\longrightarrow W$,\n$x\\mapsto\\psi(\\mu_x)$, decomposes into the direct sum of the $1$-dimensional\nrepresentations given by the characters $\\varpi_i\\bigl\\lvert_Z$. However, basic\ncharacter theory of finite groups implies that it is enough to show that for\neach $x\\in Z$, the eigenvalues of $\\psi(\\mu_x)$ on $\\operatorname{Lie}(A)^*$ are the numbers\n$\\varpi_i(x)$, counting the multiplicities, and this is precisely the statement\ndual to part (c) of Theorem \\ref{t:s} above, in view of part (a) of Theorem\n\\ref{t:s}.\n\\end{rem}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Geometric interpretation of $\\psi$}\\label{s:geometric}\n\n\nWe keep the conventions of \\S\\ref{ss:Conventions} and use the notation ($G$,\n$A$, $B$, $R$, $W$, etc.) introduced in \\S\\ref{ss:setup}. In this section we\nstudy in more detail the homomorphism $\\psi:P(R^\\vee)\/Q(R^\\vee)\\to W$\n(determined by the choice of $B$) and prove part (a) of Theorem \\ref{t:s}. The\nmain result of the section is Theorem \\ref{t:monodromy}, which provides a\ngeometric definition of $\\psi$.\n\n\n\\subsection{Classical definition of the homomorphism $\\psi$}\\label{ss:psi} We begin\nby recalling the definition of the homomorphism $\\psi$ that appears in\n\\cite{b}. Perhaps we provide a little more information than is strictly needed\nfor our purposes, but our presentation is somewhat clearer than Bourbaki's, and\nthe additional facts that we mention may help the reader understand the general\npicture.\n\n\\medbreak\n\nLet $V=Q(R)\\otimes_{\\mathbb Z}{\\mathbb R}$; note that our notation is different from the one used\nin \\cite{k}, since Kottwitz denotes by $V$ the complex space\n$Q(R)\\otimes_{\\mathbb Z}{\\mathbb C}$. With our notation, $R$ can be thought of as an abstract\nreduced and irreducible root system in the real vector space $V$ (\\cite{b},\nChapter VI, \\S1, no.~1). We introduce the corresponding affine root system\n$\\widetilde{R}$ in an ad hoc manner as follows. The root lattice of $\\widetilde{R}$ is by\ndefinition the lattice $Q(\\widetilde{R}):=Q(R)\\oplus{\\mathbb Z}\\cdot\\alpha_0$ in the vector space\n$\\widetilde{V}:=V\\oplus{\\mathbb R}\\cdot\\alpha_0$, where $\\alpha_0$ is just an auxiliary symbol. We call\n$\\alpha_0$ the {\\em affine simple root}. Let $\\widetilde{\\al}$ denote the highest root of\n$R$, and define $\\delta=\\alpha_0+\\widetilde{\\al}\\in Q(\\widetilde{R})$. We set\n\\[\n\\widetilde{R} = \\bigl\\{ n\\delta + \\alpha \\,\\lvert\\, \\alpha\\in R,\\ n\\in{\\mathbb Z} \\bigr\\} \\bigcup \\bigl\\{\nm\\delta \\,\\lvert\\, m\\in{\\mathbb Z}\\setminus\\{0\\}\\}.\n\\]\nThis is the {\\em affine root system} associated to $R$. The elements of $\\widetilde{R}$\nof the form $m\\delta$, $m\\in{\\mathbb Z}\\setminus\\{0\\}$, are called {\\em imaginary roots};\nall other elements are called {\\em real} ({\\em affine}) {\\em roots}. A real\nroot $n\\delta+\\alpha$ is called {\\em Dynkin} if $n=0$, and {\\em non-Dynkin}\notherwise. Thus $R$ is identified with the subset of $\\widetilde{R}$ consisting of Dynkin\nroots.\n\n\\medbreak\n\nThe elements $\\{\\alpha_i\\}_{i\\in I\\cup \\{0\\}}$ are the {\\em simple roots} of the\nsystem $\\widetilde{R}$. Thus $\\alpha_0$ is the unique non-Dynkin simple root. The set\n$I\\cup\\{0\\}$ is in natural bijection with the set of vertices of the extended\nDynkin diagram corresponding to the root system $R$. Moreover, let us write\n\\[\n\\widetilde{\\al} = \\sum_{i\\in I} \\delta_i \\alpha_i, \\qquad \\delta_i\\in{\\mathbb N},\n\\]\n(note that Bourbaki uses a different notation: $\\widetilde{\\al}=\\sum n_i\\alpha_i$), and set\n$\\delta_0=1$. Then we have (by construction)\n\\[\n\\delta = \\sum_{i\\in I\\cup\\{0\\}} \\delta_i \\alpha_i.\n\\]\n\n\\medbreak\n\nWe identify $V$ with a quotient of $\\widetilde{V}$, namely, $\\widetilde{V}\\bigl\/({\\mathbb R}\\cdot\\delta)\\cong\nV$.  Explicitly, this isomorphism is the inverse of the composition of the\nnatural inclusion $V=V\\oplus(0)\\hookrightarrow\\widetilde{V}$, followed by the projection. This\nisomorphism takes the image of $\\alpha_i\\in\\widetilde{V}$ in $\\widetilde{V}\\bigl\/({\\mathbb R}\\cdot\\delta)$ to\n$\\alpha_i\\in V$ for all $i\\in I$, and it takes the image of $\\alpha_0\\in\\widetilde{V}$ in\n$\\widetilde{V}\\bigl\/({\\mathbb R}\\cdot\\delta)$ to $-\\widetilde{\\al}\\in V$. Dually, we identify $V^*$ with a\nsubspace of $\\widetilde{V}^*$:\n\\[\nV^* \\cong \\{ f\\in \\widetilde{V}^* \\,\\bigl\\lvert\\, f(\\delta)=0 \\}.\n\\]\nIn particular,\n\\begin{equation}\\label{e:Rvee}\nP(R^\\vee) \\cong \\{ f\\in\\widetilde{V}^* \\,\\bigl\\lvert\\, f(R)\\subseteq{\\mathbb Z},\\ \\ f(\\delta)=0 \\}.\n\\end{equation}\nWe also define\n\\[\nE = \\{ f\\in \\widetilde{V}^* \\,\\bigl\\lvert\\, f(\\delta)=1 \\}.\n\\]\nThis is clearly an affine space for the vector space $V^*$. We now define a\nlinear action $\\mu\\mapsto t_\\mu$ of $P(R^\\vee)$ on $\\widetilde{V}^*$ as follows: if\n$\\mu\\in P(R^\\vee)$ and $f\\in\\widetilde{V}^*$, then $t_\\mu(f)=f+f(\\delta)\\cdot\\mu$. Note that\nthis action preserves $E$, thanks to \\eqref{e:Rvee}, and the induced action of\n$P(R^\\vee)$ on $E$ is simply by translations. We also define $\\mu\\mapsto\nt^*_\\mu$ to be the contragredient action of $P(R^\\vee)$ on $\\widetilde{V}$. On the other\nhand, we define an action of the Weyl group $W$ on $\\widetilde{V}$ by declaring that $W$\nacts trivially on $\\delta$ and acts by its usual action on $V$. Then we also\nobtain the contragredient action of $W$ on $\\widetilde{V}^*$, and it is easy to check\nthat the action of $W$ on $\\widetilde{V}$ (resp., on $\\widetilde{V}^*$) normalizes the action of\n$P(R^\\vee)$ on $\\widetilde{V}$ (resp., on $\\widetilde{V}^*$); in fact, e.g., if $w\\in W$ and\n$\\mu\\in P(R^\\vee)$, then $wt^*_\\mu w^{-1} = t^*_{w(\\mu)}$. This equation\nimplies that we have an action of the {\\em extended Weyl group}\n$W_e:=P(R^\\vee)\\rtimes W$ on $\\widetilde{V}$ and $\\widetilde{V}^*$, and, moreover, this action\npreserves $E$. Note also that $W_e$ contains as a subgroup the {\\em affine Weyl\ngroup} $W_a:=Q(R^\\vee)\\rtimes W$.\n\n\\medbreak\n\nThe last key ingredient is the following fact. For every real affine root\n$\\alpha\\in\\widetilde{R}$, we have the corresponding root hyperplane\n$\\operatorname{Ker}(\\alpha)\\subseteq\\widetilde{V}^*$, and the intersection $\\operatorname{Ker}(\\alpha)\\cap E$ is a\nhyperplane in $E$. The complement of the union of all such affine root\nhyperplanes in $E$ is a disjoint union of bounded connected open subsets of\n$E$, called the {\\em alcoves}. Moreover, it is clear that the action of $W_e$\npermutes the real affine roots, hence also permutes the alcoves. A fundamental\nresult (\\cite{b}, Chapter VI, \\S2, no.~1) is that the affine Weyl group $W_a$\nalready acts {\\em simply transitively} on the alcoves. This implies that if $C$\nis a fixed alcove and $\\Gamma_C$ is the group of automorphisms of $C$ in $W_e$,\nthen $\\Gamma_C$ projects isomorphically onto the quotient $W_e\/W_a\\cong\nP(R^\\vee)\/Q(R^\\vee)$. Furthermore, we do have a canonical choice for $C$: it is\nthe so-called {\\em fundamental alcove}, defined by\n\\[\nC = \\bigl\\{ v\\in E \\,\\,\\bigl\\lvert\\,\\, 0<v(\\alpha_i)<1 \\ \\forall\\,i\\in I\\cup\\{0\\}\n\\bigr\\}.\n\\]\nIf $\\xi:\\Gamma_C\\longrightarrow P(R^\\vee)\/Q(R^\\vee)$ is the corresponding isomorphism and\n$\\mathrm{pr}:W_e\\longrightarrow W$ is the natural projection, then the sought-after homomorphism\n$\\psi:P(R^\\vee)\/Q(R^\\vee) \\longrightarrow W$ is defined as the composition\n$\\psi=\\mathrm{pr}\\circ\\xi^{-1}$. Note that the construction of $\\psi$ is completely\ncanonical once the basis of simple roots $\\{\\alpha_i\\}_{i\\in I}$ has been chosen.\n\n\\medbreak\n\nWe now state the main property of the homomorphism $\\psi$ that will be used\nbelow. Let us denote by $\\{\\overline{\\omega}_i\\}_{i\\in I}$ the basis of $P(R^\\vee)$ dual to\nthe basis $\\{\\alpha_i\\}_{i\\in I}$ of $Q(R)$. For notational convenience, we set\n$\\overline{\\omega}_0=0$. Following Bourbaki (\\cite{b}, Chapter VI, \\S2, no.~2), we define a\nsubset $J\\subseteq I$ by\n\\[\nJ = \\{ i\\in I \\,\\bigl\\lvert\\, \\delta_i=1 \\}.\n\\]\nFor example, $J=I$ if $R$ is of type $A$, and $J\\subsetneq I$ otherwise.\n\\begin{lem}\\label{l:psi} The elements $\\{\\overline{\\omega}_i\\}_{i\\in J\\cup\\{0\\}}$ form a\ncomplete set of representatives for the quotient $P(R^\\vee)\/Q(R^\\vee)$.\nMoreover, for each $i\\in J\\cup\\{0\\}$, we have\n$\\xi^{-1}\\left(\\left[\\overline{\\omega}_i\\right]\\right)=t_{\\overline{\\omega}_i}\\circ\\psi(\\overline{\\omega}_i)$, where\n$\\left[\\overline{\\omega}_i\\right]$ denotes the image of $\\overline{\\omega}_i$ in $P(R^\\vee)\/Q(R^\\vee)$, and\n$\\psi(\\overline{\\omega}_i)$ is viewed as an element of $W_e$ via the obvious inclusion\n$W\\hookrightarrow W_e$.\n\\end{lem}\n\\begin{proof}\nSee \\cite{b}, Chapter VI, \\S2, no.~3, Proposition 6 and\nits Corollary.\n\\end{proof}\n\n\n\n\n\n\n\n\n\\subsection{Interpretation of $\\psi$ in terms of monodromy}\\label{ss:monodromy} The\nhomomorphism $\\psi$ described above does not appear to be very well understood,\nsince the purely algebraic definition of \\cite{b} is not very enlightening. The\nresults of this section and the previous one are a first step towards\nunderstanding the homomorphism $\\psi$ in more concrete terms. One drawback of\nTheorem \\ref{t:s} is that it only describes the conjugacy classes of each of\nthe elements $\\psi(\\mu)$ separately, rather than the conjugacy class of the\nwhole homomorphism $\\psi$. This drawback is corrected in Theorem\n\\ref{t:monodromy} below.\n\n\\medbreak\n\nWe now concentrate on the question of whether the homomorphism $\\psi$ admits a\ngeometric, rather than algebraic, description. One way to state this question\nprecisely is as follows: do there exist a smooth complex algebraic variety $Y$\nwhose fundamental group (with respect to the complex topology) is isomorphic to\n$P(R^\\vee)\/Q(R^\\vee)$ and a $W$-torsor $X$ over $Y$ such that the induced\nmonodromy homomorphism $P(R^\\vee)\/Q(R^\\vee)\\longrightarrow W$ is conjugate to $\\psi$? We\ndo not know a definite answer to this very question, but we do have a\n``semi-positive'' answer, which may be sufficient for many practical purposes.\nThe formulation of the answer is due to Ilya Zakharevich. The main idea is as\nfollows.\n\n\\medbreak\n\nLet $G_{ad}=G\/Z$ be the simple adjoint group corresponding to $G$, let\n$G^{rs}\\subset G$ and $G^{rs}_{ad}\\subset G_{ad}$ be the (Zariski open) subsets\nconsisting of the regular semisimple elements of the groups $G$ and $G_{ad}$,\nand consider the following diagram:\n\\begin{equation}\\label{e:coverings}\n\\xymatrix{\n  \\ar[dd]_{p'} \\widetilde{G_{ad}^{rs}}  &     & \\ar[ll]_{q'} \\ar[dd]^p \\widetilde{G^{rs}}   \\\\\n     &        &     \\\\\n   G_{ad}^{rs}   &     &   \\ar[ll]_q G^{rs}\n   }\n\\end{equation}\nHere $\\widetilde{G^{rs}}$ denotes the incidence variety of pairs $(g,B')$\nconsisting of an element $g\\in G^{rs}$ and a Borel subgroup $B'\\subset G$\ncontaining $g$, and $p$ is the obvious projection. It is well known that $p$ is\na $W$-torsor, and, in particular, topologically it is a covering map. The\n$W$-torsor $p':\\widetilde{G^{rs}_{ad}}\\longrightarrow G^{rs}_{ad}$ is defined in a similar\nmanner; $q$ is induced by the quotient map $G\\longrightarrow G_{ad}$, and $q'$ is induced\nby $q$ in the obvious way. All maps in the diagram above are surjective, and\n$q$ can also be thought of as a $Z$-torsor.\n\n\\medbreak\n\nThe monodromy of $p'$ yields a homomorphism $\\tau:\\pi_1(G_{ad}^{rs})\\longrightarrow W$,\nwhich is well defined up to conjugation. On the other hand, $q$ induces an\nexact sequence of groups\n\\begin{equation}\\label{e:fund-grps}\n\\pi_1(G^{rs}) \\rarab{q_*} \\pi_1(G^{rs}_{ad}) \\longrightarrow Z \\longrightarrow 1.\n\\end{equation}\nOf course, if the composition $\\tau\\circ q_*$ were trivial, then $\\tau$ would\ninduce a homomorphism $Z\\longrightarrow W$. However, this is far from being true, for the\ntriviality of $\\tau\\circ q_*$ would imply that $p$ is a trivial $W$-torsor,\nwhich is certainly false. Instead, we proceed as follows. Let us choose a\ncompact form $K\\subset G$ of $G$, and write $K_{ad}=K\/Z$, which is a compact\nform of $G_{ad}$. The diagram \\eqref{e:coverings} can be restricted to the\n``compact part'', which produces\n\\begin{equation}\\label{e:coverings-compact}\n\\xymatrix{\n  \\ar[dd]_{p'_K} \\widetilde{K_{ad}^{rs}}  &     & \\ar[ll]_{q'_K} \\ar[dd]^{p_K} \\widetilde{K^{rs}}   \\\\\n     &        &     \\\\\n   K_{ad}^{rs}   &     &   \\ar[ll]_{q_K} K^{rs}\n   }\n\\end{equation}\nOf course, $K^{rs}$ (resp., $K^{rs}_{ad}$) just stands for the set of regular\nelements of $K$ (resp., $K_{ad}$), since all elements of $K$ and $K_{ad}$ are\nautomatically semisimple. The advantage of considering this diagram lies in the following\n\\begin{lem}\\label{l:codim}\nThe complement $K\\setminus K^{rs}$ has real codimension $\\geq 3$ in $K$.\n\\end{lem}\n\\begin{proof}\nLet $T=A\\cap K$, which is a maximal torus of $K$. Let $T^{reg}=K^{rs}\\cap T$\ndenote the set of regular elements of $T$. It is well known that every element\n$y\\in K$ is conjugate to an element of $T$, and $y$ is regular if and only if\nit is conjugate to an element of $T^{reg}$. Thus the conjugation map\n\\[\nK \\times \\bigl(T\\setminus T^{reg}\\bigr) \\longrightarrow K\\setminus K^{rs}\n\\]\nis surjective. Moreover, its fibers have dimension $\\geq T+2$, because if $t\\in\nT$ is not regular, its centralizer in $K$ contains $T$ as well as a nontrivial\ncompact semisimple Lie group. Hence\n\\[\n\\operatorname{dim}(K\\setminus K^{rs}) \\leq \\operatorname{dim}(K) + \\operatorname{dim}(T\\setminus T^{reg}) - \\operatorname{dim}(T) -2 \\leq \\operatorname{dim}(K)-3,\n\\]\ncompleting the proof.\n\\end{proof}\n\nA standard fact in algebraic topology now yields\n\n\\begin{cor}\\label{c:fund}\nThe inclusions $K^{rs}\\hookrightarrow K$ and $K_{ad}^{rs}\\hookrightarrow K_{ad}$ induce isomorphisms on the fundamental groups.\n\\end{cor}\n\nThus $K^{rs}$ is simply connected, and the natural map $\\pi_1(K^{rs}_{ad})\\longrightarrow\nZ$ is an isomorphism. In particular, the monodromy of $p'_K$ does determine a\nconjugacy class of homomorphisms $\\tau_K : Z \\longrightarrow W$. Our last main result is\n\n\\begin{thm}\\label{t:monodromy}\nIn the situation of Theorem $\\ref{t:s}$, there exists a regular element $u\\in\nA\\cap K$ such that $\\psi(\\mu_x)(u)=x u$ for all $x\\in Z$. If $\\overline{u}$\ndenotes its image in $K_{ad}^{rs}$, then the corresponding monodromy\nhomomorphism $\\tau_K:Z\\cong\\pi_1(K_{ad}^{rs},\\overline{u})\\longrightarrow W$ is conjugate\nto the homomorphism $x\\mapsto\\psi(\\mu_x)$.\n\\end{thm}\n\nNote that this theorem is not quite what we were looking for, since the\nconstruction is based on the compact form $K$, and hence cannot be formulated\nin a completely algebro-geometric way. The situation can be improved slightly\nby noting that the inclusion $K^{rs}_{ad}\\hookrightarrow G^{rs}_{ad}$ induces a\nhomomorphism\n\\[\nj : Z\\rarab{\\simeq}\\pi_1(K^{rs}_{ad})\\longrightarrow\\pi_1(G^{rs}_{ad})\n\\]\nwhich is a splitting of \\eqref{e:fund-grps}. Hence the homomorphism\n$x\\mapsto\\psi(\\mu_x)$ is also conjugate to the composition of the monodromy of\nthe algebraic covering $p':\\widetilde{G^{rs}_{ad}}\\longrightarrow G^{rs}_{ad}$ with $j$.\nUnfortunately, the definition of $j$ still involves $K$; however, if we could\ncharacterize abstractly all splittings $j$ of \\eqref{e:fund-grps} that arise in\nthis way, then we would have a manageable algebro-geometric description of the\nhomomorphism $\\psi:P(R^\\vee)\/Q(R^\\vee)\\longrightarrow W$.\n\n\n\n\n\n\n\n\\subsection{Proof of Theorem \\ref{t:monodromy}}\\label{ss:proof-monodromy} The proof\nof the first statement of Theorem \\ref{t:monodromy} that we give below was\nexplained to us by R.~Kottwitz (it is much faster and more transparent than our\noriginal argument). A similar idea was suggested independently by J.-P.~Serre.\n\n\\medbreak\n\nTo simplify notation, we assume that the embedding $\\iota:{\\mathbb Q}\/{\\mathbb Z}\\hookrightarrow\\bC^\\times$ is\nchosen to be the same as the one used in Section \\ref{s:systems}, namely,\n$\\iota(q)=\\exp(-2\\pi\\sqrt{-1}\\cdot q)$. Let $T=A\\cap K$, as before. There is a\nnatural identification $T=\\operatorname{Hom}(P(R),S^1)$. If $V$, $\\widetilde{V}$ and\n$E\\subset \\widetilde{V}^*$ are as in \\S\\ref{ss:psi}, we define an ``exponential\nmap''\n\\[\n\\operatorname{Exp} : E \\longrightarrow T\n\\]\nby the formula\n\\[\n\\operatorname{Exp}(f)(v)=\\exp\\bigl(2\\pi\\sqrt{-1}\\cdot f(v)\\bigr) \\qquad \\forall\\, v\\in P(R);\n\\]\nit identifies $T$ with the quotient of $E$ by the translation action of\n$Q(R^\\vee)$. In particular, let $e\\in C$ denote the barycenter of the\nfundamental alcove $C\\subset E$ introduced in \\S\\ref{ss:psi} (i.e., $e$ is the\narithmetic mean of the vertices of $C$). Since $e$ does not lie in any of the\naffine root hyperplanes in $E$, the element $u:=\\operatorname{Exp}(e)\\in T$ is regular. On\nthe other hand, $\\Gamma_C$ fixes $e$ because it preserves $C$. According to Lemma\n\\ref{l:psi}, this means that $t_{-\\nu}(e)\\equiv\\psi(\\nu)(e)$ modulo $Q(R^\\vee)$\nfor all $\\nu\\in P(R^\\vee)\/Q(R^\\vee)$, which is equivalent to the property\n$xu=\\psi(\\mu_x)(u)$ for all $x\\in Z$. This proves the first statement of\nTheorem \\ref{t:monodromy}.\n\n\\medbreak\n\nWe now prove the second statement. Let $u$ be as above, and let $\\overline{u}$\ndenote the image of $u$ in $K^{rs}_{ad}$. For each $x\\in Z$, choose a\ncontinuous path $\\gamma_x$ from $u$ to $xu$ in $K^{rs}$, and let\n$\\overline{\\gamma}_x$ denote its image in $K^{rs}_{ad}$, which is a loop at\n$\\overline{u}$. Then the map $Z\\longrightarrow \\pi_1(K^{ad}_{rs},\\overline{u})$ given by\n$x\\mapsto[\\overline{\\gamma}_x]$ is a group isomorphism by Corollary \\ref{c:fund}.\nSince $K^{rs}$ is simply connected, the projection $p_K:\\widetilde{K}^{rs}\\to\nK^{rs}$ has a (unique) continuous section $\\sigma_K:K^{rs}\\to\\widetilde{K}^{rs}$\nsatisfying $\\sigma_K(u)=(u,B)$, where $B$ is our chosen Borel subgroup of $G$\ncontaining $A\\ni u$. The second statement of Theorem \\ref{t:monodromy} is\nequivalent to the assertion that $\\sigma_K(xu)=(xu,B^{\\psi(\\mu_x)})$ for all $x\\in\nZ$, where $B^{\\psi(\\mu_x)}$ is the Borel subgroup obtained by conjugating $B$\nwith the Weyl group element $\\psi(\\mu_x)\\in W$. However, $xu=\\psi(\\mu_x)(u)$ by\nconstruction, so the equality $\\sigma_K(xu)=(xu,B^{\\psi(\\mu_x)})$ follows from the\nstronger assertion that $\\sigma_K(gug^{-1})=(gug^{-1},B^g)$ for all $g\\in K$. To\nverify the last equality, observe that it holds for $g=1$ by assumption, and\ntherefore holds for all $g\\in K$ by connectedness. This proves Theorem\n\\ref{t:monodromy}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nWe consider the problem of uniformly tessellating sets in a high-dimensional Euclidean space using random hyperplanes. In what follows we use the following notation. For any $a\\in \\mathbb{R}^n$ and $\\tau\\in \\mathbb{R}$ let\n$$\nH(a,\\tau) = \\{x\\in \\mathbb{R}^n \\ : \\ \\langle a,x\\rangle+\\tau=0\\}\n$$\ndenote the hyperplane defined by $(a,\\tau)$. Each hyperplane $H(a,\\tau)$ divides $\\mathbb{R}^n$ into two parts---the positive and the negative ones, and as a result a collection of hyperplanes endows a tessellation of any given subset $T\\subset \\mathbb{R}^n$.\n \nLet $A:\\mathbb{R}^n\\to \\mathbb{R}^m$ be the matrix with rows $a_1^T,\\ldots,a_m^T$, let $\\tau=(\\tau_1,\\ldots,\\tau_m)\\in \\mathbb{R}^m$, and let $\\operatorname{sign}:\\mathbb{R}^m\\to \\{-1,1\\}^m$ be defined by $\\operatorname{sign}(y)=(\\operatorname{sign}(y_i))_{i=1}^m$. In this notation, for any given $x\\in T$, the bit string $\\operatorname{sign}(Ax+\\tau)\\in \\{-1,1\\}^m$ encodes the ``cell\" in which $x$ is located. Moreover, the normalized Hamming distance\n$$\n\\frac{1}{m}d_H(\\operatorname{sign}(Ax+\\tau),\\operatorname{sign}(Ay+\\tau))\n$$\nis the fraction of the $m$ hyperplanes that separate $x$ and $y$.\n\\begin{Definition} \\label{def:tess}\nThe hyperplanes $H(a_i,\\tau_i)$, $i=1,\\ldots,m$, endow a $\\delta$-tessellation of the set $T$ if, for some constant $\\kappa$,\n\\begin{equation}\n\\label{eqn:deltaTessProp}\n\\left|\\frac{\\kappa}{m}d_H(\\operatorname{sign}(Ax+\\tau),\\operatorname{sign}(Ay+\\tau)) - \\|x-y\\|_2\\right|\\leq \\delta, \\qquad \\text{for every} \\  x,y \\in T.\n\\end{equation}\n\\vspace{0.1cm}\n\\end{Definition}\n\nIn other words, a $\\delta$-tessellation implies that the Euclidean distance between any two points is proportional to the fraction of hyperplanes separating the two, up to a small additive error of $\\delta$. In particular, the Euclidean diameter of all the cells of a $\\delta$-tessellation of $T$ is at most $\\delta$. Phrased differently, such a $\\delta$-tessellation means that the map $f:\\mathbb{R}^n\\to \\{-1,1\\}^m$, $f(t)=\\operatorname{sign}(At+\\tau)$, is a \\emph{$\\delta$-embedding}; an almost-isometric embedding of $T$ into the discrete cube, with the latter endowed with the metric $\\frac{\\kappa}{m}d_H$. It is important to stress that an `almost-isometric' embedding here means isometric up to an \\emph{additive error}, rather that a multiplicative one.\n\\par\nApart from being of intrinsic geometric interest, random hyperplanes tessellations and their associated random embeddings appear naturally in at least two applications. First, a $\\delta$-embedding can clearly be used to compress a dataset $T$ into a small number of bits. Property \\eqref{eqn:deltaTessProp} ensures that one can quickly retrieve information on Euclidean distances between the original data up to a small error---an important feature for many computational tasks. In this application, the number of hyperplanes $m$ determines the degree of compression that is achieved. Second, the map $t\\mapsto\\operatorname{sign}(At+\\tau)$ occurs as a simple and popular model in signal processing. In this context, $Ax$ models the analog measurements of a signal $x$ and the map $y\\mapsto \\operatorname{sign}(y+\\tau)$ a coarse quantizer that converts $Ax$ into a digital signal. Property \\eqref{eqn:deltaTessProp} plays an essential role in proving error bounds for algorithms that aim to reconstruct a signal $x$ from its quantized measurements $\\operatorname{sign}(Ax+\\tau)$. In this second application, $m$ corresponds to the number of measurements of the signal. In both applications, it is of significant interest to understand which $m$ is necessary and sufficient to guarantee that $f$ is a $\\delta$-embedding of a given set $T$. We refer to \\cite{DM18,PlV14} and the references therein for a more extensive discussion of these applications.\\par   \nIn this work we focus on what is arguably the most natural random hyperplane tessellation---with directions selected according to the standard gaussian distribution. Motivated by the aforementioned applications, our goal is to derive sharp bounds on the number of such hyperplanes that are needed to generate a $\\delta$-tessellation. This problem was first studied by Plan and Vershynin \\cite{PlV14} for subsets of the Euclidean sphere $S^{n-1}$, and with hyperplanes that pass through the origin (i.e., $\\tau_i=0$ for $i=1,\\ldots,m$).\n\\begin{Remark}\nIt is important to stress that hyperplanes that pass through the origin cannot separate points on any ray $\\{ \\alpha x : \\alpha \\geq 0\\}$. As a result, shifts must be used in a $\\delta$-tessellation of an arbitrary set.\n\\end{Remark}\n\n\nTo formulate the result from \\cite{PlV14}, denote by $G$ the standard gaussian random vector and set\n$$\n\\ell_*(T):= \\mathbb{E} \\sup_{t \\in T} |\\inr{G,t}| \n$$\ndenote the \\emph{gaussian mean width} of a set $T\\subset \\mathbb{R}^n$. Also, let\n$$\nd_{S^{n-1}}(x,y)=\\frac{1}{\\pi}\\arccos\\left(\\frac{\\inr{x,y}}{\\|x\\|_2\\|y\\|_2}\\right)\n$$\nbe the spherical distance.\n\n\\begin{Theorem} \\label{thm:PV} \\cite{PlV14}\nThere are absolute constants $c_1$ and $c_2$ such that the following holds. Let $T \\subset S^{n-1}$ and let $A:\\mathbb{R}^n \\to \\mathbb{R}^m$ be the gaussian matrix. For $\\delta>0$, if\n$$\nm \\geq c_1 \\frac{\\ell_*^2(T)}{\\delta^6}\n$$\nthen with probability at least $1-2e^{-c_2m\\delta^2}$, for every $x,y\\in T$,\n\\begin{equation}\n\\label{eqn:PlVEmbed}\n\\left|\\frac{1}{m}d_H(\\operatorname{sign}(Ax),\\operatorname{sign}(Ay)) - d_{S^{n-1}}(x,y)\\right|\\leq \\delta.\n\\end{equation}\n\\end{Theorem}\n\\begin{Remark}\nIt was conjectured by Plan and Vershynin that the optimal bound on $m$ should be $m\\sim \\ell_*^2(T)\/\\delta^2$.\n\\end{Remark}\nAn improvement on the estimate from Theorem \\ref{thm:PV} can be found in \\cite{OyR15} (see Theorem~2.5 there). To formulate it, let ${\\cal N} (T,\\varepsilon)$ be the covering number at scale $\\varepsilon$ with respect to the Euclidean norm; that is, the smallest number of open Euclidean balls of radius $\\varepsilon$ needed to cover $T$. Moreover, set $T-T=\\{t_1-t_2 : t_1,t_2 \\in T\\}$.\n\\begin{Theorem} \\cite{OyR15}\n\\label{thm:OyR}\nThere are absolute constants $c_1,c_2,$ and $c_3$ such that the following holds. Let $T \\subset S^{n-1}$ and let $A:\\mathbb{R}^n \\to \\mathbb{R}^m$ be the gaussian matrix. For $\\delta>0$ let\n$$\n0<\\theta \\leq c_1 \\frac{\\delta}{\\sqrt{\\log(e\/\\delta)}}\n$$\nand set\n\\begin{equation} \\label{eq:est-OyR15}\nm \\geq c_2 \\left(\\frac{\\log{\\cal N}(T,\\theta)}{\\delta^2} + \\frac{\\ell_*^2((T-T)\\cap \\theta B_2^n)}{\\delta^3}\\right).\n\\end{equation}\nThen with probability at least $1-2e^{-c_3 m\\delta^2}$, for every $x,y\\in T$,\n$$\n\\left|\\frac{1}{m}d_H(\\operatorname{sign}(Ax),\\operatorname{sign}(Ay)) - d_{S^{n-1}}(x,y)\\right|\\leq \\delta.\n$$\n\\end{Theorem}\n\nAs noted previously, the hyperplanes in Theorems~\\ref{thm:PV} and \\ref{thm:OyR} cannot be used to generate a $\\delta$-tessellation of an arbitrary set  $\\mathbb{R}^n$. A natural way of addressing that issue is to consider parallel shifts---meaning that $\\tau_1,\\ldots,\\tau_m$ need not be $0$. As it happens, under relatively mild randomness assumptions on $A$, \\emph{random shifts} prove to be effective---leading to hyperplane tessellations that endow multiplicative \\emph{isomorphic} embeddings in $\\{-1,1\\}^m$.\n\nTo formulate that fact, recall that a random vector $X$ is isotropic if for every $t \\in \\mathbb{R}^n$, $\\mathbb{E} \\inr{X,t}^2 = \\|t\\|_2^2$. The random vector is $L$-subgaussian if for every $p \\geq 2$ and $t\\in \\mathbb{R}^n$,\n$$\n(\\mathbb{E} |\\langle X,t\\rangle|^p )^{1\/p} \\leq L \\sqrt{p} (\\mathbb{E} \\langle X,t\\rangle^2)^{1\/2}.\n$$\nLet $A$ be a matrix whose rows are independent copies of a symmetric, isotropic, $L$-subgaussian random vector and let $\\tau_i$ be independent random variables, which are distributed uniformly in $[-\\lambda,\\lambda]$ and are independent of $A$.\n\n\\begin{Theorem} \\label{thm:DM} \\cite{DM18}\nThere are constants $c_0,...,c_4$ that depend only on $L$ such that the following holds. Let $T \\subset \\mathbb{R}^n$ and set $R=\\sup_{t \\in T} \\|t\\|_2$. Put $\\lambda = c_0 R$ and set\n\\begin{equation} \\label{eqn:bitCompDM18}\nm \\geq c_1R \\log(eR\/\\delta) \\frac{\\ell_*^2(T)}{\\delta^3}.\n\\end{equation}\nThen with probability at least $1-8\\exp(-c_2m\\delta\/R)$, for any $x,y \\in {\\rm conv}(T)$ that satisfy $\\|x-y\\|_2 \\geq \\delta$, one has\n\\begin{equation} \\label{eq:isomorphic}\nc_3\\frac{ \\|x-y\\|_2}{R} \\leq \\frac{1}{m}d_H(\\operatorname{sign}(Ax+\\tau),\\operatorname{sign}(Ay+\\tau)) \\leq c_4\\sqrt{\\log(eR\/\\delta)} \\cdot \\frac{\\|x-y\\|_2}{R}.\n\\end{equation}\n\\end{Theorem}\n\nIt is unrealistic to hope for an almost isometric estimate in the context of Theorem \\ref{thm:DM} when using an arbitrary subgaussian matrix $A$ (because of the behaviour of the means $\\mathbb{E} d_H(\\operatorname{sign}(Ax+\\tau),\\operatorname{sign}(Ay+\\tau))$. But as it happens, our first result is that if $A:\\mathbb{R}^n \\to \\mathbb{R}^m$ is a gaussian matrix and $m$ is well-chosen, then the random mapping $t \\mapsto \\operatorname{sign}(At+\\tau)$ is an almost isometric embedding (in the additive error sense) of an arbitrary $T \\subset \\mathbb{R}^n$. Moreover, the estimate on $m$ is of a similar nature to the one from \\eqref{eq:est-OyR15}.\n\n\\begin{Theorem} \\label{thm:main-A}\nThere exist absolute constants $c_0,...,c_3$ such that the following holds. Let $T \\subset \\mathbb{R}^n$ and put $R=\\sup_{t \\in T} \\|t\\|_2$. Set $\\delta \\in (0,\\frac{R}{2}]$, $u\\geq 1$ and let\n$$\n0<\\theta \\leq c_0 \\frac{\\delta}{\\sqrt{\\log(e\\lambda\/\\delta)}}.\n$$\nConsider $\\lambda \\geq c_1 R\\sqrt{\\log(R\/\\delta)}$ and\n\\begin{equation}\n\\label{eqn:main-ABdm}\nm\\geq c_2 \\left( \\lambda^2 \\frac{\\log {\\cal N}(T,\\theta)}{\\delta^2}  +  \\lambda \\frac{\\ell_*^2((T-T)\\cap \\theta B^n_2)}{\\delta^3} \\right).\n\\end{equation}\nIf $A:\\mathbb{R}^m \\to \\mathbb{R}^n$ is the standard gaussian matrix and $\\tau$ is uniformly distributed in $[-\\lambda,\\lambda]^m$, then with probability at least $1-2\\exp(-c_3\\delta^2m\/\\lambda^2)$, the map $f(t)= \\operatorname{sign}(At+\\tau)$ satisfies\n\\begin{equation} \\label{eqn:GaussianDitheredText}\n\\sup_{x,y\\in T}\\left|\\frac{\\sqrt{2\\pi}\\lambda}{m}d_H(f(x),f(y))-\\|x-y\\|_2\\right|\\leq  \\delta.\n\\end{equation}\n\\end{Theorem}\n\nThe proof of Theorem \\ref{thm:main-A} follows from a generic embedding result, presented in Section~\\ref{sec:generic}. We show that if a deterministic matrix $A:\\mathbb{R}^n \\to \\mathbb{R}^m$ `acts well' on $T$ (in a sense that is clarified in what follows) and $\\tau$ is as in Theorem \\ref{thm:main-A}, then with high probability with respect to $\\tau$, the mapping $t \\mapsto \\operatorname{sign}(At+\\tau)$ is a $\\delta$-embedding of $T$. We then show in Section \\ref{sec:gaussian} that for any $T \\subset \\mathbb{R}^n$, a typical realization of the gaussian matrix $A$ `acts well' on $T$.\n\n\\vskip0.3cm\n\nThe choice of $\\lambda$ is very natural --- random shifts at a scale proportional to the radius of the set ensure that one can separate points that belong to the same ray $\\{\\alpha x : \\alpha \\geq 0\\}$. In fact, $x \\mapsto \\operatorname{sign}(Ax+\\tau)$ can already fail to be a $\\delta$-embedding with high probability on sets consisting of only two points on a ray if $\\lambda$ is smaller than the radius (see Appendix~\\ref{sec:minShift} for details). At the same time, the estimate \\eqref{eqn:main-ABdm} in Theorem \\ref{thm:main-A} looks anything but natural. In fact, it is reasonable to guess that it is suboptimal. It was conjectured that the right tradeoff between the accuracy parameter $\\delta$, the set $T$ and the dimension $m$ is $m \\sim \\ell_*^2(T)\/\\delta^2$---just as in the conjecture regarding subsets of the sphere.\n\\begin{tcolorbox}\nSurprisingly, the assertion of Theorem \\ref{thm:main-A} is optimal (up to logarithmic factors). And with the natural choice of $\\lambda$, the conjectured optimal estimate of $m \\sim \\ell_*^2(T)\/\\delta^2$ is simply false.\n\\end{tcolorbox}\nWe prove that fact by establishing two lower bounds, the first of which deals with properties of arbitrary random embeddings into the discrete cube.\n\n\\begin{Definition} \\label{def:random-emb}\nLet $(f,\\phi)$ be a pair of maps, where $f:B_2^n \\to \\{-1,1\\}^m$ and $\\phi:\\{-1,1\\}^m \\to \\mathbb{R}$. $(f,\\phi)$ is a \\emph{$\\delta$-inner product-preserving embedding} of $W \\subset B_2^n$ if \n\\begin{equation}\n\\label{eqn:defInnProdPres}\n\\sup_{(x,y) \\in W \\times W} \\left| \\phi(f(x),f(y)) - \\langle x,y\\rangle \\right| < \\delta.\n\\end{equation}\nA pair of random maps $(f,\\phi)$ is a \\emph{random inner product-preserving embedding into $\\{-1,1\\}^m$ with parameters $n,\\delta,\\eta,$ and $N$} if, for any $W \\subset B_2^n$ of cardinality $N$, \\eqref{eqn:defInnProdPres} holds with probability at least $1-\\eta$.\n\\end{Definition}\nIn what follows we abuse notation by omitting the dependence of the embedding on the parameters $n$, $\\eta$ and $N$ and keeping track only of the accuracy parameter $\\delta$ and the embedding dimension $m$. Using a polarization argument, it is straightforward to verify that a $\\delta$-embedding of $W\\cup(-W)$ yields a $\\delta$-inner product-preserving embedding of $W$. In particular, the map in Theorem~\\ref{thm:main-A} yields a random inner product-preserving embedding in the sense of Definition \\ref{def:random-emb} if\n\\begin{equation} \\label{eq:est-m-rand-emb}\nm\\sim \\log(e\/\\delta)\\frac{\\log(eN\/\\eta)}{\\delta^2}.\n\\end{equation}\nThe first lower bound we present shows that \\eqref{eq:est-m-rand-emb} is not improvable (up to a logarithmic factor).\n\\begin{Theorem} \\label{thm:main-B-1}\nThere are absolute constants $c_0,c_1,c_2,c_3$ such that the following holds. Let $N$ be an integer and set $0<\\eta<1$. Let $c_0(\\eta\/N)^{1\/2} \\leq \\delta \\leq c_1$ and put $n=\\frac{c_2}{\\delta^2}\\log(eN\/\\eta)$. If $(f,\\phi)$ is a random inner product-preserving embedding into $\\{-1,1\\}^m$ with parameters $n,\\delta,\\eta,N$ then\n$$\nm \\geq c_3 \\frac{\\log (eN\/\\eta)}{\\delta^2}.\n$$\n\\end{Theorem}\n\nThe proof of Theorem \\ref{thm:main-B-1} actually reveals more than is stated: for any random embedding as above, there is a `bad set' $W$ that forces the embedding dimension to be at least $\\delta^{-2} \\log (N\/\\eta)$. Moreover, there is an absolute constant $\\alpha$ so that $\\hat{W}=W\\cup(-W)$ is $\\alpha$-separated. In particular, if $\\delta$ is sufficiently small, then $(\\hat{W}-\\hat{W}) \\cap \\delta B_2^n = \\{0\\}$. As a result, for this bad set the dominating term in the upper estimate on the embedding dimension in Theorem \\ref{thm:main-A} is the entropic one, showing that it cannot be improved (up to logarithmic factors)---regardless of the choice of the random embedding.\n\nThe proof, which can be found in Section \\ref{sec:proof-B1}, uses similar ideas to the ones appearing in a construction due to Alon and Klartag \\cite{AlK17}. Alon and Klartag studied $\\delta$-distance sketches, which are data structures that allow the reconstruction of all Euclidean scalar products between points in a subset of $\\mathbb{R}^n$ of a given cardinality.\n\n\n\\vskip0.3cm\n\n\nIn the second lower bound, our focus is on the specific embedding $t \\mapsto \\operatorname{sign}(At+\\tau)$ that is used in Theorem \\ref{thm:main-A}. We show that for any convex body\\footnote{A convex body in $\\mathbb{R}^n$ is a convex, centrally-symmetric set with a nonempty interior.} $T \\subset \\mathbb{R}^n$, the best possible dimension $m$ one can hope for coincides with the estimate from Theorem \\ref{thm:main-A} (up to logarithmic factors). To formulate that lower bound, set\n$$\nd^*(T) = \\left(\\frac{\\ell_*(T)}{\\sup_{t \\in T}\\|t\\|_2}\\right)^2\n$$\nto be the \\emph{Dvoretzky-Milman dimension} of $T$.\n\\begin{Theorem} \\label{thm:main-B-2}\nThere are absolute constants $c_0$, $c_1$ and $c_2$ such that the following holds. For any convex body $T \\subset \\mathbb{R}^n$, if $\\lambda \\geq c_0 \\ell_*((T-T)\\cap \\delta B^n_2)\/\\sqrt{m}$,\n$$\nm \\leq c_1 \\lambda \\frac{\\ell_*^2((T-T)\\cap \\delta B^n_2)}{\\delta^3},\n$$\nand $f$ is as in Theorem~\\ref{thm:main-A}, then with probability at least $1-2\\exp(-c_2d^*(T))$ there are $x,y \\in T$ such that\n$$\n\\left|\\frac{\\sqrt{2\\pi}\\lambda}{m}d_H(f(x),f(y))-\\|x-y\\|_2\\right| \\geq 2\\delta.\n$$\n\\end{Theorem}\n\n\\begin{Remark}\nThe lower bound on $\\lambda$ in Theorem~\\ref{thm:main-B-2} is not really restrictive. As an example, we will present in Lemma~\\ref{lemma:lower-on-lambda} a rather general situation in which that lower bound holds. \n\\end{Remark}\n\n\nThere is a purely geometric reason for Theorem \\ref{thm:main-B-2} being true. We first show that it is hard to embed a Euclidean ball using the mapping $y \\mapsto \\operatorname{sign}(y + \\tau)$. More accurately, if there is a subset of coordinates $I \\subset \\{1,...,m\\}$ such that $P_I V=\\{ (x_i)_{i \\in I} : x \\in V\\} =r B_2^I$, then embedding $V$ in $\\{-1,1\\}^m$ by using the mapping $y \\mapsto \\operatorname{sign}(y+\\tau)$ fails unless $m$ is sufficiently large (depending on $|I|$, $r$ and $\\lambda$). We then invoke the classical Dvoretzky-Milman Theorem and show that if $T \\subset \\mathbb{R}^n$ is a convex body and $A:\\mathbb{R}^n \\to \\mathbb{R}^m$ is the standard gaussian matrix, then for a well-chosen $I$, with high probability, $P_I V=P_I A (T \\cap \\delta B_2^n)$ contains a large Euclidean ball. In particular, with high probability, the map $t \\mapsto \\operatorname{sign}(At + \\tau)$ fails to be a $\\delta$-embedding of $T$ unless $m$ is as in Theorem \\ref{thm:main-B-2}.\n\n\nThe proof of Theorem \\ref{thm:main-B-2} can be found in Section \\ref{sec:proof-B2}. The rest of that section is devoted to the conjecture we mentioned previously: that $m \\sim \\ell_*^2(T)\/\\delta^2$ suffices to ensure that $t \\mapsto \\operatorname{sign}(At + \\tau)$ is a $\\delta$-embedding of an arbitrary $T \\subset \\mathbb{R}^n$. Thanks to Theorem~\\ref{thm:main-B-2}, we construct an example which shows that the conjecture is false: a set for which $m \\sim \\ell_*^2(T)\/\\delta^3$ is required---see Theorem~\\ref{thm:counter-example}. \n\n\\section{A generic embedding result} \\label{sec:generic}\nThe first step in the proof of Theorem \\ref{thm:main-A} is actually rather general. We consider a set $T \\subset \\mathbb{R}^n$, and a (deterministic) matrix $A:\\mathbb{R}^n \\to \\mathbb{R}^m$. For a parameter $\\lambda>0$, let $\\tau$ be a random vector distributed uniformly  in $[-\\lambda,\\lambda]^m$, and the embedding function we consider is $f: \\mathbb{R}^n \\to \\{-1,1\\}^m$, defined by\n\\begin{equation} \\label{eqn:GaussBinEmdDef}\n f(x)=\\operatorname{sign} (Ax+\\tau).\n\\end{equation}\nHere, as always, the sign-function is applied component-wise. \n\nOur goal is to show that if $A$ acts on $T$ `in a regular way', then $f$ is a $\\delta$-embedding of $T$ into $\\{-1,1\\}^m$. To explain what we mean by `regularity', let $0 < \\theta < \\delta$, and set $T_\\theta \\subset T$ to be a minimal $\\theta$-net of $T$; in particular, $|T_\\theta| = {\\cal N}(T,\\theta)$. For $v \\in \\mathbb{R}^m$ and $1 \\leq s \\leq m$ set\n$$\n\\|v\\|_{[s]} = \\max_{|I|=s} \\left(\\sum_{i \\in I} v_i^2 \\right)^{1\/2},\n$$\nand assume that the following holds:\n\\begin{description}\n\\item{$(a)$} \\underline{uniform $\\ell_1$-concentration:} there is a constant $\\kappa$ such that\n\\begin{equation} \\label{eqn:ell1ell2assump}\n\\sup_{x,y\\in T_{\\theta}}\\left|\\frac{\\kappa}{m} \\|A(x-y)\\|_1 - \\|x-y\\|_2\\right|\\leq \\delta.\n\\end{equation}\n\\item{$(b)$} \\underline{$A$ maps $T$ to `well-spread' vectors:} For $s=\\lfloor\\delta m\/\\lambda\\rfloor$,\n\\begin{equation} \\label{eqn:knormbias}\n\\frac{1}{\\sqrt{s}} \\sup_{x\\in T_\\theta} \\|Ax\\|_{[s]}\\leq \\lambda,\n\\end{equation}\nand\n\\begin{equation} \\label{eqn:knormoscillations}\n\\frac{1}{\\sqrt{s}} \\sup_{x\\in (T-T)\\cap \\theta B_2^n} \\|Ax\\|_{[s]}\\leq \\delta.\n\\end{equation}\n\\end{description}\n\nDenote the normalized Hamming distance on $\\{-1,1\\}^m$ by\n\\begin{equation*}\n\\tilde{d}(x,y)= \\frac{2\\lambda \\kappa}{m}  d_H(x,y)\n\\end{equation*}\nfor a constant $\\kappa$ whose value is specified in what follows. In our application, where $A$ is the gaussian matrix, $\\kappa$ turns out to be an absolute constant. \n\\vskip0.3cm\nOnce $A$ exhibits the necessary regular behaviour, one may show that with high probability with respect to $\\tau$, $f$ is a $\\delta$-embedding of $T$.\n\n\\begin{Theorem} \\label{thm:mainGeneric}\nThere exist absolute constants $c_1,c_2,$ and $c_3$ such that the following holds. Let $\\theta \\leq \\delta$, set\n$$\nm \\geq c_1 \\lambda^2 \\kappa^2\\frac{\\log{\\cal N}(T,\\theta)}{\\delta^2}\n$$\nand assume that $A$ satisfies \\eqref{eqn:ell1ell2assump}, \\eqref{eqn:knormbias}, and \\eqref{eqn:knormoscillations}. Then with probability at least\n$$\n1-2 \\exp(-c_2 \\delta^2 m\/(\\lambda^2 \\kappa^2))\n$$\nwith respect to $\\tau$,\n\\begin{equation*}\n\\sup_{x,y\\in T}\\left|\\tilde{d}(f(x),f(y))-\\|x-y\\|_2\\right|\\leq c_3 (\\kappa+1) \\delta.\n\\end{equation*}\n\\end{Theorem}\n\n\nThe proof of Theorem \\ref{thm:mainGeneric} requires two simple observations.\n\\begin{Lemma}\\label{lem:exp}\nFix $\\lambda>0$ and let $\\tau$ be uniformly distributed in $[-\\lambda, \\lambda]$. If $\\phi_\\lambda(x)=(|x|-\\lambda)\\mathbbm{1}_{\\{|x|\\geq \\lambda\\}}$ then for every $a,b\\in \\mathbb{R}$,\n\\begin{equation*}\n\\left|2\\lambda \\mathbb{P}(\\operatorname{sign}(a+\\tau) \\neq \\operatorname{sign}(b+\\tau))- |a-b|\\right|\\leq \\phi_\\lambda(a)+\\phi_\\lambda(b).\n\\end{equation*}\n\\end{Lemma}\n\nThe proof of Lemma \\ref{lem:exp} follows from a straightforward and tedious computation. It is deferred to Appendix \\ref{app:proof-lem:exp}.\n\n\\begin{Corollary}  \\label{cor:exp}\nLet $x,y \\in T_\\theta$. If \\eqref{eqn:knormbias} holds then\n\\begin{equation*}\n\\left| \\mathbb{E}_\\tau d_H (f(x),f(y)) - \\frac{1}{2\\lambda} \\|A(x-y)\\|_1 \\right| \\leq s.\n\\end{equation*}\n\\end{Corollary}\n\n\n\\noindent {\\bf Proof.}\\ \\  Observe that\n\\begin{align*}\n\\mathbb{E}_\\tau d_H (f(x),f(y)) & =\\mathbb{E}_\\tau \\sum_{i=1}^m \\mathbbm{1}_{\\{ \\operatorname{sign}( (Ax)_i + \\tau_i) \\not= \\operatorname{sign}( (Ay)_i + \\tau_i)\\}}\n\\\\\n& = \\sum_{i=1}^m \\mathbb{P}_\\tau \\left(\\operatorname{sign}( (Ax)_i + \\tau_i) \\not= \\operatorname{sign}( (Ay)_i + \\tau_i)\\right),\n\\end{align*}\nand therefore, by Lemma \\ref{lem:exp},\n$$\n\\left| \\mathbb{E}_\\tau d_H (f(x),f(y)) - \\frac{1}{2\\lambda} \\|A(x-y)\\|_1 \\right|  \\leq \\frac{1}{\\lambda} \\sup_{z \\in T_\\theta}   \\sum_{i=1}^m |(Az)_i| \\mathbbm{1}_{\\{|(Az)_i|>\\lambda\\}}.\n$$\nTo control the right-hand side, note that by \\eqref{eqn:knormbias}, for every $z \\in T_\\theta$,\n\\begin{equation*} \n|(Az)^*_{s}| \\leq \\frac{1}{\\sqrt{s}} \\|Az\\|_{[s]}\\leq \\lambda,\n\\end{equation*}\nwhere $(Az)^*_{s}$ denotes the $s$-largest coordinate of $( |(Az)_i|)_{i=1}^m$. In particular,\n\\begin{equation} \\label{eqn:largeCoeffk-normBias}\n\\sup_{z\\in T_{\\theta}} \\left| \\left\\{i \\in \\{1,...,m\\} \\ : \\ |(Az)_i|> \\lambda \\right\\} \\right| \\leq s.\n\\end{equation}\nThus, using \\eqref{eqn:knormbias} once again,\n$$\n\\frac{1}{\\lambda}\\sup_{z \\in T_\\theta} \\sum_{i=1}^m |(Az)_i| \\mathbbm{1}_{\\{|(Az)_i|>\\lambda\\}} \\leq \\frac{1}{\\lambda}\\sup_{z \\in T_\\theta} \\sqrt{s} \\|Az\\|_{[s]} \\leq s.\n$$\n{\\mbox{}\\nolinebreak\\hfill\\rule{2mm}{2mm}\\par\\medbreak}\n\n\n\\noindent{\\bf Proof of Theorem \\ref{thm:mainGeneric}.} For every $x\\in T$ let $\\pi x  \\in T_\\theta$ satisfy that $\\|x-\\pi x\\|_2 \\leq \\theta$. By the triangle inequality,\n\\begin{align}\\label{eq:four_summands}\n& \\left|\\tilde{d}(f(x),f(y))-\\|x-y\\|_2\\right| \\nonumber\\\\\n& \\qquad \\leq   \\left|\\tilde{d}(f(x),f(y))- \\tilde{d}(f(\\pi x),f(\\pi y))\\right| \\nonumber\n\\\\\n& \\qquad \\qquad + \\left|\\tilde{d}(f(\\pi x),f(\\pi y)) - \\mathbb{E}_{\\tau} \\tilde{d}(f(\\pi x),f(\\pi y)) \\right|\n+ \\left|\\mathbb{E}_{\\tau} \\tilde{d}(f(\\pi x),f(\\pi y)) - \\frac{\\kappa}{m}\\|A(\\pi x - \\pi y)\\|_1\\right| \\nonumber\n\\\\\n& \\qquad \\qquad + \\left|\\frac{\\kappa}{m}\\|A(\\pi x - \\pi y)\\|_1 - \\|\\pi x-\\pi y\\|_2\\right|\n+ \\left| \\|\\pi x-\\pi y\\|_2-\\|x-y\\|_2\\right| \\nonumber\n\\\\\n& \\qquad = (a)+(b)+(c)+(d)+(e).\n\\end{align}\nClearly, $(e)\\leq 2\\theta \\leq 2\\delta$ and, by \\eqref{eqn:ell1ell2assump}, $(d)\\leq \\delta$. Moreover, it follows from Corollary \\ref{cor:exp} that\n\\begin{align*}\n(c)= & \\left|\\mathbb{E}_{\\tau} \\tilde{d}(f(\\pi x),f(\\pi y)) - \\frac{\\kappa}{m}\\|A(\\pi x - \\pi y)\\|_1\\right|\n\\\\\n= & \\frac{2\\lambda \\kappa}{m} \\left|\\mathbb{E}_{\\tau} d_H(f(\\pi x),f(\\pi y)) - \\frac{1}{2\\lambda}\\|A(\\pi x - \\pi y)\\|_1\\right| \\leq 2\\lambda \\kappa \\frac{s}{m} \\leq 2 \\kappa \\delta.\n\\end{align*}\n\nTurning to $(b)$, note that\n$$\n\\tilde{d}(f(x),f(y))=\\frac{2\\lambda \\kappa}{m} \\sum_{i=1}^m \\mathbbm{1}_{\\{\\operatorname{sign}(Ax+\\tau)_i \\not = \\operatorname{sign}(Ay+\\tau)_i\\}},\n$$\nand in particular it is the sum of independent, $\\{0,1\\}$-valued random variables. Hence, by Hoeffding's inequality there is an absolute constant $c$ such that,  for every $x,y\\in T_\\theta$ and $\\delta>0$,\n\\begin{equation*}\n\\mathbb{P}_{\\tau}\\left(\\left|\\tilde{d}(f(\\pi x),f(\\pi y)) - \\mathbb{E}_{\\tau} \\tilde{d}(f(\\pi x),f(\\pi y))\\right| \\geq \\delta \\right)\\leq 2 \\exp\\left(-c\\frac{\\delta^2 m}{\\lambda^2 \\kappa^2}\\right).\n\\end{equation*}\nIt follows that if\n$$\nm \\geq c_1 \\kappa^2 \\frac{\\lambda^2}{\\delta^{2}} \\log(|T_{\\theta}|),\n$$\nthen by the union bound, with probability at least $1-2 \\exp(-c_2\\delta^2 m\/(\\lambda^2\\kappa^2))$,\n\\begin{equation*}\n\\sup_{x,y\\in T} \\left|\\tilde{d}(f(\\pi x),f(\\pi y)) - \\mathbb{E}_{\\tau} \\tilde{d}(f(\\pi x),f(\\pi y))\\right|\\leq \\delta.\n\\end{equation*}\nFinally, to control $(a)$, by the triangle inequality we have that\n\\begin{align*}\n& \\left|\\tilde{d}(f(x),f(y))- \\tilde{d}(f(\\pi x),f(\\pi y))\\right|\n\\\\\n& \\qquad \\leq \\frac{2\\lambda \\kappa}{m} \\sum_{i=1}^m \\left|\\mathbbm{1}_{\\{ \\operatorname{sign}((Ax)_i+\\tau_i) \\neq \\operatorname{sign} ((Ay)_i+\\tau_i)\\}} -\\mathbbm{1}_{ \\{\\operatorname{sign}((A \\pi x)_i+\\tau_i)\\neq \\operatorname{sign}((A \\pi y)_i+\\tau_i)\\}}\\right|.\n\\end{align*}\nObserve that if\n$$\n\\operatorname{sign}((Ax)_i+\\tau_i)=\\operatorname{sign}((A \\pi x)_i+\\tau_i) \\ \\ {\\rm and} \\ \\\n\\operatorname{sign}((Ay)_i+\\tau_i)=\\operatorname{sign}((A\\pi y)_i+\\tau_i),\n$$\nthen\n$$\n\\mathbbm{1}_{\\{\\operatorname{sign}( (Ax)_i+\\tau_i) \\neq \\operatorname{sign}((Ay)_i+\\tau_i)\\}} -\\mathbbm{1}_{\\{ \\operatorname{sign} ((A\\pi x)_i+\\tau_i) \\neq \\operatorname{sign}( (A \\pi y)_i+\\tau_i)\\}}=0.\n$$\nTherefore,\n\\begin{equation}\n\\label{eq:applyDM18}\n\\sup_{x,y\\in T}\\left|\\tilde{d}(f(x),f(y))- \\tilde{d}(f(\\pi x),f(\\pi y))\\right|\n\\leq \\frac{4\\lambda \\kappa}{m} \\sup_{x \\in T} \\sum_{i=1}^m \\mathbbm{1}_{\\{\\operatorname{sign}( (Ax)_i+\\tau_i)\\neq \\operatorname{sign}( (A \\pi x)_i+\\tau_i)\\}}.\n\\end{equation}\nMoreover,\n$$\n\\mathbbm{1}_{\\{\\operatorname{sign}( (Ax)_i+\\tau_i)\\neq \\operatorname{sign}( (A \\pi x)_i+\\tau_i)\\}} = \\mathbbm{1}_{\\{\\tau_i \\in [-(Ax)_i,-(A \\pi x)_i] \\cup [-(A \\pi x)_i,-(Ax)_i]\\}}\n$$\nimplying that\n\\begin{equation} \\label{eqn:indZeroSep}\n\\mathbbm{1}_{\\{\\operatorname{sign}((Ax)_i+\\tau_i)\\neq \\operatorname{sign}( (A \\pi x)_i+\\tau_i)\\}}=0\n\\end{equation}\non the set\n$$\nA_{\\delta,i} = \\left\\{x\\in \\mathbb{R}^n \\ : \\ |(A \\pi x)_i+\\tau_i|>\\delta \\geq |(A (x-\\pi x))_i|\\right\\}.\n$$\nTherefore,\n\\begin{equation} \\label{eqn:signChangeCov}\n\\mathbbm{1}_{\\{\\operatorname{sign}((Ax)_i+\\tau_i) \\neq \\operatorname{sign}( (A \\pi x)_i+\\tau_i)} \\leq \\mathbbm{1}_{A_{\\delta,i}^c} \\leq \\mathbbm{1}_{\\{|(A \\pi x)_i+\\tau_i|\\leq \\delta\\}} + \\mathbbm{1}_{\\{| (A (x-\\pi x))_i|>\\delta\\}},\n\\end{equation}\nand in particular,\n\\begin{align}\\label{eq:proof:Gaussian:twoprocesses}\n&\\sup_{x,y\\in T}\\left|\\tilde{d}(f(x),f(y))- \\tilde{d}(f(\\pi x),f(\\pi y))\\right| \\nonumber\n\\\\\n& \\qquad \\leq \\frac{4\\lambda \\kappa}{m} \\left(\\sup_{y\\in T_{\\theta}} \\sum_{i=1}^m \\mathbbm{1}_{\\{|(Ay)_i+\\tau_i|\\leq \\delta\\}}\n  + \\sup_{z\\in (T-T)\\cap \\theta B_2^n} \\sum_{i=1}^m \\mathbbm{1}_{\\{|(Az)_i|>\\delta\\}}\\right).\n\\end{align}\nClearly, for every $\\alpha \\in \\mathbb{R}$ and $1 \\leq i \\leq m$, $\\mathbb{P}_\\tau(|\\alpha+\\tau_i| \\leq \\delta) \\leq \\delta\/\\lambda$; thus,\n$$\n\\mathbb{P}_{\\tau}\\left(| (Ay)_i+\\tau_i|\\leq \\delta\\right)\\leq \\frac{\\delta}{\\lambda}.\n$$\nBy the Chernoff bound, there is an absolute constant $c_3$ such that with $\\tau$-probability at least $1-\\exp(-c_3m\\delta\/\\lambda)$,\n$$\n\\sum_{i=1}^m \\mathbbm{1}_{\\{|(Ay)_i+\\tau_i|\\leq \\delta\\}}\\leq \\frac{2\\delta m}{\\lambda}.\n$$\nHence, if\n$$\nm \\geq c_4 \\frac{\\lambda}{\\delta} \\log|T_\\theta|\n$$\nthen by the union bound, with $\\tau$-probability at least $1-\\exp(-c_3m\\delta\/(2\\lambda))$\n$$\n\\sup_{y\\in T_{\\theta}} \\frac{4\\lambda \\kappa}{m}\\sum_{i=1}^m \\mathbbm{1}_{\\{|(Ay)_i+\\tau_i|\\leq \\delta\\}} \\leq 8\\kappa \\delta.\n$$\nFinally, by \\eqref{eqn:knormoscillations}, for every $z\\in (T-T)\\cap \\theta B_2^n$,\n\\begin{equation*}\n(Az)^*_{s} \\leq \\frac{1}{\\sqrt{s}}\\|Az\\|_{[s]}\\leq \\delta,\n\\end{equation*}\nimplying that the second term on the right-hand side of \\eqref{eq:proof:Gaussian:twoprocesses} is bounded by $c\\kappa \\delta$.\n{\\mbox{}\\nolinebreak\\hfill\\rule{2mm}{2mm}\\par\\medbreak}\n\n\\subsection{The gaussian case} \\label{sec:gaussian}\n\nWhen $A$ is the standard gaussian matrix, the application of Theorem~\\ref{thm:mainGeneric} is straightforward. The analysis is based on two well-known facts on the way that a gaussian matrix acts on an arbitrary subset of $\\mathbb{R}^n$. The first is a consequence of the standard gaussian concentration inequality, see e.g.\\ \\cite[Lemma 2.1]{PlV14}.\n\\begin{Theorem} \\label{thm:gaussian-1}\nThere is an absolute constant $c$ such that the following holds. Let $T \\subset \\mathbb{R}^n$ and $A:\\mathbb{R}^n \\to \\mathbb{R}^m$ be the standard gaussian matrix. Then for $u \\geq 1$,\n$$\n\\mathbb{P} \\left( \\sup_{t \\in T} \\left| \\|A t\\|_1 - \\mathbb{E} \\|At\\|_1 \\right| \\geq u\\sqrt{m} \\ell_*(T) \\right) \\leq 2\\exp\\left(-cu^2 d^*(T)\\right),\n$$\nwhere\n$$\nd^*(T) = \\left(\\frac{\\ell_*(T)}{{\\cal R}(T)}\\right)^2 \\ \\ {\\rm and} \\ \\ {\\cal R}(T)=\\sup_{t \\in T} \\|t\\|_2.\n$$\n\\end{Theorem}\nThe second fact required here is that a typical realization of $A$ maps $T$ to a set of `regular vectors'.\n\\begin{Theorem} \\label{thm:gaussian-good-position}\nThere are absolute constants $c$ and $C$ such that the following holds. Let $T \\subset \\mathbb{R}^n$ and $A:\\mathbb{R}^n \\to \\mathbb{R}^m$ be the standard gaussian matrix. Then for $u \\geq 1$ and $1 \\leq k \\leq m$, with probability at least $1-2\\exp(-cu^2 k\\log(em\/k))$,\n$$\n\\sup_{t \\in T} \\|At\\|_{[k]} \\leq C \\left(\\ell_*(T) + u {\\cal R}(T) \\sqrt{k \\log (em\/k)} \\right).\n$$\n\\end{Theorem}\nTheorem~\\ref{thm:gaussian-good-position} follows by a standard chaining argument. We include this proof in Appendix~\\ref{sec:knormGaussian} for the sake of completeness. Using Theorem \\ref{thm:gaussian-1} and Theorem \\ref{thm:gaussian-good-position}, it is straightforward to establish Theorem \\ref{thm:main-A}.\n\n\\noindent{\\bf Proof of Theorem \\ref{thm:main-A}.} Following Theorem \\ref{thm:mainGeneric}, we apply Theorem \\ref{thm:gaussian-good-position} for the sets $T_{\\theta}$ and $(T-T) \\cap \\theta B_2^n$. Thus, for every fixed $1 \\leq k \\leq m$, with probability at least $1-2\\exp(-c_0 u^2 k \\log(em\/k))$, for every $t \\in T_{\\theta}$,\n\\begin{equation} \\label{eq:gaussian-k-norm-1}\n\\|At\\|_{[k]} \\leq c_1\\left(\\ell_*(T_{\\theta})+ u {\\cal R}(T_{\\theta}) \\sqrt{k \\log (em\/k)} \\right)\n\\end{equation}\nand for every $z \\in (T-T) \\cap \\theta B_2^n$,\n\\begin{equation} \\label{eq:gaussian-k-norm-2}\n\\|At\\|_{[k]} \\leq c_1\\left(\\ell_*((T-T)\\cap \\theta B_2^n)+ u \\theta \\sqrt{k \\log (em\/k)} \\right).\n\\end{equation}\nUsing these inequalities for $k=\\lfloor\\delta m\/\\lambda\\rfloor$, it is straightforward to verify that \\eqref{eqn:knormbias} and \\eqref{eqn:knormoscillations} hold under the stated conditions and with the stated probability. Moreover, Theorem~\\ref{thm:gaussian-1} immediately implies that \\eqref{eqn:ell1ell2assump} holds with the wanted probability for $\\kappa=\\sqrt{\\frac{\\pi}{2}}$. The result now follows from Theorem~\\ref{thm:mainGeneric}.\n\n{\\mbox{}\\nolinebreak\\hfill\\rule{2mm}{2mm}\\par\\medbreak}\n\n\\begin{tcolorbox}\nInteresting as Theorem \\ref{thm:main-A} may be, it is not really a major surprise that the embedding $t \\mapsto \\operatorname{sign}(At+\\tau)$ is successful. What is far more surprising is that the seemingly unnatural estimate on the dimension happens to be sharp. The next two sections are devoted to the proof of that fact.\n\\end{tcolorbox}\n\n\\section{Proof of Theorem \\ref{thm:main-B-1}} \\label{sec:proof-B1}\n\nThe proof is based on two standard observations (see, e.g., \\cite{BLM13} for their proofs).\n\\begin{Lemma} \\label{lemma:standard}\nLet $X$ be uniformly distributed in $S^{n-1}$. Then for $0 < \\delta <1\/2$ and any $z \\in S^{n-1}$,\n$$\n\\mathbb{P}( |\\langle X,z\\rangle| \\geq \\delta  ) \\geq \\exp(-2\\delta^2 n),\n$$\nand\n$$\n\\mathbb{P}( \\|X-z\\|_2 \\leq \\alpha) \\leq \\frac{1}{\\sqrt{n}} \\alpha^{n-1}.\n$$\n\\end{Lemma}\n\nSet $T \\subset S^{n-1}$ to be a $1\/2$-separated subset of $S^{n-1}$ of cardinality  $|T| = \\exp(c_0n)$. Such a set exists thanks to a standard volumetric argument. The value of the constant $c_0$ remains unchanged for the rest of this section.\n\n\n\\vskip0.3cm\n\n\n\\begin{Lemma} \\label{lemma:first-obs}\nThere is an absolute constant $c_1$ such that the following holds. Let\n\\begin{equation} \\label{eq:cond-on-k-1}\n\\log k \\geq c_1 \\max\\{\\delta^2 n, \\log n\\}\n\\end{equation}\nand let $X_1,...,X_k$ be independent and uniformly distributed in $S^{n-1}$. Then with probability at least $0.99$, for every $t_1,t_2 \\in T$, there is a $1 \\leq j \\leq k$ such that\n$$\n|\\langle X_j,t_1-t_2\\rangle| \\geq \\delta.\n$$\n\\end{Lemma}\n\\noindent {\\bf Proof.}\\ \\  Define $U\\subset S^{n-1}$ by\n$$\nU=\\left\\{\\frac{t_i-t_j}{\\|t_i-t_j\\|_2} : t_i \\not = t_j, \\ t_i,t_j \\in T \\right\\},\n$$\nand let $\\gamma=2\\delta$. \nSince $\\|t_i-t_j\\|_2 \\geq 1\/2$ if $t_i \\not = t_j$, it suffices to show that, with the stated probability, for any $u\\in U$ there is some $1 \\leq j \\leq k$ such that\n$$\n|\\langle X_j,u\\rangle| \\geq \\gamma.\n$$\nLet $X$ be uniformly distributed in $S^{n-1}$ and set $p=\\mathbb{P}(|\\langle X,u\\rangle| \\geq \\gamma)$. By independence of the $X_j$,\n$$\n\\mathbb{P}( \\forall 1 \\leq j \\leq k, : \\ |\\langle X_j,u\\rangle| < \\gamma) = [\\mathbb{P}(|\\langle X,u\\rangle| < \\gamma)]^k = (1-p)^k\\leq e^{-kp}.\n$$\nBy the union bound, the wanted estimate follows if $|T|e^{-kp}\\leq 0.01$. Recalling that $|T| =\\exp(c_0n)$ and that, by Lemma~\\ref{lemma:standard}, $p \\geq \\exp(-c_2\\gamma^2 n)$, the condition holds when $k$ satisfies \\eqref{eq:cond-on-k-1}.\n{\\mbox{}\\nolinebreak\\hfill\\rule{2mm}{2mm}\\par\\medbreak}\n\nThe second observation we require follows immediately from the second part of Lemma~\\ref{lemma:standard} and the union bound.\n\\begin{Lemma} \\label{lemma:second-obs}\nThere are absolute constants $c_1$ and $c_2$ such that the following holds. Let $0 < \\alpha \\leq c_1$ and $\\log k \\leq c_2 n \\log(1\/\\alpha)$. Then with probability at least $0.99$, for every $1 \\leq i \\not = j \\leq k$,\n$\\|X_i \\pm X_j\\|_2 \\geq \\alpha$ and for every $1 \\leq i \\leq k$ and every $t \\in T$, $\\|X_i \\pm t\\|_2 \\geq \\alpha$.\n\\end{Lemma}\n\nCombining Lemma \\ref{lemma:first-obs} and Lemma \\ref{lemma:second-obs}, we find that there are (deterministic) sets $V,T \\subset S^{n-1}$ with the following properties:\n\\begin{description}\n\\item{$(1)$} $V\\cup(-V) \\cup T$ is an $\\alpha$-separated subset of $S^{n-1}$.\n\\item{$(2)$} For every $t_1,t_2 \\in T$ there is some $v \\in V$ such that $|\\langle v,t_1-t_2\\rangle| \\geq \\delta$.\n\\end{description}\nNext, set $s =\\frac{k}{N-1}$; split the set $V$ to $s$ disjoint sets, each  of cardinality $N-1$; and denote those sets by $V_j$, $1 \\leq j \\leq s$. Let $Y$ be uniformly distributed in $T$ and consider the random sets\n$$\nV_j \\cup \\{Y\\}, \\ \\ 1 \\leq j \\leq s,\n$$\nwhich are of cardinality $N$. Note that the only source of randomness in $V_j \\cup \\{Y\\}$ is $Y$. Let $(f,\\phi)$ be a random inner product-preserving embedding into $\\{-1,1\\}^m$ with parameters $n,\\frac{\\delta}{4},\\eta,$ and $N$ (see Definition~\\ref{def:random-emb}), and consider the event\n$$\n\\left\\{ (f,\\phi) \\ \\ \\delta\/4{\\rm -embeds \\ } V_j \\cup \\{Y\\} \\ {\\rm in \\ } \\{-1,1\\}^m \\ {\\rm for \\ every \\ } 1 \\leq j \\leq s \\right\\}.\n$$\nObserve that for every fixed realization of $Y$ and the union bound,\n$$\n\\mathbb{P}_{(f,\\phi)} \\left( \\left\\{ (f,\\phi) \\ \\ \\delta\/4{\\rm-embeds \\ } V_j \\cup \\{Y\\} \\ {\\rm in \\ } \\{-1,1\\}^m \\ {\\rm for \\ every \\ } 1 \\leq j \\leq s \\right\\} \\right) \\geq 1-s\\eta \\geq 1\/2\n$$\nprovided that\n\\begin{equation} \\label{eq:cond-on-k-11}\ns = \\frac{k}{N-1} \\leq \\frac{1}{2\\eta}.\n\\end{equation}\nIn that case,\n\\begin{align*}\n& \\mathbb{P} \\left( \\left\\{ (f,\\phi) \\ \\delta\/4{\\rm -embeds \\ } V_j \\cup \\{Y\\} \\ {\\rm in \\ } \\{-1,1\\}^m \\ {\\rm for \\ every \\ } 1 \\leq j \\leq s \\right\\} \\right)\n\\\\\n& \\qquad = \\mathbb{E}_Y \\left( \\mathbb{P}_{(f,\\phi)} \\left( \\left\\{ (f,\\phi) \\ \\delta\/4{\\rm -embeds \\ } V_j \\cup \\{Y\\} \\ {\\rm in \\ } \\{-1,1\\}^m \\ {\\rm for \\ every \\ } 1 \\leq j \\leq s \\right\\} \\right) \\big| Y \\right) \\geq \\frac{1}{2}.\n\\end{align*}\nHence, by Fubini's Theorem, there is some realization of $(f,\\phi)$ such that\n$$\n\\mathbb{P}_Y \\left( \\left\\{ (f,\\phi) \\ \\delta\/4{\\rm -embeds \\ } V_j \\cup \\{Y\\} \\ {\\rm in \\ } \\{-1,1\\}^m \\ {\\rm for \\ every \\ } 1 \\leq j \\leq s \\right\\} \\right) \\geq \\frac{1}{2}.\n$$\nThus, by the definition of $Y$, there is a set $T^\\prime \\subset T$ of cardinality at least $\\frac{1}{2}|T|=\\frac{1}{2}\\exp(c_0n)$, such that for every $t \\in T^\\prime$ and every $1 \\leq j \\leq s$, the set $V_j \\cup \\{t\\}$ is $\\delta\/4$-embedded in $\\{-1,1\\}^m$ by the fixed embedding $(f,\\phi)$.\n\n\n\n\\vskip0.3cm\nThe key observation is now as follows.\n\n\\begin{Lemma} \\label{lemma:lower-injection}\nThe embedding $f$ is injective on $T^\\prime$.\n\\end{Lemma}\n\n\\noindent {\\bf Proof.}\\ \\   Let $t_1, t_2  \\in T^\\prime$ such that $f(t_1)=f(t_2)$.\nBy Property $(2)$, there is a $v \\in V$ that `separates' $t_1$ and $t_2$; i.e.,\n$$\n\\left|\\langle t_1,v\\rangle - \\langle t_2,v\\rangle\\right| \\geq \\delta.\n$$\nLet $j$ be such that $v \\in V_j$, observe that $(f,\\phi)$ is a $\\delta\/4$-embedding of $V_j \\cup \\{t_1\\}$ and $V_j \\cup \\{t_2\\}$, and trivially we have that $\\phi(f(t_1),f(v))=\\phi(f(t_2),f(v))$. Because $(f,\\phi)$ is a $\\delta\/4$-embedding of both sets, it follows that\n\\begin{equation} \\label{eq:good-embedding}\n\\left| \\phi(f(t_1),f(v)) - \\langle t_1,v\\rangle \\right| \\leq \\frac{\\delta}{4} \\ \\ {\\rm and} \\ \\ \\left| \\phi(f(t_2),f(v)) - \\langle t_2,v\\rangle \\right| \\leq \\frac{\\delta}{4}.\n\\end{equation}\nHence,\n$$\n|\\langle t_1,v\\rangle - \\langle t_2,v\\rangle| \\leq \\left|\\langle t_1,v\\rangle-\\phi(f(t_1),f(v)) \\right| + \\left|\\langle t_2,v\\rangle-\\phi(f(t_2),f(v)) \\right| \\leq \\frac{\\delta}{2},\n$$\nwhich is a contradiction.\n{\\mbox{}\\nolinebreak\\hfill\\rule{2mm}{2mm}\\par\\medbreak}\n\nThanks to Lemma \\ref{lemma:lower-injection} we have that $2^m \\geq |T^\\prime| = \\frac{1}{2}\\exp(c_0n)$; thus, $m \\geq c_1 n$, and all that is left is to verify that it is possible to take $n$ sufficiently large. By \\eqref{eq:cond-on-k-11}, one may set $k=(N-1)\/2\\eta \\sim N\/\\eta$ and by Lemma~\\ref{lemma:first-obs} one has to ensure that\n$$\n\\log(c_1N\/\\eta) \\geq c_2 \\max\\{\\delta^2 n, \\log n\\}.\n$$\nSetting $n=c_3 \\frac{\\log(c_1N\/\\eta)}{\\delta^2}$, the wanted condition holds by the choice of $\\delta \\geq c_4(\\eta\/N)^{1\/2}$, and the condition in Lemma \\ref{lemma:second-obs} is satisfied for a well-chosen absolute constant $\\alpha$ provided that $\\delta\\leq c_5$.\n{\\mbox{}\\nolinebreak\\hfill\\rule{2mm}{2mm}\\par\\medbreak}\n\n\n\\section{Proof of Theorem \\ref{thm:main-B-2}} \\label{sec:proof-B2}\n\nThe proof is based on two ingredients. First, the Dvoretzky-Milman Theorem (for projections). For more information on the Dvoretzky-Milman Theorem and its central role in Asymptotic Geometry Analysis, see e.g.\\ \\cite{MR3331351}.\n\\begin{Theorem} \\label{thm:DvorMil}\nThere are absolute constants $c_1,c_2,$ and $c_3$ such that the following holds. Let $K \\subset \\mathbb{R}^n$ be a convex body, and consider $s\\leq c_1d^*(K)$. If $A:\\mathbb{R}^n \\to \\mathbb{R}^s$ is the gaussian matrix, then with probability at least $1-2\\exp(-c_2 d^*(K))$,\n$$\nc_3 \\ell_*(K) B_2^s \\subset A K.\n$$\n\\end{Theorem}\nThe proof of Theorem \\ref{thm:DvorMil} is standard. It can be found, for example, in \\cite{MR3571258}.\n\n\n\\begin{Remark}\nThe Dvoretzky-Milman Theorem actually implies a two-sided, almost isometric equivalence (i.e., $c_3$ can be made arbitrarily close to $1$, and the lower bound is complemented by an analogous upper bound). However, for our purposes, an isomorphic, one-sided bound suffices.\n\\end{Remark}\n\n\n\n\nThe second component needed here is a standard probabilistic estimate.\n\\begin{Lemma} \\label{lemma:lower-lambda}\nThere are absolute constants $c_0,c_1$ such that the following holds. Let $\\tau=(\\tau_i)_{i=1}^m$ be distributed uniformly in $[-\\lambda,\\lambda]^m$. Then for any $k \\leq m\/2$, with probability $1-c_0\\exp(-c_1k)$, there is a permutation $\\pi$ of $\\{1,...,m\\}$, such that for every $k\/2 \\leq i \\leq k$,\n$$\n\\frac{|\\tau_{\\pi(i)}|}{\\lambda} \\leq \\frac{2i}{m}.\n$$\n\\end{Lemma}\n\n\\noindent {\\bf Proof.}\\ \\  Fix $i \\leq  m\/2$. Note that $\\lambda^{-1} \\tau_j$, $j=1,\\ldots,m$, are independent and distributed uniformly in $[-1,1]$. Hence, for any $1\\leq j\\leq m$,\n$$\n\\mathbb{P}\\left( \\frac{|\\tau_j|}{\\lambda} \\leq \\frac{2i}{m}\\right) = \\frac{2i}{m}.\n$$\nBy a binomial estimate, with probability at least $1-2\\exp(-c_0 i)$,\n$$\n\\left| \\left\\{j\\in\\{1,\\ldots,m\\} : \\frac{|\\tau_j|}{\\lambda} \\leq \\frac{2i}{m} \\right\\} \\right| \\geq i.\n$$\nNow fix $k \\leq m\/2$, and the claim follows by the union bound over $k\/2 \\leq i \\leq k$.\n{\\mbox{}\\nolinebreak\\hfill\\rule{2mm}{2mm}\\par\\medbreak}\n\n\nLet us now complete the proof of Theorem~\\ref{thm:main-B-2}. Let $T$ be a convex body. Fix $\\delta>0$ and observe that for any $t \\in T \\cap \\delta B_2^n$ we have that $f(t)=\\operatorname{sign}(At + \\tau)$ and $f(0)=\\operatorname{sign}(\\tau)$. Therefore,\n$$\nd_H(f(t),f(0)) = \\sum_{i=1}^m \\mathbbm{1}_{\\{ \\operatorname{sign} ( (At)_i + \\tau_i) \\not = \\operatorname{sign}(\\tau_i)\\}}.\n$$\nClearly, $\\operatorname{sign} ( (A t)_i + \\tau_i) \\not = \\operatorname{sign}(\\tau_i)$ if and only if\n$$\n\\tau_i \\in [-(A t)_i,0) \\cup [0,-(A t)_i).\n$$\n\nLet $k \\leq m\/2$ to be specified in what follows. By Lemma~\\ref{lemma:lower-lambda}, with probability at least $1-2\\exp(-c_1 k)$ there exists a permutation $\\pi$ of $\\{1,...,m\\}$ (depending on the realization of $(\\tau_i)_{i=1}^m$) for which\n$$\n\\frac{|\\tau_{\\pi(i)}|}{\\lambda} \\leq \\frac{2i}{m} \\ \\ \\ {\\rm for \\ every } \\ \\ \\frac{k}{2} \\leq i \\leq k.\n$$\nConditioned on that event (which depends only on $\\tau$), there is an absolute constant $c_2$ such that\n$$\n\\left(\\sum_{i=k\/2}^{k} \\tau_{\\pi(i)}^2 \\right)^{1\/2} \\leq c_2 \\lambda \\frac{k^{3\/2}}{m}.\n$$\nLet $I=\\{ \\pi(i) : k\/2 \\leq i \\leq k\\}$ and let $P_I$ be the projection operator onto ${\\rm span}(e_i)_{i \\in I}$. Then $P_I A$ is a $\\frac{k}{2} \\times n$ standard gaussian matrix, and by Theorem \\ref{thm:DvorMil}, with probability at least\n$$\n1-2\\exp(-c_3d^*(T \\cap \\delta B_2^n)),\n$$\nwe have that $P_I A (T \\cap \\delta B_2^n)$ contains the Euclidean ball\n$$\nc_4 \\ell_*(T \\cap \\delta B_2^n) B_2^{|I|}\n$$\nas long as\n\\begin{equation}\n\\label{eqn:condonk1}\nk \\leq c_5 d^*(T \\cap \\delta B_2^n) \\sim \\frac{\\ell_*^2(T \\cap \\delta B_2^n)}{\\delta^2}.\n\\end{equation}\nHence, if in addition\n\\begin{equation}\n\\label{eqn:condonk2}\nc_2 \\lambda \\frac{k^{3\/2}}{m} \\leq c_4 \\ell_*(T \\cap \\delta B_2^n),\n\\end{equation}\nthen there is some $t^* \\in T \\cap \\delta B_2^n$ such that for every $i \\in I$\n$$\n\\tau_i \\in [-(A t^*)_i,0) \\cup [0,-(A t^*)_i).\n$$\nIn particular,\n$$\nd_H(f(t^*),f(0)) \\geq \\frac{k}{2} \\ \\ \\  \\text{and} \\ \\ \\  \\|t^*-0\\|_2 \\leq \\delta.\n$$\nThe required conditions \\eqref{eqn:condonk1}, \\eqref{eqn:condonk2}, and $k\\leq m\/2$ are all satisfied for\n$$\nk \\sim \\min\\left\\{ \\left(\\frac{m}{\\lambda} \\ell_*(T \\cap \\delta B_2^n)\\right)^{2\/3}, \\frac{\\ell_*^2(T \\cap \\delta B_2^n)}{\\delta^2}\\right\\},\n$$\nthanks to the assumed lower bound on $\\lambda$. \n\nTherefore,\n$$\n\\frac{\\lambda}{m} d_H(f(t^*),f(0)) \\geq \\frac{\\lambda k}{2m} \\geq c_6 \\lambda \\min\\left\\{ \\frac{\\ell_*^{2\/3}(T \\cap \\delta B_2^n)}{\\lambda^{2\/3} m^{1\/3}},\\frac{\\ell_*^2(T \\cap \\delta B_2^n)}{m \\delta^2}\\right\\}.\n$$\nand if\n$$\nm \\leq c_7\\lambda \\frac{\\ell_*^2(T \\cap \\delta B_2^n)}{\\delta^3}\n$$\nit follows that\n$$\n\\left| \\sqrt{2\\pi} \\frac{\\lambda}{m} d_{H}(f(t^*),f(0)) - \\|t^*-0\\|_2 \\right| \\geq 2\\delta.\n$$\nThe claim is now evident by noticing that if $T$ is a convex body, then $T-T = 2T$.\n{\\mbox{}\\nolinebreak\\hfill\\rule{2mm}{2mm}\\par\\medbreak}\n\n\\subsection{The lower bound on $\\lambda$}\nThe lower bound in Theorem \\ref{thm:main-B-2} comes with a caveat: that\n$$\n\\lambda \\sqrt{m} \\gtrsim \\ell_*((T-T) \\cap \\delta B_2^n).\n$$\nLet us show that this caveat is not very restrictive.\n\\begin{Lemma} \\label{lemma:lower-on-lambda}\nThere is an absolute constant $c$ and for every $\\beta>0$ there is a constant  $\\gamma=\\gamma(\\beta) \\geq 1$ such that the following holds.\nAssume that\n\\begin{equation} \\label{eq:in-lemma-lower-on-lambda}\n\\ell_*( (T-T) \\cap \\lambda B_2^n) > \\gamma \\ell_*( (T-T) \\cap \\delta B_2^n).\n\\end{equation}\nIf, with probability at least $0.9$, the mapping $t \\mapsto \\operatorname{sign}(At+\\tau)$ is a $\\delta$-embedding of a convex body $T$, then\n\\begin{equation}\n\\label{eq:in-lemma-lower-on-lambda-conclusion}\n\\beta \\lambda \\sqrt{m} \\geq \\ell_*((T-T)\\cap \\delta B^n_2).\n\\end{equation}\n\\end{Lemma}\nIt follows that if the function $r \\mapsto \\ell_*( (T-T) \\cap r B_2^n)$ exhibits a regular decay around $r=\\delta$, then the condition in Lemma \\ref{lemma:lower-on-lambda} is satisfied once $\\lambda \\gtrsim \\delta$. As is explained in Appendix~\\ref{sec:minShift}, the stronger condition $\\lambda\\gtrsim {\\cal R}(T)$ is needed for the mapping $t \\mapsto \\operatorname{sign}(At+\\tau)$ to have any chance of separating close points on rays $\\{\\alpha x : \\alpha \\geq 0\\}$.\n\n\\begin{Remark}\nThe proof of Theorem \\ref{thm:main-B-2} shows that the value of $\\beta$ that is needed is just an absolute constant. Therefore, $\\gamma$ in Lemma \\ref{lemma:lower-on-lambda} may be taken to be an absolute constant as well.\n\\end{Remark}\n\n\\vskip0.3cm\nThe proof of Lemma \\ref{lemma:lower-on-lambda} is based on the behaviour of the function\n$$\n\\psi(r)=\\frac{1}{\\sqrt{m}r}\\ell_*( (T-T) \\cap r B_2^n).\n$$\nOn the one hand, the map $r \\mapsto \\ell_*( (T-T) \\cap r B_2^n)$ is increasing. On the other hand, it is standard to verify that $\\psi$ is continuous and decreasing in $(0,\\infty)$. Also, it tends to $0$ at infinity and to $\\sqrt{n\/m}$ at $0$ (because $T-T=2T$ is a convex body and in particular has a nonempty interior). Also note that by \\eqref{eq:in-lemma-lower-on-lambda}, $\\delta < \\lambda\/\\gamma$. Indeed, \\eqref{eq:in-lemma-lower-on-lambda} and the monotonicity of $r \\mapsto \\ell_*( (T-T) \\cap r B_2^n)$ imply that $\\delta \\leq \\lambda$ (as $\\gamma\\geq 1$). The monotonicity of $\\psi$ therefore implies that $\\psi(\\lambda)\\leq \\psi(\\delta)$ and hence\n$$\n\\ell_*((T-T) \\cap \\lambda B_2^n) \\leq \\frac{\\lambda}{\\delta} \\ell_*((T-T) \\cap \\delta B_2^n) < \\frac{\\lambda}{\\delta} \\cdot \\frac{1}{\\gamma} \\ell_*((T-T) \\cap \\lambda B_2^n).\n$$\nThe monotonicity of $\\psi$ and the fact that $\\delta<\\lambda\/\\gamma$ are used frequently in the proof of Lemma~\\ref{lemma:lower-on-lambda}.\n\n\n\n\n\\noindent {\\bf Proof.}\\ \\   Let $c_0$ be an absolute constant whose value is specified in what follows.  The key to the proof is to show that if $\\rho$ satisfies\n\\begin{equation} \\label{eq:lower-fixed-1}\n\\ell_*((T-T)\\cap \\delta B_2^n) \\leq c_0 \\frac{\\rho}{\\gamma} \\sqrt{m},\n\\end{equation}\nand\n\\begin{equation} \\label{eq:lower-fixed-2}\n\\ell_*((T-T)\\cap \\rho B_2^n) \\geq  c_0 \\rho \\sqrt{m}\n\\end{equation}\nthen $\\lambda \\geq c^\\prime \\rho$ for an absolute constant $c^\\prime$. Once we establish that, the claim follows immediately: by \\eqref{eq:lower-fixed-1}, we have that\n$$\n\\ell_*((T-T)\\cap \\delta B^n_2) \\leq c_0\\frac{\\rho}{\\gamma} \\sqrt{m} \\leq  \\frac{c_0}{c^\\prime \\gamma} \\lambda\\sqrt{m} \\leq \\beta \\lambda \\sqrt{m},\n$$\nprovided that $\\gamma \\geq c_0\/(c^\\prime \\beta)$.\n\n\nThe starting point of the proof is, once again, the Dvoretzky-Milman Theorem. By \\eqref{eq:lower-fixed-2}, the Dvoretzky-Milman dimension of $(T-T)\\cap \\rho B_2^n$ satisfies\n$$\nd^*((T-T)\\cap \\rho B_2^n) \\geq c_0^2 m.\n$$\nIn particular, if $c_0$ is a sufficiently large absolute constant and $A:\\mathbb{R}^n \\to \\mathbb{R}^m$ is the standard gaussian matrix, then with probability at least $0.9$,\n$$\nc_1 \\ell_*((T-T)\\cap \\rho B_2^n) B_2^m \\subset A((T-T)\\cap \\rho B_2^n)\n$$\nfor an absolute constant $c_1$. For the rest of the proof, the values of $c_0$ and $c_1$ remain unchanged.\n\nBefore proceeding to the heart of the proof, let us show that we may assume without loss of generality that a suitable choice of $\\rho$, satisfying \\eqref{eq:lower-fixed-1} and \\eqref{eq:lower-fixed-2}, exists.\n\nConsider two cases: if $c_0 \\geq \\sqrt{n\/m}$ then trivially, for any $r>0$, $\\psi(r) \\leq c_0$. In other words, for every $r>0$,\n$$\n\\ell_*( (T-T) \\cap r B_2^n) \\leq c_0 r \\sqrt{m}.\n$$\nIn particular, for $r=\\delta$,\n$$\n\\ell_*( (T-T) \\cap \\delta B_2^n) \\leq c_0 \\delta \\sqrt{m} \\leq \\beta \\lambda \\sqrt{m},\n$$\nwhen $\\lambda \\geq c_0\\delta\/\\beta$. Recall that $\\lambda > \\gamma \\delta$, and thus it suffices that $\\gamma \\geq c_0\/\\beta$ for the wanted estimate \\eqref{eq:in-lemma-lower-on-lambda-conclusion} to hold.\n\nTurning to the second case, when $c_0 \\leq \\sqrt{n\/m}$, then since $\\psi$ is continuous and decreasing from $\\sqrt{n\/m}$ to $0$, there is some $\\rho^*$ for which\n\\begin{equation}\n\\label{eqn:rhoStarDef}\n\\ell_*((T-T)\\cap \\rho^* B_2^n) = c_0\\rho^* \\sqrt{m}.\n\\end{equation}\nIn particular, $\\rho^*$ satisfies \\eqref{eq:lower-fixed-2}.\n\nNext, observe that either $\\rho^*$ satisfies \\eqref{eq:lower-fixed-1} and also $\\lambda \\leq \\rho^*$, or, alternatively, the wanted estimate \\eqref{eq:in-lemma-lower-on-lambda-conclusion} holds. Indeed, if $\\lambda \\geq \\rho^*$ there is nothing to prove: in that case, using our assumption \\eqref{eq:in-lemma-lower-on-lambda}, $\\psi(\\lambda)\\leq \\psi(\\rho^*)$, and \\eqref{eqn:rhoStarDef} we find\n$$\n\\ell_*( (T-T) \\cap \\delta B_2^n) < \\frac{1}{\\gamma} \\ell_*( (T-T) \\cap \\lambda B_2^n) \\leq \\frac{c_0}{\\gamma} \\lambda \\sqrt{m} \\leq \\beta \\lambda \\sqrt{m}\n$$\nprovided that $\\gamma \\geq c_0\/\\beta$. On the other hand, consider the case $\\lambda \\leq \\rho^*$. If $\\rho^*$ does not satisfy \\eqref{eq:lower-fixed-1} then by \\eqref{eqn:rhoStarDef} and \\eqref{eq:in-lemma-lower-on-lambda},\n$$\n\\ell_*((T-T)\\cap \\delta B_2^n) > c_0\\frac{\\rho^*}{\\gamma} \\sqrt{m} = \\frac{1}{\\gamma} \\ell_*((T-T)\\cap \\rho^* B_2^n) \\geq \\frac{1}{\\gamma} \\ell_*((T-T)\\cap \\lambda B_2^n) > \\ell_*((T-T)\\cap \\delta B_2^n),\n$$\nwhich is impossible.\n\nTherefore, we may assume for the remainder of the proof that $c_0 \\leq \\sqrt{n\/m}$, that $\\rho^*$ satisfies both \\eqref{eq:lower-fixed-1} and \\eqref{eq:lower-fixed-2}, and that $\\lambda \\leq \\rho^*$. To complete the proof we require three observations.\n\nFirst, $\\delta \\leq \\rho^*\/\\gamma$ since $\\delta \\leq \\lambda \\leq \\rho^*$ and $\\delta \\leq \\lambda\/\\gamma$.\n\nSecond, by applying Theorem~\\ref{thm:gaussian-good-position} (with $k=m$) to the set\n$$\n(T-T) \\cap \\delta B_2^n=2T \\cap \\delta B_2^n,\n$$\nand using \\eqref{eq:lower-fixed-1}, there is an absolute constant $c_2$ such that with probability at least $0.99$, for any $t \\in T$,\n$$\nA\\left( \\left(t + \\{z \\in T : \\|t-z\\|_2 \\leq \\delta\\} \\right) \\right) \\subset At + c_2 \\left(\\delta+\\frac{\\rho^*}{\\gamma}\\right) \\sqrt{m} B_2^m \\subset At + c_3 \\frac{\\rho^*}{\\gamma} \\sqrt{m} B_2^m,\n$$\nwhere we have used that $\\delta \\leq \\rho^*\/\\gamma$.\n\nThird, as noted previously, by the choice of $c_0$ and the Dvoretzky-Milman Theorem, with probability at least $0.99$,\n$c_1 \\ell_*((T-T)\\cap \\rho^* B_2^n) B_2^m \\subset A((T-T)\\cap \\rho^* B_2^n)$. Moreover, $T-T=2T$ and $2T \\cap \\rho^* B_2^n = 2(T \\cap (\\rho^*\/2)B_2^n)$. Therefore,\n$$\nc_1\\cdot c_0 \\rho^* \\sqrt{m} B_2^m = c_1 \\ell_*((T-T)\\cap \\rho^* B_2^n) B_2^m \\subset 2 A(T \\cap (\\rho^*\/2) B_2^n).\n$$\nNow set $c_4=\\frac{c_1}{2}c_0$ and note that $c_4$ is an absolute constant. If $\\tau$ is supported in the interval $[-c_4\\rho^*\/10,c_4 \\rho^*\/10]$, then any two points in the same ``quadrant\" of the set\n$$\nV=c_4 \\rho^* \\sqrt{m} B_2^{m} \\cap \\left\\{ |x_i| \\geq \\frac{c_4 \\rho^*}{10}, {\\rm for \\ every \\ } 1 \\leq i \\leq m \\right\\},\n$$\nare indistinguishable for any realization of the function $y \\mapsto \\operatorname{sign}(y+\\tau)$. Clearly, $V \\subset A(T \\cap (\\rho^*\/2) B_2^n)$ and the diameter of each of its quadrants is at least $(c_4\/10) \\rho^* \\sqrt{m}$.\n\nFix a realization of $A$ in the intersection of the two events stated above---which has probability at least $0.98$. On that event, for every $t \\in T$,\n$$A\\left( \\left(t + \\{z \\in T : \\|t-z\\|_2 \\leq \\delta\\} \\right) \\right)$$\nhas diameter at most $2c_3 \\frac{\\rho^*}{\\gamma} \\sqrt{m}$. On the other hand, any quadrant ${\\cal Q}$ of $V$ has diameter at least $(c_4\/10) \\rho^* \\sqrt{m}$. Hence, we can pick $y_1,y_2$ in ${\\cal Q}$ with $\\|y_1-y_2\\|_2\\geq (c_4\/10) \\rho^* \\sqrt{m}$. Since $V \\subset A(T \\cap (\\rho^*\/2) B_2^n)$, $y_1=A t_1$ and $y_2=A t_2$ for some $t_1,t_2\\in T$. If $\\gamma$ is large enough so that \n$$2c_3 \\frac{\\rho^*}{\\gamma}<(c_4\/10) \\rho^*$$  \nthen $y_1,y_2$ cannot be contained in $A\\left(t + \\{z \\in T: \\|t-z\\|_2 \\leq \\delta\\} \\right)$ for a single $t \\in T$; in particular, $\\|t_1-t_2\\|_2>\\delta$. Finally, if $\\lambda\\leq c_4 \\rho^*\/10$, then since $y_1,y_2$ belong to the same quadrant of $V$ it would imply that $\\operatorname{sign}(At_1+\\tau)=\\operatorname{sign}(At_2+\\tau)$, violating the assumption that $t \\mapsto \\operatorname{sign}(At+\\tau)$ is a $\\delta$-embedding of $T$ with probability at least $0.9$. Thus, we must have $\\lambda \\geq \\frac{c_4}{10}\\rho^*$, and $c_4\/10$ is an absolute constant---as required.    \n{\\mbox{}\\nolinebreak\\hfill\\rule{2mm}{2mm}\\par\\medbreak}\n\n\n\\subsection{A counterexample}\nNext, let us explore the conjecture that $m \\sim \\ell_*^2(T)\/\\delta^2$ suffices to ensure that $f(t)=\\operatorname{sign}(At+\\tau)$ is, with high probability, a $\\delta$-embedding of an arbitrary $T \\subset \\mathbb{R}^n$. We show that this conjecture is false, and that sometimes one in fact needs $m \\geq c\\ell_*^2(T)\/\\delta^3$.\n\nFix $0<\\varepsilon<\\delta\/2$, let $r \\in \\mathbb{N}$, set $I=\\{1,...,r\\}$ and put $J=\\{r+1,...,\\eta r\/\\varepsilon^2\\}$ for an absolute constant $\\eta$ to be specified in what follows. Let $n=|I|+|J|$ and denote by $B_2^I$ and $B_2^J$ the Euclidean balls supported on ${\\rm span}(e_i, i \\in I)$ and ${\\rm span}(e_j : j \\in J)$ respectively.\nSet\n$$\n{\\cal E}=B_2^I \\otimes \\varepsilon B_2^J = \\left\\{ (x,y): x \\in B_2^I, \\ y \\in \\varepsilon B_2^J \\right\\} \\subset \\mathbb{R}^{n},\n$$\nand note that ${\\cal E}$ is a convex body. \n\nConsider our random embedding $f:\\mathbb{R}^n \\to \\{-1,1\\}^m$, $f(t)=\\operatorname{sign}(At+\\tau)$. Observe that for any $0<\\delta<\\frac{1}{4}$, ${\\cal E}$ contains two points $x,y$ located on a ray emanating from the origin such that $\\|x\\|_2=1$, $\\|y\\|_2<\\|x\\|_2$, and $\\|x-y\\|_2=2\\delta$. As we show in detail in Appendix~\\ref{sec:minShift}, this causes $f$ to fail to be a $\\delta$-embedding of ${\\cal E}$ with probability at least $0.9$ if $\\lambda\\leq c$, where $c$ is an absolute constant. Thus, to have any hope of obtaining a $\\delta$-embedding, one needs to assume that $\\lambda>c$, and $c$ is of the order of the diameter of ${\\cal E}$. We will make a slightly stronger assumption: that $\\lambda \\geq C \\sqrt{\\log(e\/\\delta)}$; it corresponds to the context of Theorem~\\ref{thm:main-A}.\n\nLet us now assume that $m$ is such that, with probability at least $0.9$, $f$ is a $\\delta$-embedding of ${\\cal E}$. Using the notation of the previous section, the proof of Theorem \\ref{thm:main-B-2} shows that for an absolute constant $\\beta$, $\\lambda$ must satisfy that\n\\begin{equation} \\label{eq:in-counter-1}\n\\beta \\lambda \\sqrt{m} \\geq \\ell_*( ({\\cal E}-{\\cal E}) \\cap \\delta B_2^n).\n\\end{equation}\nBy Lemma \\ref{lemma:lower-on-lambda}, it suffices to verify that\n\\begin{equation} \\label{eq:in-counter-2}\n\\ell_*(({\\cal E}-{\\cal E}) \\cap \\lambda B_2^n) > \\gamma(\\beta) \\ell_*(({\\cal E}-{\\cal E}) \\cap \\delta B_2^n)\n\\end{equation}\nfor a constant $\\gamma=\\gamma(\\beta)$ (and thus, an absolute constant as well) to ensure that \\eqref{eq:in-counter-1} holds.\n\n\nWithout loss of generality, we have that $\\lambda \\geq {\\cal R}({\\cal E})=\\sup_{t \\in {\\cal E}} \\|t\\|_2=1$, and to verify \\eqref{eq:in-counter-2} it is enough to ensure that\n\\begin{equation} \\label{eq:in-counter-3}\n\\ell_*(2{\\cal E}) \\geq \\gamma(\\beta) \\ell_*( 2{\\cal E} \\cap \\delta B_2^n).\n\\end{equation}\nClearly, \\eqref{eq:in-counter-3} holds if $\\ell_*(B_2^I) \\geq \\gamma(\\beta) (\\ell_*(\\delta B_2^I)+\\ell_*(2\\varepsilon B_2^J))$, i.e., that \n$$\n\\sqrt{r} \\geq c \\gamma(\\beta) (\\delta\\sqrt{ r}+\\varepsilon \\sqrt{r}\\sqrt{1+\\eta\/\\varepsilon^2}).\n$$ \nIt follows that if $\\delta$ and $\\eta$ are sufficiently small (depending on the absolute constant $\\gamma(\\beta)$), the wanted estimate holds.\n\nNow we may assume that $\\eta$ is a suitable, sufficiently small absolute constant and that $\\delta$ is also small enough---ensuring that \\eqref{eq:in-counter-3} holds. Set $\\theta \\sim \\delta\/\\log(e\\lambda\/\\delta)$, and by the choice of $\\lambda$ we have that $\\theta \\sim \\delta\/\\log(e\/\\delta)$.\n\n\n\\begin{Theorem} \\label{thm:counter-example}\nThere are absolute constants $c_0$,$c_1$, and $c_2$ such that the following holds.\nLet $\\delta\\leq c_0$ and let $A:\\mathbb{R}^n \\to \\mathbb{R}^m$ be the standard gaussian matrix. If $m \\leq c_1 \\sqrt{\\log(e\/\\delta)}\\frac{\\ell_*^2({\\cal E})}{\\delta^3}$,\nthen with probability at least $0.9$ there are $x,y \\in {\\cal E}$ such that\n$$\n\\left| \\sqrt{2\\pi} \\frac{\\lambda}{m} d_{H}(f(x),f(y)) - \\|x-y\\|_2 \\right| \\geq 2\\delta.\n$$\nMoreover, if $m \\geq c_2 \\sqrt{\\log(e\/\\delta)}\\frac{\\ell_*^2({\\cal E})}{\\delta^3}$ then with probability at least 0.9\n$$\n\\sup_{x,y \\in {\\cal E}} \\left| \\sqrt{2\\pi} \\frac{\\lambda}{m} d_{H}(f(x),f(y)) - \\|x-y\\|_2 \\right| \\leq \\delta.\n$$\n\\end{Theorem}\n\n\\noindent {\\bf Proof.}\\ \\  Set $\\varepsilon=\\theta\/2$ and let $U \\subset B_2^I\\otimes 0$ be a minimal $\\theta\/2$-cover of $B_2^I$ with respect to the Euclidean norm. Thus, $U$ is also a $\\theta$-cover of ${\\cal E}$: every $x \\in  {\\cal E}$ is supported on $I \\cup J$, and there is some $u \\in U$ such that\n$$\n\\|x-u\\|_2 = \\|P_I x -u\\|_2 + \\|P_Jx\\|_2 \\leq \\theta\/2 + \\varepsilon \\leq \\theta.\n$$\nMoreover, by a standard volumetric estimate,\n$$\n\\log {\\cal N}({\\cal E}, \\theta) \\leq \\log {\\cal N}(B_2^I, \\theta\/2) \\lesssim r \\log(5\/\\theta).\n$$\nNext, note that $\\ell_*({\\cal E})$, $\\ell_*(B_2^I)$ and $\\ell_*(\\varepsilon B_2^J)$ are all equivalent to $\\sqrt{r}$. Moreover, using the choices of $\\varepsilon$ and $\\theta$, it is straightforward to verify that there is an absolute constant $c_1$ such that \n$$\nc_1\\ell_*({\\cal E}) \\leq \\ell_*(\\varepsilon B_2^J) \\leq \\ell_*({\\cal E} \\cap \\theta B_2^n) \\leq \\ell_*({\\cal E} \\cap \\delta B_2^n) \\leq \\ell_*({\\cal E}).\n$$\nHence, by Theorem \\ref{thm:main-B-2}\n$$\nm \\geq c_2 \\sqrt{\\log(e\/\\delta)} \\frac{\\ell_*^2({\\cal E})}{\\delta^3}\n$$\nis a necessary condition for ensuring that with high probability, $t \\mapsto \\operatorname{sign}(At+\\tau)$ is a $\\delta$-embedding of ${\\cal E}$ in $\\{-1,1\\}^m$. \n\n\nAt the same time, recalling that $\\lambda\\sim \\sqrt{\\log(e\/\\delta)}$, we see that the second term in the upper bound \\eqref{eqn:main-ABdm} from Theorem \\ref{thm:main-A} (featuring the gaussian mean-width) is dominant for $\\delta$ that is sufficiently small.  Thus, the lower bound is matched by the upper one from Theorem \\ref{thm:main-A}.\n\n{\\mbox{}\\nolinebreak\\hfill\\rule{2mm}{2mm}\\par\\medbreak}\n\n\\begin{Remark}\n(Possible logarithmic improvement) If $c<\\lambda\\leq C \\sqrt{\\log(e\/\\delta)}$, then Theorem~\\ref{thm:main-A} is not applicable for $T={\\cal E}$. However, Theorem \\ref{thm:main-B-2} shows in this context that $f$ fails to be a $\\delta$-embedding with probability at least $0.9$ if $m \\leq c\\frac{\\ell_*^2({\\cal E})}{\\delta^3}$. We leave the question whether Theorem \\ref{thm:main-A} continues to hold if $\\lambda\\geq C' {\\cal R}(T)$ unanswered. If the answer is positive, then that would yield matching bounds for $T={\\cal E}$ in the slightly improved setting $\\lambda \\sim {\\cal R}({\\cal E})$.\n\\end{Remark}\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\\label{sec:intro}\n\nThe presence of multiple stellar populations (MPs) is considered today a distinctive feature of almost all globular clusters (GCs). Despite important theoretical efforts, such as the recent suggestion of the role of stellar mergers in the MP origin \\citep{Wang2020}, a clear understanding of the MP formation is still lacking \\citep[see e.g.][and references therein]{Bastian2018}. A typical feature of these  MPs is an abundance variation in light elements with little or no dispersion in heavy elements, with significant element abundance anti-correlations. \nA widely used signature to identify MPs inside a GC is the distribution of key abundance indexes such as [O\/Na] or CN-CH \\citep[e.g.][]{Carretta2009b, Carretta2015, Hong2021}. The MP identification is often based on judgments made by naked eye, guided by of apparent splits in the colour-magnitude diagram (CMD) or peaks in the element abundances. In these cases, the classification of a star into a given group, and the number of identified populations itself, is based on subjective judgement and not on statistical methods.\nAn obvious drawback of this approach is the lack of robustness of the results and severe reproducibility issues. In fact, different researchers may get different data groupings, without the possibility of comparing these alternative claims.   \n \nMoreover, basing the grouping definition on the analysis of patterns in the abundance versus abundance index plots, as is frequently done, is risky because of how individuals perceive  visual representations.  \nGestalt principles of perceptual organisation \\citep{Wertheimer1938} emphasize that organisms perceive entire patterns or configurations and not the individual components. The fundamental law that governs a Gestalt principle is that we tend to order our experience in a regular, orderly, and recognizable manner.\nThe Gestalt laws of proximity, similarity, connectedness, continuity, and common fate determine how individual graphical features are grouped together to form coherent representations \\citep{Ali2013, Pinker1990}.\nIn particular, the law of proximity states that when an individual perceives an assortment of objects, they perceive objects that are close to each other as forming a group.\nThese factors can lead to an overestimation of subgroups in a population because random gaps are perceived as representing a real boundary between subpopulations.\n\n \nFortunately, there are a number of statistically validated techniques to deal with the grouping problem. Even when robust statistics-based clustering methods are adopted to identify MPs \\citep[e.g.][]{Carretta2011, Gratton2011, Simpson2012, pasquato2019multiple}, however, several paramount aspects are usually not addressed and therefore deserve particular attention.   \nThe first aspect is the impact of the data uncertainties. This is a relevant issue because chemical abundances are affected by non-negligible uncertainties that are often comparable with the intercluster variability.\nThe second problem is that statistical clustering methods do not directly provide information about the optimal grouping, that is, whether a two-group split is better than a three-group split. While several statistical methods exist to investigate this, they are usually ignored. \n\nThe aim of this paper is to illustrate the strength of some methods, using as a case study one of the most frequently studied GCs, that is, NCG 2808. This cluster is a perfect target for this investigation because multiple stellar populations have been reported in the literature in different evolutionary stages, from the main sequence to the asymptotic giant branch \\citep[see e.g.][]{DAntona2005, Piotto2007, DAntona2008, Lee2009, Dalessandro2011, Carretta2015, Marino2017}.\n\n\n\nThe paper is organised as follows. Section~\\ref{sec:method} introduces the adopted statistical methods and the data sets we used in the analysis. Section~\\ref{sec:results} contains the results of the cluster analysis for the two data sets. Some concluding remarks are collected in Section~\\ref{sec:conclusions}. The paper contains a statistical appendix in which the adopted statistical methods are presented in some detail.\n\n\n\\section{Methods and data}\\label{sec:method}\n\n\\subsection{Statistical methods}\n\nIdentifying multiple stellar populations in GCs by their chemical abundances is a classical problem that is addressed in statistic literature in the vast area of unsupervised clustering techniques. These methods attempt to split data into groups that are inferred from data patterns, so that the intergroup difference is as small as possible, while the intragroup difference is as large as possible.\n\nThese unsupervised techniques can be broadly divided into hierarchical clustering (HC) methods and partition methods. Partition methods may additionally be split into distance-based, density-based, and model-based methods \\citep[see e.g.][]{KaufmanRousseeuw90, venables2002modern, simar, Feigelson2012,SAXENA2017}. While these methods were not developed to take measurement errors into account,  they have recently been further generalised to cover this important topic \\citep[e.g.][]{Kumar2007, Su2018, pankowska2020effect}.  \n\nIn the following we adopt three different clustering methods: a divisive HC, an agglomerative HC, and a method called partition around medoids (PAM). The rationale for using more than one clustering algorithm is that none of them is the absolute best for any data set; indeed they all have known weaknesses, and comparing their performances usually helps to better understand the underlying data structure.  For each method, we tested different possible data splits, from two to five groups, then we evaluated the average  silhouette, which is an indicator of how clearly the elements belong to a given group or cluster. An overview of the adopted methods is given in Appendix~\\ref{sec:app-tools}. The analysis was conducted using R 4.1.0 \\citep{R}\\footnote{The computational methods adopted in the paper are demonstrated at {\\url{https:\/\/github.com\/mattdell71\/cluster-analysis}} on the low-resolution spectroscopy data set}.\n\nA crucial point in this analysis is the possibility of including  observational errors. We took the errors in the abundances at two different levels into account. First, errors were used to construct the dissimilarity matrix (see Appendix~\\ref{sec:app-tools} for details): briefly, the dissimilarity matrix is the metric adopted to divide objects into groups relying on the clustering techniques described above. The objective of these methods is to group objects with low \ndissimilarities. Neglecting the data uncertainties and adopting a straightforward Euclidean distance in the abundances hyperspace would ignore that some elements are known to a better  precision; the same raw difference is therefore quite different according to the considered index. However, this is not enough to fully account for the measurement error problem. A probabilistic clustering is advised in this case. Therefore\nwe repeated the cluster analysis in the framework proposed by \\citet{Su2018}. This approach adopts a Monte Carlo (MC) resampling of the data set to assess the probability that an object might belong to a given cluster (see Appendix~\\ref{sec:error-var} for details). According to this analysis, we can only obtain the probability that an object belongs to a given cluster and not a rigid clustering. It also allows us to evaluate how the average silhouette of different grouping responds to the data uncertainty.  \n\n\nA widespread approach to the identification of MPs by chemical abundances or photometric data is the examination of the differences in key elements, such as [O\/Na] or CN-CH \\citep[e.g.][]{Carretta2009b, Carretta2015, Simpson2017, Guerco2019, Hong2021}, or differences in photometric bands or colour indices \\citep[e.g.][]{DAntona2005, Dalessandro2011, Marino2017}. MPs are then identified by comparing the element abundances with those of field stars or by peaks in the histogram of these indexes. This approach does not determine the reliability of the detected substructure, however. Furthermore, two main drawbacks hinder the adoption of peaks in the histogram as tracers of MPs, both related to binning: the bin width, and the starting position of the first bin \\citep[see e.g.][]{simar}. A better approach to approximate the empirical probability density function is adopting a kernel density estimator (see Appendix~\\ref{sec:kernel} for details). Roughly speaking, a kernel density estimator can be considered a smoothed histogram without the dependence on the starting position. The bin width is replaced by the so-called bandwidth $h$, and several well-tested approaches exist to estimate it. This parameter can be considered as a moving smoothing window. A too large $h$ results in a very smooth density estimator, which may mask interesting features, while a too low $h$ produces a noisy estimator with many spurious peaks. \nGeneral-purpose widespread algorithms exist to estimate $h$ (e.g. the SJ estimate from \\citealt{Sheathe1991} and the Wand estimate from \\citealt{Wand1994}). In addition to these,\nwe also adopted a third density estimator, the decon approach by \\citet{Wang2011}, which corrects the density estimate considering data uncertainties.\n\nThe subjective judgement of multimodality in the empirical density was further investigated by means of two statistical tests: the Dip test, which is a statistical tool for determining a departure for unimodality \\citep{Hartigan1985}, and which is implemented in the diptest R library \\citep{diptest}, and the \\citet{Ameijeiras2019} excess mass test, which is available in the multimode R library \\citep{multimode}. \n\nFinally, to show some issues arising from the adoption of an empirical density estimator as a tool for identifying MPs, we performed some MC simulations (Sect.~\\ref{sec:peaks}). \nWe simulated random points from a single population in a 2D space, and we compared the number of subpopulations inferred by peak detection and divisive HC.\n\n\n\n\\subsection{Data}\n\nAs a difference from \\citet{pasquato2019multiple}, who showed the use of several clustering methods for NCG 2808, we did not make use of the photometric chromosome map \\citep[see e.g.][]{Milone2015, Milone2017}, but we adopted element abundances for red giant branch (RBG) stars from \\citet{Carretta2015} (high-resolution spectroscopy) and \\citet{Hong2021} (low-resolution spectroscopy). However, the methods presented in this paper can be easily extended to also incorporate pseudo-colours in the analysis.\n\nAbundances of Na, O, Mg, and Si from high-resolution spectroscopy from \\citet{Carretta2015} were used, as presented in their Table~4. To perform the cluster analysis, objects with missing values in any of these elements were excluded. This resulted in a sample of 116 RGB stars. Data preprocessing was needed because some element abundances did not have an accompanying error. In these cases, the missing values were replaced by the mean of the error of the specific element over the whole data set.\nThe adoption of this flat average error is justified by the fact that the uncertainties in the selected quantities are not correlated with the element abundances, as verified with multivariate linear regressions.\n\nData from low-resolution spectroscopy were obtained from \\citet{Hong2021}, who      \nmeasured spectral indexes for CN, CH, and Ca II H\\&K lines as tracers of N, C, and\nCa abundances, respectively, from a sample of 91 RGB stars. To remove the effects of gravity and temperature from these indexes, the authors detrended them by subtracting the mean value of the regression against the magnitude V over the whole sample, obtaining the $\\delta$CN, $\\delta$CH, and $\\delta$HK' indexes, which we adopt in the present analysis.\n\n\\section{Results}\\label{sec:results}\n\nThe clustering we evaluated was obtained using two data sets: 1) the indexes $\\delta$CN, $\\delta$CH, and $\\delta$HK' obtained from low-resolution spectra, and 2) the abundances of Na, O, Mg, and Si estimated using high-resolution spectra. The analysis was performed in the same way for both data sets, and the corresponding results are discussed in detail in Sect.~\\ref{sec:res-low} and Sect.~\\ref{sec:res-high}.\n\nAs a preliminary step, we explored the data clustering by means of divisive HC, agglomerative HC, and PAM methods. The functions {\\it diana}, {\\it agnes}, and {\\it pam} in the R library cluster \\citep{cluster2021} were used for the computations. We verified that the comparisons of the silhouette values did not show relevant differences among the three adopted algorithms. In the following, we therefore only present the results obtained from divisive HC, which provided the strongest clustering in all the analysed cases. We accordingly adopted the average silhouette from divisive HC to evaluate alternative splits throughout the paper.\n\nThe capability of simultaneously taking all the available data into account is another aspect that makes these methods superior to the splitting that is obtained by only considering one index and using other elements to validate the grouping. However, before a multivariate data set is used in the analysis, the variability range or scale of the employed data has to be checked.  When variables vary in very different intervals or ranges, it is advised to rescale them to a new data set so that every variable has zero mean and unit variance \\citep[e.g.][]{Rousseeuw1987}. If the mean value of a variable is much higher than the others (which can be the case when abundances and magnitudes or effective temperatures are analysed simultaneously), it dominates the analysis and masks the effect of all the others.\nIn our case, there was no need to scale data to zero mean and unit variance before clustering. However, we also verified that scaling the data as well as the errors before the clustering did not modify the results. \n\n\\begin{figure*}\n        \\centering\n        \\includegraphics[height=16.0cm,angle=-90]{fig\/sil-dn-CN.eps}\n        \\caption{{\\it Left}: Silhouette plot of the divisive HC applied to the low-resolution spectroscopy data set for a two-group split. {\\it Right}: Same as in the left panel, but for a three-group split. The bars correspond to the silhouette values of objects clustered in the first group (red), in the second group (blue), and in the third group (green). The average silhouettes for the groups are listed along with the group sizes.}\n        \\label{fig:CN-sil}\n\\end{figure*}\n\n\\begin{figure}\n        \\centering\n        \\includegraphics[height=8.0cm,angle=-90]{fig\/boxplot-sil-dn-CN.eps}\n        \\caption{Boxplot of the average silhouette widths for two, three, and four groups obtained by MC resampling on the low-resolution spectroscopy data set.}\n        \\label{fig:CN-silMC}\n\\end{figure}\n \n\n\n\\subsection{Low-resolution spectroscopy}\\label{sec:res-low}\n\nFigure~\\ref{fig:CN-sil} shows the results for the silhouette analysis applied to divisive HC on low-resolution spectroscopy data. This first result takes measurement errors only in the construction of the dissimilarity matrix into account. The left panel displays a two-group subsetting, while a three-group split is shown in the right panel.\nDifferent colours identify the subgroups. Objects are ordered from top to bottom in every group by decreasing silhouette value. The silhouette is bounded between -1 and 1; positive values indicate that the object fits its group well, and negative values suggesting that the object might be poorly classified. For every group, the size and average silhouette are shown (e.g. the upper group in the left panel is composed of 42 stars and has an average silhouette of 0.76).\n\nThe two-group division is highly favoured by the silhouette analysis as its average silhouette is 0.73. On the other hand, in the three-group case (right panel), the average silhouette is only 0.50. In particular, the three-group substructure is severely penalised by the fact that several stars assigned to the third group (green bars) fit poorly there, showing negative silhouette indexes.\n\n\\begin{table}[ht]\n        \\caption{Average silhouette values according to the clustering algorithm (by column) and number of proposed groups (by row).}\n        \\label{tab:CN-sil} \n        \\centering\n        \\begin{tabular}{lccc}\n                \\hline\n                Groups & Agglomerative HC & Divisive HC & PAM \\\\ \n                \\hline\n2 & 0.68 & 0.73 & 0.72 \\\\\n3 & 0.60 & 0.50 & 0.62\n\\\\\n4 & 0.58 & 0.51 & 0.56 \\\\\n        \\hline\n\\end{tabular}\n\\end{table}\n\nTable~\\ref{tab:CN-sil} shows the average silhouette values obtained by the three clustering algorithms in the case of two, three, and four groups. All of them suggest that a two-group split is the optimum for the data set. However, it should be stressed that the maximum average silhouette we obtained, that is, 0.73, is quite low. As a rule of thumb, values above 0.7 are considered high enough to possibly originate from an actual population substructure and not from a random fluctuation \\citep{Rousseeuw1987,KaufmanRousseeuw90}. The obtained value is just above this soft boundary, which means that the proposed MP identification is suspiciously weak and should be considered with great care.\n\n\\begin{figure}\n        \\centering\n        \\includegraphics[height=8.0cm,angle=-90]{fig\/CN-CH-fuzzy.eps}\n        \\caption{Scatter plot in the $\\delta$CN vs $\\delta$CH plane for the low-resolution spectroscopy data set. Data are classified according to the MC resampling. Circles correspond to objects classified into group 1, and squares show objects classified into group 2. Colours were adopted to distinguish data according to the cluster membership probability.  Purple symbols correspond to data for which the classification is uncertain (see text for details). }\n        \\label{fig:CN-scatter}\n\\end{figure}\n\n\nAs detailed in Sect.~\\ref{sec:method}, we explored the effect of data uncertainty by repeating the cluster analysis and relying on MC data resampling. Figure~\\ref{fig:CN-silMC} shows a boxplot of the average silhouette values for two, three, and four groups, obtained from 50 MC experiments. For each of them, the objects were perturbed according to a multivariate Gaussian distribution that accounted for the measurement errors.   \nThis result confirms that a two-group substructure is highly preferred.\n\nFigure~\\ref{fig:CN-scatter} shows the stars in the $\\delta$CH - $\\delta$CN plane classified according to the measurement error divisive HC algorithm. A slightly different approach was adopted in this case. Every star was perturbed $k = 21$ times, and all the $91 \\times 21$ objects were retained in the data set. This full data set was then subjected to the HC analysis, as suggested by \\citet{Su2018}. Then, for every object, the probability of falling into group $i$ was obtained by dividing the times $f_i$ it fell in group $i$ by $k$: $p_i = f_i\/k$. The object was then assigned to the group corresponding to the maximum $p_i$. The two groups are identified by different symbols. The colours were adopted to code the information about the confidence in the grouping. The eight objects for which the maximum $p_i$ was lower than 0.75 are represented by purple symbols.\n \n \\begin{figure}\n        \\centering\n        \\includegraphics[height=8cm,angle=-90]{fig\/density-CN.eps}\n        \\caption{Kernel density estimators of the difference $\\delta$CN~-~$\\delta$CH, obtained by adopting the SJ bandwidth (solid red line), the Wand bandwidth (dashed blue line), and the approach from \\citet{Wang2011} to correct for measurement errors (dot-dashed green line). }\n        \\label{fig:CN-density}\n \\end{figure}\n \nThe analysis presented so far suggests a possible presence of two subpopulations from the low-resolution spectroscopy data set. However, relying on the analysis of the histogram of $\\delta$CN~-~$\\delta$CH, \\citet{Hong2021} found four distinct populations. Thus, to explore the possible sources of these different results, we devote the final part of this section to study some potential issues arising from a histogram as a tool for detecting MP.\n\nFigure~\\ref{fig:CN-density} shows three different kernel density estimators obtained by adopting the SJ and Wand bandwidths. The values of the two bandwidths are 0.046 and 0.066 for SJ and Wand, respectively. The figure shows that the Wand bandwidth produces a smoother density estimator with two clear peaks. The adoption of the smaller SJ bandwidth results in a density with more features than the previous one, with the possible presence of a third small peak at $\\delta$CN~-~$\\delta$CH $\\approx$ 0.1. This simple exercise shows that the adoption of an empirical density estimator for MP recognition is a quite dangerous approach and can lead to false-positive claims (see Sect.~\\ref{sec:peaks} for further details). The adoption of the {\\it decon} approach produces a density that looks more similar to that obtained with the Wand bandwidth than to that estimated using the SJ method.\n\nThe Dip test of multimodality showed highly significant evidence that the distribution is multimodal, producing a $p$ value of 0.01\\footnote{The $p$ value is the probability of obtaining a test result as extreme as the actually observed result, assuming that the null hypothesis $H_0$ is true. Conventionally, a $p$ value lower than 0.05 is considered statistically significant and leads to the rejection of $H_0$.}. The result was confirmed by the excess mass test, which produced a $p$ value of 0.002. \n\n\\begin{figure*}\n        \\centering\n        \\includegraphics[height=16.0cm,angle=-90]{fig\/sil-dn-carretta.eps}\n        \\caption{Same as in Fig.~\\ref{fig:CN-sil}, but for the high-resolution spectrum data set.}\n        \\label{fig:Car-sil}\n\\end{figure*}\n\n\\begin{figure}\n        \\centering\n        \\includegraphics[height=8.0cm,angle=-90]{fig\/boxplot-sil-dn-carretta.eps}\n        \\caption{Boxplot of the average silhouette widths for two, three, four, and five groups obtained by MC resampling on the high-resolution spectroscopy data set.}\n        \\label{fig:Car-silMC}\n\\end{figure}\n\n\n\\subsection{High-resolution spectroscopy}\\label{sec:res-high}\n\nThe analysis presented in Sect.~\\ref{sec:res-low} was repeated on the high-resolution spectroscopy data set. Figure~\\ref{fig:Car-sil} shows the silhouette plots for two and three subpopulations. A two-group split is preferred in this case as well, with an average silhouette width of 0.82 with respect to 0.73 for three groups. The value of the average silhouette width is higher than for low-resolution spectroscopy, which means that in this case, the detected substructure is more reliable.  \n\n\\begin{figure*}\n        \\centering\n        \\includegraphics[height=17.0cm,angle=-90]{fig\/scatter-Carretta-fuzzy.eps}\n        \\caption{{\\it Left}: Scatter plot in the [Na\/Fe] vs [O\/Fe] plane for the high-resolution spectroscopy data set. Data are classified according to the MC resampling. Circles correspond to objects classified into group 1, and squares show objects classified into group 2. Colours were adopted to distinguish data according to the cluster membership probability.  Purple symbols correspond to data for which the classification is uncertain (see text for details). {\\it Right}: Same as in the left panel, but in the [Mg\/Fe] vs [Na\/Fe] plane.}\n        \\label{fig:Car-scatter}\n\\end{figure*}\n\nThe MC resampling was adopted to estimate the effect of measurement errors on the clustering. Figure~\\ref{fig:Car-silMC} shows the boxplot of the average silhouette widths from two to five groups (the number of subpopulations suggested by \\citealt{Carretta2015} in their analysis). The median value of the silhouette width decreases monotonically. Ultimately, even in the high-resolution spectroscopy, a two-group configuration is preferred for RGB stars in NGC 2808.\n\nFigure~\\ref{fig:Car-scatter} shows two scatter plots in the [Na\/Fe] versus [O\/Fe] and [Mg\/Fe] versus [Na\/Fe] planes. The object memberships were established by the same MC procedure as described for the low-resolution spectroscopy data set. Only four objects have a questionable membership and are shown in purple in the figure. \n\n\\begin{figure}\n        \\centering\n        \\includegraphics[height=8.0cm,angle=-90]{fig\/density-Carretta.eps}\n        \\caption{Kernel density estimators of [O\/Na], obtained adopting the SJ bandwidth (solid red line), the Wand bandwidth (dashed blue line), and the approach from \\citet{Wang2011} to correct for measurement errors (dot-dashed green line).}\n        \\label{fig:Car-density}\n\\end{figure}\n\nFinally, Fig.~\\ref{fig:Car-density} shows three kernel density estimators of the [O\/Na] probability density function. The abundance ratio [O\/Na] was used by \\citet{Carretta2015} to propose the presence of five MPs by the peaks detected in the histogram. The three methods we adopted give very similar results: a strong peak at [O\/Na] $\\approx$ 0.3 and two much smaller peaks around $-0.6$ and $-1.4$. The difference in the peak heights should be carefully considered because it means that only few data are in the low [O\/Na] zone, so that fluctuations are much more probable there. The Dip test $p$ value for the presence of more than one peak was 0.94, therefore we cannot reject the hypothesis of a unimodal distribution. The conclusion was confirmed by the excess mass test ($p = 0.58$).\nUltimately, as already mentioned, the peak count cannot be considered a reliable tool for MP identification. We discuss this issue in more detail in the next section.\n    \n\\subsection{Issues in identifying MP by peaks in the histograms}\\label{sec:peaks}    \n\\begin{figure*}\n        \\centering\n        \\includegraphics[height=16.0cm,angle=-90]{fig\/peaks-SJ.eps}\n        \\caption{{\\it Left}: Empirical density function for the number of peaks detected in the MC simulation, classified according to the sample size (see text). {\\it Right}:  Boxplots of the average silhouette widths for two- and three-group splits, according to the sample size.  }\n        \\label{fig:peaks}\n\\end{figure*}\n \n To show some issues arising from the adopting an empirical density estimator such as a kernel density or a histogram, we present here a simple exercise. We simulated random points from a single population with an arc-shaped distribution in 2D space to mimic the abundance plots in Figs.~\\ref{fig:CN-scatter} and \\ref{fig:Car-scatter}, which we call $P_i = \\{x_i, y_i\\}$. As a simple choice, we sampled the points from a circle with a unit radius and angular coordinate $\\theta \\in [0, \\pi\/2]$, adopting a uniform distribution for $\\theta$. The points were then perturbed with Gaussian errors with $\\sigma = 0.06$ in both dimensions. Then, we computed the kernel density estimator of the coordinate difference $z_i = x_i - y_i$, similarly to the process of inferring peaks in the element abundances carried out in \\citet{Carretta2015} and \\citet{Hong2021}. The SJ bandwidth estimator was adopted. The peaks in the kernel density estimator were identified by the change in the sign of the first derivative. Finally, we also performed a divisive HC analysis and computed the average silhouette widths for two and three subgroups. The process was repeated 5,000 times for three different population sizes: 50, 70, and 100 objects.\n \n \nThe results we obtained by adopting a uniform distribution for the angular coordinate $\\theta$ are presented in Fig.~\\ref{fig:peaks}. The left panel shows that the detection of multiple peaks is indeed very probable in this scenario, with two peaks being the most frequent finding. As the sample size increases, the detection of two or more peaks becomes increasingly probable because the higher the number of points, the higher the occurrence of mini clusters, which in turn leads to peaks in the density estimator. The right panel of the figure shows the boxplots of the average silhouette widths for the two- and three-group split. The median of these boxplots is reassuringly low, it is about 0.60 for two groups and about 0.50 for three groups. Therefore the cluster analysis leads to the rejection of the hypothesis of the existence of MPs in the parent population.      \nMultiple peaks can even be expected and are partially due to the way in which the synthetic data were simulated. The adoption of a flat sampling enhances the possibility of a random aggregation of a few points, thus leading to peaks in the density function. Nevertheless, we remark that the peaks identified in the density estimator analysis are rejected by the silhouette plots, showing the robustness of this second method.\n  \nIn the real world, the assumptions of this exercise may not hold perfectly because the scatter plot of the element abundance may differ from a perfect arc, and the angular distribution may not be uniform. To verify the result robustness, we repeated the exercise and adopted a single-peak angular distribution: a Gaussian angular distribution peaked at $\\pi\/4$ and standard deviation $\\pi\/12$. Even in this case, we obtained a relevant probability of a multiple peak ($\\geq 2$) detection using a kernel density analysis, from 30\\% for a sample size of 50 objects to 22\\% for 100 objects. On the other hand, as in the previous case, the silhouette analysis rejected the presence of multiple groups in the synthetic data set. \n\nIn summary, the results presented in this section are very relevant when the number of subgroups is judged by counting peaks in the empirical density. They should be considered as an alarm bell showing that empirical density estimators are prone to peak detection even when no peak is present.\n   \n\n\\section{Discussion and conclusions}\\label{sec:conclusions}\n\nThis paper investigated the adoption of robust statistical clustering methods to assess the presence of MPs in a globular cluster by considering the abundances of some key elements alone.\nWe explicitly addressed the problem of clustering data in the presence of measurement errors by adopting techniques that were recently proposed in the statistical literature \\citep[e.g.][]{Kumar2007, Su2018}. \n\nThe globular cluster NGC 2808 was chosen as case study because it hosts several MP reported in the literature, from the main sequence to the asymptotic giant phase  \\citep[see e.g.][]{DAntona2005, Piotto2007, DAntona2008, Lee2009, Dalessandro2011, Carretta2015, Marino2017}. In particular, we focused our analysis on the RGB evolutionary phase, where two high-quality data sets are available:  \\citet{Carretta2015} for high-resolution spectroscopy and \\citet{Hong2021} for low-resolution spectroscopy.\n\nSeveral statistical clustering techniques were applied to the two data sets. Two HC approaches (divisive and agglomerative) and a partition method (partition around medoids) were adopted. As they gave similar results on both data sets, we only discussed those obtained by the divisive HC algorithm. The dendrograms resulting from the HC algorithm were analysed by means of the silhouette plot method to compare the performances of different numbers of subgroups \\citep{Rousseeuw1987, KaufmanRousseeuw90}. This is a general-purpose well-established method for investigating the structure of a dendrogram; adopting this has the fundamental merit of making the population split objective and reproducible among different investigations.\n\n\nFor both data sets, the analysis suggested a possible split into two subpopulations as optimal. The claim is stronger for the high-resolution spectroscopy data, with a 0.82 average silhouette width for a two-group division. In contrast, the average silhouette width for two groups from low-resolution spectroscopy was only 0.73. As a rule of thumb, values higher than 0.7 are considered due to an actual population substructure and not to a mere fluctuation. \n\nThe finding of only two subpopulations in NGB 2808 is at odds with the five subgroups reported by \\citet{Carretta2015} for high-resolution spectroscopy and the four MPs by \\citet{Hong2021} for low-resolution spectroscopy. These differences are easily due to the difference in the methods adopted to define the presence of a distinct populations. In the two quoted papers, the estimated number of MPs relied on the analysis of the empirical distribution of some key abundances by means of histograms. Peaks were considered linked to the presence of a possible subpopulation, and these findings were corroborated by analysing the scatter plot of different elements to confirm by naked eye that the proposed division holds. Several shortcomings make this approach suboptimal. First, it only adopts part of the data to define the peaks and uses the remaining part as a confirmation. A multivariate approach to classification problems has the advantage of considering all the data at once, offering a much more robust classification. This was recognised  by \\citet{Fisher1936} in his seminal paper about multivariate analysis. Second, as we directly showed here with an MC experiment, histogram analysis is prone to false-peak detection. Sampling data uniformly from an arc-shaped distribution, we showed that multiple peaks are detected by empirical distribution estimators in about 85\\% of the experiments. We also showed that the divisive HC method, followed by silhouette width analysis, did not suggest an underlying substructure in any population. Third, the presence of peaks was established by subjective judgement and not by sound statistical methods. The adoption of the Dip test and of the mass excess test \\citep{Hartigan1985, Ameijeiras2019} to verify the presence of multimodality provided significant departure from unimodality for the low-resolution spectroscopy data set, but no evidence against the hypothesis of a single mode in the high-resolution case.\n\n\nParticular emphasis  was given to the correction needed for classical clustering algorithms in the presence of measurement errors. In this paper we adopted the approach by \\citet{Su2018}, based on MC resampling and data perturbation. In the presence of data uncertainties, clustering methods cannot provide a firm membership for the various objects, but the probability that they fall into the different groups. Objects can then be assigned to the group with the highest probability. We found that for both data sets, the vast majority of stars were clearly classified into a group (membership probability higher than 75\\%); only eight stars from low-resolution spectroscopy and four from high-resolution spectroscopy had a questionable membership.     \n\nThe analysis presented in this paper only used element abundances to define the population substructure. However, photometric pseudo-colours are often adopted as a proxy to trace variation in the stellar abundances. Several pseudo-indexes were proposed in the literature that are sensitive to different element variations \\citep[see e.g.][]{Marino2008, Milone2015, Milone2017, Bastian2018}. An advantage of the techniques adopted in our analysis is that they can be easily extended to incorporate this information whenever available. A mixed data set containing both abundances and colour indexes requires some care because the ranges they span can be somewhat different. In this case, it is advised to standardise the data (and scale the associated uncertainties) to zero mean and unit variance before the dissimilarity matrix is computed.\n\nThe increasing availability of high-quality data facilitates the analysis of the presence of MPs in globular cluster more than ever. It is therefore of paramount importance to base  these analyses  on firm statistical ground and not on naked-eye assessment.  \nWhile we presented  methods that are well established in the statistical literature, many other exist and the research in the field is ongoing: as an example, we can mention clustering by Gaussian mixture models, density-based spatial clustering of applications with noise\n\\citep{Ester1996}, or clustering in generative adversarial networks \\citep[e.g.][]{He2020, Pal2021}. \nUltimately, the adoption of methods analogous to those discussed here can help to obtain more robust and reproducible researches.\n\n\n\\begin{acknowledgements}\n  The authors acknowledge the anonymous referee for the useful comments that helped to improve the clarity of the paper. ET is funded by the Czech Science Foundation GA\\u{C}R (Project: 21-16583M). He also thanks Pisa University for the hospitality during 6\/09\/2021 - 5\/10\/2021 through the visiting program.\n\\end{acknowledgements}\n\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:level0}Introduction}\nThe physics classroom today contains many forms of teaching designed to show students a different perspective on the material. Students can read a textbook, take notes in class, and observe demonstrations of the phenomenon being studied, among much else. The different routes of exposure are chosen to complement each other and provide new pathways to learn the material. \n\nA hands-on activity provides students with a personal, real, physical application of course content. This type of activity provides an embodied cognitive experience in which we make sense of the world through our body's interaction with it \\cite{ANDERSON200391}. While embodied cognitive experiences enhance learning, the activities are often plagued by variability in fidelity, cognitive load, and sacrifices of realism that distract from the concepts being taught \\cite{Finkelstein2005}. Implementing a heavily guided activity reduces these variables but also the opportunity for self-directed learning \\cite{Weiman2015}. This is where the benefits of computer simulations and virtual reality step in to help. A computer simulation is able to reduce variability by providing a controlled experience and the realism can be programmed. A computer simulation that is designed with exploration in mind can also lend itself well to self-directed learning \\cite{adams2008,Finkelstein2005}. \n\nVirtual Reality (VR) goes an extra step beyond the versatility and consistency of a computer simulation by also providing the embodied learning experience of a hands-on activity. VR brings other features that make it uniquely suited as a teaching tool in physics \\cite{Bricken:1991:VRL:126640.126657,Perone2016}. \nIt also allows for the use of inaccessible perspectives that are more immersive, and its ability to track students provides additional opportunities to monitor learning, engagement, and provide feedback in real time \\cite{won}. \n\nA brief search through Physical Review - Physics Education Research (PER) and the PERC proceedings will show that very little research has been done in PER on the impacts of VR on learning. Recently, one study performed a controlled experiment comparing student learning and attitudes due to VR, video, and static images \\cite{Smith2017}. They found no differences between conditions, except for a marginal improvement from VR for students with experience with video games. They also found significant participant preference towards VR.\n\nA significant limitation to the study, however, was that the electrostatics concepts evaluated across conditions did not take advantage of the interactive and embodied cognitive affordances of VR. The students were not able to interact with the environment, control variables, or explore the concepts - features that are key for student learning from simulations \\cite{Salehi2015,adams2008}.\n\nIn the present study, we aimed to test the impacts of a fully featured and interactive experience in VR. We compared participant learning and activity preference when engaging in equivalent activities in VR, a hands-on activity, and a desktop simulation. We found that, even with the interactive elements, participant learning was again equivalent across conditions, and that participants overwhelmingly preferred engaging in the VR condition.\n\n\\section{\\label{sec:level1}Methods}\nTo carry out our study we created three Moon phase activities; a hands-on activity as close to traditional as possible, a computer based version, and a fully featured VR version. \n\nThe traditional hands-on version of this activity involved a bright spotlight (the Sun) pointed at a participant (the Earth) who is holding a short stick with a ball on top (the Moon). If the participant holds the ball constantly at arm's length and spins around, this symbolic Sun-Earth-Moon system mimics the illumination pattern of the Moon phases so they can be viewed with the control of the participant \\cite{tutorial,newbury}. \n\nIn the desktop simulation version of the activity, participants saw realistic visualizations of the Sun, Earth, and Moon. Students used the keys on the computer keyboard to move forward or backwards in time. They could also change their viewing position by using the mouse and keyboard, for example to see a birds-eye-view of the 3-body system, or planar perspectives to more clearly see the shadows on the Moon. The time progression appropriately synced Earth's rotation, the Moon's orbit, and Earth's orbit.\n\nThe VR activity used the Oculus Rift head mounted display with two hand controllers running a simulation made using the Unity game engine. The structure of the VR activity was much the same as the desktop simulation, except that participants used the hand controllers to move forward or backwards in time. They could also use a cursor, virtually attached to their hand, to reach out and grab the Moon to drag it forwards or backwards in its orbit, in much the same way as the hands-on condition. Participants were initially placed in space above Earth's North pole, but could relocate to other viewing positions. \n\nThe desktop simulation and VR conditions also had the opportunity for participants to transport to Earth's surface to observe rise and set times. They could fluidly transition between views to observe the rise and set of the Moon on Earth and the relative positions of the bodies at the same time. \n\nParticipants were recruited from a pool of Cornell students and the instructions were kept as similar as possible for each condition. Each participant was randomly assigned to one of the three conditions. We used a pre-post model to measure participant learning and activity preference. The pre- and post-tests consisted of 14 questions each, drawn from existing assessments of student understanding of Moon phases \\cite{2002AEdRv...1a..47H,lindell2002,lindell2005,slater2014}. Due to the short duration of the activity, the pre- and post-test items were not identical. The questions were, however, matched on content, such that each test contained a isomorphic set of questions. \n\n\nAfter completing the pre-test, the researcher provided verbal instructions on how to use the equipment for their condition. The language of the instructions were matched as closely as possible between conditions. The participant then completed an activity that used guiding questions modeled after common tutorials in astronomy \\cite{tutorial,newbury}. We opted to use semi-structured activities as research has found that students can struggle to engage productively in unstructured computer-based activities \\cite{Salehi2015,adams2008}. Once again, the guiding questions were equivalent between conditions.\n\nFollowing the activity, the participant completed the post-test, which also included demographic and attitudinal questions. The participant was then shown the other two conditions and completed a short survey to indicate their preferred activity and to explain their reasoning.\n\nThere were 56 participants in the VR condition, 59 in the hands-on condition, and 57 in the desktop condition. All participants were students at Cornell University. 138 participants identified themselves as women, 31 as men, and 3 identified as other. The most common majors were social science, such as communication but there were also participants from biological science, engineering, physical science, and art. \n\nThe results presented here, as gathered from the pre- and post-tests, include: participant performance by condition, participant performance by question, and activity preference. Video recordings were also collected for each participant during their time with the teaching activity and tracked movement data was collected for participants in the VR condition. The data have not yet been analyzed. \n \n\\section{\\label{sec:level2}Results}\n\\subsection{Total score comparisons}\nThe average score on the pre-test was around 36\\% in all three conditions, with no significant differences between conditions (Fig.\\ref{fig:Histograms}). There were statistically significant increases in performance in all three conditions between pre- and post-test (\\textit{p}<0.001). Because pre-test scores were statistically equivalent, we directly compared the post-test scores across conditions. The average score on the post-test was about 58\\% in all three conditions, with no significant differences between conditions $(F(2,169)=0.86,$ $p=0.57)$. \n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=\\columnwidth]{{\"HistogramScoresV2\"}.pdf}\n\\caption{\\label{fig:Histograms} An overall view of pre- to post-test performance. Violin plots show how scores were distributed across conditions with green (left) being the pre-test and blue (right) being the post-test. Bins are 1 point wide. Average scores and standard error are indicated in white.}\n\\end{figure}\n\n\\subsection{Score comparison by topic}\nThe three conditions provided different potential affordances for teaching particular content. For example, the orbit and rotation periods are not controlled in the hands-on condition, but are appropriately controlled in the desktop simulation and VR conditions. We would expect, therefore, that participants in the hands-on condition may perform worse on questions related to time scales. Upon being broken down by question the pre- to post-test scores show some variation between conditions but no statistically or practically significant differences (Fig. \\ref{fig:prepostbyq}). A Fisher's exact test gives p-values greater than 0.1 for all pre\/post scores by question. Some topic areas show gains over 40\\% that are consistent across condition (orbit period, phase period, and illumination) while others show gains less than 10\\%  (scale, Moon rotation, phase diagram, rise\/set time), consistent with previous research on learning about Moon phases \\cite{lindell2005,wilhelm2018}. Note the negative shifts in the ``Why phases occur\" question. Upon investigation, the questions for this topic were not truly isomorphic. Though they referred to the same topic, they varied significantly in difficulty level.\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=\\columnwidth]{{\"PrePostbyQ3\"}.pdf}\n\\caption{\\label{fig:prepostbyq} The difference in participant responses from the pre-test to the post-test broken down by question topic. The percent of correct responses on the pre-test and post test for that topic are connected by a colored bar. A green bar signifies improvement, with the higher number representing the post-test score. A red bar means there were fewer correct responses on the post-test, with the higher number representing pre-test score.}\n\\end{figure}\n\n\\subsection{Participant preference}\nDespite these lack of differences between learning, we found that 78\\% of participants preferred the VR activity (Fig. \\ref{fig:Pref}) with no dependence on assigned condition. The participants who preferred VR generally found that seeing the full picture helped build their mental model. The main contributors to this were the ease of viewing different perspectives and having control over the system. For example, one participant wrote: \n\\begin{quote}\n``Having a overall space to see where everything is helps a lot. Even in class I still had a hard time understanding what they are talking about in concept. But I think I learned a lot in VR and being able to manipulate the environment on my own accord. It seems more engaging than the 2 other methods.\"\n\\end{quote}\n\nIf the participant did not prefer VR it was mostly because they felt uncomfortable with the overwhelming sensory input. They preferred either the hands-on or desktop conditions because they were more familiar. For example, a participant who preferred hands-on wrote:\n\\begin{quote}\n``I really liked the virtual reality method. And it gave me more information than the other two methods, for instance, what time of day certain moon phases would rise and set. Nevertheless, it was almost too overwhelming and it was as if I was too excited to be in space to actually commit to learning the moon phases. With the hands-on demonstration. there was nothing to distract me. And, obviously, controlling the demonstration felt about as natural as possible.\" \n\\end{quote}\n\nA participant who preferred the desktop simulation wrote: \n\\begin{quote}\n``The VR was cool but since I'm very new to it I spent most of my time just trying to figure out how it worked--it was also tough to find where the sun and moon were at times because of how `large' the environment was. The desktop game was more familiar and easy-to-use for me. Personally.\"\n\\end{quote}\n\nWe can see that even participants who preferred other methods indicated that they liked the VR activity, but some participants were either overwhelmed by the controls or found it generally too distracting. As with computers, if the technology becomes more common, the issues regarding control and novelty may one day no longer be relevant. Currently however, efforts should be made to ensure familiarity before optimal efficiency is attained. \n\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[width=0.75\\columnwidth]{{\"PrefPieChart\"}.pdf}\n\\caption{\\label{fig:Pref} Preferred choice after viewing each activity.}\n\\end{figure}\n\n\\section{Discussion}\nIn this study, we evaluated the impact of an immersive and interactive VR activity, compared with similarly interactive hands-on and desktop simulations. It is clear there was significant improvement in all conditions in overall test scores from the pre- to post-test, but differences in learning between conditions were not significant (Fig. \\ref{fig:Histograms}). Previous work using these assessments found that typical astronomy students score just between 30 and 50\\% at pre-test and after a course in astronomy score about 65\\% \\cite{lindell2004,lindell2005}. This suggests that our participants, from various majors, achieved similar learning gains on the subject of moon phases after a single 10 minute activity as to those obtained in typical instruction. While the non-significant difference between conditions does not indicate virtual reality as one that enhances learning over traditional methods, it does show that VR can perform just as well. The hands-on and desktop activities both used technology very familiar with our participants. If participants can learn the same amount from a new technology they are not familiar with and running a simulation with a low production value there is likely still room for improvement for VR as a teaching tool. \n\nWe can also see how this activity in general performs as a teaching tool for the various Moon phase topic areas. There is clear evidence that further instruction is required on the topics of scale, Moon rotation, reasoning behind phases, and the rise\/set times of certain phases (Fig. \\ref{fig:prepostbyq}). Students typically struggle with understanding the topic of rise and set times for the Moon phases \\cite{lindell2004}. Our results indicate that VR does not help in these areas any better than the traditional methods. Learning in these areas may improve with alternate teaching methods since the results also suggest that the learning activity may be the stronger factor for student learning than the modality of instruction. The population differences between conditions may also be making it difficult to determine the advantages of each condition.\n \nAs in the previous work, we saw dramatic differences in participants' preference towards the VR simulation. It is possible that VR could be a powerful tool for improving student attitudes towards learning, while obtaining the same learning benefits as other modalities. This impact should be considered as researchers aim to develop ways to recruit more STEM majors or to promote knowledge retention. \n\n \\section{Future work}\nPrevious work found no differences in student learning in VR, video, and static images, but did find interactions with gender and video game experience \\cite{Smith2017}. We are currently analyzing our data for these effects and how performance relates to previous VR experience. We also aim to further probe students' attitudes towards and experiences in the activities through analysis of the video recordings, time spent in the simulations, and tracking movement. Future publications will provided greater detail on the population, methods, analysis, and results of the work presented in this paper and explore whether the various modalities provide different benefits to different learners. \n\\newline\n\\section{Conclusions}\nA cursory glance reveals that VR is as good at teaching Moon phases as traditional methods. The inexperience of participants and the current technical limits on VR appear to be the main limitations in providing a truly immersive experience and distract users from the concepts being taught much as hands-on activities do. Unlike hands-on activities however, these distractions can be overcome by advances in the technology. Higher frame rates, and larger fields of view will eventually remove simulator sickness as an obstacle to immersive interaction with the environment. At its current stage VR is a highly favored way to show the same learning gains as the tried-and-true methods.\n\n\\acknowledgements This work was supported by Oculus Education. We would like to thank all the graduate and undergraduate students who helped develop the simulations and run participants.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \n\\vspace*{-0.25cm}\n\nHeavy-quarkonium production has attracted \nconsiderable interest in the high-energy physics community since the $J\/\\psi$ discovery, exactly forty years\nago. It indeed probes the strong interaction at the interplay of its perturbative and non-perturbative \nregimes~\\cite{review}. It can also help to understand a new dynamics of hadron collision\nwhere multiple (hard) parton scatterings (MPS) take place.\nMPS are normally very rare since already a single (hard) parton scattering (SPS) is  rare as \ncompared to soft scatterings. Owing to the high parton flux at high energies, \nMPS should be likelier at the LHC, starting with two short-distance interactions from a single \nhadron-hadron collision, usually referred to as  double parton scattering (DPS). These have been \nsearched in $4$-jets~\\cite{Akesson:1986iv,Alitti:1991rd,Abe:1993rv}, $\\gamma+3$-jets~\\cite{Abe:1997xk,Abazov:2009gc}, \n$W+2$-jets~\\cite{Aad:2013bjm,Chatrchyan:2013xxa}, $J\/\\psi+W$~\\cite{Aad:2014rua}, $J\/\\psi+Z$~\\cite{Aad:2014kba}, \n$4$-charm~\\cite{Aaij:2012dz}, $J\/\\psi+$charm~\\cite{Aaij:2012dz} and $J\/\\psi+J\/\\psi$~\\cite{Abazov:2014qba} final states. \n\nAlong these lines, $J\/\\psi$-pair hadroproduction is of great interest. First, it provides \nan original tool to study quarkonium production in conventional SPSs.\nMost of the earlier theoretical studies are based on \nSPSs~\\cite{Kartvelishvili:1984ur,Humpert:1983yj,Vogt:1995tf,Li:2009ug,Qiao:2009kg,Ko:2010xy,Berezhnoy:2011xy,Li:2013csa,Lansberg:2013qka,Sun:2014gca};\nsome using the colour-singlet model (CSM)~\\cite{CSM_hadron}, others Nonrelativistic QCD (NRQCD)~\\cite{Bodwin:1994jh}. \nMoreover, it is widely claimed that  DPSs~\\cite{Kom:2011bd,Baranov:2011ch,Berezhnoy:2012xq,Baranov:2012re} \ncould indeed be a significant source of $J\/\\psi$ pairs at the LHC in proton-proton collisions and in \nproton-nucleus\/nucleus-nucleus collisions~\\cite{d'Enterria:2013ck,d'Enterria:2014dva}. Generally, it \nremains a poorly understood process. Its measurement with both $J\/\\psi$ decaying into a muon pair is a \nclean signal, accessible to most experiments, which is complementary to the DPS studies based \non open charm mesons and hadronic jets. With respect to the latter, it allows one to investigate\nthe physics of DPS at lower scales and lower $x$ where different mechanisms may be at work \n(see {\\it e.g.}~\\cite{Blok:2013bpa}).\n\n\n\n\n\n\nThe first observation of $J\/\\psi$-pair events dates back to that of the \nCERN-NA3 Collaboration~\\cite{Badier:1982ae,Badier:1985ri}. Recently, the LHCb~\\cite{Aaij:2011yc}, \nCMS~\\cite{Khachatryan:2014iia} and D0~\\cite{Abazov:2014qba} collaborations reported their measurements \nat the LHC and the Tevatron. \n\\begin{figure*}[hbt!]\n\\centering\n\\subfloat[]{\\includegraphics[width=.295\\columnwidth,draft=false]{LO-psi-psi-1.pdf}\\label{diagram-a}}\n\\subfloat[]{\\includegraphics[width=.295\\columnwidth,draft=false]{NLOg-t-channel-psi-psi.pdf}\\label{diagram-c}}\n\\subfloat[]{\\includegraphics[width=.295\\columnwidth,draft=false]{c-frag-psi-psi-2.pdf}\\label{diagram-e}}\n\\subfloat[]{\\includegraphics[width=.295\\columnwidth,draft=false]{c-frag-psi-psi.pdf}\\label{diagram-f}}\n\\subfloat[]{\\includegraphics[width=.295\\columnwidth,draft=false]{c-nonfrag-psi-psi-1.pdf}\\label{diagram-f2}}\n\\subfloat[]{\\includegraphics[width=.295\\columnwidth,draft=false]{c-nonfrag-psi-psi-2.pdf}\\label{diagram-f3}}\n\\subfloat[]{\\includegraphics[width=.295\\columnwidth,draft=false]{g-frag-psi-psi-2.pdf}\\label{diagram-g}}\n\\caption{Representative diagrams for the hadroproduction of $J\/\\psi+J\/\\psi$ via SPSs at $\\mathcal{O}(\\alpha_s^4)$ (a), at $\\mathcal{O}(\\alpha_s^5)$ (b) and \n $\\mathcal{O}(\\alpha_s^6)$ (c-g). }\n\\label{diagrams} \\vspace*{-0.5cm}\n\\end{figure*}\nIn contrast to Kom~{\\it et al.}~\\cite{Kom:2011bd}, we recently pointed out~\\cite{Lansberg:2013qka} that \n no definite conclusion on the presence of DPSs in LHCb data~\\cite{Aaij:2011yc} should be drawn \ngiven the very large theoretical uncertainties on the SPS predictions. However, the recent \nD0~\\cite{Abazov:2014qba} study could provide the very first separation  of the DPSs from SPSs \nand a measurement of $\\sigma^{\\rm DPS}_{\\psi\\psi}$ and $\\sigma^{\\rm SPS}_{\\psi\\psi}$ by using the \nyield dependence on the (pseudo)rapidity difference between the $J\/\\psi$ pair, as it \nwas first proposed in~\\cite{Kom:2011bd}. Although such a separation relies on a good modelling \nof the DPS and SPS rapidity-difference spectra, this can reasonably be considered as the first \nobservation of a DPS signal in quarkonium-pair production, even if SPSs also contribute in a \nsignificant fraction of the D0 acceptance.  Two fundamental remaining questions are whether such \nDPS contributions are also of importance elsewhere than at large rapidity difference, $\\Delta y$,\nand whether they agree with theory.\nIn addition, the recent CMS analysis, up to large $J\/\\psi$-pair transverse momenta ($P_T^{\\psi \\psi}$) \nbrought to light a new striking puzzle. As pointed out in~\\cite{Sun:2014gca},\nthe $P_T^{\\psi \\psi}$ and invariant mass, $M_{\\psi \\psi}$, spectra measured by CMS~\\cite{Khachatryan:2014iia} \nseverely overshoot the SPS contributions --even at next-to-leading order (NLO), {\\it i.e.}~$\\alpha_s^5$ . \n\nIn this Letter, we first show that the SPS yield extracted by D0 can only be reproduced thanks to the \nadditional $\\alpha_s^5$ or feed-down contributions from $J\/\\psi+\\psi'$.\nThen, along the lines of~\\cite{Kom:2011bd}, we model the DPS spectra based on 3 parametrisations of existing \nsingle $J\/\\psi$ data and extract -- accounting for the predicted SPS yield up to  $\\alpha_s^5$-- \nfrom a fit to the CMS results~\\cite{Khachatryan:2014iia}\nthe effective cross section $\\sigma_{\\rm eff}$ which characterises the effective spatial area of \nthe parton-parton interactions. Our fit result is then found to be well compatible with the DPS D0 results~\\cite{Abazov:2014qba}\nwhich means that we {\\it de facto} provide a solution to the aforementioned puzzle, with a coherent description \nof CMS and D0 results.\n\nIn addition, we provide an original test of the DPS vs SPS dominance based on the yields involving\nexcited states. Such a test can be used to validate our explanation of the CMS puzzle. \nFinally, we evaluate the first piece of the next-to-next-to-leading-order  contributions \nfrom $gg\\to J\/\\psi J\/\\psi c \\bar c$ (denoted $\\psi c \\bar c \\psi$) which is gauge invariant\nand infrared finite. Although it was expected to be enhanced at large \n$P^\\psi_{T\\rm min}=\\min(P^\\psi_{T1},P^\\psi_{T2})$, we find it is dominant \nonly when the yields are out of reach for current experiments and we conclude that an evaluation up to\n$\\alpha_s^5$ accuracy is probably sufficient at present time.\n\n\n\\vspace*{-0.25cm}\n\\section{Theoretical frameworks}\n\\label{sec:theory}\n\\subsection{SPS Contribution to $J\/\\psi+J\/\\psi$ production}\n\nIn this section, we outline the computation of the SPS contribution in the CSM~\\cite{CSM_hadron} or \nequally NRQCD at LO in $v^2$. The amplitude to produce of a pair of $S$-wave \nquarkonia denoted ${{\\cal Q}_1}$ and  ${{\\cal Q}_2}$, of given momenta $P_{1,2}$ and of polarisation $\\lambda_{1,2}$ \naccompanied by other partons --inclusive production--, noted $k$, is then given by the product \nof (i) the amplitude to create the corresponding double heavy-quark pair, in the specific kinematical \nconfiguration where the relative momenta of these heavy quarks ($p_{1,2}$) in each pairs is zero, (ii) \ntwo spin projectors $N(\\lambda_{1,2}| s_{1,3},s_{2,4})$ and (iii) $R_{1,2}(0)$, \nthe radial wave functions at the origin in the configuration space for both quarkonia. \nAt the partonic level, the SPS amplitude thus reads: \n\\eqs{\\label{CSMderiv3}\n&{\\cal M}(ab \\to {{\\cal Q}_1}^{\\lambda_1}(P_1)+{{\\cal Q}_2}^{\\lambda_2}(P_2)+k)=\\\\\n&\\sum_{s_1,s_2,c_1,c_2}\\sum_{s_3,s_4,c_3,c_4}\\!\\!\\!\\!\\frac{N(\\lambda_1| s_1,s_2)N(\\lambda_2| s_3,s_4)}{ \\sqrt{m_{{\\cal Q}_1}m_{{\\cal Q}_2}}} \\frac{\\delta^{c_1c_2}\\delta^{c_3c_4}}{3}\\frac{R_1(0)R_2(0)}{4 \\pi}\\\\\n&\\times {\\cal M}(ab \\to Q^{s_1}_{c_1} \\bar Q^{s_2}_{c_2}(\\mathbf{p_1}=\\mathbf{0}) + Q^{s_3}_{c_3} \\bar Q^{s_4}_{c_4}(\\mathbf{p_2}=\\mathbf{0}) + k),\\nonumber}\nwhere one defines from the heavy-quark momenta, $q_{1,2,3,4}$, \n$P_{1,2}=q_{1,3}+q_{2,4}$, $p_{1,2}=(q_{1,3}-q_{2,4})\/2$, and where $s_{1,3}$,$s_{2,4}$ are the \nheavy-quark spin components and $\\delta^{c_ic_j}\/\\sqrt{3}$  is the colour projector. $N(\\lambda| s_i,s_j)$ \nis the spin projector, which has a simple expression in the non-relativistic limit: \n$\\frac{1}{2 \\sqrt{2} m_{Q} } \\bar{v} (\\frac{\\mathbf{P}}{2},s_j) \\Gamma_{S} u (\\frac{\\mathbf{P}}{2},s_i) \\,\\, $ \nwhere $m_Q$ is the heavy-quark mass and $\\Gamma_S$ is $\\varepsilon^{\\lambda}_{\\mu}\\gamma^{\\mu}$ when $S=1$ \n({\\it e.g.}~ $J\/\\psi$, $\\psi'$). Such a partonic amplitude is then squared, summed over the\ncolour and spin of external partons and convoluted with the partonic densites (PDFs) in the allowed\nkinematical phase space. \n\nIn the case of $gg\\to J\/\\psi J\/\\psi+c\\bar c$, there exist more than \n2000 graphs (see \\cf{diagrams} (c-f)) \nwhich are non-trivially zero even after the topologies with a single gluon connected to an individual \nheavy-quark lines are removed. In order to generate the amplitude for this process, and the other ones \nconsidered in this study, we use {\\sc HELAC-Onia}, described in~\\cite{Shao:2012iz}, which generates\nthe amplitude based on \\ce{CSMderiv3} using recursion methods \\cite{Kanaki:2000ey}. It can also deal with \n$P$-wave production which involves the derivative of the amplitude in the relative momentum of the heavy-quark\nforming the quarkonia.\n\n{\\sc HELAC-Onia} also performs the helicity-amplitude calculations and the convolution with the PDF \nas well as the final-state variable integration. At LO, $J\/\\psi$-pair production at colliders is \nfrom $gg\\to J\/\\psi+J\/\\psi$ at $\\alpha_s^4$ (see {\\it e.g.}~\\cf{diagram-a}). At $\\alpha_s^5$, one needs to \nconsider real-emission contribution (see {\\it e.g.}~\\cf{diagram-c}) as well as loop corrections.\nIn~\\cite{Sun:2014gca}, a full NLO computation  showed that for\\footnote{Unless specified otherwise,  $P^\\psi_T$\nis the $P_T$ of one $J\/\\psi$ randomly chosen among both and the $P_T$ cuts discussed here apply to both.}  \n$P^\\psi_T > 2$ GeV, it is sufficient \nto rely on a NLO$^\\star$~\\cite{Lansberg:2013qka} evaluation where only the LO and NLO real-emission \ncontributions are accounted for; the latter being regulated by an infrared cut-off.\nThis is easily explained by the $P_T^2$ suppression of the loop contributions.\nFor the $\\psi c \\bar c \\psi$ contribution which we compute, there is no need to apply any infrared cut-off. \nSince the LO kinematics is that of a $2\\to 2$ process (\\cf{diagram-a}), it generates a trivial \n$P_T^{\\psi \\psi}$ dependence ($\\delta(P_T^{\\psi \\psi})$) in the collinear factorisation with\nconventional PDFs. We have therefore accounted for the $k_T$'s of the initial partons with a Gaussian \ndistribution with $\\langle k_T \\rangle=2$~GeV as in~\\cite{Lansberg:2013qka} in order to obtain\nfairer comparisons of the $P_T^{\\psi \\psi}$ spectra. Technical details about the implementation\n can be found in~\\cite{Shao:2015vga}. \n\nAs regards the PDFs, we use the set CTEQ6L1 for  LO (${\\cal O}(\\alpha_s^4)$) calculations and \nCTEQ6M~\\cite{Pumplin:2002vw} for NLO$^{\\star}$ (${\\cal O}(\\alpha_s^5)$)  and the ${\\cal O}(\\alpha_s^6)$ \n$\\psi c \\bar c \\psi$ calculations. The SPS uncertainties are obtained by a common variation of $m_c$\nand $\\mu_F=\\mu_R$ as $((1.4\\hbox{ GeV},0.5 \\times m_T^{\\psi\\psi});(1.5\\hbox{ GeV}, m_T^{\\psi\\psi});(1.6\\hbox{ GeV},2 \\times m_T^{\\psi\\psi}))$\nwith $m_T^{\\psi\\psi}=\\big((\\sum m_i)^2+(P_T^{\\psi})^2\\big)^{1\/2}$ \nwith $\\sum m_i =4 m_c$, but for $\\psi c \\bar c \\psi$ where  $\\sum m_i =6 m_c$.\n\n\\subsection{DPS  Contribution to $J\/\\psi+J\/\\psi$ production}\n\\label{subsec:DPS}\n\n\nQuarkonium-pair (${\\cal Q}_1+{\\cal Q}_2$) production from DPSs is usually assumed to come \nfrom 2 independent SPSs which create each a single quarkonia. One therefore assumes\n\\begin{eqnarray}\n\\sigma^{\\rm DPS}_{{\\cal Q}_1{\\cal Q}_2}=\\frac{m}{2}\\frac{\\sigma_{{\\cal Q}_1}\\sigma_{{\\cal Q}_2}}{\\sigma_{\\rm eff}},\n\\label{eq:DPS-fact}\n\\end{eqnarray}\nwhere $m=1$ for identical final-state particles and $m=2$ otherwise, $\\sigma_{\\cal Q}$ is the cross section \nfor {\\it single} ${\\cal Q}$ production. $\\sigma_{\\rm eff}$ is expected to account for the effective\nsize of the parton-parton interaction and should thus be universal --that is process-independent--  \nas well as $\\sqrt{s}$-independent if the factorisation holds as in~\\ce{eq:DPS-fact}. \nYet, there does not exist proofs of such a factorisation. Factorisation-breaking effects\nhave been discussed in a number of recent studies (see {\\it e.g.}~\\cite{Blok:2013bpa,Kasemets:2012pr,Diehl:2014vaa}).  \nAs of today, data is needed to test it case by case. Finally $\\sigma_{\\rm eff}$ cannot be determined \nfrom first principles or from perturbative methods. \n\n\n\nAnticipating the discussion of the D0 results in the next sections, which extracted a\nsmaller $\\sigma_{\\rm eff}$ from $J\/\\psi$-pair production than usually found in previous studies\ninvolving jet observables, we note that $\\sigma_{\\rm eff}$ could very well depend\non the flavour of the initial partons (see {\\it e.g.}~\\cite{Blok:2013bpa}). In the present case, these \nare gluons only, whereas for high-$P_T$ jets with $W$, light-quark initiated processes give a \nsignificant contribution. In addition, the process considered here occurs at a rather low \nmomentum scale. Finally, we stress that if the 1v2 contributions\\footnote{Such contributions, also\nsometimes denoted $1 \\otimes 2$, arise from 2 parton-parton scatterings \nwhere 2 partons from one proton come from a single parton. In a sense, these only initially involve 3 partons.} \ndiscussed in~\\cite{Gaunt:2014rua} matter, this would result\nin larger DPS contributions, thus in a smaller extracted $\\sigma_{\\rm eff}$.\n\n\n\nFollowing the common practice, DPS yields are simply computed from the corresponding measured \nsingle-$J\/\\psi$ yields using a Monte Carlo code with as input a parametrisation of $\\sigma_{\\psi}$, \nsee {\\it e.g.}~\\cite{Kom:2011bd}. Let us stress here that if one uses a parametrisation of \nprompt, {\\it i.e.}\\ excluding $b$-decay $J\/\\psi$, or direct quarkonium data, one would predict DPS yields for \nprompt or direct quarkonium pairs. Given the level of understanding of quarkonium-production mechanisms, \nusing theoretical models to compute $\\sigma_{\\psi}$ entering \\ce{eq:DPS-fact}  would only inflate \nthe theoretical uncertainties. As for now, the objective of DPS studies is to quantify their impact \nand to verify the factorisation hypothesis in a given kinematical domain. In the present study, \nthe objective is for instance to address the apparent discrepancy between the predicted SPS yield and the CMS data.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=.875\\columnwidth,draft=false]{fit_results2-190815-crop.pdf}\n\\caption{Illustrative comparison between  3 fits of $\\sigma_\\psi$  and LHCb~\\cite{Aaij:2011jh} and CDF~\\cite{Acosta:2004yw} \ndata for prompt $J\/\\psi$ production.}\\label{fig:comparison-fit}\\vspace*{-0.5cm}\n\\end{center}\n\\end{figure}\n\nWe thus use the setup proposed in~\\cite{Kom:2011bd} for the single $J\/\\psi$ cross section, $\\sigma_\\psi$, \n(Eqs.(2-4) of~\\cite{Kom:2011bd} slightly improved since we used 3 $\\sigma_\\psi$ fits \nin order to assess the systematic uncertainties attached to the parametrisation of $\\sigma_\\psi$.\nDetails regarding these fits are given in the \\ref{appendix-fits}. As an illustrative\ncomparison, the fits to LHC and Tevatron data are shown on~\\cf{fig:comparison-fit}. We\nstress that the data used for the 3 fits are for prompt $J\/\\psi$.\nThe corresponding short-distance matrix element has been added to a specific branch of \n{\\sc HELAC-Onia}~\\cite{Shao:2012iz} with as inputs the fit parameters. This branch \nis separate from that used to compute the SPS contributions (for technical details, the reader is referred to~\\cite{Shao:2015vga}). \n\nWe stress that the purpose\nof using an event generator such as {\\sc HELAC-Onia} for DPS computations is to perform the spin-entangled decay of the $J\/\\psi$'s \nunder different polarisation hypotheses to\napply the fiducial cuts (on the muons) of a given analysis if the muon acceptance was not corrected, \nas for the D0 analysis~\\cite{Abazov:2014qba}. A simple combination of $2 \\sigma_{\\psi}$ would not allow for this. \nIn the D0 case, a variation of $\\lambda_\\theta$ within $-0.45 <\\lambda_\\theta<0.45$, which represents\n a reasonable envelope of the existing experimental measurements in similar conditions (see {\\it e.g.}~\\cite{Abulencia:2007us}),\ninduces a systematical 20$\\%$ uncertainty. This can be compared to the 25 \\% systematical uncertainty on the corrected muon \nacceptance quoted by CMS~\\cite{Khachatryan:2014iia} due to the unknown $J\/\\psi$ polarisation.\n\n\nAs in~\\cite{Kom:2011bd}, we use the MSTW 2008 NLO PDF set~\\cite{Martin:2009iq} \nand the factorisation scale $\\mu_F=m_T^\\psi=(m_{\\psi}^2+(P_T^{\\psi})^2)^{1\/2}$. \nThe fit uncertainties attached to our DPS evaluation are discussed in the next sections.\nFinally, let us mention that we have studied one factorisation-breaking effect which is the \npossible correlation between two partons from a single proton as encoded in the double PDF \n(dPDF) of \\cite{Gaunt:2009re}.\nWe did not find any relevant difference in the region considered in this study.\n\n\n\\vspace*{-0.25cm}\n\\section{Feed-down relations for the DPS and SPS yields}\n\n\\subsection{Feed-down fractions under DPS dominance}\n\\label{subsec:FD_DPS}\n\nIf one sticks to a simplistic --although widely used-- view of the DPS production mechanisms \nas the one presented above, it is possible to derive general relations between the \nfeed-down fractions of the DPS yields for double and single $J\/\\psi$ production. These can be used \nto evaluate the feed-down impact, but also, by returning the argument, to test a possible \nDPS-dominance hypothesis by directly measuring pair productions involving the excited states.\n\n\nJust as we define the fractions, $F^{\\rm direct}_\\psi$, $F^{\\chi_c}_\\psi$ and $F^{\\psi'}_\\psi$,  \nof single  $J\/\\psi$ produced directly, from $\\chi_c$ decay or from $\\psi'$ decay, \n one can  define various feed-down fractions  for $J\/\\psi+J\/\\psi$. However, \none should keep in mind that it would  probably be experimentally very challenging\nto measure (and subtract) the yield of $\\chi_c+\\chi_c$ or even $\\chi_c+\\psi'$. We therefore limit ourselves to define \n$F^{\\chi_c}_{\\psi\\psi}$ (resp. $F^{\\psi'}_{\\psi\\psi}$)\nas the fraction of  $J\/\\psi+J\/\\psi$ events from the feed-down of {\\it at least} a $\\chi_c$ (or resp.  a $\\psi'$) decay. \nIn other words, $F^{\\chi_c}_{\\psi\\psi}$ is the fraction of events including a prompt $J\/\\psi$ (direct or from \n$\\chi_c$ and $\\psi'$ feed-down) plus a $J\/\\psi$ identified as from a $\\chi_c$.  Although it is probably \nvery difficult to measure it, we also define $F^{\\rm direct}_{\\psi\\psi}$ as being the pure  direct \ncomponent, excluding all the possible feed-downs, which can be easier to predict theoretically.\n\n\n\\begin{table*}[hbt!] \n  \\begin{center} \\renewcommand{\\arraystretch}{1.8}\\setlength\\tabcolsep{5pt}\\small\n\\begin{tabularx}{17.5cm}{cp{5cm}|cccccc}\n\\hline  \\hline      & Energy and quarkonium cuts   &$\\sigma_{\\rm exp.}$ &$\\sigma^{\\rm SPS, prompt}_{\\rm LO}$ &\n$\\sigma^{\\rm SPS, prompt}_{\\rm NLO^{(\\star)}}$ & $\\sigma^{\\rm DPS, prompt}$ & $\\chi^2$ \\\\\n\\hline\\hline \nLHCb\n &  $\\sqrt{s}=7$~TeV, $P^{\\psi_{1,2}}_T < 10$~GeV, \\hspace*{0.82cm} $2<y_\\psi < 5$~\\cite{Aaij:2011yc} &$18 \\pm 5.3$ pb  & $41^{+51}_{-24}$  pb  & $46^{+58}_{-27}$ &   $31^{+11}_{-6.3}(^{+24}_{-15})$ pb &$0.5 - 1.2$\\\\\\hline \n\\multirow{2}{*}{D0} & $\\sqrt{s}=1.96$~TeV, $P^{\\psi_{1,2}}_T > 4$~GeV, & SPS: $70 \\pm 23$ fb  & $53^{+57}_{-27}$ fb \n& $170^{+340}_{-110}$ fb &  -- & --\\\\ \n                    & $|\\eta_{\\psi}|<2.0$ ~\\cite{Abazov:2014qba} (+ $\\mu$ cuts in caption)& DPS: $59 \\pm 23$ fb           &        --           & --  & \n$44^{+16}_{-9.1}(^{+7.5}_{-5.1})$ fb&    $0.06 - 0.5$                            \\\\ \\hline \nCMS  & $\\sqrt{s}=7$~TeV, $P_T^{\\psi_{1,2}}> 6.5 \\rightarrow 4.5$~GeV depending on $|y_{\\psi_{1,2}}| \\in [0,2.2]$ (see the caption)~\\cite{Khachatryan:2014iia}&$5.25\\pm 0.52$  pb & $0.35^{+0.26}_{-0.17}$  pb  & $1.5^{+2.2}_{-0.87}$  pb & $0.69^{+0.24}_{-0.14}(^{+0.039}_{-0.027})$ pb & $1.09 - 1.14$ \\\\\\hline \nATLAS & $\\sqrt{s}=7$ TeV,  $P_T^{\\psi_{1,2}}>5$ GeV and $|y_{\\psi_{1,2}}|<2.1$ (+ $\\mu$ cuts in the caption)~\\cite{private-ATLAS}& -- & $6.4^{+4.3}_{-2.6}$ fb & $36^{+49}_{-20}$ fb &  $19^{+6.8}_{-4.0}(^{+2.2}_{-1.6})$ fb& N\/A\\\\ \\hline\\hline \n\\end{tabularx}\n\\caption{Comparison for $\\sigma(pp(\\bar{p})\\to J\/\\psi+J\/\\psi+X) \\times {\\cal B}^2(J\/\\psi \\to \\mu\\mu)$ between the LHCb, CMS and D0 data and our predictions  in the relevant kinematical regions (+ that of the forthcoming ATLAS analysis).\nThe theory predictions are: the SPS prompt yields at LO and NLO$^{\\star}$ [For LHCb, the evaluation is a complete NLO~\\cite{Sun:2014gca}], \nthe DPS prompt yields with $\\sigma_{\\rm eff}$ fitted to the CMS differential distributions (see section~\\ref{subsec:CMS-data}) and the $\\chi^2$ between the sum of DPS+SPS (resp. DPS) yield and CMS and LHCb (resp. D0 DPS) data. For the DPS yields, \nthe first uncertainty is from $\\sigma_{\\rm eff}$ (see~\\ct{fitDPS}) and the second in parenthesis \nis a systematical certainty from the 3 fits (alike the variation of the central value of $\\sigma_{\\rm eff}$ \nin~\\ct{fitDPS}).\nThe range of the $\\chi^2$  also comes from the 3 fits.\n[The additional uncorrected $\\mu$ cuts are: for D0, $P_T^{\\mu}> 2$~GeV when $|\\eta_{\\mu}|<1.35$ and total momenta $|p^{\\mu}|>4$~GeV when $1.35<|\\eta_{\\mu}|<2.0$; for ATLAS: \n$P_T^{\\mu}>2.5$~GeV and  $|\\eta_{\\mu}|<2.3$ and at least one $J\/\\psi$ with two muons with $P_T^{\\mu}>4$ GeV. For CMS, the detailed cuts are $P_T^{\\psi}> 6.5$~GeV if $|y_{\\psi}|<1.2$; $P_T^{\\psi}> 6.5 \\rightarrow 4.5$~GeV \nif $1.2<|y_{\\psi}|<1.43$;$P_T^{\\psi}> 4.5$~GeV if $1.43<|y_{\\psi}|<2.2$ where in $1.2<|y_{\\psi}|<1.43$, \nthe $P_T^{\\psi}$ cutoff scales linearly with $|y_{\\psi}|$].\n }\\vspace*{-0.5cm}\n\\label{xsections}\n\\end{center}\n\\end{table*}\n\nAssuming \\ce{eq:DPS-fact} holds for all charmonia, one gets\\footnote{The derivation of \\ce{eq:FD_fraction_DPS}\nfor $\\chi_c$ follows from the decomposition of the different sources of a prompt $J\/\\psi$ + a $J\/\\psi$ from a $\\chi_c$. Namely, one has~: direct $J\/\\psi+\\chi_c$, $\\chi_c+\\chi_c$ and $\\psi'+\\chi_c$. Their cross section with the relevant branchings can then be decomposed in terms of single quarkonium cross sections using \\ce{eq:DPS-fact} taking care of not double counting $\\chi_c+\\chi_c$ ($m=1$). Their sum divided by the cross section for a pair of prompt $J\/\\psi$ decomposed likewise then reads as \\ce{eq:FD_fraction_DPS} using the standard definitions of $F^{\\chi_c}_\\psi$, $F^{\\psi'}_\\psi$ and $F^{\\rm direct}_\\psi$.}\n\\eqs{\\label{eq:FD_fraction_DPS}\nF^{\\chi_c}_{\\psi\\psi}&= F^{\\chi_c}_\\psi \\times \\big(F^{\\chi_c}_\\psi + 2 F^{\\rm direct}_\\psi + 2 F^{\\psi'}_\\psi\\big),\\\\\nF^{\\psi'}_{\\psi\\psi}&= F^{\\psi'}_\\psi \\times \\big(F^{\\psi'}_\\psi + 2 F^{\\rm direct}_\\psi + 2 F^{\\chi_c}_\\psi\\big),\\\\\nF^{\\rm direct}_{\\psi\\psi}&= (F^{\\rm direct}_\\psi)^2.}\nIn order to obtain numbers,  let us recall that the world data tell us that \n$F^{\\rm direct}_\\psi$, $F^{\\chi_c}_\\psi$ and $F^{\\psi'}_\\psi$ are close to 60\\%, 30\\% and 10\\%. \nWe then obtain $F^{\\chi_c}_{\\psi\\psi} \\simeq 50 \\%$, $F^{\\psi'}_{\\psi\\psi} \\simeq 20\\%$ \nand $F^{\\rm direct}_{\\psi\\psi} \\simeq 35 \\%$. Although $F^{\\chi_c}_{\\psi\\psi}$ and $F^{\\psi'}_{\\psi\\psi}$ \nare experimentally accessible via $\\sigma((\\chi_c \\to J\/\\psi)+J\/\\psi)$ and $\\sigma((\\psi' \\to J\/\\psi)+J\/\\psi)$, \nthey are not sufficient to determine the pure direct yield since \n$F^{\\rm direct}_{\\psi\\psi} \\neq 1 - F^{\\chi_c}_{\\psi\\psi} - F^{\\psi'}_{\\psi\\psi}$. Its extraction would require\nthe measurement of $\\sigma(\\chi_c+\\psi')$. \n\n\n\n\\subsection{Feed-down fractions under SPS dominance}\n\nOn the contrary, one expects a larger feed-down from $\\psi'$ if SPSs dominate.\nIn the CSM or NRQCD at LO in $v^2$, the hard part for $\\psi'+J\/\\psi$ and $J\/\\psi+J\/\\psi$ is identical; only $|R(0)|^2$ differ.\nTaking $|R_{\\psi'}(0)|^2=0.53~{\\rm GeV}^3$~\\cite{Eichten:1995ch}, whereas $|R_{J\/\\psi}(0)|^2=0.81~{\\rm GeV}^3$,\nand ${\\cal B}(\\psi'\\to J\/\\psi)=55 \\%$~\\cite{Nakamura:2010zzi} as well as accounting for \na factor $2$ from the final-state symmetry, the ratio of $F^{\\psi'}_{\\psi\\psi}\/F^{\\rm direct}_{\\psi\\psi}$ --defined as in section \\ref{subsec:FD_DPS}-- is expected\nto be as large as $0.53\/0.81 \\times 0.55 \\times 2 + (0.53\/0.81 \\times 0.55)^2\\simeq 0.85$. \nIt may even be a bit larger since we neglected $\\sigma(\\chi_c+\\psi')$ in this evaluation of $F^{\\psi'}_{\\psi\\psi}$. \nThe latter approximation is justified, since we checked that neither $\\sigma(\\chi_c+J\/\\psi)$ nor \n$\\sigma(\\chi_c+\\psi')$ are significant under SPS dominance. In the CSM they are absent at $\\alpha_s^4$. The colour-octet \n(CO) contributions for the production of these pairs are small because\nof the small size of the CO non-perturbative parameters (also called LDMEs)~\\cite{Shao:2014fca} \nand the absence of any kinematical enhancement.\nIn the remaining our this work, we will thus consider that $\\sigma_{\\rm SPS}^{\\rm prompt}=\n1.85 \\times \\sigma_{\\rm SPS}^{\\rm direct}$. In turn, we also have  $F^{\\psi'}_{\\psi\\psi}\\simeq 0.85\/(1+0.85) \\simeq 46 \\%$\nat any order in $\\alpha_s$.\n\n\n\n\\subsection{DPS vs. SPS}\n\nTo summarise, in the SPS case,  $F^{\\psi'}_{\\psi\\psi}$ can be as large as $46 \\%$ whereas $F^{\\chi_c}_{\\psi\\psi}$ \nis expected to be small. In the DPS case, $F^{\\psi'}_{\\psi\\psi}$ is half as small, around $20\\%$,\nand $F^{\\chi_c}_{\\psi\\psi}$ large, around $50\\%$.\nThis clearly means that the relative measurements of charmonium-pair \nproduction of different states can serve as a clear test to pin down DPS or SPS dominance since\nthey correspond to rather opposite predictions. This can reliably be done provided that\nthe single-charmonium yields are known in the same kinematical region. We stress that, for\nsuch a test, we do not need to know the value of $\\sigma_{\\rm eff}$ which does not appear in \\ce{eq:FD_fraction_DPS}.\n\n\\vspace*{-0.25cm}\n\\section{Data-theory comparisons}\n\n\\subsection{The early LHCb data at low transverse momenta}\n Let us first look at the LHCb data. We claimed in a recent work~\\cite{Lansberg:2013qka} that there was no compelling reason \nto call for significant DPS contributions in order to describe the $J\/\\psi$-pair measurement by LHCb at 7 TeV\nin the forward rapidity region ($2 < y_\\psi <4.5$). In particular, there is absolutely no difficulty\nto reproduce the measured yield with the SPS contributions alone, see the first line of \\ct{xsections}. In fact, \nthe LO~\\cite{Lansberg:2013qka} and NLO~\\cite{Sun:2014gca} prompt SPS values  \neven tends to be above the LHCb one, leaving room for a possible DPS yield {\\it only} when the uncertainties\nare accounted for. We stress that this measurement was performed \nwithout any lower $P_T$ cuts and that, in this case, the LO and NLO SPS predictions \nare in very good agreement, showing a good convergence of the perturbative series.\nWe will comment later on the DPS predictions. \n\n\n\\subsection{The D0 data up to large $\\Delta y$ and the observation of DPS contributions}\nWe now discuss the recent D0 measurement~\\cite{Abazov:2014qba}. \nThanks to a wide (pseudo)rapidity coverage of about 4 units ($|\\eta|\\leq 2$), the D0 detector made \nthe first extraction of the DPS contributions to  $J\/\\psi$-pair production possible.\nAs neatly discussed in~\\cite{Kom:2011bd}, the yield as a function of the rapidity difference $\\Delta y$ \nbetween both $J\/\\psi$'s should be a good observable to distinguish DPS and SPS events. The DPS\n events have a broader distribution in $\\Delta y$ than the SPS ones. For the latter, large values of \n$\\Delta y$ imply large momentum transfers, thus highly off-shell particles, and are strongly suppressed. It is not \nthe case for DPSs where the rapidity of both $J\/\\psi$ is independent. \nLarge rapidity differences are only suppressed because the individual yields are suppressed for increasing rapidities. \n\n\nBy fitting the  $\\Delta y$  \ndistribution of their data, D0 managed~\\cite{Abazov:2014qba} to separate out the DPS and the SPS yields, {\\it i.e.}\\\n $\\sigma^{\\rm DPS}_{\\psi\\psi}$ and $\\sigma^{\\rm SPS}_{\\psi\\psi}$.\nThey found that about half of the (prompt) yield was from SPSs, \nthe remaining half from DPSs, about 60 fb (see \\ct{xsections}).\n\n\n\nIn~\\cite{Lansberg:2013qka}, we discussed \nthe relevance of taking into account $P_T$-enhanced \ntopologies (e.g. \\cf{diagram-c}) at NLO and performed, for the first time, a partial NLO evaluation, dubbed\nas NLO$^\\star$.\nIndeed, the real $\\mathcal{O}(\\alpha_s^5)$ emissions should be dominant in the intermediate \nand high $P_T$ regimes. This was recently confirmed by a full NLO evaluation~\\cite{Sun:2014gca}: \nthe NLO$^\\star$ $P_T^{\\psi}$ spectrum indeed accurately coincides with the NLO one for $P_T^{\\psi}>2 m_c$. \nIt is important to note that the NLO$^{(\\star)}$ yield is almost one order of magnitude larger than LO one \nwhen $P_T^{\\psi} = 5~{\\rm GeV}$ and that one must use\na NLO (or NLO$^\\star$) evaluation when dealing with data sets with a $P_T$ cut as it is the case here for all\nbut LHCb data. \n\nSince both $J\/\\psi$ should have their $P_T> 4$ GeV, it is interesting to look at \nthe impact of the $\\alpha_s^5$ corrections (NLO$^\\star$). Whereas the LO SPS yield \nis a bit below the SPS D0 yield, the NLO$^\\star$ yield, which is about 3 times larger, is above. \nBoth agree with the data within the large theoretical uncertainties --mainly from $m_c$. \nAs we shall see later, the need for $\\alpha_s^5$ corrections is far more obvious in the CMS acceptance\nwith a higher $P_T$ cut.\n\n\nInjecting their measured $\\sigma_\\psi$ and $\\sigma^{\\rm DPS}_{\\psi\\psi}$ in \\ce{eq:DPS-fact},\n D0 has found $\\sigma_{\\rm eff} \\simeq 5.0 \\pm 2.75~{\\rm mb}$. This value is 3 times lower\nthan those extracted with jet observables, which means that the DPS yield seems to be \n3 times higher than what could have been naively expected -- or at least twice higher accounting\nfrom their uncertainty. Yet, as next section will show, a low value of $\\sigma_{\\rm eff}$\nallows one to solve the CMS puzzle.\n\n\n\\begin{figure}[t!]\n\\begin{center}\n\\subfloat{\\includegraphics[width=\\columnwidth,draft=false]{dPt_CMS_280815_noCO-improved.pdf}\\label{fig:dsigCMSa}}\\vspace*{-0.25cm}\\\\\n\\subfloat{\\includegraphics[width=\\columnwidth,draft=false]{dy_CMS_280815-improved.pdf}\\label{fig:dsigCMSb}}\\vspace*{-0.25cm}\\\\\n\\subfloat{\\includegraphics[width=\\columnwidth,draft=false]{dM_CMS_280815-improved.pdf}\\label{fig:dsigCMSc}}\\vspace*{-0.25cm}\\\\\n\n\\caption{Comparison of different theoretical contributions with the CMS measurement: (a)  pair transverse momentum; (b) absolute-rapidity difference ; (c) pair invariant mass.}\n\\label{fig:CompareCMS}\n\\end{center}\\vspace*{-1cm}\n\\end{figure}\n\n\\subsection{The CMS data at large momenta} \n\\label{subsec:CMS-data}\n\n\nLO and NLO$^\\star$ SPS cross sections for  prompt $J\/\\psi$ pair production in the \nCMS acceptance~\\cite{Khachatryan:2014iia} are given in \\ct{xsections}.\nAs expected because of the higher $P_T$ cut, one observes a larger  NLO$^\\star$\/LO ratio than in the D0\nacceptance. Yet the NLO$^\\star$ SPS yield is significantly below the CMS data~\\cite{Khachatryan:2014iia}\nand hint at the presence of another source of $J\/\\psi$ pairs.\nAs we whall see, the discrepancy is much more evident when one looks at \ndifferential distributions.\n\n\n\nIndeed, besides this integrated yield, CMS measured differential distributions~\\cite{Khachatryan:2014iia}\nwhich further indicate the importance of both NLO SPSs and DPSs but in different regions.\nOverall CMS released in addition 17 data points of differential cross sections as a function of \n$P_T^{\\psi \\psi}$, $|\\Delta y|$ and $M_{\\psi \\psi}$. To quantify the impact\nof the DPS contributions, we have used these experimental data to fit $\\sigma_{\\rm eff}$ via \\ce{eq:DPS-fact}\nusing the 3 fits of $\\sigma_\\psi$ discussed in section~\\ref{subsec:DPS} and subtracting our theoretical \nevaluations of the SPS NLO$^\\star$ yield acounting for their uncertainties (green band in the plot of \\cf{fig:CompareCMS}).\n\n\n\\ct{fitDPS} summarises the fit result: the $\\chi^2_{\\rm d.o.f.}$ and the values of $\\sigma_{\\rm eff}$ along with \ntheir uncertainites coming from (a) the CMS experimental uncertainties\\footnote{including\nthe 25 \\% systematical uncertainty due to the unknown $J\/\\psi$ polarisation as discussed above.}  and (b) the theoretical uncertainties on the SPS yield.\nWe have also given the $\\chi^2_{\\rm d.o.f.}$ when no DPS contribution is considered. We note that the goodness of the 3 \nfits is similar.  The dispersion of the central values of $\\sigma_{\\rm eff}$ thus allows us to assess a systematical uncertainty due to\nthe paramatrisation of $\\sigma_\\psi$. \\cf{fig:CompareCMS} shows the DPS distributions with the Fit~2. Comparison plots using the Fit~1 \\&~3 are\ngiven a supplementary materials. In addition, we note that $\\sigma_{\\rm eff}$ fitted using the Fit~3 (only Tevatron data) \nis very close to the D0 value. Whereas the $p$-values\\footnote{assuming Gaussian uncertainties which is probably not true for \ntheory and some systematical experimental uncertainties.} of our DPS fits of $\\sigma_{\\rm eff}$ are about 2\\% (see also below), the one without \nDPS is below 0.03\\% (even less without $\\alpha_s^5$ contributions).\n\n\n\\begin{table}\n\\begin{center}  \\small \n\\begin{tabular}{c|cccccccccc\n       & $\\sigma_{\\rm eff}$ [mb]& $\\chi^2_{\\rm d.o.f.}$ & d.o.f. \\\\\n\\hline\\hline\n$\\sigma_\\psi$  Fit 1 \\cite{Kom:2011bd}   & $11  \\pm 2.9$  &  1.9 & 16 \\\\\n$\\sigma_\\psi$ Fit 2                      & $8.2 \\pm 2.2$  &  1.8 & 16 \\\\\n$\\sigma_\\psi$ Fit 3                      & $5.3 \\pm 1.4$  &  1.9 & 16 \\\\\nOnly LO SPS                             & N\/A            &  7.6 & 17 \\\\\nOnly NLO$^\\star$ SPS                      & N\/A           &   2.6 & 17 \\\\\n\\end{tabular}\\vspace*{-0.5cm}\n\\end{center}\n\\caption{Result of the fit of the DPS yield via $\\sigma_{\\rm eff}$ on the 18 CMS values.\\label{fitDPS}}\\vspace*{-0.5cm}\n\\end{table}\n\n\n\n\nAs regards the $J\/\\psi$-pair $P_T$, $P_T^{\\psi \\psi}$, distribution, \\cf{fig:dsigCMSa}\nclearly emphasises the importance of $\\alpha_s^5$ QCD corrections to the SPS yield. \nPhenomenologically accounting for the initial parton $k_T$ is not sufficient to reproduce the data\nas the smeared LO curve shows. \nAt NLO, hard real-emissions tend to generate larger momentum imbalances\nand configurations with $P_T^{\\psi \\psi}\\neq 0$. In fact, the real-emission topologies (\\cf{diagram-c}) tend to produce,\nat large $P_T$, two {\\it near} $J\/\\psi$ --as opposed to back-to-back-- with a large $P_T^{\\psi \\psi}$.\nIn the case of DPSs, correlations are absent and configurations at low $P_T^{\\psi \\psi}$ are favoured. There is also no reason\nfor large $P_T^{\\psi \\psi}$ configurations to be --relatively-- enhanced. It is thus not surprising that \nthe DPS band drops faster than the NLO$^\\star$ SPS one at large $P_T^{\\psi \\psi}$. \nThe ``bump'' around $P_T^{\\psi \\psi}\\simeq12~{\\rm GeV}$ simply reflects the kinematic cuts\nin the CMS acceptance.  Overall, one obtains a good agreement ($\\chi^2_{\\rm d.o.f.}\\simeq 1.1$) with the $P_T^{\\psi \\psi}$ distribution when DPS and (NLO$^\\star$) SPS \ncontributions are considered together, but it also confirms the dominance\nof (NLO$^\\star$) SPS contributions at large  $P_T^{\\psi \\psi}$.\n\nIn addition, CMS analysed the relative-rapidity spectrum, $d\\sigma\/d|\\Delta y|$. Along the lines of the D0 \ndata discussion, the SPS contribution dominate when $|\\Delta y| \\to 0 $, \nwhile the DPS ones are several orders of magnitude larger than the SPS ones at large $|\\Delta y|$. \nA comparison with the CMS data is shown in \\cf{fig:dsigCMSb}.\nMost of the data are consistent with our results, except for the last bin, \nwhich probably explains the low $p$-value which we obtained with the DPS fit. For this data set, \n$\\chi^2_{\\rm d.o.f.}\\simeq 2.1$ with DPS for the 3 fits and about 2.6 without (only NLO$^\\star$). Note\nthe large NLO$^\\star$ SPS uncertainty which de facto reduces the corresponding $\\chi^2$.\nThe CMS acceptance with a rapidity-dependent $P_T$ cut renders $d\\sigma\/d|\\Delta y|$ \nflatter but this effect is apparently not marked enough in our theory curves. \nMore data are however needed to confirm the absence of binning effects which could have generated \na dip in the distribution. As another possible cross-check, we provide \npredictions for the ATLAS, D0, and LHCb acceptances in \\ct{xsections} and as supplementary materials.\n\n\nAt $P_T^{\\psi \\psi}=0$ --where the bulk of the yield lies--, the $J\/\\psi$-pair invariant mass, $M_{\\psi \\psi}$, \nis closely related to $|\\Delta y|$ and provides similar information (see~\\cf{fig:dsigCMSc}). One indeed has\n$M_{\\psi \\psi}=2 m_T^{\\psi}\\cosh{\\frac{\\Delta y}{2}}$. \nLarge $\\Delta y$ --{\\it i.e.}~large relative {\\it longitudinal} momenta-- correspond to \nlarge $M_{\\psi \\psi}$. [At $\\Delta y=3.5$ and $P_T=6$ GeV, $M_{\\psi \\psi}\\simeq 40$ GeV.] Without additional\ncuts, the  $M_{\\psi \\psi}$  and $d\\sigma\/d|\\Delta y|$ spectra of the CMS do reveal the same conclusion: \nthe DPS contributions dominate the region of large momentum differences. At small $M_{\\psi \\psi}$, \nSPS contributions dominate and NLO corrections are large --essentially because CMS data do not cover low $P^\\psi_T$.\nFor this data set, $\\chi^2_{\\rm d.o.f.}\\simeq 3.0$ with DPS for the 3 fits and about 5.3 without (only NLO$^\\star$).\n\n\\subsection{Back to D0 and LHCb data}\n\nAs we have just seen, the inclusion of the DPS contributions with $\\sigma_{\\rm eff}$ ranging from 5 to 11 mb, depending\non the fit used for $\\sigma_\\psi$, solves the so-called CMS puzzle found (with SPS contributions only) in~\\cite{Sun:2014gca} with a $\\chi^2_{\\rm d.o.f.}$ \nreduced for all the distributions. Using these fits of $\\sigma_{\\rm eff}$, we should also reproduce the D0 DPS rate~\\cite{Abazov:2014qba}. \nThe agreement (see \\ct{xsections}) is quite good ($\\chi^2<1$) and gives us confidence\nthat our proposed solution to the CMS puzzle is indeed correct. The comparison with the LHCb result is less instructive\nowing to the uncertainties from the data and from the SPS.\n\n\n\n\\vspace*{-0.25cm}\n\\section{A first step toward a NNLO evaluation of the SPS contributions}\nAt high $P_T^{\\psi}$,  $\\mathcal{O}(\\alpha_s^6)$ (NNLO) contributions like  $g g \\to c \\bar{c}$ \nfollowed by $c(\\to J\/\\psi +J\/\\psi+c)$ (\\cf{diagram-e}) or twice by $c(\\to J\/\\psi +c)$  (\\cf{diagram-f})\nand $g g \\to g^\\star g \\to (J\/\\psi +J\/\\psi+g) g$ (\\cf{diagram-g}) are expected to be enhanced by factors \nof $P_T^{\\psi}$ w.r.t NLO~\\cite{Lansberg:2013qka,Sun:2014gca}. A two-loop computation is \nhowever needed to evaluate them, which is beyond the state-of-the-art. \nYet, such leading-$P_T$ contributions can be evaluated via the fragmentation approximation, as done in~\\cite{Sun:2014gca} \nonly for topologies like~\\cf{diagram-f}, which were expected to dominate at large $M_{\\psi \\psi}$. \nHowever, such an approximation has been shown~\\cite{Artoisenet:2007xi} to be unjustified \nfor the similar process $gg \\to J\/\\psi c \\bar{c}$ unless $P_T$ is much larger than $m_\\psi$.\nAs discussed in section~\\ref{sec:theory}, the process $gg\\to J\/\\psi J\/\\psi+c \\bar{c}$ is infrared safe and can be computed by itself\nusing {\\sc HELAC-Onia} ``out-of-the-box''~\\cite{Shao:2012iz}. \nAs just said, it includes\na class of likely dominant NNLO corrections depicted in \\cf{diagram-e} and \\cf{diagram-f}. \nOn the contrary, the contributions from topologies \\cf{diagram-g} could be evaluated using {\\sc \\small HELAC-Onia}\nbut only with an infrared cutoff as in \\cite{Artoisenet:2008fc,Lansberg:2013qka} for the NNLO$^\\star$. \nIn the present study, we prefer to limit ourselves to $gg\\to J\/\\psi J\/\\psi+c \\bar{c}$ which does not require\nany ad-hoc prescription. \n\nThe band labelled $\\psi c \\bar c \\psi$ in \\cf{fig:CompareCMS} shows its full contribution, which is computed for the first time.\nThis partial $\\alpha_s^6$ contribution is as large as the NLO$^\\star$ ones only at the highest $M_{\\psi \\psi}$ and $\\Delta y$, \nwhere the DPS ones are anyhow dominant. The case of another variable, the {\\it sub-leading} $P_T$, $P_{T\\rm min}$, is however particular since\nthe DPS spectrum is expected to scale as $P_{T\\rm min}^{-2\\,\\times\\, n}$, $n$ being the scaling power of the single $J\/\\psi$ yield ($P_T^{-n}$).\nWe thus found $J\/\\psi J\/\\psi+c \\bar{c}$  to be dominant (see  \\cf{fig:dsigCMSd} in the \\ref{sec:Ptmin-plot})\nat very high $P_{T\\rm min}$, which  corresponds to back-to-back production as in \\cf{diagram-f}. \n\nOverall, the aforementioned missing fragmentation contributions (\\cf{diagram-g}) at $\\mathcal{O}(\\alpha_s^6)$ \nare expected to be of similar sizes. In general, a full NNLO computation is thus expected to be similar\nto one at NLO accuracy, except in kinematical regions which are not currently accessible. \nCorresponding predictions for the forthcoming ATLAS and LHCb analyses as well as the current D0 acceptance\nare given as supplementary material for comparison with forthcoming data.\n\n\\vspace*{-0.25cm}\n\\section{Possible impact of colour-octet transitions}\n\nWe have also investigated the possible impact of CO channels as discussed at LO in~\\cite{Li:2009ug,Qiao:2009kg}.\nWe found that, because of the double suppression of the CO LDMEs, CO+CO\n channels are nowhere important when $P_T^{\\psi}<50~{\\rm GeV}$, as we found out~\\cite{Lansberg:2013qka}.\nWe have evaluated the contribution from  $^3S_1^{[8]}+\\ ^3S_1^{[8]}$ and $^1S_0^{[8]}+\\ ^1S_0^{[8]}$ (see~\\cf{fig:CompareCMS}) \nusing the 1-$\\sigma$ upper value of the  $^3S_1^{[8]}$ (resp. $^1S_0^{[8]}$) LDMEs\nof the NLO prompt fit of \\cite{Butenschoen:2011yh}, {\\it i.e.}\\ 0.00283 GeV$^3$ (resp. 0.0541 GeV$^3$), these\nare compatible with the LO direct fit of \\cite{Sharma:2012dy} and are the only ones not  dramatically overshooting\nthe low-$P_T$ single $J\/\\psi$ data~\\cite{Feng:2015cba}.\nAs we look for an upper value, we disregard the  $^3P_J^{[8]}+\\ ^3P_J^{[8]}$ contribution which is negative. A complete CO study is\nbeyond the scope of this Letter and will be the object a dedicated publication.\n\nIn any case, this upper limit of the CO contributions is always smaller than the CS ones except for $|\\Delta y| > 2.5$ (last bin in \\cf{fig:dsigCMSb}) and $M_{\\psi\\psi} > 40$~GeV \n(last two bins in \\cf{fig:dsigCMSc}). In these regions, these SPS contributions are anyhow extremely suppressed as compared to DPS ones\nand the CS SPS can receive significant $\\alpha_s^6$ contributions. \nThe only distribution where the CO contributions might show up is for the $P_{T \\rm min}$ (\\cf{fig:dsigCMSd}); it has\na similar size and the same dependence as our partial NNLO evaluation. They are larger than the NLO$^\\star$ SPS and DPS yields\nonly where the cross sections are on the order of $10^{-8}$ nb before accounting for the branchings.  \nAs regards the mixed CO+CS channels, there is no $P_T^{\\psi}$ enhancement to be expected and \nthese are simply suppressed by the LDME. \n\n\\vspace*{-0.25cm}\n\\section{Conclusion}\nIn this Letter, we have focused on the explanation of the recent observations of (prompt) $J\/\\psi$-pair \nproduction made by D0 and CMS. The measurements by CMS~\\cite{Khachatryan:2014iia},\nwhich severely overshoot the theory if one solely considers SPS contributions~\\cite{Sun:2014gca},\nindicate significant DPS contributions, which we find to agree with the magnitude measured by D0~\\cite{Abazov:2014qba}. \nFor the first time, our study shows that both DPSs and the NLO \nQCD corrections to SPSs are crucial to account for the existing data. \n\n\n\nIf these experimental results are confirmed, this would provide evidence\nfor \n\\begin{enumerate}[(i)]\n\\item the dominance of $\\alpha_s^4$ (LO) contributions for the total cross section, \n\\item the dominance \nof $\\alpha_s^5$ (NLO) contributions at mid and large $P_T^{\\psi \\psi}$, \n\\item  the dominance of DPS contributions\nat  large $\\Delta y$ and at large $M_{\\psi \\psi}$. \n\\end{enumerate}\nWe have also derived generic formulae predicting feed-down contributions or, equally speaking, charmonium-pair-production rates involving excited states,\n in case DPSs dominate. These can be checked\nby measuring $J\/\\psi+\\psi'$ or $J\/\\psi+\\chi_c$ production. Such data can also therefore check a \npossible DPS dominance as found by D$0$ and CMS at large momenta. The relatively small \nvalue of $\\sigma_{\\rm eff}$ (see \\cf{fig:sigma_eff}) compared to jet-related extractions we obtained to describe \nthe CMS data --also compatible with the D0 DPS yield\nand their extracted value of $\\sigma_{\\rm eff}$-- may\nbe a first hint at the flavour dependence of this effective cross section.\n\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=\\columnwidth,draft=false]{sigmaeff2-082015.pdf}\\vspace*{-0.25cm}\\\\\n\\caption{Comparison of our CMS fit result of $\\sigma_{\\rm eff}$ ($8.2 \\pm 2.0 \\pm 2.9$~mb: first uncertainty \nfrom the data and the SPS theory uncertainty, the second from the single $J\/\\psi$ template) with \nother extractions~\\protect\\cite{Akesson:1986iv,Alitti:1991rd,Abe:1993rv,Abe:1997xk,Abazov:2009gc,Aad:2013bjm,Chatrchyan:2013xxa,Abazov:2014qba}.\\label{fig:sigma_eff}}\n\\label{fig:CompareCMS}\n\\end{center}\n\\end{figure}\n\n\nFinally, we have carried out \nthe first exact evaluation of leading-$P_T$ NNLO ($\\alpha_s^6$) \ncontributions, {\\it i.e.}~$J\/\\psi$-pair production with a $c\\bar c$ pair, which could contribute\nat large $P_{T\\rm min}$.\nOn the way, our study of the impact of all the real emissions at $\\alpha_s^5$ and some at $\\alpha_s^6$  \nalso demonstrates the absence of a significant colour-octet \ncontribution contrary to what was found at LO in~\\cite{Li:2009ug,Qiao:2009kg}. \n\n\n\\vspace*{-0.25cm}\n\\section*{Acknowledgments}\n\\vspace*{-0.25cm}\n\\begin{small} \n We thank D. Bandurin, V. Belyaev, K.-T. Chao, D.~d'Enterria, E.G. Ferreiro, S. Spanier, B. Weinert, Z. Yang and D. Zieminska for fruitful and stimulating \ndiscussions. This work is \nsupported in part by the France-China Particle Physics Laboratory (FCPPL). H.-S. Shao is \nalso supported by the National Natural Science Foundation of China (No 11075002, No 11021092) \nand ERC grant 291377 ``LHCtheory: Theoretical predictions and analyses of LHC physics: \nadvancing the precision frontier\". \n\\end{small}\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nWe study portfolio optimisation subject to drawdown constraints. A drawdown constraint specifies that the investor's wealth $V_t$ has to remain above a given function $w$ of its maximum to date: $V_t> w(\\sup_{u\\leq t} V_u)$. The motivating example is the case of a linear $w$, when the current wealth is always greater than a fixed fraction of its past maximum. Such features are often embedded in investment opportunities available in the financial markets. From the investor's perspective, they offer a partial protection of the realised gains, where the past maximum is viewed as a natural reference point. For a manager who is trading clients' money avoiding large drawdowns is crucial -- typically many investors have a \\emph{stop-loss} provision and a large drawdown would result in a sudden withdrawal of capital from the fund, see Chekhlov et al.\\ \\cite{CUZ:05}.\n\nThis problem was originally introduced by Grossman and Zhou \\cite{GZ} who considered a power utility investor in a Black-Scholes market who faces a linear drawdown constraint and maximises the long-term (asymptotic) growth rate of the expected utility of her wealth. Grossman and Zhou \\cite{GZ} applied the \\emph{forward approach} and solved the problem using the dynamic programming principle. Later Cvitani\\'c and Karatzas \\cite{CK} generalised the setting in \\cite{GZ} to a complete $n$-dimensional market with deterministic coefficients. Using martingale theory they were able to link the solution to the optimisation problem with the drawdown constraint to an unconstrained problem which they could solve using the \\emph{dual approach} as in Karatzas, Lehoczky and Shreve \\cite{KaratzasLehochkyShreve:87}. The initial motivation for our work was to see if a similar link between the constrained and  unconstrained problems could be established in a much greater generality.\n\nInline with \\cite{GZ,CK} we consider maximisation of the asymptotic growth rate of the expected utility of wealth. The idea to look at the long-run optimality has proven to be a powerful tool in solving various portfolio optimisation problems explicitly. It was used by Grossman and Vila \\cite{GrossmanVila:92} in a problem with leverage constraints and by Dumas and Luciano \\cite{DumasLuciano:91} in presence of transaction costs. It is close to the objectives studied in the \\emph{risk-sensitive control}, see Section \\ref{sec:examples}. We refer to Guasoni and Robertson \\cite{GuasoniRobertson:11} for a more detailed discussion.\n\nIn this paper, we effectively solve the long-run continuous time portfolio optimisation problem with drawdown constraints.\nMore precisely, the main contribution of the paper is an equivalence result: the $w$-drawdown constrained problem with utility $U$ has the same value function as an unconstrained portfolio optimisation problem but with utility $U\\circ F_w$, where $F_w$ is given explicitly in terms of $w$. Moreover, the optimal wealth process for the drawdown constrained problem is obtained as an explicit pathwise transformation $M^{F_w}(V^*)$ of the optimal wealth process $V^*$ for the unconstrained problem.\nBoth the function $F_w$ and the transform $M^{F_w}$ are given in terms of the drawdown constraint $w$ and are independent of the underlying semimartingale model. In consequence, endowing an investor with a drawdown constraint is an effective and model-independent way of encoding her preferences. It is equivalent to modifying her utility function, e.g. a (long-run) power utility investor with risk aversion $\\rho\\geq 0$, $U(x)=\\frac{1}{1-\\rho}x^{1-\\rho}$, and a linear drawdown $w(x)=\\alpha x$, $\\alpha\\in (0,1)$, has the same value function as an unconstrained investor with risk aversion $\\alpha + \\rho (1-\\alpha)$. Note however that the two investors employ different optimal strategies. The former uses $M^{F_w}(V^*_\\rho)$ which satisfies the $\\alpha$-drawdown constraint, where $V^*_\\rho$ would be her optimal strategy without drawdown constraints. The latter uses $V^*_{\\alpha + \\rho(1-\\alpha)}$, which may have arbitrarily large drawdowns.\n\nOur results hold in an abstract semimartingale model and the investor is endowed with a generic utility function $U$ and a drawdown constraint $w$. Specifically, we only assume that wealth processes are max-continuous (i.e.\\ have a continuous running supremum), that $U$ either behaves like a logarithm or dominates a power function, and has a finite asymptotic elasticity as in Kramkov and Schachermayer \\cite{KrSch}, and that $w(x)\/x\\in (0,1)$ is bounded away from $1$. In such a general setting there is little hope to solve portfolio optimisation problems explicitly.\nAdding a drawdown constraint, which is a path-dependent constraint on the admissible investment strategies, appears to significantly increase the complexity. Rather surprisingly, our results show that this is not the case: the constrained problem is just as easy, or just as hard, as the analogue portfolio optimisation problem with no constraints. Since we consider the long-run optimality, Guasoni and Robertson \\cite{GuasoniRobertson:11} show that the latter can be solved in a rather general diffusion setting, see Section \\ref{sec:example_incomplete}.\n\nThis paper relies in an essential way on the so-called Az\\'ema-Yor processes. They effectively provide us with a bijection between non-negative wealth processes and the wealth processes which satisfy a given drawdown constraint. Az\\'ema-Yor martingales have initially appeared in \\cite{AY} where they were used to solve the Skorokhod embedding problem. Carraro, El Karoui and Ob\\l\\'oj \\cite{CEO} introduced a more general class of Az\\'ema-Yor processes and studied them from an SDE perspective. In particular they investigated their properties in relation to drawdown constraints. These results provided crucial insights for our work. In fact, objects in Cvitani\\'c and Karatzas \\cite{CK} can be expressed using Az\\'ema-Yor processes simplifying greatly the proofs in \\cite{CK}, see Section \\ref{sec:example_complete}.\n\nWithout the methods of our paper the drawdown constraints are in general very hard to study and we are not aware of any works investigating them in the generality considered here. Nevertheless, they have received some attention in the financial literature, which one would expect given their practical significance. Magdon-Ismail and Atiya \\cite{Magdon-IsmailAtiya:04} derived results linking the maximum drawdown to the mean return. Chekhlov, Uryasev and Zabarankin \\cite{CUZ:05} analysed discrete-time portfolio optimisation where the investors maximises the expected return subject to risk constraints expressed in terms of drawdowns. They reduce the problem to a linear programming problem which can then be solved numerically. \n\nIn the continuous time framework, apart from the early contributions in \\cite{GZ} and \\cite{CK}, drawdown constraints have recently been considered in setups with consumption. Roche \\cite{Roche} investigated maximisation of expected utility of consumption over infinite time horizon for a power utility and under a linear drawdown constraint. Elie and Touzi \\cite{ET} generalised this to a general class of utility functions in the setting of zero interest rates obtaining explicit representation of the solution. Subsequently, Elie \\cite{Elie:08} analysed the problem of maximising the expected utility of consumption and terminal wealth on a finite time horizon. He did not have explicit formulae but rather represented the value function as the unique (discontinuous) viscosity solution of the Hamilton-Jacobi-Bellman equation. All the above works considered only the Black-Scholes market. It is not clear at present if, and to what extent, our methods extend to such problems.\n\nFinally, we mention that Vecer \\cite{Vecer:06} analysed options on drawdowns as a more effective way against portfolio losses than put or lookback options. Further analysis of option sensitivities to drawdowns was presented in Pospisil and Vecer \\cite{PospisilVecer:10}.\n\nThe paper is organised as follows. Firstly, in Section \\ref{sec:market_model}, we introduce the financial\nmarket, give definitions and formulate the main portfolio optimisation problems of interest. In Section \\ref{AY_processes}, we discuss the necessary results on Az\\'ema-Yor processes. Section \\ref{sec:main_num} presents the main result and its proof. It considers the problem with uniform units: the wealth in both the utility function and the drawdown constraint is discounted by the same numeraire. In Section \\ref{sec:main_dollars}, we provide our results for utility of ``wealth in dollars\" but subject to drawdown condition on the discounted wealth, as in \\cite{CK,GZ}. This requires stronger asymptotic assumptions on $U$ and $w$ as well as deterministic interest rates. Section \\ref{sec:logut} is devoted to the drawdown constrained optimisation problem with an asymptotically logarithmic utility. Finally, in Section \\ref{sec:examples} several examples are presented. We first consider a general market model which admits price deflators as in Karatzas and Kardaras \\cite{KarKar} and give sufficient conditions for \nfiniteness of the value function. Then in Section \\ref{sec:example_complete} we specialise to the complete market with deterministic coefficients and give explicit solutions, extending results in \\cite{CK}. Finally, Section \\ref{sec:example_incomplete} provides an explicit solution for an incomplete market model.\n\nThe Appendix contains some technical lemmas needed in the proofs but which are of independent interest. In particular we show continuity of the value function -- the long-term (asymptotic) growth rate of the expected utility of wealth -- in the utility function $U$ and its invariance under perturbation of $U$ on some initial interval $[0,x_0]$.\n\n\\section{Financial market model and portfolio choice problems}\n\\label{sec:market_model}\n\nWe consider a general financial market model with no frictions. \nThe dynamics of traded assets are represented by a vector $\\tilde S=(\\tilde S^0,\\tilde S^1,\\ldots,\\tilde S^d)$ of semimartingales defined on a filtered probability space $(\\Omega, \\F,(\\F_t),\\p)$ satisfying the usual conditions. We write $\\overline X_t:= \\sup_{u\\leq t} X_u$ for the running supremum of a process $X$. \nWe say that a process $X$ is \\emph{max-continuous} if $(\\overline X_t)_{t\\geq 0}$ is a continuous process.\n\nWe fix a baseline asset, or a numeraire, $N=\\tilde S^0$, and express all other quantities in units of $N$. The traded assets are then given as $S:=(1, S^1,\\ldots, S^d)$, where $S_t^i=\\tilde S^i_t\/N_t$. It is customary for $N$ to be the (domestic) savings account but this is not necessary. In particular, the market could be spanned by $\\tilde S^1,\\ldots,\\tilde S^d$ and $N$ could be the wealth process of some trading strategy in these assets. We assume that $N$ is max-continuous, strictly positive and $N_0=1$.\n Agents are allowed to invest by trading in the usual self-financing way as long as their wealth processes are \\emph{max-continuous} and strictly positive. All wealth processes are expressed in units of $N$:\n\\begin{defn}\\label{def:wealth}\nAn adapted semimartingale $(V_t)$ is called a \\emph{wealth process} if it is strictly positive, $V_t>0$ and $V_{t-}>0$ for all $t\\geq 0$ a.s., $V$ is max-continuous and there exists an $(\\mathcal{F}_t)$--predictable process $\\pi=(\\pi^1,\\ldots,\\pi^d)$ such that $V_t=V_0+\\int_0^t \\sum_{i=1}^d \\pi^i_u \\mathrm{d} S^i_u$, where the (vector) integral is assumed to be well-defined. The set of wealth processes with $V_0=v_0$ is denoted $\\mathcal{A}(v_0)$.\n\\end{defn}\nWe note that in general $\\mathcal{A}(v_0)$ depends on our choice of the numeraire $N$ since max-continuity of $V$ and $VN\/\\tilde S^i$  are not necessarily equivalent. They are if, for example, all the assets $\\tilde S^i$ are continuous. Likewise, if we consider the case when $\\tilde S^i$ may only have jumps which are negative, totally inaccessible and unbounded, then $\\mathcal{A}(v_0)$ is the same for all continuous numeraires and corresponds to wealth processes with no short selling restriction.\n\nNote that so far we have not assumed ``no-arbitrage\" in either strong sense of existence of an equivalent martingale measure $\\mathbb{Q}$, or in a weaker sense of existence of a benchmark asset $\\hat V$ for which all $V\/\\hat V$ are supermartingales, see Section \\ref{sec:examples}. Neither have we made any strong assumptions on the integrability of $(\\pi_t)$ which would make wealth processes $\\mathbb{Q}$--martingales. Instead we consider utility maximisation in a general setting. As motivated in the Introduction, we are interested in enforcing drawdown constraints.\n\\begin{defn}\\label{def:DDfunc}\nWe say that $w$ is a \\emph{drawdown function} if it is non-decreasing and\n\\begin{equation}\\label{eq:w-assumption}\n\\exists \\alpha_1: 0 < w(x)\/x \\leq \\alpha_1 < 1,\\quad x> 0.\n\\end{equation}\nWe say that $(V_t)$ satisfies the \\emph{$w$--drawdown} ($w$-DD) condition if\n$$\\min\\{V_{t-},V_t\\}>w(\\sup_{u\\leq t}V_u),\\quad t\\geq 0,\\quad a.s.$$\nThe set of $(V_t)\\in \\mathcal{A}(v_0)$ which satisfy $w$-DD is denoted $\\mathcal{A}^w(v_0)$.\n\\end{defn}\nWe stress that wealth is expressed in units of the baseline asset $N$ and hence the drawdown constraint is relative to our choice of $N$.\nThe canonical example of a drawdown function is: $w(x)=\\alpha x$, see Example \\ref{ex:lin_dd}. To the best of our knowledge this is the only example which has been considered in the literature, including \\cite{CK,Elie:08,ET,GZ,Roche}. We consider a general possibly non-linear drawdown constraint. In particular, Definition \\ref{def:DDfunc} allows for a piece-wise constant\\footnote{More precisely, we can take $w$ to be piece-wise constant on $[v_0,\\infty)$ for any $v_0>0$ and for the drawdown problem we only consider $w(x)$ for $x$ greater than the initial capital.} function $w$ which has the effect that drawdown constraint is updated discretely at times when the wealth process reaches a new threshold. Definition \\ref{def:DDfunc} also allows for $w(x)\\equiv c>0$ when $\\mathcal{A}^w$ corresponds to wealth processes bounded below by a constant $c$.\n\nOne of the main aims of this paper is to relate portfolio choice problems with and without a drawdown constraint. The investor is assumed to be maximising the long-term growth rate of her expected utility of wealth. More precisely we consider the following two problems\n\\begin{prob}\\label{pb:utmax_DD}\nGiven $v_0>0$, a drawdown function $w$ and a function $U$ compute\n\\begin{equation}\\label{eq:opt_prob_DD}\n\\begin{split}\n\\mathrm{CER}_{U}^w(v_0) &:= \\sup_{V\\in \\mathcal{A}^w(v_0)}\\mathcal{R}_{U}(V),\n\\\\ &\\quad \\textrm{where}\\quad \\mathcal{R}_{U}(V):=\\limsup_{T\\to \\infty} \\frac{1}{T}\\log \\mathbb{E}\\left[U\\left(V_T\\right)\\right],\n\\end{split}\n\\end{equation}\nalong with the optimal wealth process which achieves the supremum.\n\\end{prob}\nHereafter $\\log$ is extended to $\\mathbb{R}\\setminus \\{0\\}$ via $\\log(x)=-\\log(-x)$ for $x< 0$.\n\n\\begin{prob}\\label{pb:utmax}\nGiven $v_0>0$ and a function  $U$ compute\n\\begin{equation}\\label{eq:opt_prob}\n\\begin{split}\n\\mathrm{CER}_{U}(v_0) &:= \\sup_{X\\in \\mathcal{A}(v_0)}\\mathcal{R}_{U}(X)\n\\end{split}\n\\end{equation}\nalong with the optimal wealth process which achieves the supremum.\n\\end{prob}\n\nProblem \\ref{pb:utmax} can be solved in number of fairly general setups, see Section \\ref{sec:examples} below. \nOur aim is to build a direct link between the solution to this problem and the solution to a seemingly more complex Problem \\ref{pb:utmax_DD} which features pathwise drawdown constraints. Problem \\ref{pb:utmax_DD}, considered in Theorem \\ref{thm:main} below, is a greatly generalised version of the problem introduced by Grossman and Zhou \\cite{GZ} and analysed later by Cvitani\\'c and Karatzas \\cite{CK}. Firstly, we allow for an almost arbitrary utility function $U$ and not just the power utility.\nSecondly, we consider a general possibly non-linear drawdown constraint. Finally, we work in a general semimartingale financial market model and not a complete Black-Scholes-like model. Working in such a generality we can not hope for an explicit solution to Problem \\ref{pb:utmax_DD} as in \\cite{CK,GZ}. \nOur main result offers second best: we obtain an explicit formula for the value function and the optimal investment strategy in Problem \\ref{pb:utmax_DD} in terms of the value function and the optimal investment strategy in Problem \\ref{pb:utmax} but with a suitably modified utility function.\n\nThe idea to look at $\\mathcal{R}_U(V)$ -- the growth rate of the expected utility -- goes back to Dumas and Luciano \\cite{DumasLuciano:91}, Grossman and Vila \\cite{GrossmanVila:92} and Grossman and Zhou \\cite{GZ}. \nCER above stands for \\emph{Certainty Equivalent Rate} and is interpreted as the critical safe rate -- if the investor was offered such (or higher) rate of growth via other investment opportunities she would be happy to abandon the market and move to the alternative investment opportunities.\nSimilar criterion also appears in the \\emph{risk-sensitive control} literature e.g.\\ Fleming and Sheu \\cite{FleSheu}, see also Section \\ref{sec:example_incomplete}. The criterion is designed to capture the long-horizon optimality and is often more tractable\\footnote{We note however that it may fail to provide strategies optimal on a finite time horizon, as discussed by Klass and Nowicki \\cite{KlassNowicki:05} in the context of drawdown constraints.} than the fixed-horizon utility maximisation of terminal wealth, cf.~Guasoni and Robertson \\cite{GuasoniRobertson:11}. \n\nNote that Problem \\ref{pb:utmax_DD} has unified units: both the drawdown and the utility are applied to wealth in units of $N$. In \\cite{CK,GZ} the drawdown is relative to $N$ (which is the savings account) but the reward functional is taken of the ``wealth in dollars\": $\\mathcal{R}_U(VN)$. This introduces further inhomogeneity and is solved in Section \\ref{sec:main_dollars} under additional assumptions. Both Sections \\ref{sec:main_num} and \\ref{sec:main_dollars} consider $U$ which is either always positive or always negative. Utility functions with behaviour similar to logarithm are treated in Section \\ref{sec:logut} where \\eqref{eq:opt_prob_DD} is modified into \\eqref{eq:opt_prob_log}.\n\nFinally, we note that we consider only wealth processes which are strictly positive. However, in view of Lemma \\ref{lem:2U}, this is not restrictive and we could allow wealth processes which become zero from some point onwards. Likewise, we impose strict drawdown condition but this could be relaxed. We could allow the drawdown constraint to be hit and the process would remain constant thereafter. Lemma \\ref{lem:2U} and Theorem \\ref{thm:main} show that such wealth processes can be excluded without affecting the value of problems of long-run utility maximisation considered in this paper.\n\n\\begin{remark}\nAs stated above, in \\eqref{eq:opt_prob_DD}--\\eqref{eq:opt_prob} and throughout the paper, we extend $\\log$ to $\\mathbb{R}\\setminus \\{0\\}$ via $\\log(x)=-\\log(-x)$ for $x<0$. This implies that $\\mathcal{R}_{U_1}(X)\\geq \\mathcal{R}_{U_2}(X)$ if $U_1\\geq U_2$ are two functions of the same sign. Note also that $V_t\\equiv v_0\\in \\mathcal{A}^w(v_0)\\subset \\mathcal{A}(v_0)$ and hence $\\mathrm{CER}_{U}\\geq \\mathrm{CER}_U^w\\geq 0$.\n\\end{remark}\n\n\\section{Drawdown constraints and Az\\'ema--Yor processes} \\label{AY_processes}\nWe recall now the so--called Az\\'ema--Yor processes. We use their properties established in Carraro, El Karoui and Ob\\l\\'oj \\cite{CEO} to build an explicit and model-independent bijection between $\\mathcal{A}(v_0)$ and $\\mathcal{A}^w(v_0)$. It will be our crucial tool used to relate Problems \\ref{pb:utmax}, \\ref{pb:utmax_DD} and their solutions. \n\n\\begin{proposition}[Carraro, El Karoui and Ob\\l\\'oj \\cite{CEO}]\\label{prop:AY}\nLet $F'$ be a locally bounded function, $F(x)=F(x_0)+\\int_{x_0}^x F'(u)\\mathrm{d} u$, and $(X_t)$ a max-continuous $(\\mathcal{F}_t)$--semimartingale. The associated Az\\'ema--Yor process $M^F(X)$ is given via\n\\begin{equation}\\label{eq:AYdef}\nM^F_t(X):=F(\\overline X_t)-F'(\\overline X_t)(\\overline X_t - X_t) = F(X_0)+\\int_0^t F'(\\overline X_u)\\mathrm{d} X_u,\\ t\\geq 0,\n\\end{equation}\nwhere $\\overline X_t := \\sup_{u\\leq t} X_u$. Further\n\\begin{itemize}\n\\item if $F'\\geq 0$ then $\\overline{M^F(X)}_t=F(\\overline X_t)$, $t\\geq 0$,\n\\item if $F'>0$ then $M^K(M^F(X))=X$ with $K=F^{-1}$ the inverse of $F$,\n\\item if $F$ is concave then $M^F_t(X)\\geq F(X_t)$, $t\\geq 0$.\n\\end{itemize}\n\\end{proposition}\nThe above combines Definition 2.1 and Proposition 2.2 in \\cite{CEO} while the last property is clear (see also Proposition 4.12 point c) in \\cite{CEO}). \nIt is important to observe that Az\\'ema--Yor processes automatically satisfy a drawdown property. More precisely, consider $M^F(X)$ for $X\\geq 0$ and $F'>0$. Then from \\eqref{eq:AYdef}, using first $F'(\\overline X_t)X_t\\geq 0$ and then $F(\\overline X_t)=\\overline{M^F(X)}_t$, we obtain \n$$M^F(X)_t\\geq F(\\overline X_t)-F'(\\overline X_t)\\overline X_t = w(\\overline{M^F(X)}_t),\\quad t\\geq 0,$$\nwhere $w(x)=x-K(x)\/K'(x)$, $K:=F^{-1}$. The following result shows that we can start with $w$, solve the ODE for $K$ and hence obtain $F$. It builds an explicit and model-independent bijection between $\\mathcal{A}(v_0)$ and $\\mathcal{A}^w(v_0)$ and will be our main tool in this paper.  \n\\begin{proposition}\\label{prop:DD}\nLet $w$ be a function satisfying \\eqref{eq:w-assumption} and define\n\\begin{equation}\\label{eq:Kdef}\nK_w(x) := v_0 \\exp \\left( \\int_{v_0}^x \\frac{1}{u - w(u)} \\mathrm{d} u \\right), x \\geq v_0>0,\n\\end{equation}\nwhich is continuous and strictly increasing and has a well defined inverse $F_w:=K_w^{-1}:[v_0,\\infty)\\to [v_0,\\infty)$.\\\\\nThe mapping $V\\to M^{F_w}(V)$ is a bijection between $\\mathcal{A}(v_0)$ and $\\mathcal{A}^w(v_0)$ with its inverse given by $X\\to M^{K_w}(X)$.\\\\\n For $V\\in \\mathcal{A}(v_0)$ we have $X:= M^{F_w}(V)\\in \\mathcal{A}^w(v_0)$ satisfies\n\\begin{equation}\\label{eq:SDEwDD}\n\\mathrm{d} X_t = \\left(X_{t-}-w\\left(\\overline{X}_t\\right)\\right)\\frac{\\mathrm{d} V_t}{V_{t-}},\\ t\\geq 0.\n\\end{equation}\nFinally, if $w$ is nondecreasing then $K_w$ is convex and $F_w$ is concave.\n\\end{proposition}\n\\begin{proof}\nFirst note that in Proposition \\ref{prop:AY} to define $M^F(X)$ it suffices that $F$ is defined on the range of possible values of $\\overline X$. \nIn particular, both $M^{F_w}(V)$ and $M^{K_w}(X)$ in the statement of Proposition \\ref{prop:DD} are well defined.\n\nProperties of $K_w$ follow by a straightforward differentiation. Observe that by \\eqref{eq:w-assumption} $K_w(x)<\\infty$ for all $x>v_0$ and \n$K_w(\\infty)=\\infty$ so that the inverse $F_w$ is well defined on $[v_0,\\infty)$ and $F_w(\\infty)=\\infty$. Further $F_w$ is increasing so, by Proposition \\ref{prop:AY}, if $V\\in \\mathcal{A}(v_0)$ then the process $X=M^{F_w}(V)$ is well defined, strictly positive and max-continuous. We apply Theorem 3.4 in \\cite{CEO}  with $v^*_0 = v_0$. Note that, in the notation therein, we have $\\zeta=\\infty$ a.s. since $r_w=\\infty$ and $K_w(\\infty)=\\infty$. We conclude that $X$ satisfies the required $w$--drawdown property for all $t\\geq 0$ and \\eqref{eq:SDEwDD} holds, and hence that $X\\in\\mathcal{A}^w(v_0)$.\n\nLikewise, for the converse, $X\\in \\mathcal{A}^w(v_0)$ satisfies the $w$-DD condition for all times and hence we apply Theorem 3.4 in \\cite{CEO} with $\\zeta=\\infty$. \nFinally, the last statement is clear since when $w$ is non-decreasing then, by direct differentiation, so is $K'$.\\hfill$\\square$\n\\end{proof}\nNote that we have $w(x)=x-K_w(x)\/K_w'(x)$, $x\\geq v_0$, a property which we will use often below. Strictly speaking, in view of \\eqref{eq:Kdef}, this holds for $x>v_0$ and is used to define $K_w'(v_0)$. Likewise we take $F_w'(v_0)=1\/K_w'(v_0)$. We adopt this convention hereafter.\n\nBy \\eqref{eq:AYdef}, in the above it is sufficient to consider $F_w(v)$ for $v\\geq v_0$ since $\\overline{V}_t\\geq V_0=v_0$. We are free to define $F_w$ on $[0,v_0)$ in any way without affecting $X_t$. As we will see later, any extension which preserves the sign, monotonicity and concavity of $F_w$ will be allowed.\nFor completeness we specify one such extension by extending $F_w$ for all positive $v$ as follows\n\\begin{equation}\\label{eq:Fdef}\nF_w(v) := \\left\\{\\begin{array}{ll} K_w^{-1}(v) & \\textrm{if $v \\geq v_0$}\\\\\nF_w'(v_0+)(v-v_0) + v_0 & \\textrm{if $0 \\leq v <v_0$} \\end{array} \\right.\n\\end{equation}\nso that $F_w(0)=w(F_w(v_0))=w(v_0)>0$ and $F_w$ is increasing and concave on $[0,\\infty)$ if $w$ is nondecreasing. \n\n\\begin{example}\\label{ex:lin_dd}\nConsider a linear drawdown constraint $w(x)=\\alpha x$, $\\alpha\\in (0,1)$. Then $K_w(v)=v_0(v\/v_0)^{1\/(1-\\alpha)}$ and $F_w(x)=v_0(x\/v_0)^{1-\\alpha}$. For $V\\in \\mathcal{A}(v_0)$, Proposition \\ref{prop:DD} gives us that $X:=M^{F_w}(V)\\in \\mathcal{A}^w(v_0)$ and an explicit calculation using \\eqref{eq:AYdef} gives\n$$X_t=M^{F_w}_t(V)=v^{\\alpha}_0(\\alpha \\overline{V}_t^{1-\\alpha}+(1-\\alpha)\\overline{V}_t^{-\\alpha}V_t) ,\\ t\\geq 0,$$\nwith an analogous expression for $V$ in terms of $X$.\n\n\\end{example}\n\n\\begin{example}\\label{ex:const_dd}\nAs an extreme example, consider a constant drawdown constraint $w(x)\\equiv c \\in (0,v_0)$. Then $K_w(v)=\\frac{v_0}{v_0-c}v-\\frac{cv_0}{v_0-c}$ and $F_w(x)=c+\\frac{v_0-c}{v_0}x$. It follows that for $V\\in \\mathcal{A}(v_0)$ we have\n$$X_t=M^{F_w}_t(V)=\\frac{v_0-c}{v_0}V_t+c>c.$$\n\\end{example}\n \nThe methodology based on Az\\'ema--Yor processes, as introduced above, is perfectly suited for our analysis. We note however that it is also possible to study drawdowns, and in particular laws related to the first time a drawdown occurs, by considering $X_t=(X_t-\\overline X_t)+\\overline X_t$ and applying methods of processes of class $\\Sigma$, see e.g.\\ Cheridito, Nikeghbali and Platen \\cite{CNP}.\n\n\\section{Main results}\\label{sec:main_num}\nWe are now ready to formulate our main results. The essence of the results is simple and explicit: the $w$--drawdown problem with a utility function $U$ has the same value as the unconstrained problem with the utility function $U\\circ F_w$: $\\mathrm{CER}^w_{U}=\\mathrm{CER}_{U\\circ F_w}$, where $w$ and $F_w$ are related by \\eqref{eq:Kdef} and \\eqref{eq:Fdef}. Further, the optimal wealth process is given by $ M^{F_w}(V^*)$, where $V^*$ is the optimal wealth for the unconstrained problem. Note that these results do not depend on the underlying stochastic market model.\nWe impose\n\\begin{asmp}\\label{ass:U}\nAssume that, for some $\\varepsilon>0$, $U$ satisfies either\n\\begin{equation*}\n\\frac{U(x)}{x^{\\varepsilon}} \\vax \\infty,\\quad \\textrm{and $U$ is strictly positive on }(0,\\infty),\n\\end{equation*}\nor\n\\begin{equation*}\n U(x)x^{\\varepsilon} \\vax 0, \\quad \\textrm{and $U$ is strictly negative on }(0,\\infty).\n\\end{equation*}\n\n\\end{asmp}\nThis insures that our utility functions are of constant sign and they dominate a power utility. We will further assume that they admit positive finite Asymptotic Elasticity in the sense of Kramkov and Schachermayer \\cite{KrSch}. Throughout the paper a \\emph{utility function} simply means a non-decreasing concave function and $U'_+$ denotes its right derivative, $U'_+(x) := \\lim_{y \\downarrow x} \\frac{U(y) - U(x)}{y-x}$. Recall Definition \\ref{def:wealth}, and in particular that wealth processes are \\emph{max-continuous}, and Definition \\ref{def:DDfunc}.\n\n\\begin{theorem}\\label{thm:main} Let $w$ be a drawdown function, $v_0>0$ and $U$ a utility function satisfying Assumption \\ref{ass:U} and \n\\begin{equation}\\label{eq:AEU}\n\\limsup_{x\\to \\infty} \\frac{xU'_+(x)}{|U(x)|}<\\infty.\n\\end{equation}\nRecall that $K_w$ is given by \\eqref{eq:Kdef} and let $F_w$ be its inverse extended to $[0,\\infty)$ as in \\eqref{eq:Fdef} or in any other way which preserves monotonicity, concavity and $F_w(0)>0$. Assume that, for some $\\delta>0$, $\\mathrm{CER}_{G}(v_0)<\\infty$ where $G(x)=U \\circ F_{w}(x)$ when $U<0$ and $G(x)=(U \\circ F_{w}(x))^{1+\\delta}$ when $U>0$.\nThen\n\\begin{equation*}\n\\mathrm{CER}^w_{U}(v_0)=\\mathrm{CER}_{U\\circ F_w}(v_0)<\\infty\n\\end{equation*}\nand if $V^*\\in  \\mathcal{A}(v_0)$ achieves the maximum in the unconstrained problem then $M^{F_w}(V^*)\\in  \\mathcal{A}^w(v_0)$ achieves the maximum in the $w$-drawdown constrained problem.\n\\end{theorem}\n\n\\begin{proof} For simplicity, in the proof we write $K=K_w$ and $F=F_w$.\nLet $V\\in \\mathcal{A}(v_0)$ and $X:= M^F(V)$ which is in $\\mathcal{A}^w(v_0)$ by Proposition \\ref{prop:DD}.\nNow, by Proposition \\ref{prop:AY} we obtain directly\n\\begin{equation}\\label{eq:first_ineq}\nU\\left(X_t\\right)=U\\left(M^F_t\\left(V\\right)\\right) \\geq U \\left( F\\left(V_t\\right) \\right),\\ t\\geq 0,\n\\end{equation}\nwhich readily implies $\\mathcal{R}_{U}(X)\\geq \\mathcal{R}_{U\\circ F}(V)$. Taking supremum over all $V \\in \\mathcal{A}(v_0)$ we conclude\n\\begin{equation*}\n \\mathrm{CER}^w_{U}(v_0) \\geq \\mathrm{CER}_{U \\circ F}(v_0)\\ .\n\\end{equation*}\nIt follows also that if we had equality and the right hand side was attained by a wealth process $V^*$ then the left hand side is attained by $ M^F(V^*)$, as required.\n\nIt remains to establish the reverse inequality. The idea of the proof is to consider a sequence of unconstrained problems whose value functions all dominate $\\mathrm{CER}^w_{U}(v_0)$ and converge to $\\mathrm{CER}_{U \\circ F}(v_0)$. We will do this by relaxing the drawdown constraint $w$ to $w_n$ and considering utility functions $U\\circ F_{w_n}$. This will allow us to obtain \\eqref{eq:second_ineq} below which reverses the inequality in \\eqref{eq:first_ineq}. Let\n\\begin{equation}\\label{eq:wn_and_w}\nw_n(x):=(1+\\frac{1}{n})w(x)-\\frac{1}{n}x= w(x)-\\frac{1}{n}\\frac{K(x)}{K'(x)}=\nx-(1+\\frac{1}{n})\\frac{K(x)}{K'(x)}, \\quad x \\geq v_0,\n\\end{equation}\nwhere the equalities follow from $w(x)=x-K(x)\/K'(x)$. We take $n$ large enough so that $w_n(x)$ satisfies \\eqref{eq:w-assumption} but we note that $w_n$ may fail to be globally non-decreasing on $(0,\\infty)$.\n\nIt follows by a direct computation that\n$$K_n(x):= K_{w_n}(x)= v_0 \\exp \\left( \\int_{v_0}^x \\frac{1}{u - w_n(u)} \\mathrm{d} u \\right) = v_0^{\\frac{1}{1+n}}\\left(K(x)\\right)^{\\frac{n}{1+n}},\\quad x\\geq v_0,$$\nand $w_n(x)=x-K_n(x)\/K'_n(x)$, where by our convention this is used to define $K'_n(v_0)$.\nConsider $(X_t) \\in \\mathcal{A}^w(v_0)$ and let $Y^n_t:=M^{K_n}_t(X)$ which is an element of $\\mathcal{A}(v_0)$ by Proposition \\ref{prop:DD}. Using \\eqref{eq:AYdef} and the drawdown property of $X$ we obtain\n\\begin{equation*}\nY^n_{t} \\geq K_n\\left(\\overline{X}_t\\right) - K'_n \\left(\\overline{X}_t\\right) \\left(\\overline{X}_t - w\\left(\\overline{X}_t\\right) \\right) = h_n\\left(\\overline{X}_t\\right), \\label{eq:y_n>x}\n\\end{equation*}\nwhere $h_n(x)$ is defined for $x \\geq v_0$ as\n\\begin{equation*}\n\\begin{split}\nh_n(x) :&= K_n(x)-K'_n(x)(x-w(x))=K_n(x)\\left(1-\\frac{K'_n(x)}{K_n(x)}(x-w(x))\\right)\\\\\n&=K_n(x) \\left( 1 - \\frac{x-w(x)}{x-w_n(x)}\\right)=K_n(x)\\left(1-\\frac{1}{1+1\/n}\\right)\\\\\n&=\\frac{1}{1+n}K_n(x),\n\\end{split}\n\\end{equation*}\nwhere we used \\eqref{eq:wn_and_w} and $w_n(x) = x - K_n(x)\/K'_n(x)$.\n\nLet $F_n(v)$ be the inverse of $K_n(v)$, for $v\\geq v_0$ and extended to $[0,\\infty)$ via \\eqref{eq:Fdef}. Explicitly, we have $F_n(v)=F\\left(v_0^{-1\/n}v^{\\frac{1+n}{n}}\\right)$, $v \\geq v_0$ and $F_n(v)=F(v)- \\frac{1}{n}(F(v_0)-F(v))$, $v\\in [0,v_0)$. $F_n$ is continuous and strictly increasing on $[0,\\infty)$ and we take $n$ large enough so that $F_n(0)>0$. \n\nAssume, which we will argue later, that $U\\circ F_n$ satisfies \\eqref{eq:AEU} and hence we may apply Lemma \\ref{lemma:ae_u}. Setting $x^n_0=v_0\/(n+1)$ we deduce that there exists $\\gamma_n\\in \\mathbb{R}$, $\\gamma_n\\neq 0$ such that for all $x\\geq x^n_0$ and all $\\lambda\\geq 1$ we have $U\\circ F_n(\\lambda x)\\leq \\lambda^{\\gamma_n} U\\circ F_n(x)$. We use this with $\\lambda=(1+n)$ and $x=K_n\\left(\\overline{X}_t\\right)\/\\lambda$. Observe that $\\overline{X}_t \\geq v_0$ and that $K_n(v_0) = v_0$ so $x\\geq x^n_0$ as required. Combining all the above, we obtain\n\\begin{equation}\\label{eq:second_ineq}\n\\begin{split}\nU \\circ F_n \\left(Y^n_t \\right) & \\geq U\\circ F_n\\left(h_n\\left(\\overline{X}_t\\right)\\right) = \n U \\circ F_n \\left(\\frac{1}{1+n} K_n \\left( \\overline{X}_t\\right) \\right)\\\\&  \\geq  \\left(\\frac{1}{1+n}\\right)^{\\gamma_n} U \\left(\\overline{X}_t\\right) \\geq  \\left(\\frac{1}{1+n}\\right)^{\\gamma_n} U \\left(X_t\\right).\n\\end{split}\n\\end{equation}\nThe factor of $(1+n)^{-\\gamma_n}$ disappears when we apply $\\frac{1}{t}\\log$ and let $t\\to\\infty$:\n\\begin{equation}\\label{eq:reward_ineq_reverse}\n\\mathcal{R}_{U\\circ F_n}(Y^n)\\geq \\mathcal{R}_U(X).\n\\end{equation}\nTaking supremum over $X\\in \\mathcal{A}^w(v_0)$ we conclude that\n$$\\mathrm{CER}_{U\\circ F_n}(v_0)\\geq \\mathrm{CER}^w_{U}(v_0)$$\nand thus we indeed have a sequence on unconstrained problems with value functions all dominating $\\mathrm{CER}^w_{U}(v_0)$. \n\nAssume, which we will show later, that $U\\circ F$ and $U\\circ F_n$ satisfy Assumption \\ref{ass:U} and now argue the convergence of $\\mathrm{CER}_{U\\circ F_n}(v_0)$ to $\\mathrm{CER}_{U\\circ F}(v_0)$ using Lemma \\ref{lem:cnv_cer} in the Appendix.\nFor $v\\geq v_0$ we have $F(v)\\leq F_n(v)=F\\left(v_0^{-1\/n}v^{\\frac{1+n}{n}}\\right)$\nand for $v\\in [0,v_0)$ we have $c_n F(v)\\leq F_n(v)\\leq F(v)$ where $c_n = 1 + \\frac{1}{n}\\frac{w(v_0)-v_0}{w(v_0)}$ and we take $n>n_0$ chosen such that $c_n>c_0:=c_{n_0} >0$. Fix $\\epsilon\\in (0,\\min\\{1,v_0\\})$ and note that $v_0^{-1\/n}\\leq \\epsilon^{-1\/n}<\\epsilon$. Together the above give us\n\\begin{equation}\\label{eq:FFnbound}\nc_0F(v)\\leq F_n(v)\\leq F(v^{1+1\/n}\/\\epsilon),\\quad v\\geq \\epsilon.\n\\end{equation}\nThanks to \\eqref{eq:AEU}, we can apply Lemma \\ref{lemma:ae_u} to $U$ with $x_0=c_{0}F(\\epsilon)$ to see that there exists a non-zero $\\gamma'\\in \\mathbb{R}$ such that $U(\\frac{1}{c_0} c_0F(v))\\leq c_0^{-\\gamma'}U(c_0F(v))$, for all $v\\geq \\epsilon$. We thus obtain\n\\begin{equation*}\nc_0^{\\gamma'}U\\circ F(v)\\leq U(c_0 F(v))\\leq U\\circ F_n(v)\\leq U\\circ F\\left(\\frac{1}{\\epsilon}v^{1+1\/n}\\right),\\quad v\\geq \\epsilon.\n\\end{equation*}\nTogether with $\\mathrm{CER}_{G}<\\infty$ and Lemma \\ref{lem:2U}, Lemma \\ref{lem:cnv_cer} now yields\n$$ \\mathrm{CER}_{U\\circ F_n}(v_0) \\van \\mathrm{CER}_{U\\circ F}(v_0)<\\infty.$$\n\nIt remains to argue the properties assumed above: that $U\\circ F_n$ satisfies \\eqref{eq:AEU} and that $U\\circ F$ and $U\\circ F_n$ satisfy Assumption \\ref{ass:U}. Observe that for $x > v_0$:\n\\begin{equation}\\label{eq:FAEbound}\nx\\frac{F_n'(x)}{F_n(x)} = \\frac{x}{K'_n(F_n(x))F_n(x)} = \\frac{F_n(x) -w_n(F_n(x))}{F_n(x)} \\leq 1\n\\end{equation}\nand hence\n\\begin{equation*}\n x \\left(U \\circ F_n(x)\\right)'_+ = F_n(x) U'_+(F_n(x)) \\frac{xF_n'(x)}{F_n(x)} \\leq F_n(x) U'_+(F_n(x)). \n \\end{equation*}\nWe conclude that\n\\begin{equation*}\n\\limsup_{x\\to \\infty}\\frac{ x \\left(U \\circ F_n(x)\\right)'_+}{|U\\circ F_n(x)|}\\leq \\limsup_{x\\to \\infty}\\frac{F_n(x) U'_+(F_n(x))}{|U\\circ F_n(x)|} =\n\\limsup_{y\\to \\infty}\\frac{ y U'_+(y)}{|U(y)|},\n\\end{equation*}\nwhere we used the fact that $F$ is a strictly increasing continuous map of $[v_0,\\infty)$ onto itself so any sequence $y_m\\to \\infty$ can be represented as $y_m=F_n(x_m)$ for $x_m=K_n(y_m)\\to \\infty$ as $m\\to \\infty$.\n\nFinally, from \\eqref{eq:w-assumption} note that $(x\/v_0)\\leq K(x)\/v_0\\leq (x\/v_0)^{1\/(1-\\alpha_1)}$, $x\\geq v_0$. Suppose $U>0$ and recall $\\varepsilon$ from Assumption \\ref{ass:U}. Take $\\delta$ small enough so that $\\delta\/(1-\\alpha_1)<\\varepsilon$. Then, using $K(F(x))=x$,\n\\begin{equation*}\n\\begin{split}\n\\frac{U\\circ F(x)}{x^\\delta} &= \\frac{U\\circ F(x)}{F(x)^\\delta}\\left(\\frac{F(x)}{K(F(x))}\\right)^\\delta \\geq  \\frac{U\\circ F(x)}{F(x)^\\delta}v_0^{\\delta\/(1-\\alpha_1)} F(x)^{\\delta (1-1\/(1-\\alpha_1))}\\\\\n& \\geq \\frac{U\\circ F(x)}{F(x)^{\\delta\/(1-\\alpha_1)}}v_0^{\\delta\/(1-\\alpha_1)} \\to \\infty,\\quad \\textrm{as }x\\to \\infty.\n\\end{split}\n\\end{equation*}\nWe conclude that $U\\circ F$ satisfies Assumption \\ref{ass:U}. \nNote that we used here $\\alpha_1\\in (0,1)$. For $U<0$ we have a similar argument which uses $K(x)\\geq x$. Finally the same arguments apply to $U\\circ F_n$ since $w_n$ satisfy \\eqref{eq:w-assumption}.\\hfill$\\square$\n\\end{proof}\nFrom Lemma \\ref{lem:2U} we immediately have\n\\begin{corollary}\\label{coro:initial_wealth}\nUnder the assumptions of Theorem \\ref{thm:main} we have for any, possibly random, bounded and bounded away from zero initial capital $\\nu$ \n\\begin{equation*}\n\\mathrm{CER}^w_{U}(\\nu)=\\mathrm{CER}^w_{U}(1)=\\mathrm{CER}_{U\\circ F_w}(1)=\\mathrm{CER}_{U\\circ F_w}(\\nu)<\\infty\n\\end{equation*}\nand if $V^*\\in  \\mathcal{A}(1)$ achieves $\\mathrm{CER}_{U\\circ F_w,}(1)$ then $\\nu V^*$ achieves $\\mathrm{CER}_{U\\circ F_w}(\\nu)$ and $M^{F_w}(\\nu V^*)\\in  \\mathcal{A}^w(\\nu)$ achieves $\\mathrm{CER}^w_{U}(\\nu)$.\n\\end{corollary}\n\nWe close this section with a series of remarks about the above results, their assumptions and proofs.\n\\begin{remark}\nFrom the proof of Theorem \\ref{thm:main}, and in particular from Lemma \\ref{lem:cnv_cer} in the Appendix, it is clear that when $U<0$ we have in fact $\\mathrm{CER}^w_{U}<\\infty$ if and only if $\\mathrm{CER}_{U\\circ F_w}<\\infty$.\n\\end{remark}\n\n\\begin{remark}\nWe note first that for $U\\geq 0$ concavity implies \\eqref{eq:AEU}: $U(x)\\geq U(x)-U(0)\\geq xU'_+(x)$ so that $\\limsup_{x \\rightarrow \\infty}\\frac{xU'_+(x)}{U(x)} \\leq 1$. Assumption \\ref{ass:U} implies that this limit is also strictly positive. Indeed, we have $\\frac{U(x)}{x^\\varepsilon} \\rightarrow \\infty$, for some $\\varepsilon >0$, which implies that there exists a sequence $x_k\\to \\infty$ such that $\\left(\\frac{U(x)}{x^\\varepsilon}\\right)'_+|_{x=x_k} = \\frac{U(x_k)}{x^{\\varepsilon+1}_k} \\left(\\frac{x_k U'_+(x_k)}{U(x_k)} - \\varepsilon\\right)>0$. Thus, $\\limsup_{x \\rightarrow \\infty}\\frac{xU'_+(x)}{U(x)} \\geq \\varepsilon$. \n\nHowever \\eqref{eq:AEU}, even if we assume the limit is strictly positive, does not imply Assumption \\ref{ass:U}. This can be seen by considering a $U$ which alternates between linear and log-like behaviours. Specifically, consider $U_0$ with $U_0(0)=0$, $U'_{0}(0+)=1$, $U_0'(x)$ being continuous, with $U_0'(x) = U'_0(x_{2k})$ for $x\\in[x_{2k},x_{2k+1}]$ and $U'_0(x)=(x-x_{2k+1}+U'_0(x_{2k+1})^{-1})^{-1}$ for $x\\in [x_{2k+1}, x_{2k+2}]$ where $x_0=0$, $x_1=1$ and\n\\begin{equation}\n\\begin{split}\nx_{2k}&:=\\inf\\left\\{x \\geq x_{2k-1} + 1: U_0(x)\\leq x^{1\/k}\\right\\}, \\quad k\\geq 1,\\\\\nx_{2k+1}&:=\\inf\\left\\{x \\geq x_{2k} + 1: \\frac{xU_0'(x)}{U_0(x)}\\geq 1 - 1\/k\\right\\},\\quad k\\geq 1.\n\\end{split}\n\\end{equation}\nChoice of $x_i$ guarantees that $U_0$ does not dominate any positive power of $x$ asymptotically, but has a strictly positive asymptotic elasticity. More precisely, the asymptotic elasticity is equal to one as it is bounded from above by 1 since $U$ is a positive utility function and $\\lim_{k\\rightarrow \\infty} \\frac{x_{2k+1} U'(x_{2k+1})}{U(x_{2k+1})} = 1$.\n\nFor $U<0$ we need both \\eqref{eq:AEU} and Assumption \\ref{ass:U}. The latter implies that the limit in \\eqref{eq:AEU} is non-zero but it could be infinite, as for $U(x)=-\\mathrm{e}^{-x}$ which satisfies Assumption \\ref{ass:U}. Conversely, $-\\frac{1}{U_0(x)}$ is a utility function which satisfies \\eqref{eq:AEU} but not Assumption \\ref{ass:U}. \n\\end{remark}\n\n\\begin{remark}\\label{remark:U_shift} Observe that $\\mathrm{CER}$ in Problems \\ref{pb:utmax} and \\ref{pb:utmax_DD} are invariant under a multiplication of $U$ by a positive constant. Further, for a positive utility function $U$, they are invariant under a constant shift of $U$ which preserves the sign. More precisely, write $C=\\mathrm{CER}_{U}(v_0)$ and let $\\kappa>0$. For any $\\delta>0$, $V\\in \\mathcal{A}(v_0)$, taking $T$ large enough we have\n$$\\log \\mathbb{E}[U(V_T)+\\kappa]\\leq \\log(\\mathrm{e}^{T(C+\\delta)}+\\kappa)\\leq \\log(2\\mathrm{e}^{T(C+\\delta)}).$$\nThis yields $\\mathcal{R}_{U+\\kappa}(V)\\leq C+\\delta$ and letting $\\delta\\searrow 0$ we have $\\mathrm{CER}_{U+\\kappa}(v_0)=C$.  Finally, both problems are invariant under changes of $U$ in the neighbourhood of zero. This is clear under the drawdown constraint since $U$ is never evaluated on $x\\in (0,w(v_0))$. By Theorem \\ref{thm:main} it is also true for the unconstrained problem. A direct argument for this is given in the proof of Lemma \\ref{lem:2U} in the Appendix.\n\\end{remark}\n\n\\begin{remark}\nBy the opening arguments in the proof of Theorem \\ref{thm:main}, if $V\\in \\mathcal{A}(v_0)$ then $\\mathcal{R}_U(X)\\geq \\mathcal{R}_{U\\circ F}(V)$, where $X=M^F(V)\\in \\mathcal{A}^w(v_0)$. In particular if $V$ is nearly optimal, say $\\mathcal{R}_U(V)\\geq \\mathrm{CER}_{U\\circ F_w}(v_0)-\\epsilon$, then so is $X$: $\\mathcal{R}_U(X)\\geq \\mathrm{CER}_{U}^w(v_0)-\\epsilon$.\n\\end{remark}\n\n\\begin{remark}\nWe stress that throughout, similarly to  \\cite{GZ,CK}, drawdown constraint is imposed on the wealth expressed in units of $N$. In practice this may be adequate if the so-called hurdle rate is present, see Guasoni and Ob\\l\\'oj \\cite{GuasoniObloj:12}. However in many situations one is interested in avoiding drawdowns for the actual wealth process ``in dollars\", $XN$. Suppose for simplicity that all the assets are continuous and let $V$ be a positive wealth process, $\\mathrm{d} V_t=\\sum_{i=1}^d \\pi_t^i \\mathrm{d} S^i_t$. Using our methodology we could consider $X=M^F(VN)\/N$ which would indeed satisfy $XN\\geq w(\\overline {XN})$, but $X$ would not be a wealth process of a self-financing portfolio in Definition \\eqref{def:wealth}.\n\nMore specifically, suppose $N_t =\\exp(\\int_0^t R_u \\mathrm{d} u)$ is the savings account, where $R$ is some positive adapted process. Let $D_t=1\/N_t$.  Then, using \\eqref{eq:AYdef},\n\\begin{eqnarray*}\n\\mathrm{d} \\left( D_t M^F_t(VN)\\right) &=& D_t F'(\\overline{VN}_t) \\mathrm{d} (V_tN_t) + M^F_t(VN) \\mathrm{d} D_t \\\\\n& = & F'(\\overline{VN}_t) \\mathrm{d} V_t +  D_t V_t F'(\\overline{VN}_t) \\mathrm{d} N_t + M^F_t(VN) \\mathrm{d} D_t\\\\\n& = & - R_t \\left(F(\\overline{VN}_t) - F'(\\overline{VN}_t) \\overline{VN}_t\\right) D_t \\mathrm{d} t + F'(\\overline{VN}_t) \\sum_{i=1}^d \\pi^i_t \\mathrm{d} S^i_t.\n\\end{eqnarray*}\nIt follows that $M^F(VN)$ is the dollar value of a wealth process of a consumption and investment strategy, where the rate of instantaneous consumption is $R_t \\left(F(\\overline{VN}_t) - F'(\\overline{VN}_t) \\overline{VN}_t\\right)$. \n\nThe above calculation suggests that drawdown constraints imposed on the undiscounted wealth should be considered in the context of maximisation of utility of consumption. We note however that it is not clear how to build bijection akin to Proposition \\ref{prop:DD} above and what the adequate sets of wealth processes should be. We believe this is a challenging topic for further studies.\n\\end{remark}\n\n\n\\section{Utility of wealth in dollars}\\label{sec:main_dollars}\nWe turn now to the inhomogeneous problem as considered by Grossman and Zhou \\cite{GZ}. Assume that $N$ represents the savings account and therefore $V$ in Definition \\ref{def:wealth} represents discounted wealth process. We seek to maximise the utility of ``wealth in dollars\" $U(VN)$, but the drawdown constraint is imposed on the discounted wealth process $V$. Note that in the case of a linear constraint, $w(x)=\\alpha x$, this is equivalent to saying that the drawdown constraint is imposed on $VN$ but is growing at a hurdle rate equal to the riskless rate\\footnote{Hurdle rate $r$ means that if a new drawdown constraint is set at time $t$ then it grows at the rate $r$ for $u>t$ until a new constraint level is achieved, see Guasoni and Ob\\l\\'oj \\cite{GuasoniObloj:12} for more details.}. In analogy to Problems \\ref{pb:utmax}, \\ref{pb:utmax_DD} we define\n\\begin{equation}\\label{eq:utmax_dollars}\n    \\textnormal{\\textcolonmonetary}\\mathrm{ER}_{U}(v_0):=\\sup_{V\\in \\mathcal{A}(v_0)}\\mathcal{R}_U(VN),\\quad \\textnormal{\\textcolonmonetary}\\mathrm{ER}_U^w(v_0):=\\sup_{X\\in \\mathcal{A}^w(v_0)}\\mathcal{R}_U(XN).\n\\end{equation}\nIn order to be able to relate these problems we essentially need to go back to the homogenous case when $\\mathcal{R}_U(VN)$ is replaced by $\\mathcal{R}_U(V)$ and for this we need to be able to factor the discounting in and out of the reward functional $\\mathcal{R}_U$. This is possible when $U$ is a power utility, $w$ is linear and $N$ is deterministic as in \\cite{CK} and \\cite{GZ}. Here we need to assume this holds asymptotically.\n\\begin{asmp}\\label{ass:UwS0}\nAssume the following three conditions hold\n\\begin{itemize}\n  \\item[(i)] $U$ is either strictly positive or strictly negative on $(0,\\infty)$ and the following limit exists  $\\frac{x U'_+(x)}{U(x)} \\to \\gamma \\in (-\\infty,1)\\setminus \\{0\\}$ as $x\\to \\infty$,\n  \\item[(ii)] the following limit exists $\\frac{w(x)}{x} \\to \\alpha \\in (0,1)$ as $x\\to \\infty$,\n  \\item[(iii)] $N$ is increasing, deterministic and the following exists $r^*:=\\lim_{T\\to\\infty} \\frac{\\log N_T}{T}$.\n\\end{itemize}\n\\end{asmp}\nThe first assumption is a strengthened version of the finite asymptotic elasticity of Kramkov and Schachermayer \\cite{KrSch} which we assumed earlier in \\eqref{eq:AEU}. It follows from Lemma \\ref{lem:Uasel} in the Appendix that it implies Assumption \\ref{ass:U} holds.\nThe second condition above is in fact equivalent to saying that $K_w$ in \\eqref{eq:Kdef} also has such (converging) finite asymptotic elasticity. This is immediate since $xK_w'(x)\/K_w(x)=x\/(x-w(x))$.\nWe denote the CRRA (power) utility with $\\powut{p}(x)=\\frac{x^p}{p}$, $p\\leq 1$. We assume $p\\neq 0$, which is the case of logarithmic utility treated below in Section \\ref{sec:logut}. Finally, we denote $w_\\alpha(x)=\\alpha x$ the linear drawdown function.\n \nWe assumed above that the interest rates are deterministic and that asymptotically $U$ is a power utility and $w$ is linear. Comparing with the setup in Section \\ref{sec:main_num} these are strong assumptions and our main contribution, relative to \\cite{CK}, is that we work in a general max-continuous semimartingale market.\n\\begin{theorem} \\label{thm:main_dollars} Let $U$ be a utility function and $w$ a drawdown function for which Assumption \\ref{ass:UwS0} holds.\nAssume further that $\\textnormal{\\textcolonmonetary}\\mathrm{ER}_{\\powut{\\gamma(1-\\alpha)(1+\\delta)}}(v_0)<\\infty$ for some $\\delta>0$. Then\n\\begin{equation*}\n\\textnormal{\\textcolonmonetary}\\mathrm{ER}^w_{U}(v_0)=\\textnormal{\\textcolonmonetary}\\mathrm{ER}^{w_\\alpha}_{\\powut{\\gamma}}(v_0)=\\textnormal{\\textcolonmonetary}\\mathrm{ER}_{\\powut{\\gamma(1-\\alpha)}}(v_0)+|\\gamma|\\alpha r^*<\\infty\n\\end{equation*}\nand  if $V^*\\in \\mathcal{A}(v_0)$ achieves the maximum in the unconstrained problem then $M^{F_w}(V^*)\\in \\mathcal{A}^w(v_0)$ achieves the maximum in the $w$-drawdown constrained problem, where $F_w$ is as in Theorem \\ref{thm:main}.\n\\end{theorem}\n\\begin{remark}\nTheorem 1.1 of Grossman and Zhou \\cite{GZ} and Theorem 5.1 of Cvitani\\'c and Karatzas \\cite{CK} are consequences of the above statement. Namely, they specialise to $w=w_\\alpha$, $U=\\powut{\\gamma}$ with $\\gamma\\in (0,1)$ and a particular (deterministic, constant coefficients) market setup. Standard techniques allow then to compute explicitly $\\textnormal{\\textcolonmonetary}\\mathrm{ER}_{\\powut{\\gamma(1-\\alpha)}}(v_0)$ and find the optimal wealth process $V^*$, see Section \\ref{sec:example_complete}. Therein we also discuss how various objects in \\cite{CK} and in our paper relate explaining how methods of \\cite{CK} helped us develop intuition behind this paper.\n\\end{remark}\n\\begin{proof}\nNote that, by considering $V\\equiv v_0$ and using $N_t\\geq N_0=1$, we see that $\\textnormal{\\textcolonmonetary}\\mathrm{ER}_U\\geq \\textnormal{\\textcolonmonetary}\\mathrm{ER}^w_U\\geq 0$. Recall also that $\\mathcal{R}_{U_1}(XN)\\leq \\mathcal{R}_{U_2}(XN)$ for two functions $U_1\\leq U_2$ of the same sign.\n\nConsider $X\\in \\mathcal{A}^w(v_0)$ and a small $\\varepsilon>0$. As $N_tX_t\\geq N_tw(v_0)\\geq w(v_0)>0$, we can apply Lemma \\ref{lem:Uasel} in the Appendix to obtain\n\\begin{equation}\\label{eq:proof2eq1}\n\\mathcal{R}_{U}(NX)\\leq \\mathcal{R}_{\\powut{\\gamma(1+\\varepsilon)}}(NX) = \\mathcal{R}_{\\powut{\\gamma(1+\\varepsilon)}}(X)+|\\gamma|(1+\\varepsilon)r^*,\n\\end{equation}\nthe last equality following since $N$ is deterministic.\n\nRecall $K_w$ defined in \\eqref{eq:Kdef}. Note that $xK_w'(x)\/K_w(x)=x\/(x-w(x))$ and hence that Assumption \\ref{ass:UwS0} implies\n$$\\lim_{x\\to \\infty}\\frac{xK_w'(x)}{K_w(x)}=\\lim_{x\\to\\infty}\\frac{x}{x-w(x)}=\\lim_{x\\to\\infty}\\frac{1}{1-w(x)\/x}=\\frac{1}{1-\\alpha}.$$\n$F_w:[v_0,\\infty)\\to [v_0,\\infty)$ is the inverse of $K_w$ extended to $[0,\\infty)$ through \\eqref{eq:Fdef}. It is increasing, concave and we have $\\lim_{x\\to\\infty}xF_w'(x)\/F_w(x)=1-\\alpha$.\nLemma \\ref{lem:Uasel} then implies that $F_w(x)\\leq c_+ \\powut{(1-\\alpha)(1+\\varepsilon)}(x)$ for some $c_+\\geq 1$ and all $x\\geq v_0\/2$. \nIn consequence, for any $Y\\in \\mathcal{A}(v_0)$ with $Y\\geq v_0\/2$ \n\\begin{equation}\\label{eq:proof2eq2}\n\\begin{split}\n\\mathcal{R}_{\\powut{\\gamma(1+\\varepsilon)}\\circ F_w}(Y)& \\leq \\mathcal{R}_{\\powut{\\gamma(1-\\alpha)(1+\\varepsilon)^2}}(Y) \\\\&\n=\\mathcal{R}_{\\powut{\\gamma(1-\\alpha)(1+\\varepsilon)^2}}(NY)-|\\gamma|(1-\\alpha)(1+\\varepsilon)^2r^*,\n\\end{split}\n\\end{equation}\nwhich is finite for $\\varepsilon$ small enough by assumption. Note that by Lemma \\ref{lem:2U} it is sufficient to consider only such $Y$.\n\nBy a similar reasoning and using the assumption of the theorem we conclude that $\\mathrm{CER}_{G}(v_0)<\\infty$ for $G(x)=(\\powut{\\gamma(1+\\varepsilon)}\\circ F_w(x))^{1+\\kappa}$ with $\\kappa=\\varepsilon\\mathbf{1}_{\\gamma>0}$ and for $\\varepsilon$ small enough. Applying Theorem \\ref{thm:main} and combining \\eqref{eq:proof2eq1} with \\eqref{eq:proof2eq2} we obtain\n\\begin{equation*}\n\\begin{split}\n\\textnormal{\\textcolonmonetary}\\mathrm{ER}^w_U(v_0)&\\leq  \\mathrm{CER}^w_{\\powut{\\gamma(1+\\varepsilon)}}(v_0)+|\\gamma|(1+\\varepsilon)r^*\\\\\n&\\leq \\textnormal{\\textcolonmonetary}\\mathrm{ER}_{\\powut{\\gamma(1-\\alpha)(1+\\varepsilon)^2}}(v_0)+|\\gamma| \\alpha r^* + \\varepsilon |\\gamma| r^*(1-(2+\\varepsilon)(1-\\alpha)).\n\\end{split}\n\\end{equation*}\nTaking $\\varepsilon\\to 0$ and invoking Lemma \\ref{lem:cnv_cer} yields ``$\\leq$\" inequalities in the desired equality. The reverse inequalities are obtained in an analogous manner but exploiting the lower bound in Lemma \\ref{lem:Uasel}. \nFinally, the above also shows we can replace $w$ by $w_\\alpha$.\\hfill$\\square$\n\\end{proof}\n\\begin{remark}\nSimilarly to Theorem \\ref{thm:main} and Corollary \\ref{coro:initial_wealth}, $\\textnormal{\\textcolonmonetary}\\mathrm{ER}^w_U(v_0)$ above does not depend on $v_0$ and the optimal strategy for the unconstrained problem scales linearly in the initial wealth.\n\nAlso, similarly to Theorem \\ref{thm:main}, it follows from Lemma \\ref{lem:cnv_cer} that when $\\gamma<0$ the equality $\\textnormal{\\textcolonmonetary}\\mathrm{ER}^w_{U}=\\textnormal{\\textcolonmonetary}\\mathrm{ER}_{\\powut{\\gamma(1-\\alpha)}}+|\\gamma|\\alpha r^*$ holds without assuming that $\\textnormal{\\textcolonmonetary}\\mathrm{ER}_{\\powut{\\gamma(1-\\alpha)(1+\\delta)}}$ is finite.\n\\end{remark}\n\\begin{corollary}\nIn the setup of Theorem \\ref{thm:main_dollars} we have\n$$\\textnormal{\\textcolonmonetary}\\mathrm{ER}_{\\powut{\\gamma(1-\\alpha)}}(v_0)=\\textnormal{\\textcolonmonetary}\\mathrm{ER}_{U\\circ F_w}(v_0).$$\n\\end{corollary}\nThis Corollary follows from the proof of Theorem \\ref{thm:main_dollars}.\nNaturally, similar statements can be made relating in general $\\textnormal{\\textcolonmonetary}\\mathrm{ER}$ for power utility and for $U$ which satisfies $(i)$ in Assumption \\ref{ass:UwS0}. This is not surprising in the light of the results on the so-called turnpike theorems.\nIn this stream of literature authors study the convergence of the value function and the optimal strategy for the Merton problem of maximising utility of terminal wealth as the horizon $T$ tends to infinity.\nIn particular, Hubermann and Ross \\cite{HuRo} argue that, in the case of a complete discrete market, the convergence of optimal strategies is equivalent to the convergence of the relative risk aversion, i.e. $-\\frac{x U''(x)}{U'(x)} \\rightarrow 1-\\gamma$, which is essentially $(i)$ in Assumption \\ref{ass:UwS0}. Huang and Zariphopoulou \\cite{HuZa} study the problem for a continuous time complete market model with deterministic coefficients, as in Section \\ref{sec:example_complete}. They find sufficient conditions on $U$ for the optimal strategy to converge to the optimal strategy coming from the problem with a power utility. The analysis in a recent paper of Guasoni and Robertson \\cite{GuasoniRobertson:11} includes also incomplete markets. In comparison, our results apply in a much more general context but are also much weaker. Problem \\ref{pb:utmax} looks at maximising the long-term asymptotic growth rate of the expected utility and the above Corollary shows that the resulting value function is the same when \ntwo utility functions have the same asymptotic behaviour. It does not say anything precise about finite horizon utility maximisation and its convergence.\n\n\\section{Logarithmic Utility}\\label{sec:logut}\n\nSo far we have only considered utility functions with constant sign and which dominated a power utility, as in Assumption \\ref{ass:U}. In this section, we consider utility functions akin to $U(x)=\\log x$. The results are very close in spirit to the ones in the previous two sections, but in fact require less technicalities in the proofs. We go back to the general setup of Section \\ref{sec:market_model} and introduce a modified version of maximisation criterion in \\eqref{eq:opt_prob}.\n\\begin{prob}\\label{pb:utmax_log}\nGiven $v_0>0$, a drawdown function $w$ and function  $U$ compute\n\\begin{equation}\\label{eq:opt_prob_log}\n\\begin{split}\n\\widetilde{\\mathrm{CER}}_{U}(v_0) &:= \\sup_{V\\in \\mathcal{A}(v_0)}\\widetilde{\\mathcal{R}}_{U}(V),\\\\\n\\widetilde{\\mathrm{CER}}^w_{U}(v_0) &:= \\sup_{V\\in \\mathcal{A}^w(v_0)}\\widetilde{\\mathcal{R}}_{U}(V),\\\\\n&\\quad \\textrm{where}\\quad \\widetilde{\\mathcal{R}}_{U}(V):=\\limsup_{T\\to \\infty} \\frac{1}{T} \\mathbb{E}\\left[U\\left(V_T\\right)\\right],\n\\end{split}\n\\end{equation}\nalong with the optimal wealth processes which achieve the supremum.\n\\end{prob}\n\\begin{theorem}\\label{thm:main_log} Let $v_0>0$, $w$ be a drawdown function and $U$ a utility function satisfying\n\\begin{equation*}\n\\limsup_{x\\to\\infty} xU'_+(x)<\\infty \\quad \\textrm{and} \\quad \\liminf_{x\\to \\infty} \\frac{U(x)}{\\log(x)} > 0.\n\\end{equation*}\nLet $F_w$ be as in Theorem \\ref{thm:main} then\n\\begin{equation*}\n\\widetilde{\\mathrm{CER}}^w_{U}(v_0)=\\widetilde{\\mathrm{CER}}_{U\\circ F_w}(v_0)\n\\end{equation*}\nand if $V^*$ achieves the maximum in the unconstrained problem then $M^{F_w}(V^*)$ achieves the maximum in the $w$-drawdown constrained problem.\n\\end{theorem}\n\\begin{remark}\nThe equality between value functions in particular states that they are either both finite or both infinite.\n\\end{remark}\n\n\\begin{proof}\nWe write $K=K_w$ and $F=F_w$. The first part of the proof is identical to the first part of the proof of Theorem \\ref{thm:main} and yields\n\\begin{equation*}\n \\widetilde{\\mathrm{CER}}^w_{U}(v_0) \\geq \\widetilde{\\mathrm{CER}}_{U \\circ F}(v_0).\n\\end{equation*}\nAlso, if we had the desired equality then $M^{F}(V^*)$ is optimal for constrained problem when $V^*$ is optimal for the unconstrained one.\n\nIt follows from the assumptions that for any $y>0$ there exists $\\gamma$ such that\n$$xU'_{+}(x)<\\gamma<\\infty,\\quad x \\geq y.$$\nApplying Lemma \\ref{lemma:ae_u} to $e^{U(x)}$,\nwe deduce that \n\\begin{equation}\nU(\\lambda x) \\leq U(x) + \\gamma \\log \\lambda, \\text{  } \\lambda > 1, \\ x \\geq y. \\label{eq:U_log} \n\\end{equation}\nDefine $w_n, K_n, F_n$ as in the proof of Theorem \\ref{thm:main} and recall the computation in \\eqref{eq:FAEbound}. It follows that \n$x{U\\circ F_n}'_+(x)<\\gamma$ for $x\\geq v_0$. The same reasoning holds with $F$ in place of $F_n$. We deduce that \\eqref{eq:U_log} holds with $U\\circ F$ in place of $U$ and $\\gamma'$ instead of $\\gamma$ and likewise for $U\\circ F_n$ and $\\gamma_n$.\n\nLet $X\\in \\mathcal{A}^w(v_0)$, $Y^n=M^{K_n}(X)$ and recall from the proof of Theorem \\ref{thm:main} that\n\\begin{equation*}\nY^n_{t} \\geq \\frac{1}{1+n}K_n \\left(\\overline{X}_t\\right). \\label{eq:y_n>x_log}\n\\end{equation*}\nUsing \\eqref{eq:U_log} for $U\\circ F_n$ and $y=v_0\/(n+1)$, $\\lambda=n+1$, noting that $K_n(\\overline{X}_t) \\geq v_0$, we obtain\n\\begin{equation*}\n\\begin{split}\nU \\left(\\overline{X}_t \\right)&= U\\circ F_n\\circ K_n \\left(\\overline{X}_t \\right) \\leq U \\circ F_n\\left(\\frac{1}{1+n} K_n(\\overline{X}_t )\\right) +\\gamma_n\\log(n+1)\\\\ &\\leq U \\circ F_n \\left( Y^n_{t} \\right) +\\gamma_n\\log(n+1).\n\\end{split}\n\\end{equation*}\nTaking expectations, dividing by $t$ and passing to the limit $t\\to\\infty$ yields:\n\\begin{equation*\n\\widetilde{\\mathcal{R}}_U(X)\\leq \\widetilde{\\mathcal{R}}_U(\\overline{X}) \\leq \\widetilde{\\mathcal{R}}_{U\\circ F_n}(Y^n).\n\\end{equation*}\nTaking supremum over $X\\in \\mathcal{A}^w(v_0)$ we conclude that $$\\widetilde{\\mathrm{CER}}^w_{U}(v_0)\\leq \\widetilde{\\mathrm{CER}}_{U\\circ F_n}(v_0).$$\n\nIt remains to establish the convergence in $n$ on the LHS. An analogous argument to Lemma \\ref{lem:2U} shows that \nto compute $\\widetilde{\\mathrm{CER}}_{U\\circ F_n}(v_0)$ it suffices to consider $V\\in \\mathcal{A}(v_0)$ such that $V\\geq v_0\/2$. The bound in \\eqref{eq:FFnbound} gives\n$$U(c_0F(v))\\leq U\\circ F_n(v)\\leq U\\circ F\\left(\\frac{1}{\\epsilon} v^{1+1\/n}\\right),\\quad v\\geq \\epsilon $$\nfor arbitrary $\\epsilon\\in (0,1)$ and some $0<c_0<1$. Taking $\\epsilon<v_0\/2$ and using \\eqref{eq:U_log} for $U$ and $U\\circ F$ with $y=\\min\\{\\epsilon,c_0F(\\epsilon)\\}$ we obtain\n\\begin{equation}\\label{eq:UFlogbound}\nU\\circ F(v)+\\gamma\\log c_0\\leq U\\circ F_n(v)\\leq U\\circ F(v)+\\gamma'\\log(v^{1\/n}\/\\epsilon),\\quad v\\geq \\epsilon.\n\\end{equation}\nFinally, consider $\\log F(x)\/\\log x$ for large $x$ and let $z=F(x)$. Then, using \\eqref{eq:w-assumption},\n$$\\frac{\\log F(x)}{\\log x}=\\frac{\\log z}{\\log K(z)}=\\frac{\\log z}{\\log v_0+\\int_{v_0}^z \\frac{du}{u-w(u)}}\\geq \\frac{\\log z}{\\log v_0+\\frac{1}{1-\\alpha_1}\\log z\/v_0},$$\nwhich can be made arbitrary close to $1-\\alpha_1>0$ by considering $z$ large enough. Using the assumption on $U$ we conclude that\n$$\\liminf_{x\\to \\infty}\\frac{U\\circ F(x)}{\\log x}=\\liminf_{x\\to \\infty}\\frac{U\\circ F(x)}{\\log F(x)}\\frac{\\log F(x)}{\\log x}>0.$$\nIt follows that for some positive constants $c,c_1$ we have $cU\\circ F(x)+c_1\\geq \\log(x)$ for all $x\\geq \\epsilon$. \nCombined with \\eqref{eq:UFlogbound}, this shows that \n$$\\widetilde{\\mathrm{CER}}_{U \\circ F}(v_0)\\leq \\widetilde{\\mathrm{CER}}_{U \\circ F_n}(v_0)\\leq (1+c\\gamma'\/n)\\widetilde{\\mathrm{CER}}_{U \\circ F}(v_0).$$\nTaking $n\\to \\infty$ establishes the desired convergence.\\hfill$\\square$\n\\end{proof}\nWe close this section with a result similar to Theorem \\ref{thm:main_dollars}. The definitions of $\\widetilde{\\textnormal{\\textcolonmonetary}\\mathrm{ER}}_U$, $\\widetilde{\\textnormal{\\textcolonmonetary}\\mathrm{ER}}_U^w$ should be clear and the proof follows closely the arguments in Section \\ref{sec:main_dollars} and we omit it for the sake of brevity.\n\\begin{theorem} \\label{thm:log_dollars} Let $U$ be a utility function with $xU'_+(x) \\rightarrow \\gamma \\in (0,\\infty)$ as $x \\rightarrow \\infty$, $v_0>0$, $w$ a drawdown function, $F_w$ as in Theorem \\ref{thm:main} and assume $(ii)$ and $(iii)$ in Assumption \\ref{ass:UwS0} hold.\n Then\n\\begin{equation*}\n\\widetilde{\\textnormal{\\textcolonmonetary}\\mathrm{ER}}^w_{U}(v_0)= \\gamma(1-\\alpha) \\widetilde{\\textnormal{\\textcolonmonetary}\\mathrm{ER}}_{\\log}(v_0) + \\gamma \\alpha r^*\n\\end{equation*}\nand $M^{F_w}(V^*)\\in \\mathcal{A}^w(v_0)$ achieves the maximum in the drawdown constrained problem if $V^*\\in\\mathcal{A}(v_0)$ achieves the maximum in the unconstrained problem.\n\\end{theorem}\n\n\n\n\\section{Examples}\\label{sec:examples}\nWe discuss now some examples. Our aim is twofold. First, we want to give an example of a rather general setup in which sufficient conditions can be found which guarantee finiteness of $\\mathrm{CER}$ for the unconstrained problem, as assumed in Theorem \\ref{thm:main}. Second, we want to discuss specific examples when the unconstrained, and hence also the drawdown constrained, portfolio optimisation problem is solved explicitly. In particular we relate our results and methods to the ones in \\cite{CK}.\n\\subsection{Market with price deflators}\nWe start by assuming existence of a price deflator (or a state price density) process. In the setup of Section \\ref{sec:market_model} we further assume that all $S^i_t$ are continuous and that there exists a $\\mathbb{P}$-local martingale $(Z_t)$, $Z_t>0$ for all $t\\geq 0$, such that $(Z_tS_t^i)$ are $\\mathbb{P}$-local martingales, $i=1,\\ldots, d$. Note that we do not necessarily assume that $(Z_t)$ is a true martingale and hence that an equivalent martingale measure exists. Our setup is in fact analogous to the most general setup in which stochastic portfolio optimisation is considered, see Fernholz and Karatzas \\cite{FerKar}.\nNote that if $(V_t)\\in \\mathcal{A}(v_0)$ then\n$$\\mathrm{d} (Z_t V_t)= V_t\\mathrm{d} Z_t + Z_t \\pi_t \\mathrm{d} S_t + \\pi_t \\mathrm{d} \\langle Z, S\\rangle_t = (V-\\pi_t S_t)\\mathrm{d} Z_t + \\pi_t \\mathrm{d} (Z_t S_t),$$\nso that $(Z_t V_t)$ is a positive $\\mathbb{P}$-local martingale and hence a supermartingale.\nKaratzas and Kardaras \\cite{KarKar} show that the existence of $(Z_t)$ is equivalent to the NUPBR condition (No Unbounded Profit With Bounded Risk). This condition is weaker than the usual NFLVR condition from Delbaen and Schachermayer \\cite{DelSch} and allows for some (very mild) arbitrage opportunities, see examples constructed in \\cite{KarKar}. Recall that $\\powut{p}(x)=\\frac{1}{p}x^p$.\n\\begin{lemma}\\label{lem:suff_finite_cer}\nThe following implications hold for any $p<1$, $p\\neq 0$, and $v_0>0$\n\\begin{equation*}\n\\begin{split}\n\\mathcal{R}_{\\powut{-p\/(1-p)}}(Z)<\\infty \\quad &\\Longrightarrow \\quad \\mathrm{CER}_{\\powut{p}}(v_0)<\\infty,\\\\\n\\mathcal{R}_{\\powut{-p\/(1-p)}}(Z\/N)<\\infty \\quad &\\Longrightarrow \\quad \\textnormal{\\textcolonmonetary}\\mathrm{ER}_{\\powut{p}}(v_0)<\\infty.\n\\end{split}\n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\nLet $(V_t)\\in \\mathcal{A}(v_0)$ so that $(Z_t V_t)$ is a $\\mathbb{P}$-local martingale, as above. For $p<0$ we have\n\\begin{equation*}\n\\begin{split}\n\\mathbb{E} [V_T^p]&=\\mathbb{E}[(Z_T)^{-p}(Z_TV_T)^p]\\\\\n& \\geq \\left(\\mathbb{E}[(Z_T)^{-\\frac{p}{1-p}}]\\right)^{(1-p)}\\left(\\mathbb{E}[Z_T V_T]\\right)^{p}\\geq v_0^p \\left(\\mathbb{E}[(Z_T)^{-\\frac{p}{1-p}}]\\right)^{(1-p)},\n\\end{split}\n\\end{equation*}\nwhere we used H\\\"older's inequality, whose direction is reversed for $p<0$, and the fact that a non-negative local martingale is a supermartingale. The inequalities  above are reversed when we divide both sides by $\\frac{1}{p}<0$ and the claim follows. The case $p\\in (0,1)$ is even more straightforward --- it is enough to use H\\\"older's inequality which directly gives the desired inequality (reverse from the one displayed above). The case with numeraire is entirely analogous.\\hfill $\\square$\n\\end{proof}\nThe lemma above gives an example of sufficient conditions to apply Theorems \\ref{thm:main} and \\ref{thm:main_dollars} since they both require that $\\mathrm{CER}_{G}$ or $\\textnormal{\\textcolonmonetary}\\mathrm{ER}_{G}$ is finite. For the latter $G$ is a power utility function and Lemma \\ref{lem:suff_finite_cer} applies directly. For the former we would need to bound $G$ by a power utility.\n\nNaturally, in the current very general setup there might be little hope to compute $\\mathrm{CER}_{U}$ or find the optimal wealth process. However, one might expect this to be the simplest portfolio optimisation problem to solve. The strength of our results is to show that solving the seemingly much more complex problem with drawdown constraint on wealth paths is in fact equally simple (or hard).\n\nKaratzas and Kardaras \\cite[Theorem 4.12]{KarKar} also show that the existence of $(Z_t)$ is equivalent to the existence of a benchmark numeraire $\\tilde{N}$ such that $V\/\\tilde N$ is a supermartingale for any $V\\in \\mathcal{A}(v_0)$, see also Christensen and Larsen \\cite{ChristensenLarsen:07}. This readily implies that  $\\widetilde{\\mathrm{CER}}_{\\log}(v_0)=\\widetilde{\\mathcal{R}}_{\\log}(\\tilde N)$ and $\\widetilde{\\textnormal{\\textcolonmonetary}\\mathrm{ER}}_{\\log}(v_0)=\\widetilde{\\mathcal{R}}_{\\log}(\\tilde N N)$.\nIndeed, considering $V\\in \\mathcal{A}(v_0)$ and applying Jensen's inequality gives\n\\begin{equation*}\n\\begin{split}\n \\limsup_{T \\rightarrow \\infty} \\frac{1}{T} \\mathbb{E} \\log V_T &\\leq \\limsup_{T \\rightarrow \\infty} \\frac{1}{T} \\mathbb{E} \\log  \\tilde N_T+ \\limsup_{T \\rightarrow \\infty} \\frac{1}{T} \\mathbb{E} \\log \\frac{V_T}{\\tilde N_T}\\\\& \\leq\n \\limsup_{T \\rightarrow \\infty} \\frac{1}{T} \\mathbb{E} \\log  \\tilde N_T.\n\\end{split} \\end{equation*}\n This observation essentially goes back to Bansal and Lehmann \\cite{BansalLehmann:97}.\nIn a no-arbitrage complete market model, taking $N$ to be the savings account, $(Z_t)$ is the density $\\frac{\\mathrm{d} \\mathbb{Q}}{\\mathrm{d} \\mathbb{P}}$ where $\\mathbb{Q}$ is the equivalent martingale (risk-neutral) measure. Completeness means that $(Z_t)^{-1}$ is an admissible wealth process and thus the benchmark numeraire. In particular, in the setting of Theorem \\ref{thm:log_dollars}, we then have\n\\begin{equation*}\n \\widetilde{\\textnormal{\\textcolonmonetary}\\mathrm{ER}}^w_U (v_0) = \\gamma r^*-\\gamma (1-\\alpha) \\limsup_{T \\rightarrow \\infty} \\frac{1}{T} \\mathbb{E} \\log Z_T .\n\\end{equation*}\nIt may be natural to start modelling by simply requiring that the benchmark numeraire $\\tilde N$ exists. This is pursued in the so-called \\emph{benchmark approach}, see Platen and Heath \\cite{PlatenHeath:06}. It is clear that Lemma \\ref{lem:suff_finite_cer} remains true in this approach when we replace $DZ$ by $1\/\\tilde N$.\n\n\\subsection{Complete market model with deterministic coefficients}\\label{sec:example_complete}\nWe consider now the classical complete financial market model with deterministic coefficients. Let $W_t=(W^1_t,\\ldots,W^d_t)'$ be a standard $d$-dimensional Brownian motion and $(\\mathcal{F}_t)$ the augmentation of its natural filtration. Here $'$ denotes vector transpose. $N_t=\\exp(\\int_0^t r_u\\mathrm{d} u)$ is deterministic and $\\frac{1}{T}\\int_0^T r_u \\mathrm{d} u \\to r^*\\geq 0$ as $T\\to \\infty$. Each asset follows dynamics given by\n$$\\frac{\\mathrm{d} \\tilde S^i_t}{\\tilde S^i_t}= \\mu^i_t \\mathrm{d} t + \\sum_{j=1}^d \\sigma^{ij}_t\\mathrm{d} W^j_t,\\quad \\tilde S^i_0=s^i_0>0$$\nwhere $\\mu^i_t$ and $\\sigma^{ij}_t$ are bounded deterministic functions and $\\sigma_t$ is invertible. Recall Definition \\ref{def:wealth} of wealth process and let $\\tilde \\pi_t^i:= \\pi^i_t S^i_t\/V_t$ be the proportion of wealth invested in the $i^{\\textrm{th}}$ asset so that $\\mathrm{d} V_t= \\sum_{i=1}^d \\tilde\\pi^i_t V_t \\frac{\\mathrm{d} S^i_t}{S_t^i}$.\nThe market price of risk is given as $\\theta_t:= \\sigma^{-1}(\\mu_t-r_t\\mathbb{I})$, where $\\mathbb{I}$ is a $d$-dimensional vector with all entries equal to one. We assume $\\theta_t$ is also bounded and that\n$$||\\theta^*||^2:= \\lim_{T\\to\\infty}\\frac{1}{T} \\int_0^T ||\\theta_u||^2\\mathrm{d} u\\quad \\textrm{ is well defined and finite.}$$\nThe state price density\n$$Z_t:= \\exp\\left\\{-\\int_0^t \\theta_u' \\mathrm{d} W_u - \\frac{1}{2}\\int_0^t ||\\theta_u||^2 \\mathrm{d} u\\right\\}$$\nis a $\\mathbb{P}$--martingale which defines for every $T\\in (0,\\infty)$ a unique risk neutral measure up to time $T$ via $\\frac{\\mathrm{d} \\mathbb{Q}}{\\mathrm{d} \\mathbb{P}}|_{\\mathcal{F}_T}=Z_T$. To solve $\\mathrm{CER}_{\\powut{p}}$ one first considers the problem of maximising the expected utility of wealth at a given horizon $T$. The solution is obtained using, by now standard, convex duality arguments, see Karatzas, Lehoczky and Shreve \\cite{KaratzasLehochkyShreve:87} or Karatzas and Shreve \\cite[Sec.~3.5--3.8]{KaSh}. The optimal wealth process $V^*$ is described explicitly via\n\\begin{equation}\n\\tilde\\pi^*_t=\\frac{1}{1-p}\\theta_t'\\sigma^{-1}_t \\label{eq:pi_compl}\n\\end{equation}\nand in particular it is independent of the time horizon $T$. We conclude that it is also optimal for the long-term asymptotic growth rate optimisation. Taking limit of the value functions for the finite horizon problem we obtain\n$$\\textnormal{\\textcolonmonetary}\\mathrm{ER}_{\\powut{p}}(v_0)=\\mathcal{R}_{\\powut{p}}(NV^*)=|p|r^*+ \\frac{|p|}{2(1-p)}||\\theta^*||^2.$$\nNote that the difference of a factor $|p|$ when compared to \\cite{GZ,CK} is immediate since they consider $ \\frac{1}{|p|} \\mathcal{R}_{\\powut{p}}(NV)$ instead of $\\mathcal{R}_{\\powut{p}}(NV)$.\n\nApplying Theorem \\ref{thm:main_dollars} for a utility function $U$ and a drawdown function $w$, which satisfy Assumption \\ref{ass:UwS0}, we obtain\n$$\\textnormal{\\textcolonmonetary}\\mathrm{ER}_{U}^w(v_0)=\\textnormal{\\textcolonmonetary}\\mathrm{ER}_{\\powut{\\gamma(1-\\alpha)}}(v_0)+|\\gamma|\\alpha r^*=\n|\\gamma|\\left(r^*+\\frac{(1-\\alpha)}{2(1-\\gamma(1-\\alpha))}||\\theta^*||^2\\right)$$\nwhich is achieved by the optimal wealth process $X=M^{F_w}(V^*)$. Using \\eqref{eq:SDEwDD} we see that\n$$\\mathrm{d} X_t= \\left(X_t - w(\\overline{X}_t)\\right)\\sum_{i=1}^d \\left(\\frac{1}{1-\\gamma(1-\\alpha)}\\theta_t'\\sigma^{-1}_t\\right)^i \\frac{\\mathrm{d} S^i_t}{S^i_t}.$$\nIn particular, we recover Theorem 5.1 in \\cite{CK} by taking $U=\\powut{\\gamma}$, $\\gamma\\in (0,1)$ and $w(x)=\\alpha x$. It is insightful to understand better how the objects in \\cite{CK} relate to the tools of our paper. In fact the Auxiliary problem introduced and solved in \\cite{CK} is nothing else but $\\textnormal{\\textcolonmonetary}\\mathrm{ER}_{U\\circ F}(v_0)=\\textnormal{\\textcolonmonetary}\\mathrm{ER}_{\\powut{\\gamma(1-\\alpha)}}(v_0)$.\nIndeed, the process $N^\\pi_\\alpha$ defined in $(4.1)$ therein is simply $N M^K(X) $ and $\\hat{\\pi}_t = \\frac{1}{1 - \\gamma(1-\\alpha)} \\theta'_t \\sigma^{-1}_t$.\n\n\\subsection{Incomplete market example}\\label{sec:example_incomplete}\nIn a recent paper Guasoni and Robertson \\cite{GuasoniRobertson:11} solve the unconstrained portfolio optimisation problem for an investor with a power utility in a rather general diffusion model. Our results allow to solve $w$--drawdown constrained problem in their setting. The solution in \\cite{GuasoniRobertson:11} is involved and we do not cite the details here for the sake of brevity. Instead we propose to study an application of Theorem \\ref{thm:main_dollars} in an incomplete market example adapted from the risk-sensitive control approach in Fleming and Sheu \\cite{FleSheu}.\n\nConsider a market with constant interest rate $r$ and one risky asset $\\tilde S(t)$ evolving according to\n\\begin{eqnarray*}\n \\frac{\\mathrm{d} \\tilde S(t)}{\\tilde S(t)} &=& (\\mu_1 + \\mu_2 x(t))\\mathrm{d} t + \\sigma \\mathrm{d} W^1_t + \\rho \\mathrm{d} W^2_t, \\\\\n \\mathrm{d} x(t) &=& b x(t) \\mathrm{d} t + \\mathrm{d} W^1_t,\n\\end{eqnarray*}\nwhere $W^1,W^2$ are two independent Brownian motions and $x(t)$ has an interpretation of an economical factor.\n\nIn Theorem 3.1 in \\cite{FleSheu} the authors provide a link between Problem \\ref{pb:utmax} with a power utility function $\\powut{\\gamma}$ and a viscosity solution of the dynamic programming equation. In Theorem 4.1 the optimal investment policy is found. We refer the reader to \\cite{FleSheu} for further details of the method. In the last section of their paper Fleming and Sheu consider Vasicek interest rate model with a single stock and give an explicit solution to the utility maximisation problem. Our model above is slightly different but we are still able to use their solution.\n\nThe difference with Fleming and Sheu \\cite{FleSheu} example is that the interest rate is given by $r(t) = r$ in our work and by $r(t) = \\lambda x(t) - \\frac{b_1}{b_2}$ in theirs, which  requires us to change some coefficients in final formulae in \\cite{FleSheu}.\nMore precisely, assume $\\gamma < 0$ and $\\mu^2_2 \\geq \\sigma^2 (K^{(\\gamma)})^2$ where $K^{(\\gamma)}$ is defined below. Then the value function is equal to\n\\begin{equation*}\n\\textnormal{\\textcolonmonetary}\\mathrm{ER}_{\\powut{\\gamma}}(v_0) = \\frac{1}{2} K^{(\\gamma)} + \\frac{1}{2}|\\eta|^2 + 1\/2 \\frac{\\gamma}{1-\\gamma}\\frac{(\\mu_1-r + \\sigma \\eta)^2}{\\sigma^2 + \\rho^2} + |\\gamma| r,\n\\end{equation*}\nwhere\n\\begin{equation*}\n\\eta = - \\frac{\\gamma}{1-\\gamma} \\frac{\\mu_2 + K^{(\\gamma)} \\sigma (\\mu_1-r)}{(D^{(\\gamma)} + K^{(\\gamma)} E^{(\\gamma)})(\\sigma^2 + \\rho^2)}\n\\end{equation*}\nand\n\\begin{eqnarray*}\n E^{(\\gamma)} &=& 1+ \\frac{\\gamma}{1-\\gamma} \\frac{\\sigma^2}{\\sigma^2 + \\rho^2},\\\\\n K^{(\\gamma)} &=& - \\frac{b + \\frac{\\gamma}{1-\\gamma} \\frac{1}{\\sigma^2 + \\rho^2} \\mu_2 \\sigma}{1 + \\frac{\\gamma}{1-\\gamma}\\frac{\\sigma^2}{\\sigma^2 + \\rho^2} }\n- \\frac{1}{\\sqrt{1 + \\frac{\\gamma}{1-\\gamma}\\frac{\\sigma^2}{\\sigma^2 + \\rho^2}}} \\\\\n&& \\cdot \\left( -\\frac{\\gamma}{1-\\gamma} \\frac{\\mu^2_2}{\\sigma^2 + \\rho^2} + \\frac{(b + \\frac{\\gamma}{1-\\gamma} \\frac{1}{\\sigma^2+ \\rho^2})^2}{1+ \\frac{\\gamma}{1-\\gamma}\\frac{\\sigma^2}{\\sigma^2+\\rho^2}}\\right)^{1\/2},\\\\\n D^{(\\gamma)} &=& - \\sqrt{1+ \\frac{\\gamma}{1-\\gamma} \\frac{\\sigma^2}{\\sigma^2 + \\rho^2}} \\left( -\\frac{\\gamma}{1-\\gamma} \\frac{\\mu^2_2}{\\sigma^2+\\rho^2} + \\frac{(b+ \\frac{\\gamma}{1-\\gamma}\\frac{\\mu_2 \\sigma}{\\sigma^2+\\rho^2})^2}{1+\\frac{\\gamma}{1-\\gamma}\\frac{\\sigma^2}{\\sigma^2+\\rho^2}}\\right)^{1\/2}.\n\\end{eqnarray*}\nAnd the optimal investment policy $\\tilde \\pi_t$, which is fraction of wealth invested in risky asset at time $t$, is given by\n\\begin{equation*}\n \\tilde \\pi_t = D^{(\\gamma)} x(t) + a^{(\\gamma)},\n\\end{equation*}\nfor some constant $a^{(\\gamma)}$.\n\nIn the setting of Theorem \\ref{thm:main_dollars} we obtain that\n\\begin{equation*}\n \\textnormal{\\textcolonmonetary}\\mathrm{ER}^w_U (v_0) = \\frac{1}{2} K^{(\\gamma (1-\\alpha))} + \\frac{1}{2}|\\eta|^2 + 1\/2 \\frac{\\gamma (1-\\alpha)}{1-\\gamma (1-\\alpha)}\\frac{(\\mu_1 -r + \\sigma \\eta)^2}{\\sigma^2 + \\rho^2} + |\\gamma| (1-\\alpha)r,\n\\end{equation*}\nwhere $\\gamma<0$ and $\\alpha\\in (0,1)$ are defined in Assumption \\ref{ass:UwS0}. Note that we could also consider $\\gamma\\in (0,1)$ under the additional parameter restriction which makes appropriate $K^{(\\cdot)}$ well defined.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIf the electroweak scale is stabilized by supersymmetry (SUSY), then the higgsino masses should be around the Higgs mass, and top squarks (stops) should not be much heavier. By now, many SUSY models have been addressed either directly by ATLAS or CMS, or via reinterpretations of their searches by theorists. These studies have excluded a large range of scenarios, including many conservative ones in which stops are the only abundantly produced superpartners. However, some scenarios were identified where existing LHC searches were insensitive. Most relevant to the present note is our earlier work~\\cite{our-stops}, which classified the signatures of the various possible decays of stops in $R$-parity violating (RPV) models~\\cite{RPV-review}, and analyzed them in the context of LHC searches. The multitude of possible RPV couplings (including their flavor structure) lead to a diverse set of scenarios, many of which were found to elude the LHC searches available at the time of~\\cite{our-stops}. In this note, we will focus on the developments in LHC searches that have occurred since then and point out several additional promising search strategies that ATLAS and CMS could implement.\n\nThe RPV scenarios we will examine here are potentially challenging for LHC searches both because the missing energy is low (arising only from top or tau decays, if it is present at all), and because in some cases the only sources of electrons and muons are taus, or there are no leptons at all. On the other hand, there are also opportunities, that are not fully being used, to probe these scenarios. First, the jet multiplicity is typically very high, due to both cascade decays and RPV couplings involving quark superfields. Second, multiple $b$-jets are typically present, due to both third-generation squarks and specific RPV couplings. Finally, in most cases, some electrons and muons \\emph{are} available, even if only from top or tau decays.  We will examine how sensitive existing searches are to such scenarios and what extensions to the search strategies these scenarios motivate.\n\n\\section{Benchmark models}\n\nThe examples we will analyze involve the lightest stop (${\\tilde t}$) being pair-produced before decaying to a higgsino (${\\tilde H}$), namely\n\\begin{equation}\npp \\to {\\tilde t}\\st^\\ast \\,,\\quad {\\tilde t} \\to b {\\tilde H}^+ \\,.\n\\end{equation}\nHere we have assumed $m_{\\tilde t} - m_{\\tilde H} < m_t$ so that the decays ${\\tilde t} \\to t{\\tilde H}^0_{1,2}$ are absent. The chargino ${\\tilde H}^+$ can decay either through the lightest neutralino ${\\tilde H}^0_1$\n\\begin{equation}\n{\\tilde H}^+ \\to W^{+\\ast} {\\tilde H}^0_1 \\,,\\quad W^{+\\ast} \\to \\mbox{soft particles} \\,,\\quad {\\tilde H}^0_1 \\to \\mbox{RPV}\n\\label{eq:transition}\n\\end{equation}\n(where we assume $m_{{\\tilde H}^+} - m_{{\\tilde H}^0_1} \\lesssim$~few~GeV, so that the decay products of the off-shell $W^+$ are essentially unobservable) or directly\n\\begin{equation}\n{\\tilde H}^+ \\to \\mbox{RPV} \\,.\n\\label{eq:no-transition}\n\\end{equation}\nThe RPV decays proceed through a sfermion (slepton or squark) $\\tilde f$ and a certain $UDD$ or $LQD$ coupling.\\footnote{$LLE$ couplings are less interesting to us because they are easily excluded by multilepton searches~\\cite{our-stops}.} The process is illustrated schematically in Fig.~\\ref{fig:diagram}. Table~\\ref{tab:scenarios} describes the sfermion and coupling choices, the assumption of whether (\\ref{eq:transition}) or (\\ref{eq:no-transition}) occurs, and the resulting stop decay final states for each scenario we will discuss.  Note that in the case of (\\ref{eq:transition}), leptons from stop and antistop decays can have the same sign. In all cases, we will take\n\\begin{equation}\nm_{\\tilde H} = m_{\\tilde t} - 100\\mbox{ GeV} \\,,\\quad\nm_{\\tilde f} \\gg m_{\\tilde H}.\n\\end{equation}\nWe will also assume the mass splitting between ${\\tilde H}^+$ and ${\\tilde H}^0_1$, the size of the RPV coupling, and the mass of $\\tilde f$ to be such that all the decays are prompt.\n\n\\begin{figure}[t]\n\\begin{center}\n\\begin{picture}(0,0)(0,0)\n\\large\n\\put(-5,50){${\\tilde t}$}\n\\put(150,100){$b$}\n\\put(150,65){$f_2$}\n\\put(150,35){$f_3$}\n\\put(150,0){$f_1$}\n\\put(95,50){$\\tilde f$}\n\\put(50,25){${\\tilde H}$}\n\\normalsize\n\\end{picture}\n\\includegraphics[scale=1.3]{4bstop.pdf}\n\\caption{Stop decay processes. The higgsinos ${\\tilde H}$ are assumed to be on-shell, while the sfermion ($\\tilde f = {\\tilde t}_R$, ${\\tilde t}_L$, ${\\tilde\\tau}_L$, or ${\\tilde\\nu}_\\tau$) is assumed to be off-shell. The symbols $f_1$, $f_2$, $f_3$ denote Standard Model fermions.}\n\\label{fig:diagram}\n\\end{center}\n\\end{figure}\n\n\\begin{table}[t]\n\\begin{center}\\small{\n\\begin{tabular}{|c|c|c|c|c|c|}\\hline\n&  \\multicolumn{3}{|c|}{Scenario}                       & ${\\tilde H}^+ \\to {\\tilde H}^0_1$ & Final state \\\\\\cline{2-4}\n&  \\multicolumn{2}{|c|}{Coupling} & Mediator $\\tilde f$ & transition          & (for each stop) \\\\\\hline\nA & \\multirow{2}{*}{$UDD$}\n    & 312 & ${\\tilde t}_R$        & YES & $tbqq$ \\\\\\cline{3-6}\nB & & 323 & ${\\tilde t}_R$        & NO  & $bbbq$ \\\\\\hline\nC & \\multirow{3}{*}{$LQD$}\n    & 323 & ${\\tilde\\tau}_L$      & YES & $\\tau bbq$ \\\\\\cline{3-6}\nD & & 321 & ${\\tilde\\nu}_\\tau$    & NO  & $\\tau bqq$ \\\\\\cline{3-6}\nE & & 232 & ${\\tilde t}_L$        & NO  & $\\mu bbq$  \\\\\\hline\n\\end{tabular}}\\end{center}\n\\caption{Benchmark models (referred to later by their labels A--E).}\n\\label{tab:scenarios}\n\\end{table}\n\n\\section{Simulation of Searches}\n\\label{sec:searches}\n\nOur simulation framework includes a comprehensive set of ATLAS and CMS searches from the 7 and 8~TeV LHC runs. Among the more recent searches added (relative to~\\cite{our-stops}), we would like to mention in particular the exotic heavy quark searches from ATLAS~\\cite{ATLAS-LSST-3b} and CMS~\\cite{CMS-B2G},\\footnote{The single-lepton channel of the CMS search uses a boosted decision tree and is therefore not useful for reinterpretation studies like ours. The opposite and same-sign dilepton channels are included in our framework.} the CMS search for second-generation leptoquarks~\\cite{CMS-LQ2}, and the ATLAS search for 6-7 high-$p_T$ jets~\\cite{ATLAS-6-7}.  In addition to discussing existing searches and ways in which they could be strengthened in the context of our scenarios, we will examine the expected sensitivity of a search, proposed in~\\cite{LSST}, for high-$S_T$ events with a lepton, high jet multiplicity and a $b$ tag (without a significant $\\slash E_T$\\, requirement). We have implemented this search as in Sec.~3.3 of~\\cite{our-gluinos}, assuming 20~fb$^{-1}$ at 8~TeV.\n\nFor generating events with RPV decays, we used \\textsc{MadGraph~5}~\\cite{MG5} with the RPVMSSM model~\\cite{RPVMSSM_UFO}, and then \\textsc{Pythia~8}~\\cite{Pythia8}. For the technical details of how we simulate searches, see the appendix of~\\cite{our-gluinos}. In all cases, we have applied an efficiency threshold of $\\sim 10^{-3}$ to reduce sensitivity to the far tails of the signal distributions.\n\n\\section{Results}\n\n\\subsection{Tops\/taus + many jets (with many $b$ tags)}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[scale=0.57]{cascadehiggsinoUDD312.pdf}\n\\includegraphics[scale=0.57]{cascadehiggsinolLQD323.pdf}\n\\caption{Limits on model A, where ${\\tilde t} \\to tbqq$ (left) and model C, where ${\\tilde t} \\to \\tau bbq$ (right), from~\\cite{ATLAS-LSST-3b,CMS-SSDIL,ATLAS-SSDIL}.}\n\\label{fig:tops-taus-manyb}\n\\end{center}\n\\end{figure}\n\nIn Fig.~\\ref{fig:tops-taus-manyb} (left), we present limits on a scenario in which the stop decays often contain leptons (from top decays) and multiple jets. Note that even though SS dileptons are available, the strongest limits come from an ATLAS search~\\cite{ATLAS-LSST-3b}\\footnote{For our reinterpretation of this search, we defined the search regions as $H_T > 800$, 1000, 1200, 1400, 1600, 1800~GeV for bins with 3 and $\\geq 4$ $b$-tags (on top of ATLAS's selection).} that requires a lepton, large ($\\geq 6$) jet multiplicity and multiple $b$ tags (but not much $\\slash E_T$). It is likely that the reach of this search could be improved by using bins with more than 6 jets, since the signal contains 10 hard partons.\n\nIn Fig.~\\ref{fig:tops-taus-manyb} (right), we show that the same search~\\cite{ATLAS-LSST-3b} excludes a certain range of masses even when the leptons are only coming from tau decays. It would be useful to design a search optimized for lower values of $H_T$, to cover the gap at low stop masses.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[scale=0.62]{oshiggsinoLQD321.pdf}\n\\caption{Limits on model D, where ${\\tilde t} \\to \\tau bqq$, from~\\cite{ATLAS-SUSY-taus,ATLAS-7-10,CMS-ttbar-tau} and an expected limit (dashed) from a lepton + many jets search proposed in~\\cite{LSST,our-gluinos}.}\n\\label{fig:taus-fewb}\n\\end{center}\n\\end{figure}\n\n\\subsection{Taus + many jets (with few $b$ tags)}\n\nIn Fig.~\\ref{fig:taus-fewb}, we present results for another tau-rich scenario, but this time with only one $b$-quark per stop decay. The ATLAS search~\\cite{ATLAS-LSST-3b}, which requires 3 or 4 $b$ tags, is no longer useful. However, the proposed search for a lepton + many jets with a \\emph{single} $b$ tag~\\cite{LSST,our-gluinos} (which we mentioned in section~\\ref{sec:searches}) is not very far from being sensitive, especially at low masses. Apart from optimizing the event selection cuts of that search (which we have not attempted), one could try designing an analogous search for a lepton + hadronic tau + many jets.\n\nWe also see in that plot that the ATLAS searches for hadronic taus + jets + $\\slash E_T$\\,~\\cite{ATLAS-SUSY-taus} and large jet multiplicity + low $\\slash E_T$\\,~\\cite{ATLAS-7-10} are not far from being sensitive at intermediate masses. One could likely obtain better sensitivity by using $b$ tagging and lowering the $\\slash E_T$\\, cuts in~\\cite{ATLAS-SUSY-taus} (since the $\\slash E_T$\\, in our scenario comes only from tau decays), or alternatively by adding a hadronic tau requirement in~\\cite{ATLAS-7-10} (since all the events contain two taus).\n\nAnother approach to probing this scenario would be to modify the CMS search for third-generation leptoquarks and RPV stops decaying to $\\tau b$ (available in~\\cite{CMS-LQ3} with 7~TeV data) in a way that would account for the different kinematics of the $\\tau$s and $b$s and utilize the additional jets.\n\nFinally, note that at very low masses the limit from the $t\\bar t$ cross section measurement (in the $\\ell+\\tau_h$ channel)~\\cite{CMS-ttbar-tau}, even just with $\\sim 2$~fb$^{-1}$ at 7~TeV, is stronger than the limits from any of the new physics searches that we examined. Recasting the $t\\bar t$ cross section measurement, perhaps with the requirement of a larger-than-usual jet multiplicity, as a new physics search, would be useful.\n\n\\subsection{Opposite-sign dileptons + many jets}\n\nIn Fig.~\\ref{fig:LQ-bbb} (left), we show that the CMS search for second-generation leptoquarks~\\cite{CMS-LQ2} delivers very strong limits on scenarios with pairs of opposite-sign leptons, even though the search's motivation is a two-body rather than a four-body decay. Here we have taken advantage of the fact that CMS uses a separate set of cuts for each leptoquark mass hypothesis. In our reinterpretation, we have used all of these as potential search regions, regardless of the stop mass, in order to allow for the different kinematics. The limits could likely be strengthened by making use of the large jet multiplicity and\/or the presence of $b$ jets in the signal. Limits from the relevant opposite-sign dilepton region of the CMS search for exotic heavy quarks~\\cite{CMS-B2G}, which does not utilize the high $p_T$ of the leptons and has just a single $S_T$ bin, are somewhat weaker.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[scale=0.57]{oshiggsinoLQD232.pdf}\n\\includegraphics[scale=0.57]{oshiggsinoUDD323.pdf}\n\\caption{Left: limits on model E, where ${\\tilde t} \\to \\mu bbq$, from~\\cite{CMS-B2G,CMS-LQ2}. Right: limits on model B, where ${\\tilde t} \\to bbbq$, from~\\cite{ATLAS-6-7}.}\n\\label{fig:LQ-bbb}\n\\end{center}\n\\end{figure}\n\n\\subsection{Multiple $b$ jets}\n\nIn Fig.~\\ref{fig:LQ-bbb} (right), we consider a scenario with a jets-only final state. While all-jet scenarios with stop production cross sections are generally difficult, in this particular scenario 6 of the 8 jets are $b$-jets. Clearly, the background for such events is small, but there is no search that utilizes this fact. The ATLAS search for 6-7 high-$p_T$ jets~\\cite{ATLAS-6-7}, which almost sets a limit at low masses, uses bins requiring 0, 1 or 2 $b$-tags. Adding bins with larger numbers of $b$-tags would be beneficial. More detailed kinematic properties of the signal, such as resonant structures, could also be exploited.\n\n\\section{Conclusions}\n\nBy examining a set of new physics benchmark models, we have identified certain well-defined final states that are not optimally covered by existing LHC searches. We have proposed strategies, which are modifications on existing searches for exotic heavy quarks~\\cite{ATLAS-LSST-3b,CMS-B2G}, leptoquarks~\\cite{CMS-LQ2,CMS-LQ3}, hadronic taus + jets + $\\slash E_T$\\,~\\cite{ATLAS-SUSY-taus} and multiple high-$p_T$ jets~\\cite{ATLAS-6-7}, that would be more sensitive to these final states. We have also pointed out the value of designing a search based on the $t\\bar t$ cross section measurement in the $\\ell+\\tau_h$ channel~\\cite{CMS-ttbar-tau}.\n\nAt this moment in time, when theory does not provide a clear guidance as to how new physics will manifest itself, it is essential that all classes of final states are being covered experimentally.  Implementing the ideas presented here could greatly improve LHC reach, both for the RPV scenarios we discussed explicitly and for any new physics scenarios with similar final states.\n\n\\acknowledgments\n\nThis work was supported in part by DOE grant DE-FG02-96ER40959.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction} \\label{sec-intro}\n\n\\lremind{sec-intro}In this article we describe recent work on positive\nalgorithms for computing the structure constants of the cohomology of\nhomogeneous varieties. We also discuss extensions of these rules to\n$K$-theory, equivariant cohomology, equivariant $K$-theory and quantum\ncohomology. We have two aims. First, we would like to provide the\nexperts in the field with a compendium of recent results and\nreferences regarding positivity in Schubert calculus. Second, we would\nlike to present many examples so that the casual user can\nperform basic calculations that occur in concrete problems. \n\nHomogeneous varieties are ubiquitous in mathematics, playing an\nimportant role in representation theory, algebraic and differential\ngeometry, combinatorics and the theory of symmetric functions.  The\nstructure constants (``Littlewood-Richardson coefficients'') of the\ncohomology rings of homogeneous varieties exhibit a rich and\nsurprising structure.  For fundamental geometric reasons, the\nLittlewood-Richardson coefficients and their generalizations tend to\nbe positive (interpreted appropriately).  In cohomology, this is a\nconsequence of Kleiman's Transversality Theorem~\\ref{thm-kleiman}\n(sometimes called the Kleiman-Bertini Theorem).  In $K$-theory, the\nmost notable general positivity result is a theorem of Brion\n\\cite{brion:positivity}, and in equivariant cohomology, the key result\nis due to Graham \\cite{graham:equivariant},\nconfirming a conjecture of D.~Peterson.\nSuch positivity suggests\nthat these coefficients have a combinatorial interpretation (a\n``Littlewood-Richardson rule''), and that such an interpretation\nshould be geometric in nature.  In recent years positive algorithms\nfor computing these constants have unraveled some of this beautiful\nstructure. We now survey the techniques used in obtaining positive\ngeometric algorithms for determining these structure constants\nstarting with the case of the ordinary Grassmannians.  \n\nIn Part~\\ref{partone} we consider the case of type $A$, following a\nparticular series of degenerations (which first arose in\n\\cite{vakil:checkers}, and are described in \\S\n\\ref{degenerationorder}) that seems to be particularly fruitful in\nresolving a product of Schubert classes into a combination of other\nSchubert classes.  In most cases, we may at least conjecturally\ninterpret these degenerations in terms of generalizations of the\nelegant puzzles of A. Knutson and T. Tao.  Many of the new conjectural\nstatements in this Part are joint work of Knutson and the second\nauthor.\n\n\nIn Part \\ref{parttwo}, instead of degenerating according to a fixed\norder, we adapt our degeneration order to the Schubert problem at\nhand. This flexibility allows us to simplify the geometry. In addition\nto a new rule for Grassmannians (\\S \\ref{secflag}), we obtain\nLittlewood-Richardson rules for two-step flag varieties (\\S \\ref{pmt}) and\nthe quantum cohomology of Grassmannians (\\S \\ref{qcg}). Furthermore, the same\ndegeneration technique can be applied to Fano varieties of quadric\nhypersurfaces, thus yielding a method to calculate certain\nintersections in Type $B$ and $D$ Grassmannians (\\S \\ref{sec-orth}).\n\n\nWe will present lots of explicit examples, which are the heart of the\narticle, as they illustrate the general techniques.  In some sense,\nevery example is a generalization of a single classical example, \\S\n\\ref{fourlines}.\n\nWe emphasize that this is an active, burgeoning area of research, and\nwe are presenting only a sample of recent work.  We wish to at least\nadvertise several results closely related to those discussed here.\n(i) L. Mihalcea has an explicit statement of a geometric\nLittlewood-Richardson rule for the Lagrangian Grassmannian\n(type $C$), verified\ncomputationally in a convincing number of cases.  He has proved part\nof the statement and is working on the rest.  \n(There is already a non-geometric Littlewood-Richardson rule in this\ncase, due to Stembridge.)\n(ii) D. Davis is\npursuing ongoing work on an analogous geometric rule in type $B$.\n(iii) W. Graham and S. Kumar have explicitly computed the structure\nco-efficients of equivariant $K$-theory of projective space \\cite{grahamkumar}.\nThe description is surprisingly rich and nontrivial given the\n``simplicity'' of the space.  Earlier approaches to the equivariant\n$K$-theory of $G\/B$ are due to Griffeth and Ram \\cite{gr} and Willems\n\\cite{w}.  We haven't yet attempted to relate these results to the\nconjecture of \\S \\ref{KT}.  (iv) Thomas and Yong have recently given\na root-theoretically uniform generalization of the Littlewood-Richardson rule\nfor intersection numbers of Schubert varieties in minuscule and cominuscule\nflag varieties \\cite{thomasyong}.  It would be very interesting to \nunderstand their work geometrically.\n\nThere is an even larger amount of work on positive rules without (yet)\ndirect geometric interpretation.  A discussion of these ideas would\ntriple the length of the article, so we content ourselves with listing a\nsampling of the authors who have contributed to the area: N.~Bergeron,\nFomin, Gelfand, Knutson, Lenart, Postnikov, Robinson, Sottile, Yong,\n\\dots .\n\n\\subsection{(Type $A$) Grassmannians and positivity} Let $G(k,n)$\ndenote the Grassmannian that parametrizes $k$-dimensional subspaces\nof an $n$-dimensional vector space $W$. It is sometimes more\nconvenient to interpret $G(k,n)$ as the parameter space of\n$(k-1)$-dimensional projective linear spaces in $\\mathbb{P}^{n-1}$. When we\nwish to emphasize this point of view, we will denote $G(k,n)$ by\n$\\mathbb{G}(k-1,n-1)$.\n\n\nFix a complete flag $F_{\\bullet}$\n$$0 =F_0 \\subset F_1 \\subset \\cdots \\subset F_n = W.$$ Let $\\lambda$\nbe a partition with $k$ parts satisfying $n-k \\geq \\lambda_1 \\geq\n\\cdots \\geq \\lambda_k \\geq 0.$ The {\\em Schubert variety}\n$\\Omega_{\\lambda}(F_{\\bullet})$ of type $\\lambda$ associated to the\nflag $F_{\\bullet}$ is defined by\n$$\\Omega_{\\lambda}(F_{\\bullet}) := \\{ \\ [V] \\in G(k,n) \\ : \\ \\dim(\nV \\cap F_{n-k+i-\\lambda_i}) \\geq i \\ \\}.$$ \nThose partitions corresponding to  (non-empty) Schubert\nvarieties in $G(k,n)$ are readily seen to be those  contained in a\n$k \\times (n-k)$ rectangle. \nThe parts $\\lambda_i$\nthat are zero are often omitted from the notation. \n\n\n For example, in\n$G(2,4)= \\mathbb{G}(1,3)$ a complete flag is given by specifying a point $p$,\ncontained in a line $l$, contained in a plane $P$, contained in\n$\\mathbb{P}^3$.  The Schubert variety $\\Omega_{1}$ parametrizes the lines in\n$\\mathbb{P}^3$ that intersect the line $l$. The Schubert variety\n$\\Omega_{2,1}$ parametrizes the \nlines in $\\mathbb{P}^3$ that are contained in the plane $P$ and contain the\npoint $p$.\n\nAn alternate indexing set for Schubert varieties of $G(k,n)$ consists\nof the size $k$ subsets of $\\{ 1, \\dots, n \\}$, presented as a string\nof $n$ digits, of which $k$ are $1$ and $n-k$ are $0$.  This is known\nas {\\em string notation}.  The bijection is as follows.  The $1$'s are\nplaced at those positions $j \\in \\{ 1, \\dots, n \\}$ where $\\dim V \\cap\nF_j > \\dim V \\cap F_{j-1}$.  A more visual description is given in\nFigure~\\ref{seattlerec}: to obtain the string of $0$'s and $1$'s,\nconsider the path from the northeast corner to the southwest corner of\nthe rectangle, along the border of the partition.  This path consists\nof $n$ segments.  If the $j$th step is west (resp.\\ south), the $j$th\nelement of the string is $0$ (resp.\\ $1$).\n\\begin{figure}[ht]\n\\begin{center} \n\\setlength{\\unitlength}{0.00083333in}\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  \\selectfont}%\n\\fi\\endgroup%\n{\\renewcommand{\\dashlinestretch}{30}\n\\begin{picture}(7033,885)(0,-10)\n\\put(12,36){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}partition}}}\n\\dashline{60.000}(2712,36)(2712,336)(3012,336)\n\\dashline{60.000}(2412,636)(2412,36)\n\\path(2112,36)(2112,336)(2712,336)\n\t(2712,636)(3012,636)\n\\blacken\\path(4482.000,366.000)(4362.000,336.000)(4482.000,306.000)(4482.000,366.000)\n\\path(4362,336)(4662,336)\n\\blacken\\path(4782.000,366.000)(4662.000,336.000)(4782.000,306.000)(4782.000,366.000)\n\\path(4662,336)(4962,336)\n\\blacken\\path(4932.000,456.000)(4962.000,336.000)(4992.000,456.000)(4932.000,456.000)\n\\path(4962,336)(4962,636)\n\\blacken\\path(5082.000,666.000)(4962.000,636.000)(5082.000,606.000)(5082.000,666.000)\n\\path(4962,636)(5262,636)\n\\blacken\\path(4332.000,156.000)(4362.000,36.000)(4392.000,156.000)(4332.000,156.000)\n\\path(4362,36)(4362,336)\n\\path(612,636)(12,636)(12,336)\n\t(612,336)(612,636)\n\\dashline{60.000}(312,336)(312,636)\n\\put(1737,261){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$k=2$}}}\n\\put(2337,786){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$n-k=3$}}}\n\\put(3537,261){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$\\Longleftrightarrow$}}}\n\\put(6387,261){\\makebox(0,0)[lb]{{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}$01001$}}}\n\\put(5712,261){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$\\Longleftrightarrow$}}}\n\\put(4137,111){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$5$}}}\n\\put(4437,411){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$4$}}}\n\\put(4737,411){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$3$}}}\n\\put(5037,411){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$2$}}}\n\\put(5037,711){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(1062,261){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$\\Longleftrightarrow$}}}\n\\dashline{60.000}(2112,636)(3012,636)(3012,36)\n\t(2112,36)(2112,636)\n\\end{picture}\n}\n\n\\end{center}\n\\caption{The bijection between partitions contained in a $k \\times (n-k)$\nrectangle and strings of $k$ $1$'s and $n-k$ $0$'s.\n\\lremind{seattlerec}\n} \\label{seattlerec}\n\\end{figure}\n\nWe will use string notation interchangeably with partition notation\n(as the index set of Schubert varieties of the Grassmannian), and we\napologize for any confusion this might cause.  Partitions (of length\ngreater than  $1$) will have commas, and strings will have no commas.\n\n\n\n\nThe homology class of a Schubert variety is independent of the\ndefining flag, and depends only on the partition. We will denote the\nPoincar\\'e dual of the class of $\\Omega_{\\lambda}$ by\n$\\sigma_{\\lambda}$.  The Poincar\\'e duals of the classes of Schubert\nvarieties give an additive basis for the cohomology of $G(k,n)$.\nTherefore, given two Schubert cycles $\\sigma_{\\lambda}$,\n$\\sigma_{\\mu}$, their product is a $\\mathbb{Z}$-linear combination of Schubert\ncycles $\\sum_{\\nu} c_{\\lambda, \\mu}^{\\nu} \\sigma_{\\nu}$. The structure\nconstants $c_{\\lambda, \\mu}^{\\nu}$ of the cohomology ring with respect\nto the Schubert basis are called {\\em Littlewood-Richardson\n  coefficients}. \n\n\nThe basic problem we would like to address is finding positive\ngeometric algorithms for computing the Littlewood-Richardson\ncoefficients. A positive combinatorial rule giving  these\ncoefficients is called a {\\em Littlewood-Richardson rule}. \n\n\n The underlying strategy for the geometric rules is as follows. One\n begins with the intersection of two Schubert varieties defined with\n respect to two general flags. Each rule has a recipe for making the\n flags more special via codimension one degenerations. As one makes\n the flags more special, the intersection of the two Schubert\n varieties defined with respect to the flags become more special,\n sometimes breaking into a number of pieces, each of which is analyzed\n separately in the same way.  The pieces at the end are Schubert\n varieties.  The combinatorial objects such as puzzles or checkers\n encode the varieties that arise as a result of the degenerations. The\n following fundamental example illustrates the strategy.\n\n\n \\begin{example} \\label{fourlines}\\lremind{fourlines}How many lines in\n   $\\mathbb{P}^3$ intersect four general given lines $l_1, \\dots, l_4$?  If\n   the lines are in general position, the answer to this question is\n   not immediately clear. However, if the lines are in a special\n   position, then the answer might be easier to see. Suppose two of\n   the lines $l_1$ and $l_2$ intersect at a point $q$. There are two\n   ways that a line $l$ can intersect both $l_1$ and $l_2$. If $l$\n   passes through the point $q$, then $l$ intersects both. If not, $l$\n   must lie in the plane $P$ spanned by $l_1$ and $l_2$. In the first\n   case the only line that can also intersect $l_3$ and $l_4$, is the\n   intersection of the two planes spanned by $q$ and $l_3$ and $q$ and\n   $l_4$. In the second case the only line that intersects $l_3$ and\n   $l_4$ is the line that joins $P \\cap l_3$ and $P \\cap l_4$. We see\n   that when the lines are in a slightly special position, then the\n   answer to the question is $2$ (see Figure \\ref{lines}). \n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\begin{picture}(0,0)%\n\\includegraphics{lines.pstex}%\n\\end{picture}%\n\\setlength{\\unitlength}{3947sp}%\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  \\selectfont}%\n\\fi\\endgroup%\n\\begin{picture}(2173,1074)(781,-1198)\n\\put(2476,-1036){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}$q$}%\n}}}}\n\\put(1726,-961){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{12}{14.4}{\\familydefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}$P$}%\n}}}}\n\\end{picture}%\n\n\\end{center}\n\\caption{There are two lines in $\\mathbb{P}^3$ that intersect four general\nlines.\\lremind{lines}}\n\\label{lines}\n\\end{figure}\n\n\nAn additional argument is required to deduce that if the lines $l_1,\n\\dots, l_4$ are in general position, the answer is still $2$. Suppose\nwe begin with the lines in general position and rotate $l_1$ until it\nintersects $l_2$. By the principle of conservation of number, the\nnumber of lines intersecting $l_1, \\dots, l_4$ counted with\nmultiplicity remains constant. One may check that the multiplicities\nare one by a tangent space calculation that shows that the Schubert\nvarieties of lines intersecting $l_1$ and $l_2$ intersect generically\ntransversally even when $l_1$ and $l_2$ intersect at a point.\n\n\\end{example}\n\n \n\\subsection{Advantages of geometric rules}\nThere are many presentations of the cohomology ring of the\nGrassmannian. For instance, Pieri's rule gives an easy method to\nmultiply special Schubert cycles, those Schubert cycles where\n$\\lambda_i = 0$ except for $i=1$. Let $\\sigma_{\\lambda}$ be a special\nSchubert cycle. Suppose $\\sigma_{\\mu}$ is any Schubert cycle with\nparts $\\mu_1, \\dots, \\mu_k$. Then\n$$\n\\sigma_{\\lambda} \\cdot \\sigma_{\\mu} = \\sum_{\\substack{ \\mu_i \\leq\n\\nu_i \\leq \\mu_{i-1} \\\\ \\sum \\nu_i = \\lambda + \\sum\n\\mu_i}}\\sigma_{\\nu}.\n$$\nGiambelli's determinantal\nformula expresses any Schubert cycle as a determinant of a matrix\nconsisting of only special Schubert cycles: \n$$\\sigma_{\\lambda_1, \\dots, \\lambda_k} =\n  \\begin{vmatrix} \\sigma_{\\lambda_1} & \\sigma_{\\lambda_1 + 1} &\n  \\sigma_{\\lambda_1 + 2} & \\dots & \\sigma_{\\lambda_1 + k -1} \\\\\n  \\sigma_{\\lambda_2 - 1} & \\sigma_{\\lambda_2} & \\sigma_{\\lambda_2 + 1}\n  & \\dots & \\sigma_{\\lambda_2 + k -2} \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots  \\\\\n  \\sigma_{\\lambda_k -k+ 1} & \\sigma_{\\lambda_k - k + 2} &\n  \\sigma_{\\lambda_k - k + 3} & \\dots &\n  \\sigma_{\\lambda_k}\\end{vmatrix}.$$ These two formulae taken together\n  give a presentation for the cohomology ring of $G(k,n)$. \n However, a geometric Littlewood-Richardson rule has some\nadvantages over these\n  presentations.\n\n\n\n  Positive combinatorial rules give more efficient algorithms for\n  computing structure constants than determinantal formulae. More\n  importantly, some of the structure of the Littlewood-Richardson\n  coefficients is very hard to see from determinantal formulae. For\n  instance, Horn's conjecture was resolved through the combined work\n  of Klyachko, Knutson, and Tao, and the key final step was Knutson\n  and Tao's proof of the Saturation conjecture \\cite{knutson:puzzle}\n  using honeycombs, which are intimately related to puzzles. An\n  important consequence of their work is the following property of\n  Littlewood-Richardson coefficients, which was originally known as\n  Fulton's conjecture:\n \n\\begin{theorem}(Knutson-Tao-Woodward, \\cite{ktw})\nIf $c_{\\lambda,\n  \\mu}^{\\nu} = 1$, then $c_{N\n  \\lambda, N\\mu}^{N \\nu} = 1$.\n\\end{theorem}\n\\noindent This kind of structure is very hard to prove using\ndeterminantal formulae, but is often immediate from positive\nLittlewood-Richardson rules. \n\n\n\nAnother advantage of geometric Littlewood-Richardson rules is that\nthey apply over arbitrary fields.  Over algebraically closed fields of\ncharacteristic zero the connection between cohomology and enumerative\ngeometry is provided by Kleiman's Transversality Theorem\n\\cite{kleiman:transverse}.\n\n\\begin{theorem}(Kleiman) \\label{thm-kleiman}\n\\lremind{thm-kleiman}Let $G$ be an integral algebraic group scheme, $X$ an integral\nalgebraic scheme with a transitive $G$ action. Let $f: Y \\rightarrow\nX$ and $g: Z \\rightarrow X$ be two maps of integral algebraic\nschemes. For each rational element of $g \\in G$, denote by $gY$ the\n$X$-scheme given by $y \\mapsto g f(y)$. Then there exists a dense open\nsubset $U$ of $G$ such that for every rational element $g \\in U$, the\nfiber product $(gY) \\times_X Z$ is either empty or equidimensional of\nthe expected dimension $$\\dim Y + \\dim Z - \\dim X.$$\nFurthermore, if $Y$ and $Z$ are regular, then for a dense open set\nthis fibered product is regular.\n\\end{theorem} \n\n\\noindent In particular, applying Theorem \\ref{thm-kleiman} in the\ncase when $X$ is a homogeneous variety and $f$ and $g$ are the\ninclusions of Schubert subvarieties, we conclude that for general\ntranslates the intersections will be generically transverse of the\nexpected dimension provided the intersection is\nnon-empty. Unfortunately, Kleiman's theorem does not hold when the\ncharacteristic of the ground field is not zero, or when the\nground field is not algebraically closed.\n\nThe geometric Littlewood-Richardson rules apply even when the ground\nfield has positive characteristic. The following is a\ncharacteristic-free version of the Kleiman-Bertini Theorem proved in\n\\cite{vakil:induction}.\n\n\\begin{theorem}(Generic smoothness)\nSuppose $Q \\subset G(k,n)$ is a subvariety such that $(Q \\times Fl(n))\n\\cap \\Omega_{\\lambda}(F_{\\bullet}) \\rightarrow Fl(n)$ is generically\nsmooth for all $\\lambda$. Then $$(Q \\cap Fl(n)^m) \\cap\n\\pi_1^*\\Omega_{\\lambda_1}(F_{\\bullet}^1) \\cap \\cdots \\cap\n\\pi_m^*\\Omega_{\\lambda_m} (F_{\\bullet}^m) \\rightarrow Fl(n)^m$$ is\nalso generically smooth.\n\\end{theorem}\n\n\nLet $F_{\\bullet}^1, \\dots, F_{\\bullet}^m$ be $m$ general flags in\n$\\mathbb{C}^n$. Suppose $\\Omega_{\\lambda_1}(F_{\\bullet}^1), \\dots,\n\\Omega_{\\lambda_m}(F_{\\bullet}^m)$ are $m$ Schubert varieties in\n$G(k,n)$ whose dimension of intersection is zero. The corresponding\nSchubert problem asks for the cardinality of the intersection of these\nvarieties. More generally, a Schubert problem asks for the cardinality\nof the intersection of Schubert varieties defined with respect to\ngeneral flags in case the dimension of intersection is zero. Over the\ncomplex numbers this cardinality can be determined by computing the\ndegree of the intersection in the cohomology ring by Kleiman's\ntheorem. However, a priori, over other fields it is not clear that the\ncardinality will equal the degree.  We say that a Schubert problem is\nenumerative over a field $L$ if there exists flags defined over\n$L$ such that the cardinality of the $L$-points in the\nintersection of the Schubert varieties defined with respect to these\nflags is equal to the degree of intersection of the Schubert\nvarieties.\n\n\nOne application of geometric Littlewood-Richardson rules has been to\nresolve the question of whether Schubert problems are enumerative over\nthe real numbers, finite fields and algebraically closed fields of any\ncharacteristic. The theorems may be found in \\cite{vakil:induction}.\n\n\n\\begin{theorem}\n\\begin{enumerate}\n\\item All Schubert problems for the Grassmannians are enumerative over\n  the real numbers (in fact for any field satisfying the implicit\n  function theorem).\n\n\\item All Schubert problems are enumerative over algebraically closed\n  fields (of arbitrary characteristic).\n\n\n\\end{enumerate}\n\\end{theorem}\n\nDespite the progress many natural questions still remain.\n\n\\begin{question}\nAre Schubert problems enumerative over the rational numbers?\n\\end{question}\n\n\n\n\nThere are many other applications of geometric Littlewood-Richardson\nrules.  For instance, they can be used to compute monodromy groups of\nSchubert problems. One may construct interesting examples of Schubert\nproblems whose monodromy group is not the full symmetric group. For\nprecise details about this and other applications see\n\\cite{vakil:induction}.\n\n\nOne final appeal of the geometric rules over other combinatorial\npositive rules is that the techniques extend to other homogeneous\nvarieties. We will now detail the extension of these ideas to two-step\nflag varieties and to orthogonal and symplectic Grassmannians.\n\n\n\\subsection{Flag varieties}\n\nLet $0 < k_1 < \\cdots < k_r < n$ be an increasing sequence of $r$\npositive integers.  Let $Fl(k_1, \\dots, k_r ; n)$ denote the $r$-step flag variety of\n$r$-tuples of linear subspaces $(V_1, \\cdots, V_r)$ of an\n$n$-dimensional vector space $W$, where $V_i$ are $k_i$-dimensional\nlinear spaces and $V_i \\subset V_{i+1}$ for all $1 \\leq i \\leq r-1$.\nWhen we would like to consider the flag variety as a parameter space\nfor nested sequences of linear subspaces of projective space, we will\nuse the notation $\\mathbb{F} l(k_1 - 1, \\dots, k_r -1; n-1)$.\n\n\n\nThe cohomology groups of flag varieties are also generated by Schubert\nclasses. Usually in the literature the Schubert cycles are\nparametrized by certain permutations. More precisely, Schubert\nvarieties are parametrized by permutations $\\omega$ of length $n$ for\nwhich $\\omega(i) < \\omega(i+1)$ whenever $i \\notin \\{k_1, \\dots, k_r\n\\}$. More explicitly, the Schubert variety $X_{\\omega} (F_{\\bullet})$\nis defined by\n\\begin{align*}\n  X_{\\omega} (F_{\\bullet}) : = \\{ \\ ( V_1, \\dots, V_r) \\in Fl(k_1, \\dots, k_r ; n) \\ |\n   \\dim (V_i \\cap F_j) \\geq \\# \\{ \\alpha \\leq i : \\omega(\\alpha) > n-j\n   \\} \\ \\forall \\ i,j \\ \\}.\n\\end{align*}\nThe Poincar\\'e duals of the classes of all the Schubert varieties form\nan additive basis for the cohomology of the flag variety.\n\nFor our purposes two other notations for Schubert varieties of\n$r$-step flag varieties are useful.\n\nFirst, in analogy with the partition notation for the Grassmannians we\nwill use the notation $\\sigma_{\\lambda_1, \\cdots, \\lambda_{k_r}}^{\n  \\delta_1, \\cdots, \\delta_{k_r}}$. The bottom row denotes the usual\npartition corresponding to the $k_r$-plane $V_r$ in $W$ treated as a\nSchubert cycle in $G(k_r, n)$. The numbers $\\delta_i$ are integers\nbetween $1$ and $r$. For a Schubert cycle in $Fl(k_1, \\dots, k_r ; n)$, $k_1$ of the\nupper indices will be $1$ and $k_i - k_{i-1}$ of them will be $i$. The\nflag $F_{\\bullet}$ induces a complete flag $G_{\\bullet}$ on the\nlargest vector space $V_r$.  For each $j$, there exists a smallest $i$\nsuch that $$\\dim (V_i \\cap G_j) = \\dim (V_i \\cap G_{j-1}) + 1.\n$$\nFor a\nZariski-open subset of the Schubert variety this index will be\nconstant. In that case we write $i$ on top of $\\lambda_j$. In the case\nof Grassmannians this notation reduces to the ordinary notation with a\nsequence of $1$'s on the top row. For the complete flag variety the top\nrow becomes the permutation defining the Schubert cycle.\n\nSecond, there is a string notation for partial flag varieties.\nSchubert varieties of $Fl(k_1, \\dots, k_r ; n)$ are\nindexed by $n$-tuples of numbers $0$, \\dots, $r-1$, where $k_{i+1} -\nk_i$ of the digits are $i$ (where $k_0 := 0$).  \nTaking\nthe convention $k_{r+1} = n$, \nfor each $j$, there exists a smallest $i$\nsuch that $$\\dim (V_i \\cap F_j) = \\dim (V_i \\cap F_{j-1}) + 1.\n$$\nThen in position $j$, we place $r+1-i$.  The reader may verify that\nthis generalizes the string notation for the Grassmannian.\n\n\nWe will use both notations interchangeably. As with the Grassmannian\ncase, strings will have no commas.\n\n\\begin{example} Fix a flag $F_1 \\subset \\cdots \\subset F_6$ in $W^6$.\n  The Schubert cycle $\\sigma_{2,1,0}^{2,1,2}$ in $Fl(1,3;6)$ denotes\n  the pairs of subspaces $V_1 \\subset V_2$ where $V_1$ has dimension\n  one and $V_2$ has dimension 3. $V_2$ is required to meet $F_2$ in\n  dimension one, $F_4$ in dimension $2$ and be contained in $F_6$.\n  $V_1$ lies in the intersection of $V_2$ with $F_4$.  In string\n  notation, this is written $\\sigma_{010201}$.\n\\end{example}\n\nUsing the ideas of geometric rules for Grassmannians it is possible to\ngive an explicit rule for two-step flag varieties (see\n\\cite{coskun1:LR}, discussed in \\S \\ref{secflag}). Previously there\nwere no proven rules for partial flag varieties. Knutson conjectured a\nrule for two-step flag varieties in terms of puzzles\n(Conjecture~\\ref{allentwostep}).  A. Buch extended the conjecture to\nthree-step flag varieties, described in Section \\ref{s:anders}.  We\ngive a conjectural geometric rule \n(due in part to Knutson) based on Knutson's two-step\nconjecture in Section \\ref{s:twostep}.\n\n\\subsection{Other groups}\\label{othergps}\n\n\nSo far we have discussed the ordinary Grassmannians and flag\nvarieties. These varieties are associated to the Lie group $SL(n)$\n(Type $A$). One can consider homogeneous varieties associated to other\nclassical groups such as $SO(n)$, $Sp(2n)$ and the exceptional groups.\nIn this paper we will have nothing to say about the exceptional Lie\ngroups. However, we would like to explain the geometric point of view\nfor the other infinite families of Lie groups. \n\nLet $Q$ be a non-degenerate, symmetric (resp., alternating) bilinear\nform on an $n$-dimensional vector space $W$. The isotropic\nGrassmannians $OG(k,n)$ (resp., $SG(k,n)$) parametrize\n$k$-dimensional subspaces of $W$ which are isotropic with respect to\n$Q$. When $n$ is even, $k=n\/2$ and $Q$ is symmetric, the isotropic\nsubspaces form two isomorphic irreducible families. In that case, the\northogonal Grassmannian is one of the irreducible components.  The\nvarieties $OG(k,n)$ are quotients of the Lie group $SO(n)$ by maximal\nparabolic subgroups. They are classified into Type $B$ and Type $D$\ndepending on the parity of $n$. The varieties $SG(k,n)$ are quotients\nof $Sp(n)$ by maximal parabolic subgroups.  Since there cannot be a\nnon-degenerate alternating form on an odd-dimensional vector space,\nhere $n$ has to be even. This is the Type $C$\ncase.\n\nThe cohomology of isotropic Grassmannians is generated by Schubert\nvarieties. The Littlewood-Richardson coefficients of Type $B$ and Type\n$C$ flag varieties are equal up to an explicit power of $2$.  The\nrelation, which we will summarize in Lemma \\ref{sot}, is discussed in\n\\cite{sottile:pieri} p. 17 or \\cite{billey:polynomials}. Let $u$, $v$\nand $w$ be three permutations. For a permutation $u$ denote $s(u)$ the\nnumber of sign changes of the permutation.\n\n\n\\begin{lemma}\\label{sot}\nThe Littlewood-Richardson coefficients of Type $B$ and Type $C$ full-flag\nvarieties satisfy \n$$2^{s(u)+s(v)} b_{u,v}^w = 2^{s(w)} c_{u,v}^w,$$ where $b_{u,v}^w$\n(resp., $ c_{u,v}^w$) denotes the structure coefficients of Type $B$\n(resp., Type $C$) full-flag variety.  \\end{lemma} In view of Lemma\n\\ref{sot}, we can restrict our attention to orthogonal Grassmannians.\nThere are minor differences in the description of the Schubert\nvarieties depending on the type of the isotropic Grassmannian. For\nease of exposition, we will describe the geometric viewpoint for the\ncase $OG(k,2m+1)$. \n\nLet $s \\leq m$ be a non-negative integer. Let $\\lambda$ denote a\nstrictly decreasing partition $$ m \\geq \\lambda_1 > \\lambda_2 > \\cdots\n> \\lambda_s > 0.$$ Given $\\lambda$, there is an associated partition\n$\\tilde{\\lambda}$\n$$ m-1 \\geq \\tilde{\\lambda}_{s+1} > \\tilde{\\lambda}_{s+2} > \\cdots >\n\\tilde{\\lambda}_m \\geq 0 $$ defined by the requirement that there does\nnot exist any parts $\\lambda_i$ for which $\\tilde{\\lambda}_j +\n\\lambda_i = m$. In other words, the associated partition is obtained\nby removing the integers $m-\\lambda_1, \\dots, m-\\lambda_s$ from the\nsequence $m-1, m-2, \\dots, 0$. For example, if $m=6$, then the\npartition associated to $(6,4)$ is $(5,4,3,1)$. The Schubert varieties\nin $OG(k,2m+1)$ are parametrized by pairs $(\\lambda, \\mu)$, where\n$\\lambda$ is a strictly decreasing partition of length $s$ and $\\mu$\n$$\\mu_{s+1} > \\mu_{s+2} > \\cdots > \\mu_k \\geq 0$$ is a subpartition of\n$\\tilde{\\lambda}$ of length $k-s$. Given a pair $(\\lambda, \\mu)$ the\n{\\em discrepancy} $\\operatorname{dis}(\\lambda,\\mu)$ of the pair is defined as follows:\nSince $\\mu$ is a subpartition of $\\tilde{\\lambda}$ we can assume that\nit parts occur as $\\tilde{\\lambda}_{i_{s+1}}, \\cdots,\n\\tilde{\\lambda}_{i_k}$. The {discrepancy} is given by\n$$\\operatorname{dis}(\\lambda,\\mu) = (m-k)s + \\sum_{j=s+1}^k (m -k + j - i_j).$$ \n\n\n\nFix an isotropic flag $F_{\\bullet}$\n$$0 \\subset F_1 \\subset F_2 \\subset \\cdots \\subset F_m \\subset\nF_{m-1}^{\\perp} \\subset \\cdots \\subset F_1^{\\perp} \\subset W.$$ Here\n$F_i^{\\perp}$ denotes the orthogonal complement of $F_i$ with respect\nto the bilinear form. The Schubert variety\n$\\Omega_{\\lambda,\\mu}(F_{\\bullet})$ is defined as the closure of the\nlocus $$ \\{ [V] \\in OG(k,2m+1)\\ | \\ \\dim( V \\cap F_{m+1 - \\lambda_i})\n= i, \\ \\dim( V \\cap F_{\\mu_j}^{\\perp}) = j \\ \\}. $$ The codimension of\na Schubert variety is given by $\\sum_{i=1}^s \\lambda_i +\n\\operatorname{dis}(\\lambda,\\mu).$ We will denote the Poincar\\'e dual of the\ncohomology class of $\\Omega_{\\lambda, \\mu}$ by $\\sigma_{\\lambda,\\mu}$.\nObserve that for maximal isotropic Grassmannians $OG(m,2m+1)$, the\npartition $\\mu$ is uniquely determined by the partition $\\lambda$.\nConsequently, in the literature the sequence $\\mu$ is often omitted\nform the notation. \n\n\nGeometrically, the orthogonal Grassmannian $OG(k,2m+1)$ may be\ninterpreted as the Fano variety of $(k-1)$-dimensional projective\nlinear spaces on a smooth quadric hypersurface in $\\mathbb{P}^{2m}$. The\nnon-degenerate form $Q$ defines the smooth quadric hypersurface\n$\\mathrm{Q}$. A linear space $F_i$ is isotropic with respect to $Q$ if\nand only if its projectivization is contained in $\\mathrm{Q}$. The\nprojectivization of the orthogonal complement $F_i^{\\perp}$\ncorresponds to the linear space of codimension $i$ everywhere tangent\nto $\\mathrm{Q}$ along the projectivization of $F_i$. This geometric\nreinterpretation allows us to apply modifications of the Mondrian\ntableaux rule to perform calculations in the cohomology of\n$OG(k,2m+1)$ (see \\S \\ref{sec-orth}). \n\nIn order to adapt the previous discussion to Grassmannians $OG(k,2m)$,\nwe have to account for the existence of two isomorphic irreducible\nfamilies of $m$-dimensional isotropic linear spaces. The\n$m$-dimensional linear spaces belong to the same irreducible family if\nand only if their dimension of intersection has the same parity as $m$. \nLet $\\lambda$ be a strictly decreasing partition\n$$m-1 \\geq \\lambda_1 > \\cdots > \\lambda_s \\geq 0,$$ where $s$ has\nthe same parity as $m$. Define the associated\npartition as those integers $$m-1 \\geq \\tilde{\\lambda}_{s+1} > \\cdots\n> \\lambda_m \\geq 0,$$ whose sums with any $\\lambda_i$ is not equal to\n$m-1$.  The Schubert varieties in $OG(k,2m)$ are parametrized by\npairs of strictly decreasing partitions $(\\lambda, \\mu)$, where $\\mu$\nis a subpartition of $\\tilde{\\lambda}$ of length $k-s$. With these\nmodifications in the numerics, the discussion of the Schubert\nvarieties of $OG(k,2m+1)$ carries over to the case of $OG(k,2m)$. We\nwill leave the details to the reader.\n\n\\noindent {\\bf Acknowledgments.}  We thank Sara Billey, Anders Buch,\nDiane Davis, William Graham, Joe Harris, Allen Knutson, Shrawan Kumar,\nRobert MacPherson, Leonardo Mihalcea, Harry Tamvakis, and the\norganizers of the 2005 Seattle conference in algebraic geometry.  In\nparticular, we are grateful to Buch and Knutson for allowing us to\ndescribe their conjectures, and Graham and Kumar for allowing us to\npresent their as-yet-unpublished work \\cite{grahamkumar}.  We thank\nBuch, Knutson, Mihalcea, and Tamvakis for detailed helpful comments on the\nmanuscript.\n\n\\part{TYPE $A$ RULES, USING A SPECIFIC DEGENERATION ORDER}\n\n\\label{partone}\nIn the next section, we will describe the geometric\nLittlewood-Richardson rule \\cite{vakil:checkers}, but in the language\nof puzzles of Knutson and Tao.  In later sections in\nPart~\\ref{partone}, we will use the same degeneration order to extend\nthese ideas conjecturally to much more general situations ($K$-theory,\nequivariant cohomology, equivariant $K$-theory, $2$-step flag\nvarieties, and partial flag varieties in general).  The key\nconstruction will be varieties associated to partially completed puzzles\n(\\eqref{puzzlevariety}, and a variation \\eqref{eqPV} in the\nequivariant case).  The puzzles are very friendly to use, and the\nreader is very strongly encouraged to work through the examples.\n\n\n\\section{The Grassmannian}\n\n\\label{hhh}\\lremind{hhh}We now describe the geometric\nLittlewood-Richardson rule for the Grassmannian $G(k,n)$, following\n\\cite{vakil:checkers}.  We begin by describing the sequence of\ndegenerations.  We will be considering the intersection of Schubert\nvarieties with respect to two transverse flags $F_{\\bullet}$ and\n$M_{\\bullet}$ in $Fl(n)$ (where $F$ and $M$ stand for $F$ixed and\n$M$oving flags, respectively).  We describe a series of ``codimension\n$1$'' degenerations of $M_{\\bullet}$, beginning with $M_{\\bullet}$\ntransverse to $F_{\\bullet}$, and ending with $M_{\\bullet} =\nF_{\\bullet}$.  This degeneration order is very special; it appears\nthat the good behavior described below happens in general only for\nthis order (and the ``dual'' order).\n\n\\subsection{A key example}\n\\label{akeyexample}\\lremind{akeyexample}Before describing the order of degenerations in general, we begin with an example,\nshown in Figure~\\ref{degenorder}, for $n=4$, but shown projectively\n(in $\\mathbb{P}^3$) for convenience of visualization. \n In the first\ndegeneration, the moving plane moves (and all other\nparts of the moving flag are stationary) until ``something unusual''\nhappens, which is when it contains the fixed point.  Then the moving\nline (in the moving plane, containing the moving point) moves until it too\ncontains the fixed point.  Then the moving plane moves again (around\nthe moving line, which remains stationary) until it contains the fixed\nline.  Then the moving point moves down to agree with the fixed point,\nthen the moving line pivots to agree with the fixed line, and finally\nthe moving plane closes like a book to agree with the fixed plane.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\setlength{\\unitlength}{0.00083333in}\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  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degeneration order for $n=4$ (shown projectively,\nin $\\mathbb{P}^3$).  The puzzles will be explained shortly.\\lremind{degenorder}\n}\n\\label{degenorder}\n\\end{figure}\n\nWe continue the example, using this degeneration to deform an\nintersection of two Schubert cycles.  (The reader may recognize this\nas a variant of Example~\\ref{fourlines}.)  This example, understood\nwell enough, leads to the general rule for the Grassmannian in\ncohomology, and the many other rules (conjectural and otherwise)\nstated in Part~\\ref{partone}.  Let $\\alpha = \\beta$ be the unique Schubert\ndivisor class (the class $\\Box = (1)$ in terms of partitions), so\n$\\overline{\\Omega}_{\\alpha} = \\overline{\\Omega}_{\\beta}$ both correspond to\nthe set of lines in $\\mathbb{P}^3$ meeting a fixed line.  We will degenerate\nthe locus of lines meeting two general (skew) lines into a union of\nSchubert varieties.  This is depicted in Figure~\\ref{keyexample}.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\setlength{\\unitlength}{0.00083333in}\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  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motivating example of the degenerations\nof the geometric Littlewood-Richardson rule,\nillustrating $\\sigma_{1}^2 = \\sigma_{1,1} + \\sigma_{2}$,\nor in string notation,\n$\\sigma_{0101}^2 = \\sigma_{0110} + \\sigma_{1001}$ (equ.\\ \\eqref{basic}).  The puzzles and daggers will be explained shortly.  In the puzzles, $T$ is written instead of $10$ for convenience. \\lremind{keyexample}\n}\n\\label{keyexample}\n\\end{figure}\n\n\nConsider the locus of lines meeting both the fixed line $\\mathbb{P} F_2$\nand the moving line $\\mathbb{P} M_2$.  This is clearly a two-dimensional\ncycle (isomorphic to $\\mathbb{P} F_2 \\times \\mathbb{P} M_2 \\cong \\mathbb{P}^1 \\times\n\\mathbb{P}^1$).  After the first degeneration, the moving line has not\nmoved, so the locus remains the same.  In the second degeneration, the\nmoving line has degenerated to meet the fixed line at a point $\\mathbb{P}\nF_1$.  Then there are two irreducible families of lines meeting both\nthe fixed and moving lines.  First, the lines could contain the point\n$\\mathbb{P} F_1$ (depicted in the upper right panel).  Second, the lines\ncould lie in the plane spanned by $\\mathbb{P} F_2$ and $\\mathbb{P} M_2$ (the\ncenter left panel).  In the first case, we continue the degenerations,\nalways interpreting our locus as those lines containing the point\n$\\mathbb{P} F_1$.  At the end, we interpret this as the Schubert variety\n$\\overline{\\Omega}_{(2)}(M_{\\bullet}) = \\overline{\\Omega}_{(2)}(F_{\\bullet})$.\nAfter the next degeneration in the second case, the locus of lines in\nquestion can be restated as those lines contained in the plane $\\mathbb{P}\nM_3$; and this description continues until then end, where we\ninterpret this as the Schubert variety\n$\\overline{\\Omega}_{(1,1)}(M_{\\bullet}) = \\overline{\\Omega}_{(1,1)}(F_{\\bullet})$.\nWe have thus illustrated the identity\\lremind{basic}\n\\begin{equation}\n\\label{basic}\n\\sigma_{1} \\cdot\n\\sigma_{1} =\n\\sigma_{2} +\n\\sigma_{1,1} .\n\\end{equation}\n\n\n\\subsection{The degenerations in general}\n\\label{degenerationorder}\\lremind{degenerationorder}\nIn general, the degeneration corresponds to a path in the\nBruhat order of $S_n$.  The path corresponds to partial factorizations\nfrom the right of the longest word $w_0$:\n$$\nw_0 = e_{n-1} \\cdots e_2 e_1  \\quad \\cdots \\quad e_{n-1} e_{n-2} e_{n-3} \n\\quad e_{n-1} e_{n-2} \\quad e_{n-1}.\n$$\nWe denote the $\\binom n 2 + 1$ permutations by\n$d_0 = w_0$, \\dots, \n $d_{\\binom n 2 - 2} = e_{n-2} e_1$,\n$d_{\\binom n 2 - 1} = e_{n-1}$,\n$d_{\\binom n 2} = e$.\n\nAt each stage, we will consider cycles of the form\n$\\overline{\\Omega_{\\alpha} (M_{\\bullet}) \\cap\n  \\Omega_{\\beta}(F_{\\bullet})}$.  At the start, where $M_{\\bullet}$ and\n$F_{\\bullet}$ are transverse, we have the intersection of Schubert\ncycles that we seek to degenerate.  At the end, where $M_{\\bullet} =\nF_{\\bullet}$, we have Schubert cycles with respect to a single flag.\nAt each stage, this cycle $\\overline{\\Omega_{\\alpha} (M_{\\bullet}) \\cap\n  \\Omega_{\\beta}(F_{\\bullet})}$ (with $M_{\\bullet}$ related to\n$F_{\\bullet}$ by $d_i$, $i< \\binom n 2$) will degenerate to one or two\ncycles of the same form, except that $M_{\\bullet}$ and $F_{\\bullet}$\nare related by $d_{i+1}$.  These one or two cycles will each appear\nwith multiplicity $1$.\n\nIn \\cite{vakil:checkers}, these intermediate cycles are\ndescribed geometrically cleanly using {\\em checkerboards}.  For the\ngeneralizations we will consider, we will describe them here in terms\nof Knutson and Tao's {\\em puzzles} \\cite{knutson:puzzle}, which we\ndescribe now.  The bijection between checkerboard games and puzzles is\ngiven in \\cite[App.~ A]{vakil:checkers}.  \n\n{\\bf Littlewood-Richardson rule (puzzle version).}  The\nLittlewood-Richardson coefficient $c^{\\gamma}_{\\alpha, \\beta}$ may be\ncomputed as follows.  We write $\\alpha$, $\\beta$, and $\\gamma$ in terms of\nstrings of $n$ $0$'s and $1$'s, of which $k$ are $1$'s.  We write\nthe digits of $\\alpha$, $\\beta$, and $\\gamma$ along the sides of a lattice\nequilateral triangle of side length $n$, as shown in\nFigure~\\ref{puzzleD}. Then $c^{\\gamma}_{\\alpha, \\beta}$ is the number of ways\nof filling in this puzzle with unit triangular pieces of the sort\nshown in Figure~\\ref{puzzleD}, where two pieces can share an edge only\nif the edges have the same label.  \n\nExamples are shown in Figure~\\ref{keyexample}.  As an enlightening\nexercise, the reader may enjoy showing that\n$c^{101010}_{010101,010101}=2$.  (In \\cite{knutson:puzzle}, the third\npiece is different: as such a piece can only be glued along the $10$\nedge along another such piece, one may as well consider ``rhombi''\nconsisting of two such pieces glued along $10$-edges, see\nFig.~\\ref{rhombus}.  However, we shall see that these triangular\npieces are more suitable for generalization.  But the rhombi of\nKnutson and Tao lead to less cluttered diagrams, so we occasionally\nignore the $10$ edges in figures.)\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\setlength{\\unitlength}{0.00083333in}\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  \\selectfont}%\n\\fi\\endgroup%\n{\\renewcommand{\\dashlinestretch}{30}\n\\begin{picture}(3462,1333)(0,-10)\n\\put(3234,674){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$10$}}}\n\\path(2550,706)(2700,1006)(2850,706)(2550,706)\n\\path(75,556)(375,1156)\n\\blacken\\path(348.167,1035.252)(375.000,1156.000)(294.502,1062.085)(337.434,1080.868)(348.167,1035.252)\n\\path(825,1156)(1125,556)\n\\blacken\\path(1044.502,649.915)(1125.000,556.000)(1098.167,676.748)(1087.434,631.132)(1044.502,649.915)\n\\path(300,256)(900,256)\n\\blacken\\path(780.000,226.000)(900.000,256.000)(780.000,286.000)(816.000,256.000)(780.000,226.000)\n\\path(3150,706)(3300,1006)(3450,706)(3150,706)\n\\path(1950,706)(2100,1006)(2250,706)(1950,706)\n\\put(1050,931){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$\\beta$}}}\n\\put(0,931){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$\\alpha$}}}\n\\put(525,31){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$\\gamma$}}}\n\\put(2075,672){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(2756,841){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(2595,841){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(2001,836){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(3186,846){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(3344,842){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(2675,677){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(2150,835){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\path(150,406)(600,1306)(1050,406)(150,406)\n\\end{picture}\n}\n\n\\end{center}\n\\caption{The puzzles of Knutson and Tao (the pieces may be\n  rotated but not reflected).\\lremind{puzzleD}\n}\n\\label{puzzleD}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\setlength{\\unitlength}{0.00083333in}\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  \\selectfont}%\n\\fi\\endgroup%\n{\\renewcommand{\\dashlinestretch}{30}\n\\begin{picture}(535,436)(0,-10)\n\\path(162,380)(462,380)(312,80)\n\\path(162,380)(12,80)(312,80)\n\\put(52,189){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(351,187){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(126,31){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(255,337){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\end{picture}\n}\n\n\\end{center}\n\\caption{The original rhombus piece of Knutson and Tao (obtained by gluing\ntwo of the third piece of Fig.~\\ref{puzzleD} together).\nIt may be rotated but not reflected.\\lremind{rhombus}\n}\n\\label{rhombus}\n\\end{figure}\n\n\n\\subsection{The geometric Littlewood-Richardson rule in the guise of\npartially completed puzzles}\n\\lremind{pp}\\label{pp}The geometric Littlewood-Richardson rule may be\nunderstood in terms of partially completed puzzles.  The description\nhere is slightly cleaner than, although equivalent to, that of\n\\cite[\\S A.2]{vakil:checkers}.  To each term in the degeneration\norder, we associate the shape of a partially filled puzzle, as shown\nin Figure~\\ref{degenorder}.  Notice that there is at most one\nsouthwest-to-northeast edge in the middle of the puzzle; call this the\n{\\em leading edge}.  The cycles we will consider at step $d_i$ are as\nfollows: We label the northwestern, northern, and northeastern edges\nof the partially filled puzzle, where the left and right edges of the\nlarge triangle have labels $0$ or $1$, and the remaining northern\nborder edges are labeled $0$, $1$, or $10$.  To such a labeled\npartially filled puzzle, we define two strings of $0$'s and $1$'s (of\nwhich $k$ are $1$'s), denoted $\\alpha$ and $\\beta$ as follows.  For $\\alpha$,\nwe read off the list of $0$'s and $1$'s on the edges in the order\nshown in Figure~\\ref{readalpha}, where horizontal $10$ edges are\ninterpreted as $1$, and a $10$ leading edge is interpreted as a $0$.\nFor $\\beta$, we read off the list of $0$'s and $1$'s as shown in\nFigure~\\ref{readbeta}, where we first interpret the horizontal $10$\nedges as $0$'s, and then if the leading edge is $10$, we turn the\nfirst $1$ in $\\beta$ after the leading edge to $0$.\n\\begin{figure}[htbp]\n\\begin{center}\n\\setlength{\\unitlength}{0.00083333in}\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  \\selectfont}%\n\\fi\\endgroup%\n{\\renewcommand{\\dashlinestretch}{30}\n\\begin{picture}(4374,969)(0,-10)\n\\put(3862,593){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$10$}}}\n\\path(312,612)(912,612)\n\\blacken\\path(792.000,582.000)(912.000,612.000)(792.000,642.000)(828.000,612.000)(792.000,582.000)\n\\path(1062,912)(1662,912)\n\\blacken\\path(1542.000,882.000)(1662.000,912.000)(1542.000,942.000)(1578.000,912.000)(1542.000,882.000)\n\\path(12,12)(312,612)\n\\blacken\\path(285.167,491.252)(312.000,612.000)(231.502,518.085)(274.434,536.868)(285.167,491.252)\n\\path(912,612)(1062,912)\n\\blacken\\path(1035.167,791.252)(1062.000,912.000)(981.502,818.085)(1024.434,836.868)(1035.167,791.252)\n\\path(2862,12)(3012,312)(3312,312)\n\t(3462,612)(4062,612)(4362,12)(2862,12)\n\\put(3462,57){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$01001$}}}\n\\put(2637,312){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}e.g.}}}\n\\put(2906,143){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(3124,293){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(3338,443){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$10$}}}\n\\put(3580,585){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\path(12,12)(312,612)(912,612)\n\t(1062,912)(1662,912)(2112,12)(12,12)\n\\end{picture}\n}\n\n\\end{center}\n\\caption{Reading $\\alpha$ from a partially filled puzzle\n(horizontal $10$'s are read as $1$, leading edge $10$ is read as $0$).\nThe string  $\\alpha$ describes the $k$-plane's intersection with the moving flag $M_{\\bullet}$.\n\\lremind{readalpha}\n}\n\\label{readalpha}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\setlength{\\unitlength}{0.00083333in}\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  \\selectfont}%\n\\fi\\endgroup%\n{\\renewcommand{\\dashlinestretch}{30}\n\\begin{picture}(4374,969)(0,-10)\n\\put(4262,149){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\path(312,612)(912,612)\n\\blacken\\path(792.000,582.000)(912.000,612.000)(792.000,642.000)(828.000,612.000)(792.000,582.000)\n\\path(1062,912)(1662,912)\n\\blacken\\path(1542.000,882.000)(1662.000,912.000)(1542.000,942.000)(1578.000,912.000)(1542.000,882.000)\n\\path(1662,912)(2112,12)\n\\blacken\\path(2031.502,105.915)(2112.000,12.000)(2085.167,132.748)(2074.434,87.132)(2031.502,105.915)\n\\path(2862,12)(3012,312)(3312,312)\n\t(3462,612)(4062,612)(4362,12)(2862,12)\n\\put(3462,57){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$00011$}}}\n\\put(2637,312){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}e.g.}}}\n\\put(3113,288){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$10$}}}\n\\put(3344,443){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$10$}}}\n\\put(3588,586){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(3861,587){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(4118,430){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\path(12,12)(312,612)(912,612)\n\t(1062,912)(1662,912)(2112,12)(12,12)\n\\end{picture}\n}\n\n\\end{center}\n\\caption{Reading $\\beta$ from a partially filled puzzle\n($10$'s are read as $0$, and leading edge $10$ turns the first $1$ afterward to $0$).  The string $\\beta$ describes the $k$-plane's intersection with the fixed flag $F_{\\bullet}$.\\lremind{readbeta}}\n\\label{readbeta}\n\\end{figure}\n\nNote  that $\\alpha$ and $\\beta$ are not independent.\nThis is clearest after the last step, where the above description \nensures that $\\alpha=\\beta$.\n\nThe reader may prefer the following equivalent description of $\\alpha$\nand $\\beta$, depicted in Figure~\\ref{newab}.  To each horizontal edge\n$10$, temporarily glue on a copy of the last triangle of\nFigure~\\ref{puzzleD} (with sides $0\/1\/10$ read clockwise), so now all\nexposed edges are $0$'s and $1$'s, except possibly the leading edge.\nThen $\\alpha$ is the string visible looking northwest form the bottom\nedge of the puzzle, except a leading edge $01$ is read as $0$; and\n$\\beta$ is the string visible looking northeast from the bottom edge of\nthe puzzle, except that a leading edge $01$ turns the next $1$ into\n$0$.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\setlength{\\unitlength}{0.00083333in}\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  \\selectfont}%\n\\fi\\endgroup%\n{\\renewcommand{\\dashlinestretch}{30}\n\\begin{picture}(3924,2008)(0,-10)\n\\put(3807,609){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\path(912,1081)(1062,781)(1212,1081)\n\\blacken\\path(231.502,274.915)(312.000,181.000)(285.167,301.748)(274.434,256.132)(231.502,274.915)\n\\dashline{60.000}(312,181)(87,631)\n\\blacken\\path(531.502,274.915)(612.000,181.000)(585.167,301.748)(574.434,256.132)(531.502,274.915)\n\\dashline{60.000}(612,181)(312,781)\n\\blacken\\path(831.502,274.915)(912.000,181.000)(885.167,301.748)(874.434,256.132)(831.502,274.915)\n\\dashline{60.000}(912,181)(537,931)\n\\blacken\\path(1131.502,274.915)(1212.000,181.000)(1185.167,301.748)(1174.434,256.132)(1131.502,274.915)\n\\dashline{60.000}(1212,181)(762,1081)\n\\blacken\\path(1431.502,274.915)(1512.000,181.000)(1485.167,301.748)(1474.434,256.132)(1431.502,274.915)\n\\dashline{60.000}(1512,181)(1137,931)\n\\blacken\\path(3638.833,301.748)(3612.000,181.000)(3692.498,274.915)(3649.566,256.132)(3638.833,301.748)\n\\dashline{60.000}(3612,181)(3837,631)\n\\blacken\\path(3338.833,301.748)(3312.000,181.000)(3392.498,274.915)(3349.566,256.132)(3338.833,301.748)\n\\dashline{60.000}(3312,181)(3687,931)\n\\blacken\\path(3038.833,301.748)(3012.000,181.000)(3092.498,274.915)(3049.566,256.132)(3038.833,301.748)\n\\dashline{60.000}(3012,181)(3462,1081)\n\\blacken\\path(2738.833,301.748)(2712.000,181.000)(2792.498,274.915)(2749.566,256.132)(2738.833,301.748)\n\\dashline{60.000}(2712,181)(3162,1081)\n\\blacken\\path(2438.833,301.748)(2412.000,181.000)(2492.498,274.915)(2449.566,256.132)(2438.833,301.748)\n\\dashline{60.000}(2412,181)(2637,631)\n\\path(2562,781)(2712,481)(2862,781)\n\\dashline{60.000}(2937,931)(3012,181)\n\\blacken\\path(2970.208,297.419)(3012.000,181.000)(3029.911,303.390)(3003.642,264.583)(2970.208,297.419)\n\\path(162,781)(762,1981)(1212,1081)\n\\path(2562,781)(3162,1981)(3612,1081)\n\\path(2412,481)(2562,781)(2862,781)\n\t(3012,1081)(3612,1081)(3912,481)(2412,481)\n\\put(312,31){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(612,31){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(912,31){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(1212,31){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(1512,31){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(2337,31){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(2637,31){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(2937,31){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(3237,31){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(3537,31){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(686,1381){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$\\alpha$}}}\n\\put(3085,1381){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$\\beta$}}}\n\\put(63,620){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(289,756){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(497,908){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$10$}}}\n\\put(734,1056){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(954,915){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(1109,908){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(1016,1058){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$10$}}}\n\\put(2620,613){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(2753,620){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(2671,753){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$10$}}}\n\\put(2884,897){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$10$}}}\n\\put(3132,1052){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(3427,1053){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(3664,908){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\path(12,481)(162,781)(462,781)\n\t(612,1081)(1212,1081)(1512,481)(12,481)\n\\end{picture}\n}\n\n\\end{center}\n\\caption{Alternate (equivalent) description of how to read $\\alpha$ and $\\beta$ (cf.\\ Figures~\\ref{readalpha} and~\\ref{readbeta}).\\lremind{newab}}\n\\label{newab}\n\\end{figure}\n\n\nTo such a partially filled puzzle, we associate a subvariety\nof the Grassmannian as follows.\n(This construction will be central to the\ngeneralizations in every section of Part~\\ref{partone}!)\n  We fix two flags $F_{\\bullet}$\nand $M_{\\bullet}$ in the relative position determined by the partially\ncompleted puzzle, and we consider \\lremind{puzzlevariety}\n\\begin{equation}\\label{puzzlevariety}\n\\overline{ \\Omega_{\\alpha}(M_{\\bullet}) \\cap \\Omega_{\\beta}(F_{\\bullet})}.\n\\end{equation}\nCall such a subvariety of the Grassmannian a {\\em puzzle variety}.\nThe reader is\nstrongly encouraged to see this definition in practice by examining a\ncouple of panels of Figure~\\ref{keyexample}.\n\nThen the geometric Littlewood-Richardson rule may now be described\nquickly as follows.  Suppose we have a partially filled puzzle,\ncorresponding to two flags $F_{\\bullet}$ and $M_{\\bullet}$ in given relative\nposition, and a puzzle variety (those $k$-planes meeting the two\nreference flags as described in \\eqref{puzzlevariety}).  Then if the\nmoving flag $M_{\\bullet}$ is degenerated (to the next step in the\ndegeneration order, Figure~\\ref{degenorder}), the cycle in the\nGrassmannian degenerates to all possible ways of adding more\ntriangular pieces to the puzzle to create a ``next-larger'' partially\ncompleted puzzle.  (Note that the edges on the bottom of the puzzle\nmust be labeled $0$ or $1$, and may not be labeled $10$.)\n\nThis involves adding two adjacent triangles (forming a rhombus), and\npossibly a third triangle to ``complete a row''.  There is no choice\nfor the third triangle: if we know two sides of a puzzle piece, then\nwe know the third.  The reader may readily verify that there are nine\npossible choices for the pair of pieces, shown suggestively in\nFigure~\\ref{pairs}.  Once the northern edge and the northwestern edge\nare specified, there is usually one choice for the rhombus, and there\nmay be two.  For those familiar with the checkerboard description of\nthis rule, they correspond to the ``stay'' and ``swap'' options\nrespectively.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\setlength{\\unitlength}{0.00083333in}\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  \\selectfont}%\n\\fi\\endgroup%\n{\\renewcommand{\\dashlinestretch}{30}\n\\begin{picture}(3250,1647)(0,-10)\n\\put(1634,31){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\path(312,77)(462,377)(762,377)\n\t(612,77)(312,77)\n\\path(1512,1277)(1662,1577)(1962,1577)\n\t(1812,1277)(1512,1277)\n\\path(1512,677)(1662,977)(1962,977)\n\t(1812,677)(1512,677)\n\\path(1512,77)(1662,377)(1962,377)\n\t(1812,77)(1512,77)\n\\path(2712,677)(2862,977)(3162,977)\n\t(3012,677)(2712,677)\n\\path(2712,1277)(2862,1577)(3162,1577)\n\t(3012,1277)(2712,1277)\n\\path(462,1577)(612,1277)\n\\drawline(20,902)(20,902)\n\\path(312,1277)(462,1577)(762,1577)\n\t(612,1277)(312,1277)\n\\path(1662,1577)(1812,1277)\n\\path(2862,977)(3012,677)\n\\path(1812,677)(1662,977)\n\\path(1662,377)(1812,77)\n\\path(612,77)(462,377)\n\\path(312,677)(162,977)\n\\path(3012,1277)(2862,1577)\n\\path(612,677)(762,977)(1062,977)\n\t(912,677)(612,677)\n\\path(912,677)(762,977)\n\\put(430,1233){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(506,1390){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(662,1390){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(348,1383){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(576,1548){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(1793,1537){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(1546,1383){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(1686,1391){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$10$}}}\n\\put(1858,1393){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(1644,1234){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(2759,1396){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(2907,1397){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(3066,1393){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(2964,1537){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$10$}}}\n\\put(2821,1233){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(287,939){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(49,793){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(207,790){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(357,790){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(115,639){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$10$}}}\n\\put(649,788){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(737,639){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(823,788){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(941,798){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$10$}}}\n\\put(887,935){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(1795,938){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(1559,788){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(1704,789){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(1857,785){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(1639,635){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(2964,939){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$10$}}}\n\\put(2753,788){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(2907,788){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(3062,789){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(2805,631){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$10$}}}\n\\put(335,195){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$10$}}}\n\\put(501,195){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(651,200){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$10$}}}\n\\put(583,333){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(431,35){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(1790,348){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(1867,195){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(1704,195){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(1529,189){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$10$}}}\n\\path(12,677)(162,977)(462,977)\n\t(312,677)(12,677)\n\\end{picture}\n}\n\\end{center}\n\\caption{The puzzle interpretation of the geometric Littlewood-Richardson rule.\\lremind{pairs}}\n\\label{pairs}\n\\end{figure}\n\n\\subsection{Connection to tableaux}\nThe Littlewood-Richardson rule is \ntraditionally\ngiven in terms of tableaux.\nThe bijection to tableaux is straightforward:  whenever the\n$k$-plane changes its relationship to the moving flag $M_{\\bullet}$ (e.g.\\ the\nmoves marked $\\dagger$ in Figure~\\ref{keyexample}),\na number is put into the tableau.  See \\cite[\\S 3.1]{vakil:checkers}\nfor the precise statement of which number is placed, and in which row.\nAn elegant  bijection from puzzles to tableaux, due to Terry Tao, is\ngiven in \\cite[Fig.~11]{vakil:checkers}; whenever the puzzle piece(s)\nof Figure~\\ref{puztab} appear in the puzzle, a number is placed\nin the tableau.\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\setlength{\\unitlength}{0.00083333in}\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  \\selectfont}%\n\\fi\\endgroup%\n{\\renewcommand{\\dashlinestretch}{30}\n\\begin{picture}(546,434)(0,-10)\n\\put(284,335){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\path(162,371)(312,71)\n\\put(147,186){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$10$}}}\n\\path(12,71)(162,371)(462,371)\n\t(312,71)(12,71)\n\\put(362,186){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(53,183){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(135,31){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\end{picture}\n}\n\n\\end{center}\n\\caption{These puzzle pieces (in this orientation!) correspond to the entries in the tableau\nLittlewood-Richardson rule.\\lremind{puztab}}\n\\label{puztab}\n\\end{figure}\n\nWhile discussing tableaux, we should mention K.~Purbhoo's beautiful\n``mosaics'', with which he not only bijects puzzles and tableaux, but\nproves the commutativity and associativity of each \\cite{purbhoo}.\n\n\n\\section{The $K$-theory (or Grothendieck ring) of the Grassmannian}\n\\lremind{K}\\label{K}The Grothendieck groups or $K$-theory of the\nGrassmannian is generated by the classes of the structure sheaves of\nthe Schubert cells.  Buch gave a Littlewood-Richardson rule in\n$K$-theory in \\cite{buch:K}.\nBuch's rule states that $K$-theory Littlewood-Richardson coefficients\nenumerate certain ``set-valued tableaux''.  There is also a sign: the\nsign of $c^{\\gamma}_{\\alpha, \\beta}$ is $(-1)^{\\operatorname{codim} \\Omega_{\\gamma} - \\operatorname{codim}\n  \\Omega_{\\alpha} - \\operatorname{codim} \\Omega_{\\beta} }$, depending on the difference of the\ncodimension of $\\gamma$ from its ``expected codimension''.\n\nAs an aside, we note that Buch conjectured positivity --- with this\nsign convention --- in the Grothendieck ring of a flag variety in\ngeneral.  A beautiful proof was given by M.~Brion in\n\\cite{brion:positivity}.\n\nBuch's generalization of the tableau rule to\n$K$-theory yields a generalization to $K$-theory of the geometric\nchecker rule \\cite[Thm.~3.4]{vakil:checkers} and the puzzle rule\n\\cite[Thm.~3.6]{vakil:checkers}.  The puzzle rule is as follows: there\nis a new puzzle piece, shown in Figure~\\ref{puzzleK}, and the\n$K$-theory Littlewood-Richardson coefficient is the number of puzzles\ncompleted with the usual pieces and the $K$-theory piece, where each\npuzzle is counted with sign corresponding to the number of $K$-theory\npieces.  As an example, the reader can verify that in $K$-theory,\n\\lremind{basicK}\n\\begin{equation}\n\\label{basicK}\n\\sigma_{0101}^2 = \\sigma_{0110} + \\sigma_{1001} - \\sigma_{1010}.\n\\end{equation}\nin string notation, extending \\eqref{basic}.\n\\begin{figure}[htbp]\n\\begin{center}\n\\setlength{\\unitlength}{0.00083333in}\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  \\selectfont}%\n\\fi\\endgroup%\n{\\renewcommand{\\dashlinestretch}{30}\n\\begin{picture}(683,671)(0,-10)\n\\put(360,121){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\path(162,312)(312,612)(462,312)(162,312)\n\\put(133,572){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(437,569){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(499,412){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(63,417){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(203,119){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\path(12,612)(612,612)(312,12)(12,612)\n\\end{picture}\n}\n\n\\end{center}\n\\caption{The new $K$-theory puzzle piece.  It may {\\em not} be rotated.\n(The central edges might be plausibly labeled $10$, although this\nhas no mathematical effect.)\\lremind{puzzleK}}\n\\label{puzzleK}\n\\end{figure}\n\nThe puzzle statement is quite clean.  The checker statement (not given \nhere) suggests a geometric conjecture \\cite[Conj.~3.5]{vakil:checkers}\n(by the second author with Buch, with additional comments by\nKnutson), which is also quite clean.  We will now do something\nseemingly perverse: give a {\\em less clean} puzzle statement.  The advantage\nis that the puzzle rule will have an interpretation as a {\\em\n  conjectural $K$-theoretic geometric Littlewood-Richardson rule.}\n(When we extend our discussion to equivariant cohomology, \\S \\ref{HT},\nand equivariant $K$-theory, \\S \\ref{KT}, the checker description will\nbreak down, but the geometric puzzle description will still work.)\n\nThe alternate puzzle pieces are shown in Figure~\\ref{geopuzzleK}.\nNote that we have a new flavor of edge (labeled $10K$), which may only\nbe oriented southwest\/northeast.  The interested reader will readily verify\nthat the puzzles with the alternate $K$-puzzle pieces of\nFigure~\\ref{geopuzzleK} are in straightforward bijection with the\npuzzles using the original $K$-piece of Figure~\\ref{puzzleK}.  \n\n\\begin{figure}[htbp]\n\\begin{center}\n\\setlength{\\unitlength}{0.00083333in}\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  \\selectfont}%\n\\fi\\endgroup%\n{\\renewcommand{\\dashlinestretch}{30}\n\\begin{picture}(2348,432)(0,-10)\n\\put(1259,185){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$10K$}}}\n\\path(912,71)(1062,371)(1362,371)\n\t(1212,71)(912,71)\n\\path(1812,71)(1962,371)(2262,371)\n\t(2112,71)(1812,71)\n\\path(162,371)(312,71)\n\\path(1062,371)(1212,71)\n\\path(1962,371)(2112,71)\n\\put(284,333){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(205,188){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(150,31){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(1035,31){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(1900,33){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$10$}}}\n\\put(2164,181){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(1986,181){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$10$}}}\n\\put(1693,181){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$10K$}}}\n\\put(1186,328){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(2079,329){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(57,182){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(358,189){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$10K$}}}\n\\put(797,180){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$10K$}}}\n\\put(1119,180){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\path(12,71)(162,371)(462,371)\n\t(312,71)(12,71)\n\\end{picture}\n}\n\\end{center}\n\\caption{Alternate $K$-theory puzzle pieces.  They may {\\em not} be rotated.\nThe first piece contributes a sign of $-1$.  The labels on the internal\nedges don't matter.\\lremind{geopuzzleK}}\n\\label{geopuzzleK}\n\\end{figure}\n\n\nCorresponding to a partially-completed $K$-puzzle, there is a {\\em\n  $K$-theory puzzle variety}, which we describe by defining $\\alpha$ and\n$\\beta$ and then using the puzzle variety definition\nof \\eqref{puzzlevariety}.  To define $\\beta$, use the same recipe as before\n(Fig.\\ ~\\ref{readbeta}), treating $10K$ as $10$.  To define $\\alpha$\n(Fig.\\ ~\\ref{readalpha}), treat $10K$ as $10$, except the first $1$\nafter that position is exchanged with the $0$ immediately to its left.\n\nOnce again, when completing a puzzle from top to bottom, there is\nusually only one choice at each stage.  However, where there used to\nbe two choices, there are now three.  This leads to a conjectural\ngeometric interpretation of Buch's rule.  Recall that if a variety\ndegenerates into two pieces, say $X$ degenerates into $X_1 \\cup X_2$,\nthen in homology, $[X] = [X_1] + [X_2]$, but in $K$-theory\n$$[\\mathcal{O}_X] = [\\mathcal{O}_{X_1}] + [\\mathcal{O}_{X_2}] - [\\mathcal{O}_{X_1 \\cap X_2}],$$\nwhere the intersection is the scheme-theoretic intersection.\nThen we have a {\\em $K$-theoretic geometric Littlewood-Richardson\n  rule}: when the puzzle variety breaks in the geometric\nLittlewood-Richardson rule (the center left panel of\nFig.~\\ref{pairs}), into $X_1$ and $X_2$, say, and $X_3$ is the puzzle\nvariety corresponding to the third (new, $K$-theoretic) option,\nthen:\\lremind{Kconj}\n\n\\begin{conjecture}\n  $X_3$ is a $GL(n)$-translate of the scheme-theoretic intersection $X_1 \\cap\n  X_2$.\n\\label{Kconj}\n\\end{conjecture}\n\nThis is straightforward to check set-theoretically.\nThe reader may verify that Conjecture~\\ref{Kconj} is equivalent to\nthe $K$-theoretic conjecture \\cite[Conj.~3.5]{vakil:checkers}, and hence implies Buch's combinatorial\n$K$-theoretic Littlewood-Richardson rule.\n\nThe reader may prefer to instead examine the example of\nFigure~\\ref{keyexample}.  In the single instance when the variety\nbreaks into two pieces (the second panel in the top row breaks into\nthe union of the third panel in the top row and the first panel in the\nsecond row), the scheme-theoretic intersection corresponds to the\nlocus of lines passing through the fixed point $\\mathbb{P} F_1$ (the\ncondition of the second panel in the top row), and lying in the plane\nspanned by the moving line $\\mathbb{P} M_1$ and the fixed line $\\mathbb{P} F_1$\n(the condition of the first panel of the second row).  This is clearly\na translate of the locus of lines passing through the fixed point\n$\\mathbb{P} F_1$ and lying in the moving plane $\\mathbb{P} M_3$, which is the\npuzzle variety predicted by Conjecture~\\ref{Kconj}.\n\n\n\\section{The equivariant cohomology of the Grassmannian}\n\\lremind{HT}\\label{HT}Suppose $T$ is the natural $n$-dimensional torus\nacting on $\\mathbb{C}^n$.  Choose an order of the $T$-fixed basis $\\vec{v}_1$,\n\\dots, $\\vec{v}_n$, and let the $T$-equivariant cohomology of a point be\n$\\mathbb{Z}[y_1, \\dots, y_n]$, where $y_i$ corresponds to $\\vec{v}_i$.  We next\ngeneralize our geometric construction, at least conjecturally, to\n$T$-equivariant cohomology.  The equivariant Schubert classes are\ndefined as equivariant cohomology classes corresponding to Schubert\nvarieties with respect to the fixed flag\n$$\nF_{\\bullet} = \\{ \\{ 0 \\} \\subset \\langle \\vec{v}_1 \\rangle \\subset \\langle \\vec{v}_1, \\vec{v}_2\n\\rangle \\subset \\cdots \\subset \\langle \\vec{v}_1, \\vec{v}_2, \\dots, \\vec{v}_n\n\\rangle = \\mathbb{C}^n \\} .$$ A beautiful argument showing positivity in\nequivariant cohomology was given  by W.~Graham in \\cite{graham:equivariant},\nconfirming a conjecture of D.~Peterson.\n\nKnutson and Tao gave an elegant equivariant Littlewood-Richardson rule\nin terms of their puzzles.  Equivariant puzzles have the same pieces\nas puzzles for ordinary cohomology (Fig.~\\ref{puzzleD}), plus an additional\npiece, shown in Figure~\\ref{eqpuzzle}.  It may {\\em not}\nbe rotated or reflected.  Each puzzle appears with a certain weight,\nwhich is a product of contributions from the equivariant pieces in the\npuzzle.  The contribution of the piece shown in Figure~\\ref{eqweight},\nwhich projects to position $i$ in the southwest direction and to\nposition $j$ in the southeast direction, is $y_j-y_i$. \n\n\\begin{figure}[htbp]\n\\begin{center}\n\\setlength{\\unitlength}{0.00083333in}\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  \\selectfont}%\n\\fi\\endgroup%\n{\\renewcommand{\\dashlinestretch}{30}\n\\begin{picture}(405,639)(0,-10)\n\\texture{88555555 55000000 555555 55000000 555555 55000000 555555 55000000 \n\t555555 55000000 555555 55000000 555555 55000000 555555 55000000 \n\t555555 55000000 555555 55000000 555555 55000000 555555 55000000 \n\t555555 55000000 555555 55000000 555555 55000000 555555 55000000 }\n\\shade\\path(12,312)(162,612)(312,312)\n\t(162,12)(12,312)\n\\path(12,312)(162,612)(312,312)\n\t(162,12)(12,312)\n\\put(200,127){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(55,125){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(221,406){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(59,405){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\end{picture}\n}\n\n\\end{center}\n\\caption{The new equivariant puzzle piece  of Knutson and Tao.\nIt may {\\em not} be\n  rotated or reflected).\\lremind{eqpuzzle}}\n\\label{eqpuzzle}\n\\end{figure}\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\setlength{\\unitlength}{0.00083333in}\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  \\selectfont}%\n\\fi\\endgroup%\n{\\renewcommand{\\dashlinestretch}{30}\n\\begin{picture}(1688,1179)(0,-10)\n\\put(1212,627){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$y_j-y_i$}}}\n\\path(537,1152)(1062,102)(12,102)(537,1152)\n\\path(612,552)(387,102)\n\\blacken\\path(413.833,222.748)(387.000,102.000)(467.498,195.915)(424.566,177.132)(413.833,222.748)\n\\path(612,552)(837,102)\n\\blacken\\path(756.502,195.915)(837.000,102.000)(810.167,222.748)(799.434,177.132)(756.502,195.915)\n\\put(87,27){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(387,27){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$i$}}}\n\\put(837,27){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$j$}}}\n\\put(987,27){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$n$}}}\n\\texture{88555555 55000000 555555 55000000 555555 55000000 555555 55000000 \n\t555555 55000000 555555 55000000 555555 55000000 555555 55000000 \n\t555555 55000000 555555 55000000 555555 55000000 555555 55000000 \n\t555555 55000000 555555 55000000 555555 55000000 555555 55000000 }\n\\shade\\path(537,552)(612,702)(687,552)\n\t(612,402)(537,552)\n\\path(537,552)(612,702)(687,552)\n\t(612,402)(537,552)\n\\end{picture}\n}\n\n\\end{center}\n\\caption{The weight contributed by an equivariant puzzle piece.  See\n  the second panel of Figure~\\ref{enex} for a specific\n  example.\\lremind{eqweight}}\n\\label{eqweight}\n\\end{figure}\n\nA specific example is given in Figure~\\ref{enex}, discussed at length\nin \\S \\ref{eqex}.  This example corresponds to two Schubert condition\ncorresponding to those points lying on the fixed line $\\mathbb{P} \\langle\n\\vec{v}_1, \\vec{v}_2 \\rangle$.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\setlength{\\unitlength}{0.00083333in}\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  \\selectfont}%\n\\fi\\endgroup%\n{\\renewcommand{\\dashlinestretch}{30}\n\\begin{picture}(2788,1017)(0,-10)\n\\put(135,48){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\path(12,90)(462,990)(912,90)(12,90)\n\\path(162,390)(762,390)\n\\path(462,390)(312,90)\n\\path(1812,90)(2262,990)(2712,90)(1812,90)\n\\path(612,90)(762,390)\n\\path(1962,390)(2112,90)(2412,690)\n\t(2112,690)(2412,90)(2562,390)\n\\texture{88555555 55000000 555555 55000000 555555 55000000 555555 55000000 \n\t555555 55000000 555555 55000000 555555 55000000 555555 55000000 \n\t555555 55000000 555555 55000000 555555 55000000 555555 55000000 \n\t555555 55000000 555555 55000000 555555 55000000 555555 55000000 }\n\\shade\\path(2262,390)(2412,690)(2562,390)\n\t(2412,90)(2262,390)\n\\path(2262,390)(2412,690)(2562,390)\n\t(2412,90)(2262,390)\n\\put(491,809){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(238,350){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(546,354){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(181,493){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(346,495){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(439,648){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(641,510){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(808,205){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(640,200){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(731,43){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(2604,201){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(2450,205){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(2305,203){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(2230,43){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(2530,39){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(1939,31){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(1850,189){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(2005,187){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(2163,191){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(1990,471){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(2164,465){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(2471,484){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(2309,483){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(2235,643){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(2167,813){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(2304,811){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(357,199){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(65,210){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(496,200){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(357,806){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(434,47){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\path(612,690)(312,690)(612,90)\n\\end{picture}\n}\n\n\\end{center}\n\\caption{Example of the two equivariant puzzles associated to a single\n  equivariant Schubert problem $\\sigma_{010} \\cdot \\sigma_{010}$.  The first\n  already appears in ordinary cohomology.  The second appears with\n  weight $y_3-y_2$ (notice the equivariant rhombus in the southeast of\n  the figure).  The $10$ and $01$ edges are omitted for the sake of\n  clarity. This verifies equ.\\ \\eqref{basicHT}:  $\\sigma_{010}^2 = \\sigma_{100} + (y_3-y_2) \\sigma_{010}$. \\lremind{enex}}\n\\label{enex}\n\\end{figure}\n\nWe now give a conjectural geometric interpretation to this\ncombinatorial rule, due to Knutson and the second author, extending\nthe geometric Littlewood-Richardson rule in ordinary cohomology.  The\ntorus action ``prevents'' the equivariant cycle from degenerating (the\ndegeneration of \\S \\ref{hhh} is not equivariant), so we cannot use our\ndegeneration interpretation.  As in the example of Figure~\\ref{enex},\nthese Schubert problems are excess intersection problems: the Schubert\ncycles $\\overline{\\Omega}_{\\alpha}(F_{\\bullet})$ and\n$\\overline{\\Omega}_{\\beta}(F_{\\bullet})$ in $G(k,n)$ do not intersect\nproperly (in the ``expected'' dimension).  So we use instead the\nfollowing trick.\n\nThe degeneration order may be interpreted in the flag variety $Fl(n)$\nparametrizing the moving flag's relative position to the fixed flag.\nIt corresponds to a sequence of nested Schubert varieties in $Fl(n)$,\neach a Cartier divisor on the previous one, where the first element is\nall of $Fl(n)$, and the last is a point.  (This Cartier requirement is\na strong constraint on the degeneration order, equivalent to the fact\nthat each Schubert variety in the sequence  is smooth.)  We consider subvarieties of\n$G(k,n) \\times Fl(n)$, where $Fl(n)$ parametrizes the ``moving'' flag\n$M_{\\bullet}$, of the form \\lremind{eqPV}\n\\begin{equation}\n\\pi_1^* \\left( \\overline{ \\Omega_{\\alpha}(M_{\\bullet}) \\cap\n    \\Omega_{\\beta}(F_{\\bullet}) } \\right) \\cap \\pi_2^*\n\\overline{\\Omega}_{d_i}(F_{\\bullet})\\label{eqPV}\n\\end{equation}\nwhere $d_i$ is in the degeneration\norder (\\S \\ref{degenerationorder}).  In other words, $M_{\\bullet}$ is required to be in given relative\nposition to $F_{\\bullet}$ (or in a more degenerate position), and the\n$k$-plane is required to be in given relative position to both\n$M_{\\bullet}$ and $F_{\\bullet}$.  We will see that $\\alpha$ and $\\beta$ may force\nthe fixed and moving flags to be in more degenerate position than that\nrequired by $d_i$; an explicit example will be given in \\S \\ref{eqex}.\n\nCall such a variety an {\\em equivariant puzzle variety}.  Note that\nthe fiber of the equivariant puzzle variety \\eqref{eqPV} over a point\nof the (open) Schubert cell $\\Omega_{d_i}(F_{\\bullet})$ is a puzzle\nvariety \\eqref{puzzlevariety}.  We will shortly associate such a\nvariety to a partially completed equivariant puzzle.  Note that if\n$d_i$ is the final step in the degeneration order ($i= \\binom n 2$),\nthe equivariant puzzle variety is a Schubert variety with respect to\nthe fixed flag $F_\\bullet$.\n\nNow pull back (to a given equivariant puzzle variety) the divisor\n$\\overline{\\Omega}_{d_{i+1}}(F_{\\bullet})$ on $\\overline{\\Omega}_{d_i}(F_{\\bullet})$ corresponding\nto the next Schubert variety in the degeneration order.  We conjecture\nthat either (i) this divisor does not contain our equivariant puzzle\nvariety (i.e.\\ it pulls back to a Cartier divisor on the equivariant\npuzzle variety), in which case it is (scheme-theoretically) the union\nof one or two other equivariant puzzle varieties (corresponding to\n$d_{i+1}$), or (ii) it contains the equivariant puzzle variety.  In\nthe latter case, the excess intersection problem (intersection with\nthe zero-section of a line bundle) is simplest sort of excess\nintersection: we obtain a contribution of the equivariant first Chern\nclass of the line bundle, which is pure weight.  (This is where the\nCartier hypothesis is used.)\n\nWe now make this precise, and conclude with an example, which will\nperhaps be most helpful to the reader.\n\nHere is how to associate an equivariant puzzle variety to a partially\ncompleted equivariant puzzle.  In order to parallel our earlier\ndescription, we replace Knutson and Tao's piece of\nFigure~\\ref{eqpuzzle} with the two halves of Figure~\\ref{eqpuzzletwo}.\n(This is clearly a trivial variation of Fig.~\\ref{eqpuzzle}.)  To each\npartially completed puzzle (as in Fig.~\\ref{degenorder}), we define\n$\\alpha$ and $\\beta$ as follows.  The recipe is the same as in\nFigures~\\ref{readalpha} and ~\\ref{readbeta}, with the additional fact\nthat $01$ is read as $0$ for $\\alpha$ and $1$ for $\\beta$.\n(This may also be interpreted in the same way as Figure~\\ref{newab}.)  Then the\nconjectural equivariant geometric Littlewood-Richardson rule is as\nfollows.  To compute the product of two equivariant Schubert classes,\nwe fill a puzzle in the degeneration order  with equivariant puzzle pieces.  For each move,\nwe fill in the next two or three puzzle pieces.  There will always be\none or two choices.  If we place a copy of the second piece in\nFigure~\\ref{eqpuzzletwo} (the bottom half of Knutson-Tao's\npiece), there is one choice, and this will correspond to excess\nintersection.  The weight is as given in Figure~\\ref{eqweight}.\nOtherwise, there is no excess intersection, and the Cartier divisor\nwill be the scheme-theoretic union of the equivariant puzzle varieties\ncorresponding to the one or two choices.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\setlength{\\unitlength}{0.00083333in}\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  \\selectfont}%\n\\fi\\endgroup%\n{\\renewcommand{\\dashlinestretch}{30}\n\\begin{picture}(1280,435)(0,-10)\n\\texture{88555555 55000000 555555 55000000 555555 55000000 555555 55000000 \n\t555555 55000000 555555 55000000 555555 55000000 555555 55000000 \n\t555555 55000000 555555 55000000 555555 55000000 555555 55000000 \n\t555555 55000000 555555 55000000 555555 55000000 555555 55000000 }\n\\shade\\path(1062,74)(912,374)(1212,374)(1062,74)\n\\path(1062,74)(912,374)(1212,374)(1062,74)\n\\shade\\path(12,74)(162,374)(312,74)(12,74)\n\\path(12,74)(162,374)(312,74)(12,74)\n\\put(209,180){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(117,31){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$01$}}}\n\\put(959,181){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(1096,185){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(1012,336){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$01$}}}\n\\put(52,180){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\end{picture}\n}\n\n\\end{center}\n\\caption{An alternate equivariant puzzle piece.  It may appear only in\n  the orientations shown here; it may {\\em not} be further rotated or\n  reflected.  This is a trivial variation of Figure~\\ref{eqpuzzle}.\n\\lremind{eqpuzzletwo}}\n\\label{eqpuzzletwo}\n\\end{figure}\n\nThus to compute a product of two equivariant Schubert classes, we\nstart with an empty puzzle with labeled northwest and northeast edges,\nand complete it.  The partially completed puzzles we see en route\ncompletely describe the geometry of the successive Cartier slicing.\nNote that this rule is manifestly positive (as it is simply an\ninterpretation of the puzzle rule).  It is also clearly a\ngeneralization of the geometric Littlewood-Richardson rule in ordinary\ncohomology.  It is well-checked (for example for $n$ up to $5$, and\nfor many cases for larger $n$).\n\n\\subsection{A worked equivariant example}\n\\lremind{eqex}\\label{eqex}We conclude our equivariant discussion with\na worked example, which we hope will illustrate what is happening\ngeometrically.  Consider the problem of Figure~\\ref{enex},\ncorresponding to intersecting the locus of points meeting the fixed\nline with itself.  We will verify that\\lremind{basicHT}\n\\begin{equation}\\label{basicHT}\n\\sigma_{010}^2 = \\sigma_{100} + (y_3-y_2) \\sigma_{010}.\n\\end{equation}\nIn the Grassmannian $G(1,3)$, also known as $\\mathbb{P}^2$, this is the\nintersection of a codimension $1$ class with itself, so we expect a\ndimension $0$ answer.  In ordinary cohomology (which we recover by\nsetting the equivariant parameters to zero), we expect to see the\nclass of a point (in ordinary cohomology, lines can deform, and the\nintersection of two general lines is one point).\n\nFollowing the recipe, instead of working in $\\mathbb{P}^2$, we work in\n$\\mathbb{P}^2 \\times Fl(3) \\cong \\mathbb{P}^2 \\times \\mathbb{F} l (2)$, parametrizing\nthe moving line, the moving point (on the moving line), and the point\n$p$ of $\\mathbb{P}^2$.  On this fivefold, we consider the threefold\ncorresponding to requiring the point $p$ to lie on both the moving\nline $\\mathbb{P} M_2$ and on the fixed line $\\mathbb{P} F_2$.  The generic such\nconfiguration is depicted on the left side of Figure~\\ref{enexg}.\n\nWe now slice with our first divisor $D$, corresponding to requiring\nthe moving line $\\mathbb{P} M_2$ to pass through the fixed point $\\mathbb{P}\nF_1$.  This Cartier divisor (a fourfold) has two components $D_1$\nand $D_2$.\n\nThe first component $D_1$ is the geometrically clear one: the moving line\n$\\mathbb{P} M_2$ rotates around the fixed point $\\mathbb{P} F_1$ (yielding one\ndimension of moduli), $p$ is the fixed point $\\mathbb{P} F_1$, and the\nmoving point varies (yielding a second dimension of moduli).  This\ncorresponds to (the top two rows of) the first puzzle of\nFigure~\\ref{enex}.  The generic such configuration is depicted on the\ntop right side of Figure~\\ref{enexg}.\n\nBut there is a second two-dimensional component $D_2$: the moving line\n$\\mathbb{P} M_2$ equals the fixed line $\\mathbb{P} F_2$, $p$ varies on the\nmoving line (yielding one dimension of moduli), and the moving point\n$\\mathbb{P} M_1$ varies on the moving line (yielding the second dimension\nof moduli).  This component is not seen in the usual geometric\nLittlewood-Richardson rule.  This corresponds to (the top two rows of) the\nsecond puzzle of Figure~\\ref{enex}.  The generic such configuration is\ndepicted on the bottom right side of Figure~\\ref{enexg}.  Note that\nthe moving and fixed flags are forced to be in\nmore degenerate condition than required by the degeneration order:\nthe moving and fixed lines are forced to agree.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\setlength{\\unitlength}{0.00083333in}\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  \\selectfont}%\n\\fi\\endgroup%\n{\\renewcommand{\\dashlinestretch}{30}\n\\begin{picture}(4524,3029)(0,-10)\n\\put(3762,2707){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$\\mathbb{P} M_1$}}}\n\\put(3912,607){\\ellipse{1200}{1200}}\n\\put(3912,2407){\\ellipse{1200}{1200}}\n\\put(912,1507){\\blacken\\ellipse{36}{36}}\n\\put(912,1507){\\ellipse{36}{36}}\n\\put(4212,2407){\\blacken\\ellipse{36}{36}}\n\\put(4212,2407){\\ellipse{36}{36}}\n\\put(4212,607){\\blacken\\ellipse{36}{36}}\n\\put(4212,607){\\ellipse{36}{36}}\n\\put(612,1507){\\blacken\\ellipse{36}{36}}\n\\put(612,1507){\\ellipse{36}{36}}\n\\put(3912,607){\\blacken\\ellipse{36}{36}}\n\\put(3912,607){\\ellipse{36}{36}}\n\\put(3612,607){\\blacken\\ellipse{36}{36}}\n\\put(3612,607){\\ellipse{36}{36}}\n\\put(612,1957){\\blacken\\ellipse{36}{36}}\n\\put(612,1957){\\ellipse{36}{36}}\n\\put(3912,2857){\\blacken\\ellipse{36}{36}}\n\\put(3912,2857){\\ellipse{36}{36}}\n\\path(1362,1207)(3162,757)\n\\blacken\\path(3038.307,757.000)(3162.000,757.000)(3052.859,815.209)(3080.508,777.373)(3038.307,757.000)\n\\path(1362,1807)(3162,2257)\n\\blacken\\path(3052.859,2198.791)(3162.000,2257.000)(3038.307,2257.000)(3080.508,2236.627)(3052.859,2198.791)\n\\path(12,1507)(1212,1507)\n\\path(3312,2407)(4512,2407)\n\\path(3312,607)(4512,607)\n\\path(612,2107)(612,907)\n\\path(4429,2099)(3815,2985)\n\\put(837,1357){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$\\mathbb{P} F_1$}}}\n\\put(4137,457){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$\\mathbb{P} F_1$}}}\n\\put(3612,757){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$\\mathbb{P} M_2 = \\mathbb{P} F_2$}}}\n\\put(687,1882){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$\\mathbb{P} M_1$}}}\n\\put(462,1357){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$p$}}}\n\\put(3912,2257){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$p=\\mathbb{P} F_1$}}}\n\\put(3876,457){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$p$}}}\n\\put(3537,457){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$\\mathbb{P} M_1$}}}\n\\put(462,1057){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$M_{\\cdot}$}}}\n\\put(87,1546){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$F_{\\cdot}$}}}\n\\put(612,1507){\\ellipse{1200}{1200}}\n\\end{picture}\n}\n\n\\end{center}\n\\caption{A worked example (\\S \\ref{eqex}) showing\na reducible Cartier slice.\n\\lremind{enexg}}\n\\label{enexg}\n\\end{figure}\n\nThe reader may verify that there are no other components of $D$.\n\nIn order to see how the weight arises, we continue to follow the\nsecond case.  In the next Cartier slice, the moving point is required\nto agree with the fixed point ($\\mathbb{P} M_1 = \\mathbb{P} F_1$).  This is\nclearly an irreducible divisor: the point $p$ may still vary on the\nmoving line $\\mathbb{P} M_2$, and the moving point is now fixed (despite\nits name).  We then slice with the pullback of the next Cartier\ndivisor ($\\overline{\\Omega}_{d_3}(F_\\bullet) \\subset \\overline{\\Omega}_{d_2}(F_{\\bullet})$),\nrequiring the moving line to agree with the fixed line.  However, this\nis {\\em not} a divisor on our equivariant puzzle variety: the\ncondition of the moving line agreeing with the fixed line is satisfied\nby the entire equivariant puzzle variety.  Thus we get excess\nintersection given by the (equivariant) first Chern class of this line\nbundle corresponding to the divisor, which is readily checked to\nbe $y_3-y_2$.\n\n\\section{The equivariant $K$-theory of the Grassmannian}\n\\lremind{KT}\\label{KT}The persevering reader who has read\nSections~\\ref{K} and \\ref{HT} will realize that this begs for an extension\nto equivariant $K$-theory, and that this would ideally correspond to\nthe following: in a Cartier slice in the equivariant\ndiscussion where the variety breaks into two pieces, the\nscheme-theoretic intersection of the two pieces should be another\nequivariant puzzle variety.  In this case we do not have pre-existing\npuzzles to guide us.\n\nKnutson and the second author indeed conjecture such a rule.  We first\ndescribe the new equivariant $K$-theory puzzle pieces, which gives a\npurely combinatorial conjectural rule in equivariant $K$-theory, and\nthen describe the geometry conjecturally associated to it.  Knutson\nand the first author have verified this rule up to dimension $5$.  We\nemphasize that although this may be interpreted as a purely\ncombinatorial rule, it was induced from many geometric examples.\n\nThe equivariant $K$-theory pieces include the original pieces from\nequivariant cohomology --- those from ordinary cohomology shown in\nFigure~\\ref{puzzleD}, and the equivariant piece of Figure~\\ref{eqpuzzle}.\nThe equivariant piece in the position shown in Figure~\\ref{eqweight}\nnow contributes $1 - e^{y_i-y_j}$.  (The geometric reason\nis that this is will be the Chern class\nof a line bundle\nin equivariant $K$-theory, not in equivariant cohomology.)\n\nWe do {\\em not} use the $K$-theory pieces of Figs.~11 or~12.  There\nare instead two new pieces, shown in Fig.~\\ref{KTpieces}, each\ncontributing a sign of $-1$.  Each $K_T$-piece has an unusual pair of\nedges, each of which is $0$ on one side and $1$ on the other.  Readers\nfamiliar with \\cite{knutson:puzzle} will recognize these pairs of\nedges as {\\em gashes}.  These pieces must be placed in this\norientation (i.e.\\ may not be reflected or rotated). Another new\nfeature is that there are constraints on where the two pieces may be\nplaced.  The first piece must be placed to the right of an equivariant\nrhombus (shown in Figure~\\ref{KTplaceone}).  The second piece may only\nbe placed (when completing the puzzle from top to bottom and left to\nright as usual) if the edges to its right are a (possibly empty)\nseries of horizontal $0$'s followed by a $1$.  This is depicted in\nFigure ~\\ref{KTplace}.  This latter condition is a new, potentially\ndisturbing, nonlocality in filling a puzzle (although it is at least\nlocal to the row).  However, it is dictated by the conjectural\ngeometry.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\setlength{\\unitlength}{0.00083333in}\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  \\selectfont}%\n\\fi\\endgroup%\n{\\renewcommand{\\dashlinestretch}{30}\n\\begin{picture}(1458,670)(0,-10)\n\\put(1274,436){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\path(1212,312)(1362,612)(1062,612)(1362,12)\n\\put(162,462){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(12,162){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(1062,387){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(1212,87){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(112,90){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(265,392){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(1159,462){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(1307,162){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(138,570){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(49,427){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(1183,571){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\path(162,312)(12,612)(312,612)(12,12)\n\\end{picture}\n}\n\n\\end{center}\n\\caption{The new  (conjectural) $K_T^*$ puzzle pieces.\\lremind{KTpieces}}\n\\label{KTpieces}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\setlength{\\unitlength}{0.00083333in}\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  \\selectfont}%\n\\fi\\endgroup%\n{\\renewcommand{\\dashlinestretch}{30}\n\\begin{picture}(2429,680)(0,-10)\n\\texture{88555555 55000000 555555 55000000 555555 55000000 555555 55000000 \n\t555555 55000000 555555 55000000 555555 55000000 555555 55000000 \n\t555555 55000000 555555 55000000 555555 55000000 555555 55000000 \n\t555555 55000000 555555 55000000 555555 55000000 555555 55000000 }\n\\shade\\path(1962,312)(2112,612)(2262,312)(1962,312)\n\\path(1962,312)(2112,612)(2262,312)(1962,312)\n\\shade\\path(12,312)(162,612)(312,312)\n\t(162,12)(12,312)\n\\path(12,312)(162,612)(312,312)\n\t(162,12)(12,312)\n\\shade\\path(2262,312)(2112,12)(1962,312)(2262,312)\n\\path(2262,312)(2112,12)(1962,312)(2262,312)\n\\path(312,312)(162,612)(462,612)\n\t(162,12)(12,312)(162,612)\n\\put(2337,387){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(162,162){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(387,387){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(2112,162){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(2262,462){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(2187,87){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(312,462){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(237,87){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(290,574){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(198,423){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(54,422){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(50,126){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(2236,581){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(2151,423){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(2009,427){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(2065,276){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$01$}}}\n\\path(2262,312)(2112,612)(2412,612)(2112,12)\n\\put(2007,122){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\path(2262,312)(1962,312)(2112,612)\n\\path(1962,312)(2112,12)\n\\end{picture}\n}\n\n\\end{center}\n\\caption{Placing the first $K_T$-piece.\\lremind{KTplaceone}\nThe left description is if one is using the equivariant\npiece of Figure~\\ref{eqpuzzle}, and the right is if one\nis using the alternate equivariant piece of Figure~\\ref{eqpuzzletwo}.}\n\\label{KTplaceone}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\setlength{\\unitlength}{0.00083333in}\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  \\selectfont}%\n\\fi\\endgroup%\n{\\renewcommand{\\dashlinestretch}{30}\n\\begin{picture}(2948,786)(0,-10)\n\\put(2764,419){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\path(312,612)(1062,612)\n\\path(1812,312)(1962,612)(1662,612)(1962,12)\n\\path(1962,612)(2712,612)\n\\path(1062,612)(1212,612)\n\\path(2712,612)(2862,312)\n\\put(87,462){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(12,387){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(237,162){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(162,87){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(612,687){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$\\cdots$}}}\n\\put(1737,462){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(1662,387){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(1887,162){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(1812,87){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(2262,687){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$\\cdots$}}}\n\\put(1362,237){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}or}}}\n\\put(216,422){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(135,570){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(908,570){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(1058,570){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(1785,569){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(1861,421){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(1989,570){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(340,570){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(2549,570){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\path(162,312)(312,612)(12,612)(312,12)\n\\end{picture}\n}\n\n\\end{center}\n\\caption{Placing the second $K_T$-piece.\\lremind{KTplace}}\n\\label{KTplace}\n\\end{figure}\n\nWe now describe the conjectural geometry, by explaining how to\ndetermine the equivariant puzzle variety corresponding to a partially\ncompleted $K_T$-puzzle.\nWe first describe how to find $\\alpha$ and $\\beta$ in words, and then\ngive examples which  may prove more enlightening.  We read off $\\alpha$ and $\\beta$ from the edges as usual (see Figs.~ \\ref{readalpha} and~\\ref{readbeta}\nrespectively).  The $0$, $1$, $10$, and $01$ edges have the\nsame interpretation as before.    In addition:\n\\begin{itemize}\n\\item A gash dangling southwest, part of the first $K_T$-piece of Figure~\\ref{KTpieces}, turns  the $0$ preceding it in $\\alpha$ into a $1$.\n(Because this piece must be placed immediately after an equivariant piece, as in Fig.~ \\ref{KTplaceone}, the gash is {\\em immediately} preceded by $0$\n in $\\alpha$.)\n\\item  A gash dangling southeast, part of the second $K_T$-piece of Figure\n~\\ref{KTpieces}, turns the first $1$ following it in $\\beta$ into a $0$.\n(There is guaranteed to be such a $1$ for each such gash by the placement rule of Fig.~\\ref{KTplace}.)\n\\end{itemize}\nExamples are given in Figure~\\ref{alphabetaKT}.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\setlength{\\unitlength}{0.00083333in}\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  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$\\alpha$ and $\\beta$\nfrom partially filled $K_T$-puzzles.\\lremind{alphabetaKT}}\n\\label{alphabetaKT}\n\\end{figure}\n\n\n\\subsection{Extending Example~\\ref{eqex} to equivariant $K$-theory}\nAs an example, we continue the discussion of \\S \\ref{eqex}.  At the\nkey step of that example, we considered a Cartier divisor, and\nobserved that it had two components, which gave two equivariant cycles\nthat we analyzed further.  In equivariant $K$-theory, we are expecting\na third term, appearing with sign $-1$, corresponding to the\nscheme-theoretic intersection of these first two pieces.  This leads\nto the $K_T$-puzzle shown in Figure~\\ref{swishy}.  This\ncorresponds to the fact that\\lremind{basicKT}\n\\begin{eqnarray}\n\\label{basicKT}\n\\sigma_{010}^2 &=& \\sigma_{100} + (1- e^{y_2-y_3}) \\sigma_{010} - (1 - e^{y_2-y_3}) \\sigma_{100} \\\\\n&=&  e^{y_2-y_3} \\sigma_{100} + (1 - e^{y_2-y_3}) \\sigma_{010},\n\\nonumber\n\\end{eqnarray}\nextending \n\\eqref{basicHT}.\nThe first term on the right side of \\eqref{basicKT}\ncorresponds to the left puzzle of Figure~\\ref{enex},\nthe second term corresponds to the right side of Figure~\\ref{enex},\nand the third term corresponds to  the new $K_T$-puzzle of Figure~\\ref{swishy}.  The partially completed puzzle corresponding to this puzzle is\nthe first panel of Figure~\\ref{alphabetaKT}.\n\nWe now describe the geometry corresponding to this example.  Consider\nthe five-dimensional variety whose points generically correspond to the\nconfiguration corresponding to the left side of Figure~\\ref{enexg}.\nAs in the equivariant cohomology discussion, we consider the Cartier\ndivisor corresponding to requiring the moving line $\\mathbb{P} M_2$ to pass\nthrough the fixed point $\\mathbb{P} F_1$.  This is reducible, and the\ngeneric behaviors of the two irreducible components are shown on the\nright side of Figure~\\ref{enexg}.  These two components contributed to\nthe equivariant cohomology calculation.  We have a third term,\nappearing with sign $-1$, corresponding to the scheme-theoretic\nintersection of these two components.  This corresponds to those\nconfigurations where the moving line $\\mathbb{P} M_2$ equals the fixed line\n$\\mathbb{P} F_2$ (the condition of the lower-right panel of Fig.\\\n~\\ref{enexg}) and also the point $p$ agrees with the fixed point\n$\\mathbb{P} F_1$ (the condition of the upper-right panel of Fig.\\\n~\\ref{enexg}).  This is indeed the equivariant puzzle variety\npredicted by the conjecture.\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\setlength{\\unitlength}{0.00083333in}\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  \\selectfont}%\n\\fi\\endgroup%\n{\\renewcommand{\\dashlinestretch}{30}\n\\begin{picture}(989,999)(0,-10)\n\\texture{88555555 55000000 555555 55000000 555555 55000000 555555 55000000 \n\t555555 55000000 555555 55000000 555555 55000000 555555 55000000 \n\t555555 55000000 555555 55000000 555555 55000000 555555 55000000 \n\t555555 55000000 555555 55000000 555555 55000000 555555 55000000 }\n\\shade\\path(612,72)(462,372)(612,672)\n\t(762,372)(612,72)\n\\path(612,72)(462,372)(612,672)\n\t(762,372)(612,72)\n\\path(12,72)(462,972)(912,72)(12,72)\n\\path(162,372)(462,372)(312,72)\n\\path(612,72)(762,372)\n\\put(537,222){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(326,463){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(478,163){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(391,526){\\makebox(0,0)[lb]{{\\SetFigFont{5}{6.0}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(354,781){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(505,781){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(435,631){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(225,487){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(289,330){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(56,181){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(148,35){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(356,179){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(434,31){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(733,31){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(805,180){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(654,181){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(508,476){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\path(462,372)(612,672)(312,672)(612,72)\n\\put(659,478){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\end{picture}\n}\n\n\\end{center}\n\\caption{The new $K_T$-puzzle arising in computing\n$\\sigma_{010}^2$. (The puzzles already arising in equivariant\ncohomology are shown in Figure~\\ref{enex}.)\\lremind{swishy}}\n\\label{swishy}\n\\end{figure}\n\n\nThe reader wishing to see the other $K_T$-piece in use\nshould compute (using the conjecture)\n\\begin{eqnarray*}\n\\sigma_{101}^2 &=& \\sigma_{110} + (1 - e^{y_1-y_2}) \\sigma_{101} - (1 - e^{y_1-y_2}) \\sigma_{110} \\\\\n&=& e^{y_1-y_2} \\sigma_{110} + (1 - e^{y_1 - y_2}) \\sigma_{101}.\n\\end{eqnarray*}\nThere are three puzzles, and the reader is encouraged to follow\nthe geometry of the successive Cartier slicing corresponding\nto these puzzles.  The second panel of Figure~\\ref{alphabetaKT}\nwill appear, contributing a $K_T$-term (not appearing in the $H_T$ calculation).\n\n\n\\section{A conjectural geometric Littlewood-Richardson rule\nfor the two-step flag variety}\n\n\\label{s:twostep}\\lremind{s:twostep}We next use these same ideas to give a conjectural geometric\nLittlewood-Richardson rule for two-step flag varieties (by the Knutson\nand the second author), generalizing the geometric\nLittlewood-Richardson rule for the Grassmannian.  Recent interest in\nthe two-step flag variety is likely due to the realization by A. Buch,\nA. Kresch and H. Tamvakis \\cite{bkt} that (i) the two-step problem is\nintimately related to the quantum Grassmannian problem, and (ii) the\ntwo-step problem appears to be simpler than the $m$-step\nproblem in general.\n\nWe first give Knutson's Littlewood-Richardson puzzle conjecture.\n\n\\begin{conjecture}[Knutson]  If\n$\\alpha$, $\\beta$, and $\\gamma$ are given in string notation,\nthe two-step Littlewood-Richardson number  $c^{\\gamma}_{\\alpha, \\beta}$\nis the number of puzzles with sides $\\alpha$, $\\beta$, $\\gamma$ (written as in the\nleft panel of Fig.~\\ref{puzzleD}), filled with puzzle pieces as given in \nFigure~\\ref{twostep}.  \\label{allentwostep}\\lremind{allentwostep}\n\\end{conjecture}\n\nThis rule has been checked up to $n=16$ by Buch, Kresch, and\nTamvakis \\cite{bkt}.  Given the number of cases to check, this\nverification clearly required a great deal of ingenuity.\nConjecture~\\ref{allentwostep} was originally stated as a conjecture by\nKnutson for all partial flag varieties, but languished unpublished\nonce Knutson noted that this general version fails at $n=5$.\nBuch, Kresch and Tamvakis noted that this generalization already\nfails for three-step partial flag manifolds when $n=5$; Buch's patch\nto the three-step conjecture is given in the next section.\n\nThe pieces are most cleanly described as follows.  The edges\ncorrespond to binary trees with the nodes labeled by the integers $0$,\n$1$, and $2$, such that the labels decrease strictly from left to right.\nThese trees can be represented by sequences of integers and\nparentheses, as shown in Figure~\\ref{twostep}.  The puzzle pieces \nconsist of triangles labeled $x\/x\/x$ where $x \\in \\{ 0, 1, 2 \\}$ as\nwell as triangles labeled $a\/b\/ab$ where $a$, $b$ and $ab$ are\nacceptable edge-labels (read clockwise).  This description will be partially\nconjecturally extended to the three-step case in the next section.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\setlength{\\unitlength}{0.00083333in}\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  \\selectfont}%\n\\fi\\endgroup%\n{\\renewcommand{\\dashlinestretch}{30}\n\\begin{picture}(3851,434)(0,-10)\n\\put(3006,32){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$2(10)$}}}\n\\path(912,107)(1062,407)(1212,107)(912,107)\n\\path(1512,107)(1662,407)(1812,107)(1512,107)\n\\path(1962,107)(2112,407)(2262,107)(1962,107)\n\\path(2412,107)(2562,407)(2712,107)(2412,107)\n\\path(3012,107)(3162,407)(3312,107)(3012,107)\n\\path(3462,107)(3612,407)(3762,107)(3462,107)\n\\path(462,107)(612,407)(762,107)(462,107)\n\\put(63,223){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(203,219){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(129,59){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(596,64){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(524,223){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(677,224){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(963,223){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$2$}}}\n\\put(1113,223){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$2$}}}\n\\put(1034,64){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$2$}}}\n\\put(1563,223){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$2$}}}\n\\put(1712,226){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(1610,65){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$20$}}}\n\\put(2024,217){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(2161,225){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(2071,59){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$10$}}}\n\\put(2458,217){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$2$}}}\n\\put(2612,223){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(2510,59){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$21$}}}\n\\put(3057,229){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$2$}}}\n\\put(3171,228){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$10$}}}\n\\put(3472,217){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$21$}}}\n\\put(3664,223){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(3452,31){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$(21)0$}}}\n\\path(12,107)(162,407)(312,107)(12,107)\n\\end{picture}\n}\n\n\\end{center}\n\\caption{Knutson's $2$-step puzzle pieces, which may be rotated.  \\lremind{twostep}}\n\\label{twostep}\n\\end{figure}\n\nWe now conjecture a geometric interpretation of partially filled\npuzzles of this sort.  (This conjecture has been verified in a large\nnumber of cases.) To any such partially filled puzzle, we associate a\n{\\em two-step puzzle variety}, extending the definition \\eqref{puzzlevariety}\nfor Grassmannians,\nof the form\n$$\n\\overline{ \\Omega_{\\alpha}(M_\\bullet) \\cap \\Omega_{\\beta}(F_\\bullet)}.\n$$\nAs before, we describe how to read $\\alpha$ and $\\beta$ from the partially\nfilled puzzles, but now $\\alpha$ and $\\beta$ will be strings of $0$'s, $1$'s,\nand $2$'s.  It will be most convenient to describe the recipe using\nthe alternate visualization of Figure~\\ref{newab}.  As in that figure,\non each horizontal edge labeled $10$, we temporarily glue on the\n$1\/0\/10$ piece.  Similarly, for each horizontal edge labeled $20$,\n$21$, $2(10)$, and $(21)0$, we temporarily glue on the $2\/0\/20$,\n$2\/1\/21$, $2\/10\/2(10)$, and $21\/0\/(21)0$ piece (respectively).  Now\nwhen attempting to read off $\\alpha$ and $\\beta$, each edge visible will be\na $0$, $1$, or $2$ as desired, except for the following possibilities.\n\\begin{itemize}\n\\item As in the Grassmannian case, if the leading edge is labeled $10$, \nthis counts for $0$ in $\\alpha$, and in $\\beta$ turns the next $1$ to a $0$.\nMore generally, if a southwest\/northeast edge is labeled\n$ab$ ($a,b \\in \\{ 0,1,2 \\}$, $a>b$), this counts for $b$ in $\\alpha$, and in $\\beta$\nturns the next $a$ to a $b$.\n\\item If the leading edge is $2(10)$, then this counts as $1$  in $\\alpha$,\nand in $\\beta$, the next $2$ is turned into a $1$, and the next $1$\n(possibly the one just changed from a $2$) is turned into a $0$.\n\\item If the leading edge is $(21)0$, then this counts as $0$\nin $\\alpha$, and in $\\beta$ the next $2$ is turned into a $0$.  (The reader\nmay check that there is no $1$ before the $2$, so this could be\ninterpreted as the same statement  in the previous item:\nthe next $2$ is turned into a $1$, and the next $1$\nis turned into a $0$.)\n\\item If there is a northwest-southeast edge labeled $10$\n(this arises when temporarily gluing a $2\/10\/2(10)$-piece onto a horizontal\n$2(10)$-edge), this counts as a $1$ in $\\beta$, and in $\\alpha$, turns the next {\\em earlier} $0$ into a $1$.\n\\end{itemize}\n\n\\section{Buch's conjectural combinatorial (non-geometric) rules in the\n  three-step case, and for the two-step case in equivariant\n  cohomology}\n\n\\label{s:anders}\\lremind{s:anders}Buch has given combinatorial Littlewood-Richardson conjectures\nin two additional cases:  three-step partial flag varieties in\nordinary cohomology, and two-step partial flag varieties in \nequivariant cohomology.  It is natural\ngiven our earlier discussion to seek to understand the corresponding\nconjectural geometry.  This may shed light on possible proofs.\n\nWe begin with Buch's three-step conjecture.  As usual, each\nLittlewood-Richardson coefficient will count the number of puzzles\nwith sides corresponding to given strings (this time of $0$'s, $1$'s,\n$2$'s, and $3$'s).  The triangular pieces are as follows.  Most of the\npieces will have edges that are analogous to the two-step case: they\ncorrespond to binary trees with nodes labeled by $0$, $1$, $2$, or\n$3$, with the labels strictly decreasing from left to the right.  We\nwrite such trees as sequences of integers and parentheses.  The\ncomplete list is $0$, $1$, $2$, $3$, $10$, $20$, $30$, $21$, $31$,\n$32$, $(21)0$, $(31)0$, $(32)0$, $(32)1$, $2(10)$, $3(10)$, $3(20)$, $3(21)$, \n$((32)1)0$, $3((21)0)$, $(3(21))0$, $3(2(10))$, $(32)(10)$.  There are pieces\nof the form $x\/x\/x$ where $x \\in \\{ 0, 1, 2, 3 \\}$, and $a\/b\/ab$ where\n$a$, $b$, and $ab$ are in the list above.  However, four extra pieces\nare also required, shown in Figure~\\ref{andersD}.  \nEquivalently, an integer can be repeated in a tree, if the two copies are separated by exactly three parentheses.  \nBuch has verified this\nrule up to $n=9$.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\setlength{\\unitlength}{0.00083333in}\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  \\selectfont}%\n\\fi\\endgroup%\n{\\renewcommand{\\dashlinestretch}{30}\n\\begin{picture}(3353,406)(0,-10)\n\\put(2631,237){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$3$}}}\n\\path(1858,79)(2008,379)(2158,79)(1858,79)\n\\path(2608,79)(2758,379)(2908,79)(2608,79)\n\\path(358,79)(508,379)(658,79)(358,79)\n\\put(0,196){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$3(2(10))$}}}\n\\put(2032,195){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$(21)0$}}}\n\\put(958,31){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$(3(21))(10)$}}}\n\\put(1715,31){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$(32)((21)0)$}}}\n\\put(2495,31){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$3(((32)1)0)$}}}\n\\put(139,42){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$(3(2(10)))0$}}}\n\\put(912,180){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$3(21)$}}}\n\\put(2800,201){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$((32)1)0$}}}\n\\put(1786,231){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$32$}}}\n\\put(1296,224){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$10$}}}\n\\put(557,225){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\path(1108,79)(1258,379)(1408,79)(1108,79)\n\\end{picture}\n}\n\n\\end{center}\n\\caption{Buch's four unusual pieces in the $3$-step case. \\lremind{andersD}}\n\\label{andersD}\n\\end{figure}\n\nWe next give Buch's equivariant $2$-step conjecture.  There are puzzle\npieces that are the same as the ordinary $2$-step conjecture\n(Fig.~\\ref{twostep}).  There are also six equivariant pieces, shown in\nFigure~\\ref{andersB}, generalizing Knutson and Tao's equivariant piece\nfor the Grassmannian (Fig.~\\ref{eqpuzzle}).  Like the equivariant\nGrassmannian piece, they may not be rotated or reflected.  Each\nequivariant piece contributes a weight according to the same recipe as\nthe Grassmannian case (Fig.~\\ref{eqweight}).  Buch has verified this\nrule up to $n=7$.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\setlength{\\unitlength}{0.00083333in}\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  \\selectfont}%\n\\fi\\endgroup%\n{\\renewcommand{\\dashlinestretch}{30}\n\\begin{picture}(3443,639)(0,-10)\n\\texture{88555555 55000000 555555 55000000 555555 55000000 555555 55000000 \n\t555555 55000000 555555 55000000 555555 55000000 555555 55000000 \n\t555555 55000000 555555 55000000 555555 55000000 555555 55000000 \n\t555555 55000000 555555 55000000 555555 55000000 555555 55000000 }\n\\shade\\path(2412,312)(2562,612)(2712,312)\n\t(2562,12)(2412,312)\n\\path(2412,312)(2562,612)(2712,312)\n\t(2562,12)(2412,312)\n\\shade\\path(3012,312)(3162,612)(3312,312)\n\t(3162,12)(3012,312)\n\\path(3012,312)(3162,612)(3312,312)\n\t(3162,12)(3012,312)\n\\shade\\path(612,312)(762,612)(912,312)\n\t(762,12)(612,312)\n\\path(612,312)(762,612)(912,312)\n\t(762,12)(612,312)\n\\shade\\path(1212,312)(1362,612)(1512,312)\n\t(1362,12)(1212,312)\n\\path(1212,312)(1362,612)(1512,312)\n\t(1362,12)(1212,312)\n\\shade\\path(1812,312)(1962,612)(2112,312)\n\t(1962,12)(1812,312)\n\\path(1812,312)(1962,612)(2112,312)\n\t(1962,12)(1812,312)\n\\shade\\path(12,312)(162,612)(312,312)\n\t(162,12)(12,312)\n\\path(12,312)(162,612)(312,312)\n\t(162,12)(12,312)\n\\put(1409,428){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$2$}}}\n\\put(1256,126){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$2$}}}\n\\put(1443,131){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(1826,428){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$10$}}}\n\\put(2014,423){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$2$}}}\n\\put(1856,126){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$2$}}}\n\\put(1998,132){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$10$}}}\n\\put(2451,428){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(2592,429){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$21$}}}\n\\put(2434,120){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$21$}}}\n\\put(2609,126){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(3026,439){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$10$}}}\n\\put(3193,435){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$21$}}}\n\\put(3032,126){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$21$}}}\n\\put(3197,126){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$10$}}}\n\\put(652,120){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$2$}}}\n\\put(226,428){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(73,126){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\put(215,126){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(51,428){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(651,428){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(803,121){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$0$}}}\n\\put(809,428){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$2$}}}\n\\put(1273,429){\\makebox(0,0)[lb]{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}$1$}}}\n\\end{picture}\n}\n\n\\end{center}\n\\caption{Buch's equivariant $2$-step pieces. \nThey may not be rotated or reflected.  \\lremind{andersB}}\n\\label{andersB}\n\\end{figure}\n\n\n\\section{A less explicit conjectural geometric Littlewood-Richardson\n  rule for partial flag varieties in general}\n\n  Despite the failure of Knutson's Conjecture~\\ref{allentwostep} to\nextend to all partial flag varieties, we are still led to a geometric\nconjecture (Conj.~\\ref{bigflag} below).  This is equivalent to\nConjecture~4.9 of \\cite{vakil:checkers}.  It has been verified up to\n$n=5$, and generalizes the geometric Littlewood-Richardson rule for\nthe Grassmannian, and the two-step geometric conjecture of\nSection~\\ref{s:twostep}.\n\nFix a variety $\\mathbb{F} l(a_1, \\dots, a_m)$ of $m$-part partial flags in\n$n$-space.  Let $S$ be the set of Schubert cells of $\\mathbb{F} l(a_1, \\dots,\na_m)$ (i.e.\\ certain strings of $0$'s, $1$'s, \\dots, $m-1$'s), and let\n$\\{ d_0, \\dots, d_{\\binom n 2} \\}$ be subset of the Schubert cells of\n$\\mathbb{F} l(n)$ corresponding to the degeneration order (described in \\S\n\\ref{degenerationorder}, and used throughout Part~\\ref{partone}).  For\nany element of $(\\alpha, \\beta, d_i) \\in S \\times S \\times \\{ d_0, \\dots,\nd_{\\binom n 2} \\}$, fix two flags $M_{\\bullet}$ and $F_{\\bullet}$ in\nrelative position $d_i$ (there is a unique such pair up to\ntranslation), and as before (e.g.\\ \\eqref{puzzlevariety}) define the {\\em puzzle variety} $PV_{\\alpha,\n  \\beta, d_i}$\n$$\n\\overline{ \\Omega_{\\alpha}(M_{\\bullet}) \\cap \\Omega_{\\beta}(F_{\\bullet})}.\n$$\nNote that  \n$PV_{\\alpha, \\alpha, d_{\\binom n 2}}$ is the Schubert variety $\\overline{\\Omega}_{\\alpha}(F_{\\bullet})$\nand $PV_{\\alpha, \\beta, d_0}$ \nis the intersection of general translates of the two Schubert varieties.\n(Recall that $d_{\\binom n 2}$ corresponds to $M_{\\bullet} = F_{\\bullet}$,\nand $d_0$ corresponds to traverse $M_\\bullet$ and $F_\\bullet$.)\n\n\n\\begin{conjecture}\n{\\bf (\\cite[Conj.~4.9]{vakil:checkers}, rephrased)}\nThere exists a subset $U$ of $S \\times S \\times \\{d_0, \\dots, d_{\\binom n 2} \\} = \\{ (\\alpha, \\beta, d_i) \\}$\nsuch that \\label{bigflag}\\lremind{bigflag}\n\\begin{itemize}\n\\item $(\\alpha, \\beta, d_0) \\in U$ for all $\\alpha$ and $\\beta$.\n\\item If \n$(\\alpha, \\beta, d_{\\binom n 2}) \\in U$, then $\\alpha=\\beta$.\n\\item If $(\\alpha, \\beta, d_i) \\in U$ and $i< \\binom n 2$ (i.e.\\ $d_i$ is not the final\nelement of the degeneration order),\nthen upon degenerating $M_{\\bullet}$ from relative position $d_i$\nto $d_{i+1}$ with respect to $F_\\bullet$, the puzzle variety\n$PV_{\\alpha, \\beta, d_i}$ degenerates to a union of other\npuzzle varieties $PV_{\\alpha', \\beta', d_{i+1}}$, where each $(\\alpha', \\beta', d_{i+1})$\nis in $U$, and \neach appears with multiplicity $1$.\n\\end{itemize}\n\\end{conjecture}\n\nIf this conjecture were true, and one had an explicit description of\nwhich $\\alpha'$ and $\\beta'$ arose at each step, one would have a\ncombinatorial Littlewood-Richardson with a geometric interpretation\n(and motivation).  However, lacking even a conjectural explicit\ndescription, this conjecture is admittedly vague and speculative.  The\nmain motivation for stating such a vague rule is that it suggests\nwhere to look for a more precise rule, and also suggests how to\ninterpret more combinatorial conjectures.  It partly motivated the\nconjectures of Knutson and the first author stated earlier.\n\nOne might also speculate that such a rule could also be extended to\nequivariant $K$-theory, as in Section~\\ref{KT}.\n\n\\part{MORE GENERAL RULES, MORE GENERAL DEGENERATION ORDERS}\n\n\\label{parttwo}\n\\section{The cohomology of flag varieties}\\label{secflag}\n\nIn this section we will describe a different Littlewood-Richardson\nrule for Grassmannians and explain how it generalizes to give a\nLittlewood-Richardson rule for two-step flag varieties. These rules\nwill be in terms of combinatorial objects called Mondrian tableaux.\nMondrian tableaux are very efficient for encoding degenerations and\nare very friendly to use. We encourage the reader to follow the\nexamples with graph paper and colored pencils in hand.\n\nA {\\it Mondrian tableau} associated to a Schubert class\n$\\sigma_{\\lambda_1, \\cdots, \\lambda_k}$ in $G(k,n)$ is a collection of\n$k$ nested squares labeled by integers $1, \\dots, k$, where the $j$th\nsquare has size $n-k+j - \\lambda_j$. The labels of the squares are\ndetermined by the picture: assign $1$ to the smallest square; if a\nsquare has label $i$, assign the label $i+1$ to the next larger\nsquare.  Hence we will omit them when we depict Mondrian\ntableaux. Figure \\ref{mondrian} shows two examples of Mondrian\ntableaux for $\\sigma_{2,1}$ in $G(3,6)$.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\begin{picture}(0,0)%\n\\includegraphics{mondi.pstex}%\n\\end{picture}%\n\\setlength{\\unitlength}{3947sp}%\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  \\selectfont}%\n\\fi\\endgroup%\n\\begin{picture}(2775,763)(514,-523)\n\\put(2746,-326){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{10}{12.0}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}unit size}%\n}}}}\n\\end{picture}%\n\n\\end{center}\n\\caption{Two Mondrian tableaux associated to $\\sigma_{2,1}$ in\n  $G(3,6)$.}\n\\label{mondrian}\n\\end{figure}\n\nIn a Mondrian tableau a square of side-length $s$ denotes a vector\nspace of dimension $s$. We will denote squares in a Mondrian tableau\nby capital letters in the math font (e.g., $A_i$) and the vector\nspaces they represent by the corresponding letter in Roman font (e.g.,\n$\\mathrm{A}_i$). If a square $S_1$ is contained in a square $S_2$,\nthen the vector space $\\mathrm{S}_1$ is a subspace of $\\mathrm{S}_2$.\nThe reader should think of unit squares along the anti-diagonal of a\nMondrian tableau as a basis of the underlying vector space.  The\nvector space represented by any square centered along the\nanti-diagonal is the span of the basis elements it contains. A\nMondrian tableau associated to $\\sigma_{\\lambda}$ depicts the vector\nspaces that have exceptional behavior for the $k$-planes parametrized\nby the Schubert cycle. The $k$-planes are required to intersect the\nvector space represented by the $i$th square in dimension at least\n$i$. \n\nBefore describing the rule in detail, we repeat the key example~\\ref{akeyexample} of\nPart \\ref{partone} in terms of Mondrian tableaux. Figure \\ref{flipone}\nshows the calculation $\\sigma_1^2 = \\sigma_2 + \\sigma_{1,1}$ in\n$G(2,4) = \\mathbb{G}(1,3)$. The reader might want to refer to this example\nwhile reading the rule.\n\n \\begin{figure}[htbp]\n\\begin{center}\n\\epsfig{figure=simple.eps}\n\\end{center}\n\\caption{The product $\\sigma_1^2= \\sigma_2 + \\sigma_{1,1}$ in $G(2,4)=\n  \\mathbb{G}(1,3)$: Mondrian tableaux and the projective geometry\n  corresponding to them.}\n\\label{flipone}\n\\end{figure}\n\nThe Mondrian tableaux rule follows the same basic strategy as the\nrule of Part~\\ref{partone}. We specialize the flags defining the Schubert varieties\nuntil the intersection decomposes into Schubert varieties. However,\nthe Mondrian tableaux rule differs from the earlier rule in two\naspects.  First, the order of specialization is not pre-determined but\ndepends on the intersection problem. This added flexibility allows us\nto avoid some geometric complications.  Second, the specializations do\nnot depend on a choice of basis. Although Schubert varieties are often\ndefined in terms of a fixed full-flag, one cannot canonically\nassociate a full-flag to a Schubert variety in $G(k,n)$. One can,\nhowever, associate to it a canonical partial flag of at most\n$k$-steps. The Mondrian tableaux rule will depend only on the\ncanonical partial flags. \n\n\n\\noindent {\\bf The game.} To multiply two Schubert classes\n$\\sigma_{\\lambda}$ and $\\sigma_{\\mu}$ in $G(k,n)$ we place the tableau\nassociated to $\\lambda$ (respectively, $\\mu$) at the southwest\n(respectively, northeast) corner of an $n \\times n$ square. The\nsquares in the $\\lambda$ (respectively, $\\mu$) tableau are all left\n(respectively, right) aligned with respect to the $n \\times n$ square.\nWe will denote the squares corresponding to $\\lambda$ and $\\mu$ by\n$A_{i}$ and $B_{j}$, respectively.  Figure \\ref{initial} shows the\ninitial tableau for the multiplication $\\sigma_{2,1,1} \\cdot\n\\sigma_{1,1,1}$ in $G(3,6)$. \n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\begin{picture}(0,0)%\n\\includegraphics{initial.pstex}%\n\\end{picture}%\n\\setlength{\\unitlength}{3947sp}%\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  \\selectfont}%\n\\fi\\endgroup%\n\\begin{picture}(2248,1087)(376,-705)\n\\put(376,-547){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{11}{13.2}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}$A_1$}%\n}}}}\n\\put(376,-258){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{11}{13.2}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}$A_2$}%\n}}}}\n\\put(1468,250){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{11}{13.2}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}$B_1$}%\n}}}}\n\\put(885,250){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{11}{13.2}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}$B_3$}%\n}}}}\n\\put(1178,250){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{11}{13.2}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}$B_2$}%\n}}}}\n\\put(2040,-485){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{9}{10.8}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}Unit size}%\n}}}}\n\\put(376,-41){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{11}{13.2}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}$A_3$}%\n}}}}\n\\end{picture}%\n\n\\end{center}\n\\caption{The initial Mondrian tableau.}\n\\label{initial}\n\\end{figure}\n\nInitially the two Schubert cycles are defined with respect to two\ntransverse flags. If the intersection of the two Schubert cycles is\nnon-empty, then the Schubert cycles have to satisfy certain\nconditions. A preliminary rule (MM rule) checks that these conditions\nare satisfied. Then there are some simplifications that reduce the\nproblem to a smaller problem. The OS and S rules give these\nsimplifications.\n\n\n \n$\\bullet$ {\\bf The MM (``must meet'') rule.}  We check that $A_i$\nintersects $B_{k-i+1}$ in a square of side-length at least one for\nevery $i$ between $1$ and $k$.  If not, we stop:  the Schubert cycles\nhave empty intersection. The class of their intersection is zero.\n\n In a $k$-dimensional vector space $V^k$ every $i$-dimensional\nsubspace (such as $V^k \\cap \\mathrm{A}_i$) {\\bf M}ust {\\bf M}eet every\n$(k-i+1)$-dimensional subspace (such as $V^k \\cap \\mathrm{B}_{k-i+1}$)\nin at least a line. The intersection of two Schubert cycles is zero if\nand only if the initial tableau formed by the two cycles does not\nsatisfy the MM rule. \n\n$\\bullet$ {\\bf The OS (``outer square'') rule.} We call the intersection\nof $A_k$ and $B_k$ the {\\bf O}uter {\\bf S}quare of the tableau. We\nreplace every square with its intersection with the outer square. \n\nSince the $k$-planes are contained in both $\\mathrm{A}_k$ and\n$\\mathrm{B}_k$, they must be contained in their intersection.  Figure\n\\ref{initial2} shows an application of the OS rule for the\nintersection in Figure \\ref{initial}.  \n\n\\begin{figure}[htbp]\n\\begin{center}\n\\begin{picture}(0,0)%\n\\includegraphics{initial2.pstex}%\n\\end{picture}%\n\\setlength{\\unitlength}{3947sp}%\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  \\selectfont}%\n\\fi\\endgroup%\n\\begin{picture}(4048,1207)(376,-705)\n\\put(376,-547){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{11}{13.2}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}$A_1$}%\n}}}}\n\\put(376,-258){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{11}{13.2}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}$A_2$}%\n}}}}\n\\put(1468,250){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{11}{13.2}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}$B_1$}%\n}}}}\n\\put(885,250){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{11}{13.2}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}$B_3$}%\n}}}}\n\\put(1178,250){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{11}{13.2}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}$B_2$}%\n}}}}\n\\put(2191,-550){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{11}{13.2}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}$A_1$}%\n}}}}\n\\put(2191,-114){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{11}{13.2}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}$A_3$}%\n}}}}\n\\put(2917,104){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{11}{13.2}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}$B_1$}%\n}}}}\n\\put(2336,104){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{11}{13.2}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}$B_3$}%\n}}}}\n\\put(2627,104){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{11}{13.2}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}$B_2$}%\n}}}}\n\\put(2191,-333){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{11}{13.2}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}$A_2$}%\n}}}}\n\\put(3840,-305){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{9}{10.8}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}Unit size}%\n}}}}\n\\put(3301,378){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{10}{12.0}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}Outer square}%\n}}}}\n\\put(376,-41){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{11}{13.2}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}$A_3$}%\n}}}}\n\\end{picture}%\n\n\\end{center}\n\\caption{An application of the OS rule.}\n\\label{initial2}\n\\end{figure}\n\n\n$\\bullet$ {\\bf The S (``span'') rule.} We check that $A_i$ and\n$B_{k-i}$ touch or have a common square. If not, we remove the rows\nand columns between these squares as shown in Figure \\ref{rule1}.\n\n\nThis rule corresponds to the fact that a $k$-dimensional vector space\nlies in the {\\bf S}pan of any two of its subspaces of complementary\ndimension whose only intersection is the origin. This rule removes any\nbasis element of the ambient vector space that is not needed in\nexpressing the $k$-planes parametrized by the intersection of the two\nSchubert varieties. \n\n\n\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\begin{picture}(0,0)%\n\\includegraphics{rule1.pstex}%\n\\end{picture}%\n\\setlength{\\unitlength}{3947sp}%\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  \\selectfont}%\n\\fi\\endgroup%\n\\begin{picture}(2262,1068)(676,-1124)\n\\put(676,-638){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{9}{10.8}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}$A_2$}%\n}}}\n\\put(1599,-176){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{9}{10.8}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}$B_1$}%\n}}}\n\\end{picture}\n\n\\end{center}\n\\caption{Adjusting the span of the linear constraints.}\n\\label{rule1}\n\\end{figure}    \n \n\n\n\n\n\\noindent Once we have performed these preliminary steps, we will\ninductively build a new flag (the $D$ flag) by degenerating the two\nflags (the $A$ and $B$ flags).  At each stage of the game we will have\na partially built new flag (depicted by $D$ squares that arise as\nintersections of $A$ and $B$ squares) and partially remaining $A$ and\n$B$ flags (depicted by squares $A_i, \\dots, A_k$ and $B_k$, $B_{k-i},\n\\dots, B_1$). After nesting the $D$ squares, we will increase the\ndimension of the intersection of $\\mathrm{A}_i$ with\n$\\mathrm{B}_{k-i}$ by one in order of increasing $i$. We will depict\nthis move in the Mondrian tableau by sliding $A_i$ anti-diagonally up\nby one unit. As we specialize the flags, the intersection of the Schubert\nvarieties will break into irreducible components. Admissible Mondrian\ntableaux depict the intermediate varieties that occur during the\nprocess. \n\n\n\nA Mondrian tableau is {\\it admissible} for $G(k,n)$ if the squares\n that constitute the tableau (except for the outer square) are uniquely\n labeled as an indexed $A$, $B$ or $D$ square such that\n\n\\begin{enumerate}\n\\item \\label{1} The squares $A_k = B_k$ form the outer square.  They have\n  side-length $m \\leq n$ and contain the entire tableau.\n\n \n\n\\item \\label{2} The $A$ squares are nested, distinct, left aligned and\n  strictly contain all the $D$ squares. If the number of $D$ squares\n  is $i-1 < k$, then the $A$ squares are $A_i, A_{i+1}, \\dots, A_k$\n  with the smaller index corresponding to the smaller squares.  (In\n  particular, the total number of $A$ and $D$ squares is $k$.)\n\n\n\\item \\label{3} The $B$ squares are nested, distinct and right\n  aligned. They are labeled $B_k, B_{k-i}, B_{k-i-1}, \\dots, B_1$,\n  where a smaller square has the smaller index. The $A$ and $B$\n  squares satisfy the MM and S rules. The $D$ squares may intersect\n  $B_{k-i}$, but none are contained in $B_{k-i}$.\n\n\\item \\label{4} The $D$ squares are distinct and labeled $D_1, \\dots,\n  D_{i-1}$.  They do not need to be nested; however, there can be at\n  most one unnested $D$ square.  An {\\it unnested} $D$ square is a $D$\n  square that does not contain every $D$ square of smaller index. More\n  precisely, if $D_j$ does not contain all the $D$ squares of smaller\n  index, then it does not contain any of the $D$ squares of smaller\n  index; it is contained in every $D$ square of larger index; and $D_h\n  \\subset D_s$ for every $h < s$ as long as $h$ and $s$ are different\n  from $j$. All the $D$ squares of index lower than $j$ are to the\n  southwest of $D_j$.  $D_{j-1}$ and $D_j$ share a common square or\n  corner. Figure \\ref{Dboxes} shows a typical configuration of $D$\n  squares.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\begin{picture}(0,0)%\n\\includegraphics{Dboxes.pstex}%\n\\end{picture}%\n\\setlength{\\unitlength}{3947sp}%\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  \\selectfont}%\n\\fi\\endgroup%\n\\begin{picture}(2361,774)(1714,-1498)\n\\put(3099,-951){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{9}{10.8}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}unnested $D$ square}%\n}}}}\n\\end{picture}%\n\n\\end{center}\n\\caption{A typical configuration of $D$ squares.}\n\\label{Dboxes}\n\\end{figure}    \n\n\\item \\label{5} Let $S_1$ and $S_2$ be any two squares of the\n    tableau. If the number of squares contained in their span but not\n    contained in $S_1$ is $r$, then the side-length of $S_1$ is at\n    least $r$ less than the side-length of their span.\n\\end{enumerate} \n\n\\begin{figure}[htbp]\n\\begin{center}\n\\begin{picture}(0,0)%\n\\includegraphics{typmon.pstex}%\n\\end{picture}%\n\\setlength{\\unitlength}{3947sp}%\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  \\selectfont}%\n\\fi\\endgroup%\n\\begin{picture}(5108,1524)(-74,-748)\n\\put(2367,571){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{9}{10.8}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}There is an unnested $D$ square}%\n}}}}\n\\put(3868,-369){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}Active square}%\n}}}}\n\\put(3931,  5){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}Unnested $D$ square}%\n}}}}\n\\put(489,567){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{9}{10.8}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}The $D$ squares are nested}%\n}}}}\n\\put(-74,-306){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{8}{9.6}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}Active square }%\n}}}}\n\\end{picture}%\n\n\\end{center}\n\\caption{Two typical admissible Mondrian tableaux.}\n\\label{typmon}\n\\end{figure} \n\n\\noindent Figure \\ref{typmon} depicts two typical admissible Mondrian\ntableaux.  Observe that the labels in an admissible Mondrian tableau\nare determined from the picture.  The $B$ squares are the $k-i+1$\nsquares that are aligned with the north and east edges of the tableau.\nThe number of $A$ squares is equal to the number of $B$ squares, so\nthe $A$ squares are the largest $k-i+1$ squares that are aligned with\nthe south and west edges of the tableau. The rest of the squares are\n$D$ squares.  Consequently, when we depict Mondrian tableaux we will\nomit the labels\n\nTo every admissible Mondrian tableau $M$, we can associate an\nirreducible subvariety $\\mathrm{M}$ of $G(k,n)$.  Define $AB_h$ to be\nthe intersection of $A_h$ and $B_{k-h+1}$ for $h=i+1, \\dots, k$.\nFirst, suppose $M$ does not contain any unnested $D$ squares.\n$\\mathrm{M}$ is defined as the closure of the locus of $k$-planes $V$\nthat satisfy\n\\begin{enumerate}\n\\item  $\\dim(V \\cap \\mathrm{D}_s) = s$, for $1 \\leq s\n\\leq i-1$, \n\n\\item  $\\dim(V \\cap \\mathrm{A}_i) = i$,\n \n\\item$\\dim(V \\cap \\mathrm{AB}_h) = 1$ for $h=i+1, \\dots,\nk$,\n\n\\item $V$ is spanned by its intersection with $\\mathrm{A}_i$ and\n  $\\mathrm{AB}_h$ for $h=i+1, \\dots, k$.\n\\end{enumerate}\nIf $M$ has an unnested square $D_j$, we  modify the above by\nrequiring $\\dim(V \\cap \\mathrm{D}_j) =1$.\n\n\n\\noindent {\\bf The algorithm:} We now describe an algorithm that\nsimplifies admissible Mondrian tableaux.  Let $M$ be an admissible\nMondrian tableau with an outer square of side-length $m$. If $M$\ncontains no $A$ or $B$ squares (other than the outer square) and all\nthe $D$ squares are nested, then $M$ is a Mondrian tableau associated\nto a Schubert variety. The algorithm terminates for $M$. Otherwise,\n$M$ can be simplified as follows: If all the $D$ squares in $M$ are\nnested, define the {\\it active square} to be the smallest $A$ square\n$A_i$. If $D_j$ is the unique unnested $D$ square in $M$, define the\n{\\it active square} to be $D_{j-1}$.  Move the active square\nanti-diagonally up by one unit.  If there are any $D$ squares aligned\nwith the south and west edges of the active square, move them\nanti-diagonally up by one unit with the active square. Keep all the\nother squares fixed. Replace $M$ by the following two tableaux unless\nthe second tableau is not admissible (see Figure \\ref{move}). In the\nlatter case replace $M$ with only Tableau 1.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\begin{picture}(0,0)%\n\\includegraphics{movemove.pstex}%\n\\end{picture}%\n\\setlength{\\unitlength}{3947sp}%\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  \\selectfont}%\n\\fi\\endgroup%\n\\begin{picture}(5154,2649)(409,-1948)\n\\end{picture}%\n\n\\end{center}\n\\caption{Simplifying the Mondrian tableaux in Figure \\ref{typmon}.}\n\\label{move}\n\\end{figure} \n\n$\\bullet$ {\\bf Tableau 1.}  If the active square is $A_i$, delete\n$A_i$ and $B_{k-i}$. Draw their new intersection and label it $D_i$.\nKeep their old span as the outer square.  If $D_i$ does not intersect\nor touch $B_{k-i-1}$, slide all the $D$ squares anti-diagonally up\nuntil $D_{i}$ touches $B_{k-i-1}$.  If the active square is $D_{j-1}$,\ndelete $D_{j-1}$ and $D_j$. Draw their new intersection and label it\n$D_{j-1}$. Draw their old span and label it $D_j$. If $D_{j-2}$ does\nnot intersect or touch the new $D_{j-1}$, slide all the $D$ squares of\nindex $j-2$ anti-diagonally until $D_{j-2}$ touches $D_{j-1}$.  All\nthe remaining squares stay as in $M$. \n\n$\\bullet$ {\\bf Tableau 2.}  If the active square is $A_i$, we shrink\nthe outer square by one unit so that it passes along the new boundary\nof $A_i$ and $B_{k-i}$ and we delete the column and row that lies\noutside this square. The rest of the squares stay as in $M$.  If the\nactive square is $D_{j-1}$, we place the squares we move in their new\npositions and keep the rest of the squares as in $M$.\n\n\nObserve that if the active square is $A_i$, Tableau 2 is not\nadmissible if either $A_{i+1}$ has side-length one larger than $A_i$\nor if $B_{k-i}$ has side-length $m-i$ (informally, if $A_i$ or\n$B_{k-i}$ are as large as possible given $A_{i+1}$ and $B_{k}$).  If\nthe active square is $D_{j-1}$, then Tableau 2 is not admissible\neither if the side-length of $D_j$ is not at least $j-1$ units smaller\nthan the side-length of the span of $D_j$ and $D_{j-1}$ or if\n$D_{j-1}$ contains $D_j$ as a result of the move (informally, if\n$D_{j-1}$ and $D_j$ are as large as possible). In these cases we\nreplace $M$ with only Tableaux 1. \n\n\n\nGeometrically, moving the active square corresponds to a\nspecialization of the flags defining the Schubert varieties. Let us\ndescribe this explicitly in the case the active square is $A_i$ and\nthere are no $D$ squares abutting the south and west edges of $A_i$.\nLet $s$ be the side-length of $A_i$ and suppose that initially $A_{i}$\nand $B_{k-i}$ intersect in a square of side-length $r$. There is a\nfamily of $s$-dimensional linear spaces $\\mathrm{A}_i(t)$\nparametrized by an open subset $0 \\in U \\subset \\mathbb{P}^1$ such that over\nthe points $t \\in U$ with $ t \\not= 0$, the dimension of intersection\n$\\mathrm{A}_i (t) \\cap \\mathrm{B}_{k-i}$ is equal to $r$ and when\n$t=0$, the dimension of intersection $\\mathrm{A}_i(0) \\cap\n\\mathrm{B}_{k-i}$ is $r+1$.  Denoting the basis vectors represented by\nthe unit squares along the diagonal by $e_1, \\dots, e_n$, we\ncan take this family to be\n$$\\mathrm{A}_i (t) = \\ \\mbox{the span of} \\ \\{(te_1 + (1-t) e_{s+1},\ne_2, \\dots, e_s \\}.$$ When $t=1$, we have our original vector space\n$\\mathrm{A}_i$ represented by the old position of the square $A_i$.\nWhen $t=0$, we have the new vector space $\\mathrm{A}_i(0)$ represented\nby the new position of the square $A_i$. When $t=0$, the intersection\nof Schubert varieties defined with respect to the $A$ and $B$ flags\neither remains irreducible or breaks into two irreducible components.\nThe two tableaux record these possibilities. The Littlewood-Richardson\nrule may be phrased informally as: {\\it If the $k$-planes in the limit\n  do not intersect $\\mathrm{A}_i (0) \\cap \\mathrm{B}_{k-i}$, then they\n  must be contained in their new span.}\n\n\nMore precisely, after we apply the MM, OS and S rules, the initial\nMondrian tableau is admissible.  It is clear that the move transforms\nan admissible Mondrian tableau to one or two new admissible Mondrian\ntableaux.  Therefore, we can continue applying the move to each of the\nresulting tableau. After a cycle of moves (beginning with nested $D$\nsquares; forming a new $D$ square; and nesting the $D$ squares again),\nwe decrease the number of $A$ and $B$ squares each by one and we\nincrease the number of nested $D$ squares by one. If we continue\napplying the move to every tableau that results from the initial\nMondrian tableau, after finitely many steps all the resulting tableaux\nare tableaux associated to Schubert varieties. The\nLittlewood-Richardson coefficient $c_{\\lambda, \\mu}^{\\nu}$ of $G(k,n)$\nis equal to the number of times the tableau corresponding to\n$\\sigma_{\\nu}$ occurs at the end of this algorithm.\n\n\\begin{theorem}[\\cite{coskun1:LR}, Thm.~ 3.1] \\label{grmain} The\n  Littlewood-Richardson coefficient $c_{\\lambda, \\mu}^{\\nu}$ of\n  $G(k,n)$ equals the number of times the Mondrian tableau associated\n  to $\\sigma_{\\nu}$ results in a game of Mondrian tableaux starting\n  with $\\sigma_{\\lambda}$ and $\\sigma_{\\mu}$ in an $n \\times n$\n  square.\n\\end{theorem}\n\n\\begin{figure}[htbp]\n\\begin{center}\n  \\epsfig{figure=example1.eps}\n\\end{center}\n\\caption{The product $\\sigma_{2,1}^2 = \\sigma_{3,3} + 2\n  \\sigma_{3,2,1} + \\sigma_{2,2,2}$ in $G(3,6)$.}\n\\label{grasex}\n\\end{figure}   \n\n\nIn Figure \\ref{grasex}, we compute $\\sigma_{2,1} \\cdot \\sigma_{2,1}$\nin $G(3,6)$ using the Mondrian tableaux rule.  When we move $A_1$,\nthere are two possibilities. We replace the initial tableau by the two\ntableaux where we replace $A_1$ and $B_2$ by their new intersection\n(and slide it up) and where we restrict the tableau to the new span of\n$A_1$ and $B_2$. We continue resolving the first tableau by moving\n$A_2$ and replacing it by two new tableaux. In the second tableau,\n$B_2$ is as large as possible given the outer square. Hence, when we\nmove $A_1$ again, there is only one possibility. We then move $A_2$\nand now there are two possibilities.  We replace the tableau with the\ntableau where we take the intersection of $A_2$ and $B_1$ and with the\ntableau where we restrict the tableau to the new span of $A_2$ and\n$B_1$.  Continuing we conclude that\n$$\\sigma_{2,1}^2 = \\sigma_{3,3} + 2 \\sigma_{3,2,1} + \\sigma_{2,2,2}.$$\n\n\n\n\n\n\nMore generally, we can define a {\\it generalized Mondrian tableau} in\n$G(k,n)$ to be any collection of $k$ squares centered along the\nanti-diagonal of an $n \\times n$ square satisfying the following two\nproperties:\n\\begin{enumerate}\n\\item None of the squares are equal to the span of the squares\n  contained in them.\n\n\\item Let $S_1$ and $S_2$ be any two squares in the tableau. If the\n  number of squares contained in their span but not contained in $S_1$\n  is $r$, then the side-length of $S_1$ is at least $r$ less than the\n  side-length of their span.\n\\end{enumerate}\nWe can associate an irreducible subvariety of the Grassmannian\n$G(k,n)$ to such a tableau. Consider the locus of $k$-planes $V$\nsatisfying the following properties. For any square $S$ in the\ntableau, $V$ intersects $\\mathrm{S}$ in dimension equal to the number\nof squares contained in $S$ (including itself).  If $S_1$ and $S_2$\nare any two squares of the tableau, we further require $V \\cap\n\\mathrm{S}_1$ and $V \\cap \\mathrm{S}_2$ to intersect only along the\nsubspaces represented by squares contained in both $S_1$ and $S_2$ and\notherwise to be independent.  The variety associated to the\ngeneralized Mondrian tableau is the closure in $G(k,n)$ of the variety\nparameterizing such $k$-planes.\n\n\n\n\n\\cite{coskun1:LR} contains an algorithm for computing the classes of\nvarieties associated to generalized Mondrian tableaux. Observe that\nthe intersection of two Schubert varieties is a special case.\nReplacing the $A$ and $B$ squares in the initial tableau with the\nintersections $A_i \\cap B_{k-i+1}$ for $i=1, \\dots, k$ we obtain a\ngeneralized Mondrian tableau. Here we will not discuss the rule that\nexpresses the classes of the varieties defined by generalized Mondrian\ntableaux as a sum of Schubert varieties. However, we note that this\nflexibility of Mondiran tableaux makes it possible to adapt them to\nother contexts.\n\n\n\n\n\n\n\n\n\n\\subsection{Painted Mondrian tableaux} \\label{pmt}Mondrian tableaux are very\nwell-suited for computations in the cohomology of partial flag\nvarieties. For instance, a modification of the algorithm for the\nGrassmannians yields a Littlewood-Richardson rule for two-step flag\nvarieties. In this subsection we will describe this rule and give\nexamples of how to express classes of subvarieties of $r$-step partial\nflag varieties in terms of Schubert varieties. \n \nIn order to denote Schubert varieties in $Fl(k_1, \\dots, k_r ; n)$, we need to allow the\nsquares in a Mondrian tableau to have one of $r$ colors.  Let $C_1,\n\\dots, C_r$ be $r$ colors ordered by their indices.  A Mondrian\ntableau associated to the Schubert cycle $\\sigma_{\\lambda_1, \\dots,\n  \\lambda_{k_r}}^{\\delta_1, \\dots, \\delta_{k_r}}$ is a collection of\n$k_r$ nested squares in $r$ colors.  The $i$th square has side-length\n$n-k_r + i - \\lambda_i$ and color $C_{\\delta_i}$. (In particular,\n$k_1$ of the squares have color $C_1$; and $k_i - k_{i-1}$ of the\nsquares have color $C_i$ for $i > 1$.) Each square is labeled by an\n$r$-vector of non-negative integers, where the $j$th index denotes the\nnumber of squares of color less than or equal to $C_{j}$ in the square\n(including possibly the square itself).  Figure \\ref{paintmon} shows\ntwo examples of painted Mondrian tableaux. When depicting Mondrian\ntableaux for two-step flag varieties, we will always use dashed lines\nfor the color $C_1$ and solid lines for the color $C_2$\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\begin{picture}(0,0)%\n\\includegraphics{paintmon.pstex}%\n\\end{picture}%\n\\setlength{\\unitlength}{3947sp}%\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  \\selectfont}%\n\\fi\\endgroup%\n\\begin{picture}(3849,1449)(889,-1648)\n\\put(4051,-811){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}$C_1$  }%\n}}}}\n\\put(4051,-1036){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}$C_2$}%\n}}}}\n\\put(4051,-1261){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}$C_3$}%\n}}}}\n\\put(1951,-886){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}$C_1$  }%\n}}}}\n\\put(1951,-1111){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}$C_2$}%\n}}}}\n\\end{picture}%\n\n\\end{center}\n\\caption{Mondrian tableaux associated to $\\sigma_{1,1,0,0}^{2,1,2,1}$\n  in $Fl(2,4;6)$ and $\\sigma_{2,1,0,0}^{1,2,1,3}$ in $Fl(2,3,4;6)$.}\n\\label{paintmon}\n\\end{figure} \n\nAs in the Grassmannian case, the labels of the squares are clear from\nthe picture. The smallest square has last ($r$th) index $1$. If a\nsquare has last index $i$, the next larger square has last index\n$i+1$. The square with last index $i$ has as its $j$th index the\nnumber of squares of color at most $C_{j}$ contained in it.\nGeometrically, the vector space $V_j$ in the tuple $(V_1, \\dots, V_r)$\nparametrized by the Schubert variety is required to intersect the\nvector space represented by a square in dimension equal to the number\nof squares of color at most $C_j$ in that square. \n\n\nWe will now describe the Littlewood-Richardson rule for two-step flag\nvarieties. The strategy and many of the details are very similar to\nthe case of Grassmannians described above. Here we will focus mainly\non examples. The reader should refer to \\cite{coskun1:LR} for more\ndetails and to \n\\verb+http:\/\/math.mit.edu\/~coskun\/gallery.html+\nfor more examples. \n\n\n\nAs in the case of Grassmannians, in order to multiply two Schubert\ncycles, we place the painted Mondrian tableaux associated to the two\nSchubert cycles in opposite corners of an $n\\times n$ square. We make\nsure that the tableau at the southwest (respectively, northeast)\ncorner is left (respectively, right) justified. We refer to the\nsquares at the southwest corner as $A$ squares and the squares at the\nnortheast corner as $B$ squares. The initial position represents the\ncase when the flags defining the Schubert varieties are transverse.\nSee the left panel of Figure \\ref{paintinit} for an example.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\begin{picture}(0,0)%\n\\includegraphics{paininit.pstex}%\n\\end{picture}%\n\\setlength{\\unitlength}{3947sp}%\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  \\selectfont}%\n\\fi\\endgroup%\n\\begin{picture}(2124,924)(589,-823)\n\\end{picture}%\n\n\\end{center}\n\\caption{The initial tableau and the OS rule for the multiplication\n  $\\sigma_{1,1,1}^{2,1,2} \\cdot \\sigma_{1,1,1}^{2,1,2}$ in\n  $Fl(1,3;6)$.}\n\\label{paintinit}\n\\end{figure} \n\nIf the intersection of the two Schubert cycles is not zero, then the\ndefining flag elements have to satisfy certain intersection\nconditions. The MM (must meet) rule ensures that these conditions are\nsatisfied.  First, the $i$th square in the southwest corner has to\nintersect the $(k_2 - i +1)$th square in the northeast corner in a\nsquare of side-length at least one for every $i$. Similarly, the $i$th\nsquare of color $C_1$ in the southwest corner has to intersect the\n$(k_1 - i +1)$th square of color $C_1$ in the northeast corner. The\nintersection of two Schubert cycles is non-zero if and only if this\nrule is satisfied. After checking this, we simplify the tableau\napplying the OS (outer square) and S (span) rules. We apply the OS\nrule for squares of both colors. We first restrict the tableau to the\nintersection of the largest $A$ and $B$ squares. We then restrict all\nthe squares of color $C_1$ to the intersection of the largest $A$ and\n$B$ squares of color $C_1$. The right panel in Figure \\ref{paintinit}\nshows an example.\n\nOnce we have performed these preliminary operations, we move the $A$\nsquares in a specified order (with two exceptions the same order as\nbefore). We thus eliminate the $\\mathrm{A}$ and $\\mathrm{B}$ flags and\nbuild a new $\\mathrm{D}$ flag represented by $D$ squares. At each\nstage the painted Mondrian tableau corresponds to a subvariety of the\ntwo-step flag variety.  Very generally, a painted Mondrian tableau is\na collection of $k_2$ squares (possibly not the span of consecutive\nunit squares) of color $C_2$ that satisfy the two conditions for a\ngeneralized Grassmannian tableau and $k_1$ squares of color $C_1$\nconsisting of the spans of the squares in color $C_2$ that satisfy the\nconditions for a generalized Mondrian tableau for $G(k_1,k_2)$. The\ntableaux that occur during our process can be viewed as a very special\nsubset of these tableaux called {\\it admissible painted Mondrian\n  tableaux}. We refer the reader to \\cite{coskun1:LR} for their\nprecise description\n\nWe can associate an irreducible subvariety of the two-step flag\nvariety $Fl(k_1, k_2 ; n)$ to a painted Mondrian tableau: Take the closure of the\nlocus of pairs $(V_1, V_2)$ such that \n\\begin{enumerate}\n\\item $V_2$ intersects the vector space represented by a square of\n  color $C_2$ in dimension equal to the number of squares of color\n  $C_2$ contained in that square.\n\n\\item $V_1$ intersects the subspace of $V_2$ contained in a vector\n  space represented by a square of color $C_1$ in dimension equal to\n  the number of squares of color $C_1$ in that square.\n\\end{enumerate}\nThe dimension of such a variety has a simple expression in terms of\nthe sum of side-lengths and containment relations among the squares.\nThe dimension is the sum of two terms. The first term is the sum of\nthe side-lengths of the squares of color $C_2$ minus one for every\ncontainment relation between them. The second term is the the number\nof squares of color $C_2$ contained in every square of color $C_1$\nminus the number of containment relations among the squares of color\n$C_1$. These terms geometrically correspond to the dimensions of the\nimage and of the fiber, respectively, of the projection of the variety\nfrom $Fl(k_1,k_2;n)$ to $G(k_2,n)$.\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\epsfig{figure=izexamtwo.eps}\n\\end{center}\n\\caption{Calculating the intersection  $\\sigma_{1,0,0}^{2,1,2} \\cdot\n\\sigma_{2,1,0}^{2,1,2}$ in $Fl(1,3;6)$.}\n\\label{twost1}\n\\end{figure} \n\n\nWe demonstrate the rule by calculating the intersection\n$\\sigma_{1,0,0}^{2,1,2} \\cdot \\sigma_{2,1,0}^{2,1,2}$ in $Fl(1,3;6)$\n(see Figure \\ref{twost1}).  We first move the smallest $A$ square (the\nactive square) anti-diagonally up by one unit. (Geometrically, this\nmove corresponds to specializing the third flag element of the\n$\\mathrm{A}$ flag $F_3^{\\mathrm{A}}$.) After each move, we replace the\ninitial tableau by three new tableaux unless the dimension of the\nvariety associated to one or more of these tableaux is strictly\nsmaller than the initial dimension. If the dimension is smaller, we\ndiscard that tableau. In Tableau I, we replace the active\nsquare and the largest square of color $C_1$ by their new intersection\nin color $C_1$. We complete the rest of the tableaux so that each\nsquare continues to have the same number of squares and the same\nnumber of squares of color $C_1$ as in the initial tableau. This is\ndepicted by the tableau to the right of the initial tableau.\n(Geometrically, this possibility corresponds to the case where $V_1$\nintersects the new intersection.)  In Tableau II, we draw the\nnew intersection of the active square and the largest $\\mathrm{B}$\nsquare with which the intersection increases in color $C_2$. We\ncomplete the rest of the tableaux so that each square continues to\nhave the same number of squares and the same number of squares of\ncolor $C_1$ as in the initial tableau.  This is depicted by the\ntableau to the southeast of the initial tableau. (Geometrically, this\npossibility corresponds to the case where $V_2$ intersects the new\nintersection.)  In Tableau III, we restrict the outer square to\npass through the southwest corner of the active square. This is\ndepicted by the tableau directly below the initial tableau.\n(Geometrically, neither $V_1$ nor $V_2$ intersect the new\nintersection, so they are contained in the new span.) \n\n\nThe main new feature of the rule for two-step flag varieties is\nsquares called {\\it fillers}. Fillers are squares of color $C_2$ that\noccur as the intersection of the active square with a square of color\n$C_1$ to its northeast. The newly formed intersection in the second\ntableau above is a filler. Fillers affect the degeneration order: the\nsmallest filler takes precedence over the smallest $A$ square in the\norder. Otherwise, the order follows the same pattern as in the\nGrassmannian (see \\cite{coskun1:LR}). Moving fillers may also cause\nsome of the squares containing it to become disconnected. We refer to\nsquares that are not the span of consecutive unit squares as {\\it\n  chopped squares}. See Figure \\ref{twost2} below for an example.\n\nWe continue simplifying each of the three tableaux. Since the $D$\nsquares are nested, in the first and third tableau the active square\nis the smallest $A$ square. In the second tableau the active square is\nthe filler $D$ square. Let us trace how the first tableau simplifies\n(the top row of Figure \\ref{twost1}). When we move $A_2$, only Tableau\nII can occur. Now there is an unnested $D$ square ($D_2$). The active\nsquare is $D_1$. When we move it, only Tableaux I and III occur. Note\nthat the possibility described by Tableau II would have dimension one\nless than the initial dimension. Continuing the calculation one sees\nthat\n$$\\sigma_{1,0,0}^{2,1,2} \\cdot\n\\sigma_{2,1,0}^{2,1,2} = \\sigma_{3,1,0}^{1,2,2} +\n\\sigma_{2,2,0}^{1,2,2} + \\sigma_{3,2,0}^{2,1,2} +\n\\sigma_{2,1,1}^{1,2,2} + \\sigma_{2,2,1}^{2,1,2}.$$\n\n\n\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n  \\epsfig{figure=izcompexam.eps}\n\\end{center}\n\\caption{Calculating the intersection $\\sigma_{1,1,0,0}^{2,2,1,1} \\cdot\n  \\sigma_{2,1,1,0}^{2,2,1,1}$ in $Fl(2,4;6)$. }\n\\label{twost2}\n\\end{figure}   \n\nTo illustrate the new features of the two-step rule, we calculate\n$\\sigma_{1,1,0,0}^{2,2,1,1} \\cdot \\sigma_{2,1,1,0}^{2,2,1,1}$ in\n$Fl(2,4;6)$ (see Figure \\ref{twost2}).  Since the calculation is\nsimilar to the previous case we only point out the new features. In\nthe second panel, the newly formed $D$ square is a filler. According\nto the degeneration order, we first move the filler.  This move causes\nthe vector space corresponding to the smallest $A$ box to no longer be\nthe span of consecutive basis elements. This is depicted on the\nMondrian tableau as a chopped square. The third panel has an example\nof such a square. Continuing the calculation one sees that  \n$$\\sigma_{1,1,0,0}^{2,2,1,1} \\cdot\n\\sigma_{2,1,1,0}^{2,2,1,1}= \\sigma_{1,1,2,2}^{2,2,1,1} +\n\\sigma_{2,2,2,0}^{2,2,1,1}. $$\n\nIn general, degenerating the flags in the order described decomposes\nthe intersection of two Schubert varieties in a two-step flag variety\nto a union of Schubert varieties. This process may be encoded in an\nexplicit game of painted Mondrian tableaux. The main theorem then is\nthe following:\n\n\\begin{theorem}[\\cite{coskun1:LR}, Thm.~4.1] \\label{twostepbig}\nLet $\\sigma_{\\lambda}$ and $\\sigma_{\\mu}$ denote two Schubert cycles\nin the flag variety $Fl(k_1, k_2; n)$. Let their product be\n$\\sigma_{\\lambda} \\cdot \\sigma_{\\mu} = \\sum_{\\nu} c_{\\lambda\n\\mu}^{\\nu} \\sigma_{\\nu}$. The coefficient $c_{\\lambda \\mu}^{\\nu}$ is\nequal to the number of times the painted Mondrian tableau of\n$\\sigma_{\\nu}$ occurs in a game of Mondrian tableau played by starting\nwith the Mondrian tableaux of $\\sigma_{\\lambda}$ and $\\sigma_{\\mu}$ in\nan $n \\times n$ square.\n\\end{theorem}\n\nIt is natural to wonder about the connection of this rule to the\ngeometric two-step puzzle conjecture of \\S \\ref{s:twostep}.  Because\nin the theorem above the degenerations used are more general than in\nPart~\\ref{partone}, it is not clear if much direct connection is to be\nexpected.\n\nWe conclude this section with an example of how Mondrian tableaux can\nbe used to calculate classes of subvarieties in other partial flag\nvarieties (see Figure \\ref{3step} for an example in $Fl(1,2,4;6)$).\nInformally, the procedure may\nbe described as follows. We move the active square anti-diagonally up\nby one unit and replace the initial tableau with new tableaux. Each of\nthe new tableaux is obtained by either placing the active square in\nits new position (and shrinking the outer square if necessary) or by\ndrawing the new intersection of the active square with the closest\nsquare of color at most $C_j$ to its northeast in color $C_j$ and\nfilling the rest of the tableau so that each square has the same\nnumber of squares of color at most $C_h$ as in the initial tableau for\nevery color $C_h$. We then discard the tableaux that represent\nvarieties with strictly smaller dimension.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\begin{picture}(0,0)%\n\\includegraphics{3nstepex.pstex}%\n\\end{picture}%\n\\setlength{\\unitlength}{3947sp}%\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  \\selectfont}%\n\\fi\\endgroup%\n\\begin{picture}(6099,5972)(1039,-5271)\n\\put(1304,-4373){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{10}{12.0}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}$C_1$}%\n}}}}\n\\put(1304,-4563){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{10}{12.0}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}$C_2$}%\n}}}}\n\\put(1304,-4753){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{10}{12.0}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}$C_3$}%\n}}}}\n\\put(1178,-1652){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{10}{12.0}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}$\\cdots$}%\n}}}}\n\\put(2317,-2791){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{10}{12.0}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}$\\cdots$}%\n}}}}\n\\put(3407,-3590){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{10}{12.0}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}$\\cdots$}%\n}}}}\n\\end{picture}%\n\n\\end{center}\n\\caption{The product $\\sigma_{2,1,0,0}^{3,1,2,3} \\cdot\n  \\sigma_{1,0,0,0}^{3,2,1,3} $ in $Fl(1,2,4;6)$ equals\n  $\\sigma_{2,2,1,0}^{2,1,3,3} + \\sigma_{2,2,0,0}^{1,2,3,3} +\n  \\sigma_{2,1,1,0}^{1,2,3,3} + \\sigma_{2,2,1,0}^{1,3,2,3} +\n  \\sigma_{2,1,1,1}^{1,3,2,3} + \\sigma_{2,2,2,0}^{3,1,2,3} +\n  \\sigma_{2,2,1,1}^{3,1,2,3}$.}\n\\label{3step}\n\\end{figure} \n\n\\section{Quantum cohomology of Grassmannians and flag varieties}\n\n\\label{qcg}In this section we will describe a geometric method for computing the\nsmall quantum cohomology of Grassmannians. The method is based on the\nelegant observation of Buch, Kresch and Tamvakis that three-point\ngenus-zero Gromov-Witten invariants of certain homogeneous varieties\nmay be computed as the ordinary intersection products of convenient\nauxiliary varieties.  In the case of Grassmannians these auxiliary\nvarieties are two-step flag varieties. \n\nLet $X$ be a smooth, complex projective variety. Let\n$\\beta \\in H_2 (X, \\mathbb{Z})$ be the homology class of a curve. The\nKontsevich moduli space of $m$-pointed, genus-zero stable maps\n$\\Kgnb{0,m} (X, \\beta)$ provides a useful compactification of the\nspace of rational curves on $X$ whose homology class is $\\beta$.\nRecall that $\\Kgnb{0,m} (X, \\beta)$ is the smooth, proper\nDeligne-Mumford stack parameterizing the data of\n\\begin{enumerate}\n\\item[(i)] $C$, a proper, connected, at-worst-nodal curve of\n  arithmetic genus $0$,\n\\item[(ii)]   $p_1,\\dots,p_m$, an ordered sequence of distinct, smooth\n  points of $C$,\n\\item[(iii)] and $f:C\\rightarrow X$, a morphism with $f_*[C] = \\beta$\n  satisfying the following stability condition: every irreducible\n  component of $C$ mapped to a point under $f$ contains at least $3$\n  special points, i.e., marked points $p_i$ and nodes of $C$.\n\\end{enumerate}\n\n\nThe Kontsevich moduli space $\\Kgnb{0,m}(X, \\beta)$ admits $m$\nevaluation morphisms to $X$, where the $i$th evaluation morphism\n$ev_i$ maps $[C, p_1, \\dots, p_m, f]$ to $p_i$.  The dimension of\n$\\Kgnb{0,m} (X, \\beta)$ is $$c_1(T_X) \\cdot \\beta + \\dim(X) + m - 3.$$\nGiven the classes of $m$ subvarieties $\\gamma_1, \\dots, \\gamma_m$ of\n$X$ whose codimensions add up to this dimension, the genus-zero\nGromov-Witten invariant of $X$ associated to the classes $\\gamma_1,\n\\dots, \\gamma_m$ and the curve class $\\beta$ is defined as\n$$I_{X, \\beta} (\\gamma_1, \\dots, \\gamma_m) : =\n\\int_{\\Kgnb{0,m}(X,\\beta)} ev_1^*(\\gamma_1) \\cup \\cdots \\cup\nev_m^*(\\gamma_m).$$\n\nWhen $X$ is a homogeneous variety, the Gromov-Witten invariant is\nenumerative in the following sense:\n\n\\begin{proposition}\n\\cite[Lem.\\ ~14]{fultonpan:kontsevich} \\label{enumerative} Let\n  $\\Gamma_1, \\dots, \\Gamma_m$ be general  subvarieties of a\n  homogeneous variety $X$ representing the Poincar\\'e duals of the\n  cohomology classes $\\gamma_1, \\dots, \\gamma_m$, respectively.  The\n  scheme theoretic intersection\n  $$ev_1^{-1}(\\Gamma_1) \\cap  \\cdots \\cap ev_m^{-1}(\\Gamma_m)$$ is a\n  finite number of reduced points in $\\mathcal{M}_{0,m}(X, \\beta)$. Moreover,\n  $$I_{X,\\beta}(\\gamma_1, \\dots, \\gamma_m)= \\# \\  ev_1^{-1}(\\Gamma_1)\n  \\cap \\cdots \\cap ev_m^{-1}(\\Gamma_m).$$\n\\end{proposition} \n\n\n\nWhen $m=3$, the genus-zero Gromov-Witten invariants can be used as the\nstructure constants of a commutative and associative ring called the\nsmall quantum cohomology ring.  The main theorem of \\cite{bkt}\nequates the three-point Gromov-Witten invariants of Grassmannians with\ncertain ordinary three-point intersections of two-step flag\nvarieties. Given a Schubert cycle $\\sigma_{\\lambda}$ in $G(k,n)$, there is a\nspecial Schubert cycle $X_{\\lambda}^{(d)}(F_{\\bullet})$ in $Fl(k-d,\nk+d; n)$ defined by\n\\begin{align*}\nX_{\\lambda}^{(d)}(F_{\\bullet}) := \\{ \\ (V_1, V_2) \\ | \\  \\dim (V_1\n\\cap F_{n-i-\\lambda_{k-i}}) \\geq k-d-i ,  \\ \\dim (V_2 \\cap F_{n-k+j -\n\\lambda_j}) \\geq j \\ \\}\n\\end{align*}\nwhere $1 \\leq i \\leq k-d$ and $1 \\leq j \\leq k$.\n\n\n\n\n\\begin{proposition}\\label{two} \\cite[Cor.\\ ~1]{bkt} \n  Let $\\lambda, \\mu, \\nu$ be partitions and $d \\geq 0$ be an integer\n  satisfying\n\\begin{equation}\\label{eqn}\n|\\lambda| + |\\mu| + |\\nu| = k(n-k) + dn.\n\\end{equation}\nThen the degree $d$ three-point Gromov-Witten invariants of $G(k,n)$\nequal the ordinary three-point intersections of special Schubert\nvarieties  in the flag variety $Fl(k-d,k+d;n)$:\n$$I_d (\\sigma_{\\lambda}, \\sigma_{\\mu}, \\sigma_{\\nu}) =\n\\int_{Fl(k-d,k+d;n)} [X_{\\lambda}^{(d)}] \\cup [X_{\\mu}^{(d)}] \\cup\n[X_{\\nu}^{(d)}].$$\n\\end{proposition}\n\nCombining the discussion in \\S \\ref{secflag} and Proposition \\ref{two}\nwe obtain a Littlewood-Richardson rule for the small quantum\ncohomology ring of the Grassmannian $G(k,n)$.  Given a Mondrian\ntableau $\\sigma_{\\lambda}$ in $G(k,n)$ and an integer $d \\leq k$, we\ncan associate a painted Mondrian tableau in $Fl(k-d, k+d; n)$ as\nfollows: The Mondrian tableau associated to the Schubert variety\n$\\sigma_{\\lambda}$ consists of $k$ nested squares. We take the largest\n$k-d$ squares (those of index $d+1, \\dots, k$) and color them in\n$C_1$. We color the remaining squares in $C_2$. Finally, we add $d$\nsquares of color $C_2$ at the largest available places in the flag\ndefining the Mondrian tableau of $\\sigma_{\\lambda}$ (see Figure\n\\ref{izquant} for two examples).  We call the resulting painted Mondrian\ntableau the quantum Mondrian tableau of degree $d$ associated to\n$\\sigma_{\\lambda}$. This tableau is none other than the painted\nMondrian tableau associated to the special Schubert variety\n$X_{\\lambda}^{(d)}$ of $Fl(k-d,k+d;n)$.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\begin{picture}(0,0)%\n\\includegraphics{izquantum.pstex}%\n\\end{picture}%\n\\setlength{\\unitlength}{3947sp}%\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  \\selectfont}%\n\\fi\\endgroup%\n\\begin{picture}(5708,1351)(526,-517)\n\\put(4238,711){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{10}{12.0}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}degree 2 quantum cycle}%\n}}}}\n\\put(526,-441){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{10}{12.0}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}$\\sigma_{3,2,1} \\in G(3,6)$}%\n}}}}\n\\put(1870,-441){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{10}{12.0}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}$\\sigma_{2,1,0,0}^{2,1,1,2} \\in F(2,4;6)$}%\n}}}}\n\\put(3919,-441){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{10}{12.0}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}$\\sigma_{3,2} \\in G(3,6)$}%\n}}}}\n\\put(5070,-441){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{10}{12.0}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}$\\sigma_{1,0,0,0,0}^{2,2,2,2,1}\\in F(1,5;6)$}%\n}}}}\n\\put(1102,711){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{10}{12.0}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}degree 1 quantum cycle}%\n}}}}\n\\end{picture}%\n\n\\end{center}\n\\caption{The quantum Mondrian tableaux associated to two Schubert varieties.}\n\\label{izquant}\n\\end{figure}\n\n\n\n\n\n\n\n\n\nLet $\\sigma_{\\lambda}, \\sigma_{\\mu}$ and $\\sigma_{\\nu}$ be three\nSchubert cycles in $G(k,n)$ that satisfy the equality $$|\\lambda| +\n|\\mu| + |\\nu| = k (n-k) + dn.$$ Apply the algorithm described in the\nprevious section to the quantum Mondrian tableaux of degree $d$\nassociated to $\\sigma_{\\lambda}$ and $\\sigma_{\\mu}$ to express their\nintersection as a sum of Schubert cycles in $Fl(k-d,k+d;n)$. Then\napply the algorithm to the quantum Mondrian tableau of degree $d$\nassociated to $\\sigma_{\\nu}$ and each of the summands of the previous\nproduct. We have obtained the\nfollowing theorem.\n\\\n\n\\begin{theorem}[\\cite{coskun1:LR}, Thm.\\ 5.1] \\label{gwmon}\nThe three-pointed Gromov-Witten invariant $I_d (\\sigma_{\\lambda},\n\\sigma_{\\mu}, \\sigma_{\\nu})$ is equal to the number of times the point\nclass occurs as a result of applying the Littlewood-Richardson rule\nfor the two-step flag \nvarieties to the quantum Mondrian tableau  of degree $d$ associated to\n$\\sigma_{\\nu}$ and each outcome of the product of the quantum Mondrian\ntableaux of degree $d$ associated to $\\sigma_{\\lambda}$ and\n$\\sigma_{\\mu}$.\n\\end{theorem}\n \nWe illustrate the use of Theorem \\ref{gwmon} by computing the\nGromov-Witten invariant $$I_{G(3,6),d=1}(\\sigma_{3,2,1},\n\\sigma_{3,2,1}, \\sigma_{2,1})=2.$$ Figure \\ref{quantexamp}\ndemonstrates the computation. The quantum cycle of $d=1$ associated to\n$\\sigma_{3,2,1}$ (respectively, $\\sigma_{2,1}$) is\n$\\sigma_{2,1,0,0}^{2,1,1,2}$ (respectively,\n$\\sigma_{1,0,0,0}^{2,1,2,1}$).  In order to calculate the\nGromov-Witten invariant we have to find how many times\n$\\sigma_{2,2,2,1}^{1,2,1,2}$ (the dual of\n$\\sigma_{1,0,0,0}^{2,1,2,1}$) occurs in the square of the class\n$\\sigma_{2,1,0,0}^{2,1,1,2}$. An straightforward calculation with painted\nMondrian tableaux shows that the answer is 2.\n \n\n\\begin{figure}[htbp]\n\\begin{center}\n\\begin{picture}(0,0)%\n\\includegraphics{izquantexam.pstex}%\n\\end{picture}%\n\\setlength{\\unitlength}{3947sp}%\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  \\selectfont}%\n\\fi\\endgroup%\n\\begin{picture}(6320,3678)(589,-3523)\n\\end{picture}%\n\n\\end{center}\n\\caption{Computing the Gromov-Witten invariant $I_{G(3,6),d=1}(\\sigma_{3,2,1},\n\\sigma_{3,2,1}, \\sigma_{2,1})=2.$}\n\\label{quantexamp}\n\\end{figure}\n\n\n\nEarlier, Gromov-Witten invariants could be computed algebraically\nbased on structure theorems for the quantum ring due to Bertram\n\\cite{bertram}.  The proofs of these results were drastically\nsimplified by Buch \\cite{buch} using his powerful yet simple\n``kernel-span'' technique.  Buch defined the {\\em kernel\\\/} of a\nrational curve in the Grassmannian $G(k,n)$ to be the intersection in\n$\\mathbb{C}^n$ of all the $k$-planes corresponding to points on the curve,\nand defined its {\\em span\\\/} to be the linear span of these\n$k$-planes.  When the curve has degree $d$, the dimension of the\nkernel is at least $k-d$ while the span has dimension at most $k+d$.\nBy using this observation, Bertram's structure theorems can be proved\nby applying classical Schubert calculus to the span of a curve.\n\nThe translation between three-pointed genus-zero Gromov-Witten\ninvariants and the ordinary products in the two-step flag variety\ngiven in \\cite{bkt} is obtained by proving that the map that sends a\nrational curve to the pair $(V_1,V_2)$ of its kernel and span is\ninjective on the set of rational curves contributing to a\nGromov-Witten invariant $I_d(\\sigma_\\lambda, \\sigma_\\mu, \\sigma_\\nu)$,\nand the image of this map is exactly the set of points in an\nintersection $X^{(d)}_\\lambda \\cap X^{(d)}_\\mu \\cap X^{(d)}_\\nu$ of\nspecial Schubert varieties.  Furthermore, each rational curve can be\nexplicitly reconstructed from the pair $(V_1,V_2)$, and this\nreconstruction exhibits the curve as a scroll in $\\mathbb{P}^{n-1}$ with\nvertex $\\mathbb{P}(V_1)$ and span $\\mathbb{P}(V_2)$.  The problem of counting curves\nthus transforms to finding the pairs of vertices and spans.\n\nUnfortunately, only very special curves of degree $d \\geq k$ in\n $G(k,n)$ have a non-empty ``kernel'' (the intersection of all the\n linear spaces parametrized by the curve). We can replace the kernel\n by a more natural invariant: the sequence of minimal degree\n subscrolls associated to the curve (see \\cite{coskun:jumping}).  The\n advantage of minimal degree subscrolls is that every rational curve\n in the Grassmannian has an associated sequence of minimal degree\n subscrolls. When $d < k$ and the curve is general, we recover the\n vertex of the cone (equivalently, the kernel).\n\nConsidering minimal degree subscrolls allows one to extend the\ngeometric point of view to the big quantum cohomology of\nGrassmannians.  For instance, the geometry of scrolls leads to an\nimmediate proof of the vanishing of many $m$-pointed Gromov-Witten\ninvariants (see \\cite{coskun:jumping}). Let $m$, $k$, $d$, $n$ be positive\nintegers satisfying $m \\geq 3$, $2k \\leq n$ and $d+k \\leq n$.  Let\n$\\sigma_{\\lambda^1}$, \\dots, $\\sigma_{\\lambda^m}$ be $m$ Schubert\ncycles, where the parts $\\lambda_i^j$ of the partitions satisfy\n$$\\sum_{i,j} \\lambda_i^j = dn + k(n-k) + m-3.$$ The Gromov-Witten\ninvariant $I_{G(k,n),d}(\\sigma_{\\lambda^1}, \\dots, \\sigma_{\\lambda^m})\n= 0$ unless $$\\sum_{i,j}\\max{(\\lambda_i^j - d, 0)} \\leq\n(k+d)(n-k-d).$$\n\n\nThe geometric point of view also leads to a partial\nLittlewood-Richardson rule for the big quantum cohomology of\nGrassmannians. Using an algorithm similar to the one in\n\\cite{vakil:rationalelliptic}, one can give a positive rule for\ncomputing some Gromov-Witten invariants of $G(k,n)$ (see\n\\cite{coskun1:degenerations}, Thm.\\ 8.1). It would be interesting to\nextend these results to obtain a Littlewood-Richardson rule for\narbitrary genus-zero Gromov-Witten invariants of $G(k,n)$. It would be\nequally interesting to extend these results to other homogeneous\nvarieties, especially flag varieties.\n\n\n\\section{Linear spaces and a quadratic form}\\label{sec-orth}\nA modification of Mondrian tableaux may be used to calculate classes\nof varieties in the isotropic setting as well. A precise description\nis beyond the scope of this article. However, in the spirit of the\nrest of Part \\ref{parttwo}, we give a few illustrative examples. For\nprecise statements and details, we refer the reader to\n\\cite{coskun:quadrics}.\n\n\nRecall that the orthogonal Grassmannian $OG(k,n)$ can be interpreted\nas the Fano variety of $(k-1)$-dimensional projective spaces on a smooth\nquadric hypersurface $Q$ in $\\mathbb{P}^{n-1}$. In order to record\nspecializations of linear spaces on $Q$ we need to denote subquadrics\nof $Q$. We will depict a quadric hypersurface in $\\mathbb{P}^r$ as a square\nof side-length $r+1$ whose northwest and southeast corners have been\ndiagonally chopped off by one unit (see Figure \\ref{quadram}). We will\nrefer to such shapes as {\\it quadrams} (short for the diagram of a\nquadric) and refer to $r+1$ as the side-length of the quadram. We will\nalso need to denote the singular loci of these quadrics. Consequently,\nwe will label our quadrams and write the label of the quadram in the\nsquares denoting the linear spaces along which the quadric is singular.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\begin{picture}(0,0)%\n\\includegraphics{quadram.pstex}%\n\\end{picture}%\n\\setlength{\\unitlength}{3947sp}%\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  \\selectfont}%\n\\fi\\endgroup%\n\\begin{picture}(624,624)(514,-73)\n\\end{picture}%\n\n\\end{center}\n\\caption{A quadram of side-length $4$ depicting a smooth quadric\n  surface in $\\mathbb{P}^3$.}\n\\label{quadram}\n\\end{figure}\n\nSuppose $n$ is odd. Set $m = (n-1)\/2$.  Let $\\lambda = \\lambda_1 >\n\\cdots > \\lambda_s$ and $\\mu = \\mu_{s+1} > \\dots > \\mu_k$ be two\nstrictly decreasing partitions as in \\S \\ref{othergps}. Let\n$\\sigma_{\\lambda, \\mu}$ be the corresponding Schubert cycle in\n$OG(k,n)$. The {\\it quadric diagram} associated to $\\sigma_{\\lambda,\n  \\mu}$ is a collection of $s$ nested squares $S_1 \\subset \\dots\n\\subset S_s$ of side-lengths $m+1-\\lambda_1, \\dots, m+1 -\\lambda_s$,\n$k-s$ nested quadrams $Q_{k-s} \\subset \\cdots \\subset Q_1$ of\nside-lengths $n-\\mu_{s+1} , \\dots, n-\\mu_k$ containing all the squares\nand a function $f$. The function $f$ associates to the unit squares in\nthe anti-diagonal of the quadric diagram the set of labels of the\nquadrams which represent quadrics that are singular at that point.\nWhen $n$ is even, we make the obvious modifications to this\ndefinition. In particular, we have to distinguish between the\nhalf-dimensional linear spaces that belong to the different\nirreducible components. We will depict those that correspond to\n$\\lambda_s = 0$ in solid lines and those that correspond to $\\mu_{s+1}\n= m-1$ in dashed lines. Figure \\ref{quadramexam} shows the quadric\ndiagram associated to $\\sigma_{2}^{2,0}$ in $OG(3,7)$.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\begin{picture}(0,0)%\n\\includegraphics{quadramexam.pstex}%\n\\end{picture}%\n\\setlength{\\unitlength}{3947sp}%\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  \\selectfont}%\n\\fi\\endgroup%\n\\begin{picture}(1074,1111)(589,-410)\n\\put(601,-361){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}$2$}%\n}}}}\n\\put(751,-211){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}$2$}%\n}}}}\n\\end{picture}%\n\n\\end{center}\n\\caption{The quadric diagram associated to $\\sigma_{2}^{2,0}$ in $OG(3,7)$.}\n\\label{quadramexam}\n\\end{figure}\n\nGeometrically, the $\\mathbb{P}^2$s parametrized by the Poincar\\'e dual of\n$\\sigma_{2}^{2,0}$ intersect the line $l$ represented by the square\n$S_1$. They intersect the subquadric everywhere tangent along $l$ in a\nline. Note that this subquadric, denoted by $Q_2$, is singular along\n$l$. Hence, we place the index ``$2$'' in the unit squares contained in\n$S_1$. Finally, the $\\mathbb{P}^2$s are contained in the quadric represented\nby the largest quadram $Q_1$.\n\nOne can use quadric diagrams to calculate the classes of subvarieties\nin orthogonal Grassmannians. For instance to multiply two Schubert\nvarieties in $OG(k,n)$ where the index $\\mu$ does not contribute to\nthe discrepancy, we place the two Schubert cycles in opposite corners\nof an $n\\times n$ square. After initial manipulations, the quadric\ndiagram consists of squares that form a generalized Mondrian tableau\nand nested quadrams containing the squares. Informally, we nest the\nsquares as in the Mondrian tableaux rule for ordinary\nGrassmannians keeping track of the singularities of the quadrics\nrepresented by the quadrams. Eventually all the squares and quadrams\nare nested. However, this might still not correspond to a Schubert\nvariety. We further degenerate these varieties until the quadrics\nrepresented by each quadram is as singular as possible. We give a few\nexamples demonstrating how the procedure works in practice, referring\nthe reader to \\cite{coskun:quadrics} for details.  \n\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\begin{picture}(0,0)%\n\\includegraphics{izorth.pstex}%\n\\end{picture}%\n\\setlength{\\unitlength}{3947sp}%\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  \\selectfont}%\n\\fi\\endgroup%\n\\begin{picture}(3785,3029)(589,-2548)\n\\put(2401,-1711){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}$1$}%\n}}}}\n\\put(3751,-1261){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}$1$}%\n}}}}\n\\put(3751,-2236){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}$1$}%\n}}}}\n\\end{picture}%\n\n\\end{center}\n\\caption{Calculating the intersection $\\sigma_2^{2,0} \\cdot\n  \\sigma_2^{2,0} = 2 \\sigma_{3,1}^1$ in $OG(3,7)$.}\n\\label{orth1}\n\\end{figure}\n\n\\begin{example}\n  We calculate $\\sigma_2^{2,0} \\cdot \\sigma_2^{2,0}$ in $OG(3,7)$.\n  Figure \\ref{orth1} shows the quadric diagrams and the corresponding\n  projective geometry associated to this example.  Geometrically, this\n  corresponds to finding the class of the variety of projective planes\n  on a smooth quadric $Q$ in $\\mathbb{P}^6$ that intersect $2$ general lines\n  $l_1$ and $l_2$. These lines span a $\\mathbb{P}^3$ that intersects $Q$ in a\n  smooth quadric surface (depicted by $Q_2$ in the quadric diagram in\n  the first panel). The planes must contain a line in the ruling\n  opposite to $l_1$ and $l_2$ on this quadric surface. In order to\n  calculate the class we specialize the conditions on the planes.  We\n  move the line $l_1$ along the ruling of a quadric surface contained\n  in $Q$ that intersects $l_2$.  Once we specialize $l_1$ to intersect\n  $l_2$, the limit of the $\\mathbb{P}^3$s spanned by the two lines becomes\n  tangent to the quadric at the point of intersection of $l_1$ and\n  $l_2$. The intersection of the limit $\\mathbb{P}^3$ with the quadric $Q$ is\n  a quadric cone (denoted by $Q_2$ in the second panel). The limit of\n  the planes is the locus of planes that contain a line in this\n  quadric cone (necessarily containing the cone point). Note that\n  since the planes contain the cone point $p$, they have to be\n  contained in the tangent space to $Q$ at $p$ (hence, we shrink $Q_1$\n  by one unit in the second panel).  This is not a Schubert variety.\n  To degenerate it into a Schubert variety, we take a pencil of\n  $\\mathbb{P}^3$s that become more tangent to the quadric $Q$. In the limit\n  the quadric cones break into a union of two planes. The limit of the\n  lines is the locus of lines on each of the planes that pass through\n  the limit of the vertices of the quadric cones. We conclude that\n  $\\sigma_2^{2,0} \\cdot \\sigma_2^{2,0} = 2 \\sigma_{3,1}^1$.  It is not\n  hard to see that both limits occur with multiplicity\n  $1$. \n\n\\end{example}\n\n\n\n\n\\begin{example} \n  The orthogonal Grassmannian $OG(2,6)$ and the two-step flag variety\n  $Fl(1,3;4)$ are isomorphic. It is both instructive and fun to\n  calculate the cohomology ring of $OG(2,6)$ using both quadric\n  diagrams and the Mondrian tableaux rule for two-step flag varieties.\n  Figure \\ref{orthmon} shows two sample calculations. In the left\n  panel, we calculate the class of lines in $OG(2,6)$ that intersect\n  two general lines $l_1$, $l_2$. These are lines that are contained\n  in the intersection of $Q$ with the $\\mathbb{P}^3$ spanned by the two\n  lines. The degeneration is similar to the previous example. However,\n  when the quadric cone breaks in this case, the two planes belong to\n  two different irreducible components.  In the right panel, we\n  calculate the intersection of Schubert varieties parameterizing\n  lines that intersect a plane (one in each of the two rulings) and\n  are contained in the tangent space at a point in the plane. The\n  calculation is similar. In each panel we also show the corresponding\n  Mondrian tableaux calculation below the quadric diagrams.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\begin{picture}(0,0)%\n\\includegraphics{orthmon.pstex}%\n\\end{picture}%\n\\setlength{\\unitlength}{3947sp}%\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  \\selectfont}%\n\\fi\\endgroup%\n\\begin{picture}(7299,2874)(439,-2398)\n\\put(1651,-436){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}$1$}%\n}}}}\n\\put(6601,-211){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}$1$}%\n}}}}\n\\put(4126,-586){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}$1$}%\n}}}}\n\\put(4876,164){\\makebox(0,0)[lb]{\\smash{{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{\\color[rgb]{0,0,0}$1$}%\n}}}}\n\\end{picture}%\n\n\\end{center}\n\\caption{Calculating the intersection $\\sigma_{1}^0 \\cdot \\sigma_{1}^0\n  = \\sigma_{2,0} + \\sigma_{2}^2$ in $OG(2,6)$, equivalently\n  $\\sigma_{1,0,0}^{2,1,2} \\cdot\n  \\sigma_{1,0,0}^{2,1,2}=\\sigma_{1,1,0}^{1,2,2} +\n  \\sigma_{1,1,1}^{2,1,2}$ in $Fl(1,3;4)$ (the left panel) and the\n  intersection $\\sigma_0^1 \\cdot \\sigma^{2,1}= \\sigma_{2,0} +\n  \\sigma_{2}^2$ in $OG(2,6)$, equivalently $\\sigma_{0,0,0}^{1,2,2}\n  \\cdot \\sigma_{1,1,0}^{2,2,1} = \\sigma_{1,1,0}^{1,2,2} +\n  \\sigma_{1,1,1}^{2,1,2} $ in $Fl(1,3;4)$ (the right panel).}\n\\label{orthmon}\n\\end{figure}\n\\end{example}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nDue to weakness of traditional security methods which are based on using cryptography algorithms in upper layers, physical layer security attracts many attentions in the recent years.\nPhysical Layer Security was first addressed by Wyner's celebrated paper \\cite{Wiretap:1975} in which a wiretap channel was studied and the notion of secrecy capacity was introduced. Since then, several studies have extended the Wyner's work, such as \\cite{Gaussian:1978} for Gaussian wiretap channels, \\cite{SecureFading:2008} for fading channels and \\cite{SecrecyMIMO:2011} for Multiple-Input Multiple-Output (MIMO) channels.\n\nOne practical method in this area is using beamforming techniques in combination with emission of an Artificial Noise (AN) at the data source side (Alice) to corrupt an eavesdropper's reception during transmission of data to the destination (Bob) \\cite{Guaranteeing:2008}. During past years several works have developed this idea that most of them aim to maximize secrecy capacity. In a different way, the authors in \\cite{Outage:2012} proposed a probability based approach for employing AN to guarantee a minimum level of outage probability at an unknown Eve while satisfying a certain Quality of service (QoS) requirement at the Bob. It should be noted that all these works are based on exploiting Source Artificial Noise (SAN) that means sending a noise-like signal at the data-source side of a wireless link.\n\nRecently, thanks to the development of full-duplex communications, one node can transmit and receive data signals at the same time and the same frequency band \\cite{Co-Channel:2010}, \\cite{Breaking:2013}. By full-duplex communications, it is possible to adopt Destination Artificial Noise (DAN) at the destination along with the SAN at the source. This idea was first proposed in \\cite{SecureCommunication:2012} and then in \\cite{Improving:2013} for Single-Input Multiple-Output (SIMO) communication systems. In \\cite{Securing:2013} the MIMO communication systems when only one of the receiver antennas is used for data reception has been studied, but in \\cite{Application:2014} destination can allocate more antennas for receiving the data. Similar to SAN, most works on the DAN are also based on maximizing secrecy capacity.\n\nIn this paper unlike to the previous works, we adopt an outage probability based approach for the characterization of the physical layer security that uses both DAN and SAN. In addition, we propose an optimal power allocation method to the DAN, SAN and source information signal to ensure the security requirements. We also investigate the effect of considering a power constraint at the destination for the situation in which Bob can only cancel a certain amount of self-interference. It is shown how this power constraint increases the total required power for a certain level of security. \n\nOur simulation results show that the proposed method brings a significant reduction in the total power consumption compared to the previous works where only SAN is used to ensure a certain outage probability at the Eve. We also represent the effect of Eve's location on the performance of the proposed method. It is shown that in contrast to previous works, we are able to ensure secrecy requirements even where Eve is very close to the Bob. \n\nThis paper is organized as follow. After presenting system model in Section 2, the power allocation problem is proposed in Section 3. In Section 4, simulation results are presented for both constrained and unconstrained power scenarios at the destination. Finally in Section 5, the paper has been concluded.\n\n\\textit{Notation}: Bold symbols in small and capital letter denote vectors and matrices. In addition $\\boldsymbol{(.)}^{H}$ denotes the conjugate transpose and $\\left \\| \\boldsymbol{.} \\right \\|$ is the norm operator.\n\n\n\\section{SYSTEM MODEL}\nIn this paper, we consider a network consisting of a data source (Alice), a destination (Bob) and an eavesdropper (Eve). It is assumed that the source and the destination are equipped with $N_{A}\\geq2$ and $N_{B}\\geq2$ antennas, respectively, while Eve is equipped with $N_{E}\\geq1$ antenna. The channel gains between Alice and Bob is represented by a matrix $\\boldsymbol{H}_{AB}$ of size $N_{A}\\times N_{B}$ and it is assumed to be known to all nodes. In the other hand the channel gains between Alice and Eve is represented by a matrix $\\boldsymbol{H}_{AE}$ of size $N_{A}\\times N_{E}$ and the channel gains between Bob and Eve by a matrix $\\boldsymbol{H}_{BE}$ of size $N_{B}\\times N_{E}$ which are unknown for the legal nodes (Alice and Bob). All channel gains are modelled by independent zero-mean complex Gaussian random variables. The variances of the above channels are $\\sigma_{H_{AB}}^{2}$, $\\sigma_{H_{AE}}^{2}$ and $\\sigma_{H_{BE}}^{2}$ for the channel between Alice and Bob, Alice and Eve and Bob and Eve, respectively. If $\\boldsymbol{H}$ be a channel gains matrix (small scale fading effects), by considering the \\emph{path loss} effect, we can define $\\hat{\\boldsymbol{H}}$ as \\cite{Goldsmith},\\cite{Application:2014}\n\\begin{equation} \\label{eq:Hhat}\n\\hat{\\boldsymbol{H}}= (\\lambda_{0} r^{-\\kappa})^{1\/2} \\boldsymbol{H}\n\\end{equation}\nwhere $\\lambda_{0}$ is a constant that is used for showing the power at a reference signal and it determined by empirical measurements. In addition $r$ is the distance between two nodes and $\\kappa$ is the path loss exponent and depends on the propagation environment. The values of this parameter is typically between 2 and 6.\n\nWe assume that the Eve's location is unknown to the legal nodes and has a uniform distribution in a circle with radius $r_{ab}$, where $r_{ab}$ is the distance between Alice and Bob. Also it is assumed that the legitimate users are located at the center of the area (Fig.~\\ref{fig:model}).\n\n\\begin{figure}[t]\n\\includegraphics[width=\\columnwidth]{paper1model}\n\\caption{System model}\n\\label{fig:model}\n\\end{figure}\n\nTo enhance the security of the system, we assume that only one of the destination antennas is employed for receiving the data signal and the other antennas are assigned to the AN propagation (i.e. DAN). On the other hand, all source antennas could be used to transmit both precoded data signal and AN (i.e. SAN). To decrease the power consumption and enhance the performance at the Bob, we select its best antenna (i.e. the \\textit{j}th antenna) for receiving data from Alice. One approach for selecting this antenna is as follows: \n\\begin{equation} \\label{i-arg}\nj=\\arg \\max_{i} \\left \\{ \\left \\| \\boldsymbol{H}^{i}_{AB} \\right \\| \\right \\}\n\\end{equation}\nwhere $\\boldsymbol{H}^{i}_{AB}$ is the \\textit{i}th column of $\\boldsymbol{H}_{AB}$ which is related to the \\textit{i}th antenna at the destination and $i=1,2,...,N_{B}$. Therefore the channel assigned to the data transmission between Alice and Bob is denoted by a vector $\\boldsymbol{h}_{AB}$ of size $N_{A}$. In addition, the channel matrix between the remaining Bob's antennas and Eve is presented by a matrix $\\boldsymbol{\\tilde{H}}_{BE}$ of size $(N_{B}-1)\\times N_{E}$. \n\nIn addition assuming that Eve uses the selection combining method, the channel vector between Eve's \\textit{k}th antenna and Alice is denoted by $\\boldsymbol{h}_{AE}$ of size $N_{A}$, while $\\boldsymbol{h}_{BE}$ of size $(N_{B}-1)$ denotes the channel between Bob's remained antennas and Eve's \\textit{k}th antenna.\n\nNow let $\\boldsymbol{x}$, $y_{B}$ and $\\boldsymbol{y}_{E}$ be the precoded data signal transmitted by Alice, the received signal at Bob and the received signal at Eve, respectively. Therefore\n\\begin{equation} \\label{y:received}\ny_{B}=\\hat{\\boldsymbol{h}}^{H}_{AB}\\boldsymbol{x}+n_{B}\n\\end{equation}\n\\begin{equation} \\label{z:received}\n\\boldsymbol{y}_{E}=\\hat{\\boldsymbol{H}}^{H}_{AE} \\boldsymbol{x} + \\hat{\\boldsymbol{\\tilde{H}}}^{H}_{BE} \\boldsymbol{\\nu} + \\boldsymbol{n}_{E}\n\\end{equation}\nwhere $\\boldsymbol{\\nu}$ is the DAN emitting by Bob and we assume that its self-interference can be completely canceled at the destination. Assuming there is no knowledge about the Eve's Channel State Information (CSI), DAN power must be equally distributed among $(N_{B}-1)$ Bob's antennas that are assigned to DAN propagation. Hence we can write\n\\begin{equation} \\label{DAN}\n\\boldsymbol{\\nu} = \\sqrt{\\frac{P}{N_{B}-1}}\\sum_{\\substack{i=1 \\\\ i\\neq j}}^{N_{B}}\\eta \\boldsymbol{\\psi}_{i}\n\\end{equation}\nwhere $P\\geq 0$ is the power allocated to DAN, $\\eta$ is a complex scalar with unit magnitude and uniform random phase, $\\boldsymbol{\\psi}_{i}$ is a vector of size $N_{B}$ whose \\textit{i}th element is equal to one and all other elements are equal to zero. $n_{B}$ and $\\boldsymbol{n}_{E}$ of size $N_{E}$ are the Gaussian noise terms with zero mean and variances $\\sigma_{B}^{2}$ and $\\sigma_{E}^{2} \\boldsymbol{I}$, respectively. $\\boldsymbol{x}$ is the beamformed signal transmitted by Alice and could be written as\n\\begin{equation} \\label{x-signal}\n\\boldsymbol{x}=\\sqrt{\\varphi \\acute{P}}d\\boldsymbol{t}+\\sqrt{(1-\\varphi ) \\acute{P}} \\boldsymbol{\\eta}\n\\end{equation}\nwhere $0\\leq \\varphi \\leq 1$ is a parameter that determines the ratio of power assigned to the information signal and SAN, $\\acute{P}>0$ is Alice's total power, $d$ is the information symbol with $\\mathbb{E} \\left \\{ \\left | d \\right |^{2} \\right \\}=1$ and $\\boldsymbol{t}$ of size $(N_{A})$ is normalized $(\\left \\| \\boldsymbol{t} \\right \\|=1)$ beamforming vector. $\\boldsymbol{\\eta}$ is SAN vector of size $(N_{A})$ that is orthogonal to $\\boldsymbol{t}$ $(i.e., \\boldsymbol{t}^{H} \\boldsymbol{\\eta} =0)$ and its covariance  matrix is denoted by $\\boldsymbol{C}_{\\eta}$ as\n\\begin{equation} \\label{covariance:SAN}\n\\boldsymbol{C}_{\\eta}=\\mathbb{E}\\left \\{ \\boldsymbol{\\eta} \\boldsymbol{\\eta}^{H} \\right \\}\n\\end{equation}\nTherefore we have $Tr\\left \\{ \\boldsymbol{C}_{\\eta} \\right \\}=1$. For the beamforming at the source node, as proposed in \\cite{Physical-Layer:2011}, if $\\boldsymbol{t}_{i}$ is the \\textit{i}th eigenvector of $\\boldsymbol{h}_{AB} \\boldsymbol{h}_{AB}^{H}$ and $\\boldsymbol{t}_{1}$ is assumed to be the principal eigenvector, it is assumed that $\\boldsymbol{t}=\\boldsymbol{t}_{1}$. Based on the orthogonality of the eigenvectors of $\\boldsymbol{h}_{AB} \\boldsymbol{h}_{AB}^{H}$, $\\boldsymbol{\\eta}$ is a linear combination of $N_{A}-1$ eigenvectors, and hence it lies in the nullspace of  $\\boldsymbol{h}_{AB}$. Since Eve's CSI is not known to the legal nodes, the noise power equally is distributed to these eigenvectors as follows\n\\begin{equation} \\label{SAN}\n\\boldsymbol{\\eta} =\\sqrt{\\frac{1}{N_{A}-1}}\\sum _{i=2}^{N_{A}}\\eta \\boldsymbol{t}_{i}.\n\\end{equation}  \n\nAccording to the above beamforming model, the signal to noise ratios (SNRs) at the Bob and Eve's \\textit{k}th antenna can be derived as\n\\begin{equation} \\label{SNR:B}\n\\mathrm{SNR_{B}}=\\frac{\\varphi \\acute{P} \\Vert \\hat{\\boldsymbol{h}}_{AB} \\Vert^{2}}{\\sigma_{B}^{2}}\n\\end{equation}\nand\n\\begin{equation} \\label{SNR:E}\n\\mathrm{SNR_{E}^{k}}=\\frac{\\varphi \\acute{P}  \\hat{\\boldsymbol{h}}_{AE}^{H} \\boldsymbol{t}_{1} \\boldsymbol{t}_{1}^{H} \\hat{\\boldsymbol{h}}_{AE}  }{(1-\\varphi ) \\acute{P} \\hat{\\boldsymbol{h}}_{AE}^{H} \\boldsymbol{C}_{\\eta }  \\hat{\\boldsymbol{h}}_{AE}+\\frac{P}{N_{B}-1} \\hat{\\boldsymbol{h}}_{BE}^{H} \\hat{\\boldsymbol{h}}_{BE}}.\n\\end{equation}\n\n\n\n\\section{The Power Allocation Problem}\nIn this section, we discuss about the problem of power allocation to SAN and DAN. To ensure secrecy, $\\gamma _{b}$ and $\\gamma _{e}$ QoS constraints should be satisfied at Bob and Eve and hence the power optimization problem can be written as:\n\\begin{subequations}\n\\label{eq:opt1}\n\\begin{align}\n\\min_{P, \\acute{P}, \\varphi } \\quad & P+\\acute{P} \\\\\n\\label{eq:opt1-2}\n\\mathrm{s.t.} \\quad & \\mathrm{SNR_{B}} \\ge \\gamma _{b} \\\\\n\\label{eq:opt1-3}\n& \\mathbb{P} \\lbrack \\mathrm{SNR_{E}} \\le \\gamma _{e}\\rbrack \\ge \\beta.\n\\end{align}\n\\end{subequations}\n\nIn above optimization problem, (\\ref{eq:opt1-2}) is for satisfying a certain level of SNR at Bob and (\\ref{eq:opt1-3}) guarantees a minimum value of outage probability at Eve. \n\nAs it can be seen, our object is a joint power optimization for source and destination. Considering that Eve uses the selection combining method and the channel matrix coefficients are independent, we have\n\\begin{equation} \\label{P:SNR_Ek}\n\\mathbb{P} \\lbrack \\mathrm{SNR_{E}} \\le \\gamma _{e}\\rbrack = \\prod_{k=1}^{N_{E}} \\mathbb{P} \\lbrack \\mathrm{SNR_{E}^{k}} \\le \\gamma _{e}\\rbrack \\\\\n= \\mathbb{P} \\lbrack \\mathrm{SNR_{E}^{k}} \\le \\gamma _{e}\\rbrack^{N_{E}}\n\\end{equation}\nBy substituting (\\ref{SNR:B}), (\\ref{SNR:E}) and (\\ref{P:SNR_Ek}) into (\\ref{eq:opt1}) and considering (\\ref{eq:Hhat}), we have:\n\\begin{subequations} \n\\label{eq:opt2}\n\\begin{align}\n\\label{eq:opt2-1}\n& \\min _{P, \\acute{P}, \\varphi } \\quad  P+ \\acute{P} \\\\\n\\label{eq:opt2-2}\n& \\mathrm{s.t.} \\nonumber \\\\\n& \\quad  \\varphi \\acute{P} \\ge \\gamma _{b}\\sigma _{B}^{2} (\\lambda_{0} r_{AB}^{-\\kappa})^{-1} \\Vert {\\boldsymbol{h}}_{AB}\\Vert^{-2} \\\\\n\\label{eq:opt2-3}\n& \\mathbb{P} \\lbrack \\frac{\\varphi \\acute{P} (\\lambda_{0} r_{AE}^{-\\kappa})  {\\boldsymbol{h}}_{AE}^{H} \\boldsymbol{t}_{1} \\boldsymbol{t}_{1}^{H} {\\boldsymbol{h}}_{AE} }{(1-\\varphi ) \\acute{P} (\\lambda_{0} r_{AE}^{-\\kappa}) {\\boldsymbol{h}}_{AE}^{H} \\boldsymbol{C}_{\\eta } {\\boldsymbol{h}}_{AE}+\\frac {P} {N_{B}-1} (\\lambda_{0} r_{BE}^{-\\kappa}) {\\boldsymbol{h}}_{BE}^{H} {\\boldsymbol{h}}_{BE}} \\nonumber \\\\\n& \\le \\gamma _{e} \\rbrack  \\ge \\beta^{\\frac{1}{N_{E}}}\n\\end{align}\n\\end{subequations}\nWe can rewrite (\\ref{eq:opt2-3}) as\n\\begin{equation} \\label{eq:opt3-3}\n\\mathbb{P} \\lbrack \\boldsymbol{h}^{H}_{AE} \\boldsymbol{a} \\boldsymbol{h}_{AE} + b \\boldsymbol{h}_{BE}^{H} \\boldsymbol{h}_{BE} \\le \\sigma _{E}^{2} \\rbrack \\ge \\beta^{\\frac{1}{N_{E}}}\n\\end{equation}\nwhere $\\boldsymbol{a}$ and $b$ are defined as below\n\\begin{subequations} \n\\label{eq:AandB}\n\\begin{align}\n\\label{eq:a}\n& \\boldsymbol{a}= \\acute{P} \\lambda_{0} \\bar{r}_{ae}^{-\\kappa} (\\varphi \\gamma \n_{e}^{-1} \\boldsymbol{t}_{1} \\boldsymbol{t}_{1}^{H} - (1-\\varphi ) \\boldsymbol{C}_{\\eta }) \\\\\n\\label{eq:b}\n& b= - \\frac {P} {N_{B}-1} \\lambda_{0} \\bar{r}_{be}^{-\\kappa}  \n\\end{align}\n\\end{subequations}\nwhere $\\bar{r}_{ae}=\\mathbb{E}[r_{ae}]$ and $\\bar{r}_{be}=\\mathbb{E}[r_{be}]$. The left side in (\\ref{eq:opt3-3}) can be interpreted as a Cumulative Distribution Function (CDF). By defining \n\\begin{equation} \\label{Xhah}\nX=\\boldsymbol{h}^{H}_{AE} \\boldsymbol{a} \\boldsymbol{h}_{AE}\n\\end{equation}\n\\begin{equation} \\label{Yhh}\nY=\\boldsymbol{h}_{BE}^{H} \\boldsymbol{h}_{BE}\n\\end{equation}\nand $Z=X+bY$, we have\n\\label{eq:Fz}\n\\begin{align}\nF_{Z}(z)&= \\mathbb{P} \\lbrack X+bY \\le z \\rbrack \\nonumber \\\\\n& = \\mathbb{P} \\lbrack \\boldsymbol{h}^{H}_{AE} \\boldsymbol{a} \\boldsymbol{h}_{AE} + b \\boldsymbol{h}_{BE}^{H} \\boldsymbol{h}_{BE} \\le z \\rbrack \\nonumber \\\\\n\\label{eq:Fz-3}\n& =\\int_{-\\infty}^{+\\infty}{f_{Y}(y)}dy \\int_{-\\infty}^{z-by} {f_{X}(x)}dx \n\\end{align}\nwhere $\\int_{-\\infty}^{z-by} {f_{X}(\\tau)}d\\tau=F_{X}(z-by)$ is the CDF of $X$. On the other hand $X$ is an indefinite Hermitian quadratic form for $x \\ge 0$ \\cite{distribution:2009,Outage:2012}, so $F_{X}(x)$ can be derived as \\cite{distribution:2009}:\n\\begin{equation} \\label{eq:Fx}\nF_{X}(x)=u(x)+\\frac{\\alpha _{1}}{\\vert \\lambda _{1}\\vert }e^{(\\frac{-x}{\\lambda _{1}})} u(\\frac{x}{\\lambda _{1}})\n\\end{equation}\nwhere\n\\begin{equation} \\label{eq:alpha}\n\\alpha_{1}=-\\frac{\\lambda _{1}}{(1-\\frac{\\lambda _{2}}{\\lambda_{1}})^{(N_{A}-1)}}\n\\end{equation}\nand\n\\begin{subequations} \n\\label{eq:lambda}\n\\begin{align}\n\\label{eq:lambda-1}\n& \\lambda _{1}=\\varphi \\acute{P}\\lambda _{0} \\bar{r}_{ae}^{-\\kappa }\\gamma_{e}^{-1}\\sigma _{H_{AE}}^{2}\\ge 0 \\\\\n\\label{eq:lambda-2}\n& \\lambda _{2}=-(1-\\varphi )\\acute{P}\\lambda _{0} \\bar{r}_{ae}^{-\\kappa }\\sigma_{H_{AE}}^{2}(N_{A}-1)^{-1}\\le 0.\n\\end{align}\n\\end{subequations}\n\nIn addition, $Y$ is sum of the squares of $2(N_{B}-1)$ independent normal random variables and has chi-squared ($\\chi^{2}$) distribution as \n\\begin{equation} \\label{eq:fY}\nf_{Y}(y;2(N_{B}-1))=\\frac{(\\frac{y}{\\sigma_{H_{BE}}^{2}})^{(N_{B}-2)}.e^{(\\frac{-y}{2\\sigma_{H_{BE}}^{2}})}}\n{\\sigma _{H_{BE}}^{2}2^{(N_{B}-1)}\\Gamma (N_{B}-1)} u(y).\n\\end{equation}\n\nTherefore (\\ref{eq:Fz-3}) can be rewritten as \n\\begin{equation} \\label{eq:Fz2}\nF_{Z}(z)=\\int_{0}^{+\\infty }{f_{Y}(y) dy}\\int_{-\\infty}^{z-by}{f_{X}(x)}dx,\n\\end{equation}\nConsidering $(z-by) \\geq 0$, substituting (\\ref{eq:Fx}) and (\\ref{eq:fY}) into (\\ref{eq:Fz2}) and considering $\\sigma_{H_{BE}}^{2}=1$, we have\n\\label{eq:Fz3}\n\\begin{align}\nF_{Z}(z)& =\\int_{0}^{+\\infty}{\\frac{1}{2^{(N_{B}-1)}(N_{B}-2)!}y^{(N_{B}-2)}e^{-0.5y}dy} \\nonumber \\\\ \n\\label{eq:Fz3-2}\n& +\\int_{0}^{+\\infty}{\\frac{\\alpha _{1}e^{(\\frac{-z}{\\lambda_{1}})}}{2^{(N_{B}-1)}(N_{B}-2)!\\vert \\lambda _{1}\\vert}y^{(N_{B}-2)}e^{(\\frac{b}{\\lambda _{1}}-0.5)y}dy}\n\\end{align}\nAfter some manipulations, $F_{Z}(z)$ in (\\ref{eq:Fz3-2}) derived as\n\\begin{equation} \\label{eq:Fz4}\nF_{Z}(z)=(1+\\frac{(-1)^{N_{B}}e^{(\\frac{-z}{\\lambda_{1}})}}{2^{(N_{B}-1)}(1-\\frac{\\lambda _{2}}{\\lambda_{1}})^{N_{A}-1}(\\frac{b}{\\lambda _{1}}-0.5)^{(N_{B}-1)}}).\n\\end{equation}\n\nSubstituting (\\ref{eq:lambda}) and (\\ref{eq:Fz4}) into (\\ref{eq:opt2}), the optimization problem in (\\ref{eq:opt2}) is simplified as\n\\begin{subequations} \n\\label{eq:opt4}\n\\begin{align}\n\\label{eq:opt4-1}\n& \\min _{P, \\acute{P}, \\varphi } \\quad P+ \\acute{P} \\\\\n\\label{eq:opt4-2}\n& \\mathrm{s.t.} \\quad \\varphi \\acute{P} \\ge \\gamma _{b}\\sigma _{B}^{2} (\\lambda_{0} r_{AB}^{-\\kappa})^{-1} \\Vert {\\boldsymbol{h}}_{AB}\\Vert^{-2} \\\\\n\\label{eq:opt4-3}\n& (1-\\frac{2^{(1-N_{B})}e^{(\\frac{-\\gamma _{e}\\sigma _{E}^{2}}{\\varphi \\acute{P}\\lambda \n_{0} \\bar{r}_{ae}^{-\\kappa }\\sigma_{H_{AE}}^{2}})}}{(1+\\frac{(1-\\varphi )\\gamma _{e}}{\\varphi (N_{A}-1)})^{(N_{A}-1)}(\\frac{P \\bar{r}_{be}^{-\\kappa }\\gamma _{e}}{(N_{B}-1) \\varphi \\acute{P} \\bar{r}_{ae}^{-\\kappa }\\sigma _{H_{AE}}^{2}}+0.5)^{(N_{B}-1)}}) \\nonumber \\ge \\beta^{\\frac{1}{N_{E}}}. \\\\\n\\end{align}\n\\end{subequations}\n\nAccording to (\\ref{eq:opt4-3}), it is obvious that the optimum value for (\\ref{eq:opt4-2}) is  $\\varphi \\acute{P} = \\gamma _{b}\\sigma _{B}^{2} (\\lambda_{0} r_{AB}^{-\\kappa})^{-1} \\Vert {\\boldsymbol{h}}_{AB}\\Vert^{-2}$, therefore the optimization method needs to only be applied to (\\ref{eq:opt4-3}). We use standard numerical methods to solve the problem.\n\n\n\n\\section{Simulation and Numerical Results}\nIn this section, we evaluate the performance of our proposed method by computer simulations. In our simulations, it is assumed that $\\sigma_{H_{AB}}^{2}=\\sigma_{H_{AE}}^{2}=\\sigma_{H_{BE}}^{2}=1$, $r_{AB}=2$ \\emph{Km}, $\\gamma_{E}=0.5$ and $\\sigma^{2}_{B}=\\sigma^{2}_{N}=4 \\times 10^{-14}$. For the path loss, it is assumed that $\\lambda_{0}=10^{-1}$ and $\\kappa=3$. Also the channel between Alice and Bob, $\\boldsymbol{H}_{AB}$ is generated randomly with a distribution of $\\boldsymbol{H}_{AB} \\sim \\mathcal{CN}(0,\\sigma_{H_{AB}}^{2} \\boldsymbol{I})$.\n\nTo represent how changes in the main parameters of our problem influence the security and power consumption, we first consider a scenario in which the destination has no power constraint for DAN. Then we also investigate the effect of a power constraint on DAN at the destination.\n\n\\subsection*{Case a) Destination has no power constraint for DAN}\nAs mentioned before, we assume that the Eve's location is distributed uniformly on the area and $\\bar{r}_{ae}=\\bar{r}_{be}=1000$. In addition we first assume that $N_{E}=N_{A}=N_{B}=4$. As it is seen in Fig.~\\ref{fig:beta}, in the case that both DAN and SAN are used, the required power for AN to achieve a given outage probability at the Eve is much less than the case that only SAN is used. In addition, in both cases, with increasing $\\gamma_{B}$, more power is needed to guarantee the given outage probability at the Eve. \n\nIn Fig.~\\ref{fig:Ne}, we investigate the effect of different number of antennas at the Eve on the required power for the case $N_{A}=N_{B}=4$. It could be seen that when Eve has more capabilities, more power is needed to achieve a given outage probability. In Fig.~\\ref{fig:Nab}, the required power for AN is shown for different values of $N_{A}$ and $N_{B}$, when $N_{E}=4$.\n\nIn another scenario, we assume Alice and Bob are located at locations $(-1000,0)$ and $(1000,0)$, $N_{E}=N_{A}=N_{B}=4$, $\\gamma_{B}=0.5$ and  $\\beta=0.6$. The total power should be allocated to AN to achieve a target outage probability at Eve $(\\beta)$ while Eve moves from $(-1500,0)$ to $(15000,0)$ is shown in Fig.~\\ref{fig:distance}. From this figure, it is seen that by using DAN we need less power to guarantee a similar $\\beta$. Especially in the case that Eve is close to Bob, it is still possible to guarantee a given security.\n\n\n\\begin{figure}[h]\n\\includegraphics[width=\\columnwidth]{sim1}\n\\caption{The required power for AN (total SAN and DAN) versus the target outage probability ($\\beta$) at the Eve, for $N_{E}=4$, for $\\gamma_{E}=0.4$ and $0.6$.}\n\\label{fig:beta}\n\\end{figure}\n\n\\begin{figure}[h]\n\\includegraphics[width=\\columnwidth]{sim1Ne}\n\\caption{The required power for AN versus the number of antennas at Eve, for $\\beta=0.6$ and $0.8$.}\n\\label{fig:Ne}\n\\end{figure}\n\n\\begin{figure}[h]\n\\includegraphics[width=\\columnwidth]{sim1Nab}\n\\caption{The required power for AN versus the target outage probability ($\\beta$) at the Eve, for different sets of $N_{A}$ and $N_{B}$ when $N_{E}=4$.}\n\\label{fig:Nab}\n\\end{figure}\n\n\\begin{figure}[h]\n\\includegraphics[width=\\columnwidth]{sim2}\n\\caption{The optimal power is needed for AN (total SAN and DAN) versus Eve's location changes from $(-1500,0)$ to $(1500,0)$, for $\\beta=0.6$}\n\\label{fig:distance}\n\\end{figure}\n\n\n\n\\subsection*{Case b) Destination has a power constraint}\nAlthough \\emph{Full Duplex} technique is advancing so fast and today it is possible to cancel even high levels of \\emph{Self-Interference} at the destination (\\cite{Co-Channel:2010} and \\cite{Breaking:2013}), but it is still useful to investigate how power consumption will be changed by limiting the maximum power allocated to the DAN. This is the case that is considered in Fig.~\\ref{fig:constraint}. As we see in this figure, at the outage probability $\\beta=0.9$ and for $N_{E}=N_{A}=N_{B}=4$, when DAN power limitation decreases from $2 mW$ to $0.5 mW$, total required Power for AN increases from $15 mW$ to $20 mW$.\n\n\n\\begin{figure}[h]\n\\includegraphics[width=\\columnwidth]{sim3}\n\\caption{The optimal power is needed for AN (total SAN and DAN) versus target outage probability ($\\beta$) at Eve, while there is a power constraint at the Destination}\n\\label{fig:constraint}\n\\end{figure}\n\n\n\\section{Conclusion}\n\nIn this paper, using destination artificial noise along with source artificial noise is introduced to guarantee an intended outage probability at Eve, while ensuring a certain SNR at Bob. For both constrained and unconstrained power allocation scenarios it has been shown how using the DAN, decreases the total power which is required to guarantee an intended outage probability at Eve. Using outage probability approach instead of secrecy capacity, makes our solution appropriate for the quasi-static channels.\n\n\n\\bibliographystyle{spbasic}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nFor some years, the static QCD potential $V_{\\rm QCD}(r)$\nhas been successfully computed\nin lattice simulations.\nFor instance, the distance $r$ \nof the potential computed in some of most recent lattice computations in the quenched approximation\nranges from about 0.1~fm to beyond 2~fm.\nComputed results by different groups, when superimposed with one another, \ncome more or less on a single curve\nwith fairly good accuracy, showing very good stability of the lattice predictions\n(see Fig.~\\ref{comp-VS-lat}(a) below).\nIn particular, they show that numerically the QCD potential tends to a linear \npotential smoothly at large distances, in accord with the\nconfinement picture of the quarks.\n\nIn this paper, we analyze the potential in the distance region\nwhich is smaller than (but not too small as compared to) \nthe typical hadron scale $\\Lambda_{\\rm QCD}^{-1}$.\nMore specifically, we consider the region\n$0.1~{\\rm fm} \\hbox{ \\raise3pt\\hbox to 0pt{$<$}\\raise-3pt\\hbox{$\\sim$} } r \\hbox{ \\raise3pt\\hbox to 0pt{$<$}\\raise-3pt\\hbox{$\\sim$} } 0.5$--1~fm if expressed in physical units.\nThis is the region that is relevant to \nspectroscopy of the heavy quarkonia, such as bottomonium and charmonium.\nIn this region, it is known empirically that the static potential\ncan be approximated well by a Coulomb + linear form.\n\nIn this distance range, accuracy of the perturbative predictions for the\nstatic potential improved drastically around year 1998\n\\cite{Hoang:1998nz}.\nIt was proposed to subtract IR renormalon contained in the $r$-independent\n(constant) part of the potential, e.g.\\ by\ncomputing the total energy $2m_{\\rm pole}+V_{\\rm QCD}(r)$ after reexpressing \n$m_{\\rm pole}$\nin terms of the $\\overline{\\rm MS}$ mass or by computing the force.\nThen,\nwe find dramatic improvement in convergence of the perturbative\nseries, as well as much more stable perturbative predictions against\nscale variation.\nMoreover, once we obtain much more accurate predictions,\ngood agreement with phenomenological potentials\nand with lattice computations of the potential has been observed\nin the distance range $0.1~{\\rm fm} \\hbox{ \\raise3pt\\hbox to 0pt{$<$}\\raise-3pt\\hbox{$\\sim$} } r \\hbox{ \\raise3pt\\hbox to 0pt{$<$}\\raise-3pt\\hbox{$\\sim$} } 0.5$~fm;\nby now, this has been confirmed by several groups \\cite{Sumino:2001eh,Necco:2001xg,Pineda:2002se}.\n\nIn this article we review our recent work \\cite{Sumino:2005cq}, in which we analyze\nthe QCD potential in the framework of operator-product expansion\n(OPE) \\cite{Brambilla:1999xf}.\nWe assemble most recent developments of the perturbative\npredictions and lattice computations.\nUsing OPE, one can separate perturbative contributions\nfrom non-perturbative contributions\nin an unambiguous manner.\nIn this way, we are able to determine\n(1) the size of non-perturbative contributions to the potential, and\n(2) the relation between $\\Lambda_{\\overline{\\rm MS}}$ and lattice scale (Sommer scale).\nIn practical applications, the latter determination would be particularly\nimportant, since it can be used to determine\n$\\alpha_S(M_Z)$ with high accuracy in near future, when lattice simulations \nincorporating dynamical quarks with realistic masses will become\navailable.\n\n\\section{OPE of the QCD potential}\n\nThe OPE of the static QCD potential was developed around year 1999 \n\\cite{Brambilla:1999xf},\nwithin the framework of potential-NRQCD effective field theory \\cite{Pineda:1997bj}.\nConceptually it is close to OPEs of other physical quantities, with which one\nmay be more familiar.\nWhen there is a small parameter as compared to the typical hadron size\n(in our case the distance $r \\ll \\Lambda_{\\rm QCD}^{-1}$ \nbetween static quark and antiquark),\none constructs a series of operators as an expansion in $r$.\nEach operator has a Wilson coefficient, which is perturbatively\ncomputable, whereas a non-perturbative contribution is contained\nin the matrix element of the operator.\nThe OPE of the QCD potential reads\n\\begin{eqnarray}\nV_{\\rm QCD}(r) = V_S(r) + \\delta E_{\\rm US}(r)\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n\\delta E_{\\rm US}  \\propto \\alpha_S \\int_0^\\infty dt \\,\n\\, e^{-i \\, \nt \\, \\{ V_O(r) - V_S(r) \\} } \\,\n\\bra{0} \\vec{r}\\cdot\\vec{E}^a(t) \\,\n\\varphi(t,0)_{ab} \\,\n\\vec{r}\\cdot\\vec{E}^b(0) \\ket{0}\n+ {\\cal O}(r^3) .\n\\end{eqnarray}\n\nHere, $V_S(r)$ denotes the Wilson coefficient corresponding to the\nleading UV contribution to the potential (whose operator is the identity operator).\n$V_S(r)$ is referred to as ``singlet potential,'' is perturbatively computable,\nand has a Coulombic behavior (with logarithmic corrections)\n$\\sim -1\/|r \\log r|$ at short-distances.\nWe note that $V_S(r)$ is different from the perturbative expansion of\n$V_{\\rm QCD}(r)$;\nall IR contributions (IR divergences and IR renormalons)\nhave been subtracted and absorbed into a non-perturbative contribution,\nso that the perturbative expansion of $V_S(r)$ is more convergent\nthan that of $V_{\\rm QCD}(r)$.\nSee \\cite{Sumino:2005cq} for the precise definition of $V_S(r)$.\n\nOn the other hand, $\\delta E_{\\rm US}(r)$ denotes the\nleading non-perturbative contribution.\nIt is given as an integral over time of an operator matrix element.\nThe operator is local in space but non-local in time.\nThe behavior of $\\delta E_{\\rm US}(r)$ at small-distances is known\nexactly to be ${\\cal O}(\\Lambda_{\\rm QCD}^4 r^3)$ when $r$ is very small\n(when $V_O-V_S \\approx C_A \\alpha_S\/r \\gg \\Lambda_{\\rm QCD}$ holds).\nIt is, however, doubious whether this strong condition is met in the distance\nregion of our interest, $0.1~{\\rm fm} \\hbox{ \\raise3pt\\hbox to 0pt{$<$}\\raise-3pt\\hbox{$\\sim$} } r \\hbox{ \\raise3pt\\hbox to 0pt{$<$}\\raise-3pt\\hbox{$\\sim$} } 1$~fm.\nIn this distance region, where it is more likely that $V_O-V_S \\sim \\Lambda_{\\rm QCD}$, the behavior of\n$\\delta E_{\\rm US}(r)$ cannot be predicted model-independently.\nAccording to some models it is\nestimated to be ${\\cal O}(\\Lambda_{\\rm QCD}^3 r^2)$.\nIn any case, in the following analysis, we will use only the fact that\n$\\delta E_{\\rm US}(r) \\to 0$ as $r \\to 0$, which is true because at very\nsmall distances $\\delta E_{\\rm US}(r)={\\cal O}(\\Lambda_{\\rm QCD}^4 r^3)$.\n\nIntuitively one can interpret in the following way.\nIt is the contributions of gluons, whose wavelengths are smaller than\n$r$, that generates the Coulomb singularity of the potential\nas $r \\to 0$.\nOn the other hand, non-perturbative \ncontributions can be regarded as contributions of\ngluons whose wavelengths are larger than $r$.\nSuch gluons see only the total charge of the system, hence, as $r \\to 0$,\nthey decouple from the color-singlet system.\nTherefore,\ntheir contributions vanish as $r \\to 0$.\n\n\\section{Comparison of {\\boldmath $V_S(r)$} and recent lattice computations}\n\nIn order to make a numerical cross chek of the prediction of OPE,\nwe first compare the perturbative predictions of the leading Wilson\ncoefficient $V_S(r)$ and\nrecent lattice results.\nIn Fig.~\\ref{comp-VS-lat}(a) we plot lattice data of the QCD potential from \nthree different\ngroups \\cite{Necco:2001xg,Takahashi:2002bw}.\nIn the same figure, the perturbative predictions of $V_S(r)$ are\nplotted up to the\nnext-to-next-to-next-to-leading logarithmic order (NNNLL).\nAs a salient feature, we see that the perturbative predictions\nfollow the lattice data up to larger distances as we increase the order.\n\n\\begin{figure}\n\\hspace*{-17mm}\n\\includegraphics[width=.63\\textwidth]{comp-lat.eps}\n\\psfrag{LL}{\\footnotesize  LL}\n\\psfrag{NLL}{\\hspace{2mm}\\footnotesize  NLL}\n\\psfrag{NNLL}{\\footnotesize NNLL}\n\\psfrag{NNNLL}{\\hspace{0mm}\\footnotesize NNNLL}\n\\psfrag{Large-beta0}{\\hspace{-3mm}\\footnotesize large-$\\beta_0$ appr.}\n\\psfrag{r0}{\\hspace{-6mm} $r=r_0$}\n\\psfrag{delEUS}{\\hspace{-13mm} $[V_{\\rm latt}(r)-V_S(r)]\/\n\\Lambda_{\\overline{\\rm MS}}^{\\mbox{\\scriptsize 3-loop}}$}\n\\psfrag{r}\n{\\hspace{-3mm}\n$r \\cdot \\Lambda_{\\overline{\\rm MS}}^{\\mbox{\\scriptsize 3-loop}}$}\n\\hspace*{5mm}\n\\includegraphics[width=.4\\textwidth]{delEUS.eps}\\\\\n\\hspace{44mm}(a)\n\\hspace{68mm}(b)\n\\caption{\n(a) Comparison of the Wilson coefficient $V_S(r)$ and lattice data.\n(b) $V_{\\rm latt}(r)-V_S(r)$ vs.\\ $r$.\nWe set $r_0\\, \n\\Lambda_{\\overline{\\rm MS}}^{\\mbox{\\tiny{3-loop}}}\n=0.602$ only in these figures.\n}\n\\label{comp-VS-lat}\n\\end{figure}\n\nFollowing input parameters were chosen in depicting this figure.\nFirst, we note that in order to make a comparison between $V_S(r)$\nand lattice data,\nthe relation between $\\Lambda_{\\overline{\\rm MS}}$ and lattice scale\n(we take the Sommer scale $r_0$)  is needed.\nThis is because, a priori\neach lattice data set\nis given in units of $r_0$, whereas perturbative predictions\nare given in units of $\\Lambda_{\\overline{\\rm MS}}$.\nThus, conversion of units is needed to compare them in common units.\nHere, only in this section, we use the central value of the conventionally\nknown relation $r_0\\, \n\\Lambda_{\\overline{\\rm MS}}^{\\mbox{\\tiny{3-loop}}}\n=0.602 \\pm 0.048$ \\cite{Capitani:1998mq}.\nOther inputs are:\n$\\alpha_S(Q)=0.2$, $n_l=0$;\nat NNNLL, the 3-loop non-logarithmic term\n$\\bar{a}_3$ is not known yet, only some estimates exist,\nhence, we used\nPineda's estimate for $\\bar{a}_3$ \\cite{Pineda:2002se}\nand varied it within a range including\nother estimates as a part of our estimates of uncertainties;\nwe chose C+L scheme in computing $V_S(r)$.\n\nTo verify the prediction of OPE, a closer inspection is needed.\nData points in Fig.~\\ref{comp-VS-lat}(b) \nshow the differences of $V_S(r)$ and the lattice data \\cite{Necco:2001xg},\nwhere the vertical scale is magnified as compared to Fig.~\\ref{comp-VS-lat}(a).\nAt LL, one can still see a Coulombic contribution in the difference, but\nas the order is increased, a Coulombic contribution vanishes and\nthey tend to be regular at the origin.\nSolid lines in the same figure show fits of the data points\nby cubic polynomials,\nusing data points at \n$r  \\Lambda_{\\overline{\\rm MS}}^{\\mbox{\\tiny 3-loop}}\n<0.5$.\nOne sees that the solid lines approximate the data points \nup to larger distances as the order increases,\nwhich indicate that these data points become regular around\nthe origin as the order increases.\nThese features support the OPE prediction, that the non-perturbative\ncontribution \n$\\delta E_{\\rm US}(r)$ vanishes as $r \\to 0$.\nWe note that such behavior can be observed only when we take the\ndifference between $V_S(r)$ and the high quality lattice data, since\nin Fig.~\\ref{comp-VS-lat}(a), \nperturbative predictions become so steep toward the origin that\nany steep curve tends to go through the lattice data points.\n\n\\section{Determination of  $\\delta E_{\\rm US}(r)$ and\n$r_0\\, \\Lambda_{\\overline{\\rm MS}}^{\\mbox{\\tiny{3-loop}}}$\n}\n\nIn the light of the observation in the previous section, we\nperform a simultaneous fit to determine the non-perturbative contribution\n$\\delta E_{\\rm US}(r)$\nand the relation between lattice scale and $\\Lambda_{\\overline{\\rm MS}}$,\n$x \\equiv r_0\\, \n\\Lambda_{\\overline{\\rm MS}}^{\\mbox{\\tiny{3-loop}}}\n$.\n(In the previous section we chose a specific value for $x$ determined\nfrom another source, but in this\nsection we determine its value using the QCD potential alone.)\n\nLet us explain why we have a high sensitivity to the value of $x$.\nIt constitutes the essence of our analysis in this article.\nWe compare the lattice data for the potential and perturbative prediciton\nof $V_S(r)$ in some {\\it common units}.\nHere, we choose to compare in units of $\\Lambda_{\\overline{\\rm MS}}$, hence we convert the\nlattice data into these units by using any chosen value of $x = r_0\\, \n\\Lambda_{\\overline{\\rm MS}}^{\\mbox{\\tiny{3-loop}}}\n$.\nLet us write\n\\begin{eqnarray}\nV_{\\rm latt}(r;x)-V_S(r)\n= [ V_{\\rm latt}(r;x_{\\rm true})-\nV_S(r)]~+~[\nV_{\\rm latt}(r;x)-\nV_{\\rm latt}(r;x_{\\rm true})] .\n\\end{eqnarray}\nIt is a trivial equality, where $x_{\\rm true}$ denotes the true value of $x$.\nThe first bracket $[...]$ on the right-hand-side coincides with the\nnon-perturbative contribution $\\delta E_{\\rm US}(r)$, hence\nit vanishes as $r \\to 0$.\nOn the other hand, the second bracket on the right-hand-side takes\na ``Coulomb''+linear form as long as $x \\neq x_{\\rm true}$.\n(Here, ``Coulomb'' includes logarithmic corrections at short-distances.)\nIt follows from the fact that $V_{\\rm latt}(r;x)$ takes a ``Coulomb''+linear form\nand that $[\nV_{\\rm latt}(r;x)-\nV_{\\rm latt}(r;x_{\\rm true})]$ is merely the difference of these functions\nafter rescaling $V_{\\rm latt}(r;x)$ in the \nvertical and horizontal directions.\nHence, the second term contains a Coulombic singularity as $r \\to 0$ unless \n$x = x_{\\rm true}$.\nIt means that in order to find the true value of $x$,\nwe should adjust $x$ such that\nthe leading Coulombic behavior vanishes\nin $V_{\\rm latt}(r;x)-V_S(r)$\nas $r \\to 0$.\nThat the fit is determined by the leading singular behavior guarantees a\nhigh sensitivity to $x$.\n\nAs for $\\delta E_{\\rm US}(r)$, we simply fit it by a quadratic polynomial using\nthe data points at $r  \\Lambda_{\\overline{\\rm MS}}^{\\mbox{\\tiny 3-loop}}\n<0.5$, since $\\delta E_{\\rm US}(r)$ is regular as $r \\to 0$.\\footnote{\nWe also tried a cubic fit, where the results of the fit did not change considerably.\n}\n\nWe show the results of the fit for $\\delta E_{\\rm US}(r)$\nin Fig.~\\ref{fit-res}.\nThe central value of the coefficient of the linear potential is\nabout 1 in units of \n$(\\Lambda_{\\overline{\\rm MS}}^{\\mbox{\\tiny 3-loop}})^2$.\nWe note that the coefficient of the linear potential \n(string tension) determined from\nthe large distance behavior of lattice results is about 3.8 in the same units.\nHence, our result (central value) indicate that the non-perturbative contribution\ncontains only about one quarter of the string tension, as a \ncomponent of the linear potential\nin the short-distance region.\nThree quarters of the string tension belongs to the perturbative \ncontribution $V_S(r)$.\nThe present error is somewhat large, so it is still consistent at 95\\% confidence level\nthat either {\\it all} of the string tension is contained in  $V_S(r)$ or only\n1\/3 of it is contained in $V_S(r)$.\n\n\\begin{figure}\n\\hspace*{40mm}\n\\psfrag{A1}{$A_1$} \n\\psfrag{A2}{$A_2$} \n\\psfrag{68pcCL}{\\footnotesize 68\\%} \n\\psfrag{95pcCL}{\\footnotesize 95\\%} \n\\includegraphics[width=.4\\textwidth]{contour-plot-AB-force.eps}\n\\caption{\nAllowed regions in the parameter space $(A_1,A_2)$ at\n68\\% and 95\\% CL, when \n$\nV_{\\rm latt}(r;x)-\nV_S(r)$\nis fitted by $A_1 \\cdot \\Lambda_{\\overline{\\rm MS}}^2 r +A_2 \\cdot \\Lambda_{\\overline{\\rm MS}}^3 r^2$.\n}\n\\label{fit-res}\n\\end{figure}\n\nAnother notable feature is that (in the schemes which we examined) \n$\\delta E_{\\rm US}(r) =0$ is strongly disfavored.\nIt is seen from Fig.~\\ref{fit-res}, where the origin is excluded from the\nallowed region.\nIt means that there is a non-vanishing non-perturbative contribution.\nIn fact, this is a new result, since in a previous similar study\nby Pineda \\cite{Pineda:2002se}, \n$\\delta E_{\\rm US}(r) $ contained a larger error and\nwas still consistent with zero.\nThe major difference of the analyses, which led to the improvement of\nthe bound in our analysis, is that we determined $x$ simultaneously\nfrom the fit.\n\nWe also obtained\n\\begin{eqnarray}\nx = 0.574 \\pm 0.042 .\n\\label{value-x}\n\\end{eqnarray}\nIt should be compared with the conventional values\ndetermined by the Schr\\\"odinger functional method:\n\\begin{eqnarray}\n&&\nx = 0.602 \\pm 0.048\n~~~~~\\mbox{\\cite{Capitani:1998mq}} ,\n\\\\\n&&\nx = 0.586 \\pm 0.048\n~~~~~\\mbox{\\cite{Necco:2001xg}} .\n\\end{eqnarray}\nThe errors are of the same order of magnitude, and the\nvalues are consistent with one another with respect to\nthe errors.\nWe emphasize that our determination uses a completely independent\nmethod, and that the mutual consistency between these values\nis quite non-trivial.\nAt the moment, the error of our $x$ in eq.~(\\ref{value-x})\nis dominated by errors of the lattice data of the potential.\n\n\\section{Conclusions}\n\nWe analyzed the static QCD potential in the distance region\n$0.1~{\\rm fm} \\hbox{ \\raise3pt\\hbox to 0pt{$<$}\\raise-3pt\\hbox{$\\sim$} } r \\hbox{ \\raise3pt\\hbox to 0pt{$<$}\\raise-3pt\\hbox{$\\sim$} } 1$~fm.\nWe combined the perturbative predictions up to NNNLL and \nthe most accurate lattice computations, in the framework of\nOPE.\nIn this way, we determined the non-perturbative contribution\n$\\delta E_{\\rm US}(r) $ and \n$r_0\\, \n\\Lambda_{\\overline{\\rm MS}}^{\\mbox{\\tiny{3-loop}}}\n$,\nfor $n_l =0$ (quenched approximation).\nOur conclusions are as follows:\n\\begin{itemize}\n\\item\nMost of the linear potential at $r<1$~fm\n($\\sim 3.8~\\Lambda_{\\overline{\\rm MS}}^2\nr$)\nis included in perturbative prediction\nof the Wilson coefficient {$V_S(r)$}.\nThis is a {\\it scheme-independent} result.\\footnote{\nScheme dependence enters at ${\\cal O}(r^2)$, hence it does not\naffect the linear potential.\n}\n\\item\n$\\delta E_{\\rm US}(r)=0 $ is disfavored.\nThis is a scheme-dependent result, but it holds in different schemes\nwhich we examined (which we consider to be reasonable schemes).\n\\item\n$r_0\\, \\Lambda_{\\overline{\\rm MS}}^{\\mbox{\\tiny{3-loop}}}\n=0.574\\pm 0.042 $.\nIt is consistent with the conventional values and the error is\nof comparable size.\nThis provides a new method for its determination.\n\n\n\\end{itemize}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\n\nSince \\citet{1926ApJ....64..321H}, the bimodality in the distribution\nof galaxy properties has been one of the great curiosities in the\nfield of astronomy. Bimodality is observed in galaxy colours,\nmorphology, star formation rates and galaxy stellar masses. Modern\nspectroscopic galaxy surveys have allowed us to quantify the\nbimodality precisely, especially with the advent of the Sloan Digital\nSky Survey (SDSS)\n\\citep{2001AJ....122.1861S,2003MNRAS.341...54K,2004ApJ...600..681B}.\nEven at high redshift, evidence for bimodality in the galaxy\npopulation is being sought and found\n\\citep{2005ApJ...625..621B,2006ApJ...647..853W,2007A&A...465..711F}.\n\nRecent observations have revealed that since a redshift of around\nunity the total mass of stars living in red sequence galaxies has\nincreased by a factor of two \\citep{2004ApJ...608..752B}. At the same\ntime, the stellar mass density of the blue sequence has remained\nalmost constant. The interpretation is that some blue galaxies migrate onto\nthe red sequence after the quenching of their star formation, whilst\nthe remainder continue to form new stars \n\\citep[e.g.][]{2007ApJ...665..265F,2007A&A...476..137A}. This\nquenching of star formation is apparently occurring in galaxies of\nincreasingly lower masses as the Universe ages\n\\citep{2006ApJ...651..120B}. So-called ``dry'' (gas poor) mergers,\nwith little associated star formation, can increase individual galaxy\nmasses within the red-sequence, forming the massive ellipticals seen\ntoday \\citep{2005AJ....130.2647V,2006ApJ...640..241B}. As well as\nallowing the building of very massive ellipticals, such a scenario\nmatches the observed kinematic and photometric properties of\npresent-day ellipticals\n\\citep{2006ApJ...636L..81N,2007ApJ...658..710N}. \n\nThe overall decrease in global SFR density since $z\\sim1$ appears to\nbe caused by a gradual decline in the mean SFR of galaxies, rather\nthan a decrease in the number of starbursts\n\\citep{1997ApJ...481...49H,2007ApJ...660L..43N,2007A&A...472..403T}. However,\nthe process, or processes, responsible for this gradual decline remain\nto be determined.  What physical mechanisms cause a galaxy to stop\nforming stars, to turn into a spheroid, and thus to\nenter the red sequence?  There are many competing theories which\naddress one or both of these problems. Hot gas stripping (sometimes\ncalled strangulation) of small galaxies as they fall into overdense\nenvironments leads to a slow quenching of their star formation\n\\citep{1980ApJ...237..692L,2000MNRAS.318..703B,2008MNRAS.387...79V}. In\nregions of high external pressure, the cold gas may also be stripped\nfrom the galaxies by ram pressure stripping\n\\citep{1972ApJ...176....1G,1994AJ....107.1003C}. Gas rich major\nmergers may perform both tasks, causing a starburst that is strong\nenough to rapidly exhaust the supply of fuel, and convert the disk\ngalaxies into spheroids\n\\citep{1972ApJ...178..623T,1992ARA&A..30..705B,2003ApJ...597..893N}.\nIn this latter scenario, subsequent cooling from the inter-galactic\nmedium may cause a disk to reform and return the galaxy to the blue\nsequence. Therefore, the existence and efficient coupling to the ISM\nof additional mechanical energy from a central active nucleus (often\ncalled AGN feedback) has been proposed to prevent refueling and allow\nthe red sequence to build\n\\citep{2005ApJ...620L..79S,2007ApJ...659..976H,2008MNRAS.387...13K}.\nObservational evidence for the existence, but above all the relative\nimportance of each of these scenarios, remains scarce.\n\nMuch progress has been made on understanding galaxy mergers since the\nadvent of high resolution computer simulations. \nRealistic observational comparisons are now becoming easier through\nthe application of observational filter functions, point-spread\nfunctions and radiative transfer of starlight through dust, to\nsimulation outputs \\citep[e.g.][]{2008arXiv0805.1246L}. However, many\nareas of parameter space remain relatively unexplored with potentially\nlarge implications for our overall understanding of galaxy\nformation. For example, the comparison between simulations of galaxy\nmergers with and without mechanical AGN feedback is rarely focused upon, with\nsome groups preferring to focus on starburst feedback, and others on\nthe possible impact of an active nucleus \\citep[for an exception,\nsee][]{2008MNRAS.387...13K}. \n\nIn this paper we compare observational results directly with a suite\nof smoothed-particle hydrodynamic simulations of galaxy mergers from\n\\citet{2008arXiv0802.0210J}, both with and without AGN feedback, by\nconverting the star formation histories of the simulated galaxies into\nintegrated spectra, upon which the same analysis can be performed as\nupon the observations. Rest-frame optical spectra contain a wealth of\ninformation about a galaxy's past and present star formation rate,\noften referred to as the galaxy ``fossil record''\n\\citep[e.g.][]{2003MNRAS.343.1145P}. In particular, the 4000\\AA\\ break\nand H$\\delta$\\ absorption line strengths constrain the specific star\nformation rate and amplitude of recent bursts of star formation\n\\citep{2003MNRAS.341...33K}. At low redshift, the quality of spectra\nand models now allow the analysis of large spectral regions\n\\citep{2005MNRAS.358..363C, 2007MNRAS.tmp..840T, wild_psb,\n2008MNRAS.385.1998K}. At high redshift, spectral indices easily\nseparate blue from red sequence galaxies without the complication of\ndust \\citep{2008A&A...487...89V}. At these redshifts, where galaxy\nspectra are in general noisier, modern statistical techniques present\nus with the opportunity to vastly increase the quantity and improve\nthe quality of derived parameters. Following \\citet{wild_psb}, this\npaper presents a principal component analysis of the 4000\\AA\\ break\nregion of more than one thousand high redshift galaxy spectra culled\nfrom the VIMOS VLT Deep Survey\n\\citep[VVDS,][]{2005A&A...439..845L}. We use the information encoded\nin the spectra to recover the recent star formation history of galaxies\nat an epoch when the Universe was very different from today.\n\nFrom an observational perspective, so-called ``transition'' galaxies\nare of significant interest for pinning down which physical processes\nplay the largest role in shaping the evolution of galaxies.  Classes\nof galaxies which could be transitioning between the blue and red\nsequences are ``post-starburst'' galaxies\n\\citep[e.g.][]{2004ApJ...609..683T,\n2004ApJ...602..190Q,2006ApJ...642...48L,2007MNRAS.382..960K} and\n``green-valley'' galaxies \\citep[e.g.][]{2007ApJS..173..342M}. From\nthe build up of the red sequence in the VVDS,\n\\citet{2007A&A...476..137A} measured the net mass flux which has taken\nplace from the blue sequence to the red sequence. This amounts to\n$9.8\\times10^{-3}$M$_\\odot$\/yr\/Mpc$^3$ for a\n\\citet{2003PASP..115..763C} initial mass function (IMF), or about\n$1.4\\times10^4$\\,M$_\\odot$\/yr in the entire VVDS survey volume. As we\nshall show in this paper, galaxies which have undergone sudden\nquenching of star formation can be identified in VVDS spectra for a\nperiod of $\\sim0.35-0.6$\\,Gyr after the quenching event. Therefore, if\nthe entire build up of the red sequence were due to a physical process\nassociated with fast quenching, such as gas rich major mergers, we\ncould expect to identify a net transitioning stellar mass of\n$6-10\\times10^5$M$_\\odot$\/Mpc$^3$, or about\n$8.5-14\\times10^{11}$M$_\\odot$ in the VVDS survey, after accounting\nfor targeting rates. This could comprise, for example, a few tens of\ngalaxies of stellar mass $\\log({\\rm M\/M}_\\odot)$$=10.5$.\n\n\nThe primary questions that we wish to address in this paper are, how\nmuch mass do we see entering the red sequence after an episode of fast\nquenching?  Are gas-rich major mergers an important mechanism for the\nbuild up of the red-sequence since $z\\sim1$?  Section 2 describes the\nVVDS spectroscopic survey and Section 3 presents our method for\nidentifying post-starburst galaxies. In Section 4 we place our\nempirical results within a theoretical framework, through comparisons\nwith simple synthesised stellar populations and the star formation\nhistories of galaxy merger simulations. The results are presented in\nSection 5 and compared to the low redshift Universe in Section 6. In\nSection 7 we discuss the global importance of post-starburst galaxies\nfor developing a complete picture of the evolution of the galaxy\npopulation.\n\n\nThroughout the paper we assume a cosmology with $H_0 =\n70$\\,km\\,s$^{-1}$\\,Mpc$^{-1}$, $\\Omega_{\\rm m} = 0.3$ and\n$\\Omega_\\Lambda= 0.7$.\n\n\n\\section{The VVDS dataset}\n\n\\begin{figure}\n\\includegraphics[scale=0.4]{qsr6.5.ps}\n\\caption{The ``quality sampling rate'' between $0.5<z<1.0$ for a SNR limit\n  of 6.5 and in four bins of I$_{AB}$ apparent magnitude.}\\label{fig:qsr}\n\\end{figure}\n\nThe VIMOS VLT Deep survey (VVDS) is a deep spectroscopic redshift\nsurvey, targeting objects with apparent magnitudes in the range of\n$17.5\\le I_{AB}\\le24$. The survey is unique for high redshift galaxy\nsurveys in having applied no further colour cuts to minimise\ncontamination from stars, yielding a particularly simple selection\nfunction. In this work we make use of $\\sim9000$ spectra from the\npublicly available first epoch public data release of the VVDS-0226-04\n(VVDS-02h) field which covers an area of 1750 square arcmins\n\\citep{2005A&A...439..845L}. The spectra have a resolution (R) of 227\nand a dispersion of 7.14\\,\\AA\/pixel and a useful observed frame\nwavelength range, for our purposes, of 5500-8500\\,\\AA.\n\nThe first epoch public data release contains 8981 spectroscopically\nobserved objects in the VVDS-02h field, 8893 of which are within the\npresently defined survey mask. Of these, 7194 have secure redshifts\n(flags 2, 3, 4 and 9, see below), are not classified as type 1 AGN and\nare not secondary objects in the slit. To measure each galaxy's star\nformation history we require coverage of the rest-frame wavelength\nrange 3750--4140\\AA, which restricts the redshift range of our galaxy\nsample to $0.5<z<1.0$. As discussed in detail in Section \\ref{sec:qsr}\nbelow, we impose a signal-to-noise (SNR) limit on the spectra greater\nthan 6.5 and magnitude limits of $19<I_{AB}<23$, resulting in a final\nsample of 1246 galaxies.\n\n\nIn addition to the spectra the dataset includes multi-wavelength\nphotometry from diverse sources. Data assembled within the context of\nthe VVDS survey include deep B, V, R, and I photometry from the\nCFHT\/CFH12K camera\\footnote{http:\/\/cencosw.oamp.fr\/} with limiting\nmagnitudes of 26.5, 26.2, 25.9, and 25.0, respectively\n\\citep{2003A&A...410...17M,2004A&A...417..839L}.  Further optical\nphotometry from the Canada-France-Hawaii Telescope Legacy Survey\n(CFHTLS\\footnote{http:\/\/www.cfht.hawaii.edu\/Science\/CFHTLS}) extends\ninto the $u$ and $z$ bands.  The field has deep NIR imaging down to\nlimits of J$\\approx$ 21.50 and K$\\approx$ 20.75\n\\citep{2005A&A...442..423I} and NUV and FUV coverage as part of the\nGALEX mission \\citep{2005ApJ...619L...1M}. In order to avoid source\nconfusion due to the large point spread function (PSF) of GALEX, UV\nphotometry is based upon optical priors (Arnouts et al., in\nprep.). Infrared photometry in the IRAC 3.6, 4.5, 5.8 and 8.0 $\\mu$m\nis available from the Spitzer Wide-area InfraRed Extragalactic survey\n\\citep[SWIRE,][]{2003PASP..115..897L}. We use the source catalogue as\nreleased by the SWIRE\nteam\\footnote{http:\/\/swire.ipac.caltech.edu\/swire\/swire.html}, with\ndetection limits at 5$\\sigma$ of 5.0, 9.0, 43.0, 40.0 $\\mu$Jy,\nrespectively. In this paper the photometric data is used primarily for\nthe purpose of deriving stellar masses, as described in \\citet[][see\nSection \\ref{sec:Ms}]{2008arXiv0807.4636W}.\n\n\\subsection{Accounting for unobserved galaxies}\n\nIn order to calculate absolute quantities from galaxy redshift\nsurveys, such as galaxy numbers and mass densities, it is necessary to\naccount for several effects which cause galaxies not to be\nincluded. For our work there are three effects:\n\\begin{itemize}\n\\item The {\\it target sampling rate} (TSR): only a fraction of all\n  galaxies in the given magnitude range and survey area are targeted\n  for spectroscopic follow-up.\n\\item The {\\it spectroscopic success rate} (SSR): a fraction of\n  targeted and observed galaxies cannot be successfully assigned\n  redshifts, due to either sky line contamination or featureless\n  spectra.\n\\item The spectroscopic {\\it quality success rate} (QSR): only a\n  fraction of the targeted galaxies with successful redshifts have\n  high enough quality spectra from which star formation histories\n  can be measured.\n\\end{itemize}\nThe TSR and SSR are common to many results derived from the VVDS, such\nas luminosity functions, and are explained in detail in\ne.g. \\citet{2005A&A...439..863I}; only the QSR is new to this\nwork.\n\n\\subsubsection{The Target Sampling Rate (TSR)}\n\nAs described in \\citet{2005A&A...439..863I}, in order to maximise the\nnumber of targeted objects per VIMOS pointing, the TSR is a function\nof the size of the galaxy. We obtain the TSR from Figure 1 of\n\\citet{2005A&A...439..863I} and weight each galaxy according to its\nsize $w_i^{\\rm TSR} = 1\/{\\rm TSR(r_i)}$, where $r_i$ is the x-radius\nparameter of each galaxy.\n\n\\subsubsection{The Spectroscopic Success Rate (SSR)}\n\nThe second effect, the SSR, is potentially much more complicated,\nbecause the success of obtaining a redshift for a galaxy depends not\nonly on the apparent magnitude of the galaxy, but also on its redshift\nand spectral type.  The rest-frame spectroscopic features which allow\naccurate redshift determinations depend on the spectral type of the\ngalaxy and move through the observed-frame wavelength range depending\non the redshift of the galaxy. In the VVDS each object is assigned a\nredshift quality flag, which approximates the security of the given\nredshift. Galaxies with flags 2, 3, 4 and 9 are deemed\nsecure\\footnote{Flag 9: redshift determined from single emission line;\nFlag 4: 100\\% confidence; Flag 3: 95\\% confidence; Flag 2: 75\\%\nconfidence \\citep{2005A&A...439..863I}}. In order to account for the\nchanging SSR as a function of redshift, and following the method\npresented in Section 3.2 of \\citet{2005A&A...439..863I}, we estimate\nthe SSR by making use of the photometric redshifts calculated by\n\\citet{2006A&A...457..841I}, which are publicly available through the Terapix\nConsortium\\footnote{ftp:\/\/ftpix.iap.fr\/pub\/CFHTLS-zphot-v1\/}. We\ncalculate the SSR as a function of redshift and magnitude, as the\nnumber of objects with secure redshifts (flags 2,3,4,9) divided by the\ntotal number of objects (with flags 0,1,2,3,4 and 9).  Photometric\nredshifts are used to place the galaxies without secure spectroscopic\nredshifts in the correct redshift bin.\n\nWe calculate the SSR in 4 bins of apparent magnitude\n19--21, 21--22, 22--22.5 and 22.5--23, with bins in $\\Delta\nz_{\\rm phot}$ of width 0.05. Galaxies brighter than 19 and\nfainter than 23 magnitudes are not used in this work, the former\nbecause of our lower redshift limit of $z=0.5$ and the latter because\nof the signal-to-noise limitation we place on the spectra as described\nin the following subsection.\n\nWe weight each galaxy in our sample by $w_i^{\\rm SSR}= 1\/{\\rm\nSSR(z_{spec},} I_{AB})$, where $z_{spec}$ is the spectroscopic redshift\nof the galaxy, and $I_{AB}$ is its $I$-band apparent magnitude.\n\nIt is important that galaxies with different SEDs are treated equally\nduring this weighting procedure. Quiescent galaxies are the hardest to\nassign redshifts to, due to their lack of emission lines. To check\nthat the completeness for quiescent galaxies is accurately estimated\nusing the SED-independent method described above, we split our sample\ninto high and low SFR subsamples, where SFR was measured from SED\nfitting to multiwavelength photometry (see Section \\ref{sec:Ms}). We\nrepeated the calculation of $w_i^{\\rm SSR}$ for each subsample and\nfound them to be consistent with each other over the redshift range\n$0.5<z<1$ used in this paper. This redshift range is expected to be\nthe least problematic, due to the coverage of the 4000\\AA\\ break which\nis strong in quiescent populations.\n\n\n\\subsubsection{The Quality Success Rate (QSR)}\\label{sec:qsr}\n\nThe final selection criterium applied to the VVDS galaxies to create the sample\nstudied in this paper is that they have spectra of high enough\ncontinuum SNR to allow us to extract useful information on their star\nformation histories. In practice, this means a median\nper-pixel-SNR\\footnote{Per-pixel SNR is calculated from the median\nflux over error in the wavelength range 5500--8500\\AA.} larger than\n6.5. The SNR of the observed spectrum depends on both the apparent\nmagnitude of the galaxy, because all galaxies are observed with equal\nexposure times, and the redshift of the galaxy, because the flux of a\ngalaxy varies with wavelength. \n\nThus, in a similar manner as for the SSR, we define the quality\nsuccess rate (QSR) to be the total number of galaxies with secure\nredshifts and SNR greater than our limit, divided by the total number\nof galaxies with secure redshifts (flags 2,3,4 and 9). We calculate\nthis in the same four bins of apparent magnitude used for the SSR and\nin spectroscopic redshift bins of width $\\Delta z_{\\rm spec}=0.05$. We\nweight each galaxy in our sample by $w_i^{\\rm QSR}= 1\/{\\rm\nQSR(z_{spec},} I_{AB})$. The $w^{\\rm QSR}$ is shown in Figure\n\\ref{fig:qsr} for the four magnitude bins. In Section\n\\ref{sec:massdens} we will show that we recover stellar mass densities\nconsistent with previous work, despite the addition of this new\nselection criterion.\n\nImposing a SNR threshold on the data was found to be necessary for\nthis work. Using spectra with too low SNR causes considerable scatter\nin our spectroscopic indices, which makes difficult the robust\nidentification of the post-starburst galaxies in which we are\nprimarily interested. This problem is easily identifiable: the\nincreased scatter in the indices causes the number of post-starburst\ngalaxies to increase relative to the total sample, and biases the SNR\ndistribution of the candidate post-starbursts towards lower values\nthan typically seen in the population as a whole. Therefore the SNR\nlimit was chosen empirically, by ensuring that the distribution of SNR\nin our post-starburst galaxy sample was similar to that for galaxies\nof other types, and that our total completeness-corrected stellar mass\ndensities for each class of galaxy (i.e. quiescent, star forming,\npost-starburst, starburst) remained constant when the SNR limit was\nincreased by a small amount. We found that a per-pixel-SNR limit of 6.5\nwas required for the purposes of our study.\n\n\\subsection{Measuring Stellar masses}\\label{sec:Ms}\n\nA full description of the method used to measure the stellar masses of\nour galaxies is given in \\citet{2008arXiv0807.4636W}, to which we refer the\nreader for details. Briefly, we derive the stellar mass M$^*$ for each\ngalaxy by fitting its multi-band UV--IR spectral energy distribution\n(SED), to model stellar populations created using the\n\\citep{2003MNRAS.344.1000B} population synthesis models, assuming a\n\\citet{2003PASP..115..763C} initial mass function (IMF). We create a\nlibrary of exponentially declining star formation histories with\nsuperposed stochastic top-hat bursts (see Kauffmann et al. 2003 and\nSalim et al. 2005 for descriptions of similar\nlibraries)\\nocite{2003MNRAS.341...33K}\\nocite{2005ApJ...619L..39S}.\nThe method uses Bayesian probability distribution functions as\ndescribed in Appendix A of Kauffmann et al. (2003) to derive physical\nparameter estimates for each observed SED, including errors on the\nparameters.  For a detailed discussion of the accuracy of stellar mass\nestimates using SED fitting methods see \\citet{2007A&A...474..443P}.\n  \n\n\\subsection{Volume correction}\\label{sec:volume}\n\nTo account for the changing effective survey volume for galaxies of\ndifferent brightnesses and SED shapes, we use the V$_{\\rm max}$ method\n\\citep{1968ApJ...151..393S}. This weights galaxies according to the\nfraction of the survey volume in which they could have been observed,\ngiven their brightness and the shape of their SED. The latter point is\nimportant: galaxies with different SED shapes are visible in different\nsurvey volumes and this must be correctly accounted for in the V$_{\\rm\nmax}$ calculation \\citep{2004MNRAS.351..541I}. Using the same library\nof models as used in the stellar mass determination (see above), the\nbest fit model was scaled to the observed galaxy magnitude and shifted\nin redshift to determine the total redshift path in which the galaxy\nwould have been seen, given the survey magnitude limits.\n\nThe SNR limit alters slightly the effective $I_{AB}$ magnitude limits\nof the survey, because very few galaxies with apparent magnitudes\nabove 23 have spectra with a SNR greater than our limit of 6.5. We\ntherefore redefine the survey magnitude range to be $19<I_{AB}<23$\nand remove galaxies from our sample which fall outside of these\nlimits.\n\n\n\\subsection{The survey volume}\nThe survey area is calculated from the photometric survey mask,\ncombined with the boundaries of the spectroscopic survey region. The\ntotal survey area is 0.472 sq. degrees which corresponds to a survey\nvolume in the redshift range $0.5<z<1.0$ of $1.4\\times 10^6$\\,Mpc$^3$.\n\n\n\n\n\\section{Locating recently dead galaxies}\\label{sec:dead}\n\n\nThe galaxies in which we are interested are those in which a\ndominant A\/F star population is visible, implying that the formation\nof O and early B type stars has ceased suddenly. Such galaxies are\nidentifiable by their strong Balmer absorption lines compared to their\nmean stellar age as measured by their 4000\\AA\\ break strength.  The\ntraditional method for identifying post-starburst galaxies involves\nthe selection of objects with both a strong Balmer absorption line\n(usually H$\\beta$ or H$\\delta$) to identify a recent starburst, and no\ndetectable nebular emission (usually [O{\\sc II}] or H$\\alpha$,\nalthough see Balogh \\& Morris [2000]\\nocite{2000MNRAS.318..703B}) to\nensure no ongoing star formation. Our selection differs primarily by\nnot placing limits on nebular emission, as we do not want to bias our\nsample against narrow line AGN which are enhanced in the\npost-starburst population \\citep{2006ApJ...648..281Y,wild_psb, 2007MNRAS.382.1415S}. The additional use of\nthe 4000\\AA\\ break strength allows us to select galaxies with older\nand weaker post-starburst features. \n\nFollowing the method of \\citet{wild_psb} we employ a Principal\nComponent Analysis (PCA) of the spectra around the 4000\\AA\\ break to\nmeasure the strength of the Balmer lines with sufficient SNR. In\n\\citet{wild_psb} the new spectral indices were defined based upon\nstellar population synthesis models to avoid contamination of the\nindices by gaseous emission from the galaxies.  Because of the low\nspectral resolution of the VVDS spectra, in this work we are unable to\naccurately mask the Balmer emission lines which fall in the center of\nthe Balmer absorption lines in which we are interested. Therefore, we\nchoose to perform the PCA directly on the VVDS spectra themselves. In\nSection \\ref{sec:models} we explain how we compare a posteriori to\nmodel stellar populations created with a population synthesis code.\n\n\n\\subsection{Calculating the spectroscopic indices}\\label{sec:indices}\n\n\\begin{figure}\n\\includegraphics[scale=0.7]{espec.ps}\n\\caption{The eigenspectra of the VVDS galaxy sample in the rest-frame\n  wavelength range 3750-4140\\,\\AA. The spectral indices used in this\n  paper refer to the amount of each eigenspectrum present in an\n  individual galaxy spectrum. The first index is related to the shape\n  of the spectrum and equivalent to the common D$_n$4000 index. The\n  second index contains the Balmer absorption line series from\n  H$\\delta$ to H$\\theta$, together with the distinctive decrease in\n  flux bluewards of 3850\\,\\AA\\ seen in late B to F-star\n  spectra.}\\label{fig:espec}\n\\end{figure}\n\nAll galaxy spectra in our sample are corrected for Galactic extinction\nassuming a uniform E(B$-$V)$=0.027$ \\citep{2003A&A...410...17M}, moved\nto the galaxy rest-frame and interpolated onto a common wavelength\ngrid. Each spectrum is normalised, by dividing by the median flux\nbetween 3750 and 4140\\,\\AA. The mean spectrum is calculated and\nsubtracted and a PCA is then performed on the residuals.  Because PCA\nis a least-squares process, eigenspectra calculated directly from data\ncan easily be dominated by outliers in the input sample of spectra. In\norder to obtain robust eigenspectra that are representative of the\nmajority of the population, we use an iterative and robust PCA\nalgorithm \\citep{2008arXiv0809.0881B}. The algorithm essentially\nreplaces the least-squares minimisation solved by traditional PCA with\nthe minimisation of a new robust function of the data, which employs a\nrobust Cauchy-type function to limit the impact of outliers. The\neigenspectra are presented in Figure \\ref{fig:espec}. As described in\ndetail in \\citet{wild_psb}, the first eigenspectrum relates to the\nstrength of the 4000\\AA\\ break, and is equivalent to the well-known\nindex D$_n$4000. This has been shown to correlate most strongly with\nthe specific star formation rate of the galaxy \\citep[SSFR, star\nformation rate divided by stellar mass,][]{2004MNRAS.351.1151B}.  The\nsecond index contains the Balmer absorption lines, together with the\ncorrelated decrease in flux bluewards of 3850\\,\\AA\\ found in late B-\nto F-type stars.\n\nWe calculate the principal component amplitudes (PCs) of all galaxies\nin our sample by projecting the spectra onto the eigenspectra. The\nprojection is performed using the ``gappy-PCA'' procedure of\n\\citet{1999AJ....117.2052C}, which weights pixels by their errors\nduring the projection and gaps in the spectra due to bad pixels are\ngiven zero weight. We additionally allow the normalisation of the\nspectra to vary as a free parameter in the projection \\citep[G.~Lemson\nprivate communication, see ][]{wild_psb}. The PCs represent the amount\nof each eigenspectrum present in a galaxy spectrum and are our new\nstar formation history indices. Errors are calculated during the\nprojection of the spectra onto the eigenspectra, using the error\narrays of the individual VVDS spectra. These errors are purely\nstatistical, however, in Wild et~al. (2007) we show that for SDSS\nspectra they compare well to the scatter observed between duplicate\nobservations. \n\n\n\n\\subsection{Defining the post-starburst galaxy sample}\n\n\\begin{figure*}\n\\includegraphics[scale=0.4]{pc12.ps}\n\\includegraphics[scale=0.4]{pc12_grey.ps}\n\\caption{{\\it Left:} The first two principal components (PCs) for all\n  galaxies in our VVDS sample. PC1 is equivalent to the well known\n  index D$_n$4000, PC2 represents the excess (or lack) of Balmer\n  absorption. For analysis purposes, the sample has been split into\n  quiescent (red), starforming (cyan), star-bursting (blue) and\n  post-starburst (orange) classes. Post-starburst galaxies are defined to lie\n  within the dotted box indicated by at least $1\\sigma$. Those with\n  SSFR$<10^{-11}$\/yr are circled. The median errors of the whole\n  sample (black) and the post-starburst galaxies alone (orange) are\n  shown in the top left. {\\it Right:} The volume density of galaxies\n  with $\\log({\\rm M\/M}_\\odot)$$>9.75$ in $0.5\\times0.3$ bins in\n  PC1\/PC2. }\\label{fig:pca}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[scale=0.4]{spec1.ps}\n\\includegraphics[scale=0.4]{spec2.ps}\n\n\\caption{The spectra of post-starburst galaxies culled from the VVDS\n  galaxy sample in the rest wavelength range\n  3750--4140\\AA. Overplotted in red are the PCA fits to the\n  spectra using 10 components. }\\label{fig:spec}\n\\end{figure*}\n\n\nIn Figure \\ref{fig:pca} we show the distribution of PC1 and PC2 for\nour sample of VVDS galaxies. To guide the eye of the reader, and to\naid in further discussion, we have split the sample into four\n``classes'', with boundaries empirically defined from the distribution\nof points. We would like to emphasise that in reality the star\nformation histories of galaxies form a continuous distribution, and\nany separation into discrete classes is an over simplification,\ngreatly reducing the information contained within the full\ndistribution.  The primary division of our sample is into\n``quiescent'' and ``star-forming'' galaxies on the right and left,\nbased on the value of PC1 (i.e. the strength of their 4000\\AA\\\nbreak). The precise positioning of the boundary is arbitrary, and care\nshould be taken when comparing our results for ``red'' and ``blue''\ngalaxies to those derived from different observations.  To the bottom\nleft, systems with very blue continua and very strong emission lines\nare found, which we class as ``starburst'' galaxies.\n\nFinally, to the top center we find the ``post-starburst'' galaxies,\nwith stronger Balmer absorption lines than expected for their 4000\\AA\\\nbreak strength. We define ``post-starburst'' galaxies to have\n$-0.6<{\\rm PC1}<1.0$ and PC2$>0.6$. While these boundaries were\ndefined empirically from the distribution of data (this class are\nclear outliers in PC2), we will later compare with model stellar\npopulations to justify their classification as ``post-starburst''. We\nfind that 18 galaxies lie in this region at $>1\\sigma$ confidence.\nThe spectra of these galaxies are shown in Figure \\ref{fig:spec}. The\nselection of these 18 galaxies is robust to changes in the exact\nmethod used to create the eigenspectra.\n\n\n\n\n\\subsection{Estimating completeness limits}\n\nIt is important to estimate the completeness of the survey in terms of\ngalaxy stellar mass. This of course depends upon the mass-to-light\nratio and SED shape of the galaxies in question: flux limited surveys\nare generally complete to lower mass limits for star forming galaxies\nthan for quiescent galaxies.  While the completeness in a particular\nphotometric band can be judged empirically, this is not possible for\nthe completeness in derived parameters, such as stellar mass and star\nformation rate. We therefore use an alternative approach to estimate\nthe mass completeness of our sample, described in detail in W08.\n\nBecause we are primarily interested in the post-starburst population,\nwith mass-to-light ratios between those of star-forming and quiescent\ngalaxies, we derive the mass limits relevant for this population\nalone. From the model dataset described in Section \\ref{sec:Ms} we\nselect all model galaxies which have undergone a starburst in the last\nGyr and lie within our PSB region in PC1\/PC2. For a given stellar\nmass, we determine the fraction of model post-starburst galaxies that\nlie within our survey magnitude limits. We find that 50\\% of the model\npost-starburst galaxies at $z=0.6$ with $\\log({\\rm M\/M}_\\odot)$$>9.5$ would be detected,\nand $\\log({\\rm M\/M}_\\odot)$$>9.9$ at $z=0.9$. The stochastic model library assumed in\nthis analysis has a higher fraction of strong post-starburst galaxies\nthan appear in the real data, i.e. while the observational PC1\/PC2\nspace is fully covered by the models, the relative space density of\nthe different galaxy types is not reliable. We therefore confirmed\nthat the mass completeness limits did not vary greatly with burst mass\nfraction or burst age.\n\nIn the following sections, all results will be quoted for a mass limit\nof $\\log({\\rm M\/M}_\\odot)$$>9.75$. We have confirmed that increasing this limit to\n$\\log({\\rm M\/M}_\\odot)$$>10.0$ does not alter our conclusions. Figure \\ref{fig:pca}\n(right) shows the number density of galaxies in bins of PC1\/2 for\ngalaxies with $\\log({\\rm M\/M}_\\odot)$$>9.75$, corrected for survey volume and sampling\neffects. \n\n\n\n\\section{Comparison with models}\\label{sec:models}\n\nIn order to understand the particular properties of a starburst which\nwould lead to a post-starburst galaxy in our sample, and to measure\nthe timescale during which a post-starburst galaxy is visible, it is\nnecessary to compare our data with model stellar populations. We begin\nby creating simple toy model starbursts and subsequently turn to the\nanalysis of smoothed particle hydrodynamic (SPH) simulations of\nmerging galaxies.\n\n\n\\subsection{Toy model galaxy tracks}\n \n\\begin{figure*}\n\\includegraphics[scale=0.4]{maxpc2_n62dustem.ps}\n\\includegraphics[scale=0.4]{tpsb_n62dustem.ps}\n\\caption{By projecting evolving model stellar populations onto the\n  VVDS eigenspectra, we can measure quantitative properties relevant\n  to our PSB galaxies. After continuously forming stars for 5\\,Gyr,\n  the toy-model galaxies undergo a starburst which declines\n  exponentially with decay time $\\tau_{decay}$.  {\\it Left:} The\n  maximum PC2 attained by the models after the starburst has occurred,\n  i.e. a measure of the strength of the PSB features. Different lines\n  represent models with bursts of different $\\tau_{decay}$. {\\it\n  Right:} The time spent in the PSB phase as a function of burst mass\n  fraction.}\\label{fig:mock}\n\\end{figure*}\n\nFor a galaxy to reveal strong Balmer absorption lines, star formation\nmust switch off rapidly resulting in an excess of longer lived A- and\nF-stars over the hotter O- and early B-stars. This does not\nnecessarily have to be preceded by a short-lived starburst, the\nsudden truncation of ordinary star formation will suffice, although\nthe signature will of course be weaker. The two parameters of\nimportance for the strength and longevity of the post-starburst phase\nare the mass fraction of stars formed in a starburst and the decay\ntimescale of the starburst. In this terminology, a truncation model\nthen simply has a mass fraction of zero per cent.\n\n\nWith this in mind we have created a library of model galaxies which\nundergo exponentially decaying bursts, occurring after 5\\,Gyr of\ncontinuous star formation. These bursts are parameterised by their\ndecay time and burst mass fraction, i.e. the mass of stars formed\nwhile the star formation rate is above the continuum level.\n\nConverting the star formation histories into spectra suitable for\ncomparison with the VVDS dataset requires several steps. We input the\nstar formation histories into the {\\sc GALAXEV} code\n\\citep{2003MNRAS.344.1000B}, using Charlot \\& Bruzual (in preparation)\nsimple-stellar-populations (SSPs), a \\citet{2003PASP..115..763C} IMF\nand assuming solar metallicity. The evolution of the model galaxies in\nPC1\/2 is not greatly affected by the change between\n\\citet{2003MNRAS.344.1000B} and Charlot \\& Bruzual (in preparation)\nSSPs, which involves both a change in stellar libraries and in the\nstellar evolution tracks. Solar metallicity was chosen as being\nsuitable for our post-starburst galaxy sample, given their stellar\nmasses (Section \\ref{sec:physprop}), but the evolutionary tracks are\ninsensitive to moderate variations in metallicity. The two component\ndust prescription of \\citet{2000ApJ...539..718C} is applied to the\ncontinuum light as implemented in the {\\sc GALAXEV} code. Again, the\ninclusion or not of continuum dust has little effect on the time spent\non the PSB sequence, or the maximum PC2 observed. This is\nbecause the dominant A\/F star population in the post-starburst spectra\nis assumed to already have emerged from the dense stellar birth\nclouds.\n\nNext, because the resolution of the VVDS spectra is too low to mask\nemission lines in the data, we must add emission lines to the model\nspectra. We convert the rate of ionising photons predicted by the\nstellar population synthesis model ($Q_{ion}$) into H$\\alpha$ line flux\nusing:\n\\begin{equation}\nF_{H\\alpha} [erg\/s] =  0.45 * E_{H\\alpha}[erg] * Q_{ion}[s^{-1}]\n\\end{equation}\nwhere E$_{H\\alpha}$ is the energy of a H$\\alpha$ photon and 0.45 is\nthe fraction of ionisations which lead to the emission of an H$\\alpha$\nrecombination photon \\citep[case B, ${\\rm Te}=1\\times 10^4$K and ${\\rm\nNe}=1\\times 10^4$cm$^{-3}$,][]{1995MNRAS.272...41S}. The remaining\nBalmer line fluxes are derived from the ratios given in\n\\citet{1989agna.book.....O}. The stellar population synthesis\nmodel includes the post-AGB phase of stellar evolution which is\nbelieved to be important for producing ionising photons in older\nstellar populations \\citep{1994A&A...292...13B}. We note that no\ncontribution to the emission lines from obscured, narrow-line AGN is\nincluded in the models. Any substantial infilling of the Balmer\nabsorption lines in this way will prevent us from identifying\npost-starburst galaxies. AGN with strong enough lines to have a\nsignificant effect are extremely rare however, and their absence from\nour sample will not affect our final results.\n\nJust as dust attenuation is applied to the model stellar continua, the\nlines are similarly attenuated using the attenuation law as described\nin \\citet{wild_psb}:\n\\begin{equation}\n\\frac{\\tau_\\lambda}{\\tau_V} =(1-\\mu)(\\frac{\\lambda}{5500{\\rm\n  \\AA}})^{-1.3} + \\mu (\\frac{\\lambda}{5500{\\rm \\AA}})^{-0.7}.\n\\end{equation}\nThis law is inspired by the two-component dust model of\n\\citet{2000ApJ...539..718C}. For the birth clouds, from which\nnebular emission is believed to originate and which contain a\nfraction $1-\\mu$  of the dust, these authors adopted for\nsimplicity an attenuation law to match that of the diffuse\ninterstellar medium (i.e. of the form $\\lambda^{-0.7}$). Our slightly steeper curve,\nresulting in slightly increased attenuation, is motivated by the fact\nthat dense clouds around young stars have a more shell-like geometry\nthan the diffuse interstellar medium \\citep[see][for more\ndetails]{2008MNRAS.388.1595D}.  The attenuation of the model emission\nlines by a moderate amount ($A_V=1.0$) alters slightly the\nevolutionary tracks in PC1\/PC2 of the model during the starburst\nphase.\n\nOur ability to detect a post-starburst galaxy depends on the maximum\nPC2 reached during the evolution of the stellar population, and the\ntime spent in the post-starburst phase ($t_{\\rm PSB}$). These factors\ndepend in turn on the strength of the proceeding starburst, together\nwith the timescale of the subsequent decay of the star formation rate\n($\\tau_{\\rm decay}$). Figure \\ref{fig:pca} shows that the maximum\nPC2 reached by the VVDS galaxies is around 1.0, with 5\/18 having PC2\n$>$0.9. Using our toy models, the left panel of Figure\n\\ref{fig:mock} shows the maximum value of PC2 attained by a model galaxy\nafter the burst has occurred, as a function of the burst mass and burst\ndecay time. The models show that the descendants of starbursts with\ndecay times $\\la0.1$\\,Gyr and burst fractions above a few percent will\nappear clearly in the post-starburst phase. The right panel of Figure\n\\ref{fig:mock} shows the time spent in the post-starburst phase\n($t_{\\rm PSB}$). This time is short, $\\la$0.15\\,Gyr, for burst mass\nfractions $<5$\\%, and reaches a maximum independent of decay time of\n$\\sim$0.6\\,Gyr for mass fractions of $\\ga$30\\%. We will use this\nvalue as an upper limit on $t_{\\rm PSB}$ in later sections.\n\nWe conclude from this section that the majority of the progenitors of\nour PSB galaxies underwent a recent strong starburst, followed by a\nrapid truncation of the star formation. A small fraction may however\nbe simply the result of rapid truncation of their ongoing star\nformation.\n\n\n\n\\subsection{Simulation galaxy tracks}\\label{sec:mergers}\nThe next step in our comparison with models involves a sample of \n79 merger simulations presented in \\citet[][hereafter\nJNB09]{2008arXiv0802.0210J}. \n\n\\subsubsection{Simulation details}\n\nThe simulations were performed using the entropy conserving\nTreeSPH-code GADGET-2 \\citep{2005MNRAS.364.1105S}, which includes\nradiative cooling for a primordial mixture of hydrogen and helium\ntogether with a spatially uniform time-independent local UV\nbackground.  Star formation and the associated supernova feedback is\nimplemented using the sub-resolution multiphase model developed by\n\\citet{2003MNRAS.339..289S}.  We model the feedback from black holes\nfollowing the effective subresolution model of\n\\citet{2005MNRAS.361..776S}, in which the unresolved accretion onto\nthe black hole (BH) is related to the resolved gas distribution using\na Bondi-Hoyle-Lyttleton parameterization \\citep{1944MNRAS.104..273B}.\nFurther details concerning the feedback implementations and parameter\nchoices can be found in JNB09. In addition, for the purposes of this\npaper a new suite of simulations was performed in which the BH\nfeedback was switched off. Such a comparison is important because the\nBH feedback can cause star formation to completely shut down after the\nmerger, which may plausibly affect both the time the merger remnant\nspends in the PSB phase, and the strength of the post-starburst\nfeatures.\n\nThe simulations include mergers of gas-rich disks (Sp-Sp), of\nearly-type galaxies and disks (E-Sp, mixed mergers), and mergers of\nearly-type galaxies (E-E, dry mergers).  The progenitor galaxies have\na range of virial velocities, $v_{\\rm vir}$ of 80, 160, 320 and\n500\\,km\/s, which determines their masses and sizes (see Eqs. 1 \\& 2 in\nJNB09). All model galaxies contain an exponential disk ($d$)\ncomponent, together with a stellar bulge ($b$) embedded in a dark\nmatter halo modelled with the \\citet{1990ApJ...356..359H} profile. The\nstellar bulge has a fixed mass ratio of $M^*_{\\rm b} = 0.01367 M_{\\rm\nvir} = 1\/3 M_{\\rm d,tot}$, where the total disk mass is a sum of the\ndisk stellar and gas mass $M_{\\rm d,tot}=M^*_{\\rm d}+M_{\\rm d,gas} =\n0.041 M_{\\rm vir}$, the disk stellar mass is $M^*_{\\rm d} = 0.041\n(1-f_{\\rm gas}) M_{\\rm vir}$, and $M_{\\rm vir}$ is the virial mass of\nthe galaxy. The initial gas fraction $f_{\\rm gas}$ takes values of\n0.2, 0.4 and 0.8.  The galaxies approach each other on parabolic\norbits, which are generally motivated by statistics from N-body\nsimulations \\citep{2006A&A...445..403K}.  The resulting merger\nremnants have total stellar masses between $\\log({\\rm M\/M}_\\odot)$$=9.5-12.5$. Both\nmajor (1:1) and minor (3:1) mergers were simulated with varying\norbital and initial disk geometries \\citep{2003ApJ...597..893N}. All\nsimulations were evolved for a total of $t=3 \\ \\rm Gyr$, with the\nmerger taking place after approximately 1.5\\,Gyr, using the local\nAltix 3700 Bx2 machine hosted at the University Observatory in\nMunich. In this paper we will primarily concentrate on mergers with\nprograde in-plane orbits \\citep[``G0'', i.e. the galaxies collide\ndirectly edge on][]{2003ApJ...597..893N}. We will additionally show an\nexample in which galaxies rotate in a retrograde sense, with the disks\ntilted by 30 degrees with respect to one another (``G7'').  We will\nfocus exclusively on the Sp-Sp mergers, as all other pair combinations\nwere found not to produce the strength of PSB galaxy visible to us in\nthe data.\n\n\nAs the simulations, and thus star formation rates, are only computed\nover a total duration of 3\\,Gyr, we must assume a star formation\nhistory and galaxy age for the initial build up of stellar mass in\neach progenitor galaxy.  We assume that by the start of the simulation\neach galaxy has built both its bulge and disk stellar mass through an\nexponentially declining star formation rate: ${\\rm SFR} = {\\rm SFR}_0\n+ n \\exp(-t\/\\tau)$ where $\\tau=1{\\rm Gyr}$. SFR$_0$ is the initial SFR\nof the simulation i.e. $\\sim$1.5\\,Gyr before the merger takes\nplace. $n$ is defined such that the galaxies are 5\\,Gyr old at the\nstart of the simulation, i.e. the stellar populations are\napproximately 7.5\\,Gyr old at the onset of the merger. Both\nprogenitors in a simulation are assumed to undergo the same initial\nSFH. We note that any similar star formation history can be assumed\nwith which to build the pre-merger galaxies, without significantly\naffecting their evolution in the post-merger post-starburst phase.\nThe creation of the spectra from the SFHs of the simulations follows\nexactly the same method as for the toy burst models.\n\n\\subsubsection{Post-starburst galaxies in merger simulations}\n\n\\begin{figure*}\n\\includegraphics[scale=0.5]{simns_20.ps}\n\\includegraphics[scale=0.5]{simns_40.ps}\n\\caption{ {\\it Far left:} The combined star formation history of both\n  galaxies participating in a major merger during the 3\\,Gyr SPH\n  simulation. In these panels, the galaxies collide in a ``G0''\n  orbital configuration and BH feedback is included. Progenitor\n  galaxies have gas fractions of 20\\% and virial velocities of 80\n  (top), 160 (middle) and 320 (lower)\\,km\/s corresponding to stellar\n  masses of around $10^{10}$, $10^{11}$ and $10^{12}$M$_\\odot$. The\n  red part of the SFH depicts the ``starburst'' phase, defined to be\n  where the SFR is 5\\% greater than the continuum SFR. The values in\n  the top left of the panels are the mass fraction of stars formed\n  during the starburst and the decay timescale of the starburst (see\n  text). {\\it Next left:} the corresponding evolutionary tracks in\n  PC1\/PC2, with the red section corresponding to the red section in\n  the left panel. The dotted box shows the PSB region defined in this\n  paper. The time that the merged galaxy spends in the PSB phase is\n  given in the bottom left of the panels. {\\it Right pair of panels:} The same as the\n  left panels, but for progenitor galaxies with gas fractions of\n  40\\%. }\\label{fig:simn1}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\includegraphics[scale=0.5]{simns_orbit.ps}\n\\includegraphics[scale=0.5]{simns_nobh.ps}\n\\caption{Same as for Figure \\ref{fig:simn1} but for different initial\n conditions. {\\it Left pair of panels:} Simulations in which the\n progenitor galaxies collide with inclinations other than\n edge-on. From top to bottom the orbits are ``G7'', ``G10'' and\n ``G13''. The galaxies have gas mass fractions of 20\\% (compare to\n left hand panels of Figure \\ref{fig:simn1}. {\\it Right pair of\n panels:} Simulations in which the black hole feedback has been\n switched off. The progenitor galaxies have gas mass fractions of 40\\%\n (compare to right hand panels of Figure \\ref{fig:simn1}).}\\label{fig:simn2}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\includegraphics[scale=0.4]{maxpc2_peter.ps}\n\\includegraphics[scale=0.4]{tpsb_peter.ps}\n\\caption{ Same as Figure \\ref{fig:mock} except for SPH galaxy merger\n  simulations with ``G0'' orbits. Different shapes represent different\n  gas fractions of the progenitor galaxies, filled symbols are 1:1\n  mass ratio mergers and open symbols are 3:1 mass ratio\n  mergers. Increasing symbol size denotes increasing stellar\n  mass. Gray symbols are mergers with black hole feedback ``switched-off'',\n  40\\% gas fractions and $v_{\\rm vir}=80$, 160 and 320\\,km\/s.  {\\it\n  Left:} The maximum PC2 attained by the models after the starburst\n  has occurred, as a function of burst mass fraction. The dotted line\n  indicates our observational limit above which PSBs are\n  detected. {\\it Right:} The time spent in the PSB\n  phase.}\\label{fig:peter}\n\\end{figure*}\n\nFigure \\ref{fig:simn1} shows the star formation history and\nevolutionary tracks in PC1\/2 for three of the major merger simulations\nwith $f_{gas}=0.2$ (left) and $0.4$ (right), BH feedback and prograde,\nedge-on orbits (``G0'') over the 3\\,Gyr of the simulation. From top to\nbottom panel the mass of the progenitor galaxies increases. The SFHs\nof the major merger simulations in general follow similar patterns,\nwith an initial weak enhancement in their star formation, quickly\nfollowed by the starburst and decay to zero star formation. \n\nWe parameterise the simulations using two simple values to describe\nthe evolution of the starburst. Firstly, the burst mass fraction\n$f_{\\rm burst}$, defined as the mass of stars formed while the SFR is\nat least 1.05 times the continuum rate, divided by the total mass of\nthe final merger remnant. The continuum rate is measured in the first\nGyr of the simulation, and linearly extrapolated across the subsequent\nstarburst. The second parameter is the decay time of the starburst,\n$t_{\\rm decay}$, defined to be the time from the peak of the starburst\nuntil the SFR is only 10\\% of the maximum burst height. These values\nare shown in the top left of each SFH panel in Figure \\ref{fig:simn1}. As\nfound using the toy-models, higher gas fractions lead to stronger\npost-starburst features and longer periods of time spent in the\npost-starburst phase.\n\nUsing this suite of simulations we can investigate several other\ninitial conditions which affect the evolution of the stellar\npopulation of the merger remnant: progenitor mass ratio; orbital\ngeometry; presence or absence of a black hole. We illustrate the\nlatter two in Figure \\ref{fig:simn2}. The left two panels show the\nstar formation history and evolutionary tracks in PC1\/2 for galaxies\nwith tilted orbital configurations and $f_{gas}=0.2$. Comparing to the\nleft panels of Figure \\ref{fig:simn1} we see that in this particular\nsuite of simulations, only the prograde edge-on mergers give rise to\npost-starburst features. While the starbursts are weaker in the other\norbital configurations, it is likely that the main cause of the\ndifference is in the longer decay times of the star formation rate.\nHowever, we caution that the tilted orbital geometries tested here\nhave typical tilts of $\\ga30$ degrees and are retrograde. Other\nrelevant parameters, such as impact parameter, galactic structure and\nvariations in stellar feedback prescription\nhave not been investigated. While the simulations indicate that a\nrelatively strong interaction is required to produce a post-starburst\ngalaxy \\citep[see also][JNB09]{2008MNRAS.384..386C}, a\nfull study of the post-merger stellar populations as a function of\nimpact geometries is beyond the scope of this paper.\n\nThe right two panels of Figure \\ref{fig:simn2} show the evolution of the\nstellar population with BH feedback ``switched-off'' and\n$f_{gas}=0.4$. Comparing to the right-hand panels of Figure\n\\ref{fig:simn1} we see that the burst mass fractions and time spent in\nthe post-starburst phase are similar with or without the presence of\nBH feedback. The primary causes of the shut-down in star formation are\nsupernovae feedback and exhaustion of gas supplies, rather than any\nadditional energy supplied by the AGN.  In two out of three cases the\npost-starburst features are stronger when BH feedback aids in\nexpelling the gas, but the main effect is focused on the later stages\nof the burst where any residual star formation is halted more\nquickly. Therefore, we conclude that from these spectral indices\nalone, there is no way to confirm or refute the suggestion that strong\nBH feedback provides the mechanism for shutting off star formation in\nthe Universe.\n\n\nIn Figure \\ref{fig:peter} we summarise the results by showing the time\nspent in the post-starburst phase and maximum value of PC2 reached\nafter the merger has occurred, for all of the Sp-Sp merger simulations\nwith prograde in-plane orbits (``G0'', i.e. directly edge on).  The\nresults paint a very similar picture to that shown by the toy-models,\nbut we can now associate burst mass fractions to physically meaningful\nparameters such as gas fractions and galaxy masses. We find that\nmergers must induce burst mass fractions of at least 5\\% for them to\nbe detectable in our observationally defined PSB phase, and the time\nspent there is greater for more gas rich and more massive mergers. For\na 3:1 merger (open symbols) to appear in the PSB phase, a very high\ngas fraction of 80\\% is apparently required. The duration of the\npost-starburst phase tends to a maximum of 0.6\\,Gyr, with lower gas\nfractions leading to shorter timescales of 0.3-0.4\\,Gyr and burst mass\nfractions of 5-10\\%. Similar to the toy models, we find that decay\ntimes of the starburst must be shorter than $\\sim$0.3\\,Gyr for a\npost-starburst galaxy to be observed. The results of three simulations\nwith BH feedback ``switched-off'', $f_{gas}=0.4$ and a range of\nvirial masses, are shown as grey symbols. These can be compared\ndirectly to the filled triangles in the same figure, in which BH\nfeedback has occurred. We note that the burst mass fractions and\nstarburst decay timescales derived in this section are the same as\nthose found for luminous infra-red galaxies\n\\citep[LIRGS, ][]{2006A&A...458..369M}.\n\n\nMany of the simulated galaxy mergers in the full library of 79\nsimulations do not result in a post-starburst remnant according to our\ndemanding observational criteria. Even for gas rich major mergers, the\nstarbursts can occasionally be weak and\/or have long decay times, thus\nnot creating the sharp discontinuity in SFR required for them to\nappear in the post-starburst phase. Such galaxies may enter the red\nsequence through the ``green valley'' if their star formation rate\ncontinues to decay.\n\n\n\n\n\\section{Mass densities and mass fluxes of post-starburst galaxies at\n  z$\\sim$0.7}  \n\nIn the preceding sections we have culled a sample of post-starburst\ngalaxies with $0.5<z<1.0$ from the VVDS redshift survey. Through\ncomparisons with toy models and SPH galaxy merger simulations, we have\nshown how the spectroscopic features indicate that the galaxies have\nundergone a recent major, gas rich merger. In this section, we will\nplace the post-starburst galaxies into a global context, in terms of\ntheir stellar masses, star formation rates and their number and mass\ndensities. Finally, we will address the interesting question: how much\nmass enters the red sequence through the post-starburst phase?\n\n\\subsection{Photometrically derived physical properties}\\label{sec:physprop}\n\n\\begin{figure*}\n\\includegraphics[scale=0.5]{mass.ps}\n\\caption{The mass distribution of galaxies, classified into\n  starforming (blue), quiescent (red) and post-starburst (orange,\n  dot-filled) by their spectral type i.e. their PC1\/PC2 values shown\n  in Figure \\ref{fig:pca}. The orange line-filled histograms indicate\n  the subset of post-starburst galaxies with no ongoing residual star\n  formation.  The verticle dotted line indicates our post-starburst\n  mass completeness limit. {\\it Left:} the raw data, with no\n  correction for survey incompleteness effects. {\\it Right:} for\n  galaxies with stellar masses above our completeness limit, after\n  correction for survey incompleteness effects. }\\label{fig:mass}\n\\end{figure*}\n\nWe can now ask how the mass of the post-starburst galaxies compares to\nthat of the blue- and red-sequence galaxies? Figure \\ref{fig:mass}\nshows the mass distributions of our different classes of galaxies,\nbefore and after corrections for sample selection (TSR, QSR and SSR)\nand survey volume. The galaxies have been classified according to\ntheir spectral types i.e. position in PC1\/PC2. For clarity, we have\ncombined the starburst and starforming galaxies into a single class;\ntheir mass distributions are very similar. The left hand panel shows\nthe mass distribution before correcting for completeness effects. The\nright hand panel shows the completeness corrected mass distribution,\nindicating that the post-starburst galaxies primarily have\n$\\log({\\rm M\/M}_\\odot)$$<10.5$, i.e. to the lower end of galaxies in the red sequence\nand with a similar distribution to galaxies on the blue sequence.\nIt is also worth noting that the completeness corrected mass\ndistribution of the post-starburst galaxies continues to rise to lower\nmasses, and does not turn over before we reach the completeness limit\nof the sample. Deeper spectroscopic surveys will be required to fully\nprobe the mass distribution of post-starburst galaxies.\n\nOur spectroscopic analysis has focused on uncovering the special\nclass of post-starburst galaxies, and not on recovering further\nphysical parameters. As described in Section \\ref{sec:Ms} and more\nfully in W08, the multiwavelength (UV--IR) SEDs of these galaxies have\nbeen analysed to obtain more physical properties than just stellar\nmass. In Figure \\ref{fig:physprop} we present the light-weighted mean\nstellar age, specific star formation rate (SFR\/M$^*$), time of last\nburst and dust content of our full sample of galaxies, classified by\ntheir PC1\/PC2 values. Overall, the relative agreement between the SED\nfitting and VVDS spectral fitting is remarkable: galaxies which are\nspectroscopically classified as starforming are found, from\nmulti-wavelength photometry, to be younger, have high star formation\nrates, more recent starbursts and higher dust contents compared to\ngalaxies spectroscopically classified as quiescent. The\nmulti-wavelength photometry is able to distinguish that the\npost-starburst galaxies have ages and star formation rates\nintermediate between the quiescent and star-forming\ngalaxies. Additionally, the estimated time of their last starburst has\na lower mean value than quiescent galaxies, peaking at $10^8$--$10^9$\nyears, exactly as expected from the VVDS spectral analysis. We can see\nthat the post-starburst galaxies, identified purely through their\nstellar continuum as having undergone a recent starburst, have a range\nof specific star formation rates which place them at the lower end of\nthe star-forming sequence through to the quiescent galaxies. In Section\n\\ref{sec:massflux} we will use this result to set a lower limit on the\nmass flux onto the red sequence through the PSB phase, by identifying\nthose galaxies which have completely ceased star formation.\n\n\n\\begin{figure*}\n\\includegraphics[scale=0.4]{age.ps}\n\\includegraphics[scale=0.4]{ssfr.ps}\\\\\n\\includegraphics[scale=0.4]{tlb.ps}\n\\includegraphics[scale=0.4]{ext.ps}\n\\caption{The physical properties of our spectroscopic galaxy sample\n  above a mass limit of $\\log(M\/M_\\odot)>9.75$, derived from fitting their\n  multiwavelength SEDs \\citep[see][]{2008arXiv0807.4636W}. From top left,\n  clockwise: the light-weighted mean stellar age; the specific star\n  formation rate (SFR\/M$^*$); the time of the last starburst; the\n  extinction in the $V$ band. The galaxies\n  have been classified by their PC1\/2 values into quiescent (red),\n  starforming (cyan), starburst (blue) and post-starburst\n  (orange). The histograms represent the raw data, they are not\n  corrected for incompleteness effects.}\\label{fig:physprop}\n\\end{figure*}\n\n\n \n\\subsection{Mass and number density}\\label{sec:massdens}\n\n\\begin{table*}\n  \\begin{center}\n  \\caption{The mass and number densities of each of our classes of\n  galaxies, for a mass limit of $\\log({\\rm M\/M}_\\odot)$$>9.75$. \\label{tab:mass}}\n\n\\vspace{0.2cm}\n\n  \\begin{tabular}{cccccc} \\hline\\hline\n  Class  & Quiescent & Star-forming & Starburst  & Post-starburst & Total \\\\ \\hline\n  $\\log({\\rm Number\/Mpc}^3)$ & -2.5 & -2.9 & -3.1& -4.0& -2.3\\\\\n  $\\log(\\rho^*\/{\\rm M_\\odot\/Mpc}^3)$ & $7.89^{+0.01}_{-0.01}$ &\n  $7.88^{+0.02}_{-0.02}$ & $7.17_{-0.04}^{+0.03}$ & $6.43_{-0.04}^{+0.06}$ &\n  $8.23_{-0.01}^{+0.01}$\\\\   \n  \\hline\n  \\end{tabular}\\\\\n  \\end{center}\n\\end{table*} \n\n\nSumming up all the galaxies in our sample above the mass completeness\nlimit of $\\log M=9.75$ with the appropriate weightings, we obtain a\ntotal stellar mass density of $\\log\\rho^* =\n8.23_{-0.011}^{+0.013}$. The errors account for individual errors on\nthe stellar masses and Poisson errors from the SSR and QSR weights. As\nwe have placed a stringent criterion on the minimum signal-to-noise\nratio of the spectra, it is important to ensure that we recover the\ncorrect total stellar mass density found in other\nwork. \\citet{2007A&A...474..443P} calculate the total mass density of\ngalaxies as a function of redshift in the VVDS spectroscopic sample,\nusing the same \\citet{2003PASP..115..763C} IMF as used in the present\nwork. Integrating a Schechter function with the parameters\\footnote{We\ncompare to the $I$-band selected sample with masses determined from\ncomplex star formation histories (row 7 of their Table 2), for better\ncomparison with our own sample and masses} given by Pozzetti\net~al. for the redshift range $0.7<z<0.9$ above our mass limit of 9.75\ngives a stellar mass density of $\\log\\rho^* = 8.28$.  Although the\nstatistical errors are always $<10\\%$, Pozzetti et al. quote a typical\nscatter between different methods of mass determination and luminosity\nfunction estimates of at least $\\sim0.1$dex. Taken together with the\nslightly different redshift range, our results appear to be entirely\nconsistent with those of Pozzetti et al..\n\nTable \\ref{tab:mass} gives the mass and number densities for each of\nour galaxy classes. Note that the completeness of our survey is lower\nfor quiescent galaxies than for other classes.\n\nFocussing on the post-starburst galaxies, we find a number density of\nof $1.0\\times10^{-4}$ per Mpc$^3$ and mass density of\n$2.69\\times10^6$M$_\\odot$\/Mpc$^3$. It is instructive to compare their\nnumber density to other objects considered to be unusual in the\nUniverse at these redshifts. The number density of our post-starburst\nsample is 0.3-0.5dex above the number density of powerful X-ray\nselected AGN with 2-8\\,KeV luminosities above $10^{43}$erg\/s at\nredshifts $0.5<z<1.5$ \\citep{2005AJ....129..578B}. They are are factor\nof 10 more numerous than star-forming galaxies at these redshifts with\nradio luminosities similar to local ultra luminous infra-red galaxies\n\\citep[ULIRGS,][]{2004ApJ...603L..69C}. The post-starburst galaxies\nare considerably more common than sub-mm selected galaxies with FIR\nluminosities $\\ga6\\times10^{11}$L$_\\odot$ at $z\\sim0.9$, which have\nnumber densities of $3\\times10^{-6}$\/Mpc$^3$\/decade\n\\citep{2005ApJ...622..772C}. These comparisons are designed for\norientation only, and not to imply evolutionary sequences: the very\ndifferent duty cycles and ``on'' times for different classes of rare\nobjects make inferences about the relation between them difficult. We\nwill return to a quantitative comparison with QSO number densities and\nmajor merger rates in Section \\ref{sec:disc}.\n\n\n\\subsection{The mass flux of post-starburst galaxies onto the red sequence}\\label{sec:massflux}\n\nWe are interested in how much mass may enter the red sequence after a\nstarburst and subsequent fast quenching of the star formation as seen\nin simulations of major mergers. We estimate the total mass flux which\npasses through the post-starburst phase to be:\n\\begin{equation}\\label{eq:massflux}\n\\dot{\\rho}_{A \\rightarrow Q, PSB} =\n  \\frac{1}{t_{PSB}}\\sum_{i=1}^{N_{PSB}} \\frac{M^*_i}{w_i . \n  V_{max,i} }\n\\end{equation}\nwhere $M^*$ is the stellar mass of the galaxy in solar masses, $w$ is\nthe combination of the selection functions $w^{\\rm SSR}$, $w^{\\rm\nTSR}$ and $w^{\\rm QSR}$, and $V_{max}$ is the correction for the\nvolume in which each galaxy can be seen. The sum is over\npost-starburst galaxies above the mass completeness limit of\n$\\log({\\rm M\/M}_\\odot)$$>9.75$. $t_{PSB}$ is the time that a galaxy will be seen in the\npost-starburst phase. From Section \\ref{sec:models} we find $t_{PSB}$\nto be less than 0.6\\,Gyr. Galaxy mergers with lower gas fractions of\naround 20-40\\% result in slightly smaller burst mass fractions and\nshorter time spent in the post-starburst phase of $\\sim0.35$\\,Gyr.\n\n\nDuring the selection of our post-starburst galaxy sample, we have\nimposed no restrictions on nebular emission line strengths (see Section\n\\ref{sec:dead}).  However, galaxies which show post-starburst\nsignatures as well as nebular emission lines may be in the process of\nregenerating their star formation and may therefore subsequently return to the\nblue-sequence. In order to derive a firm lower limit on the mass flux\nentering the red sequence through the PSB phase, we elect to sum only\nthose post-starburst galaxies which show no evidence of star\nformation. Inspecting the distribution of SSFRs of our post-starburst\ngalaxies (top right panel of Figure \\ref{fig:physprop}) shows the\nmajority have SSFRs similar to the lower end of the star-forming\nsequence, 6 objects have SSFRs below $10^{-11}$ per yr, 5 of which lie\nabove our mass completeness limit. We note that these galaxies also\nhave no measurable [O{\\sc II}] emission. The mass distribution of\nthese galaxies is indicated in Figure \\ref{fig:mass} by the\nfilled-line histogram and their positions marked in PC1\/2 by the open\ncircles in Figure \\ref{fig:pca}.\n\nSumming Equation \\ref{eq:massflux} over the 5 post-starburst galaxies\nabove the mass completeness limit with negligible ongoing star\nformation as derived from their SEDs, assuming an upper limit for\n$t_{PSB}$ of 0.6\\,Gyr, results in a lower limit on the mass flux\nentering the red sequence through the post-starburst phase of\n$\\dot{\\rho}_{A \\rightarrow Q, PSB} > 0.0022^{+0.0002}_{-0.006}$\nM$_\\odot$\/Mpc$^3$\/yr.  Our best estimate for the mass flux through the\npost-starburst phase onto the red sequence, assuming\n$t_{PSB}=0.35$Gyr, is $\\dot{\\rho}_{A \\rightarrow Q, PSB} =\n0.0038^{+0.0004}_{-0.001}$ M$_\\odot$\/Mpc$^3$\/yr. We note that this is\nstill a lower limit, as we have excluded all those post-starburst\ngalaxies with detectable levels of ongoing star formation. This\nresidual star formation may still decay as the post-starburst ages, we\nnote that the younger post-starburst galaxies (those with smaller PC1)\nare found to be more likely to have residual star formation than older\npost-starburst galaxies (see circled points in Figure\n\\ref{fig:pca}). If all the post-starburst galaxies were to enter the\nred sequence, for $t_{PSB}=0.35$Gyr the mass flux would be\n$\\dot{\\rho}_{A \\rightarrow Q, PSB} = 0.0077$ M$_\\odot$\/Mpc$^3$\/yr\n\nTo determine how important this mass flux is in terms of the global\nbuild up of the red sequence, we compare to\n\\citet{2007A&A...476..137A}. They use the multiband photometric data\nin the same VVDS-02h field to measure the mass build up of the red\nsequence; comparison with this result thus limits our exposure to\ncosmic variance. The galaxy masses are measured using the same code\nand model dataset as in this paper, thus eliminating one more\npotential source of systematic error. They find\\footnote{They quote a\nvalue of $\\dot{\\rho}_{A \\rightarrow Q} = 0.017\\pm0.004$\nM$_\\odot$\/Mpc$^3$\/yr assuming a Salpeter IMF, to which they convert\ntheir Chabrier IMF stellar masses by adding 0.24\\,dex. Removing this\ncorrection results in the value quoted. } a total mass growth of the\nred sequence of $\\dot{\\rho}_{A \\rightarrow Q} = 0.0098$\nM$_\\odot$\/Mpc$^3$\/yr. If we assume that all galaxies that enter the\nred sequence subsequently remain on the red sequence, then our results\nshow that $>22^{+2}_{-6}$\\% and likely as much as $38^{+4}_{-11}$\\% of\nthe growth of the red sequence at $z<1$ takes place from galaxies which\nhave passed through the strong post-starburst phase. If all our PSB galaxies\nwith $\\log({\\rm M\/M}_\\odot)$$>9.75$ enter the red sequence, they account for $\\sim80$\\%\nof the growth of the red sequence. We note that these numbers are\nsignificant, and perhaps even surprisingly high. We will return to\nthis point in Section \\ref{sec:disc}.\n\nThe quoted errors on our mass flux include errors on the individual\ngalaxy stellar masses, calculated from the probability distribution\nfunction resulting from the Bayesian SED fit, and the Poisson errors\non the QSR and SSR weights. As described in\n\\citet{2007A&A...474..443P}, systematic errors caused by the methods\nused to estimate the masses and selection of samples lead to a typical\nscatter in stellar mass densities of 0.1-0.2\\,dex. If we discard our\nstatistical errors, and assume instead a fixed 0.15\\,dex error on the\nstellar mass density of the post-starburst galaxies, we find\n$\\dot{\\rho}_{A \\rightarrow Q, PSB} = 0.0038^{+0.001}_{-0.002}$\nM$_\\odot$\/Mpc$^3$\/yr. However, our comparison with the results of\n\\citet{2007A&A...476..137A} should be more robust due to the use of\nthe same stellar masses, similar survey selection criteria and survey\narea.\n\n\n\n\n\n\\section{Comparison at low redshift}\n\n\\begin{figure*}\n\\includegraphics[scale=0.4]{pc12_sdss_lo2.ps}\n\\includegraphics[scale=0.4]{pc12_sdss_lo_masslim2.ps}\n\\caption{The first two principal components (PCs) for all galaxies in\n  our SDSS sample (left) and for those galaxies above our mass\n  completeness limit, $\\log({\\rm M\/M}_\\odot)$$>9.75$ (right). For analysis\n  purposes, the samples have been split into quiescent (red),\n  starforming (cyan), star-bursting (blue) and post-starburst (orange)\n  classes. These classes are defined in the same way as for the VVDS\n  sample, except for the positioning of the partition between\n  quiescent and active galaxies (see Fig. \\ref{fig:pca}). The median\n  errors of the whole sample (black) and the post-starburst galaxies\n  alone (orange) are indicated in the top left. }\\label{fig:pca_sdss}\n\\end{figure*}\n\n\nTo create a comparison sample at low redshift, we combine the Sloan\nDigital Sky Survey (SDSS) spectroscopic data release 5\n\\citep[DR5,][]{2007ApJS..172..634A} with the UKIRT Infrared Deep Sky\nSurvey \\citep[UKIDSS,][]{2007MNRAS.379.1599L} large area survey (LAS),\ndata release 3 (Warren et al. in prep.). The SDSS DR5 galaxy catalogue\n\\citep{2002AJ....124.1810S} covers 5740 sq. degrees and contains more\nthan 670,000 spectra of galaxies with a median redshift of $z\\sim0.1$\nand $r_{\\rm AB}<17.77$. The accompanying SDSS photometric survey\nprovides optical $ugriz$ photometry for each object. UKIDSS is an\nongoing survey using the UKIRT Wide Field Camera\n\\citep[WFCAM,][]{2007A&A...467..777C} to obtain YJHK near-infrared\nphotometry covering 4000 sq. degrees in the same region of the sky as\nthe SDSS survey. The photometric system is described in\n\\citep{2006MNRAS.367..454H}, the pipeline processing and science\narchive are described in \\citep{2008MNRAS.384..637H}. The LAS reaches\na depth in $K$ of 18.4. \n\nFor the purposes of this paper, the SDSS provides the spectroscopic\ndata required to locate the post-starburst galaxies, and the\nnear-infrared photometry of the UKIDSS survey is combined with the\noptical photometry of SDSS to obtain accurate stellar\nmasses. Requiring the objects to be detected in all YJHK bands gives a\ncombined survey area of 916 sq. degrees (S. Maddox, private\ncommunication).\n\n\\subsection{Sample selection and incompleteness corrections}\nWe select SDSS galaxies to have extinction corrected $r$-band\npetrosian magnitudes $r_{\\rm AB}<17.7$, be primary galaxy targets\n({\\sc primtarget = target\\_galaxy} or {\\sc target\\_galaxy\\_big} or\n{\\sc target\\_galaxy\\_red}), primary catalogue objects ($mode=1$) and\nspectroscopically classified as a galaxy ({\\sc specclass=2}). They are\nmatched to UKIDSS galaxies by identifying the closest object within\n0.5\\,arcsec. We remove a small fraction of galaxies (3\\%) with bad\nspectra by imposing a per-pixel-SNR limit of 5, and account for this\nloss in our weighting scheme. Our final sample contains 14822 galaxies\nwith $0.05<z<0.1$.\n\nThe redshift range is selected with two things in mind. Firstly, we\nrequire low enough redshift that our sample is complete down to\nsimilar masses as for the VVDS survey and secondly, we require high\nenough redshift to minimise the aperture bias effect caused by the 3''\nSDSS fibres. This latter property of the SDSS survey means that the\nspectra only probe the central regions of nearby, or massive,\ngalaxies, with a strong redshift dependence. For our comparative\ncensus of post-starburst galaxies this is problematic as there is\ncurrently no known way to robustly detect post-starburst features from\noptical photometry alone. Thus, it is not possible to know for sure\nwhether the proceeding starburst was nuclear or global, the former\nbeing unlikely to be connected to the build up of the red sequence. At\n$z=0.05$ the fibre corresponds to a physical diameter of about 3\\,kpc\n($h=0.7$), which is small compared to the size of a galaxy, but is\nconsiderably larger than the extent of a nuclear starburst\n\\citep{2004AJ....127..105B}. A remaining limitation caused by the\naperture bias of SDSS, is that post-starburst populations may fail to\nbe identified due to population gradients within the\ngalaxies. Assuming disks are more likely to undergo starbursts than\nbulges, this would cause us to underestimate the number of PSB\ngalaxies.\n\n \n\nVolume corrections and stellar masses of the galaxies are measured in\nthe same way as for the VVDS sample (Section \\ref{sec:volume}), using\nthe combined UKIDSS and SDSS photometry. One final piece of\ninformation required is the target sampling rate of SDSS.  Although\nthe SDSS aims for 100\\% coverage in the spectroscopic catalogue, one\nof a pair of neighbouring galaxies is occasionally not targeted, due\nto the problem of placing fibres close together on a plate. Similarly,\ndensely populated regions of the sky may suffer from partial coverage\ndue to there being insufficient fibres available to cover the\narea. For the purposes of this study, we only require the probability\nthat a galaxy was targeted for spectroscopic follow-up, and no further\nspatial information. We use the SDSS Catalogue Archive Server (CAS) to\ncount the number of galaxies within SDSS spectroscopic sectors that\nare targeted for spectroscopic follow-up, and the number with\nspectroscopic ID numbers (i.e. that were targeted). We find that the\ntargeting success rate is 90\\%. We calculate weights for the galaxies\nby combining this value with the fraction with spectra above the SNR\nlimit described above.\n\n\n\\subsection{Calculating the Principal Components}\n\n\nTo calculate the Principal Components for the SDSS spectra, we first\ncorrect for Galactic extinction, convert to air wavelengths, convolve\nthe spectra to VVDS resolution and finally rebin onto the eigenspectra\nbinning. We then proceed as for the VVDS spectra (Section\n\\ref{sec:indices}), to obtain the PCs for the SDSS galaxies. The\ndistribution of the first two components is shown in Figure\n\\ref{fig:pca_sdss}. The left panel shows the full sample, and the\nright panel only galaxies with $\\log({\\rm M\/M}_\\odot)$$>9.75$. Comparing with Figure\n\\ref{fig:pca} there are clear differences between the SDSS and VVDS\npopulations. Firstly the tight blue sequence in SDSS contrasts greatly\nwith the cloud in the VVDS sample.  In part this is due to the smaller\nerrors on the spectral indices, but it may also be due to a more\n``quiescent'' mode of star formation in galaxies at low redshifts. The\nsecond noticeable difference is that the entire population is older\n(larger PC1). Finally, we can see that there are fewer strong\npost-starburst galaxies, despite the enormous increase in overall\nsample size.\n\n\n\n\\subsection{Mass density of post-starburst galaxies at $z\\sim0.07$}\n\nWe identify 6 post-starburst galaxies using the same criteria as for\nthe VVDS sample, although from Figure \\ref{fig:pca_sdss} these are\nclearly only the tip of a distinct and identifiable population.  The\nspectra of the three post-starburst galaxies above the mass limit of\n$\\log({\\rm M\/M}_\\odot)$$>9.75$ are shown in Figure \\ref{fig:spec_sdss}. The SDSS sample\nhas a high mass completeness even below this mass limit, but for the\npurposes of comparison we retain the same mass limit as for the VVDS\ndataset. The number density of PSBs at $z\\sim0.07$ is\n$5.1\\times10^{-7}$, i.e. a factor of 200 lower than in the VVDS survey\nat $z\\sim0.7$.  For comparison, over the same redshift range the\nnumber density of ultra-luminous infrared galaxies (ULIRGs) decreases\nby a factor of $\\sim$30\n\\citep{1998ApJS..119...41K,2004ApJ...603L..69C}. \n\n\n\\begin{figure}\n\\includegraphics[scale=0.4]{spec1_sdss.ps}\n\\caption{The spectra of post-starburst galaxies culled from the SDSS\n  galaxy sample. The original SDSS spectrum is shown in black, with\n  the convolved spectrum in blue (histogram). The red line is the overplotted\n  fit of the VVDS derived eigenspectra to the SDSS spectra. }\\label{fig:spec_sdss}\n\\end{figure}\n\nThe total completeness corrected mass of the SDSS post-starburst\ngalaxies with $\\log({\\rm M\/M}_\\odot)$$>9.75$ is $6.9^{+1.4}_{-1.3}\\times10^{10}M_\\odot$,\ngiving a mass density of $1.2\\pm0.2\\times 10^4M_\\odot\/{\\rm Mpc}^3$. This is\njust 0.43\\% of the mass density of post-starbursts at $z\\sim0.7$.\n\nAs in Section \\ref{sec:massflux} an upper limit on the time the\npost-starbursts are visible for is 0.6\\,Gyr. Following Equation\n\\ref{eq:massflux} we find a mass flux of $\\dot{\\rho}_{A \\rightarrow\nQ, PSB} > 2.0\\pm0.4\\times10^{-5}$ M$_\\odot$\/Mpc$^3$\/yr.  Assuming our\nbest estimate for the visibility of the post-starburst galaxies of\n0.35\\,Gyr results in  a mass flux of $\\dot{\\rho}_{A \\rightarrow\nQ, PSB} = 3.4\\pm0.6\\times10^{-5}$ M$_\\odot$\/Mpc$^3$\/yr.\n\nThe SED fitting results indicate that two of the low-redshift\npost-starburst galaxies have no residual star formation\n(SSFR$<10^{-11}$\/yr), while one has a SSFR of $10^{-10}$\/yr. However,\nwe caution that with only the optical and NIR bands, these values are\nless reliable than for the VVDS galaxies with UV-IR coverage. We have\ntherefore used all three galaxies to calculate the present day mass\nflux through the post-starburst phase. The masses of the three SDSS\npost-starburst galaxies are $\\log({\\rm M\/M}_\\odot)$$=$10.25, 10.15 and 10.05, similar to\nthe masses of the VVDS post-starburst galaxies.\n\nWe can compare our low redshift PSB mass flux to that calculated by\n\\citep{2007ApJS..173..342M} for galaxies in the green valley. They\nassumed that green valley galaxies defined in NUV-r colours in an\nSDSS\/GALEX matched catalogue are all entering the red sequence\n(i.e. all the mass flows from blue-to-red) to calculate a transition\nmass flux of $\\dot{\\rho}_T = 0.033$M$_\\odot$\/Mpc$^3$\/yr. Clearly, this\nvalue is considerably greater (a factor of 1,000) than our\npost-starburst mass flux. If the assumption that all green valley\ngalaxies are heading towards the red-sequence is correct, then the\npost-starburst pathway to the red sequence appears truly unimportant\nin the present-day Universe. Some of this discrepancy may be reduced\nby appealing to the differing mass limits of the two samples. For\nexample, \\citet{2004ApJ...602..190Q} studied a sample of\npost-starburst galaxies in the SDSS, using a similar magnitude limit\nto that of \\citet{2007ApJS..173..342M}, which includes many lower mass\ngalaxies than in our sample. They measured the ``rate density'' for\npost-starburst galaxies in the SDSS, i.e. the number of galaxies which\npass through the post-starburst phase, to be\n$\\sim4\\times10^{-5}$\/Mpc$^3$\/Gyr. The equivalent value for our\nhigh-mass sample is $\\sim1.5\\times10^{-6}$\/Mpc$^3$\/Gyr strongly\nsuggesting that the measured post-starburst mass flux would increase\nwere we to relax our mass limit.\n\n\n\n\n\\section{Discussion: are post-starburst galaxies important?}\\label{sec:disc}\n\nIn this paper we have presented a sample of galaxies with strong\nBalmer absorption lines and, through comparison with simulations, have\nargued that these are the likely descendants of gas-rich major\nmergers. The strength of their spectral features suggest that they are\npost-starburst galaxies with burst mass fractions of at least 5-10\\%.\nHowever, irrespective of whether a starburst has occurred, it is\ncertain that their star-formation has been quenched quickly with a\ntimescale of $\\la$0.1--0.2\\,Gyr.  As discussed briefly in Section\n\\ref{sec:intro}, such fast quenching times are generally associated\nwith supernovae feedback after periods of rapid star formation induced\nby galaxy collisions. An alternative scenario may be rapid fueling of\nstar formation in galaxies from filamentary cold-gas flows\n\\citep{2005MNRAS.363....2K,2008arXiv0803.4506O}, although direct\ncomparisons with the outputs of these simulations would be required to\ntest this. \n\n\n\\subsection{QSOs and post-starburst galaxies}\nOne popular theory for the origin of the red sequence and the relation\nbetween galaxy bulge mass and central supermassive black hole mass is\nthrough the triggering of massive outflows driven by the central black\nhole. Such outflows would quickly halt star formation and therefore\nthe theory suggests an evolutionary link between QSOs and\npost-starburst galaxies. It is therefore of interest to compare the\nnumber density of the post-starburst galaxies to that of QSOs at the\nsame redshift. Any meaningful comparison requires knowledge of the QSO\nlifetime and the visibility time of the post-starburst galaxies. The\nformer has been estimated using a variety of methods to be $t_{\\rm\nQSO} \\sim 10^7$ \\citep[see e.g.][]{2008ApJ...676..816G} and we assume\n$t_{\\rm PSB} = 0.35$\\,Gyr as previously. Setting\n\\begin{equation}\n\\frac{\\phi_{\\rm QSO}(M<M_X)}{t_{\\rm QSO}} = \\frac{\\phi_{\\rm PSB}}{t_{\\rm PSB} } \n\\end{equation}\nwe can estimate the magnitude limit of the QSOs, $M_X$, which leads to\na space density matching the observed space density of post-starburst\ngalaxies of $1\\times 10^{-4}$\/Mpc$^3$. Using the parameterised QSO\nluminosity function of \\citet{2004MNRAS.349.1397C} at $z=0.7$, we find\n$M_X \\approx M^*(z=0.7)+1.5=-22.2$ where the magnitudes are\n$b_j$--band and Vega zero-point. That the number density of strong\npost-starburst galaxies coincides with that of moderately powerful\nQSOs, 1.5 magnitudes fainter than $M^*$, does not of course\nnecessarily imply an evolutionary link. The AGN luminosity function\ncontinues to rise to fainter magnitudes \\citep{2005AJ....129.1795H}\nand uncovering the triggering mechanisms of AGN as a function of\nluminosity remains one of the biggest observational challenges for\nextragalactic astronomy.\n\nThe number density of the post-starburst galaxies evolves\nrapidly. Extrapolating the evolution of the QSO luminosity function of\n\\citet{2004MNRAS.349.1397C} to $z=0.07$, and taking $t_{\\rm QSO} \\sim\n10^7$ as before, we find a QSO number density of $3.1\\times\n10^{-5}$Mpc$^{-3}$ above the same magnitude limit of $M_X=-22.2$, a\nfactor of a few below the $z=0.7$ number density. So current results\nsuggest that the evolution in number density of QSOs is not as strong\nas the evolution in the PSB number density. To our knowledge, an\nobserved QSO luminosity function at $z\\sim0$ is not currently\navailable, and would be required for a more detailed comparison.\n\n\n\\subsection{Major mergers and post-starburst galaxies}\nIf, as we have suggested, the strong post-starburst galaxies selected\nin this paper are the result of major mergers, it is instructive to\ncompare the number of post-starburst galaxies to the number of close\ngalaxy pairs in the same survey volume. \\citet{2008arXiv0807.2578D}\nmeasure the fraction of close spectroscopic pairs to derive the merger\nrate using the same VVDS galaxy sample as in this paper. For galaxies\nwith $\\log({\\rm M\/M}_\\odot)$$>10$ they derive a merger rate of\n$2\\times10^{-4}$Mpc$^{-3}$Gyr$^{-1}$, with a magnitude difference\ncriteria such that they are sensitive to mergers with mass ratios\n$\\ga$4:1. This can be compared with the number of post-starburst\ngalaxies, divided by the time during which they are detectable\ni.e. $2.9\\times10^{-4}$Mpc$^{-3}$Gyr$^{-1}$ for $t_{\\rm\nPSB}=$0.35\\,Gyr. Given the small number statistics of both samples,\nuncertainty in both merger and PSB timescales and the problem of the\nunknown magnitude difference between the PSB progenitors, it is\nencouraging that these numbers are so close. Of course, larger samples\nand further detailed observations of the post-starburst galaxies will\ngreatly aid our understanding of their origin and possible link to\ngas-rich major mergers. As with the QSO number density, the current\nestimates of the evolution in merger rate with redshift show less\nevolution than observed in the PSB number density. Although\n\\citet{2008arXiv0807.2578D} find that the evolution with redshift is\nstronger for lower luminosity and lower mass galaxies, with the merger\nrate decreasing by a factor of two between a redshift of 1 and 0.5,\nthis is still not as strong as the evolution in PSB number\ndensity. Finally, we note that out of the 36 close galaxy pairs\nidentified by \\citet{2008arXiv0807.2578D} one galaxy is part of the\npost-starburst sample of this paper. Given the small probability of a\nchance coincidence, this suggests that in at least one case the\npost-starburst stellar population is linked with tidal disruption\ncaused by an ongoing major merger.\n\nThe strong evolution in PSB number (and mass) density compared to the\nevolution of both QSO number density and galaxy major merger rate,\nleads us to the question of what may cause such a sharp decline in the\nnumber of post-starburst galaxies. Through our comparison with merger simulations,\nwe have identified two important factors in creating a post-starburst\ngalaxy: gas mass fraction and timescale of the starburst. Larger gas\nmass fractions provide more fuel for the starburst, leading to\nstronger and more prolonged post-starburst signatures. Because gas is\nused-up in the formation of stars, it is reasonable to assume that gas\nmass fraction decreases with decreasing redshift, a fact that may play\na leading role in the global decrease of star formation rate density\nsince $z\\sim1$. A second effect is the starburst timescale, which is\ngenerally assumed to be linked to the disk dynamical timescale, which\nin turn is linked to the dynamical time of the halo. This latter value\nis known to increase with redshift, by a factor of 2 between a\nredshift of 1 and 0 \\citep[E. Neistein priv.\ncomm.,][]{2001PhR...349..125B}. A lengthening of the duration\nof the starbursts, together with a decreasing amount of fuel\navailable, may be the dominant mechanism responsible for the decrease\nin number density of post-starburst galaxies. Clearly further\nsimulations will be required to test these ideas.\n\n\n\n\\subsection{The environments of PSB galaxies}\nGiven the possible link of PSB galaxies to gas-rich major mergers, it\nis interesting to investigate the local environments of the\ngalaxies. \\citet{2006A&A...458...39C} measured the density on 5 and\n8$h^{-1}$Mpc scales around galaxies in the VVDS survey. Of the 16 PSB\ngalaxies with $\\log({\\rm M\/M}_\\odot)$$>9.75$, 15 have available density measures. A\ncontrol sample of 2122 VVDS galaxies with good quality redshift flags\nwas selected for comparison. We do not detect any significant\ndifference in the mean or median densities around the PSB galaxies, in\nagreement with the results of \\citet{2008arXiv0805.0004Y}. It is,\nhowever, interesting to see that the spread in local density values\nfor the PSB sample is very large. In detail, 3 lie in underdense\nenvironments ($\\delta_5<0$), 9 in normally overdense environments\n($0<\\delta_5<1$, while the median value for the control sample is\n$\\delta_5 = 0.44 \\pm 0.02$) and 3 in strongly overdense environments\n($\\delta_5>1$).  Furthermore, there is no clear trend between\nstar-forming vs. non-star-forming PSB galaxies. The presence of PSB\ngalaxies in all types of environment will be an important constraint\nfor understanding their origin and evolution.\n\n\n\\subsection{Building the red sequence through post-starburst galaxies}\nIn Section \\ref{sec:massflux} we compared the mass flux through the\npost-starburst phase to the mass build--up of the red sequence as\nmeasured by \\citet{2007A&A...476..137A}. We found a value of\n$\\sim$40\\% for PSBs with no residual star formation, or 80\\% for all\nPSBs with $\\log({\\rm M\/M}_\\odot)$$>9.75$. These numbers, although consistent (i.e. not\ngreater than 100\\%), perhaps still appear surprisingly high if we\nexpect other quenching mechanisms with slower timescales to also play\na role in moving mass onto the red sequence\n\\citep{2008MNRAS.387...79V}. The key question is whether there is a\none--way flow of galaxies from the blue to the red sequence?  What\nprevents a galaxy from re-starting star formation due to subsequent\ninflow of gas after experiencing a major merger? SPH simulations of\ngalaxy mergers invoke strong mechanical AGN feedback to prevent\nsubsequent star formation and allow the galaxy to remain on the\nred-sequence. The large amount of mass flowing to the red sequence\nthrough the post-starburst phase at high redshift, may in fact cause\nus to question the fact that strong mechanical feedback is effective\nin the long-term for the majority of galaxies. Semi-analytic\ncosmological models provide one method to test galaxy formation\nscenarios in a full cosmological context and could help to understand\nthe directions of mass fluxes. A full comparison of the recent star\nformation histories of the VVDS and SDSS galaxies presented in this\npaper with semi-analytic models is underway. A full census of\ndifferent types of transition galaxies as a function of redshift, when\ncombined with accurate measurements of the mass densities on the blue\nand red sequence, will help to reveal the true importance of\nmechanical gas expulsion mechanisms (feedback) on the global evolution\nof the galaxy population.\n\nWe have found that strong post-starburst galaxies have stellar masses\n  similar to those of the least massive galaxies on the red\n  sequence. This has important implications for our understanding of\n  the physical processes involved in the build-up of the red sequence,\n  and in particular the subsequent role of dry-mergers in forming the\n  shape of the red sequence mass function observed today\n  \\citep{2006ApJ...640..241B,2006ApJ...636L..81N}.\n\n\n\n\\section{Summary}\n\nUsing a PCA analysis of the spectra of the VVDS deep spectroscopic\ngalaxy survey, we have selected a sample of 16 galaxies with strong\nBalmer absorption lines at $0.5<z<1.0$ and above a mass completeness\nlimit of $\\log({\\rm M\/M}_\\odot)$$>9.75$. Through comparison with a suite of SPH merger\nsimulations and toy starburst models we have shown that these galaxies\nare likely to have undergone a strong starburst within the last few\ntenths of a Gyr, with burst mass fractions of order 10\\%, similar to\nthose derived for LIRGS at these redshifts\n\\citep{2006A&A...458..369M}.  The key requirement for the observation\nof the post-starburst galaxies, is a fast quenching timescale of\n$\\la0.1-0.2$\\,Gyr. We show that there is a maximum visible lifetime\nfor the post-starburst galaxies of 0.6\\,Gyr, but for lower gas mass\nfractions of $\\sim$20-40\\% a lifetime of 0.35\\,Gyr is more likely.\n\nThe key results of this paper are:\n\\begin{itemize}\n\\item {\\bf Number density:} PSB galaxies with $\\log({\\rm M\/M}_\\odot)$$>9.75$ have a\n  number density of $10^{-4}$ per Mpc$^3$. Assuming they are visible\n  for an average of 0.35\\,Gyr, and that a QSO shines for $10^7$ years,\n  this is equal to the number density of QSOs brighter than\n  $M^*(z=0.7)+1.5$ \\citep{2004MNRAS.349.1397C}.\n\\item{\\bf Masses and mass density:} Summing all the mass in the\n  post-starburst population gives a mass density of $\\log(\\rho^*\/{\\rm\n  M_\\odot\/Mpc}^3) = 6.43_{-0.04}^{+0.06}$. The mass distribution of\n  the PSB galaxies rises to the mass completeness limit of\n  $\\log({\\rm M\/M}_\\odot)$$>9.75$, therefore measuring the complete mass distribution for\n  post-starburst galaxies at $z\\sim0.7$ will require a deeper\n  survey. The true mass distribution is crucial for understanding the\n  relative importance of different mechanisms for causing\n  post-starburst galaxies \\citep{2007MNRAS.382..960K}.\n\\item {\\bf Mass flux:} We select the 5 PSB galaxies with no residual\n  star formation according to multiwavelength SED fitting. Taking\n  0.6\\,Gyr as an upper limit on the visibility time of the PSB\n  galaxies gives a lower limit on the mass flux of $\\dot{\\rho}_{A\n  \\rightarrow Q, PSB} > 0.0022^{+0.0002}_{-0.006}$\n  M$_\\odot$\/Mpc$^3$\/yr. Taking our best estimate for $t_{\\rm PSB}$ of\n  0.35\\,Gyr gives $\\dot{\\rho}_{A \\rightarrow Q, PSB} =\n  0.0038^{+0.0004}_{-0.001}$ M$_\\odot$\/Mpc$^3$\/yr. Comparing this to\n  the rate of build up of the red sequence\n  \\citep{2007A&A...476..137A}, we find $38^{+4}_{-11}$\\% of the growth\n  of the red sequence at $z<1$ takes place from galaxies which have\n  passed through the strong post-starburst phase, assuming all these\n  galaxies subsequently remain on the red sequence.\n\\item {\\bf Environment:} We use the local density on $5h^{-1}$Mpc scales\n  from \\citet{2006A&A...458...39C} to show that PSB galaxies are found\n  in all environments, from underdense to strongly overdense.\n\\item {\\bf Low vs. High $z$:} We compare our high-redshift results to\n  a sample of galaxies with $0.05<z<0.1$ selected from the SDSS and\n  UKIDSS surveys. Despite the enormous increase in sample size, only 3\n  post-starburst galaxies of the same strength as in the VVDS sample\n  are detected for the same mass limit. We find the mass density of\n  strong post-starburst galaxies decreases by a factor of 230 and\n  number density by a factor of 200 from $z\\sim0.7$ to $z\\sim0.07$\n  \\citep[see also][]{1997ApJ...481...49H,2006ApJ...642...48L}.\n\\end{itemize}\n\nThis paper finds that post-starburst galaxies, although rare in the\nlocal Universe, are of global importance at higher redshift. With\nlarger and deeper spectroscopic surveys, exciting new constraints\ncould be obtained on the processes which drive galaxy evolution. The\njoint analysis of imaging and spectroscopy will help to break\nobservational degeneracies, especially when combined with\n``observations'' of galaxy simulations. On the advent of an era of\nlarge broad band photometric surveys, it is important not to forget\nthe wealth of additional information on galaxy evolution available\nfrom the additional investment in spectroscopy.\n\n\n\n\n\n\n\\section*{Acknowledgments}\n\nThe authors would like to thank Celine Eminian and Steve Warren for\ntheir help with using the UKIDSS survey; Henry McCracken for all his\nhelp with the VVDS and CFHTLS surveys and the photo-zs, and Olivier\nIlbert for aiding our understanding of the photometric catalog\ngeneration; Olga Cucciati for providing the density measurements for\nthe VVDS galaxies; Christy Tremonti, Crystal Martin, John Silverman\nand Eyal Neistein for interesting discussions; the MAGPop Network and\nespecially PI Guinevere Kauffmann for providing the networking\nopportunity that led to the start of this project; and the anonymous\nreferee for their careful reading of the manuscript and constructive\ncomments for improvement. VW acknowledges an Institut d'Astrophysique\nEARA visitor grant and the Aspen center for Physics for providing the\nquiet surroundings in which this paper was mostly written. During this\nproject, VW and CJW were supported by the EU MAGPop Marie Curie\nnetwork. AP was financed by the research grant of the Polish Ministry\nof Science PBZ\/MNiSW\/07\/2006\/34A.\n\nThe numerical simulations were performed on the Munich Observatory\nSGI-Altix 3700 Bx2 machine, which was partly funded by the Cluster of\nExcellence: ``Origin and Structure of the Universe''\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe $\\Lambda$-dominated Cold Dark Matter model ($\\Lambda$CDM) has been remarkably successful in the interpretation of the large scale structure of the Universe and its evolution as probed by observations of the cosmic microwave background (CMB), the local galaxy distribution and independent probes of the dark matter power spectrum such as the Lyman-$\\alpha$ forest and weak gravitational lensing. However, significant challenges remain when the theory is confronted with observations on galaxy scales \\citep[see review by][]{weinbe2013}.\n\nA long-standing question is the apparent mismatch of the abundance of visible satellites in the Milky Way halo compared to that predicted from the steep dark matter power spectrum \\citep{klypin1999,moore1999}. Possible explanations include the likelihood that halos are dark due to the early photoionizing background \\citep{bulloc2000}, energetic feedback which suppressed growth \\citep{font2011}, or an observational bias whereby many satellites have yet to be discovered, either because they are hidden by the Galactic plane or are too faint for existing surveys \\citep{toller2008}. Recently, a more fundamental challenge relates to a surprising discrepancy between the observed and expected maximum circular velocities ($V_{max}$) for the most massive Local Group satellites. The best-studied dwarf spheroidals near the Milky Way have $12< V_{max} <25 $ km s${}^{-1}$ whereas the Aquarius \\citep{spring2008} and the Via Lactea II simulations \\citep{dieman2007, dieman2008} predict at least ten subhaloes should be visible with $V_{max} > 25$ km s${}^{-1}$ \\citep{boylan2011}. These simulations predict DM haloes five times more massive than what could be inferred given the observed densities of the satellite dwarf spheroidals. No incontrovertible explanation currently satisfies the ``Too Big to Fail\" \\citep{boylan2011} problem; some consider it a result of poorly-understood baryonic feedback effects \\citep[e.g.,][]{pontze2012}, whereas others postulate a fundamental departure from the current dark matter model \\citep[e.g.,][]{rocha2013}.\n\nBoth of the interrelated problems above rely inescapably on the question of the whether the Milky Way and its satellites are representative of those typical of the larger cosmic volumes. Furthermore, predicting satellite properties of the Milky Way is complicated by the difficulty in inferring its halo mass. For dwarf galaxies and satellites, the halo mass range $10^7 M_{\\odot}< M < 10^9 M_{\\odot}$, where resolved kinematic data are sparse, is particularly interesting. A current inventory reveals rotation curves for both Magellanic Clouds and 9 of the brightest satellites of the Milky Way. The HI Nearby Galaxy Survey (THINGS) contributes a further 4 galaxies \\citep{oh2011} to the 8 gas-rich systems from \\citet{stark2009}. \n\nA more general way to view this problem is the relationship between the stellar mass and \nhalo mass (the so-called SHM relation). This relation is well-constrained down to stellar masses of $10^9$ M${}_{\\odot}$ by various methods, including weak lensing \\citep[e.g.,][]{mandel2006,leauth2012}, abundance matching \\citep[e.g.,][]{moster2010,behroo2010} and stellar kinematics \\citep[e.g.,][]{conroy2007,more2011}. The most effective probe for lower mass systems is resolved dynamics for rotationally-supported systems, i.e. the stellar mass Tully-Fisher relation \\citep[e.g.,][]{geha2006,pizagn2007}. Various methods can be used to relate observed velocities to the halo virial circular velocity, which we discuss further as motivation for this work in the following section.\n\nThis paper presents results from a new observational program whose goal is to extend the stellar mass Tully-Fisher relation into the relevant mass range $10^7$ M${}_{\\odot} <$ M${}_{\\ast} < 10^9$ M${}_{\\odot}$ at intermediate redshifts, and constrain the stellar-to-halo mass (SHM) relation well beyond the confines of the Local Group. As we demonstrate here, such an observational program to investigate the dwarf density discrepancy described here is now feasible. \n\nA plan of the paper is as follows: in \\S 2 we describe our motivation in evaluating the SHM relation in the dwarf regime. In \\S 3, we describe the selection criteria for our sample, the DEIMOS spectroscopic data and the resolved photometry from the Hubble Space Telescope (HST) from which our stellar mass estimates are derived. Specifically \\S 3.4 discusses the techniques we use to fit kinematic models to our low mass galaxies. In \\S 4, we present our stellar mass Tully-Fisher relation and introduce our method for converting this relation to the required stellar-to-halo mass (SHM) relation. In \\S 5 we discuss our SHM results in the context of predictions from $\\Lambda$CDM cosmological simulations and abundance matching methods. Throughout the paper we adopt a $\\Omega_{\\Lambda}$ = 0.7, $\\Omega_{m}$ = 0.3, $H_0$ = 70 km sec$^{-1}$ Mpc$^{-1}$ cosmology. All magnitudes refer to the AB system. \n\n\n\\section{Motivation}\n\nThe methods used for relating velocities measured across the optical extent of galaxies to the halo virial circular velocity, or $V_{200}$ \\citep[e.g.,][]{navarr1997,bulloc2001,reyes2012}, have not been calibrated directly in the dwarf regime. Yet if the relation of \\citet{reyes2012}\\footnote{In \\citet{reyes2012}, the optical-to-virial velocity ratios are derived from both directly measured velocities from rotation curves along with $V_{200}$ values from halo masses derived with galaxy-galaxy weak lensing. A consistent and calibrated selection was used between samples rotation curve and weak lensing samples, and the constructed Tully-Fisher relations are consistent between their work at $z\\sim0$ and our previous work \\citep{miller2011} with the same mass range at higher redshift.}  to convert optical velocities to halo velocities is extrapolated to low mass, there is a serious discrepancy between the resulting halo circular velocities and those calculated based on abundance matching techniques for dwarf galaxies $<10^9$ M${}_{\\odot}$ (Fig. 1). \n\nA key question is whether this offset is a generic result for low mass dwarf galaxies beyond the Local Group. Furthermore, how do the Local Group galaxies relate to the full scatter in the stellar-to-halo mass relation of a larger, more representative sample? Although the conversion from dynamical mass to halo mass will require careful evaluation, given such scaling relations are currently only determined for higher mass systems than the focus of this work, data can hopefully provide the basis for future comparisons with numerical simulations. Nevertheless, while extrapolation of \\citet{reyes2012} is uncertain, recent work in the Next Generation Virgo Survey suggests this extrapolation to be roughly accurate (Grossauer et al. in prep.).\n\n\n\\begin{figure}\n\\includegraphics[width=3.45in]{motivation.pdf}\n\\caption{Motivating this work, there appears to be a divergence between the abundance matching result and the stellar mass Tully-Fisher relation, which has been converted to halo velocity $V_{200}$ using the extrapolated \\citet{reyes2012} relation between observed velocities as a function of stellar mass and $V_{200}$. Also plotted are the converted velocities of local group galaxies, which further suggest a discrepancy between observed velocities and abundance matching, however the scatter in the stellar-to-halo mass relation is unclear with so few data points.}\n\\label{fig:motivation}\n\\end{figure}\n\n\n\n\\begin{figure}\n\\includegraphics[width=3.5in]{i_band_cut.pdf}\n\\caption{Median magnitude curves within the \\citet{newman2012} catalog as a function of apparent $i_{AB}$ magnitude for various redshifts of interest. For the redshift range $0.2< z <0.8$ galaxies with stellar masses as low as $10^{7.2}$ to $10^{8.5}$ M${}_{\\odot}$, respectively, can be identified, thereby sampling the important range of the stellar-to-halo mass relation. Dashed lines denote the magnitude and stellar mass limits used in preparing the DEIMOS target list.}\n\\label{fig:maglimit}\n\\end{figure}\n\n\\section{Data}\n\nWe present a new spectroscopic dataset which exploits the significant multiplex gain of DEIMOS to address the kinematic properties of low mass star-forming dwarf galaxies at intermediate redshift ($0.2< z <1.0$). \nTwo advances make it practical to target dwarf galaxies at intermediate redshifts: First is our demonstrated ability to recover rotational velocities on small angular scales from our earlier work on high redshift disk galaxies \\citep{miller2011,miller2012a}, the relevant techniques of which we describe further in \\S 3.4. Second is the availability of remarkably deep Hubble Space Telescope (HST) Wide-Field Camera 3 (WFC3) infra-red imaging from the Cosmic Assembly Near-IR Deep Extragalactic Legacy Survey \\citep[CANDELS;][]{grogin2011,koekem2011}, enabling us to conduct resolved photometric models of dwarf targets and construct catalogs \\citep[i.e.,][]{newman2012} of photometric redshifts and stellar mass estimates along with deep optical Subaru SXDS data for much lower mass systems at intermediate redshifts than previously possible. Although a 10 meter class telescope and long exposures are necessary to study these low mass sources efficiently, as the target density within a DEIMOS field is substantial, the multiplex gain offers a huge advantage over modest samples of dwarf galaxies studied individually in nearby galaxies. Targets of similar mass available from SDSS have a much lower surface density and cannot be efficiently surveyed with the current multiplex gains on 4- or 10-meter aperture telescopes.\n\n\\begin{figure}\n\\includegraphics[width=3.6in]{sample.pdf}\n\\caption{The stellar mass and redshift distribution for the 82 dwarf galaxies studied spectroscopically. 23 galaxies which revealed no significant line emission are plotted as red boxes (in the colored version) at their photometric redshifts. 18 galaxies which revealed unresolved line emission are denoted with green crosses (in the colored version) at their spectroscopic redshift. The other 41 galaxies show resolved line emission are are denoted with black circular points. The dashed line marks the stellar mass limit for the overall sample.}\n\\label{fig:sample}\n\\end{figure}\n\n\\subsection{Sample Selection and DEIMOS Observations}\n\nIn this work we examine a large number of low mass galaxies at intermediate redshift drawn from the UKIRT Ultradeep Survey field (UDS) using the photometric catalog derived by  \\citet{newman2012}.\nWe adopted a magnitude limit of $i_{AB}=25$ which corresponds approximately to a median stellar mass limit of $10^{7.2}, 10^{8.0}$, and $10^{8.5}$ M${}_{\\odot}$ for $z \\simeq 0.2$, 0.5, and 0.8, respectively (Fig. \\ref{fig:maglimit}). Two further criteria were used to select targets, a photometric redshift range $0.2<z<0.8$ and a stellar mass range $\\log{(}$M${}_{\\ast}$\/M${}_{\\odot})<$ 9.3. Photometric redshifts were determined using the EAzY code \\citep{bramme2008} and stellar masses were estimated using the FAST code \\citep{kriek2009}--- for more details see \\S \\ref{sec:stmass}. In constructing the multislit DEIMOS mask, slitlets were aligned to the major axes for significantly inclined targets \\citep[based on photometry using SExtractor:][]{bertin1996}. The mask position angle was selected so as to optimally fill that portion of the UDS field with available WFC3 coverage.\n\nWe used the 1200 l mm${}^{-1}$ grating in DEIMOS centered at 7500 \\AA~so that [OIII] and H$\\beta$ line emission would be sampled for the full targeted redshift range, H$\\alpha$ emission for $z\\simeq0.2-0.4$ and [OII] for $z<0.6$. With $\\sim$30 km s${}^{-1}$ kinematic resolution, our earlier work demonstrated the ability to recover rotation curves with characteristic velocities of 50 km s${}^{-1}$ \\citep{miller2011,miller2012a}.\nOur modeling code was well-tested using simulated data to rotational velocities as low as  30 km s${}^{-1}$. This corresponds to the characteristic maximal velocity for dwarf galaxies with stellar masses of $<10^6$ M${}_{\\odot}$ in the abundance matching method, but to a stellar mass of $10^{7.3}$ M${}_{\\odot}$ in the extrapolated stellar mass TF relation of \\citet{miller2011}. Although our typical targets are only 1.5-2$\\arcsec$ across, this corresponds to the extent of more massive systems successfully targeted at $z>$1 using our code \\citep{miller2012a,miller2013}.\n\nWe successfully secured 10 hours of science-quality exposure time with the DEIMOS instrument on a single mask in the UDS field on December 11-12, 2012. A total of 82 dwarfs were targeted, but not all objects observed show ordered rotation or line emission. However we do not seek to construct a complete sample via this approach; of course such completeness is not even possible in local data. In summary we find that 28\\% (N=23) of the targeted sample reveal no significant line emission. These sources occupy the lower surface brightness portion of the photometric distribution suggesting their line emission may simply be too faint to be detectable.\nWe find 22\\% (N=18) have unresolved line emission precluding any attempt to construct rotation curves. The remaining 50\\% (N=41) have resolved emission from which we can attempt to derive rotation curves (see Fig. \\ref{fig:sample}). This mix of target properties is actually similar to that recovered in our previous higher mass samples at intermediate redshifts.\n\n\\begin{figure*}\n\\center\n\\includegraphics[width=6.5in]{dwarf_examples1.pdf}\n\\includegraphics[width=6.5in]{dwarf_examples2.pdf}\n\\caption{Examples from the new intermediate redshift dwarf sample: {\\bf (Left to right)}:  galaxy redshift and stellar mass; HST WFC3 F125W (H-band) image with the DEIMOS slitlet overlaid; a triptych of the 2-D spectrum, the modeled spectrum after blurring by both seeing and dispersion; the (data minus model) residual. Continued in Fig. \\ref{fig:ex3}.}\n\\label{fig:ex1}\n\\end{figure*}\n\n\\subsection{Stellar Mass Estimates} \\label{sec:stmass}\n\n\nStellar mass estimates were derived using the FAST (Fitting and Assessment of Synthetic Templates) code \\citep{kriek2009} which fits stellar population synthesis templates \\citep{bruzua2003} to broadband photometry adopting a \\citet{chabri2003} initial mass function (IMF). For our photometric database \\citep{newman2012}, we use $BVRiz$ from deep Subaru\/XMM-Newton Deep Survey, SXDS \\citep{furusa2008}, with mosaics prepared by \\citet{cirasu2010}; $J$ (F125W) and $H$ (F160W) from CANDELS; and K photometry by the UKIDSS UDS Data Release 6 (DR6). We chose not to include photometry available at longer wavelengths due to blending confusion given the faintness of the sources targeted. We impose a floor of $\\log{(\\tau\/}$yr) $=$ 8.5 for the timescale of the exponentially-declining star formation histories given this appears appropriate for star-forming galaxies \\citep[e.g.,][]{wuyts2011}.  Sub-solar metallicities of $Z = $0.008, 0.004 as well as solar at 0.02 are permitted, consistent with the latest work on the metallicity of dwarf galaxies in the Local Group. FAST uses the Calzetti et al. (2000) reddening law.  In summary the stellar population grid limits are:\n$\\log{(\\tau}$\/yr) $=$ 8.5 - 10.0;\n$\\log{(}$age\/yr) $=$ 7.0 - 10.1;\nA${}_V$ $=$ 0.0 - 3.0;\nmetallicity $=$ 0.004-0.02.\n\n\nWe computed best-fitting model parameters with errors derived from Monte Carlo perturbations of the photometry, as well as the marginalized mean value and standard deviation of the parameter over the likelihood. We adopt the latter as they are more robust, however there are marginal differences for this particular dataset. \n\nIt should be noted that the original set of stellar mass estimates used in the selection of the DEIMOS mask targets adopted a solar metallicity only, a Salpeter IMF, and the stellar population models of Charlot \\& Bruzual (2007) which include more contribution from TP-ABG stars. Each of these choices is suited more to a higher mass sample than the dwarfs of this work. To compare to the final adopted estimates of this work, these alternate assumptions provided a systematic offset of $+$0.14 dex. For 7 galaxies at stellar masses $\\log{(}$M${}_{\\ast}$\/M${}_{\\odot})<$ 7.7 the results are particularly sensitive to these stellar population grid differences, but all estimates in the original set still remain below $\\log{(}$M${}_{\\ast}$\/M${}_{\\odot})<$ 8.7.\n\n\\begin{figure*}\n\\center\n\\includegraphics[width=6.6in]{dwarf_examples3.pdf}\n\\caption{As in Fig. \\ref{fig:ex1}.}\n\\vspace{0.15in}\n\\label{fig:ex3}\n\\end{figure*}\n\n\\subsection{Image Measurements}\n\nExamining the HST images of our sample, although most are small in angular extent (mean half-light radius, $\\langle r_{1\/2} \\rangle \\sim$ 1.2 kpc), the majority are regular in form with modest central concentrations. A high fraction have recognizable disks but few have prominent bulges.\n\nWe measure the radii and inclination of the dwarf galaxies in our sample assuming each galaxy is represented by a circular exponential disk. Scale radii and major-to-minor axis ratios are measured using GALFIT \\citep{peng2010} with initial estimate distributions based on SExtractor \\citep{bertin1996} in the same Monte Carlo method as described in previous work \\citep{miller2011,miller2012a, miller2013}. We adopted this formalism for consistency with work at higher masses, and particularly to extract velocities at a consistent location, namely at 2.2 times the scale radius, $r_{2.2}$ \\citep{miller2011}. The fiducial velocity measured at $r_{2.2}$ is denoted by $V_{2.2}$ and represents the peak rotational amplitude in a pure exponential disk. Although all of the dwarf galaxies of our sample are not necessarily best represented by a pure exponential, $V_{2.2}$ is generally well-suited in sampling the flattened portion of the rotation curve in galaxies \\citep[see][for further discussion on this topic]{courte1997,miller2011}. For instance, allowing a free Sersic index fit in our sample produces a significantly more inconsistent set of effective radii by which to extract fiducial velocities for scaling relations.\n\nWhile exponential disks are the simplest and most appropriate model for the morphologies of the majority of our dwarf sample, we also \nextract the velocity at a radius of 1 kpc (see Table \\ref{table_of_results}).  These values are provided so that data samples analyzed with other techniques, namely predictions from numerical simulations, can more easily be compared to our values as exponential disk scale lengths via mock photometry measurements are not typically easily acquired.\n\n\\subsection{Dynamical Modeling}\n\nThe rotation curves for our sample were analyzed with the code developed in the previous studies of \\citet{miller2011,miller2012a,miller2013}. We adopt the arctangent (arctan) functional form, and extract the velocity at 2.2 times the scale radius, $V_{2.2}$, correcting for the effect of inclination. This code has been described and tested extensively in our earlier papers and includes the following features. It adopts an asymmetric Gaussian for the emission line profile of each spatial bin, and accounts for both spatially-dependent velocity dispersion and surface-brightness profiles of the traced emission line in the spectrum. Including these features, the code also accounts for the effects of blurring by atmospheric seeing in the spatial direction, and blurring by instrumental dispersion in the spectral direction.  These are crucial ingredients for  recovering the intrinsic velocity profile of galaxies, particularly those with small angular extents. Examples of our 2-D emission line spectra and model fits are given in Figures \\ref{fig:ex1} \\& \\ref{fig:ex3}. \n\n\\smallskip\n\nTable \\ref{table_of_results} lists all of the galaxies in our sample, including each of the derived physical properties, where possible, which have been described in \\S 3.\n\n\n\\section{Results}\\label{sec:results}\n\n\\begin{figure*}\n\\center\n\\includegraphics[width=4.3in]{SMTF.pdf}\n\\caption{The stellar mass Tully-Fisher (TF) relation: the new sample of low mass galaxies is shown according to two redshift intervals: blue diamonds ($0.2 < z < 0.5$) and purple squares ($0.5 < z < 0.9$). A more massive sample spanning the full redshift range is shown with green points \\citep{miller2011} and the Local Group dwarf population at the targeted mass range, including the Magellanic Clouds, is plotted with golden rings. Lines represent best fits to the dwarf stellar mass TF relation (solid black), the intermediate redshift  sample \\citep{miller2011} (dotted green), and the local Baryonic TF relation as determined by \\citet{mcgaug2012} (yellow dashed).}\n\\vspace{0.2in}\n\\label{fig:smtf}\n\\end{figure*}\n\nTo evaluate the stellar-to-halo mass relation of our dwarf galaxy sample, we first consider the stellar mass Tully-Fisher (TF) relation (Fig. \\ref{fig:smtf}), which represents the primary observational result of this paper,\nalongside earlier work on higher masses at intermediate redshift \\citep{miller2011}. For reference, we also indicate the Baryonic TF relation at $z\\sim0$ \\citep{mcgaug2012}, which includes the mass of the gas in the ordinate as well as the stellar mass. \n\nOur new data reveal a remarkably similar slope to the previously-measured Miller et al. relation at higher mass and similar redshifts (see Table \\ref{table_smtfr} for the detailed comparison). Although aspects of the dwarf relation could suggest a somewhat steeper slope toward low mass than that inferred at high mass, this possible trend is marginal and requires more data for confirmation.  More interesting is the distribution of the locally-measured dwarfs relative to our sample at intermediate redshift.  As seen in Figure \\ref{fig:smtf}, local galaxies lie mostly to the higher stellar mass (or rather slower velocity) side of the stellar mass TF relation. We will return to this topic in the next section.\n\nWe next consider the relation between stellar mass and halo mass, by converting $V_{2.2}$ to $V_{200}$ in Fig. \\ref{fig:shm} using the relation measured by \\citet{reyes2012} with weak lensing. This relation is\neffectively calibrated at log($M_{\\ast}\/M_{\\odot}$) = 9.0, where $V_{2.2}\/V_{200} = 1.05$ and has a slope of 0.53$\\pm$0.03 (in terms of the relation of velocity ratio to stellar mass). To apply this in the \nlow mass regime of interest here requires extrapolating this relationship to stellar masses of $10^7$ M${}_{\\odot}$, which clearly introduces some uncertainty. We note that preliminary abundance matching results of the Next Generation Virgo Survey (Grossauer et al. in prep.) in the dwarf regime are consistent with the extrapolation of the stellar-to-halo mass relations from higher masses using abundance matching or weak lensing. Thus since the halo velocities in the Reyes et al. result are derived from weak-lensing as well, it is a fair hypothesis that the extrapolated relations are similar to what would be derived from the method conducted explicitly in the dwarf regime. \n\nAssuming spherical symmetry, we further convert $V_{200}$ velocities (where the over-density is 200 times the critical density, $200 \\rho_{\\mathrm{crit}} = M_{\\rm 200} \/ \\frac{4}{3} \\pi R_{\\rm 200}^3$) to halo masses via the standard adopted relation where $\\log M_{200}\/[h^{-1} M_{\\odot}] = 3 \\log{(G^{-1} V_{200} [\\mathrm{km}~\\mathrm{s}^{-1}])}$. The gravitational constant $G$ here is 4.3 $\\times$ $10^{-6}$ kpc M${}_{\\odot}^{-1}$ (km\/s)${}^{2}$. While abundance matching results and calibrations based on the latest weak lensing studies \\citep[e.g.,][]{leauth2012,behroo2013} produce consistent results when compared to the stellar mass TF relation above masses of $10^9$ M${}_{\\odot}$, there is a clear divergence in the dwarf regime of these relations in Figure \\ref{fig:shm}. We discuss the implications and uncertainties of this result in the next section.\n\n\n\\begin{small}\n\\begin{longtable*}{lllllllllll}\n\\caption{The Intermediate Redshift Dwarf Galaxy Sample} \\label{table_of_results} \\\\\n\n\\hline \\hline \\\\ [-1ex]\n\\multicolumn{1}{c}{R.A.} & \n\\multicolumn{1}{c}{Dec.} &  \n\\multicolumn{1}{c}{$z$\\tablenotemark{1}} &  \n\\multicolumn{1}{c}{sin($i$)\\tablenotemark{2}} & \n\\multicolumn{1}{c}{$r_s$\\tablenotemark{3}[kpc]} & \n\\multicolumn{1}{c}{log(M${}_{\\ast}$\/M${}_{\\odot}$)\\tablenotemark{4}} & \n\\multicolumn{1}{c}{$V_{2.2}$\\tablenotemark{5} [km s${}^{-1}$]}  & \n\\multicolumn{1}{c}{$V_{1\\mathrm{kpc}}$\\tablenotemark{6}  [km s${}^{-1}$]}  & \n\\multicolumn{1}{c}{log(M${}_{h}$\/M${}_{\\odot}$)\\tablenotemark{7}} \\\\ [1ex] \\hline \\\\%[-1ex]\n\\endhead\n\\multicolumn{9}{c}{\\textbf{Sample with resolved emission (N=41; ``Tully-Fisher sample''):}} \\\\ [1ex]\n\\hline \\\\  \n34.287564 & -5.1435433 & 0.64 & 0.94 & 1.04$\\pm$0.31 & 8.74$\\pm$0.09 & 97.77$\\pm$3.82 & 83.08$\\pm$3.27 & 10.79$\\pm$0.03 \\\\\n34.303787 & -5.1458641 & 0.91 & 0.66 & 0.75$\\pm$0.20 & 8.32$\\pm$0.12 & 56.22$\\pm$11.13 & 45.29$\\pm$8.97 & 10.39$\\pm$0.15 \\\\\n34.358524 & -5.1515666 & 0.35 & 0.66 & 0.41$\\pm$0.25 & 7.84$\\pm$0.09 & 16.01$\\pm$9.71 & 12.47$\\pm$7.56 & 9.36$\\pm$0.39 \\\\\n34.398975 & -5.1525832 & 0.35 & 0.94 & 1.16$\\pm$0.48 & 8.79$\\pm$0.07 & 78.97$\\pm$14.76 & 72.21$\\pm$13.46 & 10.70$\\pm$0.14 \\\\\n34.289186 & -5.1563347 & 0.57 & 0.91 & 0.77$\\pm$0.29 & 8.31$\\pm$0.09 & 31.35$\\pm$15.67 & 31.15$\\pm$15.57 & 9.80$\\pm$0.35 \\\\\n34.359126 & -5.1571952 & 0.34 & 0.62 & 0.32$\\pm$0.27 & 7.29$\\pm$0.14 & 32.42$\\pm$16.21 & 32.30$\\pm$16.15 & 9.68$\\pm$0.30 \\\\\n34.336092 & -5.1585409 & 0.31 & 0.52 & 0.40$\\pm$0.36 & 7.83$\\pm$0.08 & 17.29$\\pm$3.26 & 17.39$\\pm$3.25 & 9.56$\\pm$0.13 \\\\\n34.489719 & -5.1596155 & 0.98 & 0.72 & 0.49$\\pm$0.30 & 8.64$\\pm$0.14 & 81.72$\\pm$40.86 & 81.06$\\pm$40.53 & 10.78$\\pm$0.32 \\\\\n34.404135 & -5.1638168 & 0.43 & 0.87 & 0.68$\\pm$0.15 & 8.43$\\pm$0.06 & 50.21$\\pm$5.89 & 48.97$\\pm$5.74 & 10.20$\\pm$0.09 \\\\\n34.310380 & -5.1642154 & 0.58 & 0.91 & 0.80$\\pm$0.18 & 8.54$\\pm$0.13 & 68.82$\\pm$17.85 & 68.36$\\pm$17.73 & 10.44$\\pm$0.20 \\\\\n34.367527 & -5.1671357 & 0.65 & 0.92 & 0.66$\\pm$0.35 & 8.41$\\pm$0.08 & 81.89$\\pm$30.77 & 79.20$\\pm$29.76 & 10.47$\\pm$0.27 \\\\\n34.440428 & -5.1676006 & 0.64 & 0.95 & 0.82$\\pm$0.23 & 8.74$\\pm$0.09 & 68.08$\\pm$19.55 & 46.02$\\pm$13.00 & 10.55$\\pm$0.21 \\\\\n34.473897 & -5.1691046 & 0.31 & 0.77 & 0.35$\\pm$0.28 & 7.76$\\pm$0.11 & 33.26$\\pm$12.27 & 32.04$\\pm$11.52 & 9.71$\\pm$0.25 \\\\\n34.446086 & -5.1700983 & 0.54 & 0.86 & 0.52$\\pm$0.11 & 8.15$\\pm$0.15 & 29.20$\\pm$3.07 & 28.60$\\pm$3.01 & 9.71$\\pm$0.08 \\\\\n34.279085 & -5.1710765 & 0.35 & 0.47 & 0.63$\\pm$0.22 & 8.30$\\pm$0.06 & 40.32$\\pm$11.63 & 28.91$\\pm$8.38 & 10.38$\\pm$0.16 \\\\\n34.350739 & -5.1712933 & 0.26 & 0.73 & 0.31$\\pm$0.10 & 7.25$\\pm$0.11 & 20.45$\\pm$5.72 & 16.88$\\pm$4.85 & 9.26$\\pm$0.18 \\\\\n34.503690 & -5.1738860 & 0.43 & 0.34 & 0.57$\\pm$0.34 & 8.27$\\pm$0.08 & 18.78$\\pm$5.41 & 13.93$\\pm$3.93 & 10.09$\\pm$0.18 \\\\\n34.432528 & -5.1760097 & 0.43 & 0.98 & 0.88$\\pm$0.12 & 8.75$\\pm$0.09 & 79.48$\\pm$5.71 & 72.14$\\pm$5.17 & 10.64$\\pm$0.06 \\\\\n34.466202 & -5.1780329 & 0.29 & 0.97 & 1.07$\\pm$0.17 & 8.76$\\pm$0.10 & 58.70$\\pm$2.42 & 59.67$\\pm$2.46 & 10.46$\\pm$0.03 \\\\\n34.429762 & -5.1785330 & 0.78 & 0.76 & 0.51$\\pm$0.39 & 8.96$\\pm$0.06 & 91.60$\\pm$24.93 & 55.58$\\pm$15.92 & 11.09$\\pm$0.17 \\\\\n34.434409 & -5.1793119 & 0.78 & 0.96 & 0.72$\\pm$0.20 & 8.50$\\pm$0.08 & 77.75$\\pm$45.98 & 69.37$\\pm$41.02 & 10.46$\\pm$0.40 \\\\\n34.517196 & -5.1793529 & 0.32 & 0.76 & 0.41$\\pm$0.09 & 7.89$\\pm$0.07 & 29.14$\\pm$7.97 & 28.12$\\pm$7.69 & 9.68$\\pm$0.20 \\\\\n34.250846 & -5.1797838 & 0.35 & 0.74 & 0.34$\\pm$0.29 & 7.60$\\pm$0.13 & 48.38$\\pm$5.91 & 47.92$\\pm$5.85 & 9.92$\\pm$0.09 \\\\\n34.415249 & -5.1820298 & 0.72 & 0.91 & 0.65$\\pm$0.23 & 8.50$\\pm$0.08 & 66.84$\\pm$4.53 & 66.25$\\pm$4.48 & 10.39$\\pm$0.06 \\\\\n34.296140 & -5.1827143 & 0.21 & 0.96 & 0.58$\\pm$0.20 & 8.13$\\pm$0.12 & 47.67$\\pm$3.51 & 49.45$\\pm$3.64 & 9.94$\\pm$0.06 \\\\\n34.362769 & -5.1829861 & 0.25 & 0.87 & 0.56$\\pm$0.28 & 8.03$\\pm$0.09 & 30.19$\\pm$3.31 & 29.87$\\pm$3.28 & 9.67$\\pm$0.08 \\\\\n34.477748 & -5.1834746 & 0.34 & 0.84 & 0.38$\\pm$0.21 & 7.46$\\pm$0.11 & 45.24$\\pm$1.52 & 43.28$\\pm$1.46 & 9.75$\\pm$0.03 \\\\\n34.423177 & -5.1861241 & 0.65 & 0.94 & 0.50$\\pm$0.06 & 7.95$\\pm$0.08 & 50.82$\\pm$16.46 & 43.36$\\pm$13.67 & 9.92$\\pm$0.23 \\\\\n34.282862 & -5.1864030 & 0.45 & 0.73 & 0.35$\\pm$0.09 & 8.02$\\pm$0.07 & 50.73$\\pm$13.39 & 34.80$\\pm$9.71 & 10.11$\\pm$0.15 \\\\\n34.520764 & -5.1871599 & 0.17 & 0.82 & 0.30$\\pm$0.24 & 7.06$\\pm$0.27 & 26.81$\\pm$13.41 & 28.01$\\pm$14.00 & 9.31$\\pm$0.31 \\\\\n34.413980 & -5.1882020 & 0.88 & 0.65 & 0.38$\\pm$0.26 & 8.20$\\pm$0.20 & 31.35$\\pm$8.27 & 29.63$\\pm$7.56 & 9.96$\\pm$0.09 \\\\\n34.523446 & -5.1899136 & 0.36 & 0.91 & 0.38$\\pm$0.37 & 7.40$\\pm$0.21 & 41.85$\\pm$12.51 & 41.70$\\pm$11.88 & 9.62$\\pm$0.21 \\\\\n34.254774 & -5.1914894 & 0.65 & 0.70 & 0.84$\\pm$0.29 & 8.50$\\pm$0.08 & 55.40$\\pm$15.34 & 51.98$\\pm$14.30 & 10.45$\\pm$0.19 \\\\\n34.359852 & -5.1916576 & 0.82 & 0.97 & 0.86$\\pm$0.14 & 8.66$\\pm$0.14 & 51.73$\\pm$25.91 & 43.39$\\pm$21.73 & 10.30$\\pm$0.35 \\\\\n34.417074 & -5.1950333 & 0.65 & 0.89 & 0.55$\\pm$0.27 & 8.36$\\pm$0.11 & 37.47$\\pm$6.58 & 36.99$\\pm$6.50 & 9.95$\\pm$0.13 \\\\\n34.370424 & -5.2035566 & 0.65 & 0.98 & 1.36$\\pm$0.33 & 8.80$\\pm$0.11 & 78.42$\\pm$1.43 & 75.90$\\pm$1.39 & 10.67$\\pm$0.02 \\\\\n34.465397 & -5.2073517 & 0.59 & 0.73 & 0.44$\\pm$0.23 & 8.82$\\pm$0.07 & 54.11$\\pm$14.78 & 50.42$\\pm$13.23 & 10.64$\\pm$0.13 \\\\\n34.446346 & -5.1498754 & 0.49 & 0.94 & 0.92$\\pm$0.29 & 9.24$\\pm$0.12 & 79.21$\\pm$8.48 & 78.91$\\pm$7.87 & 10.95$\\pm$0.09 \\\\\n34.283345 & -5.1519078 & 0.65 & 0.98 & 1.08$\\pm$0.13 & 9.06$\\pm$0.13 & 87.92$\\pm$8.62 & 82.34$\\pm$7.73 & 10.96$\\pm$0.08 \\\\\n34.392003 & -5.1590678 & 0.56 & 0.92 & 0.92$\\pm$0.09 & 8.94$\\pm$0.07 & 77.18$\\pm$17.10 & 65.16$\\pm$14.63 & 10.83$\\pm$0.17 \\\\\n34.275351 & -5.1724717 & 0.41 & 0.88 & 0.74$\\pm$0.44 & 9.08$\\pm$0.16 & 77.91$\\pm$5.11 & 76.85$\\pm$5.05 & 10.96$\\pm$0.05 \\\\ [1ex]\n\\hline \\\\ \n\\multicolumn{9}{l}{\\textbf{Sample with unresolved emission (N=18)}} \\\\ [1ex]\n\\hline \\\\ \n34.452142 & -5.1527207 & 0.62 & - & 0.57$\\pm$0.13 & 8.29$\\pm$0.10 & - & - & - \\\\\n34.490433 & -5.1591549 & 0.29 & - & 0.62$\\pm$0.47 & 8.58$\\pm$0.09 & - & - & - \\\\\n34.333114 & -5.1617036 & 0.55 & - & 0.34$\\pm$0.20 & 8.45$\\pm$0.13 & - & - & - \\\\\n34.501182 & -5.1717299 & 0.57 & - & 0.76$\\pm$0.44 & 8.88$\\pm$0.12 & - & - & - \\\\\n34.269635 & -5.1725942 & 0.52 & - & 0.91$\\pm$0.07 & 8.36$\\pm$0.08 & - & - & - \\\\\n34.321552 & -5.1734201 & 0.65 & - & 0.50$\\pm$0.14 & 8.45$\\pm$0.08 & - & - & - \\\\\n34.393462 & -5.1748474 & 0.43 & - & 0.58$\\pm$0.10 & 8.33$\\pm$0.08 & - & - & - \\\\\n34.338840 & -5.1827413 & 0.43 & - & 0.79$\\pm$0.34 & 8.83$\\pm$0.07 & - & - & - \\\\\n34.344560 & -5.1913127 & 0.39 & - & 0.29$\\pm$0.12 & 7.95$\\pm$0.11 & - & - & - \\\\\n34.496066 & -5.2060556 & 0.80 & - & 0.67$\\pm$0.13 & 8.72$\\pm$0.17 & - & - & - \\\\\n34.266036 & -5.1594535 & 0.82 & - & 0.81$\\pm$0.27 & 8.35$\\pm$0.16 & - & - & - \\\\\n34.338844 & -5.1653685 & 0.79 & - & 0.39$\\pm$0.34 & 8.50$\\pm$0.10 & - & - & - \\\\\n34.481657 & -5.1674153 & 0.75 & - & 0.67$\\pm$0.23 & 8.66$\\pm$0.06 & - & - & - \\\\\n34.370433 & -5.1823730 & 0.20 & - & 0.22$\\pm$0.15 & 7.57$\\pm$0.16 & - & - & - \\\\\n34.481431 & -5.1833789 & 0.46 & - & 0.67$\\pm$0.26 & 8.95$\\pm$0.09 & - & - & - \\\\\n34.514295 & -5.1839202 & 0.39 & - & 0.43$\\pm$0.19 & 8.23$\\pm$0.09 & - & - & - \\\\\n34.341006 & -5.1872395 & 0.44 & - & 0.27$\\pm$0.24 & 8.10$\\pm$0.08 & - & - & - \\\\\n34.375929 & -5.1970098 & 0.60 & - & 0.80$\\pm$0.07 & 8.38$\\pm$0.16 & - & - & - \\\\\n[1ex]\n\\hline \\\\ \n\\multicolumn{9}{l}{\\textbf{Sample with no emission detected (N=23)}} \\\\ [1ex]\n\\hline \\\\ \n34.295579 & -5.1471624 & (0.61) & - & 0.53$\\pm$0.11 & 8.73$\\pm$0.12 & - & - & - \\\\\n34.403433 & -5.1479603 & (0.38) & - & 0.34$\\pm$0.10 & 7.89$\\pm$0.10 & - & - & - \\\\\n34.273705 & -5.1537420 & (0.29) & - & 0.50$\\pm$0.22 & 8.07$\\pm$0.10 & - & - & - \\\\\n34.422943 & -5.1545570 & (0.58) & - & 0.73$\\pm$0.26 & 8.33$\\pm$0.16 & - & - & - \\\\\n34.354658 & -5.1553817 & (0.43) & - & 0.27$\\pm$0.22 & 8.20$\\pm$0.07 & - & - & - \\\\\n34.396846 & -5.1563103 & (0.43) & - & 0.48$\\pm$0.06 & 7.95$\\pm$0.09 & - & - & - \\\\\n34.410996 & -5.1571028 & (0.32) & - & 0.34$\\pm$0.28 & 7.85$\\pm$0.09 & - & - & - \\\\\n34.327556 & -5.1592369 & (0.22) & - & 0.25$\\pm$0.20 & 7.25$\\pm$0.25 & - & - & - \\\\\n34.454441 & -5.1590304 & (0.42) & - & 0.25$\\pm$0.20 & 7.90$\\pm$0.27 & - & - & - \\\\\n34.509970 & -5.1648048 & (0.66) & - & 0.36$\\pm$0.11 & 8.37$\\pm$0.14 & - & - & - \\\\\n34.507877 & -5.1696592 & (0.75) & - & 0.84$\\pm$0.21 & 8.62$\\pm$0.15 & - & - & - \\\\\n34.305389 & -5.1736575 & (0.28) & - & 0.38$\\pm$0.09 & 7.65$\\pm$0.16 & - & - & - \\\\\n34.313890 & -5.1745230 & (0.32) & - & 0.39$\\pm$0.26 & 7.81$\\pm$0.09 & - & - & - \\\\\n34.398413 & -5.1746398 & (0.76) & - & 0.59$\\pm$0.43 & 8.51$\\pm$0.13 & - & - & - \\\\\n34.258658 & -5.1763573 & (0.66) & - & 0.63$\\pm$0.06 & 8.73$\\pm$0.10 & - & - & - \\\\\n34.447737 & -5.1786612 & (0.52) & - & 0.46$\\pm$0.09 & 8.10$\\pm$0.16 & - & - & - \\\\\n34.474809 & -5.1845825 & (0.31) & - & 0.43$\\pm$0.38 & 7.86$\\pm$0.11 & - & - & - \\\\\n34.495861 & -5.1862426 & (0.27) & - & 0.26$\\pm$0.09 & 7.10$\\pm$0.22 & - & - & - \\\\\n34.468164 & -5.1934894 & (0.59) & - & 0.50$\\pm$0.13 & 8.05$\\pm$0.07 & - & - & - \\\\\n34.453041 & -5.1942366 & (0.39) & - & 0.39$\\pm$0.13 & 7.46$\\pm$0.20 & - & - & - \\\\\n34.438733 & -5.1996580 & (0.28) & - & 0.25$\\pm$0.24 & 8.13$\\pm$0.11 & - & - & - \\\\\n34.507926 & -5.2060951 & (0.64) & - & 0.78$\\pm$0.50 & 8.47$\\pm$0.22 & - & - & - \\\\\n34.372178 & -5.2088495 & (0.31) & - & 0.36$\\pm$0.25 & 7.88$\\pm$0.12 & - & - & - \\\\\n[1ex]\n\\hline \\\\\n\\caption{[1] Redshifts are spectroscopic except when in parentheses (photometric); [2] Sine of inclination; [3] Exponential scale radius (kpc); [4] Stellar mass in log(M${}_{\\ast}$\/M${}_{\\odot}$); [5] Rotation velocity $V_{2.2}$ km s${}^{-1}$; [6] Rotation velocity at 1 kpc (km s${}^{-1}$); [7] Halo mass inferred as described in \\S 3, log(M${}_h$\/M${}_{\\odot}$).}\n\\end{longtable*}\n\\end{small}\n\n\n\\begin{small}\n\\begin{longtable*}{lllllllllll}\n\\caption{Stellar mass TF relation fits} \\label{table_smtfr} \\\\\n \\hline \\hline \\\\ [-1ex]\n\\multicolumn{1}{c}{$z$ range} & \n\\multicolumn{1}{c}{$\\langle z \\rangle$} &  \n\\multicolumn{1}{c}{N} &  \n\\multicolumn{1}{c}{$a$\\tablenotemark{1}} & \n\\multicolumn{1}{c}{$b$\\tablenotemark{2}} & \n\\multicolumn{1}{c}{$\\sigma_{int,V}$\\tablenotemark{3}} & \n\\multicolumn{1}{c}{med $\\sigma_{V}$\\tablenotemark{4}}  & \n\\multicolumn{1}{c}{$rms_V$\\tablenotemark{5}}  & \n\\multicolumn{1}{c}{$\\sigma_{int,M}$\\tablenotemark{6}}  & \n\\multicolumn{1}{c}{med $\\sigma_{M}$\\tablenotemark{7}}  & \n\\multicolumn{1}{c}{$\\mathrm{rms}_M$\\tablenotemark{8}} \\\\ [1ex] \\hline \\\\%[-1ex]\n\\endhead\n\\multicolumn{11}{l}{\\textbf{M${}_{\\ast} > 10^9$ M${}_{\\odot}$ Relation (Miller et al. 2011):  $M_{\\ast}$  vs.  $V$($r_{2.2}$):}} \\\\ [1ex]\n\\hline \\\\  \n{0.2$<$z$\\le$1.3} & {0.64} & {129} & {1.718$\\pm$0.415} & {3.869$\\pm$0.193} & {0.058} & {0.022} & {0.083} & {0.224} & {0.091} & {0.323} \\\\[1ex] \\hline \\\\ \n\\multicolumn{11}{l}{\\textbf{Dwarf Mass Relation (this work):  $M_{\\ast}$ vs.  $V$($r_{2.2}$)}} \\\\ [1ex]\n\\hline \\\\ \n{0.2$<$z$\\le$1.0} & {0.64} & {41} & {0.566$\\pm$0.476} & {4.347$\\pm$0.623} & {0.064} & {0.078} & {0.122} & {0.375} & {0.095} & {0.532} \\\\ [1ex]\n\\hline \\\\\n\\caption{[1] y-int in $M_{\\ast}\/M_{\\odot}$ dex assuming scatter in $V$\/km s$^{-1}$ dex ; [2] slope assuming scatter in $V$\/km s$^{-1}$ dex; [3] internal scatter in $V$\/km s$^{-1}$ dex; [4] median velocity error in $V$\/km s$^{-1}$ dex; [5] total scatter in $V$\/km s$^{-1}$ dex; [6] internal scatter in $M_{\\ast}\/M_{\\odot}$ dex; [7] median stellar mass error in $M_{\\ast}\/M_{\\odot}$ dex; [8] total scatter in $M_{\\ast}\/M_{\\odot}$ dex.}\n\\end{longtable*}\n\\end{small}\n\n\\begin{figure}[h]\n\\center\n\\includegraphics[width=3.5in]{SHM.pdf}\n\\caption{The stellar-to-halo mass relation according to the abundance matching curves of \\citet{behroo2013} (aqua and red curves at z=0.1 and z=1.0, respectively) derived in $\\Lambda$CDM compared to that inferred for our sample of intermediate redshift dwarf galaxies by converting the measured $V_{2.2}$ velocities in the stellar mass Tully-Fisher relation to halo masses using the procedure discussed in \\S3. Black points represent the data, the solid grey line is the best-fit relation and the scatter is indicated with dashed grey lines. }\n\\label{fig:shm}\n\\end{figure}\n\n\n\\section{Discussion} \\label{disc}\n\n\\subsection{Is the Local Group Population Representative?}\n\nFigure \\ref{fig:smtf} shows a striking effect, namely that local galaxies appear to be drawn from a particular subset of the stellar mass TF relation defined by our larger sample. It seems unlikely that this is an evolutionary effect (relating for example to changes in the dark matter density profiles or effect of baryonic feedback) given there is no obvious trend with redshift within our sample itself. Otherwise this would result in the curious conclusion that the relation between baryons and dark matter becomes tighter over time when non-linear exchanges dominate. A more likely explanation is that we are sensitive to sources at intermediate redshift that are not typically found locally. For example, low mass galaxies generally may have a wider range of gas fractions which could explain the direction of the offset and increased scatter. Clearly we are biased in our survey towards dwarf galaxies with higher gas fractions that allow the tracing of extended emission lines for our kinematic analysis. Also the field of selection within the DEIMOS mask samples on average lower density environments compared to the group environment locally. This may proportionally affect dwarfs more than higher mass galaxies due to a satellite effect: gas is lost to interactions with other satellites and the host galaxy, whereby in the field and in voids dwarf galaxies are more likely to keep their gas.\n\nTaking this argument further, it is interesting to speculate what role gas fractions may play in reconciling the abundance matching discrepancy in Fig. \\ref{fig:shm}. The rank order for matching haloes to observed mass functions could vary widely moving into the dwarf regime with the inclusion of gas mass. Separately, could including gas and realistic feedback prescriptions in the simulations used to construct halo trees for abundance matching have an appreciable effect towards reconciling the offset? Neither of the stellar-to-halo mass relations being compared for consistency in Fig. \\ref{fig:shm} take full account of the effect of gas. Unfortunately, future progress on this front will be hindered by the difficulty in measuring gas masses in the dwarf regime as well as at higher-$z$ where they appear to be greater on average than at low redshift \\citep{taccon2010}.  \n\n\\subsection{The Role of Feedback? New Dark Matter Physics?}\n\nImplementing episodic supernova feedback in hydrodynamical simulations \\citep{govern2012} has been successful in flattening dark matter density profiles of haloes over time. This may even resolve a related, long-standing difficulty whereby simulations predicting inner density profiles that are too ``cuspy'' \\citep{pontze2012}. Although this effect can be reproduced in both adaptive mesh refinement simulations \\citep{teyssi2013}, and smoothed particle hydrodynamics, other workers have had difficulty reproducing the result in alternative prescriptions of the same feedback process. While this action may work to reconcile the offset present in Fig. \\ref{fig:shm} if the redistribution of matter was efficient out to 2.2 kpc, no amount of feedback has been effective at flattening inner density profiles below stellar masses of $10^7 $M${}_{\\odot}$ \\citep{govern2012}. Thus if the offset observed in Fig. \\ref{fig:smtf} were to continue $\\sim$1 dex lower in stellar mass, then this feedback mechanism could not alone reconcile the matched and observed relations. Additionally there are arguments that the energy of the necessary feedback for this effect exceeds what would likely occur over the typical star-formation history of most dwarfs \\citep{garris2013}. \n\nAlternatively, a combination of early feedback effects with ram pressure stripping and tidal heating by both the host halo and disk appear to extract enough energy from the gravitational potential of the host to reproduce the observations in some examples \\citep{arraki2012,zoloto2012,brooks2013}. In future work we hope to better ascertain how environmental effects may be responsible for our observed trends by determining whether targeted dwarfs are isolated or likely satellites. If most of the dwarfs are isolated, as suspected given their inferred gas-rich state, then it will be difficult to use this mechanism to explain Fig. \\ref{fig:shm}.\n\nUltimately a more fundamental adjustment to the dark matter model could be required \\citep{weinbe2013}. Alternate dark matter models that were not fully explored before the last decade of consensus around CDM include warm dark matter models \\citep[e.g.,][]{benson2012,lovell2012},  various self-interacting dark matter models \\citep{sperge2000,rocha2013,peter2013}, and flavor-mixed dark matter \\citep{medved2000,medved2012}. Any of these may potentially hold the key in resolving observed tensions in the CDM predictions for galaxy formation and evolution, however work has only just begun exploring these alternative models in detail.\n\n\n \n\\subsection{The Utility of the Stellar to Halo Mass Relation}\n\nFigure \\ref{fig:shm} is difficult to interpret without acknowledging CDM simulations generate too much substructure. When galaxies are matched to halos from the observed stellar mass functions, they are placed in halos which are too large to be consistent with the observed fiducial rotation velocities related to their stellar mass. This discrepancy increases as the (mis-)matching propagates into the dwarf galaxy regime.\n\nWe recognize that the utility of Fig. \\ref{fig:shm} in the dwarf regime relies on certain aspects of the CDM halo paradigm for us to convert observed velocities at observable radii to those of the unobservable halo. Several effects could explain the discrepancy in Fig. \\ref{fig:shm}, the most obvious of which is that the linear extrapolation of the \\citet{reyes2012} relation (connecting $V_{2.2}$ to $V_{200}$) to the dwarf regime could be inaccurate, as discussed in \\S 4. The stellar mass estimates could be incorrect, if for example the stellar population synthesis models used are inappropriate for low mass galaxies.\\footnote{Recent work has suggested that low mass galaxy stellar mass estimates are likely underestimated when using exponential star-formation histories (Dominguez et al., in prep.), exacerbating the discrepancy in Fig. \\ref{fig:shm}.}. Most importantly, the gas mass should be accounted for to create a `Baryonic' TF consistency check; this might reconcile the offset seen in Figure \\ref{fig:smtf} and affect the abundance matching method currently based only on the stellar mass distribution. Finally, only when such caveats can be laid to rest might we seriously consider powerful feedback effects which would modify dark matter density profiles out to 2.2 kpc in cosmological simulations or dark matter physics beyond the standard $\\Lambda$CDM paradigm.\n\n\\subsection{A More Direct Observational Test}\n\nNoting the considerable uncertainties in the interpretation of Figure \\ref{fig:shm}, a more direct test would be to extract quantities such as $V_{2.2}$ from halos within the cosmological simulations, in order to conduct a comparison in the observed space, i.e. Figure \\ref{fig:smtf}.  Ideally the abundance matching curve with its scatter would not be extracted in terms of the stellar mass Tully-Fisher relation, but rather the full baryonic Tully-Fisher relation with the inclusion of gas mass. Not accounting for the effects of gas in a comprehensive manner may ultimately lead to the mismatch we see between simulated CDM predictions and observations in each context:  from early versions of the ``missing satellite problem'', to the ``Too Big to Fail'' framing within the Milky Way, to the semi-empirical curves of abundance matching diverging away from the featureless TF relations. A more careful comparison, one which includes gas rather than stellar mass alone, is needed to explore these ideas further and will be addressed in future work.\n\n\\acknowledgments\n\nSHM thanks the Rhodes Trust, the British Federation of Women Graduates, the sub-department of Astrophysics and New College at the University of Oxford, the University of California, Riverside, and the California Institute of Technology for supporting her work. The spectroscopic data was secured with the W.M. Keck Observatory on Mauna Kea. We thank the observatory staff for their dedication and support. The authors recognize and acknowledge the cultural role and reverence that the summit of Mauna Kea has always had with the indigenous Hawaiian community, and we are most fortunate to have the opportunity to conduct observations from this mountain.\n\n{\\it Facilities:} \\facility{Keck II (DEIMOS)}, \\facility{HST (WFC3\/IR)}.\n\n\n\n\\bibliographystyle{apj}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nQuantum spin liquids represent nowadays one of the paradigms of strongly-correlated systems, hosting elementary excitations with fractional\nquantum numbers (e.g., $S=1\/2$ spinons), emerging gauge fields (e.g., visons or magnetic monopoles), and topological degeneracy of the\nground-state manifold (in case of a gapped spectrum)~\\cite{savary2017,zhou2017}. Their unconventional properties are ultimately triggered\nby a large entanglement between spins, which remain highly correlated down to zero temperature. In this respect, standard mean-field approaches \ncompletely fail to give a correct description of spin liquids. In order to overcome this difficulty, elegant {\\it escamotages} have been \nintroduced, based upon what is now generically known as {\\it parton construction}~\\cite{baskaran1988,affleck1988,arovas1988,read1989,wen1991}.\nHere, the spin degrees of freedom are written in terms of elementary objects (spinons), enlarging the original Hilbert space and introducing \ngauge fields (related to the redundancy of the spinon representation). Mean-field approximations of the spinon Hamiltonian can be performed, \nfixing gauge fields to a given spatial configuration; within fermionic or bosonic representations of spinons, the mean-field approximation \ngives rise to non-interacting models that can be easily handled~\\cite{baskaran1988,affleck1988,arovas1988,read1989,wen1991}. Remarkably, in \nsome very fortunate cases, the mean-field approximation gives the exact solution of the problem, most notably for the Kitaev compass model \non the honeycomb lattice~\\cite{kitaev2006}. However, for generic spin models, the accuracy of parton-based mean-field methods is questionable. \nParticularly delicate cases arise for gapless spin liquids, which are defined by mean-field theories with a gapless spinon spectrum: here, \nsmall perturbations (e.g., gauge-field fluctuations) may lead to instabilities towards some spontaneous symmetry breaking, the most celebrated \none being valence-bond order~\\cite{read1989b}. A stable spin liquid (i.e., a state that is not limited to isolated points in the phase diagram) \ncan be obtained when the gauge fields have a discrete symmetry and gapped excitations, as in the Kitaev model~\\cite{kitaev2006}. Instead, the \ncase where the gauge fields have a continuous symmetry and sustain gapless excitations is more problematic. Here, the validity of the \nmean-field approximation has been discussed only in some limiting cases~\\cite{hermele2004}, while general implications of low-energy gauge \nfluctuations on the fermionic properties are still unknown. \n\nOne technical aspect that strongly limits the use of mean-field wave functions is the fact that they are defined in the artifically enlarged\nHilbert space. For example, in the fermionic approach for $S=1\/2$ models~\\cite{wen2002}, the enlarged Hilbert space contains four states per \nsite (with zero, one, and two fermions) in contrast to the original spin Hilbert space that is limited to two states per site (up and down \nspins). Therefore, in general, these mean-field wave functions do not give reliable insights into the exact ground-state properties. In order \nto go beyond this approximation, quantum Monte Carlo methods have been very insightful. Here, starting from an auxiliary Hamiltonian for free \nfermions (not necessarily the best mean-field solution), along the Monte Carlo procedure, it is possible to restrict the calculations to the \noriginal Hilbert space of the spin model by including the Gutzwiller projector~\\cite{becca2011}. Then, genuine variational wave functions \ncan be defined, thus allowing quantitative predictions for the original spin model. \n\nThe variational approach is particularly suited within the fermionic representation, for which there are polynomial algorithms that allow to \nperform large-size calculations. This approach has been demonstrated insightful to describe $S=1\/2$ Heisenberg models on frustrated lattices \nin two spatial dimensions, such as the $J_1-J_2$ models on square~\\cite{hu2013,ferrari2020}, triangular~\\cite{iqbal2016,ferrari2019}, and \nkagome~\\cite{iqbal2013,iqbal2015} lattices. In all these cases, Gutzwiller-projected wave functions have suggested the possibility that a \n{\\it gapless} spin liquid may be stabilized in highly-frustrated regimes, the free-fermion spectrum displaying Dirac points. On the honeycomb\nlattice the gapless spin liquid does not represent the optimal state, having a slightly higher energy than a state with valence-bond \norder~\\cite{ferrari2017}. It is worth mentioning that recent density-matrix renormalization group (DMRG) calculations on the kagome~\\cite{he2017} \nand triangular~\\cite{hu2019} lattices have given further support to this conclusion. Still, obtaining a gapless spin liquid within DMRG is \nnot easy; first of all, because gapped low-entangled states are favored, thus disfavoring highly-entangled gapless phases. In addition, not \nless impactfully, DMRG simulations are usually done on cylindrical geometries, i.e., on $N$-leg ladders with a relatively large number of legs. \nWithin this choice of the clusters, the nature of the ground state can be altered with respect to the truly two-dimensional case. Indeed, only \nby using modified boundary conditions along the rungs (namely introducing a fictitious magnetic field piercing the cylinder), it was possible \nto detect gapless points in the kagome and triangular lattices~\\cite{he2017,hu2019,zhu2018}. A fictitious magnetic field has been also employed \nin the calculation of the dynamical structure factor for the Heisenberg model on the kagome lattice~\\cite{zhu2019}.\n\nGapless spin liquids represent particularly fragile states, which are expected to be stable only to a limited number of perturbations. In this \nrespect, a tiny modification to the Hamiltonian may immediately open a gap in the spectrum or lead to a spinon condensation, giving rise to \ntopological order or some sort of spontaneous symmetry breaking. For example, the gapless spin liquid of the Kitaev model, is locally stable \nfor small variations of the super-exchange couplings around the isotropic point $J_x=J_y=J_z$, but a magnetic field immediately opens a gap, \ndriving the ground state into a topological phase with anyon excitations~\\cite{kitaev2006}. In general, considering a ladder geometry, which \nexplicitly breaks point-group symmetries, represents also a relevant perturbation that may open a gap.\n\nIn this work, we study the fate of the gapless spin-liquid phase obtained on frustrated square and kagome lattice by Gutzwiller-projected \nfermionic wave functions, when cylindrical geometries are considered. We restrict to the case of {\\it even} number of legs, in order not to \nintroduce further frustration in the system. On cylindrical clusters, either gapless or gapped states can be constructed by playing with the \nboundary conditions of the fermionic operators (still keeping periodic boundary conditions on the physical spins). For the models under \ninvestigation, the best variational {\\it Ansatz} on cylindrical geometries is achieved by states with a gapped excitation spectrum in the \nthermodynamic limit, in striking constrast with the results for isotropic two-dimensional clusters, where the spin liquid states have a gapless \nspectrum and the effect of the fermionic boundary conditions is irrelevant in the thermodynamic limit (the same energy per site is achieved by \nany choice). In addition, the gapped ground state found on cylinders is unique, thus featuring neither spontaneous symmetry breaking nor \ntopological degeneracy. It is worth mentioning that this is not forbidden by the Lieb-Schultz-Mattis theorem and its generalizations to \ncylindrical geometries~\\cite{lieb1961,affleck1988b}. Our results are relevant for recent studies on both square~\\cite{mambrini2006,richter2010,jiang2012b,mezzacapo2012,hu2013,gong2014,doretto2014,morita2015,haghshenas2018,wang2018,liu2018,liao2019,choo2019,ferrari2020,nomura2020,liu2020,hasik2021}\nand kagome lattices~\\cite{yan2011,depenbrock2012,jiang2012,iqbal2013,clark2013,he2017,liao2017,hering2019}, highlighting the fact that gapless \nspin liquids, as obtained in the truly two-dimensional limit, may turn into trivial gapped phases when constrained into cylindrical geometries.\n\nThe paper is organized as follows: in section~\\ref{sec:methods}, we outline the parton construction for the Gutzwiller-projected wave function;\nin section~\\ref{sec:results}, we discuss the results for the square and kagome geometries; finally, in section~\\ref{sec:concl}, we draw our\nconclusions.\n\n\\section{Models and methods}\\label{sec:methods}\n\nIn the following, we will consider frustrated Heisenberg models\n\\begin{equation}\\label{eq:hamilt}\n{\\cal H} = \\sum_{R,R^\\prime} J_{R,R^\\prime} {\\bf S}_R \\cdot {\\bf S}_{R^\\prime},\n\\end{equation}\ndefined on either the square lattice, with primitive vectors ${\\bf a}_1=(1,0)$ and ${\\bf a}_2=(0,1)$, or the kagome lattice, with ${\\bf a}_1=(1,0)$ \nand ${\\bf a}_2=(1\/2,\\sqrt{3}\/2)$. For the square lattice, both nearest- ($J_1$) and next-nearest-neighbor ($J_2$) super-exchange couplings are \nconsidered (with $J_2\/J_1=0.5$). For the kagome lattice, only nearest-neighbor ($J$) terms are included. Clusters are identified by two vectors \n${\\bf T}_1=L_1 {\\bf a}_1$ and ${\\bf T}_2=L_2 {\\bf a}_2$, with $L_1 \\gg L_2$. The finite width $L_2$, which is taken to be even, defines the \ncylindrical geometry; periodic boundary conditions along both directions will be considered, in order not to explicitly break translational \nsymmetries.\n \nWe consider the fermionic approach, in which the $S=1\/2$ spin operators are represented by the so-called Abrikosov fermions~\\cite{wen2002}:\n\\begin{equation}\\label{eq:Sabrikosov}\n{\\bf S}_R = \\frac{1}{2} \\sum_{\\alpha,\\beta} c_{R,\\alpha}^\\dagger \n\\boldsymbol{\\sigma}_{\\alpha,\\beta} c_{R,\\beta}^{\\phantom{\\dagger}}.\n\\end{equation}\nHere $c_{R,\\alpha}^{\\phantom{\\dagger}}$ ($c_{R,\\alpha}^\\dagger$) destroys (creates) a fermion with spin $\\alpha=\\uparrow,\\downarrow$ on site $R$, and the \nvector $\\boldsymbol{\\sigma}=(\\sigma_x,\\sigma_y,\\sigma_z)$ is the set of Pauli matrices. The fermionic representation enlarges the local Hilbert\nspace, including unphysical configurations with zero and two fermions per site. \n\nThe construction of spin-liquid wave functions goes as follows. First of all, an auxiliary (fermionic) Hamiltonian is defined:\n\\begin{equation}\\label{eq:generic_mf}\n{\\cal H}_{0} = \\sum_{R,R^\\prime} \\sum_{\\alpha} t_{R,R^\\prime} c_{R,\\alpha}^\\dagger c_{R^\\prime,\\alpha}^{\\phantom{\\dagger}} +\n\\sum_{R,R^\\prime}  \\Delta_{R,R^\\prime} c_{R,\\downarrow}^{\\phantom{\\dagger}} c_{R,\\uparrow}^{\\phantom{\\dagger}} + h.c.,\n\\end{equation}\nwhere $t_{R,R^\\prime}^{\\phantom{\\dagger}}=t_{R^\\prime,R}^*$ and $\\Delta_{R,R^\\prime}=\\Delta_{R^\\prime,R}$ are singlet hopping and pairing terms. Then, the \nunprojected state is obtained by the ground state $|\\Phi_0\\rangle$ of ${\\cal H}_0$. Finally, the variational {\\it Ansatz} for the spin model \nis obtained by enforcing the constraint of one fermion per site, thus going back to the original Hilbert space of $S=1\/2$ spins. \nThis can be achieved by applying the Gutzwiller projector ${\\cal P}_G = \\prod_R (n_{R,\\uparrow} - n_{R,\\downarrow})^2$ to $|\\Phi_0\\rangle$ \n($n_{R,\\alpha}=c_{R,\\alpha}^\\dagger c_{R,\\alpha}^{\\phantom{\\dagger}}$ being the fermion density on the site $i$):\n\\begin{equation}\\label{eq:finalwf}\n|\\Psi_0\\rangle = {\\cal P}_G |\\Phi_0\\rangle.\n\\end{equation}\nThe physical properties of the projected wave function can be assessed by using standard quantum Monte Carlo techniques~\\cite{becca2017}.\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig1a.pdf}\n\\includegraphics[width=0.83\\columnwidth]{fig1b.pdf}\n\\caption{\\label{fig:square} \nTop panel: schematic illustration of the square lattice cylindrical geometry with $L_2=4$ legs. The site numbering indicates how periodic \nboundary conditions are chosen along the ${\\bf T}_2$ direction, namely how the cylinder is wrapped up. Black (red) lines represent positive \nand negative hoppings in the fermionic Hamiltonian~(\\ref{eq:generic_mf}) for the $\\pi$-flux state~\\cite{affleck1988c}. Bottom panels: cuts \nof momenta of the fermionic partons which are allowed by the cylindrical geometry in the top panel. Depending on the choice of the fermionic \nboundary conditions along ${\\bf T}_2$, the cuts can hit or avoid the Dirac points in the spinon spectra [${\\bf k}=(\\pm \\pi\/2, \\pm \\pi\/2)$, \nindicated by the crosses]. The black square represents the original Brillouin zone, while the dashed lines delimit the reduced Brillouin \nzone of the $\\pi$-flux state.}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig2a.pdf}\n\\includegraphics[width=0.85\\columnwidth]{fig2b.pdf}\n\\caption{\\label{fig:kagome}\nThe same as in Fig.~\\ref{fig:square} but for the kagome lattice and the corresponding $\\pi$-flux state~\\cite{ran2007}.}\n\\end{figure}\n\nIn the following, we will focus on specific spin-liquid {\\it Ans\\\"atze} defined by auxiliary fermionic Hamiltonians which possess a gapless\nspectrum with Dirac points in the two-dimensional limit. In particular, we will consider either simple cases with only nearest-neighbor hopping \nparameters, defining non-trivial magnetic fluxes, or cases with both hopping and pairing terms. In the former case, no variational parameters \nare present in the wave function (the hopping parameter defines the energy scale of the auxiliary Hamiltonian); in the latter case, a few \nvariational parameters are present and optimized by using standard Monte Carlo schemes~\\cite{becca2017}.\n\n\\begin{figure*}\n\\includegraphics[width=\\columnwidth]{fig3a.pdf}\n\\includegraphics[width=\\columnwidth]{fig3b.pdf}\n\\includegraphics[width=\\columnwidth]{fig3c.pdf}\n\\includegraphics[width=\\columnwidth]{fig3d.pdf}\n\\caption{\\label{fig:square_sqw} \nDynamical structure factor for the $J_1-J_2$ Heisenberg model on the square lattice with $J_2\/J_1=0.5$. The cylindrical geometry with $L_1=84$ \nand $L_2=6$ is taken. The results for the gapped state with PBC-PBC (upper panels) and the gapless one with ABC-ABC (lower panels) are reported for \ntwo values of the transverse momentum, $q_y=0$ and $q_y=\\pi$. A Gaussian smearing of the delta-functions of Eq.~(\\ref{eq:dsf}) is applied, with \n$\\sigma=0.03J_1$. In the lower-right panel, a negative excitation is observed at ${\\bf q}=(\\pi,\\pi)$. This is an artifact of the numerical method \ncaused by the fact that the gapless state is not the optimal variational {\\it Ansatz} for the $J_1-J_2$ model on the cylinder (see also the \ndiscussion in the main text).}\n\\end{figure*}\n\nDifferent choices for the boundary conditions on fermionic operators are possible, affecting the free-fermion Hamiltonian~(\\ref{eq:generic_mf})\nand, consequently, the variational wave function, but not the spin operators~(\\ref{eq:Sabrikosov}). In the following, we will restrict ourselves \nto real wave functions, which implies that either periodic boundary conditions (PBC) or anti-periodic boundary conditions (ABC) are possible:\n\\begin{eqnarray}\nc_{R+T_1,\\alpha}^{\\phantom{\\dagger}} &=& e^{i \\theta_1} c_{R,\\alpha}^{\\phantom{\\dagger}}, \\\\\nc_{R+T_2,\\alpha}^{\\phantom{\\dagger}} &=& e^{i \\theta_2} c_{R,\\alpha}^{\\phantom{\\dagger}}, \n\\end{eqnarray}\nwhere $\\theta_1, \\theta_2=0(\\pi)$ for PBC (ABC).\n\nThe important point is that the finiteness of the cylinder width $L_2$ implies a coarse discretization of the momenta along the direction of \nthe reciprocal lattice vector ${\\bf b}_2$. The discretization is present also in the limit of a infinitely long cylinder, $L_1 \\to \\infty$, \nwhere the set of allowed momenta form parallel ``cuts'' running along the ${\\bf b}_1$ direction. As a consequence, gapless points in the \nspinon spectrum of the auxiliary Hamiltonian~(\\ref{eq:generic_mf}) may be absent on cylinders, if the cuts of momenta do not run through them. \nTherefore, the unprojected spectrum will be gapped also for $L_1 \\to \\infty$, only because of the finite number of legs $L_2$. As an example, \nlet us consider the $\\pi$-flux phase on the square lattice, which has Dirac points at ${\\bf k}=(\\pm \\pi\/2,\\pm \\pi\/2)$~\\cite{affleck1988c}; \nfor $L_2=4n$ ($L_2=4n+2$) and ABC (PBC) along ${\\bf T}_2$ these k-points are not allowed and the spinon spectrum is gapped, for any value of the \ncylinder length $L_1$. By contrast, for $L_2=4n$ ($L_2=4n+2$) and PBC (ABC) along ${\\bf T}_2$, zero-energy modes are present or absent depending \non the boundary conditions along ${\\bf T}_1$: with one choice the gapless points are present for any value of $L_1$, while for the opposite \nchoice the spectrum has a finite-size gap, which vanishes in the limit of an infinitely long cylinder $L_1\\to \\infty$. The case for $L_2=4$ is \nshown in Fig.~\\ref{fig:square}. In order to have a unique and well defined variational state, zero-energy modes in Eq.~(\\ref{eq:generic_mf}) \nmust be avoided on any finite size, thus leaving to only three possibilities of boundary conditions for each choice of $L_1$ and $L_2$: two of \nthem will correspond to fully-gapped states, while the third one will correspond to a state that will become gapless for $L_1 \\to \\infty$ and \nwill be dubbed ``gapless'' for simplicity. A similar situation happens in the kagome lattice, for the $\\pi$-flux phase defined in \nRef.~\\cite{ran2007}. The case with $L_2=4$ is shown in Fig.~\\ref{fig:kagome}. Notice that the unprojected (spinon) spectrum is gauge invariant, \neven though the position of the Dirac points in $k$-space is gauge dependent. Therefore, a given choice of the boundary conditions will give \nrise to a gapped or gapless spectrum, independently on the gauge choice. In both $\\pi$-flux phases on square and kagome lattices, the fermionic \nHamiltonian~(\\ref{eq:generic_mf}) can accomodate the magnetic fluxes by breaking the translational symmetry (e.g., the unit cell is doubled \nalong ${\\bf a}_2$). Nevertheless, after Gutzwiller projection, the wave function~(\\ref{eq:finalwf}) is fully translationally \ninvariant~\\cite{wen2002}.\n\nThe properties of these variational states can be assessed by computing equal-time spin-spin and dimer-dimer correlations as a function of \ndistance along the long side of the cylinder. The former ones are defined as:\n\\begin{equation}\n{\\cal S}({\\bf R}) = \\langle \\mathbf{S}_0 \\cdot \\mathbf{S}_R \\rangle,\n\\end{equation}\nwhere $\\langle \\cdots \\rangle$ denotes the expectation value over the variational state. Instead, dimer-dimer correlations are given by:\n\\begin{equation}\n{\\cal D}({\\bf R}) = \\langle D_0 D_R \\rangle - \\langle D_0 \\rangle \\langle D_R \\rangle,\n\\end{equation}\nwhere $D_R=\\mathbf{S}_R \\cdot \\mathbf{S}_{R+a_1}$. \n\nUp to now, in the definition of gapped and gapless states, we have made reference to the unprojected spectrum of the fermionic Hamiltonian.\nThe physical excitations of the spin model may be assessed within the variational method by constructing a set of Gutzwiller-projected states \nfor each momentum ${\\bf q}$~\\cite{li2010,ferrari2018}. For the case of a Bravais lattice (e.g., the square lattice), we define a set of triplet \nstates:\n\\begin{equation}\\label{eq:qRstate}\n|q,R\\rangle =  \\mathcal{P}_G \n\\frac{1}{\\sqrt{N}} \\sum_{R^\\prime}\\sum_{\\alpha} e^{i {\\bf q} \\cdot {\\bf R^\\prime}} s_{\\alpha} \nc^\\dagger_{R^\\prime+R,\\alpha}c^{\\phantom{\\dagger}}_{R^\\prime,\\alpha} |\\Phi_0\\rangle.\n\\end{equation}\nwhere $N=L_1L_2$ and $s_{\\alpha}=1$ ($s_{\\alpha}=-1$) for $\\alpha=\\uparrow$ ($\\alpha=\\downarrow$); these states are labelled by $R$, \nwhich runs over all lattice vectors. Then, the adopted PBC or ABC are imposed to $c^\\dagger_{R^\\prime+R,\\alpha}$, whenever $R^\\prime+R$ falls \noutside of the original cluster. Low-energy excited states are given by suitable linear combinations:\n\\begin{equation}\\label{eq:psinq}\n|\\Psi_n^q\\rangle=\\sum_R A^{n,q}_R |q,R\\rangle.\n\\end{equation}\nThe parameters $\\{ A^{n,q}_R \\}$ are obtained once the spin Hamiltonian of Eq.~(\\ref{eq:hamilt}) is fully specified. In fact, for any given \nmomentum ${\\bf q}$, we can consider the Schr{\\\"o}dinger equation, restricting the form of its eigenvectors to the one of Eq.~(\\ref{eq:psinq}),\ni.e., ${ {\\cal H}|\\Psi_n^q\\rangle = E_n^q |\\Psi_n^q\\rangle }$. Then, expanding everything in terms of $\\{|q,R\\rangle\\}_R$, we get to the \nfollowing generalized eigenvalue problem\n\\begin{equation}\\label{eq:general_eig_prob}\n\\sum_{R^\\prime} \\langle q,R|{\\cal H}|q,R^\\prime \\rangle  A^{n,q}_{R^\\prime} = E_n^q \\sum_{R^\\prime} \\langle q,R|q,R^\\prime \\rangle \nA^{n,q}_{R^\\prime},\n\\end{equation}\nwhich is solved to find the expansion coefficients $A^{n,q}_R$ and the energies $E_n^q$ of the excitations. All the matrix elements,\n$\\langle q,R|{\\cal H}|q,R^\\prime \\rangle$ and $\\langle q,R|q,R^\\prime \\rangle$, are evaluated within the Monte Carlo procedure, by \nsampling according to the variational ground-state wave function~\\cite{ferrari2018}. Finally the dynamical structure factor is computed by:\n\\begin{equation}\\label{eq:dsf}\nS^{z}({\\bf q},\\omega) = \\sum_n |\\langle \\Psi_{n}^q | S^{z}_q | \\Psi_0 \\rangle|^2 \\delta(\\omega-E_{n}^q+E_0^{\\rm var}),\n\\end{equation}\nwhere $E_0^{\\rm var}$ is the variational energy of $|\\Psi_0 \\rangle$ and \n\\begin{equation}\nS^{z}_q=\\frac{1}{\\sqrt{N}} \\sum_R e^{i{\\bf q} {\\bf R}} S^{z}_R.\n\\end{equation}\nIn the case of a lattice with a basis of $n_b$ sites (e.g., the kagome lattice with $n_b=3$), the generalization is straighforward, \nwith the only modification that particle-hole excitations~(\\ref{eq:qRstate}) acquire basis indices, attached to creation and annihilation \noperators. The dynamical structure factor~(\\ref{eq:dsf}) becomes a $n_b \\times n_b$ matrix, constructed from spin operators on the elements\nof the basis (with a phase factor that depends upon the actual spatial position of each sites). Then, the physical (experimentally measurable) \nquantity is the sum of all its elements.\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig4a.pdf}\n\\includegraphics[width=\\columnwidth]{fig4b.pdf}\n\\caption{\\label{fig:piflux_spinspin} \nComparison of the absolute value of the spin-spin ($|{\\cal S}({\\bf R})|$, upper panel) and dimer-dimer ($|{\\cal D}({\\bf R})|$, lower panel) \ncorrelations for the $\\pi$-flux and the optimal spin-liquid wave functions on the square lattice for $J_2\/J_1=0.5$. Calculations are done \non the cylindrical cluster with $84 \\times 6$ sites. Empty symbols refer to the gapped state with PBC-PBC, while full symbols refer to the \ngapless state with ABC-ABC.}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig5.pdf}\n\\caption{\\label{fig:piflux_4xL} \nAbsolute value of the spin-spin correlations ($|{\\cal S}({\\bf R})|$) of the gapless $\\pi$-flux state (with ABC-PBC) on $L_1 \\times 4$ cylinders \n(square lattice). Inset: size scaling of the spin-spin correlations at the maximum distance along ${\\bf a}_1$.}\n\\end{figure}\n\n\\section{Results}\\label{sec:results}\n\n\\subsection{Preliminary considerations on the Kitaev model}\nA useful insight on how a gapless spin liquid reveals itself in a finite-size cluster comes from the investigation of the Kitaev compass \nmodel on the honeycomb lattice~\\cite{kitaev2006}. The model can be mapped to free Majorana fermions in a static magnetic field, where fluxes \npiercing the hexagons are either $0$ or $\\pi$. This auxiliary model is defined in an extended Hilbert space and the physical states are \nobtained by a projection. Most importantly, all the physical energies of the spin model also belong to the spectrum of the auxiliary Hamiltonian. \nThen, the study of the flux states of free Majorana fermions on the honeycomb lattice provides the ground state energy and the excitation \nspectrum of the Kitaev model. In the thermodynamic limit, the isotropic model ($J_x=J_y=J_z$) displays a gapless spinon branch, due to the \nexcitations of Majorana fermions, and a gapped vison spectrum, corresponding to changes in the flux pattern. On the cylindrical clusters \ndefined by ${\\bf T}_1$ and ${\\bf T}_2$ (with $L_1=4n$ and $L_2=2m$), the Lieb theorem~\\cite{lieb1994} holds, implying zero flux through all \nhexagonal plaquettes. This choice of the flux pattern leaves four possible orthogonal candidates for the ground state, corresponding to the \nfour possible boundary conditions. Zero-energy modes are present only when both $L_1$ and $L_2$ are multiple of $3$. In this case, there are \ntwo gapped states and one gapless state (for $L_1 \\to \\infty$), while the fourth one has zero-energy modes. Remarkably, the lowest energy is \nnever obtained by the gapless wave function and, therefore, on such cylindrical geometries, the Kitaev model with $J_x=J_y=J_z$ is fully gapped.\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig6a.pdf}\n\\includegraphics[width=\\columnwidth]{fig6b.pdf}\n\\caption{\\label{fig:largeL_spinspin} \nAbsolute value of the spin-spin ($|{\\cal S}({\\bf R})|$, upper panel) and dimer-dimer ($|{\\cal D}({\\bf R})|$, lower panel) correlations on \n$20L \\times L$ cylinders for the $\\pi$-flux state on the square lattice. Empty symbols refer to gapped states (with ABC-ABC for $L=4n$ and \nABC-PBC for $L=4n+2$), while full symbols refer to gapless states (with ABC-PBC for $L=4n$ and ABC-ABC for $L=4n+2$).}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig7.pdf}\n\\caption{\\label{fig:squakag}\nAbsolute value of the spin-spin correlations ($|{\\cal S}({\\bf R})|$) on the isotropic clusters for the square ($48 \\times 48$) and kagome \n($28 \\times 28$) lattices. In both cases the simple $\\pi$-flux states are considered, with ABC-ABC. Dashed lines are linear fits, compatible \nwith a power-law behavior with an exponent $\\beta \\approx 2$.}\n\\end{figure}\n\n\\subsection{Square-lattice geometry}\nHere, we present the results for the square-lattice geometry, see Fig.~\\ref{fig:square}, and the spin Hamiltonian~\\eqref{eq:hamilt} with \n${J_2\/J_1=0.5}$. In the two-dimensional limit, the Gutzwiller-projected wave function suggested the existence of a gapless spin-liquid \nphase~\\cite{hu2013,ferrari2020}. The best variational {\\it Ansatz} within this class of states has, in addition to uniform nearest-neighbor \nhopping, pairing terms with $d_{x^2-y^2}$ symmetry (at first neighbors, connected by ${\\bf a}_1$ and ${\\bf a}_2$) and $d_{xy}$ symmetry (at \nfifth neighbors, connected by $2{\\bf a}_1 \\pm 2{\\bf a}_2$). While the point-group symmetry of the hopping and pairing couplings is determined \nby the projective symmetry group of the spin liquid {\\it Ansatz}~\\cite{wen2002}, the actual values of the parameters are fully optimized by \nthe numerical minimization of the variational energy. We note that the simpler case with only hopping and nearest-neighbor pairing corresponds \nto a $U(1)$ spin liquid (gauge equivalent to the the staggered flux state~\\cite{affleck1988c}), while the addition of $d_{xy}$ pairing terms \nleads to a $Z_2$ spin liquid~\\cite{wen2002}. In both cases, the Hamiltonian~(\\ref{eq:generic_mf}) possesses Dirac points at \n${\\bf k}=(\\pm \\pi\/2, \\pm \\pi\/2)$. Variational Monte Carlo calculations of the dynamical structure factors suggested that Dirac points survive \nthe Gutzwiller projection and are a genuine feature of the low-energy spectrum~\\cite{ferrari2018b}. It is worth noting that the simple $\\pi$-flux \nstate discussed previously, despite having a slightly higher energy than the $Z_2$ spin liquid, gives a quite accurate approximation of the best \nvariational {\\it Ansatz}.\n\nLet us start by considering a cylinder with $L_1=84$ and $L_2=6$; the large aspect ratio ensures that results are indistinguishable from the \nlimit $L_1 \\to \\infty$. For this cluster, the fermionic Hamiltonian~(\\ref{eq:generic_mf}) is gapped by taking PBC along ${\\bf T}_2$ and either \nPBC or ABC along ${\\bf T}_1$. Instead, the gapless wave function corresponds to ABC along ${\\bf T}_1$ and ${\\bf T}_2$. The remaining option,\nwith ABC along ${\\bf T}_2$ and PBC along ${\\bf T}_1$, is not considered because it implies zero-energy modes in the unprojected spectrum. \n\n\\begin{figure*}\n\\includegraphics[width=\\columnwidth]{fig8a.pdf}\n\\includegraphics[width=\\columnwidth]{fig8b.pdf}\n\\caption{\\label{fig:kagome_sqw} \nDynamical structure factor for the Heisenberg model on the kagome lattice. The cylindrical geometry with $L_1=24$ and $L_2=4$ is taken. \nThe results for the gapped state with ABC-ABC (left panel) and the gapless one with ABC-PBC (right panel) are reported along the path in \nthe extended Brillouin zone indicated in the inset. A Gaussian smearing of the delta-functions of Eq.~(\\ref{eq:dsf}) is applied, with \n$\\sigma=0.03J$.}\n\\end{figure*}\n\nThe best variational energy, ${E\/J_1=-0.49837(1)}$, is obtained by the two gapped states (with PBC-PBC and ABC-PBC), which yield equivalent \nresults within the statistical errors. Indeed, these two gapped {\\it Ans\\\"atze} correspond to the same physical state, as demonstrated by the \nfact that the overlap between them (directly computed within Monte Carlo sampling) gives $1$ for the $84 \\times 6$ cylinder. On the other hand, \nthe gapless {\\it Ansatz} has a considerably higher energy, i.e., ${E\/J_1=-0.48881(2)}$. Size effects, due to the finiteness of $L_1$, are \nnegligible. This represents the first important result of this work, showing that a gapless two-dimensional spin liquid may turn into gapped \nwhen confined in a cylindrical geometry. The fact that the best variational state actually describes a gapped phase is proven by the dynamical \nstructure factor of Eq.~(\\ref{eq:dsf}), see Fig.~\\ref{fig:square_sqw}. The gapped state shows an intense and dispersive branch, which has a \nminimum at ${\\bf q}=(\\pi,\\pi)$. A weak continuum is also visible at higher energies. By contrast, the gapless state sustains low-energy \nexcitations and even a few states with negative energies are present close to ${\\bf q}=(\\pi,\\pi)$. Within this approach, negative excitation \nenergies are possible when the reference state $|\\Phi_0\\rangle$ is not the optimal variational minimum.\nThen, it may happen that some of the states defined in Eq.~(\\ref{eq:psinq}) end up having a lower energy than the one of $|\\Psi_0\\rangle$. \nA similar situation was detected in the unfrustrated Heisenberg model by using a spin-liquid state as variational \n{\\it Ansatz}~\\cite{dallapiazza2015}. However, it is worth stressing that all the wave functions~(\\ref{eq:psinq}) constructed from the gapless \n{\\it Ansatz} have much higher energies than the one of the gapped state, which remains the best variational approximation for the ground state.\n\nGapped and gapless states show also remarkably different spin-spin and dimer-dimer correlation functions, see Fig.~\\ref{fig:piflux_spinspin}. \nThe gapped state possesses rapidly decaying spin-spin and dimer-dimer correlations. The vanishingly small values of the long-range dimer-dimer \ncorrelations suggests that no valence-bond order is present. Combining this observation with the fact that the two gapped {\\it Ans\\\"atze} \n(with PBC-PBC and ABC-PBC) correspond to the same physical state, we conclude that the ground state on cylindrical geometries is consistent \nwith a trivial quantum paramagnet, with neither valence-bond nor topological order.\n\nIn order to perform a more detailed analysis of the size-dependence of the correlation functions, we adopt a simpler variational {\\it Ansatz}, \nnamely the $\\pi$-flux state of Fig.~\\ref{fig:square}, which contains only hopping terms. The results obtained by the $\\pi$-flux state are very \nsimilar to the ones given by the optimal variational wave function. For example, the energies of gapped and gapless states on the $84 \\times 6$ \ncylinder (with $J_2\/J_1=0.5$) are $E\/J_1=-0.49661(1)$ and $E\/J_1=-0.48769(1)$, respectively. Moreover, also static correlation fluctions are \nvery similar to those of the optimized $Z_2$ spin liquid, see Fig.~\\ref{fig:piflux_spinspin}. Indeed, the overlap between the optimal wave \nfunction and the simple $\\pi$-flux state is approximately $0.8$ ($0.9$) for gapped (gapless) cases. First of all, we prove that the gapless \n{\\it Ansatz} has power-law spin-spin correlations, which might be difficult to detect from the results shown in Fig.~\\ref{fig:piflux_spinspin}. \nIn order to perform a convincing size scaling, we take $L_2=4$ and perform calculations for different values of $L_1$, see Fig.~\\ref{fig:piflux_4xL}. \nFrom these results, we obtain that the spin-spin correlations decay to zero as $1\/L_1$ that is fully compatible with a power-law behavior \n$|{\\cal S}({\\bf R})| \\approx 1\/|\\mathbf{R}|^\\beta$ with an exponent $\\beta \\approx 1$.\n\nFinally, we compute the spin-spin and dimer-dimer correlations of the $\\pi$-flux state for different cylindrical geometries with $20L \\times L$ \nsites, as shown in Fig.~\\ref{fig:largeL_spinspin}. These calculations are easily affordable because the wave function, which does not correspond \nto the optimal state, does not contain any free variational parameter. Thus, no numerical optimization are required. We observe that the \nshort-distance correlations of the gapped state strengthen when increasing $L$, rapidly approaching the ones of the gapless state. Still, even \nfor cylinders with a relative large width (i.e., $L=10$), the results obtained from gapless and gapped states are quite different from each other, \nindicating a particularly slow convergenece to the truly two-dimensional limit. Interestingly, the spin-spin correlations of the gapless wave \nfunction show a crossover between a power-law behavior with $\\beta \\approx 2$ for $|\\mathbf{R}| \\lesssim L$ and $\\beta \\approx 1$ for \n$|\\mathbf{R}| \\gg L$. The former one is typical of the two-dimensional limit, see Fig.~\\ref{fig:squakag}, while the latter one is characteristic \nof quasi-one-dimensional systems, as also obtained in Fig.~\\ref{fig:piflux_4xL}.\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig9.pdf}\n\\caption{\\label{fig:kagome_4xL}\nAbsolute value of the spin-spin correlations ($|{\\cal S}({\\bf R})|$) on $L_1 \\times 4$ cylinders for the $\\pi$-flux spin-liquid on the kagome lattice. \nEmpty symbols refer to the gapped state with ABC-ABC, while full symbols refer to the gapless state with ABC-PBC.}\n\\end{figure}\n\n\\subsection{Kagome-lattice geometry}\nLet us now turn to discuss the results for the kagome-lattice geometry, see Fig.~\\ref{fig:kagome}, with nearest-neighbor super-exchange $J$.\nAlso for this highly-frustrated problem, in the two-dimensional limit, the Gutzwiller-projected fermionic state suggested that the ground\nstate corresponds to a gapless spin liquid~\\cite{ran2007,iqbal2013}. In this case, a very accurate {\\it Ansatz} has only nearest-neighbor\nhopping, such that fermions experience a $\\pi$-flux piercing the elementary unit cell (i.e., $\\pi$-flux through hexagons and $0$-flux \nthough triangles). As for the $\\pi$-flux on the square lattice, the unprojected fermionic spectrum possesses Dirac points~\\cite{ran2007}.\nThis {\\it Ansatz} corresponds to a $U(1)$ spin liquid. \n\nWe start by taking a cylinder with $L_1=24$ and $L_2=4$. Here, the fermionic Hamiltonian~(\\ref{eq:generic_mf}) is gapped by taking ABC along \n${\\bf T}_2$ and either PBC or ABC along ${\\bf T}_1$; the gapless wave function has ABC along ${\\bf T}_1$ and PBC along ${\\bf T}_2$. The fourth\npossibility with PBC along both ${\\bf T}_1$ and ${\\bf T}_2$ leads to zero-energy modes in the spectrum. As for the square lattice, the two\noptions for the gapped states correspond to the same physical wave function, after Gutzwiller projection (as verified by computing the overlap\nbetween them). \n\nThe gapped state has the best variational energy per site $E\/J=-0.43021(3)$, while the gapless wave function has a much higher energy, i.e., \n$E\/J=-0.42672(3)$, confirming that also in this case the gapped {\\it Ansatz} is favored on the cylindrical geometry, even though the \ntwo-dimensional case is gapless. In Fig.~\\ref{fig:kagome_sqw}, we show the dynamical structure factor for both gapped and gapless states. \nNotice that, in contrast to the square-lattice geometry, here no negative-energy states are obtained for the gapless {\\it Ansatz}. In addition, \nthe results for this latter case is qualitatively similar to the spectrum recently obtained in the truly two-dimensional case~\\cite{zhang2020}. \nA broad continuum of excitations is observed for both the gapped and the gapless state. Notably, in both cases, the maximum of the spectral \nintensity is concentrated around the $\\Gamma^\\prime$ point (as observed also in Ref.~\\cite{zhu2019}), within a weakly dispersive branch of \nexcitations which is not separated from the continuum at higher energy.\n\nAnother remarkable feature of this {\\it Ansatz} on the kagome lattice is the rapid decay of the spin-spin correlation functions, even when \ncomputed for the gapless wave function. In Fig.~\\ref{fig:kagome_4xL}, we report the correlations between spins on the same sublattice (a small \ndifference for the three possible sublattices is detected, not changing the qualitative behavior). Here, gapped and gapless {\\it Ans\\\"atze} \ngive almost indistinguishable correlation functions for distances larger than $4$ lattice spacings. This aspect is very different from what\nwe obtained in the square lattice, where gapped and gapless states show rather distinct correlation functions, see Fig~\\ref{fig:piflux_spinspin}.\nThe behavior is not much modified by increasing the number of legs, as reported for $10L \\times L$ cylinders in Fig.~\\ref{fig:kagome_10LxL}.\nAlso in this case, there is no apreciable difference between gapped and gapless states, at variance with the square lattice case, where the\ngapless state displays sizable correlations, substantially different from the ones of the gapped case. Quite surprisingly, also in the truly \ntwo-dimensional clusters, spin-spin correlations decay very rapidly with the distance, still being compatible with a power-law decay with an \nexponent $\\beta \\approx 2$ (similarly to the one obtained in the square lattice), see Fig.~\\ref{fig:squakag}. However, in the kagome lattice, \na small prefactor strongly renormalizes the whole behavior of correlations.\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig10.pdf}\n\\caption{\\label{fig:kagome_10LxL}\nAbsolute value of the spin-spin correlations ($|{\\cal S}({\\bf R})|$) on $10L \\times L$ cylinders for the $\\pi$-flux spin-liquid on the kagome \nlattice. Empty symbols refer to the gapped state (with ABC-ABC for $L=4n$ and ABC-PBC for $L=4n+2$), while full symbols refer to the gapless \nstate (with ABC-PBC for $L=4n$ and ABC-ABC for $L=4n+2$).}\n\\end{figure}\n\n\\section{Conclusions}\\label{sec:concl}\n\nIn conclusions, we performed a systematic study of gapless spin liquids, defined within a fermionic parton construction, on cylindrical geometries. \nWe focused on cases that have been widely investigated in the past and have been demonstrated to accurately represent the exact ground-state\nproperties of two-dimensional square and kagome lattice models~\\cite{hu2013,iqbal2013}. In both cases, Dirac points are present in the unprojected \nspectrum of the fermionic spinons. In this work, we considered $L_1 \\times L_2$ cylindrical clusters with $L_1 \\gg L_2$, in order to include a \nfinite-size gap due to the finiteness of the cylinder width $L_2$. By playing with the boundary conditions of fermionic operators, both gapless \nand gapped wave functions can be constructed. Strikingly, the gapped states have a lower variational energy, better approximating the true ground \nstate on cylinders. In addition, the gapped states are trivial, i.e., they do not possess local or non-local (topological) order, which is not in \ncontrast to the Lieb-Schultz-Mattis theorem ($L_2$ is taken to be even).\n\nOur results are important for two reasons. First of all, they show that two-dimensional gapless spin liquids are quite delicate states of matter\nand can be destabilized on constrained geometries, i.e., on cylinders. Here, the same construction that has been used in two dimensions leads\nto both gapless and gapped states, just by changing boundary conditions in the underlying parton picture. Gapped states give rise to much lower\nvariational energies, implying their stabilization on cylinders with an even number of legs. Second, they give a transparent argument to explain \nwhy previous DMRG calculations~\\cite{yan2011,depenbrock2012,jiang2012,jiang2012b,gong2014} failed to detect gapless phases in frustrated models. \nEven though our approach does not give an exact solution of the problem, it strongly suggests that the actual ground state on cylinders is gapped,\nthus giving a neat interpretation of DMRG results for the models considered here. In this regard, an important step forward has been the inclusion \nof fictitious magnetic fluxes (in the original spin model), which allowed DMRG to detect the existence of Dirac points in kagome and triangular \nlattices~\\cite{he2017,hu2019}.\n\n\\section*{Acknowledgements}\nWe thank D. Poilblanc for discussions in the early stage of the project. F.F. acknowledges support from the Alexander von Humboldt Foundation \nthrough a postdoctoral Humboldt fellowship.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe statistical features of coupled systems have attracted considerable\ninterest in many branches of physics. Random matrix theory (RMT) has\nbeen successfully used in many of those investigations. RMT was\nfounded by Wigner~\\cite{Wigner}. It is a schematic model~\\cite{Mehta}\nin which the Hamiltonian, or, more generally, the wave operator of the\nsystem, is replaced by a random matrix. The necessary prerequisite is\nthat the system be sufficiently ``complex,'' implying that the matrix\nelements of the Hamiltonian, or wave operator, calculated in an\narbitrary basis, behave like random numbers. It has been shown that\nthe spectral fluctuations in numerous different systems, if measured\non the scale of the local mean-level spacing, are very well modeled by\nRMT; see the reviews in Refs.~\\cite{GMGW,Stockmann,Haake}. Due to the\nconnection with chaos, one frequently refers to those systems as\nquantum chaotic which show correlations of RMT type.  Similarly,\nsystems are often referred to as regular if they lack spectral\ncorrelations.\n\nWe consider two coupled systems. We assume that either the two systems\nare chaotic before they are coupled or that the coupling itself\nintroduces chaoticity if the separate systems are regular. This\nscenario is equivalent to the breaking of symmetries, if only two\nvalues of the quantum number belonging to that symmetry are taken into\naccount.  The statistical features crucially depend on the strength of\nthe coupling measured on the scale of the local mean-level spacing.\nMany studies have been devoted to this issue of chaotically coupled\nsystems or, equivalently, to symmetry breaking. We mention isospin\nbreaking in nuclear physics~\\cite{values,GW}, symmetry breaking in\nmolecular physics~\\cite{Leitner}, symmetry breaking in resonating\nquartz crystals~\\cite{cristais2} and coupled microwave\nbilliards~\\cite{nos}. While these studies addressed the spectral\ncorrelation, several investigations in nuclear\nphysics~\\cite{26Al,22Na,BGH,HNSA} focused on the statistics of the\nwave functions and related observables in the presence of symmetry\nbreaking or similar effects.  In all these cases, RMT approaches in\nthe spirit of the Rosenzweig-Porter model~\\cite{Porter} were\nsuccessful.\n\nSometimes observables in the time domain such as spectral form factors\nare more appropriate than the eigenenergy correlations\nfunctions~\\cite{Mehta,GMGW,Stockmann,Haake}. This is so, for example,\nin the case of the presently much discussed fidelity; see\nRefs.~\\cite{CT,BCV,PSZ,VH} and references therein. Another example is\nthe study of the energy spread in chaotic\nsystems~\\cite{cohen00,kottos01}.  In the context of coupled systems,\nthe time evolution of wave packets was investigated in\nRef.~\\cite{BTU}.\n\nIn the present contribution, we study energy localization in two\ncoupled systems in the time domain. This problem was addressed in a\nrecent work by Weaver and Lobkis~\\cite{WL} who measured the time\ndependence of the wave intensity distribution in two coupled\nreverberation rooms. To this end, these authors recorded the time\nresponse to an elastic excitation of two coupled aluminum cubes.\nMoreover, they investigated the same problem theoretically and they\nnumerically calculated the response in coupled two-dimensional\nmembranes. In our study, we se tup and analyze an RMT model, based on\nthe approaches in Refs.~\\cite{Porter,GW,BGH}. Its general character\nmakes our model useful for similar problems in different physics\ncontexts. In particular, we expect that our RMT approach also applies\nto coupled quantum dots.\n\nThe article is organized as follows. In Sec.~\\ref{sec:exp_and_model}\nwe sketch the work of Weaver and Lobkis~\\cite{WL}. In\nSec.~\\ref{sec:RMT} we set up the RMT model and analyze it analytically\nand numerically.  We compare our results to those of Weaver and Lobkis\nin Sec.~\\ref{sec:RMT_result}. Discussion and conclusions are given in\nSec.~\\ref{sec:dic_concl}.\n\n\n\\section{Experiment and numerical calculations}\n\\label{sec:exp_and_model}\n\nAs we aim at a comparison with their findings, we present the work of\nWeaver and Lobkis~\\cite{WL} in some detail. Thereby, we also introduce\nthe notation and conventions. The system studied experimentally consists\nof two aluminum cubes coupled by a solid connection, manufactured out\nof a solid aluminum block.  The corners of the cubes were removed to\ndesymmetrize the structure. This was done to ensure ``chaotic''\nmotion. Elastomechanical wave modes were excited in one room, and the\nresponse was measured in the other room. In this way, 16\ndifferent curves of energy intensity versus time were recorded, each\nin a small region around a different frequency. The results show that\nthe energy does not always spread equally over the two rooms. If the\ncoupling is weak, then the wave intensity is higher in the room where\nthe initial excitation was performed than in the other room,\nregardless of how long one waits. Hence, the energy ratio never\napproaches unity. This deviation from the equipartition of the energy\nin the two rooms is referred to as energy localization. The resulting\ndata are shown in Fig.~\\ref{fig:WL_exp}, and as expected there is\nlocalization in the bins of larger mean-level spacing, but not in the\nbins of small mean-level spacing. We will discuss these results\nfurther in Sec.~\\ref{sec:RMT_result}.\n\n\\begin{figure}\n  \\centering \n  \\subfigure[Low frequency\n  bins.]{\\includegraphics[height=4cm]{tccs_fig_1a.eps} \n  \\label{fig:exp_loc}} \\quad \n  \\subfigure[High frequency bins.]{\\includegraphics[height=4cm]{tccs_fig_1b.eps}\\label{fig:exp_nonlocalised}} \n  \\caption{Results from the experiment by Weaver and Lobkis. The\n  energy ratio between the different rooms is plotted versus time (in\n  ms). We note the different\n  scales on the $y$ axes. Reprinted from Ref.~\\cite{WL}.}   \n\\label{fig:WL_exp}\n\\end{figure}\n\nTwo numerical studies were performed on membranes with rough\nboundaries~\\cite{WL}. The dynamics of the system was in both cases\ngoverned by discretized wave equations in each of the rooms, and the\ncoupling was realized in different ways in the two numerical\ncalculations, to which we refer as N1 and N2 in the sequel. In the\nfirst one, N1, the connection had the form of a window between the two\nmembranes similar to the situation in the experimental setup. In the\nsecond numerical calculation, N2, the rooms were separated, but\nsprings were attached to a few different sites in rooms 1 and \n2, thereby coupling those sites. In N1 and N2, a nonvanishing\ninitial condition was given to one site in one of the rooms, and the\nresponse was calculated at different sites in the other room. The\nresulting time series were cosine-bell time windowed to focus on a\nspecific instant in time. Then, the time series were Fourier\ntransformed and integrated over a small region in frequency to\naccumulate data around a certain frequency. As in the experiment,\n16 different curves of intensity versus time were obtained around\ndifferent frequencies. At different frequencies, the systems have\ndifferent effective couplings. Therefore, one expects~\\cite{WL} the\ntime behavior of the different curves to differ in the degree of\nlocalization, as well as in the way in which this asymptotic\nsaturation value is reached. Due to the differences in the coupling\nmechanism, there will also be differences between the results of N1\nand N2.\n\nMoreover, Weaver and Lobkis performed an analytical model study.  The\nelastic wave equation for the state of the system $U(t)$, say, is of\nsecond order in time. To focus on the response in a narrow interval\naround a certain frequency $\\Omega$, the ansatz $U(t) =\nu(t)\\exp(-i\\Omega t)$ is made with the assumption that $u(t)$ varies\nslowly with time. This leads to a first-order differential equation in\ntime for $u(t)$ which has the form\n\\begin{equation}\n-i\\frac{\\partial}{\\partial t} u(t) = \\frac{C+\\Omega^2}{2 \\Omega} u(t) \\ , \n\\label{eq:H_v_def}\n\\end{equation}   \nwhere $C$ is the wave operator of the original second-order equation.\nThe energies $E_1(t)$ and $E_2(t)$ in rooms 1 and 2 at time $t$\nare defined as the total probability density of finding the system\nstate $u(t)$ in one of the states $\\psi_{ik}, \\ i=1,2$, which are good\neigenstates in room $i$:\n\\begin{equation}\nE_i(t) = \\sum_k \\left| u(t) \\cdot \\psi_{ik} \\right| ^2 \\ .\n\\end{equation}\nStrictly speaking, $E_i(t)$ is no energy. Nevertheless, we find this\nterminology introduced in Ref.~\\cite{WL} appropriate and use it as\nwell, because $E_i(t)$ measures the degree of motion in room $i$.  If\nno energy dissipates into the surrounding environment, the total\nenergy $E=E_1(t)+E_2(t)$ is conserved -- i.e., independent of time. In\nthe sequel, it is always assumed that the system is excited in room\n1, and the energy is measured in room 2.  Analytical solutions for\ntwo coupled states are presented in Ref.~\\cite{WL} by employing different\nstatistical assumptions.\n\n\n\\section{Random matrix model}\n\\label{sec:RMT}\n\nWe set up the model in Sec.~\\ref{sec:RMT_model}. The connection to the\ntwo-level form factor is established in Sec.~\\ref{RMTff}, and a\n$2\\times 2$ version of the model is evaluated in Sec.~\\ref{M22}.  We\ndiscuss numerical simulations of the RMT model in\nSec.~\\ref{sec:RMT_num}. Finally, we comment on chaotically coupled\nregular systems in Sec.~\\ref{sec:RMT_pos}.\n\n\n\\subsection{Setup of the model}\n\\label{sec:RMT_model}\n\nSpectral correlations in elastomechanics have been shown to be well\ndescribed by RMT \\cite{Weaver}. This is also true in the case of\nsymmetry breaking~\\cite{cristais2}, which is of direct relevance for\nthe present study. Thus, RMT is also likely to be capable of modeling\nthe time behavior of elastomechanical systems.  As the first-order\nequation~\\eqref{eq:H_v_def} has proved to be a good approximation to\nthe experimental situation, we also base our model on this\nSchr\\\"odinger type of equation. Thus, it is more natural to\nreplace $(C+\\Omega^2)\/2\\Omega$ by the random matrix $H$ than to\nreplace $C$ itself. It turns out that this is indeed the best choice.\n\nThe appropriate RMT model is an extension of the one employed in\nRefs.~\\cite{GW,BGH}. The random matrix $H$ modeling the operator\n$(C+\\Omega^2)\/2\\Omega$ reads\n\\begin{equation}\nH = \\left[ \\begin{array}{cc} H_1 & 0 \\\\ \n                               0 & H_2 \\end{array} \\right] + \n    \\alpha \\left[ \\begin{array}{cc}  0 & V \\\\ \n                                     V^\\dagger & 0 \\end{array} \\right] \\ .\n\\label{eq:RMT_model_def}\n\\end{equation}  \nThe two matrices $H_i, \\ i=1,2$, model the uncoupled rooms 1 and\n2. They are real symmetric and have random entries. We draw them\nfrom (two independent) Gaussian orthogonal ensembles (GOE's). As the\nrooms in the experiments and the numerical calculations N1 and N2 were\nof the same size, the level densities were also the same. This can be\nadjusted in the RMT model by giving the matrices $H_i$ the same\nstatistical weights and the same dimension $N$, such that $2N$ is the\ntotal dimension of $H$. The matrix $V$ also has random entries. The\nstrength of the coupling is measured by the dimensionless parameter\n$\\alpha$. It is sufficient to always assume $\\alpha \\ge 0$. The\nstatistical weight of $V$ is chosen such that the total $H$ is in the\nGOE of $2N\\times 2N$ matrices for $\\alpha = 1$.\n\nWe write the eigenvalue equation for the total Hamiltonian $H$ in the form\n\\begin{eqnarray}\nH\\Psi_{n} = \\omega_{n}\\Psi_{n} \\ , \\qquad n=1,\\ldots,2N \\ .\n\\label{evh}\n\\end{eqnarray}\nThe eigenvalues $\\omega_{n}$ and eigenvectors $\\Psi_{n}$ are functions\nof the coupling parameter $\\alpha$. It is convenient to introduce the\nnotation\n\\begin{eqnarray}\n\\Psi_{n} =  \\left[ \\begin{array}{c} \\Psi_{1n} \\\\ \n                                    \\Psi_{2n} \\end{array} \\right] \\ ,\n\\label{psi12}\n\\end{eqnarray}\nwhere $\\Psi_{in}, \\ i=1,2$, is the projection of $\\Psi_n$ onto the\nsubspace $i$. We emphasize that $\\Psi_{in}, \\ i=1,2$, are\nfunctions of the coupling parameter $\\alpha$. The eigenvalue equations\nfor the Hamiltonians $H_i$ are written as\n\\begin{eqnarray}\nH_1\\psi_{1n} &=& \\omega_{1n}\\psi_{1n} \\ , \\qquad n=1,\\ldots,N \\ , \\nonumber\\\\\nH_2\\psi_{2n} &=& \\omega_{2n}\\psi_{2n} \\ , \\qquad n=1,\\ldots,N \\ ,\n\\label{evhi}\n\\end{eqnarray}\nwhere the eigenvalues $\\omega_{in}$ and eigenvectors $\\psi_{in}$ are\nnot functions of $\\alpha$. This difference between $\\Psi_{in}$ and\n $\\psi_{in}$ is an immediate consequence of the fact that $H$\ndepends on $\\alpha$, while $H_1$ and $H_2$ do not. We will also use\nthe notation $\\widehat{\\psi}_{2n}=(0,\\psi_{2n})^\\dagger$ for the\ncorresponding $2N$-dimensional vector with zeros in the first $N$\ncomponents.\n\nAt time $t=0$, the system is excited in room one such that the state\nof the total system can be written as\n\\begin{eqnarray}\nS =  \\left[ \\begin{array}{c} s \\\\ \n                             0 \\end{array} \\right] \\ ,\n\\label{source}\n\\end{eqnarray}\nwhere $s$ in room one is not specified in detail. We refer to $S$ as\nto the source. The time evolution of the source is then simply\n\\begin{eqnarray}\nu(t) = T(t)S \\ , \\qquad {\\rm where} \\qquad\nT(t) = \\exp\\left(iHt\\right)\n\\label{teo}\n\\end{eqnarray}\nis the time evolution operator.  Using the eigenvectors $\\widehat{\\psi}_{2n}$,\nthe energy in room 2 is thus given by\n\\begin{eqnarray}\nE_2(t,\\alpha) &=& \\sum_{n=1}^{N} \\left| u(t) \\cdot \\widehat{\\psi}_{2n} \\right|^2 \n        = \\sum_{n=1}^{N} \\left( T(t)S \\right) ^{\\dagger}\\widehat{\\psi}_{2n}\n             \\widehat{\\psi}_{2n}^\\dagger \\left( T(t)S \\right) \\nonumber \\\\\n       &=& S^{\\dagger} T^{\\dagger}(t) \n             \\left[\\begin{array}{cc} 0 & 0\\\\\n                                     0 & 1_N \\end{array} \\right]\n             T(t) S \\ .\n\\label{eq:RMT_E2}\n\\end{eqnarray}\nThe block matrix only contains the unit matrix $1_N$ for room 2.  We\nwrite $E_2(t,\\alpha)$ instead of $E_2(t)$ to emphasize the dependence\non the coupling parameter $\\alpha$ in the RMT model. We will study\naverages $\\overline{E_2}(t,\\alpha)$ over the ensemble of matrices\nintroduced in Eq.~\\eqref{eq:RMT_model_def}.  Occasionally, we will\nalso consider averages over the direction of the source. This average\nis denoted by angular brackets $\\langle \\cdots \\rangle$.\n\nThe total energy $E$ is trivially conserved in the framework of our\nmodel. We have\n\\begin{eqnarray}\nE = E_1(t,\\alpha)+E_2(t,\\alpha) = S \\cdot S = s \\cdot s \\ .\n\\label{etot}\n\\end{eqnarray}\nHence, we can always construct $E_1(t,\\alpha)$, once $E_2(t,\\alpha)$\nhas been calculated.\n\nAs all correlations have to be\nmeasured~\\cite{Mehta,GMGW,Stockmann,Haake} on the local scale of the\nmean-level spacing $D$, we introduce an unfolded time $\\tau=Dt$. This\nis also the scale on which $\\alpha$ acts; we introduce the unfolded\ncoupling parameter $\\lambda=\\alpha\/D$. The energy on the unfolded\nscale is then given by\n\\begin{eqnarray}\n\\varepsilon_2(\\tau,\\lambda) =\n            \\lim_{N\\to\\infty} \\overline{E_2}(\\tau\/D,D\\lambda).\n\\label{eun}\n\\end{eqnarray}\nSmall values of $\\alpha$ have a large effect on the correlation\nfunctions if the mean-level spacing $D$ is small as well. \n\n\n\\subsection{Relation to the two-level form factor}\n\\label{RMTff}\n\nWe now derive an estimate for the time evolution of $E_2(t,\\alpha)$\nwhich should apply to strong coupling strength-- i.e., to a parameter\n$\\alpha$ which is large on the unfolded scale of the mean level\nspacing. If one assumes that the source $S$ comprises excitations into\nall states $\\psi_{1n}$, one may average over the direction of the\nsource. We find, from Eq.~\\eqref{eq:RMT_E2},\n\\begin{eqnarray}\n\\langle E_2\\rangle (t,\\alpha) = B {\\rm tr}^{(11)}\\,T^{\\dagger}(t) \n             \\left[\\begin{array}{cc} 0 & 0\\\\\n                                     0 & 1_N \\end{array} \\right] T(t) \\ ,\n\\label{e2est1}\n\\end{eqnarray}\nwhere ${\\rm tr}^{(11)}$ is the trace over the $(11)$ sector of the\nwhole matrix -- i.e., over the upper left block. The constant $B$ results\nfrom the average. Expanding the time evolution operator in terms of\nthe eigenvectors,\n\\begin{eqnarray}\nT(t) = \\sum_{n=1}^{2N} \\Psi_n \\exp\\left(i\\omega_nt\\right) \\Psi_n^\\dagger \\ ,\n\\label{teoexp}\n\\end{eqnarray}\nwe arrive after a short calculation at\n\\begin{eqnarray}\n\\langle E_2\\rangle (t,\\alpha) = B \\sum_{n,m} \n                        \\left(\\Psi_{1m}\\cdot\\Psi_{1n}\\right)\n                        \\left(\\Psi_{2n}\\cdot\\Psi_{2m}\\right)\n                        \\exp\\left[i(\\omega_n-\\omega_m)t\\right] \\ .\n\\label{e2est2}\n\\end{eqnarray}\nThe vectors $\\Psi_{1n}$ and $\\Psi_{2n}$ are, according to\nEq.~\\eqref{psi12}, the projections of the full eigenvector $\\Psi_{n}$\nonto the subspaces corresponding to the two rooms. They depend on the\ncoupling parameter $\\alpha$. These vectors coincide with the\neigenvectors for the matrices $H_1$ and $H_2$ only for $\\alpha=0$.\nHence, they are only orthogonal in this special case. For all values\n$\\alpha>0$, the scalar products $\\Psi_{1m}\\cdot\\Psi_{1n}$ and\n$\\Psi_{2n}\\cdot\\Psi_{2m}$ are neither zero nor given by $\\delta_{nm}$.\nThe ensemble average over the eigenenergies $\\omega_n$ and over the\neigenfunctions $\\Psi_{n}$ decouples only for the parameter values\n$\\alpha=0$ and $\\alpha=1$-- i.e., if the system either falls into two\nGOE's or is represented by one GOE. We now estimate the ensemble average\nof the energy $\\langle E_2\\rangle (t,\\alpha)$ by making the\napproximation that the averages over eigenenergies and eigenfunctions\ndecouple. For strong coupling this should yield a reasonable result,\nand we will use this approximation in that limit only.  We average\nover the eigenfunctions. For each term in the double sum in\nEq.~\\eqref{e2est2} the product of scalar products gives the same\ncontribution which we can take out of the sums. Thus, we find\n\\begin{eqnarray}\n\\overline{\\langle E_2\\rangle} (t,\\alpha) \n = B \\overline{\\sum_{n,m}\\exp\\left[i(\\omega_n-\\omega_m)t\\right]} \\ ,\n\\label{e2est3}\n\\end{eqnarray}\nwhere we absorbed the contributions from the average over the scalar\nproducts into the constant $B$. The average over the\neigenvalues is yet to be performed.  Luckily, the average in\nEq.~\\eqref{e2est3} is recognized as the Fourier transform of the\ntwo--level correlation function~\\cite{Mehta,GMGW}. This yields, on the\nunfolded scale,\n\\begin{eqnarray}\n\\varepsilon_2(\\tau,\\lambda) = B \\left[\\delta(\\tau\/2\\pi) \n                            + K_2(\\tau\/2\\pi,\\lambda)\\right] \\ ,\n\\label{e2est4}\n\\end{eqnarray}\nwhere the function $K_2(\\tau,\\lambda)$ is referred to as the two-level\nform factor. The term $\\delta(\\tau)$ is due to the diagonal\ncontributions in the double sum of Eq.~\\eqref{e2est3}. We notice that\nthe conventions used here require a rescaling of time with a factor of\n$2\\pi$.  The expression~\\eqref{e2est4} is an estimate for the energy\n$\\varepsilon_2(\\tau,\\lambda)$, exclusively in terms of the \nFourier-transformed two-level spectral correlation function for the\ntransition from two GOE's to one GOE. Unfortunately, the two-level form\nfactor is not known analytically for all values of $\\lambda$.  The\nresult for one GOE, corresponding to $\\lambda\\to\\infty$,\nreads~\\cite{Mehta}\n\\begin{equation}\nK_2(\\tau,\\infty) = \\left\\{ \n            \\begin{array}{ll} 2\\tau-\\tau \\ln(2\\tau+1) & \n                                   {\\rm for} \\quad 0<\\tau\\le 1 , \\\\ \n                              2-\\tau \\ln\\frac{{\\textstyle 2\\tau+1}}\n                                             {{\\textstyle 2\\tau-1}} &\n                                   {\\rm for} \\quad \\tau\\ge 1 .\n            \\end{array} \\right. \n\\label{eq:formfactor_result}\n\\end{equation} \nWe notice that the function $\\varepsilon_2(t,\\infty)$ has the\nlimit properties\n\\begin{eqnarray}\n\\varepsilon_2(0,\\infty) &=& 0 , \\nonumber \\\\\n\\lim _{\\tau \\rightarrow \\infty} \\varepsilon_2(\\tau,\\infty) &=& B \\ .\n\\label{eq:ff_props}\n\\end{eqnarray}\nThe second property will be used to fix the scale in comparison with\nthe results of Ref.~\\cite{WL}.  The estimate~\\eqref{e2est4} should\napply to large coupling -- i.e., to parameter values $\\lambda\\gg 1$.\n\n\n\\subsection{Two-by-two model}\n\\label{M22}\n\nQuite often, one obtains surprisingly good information about an RMT\nmodel by restricting it to the smallest possible matrix dimension such\nthat the nontrivial specific characteristics of the model are still\npresent. This is successful in the case of the nearest-neighbor\nspacing distribution for the Gaussian orthogonal, unitary and\nsymplectic ensembles (GGOE, GUE and GSE, respectively) ~\\cite{GMGW,Stockmann,Haake}, but also for crossover\ntransitions~\\cite{lenz}. Here, we proceed analogously.  We obtain a\n$2\\times 2$ RMT model by setting $N=1$ in\nEq.~\\eqref{eq:RMT_model_def}.  It turns out convenient to absorb the\nparameter $\\alpha$ into the matrix element $V$ such that\n\\begin{equation}\n  H = \\left[ \\begin{array}{cc} H_1 & V \\\\ V & H_2 \\end{array} \\right]\n\\label{m1}\n\\end{equation} \nand to readjust the probability density function $P(H)$ for the\nensemble average accordingly:\n\\begin{equation}\n  P(H) = \\frac{\\sqrt{2}}{\\pi\\sqrt{\\pi\\alpha^2}} \n               \\exp\\left(-H_1^2-H_2^2-2\\frac{V^2}{\\alpha^2}\\right) \\ .\n\\label{m2}\n\\end{equation} \nAs before, the GOE is recovered for $\\alpha=1$. We introduce\neigenvalue and angle coordinates\n\\begin{equation}\n  \\left[ \\begin{array}{cc} H_1 & V \\\\ \n                             V & H_2 \\end{array} \\right] = \n  \\left[ \\begin{array}{cc} \\cos\\varphi & -\\sin\\varphi \\\\ \n                           \\sin\\varphi &  \\cos\\varphi \\end{array} \\right]\n  \\left[ \\begin{array}{cc} \\omega_1 & 0 \\\\ \n                                  0 & \\omega_2 \\end{array} \\right] \n  \\left[ \\begin{array}{cc} \\cos\\varphi & \\sin\\varphi \\\\ \n                          -\\sin\\varphi & \\cos\\varphi \\end{array} \\right] \\ ,\n\\label{m3}\n\\end{equation} \nwhich implies that the time evolution can be written in the same form\nwith eigenvalues $\\exp(i\\omega_1t)$ and $\\exp(i\\omega_2t)$. For the\nintegration measure one has\n\\begin{equation}\ndH_1 dH_2 dV = \\frac{1}{4} |\\omega_1-\\omega_2| d\\omega_1 d\\omega_2 d\\varphi \\ .\n\\label{m4}\n\\end{equation} \nThe source in this $2\\times 2$ model is simply given by $S=(1,0)$.\nThus, collecting everything, we find, with Eq.~\\eqref{eq:RMT_E2},\n\\begin{eqnarray}\n\\overline{E_2}(t,\\alpha) &=& \\frac{\\sqrt{2}}{4\\pi\\sqrt{\\pi\\alpha^2}}\n             \\int\\limits_{-\\infty}^{+\\infty}d\\omega_1\n             \\int\\limits_{-\\infty}^{+\\infty}d\\omega_2|\\omega_1-\\omega_2|   \n             \\int\\limits_0^{2 \\pi} d\\varphi \\nonumber\\\\\n    & & \\qquad \\exp\\left[-\\frac{(\\omega_1+\\omega_2)^2}{2}-\n        \\frac{(\\omega_1-\\omega_2)^2}{2} \\left(\\cos ^2 2\\varphi + \\frac{\\sin ^2\n             2\\varphi}{\\alpha^2} \\right) \\right] \\nonumber \\\\ \n    & & \\qquad \\sin^22\\varphi \\, \\sin^2\\frac{\\omega_1-\\omega_2}{2}t \\ .\n\\label{m5}\n\\end{eqnarray}\nWe introduce $x=(\\omega_1+\\omega_2)\/\\sqrt{2}$ and\n$y=(\\omega_1-\\omega_2)\/\\sqrt{2}$ as new integration variables, perform\nthe $x$ and $\\varphi$ integrals, and arrive at\n\\begin{eqnarray}\n\\overline{E_2}(t,\\alpha) &=& \\frac{\\alpha}{2 (\\alpha + 1)} - \n \\nonumber\\\\\n & & \\qquad \\alpha \\int\\limits_0^\\infty dy\\, y \\, \n                   \\exp\\left[-y^2(1+\\alpha^2)\\right] \n \\nonumber\\\\\n & & \\qquad \\qquad \n\\left[{\\rm I}_0 (y^2 (1-\\alpha^2)) - {\\rm I}_1(y^2 (1-\\alpha^2))\\right] \n \\cos 2\\alpha yt \\, .\n\\label{eq:2d_result}\n\\end{eqnarray}\nHere, ${\\rm I}_0$ and $\\rm{I}_1$ denote the modified Bessel functions\nof zeroth and first order, respectively.  \n\nFormula~\\eqref{eq:2d_result} is the final result in the framework of\nour $2\\times 2$ model. In Ref.~\\cite{WL}, solutions for two-state\nmodels based on Eq.~\\eqref{eq:H_v_def} with different statistical\nassumptions were given. Although the general behavior is similar to\nour $2\\times 2$ RMT result~\\eqref{eq:2d_result}, the analytical forms\nof these solutions in Ref.~\\cite{WL} are different.\n\nFor obvious reasons, the unfolding of formula~\\eqref{eq:2d_result} is\nmeaningless.  Nevertheless, experience with $2\\times 2$ model for the\nspacing distribution tells that meaningful statements can be achieved\nif the transition parameter -- i.e., the coupling strength $\\alpha$ in\nour case -- is interpreted properly.  Here, we can do that in the\nfollowing manner. For large times $t$, the function\n$\\overline{E_2}(t,\\alpha)$ becomes constant, because it reaches its\nsaturation limit. The latter will depend on $\\alpha$. Thus, comparing\nthe saturation limit for the $2\\times 2$ model with that of the\n$2N\\times 2N$ RMT simulation or with those of the experiment and the\nnumerical calculations N1 and N2 allows one, in principle, to\ninterpret $\\alpha$ on the unfolded scale.  The saturation limit is\neasily obtained from Eq.~\\eqref{eq:2d_result}.  Due to the\nRiemann-Lebesgue lemma~\\cite{GR}, we have\n\\begin{eqnarray}\n\\lim_{t\\to\\infty}\\overline{E_2}(t,\\alpha) = \\frac{\\alpha}{2 (\\alpha + 1)} \\ ,\n\\label{m7} \n\\end{eqnarray}\nbecause the cosine function in the integrand oscillates so rapidly\nfor large $t$, that the integral gives zero in the limit $t\n\\rightarrow \\infty$.\n\nWe notice that there is no equipartition of the energies for\n$t\\to\\infty$ at $\\alpha = 1$. This may seem a bit unexpected, because\n$H$ is a $2\\times 2$ GOE matrix for that parameter value. It simply\nreflects the need to properly interpret the parameters, as just\ndiscussed. We study a two-state system and compare the probability\nfor a transition between the two states with the probability of\nstaying in one state. There is no reason why these should be\nequal. In the $N \\rightarrow \\infty$ limit, however, we expect\nequipartition for the full GOE. As the coupling parameter has to be\nmeasured on the scale of the local mean level spacing, the limit $N\n\\rightarrow \\infty$ corresponds to the limit $\\alpha \\rightarrow \\infty$ in\nthe $2\\times 2$ version, and in that limit the right-hand side of\nEq.~\\eqref{m7} tends to $1\/2$ and therefore to equipartition.\n\n\n\\begin{figure}[htb]\n  \\centering\n  \\includegraphics[height=4cm]{tccs_fig_2.eps}\n  \\caption{Energy ratio $E_2 \/ E_1$ versus (dimensionless) time. Data\n  from N1 of Ref.~\\cite{WL}, bin 16, plotted\n  as dashed line. The result from \n    Eq.~\\eqref{e2est4} for $\\lambda \\rightarrow \\infty$, $B$ set by property 2\n    of Eq.~\\eqref{eq:ff_props} and a rescaling of time to make the two curves\n    fit for early times, is plotted as a solid line.}\n  \\label{fig:WL_num1_vs_form_factor}\n\\end{figure} \n\n\\begin{figure}[htb]\n  \\centering \\subfigure[$2 \\times 2$ RMT\n  model, fitted to N2 bins 2,3,4 from top to bottom.]{\\includegraphics[height=4cm]{tccs_fig_3a.eps}\n\\label{fig:2d_result}} \\quad \\subfigure[Data from N2, bins\n  2,3,4 from top to\n  bottom.]{\\includegraphics[height=4cm]{tccs_fig_3b.eps}\n\\label{fig:num2_for_2d}}\n  \\caption{Comparison between the $2\\times 2$ RMT model and the low\n  frequency bins of N2 of Ref.~\\cite{WL}. The energy ratio $E_2\/E_1$ is plotted versus\n  (dimensionless) time.} \n\\label{fig:2d_and_num2}\n\\end{figure} \n\n\n\\subsection{Numerical simulations}\n\\label{sec:RMT_num}\n\nThe numerical simulations of the RMT model are performed as in\nRefs.~\\cite{GW,BGH}. The unfolding of the results, however, is not\ndone in the standard way. To compare with the numerical calculations\nof Weaver and Lobkis, it is more convenient to first unfold the level\ndensities and then ``refold'' the spectra of our model with the\nlevel densities of Ref.~\\cite{WL}. Thereby we ensure that the \nmean-level densities of the RMT model acquire a form given by the Weyl\nformula for the billiardlike system of Ref.~\\cite{WL}. The\nappropriate Weyl formula for the modal density in a square membrane\nwith Dirichlet boundary conditions reads~\\cite{WL}\n\\begin{equation}\n  \\rho^{({\\rm smooth})}(\\Omega) = \\frac{\\Omega A}{2\\pi c^2} -\n                                  \\frac{L}{4c} \\ ,\n\\label{Weyl}\n\\end{equation} \ndepending on the frequency $\\Omega$. Here, $A$ is the area of the\nmembrane and $L$ is the side length of one room. Moreover, $c$ is\nthe wave velocity and is chosen as unity here. Using twice the area of a\nroom ($A = 2 \\times 198^2$) and a value of $L$ which takes the roughness\nof the boundaries into account ($L = 2 \\times 3 \\times 198$), we aquire an\nexpression for the level density of the entire system.  A comparison\nwith the Weyl formula of Ref.~\\cite{berry_sinai} shows that the terms\nfrom the extra edges introduced to model the disorder cancel. Hence,\nwe use the formula for a square membrane, which is precisely\nEq.~\\eqref{Weyl}.  This Weyl formula is employed to refold our RMT\nmodel for both numerical calculations N1 and N2. In the latter case,\nthe system is not really a billiard, due to the springs used for the\ncoupling. Nevertheless, the Weyl formula should be a good\napproximation to the real level density.\n\nFor every simulation, we generate random matrices of dimension $2N\n\\times 2N=100 \\times 100$, unfold, refold, calculate $E_2(t,\\alpha)$,\nand average over the results of 800 such simulations. As the functions\n$E_2(t,\\alpha)$ are not measured on the unfolded scale, but on that of\nthe Weyl formula, we do not introduce the notation\n$\\varepsilon_2(\\tau,\\lambda)$ in the present context.  Moreover, we\nnotice that the numerical calculations N1 and N2 depend on the mean\nfrequency $\\Omega$ in the bin under consideration. Accordingly, we\narrive at a two-parameter family of time-dependent curves\n$\\overline{E_2}(t,\\alpha,\\Omega)$. We now associate each bin in the\nnumerical calculation performed by Weaver and Lobkis with the\ncorresponding frequency $\\Omega$. This leaves us with a one-parameter\nfamily of time-dependent curves $\\overline{E_{2\\mu}}(t,\\alpha), \\\n\\mu=1,\\ldots,16$, for each bin labeled by $\\mu$.\n\n\n\\subsection{Chaotically coupled regular systems}\n\\label{sec:RMT_pos}\n\nIn the RMT model defined by Eq.~\\eqref{eq:RMT_model_def} and in its\nsubsequent analysis, we always drew the matrices $H_i, \\ i=1,2$, from\n(two independent) GOE's, implying that we model the two rooms\nindividually as chaotic systems --- before the coupling is considered.\nOne can also assume that the two rooms are regular systems before they\nare coupled. In that case, the matrices $H_i, \\ i=1,2$, would be drawn\nfrom Poisson ensembles~\\cite{GMGW}.  We also did such numerical\nsimulations. The results are qualitatively the same if the coupling --\ni.e., the matrix $V$ -- introduces enough chaos. The main effect is an\nadjustment of the scales. What matters for the qualitative behavior is\nthe chaoticity of the total system.\n\n\n\\section{Comparison with experiment and numerical calculations}\n\\label{sec:RMT_result}\n\n\\begin{figure}\n\\centering\n\\subfigure[Bin 2.]{\n\\includegraphics[height=4cm]{tccs_fig_4a.eps}\n\\label{fig:RMT_N1_bin2}}\n\\quad\n\\subfigure[Bin 5.]{\n\\includegraphics[height=4cm]{tccs_fig_4b.eps}}\n\\\\\n\\subfigure[Bin 9.]{\n\\includegraphics[height=4cm]{tccs_fig_4c.eps}}\n\\quad\n\\subfigure[Bin 13.]{\n\\includegraphics[height=4cm]{tccs_fig_4d.eps}\n}\n\\\\\n\\subfigure[Bin 16.]{\n\\includegraphics[height=4cm]{tccs_fig_4e.eps}\n\\label{fig:RMT_N1_bin16}}\n\\quad\n\\caption{Energy ratio $E_2\/E_1$ versus (dimensionless) time. Data from\n  N1 of Ref.~\\cite{WL} plotted as dashed lines and from refolded\n  numerical simulations using the RMT model as solid lines. We notice the\n  different $E_2\/E_1$ scale in plot (a).}\n\\label{fig:RMT_result_N1}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\subfigure[Bin 2.]{\n\\includegraphics[height=4cm]{tccs_fig_5a.eps}\n\\label{fig:RMT_N2_bin2}}\n\\quad\n\\subfigure[Bin 5.]{\n\\includegraphics[height=4cm]{tccs_fig_5b.eps}}\n\\\\\n\\subfigure[Bin 9.]{\n\\includegraphics[height=4cm]{tccs_fig_5c.eps}}\n\\quad\n\\subfigure[Bin 13.]{\n\\includegraphics[height=4cm]{tccs_fig_5d.eps}\n}\n\\\\\n\\subfigure[Bin 16.]{\n\\includegraphics[height=4cm]{tccs_fig_5e.eps}\n\\label{fig:RMT_N2_bin16}}\n\\quad\n\\caption{Energy ratio $E_2\/E_1$ versus (dimensionless) time. Data from\n  N2 of Ref.~\\cite{WL} plotted as dashed lines and from refolded\n  numerical simulations using the RMT model as solid lines. We notice the\n  different $E_2\/E_1$ scale in plot (a).}\n\\label{fig:RMT_result_N2}\n\\end{figure}\n\nWe now compare the results of our RMT model with those of\nRef.~\\cite{WL}. The quantity studied in Ref.~\\cite{WL} is mostly the\nfraction of energy in room 2, or $E_2(t)\/E$. In our figures, we plot\n$E_2(t)\/E_1(t)$ instead. We first consider the estimate,\nEq.~\\eqref{eq:formfactor_result}, involving the GOE form factor and the\nresult~\\eqref{e2est4} of the $2\\times 2$ RMT model and compare with\nthe numerical calculations N1 and N2.  As formula~\\eqref{e2est4} was\nderived under the assumption of strong coupling on the scale of the\nlocal mean-level spacing, it should apply to the high-frequency bins\nof N1. Anticipating the later extraction of the coupling parameter, we\nalready now mention that indeed $\\lambda>1$ in that bin. In\nFig.~\\ref{fig:WL_num1_vs_form_factor} we compare Eq.~\\eqref{e2est4}\nwith data from bin 16 (the highest frequency bin) of N1.  A good\ndescription is obtained, although no equipartition is reached.  This\nimplies that the effective coupling $\\lambda$ is large, but not very\nlarge. We recall that formally $\\lambda\\to\\infty$ corresponds to the\nstrongest coupling $\\alpha=1$ -- i.e., to one single system.  Expansion\nof the form factor for short times $\\tau$ reveals a linear short time\ndependence of the ensemble-averaged energy\n$\\varepsilon_2(\\tau,\\lambda)$. This is in agreement with the\nanalytical discussion of Ref.~\\cite{WL}.\n\n\\begin{table}\n\\caption{Coupling parameters $\\alpha$ for the $2 \\times 2$ model used in\nFig.~\\ref{fig:2d_result}. The asymptotic saturation $E_2\/E$ values for\nN2 in bins 2, 3, and 4 are taken from Ref.~\\cite{WL}. The corresponding\ncoupling parameters are calculated from Eq.~\\eqref{m7}.}\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline\nbin & $E_2\/E$ (asymptotic) & $\\alpha$ \\\\\n\\hline\n2 & 0.35 &  2.33 \\\\\n3 & 0.2  &  0.67 \\\\\n4 & 0.14 &  0.39 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\label{tab:2d_cplngs}\n\\end{table}\n\nWe turn to the $2\\times 2$ RMT model and compare to bins 2, 3, and 4 of\nN2. The values of the coupling parameter $\\alpha$ are determined from\nEq.~\\eqref{m7} and given in Table ~\\ref{tab:2d_cplngs}, together with\nthe $E_2\/E$ saturation values of N2. The $2\\times 2$ RMT model curves\nand data from N2 are shown in Figs.~\\ref{fig:2d_result}\nand~\\ref{fig:num2_for_2d}, respectively. As expected, smaller values\nof $\\alpha$ correspond to a higher degree of localization -- i.e., to a\nstronger deviation from equipartition.  The general behavior of the\nresults from the $2 \\times 2$ model stays the same for large values of\n$\\alpha$. We notice that the time scale is different, because the\ndata from N2 were not unfolded.  The similarity shown in\nFig.~\\ref{fig:2d_and_num2} between the $2\\times 2$ RMT model and the\nWeaver-Lobkis results~\\cite{WL} is remarkable. This parallels the\nsuccess of $2\\times 2$ RMT models for the spacing distributions.\n\nThe RMT simulations yield a one-parameter family of curves for each\nbin, to be compared with the numerical calculations N1 and N2. We use\nvisual inspection to determine, for each bin, which curve fits best to\nthe numerical calculations N1 and N2. Typical results for some of the\nbins are shown in Figs.~\\ref{fig:RMT_result_N1}\nand~\\ref{fig:RMT_result_N2}. In the low-frequency bins of N2 there is\na discrepancy; see Fig.~\\ref{fig:RMT_N2_bin2}. The RMT simulation does\nnot overshoot its saturation value as clearly as the data of\nRef.~\\cite{WL}. This very large overshoot within a short-time interval\nis, however, borne out in the $2\\times 2$ RMT model; see\nFig.~\\ref{fig:2d_and_num2}.  By the visual fit we determine the\ncoupling parameter $\\alpha$.  We measure it on the scale of the local\nmean-level spacing. In Table ~\\ref{tab:num_RMT}, we list the in this\nmanner obtained coupling parameter $\\lambda$ for N1 and N2 for the\nbins under consideration. The $\\lambda$ values extracted for N1 go up\nwith higher bins, because the saturation value comes closer to\nequipartition. This effect is also visible for N2, but not so\npronounced, because the saturation value does not change much for\nhigher bins.\n\n\\begin{table}\n\\caption{Coupling constants $\\lambda$ on the unfolded scale resulting \nfrom the RMT simulation, determined for some bins of the numerical\ncalculations N1 and N2 with mean frequencies $\\Omega$: see\nFigs.~(\\ref{fig:RMT_result_N1}) and (\\ref{fig:RMT_result_N2}).}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\nbin & $\\Omega$ & $\\lambda$ for N1 & $\\lambda$ for N2 \\\\\n\\hline\n2 & 0.1562 & 0.4750  & 3.2316 \\\\\n5 & 0.4686 & 2.0222  & 0.7754 \\\\\n9 & 0.8851 & 3.0586  & 0.7502 \\\\\n13 & 1.3016 & 3.9093  & 0.8441 \\\\\n16 & 1.6140 & 4.2427  & 0.8899 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\label{tab:num_RMT}\n\\end{table}\n\nWe find good qualitative agreement between the RMT simulation and N1\nand N2. Importantly, fluctuations between different random matrices\nare quite large, and the RMT curves presented here are averages of\n$E_2$ over 800 random matrices. Among those some show very much closer\nsimilarity to the curves of Weaver and Lobkis~\\cite{WL}, and we\nbelieve that most of the discrepancy in the low-frequency bins of N1\nare due to fluctuations resulting from the specific choice of the\nboundary and perhaps other parameters in the numerical calculations of\nRef.~\\cite{WL}. The reason for this interpretation of the discrepancy\nis the peculiar form of the fluctuations in bin 2 of N1; see\nFig.~\\ref{fig:RMT_N1_bin2}. Since these lower bins have a lower level\ndensity, they should also be more sensitive to this kind of\nfluctuations. In the intermediate-frequency bins, we find that the\nnumerical calculations N1 and N2 are well reproduced by the RMT model.\nIn the figures for the high-frequency bin, Figs.\n\\ref{fig:RMT_N1_bin16} and \\ref{fig:RMT_N2_bin16}, the RMT results are\nseen to deviate  \ndownwards for large times. It is conceivable that the short-time\nbehavior strongly depends on the specific realization of the coupling\nwhile the long-time behavior is universal. \n\nThe result from the experiment is shown in Fig.~\\ref{fig:WL_exp}. The\nqualitative features are well described by the RMT model.  It can be\nseen that the system localizes in the low-frequency bins, but does not\ndo so in the high-frequency bins. In other words, the system\napproaches equipartition of the energy. The localization behavior\ndepends on the strength of the coupling measured on the scale of the\nlocal mean-level spacing, and the mean-level density is significantly\nhigher in the high-frequency bins. It seems that the saturation value\nhas not yet been reached on the time scale visible in\nFig.~\\ref{fig:WL_exp}. A closer inspection shows an initial behavior\nsimilar to the one in the results from our numerical RMT calculations\nand a slight upwards trend of the energy ratio towards the end of the\ntime window studied. According to Ref.~\\cite{WL} this means that,\nfirst, the expected asymptotic value of $E_2\/E_1$ is reached in the\nmiddle of the time window studied and that, second, there is another\nunknown effect acting on longer time scales adding to the energy spread\nwhich is likely to be due to the coupling of the setup to the\nenvironment.  Thus, we do not attempt to extract a quantitative\nestimate of the coupling parameter $\\alpha$ for the experiment.\n\n\n\\section{Conclusion}\n\\label{sec:dic_concl}\n\nWe have set up and analyzed an RMT model to describe the time behavior\nof coupled reverberation rooms. This system shows localization effects\nunder certain conditions. Within our RMT model, we gave an estimate of\nthe strong coupling behavior which involved the two-level GOE\nform factor. Moreover, we studied the $2\\times 2$ version of our RMT\nmodel analytically for arbitrary coupling strength and performed numerical simulations for the\n$2N\\times 2N$ version. \n\n{}From the comparison with the work of Weaver and Lobkis, we conclude\nthat the RMT model yields a good qualitative description.  Moreover,\nwe find an interpretation of the localization effect by relating it to\nthe universal features of RMT models for crossover transitions.\n\nFormally similar RMT models have been studied in connection with\nsymmetry breaking. Then, the parameter $\\alpha$ measures the degree of\nsymmetry breaking. This is so for isospin breaking in nuclear\nphysics~\\cite{values,GW,26Al,BGH}, symmetry breaking in molecular\nphysics~\\cite{Leitner}, and symmetry breaking in resonating quartz\ncrystals~\\cite{cristais2}. The experiment that comes closest to the\npresent situation is the study of spectral correlations in coupled\nmicrowave billiards~\\cite{nos}. In this work and in the experiment of\nWeaver and Lobkis~\\cite{WL}, an excitation initially in system one\nwould stay there for all times if the coupling is zero. This is\nformally analogous to a conserved quantum number.\n\nIn the RMT model the parameter measuring the size of the connection\nhas a most natural counterpart: the root-mean-square matrix element\ndue to the coupling measured on the scale of the local mean-level\nspacing. Thus, there are two ways of making the effective,\ndimensionless coupling parameter $\\lambda$ small and to thereby\nintroduce localization: either the original coupling parameter\n$\\alpha$ which always has a dimension is small or the mean-level\nspacing $D$ is made large.\n\nIt might be surprising that no equipartition of the energy is seen for\nall coupling strengths even if one waits very long. This would be the\nexpectation if one compares the system with two water basins coupled\nby a channel. Suppose that initially the two basins are empty and the\nchannel is closed by a gate. Now one of the basins is filled with\nwater and the gate is opened at $t=0$. Obviously, the water levels in\nthe two basins will be equal after sufficiently long time. The speed\nwith which this equipartition is reached simply depends on the cross\nsection of the channel. \n\nIn the present case of the two coupled acoustic rooms the situation is\ndifferent, because the wave character of the excitations has to be\ntaken into account. Thus, the crucial parameter entering is the size\nof the coupling connection -- i.e., its geometrical width -- compared to a\ntypical wavelength. The first waves after the excitation which come\nfrom system 1 into system 2 enter a silent territory and cause the\nfirst excitations there. The next waves coming from system 1,\nhowever, encounter these first excitations in system 2 and interfere\nwith them constructively or destructively. This process continues and,\nof course, after being excited in system 2, waves also travel back\ninto system 1. For smaller times, this complicated dynamics\ncertainly depends strongly on the realization of the coupling.  This\nis clearly so in the numerical calculation of\nRef.~\\cite{WL}. Nevertheless, the loneg-time behavior, in particular\nthe saturation limit, shows universal characteristics, consistent with\ngeneral features of quantum chaotic systems; see Ref.~\\cite{GMGW}. In\nthe energy domain, this is borne out in the fact that the correlations\non smaller energy scales are described by universal RMT features,\nwhile system-specific properties show up on larger scales, leading to\ndeviations from the RMT prediction. To avoid confusion, we emphasize\nthat these universal RMT features include those in the presence of\nsymmetry breaking.  By system-specific properties we mean, most\nimportantly, the scales set by the shortest periodic orbits.\n\nIt is worthwhile to realize that the chaoticity of the individual\nsubsystems before the coupling is not crucial, if the coupling itself\nintroduces enough chaos. We tested numerically that the behavior of\nthe corresponding model for two regular subsystems coupled chaotically\nshows the same qualitative behavior. The saturation value is reached\nslightly faster, which simply means that time is rescaled.  Finally,\nwe mention that our model is not restricted to elastomechanics. It\nwould also apply to coupled quantum dots and other coupled systems.\n\n\n\\section*{Acknowledgment}\n\nThis project was prompted by a conversation between R.L.~Weaver and\none of the present authors (T.G.) during a workshop at the Centro\nInternacional de Ciencias (CIC\/UNAM), Cuernavaca, Mexico. We thank\nR.L.~Weaver for providing us with the data for the numerical\ncalculations N1 and N2. We acknowledge financial support from Det\nSvenska Vetenskapsr\\aa det.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n{\\let\\thefootnote\\relax\\footnotetext{$^*$ Equal contribution}}\n{\\let\\thefootnote\\relax\\footnotetext{$^\\dagger$ Corresponding author email: liuzy@tsinghua.edu.cn}}\n\n\n\nLarge-scale Transformer-based pretrained language models (\\plm s)~\\cite[][\\textit{inter alia}]{BERT,ALBERT,ELECTRA,CharBERT,DeBERTa} have achieved great success in recent years and attracted wide research interest, in which the tokenization plays a fundamental role.\n\nUnfortunately, current tokenization methods are mostly developed primarily for English~\\cite{BPEsuboptimal}. Almost all the current \\plm s adopt the sub-word tokenization method originating from machine translation, such as the Byte-Pair Encoding~\\cite{BPE}, WordPiece~\\cite{WordPiece,BERT} and SentencePiece based on the unigram language model~\\cite{sentencepiece}. While the idea of sub-word tokenization is intuitive and effective for morphological-rich synthetic languages, it is \\textbf{not} the case for Chinese.\n\nWe believe that it is crucial to develop tailored techniques for the languages beyond English because there can be huge differences between different languages~\\cite[\\#BenderRule]{BenderRule}. Towards this end, we devote this work to analysing three unique linguistic characteristics of Chinese (writing system) compared to English: 1) The Chinese writing system is morphemic~\\cite{hill2016typology}, which means the Chinese characters poorly reflect the pronunciation, resulting in the conventional character-based tokenization misses much more phonological information. 2) Modern Chinese words basically do not undergo morphological alternations~\\cite{morphology_chinese}, thus rendering sub-word tokenization inapplicable. However, Chinese characters are mainly logograms, which means their glyphs, the composition of stokes and radicals, also contain rich semantic information~\\cite{glyce}. 3) In Chinese writing, there is no natural word boundary like the space in English writing. Although it is possible to inject word boundaries via Chinese word Segmentation (\\cws), there is no study on how this works for Chinese \\plm s.\n\n Targeting the three factors, we then explore three corresponding tokenization strategies: 1) A pronunciation-based tokenizer family called \\camelabr{Shuo}{Wen}, which first romanizes the Chinese characters based on their pronunciations, and then constructs the vocabulary with the romanized scripts using the unigram language model~\\cite{sentencepiece}. 2) A glyph-based tokenizer family called \\camelabr{Jie}{Zi}, which decomposes characters into combinations of Chinese strokes or radicals, and then constructs the vocabulary with the stroke or radical sequences using the unigram language model. 3) A word segmented tokenizer family, which first uses a Chinese word segmenter to segment Chinese texts into words, and then constructs the vocabulary with the segmented word sequences using the unigram language model.\n\nWe pretrain BERT-style \\plm s using the proposed tokenizers from scratch and evaluate the resultant models on various downstream tasks. Through comprehensive evaluation on ten Chinese NLU tasks, we find that our pronunciation-based (\\camelabr{Shuo}{Wen}) and glyph-based (\\camelabr{Jie}{Zi}) tokenizers outperform conventional single-character tokenizers in most tasks. Furthermore, as they have the unique advantage to learn the meanings of complex characters through the composition of simpler sub-characters, they are naturally more robust on handling noisy input. \n %\nSurprisingly, we find that Chinese Word Segmentation (\\cws) has no benefit for Chinese language model pretraining. \n\nOur work suggests that linguistically informed techniques based on the characteristics of different languages need more attention. We will release the code, pretrained models, and the \\camelabr{Shuo}{Wen}-\\camelabr{Jie}{Zi}~tokenizers to serve as a better foundation for future research on Chinese \\plm .\n\n\n\n\\section{Related Work}\n\\textbf{Chinese \\plm}. Several previous works have explored techniques to improve Chinese language model pretraining. \\citet{MVP-BERT} and~\\citet{AMBERT} expanded BERT vocabulary with Chinese words apart from the single characters and incorporated them in the pretraining objectives. \\citet{ERNIE-GRAM} and~\\citet{WWM} considered coarse-grained information through masking n-gram and whole words during the masked language modeling pretraining. \\citet{ZEN} incorporated word-level information via superimposing the character and word embeddings.~\\citet{Lattice-BERT} incorporated Chinese word lattice structures in the pretraining. \n  \n\\noindent \\textbf{Linguistically Informed Techniques for Chinese}. \\cws{} is a common preprocessing step for Chinese NLP tasks~\\cite{THULAC}. \\citet{CWSnecessary} empirically analysed whether \\cws{} is helpful for downstream Chinese NLP tasks before the \\plm{} era and found that in many cases the answer is negative. We examine the impact of \\cws{} for \\plm{} instead.~\\citet{glyce} incorporated glyph information of Chinese characters though adding extra encoders to encode the images of Chinese characters and then combine them with the character embeddings. We do not intend to fuse in additional information from sources like images, but instead, \nall of our proposed tokenzation methods are drop-in replacements to the existing single-character tokenizers, without adding any extra layers or parameters. ~\\citet{wubiNMT} explore to Chinese text into Wubi sequences that represent character glyph information for the task of machine translation.\n \n\n\n\n\n\\section{Method}\n\nIn this section, we introduce our proposed tokenization methods.\n\n\n\\subsection{\\camelabr{Shuo}{Wen}: Pronunciation-based Tokenizers}\n\nThe Chinese writing system is morphemic~\\cite{hill2016typology} and barely convey phonological information. However, the pronunciation of Chinese characters also reveals semantic patterns~\\cite{phonology} and has long been widely used as input methods in China (\\textit{e.g., pinyin}).\nIn order to capture such information, we propose a pronunciation-based tokenizer named \\camelabr{Shuo}{Wen}{}.\n\nOn raw Chinese input texts (\\textit{e.g., \u9b51\u9b45\u9b4d\u9b49}), \\camelabr{Shuo}{Wen}{} performs the following steps:\n\n\\begin{enumerate}\n  \\item Romanize the text using Chinese transliteration systems. In this work, we explore two different transliteration methods: \\textit{pinyin} and \\textit{zhuyin (i.e., bopomofo)}. \\textit{Pinyin} uses the Latin alphabet and four~\\footnote{The light tone is sometimes considered as the fifth tone but we omit it for simplicity.} different tones (\\={}, \\'{}, \\v{}, \\`{}) to romanize pronunciations of characters, \\textit{e.g., \u9b51\u9b45\u9b4d\u9b49} $\\rightarrow$ \\textit{Chi\\={} Mei\\`{} Wang\\v{} Liang\\v{}}. On the other hand, \\textit{zhuyin} uses a set of self-invented characters and the same four tones to romanize the characters, \\textit{e.g., \u9b51\u9b45\u9b4d\u9b49} $\\rightarrow$ \\textit{\u3114 \u3107\u311f\\`{} \u3128\u3124\\v{} \u310c\u3127\u3124\\v{}}. Note that in \\textit{zhuyin}, the first tone mark (\\={}) is usually omitted.\n  \n  \\item Insert special separation symbols (\\textit{+}) after each character's romanized sequence, \\textit{e.g., Chi\\={}+Mei\\`{}+Wang\\v{}+Liang\\v{}+, \u3114+\u3107\u311f\\`{}+\u3128\u3124\\v{}+\u310c\u3127\u3124\\v{}+}. This prevents cases where romanized sequences of different characters are mixed together, especially when there are no tone markers to split them in \\textit{zhuyin}.\n  \n  \\item Different Chinese characters often have the same pronunciation. For disambiguation, we append different indices after the romanized sequences for the homophonic characters, so that allowing a biunique mapping between each Chinese character and its romanized sequence, \\textit{e.g., Chi\\={}33+Mei\\`{}24+Wang\\v{}25+Liang\\v{}13+, \u311410+\u3107\u311f\\`{}3+\u3128\u3124\\v{}6+\u310c\u3127\u3124\\v{}1+}. \n  \n  \\item Apply a unigram language model (\\ulm{}) as in \\citet{sentencepiece} on the romanized sequences to build the final vocabulary. \n  \n\\end{enumerate}\n\nWe do not set any constraint on the vocabulary other than the vocabulary size. The resultant vocabulary contains tokens corresponding to flexible combinations of characters and sub-characters.\n\n\n\n \n\\subsection{\\camelabr{Jie}{Zi}: Glyph-based Tokenizers}\n\nThe word shapes of Chinese characters contain rich semantic information and can help NLP models~\\cite{cw2vec}. For example, most Chinese characters can be broken down into semantically meaningful radicals. Characters that share common radicals often share related semantic information, such as the four characters \\textit{`\u9b51\u9b45\u9b4d\u9b49'} all share the same radical \\textit{`\u9b3c'} (meaning ghost), and their meanings are indeed related to ghosts and monsters.~\\footnote{Interestingly, the word \\textit{`\u9b51\u9b45\u9b4d\u9b49'} is in fact a Chinese idiom, which is now often used to refer to bad people who are like monsters.} \n\n\nHowever, the prevailing tokenization method for Chinese treats each Chinese character as a separate token and hence preventing the model to learn the shared semantics of characters with common radicals. In order to solve this problem, we propose the glyph-based tokenizer \\camelabr{Jie}{Zi}{}, which performs the following steps on raw Chinese input (\\textit{e.g., \u9b51\u9b45\u9b4d\u9b49}):\n\n\\begin{enumerate}\n \\item Convert each character into a stroke or radical sequence. To convert into stroke sequences, we use Latin alphabet to represent the basic strokes and convert the characters based on the standard stroke orders\\footnote{\\url{https:\/\/en.wikipedia.org\/wiki\/Stroke_order}}, \\textit{e.g., \u9b51} $\\rightarrow$ \\textit{\\underline{pszhshpzzn}nhpnzsszshn}; \\textit{\u9b45} $\\rightarrow$ \\textit{\\underline{pszhshpzzn}hhspn}. To convert into radical sequences, we adopt three existing glyph-based Chinese input methods: \\textit{Wubi, Zhengma, Cangjie}. These methods group strokes together in different ways to form radicals or stroke combinations, and then represent characters with them. We use Latin alphabet to represent these radicals or stroke combinations, \\textit{e.g., \u9b51\u9b45\u9b4d\u9b49} $\\rightarrow$ \\textit{Wubi: \\underline{rqc}c \\underline{rqc}i \\underline{rqc}n \\underline{rqc}w}; \\textit{Zhengma: \\underline{nj}lz \\underline{nj}bk \\underline{nj}ld \\underline{nj}oo}; \\textit{Cangjie: \\underline{hi}yub \\underline{hi}jd \\underline{hi}btv \\underline{hi}mob}.\n \n \\item Similar to the pronunciation-based tokenizers, we add the same separation symbol after each character, and also add the disambiguation indices for characters whose stroke sequences are identical (\\textit{e.g., \u4eba} (people) and \\textit{\u516b} (eight)). \n\\end{enumerate} \n\nIn the converted sequences, we can see how common radicals naturally appear (the \\underline{underlined} parts). Please refer to the Appendix~\\ref{app:inputmethod} for a detailed explanation on the differences between these different input methods involved.\n\n\n\\begin{table*}[t]\n    \\centering\n    \\small {\n    \\begin{tabular}{ lcccc|cccc }\n        \\toprule Dataset & Task & MaxLen & Batch & Epoch & \\#Train & \\#Dev & \\#Test & Domain \\\\\n        \\midrule\n            TNEWS    & DC  & 256 & 32 & 6 & 53.4K   & 10K   & 10K   & News \\\\\n            IFLYTEK  & DC  & 256 & 32 & 6 & 12.1K  & 2.6K  & 2.6K   & App Description \\\\\n            BQ       & SPM  & 256 & 32 & 6 & 100K   & 10K   & 10K    & Bank Service \\\\\n            THUCNEWS & DC  & 256 & 32 & 6 & 669K & 83.6K & 83.6K  & News \\\\\n            CLUEWSC  & WSC & 256 & 32 & 24 & 1.2K & 0.3K   & 0.3K   & Literature \\\\\n            AFQMC    & SPM  & 256 & 32 & 6 & 34.3K  & 4.3K  & 3.9K   & Financial  \\\\\n            CSL      & SPM & 256 & 32 & 6 & 20K    & 3K    & 3K     & Academic Papers \\\\\n            OCNLI    & SPM & 256 & 32 & 6 & 45.4K  & 5K    & 3K    & Mixed  \\\\\n            CHID     & MRC & 96  & 24 & 6 & 519K & 57.8K & 23K   & Mixed  \\\\\n            C3       & MRC & 512 & 24 & 6 & 12K  & 3.8K  & 3.9K  & Mixed  \\\\\n        \\bottomrule\n    \\end{tabular}}\n    \\caption{Hyper-parameters and statistics of different datasets. DC: document classification. SPM: sentence pair matching (including natural language inference). WSC: Winograd Schema Challenge. MRC: machine reading comprehension.} \n    \\label{tab:hyper_parameters}\n\\end{table*}\n\n\n\\subsection{Word Segmented Tokenizer}\n\nChinese Word Segmentation (\\cws{}) is a common technique to split Chinese chunks into a sequence of Chinese words. The resultant segmented words sometimes provide better granularity for downstream tasks~\\cite[\\textit{e.g.,}][]{CWS-NMT}. However, the impact of \\cws{} is unclear in the context of pretraining, especially how it interplays with statistical approaches like BPE and unigram LM. Hence, we study word segmented tokenizers that performs the following process on raw Chinese input, e.g., ``\u8fd9\u7bc7\u8bba\u6587\u6709\u610f\u601d\u3002(this paper is interesting)'' :\n\n\\begin{enumerate}\n  \\item We use a state-of-the-art segmenter THULAC~\\cite{THULAC} to segment the sentence into a sequence of words joined by spaces, \\textit{e.g., `\u8fd9\u7bc7\\textbackslash \u8bba\u6587\\textbackslash \u6709\u610f\u601d\\textbackslash  \u3002'} (We use \\textbackslash to indicate a blank space for easier reading.)\n\n  \\item We directly apply \\ulm{}~on these space-joined sequences to construct the vocabulary.\n  \n\\end{enumerate}\n\nIn other words, \\cws{} is used as a preprocessing step on training corpora when we build the vocabulary using above-mentioned methods. When we perform the actual pretraining and finetuning, we also perform \\cws{} as preprocessing before tokenization using the word segmented tokenizer. \n\n\n\n\n\\begin{table*}[t]\n\\centering\n\\small\n\\begin{adjustbox}{width=\\linewidth,center}\n\\setlength{\\tabcolsep}{3.5pt}{\n\\begin{tabular}{ lccccccccccl }\n\\toprule \n & TNEWS\n & IFLY\n & THUC\n & BQ \n & WSC\n & AFQMC\n & CSL\n & OCNLI\n & CHID\n & C3\n & AVG\n \\\\\n \\hline  \n\\multicolumn{11}{c}{\\textit{6-layer}} \\\\\n\\hline \nBERT-Chinese & 64.10 & 57.77 & 96.97 & 81.98 & 62.39 & 68.95 & 82.60 & 68.46 & 72.33 & 53.51 & 70.91 \\\\\n\\sentp{}-\\ulm{} & 64.26 & 55.44 & \\textbf{97.09} & 81.52 & 62.06 & 69.88 & 83.16 & 68.98 & 72.77 & 51.73 & 70.69 (-0.22) \\\\\n\\camelabr{Jie}{Zi}-CangJie & 63.86 & 59.51 & 97.04 & 81.59 & 63.27 & \\textbf{70.47} & 82.91 & 69.03 & 72.73 & 52.67 & \\textbf{71.31 (+0.40)} \\\\\n\\camelabr{Jie}{Zi}-Stroke & 63.81 & 58.74 & 96.87 & 81.55 & 62.94 & 69.66 & 82.44 & 68.02 & 72.21 & 53.35 & 70.96 (+0.05) \\\\\n\\camelabr{Jie}{Zi}-Zhengma & 63.96 & 58.74 & 96.99 & \\textbf{82.27} & 61.95 & 69.86 & \\textbf{83.46} & 68.56 & 72.12 & \\textbf{54.91} & 71.28 (+0.37) \\\\\n\\camelabr{Jie}{Zi}-Wubi & \\textbf{64.91} & 59.39 & 97.03 & 81.41 & 62.72 & 69.14 & 82.60 & 69.12 & 72.02 & 53.99 & 71.16 (+0.25) \\\\\n\\camelabr{Shuo}{Wen}-Pinyin & 63.58 & \\textbf{59.55} & 97.04 & 81.65 & 63.60 & 68.60 & 82.66 & 67.93 & \\textbf{72.81} & 53.02 & 71.04 (+0.13) \\\\\n\\camelabr{Shuo}{Wen}-Zhuyin & 64.11 & 59.16 & 97.01 & 81.64 & \\textbf{63.93} & 68.53 & 82.86 & \\textbf{69.39} & 71.48 & 54.59 & 71.27 (+0.36) \\\\\n\\hline \n\\multicolumn{11}{c}{\\textit{12-layer}} \\\\\n\\hline \nBERT-Chinese & 65.07 & 58.01 & 97.05 & 82.33 & 73.14 & 71.04 & 83.90 & 70.19 & 76.61 & 55.90 & 73.32 \\\\\n\\sentp{}-\\ulm{} & 65.01 & 58.98 & \\textbf{97.20} & 82.99 & \\textbf{73.36} & 70.93 & 83.45 & 70.46 & 77.28 & 57.70 & 73.74 (+0.42) \\\\\n\\camelabr{Jie}{Zi}-CangJie & 64.26 & 60.29 & 97.15 & \\textbf{83.48} & 71.16 & 71.48 & 83.68 & 71.50 & 76.82 & 57.99 & 73.78 (+0.46) \\\\\n\\camelabr{Jie}{Zi}-Stroke & \\textbf{65.11} & 59.75 & 97.09 & 82.88 & 70.72 & 71.64 & 83.63 & 70.03 & \\textbf{77.45} & \\textbf{59.68} & 73.53 (+0.21) \\\\\n\\camelabr{Jie}{Zi}-Zhengma & 64.51 & 60.78 & 97.14 & 83.15 & 72.15 & 70.76 & 83.68 & 71.22 & 76.72 & 57.49 & 73.76 (+0.44) \\\\\n\\camelabr{Jie}{Zi}-Wubi & 64.47 & 60.05 & 97.16 & 82.76 & 72.70 & \\textbf{72.00} & 83.62 & 70.77 & 76.34 & 58.31 & 73.82 (+0.50) \\\\\n\\camelabr{Shuo}{Wen}-Pinyin & 64.50 & \\textbf{60.40} & 97.17 & 83.13 & 70.18 & 71.37 & \\textbf{84.12} & \\textbf{71.97} & 76.11 & 58.05 & 73.70 (+0.38) \\\\\n\\camelabr{Shuo}{Wen}-Zhuyin & 64.50 & 59.98 & 97.09 & 82.99 & 73.03 & 71.83 & 83.82 & 71.74 & 76.74 & 57.23 & \\textbf{73.90 (+0.58)} \\\\\n\\bottomrule\n\\end{tabular}}\n\\end{adjustbox}\n\\caption{Results for standard evaluation. Best result on each dataset of each model size is \\textbf{boldfaced}. The numbers in brackets in the last column indicate the average difference compared to the BERT-Chinese baseline.}\n\\label{tab:main_results}\n\\end{table*}\n\n\n\\begin{table*}[t]\n\\centering\n\\small\n\\begin{adjustbox}{width=\\linewidth,center}\n\\addtolength{\\tabcolsep}{-2pt}\n\\setlength{\\tabcolsep}{2mm}{\n\\begin{tabular}{ lcccccccl }\n\\toprule \n & TNEWS\n & IFLYTEK\n & CLUEWSC\n & AFQMC\n & CSL\n & OCNLI\n & C3\n & AVG\n \\\\\n \\hline  \n\\sentp{}-\\ulm{} & 64.26 & 55.44 & 62.06 & 69.88 & 83.16 & 68.98 & 51.73 & 65.07 \\\\\n\\sentp{}-\\ulm{} + \\cws{} & 64.26 & 54.15 & 63.05 & 69.62 & 82.87 & 68.64 &  51.77 & 64.91 \\\\\n\\camelabr{Jie}{Zi}{}-Wubi & 64.91 & 59.39 & 62.72 & 69.14 & 82.60 & 69.12 & 53.99 & 65.98 \\\\\n\\camelabr{Jie}{Zi}{}-Wubi + \\cws{} & 63.66 & 59.22 & 63.16 & 68.65 & 82.21 & 68.81 & 52.76 & 65.50 \\\\\n\\camelabr{Shuo}{Wen}{}-Zhuyin & 64.11 & 59.16 & 63.93 & 68.53  & 82.86 & 69.39 & 54.59 & 66.08 \\\\\n\\camelabr{Shuo}{Wen}{}-Zhuyin + \\cws{} & 63.37 & 57.24 & 62.83 & 68.94 & 82.12 & 68.69 & 51.48 & 64.95 \\\\\n\\bottomrule\n\\end{tabular}}\n\\end{adjustbox}\n\\caption{Results of models trained with Word Segmented Tokenization. All models are 6-layer. }\n\\label{tab:cws_results}\n\\end{table*}\n\n\n\n\n\\section{Experiment}\n\n\\subsection{Baselines}\n\nWe use two strong baseline tokenizers in this work. The first one is the conventional single-character tokenizer as used in BERT-Chinese and many other follow-up Chinese \\plm{}~\\cite[\\textit{e.g.,}][]{WWM,MacBERT}. We name this tokenizer BERT-Chinese as it originated from the Chinese version of BERT.\n\nFor the second baseline, we directly apply SentencePiece with unigram LM on the raw Chinese corpus to generate the vocabulary. \nAs a result, the vocabulary contains both single characters and words (\\textit{i.e.,} character combinations). \nThis approach resembles the vocabulary of some recent multi-granularity Chinese \\plm{} variants such as AMBERT~\\cite{AMBERT} and Lattice-BERT~\\cite{Lattice-BERT}. Unlike them, we do not add any new model designs or pretraining objectives, but instead use the original BERT architecture and masked LM objective. We name this baseline \\sentp{}-\\ulm{}.\n\nTo ensure a fair comparison, we set the same vocabulary size of 22675 for all tokenizers. \nWe use the same training corpus to train all the tokenizers. We use the SentencePiece library's unigram LM implementation to train the tokenizers.\n\nIn order to evaluate the effectiveness of the tokenizers, we pretrain a BERT model using each tokenizer and compare their performance on downstream tasks. \nWhen pretraining the BERT models using each tokenizer, we use the same pretraining corpus and the same set of hyper-parameters for all models being compared. \nNotably, we also re-pretrain the BERT model using the BERT-Chinese tokenizer on our pretraining corpus instead of just loading from existing checkpoints to ensure that all baselines and proposed methods are directly comparable.\nSince our proposed tokenizers are direct drop-in replacements for the baseline tokenizers, they do not incur any extra parameters. As a result, all the models being compared have the same number of parameters, allowing for a truly apple-to-apple comparison.\n\n\\subsection{Datasets}\n\n\nWe evaluate the trained models with different tokenization methods on a total of ten different downstream datasets, including single-sentence tasks, sentence-pair tasks, as well as reading comprehension tasks. We briefly introduce each dataset below and present the dataset statistics in Table~\\ref{tab:hyper_parameters}.\n\n\\noindent \\textbf{TNEWS}~\\cite{TNEWS} is a news title  classification dataset containing 15 classes. We use the split as released in \\citet{CLUEBenchmark}.\n\n\\noindent \\textbf{IFLYTEK}~\\cite{IFLYTEK} is a long text classification dataset containing 119 classes. The task is to classify mobile applications into corresponding categories given their description.\n\n\\noindent \\textbf{BQ}~\\cite{BQ} is a sentence-pair question matching dataset extracted from an online bank customer service log. The goal is to evaluate whether two questions are semantically identical or can be answered by the same answer. \n\n\\noindent \\textbf{THUCNEWS}~\\cite{THUCNEWS} is a document classification dataset with 14 classes. The task is to classify news into the corresponding categories given their title and content.\n\n\\noindent \\textbf{CLUEWSC}~\\cite{CLUEBenchmark} is a coreference resolution dataset in the format of Winograd Schema Challenge. The task is to determine whether the given noun and pronoun in the sentence co-refer.\n\n\\noindent \\textbf{AFQMC} is the Ant Financial Question Matching Corpus for the question matching task that aims to predict whether two sentences are semantically similar\n\n\\noindent \\textbf{CSL} is the Chinese Scientific Literature dataset extracted from academic papers. Given an abstract and some keywords, the goal is to determine whether they belong to the same paper. It is formatted as a sentence-pair matching task.\n\n\\noindent \\textbf{OCNLI}~\\cite{OCNLI} is a natural language inference dataset. The task is to determine whether the relationship between the hypothesis and premise is entailment, neural, or contradiction.\n\n\\noindent \\textbf{CHID}~\\cite{CHID} is a  cloze-style reading comprehension dataset where . Given contexts where some idioms are masked, the task is to select the appropriate idiom from a list of candidates.\n\n\\noindent \\textbf{C3}~\\cite{C3} is a multiple choice machine reading comprehension dataset. The goal is to choose the correct answer for some questions given a context. \n\n\n\n\n\\begin{table*}[t]\n\\centering\n\\small\n\\addtolength{\\tabcolsep}{-2pt}\n\\setlength{\\tabcolsep}{3mm}{\n\\begin{tabular}{ lccccc }\n\\toprule \n & clean\n & 15\\%\n & 30\\%\n & 45\\% \n & 60\\%\n \\\\\n \\hline  \n\\multicolumn{6}{c}{TNEWS} \\\\\n\\hline \nBERT-Chinese & 63.99 & 62.18 & 60.49 & 57.64 & 53.97 \\\\\n\\sentp{}-\\ulm{} & 63.99 & 62.88 & 60.79 & 59.20 & 55.42 \\\\\n\\camelabr{Jie}{Zi}-Wubi & 64.05 & 62.80 & 62.56 & 62.71 & 62.81 \\\\\n\\hline \n\\multicolumn{6}{c}{OCNLI} \\\\\n\\hline \nBERT-Chinese & 68.07 & 62.60 & 56.83 & 51.37 & 46.00 \\\\\n\\sentp{}-\\ulm{} & 69.10 & 64.00 & 56.57 & 52.43 & 46.73 \\\\\n\\camelabr{Jie}{Zi}-Wubi & 68.43 & 67.10 & 65.47 & 65.40 & 64.80 \\\\\n\\bottomrule\n\\end{tabular}}\n\\caption{Results for noisy evaluation with glyph noises.}\n\\label{tab:noisy_results_glyph}\n\\end{table*}\n\n\n\\begin{table*}[t]\n\\centering\n\\small\n\\addtolength{\\tabcolsep}{-2pt}\n\\setlength{\\tabcolsep}{3mm}{\n\\begin{tabular}{ lccccc }\n\\toprule \n & clean\n & 15\\%\n & 30\\%\n & 45\\% \n & 60\\%\n \\\\\n \\hline  \n\\multicolumn{6}{c}{TNEWS} \\\\\n\\hline \nBERT-Chinese & 63.99 & 60.87 & 57.70 & 52.21 & 45.51 \\\\\n\\sentp{}-\\ulm{} & 63.99 & 61.52 & 58.60 & 53.30 & 46.81 \\\\\n\\camelabr{Shuo}{Wen}-Pinyin & 63.35 & 60.60 & 57.45 & 53.61 & 49.29 \\\\\n\\camelabr{Shuo}{Wen}-Zhuyin & 63.79 & 60.99 & 57.69 & 53.15 & 48.98 \\\\\n\\hline \n\\multicolumn{6}{c}{OCNLI} \\\\\n\\hline \nBERT-Chinese & 68.07 & 61.73 & 54.50 & 49.97 & 44.80 \\\\\n\\sentp{}-\\ulm{} & 69.10 & 62.23 & 54.77 & 50.33 & 45.70 \\\\\n\\camelabr{Shuo}{Wen}-Pinyin & 67.83 & 61.00 & 54.33 & 49.87 & 45.10 \\\\\n\\camelabr{Shuo}{Wen}-Zhuyin & 69.63 & 60.77 & 54.77 & 50.67 & 47.70 \\\\\n\\bottomrule\n\\end{tabular}}\n\\caption{Results for noisy evaluation with phonology noises.}\n\\label{tab:noisy_results_phonology}\n\\end{table*}\n\n\n\n\\subsection{Experiment Setup: Standard Evaluation}\n\n\nFor each tokenizer, we pretrain a 6-layer and 12-layer BERT style model using the Baidu Baike corpus~\\cite{CPM} which has 2.2G of raw text before processing. \nThe model configuration is exactly the same for all models: 6 or 12 layers, 12 attention heads, intermediate size 3072, hidden size 768.\nFor pretraining, we follow the original BERT paper's two-stage pretraining procedure where we first train with sequence length 128 for 8k steps and then train with sequence length 512 for 4k steps.  We only keep the masked language modeling objective in pretraining and discard the next sentence prediction objective as suggested in RoBERTa~\\cite{RoBERTa}.\nDuring fine-tuning, we use the set of hyper-parameters as shown in Table~\\ref{tab:hyper_parameters}.\nFor all experiments in this paper, we report results of the average run of three different random seeds.\n\n\\subsection{Experiment Setup: Noisy Evaluation}\n\nApart from evaluating on the standard benchmarks, we also evaluate in a noisy setting to illustrate the advantage of our proposed tokenization methods to handle noisy inputs.\nSpecifically, we inject two types of synthetic noises into both the training and test data in order to test whether the models can learn from noisy training data and also perform robustly on noisy test data. \nWe vary the ratio of noise in the data to examine the impact. The two types of noises we inject are:\n\n\\begin{itemize}\n  \\item Glyph-based noise: we replace original characters with other characters that have similar glyph but have different semantic meanings (\\textit{e.g., \u58c1} (wall) and \\textit{\u74a7} (jade)). Specifically, we obtain a substitution candidate list for each character, where the candidates are selected so that they share at least one common radical with the original character. Then, we randomly sample a certain ratio $r\\%$ of the original characters, for each of them, we randomly sample a substitution character from its candidate list for substitution.\nThis simulates common noises when people use glyph-based input methods where these similar characters could be chosen since their input encoding are similar. \n\n  \\item Pronunciation-based noise: we replace original characters with other characters that have the same pronunciation but different semantic meanings (\\textit{e.g., \u771f} (real) and \\textit{\u9488} (needle)).\n  Specifically, we obtain a substitution candidate list for each character, where all the candidates have the same pronunciation as the original character. Then, similarly, we randomly sample a certain ratio $r\\%$ of the original characters, for each of them, we randomly sample a substitution character from its candidate list for substitution.\n  This simulates the common noise when users use pronunciation-based input methods where the input encoding of these characters and their substitutions are the same.\n\\end{itemize}\n\nFor our experiment, we vary the noise ratio $r\\%$ within the range of \\{$0$, $15\\%$, $30\\%$, $45\\%$ $60\\%$\\}. For the sampled characters to be replaced with noises, we randomly sample a substitution character from their candidate list for every appearance of the character, instead of substituting with the same candidate. This induces more noise variations. Note that some substitutions are both glyph-based noises and pronunciation-based noises (\\textit{e.g., \u5feb} and \\textit{\u5757} share a same radical and also have the same pronunciation), we keep them in both types of noises.\n\nIntuitively, our \\camelabr{Shuo}{Wen}{} tokenizer could be robust to pronunciation-based noises and \\camelabr{Jie}{Zi}{} tokenizer could be robust to glyph-based noises because the substitution characters share similar pronunciation or glyph components with the original characters, which may be captured by our tokenizers. \n\nThis noisy setup is reflective of real-life use cases where user queries often contain such noises. Since most Chinese people use either glyph-based input methods (\\textit{e.g., wubi}) or pronunciation-based input methods (\\textit{e.g., pinyin, zhuyin}), such mis-typed characters can be very common. This highlights the potential impact of our work.\n \n\n\\section{Results}\n\n\\subsection{Standard Evaluation}\nWe compare the results of the baseline tokenizers (BERT-Chinese, \\sentp{}-\\ulm{}) with our proposed \\camelabr{Jie}{Zi}{} (including four variants: \\camelabr{Jie}{Zi}{}-Cangjie, \\camelabr{Jie}{Zi}{}-Stroke, \\camelabr{Jie}{Zi}{}-Zhengma, \\camelabr{Jie}{Zi}{}-Wubi) and \\camelabr{Shuo}{Wen}{} (including two variants: \\camelabr{Shuo}{Wen}{}-Pinyin and \\camelabr{Shuo}{Wen}{}-Zhuyin) tokenizers in Table~\\ref{tab:main_results}. \n\nDespite some variations across different datasets, we observe that in terms of the average score over ten datasets, all of our proposed tokenizers outperform the BERT-Chinese baseline. Notably, for the 6-layer model size, our \\camelabr{Jie}{Zi}{}-Cangjie tokenizer obtains an average of 0.40 points over the BERT-Chinese tokenizer, for the 12-layer model size, our \\camelabr{Shuo}{Wen}{}-Zhuyin tokenizer achieves an average of 0.58 points of improvement over the BERT-Chinese tokenizer.\n\nThese results indicate that on standard benchmarks, our proposed tokenizers can match or outperform the existing tokenizers for Chinese. \n\nOn the other hand, we examine the impact of \\cws{} by comparing three tokenizers with their word segmented counterparts in Table~\\ref{tab:cws_results}. We can see that adding \\cws{} as a preprocessing actually slightly decreased the average performance on downstream tasks. \n\n\\subsection{Noisy Evaluation}\n\nWe perform noisy evaluation on two datasets: TNEWS and OCNLI. For glyph-based noises, we compare baselines BERT-Chinese and \\sentp{}-\\ulm{} with our \\camelabr{Jie}{Zi}{}-Wubi. The results are presented in Table~\\ref{tab:noisy_results_glyph}. We observe that when the noise ratio increases, the advantage of \\camelabr{Jie}{Zi}{} is particularly large. For example, when $60\\%$ characters are substituted, \\camelabr{Jie}{Zi}{}-Wubi still performs close to the original performance, while other baselines suffer large drops in performance. On OCNLI, the gap can be as large as 18 points in accuracy. \n\nFor pronunciation-based noises, we compare baselines BERT-Chinese and \\sentp{}-\\ulm{} with our \\camelabr{Shuo}{Wen}{}-Pinyin and \\camelabr{Shuo}{Wen}-Zhuyin. The results are shown in Table~\\ref{tab:noisy_results_phonology}. Unlike the cases on glyph-based noises, we observe that the advantage of our \\camelabr{Shuo}{Wen}{} tokenizers are not so significant compared to the baselines. One potential reason is that there many Chinese characters with the same pronunciation. Unlike how the semantic meanings of radicals can be rather consistent across different characters, phoneme combinations can have vastly different meanings across different characters (\\textit{i.e.,} characters may pronounce the same but have totally different semantic meanings), which makes it difficult to learn these different semantic meanings in the pronunciation-based token embedding. \n\n\\section{Conclusion}\n\nIn this paper, we have explored three linguistically informed tokenization methods motivated by the unique linguistic characteristics of the Chinese writing system. Specifically, we find that pronunciation-based and glyph-based tokenizers can match or outperform the conventional Chinese tokenizers and Chinese word segmentation is not a useful addition for the tokenizer.\nMoreover, we find that our glyph-based tokenizers achieve large gains on noisy input as compared to the baselines, while our pronunciation-based tokenizers obtain limited success. This highlights the potential advantage of our proposed methods in real-life scenarios with noisy data.\nWe believe that our work sets an important example of exploiting the unique linguistic property of a language beyond English to develop more tailored techniques, which should be an important direction for the global NLP community. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Methodological approach}\n\nGiven two probability distributions, $1$ and $2$,\nwith $1$ nested within $2$,\nthe likelihood ratio test evaluates $\\hat L_{2}\/\\hat L_{1}$,\nwhere $\\hat L_{2}$ is the likelihood (at maximum)\nof the ``bigger'' or ``full'' model (either Tap or TrG)\nand $\\hat L_{1}$ corresponds to the nested or null model\n(power law in our case). Taking logarithms we get\nthe log-likelihood-ratio\n\\[\n\\mathcal{R}=\\ln \\frac{\\hat L_2}{\\hat L_1}= l_{2}-l_{1},\n\\]\nwith $ l_{j}=\\ln \\hat L_j=\\sum_{i=1}^{N} \\ln f_{j}(M_i;\\hat{\\Theta}_j) $,\nwhere $f_{i}$ denotes the probability density function of the distribution $j$\nfor every $j=1,2$, and the MLE corresponds to $\\hat \\Theta_1=\\hat\n\\beta_1$ and $\\hat\\Theta_2=\\{\\hat\\beta_2,\\hat\\theta_2\\}$. In order\nto compare the fit provided by the two distributions, it is\nnecessary to characterize the distribution of $\\mathcal{R}$.\n\nLet $n_1$ and $n_2$ be the number of free parameters in the models\n$1$ and $2$, respectively.\nIn general, if the models are nested,\nand under the null\nhypothesis that the data comes from the simpler model, the\nprobability distribution of the statistic $2 \\mathcal{R}$ in the limit\n$N\\rightarrow \\infty$ is\na chi-squared distribution\nwith degrees of freedom equal to $n_2-n_1 > 0$.\nSo,\n\\begin{equation}\n2 \\mathcal{R}>3.84\n\\end{equation}\nwith a level of risk equal to 0.05.\n{Note that the chi-squared distribution\nprovides a penalty for\n``model complexity''\nas the {``width''} of the distribution is given directly by\nthe number of the degrees of freedom.}\nThis likelihood ratio test\nconstitutes the best option to choose among models 1 and 2, in the\nsense that it has a convergence to its asymptotic distribution\nfaster\nthan any other test \\cite{mccullagh1986}.\nThe null and\nalternative hypotheses correspond to accept model $1$ or $2$,\nrespectively, although the acceptance of model $1$ does not imply the rejection\nof $2$, it is simply that\nthe ``full'' model $2$\ndoes not bring any significant improvement with respect the simpler\nmodel $1$.\n\n{However, when the nesting of distribution $1$ within $2$ takes place\nin such a way that the space of parameters of the former one\nlies within a boundary of the space of parameters of distribution $2$,\nthe approach just explained for\nthe asymptotic distribution of $2 \\mathcal{R}$\nis not valid \\cite{Self1987,Geyer1994}.\nThis is the case when testing both the Tap or the TrG distributions\nin front of the power-law distribution,\nas the $\\theta \\rightarrow \\infty$ limit of the latter\ncorresponds to the boundary of the parameter space of the two other distributions.\nBut the asymptotic theory of Refs. \\cite{Self1987,Geyer1994}\nis also invalid, as the power-law distribution lacks\nthe necessary regularity conditions,\ndue to the divergence of their moments.\nThis illustrates part of the difficulties\nof performing proper model selection when\nfractal-like distributions are involved\n\\cite{Kagan_calcutta}.\nIn order to obtain the distribution of $2\\mathcal{R}$ and from there\nthe $p-$values of the LR tests,\nwe are left to the simulation of the null hypothesis.\nWe advance that the results seem to indicate that the distribution of $2\\mathcal{R}$,\nfor high percentiles, is chi-square with one degree of freedom,\nbut we lack a theoretical support for this fact.}\n\nLet us proceed, using this {method},\nby comparing the performance of the power-law and Tap fits\nwhen applied to the global shallow seismic activity,\nfor time windows starting always in 1977 and ending in the successive times\nindexed by the abscissa in Fig. \\ref{graf:densities}(a)\n(as in Ref. \\cite{Bell_Naylor_Main}).\nThe log-likelihood-ratio of these fits (times 2),\nis shown in the figure\ntogether with the critical region of the test. In agreement with\nBell {\\it et al.} \\cite{Bell_Naylor_Main}, we find that: (i) the\npower-law fit can be safely rejected in front of the Tap\ndistribution for any time window ending between 1980 and before\n2004; and (ii) the results change drastically after the occurrence\nof the great 2004 Sumatra earthquake,\n{for which the power law cannot be rejected at the 0.05 level.\nSo, for parsimony reasons, the power law becomes preferable in front of the\nTap distribution for time windows\nending later than 2004.\nThe fact that the Tap distribution cannot be distinguished from the power law\nis also in agreement with previous results showing that the contour lines\nin the likelihood maps of the Tap distribution are highly non-symmetric\nand may be unbounded for smaller levels of risk \\cite{Kagan_Schoenberg,Kagan_gji02,Geist_Parsons_NH}.\n}\n\n\n\\begin{figure}[t!]\n\\begin{center}\n\\centerline{\\includegraphics[width=9cm]{explore_earthquake_figura2_resumlikeli_10000.pdf}}\n\\caption{ Results of likelihood ratio tests for nested and\nnon-nested models. The points denote the value of the statistic $2\n\\mathcal{R}$ or $\\mathcal{R}$ (depending on the test) and the dashed\nlines show the critical value of the corresponding test {(at level\n0.05)}. {For the nested case, boxplots show the distribution of\n$2\\mathcal{R}$ for 10000 simulations of the power-law null\nhypothesis, from which the critical value is computed.} The abscissa\ncorresponds to the ending point of a time window starting always in\n1 Jan 1977. Note that the year is considered a continuous variable\n(not a categorical variable), so, the time window ending on 31 Dec\n2004 takes value $2004.99\\dots \\simeq 2005$. (a) Tap distribution\nversus power law. (b) Truncated gamma versus power law. (c)\nTruncated gamma versus Tap {(non-nested case)}.\n\\label{graf:densities}}\n\\end{center}\n\\end{figure}\n\nWhen we compare the power-law fit with the truncated gamma, using\nthe same test, for the same data, the results are more\nsignificant, see Fig. \\ref{graf:densities}(b). The situation\nprevious to 2004 is the same, with an extremely poor performance of\nthe power law; but after 2004, despite a big jump again in the value\nof the likelihood ratio, the power law continues as being\nnon-acceptable, at the 0.05 level.\nIt is only after the great Tohoku earthquake of 2011\nthat the $p-$value of the test {enters into} the acceptance region,\nbut keeping values not far from the 0.05 limit.\nFrom here we\nconclude that, in order to find an alternative to the power-law\ndistribution, the truncated gamma distribution is a better\noption than the Tap distribution, as it is more clearly distinguishable\nfrom the power law (for this particular data).\nNevertheless, a comparison between\nthese two distributions (Tap and TrG) seems pertinent.\n\n\n\n\n\nWhen the models are not nested, as it happens if we want a direct\ncomparison between the Tap and the TrG distributions,\nthe procedure we use\nis the likelihood ratio test of Vuong for non-nested models \\cite{Vuong,Clauset}.\nIn this case the critical values\ndepend on the sample size, $N$,\nturning out to be that, when $N$ is large, $\\mathcal{R}$ is normally\ndistributed with standard deviation $s \\sqrt N$,\nwhere $s$ denotes the standard deviation of the set\n$$\n\\{\\ln f_{trg}(M_i;\\hat{\\beta}_{trg},\\hat{\\theta}_{trg})-\\ln\nf_{tap}(M_i;\\hat{\\beta}_{tap},\\hat{\\theta}_{tap})\\}\n$$\nfor $i=1,\\dots, N$ and $\\mathcal{R}= l_{trg}-l_{tap}$.\nThen, we accept that there exists a\nsignificant difference between the models if\n\\begin{equation}\\label{reg}\n|\\mathcal{R}|>\\:1.96 s \\sqrt{N}\n\\end{equation}\nat a level of risk equal to 0.05,\nwith the model with larger log-likelihood being the preferred one.\nThe critical value of the test arises because\nthe null hypothesis is that the mean value of $\\mathcal{R}$ is zero (i.e.,\nboth models are equally close to the true distribution).\nNote that the alternative hypothesis corresponds to accept that the\ndifference between the fit provided by the models is significant. As\nthe number of parameters is the same for the Tap and TrG models,\ntheir log-likelihood-ratio coincides with the difference in BIC or\nAIC, {but,} as mentioned above, the LR test incorporates\na statistical test which specifies the distribution\nof the statistic under consideration.\n\n\nFigure \\ref{graf:densities}(c) shows the evolution of the\nlog-likelihood-ratio between the two models, for different time windows (starting always in\n1977), together with the critical region of the test given by the\nEq. (\\ref{reg}). One can see how the fits provided by the Tap and TrG\ndistributions do not exhibit significant difference, although the\nTrG provides, in general, slightly higher likelihoods. After the\nmega-event in 2004 the performance of the TrG fit improves,\napproaching the limit of significance.\n{This reinforces our conclusion that the TrG distribution is preferred\nin front of the Tap and power-law distributions.}\n\nIn order to gain further insight, we simulate\nrandom samples following the truncated gamma distribution, with the\nparameters $\\hat \\beta_{trg}$ and $\\hat \\theta_{trg}$ obtained from\nML estimation of the complete dataset (Table \\ref{taula}), with the\nsame truncation parameter $a$ and number of points ($N=6150$) also.\nTo avoid that the conclusions depend on the time correlation of magnitudes,\nwe reshuffle the simulated data in such a way that the occurrence of\nthe order statistics is the same as for the empirical data; in other\nwords, the largest simulated event is assigned to take place at the\ntime of the 2011 Tohoku earthquake (the largest of the CMT catalog\n\\cite{Bell_Naylor_Main}), the second largest at the time of the 2004\nSumatra event, and so on. In this way, we model earthquake seismic\nmoments as\narising from a gamma distribution with fixed parameters and with\noccurrence times given by the empirical times and with the same\nseismic moment correlations\nas the empirical data, approximately.\n\n\n\\begin{figure}[t!]\n\\begin{center}\n\\centerline{\\includegraphics[trim = 0mm 130mm 0mm\n10mm,clip=true,width=9cm]{explore_earthquake_figura3_resumlikeli_1000.pdf}}\n\\caption{Comparison of the empirical log-likelihood-ratios between\nthe TrG and power law with those of 1000 simulations of the TrG\ndistribution, using the final parameters of Table \\ref{taula}\n{(i.e., $\\beta=0.681$ and $m_c=9.15$)}. Simulated seismic moments\nare reshuffled as explained in the text {to make the comparison\npossible}. Simulation results are displayed using boxplots,\nrepresenting the three quartiles of the distribution of\n$2\\mathcal{R}$.\n\\label{l_boxplots}}\n\\end{center}\n\\end{figure}\n\n\n\n\n\\begin{figure}\n\\begin{center}\n\\centerline{\\includegraphics[trim = 0mm 0mm 0mm\n60mm,clip,width=9cm]{explore_earthquake_final_3.pdf}}\n\\caption{Comparison of the values of the estimated parameters of the\nTrG distribution, $\\hat \\beta_{trg}$ and $\\hat m_{c\\,trg}$, for the\nempirical data and for 1000 simulations of the TrG distribution,\nusing the final parameters of Table \\ref{taula}. Simulated seismic\nmoments are reshuffled as explained in the text. The different\nstability of both parameters is apparent. \\label{gr4}}\n\\end{center}\n\\end{figure}\n\n\n\n\n\nWe simulate 1000 datasets with $N=6150$ each. The results, displayed\nin Fig. \\ref{l_boxplots}, show that the behavior of the empirical\ndata is not atypical in comparison with this gamma modeling. In\nnearly all time windows the empirical data lies in between the first\nand third quartile of the simulated data, although before 2004 the\nempirical values are close to the third quartile whereas after 2004\nthey lay just below the median. This leads us to compute the\nstatistics of the jump in the log-likelihood-ratio between 2004 and\n2005. The estimated probability of having a jump larger than the\nempirical value is around 4.5 \\%, which is not far from what one\ncould {accept} from the gamma modeling explained above.\n{Thus, a TrG distribution, with fixed parameters, is able\nto reproduce the empirical findings, if the peculiar time ordering of magnitude\nof the real events is taken into account.\nNotice also that, although the simulated data come from a TrG distribution,\nthey are not distinguishable from a power law for\nabout half of the simulations of the last time windows,\nas the critical region is close to the median\nindicated by the boxplots.}\n\n\n\nWe can also compare the evolution of the estimated parameters\nfor the empiral dataset and for the reshuffled TrG simulations, with a good agreement\nagain, see Fig. \\ref{gr4}. There, it is clear that although the exponent $\\beta$\nreaches very stable values relatively soon,\nthe scale parameter $\\theta$ (equivalent to $m_c$) is largely unstable,\nand the occurrence of the biggest events makes its value increase.\n\n\n\n{As a complementary control\nwe invert the situation, simulating\n1000 syntetic power-law datasets\nwith $\\beta=0.685$ (Table \\ref{taula}),\n$a= 5.3 \\cdot 10^{17}$ N$\\cdot$m,\nand $N=6150$, for which the same time reshuffling is perfomed,\nin such a way that the order of the order statistics is the same.\nIn this case, the results of the simulations lead to much smaller values\nof the log-ratio, for which the power-law distribution cannot be\nrejected (as expected), in contrast with the empirical data,\nsee Fig. \\ref{figlanueva}.}\n\n\\begin{figure}[t!]\n\\begin{center}\n\\centerline{\\includegraphics[trim = 0mm 70mm 0mm\n70mm,clip,width=9cm]{explore_earthquake_figura3_resumlikeli_1000.pdf}\n} \\caption{ As Fig. \\ref{l_boxplots}, but simulating a power law\nwith parameter $\\beta=0.685$ (Table \\ref{taula}) instead of a TrG\ndistribution. The reshuffling is also as in Fig. \\ref{l_boxplots},\nas explained in the text.\nThe\nsimulations cannot explain the large empirical values of the log-likelihood-ratio.\n\\label{figlanueva}}\n\\end{center}\n\\end{figure}\n\n\n\n\n\nIn summary, the truncated gamma distribution represents the best\nalternative to model global shallow earthquake seismic moments, in\ncomparison with the tapered GR distribution and the power law. The\npreponderance of the gamma model is maintained after the occurrence\nof the mega-earthquakes taking place from 2004\n{and it is only after the 2011 Tohoku earthquake\nthat it is difficult to decide between power law and TrG.\nWe have verified that these results are qualitatively similar if\nwe restrict our study to subduction zones, as defined by\nthe Flinn-Engdahl's regionalization \\cite{Kagan_pageoph99}, \nwith the main difference that\nthe values of $l_{trg}-l_{pl}$ become somewhat smaller\nand therefore the power-law hypothesis cannot be rejected\nafter the Tohoku earthquake.}\n\n\n{In order to reproduce the\ntime evolution of the statistical results\nit suffices that independent gamma seismic moments,\nwith fixed parameters, are reshuffled\nso that the peculiar empirical time correlations of magnitudes are maintained.}\nSo, although the scale parameter\n$\\theta$ is not stabilized, and the occurrence or not of more\nmega-earthquakes could significantly change its value\n\\cite{Zoller_grl},\n{the current value is enough to explain the\navailable data}.\nIt would be very interesting to investigate if\nthe high values of the likelihood ratio attained before the 2004\nSumatra event could be employed to detect the end of periods of low\nglobal seismic activity. Certainly, more data would be necessary {for that purpose}\n\\cite{Zoller_grl}.\n\nAs an extra argument in favor of the truncated gamma\ndistribution in front of the tapered GR,\nwe can bring not a statistical evidence but\nphysical plausibility; indeed,\nthe former distribution can be justified as coming from a\nbranching process that is slightly below its critical point\n\\cite{Christensen_Moloney,Corral_FontClos}.\n{Further reasons that may support\nthe truncated gamma\nare that\nthis arises\n(i) as the maximum entropy outcome\nunder the constrains of fixed (arithmetic) mean and fixed geometric mean of the seismic moment\n\\cite{Main_information};\n(ii) as the closest to the power law, in terms of the Kullback-Leibler ``distance'',\nwhen the mean seismic moment is fixed \\cite{Sornette_Sornette_bssa}; and\n(iii) as a stable distribution under a fragmentation process with a power-law\ntransition rate \\cite{Sornette_Sornette_bssa}.}\n\n\n\n\n\n\n\n\n\n\nWe are grateful to J. del Castillo, Y. Y. Kagan, I. G. Main, and M. Naylor for their feedback.\nResearch expenses were founded by projects\nFIS2012-31324 and MTM2012-31118 from Spanish MINECO,\n2014SGR-1307 from AGAUR, and the Collaborative Mathematics\nProject from La Caixa Foundation.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThemis is one of the most statistically reliable families in\nthe asteroid belt. First discovered by Hirayama (1918), it has been identified as a\nfamily in all subsequent works, and it has 550 members according to Zappal\\'{a} et al. (1995), and more than 5000 members as determined by Nesvorny (2012). \nThe Themis family is characterized by asteroids with 3.05 $ \\leq a \\leq $ 3.22 AU,  0.12 $\\leq e \\leq$ 0.19, in low inclination orbit $0.7^{o} \\leq i \\leq 2.22^{o}$ (Zappal\\'{a} et al. 1990). It is spectrally dominated by primitive C- and B-type asteroids, as reported by spectroscopic investigation of several members (Moth\\'e-Diniz et al. 2005; Florczak et al. 1999; de Leon et al. 2012; Kaluna et al., 2016). The family is old with an estimated age of $\\sim$ 2.5$\\pm$1.0 Gyr (Bro\\v{z} et al. 2013), and it was formed from a large-scale catastrophic disruption of a $\\sim$ 270 km (Bro\\v{z} et al. 2013) or a $\\sim$ 400 km sized parent body (Marzari et al. 1995; Durda et al. 2007). Nesvorny et al. (2008) confirmed that this family is quite old and formed more than 1 Gyr ago.\\\\\nInterestingly, Rivkin \\& Emery (2010) and Campins et al. (2010) found spectroscopic evidence of the presence of water ice and organics on (24) Themis. Rivkin and Emery (2010) concluded that its surface contains very fine water frost, probably in the form of surface grain coatings, and that the infrared spectral signatures can be fully explained by a mixture of spectrally neutral material, water ice, and organics. At the same time, Campins et al. (2010) suggested that water ice is evenly distributed over the entire Themis surface using spectra obtained at four different rotational phases. \nNevertheless the nature of the 3.1 $\\mu$m feature on (24) Themis is still a matter of debate, and Beck et al. (2011) proposed the hydrated iron oxide goethite as alternative interpretation of this feature, even if Jewitt \\& Guilbert-Lepoutre (2012) stress that goethite, when found in meteorites, is a product of aqueous alteration in the terrestrial environment and that extraterrestrial goethite in freshly fallen meteorites has not been detected. \\\\\nAbsorption bands in the visible region related to hydrated silicates have been detected on the surface of several Themis family members (Florczak et al. 1999). These materials are produced by the aqueous alteration process, that is a low temperature ($<$ 320 K) chemical alteration of materials by liquid water (Vilas \\& Sykes, 1996). The presence of hydrated minerals implies that liquid water was present on these asteroids in the past, and suggests that post-formation heating took place.\nThe  main belt comets 133P\/Elst-Pizarro, 176P\/LINEAR, 288P\/(300163) 2006 VW139, and possibly 238P\/Read may be related to the Themis family (Nesvorny et al. 2008; Hsieh et al. 2006, 2009, 2012; Haghighipour 2009) because of orbital proximity and spectral properties analogies. According to Licandro et al. (2011, 2013) works, the visible spectra of the main belt comets 133P, 176P, and 288P  are compatible with those of Themis and Beagle families members.\nIn particular Nesvorny et al. (2008) propose that 133P could potentially be one member of the Beagle sub-family within the Themis group. This sub-family is a young cluster with an estimates age $< 10$ Myr (Nesvorny et al. 2008).\n\nThe detection of water ice on asteroid (24) Themis, of hydrated silicates on several Themis members and the spectral and dynamical link between the Themis family and some main belt comets all indicate that Themis family is an important  reservoir of water in the outer main belt. These discoveries support the suggestion that at least some of Earth's current supply of water was delivered by asteroids some time following the collision that produced the Moon (Morbidelli et al. 2000, Lunine 2006). \\\\\nWith this paper we investigate both regular Themis family members (22) and younger ones (8), belonging to the Beagle sub-family, aiming to constrain their surface composition, to establish spectral links with meteorites, and to investigate potential space weathering effects and spectral variability between the two families members.\n\n\n\\section{Observations and data reduction} \n\nSpectroscopic observations in the visible and near infrared range of Themis and Beagle families members were made at the 3.56 m Italian Telescopio Nazionale Galileo (TNG) of the European Northern Observatory (ENO) in la Palma, Spain, in two runs, on 16-20 February and 16-18 December 2012.\nFor visible spectroscopy we used the Dolores (Device Optimized for the \nLOw RESolution) instrument equipped with the low resolution red (LRR) and blue (LRB) grisms.\nThe LRR grism covers the 0.52--0.95 $\\mu$m range with a spectral dispersion \nof 2.9 \\AA\/px, while the LRB grism covers the 0.38-0.80 $\\mu$m range with a spectral dispersion \nof 2.8 \\AA\/px. \nThe Dolores detector is a 2048 $\\times$ 2048 pixels Loral thinned and \nback-illuminated CCD, having a pixel size of \n15 $\\mu$m and a pixel scale of 0.275 arcsec\/px. \nThe red and blue spectra in the visible range were separately reduced and finally combined together to obtain spectral coverage from 0.38 to 0.95 $\\mu$m. Like most of the Loral CCDs, the Dolores chip is affected by moderate-to-strong fringing\nat red wavelengths. Despite taking as much care as possible in\nthe data reduction process, some asteroid spectra acquired with the LRR grism show residual\nfringing that impedes identification of absorption bands and an accurate analysis in the 0.81-0.95 $\\mu$m range when NICS observations are not available. \\\\\nFor the infrared spectroscopic investigation  we used the\nnear infrared camera and spectrometer (NICS) equipped with an Amici prism disperser. \nThis equipment covers the 0.78--2.40 $\\mu$m range during a single \nexposure with a spectral resolution of about 35 (Baffa et al. 2001). \nThe detector is a 1024 $\\times$ 1024 pixel Rockwell HgCdTe Hawaii array. \nThe spectra were acquired nodding the object\nalong the spatial direction of the slit, in order to obtain alternating pairs \n(named A and B) of near--simultaneous images for the  background removal. \nFor both the visible and near infrared observations we utilized a 2 arcsec \nwide slit, oriented along the parallactic angle to minimize \nthe effect of atmospheric differential refraction.\n\n\nVisible and near-infrared spectra were reduced using ordinary procedures \nof data reduction with the software packages Midas as described in\nFornasier et al. (2004, 2008). \nFor the visible spectra, the procedure includes the subtraction of the bias \nfrom the raw data, flat--field correction, cosmic ray removal, sky subtraction,\ncollapsing the two--dimensional spectra to one dimension, wavelength calibration,\nand atmospheric extinction correction, using La Palma atmospheric extinction coefficients.\nThe spectra were normalized at 5500 \\AA. \nThe reflectivity of each asteroid was \nobtained by dividing its spectrum by that of the solar analog star closest in\ntime and airmass to\nthe object. Spectra were finally smoothed with a median filter \ntechnique, using a box of 19 pixels in the spectral direction for each point of\nthe \nspectrum. The threshold was set to 0.1, meaning that the original value was\nreplaced by \nthe median value if the median value differs by more than 10\\% from the original\none.\n\nFor observations in the infrared range, spectra were first corrected for flat\nfielding, \nthen bias and sky subtraction\nwas performed by producing A-B and B-A frames. The positive spectrum of the B-A\nframe was \nshifted and added to the positive spectrum of \nthe A-B frame. The final spectrum is the result of the mean of all pairs of \nframes previously combined. The spectrum was extracted and wavelength \ncalibrated. Due to the very low resolution of the Amici prism, \nthe lines of Ar\/Xe lamps are blended and cannot be easily used for standard reduction procedures. \nFor this reason, the wavelength calibration was obtained using a look-up table\nwhich is  based on the theoretical dispersion predicted by ray-tracing and adjusted \nto best fit the observed spectra of calibration sources.  Finally, the extinction correction and solar removal was obtained by \ndivision of each asteroid spectrum by that of the solar analog star closest in\ntime and airmass to the object. The stellar and asteroid spectra\nwere cross-correlated and, if necessary, sub-pixel shifts were\nmade before the asteroid-ratio star division was done. \\\\\nThe infrared and visible spectral ranges of each asteroid were finally\ncombined by overlapping the spectra, merging the two wavelength\nregions at the common wavelengths and utilizing the zone\nof good atmospheric transmission to find the normalization factor\nbetween the two spectral parts. The\noverlapping region goes from about 0.78-0.92 $\\mu$m. For most of the data  we took the\naverage value in reflectance of the visible spectrum in the 0.89--0.91 $\\mu$m region \nto normalize the reflectance of the infrared spectrum. For the spectra where LRR grism data were not available or present but with strong fringing problems, we joint the visible and NIR data using the mean reflectance value of the visible spectrum in the 0.79-0.81 $\\mu$m region. \\\\\nAsteroids having both visible and NIR data or only visible spectra were {\\bf finally all normalized in relative reflectance at 1 at the wavelength 0.55 $\\mu$m}, while those observed only in the NIR region were normalized  at 1.25 $\\mu$m. \nDetails on the observing conditions are reported in Table~\\ref{tab1}, and the spectra of the observed asteroids are shown in Figs.~\\ref{f1}-\\ref{f4}.\n\n\n[Here Figs 1, 2, 3, 4]\n\n[Here Table 1]\n\n                          \\section{Results}\n\n                          [Here Table 2 and Figure 5]\n\nWe present new spectra of 8 Beagle and 22 Themis families members.\nMost of the asteroids were observed in the V+NIR range, except for some targets for which\nwe did not get the visible spectra (objects 621, 954, 1778, 2009, and 2203; for asteroids 621 and 954 we complete our data with the visible spectra from the\n$S^3S0^2$ survey (Lazzaro et al. 2004)), or the NIR ones (62, 383, 268, 526, 1247, and 2228; for objects 62 and 383 we got the IR spectra from Clark et al. (2010)). \nThe observed asteroids and their proper elements are reported in Table~\\ref{slope}. \\\\\nThe members of each family were derived from the family lists from analytic proper elements\n(Nesvorny et al. 2012).  A hierarchical clustering method has been applied to the asteroids proper elements to identify the dynamical families (Nesvorny et al., 2008). Themis is an old and huge family with 5425 members identified at a cutoff velocity of 60 m\/s (Nesvorny et al. 2012). Beagle is a young smaller cluster within Themis, having 149 members at a cutoff velocity of 20 m\/s, and an estimated age $<$ 10Myr (Nesvorny et al. 2008, 2012).  The cutoff velocity is related to the relative velocity of the fragments as they were ejected from the sphere of influence of the parent body, and lower values implies a higher statistical significance of the family. This means that the Beagle family is very robust.\n        \nAll the objects investigated belong to the C or B spectral types, following the Tholen (1984) classification scheme. The majority of the spectra are featureless, and in particular none of the investigated spectra show water ice absorption bands at 1.5 and 2 micron. A few Themis members ((24) Themis, (90) Antiope, (461) Saskia, and (846) Lipperta) show weak spectral absorption features in the visible range associated with hydrated minerals (Table~\\ref{hydra}, and Fig.~\\ref{f5bis}), indicating that they have experienced the aqueous alteration process, as already found for other Themis family members (Florczak et al. 1999).\nWe considered only the absorption features deeper than the peak-to-peak  scatter (that is depth $>$ 0.8\\%) in the spectrum, and we characterized them following the method described in Fornasier et al. (1999, 2014). This method includes first a smoothing of the spectra using a box of 6 pixels in the spectral direction, then the evaluation of the linear continuum at the edges of the band identified, the division of the original spectrum by the continuum, and the fitting  of the absorption band with a polynomial of order 2--4.  The band center was then calculated as the position where the minimum of the spectral reflectance curve occurs (on the polynomial fit), and the band depth as the minimum of the polynomial fit. The main absorption band identified is the one centered in the 0.68-0.73 $\\mu$m region which is attributed to $Fe^{2+}\\rightarrow Fe^{3+}$ charge transfer absorptions in phyllosilicate minerals (Vilas \\& Gaffey 1989; Vilas et al. 1993). This band is often associated with an evident UV absorption below $\\sim$ 0.5 $\\mu$m, due to a strong ferric oxide intervalence charge transfer transition (IVCT) in oxidized iron, and is often coupled with\nother visible absorption features related to the presence of aqueous alteration products (e.g. phyllosilicates, oxides, etc) (Vilas et al. 1994). We see the UV drop-off in reflectivity in most of the asteroids observed with the LRB grism, that is on Themis members (24) Themis, (268) Adorea, (461) Saskia, (492) Gismonda, (526) Jena, (1623) Vivian, (222) Lermontov, (2228) Soyuz-Apollo, (2270) Yazhi, and on Beagle members (656) Beagle, (1687) Glarona, (2519) Annagerman, (3174) Alcock, and (4903) Ichikawa,  but not on (90) Antiope and (846) Lipperta, that have a faint absorption centered at $\\sim$ 0.7 $\\mu$m. These IVCTs comprise multiple absorptions that are not uniquely indicative of phyllosilicates, but are present in the spectrum of any object\ncontaining Fe$^{2+}$ and Fe$^{3+}$ in its surface material (Vilas 1994). Thus, even if most of these spectra are featureless in the VIS-NIR range, their UV drop-off in reflectivity may indicate the presence of phillosilicates (Vilas 1994).  \\\\ \nSaskia also shows two absorption features in the NIR region centered at about 1.32 $\\mu$m and 1.82 $\\mu$m, as determined by a 4$^{th}$ order polynomial fit to the absorption bands. However, the center of these bands fall in the region of very low atmosphere transmission so their center position cannot be precise and their identification needs to be confirmed by other observations. If real, these absorptions look similar to those found on some hydrous Fe-bearing hydrated silicates (Bishop \\& Pieters 1995), thus potentially associated to the presence of hydrated minerals.  The continuum removed spectra of asteroids showing faint absorption features are shown in Fig.~\\ref{f5bis}. \\\\\nWe identified features associated to hydrated silicates only on Themis members. Thus only 18\\% of the 22 Themis members investigated show evidence of aqueous alteration in the visible and near infrared ranges. Recently Kaluna et al. (2016) investigated a sample of 22 Themis and 23 Beagle members. The asteroids they observed are all different from those of our survey, and they are all smaller than 15 km. Interestingly their analysis also indicates a lack of the 0.7 $\\mu$m features in the observed asteroids, with only one Beagle member having that absorption. It must be noted that the absence of the 0.7 $\\mu$m feature does not necessarily imply a lack of aqueous alteration, considering that this absorption is much fainter than the main absorption feature associated to hydrated minerals and centered at 3 $\\mu$m (Lebofsky 1980; Jones et al. 1990), and that not all the aqueous altered  CM\/CI meteorites show this feature (McAdam et al. 2015). Considering the results of other surveys,  Florczak et al. (1999) found  absorption bands associated to hydrated silicates in 15 out of 36 Themis members observed in the visible range, while Fornasier et al. (2014, Table 4), observed the $\\sim$ 0.7 $\\mu$m band only on 8 out of 47 Themis members spectra.\\\\\nTwo out of the three Themis members observed by Licandro et al. (2011) are in common with our observations: (62) Erato and (383) Janina. For (62) Erato Licandro et al. found a slope value (-2$\\pm$1 \\%\/(1000\\AA) similar to the one we found (Table~\\ref{slope}), while the Janina spectrum is quite different, having a positive visible slope in our data and a negative one in both Licandro et al. (2011) and  Bus \\& Binzel (2003) observations. Additional data covering different rotational phases are needed to cast light on potential surface heterogeneities of this asteroid.    \n\n[Here Table 3]\n\n\\subsection{(24) Themis: surface heterogeneities}\n\n[Here FIGURE 6]\n\n(24) Themis is a big asteroid with a diameter of  218$\\pm$1 km and a low albedo value of  6$\\pm$1\\% (Hargrove et al. 2015).  \nWe observed (24) Themis three times in the visible range and once in the NIR range during the December 2012 run, finding some spectral variability (Fig.~\\ref{f5}). Unfortunately  all the LRR spectra of (24) Themis  suffered of strong fringing problems, and they have a very bad S\/N ratio in the 0.8-0.95 $\\mu$m. We thus complete the {\\bf 18 and 19 Decem} LRB observations with the available NICS spectrum, that has a better S\/N ratio in 0.8-0.95 $\\mu$m range than the LRR data. The spectra acquired on 18 Dec. have a negative slope in the visible range and no hints of absorption bands, while the one taken on 19 Dec. shows an absorption band centered at $\\sim$ 7333$\\pm$48\\AA, having a depth of 2.6\\%, attributed to hydrated silicates. \nThe data also indicate a clear UV-drop-off of the reflectance for wavelength $<$  0.5 $\\mu$m for the observations acquired on 18 Dec. UT 22 and 19 Dec. Assuming a rotational period of 8.37677$\\pm$0.00002 h (Higley, 2008) and taking the LRB spectrum of 18 Dec. UT 02:16 as reference for the rotational phase, then the second and third LRB spectra of (24) Themis fall at rotational phases of 0.3020 and 0.7278, respectively. We thus covered different rotational phases and the observed spectral differences may be related to surface heterogeneities on (24) Themis. This conclusion is supported also by the comparison with (24) Themis spectra available in the literature (Fig.~\\ref{f5}), taken from Fornasier et al. (1999), Bus\\& Binzel (2002), and Lazzaro et al. (2004). In particular Fornasier et al. (1999) reported an absorption feature attributed to hydrated silicates, similar to the one found in this paper but centered at a shorter wavelength (6722$\\pm$39 \\AA), with a depth of 3.5\\% compared to the continuum, while the other literature spectra are featureless except for the potential presence of the UV-drop-ff in reflectivity.\n\n(24) Themis shows strong evidence of surface water ice (Campins et al. 2010; Rivkin et al. 2010), that seems to be widespread present on its surface, according to Campins et al. (2010) results. Its emissivity spectrum  exhibits a rounded 10 $\\mu$m emission feature, found also on other Themis family members (Licandro et al. 2012), attributed to small silicate grains embedded in a relatively transparent matrix, or from a very under-dense surface structure (Hargrove et al. 2015). Recently McAdam et al. (2015) interpret the emissivity behavior of (24) Themis as due to complex surface mineralogy with approximately 70 vol.\\% phyllosilicates and 25 vol.\\% anhydrous silicate. \\\\\nConsidering our data and those of the literature, (24) Themis most likely has an heterogeneous surface composition, presenting both water ice and hydrated silicates together with dark phases like those found on carbonaceous chondrites. According to Castillo-Rogez \\& Schmidt (2010) models, large Themis family objects may contain a large fraction of the parent body ice shell, and this ice may not be completely pure, but possibly contaminated with other materials such as organics, oxides and  hydrated minerals. Jewitt \\& Guilbert-Lepoutre (2012) report no detection of gas from sublimated ice on (24) Themis, and conclude that ice must cover a relatively small fraction (10\\%) of the asteroid surface, having potentially been exposed on the surfaces of (24) Themis by recent impacts. Surface heterogeneity seems to be common on the largest asteroids\/dwarf planet imaged so far, as seen for Vesta and Ceres by the Dawn mission. For instance several bright spots are observed on Ceres, and potentially attributed to water ice associated with impact craters (Reddy et al. 2015), and groundbased observations underline spectral variability on its surface (Perna et al. 2015).\\\\\nFuture observations  with high S\/N rotationally resolved spectra in the visible and near infrared region are needed to fully investigate the (24) Themis surface heterogeneity. \n\n\n\n\\subsection{Spectral slopes and physical parameters}\n\n[Here Figs. 7 and 8]\n\nTo analyze the data, we compute spectral slope values with linear\nfits to different wavelength regions: $S_{cont}$ is the spectral slope in the whole\nrange observed for each asteroid, $S_{VIS}$ is the slope in the 0.55-0.80 $\\mu$m\nrange, S$_{NIR1}$ is the slope in the 0.9-1.4 $\\mu$m range, S$_{NIR2}$ is\nthe slope in the 1.4-2.2 $\\mu$m range, and  S$_{NIR3}$ is\nthe slope in the 1.1-1.8 $\\mu$m range. Values are reported in\nTable~\\ref{slope}. The slope error bars take into account the\n$1\\sigma$ uncertainty of the linear fit plus 0.5\\%\/$10^3$\\AA\\ attributable to the spectral variation due to the use of different solar analog stars during the night. \\\\\n\n[Here Figs. 9 and 10]\n\nOur observations clearly show a range of spectral behaviors exhibited by Themis family members in the visible and near-infrared (Figs.~\\ref{f1}-~\\ref{f3}, and~\\ref{f6}), including asteroids with blue\/neutral and moderately red spectra (relative to the Sun). Our findings are consistent with, and complement previous spectroscopic studies that hinted at spectral diversity within the Themis family in the visible (Florczak et al. 1999; Kaluna et al., 2016), the near-infrared (Ziffer et al. 2011; de Le\\'on et al. 2012; Al\\'i-Lagoa et al. 2013), and the mid-infrared (Licandro et al. 2012). On the other hand, the robust and younger Beagle family members look different, with a smaller spectral slope variability. This family seems dominated by objects with a blue\/neutral spectrum in the visible range, and with a neutral to moderately red spectral behavior in the near infrared range.\n\nTo better understand the difference between the two families, we look for correlations between spectral slopes and physical parameters such as the albedo and diameter, which were derived from the WISE data (Masiero et al. 2011).\nFigure~\\ref{f7} shows the visible and NIR3 spectral slopes versus the WISE albedo ($p_v$) for the Themis and Beagle families members here investigated. It clearly appears that the young Beagle members are not only spectrally bluer but tend to have a higher albedo value (mean $p_v$ = 0.0941$\\pm$0.0055) than most of the Themis members investigated (mean $p_v$ = 0.0743$\\pm$0.0054). \\\\\nTo test if this trend is true, we extended the analysis on a larger sample including 79 Themis and Beagle families members spectra available in the literature from several surveys in the visible wavelength range (Xu et al. 1995; Bus \\& Binzel 2002; Lazzaro et al. 2004; Fornasier et al. 1999, 2014; Kaluna et al., 2016). For most of these data, the spectral slope has been evaluated in a similar way to what is done here, i. e. between 0.55-0.8 $\\mu$m. Only the spectral slope for asteroids observed in Kaluna et al. (2016) was evaluated in a different manner, between 0.49 and 0.91 $\\mu$m. However, beeing their spectra normalized at the same wavelength than us and featureless, the variation in the spectral slope values associated to the  different wavelength ranges used must be minimal. For 11 out of 23 Beagle members and 47 out of 56 Themis members of the literature data the albedo value is available from the WISE observations. Figure~\\ref{f8} shows the visible spectral slope versus albedo of this extended sample. The trend seen in our data is confirmed and it is evident that there are no Beagle members with a red spectral slope.  Conversely, Themis members show an important spectral variety, having both blue and moderately red slopes and low to moderate albedo values (4-15\\%). This spectral variety is not related to different sizes of the asteroids investigated because Themis members having diameter similar to the Beagle ones span different VIS spectral slopes (-2.5 $\\%\/(10^{3}$\\AA) to 3 $\\%\/(10^{3}$\\AA)) and albedo values (4-12.5\\%), as shown in Figure~\\ref{f9}. \\\\\nThe difference in the albedo values is less pronunced, Themis members having a slightly lower mean albedo  (0.0784$\\pm$0.0031) than Beagle ones  (0.0819$\\pm$0.0045). However, analyzing the average albedo from WISE data on a larger sample and comparing members of similar sizes,  Kaluna et al. (2016) found that the small Themis members ($D <$ 15km) have a lower albedo ($p_v$=0.068$\\pm$0.001) than the larger asteroids ($p_v$=0.075$\\pm$0.001) of the family, and that their albedo is  significantly lower than that of the Beagle population ($p_v$=0.079$\\pm$0.005). Their results clearly indicate that Beagle members have a sligthly higher albedo value than the Themis members of similar size.\n\n[Here Fig. 11]\n\nWe also analyzed the near infrared albedo (p$_{IR}$), that is the albedo at 3.4--4.6 $\\mu$m as defined in Mainzer et al. (2011a), and the geometric (p$_v$) albedo for the Themis members observed with WISE. Figure~\\ref{f10} shows the ratio of the p$_{IR}$\/p$_v$ albedo versus p$_v$ for a sample of 211 Themis and 5 Beagle members. The bright Themis members have lower p$_{IR}$\/p$_v$ ratio, indicating a blue spectrum in the NIR region and\/or the presence of absorption bands in the 3 $\\mu$m region, potentially attributed to hydrated silicates, organics and eventually water ice.\\\\\nA similar {\\it waning moon} shape of the  p$_{IR}$\/p$_v$ ratio versus p$_v$  was also noticed by Al\\'i-Lagoa et al. (2013) analyzing B-type asteroids. Some biases may partially affect the distribution we see, in particular the lack of points in the lower left corner may be attributed to faint fluxes in WISE bands W1 and W2, not allowing a proper evaluation of the reflected sunlight contribution (Mainzer et al. 2011b).  Al\\'i-Lagoa et al. (2013), in the analysis of B-type asteroids and p$_{IR}$ albedo, found that an absorption in the 3 $\\mu$m region may be common on these bodies.  \\\\\nFigure~\\ref{f10} confirms the huge spectral variability of Themis members, which have small to moderate albedo values  (3--16\\%) and a P$_{IR}$\/p$_v$ ratio varying from $\\sim$ 0.5 to 2.2.\n\n\n\n\\subsection{Meteorite analogues}\n\n[Here Figs. 12-15 and Table 4]\n\nTo constrain the possible mineralogies of the investigated asteroids, we conducted\na search for meteorite and\/or mineral spectral matches. We\nsought matches only for objects observed across the entire VIS--NIR\nwavelength range. We used the publicly available RELAB spectrum library (Pieters,\n1983). For each  spectrum in the library, a filter was applied to find relevant  \nwavelengths (0.4 to 2.45 $\\mu$m). Then, a second filter was applied to reject spectra with irrelevant albedo values (i.e. brightness at 0.55 $\\mu$m $>0.2$). A  Chi-squared value was calculated relative to the normalized input asteroid spectrum. We used the asteroid wavelength sampling to resample the laboratory spectra by linear interpolation. This allowed a least-squares calculation with the number of points equal to the wavelength sampling of the asteroid spectrum. The RELAB data files were sorted according to the corresponding Chi-squared values, and finally visually examined. A complete description of the search methodology\ncan be found in Fornasier et al. (2010, 2011).\n \nThe best matches between the observed asteroids and meteorites from the RELAB database are reported in Table~\\ref{met}, and shown in Figures~\\ref{f11} -~\\ref{f14}.\nMost of the Themis and Beagle families members are matched by CM2 carbonaceous chondrites, with few objects having CV3 or CI as best match. We do not see differences in the meteorite class matches between  the Beagle and Themis families members, except that there are no CV3 best match for Beagle members. Even if the aqueous alteration band at 0.7 $\\mu$m has been identified just on one Beagle member (Kaluna et al., 2016), the meteorite matches indicate that these asteroids may have experienced aqueous alteration in the past. As previously stated, not all the aqueous altered  CM\/CI meteorites show this feature (McAdam et al. 2015). \\\\\nThe asteroid (2519) Annagerman is spectrally perfectly matched by the unusual CI\/CM Y-86720,77 with grain size  $<$ 125 $\\mu$m, even if there are some differences in the albedo values, 11\\% for the asteroid and  6\\% for the meteorite. However it must be noted that the meteorite reflectance is usually evaluated at phase angle $>$ 7 $^o$ so it does not include the opposition surge effect. The composition and structure of the CM2 Y-86720 indicates that this meteorite was thoroughly aqueous altered (Lipschutz et al. 1999) and heated. Interestingly, some of the targets are best matched by heated or unusual meteorites. This was  already noticed by Clark et al. (2010), who found that Themis family asteroids tend to be best fitted by CM, CI, CR, CM\/CI unusual meteorites, and\/or heated CM\/CI samples, and that 50\\% of these asteroids are consistent with thermally metamorphosed material. \\\\\nOur spectral matches are also in agreement with the de L\\'eon et al. (2012) results from the  analysis of B-type asteroids, including 8 Themis members. These last fall in the G2-G5 groups as defined by de L\\'eon et al. (2012), and are spectrally best matched by meteorites who experienced different stages of the aqueous alteration,  from the less altered CV3 meteorites up to the extensively altered CM2 chondrites. \nThe spectral analysis together with the meteorite matches indicate that Themis members show a large spectral variety and different carbonaceous chondrite matches (from CV3 to CM2-CI), some of them being unusual or heated, indicating that these asteroids were thermally metamorphosed in the past. \n\n\n\n\\section{Discussion}\n\nSpectroscopic studies of asteroid families can provide information about the interior of their parent bodies (e.g., Cellino et al. 2002); this can constrain models on the thermal and collisional evolution of asteroids.  More specifically, simulations of catastrophic disruption processes suggest that the ejected material coming from coherent sections of a heterogeneous progenitor body should maintain a coherent compositional structure (Michel et al. 2004). Therefore, compositional gradients within planetesimals should expose themselves within asteroid families (e.g., Jacobson et al. 2014).  To date, no remnants of a completely disrupted differentiated body have been identified. Vesta is the only family generated from a differentiated body (e.g., Lazzaro et al. 2004; Binzel and Xu 1993), but it is not a result of a catastrophic disruption, so its members come from the surface of the parent body.\n\nThe case of the Themis family seems to be unique. For example, Jacobson et al. (2014) used estimates of the progenitor mass and the mass of the largest remnant to assess the exposed nature of asteroid families, and he suggested that the Themis family is possibly the most exposed one.   We have found clear evidence of spectral diversity within the Themis family, and between Themis and Beagle families members.  There are several possible interpretations of this spectral diversity, which include a compositional gradient with depth in the original parent body (and\/or with fragment size), space weathering effects, the survival of projectile fragments with a different composition, or a combination of these (e.g., Campins et al. 2012), that we discuss in the following. \\\\\n\n\\subsection{Scenario a): The Themis\/Beagle family parent body was heterogeneous in composition} \n\nIn this scenario, the diversity we see nowdays in Themis reflects different source regions in the parent body that was originally heterogeneous showing a compositional gradient. \nWe clarify that a compositional gradient does not necessarily mean a metal core with a silicate mantle and crust. In fact, for primitive asteroids like the Themis parent body it was more likely a differentiation of rock and ice where the core underwent mild temperatures and no high thermal metamorphism. (e.g., Castillo-Rogez \\& Schmidt 2010).\nConsistent with this mild temperature model are the visible spectroscopic results of 36 members of the Themis family presented by Florczak et al. (1999). They found that about 50\\% of their sample showed evidence of aqueous alteration, indicating that the parent body was sufficiently altered thermally to mobilize water. The results of  Florczak et al. (1999) also suggest that the percentage of asteroids showing aqueous alteration may decrease with the diameter of the objects, as expected by Vilas \\& Sykes (1996) and proven by Rivkin (2012) and Fornasier et al. (2014) on a large sample of primitive main belt asteroids. The study  presented in this paper and in Fornasier et al. (2014) indicates a much lower percentage of hydration in the Themis family ($<$ 20\\%), while just one out of the 45 Themis and Beagle small members studied by Kaluna et al. (2016) shows the 0.7 $\\mu$m absorption band. Moreover the meteorite spectral matches found by us and in the literature point towards a similarity of Themis-Beagle family members with the CV3-CM2 chondrites, the meteorites having experienced aqueous alteration processes (see section 3.3; Clark et al. 2010; de L\\'eon et al. 2012). \\\\\nThe above results support the view that at least part of the parent body of the Themis family was thermally altered and that the distribution of compositions could be attributed to fragments coming from different depths in the original parent body, which was catastrophically disrupted by a large projectile.\\\\\nIn this scenario, the Beagle family may have been originated by a fragment of a blue and bright piece of Themis parent body possibly coming from its interior.\\\\\n\n\\subsection{Scenario b): The Themis\/Beagle family parent body was homogeneous and the projectile that collided with it was different in composition}\n\n  Marzari et al. (1995) modeled the collisional evolution of the Themis family. Their best results yield parent body and projectile radii of 380 km and 190 km, respectively. They suggest that the two bodies were not formed in the same region and had different compositions. Since the projectile body was relatively large, its fragments should be a considerable part of the remnants. Interestingly, Florczak et al. (1999) argue that except for the hydration (indicated by an absorption centered near 0.7 $\\mu$m) their visible spectra were homogeneous and they did not favor a different composition between parent body and projectile or significant differentiation.  Our visible and near-infrared spectra (Figs.~\\ref{f7} and ~\\ref{f8}) and those of de Le\\'on et al. (2012) and Kaluna et al. (2016) show a wide and smooth range of behaviors suggestive of a range of compositions among the Themis family members. These results does not necessarily support different compositions between the large projectile and the Themis family parent body. Instead of a smooth spectral diversity, one might expect a bi-modal one from a different projectile and parent body composition.\n  \n\n\\subsection{Scenario c): The family parent body was homogeneous and the spectral variability is produced by space weathering effects} \n\n  Assuming that the parent body of the Themis\/Beagle families was quite homogeneous, the spectral variability seen on {\\it old} Themis members and the fact the {\\it young} Beagle members are bluer and with an higher albedo value than the Themis ones may be, at least in part, a result of space weathering effects. \n\nIn the past 20 years enormous progress has been reached in the study of space weathering (SW) effects on silicates and S-type asteroids. The so-called ordinary chondrite paradox, that is lack of asteroids similar to the ordinary chondrites, which represent 80\\% of meteorite falls, has been solved. These meteorites are now clearly related to S-type asteroids, as proved  by direct measurements of the NEAR and HAYABUSA missions on the near Earth asteroids Eros and Itokawa (Clark et al. 2001; Noguchi et al. 2011) and laboratory experiments on irradiation of silicates\/ordinary chondrites (Moroz et al. 1996; Sasaki et al. 2002; Brunetto \\& Strazzulla 2005; Noble et al. 2007). Spectral differences between S-type asteroids and ordinary chondrites are caused by space weathering effects, which produce a darkening in the albedo, a reddening of the spectra, and diminish the silicates absorption bands on the asteroids surfaces, exposed to cosmic radiation and solar wind. \\\\\nOn the other hand, our understanding of space weathering effects on primitive asteroids is still poor.\nOnly few laboratory experiments have been devoted to the investigation of SW effects on low albedo carbonaceous chondrites and on complex organic materials, and they indicate a no linear trend but much more complex effects. \nIrradiation of transparent organic materials produces  firstly  redder and lower albedo materials  that upon  further processing  turn  flatter-bluishing and darker (Kanuchova et  al. 2012; Moroz et al. 2004a). This result is consistent with a recent study that was looking for spectral differences between Ch\/Cgh-types and CM chondrites (Lantz et al. 2013). On the other hand, Lazzarin et al. (2006) expected a global reddening for the whole asteroidal population. Laboratory experiments on carbonaceous chondrites  give different results, that is reddening and\/or blueing. Some ion irradiations on CV Allende and CO Frontier Mountain 95002 (Lazzarin et al. 2006) led to reddening and darkening, and this trend  is confirmed for the Allende meteorite using both ion (Brunetto et al. 2014) and laser irradiation (Gillis-Davis et al. 2015).  \\\\\nThe laser irradiated CM Mighei resulted in spectral reddening (Moroz et al. 2004b) while laser irradiation on Tagish Lake showed a blueing effect (Hiroi et al. 2004).  Vernazza et al. (2013) performed ion irradiations on Tagish Lake and observed a strong blueing with argon, while irradiation with He produced only minor spectral changes and a slight reddening at 4 keV dose. Hiroi et al. (2013) used laser irradiations on a CO, a CK, a CM, a CI and Tagish Lake: the first two present the same behavior as ordinary chondrites while the others show a clear blueing (but reflectance decreases so that a darkening effect is seen). \\\\\nFor the CM Murchison the results after irradiation are contradictory:  Keller et al. (2015) found reddening and darkening of the sample after ion irradiation,  Matsuoka et al. (2015) found a blueing and darkening effect with laser irradiation,  Gillis-Davis et al. (2015), using again laser irradiation, found a darkening effect but without slope variations, while Lantz et al. (2015) found no clear albedo and slope variations after ion irradiation.  \n\nIt appears complicated to define a general space weathering trend for the primitive objects when considering the various results of laboratory simulations on carbonaceous chondrites. That would explain why the few studies on asteroids\nproposed different conclusions between reddening (Lazzarin et al. 2006) and blueing of the slope (Nesvorny et al. 2005; Lantz et al. 2013).\nHere we compare spectra of young and old members of the Themis group (respectively Beagle and Themis families). Starting from the hypothesis that the parent body was homogeneous, the Beagle members have been less exposed to the harsh space environment, and so less space weathered compared to Themis members. If this hypothesis is correct, then our observations seem to indicate that the surfaces of the Themis family members have the same spectral answer than the S-type asteroids: young Beagle members have globally higher albedo and bluer slopes, while most of the older Themis members are redder and have a lower albedo value. A similar conclusion is found by Kaluna et al. (2016) on their analysis of Themis and Beagle members of similar size, smaller than 15 km. From the mean slopes values of the two families, Kaluna et al. (2016) estimate a slope reddening of 0.08 $\\mu$m$^{-1}$ in 2.3 Gyr for C-complex asteroids. However, if space weathering effects are the only cause of the spectral diversity between the two families, we would expect a distinct spectral and albedo distribution between old and young asteroids, while our analysis clearly shows that several Themis members have albedo and spectral properties similar to the Beagle ones (Figure~\\ref{f8}). Some rejuvenating processes might have taken place (Shestopalov et al. 2013), but they cannot solely explain the large percentage of blue and bright Themis members observed. \\\\\nAnother important point is the timescale needed to space weathering processes to alter asteroid surfaces. Laboratory simulations indicate that solar wind may alter the surfaces in a relatively short timescale, of the order of 10$^3$--10$^5$ years, producing darkening and reddening effects on ordinary chondrites (Strazzulla et al. 2005), on S-type asteroids (Vernazza et al. 2009a, 2009b), and on the Allende carbonaceous chondrite (Brunetto et al. 2014). This has also been confirmed by the Itokawa grain analysis brought back to the Earth by the sample return mission Hayabusa (Noguchi et al. 2011). If these short timescales turn out to be correct, it is possible that Beagle members are old enought to have already experienced some space weathering processes. All this considered, we cannot firmly conclude that the observed spectral diversity is related to space weathering processes. Considering the presence of hydrated minerals on some members and the thermal evolution models which suggest a differentiation in the Themis\/Beagle parent body, we favor the compositional heterogeneity in the original parent body as explanation of the spectral diversity within Themis\/Beagle members, even if we cannot exclude that space weathering processes may be responsible of at least part of the spectral differences observed.\n\n\\section{Conclusions}\n\nIn this work we investigate the spectral properties of primitive asteroids belonging to the $\\sim$ 2 Gyr old Themis family and to the young ($<$ 10 Myr) Beagle subcluster within the Themis family.  We present new visible and near infrared spectra of 22 Themis and 8 Beagle families members. The Themis family members show different spectral behaviors including asteroids \nwith blue\/neutral and moderately red spectra, while the Beagle members have less spectral variability and they are all neutral\/blue.\nWe include in our analysis most of the data available in the literature on Themis\/Beagle families (a sample of 79 objects) for a complete analysis of their spectral properties versus physical parameters. The spectral slope versus albedo distribution clearly shows that Beagle members are less red and tend to have an higher albedo value than the Themis one. To explain the spectral differences between the two families we propose different interpretations, including heterogeneity in the parent body having a compositional gradient with depth, the survival of projectile fragments having a different composition than the parent body, space weathering effects, or a combination of these. \\\\\nThe spectral differences observed between Beagle and Themis members may be partially attributed to space weathering effects, which would produce on primitive asteroids reddening and darkening, as seen on S-type asteroids. However, space weathering effects cannot solely explain the spectral variety within the two families, first because we observe several Themis members having albedo and spectral behaviors similar to the younger Beagle members, and second because we need to assume that the parent body of the family was homogeneous. However, our and literature data show the presence of hydrated minerals on some of the Themis members, and a spectral analogy of the Themis\/Beagle members with carbonaceous chondrites having experienced different degrees of aqueous alteration. These results, together with those from thermal evolution models, indicate that the parent body of the Themis family experienced mild thermal metamorphism in the past, with a possible compositional gradient with depth. We thus favor the compositional heterogeneity in the original parent body as main source of the spectral diversity within Themis\/Beagle members. \n\n\n\n\\vspace{0.3truecm}\n{\\bf Acknowledgment} \\\\\nThis paper is based on observations made with the Italian Telescopio Nazionale Galileo (TNG) operated on the island of La Palma by the Centro Galileo Galilei of the INAF (Istituto Nazionale di\nAstrofisica), both located at the Observatorio del Roque de los Muchachos, La Palma, Spain, of the Instituto de Astrofisica de Canarias.\" This project was supported by the French Planetology National Programme (INSU-PNP). The authors gratefully acknowledge the reviewers for their comments and suggestions who help us improving the paper.\n\n\\bigskip\n\n{\\bf References} \\\\\n\n\nAl\\'i-Lagoa, V., de Le\\'on, J., Licandro, J., Delb\\`o, M., Campins, H., Pinilla-Alonso, N., Kelley, M. S., 2013. Physical properties of B-type asteroids from WISE data. Astron. Astroph. 554, id. 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Spectra are shifted by 0.4 in reflectance for clarity.}\n\\label{f1}\n\\end{figure*}\n\n\\begin{figure*}\n\\centerline{\\psfig{file=I14484_fig2.ps,width=20truecm,angle=0}}\n\\caption{Reflectance spectra of the Themis family members investigated. Spectra are shifted by 0.4 in reflectance for clarity.}\n\\label{f2}\n\\end{figure*}\n\n\\begin{figure*}\n\\centerline{\\psfig{file=I14484_fig3.ps,width=20truecm,angle=0}}\n\\caption{Reflectance spectra of the Themis family members investigated. Spectra are shifted by 0.5 in reflectance for clarity.}\n\\label{f3}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\centerline{\\psfig{file=I14484_fig4.ps,width=20truecm,angle=0}}\n\\caption{Reflectance spectra of the Beagle family members investigated. Spectra are shifted by 0.4 in reflectance for clarity.}\n\\label{f4}\n\\end{figure*}\n\n\\begin{figure*}\n\\centerline{\\psfig{file=I14484_fig5.ps,width=15truecm,angle=0}}\n\\caption{Continuum removed spectra of the 4 asteroids showing absorption features. The visible spectrum of (24) Themis is the one acquired on 19 December.}\n\\label{f5bis}\n\\end{figure*}\n\n\\begin{figure*}\n\\centerline{\\psfig{file=I14484_fig6.ps,width=15truecm,angle=0}}\n\\caption{Relative reflectance spectra in the visible region of (24) Themis taken at different aspect and rotational phases from data presented here and from the literature. The asteroid shows surface heterogeneities. }\n\\label{f5}\n\\end{figure*}\n\n\n\n\\begin{figure*}\n\\centerline{\\psfig{file=I14484_fig7.ps,width=20truecm,angle=0}}\n\\caption{NIR3 spectral slope (1.1-1.8 $\\mu$m) versus VIS spectral slope (0.5-0.8 $\\mu$m) for the Themis (black circles) and Beagle (blue stars) members. The spectral slopes are in \\%\/(10$^{3}$\\AA). The size of the circles is proportional to the asteroids' diameters.}\n\\label{f6}\n\\end{figure*}\n\n\n\\begin{figure}\n\\centerline{\\psfig{file=I14484_fig8.ps,width=15truecm,angle=0}}\n\\caption{Spectral slopes in the visible (0.55-0.8 $\\mu$m) and in the NIR (1.1-1.8$\\mu$m) ranges versus albedo. The spectral slopes are in \\%\/(10$^{3}$\\AA). Beagle and Themis family members are represented with blue stars and black circles, respectively. The size of the circles is proportional to the asteroids' diameters.}\n\\label{f7}\n\\end{figure}\n\n\\begin{figure}\n\\centerline{\\psfig{file=I14484_fig9.ps,width=15truecm,angle=0}}\n\\caption{Spectral slope in the visible wavelength range (0.55-0.8 $\\mu$m, units \\%\/(10$^{3}$\\AA)) versus the albedo for the Themis and Beagle members. Beagle family members are represented with stars, blue from our data and cyan from the literature data, while the Themis members are represented with circles, black from our data and red from the literature data. The size of the symbols is proportional to the asteroids' diameters.}\n\\label{f8}\n\\end{figure}\n\n\n\n\\begin{figure}\n\\centerline{\\psfig{file=I14484_fig10.ps,width=15truecm,angle=0}}\n\\caption{Spectral slope in the visible wavelength range (0.55-0.8$\\mu$m, units \\%\/(10$^{3}$\\AA)) versus asteroids' diameter. Beagle family members are represented with star symbol, blue from our data and cyan from the literature data,  while the Themis members are represented as circles, black from our data and red from the literature data.}\n\\label{f9}\n\\end{figure}\n\n\n\n\\begin{figure*}\n\\centerline{\\psfig{file=I14484_fig11.ps,width=20truecm,angle=0}}\n\\caption{WISE albedos for the Themis family members: ratio of the infrared over the visible albedo (p$_v$) versus the visible albedo: Themis members having higher albedo value are also bluer in the IR range.\n  The size of the symbols is proportional to the asteroids' diameters. In the upper rigth part of the plot we report the mean error bars of these measurements.}\n\\label{f10}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\centerline{\\psfig{file=I14484_fig12.ps,width=20truecm,angle=0}}\n\\caption{Best spectral matches between the observed Beagle family members and meteorites from the RELAB database.}\n\\label{f11}\n\n\\end{figure*}\n\n\\begin{figure}\n\\centerline{\\psfig{file=I14484_fig13.ps,width=20truecm,angle=0}}\n\\caption{Best spectral matches between the observed Themis family members and meteorites from the RELAB database.}\n\\label{f12}\n\\end{figure}\n\n\\begin{figure}\n\\centerline{\\psfig{file=I14484_fig14.ps,width=20truecm,angle=0}}\n\\caption{Best spectral matches between the observed Themis family members and meteorites from the RELAB database.}\n\\label{f13}\n\\end{figure}\n\n\\begin{figure}\n\\centerline{\\psfig{file=I14484_fig15.ps,width=20truecm,angle=0}}\n\\caption{Best spectral matches between the observed Themis family members and meteorites from the RELAB database.}\n\\label{f14}\n\\end{figure}\n\n\n\n\\newpage\n\n{\\bf Tables}\n\n{\\scriptsize\n       \\begin{center}\n     \\begin{longtable} {|l|l|c|c|c|c|c|c|l|}  \n\\caption[]{Observational circumstances. Solar analog stars named \"HIP\"\n  come from the Hipparcos catalogue and \"Lan\" from the\nLandolt photometric standard stars catalogue.}. \n        \\label{tab1} \\\\\n\\hline \\multicolumn{1}{|c|} {\\textbf{Object    }} & \\multicolumn{1}{c|}\n{\\textbf{Night}} & \\multicolumn{1}{c|} {\\textbf{UT$_{start}$}} &\n\\multicolumn{1}{c|} {\\textbf{T$_{exp}$ (s)}}  & \\multicolumn{1}{c|} {\\textbf{Instr.}} & \\multicolumn{1}{c|}\n{\\textbf{Grism}} & \\multicolumn{1}{c|} {\\textbf{airm.}} & \\multicolumn{1}{c|}\n{\\textbf{Solar Analog (airm.)}} \\\\  \\hline \n  \n\\endfirsthead\n\\multicolumn{8}{c}%\n{{\\bfseries \\tablename\\ \\thetable{} -- continued from previous page}} \\\\ \\hline \n\\endfoot\n\\hline \\multicolumn{1}{|c|} {\\textbf{Object    }} & \\multicolumn{1}{c|}\n{\\textbf{Night}} & \\multicolumn{1}{c|} {\\textbf{UT$_{start}$}} &\n\\multicolumn{1}{c|} {\\textbf{T$_{exp}$ (s)}}  & \\multicolumn{1}{c|} {\\textbf{Instr.}} & \\multicolumn{1}{c|}\n{\\textbf{Grism}} & \\multicolumn{1}{c|} {\\textbf{airm.}} & \\multicolumn{1}{c|}\n{\\textbf{Solar Analog (airm.)}} \\\\  \\hline \n\\endhead\n\\hline \\multicolumn{8}{r}{{Continued on next page}} \\\\ \n\\endfoot\n\\hline \\hline\n\\endlastfoot\n24 Themis & 18 Dec. 2012 & 00:38 & 20 &  NICS & Amici & 1.03 & HIP22536 (1.02) \\\\\n24 Themis & 18 Dec. 2012 & 02:16 & 10 &  Dolores & LRB & 1.22 & Lan98-978 (1.22) \\\\ \n24 Themis & 18 Dec. 2012 & 02:20 & 20 &  Dolores & LRR & 1.23 & Lan98-978 (1.22) \\\\ \n24 Themis & 18 Dec. 2012 & 21:33 & 10 &  Dolores & LRB & 1.14 & Lan115-271 (1.13) \\\\ \n24 Themis & 18 Dec. 2012 & 21:35 & 20 &  Dolores & LRR & 1.14 & Lan115-271 (1.13) \\\\ \n24 Themis & 19 Dec. 2012 & 01:07 & 60 &  Dolores & LRB & 1.07 & Lan115-271 (1.13) \\\\\n62 Erato & 19 Feb. 2012 &  05:38    & 900  &  Dolores & LRR & 1.24 & Lan98-978(1.18) \\\\\n90 Antiope & 18 Dec. 2012 & 01:48 & 30 &  Dolores & LRB & 1.20 & Lan102-1081 (1.15) \\\\ \n90 Antiope & 18 Dec. 2012 & 01:52 & 60 &  Dolores & LRR & 1.21 & Lan102-1081 (1.15) \\\\ \n90 Antiope & 17 Dec. 2012 & 23:29 & 60 &  NICS & Amici & 1.01 & HIP22536 (1.03) \\\\ \n268 Adorea & 20 Feb. 2012 &  06:08    & 300  &  Dolores & LRB & 1.33 & Hip59932 (1.32) \\\\\n268 Adorea & 20 Feb. 2012 &  06:15    & 300  &  Dolores & LRR & 1.33 & Hip59932 (1.32) \\\\\n383 Janina & 19 Feb. 2012 &  06:07    & 900  &  Dolores & LRR & 1.38 & Hip59932 (1.35) \\\\\n461 Saskia & 19 Feb. 2012 &  03:52    & 900  &  Dolores & LRB & 1.27 & Hip59932 (1.34) \\\\ \n461 Saskia & 19 Feb. 2012 &  04:09    & 900  &  Dolores & LRR & 1.33 & Hip59932 (1.35) \\\\ \n461 Saskia & 19 Feb. 2012 &  23:44    & 720  &  NICS    & Amici & 1.09 & HIP44103 (1.04) \\\\ \n468 Lina   & 19 Dec. 2012 & 06:08 & 300 &  Dolores & LRR & 1.12 & Lan115-271 (1.14) \\\\ \n492 Gismonda & 18 Dec. 2012 & 22:45 & 300 &  Dolores & LRB & 1.56 & HIP52192 (1.46) \\\\ \n492 Gismonda & 18 Dec. 2012 & 22:51 & 300 &  Dolores & LRR & 1.59 & HIP52192 (1.47) \\\\\n492 Gismonda & 16 Dec. 2012 & 20:47 & 480 &  NICS & Amici & 1.16 & Lan93-101 (1.14) \\\\\n526 Jena & 19 Dec. 2012 & 00:23 & 300 &  Dolores & LRB & 1.17 &  HIP44027 (1.17) \\\\\n526 Jena & 19 Dec. 2012 & 00:28 & 300 &  Dolores & LRR & 1.19 &  HIP44027 (1.17) \\\\ \n526 Jena & 17 Dec. 2012 & 22:54 & 240 &  NICS & Amici & 1.04 &  HIP22536 (1.03) \\\\ \n621 Werdandi  & 19 Feb. 2012 &  22:18    & 720 &  NICS    & Amici & 1.02 & HIP44103 (1.04) \\\\\n656 Beagle & 18 Dec. 2012 & 20:02 & 600 &  Dolores & LRB & 1.23 & Lan93-101 (1.23) \\\\ \n656 Beagle & 18 Dec. 2012 & 20:14 & 600 &  Dolores & LRR & 1.26 & Lan115-271 (1.14) \\\\ \n656 Beagle & 17 Dec. 2012 & 19:39 & 960 &  NICS & Amici & 1.21 & Lan115-271 (1.25) \\\\ \n846 Lipperta & 20 Feb. 2012 &  03:19    & 900  &  Dolores & LRB & 1.21 & Lan102-1081(1.31) \\\\\n846 Lipperta & 20 Feb. 2012 &  03:36    & 900  &  Dolores & LRR & 1.27 & Lan102-1081(1.31) \\\\\n846 Lipperta & 20 Feb. 2012 &  00:19    & 960  &  NICS    & Amici & 1.06 & HIP44103 (1.04) \\\\ \n954  Li      & 19 Feb. 2012 &  21:23    & 1440 &  NICS    & Amici & 1.23 & Hyades64 (1.22) \\\\ \n1027 Aesculapia & 18 Dec. 2012 & 20:42 & 1200 &  Dolores & LRR & 1.33 & Lan115-271 (1.14) \\\\ \n1027 Aesculapia & 17 Dec. 2012 & 20:07 & 1920 &  NICS & Amici & 1.29 & Lan115-271 (1.25) \\\\ \n1247 Memoria & 19 Dec. 2012 & 05:04 & 900 &  Dolores & LRR & 1.04 & Lan115-271 (1.14) \\\\ \n1623 Vivian & 20 Feb. 2012 &  05:06    & 900  &  Dolores & LRB & 1.32 & Lan102-1081(1.31) \\\\\n1623 Vivian & 20 Feb. 2012 &  05:24    & 900  &  Dolores & LRR & 1.39 & Lan102-1081(1.31) \\\\\n1623 Vivian & 20 Feb. 2012 &  01:26    & 1440 &  NICS    & Amici & 1.12 & HIP44103(1.04) \\\\ \n1687 Glarona & 19 Feb. 2012 &  03:01    & 600  &  Dolores & LRB & 1.24 & Lan98-978 (1.18) \\\\ \n1687 Glarona & 19 Feb. 2012 &  03:17    & 600  &  Dolores & LRR & 1.28 & Lan98-978 (1.18) \\\\ \n1687 Glarona & 19 Feb. 2012 &  01:26    & 720 &  NICS    & Amici & 1.04 & HIP59932 (1.09) \\\\ \n1778  Alfven & 17 Dec. 2012 & 06:04 & 1440 & NICS & Amici & 1.29 & HIP41815 (1.26) \\\\ \n1953 Rupertwildt & 18 Dec. 2012 & 04:19 & 300 &  Dolores & LRB & 1.17 & Lan102-1081 (1.15) \\\\ \n1953 Rupertwildt & 18 Dec. 2012 & 04:25 & 480 &  Dolores & LRR & 1.19 & Lan102-1081 (1.15) \\\\ \n1953 Rupertwildt & 17 Dec. 2012 & 03:48 & 960 & NICS & Amici & 1.11 & HIP22536 (1.13) \\\\ \n2009 Voloshina & 17 Dec. 2012 & 23:58 & 960 & NICS & Amici & 1.24 & Lan98-978 (1.28) \\\\ \n2203 Van Rhijn & 17 Dec. 2012 & 21:24 & 960 & NICS & Amici & 1.05 & HIP22536 (1.03) \\\\ \n2222 Lermontov  & 20 Feb. 2012 &  04:36    & 900  &  Dolores & LRB & 1.34 & Lan102-1081(1.31) \\\\\n2222 Lermontov  & 20 Feb. 2012 &  04:38    & 900  &  Dolores & LRR & 1.41 & Lan102-1081(1.31) \\\\\n2222 Lermontov  & 20 Feb. 2012 &  00:48    & 960 &  NICS    & Amici & 1.05 & Lan102-1081 (1.18) \\\\ \n2228  Soyuz-Apollo& 19 Dec. 2012 & 04:30 & 600 &  Dolores & LRB & 1.03 &  Lan115-271 (1.13) \\\\ \n2228  Soyuz-Apollo & 19 Dec. 2012 & 04:42 & 600 &  Dolores & LRR & 1.03 &  Lan115-271 (1.14) \\\\\n2264 Sabrina & 19 Dec. 2012 & 02:07 & 600 &  Dolores & LRB & 1.04 &  Lan115-271 (1.13) \\\\ \n2264 Sabrina & 18 Dec. 2012 & 06:21 & 600 &  Dolores & LRR & 1.45 &  Lan102-271 (1.15) \\\\ \n2264 Sabrina & 17 Dec. 2012 & 05:21 & 480 & NICS & Amici & 1.22 & Lan1021081 (1.16) \\\\ \n2270 Yazhi & 19 Dec. 2012 & 02:48 & 1100 &  Dolores & LRB & 1.01 &  Lan115-271 (1.13) \\\\ \n2270 Yazhi & 19 Dec. 2012 & 03:23 & 1100 &  Dolores & LRR & 1.02 &  Lan115-271 (1.13) \\\\ \n2270 Yazhi & 17 Dec. 2012 & 04:26 & 1920 &  NICS & Amici & 1.10 & HIP44027 (1.04) \\\\ \n2519    Annagerman & 19 Feb. 2012 &  04:46    & 900  &  Dolores & LRB & 1.16 & Land98-978 (1.17) \\\\ \n2519    Annagerman & 19 Feb. 2012 &  05:04    & 900  &  Dolores & LRR & 1.18 & Land98-978 (1.18) \\\\\n2519    Annagerman & 20 Feb. 2012 &  02:13    & 1920 &  NICS    & Amici & 1.18 & HIP59932 (1.09) \\\\ \n3174 Alcock & 19 Dec. 2012 & 01:22 & 600 &  Dolores & LRB & 1.26 &  Lan93-101 (1.23) \\\\ \n3174 Alcock & 19 Dec. 2012 & 01:34 & 600 &  Dolores & LRR & 1.30 &  HIP41815 (1.30) \\\\ \n3174 Alcock & 16 Dec. 2012 & 23:52 & 960 & NICS & Amici & 1.06 & HIP22536 (1.13) \\\\ \n3591 Vladimirskij & 18 Dec. 2012 & 21:53 & 2200 &  Dolores & LRR & 1.21 &  HIP44027 (1.18) \\\\ \n3591 Vladimirskij & 16 Dec. 2012 & 22:20 & 1920 & NICS & Amici & 1.36 & Lan93-101 (1.14) \\\\ \n3615 Safronov & 16 Feb. 2012 & 21:31 & 900 & Dolores & LRB & 1.21 & Lan98-978 (1.25) \\\\ \n3615 Safronov & 16 Feb. 2012 & 21:49 & 900 & Dolores & LRR & 1.21 & Lan98-978 (1.25) \\\\ \n3615 Safronov & 18 Dec. 2012 & 05:47 & 1200 & Dolores & LRR & 1.13 & Lan102-1081 (1.15) \\\\ \n3615 Safronov & 17 Dec. 2012 & 02:40 & 480 & NICS & Amici & 1.05 & HIP41815 (1.06) \\\\ \n4903 Ichikawa & 18 Dec. 2012 & 02:53 & 900 & Dolores & LRB & 1.12 & Lan102-1081 (1.15) \\\\ \n4903 Ichikawa & 18 Dec. 2012 & 03:25 & 900 & Dolores & LRR & 1.15 & Lan102-1081 (1.15) \\\\ \n4903 Ichikawa & 17 Dec. 2012 & 01:38 & 960 & NICS & Amici & 1.02 & HIP22536 (1.13) \\\\ \n\\hline\n\\hline\n\\end{longtable}\n\\end{center}\n}\n\n         \\begin{sidewaystable} \n       \\caption{Physical and orbital parameters of the Themis and Beagle families members investigated.  The albedo comes from WISE data (Masiero et al., 2011) when available, or from AKARI (indicated with the symbol $^*$, Usui et al., 2011). S$_{all}$ is the spectral slope value calculated \nin the whole {\\bf observed} wavelength range, S$_{VIS}$ for the  0.55-0.80 $\\mu$m range, S$_{NIR1}$ for the 0.9-1.4 $\\mu$m range, S$_{NIR2}$\nfor the 1.4-2.2 $\\mu$m range, and S$_{NIR3}$ for  1.1-1.8 $\\mu$m range). The family is indicated with $B$ for the Beagle, and $T$ for the Themis members. For (24) Themis we used the whole visible and NIR spectrum acquired on 18 Dec., 00:30-02:30 UT time.}\n        \\label{slope}\n        \\scriptsize{\n\\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|c|c|c|} \\hline\nAsteroid &   S$_{all}$         & S$_{Vis}$          & S$_{NIR1}$       & S$_{NIR2}$ & S$_{NIR3}$               &  D    & p$_{v}$ & Fam. & a & e & $sin(i)$ & H$_v$ \\\\          \n         &  (\\%\/$10^{3}$\\AA)   &  (\\%\/$10^{3}$\\AA) & $(\\%\/10^{3}$\\AA) & $(\\%\/10^{3}$\\AA) & $(\\%\/10^{3}$\\AA)  &  (Km) &         &      & (UA) &  &  & \\\\ \\hline\n 656        Beagle  &  1.30$\\pm$0.50  &    -0.60$\\pm$0.50  & 1.51$\\pm$0.52 & 1.29$\\pm$0.51 & 1.79$\\pm$0.51& 47.58$\\pm$0.33\t& 0.0782$\\pm$0.0222&   B &  3.15598& 0.154640 & 0.0235182&  9.95   \\\\ \n 1027    Aesculapia  &  0.56$\\pm$0.50  &    -0.72$\\pm$0.51  & 0.94$\\pm$0.52 & 1.24$\\pm$0.51 & 1.37$\\pm$0.50& 35.38$\\pm$0.34\t& 0.0814$\\pm$0.0071&   B &  3.15903& 0.152254 & 0.0240919& 10.82  \\\\ \n1687       Glarona  &  1.05$\\pm$0.50  &    -1.36$\\pm$0.51  & 1.95$\\pm$0.53 & 1.69$\\pm$0.52 & 2.32$\\pm$0.51& 42.01$\\pm$0.51  & 0.0795$\\pm$0.0130&   B &  3.15870& 0.153173 & 0.0242712& 10.53  \\\\ \n2519    Annagerman  &  0.11$\\pm$0.50  &    -0.65$\\pm$0.51  &-0.56$\\pm$0.51 & 0.60$\\pm$0.54 & 0.62$\\pm$0.55& 20.55$\\pm$0.31  & 0.1152$\\pm$0.0289&   B &  3.14765& 0.151787 & 0.0225837& 11.5    \\\\ \n3174        Alcock  &  0.19$\\pm$0.50  &    -2.88$\\pm$0.50  & 1.37$\\pm$0.51 & 0.39$\\pm$0.52 & 1.73$\\pm$0.50& 18.66$\\pm$0.80\t& 0.1020$\\pm$0.0090&   B &  3.15479& 0.154074 & 0.0239525& 11.95  \\\\\n3591  Vladimirskij  & -0.57$\\pm$0.50  &    -3.24$\\pm$0.51  & 0.16$\\pm$0.52 & 0.88$\\pm$0.52 & 0.91$\\pm$0.51& 16.42$\\pm$0.25\t& 0.0947$\\pm$0.0248&   B &  3.15704& 0.151941 & 0.0244373& 12.07   \\\\ \n3615      Safronov  & -0.79$\\pm$0.51  &    -1.63$\\pm$0.50  &-3.06$\\pm$1.30 & 1.85$\\pm$1.16 &-0.17$\\pm$0.94& 26.24$\\pm$0.10\t& 0.0852$\\pm$0.0159&   B &  3.15974& 0.152726 & 0.0238029& 11.49   \\\\ \n4903      Ichikawa  &  0.40$\\pm$0.50  &    -1.56$\\pm$0.51  & 1.51$\\pm$0.55 & 0.77$\\pm$0.62 & 2.11$\\pm$0.52& 14.79$\\pm$0.39\t& 0.1167$\\pm$0.0087&   B &  3.15055& 0.15180  & 0.0240145& 12.54  \\\\ \\hline\n  24       Themis   &  0.97$\\pm$0.51  &    -0.61$\\pm$0.50  & 1.24$\\pm$0.51 & 1.95$\\pm$0.50 & 2.15$\\pm$0.50& 202.34$\\pm$6.05 & 0.0641$\\pm$0.0157&   T &  3.13450& 0.152779 & 0.0189278&  7.21  \\\\ \n  62         Erato  & -0.06$\\pm$0.51  &    -1.73$\\pm$0.50  &-0.23$\\pm$0.50 & 0.73$\\pm$0.50 & 0.32$\\pm$0.50& 95.40$\\pm$2.0   & 0.0610$\\pm$0.0030&   T &  3.12169& 0.149807 & 0.0225411&  8.62   \\\\ \n  90       antiope  &  0.88$\\pm$0.50  &    -1.19$\\pm$0.50  & 1.45$\\pm$0.51 & 2.08$\\pm$0.51 & 2.38$\\pm$0.51& 121.13$\\pm$2.47\t& 0.0593$\\pm$0.0096&   T &  3.14619& 0.153819 & 0.0231415&  7.84  \\\\  \n268         Adorea  &  2.55$\\pm$0.50  &     1.56$\\pm$0.50  & 1.29$\\pm$0.51 & 3.76$\\pm$0.50 & 3.42$\\pm$0.50&140.59$\\pm$3.18  & 0.0436$\\pm$0.0066&   T &  3.09684& 0.167397 & 0.0243772&  8.3      \\\\\n  383        Janina  &  0.61$\\pm$0.50  &     2.97$\\pm$0.51  & 0.59$\\pm$0.51 & 0.42$\\pm$0.50 & 0.48$\\pm$0.50& 44.64$\\pm$0.74  & 0.0964$\\pm$0.0127&   T &  3.13408& 0.151863 & 0.0246743&  9.74   \\\\ \n 461        Saskia  &  3.19$\\pm$0.50  &     2.79$\\pm$0.50  & 2.49$\\pm$0.51 & 3.04$\\pm$0.51 & 3.50$\\pm$0.51& 48.72$\\pm$0.24  & 0.0479$\\pm$0.0079&   T &  3.11426& 0.155859 & 0.0239453& 10.4     \\\\ \n468           Lina  &   -             &    -1.41$\\pm$0.53  &  -            &  -            &  -           & 64.59$\\pm$1.98  & 0.0495$\\pm$0.0094&   T &  3.14189& 0.158666 & 0.0215053&  9.69 \\\\\n 492      Gismonda  &  0.79$\\pm$0.50  &    -1.42$\\pm$0.50  & 0.84$\\pm$0.51 & 1.21$\\pm$0.51 & 1.22$\\pm$0.51& 59.92$\\pm$0.35  & 0.0592$\\pm$0.0047&   T &  3.11191& 0.151069 & 0.0220835&  9.85   \\\\ \n 526          Jena  & -0.26$\\pm$0.50  &    -0.95$\\pm$0.50  &-1.60$\\pm$0.51 &-0.63$\\pm$0.50 &-0.25$\\pm$0.51& 51.03$\\pm$0.74  & 0.0580$\\pm$0.0177&   T &  3.12313& 0.156463 & 0.0249680& 10.      \\\\ \n 621      Werdandi  &  1.05$\\pm$0.50  &    -0.51$\\pm$0.51  & 2.01$\\pm$0.51 & 1.98$\\pm$0.52 & 1.91$\\pm$0.51& 30.71$\\pm$0.50$^{*}$   & 0.1240$\\pm$0.0050$^{*}$ &   T &  3.1193 & 0.150097 & 0.0253958& 10.98    \\\\ \n 846      Lipperta  &  1.39$\\pm$0.50  &     0.50$\\pm$0.50  & 1.72$\\pm$0.51 & 2.11$\\pm$0.51 & 2.83$\\pm$0.51& 51.45$\\pm$0.76$^{*}$\t& 0.0530$\\pm$0.0020$^{*}$&   T &  3.12791& 0.150758 & 0.0268551& 10.25    \\\\ \n 954            Li  &  2.83$\\pm$0.50  &     1.26$\\pm$0.50  & 3.00$\\pm$0.53 & 3.06$\\pm$0.53 & 4.15$\\pm$0.53& 52.06$\\pm$0.81  & 0.0552$\\pm$0.0073&   T &  3.13701& 0.150656 & 0.0230923& 10.07    \\\\ \n1247       Memoria  &   -             &    -0.64$\\pm$0.51  &  -            &  -            &  -           & 42.07$\\pm$0.16  & 0.0618$\\pm$0.0060&   T &  3.13875& 0.160209 & 0.0297064& 10.58 \\\\\n 1623        Vivian  &  2.53$\\pm$0.50  &     1.62$\\pm$0.50  & 1.87$\\pm$0.51 & 2.62$\\pm$0.52 & 3.20$\\pm$0.52& 29.98$\\pm$1.74$^{*}$\t& 0.0780$\\pm$0.0100$^{*}$ &   T &  3.13466& 0.15294  & 0.0239821& 11.19     \\\\\n 1778        Alfven &   0.17$\\pm$0.50  &       --           & 0.14$\\pm$0.51 & -0.47$\\pm$0.53 & 0.29$\\pm$0.50 & 20.6$\\pm$0.24 & 0.0951$\\pm$0.0069 &  T   & 3.1441 &  0.155394 &  0.0224451 & 11.7 \\\\ \n 1953   Rupertwildt  &  0.55$\\pm$0.50  &    -1.60$\\pm$0.50  & 0.79$\\pm$0.51 & 0.89$\\pm$0.52 & 2.16$\\pm$0.51& 21.97$\\pm$0.27  & 0.0697$\\pm$0.0060&   T &  3.11586& 0.147538 & 0.0242019& 11.84   \\\\ \n2009     Voloshina  &  1.78$\\pm$0.50  &      -             & 1.43$\\pm$0.51 & 1.71$\\pm$0.51 & 2.35$\\pm$0.50& 26.56$\\pm$0.48  & 0.1199$\\pm$0.0242 &   T &  3.11914 & 0.150115 & 0.0282294 & 11.16 \\\\\n2203     vanRhijn   & 1.39$\\pm$0.50   &      --            &  1.41$\\pm$0.51&  1.00$\\pm$0.51 & 1.90$\\pm$0.51 & 22.28$\\pm$0.27 & 0.0898$\\pm$0.0058 &  T &  3.10989 & 0.149153 &  0.0209356 & 11.57 \\\\\n2222     Lermontov  &  2.39$\\pm$0.50  &     1.96$\\pm$0.50  & 2.05$\\pm$0.51 & 2.87$\\pm$0.52 & 2.85$\\pm$0.51& 32.22$\\pm$0.22  & 0.0469$\\pm$0.0061&   T &  3.11619& 0.159625 & 0.0228687& 11.29     \\\\\n2228  Soyuz-Apollo  &   -             &     1.65$\\pm$0.51  &  -            &  -            &  -           & 26.08$\\pm$0.29  & 0.1134$\\pm$0.0198&   T &  3.14342& 0.160273 & 0.0241862& 11.44  \\\\\n 2264       Sabrina  &  0.08$\\pm$0.50  &    -1.49$\\pm$0.50  & 0.44$\\pm$0.52 & 0.40$\\pm$0.51 & 1.56$\\pm$0.51& 37.34$\\pm$0.38  & 0.0799$\\pm$0.0138&   T &  3.13084& 0.154135 & 0.0234201& 10.92   \\\\ \n2270         Yazhi  &  1.82$\\pm$0.50 &    -0.07$\\pm$0.51  & 1.12$\\pm$0.55 & 0.06$\\pm$0.54 & 1.65$\\pm$0.52& 26.73$\\pm$0.30  & 0.1081$\\pm$0.0225&   T &  3.15117& 0.15233  & 0.0188940& 11.43    \\\\ \n \\hline\n\\end{tabular}\n}\n         \\end{sidewaystable}\n\n         \\thispagestyle{empty}    \n\n     \n \\begin{table}\n         \\begin{center} \n         \\caption{Identification of absorption bands attributed to hydrated silicates on the Themis family members.}\n        \\label{hydra}\n                \\begin{tabular}{|l|c|c|c|c|} \\hline\n         Asteroid     & W$_{in}$ (\\AA) & W$_{fin}$ (\\AA) & Band$_{center}$ (\\AA) & Depth (\\%) \\\\ \\hline\n         24 Themis       & 5715  &  8730  &   7333$\\pm$48 &   2.6$\\pm$0.1  \\\\\n         90 Antiope      & 5358  &  8375  &   6972$\\pm$43  & 1.4$\\pm$0.1  \\\\\n         461 Saskia      & 5648  &  8338  &   6881$\\pm$123  & 1.2$\\pm$0.1  \\\\\n     \t                 & 12397  &  15126  &  13175$\\pm$55  & 2.8$\\pm$0.2  \\\\\n                         & 15965  &  21378  &  18157$\\pm$31  & 2.7$\\pm$0.2  \\\\\n         846 Lipperta     & 5487  &  8480  &    6842$\\pm$65  & 1.02$\\pm$0.1 \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n \\newpage\n \n \\begin{table}\n         \\begin{center} \n         \\caption{RELAB matches. The first eight asteroids belong to the Beagle family.}\n         \\label{met}\n          \\scriptsize{\n        \\begin{tabular}{|l|c|c|c|c|c|} \\hline\n          Asteroid & Albedo & Best fit & Met refl. & Met Class \\& name &  Grain size \\\\ \\hline\n 656           Beagle  & 0.0782   &     NAMC02      & 0.033 &  CM2 LEW90500,45 &  $<$ 500 $\\mu$m           \\\\\n 1027        Aesculapia  & 0.0814 &     C1MP61      & 0.067 &  CM2 GRO95577,6 particulate & $<$ 125 $\\mu$m           \\\\\n 1687           Glarona  & 0.0795 &     camh52      & 0.068 &  CM2  Boriskino  & $<$ 45 $\\mu$m         \\\\\n2519       Annagerman  & 0.1152   &     c1mb20      & 0.063 &  CM2  Unusual Y-86720,77 &    $<$ 125 $\\mu$m            \\\\\n 3174            Alcock  & 0.102  &     MGP094      & 0.059 &  CM2 Murchison          & whole rock  \\\\ \n3591      Vladimirskij  & 0.0947  &     c1mb19      & 0.036 &  CI  unusual, Y-82162,79 & $<$ 150 $\\mu$m     \\\\\n3615          Safronov  & 0.0852  &     BKR1MA076   & 0.042 &  CM2\/CR Kaidun            & $> $  250 $\\mu$m        \\\\\n4903          Ichikawa  & 0.1167 &      CGP090      & 0.046 &  CM2 Cold Bokkevelt      & 150 -- 500 $\\mu$m    \\\\   \\hline\n  24   Themis (17 Dec)  & 0.0641 &      NAMC02      & 0.033 &  CM2 LEW90500,45         & $<$ 500 $\\mu$m           \\\\\n  62             Erato  & 0.061  &      ccmb19      & 0.050 &  CI unusual, Y-82162,79  &    \\\\\n  90           antiope  & 0.0593 &      NAMC02      & 0.033 &  CM2 LEW90500,45         & $<$ 500 $\\mu$m           \\\\\n  268           Adorea  & 0.0436 &      ccmb64      & 0.031 &  CM2 Murchison heated 600 C & $<$ 63 $\\mu$m \\\\\n 383            Janina  & 0.0964 &      c1mb18      & 0.06  &  CM2 unusual B-7904,108     &             \\\\ \n 461            Saskia  & 0.0479 &      s1rs42      & 0.037 &  CM2 Mighei                 &   \\\\\n 492          Gismonda  & 0.0592 &      C1RS46      & 0.036 &  CM2 Boriskino             & \\\\\n 526              Jena  & 0.0580 &      CGP130      & 0.047 &  CV3 Grosnaja             & 75--150$\\mu$m             \\\\\n 621          Werdandi  & 0.124  & BKR1MP022L0      & 0.046 &  CM2 MAC88100  laser irr.             & $<$ 125 $\\mu$m       \\\\    \n 846          Lipperta  &  0.053 &      ncmp22      & 0.033 &  CM2 MAC88100,30           & $<$ 125 $\\mu$m \\\\\n 954                Li  & 0.0552 &      s1rs42      & 0.035 &  CM2 Mighei               &  \\\\\n1623            Vivian  &  0.078 &      ccmb64      & 0.031 &  CM2 Murchison heated 600 C &  $<$ 500 $\\mu$m     \\\\\n 1953       Rupertwildt  & 0.0697 &     CGP142      & 0.046 &  CM2 ColdBokkevelt         & 75-500 $\\mu$m\\\\\n2222         Lermontov  & 0.0469 &      ccmb64      & 0.031 &  CM2 Murchison heated 600 C\t& $<$ 500 $\\mu$m  \\\\\n2264           Sabrina  & 0.0799 &      CGP094      & 0.057 &  CM2 Murchison            & whole rock\\\\\n2270             Yazhi  & 0.1081 &      CBMS01      & 0.035 &  CM2 Migei Separates      & 100 -- 200 $\\mu$m   \\\\ \\hline\n        \\end{tabular}\n        }\n\\end{center}\n \\end{table}\n\n\n\\end{document}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\\label{Intro}\n\\label{sec:introduction}\nRecent advancements in the field of communication systems and computational methods necessitate improvements in the traditional cryptosystems. Substitution box (S-box) and pseudo random number generator (PRNG) play an important role in many cryptosystems such as Data Encryption Standard (DES)~\\cite{DES}, Advanced Encryption Standard (AES)~\\cite{AES}, Twofish security system~\\cite{F2}, Blowfish cryptosystem~\\cite{BF},  International Data Encryption Algorithm (IDEA)~\\cite{IDEA} and the cryptosystems developed in~\\cite{umar2, NJia, XC, SS}.\nIt has been pointed out by many researchers that the security of a cryptosystem can be improved by using dynamic S-boxes instead of a single static S-box, see for example~\\cite{RB, SK, MM, AS, MG, KK}.\nThis fact necessitates the development of new S-box generators which can generate a large number of distinct and mutually uncorrelated S-boxes with high cryptographic properties in low time and space complexity~\\cite{Azam}.\n\nMany researchers have proposed improved S-box generators and PRNGs to enhance the security of data against modern cryptanalysis.\nThese improvements are mainly based on finite field arithmetic and chaotic systems.\nKhan and Azam~\\cite{MN1, MN2} developed two different methods to generate 256 cryptographically strong S-boxes by using Gray codes, and affine mapping.\nJakimoski and Kocarev~\\cite{Jaki} used chaotic maps to develop a four-step method for the generation of an S-box. $\\rm \\ddot{O}$zkaynak and $\\rm \\ddot{O}$zer~\\cite{Ozkaynak2} introduced a new method based on a chaotic system to develop secure S-boxes. Unlike the traditional use of chaotic maps, Wang et al.~\\cite{YW} proposed an efficient algorithm to construct  S-boxes using gnetic algorithm and chaotic maps. Yin et al.~\\cite{Yin} proposed an S-box design technique using iteration of the chaotic maps. Tang and Liao~\\cite{Tang} constructed S-boxes based on an iterating discretized chaotic map.\nLambi\\'c~\\cite{Lambic} used a special type of discrete chaotic map to obtain bijective S-boxes. \\\"Ozkaynak et al.~\\cite{Ozkaynak} proposed a new S-box based on a fractional order chaotic Chen system.\n Zhang et al.~\\cite{IChing} used I-Ching operators for the construction of highly non-linear S-boxes, and the proposed approach is very efficient.\n\n\nSimilarly, chaotic systems are used to generate pseudo random numbers (PRNs), see for example~\\cite{MF, VP, CG, TS, ZF}. Francois et al.~\\cite{MF} presented a PRNG based on chaotic maps to construct multiple  key sequences. Patidar and Sud~\\cite{VP} designed a PRNG with optimal cryptographic properties using chaotic logistic maps. Guyeux et al.~\\cite{CG} developed a chaotic PRNG with the properties of topological chaos which offers sufficient security for cryptographic purposes. Stojanovski and Kocarev~\\cite{TS} analyzed a PRNG based on a piecewise linear one dimensional chaotic map. Fan et al.~\\cite{ZF} proposed a PRNG using generalized Henon map, and a novel technique is used to improve the characteristics of the proposed sequences.\n\nIt has been pointed out by Jia et al.~\\cite{NJia}\nthat the PRNs generated by a chaotic system can have small period due to the hardware computation issues and revealed that elliptic curve (EC) has high security than chaotic system.\nHowever, the computation over ECs is usually performed by group law which is computationally inefficient.\nHayat and Azam~\\cite{umar2}\nproposed an efficient S-box generator and a PRNG based on ECs by using a total order as an alternative to group law.\nThis S-box generator is efficient than the other methods over ECs, however their time and space complexity are $\\mathcal{O}(p^2)$ and $\\mathcal{O}(p)$, respectively, where $p$ is the prime of the underlying EC.\nFurthermore the S-box generator does not guarantee the generation of an S-box.\nThe PRNG proposed by Hayat and Azam~\\cite{umar2} also takes $\\mathcal{O}(p^2)$ and $\\mathcal{O}(p)$ time and space, respectively, to generate a sequence of pseudo random numbers (SPRNs) of size $m \\leq p$.\nAzam et al.~\\cite{Azam}\nproposed an improved S-box generation method to generate bijective S-boxes by using ordered Mordell elliptic curves (MECs).\nThe main advantage of this method is that its time and space complexity are $\\mathcal{O}(mp)$ and $\\mathcal{O}(m)$, respectively, where $m$ is the size of an S-box.\nAzam et al.~\\cite{ikram}\nproposed another S-box generator to generate $m \\times n$, where $m \\leq n$ injective S-boxes which can generate a large number of distinct and mutually uncorrelated S-boxes by using the concept of isomorphism on ECs.\nThe time and space complexity of this method are $\\mathcal{O}(2^np)$ and $\\mathcal{O}(2^n)$, where $n \\leq p$ and is the size of co-domain of the resultant S-box.\nA common draw back of these S-box generators is that the cryptographic properties of their optimal S-boxes are far from the theoretical optimal values.\n\nThe aim of this paper is to propose an efficient S-box generator and a PRNG based on an ordered MEC to generate a large number of distinct, mutually uncorrelated S-boxes and PRNs with optimal cryptographic properties in low time and space complexity to overcome the above mentioned drawbacks.\nThe rest of the paper is organized as follows:\nIn Section~\\ref{Prel} basic definitions are discussed. The proposed S-box generator is described in Section~\\ref{Cons}. Section~\\ref{Anal} consists of security analysis and comparison of the S-box generator.\nThe proposed algorithm for generating PRNs and some general results are given in Section~\\ref{RNG}.\nThe proposed SPRNs are analyzed in Section~\\ref{RA}, while Section~\\ref{Con} concludes the whole paper.\n\n\\section{Preliminaries}\\label{Prel}\nThroughout this paper, we denote a finite set $\\{0, 1, \\ldots, m-1\\}$ simply by $[0, m-1]$.\nA finite field over a prime number $p$ is the set $[0, p-1]$ denoted by $\\mathbb{F}_{p}$ with binary operations addition and multiplication under modulo $p$.\nA non-zero integer $\\alpha \\in \\mathbb{F}_{p}$ is said to be {\\it quadratic residue} (QR) if there exists an integer $\\beta \\in \\mathbb{F}_p$ such that $\\alpha\\equiv \\beta^2 \\pmod p$.\nA non-zero integer in $\\mathbb{F}_{p}$ which is not QR is said to be {\\it quadratic non-residue} (QNR).\n\nFor a prime $p$, non-negative $a\\in \\mathbb{F}_{p}$ and positive $b\\in \\mathbb{F}_{p}$, the EC $E_{p,a,b}$ over a finite field $\\mathbb{F}_{p}$ is defined to be the collection of identity element $\\infty$ and ordered pairs $(x, y) \\in \\mathbb{F}_p \\times \\mathbb{F}_p$ such that \\[ y^{2}\\equiv x^{3}+ax+b \\pmod p.\\]\nIn this setting, we call $p,a$ and $b$ the {\\it parameters} of $E_{p,a,b}$.\nThe number $\\#E_{p,a,b}$ of all such points can be calculated using Hasse's\ntheorem~\\cite{Schoof} \\[|\\#E_{p,a,b}-p-1|\\leq 2\\sqrt{p}.\\]\nTwo ECs $E_{p,a,b}$ and $E_{p,a',b'}$ over $\\mathbb{F}_{p}$ are isomorphic if and only if there exists a non-zero integer $t\\in \\mathbb{F}_{p}$ such that $a't^4\\equiv a \\pmod p$ and $b't^6\\equiv b \\pmod p$.\nIn this case, $t$ is called {\\it isomorphism parameter} between the ECs $E_{p,a,b}$ and $E_{p,a',b'}$.\nFor an isomorphism parameter $t$, each point $(x,y)\\in E_{p,a,b}$ is mapped on $(t^2x,t^3y)\\in E_{p,a',b'}$.\nNote that an isomorphism is an equivalence relation on all ECs over $\\mathbb{F}_{p}$, and therefore all ECs can be divided into equivalence classes~\\cite{ikram}. For the sake of simplicity we represent an arbitrary class by $\\mathcal{C}_i$ and assume that the class $\\mathcal{C}_1$ contains the EC $E_{p,0,1}$.\nA non-negative integer $b\\in \\mathbb{F}_{p}$ such that $E_{p,a,b}\\in \\mathcal{C}_i$ is called {\\it representative} $R(p, \\mathcal{C}_i)$ of the class $\\mathcal{C}_i$.\nClearly, it holds that $R(p,\\mathcal{C}_1) = 1$.\n\nAn EC $E_{p,a,b}$ with $a = 0$ is said to be a {\\it Mordel elliptic curve}.\nThe following theorem is from~\\cite[6.6 (c), p.~188]{Schoof}.\n\\begin{theorem}\\label{mordel}\nLet $p>3$ be a prime such that $p\\equiv 2\\pmod{3} $.\nFor each non-zero $b\\in \\mathbb{F}_{p}$, the MEC $E_{p,0,b}$ has exactly $p+1$\ndistinct points, and has each integer from $[0,p-1]$ exactly once as a $y$%\n-coordinate.\n\\end{theorem}\n\\noindent\nFurthermore, by~\\cite[Lemma 1]{ikram} it follows that there are only two classes of MECs when $p\\equiv 2 \\pmod 3$.\nHenceforth, we denote an MEC $E_{p,0, b}$ by simply $E_{p, b}$ and the term MEC stands for an MEC $E_{p, b}$ such that $p\\equiv 2 \\pmod 3$.\n\nFor a subset $A$ of $[0, p-1]$ and an ordered MEC $(E_{p, b}, \\prec)$, we define a total order $\\prec^*$ on $A$ w.r.t. the ordered MEC such that for any two elements $a_1, a_2 \\in A$ it holds that $a_1 \\prec^* a_2$ if and only if $(x_1, a_1) \\prec (x_2, a_2)$.\nFor any two non-negative integers $p$ and $m$ such that $1 \\leq m \\leq p$, we define an {\\em $(m, p)$-complete set} to be a set $Q$ of size $m$ such that for each element $ q \\in Q$, it holds that $0\\leq q \\leq p-1$,\nand no two elements of $Q$ are congruent with each other under modulo $m$, i.e., for each $q,q' \\in Q$, it holds that $(q \\not\\equiv q') \\pmod{m}$.\nWe denote an ordered set $A$ with a total order $\\prec$ by an ordered pair $(A, \\prec)$.\nLet $(A, \\prec)$ be an ordered set, for any two elements $a,a' \\in A$ such that $a \\prec a'$, we read as $a$ is smaller than $a'$ or $a'$ is greater than $a$ w.r.t. the order $\\prec$.\nFor simplicity, we represent the elements of $(A, \\prec)$ in the form of a non-decreasing sequence and $a_i$ denotes the $i$-th element of the ordered set in its sequence representation.\nFor an ordered MEC $(E_{p, b},\\prec)$ and an $(m,p)$-complete set $Y$, we define the \\textit{ordered $(m,p)$-complete set} $Y^*$ with ordering $\\tilde\\prec$ due to $Y$ and $\\prec$ such that for any two element $y_1, y_2 \\in Y^*$ with $y_1' \\equiv y_1 \\pmod{m}$ and $y_2' \\equiv y_2 \\pmod m$, where $y_1', y_2' \\in Y$, it holds that $y_1 \\tilde\\prec y_2$ if and only if $(x_1, y_1') \\prec (x_2, y_2')$.\n\nFor a given MEC $E_{p, b}$, Azam et al.~\\cite{Azam} defined three typical type of orderings \\textit{natural} $N$, \\textit{diffusion} $D$ and \\textit{modulo diffusion} $M$ ordering  based on the coordinates of the points on $E_{p, b}$ as\\\\\n$(x_{1},y_{1})N(x_{2},y_{2}) \\Leftrightarrow\n\\begin{cases}\n{\\rm either \\ } x_{1}<x_{2}, {\\rm \\ or} \\\\[5pt]\nx_{1}=x_{2} {\\rm \\ and \\ } y_{1}<y_{2}; %\n\\end{cases\n\\\\ [5pt]\n(x_{1},y_{1})D(x_{2},y_{2}) \\Leftrightarrow\n\\begin{cases}\n{\\rm either \\ } x_{1}+y_{1}<x_{2}+y_{2}, {\\rm \\ or} \\\\[5pt]\nx_{1}+y_{1}=x_{2}+y_{2} {\\rm \\ and \\ } x_{1}<x_{2};\n\\end{cases}\n\\\\[5pt]\n(x_{1},y_{1})M(x_{2},y_{2}) \\Leftrightarrow \\\\[5pt]\n\\begin{cases}\n{\\rm either \\ } (x_{1}+y_{1}<x_{2}+y_{2})\\pmod p, {\\rm \\ or} \\\\[5pt]\nx_{1}+y_{1}\\equiv x_{2}+y_{2}\\pmod p\n{\\rm \\ and \\ } x_{1}<x_{2}.\n\\end{cases}\n$\n\\section{The Proposed S-box Construction Scheme}\\label{Cons}\nFor an ordered MEC $(E_{p,b}, \\prec)$, an $(m,p)$-complete set $Y$ and a non-negative integer $k \\leq m -1$, we define an \\textit{$(m,p)$-complete S-box} $\\sigma(p,b,\\prec,Y, k)$ due to $(E_{p,b}, \\prec)$, $Y$ and $k$ to be a mapping from $[0, m-1]$ to $(Y^*, \\tilde\\prec)$ such that $\\sigma(p,b,\\prec,$ $Y,k)(i) = y_{(i +k)\\pmod{m}}\\pmod{m}$, where $y_{(i +k)\\pmod{m}}$ is the $(i +k)\\pmod{m}$-th element of the ordered $(m,p)$-complete set $(Y^*, \\tilde\\prec)$ in its sequence representation.\n\\begin{lemma}\\label{bijective}\nFor any ordered MEC $(E_{p,b}, \\prec)$, an $(m,p)$-complete set $Y$ and a non-negative integer $k \\leq m -1$, the $(m,p)$-complete S-box $\\sigma(p,b,\\prec,Y, k)$ is a bijection.\n\\end{lemma}\n\\begin{proof}Suppose on contrary that there exist $i,j \\in [0, m-1]$ such that $\\sigma_{p,b,\\prec,Y,k}(i) = \\sigma_{p,b,\\prec,Y,k}(j)$.\nThis implies that $y_i' = y_j'$, where $y_i' \\equiv y_i\\pmod{m}$ and $y_j' \\equiv y_j\\pmod{m}$ and $y_i', y_j' \\in Y$.\nThis leads to a contradiction to the fact that $Y$ is an $(m,p)$-complete set.\nThus, $\\sigma(p,b,\\prec,Y, k)$ is a one-one mapping on the finite sets of same order, and hence it is a bijection.\n\\end{proof}\n\nFor prime $p = 52511$ and $m = 256$, an  $(256,52511)$-complete subset $Y$ of the MEC is given in Table~\\ref{FT}, while the $(256,52511)$-complete S-box $\\sigma(52511, 1, N, Y, 0)$ due to the ordered MEC $(E_{52511,1}, N)$ is presented in Table~\\ref{SN} in hexagonal format.\nEach entry of Table~\\ref{SN} is obtained from the corresponding entry of Table~\\ref{FT} by applying modulo $256$ operator.\n\\begin{table*}[htb]\n\\caption{The  $(256,52511)$-complete set $Y$}\n\\label{FT}%\n\\centering\n\\scalebox{0.99}\n{\\begin{tabular}{llllllllllllllll}\n\\hline\n A792 & 4A5C & 9AF5 & 01C5 & 421  & 814D & B3A2 & 5CA3 & 834B & 9F90 & 1C7D & BF6A & 0A11 & 7A9D & 9E91 & 6135 \\\\%[50pt]\n1D8D & 9425 & 3F36 & 7954 & 1E1E & 5B47 & 1420 & 71CA & 8089 & 80C4 & 3150 & 12EF & 36C3 & BEA7 & 6170 & 2256 \\\\\n298A & 005E & 8032 & 0F00 & 270A & 51D0 & 421C & 942  & 6BDF & 2848 & 87FC & 4418 & 2BFB & 121B & 6F2D & 11CC \\\\\n886  & 8E53 & 6BD2 & AC14 & B65B & 062B & 37F1 & B627 & 47D4 & 59A6 & 2878 & 7D76 & 76CB & 7005 & 0CBE & 8F8F \\\\\n609E & 7A83 & 61F4 & 23C0 & 3AC2 & 3502 & BC40 & 88DE & 3645 & 2EEC & B8B3 & BBD9 & 84D5 & 165F & C061 & 0BE8 \\\\\nAE34 & 6431 & 906B & 15E4 & BE74 & 5423 & 10AA & 4D75 & B037 & 556F & 6F99 & 242E & 31AD & C9F9 & A679 & 3F82 \\\\\n749A & 7F55 & 9267 & AF29 & 33BC & 1A0E & 270D & 2312 & 7857 & B730 & 5C17 & AAA4 & 7DF7 & 698B & 7FDB & 66AC \\\\\nA203 & B46D & 7DE9 & 7E80 & 72B2 & 97E1 & 70D1 & 18D8 & 76ED & 4677 & 7A4E & 7F3F & 96B8 & 8A94 & 91D3 & 8295 \\\\\n6E0F & 7A0B & 221A & 11C7 & A7A1 & 1563 & 33BB & 15EA & 62BA & 0EB9 & 8041 & 6998 & C260 & 127C & 0B2F & 38AE \\\\\n7626 & 12A8 & 50D6 & B0CE & 67CD & 766C & 22BD & 109F & 4E4C & CBF2 & 5CA5 & 2528 & 1964 & 4724 & CAAF & 966  \\\\\nA587 & AE01 & 5584 & 0A3D & 3859 & 7504 & 063E & 5251 & 767B & 0AFF & 50E7 & 7765 & 2688 & BC58 & 972A & 0EE6 \\\\\n295A & 0BB6 & 4B43 & 5906 & 476E & C5C9 & 20A9 & 45AB & C57A & 1D07 & 694A & B57F & 0D15 & 1CBF & ABFA & C3B5 \\\\\n2096 & B138 & A671 & A262 & BAF3 & 4CB1 & 054F & C5DD & 9B85 & C144 & BBDC & 7969 & C85D & 91C6 & 0A49 & 9DE2 \\\\\nC6DA & 278C & 1C13 & 29D7 & 708E & 827E & 0FC8 & 4FEB & 4BEE & 1F97 & 20FE & 26E0 & 0E93 & 4E9C & A2E5 & 841D \\\\\nADF0 & B273 & A6E3 & 440C & AB08 & 3952 & 103A & A472 & C42C & 36CF & 9768 & 6809 & 0E22 & C439 & 291F & AEFD \\\\\nA7C1 & C23B & 2FF6 & A046 & 3BB4 & ACA0 & 5A9B & 95F8 & 7919 & 4381 & A9B0 & 7110 & 7433 & 1816 & 39B7 & 1A3C\\\\ \\hline\n\\end{tabular}}\n\\end{table*}\n\n\\begin{table}[H]\n\\caption{The  proposed $(256,52511)$-complete S-box $\\sigma(52511, 1, N, Y, 0)$}\n\\label{SN}%\n\\centering\n\\resizebox{\\columnwidth}{!}\n{\\begin{tabular}{cccccccccccccccc}\n\\hline\n92 & 5C & F5 & C5 & 21 & 4D & A2 & A3 & 4B & 90 & 7D & 6A & 11 & 9D & 91 & 35 \\\\\n8D & 25 & 36 & 54 & 1E & 47 & 20 & CA & 89 & C4 & 50 & EF & C3 & A7 & 70 & 56 \\\\\n8A & 5E & 32 & 0  & 0A & D0 & 1C & 42 & DF & 48 & FC & 18 & FB & 1B & 2D & CC \\\\\n86 & 53 & D2 & 14 & 5B & 2B & F1 & 27 & D4 & A6 & 78 & 76 & CB & 5  & BE & 8F \\\\\n9E & 83 & F4 & C0 & C2 & 2  & 40 & DE & 45 & EC & B3 & D9 & D5 & 5F & 61 & E8 \\\\\n34 & 31 & 6B & E4 & 74 & 23 & AA & 75 & 37 & 6F & 99 & 2E & AD & F9 & 79 & 82 \\\\\n9A & 55 & 67 & 29 & BC & 0E & 0D & 12 & 57 & 30 & 17 & A4 & F7 & 8B & DB & AC \\\\\n3  & 6D & E9 & 80 & B2 & E1 & D1 & D8 & ED & 77 & 4E & 3F & B8 & 94 & D3 & 95 \\\\\n0F & 0B & 1A & C7 & A1 & 63 & BB & EA & BA & B9 & 41 & 98 & 60 & 7C & 2F & AE \\\\\n26 & A8 & D6 & CE & CD & 6C & BD & 9F & 4C & F2 & A5 & 28 & 64 & 24 & AF & 66 \\\\\n87 & 1  & 84 & 3D & 59 & 4  & 3E & 51 & 7B & FF & E7 & 65 & 88 & 58 & 2A & E6 \\\\\n5A & B6 & 43 & 6  & 6E & C9 & A9 & AB & 7A & 7  & 4A & 7F & 15 & BF & FA & B5 \\\\\n96 & 38 & 71 & 62 & F3 & B1 & 4F & DD & 85 & 44 & DC & 69 & 5D & C6 & 49 & E2 \\\\\nDA & 8C & 13 & D7 & 8E & 7E & C8 & EB & EE & 97 & FE & E0 & 93 & 9C & E5 & 1D \\\\\nF0 & 73 & E3 & 0C & 8  & 52 & 3A & 72 & 2C & CF & 68 & 9  & 22 & 39 & 1F & FD \\\\\nC1 & 3B & F6 & 46 & B4 & A0 & 9B & F8 & 19 & 81 & B0 & 10 & 33 & 16 & B7 & 3C \\\\ \\hline\n\\end{tabular}}\n\\end{table}\n\n\nNext we present two efficient algorithms to compute an $(m,p)$-complete S-box.\nThe first algorithm is based on Theorem~\\ref{mordel}.\n\\begin{algorithm}[H]\n\n\t\\caption{{\\bf Constructing the proposed S-box } } \\label{algo_sbox_1}\n\\begin{algorithmic}[1]\n\\REQUIRE An ordered MEC $(E_{p,b}, \\prec)$, an $(m,p)$-complete set~$Y$ and a non-negative integer $k\\leq m-1$.\n\\ENSURE The proposed $(m,p)$-complete S-box $\\sigma(p,b,\\prec,$ $Y,k)$.\n\n\\STATE $A:=\\emptyset$; \/*A set containing the points of $E_{p,b}$ with $y$-coordinates from $Y$*\/\n\n\\FORALL {$y \\in Y$}\n\\STATE $B:= [0, p-1]$; $t:= $ {\\tt No};\n\t\\WHILE{$q=$ {\\tt No}}\n\t\t\\STATE $x \\in B$;\n\t\t\\IF{$(x^3 +b - y^2 \\equiv 0) \\pmod p$}\n\t\t\t\\STATE $q:=$ {\\tt Yes}; $A:= A \\cup \\{(x,y)\\}$\n\t\t\\ENDIF;\n\t\t\\STATE $B:=B\\setminus \\{x\\}$\n\t\\ENDWHILE\n\\ENDFOR;\n\\STATE Sort $Y^* := [0, m-1]$ w.r.t. the element of $A$, i.e., sort $Y^*$ w.r.t. the ordering $\\tilde\\prec$;\n\\STATE $\\pi := (\\pi(0), \\pi(1), \\ldots, \\pi(m-1))$;\n\\FORALL {integer $i \\in [0, m-1]$}\n\t\\STATE $\\pi(i) := y_{(i+k)\\Mod{m}}\\pmod{m}$, where $y_{(i +k)\\pmod{m}}$ is the $(i +k)\\pmod{m}$-th element of the ordered $(m,p)$-complete set $(Y^*, \\tilde\\prec)$\n\\ENDFOR;\n\\STATE Output $\\pi$ as the $(m,p)$-complete S-box $\\sigma(p,b,\\prec,Y, k)$.\n\t\n\\end{algorithmic}\n\\end{algorithm}\n\\begin{lemma}\\label{imp_1}\nFor an ordered MEC $(E_{p,b}, \\prec)$, an $(m,p)$-complete set $Y$ and a non-negative integer $k\\leq m-1$, the $(m,p)$-complete S-box $\\sigma(p,b,\\prec,Y, k)$ can be computed in $\\mathcal{O}(mp)$ time and $\\mathcal{O}(m)$ space by using Algorithm~\\ref{algo_sbox_1}.\n\\end{lemma}\n\\begin{proof}\nIn Algorithm~\\ref{algo_sbox_1} there is for-loop of size $m$ over the elements of $Y$, which has a nested while-loop to compute the subset $A$ of the MEC such that the points in $A$ has $y$-coordinate in $Y$.\nThis step is necessary to compute the ordered $(m,p)$-complete set $(Y^*, \\tilde\\prec)$ due to $Y$ and $\\prec$.\nNote that the nested while-loop will iterate for at most $p$-times, since by Theorem~\\ref{mordel}, for each $y \\in [0, p-1]$ there is a unique $x \\in [0, p-1]$ such that $(x^3 +b - y^2 \\equiv 0) \\pmod p$.\nThus, this for-loop and while-loop take $\\mathcal{O}(mp)$ time in the worst case, while the sorting of $Y^*$ takes $\\mathcal{O}(m\\log m)$ time.\nFinally, there is another independent for-loop of size $m$ to compute the sequence $\\pi$ which takes $\\mathcal{O}(m)$ time.\nThus, Algorithm~\\ref{algo_sbox_1} takes $\\mathcal{O}(mp) + \\mathcal{O}(m\\log m) + \\mathcal{O}(m)$ time to execute in the worst case.\nBy using the fact that $mp > m \\log p$, since $\\log{p} < p$ and $m \\leq p$ and by the property of $\\mathcal{O}$ notation, the time complexity of Algorithm~\\ref{algo_sbox_1} is $\\mathcal{O}(mp)$.\nFurthermore, Algorithm~\\ref{algo_sbox_1} only stores sets of size $m$, and therefore its space complexity is $\\mathcal{O}(m)$.\nThis completes the proof.\n\\end{proof}\nNext we present another algorithm for the generation of $(m,p)$-complete S-boxes on a fixed MEC.\nFor this we prove the following results.\n\nFor a fixed ordered MEC $(E_{p,b}, \\prec)$, a positive integer $m \\leq p$ and an integer $0 \\leq k \\leq m - 1$, let Num$(E_{p,b}, \\prec, m, k)$ denote the total number of $(m,p)$-complete S-boxes, possibly with repetition, generated due to the ordered MEC, $m$ and $k$.\n\\begin{lemma}\\label{LL}\nFor a fixed ordered MEC $(E_{p,b}, \\prec)$ and a positive integer $m \\leq p$,  the total number of $(m,p)$-complete S-boxes, possibly with repetition, generated due to the MEC is equal to $m(q+1)^{r}q^{m-r}$, where $p=mq+r$, $0\\leq r< m,$ and $0\\leq k \\leq m-1$.\n\\end{lemma}\n\\begin{proof}\nFor a fixed integer $0 \\leq k \\leq m-1$, it holds by the definition of $(m,p)$-complete S-box that the total number of $(m,p)$-complete S-boxes, possibly with repetition, generated due to the ordered MEC, $m$ and $k$\nis equal to the number of distinct $(m,p)$-complete sets.\nIf $p = mq + r$, where $0 \\leq r \\leq m-1$, then there are $q+1$ (resp., $q$) integers $\\ell$ (resp., $h$)\nsuch that $\\ell \\pmod m \\in [0, r-1]$ (resp., $h \\pmod m \\in [r, m-1]$).\nThus, to construct an $(m,p)$-complete set there are $q+1$ (resp., $q$) choices of an integers $a$ such that $a \\pmod m \\in [0, r-1]$ (resp., $[r, m-1]$).\nThis implies that there are $(q+1)^r q^{m-r}$ distinct $(m,p)$-complete sets.\nHence, the number of $(m, p)$-complete S-boxes due to the MEC is $m(q+1)^r q^{m-r}$, since $0 \\leq k \\leq m-1$.\n\\end{proof}\n\\begin{observation}\\label{obs_set}\nFor any subset $F$ of an MEC $E_{p,b}$ there exists a unique subset $F'$ of either MEC $E_{p, R(p,\\mathcal{C}_{1})}$ or $E_{p, R(p,\\mathcal{C}_{2})}$ and a unique integer $t \\in [1, (p-1)\/2]$ such that for each $(x,y) \\in F$ there exists a unique point $(x', y') \\in F'$ for which it holds that $x\\equiv t^2x'\\pmod p$ and $y \\equiv t^3y'\\pmod p$.\n\\end{observation}\n\\noindent\nIt is important to mention that for each subset $F$ such that the set of $y$-coordinates of its points is an $(m,p)$-complete set, the set of $y$-coordinates of the points of $F'$ is not necessarily be an $(m,p)$-complete set.\nThis is explained in Example~\\ref{E2}.\n\\begin{example}\\label{E2}\nLet $F$ be a subset of $E_{11,9}$ with an $(11,10)$-complete set $Y= \\{0,1,2,3,4,5,6,7,8,9\\}$ of $y$-coordinates, where $m = 10$.\nThen for $t = 2$, there exists $F' \\subset E_{11, 1}$ with $y$-coordinates from the set $Y'=\\{0,1,2,3,5,6,7,8,9,10\\} $ which is not an $(11,10)$-complete set.\n\\end{example}\nBy Observation~\\ref{obs_set}, we can avoid the while-loop used in Algorithm~\\ref{algo_sbox_1} to find $x$-coordinate for each element $y$ in an $(m,p)$-complete set $Y$.\n\\begin{algorithm}[H]\n\n\t\\caption{{\\bf Constructing the proposed S-box using the EC isomorphism }}\\label{algo_sbox_2}\n\\begin{algorithmic}[1]\n\\REQUIRE An MEC $E_{p, R(p,\\mathcal{C}_{i})}$, where $i \\in \\{1, 2\\}$, multiplicative inverse $t^{-1}$ of $t$ in $\\mathbb{F}_p$, where $t \\in [1, (p-1)\/2]$, a total order $\\prec$ on the MEC $E_{p,t^{6}R(p,\\mathcal{C}_{i})}$, an $(m,p)$-complete set $Y$ and an integer $k \\leq m-1$.\n\\ENSURE The proposed $(m,p)$-complete S-box $\\sigma(p, t^6R(p, \\mathcal{C}_i), \\prec, Y, k)$.\n\n\\STATE $A:=\\emptyset$; \/*A set containing the points of $E_{p,t^6R(p,\\mathcal{C}_{i})}$ with $y$-coordinates from the set $Y$*\/\n\n\\FORALL {$y \\in Y$}\n\\STATE $y':= (t^{-1})^3y$;\n\\STATE Find $x \\in [0, p-1]$ such that $(x, y) \\in E_{p,R(p,\\mathcal{C}_{i})}$;\n\\STATE $A:= A \\cup \\{(t^2x, y)\\}$;\n\\ENDFOR;\n\\STATE Sort $Y^*:=[0, m-1]$ w.r.t. the element of $A$;\n\\STATE $\\pi := (\\pi(0), \\pi(1), \\ldots, \\pi(m-1))$;\n\\FORALL {integer $i \\in [0, m-1]$}\n\t\\STATE $\\pi(i) := y_{(i+k)\\Mod{m}}\\pmod{m}$, where $y_{(i +k)\\pmod{m}}$ is the $(i +k)\\pmod{m}$-th element of the ordered $(m,p)$-complete set $(Y^*, \\tilde\\prec)$\n\\ENDFOR;\n\\STATE Output $\\pi$ as the $(m,p)$-complete S-box $\\sigma(p, t^6R(p, \\mathcal{C}_i), \\prec,$ $ Y, k)$.\n\t\n\\end{algorithmic}\n\\end{algorithm}\n\\begin{lemma}\nFor an ordered MEC $(E_{p,b}, \\prec)$, where $b= t^6R(p,\\mathcal{C}_{i})$ for some $t \\in [1, (p-1)\/2]$ and $i \\in \\{1, 2\\}$, an $(m,p)$-complete set $Y$ and a non-negative integer $k\\leq m-1$, the $(m,p)$-complete S-box $\\sigma(p, b, \\prec, Y, k)$ can be computed in $O(m\\log m)$ time and $\\mathcal{O}(m)$ space by using Algorithm~\\ref{algo_sbox_2}.\n\\end{lemma}\n\\begin{proof}\nThere is a for-loop over the set $Y$ of size $m$ for finding $x$-coordinate for each element $y \\in Y$ over the MEC $E_{p,t^{6}R(p,\\mathcal{C}_{i})}$.\nNote that at line 4 of Algorithm~\\ref{algo_sbox_2}, $x$ can be computed in constant time, i.e., $\\mathcal{O}(1)$.\nThis is due to Theorem~\\ref{mordel} the MEC $E_{p,b}$ has each element of $[0, p-1]$ uniquely as $y$-coordinate.\nThus, the for-loop over $Y$ can be computed in $\\mathcal{O}(m)$.\nThe remaining part of Algorithm~\\ref{algo_sbox_2} takes $O(m\\log m)$ time.\nHence, with the aid of the property of $\\mathcal{O}$ notion,  Algorithm~\\ref{algo_sbox_2} takes $O(m\\log m)$ time.\nMoreover, Algorithm~\\ref{algo_sbox_2} stores only a set of size $m$, other than inputs, and therefore its space complexity is  $\\mathcal{O}(m)$.\n\\end{proof}\nNote that using Algorithm~\\ref{algo_sbox_2} is practical, since Lemma~\\ref{LL} implies that for a given ordered MEC $(E_{p,b}, \\prec)$ we can generate a large number of $(m,p)$-complete S-boxes.\nHowever, $E_{p, R(p,\\mathcal{C}_{i})}$, where $i \\in \\{1, 2\\}$, $R(p, \\mathcal{C}_i)$ and $t^{-1}$ for $t \\in [0, (p-1)\/2]$ should be given as input for Algorithm~\\ref{algo_sbox_2}.\nWe know that $R(p,\\mathcal{C}_{1}) = 1$, now the next important question is how to find the representative $R(p,\\mathcal{C}_{2})$ for the class $\\mathcal{C}_{2}$ of MECs.\nFor this we prove the following results.\n\\begin{lemma}\\label{x_zero}\nAn MEC $E_{p,b}$ is an element of the class $\\mathcal{C}_1$ if and only if there exists an integer $y \\in [1, p-1]$ such that $(0, y) \\in E_{p,b}$.\n\\end{lemma}\n\\begin{proof}\nConsider the MEC $E_{p, 1}$.\nThen for $y = 1$ the equation $x^3 + 1 \\equiv 1 \\pmod p$ is satisfied by $x = 0$.\nThis implies that $(0, 1) \\in E_{p, 1}$, and hence the required statement is true for the MEC $E_{p, 1}$.\nLet $E_{p, b} \\in \\mathcal{C}_1$, where $b \\in [2, p-1]$.\nThen there exists an isomorphism parameter $t \\in [1, (p-1)\/2]$ between $E_{p, 1}$ and $E_{p, b}$ such that $(t^20, t^31) = (0, t^3) \\in E_{p, b}$.\nHence, for each MEC $E_{p,b} \\in \\mathcal{C}_1$ there exists an integer $y \\in [1, p-1]$ such that $(0, y) \\in E_{p,b}$.\n\nTo prove the converse, suppose on contrary that there is an MEC $E_{p,b}$ with a point $(0,y)$ for some $y \\in [1, p-1]$ and $E_{p,b} \\notin \\mathcal{C}_1$.\nThis implies that there does not exist an integer $t \\in  [1, (p-1)\/2]$ such that $b \\equiv t^6 \\pmod p$.\nThus, $b \\not\\equiv (t^3)^2 \\pmod p$ for all $t \\in   [1, (p-1)\/2]$.\nBut it follows from $(0,y) \\in E_{p,b}$ that $b \\equiv y^2 \\pmod p$ for some $y \\in [1, (p-1)\/2]$ which is a contradiction.\nHence $E_{p,b} \\in \\mathcal{C}_1$.\n\\end{proof}\n\n\\begin{lemma}\\label{class_2_rep}\nFor a prime $p$, the representative $R(p,\\mathcal{C}_2)$ of the class $\\mathcal{C}_2$ is a QNR integer in the field $\\mathbb{F}_{p}$.\n\\end{lemma}\n\\begin{proof}\nLet $E_{p,b} \\in \\mathcal{C}_2$.\nSuppose on contrary that $b$ is a quadratic integer in the field $\\mathbb{F}_p,$ i.e., $b \\equiv y^2 \\pmod p$ for some integer $y \\in [1, p-1]$.\nIt follows from the equation $x^3 + b \\equiv y^2 \\pmod p$ that $(0, y) \\in E_{p,b}$.\nBy Lemma~\\ref{x_zero}, it holds that $E_{p,b} \\in \\mathcal{C}_1$, which is a contradiction to our assumption.\nSo, $b$ is a QNR, and hence $R(p,\\mathcal{C}_2)$ is a QNR.\n\\end{proof}\nEuler's Criterion is a well-known method to test if a non-zero element of the field $\\mathbb{F}_p$ is a QR or not.\nWe state this test in Lemma~\\ref{EC}.\n\\begin{lemma}\\label{EC}\n\\textnormal{\\cite[p.~1797]{Sze}} An element $q\\in\\mathbb{F}_{p}$ is a QR  if and only if $q^{(p-1)\/2}\\equiv -1 \\pmod p$.\n\\end{lemma}\n\\section{Security Analysis and Comparison}\\label{Anal}\nIn this section, a detailed analysis of the proposed S-box is performed.\nMost of the cryptosystems use $8 \\times 8$ S-boxes and therefore, we use $8 \\times 8$ $(256, 525211)$-complete S-box $\\sigma(52511, 1, N, Y, 0)$ given in Table~\\ref{SN} generated by the proposed method for experiments.\nThe cryptographic properties of the proposed S-box are also compared with some of the well-known S-boxes developed by different mathematical structures.\n\\subsection{Linear Attacks}\nLinear attacks are used to exploit linear relationship between input and output bits.\nA cryptographically strong S-box is the one which can strongly resist linear attacks. The resistance of an S-box against linear attacks is evaluated by well-known tests including non-linearity~\\cite{Carlet}, linear approximation probability~\\cite{Matsui} and algebraic complexity~\\cite{Sakalli}.\nFor a bijective $n \\times n$ S-box $S$, the non-linearity NL$(S)$, linear approximation probability LAP$(S)$ can be computed by Eqs.~(\\ref{NL}) and~(\\ref{LAP}), respectively, while its algebraic complexity AC$(S)$ is measured by the number of non-zero terms in its linearized algebraic expression~\\cite{Lidl}.\n\\begin{equation}\\label{NL}\n{\\rm NL}(S)=\\min_{\\alpha ,\\beta ,\\lambda }\\{x\\in \\mathbb{F}_{2}^{n}:\\alpha \\cdot\nS(x)\\neq \\beta \\cdot x\\oplus \\lambda \\},\n\\end{equation}\n\\begin{equation}\\label{LAP}\n\\begin{split}\n&{\\rm LAP}(S)=\\frac{1}{2^{n}}\\Big\\{\\max_{\\alpha ,\\beta }\\big\\{|\\#\\{x\\in \\mathbb{F}_{2}^{n} \\mid \\\\\n & \\hspace{4cm}x \\cdot \\alpha=S(x) \\cdot \\beta \\}-2^{n-1}|\\big \\}\\Big \\},\n\\end{split}\n\\end{equation}\nwhere $\\alpha \\in \\mathbb{F}_{2}^{n}${\\normalsize , }$\\lambda \\in \\mathbb{F}_{2}$%\n{\\normalsize , }$\\beta \\in \\mathbb{F}_{2}^{n}\\backslash \\{0\\}${\\normalsize \\ and\n\\textquotedblleft }$\\cdot ${\\normalsize \\textquotedblright\\ represents the inner\nproduct over }$\\mathbb{F}_{2}.$\n\nAn S-box $S$ is said to be highly resistive against linear attacks if it has NL close to $2^{n-1} - 2^{(n\/2)-1}$, low LAP and AC close to $2^n-1$.\n\nThe experimental results of NL, LAP and AC of the proposed S-box $\\sigma(52511, 1, N, Y, 0)$ and some of the well-known S-boxes are given in Table~\\ref{allanalysis}.\nNote that the proposed S-box has NL, LAP and AC close to the optimal values.\nThe $\\rm NL$ of $\\sigma(52511, 1, N, Y, 0)$ is greater than that of the S-boxes in~\\cite{Asif, ikram, umar2, Azam, YW, GT, Jaki, Ozkaynak2,Bhattacharya,IChing,YWang,AGautam,Shi} and equal to that of~\\cite{AES}.\nThe $\\rm LAP$ of $\\sigma(52511, 1, N, Y, 0)$ is less than that of the S-boxes in~\\cite{Asif, ikram, umar2, Azam, YW, GT, Jaki, Ozkaynak2,Bhattacharya,YWang,IChing,AGautam,Shi},  and the AC of $\\sigma(52511, 1, N, Y, 0)$ attains the optimal value, which is $255$.\nThus the proposed method is capable of generating S-boxes with optimal resistance against linear attacks as compared to some of the existing well-known S-boxes.\n\\subsection{Differential Attack}\nIn this attack, cryptanalysts try to approximate the original message by observing a particular difference in output bits for a given input bits difference.\nThe strength of an $n \\times n$ S-box $S$ can be measured by calculating its differential approximation probability DAP$(S)$ using Eq.~(\\ref{DAP}).\n\\begin{equation}\\label{DAP}\n\\begin{split}\n&{\\rm DAP}(S)=\\frac{1}{2^{n}}\\Big\\{\\max_{\\Delta x,\\Delta y}\\big\\{\\#\\{x\\in \\mathbb{F}_{2}^{n} \\mid\\\\\n& \\hspace{3cm} S(x\\oplus \\Delta x)= S(x)\\oplus \\Delta y \\}\\big \\}\\Big \\},\n\\end{split}\n\\end{equation}\nwhere $\\triangle x,$\\ $\\triangle y\\in \\mathbb{F}_{2}^{n},$ and \\textquotedblleft $%\n{\\normalsize \\oplus }$\\textquotedblright\\ denotes bit-wise addition over $\\mathbb{F}_{2}$.\n\n  An S-box $S$ is highly secure against differential attack if its DAP is close to $1\/2^n$.\n  In Table~\\ref{allanalysis}, the $\\rm DAP$ of $\\sigma(52511, 1, N,$ $ Y, 0)$ and other existing S-boxes is given.\n  Note that the DAP of the proposed S-box $\\sigma(52511, 1, N, Y, 0)$ is $0.016$ which is close to the optimal value $0.0039$.\n Furthermore, it is evident from Table~\\ref{allanalysis} that the DAP of the proposed S-box is less than the S-boxes in~\\cite{Asif,ikram, umar2, Azam, YW, GT, Jaki, Ozkaynak2,Bhattacharya,YWang,IChing,AGautam,Shi}, and hence the proposed S-box scheme can generate S-boxes with high resistance against differential attack.\n\\subsection{Analysis of Boolean Functions}\nIt is necessary to analyze the boolean functions of a given S-box to measure its confusion\/diffusion creation capability.\nFor an $n \\times n$ S-box, strict avalanche criterion SAC$(S)$ and bit independence criterion BIC$(S)$ are used to analyze its boolean functions.\nThe SAC$(S)$ and the BIC$(S)$ are computed by two matrices $M(S) = [m_{ij}]$ and $B(S)=[b_{ij}]$, respectively, such that\n\\begin{equation}\\label{sAc}\n{ m_{ij}=\\frac{1}{2^{n}}\\left( \\sum_{x\\in \\mathbb{F}_{2}^{n}}{ w}%\n\\left( {S}_{i}{ (x\\oplus \\alpha }_{j}{ %\n)\\oplus S}_{i}{ (x)}\\right) \\right) { ,}}\n\\end{equation}\nand\n\\begin{equation}\\label{bIc}\n\\begin{split}\n&b_{ij}=\\frac{1}{2^{n}}\\Biggl(\\sum_{\\substack{ x\\in \\mathbb{F}_{2}^{n}\\\\%\n1\\leq r\\neq i\\leq n}}w\\biggl(S_{i}(x \\oplus \\alpha_{j})\\oplus S_{i}(x)\\oplus \\\\\n&\\hspace{4cm}S_{r}(x+\\alpha_{j})\\oplus S_{r}(x)\\biggr) \\Biggr),\n\\end{split}\n\\end{equation}\nwhere $w(y)$ is the hamming weight of $y$, $\\alpha_{j}\\in \\mathbb{F}_{2}^{n}$\nsuch that $w(\\alpha_{j})=1$, $ {S}_{i}$ and ${S}_{r}$ are $i$-th and $r$-th boolean functions of $ S$, respectively, and $1\\leq i,j,r \\leq n$.\nAn S-box $S$ satisfies the SAC and the BIC if each non-diagonal entry of $M(S)$ and $B(S)$ have value close to $0.5$.\nThe maximum and minimum values of the SAC (resp., BIC) of the proposed S-box $\\sigma(52511, 1, N, Y, 0)$ are $0.563$ and $0.438$ (resp., $0.521$ and $0.479$).\nNote that these values are closed to $0.5$, and hence the proposed S-box satisfies the SAC and the BIC.\nSimilarly, the SAC and the BIC of some other S-boxes are listed in Table~\\ref{allanalysis} and compared with the results of the proposed S-box.\nIt is evident from Table~\\ref{allanalysis} that the proposed S-box can generate more confusion and diffusion as compared to some of the listed S-boxes.\n\\begin{table}[htb]\n\\caption{Comparison of the proposed and other existing S-boxes}\n\\label{allanalysis}\n\\centering\n\\resizebox{\\columnwidth}{!}\n{\n\\bgroup\n\\def1.2{1.1}\n\\begin{tabular}{cccccccccc}\n    \\hline\n    S-boxes&Type&\\multicolumn{3}{c}{Linear}& DAP & \\multicolumn{4}{c}{Analysis of}\\\\\n           &of&\\multicolumn{3}{c}{ Attacks}&  & \\multicolumn{4}{c}{Boolean Functions}\\\\\n    \\cline{3-5}  \\cline{7-10}\n    &S-box&NL&LAP&AC & & SAC     &SAC     &BIC     &BIC\\\\\n    &     &  &   &   & & (max)   &(min)   &(max)   &(min)\\\\\n    \\cline{1-10}\n Ref.~\\cite{Bhattacharya}&&102  &0.133&254  &0.039\t&0.562&\t0.359  &0.535&0.467  \\\\\n Ref.~\\cite{IChing}&other&108          &0.133&255&0.039    &0.563 &0.493& 0.545 &0.475\\\\\n Ref.~\\cite{AES}&&112  & 0.062 & 09 &0.016  &0.562  &0.453  & 0.504 & 0.480 \\\\\n Ref.~\\cite{Shi}& &108  & 0.156 & 255 &0.046  &0.502  &0.406  & 0.503 & 0.47 \\\\\n \\cline{1-10}\n Ref.~\\cite{YW}& &108 & 0.145 &255  & 0.039 &0.578  &0.406  &0.531&0.470  \\\\\n Ref.~\\cite{GT}& &103 &0.132 &255  & 0.039 &0.570  &0.398  &0.535&0.472  \\\\\n Ref.~\\cite{Jaki}&Chaos&100  & 0.129 &255  & 0.039 & 0.594 & 0.422 &0.525&0.477  \\\\\n Ref.~\\cite{Ozkaynak2}&&100 &0.152  &255  &0.039  &0.586  &0.391  &0.537&0.468  \\\\\n Ref.~\\cite{YWang}&&110  & 0.125 & 255 & 0.039 & 0.562 &0.438  &0.555  & 0.473 \\\\\n Ref.~\\cite{AGautam}&& 74 &0.211  &253  &0.055 & 0.688 &0.109  & 0.551 & 0.402 \\\\\n \\cline{1-10}\n Ref.~\\cite{Asif}&&104&0.145&255&0.039&0.625&0.391&0.531&0.471  \\\\\n Ref.~\\cite{umar2}&&106&0.148&254&0.039&0.578&0.437&0.535&0.464  \\\\\n Ref.~\\cite{ikram}&EC&106&0.188&253&0.039 &0.609&0.406&0.527&0.465  \\\\\n Ref.~\\cite{Azam}&&106&0.148&255&0.039&0.641&0.406&0.537&0.471  \\\\\n $\\sigma(52511, 1, N, Y, 0)$&&112\t&0.063& 255   &0.016&\t0.563&\t0.438&\t0.521& 0.479 \\\\\n    \\hline\n\\end{tabular}\n\\egroup\n}\n\\end{table}\n\n\n\\subsection{Distinct S-boxes}\n An S-box generator is useful to resist cryptanalysis if it can generate a large number of distinct S-boxes~\\cite{Azam}.\nFor the parameters $p=263, b =1, m=256$ and $k=0$ the number of $(256,263)$-complete S-boxes Num$(E_{263,1}, 256, 0)$ is $128$.\nIt turned out with the computational results that all of these $(256,263)$-complete S-boxes are distinct.\nHowever this is not the case in general.\n\nAn $(m,p)$-complete S-box $\\sigma(p,b,\\prec,Y, k)$ is said to be a \\textit{natural} $(m,p)$\\textit{-complete S-box} if $Y = [0, m-1]$.\nFor a prime $p$ and ordering $\\prec$, let $p^*$ denote the largest integer such that $p^*  \\leq p-1$ and there exists at least two ordered MECs $(E_{p,b_1})$ and $E_{p,b_2}$ due to which the natural $(p^*, p)$-complete S-boxes are identical, i.e., for any fixed $m \\geq p^*$ the number of natural $(m, p)$-complete S-boxes due to all ordered MECs with prime $p$, ordering $\\prec$ and $k = 0$ is equal to $p-1$.\nA plot of primes $p \\in [11, 7817]$ and the integers $p^{*}$ is given in Fig.~\\ref{LI}, where the underlying ordering is the natural ordering $N$.\nFor the orderings $D$ and $M$, such plots are similar as that of $N$.\nIt is evident from Fig.~\\ref{LI}, that with the increase in the value of prime, there is no significant increase in the value of $p^*$ and the largest value of $p^*$ for these primes is $12$.\nHence, for each of these primes, each $m \\geq 13$ and $k = 0$, we can get $p-1$ distinct natural $(m, p)$-complete S-boxes with $k = 0$.\n\n\\begin{figure}[htb!]\n\\includegraphics [scale=0.68]{Conjecture.pdf}\n\\caption{Plot of primes $p$ and their corresponding largest integers $p^{*}$}\n\\label{LI}\n\\end{figure}\n\\begin{lemma}\\label{class1}\nLet $\\prec$ be a fixed total order on all MECs in $\\mathcal{C}_{1}$ such that for each MEC $E_{p,b} \\in \\mathcal{C}_1$ it holds that the points $(0, \\pm y)$, where $-y$ is additive inverse of $y$ in $\\mathbb{F}_p$, have indices from the set $\\{1, 2\\}$ in the sequence representation of the MEC.\nThen for a fixed integer $k \\in [0, m-1]$, the number of distinct natural $(m,p)$-complete S-boxes generated by all MECs in $\\mathcal{C}_{1}$ are at least\n\\begin{equation}\\label{L10}\n\\left\\{\n\\begin{tabular}{ll}\n$m - 1$ & \\textnormal{if} $m < (p-1)\/2$\\\\\n$(p-1)\/2 $ & \\textnormal{otherwise}.\n\n\\end{tabular}\n\\right.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nLet $E_{p,b}$ be an MEC in $\\mathcal{C}_1$, where $b \\in [1, p-1]$.\nThen by Lemma~\\ref{x_zero}, $(0, y) \\in E_{p,b}$ for some $y \\in [1, p-1]$.\nFurther by the fact that if $(x, z) \\in E_{p,b}$ then $(x, -z) \\in E_{p,b}$, where $-z$ is the additive inverse of $z$ in the field $\\mathbb{F}_p$, implies that $(0, \\pm y) \\in E_{p, b}$.\nMoreover, by the group theoretic argument exactly one of the integers $y$ and $-y$ belongs to the interval $[0, (p-1)\/2]$.\nHence, for a fixed $k \\in [0, m-1]$ and the natural $(m,p)$-complete S-box $\\sigma(p, b, \\prec, Y, k)$ it holds that $\\sigma(p, b, \\prec, Y, k)(k)\\in \\{\\pm y\\}$ if $(0, \\pm y)$ have indices from the set $\\{1,2\\}$ in the sequence representation of $E_{p,b}$.\nNote that a point $(0,z)$ cannot appear on two different MECs $E_{p,b_1}$ and $E_{p,b_2}$, otherwise this implies that $b_1 = b_2$.\nThus, for any two MECs $E_{p,b_1},E_{p,b_2}$  in $\\mathcal{C}_1$ satisfying the conditions given in the lemma it holds that the natural $(m,p)$-complete S-boxes $\\sigma(p, b_1, \\prec, Y, k)$ and  $\\sigma(p, b_2 , \\prec, Y, k)$ have different images at a fixed input $k \\in[0, m-1]$.\nThus $|\\mathcal{C}_1| = (p-1)\/2$ implies the required result.\n\\end{proof}\nFor three different primes $p$ distinct S-boxes are generated by the proposed method, and compared with the existing schemes over ECs as shown in Table~\\ref{counting}. It is evident that the proposed S-box generator performs better than other schemes.\n\\begin{table}[htb!]\n\\caption{Comparison of the number of distinct $8 \\times 8$ S-boxes generated by different schemes}\n\\label{counting}\n\\centering\n\\resizebox{\\columnwidth}{!}\n{\n\\bgroup\n\\def1.2{1.27}\n\\begin{tabular}{lcclll}\n\\cline{3-6}\n&&$p$           & 1889 & 2111 & 2141           \\\\\n\\cline{3-6}\n&&$b$           & 1888 & 1 & 7           \\\\ \\hline\nDistinct S-boxes by the& $N$ && 32768$^\\dag$ &32768$^\\dag$ &32768$^\\dag$   \\\\ \\cline{2-6}\nproposed method due to& $D$ && 31744$^\\dag$&32704$^\\dag$&30720$^\\dag$    \\\\ \\cline{2-6}\nthe ordering& $M$ && 15360$^\\dag$ & 26748$^\\dag$  & 21504$^\\dag$   \\\\ \\hline\n                       & $N$ && 944 & 1055 & 1070   \\\\ \\cline{2-6}\nDistinct S-boxes by Ref.~\\cite{ikram}& $D$ && 944 &1055  &1070    \\\\ \\cline{2-6}\n                       & $M$ && 944 &1055  &1070    \\\\ \\hline\nDistinct S-boxes by Ref.~\\cite{umar2}&&  & 50 & 654 & 663  \\\\ \\hline\nDistinct S-boxes by Ref.~\\cite{Asif}&&  & 1 & 1 & 1  \\\\ \\hline\n\\end{tabular}\n\\egroup\n}\n\n\\end{table}\n\\xdef\\@thefnmark{}\\@footnotetext{The number $h^{\\dag}$ stands for an integer greater than $h$.}\n\\subsection{Fixed Point Test}\nAn S-box construction scheme is cryptographically good if the average number of fixed points in the constructed S-boxes is as small as possible~\\cite{Azam}.  The average number of fixed points of the above generated S-boxes are shown in Table~\\ref{fixed}. The experimental results indicate that the proposed S-box generator generates S-boxes with a very small number of fixed points.\nFurthermore,  the average number of fixed points in the proposed S-boxes are comparable with that of the existing schemes over ECs.\n\\begin{table}[htb!]\n\\caption{Comparison of average number of the fixed points in the S-boxes generated by different schemes}\n\\label{fixed}\n\\centering\n\\resizebox{\\columnwidth}{!}\n{\n\\bgroup\n\\def1.2{1.1}\n\\begin{tabular}{lcclll}\n\\cline{3-6}\n&&$p$           & 1889 & 2111 & 2141           \\\\\n\\cline{3-6}\n&&$b$           & 1888 & 1 & 7           \\\\ \\hline\nAvg.  \\# fixed points by the& $N$ && 1.1298 & 1.0844 & 1.0972   \\\\ \\cline{2-6}\nproposed method due to& $D$ && 0.9471 & 0.8569  & 0.9393    \\\\ \\cline{2-6}\nthe ordering& $M$ && 0.8361 & 1.1847  & 1.0025    \\\\ \\hline\n                       & $N$ && 1.77 & 0.9735 & 0.9785   \\\\ \\cline{2-6}\nAvg.  \\# fixed points by Ref.~\\cite{ikram}& $D$ && 1.932 &0.9716  &0.9561    \\\\ \\cline{2-6}\n                       & $M$ && 1.332 &1.0019  &1.0150    \\\\ \\hline\nAvg.  \\# fixed points by Ref.~\\cite{umar2}&&  & 2.04 & 0.8976 & 0.9351  \\\\ \\hline\nAvg.  \\# fixed points by Ref.~\\cite{Asif}&&  & 2 & 3 & 0  \\\\ \\hline\n\\end{tabular}\n\\egroup\n}\n\\end{table}\n\\subsection{Correlation Test} The correlation test is used to analyze the relationship among the S-boxes generated by any scheme. A robust scheme generates S-boxes with low correlation~\\cite{Azam}.\nThe proposed method is evaluated by determining the correlation coefficients (CCs) of the designed S-boxes. The lower and upper bounds for their CCs are listed in Table~\\ref{CC}, which reveal that the proposed scheme is capable of constructing S-boxes with very low correlation as compared to the other schemes over ECs.\n\\begin{table}[htb!]\n\\caption{\\ Comparison of CCs of S-boxes generated by different schemes}\n\\label{CC}\n\\centering\n\\resizebox{\\columnwidth}{!}\n \n{\n\\bgroup\n\\def1.2{1.1}\n\\begin{tabular}{lcccccc}\n\\hline\nScheme&$p$& $b$ & Ordering &\\multicolumn{3}{c}{Correlation}  \\\\\n\\cline{5-7}\n& & & &Lower & Average & Upper\\\\\n\\hline\n        &1889 & 1888&$N$ &-0.2685&0.0508 & 0.2753     \\\\\nProposed&1889 & 1888&$D$ &-0.2263&0.0523 & 0.2986      \\\\\n        &1889 & 1888&$M$ &-0.2817&0.0506 & 0.2902     \\\\\n\\hline\n        &2111 & 1&$N$ &-0.2718&0.0504 & 0.2600     \\\\\nProposed&2111 & 1&$D$ &-0.2596&0.0531 & 0.3025      \\\\\n        &2111 & 1&$M$ &-0.2779&0.0507 & 0.2684     \\\\\n\\hline\n        &2141 & 7&$N$ &-0.2682&0.0503 & 0.2666     \\\\\nProposed&2141 & 7&$D$ &-0.2565&0.0517 & 0.2890      \\\\\n        &2141 & 7&$M$ &-0.2744&0.0503 & 0.2858     \\\\\n\\hline\n                 &1889 & 1888&$N$ &-0.2782&0.0503 & 0.2756     \\\\\nRef.~\\cite{ikram}&1889 & 1888&$D$ &-0.4637&-0.0503 & 0.2879      \\\\\n                 &1889 & 1888&$M$ &-0.2694&0.0501 & 0.4844     \\\\\n\\hline\n                 &2111 & 1&$N$ &-0.2597&0.0504 & 0.2961     \\\\\nRef.~\\cite{ikram}&2111 & 1&$D$ &-0.3679&0.0500 & 0.3996      \\\\\n                 &2111 & 1&$M$ &-0.2720&0.0499 & 0.3019     \\\\\n\\hline\n                 &2141 & 7&$N$ &-0.2984&0.0500 & 0.3301     \\\\\nRef.~\\cite{ikram}&2141 & 7&$D$ &-0.2661&0.0500 & 0.2639      \\\\\n                 &2141 & 7&$M$ &-0.2977&0.0501 & 0.2975     \\\\\n\\hline\nRef.~\\cite{umar2} & 1889& 1888 &-- &-0.0025&0.2322&0.9821\\\\                                                  \\hline\nRef.~\\cite{umar2} & 2111& 1 &-- &-0.2932&0.0785&0.9988\\\\\n\\hline\nRef.~\\cite{umar2} & 2141& 7 &-- &-0.2723 &0.0629&0.9999\\\\\n\\hline\n\\end{tabular}\n\\egroup\n}}\n\\end{table}\n\\subsection{Time and Space Complexity}\nFor a good S-box generator it is necessary to have low time and space complexity~\\cite{Azam}.\nTime and space complexity of the newly proposed method are compared with some of the existing methods in Table~\\ref{TandSCom}. It follows that for a fixed prime the proposed method can generate an S-box with low complexity and space as compared to other listed schemes.\nThis fact makes the proposed S-box generator more efficient and practical.\n\\begin{table}[htb!]\n\\caption{Comparison of time and space complexity of different S-box generators over ECs}\n\\label{TandSCom}\n\\centering\n\\resizebox{\\columnwidth}{!}\n{\\\n\\bgroup\n\\def1.2{1.2}\n{\\begin{tabular}{cccccc}\n\\hline\nS-box & Ref.~\\cite{Asif} &Ref.~\\cite{umar2} &Ref.~\\cite{ikram}&\\multicolumn{2}{c}{\\small Proposed method} \\\\\n\\cline{5-6}\n& & & &Algorithm~\\ref{algo_sbox_1}&Algorithm~\\ref{algo_sbox_2}\\\\\n\\hline\nTime complexity & $\\mathcal{O}(p^{2})$ & $\\mathcal{O}(p^{2})$ & $\\mathcal{O}(mp)$ &  $\\mathcal{O}(mp)$  & $\\mathcal{O}(m\\log m)$ \\\\\n\\hline\n\nSpace complexity & $\\mathcal{O}(p)$ & $\\mathcal{O}(p)$ & $\\mathcal{O}(m)$ &  $\\mathcal{O}(m)$  &  $\\mathcal{O}(m) $ \\\\\n \\hline\n\\end{tabular}%\n\\egroup\n}}\n\\end{table}\n\\section{The Proposed Random Numbers Generation Scheme}\\label{RNG}\nFor an ordered MEC $(E_{p,b}, \\prec)$, a subset $A \\subseteq [0, p-1]$, an integer $m \\in [1, |A|]$ and a non-negative integer $k \\in [0, m-1]$, we define a sequence of \\textit{pseudo random numbers} (SPRNs) $\\gamma(p, b, \\prec, A, m, k)$ to be a sequence of length $|A|$ whose $i$-th term is defined as $\\gamma(p, b, \\prec, A, m, k)(i) = y_{(i+k)\\pmod m}\\pmod{m}$, where $y_{(i+k)\\pmod m}$ is the $(i+k)\\pmod m$-th element of the ordered set $(A, \\prec^*)$ in its sequence representation.\\\\\nOne of the differences in the definition of an $(m,p)$-complete S-box and the proposed SPRNs is that an $(m,p)$-complete set is required as an input for the S-box generation, since an S-box of length $m$ is a permutation on the set $[0, m-1]$.\nFurthermore, Algorithm~\\ref{algo_sbox_1} and~\\ref{algo_sbox_2} can be used for the generation of the proposed SPRNs, however, we propose an other algorithm which is more efficient than Algorithm~\\ref{algo_sbox_2} for its generation.\nThis new algorithm is also based on Observation~\\ref{obs_set}, but there is no constraint on $A$ to be an $(m,p)$-complete set, and hence we can generate all proposed SPRNs for a given prime $p$ by using $E_{p, R(p, \\mathcal{C}_i)}$, where $i \\in \\{1, 2\\}$.\n\n\\begin{algorithm}[H]\n\n\t\\caption{{\\bf Constructing the proposed SPRNs using EC isomorphism}}\\label{algo_sprn}\n\\begin{algorithmic}[1]\n\\REQUIRE An MEC $E_{p, R(p,\\mathcal{C}_{i})}$, where $i\\in \\{1, 2\\}$, an integer $t \\in [1, (p-1)\/2]$, a total order $\\prec$ on the MEC $E_{p,t^{6}R(p,\\mathcal{C}_{i})}$ and a subset $Y \\subseteq [0, p-1]$.\n\\ENSURE The proposed SPRNs $\\gamma(p, t^{6}R(p,\\mathcal{C}_{i}), \\prec, t^3Y, m, k)$.\n\n\\STATE $A:=\\emptyset$; \/*A set containing the points of $E_{p,t^6R(p,\\mathcal{C}_{i})}$ with $y$-coordinates from the set $t^3Y$*\/\n\n\\FORALL {$y \\in Y$}\n\\STATE Find $x \\in [0, p-1]$ such that $(x, y) \\in E_{p,R(p,\\mathcal{C}_{i})}$;\n\\STATE $A:= A \\cup \\{(t^2x, t^3y)\\}$;\n\\ENDFOR;\n\\STATE Sort $A$ w.r.t. the element of the total order $\\prec^*$;\n\\STATE $\\pi := (\\pi(0), \\pi(1), \\ldots, \\pi(|A|-1))$;\n\\FORALL {integer $i \\in [0, |A|-1]$}\n\t\\STATE $\\pi(i) := a_{(i+k)\\Mod{m}}\\pmod{m}$, where $a_{(i +k)\\pmod{m}}$ is the $(i +k)\\pmod{m}$-th element of the ordered set $(A, \\prec^*)$\n\\ENDFOR;\n\\STATE Output $\\pi$ as the proposed SPRN $\\gamma(p, t^{6}R(p,\\mathcal{C}_{i}), \\prec, t^3Y, m, k)$\n\t\n\\end{algorithmic}\n\\end{algorithm}\n\\noindent\nNote that the time and space complexity of Algorithm~\\ref{algo_sprn} are $\\mathcal{O}(|A|\\log {|A}|$) and $\\mathcal{O}(|A|)$ respectively as obtained for Algorithm~\\ref{algo_sbox_2}.\nHowever, Algorithm~\\ref{algo_sprn} does not require $t^{-1}$ as an input parameter to compute $\\gamma(p, t^{6}R(p,\\mathcal{C}_{i}), \\prec,$ $t^3Y, m, k)$ for which we need preprocessing.\nFurthermore, Lemma~\\ref{class1} trivially holds for our proposed SPRNs.\nThis implies that the proposed PRNG can generate a large number of distinct SPRNs for a given prime.\n\n\n\n\n\n\\section{Analysis of the Proposed SPRNs Method}\\label{RA}\nWe applied some well-known tests to analyze the strength of our proposed SPRNs. A brief introduction to these tests and their experimental results are given below.\nWe used orderings $N, D$ and $M$ for these tests.\n\n\\subsection{Histogram and Entropy Test}\nHistogram and entropy are the two widely used tests to measure the extent of randomness of a RNG.\nFor a sequence $X$ over the set of symbols $\\Omega$, the histogram of $X$ is a function $f_{X}$ over $\\Omega$ such that for each $w \\in \\Omega$, $f_{X}(w)$ is equal to the number of occurrences of $w$ in $X$.\nWe call $f_X(w)$, the frequency of $w$ in $X$.\n\n\nA sequence $X$ has uniform histogram if all elements of its symbol set have same frequency.\nThe histogram test is a generalization of the Monobit test included in NIST STS~\\cite{Rukhin}.\nA sequence is said to be highly random if it has a uniformly distributed histogram.\n\nShannon~\\cite{Shannon} introduced the concept of entropy.\nFor a sequence $X$ over the set of symbols $\\Omega$,\nthe entropy H$(X)$ of $X$ is defined as\n\\begin{equation}\\label{formula}\n{\\rm H}(X)= -\\sum_{w \\in \\Omega}\\frac{f_X(w)}{|X|}\\log_{2}(\\frac{f_X(w)}{|X|}).\n\\end{equation}\nThe upper bound for the entropy is log$_{2}(|\\Omega|).$\nThe higher is the value of entropy of a sequence the higher is the randomness in the sequence.\n\\begin{remark}\n For any distinct $k_1, k_2 \\in [0, m-~1]$, the histograms of the proposed SPRNs $\\gamma(p,b,\\prec,A,m, k_1)$ and $\\gamma(p,b,\\prec,A,m,k_2)$ are the same, and hence ${\\rm H}(\\gamma(p,b,\\prec,$ $A,m,k_1)) = {\\rm H}(\\gamma(p,b,\\prec,A,m,k_2))$.\n \\end{remark\nIn the next lemmas we discuss when the proposed SPRNs has uniformly distributed histogram and its entropy approaches the optimal value.\n\\begin{lemma}\\label{opt}\nFor an $(m,p)$-complete set $A$, a positive integer $h \\leq m$ such that $m = hq + r$, a non-negative integer $k \\leq h$, and the SPRNs $X = \\gamma(p,b,\\prec,A,h,k)$ it holds that\n\\begin{enumerate}[label=\\roman*, font=\\upshape, noitemsep]\n\\item[(i)] \\begin{equation*}\n f_X(w) = \\left\\{\n \\begin{tabular}{ll}\n$q+1$ & \\textnormal{if} $w \\in [0, r-1]$,\\\\\n$q$  & \\textnormal{otherwise},\n \\end{tabular}\n \\right.\n \\end{equation*}\n \\textnormal{if}  $r \\neq 0$ \\textnormal{and} $A = [0, m-1]$,\n \\item[(ii)] \\textnormal{for each} $w \\in [0, h-1]$, $f_X(w) = q$ \\textnormal{if}  $r = 0$.\n\\end{enumerate}\n\n\n\\end{lemma}\n\\begin{proof}\nIt is trivial that the domain of the histogram of $X$ is the set $[0, h-1]$.\\\\\n(i)~If $r \\neq 0$ and $A = [0, m-1]$, then it can be easily verified that $A$ can be partitioned in $q+1$ sets $\\{ih +\\ell \\mid 0 \\leq \\ell \\leq h-1\\}$, where $0 \\leq i \\leq q-1$, and $\\{qh +\\ell \\mid 0 \\leq \\ell \\leq r -1\\}$.\nThis implies that for each $w \\in [0, h-1]$ it holds that\n\\begin{equation*}\n f_X(w) = \\left\\{\n \\begin{tabular}{ll}\n$q+1$ & \\textnormal{if} $w \\in [0, r-1]$,\\\\\n$q$  &\\textnormal{otherwise}.\n \\end{tabular}\n \\right.\n \\end{equation*}\n(ii)~If $r = 0$, then $m = hq$.\n We know that for each $a \\in A$, it holds that $a = mi + j$, where $ 0 \\leq j \\leq m- 1$.\nThus, with the fact that $m = hq$, it holds that\n\\begin{equation*}\n\\begin{split}\na \\pmod h &= ((mi) \\pmod h + j \\pmod h) \\\\\n &= j \\pmod h\n \\end{split}\n\\end{equation*}\nThis implies that $\\{a\\pmod{m} \\mid a \\in A\\}$ = $\\{(a\\pmod{m}) \\pmod {h} \\mid a \\in A\\}$.\nThus by using the same reason, we can partition $A$ into $q$ sets, since $m = hq$, and hence $f_{X}(w) = q$ for each $w \\in [0, h-1]$.\nThis completes the proof.\n\\end{proof}\nFor the parameters given in Lemma~\\ref{opt}, we can deduce that the histogram of our proposed SPRNs is either approximately uniform or exactly uniform.\n\n\\begin{corollary}\nLet $A$ be an $(m,p)$-complete set, $h \\leq m$ such that $m = hq + r$ be a positive integer, $k \\leq m-1$ be a non-negative integer, and $X$ be the proposed SPRNs $ \\gamma(p,b,\\prec,$ $A,h,k)$.\nIt holds that\n\n \\begin{equation\n \\rm {H}(\\it {X}) = \\begin{cases}\n        -r(\\frac{q+1}{|X|})\\rm{log_{2}}\\it(\\frac{q+1}{|X|})- & \\textnormal {if}\\quad r \\neq 0, A = [0, m-1],\\\\\n        (h-r)(\\frac{q}{|X|})\\rm{log_{2}}\\it(\\frac{q}{|X|}) &\\\\\n        \\log_2(h) & \\textnormal{if}\\quad r = 0.\n        \\end{cases}\n  \\end{equation}\n\\end{corollary}\n \\begin{proof\nWhen $r \\neq 0$ and $A = [0, m-1]$, then by Lemma~\\ref{opt}~(i), there are $r$ (resp., $h-r$) numbers in $[0, h-1]$ whose frequency is $q+1$ (resp., $q$), and therefore we have the result.\\\\\nWhen $r = 0$, then by Lemma~\\ref{opt}~(ii), all numbers in $[0, h-1]$ have frequency $q$ and there are $h$ elements in $[0, h-1]$ and hence the result.\n\\end{proof}\n\nTo test the efficiency of the proposed PRNG, we generated SPRNs $X_1 = \\gamma(52511, 1, N, A, 127, 0)$, $X_2 = \\gamma(52511, 1, N, A, 16, 0)$, where $A$ is the set given in Table~\\ref{FT}, $X_3 = \\gamma(101, 35, N, [0, 100], 6, 0)$ and $X_4 = \\gamma(3917, 301, N, [0, 3916], 3917, 0)$.\nThe histogram of $X_1$ is given in Fig.~\\ref{Histo1} which is approximately uniform, while by Lemma~\\ref{opt}\nthe histograms of $X_3$ and $X_4$ are uniformly distributed.\nFurthermore, the entropy of each of these SPRNs is listed in Table~\\ref{tests}.\nObserve that the newly generated SPRNs have entropy close to the optimal value.\nThus, by histogram and entropy test it is evident that the proposed method can generate highly random SPRNs.\nMoreover, the proposed SPRNs $X_4$ are compared with the SPRNs $\\mathcal{R}(3917,0,301,10,2)$ generated by the existing technique due to Hayat~\\cite{umar2} over ECs.\nBy Lemma~\\ref{opt} it holds that  $f_{X_{4}}(w) = 1$ for each $w \\in [0, 3916]$, and by Fig.~\\ref{Histo2}, it is clear that the histogram of $X_4$ is more uniform as compared to that of the SPRNs $\\mathcal{R}(3917,0,301,10,2)$.\nBy Table~\\ref{tests}, the entropy of $X_4$ is also higher than that of $\\mathcal{R}(3917,0,301,10,2)$, and hence the proposed PRNG is better than the generator due to Hayat~\\cite{umar2}.\n\\begin{table}[htb!]\n\\caption{Comparison of entropy and period of different sequence of random numbers over EC}\n\\label{tests}\n\\centering\n\\resizebox{\\columnwidth}{!}\n \n\\bgroup\n\\def1.2{1.1}\n\\begin{tabular}{llllllll}\n\\hline\n\\multicolumn{1}{c}{Random sequence $X$} & Type of A   & H$(X)$ & log$_{2}(|\\Omega|)$  & Period & Optimal\\\\\n&     &     & & &period \\\\\n\\hline\n$\\gamma(52511,1,N,A,127,0)$&Table~\\ref{FT} \t&6.6076  &6.7814&256 &256  \\\\\n$\\gamma(52511,1,N,A,16,0)$&Table~\\ref{FT} &4 &4&256 &256 \\\\\n$\\gamma(101,35,N,A,6,0)$&$[0, 100]$   & 2.5846 &2.5850& 99&101\\\\\n$\\gamma(3917, 301, N, A, 3917, 0)$ & [0, 3916] & 11.9355 &11.9355&3917 &3917\\\\\n$\\mathcal{R}(3917,0,301,10,2)$ Ref.~\\cite{umar2}&\\multicolumn{1}{c}{--} & 10.9465&11.1536&3917&3917\\\\\n\\hline\n\\end{tabular}\n\\egroup\n}\n\\end{table}\n\\begin{figure}[htb!]\n\\includegraphics [scale=0.65]{R2.pdf}\n\\caption{The histogram of $\\gamma(52511,1,N,Y,127,0)$}\n\\label{Histo1}\n\\end{figure}\n\n\\begin{figure}[htb!]\n\\includegraphics [scale=0.65]{R1.pdf}\n\\caption{The histogram of $\\mathcal{R}(3917,0,301,10,2)$}\n\\label{Histo2}\n\\end{figure}\n\\subsection{Period Test}\nPeriod test is another important test to analyze the randomness of a PRNG.\nA sequence $X = \\{a_{n}\\}$ is said to be periodic if it repeats itself after a fixed number of integers, i.e., $\\{a_{n+h}\\}=\\{a_{n}\\}$ for the least positive integer $h$.\nIn this case $h$ is called the period of the sequence $X$.\nThe maximum period that a sequence $X$ can have is $|X|$.\nThe sequence $X$ is said to be highly random if its period is long enough~\\cite{Marsaglia}.\nWe computed the period of the proposed SPRNs $X_i, i = 1, 2, \\ldots, 4$ and the SPRNs $R(3917,0,301,10,2)$ generated by the scheme proposed in \\cite{umar2} and the results are listed in Table~\\ref{tests}.\nIt is evident from Table~\\ref{tests} that the proposed SPRNs have period colse to the optimal value.\nHence, the proposed PRNG can generated highly random numbers.\n\n\n\n\n\n\n\\subsection{Time and Space Complexity}\nIt is necessary for a good PRNG to have low time and space complexity.\nThe time and space complexity of the proposed PRNG and the generator proposed by Hayat et al.~\\cite{umar2} are compared in Table~\\ref{RNcom}.\nNote that the time and space complexity of the proposed PRNG depend on the size of the input set, while the time and space complexity of PRNG due to Hayat et al.~\\cite{umar2} are $\\mathcal{O}(p^2)$ and $\\mathcal{O}(p)$, respectively, where $p$ is underlying prime.\nHence, the proposed PRNG is more efficient as compared to the PRNG due to Hayat et al.~\\cite{umar2}.\n\\begin{table}[H]\n\\caption{Comparison of time and space complexity of different PRNGs over ECs}\n\\label{RNcom}\n\\centering\n\\resizebox{\\columnwidth}{!}\n{\\\n\\bgroup\n\\def1.2{1.2}\n{\\begin{tabular}{cccc}\n\\hline\n & Input size $m$& Ref.~\\cite{umar2} & Proposed method\\\\\n \\hline\n Time complexity &\\begin{tabular}{@{}l@{}}  $m < p$ \\\\\n \t\t\t\t\t\t\t\t\t\t\t\t\t$m = p$  \\end{tabular} & $\\mathcal{O}(p^2)$ &$\\mathcal{O}(|A|\\log |A|)$\\\\\n \\hline\n Space complexity &\\begin{tabular}{@{}l@{}}  $m < p$ \\\\\n \t\t\t\t\t\t\t\t\t\t\t\t\t$m = p$  \\end{tabular} & $\\mathcal{O}(p)$ &$\\mathcal{O}(|A|)$\\\\\n \\hline\n\n\\end{tabular}%\n\\egroup\n}}\n\\end{table}\n\n\n\n\\section{Conclusion}\\label{Con}\nNovel S-box generator and PRNG are presented based on a special class of the ordered MECs.\nFurthermore, efficient algorithms are also presented to implement the proposed generators.\nThe security strength of these generators is tested by applying several well-known security tests.\nExperimental results and comparison reveal that the proposed generators are capable of generating highly secure S-boxes and PRNs as compared to some of the exiting commonly used cryptosystems in low time and space complexity.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\\section{Introduction}\n\\begin{figure}[!htb]\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/REGroup.pdf}\n    \\caption{\\textbf{Overview of REGroup.}~\\textbf{R}ank-aggregating \\textbf{E}nsemble of \\textbf{G}enerative classifiers for \\textbf{ro}b\\textbf{u}st \\textbf{p}redictions. REGroup uses a pre-trained network, and constructs layer-wise generative classifiers modeled by a mixture distribution of the positive and negative pre-activation neural responses at each layer. At test time, an input sample's neural responses are tested with generative classifiers to obtain ranking preferences of classes at each layer. These preferences are aggregated using \\textit{Borda count} based preferential voting theory to make final prediction. \\textit{Note:} construction of layer-wise generative classifiers is a one time process.\n    }\n    \\label{fig:regroup}\n\\end{figure}\n\nDeep Neural Networks (DNNs) have shown outstanding performance on many computer vision tasks such as image classification~\\cite{krizhevsky2012imagenet}, speech recognition~\\cite{hinton2012deep}, and video classification~\\cite{karpathy2014large}. Despite showing superhuman capabilities in the image classification task~\\cite{he2015delving}, the existence of \\emph{adversarial examples} \\cite{szegedy2013intriguing} have raised questions on the reliability of neural network solutions for safety-critical applications. \n\nAdversarial examples are carefully manipulated adaptations of an input, generated with the intent to fool a classifier into misclassifying them. One of the reasons for the attention that adversarial examples garnered is the ease with which they can be generated for a given model by simply maximizing the corresponding loss function. This is achieved by simply using a gradient based approach that finds a small perturbation at the input which leads to a large change in the output \\cite{szegedy2013intriguing}. This apparent instability in neural networks is most pronounced for deep networks that have an accumulation effect over the layers. This effect results in taking the small, additive, adversarial noise at the input and amplifying it to substantially noisy feature maps at intermediate layers that eventually influences the softmax probabilities enough to misclassify the perturbed input sample. This observation of amplification of input noise over the layers is not new, and has been pointed out in the past \\cite{szegedy2013intriguing}. The recent work by \\cite{xie2019feature} addresses this issue by introducing \\emph{feature denoising} blocks in a network and training them with adversarially generated examples. \n\nThe iterative nature of generating adversarial examples makes their use in training to generate defenses computationally very intensive. For instance, the adversarially trained feature denoising model proposed by \\cite{xie2019feature} takes 38 hours on 128 Nvidia V100 GPUs to train a baseline ResNet-101 with ImageNet. While we leverage this observation of noise amplification over the layers, our proposed approach \\emph{avoids any training or fine-tuning} of the model. Instead, we use a representative subset of training samples and their layer-wise pre-activation responses to construct mixture density based generative classifiers, which are then combined in an ensemble using ranking preferences. \n\nGenerative classifiers have achieved varying degrees of success as defense strategies against adversarial attacks. Recently, \\cite{Fetaya_ICLR2020} studied the class-conditional generative classifiers and concluded that it is impossible to guarantee robustness of such models. More importantly, they highlight the challenges in training generative classifiers using maximum likelihood based objective and their limitations w.r.t. discriminative ability and identification of out-of-distribution samples. While we propose to use generative classifiers, we avoid using likelihood based measures for making classification decisions. Instead, we use rank-order preferences of these classifiers which are then combined using a \\emph{Borda count}-based voting scheme. Borda counts have been used in collective decision making and are known to be robust to various manipulative attacks \\cite{rothe2019borda}. \n\nIn this paper, we present our defense against adversarial attacks on deep networks, referred to as \\emph{Rank-aggregating Ensemble of Generative classifiers for robust predictions} (REGroup). At inference time, our defense requires white-box access to a pre-trained model to collect the pre-activation responses at intermediate layers to make the final prediction. We use the training data to build our generative classifier models. Nonetheless, our strategy is simple, network-agnostic, does not require any training or fine-tuning of the network, and works well for a variety of adversarial attacks, even with varying degress of hardness. Consistent with recent trends, we focus only on the ImageNet dataset to evaluate the robustness of our defense and report performance superior to defenses that rely on adversarial training \\cite{kurakin2016adversarial} and random input transformation \\cite{raff2019barrage} based approaches. Finally, we present extensive analysis of our defense with two different architectures (ResNet and VGG) on different targeted and untargeted attacks. Our primary contributions are summarized below:\n\\begin{itemize\n    \\item We present REGroup, a retraining free, model-agnostic defense strategy that leverages an ensemble of generative classifiers over intermediate layers of the model.\n    \\item We model each layer-wise generative classifier as a simple mixture distribution of neural responses obtained from a subset of training samples. We discover that \\emph{both positive and negative} pre-activation values contain information that can help correctly classify adversarially perturbed samples. \n    \\item We leverage the robustness inherent in Borda-count based consensus over the generative classifiers.\n    \\item We show extensive comparisons and analysis on the ImageNet dataset spanning a variety of adversarial attacks.\n\\end{itemize}\n\n\n\n\n\\section{Related Work}\nSeveral defense techniques have been proposed to make neural networks robust to adversarial attacks. Broadly, we can categorize them into two approaches that: 1. Modify training procedure or modify input before testing; 2. Modify network or change hyper-parameters and optimization procedure.\n\n\n\\subsection{Modify Training\/Inputs During Testing} \n\nSome approaches of defenses in this category are mentioned below. \\textit{Adversarial training}~\\cite{zheng2016improving,moosavi2017universal,xie2020adversarial,shafahi2020universal}. \\textit{Data compression}~\\cite{bhagoji2018enhancing} suppresses the high-frequency components and presents an ensemble-based defense approach. \\textit{Data randomization} ~\\cite{wang2016learning,xie2017adversarial} based approaches apply random transformations to the input to defend against adversarial examples by reducing their effectiveness.\n\nPixelDefend \\cite{song2018pixeldefend} sets out to find the image with the highest probability within an $\\epsilon$- neighbourhood of the original image, thereby moving the image back towards distribution seen in training data. Defense-GAN \\cite{samangouei2018defense} tries to model the distribution of unperturbed images and at inference, it generates an image close to what was provided but without adversarial perturbations. These two methods use techniques to generate a clean version of the input and pass to the classifier. \n\\subsection{Modify Network\/Network Add-ons}\nDefenses under this category address the \\textit{detection} of adversarial attacks or cater to both \\textit{detection and correction} of prediction.\nThe aim of detection only defenses is to highlight if an example is adversarial and prevent it from further processing. These approaches include employing a detector sub-network~\\cite{metzen2017detecting}, training the main classifier with an outlier class~\\cite{grosse2017statistical}, using convolution filter statistics~\\cite{li2017adversarial}, or applying feature squeezing~\\cite{xu2017feature} to detect adversarial examples. However, all of these methods have shown to be ineffective against strong adversarial attacks~\\cite{carlini2017adversarial}\\cite{sharma2018bypassing}.\nFull defense approaches include applying defensive distillation~\\cite{papernot2016distillation}\\cite{papernot2017extending} to use the knowledge from the output of the network to re-train the original model and improve the resilience of a network to small perturbations. Another approach is to augment the network with a sub-network called Perturbation Rectifying Network (PRN) \\cite{akhtar2018defense} to detect the perturbations; if the perturbation is detected, then PRN is used to classify the input image. However, later it was shown that the Carlini \\& Wagner (C\\&W) attack successfully defeated the defensive distillation approach.\n\n\n\\subsection{ImageNet Focused Defense Approaches}\nA few approaches have been evaluated on the ImageNet dataset, most of which are based on input transformations or image denoising. Nearly all these defenses designed for ImageNet have failed a thorough evaluation, with a regularly updated list maintained at ~\\cite{robustml}. The approaches in ~\\cite{prakash2018deflecting} and \\cite{liao2018defense} claimed 81\\% and 75\\% accuracy respectively under adversarial attacks. But after a thorough evaluation~\\cite{athalye2018robustness} and accounting for obfuscated gradients~\\cite{athalye2018obfuscated}, the accuracy for both was reduced to 0\\%. Similarly, ~\\cite{xie2017mitigating} and ~\\cite{guo2017countering} claimed 86\\% and 75\\% respectively, but these were also reduced to 0\\%~\\cite{athalye2018obfuscated}. A different approach proposed in ~\\cite{kannan2018adversarial} claimed an accuracy 27.9\\% but later it was also reduced to 0.1\\% ~\\cite{engstrom2018evaluating}. For a comprehensive related work on attacks and defenses, we suggest reader to refer \\cite{chakraborty2018adversarial}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{REGroup~ Methodology}\n\nWell-trained deep neural networks have a hierarchical structure, where the early layers transform inputs to feature spaces capturing local or more generic patterns, while later layers aggregate this local information to learn more semantically relevant representations. In REGroup, we use many of the higher layers and learn class-conditional generative classifiers, which are simple mixture-distributions estimated from the pre-activation neural responses at each layer from a subset of training samples. An ensemble of these layer-wise generative classifiers is used to make the final prediction by performing a Borda count-based rank-aggregation. Ranking preferences have been used extensively in robust fitting problems in computer vision \\cite{chin2011accelerated,Chin_NIPS2009,tiwari2018dgsac}, and we show its effectiveness in introducing robustness in DNNs against adversarial attacks. \n\nFig. \\ref{fig:regroup} illustrates the overall working of REGroup. The approach has three main components: First, we use each layer as a generative classifier that produces a ranking preference over all classes. Second, each of these class-conditional generative classifiers are modeled using a mixture-distribution over the neural responses of the corresponding layer. Finally, the individual layer's class ranking preferences are aggregated using Borda count-based scoring to make the final predictions. We introduce the notation below and discuss each of these steps in detail in the subsections that follow. \n\n\n\n\\noindent\\textbf{Notation.} In this paper, we will {always} use $\\ell$, $i$ and $j$ for indexing the $\\ell^{th}$~ layer, $i^{th}$~ feature map and the $j^{th}$~ input sample respectively. The {true} and {predicted} class label will be denoted by $y$~ and $\\widehat{y}$~ respectively.\n\nA classifier can be represented in a functional form as $\\widehat{y}$~$=\\mathcal{F}(x)$, it takes an input $x$ and predicts its class label $\\widehat{y}$~. We define $\\boldsymbol{\\phi}^{\\ell i}$ as the $\\ell^{th}$~ layer's $i^{th}$~ {pre-activation feature map}, i.e., the neural responses \\emph{before} they pass through the activation function. For convolutional layers, this feature map $\\boldsymbol{\\phi}^{\\ell i}$ is a 2D array, while for a fully connected layer, it is a scalar value. \n\n\n\\subsection{DNN Layers as Generative Classifiers}\n\n\\label{sec:LGC}\nWe use the highest $k$ layers of a DNN as generative classifiers that use the pre-activation neural responses to produce a ranking preference\\footnote{A rank is assigned to each class based on a score. In the case of ImageNet dataset, the class with rank-1 is most preferred\/likely class, while rank-1000 is the least preferred\/likely class} over all classes. The layer-wise generative classifiers are modeled as a class-conditional mixture distribution, which is estimated using only a \\textit{pre-trained} network and a small subset $X$ of the training data.\n\nLet $X$ contain only correctly classified {training samples}\\footnote{We took 50,000 out of $\\sim$ 1.2 millions training images from ImageNet dataset, 50 per class.}, which we can further divide into $M$ subsets, one for each class i.e $X =\\{  \\cup_{y=1}^{M} X_{y}\\}$, where $X_{y}$ is the subset containing samples that have labels $y$.\n\n\\subsubsection{Layerwise Neural Response Distributions}\nOur preliminary observations indicated that while the ReLU activations truncate the negative pre-activations during the forward pass, these values still contain semantically meaningful information. Our ablative studies in Fig. \\ref{fig:ablation} confirm this observation and additionally, on occasion, we find that the negative pre-activations are complementary to the positive ones. Since the pre-activation features are real-valued, we compute the features $\\boldsymbol{\\phi}_j^{\\ell i}$ for the $j^{th}$~ sample $x_j$, and define its positive ($P^{\\ell i}_j$) and negative ($N^{\\ell i}_j$) response accumulators as $P^{\\ell i}_j = \\sum\\max(0,\\boldsymbol{\\phi}_j^{\\ell i})$, $N^{\\ell i}_j=\\sum\\max(0,-\\boldsymbol{\\phi}_j^{\\ell i})$. \n\nFor convolutional layers, these accumulators represent the overall strength of positive and negative pre-activation responses respectively, when aggregated over the spatial dimensions of the $i^{th}$~ feature map of the $\\ell^{th}$~ layer. On the other hand, for the linear layers, the accumulation becomes trivial with each neuron having a scalar response $\\boldsymbol{\\phi}_j^{\\ell i}$. We can now represent the $\\ell^{th}$~ layer by the positive and negative response accumulator vectors denoted by $P^\\ell_j$ and $N^\\ell_j$ respectively. We normalize these vectors and define the layer-wise probability mass function (PMF) for the positive and negative responses as $\\mathbb{P}^{\\ell}_j=\\frac{P^\\ell_j}{||P^\\ell_j||_1}$ and $\\mathbb{N}^{\\ell}_j=\\frac{N^\\ell_j}{||N^\\ell_j||_1}$ respectively. \n\nOur interpretation of $\\mathbb{P}^{\\ell}_j$ and $\\mathbb{N}^{\\ell}_j$ as a PMF could be justified by drawing an analogy to the softmax output, which is also interpreted as a PMF. However, it is worth emphasizing that we chose the linear rescaling of the accumulator vectors rather than directly applying a softmax normalization. By separating out the positive and negative accumulators, we obtain two independent representations for each layer, which is beneficial to our rank-aggregating ensemble discussed in the following sections. A softmax normalization over a feature map comprising of positive and negative responses would have entirely suppressed the negative responses, discarding all its constituent semantic information. An additional benefit of the linear scaling is its simple computation. Algorithm \\ref{alg:LSS} summarizes the computation of the layer-wise PMFs for a given training sample.  \n\n\\begin{algorithm}[!htb]\n\\SetAlgoLined\n\\textbf{Input:} $x_j$ pre-activation features $\\boldsymbol{\\phi}^{\\ell i}_j\\in\\mathbb{R}^{H\\times W}$\\\\\n\\For{ $\\ell \\in [1..n]$}{\n $P^{\\ell i}_j = \\sum\\max(0,\\boldsymbol{\\phi}_j^{\\ell i}),~~~~~~\\forall~ i$~~~ (sum over H, W) \\\\\n $N^{\\ell i}_j=\\sum\\max(0,-\\boldsymbol{\\phi}_j^{\\ell i}),~~~~\\forall~ i$~~ (sum over H, W) \\\\\n \n}\n$P^{\\ell}_j \\leftarrow P^{\\ell}_j+\\delta,$~~~~~$N^{\\ell}_j \\leftarrow N^{\\ell}_j+\\delta$ \\\\\n$\\mathbb{P}_{j}^{\\ell i} \\leftarrow \\frac{P^{\\ell i}_j}{\\sum_{i} \\mathbb{P}_{j}^{\\ell i}},$~~~~~$\\mathbb{N}_{j}^{\\ell i} \\leftarrow \\frac{N^{\\ell i}_j}{\\sum_{i} \\mathbb{N}_{j}^{\\ell i}}$~~~(PMFs) \n\n\\caption{Layerwise PMF of neural responses. $H\\times W$ represents the spatial dimensions of pre-activation features. For $\\ell^{th}$~ convolutional layer the dimensions of feature maps $H\\times W = r^{\\ell}\\times s^{\\ell}$, and for linear layers the dimensions of neuron output $H\\times W = 1\\times 1$. \n}\n\\label{alg:LSS}\n\\end{algorithm}\n\n\\subsubsection{Layerwise Generative Classifiers} \nWe model the layerwise generative classifiers for class $y$ as a class-conditional mixture of distributions, with each mixture component as the PMFs $\\mathbb{P}^\\ell_j$ and $\\mathbb{N}^\\ell_j$ for a given training sample $x_j\\inX_y$. The generative classifiers corresponding to the positive and negative neural responses are then defined as the following mixture of PMFs\\\\\n\\begin{equation}\n    \\mathbb{C}^{+\\ell}_y = \\sum_{j:x_j \\in X_{y}}\\lambda_j\\mathbb{P}_{j}^{\\ell},\\qquad\n    \\mathbb{C}^{-\\ell}_y = \\sum_{j:x_j \\in X_{y}}\\lambda_j\\mathbb{N}_{j}^{\\ell}\n\\end{equation}\n    \nwhere the weights $\\lambda_j$ are nonnegative and add up to one in the respective equations. Here, $\\lambda_j$ is proportional to the softmax probability of the sample $x_j$, and  $\\delta$ is the small constant used for numerical stability. We choose the weights to be proportional to the softmax probability value as predicted by the network given the input $x_j$. Using the subset of training samples $X$, we construct the class-conditional mixture distributions, $\\mathbb{C}^{+\\ell}_y$ and $\\mathbb{C}^{-\\ell}_y$ at each layer $\\ell$ only once. At inference time, we input a test sample $x_j$, from the test set $\\mathcal{T}$, to the network and compute the PMFs $\\mathbb{P}^{\\ell}_j$ and $\\mathbb{N}^{\\ell}_j$ using Algorithm \\ref{alg:LSS}. As our test input is a PMF and the generative classifier is also a mixture distribution, we simply use the KL-Divergence between the classifier model $\\mathbb{C}^{+\\ell}$ and the test sample $\\mathbb{P}^{\\ell}_j$ as a classification score as\n\\begin{equation}\n    P_{KL}(\\ell,y) = \\sum_{i} \\mathbb{C}^{+\\ell i}_y \\log \\Bigg(\\frac{\\mathbb{C}^{+\\ell i}_y}{\\mathbb{P}^{\\ell i}}\\Bigg),\\forall y\\in\\{1,\\!\\ldots,\\!M\\}\n    \\label{eq:klp}\n\\end{equation}\nand similarly for the negative PMFs\n\\begin{equation}\n    N_{KL}(\\ell,y) = \\sum_{i} \\mathbb{C}^{-\\ell i}_y  \\log \\Bigg(\\frac{\\mathbb{C}^{-\\ell i}_y}{\\mathbb{N}^{\\ell i}}\\Bigg),\\forall y \\in\\{1,\\!\\ldots,\\!M\\} \n    \\label{eq:kln}\n\\end{equation}\n\nWe use a simple classification rule and select the predicted class $\\widehat{y}$~ as the one with the smallest KL-Divergence with the test sample PMF. However, rather than identifying $\\widehat{y}$~, at this stage we are only interested in rank-ordering the classes, which we simply achieve by sorting the KL-Divergences (Eqns. (\\ref{eq:klp}) and (\\ref{eq:kln})) in ascending order. The resulting ranking preferences of classes for the $\\ell^{th}$~ layer are given below in Eqns. \\eqref{eq:rankvec_pos} and \\eqref{eq:rankvec_neg} respectively. Where, $R^{\\ell y}_{+}$ is the rank (position of $y^{th}$ class in the ascending order of KL-Divergences in $P_{KL}$) of $y^{th}$ class in the $\\ell^{th}$~ layer preference list $R^{\\ell}_{+}$. \n\\begin{eqnarray}\nR^{\\ell}_{+}=[R^{\\ell 1}_{+},R^{\\ell 2}_{+},...,R^{\\ell y}_{+},...,R^{\\ell M}_{+}] \\label{eq:rankvec_pos} \\\\\nR^{\\ell}_{-}=[R^{\\ell 1}_{-},R^{\\ell 2}_{-},...,R^{\\ell y}_{-},...,R^{\\ell M}_{-}]\n\\label{eq:rankvec_neg}\n\\end{eqnarray}\n\n\n\\subsection{Robust Predictions with Rank Aggregation}\nRank aggregation based {preferential voting} for making group decisions is widely used in selecting a winner in a democratic setup~\\cite{rothe2019borda}. The basic premise of preferential voting is that $n$ voters are allowed to rank $m$ candidates in the order of their preferences. The rankings of all $n$ voters are then aggregated to make a final prediction.\n\n{Borda count}~\\cite{black1958theory} is one of the approaches for preferential voting that relies on aggregating the rankings of all the voters to make a collective decision~\\cite{rothe2019borda,kahng2019statistical}. The other popular voting strategies to find a winner out of $m$ different choices include {Plurality voting}~\\cite{van1992borda}, and {Condorcet winner}~\\cite{young1988condorcet}. In Plurality voting, the winner would be the one who gets the maximum fraction of votes, while Condorcet winner is the one who gets the majority votes.\n\n\n\n\\subsubsection{Rank Aggregation Using Borda Count}\n\\textit{Borda count} is a generalization of the majority voting. In a two-candidates case it is equivalent to majority vote. The \\textit{Borda count} for a candidate is the sum of the number of candidates ranked below it by each voter. In our setting, while processing a test sample $x_j\\in\\mathcal{T}$, \\textit{every layer} acts as two independent voters based on $\\mP^\\ell$ and $\\mN^\\ell$. The number of classes i.e $M$ is the number of candidates. The Borda count for the $y^{th}$~ class at the $\\ell^{th}$~ layer is denoted by $B^{\\ell y}=B^{\\ell y}_{+}+B^{\\ell y}_{-}$, where $B^{\\ell y}_{+}$ and $B^{\\ell y}_{-}$ are the individual Borda count of both the voters and computed as shown in eq. \\eqref{eq:indv_bordacount}.\n\\begin{equation}\nB^{\\ell y}_{+}  = (M - R^{\\ell y}_{+}),~~~~~~~\nB^{\\ell y}_{-}  = (M - R^{\\ell y}_{-}) \n\\label{eq:indv_bordacount}\n\\end{equation}\n\n\n\\subsubsection{Hyperparameter settings} \nWe aggregate the Borda counts of highest $k$ layers of the network, which is the only hyperparameter to set in REGroup. Let $B^{:ky}$ denote the aggregated Borda count of $y^{th}$~ class from the last $k$ layers irrespective of the type (convolutional or fully connected). Here, $n$ is the total number of layers. The final prediction would be the class with maximum aggregated Borda count.\n\\begin{align}\n\\nonumber B^{:ky} & = \\sum_{\\ell=n-k+1}^{n} B^{\\ell y}\\\\\n\\nonumber & = \\sum_{\\ell=n-k+1}^{n} B^{\\ell y}_{+}+B^{\\ell y}_{-},~~\\forall y\\in\\{1..M\\}\\\\\n\\widehat{y}&= \\textit{argmax}_{y} ~~B^{:ky}\n \\label{eq:agg_borda}\n\\end{align}\nTo determine the value of $k$, we evaluate REGroup~ on 10,000 \\textit{correctly classified} samples from the ImageNet Validation set at {each layer}, using {per layer} Borda count i.e $ \\widehat{y}= \\arg\\max_{y} ~~B^{\\ell y}$. We select $k$ to be the number of later layers at which we get at-least $75\\%$ accuracy. This can be viewed in the context of the confidence of individual layers on discriminating samples of different classes. We follow the above heuristic and found $k=5$ for both the architectures ResNet-50 and VGG-19, which we use in all our experiments. An ablation study with all possible values of $k$ is included in section \\ref{sec:ablation}\n\n\n\\section{Experiments}\nIn this section, we evaluate robustness of REGroup~ against state-of-the-art attack methods. We follow the recommendations on defense evaluation in~\\cite{carlini2019evaluating}. \\\\\n\\noindent \\textbf{Attack methods.} We consider attack methods in the following two categories: \\textit{gradient-based} and \\textit{gradient-free}. \\\\ \n\\underline{{Gradient-Based Attacks}}. Within this category, we consider two variants, \\textit{restricted} and \\textit{unrestricted} attacks. The restricted attacks generate adversarial examples by searching an adversarial perturbations within the bound of $L_p$ norm, while unrestricted attacks generate adversarial example by manipulating image-based visual descriptors.  Due to restriction on the perturbation the adversarial examples generated by restricted attacks are similar to the clean original image, while unrestricted attacks generate natural-looking adversarial examples, which are far from the clean original image in terms of $L_p$ distance. We consider the following, \\textit{Restricted attacks}: PGD~\\cite{madry2017towards} , DeepFool~\\cite{moosavi2016deepfool}, C\\&W~\\cite{carlini2017towards} and Trust Region~\\cite{yao2019trust}, and \\textit{Unrestricted attack}: cAdv~\\cite{bhattad2020unrestricted} semantic manipulation attack. An example of cAdv is shown in Fig.~\\ref{fig:cadv}. \\\\\n\n\\noindent \\underline{{Gradient-Free Attacks}}. The approaches in this category does not have access to the network weights. We consider following attacks: SPSA~\\cite{uesato2018adversarial}, Boundary~\\cite{brendel2017decision} and Spatial~\\cite{engstrom2019exploring}. Refer supplementary for the attack specific hyper-parameters detail.\n\\begin{figure}[!htb]\n\\centering\n\\begin{tabular}{c}\n \\includegraphics[width=7.5cm]{figures\/cars+pretzel.pdf} \n\\end{tabular}\n\\caption{cAdv~\\cite{bhattad2020unrestricted} adversarial examples}\n\\label{fig:cadv}\n\\end{figure}\n\n\\noindent \\textbf{Network architectures.} We consider   ResNet-50\\footnote{https:\/\/download.pytorch.org\/models\/resnet50-19c8e357.pth} and VGG-19\\footnote{https:\/\/download.pytorch.org\/models\/vgg19-dcbb9e9d.pth} architectures, {pre-trained} on ImageNet dataset.\\\\\n\\noindent\\textbf{Datasets.} We present our evaluations, comparisons and analysis only on ImageNet~\\cite{imagenet} dataset. We use the subsets of full ImageNet validation set as described in Tab. \\ref{tbl:dset}. Note:  V10K, V2K and V10C would be different for ResNet-50 and VGG-19, since an image classified correctly by ResNet-50 need not be classified correctly by the VGG-19. \n\\begin{table}[h]\n\\centering\n\\resizebox{\\linewidth}{!}{%\n\\begin{tabular}{l|l}\n\\hline\nDataset & Description            \\\\\n\\hline\nV50K         & Full ImageNet validation set with 50000 images.                         \\\\\nV10K         & A subset of 10000 correctly classified images from V50K set. 10 Per class. \\\\\nV2K          & A subset of 2000 correctly classified images from V50K set. 2 Per class.\\\\\nV10C         & A subset of correctly classified images of 10 sufficiently different classes.\\\\\n\\hline\n\\end{tabular}%\n}\n\\caption{Dataset used for evaluation and analysis.}\n\\label{tbl:dset}\n\\end{table}\n\n\n\\subsection{Performance on Gradient-Based Attacks}\n\n\\noindent \\textbf{Comparison with adversarial-training\/ fine-tuning.} \nWe evaluate REGroup~on clean samples as well as adversarial examples generated using PGD $(\\epsilon = 16)$ from V50K dataset, and compare it with prior state-of-the-art works. The results are reported in Tab. \\ref{tbl:sota}, and we see that REGroup~outperforms the state-of-the-art input transformation based defense BaRT~\\cite{raff2019barrage}, both in terms of the clean and adversarial samples (except in the case of Top-1 accuracy with $\\hat{k}=10$, which is the number of input transformations used in BaRT). We see that while our performance on clean samples decreases when compared to adversarial training (Inception v3), it improves significantly on adversarial examples with a high $\\epsilon=16$. While our method is not directly comparable with adversarially trained Inception v3 and ResNet-152, because the base models are different, a similar decrease in the accuracy over clean samples is reported in their paper. The trade-off between robustness and the standard accuracy has been studied in \\cite{dohmatob2018limitations} and \\cite{tsipras2018robustness}.\n\nAn important observation to make with this experiment is, if we set aside the base models of ResNets and compare Top-1 accuracies on clean samples of full ImageNet validation set, our method (REGroup) without any \\textit{adv-training\/fine-tuning} either outperforms or performs similar to the state-of-the-art \\textit{adv-training\/fine-tuning} based methods \\cite{raff2019barrage, xie2019feature}.\n\\begin{table}[!htb]\n\\centering\n\n\\scriptsize\n\\begin{tabular}{lcccc}\n\\toprule \n     ({Dataset used:} \\text{V50K}).           &  \\multicolumn{2}{c}{Clean Images} &\\multicolumn{2}{c}{Attacked Images}\n                \\vspace{0.01cm} \\\\\n                \n                \\cline{2-5}\n                \\vspace{0.02cm}\n                    Model            & Top-1 & Top-5 & Top-1 & Top-5 \\vspace{0.01cm}\\\\\n                                \\toprule\n                                ResNet-50 & 76 & 93 & 0.0 & 0.0 \\\\\n                                Inception v3 & 78 & 94 & 0.7 & 4.4 \\\\\n                               ResNet-152 & 79 & 94 & - & - \\\\\n                               \\hline\n                               Inception v3 w\/Adv. Train & 78 & 94 & 1.5 & 5.5 \\\\\n                                ResNet-152 w\/Adv. Train & 63 & - & 45 & - \\\\\n                                ResNet-152 w\/Adv. Train w\/ denoise & 66 & - & 49 & - \\\\\n                                ResNet-50-BaRT, $\\hat{k}=5$ & 65 & 85 & 16 & 51 \\\\\n                        ResNet-50-BaRT, $\\hat{k}=10$ & 65 & 85 & 36 & 57 \\\\\n                                \\hline   \n                                ResNet-50-REGroup & 66 & 86 & {22} & {65}\\\\\n                               \\toprule\n\\end{tabular}%\n\\caption{\\textbf{Comparison with adversarially trained and fine-tuned classification models.} Top-1 and Top-5 classification accuracy (\\%) of adversarial trained (Inception V3 \\cite{kurakin2016adversarial} and ResNet-152 ~\\cite{xie2019feature}) and fine-tuned (ResNet-50 BaRT~\\cite{raff2019barrage}) classification models. Clean Images are the non-attacked original images. The results are divided into three blocks, the top block include original networks, middle block include defense approaches based on adversarial re-training\/fine-tuning of original networks, bottom block is our defense \\textit{without re-training\/fine-tuning}. Results of the competing methods are taken from their respective papers. `-' indicate the results were not provided in the respective papers. }\n\\label{tbl:sota}\n\\end{table}\n\n\n\n\\noindent \\textbf{Performance w.r.t PGD Adversarial Strength.}\nWe evaluate REGroup~ w.r.t the maximum perturbation of the adversary. The results are reported in Fig.~\\ref{fig:pgd_strength}(a).\nREGroup~ outperforms both the adversarial training \\cite{kurakin2016adversarial} and BaRT~\\cite{raff2019barrage}.\nBoth adversarial training and BaRT have shown protection against PGD adversarial attacks with a maximum perturbation strength $\\epsilon=16$ and $\\epsilon=32$ respectively, however we additionally show the results with $\\epsilon=40$ on full ImageNet validation set. \nWe also note that with increasing perturbation strength, our defense's accuracy is also strictly decreasing. This is in accordance with \\cite{carlini2019evaluating}, where transitioning from a clean image to noise should yield a downward slope in accuracy, else there could be some form of gradient masking involved. While it may seem $\\epsilon=40$ is a large perturbation budget and it will destroy the object information in the image completely, but we would like to emphasize that it is not the case when using large size images. A comparison of PGD examples generated with $\\epsilon=40$ using CIFAR-10 (32 $\\times$ 32) and ImageNet (224 $\\times$ 224) images is shown in Fig.~\\ref{fig:pgd_strength}(b). \\\\\n\n\\begin{figure}[!htb]\n    \\centering\n   \n    \\includegraphics[width=8cm]{figures\/bart_comparison1+.pdf}\n    \\caption{\\textbf{Top-1 and Top-5 accuracy(\\%) w.r.t PGD adversarial strength.} Comparison with adversarial training based method~\\cite{kurakin2016adversarial} and fine-tuning using random input transformations based method (BaRT) \\cite{raff2019barrage} with Expectation Over Transformation (EOT) steps 10 and 40, against the PGD perturbation strength ($\\epsilon$). The results of the competing methods are taken from their respective papers. Dataset used: \\text{V50K}. }\n    \\label{fig:pgd_strength}\n\\end{figure}\n\n\\begin{table}[h]\n\\centering\n\\setlength{\\tabcolsep}{1.5pt}\n\\scriptsize\n\\begin{tabular}{l|ccc|ccc|ccc}\n\\hline\n                  \\multicolumn{1}{c}{}   &\n                  \\multicolumn{1}{c}{}   & \\multicolumn{1}{c}{}   & \\multicolumn{1}{c}{}   &  \\multicolumn{3}{c|}{ResNet-50}                                                                                                                       & \\multicolumn{3}{c}{VGG-19}                                                                                                                          \\\\\n\n \\cline{5-10}   \\multicolumn{1}{c}{}   &\n \\multicolumn{1}{c}{}   &\n \\multicolumn{1}{c}{UN \/}   &\n \\multicolumn{1}{c}{}   &  \\multicolumn{1}{c}{}  & \\multicolumn{1}{c}{SMax}                          & \\multicolumn{1}{c|}{REGroup}                      & \\multicolumn{1}{c}{}   &  \\multicolumn{1}{c}{SMax}                          & \\multicolumn{1}{c}{REGroup}                           \\\\\n  \\cline{5-10} \\multicolumn{1}{c}{}   & \n     \\multicolumn{1}{c}{Data}   & \n     \\multicolumn{1}{c}{TA \/ HC}   & \n     \\multicolumn{1}{c}{$\\epsilon$}   & \\multicolumn{1}{c}{\\#S}  & \\multicolumn{1}{c}{T1(\\%)}                          & \\multicolumn{1}{c|}{T1(\\%)}                      & \\multicolumn{1}{c}{\\#S}   &  \\multicolumn{1}{c}{T1(\\%)}                          & \\multicolumn{1}{c}{T1(\\%)}                           \\\\\n \\cline{1-2}\n \\cline{3-10}\nClean &V10K&-- & --  & 10000 &      {100}                          &     88                          &10000       &        {100}                           &        76\n\\\\\nClean &V2K &--&-- &2000   & 100   &86 &2000                                  & 100       &      72                                    \\\\\nClean &V10C &--&-- &417   & 100   &84 &392                                  & 100       &      79                                    \\\\\n\\hdashline\nPGD &V10K& UN &4 ($L_\\infty$) &  9997   &  0                         & {48}                                              &9887         &        0                   &     {46}                                                                    \\\\\nDFool   &V10K&UN & 2 ($L_2$)     &  9789     &       0                                   &     {61}                            &9939         &           0                            &        {55}                                        \\\\\nC\\&W &V10K &UN &4 ($L_2$)      & 10000   &       0                                &       {40}                             &10000       &      0                                 &    {38}                                                       \\\\\nTR &V10K&UN &2 ($L_\\infty$) & 10000    &      0                       &        {41}                              &  9103     &   0                             &       {45}                                     \\\\\ncAdv &V10C&UN &-- & 417    &      0                       &        {37}                              &  392     &   0                             &       {18}                                     \\\\\n\\hdashline\n\nPGD &V2K &TA & ($L_\\infty$)&2000 &0       &47                 &2000                                           &0                           &       31                                                                 \\\\\nC\\&W  &V2K &TA &($L_2$) &2000 & 0     &46                    & 2000                       &0                                        &      38                                                   \\\\\n\n\\hdashline\nPGD  &V2K &UN+HC& ($L_\\infty$)&  2000   &  0                         & 21                                              &  2000       &        0                   &     19                                                                    \\\\\nPGD  &V2K &TA+HC& ($L_\\infty$) &  2000   &  0                         &   23                                            &  2000       &        0                   &   17                                                                      \\\\\n\\hline\n\\end{tabular}%\n\\caption{\\textbf{Performance on Gradient-Based Attacks.} Comparison of Top-1 classification accuracy between SoftMax (SMax) and REGroup~ based final classification. UN and TA indicates, un-targeted and targeted attacks respectively.  The +HC indicates adversarial examples are generated with high-confidence ( $>90\\%$) constraint, in this case $\\epsilon$ can be any value that satisfies the HC criteria. For targeted attack we select a target class uniformly at random from the 1000 classes leaving out the true class. $\\#S$ is the number of images for which the attacker is successfully able to generate adversarial examples using the respective attack models and the accuracies are reported with respect to the $\\#S$ samples, hence the 0\\% accuracies with the SoftMax (SMax). Since $\\#S$ is different for several attacks, therefore, the performance may not be directly comparable \\emph{across} different attacks. `--' indicate the information is not-applicable. For data description refer Tab. \\ref{tbl:dset}.}\n\\label{tbl:grad-based}\n\\end{table}\n\n\n\\noindent \\textbf{Performance on Un-Targeted Attacks.}\nWe evaluate REGroup~ on various untargeted attacks and report results in Tab. \\ref{tbl:grad-based}. The perturbation budgets ($\\epsilon$) and dataset used for the respective attacks are listed in the table. With the exception of the maximum perturbation allowed, we used default parameters given by FoolBox~\\cite{rauber2017foolbox}. Due to space limitations, the attack specific hyper-parameters detail are included in the supplementary. We observe that the performance of our defense is quite similar for both the models employed. This is due to the attack-agnostic nature of our defense. We achieve 48\\% accuracy (ResNet-50) for PGD attack using our defense which is significant given that PGD is considered to be one of the strongest attacks among the class of first order adversaries. \\\\\n\n\n\n\n\\noindent \\textbf{Performance on Unrestricted, Untargeted Semantic Manipulation Attacks.}\nWe consider V10C dataset for cAdv attack. We use the publicly released source code by the authors. Specifically we use $\\text{cAdv}_{4}$ variant with the parameters suggested by the authors. The results are reported in Tab. \\ref{tbl:grad-based}. \\\\\n\n\\noindent \\textbf{Performance on Targeted Attacks.}\nWe consider \\text{V2K}~dataset for targeted attacks and report the performance on PGD and C\\&W targeted attacks in Tab. \\ref{tbl:grad-based}. Target class for such attacks is chosen uniformly at random from the 1000 ImageNet classes apart from the original(ground-truth) class.  \\\\\n\n\\noindent \\textbf{Performance on PGD attack with High Confidence.}\nWe evaluate REGroup~on PGD examples on which the network makes highly confident predictions using SoftMax. We generate un-targeted and targeted adversarial examples using PGD attack with a constraint that the network's confidence of the prediction of adversarial examples is \\text{at-least 90}\\%. For this experiment we do not put constraint on the adversarial perturbation i.e $\\epsilon$. Results are reported in Tab. \\ref{tbl:grad-based}.\n\n\n\n\n\\subsection{Performance on Gradient-Free Attacks}\n\nSeveral studies \\cite{athalye2018obfuscated}, \\cite{papernot2017practical} have observed a phenomenon called\\textit{ gradient masking}. This phenomenon occurs when a practitioner unintentionally or intentionally proposes a defense which does not have meaningful gradients, either by reducing them to small values (vanishing gradients), removing them completely (shattered gradients) or adding some noise to it (stochastic gradient). \n\nGradient masking based defenses hinder the gradient computation and in turn inhibit gradient-based attacks, thus providing a false sense of security. Therefore, to establish the robustness of a defense against adversarial attacks in general, it is important to rule out that a defense relies on gradient masking. \n\nTo ensure that REGroup~ is not masking the gradients we follow the standard practice \\cite{pang2019rethinking} \\cite{zhang2020clipped} and evaluate on strong gradient-free SPSA~\\cite{uesato2018adversarial} attack. In addition to SPSA, we also show results on two more gradient-free attacks, Boundary~\\cite{brendel2017decision} and Spatial~\\cite{engstrom2019exploring} attack. The results are reported in Tab. \\ref{tbl:grad-free}. \n\n\\begin{table}[!htb]\n\\centering\n\\setlength{\\tabcolsep}{1.1pt}\n\\scriptsize\n\\begin{tabular}{l|ccc|ccc|ccc}\n\\hline\n                  \\multicolumn{1}{c}{}   &\n                  \\multicolumn{1}{c}{}   & \\multicolumn{1}{c}{}   & \\multicolumn{1}{c}{}   &  \\multicolumn{3}{c|}{ResNet-50}                                                                                                                       & \\multicolumn{3}{c}{VGG-19}                                                                                                                          \\\\\n\n \\cline{5-10}   \\multicolumn{1}{c}{}   &\n \\multicolumn{1}{c}{}   &\n \\multicolumn{1}{c}{UN \/}   &\n \\multicolumn{1}{c}{}   &  \\multicolumn{1}{c}{}  & \\multicolumn{1}{c}{SMax}                          & \\multicolumn{1}{c|}{REGroup}                      & \\multicolumn{1}{c}{}   &  \\multicolumn{1}{c}{SMax}                          & \\multicolumn{1}{c}{REGroup}                           \\\\\n  \\cline{5-10} \\multicolumn{1}{c}{}   & \n     \\multicolumn{1}{c}{Data}   & \n     \\multicolumn{1}{c}{TA \/ HC}   & \n     \\multicolumn{1}{c}{$\\epsilon$}   & \\multicolumn{1}{c}{\\#S}  & \\multicolumn{1}{c}{T1(\\%)}                          & \\multicolumn{1}{c|}{T1(\\%)}                      & \\multicolumn{1}{c}{\\#S}   &  \\multicolumn{1}{c}{T1(\\%)}                          & \\multicolumn{1}{c}{T1(\\%)}                           \\\\\n \\cline{1-2}\n \\cline{3-10}\nSPSA  &V10K&UN  &4 ($L_{\\infty}$)  & 4911 &      0                          &     {71}                          &5789       &        0                           &        {58}                                     \\\\\nBoundary &V10K&UN& 2 ($L_{2}$)    & 10000 &     0                          &   {50}                                  & 10000 &    0                &         {50}                                                             \\\\\nSpatial  &V10K&UN  & 2 ($L_{2}$) & 2624 &      0                          &     {36}                          &2634       &        0                           &        {30}                                     \\\\\n\\hline\n\\end{tabular}%\n\\caption{\\textbf{Performance on Gradient-Free Attacks.} Top-1 ( \\%) classification accuracy comparison between SoftMax (SMax) and REGroup. Legends are same as in Tab. \\ref{tbl:grad-based}. }\n\\label{tbl:grad-free}\n\\end{table}\n\nThe consistent superior performance on both gradient-based (both restricted and unrestricted) and gradient free attack shows REGroup~is not masking the gradients and is attack method agnostic.\n\n\\section{Analysis}\\label{sec:ablation}\n\n\\subsection{Accuracy vs number of layers ($k$)}\nWe report performance of REGroup~on various attacks reported in Tab. \\ref{tbl:grad-based} for all possible values of $k$. The accuracy of VGG-19 w.r.t. the various values of $k$ is plotted in Fig. \\ref{fig:acc_vs_k}. We observe a similar accuracy vs $k$ graph for ResNet-50 and note that a reasonable choice of $k$ made based on this graph does not significantly impact REGroup's performance. Refer Fig. \\ref{fig:acc_vs_k}, the `Agg' stands for using aggregated Borda count $B^{:ky}$. PGD(V10K,UN), DFool, C\\&W(V10K,UN) and Trust Region are the same experiments as reported in Tab. \\ref{tbl:grad-based}, but with all possible values of $k$. `Per\\_Layer\\_V10K' stands for evaluation using per layer Borda count i.e  $\\widehat{y}$~$= \\textit{argmax}_{y} ~~B^{\\ell y}$ on a separate 10,000 correctly classified subset of validation set. In all our experiments we choose the $k$-highest layers where `Per\\_Layer\\_V10K' has at-least $75\\%$ accuracy. A reasonable change in this accuracy criteria of $75\\%$  would not affect the results on adversarial attacks significantly. However, a substantial change (to say $50\\%$) deteriorates the performance on clean sample significantly. \n\\begin{figure}[htb]\n    \\centering\n    \\includegraphics[width=0.9\\linewidth]{figures\/vs_k.pdf}\n    \\caption{Accuracy vs no. of layers ($k$)  }\n    \\label{fig:acc_vs_k}\n\\end{figure}\n\\begin{figure*}[htb]\n    \\centering\n    \\includegraphics[width=1.0\\linewidth]{figures\/posneg_fig.pdf}\n    \\caption{ Effect of Considering Positive and Negative Pre-Activation Responses}\n    \\label{fig:ablation}\n\\end{figure*}\nThe phenomenon of decrease in accuracy of clean samples vs robustness has been studied in \\cite{dohmatob2018limitations} and \\cite{tsipras2018robustness}. \n\n\\subsection{Effect of positive and negative pre-activation responses}\nWe report the impact of using positive, negative and a combination of both pre-activation responses on the performance of REGroup~ in Fig. \\ref{fig:ablation}. We consider three variants of Borda count rank aggregation from later $k$ layers. {Pos:} $B^{:ky} = \\sum_{\\ell=n-k+1}^{n} B^{\\ell y}_{+}$, {Neg:} $B^{:ky} = \\sum_{\\ell=n-k+1}^{n} B^{\\ell y}_{-}$, and { Pos+Neg:} $B^{:ky} = \\sum_{\\ell=n-k+1}^{n} B^{\\ell y}_{+}+B^{\\ell y}_{-}$. We report the Top-1 accuracy (\\%) of the attacks experiment as set up in Tab. \\ref{tbl:grad-based} (DF: DFool, C\\&W, TR: Trust Region), in Tab. \\ref{tbl:grad-free} (BD: Boundary, SP: Spatial), and in Fig. \\ref{fig:pgd_strength} (PGD2, PGD4 and PGD8, with $\\epsilon=2$, $4$ and $8$ respectively). From the bar chart it is evident that in some experiments, {Pos} performs better than { Neg } (e.g UN\\_TR), while in others {Neg} is better than {Pos} only (e.g UN\\_DF). It is also evident that {Pos}+{Neg} occasionally improve the overall performance, and the improvement seems significant in the targeted C\\&W attacks for both the ResNet-50 and VGG-19. We leave it to the design choice of the application, if inference time is an important parameter, then one may choose either Pos or Neg to reduce the inference time to approximately half of what is reported in Tab. \\ref{tbl:inf_time}.\n    \n\n\n\n\n\\subsection{Results on CIFAR-10}\nWhile we mainly show results on large-scale dataset (ImageNet), we believe scaling down the datasets to one like CIFAR10 will not have a substantial impact on REGroup's performance. We evaluate REGroup on CIFAR10 dataset using VGG-19 based classifier. We construct generative classifiers using CIFAR-10 dataset following the same protocol as for the ImageNet case described in the Sec. \\ref{sec:LGC}. We apply PGD attack with $\\epsilon=4$ and generate adversarial examples. The results are included in the Tab. \\ref{tbl:cifar_pgd}.\n\n\\begin{table}[!htb]\n\\centering\n\\setlength{\\tabcolsep}{3pt}\n\\scriptsize\n\\begin{tabular}{lccc}\n\\hline\n                  \\multicolumn{1}{c}{}                                                                                                                        & \\multicolumn{3}{c}{VGG-19}                                                                                                                          \\\\\n\n \\cline{2-4}                       & \\multicolumn{1}{c}{}   &  \\multicolumn{1}{c}{SMax}                          & \\multicolumn{1}{c}{REGroup}                           \\\\\n  \n     \\multicolumn{1}{c}{}                       & \\multicolumn{1}{c}{\\#S}   &  \\multicolumn{1}{c}{T1(\\%)}                          & \\multicolumn{1}{c}{T1(\\%)}                           \\\\\n \\cline{1-2}\n \\cline{3-4}\n\n \\vspace{0.01cm}\nClean  &  10000       &        92                   &   88                                                                      \\\\\nPGD Untargeted  &  9243       &        0                   &   57                                                                      \\\\\n\\hline\n\\end{tabular}%\n\\caption{\\textbf{Performance on CIFAR10.} Comparison of Top-1 classification accuracy between SoftMax (SMax) and REGroup~ based final classification. $\\#S$ is the number of images for which the attacker is successfully able to generate adversarial examples using PGD attack and the accuracies are reported with respect to the $\\#S$ samples, hence the 0\\% accuracies with the SoftMax (SMax). }\n\\label{tbl:cifar_pgd}\n\\end{table}\n\n\\subsection{Inference time using REGroup} Since we suggest to use REGroup~as a test time replacement of SoftMax, we compare the inference time on both CPU and GPU in Tab. \\ref{tbl:inf_time}. We use a machine with an i7-8700 CPU and GTX 1080 GPU.\n\n\n\n\\begin{table}[!htb]\n\\centering\n\\scriptsize\n\\begin{tabular}{lcccc|cccc}\n\\hline\n                  \\multicolumn{1}{c}{}   &  \\multicolumn{4}{c|}{ResNet-50}                                                                                                                       & \\multicolumn{4}{c}{VGG-19}                                                                                                                          \\\\\n\n \\cline{2-9}&  \\multicolumn{2}{c}{SMax}&   \\multicolumn{2}{c|}{REGroup}        & \\multicolumn{2}{c}{SMax}& \\multicolumn{2}{c}{REGroup}                          \\\\\n \\cline{2-9}   \\multicolumn{1}{c}{}   &  \\multicolumn{1}{c}{GPU}  & \\multicolumn{1}{c}{CPU}& \\multicolumn{1}{c}{GPU}                          & \\multicolumn{1}{c|}{CPU}                     & \\multicolumn{1}{c}{GPU}   &  \\multicolumn{1}{c}{CPU}\n &  \\multicolumn{1}{c}{GPU}& \\multicolumn{1}{c}{CPU}                           \\\\\n \\cline{1-2}\n \\cline{3-9}\n\n \\vspace{0.01cm}\nTime(s) & 0.02    &  0.06    &0.13                     &    0.35                                         &0.03         &        0.12                   &       0.16    &  0.64                                                              \\\\\n\\hline\n\\end{tabular}%\n\\caption{\\textbf{Inference time comparison.} REGroup~vs SoftMax}\n\\label{tbl:inf_time}\n\\end{table}\n\n \n\nIn this work, we have presented a simple, scalable, and practical  defense strategy that is model agnostic and does not require any re-training or fine-tuning. We suggest to use REGroup~at test time to make a pre-trained network robust to adversarial perturbations. There are three aspects of REGroup~that justify its success. First, instead of using a maximum likelihood based prediction, REGroup~adopts a ranking preference based approach. Second, aggregation of preferences from multiple layers lead to group decision making, unlike SoftMax that relies on the output of the last layer only. Using both positive and negative layerwise responses help contribute to the robustness of REGroup. Third, there exists inherent robustness of Borda count based rank aggregation in the presence of noisy individual voters \\cite{rothe2019borda},~\\cite{kahng2019statistical}. Hence, where SoftMax fails to predict the correct class of an adversarial example, REGroup~ takes ranked preferences from multiple layers and builds a consensus using Borda count to make robust predictions. Our promising empirical results indicate that deeper theoretical analysis of REGroup~ would be an interesting direction to pursue.  One direction of analysis could be inspired from the recently proposed perspective of neurons as cooperating classifiers \\cite{davel2020dnns}. \n\n\\section{Hyper-parameters for Generating Adversarial Examples}\nWe use Foolbox's \\cite{rauber2017foolbox} implementation of almost all the adversarial attacks(except SPSA\\footnote{https:\/\/github.com\/tensorflow\/cleverhans}, Trust Region\\footnote{https:\/\/github.com\/amirgholami\/TRAttack} and cAdv\\footnote{https:\/\/github.com\/AI-secure\/Big-but-Invisible-Adversarial-Attack}) used in this work. We report the attack specific hyper-parameters in Tab.\\ref{tab:hyperparams}.\n\n\\section{Elastic-Net Attacks}\nWe evaluate REGroup~ on Elastic-Net attacks \\cite{chen2018ead}. Elastic-Net attack formulate the attack process as a elastic-net regularized optimization problem. The results are shown in the table \\ref{tbl:ead_attacks}.\n\n\\begin{table}[!htb]\n\\centering\n\\setlength{\\tabcolsep}{3pt}\n\\scriptsize\n\\begin{tabular}{l|c|cc|c|cc}\n\\hline\n                  \\multicolumn{1}{c}{}   &  \\multicolumn{3}{c|}{ResNet-50}                                                                                                                       & \\multicolumn{3}{c}{VGG-19}                                                                                                                          \\\\\n\n \\cline{2-7}   \\multicolumn{1}{c|}{}   &  \\multicolumn{1}{c|}{}  & \\multicolumn{1}{c}{SMax}                          & \\multicolumn{1}{c|}{REGroup}                      & \\multicolumn{1}{c|}{}   &  \\multicolumn{1}{c}{SMax}                          & \\multicolumn{1}{c}{REGroup}                           \\\\\n  \n     \\multicolumn{1}{c|}{Attacks}   &  \\multicolumn{1}{c|}{\\#S}  & \\multicolumn{1}{c}{T1(\\%)}                          & \\multicolumn{1}{c|}{T1(\\%)}                      & \\multicolumn{1}{c|}{\\#S}   &  \\multicolumn{1}{c}{T1(\\%)}                          & \\multicolumn{1}{c}{T1(\\%)}                           \\\\\n \\cline{1-2}\n \\cline{3-7}\n\n \\vspace{0.01cm}\nEAD-Attack  &  2000   &  0                         &   52                                            &  2000       &        0                   &   49                                                                      \\\\\n\\hline\n\\end{tabular}%\n\\caption{\\textit{Performance on EAD attacks.} Top-1 ( \\%) classification accuracy comparison between SoftMax (SMax) and REGroup. $\\#S$ is the number of images for which the attacker is successfully able to generate adversarial examples and the accuracies are reported with respect to the $\\#S$ samples, hence the 0\\% accuracies with the SoftMax (SMax).   }\n\\label{tbl:ead_attacks}\n\\end{table}\n\n\n\n\n\\section{Accuracy vs no. of layer\/voters(ResNet50)}\n\\label{sec:resnet50_vs_k}\nWe report the performance of REGroup on various attacks reported in table 2 of the main paper for all possible values of $k$. The accuracy of ResNet-50 w.r.t. the various values of $k$ is plotted in figure \\ref{fig:abl_vs_k_resnet}. \n\n\n\n\\begin{table*}[!htb]\n    \\centering\n    \\begin{tabular}{|c|c|}\n    \\hline\n    \\textbf{Attack} & \\textbf{Hyper-parameters} \\\\\n    \\hline\n    \\multirow{2}{*}{PGD (Untargeted)} & $\\epsilon=4$, Dist:$L_\\infty$, random\\_start=True,\\\\\n    & stepsize=0.01, max\\_iter=40  \\\\\n    \\hline\n    \\multirow{2}{*}{DeepFool (Untargeted)} & $\\epsilon=2$, Dist:$L_2$, max\\_iter=100, \\\\\n    & subsample=10 (Limit on the number of the most likely classes)\\\\\n    \\hline\n    \\multirow{2}{*}{CW (Untargeted)} & $\\epsilon=4$, Dist:$L_2$, binary\\_search\\_steps=5, max\\_iter=1000,\\\\\n    &  confidence=0, learning\\_rate=0.005, initial\\_const=0.01 \\\\\n    \\hline\n    Trust Region (Untargeted) & $\\epsilon=2$, Dist:$L_\\infty$, iterations=5000  \\\\\n    \\hline\n    \\multirow{7}{*}{Boundary (Untargeted)} & $\\epsilon=2$, Dist:$L_2$, iterations=500, max\\_directions=25,  starting\\_point=None, \\\\\n    &initialization\\_attack=None, \n     log\\_every\\_n\\_steps=None, \\\\\n    & spherical\\_step=0.01, source\\_step=0.01, step\\_adaptation=1.5, \\\\\n    & batch\\_size=1, tune\\_batch\\_size=True, \\\\\n    & threaded\\_rnd=True, threaded\\_gen=True \\\\\n    \\hline\n    \\multirow{3}{*}{Spatial (Untargeted)} & $\\epsilon=2$, Dist=$L_2$, do\\_rotations=True, do\\_translations=True, x\\_shift\\_limits=(-5, 5),\\\\\n    & y\\_shift\\_limits=(-5, 5), angular\\_limits=(-5, 5), granularity=10,\\\\ & random\\_sampling=False, abort\\_early=True \\\\\n    \\hline\n    \\multirow{2}{*}{PGD (Targeted)} &  Dist = $L_\\infty$, binary\\_search=True, epsilon=0.3, \\\\\n    & stepsize=0.01, iterations=40, random\\_start=True, return\\_early=True \\\\\n    \\hline\n     \\multirow{2}{*}{CW (Targeted)} & binary\\_search\\_steps=5, max\\_iterations=1000, confidence=0,  \\\\\n    & learning\\_rate=0.005, initial\\_const=0.01, abort\\_early=True \\\\\n    \\hline\n    SPSA & $\\epsilon=(4,8)$, Dist:$L_\\infty$, max\\_iter=300, batch\\_size=64, early\\_stop\\_loss\\_thresh = 0, \\\\ \n    & perturbation\\_size $\\delta=0.01$, Adam LR=0.01\\\\\n    \\hline\n    \\multirow{2}{*}{EAD} & Dist=$L_2$, binary\\_search\\_steps=5, max\\_iterations=1000, confidence=0, \\\\\n    & initial\\_learning\\_rate=0.01, regularization=0.01, initial\\_const=0.01, abort\\_early=True \\\\\n    \\hline\n    \\multirow{2}{*}{PGD (Untargeted,HC)} & min\\_conf=0.9, Dist=$L_\\infty$, binary\\_search=True, epsilon=0.3, \\\\\n    & stepsize=0.01, iterations=40, random\\_start=True, return\\_early=True \n    \\\\\n    \\hline\n    \\multirow{2}{*}{PGD (Targeted,HC)} & min\\_conf=0.9, Dist=$L_\\infty$, binary\\_search=True, epsilon=0.3, \\\\\n    & stepsize=0.01, iterations=40, random\\_start=True, return\\_early=True  \\\\\n    \\hline\n\n    \n    \\end{tabular}\n\\caption{Attack Specific Hyper-parameters.} \n\\label{tab:hyperparams}\n\\end{table*}\n\n\\begin{figure*}[!htb]\n    \\centering\n    \\includegraphics[width=17cm]{supp_figures\/resnet50_vs_k.png}\n    \\caption{\\text{Ablation study for accuracy vs no. of layers ($k$) on ResNet-50}: `Agg' stands for using aggregated Borda count $B^{:ky}$. PGD, DFool, C\\&W and Trust Region are the same experiments as reported in table 2 of the main paper, but with all possible values of $k$. \"Per\\_Layer\\_V10K\" stands for evaluation using per layer Borda count i.e  $\\widehat{y}$~$= \\textit{argmax}_{y} ~~B^{\\ell y}$ on a separate 10,000 correctly classified subset of validation set. In all our experiments we choose the $k$-highest layers where `Per\\_Layer\\_V10K' has at-least $75\\%$ accuracy. A reasonable change in this accuracy criteria of $75\\%$  would not affect the results on adversarial attacks significantly. However, a substantial change (to say $50\\%$) deteriorates the performance on clean sample significantly. The phenomenon of decrease in accuracy of clean samples vs robustness has been studied in \\cite{dohmatob2018limitations} and \\cite{tsipras2018robustness}. \\textbf{Note:} There are four down-sampling layers in the ResNet-50 architecture, hence the total 54 layers.   }\n    \\label{fig:abl_vs_k_resnet}\n\\end{figure*}\n\n\n\n\\begin{figure*}[!htb]\n\n    \\centering\n   \\hspace{-0.5cm} \\subfigure[ResNet-50 $46^{th}$ Layer (CONV)]{\n    \\includegraphics[width=5.5cm]{supp_figures\/r50_46.png}} \\hspace{-0.2cm}\n     \\subfigure[ResNet-50 $48^{th}$ Layer (CONV)]{\\includegraphics[width=5.5cm]{supp_figures\/r50_48.png}}\n    \\subfigure[ResNet-50 $50^{th}$ Layer (FC)]{\\includegraphics[width=5.5cm]{supp_figures\/r50_fc.png}}\\\\\n  \n   \\hspace{-0.5cm}     \\subfigure[VGG-19 $17^{th}$ Layer (FC)]{\\includegraphics[width=5.5cm]{supp_figures\/vgg19_fc17.png}}\n   \\subfigure[VGG-19 $18^{th}$ Layer (FC)]{\\includegraphics[width=5.5cm]{supp_figures\/vgg19_fc18.png}} \\hspace{-0.2cm}\n     \\subfigure[VGG-19 $19^{th}$ Layer (FC)]{\\includegraphics[width=5.5cm]{supp_figures\/vgg19_fc19.png}}\n   \n    \\caption{TSNE visualization of three variants of pre-activation features i.e positive only (pos), negative only (neg) and combined positive and negative (combined). Visualization of 50 samples of 5 random classes of ImageNet dataset. Class membership is color coded. The dimensions of the pos, neg and combined variants of pre-activation feature is the same for any fully connected layer, while for a CONV layer, pos and neg has the same dimension which is equal to the no. of filters\/feature maps of the respective CONV layer and for combined it is equal to the dimension we get after flattening the whole CONV layer. It can be observed in figure(b) that the cluster formed by combined pre-activation feature responses is not a tight as formed by pos and neg separately, which shows the importance of considering pos and neg re-activation responses separately.  }\n    \\label{fig:tsne_plots}\n\\end{figure*}\n\n\n\n\n\n\n\n\n\\section{Analyzing Pre-Activation Responses}\n\\label{sec:tsne}\nOne of the contributions of our proposed approach is to use both positive and negative pre-activation values separately. We observed both positive and negative pre-activation values contain information that can help correctly classify adversarially perturbed samples. An empirical validation of our statement is shown in figure 3 of the main paper. We further show using TSNE~\\cite{vanDerMaaten2008} plots that all the three variants of the pre-activation feature of a single layer i.e positive only (pos), negative only (neg) and combined positive and negative pre-activation values forms clusters. This indicates that all three contain equivalent information for discriminating samples from others. While on one hand where ReLU like activation functions discard the negative pre-activation responses, we consider negative responses equivalently important and leverage them to model the layerwise behaviour of class samples. The benefit of using positive and negative accumulators is it reduce the computational cost significantly e.g flattening a convolution layer gives a very high-dimensional vector while accumulator reduce it to number of filter dimensions.\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nDiffusion magnetic resonance imaging (dMRI) is an in vivo and non-invasive\nmedical imaging technology that uses water diffusion as a proxy to probe\nanatomical structures of biological samples.   The most important application\nof dMRI is to  reconstruct white matter fiber tracts \\revisedc{in brain} -- large \\revisedd{axonal}\nbundles with similar destinations.\nIn white matter,\n  water diffusion appears to be anisotropic as water tends to diffuse faster along the fiber bundles. Therefore,  white matter fiber structures can be deduced from the diffusion characteristics of water.\n   Mapping white matter fiber tracts is of great importance in the study of structural organization of neuronal networks and the understanding of brain functionality\n\\citep{Mori07, Sporns11}. Moreover, dMRI also \\revisedd{has} many\n  clinical applications, including detecting brain abnormality in white matter\n  due to axonal loss or deformation, which are thought to be \\revisedc{related to}\n  many neuron degenerative diseases including Alzheimer's disease, \\revisedd{and also in} surgical\n  planning by resolving complex neuronal connections  between white and gray\n  matter \\citep{nimsky2006implementation}.\n\n  \\jpeng{dMRI techniques sensitize  signal intensity with the amount of water\n    diffusion by applying pulsed magnetic gradient fields on the sample.\n    Specifically, water diffusion along the gradient field direction leads to signal\n    loss and the amount of loss at a voxel equals to the summation (across\n    locations within the voxel) of the sinusoid waves with shifted signal phases\n    weighted by the proton density at their respective locations. In other words,\n    signal loss (referred to as diffusion weighted signal) is the inverse Fourier\n    transform of the diffusion probability density function of water molecules and thus can be used to\n    recover water diffusion characteristics. The amount of signal loss is also\n    influenced by various experimental parameters including  the gradient field\n    intensity (the stronger, the more loss), the  duration of gradient fields (the\n    longer, the more loss), etc. Their effects are aggregatively reflected by an\n    experimental  parameter  called the ``$b$-value\" which is often fixed throughout\n    the experiment (though multiple $b$-values are used in Q-space imaging). Since\n    only water motion along the gradient field direction can be detected, multiple\n    gradient directions need to be applied \\citep{Mori07}.}\n\n    \\jpeng{In its raw form, dMRI\n    provides diffusion weighted signal measurements on a 3D spatial grid (of the\n    sample) along a set of predetermined gradient directions}\n\\citep{Bammer-Holdsworth-Veldhuis09, Beaulieu02, Chanraud-Zahr-Sullivan10,\n  Mukherjee-Berman-Chung08}. For example, a typical data set from the Alzheimer's Disease Neuroimaging Initiative (ADNI) has diffusion measurements along $41$ gradient directions for each voxel on a $256 \\times 256 \\times 59$ 3D grid of the brain. The first step of dMRI analysis\n  is to summarize these measurements into estimates of water diffusion at each voxel. A popular model for water diffusion is the so called single tensor model where the diffusion process is \\revisedd{modeled} as a 3D Gaussian process described by a $3 \\times 3$ positive definite matrix, referred to as a diffusion tensor; see \\citet{Mori07} for an introduction to diffusion tensor imaging (DTI) techniques.\n  \\rwong{Figure~\\ref{fig:tensormap} depicts a tensor map on a 2D grid, where each diffusion tensor is represented by an ellipsoid, estimated from diffusion weighted measurements from an ADNI data set using a single tensor model.}\n  One then extracts the\n  \\revisedd{local diffusion direction as the principal eigenvector of the (estimated) diffusion tensor}\n at each voxel and reconstructs the white matter fiber tracts by computer aided tracking algorithms via a process called tractography \\citep{Basser-Pajevic-Pierpaoli00}.\n\nHowever, DTI cannot resolve multiple fiber populations with distinct orientations, i.e., crossing fibers, within a voxel since a tensor only has one principal direction. Consequently, in crossing fiber regions, estimated diffusion tensors may lead to low anisotropy estimation or oblate tensor estimation.  Poor tensor estimation results in poor direction estimation which adversely affects fiber reconstruction; e.g., early termination of or biased fiber tracking.\n\nIn order to resolve intravoxel orientational heterogeneity, several approaches have been proposed.  \\citet{Tuch-Reese-Wiegell02} propose a multi-tensor model which assumes a finite number of homogeneous fiber directions within a voxel.  However, it has been shown that the parameters in the multi-tensor model are not identifiable \\citep{Scherrer-Warfield10}.\n\\jpeng{Imaging techniques such as Q-ball and Q-space and the corresponding nonparametric methods have also been proposed}\n\\citep{Tuch04, Descoteaux-Angelino-Fitzgibbons07}.  However such methods often require high angular resolution  diffusion imaging (HARDI) \\citep{Tuch-Reese-Wiegell02, Hosey-Williams-Ansorge05} where a large number of gradients is sampled.\n\\revisedc{In light of these facts},  the goal of this paper is to develop a new fiber \\tlee{direction estimation}\nand tracking method that can handle crossing fibers without requiring any high\nresolution techniques.  The proposed method, named DiST, short for\n\\revisedd{\\textbf{Di}ffusion Direction \\textbf{S}moothing and \\textbf{T}racking},\nis completely \\revisedd{automated} and improves existing methods in several aspects.\n\\revisedc{Particularly, it} is applicable either when there is a large number\nof gradient directions (\\revisedc{as} in the HARDI setting) or when only a\nrelatively small number of gradient directions \\revisedc{are} available (\\revisedc{as} in most clinical\nsettings).\n\n\nThe DiST method can be divided into three major steps.\n\n{\\bf Step 1:}\n\\revisedd{Estimate} the tensor directions within each voxel under a multi-tensor model.  A new parametrization is proposed which makes the tensor directions identifiable.  An efficient and numerically stable computational procedure is developed to obtain the maximum likelihood (ML) estimate of the tensor directions.  \\tlee{Here we highlight that, this method focuses on the estimation of the tensor directions rather than the actual tensors themselves.}\n\n{\\bf Step 2:}\n\\revisedd{Using the voxel-wise tensor direction estimates from Step~1 as input,}\na new direction smoothing procedure is applied to further improve the diffusion direction estimates by borrowing information from neighboring voxels.  A distinctive and unique feature of this smoothing procedure is that it handles crossing fibers through the clustering of directions into homogeneous groups.  We note that, although various tensor smoothing methods have been proposed\n\\citep[e.g.,][]{Pennec-Fillard-Ayache06,Arsigny-Fillard-Pennec06,Fillard-Pennec-Arsigny07,Fletcher-Joshi07,Yuan-Zhu-Lin12,Carmichael-Chen-Paul13},\nlittle work has been done on direct diffusion direction smoothing.\n\\revisedb{\nOne notable exception is \\citet{Schwartzman-Dougherty-Taylor08}, which\nharnesses diffusion\ndirections directly to construct a map of test statistics for detecting differences\nbetween diffusion direction maps from two groups of subjects,\n\\revisedc{while the spatial smoothness of the test statistics is being considered.}\nAlso note that approaches to averaging unsigned directions in the real\nprojective space are known in the directional statistics literature.\n}\n\n{\\bf Step 3:}\nLastly, a fiber tracking algorithm is applied to reconstruct fiber tracts using the smoothed diffusion direction estimates obtained in Step~2.  This tracking algorithm is designed to explicitly allow for multiple directions within a voxel.\n\nWe apply DiST to an ADNI data set measured on a healthy elderly person with 41-direction dMRI scan on a 3 Tesla GE\nMedical Systems MRI scanner. ADNI is a longitudinal study (since 2005) that collects serial\nMRI, cognitive assessments, and numerous additional measurements approximately twice per year\nfrom hundreds of elderly individuals spanning a range from cognitive health to clinically-diagnosed Alzheimer's\ndisease. We also examine DiST using simulated data sets which mimic the most commonly encountered experimental situations in terms of number of gradient directions and signal to noise ratio. \\tlee{DiST is shown} to lead to  superior\nresults than those based on the single tensor model in the simulation study, as well as more biologically sensible results in the real data application.\n\nThe rest of the paper is organized as follows.  Section~\\ref{sec:models} provides background material for some common tensor models.  The proposed methods for tensor direction estimation, smoothing of estimated directions, and fiber tracking are presented in, respectively, Sections~\\ref{sec:voxelwise}, \\ref{sec:smoothing} and~\\ref{sec:track}.\nSection~\\ref{sec:sim1} summarizes simulation results.  \nThe application to an ADNI data set is presented in Section~\\ref{sec:real}.  Section~\\ref{sec:discuss} provides some concluding remarks, while additional \\rwong{simulation} results and technical details are collected in an online Supplemental Material.\n\n\\begin{figure}[htpb]\n\\begin{center}\n  \\includegraphics[height=5cm]{tensormap.png}\n\\end{center}\n  \\caption{\\rwong{An example of a tensor map on a 2D grid, where each diffusion tensor is represented by an elliposid.}}\\label{fig:tensormap}\n\\end{figure}\n\n\n\\section{Tensor models}\\label{sec:models}\nSuppose dMRI measurements are made on $N$ voxels on a 3D grid representing a brain.  For each voxel, we have measurements of diffusion weighted signals along a fixed set (i.e., the same for all voxels) of unit-norm gradient vectors $\\mathcal{U}=\\{{\\brm{u}}_i:i=1,\\dots,m\\}$.\nWe write the set of measurements as $\\{S({\\brm{s}}, {\\brm{u}}): {\\brm{u}}\\in \\mathcal{U} \\}$, where ${\\brm{s}}$ is the 3D coordinate of the center of this voxel.\n\nAssuming Gaussian diffusion, the noiseless signal intensity is given by \\citep[e.g.,][]{Mori07}\n\\[\n\\bar{S} ({\\brm{s}}, {\\brm{u}}) = S_0({\\brm{s}}) \\exp\\left\\{-b{\\brm{u}}^\\intercal {\\brm{D}}({\\brm{s}}) {\\brm{u}}\\right\\},\n\\]\nwhere $S_0({\\brm{s}})$ is the non-diffusion-weighted intensity, $b>0$ is an\nexperimental constant referred to as the $b$-value and $\\brm{D}({\\brm{s}})$ is a\n$3\\times 3$ covariance matrix referred to as the diffusion tensor.  This model\nis called the single tensor model and suits for the case of at most one\ndominant diffusion direction within a voxel.\n\nAlthough the single tensor model is the most widely used tensor model in practice, it is not suitable for crossing fiber regions.  To deal with crossing fibers, this model has been extended to a multi-tensor model\n\\citep[e.g.,][]{Tuch02,\n  Behrens-Woolrich-Jenkinson03,\n  Behrens-Berg-Jbabdi07,\n  Tabelow-Voss-Polzehl12}:\n\\begin{align}\n\\bar{S}({\\brm{s}}, {\\brm{u}}) = S_0({\\brm{s}})\\sum^{J(s)}_{j=1} p_j({\\brm{s}}) \\exp\\left\\{-b{\\brm{u}}^\\intercal\n  {\\brm{D}}_j({\\brm{s}}) {\\brm{u}}\\right\\}, \\label{eqn:truedti}\n\\end{align}\nwhere $\\sum^{J({\\brm{s}})}_{j=1} p_j({\\brm{s}})=1$ and $p_j({\\brm{s}})>0$ for $j=1,\\dots,J({\\brm{s}})$.  Here $J({\\brm{s}})$ represents the number of fiber populations and $p_j({\\brm{s}})$'s denote weights of the corresponding fibers.\n\n\n\n\\section{Voxel-wise estimation of diffusion directions}\n\\label{sec:voxelwise}\nOne important goal of \\revisedc{dMRI} studies is to estimate principal diffusion directions, referred to as diffusion directions hereafter, at each voxel.  They may be interpreted as tangent directions along fiber bundles at the corresponding voxel.  The estimated diffusion directions are then used as an input for tractography algorithms to reconstruct fiber tracts.  This section explores the diffusion direction estimation within a single voxel.  For notational simplicity, dependence on voxel index ${\\brm{s}}$ is temporarily dropped.  Moreover, for ease of exposition, we assume that $\\sigma$ and $S_0({\\brm{s}})$ are known {and delay the discussion of their estimation to Section~\\ref{sec:real}.\n\nUnder the single tensor model, various methods for tensor estimation have been proposed including linear regression, nonlinear regression and ML estimation; e.g., see \\citet{Carmichael-Chen-Paul13} for a comprehensive review.  Then diffusion directions are derived as principal eigenvectors of (estimated) diffusion tensors.  However, for the estimation of multi-tensor models, severe computational issues have been observed and additional prior information and additional assumptions are usually imposed to tackle these issues.  For instance, \\citet{Behrens-Woolrich-Jenkinson03,Behrens-Berg-Jbabdi07} use shrinkage priors and \\citet{Tabelow-Voss-Polzehl12} assume all tensors to be axially symmetric (i.e., the two minor eigenvalues are the same) and have the same set of eigenvalues.  \\citet{Scherrer-Warfield10} show that the multi-tensor model is indeed non-identifiable \\tlee{in the sense that there exist multiple parameterizations that are observationally equivalent}.  These authors suggest to use multiple $b$-values in data acquisition to make the model identifiable.  However, due to practical limitations, most of the current dMRI studies are obtained under a fixed $b$-value and so render their suggestion inapplicable.  Below we show that the identifiability issue does not prevent one from estimating the diffusion directions and so neither strong assumptions nor special experimental settings are necessary if one is only interested in diffusion directions rather than the diffusion tensors themselves.\n\n\n\\subsection{Identifiability of multi-tensor model}\nModel~(\\ref{eqn:truedti}) can be re-written as\n\\[\n  \\bar{S}({\\brm{u}}) = S_0 \\sum^{J}_{j=1} p_j a_j \\exp \\left\\{\n  -b {\\brm{u}}^\\intercal\\left( {\\brm{D}}_j + \\frac{\\log a_j}{b} \\brm{I}_3\\right) {\\brm{u}}\n  \\right\\},\n\\]\nwhere $a_j>0$ for $j=1,\\dots,J$ such that $p_ja_j>0$, $\\brm{D}_j +\n(\\log a_j\/b)\\brm{I}_3$ is positive definite and $\\sum^J_{j=1}{p_ja_j}=1$. When\n$J=2$, one can easily derive the explicit conditions for $a_j$ to fulfill\nthese criteria, and see that there are infinite sets of such $a_j$'s.\nHowever, note that $\\brm{D}_j +\n(\\log a_j\/b)\\brm{I}_3$ shares the same set of eigenvectors with ${\\brm{D}}_j$. Thus,\none may still be able to estimate diffusion directions, which correspond to the\nmajor eigenvectors of the tensors. This motivates us to consider estimating\ndiffusion directions directly instead of the tensors themselves.\n\nNow we assume that ${\\brm{D}}_j$'s are axially symmetric; that is, the two minor eigenvalues of ${\\brm{D}}_j$ are equal.\nThis is a common assumption\nfor modeling dMRI data and it implies that\ndiffusion is symmetric around the principal\ndiffusion direction\n\\citep{Tournier-Calamante-Gadian04, Tournier-Calamante-Connelly07}.\nBy not differentiating the two minor eigenvectors, we obtain a\nclear meaning of diffusion direction.  In addition, this reduces the number of\nunknown parameters by one \\reviseda{for each tensor in\nthe multiple tensor model} and thus facilitates estimation.\nIn the following, we propose a new parametrization of the multi-tensor model which is identifiable and thus can be used for direction estimation.\n\n\nWrite $\\mathcal{M}$ as the space of the unit principal eigenvector; i.e., the  3D unit sphere with equivalence relation\n${\\brm{m}} \\sim -{\\brm{m}}$.\nLet $\\alpha_j\\ge0$, $\\xi_j>0$ and ${{\\brm{m}}}_j\\in\\mathcal{M}$\nbe the difference between the larger and smaller eigenvalue, smaller eigenvalue\nand the standardized principal eigenvector of ${\\brm{D}}_j$, respectively.\nSince $\\brm{D}_j =\\alpha_j {{\\brm{m}}}_j {{\\brm{m}}}_j^\\intercal + \\xi_j \\brm{I}_3 $,\nmodel~(\\ref{eqn:truedti}) becomes\n\\begin{align}\n  \\bar{S}({\\brm{u}}) &= S_0 \\sum^J_{j=1}p_j \\exp \\left\\{ -b {\\brm{u}}^\\intercal \\left(\n    \\alpha_j {{\\brm{m}}}_j {{\\brm{m}}}_j^\\intercal + \\xi_j \\brm{I}_3\\right) {\\brm{u}} \\right\\}\\nonumber\\\\\n&= S_0 \\sum^J_{j=1} \\tau_j \\exp \\left\\{ -b \\alpha_j ({\\brm{u}}^\\intercal\n  {{\\brm{m}}}_j)^2\\right\\},\\label{eqn:newpar}\n\\end{align}\nwhere $\\tau_j = p_j \\exp(-b\\xi_j) \\in (0,1)$.\n From the above, one can see that\n$p_j$ and $\\xi_j$ are not simultaneously identifiable, so\nwe cannot estimate the tensors.\nHowever, as stated in the following theorem, $\\tau_j, \\alpha_j, {\\brm{m}}_j^\\intercal$ are identifiable and hence we can estimate the principal diffusion directions ${\\brm{m}}_j$'s.\n\n\\begin{thm}\\label{thm:identi}\nUnder model~(\\ref{eqn:newpar}), \\tlee{for any arbitrary $J$}, the parameters $\\bm{\\gamma}=(\\bm{\\gamma}_1^\\intercal,\\dots,\\bm{\\gamma}_J^\\intercal)^\\intercal$ are identifiable,\nwhere\n$\\bm{\\gamma}_j=(\\tau_j, \\alpha_j, {\\brm{m}}_j^\\intercal)^\\intercal$ for $j=1, \\dots, J$.\n\\end{thm}\n\\revisedb{The proof of this theorem can be found in \\suppref{Section~S5.1} of the Supplemental Material. Note that,\n  \\revisedc{compared to the model in} \\citet{Tabelow-Voss-Polzehl12},\n  \\revisedc{model (\\ref{eqn:newpar})} allows for different eigenvalues and shapes of the\n  tensors within a voxel, \\revisedc{and thus is much more flexible}.}\n\n\\subsection{Parameter estimation using \\revisedc{maximum likelihood (ML)}}\nWe first consider the case when $J$ is known and delay the selection of $J$ to Section~\\ref{sec:J}.\n\\revisedb{By assuming Gaussian additive noise on both real and imaginary parts of the\n  \\revisedc{complex} signal, the observed signal intensity can be modeled as \\reviseda{\\citep[see, e.g.,][]{Zhu-Zhang-Ibrahim07}}\n\\[\nS({\\brm{u}}) = \\| \\bar{S}( {\\brm{u}}) \\bm{\\phi}( {\\brm{u}}) + \\sigma\n\\bm{\\epsilon}({\\brm{u}})\\|,\n\\]\nwhere $\\bar{S}( {\\brm{u}})$ is the intensity of the noiseless signal, $\\bm{\\phi}({\\brm{u}})$ is a unit vector in $\\mathbb{R}^2$ representing the phase of the signal, $\\bm{\\epsilon}({\\brm{u}})$ is the noise random variable following $\\mathcal{N}_2(\\brm{0},\\brm{I}_2)$ and $\\sigma>0$ denotes the noise level.\nNote that both $\\phi$ and $\\bm{\\epsilon}$ \\revisedc{may} depend on ${\\brm{s}}$.\nThe observed signal intensity then follows a Rician distribution \\citep{Gudbjartsson-Patz95}:\n\\[\nS({\\brm{u}}) \\sim \\mathrm{Rician}(\\bar{S}({\\brm{u}}),\\sigma).\n\\]\nMoreover, we assume the noise $\\bm{\\epsilon}({\\brm{u}})$'s are independent across different voxels and gradient directions.}\n\nUnder the Rician noise assumption, the log-likelihood of $\\bm{\\gamma}$ in model~(\\ref{eqn:newpar}) is:\n\\begin{align}\nl(\\bm{\\gamma}) &= \\sum_{{\\brm{u}}\\in\\mathcal{U}}\\log\\left[\n  \\frac{S({\\brm{u}})}{\\sigma^2} \\exp \\left\\{ -\\frac{S^2({\\brm{u}}) +\n      \\bar{S}^2({\\brm{u}})}{2\\sigma^2}  \\right\\} I_0 \\left\\{\\frac{S({\\brm{u}})\n      \\bar{S}({\\brm{u}})}{\\sigma^2}\\right\\} \\right] \\notag\\\\\n&= \\sum_{{\\brm{u}}\\in\\mathcal{U}} \\left[ \\log\\left\\{\\frac{S({\\brm{u}})}{\\sigma^2}\\right\\}\n  -\\frac{S^2({\\brm{u}}) + \\bar{S}^2({\\brm{u}})}{2\\sigma^2} + \\log I_0 \\left\\{\\frac{S({\\brm{u}})\n      \\bar{S}({\\brm{u}})}{\\sigma^2}\\right\\} \\right], \\label{eqn:likelihood}\n\\end{align}\nwhere $I_0(x) = \\int^{\\pi}_0 \\exp(x\\cos\\phi) d\\phi\/ \\pi$ is the zeroth order modified Bessel function of the first kind.\nThe ML estimate is obtained through maximizing~(\\ref{eqn:likelihood}). Although the above new parametrization avoids the identifiability issue, the likelihood function usually has multiple local maxima, which makes the computation of ML estimate difficult and unstable.\n\nThe method that we used to overcome this issue can be briefly described as follows.  We first develop an approximation of model~(\\ref{eqn:newpar}) whose likelihood can be globally maximized via a grid search.  We utilize the geometry of the problem so that the grid search can be done \\revisedc{efficiently}.  Then we use the ML estimate of this approximated model as the initial value in a gradient method to obtain the ML estimate of model~(\\ref{eqn:newpar}).  This method provides very reliable estimates.  To speed up the pace of this article, its full description is given in \\suppref{Section~S1} of the Supplemental Material.\n\n\\ignore{\nIn \\reviseda{an} attempt to find the global maximizer of~(\\ref{eqn:likelihood}), we develop an efficient algorithm through an approximation of model~(\\ref{eqn:newpar}).  This algorithm essentially performs a grid search, but it makes use of the geometry of the problem so it is fast.  It includes three major steps: (i) lay down a grid for $(\\alpha_j, {\\brm{m}}_j^\\intercal)$'s, (ii) evaluate the \\reviseda{maximized} likelihood function \\reviseda{w.r.t. $\\tau_j$'s} on the grid, and (iii) return the grid point that maximizes the likelihood function.  One can then use this returned grid point as a starting value in a gradient method for obtaining ML estimation of model~(\\ref{eqn:newpar}).  Such a strategy results in better numerical stability and accuracy in finding ML estimates.\n\n\\subsubsection{An approximation of model~(\\ref{eqn:newpar})}\nLet $\\brm{c}_j=(\\alpha_j,{\\brm{m}}_j^\\intercal)^\\intercal$, $\\brm{c}=(\\brm{c}_1^\\intercal,\\dots, \\brm{c}_J^\\intercal)^\\intercal$ and $\\mathcal{C}_j$ be the set of grid points for $\\brm{c}_j$.  For simplicity, we take the same set of grid points, $\\mathcal{C}$, for all $j$.  To lay down a grid for ${\\brm{m}}_j$'s, we apply the sphere tessellation using Icosahedron, which is depicted in Figure~\\ref{fig:tess}.  Here, we only pick unique vertices up to a sign for the formation of the grid. In our implementation, we utilize randomly rotated versions of the tessellation with two subdivisions, which results in a grid with 321 directions \\reviseda{corresponding to those unique vertices (up to a sign change) in Figure~\\ref{fig:tess} (Right)}.  If $\\brm{c}\\in \\prod^J_{j=1} \\mathcal{C}_j =\\mathcal{C}^J$, model~(\\ref{eqn:newpar}) can be rewritten as\n\\begin{equation}\n\\bar{S}({\\brm{u}}) = \\sum_{k=1}^{K} \\tilde{\\beta}_k x({\\brm{u}}, \\tilde{{\\brm{m}}}_k, \\tilde{\\alpha}_k),\n\\label{eqn:lm0}\n\\end{equation}\nwhere $K=|\\mathcal{C}|$, $x({\\brm{u}},\\tilde{{\\brm{m}}}_k, \\tilde{\\alpha}_k) = S_0\\exp\\{-b\\tilde{\\alpha}_k({\\brm{u}}^\\intercal \\tilde{{\\brm{m}}}_k)^2\\}$, $(\\tilde{\\alpha}_k, \\tilde{{\\brm{m}}}_k) \\in \\mathcal{C}$ and $\\tilde{\\beta}_k\\in[0,1)$.  One may notice that, in this reformulation, the non-zero $\\tilde{\\beta}_k$'s \\reviseda{are} $\\tau_j$'s in model~(\\ref{eqn:newpar}).  If $\\brm{c} \\not\\in \\mathcal{C}^J$, i.e. the set of parameters is not a grid point, then equation~(\\ref{eqn:lm0}) serves as an approximation to $\\bar{S}({\\brm{u}})$ in model~(\\ref{eqn:newpar}) as long as the grid is dense enough in the parameter space.\n\nFurthermore, under the commonly used scales of $b$-values and tensors, $x({\\brm{u}},\\tilde{{\\brm{m}}}_k, \\tilde{\\alpha}_k )$ and $x({\\brm{u}}, \\tilde{{\\brm{m}}}_{k'}, \\tilde{\\alpha}_{k'})$ are highly correlated if $\\tilde{{\\brm{m}}}_k = \\tilde{{\\brm{m}}}_{k'}$.\n\\reviseda{\n  Thus, $x({\\brm{u}},\\tilde{{\\brm{m}}}_k, \\tilde{\\alpha}_k )$ is proportional to\n  $x({\\brm{u}},\\tilde{{\\brm{m}}}_k, \\tilde{\\alpha}_k' )$ approximately.\n  Note that the proportional constant can be combined with\n  $\\tilde{\\beta}_k$ to form a new coefficient in linear model~(\\ref{eqn:lm0}).\n}\nInspired by this observation, we reduce the grid size by setting $\\tilde{\\alpha}_k=\\tilde{\\alpha}$ for all $k$ to a common value $\\tilde{\\alpha}$ \\reviseda{and using new coefficients $\\beta_k$'s to take care of the proportional constants due to the discrepancy between ${\\alpha}_j$'s and $\\tilde{\\alpha}$.}  From our experience, we set $\\tilde{\\alpha}=2\/b$.  With all these approximations, we consider fitting the following model:\n\\begin{equation}\n  \\bar{S}(u) = \\sum^K_{k=1} \\beta_k x_k({\\brm{u}}), \\label{eqn:lm}\n\\end{equation}\nwhere $x_k({\\brm{u}})=x({\\brm{u}}, \\tilde{{\\brm{m}}}_k, \\tilde{\\alpha})$ and $\\beta_k\\ge 0$.  For our purpose, we want to identify non-zero $\\beta_k$'s because those $\\tilde{{\\brm{m}}}_k$'s associated with non-zero $\\hat{\\beta}_k$'s  can be regarded as selected diffusion directions. Note that model~(\\ref{eqn:lm}) converts the expensive grid search to an estimation problem of a linear model (with respect to ${\\beta}_k$'s) with non-negative constraints.  A fast algorithm for fitting this model with Rician noise assumption is given in \\suppref{Section~S1} of the Supplemental Material.  As it turns out, the non-negativity constraints often result in a sparse estimate of $\\bm{\\beta}=(\\beta_1,\\dots, \\beta_K)^\\intercal$; i.e., only a subset of directions is selected.  In particular, if the estimate of the unconstrained problem (i.e., $\\beta_k$'s are allowed to be negative) is not located in the first quadrant of the parameter space, the corresponding constrained solution will be sparse.\n\nEven though the solution is often sparse, the number of selected directions is usually  larger than $J$, the true number of tensor components. This is partly due to colinearity of $x_k({\\brm{u}})$'s resulting from the use of a dense grid on the\ndirections $\\tilde{{\\brm{m}}}_k$'s.\n\nIn the following, we propose to first divide the selected directions into $I$ groups and then generate stable estimates of ${\\brm{m}}_j$'s via gradient methods (Section~\\ref{sec:grouping}).  Finally, Bayesian information criterion (BIC) \\citep{Schwarz78} is used to choose an appropriate $I$ as the estimate for $J$ (Section~\\ref{sec:J}).\n\n\\begin{figure}[htpb]\n\\begin{center}\n\\vspace*{-0.2cm}\n\\begin{tabular}{ccc}\n  \\includegraphics[height=4cm]{ico-1.png} &\n\\hspace*{-1cm}\n  \\includegraphics[height=4cm]{ico-2.png} &\n\\hspace*{-1cm}\n  \\includegraphics[height=4cm]{ico-3.png}\n\\end{tabular}\n\\end{center}\n\\vspace*{-0.5cm}\n  \\caption{Sphere tessellations through triangulation using Icosahedron with level of subdivisions: 0 (Left), 1 (Middle) and 2 (Right).}\\label{fig:tess}\n\\end{figure}\n\n\n\\subsubsection{Clustering of the selected directions}\\label{sec:grouping}\n\nWrite the above ML estimate of $\\beta_k$ as $\\hat{\\beta}_k$ for $k=1,\\dots, K$.  Suppose there are $L>0$ non-zero $\\hat{\\beta}_k$'s, without loss of generality, $k=1,\\dots, L$.  Thus, $\\tilde{{\\brm{m}}}_1,\\dots,\\tilde{{\\brm{m}}}_L$ are the selected directions. Now, we develop a strategy to cluster the selected directions into $I$ groups, for a set of $I \\in \\{1,\\dots,L\\}$.  To perform clustering, we require a metric measure on the space of directions $\\mathcal{M}$. A natural metric is\n\\begin{equation}\n  d^{*} ({\\brm{u}},{\\brm{v}}) = \\mathrm{arccos}(|{\\brm{u}}^\\intercal {\\brm{v}}|), \\label{eqn:geodist}\n\\end{equation}\nwhere ${\\brm{u}}, {\\brm{v}} \\in \\mathcal{M}$.  Note that, $d^{*} ({\\brm{u}},{\\brm{v}})$ is the acute angle between ${\\brm{u}}$ and ${\\brm{v}}$.  With this distance metric, one can define dissimilarity matrix for a set of directions and make use of a generic clustering algorithm.  Our choice is the Partition Around Medoids (PAM) \\citep{Kaufman-Rousseeuw90} due to its simplicity. The detailed procedure is described in \\suppref{Algorithm~S1} in the Supplemental Material, where the input vectors are the selected directions.  Due to the sparsity of $\\hat{\\beta}_j$'s and efficient algorithms of PAM, this clustering strategy is practically fast.  Let $\\check{{\\brm{m}}}_1, \\dots, \\check{{\\brm{m}}}_I$ be the resulting group (Karcher) means.  They are used as the starting value for gradient-based methods, such as L-BFGS-B algorithm \\citep{Byrd-Lu-Nocedal95}, for obtaining $\\hat{\\bm{\\gamma}}(I)$, the ML estimate of $\\gamma$ under model~(\\ref{eqn:newpar}) with $I$ tensor components. More specifically, the starting value is set as $((1\/I, \\tilde{\\alpha}, \\check{{\\brm{m}}}^\\intercal_1),\\dots,(1\/I, \\tilde{\\alpha},\n\\check{{\\brm{m}}}^\\intercal_I))^\\intercal$.\n}\n\n\n\\subsection{Selecting the number of tensor components $J$}\\label{sec:J}\nCommon model selection methods can be applied to select the number of components $J$.  Results from extensive numerical experiments suggest that the Bayesian information criterion (BIC) \\citep{Schwarz78} is good choice; see \\suppref{Section~S2} of the Supplemental Material.\n\n\\reviseda{\n  Under model~(\\ref{eqn:newpar}),\n  each \\revisedc{tensor} corresponds to four free scalar parameters since\n  $\\brm{m}_j$ is characterized by two free scalar parameters.\n}\nThe BIC for a model with $I$ \\revisedc{tensors} is\n\\begin{equation}\n  \\mathsf{BIC}(I) = -2l(\\hat{\\bm{\\gamma}}(I)) + 4I\\log(m), \\label{eqn:bic}\n\\end{equation}\nwhere $m$ is the number of gradient directions and $\\hat{\\bm{\\gamma}}(I)$ is the ML estimate of $\\bm{\\gamma}$ under $I$ tensors.\nThen $J$ is chosen as\n$\n  \\hat{J} = \\mathrm{argmin}_{I \\in \\{1, \\dots, \\tilde{I} \\}}\n  \\mathsf{BIC}(I),\n$\nwhere $\\tilde{I}$ is a pre-specified upper bound for the number of components.\nBased on our experience, $\\tilde{I}=4$ is a reasonable choice.\n\nIn practice, there are voxels with no major diffusion directions.\n\\revisedc{This corresponds to the case where there is only one isotropic tensor.}\nIn the case of isotropic tensor,\n(\\ref{eqn:newpar}) reduces to\n$\n  \\bar{S}({\\brm{u}}) = \\brm{S}_0\\tau_1.\n$\nThus there is only one parameter $\\tau_1$. We write\nthe corresponding likelihood function as $\\tilde{l}$\nand denote the ML estimate of $\\tau_1$ by $\\hat{\\tau}_1$,\nwhich can be obtained by a generic gradient method.\nThe corresponding\nBIC criterion is\n\\[\n  \\mathsf{BIC}(0) = -2 \\tilde{l}(\\hat{\\tau}_1) + \\log(m),\n\\]\nwhere 0 represents no diffusion direction.\nCombined with the previous BIC formulation~(\\ref{eqn:bic}), one has a comprehensive\nmodel selection rule, which handles voxels with from zero to up to $\\tilde{I}$ (here 4) fiber\npopulations.\n\n\\revisedb{In practice, we follow the convention and use\nfractional anisotropy (FA) \\citep[see, e.g.,][]{Mori07},\n\\begin{equation}\n   FA = \\sqrt{\\frac{ (\\lambda_1-\\lambda_2)^2 + (\\lambda_2-\\lambda_3)^2 +\n   (\\lambda_3-\\lambda_1)^2 }{{2(\\lambda_1^2 + \\lambda_2^2 + \\lambda_3^2})}}, \\label{eqn:FA}\n\\end{equation}\nwhere $\\lambda_1$, $\\lambda_2$ and $\\lambda_3$ are the eigenvalues of the corresponding tensor in the single tensor model,\n\\revisedc{to conduct an initial screening} to speed up the whole procedure.\nThe FA value lies between zero and one and the larger it is, the more anisotropic the water diffusion is at the corresponding voxel.\nThus we\nfirst remove voxels with very small FA values and then apply the BIC\napproach over those suspected anisotropic\nvoxels.}\n\\rwong{Note that such removal is mainly for reducing computational cost as a typical\ndMRI data set consists of hundreds of thousands of voxels. From our experiences,\nthis has little effect on the final tracking results.\nWe also note that the proposed framework including selection of $J$ can be applied\nwithout such removal if enough computational resources are available.}\n\nWe summarize our voxel-wise estimation procedure in \\suppref{Algorithm~S2} in the Supplemental Material.  A simulation study is conducted and the corresponding results are presented in \\suppref{Section~S2} of the Supplemental Material.  These numerical results suggest that our voxel-wise estimation procedure provides extremely stable and reliable results under various settings.\n\n\n\\section{Spatial smoothing of diffusion directions}\n\\label{sec:smoothing}\nAlthough model~(\\ref{eqn:newpar}) provides a better modeling than the single tensor model for crossing fiber regions, it also leads to an increase in the number of parameters and thus the variability of the estimates.\nTo further improve estimation, we consider borrowing information from neighboring voxels and develop a novel smoothing technique for diffusion directions.\n\n\n\n\n\\revisedc{In many brain regions, it is reasonable to model the fiber tracts as smooth curves at the resolution of voxels in dMRI ($\\sim$ 2mm).\nTherefore, we shall assume that the tangent directions of fiber bundles change smoothly.  This leads to the spatial smoothness of diffusion directions that belong to the same fiber bundle.}\n\n\n\\subsection{Smoothing along a single fiber}\\label{sec:single_smooth}\n\nThis subsection considers the simpler situation where there is only one homogeneous population of diffusion directions; i.e., there is only one single fiber bundle without crossing.\nWrite $T$ as the total number of estimated diffusion directions from all voxels and $\\{\\hat{{\\brm{m}}}_k: k=1,\\dots, T\\}$ as the set of all estimated diffusion directions.  Also write ${\\brm{s}}_k$ as the corresponding voxel location associated with $\\hat{{\\brm{m}}}_k$.  Note that some ${\\brm{s}}_k$'s share the same value, as some voxels contain multiple estimated directions.\nFollowing the idea of kernel smoothing on Euclidean space \\citep[e.g.,][]{Fan-Gijbels96}, the smoothing estimate at voxel ${\\brm{s}}_0$ is defined as a weighted Karcher mean of the neighboring direction vectors:\n\\begin{equation}\n  \\operatornamewithlimits{arg\\ min}_{{\\brm{v}}\\in\\mathcal{M}} \\sum^T_{i=1}\n   w_id^{*2}(\\hat{{\\brm{m}}}_i,{{\\brm{v}}}), \\label{eqn:general_smooth}\n\\end{equation}\nwhere $w_i = K_{\\brm{H}}({\\brm{s}}_i-{\\brm{s}}_0)$'s are spatial weights\nand the metric $d^*$ is defined as\n\\begin{equation}\n  d^{*} ({\\brm{u}},{\\brm{v}}) = \\mathrm{arccos}(|{\\brm{u}}^\\intercal {\\brm{v}}|), \\quad\n{\\brm{u}}, {\\brm{v}} \\in \\mathcal{M}; \\label{eqn:dist2}\n\\end{equation}\ni.e., $d^{*} ({\\brm{u}},{\\brm{v}})$ is the acute angle between ${\\brm{u}}$ and ${\\brm{v}}$.\nThe weights $w_i$'s place more emphasis on spatially closer observations.\nHere $K_{\\brm{H}}(\\cdot)=|\\brm{H}|^{-1\/2}K({\\brm{H}}^{-1\/2}\\cdot)$ with $K(\\cdot)$ as a\n3D kernel function satisfying $\\int K({\\brm{s}}) d{\\brm{s}} = 1$, and\n$\\brm{H}$ is a $3 \\times 3$ bandwidth matrix.  In our numerical work, we choose\n$K(\\cdot)$ as the standard Gaussian density, and set $\\brm{H} = h\\brm{I}_3$, where\n$h$ is \\revisedd{chosen using} the cross-validation (CV) approach described in\n\\suppref{Section~S3} of the Supplemental Material.\n\\revisedb{We adopt the leave-one-out CV idea to\ndevelop an ordinary CV score and two robust CV scores.\n\\rwong{Their practical performances are reported\nin \\suppref{Section~S6} of the Supplemental Material.\n}}\n\n\\subsection{Smoothing over multiple fibers}\\label{sec:mult_smooth}\nWhen there are crossing fibers in a voxel ${\\brm{s}}_0$, the above smoothing procedure will not work well.  To \\revisedc{address} this issue, we first cluster the neighboring estimated directions of ${\\brm{s}}_0$ into \\revisedc{groups} that correspond to different fiber populations.  Then we apply the above smoothing procedure to each individual \\revisedc{cluster}.  This subsection describes this procedure in details.\n\n\nFirst we define neighboring voxels for ${\\brm{s}}_0$.  We begin with\ncomputing the spatial weights defined in Section~\\ref{sec:single_smooth}.  We\nthen remove those voxels with weights smaller than a threshold.  By filtering\nout these voxels, we obtain tighter and better separated clusters of directions.\nMoreover,\nsuch voxels have little effects on smoothing due to their small weights.  The\nartificial data set displayed in Figure~\\ref{fig:separate} provides an\nillustrative example.  Each black dot in the left panel represents an\nestimated direction (from the center of the sphere).  In the middle panel, the\nsize of each dot is proportional to its spatial weight in equation\n(\\ref{eqn:general_smooth}).  Lastly, the right\npanel shows all dots with spatial weights larger than a threshold.  Notice that\nsuch a trimming operation leads to two obvious clusters of directions, which makes\nthe subsequent task of clustering the directions much easier.\n\n\n\\begin{figure}[htpb]\n\\[\n  \\includegraphics[height=3cm]{neighbor-all.png}\n  \\includegraphics[height=3cm]{neighbor-weight.png}\n  \\includegraphics[height=3cm]{neighbor-thres.png}\n\\]\n\\vspace*{-0.5cm}\n\\caption{\n  \\revisedc{Direction clustering.}\n  Left: all estimated directions.  Middle: sizes of all estimated directions proportional to weights.  Right: estimated directions with weights larger than a threshold. Red lines represents underlying true directions.}\\label{fig:separate}\n\\end{figure}\n\n\nNext we need a clustering strategy\nto choose the number of clusters adaptively.\nWith the distance metric (\\ref{eqn:dist2}), one can define dissimilarity matrix for a set of directions and make use of a generic clustering algorithm. Our choice is the Partition Around Medoids (PAM) \\citep{Kaufman-Rousseeuw90} due to its simplicity.\nAlso, we apply the\naverage silhouette \\citep{Rousseeuw87} to choose the number of clusters; see \\suppref{Algorithm~S3} of the Supplemental Material.\nThe silhouette of a datum $i$ measures the strength of its membership to its cluster, as compared to the neighboring cluster.  Here, the neighboring cluster is the one, apart from cluster of datum $i$, that has the smallest average dissimilarity with datum $i$.  The corresponding silhouette is defined as\n$(b_i-a_i)\/(\\max\\{a_i,b_i\\})$,\nwhere $a_i$ and $b_i$ represent the average dissimilarities of datum $i$ with all other data in the same cluster and that with the neighboring cluster respectively.\nThe average silhouette of all data gives a measure of how good the clustering\nis.  Thus we select the number of clusters via maximizing the average silhouette.\nThe detailed smoothing procedure is given in \\suppref{Algorithm~S4}.\n\n\n\\subsection{Theoretical results}\\label{sec:theory}\nThis subsection derives asymptotic properties of the proposed direction\nsmoothing estimator.\nNote that, since the space of direction vectors has a non-Euclidean geometry and\nso the theoretical framework is different from that of classical smoothing estimators.\nWithout loss of generality, suppose we observe ${\\brm{v}}_1,\\dots,{\\brm{v}}_n \\in \\mathcal{M}$\nat spatial locations ${\\brm{s}}_1,\n\\dots, {\\brm{s}}_n$ respectively.\nLet $\\mathcal{V}$ be the 3D unit sphere.\nThen $\\mathcal{M}$ is the quotient space of $\\mathcal{V}$ with equivalence relation ${\\brm{v}} \\sim -{\\brm{v}}$\nfor any ${\\brm{v}}\\in\\mathcal{V}$. This space is also identified with the so-called\nreal projective space $\\mathbb{R} P^2$.\n\n\nThe theoretical results below were derived under \\revisedb{the more convenient\n  random design} where\n${\\brm{s}}_i$'s are independently and identically sampled from a distribution with\ndensity $f_S$.}\n  \\dpaul{The below theorem (Theorem \\ref{thm:con+norm}) remains valid even\n    under a fixed, regular design setting, with the\n    number of grid points increasing to infinity. In this case, in the statement of the asymptotic\n    formulae and their proofs, the density function $f_S$ is replaced with a\n    constant-valued function, representing a regular grid, with corresponding changes\n    wherever derivatives of $f_S$ appear.}\n\n  Given a spatial location ${\\brm{s}}_0$, our target is to estimate ${\\brm{v}}_0$, namely\nthe diffusion direction at ${\\brm{s}}_0$,  \\jpeng{which is defined as the minimizer of}\n$\n  \\mathbb{E} \\left\\{d^{*2}(\\brm{V}, {\\brm{v}}) | \\brm{S}={\\brm{s}}_0 \\right\\}\n$\n\\rwong{over ${\\brm{v}}$,}\nwhere ${d}^{*}({\\brm{u}},{\\brm{v}})=\\mathrm{arccos}(|{\\brm{u}}^\\intercal {\\brm{v}}|)$.\n\\rwong{ Here $\\bm{\\mathrm{V}}$\nis a random unit vector representing a random diffusion direction\nand the expectation is taken over\n$\\bm{\\mathrm{V}}$ conditional on $\\bm{\\mathrm{S}}=\\bm{\\mathrm{s}}_0$, where\n$\\bm{\\mathrm{S}}$ represents the location of where\n$\\bm{\\mathrm{V}}$ is observed.}\nFor simplicity, we assume ${\\brm{s}}_i\\in\\mathbb{R}$ and write it as $s_i$\nthereafter. Thus, our estimator~(\\ref{eqn:general_smooth}) at $s_0$ can be written as\n\\[\n  \\hat{{\\brm{v}}}(s_0) = \\operatornamewithlimits{arg\\ min}_{{\\brm{v}}\\in\\mathcal{M}} \\sum^n_{i=1} K_h(s_i-s_0) d^{*2}({\\brm{v}}_i,{\\brm{v}}),\n\\]\nwhere \\rwong{$n$ is the number of diffusion direction vectors} and\n$K_h(\\cdot)=K(\\cdot\/h)\/h$. Here, with\nslight notation abuse, $K(\\cdot)$ represents a one dimensional kernel function throughout\nthe theoretical developments.\n\nWe first describe a working coordinate system.\nFor each ${\\brm{p}}\\in\\mathcal{V}$, one can endow a tangent space\n$T_{{\\brm{p}}} \\mathcal{V}=\\{{\\brm{v}}\\in\\mathbb{R}^3:{\\brm{v}}^\\intercal {\\brm{p}}=0\\}$ with the metric tensor\n$g_{{\\brm{p}}} : T_{{\\brm{p}}} \\mathcal{V} \\times T_{{\\brm{p}}} \\mathcal{V} \\rightarrow \\mathbb{R}$ defined as\n$g_{{\\brm{p}}}({\\brm{u}}_1, {\\brm{u}}_2)= {\\brm{u}}_1^\\intercal {\\brm{u}}_2$.\nNote that the tangent space is identified with $\\mathbb{R}^2$.\nThe geodesics are great circles and the geodesic\ndistance is $\\mathrm{arccos}({\\brm{p}}_1^\\intercal {\\brm{p}}_2)$, for any ${\\brm{p}}_1, {\\brm{p}}_2 \\in \\mathcal{V}$. The corresponding exponential\nmap at ${\\brm{p}}\\in\\mathcal{V}$, $\\mathrm{Exp}_{{\\brm{p}}}:T_{{\\brm{p}}}\\mathcal{V}\\rightarrow\\mathcal{V}$, is given by\n\\begin{align*}\n  \\mathrm{Exp}_{{\\brm{p}}}(\\bm{0}) = {\\brm{p}} \\quad \\mbox{and} \\quad\n  \\mathrm{Exp}_{{\\brm{p}}} ({\\brm{u}}) = \\cos(\\|{\\brm{u}}\\|) {\\brm{p}} + \\frac{\\sin(\\|{\\brm{u}}\\|)}{\\|{\\brm{u}}\\|}\n  {\\brm{u}} \\quad \\mbox{when} \\quad {\\brm{u}}\\neq \\bm{0},\n\\end{align*}\nwhile the corresponding logarithm map at ${\\brm{p}}\\in\\mathcal{V}$, $\\mathrm{Log}_{{\\brm{p}}}:\n\\mathcal{V}\\backslash\\{-{\\brm{p}}\\}\\rightarrow T_{{\\brm{p}}}\\mathcal{V}$, is\ngiven by\n\\begin{align*}\n  \\mathrm{Log}_{{\\brm{p}}}({\\brm{p}}) = \\bm{0} \\quad \\mbox{and} \\quad\n  \\mathrm{Log}_{{\\brm{p}}}({\\brm{v}}) = \\frac{\\mathrm{arccos}({\\brm{v}}^\\intercal {\\brm{p}})}{\\sqrt{1-({\\brm{v}}^\\intercal {\\brm{p}})^2}}\n  [ {\\brm{v}} - ({\\brm{v}}^\\intercal {\\brm{p}}) {\\brm{p}} ] \\quad \\mbox{when} \\quad\n  {\\brm{v}}\\neq {\\brm{p}}.\n\\end{align*}\nOne can use the exponential map and the logarithm map to\ndefine a coordinate system for the $\\mathcal{V}\\backslash\\{-{\\brm{v}}_0\\}$ in the following way.\nGiven ${\\brm{v}} \\in\\mathcal{V}$, we\ndefine the logarithmic coordinate as\n\\[\\omega_1 = \\brm{e}_1^\\intercal \\mathrm{Log}_{{\\brm{v}}_0} ({\\brm{v}}) \\quad \\mbox{and} \\quad \\omega_2 = \\brm{e}_2^\\intercal \\mathrm{Log}_{{\\brm{v}}_0} ({\\brm{v}}),\n\\]\nwhere $\\brm{e}_1, \\brm{e}_2 \\in T_{{\\brm{v}}_0}\\mathcal{V}$ and $\\{\\brm{e}_1, \\brm{e}_2\\}$ forms an\northonormal basis for $T_{{\\brm{v}}_0}\\mathcal{V}$.\nWrite ${\\phi}({\\brm{v}}) = (\\omega_1, \\omega_2)^\\intercal$.\nIn addition, we define\n\\[\n  \\rho_{{\\brm{v}}_0}({\\brm{v}}) =\\begin{cases}\n    \\mathrm{sign}({\\brm{v}}_0^\\intercal {\\brm{v}}) {\\brm{v}} & {\\brm{v}}_0^\\intercal {\\brm{v}} \\neq 0\\\\\n    {\\brm{v}} & {\\brm{v}}_0^\\intercal {\\brm{v}} = 0\n  \\end{cases},\n\\]\n\\reviseda{which aligns ${\\brm{v}}$ with ${\\brm{v}}_0$, and\n${d}(\\bm{\\omega},\\bm{\\theta}) = d^*({\\phi}^{-1} (\\bm{\\omega}),{\\phi}^{-1}\n(\\bm{\\theta}))$ for $\\bm{\\omega}, \\bm{\\theta}\\in \\mathbb{R}^2$.\nNote that for any ${\\brm{v}}, {\\brm{p}}\\in\\mathcal{V}$,\nwe have $d(\\tilde{\\phi}({{\\brm{v}}}), \\tilde{\\phi}({\\brm{p}})) = d^*({\\brm{v}}, {\\brm{p}})$\nwhere $\\tilde{\\phi} = {\\phi}\\circ \\rho_{{\\brm{v}}_0}$.}\n\\revisedb{\n  Here $\\tilde{\\phi}({\\brm{v}})$ first aligns a direction ${\\brm{v}}$ with\n  the true diffusion direction ${\\brm{v}}_0$ and then represents it by its\n  logarithmic coordinate.\n}\n\nWe now present the asymptotic results.\nNow, write\n$\\bm{\\theta}_i=\\tilde{\\phi}({\\brm{v}}_i)$\n for $i=1,\\dots, n$, and $\\psi(\\bm{\\omega},\\bm{\\theta}) =\nd^2(\\bm{\\omega},\\bm{\\theta})$. We have $\\bm{\\theta}_0 = \\tilde{\\phi}({\\brm{v}}_0)=\\bm{0}$.\n\\rwong{Also, let $\\bm{\\psi}_j(\\bm{\\omega},\\bm{\\theta})$ be the $j$-th order derivative of $\\psi$ with\nrespect to $\\bm{\\theta}$ for $j=1,2$.}\nLet $\\brm{m}(s) = (m_1(s), m_2(s))^\\intercal = \\mathbb{E}(\\bm{\\theta}_1|S_1=s)$ and $\\bm{\\Sigma}(s) =\n[\\Sigma_{jk}(s)]_{1\\le j, k \\le 2} =  \\mathrm{Var}(\\bm{\\theta}_1 | S_1 =s)$. Also, denote\n$\\bm{\\Psi}(s)= [\\Psi_{jk}(s)]_{1\\le j,k \\le 2} = \\mathbb{E}[\\bm{\\psi}_2(\\bm{\\theta}_1,\n\\bm{\\theta}_0)|S_1=s]$.\n\nUnder the assumptions 1-10 laid out in \\suppref{Section S5.2} of the Supplemental Material\n\\revisedc{which are all standard technical conditions (except for Assumption 1\n  which is to ensure the representation of the geodesic distance as a function\n  of the working coordinate system)},\nwe have the following theorem.\n\\begin{thm}\\label{thm:con+norm}\n  Let $M_n(\\bm{\\theta}) = \\sum^n_{i=1} h K_h(S_i-s_0) d^2(\\bm{\\theta}_i, \\bm{\\theta})$, and assume Assumptions 1-10 hold.\n\n  \\begin{enumerate}[(a)]\n    \\item There exists a sequence of solutions, $\\hat{\\bm{\\theta}}_n(s_0)$, to\n      $M_n^{(1)}(\\bm{\\theta}) = 0$,\n  such that $\\hat{\\bm{\\theta}}_n(s_0)$ converges in probability to $\\bm{\\theta}_0$.\n\\item $\\hat{\\bm{\\theta}}_n$ is asymptotically normal:\n  \\[\n    \\sqrt{nh}\\left\\{ (\\hat{\\bm{\\theta}}_n-\\bm{\\theta}_0) - h^2 \\bm{\\eta}\\right\\} \\implies\n    \\mathcal{N}_2(\\bm{0},\\bm{\\Omega}),\n  \\]\n  where\n  \\[\n    \\bm{\\eta} = 2\\int x^2 K(x) dx \\bm{\\Psi}^{-1}(s_0) \\left\\{\n    \\frac{f_{S}^{(1)}(s_0)}{f_{S}(s_0)}m^{(1)}(s_0) +\n    \\frac{1}{2} m^{(2)}(s_0)\\right\\}\n  \\]\n  and\n  \\[\n    \\bm{\\Omega} = 4 \\int K^2(x) dx \\bm{\\Psi}^{-1}(s_0) \\bm{\\Sigma}(s_0).\n  \\]\n\\end{enumerate}\n\\end{thm}\nThe proof of the Theorem~\\ref{thm:con+norm} can be found in \\revisedb{\\suppref{Section~S5.2}} of the Supplemental Material.\n\n\n\\section{Fiber tracking}\n\\label{sec:track}\nFor dMRI, fiber tractography can be \\revisedc{classified as} deterministic and probabilistic methods.  Deterministic methods \\citep[e.g.][]{Mori-Crain-Chacko99, Weinstein-Kindlmann-Lundberg99, Mori-Zijl02} track fiber bundles by utilizing the principal eigenvectors of tensors, while probabilistic methods \\citep[e.g.][]{Koch-Norris-Hund-Georgiadis02,Parker-Alexander03,Friman-Farneback-Westin06} use the probability density of diffusion orientations.\nMost \\revisedc{deterministic} methods \\revisedd{assume one} single diffusion tensor in each voxel, and hence \\revisedd{are unable} to handle voxels with crossing fibers.  In view of this, this section develops a \\revisedc{deterministic} tracking algorithm that allows for multiple or no principal diffusion directions in a voxel.\n\n\nThe proposed algorithm can be seen as a generalization of the popular Fiber Assignment by Continuous Tracking (FACT) \\citep{Mori-Crain-Chacko99} algorithm.  A brief description of FACT is as \\revisedc{follows.}\nTracking starts at the center of a voxel (Voxel~1 in Figure~\\ref{fig:tract} left panel) and continues in the direction of the estimated diffusion direction.  When it enters the next voxel (Voxel~2 in Figure~\\ref{fig:tract} left panel), the track changes its direction to align with the new diffusion direction and so on.  This tracking rule may produce many short and fragmented fiber tracts due to either a wrongfully identified isotropic voxel or spurious directions which go nowhere.  In addition, it cannot determine which direction to follow in case there are multiple directions in a voxel, which happens in crossing fiber regions.\n\nTo address these issues, we modify the above procedure in the following manner.  Given a current diffusion direction (we refer to the corresponding voxel as the current voxel), the voxel that it points to (we refer to this voxel as the destination voxel) may have (i) at least one direction; (ii) no direction (i.e., isotropic).  In case (i), we will first identify the direction with the smallest angular difference with\nthe current direction. If its separation angle is smaller than a pre-specified\nthreshold (e.g., $\\pi\/6$), we enter the destination voxel and tracking\nwill go on along this direction. See Figure~\\ref{fig:tract} (Middle).\nOn the other hand, if the separation angle is\ngreater than the threshold, or case (ii) happens, we deem that the destination\nvoxel does not have a viable direction. In this case, tracking will go along\nthe current direction if it finds a viable direction within a pre-specified\nnumber of voxels.  The number of voxels that are allowed to be skipped is\nset to be $1$ in our numerical illustrations.\nSee Figure~\\ref{fig:tract} (Right). On the other hand,\nthe tracking stops at the current voxel if no viable directions within a\npre-specified number of voxels can be found.\nThe detailed tracking algorithm is described in \\suppref{Algorithm~S5} in the\nSupplemental Material.\n\n\n\n\nAs for the choice of starting voxels, also known as seeds, there are two common strategies. One can choose seeds based on tracts of interest and start the tracking from a region of interest (ROI). This approach is based on knowledge on ROI and may not give a full picture of the tracts of interest if there are diverging branches. The other approach is \\revisedc{the} brute-force approach, where tracking starts from every voxel. It usually leads to a more comprehensive picture of tracts at a higher computational cost.  The proposed algorithm can be coupled with either strategy.\n\n\n\nCombining the voxel-wise estimation method in Section~\\ref{sec:voxelwise} and the direction smoothing procedure in Section~\\ref{sec:smoothing} gives the proposed DiST method.\n\n\n\n\n\\begin{figure}[htpb]\n  \\[\n  \\includegraphics[height=3.5cm]{tract_algorithm.png}\n  \\hspace*{0.2cm}\n  \\includegraphics[height=3.5cm]{tract_illustrate1.pdf}\n  \\hspace*{0.2cm}\n  \\includegraphics[height=3.5cm]{tract_illustrate2.pdf}\n  \\]\n  \\caption{Left: Demonstration of the proposed algorithm in single fiber region. Middle:\n    Demonstration of the proposed algorithm in crossing fiber region. Right:\n    Demonstration of the proposed algorithm in case of absence of viable\n    directions.}\\label{fig:tract}\n\\end{figure}\n\n\\section{Simulation \\revisedc{study}}\\label{sec:sim1}\n\\tlee{Extensive simulation experiments have been conducted to evaluate the practical performances of DiST.  They are reported in \\suppref{Section~S6} of the Supplemental Material.  Overall, the DiST method provided highly promising results.}\n\n\n\\section{Real data application}\\label{sec:real}\nIn this section, we apply the proposed methodology to a real dMRI data set,\nwhich was obtained from the\nAlzheimer's\nDisease Neuroimaging Initiative (ADNI) database (www.loni.ucla.edu\/ADNI).\nThe primary\ngoal of ADNI has been to test whether serial MRI,\npositron emission tomography (PET), other biological markers, and clinical and\nneuropsychological assessment can be combined to measure the progression of\nmild cognitive impairment (MCI) and onset of Alzheimer's disease (AD).\nIn the following, we use an eddy-current-corrected ADNI data set of a normal\nsubject for illustration of our technique.\n\nThis data set contains 41 distinct gradient directions with\n$b$-value set as $1000s\/mm^2$. In addition, there are 5 $b0$ images (corresponding to\n$b=0$), forming\nin total 46 measurements for each of the $256\\times256\\times59$ voxels.\nTo implement our technique, we require estimates of $S_0({\\brm{s}})$'s and $\\sigma$.\nWe first estimate\n$S_0({\\brm{s}})$ and $\\sigma({\\brm{s}})$ for each voxel by ML estimation based on the\n5 $b0$ images.\nThen we fix\n$\\sigma$  as the median of estimated $\\sigma({\\brm{s}})$'s for voxel-wise estimation\nof the diffusion directions.\nSince the original $256\\times256\\times59$ voxels contain volume outside the brain,\nwe only take median over a human-chosen set of $81\\times81\\times20$ voxels.\nThe estimated $\\sigma$ is 56.9.\n\nIn this analysis, we focus on a subset of voxels ($15\\times15\\times5$), which contains the intersection of corpus callosum (CC) and corona radiata (CR).  This region is known to contain significant fiber crossing \\citep{Wiegell-Larsson-Wedeen00}.\n\\revisedb{See Figure~\\ref{fig:project} (Left) for a fiber orientation color map of one of the five $xy$-planes.\nWithin the whole focused region, $S_0({\\brm{s}})$'s have mean 1860.1 and standard deviation 522.7.}\n\n\n\nWe then apply voxel-wise estimation to individual voxels followed by the \\textsf{DiST-mcv}\nprocedure.\nDistributions of the estimated number of diffusion directions are summarized in Table\n\\ref{tab:real:hatJ}.\nFor comparison purposes, we also fit the single tensor model with the commonly\nused regression estimator \\citep[e.g.,][]{Mori07}.\n\nThe tracking results are produced by applying the proposed tracking algorithm to the estimated diffusion directions from \\textsf{DiST-mcv},\n\\rwong{which represents the DiST procedure with $h$ chosen by the median cross-validation score\n(see Sections S3 and S6 of the Supplemental Material),}\nand those from the single tensor model estimation.\n\\revisedb{For visualization purposes, we present the longest 900 tracts in Figure~\\ref{fig:corpus_track_dsmooth_level}.}\nFrom anatomy, the CC has a mediolateral direction\nwhile the CR has a superoinferior orientation. They are clearly shown in both\ntracking results.\nIn these figures, reconstructed fiber tracts are colored by a RGB color model\nwith red for left-right, green for anteroposterior, and blue for superior-inferior.\nThus, one can easily locate the CC and the CR as the red fiber bundle and the\nblue fiber bundle respectively.\nTracking result based on \\textsf{DiST-mcv} shows clear crossing between\nmediolateral fiber and the superoinferior fiber (in the figure, the crossing of\nred and blue fiber tracts).\nFrom neuroanatomic\natlases and previous studies, \\citet{Wiegell-Larsson-Wedeen00} conclude that\nthere are several fiber populations with crossing structure in this\nconjunction region of CC and CR,\nwhich matches with the tracking based on \\textsf{DiST-mcv}.\nHowever, the single tensor model estimation can only\nreconstruct one major diffusion direction in each voxel and thus\nthe corresponding tracking result does not show crossing structure.\nInstead, the CC (red fiber bundle) is blocked by the CR (blue fiber bundle) and this\nleads to either termination of the CC fiber tracts or significant merging of the CC and\nthe CR fiber tracts instead of the known crossing structure.\nTo give further illustration,\nFigure~\\ref{fig:project} shows the locations of the CC, the CR and the region of\ncrossing fibers (Cross).\nOne can see that estimated directions based on \\textsf{DiST-mcv} reproduces the crossing\nfiber structures between the CC and the CR, while the result based on single tensor\nmodel tends to connect the CC and the CR fibers.\n\nMoreover, the green fiber on top of the CC represents the cingulum bundle.\nBoth fiber tracking based on {DiST} and single fiber model\nproduce clear and sensible reconstruction of cingulum bundle.\nAll these features match with neuroanatomic atlases and provide\na good demonstration of our proposed method.\n\n\\jpeng{As shown by Figures~\\ref{fig:project} and~\\ref{fig:corpus_track_dsmooth_level}, when comparing with the results obtained by the single tensor model, {DiST} produces more biologically sensible and interpretable tracking results.  This provides more reliable information on brain connectivity and in turn could lead to better understanding of neuro-degenerative diseases such as Alzheimer's disease and autism as well as  better detection of brain abnormality, such as deformation and neuron loss in white matter regions.}\n\n\\renewcommand{\\baselinestretch}{1}\n\\begin{table}[htpb]\n  \\centering\n  \\caption{Number of voxels with different estimated number of diffusion\n    directions.}\\label{tab:real:hatJ}\n  \\vspace{0.3cm}\n  {\\small\n    \\begin{tabular}{c|ccccc|c}\n      \\hline\\hline\n      & \\multicolumn{5}{c|}{Number of diffusion directions}\\\\\n      & 0 & 1 & 2 & 3 & 4 & \\rwong{total} \\\\\n      \\hline\\hline\n      Voxel-wise estimation & 37 & 476 & 589 & 23 & 0 & \\rwong{1125} \\\\\n      Smoothing &  37  & 476 & 593 & 19 & 0 & \\rwong{1125}\\\\\n      \\hline\\hline\n    \\end{tabular}\n  }\n\\end{table}\n\\renewcommand{\\baselinestretch}{1}\n\n\n\\begin{figure}[htpb]\n  \\centering\n  \\includegraphics[height=4.1cm]{fa_z2.png}\n  \\includegraphics[height=4.1cm]{proj_z2_col_text.png}\n  \\includegraphics[height=4.1cm]{proj_z2_col_text-single.png}\n  \\caption{\\revisedb{Left: the fiber orientation color map (based on the single tensor\n    model). The focused region is indicated by white rectangular box. Middle \\revisedc{(from \\textsf{DiST-mcv})}\n    and right \\revisedc{(from single tensor model)}: The projection of fiber\n    directions to the $xy$-plane\n    at $z=102.6$\n    for illustration of crossing fibers.\n    \\rwong{(The five $xy$-planes that we focus on have reference values $z=99.9, 102.6, 105.3, 108, 110.7$ from bottom to top.)}\n    The plot also shows the location of corpus callosum\n    (CC), corona radiata (CR) and crossing region (Cross).\n  The fiber orientation color map is overlaid as the background.}\n}\\label{fig:project}\n\\end{figure}\n\n\n\\begin{figure}[htpb]\n  \\vspace{-0.5cm}\n  \\[\n    \\includegraphics[height=5cm]{track_900.png}\n    \\hspace*{-0.5cm}\n    \\includegraphics[height=5cm]{track_900-2.png}\n  \\]\n  \\vspace{-0.5cm}\n  \\[\n    \\includegraphics[height=5cm]{track_900-single.png}\n    \\hspace*{-0.5cm}\n    \\includegraphics[height=5cm]{track_900-single-2.png}\n  \\]\n  \\caption{Top: The longest 900 tracks using \\textsf{DiST-mcv}.\n    Bottom: \\revisedc{The longest 900 tracks using the} single tensor model.\n    The left and right figures correspond to different viewing angles.\n  }\n  \\label{fig:corpus_track_dsmooth_level}\n\\end{figure}\n\n\n\n\\section{Discussion}\n\\label{sec:discuss}\nUsing tensor estimation to resolve crossing fiber can be problematic,\n\\revisedc{due to the inability of estimating multiple diffusion directions by the single tensor model and}\nthe non-identifiability issue in multi-tensor model.  In this paper, we take a\ndifferent route by focusing on the estimation of diffusion directions rather\nthan the diffusion tensors.  We develop the corresponding direction smoothing\nprocedure and fiber tracking strategy, together called DiST, along this route.\nOur technique gives promising empirical results in both simulation study\n\\rwong{(see Section S6 of the Supplemental Material)}\nand real data analysis.\n\nThe procedure we presented works well  even with moderate number of gradient directions (a few tens), as long as the number of distinct crossing fibers within a voxel is not \\tlee{larger than three}. With HARDI data, which can have up to a couple of hundreds gradient directions, rather than modeling the direction distribution within a tensor framework, we can estimate the fiber orientation distribution nonparametrically \\citep{Tuch04, Descoteaux-Angelino-Fitzgibbons07}.\n\n\\ignore{In that case, we can potentially extend the fiber tracking procedure presented here by adopting a probabilistic approach in which the directions for moving from one voxel to another are sampled from the fiber orientation distribution. Such a probabilistic fiber tracking  has the additional advantage of giving a  measure of uncertainty of the fiber tracts extracted from the data. This is a topic of future research.}\n\n\\jpeng{Applying {DiST} to multiple images from ADNI (either\n  from the same subject over time or from multiple subjects) and then relating\n  the tracking results with clinical outcomes such as cognitive measures would\n  provide valuable information about the role of white matter connectivity in\n  initiation and progression of Alzheimer's disease and dementia. Although this\n  is an important direction of research, it is beyond the scope of this paper\n  which focuses on developing a statistical procedure to denoise dMRI data and\n  to provide better tracking results. We plan to explore more sophisticated\n  applications of the proposed procedure in our future research.}\n\n\\section*{Acknowledgement}\nData collection and sharing for this project was funded by the Alzheimer's\nDisease Neuroimaging Initiative (ADNI) (National Institutes of Health Grant U01\nAG024904). ADNI is funded by the National Institute on Aging, the National\nInstitute of Biomedical Imaging and Bioengineering, and through generous\ncontributions from the following: Abbott, AstraZeneca AB, Bayer Schering Pharma\nAG, Bristol-Myers Squibb, Eisai Global Clinical Development, Elan Corporation,\nGenentech, GE Healthcare, GlaxoSmithKline, Innogenetics, Johnson and Johnson,\nEli Lilly and Co., Medpace, Inc., Merck and Co., Inc., Novartis AG, Pfizer Inc,\nF. Hoffman-La Roche, Schering-Plough, Synarc, Inc., as well as non-profit\npartners the Alzheimer's Association and Alzheimer's Drug Discovery Foundation,\nwith participation from the U.S. Food and Drug Administration. Private sector\ncontributions to ADNI are facilitated by the Foundation for the National\nInstitutes of Health (www.fnih.org). The grantee organization is the Northern\nCalifornia Institute for Research and Education, and the study is coordinated\nby the Alzheimer's Disease Cooperative Study at the University of California,\nSan Diego. ADNI data are disseminated by the Laboratory for Neuro Imaging at\nthe University of California, Los Angeles. This research was also supported by\nNIH grants P30 AG010129, K01 AG030514, and the Dana Foundation.\nThe authors would like to thank Professor Owen Carmichael for making available\nthe data and his valuable comments.\n\n\n\n\\bibliographystyle{imsart-nameyear}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n \\let\\thefootnote\\relax\\footnotetext{The work of A. G. Burr and K. Cumanan was supported by H2020- MSCA-RISE-2015 under grant number 690750. The work on which this paper is based was carried out in collaboration with COST Action CA15104 (IRACON).}\nMassive multiple-input multiple-output (MIMO) is a promising\ntechnique to achieve high data rate \\cite{5gdebbah,massivetddMarzetta14}. However, high performance multiuser MIMO (MU-MIMO) uplink techniques rely on the availability of full channel state information (CSI) of all user terminals at the base station (BS) receiver, which presents a major challenge to their practical implementation. This paper considers an uplink multiuser system where the BS is equipped with $M$ antennas and serves $K_s$ decentralized single antenna users ($M\\gg K_s$). In the uplink mode, the BS estimates the uplink channel and uses linear receivers to separate the transmitted data. The BS receiver uses the estimated channel to implement the zero-forcing (ZF) receiver which is suitable for Massive MIMO systems \\cite{MarzettaMRC13}. To investigate the performance of MIMO systems, an accurate small scale fading channel model is necessary. \n\nMost standardized MIMO channel models such as IEEE $802.11$, the 3GPP spatial model, and the COST 273 model rely on clustering \\cite{standard}. Geometry-based stochastic channel models (GSCMs) are mathematically tractable models to investigate the performance of MIMO systems \\cite{Molish_tufvesson}. The concept of clusters has been introduced in GSCMs to model scatterers in the cell environments \\cite{Molish_tufvesson}. In \\cite{Rappaport_globe15}, the authors use clusters to characterize an accurate statistical spatial channel model (SSCM) in millimeter-wave (mmWave) bands by grouping multipath components (MPCs) into clusters. {MmWave communication suffers from very large path losses, and hence requires large antenna arrays in compensation.} \\cite{Molish_tufvesson}. This paper investigates the throughput in the uplink for the Massive MIMO with carrier frequency in the order of 2 GHz, but the principles can also apply to other frequency bands, including mmWave.\n\nMost existing Massive MIMO techniques rely on the availability of the full CSI of all users at the BS, which presents a major challenge in implementing Massive MIMO. As a result, Massive MIMO techniques with reduced CSI requirement are of great interest. An important issue in Massive MIMO systems is investigating user scheduling in which multiuser diversity gain with imperfect CSI is considered \\cite{Liu_selection}. Recently, a range of user scheduling schemes have been proposed for large MIMO systems. Most of these, such as that described in \\cite{Lee14user}, require accurate knowledge of the channel from all potential users to the BS -which in the frequency division duplex (FDD) Massive MIMO case is completely infeasible to obtain. In \\cite{XuFDDuser}, the authors proposed a greedy user selection scheme by exploiting the instantaneous CSI of all users. However, in this paper we focus on a simplified and robust user scheduling algorithm, by considering Massive MIMO simplifications and the effect of the cell geometry.\n\\subsection{Contributions of This Work}\nThis work investigates a new user selection algorithm for high frequency stochastic geometry-based channels with large numbers of antennas at the BS receiver. We investigate user scheduling by considering the Massive MIMO assumption. The proposed geometry-based user scheduling (GUS) is similar to the greedy weight clique\n(GWC) algorithm but with a different cost function. In the GUS algorithm, the BS selects users based only on the geometry of the area, while in the GWC, the BS uses the channel of the users for user scheduling. Given a map of the area of the micro-cell, we perform efficient user scheduling based only on the position of users and clusters in the cell. In GSCMs, MPCs from common clusters cause high correlation which reduces the rank of the channel. In this paper, we investigate the effect of common clusters on the system performance. Moreover, we assume that the space-alternating generalized expectation (SAGE) algorithm \\cite{crlbc,crlbj} is used (offline) to estimate the direction of arrival (DoA) and the delay of the path. The performance analysis shows the significant effect of the distinct clusters on the system throughput. We prove that to maximize the capacity of system, it is required to select users with visibility of the maximum number of distinct clusters in the area. Next, we show that the position of clusters in the area can be given by geometrical calculation. \nOur results and contributions are summarized as follows: \n\n\\begin{itemize}\n\\item\n Close analytical approximations for Massive MIMO systems are found.\n\\item\n Using the map of the area and positions of users, a new user scheduling scheme is proposed \\textit{under the assumption of no CSI at the BS, other than the location of clusters}.\n Since the position of clusters in the area are fixed, we assume that cluster localization can be done offline. \n\\item \n Simulation results show that the proposed scheme significantly reduces the overhead channel estimation in Massive MIMO systems compared to conventional user scheduling algorithms, especially for indoor and outdoor of micro-cells.\n\\item\nTo investigate the robustness of the proposed algorithm to cluster localization, the performance degradation is shown for different values of the error in cluster localization and simulation results show the robustness of the proposed user scheduling algorithm to poor cluster localization.\n\\end{itemize}\n\n\\subsection{Outline}\nThe rest of the paper is organized as follows. Section II describes\nthe system model. The proposed user scheduling scheme is presented in Section III. Section IV presents performance analysis of the\nproposed user scheduling with no estimated CSI. {The robustness of the proposed user scheduling algorithm  to cluster localization errors is investigated in Section V.} Numerical\nresults are presented in Section VI. Finally, Section VII concludes\nthe paper. \n\\subsection{Notation}\nNote that in this paper, uppercase and lowercase\nboldface letters are used for matrices and vectors, respectively.\nThe notation $\\mathbb{E}\\{\\cdot\\}$ denotes expectation. Moreover, $|\\cdot|$ stands for absolute value. Conjugate\ntranspose of vector $\\textbf{x}$ is $\\textbf{x}^{H}$. Finally $\\textbf{x}^T$ and $\\textbf{X}^{\\dag}$\ndenote the\ntranspose of vector $\\textbf{x}$ and the pseudo-inverse of matrix $\\textbf{X}$, respectively.\n\\section{SYSTEM MODEL}\nWe consider uplink transmission in a single cell Massive MIMO system with $M$ antennas at the BS and $K> M$ single antenna users. The $M \\times 1$ received signal at the BS when $K_s\\, (K_s\\ll M)$ users have been selected from the pool of $K$ users, is given by \n\\begin{equation}\n\\textbf{r}= \\sqrt{p_k}\\textbf{H}\\textbf{x}+n, \n\\end{equation}\nwhere $\\textbf{x}$ represents the symbol vector of $K_s$ users, $p_k$ \nis the average power of the $k$th user and $\\textbf{H}$ denotes the aggregate $M \\times K_s$ channel of all selected users. The BS is assumed to have CSI only of the selected users \\cite{my-master-vtc,myiet_master}. We are interested in a linear ZF receiver which can be provided by evaluating the pseudo-inverse of $\\textbf{H}$, the aggregate channel of all selected users according to \n\\begin{equation}\n\\textbf{W}=\\textbf{H}^{\\dag}=\\left(\\textbf{H}^{H}\\textbf{H}\\right)^{-1}\\textbf{H}^{H}.\n\\end{equation}\n Then after using the detector, the received signal at the BS is\n\\begin{equation}\n\\textbf{y}= \\sqrt{p_k}\\textbf{W}\\textbf{H}\\textbf{x}+\\textbf{W}\\textbf{n}.\n\\end{equation}\nLet us consider equal power allocation between users, i.e. $p=\\frac{P_t}{k}$, in which $P_t$ denotes the total power. The achievable sum-rate of the system is obtained as \\cite{my-master-vtc,myiet_master}\n\\begin{equation}\nR=\\sum_{k=1}^{K_s}{\\log_{2}\n \\bigg({{  1+\\frac{p|{\\mathbf{w}_k}{\\mathbf{h}_k}|^2}{1+\\sum_{i=1,i\\ne{k}}^{K}p|\\mathbf{w}_k\\mathbf{h}_i|^2}}}\\bigg)},\n\\end{equation}\nwhere $\\textbf{w}_k$ and $\\textbf{h}_k$ are respectively the $k$th rows of the matrix $\\textbf{W}=[\\textbf{w}_1^T,\\textbf{w}_2^T,\\cdots,\\textbf{w}_{K_s}^T]^T$, and  the \\textit{k}th column of  $\\textbf{H}=[\\textbf{h}_1,\\textbf{h}_2,\\cdots,\\textbf{h}_{K_s}]$.\n\\subsection{Geometry-based Stochastic Channel Model}\n\n\\begin{figure}\n\\center\n\\includegraphics[width=91mm]{cluster.eps}\n\\caption{The general description of the cluster model. The spatial spreads for $c$th cluster are given.}\n\\label{cluster}\n\\end{figure}\nIn GSCMs, the double directional channel impulse response is a superposition of MPCs. The channel is given by \\cite{Costaction}\n\\begin{equation}\nh(t,\\tau,\\phi,\\theta)=\\sum_{j=1}^{N_C}\\sum_{i=1}^{N_p}a_{i,j}\\delta(\\phi-\\phi_{i,j})\\delta(\\theta-\\theta_{i,j})\\delta(\\tau-\\tau_{i,j}),\n\\label{h1}\n\\end{equation}\nwhere $N_p$ denotes the number of multipath components, $t$ is time, $\\tau$ denotes the delay, $\\delta$ denotes the Dirac delta function, and $\\phi$ and $\\theta$ represent the direction of arrival (DoA) and direction of departure (DoD) respectively. Similar to \\cite{Costaction,our_ew}, we group the multipath components with similar delay and directions into clusters. Three kinds of clusters are defined; local clusters, single clusters and twin clusters. Local clusters are located around users and the BS while single clusters are represented by one cluster and twin clusters are characterized by two clusters related respectively to the user and BS side as shown in Fig. \\ref{cluster}. A local cluster is a single cluster that surrounds a user: single clusters can also occur in a different position. Twin clusters consist of a linked pair of clusters, one of which defines the angles of departure of multipaths from the transmitter, while the other defines the angles of arrival at the receiver \\cite{Costaction}. There is a large number of clusters in the area, however just some of them can contribute to the channel. The circular visibility region (VR) determines whether the cluster is active or not for a given user. The MPC's gain scales by a transition function that is given by \n\\begin{equation}\nA_{VR}(\\bar{r}_{MS})=\\dfrac{1}{2}-\\dfrac{1}{\\pi}\\arctan \\left(\\dfrac{2\\sqrt{2}\\left(L_c+d_{MS,VR}-R_C\\right)}{\\sqrt{\\lambda L_c}}\\right),\n\\label{vr}\n\\end{equation}\nwhere $\\bar{r}_{MS}$ is the centre of the VR, $R_C$ denotes the VR radius, $L_C$ represents the size of the transition region and $d_{MS,VR}$ refers to the distance between the mobile stations (MS)s and the VR centre. For a constant expected number of clusters $N_C$, the area density of VRs is given by\n\\begin{equation}\n\\rho_C=\\frac{N_C-1}{\\pi \\left(R_C-L_C\\right)^2}.\n\\label{rho_C}\n\\end{equation}\nAll clusters are ellipsoids in the environment and can be characterized by the cluster spatial delay spread, elevation spread and azimuth spread. Once the position of the BS and users are fixed, we need to determine the positions of the clusters in the area by geometrical calculations. For the local clusters, we consider a circle around the users and  the BS, so that the size of the local cluster can be characterized by the cluster delay spread ($a_C$), elevation spread ($h_C$) and the position of MPCs \\cite{Costaction}. For local clusters the cluster delay, azimuth and elevation spreads can be given by \n\\begin{subequations}\n\\begin{eqnarray}\n&a_C=\\dfrac{\\Delta\\tau c_0}{2},\\\\\n& b_C=a_C,\\\\\n& h_C=d_{C,BS}\\tan \\theta_{BS},\n\\end{eqnarray}\n\\end{subequations}\nwhere $c_0$ denotes the speed of light, $d_{C,BS}$ is the distance between the cluster and the BS, $\\Delta\\tau$ refers to the delay spread and $\\theta_{BS}$ is the elevation spread seen by the BS. The delay spread, angular spreads and shadow fading are correlated random variables and for all kinds of clusters are given by \\cite{corria}\n\\begin{subequations}\n\\begin{eqnarray}\n& \\Delta\\tau_c=\\mu_\\tau(\\frac{d}{1000})^{\\frac{1}{2}} 10^{\\sigma_\\tau \\frac{Z_c}{10}},\\\\\n& \\beta_c= \\tau_\\beta 10^{\\sigma_\\beta \\frac{Y_c}{10}},\\\\\n& S_m = 10^{\\sigma_s \\frac{X_c}{10}},\n\\end{eqnarray}\n\\end{subequations}\nwhere $\\Delta\\tau_c$ refers to the delay spread, $\\beta_c$ denotes angular spread, and $S_m$ is the shadow fading of cluster $c$. Moreover, $X_c$, $Y_c$ and $Z_c$ denote correlated random variables with zero mean and unit variance. Correlated random process can be computed by Cholesky factorization \\cite{corria}. Cholesky factorization can be used to generate a random vector with a desired covariance matrix \\cite{matrixbook}. The MPCs' positions can be drawn from the truncated Gaussian distribution given by \\cite{Costaction}\n\\begin{equation}\nf(r)\\!=\\!\\left\\{\n\\begin{array}{rl}\\dfrac{1}{\\sqrt{2\\pi\\sigma_{r,o}^2}}\\exp\\left(\\!-\\!(\\dfrac{r\\!-\\!\\mu_{r,o}}{\\sqrt{2}\\sigma_{r,o}})^2\\right)& |r|\\le {r}_{T},\\\\\n0~~~~~~~~~~~~~~~~~~~~~~~~~~ &\\text{otherwise},\n\\end{array} \\right.\n\\label{fr1}\n\\end{equation}\nwhere $r_T$ denotes the truncation value. For single clusters, the cluster delay, azimuth and elevation spreads can be given by \n\\begin{subequations}\n\\begin{eqnarray}\n&a_C=\\Delta\\tau c_0\/2,\\\\\n&b_C=d_{C,BS}\\tan \\phi_{BS},\\\\\n&h_C=d_{C,BS}\\tan \\theta_{BS}.\n\\end{eqnarray}\n\\end{subequations}\nTo get the fixed positions of the single clusters, the radial distance of the cluster from the BS drawn from the exponential distribution \\cite{Costaction}\n\\begin{equation}\nf(r)=\\left\\{\n\\begin{array}{rl}\n0~~~~~~~~~~~~~~~~~~~~~~~~~~ & r<r_{min},\\\\\n\\dfrac{1}{\\sigma_r}\\exp\\left(-(r-\\dfrac{r_{min}}{\\sigma_r})\\right)& \\text{otherwise}.\n\\end{array} \\right.\n\\label{fr}\n\\end{equation}\nTo determine the fixed position of the cluster, the angle of the cluster can be drawn from the Gaussian distribution with a standard deviation $\\sigma_{\\phi,C}$. For the twin clusters, for both the BS and user side clusters we have \n\\begin{subequations}\n\\begin{eqnarray}\n& a_C=\\dfrac{\\Delta\\tau c_0}{2},\\\\\n& b_C=d_{C,BS}\\tan \\phi_{BS}.\n\\end{eqnarray}\n\\end{subequations}\nFor the BS side cluster, the elevation spread can be given by \n\\begin{equation}\nh_C=d_{C,BS}\\tan \\theta_{BS},\n \\end{equation}\n  while for the MS side cluster, we have\n\\begin{equation}\nh_C=d_{C,MS}\\tan \\theta_{MS}.\n  \\end{equation}\n   Fig. \\ref{cluster} gives an example of the geometry of the $C$th cluster. For twin clusters, the distance between the cluster and the BS and the distance from the VR center and the MS is given by \\cite{Costaction}\n\\begin{equation}\nd_{C,BS}\\tan{\\Phi_{C,BS}}=d_{C,MS}\\tan{\\Phi_{C,MS}}.\n\\end{equation}\nThe delay of  a cluster is represented by \\cite{Costaction}\n\\begin{equation}\n\\tau_C={(d_{C,BS}+d_{C,MS}+d_C)} \/ {c_0}+\\tau_{C,link},\n\\end{equation}\nwhere the geometrical distance between twin clusters is represented by $d_C$, $d_{C,MS}$ denotes the geometrical distance between the user and the center of the visibility region, $d_{C,BS}$ refers to the distance between the BS and the cluster, and finally $\\tau_{C,link}$ is the cluster link delay between the twin clusters. Hence, the cluster power attenuation is given by \\cite{Costaction}\n\\begin{equation}\nA_C=\\max \\left(\\exp\\left[-k_\\tau (\\tau_C-\\tau_0) \\right], \\exp\\left[-k_\\tau (\\tau_B-\\tau_0) \\right]\\right),\n\\label{Ac}\n\\end{equation}\nwhere $k_\\tau$ denotes the decay parameter, and $\\tau_B$ is the cut-off delay. We assume Rayleigh fading for the MPCs within each cluster. Hence, the complex amplitude of the $i$th MPC in the $j$th cluster in (\\ref{h1}) is given by\n \\begin{equation}\na_{i,j}=\\sqrt{L_p}A_{VR}\\sqrt{A_C A_{MPC}}~e^{-j2\\pi f_c \\tau_{i,j}},\n\\label{a}\n\\end{equation} \nwhere $L_p$ is the channel path loss, $A_{MPC}$ is the power of each MPC which is characterized by the Rayleigh fading distribution and $\\tau_{i,j}$ is the delay of the $i$th MPC in cluster $j$ given by \\cite{Costaction}\n\\begin{equation}\n\\tau_{i,j}=\\frac{\\left(d_{MPC_{i,j},BS}+d_{MPC_{i,j},MS}\\right)}{c_0}+\\tau_{i,C,link} .\n\\label{tau2}\n\\end{equation}\nBy assuming a fixed OFDM subcarrier, we can drop the variable $\\tau_{i,j}$ from (\\ref{a}). For the non-line-of-sight (NLoS) case of the micro-cell scenario, the path loss expression can be given by \\cite{pathloss}\n\\begin{IEEEeqnarray}{rCl}\nL = 26 \\log_{10}d+20\\log_{10}\\left(\\dfrac{4\\pi}{\\lambda}\\right),\n\\label{d}\n\\end{IEEEeqnarray}where $d$ and $\\lambda$ denote the distance (in meter) and the wavelength (in meter), respectively.\\\\\n\\section{Geometry-based User Scheduling}\nIn this section, we consider user scheduling with ZF based on the position of clusters and users in the area. In order to avoid a huge channel estimation load in the uplink of a Massive MIMO system with many users and antennas, we propose to estimate only the channels of the selected users. The reduction in the amount of channel estimation required between each transmit and receive antenna is the important result of the proposed scheme. The gain achieved by selecting users with the strongest channel is referred to as multiuser diversity and requires CSI of all users \\cite{GoldsmithJurnal}. However, we propose a new user selection scheme which relies on maximizing the number of distinct clusters seen by the scheduled users. In the next subsections, we prove that the proposed scheme results in less inter-user interference (IUI) and increases the users' SINR and the system's sum-rate. In the following subsection, we present a scheme to select users which maximizes the long term sum-rate and as it is based on the position of the users and does not need the estimated channel of all users in the uplink, and hence can be a practical user selection scheme for large MIMO systems. For this case, the performance analysis are found in the next subsection.\n\\subsection{Proposed Geometry-based User Scheduling (GUS)}\nIn this section, an algorithm is proposed for increasing the system throughput based on the geometry of the system and without estimating the channels of all users in the area. Once the set of active users has been determined, the receiver BS estimates the channels of the selected users and the users transmit data. Next, the performance of the proposed user selection algorithm to maximize the sum-rate is evaluated. In large MIMO systems with large numbers of users estimating the channels of all users is practically difficult. So the proposed user scheduling algorithm can be an efficient way to reduce the overhead of channel estimation.\\\\   \nFirst, we generate the matrix $\\textbf{V}$, as the following \n\\begin{equation}\n\\textbf{{V}}=\n\\begin{bmatrix} \nv_{1}^1 & v_{2}^1 & \\ldots & {v}_{N_C}^1 \\\\\nv_{1}^2 & {v}_{2}^2 & \\ldots & {v}_{N_C}^2 \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\nv_{1}^K & v_{2}^K & \\ldots & v_{N_C}^K\n\\end{bmatrix},\n\\end{equation}\nwhere \n\\begin{equation}\nv_{i}^j=\\sqrt{L_{p,i}^j}A_{VR,i}^j\\sqrt{A_{C,i}^j},\n\\label{a1}\n\\end{equation} \nwhere $L_{p}^j$ denotes the channel path loss for user $j$, $A_{VR,i}^j$ is the MPC power attenuation which is a function of the distance between the user $i$ and the centre of the visibility region related to the $j$th cluster and is given by (\\ref{vr}), and $A_{C,i}^j$ denotes the cluster power attenuation given by (\\ref{Ac}) for the user $j$ and the $i$th cluster. So, the matrix $\\textbf{V}$ is a function of the distance from the BS to users, the distance of the BS from clusters and from users to the center of the visibility region.\n\n\n\\begin{algorithm}[t!]\n\\caption{{\\small Geometry-based User Scheduling (GUS) Algorithm}}\nStep 1) Initialization:~~$\\mathcal{W}_{0}=[1,\\cdots,K]$, $\\mathcal{S}_{0}=\\emptyset$, $i=1$,\\\\\nStep 2) Load position of users, for example by means of GPS,\\\\\nStep 3) Generate matrix $\\textbf{V}$,\\\\\nStep 4) Greedy Algorithm:\n\\begin{itemize}\n\\item{4.1}\\begin{small}\n$\\pi(i)=\\operatornamewithlimits{argmax}\\limits_{k \\in \\mathcal{W}_{0}}f_1(||\\textbf{v}_k||)=\\operatornamewithlimits{argmax}\\limits_{k \\in \\mathcal{W}_{0}}||\\textbf{v}_k||$,\\\\ $\\textcolor{blue}{\\mathcal{S}_{0}\\gets \\mathcal{S}_{0}\\cup \\{\\pi(i)\\}}$, $\\hat{\\textbf{v}}_{(i)}=\\textbf{v}_{(\\pi(i))}$,\\\\\n\\end{small}\n\\item{4.2}~If $|\\mathcal{S}_{0}|<K_s$, $\\mathcal{W}_{i}=\\{k \\in \\mathcal{W}_{i-1}, k \\ne \\pi(i) \\}$,\\\\\n\\item{4.3}\n\\begin{small}\n$\\pi(i)=\\operatornamewithlimits{argmin}\\limits_{k \\in \\mathcal{W}_{i-1}}f_2(\\textbf{v}_k,\\hat{\\textbf{v}}_{(i)})=\\operatornamewithlimits{argmin}\\limits_{k \\in \\mathcal{W}_{i-1}}\\frac{  |\\textbf{v}_k\\hat{\\textbf{v}}_{(i)}^{*}|  }{  ||\\textbf{v}_k||||\\hat{\\textbf{v}}_{(i)}||  } \\}$,~$\\mathcal{S}_{0}\\gets \\mathcal{S}_{0}\\cup \\{k\\}$, $\\hat{\\textbf{v}}_{(i)}=\\textbf{v}_{(\\pi(i))}$,\n\\end{small}\n\\item{4.4}~$i=i+1$\n\\end{itemize}\nStep 5) The BS estimates the channels of the selected users.\n\\end{algorithm}\n\n\n\\begin{algorithm}[t!]\n\\caption{\\small Geometry-based User Scheduling (GUS) Algorithm}\n\\label{aluserselectiond}\n\\hrulefill\n\n\\textbf{1.} Initialize\n$\\mathcal{W}_0=[1,\\cdots,K]$, $\\mathcal{S}_0=\\emptyset$, $i=1$\n\n\\textbf{2.} Repeat until $|\\mathcal{S}_{0}|=K_s$\n\n\\textbf{3.} $i=i+1$\n\n\\textbf{4.} $\\pi(i)=\\argmax_{k \\in \\mathcal{W}_{i-1}}f(||\\textbf{v}_{k}||)=\\argmax_{k \\in \\mathcal{W}_{i-1}}||\\textbf{v}_{k}||$, \\\\\n$\\mathcal{S}_{0}\\gets \\mathcal{S}_{0}\\cup \\{\\pi(i)\\}$, $\\hat{\\textbf{v}}_{(i)}=\\textbf{v}_{(\\pi(i))}$\n\\\\\n\\textbf{5.} $\\mathcal{W}_{i}=\\{k \\in \\mathcal{W}_{i-1}, k \\ne \\pi(i), \\frac{|\\textbf{v}_{k}\\hat{\\textbf{v}}_{(i)}^{T}|}{||\\textbf{v}_{k}||||\\hat{\\textbf{v}}_{(i)}||}<\\epsilon_h \\}$\n\n\\textbf{6.} If $|\\mathcal{W}|=0$, end\n\\end{algorithm}\n\n\n\nNote that increasing $\\epsilon_h$ allows the users to have a larger number of shared clusters. If the value of $\\epsilon_h$ is too high, Algorithm selects users with a large normalized correlation which can reduce the sum rate due to the interference in the number of selected users. For a low $\\epsilon_h$, the number of users in set $\\mathcal{V}_0$ in step 5\ndecreases and Algorithm \\ref{aluserselectiond} selects a small number of users.\n{Suppose $W_{0}$ contains user indices considered in the proposed algorithm.  Finally, $S_{0}$ contains $K_s=|S_{0}|$ indices of the selected users.}  \n\\begin{figure}[t!]\n\\center\n\\includegraphics[width=75mm]{common.eps}\n\\caption{As example of users common cluster which causes correlation.}\n\\label{common}\n\\end{figure}\n\\section{Performance Analysis}\nIf perfect CSI is available at the BS, and assuming Gaussian input, the ergodic capacity is given by\n\\begin{IEEEeqnarray}{rCl}\nC = \\mathbb{E} \\left\\{\\log_2\\det  \\left(\\textbf{I}+\\dfrac{P_t}{K_s}\\textbf{H}\\textbf{H}^H\\right)\\right\\},\n\\end{IEEEeqnarray}\nwhere the term $\\frac{P_t}{K_s}$ is due to the equal-power allocation, $\\textbf{I}$ is an identity matrix and the channel matrix is given at the bottom of this page, where $\\textit{C}(K)$ denotes the clusters seen by the $k$th user and $\\alpha=-2\\pi\\frac{d}{\\lambda}$, where $d$ denotes the spacing between two antenna elements. In GSCMs, common clusters can reduce the rank of the channel and the capacity of the system, especially at finite SNR. These common clusters also affect the multiplexing gain of the system. {Fig. \\ref{common} illustrates the concept of common and distinct clusters.} When the number of objects is less than the number of BS antennas and all objects are shared between the users, achieving maximum multiplexing gain is impossible \\cite{Alister10ISWCS,burrijas}. \n\nFor ease of mathematical tractability, we analyse the capacity of a correlated three-user uplink using an upper bound. In the case of a large number of antennas at the BS, the capacity upper bound can be achieved in the case of distinct clusters. Note that in the case of a large number of transmit antennas, the elements of $\\textbf{H}\\textbf{H}^H$ converge to the correlation matrix so that $\\textbf{R}\\approx\\textbf{H}\\textbf{H}^H$. Hence, we have\n\\begin{IEEEeqnarray}{rCl}\n\\label{Capp}\nC&=&\\mathbb{E}\\left\\{\\log_2\\det\\left(\\textbf{I}+\\dfrac{P_t}{K}\\textbf{H}^H\\textbf{H}\\right)\\right\\}\\nonumber \\\\\n&\\approx&\\log_2\\det \\left(\\textbf{I}+\\dfrac{P_t}{K}\\textbf{R}\\right),\n\\end{IEEEeqnarray}\n\\begin{figure*}[!b]\n\\begin{small}\n\\hrulefill\n\\begin{align}\n\\begin{split}\n\\!\\!\\textbf{H}\\!\\!=\\!\\!\\!\\!\n\\begin{bmatrix} \n\\sum_{j\\in \\textit{C}(1)}\\sum_{i=1}^{N_p}a_{i,j} & \\sum_{l\\in \\textit{C}(2)}\\sum_{i=1}^{N_p}a_{i,l}  & \\ldots & \\!\\!\\sum_{m\\in \\textit{C}(K)}\\sum_{i=1}^{N_p}a_{i,m} \\\\\n\\sum_{j\\in \\textit{C}(1)}\\sum_{i=1}^{N_p}a_{i,j}e^{j\\alpha\\sin\\phi_{i,j}} & \\sum_{l\\in \\textit{C}(2)}\\sum_{i=1}^{N_p}a_{i,l}e^{j\\alpha\\sin\\phi_{i,l}} & \\ldots & \\!\\!\\sum_{m\\in \\textit{C}(K)}\\sum_{i=1}^{N_p}a_{i,m}e^{j\\alpha\\sin\\phi_{i,m}} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n\\sum_{j\\in \\textit{C}(1)}\\sum_{i=1}^{N_p}a_{i,j}e^{j\\alpha(M-1)\\sin\\phi_{i,j}} & \\sum_{l\\in \\textit{C}(2)}\\sum_{i=1}^{N_p}a_{i,l}e^{j\\alpha(M-1)\\sin\\phi_{i,l}} & \\ldots & \\!\\!\\sum_{m\\in \\textit{C}(K)}\\sum_{i=1}^{N_p}a_{i,m}e^{j\\alpha(M-1)\\sin\\phi_{i,m}}\\!\\!\\!\n\\end{bmatrix}\\!\\!,\n\\label{H3}\n\\end{split}\n\\end{align}\n\\end{small}\n\\end{figure*}\nwhere $R$ is the channel correlation matrix and is given by\n\\begin{equation}\n\\textbf{R}=\\mathbb{E}\\left\\{\\textbf{H}^H\\textbf{H}\\right\\}=\n\\begin{bmatrix} \n1 & r_{12}  & r_{13} \\\\ \nr_{12}^* & 1 & r_{23} \\\\\nr_{13}^* & r_{23}^* & 1\n\\end{bmatrix},\n\\label{R}\n\\end{equation}\nwhere\n\\begin{subequations}\n\\begin{eqnarray}\n&r_{12}=\\mathbb{E}\\left\\{\\textbf{h}_1^H\\textbf{h}_2\\right\\}=\\zeta_{12}e^{j\\beta_{12}},\\\\\n&r_{13}=\\mathbb{E}\\left\\{\\textbf{h}_1^H\\textbf{h}_3\\right\\}=\\zeta_{13}e^{j\\beta_{13}},\\\\\n&r_{23}=\\mathbb{E}\\left\\{\\textbf{h}_2^H\\textbf{h}_3\\right\\}=\\zeta_{23}e^{j\\beta_{23}}.\n\\end{eqnarray}\n\\end{subequations}\nThe term $r_{12}$ can be given by\n\\begin{small}\n\\begin{IEEEeqnarray}{rCl}\nr_{12}\\!&\\!=\\!&\\!\\mathbb{E}\\left\\{\\textbf{h}_1^H\\textbf{h}_2\\right\\}= \\mathbb{E}\\Big\\{\\sum_{j\\in \\textit{C}(1)}\\sum_{i=1}^{N_p}a_{i,j} \\sum_{l\\in \\textit{C}(2)}\\sum_{g=1}^{N_p}a_{g,l}^* \\\\&\n+&\\sum_{j\\in \\textit{C}(1)}\\sum_{i=1}^{N_p}a_{i,j}e^{j\\alpha\\sin\\phi_{i,j}}\\sum_{l=1\\in\\textit{C}(2)}\\sum_{g=1}^{N_p}a_{g,l}^*e^{-j\\alpha\\sin\\phi_{g,l}}+\\ldots\\!+\\! \\nonumber\\\\&  \n&\\sum_{j\\in \\textit{C}(1)}\\sum_{i=1}^{N_p}a_{i,j}e^{j\\alpha(M-1)\\sin\\phi_{i,j}}\\sum_{l\\in\\textit{C}(2)}\\!\\sum_{g=1}^{N_p}a_{g,l}^*e^{-j\\alpha(M-1)\\sin\\phi_{g,l}}\\Big\\}\\nonumber,\n\\label{r1}\n\\end{IEEEeqnarray}\n\\end{small}\nwhere $a_{i,j}$, the amplitude of the $i$ MPCs in cluster $j$, is given by (\\ref{a}), and the terms $r_{13}$ and $r_{23}$ can be derived in the same way. By substituting the terms $r_{12}$, $r_{13}$ and $r_{23}$ into (\\ref{Capp}), the capacity maximization problem  in a three-user scenario can be formulated as \n\\begin{IEEEeqnarray}{rCl}\n\\label{maxc}\nC\\!&=\\!&\\!\\!\\!\\max_{\\zeta_{12},\\zeta_{13},\\zeta_{23}\\beta_{12},\\beta_{13},\\beta_{23}}\\!\\log_2\\!\\Big[(1+p)^3-p^2\\left(\\zeta_{12}^2\\!+\\!\\zeta_{13}^2\\!+\\!\\zeta_{23}^2\\right)\\\\&\n&+\\!p^3\\Big(2\\zeta_{12}\\zeta_{13}\\zeta_{23}\\sin(\\beta_{12}\\!-\\!\\beta_{13}+\\beta_{23})\\!-\\!\\zeta_{12}^2-\\zeta_{13}^2-\\zeta_{23}^2\\Big)\\Big]\\nonumber,\n\\end{IEEEeqnarray}\nwhere $p=\\frac{P_t}{K}$. To maximize (\\ref{maxc}), the gradient search (GS) method results in $\\zeta_{12}=\\zeta_{13}=\\zeta_{23}=0$, for different values of $\\beta_{12}$, $\\beta_{13}$ and $\\beta_{23}$, which is the case when common clusters do not occur between the users in the cell. In the case of distinct clusters between user $m$ and user $n$, we have \n\\begin{IEEEeqnarray}{rCl}\n\\zeta_{mn} = \\mathbb{E}\\left\\{\\sum_{j\\in \\textit{C}(n)}\\sum_{i=1}^{N_p}a_{i,j} \\sum_{l\\in \\textit{C}(m)}\\sum_{g=1}^{N_p}a_{g,l}^*\\right\\}=0.\n \\label{E}\n\\end{IEEEeqnarray}\nThe equation (\\ref{E}) yields $\\zeta_{mn}=0$, which maximizes the capacity given by (\\ref{maxc}).\n\nFor the case of Massive MIMO systems with a large number of users in the cell, having distinct clusters for all users is practically difficult. In a real scenario, it is not possible to force the term $\\zeta_{mn}$ zero. The proposed user scheduling algorithm selects users which do not have common clusters and consequently forces the variable $\\zeta_{mn}$ to be small. A threshold is set for the power of a cluster to be considered active. In the COST channel model, each user interacts with several clusters in the area and the cluster power depends on the distance between the user and the center of visibility region and also the distance between the cluster and the BS. We define a threshold which can determine the minimum power that a cluster may have relative to the total powers. As in \\cite{Tufvesson_Cost_th}, we set the cluster power threshold to $0.01\\times$ total power for a cluster to be active.\nA cluster is shared between two users if contributes to both users, which means the cluster powers seen by the users are more than the threshold. Hence, the optimum value of $\\zeta_{mn}$ can be achieved only when there is no common cluster in the cell.\nMoreover, investigating the effectiveness of the proposed user scheduling scheme in the distributed Massive MIMO systems \\cite{ourjournal1,ourjournal2,ouricc1,ouricc2,ourcommletter,ourvtc18} is an interesting topic for future work.\n\\section{Robustness of the Proposed User Scheduling Algorithm}\n\\subsection{Cluster Localization}\n The BS can estimate the direction of arrival \\cite{handbook}, and hence the direction of the scattering objects should be available at the BS.\nThere is a well-known algorithm to estimate the delay, direction of arrival and the direction of departure of the channel paths; SAGE-based algorithm \\cite{crlbc,crlbj}. As a result, the BS can identify the direction of the clusters which can be seen by the users in the cell area, and hence build up a map of the location of the scattering objects. The convenient tool that has overcame the challenge of making the position of the scatterers available is the use of environment maps \\cite{Costaction}, which also shows how measured angles of arrival can be identified with physical objects in the environment, and hence can be located on the map. Successive interference cancellation has also been introduced in \\cite{molish-tuf-position} for scattering object identification: it uses the channel impulse response peaks in the delay domain to map scatterers to 2-D coordinates.\n\\subsection{{Robustness}}\nIn order to study the robustness of the proposed algorithm to cluster localization error, we use the well-known SAGE algorithm \\cite{crlbc,crlbj}, operating offline, as mentioned above. In cluster localization, we consider a receiver BS with a antenna array consisting of $M$ sensors located at a reference point \\cite{crlbc,crlbj}. Moreover, we consider planar wavefronts. \nThe closed-form Cramer-Rao lower bound (CRLB) for the delay, azimuth ($\\nu$) and elevation ($\\theta$) of the path are given by \\cite{crlbc}\n\\begin{subequations}\n\\begin{eqnarray}\n& CRLB(\\tau) =\\dfrac{1}{\\gamma_O}\\dfrac{1}{8\\pi^2BW} \\label{crlb1}\\\\\n& CRLB(\\theta) = \\dfrac{1}{\\gamma_O}\\dfrac{M}{2\\Delta \\cos (\\nu)} \\label{crlb2}\\\\\n& CRLB(\\nu) = \\dfrac{1}{\\gamma_O}\\dfrac{M}{2\\Delta}\\label{crlb3},\n\\end{eqnarray}\n\\label{crlb}\n\\end{subequations}\nwhere $BW$ is the bandwidth, and\n\\begin{IEEEeqnarray}{rCl}\n\\Delta&=&4\\pi^2(\\dfrac{d}{\\lambda})^2(\\dfrac{7}{3}M_x^3-8M_x^2+\\dfrac{29}{3}M_x-4),\n\\end{IEEEeqnarray} \nand\n\\begin{IEEEeqnarray}{rCl}\n\\gamma_O=M \\times I\\times N\\times |f(\\nu)|^2\\gamma_I,\n\\end{IEEEeqnarray}\n where $I$ is the number of periods of the received signal, $N$ denotes the length of the used pseudonoise\n(PN) sounding sequence available at the receiver and $\\gamma_I$ is the SNR at the input of each antenna \\cite{crlbc,crlbj}. Moreover, the antenna electric field pattern can be given by \\cite{crlbc} \\begin{equation}\nf(\\nu)=0.67+2.67\\nu-6.79\\nu^2+5.7\\nu^3-1.71\\nu^3.\n\\end{equation}\n The distance between the BS and cluster ($d_{BS,C}$) is given by geometrical calculation:\n\\begin{IEEEeqnarray}{rCl}\n\\left(c\\tau-d_{BS,C}\\right)^2 & = &\\left(h_{BS}-h_{MS}+d_{BS,C}\\sin(\\nu)\\right)^2+ \\\\ &  \n&\\left(d_{BS,MS}-d_{BS,C}\\cos(\\nu)\\cos(\\theta)\\right)^2,\\nonumber\n\\label{dis}\n\\end{IEEEeqnarray}\nwhere $c$ denotes the velocity of light, $d_{BS,MS}$ is the distance between the user and the BS in $x-y$ plane, and $h_{BS}$ and $h_{MS}$ are the BS and user heights. The distance between the user and cluster is easily given by \n\\begin{equation}\nd_{MS,C}+d_{BS,C}=c\\tau.\n\\end{equation}\n After the offline localization, the BS can build up the matrix $\\tilde{\\textbf{V}}$ at the beginning of each time-slot, as the following \n\\begin{equation}\n{\\tilde{\\textbf{V}}}=\n\\begin{bmatrix} \n\\tilde{v}_{1}^1 & \\tilde{v}_{2}^1 & \\ldots & \\tilde{v}_{N_C}^1 \\\\\n\\tilde{v}_{1}^2 & \\tilde{v}_{2}^2 & \\ldots & \\tilde{v}_{N_C}^2 \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n\\tilde{v}_{1}^K & \\tilde{v}_{2}^K & \\ldots & \\tilde{v}_{N_C}^K\n\\end{bmatrix},\n\\end{equation}\nwhere\n\\begin{equation}\n\\tilde{v}_{i}^j=\\sqrt{L_{p}^j}\\tilde{A}_{VR,i}^j\\sqrt{\\tilde{A}_{C,i}^j},\n\\label{a2}\n\\end{equation} \nwhere $\\tilde{A}_{VR,i}^j$ and $\\tilde{A}_{C,i}^j$ can be calculated by the distances obtained in (\\ref{dis}). Finally, for the matrix \\textbf{V}, the following equation holds\n\\begin{equation}\n\\textbf{V} = \\tilde{\\textbf{V}}+\\textbf{E},\n\\end{equation}\nwhere $\\textbf{E}$ is due to the estimation error in cluster localization. Then, we use $\\tilde{\\textbf{V}}$ instead of $\\textbf{V}$ in the proposed algorithm. The numerical results verifies the robustness of the proposed algorithm to this error. \n\\section{Numerical Results and Discussion}\nIn this section, simulation results have been provided to validate the performance of the proposed schemes with different parameters.\n\\subsection{Simulation Parameters for COST 2100 Channel Model}\nWe evaluate the throughput of the system, averaging over 50 iterations. \nA square cell with a side length of $2\\times R$ has been considered so that we call $R$ the cell size and also assume users are uniformly distributed in the cell. As in \\cite{MarzettaMRC13}, we assume that there is no user closer than $R_{th}=0.1\\times R$ to the BS. We simulate a micro-cell environment for the NLoS case and set the operating frequency $f_C=2$GHz. The external parameters and stochastic parameters are extracted from chapter 6 of \\cite{corria} and chapter 3 of \\cite{Costaction}. The BS and user heights are assumed to be $h_{BS}=5$ and $h_{MS}=1.5$, respectively. In (\\ref{rho_C}) $N_C=3$, $R_C=50$, and $L_C=20$. Moreover, we consider $N_{P}=6$ paths per cluster.\n\\begin{figure}[t!]\n\\center\n\\includegraphics[width=95mm]{vsRnew.eps}\n\\caption{The average sum-rate vs. the cell size for different values of $M=200$, $M=100$, $K_s=50$, and $K_s=40$. We set the total number of users in the cell $K=400$.}\n\\label{vsR}\n\\end{figure} \n\\begin{figure}[t!]\n\\center\n\\includegraphics[width=95mm]{vsKnew.eps}\n\\caption{The average sum-rate vs. total number of users for different values of $M=200$, $M=100$, $K_s=50$, $K_s=40$, $R=600$m and $R=1000$m.}\n\\label{vsK}\n\\end{figure}\n\\begin{figure}[t!]\n\\center\n\\includegraphics[width=95mm]{vsM.eps}\n\\caption{The channel estimation load vs. value of error of antennas at the receiver BS for different values of total number of users in the cell.}\n\\label{vsM}\n\\end{figure}\n\\begin{figure}[t!]\n\\center\n\\includegraphics[width=95mm]{e.eps}\n\\caption{{The average sum-rate vs. the estimation error for different values of total number of selected users in the cell and the cell size.}}\n\\label{e}\n\\end{figure}\n\\subsection{Simulation Results}\nFor this network setup, the average sum-rate is evaluated for the three scenarios. In the GUS scheme, it has been proposed that the receiver BS selects users that maximize the number of distinct clusters in the cell. We evaluate the average throughput of the GWC scheme \\cite{SUSGoldsmithGlobcom,ITC09_Userselection_GWC} and random selection (RS) of users. For the case of GWC, similar to \\cite{ITC09_Userselection_GWC}, we set the optimal channel direction\nconstraint to achieve the best performance for GWC, so the complexity of GWC is much higher than GUS.\n\nFig. \\ref{vsR} depicts the average sum-rate with total number of receive antennas at the BS $M=100$ and $M=200$, and two values of the number of selected users $K_s=40$ and $K_s=50$ while adopting the proposed scheme with ZF receiver. As expected, since GWC exploits perfect CSI, it has the best throughput. As seen in Fig.\\ref{vsR}, the performance of the proposed algorithm is slightly lower than the case in which the BS exploits full CSI and performs GWC. Interestingly, for bigger cells, the superiority of the proposed scheme is more obvious in terms of achieving performance close to that of the GWC scheme. \n\nIn Fig. \\ref{vsK}, we have plotted the average sum-rate for the case of GWC and GUS versus total number of users in the cell ($K$) with different numbers of receive antennas at the BS, $M$ and of selected users, $K_s$. In terms of average sum-rate, Fig. \\ref{vsK} shows that the proposed scheme results in only a small sum-rate reduction even with a smaller total number of users.\n\nThe amount of channel estimation load required in both GWC and the proposed GUS is presented in Fig. \\ref{vsM}. As the figure shows the channel estimation load of the proposed GUS is far less than that of the GWC scheme.\n\nTo investigate the robustness of the proposed scheme to different values of the error, we set \n\\begin{IEEEeqnarray}{rCl}\n|e|=\\Omega\\times \\sqrt{CRLB(\\rho)}, \n\\end{IEEEeqnarray}\nwhere $|e|$ denotes the absolute value of the estimation error, $\\Omega$ is an integer number and $CRLB(\\rho)$ is given by (\\ref{crlb1})-(\\ref{crlb3}), where the parameter $\\rho$ can be the delay, azimuth and elevation. Fig. \\ref{e} shows the average sum-rate with total number of receive antennas at the BS $M=400$, and two values of the number of selected users $K_s=10$ and $K_s=40$ versus the value of the estimation error. We set the SNR at the input of each antenna $\\gamma_I=20$dB and the bandwidth $BW=20$MHz. Moreover, in equations (\\ref{crlb1}) to (\\ref{crlb3}), $M_x=5$, $N=127$, which are extracted from \\cite{crlbc}. The figure shows the robustness of the proposed algorithm to poor cluster localization.\n\\section{Conclusions}\n{We have investigated the user scheduling problem in Massive MIMO systems and proposed a new GUS scheme which maximizes the uplink throughput of the users, considering the FDD mode.} By applying knowledge of the location of clusters and users and the geometry of the system, we suppose that the BS does not need to estimate the channels of all users and selects users based only on the location of users and clusters in the area. {Next, exploiting Cramer-Rao lower bound, we have developed a robustness analysis for the proposed scheme.} The results show that while sum-rate slightly decreases along with the reduced overhead of channel estimation, the proposed algorithm can be an efficient scheme to reduce the complexity of user scheduling in Massive MIMO systems. {In addition, the simulation results demonstrate good robustness against the estimation error.}\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn the last years, diffractive scattering processes have received\nincreasing attention for several reasons. These processes may, for\nexample, explain many features of particle production and, in\nparticular, heavy flavour production \\cite{HF} and Centauro events\n\\cite{CAP}. They are also related to the large rapidity gap physics\nand the structure of the Pomeron \\cite{POMERON}. In a diffractive\nscattering, one of the incoming hadrons emerges from the collision\nonly slightly deflected and there is a large rapidity gap between it\nand the other final state particles resulted from the other excited\nhadron. In some models diffraction is due to the Pomeron exchange \nbut the exact\nnature of the Pomeron in QCD is not elucidated yet. The first test of\na theory (or a model) of diffractive dissociation (DD) is the ability\nto properly describe the mass ($M_X$) distribution of diffractive\nsystems, which has been measured in many experiments \\cite{DATA} and\nparametrized as $(M_X^2)^{-\\alpha}$ with $\\alpha \\simeq 1$. Data\npresented in \\cite{DATA} were taken at the Tevatron collider\n($\\sqrt{s} = 1.8$ TeV). They allow us to make comparison with CERN\n\\cite{Bozzo} lower energies data and observe the energy dependence of\nthe mass spectrum. In the old Regge theory, the assumption of Pomeron\ndominance implies that the mass spectrum behaves like $1\/M^2_X$ and\ndoes not depend on the energy \\cite{REGGE} whereas in \\cite{DATA} a\nslight deviation from this behaviour was reported.\\\\\n\nIn this work we intend to study diffractive mass distributions\nusing the Interacting Gluon Model (IGM) developed by us recently\n\\cite{IGM,DNW94,DNW93}. In particular, we are interested in the\nenergy dependence of these distributions and their connection with\ninelasticity distributions \\cite{INEL,DNW94}. One advantage of the\nIGM is that it was designed in such a way that the energy-momentum\nconservation is taken care of before all other dynamical aspects.\nThis feature makes it very appropriate for the study of energy flow\nin high energy hadronic and nuclear reactions \\cite{IGM,DNW94,DNW93}\nand in cosmic ray studies \\cite{INEL}. In particular, as shown in\n\\cite{DNW94,DNW93}, the IGM was very usefull in analysing data and\nmaking predictions on the behaviour of inelasticities and leading\nparticle spectra, including leading charm production \\cite{DNW96}.\nThe aim of the present work is to demonstrate that the main\ncharacteristics of the DD processes mentioned above emerge\nnaturally from the standard IGM, enlarging profoundly its range\nof applications.\\\\\n \nIn the following section we present for readers convenience the basic\nelements of the IGM. It is then applied to DD processes in Section 3 \nwhere we also carefully explain what DD means in terms of the IGM and\nhow it differs from the conventional Regge Pomeron approach. Section 4 \ncontains our numerical results and comparison with data and the last\nsection is devoted to our conclusions.\\\\\n\n\n\\section{Interacting Gluon Model}\n\\subsection{General ideas}\nThe Interacting Gluon Model is based on the following  idea \\cite{POK} :\nsince about half of a hadron momentum is carried by gluons and since\ngluons interact more strongly than quarks, during a collision there\nis a separation of constituents. Valence quarks tend to be fast forming\nleading particles whereas gluons tend to be stopped in the central\nrapidity region. This picture is consistent with string formation and\nfragmentation as it is formulated in the Lund Model or in the Dual \nParton Model. These models are based on the concept of string and were\nconstructed to work at low average transverse momenta. When, in the\nlate eighties, the energy of hadronic collisions increased by more\nthan one order of magnitude it became necessary to incorporate the\nconcept of parton and of hard and semi-hard collisions. The latter\nare collisions between partons at a moderate scale \n($ Q^2 \\simeq (2\\, GeV)^2 $) which however still allows for the use of\nperturbative QCD. The scattered partons form the so-called minijets.\nAt $\\sqrt{s}= 540 $ GeV the minijet cross section is already $25 \\%$\nof the total inelastic cross section. A new generation of models \nappeared using the concept of parton and the QCD parton model formalism\nand trying to understand minimum-bias multiparticle physics extending\nthe parton model to semihard (moderate) energy scales. Among these\nmodels we mention those presented by Gaisser and Stanev \\cite{gs},\nSjostrand \\cite{sj}, Wang \\cite{wang} and Geiger \\cite{geiger}. The\nIGM belongs to this class of models. In all these models one finds\nat a certain point expresssions of the type\n\\begin{eqnarray}\n \\int_{Q^2}^1\\!dx_1\\, \\int_{\\frac{Q^2}{x_1}}^1\\! dx_2\\,\nf(x_1,Q^2)\\,f(x_2,Q^2)\\,\\sigma(x_1,x_2,Q^2) \n\\end{eqnarray}\nwhere f and $\\sigma$ are parton momentum distribution and parton-parton\nelementary cross section respectively. $Q^2$ is the scale. Apart from\nsome ambiguity in choosing the scale, these models have to face the\nproblem that even at very high energies a significant part of a hadronic\ncollision occurs at scales lower than the semihard one. At this point\nf and $\\sigma$ are not very well known. The atitude taken in HIJING\n\\cite{wang}, in the Parton Cascade Model \\cite{geiger} and also in the\nIGM is to extrapolate these quantities to lower scales. These \nextrapolations can be continuously improved, especially in view of the\nadvance of our knowledge on non-perturbative effects. There are, for\nexample, models for distribution functions\nwhich work at scales as low as $ 0.3\\, GeV^2$ \\cite{grv}. As for $\\sigma$\none can compute non-perturbative effects in the context of an operator\nproduct expansion \\cite{nos}. Inspite of these limitations these models\nhave the advantage of dealing with partons and being thus prepaired to\nincorporate perturbative QCD in a natural way. This is welcome since \nperturbative processes are expected to be increasingly important at\nhigher energies.  Compared to the other models mentioned above, the IGM\nis simpler because it is designed to study energy flow and makes no\nattempt to calculate cross sections or to follow hadronization in great\ndetail. This simplifies the calculations and avoids time consuming numerical\nsimulations. The most important aspect of the IGM, shared with those models,\n is the assumption of multiple parton-parton incoherent scattering which is\nimplicit  in the Poissonian distribution of the number of parton-parton\ncollisions (which is also used in refs. \\cite{sj,gs,wang}) used below. In \ngoing to lower resolution scales this independent collision approximation\nbecomes questionable. On the other hand lattice QCD calculations \\cite{gia}\nin the strong coupling regime indicate that the typical correlation length of\nthe soft gluon fields is around $0.2$ fm. This is still smaller than the typical\nhadronic size. Therefore the independent collision approximation may still be a\nreasonable one. From the practical point of view, it was shown in \\cite{IGM}\nthat replacing the Poissonian distribution by a broader one does not affect the\nresults significantly as long as some mass scale is introduced to cut off the\nvery low x region.\n\\subsection{Formulation of the model}\n\nThe IGM is based on the assumed dominance \nof hadronic collisions by gluonic interactions \\cite{POK} and can be\nsummarized as follows \\cite{IGM} (cf. Fig. 1a):\n\\begin{itemize}\n\\item[$(i)$] The two colliding hadrons are represented by valence\nquarks carrying their quantum numbers (charges) plus the accompanying\nclouds of gluons (which represent also the sea $q\\bar{q}$ pairs and\ntherefore should be regarded as effective ones).\n\\item[$(ii)$] In the course of a collision gluonic clouds interact\nstrongly and form a gluonic {\\it central fireball} (CF) located in the\ncentral region of the reaction.\n\\item[$(iii)$] The valence quarks (plus those gluons which did not\ninteract) get excited and form {\\it leading jets} (LJ's) (or {\\it beam\njets}) which then populate mainly the fragmentation regions of the\nreaction.\n\\end{itemize}\nIt should be stressed that the IGM has been \nformulated originally in order to\ngive the initial conditions for hydrodynamical models by providing in a\ndynamical way the so called {\\it inelasticity} of the reaction understood\nusually as the (invariant) fraction of energy stored in the CF:\n\\begin{equation}\nK\\, =\\, \\sqrt{x\\cdot y} \\label{eq:K}\n\\end{equation}\nwith $x$ and $y$ being the fractions of the initial energy-momenta of\nthe respective projectiles allocated to the CF. This variable and, in\nparticular, its energy dependence is of vital importance in cosmic\nray studies as it is the  necessary ingredient allowing to deduce any\nelementary information from cosmic ray experiments \\cite{INEL}.\n\n\nAccording to the IGM, pairs of gluons collide and form ``minifireballs''\n(MF's). In collisions at higher scales a minifireball is the same as a\npair of minijets or jets. In the study of energy flow the details of\nfragmentation and hadron production are not important. Most of the MF's\nwill be in the central region and we assume that they coalesce forming\nthe CF. The probability to form a fireball carrying momentum fractions \n $x$ and $y$  of two colliding hadrons is defined as the sum over\n(an undefined number $n$ of) MF's \\cite{IGM}:\n\\begin{eqnarray}\n\\chi(x,y) &=& \\sum_{\\{n_i\\}} \\delta \\left( x-\\sum_i n_i x_i \\right)\n\\delta \\left( y-\\sum_i n_i y_i \\right) \\prod_{\\{n_i\\}} P(n_i)\n\\end{eqnarray}\n(all masses and transverse momenta are neglected in what follows).\nThe distribution of the number of MF's is given by $P(n_i)$ for which\nwe use Poisson distributions:\n\\begin{eqnarray}\nP(n_i) &=& \\frac{(\\overline n_i)^{n_i} exp(-\\overline n_i)}{n_i\\!}\n\\end{eqnarray}\ncorresponding to independent production. Expressing the delta functions\nvia Fourier integrals one can perform all summations, transform certain\nsummations over $\\overline n_i$ into integrals and make the replacement\n$\\ d \\overline n_i \/ dx dy = \\omega(x,y)$ arriving at the \ngeneral formula  \\cite{IGM}:\n\\begin{eqnarray}\n\\chi(x,y) &=& \\frac{\\chi_0}{2\\pi\\sqrt{D_{xy}}}\\cdot \\label{eq:CHI}\\\\\n&&\\cdot \\exp \\left\\{ - \\frac{1}{2D_{xy}}\\,\\left[\n  \\langle y^2\\rangle (x - \\langle x\\rangle )^2 +\n  \\langle x^2\\rangle (y - \\langle y\\rangle )^2 -\n  2\\langle xy\\rangle (x - \\langle x\\rangle )(y - \\langle y\\rangle )\n  \\right] \\right\\} ,\\nonumber\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nD_{xy} &=& \\langle x^2\\rangle \\langle y^2\\rangle - \n           \\langle xy\\rangle ^2  \\nonumber \n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\langle x^ny^m\\rangle &=& \\int_0^1\\! dx\\,x^n\\, \\int_0^1\\! dy\\, \ny^m\\, \\omega (x,y), \\label{eq:defMOM}\n\\end{eqnarray}\nwith $\\chi_0$ being a normalization factor defined by the condition\nthat\n\\begin{eqnarray}\n \\int_0^1\\!dx\\, \\int_0^1\\! dy\\, \\chi(x,y) \\Theta(xy - K_{min}^2) = 1\n\\end{eqnarray}\nwith $K_{min} = \\frac{m_0}{\\sqrt{s}}$ being the minimal inelasticity\ndefined by the mass $m_0$ of the lightest possible CF. \nThe, so\ncalled, spectral function $\\omega(x,y)$ contains all the dynamical\ninput of the IGM in the general form of (cf. \\cite{DNW93}) \n\\begin{equation}\n\\omega(x,y)\\, =\\, \\frac{\\sigma_{gg}(xys)}{\\sigma(s)}\n   \\, G(x)\\, G(y)\\, \\Theta\\left(xy - K^2_{min}\\right),\n   \\label{eq:OMEGA}\n\\end{equation}\nwhere $G$'s denote the effective number of gluons from the\ncorresponding projectiles (approximated by the respective gluonic \nstructure functions) and $\\sigma_{gg}$ and $\\sigma$ the gluonic and\nhadronic cross sections, respectively.\nThese are basic formulae of the IGM from which all its \napplications are derived \\cite{APPLIC}.\\\\\n\n\\section{Diffractive Dissociation in the IGM approach}\n\nIt is now quite straightforward to extend the IGM also to diffractive\ndissociation processes. In Fig. 1b we show schematically the IGM \npicture of a diffractive dissociation event. Unlike in Fig. 1a \nhere only one of the protons\nlooses fraction $x$ of its original momentum and gets excited forming a\nLJ carrying $x_L= 1 -x$ fraction\nof the initial momentum. The other one, which we shall call here the\ndiffracted proton, looses only a fraction $y$ of its momentum but\notherwise remains intact \\cite{FOOT}. In the standard IGM\npresented in the previous section we were computing the \nprobability $\\chi(x,y)$ of\ndepositing energy fractions $x$ and $y$ in the central region in the\nform of gluonic CF of mass\n$M = \\sqrt{xy\\, s}$, which subsequently was decaying and producing\nparticles. Here we shall be rather interested in the mass $M_X$, a\nnew variable in our problem, which, as can be seen in \nFig. 1b, is just the invariant mass of a system composed of the \nCF and the leading jet\nformed by one of the colliding protons (we shall call it also\n{\\it diffractive mass}). Denoting by $E_L$ and $P_L$ the energy and\nmomentum of the upper (in Fig. 1b ) proton and by $W$ and $P$ the\nenergy and momentum of the CF,\n\\begin{equation}\nP_L = E_L\\, =\\, \\frac{\\sqrt{s}}{2}\\, ( 1 - x ), \\quad\nP   = \\frac{\\sqrt{s}}{2}\\, ( x - y ), \\quad\nW   = \\frac{\\sqrt{s}}{2}\\, ( x + y ), \\label{eq:PLPW}\n\\end{equation}\nthe energy $E_X$ and momentum $P_X$ of the diffractive cluster are\ngiven by:  \n\\begin{equation}\nE_X = E_L + W\\, =\\, \\frac{\\sqrt{s}}{2}\\, (1 + y) \\quad {\\rm and} \\quad\nP_X = P_L + P\\, =\\, \\frac{\\sqrt{s}}{2}\\, (1 - y) \\label{eq:EXPX}\n\\end{equation}\nleading to the following expressions for the mass of our diffractive\ncluster, $M_X$, and its rapidity, $Y_X$:\n\\begin{eqnarray}\nM_X\\, &=&\\, \\sqrt{E_X^2\\, -\\, P_X^2}\\, \\, =\\, \n                  \\sqrt{s\\cdot y}, \\label{eq:MASS}\\\\\nY_X\\, &=&\\, \\frac{1}{2}\\, \\ln \\frac{E_X\\, +\\, P_X}{E_X\\, -\\, P_X}\\,\n          \\, =\\, \\frac{1}{2}\\, \\ln \\frac{1}{y}, \\label{eq:RAP}\n\\end{eqnarray}         \nwhere $\\sqrt{s}$ is the invariant energy of the $pp$ system. We are\nworking in the cm frame of the incoming nucleons. All masses have \nbeen neglected \\cite{FOOT1}.\\\\\n\nIn the limit $y\\rightarrow 1$, the whole available energy is stored in\n$M_X$ which remains then at rest, i.e., $Y_X = 0$. For small values\nof $y$ we have small masses $M_X$ located at large rapidities $Y_X$.\nIn order to regard our process as being trully of the DD type we \nmust assume that all gluons from the target proton participating in \nthe collision (i.e., those emitted from the lower vertex in Fig. 1b) \n{\\it have to form a colour singlet}. Only then a large rapidity gap \nwill form separating the diffracted proton (in the lower part of our \nFig. 1) and the $M_X$ system (in its upper part), which is the\nexperimental requirement defining a diffractive event. Otherwise a \ncolour string would develop, connecting the diffracted proton and the\ndiffractive cluster, and would eventually decay, filling the\nrapidity gap with produced secondaries. In this way we are\neffectively introducing an object resembling closely \n(but by no means identical to) what is known as\nPomeron ($I\\!\\!P$) and therefore in what follows we shall \nuse this notion as a handy abbreviation of notation \\cite{FOOTPOM}.\nThe probability of finding the ``bunch'' of gluons (that forms the\nPomeron) mentioned above in a colour singlet configuration is not\ncomputable at a more fundamental level because the model is simple\nand does not involve colour quantum numbers explicitely. However\nthis probability is represented indirectly in the cross section\nbetween the gluonic bunch and the other proton, the parameter\n$\\sigma$ discussed below in eq. (\\ref{eq:defsig}).\n\nIn our approach the definition of the object $I\\!\\!P$ is essentially \nonly kinematical \\cite{FOOT2}, very much\nin the spirit of those used in all other works which deal with\ndiffractive processes in the parton and\/or string language\n\\cite{DESAI,INNO,LON,COLL}. We shall therefore try to derive the\nwhole $M_X^2$ dependence directly from the IGM. \nOur first step\nnecessary to adapt the standard IGM to DD collisions will be the\nintroduction a kinematical restriction in the formula for the moments\neq.(\\ref{eq:defMOM}) preventing gluons coming from the\ndiffracted proton (and forming our object $I\\!\\!P$) to carry more\nenergy than the one released in the diffractive system. \nTherefore we shall write it as\n\\begin{eqnarray}\n\\langle x^ny^m\\rangle &=& \\int_0^1\\! dx\\,x^n\\, \\int_0^{y_{max}}\\! dy\\, \ny^m\\, \\omega (x,y), \\label{eq:defMOMD}\n\\end{eqnarray}\nwhere $y_{max} = \\frac{M^2_X}{s}$ and the meaning of the spectral function\n$\\omega(x,y)$ remains the same as before.\\\\\n\nWe can now calculate the diffractive mass distribution $M_X$ using\nthe $\\chi(x,y)$ function by simply performing a change of variables (cf.\neq.(\\ref{eq:MASS})),  \n\\begin{eqnarray}\n\\frac{dN}{dM_X^2}\\, &=&\\, \\int_0^1\\! dx\\, \\int_0^1\\! dy\\, \n                       \\chi(x,y)\\, \\delta\\left( M^2_X - sy\\right)\\, \n                       \\Theta \\left( xy - K^2_{min}\\right)\n                       \\nonumber \\\\\n                    &=& \n          \\frac{1}{s}\\, \\int_{_{\\frac{m_0^2 }{M_X^2}}}^1\\! dx\\,\n          \\chi\\left( x,\\frac{M_X^2}{s}\\right), \\label{eq:RES}\n\\end{eqnarray}                       \nIn the IGM \\cite{IGM,DNW93} the distribution $\\chi(x,y)$ is a wide\ngaussian in the variables $x$ and $y$ changing slowly with the energy\n$\\sqrt{s}$. Substituting now eq.(\\ref{eq:CHI}) into eq.(\\ref{eq:RES})\nwe arrive at the following simple  expression for the diffractive\nmass distribution:  \n\\begin{equation}\n\\frac{dN}{dM^2_X}\\, =\\, \\frac{1}{s}\\, \\cdot\\, F(M^2_X,s) \n                   \\cdot H(M_X^2,s) \\label{eq:DIS}\n\\end{equation}           \nwhere \n\\begin{eqnarray}\nF(M^2_X,s) &=&  \\exp \\left[ - \\frac{\\langle x^2\\rangle}{2D_{xy}}\\,\n                 \\left( \\frac{M^2_X}{s} - \\langle y\\rangle \\right) ^2\n                 \\right] \\label{eq:FFF}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\nH(M^2_X,s) &=& \\frac{\\chi_0}{2\\pi \\sqrt{D_{xy}}}\\, \n  \\int^1_{\\!_{\\frac{m_0^2 }{M_X^2}}}\\! dx\n         \\cdot \\label{eq:H}\\\\\n      && \\cdot \\exp \\left\\{ - \\frac{1}{2D_{xy}}\\,\n               \\left[ \\langle y^2\\rangle (x - \\langle x\\rangle )^2 -\n               2\\langle xy\\rangle (x - \\langle x\\rangle) \n               \\left( \\frac{M_X^2}{s} - \\langle y \\rangle \\right)\n               \\right] \\right\\} .\\nonumber \n\\end{eqnarray}\nThe moments  $\\langle q^n\\rangle,~~q=x,y$ and $n=1,2$ are given by\n(\\ref{eq:defMOMD}) and are the only place where dynamical quantities\nlike the gluonic and hadronic cross sections appear in the IGM. At this\npoint we emphasize that we are all the time dealing with a proton-\nproton scattering. However, as was said above, we are in fact\nselecting a special class of events and therefore we must choose the\ncorrect dynamical inputs in the present situation, specially the\ngluon distribution inside the diffracted proton and the hadronic\ncross section $\\sigma$ appearing in $\\omega$. As a first\napproximation we shall take $G^{\\rm I\\!\\!P}(y) = G^p(y) = G(y)$ (cf.\n\\cite{IS}), with $G(x) = p(m+1)\\frac{(1-y)^m}{y}$, with m=5, the same\nalready used by us before \\cite{DNW93}. The amount of\nthe diffracted nucleon momentum, {\\it p}, allocated specifically to the\n$I\\!\\!P$ gluonic cluster and the hadronic cross section $\\sigma$ are\nboth unknown. However, they always appear as a ratio\n($\\frac{p}{\\sigma}$) of parameters in $\\omega$ and different choices\nare possible. Just in order to make contact with the present\nknowledge about the Pomeron, we shall choose\n\\begin{eqnarray}\n\\sigma(s) \\,&=& \\, \\sigma^{\\rm I\\!\\!P p} \\, = \\, a + \nb \\, \\ln \\frac{s}{s_0}\n\\label{eq:defsig}\n\\end{eqnarray}\nwhere $s_0 = 1$ $GeV^2$ and a and b are parameters \nto be fixed from data\nanalysis. As it will be seen, $\\sigma(s)$ turns out to be a very\nslowly varying function of $\\sqrt{s}$ assuming values between 2.6\nand 3.0 mb, which is a well accepted value for the\nPomeron-proton cross section, and $ p \\simeq 0.05 $.\\\\ \n\nBefore performing a full numerical calculation let us estimate\neqs.(\\ref{eq:DIS}) and (\\ref{eq:H}) keeping only the most singular\nparts of the gluonic distributions used (i.e., $G(x)\\simeq 1\/x$) and\ncollecting all other factors in eq.(\\ref{eq:OMEGA}) in a single\nparameter $c$. Let us first assume that the ratio of the cross\nsections $\\frac{\\sigma(xys)}{\\sigma(s)}$ does not depend on $x$ and\n$y$. Neglecting all terms of the order of $\\frac{m_0^2}{s}$ and\n$\\frac{m_0^2}{M_X^2}$ we arrive at the following expressions for the\nmoments calculated in eq.(\\ref{eq:defMOMD}): \n\\begin{eqnarray}\n\\langle x\\rangle \\, &=& 2\\, \\langle x^2\\rangle \\simeq\\,\n                    c\\cdot \\ln \\frac{M_X^2}{m_0^2};  \\label{eq:momX}\\\\\n\\langle y\\rangle \\, &=& 2\\, \\frac{s}{M_X^2}\\, \\langle y^2\\rangle\n                       \\simeq \\, c\\cdot \\frac{M_X^2}{s}\\cdot \n                    \\ln \\frac{M_X^2}{m_0^2}; \\label{eq:newY}\\\\\n\\langle x\\cdot y\\rangle \\, &\\simeq& \\, c\\, \\left(\\frac{M_X^2}{s}\\, -\n           \\frac{m_0^2}{s}\\cdot \\ln\\frac{M_X^2}{m_0^2}\\right).\n            \\label{eq:momXY}\n\\end{eqnarray}           \nNotice that in all cases of interest $\\langle x\\cdot y\\rangle$ is\nmuch smaller than other moments (by a factor\n$\\ln\\frac{M_X^2}{m_0^2}$, at least). It means that $D_{xy}\\, \\simeq\\,\n\\langle x^2\\rangle\\langle y^2\\rangle $ and consequently\n\\begin{eqnarray}\nF(M^2_X,s)\\, &\\simeq&\\, \\exp\\left[ - \\frac{\\left(\\frac{M_X^2}{s}\\, -\\, \n                       \\langle y\\rangle\\right)^2}\n                       {2\\, \\langle y^2\\rangle}\\right] \\nonumber\\\\\n\t     &\\simeq&\\, \\exp\\left[ - \\frac{\\left(1\\, -\\, \n                        c\\cdot \\ln\\frac{M_X^2}{m_0^2}\\right)^2}\n                        {c\\cdot \\ln\\frac{M_X^2}{m_0^2}}\\right]\n\\label{eq:F}\n\\end{eqnarray}\nand \n\\begin{eqnarray}\nH(M_X^2,s)\\, &\\simeq&\\, \\frac{\\chi_0}{2\\pi \\sqrt{D_{xy}}}\\, \n                  \\int^1_{\\!_\\frac{m_0^2}{M_X^2}}\\! dx\n                   \\exp\\left[ - \\frac{(x - \\langle x\\rangle\n                    )^2}{2\\langle x^2\\rangle}\\right] \n            \\simeq\\, {\\rm const}\\, \\cdot\\, \n            \\frac{\\sqrt{\\langle x^2\\rangle}}{\\sqrt{D_{xy}}} \n            \\, =\\, {\\rm const}\\, \\cdot \\, \n            \\frac{1}{\\sqrt{\\langle y^2\\rangle}}\\nonumber\\\\\n             &\\simeq&\\, {\\rm const}\\cdot \\frac{s}{M_X^2\n                \\cdot \\sqrt{c\\cdot \\ln\\frac{M_X^2}{m_0^2}}} \\label{eq:newapprH}\n\\end{eqnarray}\nleading to\n\\begin{eqnarray}\n\\frac{dN}{dM_X^2}\\, &\\simeq&\\, \\frac{1}{s}\\, \\cdot\\, H(M_X^2,s)\\cdot\n\t\t\t       F(M^2_X,s) \\nonumber\\\\\n              &\\simeq&\\, \\frac{\\rm const}{M_X^2}\\cdot \n                        \\frac{1}{\\sqrt{c\\cdot \\ln\\frac{M_X^2}{m_0^2}}} \\cdot \n                        \\exp\\left[ - \\frac{\\left(1\\, -\\, \n                        c\\cdot \\ln\\frac{M_X^2}{m_0^2}\\right)^2}\n                        {c\\cdot \\ln\\frac{M_X^2}{m_0^2}}\\right].\n                        \\label{eq:result}\n\\end{eqnarray}                        \n\nThe expression above is governed by the $\\frac{1}{M^2_X}$ term. The\nother two terms have a weaker dependence on $M^2_X$. They distort the\nmain ($\\frac{1}{M^2_X}$) curve in opposite directions and tend to \ncompensate each other. It is therefore very interesting to note\nthat even before choosing a very detailed form for the gluon \ndistributions and hadronic cross sections we obtain analytically the\ntypical shape of a diffractive spectrum.\n\n\\section{Comparison with experimental data}\n In Fig. 2a we show our diffractive mass spectrum and compare \nit to experimental data from the CERN-ISR \\cite{ISDATA}, which are\nusually parametrized by the form $\\frac{1}{M^2_X}$. These spectra were \ncalculated with expression (\\ref{eq:DIS}) with the same gluon \ndistributions, cross sections and the mass parameter $m_0$ \n($m_0 = 350 $ MeV) used in previous works \\cite{IGM,DNW94,DNW93}.\nAs it can be seen, the agreement between our curves and data is\nreasonable. At large values of $\\frac{M^2_X}{s}$, experimental points\nstart to flatten out, deviating from the $\\frac{1}{M^2_X}$ behaviour.\nThis may be due to the contribution of non-diffractive events. We\nexpect therefore some discrepancy between theory and experiment in\nthis region. At very low values of $\\frac{M^2_X}{s}$ and lower\nenergies $\\sqrt{s}$ our model does not give a good description of\ndata. Here again some discrepancy should be expected because we\nare neglecting all resonance effects. In principle a better agreement\nbetween theory and data could be achieved in this region. It would\nbe enough to choose $ m_0 = 550 $ MeV keeping everything else as\nbefore. The result is shown in Fig. 2b. A discrepancy persists at\nvery low masses at the lowest center of mass energy $\\sqrt{s}$. \nHowever this region is almost beyond the validity domain of the model. \nInstead of changing $m_0$ (which would also require a change\nin our previous works), we prefer to keep its usual value. Given the\nqualitative nature of the present work, we choose to be consistent with\nour previous works in prejudice of the quality of the fits. \n\nFig.3 and 4 show similar comparisons for $\\sqrt{s}=\n546$ and $ 1800  $ GeV respectively. Data are from refs. \\cite{Bozzo}\nand \\cite{DATA}. Again we find reasonable agreement with experiment.\nIn the low $\\frac{M^2_X}{s}$ region at higher $\\sqrt{s}$ the\nagreement is very good. All the curves above were obtained with\n$a = 2.6 $~mb and $b = 0.01$~mb. \\\\\n\nAs a straightforward extension of our calculation we now apply\nexpression (\\ref{eq:DIS}) to the study of diffractive pion-proton and\nkaon-proton scattering. We first consider the cases \n$ p+\\pi \\rightarrow p + X$ and $p + K \\rightarrow p +X$.\n This corresponds to replace the proton by a\npion or a kaon in the upper line of Fig. 1b, everything else remaining\nthe same. We must also substitute the gluon distributions in the\nproton, $G(x)$, by the corresponding gluon distributions in the pion\nand kaon, taken from\n\\cite{MRS}. Here, for simplicity, we take $ G^{\\pi}(x) = G^{K}(x)$.\nThis is supported by an ACCMOR collaboration data analysis\n\\cite{ACC}. We also assume that $\\sigma^{\\rm I\\!\\!P p} = \\sigma^{\\rm\nI\\!\\!P \\pi} = \\sigma^{\\rm I\\!\\!P K} $. The comparison between our\nresults and data from the EHS\/NA22 collaboration \\cite{EHS} is shown\nin Figs. 5a (pions) and 5b (kaons). \nWe may also have diffracted mesons, which undergo\nreactions of the type $\\pi +p \\rightarrow \\pi + X$ and \n$K + p \\rightarrow K + X$. This corresponds to replace the proton by\na pion or a kaon in the lower line of Fig. 1b, substituting also the\ncorresponding gluon distributions. The comparison between our results\nand experimental data \\cite{EHS} is shown in Fig. 6a (pions) and \n6b (kaons). As it can be seen, a good description of data is\nobtained. \\\\\n\nLet us consider now the energy dependence of our results.\nThe CDF collaboration studied single diffractive events and found\nsome energy dependence in the diffractive mass spectrum.  This fact\nis illustrated by writing \n\\begin{equation}\n\\frac{s}{\\sigma_{SD}}\\, \\frac{d\\sigma_{SD}}{dM^2_X} \\propto\n           \\frac{1}{\\left( M^2_X\\right) ^{1+\\epsilon}} \\label{eq:ED}\n\\end{equation}\nwhere the factor $\\epsilon$ which has been reported to be \\cite{GOUL}\n$\\epsilon = 0.121\\pm 0.011$ at $\\sqrt{s} = 546$ GeV and $\\epsilon =\n0.103\\pm 0.017$ at $\\sqrt{s} = 1800$ GeV, respectively. Considering\nthe error bars one might say that this value is just constant (and\nthat there would be no indication of energy dependence), but a real\n(albeit small one) change in $\\epsilon$ is not excluded. Therefore,\nif confirmed, it would mean that the distribution becomes slightly\nbroader.\\\\ \n\nIn the IGM everything is from the beginning energy dependent and so\nshould be the diffractive mass distribution. We start analysing the\nanalytical approximation eqs. (\\ref{eq:DIS},\\ref{eq:F},\n\\ref{eq:newapprH}). In (\\ref{eq:newapprH}) we see that the $s$-dependence\nfactorizes and the function $H(M^2_X)$ has the same shape for all energies, \nthe difference being only a multiplying factor (in numerical calculations\nthis behaviour is slightly violated). In (\\ref{eq:F}) the\n$s$-dependence does not factorize and remains in the moments or, \nequivalently,\nin the variable $c$ . $F(M^2_X)$ is a broad  function with \nmaximum value determined by the moment~$\\langle y \\rangle $ which \nincreases with the\nenergy, making $F$ to \"rotate\" in a way that it becomes higher at \nlower values\nof $M^2_X$ and becomes deeper at larger values of $M^2_X$. When\nwe multiply $H$ (which goes essntially like $1 \/ M^2_X$) by $F$ it becomes\nsteeper. This behaviour of $F$ and $H$ is illustrated in Fig. 7a and 7b \nrespectively, which show the numerical evaluation of eqs. (\\ref{eq:FFF})\nand (\\ref{eq:H}).\nThe result of the numerical evaluation of eq. (\\ref{eq:DIS})\nis presented in Fig. 8. There we show the energy\ndependence of our diffractive mass spectra in proton-proton\nscattering. Fig. 8a shows diffractive mass spectra for $\\sqrt{s}=\n23.5$ GeV (solid lines), $44.6$ GeV (dashed lines) and $62.4$ GeV\n(dotted lines). Fig. 8b shows spectra at $\\sqrt{s}=0.54$ TeV (solid\nlines), $0.9$ TeV (dashed lines) and $1.8$ TeV (dotted lines). Finally\nFig. 8c shows our prediction for the diffractive mass spectrum at\nLHC ($\\sqrt{s}= 14 $ TeV) compared to the Tevatron one. The spectra\nin Fig. 8a and 8b are the same of Figs. 2, 3 and 4. It is interesting to\nnote that the behaviour that we find is not in contradiction with\ndata. In all curves we observe a modest \nnarrowing as the energy increases. \nThis\nsmall effect means that the diffractive mass becomes a smaller fraction\nof the available energy $\\sqrt{s}$. In other words, the \"diffractive\ninelasticity\" decreases with energy and consequently the \"diffracted\nleading particles\" follow a harder $x_F$ spectrum. Physically, in\nthe context of the IGM, this means that the deposited energy is\nincreasing with $\\sqrt{s}$ but it will be mostly released outside the \nphase space region that we are selecting. A measure of the\n\"diffractive inelasticity\" is the quantity $\\xi=\\frac{M^2_X}{s}$.\nMaking a trivial\nchange of variables in eq.(\\ref{eq:DIS}) we can calculate its \naverage value $\\langle \\xi \\rangle$ :\n\\begin{eqnarray}\n\\langle \\xi \\rangle \\left( s \\right) &=& \\int_{\\xi_{min}}^{\\xi_{max}}\nd\\xi\\, \\frac{d N}{d \\xi}\\,  \\xi \\label{eq:XI}\n\\end{eqnarray}                       \nwhere $\\xi_{min}$ ($=\\frac{1.5}{s}$) and $\\xi_{max}$ ($= 0.1$) are the\nsame used in (\\cite{GOUL}) for the purpose of comparison.\nIn Fig. 9 we\nplot $\\langle \\xi \\rangle $ against $\\sqrt{s}$. As it can be seen\n$\\langle \\xi \\rangle$ decreases with $\\sqrt{s}$ not only because\n$\\xi_{min}$ becomes smaller but also because $\\frac{d N}{d \\xi}$\nchanges with the energy, falling faster. This qualitative \nbehaviour of\n$\\langle \\xi \\rangle$ is in agreement with the estimate of the\nsame quantity extracted from cosmic ray data analysis \\cite{COV}.\nAlso shown in Fig. 9 is the quantity \n$\\langle \\xi^{\\varepsilon} \\rangle$ (which\nhas been discussed in \\cite{GOUL} in connection with the energy\ndependence of the single diffractive cross-section) for \n$\\varepsilon= 0.08$~(dashed lines) and~$\\varepsilon= 0.112$\n~(dotted lines).\\\\\n\nThe energy behaviour of $\\frac{d N}{d M^2_X}$ is determined by the\nmoments (\\ref{eq:defMOMD}) and (\\ref{eq:OMEGA}). \nSince $\\sigma_{gg}(s)$ and $G(x)$ are the same as in \nprevious works, being thus fixed, the only source of uncertainty in\nthe $s$-dependence of the results is in the ratio $\\frac{p}{\\sigma}$, \nwhich is the only \nfree parameter in the model. All curves presented above were obtained \nwith the \nchoice (\\ref{eq:defsig}) made for $\\sigma$ and with $p \\simeq 0.05$.\nWe have checked that\nfor a stronger growth of $\\sigma^{\\rm I\\!\\!P p}$  with $\\sqrt{s}$ the\nenergy\nbehaviour of $\\frac{d N}{d M^2_X}$ might become even the opposite of the\none found here, i.e., the diffractive mass distribution would become\nbroader at higher energies. However these strongly $s$-dependent \nparametrizations do not give an acceptable description of  the\nexisting data and were therefore excluded. Considering what was\nsaid above one might think that we can discriminate between\ndifferent Pomeron-proton cross sections and we could use this model\nto extract $\\sigma^{\\rm I\\!\\!P p}$ from data. We stress however that, in\nthis model, only the ratio $\\frac{p}{\\sigma}$ enters effectively in\nthe calculations and it is impossible to completely disentangle\nthese two variables. In this sense, the almost constancy (with\n$\\sqrt{s}$) of $\\sigma^{\\rm I\\!\\!P p}$  may be just an indication\nof some increase of $p$ with $\\sqrt{s}$ in a way that the ratio\n$\\frac{p}{\\sigma}$ \"scales \" with the energy.\\\\ \n\n\\section{Conclusions}\nTo conclude, we have shown that:\n\\begin{itemize}\n\\item[$(i)$] The original IGM\n(\\cite{IGM,DNW94,DNW93}) is reasonably successful in describing\nnon-diffractive events. With only two natural changes, namely\nthe introduction of the kinematical cut-off \n$y_{max}=\\frac{M_X^2}{s}$ and\nthe multiplication of the function $\\omega$ by a constant factor\nreflecting the essentially unknown combination of $I\\!\\!P$-proton\ncross section $\\sigma$ and the amount of gluonic energy-momentum of\nthe diffractive proton allocated to the object $I\\!\\!P$, $p$ (in form of\nthe ratio $\\frac{p}{\\sigma}$), (with all other parameters kept as in \nprevious applications of IGM) it turns out to be also able to provide\na reasonable description of diffractive events \\cite{FOOT4} and their\nenergy dependence. We predict that, at higher energies, a narrowing\nof the $M^2_X$ distribution may be observed.\n\n\n\\item[$(ii)$] As is obvious from Fig. 1b, it provides a detailed\ndescription of the diffractive cluster (of mass $M_X$), {\\it\ndisentangling} it in a natural way into gluonic cluster of mass $M$\ncontaining the majority of produced secondaries (which corresponds to\nCF in the original IGM) and leading jet (inside the diffractive system)\nwith momentum fraction $x_L$ carrying quantum numbers of the\ndiffractively excited projectile. This fact may be very usefull for\nthe cosmic ray applications of the IGM, in particular to studies\nusing DD like those presented in \\cite{CAP,COSMIC} (where such\ndisentanglement seems to be important and so far was introduced in an\n{\\it ad hoc} way only).\n\\end{itemize}\n\n\n\n\\vspace{0.5cm}\n\\underline{Acknowledgements}: This work has been supported by FAPESP,\nCNPQ (Brazil) and KBN (Poland). F.S.N. is deeply indepted to his Polish\ncollegaues from SINS, Warsaw, for the hospitality extended to\nhim during his stay there. We would like to warmly thank R. Covolan and\nY. Hama for many fruitful discussions. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nIn the present paper we prove quantitative symmetry results for overdetermined problems involving the fractional Laplacian in unbounded exterior sets or bounded annular sets. These problems originate from the study of capacity of a set and relative capacity which, in the classical setting, are given by\n\\begin{equation*}\n    {\\rm cap}(\\Omega) := \\inf \\left\\{ \\frac12 \\int_{\\mathbb{R}^n}|\\nabla v|^2 dx\\ :\\ v\\in C^{\\infty}_c(\\mathbb{R}^n),\\,\\,v{|_\\Omega} \\geq 1 \\right\\}\\,,\n\\end{equation*}\nand\n\\begin{equation*}\n{\\rm cap}(\\Omega;D) := \\inf \\left\\{ \\frac{1}{2} \\int_{D} |\\nabla v|^2 dx:\\ v \\in C_c^\\infty (\\Omega),\\ v_{|_D} \\geq 1\\right\\},\n\\end{equation*}\nrespectively; here $D$ and $\\Omega$ are bounded open sets, with $\\overline D \\subset \\Omega \\subset \\mathbb{R}^n$, $n \\geq 3$, and $\\nabla v$  is the gradient of the function $v$.\n\nInstead of the classical notion of capacity, in this paper we consider the capacity in a fractional setting. For a parameter $s \\in (0,1)$, the $\\textit{fractional capacity of order s}$ (or $\\textit{s-capacity}$) of the set $\\Omega$ is defined as follows:\n\\begin{equation} \\label{caps_def}\n    {\\rm cap}_s(\\Omega) := \\inf\\{ [v]_s^2 \\ | \\ v \\in C_c^\\infty (\\mathbb{R}^n),\\ v_{|_\\Omega} \\geq 1\\} \\,,\n\\end{equation}\nwhere $[v]_s$ is the $\\textit{Gagliardo seminorm}$ of $v$ which is defined by\n\\begin{equation*}\n    [v]_s^2 := \\int_{\\mathbb{R}^{2n}} \\frac{|v(x)-v(y)|^2}{|x-y|^{n+2s}} dx dy \\,.\n\\end{equation*}\nAnalogously, one can define the $\\textit{relative fractional capacity of order s}$ of the couple of sets $(\\Omega,D)$ by\n\\begin{equation} \\label{caps_rel_def}\n    {\\rm cap}_s(\\Omega;D) := \\inf\\{ [v]_s^2 \\ | \\ v \\in C_c^\\infty (\\Omega),\\ v_{|_D} \\geq 1\\} \\,.\n\\end{equation}\n\nThe Euler-Lagrange equations associated to \\eqref{caps_def} and \\eqref{caps_rel_def} are both related to the so-called fractional Laplacian of order $s$ (or $s$-Laplacian), which is denoted by $(-\\Delta)^s$ and it is given by  \n\\begin{equation*}\n    (-\\Delta)^s u(x) := c_{n,s} \\, P.V. \\int_{\\mathbb{R}^n} \\frac{u(x) - u(z)}{|x-z|^{n+2s}} dz,\n\\end{equation*} \nfor $u \\in C^{\\infty}_c (\\mathbb{R}^n)$, where\n\\begin{equation} \\label{cns}\n    c_{n,s} = s \\, (1-s) \\, 4s \\pi^{-n\/2} \\frac{\\Gamma(n\/2 + s)}{\\Gamma(2-s)} \n\\end{equation}\n(see for example \\cite{di2012hitchhiker}). \nIt can be proved that ${\\rm cap}_s(\\Omega)$ and  ${\\rm cap}_s(\\Omega;D)$ are uniquely achieved by two functions $u_\\Omega \\in  H^s(\\mathbb{R}^n)$ and $u_{\\Omega,D} \\in  H^s(\\Omega)$,\\footnote{Given an open set $E \\subseteq \\mathbb{R}^n$, we denote by $H^s(E)$ the homogeneous fractional Sobolev space of $E$, which defined as the completion of $C_c^\\infty (E)$ with respect to the Gagliardo seminorm of order $s$.\n} which satisfy\n\\begin{equation}\n\\label{a2s1eq10}\n    \\begin{cases}\n    (-\\Delta)^s u_\\Omega = 0 \\qquad &\\text{in} \\ \\mathbb{R}^n \\setminus \\overline\\Omega,\\\\\n    u_\\Omega = 1 \\qquad &\\text{in} \\ \\overline \\Omega,\\\\\n    u_\\Omega(x) \\to 0 \\qquad &\\text{as} \\ |x| \\to +\\infty\\,,\n    \\end{cases}\n\\end{equation}\nand\n\\begin{equation}\n\\label{a2s1eq12}\n    \\begin{cases}\n    (-\\Delta)^s u_{\\Omega,D} = 0 \\qquad &\\text{in} \\ A:= \\Omega \\setminus \\overline D,\\\\\n    u_{\\Omega,D} = 1 \\qquad &\\text{in} \\ \\overline D,\\\\\n    u_{\\Omega,D} = 0 \\qquad &\\text{in} \\ \\mathbb R^n \\setminus \\Omega \\,,\n    \\end{cases}\n\\end{equation}\nrespectively. The function $u_\\Omega$ is sometimes called the \\textit{$s$-capacitary potential}.\n\nOverdetermined problems for \\eqref{a2s1eq10} and \\eqref{a2s1eq12} have been considered in \\cite{soave2019overdetermined} where the overdetermined condition is given on the normal \\emph{$s$-derivative} at the boundary, which is assumed to be constant in the spirit of Serrin's overdetermined problem. \n\nIn this paper we consider a somehow discrete version of Serrin's overdetermined condition, and we instead assume that the solution is constant on a surface parallel to the boundary.\\footnote{Regarding problem \\eqref{a2s1eq12}, it is more precise to say that the solution is constant on each connected component of the parallel surface, see Theorem \\ref{a2theorem3} below.} In this setting, our main results can be considered as the generalization of the results in \\cite{ciraolo2021symmetry} to exterior and annular domains.\n\nIn order to clearly state our results, we recall that the Minkowski sum of two sets $A$ and $B$ is defined by\n$$\nA+B = \\{ x + y \\ | \\ x \\in A \\ \\, y \\in B \\}.\n$$\n\n\n\nOur first result deals with solutions of problem \\eqref{a2s1eq10} with the overdermining assumption that the solution is constant on a surface parallel to $\\partial \\Omega$.\n\n\n\n\\begin{theorem}\n\\label{a2s1theorem1}\nLet $\\Omega$ be a bounded domain in $\\mathbb{R}^n$. Let $R>0$ and assume that  $G:= \\Omega + B_R$ is such that $\\partial G$ of class $C^1$. Then, there exists a solution $u \\in C^s(\\mathbb{R}^n)$ of \\eqref{a2s1eq10} such that \n\\begin{equation}\\label{a2s1eq16}\n    u = c \\qquad \\text{on} \\ \\partial G\n\\end{equation}\nfor some constant $c$ if and only if $G$ and $\\Omega$ are concentric balls and $u$ is radially symmetric.\n\\end{theorem}\n\nWe will prove Theorem \\ref{a2s1theorem1} by using the method of moving planes. Once symmetry is established, one can investigate the quantitative stability result for Theorem \\ref{a2s1theorem1}. The idea is to assume that the overdetermined condition \\eqref{a2s1eq16} is replaced by a weaker condition which implies that the solution is close to a constant on $\\partial G$. In this direction, it is useful to consider the Lipschitz seminorm $[u]_{\\Gamma}$ of $u$ on $\\Gamma=\\partial G$, which is given by\n\\begin{equation*}\n    [u]_{\\Gamma} := \\sup_{x,y \\in \\Gamma, \\, x \\neq y} \\frac{|u(x) - u(y)|}{|x-y|}\n\\end{equation*}\nand the parameter\n\\begin{equation}\\label{def:outradius-inradius}\n    \\rho (\\Omega) := \\inf \\{ |t - s| \\ | \\ \\exists p \\in \\Omega \\ \\mathrm{such \\ that} \\ B_s(p) \\, \\subset \\Omega \\subset B_t(p) \\} \\,,\n\\end{equation}\nwhich controls how much the set $\\Omega$ differs from a ball (clearly, $\\rho(\\Omega) = 0$ if and only if $\\Omega$ is a ball). \n\nAnother relevant quantity which we need to quantify the stability results is the radius of the touching ball condition. More precisely, given a set $E$ we denote the optimal exterior and interior radii in the touching ball condition by $\\mathfrak{r}_{E}^e$ and $\\mathfrak{r}_{E}^i$, respectively.\n \nHence, our main goal is to obtain quantitative bounds on $\\rho (\\Omega)$ in terms of $[u]_{\\partial G}$, as done in the following theorem.\n\n\\begin{theorem}\n\\label{a2theorem2}\nLet $\\Omega$ be a bounded domain of $\\mathbb{R}^n$ with $\\partial \\Omega$ of class $C^2$. Let $R>0$ and let $G =\\Omega + B_R$ be such that $\\partial G$ is of class $C^2$. Let $u \\in C^s(\\mathbb{R}^n)$ be a solution of \\eqref{a2s1eq10}. Then, we have that\n\\begin{equation}\n\\label{a2s1eq8}\n\\rho (\\Omega) \\leq C_* \\, [u]_{\\partial G}^{\\frac{1}{s +2}} ,\n\\end{equation}\nwith $C_* =C_*(n,s,R,\\mathrm{diam}(\\Omega),|\\Omega|,\\mathfrak{r}_{\\Omega}^e)> 0$, where $\\mathrm{diam}(\\Omega)$ and $|\\Omega|$ denote the diameter and the volume of $\\Omega$, respectively, and $\\mathfrak{r}_{\\Omega}^e$ is the radius of the exterior touching ball condition at $\\Omega$. \n\\end{theorem}\n\n\nIn the second part of the article we consider an overdetermined problem involving \\textit{annular sets}. More precisely, let $D,\\Omega \\subset \\mathbb{R}^n$ be bounded open domains such that $\\overline D \\subset \\Omega$, set\n\\begin{equation} \\label{A_def}\nA:= \\Omega \\setminus \\overline{D} \\,,\n\\end{equation}\nand we consider solutions to \\eqref{a2s1eq12}.\nIt is clear that, since $\\partial \\Omega$ and $\\partial D$ do not touch, we have that \n\\begin{equation} \\label{dbar}\n\\overline{d}:= {\\rm dist} (\\overline{D}, \\mathbb{R}^n \\setminus \\Omega) >0 \\,.\n\\end{equation}\nBy choosing a positive parameter $R < \\overline{d}\/2$ we have that the set \n\\begin{equation} \\label{Gamma_RA}\n\\Gamma_R^A := \\{ x \\in A \\ | \\ {\\rm dist} (x,\\partial A) = R \\}\n\\end{equation}\ncan be written as\n\\begin{equation} \\label{Gamma_union}\n\\Gamma_R^A = \\Gamma_R^D \\cup \\Gamma_R^\\Omega\\,,\n\\end{equation}\nwith \\footnote{Notice that $\\Gamma_R^A = \\partial ( (\\Omega^c + B_R) \\setminus (D+B_R))$.}\n\\begin{align*}\n    \\Gamma_R^D := \\{ x \\in A \\ | \\ {\\rm dist}(x,\\partial D) = R \\},\\\\\n    \\Gamma_R^\\Omega := \\{ x \\in A \\ | \\ {\\rm dist}(x,\\partial \\Omega) = R \\},\n\\end{align*}\nwith $\\Gamma_R^D  \\cap  \\Gamma_R^\\Omega = \\emptyset$. On each of these hypersurfaces we assume that the solution satisfies the overdetermined condition\n\\begin{equation}\n\\label{a2s1eq15}\n    \\begin{aligned}\n    u = \\alpha \\qquad &on \\ \\Gamma_R^D,\\\\\n    u = \\beta \\qquad &on \\ \\Gamma_R^\\Omega,\n\\end{aligned}\n\\end{equation}\nwhere $\\alpha$ and $\\beta$ are two positive constants.\n\n\\bigskip\n\nWe have the following symmetry result. \n\n\\begin{theorem}\n\\label{a2theorem3}\nLet $A$ and $\\Gamma^A_R$ be given by \\eqref{A_def} and \\eqref{Gamma_RA}, respectively, where $R$ is such that $\\Gamma^A_R$ is of class $C^1$. \n\nLet $u \\in C^s(A)$ be a solution of \\eqref{a2s1eq12} satisfying the overdetermined conditions \\eqref{a2s1eq15}. \nThen, $D$ and $\\Omega$ are concentric balls and $u$ is radially symmetric.\n\\end{theorem}\n\n\nNow we describe the quantitative stability result that we obtain for Theorem \\ref{a2theorem3}. In this case, we replace the overdetermined condition \\eqref{a2s1eq12} by assuming that the solution has small Lipschitz seminorm on each connected component of $\\Gamma_R^A$. For this reason we define the following deficit \n\\begin{equation} \\label{deficit_A}\n    \\mathrm{def}_A(u):= \\max \\{ [u]_{\\Gamma_R^D}, [u]_{\\Gamma_R^\\Omega} \\},\n\\end{equation}\nand we have the following result. \n\n\n\\begin{theorem}\n\\label{a2theorem4}\nLet $A$ and $\\Gamma^A_R$ be given by \\eqref{A_def} and \\eqref{Gamma_RA}, respectively, and assume that $\\partial A$ and $\\Gamma^A_R$ are of class $C^2$.\n\n\nLet $u \\in C^s(\\mathbb{R}^n)$ be a solution of \\eqref{a2s1eq12}. Then\n\\begin{equation}\n\\label{a2s1eq18}\n\\rho(D) + \\rho(\\Omega) \\leq C_* \\mathrm{def}_A(u)^{\\frac{1}{s +2}} \\,\n\\end{equation}\nwith $\\rho$ given by \\eqref{def:outradius-inradius} and $C_*=C_*(n,s,R,{\\rm diam}(\\Omega), |\\Omega|, |D|, \\mathfrak{r}^e_{D}, \\mathfrak{r}^i_{\\Omega}) > 0$, where $\\mathfrak{r}^e_{D}$ and $\\mathfrak{r}^i_{\\Omega}$ are the radius of the uniform exterior touching ball to $D$ and of the interior touching ball to $\\Omega$, respectively. \n\\end{theorem}\n\n\nTheorems \\ref{a2theorem2} and \\ref{a2theorem4} are the main results of this paper, and they are obtained by using a quantitative approach to the method of moving planes, which was originally developed in \\cite{aftalion1999approximate} (see also \\cite{CiraoloVezzoni}, \\cite{ciraolo2016solutions}, \\cite{CMS}, \\cite{ciraolo2018rigidity}, \\cite{ciraolo2021symmetry}). This approach presents many differences when applied in a classical local settings and in a fractional framework, and in this paper we prefer to tackle fractional problems. We mention that all the symmetry and quantitative symmetry results in this paper have their classical local counterpart, which can be still attacked by using the method of moving planes and it will be considered in a future work. \n\nWe finally notice that, in Theorems \\ref{a2s1theorem1}-\\ref{a2theorem4}, we assumed that $\\Omega$ and $D$ are domains and then they are connected. The connectedness assumption is not necessary and it can be easily removed, and hence our results can be extended in that setting. However, this has a cost in managing the notation and it would worsen the presentation and clarity of the paper. For this reason, we preferred to assume that $\\Omega$ and $D$ are connected.\n\n\n\\medskip\n\nThe paper is organized as follows. In Section 2 we present some preliminary notions and results, including a weak maximum principle for $s$-harmonic functions in an unbounded domain. Section 3 is devoted to the results for exterior sets and includes the standard machinery for the method of moving planes. In Section 4 we consider the problems involving annular domains.\n\n\n\n\n\\subsection*{Acknowledgements}\nThe authors have been partially supported by the ``Gruppo Nazionale per l'Analisi Matematica, la Probabilit\\`a e le loro Applicazioni'' (GNAMPA) of the ``Istituto Nazionale di Alta Matematica'' (INdAM, Italy). \n\n\n\n\n\n\\section{Preliminaries and notation}\n\nIn this section we introduce some notation and recall some results which will be useful in the rest of the paper. \n\n\nWe recall that given two functions $u,v \\in H^{s}(\\mathbb{R}^n)$ the $\\textit{Gagliardo seminorm}$ of $u$ is defined as\n\\begin{equation*}\n    [u]_s^2 := \\frac{c_{n,s}}{2} \\int_{\\mathbb{R}^n} \\int_{\\mathbb{R}^n} \\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}} dx dy,\n\\end{equation*}\nand the scalar product in $H^s(\\mathbb{R}^n)$ between $u$ and $v$ is defined as\n\\begin{equation*}\n    \\mathcal{E}(u,v) = \\frac{c_{n,s}}{2} \\bigg( \\int_{\\mathbb{R}^n}\\int_{\\mathbb{R}^n}\\frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{n+2s}} dx dy \\bigg)^{1\/2} \\,,\n\\end{equation*}\nwhere $c_{n,s}$ is given by \\eqref{cns}.\n\n\n\n\nIn order to write a Hopf's boundary point lemma in a quantitative form, it is useful to consider the solution $\\psi_{B_r(x_0)}$ to the fractional torsion problem in a ball of radius $r>0$ centered $x_0$, i.e. $\\psi_{B_r(x_0)}$ satisfies\n\\begin{equation} \\label{torsion_pb} \n\\begin{cases}\n(-\\Delta)^s \\psi_{B_r(x_0)} = 1 & \\text{ in } B_r(x_0) \\\\\n\\psi_{B_r(x_0)} = 0 & \\text{ in } \\mathbb{R}^n \\setminus B_r(x_0)\\,,\n\\end{cases}\n\\end{equation}\nand it is given by \n\\begin{equation} \\label{psi_def}\n\\psi_{B_r(x_0)}(x):= \\gamma_{n,s} (r^2 - |x-x_0|^2)^s_+\n\\end{equation}\nfor any $x \\in \\mathbb{R}^n$, where $\\gamma_{n,s}$ is a constant depending only on $n$ and $s$.\n\n\\begin{lemma}\n\\label{a2s3theorem1}\nLet $\\Omega \\subset \\mathbb{R}^n$ be an open set and let $u \\in C^s (\\mathbb{R}^n)$ be a solution of\n\\begin{equation*}\n    \\begin{cases}\n    (-\\Delta)^s u \\geq 0 \\quad & \\text{in} \\ \\Omega,\\\\\n    u \\geq 0 \\quad &\\text{in} \\ \\mathbb{R}^n \\setminus \\Omega \\,.\n    \\end{cases}\n\\end{equation*}\nLet $x_0 \\in \\Omega$ and $r>0$ be such that $B_r(x_0) \\subseteq \\Omega$. Let $K\\subset \\mathbb{R}^n$ be a compact set such that \n$$\n|K| > 0 \\,, \\quad {\\rm dist}(K,B_r(x_0)) > 0 \\,,\\quad  \\inf_K u > 0 \\,.\n$$ \nThen \\begin{equation*}\n    u \\geq C_H \\, \\psi_{B_r(x_0)} \\qquad \\text{in} \\  \\, B_r(x_0)\\,,\n\\end{equation*}\nwhere \n\\begin{equation} \\label{CHopf}\n   C_H := c_{n,s} \\frac{|K| \\, \\inf_K u}{(2r + {\\rm dist}(K,B_r(x_0)) + {\\rm diam}(K))^{n+2s}} \\,,\n\\end{equation}\nwith $c_{n,s}$ and $\\psi_{B_r(x_0)}$ given by \\eqref{cns} and \\eqref{psi_def}, respectively.\n\\end{lemma}\n\nLemma \\ref{a2s3theorem1} was already proved in \\cite{greco2016hopf} and \\cite{ros2014dirichlet}. Here, inspired by  \\cite{fall2015overdetermined} and \\cite{ciraolo2021symmetry}, we give a proof which allows us to explicitly write the constant $C_H$ given by \\eqref{CHopf}, and to show its dependency on the parameters which are relevant in our problem. This will be useful when we will prove the quantitative results.\n\n\\begin{proof}[Proof of Lemma \\ref{a2s3theorem1}] \nWe consider the barrier function \n$$\nw(x) := \\psi_B (x) + \\delta \\, \\chi_K (x) \\,,\n$$ \nwhere $B=B_r(x_0)$, $\\chi_K$ is the characteristic function of $K \\subset \\mathbb{R}^n$ and $\\delta > 0$ is a constant that will be chosen later.\n\nLet $\\varphi \\in H^s_0 (\\Omega)$ be a nonnegative test function. We have\n\\begin{align*}\n    \\mathcal{E} (w,\\varphi) &= \\mathcal{E} (\\psi_B, \\varphi) + \\delta \\, \\, \\mathcal{E} (\\chi_K, \\varphi) = \\int_B \\varphi - \\delta \\, c_{n,s} \\, \\int_K \\int_B \\frac{\\varphi(y)}{|x - y|^{n+2s}} \\, dy \\, dx \\leq \\\\\n    &\\leq (1 -\\delta \\, C) \\int_B \\varphi,\n\\end{align*}\nwhich is less or equal than zero if we choose $\\delta \\geq C^{-1}$ with\n\\begin{equation*}\n    C  = c_{n,s} \\, |K| \\, \\inf_{x \\in K, y \\in B} \\frac{1}{|x - y|^{n+2s}}. \n\\end{equation*}\n\nBy setting \n$$\n\\tau := \\inf_K u \/ \\delta = C \\, \\inf_K u\n$$ \nand applying the weak maximum principle for $s$-harmonic functions to \n$$\nv := u - \\tau \\, w \\,,\n$$ \nwe get that\n\\begin{equation*}\n    u \\geq c_{n,s} \\frac{|K| \\, \\inf_K u}{({\\rm diam}(B) + {\\rm dist}(K,B) + {\\rm diam}(K))^{n+2s}} \\, \\psi_B \\quad \\text{in} \\, B,\n\\end{equation*}\nwhich is the desired result. \n\\end{proof}\n\n\n \n\n\nSince our approach is based on the method of moving planes, a particular attention must be given to antisymmetric $s$-harmonic functions. More precisely, we will have to consider functions which are antisymmetric with respect to a hyperplane which can be chosen to be $\\{x_1 = 0\\}$ (up to a translation and rotation). \n\nIn order to list these results, we need to introduce some notation: we set $H^+ := \\{x_1 > 0\\}$, $H^- := \\{ x_1 < 0 \\}$ and $T\\coloneqq \\{ x_1 = 0 \\}$. Let \n$$\n\\mathcal Q: \\mathbb{R}^n \\to \\mathbb{R}^n\\,, \\ \\  y \\mapsto y' = (-y_1,y_2,\\dots,y_n) \\,,\n$$ \nbe the reflection with respect to $T$ and, for a given set $E$ we call $E^+:= E \\cap H^+$ and $E^-:= E \\cap H^-$.\n\n\nThe first result is a weak maximum principle for $s$-harmonic antisymmetric functions, which is stated in \\cite[Proposition 3.1]{fall2015overdetermined} on domains that are bounded, although for homogeneous equations this condition is not needed. We report this proposition here and we sketch a proof.\n\n\n\\begin{lemma}[Weak maximum principle for antisymmetric functions]\n\\label{a2s1lemma1}\nLet $\\Omega \\subset \\mathbb{R}^n$ be a compact set and let $u \\in H^s(\\mathbb{R}^n)$ be an antisymmetric (w.r.t. $T=\\{ x_1 = 0 \\}$) solution of\n\\begin{equation*}\n    \\begin{cases}\n    (-\\Delta)^s u = 0 \\qquad &\\text{in} \\ \\Omega^c,\\\\\n    u \\geq 0 \\qquad &\\text{in} \\ \\Omega^+.\n    \\end{cases}\n\\end{equation*}\nThen, $u \\geq 0$ a.e. in $H^+$.\n\\end{lemma}\n\n\\begin{proof}\nSince $u$ is $s$-harmonic in $\\Omega^c$ then for every $ \\varphi \\in H^s_0(\\Omega^c)$ we have\n\\begin{equation}\n    \\mathcal{E}(u,\\varphi) = 0.\n\\end{equation}\n\nLet $\\varphi = u_- \\chi_{H^+} \\in H^s_0(\\Omega^c)$, where $\\chi_{H^+}$ is the characteristic function of $H^+$. Following the same computations as in \\cite[Proposition 3.1]{fall2015overdetermined} we get \n\\begin{equation}\n    0 \\leq \\mathcal{E}(u,\\varphi) \\leq - \\mathcal{E}(\\varphi,\\varphi) = - [\\varphi]_s^2 \\,,\n\\end{equation}\nwhich immediately implies that $\\varphi=0$ a.e. and hence $u_-=0$ a.e. in $H^+$.\n\\end{proof}\n\nAn analogous weak maximum principle holds for nonnegative functions in $H^s(\\mathbb{R}^n)$. More precisely we have\n\n\\begin{lemma}[Weak maximum principle]\n\\label{a2s1lemma1bis}\nLet $\\Omega \\subset \\mathbb{R}^n$ be a compact set and let $u \\in H^s(\\mathbb{R}^n)$ be a solution of\n\\begin{equation*}\n    \\begin{cases}\n    (-\\Delta)^s u = 0 \\qquad &\\text{in} \\ \\Omega^c,\\\\\n    u \\geq 0 \\qquad &\\text{in} \\ \\Omega.\n    \\end{cases}\n\\end{equation*}\nThen, $u \\geq 0$ a.e. in $\\mathbb{R}^n$.\n\\end{lemma}\n\n\\begin{proof}\nThe proof is analogous to the one of Lemma \\ref{a2s1lemma1}, since it is enough to consider $ \\varphi = u_-$.\n\\end{proof}\n \nAs an immediate consequence we have the following comparison principle for $s$-capacitary functions.  \n\n\\begin{corollary} \\label{corol_monot}\nLet $E\\subset F \\subset \\mathbb{R}^n$ be open bounded domains, and let $u_E$ and $u_F$ be the corresponding capacitary functions, i.e. the solutions to \\eqref{a2s1eq10} for $\\Omega=E$ and $\\Omega=F$, respectively. Then we have \n\\begin{equation} \\label{monotonicity}\nu_E \\leq u_F\n\\end{equation}\nin $\\mathbb R^n$.\n\\end{corollary}\n\n\\begin{proof}\nSince $u_E$ is a $s$-capacitary function, from \\cite[Lemmas 2.6 and 2.7]{warma2015fractional} we have that $0 \\leq u_E \\leq 1$ in $\\mathbb{R}^n \\setminus E$. Then, by applying Lemma \\ref{a2s1lemma1bis} to $v:= u_F - u_E$ we obtain the result.\n\\end{proof}\n\n\nFrom Lemma \\ref{a2s1lemma1}, we can also recover a quantitative version of the Hopf lemma for antisymmetric functions as proved in \\cite{ciraolo2021symmetry}, that we recall below.\n\n\\begin{lemma} [Lemma 4.1 in \\cite{ciraolo2021symmetry}]\n\\label{s3lemma2}\nLet $\\Omega$ be an open set in $H^-$ and $B \\subset \\Omega$ a ball of radius $R > 0$ such that $\\mathrm{dist}(B,H^+) > 0$. Let $v \\in C^s(\\Omega)$ be antisymmetric and a solution of\n\\begin{equation*}\n    \\begin{cases}\n    (-\\Delta)^s v \\ge 0 \\quad & \\text{in} \\ \\Omega,\\\\\n    v \\geq 0 \\quad & \\text{in} \\ H^-.\n    \\end{cases}\n\\end{equation*}\nLet $K \\subset H^-$ be a set of positive measure such that $\\overline{K} \\subset (H^- \\setminus \\overline{B})$ and $\\inf_K v > 0$. Then we have that\n\\begin{equation}\n\\label{s3eq19}\nv \\geq C \\big[  \\mathrm{dist} (K,H^+) \\, |K| \\, \\inf_K v \\big] \\psi_B \\quad \\text{in} \\ B,\n\\end{equation}\nwith\n\\begin{equation*}\n    C:=  \\frac{ 2 (n+2s) \\, C(n,s) \\, \\mathrm{dist} (B,H^+)^{n+2s+1}}{(\\mathrm{dist}(B,H^+)^{n+2s}+\\gamma_{n,s} \\, C(n,s) \\, |B| \\, R^{2s}) \\, ( \\mathrm{diam} (B) + \\mathrm{diam}(K) + \\mathrm{dist}(\\mathcal{Q}(K),B))^{n+2s+2}}.\n\\end{equation*}\n\\end{lemma}\n\nIt is clear that Lemma \\ref{s3lemma2} provides a quantitative version of the strong maximum principle for antisymmetric $s$-harmonic functions, which still holds when $\\Omega$ is not bounded, as already noted in \\cite[Proposition 2.1]{soave2019overdetermined}. \n\n\\begin{lemma}[Strong maximum principle for antisymmetric functions] \\label{lemma_SMP}\nLet $\\Omega$ be an open set with $\\Omega \\subset H^-$ and let $v \\in C^s(\\Omega)$ be antisymmetric and a solution of\n\\begin{equation*}\n    \\begin{cases}\n    (-\\Delta)^s v \\ge 0 \\quad & \\text{in} \\ \\Omega,\\\\\n    v \\geq 0 \\quad & \\text{in} \\ H^-.\n    \\end{cases}\n\\end{equation*}\nThen, either $v > 0$ in $\\Omega$ or $v \\equiv 0$ in $\\Omega$.\n\\end{lemma}\n\n\\begin{proof}\nFrom the weak maximum principle in Lemma \\ref{a2s1lemma1} we have that $v \\geq 0$ in $\\Omega$. Now assume there exists $x_0 \\in \\Omega$ such that $v(x_0) = 0$ and choose a ball $B$ centered in $x_0$ and such that $\\overline{B} \\subset \\Omega$. Let $K \\subset \\Omega$ be a compact set  such that ${\\rm dist}(B,K) > 0$ and $|K| > 0$. If we furthermore choose $B$ and $K$ such that $\\inf_K v > 0$, by applying Lemma \\ref{s3lemma2} we have\n\\begin{equation*}\nv \\geq C \\big[  {\\rm dist} (K,H^+) \\, |K| \\, \\inf_K v \\big] \\psi_B \\quad in \\ B,\n\\end{equation*}\nand in particular $v(x_0) > 0$, which is a contradiction.\n\\end{proof}\n\n\nAnother tool from \\cite{ciraolo2021symmetry} that we will need in our proof is the boundary Harnack inequality for $s$-harmonic antisymmetric functions that we report here for clarity. \n\n\\begin{lemma}[Lemma 2.1 in \\cite{ciraolo2021symmetry}]\n\\label{a2s1lemma5}\nLet $u\\in C^2(B_R) \\cap C(\\mathbb{R}^n)$ be a solution of\n\\begin{equation*}\n    \\begin{cases}\n    (-\\Delta)^s u = 0 \\quad & \\text{in }  B_R,\\\\\n    u(x') = - u(x) \\quad &\\text{for  every }  x \\in \\mathbb{R}^n,\\\\\n    u \\geq 0 \\quad &  \\text{in }  H^+.\n    \\end{cases}\n\\end{equation*}\nThere exists a constant $K > 1$ only depending on $n$ and $s$ such that, for every $z \\in B_{R\/2}^+$ and for every $x \\in B_{R\/4}(z) \\cap B_R^+$ we have\n\\begin{equation}\n\\label{s3eq2}\n    \\frac{1}{K} \\frac{u(z)}{z_1} \\leq \\frac{u(x)}{x_1} \\leq K \\frac{u(z)}{z_1}.\n\\end{equation}\n\\end{lemma}\n\n\n\n\n\n\n\n\n\n\\section{Exterior sets}\nIn this section we consider the exterior overdetermined problem and prove Theorems \\ref{a2s1theorem1} and \\ref{a2theorem2}.\n\n\n\\subsection{The method of moving planes: notation}  \\label{subsect_MMP}\nWe introduce some notation in order to exploit the moving planes method. Given $e \\in \\mathbb{S}^{n-1}$, a set $E \\subset \\mathbb{R}^n$ and $\\lambda \\in \\mathbb{R}$, we define\n\\begin{align*}\n    &T_\\lambda = T_\\lambda^e = \\{ x \\in \\mathbb{R}^n \\, | \\, x \\cdot e = \\lambda \\} & &\\textrm{a hyperplane orthogonal to } e,\\\\\n    &H_\\lambda = H_\\lambda^e = \\{ x \\in \\mathbb{R}^n \\, | \\, x \\cdot e > \\lambda \\} & &\\textrm{the ``positive'' half space with respect to } T_\\lambda \\\\\n    &E_\\lambda = E \\cap H_\\lambda & &\\textrm{the ``positive'' cap of } E,\\\\\n    &x_\\lambda' = x -2(x\\cdot e - \\lambda) \\, e & &\\textrm{the reflection of } x \\textrm{ with respect to } T_\\lambda,\\\\\n    &\\mathcal{Q}= \\mathcal{Q}_\\lambda^e : \\mathbb{R}^n \\to \\mathbb{R}^n, x \\mapsto x_\\lambda' & &\\textrm{the reflection with respect to } T_\\lambda.\n\\end{align*}\n\nIf $E \\subset \\mathbb{R}^n$ is an open bounded set with boundary of class $C^1$ then we define \n$$\n\\Lambda_e := \\sup \\{ x \\cdot e \\, | \\, x \\in E \\}\n$$ \nand\n\\begin{equation*}\n    \\lambda_e = \\inf \\{ \\lambda \\in \\mathbb{R} \\, | \\, \\mathcal{Q}(E_{\\Tilde{\\lambda}}) \\subset E, \\textrm{for all} \\, \\Tilde{\\lambda} \\in (\\lambda, \\Lambda_e) \\}.\n\\end{equation*}\n\n{F}rom this point on, given a direction $e \\in \\mathbb{S}^{n-1}$, we will refer to $T_{\\lambda_e} = T^e$ and $E_{\\lambda_e} = \\widehat{E}$ as the \\textit{critical hyperplane} and the \\textit{critical cap} with respect to $e$, respectively, and will call $\\lambda_e$ the \\textit{critical value} in the direction $e$. We now recall from \\cite{serrin1971symmetry} that, for any given direction $e$, at least one of the following two conditions holds:\\\\\n\n\\textbf{Case 1} - The boundary of the reflected cap $\\mathcal{Q}^e(\\widehat{E})$ becomes internally tangent to the boundary of $E$ at some point $P \\not \\in T^e$;\\\\\n\n\\textbf{Case 2} - the critical hyperplane $T^e$ becomes orthogonal to the boundary of $E$ at some point $Q \\in T^e$.\n\n\n\n\\subsection{The symmetry result}\n\nWe start with the symmetry result given in Theorem \\ref{a2s1theorem1}. \n\n\\begin{proof}[Proof of Theorem \\ref{a2s1theorem1}]\nLet $e \\in \\mathbb{S}^{n-1}$ be a fixed direction. We recall that we are considering a solution $u \\in C^s(\\mathbb{R}^n)$ of \\eqref{a2s1eq10} satisfying \\eqref{a2s1eq16} and that $G=\\Omega + B_R$, with $\\partial G$ of class $C^1$. \n\nWe apply the method of moving planes described in Subsection \\ref{subsect_MMP} by letting $E=G$. Without loss of generality, we can assume that $\\lambda_e = 0$ (that is, the critical hyperplane $T$ goes through the origin), and we simplify the notation by setting $H^- := \\{ x \\cdot e < 0 \\}$, $\\Omega^- := H^- \\cap \\Omega$ and considering\n\\begin{equation*}\n    v(x):= u(x) - u(\\mathcal Q(x)) \\quad \\text{for } \\ x \\in \\mathbb{R}^n,\n\\end{equation*}\nwhere $\\mathcal Q: \\mathbb{R}^n \\to \\mathbb{R}^n, x \\mapsto x'$ is the reflection with respect to $T$. We have\n\\begin{equation*}\n    \\begin{cases}\n    (-\\Delta)^s v = 0 \\quad &\\text{in} \\ H^- \\setminus \\Omega^- \\\\\n    v \\geq 0 \\quad &\\text{in} \\ \\Omega^-\\\\\n    v(\\mathcal Q(x)) = - v (x) \\quad &\\text{for every } x \\in \\mathbb{R}^n.\n\\end{cases}\n\\end{equation*}\n\n\nBy using Lemma \\ref{a2s1lemma1} we know that $v \\geq 0$ in $H^-$ and then Lemma \\ref{lemma_SMP} tells us that either $v > 0$ in $H^- \\setminus \\Omega^-$ or $v \\equiv 0$ in $\\mathbb{R}^n$. Now we show that if we assume that $v > 0$ in $H^-\\setminus \\Omega^-$ then we obtain a contradiction.\n\n\\textbf{Case 1} - Let $P$ be a critical point on $\\partial G$. Since both $P$ and its reflection $P'$ belong to $\\partial G$ and \\eqref{a2s1eq16} holds, we immediately get\n\\begin{equation*}\n    v(P) = u(P) - u(P') = 0,\n\\end{equation*}\nwhich is a contradiction.\\\\\n\n\\textbf{Case 2} - In this case $e$ is tangent to $\\partial G$ at a point $Q \\in \\partial G$, and therefore we have that $\\partial_e v (Q) = 0$. On the other hand, since $Q$ is far away from the boundary $\\partial \\Omega$, we can use Lemma \\ref{a2s1lemma5} to show that $\\partial_e v (Q) < 0$, which is a contradiction.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\subsection{Almost symmetry in one direction}\nNow we consider the quantitative stability result and prove Theorem \\ref{a2theorem2}. This will be done in two subsequent steps: we first prove the quantitative stability estimate in one direction and then, in the proof of Theorem \\ref{a2theorem2} we will sketch a general idea of how to use the result in one direction to obtain the final quantitative estimate; the proof can be found in details in Section 6 of \\cite{ciraolo2021symmetry}. \n\nWe start by proving a preliminary result which gives the behaviour of the solution to \\eqref{a2s1eq10} close to the boundary. \n\n\\begin{lemma} \\label{lemma_conticino}\nUnder the assumptions of Theorem \\ref{a2theorem2}, let $u$ be a solution of \\eqref{a2s1eq10} and let $v:= 1- u$. For any $r \\leq \\mathfrak{r}_{\\Omega}^e$ we have\n\\begin{equation}\n\\label{a2s3eq35}\n     v(x) \\geq  C_{cap} \\, ({\\rm dist} (x,\\partial \\Omega))^s \\quad \\text{in} \\ (\\Omega + B_r)\\setminus \\Omega,\n\\end{equation}\nwhere\n\\begin{equation*}\n    C_{cap} := \\frac{c_{n,s} \\, \\gamma_{n,s} \\, \\omega_{n}}{4} \\, \\frac{r^{n+s}}{(2r + r_0 \\, {\\rm diam} (\\Omega))^{n+2s}}\n\\end{equation*}\nand $r_0 > 0$ is a constant depending on $n$ and $s$.\n\\end{lemma}\n\n\\begin{proof}\nWithout loss of generality, we can assume that the origin $O$ is contained in $\\Omega$ and consider the $s$-capacitary solution $\\tilde{u}$ of the ball $B_{{\\rm diam}(\\Omega)}$ centered at the origin and of radius ${\\rm diam}(\\Omega)$:  \n\\begin{equation*}\n    \\begin{cases}\n    (-\\Delta)^s \\tilde{u} = 0 \\quad &\\text{in} \\ \\mathbb{R}^n \\setminus B_{{\\rm diam}(\\Omega)},\\\\\n    \\tilde{u} = 1 \\quad &\\text{in} \\ B_{{\\rm diam}(\\Omega)},\\\\\n    \\tilde{u}(x) \\to 0 \\quad &\\text{as} \\ |x| \\to + \\infty.\n    \\end{cases}\n\\end{equation*}\nSince $0 \\leq \\tilde{u} \\leq 1$ and $\\tilde{u}$ is radial, non-increasing and continuous (see for instance \\cite[Theorem 1.10]{soave2019overdetermined}, there exists a radius $\\tilde{R} = \\tilde{R}({\\rm diam} (\\Omega)) > 0$ such that\n\\begin{equation*}\n    \\tilde{u} < 1\/2 \\quad \\text{in} \\quad \\mathbb{R}^n \\setminus B_{\\tilde{R}}.\n\\end{equation*}\nMoreover, from Corollary \\ref{corol_monot} we have that $\\tilde{u} \\geq u$ in the whole space. From this we get that\n\\begin{equation}\n    v = 1 - u \\geq 1 - \\tilde{u} \\geq 1\/2 \\quad \\text{in} \\ \\mathbb{R}^n \\setminus B_{\\tilde{R}}.\n\\end{equation}\n\nWe now choose $K=\\overline{B_{\\mathfrak{r}^e_\\Omega}((\\tilde{R}+\\mathfrak{r}^e_\\Omega) \\,e_1)}$. For $x_0 \\in \\partial \\Omega$, we now apply Lemma \\ref{a2s3theorem1} to $v$ with $B = B_{\\mathfrak{r}^e_\\Omega}(x_0)$ and $K$ and get\n\\begin{equation*}\n    v(x) \\geq \\frac{c_{n,s} \\, \\gamma_{n,s} \\, \\omega_n}{2} \\, \\frac{R^{n+s}}{(4R + {\\rm dist}(K,B))^{n+2s}} \\, (R - |x|)^s \\quad \\text{in} \\ B.\n\\end{equation*}\n\nWe now repeat the same argument on the whole boundary $\\partial \\Omega$ by keeping each time the same fixed set $K$. We notice that in every case we have ${\\rm dist}(K,B) \\leq 2\\tilde{R}$, and by using the previous inequality we obtain \\eqref{a2s3eq35}, where the constant $C_{cap} > 0$ can be written as \n\\begin{equation*}\n   C_{cap} := \\frac{c_{n,s} \\, \\gamma_{n,s} \\, \\omega_n}{4} \\, \\frac{R^{n+s}}{(2R + \\tilde{R})^{n+2s}}.\n\\end{equation*}\nIn order to complete the proof, we show how $\\tilde R$ depends on ${\\rm diam}(\\Omega)$. We consider the solution $u_{B_1}$ to the capacitary problem \\eqref{a2s1eq10} with $\\Omega = B_1$, and we set\n$$\nr_0=\\inf\\{ \\, |x| \\ | \\ u_B(x)< 1\/2 \\} \\,. \n$$ \nBy scaling properties, it is clear that $\\tilde R = r_0 \\, {\\rm diam} (\\Omega)$. This completes the proof.\n\\end{proof}\n\nWith this result at hand, we can prove a quantitative estimate which involves the measure of $\\Omega^- \\setminus \\mathcal Q(\\widehat{\\Omega})$.\n\n\\medskip\n\nWe fix a direction $e \\in \\mathbb{S}^{n-1}$. Without loss of generality, we can assume that $e=e_1$ and that the associated critical hyperplane is $T = \\{x_1 = 0\\}$, with $\\mathcal Q: \\mathbb{R}^n \\to \\mathbb{R}^n, x \\mapsto x'$ the reflection with respect to $T$. For the proof of the next lemma we will use the following notation: we set for $t \\geq 0$\n\\begin{equation*}\n    \\Omega_t \\coloneqq \\Omega + B_t(0), \\quad \\widehat{\\Omega_t} \\coloneqq \\Omega_t \\cap H^+, \\quad \n    \\Omega_t^- \\coloneqq \\Omega_t\\cap H^- \\quad U_t \\coloneqq \\mathcal Q(\\widehat{\\Omega_t}).\n\\end{equation*}\nNote that $G = \\Omega_R$.\n\n\\begin{lemma}\n\\label{a2s3lemma1}\nGiven $P \\in \\overline{U_R}$ with $B=B_{R\/8} (P)$ such that $\\mathrm{dist}(B,\\partial U_0) \\ge R\/8$, for $\\delta > 0$, we have that\n\\begin{equation}\n\\label{a2s3eq39}\n|\\Omega^- \\setminus \\mathcal Q(\\widehat{\\Omega}) \\ | \\ \\leq \\Tilde{C} \\, (\\delta^{-(1+s)} v(P) + \\delta),\n\\end{equation}\nwhere $\\Tilde{C} > 0$ is a constant depending only on $n$, $s$, $R$, $\\mathfrak{r}^e_\\Omega$ and $\\mathrm{diam}(\\Omega)$.\n\\end{lemma}\n\n\\begin{proof}\nWe set $K_\\delta := (\\Omega^- \\setminus \\mathcal Q(\\widehat{\\Omega})) \\setminus (E_\\delta \\cup F_\\delta)$,\nwhere\n$$\nE_\\delta := \\mathcal Q (A_\\delta) \\cap (\\Omega^- \\setminus \\mathcal Q(\\widehat{\\Omega})) \\quad \\text{ with } A_\\delta := \\{ x \\in \\Omega^c \\ | \\ {\\rm dist} (x, \\partial \\Omega) < \\delta  \\},  \n$$\n$$\nF_\\delta:= \\{ x \\in \\Omega^- \\setminus \\mathcal{Q}(\\widehat{\\Omega}) \\, | \\, \\mathrm{dist}(x, T) < \\delta  \\}.\n$$\n\nUsing Lemma \\ref{s3lemma2} with $B := B_{R\/8} (P)$ and $K := K_\\delta$ we obtain\n\\begin{equation*}\n\tv \\geq \\overset{\\star}{C} \\,  \\big[  \\mathrm{dist} (K_\\delta ,H^+) \\, | K_\\delta | \\, \\inf_{K_\\delta} v \\big] \\psi_B \\quad \\text{in} \\ B,\n\\end{equation*}\nwhere $\\overset{\\star}{C} > 0$ is an explicit constant depending on $n$, $s$, $R$ and ${\\rm diam}(\\Omega)$. Here we used that, in the present situation, we have $K \\subset \\Omega$ and that ${\\rm dist} (B, U_0) \\le R$.\n\n\nSince $K_\\delta \\subseteq (\\Omega^- \\setminus \\mathcal  Q(\\widehat{\\Omega})) \\setminus F_\\delta$, then \n\\begin{equation*}\n\\mathrm{dist} (K_\\delta,H^+) \\geq \\delta .\n\\end{equation*}\n\nWe now point out that in $K_\\delta$ we have $v = u - u' = 1 - u'$; we can therefore apply Lemma \\ref{lemma_conticino} and get\n\\begin{equation*}\n    v(x) \\geq C_{cap} (n,s,\\mathfrak{r}^e_{\\Omega}) \\, \\delta^s =  C_{cap} \\, \\delta^s \\quad \\text{for every } \\ x\\in K_\\delta.\n\\end{equation*}\n\nMoreover, we have\n\\begin{equation}\n\\label{a2s3eq43}\n|K_\\delta| = |\\Omega^- \\setminus U_R| - |E_\\delta \\cup F_\\delta| \\ge |\\Omega^- \\setminus U_R| - (|E_\\delta| + | F_\\delta |) .\n\\end{equation}\nBy definition of $F_\\delta$, we have that\n\\begin{equation}\n\\label{a2s3eq44}\n\t|F_\\delta| \\le  \\mathrm{diam}( \\Omega )^{n-1} \\delta.\n\\end{equation}\nBy using Lemma 5.2 in \\cite{ciraolo2021symmetry}, since $E_\\delta \\subseteq A_\\delta$, we have\n\\begin{equation}\n\\label{a2s3eq45}\n    |E_\\delta| \\leq \\left[\\frac{2 n |\\Omega|}{R} \\right]  \\delta.\n\\end{equation}\nPutting together \\eqref{a2s3eq43}, \\eqref{a2s3eq44} and \\eqref{a2s3eq45} we get\n\\begin{equation*}\n\t|K_\\delta| \\ge |\\Omega^- \\setminus U_R| - \\tilde{c} \\,  \\delta,\n\\end{equation*}\nwhere $\\tilde{c}$ is a positive constant depending on $n$, $\\mathrm{diam}(\\Omega)$ and $\\mathfrak{r}^e_{\\Omega}$.\n\n\\medskip\n\nHence we have proved that\n\\begin{equation*}\n    v(P) \\geq \\overset{\\star}{C} \\, C_{cap} \\, (R\/8)^{2s} \\, \\gamma_{n,s} \\delta^{1+s} \\big( |\\Omega^- \\setminus U_R| - \\tilde{c} \\, \\delta \\big) \\,,\n\\end{equation*}\nand, by choosing\n\\begin{equation*}\n    \\Tilde{C} := \\max \\left\\lbrace \\frac{8^{2s}}{ C_{cap} \\, R^s \\, \\overset{\\star}{C}} \\, , \\, \\, \\tilde{c} \\right\\rbrace ,\n\\end{equation*}\nwe get the desired inequality \\eqref{a2s3eq39}.\n\\end{proof}\n\nOnce Lemma \\ref{a2s3lemma1} is proved we can follow \\cite{ciraolo2021symmetry} to get the almost symmetry in one direction and, with the same reasoning as in \\cite[Section 6]{ciraolo2021symmetry}, obtain the same quantitative stability estimate required for the proof of Theorem \\ref{a2theorem2}. For this reason we only sketch the main ideas in the following proof.\n\n\\begin{proof}[Proof of Theorem \\ref{a2theorem2}]\nOnce we have inequality \\eqref{a2s3eq39}, we can argue as in Lemma 5.6 of \\cite{ciraolo2021symmetry} to obtain the estimate for the almost symmetry in one direction, namely\n\\begin{equation}\n\\label{a2s3eq49}\n    |\\Omega^- \\setminus \\mathcal Q(\\widehat{\\Omega}) | \\leq \\overline{C} [u]_{\\partial G}^{\\frac{1}{s+2}},\n\\end{equation}\nwhere $\\overline{C}:= \\max\\{ 1, {\\rm diam}(\\Omega), K \\, (R\/2) \\} \\, \\tilde{C}$, with $\\tilde{C}$ as in \\eqref{a2s3eq39} and $K = K(n,s) \\geq 1$ is the constant that appears in the boundary Harnack inequality in Lemma \\ref{a2s1lemma5}.\n\n\\medskip\n\nNow, up to a translation we can assume that the critical hyperplanes with respect to the $N$ coordinate directions  $T^{e_j}$ coincide with $\\{ x_j = 0 \\}$ for each $j=1, \\dots , N$, that is, they all intersect at the origin. \n\n\\medskip\n\nThe idea is then the following: for a given direction $e \\in \\mathcal{S}^{n-1}$ we slice the positive cap of $\\Omega$ in (a finite number of) sections depending on the critical value $\\lambda_e$ and consider their measure, namely\n\\begin{equation*}\n    m_k := | \\, \\{ x \\in \\Omega \\ | \\ (2k-1) \\, \\lambda_e \\leq x \\cdot e \\leq (2k+1) \\, \\lambda_e \\} \\, |, \\quad \\text{for } k\\geq 1.\n\\end{equation*}\n\nSince $\\Omega$ is bounded, $m_k > 0$ only up to an index $k_0$ which behaves like the inverse of $\\lambda_e$. The key observation is that, by reflecting with respect to the origin and using \\eqref{a2s3eq49}, one has\n\\begin{equation}\n\\label{a2s3eq50}\n    m_1 = | \\, \\{ x \\in \\Omega \\ | \\ -\\lambda_e \\leq x \\cdot e \\leq \\lambda_e \\} \\, | \\leq (n+3) \\, \\overline{C} \\, [u]_{\\partial G}^{\\frac{1}{s+2}};\n\\end{equation}\nmoreover, by the moving plane procedure, $m_k \\leq m_1$ for every $k$ up to $k_0$ and therefore one can then write the expression \n\\begin{equation}\n\\label{unaltraequazione}\n    |\\Omega_{\\lambda_e}| \\leq \\sum_{k=1}^{k_0} m_k \\leq k_0 \\, m_1 \\leq (n+3) \\, {\\rm diam}(\\Omega) \\, \\overline{C} \\, \\frac{1}{\\lambda_e} \\, [u]_{\\partial G}^{\\frac{1}{s+2}}.\n\\end{equation}\n\nInequalities \\eqref{unaltraequazione} and some further calculations (see section 4 in \\cite{ciraolo2021symmetry}) yield\n\\begin{equation}\n\\label{a2s3eq52}\n    |\\lambda_e| \\leq 4 \\, (n+3) \\, \\frac{{\\rm diam}(\\Omega)}{|\\Omega|} \\, \\overline{C} [u]_{\\partial G}^{\\frac{1}{s+2}}.\n\\end{equation}\n\nNow it remains to establish a relationship between $|\\lambda_e|$ and $\\rho(\\Omega)$. We set $\\rho_{min}:= \\min_{z \\in \\partial \\Omega} |z|$, $\\rho_{max}:= \\max_{z \\in \\partial \\Omega} |z|$ and choose $x, y \\in \\partial \\Omega$ such that $|x| = \\rho_{min}$ and $|y| = \\rho_{max}$. We then consider the unit vector\n\\begin{equation*}\n    e := \\frac{x - y}{|x - y|}\n\\end{equation*}\nand the corresponding critical hyperplane $T^e$. By construction, we know that ${\\rm dist} (x,T^e) \\geq {\\rm dist} (y,T^e)$ and therefore some simple calculations lead to \n\\begin{equation}\n\\label{a2s3eq53}\n    \\rho(\\Omega) \\leq \\rho_{max} - \\rho_{min} = |y| - |x| \\leq 2 \\, {\\rm dist} (0,T^e) = 2 |\\lambda_e|.\n\\end{equation}\n\nCombining \\eqref{a2s3eq52} and \\eqref{a2s3eq53} leads to \\eqref{a2s1eq8}. It is worth pointing out that the new constants appearing in \\eqref{a2s3eq50} and onward only depend on the dimension $n$, the diameter ${\\rm diam}(\\Omega)$ and the volume $|\\Omega|$.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\section{Annular sets}\nIn this section we consider annular domains and prove Theorems \\ref{a2theorem3} and \\ref{a2theorem4}. The strategy that we use is still via the method of moving planes and it is similar to the previous one; nevertheless, the method has to be carefully adapted to this situation. We recall that we are considering solutions to \\eqref{a2s1eq12}, where $A=\\Omega \\setminus \\overline D$, with $\\overline D \\subset \\Omega$ bounded open domains.\n\n\n\n\\bigskip\n\nNow for a fixed direction $e$ and a parameter $\\lambda \\in \\mathbb{R}$ we let $T_\\lambda$, $H_\\lambda$, $\\mathcal Q_\\lambda$ be as in the previous section. We now consider \n$$\n\\Sigma_\\lambda := (\\Omega \\cap H_\\lambda) \\setminus \\mathcal Q_\\lambda(\\overline{D})\n$$\nwhich is the cap of the annulus $\\Omega \\setminus D$. Moreover, for a given set $E$, we define \n\\begin{align*}\n    &d_{E}:= \\inf \\{ \\lambda \\in \\mathbb{R} \\ | \\ T_\\mu \\cap \\overline{E} = \\emptyset  \\}\\\\\n    &\\overline{\\lambda}_E := \\inf \\{ \\lambda \\leq d_E \\ | \\ \\text{for every } \\ \\mu > \\lambda, (\\overline{E} \\cap H_\\mu)^\\mu \\subset ( E \\cap H_\\mu^\\mu ) \\  \\text{ and } \\ \\nu(x) \\cdot e > 0 \\ \\forall x \\in T_\\mu \\cap \\partial E \\},\n\\end{align*}\nand the critical parameter $\\overline \\lambda$ is given by\n\\begin{equation*}\n\\overline{\\lambda} := \\max \\{ \\overline{\\lambda}_D, \\ \\overline{\\lambda}_\\Omega  \\}.\n\\end{equation*}\n\n\nWe mention that $\\Sigma_\\lambda$ is chosen in such a way that both the function $u$ and its reflection $u'$ are $s$-harmonic, as we will see in the proof of Theorem \\ref{a2theorem3}. We also notice that, thanks to our choices, $\\overline{\\lambda}$ is the critical value for $A$ with respect to the direction $e$, and now the critical position can occur in four possible cases (namely, Cases 1 and 2 in Subsection \\ref{subsect_MMP} for both $D$ and $\\Omega$).\n\n\\medskip\n\nIn order to avoid further technicalities we ask for the domains $D$ and $\\Omega$ to be regular (namely, with boundaries $\\partial G$ and $\\partial \\Omega$ of class $C^2$); the proof works in the same way if we instead assume that $\\partial A$ is just of class $C^1$, and $\\Gamma_R^G$ and $\\Gamma_R^\\Omega$ of class $C^2$.\n\n\\medskip\n\n\n\\subsection{Symmetry result}\n\nWith this setting, we are now ready to give a proof of the symmetry result for annular sets.\n\n\\begin{proof}[Proof of Theorem \\ref{a2theorem3}]\nWe fix a direction $e=e_1$ and reach the critical value $\\overline{\\lambda}$. Without loss of generality, we assume that $T = \\{ e_1 = 0 \\}$ and define the function $w(x) := u(x) - u(x')$ for every $x \\in \\mathbb{R}^n$. To simplify the notation we set $\\mathcal{Q} = \\mathcal{Q}_{\\overline{\\lambda}}$. Our aim is to show that $w$ is actually identically zero in $\\mathcal Q (\\Sigma_{\\overline{\\lambda}})$. This implies that both the function $w$ and the set $A$ itself are symmetric with respect to direction $e$; since the direction $e$ can be chosen arbitrarily, the proof is then complete.\n\nHence we have to show that $w \\equiv 0$ in $\\mathcal Q(\\Sigma_{\\overline{\\lambda}})$. We notice that the function $w$ is antisymmetric with respect to $e=e_1$ and\n\\begin{align*}\n    (-\\Delta)^s w(x) = 0 \\qquad \\qquad &\\text{for} \\ x \\in \\mathcal Q(\\Sigma_{\\overline{\\lambda}}),\\\\\n    w(x) = u(x) - u(x') = 1 - u(x') \\geq 0 \\qquad \\qquad &\\text{for} \\ x \\in \\overline{D} \\cap \\mathcal Q(\\widehat{\\Omega}),\\\\\n    w(x) = u(x) - u(x') = u(x) \\geq 0 \\qquad \\qquad &\\text{for} \\ x \\in \\Omega^- \\setminus \\mathcal Q(\\widehat{\\Omega}),\\\\\n    w(x) = 0 \\qquad \\qquad &\\text{for} \\ x \\in H^- \\setminus \\Omega^-.\n\\end{align*}\nIn particular, the last three inequalities tell us that $w \\geq 0$ in $H^- \\setminus \\mathcal Q(\\Sigma_{\\overline{\\lambda}})$. The weak maximum principle for antisymmetric solutions in Lemma \\ref{a2s1lemma1} implies that $w \\geq 0$ in $\\mathcal Q(\\Sigma_{\\overline{\\lambda}})$; then, from the strong maximum principle in Lemma \\ref{lemma_SMP} we get that either $w > 0$ in $\\mathcal Q(\\Sigma_{\\overline{\\lambda}})$ or $w \\equiv 0$ in $\\mathcal Q(\\Sigma_{\\overline{\\lambda}})$.\n\nIn order to conclude, we notice that in this case we have four possible critical cases, all of which can be treated as in the proof of Theorem \\ref{a2s1theorem1}. The conclusion then follows straightforwardly. \n\\end{proof}\n\n\\subsection{Almost symmetry in one direction}\n\nIn order to prove almost symmetry in one direction for the annular set, we need to make use again of the quantitative Hopf's type lemma (\\cite[Lemma 4.1]{ciraolo2021symmetry}) and adapt it to the current problem. \n\nWe start with a lemma which gives the behaviour of the solution $u$ of \\eqref{a2s1eq12} close to the boundary of the annulus $A$. For this reason, for $t>0$ we define\n\\begin{equation} \\label{At_def}\nA_t = \\{x \\in A :\\ {\\rm dist}(x,\\partial A) > t \\}\\,.\n\\end{equation}\n\nWe start with a simple remark.\n\n\\begin{remark}\nIf $u \\in C^s (\\mathbb{R}^n)$ solves \\eqref{a2s1eq12} with $\\partial A \\in C^1$, then we have\n\\begin{equation}\n\\label{a2s4eq3}\n    0 < u < 1 \\qquad \\text{in} \\ A.\n\\end{equation}\n\\end{remark}\nIndeed, applying the maximum principles for an $s$-harmonic function in $A$ we get that $u$ has to be strictly positive in $A$. By using the same argument for $\\Tilde{u} := 1 - u$ we get the latter part of \\eqref{a2s4eq3}.\n\n\\bigskip\n\nWe have the following lemma.\n\n\n\\begin{lemma}\n\\label{a2s4lemma4}\nUnder the assumptions of Theorem \\eqref{a2theorem4}, let $u$ be a solution of \\eqref{a2s1eq12} and let $r_{ann}:= \\min \\{ \\overline{d}\/4, \\, \\mathfrak{r}^i_\\Omega, \\, \\mathfrak{r}^e_D \\}$. Then\n\\begin{equation}\n\\label{a2s4eq36}\n    \\min \\{u, 1 - u \\} (x) \\geq C_{ann} \\, ({\\rm dist} (x,\\partial A))^s \\quad \\text{in} \\ A \\setminus A_{r_{ann}},\n\\end{equation}\nwhere $A_{r_{ann}}$ is given by \\eqref{At_def} and $C_{ann} > 0$ is a constant depending on $n$, $s$, $\\mathfrak{r}^e_{D}$, $\\mathfrak{r}^i_{\\Omega}$, ${\\rm diam}(\\Omega)$ and $|D|$.  \n\\end{lemma}\n\n\\begin{proof}\nWe start by proving \\eqref{a2s4eq36} for points which are close to the boundary of $\\partial \\Omega$. Let $R_1 := \\min \\{ \\overline{d}\/4, \\, \\mathfrak{r}^i_\\Omega \\}$; for a given $x_0 \\in \\partial \\Omega$ we choose $\\overline{x} \\in A$ such that $\\partial B_{R_1}(\\overline{x}) \\cap \\partial \\Omega = \\{ x_0 \\}$. By using Lemma \\ref{a2s3theorem1}, with $B = B_R (\\overline{x})$ and $K = \\overline{D}$, some easy computations lead us to\n\\begin{equation*}\n    u(x) \\geq \\frac{c_{n,s} \\, \\gamma_{n,s}}{4^{n+2s}} \\frac{|D| \\, R_1^s}{{\\rm diam} (\\Omega)^{n+2s}} \\, (R_1 - |x|)^s \\quad \\text{in} \\, B_{R_1}(\\overline{x}).\n\\end{equation*}\n\nSince $\\partial \\Omega$ is of class $C^2$, then the radius $R_1$ can be chosen uniformly with respect to $\\overline x$ and then the above inequality does not depend on the center $\\overline{x}$. Therefore, we can repeat the same argument on the whole boundary $\\partial \\Omega$ and get\n\\begin{equation}\n\\label{a2s4eq41}\n    u(x) \\geq C_{ext} \\, {\\rm dist}(x, \\partial \\Omega)^s \\quad \\text{ for any} \\ x \\in \\{ x \\in A \\ | \\ {\\rm dist} (x,\\partial \\Omega) < R_1 \\},\n\\end{equation}\nwhere  \n\\begin{equation*}\n    C_{ext} := \\frac{c_{n,s} \\, \\gamma_{n,s}}{4^{n+2s}} \\frac{|D| \\, R_1^s}{{\\rm diam} (\\Omega)^{n+2s}}.\n\\end{equation*}\n\nNow we prove \\eqref{a2s4eq36} for points which are close to the boundary of $\\partial D$. In this case we apply the Hopf's lemma to $v := 1 - u$. We set $R_2:= \\{ \\overline{d}\/4, \\mathfrak{r}^e_D \\}$ and choose $x_\\star \\in \\mathbb{R}^n \\setminus \\Omega$ such that $K:= \\overline{B_{R_2}}(x_\\star)$ is tangent to $\\partial \\Omega$. For each $x_0 \\in \\partial D$, let $\\overline{x} \\in A$ be such that $\\partial B_{R_2}(\\overline{x}) \\cap \\partial \\Omega = \\{ x_0 \\}$. We apply Lemma \\ref{a2s3theorem1} to $B_{R_2}(\\overline{x})$ and $K$ as before and repeat the argument for every boundary point in $\\partial D$. By putting all together we will have \n\\begin{equation}\n\\label{a2s4eq47}\n    v(x) \\geq C_{int} \\, ({\\rm dist} (x,\\partial D))^s \\quad in \\, (D + B_{R_2}) \\setminus D,\n\\end{equation}\nwhere\n\\begin{equation*}\n    C_{int} := c_{n,s} \\, \\gamma_{n,s} \\, \\omega_n \\, \\frac{R_2^{n+s}}{(4R_2 + {\\rm diam} (\\Omega))^{n+2s}}.\n\\end{equation*}\n\n\\bigskip\n\nWe now see that for $r_{ann} = \\min \\{ \\overline{d}\/4, \\, \\mathfrak{r}^i_\\Omega, \\, \\mathfrak{r}^e_D \\}$ both inequalities \\eqref{a2s4eq41} and \\eqref{a2s4eq47} hold; therefore, by setting\n\\begin{equation*}\n    C_{ann}:= \\min \\{ C_{ext}, \\, C_{int} \\},\n\\end{equation*}\ninequality \\eqref{a2s4eq36} holds and the proof is complete.\n\\end{proof}\n\nWe are now ready to state a version of Lemma \\ref{a2s3lemma1} for the annulus, under the assumptions of Theorem \\ref{a2theorem4}. The ball $B$ will be chosen inside of a set where the antisymmetic function $w:= u - u'$ is $s$-harmonic (this time, it will be $Q(\\Sigma_{\\overline{\\lambda}})$); in this case, the compact set $K$ consists of two components, in such a way that we can take into account the symmetric differences between the sets $\\Omega$ and $G$ and their respective reflections via the moving plane method. We define\n\\begin{align*}\n    &L := \\Omega^- \\setminus \\mathcal Q(\\widehat{\\Omega}) \\\\\n    &M := D^- \\setminus \\mathcal Q(\\widehat{D})\\\\\n    &\\Tilde{K} := L \\cup (M \\cap \\mathcal Q(\\widehat{\\Omega})).\n\\end{align*}\n\nWe have the following lemma.\n\n\\begin{lemma}\nGiven $P \\in \\mathcal Q(\\Sigma_{\\overline{\\lambda}})$ with $B=B_{R\/8} (P)$ such that $\\mathrm{dist}(B, \\mathcal Q(\\Sigma_{\\overline{\\lambda}})) \\ge R\/8$ and given $\\delta > 0$, we have that\n\\begin{equation}\n\\label{a2s4eq52}\n|\\Tilde{K}| \\leq \\Tilde{C} \\, (\\delta^{-(1+s)} v(P) + \\delta),\n\\end{equation}\nwhere $\\Tilde{C} > 0$ is a constant depending only on $n$, $s$, $R$, $\\mathfrak{r}^e_{D}$, $\\mathfrak{r}^i_{\\Omega}$, ${\\rm diam}(\\Omega)$, $|D|$ and $|\\Omega|$.\n\\end{lemma}\n\n\\begin{proof}\nWe set $K_\\delta := \\Tilde{K} \\setminus (E_\\delta \\cup F_\\delta)$ where \n\\begin{align*}\n    &E_\\delta := [ \\, \\{ x \\in \\Omega \\ | \\ {\\rm dist}(x,\\partial \\Omega) < \\delta \\} \\cap (\\Omega^- \\cup \\mathcal Q(\\widehat{\\Omega}) ) \\, ] \\, \\cup \\, [ G^- \\setminus \\mathcal Q(\\widehat{G} + B_\\delta) ) \\, ],\\\\\n    &F_\\delta := \\{ x \\in \\Omega \\ | \\ {\\rm dist} (x,H^+) < \\delta \\}.\n\\end{align*}\nWe apply Lemma \\ref{s3lemma2} with $B := B_{R\/8} (P)$ and $K := K_\\delta$ to get \\eqref{s3eq19}, i.e.\n\n\\begin{equation}\n\\label{a2s4eq55}\n\tv \\geq \\overset{\\star}{C} \\,  \\big[  \\mathrm{dist} (K_\\delta ,H^+) \\, | K_\\delta | \\, \\inf_{K_\\delta} v \\big] \\psi_B \\quad \\text{in} \\ B.\n\\end{equation}\n\nBy arguing as done for \\eqref{a2s3eq43}, \\eqref{a2s3eq44} and \\eqref{a2s3eq45} we get that\n\\begin{equation}\n\\label{a2s4eq56}\n    |\\tilde{K}_\\delta| \\geq |\\Omega^- \\setminus \\mathcal Q(\\widehat{\\Omega})| + |G^- \\setminus \\mathcal Q(\\widehat{G})| - \\tilde{c}\\, \\delta \\,,\n\\end{equation}\nand plugging \\eqref{a2s4eq56} into \\eqref{a2s4eq55}, together with Lemma \\ref{a2s4lemma4} and the fact that ${\\rm dist}(\\tilde{K}_\\delta, H^+) \\geq \\delta$ we get \\eqref{a2s4eq52}.\n\\end{proof}\n\nThe way to use Lemma \\ref{a2s4lemma4} to establish almost symmetry in one direction and then prove Theorem \\ref{a2theorem4} is again the one sketched in the proof of Theorem \\ref{a2theorem2}. We just need to highlight some minor differences with the annular case, that we report below.\n\n\\begin{proof}\nThe first goal is to obtain the almost symmetry in one direction from \\eqref{a2s4eq52}. While in the proof of Theorem \\ref{a2theorem2} we need to take into account the two possible critical Cases 1 and 2 for the moving plane method, with the first one being further divided into cases 1a and 1b, now the critical position can be reached for both the set $D$ and the set $\\Omega$, resulting in a total of six possible critical cases. Nonetheless, they are tackled in the same exact way; the only thing that we need to point out is that in each of the critical cases we can write\n\\begin{equation}\n\\label{a2s4eq58}\n    v(P) \\leq  c_\\star  \\max \\{ [u]_{\\Gamma^R_{\\Omega}}, [u]_{\\Gamma^R_{G}} \\} = c_\\star \\, \\mathrm{def}_A(u) \\,,\n\\end{equation}\nwhere $c_\\star := \\max\\{1, {\\rm diam}(\\Omega), K \\, R\/2 \\}$.\nFrom \\eqref{a2s4eq58} we can then recover the inequality\n\\begin{equation}\n    |\\Omega^- \\setminus \\mathcal Q(\\widehat{\\Omega})| + |G^- \\setminus \\mathcal Q(\\widehat{G})| \\leq \\overline{C} \\, \\mathrm{def}_A(u)^{\\frac{1}{s+2}},\n\\end{equation}\nwhere $\\overline{C} = c_\\star \\tilde{C}$. The slicing of the two sets can then be performed in the same way, which leads to an estimate of type\n\\begin{equation}\n\\label{a2s4eq60}\n    |\\lambda_e| \\leq 4 \\, (n+3) \\frac{{\\rm diam}(\\Omega)}{|\\Omega|} \\, \\overline{C} \\, \\mathrm{def}_A(u)^{\\frac{1}{s+2}},\n\\end{equation}\nwhere now again the bound depends on the seminorms on both of the parallel surfaces. We now only need to make sure that formula \\eqref{a2s3eq53} still applies. Again, for the set $\\Omega$ we define $\\rho_{min}:= \\min_{z \\in \\partial \\Omega} |z|$, $\\rho_{max}:= \\max_{z \\in \\partial \\Omega} |z|$, choose $x, y \\in \\partial \\Omega$ such that $|x| = \\rho_{min}$ and $|y| = \\rho_{max}$ and consider the direction $e= y - x$ up to normalization with its critical hypeplane $T^e$. Since we are now in the annular case $\\overline{\\lambda}^e = \\max \\{ \\overline{\\lambda}^e_D, \\overline{\\lambda}^e_\\Omega \\}$ and therefore the moving plane might stop before reaching the cricial position for the set $\\Omega$ itself. However we can still write\n\\begin{equation}\n\\label{a2s4eq62}\n    \\rho(\\Omega) \\leq \\rho_{max} - \\rho_{min} = |y| - |x| \\leq 2 \\, {\\rm dist} (0,T^e) = 2 |\\overline{\\lambda}^e| \\leq 2 \\, |\\overline{\\lambda}^e_\\Omega|.\n\\end{equation}\nCombining \\eqref{a2s4eq60} and \\eqref{a2s4eq62} and repeating the same argument for $G$ lead us to \\eqref{a2s1eq18}.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n  Consider the space $\\mathbf{L}$ of planar polygons with prescribed edge lengths\\footnote{The space $\\mathbf{L}$ appears in the literature as ``configuration space of a flexible polygon'', or ``configuration space of a polygonal linkage'', or just as ``space of polygons''.}  and the oriented area $\\mathcal{A}$  as a Morse function  defined on it.\n  It is known that  generically:\n  \\begin{itemize}\n    \\item $\\mathbf{L}$  is a smooth closed manifold whose diffeomorphic type depends on the edge lengths \\cite{Farber, MilKap}.\n    \\item  The oriented area $\\mathcal{A}$ is a Morse function whose critical points are cyclic configurations (that is, polygons with all the vertices lying on a circle), whose Morse indices are known, see Theorem \\ref{Thm_Morse_closed_plane}, \\cite{khipan,panzh, zhu}. The Morse index depends not only on the combinatorics of a cyclic polygon, but also on some metric data. Direct computations of the Morse index proved to be quite involved, so the existing proof  comes from bifurcation analysis combined with a number of combinatorial tricks.\n    \\item Bifurcations of $\\mathcal{A}$ are captured by  cyclic polygons $P$ whose dual polygons $P^*$ have zero perimeter \\cite{panzh}; see also Lemma \\ref{LemmaBifurc}. That is, in a generic one-parametric family of edge lengths a critical point $P$ bifurcates whenever the perimeter of the dual tangential polygon $P^*$  vanishes.\n  \\end{itemize}\nThe polygon $P^*$ is \\textit{tangential} (see Definition \\ref{DefTang}),  so tangential polygons with zero perimeter play a special role in the framework of flexible polygons and oriented area. The initial motivation of the present paper was to clarify\nthis role.\n\n\n  In the paper we consider the following problem: instead of prescribing edge lengths, we prescribe the slopes of the edges. Instead of taking the oriented area as a Morse function, we take the oriented perimeter. We prove:\n \\begin{itemize}\n    \\item The space $\\mathbf{S}$ of polygons with prescribed edge slopes  is a smooth non-compact  manifold (see Theorem \\ref{ThmConfSpace} for its diffeomorphism type).\n    \\item The (oriented) perimeter $\\mathcal{P}$ is a Morse function with either zero or  two critical points (Theorem \\ref{ThmCritPer}).\n    Critical points of $\\mathcal{P}$ are{ tangential} polygons.\n    \\item  The absence of critical points is captured by existence of tangential polygons with zero perimeter (Corollary  \\ref{CorCritPer}). That is, in a generic one-parametric family of slopes critical points disappear whenever the perimeter of the (uniquely defined) tangential polygon vanishes.\n\n        \\item  Although there are at most two critical points, these are not necessarily maximum and minimum of the perimeter function. The Morse index of a tangential polygon is expressed in Theorem \\ref{ThmMorseTangential}. The proof is based on  direct computation of the leading principal minors of the Hessian matrix.\n            \\item The Morse index of a tangential polygon depends on the combinatorics of the polygon and the sign of its perimeter only.\n            \\item Local projective duality provides an alternative proof of Theorem \\ref{Thm_Morse_closed_plane}, that is, the formula for the Morse index of a cyclic polygon (rel the area function).\n  \\end{itemize}\n\n  Yet another motivation of this research is projective duality. Oversimplifying, assume that  the ambient space of the polygons is the sphere $S^2$.\nThen projective duality takes polygons with prescribed edge lengths to polygons with prescribed angles. It also takes area to a linear function of perimeter, so the critical polygons in the two settings are mutually projectively dual and have related Morse indices.\n(A necessary warning:  there exists only  a local version of projective duality. However it is sufficient for purposes.)\n\n\n\n\n\n\n\n\\section*{Acknowledgments}\nSections 3,4, and 5 are supported by the Russian Science Foundation grant N 14-21-00035.\n\nGaiane Panina is\n supported by the RFBR grant 17-01-00128  and   the Program of the Presidium of the Russian\n Academy of Sciences N 01 'Fundamental Mathematics and its Applications'\n under\n grant PRAS-18-01.\n\n We are indebted to Alexander Gaifullin who was the first to point out the vanishing perimeter of a bifurcating polygon.\nWe also thank Mikhail Khristoforov for useful discussions.\n\n\n\\section{Definitions and setups}\n\nA \\textit{polygon} is an oriented closed broken line in the plane. We assume that its vertices are numbered in the cyclic order and thus induce an orientation of the polygon.\n\n\n\n\\begin{definition} \\label{Dfn_area} The \\textit{oriented area} of a polygon $P$ with the vertices \\newline $v_i = (x_i,\ny_i)$  is defined by\n$$2\\mathcal{A}(P) = (x_1y_2 - x_2y_1) + \\ldots + (x_ny_1 - x_1y_n).$$\n\n\nEquivalently, one defines    $$\\mathcal{A}(P):=\\int_{\\mathbb{R}^2} w(P,x)dx,$$\n\nwhere $w(P,x)$  is the winding number of $P$ around  the point $x$.\n\n\n\n\\end{definition}\n\n\n\\subsection{Polygons with prescribed edge lengths. Oriented area as a Morse function.}  Assume that a generic $n$-tuple of positive numbers $(l_1,...,l_n)$ is given.\nThe space of all planar polygons whose consecutive edge lengths are  $(l_1,...,l_n)$ (modulo translations and rotations) is called the \\textit{configuration space\nof polygons with prescribed edge lengths.} We denote it by $\\mathbf{L}=\\mathbf{L}(l_1,...,l_n)$.\n\nGenerically, $\\mathbf{L}$ is a smooth closed manifold  \\cite{Farber, MilKap}, and\n the oriented area $\\mathcal{A}$  is a  Morse function on $\\mathbf{L}$.\n\n\n\n\n\\begin{definition}\n    A\npolygon  $P$  is  \\textit{cyclic} if all its vertices $v_i$\nlie on a circle.\n\n\\end{definition}\n\n\n\n\\begin{theorem}\\label{Thm_crirical_are_cyclic}\\cite{khipan}  Generically, $\\mathcal{A}$ is a Morse function.\n At smooth points of the space  $\\mathbf{L}$, a polygon $P$ is a critical point of the\noriented area  $\\mathcal{A}$  iff $P$ is a cyclic configuration.\n        \\qed\n\\end{theorem}\n\n\nBefore we recall a formula for the Morse index of a cyclic configuration from \\cite{zhu}, \\cite{khipan1},\nlet us fix the following notation for a cyclic polygon $P$.\n\n$\\omega_P=w(P,O)$ is the winding number of $P$ with respect to the center $O$ of the circumscribed circle.\n\n$\\alpha_i$  is the half of the angle between the vectors\n$\\overrightarrow{Ov_i}$ and $\\overrightarrow{Ov_{i+1}}$. The angle\nis defined to be positive, orientation is not involved.\n\n\n\n\n$\\varepsilon_i$ is the\norientation of the edge $v_iv_{i+1}$, that is,\n\n $\\varepsilon_i=\\left\\{\n                       \\begin{array}{ll}\n                         1, & \\hbox{if the center $O$ lies to the left of } \\overrightarrow{v_iv_{i+1}};\\\\\n                         -1, & \\hbox{if  $O$ lies to the right of } \\overrightarrow{v_iv_{i+1}}.\n                       \\end{array}\n                     \\right.$\n\n\n\n$e(P)$ is the number of positive entries in $\\varepsilon_1,...,\\varepsilon_n$, that is, $e(P)$ is the number of \\textit{positively oriented }edges .\n\n\n\n$\\mu_P=\\mu_P(\\mathcal{A})$ is the Morse index of the function $\\mathcal{A}$ at the point\n$P$. That is, $\\mu_P(\\mathcal{A})$ is the number of negative eigenvalues of the\nHessian matrix $Hess_P(\\mathcal{A})$.\n\n\n\\begin{theorem}\\label{Thm_Morse_closed_plane}\\cite{khipan1}, \\cite{panzh}, \\cite{zhu} Generically,\nfor a cyclic polygon $P$,\n$$\\mu_P( \\mathcal{A})= e(P)-1-2\\omega_P-\\left\\{\n       \\begin{array}{ll}\n    0    &\\hbox{if }\\   \\sum_{i=1}^n \\varepsilon_i \\tan \\alpha_i>0; \\\\\n        1 & \\hbox{otherwise}.\\qed\n       \\end{array}\n     \\right.$$\n\\end{theorem}\n\nIn the present paper we give an alternative proof of this theorem: we prove a slightly stronger claim, see Corollary \\ref{RemFinal}.\n\\begin{remark}\n In a continuous one-parametric family of  cyclic polygons  with non-vanishing edge lengths, $\\mu_P$ changes iff  $\\sum_{i=1}^n \\varepsilon_i \\tan \\alpha_i$  vanishes. Although $e(P)$ and $\\omega_P$  can vary, the sum $e(P)-1-2\\omega_P$ is constant.\n\\end{remark}\n\\bigskip\n\n\n\\subsection{Polygons with prescribed edge slopes}\n\nFix $n$ pairwise non-parallel straight lines  \\(s_1,\\ldots,s_n\\) in \\(\\R^2\\) passing through the origin and call them \\textit{slope lines}.\n\nEach \\(n\\)-tuple of  lines \\(e_1,\\ldots,e_n\\subset \\R^2\\)\n with  $e_i$ parallel to $s_i$ yields a polygon $P$ whose consecutive vertices  are \\(v_1=e_1\\cap e_2, \\ldots, v_n=e_n\\cap e_1 \\). We will denote this polygon by \\(Q=Q\\left(e_1,\\ldots,e_n\\right)\\)  and say that $Q$ is a \\textit{polygon with edge slopes} \\(s_1,\\ldots,s_n\\) .\n\nThe space of all  polygons $Q$ (modulo translations)\n with edge slopes \\(s_1,\\ldots,s_n\\),\n  is denoted  by $\\widetilde{\\mathbf{S}}=\\widetilde{\\mathbf{S}}(s_1,...,s_n)$.\n\n\nThe subspace of $\\widetilde{\\mathbf{S}}(s_1,...,s_n)$ consisting of polygons with $|\\mathcal{A}(Q)|=1$\nis called the    \\textit{configuration space\nof polygons  with prescribed edge slopes}. We denote it by $\\mathbf{S}=\\mathbf{S}(s_1,...,s_n)$. It splits into a disjoint union \\(\\mathbf{S}_-\\sqcup \\mathbf{S}_+\\) where the index indicates the sign of \\(\\mathcal{A}\\).\n\n\\begin{figure}[h]\n\\centering \\includegraphics[width=10 cm]{SPolygon.eps}\n\\caption{Slope lines (left)  and  a polygon with these edge slopes (right).}\\label{FigSlopes}\n\\end{figure}\n\nNote that: (1) the condition $|\\mathcal{A}(Q)|=1$ means that we  factor out dilations; (2) fixing slopes is the same as fixing angles and factoring out rotations.\n\\bigskip\n\n\nNow fix a direction on each of the slope lines $\\vec{s}_i$. Take $Q=Q(e_1,...,e_n)\\in \\mathbf{S}(s_1,...,s_n)$, and orient the lines $e_1,...,e_n$ consistently. Denote the oriented lines by $\\vec{e}_1,\\ldots,\\vec{e}_n$.\n\n\\begin{definition}\\label{DefPerim}\n  The  perimeter of a polygon $Q\\in \\widetilde{\\mathbf{S}}(s_1,...,s_n)$  is defined as follows:\n\\[\n\\mathcal{P}(Q)=\\sum_{i=1}^n \\sign_Q(i)|v_iv_{i+1}|,\n\\]\n\nwhere \\(\\sign_Q(i)=\\left\\{\n                                                   \\begin{array}{ll}\n                                                     1, & \\hbox{if $\\vec{e}_i$ is codirected with $\\overrightarrow{v_iv}_{i+1}$;} \\\\\n                                                     -1, & \\hbox{otherwise.}\n                                                   \\end{array}\n                                                 \\right.\\)\n\n\\end{definition}\n\nThus defined, perimeter may be negative or vanish.\n\n\\section{Topology of the configuration space  of polygons with prescribed slopes}\n\n\n\nLet the \\textit{angle \\(\\angle(r,s)\\) between two lines} $r$ and $s$ be the minimal positive angle such that the counterclockwise rotation by \\(\\angle(r,s)\\) takes \\(r\\) to \\(s\\).\n\nAssuming that an $n$-tuple of slope lines  \\(s_1,\\ldots,s_n\\) in \\(\\R^2\\) is fixed, set\n\\newline \\(t(s_1,\\ldots,s_n)\\defeq\\sum_{i=1}^{n-1}\\angle(s_i,s_{i+1})+\\angle(s_n,s_1)\\).\n\n\\medskip\n\nThe example for \\(n=3\\) will be useful in the sequel:\n\\begin{example}\\label{triangle}\n\\(t(s_1,s_2,s_3)\\) is either \\(\\pi\\) or \\(2\\pi\\). In the first (respectively, second) case the area of each nondegenerate triangle in \\(\\widetilde{\\mathbf{S}}\\) is negative (respectively, positive).\n\n\\end{example}\n\n\n\\begin{lemma}\n\\label{turn}\n\\(\\,\\)\n\\begin{enumerate}\n\n    \\item \\(t(s_1,\\ldots,s_n)\\) takes values in $\\{\\pi,2\\pi, \\dots, (n-1)\\pi\\}$.\n    \\item \\(t(s_1,\\ldots,s_n)=t(s_1,\\ldots,s_{n-1})+t(s_1,s_{n-1},s_n)-\\pi\\).\\qed\n\n\\end{enumerate}\n\n\\end{lemma}\n\nThe informal meaning of the following lemma is: the area and perimeter behave additively with respect to homological sum of polygons.\n\\begin{lemma}\n\\label{sum}\n\\(\\,\\)\nFor a polygon  \\(Q=Q(e_1,\\ldots,e_n)\\in\\widetilde{\\mathbf{ S}}(s_1,...,s_n)\\), we have\n\\begin{enumerate}\n    \\item \\(\\mathcal{A}(Q)= \\mathcal{A}(Q_{1,\\ldots,n})=\\mathcal{A}(Q_{1,\\ldots,n-1})+\\mathcal{A}(Q_{1,n-1,n})\\);\n\t\t\\item  \\(\\mathcal{P}(Q)= \\mathcal{P}(Q_{1,\\ldots,n})=\\mathcal{P}(Q_{1,\\ldots,n-1})+\\mathcal{P}(Q_{1,n-1,n})\\),\n\n\\end{enumerate}\nwhere $Q_{i_1,...,i_k}=Q(e_{i_1},...e_{i_k})\\in \\widetilde{\\mathbf{S}}(s_{i_1},...,s_{i_k})$.\\qed\n\\end{lemma}\n\n\n\\begin{figure}[h]\n\\centering \\includegraphics[width=8 cm]{coord.eps}\n\\caption{A summand of $\\mathcal{A}(Q_{1,\\ldots,n})$.}\\label{FigCoordinates}\n\\end{figure}\n\n\n\\begin{theorem}\\label{ThmConfSpace}  The configuration space $\\mathbf{S}(s_1,\\ldots,s_n)$ is homeomorphic to disjoint union of products of a sphere and a disc:\n$$\n S^{n-k-2}\\times \\D^{k-1}\\sqcup S^{k-2} \\times \\D^{n-k-1}\n,$$ where $t(s_1,\\ldots,s_n)=k\\pi$.\n\nThe left-hand  part corresponds to \\(\\mathbf{S}_-\\) and the right-hand part corresponds to \\(\\mathbf{S}_+\\).\n\\end{theorem}\n\n\\begin{proof}\nWe shall prove that \\(\\mathbf{S}_+\\) is homeomorphic to \\(S^{k-2} \\times \\D^{n-k-1}\\). The proof  for \\(\\mathbf{S}_-\\) is analogous.\n\n\nBy Lemma \\ref{sum},\n\\[\n\\mathcal{A}(Q_{1,\\ldots,n})=\\sum_{i=2}^{n-1}\\mathcal{A}(Q_{1,i,i+1})=\\sum_{i=2}^{n-1}\\sign\\Big(t(s_1,s_i,s_{i+1})-\\frac{3\\pi}2\\Big)\\cdot c_i\\cdot \\dist(v_i,e_1)^2,\n\\]\nwhere \\(c_i=c_i(s_1,s_i,s_{i+1})\\) is some positive constant depending only on the edge slopes (see Example \\ref{triangle} and Fig. \\ref{FigCoordinates}).  Thus we can parameterize $\\widetilde{\\mathbf{S}}(s_1,...,s_n)$ by \\(\\{x_i=\\sqrt{c_{i+1}}\\dist(v_{i+1},e_1)\\}_{i=1}^{n-2}\\). That is,\n\\[\\mathbf{S}_+=\\{x\\in\\R^{n-2} \\mid \\sum_{j\\in A}x_j^2 - \\sum_{j\\in B} x_j^2 =1\\},\\] where\n\\(|A|=k-1\\), \\(A\\sqcup B=\\{1,\\ldots,n-2\\}\\).\nThis is homeomorphic to \\[\\{x\\in\\R^{n-2} \\mid \\sum_{j\\in A}x_j^2 =1, \\sum_{j\\in B}x_j^2 <1\\}.\\]\nIndeed, for \\(k>1\\) the homeomorphism \\(h: x\\mapsto \\frac{x}{\\sqrt{\\sum_{j\\in A}x_j^2}}\\) is well defined and appropriate, whereas for \\(k=1\\) both sets are empty.\nThe claim follows.\n\\end{proof}\n\n\n\\section{Critical points of the perimeter }\nAssume that an $n$-tuple of directed slope lines  \\(\\vec{s}_1,\\ldots,\\vec{s}_n\\) in \\(\\R^2\\) is fixed.\n\n\n\\begin{definition}\\label{DefTang} \\begin{enumerate}\n                                 \\item A polygon \\(Q(e_1,\\ldots,e_n)\\in \\widetilde{\\mathbf{S}}(s_1,...,s_n)\\)  is \\textit{tangential} if there exists a circle  $\\sigma$ such that\n\n                                      (a)  each of $e_i$  is tangent to $\\sigma$, and\n\n\n(b) either $\\sigma$ lies on the left with respect to all of $\\vec{e}_i$, or $\\sigma$ lies on the right with respect to all of $\\vec{e_i }$, see Fig. \\ref{FigSuperscribed}.\n\nIn this case we say that the circle $\\sigma$ is \\textit{inscribed} in $Q$  and write $\\sigma=\\sigma(Q)$.\n                                 \\item By the \\textit{radius} $r=r(\\sigma(Q))$  of the inscribed circle $\\sigma$ we mean the usual radius taken with the  sign  ''$+$'' if $\\sigma$ lies on the left of $\\vec{e}_i$, and with the sign ''$-$'' otherwise.\n                               \\end{enumerate}\n\n\n\n\\end{definition}\n\n\\begin{figure}[h]\n\\centering \\includegraphics[width=14 cm]{Twosuperscribed.eps}\n\\caption{(a) A tangential polygon $Q$ with the inscribed circle of positive radius. Here $\\mathcal{A}(Q)<0, \\mathcal{P}(Q)<0$.\n(b) The tangential polygon $-Q$ with $\\mathcal{A}(-Q)=\\mathcal{A}(Q)<0, \\mathcal{P}(-Q)=-\\mathcal{P}(Q)>0$. The inscribed circle has negative radius. }\\label{FigSuperscribed}\n\\end{figure}\n\n\\begin{lemma}\\label{APr}\n  The area  and perimeter of a tangential polytope $Q$ satisfy:\n  $$\\mathcal{A}(Q)=\\frac12\\mathcal{P}(Q)\\cdot r(\\sigma(Q)).$$\n\n\\end{lemma}\n\n\\begin{theorem}\\label{ThmCritPer}\n \\(Q\\in\\mathbf{S}\\) is a critical point of \\(\\mathcal{P}\\) iff $Q$ is tangential.\n\\end{theorem}\n\n\n\n\\begin{proof}\n\nLet \\(r_i\\) be the radius of the (uniquely defined) circle inscribed\\footnote{in the sense of Definition \\ref{DefTang}.}  in the triangle \\(Q_{1,i+1,i+2}\\), \\ \\(i=1,\\ldots,n-2\\).  Denote by \\(p_i\\) the perimeter of the triangle homothetic to $Q_{1,i+1,i+2}$ (that is, with the same edge slopes) whose inscribed radius equals $1$. If $r_i\\neq 0$, \\(p_i\\) is the perimeter of is the triangle\n \\(\\frac{1}{r_i}Q_{1,i+1,i+2}\\).\nThen by Lemma~\\ref{sum} and Lemma~\\ref{APr} we have\n$$\n                                  \\mathcal{P}(Q)=\\sum_{i=1}^{n-2}p_i\\cdot  r_i,  \\ \\ \\ \\ \\hbox{and}$$\n                                 $$ \\pm1=\\mathcal{A}(Q)=\\frac{1}{2}\\sum_{i=1}^{n-2}p_i \\cdot r_i^2.$$\n\nThe collection of radii gives a coordinate system on $\\widetilde{\\mathbf{S}}$.\nW.l.o.g. we assume \\(r_1>0\\). Locally on $\\mathbf{S}$, the second row implicitly defines a function \\(r_1(r_2,\\ldots,r_{n-2})\\). Let us take \\(j\\)-th partial derivatives of the both rows, \\(j=2,\\ldots,n-2\\).\n $$             \\hbox{Since } \\         0=\\diffp{\\mathcal{A}}{{r_j}}(Q)=p_1 r_1\\diffp{r_1}{{r_j}}+p_j r_j,\n$$\n\n$$ \\hbox{we have } \\\n                                  \\diffp{r_1}{{r_j}}=-\\frac{p_j r_j}{p_1 r_1} .$$\n$$ \\hbox{Therefore, } \\\n                                  \\diffp{\\mathcal{P}}{{r_j}}(Q)=p_1 \\diffp{r_1}{{r_j}}+p_j=p_j(1-\\frac{r_j}{r_1}). $$\n\n\n\nThus the gradient of the perimeter is zero iff \\(r_1=r_2=\\ldots=r_{n-2}=r\\)\n\\end{proof}\n\nThere exists exactly one (up to a dilation) pair of mutually symmetric tangential polygons  $Q$ and $-Q$. For them we have: $\\mathcal{A}(-Q)=\\mathcal{A}(Q),\\  \\mathcal{P}(-Q)=-\\mathcal{P}(Q)$, and $r(-Q)=-r(Q)$, see Fig.~\\ref{FigSuperscribed} for example. If the area is non-zero, scaling gives $|\\mathcal{A}|=1$.\n\n\\begin{corollary}\\label{CorCritPer}\nIf the area of a tangential polygon is zero, there are no critical points of \\(\\mathcal{P}\\) on the configuration space $\\mathbf{S}$.\nOtherwise there are exactly two critical points on the configuration space $\\mathbf{S}$. Either they both lie in $\\mathbf{S}_+$, or they both lie in $\\mathbf{S}_-$.\\qed\n\n\\end{corollary}\n\n\n\\medskip\n\nA configuration space $\\mathbf{S}(s_1,...,s_n)$ with no critical points is called\\textit{ exceptional}.\n\n\\section{Morse index of a tangential polygon}\nNow compute the Hessian matrix of $\\mathcal{P}$ at a critical point. Assume that $Q$ is a tangential polygon. In notation of the previous section, we have:\n\\[\n\\diffp[2]{\\mathcal{P}}{{r_j}}(Q)=p_1\\diffp[2]{r_1}{{r_j}}=p_1\\diffp{}{{r_j}}\\Big(-\\frac{p_j r_j}{p_1 r_1}\\Big)=-p_j\\Big(\\frac1{r_1}-\\frac{r_j}{r_1^2}\\diffp{r_1}{{r_j}}\\Big)=-\\frac{p_j}{r_1}\\Big(1+\\frac{p_j r_j^2}{p_1 r_1^2}\\Big).\n\\]\n\nSince at the critical point all \\(r_i\\) are equal,\n\\[\nH_{jj}(Q)=\\diffp[2]{\\mathcal{P}}{{r_j}}(Q)=-\\frac{p_j}{r p_1}(p_1+p_j).\n\\]\nIn the same way, for \\(j\\neq k\\)\n\\[\n\\diffp{\\mathcal{P}}{{r_j}{r_k}}=p_1\\diffp{r_1}{{r_j}{r_k}}=p_1\\diffp{}{{r_k}}\\Big(-\\frac{p_j r_j}{p_1 r_1}\\Big)=-p_j r_j\\Big(-\\frac1{r_1^2}\\cdot \\diffp{r_1}{{r_k}}\\Big)=-\\frac{p_j r_j}{r_1^2}\\cdot \\frac{p_k r_k}{p_1 r_1}.\n\\]\n\nThus: $$ H_{jk}(Q)=\\diffp{\\mathcal{P}}{{r_j}{r_k}}=-\\frac{p_j p_k}{r p_1}.$$\n\nTo compute its determinant we do the following:\n\\begin{enumerate}\n\\item add all the columns to the first one;\n\\item subtract the first row from the \\(i\\)-th row $(i=2,3,...,n-2)$ taken with the coefficient \\(\\frac{p_{i+1}}{p_2}\\) ;\n\\item subtract all rows from the first row with coefficient \\(\\frac{p_2}{p_1}\\).\n\\end{enumerate}\n\nWe get:\n\n\\begin{equation}\\label{det}\nr^{n-3}\\det(H)=\n\\begin{vmatrix}\n-\\frac{p_2}{p_1}\\Pi & 0 & \\ldots & 0 \\\\\n0 & -p_3 & \\ldots & 0 \\\\\n& & \\ddots \\\\\n0 & \\ldots & 0 & -p_{n-2}\n\\end{vmatrix},\n\\end{equation}\n\nwhere \\(\\Pi=p_1+\\ldots+p_{n-2}=\\frac{\\mathcal{P}(Q)}r\\).\n\n\\begin{definition}\n  For an ordered pair of  slopes $\\vec{s}_i$ and $\\vec{s}_j$ we say that we have the \\textit{right turn}  (\\textit{left turn}, respectively), if $\\vec{s}_j$ is obtained from $\\vec{s}_i$  by a clockwise (counterclockwise, respectively)  turn by an angle smaller than $\\pi$, see Fig. \\ref{FigTurn}.\n\n  The\\textit{ number of right turns} \\(RT=RT(\\vec{s_1},\\ldots,\\vec{s_n})\\) for a slope collection $(\\vec{s_1},\\ldots,\\vec{s_n})$ is\nthe number of right turns of the pairs $(\\vec{s_1},\\vec{s_2}),(\\vec{s_2},\\vec{s_3}),...,(\\vec{s_{n}},\\vec{s_1}) . $\nThe\\textit{ number of left turns} \\(LT\\) is defined analogously.\n\\end{definition}\n\n\n\n\n\\begin{figure}[h]\n\\centering \\includegraphics[width=8 cm]{Turn.eps}\n\\caption{Right and left turns}\\label{FigTurn}\n\\end{figure}\n\n\n\\begin{theorem}\\label{ThmMorseTangential}\nAssume that for a tangential polygon $Q\\in \\mathbf{S}$,  the radius \\(r\\) of the inscribed circle $\\sigma(Q)$ is positive.\n{Then $Q$ is a Morse point, and its Morse index of $\\mathcal{P}$ is equal to}\n\\begin{equation}\\label{index}\n\\mu_Q(\\mathcal{P})= RT-1+2\\omega_Q-\\left\\{\n       \\begin{array}{ll}\n    1    &\\hbox{if }\\   \\mathcal{P}(Q)>0; \\\\\n        0 & \\hbox{otherwise}.\n       \\end{array}\n     \\right.\n\\end{equation}\nIn the case of negative radius it is equal to\n\n\\[\n\\mu_Q(\\mathcal{P})=n-3-\\mu_{-Q}(\\mathcal{P})=LT-1-2\\omega_Q-\\left\\{\n       \\begin{array}{ll}\n    1    &\\hbox{if }\\   \\mathcal{P}(Q)>0; \\\\\n        0 & \\hbox{otherwise}.\n       \\end{array}\n     \\right.\n\\]\n\\end{theorem}\n\n\\begin{example}\n  For the polygon depicted in Fig. \\ref{FigSuperscribed}  (a), we have $\\omega=0$, $RT=2$, and $m=0$.\n\n\tFor the polygon depicted in Fig. \\ref{FigTanMorse}, we have $\\omega=1$, $RT=1$, and $m=1$.\n\\end{example}\n\n\n\\begin{figure}[h]\n\\centering \\includegraphics[width=8 cm]{tangentialMorse.eps}\n\\caption{This tangential  polygon is a saddle critical point of the perimeter.}\\label{FigTanMorse}\n\\end{figure}\n\\begin{proof}\n\nFor now denote the right-hand side of (\\ref{index}) as \\(m(Q)\\) and assume \\(r>0\\).\n\nLet us prove (\\ref{index}) by induction on \\(n\\).\nFor \\(n=3\\) the value of \\(m\\) is always zero, so is the Morse index.\n\nProve the claim for \\(n+1\\) assuming it is true for all the numbers smaller or equal than \\(n\\geq 3\\).\n\nRecall that the number of negative eigenvalues of the matrix equals the number of sign changes in the sequence of its leading principal minors. The \\(k\\)-th leading principal minor of \\(H(Q)\\) is the determinant of \\(H(Q_{1,\\ldots,k+2})\\).\n\nSo we have:\n\\begin{itemize}\n\\item \\(m(Q)=m(Q_{1,\\ldots,n})+1\\) whenever the sign of the determinant (\\ref{det}) is different for \\(n\\) and \\(n+1\\);\n\\item \\(m(Q)=m(Q_{1,\\ldots,n})\\) whenever the sign of the determinant (\\ref{det}) is the same for \\(n\\) and \\(n+1\\).\n\\end{itemize}\n\nThe change of sign of the determinant (\\ref{det}) depends only on the sign of \\(p_{n-1}\\) and the sign change of \\(\\Pi\\).\n\nNote that\n\\[\nRT(s_1,\\ldots,s_{n+1})=RT(s_1,\\ldots,s_n)+RT(s_1,s_n,s_{n+1})-1\n\\]\nand\n\\[\n\\omega_{Q_{1,\\ldots,n+1}}=\\omega_{Q_{1,\\ldots,n}}+\\omega_{Q_{1,n,n+1}}.\n\\]\n\n\nTherefore,\n\n\n\\[\nm(Q_{1,\\ldots,n+1})-m(Q_{1,\\ldots,n})=m(Q_{1,n,n+1})+\\mathbbm{1}_{\\mathcal{P}(Q_{1,n,n+1})>0}-\\mathbbm{1}_{\\mathcal{P}(Q)>0}+\\mathbbm{1}_{\\mathcal{P}(Q_{1,\\ldots,n})>0}.\n\\]\n\nThe first summand is zero by the base of induction. So we get:\n\\[\nm(Q_{1,\\ldots,n+1})-m(Q_{1,\\ldots,n})=\\mathbbm{1}_{p_{n-1}>0}-\\mathbbm{1}_{p_1+\\ldots+p_{n-1}>0}+\\mathbbm{1}_{p_1+\\ldots+p_{n-2}>0},\n\\]\n\nwhich is exactly what we require.\n\nThe case of \\(r<0\\) follows from \\(\\mathcal{P}(-Q)=-\\mathcal{P}(Q)\\).\n\n\n\\end{proof}\n\n\\section{Cyclic polygons and tangential polygons meet}\nTheorem \\ref{Thm_Morse_closed_plane} motivates the following definition:\n\n\\begin{definition}\nA cyclic polygon is a \\textit{bifurcating polygon} if $\\sum_{i=1}^n \\varepsilon_i \\tan \\alpha_i=0$.\n\\end{definition}\nBifurcating polygons correspond to bifurcations of the area function.\n\n\\begin{definition}\nGiven a cyclic polygon $P=\\{v_1,...,v_n\\}$, define its \\textit{dual  polygon} $P^*$ (Fig. \\ref{FigRelation}) as a closed broken line with orientations on the edges  constructed as follows:\n\n\\begin{enumerate}\n  \\item  Take the lines $e_1,...,e_n$ tangential to the circle at the points $\\{v_1,...,v_n\\}$.\n  \\item Take the intersection points of $e_i$ and $e_{i+1}$.\n  \\item  Orient each of the lines so that the circle lies to the left of the line.\n\\end{enumerate}\n\n\nThe construction can be easily reversed: given a tangential polygon $Q$,  there exists a cyclic polygon $P$ such that $P^*=Q$.\n\\begin{comment}\nHere is the reversing construction:\n\\begin{enumerate}\n  \\item Take the tangential points of the edges.\n  \\item Connect them into a closed broken line according to the ordering of the slopes.\n\\end{enumerate}\n\\end{comment}\n\\end{definition}\n\n\\begin{lemma}\\label{LemmaBifurc}  \\begin{enumerate}\n                 \\item For a cyclic polygon $P$, the perimeter of $P^*$ (in the sense of Definition \\ref{DefPerim}) equals $\\sum_{i=1}^n \\varepsilon_i \\tan \\alpha_i$.\n                 \\item In particular, the perimeter of $P^*$ vanishes iff $P$ is a bifurcating polygon.\n                 \\item  The oriented area of $P^*$ vanishes iff $P$ is a bifurcating polygon.\\qed\n               \\end{enumerate}\n\\end{lemma}\n\nTo summarize, if a polygon $P$ is a bifurcation of the area, then its dual $Q=P^*$  yields an exceptional configuration space $\\mathbf{S}(s_1,...,s_n)$.\n\n\n\n\\begin{figure}[h]\n\\centering \\includegraphics[width=8 cm]{Relation.eps}\n\\caption{ A cyclic polygon $P$, the polygon $P^*$ (bold), and the projectively dual polygon $P^o$ (dashed).}\\label{FigRelation}\n\\end{figure}\n\n\\section{Shperical polygons, local projective duality, and an alternative proof of Theorem \\ref{Thm_Morse_closed_plane}}\n\n\\subsection{Shperical polygons}\n\nA \\textit{spherical polygon} is an oriented closed broken line lying on the sphere of radius $R$. We always assume that its edges are the unique shortest geodesics, that is,  $l_i<\\pi R$, so a spherical polygon is uniquely defined by the (circular) sequence of its vertices.\n\nOne fails to  correctly define the area function on the space of spherical polygons with  prescribed edge lengths. One also fails to define  the space of spherical polygons with  prescribed angles  together with the perimeter function.  However we shall make use of their local versions.\n\nHere is how it goes:\n\n\\begin{definition}\n  Let $P_0, Q_0$ be  spherical polygons.\n  \\begin{enumerate}\n    \\item Consider the space of all spherical polygons with the same edge lengths lying close\\footnote{With respect to any reasonable metric} to $P_0$ subject to rotation of the sphere.  This space is  called the \\textit{local configuration space }\\textit{of spherical polygons with prescribed edge lengths} $\\mathbf{L}^{loc}(P_0)$.\n    \\item Elimination of a point from the sphere allows to define the winding numbers for curves  in the sphere. So fix a point $\\infty\\notin P_0$ (that is, not lying on the broken line) and define the oriented area  of $P\\in \\mathbf{L}^{loc}(P_0)$  as the integral\n        $$A(P):=\\int_{S^2} w(P,x)dx.$$\n    \\item Analogously, we define the \\textit{local configuration space of spherical polygons with prescribed angles} $\\mathbf{S}^{loc}(Q_0)$. Once we set  some fixed orientations on the edges, we have a well defined perimeter function $\\mathcal{P}$  on the space $\\mathbf{S}^{loc}(Q_0)$. Note that the area $\\mathcal{A}$ is constant on $\\mathbf{S}^{loc}(Q_0)$.\n  \\end{enumerate}\n\\end{definition}\n\n\n\\begin{proposition}\\label{PropSphCrit}\n  Assume that a spherical polygon $P$ fits in a hemisphere not containing $\\infty$. $P$  is a critical point of the area function $\\mathcal{A}$ iff it is a cyclic polygon, that is, its vertices lie on a circle.\n\\end{proposition}\nProof.\n(1) Prove first the statement for  polygons with four edges.\nIf a polygon bounds a spherically convex  region, then the statement is classical: a cyclic convex  $4$-gon exhibits either the maximum or the minimum point of the area, depending on orientation of $P$.\n\n(''If'') Assume  that $P$ intersects itself  and is a cyclic polygon with vertices $1,2,3,4$. Add a new point $5$ on the circle  together with two new bars as is shown in Fig. \\ref{fourgon}.  Then $\\mathcal{A}(1234)=\\mathcal{A}(1254)-\\mathcal{A}(2543)$.  A local flex of the polygon $P$ induces flexes of the polygons\n$(1254)$ and $(2543)$. Since the latter are critical, the claim follows.\n\n(''Only if'') Assume that  $P$ intersects itself  and is a  critical point of $\\mathcal{A}$, but not a cyclic polygon.  Take a circle superscribing $412$ and add  a new point $5$ on the circle  together with two new bars as we did above.  Now $(125 4)$ is a critical polygon, and  $(2543)$  is not.\n $\\mathcal{A}(1234)=\\mathcal{A}(1254)-\\mathcal{A}(2543)$ completes the proof.\n\n\n(2) The general case (any number of edges) is obtained by verbatim repeating  the reasonings from \\cite{khipan}.\\qed\n\\begin{figure}[h]\n\\centering \\includegraphics[width=4 cm]{Crit4gon.eps}\n\\caption{Notation for Proposition \\ref{PropSphCrit}.}\\label{fourgon}\n\\end{figure}\n\n\n\n\\subsection*{Local duality of $\\mathbf{L}^{loc}(P)$  and $\\mathbf{S}^{loc}(P^o)$. }\nAssume that $P$ is a spherical cyclic polygon. It fits in a hemisphere, and so do all the polygons from $\\mathbf{L}^{loc}(P)  $.\nAssume also that a point  $\\infty$ lies beyond the hemisphere, so $\\mathcal{A}$ is well-defined on $\\mathbf{L}^{loc}(P)  $,\nand $P$ is a critical point.\n\n\\begin{definition}\n\\begin{enumerate}\n  \\item In the above setting, define\nthe dual  polygon $P^0$:\n\n\\begin{enumerate}\n\\item Fix the hemisphere containing P centered at the center of the superscribed circle.\n  \\item Assume that $P=\\{p_1,...,p_n\\}$. Take the big circles $C_1,...,C_n$  projectively dual to the points $\\{p_1,...,p_n\\}$.\n  \\item Connect the intersection points $q_i$ of $l_i$ and $l_{i+1}$ lying in the hemisphere  by short geodesics.\n  \\item  Orient each of the lines such that the circle lies on the  {left} from each of the lines.\n\\end{enumerate}\n\n  \\item Continuously extend the duality to $\\mathbf{L}^{loc}(P)  $. The extension  is uniquely defined by the condition that edges of the dual polygon lie on dual lines to the vertices of the initial polygon.\n\\end{enumerate}\n\n\n\\end{definition}\n\\begin{lemma}\\label{LemmaRelAP}For a cyclic $P$,\n  \\begin{enumerate}\n    \\item The polygon $P^o$ is  tangential.\n   \n  \\item \\(\\mu_{P}(\\mathcal{A})=n-3-\\mu_{P^o}(\\mathcal{P}).\\)\n  \n\n  \\end{enumerate}\n\\end{lemma}\nProof. (1) is straightforward.\n\n  Projective duality (on the unit sphere) takes edge lengths of a polygon to the exterior angles of the dual, and vice versa.\n  Since\n  $$\\mathcal{A}(P)=Const -R\\cdot \\mathcal{P}(P^o),$$\n  the claims (2) follows.\\qed\n\n\n\n\\bigskip\n\n\n\\subsection*{Morse indices: planar vs spherical.}\n\nLet $P_0$ be a planar cyclic polygon. It is uniquely defined by the circumscribed circle $\\sigma$ and the ordered sequence of its points. Put the circle $\\sigma$ with the $n$ points on the sphere of radius $R$, provided that $R>r(\\sigma)$. It defines a spherical cyclic polygon $P_0^R$ fitting in the hemisphere centered at the center of  $\\sigma$. In turn, the spherical polygon $P_0^R$  gives rise to the local configuration space $\\mathbf{L}^{loc}(P_0^R)$ of polygons  with prescribed edge lengths. Clearly, the bigger $R$ is, the smaller is the distortion of edge lengths and angles.\n\n\n\n\\begin{lemma}\\label{LemmaBifurkFree}\nThe Morse index of  the spherical polygon $P_0^R$ with respect to the area function $\\mathcal{A}_R$ is the same as the Morse index of $P_0$.\n\\end{lemma}\n\n {Proof.\n\\begin{comment} keeps all the angles of the polygon except for the two last ones, and $\\phi$ keeps the edge lengths except for the last one.\n\nThen in the neighborhood $U(Q_0)$, one has  $\\mathcal{P}(\\phi(Q))=\\mathcal{P}(Q)+\\frac{1}{R}\\cdot \\mathbf{R}(Q,R)$, where $\\mathbf{R}$ and its second derivatives are uniformly bounded.\n\\end{comment}\nWe have the one parametric family of spherical polygons and their local configuration spaces $P_0^R,\\mathbf{L}^{loc}(Q_0^R)$. The area function $\\mathcal{A}_R$  is well defined  on $\\mathbf{L}^{loc}(P_0^R)$, and $P_0^R$ is its critical point.   As the radius $R$ tends to infinity, the polygon  $P_0^R$ deforms and tends to $P_0$.  Since $P_0^R$ is  the unique critical point in the neighborhood, $P_0^R$ does not bifurcate, so its Morse index does not change.  Besides, by standard arguments, the Morse index of $P_0^R$  converges to the Morse index of the planar polygon $P_0$. \\qed\n\n\\bigskip\n\n\n\n\\bigskip\nNow we are ready to prove  Theorem \\ref{Thm_Morse_closed_plane}. Take a planar cyclic polygon $P$.\nTake $P^R$ for some big $R$.  By the above lemma, $\\mu_{P^R}(\\mathcal{A}_R)= \\mu_{P}(\\mathcal{A})$. Gradually make $R$ smaller, such that the circumscribed circle tends to an equator of the sphere. The Morse index stays the same. Now take the projectively dual polygon  $Q^R:=(P^R)^o$. It is a small (and therefore, almost planar) tangential polygon.\n\nBy Lemma \\ref{LemmaRelAP}, $$\\mu_{P^R}(\\mathcal{A}_R)=n-3-\\mu_{Q^R}(\\mathcal{P}_R).$$\nReplace  $Q^R$ by a planar polygon $Q$. On the one hand, the Morse index $\\mu_{Q}(\\mathcal{P})$ stays the same. On the other hand, we know the Morse index by\nTheorem \\ref{Thm_Morse_closed_plane}. It remains to observe  that\n  local  projective duality  maintains the winding number, and\n               takes  {left turns} to positively oriented edges.\n            \\qed\n\nThis approach also gives the following fact which exceeds Theorem \\ref{Thm_Morse_closed_plane}:\n\\begin{corollary}\\label{RemFinal}\nAssume we have a cyclic polygon $P$ such that (1) no two consecutive vertices are antipodal (with respect to the superscribed circle), and (2) the polygon does fit in a straight line. Then\n$P$ is a Morse point of the oriented area function iff it is not a bifurcating polygon.\n\\end{corollary}\nProof. The dual polygon is a non-degenerate Morse point, see Theorem \\ref{ThmMorseTangential}. \\qed\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{secI}Introduction}\nOne of the most studied phenomenon in low-energy nuclear physics, both experimentally and \ntheoretically, is the way nucleonic matter organises itself to support a variety of \nshapes observed in finite nuclei. The occurrence of different shapes, shape coexistence, \nand shape transitions have their origin in the evolution of single-nucleon shell structure \nwith nuclear deformation, angular momentum, temperature and number of valence nucleons. \nThe manifestation of shells is a generic property of finite fermion systems and shell closures, \nin particular, are characteristic of the confining single-particle potential. When nucleons \ncompletely fill a major shell (single or doubly closed-shell nuclei), the relatively large energy gap \nto the next shell stabilises a spherical shape, whereas long-range correlations between \nvalence nucleons in open-shell nuclei drive the nucleus toward deformed (quadrupole, octupole) \nequilibrium shapes. \n\nCoexistence of different shapes in a single nucleus, and shape (phase) transitions as a function of \nnucleon number, present universal phenomena that occur in light, medium-heavy, heavy and \nsuperheavy nuclei, and reflect the organisation of nucleons in finite nuclei \\cite{HW.11,BN.96,SP.08,CJC.10}. \nA unified description of shape evolution and shape coexistence over the entire chart of nuclides \nnecessitates a universal theory framework that can be applied to different mass regions. \nNuclear energy density functionals (EDF) provide an economic, global and accurate \nmicroscopic approach to nuclear structure that can be extended  \nfrom relatively light systems to superheavy nuclei, and from the\nvalley of $\\beta$-stability to the particle drip-lines \\cite{BHR.03,VALR.05,LNP.641,SR.07}. \nThis is particularly important for extrapolations to regions far from stability where not enough \ndata are available to determine the parameters of a more local approach such as, for \ninstance, the interacting shell model. \n\nThe basic implementation of the EDF framework is in terms of\nself-consistent mean-field (SCMF) models, in which an EDF is constructed\nas a functional of one-body nucleon density matrices that correspond to\na single product state. Nuclear SCMF models\neffectively map the many-body problem onto a one-body\nproblem, and the exact EDF is approximated by simple   \nfunctionals of powers and gradients of ground-state nucleon densities and currents. \nSome of the advantages of the EDF approach are apparent already at the \nSCMF level: an intuitive interpretation of mean-field results in terms of intrinsic \nshapes and single-nucleon states, and the possibility to formulate structure models \nin the full model space of occupied states, with no distinction between core and \nvalence nucleons. \n\nQuantitative studies of low-energy structure phenomena related to shell \nevolution and coexistence usually start from a constrained Hartree-Fock\nplus BCS (HFBCS), or Hartree-Fock-Bogoliubov (HFB) calculation of the\ndeformation energy surfaces with mass multipole moments as\nconstrained quantities. When based on microscopic EDFs or effective\ninteractions, such calculations comprise many-body correlations \nrelated to the short-range repulsive inter-nucleon interaction and \nlong-range correlations mediated by nuclear resonance modes.\nThe result are static symmetry-breaking product many-body states. \nThe basic idea of a deformation energy surface is that, even though the quantum  \nmany-body system is determined by a very large number of microscopic states, \nthese can be organised in a collection of basins that are robust to small external \nperturbations \\cite{Laughlin.00}. The basins can be structurally distinct and \ndistant from each other, but they can occur at comparable energies. \n\nThe constrained SCMF method, however, produces semi-classical \ndeformation energy surfaces. The static nuclear mean-field is characterised by the \nbreaking of symmetries of the underlying Hamiltonian -- translational, rotational, particle \nnumber and, therefore, includes important static correlations, e.g. deformations and \npairing. To calculate excitation spectra and electromagnetic transition rates\nit is necessary to extend the SCMF scheme to include collective correlations that\narise from symmetry restoration and fluctuations around the mean-field\nminima. Collective correlations are sensitive to shell effects, display\npronounced variations with particle number and, therefore, cannot be\nincorporated in a universal EDF but rather require an explicit \ntreatment \\cite{BHR.03,NVR.11,LNP.879}. \n\nOn the second level of implementation of nuclear EDFs that takes into\naccount collective correlations through the restoration of broken symmetries\nand configuration mixing of symmetry-breaking product states, the\nmany-body energy takes the form of a functional of all transition\ndensity matrices that can be constructed from the chosen set of\nproduct states. This set is chosen to restore symmetries or\/and to\nperform a mixing of configurations that correspond to specific collective\nmodes using, for instance, the (quasiparticle) random-phase approximation\n(QRPA) or the Generator Coordinate Method (GCM) \\cite{RS.80}.\nThe latter includes correlations related to finite-size fluctuations in a collective\ndegree of freedom and presents the most effective approach for\nconfiguration mixing calculations, with multipole moments used as coordinates \nthat generate the intrinsic wave functions.\n\nMany interesting phenomena related to shell evolution have been investigated \nover the last decade by employing GCM configuration mixing of angular-momentum \nand particle-number projected states based on energy density functionals or effective interactions,\nbut the extension of this method to non-axial shapes and\/or heavy nuclei still presents \nformidable conceptual and computational challenges \\cite{LNP.879,BH.08,RE.10,Yao.10,BABH.14}. \nIn an alternative approach to nuclear collective dynamics that restores rotational symmetry and \nallows for fluctuations around mean-field minima, a collective Hamiltonian\ncan be formulated, with deformation-dependent parameters determined by\nself-consistent mean-field calculations \\cite{RG.87,PR.09}. \nThe dynamics of the collective Hamiltonian is governed by the\nvibrational inertial functions and the moments of inertia \\cite{GR.80}, \nand these functions are\ndetermined by the microscopic nuclear energy\ndensity functional and the effective interaction in the pairing channel. \nFive-dimensional collective Hamiltonian models for quadrupole vibrational and rotational\ndegrees of freedom, with parameters determined by constrained triaxial SCMF calculations based \non the Gogny effective interaction \\cite{Delaroche10}, the Skyrme density functional \\cite{PR.09}, and \nrelativistic density functionals \\cite{NVR.11}, have been developed over the last decade and \napplied in a number of studies of structure phenomena related to shape coexistence and \nshape transitions. \n\nThe present study is based on the framework of relativistic energy density functionals, extended \nto include the treatment of collective correlations using the collective \nHamiltonian model \\cite{NVR.11}. To emphasise the universality of the EDF approach, all \nillustrative calculations performed in this study, from relatively light \nsystems to very heavy nuclei, have been carried out using a single energy density functional --\nDD-PC1 \\cite{Nik.08}. Starting from microscopic nucleon self-energies in nuclear matter, \nand empirical global properties of the nuclear matter equation of state, the coupling parameters \nof DD-PC1 were fine-tuned to the experimental masses of a set of 64 deformed nuclei in the \nmass regions $A \\approx 150 -180$ and $A \\approx 230 - 250$. The functional has been further \ntested in a number of mean-field and beyond-mean-field calculations in different mass \nregions. For the examples considered here, pairing correlations have been taken into account \nby employing an interaction that is separable in momentum space, and is completely determined \nby two parameters adjusted to reproduce the empirical bell-shaped pairing gap in symmetric \nnuclear matter \\cite{Tian_PLB.09}. For the details of the particular implementation of the \nEDF-based collective Hamiltonian used in the present study, \nwe refer the reader to Ref.~\\cite{NVR.11}. \n\\section{\\label{secII}Beyond the relativistic mean-field approximation: collective correlations}\n\nFor a self-consistent description of collective excitation spectra \nand electromagnetic transition rates, the framework of (relativistic) \nenergy density functionals has to be extended to take into account \ncollective correlations in relation to  \nrestoration of broken symmetries and fluctuations in collective coordinates. \nBoth types of correlations can be included simultaneously by mixing symmetry-projected states \ncorresponding to different values of chosen collective coordinates. \nThe most effective approach for configuration mixing calculations is the generator \ncoordinate method (GCM), with multipole moments used as coordinates that generate the \nintrinsic wave functions.\nThe GCM is based on the assumption\nthat, starting from a set of mean-field states $\\ket{\\Phi (q)}$ that \ndepend on a collective coordinate $q$, one can build\napproximate eigenstates of the nuclear Hamiltonian:\n\\begin{equation}\n\\ket{\\Psi_\\alpha} =  \\int d q {f_\\alpha(q)\\ket{\\Phi (q)}}\\;.\n\\label{GCM-state}\n\\end{equation}\nHere the basis states $\\ket{\\Phi (q)}$ are Slater determinants of single-nucleon states \ngenerated by constrained SCMF calculations. Several advanced implementations of the GCM have been \ndeveloped recently, fully based on the microscopic EDF framework. For the   \nrelativistic SCMF approach, in particular, the most advanced model performs \nconfiguration mixing of angular-momentum and particle-number\nprojected wave functions generated by constraints on quadrupole deformations \\cite{Yao.10,Yao.14}\n\\begin{equation}\n|JNZ;\\alpha \\rangle = \\int{dq\\sum_{K}{f_{\\alpha}^{JK} \\hat{P}^J_{MK} \\hat{P}^N\\hat{P}^Z |q\\rangle } },\n\\end{equation}\n\n\\noindent\nwhere $\\alpha = 1,2,\\dots$ denotes different collective states for a given angular momentum $J$,\nand $|q\\rangle \\equiv |\\beta, \\gamma\\rangle$ denotes a set of intrinsic SCMF states with deformation parameters\n$(\\beta,\\gamma)$. \n$\\hat{P}_{MK}^J$ is the angular momentum projection operator, and \nthe operators $\\hat{P}^N$ and $\\hat{P}^Z$ project onto states with \ngood neutron and proton number, respectively. \nThis implementation is equivalent to a seven-dimensional GCM calculation, mixing all five degrees of freedom of the \nquadrupole operator and the gauge angles for protons and neutrons. \nThe weight functions $f^{JK}_{\\alpha}(q)$ in the collective wave function \nare determined from the variational equation:\n\\begin{equation}\n \\delta E^{J} =\n \\delta \\frac{\\bra{\\Psi_\\alpha^{JM}} \\hat{H} \\ket{\\Psi_\\alpha^{JM}}}\n            {\\bra{\\Psi_\\alpha^{JM}}\\Psi_\\alpha^{JM}\\rangle} = 0 \\; ,\n\\label{variational}\n\\end{equation}\nthat is, by requiring that the expectation value of the Hamiltonian is\nstationary with respect to an arbitrary variation $\\delta\nf_{\\alpha}^{JK}$. This leads to the  Hill-Wheeler-Griffin (HWG)\nintegral equation:\n\\begin{equation}\n \\label{HWEq}\n \\int dq^\\prime\\sum_{K^\\prime\\geq 0}\n \\left[\\mathcal{H}^J_{KK^\\prime}(q,q^\\prime)\n - E^J_\\alpha\\mathcal{N}^J_{KK^\\prime}(q,q^\\prime)\\right]\n  f^{JK^\\prime}_\\alpha(q^\\prime)=0,\n\\end{equation}\n where $\\mathcal{H}$ and $\\mathcal{N}$ are the projected GCM\n kernel matrices of the Hamiltonian and the norm, respectively \\cite{BH.08,RE.10,Yao.10,Yao.14}.\n\nMultidimensional GCM calculations involve a number of technical and computational issues \\cite{LNP.879,Yao.14,BABH.14}, \nthat have so far impeded systematic applications to medium-heavy and heavy nuclei. \nCollective dynamics can also be described using an alternative method in which \na collective Hamiltonian is constructed, with deformation-dependent parameters determined from\nmicroscopic SCMF calculations \\cite{RG.87,PR.09}. \nThe collective Hamiltonian can be derived in\nthe Gaussian overlap approximation (GOA)~\\cite{RS.80} to the full\nmulti-dimensional GCM. With the assumption that the GCM overlap kernels can be\napproximated by Gaussian functions, the local expansion of the\nkernels up to second order in the non-locality transforms the\nGCM Hill-Wheeler equation into a second-order differential equation -\nthe Schr\\\"odinger equation for the collective Hamiltonian. \nFor instance, in the case of quadrupole degrees of freedom:\n\\begin{equation}\n\\label{hamiltonian-quad}\n\\hat{H}_{\\rm coll} = \\hat{T}_{\\rm{vib}}+\\hat{T}_{\\rm{rot}}\n              +V_{\\rm{coll}} \\; ,\n\\end{equation}\nwhere the vibrational kinetic energy is parameterized by the the mass parameters\n$B_{\\beta\\beta}$, $B_{\\beta\\gamma}$, $B_{\\gamma\\gamma}$\n\\begin{eqnarray}\n\\hat{T}_{\\textnormal{vib}} =-\\frac{\\hbar^2}{2\\sqrt{wr}}\n   \\left\\{\\frac{1}{\\beta^4}\n   \\left[\\frac{\\partial}{\\partial\\beta}\\sqrt{\\frac{r}{w}}\\beta^4\n   B_{\\gamma\\gamma} \\frac{\\partial}{\\partial\\beta}\n   - \\frac{\\partial}{\\partial\\beta}\\sqrt{\\frac{r}{w}}\\beta^3\n   B_{\\beta\\gamma}\\frac{\\partial}{\\partial\\gamma}\n   \\right]\\right.\n   \\nonumber \\\\\n   +\\frac{1}{\\beta\\sin{3\\gamma}}\\left.\\left[\n   -\\frac{\\partial}{\\partial\\gamma} \\sqrt{\\frac{r}{w}}\\sin{3\\gamma}\n      B_{\\beta \\gamma}\\frac{\\partial}{\\partial\\beta}\n    +\\frac{1}{\\beta}\\frac{\\partial}{\\partial\\gamma} \\sqrt{\\frac{r}{w}}\\sin{3\\gamma}\n      B_{\\beta \\beta}\\frac{\\partial}{\\partial\\gamma}\n   \\right]\\right\\} \\; ,\n \\end{eqnarray}\n\nthe three moments of inertia $\\mathcal{I}_k$ determine the \nrotational kinetic energy \n\\begin{equation}\n\\hat{T}_{\\textnormal{\\textnormal{\\textnormal{rot}}}} = \n\\frac{1}{2}\\sum_{k=1}^3{\\frac{\\hat{J}^2_k}{\\mathcal{I}_k}} \\; ,\n\\end{equation}\nand $V_{\\rm{coll}}$ is the collective potential that includes zero-point\nenergy (ZPE) corrections. $w$ and $r$ are products of mass parameters \nand moments of inertia, respectively, that specify the volume element in  \ncollective space \\cite{Nik.09}. \nThe self-consistent mean-field solution for the \nsingle-quasiparticle energies and wave functions for\nthe entire energy surface, as functions of the quadrupole\ndeformations $\\beta$ and $\\gamma$, provide the microscopic input for \ncalculation of the mass parameters, moments of inertia and the collective \npotential. The Hamiltonian describes quadrupole vibrations,\nrotations, and the coupling of these collective modes.\n \nThe dynamics of the collective Bohr Hamiltonian is governed by the\nvibrational inertial functions and the moments of inertia \\cite{GR.80}.\nFor these quantities either the GCM-GOA \nor the adiabatic approximation to the time-dependent HFB (ATDHFB) expressions \n(Thouless-Valatin masses) can be used. The Thouless-Valatin masses\nhave the advantage that they also include the time-odd components of\nthe mean-field potential and, in this sense, the full dynamics of a\nnuclear system. In the GCM approach these components can only be included if,\nin addition to the coordinates  $q_i$, the corresponding\ncanonically conjugate momenta $p_i$ are also taken into account. \nIn many applications, including the collective Hamiltonians considered in the present study, a further\nsimplification is thus introduced in terms of cranking formulas,\ni.e. the perturbative limit for the Thouless-Valatin masses, and the\ncorresponding expressions for ZPE corrections \\cite{Nik.09}. In the present implementation of the \ncollective Hamiltonian model the moments of inertia and \nmass parameters do not include the contributions of time-odd mean-fields (the so \ncalled dynamical rearrangement contributions) and, to a certain extent, this breaks \nthe self-consistency of the approach \\cite{Hin.12}. \n\nThe diagonalization of the collective Hamiltonian gives the \nenergy spectrum $E_\\alpha^I$ and the corresponding eigenfunctions\n\\begin{equation}\n\\label{wave-coll}\n\\Psi_\\alpha^{IM}(\\beta,\\gamma,\\Omega) =\n  \\sum_{K\\in \\Delta I}\n           {\\psi_{\\alpha K}^I(\\beta,\\gamma)\\Phi_{MK}^I(\\Omega)}\\; , \n\\end{equation}\nthat are used to calculate various observables, for instance the E2 reduced\ntransition probabilities. The shape of a nucleus can be characterized in a qualitative way\nby the expectation values of invariants $\\beta^2$, $\\beta^3\\cos{3\\gamma}$, as well as their\ncombinations.\n\nNuclear excitations characterised by quadrupole and octupole vibrational and rotational degrees of\nfreedom can be simultaneously described by considering quadrupole and octupole collective coordinates\nthat specify the surface of a nucleus $R=R_0\\left[1+\\sum_\\mu{\\alpha_{2\\mu}Y^*_{2\\mu}\n  + \\sum_\\mu{\\alpha_{3\\mu}Y_{3\\mu}^*} } \\right]$. In addition, when axial symmetry is imposed, \n  the collective coordinates\n  can be parameterized in terms of two deformation parameters $\\beta_2$ and $\\beta_3$, and three Euler\n  angles  $ \\Omega\\equiv(\\phi,\\theta,\\psi)$. After quantization the collective Hamiltonian takes the form\n\\begin{eqnarray}\n{\\hat H}_{\\rm coll} &= -\\frac{\\hbar^2}{2\\sqrt{w{\\cal I}}}\n              \\left[\\frac{\\partial}{\\partial\\beta_2}\\sqrt{\\frac{{\\cal I}}{w}}B_{33}\\frac{\\partial}{\\partial\\beta_2}\n                      -\\frac{\\partial}{\\partial\\beta_2}\\sqrt{\\frac{{\\cal I}}{w}}B_{23}\\frac{\\partial}{\\partial\\beta_3} \n            -\\frac{\\partial}{\\partial\\beta_3}\\sqrt{\\frac{{\\cal I}}{w}}B_{23}\\frac{\\partial}{\\partial\\beta_2}\\right.\\nonumber\\\\\n&\n                       \\left. +\\frac{\\partial}{\\partial\\beta_3}\\sqrt{\\frac{{\\cal I}}{w}}B_{22}\\frac{\\partial}{\\partial\\beta_3}\\right.\n           +\\frac{\\hat{J}^2}{2{\\cal I}}+{V}(\\beta_2, \\beta_3),\n   \\label{eq:CH2}\n\\end{eqnarray}\nwhere the mass parameters $B_{22}$, $B_{23}$ and $B_{33}$, and the moment of inertia\n$\\mathcal{I}$, are functions of the quadrupole  $\\beta_2$ and octupole $\\beta_3$ deformations.\n$w=B_{22}B_{33}-B_{23}^2$. \n\nJust as in the case of the quadrupole five-dimensional collective Hamiltonian, the moments of inertia \nare\ncalculated from the Inglis-Belyaev formula:\n\\begin{equation}\n\\label{Inglis-Belyaev}\n\\mathcal{I}= \\sum_{i,j}{\\frac{| \\langle ij |\\hat{J} | \\Phi \\rangle |^2}{E_i+E_j}}\\;,\n\\end{equation}\nwhere $\\hat{J}$ is the angular momentum along the axis perpendicular to the symmetric axis, \nthe summation runs over\nproton and neutron quasiparticle states\n$|ij\\rangle=\\beta^\\dagger_i\\beta^\\dagger_j|\\Phi\\rangle$, and\n$|\\Phi\\rangle$ represents the quasiparticle vacuum. The quasiparticle\nenergies $E_i$ and wave functions are determined by SCMF calculations of \ndeformation energy surfaces with constraints on the quadrupole and octupole \ndeformation parameters. \nThe mass parameters associated with the collective \ncoordinates $q_2=\\langle\\hat{Q}_{2}\\rangle$ and $q_3=\\langle\\hat{Q}_{3}\\rangle$\nare calculated in the cranking approximation, as well as the vibrational and rotational   \nzero-point energy corrections to the collective energy surface \\cite{GG.79}. \n\\section{\\label{secIII}Evolution of shapes and coexistence in $\\bm{N=28}$ isotones}\nNuclei with closed major proton and\/or neutron shells are usually characterised by spherical \nequilibrium shapes. However, this is not necessarily the case for nuclei away from \nthe $\\beta$-stability line in which energy spacings between single-particle levels \ncan undergo considerable changes \nwith the number of neutrons and\/or protons. This can result in\nreduced spherical shell gaps, modifications of shell structure,\nand in some cases spherical magic numbers may disappear.\nThe reduction of a spherical shell closure is associated with the\noccurrence of deformed ground states and, in a number of cases,\nwith the phenomenon of shape coexistence. Because of the low density of \nsingle-particle states close to the Fermi surface, in relatively light nuclei coexistence \noccurs only when proton and neutron density distributions favour different \nequilibrium deformations, e.g. prolate {\\em vs} oblate shapes. \nHere we consider the well known example of neutron-rich $N=28$ isotones, which \nexhibit rapid shape variations and shape coexistence. \n\nIn Fig.~\\ref{fig:n28_pes} we plot the self-consistent triaxial \nquadrupole deformation energy surfaces of $N=28$ isotones \\cite{Li.11}. \nThe equilibrium shape of the doubly-magic nucleus $^{48}$Ca is, of course, \nspherical but the $N=28$ spherical shell is strongly reduced in the isotones \nwith a smaller number of protons. This leads to rapid transitions between deformed equilibrium \nshapes and shape coexistence in $^{44}$S. \nThe energy surface of $^{46}$Ar is \nsoft both in $\\beta$ and $\\gamma$ directions, with a shallow extended minimum\nalong the oblate axis. Only four protons away from the doubly magic $^{48}$Ca,\nin $^{44}$S the self-consistent mean-field calculation predicts a coexistence of \nprolate and oblate minima separated by a rather low barrier ($ < 1$ MeV). \nFor $^{42}$Si the energy surface displays a deep oblate minimum at\n$(\\beta,\\gamma)=(0.35, 60^\\circ)$, whereas a  prolate equilibrium minimum \nat $(\\beta,\\gamma)=(0.45, 0^\\circ)$ is predicted in the very neutron-rich nucleus $^{40}$Mg.\nSimilar results for the quadrupole deformation energy surfaces\nwere also obtained in self-consistent Hartree-Fock-Bogoliubov (HFB) studies \nbased on the finite-range and density-dependent Gogny interaction D1S~\\cite{Delaroche10,RE.11}.\n\n\\begin{figure}[]\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[scale=0.42]{Figs\/N=28\/pes_ar46.eps}&\n\\hspace{-1cm}\\includegraphics[scale=0.42]{Figs\/N=28\/pes_s44.eps} \\\\\n\\includegraphics[scale=0.42]{Figs\/N=28\/pes_si42.eps}&\n\\hspace{-1cm}\\includegraphics[scale=0.42]{Figs\/N=28\/pes_mg40.eps} \n\\end{tabular}\n\\caption{\\label{fig:n28_pes} Self-consistent triaxial\nquadrupole constrained energy surfaces of $N=28$ isotones in the $\\beta-\\gamma$ plane\n($0\\le \\gamma \\le 60^0$). For each nucleus energies are normalized with respect to the\nbinding energy of the global minimum. The contours join points on\nthe surface with the same energy, and the spacing between \nneighbouring contours is 0.5 MeV.}\n\\end{center}\n\\end{figure}\n\nRapid transitions between equilibrium shapes in a chain of isotones (isotopes)  \nare governed by the evolution of the shell structure of single-nucleon\norbitals. In particular, one or more close-lying deformed minima can develop  \nas a result of the occurrence of gaps or regions of low single-particle level density\naround the Fermi surface at finite deformation. In the present analysis we illustrate \nthis phenomenon with the example of shape coexistence in $^{44}$S.\n\nThere are many experimental indications  for the transitional nature of $^{44}$S:\nthe low-lying $0_2^+$ and $2_1^+$ excited states, the large value of the reduced \ntransition probability $B(E2, 2_1^+ \\to 0_1^+)$~\\cite{Grevy.05,Glasmacher.97}, \nthe monopole strength $\\rho^2(E0, 0_2^+ \\to 0_1^+)$, and the \nreduced transition probability $B(E2, 2_1^+ \\to 0_2^+)$ determined in a recent \nexperiment~\\cite{Force.11}. Based on these results, shell model calculations and\na simple two-level mixing model have shown that $^{44}$S exhibits \na shape coexistence between a prolate ground state and a spherical $0_2^+$ excited state.\nOn the other hand, spectroscopic calculations based on the self-consistent mean-field \napproach indicate a coexistence of prolate and\noblate shapes in $^{44}$S~\\cite{RE.11,Li.11}.\nIn a very recent shell model study several inconsistencies in previous interpretations\nof data for $^{44}$S have been resolved~\\cite{Chevrier.14}. Using quadrupole invariants\nto determine axial and triaxial shape parameters from a shell model calculation, \na pronounced triaxial\nshape for the ground state and a slightly more pronounced prolate shape for the excited\n$0_2^+$ state have been predicted. It has been shown that these states display a \nlarge overlap in the $(\\beta,\\gamma)$\nplane. Higher-lying members of the ground-state band show a tendency towards prolate deformation,\nwhereas strong fluctuations have been predicted in the band built on top of the  state $0_2^+$.\nAn isomeric state $4_1^+$ was recently observed in $^{44}$S~\\cite{Santiago-Gonzalez.11},\nand the nature of this state has been explored in a shell model study~\\cite{Utsuno.15} which has\nshown that this state most probably corresponds to a two-quasiparticle $K=4$ configuration. \n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[scale=0.6]{Figs\/N=28\/s44_levels.eps}\n\\caption{\\label{fig:s44_levels}Proton (upper panel) and neutron (lower panel) \nsingle-nucleon energy levels of $^{44}$S, as functions of the\ndeformation parameters along a closed path in the $\\beta - \\gamma$ plane.\nSolid (blue) curves correspond to levels with negative parity, and\n(red) dashed curves denote positive-parity levels.\nThe dot-dashed (green) curves corresponds to the Fermi levels.\nThe panels on the left and right display prolate ($\\gamma =0^\\circ$) and oblate\n ($\\gamma =60^\\circ$) axially-symmetric single-particle levels, respectively.\n In the middle panel the proton and neutron levels are\n plotted as functions of $\\gamma$ for a fixed value \n $|\\beta|=0.35$. }\n\\end{center}\n\\end{figure}\n\n\nIn Fig.~\\ref{fig:s44_levels} we plot the proton (upper panel) and neutron (lower panel) single-particle levels in the canonical basis for $^{44}$S. \nSolid (blue) curves correspond to levels with negative parity, and dashed (red) curves denote positive-parity levels. \nThe dot-dashed (green) curves correspond to \nthe Fermi levels. The proton and neutron levels are plotted as functions of the deformation parameters along a closed \npath in the $\\beta$-$\\gamma$ plane. The panels on the left and right display prolate \n($\\gamma = 0^\\circ$) and oblate ($\\gamma = 60^\\circ$) axially symmetric single-particle levels, respectively. \nIn the middle panel of each figure the neutron and proton levels are plotted as functions of $\\gamma$ \nfor a fixed value of the axial deformation $|\\beta| = 0.35$ which corresponds to the position of the prolate mean-field minimum \nin $^{44}$S isotope (cf. Fig.~\\ref{fig:n28_pes}). Starting from the spherical configuration, we follow the single-nucleon levels on a \npath along the prolate axis up to the approximate position of the minimum (left panel), then for this fixed value of $\\beta$ the path from \n$\\gamma = 0^\\circ$ to $\\gamma=60^\\circ$ (middle panel) and, finally, back to\nthe spherical configuration along the oblate axis (right panel). \nAxial deformations with $\\gamma=60^\\circ$ are denoted by negative values of $\\beta$. \nThis figure illustrates the principal\ncharacteristics of structural changes in neutron-rich $N=28$ nuclei: the near degeneracy\nof the $d_{3\/2}$ and $s_{1\/2}$ proton orbitals, and the reduction of the size of the\n$N=28$ shell gap \\cite{Sorlin.10}. Between the doubly magic $^{48}$Ca and $^{44}$S the \nspherical gap $N=28$ decreases from 4.73 MeV to 3.86 MeV, respectively. \nConsequently, the largest gap\nbetween neutron states at the Fermi surface \nis located on the oblate axis (lower panel of Fig.~\\ref{fig:s44_levels}), \nand we also notice the increased density of single-neutrons levels close to the\nFermi surface at $\\gamma \\approx 30^\\circ$ which leads to formation of \nthe potential barrier in the triaxial region.\nFor the protons (upper panel of Fig.~\\ref{fig:s44_levels}), the largest gap\nis located on the prolate axis. The competition between pronounced \nproton-prolate and neutron-oblate energy gaps is at the origin\nof the coexistence of deformed shapes in $^{44}$S.\n\n\\begin{figure}[]\n\\centering\n\\includegraphics[scale=0.65]{Figs\/N=28\/s44_spectrum.eps}\n\\caption{\\label{fig:s44_spec}The theoretical excitation spectrum of $^{44}$S (left), compared to \ndata~\\cite{Glasmacher.97,Force.11,Santiago-Gonzalez.11} (right). The\n$B(E2)$ values are in units of $e^2$fm$^4$. The comparison of\nthe monopole transition strength $\\rho^2(E0;0_2^+ \\to 0_1^+)$ \nwith the experimental value is also included.  }\n\\end{figure}\n\nStarting from constrained self-consistent solutions of the relativistic \nHartree-Bogoliubov (RHB) equations at each point on the \nenergy surfaces (Fig.~\\ref{fig:n28_pes}), we calculate \nthe mass parameters $B_{\\beta \\beta}$, $B_{\\beta \\gamma}$, $B_{\\gamma \\gamma}$,\nthe three moments of inertia $\\mathcal{I}_k$, as well as the zero-point energy corrections,  \nthat determine the collective Hamiltonian~(\\ref{hamiltonian-quad}).\nThe diagonalization of the resulting Hamiltonian yields the excitation energies and reduced transition\nprobabilities. Physical observables are calculated in the full configuration space and there are no \neffective charges in the model. In Fig.~\\ref{fig:s44_spec} we display the low-energy spectrum of $^{44}$S \nin comparison to available data for the excitation energies, reduced electric quadrupole\ntransition probabilities $B(E2)$ (in units of $e^2$fm$^4$), and the electric monopole transition\nstrength $\\rho^2(E0;0_2^+ \\to 0_1^+)$. The model reproduces both the excitation energy and\nthe reduced transition probability $B(E2; 2_1^+ \\to 0_1^+)$ for the first excited\nstate $2_1^+$. The theoretical value for $B(E2;0_2^+ \\to 2_1^+)$ is also in good agreement with\nthe data. However, the calculated excitation energy of the $0_2^+$ state is much \nhigher than the experimental counterpart, and the monopole\ntransition strength $\\rho^2(E0;0_2^+ \\to 0_1^+)$ overestimates the\nexperimental value considerably. This indicates that there is probably more mixing between the \ntheoretical states $0_1^+$ and $0_2^+$ than what can be inferred from the data. We also \nnote that very recently the low-lying state $4_1^+$ has been interpreted \nas a $K = 4$ isomer dominated by the two-quasiparticle configuration \n$\\nu \\Omega^\\pi  = 1\/2^-  \\otimes  \\nu \\Omega^\\pi  = 7\/2^-$ \\cite{Utsuno.15}, a configuration \nnot included in our collective model space.\n\n\\begin{figure}[t]\n\\begin{center}\n\\begin{tabular}{ccc}\n\\hspace{-1cm}\\includegraphics[scale=0.3]{Figs\/N=28\/s44_wave_0_1.eps}&\n\\hspace{-1cm}\\includegraphics[scale=0.3]{Figs\/N=28\/s44_wave_2_2.eps}&\n\\hspace{-1cm}\\includegraphics[scale=0.3]{Figs\/N=28\/s44_wave_0_2.eps} \\\\\n\\hspace{-1cm}\\includegraphics[scale=0.3]{Figs\/N=28\/s44_wave_2_1.eps}&\n\\hspace{-1cm}\\includegraphics[scale=0.3]{Figs\/N=28\/s44_wave_3_1.eps}&\n\\hspace{-1cm}\\includegraphics[scale=0.3]{Figs\/N=28\/s44_wave_2_3.eps} \\\\\n\\hspace{-1cm}\\includegraphics[scale=0.3]{Figs\/N=28\/s44_wave_4_1.eps}&\n\\hspace{-1cm}\\includegraphics[scale=0.3]{Figs\/N=28\/s44_wave_4_2.eps}&\n\\hspace{-1cm}\\includegraphics[scale=0.3]{Figs\/N=28\/s44_wave_4_4.eps}\n\\end{tabular}\n\\caption{\\label{fig:s44_waves} Probability distributions Eq.~(\\ref{eq:probability})\nin the $\\beta-\\gamma$ plane for the lowest collective states of $^{44}$S.}\n\\end{center}\n\\end{figure}\n\nTo illustrate the degree of configuration mixing and shape coexistence \nin $^{44}$S, in Fig.~\\ref{fig:s44_waves} we plot the\nprobability density distributions for the lowest three states in the ground-state band (left), \nthe (quasi) $\\gamma$-band (middle), and the excited band built on the state $0_2^+$ (right). \nFor a given collective state, the probability distribution\nin the $(\\beta,\\gamma)$ plane is defined as\n\\begin{equation}\n\\rho_{I\\alpha}(\\beta,\\gamma) = \\sum_{K \\in \\Delta I}{ \\left| \\psi_{\\alpha K}^I(\\beta,\\gamma)\\right|^2\\beta^3 },\n\\label{eq:probability}\n\\end{equation}\nwith the summation over the allowed set of values of the projection $K$ of the angular \nmomentum $I$ on the body-fixed symmetry axis, and \nwith the normalization\n\\begin{equation}\n\\int_0^\\infty{\\beta d\\beta \\int_0^{2\\pi}{ \\rho_{I\\alpha}(\\beta,\\gamma)|\\sin{3\\gamma}|d\\gamma }}=1.\n\\end{equation}\n\n\\begin{table}\n\\caption{\\label{tab:s44_K_spec}Percentage of the $K=0$ and $K=2$ components (projection of the\nangular momentum on the body-fixed symmetry axis) for the collective wave functions of the\nthree lowest $2^+$ states in $^{44}$S, and the corresponding spectroscopic quadrupole moments\n(in $e$fm$^2$).}\n\\centering\n\\begin{tabular}{cccc}\n\\hline\\hline\n &  $K = 0$ & $K = 2$ & $Q_{spec}$ \\\\ \\hline\n $2_1^+$ &  $89\\%$ & $11\\%$& $-10.8$\\\\\n $2_2^+$ & $21\\%$ & $79\\%$ & $8.2$\\\\\n$2_3^+$  & $78\\%$ & $22\\%$ & $-7.3$\\\\\n \\hline \\hline\n\\end{tabular}\n\\end{table}\n\nThe probability distribution of the ground state $0_1^+$ displays a deformation $|\\beta| \\ge 0.3$,\nextended in the $\\gamma$ direction from the prolate $\\gamma=0^\\circ$ to the oblate\n$\\gamma=60^\\circ$ shape. The $\\gamma$ softness of the $0_1^+$ state reflects the \nground-state mixing of configurations based on the prolate and oblate minima of the \nmean-field potential (cf. Fig.~\\ref{fig:n28_pes}).\nStates of the ground-state band with higher angular momenta are progressively concentrated\non the prolate axis and the average $\\beta$ deformation gradually increases because of \ncentrifugal stretching. Although the $0_2^+$ state is predominantly prolate, one notices \noblate admixtures and,\nconsequently, a relatively large overlap between the wave functions of $0_1^+$ and \n$0_2^+$. The mixing between these states leads to a pronounced level repulsion which \nis probably the cause for the too high excitation energy of the \ntheoretical state $0_2^+$. \nThe low-lying $0_2^+$ state at $1.365$ MeV and the monopole strength \n$\\rho^2(E0;0_2^+ \\to 0_1^+)=8.7(7)\\times 10^{-3}$ have been regarded as signatures of\nprolate-spherical shape coexistence in $^{44}$S~\\cite{Force.11}. \nHowever, recent studies have re-examined the structure of $^{44}$S~\\cite{Chevrier.14},\nemphasizing the effect of the triaxial degree of freedom on the low-lying excitation \nstructure. The probability distribution of \n$2_3^+$ is concentrated on the prolate axis and the transition strength $B(E2;2_3^+\\to 0_2^+)$\nis comparable to $B(E2;2_1^+\\to 0_1^+)$. \nFor the three lowest $2^+$ levels, in Table~\\ref{tab:s44_K_spec} we list the percentage of the\n$K=0$ and $K=2$ components in the collective wave functions, together with the spectroscopic\nquadrupole moments. The wave functions of the states $2_1^+$ and $2_3^+$ are dominated by\nthe $K=0$ components, and the spectroscopic quadrupole moments indicate prolate configurations.\nIn contrast the positive quadrupole moment of $2_2^+$ state points to a predominantly oblate\nconfiguration, while the $\\approx 80\\%$ contribution of the $K=2$ component in the wave function\nshows that this state is the band-head of a (quasi) $\\gamma$-band. One also notices the\nclose-lying doublet $3_1^+$ and the $4_2^+$, characteristic for a $K=2$ band in a $\\gamma$-soft \npotential.\n\nThe example considered in this section and similar calculations reported recently have shown that  \nthe EDF approach provides an accurate microscopic interpretation of the reduction of the $N=28$\nspherical energy gap in neutron-rich nuclei, and a quantitative description of the evolution\nof shapes in $N=28$ isotones in terms of single-nucleon orbitals as functions of the\nquadrupole deformation parameters $\\beta$ and $\\gamma$. In particular,  \nthe formation of the oblate neutron and prolate proton gaps in $^{44}$S, illustrated in Fig.~\\ref{fig:s44_levels}, \nis at the origin of the predicted shape coexistence, in very good agreement with recent data.\n\n\\section{\\label{secIV}Lowest $\\bm{0^+}$ excitations in $\\bm{N \\approx 90}$ rare-earth nuclei}\n\nRare-earth nuclei with neutron number  $N\\approx 90$ present some of the best examples \nof rapid shape evolution and shape phase transitions \\cite{HW.11,CJC.10,CMC.07}. Employing a \nconsistent framework of structure models (GCM, quadrupole collective Hamiltonian) based on \nrelativistic energy density functionals, in a series of studies \\cite{NVL.07,LNV.09,Li.09} we \nanalysed microscopic signatures of ground-state shape phase transitions in this \nregion of the nuclear mass table. Phase transitions in equilibrium shapes of atomic nuclei correspond\nto first- and second-order quantum phase transitions (QPT) between\ncompeting ground-state phases induced by variation of a non-thermal\ncontrol parameter (number of nucleons) at zero temperature. In general, one observes \na a gradual evolution of shapes with the number of nucleons, and these transitions \nreflect the underlying modifications of shell structure and interactions between valence nucleons. \nA phase transition, on the other hand, is characterised by a significant variation of one or more order\nparameters as functions of the control parameter. Even though in systems composed\nof a finite number of particles phase transitions are actually smoothed out, in many cases clear \nsignatures of abrupt changes of structure properties are observed. A number of experiments \nover the last two decades, as well as many theoretical studies of deformation energy surfaces and \nalso direct computation of observables related to order parameters, have shown that \ntwo-neutron separation energies, isotope shifts, energy gaps between the ground state and the \nexcited vibrational states with zero angular momentum,\nisomer shifts, and monopole transition strengths, exhibit sharp discontinuities at neutron number\n$N=90$. \n\n\\begin{figure}[]\n\\centering\n\\includegraphics[scale=0.65]{Figs\/N=90\/sm152_spec_corr.eps}\n\\caption{\\label{fig:sm152_spec}The theoretical excitation spectrum of $^{152}$Sm \n(left), compared to data~\\cite{data}. The intraband \nand interband B(E2) values (thin solid arrows) are in Weisskopf units(W.u.), \nand (red) dashed arrows denote E0 transitions with the \ncorresponding $\\rho^2(E0) \\times 10^3$ values.}\n\\end{figure}\n\\begin{figure}[]\n\\centering\n\\includegraphics[scale=0.65]{Figs\/N=90\/gd154_spec_corr.eps}\n\\caption{\\label{fig:gd154_spec} Same as in the caption to Fig.~\\ref{fig:sm152_spec} \nbut for $^{154}$Gd.}\n\\end{figure}\n\\begin{figure}[]\n\\centering\n\\includegraphics[scale=0.65]{Figs\/N=90\/dy156_spec.eps}\n\\caption{\\label{fig:dy156_spec}Same as in the caption to Fig.~\\ref{fig:sm152_spec} \nbut for $^{156}$Dy.}\n\\end{figure}\n\nIn the present study we focus on the low-lying $0^+$ excitations in the deformed \n$N=90$ isotones and examine the mixing between the lowest $K=0$ bands. Traditionally the \nfirst excited $0^+$ level in deformed nuclei has been interpreted as a $\\beta$-vibrational state, \nwith the associated $K=0$ rotational $\\beta$-band. However, many experimental studies \nhave shown that most of the $0^+_2$ excitations are not $\\beta$ vibrations. In the \nexhaustive review of properties of the lowest-lying $0^+$ states in deformed rare-earth \nnuclei \\cite{Garr.01}, it was emphasized that there is no {\\it a priori} reason to associate \nthe $\\beta$ vibration with the lowest $0^+$ excited state. A set of properties was suggested that \nthe first excited $0^+$ level should exhibit in order to be labelled as a $\\beta$ vibration. \nAmong those: $B(E2; 0^+_\\beta \\to  2^+_1)$ values of $12 - 33$ W.u. or conversely \n$B(E2; 2^+_\\beta \\to  0^+_1)$ of $2.5 - 6$ W.u., and $\\rho^2(E0;{0^+_2 \\to 0^+_1}) \\times 10^3$ values of $85-230$.\nIn our microscopic analysis of order parameters in nuclear quantum phase transitions \\cite{Li.09}, \nin particular for the Nd isotopic chain, it was shown that \nthe excitation energies of both $0^+_2$ and $0^+_3$ exhibit a pronounced dip at\n$N=90$, which can be attributed to the softness of the potential with respect to $\\beta$\ndeformation in $^{150}$Nd. The calculated monopole transition strengths \nexhibit a pronounced increase toward $N=90$, and the \n$\\rho^2(E0;{0^+_2 \\to 0^+_1})$ values remain rather large\nin the deformed nuclei $^{152,154,156}$Nd, a behaviour \ncharacteristic for an order parameter at the point of first-order QPT.\n\nIn Figs.~\\ref{fig:sm152_spec} - \\ref{fig:dy156_spec} we plot the theoretical low-energy \nspectra of the $N=90$ nuclei: $^{152}$Sm, $^{154}$Gd and $^{156}$Dy, in comparison \nwith data~\\cite{data}. The ground-state bands, lowest $K=0$ and $K=2$ bands \nare compared to their experimental counterparts: excitation energies, intraband \nand interband $B(E2)$ values, and $E0$ transition strengths. The theoretical spectra \ncomprise eigenstates of the five-dimensional quadrupole collective Hamiltonian~(\\ref{hamiltonian-quad}),  \nbased on triaxial SCMF solutions obtained with the density functional DD-PC1 and \na finite-range pairing force separable in momentum space. No additional parameters \nare adjusted to data, and transition rates have been calculated with bare \ncharges, that is, $e_p = e$ and $e_n =0$. \n\n\\begin{figure}[]\n\\begin{center}\n\\begin{tabular}{ccc}\n\\hspace{-1cm}\\includegraphics[scale=0.3]{Figs\/N=90\/sm152_waves_0_1.eps}&\n\\hspace{-1cm}\\includegraphics[scale=0.3]{Figs\/N=90\/gd154_waves_0_1.eps}&\n\\hspace{-1cm}\\includegraphics[scale=0.3]{Figs\/N=90\/dy156_waves_0_1.eps}\\\\\n\\hspace{-1cm}\\includegraphics[scale=0.3]{Figs\/N=90\/sm152_waves_0_2.eps}&\n\\hspace{-1cm}\\includegraphics[scale=0.3]{Figs\/N=90\/gd154_waves_0_2.eps}&\n\\hspace{-1cm}\\includegraphics[scale=0.3]{Figs\/N=90\/dy156_waves_0_2.eps}\\\\\n\\hspace{-1cm}\\includegraphics[scale=0.3]{Figs\/N=90\/sm152_waves_0_3.eps}&\n\\hspace{-1cm}\\includegraphics[scale=0.3]{Figs\/N=90\/gd154_waves_0_3.eps}&\n\\hspace{-1cm}\\includegraphics[scale=0.3]{Figs\/N=90\/dy156_waves_0_3.eps}\\\\\n\\end{tabular}\n\\caption{\\label{fig:N=90_0_states} Probability distributions Eq.~(\\ref{eq:probability})\nin the $\\beta-\\gamma$ plane for the lowest collective $0^+$ states of $^{152}$Sm, $^{154}$Gd and $^{156}$Dy.}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[t]\n\\begin{center}\n\\begin{tabular}{ccc}\n\\hspace{-1cm}\\includegraphics[scale=0.3]{Figs\/N=90\/sm152_wave_2_3.eps}&\n\\hspace{-1cm}\\includegraphics[scale=0.3]{Figs\/N=90\/gd154_wave_2_3.eps}&\n\\hspace{-1cm}\\includegraphics[scale=0.3]{Figs\/N=90\/dy156_wave_2_3.eps}\\\\\n\\end{tabular}\n\\caption{\\label{fig:N=90_2_3_states} Probability distributions Eq.~(\\ref{eq:probability})\nin the $\\beta-\\gamma$ plane for the band-heads of the $K=2$ $\\gamma$-bands \nof $^{152}$Sm, $^{154}$Gd and $^{156}$Dy.}\n\\end{center}\n\\end{figure}\n\nFor the ground-state bands the theoretical excitation energies and $B(E2)$ values for \ntransitions within the band are in very good agreement with data, except for the fact that the \nempirical moments of inertia are systematically larger than those calculated with the\ncollective Hamiltonian. This is a well known effect of using the simple Inglis-Belyaev \napproximation for the moments of inertia, and is also reflected in the excitation energies \nof the excited $K=0$ and $K=2$ bands \\cite{DGL.10}. The wave functions, however, are not \naffected by this approximation and we note that the model \nreproduces both the intraband and interband $E2$ \ntransition probabilities. The $K=2$ $\\gamma$-bands are predicted at somewhat  \nhigher excitation energies compared to their experimental counterparts, and this is most \nprobably due to the potential energy surfaces being too stiff in $\\gamma$.  \nThe deformed rare-earth $N=90$ isotones are characterised by very low $K=0$ bands \nbased on the $0^+_2$ states. In $^{152}$Sm, for instance, this state is found \nat 685 keV excitation energy, considerably below the $K=2$ $\\gamma$-band. Nevertheless, \nthis state has been interpreted as the band-head of the $\\beta$-band \\cite{Garr.01,DGL.10}. \n\nIn $^{152}$Sm the excited $K=0$ band is calculated at moderately higher energy \ncompared to data, while the agreement with experiment is very good for $^{154}$Gd and $^{156}$Dy. \nWe note that a very similar excitation spectrum for $^{152}$Sm was also obtained \nwith the collective Hamitonian based on the D1S Gogny interaction \\cite{DGL.10}. \nParticularly important for the present study are the $E0$ transitions between the two lowest \n$K=0$ bands, and the $B(E2; 0^+_2 \\to  2^+_1)$ and $B(E2; 2^+_2 \\to  0^+_1)$ values. \nThe available data are very accurately reproduced by the calculation and, in particular \nfor $^{152}$Sm, the $E0$ transition strengths and $B(E2)$ values seem to match the \ncriteria for a $\\beta$-vibrational state \\cite{Garr.01}.  \n\nThe $E0$ transitions strengths reflect the degree of mixing between the two lowest $K=0$ bands, \nand Figs.~\\ref{fig:sm152_spec} - \\ref{fig:dy156_spec} show that the theoretical values that \ncorrespond to transitions between the eigenstates of the collective Hamiltonian reproduce \nthe empirical $\\rho^2(E0)$ values. The structure of the low-lying $0^+$ states is \nanalysed in Fig.~\\ref{fig:N=90_0_states}, in which the probability density distributions \nare plotted in the $\\beta-\\gamma$ plane for the three lowest collective \n$0^+$ states of $^{152}$Sm, $^{154}$Gd and $^{156}$Dy. We note that the probability \ndistributions for these states are concentrated on the prolate axis $\\gamma = 0^\\circ$, \nin contrast to the band-heads of the $K=2$ $\\gamma$-bands, for which the probability \ndensity distributions are shown in Fig.~\\ref{fig:N=90_2_3_states}. The dynamical  \n$\\gamma$-deformations of the latter clearly point to the $\\gamma$-vibrational nature \nof these states. The average values of the deformation parameter \n$\\beta$ for the collective ground-state wave functions of $^{152}$Sm: $<\\beta> = 0.32$, \n$^{154}$Gd: $<\\beta> = 0.31$, and $^{156}$Dy: $<\\beta> = 0.30$, correspond to the \nminimum of the respective deformation energy surface. The corresponding values for the first \nexcited $0^+$ states are: $<\\beta> = 0.33$ for $^{152}$Sm, $<\\beta> = 0.34$ for \n$^{154}$Gd, and $<\\beta> = 0.37$ for $^{156}$Dy. For a pure harmonic $\\beta$-vibrational \nstate one expects that the average deformation is the same as for the ground-state, that \nthe ratio of $\\Delta \\beta$ values for $0^+_\\beta$ with respect to $0^+_1$ is $\\sqrt{3}$, and that \nthe probability density distribution displays one node at $<\\beta>_{\\rm g.s.}$ \nand two peaks of the same amplitude. The ratio of $\\Delta \\beta$ values for $0^+_2$ with respect to \n$0^+_1$ is: 1.6 for $^{152}$Sm, 1.53 for $^{154}$Gd, and 1.42 for $^{156}$Dy. Considering \nall these quantities, it appears that the best candidate for the $\\beta$-vibrational state is \n$0^+_2$ in $^{152}$Sm. However, even in this case the probability distribution does not \ndisplay two peaks of equal amplitude, and the shift to larger deformation is more \npronounced in $^{154}$Gd and $^{156}$Dy. Note that for the latter two nuclei the calculated \nexcitation energy of the $0^+_2$ states are in even better agreement with data. The \nprobability distributions for the $0^+_3$ levels, plotted in the third row of \nFig.~\\ref{fig:N=90_0_states}, indicate the development of a second node and third peak, \nthat is, the appearance of two-phonon states. One notices, however, the mixing \nwith states based on $\\gamma$ vibrations, which becomes even more pronounced \nfor higher lying $0^+$ states, not shown in the figure. An experimental exploration \nof a possible occurrence of multiple (two) phonon intrinsic collective excitations \nin $^{152}$Sm did not find evidence for two-phonon $K^\\pi = 0^+$ quadrupole \nvibrations \\cite{Kulp.08}. In fact, it has been argued that an emerging pattern of repeating \nexcitations built on $0^+_2$, similar to those based on the ground state, \nshows that $^{152}$Sm is an example of shape coexistence \\cite{Garr.09}.\n \n\nThe simple analysis presented in this section illustrates the complex structure of \nexcited $0^+$ levels in deformed nuclei, and the difficulties in classifying these \nstates as simple collective vibrational states, that is, as one and two-phonon \n$\\beta$ vibrations. A more quantitative theoretical investigation should involve \nadditional effects not included in our collective Hamiltonian model, such as \nare the coupling between shape oscillations and pairing vibrations \\cite{PP.93} and, \nin general, the coupling between collective and intrinsic two-quasiparticle \nexcitations \\cite{Ber.11}, that can lower the collective energy levels \nand improve the agreement with data \\cite{data,Garr.01}. In particular, a more \nadvanced model that includes coupling between collective and intrinsic two-quasiparticle \nexcitations can be used to analyse excited rotational bands \nbased on pairing isomers, such as those identified in \n$^{154}$Gd \\cite{Kulp.03} and $^{152}$Sm \\cite{Kulp.05}.\n\n\n\\section{\\label{secV}Quadrupole and octupole shape transition in Thorium}\n\nMost deformed medium-heavy and heavy nuclei display quadrupole equilibrium shapes, \nbut there are also regions of the mass table \nin which octupole deformations (reflection-asymmetric, pear-like shapes) occur.\nReflection-asymmetric shapes are characterized by the presence \nof negative-parity bands, and by pronounced electric dipole and octupole\ntransitions. In the case of static octupole deformations, for instance, the lowest \npositive-parity even-spin states and the negative-parity odd-spin states form an\nalternating-parity band, with states connected by the enhanced\nE1 transitions. In a simple microscopic picture strong octupole correlations \narise through a coupling of orbitals near the Fermi surface with quantum numbers \n($l$, $j$) and ($l+3$, $j+3$). This leads \nto reflection-asymmetric intrinsic shapes that develop either dynamically \n(octupole vibrations) or as static octupole equilibrium deformations \\cite{BN.96,BW.15}.\nFor example, in the case of $N\\approx 134$ and $Z\\approx 88$\nnuclei in the region of light actinides, the coupling of the neutron orbitals based on \n$g_{9\/2}$ and $j_{15\/2}$, and that of the proton single-particle\nstates arising from $f_{7\/2}$ and $i_{13\/2}$, can give rise to octupole mean-field deformations. \n\nAn interesting phenomenon are simultaneous quadrupole and octupole shape transitions. \nIn a series of recent studies \\cite{Nom.13,Li.13,Nom.14} we have analyzed the evolution of \nquadrupole and octupole shapes using a consistent microscopic framework based on \nrelativistic EDFs. In thorium isotopes, in particular, the calculated triaxial quadrupole and \naxial quadrupole-octupole energy surfaces, and predicted \nobservables (excitation energies, isotope shifts of charge radii, electromagnetic\ntransition rates) point to the occurrence of a simultaneous phase\ntransition between spherical and quadrupole-deformed prolate shapes,\nand between non-octupole and octupole-deformed  shapes, with $^{224}$Th being\nclosest to the critical point of the double shape phase transition \\cite{Li.13}. \n\n\\begin{figure}[]\n\\begin{center}\n\\begin{tabular}{ccc}\n\\includegraphics[scale=0.275]{Figs\/Th-PES\/th220_pes.eps}\n\\includegraphics[scale=0.275]{Figs\/Th-PES\/th222_pes.eps}\n\\includegraphics[scale=0.275]{Figs\/Th-PES\/th224_pes.eps} \\\\\n\\includegraphics[scale=0.275]{Figs\/Th-PES\/th226_pes.eps}\n\\includegraphics[scale=0.275]{Figs\/Th-PES\/th228_pes.eps}\n\\includegraphics[scale=0.275]{Figs\/Th-PES\/th230_pes.eps}\n\\end{tabular}\n\\caption{\\label{fig:th_pes} Axially symmetric energy surfaces of the isotopes\n $^{220-230}$Th in the $(\\beta_{2},\\beta_{3})$ \n plane. The contours join points on the surface with the same energy\n and the energy difference between neighboring contours is\n 1 MeV. Positive (negative) values of $\\beta_{2}$ correspond to prolate (oblate) \n configurations. Energy surfaces are symmetric with respect to the\n $\\beta_{3}=0$ axis.}\n\\end{center}\n\\end{figure}\n\nFigure~\\ref{fig:th_pes} displays the deformation energy surfaces in the plane \nof axial quadrupole and octupole deformation parameters \n($\\beta_{2}$, $\\beta_{3}$) for the isotopes $^{220 - 230}$Th. \nThis isotopic chain exhibits an interesting structural evolution, visible already \nat the SCMF level. A rather soft energy surface is calculated for $^{220}$Th with the \nminimum at $(\\beta_{2}, \\beta_{3})\\approx (0, 0)$, and this will give rise to quadrupole \nvibrational excitation spectra. Quadrupole deformation becomes more pronounced in $^{224}$Th, \nand one also notices \nthe emergence of octupole deformation. The energy minimum is found in the \n$\\beta_{3}\\neq 0$ region, located at $(\\beta_{2}, \\beta_{3})\\approx (0.1, 0.1)$ . \nFrom $^{224}$Th to $^{228}$Th the occurrence of a rather strongly marked \noctupole minimum is predicted. Starting from $^{228}$Th, the minimum \nbecomes softer in the octupole $\\beta_{3}$ direction. An octupole-soft surface, \nalmost completely flat in $\\beta_{3}$ for $\\beta_{3} \\leq 0.3$,\nis calculated for $^{232}$Th. \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.6]{Figs\/Th-PES\/states_th.eps}\n\\caption{\\label{fig:th_spec}Excitation energies of low-lying yrast positive-parity (left)\nand negative-parity (right) collective states of $^{224-232}$Th. Lines and symbols denote the \n theoretical and experimental \n \\cite{data} levels, respectively.}\n\\end{center}\n\\end{figure}\n\nIn Fig.~\\ref{fig:th_spec} we analyse the systematics of energy spectra of the positive-parity \nground-state band ($K^{\\pi}=0^{+}$) (left) and the lowest negative-parity ($K^\\pi =0^-$) sequences \n(right) in $^{224-232}$Th. The theoretical values calculated using the quadrupole-octupole collective \nHamiltonian~(\\ref{eq:CH2}) are shown in comparison to available data \\cite{data}.  The excitation energies of \npositive-parity states systematically decrease with mass number, reflecting the increase of \nquadrupole collectivity. $^{220,222}$Th exhibit a quadrupole vibrational structure, whereas \npronounced ground-state rotational bands with\n$R_{4\/2}=E(4^{+}_{1})\/E(2^{+}_{1})\\approx 3.33$ are calculated in\n$^{226-232}$Th. For the lowest negative-parity bands\nthe excitation energies display a parabolic \nstructure centered between $^{224}$Th and $^{226}$Th. \nThe approximate parabola of $1^{-}_{1}$ states has a minimum \nat $^{226}$Th, in which the octupole deformed minimum \nis most pronounced. Starting from $^{226}$Th the energies of negative-parity \nstates systematically increase and the band becomes more \ncompressed. A rotational-like collective band based on the octupole vibrational\nstate, i.e., the $1^{-}_{1}$ band-head, develops. The parabolas of negative-parity \nstates calculated with the quadrupole-octupole Hamiltonian are in qualitative agreement \nwith data, although the minima are predicted to occur at $^{228}$Th rather than $^{226}$Th. \nNote, however, that all levels shown in Fig.~\\ref{fig:th_spec} are below 1 MeV excitation \nenergy, so that the differences between calculated and experimental levels are rather \nsmall, especially considering that no parameters were adjusted to data. The approximations \ninvolved in the calculation of the Hamiltonian parameters (perturbative cranking for the \nmass parameters and the Inglis-Belyaev formula for the moments of inertia) determine \nthe level of quantitative agreement with experiment. We note that the theoretical \nB(E2) values for transitions within the ground state bands are in agreement \nwith availabe data, while the calculated $B(E3; 3^-_1\\to 0^+_1)$: \n61 W.u. for $^{230}$Th and 41 W.u. for $^{232}$Th, are somewhat larger than \nthe experimental values: 29(3) W.u. for $^{230}$Th and 24(3) W.u. for $^{232}$Th \\cite{Kib.02}. \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.6]{Figs\/Th-PES\/th226-single-particle.eps}\n\\caption{\\label{fig:th226-single-particle} $^{226}$Th single-neutron (upper panel) and single-proton\n(lower panel) levels in the canonical basis as functions of the deformation\nparameters. The path follows the quadrupole deformation parameter\n$\\beta_2$ up to the position of the equilibrium minimum $\\beta_2=0.2$, with \nthe constant octupole deformation $\\beta_3=0$ (left panels). For $\\beta_2=0.2$ \nthe panels on the right display the single-nucleon energies from $\\beta_3=0$ to $\\beta_3=0.3$. \nDashed curves denote the position of the Fermi level at each deformation.}\n\\end{center}\n\\end{figure}\n\nLet us consider in more detail the structure of $^{226}$Th which, at the SCMF level, exhibits a nice \nexample of coexistence of axial quadrupole and octupole minima. The energy \ndifference between the local minimum at $\\beta_3 = 0$ and the equlibrium minimum \nat $\\beta_3 = 0.16$ is 1.8 MeV. We have also carried out a constrained triaxial \nquadrupole SCMF calculation which confirms that the quadrupole minimum is indeed at \n$\\gamma = 0^\\circ$, that is, axial prolate. The microscopic origin of coexistence \nof the two minima becomes apparent from the dependence of the single-nucleon\nlevels on the two deformation parameters. Figure \\ref{fig:th226-single-particle}\ndisplays the single neutron and proton levels\nof $^{226}$Th along a path in the $\\beta_2 - \\beta_3$ plane. Starting from the spherical configuration,\nthe path follows the quadrupole deformation parameter $\\beta_2$\nup to the position of the equilibrium minimum $\\beta_2 = 0.2$, with the\noctupole deformation parameter kept constant at zero value. Then,\nfor the constant value $\\beta_2 = 0.2$, the path continues from\n$\\beta_3 = 0$ to $\\beta_3 = 0.3$. The necessary condition for the occurrence of \nlow-energy octupole collectivity is the presence of pairs of orbitals near the Fermi level \nthat are strongly coupled by the octupole interaction. In the panels on the left of Fig.~\\ref{fig:th226-single-particle} \nwe notice states of opposite parity that originate from the spherical levels $g_{9\/2}$ and $j_{15\/2}$ for neutrons, \nand $f_{7\/2}$ and $i_{13\/2}$ for protons. The total energy can be related to the level density \naround the Fermi surface, that is, a lower-than-average density of single-particle levels results in extra \nbinding. Therefore, the local quadrupole minimum seen on the axial energy surface \nof $^{226}$Th reflects the $\\beta_2$-dependence of the levels of the Nilsson diagram \nfor $\\beta_3 = 0$.  \nFor the levels in the panels on the right of Fig.~\\ref{fig:th226-single-particle} \nparity is not conserved, and the only quantum number that \ncharacterises these states is the projection of the angular momentum on the symmetry axis. \nThe octupole minimum, rather soft along the $\\beta_3$-path in $^{226}$Th, \nis attributed to the low density of both proton and neutron states close to the \ncorresponding Fermi levels in the interval of deformations around $\\beta_3 = 0.16$.\n\nFully microscopic analyses of coexistence of quadrupole and octupole shapes and, in general, the \nevolution of octupole correlations in heavy nuclei, will be particularly important for future experimental  \nstudies of reflection-asymmetric shapes using accelerated radioactive beams \\cite{Gaffney.13},  \nand in searches for new symmetry violating interactions beyond the standard model \\cite{BW.15}. \n\\section{\\label{secVI}Summary}\n\nThe framework of nuclear energy density functionals provides an intuitive and yet accurate \nmicroscopic interpretation of the evolution of single-nucleon shell structure and the related \nphenomena of deformations, shape transitions and shape coexistence. Self-consistent mean-field \ncalculations of deformation energy surfaces produce symmetry-breaking many-body states \nthat include important static correlations such as deformations and pairing. Intrinsic \nshapes that correspond to minima on the deformation energy surface are determined by the \nstatic (constrained) collective coordinates. Dynamical correlations are included by  \ncollective structure models that restore symmetries broken by the static mean field \nand take into account quantum fluctuations of collective variables.\nThe microscopic input \nfor the generator coordinate method or the collective Hamiltonian model are completely \ndetermined by the choice of the energy density functional and pairing interaction. These models \ncan be used to calculate observables that characterise the evolution and eventual \ncoexistence of different shapes: low-energy excitation spectra, electromagnetic transition rates, \nchanges in masses (separation energies), isotope and isomer shifts, and that can be directly \ncompared to data.  \n\nUsing a single relativistic energy density functional and a finite-range pairing interaction \nseparable in momentum space, in the present study we have analysed illustrative examples \nof diverse phenomena related to evolution of shell structure: shape transition and \ncoexistence in neutron-rich $N=28$ isotones, the structure of\nlowest $0^+$ excitations in deformed $N = 90$ rare-earth nuclei,\nand quadrupole and octupole shape transitions \nin thorium isotopes. Spectroscopic properties have been calculated using the \nfive-dimensional quadrupole and axial quadrupole-octupole collective Hamiltonian models. \nThe very good agreement between theoretical predictions and available data, and especially \nthe fact that very different regions of the mass table could be considered with no \nneed to adjust or fine-tune model parameters to specific data, demonstrate that the approach \nbased on universal density functionals is a method of choice for studies of shape transitions \nand coexistence over the entire table of nuclides, including regions of exotic short-lived \nnuclei far from stability.  \n\nFuture developments of structure methods based on nuclear energy density functionals \ninclude a number of major challenges. One of the most important and certainly most \ndifficult is the construction of a consistent set of approximations for the exchange-correlation \nenergy functional. In the context of phenomena discussed in the present analysis, \nit would be very interesting to try to develop microscopic functionals that, in addition to the \ndependence on ground-state densities and currents composed of occupied Kohn-Sham \norbitals, also include a dependence on unoccupied orbitals. This is particularly important \nfor studies of the evolution of shell structure and modification of gaps in nuclei far from \nstability and\/or superheavy nuclei. For quantitative comparison with available \nspectroscopic data and predictions in new regions of the chart of nuclides, \naccurate and efficient algorithms have to be developed that perform a complete \nrestoration of symmetries broken by self-consistent mean-field solutions for general \nquadrupole and octupole shapes. Deformation-dependent parameters of  \ncollective quadrupole (-octupole) Hamiltonians (vibrational inertial functions and moments \nof inertia) need to be determined using methods that go beyond the simple cranking \nformulas and include the full dynamics of a nuclear system. Finally, theoretical studies \nof low-energy spectroscopic properties that characterise shape coexistence will also have to \ninclude uncertainty estimates, quantify theoretical errors and evaluate correlations \nbetween observables. \n\n\n\\ack{\nThe authors acknowledge discussions with Bing-Nan Lu,  P. Marevi\\'{c}, K. Nomura, P. Ring and J. M. Yao.\nThis work was supported in part by the Chinese-Croatian project \"Universal models of exotic\nnuclear structure\", the Croatian Science Foundation under the project \"Structure and Dynamics\nof Exotic Femtosystems\" (IP-2014-09-9159), and the NSFC project No. 11475140.}\n\\bigskip\n\\bigskip\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Conclusion and Future Work}\nThis paper presented a method for conducting online merge planning with awareness of the latent behaviors of vehicles on the main lane. We modeled the behavior of agents on a freeway using the Cooperative Intelligent Driver Model and demonstrated accurate estimation of the latent cooperation level using a particle filter. Our estimate of the cooperation level informed a behavior-aware POMDP model that was solved online for a merging policy. The performance of the proposed strategy closely matched the results achieved by baseline strategies that assumed full knowledge of other agents' latent characteristics. Furthermore, our driving strategy greatly outperformed a rule-based driving strategy that did not consider the cooperation level of other agents. Although the application described in this paper is specific to freeway merge scenarios, the parameter estimation and online planning framework is quite general and can easily be extended to other driving scenarios or robotic applications.\n\nThis work assumed that the latent behavior of vehicles was time-invariant; future work will relax this assumption to consider agents that change their behavior during a driving scene. \nFurthermore, the safety and efficiency of a merging ego vehicle will be improved by quantifying the position and intention uncertainty of agents on a freeway. Uncertainty-aware models of freeway agents will inform the construction of probabilistically safe ego trajectory envelopes.\n\\section{Driver Models}\n\nRule-based driver models can provide reasonable approximations of microscopic driver behavior with a relatively small number of parameters, making them ideally suited to the online parameter estimation framework. In this section we discuss the ubiquitous IDM \\cite{treiber2000congested} and an IDM extension for merge scenarios, the Cooperative IDM \\cite{bouton2019cooperation}. We also present approaches for learning driver model parameters.\n\n\\subsection{Intelligent Driver Model}\nThe Intelligent Driver Model is an adaptive cruise control model that governs longitudinal vehicle motion. The IDM specifies a mathematical model that balances an agent's desire to travel at a particular speed with the need to maintain adequate separation distance between a leader vehicle. \nGiven a current speed $v(t)$, a distance headway $d(t)$, and a desired speed $v_{\\mathrm{des}}$, the IDM computes a desired acceleration according to\n\n\\begin{equation}\n    {a_{{\\mathrm{IDM}}}} = {a_{\\max }}\\left( {1 - {{\\left( {\\frac{{v(t)}}{{{v_{{\\mathrm{des}}}}}}} \\right)}^4} - {{\\left( {\\frac{{{d_{{\\mathrm{des}}}}}}{{d(t)}}} \\right)}^2}} \\right).\n\\end{equation}\nThe desired separation distance $d_{\\mathrm{des}}$ is defined as\n\\begin{equation}\n    {d_{{\\mathrm{des}}}} = {d_{{\\mathrm{min}}}} + \\tau \\cdot v(t) - \\frac{{v(t) \\cdot r(t)}}{{2\\sqrt {{a_{{\\mathrm{max}}}} \\cdot {b_{{\\mathrm{pref}}}}} }}.\n\\end{equation}\n\nModel inputs include the minimum allowable separation distance $d_{\\mathrm{min}}$; the minimum allowable separation time $\\tau$; the acceleration limit $a_{\\mathrm{max}}$; and the deceleration limit $b_{\\mathrm{pref}}$.\n\n\n\n\\subsection{Cooperative Intelligent Driver Model}\n\nThe Cooperative Intelligent Driver Model is an extension of the IDM developed by \\citet{bouton2019cooperation} for merge scenarios. The model governs the longitudinal acceleration of a vehicle on the main lane while considering the time to reach the merge point ($TTM$) for a merging agent. The C-IDM is parameterized by the IDM parameters as well as the \\textit{cooperation level} $c \\in [0,1]$ that determines the extent to which the driver cooperates with the merging vehicle. A cooperation level of $c = 1$ corresponds to a \\textit{cooperative} agent that yields to the merging vehicle, while a cooperation level of $c = 0$ corresponds to a \\textit{non-cooperative} agent that ignores the merging vehicle. \n\nConsider the merge scenario shown in \\cref{fig:cidm_setup} with an ego vehicle $V_{\\mathrm{E}}$, a leading agent on the main lane $V_{\\mathrm{L}}$, and a trailing agent on the main lane $V_{\\mathrm{T}}$. Let $TTM(V_{i})$ represent the time required for the $i$th actor to reach the specified merge point. C-IDM outputs the longitudinal acceleration for the trailing agent according to the following rule:\n\n\\begin{itemize}\n    \\item If $TTM(V_{\\mathrm{E}}) < c \\times TTM(V_{\\mathrm{T}})$, then $V_{\\mathrm{T}}$ follows IDM while considering the projection of the ego on the main lane as its leading vehicle.\n    \\item If $TTM(V_{\\mathrm{E}}) \\geq c \\times TTM(V_{\\mathrm{T}})$ or if no merging vehicle exists, then $V_{\\mathrm{T}}$ follows IDM while considering the next actor on the main lane, $V_{\\mathrm{L}}$, as its leading vehicle.\n\\end{itemize}\n\\begin{figure}[h!]\n\\centering\n\\include{figs\/cidm_setup}\n\\label{fig:cidm_setup}\n\\end{figure}\n\n\\subsection{Rule-Based Driver Model Parameter Estimation}\n\nRule-based driver models are governed by a relatively small number of parameters, which can be estimated in either an \\textit{offline} or an \\textit{online} fashion. Offline methods typically aggregate driving data in the training dataset and thus lose information on individual traffic participants. However, such methods can use arbitrarily large computation times and dataset sizes to maximize accuracy. Conversely, online methods must operate with constraints on computation time and data availability but capture the behavior of individual drivers. Offline strategies for rule-based driver model parameter estimation include constrained nonlinear optimization \\cite{lefevre2014comparison}, the Levenberg-Marquardt algorithm \\cite{morton2016analysis}, and genetic algorithms \\cite{kesting2008calibrating}. Online strategies include recursive Bayesian filters such as the Extended Kalman filter \\cite{monteil2015real} and the particle filter \\cite{bhattacharyya2021hybrid, buyer2019interaction, schulz2018interaction}. We employ a particle filter to conduct latent parameter estimation due to its ability to represent complex belief states.\n\\section{Experiments}\n\nIn this work, we consider a freeway merge scenario as visualized in \\cref{fig:cidm_setup}. An ego vehicle---representing either an autonomous agent or a human-driven vehicle with a guardian software system---seeks a gap on the main freeway lane. The vehicles are positioned such that the projection of the ego vehicle is in front of the trailing agent. Furthermore, we induce a conflict between the ego and the trailing agent by selecting initial velocities that place them on an intercept trajectory at the merge point. \\Cref{tab:experimental_setup} shows key values for the POMDP reward functions. The POMDP was solved online using partially observable Monte Carlo planning with observation widening (POMCPOW) \\cite{sunberg2018online}.\n\n\\begin{table}\\caption{\\label{tab:experimental_setup} POMDP Reward Function Parameters}\n\\centering\n\\begin{tabular}{@{}lr@{}} \\toprule\n    \\textbf{Parameter} & \\textbf{Value}  \\\\\n    \\midrule\n    $\\lambda_1$ (speed deviation) & $1$ \\\\\n    $\\lambda_2$ (control effort)& $1$ \\\\\n    $\\lambda_3$ (collision) & $100$ \\\\\n    $d_{\\mathrm{safety}}$ & $\\SI{15.0}{\\meter}$ \\\\\\bottomrule\n\\end{tabular}\n\\vspace{-2.5mm}\n\\vspace{-2.5mm}\n\\end{table}\n\n\\subsection{Benchmark Methods}\nWe tested our approach for learning the cooperation parameter against two other POMDP-based planning approaches that make \\textit{a priori} assumptions about the trailing agent's cooperation level: first, a cautious approach where the trailing agent is always assumed to be non-cooperative (e.g., the ego always assumes $c = 0.0$), and second, a risky approach where the trailing agent is always assumed to be cooperative (e.g., the ego always assumes $c = 1.0$). We also benchmarked against a scenario wherein the ego acceleration was governed by the stochastic IDM (SIDM)---an extension of the IDM that outputs noisy acceleration values \\cite{treiber2017intelligent}. We reproduced the merge scene shown in \\cref{fig:merge_scene} using the Applied Intuition simulation engine and ran 500 trials for each driving strategy; the trailing agent was assigned a ground truth cooperation level of $c = 0.0$ in 250 trials and a ground truth cooperation level of $c = 1.0$ in the other 250 trials.\n\\subsection{Filtering}\n\\Cref{fig:parameter_estimation} shows two examples of cooperation level parameter estimation drawn from the batch of experiments. The red line indicates the true cooperation level of the trailing agent while the blue line represents the ego's estimate of the cooperation level. The light blue regions show the 90\\% empirical error bounds. The parameter estimate is initialized to $c=0.5$ with diffuse error bounds. The ego vehicle observes the trailing agent either slow down to facilitate a merge (\\cref{fig:est_coop}) or maintain its current speed while ignoring the ego (\\cref{fig:est_noncoop}) and updates its estimate of the cooperation level accordingly. \\Cref{fig:parameter_estimation} indicates that the ego is able to quickly identify whether a trailing agent is behaving in a generally cooperative or uncooperative manner. The fact that the parameter estimates do not converge exactly to the true value is not unexpected; since the C-IDM relates the time to merge of the ego and the trailing agent with an inequality, there are in general a range of cooperation levels that will yield the same behavior from the trailing agent. A visualization of parameter estimation during merge scenarios was created using Applied Intuition simulation tools and can be found on the lab video channel at \\url{https:\/\/youtu.be\/KnM3azGH_Sg}.\n\n\\begin{figure*}[h]\n    \\begin{subfigure}[b]{0.48\\textwidth}\n       \\centering\n       \\input{figs\/EST_COOP}\n        \\caption{Sample cooperation level estimation when the trailing agent is fully cooperative, e.g., $c = 1.0$}\n        \\label{fig:est_coop}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{0.48\\textwidth}\n        \\centering\n        \\input{figs\/EST_NONCOOP}\n        \\caption{Sample cooperation level estimation when the trailing agent is fully non-cooperative, e.g., $c = 0.0$}\n        \\label{fig:est_noncoop}\n    \\end{subfigure}\n    \\hfill\n\\caption{Examples of online latent parameter estimation for the trailing agent's cooperation level. The estimate of $c$ quickly approaches the true value, indicating that the ego is aware of the trailing agent's cooperation level as it nears the merge point.}\n\\label{fig:parameter_estimation}\n\\vspace{-2.5mm}\n\\end{figure*}\n\n\\subsection{Performance Results}\nWe count the occurrences of hard braking and divide by the total number of trials to obtain the hard brake rate across all experimental configurations. We also measure \\textit{time-to-collision} (TTC), which is a safety metric, and \\textit{time-to-merge}, which indicates how efficient a merge maneuver is.\nHigher TTC values are associated with safer driving scenes \\cite{minderhoud2001extended}, while excessively high time-to-merge values could be indicative of the \\textit{freezing robot problem} wherein the robot freezes in place to avoid a collision \\cite{trautman2010unfreezing}.\n\n\\Cref{tab:hard-brake-rate} displays the rate of hard brake incidents across all driving methods. Hard brake rates remain predictably low across experiments with a cooperative trailing agent since the trailing agent avoids conflict with the merging ego by considering its projection on the main lane. When the trailing agent is non-cooperative, an upper bound on performance is given by the POMDP-based planner that assumes full knowledge of the true cooperation level of $c = 0.0$. Our proposed method for learning $c$ and the strategy of fixing $c = 0.0$ complete all trials without a hard brake incident. Planning with a fixed $c = 1.0$ expresses overconfidence in the desire of the vehicle on the main lane to cooperate and results in a hard brake rate of over $65\\%$. The SIDM driving strategy, which does not consider the cooperation level of the vehicle on the main lane, must conduct hard braking on every trial to avoid a collision.\n\n\\begin{table}\\caption{\\label{tab:hard-brake-rate}Hard Brake Rate}\n\\centering\n\\begin{tabular}{@{}lcr@{}} \\toprule\n    \\textbf{Driving Strategy} & \\multicolumn{2}{c}{\\textbf{Hard Brake Rate (\\%)}}  \\\\\n    \\midrule\n   \n    & $c_T = 1.0$ & $c_T = 0.0$ \\\\\n    \\midrule\n    Learned $c$ & $0.0$ & $0.0$\\\\ \n    Fixed $c = 0.0$ & $0.0$ & $0.0$ \\\\\n    Fixed $c = 1.0$ & $0.4$ & $65.2$\\\\\n    SIDM & $4.0$ & $100.0$\\\\\\bottomrule\n\\end{tabular}\n\\vspace{-2.5mm}\n\\vspace{-2.5mm}\n\\end{table}\n\\Cref{fig:metrics_a1} presents the results for the trials with a cooperative trailing agent. The POMDP-based driving strategies achieve average TTC values greater than $10$ seconds. Many SIDM trials also achieve high TTC values; however, the SIDM trials that result in hard brake incidents are associated with small TTC values. Our strategy for learning $c$ achieves comparable results to the planner that assumes full knowledge of the trailing agent's true cooperation level of $c = 1.0$. Note that assuming non-cooperative behavior does not underperform in these trials since the scenario is designed to induce conflict. In fact, assuming a non-cooperative agent leads to smaller time-to-merge values as the ego accelerates to reach the gap ahead of the trailing agent, as shown in \\cref{fig:ttm_a1}.\n\\begin{figure*}[h]\n    \\begin{subfigure}[t]{0.48\\textwidth}\n    \\centering\n       \\input{figs\/MIN_TTC_A1}\\caption{Distribution over minimum time-to-collision values achieved in trials with a cooperative agent.}\n        \\label{fig:min_ttc_a1}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[t]{0.48\\textwidth}\n    \\centering\n        \\input{figs\/TTM_A1}\\caption{Distribution over time required to merge in trials with a cooperative agent.}\n        \\label{fig:ttm_a1}\n    \\end{subfigure}\n    \\hfill\n\\caption{Experimental results given a cooperative agent on the main lane. Our strategy for learning $c$ achieves comparable results to the nominal planner that assumes full knowledge of the trailing agent's true cooperation level of $c = 1.0$.}\n\\label{fig:metrics_a1}\n\\end{figure*}\n\\Cref{fig:metrics_a0} displays the results for the trials with a non-cooperative trailing agent. Both the SIDM and fixed $c = 1.0$ methods suffered from elevated hard brake rates and correspondingly low average TTC values. Our proposed method and the nominal benchmark obtained by fixing $c = 0.0$ achieved higher TTC values as shown in \\cref{fig:min_ttc_a0}. An explanation for the difference in TTC values is clear from \\cref{fig:ttm_a0}. Our proposed approach and the nominal baseline experience decreased time-to-merge values as they accelerate into the gap in front of the trailing agent. The POMDP-based driving strategy that assumes cooperative behavior takes more time to merge and thus encounters a conflict with the trailing agent at the merge point. The SIDM approach does not consider the behavior of the trailing agent at all and thus takes the longest time to merge on average.\n\\begin{figure*}[h]\n    \\begin{subfigure}[t]{0.48\\textwidth}\n    \\centering\n       \\input{figs\/MIN_TTC_A0}\\caption{Distribution over minimum time-to-collision values achieved in trials with a non-cooperative agent.}\n        \\label{fig:min_ttc_a0}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[t]{0.48\\textwidth}\n    \\centering\n        \\input{figs\/TTM_A0}\\caption{Distribution over time required to merge in trials with a non-cooperative agent.}\n        \\label{fig:ttm_a0}\n    \\end{subfigure}\n    \\hfill\n\\caption{Experimental results given a non-cooperative agent on the main lane. Our strategy for learning $c$ achieves comparable results to the nominal planner that assumes full knowledge of the trailing agent's true cooperation level of $c = 0.0$.}\n\\label{fig:metrics_a0}\n\\vspace{-2.5mm}\n\\end{figure*}\n\n\n\n\n\n\n\n\n\n\n\\section{Introduction}\nEnsuring safe interactions between automated and human traffic participants is a crucial step to making autonomous driving in urban environments a reality. Autonomous vehicles (AVs) must reason about the evolution of highly stochastic driving scenarios to provide protection for human passengers. Predicting the behavior of other traffic participants is an essential step to modeling scene evolution. However, forecasting human behaviors is a challenging task, as myriad psychological and physiological factors determine a traffic participant's driving tendencies \\cite{bhattacharyya2020online}. Furthermore, urban driving scenarios induce complex multi-agent interactions as traffic participants seek to complete individual objectives. For example, vehicles attempting to merge on a freeway must identify a gap in the traffic, while vehicles on the freeway may exhibit cooperative or non-cooperative behavior depending on their own latent characteristics. Consequently, reasoning about the intentions and future actions of agents is important for multi-agent driving interactions such as merging \\cite{bouton2019cooperation}, intersection navigation \\cite{bouton2017belief}, and freeway driving \\cite{sunberg2017value}.\n\n\\begin{figure}[t]\n\\centering\n\\include{figs\/merge_scene}\n\\label{fig:merge_scene}\n\\end{figure}\n\n\nHuman driving behavior can be modeled using \\textit{data-driven} or \\textit{rule-based} methods. Data-driven models are trained on expansive driving datasets and are highly parameterized to capture nuanced driver characteristics. For example, generative adversarial networks and imitation learning techniques have been used to construct expressive models of human driving behavior \\cite{bhattacharyya2020modeling}. However, data-driven approaches are \\textit{black-box} in nature and difficult to interpret. Furthermore, data-driven models might induce dangerous behavior in driving scenes that do not appear in the training dataset \\cite{bhattacharyya2020online}. An alternative to data-driven models is rule-based driver models such as the Intelligent Driver Model \\cite{treiber2000congested}. Rather than extracting driving characteristics from data, rule-based models specify a mathematical model using a set of rules derived from expert knowledge. Although such models rely on assumptions of human driving characteristics and can fail to generalize across a range of scenarios, they are highly interpretable and provide useful insights into microscopic human driving behavior. \\citet{brown2020taxonomy} present an extensive survey of human driver models.\n\nOnce the parameters of a driver model have been specified, the behavior of traffic participants can be forecasted and input to a decision-making system such as open-loop planing, or \\textit{model predictive control} \\cite{brown2017safe, leung2018infusing}. Game-theoretic frameworks \\cite{wang2021game} and reinforcement learning \\cite{ma2021reinforcement, bouton2019cooperation} have also been used for automotive decision making, but these strategies are computationally expensive. Partially observable Markov decision processes (POMDPs) are well-suited for AV decision making because they account for uncertainty in the environment and latent behaviors of traffic participants, and have been used with great success in the freeway driving context \\cite{sunberg2017value, sunberg2020improving}. In this work, we model the merge scenario shown in \\cref{fig:merge_scene} as a POMDP and solve the POMDP online for a driving policy. \\textit{Online} driving strategies choose the next action using real-time sensor information, while \\textit{offline} driving strategies plan all actions in advance. \n\nSeveral papers have investigated interaction-aware planning for merge scenarios. \\citet{ward2017probabilistic} considered merging on freeways and T-junctions; they fit parameters to an extension of the IDM using a nonlinear least-squares procedure. \\citet{bouton2019cooperation} and \\citet{ma2021reinforcement} conducted reinforcement learning-based techniques to learn merging policies. These works represent offline strategies for parameter estimation; we instead pursue an online strategy to capture the idiosyncrasies of individual drivers in real-time. The strategy presented by \\citet{sunberg2020improving} is most similar to our approach; however, they only consider lane change merging. We investigate freeway merging, which has been qualitatively described as one of the riskiest and complex driving maneuvers frequently encountered in day-to-day driving \\cite{ward2017probabilistic}.\n\nIn this paper, we perform online parameter estimation for a rule-based driver model---the Cooperative Intelligent Driver Model (C-IDM) \\cite{bouton2019cooperation}---that explicitly models the cooperation level of vehicles on a main freeway lane encountering a merging vehicle. The estimated cooperation level informs an online POMDP-based planner that executes a merge maneuver with awareness of latent driver behaviors. Our specific contributions are:\n\\begin{itemize}\n\\item We formulate the latent behavior prediction task as a recursive Bayesian estimation problem and estimate the cooperation level of vehicles using a particle filter.\n\\item We use the estimates of latent driver behaviors to inform an online POMDP-based planner which controls the merging vehicle. We thus account for two levels of uncertainty, as we propagate uncertainty over latent behavior through the planning framework.\n\\item We evaluate the safety and efficiency of our driving strategy in a high-fidelity automotive simulator and benchmark our planning strategy against methods that do not model latent behaviors or rely on \\textit{a priori} assumptions about agent behavior. \n\\end{itemize}\n\n\\section{Methodology}\n\nWe model the merge scenario as a partially observable Markov decision process, which can be defined by the tuple $(\\mathcal{S},\\mathcal{A},T,R,\\mathcal{O},Z,\\gamma)$, with $\\mathcal{S}$ the state space, $\\mathcal{A}$ the action space, $T$ the transition model, $R$ the reward function, $\\mathcal{O}$ the observation space, $Z$ the observation likelihood function, and $\\gamma$ the discount factor. An agent in state $s \\in \\mathcal{S}$ takes action $a \\in \\mathcal{A}$ and transitions to a successor state $s^{\\prime}$ according to the transition model $T(s^{\\prime},s,a) = \\mathrm{Pr}(s^{\\prime} \\mid s, a)$. The agent receives a reward according to the reward function $R(s,a,s^{\\prime})$. Rather than observing the true state, the agent receives a potentially imperfect observation $o \\in \\mathcal{O}$ at each time step. The likelihood of receiving observation $o$ is governed by $Z(o,s,a) = \\mathrm{Pr}(o \\mid s, a)$. Finally, the discount factor $\\gamma$ can be adjusted to make the agent more or less myopic. We model the merge scenario with the following definitions:\n\n\\subsection{The State Space}\n\nThe state of a given driving scene,\n\\begin{equation*} s=(V_{E},\\ V_{T},\\ c_{T},\\ V_{L})\\in \\mathcal{S}, \\end{equation*}\nconsists of the observable states $V_{E}$, $V_{T}$, $V_{L}$ of the ego, trailing agent, and lead agent, respectively, and the latent cooperation level $c_{T}$ which governs the C-IDM for the trailing agent. The observable vehicle state\n\\begin{equation*} V_{i}=(x_{i},\\ y_{i},\\ v_{i},\\ \\dot{v}_{i},\\ \\theta_{i}) \\end{equation*}\nconsists of the $i$th vehicle's position $(x_{i},\\ y_{i})$ on the simulator map, longitudinal velocity $v_{i}$, longitudinal acceleration $\\dot{v}_{i}$, and heading angle $\\theta_{i}$.\n\n\\subsection{The Action Space}\n\nAt each time step, the ego takes an action $a \\in \\mathcal{A}$ which consists of a longitudinal jerk value. Planning occurs in jerk space to produce smooth velocity profiles. Jerk values are restricted to the range $\\SI{-0.6}{\\meter\/\\second^3} \\leq a \\leq \\SI{0.6}{\\meter\/\\second^3}$ to prevent unrealistic or unsafe motion.\n\n\\subsection{The Transition Model}\n\nThe transition model is implicitly defined by a generative model, $s^{\\prime} \\sim T(s,a)$, which generates successor states from the distribution  $\\mathrm{Pr}(s^{\\prime} \\mid s, a)$. Vehicles on the main lane propagate according to one-dimensional point mass dynamics:\n\\begin{align*} & {v^{t + 1}} = {v^t} + {\\dot{v}^t}\\Delta t \\\\ & {x^{t + 1}} = {x^t} + {v^t}\\Delta t + \\frac{1}{2}{\\dot{v}^t}\\Delta {t^2} \\end{align*}\n\nLateral speed is assumed to be zero. The trailing agent's acceleration is governed by the C-IDM, while the lead agent moves with constant velocity. The ego vehicle propagates according to a state transition function \n\\begin{align*}  & {\\dot{v}^{t + 1}} = {\\dot{v}^t} + \\ddot{v}^t\\Delta t \\\\ & {v^{t + 1}} = {v^t} + {\\dot{v}^t}\\Delta t \n\\end{align*}\nThe applied jerk $\\ddot{v}^t$ is the action $a$ extracted at the current time step. We assume that the ego travels a distance ${v^t}\\Delta t$ along the merge lane curve during each discrete time step.\n\n\\subsection{The Reward Function}\n\nAutonomous vehicles must balance multiple and occasionally competing objectives such as safety, control effort, and rider comfort. We construct the following reward function to promote safe and efficient merge maneuvers:\n\\begin{equation} R(s)= -\\lambda_1 \\Vert v_{E} - v_{\\text{ref}}\\Vert - \\lambda_2 \\Vert \\dot{v}_{E} \\Vert  - \\lambda_3 \\mathbf{1}_{b_{\\text{hard}}} \\end{equation}\nwhere $v_{\\text{ref}}$ is the ego vehicle's desired reference velocity and $\\mathbf{1}_{b_{\\text{hard}}}$ is an indicator function of hard braking:\n\\begin{equation*}\n\\mathbf{1}_{b_{\\text{hard}}} = \\begin{cases}\n{1}&{\\text{if}\\;\\text{$\\Vert(x_E,\\ y_E) - (x_T,\\ y_T) \\Vert < d_{\\mathrm{safety}} $}}\\\\\n{0}&{\\text{otherwise}}\\\\\n\\end{cases}\n\\end{equation*}\nwhere $d_{\\mathrm{safety}}$ is a user-defined safety threshold. We define \\textit{hard braking} as an uncomfortably abrupt deceleration that can be applied to prevent a collision. The hard brake rate is thus a proxy metric for AV safety and rider comfort across simulations. The weights $\\lambda_i$ can be adjusted to modify the ego's preferences for safety, efficiency, and control effort. \n\n\\subsection{Online Parameter Estimation}\n\n\nGiven a sequence of observations of the trailing agent's position, we wish to obtain a distribution over the latent cooperation level $c_T$. We assume that the cooperation level is time-invariant during the brief interaction between the trailing agent and the merging ego. Our proposed approach frames the task of estimating latent driver behavior as a state estimation problem and solves the inference problem using recursive Bayesian estimation. The general form of the recursive Bayesian update equation is \n\\begin{equation}\n    p(\\theta_T \\mid \\mathbf{y}_{1:t}) = \\frac{p(\\mathbf{y}_t \\mid \\theta_T)p(\\theta_T \\mid \\mathbf{y}_{1:t-1})}\n    {\\int_{\\theta_T} p(\\mathbf{y}_t \\mid \\theta_T)p(\\theta_T \\mid \\mathbf{y}_{1:t-1}) \\mathrm{d}\\theta_T} \\text{.}\n    \\label{eqn:recursive_state_estimation}\n\\end{equation}\n\nBecause \\cref{eqn:recursive_state_estimation} cannot be solved analytically \\cite{bhattacharyya2020online}, we use particle filtering to provide an approximate solution to the inference problem. \\Cref{alg:particle_filtering} presents our strategy for estimating the cooperation level of the trailing agent. The particles are initially sampled from a uniform distribution and are then propagated according to the POMDP transition model. To prevent particle deprivation, we add noise $d_{c_T}$ sampled from a discrete uniform distribution $D = \\mathcal{U}(\\{-0.05,\\ 0,\\ 0.05 \\})$.\n\n\\begin{algorithm}[htb]\n\n\\setstretch{1.2}\n\n\\textbf{Input:} Initial particle set $\\Theta$, ego $V_E$, trailing agent $V_T$, leading agent $V_L$, POMDP model $\\mathcal{P}$ \\\\\n$c_{T}^{0} \\leftarrow 0.5$ (initialize cooperation parameter)\\\\\n\\For{$t \\gets 0, 1, \\ldots, N\\ \\mathrm{(time \\ steps)}$}{\n    $x_T^{t+1} \\leftarrow \\mathrm{true \\ position \\ of \\ trailing \\ agent}$\\\\\n    \\For{$i \\leftarrow 1, 2, \\ldots, K\\ \\mathrm{(particles)}$}{\n        \\uIf{$TTM(V_{\\mathrm{E}}) < c \\times TTM(V_{\\mathrm{T}})$}{\n            $\\dot{v}_{T}^{t+1} \\leftarrow$ IDM with ego projection as leader\\\n        }\n       \n       \n       \n        \\Else{\n            $\\dot{v}_{T}^{t+1} \\leftarrow$ IDM with lead agent as leader\\\n        }\n        Propagate vehicles according to $\\mathcal{P}.T(s^{\\prime}, a, s)$ \\\\\n        $w_{i} \\gets Z\\left(x_T^{t+1} \\ | \\ x_{T,i}^{t+1}\\right)$\\\\\n        $d_{c_T} \\sim D$ (dithering)\\\\\n       \n        $c_{T}^{t+1} \\leftarrow c_{T}^{t} + d_{c_{T}}$, clamped between $0$ and $1$\n    }\n  \n    $\\Theta \\leftarrow \\mathrm{Resample \\ from \\ } \\Theta \\propto \\left[ w_{1}, \\ldots, w_{K}\\right]$\n}\\caption{Online Parameter Estimation}\n\\label{alg:particle_filtering}\n\\end{algorithm}\n\\section{Related Work}\n\nThe Intelligent Driver Model (IDM), developed by \\citet{treiber2000congested}, is a rule-based adaptive cruise control model. The IDM is collision-free by construction and used for longitudinal motion modeling. It has inspired numerous extensions to capture microscopic driver behaviors. The Enhanced IDM prevents excessive breaking reactions due to lane changes, resulting in more relaxed and natural driving behavior \\cite{kesting2010enhanced}. The Foresighted Driver Model is a risk-aware driver model that considers hazards such as obstacles and roadway curves \\cite{eggert2015foresighted}. The Stochastic IDM attempts to model the stochasticity of human drivers by randomly sampling an output acceleration from a Gaussian distribution \\cite{treiber2017intelligent}. In this work, we are concerned with freeway merge scenarios and thus consider the Cooperative Intelligent Driver Model proposed by \\citet{bouton2019cooperation}. The C-IDM uses a simple deterministic model to represent the behavior of vehicles on the main lane in the presence of a merging vehicle. Both the The C-IDM and its foundational IDM framework are presented in more detail in the next section.\n\nRule-based driver models are governed by a relatively small number of parameters, which can be estimated in either an \\textit{offline} or an \\textit{online} fashion. Offline methods typically aggregate driving trajectory data in the training dataset and thus lose information on individual traffic participants. However, such methods can use arbitrarily large computation times and dataset sizes to maximize accuracy. Conversely, online methods must operate with constraints on computation time and data availability but capture the behavior of individual drivers. Offline strategies for rule-based driver model parameter estimation include constrained nonlinear optimization \\cite{lefevre2014comparison}, the Levenberg-Marquardt algorithm \\cite{morton2016analysis}, and genetic algorithms \\cite{kesting2008calibrating}. Online strategies include Bayesian changepoint detection \\cite{galceran2017multipolicy} and recursive Bayesian filters such as the Extended Kalman filter \\cite{monteil2015real} and the particle filter \\cite{bhattacharyya2021hybrid, buyer2019interaction, schulz2018interaction}. We employ a particle filter to conduct latent parameter estimation due to its ability to represent complex belief states.\n\nSeveral papers have investigated interaction-aware planning for merge scenarios. \\citet{ward2017probabilistic} consider merging on freeways and T-junctions; they fit parameters to an extension of the IDM using a nonlinear least-squares procedure. \\citet{bouton2019cooperation} and \\citet{ma2021reinforcement} conduct reinforcement learning-based techniques to learn merging policies. These works represent offline strategies for parameter estimation; we instead pursue an online strategy to capture the idiosyncrasies of individual drivers in real-time. The online parameter strategy presented by \\citet{sunberg2020improving} is most similar to our approach; however, they only consider merging in lane change scenarios. We investigate freeway merging, which has been qualitatively described as one of the riskiest and complex driving maneuvers frequently encountered in day-to-day driving \\cite{ward2017probabilistic}.\n\n\n\\section*{Acknowledgments}\nToyota Research Institute (TRI) provided funds to assist the authors with their research, but this article solely reflects the opinions and conclusions of its authors and not TRI or any other Toyota entity. This research was supported in part by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-1656518. We thank Applied Intuition for support with the simulation platform.\n\n\\renewcommand*{\\bibfont}{\\footnotesize}\n\\printbibliography\n\n\\end{document}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nGenome assembly is one of the most complex and computationally exhaustive tasks in genomics projects~\\cite{PadovanideSouza2018}. Additionally, it is among the most important tasks which allow genome sequences analyses. The main goal of genome assembly is to reconstruct a genome using an assembler software, that does it by the analysis of several small fragments coming from a genome --- commonly referred to as \\emph{target genome}~\\cite{Li2011}. These fragments, named \\emph{reads}, are obtained from an equipment, called the DNA sequencer.\n\nA high-performance assembler is something highly desired among researchers, as it will imply more accurate genomes, allowing researchers to reach a better understanding of the traits and functions of living organisms~\\cite{Manzoni2016}. As a result, the knowledge acquired from these whole genomes produces positive impacts in several fields, such as medicine, biotechnology and biological sciences.\n\nA DNA sequencer is a machine responsible for the initial, but fractional, reading of the genetic code of living organisms~\\cite{Paszkiewicz2010}. However, the genomes of most organisms --- even microorganisms --- are too long for being read in a single run in the sequencer~\\cite{Heather2016}. To surpass this limitation, a technique called Shotgun is applied, consisting of cutting up the genome into small pieces and producing small fragments of DNA which can be entirely (or mostly) read by the sequencer, representing the corresponding genetic information in text sequences (the \\emph{reads})~\\cite{Rodriguez2012}. \n\nSince DNA molecules are formed by sequential pairs of complementary nucleotides (each of them composed by Adenine-Thymine or Guanine-Cytosine), \\emph{reads} represent only a single nucleotide from each pair, sequentially written as a character in the text. In biological terms, we can say that nucleotides of only one (out of two) strand of each DNA fragment is read. The reading of each nucleotide is represented by its corresponding initial (A, C, G or T). Thus, the number of characters in each \\emph{read} is commonly referred to as base pairs (or $bp$). \n\nThe genome of an organism is the sequence of all nucleotides from its DNA molecules, represented by letters (A, C, T, G). Each nucleotide isolated does not represent any relevant biological information, however, when we put them all together, the corresponding sequences provide a deep knowledge about this organism. Within the organism's genome, for example, there are (among other information) the species genes. Genes are continuous fragments of the genome whose nucleotide sequences define species traits and behaviors (e.g. the human eye color)~\\cite{Portin2017}. A single \\emph{read}, however, generally cannot represent the complete information from even a gene, thus genome assembly is commonly required to obtain the whole genome~\\cite{Heather2016}.\n\nGenome sequencing technology defines the maximum number of base pairs will be read from each DNA fragment~\\cite{Heather2016}, which, in turn, defines the size of the produced \\emph{reads} and directly affects the number of \\emph{reads} produced during the sequencing process. As genome assembly is a computational task aiming to order \\emph{reads} in an attempt to reconstruct the original DNA sequence, the number of \\emph{reads} and their sizes directly impact the complexity of the assembly process --- the more and smaller the \\emph{reads}, the higher the complexity for assembling them.\n\nFrom a computational perspective, given a set, $R={read_1,read_2,...,read_n}$, with $n$ small text sequences, called \\emph{reads}, the assembly problem is defined as the search for the superstring that originated the $n$ reads of $R$ by analysing the existing overlaps between them.\n\nGenome assembly is usually carried out by assemblers adopting \\emph{de novo} strategy and\/or the comparative strategy. The comparative approach is relatively simpler and computationally treatable, however it requires a previously assembled genome as reference (e.g. the genome of a similar species) to guide the assembly process by comparing the produced \\emph{reads} with the reference genome, meanwhile the \\emph{de novo} approach presents no such dependence~\\cite{Ji2017}.\n\n\\emph{De novo} strategy is particularly important given that only a small number of reference genomes are currently available, compared to the number of existing and non-sequenced genomes - it is estimated that the vast majority of microorganisms' genomes are still unknown~\\cite{Wong2020}. However, although \\emph{de novo} assembly approach allows the assembly of new genomes without requiring a reference genome, it is considered a highly complex combinatorial problem, falling into the theoretically intractable class of computational problems, referred to as NP-hard~\\cite{Medvedev2007}.\n\nIn computer  science, the commonly applied strategies for \\emph{de novo} genome assembly process are based on heuristics and graphs, and they are known as \\emph{Greedy}, \\emph{Overlap-Layout-Consensus (OLC)}, and \\emph{De Bruijn graph}~\\cite{Pop2009}. For instance, in OLC strategy, each \\emph{read} is represented as a node in a graph (named overlap graph) and edges represent the overlap between \\emph{reads}. Thus, the reconstructed genome corresponds to the \\emph{reads} along the path traversing all the nodes. This algorithm corresponds to the Hamiltonian path algorithm. Another computational formulation for genome assembly is to find the shortest common superstring (SCS) formed by the \\emph{reads} --- which can also be polynomially reduced to the Travelling Salesman Problem (TSP)~\\cite{Springer2007}.\n\nRegardless the difficulties and limitations, current \\emph{de novo} assemblers are capable of producing acceptable solutions. However, the use and application of \\emph{de novo}-based assemblers normally require specific bioinformatics knowledge in order to correctly set configurations and parameters for the assembler. Nevertheless, optimal results are not always guaranteed --- given the high complexity enclosed to this theme~\\cite{Gurevich2013}.\n\nDespite the great contributions of the current assemblers for developing scientific discoveries on the genomics analyses of organisms, genome assembly is not yet a fully solved problem. So, it is very important to continue the development of new and more robust assemblers, in order to assemble DNA sequences faster and with improved accuracy~\\cite{bocicor2011}. This challenge has been the aim of numerous currently ongoing researches, which apply computational techniques to genomics in the search for better solutions, including the use of machine learning (ML)~\\cite{PadovanideSouza2018}.\n\nAlthough machine learning is an alternative approach to heuristics for dealing with high complex computational problems (as it is the case of NP-hard problems), few approaches apply machine learning for dealing with the assembly problem. According to the literature review presented in Souza et al.~\\cite{PadovanideSouza2018}, the few scientific investigations evaluating the application of machine learning techniques for genome assembly were recently published, and only one study dated to the end of the previous century while 12 others date from later. For comparison, in a similar review~\\cite{Henrique2019}, Henrique et al. reported 140 publications applying machine learning techniques into the problem of financial credit risk.\n\nWith the recent access to computational advances --- including increased processing and storage power of computers, the investigation of machine learning application for complex and high-scale computational problems has started to increase in the scientific community and some good results have been reported~\\cite{LeCun2019}. This larger resources availability has also allowed the return of reinforcement learning application for these problems~\\cite{Botvinick2019}. \n\nReinforcement learning (RL), alongside supervised learning and unsupervised learning, is a basic machine learning paradigm that works with intelligent agents which take actions in an environment that represents a task. Ideally, this agent is supposed to solve this task if it is able to learn how to maximize the rewards received from its actions taken~\\cite{sutton2018}.\n\nAlthough scarcely applied in machine learning development, RL has shown some surprisingly positive results, especially for games~\\cite{Botvinick2019}. Sutton et. al, by the way, recently suggested in \\cite{Silver2021} that RL, unlike supervised and unsupervised learning, may be the path to the development of general artificial intelligence. Two great (and commonly cited) examples of the application of reinforcement learning are the training of agents for playing the classic board game Go, as well as several Atari games, showing better results when compared to those from previous approaches and showing superior performances even when compared to trained humans~\\cite{Mnih2015,Silver2017,Vinyals2019}.\n\nHowever, RL successful applications are predominant in problems that rely on accurate environment simulators, such as games, where the rules and environments are known and allow the development of simulations for intensive training of intelligent agents~\\cite{Nian2020}. Despite the importance of games to both society and computer science, there is a growing expectation, and some initial efforts have been made, for extending the success of RL in games into real world problems --- for which it is generally impossible or impracticable to create the required simulators that would provide appropriate training for the agents~\\cite{dulac2019}.\n\nSuch low use of RL in real world problems is also observed in the specific scenario of the genome assembly, where only a single study can be found, proposed by Bocicor et al.~\\cite{bocicor2011, PadovanideSouza2018}. This approach, which is also the object of this study and will be hence forward referred to as seminal approach, proposes the use of an episodic trained agent (i.e. whose training has been divided into episodes) applying the Q-learning reinforcement learning algorithm for learning the correct order of a set of \\emph{reads} and, consequently, for reaching the corresponding genome.\n\nOne of the most valued characteristics of the reinforcement learning is that it allows the agent's autonomy for learning. For example, in supervised learning, a great deal of intervention is required (usually by human specialists) during the learning process, given that all information provided for the machine is previously and properly labeled. In reinforcement learning, the agent learns through the consequences of its successive failures and successes.\n\nThis is particularly useful for solving tasks whose solutions extrapolate the human knowledge and capacity, such as genome assembly. The assembly complexity starts from the assembler's choice --- as assemblers using similar strategies may produce different results~\\cite{Vollmers2017, vanderWalt2017} -- and extends until the assembler's configuration following the user's decisions. Obtaining intelligent and trained agents by reinforcement learning is important in this scenario as it could eliminate the need for human specialists.\n\nAnother relevant aspect for the application of reinforcement learning is the capacity of agents to deal with large volumes of data and extract new rules associated with the main task, which were not explicitly provided before and, in some cases, were not identified by humans. In games, as above mentioned~\\cite{Mnih2015, Silver2017}, agents were able to play new games on their own - without human supervision - and in some cases, they were able to outperform the best known human players.\n\nConsidering that Q-learning algorithm requires a Markov decision process definition with established parameters of states and actions, together with a reward system to be achieved by the agent at each action in every state~\\cite{sutton2018}, the problem was modeled by the authors~\\cite{bocicor2011} through a space of states capable of representing all possible \\emph{read} permutations, so that only one initial state there existed and, in each state, there is one action to be taken by the agent for each \\emph{read} in the pool to be assembled.\n\nFollowing these definitions, from the graph theory perspective, the proposed states space for $n$ \\emph{reads} can be visualized as a complete $n$-ary tree, with height equal to $n$, as the set of states presents only one initial state and forms a connected and acyclic graph~\\cite{cormen2009}. Thus, we can reach the number of existing states in the states space, represented by Equation \\ref{eq:04}.\n\n\\begin{equation}\n\\label{eq:04}\n\\textrm{number of states}=\\frac{n^{n+1}-1}{n-1}\n\\end{equation}\n\nThe proposed reward system depends, first, on the type of the state reached after taking each action, which can be an absorbing or a non-absorbing state. An absorbing state, once reached, does not allow any other state to be reached~\\cite{grinstead2012,sutton2018}. The authors have defined that each state requiring $n$ actions to be reached (being $n$ the number of \\emph{reads} to be assembled) is an absorbing state. \n\nA small and constant reward (e.g. 0.1) was assigned as reward for actions reaching non-absorbing states. This reward was also set to every action leading to absorbing states where repeated \\emph{reads} were used to achieve them. Finally, actions leading to other absorbing states (the final states) produce a reward corresponding to the sum of overlaps between all pairs of consecutive \\emph{reads} used to reach these states.\n\nFor your better understanding, Figure \\ref{fig:02} presents a simple example of a space of states for a set of only 2 \\emph{reads}, identified as A and B. In this example, we can observe the existence of a single initial state, two actions associated with non-absorbing states and four absorbing states (highlighted in black), achieved after taking two any actions.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=6cm,keepaspectratio]{fig2}\n\\caption{Example of state space for a set of two \\emph{reads}, here referred to by A and B.}\n\\label{fig:02}\n\\end{figure}\n\nIn this figure two of the absorbing states are highlighted by the letter $X$. These states, unlike the previous ones, are final states, as they are the only ones in the space of states reached directly from the initial state without repeated actions --- one is achieved from actions referred to as \\emph{read A} and \\emph{read B}, respectively, and the other from the actions of \\emph{read B} and, then, \\emph{read A}. \n\nThe Smith-Waterman algorithm (SW) was applied for obtaining the overlaps between pairs of reads, which were added for obtaining the rewards of actions that led to final states~\\cite{smith1981}. The sum of overlaps when reaching a final state $s$, here referred to as Performance Measure (PM), is described in Equation \\ref{eq:03}, where $read_s$ corresponds to the sequence of reads associated with the actions taken for achieving $s$. In an optimal solution, repeated reads tend to completely overlap, as the pairs reach maximum PM.\n\n\\begin{equation}\n\\label{eq:03}\nPM(s) = \\sum_{i=1}^{n-1}sw(read_s[i], read_s[i+1])\n\\end{equation}\n\nBy using these definitions, the seminal approach produced attractive results, however, it has been evaluated by the authors against only two small sets of \\emph{reads}, one with 4 \\emph{reads} with less than 10$bp$ and the second with 10 \\emph{reads} of 8$bp$ each. These \\emph{reads} were obtained by simulating the sequencing process, assuming as the target genome only a small fragment (a microgenome) of the real genome of the bacterium \\emph{Escherichia coli}.\n\nIn order to perform a scalability analysis of the seminal approach, Xavier et al.~\\cite{xavier2020} evaluated the performance of this approach against 18 datasets. These 18 datasets were produced following the same simulation methods. The first dataset corresponds to the set of 10 \\emph{reads} with 8$bp$ from the seminal approach, originated from a 25$bp$ microgenome. From this same microgenome and from a 50$bp$ microgenome, a total of 17 new datasets were generated (8 from the minor microgenome and 9 from the major one) each containing 10, 20 or 30 \\emph{reads}, with 8$bp$, 10$bp$ or 15$bp$.\n\nAlmost all definitions made by Bocicor et al. were replicated by Xavier et al. in this analysis, but they experimentally set $\\alpha$ and $\\beta$ to $0.8$ and $0.9$, respectively, and the space of actions was slightly reduced, so that actions associated with \\emph{reads} that has already been taken previously were removed from the available actions. In the states space depicted in Figure \\ref{fig:02}, for example, the leftmost and rightmost leaves (i.e. absorbing states) would be removed after this change. Although this change reduces the number of states, the size of the space of states continues to grow exponentially, as we can observe in Equation \\ref{eq:08}.\n\n\\begin{equation}\n\\label{eq:08}\n\\textrm{number of states}=\\sum_{i=0}^{n}\\frac{n!}{(n-i)!}\n\\end{equation}\n\nThis study confirmed the previously positive results found in the seminal approach with the first dataset, however, as the dataset size increases the performance of the seminal approach decreases considerably, reaching the target microgenome in only 2 out of 17 major datasets. According to the authors, such bad results may be related to the high cost required by the agent to explore such a vast states space --- as seen in Equation \\ref{eq:04}, this space grows exponentially --- and also from the failures in the proposed reward system.\n\nIn order to continue the investigation of the application of reinforcement learning in the genome assembly problem and targeting the current challenge of applying the reinforcement learning into real-world problems~\\cite{dulac2019}, here we propose to analyze the limits of reinforcement learning into a real-world problem, which is also a key problem for the development of science. This analysis was carried out by evaluating the performance of strategies that are complementary to those previously studied, and that could be incorporated into the seminal approach for obtaining improved genome assembly.\n\n\\section{Methods}\n\nIn this study, 7 experiments were evaluated against the seminal approach --- referred to as approaches 1.1, 1.2, 1.3, 1.4, 2, 3.1, and 3.2 --- and their methodologies are described in the next 3 subsections. In each approach, as in the seminal approach, the main goal is to reach an agent trained by reinforcement learning capable of identifying the correct order of \\emph{reads} from a sequenced genome. Figure 2 illustrates this proposal, where the set of \\emph{reads} from the sequencing process is initially represented as the agent's interaction environment; the agent learns the order of \\emph{reads} by observing the environment's current state and the rewards produced from the successive actions taken --- until it (ideally) reaches the correct order for the analyzed \\emph{reads}.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=8cm,keepaspectratio]{fig7}\n\\caption{Illustration that demonstrates the application of reinforcement learning in the genome assembly problem. The set of \\emph {reads}, which are obtained in random order by the sequencer, is represented computationally by a reinforcement learning environment. Through successive interactions with the environment, caused by taking actions, the agent ideally learns the correct order of \\emph {reads} --- thus allowing the target genome to be reached.}\n\\label{fig:07}\n\\end{figure}\n\nThese approaches will consider the findings of the scalability analysis from the previously mentioned work of Xavier et. al. Efforts were then made for improving the reward system adopted in the seminal approach --- especially in the approaches described in the next subsection - and to optimize the agent's exploration --- in the approaches described in the last two subsections.\n\n\\subsection*{Approaches 1: Tackling sparse rewards}\n\nIn approaches 1, the reward system, originally defined by Equation 3, has been improved for preventing that permutations of \\emph{reads} that are inconsistent, in terms of alignment, produce high reward values for the agent. This is an undesirable behavior as the agent's learning process is based on maximizing the accumulated rewards. Therefore, ideally, high rewards are expected to be associated with high quality responses.\n\n\\begin{equation}\n\\label{eq:01}\nr(s,a, s')=\\left\\{\n\\begin{array}{ll}\nPM(s') & \\mbox{if s' is a final state},\\\\\n0.1 &\\mbox{otherwise}\n\\end{array}\n\\right.\n\\end{equation}\n\nThis observed inconsistency stem from the \\emph{Smith-Waterman} (SW) algorithm, chosen for calculating the local alignment between two given sequences. With this algorithm, a numeric score is calculated to represent the major alignment size (even if partial) between two sequences. However, the SW algorithm has no constraint for the order between sequences. Therefore, it does not previously consider that a sequence (in our case, a \\emph{read}) must be aligned either left or right to the other one.\n\nIn the case of the genome assembly problem, where the alignment between subsequent \\emph{reads} is expected to follow an order, the overlap score obtained from the SW algorithm may induce the agent to find \\emph{reads} permutations with high overlap values in pairs, however, without presenting a consecutive alignment (suffix-prefix) between \\emph{reads} in the set.\n\nIn Figure \\ref{fig:03} we can observe this type of inconsistency through an example, which presents two different permutations for a set of \\emph{reads} obtained from a given (and known) microgenome --- identified by letters ranging from A to J. The first permutation is formed, in this order, by the \\emph{reads} A, B, C, D, E, F, G, H, I, and J, and, according to SW algorithm, the accumulated overlap (aka PM) is $40.34$. Such score was obtained by summing the maximum overlap, calculated by SM algorithm, between \\emph{reads} A and B with the maximum overlap between \\emph{reads} B and C, and so on. Table \\ref{tab:04} presents the maximum overlap between all pairs of subsequent \\emph{reads} in this permutation.\n\n\\begin{table}[ht]\n\\centering\n\\begin{tabular}{cccccccccc}\n\\hline\n\\textbf{A-B} & \\textbf{B-C} & \\textbf{C-D} & \\textbf{D-E} & \\textbf{E-F} & \\textbf{F-G} & \\textbf{G-H} & \\textbf{H-I} & \\textbf{I-J} \\\\\n\\hline\n4 & 5 & 4 & 5 & 5 & 3.67 & 4 & 2.67 & 7 \\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{tab:04}Maximum overlap between each pair of subsequent \\emph{reads} of the permutation A-B-C-D-E-F-G-H-I-J, obtained by using the SW algorithm (using $match=1.0$, $mismatch=-0.33$, and $gap=-1.33$), thus producing a cumulative overlap of $40.34$.}\n\\end{table}\n\nThe aforementioned permutation (from A to J) corresponds to the optimal permutation, because the union of subsequent \\emph{reads} through the maximum exact overlap between the suffix of the previous read and the prefix of the next read produces exactly the target microgenome (provided at the bottom of the figure as the output genome).\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=8cm,keepaspectratio]{fig5b}\n\\caption{Illustration showing that the measure used as a reward to train agents does not produce maximum values for optimal outputs in some cases. Above, we have an optimal permutation of \\emph{reads}, for which the PM is $40.34$ and whose corresponding genome is equal to the target genome itself; and, below, we have another permutation whose output differs from the target genome, but the corresponding PM is greater than the PM of the optimal permutation.}\n\\label{fig:03}\n\\end{figure}\n\nWe can see in Table \\ref{tab:04} that the overlap between \\emph{reads} I and J is equal to 7, and it is easy to understand such value when we observe the alignment of the corresponding \\emph{reads} in Figure \\ref{fig:03} --- we have 7 letters matching between the sequences (i.e. GACACCC). However, it is not that trivial to understand why the SW algorithm reached a maximum overlap of 4 letters between \\emph{reads} A and B --- since the way they are aligned in Figure \\ref{fig:03} suggests only 1 overlapping letter (i.e. A). We will return to this case and explain it further later.\n\nSimilarly, we can analyze the second permutation, formed by the \\emph{reads} H, G, F, E, C, B, D, A, J, and I, respectively, is not optimal, as the corresponding output genome (obtained using the same read union procedure mentioned above) is quite different (and also larger) from the target genome. However, for this permutation, the SW algorithm presents an accumulated overlap of $43.02$, which is greater than that obtained for the optimal permutation and can be achieved by summing up the overlaps between each pair of subsequent \\emph{reads}, presented in Table \\ref{tab:05}.\n\n\\begin{table}[!ht]\n\\centering\n\\begin{tabular}{cccccccccc}\n\\hline\n\\textbf{A-B} & \\textbf{B-C} & \\textbf{C-D} & \\textbf{D-E} & \\textbf{E-F} & \\textbf{F-G} & \\textbf{G-H} & \\textbf{H-I} & \\textbf{I-J} \\\\\n\\hline\n4 & 3.67 & 5 & 6 & 5 & 4.01 & 4.67 & 3.67 & 7 \\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{tab:05}Maximum overlap between each pair of subsequent \\emph{reads} of the permutation H-G-F-E-C-B-D-A-J-I, obtained by using the SW algorithm (using $match=1.0$, $mismatch=-0.33$, and $gap=-1.33$), thus producing a cumulative overlap of $43.02$.}\n\\end{table}\n\nWe can see in Table \\ref{tab:05} that the overlapping score calculated by SW algorithm considering \\emph{reads} I and J is 7; but, again, it may be not clear to understand it, because the way they are aligned in Figure \\ref{fig:03} suggests no overlapping between them. The reason SW algorithm produces an overlapping score equals to 7 is that such algorithm does not constraint an order between the pair of sequences. \n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=4cm,keepaspectratio]{fig9}\n\\caption{Illustration to emphasize that the overlap score calculated by SW algorithm does not take in to account the relative order of the pair of sequences.}\n\\label{fig:09}\n\\end{figure}\n\nIn that case, it will return the maximum overlap, whether obtained considering \\emph{read} I to the left of read J, or vice versa --- and, as we mentioned above, in the first permutation, we have a 7-letter overlap when considering that \\emph{read} I is to the left of \\emph{read} J. This is the same reason why we have, even in the first permutation, a 4-letter overlap between \\emph{reads} A and B, because, considering \\emph{read} B is to the left of A (and not in reverse), it is possible to achieve a 4-letter overlap between the sequences (i.e. TAAC), as illustrated in Figure \\ref{fig:09}.\n\nAs the general goal of reinforcement learning agents is to learn actions that \\emph{maximize} the accumulated rewards, at the ending of the training, it is expected that the agent learns sequences of actions whose rewards are the highest possible. Thus, it is common and expected that the reward system will yield maximum rewards for optimal solutions. So that, using PM score as a reward for training, may be ineffective for some datasets. \n\nAnother aspect to be considered in the reward system of the seminal approach is that the agent is able to receive high rewards only when it takes actions leading to sparse final states in the states space. That is, considering that the applied training is episodic, in each training episode, the agent receives a non-constant and high reward for only 1 of the $n$ actions taken.\n\nGiven that, the reward system was adjusted in 4 different ways, in order to explore two aspects: (a) the use of overlap score that considers the relative order of \\emph{reads} and\/or (b) the use of dense rewards. These new reward systems are referred here by approaches 1.1, 1.2, 1.3 and 1.4, as follows.\n\nAs proposed in the seminal approach, the reward system of the 1.1 approach defines that actions leading to final states produce a bonus reward (of 1.0) that is added to another numerical overlap score between all subsequent \\emph{reads} used since the initial state. Thus, these actions produces a reward corresponding to the sum of the normalized overlap score (ranging from 0 to 1) of each pair of \\emph{reads}, taking into account the relative order of them.\n\nStill, every action that leads to a non-final state produces a constant and low reward (0.1). Equation \\ref{eq:05} formalizes the reward system for Approach 1.1, with $PM_{norm}(s')$ representing the normalized overlap between the \\emph{reads} used to reach the $s'$ (see more information on the normalized overlap calculation in Section 2 of the supplementary material).\n\n\\begin{equation}\n\\label{eq:05}\nr(s,a, s')=\\left\\{\n\\begin{array}{ll}\nPM_{norm}(s') + 1.0 & \\mbox{if s' is a final state},\\\\\n0.1 &\\mbox{otherwise}\n\\end{array}\n\\right.\n\\end{equation}\n\nDespite the use of overlap score that considers the order of the \\emph{reads} in approach 1.1, it is susceptible to the sparse rewards problem --- as well as in the seminal approach. Although it often produces a small, constant and usually positive reward, and not a zero-value reward as traditionally applied by sparse reward systems, only few and sparse state-action pairs would produce higher rewards.\n\nWe can observe in both systems (from Equations \\ref{eq:01} and \\ref{eq:05}) that there is no rewards provided during learning process to guide the agent towards its goal (since any \\emph{read} incorporated would produce a  reward $0.1$). Thus, the agent's learning process would depend exclusively on the sparse actions taken during the exploration of this vast space of states.\n\nConsidering that the agent's learning process tends to take longer in environments suffering from the sparse reward problem~\\cite{trott2019}, the proposed changes in approaches 1.2, 1.3 and 1.4 mainly focused on improving this aspect and, for this, higher rewards, previously obtained only at the end of the episode, were distributed for each action taken in each episode.\n\nThus, these approaches \u2013-- in addition to making the reward system dense, instead of sparse as originally proposed \u2013-- focused on reducing or eliminating the occurrence of inconsistencies, which, from the genome assembly perspective, would allow permutations of unaligned \\emph{reads} to produce maximum accumulated rewards.\n\nEquations \\ref{eq:07}, \\ref{eq:06} and \\ref{eq:02} represent, respectively, the reward systems for approaches 1.2, 1.3 and 1.4 --- so that $ol_{norm}(s,s')$ represents the normalized overlap between the two subsequent \\emph{reads} that allowed to reach state $s$ and then $s'$.\n\n\\begin{equation}\n\\label{eq:07}\nr(s,a, s')=PM_{norm}(s')\n\\end{equation}\n\n\\begin{equation}\n\\label{eq:06}\nr(s,a, s')=\\left\\{\n\\begin{array}{ll}\nPM_{norm}(s') + 1.0 & \\mbox{if s' is a final state},\\\\\nol_{norm}(s, s') &\\mbox{otherwise}\n\\end{array}\n\\right.\n\\end{equation}\n\n\\begin{equation}\n\\label{eq:02}\nr(s,a, s')=\\left\\{\n\\begin{array}{ll}\nol_{norm}(s, s') + 1.0 & \\mbox{if s' is a final state},\\\\\nol_{norm}(s, s') &\\mbox{otherwise}\n\\end{array}\n\\right.\n\\end{equation}\n\n\\subsection*{Approach 2: Pruning-based action elimination}\n\nOne of the great challenges for applying reinforcement learning in real-world problems is the high dimensionality of the states space that must be explored by the agents~\\cite{zahavy2018,dulac2020}. The Q-learning algorithm is especially susceptible to the dimensionality curse, as the number of states (as well as the number of actions) directly affects the data structure required for the agent's  learning ~\\cite{wiele2020}. Given that the number of actions in each state directly affects the number of states in the seminal proposal, eliminating actions are a strategy for dealing with the high dimensionality~\\cite{zahavy2018}.\n\nTo reduce even more the states space proposed by the seminal approach, a heuristic procedure was applied to eliminate actions where states reached directly or indirectly had already been fully explored and where the maximum cumulative reward achieved is smaller than the cumulative reward obtained by taking any other action available in the state. \n\nIn Figure \\ref{fig:04}, we have an example of this action elimination in the given states space considering 3 reads. Looking again at the  changed states space as a tree --- so removing actions associated with \\emph{reads} that have already been used, we see 16 states in this illustration, 6 of them are absorbing states (in the base of the tree) and also correspond to final states following the proposed modeling approach.\n\nFor better understanding the pruning process, note that 3 out of 6 final states are highlighted in black, while the remaining states are in gray and white. Black states correspond to explored final states (i.e. already visited by the agent). Gray states, such as the one reached by taking action $a$ in the initial state, represent the states in which all children states have been fully visited in the learning process. Finally, white states (final or not) are those not yet explored and\/or that have unexplored children --- e.g. the initial state, where one child is not explored and the other one is partially explored.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=8cm,keepaspectratio]{fig3}\n\\caption{Illustration of the pruning procedure for a state space referring to the assembly of 3 \\emph{reads}, referred by \\emph{a}, \\emph{b} and \\emph{c}. The generic pruning procedure is defined in detail by Algorithm \\ref{alg:prunning}}\n\\label{fig:04}\n\\end{figure}\n\nWhen reaching an unexplored final state, such as the rightmost final state in Figure \\ref{fig:04}, the accumulated rewards achieved since the initial state is maintained and propagated for its predecessors, maintaining only the highest value propagated for the children. Each reward is represented by integer numbers within the states in the figure. Thus, each non-final state stores the highest accumulated reward achieved from it during the training process.\n\nBased on this information, it is possible to prune irrelevant actions, those actions that, after taken, do not produce the maximum accumulated reward. This type of action can be found in action \\emph{a} of the initial state in Figure \\ref{fig:04}. Note that all possible achievable states after taking this action have been explored and the maximum cumulative reward obtained is 6, obtained from consecutively taking the actions \\emph{a}, \\emph{b}, \\emph{c}. Also note that action \\emph{c} in the initial state, even that not fully explored, is capable of producing a higher reward, equals to 8, and obtained by taking the actions \\emph{c}, \\emph{a} and \\emph{b}, in that order. \n\nThus, when the agent first goes through the sequence of states corresponding to the actions \\emph{c}, \\emph{a} and \\emph{b}, the pruning mechanism propagates the maximum reward value up to the initial state and, at that moment, it cuts the action \\emph{a} from the initial state. The pseudo code presented in Algorithm \\ref{alg:prunning} presents the procedure to update pruning process when the last explored final state (referred to as $state$) is reached obtaining the corresponding accumulated reward achieved (referred to as $newReward$).\n\n\\begin{algorithm}\n\n\t\\caption{Pruning's algorithm}\\label{alg:prunning}\n\t\\begin{algorithmic}[1]\n\t\t\\Procedure{Prune}{$state:treeNode,newReward:float$}\n\n            \\If{$state \\not= null$ \\textbf{and} $(state.unseen$ \\textbf{or} $newReward > state.maxReward)$}\\Comment{$state.unseen$ starts $true,\\forall_{state}$}\n            \\State $state.unseen \\gets false$\n            \\State $state.maxReward \\gets newReward$\n            \\If{$state.final$}\n            \\State\n            \\textsc{PruneUselessChildren}($state$)\\Comment{i.e. prune all fully explored children where $maxReward < newReward$ }\n            \\EndIf\n            \\State \\textsc{Prune}($state.parent, newReward$)\n            \\EndIf\n\n\t\t\\EndProcedure\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\\subsection*{Approaches 3: Evolutionary-based exploration}\n\nIn this proposal, the potential for mutual collaboration between reinforcement learning and evolutionary computing was investigated --\u2013 by applying the elitist selection of the genetic algorithm ~\\cite{Baluja95, Konar2005} ---  to optimize the exploration of the states space. For assessing the individual contribution of the genetic algorithm in this hybrid proposal, this approach has been divided into two smaller approaches, referred to as approaches 3.1 and 3.2 and presented in the following subsections.\n\n\\subsubsection*{Approach 3.1: Evolutionary-aided reinforcement learning assembly}\n\nThe strategy of applying $\\epsilon$-greedy --- and its variations --- is a classic solution for expanding the exploration of agents trained by the Q-learning algorithm, as it allows a broader initial exploration, achieving the optimal policy once the states space has been sufficiently explored~\\cite{sutton2018}. However, the existing trade-off between exploitation and exploration remains a major and challenging problem for reinforcement learning in high-dimensional environments~\\cite{gimelfarb2020, peterson2019}.\n\nSearching for a more efficient exploration process and also considering the good performance of genetic algorithms in a similar genome assembly approach carried out by Oliveira et al.\\cite{oliveira2017}, here, the interaction between reinforcement learning and evolutionary computation was introduced into the exploration process.\n\nThis approach is based on the traditional operation of the Q-learning algorithm. However, in each Q-learning episode, the sequence of actions is stored, and at the end of the episode it is transformed into a chromosome of an initial population, that later will evolve. This procedure is presented in Figure \\ref{fig:05}, where we can see the list of actions made in each episode being used for constructing a new individual.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=8cm,keepaspectratio]{fig4}\n\\caption{Illustration of the proposed interaction between reinforcement learning and genetic algorithm. At each reinforcement learning episode, the actions taken by the agent are converted into the chromosome (having each action as a gene) of an individual of the initial population of the genetic algorithm, whose size $n$ is predefined. After $n$ episodes (and thus $n$ individuals in the initial population), this population evolves for an also predefined number of generations through the genetic algorithm. Then, the most adapted individual of the last generation is obtained. In the end, that individual's chromosome genes are used as actions in the next episode of reinforcement learning.}\n\\label{fig:05}\n\\end{figure}\n\nNew chromosomes are inserted into the initial population until the amount of chromosomes reaches the predefined and expected size for this population. At this point, agent training is interrupted and $m$ genetic generations are carried out \u2013-- being $m$ also predetermined (for details see Section 4 of supplementary material) and applying the normalized sum of overlaps between \\emph{reads} as the adaptive function --- the same applied in Equation \\ref{eq:02} and detailed in Section 2 of the supplementary material.\n\nAfter $m$ generations, according to the objective function, the most fit individual is used for conducting the next episode in the agent's reinforcement learning training, hitherto interrupted. As each gene of the individual's chromosome corresponds to one possible action, the complete gene sequence will contain distinct successive actions to be taken by the agent in the current episode, producing then a mutual collaboration between reinforcement learning and the genetic algorithm --- the initial populations of the genetic algorithm are produced by reinforcement learning and, as a counterpart, the results from the evolution of the genetic algorithm is introduced in an episode of reinforcement learning.\n\nFor your better understanding of all the aforementioned approaches, considering they share the common basis of the Q-learning algorithm, Figure \\ref{fig:06} presents a flowchart where we can see the proposed updates for these approaches. The elements highlighted in gray represent the procedures and conditions of Q-learning algorithm --- which, briefly, is based on the action choice, action taking and the update of Q-values in the Q-table during several episodes, all of them starting in an initial state and ending when an absorbing state is reached.\n\nAs a consequence, these elements (in gray) fully represent Approaches 1.1, 1.2, 1.3, and 1.4, as the proposal is focused on replacing the reward system proposed by the seminal approach. Approaches 2 and 3.1 emerge as complementary to this common basis of procedures with specific procedures, namely the insertion of the dynamic pruning mechanism in approach 2 --- represented by the dashed element with double edges --- and the introduction of mutual contribution with the genetic algorithm to improve the agent's exploration performance --- represented by the dashed elements with simple edges. \n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=8cm,keepaspectratio]{fig6}\n\\caption{Flowchart representing Approaches 1.1, 1.2, 1.3, 1.4, 2 and 3.1, so that Approaches 1.1, 1.2, 1.3, and 1.4 are defined by the elements in gray, Approach 2 by the dashed element with double edges and Approach 3.1 by the dashed elements with single border.}\n\\label{fig:06}\n\\end{figure}\n\n\\subsubsection*{Approach 3.2: Evolutionary-based assembly}\n\nFinally, to estimate the contribution of the genetic algorithm in Approach 3.1, which applies a mutual collaboration between reinforcement learning and the genetic algorithm, the genetic algorithm assembling performance was evaluated separately, following the same configurations set of the previous approach, but this time, adopting as starting population a set of individuals whose chromosomes were built from random permutations without repetition of \\emph{reads}.\n\n\\subsection*{Datasets}\n\nTo assess the performance of all approaches (including seminal approach), in addition to the 18 datasets proposed and made available by Xavier et al.~\\cite{xavier2020}, 5 novel datasets derived from other microgenomes extracted from the genome of the previous studies~\\cite{bocicor2011, xavier2020} were complementarily created here. These microgenomes are not arbitrary genome fragments, as the previously used microgenomes (which had 25$bp$ and 50$bp$), but represent larger fragments of previously annotated genes from the corresponding organism (i.e. \\emph{E. coli}). As the datasets are simulated data, it was considered that there are no cycles in the genome, thus being a limitation of the approach. The experiments were then carried out on 23 datasets, whose microgenomes sizes, number of \\emph{reads} and sizes of \\emph{reads} are presented in Table \\ref{tab:03} --- the last 5 lines correspond to the 5 datasets derived from genes.\n\n\\begin{table}[h]\n\\centering\n\\begin{tabular}{cccc}\n\\hline\n\\textbf{$\\mu$gen.} & \\textbf{\\# of} & \\textbf{read} & \\textbf{Gym} \\\\\n\\textbf{size} & \\textbf{reads} & \\textbf{size} & \\textbf{environment name} \\\\\n\\hline\n25 & 10 & 8 & GymnomeAssembly\\_25\\_10\\_8-v2 \\\\\n25 & 10 & 10 & GymnomeAssembly\\_25\\_10\\_10-v2 \\\\\n25 & 10 & 15 & GymnomeAssembly\\_25\\_10\\_15-v2 \\\\\n50 & 10 & 8 & GymnomeAssembly\\_50\\_10\\_8-v2 \\\\\n50 & 10 & 10 & GymnomeAssembly\\_50\\_10\\_10-v2 \\\\\n50 & 10 & 15 & GymnomeAssembly\\_50\\_10\\_15-v2 \\\\\n25 & 20 & 8 & GymnomeAssembly\\_25\\_20\\_8-v2 \\\\\n25 & 20 & 10 & GymnomeAssembly\\_25\\_20\\_10-v2 \\\\\n25 & 20 & 15 & GymnomeAssembly\\_25\\_20\\_15-v2 \\\\\n50 & 20 & 8 & GymnomeAssembly\\_50\\_20\\_8-v2 \\\\\n50 & 20 & 10 & GymnomeAssembly\\_50\\_20\\_10-v2 \\\\\n50 & 20 & 15 & GymnomeAssembly\\_50\\_20\\_15-v2 \\\\\n25 & 30 & 8 & GymnomeAssembly\\_25\\_30\\_8-v2 \\\\\n25 & 30 & 10 & GymnomeAssembly\\_25\\_30\\_10-v2 \\\\\n25 & 30 & 15 & GymnomeAssembly\\_25\\_30\\_15-v2 \\\\\n50 & 30 & 8 & GymnomeAssembly\\_50\\_30\\_8-v2 \\\\\n50 & 30 & 10 & GymnomeAssembly\\_50\\_30\\_10-v2 \\\\\n50 & 30 & 15 & GymnomeAssembly\\_50\\_30\\_15-v2 \\\\\n381 & 20 & 75 & GymnomeAssembly\\_381\\_20\\_75-v2 \\\\ \n567 & 30 & 75 & GymnomeAssembly\\_567\\_30\\_75-v2 \\\\ \n726 & 40 & 75 & GymnomeAssembly\\_728\\_40\\_75-v2 \\\\ \n930 & 50 & 75 & GymnomeAssembly\\_930\\_50\\_75-v2 \\\\ \n4224 & 230 & 75 & GymnomeAssembly\\_4224\\_230\\_75-v2 \\\\ \\hline\n\\end{tabular}\n\\caption{\\label{tab:03}Data sets used in the experiments. The first column shows the size (in $bp$) of the microgenome used to generate the \\emph{reads} of each set; the second column shows the number of \\emph{reads} generated; the third column shows the size of the generated \\emph{reads}; and the fourth column shows the name of the environment built for each set in the OpenAI Gym toolkit (accessible in \\url{http:\/\/github.com\/kriowloo\/gymnome-assembly}).}\n\\end{table}\n\nAs each of these datasets correspond to a reinforcement learning environment, an environment for each of them was created in the OpenAI Gym toolkit~\\cite{brockman2016}, in order to share such reinforcement learning challenges in a simple way. These environments are provided in \\url{http:\/\/github.com\/kriowloo\/gymnome-assembly} (see section 1 of the supplementary material for additional technical information) and use the reward system proposed in Approach 1.4. The identification names of each environment are presented in the last column of Table \\ref{tab:03}. The seminal reward system is also implemented and available \u2013-- for running it, use the version 1, replacing \\emph{v2} by \\emph{v1} in the environment name field.\n\nTwo experiments were then carried out for evaluating the approaches. In each experiment, 20 successive runs of each evaluated approach was performed for all the 23 existing datasets; totalizing 460 runs per approach. Given that each approach has different levels of complexity, the real execution time for each approach was considered for comparing them. To reduce the interference of external factors in execution time, all experiments were individually and sequentially performed in the same station (with Ubuntu 16.04 in an AWS EC2 instance of the \\emph{r5a.large} type, dual core, with 16GB RAM and 30GB of storage).\n\nIn the first experiment, here referred to as \\emph{Experiment A}, the objective was to verify the impact of progressively including the previously described strategies. For this, the performance of the seminal approach was evaluated (according to \\cite{bocicor2011}) against approaches 1.1, 1.2, 1.3, 1.4 (which modify the reward system), 2 (which includes the pruning dynamic) and 3.1 (which uses the AG as an complementary factor).\n\nIn the second experiment, referred to as \\emph{Experiment B}, the main objective was to compare the performance of the new RL-based approaches against the performance of the AG alone. Therefore, in addition to Approaches 1.1, 1.2, 1.3, 1.4, 2 and 3.1, the approach 3.2 (which explores GA alone) was performed in an equivalent time.\n\nFor performance measure in each experiment, two percentage measures were calculated, called distance-based measure (DM) and reward-based measure (RM). Evaluations of \\emph{de novo} assembly are commonly performed using proper metrics, such as the N50~\\cite{Bradnam2013}. These metrics were created because, as previously indicated, \\emph{de novo} assemblies are not supported by a reference genome. This way, in some scenarios, it is not possible to accurately assess the results obtained from the assemblers \u2013-- as the optimal output is unknown. \n\nHere, although a \\emph{de novo} assembler is evaluated, its assessment environment is restricted and the target genomes are known, and this scenario allows the use of specific (and exact) evaluations, such as the DM and RM metrics. \n\nDM considers that a run was successful when the consensus sequence resulting from the orders of \\emph{reads} produced in that run is exactly identical to the expected sequence. RM, however, consider any run as successful when the proposed order of \\emph{reads} presents the corresponding sum of PM$_{norm}$ higher or equal to the sum of PM$_{norm}$ from the optimal \\emph{reads} sequence (for details, see Section 3 of the supplementary material).\n\n\\section{Results}\n\nThe results obtained from \\emph{Experiment A} are presented in Table \\ref{tab:01}, where it is possible to observe that the seminal approach, although consuming the longest running time (23 hours and 34 minutes), also presented the lowest average performances, obtaining an optimal response in only 16.96\\% of the runs (i.e. 78 out of 460 executions) in terms of distance between the produced and the expected genome (DM) and 21.30\\% (98 out of 460) in terms of maximum reward (RM). This difference is based on the previously mentioned inconsistency of the proposed reward system, which allowed non-optimal permutations to produce maximum accumulated rewards.\n\nBeyond that, following the updates in the rewards system, the DM and MR performances in Approaches 1.2, 1.,3, and 1.4 surpassed those of the previous approach, and they also consumes about 4 hours less (19 hours and 38 minutes of execution time). In such experiments the part of the gains are due to the improved agent's performance in one of the sets where the sum of rewards for the optimal permutation of \\emph{reads} were not maximum in the previous reward system (as presented in Figure \\ref{fig:03}).\n\nDespite the gains obtained from the updated reward system, it was possible to note, based on the results, that the previously mentioned inconsistencies were not completely resolved. In some of the datasets the agent reached and even surpassed the maximum expected accumulated rewards, however, without obtaining the target genome. \n\nA minor improvement is observed after the application of the approach 2, with the incorporation of the agent's dynamic pruning, presenting a performance slightly superior to that of the agent in Approaches 1, thus requiring approximately one hour less of processing. The great highlight for this comparison, however, is presented in Approach 3.1, which, while benefiting from the environment exploration improved by AG, has reached a significantly improved result in a reduced amount of time for execution.\n\n\\begin{table}[ht]\n\\centering\n\\begin{tabular}{cccc}\n\\hline\n\\textbf{Experiment A} & \\textbf{Average} & \\textbf{Average} &\\textbf{Total} \\\\\n\\textbf{(Approach)} & \\textbf{DM} & \\textbf{RM} &\\textbf{runtime} \\\\\\hline\nSeminal & 16.96\\% & 21.30\\% & 23h34m \\\\ \n1.1 & 9.57\\% & 13.70\\% & 19h38m \\\\ \n1.2 & 18.48\\% & 21.30\\% & 19h38m \\\\ \n1.3 & 20.00\\% & 24.35\\% & 19h38m \\\\ \n1.4 & 20.43\\% & 24.78\\% & 19h38m \\\\ \n2 & 20.65\\% & 25.00\\% & 18h41m \\\\ \n3.1 & \\textbf{73.91\\%} & \\textbf{80.87\\%} & 17h03m \\\\ \\hline\n\\end{tabular}\n\\caption{\\label{tab:01}Results of Experiment A, which compares the performances of trained agents with different reinforcement learning strategies. The performances of each approach are expressed using distance-based (DM) and reward-based (RM) metrics (see \\emph{Methods} for details).}\n\\end{table}\n\nIn order to analyze the contribution of each technique for the gain of performance in the approach 3.1, a second experiment was performed, here referred to as \\emph{Experiment B}, to compare the performance of RL-based approaches (approaches 1.4, 2 and 3) and the isolated use of AG, but using random initial populations (approach 3.2). The results of Experiment B are presented in Table \\ref{tab:02}.\n\n\\begin{table}[ht]\n\\centering\n\\begin{tabular}{cccc}\n\\hline\n\\textbf{Experiment B} & \\textbf{Average} & \\textbf{Average} &\\textbf{Total} \\\\\n\\textbf{(Approach)} & \\textbf{DM} & \\textbf{RM} &\\textbf{runtime} \\\\ \\hline\n1.4 & 13.91\\% & 17.61\\% & 01h36m \\\\ \n2 & 12.39\\% & 16.30\\% & 01h36m \\\\\n3.1 & 14.78\\% & 14.78\\% & 01h42m \\\\\n3.2 & \\textbf{87.83\\%} & \\textbf{95.65\\%} & 01h34m \\\\\\hline\n\n\\end{tabular}\n\\caption{\\label{tab:02}Experimental performances considering similar running times (RT). Performances were expressed using Distance-based Measure (DM) and Reward-based Measure (RM) (see \\emph{Methods})}.\n\\end{table}\n\n\nGiven the remarkable performance of Approach 3.2, Experiment B applied as reference the time taken by the AG to find an optimal solution in terms of RM for 22 out of the 23 datasets used (i.e. $95.65\\%$), which corresponds to 1 hour and 34 minutes. Given this results, it is evident the superiority of the results obtained from Approach 3.2 when compared to Approaches 1.4, 2 and 3.1.\n\nFinally, after verifying the dominance of Approach 3.2, an additional experiment was carried out, aiming to verify the limits of this approach. It considers the dataset for which no optimal response was achieved. This dataset corresponds to the reads from the largest gene (4Kbp) selected for the production of simulated datasets.\n\nFor that experiment, the running time was considerably increased, reaching approximately 38 hours of running --- surpassing the total time allocated for running all datasets in the other experiments, considering about 30 minutes per run (against less than 2 minutes for the same dataset in the tests run for approach 3.2 of \\emph{Experiment B}).\n\nAlthough with an expressive increase in execution time, no optimal solution has yet been reached for any of the 20 runs of this dataset. However, as presented Figure \\ref{fig:01}, through violinplots, it is possible to observe a consistent gain in performance, both in terms of DM (where longer runs had smaller distances than most distances reached by shorter runs) and RM (which was higher accumulated rewards in all longer time runs).\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=\\linewidth]{fig1c}\n\\caption{Violin plots demonstrating the performances obtained by the GA in experiments with short (1h34m) and long (37h58m) execution times in terms of sums of DM and RM. Gray dots represent the distance\/reward obtained for all runs; the black line in the middle indicates interquartile range; and the violin curves shows the distribution density, where the wider the section, the greater the probability of the observations take the corresponding value.}\n\\label{fig:01}\n\\end{figure}\n\n\\section{Discussion}\n\nAs mentioned, genome assembly is among the most complex problems confronted by computer scientists within the context of genomics projects, regardless the importance of its results for scientific development. This complexity, in computational terms, allocates the problem of finding an optimal permutation of sequenced \\emph{reads} and reaching the target genome into a class of problems called NP-hard which comprises the most difficult problems in computer science~\\cite{roughgarden2020}.\n\nThis high complexity is particularly expressed in the vast space of states required for representing the assembly problem into the modeling proposed by the seminal approach and, consequently, into the approaches proposed here. To achieve the optimal solution in sets of \\emph{reads} of only 30 sequences in the seminal approach, for example, the agent should explore a states space composed of approximately $2e44$ states~\\cite{bocicor2011} (this number exceeds the estimated number of stars in the universe and corresponds to the number of sand grains on Earth). It is also worth noting that, in real-world scenarios, it is common to sequence millions of \\emph{reads} per genome --- which increases even more the corresponding states space. So, applying RL combined with a heuristic is a strategy to deal with the complexity of the problem, since it is not feasible to map all the state space. Therefore, this approach seeks map actions into states that tend to maximize their reward, thus decreasing the computational complexity of the problem.\n\nThe approaches proposed in this study aimed to expand the agent's learning based on two difficulties observed in the seminal approach for applying reinforcement learning into the genome assembly problem: (1) the reward system and (2) the agent's exploration strategies. Both the updates in the rewards system and the incorporation of new explorative strategies improved the agent's learning performance, as demonstrated by the results.\n\nThe definition of extrinsic rewards is one of the most challenging tasks for the construction of environments suitable for the agent's learning. The updates into the reward systems proposed here favored the agent's improved learning. However, it is not yet an optimal solution for the problem, as some non-optimal solutions are still present in datasets that produced maximum accumulated rewards. This would justify the occurrence of RM percentages higher than DM percentages in several experiments.\n\nThe dynamic pruning mechanism showed a discreet improvement for the learning process. However, the relationship between the additional processing cost resulting from the application of this mechanism and the benefit obtained from its implementation did not indicate a reasonable net gain from its use as bypass for the problem emerging from the high dimensionality of the space of states.\n\nIn this sense, the application of the RL strategy combined with the AG, in the hybrid approach, presented a much more attractive performance for supporting the exploration process in the long term. This combination was proved to be advantageous, probably given the dimensionality curse encountered by the Q-learning algorithm, as a strong AG support was observed for the agent while conduct the RL exploration.\n\nDespite the performance improvements, it remains evident the insufficiency of the approaches when applied to real-world scenarios. This insufficiency is more evident in the experiments performed with the largest dataset analyzed, with a size that corresponds to a gene of approximately $4Kbp$. Despite being the largest dataset employed, this gene remains much smaller than the smallest genomes from living organisms. None of the applied approaches presented an optimal response in this scenario, not even when applying the isolated genetic algorithm for a time that was largely longer than the time applied for the tests with the other datasets.\n\nThis finding, together with the superior results obtained from the GA alone, allow us to conclude on the infeasibility of applying the Q-learning algorithm to solve the genomes assembly problem in search for an optimal \\emph{reads} permutation, as originally proposed in the seminal approach.\n\nHowever, given the absence of approaches in the literature for tackling the problem through reinforcement learning and also considering the optimistic results obtained from RL, especially when combined to deep learning, further investigations on the applicability of reinforcement learning are required, including the use of different modeling approaches and algorithms.\n\nConsidering the importance of reproducing scientific studies and with the special intention of supporting future investigations, all experiments performed in this study, as well as the reinforcement learning environments applied to simulate the assembly problems, are available at \\url{https:\/\/osf.io\/tp4zj\/?view_only=8bd12290d9d4480f95bc2c6e49c73ff7}, and are open for reuse (for details on how to reproduce the experiments see Section 5 of the supplementary material).\n\nOne of the major challenges for applying reinforcement learning to real-world problems is the low sample efficiency of the algorithms~\\cite{yang2018}; and it is not different in the genome assembly problem treated here. From the time required by the agent trained by the Q-learning algorithm to reach an optimal solution, it is possible to perceive a high need for numerous interactions with the data. Considering that real-world inputs are even bigger than those experimentally applied here, obtaining a sample efficient algorithm for the problem would be required for supporting the development of solutions applicable into real-world problems.\n\nOne aspect that directly interferes with the agent's sample efficiency is optimizing the exploration of the space of states. In this sense, the application of techniques that remove duplicate \\emph{reads} --- due to repeats --- and the use of an intrinsic motivation could be alternatively investigated to bypass the exploration problem, given the high dimensionality of the proposed space of states~\\cite{barto2012, yang2018}.\n\nAs previously mentioned, it is important to continue the investigation for updating and replacing the Q-learning algorithm, as well as the use of distributed approaches and\/or algorithms using eligibility traces. Although simple, Q-learning is a very powerful reinforcement learning method for solving tasks and remains applied for real-world applications~\\cite{Chakole2021, Abdulhai2003}. As is the case with other RL methods, however, its use in real-world problems where the space of states is too vast is not recommended. The use of approximating methods for the Q function may prove opportune in such cases, especially given the promising results obtained in this study through the combination of RL with deep learning~\\cite{fjelland2020}.\n\nAlthough it faces some obstacles for being implemented into commercial applications, it is reasonable to consider the recent achievements of deep reinforcement learning when applied to games, which presents an equivalent computational problem to that of the genome assembly~\\cite{Vinyals2019,Silver2017,fjelland2020}. One of the great challenges for production equivalent proposals for the assembly problem is that several promising works apply convolutional artificial networks, which use images as inputs.\n\nThe transformation of real problems into games is a possibility for reusing the promising technology developed focusing on games~\\cite{Reis2020}. In this sense, the modeling of the genome assembly problem through a game can work as an alternative representation of the problem and may reduce the space of states explored by the agent. Representing the assembly problem as a maze with multiple targets\/objectives could be an example for such modeling approach. \n\nOne of the main benefits of representing the problem as a game is the reduction of the space of actions, which, in the proposed modeling approach, increases with the number of \\emph{reads}. Mnih et. al~\\cite{Mnih2015}, for example, were capable of achieving a common deep reinforcement learning architecture capable of resolving several Atari games, being remarkable that the best performances were achieved in games with fewer actions, as in the game \\emph{Breakout}, which requires only two actions (move right or left). In this game the agent was able to learn, from its performance and without instruction, that producing a gap is a valid strategy for optimizing its results (illustrated in Figure \\ref{fig:08}).\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=8cm,keepaspectratio]{fig8}\n\\caption{Example of application of deep reinforcement learning in games where the agent was able to find strategies to optimize accumulated rewards. Here, the agent was able to discover that caving a tunnel could maximize rewards gain (images adapted from a video demonstration of learning progress in ~\\cite{Mnih2015}).}\n\\label{fig:08}\n\\end{figure}\n\nThe use of \\emph{Graph Embedding} may act as an option of modeling approach allowing the use of deep reinforcement learning without requiring the conversion of the problem into an image \u2013-- especially when considering that the genome assembly problem may be represented through a graph for optimizing the problem, in the shape of the Traveling Salesman Problem (TSP)~\\cite{Cook2012, Zhenyu2011}. As presented by Vesselinova et. al, numerous studies investigated the application of deep reinforcement learning for solving graph problems, including TSP~\\cite{Vesselinova2020}.\n\nFinally, one other aspect to be considered before the adoption of reinforcement learning into the genome assembly problem is the generalization of the agent\u00b4s learning --- a major challenge for the use of RL in real-world problems~\\cite{Ponsen2010}. As designed into the RL environment for the genome assembly problem, the learning acquired by the agent when assembling a set of \\emph{reads} will hardly be applied for the assembly of a new set.\n\n\\section{Additional information}\n\nAll data generated or analysed during this study are included in this published article. Supplementary material and reproduction codes are available at \\url{https:\/\/osf.io\/tp4zj\/?view_only=8bd12290d9d4480f95bc2c6e49c73ff7}.\n\n\\section{Competing interests}\nThe authors declare no competing interests.\n\n\\section{Author contributions statement}\n\nK.P., R.X and R.A. conceived the experiments,  K.P. and R.X. conducted the experiments, K.P., R.A., A.C. and A.R. analysed the results. All authors reviewed the manuscript. \n\n\\section{Acknowledgments}\nThis study was financed in part by the Coordena\u00e7\u00e3o de Aperfei\u00e7oamento de Pessoal de N\u00edvel Superior \u2013 Brasil (CAPES) \u2013 Finance Code 001.\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nJamison~\\cite{Jamison1983} initiated the study of the mean subtree order of a tree.  A number of extensions of this mean to other (connected) graphs have recently been considered:\n\\begin{itemize}\n\\item the mean order of the sub-$k$-trees of a $k$-tree~\\cite{StephensOellermann2018},\n\\item the mean order of the subtrees (i.e., minimally connected subgraphs) of a graph~\\cite{ChinGordonMacpheeVincent2018}, and\n\\item the mean order of the connected induced subgraphs of a graph~\\cite{KroekerMolOellermann2018}.\n\\end{itemize}\nFor a tree $T$, all of these means equal the mean subtree order of $T$. However, for connected graphs in general, the last two means have rather different behaviour.\n\nIn this article, we continue the study of the average order of the connected induced subgraphs of a graph $G$, called the {\\em mean connected induced subgraph (CIS) order} of $G$.  An in-depth study of the mean CIS order of cographs was undertaken in~\\cite{KroekerMolOellermann2018}, where the connected cographs of order $n$ having largest and smallest mean CIS order were determined (both the maximum and minimum values tend to $n\/2$ asymptotically).  Here, we focus on the mean CIS order of block graphs, i.e., graphs for which every block is complete.  We extend several of Jamison's results~\\cite{Jamison1983} on the mean subtree order of trees to the more general setting of the mean CIS order of connected block graphs (note that every tree is a connected block graph).\n\nIn particular, Jamison~\\cite{Jamison1983} demonstrated that among all trees of order $n$, the path $P_n$ has minimum mean subtree order (or equivalently, mean CIS order).  Our main result is that the path $P_n$ has minimum mean CIS order among all connected block graphs of order $n$.  This supports the conjecture of Kroeker, Mol, and Oellermann~\\cite{KroekerMolOellermann2018} that the path $P_n$ has minimum mean CIS order among all connected graphs of order $n$, which has been verified for all $n\\leq 9$.\n\nA key tool in the proof of our main result is an extension of the ``local-global mean inequality'' proven by Jamison~\\cite{Jamison1983} for trees.  For a given tree $T$, and every vertex $v$ of $T$, Jamison demonstrated that the mean order of all connected induced subgraphs of $T$ containing $v$ (i.e., the ``local'' mean CIS order of $T$ at $v$) is at least as large as the mean CIS order of $T$ (i.e., the ``global'' mean CIS order of $T$). It is known that this inequality between local and global mean CIS orders does not extend to all connected graphs (at least not at every vertex)~\\cite{KroekerMolOellermann2018}.  However, we note that it was recently proven, in a more general context, that every graph with nonempty edge set contains at least one vertex at which the local mean CIS order is larger than the global mean CIS order (apply~\\cite[Theorem 3.1]{AMW} to the collection of vertex sets that induce connected subgraphs of $G$).  In other words, while the local-global mean inequality does not necessarily hold at \\emph{every} vertex of a connected graph $G$, it must hold at \\emph{some} vertex of $G$.  In this article, we demonstrate that the local-global mean inequality does hold at every vertex of a connected block graph.  This fact is essential to the proofs of the three key lemmas used to establish our main result.  \n\nWe now give a brief description of the layout of the article.  In Section~\\ref{Preliminaries}, we provide notation and preliminaries that will be used throughout the article.  In Section~\\ref{Main_Result}, we state three key lemmas (the Vertex Gluing Lemma, the Edge Gluing Lemma, and the Stetching Lemma), and we use them to prove our main result.  We then describe an interesting connection between the mean CIS order of block graphs and the mean sub-$k$-tree order of $k$-trees, and explain the implications of our main result in this setting.  In Section~\\ref{LocalGlobalSection}, we prove the local-global mean inequality for the mean CIS order of block graphs.  In Section~\\ref{KeyLemmas}, we prove the Vertex Gluing Lemma, the Edge Gluing Lemma, and the Stretching Lemma.  We conclude with some open problems.\n\t\n\\section{Notation and Preliminaries}\\label{Preliminaries}\n\nFor a graph $G$, the vertex and edges sets of $G$ are denoted by $V(G)$ and  $E(G)$, respectively.  The \\emph{order} of $G$ is $|V(G)|$ and the \\emph{size} of $G$ is $|E(G)|$.  For $U\\subseteq V(G)$, the subgraph of $G$ induced by $U$ is denoted $G[U]$.  The (open) neighbourhood of  a vertex $v$ of $G$ is denoted $N_G(v)$.\n\nLet $G$ be a graph of order $n$.  Let $\\mathcal{C}_G$ denote the collection of connected induced subgraphs of $G$. The \\emph{CIS polynomial of $G$} is given by\n\\[\n\\Phi_G(x)=\\sum_{H\\in \\mathcal{C}_G}x^{|V(H)|}=\\sum_{i=1}^na_ix^i,\n\\]\nwhere $a_i$ is the number of connected induced subgraphs of $G$ of order $i$ for each $i\\in\\{1,\\dots,n\\}$.\nOne easily verifies that $\\Phi_G(1)$ is the total number of connected induced subgraphs of $G$, and that $\\Phi'_G(1)$ is the sum of the orders of all connected induced subgraphs of $G$.  Throughout, we use the shorthand notation $N_G=\\Phi_G(1)$ and $W_G=\\Phi'_G(1)$.  The \\emph{mean CIS order of $G$}, denoted $M_G$, is given by\n\\[\nM_G=\\frac{\\Phi'_G(1)}{\\Phi_G(1)}=\\frac{W_G}{N_G}.\n\\]\nFor a vertex $v\\in V(G)$, let $\\mathcal{C}_{G,v}$ denote the collection of connected induced subgraphs of $G$ containing $v$.  The \\textit{local CIS polynomial of $G$ at $v$} is given by\n\\[\n\\Phi_{G, v}(x)=\\sum_{H\\in\\mathcal{C}_{G,v}}x^{|V(H)|}=\\sum_{i=1}^nb_ix^i,\n\\]\nwhere $b_i$ is the number of connected induced subgraphs of $G$ of order $i$ containing $v$ for each $i\\in\\{1,\\dots,n\\}.$  So $N_{G,v}=\\Phi_{G,v}(1)$ denotes the total number of connected induced subgraphs of $G$ containing $v$, and  $W_{G,v}=\\Phi'_{G,v}(1)$ denotes the sum of the orders of all connected induced subgraphs of $G$ containing $v$.  The \\emph{local mean CIS order of $G$}, denoted $M_{G,v}$, is given by\n\\[\nM_{G,v}=\\frac{\\Phi'_{G,v}(1)}{\\Phi_{G,v}(1)}=\\frac{W_{G,v}}{N_{G,v}}.\n\\]\n\nThe next lemma gives a recursion for the local mean CIS order of a block graph at a cut vertex $v$.  It holds trivially if $v$ is not a cut vertex.\n\n\\begin{lemma}\\label{LocalSum}\nLet $G$ be a block graph with vertex $v$, and let $H_1,\\dots, H_k$ be the components of $G-v$.  For $i\\in\\{1,\\dots,k\\},$ let $G_i=G[V(H_i)\\cup \\{v\\}]$.   Then\n\\[\nM_{G,v}=\\left[\\sum_{i=1}^k M_{G_i,v}\\right]-(k-1).\n\\]\nFurther, we have\n\\[\nM_{G,v}\\geq M_{G_i,v}\n\\]\nfor all $i\\in\\{1,\\dots,k\\}.$\n\\end{lemma}\n\n\\begin{proof}\nBy a straightforward counting argument,\n\\[\n\\Phi_{G,v}(x)=\\tfrac{1}{x^{k-1}}\\prod_{i=1}^k\\Phi_{G_i,v}(x).\n\\]\nTaking the natural logarithm on both sides and differentiating with respect to $x$, we obtain\n\\[\n\\frac{\\Phi'_{G,v}(x)}{\\Phi_{G,v}(x)}=\\left[\\sum_{i=1}^k\\frac{\\Phi'_{G_i,v}(x)}{\\Phi_{G_i,v}(x)}\\right]-\\frac{k-1}{x}.\n\\]\nSubstituting $x=1$ yields\n\\[\nM_{G,v}=\\left[\\sum_{i=1}^k M_{G_i,v}\\right]-(k-1).\n\\]\nSince $M_{G_i,v}\\geq 1$ for all $i\\in\\{1,\\dots,k\\},$ it follows that $M_{G,v}\\geq M_{G_i,v}$ for all $i\\in\\{1,\\dots,k\\}$.\n\\end{proof}\n\nWe extend the notion of the local mean CIS order of a graph $G$ in two natural ways.  For a subset $U$ of $V(G)$, we let $M_{G,U}$ denote the mean order of all connected induced subgraphs of $G$ containing every vertex of $U$.  We let $\\Phi_{G,U}(x)$ denote the corresponding generating polynomial.  We let $M^*_{G,U}$ denote the mean order of all connected induced subgraphs of $G$ containing at least one vertex of $U$.  We let $\\Phi^*_{G,U}(x)$ denote the corresponding generating polynomial, and $N^*_{G,U}=\\Phi^*_{G,U}(1)$ and $W^*_{G,U}(x)=\\Phi^{*'}_{G,U}(1).$  Note that if $U$ contains only a single vertex $u$, then $M_{G,U}=M^*_{G,U}=M_{G,u}.$\n\n\\begin{lemma}\\label{LocalBlockLemma}\nLet $G$ be a block graph, and let $U=\\{u_1,\\dots,u_k\\}$ be the vertex set of a single block $B$ of $G$.  For each $i\\in\\{1,\\dots,k\\},$ let $G_i$ be the connected component of $G-E(B)$ containing $u_i$.  Then\n\\begin{align}\\label{LocalAtBlock}\nM^*_{G,U}=\\frac{N^*_{G,U}+1}{N^*_{G,U}}\\sum_{i=1}^k \\frac{W_{G_i,u_i}}{N_{G_i,u_i}+1}.\n\\end{align}\n\\end{lemma}\n\n\\begin{proof}\nBy a straightforward counting argument,\n\\[\n\\Phi^*_{G,U}(x)+1=\\prod_{i=1}^k \\left[ 1+\\Phi_{G_i,u_i}(x)\\right].\n\\]\nTaking the natural logarithm on both sides and differentiating, we obtain\n\\[\n\\frac{\\Phi^{*'}_{G,U}(x)}{\\Phi^*_{G,U}(x)+1}=\\sum_{i=1}^k\\frac{\\Phi'_{G_i,u_i}(x)}{\\Phi_{G_i,u_i}(x)+1}.\n\\]\nSubstituting $x=1$ gives\n\\[\n\\frac{W^*_{G,U}}{N^*_{G,U}+1}=\\sum_{i=1}^k\\frac{W_{G_i,u_i}}{N_{G_i,u_i}+1}.\n\\]\nMultiplying both sides by $\\frac{N^*_{G,U}+1}{N^*_{G,U}}$ and noting that $M^*_{G,U}=\\frac{W^*_{G,U}}{N^*_{G,U}},$ we obtain~(\\ref{LocalAtBlock}).\n\\end{proof}\n\nA key idea that we use in many of our arguments states that if the connected induced subgraphs of a graph $G$ can be partitioned into two or more sets, then $M_G$ is a convex combination (or weighted average) of the mean orders of each of the sets in the partition. This tool was used by Jamison~\\cite[Lemma 3.8]{Jamison1983} for the mean subtree order of a tree.   For example, since every connected induced subgraph of $G$ either contains a given vertex $v$, or does not contain $v$, we can write\n\\[\n\\Phi_G(x)=\\Phi_{G,v}(x)+\\Phi_{G-v}(x).\n\\]\nIt follows that $M_G$ is a convex combination of $M_{G,v}$ and $M_{G-v}$.  Another useful application of this principle is to disconnected graphs.  If $G$ is a disconnected graph with components $G_1,\\dots,G_k$, then $M_G$ is a convex combination of $M_{G_1},\\dots,M_{G_k}$.  It follows that $\\min\\{M_{G_i}\\}\\leq M_G\\leq \\max\\{M_{G_i}\\}.$\n\n\\section{Proof of the Main Result}\\label{Main_Result}\n\nIn this section, we show that among all block graphs of order $n$, the path has minimum mean CIS order.  We use three key lemmas, namely the Vertex Gluing Lemma, the Edge Gluing Lemma, and the Stretching Lemma, which are proven in Section~\\ref{KeyLemmas}.  We state these lemmas here, and provide illustrations depicting how they are used in the proof of the main result (see Figure~\\ref{LemmaPictures}). By \\emph{gluing} two vertices from disjoint graphs, we mean the process of identifying these two vertices.\n\n\n\\medskip\n\n\\noindent{\\bf The Vertex Gluing Lemma} (Lemma \\ref{vertex_gluing})\\\\\n\\textit{Let $H$ be a connected block graph of order at least $2$ having vertex $v$.  Fix a natural number $n\\geq 3$.  Let $P:u_1\\dots u_n$ be a path of order $n$. For $s \\in \\{1, \\dots, n\\}$, let $G_s$ be the block graph obtained from the disjoint union of $P_n$ and $H$ by gluing $v$ to $u_s$.  If $1\\leq i<j\\leq \\tfrac{n+1}{2}$, then $M_{G_i}<M_{G_j}.$}\n\\medskip\n\nIn the notation of the Vertex Gluing Lemma, $G_s \\cong G_{n-s+1}$.  Thus, it follows from the Vertex Gluing Lemma that $G_1\\cong G_n$ has strictly smaller mean CIS order than $G_s$ for all $2\\leq s\\leq n-1$ (see Figure~\\ref{VertexPicture}).  We note that the Vertex Gluing Lemma extends the Gluing Lemma of~\\cite{MolOellermann2017} from trees to block graphs.\n\nTo simplify the statement of the Edge Gluing Lemma, we will refer to a leaf as a vertex of degree at most $1$ (so that the single vertex of the path of order $1$ is considered a leaf).\n\n \\medskip\n\n\\noindent{\\bf The Edge Gluing Lemma} (Lemma \\ref{edge_gluing})\\\\\n\\textit{Let $H$ be a connected block graph of order at least $3$ with adjacent non-cut vertices $u$ and $v$.  Fix a natural number $n\\geq 4$.  For $s\\in\\{1,2,\\dots,n-1\\}$, let $G_s$ be the graph obtained from $H\\cup P_s\\cup P_{n-s}$ by gluing a leaf of $P_s$  to $u$ and a leaf of $P_{n-s}$ to $v$.  If $1\\leq i<j\\leq \\tfrac{n}{2}$, then $M_{G_i}<M_{G_j}.$}\n\n\n\\medskip\n\nIn the notation of the Edge Gluing Lemma, $G_s \\cong G_{n-s}$. Thus, it follows from the Edge Gluing Lemma that $G_1\\cong G_{n-1}$ has strictly smaller mean CIS order than $G_s$ for all $2\\leq s\\leq n-2$ (see Figure~\\ref{EdgePicture}).\n\nWe now introduce notation used in the statement of the Stretching Lemma. For a fixed integer $n\\geq 3$ and $s\\in\\{1,2,\\dots,n-1\\}$, let $F_{s,n-s}$ (or $F_s$ for short) denote the graph obtained from the disjoint union of $K_{s}$ and $P_{n-s}$ by joining a leaf of $P_{n-s}$ to every vertex of $K_s$.  Note that if $s=1$, then $F_s\\cong P_n$, while if $s=n-1$, then $F_s\\cong K_n$.\n\n\\medskip\n\n\n\\noindent{\\bf The Stretching Lemma} (Lemma \\ref{stretching_lemma})\n\\textit{Let $H$ be a connected block graph of order at least $2$ having vertex $u$.  Fix a natural number $n\\geq 3$.  Let $v$ be a vertex of $F_s=F_{s,n-s}$ belonging to the initial $K_s$.  For $s \\in \\{1, \\dots, n-1\\}$, let $G_s$ be the block graph obtained from the disjoint union of $H$ and $F_{s,n-s}$ by identifying $u$ and $v$.  If $1\\leq i<j\\leq n-1$, then $M_{G_i}<M_{G_j}.$}\n\n\\medskip\n\nIn particular, in the notation of the Stretching Lemma, we see that $G_1$ has strictly smaller mean CIS order than $G_s$ for all $2\\leq s\\leq n-1$ (see Figure~\\ref{StretchingPicture}).\n\n\\begin{figure}\n\\centering{\n\\begin{subfigure}[b]{\\textwidth}\n\\centering{\n\\begin{tikzpicture}\n\\node[vertex] (0) at (0,0) {};\n\\node[vertex] (1) at (1,0) {};\n\\node[vertex] (2) at (2.5,0) {};\n\\node[vertex] (3) at (4,0) {};\n\\node[vertex] (4) at (5,0) {};\n\\path\n(0) edge (1)\n(3) edge (4);\n\\draw (1,0) -- (1.4,0);\n\\draw[dotted] (1.4,0)--(2.1,0);\n\\draw (2.1,0) -- (2.5,0);\n\\draw (2.5,0) -- (2.9,0);\n\\draw[dotted] (2.9,0)--(3.6,0);\n\\draw (3.6,0) -- (4,0);\n\\draw (2.5,-1) circle [radius=1];\n\\node[above,yshift=2pt] at (0,0) {\\footnotesize $u_1$};\n\\node[above,yshift=2pt] at (1,0) {\\footnotesize $u_2$};\n\\node[above,yshift=2pt] at (2.5,0) {\\footnotesize $u_s$};\n\\node[above,yshift=2pt] at (4,0) {\\footnotesize $u_{n-1}$};\n\\node[above,yshift=2pt] at (5,0) {\\footnotesize $u_n$};\n\\node[below,yshift=-2pt] at (2.5,0) {\\footnotesize $v$};\n\\node at (2.5,-1) {$H$};\n\\end{tikzpicture}\n\\hspace{1cm}\n\\begin{tikzpicture}\n\\node[vertex] (3) at (3,0) {};\n\\node[vertex] (4) at (4,0) {};\n\\node[vertex] (5) at (5.5,0) {};\n\\node[vertex] (6) at (6.5,0) {};\n\\path\n(3) edge (4)\n(5) edge (6);\n\\draw (4,0) -- (4.4,0);\n\\draw[dotted] (4.4,0)--(5.1,0);\n\\draw (5.1,0) -- (5.5,0);\n\\draw (3,-1) circle [radius=1];\n\\node[above,yshift=2pt] at (3,0) {\\footnotesize $u_1$};\n\\node[above,yshift=2pt] at (4,0) {\\footnotesize $u_{2}$};\n\\node[above,yshift=2pt] at (5.5,0) {\\footnotesize $u_{n-1}$};\n\\node[above,yshift=2pt] at (6.5,0) {\\footnotesize $u_n$};\n\\node[below,yshift=-2pt] at (3,0) {\\footnotesize $v$};\n\\node at (3,-1) {$H$};\n\\end{tikzpicture}\n\\caption{The Vertex Gluing Lemma: The graph $G_1$ (right) has smaller mean CIS order than the graph $G_s$ (left) for all $2\\leq s\\leq n-1$.}\\label{VertexPicture}}\n\\end{subfigure}\\\\\n\\bigskip\n\\begin{subfigure}[b]{\\textwidth}\n\\centering{\n\\begin{tikzpicture}\n\\node[vertex] (0) at (0,0) {};\n\\node[vertex] (1) at (1,0) {};\n\\node[vertex] (2) at (2.5,0) {};\n\\node[vertex] (3) at (3.5,0) {};\n\\node[vertex] (4) at (5,0) {};\n\\node[vertex] (5) at (6,0) {};\n\\path\n(0) edge (1)\n(2) edge (3)\n(4) edge (5);\n\\draw (1,0) -- (1.4,0);\n\\draw[dotted] (1.4,0)--(2.1,0);\n\\draw (2.1,0) -- (2.5,0);\n\\draw (3.5,0) -- (3.9,0);\n\\draw[dotted] (3.9,0)--(4.6,0);\n\\draw (4.6,0) -- (5,0);\n\\draw (3,-0.85) circle [radius=1];\n\\draw [decorate,decoration={brace,amplitude=4pt},xshift=0pt,yshift=6pt]\n(-0.2,0) -- (2.7,0) node [above,black,midway,yshift=4pt]\n{\\footnotesize $s$ vertices};\n\\draw [decorate,decoration={brace,amplitude=4pt},xshift=0pt,yshift=6pt]\n(3.3,0) -- (6.2,0) node [above,black,midway,yshift=4pt]\n{\\footnotesize $n-s$ vertices};\n\\node[below,yshift=-2pt] at (2.5,0) {\\footnotesize $u$};\n\\node[below,yshift=-2pt] at (3.5,0) {\\footnotesize $v$};\n\\node at (3,-0.85) {$H$};\n\\end{tikzpicture}\n\\hspace{1cm}\n\\begin{tikzpicture}\n\\node[vertex] (3) at (3,0) {};\n\\node[vertex] (4) at (4,0) {};\n\\node[vertex] (5) at (5.5,0) {};\n\\node[vertex] (6) at (6.5,0) {};\n\\path\n(3) edge (4)\n(5) edge (6);\n\\draw (4,0) -- (4.4,0);\n\\draw[dotted] (4.4,0)--(5.1,0);\n\\draw (5.1,0) -- (5.5,0);\n\\draw (3.5,-0.85) circle [radius=1];\n\\draw [decorate,decoration={brace,amplitude=4pt},xshift=0pt,yshift=6pt]\n(2.8,0) -- (6.7,0) node [above,black,midway,yshift=4pt]\n{\\footnotesize $n$ vertices};\n\\node[below,yshift=-2pt] at (3,0) {\\footnotesize $u$};\n\\node[below,yshift=-2pt] at (4,0) {\\footnotesize $v$};\n\\node at (3.5,-0.85) {$H$};\n\\end{tikzpicture}\n\\caption{The Edge Gluing Lemma: The graph $G_1$ (right) has smaller mean CIS order than the graph $G_s$ (left) for all $2\\leq s\\leq n-2$. Note: $u$ and $v$ must be adjacent in $H$, and must not be cut vertices of $H$.}\\label{EdgePicture}}\n\\end{subfigure}\\\\\n\\bigskip\n\\begin{subfigure}[b]{\\textwidth}\n\\centering{\n\\begin{tikzpicture}\n\\draw (-0.5,0) circle[radius=1];\n\\node at (-0.5,0) {$H$};\n\\draw (1.25,0) circle[radius=0.75];\n\\node at (1.25,0) {\\footnotesize $K_{s+1}$};\n\\draw [decorate,decoration={brace,amplitude=4pt},xshift=0pt,yshift=6pt]\n(1.8,0) -- (5.7,0) node [above,black,midway,yshift=4pt]\n{\\footnotesize $n-s$ vertices};\n\\node[vertex] (1) at (0.5,0) {};\n\\node[vertex] (2) at (2,0) {};\n\\node[vertex] (3) at (3,0) {};\n\\node[vertex] (4) at (4.5,0) {};\n\\node[vertex] (5) at (5.5,0) {};\n\\path\n(2) edge (3)\n(4) edge (5);\n\\draw (3,0) -- (3.4,0);\n\\draw[dotted] (3.4,0)--(4.1,0);\n\\draw (4.1,0) -- (4.5,0);\n\\end{tikzpicture}\n\\hspace{1cm}\n\\begin{tikzpicture}\n\\draw (1,0) circle[radius=1];\n\\node at (1,0) {$H$};\n\\draw [decorate,decoration={brace,amplitude=4pt},xshift=0pt,yshift=6pt]\n(1.8,0) -- (5.7,0) node [above,black,midway,yshift=4pt]\n{\\footnotesize $n$ vertices};\n\\node[vertex] (2) at (2,0) {};\n\\node[vertex] (3) at (3,0) {};\n\\node[vertex] (4) at (4.5,0) {};\n\\node[vertex] (5) at (5.5,0) {};\n\\path\n(2) edge (3)\n(4) edge (5);\n\\draw (3,0) -- (3.4,0);\n\\draw[dotted] (3.4,0)--(4.1,0);\n\\draw (4.1,0) -- (4.5,0);\n\\end{tikzpicture}\n\\caption{The Stretching Lemma: The graph $G_1$ (right) has smaller mean CIS order than the graph $G_s$ (left) for all $2\\leq s\\leq n-1$.}\\label{StretchingPicture}}\n\\end{subfigure}\n}\n\\caption{An illustration of the use of the Vertex Gluing Lemma, the Edge Gluing Lemma, and the Stretching Lemma, where $H$ is a block graph.}\n\\label{LemmaPictures}\n\\end{figure}\t\n\t\n\t\n\n\\medskip\n\n\n\nWe now give some terminology and some basic facts used in the proof of our main result.  Let $G$ be a  graph that has at least two blocks.  An \\emph{end-block} is a block that contains exactly one cut vertex. A \\emph{cyclic block} is a block that contains at least $3$ vertices. Thus, cyclic blocks in block graphs are complete graphs of order at least $3$. The \\emph{block cut vertex tree} $T$ of a connected graph $G$ is the graph whose vertex set is the union of the set of blocks of $G$ and the set of cut vertices of $G$, where a block is joined to a cut vertex if and only if the cut vertex is contained in the block.\nIf $G$ is a block graph that is not a path, and if $G$ has a leaf $w$, then the shortest path $P$ from $w$ to a vertex $v$ of degree at least $3$ is called an {\\em antenna} of $G$ {\\em incident} with $v$. If $B$ is a block containing $v$ and no other vertex of $P$, we say that the antenna $P$ is {\\em incident} with $B$.  Finally, we use the fact that $M_{P_n} = \\frac{n+2}{3}$ (see~\\cite{Jamison1983}), and the fact that $M_{K_n} = \\frac{n2^{n-1}}{2^n -1}$ (see~\\cite{KroekerMolOellermann2018}).  We are now ready to prove our main result.\n\n\\begin{theorem}\nIf $G$ is a connected block graph of order $n$, then $M_G\\geq \\frac{n+2}{3}$, with equality if and only if $G\\cong P_n$.  In other words, the path $P_n$ is the unique connected block graph of order $n$ having minimum mean CIS order $\\frac{n+2}{3}.$\n\\end{theorem}\n\\begin{proof}\nLet $G\\not\\cong P_n$ be a connected block graph of order $n$.  We demonstrate that there is a connected block graph of order $n$ that has smaller mean CIS order than $G$, from which the statement follows.  Since the only connected block graphs of order $1$ and $2$ are paths, we may assume that $n \\ge 3$. If $G$ has only one block, then $G \\cong K_n$. Since $M_{P_n}=\\frac{n+2}{3}< \\tfrac{n2^{n-1}}{2^n-1}=M_G$ for $n \\ge 3$, our claim follows.  Thus, we may assume that $G$ has at least two blocks.\nFor the remainder of the proof we consider two cases that depend on the structure of the block cut vertex tree $T$ of $G$.\n\nIf $T$ is a path, then it follows, since $G$ is not a path, that $G$ has a cyclic block. So either $G$ has a cyclic end-block or $G$ has a cyclic block that is incident with an antenna and contains exactly two cut-vertices. We can thus apply the Stretching Lemma to $G$ to obtain a connected block graph that has smaller mean CIS order.\n\nAssume next that $T$ is not a path. Then $T$ contains a vertex of degree at least 3. So $T$ has a vertex  $u$ of degree at least 3, such that all but possibly one connected component of $T-u$ is isomorphic to a path. Let $\\mathcal{C}$ be the family of components of $T-u$ that induce paths in $T-u$.  Suppose that at least one of the members of $\\mathcal{C}$, $Q$ say,  contains a  block that is cyclic in $G$. In this case we can apply the Stretching Lemma to obtain a connected block graph with smaller mean CIS order than $G$. We may thus assume that all members of $\\mathcal{C}$ correspond to antenna in $G$ that are incident with $u$.\n\nNow $u$ is either a cut vertex of $G$, or a block of $G$. If $u$ is a cut vertex of $G$, then we can apply the Vertex Gluing Lemma to obtain a connected block graph whose mean CIS order is less than that of $G$.  So we may assume that $u$ is a block of $G$.  Let $v_1, v_2, \\ldots, v_k$ be the cut vertices of $G$ that belong to the block $u$. By assumption, $k \\ge 3$. Moreover, we may assume that, for $1 \\le i \\le k-1$, the component of $T-u$ containing $v_i$ is a path, i.e., belongs to $\\mathcal{C}$.  By the argument of the previous paragraph, the members of $\\mathcal{C}$ contain no cyclic blocks, and hence they correspond to antenna in $G$. Let $P_1$ and $P_2$ be the antenna of $G$ incident with $v_1$ and $v_2$, respectively. If we delete the vertices of $V(P_1) -\\{v_1\\}$ and $V(P_2)-\\{v_2\\}$ from $G$, neither $v_1$ nor $v_2$ is a cut vertex of the resulting graph.  By applying the Edge Gluing Lemma, we obtain a connected block graph with smaller mean CIS order than $G$.\n\\end{proof}\n\nWe close this section by describing a relationship between the mean sub-$k$-tree order of $k$-trees (see \\cite{StephensOellermann2018}) and the mean CIS order of connected block graphs, and the implications of our main result in this setting.   The \\emph{dual} $T'$ of a $k$-tree $T$ of order $n$ is defined to be the graph of order $n-k$ whose vertex set consists of the $(k+1)$-cliques in $T$, such that two vertices of $T'$ are adjacent if and only if the corresponding $(k+1)$-cliques in $T$ share a $k$-clique. It is not difficult to see that this dual of a nontrivial $k$-tree is a connected block graph. It was demonstrated in \\cite{StephensOellermann2018} that there is a correspondence between the number of nontrivial sub-$k$-trees of a $k$-tree, and the number of connected induced subgraphs of its dual. Let $\\mu(T)$ denote the mean order of the sub-$k$-trees of the $k$-tree $T$. Then the argument used in \\cite{StephensOellermann2018}, for a certain type of $k$-tree, called a simple clique $k$-tree, can be used to show that for any $k$-tree $T$,\n\\[\\mu(T) = \\frac{W_{T'}}{N_{T'}+(n-k)k+1} +k.\\]\n Since $T'$ has order $n-k$, it follows from our main result that $M_{T'} \\ge M_{P_{n-k}} = \\frac{n-k+2}{3}$. Moreover, $N_{T'} \\ge N_{P_{n-k}}= \\binom{n-k+1}{2}$. So, for fixed $k$, $\\lim_{n\\rightarrow \\infty}\\tfrac{\\mu(T)}{M_{T'}} = 1$. In other words, $\\mu(T) = M_{T'} + \\mathrm{o}(n)$.  We conclude that understanding the behaviour of the mean CIS order of block graphs offers insight into the mean order of sub-$k$-trees of $k$-trees for large orders.\n\n\n\n \t\n\\section{The Local-Global Mean Inequality}\\label{LocalGlobalSection}\n\nIn this section, we prove the local-global mean inequality for connected block graphs.  That is, we show that the local mean CIS order of a connected block graph $G$ at any vertex $v$ is greater than the (global) mean CIS order of $G$, extending the result of Jamison for trees~\\cite[Theorem 3.9]{Jamison1983}.  We actually prove the stronger result that $M^*_{G,U}\\geq M_G,$ where $U$ is either a single vertex, or the vertex set of a block of $G$.  In the process, we achieve several intermediate results which are used again later.\n\nWe begin by presenting a series of short lemmas that give inequalities between the number and\/or total order of connected induced subgraphs of certain types in a given graph.  Some of these results hold for connected graphs in general.  First we note that \\cite[Lemma 2.1]{WagnerWang2016} extends to the mean CIS order of connected graphs.  The proof is analogous to the proof of~\\cite[Lemma 2.1]{WagnerWang2016}, and is omitted.\n\n\\begin{lemma}\\label{WagLem}\nLet $G$ be a connected graph and let $v$ be any vertex of $G$.  Then\n$$W_{G, v}\\leq\\frac{N_{G, v}^2+N_{G, v}}{2},$$\nwith equality if and only if $G\\cong P_n$ and $v$ is a leaf of $G$. \\hfill\\qed\n\\end{lemma}\n\nWe continue with three short lemmas whose proofs all use a similar technique.  Recall that $\\mathcal{C}_{G}$ denotes the collection of connected induced subgraphs of $G$, and $\\mathcal{C}_{G,v}$ denotes the collection of connected induced subgraphs of $G$ containing $v$. So $|\\mathcal{C}_G|=N_G$ and $|\\mathcal{C}_{G,v}|=N_{G,v}$.\n\n\\begin{lemma}\\label{AdjacentVertexNumbers}\nLet $G$ be a connected graph with adjacent vertices $u$ and $v$.  Then\n\\[\nN_{G-v,u}\\leq N_{G,v}-1,\n\\]\nwith equality if and only if $v$ is a leaf of $G$.\n\\end{lemma}\n\n\\begin{proof}\nDefine $f\\colon\\ \\mathcal{C}_{G-v,u}\\rightarrow \\mathcal{C}_{G,v}$ by $f(H)=G[V(H)\\cup\\{v\\}].$  One easily verifies that $f$ is well-defined and injective.  Further, the trivial graph on singleton vertex set $\\{v\\}$ is not in the image of $f$, hence\n\\[\nN_{G-v,u}=|\\mathcal{C}_{G-v,u}|\\leq |\\mathcal{C}_{G,v}|-1= N_{G,v}-1.\n\\]\nIf $v$ is a leaf of $G$ then every nontrivial connected induced subgraph of $G$ containing $v$ also contains $u$, and thus is in the image of $f$.  This gives equality.  Otherwise, $v$ has a neighbour $w\\neq u$, and the nontrivial connected subgraph of $G$ induced by $\\{v,w\\}$ is not in the image of $f$, giving strict inequality.\n\\end{proof}\n\n\\begin{lemma}\\label{NumCom}\nLet $G$ be a connected block graph with non-cut vertex $v$.  Then\n\\[\nN_{G, v}\\leq N_{G-v}+1,\n\\]\nwith equality if and only if $G\\cong K_n$.\n\\end{lemma}\n\\begin{proof}\nLet $\\mathcal{C}^+_{G,v}$ denote the collection of \\emph{nontrivial} connected induced subgraphs of $G$ containing $v$.  Define a map $f\\colon\\ \\mathcal{C}^+_{G,v}\\rightarrow \\mathcal{C}_{G-v}$ by $f(H)=H-v$. First we show that $f$ is well-defined.  Let $H\\in \\mathcal{C}^+_{G,v}.$  Since $\\mathcal{C}^+_{G,v}$ contains only \\emph{nontrivial} graphs, we note that $H-v$ has at least one vertex.  It remains to show that $H-v$ is connected.  Since $v$ is a non-cut vertex of $G$, and $G$ is a block graph, the open neighbourhood $N_H(v)$ induces a complete subgraph of $H$.  So the deletion of $v$ from $H$ does not separate any two neighbours of $v$.  Therefore, $H-v$ is connected and thus $f$ is well-defined.  Further, note that $f$ is injective, since if $f(H_1)=f(H_2)$, then $H_1$ and $H_2$ have the same vertex set.  Therefore,\n\\[\nN_{G,v}=\\left|\\mathcal{C}^+_{G,v}\\right|+1\\leq \\left|\\mathcal{C}_{G-v}\\right|+1=N_{G-v}+1.\n\\]\nEquality is easily verified if $G\\cong K_n$.  On the other hand, if $G\\not\\cong K_n$, then there is some vertex $u\\in V(G)$ that is not adjacent to $v$.  Now $G[\\{u\\}]\\in \\mathcal{C}_{G-v},$ but $G[\\{u,v\\}]\\not\\in \\mathcal{C}^*_{G,v},$ so we conclude that $f$ is not onto.  This gives strict inequality.\n\\end{proof}\n\n\\begin{lemma}\\label{WeiNum}\nLet $G$ be a connected graph of order $n \\geq 2$ and let $u$ be any vertex of $G$.  Then\n\\[\nN_G\\leq W_{G, u}.\n\\]\n\\end{lemma}\n\\begin{proof}\nLet\n\\[\n\\mathcal{D}_{G,u}=\\{(X,x)\\colon\\ X\\in\\mathcal{C}_{G,u}, x\\in V(X)\\}.\n\\]\nObserve that $|\\mathcal{D}_{G,u}|=W_{G,u}$.  Therefore, it suffices to show that there is an injective function $f\\colon\\ \\mathcal{C}_G\\rightarrow \\mathcal{D}_{G,u}$.\n\t\t\nAssign a fixed ordering to the vertices of $G$.  For each vertex $v\\in V(G)$, assign a fixed ordering to the shortest $u$--$v$ paths in $G$.  Among all vertices of $H$ closest to $u$, let $x_H$ be the vertex that appears first in the given ordering.\nLet $P_H$ be the first path in the fixed ordering of shortest $u$--$x_H$ paths (if $H$ contains $u$ then $P_H$ consists of the single vertex $u$). Let $X_H$ be the subgraph of $G$ induced by $V(H) \\cup V(P_H)$.  Clearly $X_H\\in \\mathcal{C}_{G,u}$. Define $f(H) = (X_H, x_H)$. Suppose $f(H_1)=f(H_2)=(X,x)$.  Let $P$ be the first shortest $u$--$x$ path in the fixed ordering of $u$--$x$ paths.  So $X=G[V(H_1)\\cup V(P)]=G[V(H_2)\\cup V(P)]$.  Note that no vertex of $P$ other than $x$ can lie in $H_1$ or $H_2$, since $P$ is a shortest path from $u$ to $H_1$, and from $u$ to $H_2$.  Therefore,\n\\[\nH_1=X-(V(P)-x)=H_2.\n\\]\nThus, we have shown that $f$ is injective, and the desired conclusion follows.\n\\end{proof}\n\nTo prove the next lemma, we use an inductive argument and several lemmas already proven in this section.\n\n\\begin{lemma}\\label{WeightBound}\nLet $G$ be a connected block graph of order $n\\geq 1$, and let $v$ be a vertex of $G$.  Then\n\\[\nW_{G-v} \\leq \\frac{N_{G,v}N_{G-v}}{2},\n\\]\nwhich is equivalent to\n\\[\nM_{G-v}\\leq \\frac{N_{G,v}}{2}\n\\]\nwhen $n\\geq 2$.\n\\end{lemma}\n\n\\begin{proof}\nWe proceed by induction on $n$.  If $n=1$, then the statement holds trivially since $W_{G-v}=0$.  Now let $n\\geq 2$ and suppose that the statement holds for all graphs of order less than $n$.  First of all, if $v$ is a cut vertex of $G$, then $M_{G-v}\\leq M_H$, where $H$ is a component of $G-v$ of largest mean CIS order.  Let $H'=G[V(H)\\cup \\{v\\}]$.  By the induction hypothesis applied to $H'$ at $v$, we have\n\\[\nM_{G-v}\\leq M_{H}=M_{H'-v}\\leq \\frac{N_{H',v}}{2}<\\frac{N_{G,v}}{2}.\n\\]\n\nSo we may assume that $v$ is  not a cut vertex of $G$.  Let $u$ be a neighbour of $v$ in $G$, and write\n\\begin{align}\\label{WeightDecomp}\nW_{G-v}=W_{G-v,u}+W_{G-\\{u,v\\}}.\n\\end{align}\nBy Lemma \\ref{WagLem}, we have\n\\[\nW_{G-v,u}\\leq \\frac{N^2_{G-v,u}+N_{G-v,u}}{2},\n\\]\nand by the induction hypothesis applied to $G-v$ at $u$, we have\n\\[\nW_{G-\\{u,v\\}}\\leq \\frac{N_{G-v,u}N_{G-\\{u,v\\}}}{2}.\n\\]\nSubstituting these inequalities into~(\\ref{WeightDecomp}), we obtain\n\\begin{align*}\nW_{G-v} &\\leq \\frac{N^2_{G-v,u}+N_{G-v,u}}{2}+\\frac{N_{G-v,u}N_{G-\\{u,v\\}}}{2}\\\\\n&=\\frac{N_{G-v,u}\\left[N_{G-v,u}+1+N_{G-\\{u,v\\}}\\right]}{2}\\\\\n&=\\frac{N_{G-v,u}\\left[N_{G-v}+1\\right]}{2}.\n\\end{align*}\nSince $u$ and $v$ are adjacent in $G$, we may apply Lemma \\ref{AdjacentVertexNumbers}, which gives\n\\begin{align*}\nW_{G-v}&\\leq \\frac{\\left[N_{G,v}-1\\right]\\left[N_{G-v}+1\\right]}{2}\\\\\n&=\\frac{N_{G,v}N_{G-v}+N_{G,v}-N_{G-v}-1}{2}\\\\\n&\\leq \\frac{N_{G,v}N_{G-v}}{2},\n\\end{align*}\nas $N_{G, v}-N_{G-v}-1\\leq 0$ by Lemma \\ref{NumCom} (which applies since $v$ is not a cut vertex of $G$).\n\\end{proof}\n\nFor the next lemma, we introduce some new notation.  For a connected block graph $G$ of order at least $2$ with vertex $v$, define\n\\[\n\\mu_{G,v}=\\frac{W_{G,v}-N_{G,v}}{N_{G,v}-1}.\n\\]\nNote that $\\mu_{G,v}$ is the average order of the connected induced subgraphs of $G-v$ containing at least one neighbour of $v$.  That is, $\\mu_{G,v}=M^*_{G-v,N_G(v)}$.  In the next lemma, we show that if $v$ is not a cut vertex, then $\\mu_{G,v}$ is at least as large as $M_{G-v}$.  We obtain our extension of the local-global mean inequality as a straightforward consequence of this result.\n\n\\begin{lemma}\\label{MuLem}\nLet $G$ be a connected block graph of order $n\\geq 2$ with non-cut vertex $v$.\nThen\n\\[\n\\mu_{G,v}\\geq M_{G-v},\n\\]\nwith equality if and only if $G\\cong K_n$.\n\\end{lemma}\n\n\\begin{proof}\nWe proceed by induction on the number of blocks $k$ of $G$.  If $k=1$, then $G\\cong K_n$, and we verify that\n\\[\n\\mu_{G,v}=M_{G-v}.\n\\]\nNow let $k>1$, and suppose that the statement holds for all connected block graphs of order at least $2$, with less than $k$ blocks.  Since $v$ is not a cut vertex, it is contained in only one block $B$ of $G$, and $N_G(v)=V(B)-v$.  Since we can partition the connected induced subgraphs of $G-v$ into all those that contain at least one neighbour of $v$ and all those that do not, we can write $M_{G-v}$ as a convex combination of $\\mu_{G,v}$ and $M_{G-B}$.  So it suffices to show that $\\mu_{G,v}\\geq M_{G-B}.$\n\t\nLet $U=V(B)-v=\\{v_1,v_2,\\dots, v_k\\}$.  Note that $U$ is either a singleton, or induces a block in $G-v$.  For each $i\\in\\{1,\\dots,k\\}$, let $G_i$ be the connected component of $(G-v)-E(B)$ containing $v_i$.  By Lemma~\\ref{LocalBlockLemma},\n\\[\n\\mu_{G,v}=M^*_{G-v,U}=\\frac{N^*_{G-v,U}+1}{N^*_{G-v,U}}\\sum_{i=1}^k \\frac{W_{G_i,v_i}}{N_{G_i,v_i}+1}>\\sum_{i=1}^k \\frac{W_{G_i,v_i}}{N_{G_i,v_i}+1}.\n\\]\n\t\nNow $G-B$ may be a disconnected graph, so $M_{G-B}$ is at most $M_H$, where $H$ is a connected component of $G-B$ of largest mean CIS order.  In particular, $H$ is a subgraph of $G_i-v_i$ for some $i\\in\\{1,\\dots,k\\}$.  Without loss of generality, suppose $H$ is a subgraph of $G_1-v_1$.  Let $H_1=G[V(H)\\cup\\{v_1\\}]$.  By Lemma~\\ref{LocalSum}, we have\n\\[\nM_{G_1,v_1}\\geq M_{H_1,v_1},\n\\]\nand together with $W_{G_1,v_1}\\geq W_{H_1,v_1},$ which clearly holds since every connected induced subgraph of $H_1$ containing $v_1$ is a connected induced subgraph of $G_1$ containing $v_1$, this implies\n\\[\n\\frac{W_{G_1,v_1}}{1+N_{G_1,v_1}}\\geq \\frac{W_{H_1,v_1}}{1+N_{H_1,v_1}}.\n\\]\nAltogether, we have\n\\[\n\\mu_{G,v}>\\frac{W_{G_1,v_1}}{1+N_{G_1,v_1}}\\geq \\frac{W_{H_1,v_1}}{1+N_{H_1,v_1}}=\\frac{\\mu_{H_1,v_1}(N_{H_1,v_1}-1)+N_{H_1,v_1}}{1+N_{H_1,v_1}}.\n\\]\nNote that $H_1$ has order at least $2$, and fewer than $k$ blocks, and that $v_1$ is a non-cut vertex of $H_1$.  Hence, by the induction hypothesis, applied to $H_1$ at $v_1$,\n\\[\n\\mu_{G,v}> \\frac{M_{H_1-v_1}(N_{H_1,v_1}-1)+N_{H_1,v_1}}{1+N_{H_1,v_1}}.\n\\]\nFinally,\n\\[\n\\frac{M_{H_1-v_1}(N_{H_1,v_1}-1)+N_{H_1,v_1}}{1+N_{H_1,v_1}}\\geq M_{H_1-v_1}\n\\]\nis equivalent to\n\\[\nW_{H_1-v_1}\\leq \\frac{N_{H_1,v_1}N_{H_1-v_1}}{2},\n\\]\nwhich holds, by Lemma \\ref{WeightBound}.  So we have shown that\n\\[\n\\mu_{G,v}> M_{H_1-v_1}=M_H\\geq M_{G-B},\n\\]\nand it follows, from our earlier observation, that $\\mu_{G,v}> M_{G-v}.$\n\\end{proof}\n\n\\begin{theorem}[The Local-Global Mean Inequality]\\label{LocalGlobal}\nLet $G$ be a connected block graph.\n\\begin{enumerate}\n\\item \\label{CorA} If $v$ is a vertex of $G$, then $M_{G}\\leq M_{G,v}$.\n\\item \\label{CorB} If $B$ is a block of $G$, then $M_{G}\\leq M^*_{G,B}$.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\nFor \\ref{CorA}, let $G'$ be the graph obtained from $G$ by adding a new leaf vertex $u$ to $v$.  Note that $\\mu_{G',u}=M_{G,v}.$  Thus, by Lemma~\\ref{MuLem},\n\\[\nM_{G,v}=\\mu_{G',u}\\geq M_{G'-u}=M_{G}.\n\\]\n\nFor \\ref{CorB}, let $G'$ be the graph obtained from $G$ by adding a new vertex $u$ and joining it to all vertices of $B$.  Note that $\\mu_{G',u}=M^*_{G,B}.$  Thus by Lemma~\\ref{MuLem},\n\\[\nM^*_{G,B}=\\mu_{G',u}\\geq M_{G'-u}=M_G.  \\qedhere\n\\]\n\\end{proof}\n\n\\section{The Vertex Gluing Lemma, the Edge Gluing Lemma, and the Stretching Lemma} \\label{KeyLemmas}\n\nIn this section, we prove the Vertex Gluing Lemma, the Edge Gluing Lemma, and the Stretching Lemma.  The Vertex Gluing Lemma extends the Gluing Lemma of~\\cite{MolOellermann2017} from trees to connected block graphs, and the proof is very similar once the local-global mean inequality is established for connected block graphs.  The proof is included in Appendix~\\ref{Appendix} for completeness.\n\n\\begin{lemma}[The Vertex Gluing Lemma] \\label{vertex_gluing}\nLet $H$ be a connected block graph of order at least $2$ having vertex $v$.  Fix a natural number $n\\geq 3$.  Let $P:u_1\\dots u_n$ be a path of order $n$. For $s \\in \\{1, \\dots, n\\}$, let $G_s$ be the block graph obtained from the disjoint union of $P_n$ and $H$ by gluing $v$ to $u_s$.  If $1\\leq i<j\\leq \\tfrac{n+1}{2}$, then $M_{G_i}<M_{G_j}.$  \\hfill \\qed\n\\end{lemma}\n\nThe proof of the Edge Gluing Lemma uses a similar technique. We use the fact that if $P_n$ is a path of order $n$, then $N_{P_n} = \\binom{n+1 }{2}$ and $W_{P_n} = \\binom{n+2}{3}$ (see  \\cite{Jamison1983}).\n\n\\begin{lemma}[The Edge Gluing Lemma] \\label{edge_gluing}\nLet $H$ be a connected block graph of order at least $3$ with adjacent non-cut vertices $u$ and $v$.  Fix a natural number $n\\geq 4$.  For $s\\in\\{1,2,\\dots,n-1\\}$, let $G_s$ be the graph obtained from $H\\cup P_s\\cup P_{n-s}$ by gluing a leaf of $P_s$ to $u$ and a leaf of $P_{n-s}$ to $v$.  If $1\\leq i<j\\leq \\tfrac{n}{2}$, then $M_{G_i}<M_{G_j}.$\n\\end{lemma}\n\n\\begin{proof}\nFor ease of notation, let $u$ and $v$ also denote the glued vertices in $P_s$ and $P_{n-s}$, respectively.  Let $w$ be the vertex adjacent to $u$ in $P_s$ and let $z$ be the vertex adjacent to $v$ in $P_{n-s}$.  We have\n\\begin{align*}\n\\Phi_{G_s}(x)&=\\Phi_{G_s,\\{u,v\\}}(x)+\\Phi_{G_s-v,u}(x)+\\Phi_{G_s-u,v}(x)+\\Phi_{G_s-\\{u,v\\}}(x)\\\\\n&=\\Phi_{H,\\{u,v\\}}(x)\\left(1+\\Phi_{P_{s-1},w}(x)\\right)\\left(1+\\Phi_{P_{n-s-1},z}(x)\\right)+\\Phi_{H-v,u}(x)\\left(1+\\Phi_{P_{s-1},w}(x)\\right)\\\\\n& \\ \\ \\ \\ +\\Phi_{H-u,v}(x)\\left(1+\\Phi_{P_{n-s-1},z}(x)\\right)+\\Phi_{H-\\{u,v\\}}(x)+\\Phi_{P_{s-1}}(x)+\\Phi_{P_{n-s-1}}(x)\\\\\n&=x\\Phi_{H-v,u}(x)\\left(1+\\Phi_{P_{s-1},w}(x)\\right)\\left(1+\\Phi_{P_{n-s-1},z}(x)\\right)+\\Phi_{H-v,u}(x)\\left(1+\\Phi_{P_{s-1},w}(x)\\right)\\\\\n& \\ \\ \\ \\ +\\Phi_{H-v,u}(x)\\left(1+\\Phi_{P_{n-s-1},z}(x)\\right)+\\Phi_{H-\\{u,v\\}}(x)+\\Phi_{P_{s-1}}(x)+\\Phi_{P_{n-s-1}}(x)\\\\\n&=\\Phi_{H-v,u}(x)\\left[x\\left(1+\\Phi_{P_{s-1},w}(x)\\right)\\left(1+\\Phi_{P_{n-s-1},z}(x)\\right)+\\left(1+\\Phi_{P_{s-1},w}(x)\\right)+\\left(1+\\Phi_{P_{n-s-1},z}(x)\\right)\\right]\\\\\n& \\ \\ \\ \\ +\\Phi_{H-\\{u,v\\}}(x)+\\Phi_{P_{s-1}}(x)+\\Phi_{P_{n-s-1}}(x).\n\\end{align*}\nDifferentiating with respect to $x$, we obtain\n\\begin{align*}\n\\Phi'_{G_s}(x)&=\\Phi'_{H-v,u}(x)\\left[x\\left(1+\\Phi_{P_{s-1},w}(x)\\right)\\left(1+\\Phi_{P_{n-s-1},z}(x)\\right)+\\left(1+\\Phi_{P_{s-1},w}(x)\\right)+\\left(1+\\Phi_{P_{n-s-1},z}(x)\\right)\\right]\\\\\n&\\ \\ \\ \\ +\\Phi_{H-v,u}(x)\\bigg[\\left(1+\\Phi_{P_{s-1},w}(x)\\right)\\left(1+\\Phi_{P_{n-s-1},z}(x)\\right)+x\\Phi'_{P_{s-1},w}(x)\\left(1+\\Phi_{P_{n-s-1},z}(x)\\right)\\\\\n& \\hspace{3cm} +x\\left(1+\\Phi_{P_{s-1},w}(x)\\right)\\Phi'_{P_{n-s-1},z}(x)+\\Phi'_{P_{s-1},w}(x)+\\Phi'_{P_{n-s-1},z}(x)\\bigg]\\\\\n& \\ \\ \\ \\ +\\Phi'_{H-\\{u,v\\}}(x)+\\Phi'_{P_{s-1}}(x)+\\Phi'_{P_{n-s-1}}(x).\n\\end{align*}\nEvaluating at $x=1$ and letting $F=H-v$,\n\\begin{align*}\n\\Phi_{G_s}(1)=N_{F, u}\\left(s(n-s)+n\\right)+N_{F-u}+\\tbinom{s}{2}+\\tbinom{n-s}{2},\n\\end{align*}\nand\n\\begin{align*}\n\\Phi'_{G_s}(1)&=W_{F, u}\\left(s(n-s)+n\\right)+N_{F, u}\\left(s(n-s)+\\tbinom{s}{2}(n-s+1)+(s+1)\\tbinom{n-s}{2}\\right)\\\\\n& \\ \\ \\ \\ +W_{F-u}+\\tbinom{s+1}{3}+\\tbinom{n-s+1}{3}.\n\\end{align*}\nBy a straightforward computation,\n\\begin{align}\\label{NGs}\n\\tfrac{d }{ds}N_{G_s}=\\tfrac{d}{ds}\\Phi_{G_s}(1)=N_{F, u}(n-2s)-(n-2s)=(n-2s)(N_{F, u}-1),\n\\end{align}\nand\n\\begin{align}\\label{WGs}\n\\tfrac{d}{ds}W_{G_s}=\\tfrac{d}{ds}\\Phi'_{G_s}(1)&=W_{F, u}(n-2s)+N_{F, u}\\tfrac{n-2}{2}(n-2s)-\\tfrac{n}{2}(n-2s)\\nonumber \\\\\n&=(n-2s)\\left[W_{F, u}+\\tfrac{n-2}{2}N_{F, u}-\\tfrac{n}{2}\\right]\\nonumber \\\\\n&=(n-2s)\\left[W_{F, u}+\\tfrac{n}{2}(N_{F, u}-1)-N_{F, u}\\right].\n\\end{align}\nNow we consider the mean $M_{G_s}=\\tfrac{W_{G_s}}{N_{G_s}}$ as a continuous function of $s$. By the quotient rule, $\\tfrac{d}{ds}M_{G_s}$ will have the same sign as $\\tfrac{d}{ds}\\left[W_{G_s}\\right]N_{G_s}-W_{G_s}\\tfrac{d}{ds}\\left[N_{G_s}\\right]$ on $\\left[1, \\frac{n}{2}\\right)$, since the denominator of $\\frac{d}{ds}M_{G_s}$ is strictly positive on $\\left[1, \\frac{n}{2}\\right)$.  Since both $\\tfrac{d}{ds}N_{G_s}$ and $\\tfrac{d}{ds}W_{G_s}$ have a factor of $n-2s$, the expression\n\\begin{align}\\label{SameSignAsDeriv}\n\\frac{\\tfrac{d}{ds}\\left[W_{G_s}\\right]N_{G_s}-W_{G_s}\\tfrac{d}{ds}\\left[N_{G_s}\\right]}{n-2s},\n\\end{align}\nwill have the same sign as $\\frac{d}{ds}M_{G_s}$ for $s\\in[1,n\/2)$.  We show that (\\ref{SameSignAsDeriv}) is strictly positive for $s\\in[1,n\/2)$.  Substituting (\\ref{NGs}) and (\\ref{WGs}) into (\\ref{SameSignAsDeriv}), and then simplifying, we obtain\n\\begin{align}\n&\\left[W_{F, u}+\\tfrac{n}{2}(N_{F, u}-1)-N_{F, u}\\right]\\left[N_{F, u}\\left(s(n-s)+n\\right)+N_{F-u}+\\tbinom{s}{2}+\\tbinom{n-s}{2}\\right]\\nonumber\\\\\n& \\ \\ \\  -(N_{F, u}-1)\\bigg[W_{F, u}\\left(s(n-s)+n\\right)+N_{F, u}\\left(s(n-s)+\\tbinom{s}{2}(n-s+1)+(s+1)\\tbinom{n-s}{2}\\right)\\nonumber\\\\\n& \\hspace{3cm} +W_{F-u}+\\tbinom{s+1}{3}+\\tbinom{n-s+1}{3}\\bigg]\\nonumber\\\\\n&=W_{F,u}\\tbinom{n+1}{2}-N_{F, u}\\tfrac{n^2}{2}-\\tfrac{n}{2}N_{F-u}+\\tfrac{n-2}{2}N_{F, u}N_{F-u}-N_{F, u}^2\\tfrac{n}{2}\\nonumber\\\\\n& \\ \\ \\ \\ +W_{F, u}N_{F-u}-W_{F-u}N_{F, u}+W_{F-u}+(N_{F, u}-1)\\tfrac{n(n-1)(n-2)}{12}\\nonumber \\\\\n\\begin{split}\\label{eqwe}\n&=\\tfrac{n^2}{2}(W_{F, u}-N_{F, u})+\\tfrac{n}{2}(W_{F, u}-N_{F,u}-N_{F-u})+\\tfrac{n}{2}N_{F,u}(N_{F-u}+1-N_{F,u})\\\\\n& \\ \\ \\ \\ +\\left(W_{F, u}N_{F-u}-W_{F-u}N_{F, u}-N_{F, u}N_{F-u}+W_{F-u}\\right)+(N_{F, u}-1)\\tfrac{n(n-1)(n-2)}{12}.\n\\end{split}\n\\end{align}\nWe now explain why each term of (\\ref{eqwe}) is nonnegative (and in fact two terms are positive).  The strict inequality $W_{F,u}-N_{F,u}>0$ is obvious. The inequality $W_{F,u}-N_{F,u}-N_{F-u}=W_{F,u}-N_F\\geq 0$ follows by Lemma~\\ref{WeiNum}, and the inequality $N_{F-u}+1-N_{F,u}\\geq 0$ follows by Lemma~\\ref{NumCom}.  The inequality $(N_{F, u}-1)\\frac{n(n-1)(n-2)}{12}>0$ is immediate since $N_{F,u}\\geq 2$ and $n\\geq 4$.\nFinally,\n\\begin{align}\\label{MuIneq}\nW_{F, u}N_{F-u}-N_{F, u}N_{F-u}-N_{F, u}W_{F-u}+W_{F-u}\\geq 0 \\ \\ \\ &\\Longleftrightarrow \\ \\ \\   \\frac{W_{F,u}-N_{F,u}}{N_{F,u}-1}\\geq \\frac{W_{F-u}}{N_{F-u}}\\nonumber \\\\\n&\\Longleftrightarrow \\ \\ \\ \\mu_{F,u}\\geq M_{F-u},\n\\end{align}\nand (\\ref{MuIneq}) holds by Lemma \\ref{MuLem}.  Therefore, the function $\\frac{d}{ds}M_{G_s}$ is strictly positive on the interval $\\left[1, \\frac{n}{2}\\right)$, and we conclude that $M_{G_1}<M_{G_2}<... <M_{G_{\\lfloor n\/2\\rfloor}}$.\n\\end{proof}\n\t\nFinally, we prove the Stretching Lemma.  We recall some notation used in the statement.  Fix an integer $n\\geq 3$.  For $s\\in\\{1,2,\\dots,n-1\\}$, let $F_{s,n-s}$ (or $F_s$ for short) denote the graph obtained from the disjoint union of $K_{s}$ and $P_{n-s}$ by joining a leaf of $P_{n-s}$ to every vertex of $K_s$.  Note that if $s=1$, then $F_s\\cong P_n$, while if $s=n-1$, then $F_s\\cong K_n$.\n\nLet $v$ be a vertex of $F_s$ belonging to the initial $K_s$.  By straightforward counting arguments,\n\\[\n\\Phi_{F_s,v}(x)=x(1+x)^{s-1}\\sum_{i=0}^{n-s}x^i\n\\]\nand\n\\begin{align*}\n\\Phi_{F_s-v}(x)\n&=(1+x)^{s-1}\\left(\\sum_{i=0}^{n-s}x^i\\right)+\\Phi_{P_{n-s-1}}(x)-1.\n\\end{align*}\n\n\\begin{lemma}[The Stretching Lemma] \\label{stretching_lemma}\nLet $H$ be a connected block graph of order at least $2$ having a vertex $u$.  Fix a natural number $n\\geq 3$.  Let $v$ be a vertex of $F_s=F_{s,n-s}$ belonging to the initial $K_s$.  For $s \\in \\{1, \\dots, n-1\\}$, let $G_s$ be the block graph obtained from the disjoint union of $H$ and $F_{s}$ by identifying $u$ and $v$.  If $1\\leq i<j\\leq n-1$, then $M_{G_i}<M_{G_j}.$\n\\end{lemma}\n\n\\begin{proof}\nThe general technique is to write $M_{G_s}$ as a function of $s$, treat $s$ as a real variable, and show that $M_{G_s}$ is increasing for $s\\in[1,n-1]$.  In other words, we show that $\\tfrac{d}{ds}\\left[M_{G_s}\\right]=\\tfrac{d}{ds}\\left[W_{G_s}\/N_{G_s}\\right]>0$ for all $s\\in [1,n-1]$.  By the quotient rule, $\\tfrac{d}{ds}\\left[M_{G_s}\\right]$ has the same sign as\n\\begin{align}\\label{derivative}\n\\tfrac{d}{ds}\\left[W_{G_s}\\right]N_{G_s}-W_{G_s}\\tfrac{d}{ds}\\left[N_{G_s}\\right],\n\\end{align}\nso it suffices to show that this expression is positive for all $s\\in[1,n-1]$.\n\nLet $s\\in[1,n-1]$.  We first derive expressions for $N_{G_s}$, $W_{G_s}$, $\\tfrac{d}{ds}[N_{G_s}]$, and $\\tfrac{d}{ds}[W_{G_s}]$.  We have\n\\begin{align*}\n\\Phi_{G_s}(x)&=\\frac{\\Phi_{F_s,v}(x)\\Phi_{H,u}(x)}{x}+\\Phi_{F_s-v}(x)+\\Phi_{H-u}(x)\\\\\n&=(1+x)^{s-1}\\left(\\sum_{i=0}^{n-s}x^i\\right)\\left[1+\\Phi_{H,u}(x)\\right]+\\Phi_{H-u}(x)+\\Phi_{P_{n-s-1}}(x)-1.\n\\end{align*}\nSubstituting $x=1$ yields\n\\begin{align}\\label{NG}\nN_{G_s}=2^{s-1}(n-s+1)(N_{H,u}+1)+N_{H-u}+\\tbinom{n-s}{2}-1.\n\\end{align}\nWe also find\n\\begin{align*}\n\\Phi'_{G_s}(x)&=\\left[(s-1)(1+x)^{s-2}\\left(\\sum_{i=0}^{n-s}x^i\\right)+(1+x)^{s-1}\\left(\\sum_{i=0}^{n-s}ix^{i-1}\\right)\\right]\\left[1+\\Phi_{H,u}(x)\\right]\\\\\n& \\ \\ \\ +(1+x)^{s-1}\\left(\\sum_{i=0}^{n-s}x^i\\right)\\left[\\Phi'_{H,u}(x)\\right]+\\Phi'_{H-u}(x)+\\Phi'_{P_{n-s-1}}(x).\n\\end{align*}\nSubstituting $x=1$ and simplifying yields\n\\begin{align}\nW_{G_s}&=\\left[(s-1)2^{s-2}(n-s+1)+2^{s-1}\\tbinom{n-s+1}{2}\\right]\\left[N_{H,u}+1\\right]\\nonumber\\\\\n& \\ \\ \\ +2^{s-1}(n-s+1)W_{H,u}+W_{H-u}+\\tbinom{n-s+1}{3}\\nonumber \\\\\n&=2^{s-2}(n-s+1)\\left[(n-1)\\left(N_{H,u}+1\\right)+2W_{H,u}\\right]+W_{H-u}+\\tbinom{n-s+1}{3}.\\label{WG}\n\\end{align}\nDifferentiating (\\ref{NG}) and (\\ref{WG}) with respect to $s$, and letting $L=\\ln(2)$ for ease of reading, we find\n\\begin{align}\n\\tfrac{d}{ds}\\left[N_{G_s}\\right]&=2^{s-1}L\\left(n-s+1-\\tfrac{1}{L}\\right)(N_{H,u}+1)-\\tfrac{2(n-s)-1}{2}, \\mbox{ and }\\label{NGprime}\\\\\n\\tfrac{d}{ds}\\left[W_{G_s}\\right]&=2^{s-2}L\\left(n-s+1-\\tfrac{1}{L}\\right)\\left[\\left(n-1\\right)\\left(N_{H,u}+1\\right)+2W_{H,u}\\right]-\\tfrac{3(n-s)^2-1}{6}\\label{WGprime}\n\\end{align}\n\nFor convenience, we let $t=n-s$ and rewrite (\\ref{NG}), (\\ref{WG}), (\\ref{NGprime}), and (\\ref{WGprime}) below.  Since $s\\in[1,n-1]$, we have $t\\in[1,n-1]$ as well.\n\\begin{align*}\nN_{G_s}&=2^{s-1}\\left(t+1\\right)(N_{H,u}+1)+N_{H-u}+ \\tbinom{t}{2}-1,\\\\\nW_{G_s}&=2^{s-2}(t+1)\\left[(n-1)\\left(N_{H,u}+1\\right)+2W_{H,u}\\right]+W_{H-u}+\\tbinom{t+1}{3},\\\\\n\\tfrac{d}{ds}[N_{G_s}]&=2^{s-1}L\\left(t+1-\\tfrac{1}{L}\\right)(N_{H,u}+1)-\\tfrac{2t-1}{2}, \\mbox{ and }\\\\\n\\tfrac{d}{ds}[W_{G_s}]&=2^{s-2}L\\left(t+1-\\tfrac{1}{L}\\right)\\left[\\left(n-1\\right)\\left(N_{H,u}+1\\right)+2W_{H,u}\\right]-\\tfrac{3t^2-1}{6}.\n\\end{align*}\nBy substituting these expressions into (\\ref{derivative}), expanding, and regrouping (and confirming with a computer algebra system), we find\n\\begin{align*}\n\\tfrac{d}{ds}\\left[W_{G_s}\\right]N_{G_s}-W_{G_s}\\tfrac{d}{ds}\\left[N_{G_s}\\right]\n&=\\sum_{i=1}^5 E_i,\n\\end{align*}\nwhere\n\\begin{align*}\nE_1&=2^{s-2}\\left(Lt+L-1\\right)\\left[(N_{H,u}+1)N_{H-u}-2W_{H-u}\\right],\\\\\nE_2&=2^{s-1}\\left(Lt+L-1\\right)\\left[W_{H,u}N_{H-u}-N_{H,u}W_{H-u}\\right],\\\\\nE_3&=2^{s-3}(t+1)^2\\left[(N_{H,u}+1)\\left((n-1)(Lt-2L+1)-\\tfrac{2}{3}(Lt^2+(2-L)t-1)\\right)+2W_{H,u}(Lt-2L+1)\\right],\\\\\nE_4&=-\\tfrac{1}{12}(t+1)^2(t^2-4t+2), \\mbox{ and }\\\\\nE_5&=2^{s-2}\\left(Lt+L-1\\right)(n-2)(N_{H,u}+1)N_{H-u}+\\tfrac{2t-1}{2}W_{H-u}-\\tfrac{3t^2-1}{6}N_{H-u}.\n\\end{align*}\n\nWe now show that $\\sum_{i=1}^5 E_i>0$.  We first demonstrate that $E_1>0$ and $E_2\\geq 0$.  Note that since $t\\geq 1$, the factor $Lt+L-1\\geq 2L-1>0$.  By Lemma~\\ref{WeightBound}, $N_{H,u}N_{H-u}\\geq 2W_{H-u}$.  Therefore,\n\\[\nE_1\\geq 2^{s-2}(Lt+L-1)N_{H-u}>0.\n\\]\nBy Theorem~\\ref{LocalGlobal}\\ref{CorA}, $M_{H,u}\\geq M_{H-u}$, or equivalently $W_{H,u}N_{H-u}\\geq W_{H-u}N_{H,u}$.  It follows immediately that $E_2\\geq 0$.\n\nNext we show that $E_3+E_4> 0$.  We begin by bounding $E_3$.  Since $H$ has order at least $2$, we have $W_{H,u}\\geq N_{H,u}+1$.  This gives\n\\[\nE_3\\geq 2^{s-3}(t+1)^2(N_{H,u}+1)\\left[(n+1)(Lt-2L+1)-\\tfrac{2}{3}(Lt^2+(2-L)t-1)\\right]\n\\]\nNow we use the fact that $t\\leq n-1$, or equivalently $n+1\\geq t+2$.  This gives\n\\begin{align*}\nE_3&\\geq 2^{s-3}(t+1)^2(N_{H,u}+1)\\left[(t+2)(Lt-2L+1)-\\tfrac{2}{3}(Lt^2+(2-L)t-1)\\right]\\\\\n&=2^{s-3}(t+1)^2(N_{H,u}+1)\\tfrac{1}{3}\\left[Lt^2+(2L-1)t+8-12L\\right]\n\\end{align*}\nFinally, since $H$ has order at least $2$, we have $N_{H,u}\\geq 2$.  Applying this inequality along with $s\\geq 1$ gives\n\\[\nE_3\\geq \\tfrac{1}{4}(t+1)^2\\left[Lt^2+(2L-1)t+8-12L\\right].\n\\]\nTherefore,\n\\begin{align*}\nE_3+E_4&\\geq \\tfrac{1}{4}(t+1)^2\\left[Lt^2+(2L-1)t+8-12L\\right]-\\tfrac{(t+1)^2(t^2-4t+2)}{12}\\\\\n&=\\tfrac{1}{12}(t+1)^2\\left[(3L-1)t^2+(6L+1)t+22-36L\\right].\n\\end{align*}\nRecalling that $L=\\ln(2)$, one can verify that the quadratic in the square brackets of this last expression is positive for all $t\\geq 1$.  Thus, we conclude that $E_3+E_4>0$.\n\nFinally, we show that $E_5>0$.  We use the inequalities $W_{H-u}\\geq N_{H-u}$, $N_{H,u}\\geq 2$, $s\\geq 1$, and $n-2\\geq t-1$.\n\\begin{align*}\nE_5&\\geq \\tfrac{3}{2}\\left(Lt+L-1\\right)(t-1)N_{H-u}+\\tfrac{2t-1}{2}N_{H-u}-\\tfrac{3t^2-1}{6}N_{H-u}\\\\\n&=\\tfrac{1}{6}\\left[(9L-3)t^2-(9L+3)t+7\\right]\n\\end{align*}\nRecalling that $L=\\ln(2)$, it is straightforward to verify that this last expression is strictly positive for all $t$, and hence $E_5>0$.\n\nWe have shown that $E_1>0,$ $E_2\\geq 0$, $E_3+E_4>0$, and $E_5>0$ for all $s\\in[1,n-1]$.  It follows that $\\tfrac{d}{ds}\\left[W_{G_s}\\right]N_{G_s}-W_{G_s}\\tfrac{d}{ds}\\left[N_{G_s}\\right]=\\sum_{i=1}^5 E_i>0$ for all $s\\in[1,n-1]$, as desired.\n\\end{proof}\t\n\t\n\t\n\n\\section{Concluding Remarks}\n\nIn this article, we demonstrated that among all connected block graphs of order $n$, the path has smallest mean CIS order.  This extends Jamison's result: {\\em Among all trees of order $n$, the path has smallest mean subtree order}. Moreover, our main result lends support to the conjecture made in \\cite{KroekerMolOellermann2018}:  {\\em Among all connected graphs of order $n$, the path has minimum mean CIS order}.\n\n\\medskip\n\nThe problem of determining the structure of those block graphs, of a given order, with maximum mean CIS order remains open. It was conjectured by Jamison \\cite{Jamison1983} that a tree with maximum mean subtree order among all trees of order $n$, called an {\\em optimal} tree of order $n$, is a caterpillar. This is known as Jamison's Caterpillar Conjecture. This conjecture has been verified for all $n\\leq 24$ (see~\\cite{Jamison1983, MolOellermann2017}).  Mol and Oellermann~\\cite{MolOellermann2017} made some progress on describing the structure of optimal trees. They proved that in any optimal tree of order $n$, every leaf is adjacent with a vertex of degree at least $3$, and that the number of leaves in an optimal tree of order $n$ is $\\mathrm{O}(\\log_2n)$ (moreover, the number of leaves is $\\Theta(\\log_2n)$ if Jamison's Caterpillar Conjecture is true).\n\nTurning to block graphs, for $n\\in\\{3,4\\}$, the complete graph has maximum mean CIS order among all block graphs of order $n$. We have verified that for $5\\leq n\\leq 11$, the block graph of order $n$ with maximum mean CIS order is a tree (more specifically, a caterpillar). We make the following conjecture, which strengthens Jamison's Caterpillar Conjecture.\n\n\\begin{conjecture}\nFor $n\\geq 5$, if $G$ has maximum mean CIS order among all block graphs of order $n$, then $G$ is a caterpillar.\n\\end{conjecture}\n\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n\\providecommand{\\MR}{\\relax\\ifhmode\\unskip\\space\\fi MR }\n\\providecommand{\\MRhref}[2]{%\n  \\href{http:\/\/www.ams.org\/mathscinet-getitem?mr=#1}{#2}\n}\n\\providecommand{\\href}[2]{#2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe ongoing transition to 'green' energy urges us to develop advanced electrochemical devices for energy conversion and storage and, consequently, increases the need for a deeper theoretical understanding of such systems. The electrode\/solution interface, which is an essential element of electrochemical devices, is especially interesting but presents a challenge for researchers. At the charged electrode\/electrolyte interface the concentration of ions may be many times higher than the bulk solution value. That, of course, strongly affects the rate of the electrochemical reactions involving ionic species. Thus, tuning the electric double layer (EDL) composition may be of use in a number of electrochemical devices. For instance, in alkali-metal batteries (including Li-ion ones) the so-called solid-electrolyte interphase (SEI) layer is formed at the surface of both the anode and the cathode active materials. SEI consists of insoluble and partially soluble reduction products of electrolyte components and has a great impact on battery performance \\cite{peled2017sei}. Particularly, the SEI structure affects the electron transfer kinetics and the battery cycle life. Controlling the EDL composition would be an extra tool for managing the SEI formation process. In an alkali-salt electrolyte the interface structure can be altered by introducing organic cations that would behave differently inside the EDL due to the factors arising from its complex molecular structure, e.g. steric limitations, dipole moment of the functional groups and specific interaction with the surface such as $\\pi-\\pi$ stacking. The remarkable modifiability of the organic cation implies that a compound can be designed to efficiently alter the EDL structure. Furthermore, the mixed electrolyte composition including both simple (alkali) and organic cations is already becoming more common as researchers and engineers are exploring the benefits of ionic liquids when applied to batteries, fuel cells, supercapacitors, solar cells, {\\sl etc}.  \\cite{fedorov2014ionic}. Thus, it is also becoming more relevant to make a theoretical description of the EDL incorporating both simple and organic cations.     \n\n\nAnother bright example confirming the importance of the EDL structure is $Li$-air ($Li-O_2$) battery which is a promising electrochemical storage technology that hypothetically can provide several times higher specific energy than that of conventional $Li$-ion batteries \\cite{aurbach2016advances}. Unfortunately, the cell capacity values achieved in the experiments so far are way below the theoretical expectations mostly due to the electrode surface passivation by the discharge product itself, i.e. $Li_2O_2$. There are two possible pathways of $Li_2O_2$ formation during the discharge process \\cite{johnson2014role}. Relatively large micron-size $Li_2O_2$ particles which do not consume much of the electrode surface can be formed as a result of the chemical disproportionation reaction of $LiO_2$ intermediate in the bulk electrolyte \\cite{zhai2013disproportionation, mitchell2013mechanisms}. Alternatively, it is possible to grow a thin $Li_2 O_2$ film through $Li O_2$ electrochemical reduction \\cite{wen2013situ, johnson2014role} at the surface with the consumption of extra $Li^{+}$ ions. The predominance of the second surface-mediated pathway leads to complete electrode surface passivation (and cell \"death\") while most of the available space inside the porous cathode is still unfilled with the discharge product. That drastically decreases the cell specific capacity. Exclusion of $Li^{+}$ ions from the EDL could inhibit the undesired surface-mediated pathway and slow down the passivation process thus enabling the accumulation of a larger amount of the discharge product inside the porous electrode. Introduction of organic cations into the electrolyte solution would be an easy practical way of controlling the EDL structure.\n\nFrom the physics point of view, such organic cations should be attracted by the cathode more strongly than $Li^{+}$ ions. \nThus, the greater attractive force acting on each of the organic cations must cause the lithium cations to be expelled from the cathode surface at its sufficiently large surface charge density. One of the possible ways to make the attraction of the organic cations stronger is using of an organic salt with highly polar cations. It can be, for instance, some room temperature ionic liquid, whose organic cations usually possess a rather high dipole moment or electronic polarizability. As is well known, in an inhomogeneous electric field a polar particle suffers a dielectrophoretic force (see, for instance, \\cite{jones1979dielectrophoretic}) directed to the highest electric field. In our case, the electric field inhomogeneity takes place due to the electrostatic screening of the electrode charge by the mobile electrolyte ions, so that the maximal electric field is reached on the electrode surface. Therefore, the occurrence of a dipole moment on the organic cation should lead to its higher attraction to the electrode with respect to $Li^{+}$. The latter will provoke the expelling of lithium cations from the electrode surface  due to the excluded volume interactions and their complete replacement with the electrochemically inactive organic cations at a sufficiently large electrode surface charge density (see Fig. \\ref{Schematic}). \n\n\\begin{figure}[h!]\n\\center{\\includegraphics[width=0.8\\linewidth]{FIG_0.PNG}}\n\\caption{Schematic representation of the electrode\/solution interface in mixed electrolyte containing simple (red circle) and molecular (large red circle with a small blue one) cations: bulk solution \u2013 dipole moments of molecular cations are disordered, equimolar cations concentration; EDL \u2013 dipoles are highly ordered according to electric field, molecular cations dominate.}\n\\label{Schematic}\n\\end{figure}\n\nIn order to understand whether it is possible to replace (at least partially) the lithium cations with organic dipolar cations on the cathode surface at physically reasonable parameters of the system, such as surface charge density, dielectric permittivity of the solvent, dipole moment of the cation, and ionic concentrations in the bulk, it would be useful to have a simple analytical model of a two-component electrolyte solution with one type of anions and two types of cations on the negatively charged metal electrode. Within such a theory, one of the cations would be a simple structureless particle with a charge $+e$, whereas the other one -- a particle possessing not only a charge $+e$, but also a certain permanent dipole moment $p$. A self-consistent field (SCF) theory, based on the modified Poisson-Boltzmann equation for the electrostatic potential and taking into account the steric interactions of ions and polarity of one type of cations could be such a theory. Despite the fact that there are a lot of SCF theories of electrolyte solutions taking into account polarizable\/polar additives \\cite{coalson1996statistical,abrashkin2007dipolar,gongadze2011langevin,frydel2011polarizable,budkov2018nonlocal,budkov2019statistical,Sin2017,Misra2013,Hatlo2012,Ben-Yaakov2011,Wei1993,Das2012,Jiang2014,gongadze2011generalized,budkov2016theory,budkov2018theory,budkov2015modified,nakayama2015differential,lopez2018diffuse,lopez2018multiionic}, to the best of our knowledge, up to present, the issues raised above have not been systematically discussed in the literature. Therefore, below we will formulate such a SCF theory and study in its framework the possibility, in principle, to exclude simple cations (which mimic the $Li^{+}$ ions) from the cathode surface by adding some dipolar cations (which mimic organic cations) to the electrolyte solution. In order to confirm our theoretical observations, we will compare the obtained results with those obtained by molecular dynamics simulation.        \n\n\n\\section{Theory}\nLet us consider an interface between a flat charged electrode with a surface charge density $\\sigma$ and an electrolyte solution with one type of monovalent anions and two types of monovalent cations. Let us also assume that the cations of the first type (simple cations) carry an electric charge $e$, whereas the cations of the second type (molecular cations) in addition to the same charge possess a permanent dipole moment $p$. We start from the grand thermodynamic potential of the electrolyte solution in the local density approximation, which can be written in the form \\cite{budkov2016theory,budkov2018theory}\n\\begin{equation}\n\\label{Grand_pot}\n\\Omega=\\int\\limits_{0}^{\\infty}\\left(-\\frac{\\varepsilon\\varepsilon_0\\mathcal{E}^2}{2}+ \\rho_{c}\\psi-c_{+}^{(2)}\\Psi +f-\\mu_{+}^{(1)}c_{+}^{(1)}-\\mu_{+}^{(2)}c_{+}^{(2)}-\\mu_{-}c_{-}\\right)dz,\n\\end{equation} \nwhere $f=f(T,c_{+}^{(1)},c_{+}^{(2)},c_{-})$ is the Helmholtz free energy density of a reference system (a system without electrostatic interactions) and the auxiliary function\n\\begin{equation}\n\\Psi(z)=k_{B}T\\ln\\frac{\\sinh{\\beta p\\mathcal{E}(z)}}{\\beta p\\mathcal{E}(z)},\n\\end{equation}\nis introduced. $\\mathcal{E}(z)=-\\psi^{\\prime}(z)$ is the local electric field and $\\rho_{c}(z)=e\\left(c_{+}^{(1)}(z)+c_{+}^{(2)}(z)-c_{-}(z)\\right)$ is the local charge density of the ions; $T$ is the temperature and $k_B$ is the Boltzmann constant. Using the local Legendre transformation \\cite{maggs2016general,budkov2016theory,budkov2018theory,mceldrew2018theory}, we can rewrite the grand thermodynamic potential in the form\n\\begin{equation}\n\\label{Grand_pot_2}\n\\Omega=-\\int\\limits_{0}^{\\infty}dz\\left(\\frac{\\varepsilon\\varepsilon_0\\mathcal{E}^2(z)}{2}+P(T,\\bar{\\mu}_{+}^{(1)},\\bar{\\mu}_{+}^{(2)},\\bar{\\mu}_{-})\\right),\n\\end{equation}\nwhere the local chemical potentials of the species\n\\begin{equation}\n\\bar{\\mu}_{+}^{(1)}=\\mu_{+}^{(1)}-e\\psi,~\\bar{\\mu}_{+}^{(2)}=\\mu_{+}^{(2)}-e\\psi+\\Psi,\n\\end{equation}\n\\begin{equation}\n\\bar{\\mu}_{-}=\\mu_{-}+e\\psi\n\\end{equation}\nare introduced. A variation of functional (\\ref{Grand_pot_2}) with respect to the potential $\\psi(z)$ leads to the following self-consistent field equation \\cite{budkov2016theory}:\n\\begin{equation}\n\\frac{d}{dz}\\left(\\epsilon(z)\\psi^{\\prime}(z)\\right)=-e\\left(\\bar{c}_{+}^{(1)}(z)+\\bar{c}_{+}^{(2)}(z)-\\bar{c}_{-}(z)\\right), \\end{equation}\nwhere\n\\begin{equation}\n\\bar{c}_{+}^{(1,2)}=\\frac{\\partial{P}}{\\partial{\\bar{\\mu}}_{+}^{(1,2)}}~,\\bar{c}_{-}=\\frac{\\partial{P}}{\\partial{\\bar{\\mu}}_{-}},\n\\end{equation}\nare the local equilibrium concentrations of the ions and\n\\begin{equation}\n\\epsilon(z)=\\varepsilon\\varepsilon_0+\\frac{p^2}{k_{B}T}\\frac{L(\\beta p\\mathcal{E})}{\\beta p\\mathcal{E}}\\bar{c}_{+}^{(2)}(z)\n\\end{equation}\nis the local dielectric permittivity; $L(x)=\\coth{x}-x^{-1}$ is the Langevin function. The first integral of the self-consistent field equation determining the condition of the solution mechanical equilibrium \\cite{budkov2016theory,budkov2015modified} takes the form\n\\begin{equation}\n\\label{mech_eq_cond}\n-\\frac{\\varepsilon\\varepsilon_0\\mathcal{E}^2}{2}-p\\mathcal{E}\\bar{c}_{+}^{(2)}L(\\beta p\\mathcal{E})+P(T,\\bar{\\mu}_{+}^{(1)},\\bar{\\mu}_{+}^{(2)},\\bar{\\mu}_{-})=P(T,{\\mu}_{+}^{(1)},{\\mu}_{+}^{(2)},{\\mu}_{-}).\n\\end{equation}\nSolving equation (\\ref{mech_eq_cond}) with respect to $\\mathcal{E}$ at different $\\psi$, we obtain the implicit function $\\mathcal{E}=\\mathcal{E}(\\psi)$. In order to obtain the potential profile $\\psi(z)$, it is necessary to solve the ordinary first-order differential equation $\\psi^{\\prime}(z)=-\\mathcal{E}(\\psi)$ with the initial condition $\\psi(0)=\\psi_0(\\sigma)$, where $\\psi_0(\\sigma)$ is the surface potential of the electrode corresponding to the fixed surface charge density $\\sigma$. In order to obtain the function $\\psi_0(\\sigma)$, it is necessary to use the boundary condition $\\epsilon_s\\mathcal{E}_0=\\sigma$,\nwhere the local dielectric permittivity at the electrode is $\\epsilon_s=\\epsilon(0)$ and the electric field at the electrode $\\mathcal{E}_{0}=\\mathcal{E}(0)$ are introduced. The latter is an implicit function of the surface potential $\\psi_0$, determined by eq. (\\ref{mech_eq_cond}).\n\n\nAs a reference system, let us consider three-component symmetric lattice gas model \\cite{maggs2016general,budkov2016theory,kornyshev2007double} with the equation of state\n\\begin{equation}\nP(T,{\\mu}_{+}^{(1)},{\\mu}_{+}^{(2)},{\\mu}_{-})=\\frac{k_B T}{v}\\ln\\left(1+e^{\\beta\\mu_{+}^{(1)}}+e^{\\beta\\mu_{+}^{(2)}}+e^{\\beta\\mu_{-}}\\right), \n\\end{equation}\nwhere $\\beta=(k_{B}T)^{-1}$. The local concentrations of the ions within the three-component lattice gas model are determined by the following relations\n\\begin{equation}\n\\bar{c}_{+}^{(1)}(z)v=\\frac{e^{\\beta(\\mu_{+}^{(1)}-e\\psi(z))}}{1+e^{\\beta(\\mu_{+}^{(1)}-e\\psi(z))}+e^{\\beta(\\mu_{+}^{(2)}-e\\psi(z)+\\Psi(z))}+e^{\\beta(\\mu_{-}+e\\psi(z))}},\n\\end{equation}\n\\begin{equation}\n\\bar{c}_{+}^{(2)}(z)v=\\frac{e^{\\beta(\\mu_{+}^{(2)}-e\\psi(z)+\\Psi(z))}}{1+e^{\\beta(\\mu_{+}^{(1)}-e\\psi(z))}+e^{\\beta(\\mu_{+}^{(2)}-e\\psi(z)+\\Psi(z))}+e^{\\beta(\\mu_{-}+e\\psi(z))}},\n\\end{equation}\n\\begin{equation}\n\\bar{c}_{-}(z)v=\\frac{e^{\\beta(\\mu_{-}+e\\psi(z))}}{1+e^{\\beta(\\mu_{+}^{(1)}-e\\psi(z))}+e^{\\beta(\\mu_{+}^{(2)}-e\\psi(z)+\\Psi(z))}+e^{\\beta(\\mu_{-}+e\\psi(z))}}.\n\\end{equation}\n\nUsing the electrical neutrality condition for the bulk solution $c_{-,b}=c_{+,b}^{(1)}+c_{+,b}^{(2)}$ and \nintroducing the bulk concentrations of electrolytes $c_{1}=c_{+,b}^{(1)}$ and $c_{2}=c_{+,b}^{(2)}$, we obtain the relations for the species chemical potentials\n\\begin{equation}\n\\mu_{+}^{(1,2)}=k_{B}T\\ln\\left(\\frac{c_{1,2}v}{1-2v(c_{1}+c_{2})}\\right),~\\mu_{-}=k_{B}T\\ln\\left(\\frac{(c_{1}+c_{2})v}{1-2v(c_{1}+c_{2})}\\right).\n\\end{equation}\n\\section{Computational details: Molecular dynamics simulations}\nAll the molecular dynamics simulations were performed with the help of the LAMMPS \\cite{plimpton1993fast} simulation package. The cutoff distance for the electrostatic and van der Waals interactions was set to $2~nm$. The long range electrostatic interactions were calculated using the pppm scheme \\cite{hockney1988computer}. The simulations we carried out in an $NVT$-ensemble at $300~K$ using a Langevin thermostat (dump parameter $100~fs$) wan integration time-step of $1~fs$. Periodic boundary conditions were applied along the $x$ and $y$ directions, while the unwanted interactions between the replicas along the $z$ direction were eliminated by using special algorithms provided within the LAMMPS package \n\\cite{yeh1999ewald,ballenegger2009simulations}.\nThe simulation cell contained two parallel plates (parallel to $xy$ plane) constructed from frozen atoms ordered according to the hexagonal $2-d$ lattice with the parameter $a = 0.293~nm$ (that resembles Au(111) surface topology). The plate size was $58.6 \\times 60.90$ $\\AA$ ($xy$) with an $8~nm$ distance (along the $z$ axis) between them. The plates were charged oppositely by setting the required atomic charges. The space between the 'electrode plates' was filled with ions. The solvent effect was taken into account implicitly by setting the relative dielectric permittivity to 40 (close to that of commonly used organic solvents such as acetonitrile, dimethylsulfoxide, {\\sl etc.}). The solvent viscosity effect was simulated by a Langevin thermostat. The van der Waals interactions were described by the Lennard-Jones 6-12 potential with the depth $\\varepsilon=0.1~Kcal\/mol$ and the repulsion distance equal to $0.3~nm$ and $0.38~nm$ for the ions and 'electrode' atoms, respectively. The simple cations (and anions) were represented by individual atoms with the charge $+1e$ ($-1e$) and mass $10~g\/mol$. The organic cations were designed in a Drude-particle fashion with a constant dipole moment. It was composed of a Drude-core (mass $9.6~g\/mol$, charge $+3e$) and a pseudo electron (mass $0.4~g\/mol$, charge $-2e$). The distance between the core and the 'electron' was kept fixed during the simulation at the value of $0.4~\\AA$ or $0.2~\\AA$ that yield the dipole moment about $4~D$ ($3.84$) and $2~D$ ($1.92$), respectively. The pseudo-electrons were thermalized separately \\cite{jiang2011high} at $T=1~K$ temperature by a special Langevin thermostat with a dump parameter of $20~fs$. \n\nAt higher surface charge values the EDL capacity turned out to exceed the number of ions between the plates at the desired concentration ($1.0$ or $0.5~M$ of the cation-anion pairs). Therefore, prior to the main simulation an iterative procedure was employed to continuously add (or remove, in case of overshoot) ion pairs with cations of the required type to achieve the target bulk concentration value. \\footnote{Here 'bulk' concentration is attributed to the central $60~\\AA$ thick slab.} After the EDL was completely saturated and the target bulk concentrations were reached, the refiling procedure was stopped. The following productive run was performed to obtain the $1~ns$ long trajectory.       \n\n\\begin{figure}[h!]\n\\center{\\includegraphics[width=0.8\\linewidth]{Fig_1AB.png}}\n\\caption{The concentration profiles of the simple cations calculated within the mean field theory at different values of the cathode surface charge density. The concentration profile of the molecular cations is also shown only for one case. The data are shown for the effective size $v^{1\/3}=3.7~\\AA$, temperature $T=300~K$, bulk concentrations of the cations $c_{1}=c_{2}=0.5~M$, and two dipole moments of the molecular cations $p=2~D$ (A) and $p=4~D$ (B).}\n\\label{Profiles_1}\n\\end{figure}\n\n\n\\section{Results and discussions}\nFirst, we discuss the cation concentration profiles near the electrode surface, which can be predicted by the formulated above mean field theory at different values of dipole moment of the molecular cation $p$ and the cathode surface charge density $\\sigma <0$. Let us consider the equimolar electrolyte compostion, i.e. $c_{1}=c_{2}=0.5~M$ (total $1~M$ salt concentration). Such value is within the typical range ($0.1-1.0~M$) for the real electrochemical devices discussed in the introductory section. We also assume that the dielectric permittivity $\\varepsilon=40$ that approximately corresponds to the value of common organic solvents such as dimethylsulfoxide or acetonitrile. Figure \\ref{Profiles_1} (A,B) shows the theoretical concentration profiles of the simple cations calculated for two values of the dipole moments of molecular cations at different values of the cathode surface charge density $|\\sigma|$. As is seen, at a sufficiently small $|\\sigma|$, the concentration of the simple cations monotonically decreases together with the distance from the electrode, so that its maximal value is attained at the electrode surface (at $z=0$). Moreover, higher absolute value of the surface charge density $|\\sigma|$ causes the increase in the local concentration of the simple cations on the electrode. However, when the $|\\sigma|$ exceeds a certain threshold value, the local concentration monotonically decreases, while the concentration profiles exhibit a pronounced maximum at a certain  distance $z>0$. This maximum can be shifted to higher distances by a further increase in the surface charge density. As is seen in Fig. \\ref{Profiles_1} (B), the described effects become stronger at a larger dipole moment of the molecular cations. Moreover, at a rather large dipole moment of the molecular cations the maxima on the concentration profiles become sharper. Fig. \\ref{Profiles_1} (B) also shows the concentration profile of the molecular cations only for one case. As is seen, at $|\\sigma|=86~\\mu C\/cm^2$ the simple cations are depleted near the electrode surface, while the concentration of the molecular cations reaches the maximal value there. The latter clearly demonstrates the replacement of the simple cations by the molecular cations.    \n\n\\begin{figure}[h!]\n\\center{\\includegraphics[width=0.8\\linewidth]{Fig_2.png}}\n\\caption{The concentration profiles of the simple cations obtained by the MD simulations for the different cathode surface charge density. The data are shown for $p=4~D$.}\n\\label{Profiles_2}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\center{\\includegraphics[width=0.8\\linewidth]{Fig_3.png}}\n\\caption{The local concentration of the simple cations on the electrode as a function of the cathode surface charge density calculated within the mean field approximation. The local concentration of molecular cations is also shown for only one case. The data are shown for different dipole moments and concentrations of the simple and molecular cations in the bulk.}\n\\label{surf_conc_1}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\center{\\includegraphics[width=0.8\\linewidth]{Fig_4.png}}\n\\caption{The local concentration of the simple cations on the electrode as a function of the cathode surface charge density obtained by the MD simulations. The local concentration of molecular cations is also shown for only one case. The data are shown for dipole moment $p=4~D$ of the molecular cation and different bulk concentrations of all cations.}\n\\label{surf_conc_2}\n\\end{figure}\n\nSuch a behavior can be interpreted as follows. At a rather small $|\\sigma|$ there is a lot of free space for both types of the cations near the electrode surface, a higher surface charge density increases the local concentration of the simple cations on the electrode, while the concentration profiles are described by the monotonically decreasing functions of the distance. In this case, the maximum of concentration is reached on the electrode surface. At a sufficiently strong surface charge density the molecular cations are more strongly attracted to the electrode than the simple cations due to the occurrence of an additional dielectrophoretic force \\cite{jones1979dielectrophoretic,budkov2018theory,budkov2016theory} acting on each dipolar cation. Therefore, at a rather large $|\\sigma|$ the presence of the molecular cations near the electrode is more thermodynamically favorable, so that the simple cations are expelled from the $\"$near-surface$\"$ layer due to the steric interactions. The latter explains the occurrence of a pronounced maximum on the concentration profiles of the simple cations, which can be shifted by the increase in the surface charge density (see Figs. \\ref{Profiles_1} (A,B)). The shift in the concentration maximum due to the increase in the surface charge density can also be explained by the simple cations exclusion the layers nearest to the electrode and their replacement with molecular cations. We would like to note that similar replacement from the cathode surface of alkaline cations by another alkaline cations due to a difference in their excess polarizabilities in aqueous medium was discussed within the mean-field theory in paper \\cite{lopez2018diffuse,lopez2018multiionic}.\n\nIn order to confirm our theoretical findings, we performed MD simulations (for the details of the simulation setup, see the previous section). Fig. \\ref{Profiles_2} shows the concentration profiles of the simple cations obtained by means of the MD simulations for the case $p=4~D$. The first concentration peak (which can be considered as a local concentration at the electrode surface) grows as the surface charge density increases up to about 65 $\\mu C\/cm^2$. A further increase in the electrode surface charge density leads to a decrease in the first concentration peak and simultaneous growth in the second one. Thus, the simple cations are expelled into the second ionic layer. The difference in comparison with the theoretical model is that the simple cations are pushed away from the surface not continuously (along the z coordinate) but layer by layer. Nevertheless, the general behavior of the concentration profiles predicted by the theoretical model is in qualitative agreement with that obtained by MD simulations. \n\nLet us now analyze the local concentration of the simple cations at the electrode surface as a function of the surface charge density for several electrolyte compositions. The results obtained within our mean field theory are presented in Fig. \\ref{surf_conc_1}. The grey dotted line represents the surface concentration of the molecular cations. Comparing it with the simple cations concentration in the same case (black line) one can see how the surface ionic composition depends on the electrode charge density. Considering the equimolar compositions ($c_{1}\/c_{2}=0.5\/0.5$, i.e. $1~M$ total and $c_{1}\/c_{2}=0.25\/0.25$, i.e. $0.5~M$ total), it should be noted that the surface concentration of the simple (as well as molecular) cations weakly depends on the total bulk electrolyte concentration. The higher dipole moment of the molecular cations leads to a lower surface concentration of the simple cations, as the dielectrophoretic force is stronger in that case and the molecular cations are more strongly attracted to the cathode surface. The decline from the equimolar compositions in case of the higher bulk concentration of the simple cations ($c_{1}\/c_{2}=0.5\/0.25$, i.e. $0.75~M$ total) leads to higher surface concentration of simple cations, as there are less molecular cations to compete for the place within the EDL. However, the first ionic layer still consists mostly of molecular cations. It should be also noted that the position of the maximum of the function $c_{surf}=c_{surf}(\\sigma)$ depends on the dipole moment $p$, but does not depend on the bulk electrolyte composition. Fig. \\ref{surf_conc_2} presents the surface concentration dependences for $p = 4~D$ obtained by means of MD simulations. Although there are some quantitative differences, the qualitative behavior of the system is in good agreement with the theoretical predictions.\n\n\nWe would like to note that in the present theoretical mean field model as well as in the MD simulations we modelled the solvent as a continuous dielectric medium with constant dielectric permittivity. Thereby, we neglected the effects of solvent polarization causing a reduction in the dielectric permittivity near the charged electrode relative to the bulk solution \\cite{yeh1999dielectric,gongadze2011langevin,gongadze2011generalized}. It is clear that accounting for this effect will not change qualitatively the simple cations exclusion effects, but can still significantly change the region of the surface charge densities, where this effect may take place. We neglected also an influence of ions on the dielectric permittivity of solution (so-called, dielectric decrement) \\cite{Ben-Yaakov2011}. Moreover, in this study we did not take into account the short-range specific interactions between the ionic species and the electrode \\cite{budkov2018theory,goodwin2017mean}. However, in this study we aimed to investigate the possibility, in principle, of excluding simple cations from the charged cathode by adding an organic salt with dipolar cations to the electrolyte solution at the simplest level of the theoretical model. The reasonable values of the surface charge densities, bulk concentrations of the electrolytes, and dipole moment of the organic cations, predicted by our mean field theory and qualitative agreement of the theoretical predictions with the MD simulations support the correctness of the obtained results.\n\n\n\\section{Conclusions}\nIn this work, we have formulated a mean field theory of the flat electric double layer on the interface between a metallic cathode and an electrolyte solution with two types of monovalent cations. One type of the cations is described as structureless charged particles (simple cation, e.g. alkali ion), whereas another one \u2013 as charged particle carrying polar groups with a considerable dipole moment (molecular cations). In practice the latter can be an organic cation (e.g. quaternary ammonium $NR_4^{+}$) with an attached acceptor group, such as trifluoromethyl ($-CF_3$), nitrile ($-CN$), carbonyl ($-COOR$), sulfonyl ($-SO_3 R$), {\\sl etc}. We have shown that at a sufficiently high absolute value of the cathode surface charge density (about $30~\\mu C\/cm^2$), the simple cations are expelled from the cathode surface by the molecular cations due to the stronger attraction of the latter to the electrode. We have shown that the stronger attractive force is realized due to the occurrence of the dielectrophoretic force acting on each molecular cation in the inhomogeneous electrostatic field. We have confirmed our theoretical predictions by molecular dynamics simulations. The reported study estimates the strength of the cation replacement effect for various sets of parameters, which can be further used for experiment design.  \n\n\n\\begin{acknowledgement}\nThe reported study was partially supported by the RFBR according to research project no. 18-31-20015. The MD simulation part of the study carried out by Artem Sergeev was supported by Russian Science Foundation grant no. 19-43-04112.\n\\end{acknowledgement}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn the standard model (SM), the weak phase of $\\bs$-$\\bsbar$ mixing,\n$\\beta_s$, is $\\approx 0$. Thus, if its value is measured to be\nnonzero, this is a clear sign of new physics (NP). Indeed, experiments\nhave already started measuring $\\beta_s$ in $\\Bsdecay$. The results of\nthe CDF \\cite{CDF} and D\\O\\ \\cite{D0} collaborations hint at NP, but\nthe errors are very large. On the other hand, the LHCb collaboration\n\\cite{LHCb_betas} finds a central value for $\\beta_s$ which is\nconsistent with zero: $\\beta_s = (-0.03 \\pm 2.89~({\\rm stat}) \\pm\n0.77~({\\rm syst}))^\\circ$, implying that, if NP is present in\n$\\bs$-$\\bsbar$ mixing, its effect is small.\n\nIn the $\\bd$ system, the phase of $\\bd$-$\\bdbar$ mixing, $\\beta$, was\nfirst measured in the ``golden mode'' $\\bd\\to J\/\\psi \\ks$, and\nsubsequently in many other modes such as $\\btos$ penguin decays\n(e.g.\\ $\\bd\\to \\phi \\ks$), ${\\bar b} \\to {\\bar c}c{\\bar d}$ decays\n(e.g.\\ $\\bd\\to J\/\\psi \\pi^0$), etc. In the same vein, it is important\nto measure $\\beta_s$ in many different decay modes.\n\nOne process which is potentially a good candidate for measuring\n$\\beta_s$ is the pure $\\btos$ penguin decay $\\bskk$. Its amplitude can\nbe written\n\\beq\n{\\cal A}_s = V_{ub}^* V_{us} P'_{uc} + V_{tb}^* V_{ts} P'_{tc} ~.\n\\eeq\nNow, we know that $|V_{ub}^* V_{us}|$ and $|V_{tb}^* V_{ts}|$ are\n$O(\\lambda^4)$ and $O(\\lambda^2)$, respectively, where $\\lambda=0.23$\nis the sine of the Cabibbo angle. This suggests that the $V_{ub}^*\nV_{us} P'_{uc}$ term is possibly negligible compared to $V_{tb}^*\nV_{ts} P'_{tc}$. If this is justified, then there is essentially only\none decay amplitude, and $\\beta_s$ can be cleanly extracted from the\nindirect CP asymmetry in $\\bskk$.\n\nThe difficulty is that it is not completely clear whether $V_{ub}^*\nV_{us} P'_{uc}$ is, in fact, negligible. This term has a different\nweak phase than that of $V_{tb}^* V_{ts} P'_{tc}$, so that its\ninclusion will ``pollute'' the extraction of $\\beta_s$. That is, if it\ncontributes significantly to the amplitude, the value of $\\beta_s$\nmeasured in $\\bskk$ will deviate from the true value of $\\beta_s$, and\nthis theoretical error is directly related to the relative size of the\ntwo terms.\n\nThis issue has been examined by Ciuchini, Pierini and Silvestrini\n(CPS) in Ref.~\\cite{CPS}. In order to get a handle on the size of\n$P'_{uc}$, CPS proceeded as follows. They considered the\nU-spin-conjugate decay, $\\bdkk$, focusing specifically on $\\bd \\to\nK^{*0} {\\bar K}^{*0}$.  This is a pure $\\btod$ penguin decay, whose\namplitude is\n\\beq\n{\\cal A}_d = V_{ub}^* V_{ud} P_{uc} + V_{tb}^* V_{td} P_{tc} ~.\n\\eeq\nIf one takes the values for the CKM matrix elements, including the\nweak phases, from independent measurements, then this amplitude\ndepends only on three unknown parameters: the magnitudes of $P_{uc}$\nand $P_{tc}$, and their relative strong phase. But there are three\nexperimental measurements one can make of this decay -- the branching\nratio, the direct CP asymmetry, and the indirect (mixing-induced) CP\nasymmetry. It is therefore possible to solve for all the unknown\nparameters. In particular, one can obtain $|P_{uc}|$. This quantity\ncan be related to $|P'_{uc}|$ by an SU(3)-breaking factor. Now, in\n2007, when Ref.~\\cite{CPS} was written, there were no experimental\nmeasurements of $B_{d,s}^0 \\to K^{*0} {\\bar K}^{*0}$. Instead, CPS\nassumed values for these measurements, inspired by QCD factorization\n(QCDf) \\cite{BBNS}. They found that, even allowing for 100\\% SU(3)\nbreaking, the value of $|P'_{uc}|$ is such that the error on $\\beta_s$\ndue to the inclusion of a nonzero $V_{ub}^* V_{us} P'_{uc}$ term is\nless than $1^\\circ$. This inspired CPS to dub $\\bskk$ the {\\it golden\n  channel} for measuring $\\beta_s$.\n\nIn this paper, we re-examine the method of CPS. In particular, we want\nto establish to what extent CPS's conclusion is dependent on the\nvalues chosen for the $\\bdkk$ experimental observables. As we will\nsee, the CPS result holds for a significant subset of the input\nvalues. However, it also fails for other choices of the inputs -- the\nerror on $\\beta_s$ due to the presence of the $V_{ub}^* V_{us}\nP'_{uc}$ term can be as large as $18^\\circ$. It is therefore not\ncorrect to say that the $V_{ub}^* V_{us} P'_{uc}$ term has little\neffect, i.e.\\ that $\\beta_s$ can always be measured cleanly in\n$\\bskk$. On the other hand, it is true that $|P_{uc}|$ can be\nextracted from $\\bdkk$. This can then be used to obtain information\nabout $|P'_{uc}|$ if the SU(3)-breaking factor were known reasonably\naccurately. We discuss different ways, both experimental and\ntheoretical, of learning about the size of the SU(3) breaking.\n\nIn Sec.~2, we examine $B^0_{d,s} \\to K^{(*)0} {\\bar K}^{(*)0}$, and\nshow how the $\\bd$ decay can be used to obtain information about the\n$\\bs$ decay. We allow for all values of the observables in the $\\bd$\ndecay, and compute the theoretical error on $\\beta_s$, allowing for\n100\\% SU(3) breaking. It turns out that this error can be\nsubstantial. In Sec.~3, we discuss ways, both experimental and\ntheoretical, of determining the SU(3) breaking. If this breaking is\nknown with reasonable accuracy, this greatly reduces the theoretical\nerror on $\\beta_s$, and allows this mixing quantity to be extracted\nfrom $B^0_{d,s} \\to K^{(*)0} {\\bar K}^{(*)0}$ decays. We conclude in\nSec.~4.\n\n\\section{\\boldmath $B^0_{d,s} \\to K^{(*)0} {\\bar K}^{(*)0}$}\n\n\\subsection{\\boldmath $\\bskk$}\n\\label{bskkSec}\n\n$\\bskk$ is a pure $\\btos$ penguin decay. That is, its amplitude\nreceives contributions only from gluonic and electroweak penguin (EWP)\ndiagrams.  There are three contributing amplitudes, one for each of\nthe internal quarks $u$, $c$ and $t$ (the EWP diagram contributes only\nto $P'_t$):\n\\bea\n{\\cal A}_s &=& \\lambda^{(s)}_u P'_u + \\lambda^{(s)}_c P'_c + \\lambda^{(s)}_t P'_t \\nn\\\\\n         &=& |\\lambda^{(s)}_u| e^{i\\gamma} P'_{uc} - |\\lambda^{(s)}_t| P'_{tc} ~,\n\\label{Bsamp}\n\\eea\nwhere $\\lambda^{(q')}_q \\equiv V_{qb}^* V_{qq'}$.  (As this is a\n$\\btos$ transition, the diagrams are written with primes.) In the\nsecond line, we have used the unitarity of the\nCabibbo-Kobayashi-Maskawa (CKM) matrix ($\\lambda^{(s)}_u +\n\\lambda^{(s)}_c + \\lambda^{(s)}_t = 0$) to eliminate the $c$-quark\ncontribution: $P'_{uc} \\equiv P'_u - P'_c$, $P'_{tc} \\equiv P'_t -\nP'_c$. Also, above we have explicitly written the weak-phase\ndependence (including the minus sign from $V_{ts}$ in\n$\\lambda^{(s)}_t$), while $P'_{uc}$ and $P'_{tc}$ contain strong\nphases. (The phase information in the CKM matrix is conventionally\nparametrized in terms of the unitarity triangle, in which the interior\n(CP-violating) angles are known as $\\alpha$, $\\beta$ and $\\gamma$\n\\cite{pdg}.)  The amplitude ${\\bar {\\cal A}}_s$ describing the\nCP-conjugate decay $\\bsbar\\to K^{(*)0} {\\bar K}^{(*)0}$ can be\nobtained from the above by changing the signs of the weak phases (in\nthis case, $\\gamma$).\n\nThere are three measurements which can be made of $\\bskk$: the\nbranching ratio, and the direct and indirect CP-violating asymmetries.\nThese yield the three observables\n\\bea\n\\label{'observables}\nX' & \\equiv & \\frac{1}{2} \\left( |{\\cal A}_s|^2 + |{\\bar{\\cal A}}_s|^2 \\right) ~, \\nn \\\\\nY' & \\equiv & \\frac{1}{2} \\left( |{\\cal A}_s|^2 - |{\\bar{\\cal A}}_s|^2 \\right) ~, \\nn\\\\\nZ'_I & \\equiv & {\\rm Im}\\left( e^{-2i \\beta_s} {\\cal A}^*_s {\\bar {\\cal A}}_s \\right) ~.\n\\eea\nAssuming one takes the values for $|\\lambda^{(s)}_u|$,\n$|\\lambda^{(s)}_t|$ and $\\gamma$ from independent measurements, ${\\cal\n  A}_s$ then depends only on the magnitudes of $P'_{uc}$ and\n$P'_{tc}$, and their relative strong phase $\\delta'$. With $\\beta_s$,\nthis makes a total of four unknown parameters. These cannot be\ndetermined from only three observables -- additional input is needed.\n\nNote that, if $\\lambda^{(s)}_u P'_{uc}$ were negligible, we would only\nhave two unknowns -- $|P'_{tc}|$ and $\\beta_s$. These could be\ndetermined from the measurements of $X'$ and $Z'_I$ ($Y'$ would\nvanish). This demonstrates that if one extracts $\\beta_s$ from $Z'_I$\nassuming that $\\lambda^{(s)}_u P'_{uc}$ is negligible, and it is not,\nthen one will obtain an incorrect value for $\\beta_s$. The size of\nthis error is directly related to the size of $\\lambda^{(s)}_u\nP'_{uc}$. Here, the possibility of an error is particularly\nimportant. Since $\\beta_s \\approx 0$ in the SM, a nonzero measured\nvalue of $\\beta_s$ would indicate NP\\footnote{Note that, if there is\n  an indication of NP, we will know that it is in $\\btos$\n  transitions. However, we will not know if $\\bs$-$\\bsbar$ mixing\n  and\/or the $\\btos$ penguin amplitude is affected.}. It is therefore\ncrucial to have this theoretical uncertainty under control.\n\n\\subsection{\\boldmath $\\bdkk$}\n\nIn order to deal with the $P'_{uc}$ problem in $\\bskk$, in\nRef.~\\cite{CPS}, CPS use its U-spin-conjugate decay $\\bdkk$ . This is\na pure $\\btod$ penguin decay, whose amplitude can be written\n\\beq\n{\\cal A}_d = |\\lambda^{(d)}_u| e^{i\\gamma} P_{uc} + |\\lambda^{(d)}_t| e^{-i\\beta} P_{tc} ~.\n\\eeq\nAs with ${\\cal A}_s$, we take the values for the magnitudes and weak\nphases of the CKM matrix elements from independent measurements. This\nleaves three unknown parameters in ${\\cal A}_d$: the magnitudes of\n$P_{uc}$ and $P_{tc}$, and their relative strong phase $\\delta$.  And,\nas with $\\bskk$, there are three measurements which can be made of\n$\\bdkk$: the branching ratio, and the direct and indirect CP-violating\nasymmetries. Given an equal number of observables and unknowns, we can\nsolve for $|P_{uc}|$, $|P_{tc}|$ and $\\delta$.\n\nThe key point is that $|P_{uc}|$ and $|P'_{uc}|$ are equal under\nflavor SU(3) symmetry. Thus, given a value for $|P_{uc}|$ and a value\n(or range) for the SU(3)-breaking factor, one obtains the value (or\nrange) of $|P'_{uc}|$. With this, one can extract the true value (or\nrange) of $\\beta_s$ from the $\\bskk$ experimental data.\n\nNow, CPS focused mainly on the decays $B_{d,s}^0 \\to K^{*0} {\\bar\n  K}^{*0}$. As mentioned in the introduction, there were no\nexperimental measurements of these decays when their paper was\nwritten, so it was necessary to assume experimental values in order to\nextract $|P_{uc}|$. CPS chose values roughly based on the QCDf\ncalculation of Ref.~\\cite{BRY}. They found that the value of\n$|P_{uc}|$ is such that, even allowing for 100\\% SU(3) breaking,\n$|\\lambda^{(s)}_u P'_{uc}|$ is indeed small. The upshot is that the\ntheoretical uncertainty in the extraction of $\\beta_s$ is less than\n$1^\\circ$.\n\nThere are several reasons not to take this result at face value.\nFirst, although QCDf has been very successful at describing the\n$B$-decay data, it is still a model. Indeed, the predictions and\nexplanations of other models of QCD -- perturbative QCD \\cite{pQCD}\n(pQCD) and SCET \\cite{SCET}, for example -- do not always agree with\nthose of QCDf. Second, QCDf assumes that factorization holds to\nleading order for all $B$ decays. However, $B^0_{d,s} \\to K^{(*)0}\n{\\bar K}^{(*)0}$ are penguin decays, and it has been argued that\nnon-factorizable effects are important for such decays. It may be that\nsub-leading effects in QCDf are, in fact, important for $\\bd \\to\nK^{*0} {\\bar K}^{*0}$. We therefore re-examine the CPS method taking a\nmore model-independent approach.\n\n\\subsection{\\boldmath Theoretical Uncertainty on $\\beta_s$}\n\nIn this subsection, we generalize the CPS method. First, we consider\nall final states in $B^0_{d,s} \\to K^{(*)0} {\\bar K}^{(*)0}$. Second,\nwe scan over a large range of experimental input values.\n\nWe proceed as follows. The three experimental measurements of the\n$\\bd$ decay correspond to the three observables\n\\bea\n\\label{XYZdefs}\nX & \\equiv & \\frac{1}{2} \\left( |{\\cal A}_d|^2 + |{\\bar{\\cal A}}_d|^2 \\right) \\nn\\\\\n&& \\hskip0.4truecm \n=~|\\lambda^{(d)}_u|^2 |P_{uc}|^2 + |\\lambda^{(d)}_t|^2 |P_{tc}|^2 - 2 |\\lambda^{(d)}_u||\\lambda^{(d)}_t||P_{uc}| |P_{tc}| \\cos\\delta \\cos\\alpha ~, \\nn \\\\\nY & \\equiv & \\frac{1}{2} \\left( |{\\cal A}_d|^2 - |{\\bar{\\cal A}}_d|^2 \\right) =\n- 2 |\\lambda^{(d)}_u||\\lambda^{(d)}_t||P_{uc}| |P_{tc}| \\sin\\delta \\sin\\alpha ~, \\\\\nZ_I & \\equiv & {\\rm Im}\\left( e^{-2i \\beta} {\\cal A}_d^* {\\bar {\\cal A}}_d \\right)\n= |\\lambda^{(d)}_u|^2 |P_{uc}|^2 \\sin 2\\alpha - 2 |\\lambda^{(d)}_u||\\lambda^{(d)}_t||P_{uc}| |P_{tc}| \\cos\\delta \\sin\\alpha ~. \\nn\n\\eea\nIt is useful to define a fourth observable:\n\\bea\n\\label{ZRdef}\nZ_R & \\equiv & {\\rm Re}\\left( e^{-2i \\beta} {\\cal A}_d^* {\\bar {\\cal A}}_d \\right) \\\\\n&& \\hskip0.4truecm \n=~|\\lambda^{(d)}_u|^2 |P_{uc}|^2 \\cos 2\\alpha + |\\lambda^{(d)}_t|^2 |P_{tc}|^2 - 2 |\\lambda^{(d)}_u||\\lambda^{(d)}_t| |P_{uc}| |P_{tc}| \\cos\\delta \\cos\\alpha ~. \\nn\n\\eea\nThe quantity $Z_R$ is not independent of the other three observables:\n\\beq\nZ_R^2 = X^2 - Y^2 - Z_I^2 ~.\n\\label{eq:ZR}\n\\eeq\nThus, one can obtain $Z_R$ from measurements of $X$, $Y$ and $Z_I$, up\nto a sign ambiguity.\n\n$X$, $Y$ and $Z_I$ are related to the branching ratio\n($\\mathcal{B}_d$), the direct CP asymmetry ($C_d$) and the indirect CP\nasymmetry ($S_d$) of $\\bdkk$ as follows:\n\\beq\nX = \\kappa_d \\mathcal{B}_d ~~,~~~~ \nY = \\kappa_d \\mathcal{B}_d C_d ~~,~~~~\nZ_I = \\kappa_d \\mathcal{B}_d S_d ~,\n\\eeq\nwhere\n\\beq\n\\kappa_d = \\frac{8 \\pi m_{B_d}^2}{\\tau_d p_c} ~.\n\\eeq\nIn the above, $m_{B_d}$ and $\\tau_d$ are the mass and the lifetime of\nthe decaying $\\bd$ meson, respectively, and $p_c$ is the momentum of\nthe final-state mesons in the rest frame of the $\\bd$. \n\n{}From Eqs.~(\\ref{XYZdefs}) and (\\ref{ZRdef}), the quantity $|P_{uc}|$\ncan then be written in terms of the observables as\n\\beq\n|P_{uc}|^2 = \\frac{1}{|\\lambda^{(d)}_u|^2} \\, \\frac{Z_R - X}{\\cos 2\\alpha - 1}\n= \\frac{\\kappa_d \\mathcal{B}_d}{|\\lambda^{(d)}_u|^2} \\, \\frac{\\pm \\sqrt{1-C_d^2-S_d^2} - 1}{\\cos 2\\alpha - 1} ~.\n\\label{eq:Puc2}\n\\eeq\nThe value of $\\alpha$ is not known exactly, but we know from\nindependent measurements that it is approximately $90^\\circ$. In what\nfollows, we fix $\\alpha$ to $90^\\circ$ for simplicity.  Note that any\ndeviation of $\\alpha$ from this value decreases the denominator in\nEq.~(\\ref{eq:Puc2}), and thus makes $|P_{uc}|$ larger. The above\nexpression allows us to calculate $|P_{uc}|$ for a given set of\nobservables.\n\nOn the whole, the decays $\\bdkk$ have not yet been measured. One\nexception is $\\bd \\to K_S K_S$. From BaBar \\cite{Aubert:2006gm}, we\nhave\n\\beq\n\\mathcal{B}_d = (1.08 \\pm 0.28 \\pm 0.11) \\times 10^{-6} ~~,~~~~\nS_d = -1.28^{+0.80+0.11}_{-0.73-0.16} ~~,~~~~\nC_d = -0.40 \\pm 0.41 \\pm 0.06 ~,\n\\eeq\nwhile Belle finds \\cite{Nakahama:2007dg}\n\\beq\n\\mathcal{B}_d = (0.87^{+0.25}_{-0.20} \\pm 0.09) \\times 10^{-6} ~~,~~~~\nS_d = -0.38^{+0.69}_{-0.77} \\pm 0.09 ~~,~~~~\nC_d = 0.38 \\pm 0.38 \\pm 0.05 ~.\n\\eeq\nWe see that essentially all values of $\\sqrt{C_d^2 + S_d^2}$ are still\nexperimentally allowed.\n\nHere is an example of the calculation of $|P_{uc}|$ using\nEq.~(\\ref{eq:Puc2}).  We take the QCDf-inspired central value of CPS\nfor the branching ratio ($\\mathcal{B}_d = 5 \\times\n10^{-7}$)\\footnote{In fact, the branching ratio for $\\bd \\to K^{*0}\n  {\\bar K}^{*0}$ has been measured \\cite{BdK*K*bar}. The world average\n  is $\\mathcal{B}_d = (8.1 \\pm 2.3) \\times 10^{-7}$ \\cite{hfag}. In\n  order to make the generalization of the CPS method more direct, in\n  our analysis we use the CPS value for $\\mathcal{B}_d$ (which differs\n  from the experimental value by only a little more than $1\\sigma$).},\nand also take $0 \\le \\sqrt{C_d^2 + S_d^2} \\le 1$. We compute\n$|\\lambda^{(d)}_u|$ using values for the various quantities taken from\nthe Particle Data Group \\cite{pdg}.  Including the errors on these\nquantities, we find that $|P_{uc}|$ can be as large as $1460 \\pm 170$\neV ($Z_R$ positive) or $2060 \\pm 240$ ($Z_R$ negative).  For\ncomparison, $|P_{uc}|$ is only about 180 eV if the CP asymmetries are\nalso fixed at the QCDf-inspired central values of CPS.  Note that the\ndiscrete ambiguity with $Z_R$ negative corresponds to the case for\nwhich $\\bd$ decays are dominated by $P_{uc}$. On the other hand, we\nnaively expect $P_{tc}$ to be larger. Still, even if this solution\nwere discarded, the results of our analysis below would not be changed\nfundamentally. The bottom line is that $|P_{uc}|$ can, in fact, be\nlarge in $\\bdkk$ decays.\n\nWe now return to $\\bskk$ decays.  Even if $|P_{uc}|$ is large in $\\bd$\ndecays, because of the $|\\lambda^{(s)}_u|$ CKM suppression it is not\nclear whether or not $|\\lambda^{(s)}_u P'_{uc}|$ really plays a\nsignificant role in the $\\bs$ decays. In order to ascertain this, we\nproceed as follows.  We apply the CPS method, but consider all\npossible values of the observables in both $\\bd$ and $\\bs$\ndecays\\footnote{In fact, the branching ratio for $\\bs \\to K^{*0} {\\bar\n    K}^{*0}$ has been measured \\cite{BsK*K*bar}. Its value is $(2.81\n  \\pm 0.46~({\\rm stat}) \\pm 0.45~({\\rm syst}) \\pm 0.34~(f_s\/f_d))\n  \\times 10^{-5}$. The CPS value for $\\mathcal{B}_s$, which we use in\n  our analysis, is $1.18 \\times 10^{-5}$.}.  Thus, we use flavor SU(3)\nsymmetry to relate $|P_{uc}|$ and $|P'_{uc}|$, allowing for a 100\\%\nsymmetry breaking. In order to study the worst-case scenario (the\nlargest possible value of $|P'_{uc}|$ within 100\\% breaking), we fix\n$|P'_{uc}| = 2 |P_{uc}|$. Thus, for example, for the case where the\nbranching ratio $\\mathcal{B}_d$ is taken to be the QCDf-inspired\ncentral value of CPS, but the CP asymmetries take all possible values,\n$|P'_{uc}|$ can be as large as 2920 eV ($Z_R$ positive) or 4120 eV\n($Z_R$ negative).\n\nGiven the worst-case value of $|P'_{uc}|$, assuming the CKM phases to\nbe known, and fixing $\\beta_s = 0$ (in order to study the worst-case\nprediction in the SM), only two parameters are left unknown in the\n$\\bskk$ decay.  These can be extracted from the branching ratio\n$\\mathcal{B}_s$ and the direct CP asymmetry $C_s$ (up to discrete\nambiguities, but this does not affect the following discussion).  Once\nthis is done, all the theoretical parameters in $\\bskk$ are known, and\nwe can compute the time-dependent CP asymmetry $S_s$ and the effective\nphase $\\beta_s^{eff}$ ($\\arg{(\\bar\\mathcal{A}_s \/ \\mathcal{A}_s)}$).\nThus we get an evaluation of the (worst-case) theoretical uncertainty\nof $\\beta_s$ as extracted from the time-dependent CP asymmetry of\n$\\bskk$ decays.\n\n\\begin{figure}[htb]\n\t\\centering\n\t\t\\includegraphics[height=7.2cm]{fig1.eps}\n\\caption{Worst-case values of $\\beta_s^{eff}$ (in degrees) as a\n  function of $|P'_{uc}|$ and the direct CP asymmetry $C_s$.  The\n  branching ratios are fixed to $\\mathcal{B}_d = 5 \\times 10^{-7}$ and\n  $\\mathcal{B}_s = 11.8 \\times 10^{-6}$ (central values of CPS).}\n\\label{fig1}\n\\end{figure}\n\nWe now present figures showing the worst-case $\\beta_s^{eff}$ in\nvarious situations.  The aim is to scan over the whole observable\nspace in order to ascertain how large $\\beta_s^{eff}$ can be within\nthe SM.  In Fig.~\\ref{fig1}, we fix both branching ratios to the CPS\ncentral values, and give the worst-case value of $\\beta_s^{eff}$ as a\nfunction of $|P'_{uc}|$ and the direct CP asymmetry $C_s$.  The\neffective phase is roughly proportional to $|P'_{uc}|$ and can be up\nto $10^\\circ$ in this restricted scenario. In Fig.~\\ref{fig2}, we\nrepeat the calculation but also allow the branching ratios to vary,\npresenting $\\beta_s^{eff}$ as a function of $C_s$ and the ratio of\nbranching ratios ($\\mathcal{B}_s\/\\mathcal{B}_d$).  In this case, for\nthe central maximum value of $|P_{uc}|$, an effective phase of up to\n$12^\\circ$ ($Z_R$ positive) or $18^\\circ$ ($Z_R$ negative) is\nobtained.  In Fig.~\\ref{fig3}, $\\beta_s^{eff}$ is given as a function\nof $\\sqrt{C_d^2+S_d^2}$ and $\\mathcal{B}_s\/\\mathcal{B}_d$.\n\n\\begin{figure}\n\t\\centering\n\t\t\\includegraphics[height=7.2cm]{fig2A.eps}\n\t\t\\includegraphics[height=7.2cm]{fig2B.eps}\n\\caption{Worst-case values of $\\beta_s^{eff}$ (in degrees) as a\n  function of the direct CP asymmetry $C_s$ and the ratio of branching\n  ratios ($\\mathcal{B}_s\/\\mathcal{B}_d$).  The plot on the left\n  (right) is for $Z_R$ positive (negative) in Eq.~(\\ref{eq:ZR}).}\n\\label{fig2}\n\\end{figure}\n\n\\begin{figure}\n\t\\centering\n\t\t\\includegraphics[height=7.2cm]{fig3A.eps}\n\t\t\\includegraphics[height=7.2cm]{fig3B.eps}\n\\caption{Worst-case values of $\\beta_s^{eff}$ (in degrees) as a\n  function of $\\sqrt{C_d^2+S_d^2}$ and the ratio of branching ratios\n  ($\\mathcal{B}_s\/\\mathcal{B}_d$).  The plot on the left (right) is\n  for $Z_R$ positive (negative) in Eq.~(\\ref{eq:ZR}).}\n\\label{fig3}\n\\end{figure}\n\n{}From the above figures, it is clear that $\\beta_s^{eff}$ can be\nlarge within the SM, and that the conclusions of CPS hold only for\ncertain sets of values of the experimental inputs.  Still, it is\ninteresting to note that a small theoretical error (say $\\beta_s^{eff}\n\\le 5^\\circ$) is found for a non-negligible subset of the input\nnumbers.  The general behavior of solutions is as follows:\n\\begin{enumerate}\n\n\\item for $Z_R$ positive, $\\beta_s^{eff}$ is smaller for smaller\n  values of $\\sqrt{C_d^2+S_d^2}$ (it's the opposite for $Z_R$\n  negative),\n\n\\item $\\beta_s^{eff}$ is smaller for larger values of $|C_s|$ for\n  fixed $|P'_{uc}|$,\n\n\\item $\\beta_s^{eff}$ is smaller for larger values of\n  $\\mathcal{B}_s\/\\mathcal{B}_d$,\n\n\\item $\\beta_s^{eff}$ is smaller for smaller values of SU(3) breaking.\n\n\\end{enumerate}\nFor the first three points we cannot do anything -- the measurements\nof the observables are what they are.  The fourth point can be\nunderstood as follows. The theoretical error $\\beta_s^{eff}$ is due to\nthe presence of a nonzero $P'_{uc}$ in ${\\cal A}_s$\n[Eq.~(\\ref{Bsamp})]. This error is roughly proportional to\n$|P'_{uc}|$, which is itself equal to the product of $|P_{uc}|$ and an\nSU(3)-breaking factor. For a given value of $|P_{uc}|$,\n$\\beta_s^{eff}$ is smaller if the SU(3)-breaking factor is\nsmaller. Thus, the assumption of CPS of 100\\% breaking often leads to\na large $\\beta_s^{eff}$.  The precise knowledge of the SU(3) breaking\nbetween $|P_{uc}|$ and $|P'_{uc}|$ would therefore considerably reduce\nthe theoretical uncertainty on the extracted value of $\\beta_s$ using\nthis method. The determination of the size of SU(3) breaking is\ndiscussed in the next section.\n\n\\section{SU(3) Breaking}\n\nAs we have seen, the idea of obtaining information on $|P'_{uc}|$ by\nrelating it to $|P_{uc}|$ using flavor SU(3) is tenable. However, if\none simply takes an SU(3)-breaking factor of 100\\%, this can lead to a\ntheoretical error on the extraction of $\\beta_s$ of up to $18^\\circ$.\nThus, in order to use this method, a better determination of the size\nof SU(3) breaking must be found. In this section, we discuss ways,\nboth experimental and theoretical, of getting this information.\n\n\\subsection{Experimental Measurement of SU(3) Breaking}\n\n\\subsubsection{\\boldmath $B^0_{d,s} \\to K^{*0} {\\bar K}^{*0}$}\n\\label{BKKSU3break}\n\nThe decays $B^0_{d,s} \\to K^{(*)0} {\\bar K}^{(*)0}$ really represent\nthree types of decay -- the final state can consist of $PP$, $PV$ or\n$VV$ mesons ($P$ is pseudoscalar, $V$ is vector). Now, the CPS method\napplies when the final state is a CP eigenstate. For $PP$ and $VV$\ndecays, this holds. However, $PV$ decays do not satisfy this\ncondition. Still, these decays can be used if the $K^{*0}$\/${\\bar\n  K}^{*0}$ decays neutrally. That is, we have\n\\bea\nB^0 & \\to & \\frac{1}{\\sqrt{2}} \\left( K^0 {\\bar K}^{*0} + K^{*0} {\\bar K}^0 \\right) ~~~~~~ {\\hbox{(CP eigenstate)}} \\nn\\\\\n& \\to & K^0 {\\bar K}^0 \\pi^0 ~.\n\\label{PVdecay}\n\\eea\nOn the other hand, the CPS method cannot be used if the\n$K^{*0}$\/${\\bar K}^{*0}$ decays to charged particles. This is because,\nin this case, one cannot extract $|P_{uc}|$ from the $\\bd$ decay --\nthere are more theoretical unknowns than observables.\n\nThe SM value of SU(3) breaking can be found from any single pair of\ndecays -- $|P_{uc}|$ and $|P'_{uc}|$ can be extracted from the $\\bd$\nand $\\bs$ decays, respectively. (As we are interested in SU(3)\nbreaking in the SM, we set $\\beta_s$ to zero.) In principle, this\nvalue of SU(3) breaking ($|P'_{uc}|\/|P_{uc}|$) can then be used in a\ndifferent decay, and the method of the previous section can be\napplied. The problem here is that this approach is applicable only if\nthe SU(3) breaking in the two decays is expected to be\nsimilar. However, $PP$, $PV$ and $VV$ decays are all different\ndynamically, so that there is no a-priori reason to expect this to\nhold. For example, the decay of Eq.~(\\ref{PVdecay}) is very different\nfrom the $PP$ decay $B^0 \\to K^0 {\\bar K}^0$, and so the $PV$ and $PP$\nSU(3) breakings are not likely to be similar.\n\nThere is one exception, and it involves the $VV$ decays $B^0_{d,s} \\to\nK^{*0} {\\bar K}^{*0}$. Since the final-state particles are vector\nmesons, when the spin of these particles is taken into account, these\ndecays are in fact three separate decays, one for each polarization.\nThe polarizations are either longitudinal ($A_0$), or transverse to\ntheir directions of motion and parallel ($A_\\|$) or perpendicular\n($A_\\perp$) to one another. By performing an angular analysis of these\ndecays, the three polarization pieces can be separated. \n\nIt is also possible to express the polarization amplitudes using the\nhelicity formalism. Here, the transverse amplitudes are written as\n\\bea\nA_\\| &=& \\frac{1}{\\sqrt{2}} (A_+ + A_-) ~, \\nn\\\\\nA_\\perp &=& \\frac{1}{\\sqrt{2}} (A_+ - A_-) ~.\n\\eea\nHowever, in the SM, the helicity amplitudes obey the hierarchy\n\\cite{BRY,Kagan}\n\\beq\n\\left\\vert \\frac{A_+}{A_-} \\right\\vert = \\frac{\\Lambda_{QCD}}{m_b} ~.\n\\eeq\nThat is, in the heavy-quark limit, $A_+$ is negligible compared to\n$A_-$, so that $A_\\| = -A_\\perp$. Thus, one expects the SU(3) breaking\nfor the $\\|$ and $\\perp$ polarizations to be approximately equal.\nOne can therefore extract $|P_{uc}|$ and $|P'_{uc}|$ from the $\\bd$\nand $\\bs$ decays for one of the transverse polarizations, compute the\nSU(3) breaking ($|P'_{uc}|\/|P_{uc}|$), and apply this value of SU(3)\nbreaking to the other transverse polarization decay pair.  In this way\nthe SU(3)-breaking factor can be measured experimentally, and can be\nused to determine the theoretical uncertainty in the extraction of\n$\\beta_s$.\n\n\\subsubsection{\\boldmath $B^+ \\to K^+ \\bar K^0$ and $B^+ \\to \\pi^+ K^0$}\n\nOther decays which can be used to measure SU(3) breaking are the\nU-spin-conjugate pair $B^+ \\to K^+ \\bar K^0$ and $B^+ \\to \\pi^+\nK^0$. While it is true that these do not involve $\\bs$ and $\\bd$\nmesons, both are pure penguin decays, just like $B^0_{d,s} \\to\nK^{(*)0} {\\bar K}^{(*)0}$.  Restricting ourselves to the $PP$ final\nstates, we then expect that the SU(3) breaking in $B^+ \\to K^+ \\bar\nK^0$ and $B^+ \\to \\pi^+ K^0$ is similar (though not necessarily equal)\nto that in $\\bd \\to K^0 {\\bar K}^0$ and $\\bs \\to K^0 {\\bar K}^0$.\nThe measurement of SU(3) breaking can therefore be done using the\n$B^+$ decays and applied to the $\\bd$\/$\\bs$ decays.\n\nThere is a difference compared to the previous example. Since there\nare no indirect CP asymmetries in $B^+$ decays, one cannot measure\n$|P'_{uc}|\/|P_{uc}|$. The SU(3) breaking probed in $B^+ \\to K^+ \\bar\nK^0$ and $B^+ \\to \\pi^+ K^0$ is\n\\bea\n\\label{Y'\/Y}\n-\\frac{Y'}{Y} & = &\n\\frac{|\\lambda^{(s)}_u||\\lambda^{(s)}_t|}{|\\lambda^{(d)}_u||\\lambda^{(d)}_t|} \\,\n\\frac{\\sin{\\gamma}}{\\sin{\\alpha}} \\, \n\\frac{|P'_{uc}|}{|P_{uc}|} \\, \\frac{|P'_{tc}|}{|P_{tc}|} \\, \\frac{\\sin\\delta'}{\\sin\\delta} \\nn\\\\\n& = &\n\\frac{|P'_{uc}|}{|P_{uc}|} \\, \\frac{|P'_{tc}|}{|P_{tc}|} \\, \\frac{\\sin\\delta'}{\\sin\\delta} ~.\n\\eea\nIn the second line, all the CKM factors cancel due to the sine law\nassociated with the unitarity triangle.  Thus, if the $B^+$ decay pair\nis used to measure the SU(3) breaking, the theoretical error in the\nextraction of $\\beta_s$ must be calculated relating $|P'_{uc}|\n|P'_{tc}| \\sin\\delta'$ of $\\bs \\to K^0 {\\bar K}^0$ to $|P_{uc}|\n|P_{tc}| \\sin\\delta$ of $\\bd \\to K^0 {\\bar K}^0$.\n\n\\subsubsection{Other SU(3) pairs}\n\nThere are many other pairs of decays that are related by U spin or\nSU(3): $\\bd \\to \\pi^+ \\pi^-$ and $\\bs \\to K^+ K^-$, $\\bd \\to \\pi^0K^0$\nand $\\bs \\to \\pi^0 \\kbar$, etc. A complete list of two- and three-body\ndecay pairs, as well as a discussion of the measurement of\nU-spin\/SU(3) breaking, is given in Ref.~\\cite{SU3break}.  For some of\nthem we already have measurements of the breaking.\n\nFor example, consider the pair $\\bs \\to \\pi^+ K^-$ and $\\bd \\to \\pi^-\nK^+$. The measurement of the SU(3) breaking of Eq.~(\\ref{Y'\/Y}) gives\n\\cite{SU3break}\n\\beq\n-\\frac{Y'}{Y} = 0.92 \\pm 0.42 ~.\n\\eeq\nAlthough the error is still substantial, we see that the central value\nimplies small SU(3) breaking.  The problem is that $\\bs \\to \\pi^+ K^-$\nand $\\bd \\to \\pi^- K^+$ are not pure-penguin decays, so that it is not\nclear how the above SU(3) breaking is related to that in $B^0_{d,s}\n\\to K^{(*)0} {\\bar K}^{(*)0}$, if at all.  Still, if one measures the\nSU(3) breaking in several different decay pairs, it can give us a\nrough indication as to what to take for $B^0_{d,s} \\to K^{(*)0} {\\bar\n  K}^{(*)0}$.\n\n\\subsection{Theoretical Input on SU(3) Breaking}\n\nConsider again the $\\bskk$ amplitude, Eq.~(\\ref{Bsamp}). If the\n$t$-quark contribution is eliminated using the unitarity of the CKM\nmatrix, we have\n\\beq\n{\\cal A}_s = T' \\lambda^{(s)}_u + P' \\lambda^{(s)}_c ~,\n\\eeq\nwhere $T' \\equiv P'_u - P'_t$, $P' \\equiv P'_c - P'_t$.  Now, in QCDf\n$T'$ and $P'$ are calculated using a systematic expansion in\n$1\/m_b$. However, a potential problem occurs because the higher-order\npower-suppressed hadronic effects contain some chirally-enhanced\ninfrared (IR) divergences. In order to calculate these, one introduces\nan arbitrary infrared (IR) cutoff. The key observation here is that\nthe difference $T' - P'$ is free of these dangerous IR divergences\n\\cite{DMV}.  And, although the calculation of various hadronic\nquantities in pQCD is different than in QCDf, the difference $T'-P'$\nis the same in both formulations. This also holds for $T - P$ in the\n$\\bdkk$ amplitude.\n\nSince $T' - P' = P'_{uc}$ and $T - P = P_{uc}$, this suggests that the\ncalculation of $|P'_{uc}|$ and $|P_{uc}|$ is under good control\ntheoretically. This allows us to calculate $|P'_{uc}|\/|P_{uc}|$, which\ngives us the theoretical prediction of SU(3) breaking. There are many\nquantities which enter into the calculation of $|P'_{uc}|$ and\n$|P_{uc}|$ -- the renormalization scale $\\mu$, the Gegenbauer\ncoefficients in the light-cone distributions, the quark masses,\netc.\\ -- and the errors on these quantities are quite large at\npresent. However, most of these quantities and their errors cancel in\nthe ratio $|P'_{uc}|\/|P_{uc}|$. For the various $B^0_{d,s} \\to\nK^{(*)0} {\\bar K}^{(*)0}$ decays we find\n\\bea\n\\label{SU3break}\nPP &:& \\frac{|P'_{uc}|}{|P_{uc}|} =\n\\frac{M_{B_s}^2 F_0^{\\bs \\to K}(M_K^2)}{M_{B_d}^2 F_0^{\\bd \\to K}(M_K^2)} = 0.86 \\pm 0.15 ~,\\nn\\\\\nPV &:& \\frac{|P'_{uc}|}{|P_{uc}|} = \\frac{M_{B_s}^2 F_+^{\\bs \\to K}(M_{K^*}^{2})}{M_{B_d}^2 F_+^{\\bd \\to K}(M_{K^*}^{2})} = 0.86 \\pm 0.15 ~,\\nn\\\\\nVP, VV_0 &:& \\frac{|P'_{uc}|}{|P_{uc}|} = \\frac{M_{B_s}^2 A_0^{\\bs \\to K^*}(M_{K^{(*)}}^{2})}{M_{B_d}^2 A_0^{\\bd \\to K^*}(M_{K^{(*)}}^{2})} = 0.87 \\pm 0.19 ~,\\\\\nVV_{\\|}, VV_{\\perp} &:& \\frac{|P'_{uc}|}{|P_{uc}|} =  \\frac{M_{B_s} ( F_-^{\\bs \\to K^*}(M_{K^*}^{2}) \\pm F_+^{\\bs \\to K^*}(M_{K^*}^{2}))}{M_{B_d} (F_-^{\\bd \\to K^*}(M_{K^*}^{2}) \\pm F_+^{\\bd \\to K^*}(M_{K^*}^{2}))} = 0.79 \\pm 0.16~. \\nn\n\\eea\n\n\n\n\n\n\n\nAbove we have taken $F^{B \\to K^{(*)}}(M_{K^{(*)}}^2) \\simeq F^{B\\to\n  K^{(*)}}(0)$ since the variation of the ratio of form factors over\nthis range of $q^2$ falls well within the errors of their calculation\n\\cite{BRY, LCSR}. For $PV$ and $VP$ decays, the spectator quark goes\nin the first meson. In the last expression, we have $F_+^{B \\to K^*} =\n0.00 \\pm 0.06$ \\cite{BRY}, so that one has the same SU(3) breaking for\nthe $\\|$ and $\\perp$ polarizations. As discussed in\nSec.~\\ref{BKKSU3break}, this is to be expected.\n\nNow, the QCDf calculation is to $O(\\alpha_s)$, and the above\nexpression indicates that, to this order, the SU(3)-breaking term is\nfactorizable. Thus, the theoretical prediction is fairly robust.  On\nthe other hand, SCET says that there are long-distance contributions\nto $P'_c$ and $P_c$. Although this could introduce some uncertainty\ninto $|P'_{uc}|\/|P_{uc}|$, there might also be a partial cancellation\nin the ratio. Our point here is that, though one generally wants to\navoid theoretical input, since this is largely based on models, the\nSU(3) breaking in $|P'_{uc}|\/|P_{uc}|$ may be theoretically clean.\n\nIn Sec.~\\ref{BKKSU3break}, it was noted that the CPS method can be\nused when the final state in $B^0_{d,s} \\to K^{(*)0} {\\bar K}^{(*)0}$\nis a CP eigenstate. Thus, if one wishes to use the theoretical input\nof Eq.~(\\ref{SU3break}), one can simply apply it to $PP$ or $VV$\ndecays. However, this does not hold for $PV$ or $VP$ decays, which are\nnot CP eigenstates. Still, one can use the CPS method on the decay of\nEq.~(\\ref{PVdecay}), which is a linear combination of $PV$ and $VP$\nstates. And, since the theoretical $PV$ SU(3) breaking in\nEq.~(\\ref{SU3break}) is about equal to that of $VP$, this theoretical\ninput can be applied to the $PV + VP$ decay.\n\nFinally, in Sec.~\\ref{bskkSec} we noted that, in the presence of a\nnonzero $P'_{uc}$, one cannot cleanly extract $\\beta_s$ from $\\bskk$\n-- one needs additional input. In Ref.~\\cite{DMV2} it was the above\ntheoretical calculation of $|P'_{uc}|$ which was taken as the\ninput. We note that the method using $\\bskk$ and $\\bdkk$ is somewhat\nmore precise since most of the errors in the calculation of\n$|P'_{uc}|$ cancel in the SU(3)-breaking ratio of\nEq.~(\\ref{SU3break}).\n\n\\section{Conclusions}\n\nThe pure $\\btos$ penguin decay $\\bskk$ is potentially a good candidate\nfor measuring the $\\bs$-$\\bsbar$ mixing phase, $\\beta_s$. If its\namplitude were dominated by $V_{tb}^* V_{ts} P'_{tc}$, the indirect CP\nasymmetry would simply measure $\\beta_s$. Unfortunately, although the\nsecond contributing amplitude, $V_{ub}^* V_{us} P'_{uc}$, is expected\nto be small, it is not clear that it is completely negligible. A\nnonzero $V_{ub}^* V_{us} P'_{uc}$ can change the extracted value of\n$\\beta_s$ from its true value, i.e.\\ it can lead to a theoretical\nerror. Since the measurement of $\\beta_s$ is an important step in the\nsearch for new physics, the size of this theoretical error is\nimportant.\n\nThe size of $P'_{uc}$ has been examined by Ciuchini, Pierini and\nSilvestrini (CPS). They note that the amplitude $P_{uc}$ can be\nextracted from the U-spin-conjugate decay, $\\bdkk$, and can be related\nto $P'_{uc}$ by SU(3). They choose values for the $\\bdkk$ experimental\nobservables inspired by QCDf, allow for 100\\% SU(3) breaking, and\ncompute $P'_{uc}$. They find that the theoretical error on $\\beta_s$\nis very small, i.e.\\ that the presence of the $V_{ub}^* V_{us}\nP'_{uc}$ amplitude has little effect on the extraction of $\\beta_s$.\n\nIn this paper, we revisit the CPS method. In particular, we consider\nmost values of the $\\bdkk$ observables, still allowing for 100\\% SU(3)\nbreaking. We find that, although the theoretical error remains small\nfor a significant subset of these input values, it can be large for\nother values. We find that an error of up to $18^\\circ$ is possible,\nwhich makes the extraction of $\\beta_s$ from $\\bskk$ problematic.\n\nThis issue can be resolved if we knew the value of SU(3) breaking. We\ntherefore discuss different ways, both experimental and theoretical,\nof determining this quantity. From the experimental point of view, the\nsize of SU(3) breaking can be measured using a different $\\bd$\/$\\bs$\ndecay pair. We show that the $VV$ decay $B^0_{d,s} \\to K^{*0} {\\bar\n  K}^{*0}$ or $B^+ \\to K^+ \\bar K^0$\/$B^+ \\to \\pi^+ K^0$ can be used\nin this regard. It is also possible to use theoretical input. Within\nQCDf, the SU(3)-breaking term is factorizable, and so the theoretical\nprediction for this quantity may be reasonably clean.\n\n\\bigskip\n\\noindent\n{\\bf Acknowledgments}: We thank Tim Gershon for helpful\ncommunications. This work was financially supported by NSERC of Canada\n(BB, DL), by the US-Egypt Joint Board on Scientific and Technological\nCo-operation award (Project ID: 1855) administered by the US\nDepartment of Agriculture and in part by the National Science\nFoundation under Grant No.\\ NSF PHY-1068052 (AD), and by FQRNT of\nQu\\'ebec (MI).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:intro}\n\nAfter formation of a young star and its circumstellar disk due to the gravitation collapse of the protostellar cloud the process of accretion of matter from the nearest surrounding the star can continue and have a form of the clumpy accretion. \\citet{1992PASP..104..479G} was probably the first who used this term. He tried to explain by such a way  the observations of the strong extinction events observed in  some young variables. Later this type of accretion has been considered by \\citet{1996ARA&A..34..207H} for explanation of the FUOR's  phenomenon.  This idea is quite popular up to now \\citep{2010ApJ...713.1143Z, 2013ApJ...764..141B, 2018MNRAS.474...88H}. The mechanism of formation of chondrule as a result of the clumpy accretion was discussed  \\citet{1998Icar..134..137T}. Obviously, the fall of the clump on the disk should cause disturbances at the place of the fall. It is interesting to trace how this disturbance will develop and what structures on the disk can be caused by it.\n \nThe detection of various structures on images of protoplanetary disks is one of the most interesting results obtained with the ALMA interferometer \\citep[see, e.g.][]{2018A&A...619A.161C, 2018ApJ...869L..43H,2018ApJ...869L..42H, 2018ApJ...869L..50P}. Ring and spiral structures are most often observed. Less commonly, structures resembling highly elongated vortices are observed. A number of papers are devoted to theoretical studies of the formation of such structures. Their formation is associated with perturbations in the disks caused by the motion of the forming planets \\citep[e.g.,][]{2013A&A...549A..97R, 2015A&A...579A.106V, 2015ApJ...809...93D,  2016MNRAS.463L..22D, 2016ApJ...818...76J, 2018ApJ...866..110D}, the development of various kinds of instabilities in the disks ~\\citep[e.g.,][]{2015ApJ...815L..15B, 2015ApJ...806L...7Z, 2015ApJ...813L..14B, 2016ApJ...821...82O,2009ApJ...697.1269J, 2014ApJ...796...31B, 2014ApJ...794...55T, 2018A&A...609A..50D}, a large\nscale vertical magnetic field \\citep{2018MNRAS.477.1239S} or with the destruction of large bodies in collisions \\citep{2019ApJ...887L..15D,2020MNRAS.495..285N}. In all these papers, the source of disturbance is in the disk itself.\n\nIn our paper we discuss an alternative scenario for the formation of the observed structure. We investigate in the first time the dynamical response  of circumstellar disk on the perturbation associated  with the  clumpy accretion events. Using  the hydrodynamic simulations we calculate the disk images  at $1$ mm and discuss the results  in the context  with interferometric  observations  of  the protoplanetary disks with  ALMA.  \n\n\\section{Initial condition}\nA model consists of a young star of solar mass ($ M_{\\ast} =\nM_{\\odot} $) embedded in a gas disk with total mass is $ M_{disk}\n= 0.01M_{\\odot} $. At the beginning of simulation the disk matter was distributed\nazimuthally symmetrical within the radii $ r_{in} = 0.5$ and\n$r_{out} = 50$ AU. The initial density distribution of the disk\nis\n\\begin{equation}\n\\rho(r,z,0)=\\frac{\\Sigma_0}{\\sqrt{2\\pi}H(r)}\\frac{r_{in}}{r}e^{-\\frac{z^2}{2H^2(r)}},\n\\end{equation}\nwhere $\\Sigma_0$ is arbitrary scale parameter, which is determined by\ndisk mass. Hydrostatic scale height is $H(r)=\\sqrt{\\frac{\\kappa T_{mid}(r) r^3}{GM_{\\ast} \\mu m_H}}$, where $\\kappa$, $G$ and $m_H$ are\nthe Boltzmann constant, the gravitational constant and the mass of\na hydrogen atom and $\\mu=2.35$ is the mean molecular weight\n\\citep{1994A&A...286..149D}. Following~\\citet{1997ApJ...490..368C}\nwe determine the law of midplane temperature distribution\n$T_{mid}(r)=\\sqrt[4]{\\frac{\\gamma}{4}}\\sqrt{\\frac{R_{\\ast}}{r}}T_{\\ast}$,\nwhere $\\gamma=0.05$ \\citep{2004A&A...421.1075D}. The calculations were performed in the local thermodynamic equilibrium approximation  $P(r,z,t)=c^2(r)\\rho(r,z,t)$, where $P$ is local pressure at the moment $t$ and $c$ is a sound speed. It was assumed that in the vertical direction along $z$ the disk is isothermal. The temperature of the star was assumed to be $T_{\\ast} = 5780$ K and star radius is $R_{\\ast}=R_{\\odot}$. The disk relaxed during $600$ years, and then the remnant of the fallen gas clump was added into it.\n\n\\subsection{Impulse approximation to the clump-disk collision}\n\nWhen a clump falls onto a disk, part of its kinetic energy is converted into thermal energy. An infrared spot should appear on the image of the disk where the clump fell. The thermal relaxation time in protoplanetary disks at the distance $\\geq20$  AU from the star is much less than the local orbital period \\citep{2017A&A...605A..30M}.  Therefore, the remnant of the fallen clump quickly comes to thermodynamic equilibrium with the matter of the disk.  It participates in the Keplerian motion of the matter, while maintaining the residual velocity component orthogonal to the plane of the disk. \n\nAt the initial moment of time, we assume that the remnant of the clump has already reached thermodynamic equilibrium with the matter of the disk. The remnant was generated as a density perturbation on the disk in the form of a disk segment bounded by radii $R_0$ and step $dR$ and distributed over the azimuthal angle $\\phi$ with the axis of symmetry along negative part of x-axis ($\\phi=30^{\\circ}$ for all models). The density of matter in the disturbance exceeded the local density of the disk by a factor of $\\displaystyle K=\\frac{\\Sigma_{cl}}{\\Sigma_d}$, where $\\Sigma_{cl}$ and $\\Sigma_d$ are local surface density of remnant and disk respectively. The remnant moves prograde. The particle velocity of the remnant is equal to  $\\displaystyle V(R)=L\\cdot V_k(R)$, where $V_k(R)$ is Keplerian velocity at a given distance from the star and $L$ is parameter of the problem. The velocity vector had a residual inclination to the disk plane $\\displaystyle sin(I)=\\frac{V_z(R)}{V(R)}$ (Fig.~\\ref{fig:disk}). The residual angle of inclination of the remnant depends on the initial angle of the fall and the amount of kinetic energy that is spent on heating the disk in the region of fall. We considered a number of the possible options for the value and inclination of the velocity vector. The parameters of all calculated models are given in the Table~\\ref{tab:models}.\n\n\\begin{figure}[ht!]\n\\plotone{disk.eps}\n\\caption{\\normalsize Particle distribution at the initial moment of the remnant of the clump motion. Top is the projection onto the plane of the disk, bottom is a section along the axis $y$. \\label{fig:disk}}\n\\end{figure}\n\n\\begin{figure*}[ht!]\n\\plotone{Fig2.eps}\n\\caption{\\normalsize  \nThe average value of the $z$ coordinate of the particles in the cells of $R$, $\\phi$. The model parameters are $K=3$, $I=30^\\circ$ and $L=0.8$.  The time in years is in the upper right corner of the pictures. \\label{fig:08inclImg}}\n\\end{figure*} \n\n\\begin{deluxetable*}{ccccccccc}\n\\tablenum{1}\n\\tablecaption{The models parameters \\label{tab:models}}\n\\tablewidth{0pt}\n\\tablehead{\n\\colhead{L} & \\colhead{I} & \\colhead{K} & \\colhead{R} & \\colhead{dR} & \\colhead{$\\phi$} & \\colhead{Remnant mass} & \\nocolhead{} & \\colhead{Lifetime}\\\\\n\\colhead{Float} & \\colhead{Degrees} & \\colhead{Number} & \\colhead{AU} &\n\\colhead{AU} & \\colhead{Degrees} & \\colhead{Jupiter mass} & \\colhead{Sructures} & \\colhead{yrs}\n}\n\\decimalcolnumbers\n\\startdata\n0.8 & 5 & 3 & 20 & 5 & 30 & 0.11 & Arc, One-hand spiral, Horseshoe & $> 600$ \\\\\n0.8 & 10 & 3 & 20 & 5 & 30 & 0.11 & Arc, One-hand spiral, Horseshoe & $> 600$\\\\\n0.8 & 20 & 3 & 20 & 5 & 30 & 0.11 & Arc, Faint two-arm spiral, Horseshoe & $> 600$\\\\\n0.8 & 30 & 3 & 20 & 5 & 30 & 0.11 & Arc, Faint two-arm spiral, Horseshoe & $> 600$\\\\\n0.8 & 10 & 5 & 20 & 5 & 30 & 0.19 & Arc, One-hand spiral, Horseshoe & $> 600$\\\\\n\\hline\n1 & 5 & 3 & 20 & 5 & 30 & 0.11 & Arc, One-arm spiral, Multi Rings, Ring & $\\sim 1000$ \\\\\n1 & 10 & 3 & 20 & 5 & 30 & 0.11 & Arc, One-arm spiral, Ring & $> 600$\\\\\n1 & 20 & 3 & 20 & 5 & 30 & 0.11 & Arc, One-arm spiral, Faint two-arm spiral  & $> 600$\\\\\n1 & 30 & 3 & 20 & 5 & 30 & 0.11 & Arc, One-arm spiral, Faint two-arm spiral & $> 600$\\\\\n1 & 5 & 5 & 20 & 5 & 30 & 0.19 & Arc, One-arm spiral, Multi Rings, Ring & $> 600$\\\\\n1 & 5 & 10 & 20 & 5 & 30 & 0.38 & Arc, One-arm spiral, Multi Rings, Ring & $> 600$\\\\\n\\hline\n1.2 & 5 & 3 & 20 & 5 & 30 & 0.11 & Arc, One-arm spiral, Multi Rings, Ring & $> 600$ \\\\\n1.2 & 10 & 3 & 20 & 5 & 30 & 0.11 & Arc, Bright two-arm spiral & $> 600$\\\\\n1.2 & 20 & 3 & 20 & 5 & 30 & 0.11 & Arc, Bright two-arm spiral & $> 600$\\\\\n1.2 & 30 & 3 & 20 & 5 & 30 & 0.11 & Arc, Bright two-arm spiral, Asymmetric ring & $\\sim 2000$\\\\\n1.2 & 30 & 1 & 20 & 5 & 30 & 0.04 & Arc, Bright two-arm spiral & $> 600$\\\\\n\\hline\n0.8 & 30 & 3 & 10 & 2 & 30 & 0.04 & Arc & $\\sim 100$\\\\\n1 & 30 & 3 & 10 & 2 & 30 & 0.04 & Arc & $\\sim 100$\\\\\n1.2 & 30 & 3 & 10 & 2 & 30 & 0.04 & Arc, Faint two-arm spiral, Multi Rings & $\\sim 450$\\\\\n\\enddata\n\\tablecomments{The column of ``Structures'' lists the types of observed asymmetries in the order of their appearance in the disk images. The column of ``Lifetimes'' is a long-lived structures lifetime. }\n\\end{deluxetable*}\n\n\\begin{figure*}[ht!]\n\\plotone{Fig1.eps}\n\\caption{\\normalsize The surface density multiplied by the distance from the center of mass ($\\Sigma R$). \nThe model parameters are $K=3$, $I=10^\\circ$ and $L=0.8$.  The time in years is in the upper right corner of the pictures. \\label{fig:08sig}}\n\\end{figure*}\n\n\\begin{figure}[ht!]\n\\plotone{Fig3.eps}\n\\caption{\\normalsize  \nThe average value of the $z$ coordinate of the particles in the cells of $R$, $\\phi$ along $x$ (left) and $y$ (right) axes after $600$ years. The model parameters are $K=3$, $L=0.8$ and the angles are in the upper right corner of the pictures. \\label{fig:08inclxy}}\n\\end{figure}\n\\begin{figure*}[h]\n\\plotone{Fig4.eps}\n\\caption{\\normalsize  \nThe images at wavelength 1 mm. The color shows the flux of radiation ($F_\\nu$) multiplied by $R^2$ in conventional units. The model parameters are $K=3$, $I=10^\\circ$ and $L=0.8$.  The time in years is in the upper right corner of the pictures. \\label{fig:08img}}\n\\end{figure*}  \n\n\\section{Methods}\nThe evolution of the remnant in gas disk was simulated by\nthe SPH method (smooth particle hydrodynamics). The calculations\nwere performed using the code Gadget-2~\\citep{2001NewA....6...79S,\n2005MNRAS.364.1105S} modified by us \\citep{2016Ap.....59..449D}. In total,\nfrom $5\\cdot 10^5$ to $2.5\\cdot 10^6$ particles of the gas disk and from $5\\cdot 10^3$ to $2\\cdot 10^5$ particles of the perturbation were involved in the simulations. The calculations took into account the self-gravity of the disk. \n\nThe simulated region was divided by $200\\times30\\times90$ cells in spherical coordinates ($R,\\theta,\\phi$), in which the average values of the SPH particle density were determined. We assume that dust particles with a size of 1, 10, 100 microns and 1 mm are well mixed with the gas, and are distributed according to the law$\\frac{dn(s)}{ds}\\propto s^{-3.5}$, where n is the concentration and s is the size of the dust grain \\citep{1969JGR....74.2531D}. The dust to gas mass ratio in the disk was $0.01$ as the average in interstellar medium. The dust opacity is calculated using Mie theory for Magnesium-iron silicates~\\citep{1995A&A...300..503D}. RADMC-3D~\\citep{2012ascl.soft02015D} code was used for the 3-D radiative transfer calculations.  \n\n\\begin{figure*}[ht!]\n\\plotone{Fig5.eps}\n\\caption{\\normalsize The same as in Fig.~\\ref{fig:08sig} \nfor model parameters $K=3$, $I=10^\\circ$ and $L=1$. \\label{fig:1sig}}\n\\end{figure*}\n\\begin{figure*}[ht!]\n\\plotone{Fig6.eps}\n\\caption{\\normalsize The same as in Fig.~\\ref{fig:08img} \nfor model parameters $K=3$, $I=10^\\circ$ and $L=1$.  \\label{fig:1img}}\n\\end{figure*}  \n\\begin{figure*}[ht!]\n\\plotone{Fig7.eps}\n\\caption{\\normalsize The same as in Fig.~\\ref{fig:08img} \nfor model parameters $K=3$, $L=1$ after $600$ years. The angle of $I$ is in the upper right corner of the pictures.  \\label{fig:img600}}\n\\end{figure*} \n \\begin{figure*}[ht!]\n\\plotone{Fig8.eps}\n\\caption{\\normalsize The same as in Fig.~\\ref{fig:08sig} \nfor model parameters $K=3$, $I=10^\\circ$ and $L=1.2$. \\label{fig:1.2sig}}\n\\end{figure*}\n\\begin{figure}[ht!]\n\\plotone{Fig9.eps}\n\\caption{\\normalsize The same as in Fig.~\\ref{fig:08inclxy} \nfor model parameters $K=3$, $L=1.2$.\\label{fig:12inclxy}}\n\\end{figure} \n\\begin{figure*}[ht!]\n\\plotone{Fig10.eps}\n\\caption{\\normalsize The same as in Fig.~\\ref{fig:08img} \nfor model parameters $K=3$, $L=1.2$ after $600$ years. The angle of $I$ is in the upper right corner of the pictures. \\label{fig:img1_600}}\n\\end{figure*}  \n\\begin{figure}[ht!]\n\\plotone{Fig11.eps}\n\\caption{\\normalsize The same as in Fig.~\\ref{fig:08img} \nfor model parameters $K=1$, $I=30^\\circ$, $L=1.2$ after $600$ years. \\label{fig:K1I30}}\n\\end{figure}\n\\begin{figure*}[ht!]\n\\plotone{Fig12.eps}\n\\caption{\\normalsize The same as in Fig.~\\ref{fig:08img} \nfor two models with $2.5\\cdot 10^6$ particles. The models parameters $K=3$, $I=5^\\circ$, $L=1$ (left) and $K=3$, $I=30^\\circ$, $L=1.2$ (right) at the time $1000$ years. \\label{fig:longImg}}\n\\end{figure*} \n\\begin{figure}[ht!]\n\\plotone{Fig13.eps}\n\\caption{\\normalsize The same as in Fig.~\\ref{fig:longImg} for model with parameters $K=3$, $I=30^\\circ$, $L=1.2$ at the time $2000$ years. \\label{fig:im2000}}\n\\end{figure} \n\\begin{figure}[ht!]\n\\plotone{Fig14.eps}\n\\caption{\\normalsize The same as in Fig.~\\ref{fig:08inclxy} \nfor models with parameters $K=3$, $I=5^\\circ$, $L=1$ (top) and $K=3$, $I=30^\\circ$, $L=1.2$ (bottom). The time in years is in the upper right corner of the pictures. \\label{fig:longIncl}}\n\\end{figure} \n\\begin{figure*}[ht!]\n\\plotone{Fig15.eps}\n\\caption{\\normalsize The same as in Fig.~\\ref{fig:08img} \nfor models with parameters $K=3$, $I=30^\\circ$, $R_0=10$ AU and $dR=2$ AU after $36.5$ years. The parameter of $L$ is in the upper right corner of the pictures. \\label{fig:closeImg}}\n\\end{figure*} \n\\begin{figure}[ht!]\n\\plotone{Fig16.eps}\n\\caption{\\normalsize The same as in Fig.~\\ref{fig:closeImg} \nfor models with parameters $K=3$, $I=30^\\circ$, $L=1.2$ for the time $400$ years. \\label{fig:close400}}\n\\end{figure} \n\\begin{figure}[ht!]\n\\plotone{Fig17.eps}\n\\caption{\\normalsize The average value of the $z$ along $y$ axes after $365$ (left) and $1460$ (right) years. The model parameters are $K=3$, $I=30$. The values of $L$ are in the upper right corner of the pictures. \\label{fig:yincl}}\n\\end{figure} \n\n\\section{Results} \nThe emergence of the remnant of the clump of matter in the disk leads to the propagation of density waves in horizontal and vertical directions.  The strongest perturbations arise at large values of the parameters $K$, $L$, and $I$, as expected. Due to the differential rotation of the Keplerian disk the remnant stretches and transform during the time.  A local increase in the surface density leads to the appearance of corresponding large-scale inhomogeneities in the disk images.\n\n\\subsection{Perturbations at large radii} \nFor the models discussed here, the initial position of the disturbing remnant of the clump was set equal to $R_0 = 20$ AU with a step of $dR = 5$ AU. Calculations have shown that the parameter $L$, which characterizes the kinetic energy of the falling clump, has the greatest influence on the type of disturbance in the disk. Therefore, we will sequentially discuss the three energy regimes considered in our models. \n\n\\subsubsection{Sub Keplerian perturbations}\nIn this case, when the parameter is $L = 0.8$, due to the differential rotation of the Keplerian disk the remnant stretches and transform to a piece of arc reembling a cyclonic vortex and then turn into a spiral during one revolution around the star ($\\sim 125$ years).  Since the matter of the disk is involved in its motion the spiral dips and thickening of the density are visible in the disk (Fig.~\\ref{fig:08sig}). In the central part of the disk, the spiral splits into two branches. The spiral structure quickly ($\\sim 300$ years) twists into an asymmetric ring, which, scattering in the disk, retains the asymmetry until the end of the calculations ($600$ years). This asymmetry rotates with the disk. \n\nThe wave also passes in the vertical direction along the disk, which extends inside and to the edge of the disk. Perturbation twists the central plane of the disk. The maximum distortion of the disk plane occurs near the axis $y$, but does not coincide with its position. The inner parts of the disk inclines relative to the periphery. Over time, the radius of the outer boundary of the inclined area increases. It distorts 30 AU at the time of $600$ years  (Fig.~\\ref{fig:08inclImg}). In this case, the tilt of the disk plane relative to the initial position is approximately equal to $0.2^\\circ$ at $I = 5^\\circ $ and $0.9^\\circ$ in the case of $I = 30^\\circ$. An increase in the angle of $I$ does not affect the speed of propagation of global perturbation to the edge of the disk (Fig.~\\ref{fig:08inclxy}). Increasing the initial density (parameter K) of the clump increases the vertical distortion of the disk.  \n\nThe perturbations described above show themselves on the images of protoplanetary disks. The form of asymmetric structures in images corresponds to the perturbations of the density of the disk matter. However, on the periphery of the disk is visible a shadow from the matter above the disk plane. The asymmetric ring-shaped structure on the images has a horseshoe-shaped form (Fig.~\\ref{fig:08img}). Calculations have shown that the minimum value of $K$ in which this structure is visible on the images, equal to $3$ (about $0.1$ of Jupiter mass). An increase in the parameter $K$ to $5 $ increases the brightness of the structure and their lifetime, but its horseshoe form is preserved.  \n\n\\subsubsection{Keplerian perturbations}\nFor this class of models of the phase of the disintegration of the clump remnant during the first orbit similar to the previous one (Fig.~\\ref{fig:1sig}). A piece of arc is converted to the one-hand spiral. It twisted into a symmetric ring-shaped structure during the next few revolutions ($\\sim 500$ years). \n\nAn increase of the radius of distortion of the central plane of the disk relative to the initial position is faster than in the previous case. It reaches $40$ AU at the time $600$ years. The maximum inclination angle is $0.2^\\circ$ in the case $I=5^\\circ$ and $0.8^\\circ$ at $I = 30^\\circ$. The direction of maximum distortion in the vertical direction is also close to the axis $y$, but does not coincide with it.\n\nThe visible asymmetry on the images of the disk are also appropriate for surface density (Fig.~\\ref{fig:1img}).  In this case, the propagation of the density wave gives multi-lane image of the protoplanetary disk at a certain point in time (right image of Fig.~\\ref{fig:1img}). \nAn increase in the angle of $I$ affects the image of the ring-shaped structure. Two symmetric weakly pronounced spirals are visible on the images instead of the ring, if the value of $I\\geq20$ (Fig.~\\ref{fig:img600}). \n\n\\subsubsection{Super Keplerian perturbations}\nIn this case, the clump matter motion in the protoplanetary disk causes severe density perturbations and significantly distorts the disk in the vertical direction. As in previous cases, during one convolution of the clump, the vortex-like structure is stretched into a spiral, which is converted into two spirals during the next revolution (Fig.~\\ref{fig:1.2sig}). Each of the spirals is logarithmic, they are shifted by phase relative to each other by $180^\\circ$. The form of the spirals weakly depends on the angle of inclination of $I$. \n\nDisk distortion in the vertical direction differs from the models described above for this case. The periphery of the disk is distorted, and the inner parts of the disk deforms weaker. The waves are still propagating along the disk in a vertical direction at the time of 600 years as seen from Fig.~\\ref{fig:12inclxy}. In this case, the distortion of the inner region depends on the inclination angle and has the opposite character for $I<20$ and $I\\geq 20$.  \n\nIt become seen noticeable asymmetry of the spirals on the periphery of the disk in images with an increase in the angle of inclination of $I$ (Fig.~\\ref{fig:img1_600}). \n\nFor this class of models, the case $K = 1$ (corresponds to the mass of the remnant $\\sim 12 M_{\\oplus}$) was considered. In this case, the perturbation is weaker; however, the two-arm spiral can also be identified in the disk images. It is more pronounced at a larger inclination angle $I$ (Fig.~\\ref{fig:K1I30}). \n\n\\subsubsection{Long-term dynamics} \nFor the models described above, the number of SPH particles was $5\\cdot 10^5$. These models have a lower resolution compared to the models, the calculations of which involved $2.5\\cdot 10^6$ particles. However, the calculations showed that at the initial phases of clump destruction, the images of disks obtained on the basis of models with a small number of particles show the same structures as for more accurate models. However, to study the long-term dynamics of the fallen clump remnant, higher resolution is required. We have calculated two limiting cases that correspond to the parameters that cause the minimum ($K=3,I=5,L=1$) and maximum ($K=3,I=30,L=1.2$) disturbance in the disk.  \n\nOver time, density waves scattered and the all structure settles down to the plane of the disk. The ring of the first model stretches along the radius and loses brightness mixing with the matter of the protoplanetary disk with time. It is faintly noticeable at $1000$ years after the fall of the clump (Fig.~\\ref{fig:longImg} left). Than it is not visible against the background of  the disk matter. For the second model spiral waves are still noticeable after $1000$ years (Fig.~\\ref{fig:longImg} right), but after $\\sim 2000$ years they completely disappear. An asymmetric ring structure can be seen on the disk by the time of 2000 years (Fig.~\\ref{fig:im2000}). \n\nThe charateristic time of the disk dynamic relaxation after the fall of the clump also depends on the place of its fall. For example, with $R_0 = 30$ AU and the same clump parameters as in the previous model, the lifetime of spirals and ring structures generated by its fall increases to $4 \\times 10^3$ years. At $R_0 = 50$ AU, the characteristic relaxation time of disturbances on the disk is even longer: $\\sim 10^4$ years.\n\nOn the Fig.~\\ref{fig:longIncl} it is seen that the plane of the disk settles to its original position over time. However, even 2000 years after the fall of the clump, a slight inclination of the disk plane near the axis $y$ remains. For the first model, it is equal to $\\sim 0.14^\\circ$, and for the second, it is $\\sim 0.72^\\circ$. \n\n\\subsection{Perturbations at small radii} \n\nIn this class of models the perturbation was located near the star at the distance $R_0 = 10$ AU with the step of $dR = 2$ AU. The clump had parameters $\\phi=30^\\circ$, $K=3$ and $I=30^\\circ$. The parameter $L$ was varied. The mass of the clump was about $13 M_{\\oplus}$. \n\nFig.~\\ref{fig:closeImg} shows images of the disk for three models with the L parameter $0.8$, $1$ and $1.2$ after one period ($36.5$ years). Bright dense structures and areas of shadow, which are caused by matter rising above the plane of the disk, are visible on the disk. But all structures are scattered after next period for models with $L \\leq 1$. For case $L=1.2$ waves propagate along the disk, which in time from 250 to 500 years can be seen in the images as a multi-lane structure (Fig.~\\ref{fig:close400}). \n\nIn the vertical direction, the disk is distorted in all considered cases. The maximum distortion is achieved near the $y$ axis as for models distant from the star. The Fig.~\\ref{fig:yincl} shows the average values of $z$ along the y-axis for two points in time $365$ and $1460$ years. One can see in all cases, the perturbation propagates outward from the inner part of the disk, tilting its central plane. With an increase in L, the final inclination of the disk increases, but remains within $0.5^\\circ$. \n\n\\section{Conclusion} \nCalculations have shown that at the initial stages of the remnant of the clump disintegration, the structures that are visible in the images of the protoplanetary disk are similar for the entire set of the models. However, the shape of the final long-lived structure primarily depends on the kinetic energy of the falling clump. \n\nDuring the first revolution of the center of the remnant (at the initial moment of time), an arc-like structure resembling a vortex is visible in the disk image. Similar structures are observed, for example, in objects HD 135344B \\citep{2018A&A...619A.161C} and HD 143006 \\citep{2018ApJ...869L..50P}. At the next stage of evolution, the image shows a tightly wound spiral, as, for example, in the case of object HD 163296 \\citep{2018ApJ...869L..42H}. In the case when the residual velocity of the remnant does not exceed the Keplerian velocity, a ring is a long-lived structure, which can also be asymmetric. In this case, at a certain moment in time, the passage of a wave over the disk can give a multi-lane structure in the image. The ring-shaped structure is visible in the images of a number of objects, for example, HD 169142 \\citep{2017A&A...600A..72F}, HD 97048 \\citep{2017A&A...597A..32V}, RU Lup, Elias 24, AS 209, GW Lup \\citep{2018ApJ...869L..42H}. In the case of a high kinetic energy of a clump, a two-armed spiral appears on the disk image, which, after several thousand years, transforms into an asymmetric ring. Two-arm spirals were obtained on images of objects Elias 27, IM Lup, WaOph 6 \\citep{2018ApJ...869L..43H}. The median age of these sources is $\\sim 1$ Myr~\\citep{2018ApJ...869L..42H}, and the youngest  sources have  estimated ages about a few tenths of Myr~\\citep{2018ApJ...869L..43H}.\n\nSince the velocity vector of the remnant has a residual inclination relative to the plane of the disk, it is distorted. However, over time, the inclination of the plane of the disk relative its original position decreases. The tilt angle at the end of calculations does not exceed $1^\\circ$. Probably, for a more noticeable change in the inclination of the disk, a large mass of the clump remnant is required. In this work, we were looking for the minimum mass of the remnant, which develops into a structure visible in the image of the protoplanetary disk. It turned out that in the case of a high-energy fall, the minimum mass of the remnant of the clump is $\\sim 10 M _{\\oplus}$. If the residual velocity of the remnant does not exceed Kepler one, then the minimum mass is $\\sim 0.1M_J$. \n\nRing-shaped structures formed from the material of the remnant of the fallen clump and the protoplanetary disk are long-living structures and can exist for more than $3000$ years. Thus with a sufficiently large mass of the falling clump, the evolution of the disturbance can lead to the formation of a planet on an inclined orbit. It should be noted that Toomre parameter in the disks under consideration is equal to $\\sim 40$  at a distance of $20$ AU. Consequently, in a more massive disk, the fall of a clump several times denser than the material of the disk can trigger the process of gravitational collapse and the formation of a planet.\n\nThe fall of the clump near the star can cause not only a FUOR flare, but also a strong increase in circumstellar extinction, leading to a deep and prolonged weakening of the optical brightness of the star and an increase in its infrared radiation. Such events were observed in three young objects: V1184 Tau~\\citep{2009AstL...35..114G}, RW Aur~\\citep{2015IBVS.6143....1S,2019A&A...625A..49K} and AA Tau~\\citep{2021AJ....161...61C}. \n\nLet us make a rough estimate of the additional mass of circumstellar matter, which can be added to the mass of the disk during the lifetime of a star due to episodic falls of clumps. Suppose that the average age of stars with protoplanetary disks $10^6$ years~\\citep{2018ApJ...869L..42H} and the average lifetime of one disturbance $3 \\times 10^3$ years. This will give the probability of observing one event $P_1 \\sim 3\\times 10^{-3}$. The probability close to unity will be obtained when $\\sim 3\\times 10^3$ clumps fall during the disk lifetime ($\\sim 1$~Myr). If the clump mass of $0.1 M_J$ is necessary to create a strong disturbance (see above), the disk mass will increase during this time by $0.03 M_\\odot$ at an average accretion rate on the disk of $3\\times 10^{-8} M_\\odot yr^{-1}$. Such an increase in mass is not critical for a typical disk mass of $0.01-0.2 M_\\odot$ and an accretion rate onto a young star of $\\sim 10^{-7}-10^{-8} M_\\odot yr^{-1}$.\n\nThus, the mechanism of the clumpy accretion in protoplanetary disks can explain the formation of the main types of structures identified in images of protoplanetary disks. In addition, one clump falling at an angle onto a protoplanetary disk can produce multi-lane bright ring-shaped structures. So far, we have demonstrated here the fundamental possibility of obtaining the observed structures on protoplanetary disks in the model of the clumpy accretion. Appreciate the complexity of this process, it needs a more detailed consideration, taking into account the thermal regime in the perturbed region of the disk. \n\n{\\bf Acknowledgments.} It is a pleasure to thank the referee for valuable and useful remarks. Authors acknowledge the support of Ministry of Science and Higher Education of the Russian Federation under the grant 075-15-2020-780 (N13.1902.21.0039).\n\n\\software{Gadget-2~\\citep{2001NewA....6...79S,\n2005MNRAS.364.1105S}, RADMC-3D~\\cite{2012ascl.soft02015D}}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{I. Introduction}\n\nBose Einstein condensates (BECs) provide\n an excellent environment to study experimentally and theoretically\n a rich variety of macroscopic\nquantum phenomena. In a rotating single component condensate,\ntopological defects to the order parameter often manifest themselves as vortices that\ncorrespond to a zero of the order parameter with a circulation of\nthe phase. When they get numerous, these vortices  arrange\nthemselves on a triangular lattice.\nIn fact, vortices were first\nobserved in two-component BEC's \\cite{matthews}. Since then two-component BECs and the topological excitations within have been experimentally realised in a number of configurations: a single isotope that is in two different\nhyperfine spin states \\cite{hall,ander,myatt,delannoy,madd,matthews,sch}, two different isotopes of the same\natom \\cite{papp} or isotopes of two different atoms \\cite{modugno,thall,ferrari}.\n\nWhen a two-component\ncondensate is under rotation, topological defects of both order\nparameters are created which lead to more exotic defects such as\nsingly or multiply quantised skyrmions\n \\cite{wh,sch,mertes,cooper,savage,ruo}. Analogy with the Skyrme model\n  from particle physics is often invoked to represent the defects \\cite{manton,cho}. The\nsingly quantised skyrmions contain a vortex in one component which\nhas the effect of creating a corresponding density peak in the other\ncomponent. These singly quantised skyrmions are often referred to as\ncoreless vortices. Once numerous, these vortices and peaks arrange\nthemselves in (coreless vortex) lattices, that can be either\ntriangular or square. Other defects such as vortex sheets or giant\nskyrmions can also be observed \\cite{ktu,ktu1,ywzf}.  The aim of this\npaper is to classify the type of defects according to the different\nparameters of the problem restricting ourselves to two-dimensions. While this\npaper will only concern itself with magnetically trapped\ntwo-component BEC's, there is active research into\nspinor condensates (for a recent review see\n\\cite{uk}).\n\n\n   In the mean-field regime, the two-component Bose-Einstein condensate\nat zero temperature is described in terms of two  wave functions (order parameters),\n$\\Psi_1$ and $\\Psi_2$, respectively representing component-1 and\ncomponent-2.\n The two-component condensate is placed into rotation about the $z$-axis with $\\bm{\\bar{\\Omega}}=\\bar{\\Omega}\\bm{\\hat{z}}$ where $\\bar{\\Omega}$ is the rotation frequency assumed to apply equally to both components. The Gross-Pitaevskii (GP) energy functional of the rotating two-component, two-dimensional BEC is then given by\n\\begin{equation}\n\\begin{split}\n\\label{enfn}\nE[\\Psi_1,\\Psi_2]=\\int \\sum_{k=1,2}\\Bigg(\\frac{\\hbar^2}{2m_k}|\\nabla\\Psi_k|^2+V_k(\\bm{r})|\\Psi_k|^2\\\\\n-\\hbar\\bar{\\Omega}\\Psi^*_k L_z \\Psi_k +\\frac{U_k}{2}|\\Psi_k|^4\\Bigg)+U_{12}|\\Psi_1|^2|\\Psi_2|^2\\quad d^2r\n\\end{split}\n\\end{equation}\nwhere $r^2=x^2+y^2$, and  $V_k(\\bm{r})=m_k\\omega_k^2r^2\/2$ are the\nharmonic trapping potentials with trapping frequencies $\\omega_k$,\ncentered at the origin. Here $m_1$ and $m_2$ are the masses of the\nbosons in component-1 and component-2 respectively, and $\\hbar$ is\nPlanck's constant. The angular momentum is in the $z$-axis and is\ndefined as $L_z=i[\\bm{\\hat{z}}\\cdot(\\bm{r}\\times\\bm{p})]$ for linear\nmomentum $\\bm{p}$.\n    The energy functional (\\ref{enfn})\n contains three interaction constants:\n $U_k$ representing the internal interactions in  component $k$, and $U_{12}$\n  representing the interactions between the two components.\n\n\n\n    The time-dependent coupled Gross-Pitaevskii (GP) equations\n    are obtained from a variational procedure, $i\\hbar\\partial\\Psi_k\/\\partial t=\\delta E\/\\delta\\Psi^*_k$,\n     on the energy functional (\\ref{enfn}).\nLet $\\tilde{\\omega}=(\\omega_1+\\omega_2)\/2$ be the average of the\ntrapping frequencies of the two components and introduce the reduced mass $m_{12}$  such that\n$m_{12}^{-1}=m_1^{-1}+m_2^{-1}$. The GP energy and the coupled GP\nequations  can be non-dimensionalised by choosing\n$\\tilde{\\omega}^{-1}$, $\\hbar\\tilde{\\omega}$ and\n$r_0=\\sqrt{\\hbar\/(2m_{12}\\tilde{\\omega})}$ as units of time, energy\nand length respectively.  On defining the non-dimensional\nintracomponent coupling parameters $g_k=2U_km_{12}\/\\hbar^2$ and the\nintercomponent coupling parameter $g_{12}=2U_{12}m_{12}\/\\hbar^2$\n($\\equiv g_{21}$), the dimensionless coupled GP equations read\n    \\begin{equation}\n    \\begin{split}\n\\label{gp11}\ni\\frac{\\partial\\psi_k}{\\partial t}=&-\\frac{m_{12}}{m_k}\\left(\\nabla-i\\bm{A}_k\\right)^2\\psi_k+\\frac{m_k}{4m_{12}}\\left(\\frac{\\omega_k^2}{\\tilde{\\omega}^2}-\\Omega^2\\right)r^2\\psi_k\\\\\n&\\quad+g_k|\\psi_k|^2\\psi_k+g_{12}|\\psi_{3-k}|^2\\psi_k,\n\\end{split}\n\\end{equation}\nfor $\\Omega=\\bar{\\Omega}\/\\tilde{\\omega}$ and where\n\\begin{equation}\n    \\bm{A}_k=\\frac{1}{2}\\frac{m_k}{m_{12}}\\bm{\\Omega}\\times\\bm{r},\n\\end{equation}\nfor $\\bm{\\Omega}=(0,0,\\Omega)$ and $\\bm{r}=(x,y,0)$.  The ground\nstate of the energy (\\ref{enfn}) or the GP equations (\\ref{gp11}) is\ndetermined by preserving the normalisation condition which in this\npaper is either taken to be\n\\begin{equation}\n\\label{norm} \\int |\\psi_k|^2\\quad d^2r=N_k,\n\\end{equation}\nwhere $N_k$ is the total particle number of the $k$th-component, or\n\\begin{equation}\n\\int \\left (|\\psi_1|^2+|\\psi_2|^2\\right )\\quad d^2r=N_1+N_2.\n\\label{norm12}\n\\end{equation}\n\nThe following sections will only consider repulsive interactions so\nthat $g_1$, $g_2$ and $g_{12}$ are always non-negative.\nIn order to separate the regimes of interest, a non-dimensional\nparameter combining these $g_k$ and $g_{12}$ will appear:\n\\begin{equation}\\label{gamma12}\n    \\Gamma_{12}=1-\\frac{g_{12}^2}{g_1g_2}.\n\\end{equation}\nFurthermore to simplify matters, we introduce ratio parameters $\\eta$ and $\\xi$ such that\n\\begin{equation}m_1=\\eta m_2\\hbox{  and }\\omega_1=\\xi\\omega_2\\hbox{ with\n}\\{\\eta,\\xi\\}>0.\\end{equation} The effective trapping potentials\n for each component coming from equation (\\ref{gp11}) are then,\nrespectively,\n\\begin{subequations}\n\\begin{align}\n    V_1^{\\text{eff}}(r)=&(\\eta+1)\\left(\\frac{\\xi^2}{(\\xi+1)^2}-\\frac{1}{4}\\Omega^2\\right)r^2,\\\\\n    V_2^{\\text{eff}}(r)=&\\frac{(\\eta+1)}{\\eta}\\left(\\frac{1}{(\\xi+1)^2}-\\frac{1}{4}\\Omega^2\\right)r^2.\n\\end{align}\n\\end{subequations}\nIt follows that the limiting rotational frequency for each component\nis $\\Omega_1^{\\text{lim}}=2\\xi\/(\\xi+1)$ and\n$\\Omega_2^{\\text{lim}}=2\/(\\xi+1)$, so that it is necessary to\nconsider a rotational frequency  such that\n$0<\\Omega<\\Omega^{\\text{lim}}=\\min\\{{\\Omega_1^{\\text{lim}},\\Omega_2^{\\text{lim}}}\\}$.\n\n\n\n\n\n\n\n\n\nExperimentally, it is often the case that the $U_k$, $m_k$,\n$\\omega_k$ and $N_k$  are of the same order, so that much of the\ntheoretical and numerical analysis concerning two-component\ncondensates has assumed equality of these parameters.\n In the case where the\nintracomponent coupling strengths are equal, Kasamatsu et al.\n\\cite{ktu,ktu1} produced a numerical phase diagram in terms of the\nrotation and the intercomponent coupling strengths: they found\n phase separation regions with either vortex sheets\nor droplet behavior and regions of coexistence of the components\nwith coreless vortices, arranged in triangular or square lattices.\n\nIn this paper, we present a  classification of the ground states and\nvarious types of topological defect when the intracomponent coupling\nstrengths are distinct. Depending on the magnitude of the various\nparameters the components can either coexist, spatially separate\nor exhibit symmetry breaking. A richer phenomenology of topological\ndefects is then found.\n\nMuch of the analysis carried out to investigate the ground states and topological defects will use a\n a nonlinear sigma model.\n It has been introduced previously in the literature \\cite{ktu,ktu3} for $\\eta=\\xi=1$ but\nwe will generalise this to the cases $\\eta$ and $\\xi$ different from\n1. This  involves writing the total density\nas\n\\begin{equation}\\label{rhot}\n    \\rho_T=|\\psi_1|^2+\\eta|\\psi_2|^2.\n\\end{equation}\n  A normalised complex-valued spinor\n$\\bm{\\chi}=[{\\chi_1},{\\chi_2}]^T$ is introduced from which the\nwave functions are decomposed as $\\psi_1=\\sqrt{\\rho_T}\\chi_1$,\n$\\psi_2=\\sqrt{\\rho_T\/\\eta}\\chi_2$ where $|\\chi_1|^2+|\\chi_2|^2=1$.\n Then we define the spin density $\\bm{S}=\\bar{\\bm{\\chi}}\\bm{\\sigma}\\bm{\\chi}$ where\n$\\bm{\\sigma}$ are the Pauli matrices. This gives the components of $\\bm{S}$ as\n\\begin{subequations}\n    \\label{seqs}\n\\begin{align}\n    \\label{sx}\n    S_x=&\\chi^*_1\\chi_2+\\chi_2^*\\chi_1,\\\\\n    \\label{sy}\n    S_y=&-i(\\chi^*_1\\chi_2-\\chi_2^*\\chi_1),\\\\\n    S_z=&|\\chi_1|^2-|\\chi_2|^2,\n    \\label{sz}\n\\end{align}\n\\end{subequations}\nwith $|\\bm{S}|^2 = 1$ everywhere. We will write the GP energy (\\ref{enfn}) in terms of\n $\\rho_T$ and $\\bm{S}$ that will allow us to derive information on\n the ground state of the system.\n\nThe paper is organised as follows. The different regions of the\n$\\Omega-\\Gamma_{12}$ phase diagrams are outlined in Sect. II with a\ndetailed analysis of the range of ground states and topological\ndefects. Then the non-linear $\\sigma$-model is developed in Sect.\nIII to analyse the ground states in terms of the total density. Lastly\nSect. IV  takes the second normalisation condition (\\ref{norm12}) and presents an example $\\Omega-\\Gamma_{12}$ phase diagram.\n\n\\section*{II. The $\\Gamma_{12}-\\Omega$ Phase Diagram}\n\n\\subsection*{A. Numerical Parameter Sets}\n\n In this paper,\nthree different configurations are considered, two of which relate directly to experimental configurations.\n Firstly, we analyze a\n$^{87}$Rb-$^{87}$Rb mixture  with one isotope in spin state $|F=2$,\n$m_f=1\\rangle$ and the other in state $|1$, $-1\\rangle$. The complex\nrelative motions between these two isotope components of rubidium\nwere considered experimentally by Hall et al \\cite{hall}. Here, the\nmasses are equal ($\\eta=1$) and  the transverse trapping potentials\n are equal, $\\omega_1=\\omega_2$ ($\\xi=1$). The scattering lengths for each\ncomponent are $a_1=53.35\\AA$ and $a_2=56.65\\AA$.\n  The intracomponent coupling strengths\n   $U_k$ for a two-dimensional model are  defined as\n$U_k={\\sqrt{8\\pi}\\hbar^2a_k}\/[{m_ka_{zk}}]$ ($k=1,2$),\nand the intercomponent coupling strength is defined as\n$U_{12}={\\sqrt{2\\pi}\\hbar^2a_{12}}\/[{m_{12}\\tilde{a}_z}]$,\nwhere\n $m_{12}$ is the reduced mass, given by $m_{12}^{-1}=m_1^{-1}+m_2^{-1}$,\n$a_k$, $a_{12}$ are the s-wave\n(radial) scattering lengths,\n$a_{zk}$ is the characteristic size of the condensate in the\n  $z$ direction,\n \n\nand $\\tilde{a}_z=(a_{z1}+a_{z2})\/2$.\n In the experiment \\cite{hall}, the computations of approximate values of $a_{z1}$\n and $a_{z2}$ can be made using a Thomas-Fermi approximation in the $z$ direction\n  as explained in \\cite{PS}, chapter 17. This relies on the assumption\n  that $N_k a_k\/a_{zk}$ are large. Then, it is reasonable to describe the experiments  through a two-dimensional model.\n For our simulations, we choose $g_1=0.0078$ and\n$g_2=0.0083$, which are consistent with the experimental data and we set $N_1=N_2=10^5$.  We denote\nthis parameter set as `Experimental Set 1' (ES1).\n\nSecondly, we tackle a $^{41}$K-$^{87}$Rb mixture with both isotopes\nin spin state $|2$, $2\\rangle$, as was considered by Modugno et al\n\\cite{modugno}. Component-1 is identified with\nthe $^{41}$K isotope and component-2 with the $^{87}$Rb isotope. In the experiment,\n  the masses are different ($\\eta\\sim 0.48$),\n  but since $\\eta\\xi^2=1$, then $m_1 \\omega_1^2= m_2\\omega_2^2$\n   so that  both components experience the\nsame trapping potential.\nHere the scattering lengths for each component are $a_1=31.75\\AA$\nand $a_2=52.39\\AA$ and we  choose intracomponent coupling  strengths\nas $g_1=0.0067$ and $g_2=0.0063$. This set is denoted ES2.\n\n Lastly, in order to make an\nanalogy with previous theoretical studies, we consider the  set\n(ES3) which contains a mixture chosen such that all the parameter\ngroups are equal, i.e. here $g_1=g_2=0.0078$ and $N_1=N_2=10^5$ with\nequality of the $m_k$ and\n$\\omega_k$ ($\\eta=\\xi=1$).\n\nIn each case, the features of the ground states will be explained as\n$g_{12}$ and $\\Omega$ are varied. While the value of the $g_k$ are\nnominally fixed, the value of the intercomponent coupling strength,\n$g_{12}$, can be altered by Feshbach resonance (see for instance\n\\cite{tojo,theis,bauer,kempen}), which allows for an\nextensive experimental range in $\\Gamma_{12}$, given by (\\ref{gamma12}) (keeping $\\Gamma_{12}\\le1$). Simulations\nare thus performed on the coupled GP equations (\\ref{gp11}) in\nimaginary time. In general, it is difficult to find the true minimizing energy state. But the use of various initial data converging to the same final state allows us to state that the true ground state will be of the same pattern as the one we exhibit.\n\n We would like to note that while we have chosen these particular parameters, we have conducted extensive numerical simulations over a range of different parameters sets and find the sets presented here illustrate well the possible ground states.\n\n\\subsection*{B. Classification of the Regions of the Phase Diagram}\n\nThe ground states can be classified according to\n\\begin{enumerate}\n\\item the symmetry properties\n\\item the properties of coexistence of the condensates or spatial\nseparation.\n\\end{enumerate}\n\nWhen there is no rotation ($\\Omega=0$), as illustrated\n in Fig. \\ref{gs}, the geometry of the ground state can either be\n \\begin{itemize}\\item two disks (coexistence of the components and symmetry preserving state)\n \\item a disk and an annulus (symmetry preserving with  spatial\n separation of the components), which depends strongly on the fact that the $g_k$, $m_k$, $\\omega_k$ are\nnot equal\n \\item droplets (symmetry breaking and spatial separation).\n \\end{itemize}\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.5]{gs.eps}\n\\end{center}\n\\caption{Density (divided by $10^4$) plots along\n$y=0$ for component-1 (dashed lines), component-2 (dotted lines) and\nthe total density (solid lines) in which the components (a) coexist\nboth being disks ($\\Gamma_{12}=0.1$), (b) spatially separate to be a\ndisk and an annulus ($\\Gamma_{12}=0$) and (c) have symmetry broken\nto be two droplets ($\\Gamma_{12}=-0.3$). The parameters are\n$g_1=0.0078$, $g_2=0.0083$ and $N_1=N_2=10^5$ with $\\eta=\\xi=1$ (set ES1). The\nangular velocity of rotation is $\\Omega=0$. Distance is measured in units of $r_0$ and density in\nunits of $r_0^{-2}$.} \\label{gs}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.5]{05.eps}\n\\end{center}\n\\caption{(Color online) A series of density plots for\ncomponent-1 (left column) and component-2 (right column) in which\nthe components coexist and are both disks. The parameters are\n$g_1=0.0078$, $g_2=0.0083$ and $N_1=N_2=10^5$ and with $\\eta=\\xi=1$ (set ES1)\nand $\\Gamma_{12}=0.5$ (which gives $g_{12}=0.0057$). The angular\nvelocity of rotation is $\\Omega$ and it takes the values (a) $0.25$,\n(b) $0.5$, and (c) $0.75$. Vortices in one component have a\ncorresponding density peak in the other component (coreless\nvortices). In  (c) the coreless vortex lattice is square. At these\nparameters the components are in region 1 of the phase diagram of\nFig. \\ref{sch_2}. Distance is measured in units of $r_0$ and density in\nunits of $r_0^{-2}$.} \\label{05}\n\\end{figure}\n\n\n\n\nUnder rotation, the different ground states\ncan be classified according to the parameters  $\\Gamma_{12}$ and\n$\\Omega$. When the condensates are two disks,  or a disk and annulus, defects may break some symmetry of the system as $\\Omega$ is increased. But the wave function retains some non trivial rotational symmetry. We will not refer to this as symmetry breaking since the bulk\n condensate keeps some symmetry. We will use the terminology symmetry breaking when the bulk completely breaks the symmetry of the system and is a droplet or has vortex sheets.\n We find that there are four distinct regions, determined by the geometry of the ground state.\n\n{\\em Region 1}: Both components are disk shaped, with no spatial separation. Above some critical velocity\n$\\Omega$, coreless vortices appear:\n a vortex in one component which has the effect of\ncreating a corresponding density peak in the other component. They\narrange themselves either on a triangular or on a square lattice.\nFigure \\ref{05} provides a typical example of the density profiles.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.5]{03neg.eps}\n\\end{center}\n\\caption{(Color online) A series of density plots for\ncomponent-1 (left column) and component-2 (right column) in which\nthe components have spatially separated. The parameters are\n$g_1=0.0078$, $g_2=0.0083$ and $N_1=N_2=10^5$ and with $\\eta=\\xi=1$ (set ES1)\n and $\\Gamma_{12}=-0.3$ (which gives\n$g_{12}=0.0092$). The angular velocity of rotation is $\\Omega$ and\nit takes the values (a) $0.1$, (b) $0.5$, and (c) $0.9$. In (a) the\ncomponents are rotating droplets (region 4 of Fig. \\ref{sch_2}), in\n(b) the components have spatially separated (but keep some symmetry) and have isolated density peaks (region 3 of Fig.\n\\ref{sch_2}) and in (c) there are vortex sheets present (region 2 of\nFig. \\ref{sch_2}). Distance is measured in units of $r_0$ and density in\nunits of $r_0^{-2}$.} \\label{03neg}\n\\end{figure}\n\n{\\it Region 2}: Vortex sheets. Under the effect of strong rotation,\nthe components spatially separate and completely break the symmetry of the system. Nevertheless, they are approximately\n disk shaped with similar radii. Many vortices are nucleated that\narrange themselves into rows  that can have various patterns: either\nthey can be striped or bent and are often disconnected from each\nother. A vortex sheet in one component corresponds to a region of\nmacroscopic density in the other component. These features can be\nseen in the density plots of Fig. \\ref{03neg}(c).\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.5]{13neg.eps}\n\\end{center}\n\\caption{(Color online) A series of density plots for\ncomponent-1 (left column) and component-2\n (right column). The parameters are $g_1=0.0078$, $g_2=0.0083$ and $N_1=N_2=10^5$ and with\n  $\\eta=\\xi=1$ (set ES1) and $\\Gamma_{12}=-1.3$ (which gives $g_{12}=0.0122$). The angular velocity of rotation is $\\Omega$ and it takes the values (a) $0$ and (b) $0.9$. In (a) the components are rotating droplets (region 4 of Fig. \\ref{sch_2}) while in (b) the components have spatially separated (but keep some symmetry) and there are no isolated density peaks (region 3 of Fig. \\ref{sch_2}). Distance is measured in units of $r_0$ and density in\nunits of $r_0^{-2}$.}\n\\label{13neg}\n\\end{figure}\n\n{\\em Region 3}: Spatial separation preserving some symmetry. Here\none component is a disk while the other component is an annulus and the disk\n fits within the annulus with a\nboundary layer region in which both components have microscopically\nsmall density as illustrated in Figure \\ref{03neg}(b) and \\ref{13neg}(b). Under rotation, the topological defects can\neither be coreless vortex lattices  in the disk and (but not always)\ncorresponding isolated density peaks in the annulus, and\/or\n a giant skyrmion at the boundary interface between the two\ncomponents, as will be described later.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.5]{drop.eps}\n\\end{center}\n\\caption{(Color online) A density plot for component-1 (left column) and component-2 (right column) showing examples of vortex nucleation in rotating droplets. The parameters are $g_1=0.0078$, $g_2=0.0083$ and $N_1=N_2=10^5$ and with $\\eta=\\xi=1$ (set ES1) and $\\Gamma_{12}=-9$ and $\\Omega=0.5$. At these parameters the components are in region 4 of the phase diagram of Fig. \\ref{sch_2}. Distance is measured in units of $r_0$ and density in\nunits of $r_0^{-2}$.}\n\\label{drop}\n\\end{figure}\n\n{\\it Region 4}: Rotating droplets. The components spatially separate\nand have broken symmetry such that the centres of each condensate\nare different and each component contains a single patch of density\n as illustrated in Figures  \\ref{03neg}(a) and \\ref{13neg}(a). In the rotating droplets, vortices can appear provided the\nrotation is greater than some critical value. These features can be seen in the density plots\nof Fig. \\ref{drop}.\n\n\nThe $\\Omega-\\Gamma_{12}$ phase diagrams corresponding to the\nexperimental parameter sets introduced above (sets ES1, ES2 and ES3)\nare shown in Fig.'s \\ref{sch_2}, \\ref{sch_3}  and \\ref{sch_1}.  The\nlast one is of similar type as the one reported by Kasamatsu et al \\cite{ktu1}\n(there $g_1=g_2=4000$ and $N_1=N_2=1\/2$).\n\nNew features can be observed in Fig \\ref{sch_2} and \\ref{sch_3},\nsuch as isolated density peaks that eventually vanish as\n$\\Gamma_{12}$ is made more negative and the multiply quantised\nskyrmions at the interface between the disk component and the\nannular component in region 3. When $\\{\\eta,\\xi\\}\\neq1$ (Fig. \\ref{sch_3}), no region 2\nis found to exist.  This absence is easily explained by two factors:\nthe onset of region 3 for (large) positive values of $\\Gamma_{12}$\nand the lack of vortices nucleated in component-1 (and to some\nextent in component-2).\n\nWe will now analyze in more detail some of\nthe features of regions 1, 3 and 4 of the phase diagrams.  Vortex\nsheets (region 2) have been analysed (mainly in the case of equal $g_k$, $m_k$\nand $\\omega_k$) in \\cite{kt}.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.4]{sch_2a.eps}\\\\\n\\includegraphics[scale=0.4]{sch_2b.eps}\n\\end{center}\n\\caption{(Color online) $\\Omega-\\Gamma_{12}$ phase diagrams for parameters\n$g_1=0.0078$, $g_2=0.0083$, $N_1=N_2=10^5$ with $\\eta=\\xi=1$ (set ES1)\nwith normalisation taken over the individual components (Eq.\n(\\ref{norm})). (a) Numerical simulations where triangles (squares) indicate\nthat the vortex lattice in both components is triangular (square) and diamonds that no vortices\nhave been nucleated. Filled triangles,\nsquares and diamonds are where the two components are disk-shaped\nand coexist. The empty triangles and empty diamonds\nare where the two components have spatially separated: one component\nis a disk and the other an annulus with a giant skyrmion at the\nboundary layer; those triangles with a dot in the centre\nrepresent the appearance of coreless vortices inside the annulus.\nThe crosses `x' are where the two components\nhave broken symmetry and are vortex sheets and the crosses `+' are rotating droplets.\n(b) A schematic representation of the numerical simulations. The\nsolid lines indicate the boundary between different identified\nregions (determined by the geometry of the ground state)\nand the dashed lines the boundary between triangular and\nsquare lattices in region 1 and the boundary between peaks and no\npeaks in region 3. The boundary between region 1 and the others can\nbe calculated analytically by Eq. (\\ref{spatsep}) to give\n$\\Gamma_{12}=0.008$. The unit of rotation is $\\tilde{\\omega}$.}\\label{sch_2}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.4]{sch_3a.eps}\\\\\n\\includegraphics[scale=0.4]{sch_3b.eps}\n\\end{center}\n\\caption{(Color online) $\\Omega-\\Gamma_{12}$ phase diagrams for parameters $g_1=0.0067$, $g_2=0.0063$, $N_1=N_2=10^4$ with $\\eta=0.48$ and $\\eta\\xi^2=1$ (set ES2) with normalisation taken over the individual components (Eq. (\\ref{norm})). (a) Numerical simulations where triangles indicate that the vortex lattice in component-2 is triangular, circles that component-2 contains rings of vortices and diamonds where no vortices have been nucleated. Filled triangles, circles and diamonds are where the two components are disk-shaped and coexist. The empty circles and empty diamonds are where the two components have spatially separated: one component is a disk and the other an annulus with a giant skyrmion at the boundary layer; those circles with a dot in the centre represent the appearance of coreless vortices inside the annulus. The crosses `+' are where the two components have broken symmetry and are rotating droplets.\n(b) A schematic representation of the numerical simulations.\nThe solid lines indicate the boundary between different identified regions (determined by the geometry of the ground state).\nThe boundary between region 1 and region 3 is also calculated analytically by Eq. (\\ref{spatsep}) (dotted line).\nFor these parameters there is no region 2. The unit of rotation is $\\tilde{\\omega}$.}\\label{sch_3}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.4]{sch_1.eps}\n\\end{center}\n\\caption{A schematic $\\Omega-\\Gamma_{12}$ phase diagram for $g_1=g_2\\equiv g=0.0078$, $N_1=N_2=10^5$ where $\\eta=\\xi=1$ (set ES3) and with the normalisation taken over the individual components (Eq. (\\ref{norm})). The solid lines indicate the boundary between different identified regions (determined by the geometry of the ground state) and the dashed lines the boundary between triangular and square lattices. For these parameters there is no region 3. The unit of rotation is $\\tilde{\\omega}$.}\\label{sch_1}\n\\end{figure}\n\n\\subsection*{C. Analysis of the Features of the Phase Diagrams}\n\n\n\\subsubsection*{1. Square lattices}\n\nIt is a specific property of two components to stabilize square\nlattices. A requirement for the existence of square lattices is the\nnucleation of many vortices in both components, something not\npermitted in ES2 (Fig. \\ref{sch_3}), where only a small number of vortices\nare ever nucleated in component 1.\n\nSquare lattices generally occur at high rotational velocities and\nexamples have been observed experimentally \\cite{sch} and\nnumerically \\cite{wcbb,ktu,ktu1}. Mueller and Ho \\cite{mh} and later\nKe{\\c c}eli and \\\"Oktel \\cite{ko} have analysed the transition from\ntriangular to square lattices as $\\Gamma_{12}$ is varied in\ntwo-component condensates when $g_1\\sim g_2$ and $\\Omega$ is such that the condensate is in the lowest Landau\nlevel (LLL). Providing $\\Omega$ is large, according to \\cite{ko,mh}\nthe value of $\\alpha=\\sqrt{1-\\Gamma_{12}}$ determines whether the\nvortex lattices are triangular ($0<\\alpha<0.172$), in transition to\nbecoming square ($0.172<\\alpha<0.373$) or square ($\\alpha>0.373$).\nTherefore, assuming that, in the example parameters of $g_1=0.0078$,\n$g_2=0.0083$ and $\\eta=\\xi=1$ (set ES1), the $g_k$ are sufficiently close\nenough, the square lattice should first appear at $\\alpha=0.37$, or\nequivalently $\\Gamma_{12}=0.86$. Comparing this to the numerically\nobtained values, where the square lattice first appears for\n$0.8<\\Gamma_{12}<0.9$ in Fig. \\ref{sch_2}, the agreement seems good.\n It will be interesting to see how the\nanalysis of \\cite{ko,mh} has to be modified when $g_1$ becomes distant from $g_2$.\n\nNevertheless, the square lattices are present for lower rotational\nvelocities at which the LLL is not necessarily valid. In a\nforthcoming paper, using the nonlinear sigma model\npresented below, we hope to derive a vortex energy\n in terms\nof the positions of vortices. The ground state of this energy indeed\nleads to square lattices in some range of parameters (see also Section III.E). It turns out\nthat this vortex energy is similar to that of Barnett et al\n\\cite{barnett,barnett2}.\n\n\n\n\n\n\\subsubsection*{2. Rotating droplets}\n\n\n\nIn the parameter range of region 4, symmetry breaking  with spatial\nseparation occurs. When the condensate is not under rotation, the\nground states of the two components will be `half ball'-like\nstructures, as can be seen in Fig. \\ref{13neg}(a) (if $g_1=g_2$ and\n$\\eta=\\xi=1$, then the two components are exactly half-balls). The\ndifference between the intercomponent strengths introduces a\ncurvature to the inner boundary of each component with the component\nthat corresponds to larger $g_k$ having positive curvature and the\nother component having negative curvature. This curvature depends on\nboth $\\Gamma_{12}$ and $\\Omega$. If $\\Gamma_{12}$ is held constant,\nthen the curvature increases as $\\Omega$ increases. When\n$\\Gamma_{12}$ goes to $-\\infty$, the droplets approach half-balls.\n\n\n\nVortex nucleation is also seen in region 4; see Fig. \\ref{drop}. In this figure there are four vortices in component-1 and three vortices in component-2. The number of vortices in each component will increase as $\\Omega$ is increased but this increase will also increase the curvature of the inner boundaries of the components, thus preventing the vortices aligning themselves into vortex sheets.\nExamples of the rotating droplet ground states can be seen in Fig. \\ref{13neg}(a), and further examples can be found in  \\cite{kyt,ohberg,ohberg1,navarro,chris}.\n\n\n\n\\subsubsection*{3. Spatial separation preserving some symmetry: disk plus annulus}\n\n\n\n\n\n Region 3 is defined by the $\\Gamma_{12}$ and $\\Omega$ in which the\ncondensates have spatially separated components but still possess some symmetry about the origin: the disk component-1 sits securely within the annular\ncomponent-2 (see Fig.'s \\ref{03neg}(b) and \\ref{13neg}(b)).\nThere is a boundary layer evidenced where the outer edge\nof the disk overlaps the inner edge of the annulus.\n\n Let us describe the onset of region 3 from region 4, captured in Fig. \\ref{03neg} (sub-image (a) to (b) or (c) to (b)). For\n a particular $\\Gamma_{12}$,  as $\\Omega$ is increased, then the curvature increases  to such an extent  that the components develop constant non-zero curvature, and are identified as a disk and an annulus.\nConversely, if $\\Omega$ is held constant and $\\Gamma_{12}$ is pushed to more negative values, then the curvature decreases.\n\nUnder rotation, defects can be observed: coreless vortices and giant\nskyrmions. Coreless vortices sit inside the disk while giant\n skyrmions are observed in the boundary layer.\n\n \\subsubsection*{4. Coreless Vortices in the disk plus annulus case}\n\n In the geometry of disk plus annulus, the vortices in the inside disk have corresponding isolated\ndensity peaks  inside the annulus, hence in the microscopic density\nregions, as illustrated in Fig. \\ref{03neg}(b). Thus any vortex\nlattice structure produced in component-1 is replicated by a peak\nlattice structure in component-2, i.e. there is the continuing\npresence of coreless vortex lattices in the spatially separated\ncondensates \\cite{note2}.\n\n As the number of\nvortices in the disk  increases with increasing angular\nvelocity, the number of density peaks inside the central hole of the\nannulus likewise increases. This has the effect that the central hole\nof the annulus can be masked at high angular velocities by the\nmultiple occurrence of the density peaks.  A recent analytical\nunderstanding of the interaction between vortices and peaks\n has been provided by \\cite{kasa}. It may be  extended to the case of the disk and annulus, for which the average density in the central hole is very small for one component.\n  \nPushing $\\Gamma_{12}$ to lower values has the effect of reducing the\nsize of the boundary layer between the disk and annular components,\nbut also the isolated density peaks disappear; see Fig.\n\\ref{13neg}(b). A recent work has analysed, from an energy perspective, the preference for the ground state to contain or not contain density peaks \\cite{cate}.\n\n\n Figure \\ref{peaks} plots the maximum peak density of\nthe density peaks in component-2 as a function of $\\Gamma_{12}$ for\nthe parameters of Fig. \\ref{sch_2} and with $\\Omega=0.5$ and\n$\\Omega=0.75$.  The disjoint region when $\\Omega=0.75$ for\n$-0.4<\\Gamma_{12}<0.1$ on Fig. \\ref{peaks} is due to the absence of\ndensity peaks in component-2 as a result of the appearance of vortex\nsheets (region 4). We see from Fig. \\ref{peaks},\n that the maximum peak density  occurs when\n$\\Gamma_{12}\\sim0.01$ which is the value at which the components\nbegin to spatially separate (the transition between region 1 and\nregion 3) in set ES1.  For $\\Gamma_{12}$ and $\\Omega$ that take\nvalues outside of region 1, the maximum density of the peaks\ndecreases linearly as $\\Gamma_{12}$ increases. The maximum density approaches microscopically small values for\n$\\Gamma_{12}\\sim-1.1$ when $\\Omega=0.5$ and for $\\Gamma_{12}\\sim-1$\nwhen $\\Omega=0.75$. An example of the transition can be seen in Fig.'s\n\\ref{03neg}(b) and \\ref{13neg}(b).\n\n\\begin{figure}\\begin{center}\n\\includegraphics[scale=0.5]{peaks.eps}\\end{center}\n\\caption{Plots of maximum peak density in component-2 against\n$\\Gamma_{12}$ for $\\Omega=0.5$ (dashed line) and $\\Omega=0.75$\n(solid line) when $g_1=0.0078$, $g_2=0.0083$, $N_1=N_2=10^5$ and\nwith $\\eta=\\xi=1$ (set ES1) . The disjoint region when $\\Omega=0.75$\nfor $-0.4<\\Gamma_{12}<0.1$ is due to the absence of isolated density\npeaks in component-2 as a result of the appearance of vortex sheets.\nDensity is measured in units of\n$r_0^{-2}$.}\\label{peaks}\\end{figure}\n\n\n\n\\subsubsection*{5. Giant skyrmion}\n\nA boundary layer between the overlap of component-1 and component-2\n is present for all values of $\\Gamma_{12}$ in region 3 but reduces\nin width as $\\Gamma_{12}$ becomes more negative (indeed the boundary\nlayer disappears only as $\\Gamma_{12}\\rightarrow-\\infty$). There are\nadditional topological defects that occur in the boundary layer that\nare not discernible on a traditional density plot. These\ntopological defects' presence can be observed in a phase profile,\nhowever a better visualisation is to use the pseudospin\nrepresentation (\\ref{rhot})-(\\ref{seqs}).\n    One can plot the functions $S_x$, $S_y$ and\/or $S_z$\n    which reveal the presence of all the topological defects -\n     the coreless vortices (singly quantised skyrmions) that have already been visualised on the plots,\n      and the new defect, a multiply quantised skyrmion.\n\nThe distinct nature of the two types of topological defect can be\nclearly seen in Fig. \\ref{topdef}(a,b) where an $S_x$ plot and an\n$(S_x,S_y)$ vectorial plot are shown  respectively. A density plot of each component for these same parameters is seen in\nFig. \\ref{03neg}(b). The coreless vortices evident in Fig.\n\\ref{03neg}(b) are again clearly evident in Fig. \\ref{topdef}(a) and\n(b). A blow-up of the region close to one of (the three) coreless\nvortices in the $(S_x,S_y)$ vectorial plot, centred at $(1,-1)$ and\nexhibiting circular disgyration is shown in Fig. \\ref{topdef}(c).\nThe texture of $\\bm{S}$ can also exhibit cross- and radial-\ndisgyration \\cite{ohmi}. Conversely, the multiply quantised\nskyrmion, not present in the density plots of Fig.\n\\ref{03neg}(b), can clearly be seen in both Fig. \\ref{topdef}(a) and\n(b). The multiply quantised skyrmion forms a ring along the boundary\nlayer. A blow-up around $(0,-3.5)$ for the $(S_x,S_y)$ vectorial\nplot again shows the multiply quantised nature of this defect. A\nmultiply quantised skyrmion was evidenced by \\cite{ywzf} who termed\nit a giant skyrmion, a terminology retained in this paper.\n Increasing the rotation results in an increase in both the number of\ncoreless vortices and in the multiplicity of the giant skyrmion.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.5]{topdef.eps}\n\\end{center}\n\\caption{(Color online) Plots of (a) $S_x$ and (b) a vectorial plot of $(S_x,S_y)$ for $\\Gamma_{12}=-0.3$ and $\\Omega=0.5$ when $g_1=0.0078$, $g_2=0.0083$, $N_1=N_2=10^5$ and with $\\eta=\\xi=1$ (set ES1). The ring of skyrmions, at $r=r_s$, traces the boundary layer and the coreless vortices exist for $r<r_s$. Blow-ups (c) and (d) of (b) highlight the nature of the two types of topological defect; (c) a coreless vortex at $(1,-1)$ and (d) a section of the multiply quantised skyrmion. The respective density plots for this case are shown in Fig. \\ref{03neg}(b). Distance is measured in units of $r_0$.}\n\\label{topdef}\n\\end{figure}\nRef \\cite{ywzf} gave a relationship, $m=n+1$, between the\nmultiplicity of the giant skyrmion $n$ and the total circulation $m$\nin the central hole of component-2. However the numerical\nsimulations here suggest that a more appropriate correlation is\nprovided by $m=n+l$ where $l$ is the number of isolated coreless\nvortices. As an example, consider Fig. \\ref{topdef} where\n$\\Omega=0.5$ and $n=8$, $l=3$ and $m=11$, satisfying $m=n+l$. It\nshould be noted that the relationship $m=n+l$ still holds as the\nisolated coreless density peaks disappear from the density\nprofile of component-2; this is because the phase imprints due to the vortices in component-1,\nwhich\nconstitute $l$, are still present.\n\n\\section*{III. A Non-Linear $\\sigma$-Model}\n\\label{tfprof}\n\nThe ground states of the rotating two-component\ncondensate can be recovered from a `non-linear $\\sigma$-model'\nformulation of the energy functional in terms of the total density\n$\\rho_T$ and spin vector $\\bf S$ (Eq.'s\n(\\ref{rhot})-(\\ref{seqs})). We will analyze in detail the possible regimes in this section and show\nhow a Thomas Fermi approximation can hold in the case $\\Gamma_{12}>0$\n or can be generalized and provide relevant information in the case $\\Gamma_{12}<0$.\n\n\\subsection*{A. Energy Functional Representation}\n\nWe write the energy functional of the two wave functions  (Eq.\n(\\ref{enfn}), non-dimensionalised) as\n$E[\\psi_1,\\psi_2]=E_{\\text{KE}}+E_{\\text{PE}}+E_{\\text{I}}$ where\n\\begin{subequations}\n    \\label{energyps}\n    \\begin{align}\n   \\begin{split} E_{\\text{KE}}&=\\frac{1}{\\eta+1}\\int\\left|\\left(\\nabla-\\frac{i}{2}(\\eta+1)\\bm{\\Omega}\\times\\bm{r}\\right)\\psi_1\\right|^2\\\\\n    &\\quad+\\eta\\left|\\left(\\nabla-\\frac{i}{2\\eta}(\\eta+1)\\bm{\\Omega}\\times\\bm{r}\\right)\\psi_2\\right|^2\\quad d^2r,\n\\end{split}\\\\\n    E_{\\text{PE}}&=\\int2j_1r^2|\\psi_1|^2+2j_2r^2|\\psi_2|^2\\quad d^2r,\\\\\n    E_{\\text{I}}&=\\int\\frac{1}{2}g_1|\\psi_1|^4+\\frac{1}{2}g_2|\\psi_2|^4+g_{12}|\\psi_1|^2|\\psi_2|^2\\quad d^2r.\n\\end{align}\n\\end{subequations}\nwith\n\\begin{subequations}\n\\begin{align}\n    j_1=&\\frac{1}{2}\\frac{(1+\\eta)\\xi^2}{(1+\\xi)^2}-\\frac{1}{8}(1+\\eta)\\Omega^2,\\\\                  j_2=&\\frac{1}{2}\\frac{(1+\\eta)}{\\eta(1+\\xi)^2}-\\frac{1}{8\\eta}(1+\\eta)\\Omega^2.\n\\end{align}\n\\end{subequations}\nIt is assumed that the $g_k$, $m_k$, $\\omega_k$ and $N_k$ are distinct\nso that a weighted total density can be defined as (\\ref{rhot}) with\n $ \\psi_1=\\sqrt{\\rho_T}\\chi_1$ and $\n\\psi_2=\\sqrt{\\rho_T\/\\eta}\\chi_2$. The spin density vector $\\bm{S}$,\nwhich satisfies $|\\bm{S}|^2=1$ everywhere, has components given by\nEq.'s (\\ref{seqs}). We then have\n\\begin{equation}\n    |\\psi_1|^2=\\frac{1}{2}\\rho_T(1+S_z),\\quad|\\psi_2|^2=\\frac{1}{2\\eta}\\rho_T(1-S_z).\n\\end{equation}\nWe introduce the phases $\\theta_1$ and $\\theta_2$ defined by $\\chi_1=|\\chi_1|\\exp(i\\theta_1)$ and $\\chi_2=|\\chi_2|\\exp(i\\theta_2)$.\n The energy functional is\nexpressed in terms of 4 variables: the total density $\\rho_T$, the component $S_z$\n and the angles $\\theta_1$, $\\theta_2$. We see that $E_{\\text{KE}}= E_{\\rho_T}+E_{\\text{Sz}}+E_{\\theta_1,\\theta_2}$ where\n\\begin{subequations}\n    \\begin{align}\n        E_{\\rho_T}=&\\int\\frac{1}{(\\eta+1)}(\\nabla\\sqrt{\\rho_T})^2\\quad d^2r,\\\\\n        E_{\\text{Sz}}=&\\int \\frac{1}{4}\\frac{\\rho_T}{(\\eta+1)}\\frac{(\\nabla{{S_z}})^2}{(1-S_z^2)}\\quad d^2r,\\\\\n        E_{\\theta_1,\\theta_2}=&\\int \\frac{1}{2}\\frac{\\rho_T}{(1+\\eta)}\\Bigg[(1+S_z)\\left(\\nabla\\theta_1-\\frac{1}{2}(1+\\eta)\\bm{\\Omega}\\times\\bm{r}\\right)^2\\nonumber\\\\\n        &\\quad+(1-S_z)\\left(\\nabla\\theta_2-\\frac{1}{2\\eta}(1+\\eta)\\bm{\\Omega}\\times\\bm{r}\\right)^2\\Bigg]\\quad d^2r.\n    \\end{align}\n\\end{subequations}\nThe other terms of the energy straightforwardly become\n\\begin{subequations}\n    \\begin{eqnarray} E_{\\text{PE}}=&&\\int\\left[(j_1+j_2\/\\eta)+(j_1-j_2\/\\eta)S_z\\right]r^2\\rho_T\\quad d^2r,\\\\     E_{\\text{I}}=&&\\int\\frac{\\rho_T^2}{2}\\left(\\bar{c}_0+\\bar{c}_1S_z+\\bar{c}_2S_z^2\\right)\\quad d^2r,\n    \\end{eqnarray}\n\\end{subequations}\nwith\n\\begin{subequations}\n\\begin{align}\n\\label{c00}\n\\bar{c}_0=&\\frac{1}{4\\eta^2}(\\eta^2g_1+g_2+2\\eta g_{12}),\\\\\n    \\bar{c}_1=&\\frac{1}{2\\eta^2}(\\eta^2g_1-g_2),\\\\\n\\label{c22}\n    \\bar{c}_2=&\\frac{1}{4\\eta^2}(\\eta^2g_1+g_2-2\\eta g_{12}).\n\\end{align}\n\\end{subequations}\n\nThus the complete energy is\n\\begin{equation}\n    \\label{sig}\n\\begin{split}\nE&=\\int \\frac{1}{(\\eta+1)}(\\nabla\\sqrt{\\rho_T})^2+\\frac{\\rho_T}{4(\\eta+1)}\\frac{(\\nabla{{S_z}})^2}{(1-S_z^2)}\\\\\n&+\\frac{\\rho_T}{2(1+\\eta)}\\times\\\\\n&\\Bigg[(1+S_z)\\left(\\nabla\\theta_1-\\frac{1}{2}(1+\\eta)\\bm{\\Omega}\\times\\bm{r}\\right)^2\\\\\n&+(1-S_z)\\left(\\nabla\\theta_2-\\frac{1}{2\\eta}(1+\\eta)\\bm{\\Omega}\\times\\bm{r}\\right)^2\\Bigg]\\\\\n&+\\left[(j_1+j_2\/\\eta)+(j_1-j_2\/\\eta)S_z\\right]r^2\\rho_T\\\\\n&+\\frac{\\rho_T^2}{2}\\left(\\bar{c}_0+\\bar{c}_1S_z+\\bar{c}_2S_z^2\\right)\\quad\nd^2r.\n\\end{split}\n\\end{equation}\nEnergy (\\ref{sig}) is subject to the constraints (\\ref{norm}) that can be rewritten\nas \\begin{subequations}\n    \\label{nn2}\n\\begin{align}\n    \\label{nn2a}\n    \\int\\rho_T(1-S_z)\/2\\quad d^2r=&N_2\\eta,\\\\\n    \\int\\rho_T(1+S_z)\/2\\quad d^2r=&N_1.\n    \\label{nn2b}\n\\end{align}\n\\end{subequations}or equivalently\n\\begin{subequations}\n    \\label{nn1}\n\\begin{align}\n    \\label{nn1a}\n    \\int\\rho_T\\quad d^2r=&N_1+N_2\\eta,\\\\\n    \\int\\rho_TS_z\\quad d^2r=&N_1-N_2\\eta,\n    \\label{nn1b}\n\\end{align}\n\\end{subequations}\nThe minimization of the energy under the constraints (\\ref{nn1})\nyields two coupled equations with two Lagrange multipliers, $\\mu$ and $\\lambda$. We will\nwrite them\n when the phase is constant (the gradient of the phases\n $\\theta_1$ and $\\theta_2$ can be ignored) and the effect of rotation is contained in an effective centrifugal dilation of the trapping potential included in $j_1,j_2$:\n\\begin{equation}\n    \\begin{split}\n    \\label{muu1}\n    \\mu+\\lambda S_z=&-\\frac 1{(1+\\eta)}\\frac{\\Delta \\sqrt{\\rho_T}}{\\sqrt{\\rho_T}}\n    +\\frac 1{4(1+\\eta)}\\frac{(\\nabla S_z)^2}{(1-S_z^2)}\\\\&\\quad+\\left[(j_1+j_2\/\\eta)+(j_1-j_2\/\\eta)S_z\\right]r^2\\\\\n&\\qquad+\\rho_T(\\bar{c}_0+\\bar{c}_1S_z+\\bar{c}_2S_z^2),\n\\end{split}\n\\end{equation}\nand\n\\begin{equation}\n    \\begin{split}\n    \\label{szeq1}\n    \\lambda=&-\\frac 1{4(1+\\eta)}\\frac{\\Delta S_z+\\nabla \\rho_T \\cdot\\nabla S_z}{(1-S_z^2)}+\\frac 1{2(1+\\eta)}\\frac{S_z(\\nabla S_z)^2}{(1-S_z^2)}\\\\\n&\\quad +(j_1-j_2\/\\eta)r^2+\\left({\\bar{c}_1}+2\\bar{c}_2S_z\\right)\\rho_T\/2.\n\\end{split}\n\\end{equation}\n\nAs pointed out in \\cite{cooper}, in the general case where $\\eta$ and $\\xi$ are\n different from 1, we have seen from the expression $E_{\\theta_1,\\theta_2}$ that the effective velocity in each component is different. Nevertheless, in the case when $\\eta=\\xi=1$, it is possible to define an effective velocity shared by both components,\n\\begin{equation}\n    v_{\\text{eff}}=\\frac{\\nabla\\Theta}{2}+\\frac{\\bm{R}S_z}{2(1-S_z^2)}\n\\end{equation} where $\\Theta=\\theta_1+\\theta_2$ and $\\bm{R}=S_y\\nabla S_x-S_x\\nabla\nS_y$. We note the identity\n\\begin{equation}\n    \\frac{(\\nabla S_z)^2}{(1-S_z^2)}=(\\nabla \\bm{S})^2-\\frac{R^2}{(1-S_z^2)},\n\\end{equation} where $(\\nabla \\bm{S})^2=(\\nabla S_x)^2+(\\nabla S_y)^2+(\\nabla S_z)^2$.\nExpansion of the square in $E_{\\theta_1,\\theta_2}$,\n substituting in the $v_{\\text{eff}}$ and using the identity\nfrom above reduces the energy to the simple form found in \\cite{ktu,ktu3}.\n\\begin{equation}\n\\begin{split}\n\\label{sig2}\n    E=&\\int\\frac{1}{2}(\\nabla\\sqrt{\\rho_T})^2+\\frac{\\rho_T}{8}(\\nabla\\bm{S})^2\\\\\n\n&\\quad+\\frac{\\rho_T}{2}\\left(v_{\\text{eff}}-\\bm{\\Omega}\\times\\bm{r}\\right)^2+\\frac{1}{2}r^2(1-\\Omega^2)\\rho_T\\\\\n&\\qquad+\\frac{\\rho_T^2}{2}\\left(c_0+c_1S_z+c_2S_z^2\\right)\\quad d^2r,\n\\end{split}\n\\end{equation}\nwhere ${c}_0$, ${c}_1$ and ${c}_2$ are equal to $\\bar{c}_0$, $\\bar{c}_1$ and $\\bar{c}_2$ with $\\eta$ set equal to unity.\n\n\\subsection*{B. A Thomas-Fermi approximation}\n\nThe typical Thomas-Fermi (TF) approximation to Eq.'s (\\ref{muu1})-(\\ref{szeq1}) is\nto assume that derivatives in $\\rho_T$ and $S_z$ are negligible in\nfront of the other terms. If we apply the TF approximation we then get\n\\begin{equation}\n    \\begin{split}\n    \\label{muu}\n    \\mu+\\lambda S_z=&\\left[(j_1+j_2\/\\eta)+(j_1-j_2\/\\eta)S_z\\right]r^2\\\\\n&\\quad+\\rho_T(\\bar{c}_0+\\bar{c}_1S_z+\\bar{c}_2S_z^2),\n\\end{split}\n\\end{equation}\nand\n\\begin{equation}\n    \\label{szeq}\n    \\lambda=(j_1-j_2\/\\eta)r^2+\\frac{1}{2}\\left({\\bar{c}_1}+2\\bar{c}_2S_z\\right)\\rho_T.\n\\end{equation}\nThe TF energy is then\n \\begin{equation}\n    \\label{sigTF}\n\\begin{split}\nE_{\\text{TF}}=&\\int \\left[(j_1+j_2\/\\eta)+(j_1-j_2\/\\eta)S_z\\right]r^2\\rho_T\\\\\n&\\quad+\\frac{\\rho_T^2}{2}\\left(\\bar{c}_0+\\bar{c}_1S_z+\\bar{c}_2S_z^2\\right)-\\mu\\rho_T\n-\\lambda \\rho_T S_z\\quad d^2r.\n\\end{split}\n\\end{equation}It is important to point out that the reduction of this quadratic form in $\\rho_T$ and\n$\\rho_T S_z$ yields\n\\begin{equation}\n    \\label{sigTFred}\n\\begin{split}\nE_{\\text{TF}}=&\\int \\frac{\\bar c_2}2 \\left(\\rho_TS_z+\\frac{\\bar\nc_1}{2\\bar c_2}\\rho_T+\n\\frac 1{\\bar c_2}(j_1-j_2\/\\eta)r^2-\\frac{\\lambda}{\\bar c_2}\\right )^2\\\\\n&\\quad+\\frac 12 \\left(\\bar c_0-\\frac {\\bar c_1^2}{4\\bar c_2}\\right)\\times\\\\\n&\\qquad \\left[ \\rho_T-\\frac 1 {\\left(\\bar c_0-\\bar c_1^2\/4\\bar\nc_2\\right)}(\\mu -(j_1+j_2\/\\eta)r^2) \\right .\\\\\n &\\quad\\quad\\left . +\\frac 12\\frac{\\bar c_1}{(\\bar c_0\\bar c_2- \\bar c_1^2\/4)}(\\lambda\n-(j_1+j_2\/\\eta)r^2)\\right ]^2\\ d^2r\\\\\n&\\qquad\\quad+\\text{constant terms}.\n\\end{split}\n\\end{equation} The existence of a minimum for this quadratic form\nimplies that $\\bar c_2 \\geq 0$ and $\\bar c_0- \\bar c_1^2\/(4\\bar c_2)\n\\geq 0$. Since $\\bar c_0- \\bar c_1^2\/(4\\bar c_2)\n=g_1g_2\\Gamma_{12}\/(4\\eta^2\\bar c_2)$, this implies in particular\nthat $\\Gamma_{12}>0$.\n\nTherefore, if $\\Gamma_{12}>0$, (which implies $\\bar c_2>0$), a\nThomas Fermi approximation can be performed as has previously been considered,\n generally taking an  approximation on the individual component\nwave functions $\\psi_1$ and $\\psi_2$\n\\cite{ohberg1,tripp,corro,hs,kyt,pb,riboli,jc}. Nevertheless, as we\nwill see below, if $\\bar c_2<0$ and $\\Gamma_{12}<0$, then $(\\bar c_0-\\frac {\\bar c_1^2}{4\\bar c_2})>0$\n and a TF\napproximation can still be performed on $\\rho_T$, provided we keep\ngradient terms in $S_z$.\n\n\n Multiplying (\\ref{szeq}) by $S_z$ and subtracting (\\ref{muu})\n leads to\n \\begin{equation}\n    \\mu=(j_1+j_2\/\\eta)r^2+\\rho_T\\left(\\frac{\\bar{c}_1}2S_z+\\bar{c}_0\\right).\n\\end{equation}\n Simplification by $\\rho_T S_z$ with (\\ref{szeq}) yields, when\n $\\psi_1\\times\\psi_2\\neq0$,\n  \\begin{equation}\n    \\label{rhoty}\n\\rho_T=\\frac{a_3+a_4r^2}{g_1g_2\\Gamma_{12}},\n\\end{equation}\nand\n\\begin{equation}\n    \\label{szform}\nS_z=\\frac{a_1+a_2r^2}{a_3+a_4r^2},\n\\end{equation}\nwhere\n\\begin{subequations}\n    \\label{as}\n\\begin{align}\n    &a_1=4\\eta^2(\\lambda\\bar{c}_0-\\mu\\bar{c}_1\/2),\\\\\n&a_2=\\eta g_1h_2-g_2h_1,\\\\\n&a_3=4\\eta^2(\\mu\\bar{c}_2-\\lambda\\bar{c}_1\/2),\\\\\n&a_4=-(\\eta g_1h_2+g_2h_1),\n\\end{align}\n\\end{subequations}\nand where we define\n\\begin{equation}\n    h_k=2\\left(j_k-\\frac{g_{12}}{g_{3-k}}j_{3-k}\\right).\n\\end{equation}\nWhen only one component is present ($\\psi_1\\times\\psi_2=0$), this\nsimplifies to\n\\begin{equation}\\label{rhoonly}\n    \\rho_T=\n\\begin{cases}\n\\frac{1}{g_1}\\left[\\mu+\\lambda-2j_1 r^2\\right]&\\hbox{ if } \\psi_{2}=0\\\\\n\\frac{\\eta}{g_2}\\left[\\eta(\\mu-\\lambda)-2j_2r^2\\right]&\\hbox{ if } \\psi_1=0\n\\end{cases}\n\\end{equation}\nsince $S_z=+1$ when $\\psi_2=0$ and $S_z=-1$ when $\\psi_1=0$.\n\n\n\nTo begin, we note from Eq. (\\ref{szform}) that\n$r^2=(a_1-a_3S_z)\/(a_4S_z-a_2)$, so that\n\\begin{equation}\\label{rhoSz}\n    \\rho_T=\\frac{a_1a_4-a_2a_3}{g_1g_2\\Gamma_{12}(a_4S_z-a_2)}.\n\\end{equation}\nSince the intracomponent coupling strengths are chosen to be\ndistinct, the support of each component will not necessarily be\nequal. In order to\n lead the computations, we have to assume a geometry for the\n components: two disks, a disk and annulus or two half-balls (droplets)\n and a sign for $\\Gamma_{12}$.\n\n\\subsection*{C. $\\Gamma_{12}>0$}\n\n\\subsubsection*{1. Both components are disks}\n\nWe start when both components are disks.  Without loss of\ngenerality, we can assume that the outer boundary of component-2 (at\n$r=R_2$) is larger than that of component-1 (at $r=R_1$). Therefore,\n$S_z=-1$ and (\\ref{rhoonly}) holds in the annulus, while (\\ref{rhoSz}) holds in the coexisting region, which is the inside disk. The integrals from Eq.'s (\\ref{nn2}) then\ngive\n\\begin{subequations}\n    \\begin{equation}\n    \\begin{split}\n            \\label{szp1}\n    \\frac{{N}_1}{2\\pi}=&\\int_0^{R_1}\\rho_T\\frac{(1+ S_z)}{2}\\quad rdr\\\\\n    =&-\\frac{(a_1a_4-a_2a_3)^2}{4g_1g_2\\Gamma_{12}}\\int_{s_0}^{-1}\\frac{(1+ S_z)}{(a_4S_z-a_2)^3}\\quad dS_z,\n\\end{split}\n\\end{equation}\nwhere $s_0=S_z|_{r=0}=a_1\/a_3$ and\n    \\begin{equation}\n    \\begin{split}\n        \\label{szp1b}\n    \\frac{\\eta{N}_2}{2\\pi}=&\\int_0^{R_1}\\rho_T\\frac{(1- S_z)}{2}\\quad rdr+\\int_{R_1}^{R_2}\\rho_T\\quad rdr\\\\\n    =&-\\frac{(a_1a_4-a_2a_3)^2}{4g_1g_2\\Gamma_{12}}\\int_{s_0}^{-1}\\frac{(1- S_z)}{(a_4S_z-a_2)^3}\\quad dS_z\\\\\n    &\\quad-\\frac{g_2}{4\\eta j_2}\\int_{\\rho_d}^0\\rho_T\\quad d\\rho_T,\n\\end{split}\n\\end{equation}\n\\end{subequations}\nwith\n    \\begin{equation}\n    \\begin{split}\n    \\rho_d=&\\frac{a_3+a_4r^2}{g_1g_2\\Gamma_{12}}\\Big|_{r=R_1}\\\\\n    =&\\frac{a_2a_3-a_1a_4}{g_1g_2\\Gamma_{12}(a_2+a_4)},\n\\end{split}\n\\end{equation}\nsince $S_z=-1$ at $r=R_1$.\n\nCompletion of these integrals and noting that $\\{a_2+a_4,a_2-a_4\\}=\\{-2g_2h_1,2\\eta g_1h_2\\}$ gives\n\\begin{subequations}\n    \\label{all}\n\\begin{equation}\n    \\label{szp2}\n    a_3^2=\n\\frac{8N_1g_1g_2^2\\Gamma_{12}h_1}{\\pi(s_0+1)^2}\\\\%\\qquad &k=1\\\\\n\\end{equation}\nwhich must necessarily be positive, i.e. $h_1\\Gamma_{12}>0$, and\n\\begin{equation}\n    \\label{qq1}\n    \\begin{split}\nN_1(a_2-s_0a_4)^2=&2\\eta^2N_2g_1g_2j_2h_1\\Gamma_{12}(1+s_0)^2\\\\\n&\\quad+2N_1(1+s_0)\\eta j_2g_1\\Gamma_{12}\\times\\\\\n&\\qquad[\\eta g_1h_2(1+s_0)+2g_2h_1(s_0-1)]\n\\end{split}\n\\end{equation}\n\\end{subequations}\nwhere in Eq. (\\ref{qq1}) the expression for $a_3$ from Eq.\n(\\ref{szp2}) has been substituted. This equation can always be\nsolved in terms of $s_0$ when $h_1 \\Gamma_{12}$ is positive since\nthe discriminant is equal to\n$8N_1\\eta^2g_2^2g_1j_2h_1\\Gamma_{12}(N_1g_{12}+N_2 g_2)$.\n We find that $$1+s_0=\\frac{2N_1g_2}{N_1g_2-\\eta N_1  g_{12}+\\sqrt{2N_1\\eta^2g_1j_2\\Gamma_{12}(N_1g_{12}+N_2 g_2)\/h_1}}.$$\n\nA similar calculation can be completed if the outer boundary of\ncomponent-2 is larger than that of component-1. This then gives that\n${h_2\\Gamma_{12}>0}$ (although expressions (\\ref{all}) change\nslightly).\n\n\n\n If an annulus develops  in\n component $3-k$, it means that $s_0=1$ ($s_0=-1$) for $k=1$ ($k=2$).\n  Inputting this choice into Eq. (\\ref{qq1})  gives\n\\begin{equation}\n    \\begin{split}\n    \\label{spatsep}\n&\\bar{g}_{12}=\\frac{N_kg_kj_{3-k}}{2(N_1j_1+N_2j_2)}\\\\\n&\\quad+\\frac{1}{2}\\left(\\left[\\frac{N_kg_kj_{3-k}}{N_1j_1+N_2j_2}\\right]^2+\\frac{4N_{3-k}g_1g_2j_{3-k}}{N_1j_1+N_2j_2}\\right)^{1\/2}\n    \\end{split}\n\\end{equation}\nas the critical $g_{12}$ at which an annulus forms in\ncomponent-$\\{3-k\\}$. In the case of Fig. \\ref{sch_3}, the curve (\\ref{spatsep}) has been\nplotted in dashed lines and is close to the numerical curve.\n Notice that if $\\xi^2=1$ (equal trapping\nfrequencies for both components), then Eq. (\\ref{spatsep}) becomes\nindependent of $\\Omega$, as in the phase diagrams of\nFig. \\ref{sch_2}, where it yields\n $\\Gamma_{12}=0.008$.\n\nLet us point out that before the transition to the disk plus annulus takes place, there is\n a subregion of region 1, where there are 2 disks, but in one component the wave function has a local minimum\n  at the origin. For instance, in the case of Fig. \\ref{sch_2}, it corresponds to $\\Gamma_1$ changing sign.\n\n\nAs a conclusion, in order for the ground state to be composed of two\ndisks, assuming that component-$k$ is the component with smaller\nsupport, we need that $h_k>0$, $\\Gamma_{12}>0$ and\n$g_{12}<\\bar{g}_{12}$. These three conditions can be summarised as\n\\begin{subequations}\n\\begin{align}\n\\label{con1a}\ng_{12}<\\min\\left(\\frac{j_k}{j_{3-k}}g_{3-k},\\frac{j_{3-k}}{j_{k}}g_{k},\\bar{g}_{12},\\sqrt{g_1g_2}\\right)\n\\hbox{ if } &h_1,\\ h_2>0,\\\\\n\\frac{j_{3-k}}{j_{k}}g_{k}<\ng_{12}<\\min\\left(\\frac{j_k}{j_{3-k}}g_{3-k},\\bar{g}_{12},\\sqrt{g_1g_2}\\right)\\hbox{\nif } &h_{3-k}<0.\n\\label{con1b}\n\\end{align}\n\\end{subequations}\n\n\\subsubsection*{2. A disk and an annular component}\n\nIn this case, we can assume a disk in component-1 and an annulus in component-2; there are three regions:\n an inner disk where only component-1 is present and (\\ref{rhoonly}) holds, an outer\n  annulus where only component-2 is present and (\\ref{rhoonly}) holds and an inner\n  annulus where both components coexist and (\\ref{rhoSz}) holds.\n\nIn order to use the TF approximation when one component is annular,\n computations similar to\n(\\ref{szp2})-(\\ref{qq1}) lead to $h_1h_2<0$ and\n$g_{12}>\\bar{g}_{12}$. This can be summarized (for an annulus in\ncomponent-$\\{3-k\\}$) as\n\\begin{equation}\n    \\label{an}\n    \\max\\left(\\frac{j_{3-k}}{j_{k}}g_{k},\\bar{g}_{12}\\right)< g_{12}<\\min\\left(\\frac{j_k}{j_{3-k}}g_{3-k},\\sqrt{g_1g_2}\\right),\n\\end{equation}\nwith $h_{3-k}<0$.\nEquation (\\ref{an}) also places the restriction that\n\\begin{align}\n    g_k<&g_{3-k}\\left(\\frac{j_k}{j_{3-k}}\\right)^2\\nonumber\\\\\n    =&g_{3-k}\\Lambda_k,\n    \\label{toget}\n\\end{align}\nwhere $\\Lambda_k=({j_k}\/{j_{3-k}})^2$. Note that $\\Lambda_1\\Lambda_2=1$ such that an annulus develops in component-2 (-1) if $g_2 >$ ($<$) $g_1\\Lambda_2$.\n\n\n\n\\subsubsection*{3. Orders of Intracomponent Strengths and Special Cases}\n\n The effect that changing the order of the intracomponent\nstrengths and particle numbers has on $\\bar{g}_{12}$, and thus on\nthe phase diagrams, is now investigated. There are two cases to\nconsider, depending on the relative orders of the particle numbers.\nDrawing aid from experimental values, it is always expected that\n$\\min \\{N_1, N_2\\}\\gg \\max \\{ g_1,\\ g_2\\}$. Throughout it will be\nassumed that $g_2>\\Lambda_2 g_1$, so the annulus develops in\ncomponent-2 and that $j_1$ and $j_2$ are of order unity.\n\nIn the first case when the particle number of  component-1 is much\ngreater than the particle number of component-2 ($N_1\\gg N_2$), it\nfollows that  $\\bar{g}_{12}\\sim g_1j_2\/j_1$.\nThe boundary between  region 1 and region 3 is then directly\ndependent on the value of the ratio $j_2g_1\/[j_1g_2]$. Notice that\nif $g_2\\gg g_1$, $\\Gamma_{12}$ evaluated at $g_{12}=\\bar{g}_{12}$\ntends to unity and as such an annulus would always be present in\ncomponent-2, whatever the value of $g_{12}$.\n\nConversely in the second case when the particle  number of\ncomponent-1 is much smaller than the particle number of component-2\n($N_1\\ll N_2$), it follows that  $\\bar{g}_{12}\\sim\\sqrt{g_1g_2}$\nwhich implies that the annulus will only develop near\n$\\Gamma_{12}=0$.\n\n\n\nWe can also look at some special cases - there are four that can be considered:\n\n{\\it Case (i)}. $\\Lambda_k g_{3-k}=g_k$. When $\\Lambda_2$\n(equivalently $\\Lambda_1$) is such that this equality is made, the\ntwo components are both disks and no annulus develops.\n\n{\\it Case (ii)}. $\\eta=\\xi=1$. Then $\\Lambda_k=1$\nand $R_2\\gtrless R_1\\iff N_2g_2\\Gamma_1\\gtrless N_1g_1\\Gamma_1$ when\nthere are two disks and the annulus develops in the component which\nhas the larger interaction strength.\n\n{\\it Case (iii)}. $\\eta=\\xi=1$ and $g_1=g_2\\equiv g$. When the\nintracomponent coupling strengths are equal, $\\bar{g}_{12}=g$ and\nthere are always two disks with $R_2\\gtrless R_1\\iff N_2\\gtrless\nN_1$.\n\n{\\it Case (iv)}. $\\eta=\\xi=1$, $g_1=g_2\\equiv g$ and $N_1=N_2\\equiv\nN$. When the particle numbers and intracomponent coupling strengths\nare equal, $\\bar{g}_{12}=g$  and there are always two disks with\n$R_2=R_1$. A detailed analysis of this case, explored numerically in\na phase diagram for all $\\Gamma_{12}$ and analytically in the TF\nlimit, was considered by \\cite{ktu1}.\n\n\\subsubsection*{4. Justification of the Thomas-Fermi approximation}\n\n  The gradient terms in\n(\\ref{muu1})-(\\ref{szeq1}) can be neglected if $d_{hl}$,\n the characteristic length of variation of  $\\rho_T$ and $S_z$ is much smaller than\n$d_c$, the characteristic size of the condensates. We have that\n$1\/d_{hl}^2$ is of order of  $\\mu$ and $\\lambda$, which are of order\n$\\sqrt{N_kg_k}$, $\\sqrt{N_kg_{3-k}}$. Hence\n $d_{hl}$ is bounded above by the maximum\n of $(N_kg_k)^{-1\/4}$, $(N_kg_{3-k})^{-1\/4}$. From the expression of $a_3$,\nthe characteristic size of the condensate is of order the minimum of\n$(g_kN_k \\Gamma_{12})^{1\/4}$. Therefore, the Thomas Fermi approximation holds\n if $N_kg_k\\sqrt{\\Gamma_{12}}$, $N_kg_{3-k}\\sqrt{\\Gamma_{12}}$ are large. This requires\n the usual Thomas Fermi criterion that $N_kg_k$, $N_kg_{3-k}$ are large, but\n breaks down if $\\Gamma_{12}$ is too small.\n\n\n\nFor $\\Gamma_{12}=0$,\n the\n equations lead to spatial separation: either the radii of the disks tend to 0\n in the case of two disks or the outer radius of the inner disk tends\n to the inner radius of the annulus in the case of disk plus annulus\n so that $\\psi_1\\times\\psi_2=0$ everywhere. Another analysis\n has to be carried out to understand the region of coexistence,\n  which is of small size and has strong gradients.\n\n\n\n\\subsection*{D. $\\Gamma_{12}<0$, beyond the TF approximation}\n\nFor negative $\\Gamma_{12}$, and if $N_kg_k$ are large, the  Thomas Fermi\n  approximation can be extended provided some model takes into account the small region where\nthe condensates coexist.\n\nIndeed, if we go back to (\\ref{sigTFred}), and have both $\\bar c_2<0$ and $\\Gamma_{12}<0$,\n then the coefficient in front of the second square is positive and the optimal situation\n is to have the square equal to 0, which leads to the inverted parabola (\\ref{rhoty}). On the other\n hand, the coefficient in front of the first square is negative, and the ground state involves\n  derivatives  in $S_z$ to compensate it. Under the assumption\n   that the boundary layer where $S_z$ varies is small,\n we are going to derive a TF model with jump  for $\\rho_T$. We will analyze it\n   for the different geometries (disk plus annulus, droplets and vortex sheets)\n    and show that it provides  information consistent  with the numerics.\n\n\n\n\n\n\\subsubsection*{1. Disk Plus Annulus}\n\n\n Assuming that the\nboundary layer is present only at some $r=r_s$, then $S_z=+1$ in the region in which component-2 is zero\n($r\\le r_s^-$) and $S_z=-1$ in the region in which component-1 is\nzero ($r\\ge r_s^+$). The\ntransition from $S_z=+1$ to $S_z=-1$ is not smooth, therefore\ncreating the jump in density.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.5]{sig_1.eps}\n\\end{center}\n\\caption{Total density profiles (divided by $10^4$) obtained numerically (solid lines) and analytically (dashed lines) for two spatial separation cases with $g_1=0.0078$, $g_2=0.0083$,  $N_1=N_2=10^5$, $\\eta=\\xi=1$ (set ES1) and $\\Gamma_{12}=-10$ and $\\Omega=0$: (a) annulus plus disk,  analytical estimate coming from (\\ref{eann}) and (b) droplet, analytical estimate coming from (\\ref{edrop}). The inset in (a) shows the discontinuity of density at $r_s=3.63$.  Distance is measured in units of $r_0$ and density in units of $r_0^{-2}$.}\\label{sig1}\n\\end{figure}\nTherefore, we are lead to minimise the integral\n\\begin{equation}\n    \\label{eann}\n    \\begin{split}\nI&=\\int_{B_{r_s}}2j_1r^2\\rho_T+\\frac{g_1}{2}\\rho_T^2 \\quad d^2r\\\\\n&\\quad+\\int_{B_{R\\backslash r_s}}2\\frac{j_2}{\\eta}r^2\\rho_T+\\frac{g_2}{2\\eta^2}\\rho_T^2\\quad  d^2r\n\\end{split}\n\\end{equation}\nwith respect to $r_s$. Here $B_{r_s}$ is a ball of radius $r_s$ and $B_{R\\backslash r_s}$ is a torus with outer boundary at $r=R$ and inner boundary at $r=r_s$. Thus\n\\begin{equation}\n    \\begin{split}\n&2\\pi r_s\\left[2j_1r^2\\rho_T+\\frac{g_1}{2}\\rho_T^2\\right]\\Bigg|_{r=r_s^-}\\\\\n&\\qquad-2\\pi r_s\\left[2\\frac{j_2}{\\eta}r^2\\rho_T+\\frac{g_2}{2\\eta^2}\\rho_T^2\\right]\\Bigg|_{r=r_s^+}=0,\n\\label{jumpp}\n\\end{split}\n\\end{equation}\nwhich implies that\n\\begin{subequations}\n    \\label{rhopm}\n\\begin{align}\n\\rho^-=&\\frac{1}{g_1}\\left[\\mu_1-2j_1r^2\\right]\\quad r\\in B_{r_s}\\\\\n\\rho^+=&\\frac{\\eta}{g_2}\\left[\\mu_2-2j_2r^2\\right]\\quad r\\in B_{R\\backslash r_s}.\n\\end{align}\n\\end{subequations}\nThen using the normalisation conditions $\\int\\rho^-d^2r=N_1$ and $\\int\\rho^+d^2r=\\eta N_2$ we get an outer radius\n\\begin{subequations}\n    \\begin{equation}\n        R=\\left(r_s^2+\\sqrt{\\frac{g_2N_2}{\\pi j_2}}\\right)^{1\/2},\n    \\end{equation}\nand chemical potentials\n\\begin{align}\n\\mu_1=&\\frac{N_1g_1}{\\pi r_s^2}+j_1r_s^2,\\\\\n\\mu_2=&2j_2\\left(r_s^2+\\sqrt{\\frac{g_2N_2}{\\pi j_2}}\\right).\n\\end{align}\n\\end{subequations}\nIt remains to find $r_s$. But, from (\\ref{jumpp}), it follows that\n\\begin{equation}\nr_s^2=\\frac{\\frac{4N_2j_2}{\\pi}-g_1\\left(\\frac{N_1}{\\pi r_s^2}-\\frac{j_1r_s^2}{g_1}\\right)^2}{4\\left[j_1\\left(\\frac{N_1}{\\pi r_s^2}-\\frac{j_1 r_s^2}{g_1}\\right)-2j_2\\sqrt{\\frac{N_2j_2}{\\pi g_2}}\\right]}.\n\\end{equation}\nThe above leaves a quartic equation in $r_s^2$, the solution of which can be found numerically.\n\nWe can then complete the energy calculation to give\n\\begin{equation}\n        \\label{enda}\nI=\\frac{g_1N_1^2}{2\\pi r_s^2}+(N_1j_1+2N_2j_2)r_s^2-\\frac{\\pi j_1^2}{6g_1}r_s^6+\\frac 43\\left(\\frac{N_2^3g_2j_2}{\\pi}\\right)^{1\/2}.\n\\end{equation}\nA check plot of the density profile with the parameters $g_1=0.0078$, $g_2=0.0083$ and $N_1=N_2=10^5$ where $\\eta=\\xi=1$\nagrees well with this model\n where the boundary is calculated to be at $r_s=3.62$ (see Fig.\\ref{sig1}(a)).\n\n\n\\subsubsection*{2. Droplets}\n\nThis formalism can also be extended to study the droplet case. As before, assuming a thin boundary layer, one can allow for a jump in density. We thus need to minimise the Thomas-Fermi energy for two droplets described by $B_{1}$ and $B_{2}$ that exist in the regions $0\\le\\theta\\le\\alpha$ and $\\alpha\\le\\theta\\le2\\pi$ respectively. The energy is then\n\\begin{equation}\n    \\label{edrop}\n    \\begin{split}\nI&=\\alpha\\int_{0}^{R_1}2j_1r^2\\rho_T+\\frac{g_1}{2}\\rho_T^2 \\quad rdr\\\\\n&\\quad+(2\\pi-\\alpha)\\int_{0}^{R_2}2\\frac{j_2}{\\eta}r^2\\rho_T+\\frac{g_2}{2\\eta^2}\\rho_T^2\\quad  rdr.\n\\end{split}\n\\end{equation}\nThe expressions for the density in each domain (\\ref{rhopm}) and the normalisation conditions allow completion of this integral to give\n\\begin{equation}\nI=\\frac{4\\sqrt{2}}{3}\\left[\\left(\\frac{N_1^3g_1j_1}{\\alpha}\\right)^{1\/2}+\\left(\\frac{N_2^3g_2j_2}{(2\\pi-\\alpha)}\\right)^{1\/2}\\right].\n\\end{equation}\n\nIt remains to find the optimum $\\alpha$. This is achieved through the condition $dI\/d\\alpha=0$ which gives\n\\begin{equation}\n    \\begin{split}\n\\frac{(N_1^3g_1j_1)^{1\/2}}{\\alpha^{3\/2}}&=\\frac{(N_2^3g_2j_2)^{1\/2}}{(2\\pi-\\alpha)^{3\/2}}\\\\\n\\Rightarrow \\alpha&=2\\pi\\frac{\\bar{N}(\\bar{g}\\bar{j})^{1\/3}}{(1+\\bar{N}(\\bar{g}\\bar{j})^{1\/3})}\n\\end{split}\n\\end{equation}\nwhere we have set $\\bar{N}=N_1\/N_2$, $\\bar{g}=g_1\/g_2$ and $\\bar{j}=j_1\/j_2$. As expected, equality of the $N_k$, $g_k$ and setting $\\eta=\\xi=1$ ($\\Rightarrow j_1=j_2$) gives $\\alpha=\\pi$ and the condensate is then composed of two half-balls. Otherwise, a curvature is present. A check plot of the density profile with the parameters $g_1=0.0078$, $g_2=0.0083$ and $N_1=N_2=10^5$ where $\\eta=\\xi=1$ is given in Fig \\ref{sig1}(b).\n\nFinally we can note the energy for the droplets is\n\\begin{equation}\n\t\\label{endd}\n    \\begin{split}\nI&=\\frac{4}{3\\sqrt{\\pi}}\\left(1+\\bar{N}(\\bar{g}\\bar{j})^{1\/3}\\right)^{-1\/2}\\times\\\\\n&\\quad\\left[(N_1^3g_1j_1)^{1\/2}+(N_1N_2^2(g_1g_2^2j_1j_2^2)^{1\/3})^{1\/2}\\right].\n\\end{split}\n\\end{equation}\nThis energy can be compared to the energy of the disk plus annulus (\\ref{enda}) to determine which is the optimum geometry.\nIndeed, in the numerical cases studied before, the droplets are preferred states for small $\\Omega$.\n\n\n\\subsubsection*{3. Regions of Vortex Sheets}\n\nIn the case of vortex sheets,\n we can assume that the global profile of the total density is TF-like, obeying Eq. (\\ref{rhoty}) in the bulk of the condensate. By working with the total density, we do not require any information on the vortex sheets themselves (and consequently $S_z$).\n\n\nWe thus take the form of $\\rho_T$ from Eq. (\\ref{rhoty}) from which we note that the outer boundary at $r=R$ satisfies $R=\\sqrt{-a_3\/a_4}$ and that completion of the normalisation condition ($\\ref{nn1a}$) gives\n\\begin{equation}\na_3=\\sqrt{-\\frac{2(N_1+\\eta N_2)g_1g_2\\Gamma_{12} a_4}{\\pi}}.\n\\end{equation}\nThis expression evidently requires $a_4\\Gamma_{12}<0$. We however expect the vortex sheets to be present only in the $\\Gamma_{12}<0$ domain, thus we can be more specific on the condition, namely that $a_4>0$, or\n\\begin{equation}\n    g_{12}>\\frac{\\eta g_1j_2+g_2j_1}{(\\eta j_1+j_2)}.\n\\end{equation} We point out that this critical number corresponds to $\\Gamma_{12}=0$ in the cases studied above.\nWith $a_3$ as above and $a_4$ given by the parameters of the system, the density profile is then fully accessible.\n\n\n\\subsection*{E. Analysis of defects}\n\nThe advantage of the Thomas Fermi analysis of $\\rho_T$ in the nonlinear sigma\nmodel is that it allows for analysis of\n defects as a perturbation calculation when $N_k g_k$, $N_k g_{3-k}$ and $N_k g_{12}$ are large,\n  in the spirit of what has been done for a single condensate in \\cite{ad,ar,cd,wei}\n  or for two condensates in \\cite{wei2}.\n   We do not need\n   to analyze specifically the peaks because they are taken into account in the $S_z$ formulation.\n   This has\nbeen attacked in a more complicated fashion in the case of a single coreless vortex in \\cite{ktu}.\n   We will develop our ideas  in a later work but let us point out that the starting point\n   is the energy (\\ref{sig}) and if we call $p_i$ and $q_i$\n    the location of the vortex in each component, the main terms coming from the vortex contributions lead to:\n   \\begin{itemize}\\item Each vortex core of each component providing a kinetic energy term\n    proportional to $\\rho_T(0) \\log d_{hl}\/(2(1+\\eta))$ where $d_{hl}$ is the healing length, the characteristic\n     size of the vortex core, here of order $1\/\\sqrt {N_kg_k}$.\n     \\item the rotational term providing a term in $-c\\Omega (\\rho_T(0))^2$, where $c$\n        is a numerical constant. The balance of these two terms allows us to compute the critical velocity\n         for the nucleation of the first vortex.\n         \\item the kinetic energy then yielding a term in $-\\log |p_i-p_j|-\\log|q_i-q_j|$.\n         \\item the rotation term yielding a term in $\\Omega (|p_i|^2+|q_i|^2)$.\n     \\item the interaction term providing  a term $\\rho_T^2 S_z^2$ for the perturbation of\n      $S_z$ close to a coreless vortex. The ansatz can be made in several ways\n       leading to an interaction term in $e^{-|p_i-q_i|^2}$.\n           \\end{itemize}\n  This leads to a point energy of the type\n  $$\\sum_i a (|p_i|^2+|q_i|^2+be^{-|p_i-q_i|^2} )-\\sum_{i,j}(\\log|p_i-p_j|+\\log|q_i-q_j|)$$ where\n   $a$ and $b$ are related to the parameters of the problem.\n  The ground state of such a point energy leads to a square lattice for a sufficient number of points\n   and for some range of $a$ and $b$.\n\n\n\\section*{IV. Phase Diagram Under Conserved Total Particle Number}\n\n In the experiment of Hall\net al \\cite{hall} a single component BEC of $^{87}$Rb in the\n$|1$,$-1\\rangle$ state was initially created, before a transfer of\nany desired fraction of the atoms from this $|1$,$-1\\rangle$ state\nto the $|2$,$1\\rangle$ state created the two-component BEC. Thus the\nratio of particle numbers $N_1\/N_2$ is controllable experimentally,\nwith the constraint that $N_1+N_2$ is constant (in the case of the\nexperiment of \\cite{hall}, $N_1+N_2=5\\times10^5$). As such,\nexperimentally, it is possible to keep the individual particle\nnumbers constant (as in the normalisation condition (\\ref{norm})) or\nto keep the total particle number $N_1+N_2$ constant, allowing $N_1$\nand $N_2$ to vary (as in (\\ref{norm12})). We produce an\n$\\Omega-\\Gamma_{12}$ phase diagram for the parameters $g_1=0.003$,\n$g_2=0.006$ and $\\eta=\\xi=1$ using\nthe normalisation condition given by (\\ref{norm12}) with $N_1+N_2=2.1\\times 10^5$.\nNote that if $g_1=g_2$, $m_1=m_2$ and\n$\\omega_1=\\omega_2$ ($\\eta=\\xi=1$), then the phase diagram would be\nidentical to that of Fig. \\ref{sch_1} (i.e. the normalisation\ncondition in this case would not be important). The $\\Omega-\\Gamma_{12}$ phase diagram is presented in Fig. \\ref{sketchv}. There are three distinct regions (determined by the geometry of the ground state):\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.4]{pp_midgv.eps}\\\\\n\\includegraphics[scale=0.4]{pp_midg2v.eps}\n\\end{center}\n\\caption{(Color online) $\\Omega-\\Gamma_{12}$ phase diagram for parameters $g_1=0.003$ and $g_2=0.006$\nwith $\\eta=\\xi=1$ using the normalisation condition given by (\\ref{norm12}). (a) Numerical simulations where triangles indicate that the lattice in both components is triangular, squares that the lattice in both components is square and diamonds where no vortices have been nucleated. Filled triangles, squares and diamonds are where the two components are disk-shaped and coexist, empty triangles, squares and diamonds are where only component-1 exists; those with a dot in the centre represent the appearance of coreless vortices in component-2 and those without a dot in the centre represent the complete disappearance of component-2. (b) A schematic representation of the numerical simulations.\nThe solid lines indicate the boundary between different identified regions (determined by the geometry of the ground state) and the dashed lines the boundary between triangular and square lattices. The unit of rotation is $\\tilde{\\omega}$.}\\label{sketchv}\n\\end{figure}\n\n{\\it Region 1}. In the first region, both components are disk-shaped. As before, the coreless vortices can either form a triangular or a square lattice depending on the values of $\\Gamma_{12}$ and $\\Omega$. Figure \\ref{7posv} shows this case for $\\Gamma_{12}=0.7$ and $\\Omega=0.65$ (where a triangular lattice is present) and $\\Omega=0.9$ (where a square lattice is present). In region 1, both components have the same radii.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.5]{7posv_1.eps}\n\\end{center}\n\\caption{(Color online) A series of density plots for component-1 (left column) and component-2 (right column). The parameters are $g_1=0.003$ and $g_2=0.006$ with $\\eta=\\xi=1$ and $\\Gamma_{12}=0.7$ (which gives $g_{12}=0.0023$) and normalisation taken over the total density (Eq. (\\ref{norm12})). The angular velocity of rotation is $\\Omega$ and it takes the values (a) $0.65$ and (b) $0.9$. There is a triangular lattice in (a) and a square lattice in (b). At these parameters the components are in region 1 of the phase diagram Fig. \\ref{sketchv}. Distance is measured in units of $r_0$ and density in\nunits of $r_0^{-2}$.}\n\\label{7posv}\n\\end{figure}\n\n{\\it Region 2}. In the second region, only component-1 exists except for isolated density peaks that exist in component-2. These isolated density peaks occur at the same location as the vortices do in component-1, and are thus identical to the isolated coreless vortices described in detail in Sect. IV. Figure \\ref{3posv} shows this case $\\Gamma_{12}=0.3$ with $\\Omega=0.5$ and $\\Omega=0.9$.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.5]{3posv_1.eps}\n\\end{center}\n\\caption{(Color online) A series of density plots for component-1 (left column) and component-2 (right column). The parameters are $g_1=0.003$ and $g_2=0.006$ with $\\eta=\\xi=1$ and $\\Gamma_{12}=0.3$ (which gives $g_{12}=0.0035$) and normalisation taken over the total density (Eq. (\\ref{norm12})). The angular velocity of rotation is $\\Omega$ and it takes the values (a) $0.5$ and (b) $0.9$. At these parameters the components are in region 2 of the phase diagram Fig. \\ref{sketchv}. Distance is measured in units of $r_0$ and density in\nunits of $r_0^{-2}$.}\n\\label{3posv}\n\\end{figure}\n\n{\\it Region 3}. In the third region, only component-1 exists (the density peaks in component-2 that were present in region 2 are no longer present). Furthermore, only triangular vortex lattices are observed in this effective one component condensate (in component-1).\n\n\nComputations similar to Section III hold, except that now we have to\n set $\\lambda=0$. This leads to $a_1\/a_3=-\\bar c_1\/(2\\bar c_2)$. We  see that this ratio (which\n is $S_z(0)$), reaches 1 or -1 when $\\bar c_1=\\pm2\\bar c_2$. In our numerical case, this leads to $\\bar \\Gamma_{12}=0.5$.\n We see clearly 3 regimes: $\\Gamma_{12}>\\bar\\Gamma_{12}$, where the condensates\nare 2 disks, $\\Gamma_{12}<0$, which is phase separation, in which case $S_z=1$ is the preferred state and\n $0<\\Gamma_{12}<\\bar \\Gamma_{12}$, in which case the TF approximation leads to a computation of $\\rho_T$ with\n coreless vortex lattices and variations in $S_z$ which improve the energy and lead to this intermediate state, still to be studied in more detail.\n\n\\section*{V. Conclusion}\n\nWe have presented phase diagrams of rotating two component condensates in terms of\n the angular velocity $\\Omega$ and a nondimensionnalized parameter related to\n the coupling strengths\n$\\Gamma_{12}=1-g_{12}^2\/(g_1g_2)$. We have analyzed\nthe various ground states and topological defects and have found\n four sets characterized by the\n symmetry preserving\/symmetry breaking,\n coexistence or spatial separation of the components.\n When the geometry of the ground\nstates is either two disks (coexistence of components, region 1) or a disk\nand an annulus (spatial separation keeping some symmetry, region 3), the topological\ndefects are coreless vortex lattices (with possible stabilization\nof the square lattice) or giant skyrmions at the boundary\ninterface between the two components. In the complete symmetry breaking case, we have found vortex\n sheets and droplets. The difference of masses or coupling strengths between\n the components can induce very different patterns.\n\nWe have introduced an energy (\\ref{sig}) related to the total density and a pseudo spin vector.\n The minimization in a generalized Thomas Fermi approximation provides a lot of information\n on the ground states for general masses, trapping frequencies and coupling strengths.\n Some parts of the phase diagrams can be justified rigorously, both in the case\n  $\\Gamma_{12}>0$, which had been studied before, but also in the case $\\Gamma_{12}<0$\n  with generalized models. This formulation of the energy\nshould bring in the future more information on the defects.\n\n\n\n\\section*{Acknowledgments}\n\nThe authors wish to thank Thierry Jolicoeur  for useful discussions\nthat took place for the duration of this work. They are very grateful to the referee for  his careful reading of the manuscript and his appropriate comments. We acknowledge\nsupport from the French ministry Grant ANR-BLAN-0238, VoLQuan.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nIn this paper, we consider the array low-density parity-check (LDPC) codes, originally introduced by Fan in \\cite{fan00},  and their minimum\/stopping distance.  Array LDPC codes are specified by two integers $q$ and $m$, where $q$ is an odd prime and $m \\leq q$. Furthermore, in this work, $\\mathcal{C}(q,m)$ will denote the array LDPC code with parameters $q$ and $m$, and $d(q,m)$ (resp.\\ $h(q,m)$)  its minimum (resp.\\ stopping) distance. \n\nSince the original work by Fan, several authors have consider the \\emph{structural} properties of these codes (see, e.g., \\cite{mit02,yan03,sug08,esm09,esm11,liu10,dol10}). For high rate and\nmoderate length, these codes perform well under iterative decoding, and they are also well-suited for practical implementation due to their regular structure \\cite{olc03,bha05}.\n\n\n\n\nThe minimum distance of these codes was first analyzed by Mittelholzer in \\cite{mit02}, where general (i.e., independent of $q$) minimum distance upper bounds for $m \\leq 6$ were provided. Subsequently, Yang and Helleseth \\cite{yan03} investigated the minimum\ndistance of these codes in an algebraic way by first proving that the codes are invariant under a doubly transitive group of ``affine'' permutations. Then, they proved the general lower bound $d(q,4) \\geq 10$, for $q > 7$, on the minimum distance. In \\cite{sug08}, the general upper bounds $d(q,4) \\leq 10$  and $d(q,5) \\leq 12$\non the minimum distance were proved. Furthermore, by combining these bounds with the results in \\cite{yan03}, it follows that $d(q,4)=10$  and that $d(q,5)$ is either $10$ or $12$, for $q > 7$. In summary,\n\\begin{displaymath}\nd(q,m) \\leq \\begin{cases}\n6, & \\text{if $m=3$, with equality for $q \\geq 5$ \\cite{yan03}}\\\\\n10, & \\text{if $m=4$, with equality for $q > 7$ \\cite{yan03,sug08}}\\\\\n12, & \\text{if $m=5$, with exact value either} \\\\\n& \\text{$10$ or $12$ for $q > 7$ \\cite{yan03,sug08}}\\\\\n32, & \\text{if $m=6$ \\cite{mit02}.} \\end{cases}\n\\end{displaymath}\n\nThe case $m=6$ has not been treated in the literature before, except for in the initial work of Mittelholzer \\cite{mit02}. In this work, we will consider this case in more detail as well as the case $m=7$, both from an experimental point of view and by deriving an improved upper bound on $d(q,6)$ and a new upper bound on $d(q,7)$.\n\nThis paper is organized as follows. \nIn Section~\\ref{sec:prelim}, some of the basic notation is introduced and the definition of array LDPC codes is given.\nThe concept of a \\emph{template support matrix} is also introduced.\nIn Section~\\ref{sec:algo}, an heuristic is presented that will be used to infer a \\emph{candidate} template support matrix.\nThe heuristic analyzes the \\emph{graphical cycle structure} of support matrices of codewords\/stopping sets for different values of $q$, with $m$ fixed.\nIn Section~\\ref{sec:bound}, we use the template support matrix found in Section~\\ref{sec:algo} to formally prove the improved upper bound $d(q,6) \\leq 20$.\nFurthermore, in Section~\\ref{sec:bound_m7}, we present a template support matrix for $m=7$, found by using the heuristic of Section~\\ref{sec:algo}, which is used to formally prove the new upper bound $d(q,7)\\leq24$.\nIn Section~\\ref{sec:results}, new minimum\/stopping distance results are presented for fixed values of $m \\leq 7$ and $q \\leq 79$. \nFinally, in Section~\\ref{sec:conclu}, we draw the conclusions.\n\n\n\\begin{comment}\nFurthermore, a submatrix of the codeword \\emph{support matrix} is\n\n\\begin{displaymath}\n\\begin{pmatrix}\n0 & 0 & -1 & -2 & 5x & 5x & -2 &  3^{-1}y+y & -1 \\\\\n0 & z & 0 & -1 & 4x & z & y & y & 3^{-1}-1 \\\\\n0 & 2z & 1 & 0 & 3x & 2z-5x & 2y+2 & -3^{-1}y+y & 2 \\cdot 3^{-1}-1 \\\\\n0 & 3z & 2 & 1 & 2x & 3z-10x & 3y+4  & -2 \\cdot 3^{-1}y+y  & 0 \\\\\n0 & 4z & 3 & 2 & x & 4z-15x & 4y+6 & 0 & 4 \\cdot 3^{-1}-1 \\\\\n0 & 5z & 4 & 3 & 0 & 5z-20x & 5y+8 & -4 \\cdot 3^{-1}y+y  & 5 \\cdot 3^{-1}-1 \\\\\n\\end{pmatrix}\n\\end{displaymath}\nfor a codeword of weight $20$.\n\n\\end{comment}\n\n\n\\section{Preliminaries} \\label{sec:prelim}\n\nThe array LDPC code $\\mathcal{C}(q,m)$, with parameters $q$ and $m$, has length $q^2$ and can be defined by the parity-check matrix\n\\begin{equation} \\label{eq:pcmatrix}\n\\mathbf{H}(q,m) = \\begin{pmatrix}\n\\mathbf{I} & \\mathbf{I} & \\mathbf{I} & \\cdots & \\mathbf{I} \\\\\n\\mathbf{I} & \\mathbf{P} & \\mathbf{P}^2 & \\cdots & \\mathbf{P}^{q-1} \\\\\n\\mathbf{I} & \\mathbf{P}^2 & \\mathbf{P}^4 & \\cdots & \\mathbf{P}^{2(q-1)} \\\\\n& & \\vdots  & & \\vdots \\\\\n\\mathbf{I} & \\mathbf{P}^{m-1} & \\mathbf{P}^{2(m-1)} & \\cdots & \\mathbf{P}^{(m-1)(q-1)} \\end{pmatrix}\n\\end{equation}\nwhere $\\mathbf{I}$ is the $q \\times q$ identity matrix and $\\mathbf{P}$  is a $q \\times q$ permutation matrix defined by\n\\begin{displaymath}\n\\mathbf{P} = \\begin{pmatrix}\n0 & 0 & \\cdots & 0 & 1 \\\\\n1 & 0 & \\cdots & 0 & 0 \\\\\n0 & 1 & \\cdots & 0 & 0 \\\\\n& \\vdots & &  \\vdots & \\\\\n0 & 0 & \\cdots & 1 & 0 \\end{pmatrix}.\n\\end{displaymath}\nSince the number of ones in each row of the matrix in (\\ref{eq:pcmatrix}) is $q$ and the number of ones in each column is $m$, the array LDPC codes are $(m,q)$-regular codes. Furthermore, it is not hard to see that the parity-check matrix in (\\ref{eq:pcmatrix}) has rank $qm-m+1$, from which it follows that the dimension of $\\mathcal{C}(q,m)$ is $q^2-qm+m-1$. \n\nIn \\cite{yan03}, a new representation for $\\mathbf{H}(q,m)$ was introduced. In particular, since each column of the parity-check matrix $\\mathbf{H}(q,m)$ has $m$ blocks and each block is a permutation of $(1,0,0,\\dots,0,0)^T$, we can represent each column as a vector of integers between $0$ and $q-1$, where\n\\begin{equation} \\label{eq:rep}\ni \\triangleq \\left(\\overbrace{0,\\dots,0}^{i},1,\\overbrace{0,\\dots,0}^{q-i-1} \\right)^T\n\\end{equation}\nand  $(\\cdot)^T$ denotes the transpose of its argument, i.e., the $1$-positions are associated with the integers modulo $q$.\n Furthermore, it follows from (\\ref{eq:pcmatrix}) and the integer representation in (\\ref{eq:rep})  that any column in an array LDPC code parity-check matrix is of the form \n\\begin{equation} \\label{eq:form}\n(x,x+y,x+2y,\\dots,x+(m-1)y)^T \\pmod q\n\\end{equation}\nwhere $x$ and $y$ are integers between $0$ and $q-1$. Thus, a column can be specified by two integers $x$ and $y$. Also, note that since there are $q^2$ distinct columns in an array LDPC code parity-check matrix, any pair $(x,y) \\in \\{0,\\dots,q-1\\}^2$ specifies a valid column.\n\nIn the following, the \\emph{support matrix} of a codeword\/stopping set will be the submatrix of $\\mathbf{H}(q,m)$ corresponding to the \\emph{support set} of the codeword\/stopping set, i.e., we keep the columns of  $\\mathbf{H}(q,m)$ whose column indices coincide with the support set of the codeword\/stopping set. Also, we will use the integer representation in (\\ref{eq:rep}) for the columns of the submatrix.\n\nFurthermore, a \\emph{template} support matrix with parameters $m$, $q$, $w$, and $q_0$  is formally defined as an $m \\times w$ matrix with entries that are functions of $q$ and such that it is the support matrix (possibly column-permuted) of a codeword\/stopping set of weight\/size $w$ of  $\\mathcal{C}(q,m)$ for all $q \\geq q_0$.\nThe specific matrix which results when a template support matrix is evaluated for a specific value of $q$ is called an \\emph{instance} of the template support matrix.\n\n\n\\begin{comment}\n\n\\begin{figure*}[!t]\n\\normalsize\n\\begin{displaymath}\n{\\tiny {\n\\left[ \\begin{smallmatrix}\n0 & 0 & 1 & 1 & -11 \\cdot 3^{-1} & -11 \\cdot 3^{-1}  & -8 \\cdot 3^{-1} & -8 \\cdot 3^{-1}  & -2 \\cdot 3^{-1} & -2 \\cdot 3^{-1} & 2 \\cdot 3^{-1} & 2 \\cdot 3^{-1} & -4 \\cdot 3^{-1} & -4 \\cdot 3^{-1} & 5 \\cdot 3^{-1} & 5 \\cdot 3^{-1} & -3 & -3 & -2 & -2 &  -1 & -1 \\\\\n0 & 6^{-1} & 3 \\cdot 2^{-1} & 5 \\cdot 6^{-1} & -11 \\cdot 6^{-1} & -2 & -11 \\cdot 6^{-1} & -4 \\cdot 3^{-1} & - 2^{-1} &  6^{-1} & 5 \\cdot 6^{-1} & 4 \\cdot 3^{-1} & 0 & -2 \\cdot 3^{-1} & 4 \\cdot 3^{-1} & 3 \\cdot 2^{-1} & -2  & -4 \\cdot 3^{-1} & -2 \\cdot 3^{-1} & -2^{-1} & 0 & -2^{-1} \\\\\n0 & 3^{-1} & 2 & 2 \\cdot 3^{-1} & 0 & -3^{-1} & -1 & 0 & -3^{-1} & 1 & 1 & 2 & 4 \\cdot 3^{-1} & 0 & 1 & 4 \\cdot 3^{-1} & -1 & 3^{-1} & 2 \\cdot 3^{-1} & 1 & 1 & 0 \\\\\n0 & 2^{-1} & 5 \\cdot 2^{-1} & 2^{-1} & 11 \\cdot 6^{-1} & 4 \\cdot 3^{-1} & -6^{-1} & 4 \\cdot 3^{-1} & -6^{-1} & 11 \\cdot 6^{-1} &  7 \\cdot 6^{-1} & 8 \\cdot 3^{-1} & 8 \\cdot 3^{-1} & 2 \\cdot 3^{-1} & 2 \\cdot 3^{-1} & 7 \\cdot 6^{-1} & 0 & 2 & 2 & 5 \\cdot 2^{-1} & 2 & 2^{-1}\\\\\n0 & 2 \\cdot 3^{-1} & 3 & 3^{-1} & 11 \\cdot 3^{-1} & 3 & 2 \\cdot 3^{-1} & 8 \\cdot 3^{-1} & 0 & 8 \\cdot 3^{-1} & 4 \\cdot 3^{-1} & 10 \\cdot 3^{-1} & 4 & 4 \\cdot 3^{-1} & 3^{-1} & 1 & 1 & 11 \\cdot 3^{-1} & 10 \\cdot 3^{-1} & 4 & 3 & 1 \\\\\n0 & 5 \\cdot 6^{-1} & 7 \\cdot 2^{-1} & 6^{-1} & 11 \\cdot 2^{-1} & 14 \\cdot 3^{-1} & 3 \\cdot 2^{-1} & 4 & 6^{-1} & 7 \\cdot 2^{-1} & 3 \\cdot 2^{-1} & 4 & 16 \\cdot 3^{-1} & 2 & 0 & 5 \\cdot 6^{-1} & 2 & 16 \\cdot 3^{-1} & 14 \\cdot 3^{-1} & 11 \\cdot 2^{-1} &4 & 3 \\cdot 2^{-1} \n\\end{smallmatrix} \\right]}}\n\\end{displaymath}\n \\hrulefill\n\\vspace*{-2mm}\n\\end{figure*}\n\n\\end{comment}\n\n\n\n\n\n\n\\section{Deriving Upper Bounds on $d(q,m)$} \\label{sec:algo}\n\nIn this section, we describe an heuristic which can be used to derive upper bounds on the minimum\/stopping distance of array LDPC codes. For simplicity, we will only consider the codeword case (the stopping set case is similar). \nThe heuristic is a three-step procedure:\n\\begin{enumerate}\n\\item In the first step, pairs of codewords $\\mathbf{c}_1 \\in \\mathcal{C}(q_1,m)$ and $\\mathbf{c}_2 \\in \\mathcal{C}(q_2,m)$, $q_1 < q_2$ and $m$ fixed, where $\\mathbf{c}_1$ and $\\mathbf{c}_2$ have the same \\emph{graphical cycle structure} (a concept to be defined later), are identified.\n\\item The second step is to infer a \\emph{candidate} template support matrix (which may or may not exist) such that the instances for $q=q_1$ and $q=q_2$ are the support matrices (possibly column-permuted) of the two codewords $\\mathbf{c}_1$ and $\\mathbf{c}_2$, respectively. \nWe emphasize here that the inferred matrix is only a \\emph{candidate} template support matrix, since a formal proof is needed to show that all instances for $q \\geq q_0$, for some $q_0$, are in fact valid (possibly column-permuted) support matrices. \n\\item The third step is a formal proof that the instances of the candidate template support matrix are indeed valid  \n(possibly column-permuted)  \nsupport matrices of codewords for all possible values of $q$ larger than or equal to $q_0$. \n\\end{enumerate}\nNote that all instances of a template support matrix may not have their columns in the order implied by the parity-check matrix in (\\ref{eq:pcmatrix}). \nThis is obviously not important, since the order of the columns in a support matrix is not relevant (independent of the order, it will represent the same codeword\/stopping set).\n\n\\subsection{First Step: Graphical Cycle Structure} \\label{sec:structure}\n\nNote that for the array LDPC codes there exists a subgroup of the automorphism group which is doubly transitive \\cite{yan03}. This means that (by definition), for each codeword $\\mathbf{c} \\in \\mathcal{C}(q,m)$, there exists a codeword $\\rho(\\mathbf{c})$ (obtained by permuting the coordinates of $\\mathbf{c}$ according to the permutation $\\rho$ from this subgroup) having the coordinates $(p_1,p_2)$, for any $0 \\leq p_1 < p_2 \\leq q^2-1$, in its support set. Thus, it is always possible to permute any codeword (using permutations from this subgroup) such that the corresponding support matrix contains the columns $(0,0,0,\\dots,0)^T$ and $(q-1,0,1,\\dots,q-2)^T$. This is the case since these columns will always be in the parity-check matrix $\\mathbf{H}(q,m)$ for all valid values of $q$ and $m$. In particular, the column $(0,0,0,\\dots,0)^T$ (resp.\\ $(q-1,0,1,\\dots,q-2)^T$) is generated by $x=0$ and $y=0$ (resp.\\ $x=q-1$ and $y=1$) using the representation in (\\ref{eq:form}).\n\nAs argued above, the support matrix can be regarded as an $m \\times w$ matrix of integers modulo $q$, where $w$ is the weight  of the underlying codeword.  From this matrix we can make a bipartite graph, denoted by $G^{(i,j)}=G(V^{(i,j)},E^{(i,j)})$, for each pair of rows $(i,j)$, $i<j$. The vertex set $V^{(i,j)}$ partitions into two distinct sets which we denote by $V^{(i)}$ and $V^{(j)}$, respectively. Now, for each \\emph{distinct} entry in the $i$th row of the support matrix we associate a node in the vertex set $V^{(i)}$. Thus, if there are two (or more) identical entries in the $i$th row of the support matrix, then they will correspond to the same vertex in  $V^{(i)}$. Similarly, for each \\emph{distinct} entry in the $j$th row of the support matrix we associate a node in the vertex set $V^{(j)}$. Furthermore, there will be an edge from a vertex $v^{(i)} \\in V^{(i)}$ to a vertex $v^{(j)} \\in V^{(j)}$ if and only if there exists a column in the support matrix in which the entry corresponding to $v^{(i)}$ appears as the $i$th element and the entry corresponding to $v^{(j)}$ appears as the $j$th element. \nIn the following, we will refer to the graphs $G^{(i,j)}$ as the  \\emph{support matrix graphs}.\nFor convenience, we let $v^{(i)}_{\\alpha}$ denote the vertex in $V^{(i)}$ representing the entry (or entries) with value $\\alpha$ in the $i$th row of the support matrix of a codeword. \nAlso,\ndue to the automorphism, \nwe will assume that the support matrix of a codeword  contains the columns $(0,0,0,\\dots,0)^T$ and $(q-1,0,1,\\dots,q-2)^T$.\n\n\n\nLet  $\\mathbf{c}_1 \\in \\mathcal{C}(q_1,m)$ and $\\mathbf{c}_2 \\in \\mathcal{C}(q_2,m)$, $q_2 > q_1$, be two distinct \\emph{minimal} codewords of the same Hamming weight, where a \\emph{minimal} codeword is a codeword that does not have the support set of a nonzero codeword as a proper subset of its own support set. From each of the corresponding support matrices we build the support matrix graphs $G^{(i,j)}$ for each pair of rows $(i,j)$, $0 \\leq i < j \\leq m-1$, as outlined above. The graphs corresponding to $\\mathbf{c}_1$ and $\\mathbf{c}_2$ are denoted by  $G_{\\mathbf{c}_1}^{(i,j)}$ and $G_{\\mathbf{c}_2}^{(i,j)}$, respectively. Now, $\\mathbf{c}_1$ and $\\mathbf{c}_2$ are said to have the same \\emph{graphical cycle structure} (by definition) if and only if the graphs  $G_{\\mathbf{c}_1}^{(i,j)}$ and  $G_{\\mathbf{c}_2}^{(i,j)}$, for each pair $(i,j)$,\nhave the same number of cycles of a given length containing the edge $(v^{(i)}_{q-1+i \\pmod q}, v^{(j)}_{q-1+j \\pmod q})$ and also the same number of cycles of a given length containing the edge $(v^{(i)}_0,v^{(j)}_0)$.\n\n\n\n\n\n\nThe basic idea is to identify pairs of (minimal) codewords $\\mathbf{c}_1 \\in \\mathcal{C}(q_1,m)$ and $\\mathbf{c}_2 \\in \\mathcal{C}(q_2,m)$, $q_2 > q_1$ and $m$ fixed, with the same graphical cycle structure, since if they do not have the same graphical cycle structure, then it is likely (although not impossible when $q_1$ or $q_2$ is small)\nthat their support matrices \ncan not be instances of the same template support matrix. Then, for a pair of (minimal) codewords with the same graphical cycle structure, we would like to infer a template support matrix\nsuch that the instances for $q=q_1$ and $q=q_2$ are the support matrices (possibly column-permuted) of the codewords $\\mathbf{c}_1$ and $\\mathbf{c}_2$, respectively. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{comment}\n\n\\begin{example} \\label{ex:1}\nConsider the case $q=29$ and $m=6$. Using a computer search, we have found a stopping set of size $22$. The corresponding support matrix is\n\\begin{equation} \\label{eq:supportmatrix1}\n{\\tiny {\n\\left[ \\begin{smallmatrix}  \n0 & 0 & 1 & 1 & 6 & 6 & 7 & 7 & 9 & 9 & 20 & 20 & 18 & 18  & 21 & 21 & 26 & 26 & 27 & 27 & 28 & 28 \\\\\n0 & 5 & 16 & 25 & 3 & 27 & 3 & 18 & 14 & 5 & 25 & 11 & 0 & 9  & 11 & 16 & 27 & 18 & 9 & 14 & 0 & 14 \\\\\n0 & 10 & 2 & 20 & 0 & 19 & 28 & 0 & 19 & 1 & 1 & 2 & 11 & 0  & 1 & 11 & 28 & 10 & 20 & 1 & 1 & 0 \\\\\n0 & 15 & 17 & 15 & 26 & 11 & 24 & 11 & 24 & 26 & 6 & 22 & 22 & 20  & 20 & 6 & 0 & 2 & 2 & 17 & 2 & 15 \\\\\n0 & 20 & 3 & 10 & 23 & 3 & 20 & 22 & 0 & 22 & 11 & 13 & 4 & 11  & 10 & 1 & 1 & 23 & 13 & 4 & 3 & 1 \\\\\n0 & 25 & 18 & 5 & 20 & 24 & 16 & 4 & 5 & 18 & 16 & 4 & 15 & 2  & 0 & 25 & 2 & 15 & 24 & 20 & 4 & 16 \n\\end{smallmatrix}  \\right]}}\n\\end{equation}\nand the support matrix graph $G^{(0,1)}$ (corresponding to the first two rows) is shown in Fig.~\\ref{fig:graph1}. \nThere are two distinct cycles in the graph containing the edge $(v^{(0)}_{28},v^{(1)}_0)$, namely the cycle \n\\begin{displaymath}\n\\left(v^{(0)}_{28},v^{(1)}_0,v^{(0)}_{18},v^{(1)}_9,v^{(0)}_{27},v^{(1)}_{14},v^{(0)}_{28} \\right)\n\\end{displaymath}\n (indicated with dashed edges in Fig.~\\ref{fig:graph1}) and the cycle \n\\begin{displaymath}\n\\left( v^{(0)}_{28},v^{(1)}_0,v^{(0)}_0,v^{(1)}_5,v^{(0)}_9,v^{(1)}_{14},v^{(0)}_{28} \\right)\n\\end{displaymath}\n both of length $6$. Furthermore, for the support matrix\n\\begin{equation} \\label{eq:supportmatrix2}\n{\\tiny {\n\\left[ \\begin{smallmatrix}\n0 & 0 & 1 & 1 &  21 & 21 & 22 & 22 & 24 & 24 & 13 & 13 & 11 & 11 & 14 & 14  & 34 & 34 & 35 & 35 & 36 & 36 \\\\\n0 & 31 & 20 &  7 & 29 & 35 & 29 & 11 & 18 & 31 & 7 & 26 & 0 & 24 & 26 & 20  & 35 & 11 & 24 & 18 & 0 & 18 \\\\\n0 & 25 & 2 & 13 &  0 & 12 & 36 & 0 & 12 & 1 & 1 & 2 & 26 & 0  & 1 & 26 & 36 & 25 & 13 & 1 & 1 & 0 \\\\\n0 & 19 & 21 & 19 & 8 & 26 & 6 & 26 & 6 & 8 & 32 & 15 & 15 & 13  & 13 & 32  & 0 & 2 & 2 & 21 & 2 & 19 \\\\\n0 & 13 & 3 & 25 & 16 & 3 & 13 & 15 & 0 & 15 &  26 & 28 & 4 & 26  & 25 & 1  & 1 & 16 & 28 & 4 & 3 & 1 \\\\\n0 & 7 & 22 & 31 & 24 & 17 & 20 & 4 & 31 & 22 & 20 & 4 & 30 & 2 &  0 & 7  & 2 & 30 & 17 & 24 & 4 & 20 \n\\end{smallmatrix} \\right]}}\n\\end{equation}\ncorresponding to a stopping set of size $22$ for $q=37$, the corresponding cycles are \n\\begin{displaymath}\n\\left(v^{(0)}_{36},v^{(1)}_0,v^{(0)}_{11},v^{(1)}_{24},v^{(0)}_{35},v^{(1)}_{18},v^{(0)}_{36} \\right)\n\\end{displaymath}\nand \n\\begin{displaymath}\n\\left(v^{(0)}_{36},v^{(1)}_0,v^{(0)}_0,v^{(1)}_{31},v^{(0)}_{24},v^{(1)}_{18},v^{(0)}_{36} \\right)\n\\end{displaymath}\n both of length $6$. Thus, we get the same cycle lengths. Continuing with the remaining pairs of rows, $(i,j)=(0,2),(0,3),\\dots,(4,5)$, we get\nthe same cycle lengths for cycles containing the edge $(v^{(i)}_{q-1+i \\pmod{q}},v^{(j)}_{q-1+j \\pmod{q}})$ or the edge $(v^{(i)}_{0},v^{(j)}_{0})$  for both support matrices. Thus, we would expect that there exists a template support matrix which is consistent with both support matrices for $q=29$ and $37$, respectively.\nIn fact, by carefully looking at the support matrices in (\\ref{eq:supportmatrix1}) and (\\ref{eq:supportmatrix2}), we can observe that they are consistent with the template matrix shown in (\\ref{eq:supportm}) at the top of the  page where all entries should be reduced modulo $q$.\n\\end{example}\n\n\\end{comment}\n\n\\begin{figure}[tbp]\n  \\centerline{\\includegraphics[height=0.5\\columnwidth]{Graph3}}\n    \\caption{Support matrix graph $G^{(0,1)}$ for the support matrix in (\\ref{eq:supportmatrix1}). The cycle with dashed edges is the cycle $(v^{(0)}_{46},v^{(1)}_0,v^{(0)}_{0},v^{(1)}_{28},v^{(0)}_{5},v^{(1)}_{8},v^{(0)}_{11},v^{(1)}_{32},v^{(0)}_{6},v^{(1)}_{4},v^{(0)}_{46})$  of length $10$.}\n    \\label{fig:graph1}\n\\end{figure}\n\n\n\\begin{example} \\label{ex:1}\nConsider the case $q=47$ and $m=6$. Using a computer search, we have found a (minimal) codeword of weight $20$. The corresponding support matrix is\n\\begin{equation} \\label{eq:supportmatrix1}\n{\\tiny {\n\\left[ \\begin{smallmatrix}  \n0&\t42&\t46&\t5&\t36&\t46&\t37&\t31&\t11&\t5&\t43&\t6&\t37&\t0&\t43&\t42&\t36&\t31&\t11&\t6\\\\\t\n0&\t43&\t0&\t8&\t39&\t4&\t43&\t39&\t32&\t28&\t20&\t32&\t16&\t28&\t24&\t24&\t20&\t16&\t8&\t4\\\\\t\n0&\t44&\t1&\t11&\t42&\t9&\t2&\t0&\t6&\t4&\t44&\t11&\t42&\t9&\t5&\t6&\t4&\t1&\t5&\t2\\\\\t\n0&\t45&\t2&\t14&\t45&\t14&\t8&\t8&\t27&\t27&\t21&\t37&\t21&\t37&\t33&\t35&\t35&\t33&\t2&\t0\\\\\t\n0&\t46&\t3&\t17&\t1&\t19&\t14&\t16&\t1&\t3&\t45&\t16&\t0&\t18&\t14&\t17&\t19&\t18&\t46&\t45\\\\\t\n0&\t0&\t4&\t20&\t4&\t24&\t20&\t24&\t22&\t26&\t22&\t42&\t26&\t46&\t42&\t46&\t3&\t3&\t43&\t43\n\\end{smallmatrix}  \\right]}}\n\\end{equation}\nand the support matrix graph $G^{(0,1)}$ (corresponding to the first two rows) is shown in Fig.~\\ref{fig:graph1}. \nThere is one distinct cycle in the graph containing the edge $(v^{(0)}_{46},v^{(1)}_0)$, namely the cycle \n\\begin{equation} \\label{eq:cycle1}\n\\left(v^{(0)}_{46},v^{(1)}_0,v^{(0)}_{0},v^{(1)}_{28},v^{(0)}_{5},v^{(1)}_{8},v^{(0)}_{11},v^{(1)}_{32},v^{(0)}_{6},v^{(1)}_{4},v^{(0)}_{46} \\right)\n\\end{equation}\n (indicated with dashed edges in Fig.~\\ref{fig:graph1}) of length $10$.\nFurthermore, for the support matrix\n\\begin{equation} \\label{eq:supportmatrix2}\n{\\tiny {\n\\left[ \\begin{smallmatrix}\n0&\t54&\t58&\t5&\t48&\t58&\t49&\t43&\t11&\t5&\t55&\t6&\t49&\t0&\t55&\t54&\t48&\t43&\t11&\t6\\\\\t\n0&\t55&\t0&\t8&\t51&\t4&\t55&\t51&\t38&\t34&\t26&\t38&\t22&\t34&\t30&\t30&\t26&\t22&\t8&\t4\\\\\t\n0&\t56&\t1&\t11&\t54&\t9&\t2&\t0&\t6&\t4&\t56&\t11&\t54&\t9&\t5&\t6&\t4&\t1&\t5&\t2\\\\\t\n0&\t57&\t2&\t14&\t57&\t14&\t8&\t8&\t33&\t33&\t27&\t43&\t27&\t43&\t39&\t41&\t41&\t39&\t2&\t0\\\\\t\n0&\t58&\t3&\t17&\t1&\t19&\t14&\t16&\t1&\t3&\t57&\t16&\t0&\t18&\t14&\t17&\t19&\t18&\t58&\t57\\\\\t\n0&\t0&\t4&\t20&\t4&\t24&\t20&\t24&\t28&\t32&\t28&\t48&\t32&\t52&\t48&\t52&\t56&\t56&\t55&\t55\t\n\\end{smallmatrix} \\right]}}\n\\end{equation}\ncorresponding to a (minimal) codeword of weight $20$ for $q=59$ (and $m=6$), the corresponding cycle (also of length $10$) is \n\\begin{equation} \\label{eq:cycle2}\n\\left(v^{(0)}_{58},v^{(1)}_0,v^{(0)}_{0},v^{(1)}_{34},v^{(0)}_{5},v^{(1)}_{8},v^{(0)}_{11},v^{(1)}_{38},v^{(0)}_{6},v^{(1)}_{4},v^{(0)}_{58} \\right).\n\\end{equation}\nThus, we get the same cycle lengths. Continuing with the remaining pairs of rows, $(i,j)=(0,2),(0,3),\\dots,(4,5)$, we get\nthe same cycle lengths for cycles containing the edge $(v^{(i)}_{q-1+i \\pmod{q}},v^{(j)}_{q-1+j \\pmod{q}})$ or the edge $(v^{(i)}_{0},v^{(j)}_{0})$  for both support matrices. Thus, we would expect that there might exist a template support matrix whose instances (possibly column-permuted) for $q=47$ and $q=59$ are the support matrices in (\\ref{eq:supportmatrix1}) and (\\ref{eq:supportmatrix2}), respectively.\n\\end{example}\n\n\n\n\\begin{algorithm}[tbp] \n\\caption{Template Support Matrix Inference}\n\\label{alg:tsmi}\n\\begin{algorithmic}[1]\n\\small\n\\STATE $\/*$ Fill in entries in the candidate template support matrix in (\\ref{eq:support_matrix_gen}) based on the support matrices of two (minimal) codewords $\\mathbf{c}_1 \\in \\mathcal{C}(q_1,m)$ and $\\mathbf{c}_2 \\in \\mathcal{C}(q_2,m)$, $q_2 > q_1$, of the same Hamming weight and with the same graphical cycle structure.\\\\\n{\\bf Input:} Row indices $i$ and $j$, and a pair of cycles $(\\mathbf{v}_1,\\mathbf{v}_2)$ (of the same length $2l$) as defined in (\\ref{eq:cycle1gen}) and (\\ref{eq:cycle2gen}).\\\\\n{\\bf Output:} A (partial) candidate template support matrix as defined in (\\ref{eq:support_matrix_gen}), and a (partial) permutation $\\pi(\\cdot)$. $*\/$\n\\STATE Assign to $\\mathcal{I}$ all integers in $\\{1,\\dots,I\\}$ for some integer $I \\geq 1$.\n\\FOR{$r \\leftarrow 0$ to $2l-1$}\n\\STATE Find an index pair $(a,b)$ (which is also unique) such that $a \\in \\psi_1(v^{(\\gamma)}_{\\alpha_{1,r}}) \\cap \\psi_1(v^{(\\delta)}_{\\alpha_{1,r+1}})$ and $b \\in \\psi_2(v^{(\\gamma)}_{\\alpha_{2,r}}) \\cap \\psi_2(v^{(\\delta)}_{\\alpha_{2,r+1}})$, where $\\gamma = i$ and $\\delta=j$ if $r$ is even, and $\\gamma = j$ and $\\delta=i$ if $r$ is odd.\n\\STATE Solve the two systems of equations\n\\begin{align}\nx^{(1)}_a+\\gamma y^{(1)}_a \\pmod {q_1} &= \\alpha_{1,r} \\notag \\\\\nx^{(1)}_a+\\delta y^{(1)}_a \\pmod {q_1} &= \\alpha_{1,r+1} \\notag\n\\end{align}\nand\n\\begin{align}\nx^{(2)}_b+\\gamma y^{(2)}_b \\pmod {q_2} &= \\alpha_{2,r} \\notag \\\\\nx^{(2)}_b+\\delta y^{(2)}_b \\pmod {q_2} &= \\alpha_{2,r+1} \\notag\n\\end{align}\n\\STATE Find the integers $k_x$ and $k_y$ in $\\mathcal{I}$ that give the \\emph{simplest} solutions (for $x$ and $y$, modulo $q_1 q_2$) to the two systems of congruences\n\\begin{align}\nx &\\equiv k_x \\cdot x^{(1)}_a \\pmod {q_1}  \\notag \\\\\nx &\\equiv k_x \\cdot x^{(2)}_b \\pmod {q_2}  \\notag \n\\end{align}\nand\n\\begin{align}\ny &\\equiv k_y \\cdot y^{(1)}_a \\pmod {q_1}  \\notag \\\\\ny &\\equiv k_y \\cdot y^{(2)}_b \\pmod {q_2}  \\notag \n\\end{align}\n\\IF{$|x| \\leq |x-q_1 q_2|$}\n\\STATE $\\tilde{x} \\leftarrow x \\cdot k_x^{-1}$\\\\ \n\\ELSE \n\\STATE $\\tilde{x} \\leftarrow (x-q_1 q_2) \\cdot k_x^{-1}$\\\\ \n\\ENDIF\n\\IF{$|y| \\leq |y-q_1 q_2|$}\n\\STATE $\\tilde{y} \\leftarrow y \\cdot k_y^{-1}$ \\\\%F_{v,2}(y^{(1)}_b,y^{(2)}_b; q_1,q_2,\\mathcal{P})$ \\\\\n\\ELSE\n\\STATE $\\tilde{y} \\leftarrow (y -q_1 q_2) \\cdot k_y^{-1}$ \\\\%F_{v,2}(y^{(1)}_b,y^{(2)}_b; q_1,q_2,\\mathcal{P})$ \\\\\n\\ENDIF\n\\IF{$x_a = \\ast$ (and $y_a = \\ast$)}\n\\STATE $x_a \\leftarrow \\tilde{x}$, $y_a \\leftarrow \\tilde{y}$, \n$\\pi(b) \\leftarrow a$, and go to Step 3.\n\\ELSIF{$x_a \\neq x$ or $y_a \\neq  y$}\n\\STATE an inconsistency has occurred. Exit.\n\\ENDIF\n\\ENDFOR\n\\end{algorithmic}\n\\end{algorithm} \n\n\n\\subsection{Second Step: Inferring a Candidate Template Support Matrix} \\label{sec:infer}\n\nIn this subsection, we consider the second step of the procedure, i.e., to infer a candidate template support matrix from two minimal codewords with the same graphical cycle structure. This is done by solving simple $2$-by-$2$ equation systems and congruences.\nWe remark that this procedure will give a \\emph{candidate} template support matrix, since we formally need to prove that the resulting matrix is a template support matrix.\n\nNow, let \n\\begin{equation} \\label{eq:cycle1gen}\n\\mathbf{v}_1 = \n(v_{\\alpha_{1,0}}^{(i)}, v_{\\alpha_{1,1}}^{(j)}, \\dots, v_{\\alpha_{1,2l-2}}^{(i)}, v_{\\alpha_{1,2l-1}}^{(j)}, v_{\\alpha_{1,2l}}^{(i)})\n\\end{equation}\n denote a cycle of length $2l$, where $\\alpha_{1,0} = \\alpha_{1,2l}$, in the support matrix graph $G_{\\mathbf{c}_1}^{(i,j)}$\ncomputed from a given minimal codeword $\\mathbf{c}_1 \\in \\mathcal{C}(q_1,m)$. In a similar manner, we denote by \n\\begin{equation} \\label{eq:cycle2gen}\n\\mathbf{v}_2 = \n(v_{\\alpha_{2,0}}^{(i)}, v_{\\alpha_{2,1}}^{(j)}, \\dots, v_{\\alpha_{2,2l-2}}^{(i)}, v_{\\alpha_{2,2l-1}}^{(j)}, v_{\\alpha_{2,2l}}^{(i)})\n\\end{equation}\nwhere $\\alpha_{2,0} = \\alpha_{2,2l}$,  cycle of length $2l$ in the support matrix graph $G_{\\mathbf{c}_2}^{(i,j)}$\ncomputed from a given minimal codeword $\\mathbf{c}_2 \\in \\mathcal{C}(q_2,m)$, where $q_2 > q_1$. We assume here that $\\mathbf{c}_1$ and $\\mathbf{c}_2$ have the same Hamming weight and also the same graphical cycle structure. Now, the purpose is to infer the entries in a matrix\n\\begin{equation} \\label{eq:support_matrix_gen}\n\\left[ \\begin{smallmatrix}\nx_0 & x_1 & \\cdots & x_{w-1} \\\\\nx_0+y_0 & x_1+y_1 & \\cdots &x_{w-1}+y_{w-1} \\\\\n\\cdots & \\cdots & \\cdots & \\cdots \\\\\nx_0+(m-1)y_0 & x_1+(m-1)y_1 & \\cdots &x_{w-1}+(m-1)y_{w-1}\n\\end{smallmatrix}\n\\right]\n\\end{equation}\nwhere $w$ is the Hamming weight of $\\mathbf{c}_1$ and $\\mathbf{c}_2$, and \nsuch that the instances of the matrix for $q=q_1$ and $q=q_2$ are the support matrices (possibly column-permuted)  of $\\mathbf{c}_1$ and $\\mathbf{c}_2$, respectively. \n\n\n\n\n\n\n\nAlgorithm~\\ref{alg:tsmi} presents such an algorithm, where $\\psi_1(v^{(i)}_{\\alpha_1})$ (resp.\\ $\\psi_2(v^{(i)}_{\\alpha_2})$) denotes the set of column indices of the support matrix of $\\mathbf{c}_1$ (resp.\\ $\\mathbf{c}_2$) containing the entry ${\\alpha_1}$ (resp.\\ ${\\alpha_2}$) in the $i$th row. \nAll entries in the resulting matrix (after applying Algorithm~\\ref{alg:tsmi}) should be reduced modulo $q$ to get an instance for a specific value of $q$. The algorithm works on two cycles of the same length, one from the support matrix graph of a minimal codeword $\\mathbf{c}_1 \\in \\mathcal{C}(q_1,m)$ and the other from the support matrix graph of a minimal codeword $\\mathbf{c}_2 \\in \\mathcal{C}(q_2,m)$, where $q_2 > q_1$. The purpose is to fill in the entries in a candidate template support matrix, which initially is filled  with erasures denoted by $\\ast$. Furthermore, the algorithm also updates a permutation $\\pi(\\cdot)$ which gives the index mapping that should be applied to the columns of the support matrix of $\\mathbf{c}_2$ to get the instance of the candidate template support matrix for $q=q_2$. The algorithm is run on pairs of cycles (both containing either the edge $(v^{(i)}_{q-1+i \\pmod q},v^{(j)}_{q-1+j \\pmod q})$ or the edge $(v^{(i)}_0,v^{(j)}_0)$ as the left-most or first edge in the cycle) until all entries are filled in. In Step 4 of the algorithm, an index pair $(a,b)$ is identified, where $a$ (resp.\\ $b$) is the index of the column in the support matrix of $\\mathbf{c}_1$ (resp.\\ $\\mathbf{c}_2$) containing $\\alpha_{1,r}^{(\\gamma)}$ (resp.\\ $\\alpha_{2,r}^{(\\gamma)}$) as the $\\gamma$th entry and $\\alpha_{1,r+1}^{(\\delta)}$ (resp.\\ $\\alpha_{2,r+1}^{(\\delta)}$) as the $\\delta$th entry. Later in Step 18 of the algorithm, these two indices are used to fill the permutation $\\pi$ ($\\pi(b) \\leftarrow a$). Actually, the index pairs $(a,b)$ can be computed in a preprocessing stage before the algorithm is even run, since they are available by simple cycle analysis.\n\nIn Steps 5 and 6 of the algorithm we determine the entries in column $a$ (of the candidate template support matrix) based on the two cycles. In Step 5 we first determine the actual values for $x$ and $y$ modulo $q_1$ (denoted by $x_a^{(1)}$ and $y_a^{(1)}$, respectively) for column $a$ of the support matrix of $\\mathbf{c}_1$, and then the corresponding values modulo $q_2$, now in column $b$ (denoted by $x_b^{(2)}$ and $y_b^{(2)}$, respectively), of the support matrix of $\\mathbf{c}_2$. Then, in Step 6, we find the \\emph{simplest} template solutions, i.e., the \\emph{simplest}  expressions for both $x$ and $y$ such that they both evaluate modulo $q$ (for $q=q_1$ and $q=q_2$) to the correct values as given by the support matrices of the codewords $\\mathbf{c}_1$ and $\\mathbf{c}_2$, respectively, and then the entries for $x_a$ and $y_a$ are filled in the candidate template support matrix as defined in (\\ref{eq:support_matrix_gen})\nand \nas indicated in Steps 7 to 18. Note that in Steps 8 and 10 neither the inverse nor the product operation are performed and the formal string of three characters;  $x$ (or $x-q_1q_2$ in Step 10) (with a specific value inserted for $x$ (or $x-q_1 q_2$ evaluated for a specific value of $x$ in Step 10)), $\\cdot$, and $k_x^{-1}$  (with a specific value inserted for $k_x$), is assigned to $\\tilde{x}$. A similar comment applies to the assignments in Steps 13 and 15. \nBy the \\emph{simplest} solutions we mean the values for $k_x \\in \\mathcal{I}$ and $k_y \\in \\mathcal{I}$ such that, respectively, $\\max(|k_x|, \\min(|x|, |x-q_1 q_2|))$ and  $\\max(|k_y|, \\min(|y|, |y-q_1 q_2|))$ are as small as possible. We remark here that this is only to find a candidate template support matrix with a nice\/compact representation, which also makes it easier to prove analytically that all instances (possibly column-permuted) are indeed valid support matrices of codewords for all values of $q$ larger than or equal to some $\\tilde{q}$, when $\\tilde{q}$ is small. In any case, for any practical value of $\\tilde{q}$, regardless of the representation, a simple and fast computer search can be used to prove that the candidate template support matrix indeed gives a valid support matrix for all values of $q_0 \\leq q < \\tilde{q}$, for some $q_0$. For details, see the proofs of Theorems~\\ref{th:1} and \\ref{th:2} in Sections~\\ref{sec:bound} and \\ref{sec:bound_m7}, respectively.  Note that if a specific pair of values $(k_x, k_y) \\in \\mathcal{I}^2$ gives a valid template support matrix (which of course requires a formal proof), then all pairs $(k_x,k_y) \\in \\mathcal{I}^2$ will give a valid template support matrix, possibly with different values for $q_0$. Interestingly, however, is the fact that the template support matrices will  not necessarily be equivalent, in the sense that the instances of the different templates, for a given value of $q$ different from  $q_1$ and $q_2$, will not necessarily give the same codeword from $\\mathcal{C}(q,m)$.  In conclusion, a template support matrix is \\emph{not} unique, even if it is based on the same pair of codewords.\n\nNote that Algorithm~\\ref{alg:tsmi} does in fact identify a one-to-one mapping (through the permutation $\\pi(\\cdot)$) between the columns of the support matrices of the codewords $\\mathbf{c}_1 \\in \\mathcal{C}(q_1,m)$ and $\\mathbf{c}_2 \\in \\mathcal{C}(q_2,m)$ by \\emph{matching} cycles in the corresponding support matrix graphs. Then, the template values for $x$ and $y$ are established by \\emph{matching} columns (and solving equations and congruences independently for each column) through this one-to-one mapping. It is in fact this particular one-to-one mapping (as opposed to an arbitrary mapping) that make sure that the resulting candidate template support matrix have entries that appear an even number of times (the codeword case) or at least two times (the stopping set case) in each row.\n\nIn principle, one type of error condition can occur, i.e., we can exit in Step 20. \nThis happens when a previous pair of cycles has determined the entries in column $a$ and then the current pair of cycles gives different values.\nIf the algorithm exits in Step 20, we need to start from scratch by considering a  different pair of minimal codewords $\\mathbf{c}_1 \\in \\mathcal{C}(q_1,m)$ and $\\mathbf{c}_2 \\in \\mathcal{C}(q_2,m)$ of the same Hamming weight and with the same graphical cycle structure, and revert (back to erasures) all the entries filled in so far in the candidate template support matrix.\n\nIn Step 5 of Algorithm~\\ref{alg:tsmi} two systems of equations need to be solved. They have the following solutions:\n\\begin{align}\nx^{(1)}_a &= \\alpha_{1,r}-\\gamma (\\delta-\\gamma)^{-1} (\\alpha_{1,r+1}-\\alpha_{1,r}) \\pmod {q_1} \\notag \\\\\ny^{(1)}_a &= (\\delta-\\gamma)^{-1}(\\alpha_{1,r+1}-\\alpha_{1,r}) \\pmod {q_1} \\notag \\\\\nx^{(2)}_b &= \\alpha_{2,r}-\\gamma (\\delta-\\gamma)^{-1} (\\alpha_{2,r+1}-\\alpha_{2,r}) \\pmod {q_2} \\notag \\\\\ny^{(2)}_b &= (\\delta-\\gamma)^{-1}(\\alpha_{2,r+1}-\\alpha_{2,r}) \\pmod {q_2} \\notag\n\\end{align}\nwhich also gives the rationale behind the assignment to the set of integers $\\mathcal{I}$ in Step 2 of the algorithm,\n since the solutions involve a multiplication by $(\\delta-\\gamma)^{-1}$.\n\nIn Step 6 of Algorithm~\\ref{alg:tsmi} two systems of congruences need to be solved. They have the following solutions:\n\\begin{align}\nx &= k_x (x_a^{(1)}+q_1 \\cdot \\kappa (x_b^{(2)}-x_a^{(1)})) \\pmod {q_1 q_2} \\label{eq:congu1} \\\\\ny &= k_y (y_a^{(1)}+q_1 \\cdot \\kappa (y_b^{(2)}-y_a^{(1)})) \\pmod {q_1 q_2} \\label{eq:congu2} \n\\end{align}\nmodulo $q_1 q_2$,\nwhere $\\kappa$ can be found using the extended Euclidean algorithm which yields integers $\\kappa$ and $\\eta$ such that $\\kappa \\cdot q_1 + \\eta \\cdot q_2 = \\gcd(q_1,q_2)=1$.\n\nWe will illustrate the procedure in Example~\\ref{ex:2} below.\n\n\n\n\n\n\n\\begin{example} \\label{ex:2}\nConsider the two cycles in (\\ref{eq:cycle1}) and (\\ref{eq:cycle2}) for $q=47$ and $q=59$, respectively. Here, $i=0$ and $j=1$, and $\\kappa = -5$ and $\\eta=4$ (since $-5 \\cdot 47 + 4 \\cdot 59 = 1$). For $r=0$ (see Step 3 in Algorithm~\\ref{alg:tsmi}), $\\alpha_{1,r} = \\alpha_{1,0} = 46$, $\\alpha_{1,r+1} = \\alpha_{1,1} = 0$, $\\alpha_{2,r} = \\alpha_{2,0} = 58$, $\\alpha_{2,r+1} = \\alpha_{2,1} = 0$, $\\gamma=i=0$, and $\\delta=j=1$. Since $46$ appears in the first row and $0$ in the second row of the third column (column index $2$) of the support matrix in (\\ref{eq:supportmatrix1}), $a=2$. Similarly,  $b=2$, since $58$ appears in the first row and $0$ in the second row of the third column  of the support matrix in (\\ref{eq:supportmatrix2}). This completes Step 4 of the algorithm, and we get the  solutions\n\\begin{align}\nx_{2}^{(1)} &= 46 - 0 \\cdot (1-0)^{-1} (0-46) \\pmod {47} = 46 \\notag \\\\\ny_{2}^{(1)} &= (1-0)^{-1} (0-46) \\pmod {47} = 1 \\notag \\\\\nx_{2}^{(2)} &= 58 - 0 \\cdot (1-0)^{-1} (0-58) \\pmod {59} = 58 \\notag \\\\\ny_{2}^{(2)} &= (1-0)^{-1} (0-58) \\pmod {59} = 1 \\notag\n\\end{align}\nin Step 5, \nfrom which we can calculate the following solutions for $x$ and $y$ in Step 6 (with $I=m-1=5$), using (\\ref{eq:congu1}) and (\\ref{eq:congu2}), respectively:\n\\begin{displaymath}\n\\begin{tabular}{r|rrrrrr}\n$k_x$\/$k_y$ & $1$ & $2$ & $3$ & $4$ & $5$ \\\\\n\\hline\n$x$ & $2772$ & $2771$ & $2770$ & $2769$ & $2768$ \\\\\n$x-q_1 q_2$ & $-1$ & $-2$&$-3$ &$-4$ & $-5$ \\\\\n$y$ & $1$ & $2$&$3$ &$4$ & $5$ \\\\\n$y-q_1 q_2$ & $-2772$ & $-2771$ & $-2770$ & $-2769$ & $-2768$\n\\end{tabular}\n\\end{displaymath}\nThus, we can fill in $\\pi(2)=2$, $x_{2}=-1$, and $y_{2}=1$ (corresponding to the values $k_x=1$ and $k_y=1$, which give the simplest solutions).\n\nIn a similar manner, for instance for $r=3$, we get $\\alpha_{1,r} = \\alpha_{1,3} = 28$, $\\alpha_{1,r+1} = \\alpha_{1,4} = 5$, $\\alpha_{2,r} = \\alpha_{2,3} = 34$, $\\alpha_{2,r+1} = \\alpha_{2,4} = 5$, $\\gamma=j=1$, and $\\delta=i=0$. For this case we have $a=9$ and $b=9$ (from Step 4), and the solutions\n\\begin{align}\nx_{9}^{(1)} &= 28 - 1 \\cdot (0-1)^{-1} (5-28) \\pmod {47} = 5 \\notag \\\\\ny_{9}^{(1)} &= (0-1)^{-1} (5-28) \\pmod {47} = 23 \\notag \\\\\nx_{9}^{(2)} &= 34 - 1 \\cdot (0-1)^{-1} (5-34) \\pmod {59} = 5 \\notag \\\\\ny_{9}^{(2)} &= (0-1)^{-1} (5-34) \\pmod {59} = 29 \\notag\n\\end{align}\n in Step 5, \nfrom which we can calculate the following solutions for $x$ and $y$ in Step 6 (with $I=m-1=5$), using (\\ref{eq:congu1}) and (\\ref{eq:congu2}), respectively:\n\\begin{displaymath}\n\\begin{tabular}{r|rrrrrr}\n$k_x$\/$k_y$ & $1$ & $2$ & $3$ & $4$ & $5$ \\\\\n\\hline\n$x$ & $5$ & $10$ & $15$ & $20$ & $25$ \\\\\n$x-q_1 q_2$ & $-2768$ & $-2763$ & $-2758$ & $-2753$ & $-2748$ \\\\\n$y$ & $1386$ & $2772$ & $1385$ & $2771$ & $1384$ \\\\\n$y-q_1 q_2$ & $-1387$ & $-1$ & $-1388$ & $-2$ & $-1389$ \n\\end{tabular}\n\\end{displaymath}\nThus, we can fill in $\\pi(9)=9$, $x_{9}=5$, and $y_{9}=-2^{-1}$ (corresponding to the values $k_x=1$ and $k_y=2$, which give the simplest solutions).\n\nContinuing with the rest of the values for $r$ (see Step 3 in Algorithm~\\ref{alg:tsmi}) a total of $10$ (the cycle length) columns of the candidate template support matrix can be determined. To determine the rest of the entries in the  matrix, other cycle pairs must be considered. \nFor instance, by looking at the graphs $G^{(0,2)}$, we find the cycles\n\\begin{equation} \\notag\n\\left(v^{(0)}_{46},v^{(2)}_1,v^{(0)}_{31},v^{(2)}_{0},v^{(0)}_{0},v^{(2)}_{9},v^{(0)}_{46} \\right)\n\\end{equation}\nand\n\\begin{equation} \\notag\n\\left(v^{(0)}_{58},v^{(2)}_1,v^{(0)}_{43},v^{(2)}_{0},v^{(0)}_{0},v^{(2)}_{9},v^{(0)}_{58} \\right)\n\\end{equation}\nfor $q=47$ and $q=59$, respectively. Choose $r=1$, from which we get  $\\alpha_{1,r} = \\alpha_{1,1} = 1$, $\\alpha_{1,r+1} = \\alpha_{1,2} = 31$, $\\alpha_{2,r} = \\alpha_{2,1} = 1$, $\\alpha_{2,r+1} = \\alpha_{2,2} = 43$, $\\gamma=j=2$, and $\\delta=i=0$. For this case we have $a=17$ and $b=17$ (from Step 4), and the solutions\n\\begin{align}\nx_{17}^{(1)} &= 1 - 2 \\cdot (0-2)^{-1} (31-1) \\pmod {47} = 31 \\notag \\\\\ny_{17}^{(1)} &= (0-2)^{-1} (31-1) \\pmod {47} = 32 \\notag \\\\\nx_{17}^{(2)} &= 1 - 2 \\cdot (0-2)^{-1} (43-1) \\pmod {59} = 43 \\notag \\\\\ny_{17}^{(2)} &= (0-2)^{-1} (43-1) \\pmod {59} = 38 \\notag\n\\end{align}\nin Step 5, \nfrom which we can calculate the following solutions for $x$ and $y$ in Step 6 (with $I=m-1=5$), using (\\ref{eq:congu1}) and (\\ref{eq:congu2}), respectively:\n\\begin{displaymath}\n\\begin{tabular}{r|rrrrrr}\n$k_x$\/$k_y$ & $1$ & $2$ & $3$ & $4$ & $5$ \\\\\n\\hline\n$x$ & $2757$ & $2741$ & $2725$ & $2709$ & $2693$ \\\\\n$x-q_1 q_2$ & $-16$ & $-32$ & $-48$ & $-64$ & $-80$ \\\\\n$y$ & $1395$ & $17$ & $1412$ & $34$ & $1429$ \\\\\n$y-q_1 q_2$ & $-1378$ & $-2756$ & $-1361$ & $-2739$ & $-1344$\n\\end{tabular}\n\\end{displaymath}\nThus, we can fill in $\\pi(17)=17$, $x_{17}=-16$, and $y_{17}=17 \\cdot 2^{-1}$ (corresponding to the values $k_x=1$ and $k_y=2$, which give the simplest solutions).\n Continuing (by considering more cycle pairs) we can determine the rest of the columns,  and we end up with the candidate template support matrix shown in (\\ref{eq:supportm}) at the top of the page, where all entries should be reduced modulo $q$ to get an instance for a specific value of $q$. The remaining detailed calculations are omitted for brevity.\n\n\n\n\\end{example}\n\n\n\\begin{figure*}[!t]\n\\normalsize \\setcounter{mytempeqncnt}{\\value{equation}}\n\\setcounter{equation}{12}\n\\begin{equation} \\label{eq:supportm}\n{\\tiny {\n\\left[ \\begin{smallmatrix}\n0 & -5 & -1 & 5 & -11 & -1 & -10 & -16 & 11 & 5 & -4 & 6 & -10 & 0 & -4 & -5 & -11 & -16 & 11 & 6 \\\\\n0 & -4 &  0 & 8 & -8 & 4 & -4 & -8 & 17 \\cdot 2^{-1} & 9 \\cdot 2^{-1} & -7 \\cdot 2^{-1} & 17 \\cdot 2^{-1} & -15 \\cdot 2^{-1} & 9 \\cdot 2^{-1} & 2^{-1} & 2^{-1} & -7 \\cdot 2^{-1} & -15 \\cdot 2^{-1} & 8 & 4\\\\\n0 & -3 & 1 & 11 & -5 & 9 & 2 & 0 & 6 & 4 & -3 & 11 & -5 & 9 & 5 & 6 & 4 & 1 & 5 & 2 \\\\\n0 & -2 & 2 & 14 & -2 & 14 & 8 & 8 & 7 \\cdot 2^{-1} & 7 \\cdot 2^{-1} & -5 \\cdot 2^{-1} & 27 \\cdot 2^{-1} & -5 \\cdot 2^{-1} & 27 \\cdot 2^{-1} & 19 \\cdot 2^{-1} & 23 \\cdot 2^{-1} & 23 \\cdot 2^{-1} & 19 \\cdot 2^{-1} & 2&  0\\\\\n0 & -1 & 3 & 17 & 1 & 19 & 14 & 16 & 1 & 3 & -2 & 16 & 0 & 18 & 14 & 17 & 19 & 18 & -1 & -2 \\\\\n0 & 0 & 4 & 20 & 4 & 24 & 20 & 24 & -3 \\cdot 2^{-1} & 5 \\cdot 2^{-1} & -3 \\cdot 2^{-1} & 37 \\cdot 2^{-1} & 5 \\cdot 2^{-1} & 45 \\cdot 2^{-1} & 37 \\cdot 2^{-1} & 45 \\cdot 2^{-1} & 53 \\cdot 2^{-1} & 53 \\cdot 2^{-1} & -4 & -4\n\\end{smallmatrix} \\right]}}\n\\end{equation}\n\\setcounter{equation}{\\value{mytempeqncnt}}\n \\hrulefill\n\\vspace*{-2mm}\n\\end{figure*}\n\n\n\n\\subsection{Third Step: A Formal Proof} \\label{sec:formal}\n\nThe third step is showing that the candidate template support matrix is indeed a valid template support matrix for some parameter $q_0$, i.e., the instances for $q \\geq q_0$ (possibly column-permuted) are all valid support matrices of codewords from $\\mathcal{C}(q,m)$. In fact it is sufficient (to prove an upper bound on the minimum distance) to show that the instances for $q \\geq q_0$ (possibly column-permuted) all contain as submatrices valid support matrices of codewords from $\\mathcal{C}(q,m)$. In the case an instance (possibly column-permuted) contains as a proper submatrix a valid support matrix of a codeword, the established upper bound is obviously not tight. In particular, we need to show, for any value of $q \\geq q_0$, for some $q_0$, that\n\\begin{enumerate}\n\\item all entries in a row occur an even number of times, \n\\item all columns in the matrix are in fact valid columns in an array LDPC code parity-check matrix, and \n\\item at least two columns are distinct modulo $q$ and appear an odd number of times.\n\\end{enumerate}\nNote that the first item above will always be satisfied if Algorithm~\\ref{alg:tsmi} indeed produces a complete candidate template support matrix, since the entries in the matrix are filled in by solving systems of equations and congruences that are based on cycles (see the discussion in Section~\\ref{sec:infer}). Furthermore, the second item is also always satisfied, since by construction all columns are of the form in (\\ref{eq:form}), for some $x$ and $y$, and all possible values for $x$ and $y$ will give a valid column\n(see the discussion following (\\ref{eq:form})). Thus, only the third item above needs to be explicitly checked if in fact the candidate template support matrix was produced by Algorithm~\\ref{alg:tsmi}.\n\n\n\n\\subsection{Applicability}\n\nThe heuristic outlined above in Sections~\\ref{sec:structure} through \\ref{sec:formal} is very general and can be applied for any pair of values $(q,m)$. However, the difficult part is finding low-weight\/small-size candidate codewords\/stopping sets for different values of $q$, which is increasingly difficult when $m$ grows, since the minimum\/stopping distance increases with $m$. For this we have used the algorithm in \\cite{ros09,ros12}, and the minimum distance probabilistic algorithm in \\cite{tom07}.\n\nIn this work, we have applied the heuristic for $m=6$ and $m=7$, but remark that it will easily provide the upper bounds $d(q,4) \\leq 10$ and $d(q,5) \\leq 12$ which can be found in the literature \\cite{sug08}.\n\n\n\\subsection{Improved Algorithm} \\label{sec:improved}\n\nThe basic algorithm from Sections~\\ref{sec:structure} and \\ref{sec:infer} can be improved in the sense of increasing its probability of success, i.e., finding a valid template support matrix. The key observation in this respect is that even though the two codewords $\\mathbf{c}_1 \\in \\mathcal{C}(q_1,m)$ and $\\mathbf{c}_2 \\in \\mathcal{C}(q_2,m)$ do not have the same graphical cycle structure, their support matrices (possibly column-permuted) may still be instances of the same template matrix. The reason is that different entries in the template matrix may reduce to the same value modulo $q$ for different values of $q$. This typically happens when either $q_1$ or $q_2$ is small. A simple way to deal with such scenarios is by relaxing the condition that $\\mathbf{c}_1$ and $\\mathbf{c}_2$ should have \\emph{exactly} the same graphical cycle structure. In particular, it may be sufficient to require that the \\emph{minimum} cycle length of all cycles  containing the edge $(v^{(i)}_{q-1+i \\pmod q},v^{(j)}_{q-1+j \\pmod q})$ and the \\emph{minimum} cycle length of all cycles containing the edge $(v^{(i)}_0,v^{(j)}_0)$ are the same for both support matrix graphs $G_{\\mathbf{c}_1}^{(i,j)}$ and $G_{\\mathbf{c}_2}^{(i,j)}$, $0 \\leq i < j \\leq m-1$, and then run Algorithm~\\ref{alg:tsmi} on such pairs of cycles (which have the same length).\n\n\n\\section{Upper Bound on $d(q,6)$} \\label{sec:bound}\n\nBy using the heuristic from Section~\\ref{sec:algo}\nwe have found the \\emph{candidate} template support matrix in (\\ref{eq:supportm}) shown at the top of the previous page. All entries in the matrix should be reduced modulo $q$. At this stage we emphasis that this is a \\emph{candidate} template support matrix, since we need to formally prove that the matrix is a template support matrix.  In particular, we have used the procedure from Section~\\ref{sec:infer} to infer the  matrix in (\\ref{eq:supportm})  from the codewords of Example~\\ref{ex:1}, which have the same graphical cycle structure. Also, in Example~\\ref{ex:2}, some of the columns in the matrix in (\\ref{eq:supportm}) were explicitly determined. The rest of the columns can be determined in a similar manner. Details are omitted for brevity.\nWe can now prove the following theorem.\n\n\\begin{theorem} \\label{th:1}\nThe minimum distance $d(q,6)$ is upper-bounded by $20$ for $q > 7$.\n\\end{theorem}\n\n\\begin{IEEEproof}\nThe proof is based on the candidate template support matrix in (\\ref{eq:supportm}). As explained in Section~\\ref{sec:formal}, there are three conditions that need to be checked. Also, if the candidate template support matrix was indeed produced by Algorithm~\\ref{alg:tsmi}, only the third condition needs to be explicitly checked. For completeness and for providing a formal proof, we will however check all three conditions. Obviously, computing an upper bound on the minimum distance from a template support matrix based on a codeword is easy; the upper bound is just the number  of columns in the matrix. Thus, establishing that the matrix in  (\\ref{eq:supportm}) is a valid template support matrix, in the sense that all instances (possibly column-permuted) for $q > 7$ contain the support matrix of a codeword as a submatrix, establishes the upper bound of $20$, since there $20$ columns in the matrix.\n\nIt is easy to check  that each entry in each row of the matrix appears exactly twice, which means that the result is true if for any value of $q > 7$ \n\\begin{enumerate}\n\\item[2)] all columns in the matrix are in fact valid columns in an array LDPC code parity-check matrix, and\n\\item[3)] at least two columns are distinct modulo $q$ and appear an odd number of times.\n\\end{enumerate}\n\nSince all columns in the matrix in (\\ref{eq:supportm})  are of the form in (\\ref{eq:form}), it follows that they are all valid columns in an array LDPC code parity-check matrix (see the discussion following (\\ref{eq:form})). \nIn particular, the values for $x,y$ for the first $6$ columns are\n\n\\begin{displaymath}\n\\begin{tabular}{c|cccccc}\n$x$&$0$&$-5$&$-1$&$5$&$-11$&$-1$\\\\\n\\hline\n$y$&$0$&$1$&$1$&$3$&$3$&$5$\n\\end{tabular}.\n\\end{displaymath}\n\nFor the third part of the proof, we need to show, for any value of $q > 7$, that there exist (at least two) columns in the parity-check matrix which are not identical modulo $q$ and appear an odd number of times. This is simple  (and very fast) to check by a computer search for any finite value of $q$ that would be of any practical value.  It is only for large values of $q$ that the theoretical proof below is needed.\n\nNote that the maximum absolute value of the entries in the first row of the  matrix in (\\ref{eq:supportm})  is $16$. \nThus, the only possibility for repeated columns,  when $q > 2 \\cdot 16 = 32$, is for two \\emph{neighboring} columns (with identical entries in the first row) to be the same. \nHowever, by looking at the third row in the matrix, this possibility can be ruled out by requiring that $q$ is larger than twice the maximum absolute value of the entries in the third row, i.e., by requiring $q > 2 \\cdot 11 = 22$. \nIn summary, it follows that there are no identical columns in the matrix in (\\ref{eq:supportm})  if $q > \\max(32,22)=32$. \nFurthermore, for values of $7 < q < 32$, it can be checked numerically that there are no repeated columns in (\\ref{eq:supportm}), and the result follows. \n\\end{IEEEproof}\n\n\n\n\\setcounter{equation}{13}\nWe remark that for $q=7$, the matrix in (\\ref{eq:supportm}) reduces to\n\\begin{equation} \\label{eq:supportmq7}\n\\left[ \\begin{smallmatrix}\n 0&  2&  6&  5&  3&  6&  4&  5&  4&  5&  3&  6&  4&  0&  3&  2&  3&  5&  4&  6\\\\ \n 0&  3&  0&  1&  6&  4&  3&  6&  5&  1&  0&  5&  3&  1&  4&  4&  0&  3&  1&  4 \\\\\n 0&  4&  1&  4&  2&  2&  2&  0&  6&  4&  4&  4&  2&  2&  5&  6&  4&  1&  5&  2 \\\\\n 0&  5&  2&  0&  5&  0&  1&  1&  0&  0&  1&  3&  1&  3&  6&  1&  1&  6&  2&  0 \\\\\n 0&  6&  3&  3&  1&  5&  0&  2&  1&  3&  5&  2&  0&  4&  0&  3&  5&  4&  6&  5 \\\\\n 0&  0&  4&  6&  4&  3&  6&  3&  2&  6&  2&  1&  6&  5&  1&  5&  2&  2&  3&  3 \n\\end{smallmatrix} \\right].\n\\end{equation}\nWe observe that there are indeed some identical columns when $q=7$. \nHowever, the bound in Theorem~\\ref{th:1} is still valid, since these columns can just be removed from (\\ref{eq:supportmq7}) and we will end up in the valid (but column-permuted) support matrix\n\\setcounter{equation}{15}\n\\begin{equation}  \\label{eq:supportmq7red}\n\\left[ \\begin{smallmatrix}\n 0&  2&  6&    3&       5&  4&        6&     0&  3&  2&    5&  4  \\\\ \n 0&  3&  0&    6&        6&  5&        5&     1&  4&  4&    3&  1   \\\\\n 0&  4&  1&    2&        0&  6&        4&     2&  5&  6&    1&  5   \\\\\n 0&  5&  2&    5&        1&  0&        3&     3&  6&  1&    6&  2   \\\\\n 0&  6&  3&    1&        2&  1&        2&     4&  0&  3&    4&  6   \\\\\n 0&  0&  4&    4&        3&  2&        1&     5&  1&  5&    2&  3   \n\\end{smallmatrix} \\right]\n\\end{equation}\nwhich corresponds to a codeword of weight $12$, but the bound $d(7,6) \\leq 20$ is of course not tight in this case. In fact, we found by exhaustive search that the codeword corresponding to the matrix in (\\ref{eq:supportmq7red}) is indeed a minimum-weight codeword.\n\nFinally, we remark that the template support matrix in (\\ref{eq:supportm}) for $q=7$, $11$, $13$, $17$, and $19$ do not give instances with columns in the order as implied by the parity-check matrix in (\\ref{eq:pcmatrix}). This can easily be seen from the sequence of $y$-values for the matrix in (\\ref{eq:supportm}), which should be nondecreasing.\nFurthermore, if two $y$-values are the same, then the corresponding sequence of $x$-values  should be nondecreasing. For $q > 19$, it can easily be proved that the order is always according to  (\\ref{eq:pcmatrix}).  However, as argued previously, this is not important (independent of the order, a support matrix will represent the same codeword\/stopping set).\n\n\n\\begin{figure*}[!t]\n\\normalsize \\setcounter{mytempeqncnt}{\\value{equation}}\n\\setcounter{equation}{14}\n\\begin{equation} \\label{eq:supportm7}\n{\\tiny {\n\\begin{split}\n&\\left[ \\begin{smallmatrix}\n0 & 3 \\cdot 2^{-1} & 0 & -9 \\cdot 2^{-1} & -7 \\cdot 2^{-1} & -1 & -11 \\cdot 2^{-1} & -5 & 2 & -2 & -5 & -1 & 2 &  5 \\cdot 2^{-1} & 2^{-1} &  3 \\cdot 2^{-1} & -3 & -2 & -9 \\cdot 2^{-1} & \\\\\n0 & 3 \\cdot 2^{-1} & 1 & -7 \\cdot 2^{-1} & -5 \\cdot 2^{-1} & 0 & -7 \\cdot 2^{-1} & -3 & 9\\cdot 2^{-2} & -7 \\cdot 2^{-2} & -15\\cdot 2^{-2} & 2^{-2} & 3 \\cdot 2^{-1} & 2 &  1 &  2 &   -5 \\cdot 2^{-1} & -3 \\cdot 2^{-1} & -3  & \\\\\n0 & 3 \\cdot 2^{-1} & 2 & -5 \\cdot 2^{-1} & -3 \\cdot 2^{-1} &  1 & -3 \\cdot 2^{-1} &  -1 & 5 \\cdot 2^{-1} &  -3 \\cdot 2^{-1} & -5 \\cdot 2^{-1} & 3 \\cdot 2^{-1} &  1 &   3 \\cdot 2^{-1} & 3 \\cdot 2^{-1} & 5 \\cdot 2^{-1} & -2 &  -1 &   -3 \\cdot 2^{-1} & \\\\\n0 & 3 \\cdot 2^{-1} & 3 & -3 \\cdot 2^{-1} & -2^{-1} &   2 &  2^{-1}  &   1 & 11\\cdot 2^{-2} &  -5\\cdot 2^{-2}\u00a0& -5\\cdot 2^{-2} &  11\\cdot 2^{-2} & 2^{-1} &  1 & 2 &   3 &   -3 \\cdot 2^{-1} & -2^{-1} &  0  & \\\\\n0 & 3 \\cdot 2^{-1} & 4 & -2^{-1} &   2^{-1}  &  3 &  5 \\cdot 2^{-1} &   3 & 3 &    -1 &    0 &    4 &   0 &   2^{-1} &  5 \\cdot 2^{-1} & 7 \\cdot 2^{-1} & -1 &   0 &    3 \\cdot 2^{-1}  & \\\\\n0 & 3 \\cdot 2^{-1} & 5 &  2^{-1} &   3 \\cdot 2^{-1} &  4 &  9 \\cdot 2^{-1} &   5 & 13\\cdot 2^{-2} & -3\\cdot 2^{-2} &  5\\cdot 2^{-2} &  21 \\cdot 2^{-2} & -2^{-1} & 0 &   3 &   4 &  -2^{-1}  & 2^{-1}  &  3 & \\\\\n0 & 3 \\cdot 2^{-1} & 6 &  3 \\cdot 2^{-1} &  5 \\cdot 2^{-1} &  5 &  13 \\cdot 2^{-1} & 7 & 7 \\cdot 2^{-1} &  -2^{-1} &   5 \\cdot 2^{-1} &  13 \\cdot 2^{-1} & -1 & -2^{-1} &  7 \\cdot 2^{-1} & 9 \\cdot 2^{-1} & 0 &    1 &    9 \\cdot 2^{-1}  &\n\\end{smallmatrix} \\right.  \\\\\n& \\left. \\begin{smallmatrix}\n& -3   & 2^{-1} &    5 \\cdot 2^{-1} & -11 \\cdot 2^{-1} & -7 \\cdot 2^{-1}  \\\\\n& -3 \\cdot 2^{-1} & 2^{-2} &   9\\cdot 2^{-2} & -15\\cdot 2^{-2} & -7\\cdot 2^{-2} \\\\\n& 0 &    0 &    2 &   -2 &    0 \\\\\n\\hspace{10.0cm} &    3 \\cdot 2^{-1} &  -2^{-2} &   7\\cdot 2^{-2} & -2^{-2} &   7\\cdot 2^{-2} \\\\\n&  3 &    -2^{-1} &  3 \\cdot 2^{-1} & 3 \\cdot 2^{-1} &   7 \\cdot 2^{-1} \\\\ \n&     9 \\cdot 2^{-1} &  -3\\cdot 2^{-2} & 5 \\cdot 2^{-2} & 13 \\cdot 2^{-2} &  21 \\cdot 2^{-2}  \\\\\n&  6 &   -1 &   1 &   5 &    7  \\end{smallmatrix} \\right] \n\\end{split} }}\n\\end{equation}\n\\setcounter{equation}{\\value{mytempeqncnt}}\n \\hrulefill\n\\vspace*{-2mm}\n\\end{figure*}\n\n\n\\section{Upper Bound on $d(q,7)$} \\label{sec:bound_m7}\n\nFor the case $m=7$ we have found, using the algorithm from \\cite{tom07}, the support matrices\n\\begin{equation}  \\notag\n\\left[ \\begin{smallmatrix}\n0 & 13 & 0 & 7 & 8 & 22 & 6 & 18 & 2 & 21 & 18 & 22 & 2 & 14 & 12 & 13 & 20 & 21 & 7 & 20 & 12 & 14 & 6 & 8 \\\\\n0 & 13 & 1 & 8 & 9 & 0 & 8 & 20 & 8 & 4 & 2 & 6 & 13 & 2 & 1 & 2 & 9 & 10 & 20 & 10 & 6 & 8 & 2 & 4 \\\\\n0 & 13 & 2 & 9 & 10 & 1 & 10 & 22 & 14 & 10 & 9 & 13 & 1 & 13 & 13 & 14 & 21 & 22 & 10 & 0 & 0 & 2 & 21 & 0 \\\\\n0 & 13 & 3 & 10 & 11 & 2 & 12 & 1 & 20 & 16 & 16 & 20 & 12 & 1 & 2 & 3 & 10 & 11 & 0 & 13 & 17 & 19 & 17 & 19 \\\\\n0 & 13 & 4 & 11 & 12 & 3 & 14 & 3 & 3 & 22 & 0 & 4 & 0 & 12 & 14 & 15 & 22 & 0 & 13 & 3 & 11 & 13 & 13 & 15 \\\\\n0 & 13 & 5 & 12 & 13 & 4 & 16 & 5 & 9 & 5 & 7 & 11 & 11 & 0 & 3 & 4 & 11 & 12 & 3 & 16 & 5 & 7 & 9 & 11 \\\\\n0 & 13 & 6 & 13 & 14 & 5 & 18 & 7 & 15 & 11 & 14 & 18 & 22 & 11 & 15 & 16 & 0 & 1 & 16 & 6 & 22 & 1 & 5 & 7\n\\end{smallmatrix} \\right]\n\\end{equation}\nand\n\\begin{equation}  \\notag\n\\left[ \\begin{smallmatrix}\n0 & 16 & 0 & 10 & 11 & 28 & 9 & 24 & 15 & 17 & 9 & 11 & 2 & 17 & 15 & 16 & 26 & 27 & 10 & 26 & 2 & 27 & 24 & 28 \\\\\n0 & 16 & 1 & 11 & 12 & 0 & 11 & 26 & 22 & 24 & 18 & 20 & 16 & 2 & 1 & 2 & 12 & 13 & 26 & 13 & 24 & 20 & 18 & 22 \\\\\n0 & 16 & 2 & 12 & 13 & 1 & 13 & 28 & 0 & 2 & 27 & 0 & 1 & 16 & 16 & 17 & 27 & 28 & 13 & 0 & 17 & 13 & 12 & 16 \\\\\n0 & 16 & 3 & 13 & 14 & 2 & 15 & 1 & 7 & 9 & 7 & 9 & 15 & 1 & 2 & 3 & 13 & 14 & 0 & 16 & 10 & 6 & 6 & 10 \\\\\n0 & 16 & 4 & 14 & 15 & 3 & 17 & 3 & 14 & 16 & 16 & 18 & 0 & 15 & 17 & 18 & 28 & 0 & 16 & 3 & 3 & 28 & 0 & 4 \\\\\n0 & 16 & 5 & 15 & 16 & 4 & 19 & 5 & 21 & 23 & 25 & 27 & 14 & 0 & 3 & 4 & 14 & 15 & 3 & 19 & 25 & 21 & 23 & 27 \\\\\n0 & 16 & 6 & 16 & 17 & 5 & 21 & 7 & 28 & 1 & 5 & 7 & 28 & 14 & 18 & 19 & 0 & 1 & 19 & 6 & 18 & 14 & 17 & 21 \n\\end{smallmatrix} \\right]\n\\end{equation}\n of (minimal) codewords of weight $24$ for $q=23$ and $q=29$, respectively. \nNote that in the matrix for $q=23$ (the first matrix) the entries $2$ and $8$ appear four times in the second row, while in the matrix for $q=29$ (the second matrix) all entries appear twice in the second row. In the first row, however, all entries appear twice for both matrices. Thus, there are several cycles containing either the edge $(v^{(0)}_{q-1},v^{(1)}_{0})$ or the edge $(v^{(0)}_0,v^{(1)}_0)$ in the support matrix graph $G_{\\mathbf{c}_1}^{(0,1)}$ (corresponding to the first matrix), while there is only a single such cycle in the support matrix graph  $G_{\\mathbf{c}_2}^{(0,1)}$ (corresponding to the second matrix), and  as a consequence the codewords $\\mathbf{c}_1$ and $\\mathbf{c}_2$ do \\emph{not} have the same graphical cycle structure. Note, however, that the \\emph{minimum} cycle lengths are the same, and this is also the case for all the other pairs of  graphs  $G_{\\mathbf{c}_1}^{(i,j)}$ and  $G_{\\mathbf{c}_2}^{(i,j)}$, $0 \\leq i < j \\leq m-1$.  Following the discussion in Section~\\ref{sec:improved}, we may apply Algorithm~\\ref{alg:tsmi}, which infers the candidate template support matrix shown in (\\ref{eq:supportm7}) at the top of the previous page. Details are omitted for brevity.\n\nWe can now prove the following theorem.\n\n\\begin{theorem} \\label{th:2}\nThe minimum distance $d(q,7)$ is upper-bounded by $24$ for $q > 7$.\n\\end{theorem}\n\n\\begin{IEEEproof}\n  The proof is based on the candidate template support matrix shown in (\\ref{eq:supportm7}) at the top of the previous page and is almost identical to the proof of Theorem~\\ref{th:1}. In particular, it is easy to check  that each entry in each row of the matrix appears an even number of times and that all columns in the matrix are in fact valid columns in an array LDPC code parity-check matrix (all columns are of the form in (\\ref{eq:form})).\n\nFor the third part of the proof, we need to show, for any value of $q > 7$, that there exist columns in the parity-check matrix which are not identical modulo $q$ and appear an odd number of times. Again, this is simple  (and very fast) to check by a computer search for any finite value of $q$ that would be of any practical value. It is only for large values of $q$ that the theoretical proof below is needed. \n\nNow, let the largest absolute value of the entries in the $i$th row of the matrix in (\\ref{eq:supportm7}) which do not involve a multiplication by $2^{-1}$ be denoted $\\lambda_i$, and let the largest absolute value of the factor in front of $2^{-1}$ of the remaining entries in the $i$th row be denoted by $\\mu_i$. Since $a \\cdot 2^{-1} \\pmod{q}$, when $a$ is odd (which is always the case in  (\\ref{eq:supportm7})), can be written as $(q+a)\/2$, it follows easily that for a row $i$ where all entries are of the form $a$ or $a \\cdot 2^{-1}$, different template entries can never be the same modulo $q$ when $q > 2\\lambda_i+\\mu_i$. For the first row this bound is $2 \\cdot 5 + 11 = 21$, and for the third row, this bound is $2 \\cdot 2 + 5 = 9$. Thus, looking at the first row, the only possibility for repeated columns,  when $q > 21$ (the bound for the first row), is for two \\emph{neighboring} columns (with identical entries in the first row) to be the same. \nHowever, by looking at the third row in the matrix, this possibility can be ruled out by requiring that $q > 9$ (the bound for the third row). \nIn summary, it follows that there are no identical columns in the matrix in (\\ref{eq:supportm7})  if $q > \\max(21,9)=21$. \nFurthermore, for values of $7 < q < 21$, it can be checked numerically that there are no repeated columns in (\\ref{eq:supportm7}), and the result follows. \n\\end{IEEEproof}\n\n\n\n\n\n\\section{Numerical Results} \\label{sec:results}\n\nIn addition to the analytic results of Theorems~\\ref{th:1} and \\ref{th:2}, we have performed a computer search to compute the exact values for $d(q,m)$ and $h(q,m)$ for small values of $q$ and $m$. The results are summarized in Table~\\ref{table:arrayLDPC}, where the entries that appear in bold are new results. Results from the literature have also been included with an explicit reference. \n\nFor $m=6$, we have computed the exact value of $d(q,6)$ for $q \\leq 19$. For larger values of $q$, we have run the exhaustive algorithm from \\cite{ros09,ros12} with an upper weight threshold of $16$ without finding any codewords. From the upper bound from Theorem~\\ref{th:1} and the fact that these codes are even-weight codes, we can conclude that  the minimum distance, for $23 \\leq q \\leq 79$, is either $18$ or $20$. Furthermore, extensive minimum distance calculations using the probabilistic algorithm from \\cite{tom07} for several values of $q \\geq 23$, indicate that the minimum distance is indeed $20$ for $q \\geq 23$, from which it follows that  the upper bound from Theorem~\\ref{th:1} appears to be tight.\n\nFor $m=7$, we have only been able to compute the exact value for $q=7$ and $11$ using the exhaustive algorithm from \\cite{ros09,ros12}. For $q=13$, we were able to run the exhaustive algorithm from \\cite{ros09,ros12} with an upper weight threshold of $16$. In addition, we found a codeword of weight $20$ using the probabilistic algorithm from \\cite{tom07}, from which (and the fact that the array LDPC codes are even-weight codes) we can conclude that the minimum distance is  either $18$ or $20$. \nFor larger values of $q$, $17 \\leq q \\leq 29$, the probabilistic algorithm from \\cite{tom07} has provided the estimates  (in the form of upper bounds) in Table~\\ref{table:arrayLDPC}. Note that even if the results are formally stated as upper bounds, the algorithm from \\cite{tom07} indicates that the estimates  are indeed likely to give the exact values, which indicates that the bound from Theorem~\\ref{th:2} is in fact tight (for instance, $q=17$ gives a minimum distance of $24$ with very high probability).\nFor the high values of $q$ ($31 \\leq q \\leq 79$), Theorem~\\ref{th:2} has provided the estimates (in the form of upper bounds).\n\n\n\n\n\\begin{table}[t]\n\\scriptsize \\centering \\caption{Minimum\/Stopping Distance Results for Array LDPC Codes for Different Values of $q$ and $m$}\\label{table:arrayLDPC}\n\\def\\Hline{\\noalign{\\hrule height 2\\arrayrulewidth}}\n\\vskip -3.0ex\n\\begin{tabular}{cllllll}\n\\Hline \\\\ [-2.0ex]\n   $q$ & $d(q,7)$ & $d(q,6)$ & $h(q,5)$ & $d(q,5)$ & $h(q,4)$ & $d(q,4)$\\\\\n\\hline\n\\\\ [-2.0ex] \\hline  \\\\ [-2.0ex]\n7 & $\\mathbf{14}$ & $12$ \\cite{sug08} & $\\mathbf{9}$ & $12$ \\cite{sug08} & 8 \\cite{esm11} & 8 \\cite{sug08}\\\\\n11 & $\\mathbf{20}$ & $16$ \\cite{sug08} & $10$ \\cite{esm11} & $10$ \\cite{sug08} & 10 \\cite{esm11} & 10 \\cite{sug08}\\\\\n13 & $\\mathbf{18}$ or $\\mathbf{20}$ & $14$ \\cite{sug08} & $\\mathbf{12}$ & $12$ \\cite{sug08} & $\\mathbf{10}$ & 10 \\cite{sug08}\\\\\n17 & {\\bf $\\leq \\mathbf{24}$} &  $\\mathbf{16}$ & $\\mathbf{12}$ & $12$ \\cite{sug08} & $\\mathbf{10}$ & 10 \\cite{sug08}\\\\\n19 & {\\bf $\\leq \\mathbf{20}$} & {\\bf $\\mathbf{18}$} & $\\mathbf{12}$ & $12$ \\cite{sug08} & $\\mathbf{10}$ & 10 \\cite{sug08}\\\\\n23 & {\\bf $\\leq \\mathbf{22}$} & $\\mathbf{18}$ or $\\mathbf{20}$& $\\mathbf{12}$ & $\\mathbf{12}$ & $\\mathbf{10}$ & 10 \\cite{sug08}\\\\\n29 & {\\bf $\\leq \\mathbf{24}$} & $\\mathbf{18}$ or $\\mathbf{20}$& $\\mathbf{12}$ & $\\mathbf{12}$ & $\\mathbf{10}$ & 10 \\cite{sug08}\\\\\n31 & {\\bf $\\leq \\mathbf{24}$} & $\\mathbf{18}$ or $\\mathbf{20}$& $\\mathbf{12}$ & $\\mathbf{12}$ & $\\mathbf{10}$ & 10 \\cite{sug08}\\\\\n37 & {\\bf $\\leq \\mathbf{24}$} & $\\mathbf{18}$ or $\\mathbf{20}$& $\\mathbf{12}$ & $\\mathbf{12}$ & $\\mathbf{10}$ & 10 \\cite{sug08}\\\\\n41 & {\\bf $\\leq \\mathbf{24}$} & $\\mathbf{18}$ or $\\mathbf{20}$& $\\mathbf{12}$ & $\\mathbf{12}$ & $\\mathbf{10}$ & 10 \\cite{sug08}\\\\\n43 & {\\bf $\\leq \\mathbf{24}$} & $\\mathbf{18}$ or $\\mathbf{20}$& $\\mathbf{12}$ & $\\mathbf{12}$ & $\\mathbf{10}$ & 10 \\cite{sug08}\\\\\n47 & {\\bf $\\leq \\mathbf{24}$} & $\\mathbf{18}$ or $\\mathbf{20}$& $\\mathbf{12}$ & $\\mathbf{12}$ & $\\mathbf{10}$ & 10 \\cite{sug08}\\\\\n53 & {\\bf $\\leq \\mathbf{24}$} & $\\mathbf{18}$ or $\\mathbf{20}$& $\\mathbf{12}$ & $\\mathbf{12}$ & $\\mathbf{10}$ & 10 \\cite{sug08}\\\\\n59 & {\\bf $\\leq \\mathbf{24}$} & $\\mathbf{18}$ or $\\mathbf{20}$& $\\mathbf{12}$ & $\\mathbf{12}$ & $\\mathbf{10}$ & 10 \\cite{sug08}\\\\\n61 & {\\bf $\\leq \\mathbf{24}$} & $\\mathbf{18}$ or $\\mathbf{20}$& $\\mathbf{12}$ & $\\mathbf{12}$ & $\\mathbf{10}$ & 10 \\cite{sug08}\\\\\n67 & {\\bf $\\leq \\mathbf{24}$} & $\\mathbf{18}$ or $\\mathbf{20}$& $\\mathbf{12}$ & $\\mathbf{12}$ & $\\mathbf{10}$ & 10 \\cite{sug08}\\\\\n71 & {\\bf $\\leq \\mathbf{24}$} & $\\mathbf{18}$ or $\\mathbf{20}$& $\\mathbf{12}$ & $\\mathbf{12}$ & $\\mathbf{10}$ & 10 \\cite{sug08}\\\\\n73 & {\\bf $\\leq \\mathbf{24}$} & $\\mathbf{18}$ or $\\mathbf{20}$& $\\mathbf{12}$ & $\\mathbf{12}$ & $\\mathbf{10}$ & 10 \\cite{sug08}\\\\\n79 & {\\bf $\\leq \\mathbf{24}$} & $\\mathbf{18}$ or $\\mathbf{20}$& $\\mathbf{12}$ & $\\mathbf{12}$ & $\\mathbf{10}$ & 10 \\cite{sug08}\n\\end{tabular}\n\\end{table}\n\n\n\\section{Conclusion} \\label{sec:conclu}\nIn this paper, the minimum\/stopping distance of array LDPC codes have been studied. We have presented an improved general (i.e., independent of $q$) upper bound on the minimum distance for the case $m=6$, using the concept of a template support matrix of a codeword\/stopping set, which significantly improves the currently best know bound.   The bound appears to be tight with high probability in the sense that we have not found  codewords of strictly lower weight  for several values of $q$ using a minimum distance probabilistic algorithm.  In addition, we have provided the new upper bound $d(q,7) \\leq 24$ which also (from extensive numerical computations) appear to be tight. Finally, we have provided several new specific minimum\/stopping distance results for $m \\leq 7$ and low-to-moderate values of $q \\leq 79$.\n\n\n\\balance\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe expression ``$\\omega$-automata'' generally refers to automata that accept or reject $\\omega$-words---infinite sequences of letters from some alphabet. They define $\\omega$-languages---sets of $\\omega$-words---just as ordinary automata define languages of finite words; they are means for working with $\\omega$-languages, just as ordinary automata are means for working with languages of finite words.\n\nThe fundamental questions about $\\omega$-automata are similar to the fundamental questions about automata on finite words. Can operations on languages, such as intersection, complementation, and projection, be performed on automata? What is the difference between nondeterminism and determinism? What is the descriptional complexity of various types of automata? Which types of automata can be transformed into other types and at which cost? Can automata be minimized efficiently? How can automata be compared? \\dots\\ \n\nSome of the techniques developed for finite words can be adopted in the infinite-word setting; some of the theory of $\\omega$-automata is very similar to ordinary automata theory. There are, however, many interesting new aspects, which, most often, have something to do with what happens ``in the infinite''. As this interesting behavior ``in the infinite'' is already present in $\\omega$-automata with a finite state space, this paper is limited to the theory of finite-state $\\omega$-automata.\n\nTo convey the core ideas as crisp and clear as possible within the given space limits, some of the material is presented in an uncommon way, at the expense of continuity with prior work.\n\nExcellent surveys that cover $\\omega$-languages and $\\omega$-automata in their\nentire breadth, especially their relationship with mathematical logic, have been\nwritten by Wolfgang Thomas, one in the late eighties \\cite{Thomas1990}, and one\nin the nineties \\cite{thomas-languages-automata-logic-handbook-1997}. There is\nalso a comprehensive monograph by Dominique Perrin and Jean-\\'{E}ric Pin\n\\cite{perrin-pin-infinite-words-2004}. This paper tries to be a concise introduction into the \\emph{theory} of $\\omega$-automata.\n\n\\subsection{\\texorpdfstring{$\\omega$}{\u03c9}-Words}\n\\label{sec:omega-words}\n\n$\\omega$-Automata are devices that work on $\\omega$-words rather than finite ones. Technically, an $\\omega$-word \\index{$\\omega$-word} over an alphabet $A$ is a function $\\omega \\to A$, where $\\omega$ stands for the set of natural numbers. In contrast, a finite word over $A$ is a function $[n] \\to A$, where $n$ is a natural number and $[n]$ denotes the set $\\{0, \\dots, n-1\\}$. When $u$ is a word, finite or infinite, then $\\Occ u$ denotes the set of letters occurring in $u$; when $u$ is an $\\omega$-word, then $\\inf(u)$ denotes the set of letters occurring infinitely often in $u$.\n\nThere are essentially three concatenation operations involving $\\omega$-words. First, given a finite word $u$ and an $\\omega$-word $v$, the $\\omega$-word obtained by appending $v$ to~$u$ is denoted $u \\cdot v$---we speak of $\\omega$-concatenation\\index{$\\omega$-concatenation}. Second, when $\\langle u_0, u_1, \\dots\\rangle$ is an infinite sequence of finite nonempty words, then $u_0 \\cdot u_1 \\cdot \\dots$ denotes the $\\omega$-word obtained by concatenating all the $u_i$'s in the given order---we speak of $\\omega$-product\\index{$\\omega$-product}. Finally, when $u$ is a finite nonempty word, then $u^\\omega$ denotes $u \\cdot u \\cdot \\dots$---we speak of $\\omega$-power\\index{$\\omega$-power}. Note that it is legitimate to use the same symbol $\\cdot$ for ordinary concatenation, $\\omega$-concatenation, and $\\omega$-product, because there are various laws of associativity that hold. As usual, the symbol ``$\\cdot$'', representing the different forms of concatenation, is omitted in many contexts, and these operations are extended to sets of words in a straightforward fashion. An $\\omega$-word $u$ is periodic if $u = v^\\omega$ for some finite nonempty word $v$; it is ultimately periodic\\index{ultimately periodic}\\index{word!ultimately periodic} if $u = v w^\\omega$ for finite words $v$ and $w$ with $w$ being nonempty. The convolution of $\\omega$-words $u$ and $v$ over alphabets $A$ and $B$, respectively, is an $\\omega$-word over the alphabet $A \\times B$, denoted $u * v$ and defined by $(u * v)(i) = \\langle u(i), v(i)\\rangle$ for every $i$. \n\nOne of the reasons why $\\omega$-automata are applicable in various situations is that $\\omega$-words can represent infinite objects and therefore $\\omega$-automata can represent sets of infinite objects or even transform infinite objects into other infinite objects.  \n\nWhen the alphabet $A$ is the binary alphabet, $[2]$, then an $\\omega$-word can be identified with a subset of $\\omega$, that is, with a set of natural numbers, more precisely, a word $u$ can be identified with $\\{i \\in \\omega \\mid u(i) = 1\\}$. When the alphabet is $\\bigtimes_{i<k} [2]$, the $k$-fold cartesian product of $[2]$, then an $\\omega$-word can be identified with a $k$-tuple of sets of natural numbers.\n\nEvery $\\omega$-word over $[2]$ represents a real number from the interval $[0,1]$ in a natural fashion, more precisely, $u \\in [2]^\\omega$ represents the number $\\sum_i u(i) \\,  2^{-i-1}$. Observe that some numbers are represented twice, for instance, $1\/2$, which is represented by $1 0^\\omega$ and by $01^\\omega$. \n\nThere are various ways to represent any real number by an $\\omega$-word. One way is to consider only words $u$ over $[2]$ where $u(2i+2) = 0$ for almost all $i$ and then let $u$ represent $(-1)^{u(0)} (\\sum_i u(2i+2) 2^i  + \\sum_i u(2i+1) 2^{-i-1})$. So the letters at even positions determine the integer part, including the sign, and the letter at odd positions determine the fractional part. Another way is to use a larger alphabet, for instance, $[2] \\times [2]$, and represent the integer part in one dimension and the fractional part in the other dimension.\n\nIn this paper, a binary tree is a prefix-closed subset of $[2]^*$; level $i$ of such a tree~$T$ is the set of vertices $T \\cap [2]^i$; its width is $\\sup_i |T \\cap [2]^i|$. Given some $k$, the set of trees of width at most $k$ can be represented by $\\omega$-words over a fixed alphabet. A simple such representation is the sequence of (representations of) its slices, where a slice is two consecutive levels together with their interconnections, that is, a slice looks like this: \n\\begin{center}\n \\small\n \\begin{tikzpicture}[level distance=10mm, sibling distance=7mm, every node\/.style={draw,circle,fill,inner sep=1pt},xscale=0.8]\n\t \\node at (-3,0) {} \n\t child {node {}}\n\t child {node {}};\n\t \\node at (-1.5,0) {};\n\t \\node at (0,0) {} \n\t child {node {}}\n\t child {node {}};\n\t \\node at (1.5,0) {} \n\t child {node {}}\n\t child[missing];\n\t \\node at (3,0) {} \n\t child[missing]\n\t child {node {}};\n\t \\node at (4.5,0) {}; \n\t \\node [right,fill=white,draw=none] at (5.5,0) {upper level};\n\t \\node [right,fill=white,draw=none] at (5.5,-1) {lower level};\n \\end{tikzpicture}\n\\end{center}\nTo represent this slice one could use the single ``letter'' $\\langle 1,1\\rangle\\langle 0,0\\rangle\\langle 1,1\\rangle\\langle 1,0\\rangle\\langle 0,1\\rangle\\langle 0,0\\rangle$. \n\nNot all $\\omega$-words over the respective alphabet represent a tree, but every tree (of a given maximum width~$k$) can be represented. For instance, the tree denoted by $0^*(\\epsilon + 1 + 11)$, which looks like a comb, is represented by $\\langle 1,1\\rangle \\, (\\langle 1,1 \\rangle \\langle 0,1 \\rangle) \\, (\\langle 1,1 \\rangle \\langle 0,1 \\rangle \\langle 0,0 \\rangle)^\\omega$.\n\nA labeled binary tree is a function $T \\to A$ from a binary tree to an alphabet, and such trees, if restricted in their width, can also be represented by $\\omega$-words, simply by augmenting the above representation by information about the labels. It is enough to encode in the representation of one slice the labels of the vertices in the ``upper'' level. \n\nWhat has just been said for infinite trees is also true for graphs to a certain extent. A leveled DAG \\index{leveled DAG}\\index{DAG!leveled} is a directed acyclic graph together with a partition of its vertex set into levels, more precisely, such a graph is given by a family $\\{V^{(i)}\\}_{i \\in \\omega}$ of pairwise disjoint vertex sets and an edge set $E \\subseteq \\bigcup_i V^{(i)} \\times V^{(i+1)}$. The set of all vertices, $\\bigcup_i V^{(i)}$, is denoted $V$; the elements of $V^{(i)}$ are the vertices on level $i$. Similarly to above, the width of such a DAG is $\\sup_i |V^{(i)}|$. \n\nGiven a natural number~$k$, leveled DAG's of width at most $k$ can be represented over a fixed alphabet, again by spelling out the sequence of its slices, where a slice represents a subgraph induced by two consecutive levels. Such a subgraph can, for instance, look like this:\n\\begin{center}\n \\small\n \\begin{tikzpicture}[every node\/.style={circle, draw, fill, inner sep=1pt}]\n\t \\node (a) at (0,0) {};\n\t \\node (b) at (1,0) {};\n\t \\node (c) at (2,0) {};\n\t \\node (d) at (3,0) {};\n\t \\node (e) at (4,0) {};\n\t \\node (f) at (5,0) {};\n\n\t \\node (A) at (0,-1) {};\n\t \\node (B) at (1,-1) {};\n\t \\node (C) at (2,-1) {};\n\t \\node (E) at (3,-1) {};\n\t \\node (D) at (4,-1) {};\n\t \\node[fill=white,draw=none] (F) at (5,-1) {};\n\n\t \\node [fill=white,draw=none,right of=f,xshift=10mm] {upper level};\n\t \\node [fill=white,draw=none,right of=F,xshift=10mm] {lower level};\n\t \\node [fill=white,draw=none,left of=a,xshift=-20mm,right] {level $i$};\n\t \\node [fill=white,draw=none,left of=A,xshift=-20mm,right] {level $i{+}1$};\n\n\t \\draw (a) -- (A) {};\n\t \\draw (a) -- (C) {};\n\t \\draw (c) -- (C) {};\n\t \\draw (b) -- (D) {}; \n\t \\draw (d) -- (A) {};\n\t \\draw (d) -- (B) {};\n\t \\draw (e) -- (B) {};\n \\end{tikzpicture}\n\\end{center}\nThis slice could be represented by the ``letter'' $\\langle \\{0, 2\\}, \\{4\\}, \\{2\\}, \\{0, 1\\}, \\{1\\}, \\{\\}\\rangle$, an enumeration of the adjacency sets of the vertices on the the upper level. We are only interested in leveled graphs up to isomorphism; so there are, in general, many representations for the same graph. Labeled leveled DAG's of bounded width can also be represented by $\\omega$-words.\n\n\\section{Types of \\texorpdfstring{$\\omega$}{\u03c9}-automata}\n\nA finite-state nondeterministic $\\omega$-automaton \\index{finite-state $\\omega$-automaton}\\index{$\\omega$-automaton!finite-state} over a given alphabet $A$ consists of\n\\begin{itemize}\n\\item a finite set of states, $Q$,\n\\item a set of initial states, $Q_I \\subseteq Q$,\n\\item a set of transitions, $\\Delta \\subseteq Q \\times A \\times Q$, and\n\\item a representation of a recurrence condition.\\index{recurrence condition}\\index{condition!recurrence}\n\\end{itemize}\nThe last ingredient is the one that distinguishes an $\\omega$-automaton from an ordinary finite-state automaton that works on finite words; it replaces the set of final states. This is necessary, because $\\omega$-words have no end. But before recurrence conditions can be explained in detail, the notion of a run needs to be adapted to $\\omega$-words. \n\nA run of an $\\omega$-automaton on a given word $u \\in A^\\omega$ is an $\\omega$-word $r \\in Q^\\omega$ such that $\\langle r(i), u(i), r(i+1)\\rangle \\in \\Delta$ holds for every $i \\in \\omega$. It is initial if $r(0) \\in Q_I$ holds; it is accepting if it is initial and recurring, and what it means for $r$ to be recurring depends on the recurrence condition, as described in what follows. \n\nA simple type of recurrence condition is the B\\\"{u}chi type, which is represented as a set $B \\subseteq Q$; a run $r$ is recurring if one of the states from $B$ occurs infinitely often in it, that is, if $\\inf(r) \\cap B \\neq \\emptyset$ holds. For instance, the language $L_\\text{fin1}$ defined by $L_\\text{fin1} = \\{u \\in [2]^\\omega \\mid \\inf(u) = \\{0\\}\\}$ (``$1$ occurs only finitely often'') is recognized by a B\\\"{u}chi automaton with two states, see Figure~\\ref{fig:simple-buechi}. (As usual, an automaton recognizes the language consisting of the words the automaton has an accepting run for; when an automaton is denoted $\\Aut A$, this language is denoted $\\text L(\\Aut A)$.)\n\nObviously, using a B\\\"{u}chi condition one can neither specify that two particular states occur infinitely often nor that a specific state occurs only finitely often. A type of recurrence condition which is sufficiently expressive in this respect is the Muller type. Such a condition is represented by a set $\\mathscr M \\subseteq \\powerset(Q)$; a run $r$ is recurring if $\\inf(r) \\in \\mathscr M$. In other words, one explicitly specifies which states occur infinitely often in a run and which are the ones that occur only finitely many times. The language $L_\\text{fin1}$ can also be recognized by a Muller automaton with two states, see Figure~\\ref{fig:simple-buechi}.\n\n\\begin{figure}\n  \\small\n  \\centering\n \\begin{tikzpicture}\n\t \\node[] (q) {};\n\t \\node (q0) [right of=q] {$q_0$};\n\t \\node (q1) [right of=q0] {$q_1$}; \n\n\t \\draw[->] (q) -- (q0);\n\t \\draw[->] (q0) to node[above] {$0$} (q1);\n\n\t \\draw[my loop] (q0) to node[above] {$0,1$} (q0);\n\t \\draw[my loop] (q1) to node[above] {$0$} (q1);\n\n\n\t \\node[] (p) [right of=q,xshift = 25mm] {};\n\t \\node (p0) [right of=p] {$q_0$};\n\t \\node (p1) [right of=p0] {$q_1$}; \n\n\t \\draw[->] (p) -- (p0);\n\t \\draw[->,bend left] (p0) to node[above] {$1$} (p1);\n\t \\draw[->,bend left] (p1) to node[below] {$0$} (p0);\n\n\t \\draw[my loop] (p0) to node[above] {$0$} (p0);\n\t \\draw[my loop] (p1) to node[above] {$1$} (p1);\n\n\n\t \\node[] (r) [right of=p,xshift = 25mm,yshift=-8mm] {};\n\t \\node (r0) [right of=r] {$q_0$};\n\t \\node (r1) [right of=r0] {$q_1$}; \n\t \\node (r2) [right of=r1] {}; \n\n\t \\draw[->] (r) -- (r0);\n\t \\draw[->] (r0) to node[above] {$1$} (r1);\n\t \\draw[->] (r2) -- (r1);\n\n\t \\draw[my loop] (r0) to node[above, left=1mm] {$0,1$} (r0);\n\t \\draw[my loop] (r1) to node[above, right=1mm] {$0$} (r1);\n\n\t \\node[] (s) [above of=r,yshift = 4mm] {};\n\t \\node (s0) [right of=s] {$q_2$};\n\t \\node (s1) [right of=s0] {$q_3$}; \n\n\t \\draw[->,bend left] (s0) to node[above] {$0$} (s1);\n\t \\draw[->,bend left] (s1) to node[below] {$1$} (s0);\n\n\t \\draw[my loop] (s0) to node[above, left=1mm] {$0$} (s0);\n\t \\draw[my loop] (s1) to node[above, right=1mm] {$1$} (s1);\n\n\n\t \\node[] (t) [right of=r,xshift = 25mm,yshift=8mm] {};\n\t \\node (t0) [right of=t] {$q_0$};\n\n\t \\draw[->] (t) -- (t0);\n\n\t \\draw[my loop] (t0) to node[above] {$0,1$} (t0);\n \\end{tikzpicture}\n \\caption{Different types of $\\omega$-automata for the set of all $\\omega$-words over $[2]$ with a finite number of occurrences of $1$, denoted $L_\\text{fin1}$: the first automaton is forward nondeterministic and, if augmented with the B\\\"{u}chi condition $\\{q_1\\}$, recognizes $L_\\text{fin1}$; the second one is forward deterministic and, if augmented with the co-B\\\"{u}chi condition $\\{q_0\\}$, the parity condition $\\pi \\colon q_0 \\mapsto 2, q_1 \\mapsto 1$, the Muller condition $\\{\\{q_0\\}\\}$, or a suitable Streett or Rabin condition, recognizes $L_\\text{fin1}$; the third one is backward deterministic if used with the B\\\"{u}chi condition $\\{q_1, q_3\\}$ and then recognizes $L_\\text{fin1}$; the fourth one is forward deterministic and recognizes $L_\\text{fin1}$ if augmented with the transition-Muller condition $\\{\\{(q_0, 0, q_0)\\}\\}$.}\n \\label{fig:simple-buechi}\n\\end{figure}\n\n\\newcommand{w.\\,r.\\,t.\\xspace}{w.\\,r.\\,t.\\xspace}\n\n\\index{B\u00fcchi recurrence condition}\n\\index{recurrence condition!B\u00fcchi}\n\\index{parity recurrence condition}\n\\index{recurrence condition!parity}\n\\index{Rabin recurrence condition}\n\\index{recurrence condition!Rabin}\n\\index{Streett recurrence condition}\n\\index{recurrence condition!Streett}\n\\index{Muller recurrence condition}\n\\index{recurrence condition!Muller}\n\\index{weak recurrence condition}\n\\index{recurrence condition!weak}\n\\index{co-B\u00fcchi recurrence condition}\n\\index{recurrence condition!co-B\u00fcchi}\n\\index{generalized B\u00fcchi recurrence condition}\n\\index{recurrence condition!generalized B\u00fcchi}\n\\index{transition recurrence condition}\n\\index{recurrence condition!transition}\nThere are essentially five different types of recurrence conditions that have been investigated traditionally, all explained in the upper part of Table~\\ref{tab:acceptance-conditions} and named after their originators  \\cite{buechi-decision-method-restricted-second-order-arithmetic-1962,rabin-decidability-of-second-order-theories-and-automata-on-infinite-trees-1969,Streett82,muller-infinite-sequences-and-finite-machines-1963} except for the parity condition \\cite{mostowski-regular-expressions-for-infinite-trees-and-a-standard-form-of-automata-1984}. In the lower part of the table, there are three types of conditions derived from the B\\\"{u}chi condition \\cite{muller-saoudi-schupp-alternating-automata-weak-monadic-theory-tree-its-complexity-1992,klarlund-progress-measures-complementation-omega-automata-applications-temporal-logic-1991,courcoubetis-vardi-wolper-yannakakis-memory-efficient-algorithms-verification-temporal-properties-1992}. \n\nThe trivial recurrence condition, which is not mentioned in the table, considers every run recurrent. For instance, all representations of binary trees of a fixed width (over the alphabet indicated in Section~\\ref{sec:omega-words}) are recognized by an automaton with trivial recurrence condition.\n\n\\begin{table}[!t]\n \\renewcommand{\\arraystretch}{1.2}\n \\begin{tabularx}{\\textwidth}{lll>{\\raggedright\\arraybackslash}X}\n\t \\hline\n\t Name & & Format & Semantics\\\\ \\hline \n\t B\\\"{u}chi & $B$ & $B \\subseteq Q$ & $\\inf(r) \\cap B \\neq \\emptyset$\\\\\n\t parity & $\\mathscr P$ & $\\pi \\colon Q \\to [2n]$, $n \\in \\omega$ & $\\min(\\inf(\\pi \\circ r)) \\text{ mod } 2 = 0$\\\\\n\t Rabin & $\\mathscr R$ & $\\mathscr R \\subseteq \\powerset(R) \\times \\powerset(R)$ & there exists $\\langle L, U\\rangle \\in \\mathscr R$ such that $\\inf(r) \\cap L = \\emptyset$ and $\\inf(r) \\cap U \\neq \\emptyset$\\\\\n\t Streett & $\\mathscr S$ & $\\mathscr S \\subseteq \\powerset(R) \\times \\powerset(R)$ & for every $\\langle R, G\\rangle \\in \\mathscr S$, if $\\inf(r) \\cap R \\neq \\emptyset$, then $\\inf(r) \\cap G \\neq \\emptyset$\\\\\n\t Muller & $\\mathscr M$ & $\\mathscr M \\subseteq \\powerset(Q)$ & $\\inf(r) \\in \\mathscr M$\\\\\n\t \\hline\n\t weak & $W$ & $W \\subseteq Q$, union of SCC's & $\\inf(r) \\subseteq W$\\\\\n\t co-B\\\"{u}chi & $C$ & $C \\subseteq Q$ & $\\inf(r) \\subseteq C$\\\\\n\t gen.\\ B\\\"{u}chi & $\\mathscr G$ & $\\mathscr G \\subseteq \\powerset(Q)$ & $B \\cap \\inf(r) \\neq \\emptyset$ for every $B \\in \\mathscr G$\\\\\n\t \\hline\n \\end{tabularx}\n \\caption{Recurrence conditions, ordered according their expressive power; ``gen.\\ B\\\"{u}chi'' and ``SCC'' are abbreviations of ``generalized B\\\"{u}chi'' and ``strongly connected component'', respectively.}\n \\label{tab:acceptance-conditions}\n\\end{table}\n\nIt is convenient to give names to the elements of the recurrence conditions: a state $q \\in B$ is called a B\\\"{u}chi state, a pair $\\langle L, U\\rangle \\in \\mathscr R$ is called a Rabin pair, a pair $\\langle R, G\\rangle \\in \\mathscr S$  is called a Streett pair, a set $M \\in \\mathscr M$ is called a Muller set, a state $q \\in W$ is called a weak state, a state $q \\in C$ is called a co-B\\\"{u}chi state, and a set $B \\in \\mathscr G$ is called a B\\\"{u}chi set. The function $\\pi$ is called priority function. Only weak, co-B\\\"{u}chi, and B\\\"{u}chi conditions have straightforward representations of size polynomial in the number of states of a given automaton.\n\nEvery type of recurrence condition is also considered in a transition variant, where states from~$Q$ are replaced by transitions from $\\Delta$ and $\\inf(r)$ is replaced by the set of triples $\\langle q, a, q'\\rangle$ for which there exist an infinite number of $i$ such that $\\langle r(i), u(i), r(i+1)\\rangle = \\langle q, a, q'\\rangle$. Transition variants come in more handy in certain situations, for instance, $L_\\text{fin1}$ is recognized by a single-state transition-Muller automaton, see Figure~\\ref{fig:simple-buechi}.\\footnotemark[1]\n\nTable~\\ref{tab:conversions} shows how conditions of various types can be expressed in terms of conditions of other types: every B\\\"{u}chi condition may be viewed as a parity condition, which, in turn, can be viewed as a Rabin or Streett condition, and these can be viewed as Muller conditions. \n\nUnlike finite words, $\\omega$-words are not symmetric (in the sense that there is no order isomorphism from the order of the natural numbers to its inverse). So when talking about determinism the direction makes a difference.%\n\\footnotetext[1]{In hindsight this paper should have been written using transition conditions throughout, in particular, Section~\\ref{run trees} would profit much from this.}\n\\refstepcounter{footnote}\nA forward deterministic automaton is one where $Q_I$ consists of exactly one state and $|\\Delta(q, a)| = 1$ for all $q \\in Q, a \\in A$. (As usual, $\\Delta(q,a)$ is used as an abbreviation of $\\{q' \\in Q \\mid \\langle q, a, q'\\rangle \\in \\Delta\\}$.) A backward deterministic automaton%\n\\footnote{In \\cite{carton-michel-unambiguous-buechi-automata-2003}, where these automata were introduced, they are called ``complete unambiguous''; in \\cite{perrin-pin-infinite-words-2004}, the attribute ``prophetic'' is used; in \\cite{preugschat-wilke-effective-characterizations-of-simple-fragments-of-temporal-logic-using-carton-michel0automata-2013}, they are referred to as ``Carton-Michel automata''. The terminology used in this paper tries to be systematic.} \n\\index{Carton-Michel automaton}\n\\index{prophetic automaton}\n\\index{automaton!Carton-Michel}\n\\index{automaton!prophetic}\nis one where for every $\\omega$-word over $A$, there is exactly one recurring run and $|\\Delta(a, q')| = 1$ for all $a \\in A, q' \\in Q$. (Here, $\\Delta(a, q')$ stands for $\\{q \\in Q \\mid \\langle q, a, q'\\rangle \\in \\Delta\\}$.) At times, when deterministic automata are used, the transition relation $\\Delta$ is replaced by a transition function $\\delta \\colon Q \\times A \\to Q$ (forward automata) or $\\delta \\colon A \\times Q \\to Q$ (backward automata).\n\nIn general, a type of an $\\omega$-automaton is given by a type of recurrence condition \\emph{and} a type of mode, with the following modes being considered: nondeterministic (default), forward deterministic, backward deterministic, and alternating, defined in Section~\\ref{sec:alternation}.\n\n\\begin{table}[b]\n \\centering\n \\renewcommand{\\arraystretch}{1.2}\n \\begin{tabularx}{\\textwidth}{llX}\n\t \\hline\n\t From & To & Conversion \\\\ \\hline\n\t B\\\"{u}chi $B$ & parity $\\pi$ & $\\pi(q) = 0$ for $q \\in B$ and $\\pi(q) = 1$ for $q \\notin B$\\\\\n\t parity $\\pi$ & Rabin $\\mathscr R$ & $\\mathscr R = \\{(\\pi^{-1}(\\{0, \\dots, 2i-1\\}), \\pi^{-1}(\\{0, \\dots, 2i\\})) \\mid i + 1 < n\\}$ \\\\\n\t parity $\\pi$ & Streett $\\mathscr S$ & $\\mathscr S = \\{(\\pi^{-1}(\\{0, \\dots, 2i+1\\}), \\pi^{-1}(\\{0, \\dots, 2i\\})) \\mid i + 1 < n\\}$ \\\\\n\t Rabin $\\mathscr R$ & Muller $\\mathscr M$ & $\\mathscr M = $ set of all $Q' \\subseteq Q$ such that there exists $\\langle L, U\\rangle \\in \\mathscr R$ satisfying $Q' \\cap L = \\emptyset$ and $Q' \\cap U \\neq \\emptyset$ \\\\\n\t Streett $\\mathscr S$ & Muller $\\mathscr M$ & $\\mathscr M = $ set of all $Q' \\subseteq Q$ such that, for every $\\langle R, G\\rangle \\in \\mathscr S$, if $Q' \\cap R \\neq \\emptyset$, then $Q' \\cap G \\neq \\emptyset$ \\\\ \\hline\n \\end{tabularx}\n \\caption{Conversions between recurrence conditions}\n \\label{tab:conversions}\n\\end{table}\n\nThe most fundamental result about $\\omega$-automata compares the different types of $\\omega$-automata with respect to their expressive power. As a yardstick, nondeterministic B\\\"{u}chi automata are used; the $\\omega$-languages recognized by them are called regular $\\omega$-languages, see Theorem~\\ref{thm:regexps} for the origin of this terminology.\n\n\\begin{theorem}[equivalence of types of $\\omega$-automata]\n \\label{thm:fundamental}\n For every type of $\\omega$-automaton, consider the class of $\\omega$-languages recognized by automata of this type. Then all these classes coincide with the class of regular $\\omega$-languages except for the classes corresponding to the following types: \n \\begin{inparaitem}[]\n \\item forward deterministic generalized B\\\"{u}chi;\n \\item forward deterministic B\\\"{u}chi; \n \\item forward deterministic, backward deterministic, and nondeterministic co-B\\\"{u}chi; \n \\item forward deterministic, backward deterministic, and nondeterministic weak.\n \\end{inparaitem}\n\\end{theorem}\n\nMuch of $\\omega$-automata theory revolves around Theorem~\\ref{thm:fundamental}. The quest for good proofs of this theorem---efficient language-preserving transformations between automata of different types---has led to many interesting results. All types of $\\omega$-automata are interesting in their own right; each one has its advantages and applications in specific contexts. \n\n\n\\section{Basic properties of B\\\"{u}chi automata}\n\nSome basic insights into $\\omega$-automata can be derived from analyzing runs in a straightforward fashion, for instance, that regular languages can be defined by regular expressions of a certain type, that deterministic B\\\"{u}chi automata are less expressive than nondeterministic ones, and that complementation is problematic for B\\\"{u}chi automata.\n\n\\subsection{\\texorpdfstring{$\\omega$}{\u03c9}-Regular expressions}\n\nAn $\\omega$-regular expression \\index{$\\omega$-regular expression}\\index{expression!$\\omega$-regular} \\cite{muller-infinite-sequences-and-finite-machines-1963} is of the form \n\\begin{align}\n r_0 \\cdot s_0^\\omega + \\dots + r_{n-1} \\cdot s_{n-1}^\\omega \\enspace,\n\\end{align}\nwith $n$ being a natural number and the $r_i$'s and the $s_i$'s being ordinary regular expressions. The semantics is the obvious one.\n\nSince the empty set can be denoted by an empty expression ($n = 0$) and since our definition of $\\omega$-power is only defined for (sets of) nonempty finite words (see Section~\\ref{sec:omega-words}), it is reasonable to require that the $s_i$'s be  built from the letters of the alphabet, ``$+$'' (for union), ``$\\cdot$'' (for concatenation), and ``${}^+$'' (for finite positive iteration). It is also reasonable to allow that individual $r_i$'s are omitted.\n\n\\begin{theorem}[$\\omega$-regular expression \\cite{buechi-decision-method-restricted-second-order-arithmetic-1962}]\n \\label{thm:regexps}\n Every $\\omega$-language recognized by a B\\\"{u}chi automaton is denoted by an $\\omega$-regular expression and vice versa.\n\\end{theorem}\n\nFor the proof, assume a B\\\"{u}chi automaton is given. The insight needed is that for every accepting run $r$ there are some state $q \\in B$ and an infinite sequence $\\langle i_0, i_1, \\dots\\rangle$ of positions such that $r(0)$ is initial, $i_0 < i_1 < \\dots$, and $r(i_j) = q$ for every $j$. This motivates the following definition. For states $q, q'$, let $L_{q,q'}$ be the language recognized by the ordinary automaton on finite words with $q$ as initial and $q'$ as final state. Then the language recognized by the B\\\"{u}chi automaton is \n\\begin{align} \n \\bigcup_{q \\in Q_I, q' \\in B} L_{q,q'} (L_{q',q'} \\setminus \\{\\epsilon\\})^\\omega\\enspace.\n\\end{align}\nThis representation can be turned into an $\\omega$-regular expression using techniques known from finite-state automata on finite words.\n\n\\newcommand{\\New}[1]{q_\\text{new}}\n\nFor the proof of the converse, first observe that it is enough to show that an expression of the form $r \\cdot s^\\omega$ (meaning $n = 1$) denotes a language recognized by a B\\\"{u}chi automaton, because the class of languages recognized by B\\\"{u}chi automata is closed under union: the disjoint union of two given B\\\"{u}chi automata is a B\\\"{u}chi automaton recognizing the union of the languages recognized by the given automata. So assume $\\mathscr A$ and $\\mathscr B$ are finite-state automata on finite words recognizing the languages denoted by $r$ and $s$, respectively. A B\\\"{u}chi automaton for $r \\cdot s^\\omega$ is obtained by modifying the disjoint union of $\\mathscr A$ and $\\mathscr B$ as follows. First, an additional state $\\New q$ is added. Second, for every transition $\\langle q, a, q'\\rangle$ in~$\\mathscr A$ where $q'$ is final, the transition $\\langle q, a, \\New q\\rangle$ is added. Third, for every transition $\\langle q, a, q'\\rangle$ in~$\\mathscr B$ where $q$ is initial, the transition $\\langle \\New q, a, q'\\rangle$ is introduced. Fourth, for every transition $\\langle q, a, q'\\rangle$ in $\\mathscr B$ where $q'$ is final, the transition $\\langle q, a, \\New q\\rangle$ is added. Finally, every final state looses its status as final state; every initial state of $\\mathscr B$ looses its status as initial state; the state $\\New q$ becomes the only B\\\"{u}chi state; if one of the initial states of $\\Aut A$ was final (the empty word was accepted), then $\\New q$ becomes initial, too.\\qed\n\nFrom the above proof, it immediately follows:\n\n\\begin{remark}\n \\label{thm:basic}\n \\begin{inparaenum}\n \\item Every nonempty regular $\\omega$-language contains an ultimately\n\t periodic word.\n \\item The emptiness problem for B\\\"{u}chi automata is decidable nondeterministically in logarithmic space, by a simple graph search.\n \\end{inparaenum}\n\\end{remark}\n\n\\subsection{Co-B\\\"{u}chi and deterministic B\\\"{u}chi automata}\n\\label{sec:deterministic-buchi}\n\nDis- and reassembling runs is a simple but powerful technique in the context of $\\omega$-automata, which can, for instance, be used to show that co-B\\\"{u}chi automata and deterministic B\\\"{u}chi automata are weaker than nondeterministic ones:\n\n\\begin{proposition}\n \\label{thm:co-buchi}\n \\begin{inparaenum}\n \\item The language denoted by $((0+1)^* 0)^\\omega$ cannot be recognized by a co-B\\\"{u}chi automaton.\n \\item The language denoted by $(0+1)^* 1^\\omega$, which is the complement of the language denoted by $((0+1)^* 0)^\\omega$, cannot be recognized by a deterministic B\\\"{u}chi automaton. \n \\end{inparaenum}\n\\end{proposition}\n\nFor the proof of the first part, assume a co-B\\\"{u}chi automaton with $n$ states recognizes the language. Then it accepts the word $(1^n0)^\\omega$, say $r$ is an accepting run. There is some~$k$ such that all letters in the segment $r[(n+1)k,(n+1)(k+1))$ are co-B\\\"{u}chi states. (As usual, if $u$ denotes an $\\omega$-word, then $u[i,j)$ denotes $u(i)u(i+1) \\dots u(j-1)$.) Because this segment has $n+1$ positions, there are $i$ and $j$ such that $i<j \\leq n$ and $r((n+1)k+i) = r((n+1)k+j)$, which means $r[0,(n+1)k+i)r[(n+1)k+i,(n+1)k+j)^\\omega$ is an accepting run of the automaton on $(1^n0)^k1^\\omega$---a contradiction.\n\nFor the proof of the second part, assume a deterministic B\\\"{u}chi automaton recognizes the language denoted by $(0+1)^* 1^\\omega$. By complementing its B\\\"{u}chi set and viewing it as a co-B\\\"{u}chi set, one obtains a co-B\\\"{u}chi automaton for the language denoted by $((0+1)^* 0)^\\omega$---a contradiction to the first part.\\qed\n\nProposition~\\ref{thm:co-buchi} shows that the complementation procedure known from finite-state automata (first determinize, then negate the ``final condition''---does not work for B\\\"{u}chi automata, because the following two transformations are not possible in general: \n\\begin{inparaitem}[]\n\\item from a nondeterministic B\\\"{u}chi automaton to an equivalent deterministic B\\\"{u}chi automaton;\n\\item from a deterministic B\\\"{u}chi automaton to a deterministic B\\\"{u}chi automaton for the complement of the language recognized.\n\\end{inparaitem}\nThere are fundamental differences between $\\omega$-automata and ordinary ones. \n\nAs complementation and determinization are important operations on automata in general, much of the work on $\\omega$-automata deals with them and so does this paper. B\\\"{u}chi was the first to show that nondeterministic B\\\"{u}chi automata are closed under complementation \\cite{buechi-decision-method-restricted-second-order-arithmetic-1962}; Safra's construction \\cite{safra-complexity-omega-automata-1988}\\index{Safra's construction} was the first with a worst-case state complexity of $\\theta(n)^n$, which is optimal \\cite{Michel88}. The first determinization construction for B\\\"{u}chi automata, transforming a nondeterministic B\\\"{u}chi automaton into an equivalent forward deterministic Rabin automaton, was given by McNaughton~\\cite{mcnaughton-testing-and-generating-infinite-sequences-1966}; again, Safra's construction was the first with a worst-case state complexity of $\\theta(n)^n$, which is, again, optimal \\cite{loeding-optimal-bounds-for-transformations-omega-automata-1999}. \nThe development with regard to complementation up to the year 2007 is described very nicely in~\\cite{vardi-buechi-complementation-saga-2007}.\n\n\\section{Basic constructions}\n\nIn this section of introductory technical nature, some important basic constructions are described. They exhibit parallels to the situation with finite words, but demonstrate also distinctive features of $\\omega$-automata. \n\n\\subsection{Products of B\\\"{u}chi automata}\n\\label{sec:products}\n\nA simple operation known from nondeterministic finite-state automata is the disjoint union of two automata, which yields a non-deterministic automaton recognizing the union of the two languages recognized by the two given automata. This works for $\\omega$-automata exactly in the same way, provided the two automata have recurrence conditions of the same type.\n\nAnother simple operation known from finite-state automata is the product of two automata: it can be used to construct an automaton recognizing the intersection or the union of the two languages recognized by the given automata. The adaptation to $\\omega$-words is possible, but not straightforward. In particular, if the recurrence condition is a B\\\"{u}chi condition, the problem arises that B\\\"{u}chi states may not be visited simultaneously, which, in the worst case, may result in an automaton not accepting a single word, while the intersection of the two languages may be the set of all $\\omega$-words over the given alphabet.\n\nThe problem can be overcome by adding one bit to the state space. More precisely, a state in the adjusted product is of the form $\\langle q_0, q_1, b\\rangle$, where $b \\in [2]$. The transition relation is chosen in such a way that $b = 1$ if there was a prior position with a B\\\"{u}chi state in the second component, but no position in between with a B\\\"{u}chi state in the first component. The word is accepted if a state $\\langle q_0, q_1, 1\\rangle$ with $q_0 \\in B_0$ occurs infinitely often, that is, $B_0 \\times Q_1 \\times \\{1\\}$ is the B\\\"{u}chi set.\n\nMore precisely, assume there is a transition $\\langle q_0, a, q_0'\\rangle$ in the first automaton and a transition $\\langle q_1, a, q_1'\\rangle$ in the second one. This gives rise to a transition from $\\langle q_0, q_1, b\\rangle$ to $\\langle q_0', q_1', b'\\rangle$, where $b'$ is defined by: if $q_1 \\in B_1$, then $b' = 1$; if $b = 1$ and $q_0 \\in B_0$ and $q_1 \\notin B_1$, then $b' = 0$; in all other cases, $b' = b$.\n\nThe constructed automaton is forward deterministic, provided the given automata are forward deterministic. When the transitions are reversed and the given automata are backward deterministic, then it is backward deterministic \\cite{carton-michel-unambiguous-buechi-automata-2003}. The construction can also be used to turn a generalized B\\\"{u}chi automaton into an ordinary one, resulting in an automaton with $kn$ states, assuming the given automaton has $n$ states and $k$ B\\\"{u}chi sets.\n\n\\subsection{Automata with output and cascades}\n\nIn various situations, it is very helpful to consider $\\omega$-automata with output\\index{$\\omega$-automaton!with output}, which have an extra output function $\\lambda \\colon \\Delta \\to O$, where $O$ is some alphabet. The relation defined by such an automaton is the relation between $A^\\omega$ and $O^\\omega$ which contains a pair $\\langle u, v\\rangle$ if there is an accepting run $r$ of the automaton on $u$ such that $v(i) = \\lambda(\\langle r(i), u(i), r(i+1)\\rangle)$ for every~$i$.\n\nA simple example is the relation which holds between a binary tree and an $\\omega$-word if, and only if, the prefixes of the $\\omega$-word form an infinite rooted path in the tree.  When the trees considered are of width at most $k$, then the states of a suitable automaton can be chosen to be elements of $[k]$, representing the vertex on the current level that is chosen to be part of the path. There is a transition $\\langle i, a, j\\rangle$ if vertex $j$ is a successor of vertex $i$ in slice $a$, and $\\lambda(\\langle i, a, j\\rangle) = 0$ if $j$ is a left successor and else $\\lambda(\\langle i, a, j\\rangle) = 1$. \n\nJust as in the theory of finite-state automata on finite words, an $\\omega$-automaton with output can be composed with an $\\omega$-automaton (with or without output) by using the output of the first automaton as input for the second automaton---one speaks of a cascade\\index{cascade!of$\\omega$-automata} or of cascading. For instance, if the above automaton is cascaded with a B\\\"{u}chi automaton recognizing $(1^*0)^\\omega$, then the resulting automaton recognizes the trees that have a rooted path with infinitely many left successors. \n\nWhen cascading two finite-state automata on finite words a state of the resulting automaton is a pair consisting of a state of the first automaton and a state of the second one; when cascading $\\omega$-automata one needs to be careful about the recurrence condition. For instance, if two B\\\"{u}chi automata are cascaded, then the result can be chosen to be a generalized B\\\"{u}chi automaton or a B\\\"{u}chi automaton with a third component, consisting of a single bit, for combining the two B\\\"{u}chi recurrence conditions as described in~Section~\\ref{sec:products}.\n\n\\subsection{The breakpoint construction}\n\\label{sec:breakpoint}\n\\index{breakpoint construction}\n\nA more important example for a B\\\"{u}chi automaton with output is an automaton which defines the function that maps each leveled DAG to the subgraph which is composed of the finitary vertices of the DAG, or, dually, the infinitary vertices. A vertex is called finitary\\index{finitary!vertex of a DAG}\\index{infinitary!vertex of a DAG} if it has only a finite number of descendants, else it is called infinitary. For DAG's of finite width, which we only consider, being infinitary is equivalent to being on an infinite path.\n\nA B\\\"{u}chi automaton can guess which vertices on a level of a given DAG are finitary and which are not, and it can check that vertices guessed infinitary are indeed infinitary ones by forcing, via the transition relation, each infinitary vertex to have a successor. The problem is to verify that every vertex guessed finitary is indeed finitary. All successors of a vertex guessed finitary must be guessed finitary and this can be enforced by the transition relation, but this condition is only necessary and not sufficient. \n\nTo solve the problem a construction referred to as breakpoint construction can be used. A ``breakpoint automaton'' works in phases. When a phase starts, all vertices on the current level guessed to be finitary are stored in some set, the verification set. During a phase, the verification set is updated from level to level by replacing the vertices in it by their successors. If the guesses were correct, the verification set becomes empty at some point and the phase ends successfully. During a phase, all vertices newly guessed finitary are stored in some other set---they are put on hold. When a new phase starts, the vertices put on hold previously are moved into the verification set. For all guesses to be correct, every phase has to end successfully.\n\nA state of a breakpoint automaton is of the form $\\langle V, H\\rangle$, where $V$ is the current verification set and $H$ is the set of states currently put on hold. A transition is of the form $\\langle\\langle V, H\\rangle, a, \\langle V', H'\\rangle\\rangle$ and must satisfy the following conditions, which are all phrased with respect to the slice $a$ being read:\n\\begin{compactitem}\n\\item The sets $V$ and $H$ are disjoint sets of vertices of the upper level.\n\\item The sets $V'$ and $H'$ are disjoint sets of vertices of the lower level.\n\\item Every vertex on the upper level not in $V \\cup H$ has a successor on the lower level not in $V' \\cup H'$.\n\\item If $V \\neq \\emptyset$, then $V'$ is the set of all successors of the vertices in $V$, and $H'$ contains at least all the successors of the vertices in $H$ which do not belong to $V'$.\n\\item If $V = \\emptyset$, then $V'$ is the set of all successors of the vertices in $H$.\n\\end{compactitem}\nA state is initial if $H = \\emptyset$; it is a B\\\"{u}chi state if $V = \\emptyset$. \n\nTo a transition as above, the output function assigns the slice which is obtained from~$a$ by restricting it to the vertices from $V \\cup V' \\cup H \\cup H'$ or, dually, to the other vertices.\n\n\\begin{theorem}[breakpoint construction \\cite{miyano-hayashi-alternating-finite-automata-on-omega-words-2-1984}]\n \\label{thm:breakpoint}\n For every $k$, the breakpoint construction yields a B\\\"{u}chi automaton with $3^k$ states outputting, for every leveled DAG of width at most $k$, the subgraph of its finitary [infinitary]  vertices.\n\\end{theorem}\n\nThe breakpoint construction is used in Sections~\\ref{sec:complementation}, \\ref{sec:disambiguation}, \\ref{thm:weak-characterization}, and \\ref{sec:weak-alternation}.\n\n\\subsection{The lift construction}\n\\index{lift construction}\n\nFor backward deterministic automata, the previous task---computing the finitary or infinitary vertices in a leveled DAG---can be solved using a construction here referred to as lift construction. If one knows the finitary vertices of a leveled DAG on one level, one can determine the finitary vertices on the previous level in a deterministic fashion: a vertex on the previous level is finitary if, and only if, all its successors are finitary (in particular, a vertex without successors is finitary). The naive approach for constructing a backward deterministic automaton for determining the finitary vertices is to use states which have one bit for every vertex on the current level, indicating whether the vertex is finitary or not, and to use the above rule as a backward transition function. The problem is that this construction may overapproximate, because when in a run on a graph with only infinitary vertices all vertices are assumed to be finitary the above rule is obeyed.\n\n\nTo overcome the problem it is important to realize that for every finitary vertex $v$, say on level $l$, there is some smallest level $i > l$ without a descendant of $v$. We call this level the extinction level of $v$, denote it by $\\text{el}(v)$, and use it to rank~$v$. \n\nTo this end, assume a vertex $v$ is on some level $l$. The rank of $v$ measures how difficult it is to get from $v$ to its extinction level, more concretely, how ``wide'' the part of the run DAG is which one needs to pass by while moving from $v$ to its extinction level. For every level $i$ between $l$ and the extinction level of $v$, that is, for every $i$ with $l \\leq i < \\text{el}(v)$, let $W_i$ contain all extinction levels of vertices on level~$i$, but only the ones which are before the extinction level of~$v$, that is, $W_i = \\{\\text{el}(w) \\mid w \\in V^{(i)} \\text{ and } \\text{el}(w) < \\text{el}(v)\\}$. The maximum of the cardinalities of the $W_i$'s is the extinction rank of~$v$ and denoted $\\text{er}(v)$. Formally, $\\text{er}(v) = \\max\\{|W_i| \\mid l \\leq i < el(v)\\}$. For an illustration, see Figure~\\ref{fig:ext-ranks}.\n\n\\begin{figure}[t]\n \\small\n \\hfill\n \\protect\\begin{minipage}[t]{0.5\\linewidth}\n \\begin{tikzpicture}[baseline=(current bounding box.north),\n\t node distance=5mm, every node\/.style={circle,inner sep=1pt}]\n\t \\node (a0) {0};\n\t \\node[xshift=5mm] (b0) [right of=a0] {0};\n\t \\node[xshift=5mm] (c0) [right of=b0] {\\bf 1};\n\t \\node[xshift=5mm] (d0) [right of=c0] {0};\n\t \\node[xshift=5mm] (e0) [right of=d0] {0};\n\t \\node[xshift=5mm] (f0) [right of=e0] {};\n\n\t \\node[xshift=-2mm,yshift=3mm] (aa) [below of=a0] {};\n\t \\node[xshift=2mm,yshift=3mm] (fa) [below of=f0] {};\n\t \\draw (aa) -- (fa);\n\n\t\n\t\n\t\n\t\n\t\n\t\n\n\t \\node (ad) [below of=a0] {};\n\t \\node (bd) [below of=b0] {};\n\t \\node (cd) [below of=c0] {2};\n\t \\node (dd) [below of=d0] {1};\n\t \\node (ed) [below of=e0] {0};\n\t \\node (fd) [below of=f0] {};\n\n\t \\node (a1) [above of=a0] {2};\n\t \\node (a2) [above of=a1] {0};\n\n\t \\node (b1) [above of=b0] {0};\n\t \\node (b2) [above of=b1] {0};\n\t \\node (b3) [above of=b2] {2};\n\n\t \\node (c1) [above of=c0] {0};\n\t \\node (c2) [above of=c1] {\\bf 1};\n\t \\node (c3) [above of=c2] {\\bf 2};\n\n\t \\node (d1) [above of=d0] {\\bf 1};\n\t \\node (d2) [above of=d1] {0};\n\n\t \\node (e1) [above of=e0] {?};\n\t \\node (e2) [above of=e1] {?};\n\t \\node (e3) [above of=e2] {0};\n\n\t \\node (f1) [above of=f0] {};\n\t \\node (f2) [above of=f1] {};\n\t \\node (f3) [above of=f2] {};\n\n\t \\node (g1) [right of=f1] {\\dots};\n\n\n\t \\draw[->] (a0) -- (b1);\n\t \\draw[->] (a1) -- (b3);\n\t \\draw[->] (a2) -- (b2);\n\n\t \\draw[->] (b0) -- (c1);\n\t \\draw[->] (b1) -- (c1);\n\t \\draw[->] (b2) -- (c1);\n\t \\draw[->] (b3) -- (c3);\n\n\t \\draw[->] (c0) -- (d2);\n\t \\draw[->] (c2) -- (d2);\n\t \\draw[->] (c2) -- (d0);\n\t \\draw[->] (c3) -- (d1);\n\t \\draw[->] (c3) -- (d2);\n\n\t \\draw[->] (d1) -- (e0);\n\n\t \\draw[->] (e1) -- (f1);\n\t \\draw[->] (e2) -- (f2);\n\t \\draw[->] (e1) -- (f3);\n \\end{tikzpicture}  \n \\end{minipage}\n \\hfill\n \\protect\\begin{minipage}[t]{0.4\\linewidth}\n\t \\protect\\caption{Beginning of a leveled DAG with vertices labeled by their extinction ranks and critical values at the bottom.  Lifts are in bold; question marks indicate vertices with ranks which cannot be determined from the visible part of the DAG.}\n\t \\label{fig:ext-ranks}\n\t \\vspace*{\\fill}\n \\end{minipage}\n \\hspace*{\\fill}\n\\end{figure}\n\n\\begin{remark}\n \\label{thm:ext-ranks}\n \\begin{inparaenum}\n \\item The extinction rank of a vertex in a leveled graph of width at most $k$ is at most $k-1$.\n \\item On every maximum path starting in a finitary vertex the extinction rank is monotone descending and eventually reaches~$0$.\n \\end{inparaenum}\n\\end{remark}\n\nIn the lift construction, the backward deterministic automaton determines, for each vertex, its extinction rank, that is, a state of the automaton maps each vertex on the current level to a value in $[k] \\cup \\{\\infty\\}$, where $\\infty$ is used for infinitary vertices. It is possible to define a backward deterministic transition function accordingly, because the extinction ranks of the vertices on one level can be determined from the structure of the respective slice and the extinction ranks of the vertices on the next level, as described in what follows. \n\nLet $U$ be the set of vertices on the upper level of a slice~$a$. For every $v \\in U$, let $m_v$ be the maximum of all values $\\text{er}(v')$ where $v'$ is a successor of $v$ in the slice $a$; by convention, $m_v = -1$ if $v$ has no successor. If there is no $v \\in U$ with $m_v = -1$, then $\\text{er}(v) = m_v$ for all $v \\in U$. If there is such a $v$, then the largest number $c$ such that $[c] \\subseteq \\{m_v \\mid v \\in U\\}$ is called the critical value of the upper level. For every $v \\in U$ with $m_v \\geq c$, the equation $\\text{er}(v) = m_v$ still holds. For every other $v \\in U$, the values are ``lifted'': $\\text{er}(v) = m_v + 1$.\n\nIf extinction ranks are used, then an overapproximation as described above can be avoided by adding an appropriate recurrence condition, more precisely, a generalized transition-B\\\"{u}chi condition. For every rank $i$, there is a transition-B\\\"{u}chi set $B_i$ which includes all transitions in which $i$ does not occur as a value on the upper level or $i$ is less than the critical value (and thus lifted), see Remark~\\ref{thm:ext-ranks}(2) and Figure~\\ref{fig:ext-ranks}. \n\n\\begin{theorem}[lift construction   \n \\cite{carton-michel-unambiguous-buechi-automata-2003}]\n \\label{thm:shift}\n The lift construction yields, for every $k$, a backward deterministic generalized transition-B\\\"{u}chi automaton with at most $(k+1)^n$ states outputting, for every leveled DAG of width at most $k$, the subgraph of its finitary [infinitary] vertices.  \n\\end{theorem}\n\nNote that the above approach is very versatile. If, for instance, one wants to determine the vertices which have at least one descendant with no successors, which one could call weakly finitary vertices, then one can take the same approach, replacing maximization by minimization. \n\nThe above description of the lift construction is somewhat technical because of the measure introduced; a more ``automatic'' description follows. A state is a sequence $P_0 \\dots P_{m-1}$ of nonempty pairwise disjoint sets of vertices. Assume a letter $a$ (a slice) is read backwards. Then the new state is determined in two steps. First, the sequence $P'_{-1} P'_0 \\dots P'_{m-1}$ is determined where \n\\begin{inparaenum}[(i)]\n\\item $P_{-1}'$ consists of all vertices $v$ on the upper level of $a$ without\n  successors and \n\\item $P_i'$, for $i \\geq 0$, consists of all such vertices with\n  some successor in $P_i$, but no successor in $P_{i'}$ for any $i' < i$.\n\\end{inparaenum}\nSecond, the new state is obtained from $P'_{-1} P'_0 \\dots P'_{m-1}$ by removing all empty entries. The recurrence condition is, again, a generalized transition-B\\\"{u}chi condition: for every~$i$ there are infinitely many transitions with $i>m$ or $P_{i'}' \\neq \\emptyset$ for $i' < i$. The vertices that occur in the states are exactly the finitary ones.\n\n\\enlargethispage{\\baselineskip}\n\nThe lift construction is used for different purposes in Section~\\ref{sec:backward-determinization}.\n\n\\subsection{Latest appearance records}\n\\label{sec:lar}\n\\index{latest appearance record}\\index{LAR!latest appearance record}\\index{latest appearance automaton}\\index{LAA!latest appearance automaton}\n\nGiven an alphabet $A$ and a special symbol $\\$$ not in $A$, the latest appearance automaton (LAA) is a forward deterministic automaton with states being words over $A \\cup \\{\\$\\}$ where every letter from $A$ occurs at most once and $\\$$ occurs exactly once. One such word is called a latest appearance record (LAR) and the part to the right of ``\\$'' is its frame. \n\nThe initial state of the LAA is the one-letter word $\\$$; the recurrence condition is trivial; the transition function $\\delta$ is defined as follows.  When $u$ is a state of the form $v\\$v'$ and $a$ is a letter of the alphabet occurring in $vv'$, say $vv' = waw'$, then $\\delta(u,a) = w\\$w'a$. When~$a$ does not occur in $vv'$, then $\\delta(u,a) = vv'\\$a$. So the order in which the letters occur in the current state of the automaton is the order of their latest appearances in the prefix of the given word read so far, with all letters in the frame of the current state being the ones that have occurred since the previous occurrence of the letter just read. From this, the following can be derived.\n\n\\begin{remark}\n \\cite{buchi-landweber-1969-a,gurevich-harrington-1982}\n Consider the frames of maximal length among all frames occurring infinitely often in the run of the LAA on a given word. Then all theses frames contain the same letters and these are exactly the ones occurring infinitely often in the given word.\n\\end{remark}\n\nAn interesting application of the latest appearance record is the transformation of a given Muller automaton into an equivalent parity automaton. First, the Muller condition is removed (and replaced by the trivial recurrence condition). Second, the automaton is augmented by the trivial output function, which simply outputs the current state. Third, the generated automaton is cascaded with the LAA over the state set of the Muller automaton. Finally, assuming the automaton has $n$ states, a priority function is added that assigns each state $\\langle q, v\\$v'\\rangle$ the priority $2n - 2|v'|$ if $\\Occ{v'}$ is a Muller set and else $2n - 2|v'| + 1$.\n\n\\begin{theorem}[Muller to parity]\n \\label{thm:muller-to-parity}\n \\cite{mostowski-regular-expressions-for-infinite-trees-and-a-standard-form-of-automata-1984}\n For every forward deterministic [non-deterministic] Muller automaton with $n$ states there is an equivalent forward deterministic [non-deter\\-min\\-is\\-tic] parity automaton with $(n+1)!$ states and at most $2n$ priorities.\n\\end{theorem}\n\nA refined construction, saving priorities if possible, is presented in Section~\\ref{sec:parity-index}.\n\n\\section{Run DAG's of B\\\"{u}chi automata}\n\\label{sec:dags}\n\nB\\\"{u}chi automata, in general, are nondeterministic automata, in other words, there may be several runs of a given B\\\"{u}chi automaton on a given word. These runs have to be considered at the same time if, for instance, one wants to turn a B\\\"{u}chi automaton into a B\\\"{u}chi automaton for the complement of the language recognized, because not to accept means \\emph{all} initial runs are not recurrent. \n\nThere are essentially two global structures that have been investigated for arranging all runs of a B\\\"{u}chi automaton in a concise way: DAG's and trees. The former are treated in this section, the latter in the next one. Applications are complementation, determinization, and disambiguation (defined in Section~\\ref{sec:disambiguation}). \n\nAssume a B\\\"{u}chi automaton is given. The run DAG \\index{run DAG}\\index{DAG!run} of a given $\\omega$-word $u$ is the leveled graph with levels $\\{Q \\times \\{i\\}\\}_{i \\in \\omega}$ and edges $\\langle \\langle q, i \\rangle, \\langle q', i+1\\rangle \\rangle$ for $\\langle q, u(i), q' \\rangle \\in \\Delta$. Its width is the number of states of the given automaton. \n\nOften, it is useful to think of a run DAG as a graph labeled with elements from $Q$; in this section, it is sufficient to think of it as providing only information about whether the state component of a vertex is an initial or a B\\\"{u}chi state. Technically, the DAG is labeled with elements from $\\powerset(\\{I, B\\})$ and we say it is $\\{I,B\\}$-tagged; if a vertex is labeled with a letter $a$ and $I \\in a$, we say it is $I$-tagged, and, analogously, if $B \\in a$, we say it is $B$-tagged.\n\nA vertex of an $\\{I,B\\}$-tagged DAG is called $B$-recurring if a path with an infinite number of $B$-tagged vertices starts in it; it is called $B$-free if none of its descendants (including itself) is $B$-tagged. The ultimate width of such a DAG is the limes inferior of the number of non-$B$-recurring infinitary vertices on a given level.\n\n\\begin{remark}\n\tAn $\\omega$-word is accepted by a B\\\"{u}chi automaton if, and only if, there is an $I$-tagged $B$-recurring vertex on level~$0$ of the run DAG of the word.\n\\end{remark}\n\nThe main insight needed about $\\{I,B\\}$-tagged DAG's (or simply $\\{B\\}$-tagged DAG's) of finite width is that they can be decomposed in a simple manner. Consider the following operation, here called peeling. First, remove all finitary vertices; second, remove all $B$-free vertices. Peeling does not remove any $B$-recurring vertex, and if it does not change the DAG at all, then all vertices are $B$-recurring, because every vertex has a strict $B$-tagged descendant. \nMoreover, if there are non-$B$-recurring infinitary vertices, then peeling decreases the ultimate width by at least one, as explained in what follows.\n\nConsider a non-$B$-recurring infinitary vertex. By K\\\"{o}nig's lemma \\cite{koening-theorie-der-endlichen-und-unendlichen-graphen-1936}, there is an infinite path starting in it. Assume that every $B$-tagged strict descendant of the vertex is finitary. Then, after removing the finitary vertices, each successor of the vertex is $B$-free, but the infinite path is still there and all of its vertices (except, maybe, the first one) are removed in the second step, decreasing the ultimate width by one. If there is a strict $B$-tagged infinitary descendant of the vertex, apply the same argument to it. This cannot go ad infinitum, because a path with an infinite number of $B$-tagged vertices would be constructed. \n\nThis all implies:\n\n\\begin{lemma}[peeling \\cite{kupferman-vardi-weak-alternating-automata-are-not-that-weak-2001}]\n For every B\\\"{u}chi automaton with $n$ states, peeling the run DAG of any $\\omega$-word $n$ times yields the subgraph induced by the $B$-recurring vertices.\\qed\n\\end{lemma}\n\n\\index{peeling!of a leveled DAG}\n\nThis can be used in various ways, in particular, it can be used for complementing B\\\"{u}chi automata, see Section~\\ref{sec:complementation}, determinizing them backward, see Section~\\ref{sec:backward-determinization}, and showing that alternating B\\\"{u}chi automata can easily be converted into weak alternating automata, see Section~\\ref{sec:weak-alternation}.\n\nTo describe these applications, it is useful to have some notation and terminology at hand. By the above, each vertex $v$ in a $\\{B\\}$-tagged DAG of finite width can be assigned a value in $\\omega \\cup \\{\\infty\\}$ according to when the vertex is removed by peeling the DAG successively. More precisely, when $i$ is a natural number and all vertices with value $< 2i$ are removed from the given DAG, the finitary vertices in the remaining DAG get assigned~$2i$; when all vertices with value $< 2i+1$ are removed, the $B$-free vertices in the remaining DAG get assigned~$2i+1$. The $B$-recurring vertices get assigned $\\infty$. The number assigned to a vertex $v$ is called its canonical rank, it is denoted $c(v)$, and, according to the above, it is $\\infty$ or $< 2n$, when $n$ is the width of the DAG.\\index{canonical rank function}\\index{rank function!canonical}\n\n\\begin{corollary} \n \\label{three}\n For a B\\\"{u}chi automaton with $n$ states, let $c$ be the canonical rank function of the run DAG of some $\\omega$-word. \n \\begin{inparaenum}\n \\item The word is accepted if, and only if, $c(v) = \\infty$ for some $I$-tagged vertex $v$ on level~$0$. \n \\item Equivalently, the word is not accepted if, and only if, $c(v) < 2n$ for every $I$-tagged vertex $v$ on level~$0$.\n \\end{inparaenum}\n\\end{corollary}\n\n\\subsection{Complementation via canonical ranks}\n\\label{sec:complementation}\n\nThe idea of using ranks or ``progress measures'' for complementing $\\omega$-automata goes back to \\cite{klarlund-progress-measures-complementation-omega-automata-applications-temporal-logic-1991} and has been improved and refined over the years, especially in \\cite{kupferman-vardi-weak-alternating-automata-are-not-that-weak-2001}. The basic idea is to implement Corollary~\\ref{three}(2). The starting point is a compilation of properties of the canonical rank function of a given $\\{I, B\\}$-tagged leveled DAG.\n\n\\begin{ennote}\n \\label{one} \n Let $v$ be any vertex. If $v$ does not have any successor, let $M = 0$, else let~$M$ be the maximum of all values $c(v')$ for successors~$v'$ of~$v$. If $v$ is not $B$-tagged or if $M$ is even, then $c(v) = M$; if $v$ is $B$-tagged and $M$ is odd, then $c(v) = M + 1$.\n\\end{ennote}\n\n\\begin{ennote}\n \\label{two} \n For any vertex with an even rank, the number of its descendants with the same rank is finite.\n\\end{ennote}\n\nIn general, a rank function of a leveled DAG of width~$n$ with vertex set~$V$ is a function $f \\colon V \\to [2n]$ satisfying Properties~\\ref{one} and~\\ref{two} with $f$ instead of $c$.\\index{rank function!of a leveled DAG}\\index{leveled DAG!rank function}\n\n\\begin{remark}\n Any rank function is pointwise greater or equal to the canonical rank function.\n\\end{remark}\n\nA complementation construction for B\\\"{u}chi automata can now be based on Corollary~\\ref{three}(2) and the following observations. First, there is a forward deterministic automaton with trivial recurrence condition that outputs the part of the $\\{I, B\\}$-tagged run DAG of a given word which is reachable from the $I$-tagged vertices on level~$0$. Second, there exists a nondeterministic B\\\"{u}chi automaton that produces for every $\\{I, B\\}$-tagged leveled graph of width at most~$n$ the same graph, but with any labeling with numbers from $[2n]$ such that Property~\\ref{one} is satisfied. Third, using a variant of the breakpoint construction, see Theorem~\\ref{thm:breakpoint}, a B\\\"{u}chi automaton can be constructed that checks Property~\\ref{two} for a $[2n]$-labeled DAG. In other words, a suitable cascade yields a B\\\"{u}chi automaton for the complement of the language recognized by a given B\\\"{u}chi automaton.\n\n\\begin{theorem}[complementation via ranks\n \\cite{kupferman-vardi-weak-alternating-automata-are-not-that-weak-2001}]\n \\label{thm:kupferman-vardi}\n Complementation via canonical ranks yields, for every B\\\"{u}chi automaton with $n$ states, a B\\\"{u}chi automaton with at most $(6n)^n$ states.\n\\end{theorem}\n\nIn \\cite{friedgut-kupferman-vardi-buechi-complementation-made-tighter-2006}, the above approach is improved, resulting in an asymptotic upper bound of $(0.96\\, n)^n$ for the number of states, and in \\cite{schewe-buechi-complementation-made-tight-2009} a further improvement leads to a construction which is optimal within a factor of $O(n^2)$ and has an asymptotic upper bound of $(0.76n)^n$.\n\n\\subsection{Backward determinization via canonical ranks}\n\\label{sec:backward-determinization}\n\nA second application of canonical ranks is the conversion of a given nondeterministic B\\\"{u}chi automaton into an equivalent backward deterministic generalized transition-B\\\"{u}chi automaton. The idea, which is due to \\cite{carton-michel-unambiguous-buechi-automata-2003}, is to use Corollary~\\ref{three}(1) and to construct an automaton which labels the run DAG in a backward deterministic fashion with the values of the canonical rank function.\n\nThe key to designing such an automaton is the fact that the canonical rank function is the only function on an $\\{I,B\\}$-tagged DAG satisfying Property~\\ref{one} and the following one, Property~\\ref{twostar}.\n\\begin{ennote}\n \\label{twostar}\n \\begin{inparaenum}\n \\item For every even number $i$, the set $c^{-1}(\\{i\\})$ is exactly the set of finitary vertices in the sub-DAG without the vertices in $c^{-1}([i])$.\n \\item For every vertex $v \\in V$, if $c(v) > 1$ and $c(v)$ is odd, then $v$ has a descendant $v'$ with $c(v') = c(v) - 1$.\n \\item In the sub-DAG consisting of the vertices in $c^{-1}(\\{\\infty\\})$, every vertex has a strict $B$-tagged descendant.\n \\end{inparaenum}\n\\end{ennote}\n\nThis means a backward deterministic generalized transition-B\\\"{u}chi automaton computing the rank function for a run DAG can be constructed as a cascade of two automata: \n\\begin{inparaenum}[(i)]\n\\item an automaton with a backward deterministic transition function and a trivial recurrence condition outputting the run DAG of a given word and an assignment to the vertices satisfying Property~\\ref{one}; \n\\item a backward deterministic automaton checking Property~\\ref{twostar} using adaptations of the lift construction, see Theorem~\\ref{thm:shift} and also the subsequent remark on weakly finitary vertices.\n\\end{inparaenum}\nAn automaton equivalent to the given B\\\"{u}chi automaton is obtained when the states which assign $\\infty$ to an $I$-tagged vertex are chosen to be initial. \n\n\\begin{theorem}[backward determinization via canonical ranks\n \\cite{carton-michel-unambiguous-buechi-automata-2003}]\n Backward determinization via canonical ranks yields, for every B\\\"{u}chi automaton with $n$~states, a generalized transition-B\\\"{u}chi automaton with at most $(3n)^n$ states.\n\\end{theorem}\n\n\\section{Run trees of B\\\"{u}chi automata}\n\\label{run trees}\n\nRun DAG's are one way to represent the set of all runs of a B\\\"{u}chi automaton on an $\\omega$-word. A different approach, which can serve as a basis for complementation, disambiguation, and forward determinization, is to use compressed run trees.\n\nIn a first step towards the definition of the compressed run tree for a given word $u$ with respect to a given B\\\"{u}chi automaton, a labeled binary tree $t$ is defined, using a refined subset construction. Just as in the subset construction, all the states reachable from the initial states are tracked at the same time. The difference is that in each step the set of states reachable by reading the next letter is split into the B\\\"{u}chi states and the non-B\\\"{u}chi states: a binary tree emerges. To keep this tree compact, only one occurrence of each state---more precisely, its leftmost occurrence---is kept, that is, the tree is pruned in a straightforward fashion.\n\nIn the following, when a vertex $v$ is said to be to the left of another vertex $v'$, then this means that $v$ and $v'$ are on the same level, that is, $|v| = |v'|$, and there exists $i < |v|$ such that $v(j) = v'(j)$ for all $j<i$, $v(i) = 0$, and $v'(i) = 1$. The corresponding ordering is denoted by $<_{\\text{lft}}$.\n\nThe definition of $t$ is by induction on the levels. The root (level~$0$) of $t$ is labeled with~$Q_I$. Assume all vertices on level $l$ have already been constructed and assigned labels, say these vertices form the set $W$, and let $a$ stand for the next letter, $u(l)$. For each $v \\in W$, let $R_v = \\bigcup \\{\\Delta(q,a) \\mid q \\in t(v)\\}$, which means $R_v$ is the set of states reached from any state in the label of $v$ by reading~$a$. Set\n\\begin{align}\n Q_v^0 & = (R_v \\cap B) \\setminus  \\bigcup_{w <_{\\text{lft}} v} R_w\\enspace,&\n Q_v^1 & = R_v  \\setminus (B \\cup \\bigcup_{w <_{\\text{lft}} v} R_w) \\enspace.\n\\end{align} \nIn other words, $Q_v^0$ is the set of B\\\"{u}chi states reached from $t(v)$ by reading letter~$a$, but not including the states that are reached from any state in a label of a vertex to the left of~$v$. Similarly, $Q_v^1$ is the set of non-B\\\"{u}chi states reached from $t(v)$ by reading letter~$a$, but not including the states that are reached from any state in a label of a vertex to the left of~$v$. The definition of the tree $t$ now says that, for $v \\in W$ and $i<2$, if $Q_v^i \\neq \\emptyset$, then $vi \\in t$ and $t(vi) = Q_v^i$. For an illustration, see Figure~\\ref{fig:run-tree}. \n\n\\begin{figure}\n \\centering\n \\small\n \\begin{tikzpicture}[%\n\t level distance=5mm,\n\t level 1\/.style={sibling distance=13mm},\n\t level 2\/.style={sibling distance=7mm},\n\t level 3\/.style={sibling distance=5mm},\n\t level 4\/.style={sibling distance=3mm}]\n\n\t \\begin{scope}[every node\/.style={draw,circle,fill,inner sep=1pt}]\n\t \\node {}\n\t\t child {node {} \n\t\t\t child {node {}\n\t\t\t\t child {node {}\n\t\t\t\t\t child {node {}}\n\t\t\t\t\t child[missing]}\n\t\t\t\t child[missing]}\n\t\t\t child {node {}\n\t\t\t }}\n\t\t child {node {} \n\t\t\t child {node {}\n\t\t\t\t child {node {}\n\t\t\t\t\t child {node {}}\n\t\t\t\t\t child {node {}}}\n\t\t\t\t child[missing]\n\t\t\t }\n\t\t\t child {node {}\n\t\t\t\t child {node {}\n\t\t\t\t\t child {node {}}\n\t\t\t\t\t child {node {}}} \n\t\t\t\t child {node {}}}};\n\t \\end{scope}\n\n\t \\begin{scope}[every node\/.style={inner sep=1pt},\n\t\t level distance=5mm,\n\t\t level 1\/.style={sibling distance=13mm},\n\t\t level 2\/.style={sibling distance=8mm},\n\t\t level 3\/.style={sibling distance=6mm},\n\t\t level 4\/.style={sibling distance=6mm}]\n\t\t \\node at (-3.7,0) {$q_0$}\n\t\t child {node {$q_1, q_2$} \n\t\t\t child {node {$q_1, q_5$}\n\t\t\t\t child {node {$q_2$}\n\t\t\t\t\t child {node {$q_1$}}\n\t\t\t\t\t child[missing]}\n\t\t\t\t child[missing]}\n\t\t\t child {node {$q_3$}\n\t\t\t }}\n\t\t child {node {$q_3, q_4$} \n\t\t\t child {node {$q_2$}\n\t\t\t\t child[sibling distance=7mm] {node {$q_1$}\n\t\t\t\t\t child {node {$q_5$}}\n\t\t\t\t\t child {node {$q_0$}}}\n\t\t\t\t child[missing]\n\t\t\t }\n\t\t\t child {node {$q_0$}\n\t\t\t\t child[sibling distance=3mm] {node {$q_5$}\n\t\t\t\t\t child {node {$q_2$}}\n\t\t\t\t\t child {node {$q_3, q_4$}}} \n\t\t\t\t child[sibling distance=5mm] {node {$q_4$}}}};\n\t \\end{scope}\n\n\t \\begin{scope}[every node\/.style={inner sep=1pt},\n\t\t\t\t level 1\/.style={sibling distance=16mm},\n\t\t\t\t level 2\/.style={sibling distance=15mm},\n\t\t\t\t level 3\/.style={sibling distance=5mm},\n\t\t\t\t level 4\/.style={sibling distance=4mm}]\n\t \\node at (-8.9,0) {$q_0$}\n\t\t child[sibling distance = 10mm] {node {$q_1$} \n\t\t\t child[sibling distance = 4mm] {node {$q_3$}\n\t\t\t\t child {node {$q_2$}\n\t\t\t\t\t child {node {$q_1$}}}}\n\t\t\t child[sibling distance = 4mm] {node {$q_5$} \n\t\t\t\t child {node {$q_2$}\n\t\t\t\t\t child {node {$q_1$}}}}} \n\t\t child {node {$q_2$} \n\t\t\t child {node {$q_1$}\n\t\t\t\t child {node {$q_2$}\n\t\t\t\t\t child {node {$q_1$}}}}} \n\t\t child {node {$q_3$} \n\t\t\t child {node {$q_0$}\n\t\t\t\t child {node {$q_4$}}\n\t\t\t\t child {node {$q_5$}\n\t\t\t\t\t child {node {$q_2$}}\n\t\t\t\t\t child {node {$q_3$}}\n\t\t\t\t\t child {node {$q_4$}}}} \n\t\t\t child {node {$q_3$}\n\t\t\t\t child {node {$q_1$}\n\t\t\t\t\t child {node {$q_0$}}\n\t\t\t\t\t child {node {$q_5$}}}\n\t\t\t\t child {node {$q_2$}\n\t\t\t\t\t child {node {$q_1$}}}}} \n\t\t child {node {$q_4$} \n\t\t\t child {node {$q_2$}\n\t\t\t\t child {node {$q_1$}\n\t\t\t\t\t child {node {$q_0$}}\n\t\t\t\t\t child {node {$q_5$}}}}} ;\n\t \\end{scope}\n \\end{tikzpicture}\n \\caption{The first five levels of the ordinary run tree of a word; the corresponding part of the labeled compressed run tree; the corresponding part of the compressed run tree. For ease in notation, curly brackets to denote sets are omitted.}\n \\label{fig:run-tree}\n\\end{figure}\n\nThe tree $t$ is called the labeled compressed run tree of~$u$, while the unlabeled tree, that is, the domain of $t$, is called the compressed run tree of~$u$ and denoted~$T$.\\index{labeled compressed run tree}\\index{run tree!labeled compressed}\n\n\\begin{lemma} \n \\label{thm:run-trees}\n A B\\\"{u}chi automaton accepts an $\\omega$-word if, and only if, its compressed run tree has a path with infinitely many left successors, also called left-recurring path.\\index{left-recurring!path}\\index{path!left-recurring}\n\\end{lemma}\n\nFor the proof, assume $u$ is a word which is accepted by a given B\\\"{u}chi automaton. We construct an infinite path $\\langle v_0, v_1, \\dots\\rangle$ in $t$ in such a way that for every $l$ the following conditions hold.\n\\begin{inparaenum}[(i)]\n\\item There exists a recurring run on $u[l,\\infty)$ starting in some $q \\in t(v_l)$.\n\\item There is no vertex $v <_{\\text{lft}} v_l$ satisfying the same condition.\n\\end{inparaenum}\nFor the induction base, we choose $v_0 = \\epsilon$, which obviously works. Assume $v_l$ has already been defined. By~(i), there is a recurring run on $u[l,\\infty)$ starting in some state of $t(v_l)$. If there is such a run~$r$ with $r(1) \\in B$, we set $v_{l+1} = v_l 0$ or else $v_{l+1} = v_l 1$. To show $v_{l+1} \\in T$, we fix a state $q \\in t(v_l)$ and a recurring run $r$ on $u[l+1,\\infty)$ such that $qr$ is a recurring run on $u[l,\\infty)$. By way of contradiction, assume $v_{l+1} \\notin T$. Then there are a vertex $v <_{\\text{lft}} v_l$ and a state $q' \\in t(v)$ such that $r(0) \\in \\Delta(q', u(l))$. This means $q'r$ is a recurring run on $u[l,\\infty)$---a~contradiction to~(ii). \n\nAssume the constructed path is not left-recurring. Then there exists $i$ such that $v_j \\in (0+1)^*1$ for all $j \\geq i$. Let $r$ be a recurring run on $u[i,\\infty)$ starting with a state in $t(v_i)$. Then there is some $k>0$ such that $r(k)$ is a B\\\"{u}chi state. By adjusting $i$, we can assume $k=1$. If, on one hand, $r(1) \\notin t(v_{i+1})$, we can obtain a contradiction similar to above. If, on the other hand, $r(1) \\in v_{i+1}$, then, by definition, $v_{i+1} = v_{i+1-1}0$---a contradiction.\n\nFor the converse, assume $\\langle v_0, v_1, \\dots\\rangle$ is a left-recurring path in $T$. For every $i < \\omega$ and every $q \\in t(v_i)$, there is an initial run of the automaton on $u[0,i)$ such that $r(i) = q$ and $r(j) \\in t(v_j)$ for every $0 < j < i$. All these runs can be organized in a straightforward fashion in an infinite tree with branching degree at most the number of states of the given automaton. By K\\\"{o}nig's lemma, this tree has an infinite rooted path, and, by construction, the labeling of this path is an initial run of the automaton on~$u$. Further, for every $i>0$, the state in position~$i$ of this run belongs to~$B$ if, and only if, $v_i$ is a left successor.\\qed\n\nFrom an automata-theoretic point of view the important observation is that compressed run trees of a given B\\\"{u}chi automaton have width at most $n$ and can be constructed in a forward deterministic fashion by an $\\omega$-automaton with output and trivial recurrence condition. One way to realize such an automaton is to use states of the form $\\langle Q_0, \\dots, Q_{m-1}\\rangle$ with the $Q_i$'s being pairwise disjoint, nonempty subsets of $Q$, representing the labeling of the current level of the labeled compressed run tree. An upper bound on the number of such states can be obtained using ordered Bell numbers.\n\n\\begin{remark}\n \\label{thm:run-trees-deterministic}\n For every B\\\"{u}chi automaton with $n$ states, there is a forward deterministic automaton that outputs, for every $\\omega$-word, a representation of its compressed run tree, and has a number of states which is asymptotically bounded  from above by $(0.54 \\, n)^n$.\n\\end{remark}\n\n\\subsection{Complementation and disambiguation via compressed run trees}\n\\label{sec:disambiguation}\n\nCompressed run trees can be used for complementation. To see this, consider the subtree of a compressed run tree which contains only the infinitary vertices and call it the core of the run tree.\\index{core!of a run tree}\\index{run tree!core of} From Lemma~\\ref{thm:run-trees} and the fact that the run tree has finite width it follows that an $\\omega$-word is not accepted if, and only if, in the core of its run tree there are only finitely many slices with a left successor. In other words, a B\\\"{u}chi automaton for the complement is obtained as a cascade of the following automata: \n\\begin{inparaenum}[(i)]\n\\item the automaton from Remark~\\ref{thm:run-trees-deterministic}; \n\\item an automaton based on the breakpoint construction removing the finitary vertices; \n\\item a two-state automaton checking that from some level onward, no slice with a left successor occurs anymore.\n\\end{inparaenum}\nA careful implementation leads to state spaces similar in size to those described in~\\cite{schewe-buechi-complementation-made-tight-2009}.\n\nAn interesting observation is that the cascade of the first and the second automaton from above yields a nondeterministic B\\\"{u}chi automaton which, for every $\\omega$-word over the given alphabet, outputs the core of its compressed run tree and has exactly one accepting run. In general, an automaton which has at most one accepting run for each word is called an unambiguous automaton.\\index{unambiguous!$\\omega$-automaton}\\index{$\\omega$-automaton!unambiguous} In other words, the above automaton is an unambiguous automaton for the set of all $\\omega$-words over the given alphabet. Moreover, it can be modified in two ways.\n\\begin{inparaenum}[(i)]\n\\item By cascading it with a two-state deterministic automaton checking\n that there are infinitely many slices with left successors, one\n obtains an unambiguous automaton for the language recognized by the given automaton.\n\\item By cascading it with a two-state unambiguous automaton checking that there are only finitely many slices with left successors, one obtains an unambiguous automaton for the complement.\n\\end{inparaenum}\nIn the terminology of~\\cite{colcombet-forms-determinism-automata-2012}, such an automaton could be called strongly unambiguous.\\index{strongly unambiguous!$\\omega$-automaton}\\index{$\\omega$-automaton!strongly unambiguous}\n\n\n\n\n\n\n\\subsection{Forward determinization via compressed run and history trees}\n\nTheorem~\\ref{thm:fundamental} states in particular that every nondeterministic B\\\"{u}chi automaton is equivalent to a forward deterministic parity, Rabin, Streett, or Muller automaton. The quest for good constructions establishing this---determinization constructions---has resulted in different approaches. The approach followed in this section is motivated by \\cite{muller-schupp-alternating-automata-infinite-trees-1987}, but it is also closely related to Safra-like constructions, as explained towards the end.\n\nIn view of Lemma~\\ref{thm:run-trees} and Remark~\\ref{thm:run-trees-deterministic}, a determinization construction has been established once it has been shown that the set of all binary trees of width at most $k$ which have a left-recurring path is recognized by a forward deterministic automaton. Therefore, the objective in what follows is exactly to describe such an automaton.\n\nTo understand the mechanics of infinite trees of finite width better, we associate with every vertex $v$ in a binary tree its origin.\\index{origin!of a vertex in a binary tree} This is the earliest  ancestor of~$v$ (shortest prefix of $v$) with the property that no vertex to the right of $v$ has the same ancestor; it is denoted $\\text{or}(v)$. For an illustration, see Figure~\\ref{fig:origin}. \n\nObserve, for instance, that\n\\begin{inparaenum}[(i)]\n\\item the root is the origin of the rightmost vertex on each level; \n\\item vertices on the same level have distinct origins; \n\\item if a vertex has a left and a right successor, then the left successor is its own origin.\n\\end{inparaenum}\n\nThe important definition specifies that an origin moves left in one slice if it is the origin of a vertex $v$ on the upper level of the slice and of a vertex $v'$ on the lower level of the slice and $v'$ is not the right successor of $v$, that is, $v'$ is to the left of $v1$. (Note that, by definition, $v'$ cannot be to the right of $v1$.) For an illustration, see Figure~\\ref{fig:origin}.\n\nThe key for the construction to be presented is:\n\n\\begin{lemma}\n A binary tree of finite width contains a left-recurring path if, and only if, there is some origin which moves left in infinitely many slices.\n\\end{lemma}\n\nTo prove the lemma, assume $\\langle v_0, v_1, \\dots\\rangle$ is a left-recurring rooted path. Let $i$ be minimal such that there is no other infinite path $\\langle v_0, \\dots, v_j, v_j1, \\dots\\rangle$ for any $j \\geq i$. (This number $i$ exists because there are at most $k$ rooted infinite paths in any tree of width $k$.) For every $j \\geq i$, consider the rightmost vertex $w_j$ on level $j$ such that $v_j \\LeftLeq w_j$ and there is no infinitary vertex $v'$ with $v_j <_{\\text{lft}} v' \\LeftLeq w_j$. Then $v_i$ is the origin of every $w_j$ and moves left in infinitely many slices. \n\nFor the converse, let $v$ be an origin which moves left in infinitely many slices. Consider, for every level $i \\geq |v|$, the vertex $w_i$ with origin $v$. By K\\\"{o}nig's lemma, there is a rooted path $\\langle v_0, v_1, \\dots\\rangle$ in the tree which consists of all vertices $w_i$ and their ancestors. This path is left-recurring, because otherwise there would be some $j$ with $v_k \\in v_j1^*$ for all $k \\geq j$, which is a contradiction to $v$ moving left in infinitely many slices.\\qed \n\nThere are several ways for an automaton to check whether there is an origin which moves to the left in infinitely many slices.  One is explained in what follows, another one is sketched later.\n\nWe use the notion of military ordering, denoted $<_\\text{mil}$, and defined by $v <_\\text{mil} v'$ if, and only if, either $|v| < |v'|$ or $|v| = |v'|$ and $v <_{\\text{lft}} v'$.\\index{military ordering}\\index{ordering!military}\n\nFrom level to level, the automaton determines, for each vertex, its origin number,\\index{origin number!of a vertex in a binary tree} which is defined as follows. For a given level $l$, let $v_0, \\dots, v_{r-1}$ be an enumeration of the vertices on level~$l$, ordered according to the military order of their origins, that is,  $\\text{or}(v_0) <_\\text{mil} \\dots <_\\text{mil} \\text{or}(v_{r-1})$. The index $i$ is the origin number of $v_i$ and denoted $\\text{on}(v_i)$. The index $i$ is said to refer to the origin $\\text{or}(v_i)$ on level $l$. For an illustration, see Figure~\\ref{fig:origin}. \n\n\\begin{figure}[t]\n \\centering\n \\small\n \\begin{tikzpicture}[%\n\t level distance=5mm,\n\t level 1\/.style={sibling distance=13mm},\n\t level 2\/.style={sibling distance=7mm},\n\t level 3\/.style={sibling distance=5mm},\n\t level 4\/.style={sibling distance=3mm}]\n\n\t \\begin{scope}[every node\/.style={draw,circle,fill,inner sep=1pt}]\n\t \\node {}\n\t\t child {node {} \n\t\t\t child {node {}\n\t\t\t\t child {node {}\n\t\t\t\t\t child {node {}}\n\t\t\t\t\t child[missing]}\n\t\t\t\t child[missing]}\n\t\t\t child {node {}\n\t\t\t }}\n\t\t child {node {} \n\t\t\t child {node {}\n\t\t\t\t child {node {}\n\t\t\t\t\t child {node {}}\n\t\t\t\t\t child {node {}}}\n\t\t\t\t child[missing]\n\t\t\t }\n\t\t\t child {node {}\n\t\t\t\t child {node {}\n\t\t\t\t\t child {node {}}\n\t\t\t\t\t child {node {}}} \n\t\t\t\t child {node {}}}};\n\t \\end{scope}\n\n\t \\begin{scope}[every node\/.style={draw=none,fill=white,inner sep=1pt}]\n\t \\node (o3) at (4,0) {0}\n\t\t child {node (o2) {1} \n\t\t\t child {node {2}\n\t\t\t\t child {node {1}\n\t\t\t\t\t child {node (l0) {1}}\n\t\t\t\t\t child[missing]}\n\t\t\t\t child[missing]}\n\t\t\t child {node {1}\n\t\t\t }}\n\t\t child {node {0} \n\t\t\t child {node (o4) {3}\n\t\t\t\t child {node {2}\n\t\t\t\t\t child {node (l1) {3}}\n\t\t\t\t\t child {node (l2) {2}}}\n\t\t\t\t child[missing]\n\t\t\t }\n\t\t\t child {node {0}\n\t\t\t\t child {node {3}\n\t\t\t\t\t child {node (l3) {4}} \n\t\t\t\t\t child {node (l4) {0}}} \n\t\t\t\t child {node {0}}}};\n\t\t \\draw[densely dashed,->] (l2) .. controls +(45:0.5) and +(300:0.5) .. (o4);\n\t\t \\draw[densely dashed,->] (l4) .. controls +(0:1.5) and +(0:1.5) .. (o3);\n\t\t \\draw[densely dashed,->] (l1) .. controls +(240:0.7) and +(300:0.7) .. (l1);\n\t\t \\draw[densely dashed,->] (l3) .. controls +(240:0.7) and +(300:0.7) .. (l3);\n\t\t \\draw[densely dashed,->] (l0) .. controls +(170:0.9) and +(190:0.9) .. (o2);\n\t \\end{scope}\n\n\t \\begin{scope}[every node\/.style={draw,circle,fill,inner sep=1pt}]\n\t \\node at (8,0) {}\n\t\t\tchild {node {}}\n\t\t\tchild {node {}\n\t\t\t\tchild {node {}}}\n\t\t\tchild {node {}};\n\t\t\\end{scope}\n \\end{tikzpicture}\n \\caption{The first five levels of a compressed run tree; the same tree with the origin numbers of each vertex and the origins of the vertices on level $4$; the tree induced by the origins of the vertices on level~$4$. There are two origins that move left in the last slice depicted.}\n \\label{fig:origin}\n\\end{figure}\n\nWhen a forward deterministic automaton computes the origin numbers, it can be augmented by a transition-Rabin condition to check for a left-recurring path. To see this, assume $v$ is an origin on some level $l$ that moves infinitely often to the left and let $v_0, v_1, \\dots$ be the list of vertices such that $v_i$ is on level $l+i$ and $\\text{or}(v_i) = v$, in particular, $v_0 = v$. Then there is some $i$ such that $\\text{on}(v_i) = \\text{on}(v_{i+1}) = \\dots$. So an appropriate transition-Rabin recurrence condition can be chosen to have, for each $i < k$ (maximum width of the trees considered), a Rabin pair $\\langle L_i, U_i\\rangle$ as follows. The set $L_i$ contains all transitions where for every $j \\leq i$ the index $j$ does not refer to the same origin in the upper and the lower level of the current slice. The set $U_i$ contains all transitions where the origin which $i$ refers to on the upper level moves left in the slice.\n\nAs states of a forward deterministic automaton computing the origin numbers one can choose bijections $g \\colon [m] \\to [m]$ where $m \\leq k$. The meaning of such a state $g$ would be that if $v$ is vertex $i$ on the current level in the order from left to right, then $\\text{on}(v) = g(i)$. The actual definition of the transition function is somewhat technical and omitted. \n\nIn the determinization constructions presented in \\cite{safra-complexity-omega-automata-1988, Piterm2007a, schewe-tighter-bounds-for-determinization-buechi-automata-2009}, states are trees. For instance, in \\cite{schewe-tighter-bounds-for-determinization-buechi-automata-2009}, so-called history trees\\index{history tree} are used and defined as trees where\n\\begin{inparaenum}[(i)]\n\\item each node is labeled by a nonempty set of states, \n\\item the label of every node is a strict superset of the union of the labels of its children, and \n\\item the labels of siblings are disjoint.\n\\end{inparaenum}\nObserve that, alternatively, one could require that the vertices of a history tree are labeled with pairwise disjoint, nonempty sets of states. Such a tree is obtained from the labeled compressed run tree of a given word for each level in a straightforward fashion: move every label of a vertex on the respective level to its origin and then remove all vertices except for these origins and contract edges accordingly. For an illustration, see Figure~\\ref{fig:origin}.\n\nIf one constructs a deterministic automaton which only keeps track of the history trees just described, one arrives at a fundamental construction, which is also known to be optimal in a certain sense \\cite{colcombet-zdanowski-tight-lower-bound-for-determinization-transition-labeled-buechi-automata-2009}:\n\n\\begin{theorem}[determinization via history trees\n \\cite{schewe-tighter-bounds-for-determinization-buechi-automata-2009}]\n \\label{thm:schewe}\n Determinization via history trees yields, for every B\\\"{u}chi automaton with $n$ states, an equivalent deterministic transition-Rabin automaton with an asymptotic upper bound of $(1.65\\,n)^n$ for the number of states and $2^n-1$ Rabin pairs.\n\\end{theorem}\n\n\\enlargethispage{0\\baselineskip}\n\nThe transformations on different types of $\\omega$-automata discussed in this section and the previous one are fundamental transformations, but not all one can consider. Optimal solutions for most of the basic transformation tasks can be found in \\cite{safra-complexity-omega-automata-1988} and a later paper by the same author \\cite{safra-exponential-determinization-for-omega-automata-with-a-strong-fairness-acceptance-condition-2006}. Here, ``optimal'' means with respect to a rough measure of complexity: polynomial, exponential, doubly exponential. The ``optimal'' results stated subsequent to Theorem~\\ref{thm:kupferman-vardi} and before Theorem~\\ref{thm:schewe} are with respect to much finer measures and based on very good lower bounds. A breakthrough with regard to lower bounds on $\\omega$-automata is \\cite{yan-lower-bounds-complementation-omega-automata-full-automata-technique-2006}.\n\n\\section{Congruence relations}\n\nCongruence relations are a useful tool for working with finite-state automata on finite words. For instance, the minimum-state deterministic finite-state automaton for a given regular language of finite words can be derived from the Myhill-Nerode congruence relation for the language---it merely is this congruence relation.\n\nFor $\\omega$-automata, congruences are also useful, but the situation is more complex. \n\n\\subsection{Right (and left) congruence relations}\n\\label{sec:finite-state}\n\nThe straightforward adaptation of the Myhill-Nerode congruence relation (see, for instance, \\cite{MN}) to $\\omega$-lan\\-guages is the initial syntactic congruence relation.\\index{initial syntactic congruence relation!of an $\\omega$-language}\\index{congruence relation!initial syntactic! of an $\\omega$-language} For a given $\\omega$-lan\\-guage~$L$, it considers finite words $u$ and $v$ equivalent if, and only if, for every  $\\omega$-word $w$, either $\\{uw, vw\\} \\subseteq L$ or $\\{uw,vw\\} \\cap L = \\emptyset$. \n\nOn the automata-theoretic side, there is a corresponding notion. Given an $\\omega$-au\\-tom\\-a\\-ton of any type, its initial congruence relation\\index{initial congruence relation!of an $\\omega$-automaton}\\index{congruence relation!initial! of an $\\omega$-automaton} considers finite words $u$ and $v$ equivalent if, and only if, for every initial run of the automaton on~$u$ ending in some state~$q$ there is such a run on~$v$, and vice versa.\n\nThe analogy to the Myhill-Nerode congruence relation is as follows.\n\n\\begin{remark}\n \\begin{inparaenum}\n \\item The initial syntactic congruence relation for an $\\omega$-language is a right congruence relation. \n \\item For an $\\omega$-automaton of any type, the initial congruence relation is a right congruence relation and equal to or finer than the initial syntactic congruence relation for the language recognized and has a finite number of congruence classes.\n \\end{inparaenum}\n\\end{remark}\n\n(Here, as usual, an equivalence relation on words over a given alphabet is a right congruence relation if $uw$ and $vw$ are equivalent whenever $u$ and $v$ are and $w$ is a finite word over the alphabet. A relation is finer than another one if it is a subset of it.)\n\nFor a simple language such as the set of all ultimately periodic words over a given alphabet, which is not regular, the initial syntactic congruence relation has only one equivalence class. Hence, it cannot serve as a vehicle to define regularity. It can neither be used for classifying regular $\\omega$-languages: the languages denoted by $(0+1)^\\omega$ and by $(0^*1)^\\omega$ have the same initial syntactic congruence relation, but they are completely different in nature.\n\nThe initial syntactic congruence relation can, however, take over the role of the Myhill-Nerode congruence relation for the small class of $\\omega$-languages which are recognized by forward deterministic weak automata (see also Section~\\ref{thm:weak-characterization}):\n\n\\begin{theorem}[minimization of forward deterministic weak automata\n \\cite{staiger-finite-state-omega-languages-1983,loeding-efficient-minimization-deterministic-weak-omega-automata-2001}]\n \\label{thm:initial-congruence} \n Let $L$ be an $\\omega$-language recognized by a forward deterministic weak automaton $\\Aut A$ and let $\\Aut D$ be the DFA (without final state set) corresponding to the initial syntactic congruence relation for~$L$.\n \\begin{enumerate}\n \\item The automaton $\\Aut D$ can be augmented by a weak acceptance condition in such a way that the resulting automaton recognizes $L$.\n\n \\item The automaton from (1) is, up to isomorphism, the smallest forward\n deterministic automaton recognizing~$L$ and can be computed from $\\Aut A$ by\n DFA minimization, see, for instance, \\cite{HMU}.\n \\end{enumerate}\n\\end{theorem}\n\nThe role that the initial syntactic congruence relation of a language~$L$ plays in the context of forward deterministic automata is taken over by the final syntactic congruence relation in the context of backward deterministic automata. This relation, which is a left congruence relation, considers $\\omega$-words $v$ and $w$ congruent if, and only if, for every  finite word $u$, either $\\{uv, uw\\} \\subseteq L$ or $\\{uv,uw\\} \\cap L = \\emptyset$.\n\n\\subsection{Two-sided congruence relations}\n\\label{sec:saturation}\n\nThere are essentially two straightforward adaptations of the two-sided syntactic\ncongruence relation for languages of finite words (see, for instance, \\cite{SC}) to $\\omega$-languages.\n\nLet $L$ be an $\\omega$-language over some alphabet $A$. In the first adaptation, finite words $u$ and $v$ are congruent if, and only if, for all $x \\in A^*$ and $y \\in A^\\omega$, either $\\{xuy, xvy\\} \\subseteq L$ or $\\{xuy, xvy\\} \\cap L = \\emptyset$. In the second adaptation, nonempty finite words $u$ and $v$ are congruent if, and only if, whenever $u_0, u_1, \\dots$ and $v_0, v_1, \\dots$ are sequences of nonempty finite words over the given alphabet such that $u_i = v_i$ or $\\{u_i, v_i\\} \\subseteq \\{u,v\\}$, then either $\\{u_0 u_1 \\dots, v_0 v_1 \\dots\\} \\subseteq L$ or $\\{u_0 u_1 \\dots, v_0 v_1 \\dots\\} \\cap L = \\emptyset$.\n\nThe two adaptations try to capture what it means for two finite words to behave equally in the same context, and they both yield two-sided congruence relations. The first one is finer or equal to the initial syntactic congruence relation; the second one is finer or equal to the first one and called the syntactic congruence relation of~$L$.\\index{syntactic congruence relation!of an $\\omega$-language}\\index{congruence relation!syntactic! of an $\\omega$-language}\n\nIn the following, the syntactic congruence relation is further discussed, because it provides a finer means of characterization.\n\nCorresponding to the syntactic congruence relation one can define, for every $\\omega$-au\\-tom\\-a\\-ton of any type, a suitable two-sided congruence relation. For a B\\\"{u}chi automaton, this relation considers nonempty finite words $u$ and $v$ congruent if the following two conditions hold for all states $q, q' \\in Q$.\n\\begin{inparaenum}[(i)]\n\\item There is a run of the automaton on $u$ starting with $q$ and ending in $q'$ if, and only if, this is true for $v$.\n\\item There is a run $r$ of the automaton on $u$ starting with $q$, ending in $q'$, and passing through an element of $B$, that is, $r(i) \\in B$ for some $i < |r|$, if, and only if, this is true for~$v$.\n\\end{inparaenum}\n\nJust as before, the syntactic congruence relation of an $\\omega$-language does not characterize regularity. Consider%\n\\footnote{Slides of a presentation given by Miko\\l aj Boja\\'nczyk.} the language $L_\\text{ub0}$ of all $\\omega$-words of the form $0^{i_0}10^{i_1}\\dots$ where $\\limsup_{j \\to \\infty} i_j = \\infty$. The language $L_\\text{ub0}$ and the one denoted by $(0^*1)^\\omega$ have the same syntactic congruence relation, and this has just two equivalence classes.\n\nStill, the syntactic congruence relation has interesting properties. One is phrased in terms of saturation, where an equivalence relation on finite words is said to saturate \\index{saturation!of an $\\omega$-language} an $\\omega$-language $L$ if, for all sequences $\\langle V_0, V_1 \\dots\\rangle$ of equivalence classes, either $V_0V_1 \\dots \\subseteq L$ or $V_0 V_1 \\dots \\cap L = \\emptyset$ holds.\n\n\\begin{theorem}[saturation \\cite{buechi-decision-method-restricted-second-order-arithmetic-1962,arnold-syntactic-congruence-omega-languages-1983}]%\n \\label{thm:syntactic-congruence}\n \\begin{enumerate}\n\t \\item The two-sided au\\-tom\\-a\\-ton congruence relation of a B\\\"{u}chi automaton with $n$ states has at most $2^{2n^2}$ congruence classes and saturates the language recognized by the automaton.\n \\item The syntactic congruence relation of a given regular $\\omega$-language is the coarsest congruence relation saturating the language.\n \\item An $\\omega$-language is regular if, and only if, there exists a congruence relation saturating it and having a finite number of congruence classes.\n \\end{enumerate}\n\\end{theorem}\n\nThe proofs of (1) and (2) are straightforward; one direction of~(3) follows from~(1). The other direction of~(3) can be proved on the basis of Ramsey's Theorem~A\\index{Ramsey's Theorem A} \\cite{ramsey-problem-formal-logic-1929}, which says that, for a given equivalence relation on finite words with a finite number of equivalence classes and an infinite sequence of nonempty finite words $u_0,u_1, \\dots$, there is a strictly monotone infinite sequence $\\langle i_0, i_1, \\dots\\rangle$ of natural numbers such that all finite words of the form $u_{i_j} u_{i_j+1} \\dots u_{i_k-1}$ with $j < k$ are equivalent. In the context of $\\omega$-languages, this means the following.\n\n\\begin{remark}\n \\label{thm:ramsey}\n\\cite{buechi-decision-method-restricted-second-order-arithmetic-1962}\n Given an alphabet $A$, a congruence relation on $A^+$ having a finite number of congruence classes, and $u \\in A^\\omega$, there are congruence classes $U$ and $V$ satisfying $UV \\subseteq U$, $V^2 \\subseteq V$, and $u \\in UV^\\omega$.\n\\end{remark}\n\nTo prove the other direction of Theorem~\\ref{thm:syntactic-congruence}(3), observe that from the previous remark it follows that if a congruence relation with a finite number of equivalence classes saturates a given $\\omega$-language~$L$, then $ L = \\bigcup UV^\\omega$ with $U$ and $V$ ranging over equivalence classes with \\mbox{$UV^\\omega \\subseteq L$}. Based on this, a construction such as the one described in the proof of Theorem~\\ref{thm:regexps} can be used to arrive at a B\\\"{u}chi automaton recognizing~$L$.\\qed\n\nThe procedure just described is far from being as natural as the procedure that turns the Myhill-Nerode congruence relation for a given regular language of finite words into the minimum-state DFA for the language. In fact, nothing which would come close to this is known for regular $\\omega$-languages. Still, two-sided congruence relations for $\\omega$-languages are useful in several contexts, for instance, when it comes to classifying regular $\\omega$-languages, see~\\cite{perrin-pin-infinite-words-2004}. Another application, described in what follows, is complementation.\n\n\\subsection{Complementation via two-sided congruence relations}\n\nWhen a two-sided congruence relation saturates a given language, then, by definition, it also saturates the complement of the language, which establishes once again that the class of $\\omega$-languages recognized by B\\\"{u}chi automata is closed under complementation. In fact, the first proof of this fact was along these lines \\cite{buechi-decision-method-restricted-second-order-arithmetic-1962}.\n\nIn view of the bound stated in Theorem~\\ref{thm:syntactic-congruence}(1), using congruences for complementation leads to much larger automata than the ones described in Sections~\\ref{sec:complementation} and~\\ref{sec:disambiguation}. To obtain smaller automata with this approach, the following modification suggests itself. \n\nWrite the complement of the language recognized by a given B\\\"{u}chi automaton again in the form $\\bigcup_{i < k} U_i V_i^\\omega$, but choose the $U_i$'s and $V_i$'s to be unions of congruence classes in a way such that the resulting B\\\"{u}chi automaton is small. Observe that if $m$ is the number of states of an automaton recognizing all $U_i$'s (with a different set of final states for each~$i$) and $m'$ is an upper bound on the number of states needed in automata recognizing the $V_i$'s, then $m + k (m' +1)$ is an upper bound for the number of states in the resulting B\\\"{u}chi automaton.\n\nTo see how the indicated approach works, assume a B\\\"{u}chi automaton is given as usual, with $n$ states and recognizing a language $L$. For every set $P \\subseteq Q$, let $U_P$ be set of words $u$ such that every run of the automaton on $u$ starting in some initial state ends in some state of~$P$. Then each set $U_P$ is a union of equivalence classes of the automaton congruence relation and can be recognized by a deterministic automaton with $2^n$ states.\n\nFor every nonempty sequence $\\sigma = \\langle P_0, \\dots, P_{k-1}\\rangle$ of nonempty, pairwise disjoint sets of states, let the set $P(\\sigma)$ be defined by $P(\\sigma) = P_0 \\cup \\dots \\cup P_{k-1}$, and let $V_\\sigma$ be the set of all finite words $u$ satisfying the following two conditions for every run $r$ of the automaton on~$u$.\n\\begin{inparaenum}[(i)]\n\\item If $r(0) \\in P_i$ for some $i$, then $r(|u|) \\in P_j$ for some $j$ with $i \\leq j < k$.\n\\item If $r(0) \\in P_i$ and $r$ contains a B\\\"{u}chi state, then $r(|u|) \\in P_j$ for some $j$ with $i < j < k$. \n\\end{inparaenum}\n\n\\begin{theorem}[complementation by saturation \\cite{breuers-loeding-olschewski-improved-ramsey-based-complementation-2012}]\n For every B\\\"{u}chi automaton with $n$ states, the language $\\bigcup_\\sigma U_{P(\\sigma)} V_\\sigma^\\omega $\n is the complement of the language recognized by the B\\\"{u}chi automaton, and a conversion of this expression into a B\\\"{u}chi automaton yields an automaton with $2^{\\theta(n \\log n)}$ states.\n\\end{theorem}\n\nFor the proof, first observe that each set $V_\\sigma$ is a union of equivalence classes of the two-sided automaton congruence relation: compare (i) and (ii) above with (i) and (ii) in the definition of the two-sided automaton congruence relation. \n\nNext, let $\\sigma$ and $P(\\sigma)$ be as above. To see that $U_{P(\\sigma)} V_\\sigma^\\omega \\cap L = \\emptyset$ holds, let $r$ be an initial run on any word $u \\in U_{P(\\sigma)} V_\\sigma^\\omega$ and $i_0 < i_1 < \\dots$ be such that $u[0,i_0) \\in U_{P(\\sigma)}$ and $u[i_j, i_{j+1}) \\in V_\\sigma$ for every $j$. From the definition of $V_\\sigma$ we can conclude that $r(i_0) \\in P(\\sigma)$ holds and that there are $i$ and $k$ such that $r(i_j) \\in P_i$ holds for every $j \\geq k$. This implies that, for every $j \\geq k$, there is no B\\\"{u}chi state in $r[i_j, i_{j+1})$, which means $u \\notin L$.\n\nConversely, if an $\\omega$-word $u$ is not accepted by the given B\\\"{u}chi automaton, then, by Remark~\\ref{thm:ramsey}, there are classes $U$ and $V$ of the automaton congruence relation such that $u \\in UV^\\omega$, $UV \\subseteq U$, and $V^2 \\subseteq V$. Let $P$ be the set of states which can be reached by reading some word from $U$ from some initial state. Consider the graph with vertex set~$P$ and an edge between $q$ and $q'$ if, and only if, there is a run of the automaton on some word $v \\in V$ starting in $q$ and ending in $q'$. Let $\\sigma = \\langle P_0, \\dots, P_{k-1} \\rangle$ be a list of the SCC's of this graph in topological order. Then $V \\subseteq V_\\sigma$, which means $u \\in U_{P(\\sigma)} V_\\sigma^\\omega$. \n\nTo prove the claim about the size of the resulting $\\omega$-automaton, we describe how to construct a deterministic automaton of size $(k+1)^n$ for a language $V_\\sigma$ as above. The states are functions $Q \\to \\{-\\infty\\} \\cup [k]$. The transition function is defined in a way such that if by reading a finite word $v$ the automaton reaches state $f$, then the following holds for every $q \\in Q$. If in the B\\\"{u}chi automaton there is no run on $v$ starting in $P(\\sigma)$ and ending in $q$, then $f(q) = - \\infty$; if there are such runs, then $f(q)$ is the greatest index $i$ such that a run on $v$ starting in some state from $P_i$ ends in $q$.\\qed\n\nThe construction described above can be generalized so as to improve the construction of a B\\\"{u}chi automaton from a saturating congruence relation.\n\n\\section{Loop structure}\n\nAs the set of states visited infinitely often in a run of an $\\omega$-automaton determines whether the run is recurring, it is only natural to investigate the structure of the strongly connected subsets in a given $\\omega$-automaton.\n\nA loop\\index{loop!in an $\\omega$-automaton} at some state $q$ is a word $q_0 a_0 q_1 a_1 \\dots a_n q_{n+1}$ where $\\langle q_i, a_i, q_{i+1}\\rangle \\in \\Delta$ for every $i \\in [n]$ and $q_0 = q_{n+1} = q$. The word $a_0 \\dots a_n$ is the label of the loop, the set $\\{q_0, \\dots, q_n\\}$ is the loop set. The loop is positive if it satisfies the recurrence condition of the given automaton (for a B\\\"{u}chi condition, this means $\\{q_0, \\dots, q_n\\} \\cap B \\neq \\emptyset$), it is negative if it does not---we speak of the sign of the loop. In a deterministic automaton (forward or backward), $q$ and the label determine the loop.\n\nIn forward deterministic automata, the nesting depth of positive and negative loops sets is an interesting measure for the complexity of the language recognized, explained in  Section~\\ref{sec:wagner}, whereas in backward deterministic automata, the distribution of the labels of positive loops is interesting, as explained in Section~\\ref{sec:backward-loops}. \n\nIn the following, we assume, without loss of generality, that in forward deterministic automata every state is reachable from the initial state.\n\n\\subsection{Alternating loops in forward deterministic automata}\n\\label{sec:wagner}\n\nA tower\\index{tower!in an $\\omega$-automaton} is a nonempty sequence $\\langle C_0, \\dots, C_{m-1}\\rangle$ of loop sets such that $C_0 \\supseteq \\dots \\supseteq C_{m-1}$ and the signs alternate; the sign of the last loop is the sign of the tower, the number~$m$ is the height of the tower. A maximal tower is one of maximal height. A wall\\index{wall!in an $\\omega$-automaton} is a sequence of maximal towers where each one is reachable from the previous one and the signs alternate; the sign of the wall is the sign of the first tower, the number of towers in the sequence is the length of the wall.\n\nThe types of towers and walls in a given forward deterministic $\\omega$-automaton are invariants of the language recognized:\n\n\\begin{theorem}[towers and walls \\cite{wagner-1977}]\n \\label{thm:wagner}\n All forward deterministic $\\omega$-automata recognizing the same language have the same types of towers and walls in the sense that if one of them has a tower of a certain height and sign or a wall of a certain length and sign, then the other has so, too. \n\\end{theorem}\n\nTo illustrate this theorem we prove the claim for towers and start with a useful remark.\n\n\\begin{remark}\n \\label{thm:loops}\n Consider a forward deterministic automaton with~$n$ states over an alphabet~$A$. For $u \\in A^*$, $v \\in A^+$ and $k \\geq n$, some power of $v^k$ is the label of a loop at $\\delta^*(q_I, uv^k)$, and this loop is positive if, and only if, $uv^\\omega$ is accepted. (As usual, $\\delta^*$ stands for the extended transition function, defined by $\\delta^*(q, \\epsilon) = q$ and $\\delta^*(q, ua) = \\delta(\\delta^*(q,u),a)$ for all $q \\in Q$, $u \\in A^*$, and $a \\in A$.)\n\\end{remark}\n\nFor the proof of the claim on towers, assume equivalent forward deterministic $\\omega$-automata $\\Aut A$ and $\\Aut A'$ are given and consider any tower $\\langle C_0, \\dots, C_{m-1}\\rangle$ in $\\Aut A$, say a positive one; the argument is symmetric for a negative one. Let $q$ be a state in $C_{m-1}$ and, for every $i < m$, let the word $v_i$ be a label for a loop at $q$ with loop set $C_i$. Further, let $u$ be a word such that $\\delta(q_I, u) = q$. Then $\\delta(q_I, uw) = q$ for every word $w \\in (v_0 + \\dots + v_{m-1})^*$. Moreover, whether a word $u v_{i_0} v_{i_1} \\dots$ is accepted is determined by the least index occurring infinitely often among $i_0, i_1, \\dots$. Let $k$ be greater than the number of states of $\\Aut A'$ and consider the words $w_i$ defined inductively by $w_0 = \\epsilon$ and $w_{i+1} = (v_{m-i-1} w_i)^k$, for $i < m$. Then, using Remark~\\ref{thm:loops}, we find that for each $i < m$, some power of $w_{i+1}$ is the label of a loop at $\\delta^*(q_I', u w_m)$ and has the same sign as $C_{m-i-1}$. So the reverse sequence of the loop sets forms a positive tower in~$\\Aut A'$ of height~$m$.\\qed\n\nThere is a strong relationship between towers and walls on one side and topological aspects of $\\omega$-languages on the other side, see, for instance, \\cite{perrin-pin-infinite-words-2004}.\n\n\\subsection{The parity index}\n\\label{sec:parity-index}\n\nFrom Theorem~\\ref{thm:wagner} it follows that, in particular, the greatest height of a tower in a forward deterministic automaton is characteristic for the language recognized. This number is intimately connected with the number of priorities needed by a forward deterministic parity automaton to recognize the same language. To make this more precise we say a parity automaton uses $n$ priorities if $n$ is the maximum of the number of priorities occurring in any strongly connected component of the automaton. Given a regular $\\omega$-language the smallest number of priorities used in any forward deterministic parity automata recognizing the language is its parity index.\\index{parity index!of an $\\omega$-language} \n\n\\begin{corollary}[parity index]\n The greatest height of a tower in a given forward deterministic $\\omega$-automa\\-ton is exactly the parity index of the language recognized.\n\\end{corollary}\n\nThat the parity index is at least the greatest height of a tower follows from Theorem~\\ref{thm:wagner}. For the converse, reconsider the construction from Section~\\ref{sec:lar} that turns a Muller automaton into an equivalent parity automaton. Essentially, the Muller automaton without recurrence condition is cascaded with the LAA (latest appearance automaton) and augmented by a parity condition. It is enough to adjust the latter as follows. A state $\\langle q, v\\$v'\\rangle$ is assigned the value $n - l + o$ where \n\\begin{inparaenum}[(i)]\n\\item the number $l$ is the greatest height of a tower ending in a loop with loop\n set $\\Occ{v'}$ and \n\\item the number $o < 2$ is chosen in a way such that $n - l + o$ is even if the loop is positive and else odd.\\qed \n\\end{inparaenum}\n\nThe Rabin index of a regular $\\omega$-language \\cite{wagner-1977} is a similar but somewhat coarser measure.\\index{Rabin index!of an $\\omega$-language}\n\n\\subsection{Forward deterministic weak automata}\n\nIn terms of the above complexity measure---parity index---the simplest forward deterministic automata that can be considered are the ones with parity index~$1$; these automata are exactly the forward deterministic weak automata. \n\nOn one hand, weak automata are indeed weak in the sense that the class of languages recognized by them is small, for instance, $(01)^\\omega$ cannot even be recognized by such automata. In fact, there is a simple characterization of languages recognized by forward deterministic weak automata.\n\n\\begin{remark}\\cite{staiger-wagner-automatentheoretische-automatenfreie-charakterisierungen-1974}\n \\label{thm:weak-characterization}\n An $\\omega$-language can be recognized by a forward deterministic weak automaton over some alphabet $A$ if, and only if, it is a boolean combination of languages of the form $UA^\\omega$ where $U$ is a regular language of finite words.\n\\end{remark}\n\nOn the other hand, weak automata have some properties which general $\\omega$-automata are lacking. One interesting property is described in Theorem~\\ref{thm:initial-congruence}. Another property has to do with their determinization:\n\n\\index{conditional determinization!of an $\\omega$-automaton}\n\\begin{theorem}[conditional determinization \\cite{boigelot-jodogne-wolper-effective-decision-procedure-linear-arithmetic-over-integers-reals-2005}]\n \\label{thm:forward-deterministic-weak} \n If an $\\omega$-language is recognized by some forward deterministic weak automaton, then a variant of the breakpoint construction can be used to transform a forward nondeterministic weak automaton recognizing the language into an equivalent forward deterministic weak automaton.\n\\end{theorem}\n\n\\subsection{Loops in backward deterministic automata}\n\\label{sec:backward-loops}\n\nThe requirement that in a backward deterministic automaton there is exactly one recurring run for every $\\omega$-word over the given alphabet is a very strong one, which has interesting implications.\n\n\\begin{proposition}\n \\cite{carton-michel-unambiguous-buechi-automata-2003}\n An $\\omega$-automaton is backward deterministic if, and only if, its transition relation is backward deterministic and every nonempty finite word is the label of a positive loop at exactly one state.\n\\end{proposition}\n\nFor the proof, assume a backward deterministic automaton is given and let $u$ be a nonempty finite word. If it is the label of a positive loop at two distinct states $q$ and $q'$, then there are at least two recurring runs of the automaton on $u^\\omega$---a contradiction. So $u$ can only be the label of a positive loop at at most one state. Since there is a recurring run of the automaton on $u^\\omega$, there is some $k$ such that $u^k$ is the label of a loop at state $q$. If $\\delta(u,q) \\neq q$, then $u^k$ would also be the label of a loop at $\\delta(u,q)$ and there would be two recurring runs for $u^\\omega$---a contradiction. So $u$ is the label of a positive loop at at least one state. \n\nFor the converse, assume every nonempty finite word is the label of a positive loop at exactly one state. Then every periodic word over the given alphabet has a recurring run and hence every ultimately periodic word over the same alphabet has so, too. In other words, the set of all words without recurring run is a regular $\\omega$-language without ultimately periodic words. From Remark~\\ref{thm:basic}(1), we can conclude this set is empty. This shows that for every $\\omega$-word there is at least one recurring run. By way of contradiction, assume there are two distinct recurring runs on a given $\\omega$-word~$u$, say $r$ and $r'$. Because of the backward deterministic transition relation there must be some $i$ such that $r(j) \\neq r'(j)$ for all $j \\geq i$.\nAs a consequence, there are positions $i$ and $j$ such that \n\\begin{inparaenum}[(i)]\n\\item $i<j$, $r(i) = r(j)$, $r'(i) = r'(j)$, and $r(i) \\neq r'(i)$, and \n\\item $r(i)u(i)r(i+1) \\dots r(j)$ as well as $r'(i)u(i)r'(i+1) \\dots r'(j)$ are positive loops at different states with the same label.\n\\end{inparaenum}\nThis is a contradiction to the assumption.\\qed\n\nThe above proposition, in combination with the final syntactic congruence (defined subsequent to Theorem~\\ref{thm:initial-congruence}), can be used to classify regular $\\omega$-languages using backward deterministic $\\omega$-automata, see \\cite{preugschat-wilke-effective-characterizations-of-simple-fragments-of-temporal-logic-using-carton-michel0automata-2013}.\n\n\\section{Alternation}\n\n$\\omega$-Automata are often used in the context of two-player games of infinite duration played on graphs, and results on such games are useful tools for obtaining results on $\\omega$-automata. For infinite trees, alternation is an even more important concept.\n\n\\subsection{Games of infinite duration with regular winning conditions}\n\\label{sec:games}\n\nIn this section, the fundamentals of games of infinite duration with regular winning conditions are recalled. Remark~\\ref{thm:regular-winning} is one of the prime applications of \\emph{forward deterministic} $\\omega$-automata; in combination with Theorem~\\ref{thm:positional}, it explains why the parity condition is so important.\n\nThe players of a two-player game of infinite duration\\index{game!two-player!of infinite duration} played on graphs are called Zero and One; a game is given by a set $V$ of vertices, a set $E \\subseteq V \\times V$ of edges, a set $V_0$ of vertices owned by Zero, and a winning condition $W \\subseteq V^\\omega$. A play of such a game starting in some vertex $v_I$ is a maximal path through the graph starting with the vertex $v_I$; the idea is that a pebble is moved over the edges of the graph from one vertex to the next, starting with the pebble on vertex $v_I$, and Zero moving in her vertices and One moving in the vertices owned by him, which are the ones in $V \\setminus V_0$. A play\\index{play!of a two-player game of infinite duration} is winning for Zero if the path is either finite and its last vertex belongs to One (that is, One cannot move anymore) or infinite and belongs to $W$; else it is winning for One.\n\nWhen a player has a strategy for winning the plays starting in a particular vertex, the player is said to win the game starting in this vertex.\\index{win!of a two-player game of infinite duration} The set of such vertices is called his or her winning region.\n\nOften, a winning condition is a regular $\\omega$-language. More precisely, a coloring function $c \\colon V \\to C$ into a finite set of so-called colors and a regular $\\omega$-language~$L$ over~$C$ are given; the winning condition $W$ is determined by $W = \\{u \\in V^\\omega \\mid c \\circ u \\in L\\}$. One speaks of a regular winning condition.\\index{regular winning condition!of a two-player game of infinite duration}\n\nA consequence of Martin's theorem \\cite{martin-borel-determinacy-1975} and McNaughton's result \\cite{mcnaughton-testing-and-generating-infinite-sequences-1966} is:\n\n\\index{determinacy!regular}\\index{regular determinacy}\n\\begin{theorem}[regular determinacy]\n \\label{thm:determinacy}\n Given a game with a regular winning condition and a vertex in this game, either Zero or One wins the game starting in this vertex. The game is said to be determined in the vertex.\n\\end{theorem}\n\nA game is called a parity game if there is a function $\\pi \\colon V \\to [n]$ such that $u \\in W$ if, and only if, $\\liminf_i \\pi(u(i))$ is even. Hence, parity games can be viewed as games with a regular winning condition. From the fact that every regular $\\omega$-language is recognized by some forward deterministic parity automaton, one can derive:\n\n\\index{embedding!of a game into a parity game}\n\\begin{remark}\n \\label{thm:regular-winning}\n Every game with a regular winning condition can be embedded into a game with a parity winning condition.\n\\end{remark}\n\nTo understand what exactly this means assume a game with a regular winning condition as described above and a forward deterministic parity automaton $\\Aut A$ recognizing the language $L$ are given. Consider the modified game with vertex set $V \\times Q$, Zero's vertex set $V_0 \\times Q$, edge set $\\{\\langle \\langle v,q\\rangle, \\langle v', \\delta(q, c(v'))\\rangle \\rangle \\mid \\langle v, v'\\rangle \\in V \\text{ and } q \\in Q\\}$, and priority function $\\langle v, q\\rangle \\mapsto \\pi(q)$. Because the automaton $\\mathscr A$ is forward deterministic, playing in the original game starting from a vertex $v$ is exactly the same as playing in the new game starting from the vertex $\\langle v, q_I\\rangle$. In particular, Zero wins the former game in a vertex $v$ if, and only if, she wins the latter game in $\\langle v, q_I\\rangle$. In other words, without loss of generality, only games with parity winning conditions need to be considered when regular winning conditions are used.\n\nIn general, regular winning conditions may require a player to remember a certain amount of information in order to win. For instance, if the winning condition demands that Zero visits the vertices $v_1$ and $v_2$ infinitely often in the graph \n\\begin{center}\n \\begin{tikzpicture}[baseline=(b.center)]\n\t \\node (a) {$v_0$};\n\t \\node[left of=a,node distance = 15mm] (b) {$v_1$};\n\t \\node[right of=a,node distance = 15mm] (c) {$v_2$};\n\t \\draw[->] (a) to [bend right] (b);\n\t \\draw[->] (b) to [bend right] (a);\n\t \\draw[->] (c) to [bend left] (a);\n\t \\draw[->] (a) to [bend left] (c);\n \\end{tikzpicture} \\enspace,\n\\end{center}\nwhere $v_0$ is her vertex, then she cannot base her decision what to do in vertex $v_0$ only on the fact that she is in that vertex. (Of course, when she remembers where she moved previously, she can alternate and win.) Opposed to this, if the winning condition demands that Zero visits $v_2$ infinitely often, she only needs to follow the rule ``if in vertex $v_0$, go to vertex~$v_2$''---her decision what to do next is only based on the current vertex.\n\nA uniform positional winning strategy for Zero is a function $(W_0 \\cap V_0) \\to W_0$, where $W_0$ is Zero's winning region, such that no matter where in $W_0$ a play starts, if Zero moves as determined by the function, then the resulting play is winning for Zero. For One, the definition is symmetric.\n\n\\index{positional strategy!in a two-player game of infinite duration}\n\\begin{theorem}[positional strategies \\cite{emerson-jutla-tree-automata-mu-calculus-determinacy-1991}]\n \\label{thm:positional}\n In every parity game, both players have a uniform positional winning strategy.\n\\end{theorem}\n\n\\subsection{State- and transition-controlled alternating automata}\n\\label{sec:alternation}\n\nIn general, an alternating automaton is an automaton where acceptance depends on the full computation tree on a given word, more precisely, such an automaton provides means for specifying that a given word is accepted if, and only if, a certain subgraph of the full computation tree exists. At one extreme, when this subgraph is required to be a rooted path, then the automaton is nothing else than a conventional automaton.\n\n\\index{alternating $\\omega$-automaton}\\index{$\\omega$-automaton!alternating}\nFor $\\omega$-automata, essentially two variants of alternating automata have been studied: in one variant, alternation is specified by partitioning the state space \\cite{miyano-hayashi-alternating-finite-automata-on-omega-words-2-1984}; in the other variant, alternation is specified by complex transition formulas \\cite{muller-schupp-alternating-automata-infinite-trees-1987}.\n\n\\index{state-controlled alternating $\\omega$-automaton}\\index{$\\omega$-automaton!alternating!state-controlled}\nIn the state-controlled variant the state space $Q$ is partitioned into a set $E$ of existential states and a set $U$ of universal states (where either set could be empty) and the set of initial states is either a subset of $E$ or of $U$. A run of the automaton on a word $u$ is a prefix-closed set $T \\subseteq Q^*$, which should be thought of as a tree satisfying the following properties for every vertex $vq \\in T$:\n\\begin{asparaitem}\n\\item if $q \\in E$, then there exists a state $q'$ such that $vqq' \\in T$ and $\\langle q, u(|v|), q'\\rangle \\in \\Delta$; \n\\item if $q \\in U$, then $vqq' \\in T$ for every $q'$ such that $\\langle q, u(|v|), q'\\rangle \\in \\Delta$.\n\\end{asparaitem}\nThe run is initial if either $Q_I \\subseteq E$ and $Q_I \\cap T \\neq \\emptyset$ or $Q_I \\subseteq U$ and $Q_I \\subseteq T$; it is recurring if every word $r \\in Q^\\omega$ whose finite prefixes all belong to $T$ (every infinite rooted path through~$T$) is recurring in the sense of the given transition condition. This means, in particular, if $E = Q$ and the set of initial states is existential, then the automaton can be viewed as an ordinary $\\omega$-automaton. It is said to be a universal automaton if $U = Q$.\\index{universal $\\omega$-automaton}\\index{$\\omega$-automaton!universal}\n\nFrom the closure under complementation of the class of $\\omega$-languages recognized by B\\\"{u}chi automata, one obtains immediately:\n\n\\begin{remark}\n \\label{thm:universal-co-buechi}\n Every regular $\\omega$-language is recognized by a universal co-B\\\"{u}chi automaton.\n\\end{remark}\n\n\\index{transition-controlled alternating $\\omega$-automaton}\\index{$\\omega$-automaton!alternating!transition-controlled}\nIn the transition-controlled variant, the transition relation is replaced by a transition function $\\delta \\colon Q \\times A \\to M(Q)$, where $M(Q)$ is the set of all expressions built from states, the connectives $\\vee$ (``or'') and $\\wedge$ (``and''), and the boolean constants $0$ (``false'') and $1$ (``true''). For instance, $q \\wedge (q' \\vee q'')$ could be a value of the transition function. The set of initial states is replaced by an expression from $M(Q)$. Again, a run is a prefix-closed set \\mbox{$T \\subseteq Q^*$}, but this time satisfying the following condition. For each vertex $vq \\in T$, the set $\\{q' \\mid vqq' \\in T\\}$ satisfies the expression $\\delta(q, u(|v|))$, where satisfaction is defined in the obvious way. A run is initial if $\\{q \\in Q \\mid q \\in T\\}$ satisfies the initial condition. Being recurring is defined as above.\n\n\\begin{remark}\n A state-controlled alternating $\\omega$-automaton can be viewed as a transition-controlled alternating $\\omega$-automaton.\n\\end{remark}\n\nMore precisely, for every existential state $q$ one sets $\\delta(q, a) = \\bigvee \\{q' \\mid \\langle q, a, q'\\rangle \\in \\Delta\\}$, and for every universal state $q$ one sets $\\delta(q, a) = \\bigwedge \\{q' \\mid \\langle q, a, q' \\rangle \\in \\Delta\\}$; the set of initial states is converted into an initial condition in the same way; the recurrence condition does not need to be changed.\n\n\\subsection{Alternating automata and games}\n\nGiven a state-controlled alternating automaton as above and an $\\omega$-word $u$ over the same alphabet, the question whether $u$ is accepted by the automaton can be viewed as the question whether Zero wins a certain game, the so-called automaton game for $u$. The vertices of this game are pairs of the form $\\langle q, u'\\rangle$, where $u'$ is a suffix of $u$; such a vertex belongs to Zero if, and only if, $q$ is existential; there is an edge from $\\langle q,u'\\rangle$ to $\\langle q', u''\\rangle$ if $\\langle q,u'(0), q'\\rangle \\in \\Delta$ and $u'' = u'(1)u'(2) \\dots$; the recurrence condition is adapted in the straightforward fashion, based on the state in the first component. \n\n\\begin{remark}\n \\label{thm:alternation-game}\n A state-controlled alternating $\\omega$-automaton with existential [universal] initial states accepts a word $u$ if, and only if, Zero wins the automaton game for $u$ in some [every] vertex in $Q_I \\times \\{u\\}$.\n\\end{remark}\n\nFrom Theorem~\\ref{thm:determinacy}, which states that the games that occur in this fashion are determined, one can derive that a word is not accepted if, and only if, One has a winning strategy. This is equivalent to saying that the dual automaton accepts the word, where dualizing an automaton has the obvious meaning:\\index{dual!of an alternating $\\omega$-automaton}\\index{$\\omega$-automaton!alternating!dual of} existential and universal states exchange their roles and the recurrence condition is replaced by its negation. In other words, complementation is a trivial problem for alternating automata.\n\n\\begin{proposition}[complementing alternating automata]\n \\label{thm:complementation-alternation}\n The dual of a state-controlled alternating $\\omega$-automaton recognizes the complement of the language recognized by the given automaton.\n\\end{proposition}\n\nRemark~\\ref{thm:alternation-game} and Proposition~\\ref{thm:complementation-alternation} hold true for transition-controlled alternating $\\omega$-au\\-tom\\-a\\-ta as well, but the definition of the automaton game and the dualization process need to be adapted. The vertices of the automaton games are of the form $\\langle \\phi, u'\\rangle$ where $\\phi$ is a subformula of some value of the transition function. In the dualization process, the values of the transition function and the initial condition are dualized.\n\n\\subsection{From alternating automata to nondeterministic ones}\n\nAlternating $\\omega$-automata can be exponentially more concise than ordinary ones, just as in the finite-word setting~\\cite{drusinsky-harel-on-the-power-of-bounded-concurrency-i-finite-automata-1994}, but with regard to expressive power there is no difference. This is a major application of complementing $\\omega$-automata.\n\n\\begin{theorem}[from alternating to nondeterministic \\cite{miyano-hayashi-alternating-finite-automata-on-omega-words-2-1984}]\n For every alternating $\\omega$-au\\-tom\\-a\\-ton there exists an equivalent nondeter\\-min\\-istic B\\\"{u}chi automaton.\n\\end{theorem}\n\nTo prove this, first observe that it is enough to consider alternating parity automata, because any Muller condition can be turned into a parity condition as described in the proof of Theorem~\\ref{thm:muller-to-parity}. \n\nBy Theorem~\\ref{thm:positional}, parity games have uniform positional winning strategies. It follows that if there is an accepting run (recall that runs are trees) of an alternating parity automaton on a given word, then there is also an accepting subgraph of the run DAG, where this is defined in the obvious way. Checking that in a subgraph of a run DAG \\emph{all} rooted paths are recurring can be done using an appropriate $\\omega$-automaton, as explained in what follows. \n\nConsider the nondeterministic parity automaton over the alphabet $\\wp(Q \\times Q)$, with state set $Q$, initial set $Q_I$, transition relation $\\{\\langle q, a, q'\\rangle \\mid \\langle q,q'\\rangle \\in a\\}$, and parity condition $\\pi + 1$. This automaton accepts a word $u$ if the DAG which is obtained by collating the letters of $u$ contains \\emph{some} initial rooted path starting in an initial state and not satisfying the parity condition of the original automaton. Any $\\omega$-automaton recognizing the complement of the language recognized by this automaton is one that can check the DAG's. \n\nTo sum up, cascading \n\\begin{inparaenum}[(i)]\n\\item an automaton producing a subgraph of a run DAG of a given $\\omega$-word satisfying the transition relation and \n\\item the above automaton\n\\end{inparaenum}\nyields the desired automaton.\\qed\n\n\\subsection{Weak alternating automata}\n\\label{sec:weak-alternation}\n\n\\index{weak alternating $\\omega$-automaton}\\index{$\\omega$-automaton!weak alternating}\nRemark~\\ref{thm:weak-characterization} states that weak deterministic $\\omega$-automata only recognize fairly simple $\\omega$-languages. This is different for alternating automata:\n\n\\begin{theorem}[from alternating to weak alternating]\n \\cite{kupferman-vardi-weak-alternating-automata-are-not-that-weak-2001}\n For every alternating B\\\"{u}chi automaton with $n$ states there exists an equivalent weak alternating automaton with $2n^2$ states.\n\\end{theorem}\n\nBy dualization, it is enough to consider alternating co-B\\\"{u}chi automata. Theorem~\\ref{thm:positional} says that runs of alternating co-B\\\"{u}chi (and B\\\"{u}chi) automata can be thought of as run DAG's. The use of rank functions from Section~\\ref{sec:dags} leads to the following characterization of when an $\\omega$-word $u$ is accepted by a transition-controlled co-B\\\"{u}chi alternating automaton with $n$ states. There exists a tree $T \\subseteq (Q \\times [2n])^*$ satisfying the following conditions.\n\\begin{inparaenum}[(i)]\n\\item The set $\\{q \\in Q \\mid \\langle q,c \\rangle \\in T \\text{ for some $c< 2n$}\\}$ satisfies the initial condition.\n\\item Whenever $v \\langle q, c \\rangle \\in T$, then $\\{q' \\in Q \\mid v \\langle q, c \\rangle \\langle q', c'\\rangle \\in T \\text{ for some $c' < 2n$}\\}$ satisfies $\\delta(q,u(|v|))$. \n\\item There is no vertex $v \\langle  q, 2j+1 \\rangle \\in T$ with $q \\in B$. \n\\item When $v \\langle q, c \\rangle \\langle q', c'\\rangle \\in T$, then $c \\geq c'$.\n\\item For every rooted path $\\langle q_0, c_0 \\rangle \\langle q_1, c_1 \\rangle \\dots$ there exists some $i$ such that $c_i, c_{i+1}, \\dots$ are all odd. \n\\end{inparaenum}\nThis can be used to construct a transition-controlled weak alternating automaton with state set $Q \\times [2n]$; the initial condition and the transition function are adapted from the given automaton in a straightforward fashion; the B\\\"{u}chi set consists of all states with an odd second component.\\qed\n\nIt should be noted that the breakpoint construction can be used to convert a weak alternating B\\\"{u}chi automaton into an equivalent nondeterministic one.\n\n\\subsection{Simulation relations and simulation games}\n\nOne way to compare automata with each other, more precisely, to compare their internal structure, is to use simulation relations, or, more generally, simulation games.\\index{simulation relation!for $\\omega$-automata}\\index{simulation game!for $\\omega$-automata}\n\nA simple approach is to say that a B\\\"{u}chi automaton $\\mathscr A'$ forwardly simulates a B\\\"{u}chi automaton $\\mathscr A$\\index{forward simulation!of $\\omega$-automata} if there is a relation $\\sigma \\subseteq Q \\times Q'$ such that the following three conditions are satisfied.\n\\begin{inparaenum}[(i)]\n\\item For every $q \\in Q_I$ there is some $q' \\in Q_I'$ such that $\\langle q , q' \\rangle \\in \\sigma$.\n\\item For all $\\langle q, q'\\rangle \\in \\sigma$ and $\\langle q, a, r\\rangle \\in \\Delta$ there is some $r' \\in Q'$ such that $\\langle r, r'\\rangle \\in \\sigma$ and $\\langle q', a, r'\\rangle \\in \\Delta'$.\n\\item For all $\\langle q, q'\\rangle \\in \\sigma$, if $q \\in B$, then $q' \\in B'$.\n\\end{inparaenum}\n\nThe important observations concerning this definition are:\\index{direct simulation!of $\\omega$-automata}\n\\begin{theorem}[direct simulation \n \\cite{dill-hu-wong-toi-checking-for-language-inclusion-using-simulation-preorder-1991}]\n \\label{thm:simulation}\n \\begin{compactenum}\n \\item If a B\\\"{u}chi automaton $\\mathscr A'$ simulates a B\\\"{u}chi automaton $\\mathscr A$, then the language recognized by $\\mathscr A$ is a subset of the language recognized by $\\mathscr A'$.\n \\item Whether a B\\\"{u}chi automaton $\\mathscr A'$ simulates a B\\\"{u}chi automaton $\\mathscr A$ can be determined in time linear in the product of the sizes of $\\mathscr A$ and $\\mathscr A'$.\n \\end{compactenum}\n\\end{theorem}\n\nAs a consequence, simulation relations can be used for efficient (but incomplete) inclusion tests.\n\nThe requirement that a B\\\"{u}chi state in the simulating automaton match a B\\\"{u}chi state in the simulated automaton right away is very strong. For inclusion to hold, it would be enough if a B\\\"{u}chi state in the simulated automaton is matched by a B\\\"{u}chi state in the simulating automaton at a later position. This is captured by the notion of delayed simulation,\\index{delayed simulation!of $\\omega$-automata} which is best phrased in terms of a certain two-player game, where one of the players is called Duplicator and tries to show that simulation is given, whereas the other is called Spoiler and tries to show that this is not the case. \n\nMore precisely, the game determines whether a state in a B\\\"{u}chi automaton $\\mathscr A'$ delayed simulates a state in a B\\\"{u}chi automaton $\\mathscr A$. When a play of the game starts, there is a pebble on each of the two states in question. In every round of the game, first Spoiler is required to move the pebble on $\\mathscr A$ over some transition and then Duplicator is required to move the other pebble (the pebble  in $\\mathscr A'$) over some transition with the same label. If one of the players cannot move anymore, this player looses early. If an infinite play emerges, then Duplicator wins if, and only if, the following holds: whenever Spoiler visits a B\\\"{u}chi state in some round, Duplicator visits a B\\\"{u}chi state in the same or in a later round. The state in $\\Aut A'$ delayed simulates the state in $\\Aut A$ if Duplicator has a winning strategy in the game just described. The automaton $\\mathscr A'$ delayed simulates the automaton $\\mathscr A$ if every initial state of $\\mathscr A$ is simulated by some initial state of $\\mathscr A'$. Observe that the above game can be viewed as a game of infinite duration with a regular winning condition as described in Section~\\ref{sec:games}.\n\nTheorem~\\ref{thm:simulation} carries over to delayed simulation, only the complexity of computing delayed simulation is higher \\cite{etessami-schuller-wilke-fair-simulation-relations-parity-games-and-state-space-reduction-for-buechi-automata-2005}.\n\nFor purposes of state-space reduction, it useful to study simulation in both directions: if one state [delayed] simulates another one and vice versa, the states are said to mutually [delayed] simulate each other. These relations are, indeed, equivalence relations and have a useful property:\n\n\\begin{theorem}[quotienting   \\cite{etessami-schuller-wilke-fair-simulation-relations-parity-games-and-state-space-reduction-for-buechi-automata-2005}]\nIf, in a quotient of a B\\\"{u}chi automaton with regard to the mutual [delayed] simulation relation, initial and B\\\"{u}chi states are chosen appropriately, then the resulting automaton is equivalent to the given one.\n\\end{theorem}\n\nThis gives, in effect, two polynomial-time algorithms for reducing the state space of B\\\"{u}chi automata, one less efficient than the other, but producing smaller automata. Finding and even approximating minimum-size B\\\"{u}chi automata is PSPACE-hard, in fact, this is independent of the type of the automaton, because results from finite-state automata on finite words ~\\cite{gramlich-schnitger-minimizing-nfas-and-regular-expressions-2007} carry over in a straightforward fashion.\n\nIn principle, one could also work with bisimulation rather than mutual simulation, but this gives, in general, worse reductions. \n\nMuch effort has gone into finding coarser relations for state-space reductions, and there are various ways of approaching this: letting Duplicator match with more than just one pebble, relaxing the winning condition for Duplicator further, considering backward simulation, and so on.\n\n\\section{Applications in logic}\n\n$\\omega$-Automata were introduced in the late fifties in the context of mathematical logic, more precisely, B\\\"{u}chi automata first showed up in \\cite{buechi-decision-method-restricted-second-order-arithmetic-1962} (in disguise) and were used there as a tool for proving that theories of specific structures are decidable. From a modern point of view, B\\\"{u}chi showed that the structures are $\\omega$-automatic~\\cite{hodgson-decidabilite-par-automata-fini-1983}.\n\n\\subsection{\\texorpdfstring{$\\omega$}{\u03c9}-Automatic structures}\n\nAssume a first-order structure $\\mathfrak S$ consisting of a universe $U$ and a family $\\{R_i\\}_{i \\in I}$ of relations, say $R_i$ having arity $n_i$, are given; the question is whether the theory of this structure is decidable. A good example are the real numbers with the ternary relation ``addition'', the predicate ``is positive'', and the predicate ``is power of 2''. \n\nAn $\\omega$-automatic presentation\\index{$\\omega$-automatic}\\index{structure!$\\omega$-automatic}\\index{presentation!$\\omega$-automatic}\\index{$\\omega$-automatic presentation} of a structure $\\mathfrak S$ as above is given by an alphabet $A$, an $\\omega$-automaton $\\mathscr U$ over $A$, an $\\omega$-automaton $\\mathscr E$ over $A \\times A$, and, for each $i \\in I$, an $\\omega$-automaton $\\mathscr R_i$ over $\\bigtimes_{i<n_i} A$. It is required that there exists an onto function $f \\colon \\text L(\\mathscr U) \\to U$ such that the following conditions are satisfied:\n\\begin{asparaitem}\n\\item For all $u, v \\in L(\\mathscr U)$, $f(u) = f(v)$ if, and only if, $u * v \\in L(\\mathscr E)$.\n\\item For all $i \\in I$ and $u_0, \\dots, u_{n_i - 1} \\in L(\\mathscr U)$, $u_0 * \\dots * u_{n_i-1} \\in L(\\mathscr R_i)$ if, and only if, $\\langle f(u_0), \\dots, \\linebreak f(u_{n_i-1})\\rangle \\in R_i$.\n\\end{asparaitem}\nTo extend the above example, one can start with an $\\omega$-automaton $\\mathscr U$ that accepts exactly the $\\omega$-words representing real numbers as described in Section~\\ref{sec:omega-words}. Then $\\mathscr E$ must be constructed in a way such that it identifies representations of identical numbers. Finally,  $\\omega$-automata representing the three respective relations must be found. A simple automaton is the ``is power of $2$'' automaton, which only checks that there is exactly one occurrence of~$1$ and that this occurrence is not in position~$0$ (because otherwise the number represented would be $0$, more precisely, $-0$, which is not a power of~$2$).\n\nThe fundamental result about $\\omega$-automatic structures is:\n\n\\index{first-order theory!of $\\omega$-automatic structures}\n\\begin{theorem}[$\\omega$-automatic structures\n \\cite{hodgson-decidabilite-par-automata-fini-1983}]\n \\label{thm:automatic}\n The first-order theory of every $\\omega$-automatic structure is decidable.\n\\end{theorem}\n\nThe reason for this is that, by induction, one can show that for every first-order formula in the respective vocabulary one can construct an $\\omega$-automaton that recognizes the representations of the satisfying assignments. When $\\phi = \\phi(x_0, \\dots, x_{n-1})$ is a formula with all of its free variables among $x_0, \\dots, x_{n-1}$, then a word of the form $u_0 * \\dots * u_{n-1}$ represents a satisfying assignment if $u_i \\in L(\\mathscr U)$ for every $i<n$ and $\\mathfrak S, f(u_0), \\dots, f(u_{n-1}) \\models \\phi(x_0, \\dots, x_{n-1})$.\n\nFor the base case, there is almost nothing to show, because this is part of the definition of $\\omega$-automatic structure. For the induction itself it should be noted that disjunction can essentially be viewed as union, negation as complementation, and existential quantification as projection. All these operations can easily be implemented effectively on $\\omega$-automata. In other words, there is an effective procedure that, given a closed formula, constructs an $\\omega$-automaton over the unary alphabet, $\\bigtimes_{i<0} A$, which accepts some word, more precisely, the word $\\langle \\rangle^\\omega$, if, and only if, the formula is true in the given structure. Nonemptiness can be verified effectively for $\\omega$-automata, see Remark~\\ref{thm:basic}(2).\\qed\n\nAn important example for this theorem, already mentioned in B\\\"{u}chi's seminal paper, is the one described above: the real numbers with addition and the ``is positive'',  ``is power of two'', and ``is an integer'' predicates.\n\nAnother example from B\\\"{u}chi's original work is the monadic second-order theory of the natural numbers with successor, more precisely: the structure is the set of natural numbers endowed with the successor predicate; in the vocabulary of the logical language there are, in addition to what is part of a suitable first-order language (symbols for disjunction, negation, existential quantification, the binary successor relation, variables for natural numbers),  variables for sets of natural numbers, a symbol for ``is element of'', and a symbol for existential quantification of set variables. At first glance, this does not look like a situation where Theorem~\\ref{thm:automatic} can be applied, but it actually can: a formula in the above logic can be translated in a straightforward fashion into a first-order formula for the structure with the power set of the natural numbers as universe and endowed with the ``is singleton'' predicate, the binary relation ``is subset of'', and the binary relation ``every element of \\dots\\ has a successor in \\dots''.\n\n\\index{S1S}\\index{monadic second-order theory!of one successor}\n\\begin{theorem}[decidability of S1S\n \\cite{buechi-decision-method-restricted-second-order-arithmetic-1962}]\n The monadic second-order theory of the natural numbers with the successor predicate is decidable. \n\\end{theorem}\n\nFrom the point of view of $\\omega$-automata theory, there are several applications in logic which are of particular interest. Two of them are discussed in what follows.\n\n\\subsection{Temporal logic}\n\nTemporal logic\\index{temporal logic} comes in many different flavors. The version that is most often considered and also most amenable to being dealt with using $\\omega$-automata is the one where the temporal operators used are future operators (next, X; eventually in the future, F; always in the future, G; until, U; release, R) and the time domain is discrete, more precisely, where the time domain is $\\omega$, the set of natural numbers. In such a context, a temporal variable, here denoted $p_i$, is assigned a set of natural numbers, the points in time where the variable is true. So if the variables occurring in a given formula $\\phi$ are among $p_0, \\dots, p_{n-1}$, then the models of this formula can be viewed as $\\omega$-words over the alphabet $\\wp(\\{p_0, \\dots, p_{n-1}\\})$. \n\nFor instance, the set of models of the formula $\\text G(p_0 \\rightarrow \\text F p_1)$, which is read  ``now and always in the future, if $p_0$, then $p_1$ at the same time or some point later'', can be viewed as the set of $\\omega$-words over $\\wp(\\{p_0, p_1\\})$ with the property that whenever $p_0$ is an element of a letter at some position, then $p_1$ belongs to the same letter or some other letter in a position to the right. \n\nThe starting point for constructing an $\\omega$-automaton recognizing the set of models of a given formula  is the observation that whether a formula of temporal logic is true in some point in time only depends on \n\\begin{inparaenum}[(i)]\n\\item which of its strict subformulas hold true in this and the next point in time and\n\\item whether the formula itself holds true in the next point in time.\n\\end{inparaenum}\nSo a suitable automaton can guess, for each point in time, which subformulas are true and then verify its guessing locally in a backward deterministic fashion. For some temporal operators, it is important though to also verify certain conditions globally. For instance, it is true that the formula $\\text F \\phi$ holds true in position~$i$ if, and only if, $\\phi$ holds true in position~$i$ or $\\text F \\phi$ holds true in position $i+1$, and $\\text F \\phi$ holds true in position $i+1$ if, and only if, $\\phi$ holds true in position $i+1$ or $\\text F \\phi$ holds true in position $i+2$, and so on, but, clearly, the formula $\\phi$ must become true at some point. Such a global condition can be captured by an appropriate recurrence condition. The initial states are the ones where the automaton guesses the entire formula to be true. \n\nThe general theorem is as follows:\n\n\\begin{theorem}[from temporal logic to automata\n\\cite{vardi-wolper-sistla-reasoning-infinite-computation-paths-1983,vardi-wolper-reasoning-infinite-computation-paths-1994}]\n Every temporal formula with~$n$ subformulas can be translated into an equivalent backward deterministic generalized B\\\"{u}chi automaton with at most $2^n$ states and as many B\\\"{u}chi sets as there are subformulas with leading temporal operator $\\rm F$ or $\\rm U$.\n\\end{theorem}\n\nThis implies, in particular, that satisfiability and validity of temporal formulas as well as model checking temporal formulas over finite-state system with fairness conditions are problems in PSPACE \\cite{sistla-clarke-complexity-propositional-linear-temporal-logics-1985}. \n\nFuture linear-time temporal formulas can be translated directly into weak alternating automata (see Section~\\ref{sec:weak-alternation}) of a very specific structure; the resulting number of states is the number of subformulas.\n\n\\subsection{The additive theory of the reals}\n\\index{additive theory!of the reals}\n\nRegarding the aforementioned example of the real numbers one can show (by other means than automata-theoretic ones) that the relations definable by formulas in the underlying first-order language are all recognizable by forward deterministic weak automata as introduced in Section~\\ref{sec:wagner} (when real numbers are represented as described in Section~\\ref{sec:omega-words}). In view of Theorem~\\ref{thm:forward-deterministic-weak}, this means that a recursive procedure for constructing automata representing such relations can take advantage of conditional determinization, which is much less complicated than ordinary determinization, and of simple and fast minimization procedures as pointed out in Theorem~\\ref{thm:initial-congruence}. This, in the end, leads to feasible decision procedures \\cite{boigelot-jodogne-wolper-effective-decision-procedure-linear-arithmetic-over-integers-reals-2005}.\n\n\\section{More complex recurrence conditions}\n\nMuch effort can and has been put into extending $\\omega$-automata like the ones dealt with in this paper, that is, the ones with a finite state space and a recurrence condition based on the states occurring infinitely often in a run. There are finite-state automata working on other infinite objects: other ordinals, the integers, linear orderings in general, and, most notably, infinite trees; there are $\\omega$-automata using more complex storage, for instance, $\\omega$-automata with stacks; there are probabilistic $\\omega$-automata, that is, $\\omega$-automata where transitions are taken with certain probabilities; there are timed $\\omega$-automata, using clocks and processing infinite sequences of events having a duration; and so on.\n\nBeside all this, Boja\\'nczyk and Colcombet suggest in \\cite{bojanczyk-colcombet-bounds-in-omega-regularity-2006}  to strengthen the models discussed in this chapter by a more powerful mechanism for defining recurrence, allowing a finer analysis of what happens ``in the infinite''. In their model, every automaton has a finite number of counters. A transition is of the form $\\langle q, a, \\alpha, q'\\rangle$ where $\\alpha$ is a function assigning to each counter no action or one of the following two:\n\\begin{inparaitem}[]\n\\item ``inc''---increment the counter by one;\n\\item ``prt\\&res''---output (print) the counter value and then reset the counter.\n\\end{inparaitem}\nSo, for every counter, a finite or infinite sequence of natural numbers, its recurrence sequence, is produced in each run. The recurrence condition is a boolean combination of conditions of the form $\\liminf c = \\infty$ and $\\limsup c = \\infty$, with $c$ standing for a counter. A run of such an automaton is recurring if every recurrence sequence is infinite and they all satisfy the recurrence condition.\n\nA good example for a non-regular $\\omega$-language which can be recognized by such an automaton is the language $L_\\text{ub0}$ (see Section~\\ref{sec:saturation}) of all $\\omega$-words of the form $0^{i_0}10^{i_1}1\\dots$ where $\\limsup_{j \\to \\infty} i_j = \\infty$. This language is recognized by an automaton with one counter, say~$c$:\\\\ \n\\hspace*{\\fill}\n \\begin{small}\n\t \\begin{tikzpicture}[node distance = 20mm]\n\t\t \\node[] (s) {}; \n\t\t \\node (s0) [right of=s,node distance=10mm] {$q_0$}; \n\t\t \\draw[->] (s) -- (s0);     \n\t\t \\draw[scale=1.5,my loop] (s0) to node[above, left=3mm, align=center] {$0$\\\\[-1mm] inc $c$} (s0); \n\t\t \\draw[scale=1.5,rotate=180,my loop] (s0) to node[above, right=1mm, align=center] {$1$\\\\[-1mm] prt\\&res $c$} (s0); \n\t\t \\node [right of=s, right] {$\\limsup c = \\infty$ (recurrence condition)};\n\t \\end{tikzpicture}\n \\end{small}\n\\hspace*{\\fill}\n\n\\smallskip\\noindent\n\\textbf{Acknowledgment} \\ I am grateful to Christof, Olivier, Sebastian, and my master students for insightful comments, to Wolfgang for his constant support, and to Jean-\\'{E}ric for making me write this paper.\n\n\\bibliographystyle{abbrv}\n\\addcontentsline{toc}{section}{References}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nDynamics in concentrated protein systems are of fundamental interest in fields such as protein crystallization \\cite{Durbin1996}, phase separation \\cite{Anderson:2002aa}, the glass transition \\cite{PhysRevLett.99.118301} or diffusion in crowded environments \\cite{ELLIS2001597}, to name just a few. These systems display relatively slow and heterogeneous dynamics ranging from micro-seconds to seconds on length scales ranging from micrometers down to the single-particle nanometer scale. X-ray photon correlation spectroscopy (XPCS) is well suited to cover this length scale and time window employing coherent X-ray beams and tracing fluctuations in X-ray speckle patterns \\cite{sutton_coherence,Gruebel2008,Perakis2017,Madsen2018}. \nHowever, the highly intense X-ray beams of synchrotron storage rings are also the cause of considerable radiation damage to the samples. Atomic scale XPCS experiments use X-ray doses of MGy and beyond, which can lead to beam induced dynamics, even in hard condensed matter samples \\cite(Ruta2017). Soft and biological matter samples are much more sensitive to radiation damage requiring flowing samples \\cite{Fluerasu:ot5581} \\cite{Vodnala2018} or scanning samples with optimized data taking strategies \\cite{Verwohlt2018}. Radiation damage of bio-molecules in solution is caused mainly by two effects: either via direct damage to the protein structure itself by photo-ionization or by indirect damage via hydrolysis of the surrounding water molecules (see e.g. \\cite{Garman:ba5150}, \\cite{Holton_A_2009}). In both cases, the damage becomes apparent by a characteristic change to the SAXS pattern indicating an increase of the radius of gyration, mostly due to aggregation. Typical critical X-ray doses for protein molecules in solution range from 7-10 kGy (BSA) to 0.3 kGy (Rnase) after which a degradation of the SAXS patterns become visible \\cite{Jeffries_Limiting_2015}. These doses are easily reached within ms when using focused beams of modern synchrotron sources. While in protein crystallography cryogenic cooling helps to prevent the diffusion of radicals, such an approach is obviously impossible when studying the dynamics of proteins in solution. \n\nXPCS requires a coherent X-ray beam and the signal to noise ratio (SNR) in XPCS experiments ideally scales linear with the source brilliance $B$ \\cite{Lumma2000}. The fastest accessible time scale then scales with $B^2$ promising four orders of magnitude faster temporal resolution at the upgraded sources of ESRF  and PETRA IV \\cite{Einfeld:xe5006}\\cite{Weckert:it5005} \\cite{Schroer_2018} which is one of the key drivers for XPCS at DLSR sources \\cite{Shpyrko_X_2014}. These arguments, however, only hold if radiation damage is no issue. Thus, the question arises of how much XPCS experiments of biological \/ radiation sensitive samples could really benefit from the gain in coherence performance of DLSR rings. Here, we show that the combination of (i) larger coherence lengths, (ii) higher photon energy and (iii) the increased coherent photon flux yields indeed an increase in SNR of up to one order of magnitude when compared to standard XPCS setups at today's storage rings. We calculate explicitly, using the boundary conditions set by the maximum tolerable X-ray doses of a  lysozyme solution, the XPCS speckle contrast, speckle intensities and maximum number of images per spot. We come to the conclusion, that DLSR rings hold the promise to measure dynamics of biological samples at length scales of a single protein molecule. \\\\\n\n\n\\section{XPCS on protein solutions}\n\nXPCS experiments track fluctuations in X-ray speckle patterns yielding access to the intermediate scattering function $f(q,\\tau ) = S(q,\\tau)\/S(q)$ by correlating intensities per detector pixel \\cite{GRUBEL20043}.\nThe measured signal in such experiments is the normalized intensity autocorrelation function\n\\begin{equation}\ng_2(q,\\tau)=\\frac{ \\langle I_{pix}(q,t')I_{pix}(q,t'+\\tau) \\rangle }{\\langle I_{pix}(q,t') \\rangle ^2}=1+\\beta |f(q,\\tau)|^2,\\label{eq:g2}\n\\end{equation}\nwith $\\beta$ denoting the speckle contrast and $q = 4 \\pi \\sin (\\Theta\/2)\/\\lambda$ being the scattering vector, depending on the wavelength $\\lambda$ and the scattering angle $\\Theta$. The time delay between two consecutive time frames is denoted $\\tau$ and $\\left\\langle \\ldots \\right\\rangle$ is the ensemble average over all equivalent delay times $\\tau$ and pixels within a certain range of the absolute value $\\left|\\vec{q}\\right|$.\\\\\n\nThe scattering intensity per pixel from a protein solution is given by\n\\begin{equation}\nI_{pix}(q)  = F_c \\cdot t_{fr} \\cdot T_{sample} \\cdot d \\cdot \\frac{\\mathrm{d}\\Sigma}{\\mathrm{d}\\Omega}(q) \\cdot \\Delta \\Omega_{pix},\\label{eq:I}\n\\end{equation}\nwith $F_c$ denoting the incident coherent flux (ph\/second), $t_{fr}$ the exposure time for one frame, $T_{sample}$ the sample's transmission and $\\Delta \\Omega_{pix} = (P \/ L)^2$ the solid angle covered by a single pixel, with $P$ being the pixel size and $L$ the sample-detector distance. In the following, we will set the thickness of the sample $d(E)$ to be equal to the absorption length of water $d(E) = 1\/\\mu(E)$ at each respective photon energy $E$, with the transmission following as $T_{sample} = \\exp(- \\mu d) \\approx 0.368$. \\\\\nThe differential scattering cross section per unit volume or absolute scattering intensity in 1\/m of a protein solution is defined as\n\\begin{equation}\n\\frac{\\mathrm{d}\\Sigma}{\\mathrm{d}\\Omega} (q) = C \\cdot M \\cdot \\bar{v}^2 \\cdot \\Delta \\rho^2 \\cdot P(q) \\cdot S_{eff}(q),\n\\end{equation}\nwith $P(q)$ the form and $S_{eff}(q)$ the effective structure factor and $C$ the protein concentration. We will calculate the SNR for lysozyme as model protein with a molar mass of $M=14.3$ kDa and specific volume $\\bar{v} = 0.74$ cm${}^2$\/g. The scattering contrast $\\Delta \\rho$ follows from the chemical composition of lysoszyme showing almost no dependence on energy in the energy range of interest here. With this, the absolute scattering intensity can be expressed as \n\\begin{equation}\n\\frac{\\mathrm{d}\\Sigma}{\\mathrm{d}\\Omega} (q) = C \\cdot 1.02 \\mathrm{m^2 \\over kg} \\cdot P(q) \\cdot S_{eff}(q),\n\\end{equation}\nin good agreement with measured values of $(1.03 \\pm 0.06) \\mathrm{m^2 \\over kg}$ ZITAT\\cite{}.\\\\\n\n\\begin{figure}\n\\centering\n\t\t\\includegraphics[width=1\\textwidth]{SAXS.png}\n\t\\caption{Form factor $P(Q)$ (black line) and effective structure factor $P(q) \\cdot S_{eff}$ (red dashed line) of a diluted and concentrated lysozyme solution, respectively. The inset shows the relaxation rate $\\Gamma(q)$ as a function of $q$ for both cases.}\n\t\\label{fig:BioSAXS}\n\\end{figure}\n\n The form and effective structure factor $P(q) \\cdot S_{eff}(q)$ are modeled following \\cite{Moller:2012aa} and displayed in Fig. \\ref{fig:BioSAXS} for a diluted ($10$ mg\/ml) and concentrated ($250$ mg\/ml) lysozyme solution. The $q$ values of interest are within $q = 0.5 \\ \\mathrm{nm}^{-1} - 1.5 \\ \\mathrm{nm}^{-1}$, which corresponds to length scales of $4-12$ nm.\\\\\n The dynamics of the low concentrated protein solution can be described as Brownian diffusion with a single exponential autocorrelation function\n \\begin{equation}\n     g_2(q,t)-1 = \\beta \\exp(-2\\Gamma(q)t)\n \\end{equation}\nand relaxation rate\n \\begin{equation}\n     \\Gamma = D_0 q^2\n \\end{equation}\nwhich is proportional to the Stokes-Einstein diffusion constant\n \\begin{equation}\n     D_0 = \\frac{k_B T}{6 \\pi \\eta R_H},\n \\end{equation}\n where $T$, $\\eta$, $R_H$ and $k_B$ are the temperature, the viscosity of the suspending medium, the hydrodynamic radius of the protein and the Boltzmann constant, respectively. The $q$-dependence of the relaxation rate is plotted in the upper right inset of Fig. \\ref{fig:BioSAXS} for diluted and concentrated lysozyme solutions. For the diluted case, we take the viscosity of water and a hydrodynamic radius of $R_H = 1.9$ nm. In order to illustrate the expected timescales for XPCS experiments on concentrated protein solutions we use an increased effective solution viscosity by a factor of 15 \\cite{Godfrin2015}. The time scales of interest are here ranging from 100 $\\mu s$ to seconds.\\\\\nIn practice, XPCS correlation functions are averaged over many pixels in a narrow range of $q$ values. Typical regions of interested are sketched as colored areas in Fig. \\ref{fig:BioSAXS}. The same set of regions is additionally depicted in the lower left inset, showing the location of the corresponding pixels on an EIGER 4M detector for $E = 8$ keV and a sample to detector distance of $L=2$ m. In the following, we will always calculate the SNR at the maximum of the structure factor peak at $q=0.9 \\ \\mathrm{nm}^{-1}$.\\\\\n\n\\section{Signal to noise ratio}\n\nThe signal to noise ratio for the autocorrelation function $g_2(q,\\tau)$ depends on the average intensity per pixel $I_{pix}$, the contrast $\\beta$, the number of pixels $N_{pix}$, the number of frames $N_{fr}$ and the number of repetitions $N_{rep}$ via\n\\begin{equation}\nSNR = \\beta \\cdot I_{pix} \\cdot \\sqrt{N},\n\\end{equation}\nwith $N = N_{pix} \\cdot N_{fr} \\cdot N_{rep}$.\\\\\nConsidering $N_{fr} = T\/t_{fr}$ with $t_{fr}$ being the single frame exposure time and $T$ the total accumulated time for $N_{fr}$ frames yields in combination with Eq. \\ref{eq:I} $SNR \\propto F_c \\sqrt{t_{fr} \\cdot T}$. This scaling implies that an increase in coherent flux by one order of magnitude gives access to two orders of magnitude faster dynamics for the same SNR. However, this argument only holds when the sample is capable of handling the increased photon flux. If a critical dose $D_c$ exists, beyond which radiation induced damage starts to degrade the sample, the longest overall exposure time $T$ depends on $F_c$ and the increase of coherent flux might be less or not beneficial at all for studying radiations sensitive samples. \n\nThe dose per second delivered to the sample depends on the photon flux as well as the photon energy which both also influences the achievable SNR. Here, we take all those parameters into account and calculate the benefit to the SNR from the increased coherent flux of DLSRs.We identify three parameter, which we will assume to be nearly free of choice over a wide range of values. These are the photon energy $E = \\hbar c \/ \\lambda$, the diameter $a$ of the X-ray beam spot size on the sample and the distance $L$ between sample and detector. In the following, we will establish the dependencies of the different contributions on the SNR, and determine the optimal set of $a$, $\\lambda$, and $L$ values for XPCS experiment using radiation sensitive samples.\\\\\n\n\n\n\\begin{figure}\n\\centering\n\t\t\\includegraphics[width=10cm]{Brilliance.png}\n\t\\caption{a) Brilliance taken from \\cite{Schroer_2018} b) Coherent Flux calculated from fig. a), assuming a bandwidth of $0.01\\%$, corresponding to a Si(111) monochromator with $\\Delta \\lambda \/ \\lambda = 10^{-4}$.}\n\t\\label{fig:bril}\n\\end{figure}\n\nFig. \\ref{fig:bril} a) shows the expected increase of Brilliance as a function of photon energy for an U29 undulator at PETRA III and IV. Additionally, the case of an U18 with $5$ and $10$ m length will be investigated. The data shown is taken from \\cite{Schroer_2018}. From this, the coherent flux can be calculated as \n\\begin{equation}\nF_c[ph \/ s \/ 0.1\\%] = 10^{-8} Br[ph \/ s \/ 0.1\\% \/ mm^2 \/ mr^2] \\left(\\lambda[\\AA] \\over 2\\right)^2,\n\\end{equation}\nwhich is also depicted in Fig. \\ref{fig:bril} b). Using the given brilliance we calculate the coherent flux for $8$ keV at PETRA III as $3.8 \\cdot 10^{11}$ ph\/s. This is in good agreement with measured values of $2.3 \\cdot 10^{11}$ ph\/s, taking into account transmission effects of beamline components and optics. In the following, the actual coherent flux on the sample will be calculated by taking into account the same beamline transmission factor for all undulators.\n\n\\subsection{Limitations due to radiation damage}\nWe assume a critical dose $D_c$ beyond which radiation induced damage starts to degrade the sample which can be expressed as (Meisburger (2013). Biophys. J.  104, 227\u2013236)\n\\begin{equation}\nD_c = \\frac{F_c E (1-T_{sample}) T}{d(E) a^2 \\rho},\n\\end{equation}\nwith $F_c$ the photon flux on the sample, the product of energy dependent sample thickness $d(E)$ and beam area $a^2$, the sample absorption $(1-T_{sample})$, photon energy $E$ and exposure time $T$. From this we derive the maximum number of frames which can be measured before radiation damage occurs to be\n\\begin{equation}\n N_{fr} = \\frac{d(E) a^2 \\rho D_c}{t_{fr}  F_c E (1-T_{sample})}, \\label{N}\n\\end{equation}\nignoring the latency time of the detector and absorption within the sample container walls. The sample thickness $d(E)$ is always adapted to the energy dependent absorption length of water. One important conclusion from equation \\ref{N} is that the SNR scales via SNR $ \\propto F_c \\sqrt{N_{fr}} \\propto \\sqrt{F_c}$ for radiation sensitive samples.  Moreover, with the scalings $d(E) \\propto E^3$ and $F_c \\propto Br(E) \/E^{2}$ we also find the peculiar relation of $N_{fr} \\propto E^4$ favoring higher photon energies if a  large number of frames is required. \\\\\nWe illustrate this with the example of a typical spot size for XPCS experiments of $a = 4 \\ \\mathrm{\\mu m}$, an exposure time of a single frame of $t_{fr} = 1$ ms and a critical Dose limit for a concentrated lysozyme solution of $D_c = 1$ kGy. \n\\begin{figure}\n\\centering\n\t\t\\includegraphics[width=10cm]{Nfr.png}\n\t\\caption{Maximum number of frames which can be measured on one spot before the onset of radiation damage for a lysozyme solution with beamsize $a = 4 \\ \\mathrm{\\mu m}$ and exposure time per frame of 1 ms. The vertical black line depicts the threshold of at least two consecutive frames.}\n\t\\label{fig:Nfr}\n\\end{figure}\nFig. \\ref{fig:Nfr} displays the possible number of consecutive frames as a function of photon energy.  A prerequisite for correlation spectroscopy is obviously that the number of consecutive frames is at least two (i.e. $N_{fr} \\geq 2$), indicated via filled symbols. Already with the coherent flux of PETRA III the critical dose is exceeded after or during the first image and beam damage is occurring between two images for photon energies below $10$ keV. At this energy, an increase in coherent flux would therefore not be usable for XPCS experiments on protein samples. However, it can also be seen that $N_{fr}$ increases with photon energy due to the increasing absorption length of the X-rays. Effectively, the radiation dose is spread over a larger sample volume with increasing photon energy. However, many properties like the speckle size, the coherent flux as well as the longitudinal and transversal coherence lengths decrease with increasing photon energy. Therefore, the disadvantageous influence of these properties on the speckle contrast $\\beta$ and consequently on the SNR of XPCS experiments need to be taken into account as well.\n\n\\subsection{Speckle contrast $\\beta$}\nThe speckle contrast depends on nearly all experimental parameters such as pixel size $P$, speckle size $S\\approx \\lambda L\/a$, beam size $a$, sample thickness $d$, wavevector transfer $q$, and the transverse and longitudinal coherence lengths. It can be written as a product, \n\\begin{equation}\n\\beta(a, d, q, \\lambda, L) = \\beta_{cl}(a,d,q,\\lambda) \\beta_{res}(a,L,\\lambda)\n\\end{equation}\nin which the first factor $\\beta_{cl}$ corresponds to the reduction of the contrast from unity due to the finite coherence lengths in transverse and longitudinal direction. The second factor $\\beta_{res}$ corresponds to a finite angular resolution of the experimental setup. This results in a reduction of contrast if the pixel size of the detector $P$ exceeds the size of the speckle $S$:\n\\begin{equation}\n\\beta_{res}(a,L,\\lambda) = \\left( {2 \\over w^2} \\int_0^w (w-v) \\left( \\frac{\\sin(v\/2)}{v\/2}\\right)^2 \\mathrm{d}v\\right)^2 \\label{eq:betares} \n\\end{equation}\nwith $w= 2 \\pi P a\/L\\lambda = 2 \\pi P\/S$. Fig. \\ref{fig:betares} displays the speckle contrast $\\beta_{res}$ as a function of beamsize $a$ for sample-detector distances of $L=5$ m and $L=100$ m, respectively, pixel size $P=75 \\ \\mu$m  and photon energies of 8, 15 and 25 keV.  The maximum $\\beta_{res}$ is obtained in a high resolution configuration with $S \\geq P$ and scales as $\\beta \\approx \\lambda^2 L^2 \/a^2P^2$ in the low resolution configuration, when $S \\ll  P$. Therefore, XPCS experiments with large beamsizes require long sample-detector distances in order to resolve the smaller speckles.  \n\\begin{figure}\n\\centering\n\t\t\\includegraphics[width=10cm]{beta_ps.png}\n\t\\caption{Speckle contrast $\\beta_{res}$ as a function of beam size $a$ on the sample according to eq. \\ref{eq:betares}. Calculated for photon energies of 8 keV (red), 15 keV (green) and 24 keV (blue) and for sample-detector distances L=100 m (top lines) and L=5 m (bottom lines), respectively.}\n\t\\label{fig:betares}\n\\end{figure}\n\nThe dependence of $\\beta_{cl}$ on beamsize $a$, sample thickness $d$, transverse $\\xi_h$, bandwidth $\\Delta \\lambda \\over \\lambda$ and q-value is taken into account via {\\cite{sutton_coherence}\n\\begin{equation} \\label{eq:betacl}\n\\begin{split}\n\\beta_{cl}(a,d,q,\\lambda) & = \\frac{2}{(a \\cdot d)^2} \\int_0^{a} \\mathrm {d}x \\int_0^d \\mathrm{d}z (a-x)(d-z) \\exp(-x^2\/\\xi_h^2) \\\\\n &  \\times \\left( \\exp(-2\\left|Ax+Bz\\right|)+\\exp(-2\\left|Ax-Bz\\right|) \\right), \n\\end{split}\n\\end{equation}\nwith $A = { \\Delta \\lambda \\over \\lambda} q \\sqrt{1- {1 \\over 4} q^2 \/ k^2}$ and $B = - {\\Delta \\lambda \\over 2 \\lambda}{q^2 \\over k}$. In vertical direction we assume a completely coherent beam and in horizontal direction, the coherence length is estimated as \n\\begin{equation}\n\\xi_h = {R \\cdot \\lambda \\over 2\\pi \\sigma} , \\label{eq:xih}\n\\end{equation}\nwith $R$ being the distance between source and beam defining aperture and $\\sigma$ the RMS source size. With $\\sigma_h = 36 \\ \\mu$m (P10, low-$\\beta$ source, 10 keV, $R = 90$ m), this results in a horizontal coherence length at $E=10$ keV of $\\xi_h = 49 \\ \\mu$m. A reduced horizontal source size at PETRA IV of $\\sigma_h = 12 \\ \\mu$m would result in an increased horizontal coherence length of $\\xi_h = 147 \\ \\mu$m at the same energy. These values reduce to $20 \\ \\mu$m and $59 \\ \\mu$m at an energy of $E=25$ keV, respectively. The full energy dependence of $\\xi_h$ is shown in Fig. \\ref{fig:betacl} a).\\\\\nUsing a partially coherent source like a undulator for coherent scattering experiments, cutting of the incident X-ray beam is required in order to obtain a nearly fully transversely coherent beam. Therefore, a beam defining aperture is set to an opening size equal to the transversal coherence length. Smaller beam sizes can be achieved with additional focussing elements. For our calculations, we will consider the resulting focussed beam as fully coherent with a $\\xi_h$ being equal to the beam size. For larger beamsizes, $\\xi_h$ is calculated following Eq. \\ref{eq:xih}.\\\\\nThe temporal or longitudinal coherence length can be calculated as\n\\begin{equation}\n\\xi_l = {\\lambda \\over 2} {\\lambda \\over \\Delta \\lambda},\n\\end{equation}\ndepending on the bandwidth of the used monochromator ( $\\Delta \\lambda \/ \\lambda \\approx 1.4 \\cdot 10^{-4}$ for a Si(111) monochromator and $\\Delta \\lambda \/ \\lambda \\approx 3 \\cdot 10^{-5}$ for Si(311)). \\\\\nThe results for $\\beta_{cl}$ as a function of beam size and X-ray energy are shown in figure \\ref{fig:betacl} b) for a q-value of $q=0.9 $ nm$^{-1}$, corresponding to the peak of the structure factor shown in Fig. \\ref {fig:BioSAXS}.\n\\begin{figure}\n\\centering\n\t\t\\includegraphics[width=10cm]{beta_q.png}\n\t\\caption{a) Horizontal coherence length calculated from the source properties of PETRA III and PETRA IV as a function of photon energy. b) Speckle contrast $\\beta_{cl}$ as a function of beam size calculated according to eq. \\ref{eq:betacl} (photon energies 8 keV (red), 15 keV (green) and 25 keV (blue)). The dashed line corresponds to the horizontal coherence length at P10 PETRA III, the solid line represents the horizontal coherence length expected with PETRA IV. The q-value is $q=0.9$ nm$^{-1}$ and the sample thicknesses are $d$=1.0, 6.5 and 23 mm corresponding to the absorption length of water at the respective photon energies.}\n\t\\label{fig:betacl}\n\\end{figure}\nWe observe a reduction of speckle contrast with increasing beamsize and a reduced contrast for smaller beamsizes as a function of photon energy. Both reductions can be explained by the scattering volume, defined by spot size $a$ and sample thickness $d$, exceeding the coherence volume defined by the longitudinal and transversal coherence length. \n\n\\subsection{Number of pixel}\nChanging the photon energy and sample detector distance has direct implications on the number of pixels which can be covered within an area of a certain $q$-range. The scattering signal may be in a circular region of interest on the detector of width $\\Delta q$ and radius $q$. In the SAXS regime, $q = (4 \\pi\/  \\lambda) \\theta $ and $\\Delta q = (4 \\pi\/  \\lambda) \\Delta \\theta$, and the diffraction ring has a width on the detector of $\\Delta \\theta \\cdot L$ and a circumference of $2\\pi (2\\theta) L$. The number of illuminated pixels is thus\n\\begin{equation}\nN_{pix} = \\frac{q \\Delta q \\lambda^2 L^2}{4 \\pi P^2}.\n\\end{equation}\n\n\\section{XPCS of protein solutions}\n\nHaving established the dependence of the SNR on the experimental parameters we can use the expression\n\\begin{equation}\nSNR = \\beta(a, \\lambda, L) \\cdot I_{pix}(\\lambda, L)  \\cdot \\sqrt{N_{fr}(a, \\lambda)\\cdot N_{pix}(\\lambda, L)}\n\\end{equation}\nto characterize the influence of the improved brilliance of the new generation of X-ray sources on XPCS experiments with radiation sensitive samples.\\\\\nIn Fig. \\ref{fig:StandXPCS}, we display the SNR for a standard XPCS setup. It was assumed that an EIGER 4M detector  \\cite{Johnson2014} is used, with a sample detector distance of $L=5$ m, which corresponds at a photon energy of $E=8$ keV to the inset of Fig. \\ref{fig:BioSAXS}. In order to match the speckle size to the pixel size, an X-ray spot size of $a=4$ $\\mu$m is required, corresponding to the calculations shown in Fig. \\ref {fig:Nfr}.\\\\\nFurther parameters are:\n\\begin{table}\n\\caption{Parameters fixed for the calculations of the SNR}\n\\begin{tabular}{ll}  \n$q $&$ 0.9 \\mathrm{nm}^{-1}$\\\\\n$\\Delta q $&$ 0.1 \\mathrm{nm}^{-1}$\\\\\n$C $&$ 250 \\mathrm{mg\/ml}$\\\\\n$P(q) \\cdot S(q) $&$ \\approx 0.3$\\\\\n$D_c $&$ 1,000 \\mathrm{J\/kg} = 1 \\mathrm{kGy}$\\\\\n$P $&$ 75 \\mathrm{\\mu m}$\\\\\n$t $&$ 1 \\mathrm{ms}$\\\\\n\\end{tabular}\n\\end{table}\n\n\n\\begin{figure}\n\\centering\n\t\t\\includegraphics[width=10cm]{SNR_standardXPCS.png}\n\t\\caption{a) Signal to Noise ratio (SNR) calculated as a function of photon energy for a setup with ($a= 4 \\mu$m, $L=2$ m) and for different undulators. Open symbols correspond to experimental conditions which are not accesible due to beam damage effects. b) Speckle contrast of this setup as a function of photon energy.}\n\t\\label{fig:StandXPCS}\n\\end{figure}\n\nThe red data points correspond to the photon beam properties of PETRA III, the green, blue and cyan points to the improved coherent flux $F_c$ offered by PETRA IV with different undulators. As can be seen, the increasing coherent flux offers theoretically improved SNR values of more than one order of magnitude. However, as marked with open symbols, the highest theoretically possible SNR of each configuration corresponds to experimental conditions where the critical dose limit of the sample is reached within two sequential acquisitions (i.e. $N_{fr} \\leq 2)$. Therefore, the maximum increase in SNR can not be reached in practice and the upgrade to PETRA IV would not lead to such a significant increase in SNR for this setup.\\\\\nData points which correspond to beam conditions where at least two sequential acquisitions are possible are displayed as filled symbols. It is evident that higher beam energies with also thicker samples would ease the effect of a higher flux and make XPCS experiments possible also with a standard configuration ($L=5$ m, $a=4 \\ \\mu$m). However, as displayed in Fig. \\ref{fig:StandXPCS} b), this also results in much reduced speckle contrasts and therefore the beneficial effect of an increased coherent flux on the SNR is largely lost due to the strongly reduced speckle contrast $\\beta$. \n\n\\subsection{Optimizing the experimental setup}\nIn order to use the increased coherent flux for XPCS experiments, one has to adapt the experimental setup in terms of focussing, photon energy and sample detector distance.\\\\\nTherefore, we repeat the previously presented calculations for a set of different beamsizes $a$ and sample-detector distances $L$. At each point in the $a-L$ plane, the SNR is calculated as a function of photon energy and the maximum is calculated. However, only values are considered which correspond to $N_{fr} \\geq 2$ at 1 ms exposure. The maximum SNR for each pair of $a$ and $L$ values is displayed in Fig. \\ref{fig:contour}.\\\\\nIt can be seen that the previously discussed setup with a small beam and large speckle (marked by a red dot) does not give the best SNR already for the case of PETRA III. With a sample-detector distance of $L=5.5$ m and an X-ray spot size of $a=9 \\ \\mu$m, the expected SNR increases by $25 \\%$.\\\\\n\\begin{figure}\n\\centering\n\t\t\\includegraphics[width=10cm]{contour.png}\n\t\\caption{The maximum value of SNR as a function of sample-detector distance $L$ and beamsize $a$ for DLSR. The highest SNRs are $SNR_{P3}=0.46$; $SNR_{P4}=1.7$; $SNR_{P4 U18}=2.7$; $SNR_{P4 U18-10m}=3.9$ }\n\t\\label{fig:contour}\n\\end{figure}\nHowever, in the case of PETRA IV (U18-10m), an overall increase in SNR by about one order of magnitude can be achieved, without exceeding the critical radiation dose of the sample. This setup would feature a sample detector distance of $L=26 \\ \\mathrm{m}$ and a spot size of $a=24 \\ \\mathrm{\\mu m}$ at $E=14.7$ keV.\\\\\n\n\\begin{table}\n\\caption{Parameters optimized setup for $N_{fr}=2$ using a Si(111) monochromator \\label{tab2}}\n\\begin{tabular}{lllll}  \nparamters &U29 (PIII)               & U29 & U18 5m & U18 10m \\\\\n\\hline\nsignal to noise ratio SNR           & $0.4$ &$1.7$& $2.6$ &$3.7$  \\\\\n\\hline \nbeam size $a$ \/ $\\mu$m              & $7.5$ &$13.3$ & $17.7$ &$23.7$    \\\\\n\\hline\nsample detector distance $L$ \/ m    & $3.8$ &$10.0$ & $14.7$ &$21.5$ \\\\\n\\hline \nbeam energy $E$ \/ keV               & $8.1$ &$12.2$& $13.6$ &$14.7$  \\\\\n\\hline\ncoherent Flux $F_c$ \/ ph\/s          & $2.1  \\ 10^{11}$ & $1.5  \\ 10^{12}$ &$3.4 \\ 10^{12}$ &$7.2 \\ 10^{12}$  \\\\\n\\hline\ncontrast $\\beta$                    & $0.20$ & $0.12$ & $0.10$ & $0.08$  \\\\\n\\hline\nspeckle size $S$  \/ $\\mu$m          & $78$& $76$ & $75$ &$76$  \\\\\n\\hline\nintensity per pixel $I_{pix}$ \/ ph\/ms& $2.3 \\ 10^{-3}$& $8.8.0 \\ 10^{-3}$ & $1.2 \\ 10^{-2}$ & $1.5 \\ 10^{-2}$ \\\\\n\\hline\nnumber of pixel in q-range $N_{pix}$ & 0.4 M& 1.3 M& 2.2 M& 4.2 M \\\\\n\\hline\nnumber of frames $N_{fr}$           &2 & 2& 2 & 2 \\\\\n\\hline\nsample thickness $d$ \/ mm           &$1.0$& $3.6$ & $4.9$ & $6.3$\\\\\n\\hline\nexposed sample volume \/ nL          & $0.05$& $0.6$&  $1.6$ & $3.5$ \\\\\n\\end{tabular}\n\\end{table}\n\nThe resulting parameter for the optimized experimental setups are summarized in Tab. \\ref{tab2} for each of the considered undulators. We note that for higher coherent flux setups the optimized setups feature an increase of  beam size $a$, sample detector distance $L$ and photon energy $E$. \\\\\n\nAs a general trend, it is evident that the sample volume, spanned by the sample thickness $d$ and spot size $a$ needs to be increased when the coherent flux increases. In order to compensate for the consequently decreasing angular speckle size, the sample detector distance needs to increase so that the speckle size can maintain its value of $S \\approx 75$ $\\mu$m. However, it can be seen that one can still observe a decrease in speckle contrast $\\beta$, even though the speckle have the same size on the detector for all four presented setups. This effect is due to the second contribution to the speckle contrast $\\beta_{cl}$, see Eq. \\ref{eq:betacl}, originating from the limited longitudinal coherence length of the X-ray beam.\n\n\\subsection{Si(311)-Mono}\nHere, we investigate how an additional increase of the longitudinal coherence length by using a Si(311) monochromator benefit the achievable SNR. We repeat the calculations with a reduced bandwidth of $3 \\cdot 10^{-5}$ and a reduced flux compared to the Si(111) calculations by $74 \\ \\%$. The resulting SNRs are displayed in Fig. \\ref{fig:contour_311} and Tab.' \\ref{tab3}. We find that the use of a Si(311) monochromator improves the SNR by an additional $30 \\%$ compared to the Si(111) thus leading to an overall SNR gain of a factor of 13 when comparing PETRA III with PETRA IV.\n\n\n\\begin{figure}\n\\centering\n\t\t\\includegraphics[width=10cm]{contour_311.png}\n\t\\caption{Best combination of sample-detector distance $L$ and beamsize $a$ for DLSR using an Si(311) monochromator.}\n\t\\label{fig:contour_311}\n\\end{figure}\n\n\n\\begin{table}\n\\caption{optimized setup using a Si(311) monochromator \\label{tab3}}\n\\begin{tabular}{lllll}  \nparameters &U29 (PIII)               & U29 & U18 5m & U18 10m \\\\\n\\hline\nsignal to noise ratio SNR           & $0.37$ &$1.8$& $3.0$ &$4.9$  \\\\\n\\hline \nbeam size $a$ \/ $\\mu$m              & $7.5$ &$10.0$ & $13.3$ &$17.8$    \\\\\n\\hline\nsample detector distance $L$ \/ m    & $5.6$ &$8.3$ & $10.0$ &$14.7$ \\\\\n\\hline \nbeam energy $E$ \/ keV               & $12.5$ &$14.1$& $12.5$ &$13.8$  \\\\\n\\hline\ncoherent Flux $F_c$ \/ ph\/s          & $1.5  \\ 10^{10}$ & $2.9  \\ 10^{11}$ &$1.1 \\ 10^{12}$ &$2.3 \\ 10^{12}$  \\\\\n\\hline\ncontrast $\\beta$                    & $0.23$ & $0.20$ & $0.23$ & $0.20$  \\\\\n\\hline\nspeckle size $S$  \/ $\\mu$m          & $74$& $73$ & $74$ &$74$  \\\\\n\\hline\nintensity per pixel $I_{pix}$ \/ ph\/ms& $3.0 \\ 10^{-3}$& $3.9 \\ 10^{-3}$ & $6.9 \\ 10^{-3}$ & $9.1 \\ 10^{-3}$ \\\\\n\\hline\nnumber of pixel in q-range $N_{pix}$ & 0.06 M& 0.7 M& 1.3 M& 2.2 M \\\\\n\\hline\nnumber of frames $N_{fr}$           &69 & 8& 3 & 3 \\\\\n\\hline\nsample thickness $d$ \/ mm           &$3.8$& $5.5$ & $3.8$ & $5.2$\\\\\n\\hline\nexposed sample volume \/ nL          & $0.21$& $0.55$&  $0.7$ & $1.6$ \\\\\n\\end{tabular}\n\\end{table}\n\n\n\n\n\\subsection{Multiple frame XPCS and two time correlation functions}\nIt becomes evident that with SNR values $\\propto 3-5$ XPCS from protein solutions is indeed possible at DLSRs with adapted experimental setups. As a direct consequence of the presented results, the optimized data acquisition scheme differs from conventional XPCS measurements. Instead of taking many hundreds to thousands of images at one spot, the scheme with maximum SNR for protein XPCS rather consists of \"double-shot\" exposures. This would not give a full correlation function from one spot on the sample, but rather one data point of $g_2$ for each illuminated sample spot. In consequence, the correlation function would be constructed from many of such double-shot exposures, which each can be done on a new sample spot and with a different delay time $\\tau$ between the two frames (see e.g. \\cite{Verwohlt2018}). The required sample volume therefore scales with the desired number of data points of $g_2$.\\\\\nHowever, this acquisition scheme is not suitable for samples displaying heterogeneous dynamics or aging effects. In such cases a movie-mode acquisition scheme with more than two frames per spot is needed. Fig. \\ref{fig:contour_tt} displays the resulting SNR values in the $a-L$ plane for $N_{fr}=2, 5, 25, 100$ for the case of PETRA IV U18 10m.\nWe find that with increasing number of images $N_{fr}$ the value of the maximum SNR decreases and its position in the $a-L$ plane shifts towards larger beamsizes $a$ and larger sample-detector distances $L$. For realizing the higher number of frames, an increase of the photon energy and of the beamsize is required (from Eq. 11 we find the scaling $a \\propto \\sqrt{N_{fr}}$). The resulting degradation of speckle contrast is partially counterbalanced by improving the angular resolution via a larger sample-detector distance. For example for $N=100$ frames the optimum SNR is 3.2 at $a=75 \\mu m$, $L=82$m and $E=15.4$ keV. Generally speaking we find  at the maximum of the SNR a scaling of $L\\propto a \\propto \\sqrt{N_{fr}B(E)}$.\n\n\\begin{figure}\n\\centering\n\t\t\\includegraphics[width=10cm]{contour_tt.png}\n\t\\caption{The maximum value of the SNR as a function of sample-detector distance $L$ and beamsize $a$ for PETRA IV (U18 10m), for $N_{fr}=2, 5, 25, 100$ number of frames taken on the same sample spot (Concentrated lysozyme solution, q=1 nm$^{-1}$ and critical dose of 1 kGy). The white color indicates $a-L$ combinations in which the required number of frames cannot be realized with the photon flux.}\n\t\\label{fig:contour_tt}\n\\end{figure}\n\nIn reality it might become difficult to realize a beamline with up to $100$ m sample-detector distance, which consequently would also require a detector with a very large number of pixels. However, it can be seen from Fig. 9 that also at shorter sample-detector distances $L$, the SNR is still significantly larger than 1 for up to $N_{fr}=100$. Therefore, we investigate how the SNR can be optimized, if the length of the beamline is limited to a fixed value of $L$ and on the same time a certain number of frames $N_{fr}$ is required to track the physics of the protein solution.\n\nWe demonstrate this by fixing the sample-detector distance at $L=30$ m and using a Si(311) monochromator. We plot both the SNR and the maximum number of frames possible as a function of beamsize $a$ (Fig. 10 left) for photon energies of $E=13.1$ keV (solid), $E=14.9$ keV (dashed) and $E=17$ keV (dash-dotted), respectively. Fig. 10 right displays the SNR as a function of $N_{fr}$ for the different photon energies. The benefit of using slightly higher photon energies than 13 keV is obvious as it allows to either increase the SNR value at fixed $N_{fr}$ or to record more images at fixed value of the SNR.\nWe find that with the source parameters of PETRA IV (U18 10m) the resulting values of the SNR are on the order of single digits. Specifically, we may take the example of $N_{fr}=100$ and find an SNR value of 2.5. Thus with 100 repeats (i.e. $N_{rep}=100)$ we could obtain an SNR of 25 of an averaged correlation function.  \n\n\n\n\\begin{figure}\n\\centering\n\t\t\\includegraphics[width=10cm]{multiframe.png}\n\t\\caption{Left: SNR (green lines) as a function of beamsize a for photon energies of 13.1, 14.9 and 17.0 keV (solid, dashed, dashed-dotted lines). Red lines indicates the maximum number of possible frames. Right: SNR as a function of the maximum number of frames displayed for the same photon energies. }\n\t\\label{fig:multiframe}\n\\end{figure}\n\n\\section{Conclusion}\nWe determined the signal to noise ratios (SNR) for XPCS experiments of a concentrated lysozyme solution at length scales of the hydrodynamic radius of a single protein molecule. The results show that the SNR values can at least increase up to one order of magnitude at future upgraded storage rings when compared to existing facilties. With this, the required measuring time would reduce by two orders of magnitude making dynamic studies of protein solutions at nanometer lengthscales feasible. However, in order to take full advantage of the properties of the future sources, XPCS experiments require adapted experimental setups with larger beamsizes and longer sample-detector distances than usually available at standard XPCS beamlines. \n\n\\section{Acknowledgements}\nThe authors would like to thank the PETRA IV project team, and especially C.\nSchroer, M. Tischer, and S. Klumpp for useful discussions and support. C.G. acknowledges funding by BMBF via project 05K19PS1.\n\n    \n    \n    \n\n\n    \n    \n    \n\n    \n\n\n    \n    \n\n\n\n\n\\printbibliography\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe recent success in deep learning is attributed to the availability of large labeled data~\\cite{krizhevsky2012imagenet,zhou2018brief} and the assumption of i.i.d. between training and test datasets. Such assumptions could be violated when test data features a drastic difference from the training data, e.g. training on synthetic images and test on real ones, and this is often referred to as domain shift~\\cite{quinonero2008dataset,ben2010theory}. To tackle this issue, domain adaptation~(DA)~\\cite{wang2018deep} emerges and the labeled training data and unlabeled testing data are often referred to as source and target data\/domains respectively.\n\nThe existing DA works either require the access to both source and target domain data during training~\\cite{ganin2015unsupervised} or training on multiple domains simultaneously~\\cite{zhou2021domain}. The former approach renders the methods restrictive to limited scenarios where source domain data is always available during adaptation while the latter ones are computationally more expensive. \nTo alleviate the reliance on source domain data, which may be inaccessible due to privacy issues or storage overhead, source-free domain adaptation (SFDA) emerges which handles DA on target data without access to source data~\\cite{pmlr-v119-liang20a,kundu2020universal,yang2021generalized,xia2021adaptive,liu2021ttt++}. SFDA is often achieved through self-training~\\cite{pmlr-v119-liang20a}, self-supervised learning~\\cite{liu2021ttt++} or introducing prior knowledge~\\cite{pmlr-v119-liang20a} and it requires multiple training epochs on the full target data to allow convergence. Despite easing the dependence on source data, SFDA has major drawbacks in a more realistic domain adaptation scenario where test data arrives in a stream and inference or prediction must be taken instantly, and this setting is often referred to as test-time training (TTT) or adaptation (TTA)~\\cite{sun2020test,wang2020tent,iwasawa2021test,liu2021ttt++}. Despite the attractive feature of adaption at time test, we notice a confusion of what defines a test-time training and as a result comparing apples and oranges happens frequently in the community. In this work, we first categorize TTT by two key factors after summarizing various definitions made in existing works. First, under a realistic TTT setting, test samples are sequentially streamed and prediction must be made instantly upon the arrival of a new test sample. More specifically, the prediction of test sample $X_T$, arriving at time stamp $T$, should not be affected by any subsequent samples, $\\{X_t\\}_{t=T+1\\cdots\\infty}$. Throughout this work, We refer to the sequential streaming as \\textbf{one-pass adaptation} protocol and any other protocols violating this assumption are called \\textbf{multi-pass adaptation} (model may be updated on all test data for multiple epochs before inference). Second, we  notice some recent works must \\textbf{modify source domain training loss}, e.g. by introducing additional self-supervised branch, to allow more effective TTT~\\cite{sun2020test,liu2021ttt++}. This will introduce additional overhead in the deployment of TTT because re-training on some source dataset, e.g. ImageNet, is computationally expensive.  \nIn this work, we aim to tackle on the most realistic and challenging TTT protocol, i.e. one-pass test time training with no modifications to training objective. This setting is similar to TTA proposed in~\\cite{wang2020tent} except for not restricting access to a light-weight information from the source domain. Given the objective of TTT being efficient  adaptation at time-time, this assumption is computationally efficient and improves TTT performance substantially. We name this new TTT protocol as \\textbf{sequential test time training (sTTT)}. \n\nWe propose two techniques to enable efficient and accurate sTTT. i) We are inspired by the recent progresses in unsupervised domain adaptation~\\cite{tang2020unsupervised} that encourages testing samples to form clusters in the feature space. However, separately learning to cluster in the target domain without regularization from source domain does not guarantee improved adaptation~\\cite{tang2020unsupervised}. \nTo overcome this challenge, we identify clusters in both the source and target domains through a mixture of Gaussians with each component Gaussian corresponding to one category. Provided with the category-wise statistics from source domain as anchors, we match the target domain clusters to the anchors by minimizing the KL-Divergence as the training objective for sTTT. Therefore, we name the proposed method  \\textit{test-time anchored clustering (TTAC)}. Since test samples are sequentially streamed, we develop an exponential moving averaging strategy to update the target domain cluster statistics to allow gradient-based optimization. ii) Each component Gaussian in the target domain is updated by the test sample features that are assigned to the corresponding category. Thus, incorrect assignments (pseudo labels) will harm the estimation of component Gaussian. To tackle this issue, we are inspired by the correlation between network's stability and confidence and pseudo label accuracy~\\cite{lee2013pseudo,sohn2020fixmatch}, and propose to filter out potential incorrect pseudo labels. Component Gaussians are then updated by the samples that have passed the filtering. To exploit the filtered out samples, we  incorporate a global feature alignment~\\cite{liu2021ttt++} objective. \nWe also demonstrate TTAC is compatible with existing TTT techniques, e.g. contrastive learning branch~\\cite{liu2021ttt++}, if source training loss is allowed to be modified. The contributions of this work are summarized as below.\n\n\n\n\n\\begin{itemize}\n\\item In light of the confusions within TTT works, we provide a categorization of TTT protocols by two key factors. Comparison of TTT methods is now fair within each category.\n\\item We adopt a realistic TTT setting, namely sTTT. To improve test-time feature learning, we propose TTAC by matching the statistics of the target clusters to the source ones. The target statistics are updated through moving averaging with filtered pseudo labels. \n\\item The proposed method is complementary to existing TTT method and is demonstrated on five TTT datasets, achieving the state-of-the-art performance under all categories of TTT protocols.\n\\end{itemize}\n\n\n\\section{Related Work}\n\n\\noindent\\textbf{Unsupervised Domain Adaptation}. \nDomain adaptation aims to improve model generalization when source and target data are not drawn i.i.d. When target data are unlabeled, UDA~\\cite{ganin2015unsupervised,tzeng2014deep} learns domain invariant feature representations on both source and target domains to improve generalization. Follow-up works improve DA by minimizing a divergence~\\cite{gretton2012kernel,sun2016deep,zellinger2017central}, adversarial training~\\cite{hoffman2018cycada} or discovering cluster structures in the target data~\\cite{tang2020unsupervised}. Apart from formulating DA within a task-specific model, re-weighting has been adopted for domain adaptation by selectively up-weighting conducive samples in the source domain~\\cite{jiang2007instance, yan2017mind}. Despite the efforts in UDA, it is inevitable to access the source domain data which may be not accessible due to privacy issues, storage overhead, etc. Deploying DA in more realistic scenarios has inspired research into source-free domain adaptation and test-time training\/adaptation.\n\n\\noindent\\textbf{Source-Free Domain Adaptation}. \nWithout the access to source data, source-free domain adaptation (SFDA) develops domain adaptation through self-training~\\cite{pmlr-v119-liang20a, kundu2020universal, iwasawa2021test}, self-supervised training~\\cite{liu2021ttt++} or clustering in the target domain~\\cite{yang2021generalized}. It has been demonstrated that SFDA performs well on seminal domain adaptation datasets even compared against UDA methods~\\cite{tang2020unsupervised}. Nevertheless, SFDA still requires access to all testing data and training must be carried out iteratively on the testing data. In a more realistic DA scenario where inference and adaptation must be implemented simultaneously, SFDA will no longer be effective. Moreover, some statistical information on the source domain does not pose privacy issues and can be exploited to further improve adaptation on target data.\n\n\\noindent\\textbf{Test-Time Training}. \nCollecting enough samples from target domain and adapt models in an offline manner restricts the application to adapting to static target domain. To allow fast and online adaptation, test-time training (TTT)~\\cite{sun2020test,wang2022continual} or adaptation (TTA)~\\cite{wang2020tent} emerges. Despite many recent works claiming to be test-time training, we notice a severe confusion over the definition of TTT. In particular, whether training objective must be modified~\\cite{sun2020test,liu2021ttt++} and whether sequential inference on target domain data is possible~\\cite{wang2020tent,iwasawa2021test}. Therefore, to reflect the key challenges in TTT, we define a setting called sequential test-time training (sTTT) which neither modifies the training objective nor violates  sequential inference. Under the more clear definition, some existing works, e.g. TTT~\\cite{sun2020test} and TTT++~\\cite{liu2021ttt++} is more likely to be categorized into SFDA. Several existing works~\\cite{wang2020tent,iwasawa2021test} can be adapted to the sTTT protocol. Tent~\\cite{wang2020tent} proposed to adjust affine parameters in the batchnorm layers to adapt to target domain data. The high TTA efficiency inevitably leads to limited performance gain on the target domain. T3A~\\cite{iwasawa2021test} further proposed to update classifier prototype through pseudo labeling. Despite being efficient, updating classifier prototype alone does not affect feature representation for the target domain. Target feature may not form clusters at all when the distribution mismatch between source and target is large enough. In this work we propose to simultaneously cluster on the target domain and match target clusters to source domain classes, namely anchored clustering. To further constrain feature update, we introduce additional global feature alignment and pseudo label filtering. Through the introduced anchored clustering, we achieve test-time training of more network parameters and achieve the state-of-the-art performance.\n\n\n\\section{Methodology}\n\nIn this section we first introduce the anchored clustering objective for test-time training through pseudo labeling and then describe an efficient iterative updating strategy. An overview of the proposed pipeline is illustrated in Fig.~\\ref{fig:overview}.\n\n\\begin{figure}[!htb]\n    \\centering\n    \\includegraphics[width=0.99\\linewidth]{.\/Figure\/Overview_v1.pdf}\n    \\caption{Overview of TTAC pipeline. i) In the source domain, we calculate category-wise and global statistics as anchors. ii) In the testing stage, samples are sequentially streamed and pushed into a fixed-length queue. Clusters in target domain are identified through anchored clustering with pseudo label filtering. Target clusters are then matched to the anchors in source domain to achieve test-time training.}\n   \n   \n    \\label{fig:overview}\n\\end{figure}\n\n\\subsection{Anchored Clustering for Test-Time Training}\n\n\nDiscovering cluster structures in the target domain has been demonstrated effective for unsupervised domain adaptation~\\cite{tang2020unsupervised} and we develop an anchored clustering on the test data alone\nWe first use a mixture of Gaussians to model the clusters in the target domain, here each component Gaussian represents one discovered cluster. \nWe further use the distributions of each category in the source domain as anchors for the target distribution to match against. In this way, test data features can simultaneously form clusters and the clusters are associated with source domain categories, resulting in improved generalization to target domain. Formally, we first write the mixture of Gaussians in the source and target domains $ p_s(x)=\\sum_k \\alpha_k \\mathcal{N}(\\mu_{sk},\\Sigma_{sk}),\\quad p_t(x)=\\sum_k \\beta_k \\mathcal{N}(\\mu_{tk},\\Sigma_{tk})$, where $\\{\\mu_k\\in\\mathbb{R}^d,\\Sigma_k\\in\\mathbb{R}^{d\\times d}\\}$ represent one cluster in the source\/target domain and $d$ is the dimension of feature embedding.\n\nAnchored clustering can be achieved by matching the above two distributions and one may directly minimize the KL-Divergence between the two distribution.\nNevertheless, this is non-trivial because the KL-Divergence between two mixture of Gaussians has no closed-form solution which prohibits efficient gradient-based optimization. Despite some approximations exist~\\cite{hershey2007approximating}, without knowing the semantic labels for each Gaussian component, even a good match between two mixture of Gaussians does not guarantee target clusters are aligned to the correct source ones and this will severely harm the performance of test-time training.\nIn light of these challenges, we propose a category-wise alignment.  Specifically, we allocate the same number of clusters in both source and target domains and each target cluster is assigned to one source cluster. We can then minimize the KL-Divergence between each pair of clusters as in Eq.~\\ref{eq:KLD}. \n\\vspace{-0.5cm}\n\n\\begin{equation}\\label{eq:KLD}\n\\begin{split}\n    \\mathcal{L}_{ac}&=\\sum_k D_{KL}(\\mathcal{N}(\\mu_{sk},\\Sigma_{sk})||\\mathcal{N}(\\mu_{tk},\\Sigma_{tk}))\\\\\n    &=\\sum_k -H(\\mathcal{N}(\\mu_{sk},\\Sigma_{sk})) + H(\\mathcal{N}(\\mu_{sk},\\Sigma_{sk}),\\mathcal{N}(\\mu_{tk},\\Sigma_{tk}))\n\\end{split}\n\\end{equation}\n\nThe KL-Divergence can be further decomposed into the entropy $H(\\mathcal{N}(\\mu_{sk},\\Sigma_{sk}))$ and cross-entropy $H(\\mathcal{N}(\\mu_{sk},\\Sigma_{sk}),\\mathcal{N}(\\mu_{tk},\\Sigma_{tk}))$. It is commonly true that the source reference distribution $P_s(x)$ is fixed thus the entropy term is a constant $C$ and only the cross-entropy term is to be optimized.\nGiven the closed-form solution to the KL-Divergence between two Gaussian distributions, we now write the anchored clustering objective as,\n\n\\begin{equation}\\label{eq:anchored_clustering_loss}\n    \\mathcal{L}_{ac}= \\sum_k \\{\\log \\sqrt{2\\pi^d|\\Sigma_{tk}|} + \\frac{1}{2}(\\mu_{tk}-\\mu_{sk})^\\top\\Sigma_{tk}^{-1}(\\mu_{tk}-\\mu_{sk}) + tr(\\Sigma_{tk}^{-1}\\Sigma_{sk})\\} + C\n\\end{equation}\n\nThe source cluster parameters can be estimated in an offline manner. These information will not cause any privacy leakage and only introduces a small computation and storage overheads. In the next section, we elaborate clustering in the target domain.\n\n\\subsection{Clustering through Pseudo Labeling}\n\nIn order to test-time train network with anchored clustering loss, one must obtain target cluster parameters $\\{\\mu_{tk},\\Sigma_{tk}\\}$. \nFor a minibatch of target test samples $\\set{B}^t=\\{x_i\\}_{i=1\\dots N_B}$ at timestamp $t$, we first denote the predicted posterior as $P^t=softmax(h(f(x_i)))\\in[0,1]^{B\\times K}$ where $softmax(\\cdot)$, $h(\\cdot)$ and $f(\\cdot)$ respectively denote a standard softmax function, the classifier head and backbone network. The pseudo labels are obtained via $\\hat{y}_i=\\arg\\max_k P_{ik}^t$. Given the predicted pseudo labels we could estimate the mean and covariance for each component Gaussian with the pseudo labeled testing samples.\nHowever, pseudo labels are always subject to model's discrimination ability. The error rate for pseudo labels is often high when the domain shift between source and target is large, directly updating the component Gaussian is subject to erroneous pseudo labels, a.k.a. confirmation bias~\\cite{arazo2020pseudo}. To reduce the impact of incorrect pseudo labels, we first adopt a light-weight temporal consistency (TC) pseudo label filtering approach. Compared to co-teaching~\\cite{han2018co} or meta-learning~\\cite{li2019learning} based methods, this light-weight method does not introduce additional computation overhead and is therefore more suitable for test-time training.\nSpecifically, to alleviate the impact from the noisy predictions, we calculate the temporal exponential moving averaging posteriors $\\widetilde{P}^t \\in [0,1]^{N \\times K}$ as below,\n\n\\vspace{-0.5cm}\n\n\\begin{equation}\n   \n    \\widetilde{P}^t_i = \n            (1 - \\xi) * \\widetilde{P}^{t-1}_i + \\xi * P^t_i\\quad,\\quad s.t.\\quad\\widetilde{P}^{0}_i=P^0_i\n\\end{equation}\n\nThe temporal consistency filtering is realized as in Eq.~\\ref{eq:tc_filter} where $\\tau_{TC}$ is a threshold determining the maximally allowed difference in the most probable prediction over time. If the posterior deviate from historical value too much, it will be excluded from target domain clustering.\n\\vspace{-0.2cm}\n\n\\begin{equation}\\label{eq:tc_filter}\n    F_i^{TC} = \\mathbbm{1}((P_{i\\hat{k}}^t - \\widetilde{P}^{t-1}_{i\\hat{k}}) > \\tau_{TC}),\\quad s.t. \\quad \\hat{k} = \\arg\\max_k(P_{ik}^t)\n\\end{equation}\n\n\nDue to the sequential inference, test samples without enough historical predictions may still pass the TC filtering.  So, we further introduce an additional pseudo label filter directly based on the posterior probability as,\n\\vspace{-0.5cm}\n\n\\begin{equation}\n    F_i^{PP}=\\mathbbm{1}(\\widetilde{P}^{t}_{i\\hat{k}}>\\tau_{PP})\n\\end{equation}\n\nBy filtering out potential incorrect pseudo labels, we update the component Gaussian only with the leftover target samples as below.\n\\vspace{-0.5cm}\n\n\\begin{equation}\n    \\mu_{tk} = \\frac{\\sum\\limits_{i} F^{TC}_iF^{PP}_i\\mathbbm{1}(\\hat{y}_i=k)f(x_i)}{\\sum\\limits_i F^{TC}_iF^{PP}_i\\mathbbm{1}(\\hat{y}_i=k)},\\quad\n    \\Sigma_{tk} = \\frac{\\sum\\limits_i F^{TC}_iF^{PP}_i\\mathbbm{1}(\\hat{y}_i=k) (f(x_i)-\\mu_{tk})^\\top(f(x_i)-\\mu_{tk})}{\\sum\\limits_i F^{TC}_iF^{PP}_i\\mathbbm{1}(\\hat{y}_i=k)}\n\\end{equation}\n\\vspace{-0.5cm}\n\n\\subsection{Global Feature Alignment}\n\nAs discussed above, test samples that do not pass the filtering will not contribute to the estimation of target clusters. Hence, anchored clustering may not reach its full potential without the filtered test samples. To exploit all available test samples, we propose to align global target data distribution to the source one. We define the global feature distribution of the source data as $\\hat{p}_s(x)=\\mathcal{N}(\\mu_s,\\Sigma_s)$ and the target data as $\\hat{p}_t(x)=\\mathcal{N}(\\mu_t,\\Sigma_t)$. To align two distributions, we again minimize the KL-Divergence as,\n\\vspace{-0.5cm}\n\n\\begin{equation}\\label{eq:global_loss}\n    \\mathcal{L}_{ga}=D_{KL}(\\hat{p}_s(x)||\\hat{p}_t(x))\n\\end{equation}\n\nSimilar idea has appeared in~\\cite{liu2021ttt++} which directly matches the moments between source and target domains~\\cite{zellinger2017central} by minimizing the F-norm for the mean and covariance, i.e. $||\\mu_t-\\mu_s||^2_2+||\\Sigma_t-\\Sigma_s||^2_F$. However, designed for matching complex distributions represented as drawn samples, central moment discrepancy~\\cite{zellinger2017central} requires summing infinite central moment discrepancies and the ratios between different order moments are hard to estimate.  For matching two parameterized Gaussian distributions KL-Divergence is more convenient with good explanation from a probabilistic point of view. Finally, we add a small constant to the diagonal of $\\Sigma$ for both source and target domains to increase the condition number for better numerical stability.\n\n\n\n\\subsection{Efficient Iterative Updating}\n\n\nDespite the distribution for source data can be trivially estimated from all available training data in a totally offline manner, estimating the distribution for target domain data is not equally trivial, in particular under the sTTT protocol.\nIn a related research~\\cite{liu2021ttt++}, a dynamic queue of test data features are preserved to dynamically estimate the statistics, which will introduce additional memory footprint~\\cite{liu2021ttt++}.\nTo alleviate the memory cost we propose to iteratively update the running statistics for Gaussian distribution. \nFormally, we define $t$-th test minibatch as $\\set{B}^t=\\{x_i\\}_{i=1\\cdots N_{B}}$. Denoting the running mean and covariance at step $t$ as $\\mu^t$ and $\\Sigma^t$, we present the rules to update the mean and covariance in Eq.~\\ref{eq:runningstatistics}. More detailed derivations and update rules for per cluster statistics are deferred to the Appendix. \n\n\\vspace{-0.5cm}\n\\begin{equation}\\label{eq:runningstatistics}\n\\begin{split}\n   \n   \n   \n   \n    %\n    & \\mu^t = \\mu^{t-1} + \\delta^t, \\quad\n    \\Sigma^t=\\Sigma^{t-1}+a^t{\\sum_{x_i\\in\\set{B}}\\{(f(x_i)-\\mu^{t-1})^\\top(f(x_i)-\\mu^{t-1})-\\Sigma^{t-1}\\}} - {\\delta^t}^\\top\\delta^t \\\\\n    & \\delta^t=a^t{\\sum\\limits_{x_i\\in\\set{B}}(f(x_{i}) - \\mu^{t-1})},\\quad \n    N^t = N^{t-1} + |\\set{B}^t|, \\quad \n    a^t = \\frac{1}{N^t}, \n\\end{split}\n\\end{equation}\n\n\n\nAdditionally, $N^t$ grows larger overtime. New test samples will have smaller contribution to the update of target domain statistics when $N^t$ is large enough. As a result, the gradient calculated from current minibatch will vanish. To alleviate this issue, we impose a clip on the value of $\\alpha^t$ as below. As such, the gradient can maintain a minimal scale even if $N^t$ is very large. \n\n\n\\vspace{-0.3cm}\n\n\\begin{equation}\n    a^t = \\left \\{\n        \\begin{array}{lcl}\n            \\frac{1}{N^t} & & N^t < N_{clip} \\\\\n            \\frac{1}{N_{clip}} & & others\n        \\end{array}\n        \\right.\n   \n   \n   \n   \n   \n   \n\\end{equation}\n\\vspace{-0.4cm}\n\n\n\n\n\n\n\n\n\n\n\\subsection{TTAC Training Algorithm}\nWe summarize the training algorithm for the TTAC in Algo.~\\ref{alg:main}. For effective clustering in target domain, we allocate a fixed length memory space, denoted as $\\set{C} \\in \\set{R}^{N_{C} \\times H \\times W \\times \\texttt{3}}$, to store the recent testing samples. In the sTTT protocol, we first make instant prediction on each testing sample, and only update the model when $N_B$ testing samples are accumulated. TTAC can be efficiently implemented, e.g. with two devices, one is for continuous inference and another is for model updating\n\n\\vspace{-0.4cm}\n\\begin{algorithm}\n\\caption{Test-Time Anchored Clustering Training Algorithm }\\label{alg:main}\n\\SetKwInOut{Input}{input}\n\\SetKwInOut{Return}{return}\n\\Input{A new testing sample batch $\\set{B}^t=\\{x_i\\}_{i=1\\dots N_B}$.}\n\n\\textcolor{gray}{\\# Update the testing sample queue $\\set{C}$.}\n\n$\\set{C}^t=\\set{C}^t\\setminus \\set{B}^{t-N_C\/N_B}$,\\quad\n$\\set{C}^t=\\set{C}^t\\bigcup \\set{B}^{t}$\n\n\\For{$ 1 $ \\KwTo $N_{itr}$}\n{\n    \\For{minibatch $\\{x^t_k\\}^N_{k=1}$ in  $\\set{C}^t$}\n    {\n       \n        \n       \n        \n        \\textcolor{gray}{\\# Obtain the predicted posterior and pseudo labels}\n        \n        $P^t_i=softmax(h(f(x^t_i)))$,\\quad\n        $\\hat{y}^t_i = \\arg\\max_k(P^t_{ik})$\n        \n        \\textcolor{gray}{\\# Calculate the global and per-cluster running mean and covariance by Eq.~\\ref{eq:runningstatistics}}\n        \n        $\\mu^t$,\\quad $\\Sigma^t$,\\quad $\\{\\mu_k^t\\}$,\\quad $\\{\\Sigma_k^t\\}$\n        \n       \n        \n       \n        \n       \n        \n       \n        \n        \\textcolor{gray}{\\# Optimize the combined loss by Eq.~\\ref{eq:anchored_clustering_loss} and Eq.~\\ref{eq:global_loss}}\n        \n        $\\mathcal{L}=\\mathcal{L}_{ac}+\\lambda\\mathcal{L}_{ga}$\n        \n       \n        \n        update network $f$ to minimize $\\mathcal{L}$\n    }\n}\n\n\n\\end{algorithm}\n\\vspace{-0.5cm}\n\n\n\n\n\n\n\n\n\\section{Experiment}\n\nIn this section, we first compare various existing methods based on the two key factors. Evaluation is then carried out on five test-time training datasets. We then ablate the components of TTAC. Further analysis on the cumulative performance, qualitative insights, etc. are provided at the end.\n\n\\vspace{-0.2cm}\n\\subsection{Datasets}\nWe evaluate on 5 test-time training datasets and report the classification error rate~($\\%$) throughout the experiment section. To evaluate the test-time training efficacy on corrupted target images, we use \\textbf{CIFAR10-C\/CIFAR100-C}~\\cite{hendrycks2018benchmarking}, each consisting of 10\/100 classes with 50,000 training samples of clean data and 10,000 corrupted test samples. \nWe further evaluate test-time training on hard target domain samples with \\textbf{CIFAR10.1}~\\cite{pmlr-v97-recht19a}, which contains around 2,000 difficult testing images sampled over years of research on the original CIFAR-10 dataset.\nTo demonstrate the ability to do test-time training for synthetic data to real data transfer we further use \\textbf{VisDA-C}~\\cite{VisDA}, which is a challenging large-scale synthetic-to-real object classification dataset, consisting of 12 classes, 152,397 synthetic training images and 55,388 real testing images. Finally, to evaluate test-time training on 3D point cloud data, we choose \\textbf{ModelNet40-C}~\\cite{ModelNet40-C}, which consists of 15 common and realistic corruptions of point cloud data, with 9,843 training samples and 2,468 test samples.\n\n\n\n\n\\vspace{-0.2cm}\n\\subsection{Experiment Settings}\n\n\n\n\\noindent\\textbf{Hyperparameters}. We use the ResNet-50~\\cite{he2016deep} for image datasets and the DGCNN~\\cite{wang2019dynamic} on ModelNet40-C. We optimize the backbone network $f(\\cdot)$ by SGD with momentum on all datasets. On CIFAR10-C\/CIFAR100-C and CIFAR10.1, we set BS = 256 and LR = 0.01, 0.0001, 0.01 respectively. On VisDA-C we set BS = 128 and LR = 0.0001, and on ModelNet40-C we set BS = 64 and LR = 0.001. More details of hyperparameters can be found in the Appendix. \n\n\\noindent\\textbf{Test-Time Training Protocols}.\nWe categorize test-time training based on two key factors. First, whether the training objective must be changed during training on the source domain, we use Y and N to indicate if training objective is allowed to be changed or not respectively. Second, whether testing data is sequentially streamed and predicted, we use O to indicate a sequential \\textbf{O}ne-pass inference and M to indicate non-sequential inference, a.k.a. \\textbf{M}ulti-pass inference. With the above criteria, we summarize 4 test-time training protocols, namely N-O, Y-O, N-M and Y-M, and the strength of the assumption increases from the first to the last protocols. \nOurs sTTT setting  makes the weakest assumption, i.e. N-O. Existing methods are categorized by the four TTT protocols, we notice that some methods can operate under multiple protocols\n\n\\noindent\\textbf{Competing Methods}.\nWe compare the following test-time training methods. Direct testing (\\textbf{TEST}) without adaptation simply do inference on target domain with source domain model.\nTest-time training (\\textbf{TTT-R})~\\cite{sun2020test} jointly trains the rotation-based self-supervised task and the classification task in the source domain, and then only train the rotation-based self-supervised task in the streaming test samples and make the predictions instantly. The default method is classified into the Y-M protocol.\nTest-time normalization (\\textbf{BN})~\\cite{ioffe2015batch} moving average updates the batch normalization statistics by streamed data. The default method follows N-M protocol and can be adapted to N-O protocol.\nTest-time entropy minimization (\\textbf{TENT})~\\cite{wang2020tent} updates the parameters of all batch normalization by minimizing the entropy of the model predictions in the streaming data. By default, TENT follows the N-O protocol and can be adapted to N-M protocol.\nTest-time classifier adjustment (\\textbf{T3A})~\\cite{iwasawa2021test} computes target prototype representation for each category using streamed data and make predictions with updated prototypes. T3A follows the N-O protocol by default.\nSource Hypothesis Transfer (\\textbf{SHOT})~\\cite{pmlr-v119-liang20a} freezes the linear classification head and trains the target-specific feature extraction module by exploiting balanced category assumption and self-supervised pseudo-labeling in the target domain. SHOT by default follows the N-M protocol and we adapt it to N-O protocol.\n\\textbf{TTT++}~\\cite{liu2021ttt++} aligns source domain feature distribution, whose statistics are calculated offline, and target domain feature distribution by minimizing the F-norm between the mean covariance. TTT++ follows the Y-M protocol and we adapt it to N-O (removing contrastive learning branch) and Y-O protocols. Finally, we present our own approach, \\textbf{TTAC}, which only requires a single pass on the target domain and does not have to modify the source training objective. We further modify TTAC for Y-O, N-M and Y-M protocols, for Y-O and Y-M we incorporate an additional contrastive learning branch~\\cite{liu2021ttt++}. We could further combine TTAC with additional diversity loss and entropy minimization loss introduced in SHOT~\\cite{pmlr-v119-liang20a}, denoted as TTAC+SHOT.\n\n\\vspace{-0.4cm}\n\n\\subsection{Test-Time Training on Corrupted Target Domain}\n\nWe present the test-time training results on CIFAR10C\/100C and ModelNet40C datasets in Tab.~\\ref{tab:categorization_table}. We make the following observations from the results.\n\n\\noindent\\textbf{sTTT (N-O) Protocol}. \nWe first analyze the results under the proposed sTTT (N-O) protocol. Our method outperforms all competing ones by a large margin, e.g. $3\\%$ improvement is observed on both CIFAR10-C and CIFAR100-C from the previous best (TTT++). We further combine TTAC with the class balance assumption made in SHOT (TTAC+SHOT). With the stronger assumptions out method can further improve upon TTAC alone, in particular on ModelNet40-C dataset. This result demonstrates TTAC's compatibility with existing methods.\n\n\\noindent\\textbf{Alternative Protocols}.\nWe further compare different methods under N-M, Y-O and Y-M protocols. Under the Y-O protocol, TTT++~\\cite{liu2021ttt++} modifies the source domain training objective by incorporating a contrastive learning branch~\\cite{chen2020simple}. To compare with TTT++, we also include the contrastive branch and observe a clear improvement on both CIFAR10-C and CIFAR100-C datasets. More TTT methods can be adapted to the N-M protocol which allows training on the whole target domain data multiple epochs. Specifically, we compared with BN, TENT and SHOT. With TTAC alone we observe substantial improvement on all three datasets and TTAC can be further combined with SHOT demonstrating additional improvement. Finally, under the Y-M protocol, we demonstrate very strong performance compared to TTT-R and TTT++. It is also worth noting that TTAC under the N-O protocol can already yield results close to TTT++ under the Y-M protocol, suggesting the strong test-time training ability of TTAC even under the most challenging TTT protocol.\n\n\n\n\n\n\n\n\\begin{table*}[htbp]\n    \\centering\n   \n   \\caption{Comparison under different TTT protocols. Y\/N indicates modifying source domain training objective or not. O\/M indicate one pass or multiple passes test-time training. C10-C, C100-C and MN40-C refer to CIFAR10-C, CIFAR100-C and ModelNet40-C datasets respectively. All numbers indicate error rate in percentage.}\n    \\resizebox{0.75\\linewidth}{!}{\n        \\begin{tabular}{l|cc|ccc}\n        \\toprule\n        Method  & TTT Protocol & Assum. Strength & C10-C & C100-C & MN40-C \\\\\n        \\midrule\n        TEST &  - & - & 29.15 & 60.34 & 34.62 \\\\\n        \\midrule\n        BN~\\cite{ioffe2015batch} &  N-O & Weak & 15.49 & 43.38 & 26.53 \\\\\n        TENT~\\cite{wang2020tent} &  N-O & Weak & 14.27 & 40.72 & 26.38 \\\\\n        T3A~\\cite{iwasawa2021test} &  N-O & Weak & 15.44 & 42.72 & 24.57 \\\\\n        SHOT~\\cite{pmlr-v119-liang20a} &  N-O & Weak & 13.95 & 39.10 & 19.71 \\\\\n        TTT++~\\cite{liu2021ttt++} &  N-O & Weak & 13.69 & 40.32 & - \\\\\n        TTAC~(Ours)  &  N-O & Weak & \\textbf{10.94} & 36.64 & 22.30 \\\\\n        TTAC+SHOT~(Ours) & N-O & Weak & 10.99  & \\textbf{36.39} & \\textbf{19.21} \\\\\n        \\midrule\n        TTT++~\\cite{liu2021ttt++} & Y-O & Medium & 13.00 & 35.23 & -  \\\\\n        TTAC~(Ours) & Y-O & Medium & \\textbf{10.69} & \\textbf{34.82} & - \\\\\n        \\midrule\n        BN~\\cite{ioffe2015batch} & N-M & Medium & 15.70 & 43.30 & 26.49 \\\\\n        TENT~\\cite{wang2020tent} & N-M & Medium & 12.60 & 36.30 & 21.23 \\\\\n        SHOT~\\cite{pmlr-v119-liang20a} & N-M & Medium & 14.70 & 38.10 & 15.99 \\\\\n       \n        TTAC~(Ours) & N-M & Medium & \\textbf{9.42} & 33.55 & 16.77 \\\\\n        TTAC+SHOT~(Ours) & N-M & Medium & 9.54 & \\textbf{32.89} & \\textbf{15.04} \\\\\n        \\midrule\n        TTT-R~\\cite{sun2020test} & Y-M & Strong & 14.30 & 40.40 & - \\\\ \n        TTT++~\\cite{liu2021ttt++} & Y-M & Strong & 9.80 & 34.10 & - \\\\\n        TTAC~(Ours) & Y-M & Strong & \\textbf{8.52} & \\textbf{30.57} & - \\\\\n        \\bottomrule\n        \\end{tabular}\n    }\n    \\label{tab:categorization_table}\n\\end{table*}\n\\vspace{-0.35cm}\n\n\\subsection{Additional Datasets}\n\n\\noindent\\textbf{TTT on Hard Samples}. CIFAR10.1 contains roughly 2,000 new test images that were re-sampled after the research on original CIFAR-10 dataset, which consists of some hard samples and reflects the normal domain shift in our life. The results in Table.~\\ref{tab:cifar101} demonstrate our method is better able to adapt to the normal domain shift. \n\n\n\\noindent\\textbf{TTT on Synthetic to Real Adaptation}. VisDA-C is a large-scale benchmark of synthetic-to-real object classification dataset. The setting of training on a synthetic dataset and testing on real data fits well with the real application scenario. On this dataset, we conduct experiments with our method under the N-O, Y-O and Y-M protocols and other methods under respective protocols, results are presented in Table.~\\ref{tab:visda}. We make the following observations. First, our method (TTAC Y-O)\noutperforms all methods except TTT++ under the Y-M protocol. This suggests TTAC is able to be deployed in the realistic test-time training protocol. Moreover, if training on the whole target data is allowed, TTAC (Y-M) further beats TTT++ by a large margin, suggesting the effectiveness of TTAC under a wide range of TTT protocols.\n\n\n\n\\begin{table}[htbp]\n \\begin{minipage}{0.47\\textwidth}\n    \\centering\n    \\caption{Test-time training on CIFAR10.1.}\n        \\resizebox{1\\linewidth}{!}{\n    \\begin{tabular}{ccccccc}\n    \\toprule\n    TEST & BN & TTT-R & TENT & SHOT & TTT++ & TTAC \\\\\n    \\midrule\n    12.1 & 14.1 & 11.0 & 13.4 & 11.1 & 9.5 & \\textbf{9.2} \\\\ \n    \\bottomrule\n    \\end{tabular}\n    }\n    \\label{tab:cifar101}\n          \\end{minipage}\n \\hfill\n        \\begin{minipage}{0.53\\textwidth}\n         \\centering\n  \\caption{Source-free sTTT on CIFAR10-C.}\n  \\resizebox{1\\linewidth}{!}{\n    \\begin{tabular}{ccccccc}\n    \\toprule\n    TEST  & BN & TENT  & T3A & SHOT & TTAC & TTAC+SHOT \\\\\n    \\midrule\n    29.15 & 15.49 & 14.27 & 15.44 & 13.95 & 13.74 & \\textbf{13.35} \\\\\n    \\bottomrule\n    \\end{tabular}%\n    }\n  \\label{tab:source_blind}%\n  \\end{minipage}\n\\end{table}\n\n\n\n\n\\vspace{-0.5cm}\n\n\\begin{table}[htbp]\n    \\centering\n    \\caption{Test-time training on VisDA. The numbers for competing methods are inherited from \\cite{liu2021ttt++}.}\n    \\resizebox{\\linewidth}{!}{\n        \\begin{tabular}{l|cccccccccccc|cc}\n        \\toprule\n        Method & Plane & Bcycl & Bus & Car & Horse & Knife & Mcycl & Person & Plant & Sktbrd & Train & Truck & Per-class\\\\\n        \\midrule\n        TEST & 56.52 & 88.71 & 62.77 & 30.56 & 81.88 & 99.03 & 17.53 & 95.85 & 51.66 & 77.86 & 20.44 & 99.51 & 65.19\\\\\n        BN (N-M)~\\cite{ioffe2015batch} & 44.38 & 56.98 & 33.24 & 55.28 & 37.45 & 66.60 & 16.55 & 59.02 & 43.55 & 60.72 & 31.07 & 82.98 & 48.99\\\\\n        TENT (N-M)~\\cite{wang2020tent} & 13.43 & 77.98 & 20.17 & 48.15 & 21.72 & 82.45 & 12.37 & 35.78 & 21.06 & 76.41 & 34.11 & 98.93 & 45.21\\\\\n        SHOT (N-M)~\\cite{pmlr-v119-liang20a} & 5.73 & \\textbf{13.64} & 23.33 & 42.69 & 7.93 & 86.99 & 19.17 & 19.97 & 11.63 & 11.09 & 15.06 & \\textbf{43.26} & 25.04 \\\\\n        TFA (N-M)~\\cite{liu2021ttt++} & 28.25 & 32.03 & 33.67 & 64.77 & 20.49 & 56.63 & 22.52 & 36.30 & 24.84 & 35.20 & 25.31 & 64.24 & 37.02 \\\\\n        TTT++ (Y-M)~\\cite{liu2021ttt++} & 4.13 & 26.20 & 21.60 & \\textbf{31.70} & 7.43 & 83.30 & 7.83 & 21.10 & 7.03 & \\textbf{7.73} & \\textbf{6.91} & 51.40 & 23.03\\\\\n        \\midrule\n       \n        TTAC~(N-O) & 18.54 & 40.20 & 35.84 & 63.11 & 23.83 & 39.61 & 15.51 & 41.35 & 22.97 & 46.56 & 25.24 & 67.81 & 36.71 \\\\\n        TTAC~(Y-O) & 7.19 & 29.99 & 22.52 & 56.58 & 8.14 & 18.41 & 8.25 & 22.28 & 10.18 & 23.98 & 13.55 & 67.02 & 24.01 \\\\\n        TTAC~(Y-M) & \\textbf{2.74} & 17.73 & \\textbf{18.91} & 43.12 & \\textbf{5.54} & \\textbf{12.24} & \\textbf{4.66} & \\textbf{15.90} & \\textbf{4.77} & 10.78 & 9.75 & 62.45 & \\textbf{17.38} \\\\\n        \\toprule\n        \\end{tabular}\n    }\n    \\label{tab:visda}\n\\end{table}\n\n\n\n\\vspace{-0.4cm}\n\\subsection{Ablation Study}\n\\vspace{-0.1cm}\n\nWe conduct ablation study on CIFAR10-C dataset for individual components, including anchored clustering, pseudo label filtering, global feature alignment and finally the compatibility with contrastive branch~\\cite{liu2021ttt++}. For anchored clustering alone, we use all testing samples to update cluster statistics. For pseudo label filtering alone, we implement as predicting pseudo labels followed by filtering, then pseudo labels are used for self-training. We make the following observations from Tab.~\\ref{tab:ablation}. Under both N-O and N-M protocols, introducing anchored clustering or pseudo label filtering alone improves over the baseline, e.g. under N-O $29.15\\%\\rightarrow 14.32\\%$ for anchored clustering and $29.15\\%\\rightarrow15.00\\%$ for pseudo label filtering. When anchored clustering is combined with pseudo label filtering, we observe a significant boost in performance. This is due to more accurate estimation of category-wise cluster in the target domain {and this reflects matching directly in the feature space \nmay be better than minimizing \ncross-entropy with pseudo labels}. We further evaluate aligning global features alone with KL-Divergence. This achieves relatively good performance and obviously outperforms the L2 distance alignment adopted in \\cite{liu2021ttt++}. Finally, we combine all three components and the full model yields the best performance. When contrast learning branch is included, TTAC achieves even better results.\n\n\n\n\n\n\n\n\n\\vspace{-0.5cm}\n\n\\begin{table}[htbp]\n  \\centering\n  \\caption{Ablation study for individual components on CIFAR10-C dataset.}\n    \\resizebox{1\\linewidth}{!}{\n    \\begin{tabular}{lcccccrrrrrrrr}\n    \\toprule\n    TTT Protocol & -     & \\multicolumn{5}{c}{N-O}               & \\multicolumn{1}{c}{Y-O} & \\multicolumn{5}{c}{N-M}               &  \\multicolumn{1}{c}{Y-M}\\\\\n    \\cmidrule(lr){1-1} \\cmidrule(lr){2-2} \\cmidrule(lr){3-7} \\cmidrule(lr){8-8} \\cmidrule(lr){9-13} \\cmidrule(lr){14-14} \n    Anchored Cluster. & -     & \\checkmark & -     & \\checkmark & -     & \\multicolumn{1}{c}{\\checkmark} & \\multicolumn{1}{c}{\\checkmark} & \\multicolumn{1}{c}{\\checkmark} & \\multicolumn{1}{c}{\\checkmark} & \\multicolumn{1}{c}{-} & \\multicolumn{1}{c}{-} & \\multicolumn{1}{c}{\\checkmark} & \\multicolumn{1}{c}{\\checkmark} \\\\\n    Pseudo Label Filter. & -     & -     & \\checkmark & \\checkmark & -     & \\multicolumn{1}{c}{\\checkmark} & \\multicolumn{1}{c}{\\checkmark} & \\multicolumn{1}{c}{-} & \\multicolumn{1}{c}{\\checkmark} & \\multicolumn{1}{c}{-} & \\multicolumn{1}{c}{-} & \\multicolumn{1}{c}{\\checkmark} & \\multicolumn{1}{c}{\\checkmark} \\\\\n    Global Feat. Align. & -     & -     & -     & -     & KLD   & \\multicolumn{1}{c}{KLD} & \\multicolumn{1}{c}{KLD} & \\multicolumn{1}{c}{-} & \\multicolumn{1}{c}{-} & \\multicolumn{1}{c}{L2 Dist.\\cite{liu2021ttt++}} & \\multicolumn{1}{c}{KLD} & \\multicolumn{1}{c}{KLD} & \\multicolumn{1}{c}{KLD} \\\\\n    Contrast. Branch~\\cite{liu2021ttt++} & -     & -     & -     & -     & -     & \\multicolumn{1}{c}{-} & \\multicolumn{1}{c}{\\checkmark} & \\multicolumn{1}{c}{-} & \\multicolumn{1}{c}{-} & \\multicolumn{1}{c}{-} & \\multicolumn{1}{c}{-} & \\multicolumn{1}{c}{-} & \\multicolumn{1}{c}{\\checkmark} \\\\\n    Avg Acc & 29.15 & 14.32 & 15.00 & 11.33 & 11.72 & 10.94 & 10.69 & 11.11 & 10.01 & \\multicolumn{1}{c}{11.87} & 10.8  & 9.42  & 8.52 \\\\\n    \\bottomrule\n    \\end{tabular}%\n    }\n  \\label{tab:ablation}%\n\\end{table}%\n\\vspace{-0.5cm}\n\n\n\n\n\n\n\n\n\n\n\\subsection{Additional Analysis}\n\n\\noindent\\textbf{Cumulative performance under sTTT}. We illustrate the cumulative error under the sTTT protocol in Fig.~\\ref{fig:add_study}~(a). For both datasets TTAC outperforms competing methods from the early stage of test-time training. The advantage is consistent throughout the TTT procedure.\n\n\n\\begin{figure}\n\\subfloat[Test-time cumulative error]{\\includegraphics[width=0.45\\linewidth]{.\/Figure\/OnePass4.pdf}}\n\\subfloat[TTT++ Feature]{\\includegraphics[width=0.25\\linewidth]{.\/Figure\/GlobalAlignmentTSNE.png}}\n\\subfloat[TTAC Feature]{\\includegraphics[width=0.25\\linewidth]{.\/Figure\/OursTSNE.png}}\n\n\\caption{(a) Comparison of test-time cumulative error under one-pass protocol. (b) T-SNE visualization of TTT++ feature embedding. (c) T-SNE visualization of TTAC feature embedding.}\\label{fig:add_study}\n\\vspace{-0.5cm}\n\n\\end{figure}\n\n\n\n\n\n\\noindent\\textbf{TSNE Visualization of TTAC features}.\nWe provide qualitative results for test-time training by visualizing the adapted features through T-SNE~\\cite{van2008visualizing}. In Fig.~\\ref{fig:add_study}~(b) and Fig.~\\ref{fig:add_study}~(c), we compared the features learned by TTT++~\\cite{liu2021ttt++} and TTAC~(Ours). We observe a better separation between classes by TTAC, implying an improved classification accuracy.\n\n\n\n\\noindent\\textbf{Source-Free Test-Time Training}. \nTTT aims to adapt model to target domain data by doing simultaneous training and sequential inference. It has been demonstrated some light-weight information, e.g. statistics, from source domain will greatly improve the efficacy of TTT. Nevertheless, under a more strict scenario where source domain information is strictly blind, TTAC can still exploit classifier prototypes to facilitate anchored clustering. Specifically, we normalize the category-wise weight vector with the norm of corresponding target domain cluster center as prototypes. Then, we build source domain mixture of Gaussians by taking prototypes as mean with a fixed covariance matrix. The results on CIFAR10-C are presented in Tab.~\\ref{tab:source_blind}. It is clear that even without any statistical information from source domain, TTAC still outperforms all competing methods.\n\n\n\n\\vspace{-0.2cm}\n\\section{Conclusion}\nTest-time training~(TTT) tackles the realistic challenges of deploying domain adaptation on-the-fly. In this work, we are first motivated by the confused evaluation protocols for TTT and propose two key criteria, namely modifying source training objective and sequential inference, to further categorize existing methods into four TTT protocols. Under the most realistic protocol, i.e. sequential test-time training (sTTT), we develop a test-time anchored clustering (TTAC) approach to align target domain features to the source ones. Unlike batchnorm and classifier prototype updates, anchored clustering allows all network parameters to be trainable, thus demonstrating stronger test-time training ability. We further propose pseudo label filtering and an iterative update method to improve anchored clustering and save memory footprint respectively. Experiments on five datasets verified the effectiveness of TTAC under sTTT as well as other TTT protocols.\n\n\n{\n\\small\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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n-d(d+1)\/2)}\n\\nc\\Fix{\\GM^T}\n\\nc\\Fixt{\\GMt^T}\n\\newcommand\\ic{\\operatorname{IC}}\n\\nc\\cm{\\operatorname{CM}}\n\\nc\\oCM{{^D{\\mathsf M}}}\n\\nc\\icm{\\ic(\\oCM^n)}\n\\nc\\tlam{{\\bar\\lam}}\n\\nc\\Sch{{\\mathsf{Sch}}}\n\\nc\\pg{{p\\ccirc\\gamma}}\n\\newcommand\\cd{\\!\\cdot\\!}\n\\nc\\ccirc{{{}_{\\,{}^{^\\circ}}}}\n\\newcommand{{\\overline{\\mathsf {Sch}}}{}}{{\\overline{\\mathsf {Sch}}}{}}\n\\newcommand{{\\mathsf {Gr}}}{{\\mathsf {Gr}}}\n\\newcommand{{\\mathsf {Mat}}}{{\\mathsf {Mat}}}\n\\newcommand{_{\\sf{reg}}}{_{\\sf{reg}}}\n\\newcommand{{^D{\\mathsf M}}}{{^D{\\mathsf M}}}\n\n\\newcommand{\\mathsf{R}_\\tau}{\\mathsf{R}_\\tau}\n\\newcommand{{}^A\\mathsf{M}_\\tau}{{}^A\\mathsf{M}_\\tau}\n\\newcommand{{}^A\\mathsf{R}_\\tau}{{}^A\\mathsf{R}_\\tau}\n\n\\newcommand{\\mathop{\\mathrm{alt}}}{\\mathop{\\mathrm{alt}}}\n\n\n\\setcounter{tocdepth}{1}\n\\begin{document}\n\\title{Intersection cohomology of the Uhlenbeck compactification \nof the Calogero--Moser space}\n\\author{Michael Finkelberg, Victor Ginzburg, Andrei Ionov and Alexander\nKuznetsov}\n\n\\dedicatory{To Joseph Bernstein on his 70th birthday,\nwith gratitude and admiration}\n\n\n\n\\begin{abstract}\nWe study the natural Gieseker and Uhlenbeck compactifications of the rational\nCalogero--Moser phase space. \nThe Gieseker compactification is smooth and provides a small\nresolution of the Uhlenbeck compactification. We use the resolution\nto  compute\nthe stalks of the IC-sheaf of the Uhlenbeck compactification.\n\\end{abstract}\n\\maketitle\n\n\\begin{flushright}\n{\\small{\\textit{I'd\nsay that if one can compute the Poincar\\'e polynomial\\\\\n for  intersection \ncohomology  without a computer then,\\\\ probably, there is a small\nresolution which gives it.}}\\\\\n(J. Bernstein)\\hskip 20mm\\hphantom{x}}\n\\end{flushright}\n\n\n\\sec{int}{Introduction}\n\n\\ssec{calmo}{The Calogero--Moser space}\n\nThe {\\sf Calogero--Moser space} $\\sM^n$~\\cite{kks} is the quotient modulo a free action\nof $\\PGL_n$ of the space of pairs of complex $n\\times n$-matrices $(X,Y)$ such that\n$[X,Y]-\\on{Id}$ has rank 1.\nThe Calogero--Moser space is a smooth connected affine algebraic variety of\ndimension $2n$~\\cite{W}.\n\n\n\n\n\\ssec{uhgi}{The Gieseker and Uhlenbeck compactifications}\nMore generally, for a parameter $\\tau\\in{\\mathbb C}^\\times$,\n we consider a\ngraded algebra $A^\\tau$ with generators\n$x,y,z$, of degree 1, and the following commutation relations \n\\begin{equation}\\label{A-relations}\n[x,z]=[y,z]=0,\\quad [x,y]=\\tau z^2.\n\\end {equation}\nThis algebra is a very special case of the Sklyanin algebras\nstudied in~\\cite{ns}, specifically, it corresponds to the case \nof a degenerate plane cubic curve equal\nto a triple line.\nWe set ${\\mathbb P}^2_\\tau=\\mathsf{Proj}(A^\\tau)$, a non-commutative $\\mathsf{Proj}$\nin the sense of~\\cite{AZ}, see also~\\cite{KKO}, and\nwrite $\\mathsf{coh}({\\mathbb P}^2_\\tau)=\\mathsf{qgr}(A^\\tau)$ for the corresponding\nabelian category $\\mathsf{coh}(\\BPt)$ of ``coherent\nsheaves''. Associated with an object\n$E\\in \\mathsf{coh}(\\BPt)$ there is a well defined triple\n$(r=\\rk E,\\,d=\\deg E,\\,n=c_2(E))$,   of nonnegative integers,\nthe rank, the degree, and the\nsecond Chern class of $E$, respectively.\n\nGiven a triple $(r,d,n)$, where $r$ and $d$ are coprime,\nwe introduce  two different  moduli spaces,  $\\GMt(r,d,n)$ and\n$\\UMt(r,d,n)$, of coherent sheaves on\n${\\mathbb P}^2_\\tau$. These moduli spaces are\ndefined by  stability\nconditions. The moduli space\n$\\GMt(r,d,n)$,\n the {\\em Gieseker} moduli space, is defined using   Gieseker stability.\n The moduli space $\\UMt(r,d,n)$,  the {\\em Uhlenbeck}  moduli space,\n is defined using  Mumford stability.\nThese moduli spaces are projective varieties\nwhich provide two different compactifications of the\nmoduli space of locally free sheaves.\nThe variety $\\GMt(r,d,n)$ is a particular\ncase of  moduli spaces studied in~\\cite{ns} (cf. also~\\cite{NV})\nin greater generality.\nThe  variety  $\\UMt(r,d,n)$ is more mysterious; it does not\nfit into the framework of ~\\cite{ns} and it has not been considered\nthere. In fact,  even in the commutative case,\na satisfactory construction of the  Uhlenbeck compactification of \nthe moduli space of  locally free sheaves on an arbitrary smooth \nsurface\nis not known so far, cf. \\cite{BFG}.\nIn the case we are interested in, i.e. in the case of the noncommutative \nsurface $\\BPt$, the variety  $\\UMt(r,d,n)$ will be studied in  section\n2. In particular, using an interpretation of our moduli spaces\nin terms of certain moduli spaces of quiver representations,\nwe construct a projective morphism \n$\\gamma_\\tau:\\ \\GMt(r,d,n)\\to \\UMt(r,d,n)$.\nThis morphism turns out to be  a resolution of singularities,\nprovided  $r$ and $d$ are coprime. \n\nIn  this paper, we will mostly be interested \nin the case where   $r=1,\\ d=0$, and  $\\tau\\neq 0$.\nThe moduli space of locally free sheaves $\\CE$ on $\\BPt$ such that\n$\\rk\\CE=1,\\ \\deg\\CE=0$, and $c_2(\\CE)=n$ has an ADHM type description.\nSpecifically, according   to~\\cite{NS} and \\cite{KKO}, this moduli space is\nisomorphic to the variety $\\sM^n_\\tau$ defined as\na  quotient  of the space of pairs  $(X,Y)$,\nof $n\\times n$-matrices such that\n$\\rk([X,Y]-\\tau\\on{Id})=1$, by  the (free) action\nof the group $\\PGL_n$  by conjugation.\n Note that the rescaling map $(X,Y)\\mapsto(\\frac{1}{\\tau} X,Y)$\ngives  a canonical isomorphism of $\\sM^n_\\tau$\nwith the Calogero--Moser space\n$\\sM^n$. Therefore, the varieties $\\GM^n_\\tau=\\GMt(1,0,n)$ and\n$\\UM^n_\\tau=\\UMt(1,0,n)$ provide two different  compactifications of the\nCalogero--Moser space. Since $1$ and $0$ are coprime,\n the corresponding  morphism $\\gamma_\\tau:\\ \\GM^n_\\tau\\to\\UM^n_\\tau$ is\n a resolution of singularities.\nMoreover, we show that this morphism is {\\em small} in the sense\nof Goresky--MacPherson. \n\n\nOne can allow the parameter $\\tau$ to vary in ${\\mathbb A}^1$.\nSimilarly to the above, one \nconstructs the family of Gieseker, rwesp. Uhlenbeck, compactifications\n$\\GM^n$,\nresp. $\\UM^n$,\nequipped with maps to $\\AA^1$ such that the fibers over the point $\\tau \\in \\AA^1 \\setminus \\{0\\}$\nare $\\GM^n_\\tau$, resop. $\\UM^n_\\tau$. Furthermore, we construct\na small resolution of singularities $\\gamma:\\ \\GM^n \\to \\UM^n$. In fact, over $\\AA^1 \\setminus \\{0\\}$\nthe maps $\\gamma_\\tau:\\ \\GM^n_\\tau\\to\\UM^n_\\tau$ are identified with the maps discussed above, \nwhile the fiber over $\\tau=0$ is\nthe Hilbert--Chow morphism \n$\\gamma_0:\\ \\on{Hilb}^n{\\mathbb P}^2\\to S^n{\\mathbb P}^2=({\\mathbb P}^2)^n\/\\fS_n$.\n\nIt is well known that $\\gamma_0$ is only {\\em semismall}. The reason of this\ndifference between $\\gamma_0$ and $\\gamma_\\tau,\\ \\tau\\ne0$, is due to the\ndifference between  the stratifications of the commutative and noncommutative\nUhlenbeck compactifications, respectively. Namely, we have a distinguished (classical,\ncommutative) projective line subscheme ${\\mathbb P}^1\\subset{\\mathbb P}^2_\\tau$, and\nsimilarly,\nwe have ${\\mathbb P}^1\\subset{\\mathbb P}^2$ such that ${\\mathbb P}^2\\setminus{\\mathbb P}^1={\\mathbb A}^2$.\nThere is a stratification\n\\begin{equation*}\n\\UM^n_\\tau=\\bigsqcup_{0\\leq m\\leq n}\\sM^m_\\tau\\times S^{n-m}{\\mathbb P}^1,\\qquad\\text{for $\\tau\\ne0$},\n\\end{equation*}\nand $\\gamma_\\tau$ is an isomorphism over the open part $\\sM^n_\\tau$.\nSimilarly, we have a stratification\n\\begin{equation*}\n\\UM^n_0=S^n{\\mathbb P}^2=\\bigsqcup_{0\\leq m\\leq n}S^m{\\mathbb A}^2\\times S^{n-m}{\\mathbb P}^1,\n\\end{equation*}\nbut $\\gamma_0$ is {\\em not} an isomorphism over $S^n{\\mathbb A}^2$, only a semismall\nresolution of singularities.\n\nLet us remark that the readers experienced with the classical Uhlenbeck\ncompactifications might expect another stratification with strata of $\\UM^n_\\tau$ being\n$\\sM^m_\\tau\\times S^{n-m}{\\mathbb P}^2_\\tau$\n(the reason of semismallness of the classical Gieseker resolution); however ${\\mathbb P}^2_\\tau$\nis {\\em not} a classical scheme, so only its ``classical part'' ${\\mathbb P}^1$\nsurvives in the classical moduli scheme $\\UM^n_\\tau$, yielding the\nstratification of the previous paragraph.\n\nNote also that the Gieseker moduli spaces of~\\cite{ns} carry a natural\nPoisson structure. We expect it to descend to the Uhlenbeck compactification.\nHowever, even normality of $\\UM^n_\\tau$ seems to be a hard question and it\nis out of the scope of this paper.\n\n\n\\newcommand{\\operatorname{\\mathbf{h}}}{\\operatorname{\\mathbf{h}}}\n\n\\ssec{gormac}{The main Theorem}\n\n\nLet $\\frP(n)$ denote the set of partitions of an integer $n\\geq0$ and for an algebraic variety $T$ and a partition $\\lambda = (\\lambda_1,\\dots,\\lambda_l)$ put\n\\begin{equation*}\\label{diagonal-stratification}\nS_\\lambda T = \\{ \\textstyle\\sum \\lambda_i P_i\\ |\\ P_1 \\ne P_2 \\ne \\dots \\ne P_l \\in T \\} \\subset S^nT=T^n\/\\fS_n.\n\\end{equation*}\nso that \n$S^n T = \\bigsqcup_{\\lambda \\in \\frP(n)} S_\\lambda T$\nis a stratification, which we call the {\\sf diagonal stratification}. \nLet $\\ic(\\UM^n_\\tau)$ be the IC sheaf (see~\\cite{bbd}) \nof the Uhlenbeck compactification.\nOur main result is the computation of the stalks of the IC sheaf.\n\n\n\n\\th{main}\nThe IC sheaf of the Uhlenbeck compactification is\nsmooth along the stratifications\n\\begin{equation*}\n\\UM^n_\\tau = \\bigsqcup_{\\substack{0 \\le m \\le n \\\\[.5ex] \\lambda \\in \\frP(n-m)}}  \\left(\\sM^m \\times S_\\lambda\\PP^1\\right).\n\\end{equation*}\nFor $0 \\le m \\le n$ and $\\lambda = (\\lambda_1,\\dots,\\lambda_k) \\in \\frP(n-m)$,\nthe stalk of the sheaf $\\ic(\\UM^n_\\tau)$ at a point of a stratum \n$\\sM^m\\times S_\\lambda{\\mathbb P}^1$ is isomorphic to\n\\begin{equation}\\label{heis}\n\\bigotimes_{i=1}^k \\left( \\bigoplus_{\\mu \\in \\frP(\\lambda_i)} {\\mathbb C}[2l(\\mu)] \\right)[2m]\n\\end{equation}\nas a graded vector space.\n\\eth\n\nThe proof employs the small resolution of the family\n$\\gamma:\\ \\GM^n_\\tau\\to\\UM^n_\\tau$ and reduces the study of the fibers\nfor $\\tau\\ne0$ to the well known properties of the fibers of the Hilbert--Chow\nmorphism for $\\tau=0$. \n\n\\begin{Rem} Given a complex semisimple simply connected group $G$ one can \nconsider the moduli\nspace of $G$-bundles on ${\\mathbb P}^2$ equipped with a trivialization\nat the infinite line ${\\mathbb P}^1\\sset{\\mathbb P}^2$.\nThere is also an Uhlenbeck space  ${\\mathcal U}_G({\\mathbb A}^2)$ that contains\nthe above moduli space as a Zariski open subset\n(the variety  ${\\mathcal U}_G({\\mathbb A}^2)$ is not proper in this setting).\nAssume that the group $G$ is  almost simple,\nand let $G_{\\on{aff}}$ be the affinization  of $G$, a Kac-Moody group\nsuch that ${\\mathfrak g} _{\\on{aff}}=\\textrm{Lie}(G_{\\on{aff}})$ is an affine Lie algebra.\nThen, the  Uhlenbeck space  ${\\mathcal U}_G({\\mathbb A}^2)$ may be viewed\nas  a slice in the affine Grassmannian for the group  $G_{\\on{aff}}$,\nthat is, in the  double affine Grassmannian for $G$~\\cite{bf}.\nThe IC stalks of ${\\mathcal U}_G({\\mathbb A}^2)$  may be identified, in\naccordance with the predictions based on the geometric Satake\ncorrespondence, with certain graded versions\nof the weight spaces of the basic integrable representation of\n${\\mathfrak g}_{\\on{aff}}^\\vee$, the Langlands dual of the Lie algebra ${\\mathfrak g} _{\\on{aff}}$.\nIn the simply-laced case, the Dynkin diagram of the  Lie algebra\n ${\\mathfrak g} _{\\on{aff}}$ is an  affine  Dynkin diagram of types\n $\\widetilde A,\\,\\widetilde D,\\,\\widetilde E$,\nand we have ${\\mathfrak g}_{\\on{aff}}^\\vee={\\mathfrak g}_{\\on{aff}}$.\n\n\nIt is often useful to view the graph with one vertex and one edge-loop\nat that vertex as a  Dynkin diagram  of type $\\widetilde A_0$.\nIt is known that the   Kac-Moody  Lie algebra associated with \n $\\widetilde A_0$ is  the Heisenberg Lie algebra  ${\\mathfrak H}$.\nBy definition, we have ${\\mathfrak H} := \\C\\delta\\ltimes\n\\widehat{\\C(\\!(t)\\!)}$,\nwhere $\\widehat{\\C(\\!(t)\\!)}$ is a central extension of the abelian  Lie algebra\n$\\C(\\!(t)\\!)$  and $\\delta:=t\\frac{d}{dt}$, a derivation.\nThe Fock representation of ${\\mathfrak H}$ plays the role of the basic\nintegrable representation of an affine Lie algebra.\nThe tensor factors of the graded vector space in \\eqref{heis}\nmay be identified in a natural way with \ncertain weight spaces of the Fock space (the action of the derivation\n$\\delta$ \ngives a grading\non the  Fock space).\nThis suggests, in view of the above, that\nour variety\n $\\UM^n_\\tau$ might play the role of a slice in\nsome kind of an affine Grassmannian for the Heisenberg group\nand Theorem \\ref{T:main} is\na manifestation of (a certain analogue  of) the geometric Satake\ncorrespondence in the case of   Dynkin diagram  of type $\\widetilde A_0$.\n\\end{Rem} \n\n\n\n\n\\ssec{orga}{Organization of the paper}\nIn~\\refs{ch5} we study coherent sheaves on a noncommutative projective plane\nand the corresponding representations of a Kronecker-type quiver.\nWe introduce Gieseker and Mumford stabilities of sheaves and interpret them\nas stabilities of quiver representations. We construct the Gieseker and\nthe Uhlenbeck moduli spaces of sheaves as GIT moduli spaces of quiver\nrepresentations and a map $\\gamma$ between the moduli spaces as the map\ncoming from a variation of GIT quotients.\nIn~\\refs{ch6} we discuss the special case of sheaves of rank 1 and degree 0.\nIn this case the Gieseker and the Uhlenbeck moduli spaces are compactifications\nof the Calogero--Moser space. We investigate in detail the map $\\gamma$ between\nthe compactifications and compute the stalks of the IC sheaf on the Uhlenbeck compactification.\nIn the Appendix we provide proofs of some of the results of~\\refs{ch5}.\n\n\\medskip\n\n\\noindent\n{\\textbf{Notation.}}\\en Given  a vector space $V$, we write $V^\\vee$ for\nthe dual  vector space and $S^\\bullet V=\\oplus_{i\\geq 0}\\ S^i V$  for the\nSymmetric algebra of $V$.\n\n\\ssec{ackno}{Acknowledgments}\nThe study of M.F. and A.K. has been funded by the Russian Academic Excellence\nProject `5-100'. \nV.G.  was supported in part by the NSF grant DMS-1303462.\nA.I. was supported by the grants NSh-5138.2014.1 and RFBR 15-01-09242.\nA.K. was partially supported by RFBR 14-01-00416, 15-01-02164, 15-51-50045 and NSh-2998.2014.1.\n\n\n\n\n\n\n\n\n\n\\newcommand{{\\widetilde{H}}}{{\\widetilde{H}}}\n\n\n\n\\sec{ch5}{Sheaves on the noncommutative plane $\\BPt$ and quiver representations}\n\nTo construct the Gieseker and the Uhlenbeck compactifications of the \nCalogero-Moser space we use an interpretation of the latter as moduli spaces \nof coherent sheaves on a noncommutative projective plane.\n\n\n\n\\ssec{reco}{Sheaves on the noncommutative projective plane}\n\n\\newcommand{\\mathfrak{a}}{\\mathfrak{a}}\n\nWe start with a slightly more invariant definition of the Calogero--Moser space.\nWe consider a symplectic vector space $H$ of dimension 2 with a symplectic form $\\omega \\in \\Lambda^2H^\\vee$,\na vector space $V$ of dimension $n$, a nonzero complex number $\\tau \\in \\C^\\times$,\nand consider the subvariety\n$\\widetilde\\sM_\\tau(V) \\subset \\Hom(V,V\\otimes H)$\ndefined by\n\\begin{equation*}\n\\widetilde\\sM_\\tau(V) = \\{ \\mathfrak{a} \\in \\Hom(V, V\\otimes H)\\ |\\ \\mathrm{rank}(\\omega( \\mathfrak{a} \\circ \\mathfrak{a}) - \\tau\\id_V) = 1 \\},\n\\end{equation*}\nwhere $\\omega(\\mathfrak{a}\\circ \\mathfrak{a})$ is defined as the composition\n$V \\xrightarrow{\\ \\mathfrak{a}\\ } V \\otimes H \\xrightarrow{\\ \\mathfrak{a} \\otimes \\id_H\\ } V \\otimes H \\otimes H \\xrightarrow{\\ \\id_V \\otimes \\omega\\ } V$,\nconsider the action of $\\PGL(V)$ on $\\widetilde\\sM_\\tau(V)$ by conjugation,\nand define\n\\begin{equation*}\n\\sM_\\tau(V) = \\widetilde\\sM_\\tau(V)\/\\PGL(V).\n\\end{equation*}\nA choice of a symplectic basis in $H$ allows to rewrite $\\mathfrak{a}$ as a pair of operators $(X,Y)$,\nthen $\\omega(\\mathfrak{a}\\circ \\mathfrak{a})$ becomes as $[X,Y]$, and so this definition\nagrees with the standard one. \n\n\nDenote ${\\widetilde{H}} := H \\oplus {\\mathbb C}$. We define a {\\sf twisted symmetric algebra of ${\\widetilde{H}}$} by\n\\begin{equation*}\nA^\\tau =\nS^\\bullet_\\tau{\\widetilde{H}} =\n{\\mathbb C}\\langle H \\oplus {\\mathbb C} z\\rangle\/\\langle [H,z]=0,\\ [h_1,h_2]=\\tau \\omega(h_1,h_2) z^2 \\rangle.\n\\end{equation*}\nChoosing a symplectic basis $x,y$ in $H$ the defining relations in\n$A^\\tau$\ntake the form \\eqref{A-relations}.\n\n\nThe algebra $A^\\tau$ is a  graded noetherian  algebra \nand we let\n\\begin{equation*}\n{\\mathbb P}^2_\\tau := \\mathsf{Proj}(A^\\tau)\n\\end{equation*}\nbe the non-commutative ``projective spectrum'' of \n$A^\\tau$ in the sense of~\\cite{AZ}. \nThe category of ``coherent sheaves'' on the non-commutative scheme\n${\\mathbb P}^2_\\tau$ is defined as\n$\\mathsf{coh}({\\mathbb P}^2_\\tau):=\\mathsf{qgr}(A^\\tau)$,  a quotient of the\nabelian  category\nof finitely generated graded $A^\\tau$-modules by the \nSerre subcategory of finite-dimensional modules.\nNote that the group $\\SL(H)$ acts on the algebra $A^\\tau$ by automorphisms.\nThe action on $A^\\tau$ induces an $\\SL(H)$-action \non the category of coherent sheaves $\\mathsf{coh}(\\BPt)$.\n\nAs it was shown in~\\cite{AZ,KKO,BGK,NS} and other papers, coherent sheaves\non such a noncommutative projective plane behave very similarly to those\non the usual (commutative) plane~${\\mathbb P}^2$. For instance, one can define\nthe cohomology spaces of sheaves, local $\\underline{\\Ext}$ sheaves,\nthe notions of torsion free and locally free sheaves, one has the sequence\n$\\{\\CO(i)\\}_{i\\in\\ZZ}$ of ``line bundles'', one can prove Serre duality and\nconstruct the Beilinson spectral sequence.\n\nThe main differences from the commutative case are\n\\begin{itemize}\n\\item\nin general there is {\\em no tensor product of sheaves} (due to the noncommutativity);\nhowever, one can tensor with $\\CO(i)$ and thus define the twist functors $F \\mapsto F(i)$\nsince sheaves $\\CO(i)$ correspond to graded $A^\\tau$-modules having a natural bimodule structure\n(alternatively, the twist functor can be thought of as the twist of the grading functor\nin the category of graded $A^\\tau$-modules);\n\\item\nthe dual of a sheaf on $\\PP^2_\\tau$ is a sheaf on\n$\\mathsf{Proj}((A^\\tau)^{\\mathrm{opp}})$,\nthe ``opposite'' noncommutative projective plane;\nin fact, one has $\\mathsf{Proj}((A^\\tau)^{\\mathrm{opp}})=\\PP^2_{-\\tau}$ since $(A^\\tau)^{\\mathrm{opp}} \\cong A^{-\\tau}$.\n\n\\item\nthe noncommutative projective plane ${\\mathbb P}^2_\\tau$ {\\em has less points} than the usual plane ${\\mathbb P}^2$,\nand as a consequence the category $\\mathsf{coh}({\\mathbb P}^2_\\tau)$ {\\em has more locally free sheaves} than\n$\\mathsf{coh}({\\mathbb P}^2)$.\n\\end{itemize}\n\n\n\n\n\n\n\nBelow, we summarize the results of \\cite{AZ,BGK,KKO,NS} that we are\ngoing to\nuse later in the paper.\n\nBy \\cite[Theorem 8.1(3)]{AZ},\nthe cohomology groups of the sheaves $\\CO(i)$ are given by the\nfollowing formulas (similar to those in the commutative case):\n\n\\begin{equation*}\nH^p(\\BPt,\\CO(i))=\n\\begin{cases}\nA^\\tau_i = S^i{\\widetilde{H}}, &\\text{if $p=0$ and $i\\ge0$}\\\\\n(A^\\tau_{-i-3})^\\vee = S^{-i-3}{\\widetilde{H}}^\\vee, &\\text{if $p=2$ and $i\\le-3$}\\\\\n0 &\\text{otherwise}\n\\end{cases}\n\\end{equation*}\n\nOne has  a functorial Serre duality isomorphism\n\\begin{equation*}\n\\Ext^i(E,F)\\cong\\Ext^{2-i}(F,E(-3))^\\vee.\n\\end{equation*}\n\nThe sheaves  $(\\CO(-2),\\CO(-1),\\CO)$ form a full exceptional collection\nin the derived category and there is an associated Beilinson type spectral sequence. \nThe construction of the  spectral sequence involves the sheaves $Q_0$,\n$Q_1$ and $Q_2$ on $\\BPt$ defined by\n\\begin{equation}\\label{defq}\nQ_0 = \\CO,\n\\qquad\n0 \\to \\CO \\xrightarrow{ (x,y,z) } \\CO(1) \\oplus \\CO(1) \\oplus \\CO(1) \\to Q_1 \\to 0,\n\\qquad\nQ_2 = \\CO(3).\n\\end{equation}\nSometimes another resolution for $Q_1$ is more convenient\n\\begin{equation}\\label{anotherq1}\n0 \\to Q_1 \\to \\CO(2) \\oplus \\CO(2) \\oplus \\CO(2) \\xrightarrow{ (x,y,z) } \\CO(3) \\to 0\n\\end{equation}\nWe remark that each of the two  sequences above is a truncation of the Koszul complex.\n\nThe Beilinson spectral sequence \nhas the form\n\\begin{equation*}\nE^{-p,q}_1=\\Ext^q(Q_{p}(-p),E)\\otimes \\CO(-p)\\Longrightarrow E^i_\\infty=\\begin{cases} E, &\\text{for $i=0$}\\\\\n0, &\\text{otherwise}\n\\end{cases}\n\\end{equation*}\nwhere $p=0,1,2$. \nUsing the Beilinson spectral sequence one shows, cf. \\cite[\\S7.2]{BGK}, that\nany coherent sheaf $E$ on $\\BPt$ admits a resolution of the form\n\\eq{resol1}\n0\\to V'\\otimes \\CO(k-2)\\to V\\otimes \\CO(k-1)\\to V'' \\otimes \\CO(k)\\to E\\to0\n \\end{equation} \nfor some $k\\in{\\mathbb Z}$ and  vector spaces $V',V,V''$.\n\nThe dual\n$\nE^* := \\underline{\\Hom}(E,\\CO)$,\nof any sheaf $E$ is a  sheaf on the opposite plane $\\PP^2_{-\\tau}$.\nThe sheaf $E$ is called {\\itshape locally free} if\n$\\underline{\\Ext}^i(E,\\CO) = 0$ for $i > 0$. \n\nThe following statements are proved in \\cite[Proposition 2.0.4]{BGK}.\nFor any sheaf $E$, the sheaf $E^*$ is locally free, furthermore,\n$E$  is locally free if and only if its canonical map $E\\to E^{**}$\nis an isomorphism. The kernel of a morphism of locally free sheaves is always locally free.\n\nLet $\\CE$ be a  locally free sheaf.\nWriting~\\refe{resol1} for $\\CE^*$ and dualizing, one deduces that\nany locally free sheaf $\\CE$ has a resolution of the form\n\\eq{resol2}\n0\\to\\CE\\to U' \\otimes \\CO(-k)\\to U \\otimes \\CO(1-k)\\to U'' \\otimes \\CO(2-k)\\to0.\n \\end{equation} \n\n\nA sheaf $E$ is called {\\itshape torsion free} if it can be embedded in a locally free sheaf.\nThis can be shown, e.g. using \\cite[Proposition\n2.0.6]{BGK},  to be equivalent to \nthe injectivity of the canonical map $E \\to E^{**}$.\n\n\n\n\n\n\nFor a coherent sheaf $E$ its {\\itshape Hilbert polynomial} is defined by\nthe  usual formula\n\\begin{equation*}\nh_E(t)=\\sum_{i=0}^2(-1)^i\\dim H^i(\\BPt,E(t)).\n\\end{equation*}\nFor sheaves $\\CO(i)$ it is the same as in the commutative case\n$h_{\\CO(i)}(t)=(t+i+1)(t+i+2)\/2$. So, using~\\refe{resol1} one sees\nthat the Hilbert polynomial of any sheaf can be written as\n\\eq{david}\nh_E(t)=r(E)\\frac{(t+1)(t+2)}{2}+\\deg(E)\\frac{2t+3}{2}+\\frac{\\deg(E)^2}{2}-c_2(E)\n\\end{equation}\nfor some integers $r(E),\\deg(E)$ and $c_2(E)$ defined by this equality and\ncalled the rank, degree and second Chern class of $E$ respectively.\nIt is clear from the definition that the Hilbert polynomial as well as the rank\nand the degree are additive in exact sequences. Further, one can check\nthat they behave naturally with respect to dualization\n\\begin{itemize}\n\\item for any sheaf $E$ one has $r(E^*) = r(E)$;\n\\item for a torsion free sheaf $E$ one also has $\\deg(E^*) = - \\deg(E)$;\n\\item for a locally free sheaf $\\CE$ one also has $c_2(E^*) = c_2(E)$.\n\\end{itemize}\n\\newcommand{\\mathop{\\mathrm{ch}}\\nolimits}{\\mathop{\\mathrm{ch}}\\nolimits}\n\nSometimes, instead of the second Chern class $c_2(E)$ it is more convenient to use\n\\begin{equation*}\n\\mathop{\\mathrm{ch}}\\nolimits_2(E) := \\deg(E)^2\/2 - c_2(E)\n\\end{equation*}\n(this can be thought of as the second coefficient of the Chern character).\nIts obvious advantage is additivity in exact sequences.\n\n\nFor any sheaf $E$ the rank $r(E)$ is nonnegative. If $E$ is torsion free and nonzero\nthen $r(E) > 0$, moreover, if $r(E) = 0$ then the degree $\\deg(E)$ is nonnegative.\nThe sheaf $F$ is called {\\itshape Artin sheaf of length} $n = h_F =\n\\mathop{\\mathrm{ch}}\\nolimits_2(F)$\nif  both the rank and  degree of  $F$ are equal to zero, equivalently, \n the Hilbert polynomial\nof $F$ is constant. In this case,\n the integer $n := h_F =\n\\mathop{\\mathrm{ch}}\\nolimits_2(F)$ is  nonnegative and it is called  the length of $F$.\n\n\n\n\n\n\nA special feature of the noncommutative plane $\\BPt$\nis that it has less points than the commutative ${\\mathbb P}^2$: all points of\n $\\BPt$\nare contained, in a sense, in the projective line $\\PP^1$ `at infinity'.\nIn more detail, note we have  $\\Proj(S^\\bullet(H)) = \\PP(H^\\vee) \\cong \\PP(H) =\n\\PP^1$,\nwhere we identify\n$H^\\vee = H$ via $\\omega$.\nHeuristically, one may view the  graded algebra morphism \n\\begin{equation*}\nA^\\tau\\,\\,\\twoheadrightarrow\\,\\, A^\\tau\/\\langle z\\rangle\\cong S^\\bullet(H) \\cong {\\mathbb C}[x,y].\n\\end{equation*}\nas being induced by a `closed imbedding' $\\PP^1\\,\\hookrightarrow\\, \\BPt$',\nof the projective line `at infinity'.\nSpecifically, there is a  pair of\nadjoint  functors $i_*: \\mathsf{coh}({\\mathbb P}(H)) \\to \\mathsf{coh}(\\BPt)$ and\n$i^*: \\mathsf{coh}(\\BPt) \\to \\mathsf{coh}({\\mathbb P}(H))$.\nThe  pushforward functor $i_*$ extends a graded $S^\\bullet(H)$-module structure to a graded $A^\\tau$-module structure\nby setting the action of $z$ to be zero.\nThe  pullback functor $i^*$  takes a graded $A^\\tau$-module $M$ to $M\/Mz$.\nThe projection $A^\\tau\\,\\,\\twoheadrightarrow\\,\\, S^\\bullet(H)$ is clearly $\\SL(H)$-equivariant, hence so are the functors $i_*$ and $i^*$.\nThe  functor $i_*$ is exact. The functor $i^*$ is right exact, and it has\n a sequence of left derived functors $L_pi^*$, $p > 0$. In fact\n\\begin{itemize}\n\\item for any sheaf $E$ one has $L_{>1}i^*E = 0$;\n\\item for a torsion free sheaf $E$ one also has $L_1i^*E = 0$;\n\\item for a locally free sheaf $\\CE$ the sheaf $i^*\\CE$ is also locally free.\n\\end{itemize}\n\n\nWe will use the following result.\n\n\\prop{morph2}(\\cite[Proposition 3.4.14]{BGK})\nFor any $\\tau\\ne0$ one has\n\n(1) If $i^*E=0$ then $E=0$.\n\n(2) If $\\phi\\in\\Hom(E,F)$ and $i^*\\phi$ is an epimorphism, then $\\phi$ is an epimorphism.\n\n(3)  If $\\phi\\in\\Hom(E,F)$ and both $i^*\\phi$ and $L_1i^*\\phi$ are isomorphisms then $\\phi$ is an isomorphism.\n\n(4)  If $\\phi\\in\\Hom(E,F), i^*\\phi$ is a monomorphism and $L_1i^*F=0$ then $\\phi$ is a monomorphism.\n\n(5) A sheaf $E$ is locally free iff $L_{>0}i^*E=0$ and $i^*E$ is locally free.\n\n\\end{propos}\n\n\n\n\nWe deduce the following properties of Artin sheaves:\n\n\\prop{Art0} Let $F$ be an Artin sheaf and  $h_F(t)=n$.\n\n(1) For sufficiently general $h \\in H \\subset {\\widetilde{H}} = H^0(\\BPt,\\CO(1))$,\n the map $h\\colon F(-1)\\to F$, of\nright multiplication by $h$,  is an isomorphism.\n\n(2) For any  locally free sheaf $\\CE$ we have\n\\begin{equation*}\n\\dim\\Ext^m(\\CE,F)=\\begin{cases}\n0, &\\text{if $m>0$}\\\\\nnr(\\CE), &\\text{if $m=0$}.\n\\end{cases}\n\\end{equation*}\n\n\n(3) The sheaf $F$ has a filtration\n$0=F_0\\subset F_1\\subset\\ldots\\subset F_n=F$ with $F_k\/F_{k-1}=i_*\\CO_{P_k}$\nfor some points $P_1,\\dots,P_n \\in \\PP(H)$ on the line at infinity.\nIn particular, if $h_F(t)=1$ then $F\\cong i_*\\CO_P$\nfor some point $P\\in\\PP(H)$.\n\\end{propos}\n \\begin{proof} \n(1) Since both $i^*F$ and $L_1i^*F$ are torsion sheaves on $\\PP(H)$ the maps $i^*h$ and $L_1i^*h$ are isomorphisms for generic $h$.\nHence $h\\colon F(-1) \\to F$ is also an isomorphism for generic $h$ by \\refp{morph2}(3).\n\n(2) By~\\refe{resol2} it is enough to consider the case $\\CE = \\CO(p)$ for some $p \\in \\ZZ$.\nIn this case for $p \\ll 0$ the result is clear and for arbitrary $p$ it follows from (1).\n\n\n(3) The map $F\\to i_*i^*F$ is an epimorphism. On the other hand, $i^*F$ is a nontrivial sheaf on $\\PP(H)$,\nhence there is an epimorphism $i^*F\\to\\CO_P$ for some $P\\in\\PP(H)$. The composition gives an epimorphism $F\\to i_*\\CO_P$.\nIts kernel is an Artin sheaf on $\\BPt$ of length $n-1$ and we can apply induction in $n$.\n \\end{proof} \n\n\nLet $F$ be an Artin sheaf and take an arbitrary $p \\in \\ZZ$. Consider the canonical map\n\\begin{equation*}\nH^0(\\BPt,F(p)) \\otimes H \\to H^0(\\BPt,F(p+1))\n\\end{equation*}\ninduced by the embedding $H \\subset {\\widetilde{H}} = H^0(\\BPt,\\CO(1))$.\nLet $n = h_F$ be the length of $F$, so that both cohomology spaces above are $n$-dimensional.\nA component of the $n$-th wedge power of the above map is a map\n\\begin{equation*}\n\\det (H^0(\\BPt,F(p))) \\otimes S^n H \\to \\det(H^0(\\BPt,F(p+1))).\n\\end{equation*}\nIts partial dualization gives a map\n\\begin{equation}\\label{equation-support}\n\\det (H^0(\\BPt,F(p))) \\otimes \\det(H^0(\\BPt,F(p+1)))^\\vee \\to S^nH^\\vee.\n\\end{equation}\nWe consider the projectivization of the right hand side as the space of degree $n$ divisors on $\\PP(H)$\nand denote by\n\\begin{equation*}\n\\supp(F)  \\in \\PP(S^nH^\\vee) = S^n\\PP(H^\\vee)\n\\end{equation*}\nthe image of the map. The next Lemma shows it is well defined.\n\n\\lem{artin-support}\nFor any Artin sheaf $F$ of length $n$ the map~\\eqref{equation-support} is injective,\n$\\supp(F)$ is well defined and is independent of the choice of $p$.\nIf $0 \\to F_1 \\to F_2 \\to F_3 \\to 0$ is an exact sequence of Artin sheaves then\n\\begin{equation*}\n\\supp(F_2) = \\supp(F_1) + \\supp(F_3).\n\\end{equation*}\nIf $0 \\ne h \\in H$ then $h:F(-1) \\to F$ is an isomorphism if and only if $h \\not\\in \\supp(F)$.\n\\end{lemma}\n \\begin{proof} \nClearly, evaluation of the image of~\\eqref{equation-support} on $h \\in H$ is the determinant of the map $H^0(\\BPt,F(p)) \\to H^0(\\BPt,F(p+1))$ induced by $h$.\nWe know that for generic $h$ the map is an isomorphism, hence its determinant is nonzero. This means that the image of~\\eqref{equation-support} is not\nidentically zero and proves the first claim of the Lemma. It also proves the ``only if'' part of the last claim. Moreover, the ``if'' part also follows\nfor Artin sheaves of length~1. The additivity of the support under extensions is evident (the determinant of a block upper triangular matrix is the product\nof the determinants of blocks). It follows that if $F_\\bullet$ is a filtration on $F$ with $F_k\/F_{k-1} \\cong i_*\\CO_{P_k}$ then $\\supp(F) = \\sum P_k$.\nIn particular, $\\supp(F)$ does not depend on the choice of the integer $p$. Finally, this observation and the additivity of the support also proves\nthe ``if'' part of the last claim in general.\n \\end{proof} \n\n\n\n\n\n\n\n\n\nTo finish this introductory section let us mention that one can consider families of sheaves on a noncommutative\nplane $\\BPt$. More precisely, for each affine scheme $S$ one can define a notion of a coherent\nsheaf on $S \\times \\BPt$ (see~\\cite{ns}) which is the standard way to think about $S$-families of sheaves on $\\BPt$.\nThis allows to define moduli spaces of sheaves on $\\BPt$ with appropriate stability conditions, which is the goal of this section.\n\nIn fact, a significant part of the results of this section are proved in a more general setting\n(i.e., for an arbitrary Artin--Schelter algebra instead of $A_\\tau$) in~\\cite{ns}, in particular,\nthe Gieseker moduli space we construct coincides with the moduli space of Nevins--Stafford.\nHowever, the case we consider is significantly simpler than the general case, this is the reason\nfor us to present most of the constructions here, while suppressing some proofs. The really new content\nof the section is the definition, construction, and investigation of the Uhlenbeck moduli space.\nTo make its relation to the Gieseker moduli space more clear, we use a GIT construction of the latter\nmoduli space which is different from that of~\\cite{ns}.\n\n\n\n\\ssec{quive}{Coherent sheaves and quiver representations}\nLet  $A^!_\\tau$ be\nthe quadratic dual algebra  of $A^\\tau$. From the quadratic relations \nfor $A^\\tau$ we deduce that $A^!_\\tau$ is isomorphic\nto a \n{\\sf twisted exterior algebra} $\\Lambda^\\bullet_\\tau({\\widetilde{H}}^\\vee)$ of \nthe vector space ${\\widetilde{H}}^\\vee = H^\\vee \\oplus {\\mathbb C}\\zeta$.\nSpecifically, writing\n $\\{-,-\\}$  for the anticommutator,\nwe have\n\\begin{equation*}\nA^!_\\tau \\cong\n\\Lambda^\\bullet_\\tau({\\widetilde{H}}^\\vee) =\n{\\mathbb C}\\langle H^\\vee \\oplus {\\mathbb C}\\zeta \\rangle\/\\langle \\{H^\\vee, H^\\vee\\} = \\{H^\\vee,\\zeta\\} = \\zeta^2 + \\tau\\omega = 0 \\rangle.\n\\end{equation*}\nThe group $\\SL(H)$ acts on~$A^!_\\tau$ by algebra automorphisms.\n\nChoosing a symplectic basis $\\xi,\\eta$ in $H^\\vee$ we can rewrite the\nabove as follows\n\\begin{equation*}\nA^!_\\tau = {\\mathbb C}\\langle \\xi,\\eta,\\zeta \\rangle\/\\langle \\xi^2=\\eta^2=\\eta\\xi+\\xi\\eta=\\zeta\\xi + \\xi\\zeta = \\eta\\zeta + \\zeta\\eta = \\zeta^2 + \\tau(\\xi\\eta - \\eta\\xi) = 0 \\rangle.\n\\end{equation*}\nHere, the grading on $A^!_\\tau$ corresponds to the grading $\\deg\\xi=\\deg\\eta=\\deg\\zeta=1$.\nLet $\\bpt$ be the following quiver:\n\\begin{equation*}\n\\xymatrix{\n*+[o][F]{1} \\ar[rrr]_{(A_\\tau^!)_1}\\ar@\/^1pc\/[rrrrrr]^{(A_\\tau^!)_2} &&&\n*+[o][F]{2} \\ar[rrr]_{(A_\\tau^!)_1} &&&\n*+[o][F]{3} }\n\\end{equation*}\nwith the spaces of arrows given by the components $(A_\\tau^!)_1$ and $(A_\\tau^!)_2$ of the dual algebra\nand the composition of arrows given by the multiplication $(A^!_\\tau)_1 \\otimes (A^!_\\tau)_1 \\to (A^!_\\tau)_2$ in $A^!_\\tau$.\nThe $\\SL(H)$ action on $A^!_\\tau$ induces an action on the quiver $\\bpt$,\non the category of its representations $\\Rep(\\bpt)$, and on its derived category $\\mathbf{D}(\\Rep(\\bpt))$.\n\n\\prop{derequ}\nThe functors between the bounded derived categories\n\\begin{align*}\n&\\mathbf{D}(\\mathsf{coh}(\\BPt)) \\to \\mathbf{D}(\\Rep(\\bpt)),\n&&\nE \\mapsto (\\Ext^\\bullet(Q_2(-1),E), \\Ext^\\bullet(Q_1,E), \\Ext^\\bullet(Q_0(1),E)),\\\\\n&\\mathbf{D}(\\Rep(\\bpt)) \\to \\mathbf{D}(\\mathsf{coh}(\\BPt)),\n&&\nR_\\bullet \\mapsto \\{ R_1\\otimes\\CO(-1) \\to R_2\\otimes\\CO \\to R_3\\otimes\\CO(1)\\}\n\\end{align*}\nare mutually inverse $\\SL(H)$-equivariant equivalences.\n\\end{propos}\n \\begin{proof} \nFollows from the fact that $(\\CO(-1),\\CO,\\CO(1))$ is a strong exceptional collection in $\\mathbf{D}(\\mathsf{coh}(\\BPt))$,\nand $(Q_2(-1),Q_1,Q_0(1))$ is its dual collection. The quiver $\\bpt$ is in fact\nthe quiver of morphisms of the latter sequence.\n \\end{proof} \n\nWe consider the restrictions of these functors to the abelian categories.\nGiven a representation $R_\\bullet = (R_1,R_2,R_3)$ of $\\bpt$ one constructs a complex of sheaves\n\\begin{equation}\\label{comrep}\n\\mathcal{C}(R_\\bullet) := \\{R_1\\otimes\\CO(-1) \\to R_2\\otimes \\CO \\to R_3\\otimes\\CO(1)\\}.\n\\end{equation}\nDenote by~$\\CH^i(R_\\bullet)$, $i =1,2,3$, its cohomology sheaves. Recall that a three-term complex\nis a {\\sf monad} if its cohomology at the first and  last terms vanish.\n\nAnalogously, given a sheaf $E$ we consider a representation of $\\bpt$\n\\begin{equation}\\label{repcom}\nV_\\bullet(E) = (\\Ext^1(Q_2(-1),E), \\Ext^1(Q_1,E), \\Ext^1(Q_0(1),E)).\n\\end{equation}\nThis is equivalent to applying the functor of~\\refp{derequ} and then taking\nthe first cohomology in the derived category of quiver representations.\n\n\\lem{monad-equivalence}\nIf $\\mathcal{C}(R_\\bu)$ is a monad and $E = \\CH^2(\\mathcal{C}(R_\\bu))$ is its middle cohomology sheaf then $V_\\bu(E) \\cong R_\\bu$\nand $\\Ext^i(Q_p(1-p),E) = 0$ for $i \\ne 1$ and $p = 0,1,2$.\n\nVice versa, if $E$ is a coherent sheaf on $\\BPt$ and $\\Ext^i(Q_p(1-p),E) = 0$ for $i \\ne 1$ and $p = 0,1,2$\nthen $\\mathcal{C}(V_\\bu(E))$ is a monad and $\\CH^2(V_\\bu(E)) = E$.\n\\end{lemma}\n \\begin{proof} \nIf $\\mathcal{C}(R_\\bu)$ is a monad then $E = \\CH^2(\\mathcal{C}(R_\\bu))$ is isomorphic\nto the complex $\\mathcal{C}(R_\\bu)$ in the derived category $\\mathbf{D}(\\mathsf{coh}(\\BPt))$,\nhence the complex can be used to compute $\\Ext^i(Q_p(1-p),E)$.\nThe computation gives the required result.\n\nVice versa, under the conditions of the Lemma we have $V_\\bu(E)$ is the image of $E$ in $\\mathbf{D}(\\Rep(\\bpt))$\nunder the equivalence of~\\refp{derequ}, hence $\\mathcal{C}(V_\\bu(E)) \\cong E$ in $\\mathbf{D}(\\mathsf{coh}(\\BPt))$.\nThis means that the complex is a monad and its middle cohomology is $E$.\n \\end{proof} \n\n\n\n\n\n\n\n\n\n\\ssec{stabilo}{Stability of sheaves and quiver representations}\n\nThe notions of Gieseker and Mumford (semi)stability of coherent sheaves are  standard in the commutative context.\nWe refer to~\\cite{HL} for more details and for proofs of standard\nfacts. These notions \nhave generalizations\nfor sheaves on $\\BPt$.\n\nGiven a sheaf $E$ on $\\BPt$ with $r(E)>0$,\nwe define its Mumford and Gieseker slopes as\n\\begin{align*}\n\\mm(E) & =\\frac{\\deg(E)}{r(E)}\\in\\BQ,\\\\\n\\mg(E) & =\\frac{h_E(t)}{r(E)}=\\frac{(t+1)(t+2)}{2}+\\mm(E)\\frac{2t+3}{2}+\\frac{\\deg(E)^2-2c_2(E)}{2r(E)}\\in\\BQ[t].\n\\end{align*}\nLet $p(t)$ and $q(t)$ be polynomials. We say that $p<q$ (resp. $p\\le q$) if for all $t\\gg 0$ we have $p(t)<q(t)$ (resp. $p(t)\\le q(t)$).\n\n\n\\defe{stability}\nA sheaf $E$ is {\\itshape Gieseker stable} (resp.\\ {\\itshape Gieseker semistable})\nif $E$ is torsion free and for any subsheaf $0 \\subsetneq F \\subsetneq E$ we have $\\mg(F)<\\mg(E)$ (resp.\\ $\\mg(F) \\le \\mg(E)$).\nSheaves $E$ and $F$ are called Gieseker $S$-equivalent if both of them are Gieseker semistable and have isomorphic composition factors in the category of Gieseker semistable sheaves.\n\nSimilarly, a sheaf $E$ is {\\itshape Mumford stable} (resp.\\ {\\itshape Mumford semistable})\nif any torsion subsheaf in $E$ is Artin and for any $F \\subset E$ such that $0<r(F)<r(E)$ we have $\\mm(F)<\\mm(E)$ (resp.\\ $\\mm(F)\\le\\mm(E)$).\nA pair of sheaves $E$ and $F$ are called Mumford $S$-equivalent if both of them are Mumford semistable and have isomorphic composition factors in the category of Mumford semistable sheaves.\n\\end{defeni}\n\nBoth Gieseker and Mumford stabilities of sheaves on $\\BPt$ behave analogously to those on the commutative projective plane ${\\mathbb P}^2$.\nFor example, by~\\cite{Ru} each sheaf $F$ has a {\\sf Harder--Narasimhan filtration}, i.e.\\ a filtration \n\\begin{equation*}\n0 = F_n \\subset F_{n-1} \\subset \\dots \\subset F_1 \\subset F_0 = F\n\\end{equation*}\nsuch that $F_{i-1}\/F_i$ are stable and $\\mu(F_{n-1}\/F_n) > \\mu(F_{n-2}\/F_{n-1}) > \\dots > \\mu(F_0\/F_1)$.\n\n\\medskip\n\n\nTo check Gieseker stability (semistability) it is enough to consider only subsheaves $F \\subset E$ such that\n$E\/F$ is torsion free (in particular, $r(F) < r(E)$). So, the following is clear.\n\n\\lem{Q}\nAny torsion free sheaf of rank $1$ is Gieseker stable and Mumford stable.\n\\end{lemma}\n\n\nNote also that $\\mm(E)>\\mm(F)$ implies $\\mg(E)>\\mg(F)$ and $\\mg(E)\\ge\\mg(F)$\nimplies $\\mm(E)\\ge\\mm(F)$. It follows that Gieseker semistability implies\nMumford semistability, while Mumford stability for tosion free sheaves implies Gieseker stability.\nMoreover, if the rank and the degree of a torsion free sheaf are coprime then\nsemistablity implies stability.\n\n\n\nThe following Lemma is standard\n\\lem{hom}\n\n(1) If $E$, $F$ are Mumford semistable sheaves with $F$ torsion free and $\\mu_M(E)>\\mu_M(F)$ then $\\Hom(E,F)=0$.\n\n(2) If $E,F$ are Mumford stable sheaves, $E$ is locally free and $\\mu_M(E) \\ge \\mu_M(F)$\nthen any nontrivial homomorphism $E\\to F$ is an isomorphism.\n\\end{lemma}\n\n\nThe notion of stability for a representation of a quiver depends on a choice of a polarization, see~\\cite{K}.\nA {\\sf polarization} in case of the quiver $\\bpt$ amounts to a triple $\\tet=(\\tet_1,\\tet_2,\\tet_3)$ of real numbers.\nThe {\\sf $\\tet$-slope} of a representation $R_\\bullet = (R_1,R_2,R_3)$ of $\\bpt$ is defined as\n\\begin{equation*}\n\\mt(R_\\bullet) = \\langle \\tet, \\dim R_\\bu \\rangle := \\tet_1\\dim R_1+\\tet_2\\dim R_2+\\tet_3\\dim R_3.\n\\end{equation*}\n\n\\defe{stability-reps}\nA representation $R_\\bullet$ is $\\tet$-stable\n(resp. $\\tet$-semistable) if $\\mt(R_\\bullet)=0$ and for any\nsubrepresentation $R'_\\bullet\\subset R_\\bullet$ such that $0\\ne\nR'_\\bullet\\ne R_\\bullet$ we have $\\mt(R'_\\bullet)>0$\n(resp. $\\mt(R'_\\tet)\\ge 0$). Representations $R_\\bullet$ and\n$R'_\\bullet$ are called $S$-equivalent with respect to $\\tet$ if both of\nthem are $\\tet$-semistable and have isomorphic composition factors in the\ncategory of $\\tet$-semistable representations.\n\\end{defeni}\n\n\n\n\nLet $\\tet,\\tet'$ be a pair of polarizations. It is well known (e.g.~\\cite{DH})\nthat, for all sufficiently small and positive $\\eps\\in \\mathbb{R}$,\n stability, semistability and $S$-equivalence with respect to\n $\\tet+\\eps\\tet'$ does not depend on $\\eps$.\n\n\\defe{stability-reps-2}\nA representation $R_\\bullet$ is $\\ttt$-stable (resp. $\\ttt$-semistable) if $R_\\bullet$ is\n$(\\tet+\\eps\\tet')$-stable (resp. semistable) for sufficiently small positive\n$\\eps$.\n\\end{defeni}\n\nThere is an analogue of~\\refl{hom} for representations of the quiver $\\bpt$.\n\n\n\n\\ssec{sheaves-reps}{From sheaves to quiver representations}\nLet ${\\mathcal{TF}}_{r,d}$ be the  The following result is essentially a combination of Lemma~6.4 and Theorem~5.6 from~\\cite{ns}.\nThe only new statement is the exactness claim. We provide a proof for\nthe\nreader's convenience.\n\\th{mon}\nLet $-r \\le d < r$.\nThen, the assignment $E\\mapsto V_\\bu(E)$ gives an exact functor  from the\ncategory of Mumford semistable torsion free sheaves $E$\non $\\BPt$ such that  $r(E)=r$ and $\\deg(E)=d$\nto the category of representations\nof the quiver $\\bpt$.\nFor such a  sheaf $E$,\nthe representation $V_\\bu(E)$ gives  a monad\n\\begin{equation}\\label{monad-of-e}\nV_1(E) \\otimes \\CO(-1) \\to V_2(E) \\otimes \\CO \\to V_3(E) \\otimes \\CO(1)\n\\end{equation}\nsuch that its cohomology at the middle term is isomorphic to\n$E$. Furthermore,  we have   $\\dim V_\\bullet(E) =\n(n-d(d-1)\/2,2n-d^2+r,n-d(d+1)\/2)$,\nwhere $n=c_2(E)$; in particular, $c_2(E) \\ge d(d+1)\/2$.\n\\eth\n \\begin{proof} \nFirst we note that all $Q_p$ are Mumford stable of slopes equal to $0$, $3\/2$ and $3$ respectively.\nIndeed, for $p = 0$ and $p = 2$ this follows from~\\refl{Q} and the definitions of $Q_p$.\nSo let $p = 1$. The sheaf $Q_1$ is locally free because by~\\eqref{anotherq1} it is the kernel\nof a morphism of locally free sheaves, and moreover $r(Q_1) = 2$, $\\deg(Q_1) = 3$.\nSo, it is enough to check that if $F \\subset Q_1$ is a subsheaf of rank 1 with $Q_1\/F$ torsion free then $\\deg(F) \\le 1$.\nAssume $\\deg(F) \\ge 2$. As $Q_1\/F$ is torsion free, $F$ is locally free.\nSince $Q_1$ is a subsheaf in $\\calO(2)^{\\oplus3}$ there is a nontrivial homomorphism from $F$ to $\\calO(2)$.\nOn the other hand both $F$ and $\\calO(2)$ are Mumford stable by (1), $F$ is locally free, and\n$\\mm(F)\\ge2=\\mm(\\calO(2))$. Hence $F\\simeq\\CO(2)$ by \\refl{hom}(2). But applying\nthe functor $\\Hom(\\calO(2),-)$ to~\\eqref{anotherq1} we see that $\\Hom(\\calO(2),Q_1)=0$.\n\nThe proved stability implies that\n\\begin{equation*}\n\\Hom(Q_0(1),E) = \\Hom(Q_1,E) = \\Hom(Q_2(-1),E) = 0.\n\\end{equation*}\nIndeed, the slopes of the first arguments are $1$, $3\/2$, and $2$ respectively, while the slope\nof the second argument is $d\/r < 1$, so~\\refl{hom}(1) applies. Analogously,\n\\begin{equation*}\n\\Hom(E(3),Q_0(1)) = \\Hom(E(3),Q_1) = \\Hom(E(3),Q_2(-1)) = 0\n\\end{equation*}\nsince the slope of the first argument is $d\/r+3 > 2$. By Serre duality we then have\n\\begin{equation*}\n\\Ext^2(Q_0(1),E) = \\Ext^2(Q_1,E) = \\Ext^2(Q_2(-1),E) = 0.\n\\end{equation*}\nTherefore, \\refl{monad-equivalence} applies to $E$ and shows that~\\eqref{monad-of-e} is a monad\nand $E$ is its cohomology.\nThe dimensions of the spaces $V_p(E)$ are computed directly by using the formula~\\refe{david} for the Hilbert polynomial of a sheaf.\nThe exactness of the functor $V_\\bu$ is clear from its definition and vanishing of $\\Hom$ and $\\Ext^2$ spaces.\n \\end{proof} \n\n\\prop{artin-mon}\nThe functor $F\\mapsto {\\mathcal\n  C}(R_\\bullet(F))$\nyields, for an Artin sheaf $F$,  a canonical exact sequence\n\\begin{equation*}\n0 \\to W_1(F) \\otimes \\CO(-1) \\to W_2(F) \\otimes \\CO \\to W_3(F) \\otimes \\CO(1) \\to F \\to 0\n\\end{equation*}\nThe resulting functor $W_\\bu$ from the category of Artin sheaves on $\\BPt$ to the category\nof representations of the quiver $\\bpt$ is exact\nand we have  $\\dim W_\\bullet(F) = (n,2n,n)$, where $n$ is the length of $F$.\n\\end{propos}\n \\begin{proof} \nThe proof is analogous to the proof of~\\refl{monad-equivalence}. We apply the equivalence of~\\refp{derequ} to the sheaf~$F$.\nBy~\\refp{Art0}(2) applying the functor of~\\refp{derequ} to $F$ we obtain nothing but representation\n\\begin{equation*}\nW_\\bullet(F) = (\\Hom(Q_2(-1),F),\\Hom(Q_1,F),\\Hom(Q_0(1),F))\n\\end{equation*}\nand its dimension vector is $(n,2n,n)$. Since the functor is an equivalence, it follows that\nthe complex $\\mathcal{C}(W_\\bu(F))$ is left exact and $\\CH^3(\\mathcal{C}(W_\\bullet(F))) \\cong F$, which amounts to the above exact sequence.\nExactness of the functor $W_\\bu$ follows from the vanishing of $\\Ext^1(Q_p(1-p),F)$ by~\\refp{Art0}(2).\n \\end{proof} \n\nIf $0 \\ne h \\in H$, $P \\in \\PP(H)$ is the corresponding point and $F = \\CO_P$, then\n\\begin{equation}\\label{monad-point}\nW_\\bullet(\\CO_P) = \\{ \\C \\xrightarrow{\\ \\left(\\begin{smallmatrix} h \\\\ \\zeta \\end{smallmatrix}\\right)\\ } \\C^2 \\xrightarrow{\\ (-\\zeta,h)\\ } \\C \\}.\n\\end{equation}\n\n\n\n\n\n\\ssec{relat stab}{From sheaf stability to quiver stability}\n\nIn this section we show that Gieseker and Mumford semistability correspond to semistablity of quiver representations.\n\nFrom now on we fix a triple $(r,d,n)$ such that\n\\begin{equation}\\label{assumptions-r-d-n}\n0 \\le d < r\n\\qquad\\text{and}\\qquad\nn \\ge d(d+1)\/2.\n\\end{equation} Put\n\\begin{equation}\\label{alpha}\n\\alp(r,d,n)=(n-d(d-1)\/2, 2n-d^2+r, n-d(d+1)\/2).\n\\end{equation}\nAccording to~\\reft{mon} if $E$ is a Mumford semistable sheaf such that $r(E)=r,\n\\deg(E)=d$ and $c_2(E)=n$ then $\\dim V_\\bullet(E) = \\alp(r,d,n)$.\n\nWe choose the following pair of polarizations\n\\begin{equation}\\label{thetas}\n\\begin{split}\n&\\tet^0=(-r-d,d,r-d),\\\\\n&\\tet^1=(2n-d^2+r,d^2-2n,2n-d^2+r).\n\\end{split}\n\\end{equation}\nNote that $\\tet^0$ does not depend on $n$. Note also that\n\\begin{equation*}\n\\langle \\tet^0,\\alp(r,d,n) \\rangle = \\langle \\tet^1,\\alp(r,d,n) \\rangle = 0.\n\\end{equation*}\nIn what follows we frequently consider $\\tet^0$-stability and $\\tnt$-stability of representations.\nIn fact the notion of $\\tnt$-(semi)stablity of a quiver representation is equivalent to the notion\nof (semi)stability of a Kronecker complex considered in \\cite{ns}.\n\n\\lem{kronecker-stability}\nLet $V_\\bu$ be an $\\alp(r,d,n)$-dimensional representation of the quiver $\\bpt$ and let $\\mathcal{C}(V_\\bu)$\nbe the associated complex~\\eqref{comrep}. Then $V_\\bu$ is $\\tnt$-(semi)stable if and only if $\\mathcal{C}(V_\\bu)$\nis (semi)stable in the sense of~\\cite[Def.~6.8]{ns}.\n\\end{lemma}\n \\begin{proof} \nJust note that a subcomplex in $\\mathcal{C}(V_\\bu)$ always corresponds to a subrepresentation $U_\\bu \\subset V_\\bu$,\nand the expression (1) from \\cite[Defenition~6.8]{ns} for $\\mathcal{C}(U_\\bu)$ equals $\\mu_{\\tet^0}(U_\\bu)$,\nwhile the expression (2) equals $\\mu_{\\tet^1}(U_\\bu)$,\n \\end{proof} \n\nThe following crucial observation relating stability for sheaves and for\nquiver representations, respectively,\nis due to Sergey Kuleshov. Parts (2) and (3) (as well as a part of~\\refl{monadic-sheaf} below)\nare also proved in Proposition~6.20 of~\\cite{ns}.\n\n\\lem{eq\nLet $E$ be a torsion free sheaf with $r(E)=r, \\deg(E)=d, c_2(E)=n$ and let\n$V_\\bullet(E)$ be the corresponding representation of the quiver $\\bpt$. Then\n\n(1) if $E$ is Mumford semistable then $\\VE$ is $\\tet^0$-semistable;\n\n(2) if $E$ is Gieseker semistable then $\\VE$ is $\\tnt$-semistable;\n\n(3) if $E$ is Gieseker stable then $\\VE$ is $\\tnt$-stable.\n\\end{lemma}\n\n \\begin{proof} \n(1) Assume that $E$ is Mumford semistable. Let $U_\\bullet$ be a subrepresentation\nin $\\VE$ and $W_\\bullet = V_\\bullet(E) \/ U_\\bullet$ be the quotient representation,\nand put $u_i=\\dim U_i$. Then we have a short exact sequence of complexes\n\\begin{equation}\\label{uvw}\n\\vcenter{\\xymatrix{\n0 \\ar[r] &\nU_1 \\otimes \\CO(-1) \\ar[r] \\ar[d] &\nU_2 \\otimes \\CO \\ar[r] \\ar[d] &\nU_3 \\otimes \\CO(1) \\ar[r] \\ar[d] &\n0\n\\\\\n0 \\ar[r] &\nV_1(E) \\otimes \\CO(-1) \\ar[r] \\ar[d] &\nV_2(E) \\otimes \\CO \\ar[r] \\ar[d] &\nV_3(E) \\otimes \\CO(1) \\ar[r] \\ar[d] &\n0\n\\\\\n0 \\ar[r] &\nW_1 \\otimes \\CO(-1) \\ar[r] &\nW_2 \\otimes \\CO \\ar[r] &\nW_3 \\otimes \\CO(1) \\ar[r] &\n0\n}}\n\\end{equation}\nConsidering it as an exact triple (with respect to the vertical maps) of 3-term complexes\nand applying the Snake Lemma we obtain a long exact sequence of cohomology sheaves\n\\begin{equation*}\n0 \\to\n\\CH^1(U_\\bullet) \\to 0 \\to \\CH^1(W_\\bullet) \\to\n\\CH^2(U_\\bullet) \\to E \\to \\CH^2(W_\\bullet) \\to\n\\CH^3(U_\\bullet) \\to 0 \\to \\CH^3(W_\\bullet) \\to 0\n\\end{equation*}\nof these complexes. In particular, we have $\\CH^1(U_\\bullet) = \\CH^3(W_\\bullet) = 0$.\nDenote\n\\begin{equation*}\nr_i^U = r(\\CH^i(U_\\bullet)), \\quad\nd_i^U = \\deg(\\CH^i(U_\\bullet)), \\quad\nr_i^W = r(\\CH^i(W_\\bullet)), \\quad\nd_i^W = \\deg(\\CH^i(W_\\bullet)).\n\\end{equation*}\nLet $I$ be image of the morphism $\\CH^2(U_\\bullet) \\to E$ in the above sequence\nand let\n\\begin{equation*}\nr_I=r(I), \\quad d_I=\\deg(I).\n\\end{equation*}\nThen using additivity of rank and degree we can rewrite the slope of $U_\\bullet$ as\n\\begin{multline*}\n\\mu_{\\tet^0}(U_\\bullet)\n = r(u_3-u_1)+d(u_2-u_1-u_3)\n =r(d_3^U - d_2^U)+d(r_2^U-r_3^U) \\\\\n =r(d_3^U - d_1^W - d_I)+d(r_1^W+r_I-r_3^U)\n=(rd_3^U - r_3^U d)+(r_1^W d-r d_1^W)+(r_Id-rd_I).\n\\end{multline*}\nNow we will show that all three summands in the right-hand-side are nonnegative.\n\nFirst, note that $\\CH^3(U_\\bullet)$ is a quotient of $U_3\\otimes\\CO(1)$.\nThe latter sheaf is semistable by \\refl{Q}(1) and\nwe have $\\mm(U_3\\otimes\\calO(1))=1$. Therefore, $d_3^U \\ge r_3^U$ and hence\n\\begin{equation*}\nrd_3^U-r_3^Ud \\ge r_3^U(r-d)\\ge0.\n\\end{equation*}\nNote that the inequality $rd_3^U-r_3^Ud \\ge0$ is strict unless $r_3^U=d_3^U=0$,\nthat is unless $\\CH^3(U_\\bullet)$ is an Artin sheaf .\n\nFurther, note that $\\CH^1(W_\\bullet)$ is a subsheaf of $W_1\\otimes\\CO(-1).$\nThe latter sheaf is semistable and one has $\\mm(W_1\\otimes\\CO(-1))=-1$. Therefore,\n$d_1^W \\le -r_1^W \\le 0$ and hence\n\\begin{equation*}\nr_1^W d-r d_1^W \\ge dr_1^W \\ge 0.\n\\end{equation*}\nNote that this inequality is strict unless $r_1^W = d_1^W = 0$, that is unless\n$\\CH^1(W_\\bullet)$ is an Artin sheaf. But since it is a subsheaf in $W_1\\otimes\\CO(-1)$\nit is torsion free, hence this is equivalent to $\\CH^1(W_\\bullet) = 0$.\n\n\nFinally, $I$ is a subsheaf of the Mumford semistable torsion free sheaf $E$.\nHence either $I=0$, or else $r_I>0$ and $\\mm(I)\\le\\mm(E)$. In both cases we have\n$$r_Id-d_Ir\\ge0.$$\nNote that this inequality is strict unless $I=0$ or $\\mm(I)=\\mm(E).$\n\nCombining all these inequalities we see that any subrepresentation in $\\VE$\nhas a non-negative $\\tet^0$-slope. Thus $\\VE$  is $\\tet^0$-semistable.\n\n(2) Assume that the sheaf $E$ is Gieseker semistable but $V_\\bullet(E)$ is not $(\\tet^0,\\tet^1)$-semistable.\nLet $U_\\bullet \\subset V_\\bullet(E)$ be a destabilizing subrepresentation. Since\n$E$ is automatically Mumford semistable and hence $V_\\bullet(E)$ is $\\tet^0$-semistable by (1),\nwe conclude that $\\mu_{\\tet^0}(U_\\bullet) = 0$. Then as we observed in the above argument\n$\\CH^3(U_\\bullet)$ is an Artin sheaf, $\\CH^1(W_\\bullet) = 0$ and\nhence $\\CH^2(U_\\bullet) = I$ is a subsheaf in $E$.\n\nLet $c = \\mathop{\\mathrm{ch}}\\nolimits_2(E) = d^2\/2 - n$ and $c_i^U = \\mathop{\\mathrm{ch}}\\nolimits_2(\\CH^i(U_\\bullet))$.\nThen\n\\begin{multline*}\n\\mu_{\\tet^1}(U_\\bullet)=(r-2c)u_1+2cu_2+(r-2c)u_3=r(u_1+u_3)+2c(u_2-u_1-u_3) \\\\\n=2r(c_3^U-c_2^U) + 2c(r_2^U-r_3^U)\n=2(rc_3^U-r_3^U c)+2(r_2^Uc-r c_2^U).\n\\end{multline*}\nSince $\\CH^3(U_\\bullet)$ is an Artin sheaf we have $r_3^U = 0$ and $c_3^U \\ge 0$,\nhence\n\\begin{equation*}\nrc_3^U-r_3^U c = rc_3^U \\ge 0.\n\\end{equation*}\nOn the other hand, $\\CH^2(U_\\bullet) = I$ and hence\neither $\\CH^2(U_\\bullet) = 0$, or $\\mm(\\CH^2(U_\\bullet)) = \\mm(E)$.\nIn both cases\n\\begin{equation*}\nr_2^Uc-r c_2^U = rr_2^U(\\mg(E) - \\mg(\\CH^2(U_\\bullet))) \\ge 0\n\\end{equation*}\nbecause $E$ is Gieseker semistable.\nThus $\\mu_{\\tet^1}(U_\\bullet) \\ge0$ which contradicts to the assumption that $U_\\bullet$\ndestabilizes $V_\\bullet(E)$.\n\n(3) In the notation of (2) assume that $\\mu_{\\tet^0}(U_\\bu) = \\mu_{\\tet^1}(U_\\bu) = 0$.\nThen $c_3^U=0$, hence $\\CH^3(U_\\bullet)=0$. Moreover,\neither $\\CH^2(U_\\bullet)=0$, or $\\mg(\\CH^2(U_\\bullet))=\\mg(E)$, and so $\\CH^2(U_\\bullet)=E$\nsince $E$ is Gieseker stable. In the first case the first line of~\\eqref{uvw} is exact,\nhence $U_\\bullet=0$ by~\\refp{derequ}. In the second case the first line of~\\eqref{uvw} is a resolution of $E$,\nhence $U_\\bullet=\\VE$.\n \\end{proof} \n\n\\lem{cfart}\nLet $F$ be an Artin sheaf. Then $W_\\bu(F)$ is $\\tet^0$-semistable. If the length of $F$ is equal to $1$ then $W_\\bu(F)$ is $\\tet^0$-stable.\n\\end{lemma}\n \\begin{proof} \nSince by~\\refp{Art0}(3) any Artin sheaf is an extension of the structure sheaves of points and the functor $W_\\bullet$\nis exact, it is enough to verify the $\\tet^0$-stability of $W_\\bullet(\\CO_P)$. The latter is clear from the explicit\nform~\\eqref{monad-point} of the monad --- it is easy to see that the only nontrivial subrepresentations of $W_\\bu(\\CO_P)$\nhave dimension $(0,0,1)$, $(0,1,1)$ and $(0,2,1)$, and their $\\tet^0$-slope is clearly positive with our assumptions on $d$ and $r$.\n \\end{proof} \n\n\n\n\\ssec{quiver-sheaf}{From quiver stability to sheaf stability}\n\nIn this section we show that stable representations of the quiver, in their turn,\ngive rise to stable sheaves.\n\n\\defe{red}\nA representation $V_\\bullet$ is called:\n\\begin{itemize}\n\\item\n{\\sf Artin}, if $\\CH^1(V_\\bu) = \\CH^2(V_\\bu) = 0$ and $\\CH^3(V_\\bu)$ is an Artin sheaf;\n\\item\n{\\sf monadic}, if $\\CH^1(V_\\bullet) = \\CH^3(V_\\bullet) = 0$;\n\\item\n{\\sf supermonadic}, if both $V_\\bu$ and $V_\\bu^\\vee$ are monadic.\n\\end{itemize}\n\\end{defeni}\n\n\\lem{smlf}\nA monadic representation $V_\\bullet$ is supermonadic iff $\\CH^2(V_\\bullet)$ is locally free.\n\\end{lemma}\n \\begin{proof} \nSince $V_\\bullet$ is monadic, the complex $\\mathcal{C}(V_\\bullet)$ is isomorphic to $\\CH^2(V_\\bullet)$ in the derived category $\\mathbf{D}(\\mathsf{coh}(\\BPt))$.\nTherefore the complex $\\mathcal{C}(V_\\bullet^\\vee) = \\mathcal{C}(V_\\bullet)^*$ is isomorphic to the derived dual of $\\CH^2(V_\\bu)$. In other words,\n$\\CH^i(\\mathcal{C}(V^\\vee_\\bu)) \\cong \\underline{\\Ext}^{i-2}(\\CH^2(V_\\bullet),\\CO)$. So, $V_\\bullet$ is supermonadic iff\n$\\underline{\\Ext}^i(\\CH^2(V_\\bullet),\\CO) = 0$ for $i \\ne 0$, i.e., iff $\\CH^2(V_\\bullet)$ is locally free.\n \\end{proof} \n\n\\lem{monadic-sheaf}\nLet $V_\\bu$ be a monadic representation of $\\bpt$ and let $E = \\CH^2(V_\\bu)$.\nIf $V_\\bu$ is $\\tet^0$-semistable then $E$ is Mumford semistable and\nif $V_\\bu$ is $\\tnt$-semistable then $E$ is Gieseker semistable.\nIn both cases $V_\\bu = V_\\bu(E)$.\n\\end{lemma}\n \\begin{proof} \nAssume $E$ is not Mumford semistable and consider its Harder--Narasimhan filtration.\nBreaking it up at slope $\\mu_M(E)$ we can represent $E$ as an extension\n\\begin{equation*}\n0 \\to E' \\to E \\to E'' \\to 0,\n\\end{equation*}\nsuch that the slopes of all quotients in the Harder--Narasimhan filtration of $E'$ (resp.\\ $E''$) are greater than (resp.\\ less than or equal to) $\\mu_M(E)$.\nLet $(r',d',n')$\nbe the rank, the degree and the second Chern class of $E'$.\nNote that both $E'$ and $E''$ are the cohomology sheaves of the monads $V_\\bu(E')$ and $V_\\bu(E'')$ respectively.\nIndeed, for $E'$ the argument of~\\reft{mon} shows that $\\Ext^2(Q_p(1-p),E') = 0$.\nOn the other hand, since $E'$ is a subsheaf in $E$ we have $\\Hom(Q_p(1-p),E') \\subset \\Hom(Q_p(1-p),E) = 0$.\nAnalogously, for $E''$ the argument of~\\reft{mon} gives the vanishing of $\\Hom$'s, while the surjectivity\nof the map from $E$ gives the vanishing of $\\Ext^2$. It follows that we have an exact sequence of monads\n\\begin{equation*}\n0 \\to V_\\bu(E') \\to V_\\bu \\to V_\\bu(E'') \\to 0.\n\\end{equation*}\nFinally, note that\n\\begin{equation*}\n\\langle \\tet^0, \\alpha(r',d',n') \\rangle = r'd - rd' = rr'\\left(\\frac{d}{r} - \\frac{d'}{r'}\\right) = rr'(\\mu_M(E) - \\mu_M(E')) < 0.\n\\end{equation*}\nHence the subrepresentation $V_\\bu(E') \\subset V_\\bu$ violates\nthe $\\tet^0$-semistability of $V_\\bu$. This proves the first part.\n\nIf $E$ is Mumford semistable but not Gieseker semistable, we take again $E'$ to be the part of the Harder--Narasimhan\nfiltration of $E$ with the slopes greater than $\\mu_G(E)$. Then $\\langle \\tet^0, \\alpha(r',d',n') \\rangle = 0$ but\n\\begin{equation*}\n\\langle \\tet^1, \\alpha(r',d',n') \\rangle = r'(d^2-2n) - r(d'^2 - 2n') = 2rr'\\left(\\frac{d^2-2n}{2r} - \\frac{d'^2-2n'}{2r'}\\right) < 0\n\\end{equation*}\nHence the subrepresentation $V_\\bu(E') \\subset V_\\bu$ violates the $\\tnt$-semistability of $V_\\bu$. This proves the second part.\n\nFinally, we have $V_\\bu = V_\\bu(E)$ by~\\refl{monad-equivalence}.\n \\end{proof} \n\n\\prop{tet0-ss}\nLet $V_\\bu$ be a $\\tet^0$-semistable representation of $\\bpt$ of dimension $\\alpha(r,d,n)$.\nThen $V_\\bu$ is S-equivalent to a direct sum $U_\\bu \\oplus W_\\bu$, where $U_\\bu$ is supermonadic of dimension $\\alpha(r,d,n-k)$\nand $W_\\bu$ is Artin of dimension $(k,2k,k)$ for some $0 \\le k \\le n$.\n\\end{propos}\n\n \\begin{proof} \nAssume that $\\CH^3(V_\\bu) \\ne 0$. Then $i^*(\\CH^3(V_\\bullet)) \\ne 0$ by~\\refp{morph2}(1), hence\nthere is a surjective morphism $\\CH^3(V_\\bullet) \\to i_*\\CO_P$ for some point $P \\in {\\mathbb P}(H)$.\nSince $\\CH^3(V_\\bullet)$ is the top cohomology of the complex $\\mathcal{C}(V_\\bullet)$, there is\na canonical morphism $\\mathcal{C}(V_\\bullet) \\to \\CH^3(V_\\bullet)$.\nComposing these morphisms we get a nontrivial morphism $\\mathcal{C}(V_\\bullet) \\to \\CO_P$.\nBy \\refp{derequ} this corresponds to a nontrivial morphism $V_\\bullet \\to W_\\bullet(\\CO_P)$.\nSince $V_\\bu$ is $\\tet^0$-semistable and $W_\\bu(\\CO_P)$ is $\\tet^0$-stable, the morphism is surjective.\nTaking $V'_\\bu$ to be the kernel of the morphism, we see that $V'_\\bu$ is $\\tet^0$-semistable and $V_\\bu$ is S-equivalent to\n$V'_\\bu \\oplus W_\\bu(\\CO_P)$. The dimension of $V'_\\bu$ is strictly less than that of $V_\\bu$,\nso iterating the construction we reduce to the case when $\\CH^3(V_\\bu) = 0$.\n\n\n\nAssume now that $\\CH^3(V_\\bu) = 0$, but $\\CH^3(V^\\vee_\\bu) \\ne 0$. Then applying the same argument to $V^\\vee_\\bu$\nwe obtain an injection $W_\\bu(\\CO_P) \\to V_\\bu$. Taking $V'_\\bu$ to be the cokernel of this morphism,\nwe see that $V'_\\bu$ is $\\tet^0$-semistable and $V_\\bu$ is S-equivalent to $V'_\\bu \\oplus W_\\bu(\\CO_P)$.\nIterating the construction we reduce to the case when $\\CH^3(V_\\bu) = \\CH^3(V^\\vee_\\bu) = 0$.\n\nFinally, assume that $\\CH^3(V_\\bu) = 0$ and $\\CH^3(V^\\vee_\\bu) = 0$. Let us show that $V_\\bu$ is supermonadic.\nIndeed, if ${\\mathcal F} := \\CH^1(V_\\bu) \\ne 0$ then ${\\mathcal F}$ is a locally free sheaf and we have a left exact sequence\n$0 \\to {\\mathcal F} \\to V_1 \\otimes \\CO(-1) \\to V_2 \\otimes \\CO$. After dualization we get a complex\n\\begin{equation*}\nV_2^\\vee \\otimes \\CO \\to V_3^\\vee \\otimes \\CO(1) \\to {\\mathcal F}^*\n\\end{equation*}\nin which the second arrow is nontrivial (since after second dualization it gives back the embedding ${\\mathcal F} \\to V_1\\otimes\\CO(-1)$).\nThis means that $\\CH^3(V^\\vee_\\bu) \\ne 0$, thus contradicting the assumption.\nTherefore $\\CH^1(V_\\bu) = 0$.\nAnalogous argument with $V_\\bu$ replaced by $V^\\vee_\\bu$ shows that $\\CH^1(V^\\vee_\\bu) = 0$, so $V_\\bu$ is indeed supermonadic.\n\nAs at each step of the above procedure the dimension of the representation\nhas decreased by $(1,2,1)$,\nthe dimension of the supermonadic part we end up with is equal to\n\\begin{equation*}\n\\alpha(r,d,n) - (k,2k,k) = \\alpha(r,d,n-k).\n\\end{equation*}\nSince all the components of a dimension vector are nonnegative, we have $n-k \\ge d(d+1)\/2$, in particular $k \\le n$.\n \\end{proof} \nNote that we have\n\\[(n,2n,n) = \\alpha(0,0,n).\\]\n\\cor{tet-ss-artin}\nLet $W_\\bu$ be a $\\tet^0$-semistable representation of $\\bpt$ of dimension $(n,2n,n)$. Then $W_\\bu$ is an Artin representation.\n\\end{corol}\n \\begin{proof} \nApplying~\\refp{tet0-ss} we see that $W_\\bu$ is S-equivalent to a sum of\nan Artin representation and a supermonadic representation $U_\\bu$ of dimension $\\alpha(0,0,n-k)$ for some $k$.\nBy definition and~\\refl{smlf} the corresponding complex $\\mathcal{C}(U_\\bu)$ is a monad and its middle cohomology\nis a locally free sheaf of rank $0$. Thus the cohomology is zero and the complex $\\mathcal{C}(U_\\bu)$ is acyclic.\nBy~\\refp{derequ} this means that $U_\\bu = 0$, so we have no supermonadic part. It follows that $W_\\bu$\nis S-equivalent to an Artin representation. It follows immediately that $\\CH^1(W_\\bu) = \\CH^2(W_\\bu) = 0$,\nand $\\CH^3(W_\\bu)$ is an iterated extension of Artin sheaves. Hence it is an Artin sheaf itself,\nand so $W_\\bu$ is also an Artin representation.\n \\end{proof} \n\n\nThe first part of the following result can be found in Lemma~6.14 of \\cite{ns}.\n\n\\prop{tnt-ss}\nLet $V_\\bu$ be a $\\tnt$-semistable representation of $\\bpt$ of dimension $\\alpha(r,d,n)$ with $0 \\le d < r$.\nThen $V_\\bu$ fits into a short exact sequence\n\\begin{equation*}\n0 \\to W_\\bu \\to V_\\bu \\to U_\\bu \\to 0,\n\\end{equation*}\nwhere $U_\\bu$ is supermonadic and $W_\\bu$ is Artin.\nMoreover, $V_\\bu = V_\\bu(E)$, where $E$ is a Gieseker semistable sheaf of rank $r$, degree $d$ and $c_2 = n$,\n$U_\\bu = V_\\bu(E^{**})$, and $W_\\bu = W_\\bu(E^{**}\/E)$.\n\\end{propos}\n\n \\begin{proof} \nThe argument of~\\refp{tet0-ss} proves that there is a filtration on $V_\\bu$ in which there are several factors\nwhich are Artin representations of dimension $(1,2,1)$ and one supermonadic factor of dimension $\\alpha(r,d,n-k)$\nfor some $0 \\le k \\le n$. But\n\\begin{equation*}\n\\langle \\tet^1, (1,2,1) \\rangle = 2r > 0,\n\\end{equation*}\nhence Artin factors can appear only before the supermonadic factor. This proves that the filtration\ngives the required exact sequence.\n\nApplying the functor $\\mathcal{C}$ to the exact sequence and taking into account $\\CH^2(W_\\bu) = \\CH^3(U_\\bu) = 0$,\nwe get the long exact sequence of cohomology sheaves\n\\begin{equation*}\n0 \\to \\CH^2(V_\\bu) \\to \\CH^2(\\CU_\\bu) \\to \\CH^3(W_\\bu) \\to \\CH^3(V_\\bu) \\to 0.\n\\end{equation*}\nIf $\\CH^3(V_\\bu) \\ne 0$ then the argument of the proof of~\\refp{tet0-ss} shows that\nthere is a surjection $V_\\bu \\to W_\\bu(\\CO_P)$ which, as we observed, contradicts to $\\tnt$-semistability of $V_\\bu$.\nThus $V_\\bu$ is monadic. Denote $E = \\CH^2(V_\\bu)$, $\\CE = \\CH^2(U_\\bu)$ and $F = \\CH^3(W_\\bu)$. Then the above sequence\ncan be rewritten as\n\\begin{equation*}\n0 \\to E \\to \\CE \\to F \\to 0.\n\\end{equation*}\nNote that $\\CE$ is locally free by~\\refl{smlf}, and $F$ is Artin since $W_\\bu$ is. Dualizing the sequence and taking into account\nthat $\\underline{\\Hom}(F,\\CO) = \\underline{\\Ext}^1(F,\\CO) = 0$ since $F$ is Artin,\nwe deduce that $E^* = \\CE^*$. Therefore $E^{**} = \\CE^{**} = \\CE$ since $\\CE$ is locally free and the map $E \\to \\CE = E^{**}$\nis the canonical embedding. Thus $E$ is torsion free and $F \\cong E^{**}\/E$. Moreover, by~\\refp{derequ}\nit follows that $V_\\bu = V_\\bu(E)$ and $U_\\bu = V_\\bu(E^{**})$, while $W_\\bu = W_\\bu(E^{**}\/E)$.\n\nWe finish by noting that $E$ is Gieseker semistable by~\\refl{monadic-sheaf}.\n \\end{proof} \n\n\n\n\\cor{coprime-stable}\nIf $r$ and $d$ are coprime then a $\\tet^0$-semistable representation is $\\tnt$-semistable if and only if it has no Artin quotients,\ni.e., if it is monadic.\n\\end{corol}\n \\begin{proof} \nFirst let us show that if $r$ and $d$ are coprime and $V_\\bu$ is a supermonadic $\\tet^0$-semistable representation then it is $\\tet^0$-stable.\nIndeed, $V_\\bu = V_\\bu(E)$ for a Mumford semistable sheaf $E$ by~\\refl{monadic-sheaf}, and moreover, the sheaf $E$ is locally free\nby~\\refl{smlf}. So, if $0 \\to V'_\\bu \\to V_\\bu \\to V''_\\bu \\to 0$ is an exact sequence of representations with both $V'_\\bu$ and $V''_\\bu$ nonzero and\n$\\mu_{\\tet^0}(V'_\\bu) = \\mu_{\\tet^0}(V''_\\bu) = 0$ then by~\\refl{eq} we have an exact sequence\n\\begin{equation*}\n0 \\to \\CH^2(V'_\\bu) \\to E \\to \\CH^2(V''_\\bu) \\to \\CH^3(V'_\\bu) \\to 0,\n\\end{equation*}\nand moreover, $V''_\\bu$ is monadic, $\\mu_M(\\CH^2(V''_\\bu)) = \\mu_M(E) = d\/r$, and $\\CH^3(V'_\\bu)$ is Artin.\nBut since $r$ and $d$ are coprime either the rank and the degree of $\\CH^2(V''_\\bu)$ are zero, or equal to $r$ and $d$ respectively.\nThe first is impossible since then $F := \\CH^2(V''_\\bu)$ is an Artin sheaf and $V''_p = \\Ext^1(Q_p(1-p),F) = 0$, so $V''_\\bu = 0$.\nThe second is impossible since then the rank of $\\CH^2(V'_\\bu)$ is zero, and as $\\CH^2(V'_\\bu)$ being a subsheaf in $E$\nis torsion free, it should be zero. Thus $V'_\\bu$ is a nonzero Artin subrepresentation in $V_\\bu$ which means that $V^\\vee_\\bu$\nhas a nonzero Artin quotient representation and hence cannot be monadic.\n\nNow let $V_\\bu$ be an arbitrary $\\tet^0$-semistable but not $\\tnt$-semistable representation. Consider all composition factors of $V_\\bu$\nin the category of $\\tet^0$-semistable representations. By~\\refp{tet0-ss} and the above argument these are Artin representations and\nthe supermonadic part of $V_\\bu$. Among those Artin representations have positive $\\tet^1$-slope, hence the supermonadic part\nis the only composition factor with negative $\\tet^1$-slope. So, the only way how $V_\\bu$ can be not $\\tnt$-semistable\nis when $V_\\bu$ has an Artin quotient.\n \\end{proof} \n\n\n\n\n\n\\ssec{moduli-spaces}{Moduli spaces}\n\nLet $\\CM^\\tet_\\tau(r_1,r_2,r_3)$ denote the moduli space of $\\tet$-semistable\n$(r_1,r_2,r_3)$-dimensional representations of the quiver $\\bpt$, as\ndefined by\nKing ~\\cite{K}.\nIt is a coarse moduli space for families of $\\tet$-semistable representations\nof the quiver $\\bpt$ of dimension $(r_1,r_2,r_3)$. In particular, its closed points\nare in a bijection with S-equivalence classes of $\\tet$-semistable representations.\n\nFor rational $\\tet$ there is an explicit GIT construction of the moduli space. One starts with {\\sf the representation space} of $\\bpt$:\n\\begin{equation}\\label{rt-embedded}\n\\mathsf{R}_\\tau(r_1,r_2,r_3) \\subset \\Hom(\\C^{r_1}\\otimes (A_\\tau^!)_1,\\C^{r_2}) \\times \\Hom(\\C^{r_2}\\otimes (A_\\tau^!)_1,\\C^{r_3}),\n\\end{equation}\nconsisting of those pairs of maps $f:\\C^{r_1} \\otimes (A_\\tau^!)_1 \\to \\C^{r_2}$, $g:\\C^{r_2} \\otimes (A_\\tau^!)_1 \\to \\C^{r_3}$ such that the composition\n$g\\circ (f \\otimes \\id):\\C^{r_1} \\otimes (A_\\tau^!)_1 \\otimes (A_\\tau^!)_1 \\to \\C^{r_3}$ factors through the multiplication map\n$\\C^{r_1} \\otimes (A_\\tau^!)_1 \\otimes (A_\\tau^!)_1 \\to \\C^{r_1} \\otimes (A_\\tau^!)_2$. Clearly, \\eqref{rt-embedded} is a Zarisky closed subset\nin an affine space.\nThe group\n\\begin{equation*}\n\\GL(r_1,r_2,r_3)=\\GL(r_1)\\times \\GL(r_2)\\times \\GL(r_3)\n\\end{equation*}\nacts naturally on  $\\mathsf{R}_\\tau(r_1,r_2,r_3)$.\nGiven a rational polarization $\\tet$, in the trivial\nbundle,\nlet $\\C[\\mathsf{R}_\\tau(r_1,r_2,r_3)]^{\\GL(r_1,r_2,r_3),p\\tet}$\nbe the vector space of polynomial  $\\GL(r_1,r_2,r_3)$-semiinvariants of\nweight $p\\tet$ (this space is  declaired to be zero  unless\n$p\\tet$ is an integral weight).\nOne defines an associated GIT quotient by\n\\begin{equation*}\n\\mathsf{R}_\\tau(r_1,r_2,r_3) \/\\!\/_\\tet \\GL(r_1,r_2,r_3) := \\Proj\\left(\\bigoplus_{p = 0}^\\infty \\C[\\mathsf{R}_\\tau(r_1,r_2,r_3)]^{\\GL(r_1,r_2,r_3),p\\tet}\\right).\n\\end{equation*}\nThen,\naccording to ~\\cite{K}, one has \n$\\CM^\\tet_\\tau(r_1,r_2,r_3) \\cong \\mathsf{R}_\\tau(r_1,r_2,r_3) \/\\!\/_\\tet\n\\GL(r_1,r_2,r_3)$.\nFurther,\nit turns out that the space of all polarizations $\\tet$ has a chamber structure\nand the moduli space $\\CM^\\tet_\\tau(r_1,r_2,r_3)$ depends only on the chamber in which $\\tet$ sits.\nThis allows to define $\\CM^\\tet_\\tau$ for arbitrary (real) polarization $\\tet$ by taking rational $\\tet'$ in the same chamber as $\\tet$\nand setting $\\CM^\\tet_\\tau(r_1,r_2,r_3) := \\CM^{\\tet'}_\\tau(r_1,r_2,r_3)$.\n\nAnalogously one constructs a coarse moduli space $\\Mtt(r_1,r_2,r_3)$ for a pair of polarizations $(\\tet,\\tet')$ by taking\nan arbitrary polarization in the chamber containing $\\tet + \\eps\\tet'$ for all sufficiently small and positive $\\eps$.\n\nIt has been shown in \\cite{K} that  the moduli space of\nsemistable representations  of any  quiver that has no\n oriented cycles is a projective variety. It follows,\n since\nthe quiver $\\bpt$ has no oriented cycles,\nthat each of the above moduli spaces $\\CM^\\tet_\\tau(r_1,r_2,r_3)$\nis a projective variety. This variety comes equipped\nwith a  natural $\\SL(H)$-action.\nFinally, we remark that  if the dimension vector\n$(r_1,r_2,r_3)$ is primitive, i.e.,  indivisible, then \n $\\CM^\\tet_\\tau(r_1,r_2,r_3)$ is a {\\em fine}  moduli space.\n\\medskip\n\n\n\n\n\n\n\n\n\n\n\n\n\\nc{\\reduced}{{\\mathrm{red}}}\n\nBelow we discuss moduli spaces of several classes of representations of the quiver $\\bpt$.\nFirst, recall that by~\\refc{tet-ss-artin} any $\\tet^0$-semistable representation of dimension $(n,2n,n)$\nis an Artin representation. So, we refer to the corresponding moduli space as to {\\em the moduli space of Artin represnetations}\nand denote it by ${}^A\\mathsf{M}_\\tau(n,2n,n)$. Thus we have\n\\begin{equation*}\n{}^A\\mathsf{M}_\\tau(n,2n,n) := \\CM^{\\tet^0}_\\tau(n,2n,n) = \\mathsf{R}_\\tau(n,2n,n)\/\\!\/_{\\tet^0} \\GL(n,2n,n).\n\\end{equation*}\nThe moduli space of Artin representations is highly non-reduced. In what follows, however, we will\nonly need a description of the underlying reduced scheme which we denote by ${}^A\\mathsf{M}_\\tau(n,2n,n)_\\reduced$.\nThe proof of the following Proposition can be found in the Appendix.\n\n\\prop{artin-moduli-k}\nThe map $W_\\bu \\mapsto \\supp(\\CH^3(W_\\bu)))$ gives an $\\SL(H)$-equivariant isomorphism\n\\begin{equation*}\n{}^A\\mathsf{M}_\\tau(n,2n,n)_\\reduced \\cong S^n(\\PP(H)).\n\\end{equation*}\n\\end{propos}\n\n\n\n\nDimension vector $(n,2n,n)$ considered here is a special case of the vector $\\alpha(r,d,n)$ for $r = d = 0$.\nWe also consider a general moduli space of $\\tet^0$-semistable $\\alpha(r,d,n)$-dimensional representations\nof $\\bpt$ which we call {\\sf the Uhlenbeck moduli space}\n(the reasons behind this choice of a name will become clear later) of sheaves on $\\BPt$. We denote it\n\\begin{align*}\n\\UMt(r,d,n) := \\CM^{\\tet^0}_\\tau(\\alp(r,d,n)) = \\mathsf{R}_\\tau(\\alp(r,d,n))\/\\!\/_{\\tet^0} \\GL(\\alp(r,d,n)).\n\\end{align*}\nThe last moduli space we consider is the moduli space of $\\tnt$-semistable $\\alpha(r,d,n)$-dimensions representations\n\\begin{align*}\n\\GMt(r,d,n) := \\Mt(\\alp(r,d,n)) = \\mathsf{R}_\\tau(\\alpha(r,d,n))\/\\!\/_{\\tnt}\\GL(\\alpha(r,d,n)).\n\\end{align*}\nWe call this moduli space {\\sf the Gieseker moduli space} of sheaves on $\\BPt$. The reason for this is the following\n\n\\prop{nevins}\nThe Gieseker moduli space $\\GMt(r,d,n)$ is isomorphic to the moduli space of Gieseker\nsemistable sheaves on $\\BPt$ constructed in~\\cite{ns}. Moreover, the open subset\nof $\\GMt(r,d,n)$ of $\\tnt$-stable representations corresponds, via the\nisomorphism,\nto the open set of\nGieseker stable sheaves    on $\\BPt$.\n\\end{propos}\n \\begin{proof} \nThis follows immediately from~\\refl{kronecker-stability}, as the functor of $\\tnt$-(semi)stable representations of the quiver $\\bpt$\nis isomorphic to the functor of (semi)stable Kronecker complexes considered in~\\cite{ns}.\n \\end{proof} \n\n\\cor{sm}\nIf $\\mathsf{gcd}(r,d,n)=1$, then $\\GMt(r,d,n)$ is a fine moduli space,\nmoreover,\nthis moduli space is smooth.\n\\end{corol}\n \\begin{proof} \nBy~\\cite[Prop.~7.15]{ns} the moduli space of semistable Kronecker complexes is fine. As the functor of semistable Kronecker complexes\nis isomorphic to the functor of $\\tnt$-semistable representations of $\\bpt$, we conclude that $\\GMt(r,d,n)$ is also a fine moduli space.\nMoreover, from $\\mathsf{gcd}(r,d,n) = 1$ it follows that all $\\tnt$-semistable representations of the quiver are $\\tnt$-stable,\nhence all Gieseker semistable sheaves are stable, and so the smoothness of the moduli space is proved in~\\cite[Thm.~8.1]{ns}.\n \\end{proof} \n\n\n\n\n\n\n\n\n\\ssec{stratifications}{Stratifications}\n\nRecall that by~\\refp{tnt-ss} any $\\tnt$-semistable representation $V_\\bu$ can be written as $V_\\bu(E)$\nfor a Gieseker semistable sheaf $E$.\nThis gives a decomposition of the moduli space $\\GMt(r,d,n)$ into pieces by the length of $E^{**}\/E$.\nIt will be shown in the Appendix at the end of the paper that\nthis  decomposition is, in fact, an algebraic stratification. \n\nThe proofs of   other results  of this subsection stated below\nare also deferred to  section 4.\n\n\\lem{stratification}\nThe Gieseker moduli space $\\GMt(r,d,n)$ is naturally stratified by locally closed $\\SL(H)$-invariant subsets\n\\begin{equation*}\n\\GMt(r,d,n) = \\bigsqcup_{0\\le k\\le n} \\GMt^k(r,d,n),\n\\end{equation*}\nwhere the stratum $\\GMt^k(r,d,n)$\ncorresponds to the locus of Gieseker semistable sheaves $E$ on $\\BPt$ with $c_2(E^{**}\/E)=k$.\n\\end{lemma}\n\nIn particular, the open stratum $\\GMt^0(r,d,n) \\subset \\GMt(r,d,n)$ parameterizes locally free Gieseker semistable sheaves.\n\nThere is also an analogous stratification of the Uhlenbeck moduli space.\n\n\\lem{stratification-uhlenbeck}\nThe Uhlenbeck moduli space $\\UMt(r,d,n)$ is naturally stratified by locally closed $\\SL(H)$-invariant subsets\n\\begin{equation*}\n\\UMt(r,d,n) = \\bigsqcup_{0\\le k\\le n} \\UMt^k(r,d,n),\n\\end{equation*}\nwhere the stratum $\\UMt^k(r,d,n)$\ncorresponds to the locus of Mumford semistable sheaves $E$ on $\\BPt$ with $c_2(E^{**}\/E)=k$.\n\\end{lemma}\n\n\n\n\n\n\n\n\nThe {\\em natural} stratifications of the Gieseker and the Uhlenbeck moduli spaces have highly nonreduced\nstrata. The reason for that is the nonreducedness of the moduli space of Artin sheaves, or going one step\ndeeper, the nonreducedness of the scheme of ``commutative points'' of $\\BPt$. However, as we have only\ntopological applications in mind, the nilpotents in the structure sheaves of the strata are irrelevant\nfor our purposes, so by this reason we replace each stratum of both moduli spaces by the reduced\nscheme underlying the corresponding natural stratum. \nThus, from now on we will abuse the notation and write\n$\\GMt^k(r,d,n)$, resp. $\\UMt^k(r,d,n)$, for the stratum of the\ncorresponding stratification equipped with {\\textsf{reduced scheme structure}}.\n\n\n\nIt follows from standard results of geometric invariant theory\n(see~\\cite{DH}) that there is a canonical $\\SL(H)$-equivariant \nprojective  morphism\n\\begin{equation*}\n\\gam_\\tau\\colon\\GMt(r,d,n)\\to\\UMt(r,d,n),\n\\end{equation*}\nresulting from a specialization of $\\tnt$-semistability\nto $\\tet^0$-semistability.\n\n\\begin{Rem} We do not know how to define the morphism $\\gam_\\tau$ in\n    terms\nof coherent sheaves on $\\BPt$, without using identifications of \nmoduli spaces of coherent sheaves with the corresponding moduli spaces\nof\nquiver representations.\n\\end{Rem}\n\n\nThe main result of this section establishes a compatibility between the constructed statifications\nof the Gieseker and Uhlenbeck moduli spaces and describes the relation between the strata.\n\n\n\n\\th{str}\n(1) The map $\\gamma_\\tau:\\GMt(r,d,n) \\to \\UMt(r,d,n)$ is compatible with the stratifications, i.e.,\n$\\gamma_\\tau(\\GMt^k(r,d,n)) \\subset \\UMt^k(r,d,n)$.\n\nIn the case where  $\\tau\\neq 0$ and the integers $r$ and $d$ are coprime, \nthe  Gieseker compactification is smooth and the following holds:\n\\vskip 2pt \n\n(2) The  open set $\\GMt^0(r,d,n)$ is the locus of  Gieseker stable\nsupermonadic representations; furthermore, this  open set\ncorresponds, via the isomorphism of Proposition \\ref{P:nevins},\nto the\nlocus of locally free Gieseker stable sheaves on $\\BPt$.\nMoreover, the\nmap  $\\gamma_\\tau$ yields an isomorphism $\\GMt^0(r,d,n){\\stackrel{\\sim}{\\longrightarrow}}\n\\UMt^0(r,d,n)$.\n\n\n\n(3) For any\n$k > 0$ one has  an $\\SL(H)$-equivariant isomorphism\n\\begin{equation*}\n\\UMt^k(r,d,n) \\cong \\GMt^0(r,d,n-k) \\times {}^A\\mathsf{M}_\\tau(k,2k,k)_\\reduced \\cong \\GMt^0(r,d,n-k) \\times S^k{\\mathbb P}(H).\n\\end{equation*}\nUsing this isomorphism, for  $E\\in\n\\GMt^k(r,d,n)$ we have\n\\begin{equation*}\n\\gamma_\\tau(E) = (E^{**},\\supp(E^{**}\/E)).\n\\end{equation*}\nIn particular, \nthe fiber of $\\gamma_\\tau$ over a point $(\\CE,D) \\in \\GMt^0(r,d,n-k) \\times S^k{\\mathbb P}^1 \\subset \\UMt(r,d,n)$\nis the underlying reduced scheme for the moduli space of subsheaves $E \\subset \\CE$ with $\\supp(\\CE\/E) = D$.\n\\eth\n\n\n\\begin{Rem} The relation between the moduli spaces $\\GMt(r,d,n)$ and $\\UMt(r,d,n)$\nis completely analogous to that of the Gieseker and Uhlenbeck compactifications of the moduli\nspaces of vector bundles on commutative algebraic surfaces (this justifies\nthe use of these names in our situation).\nAn important difference between commutative and noncommutative cases\ndifference is in the dimensions of the strata: in the commutative case the second factors in the product\nexpression for Uhlenbeck strata are symmetric powers of the surface, while in the noncommutative\ncase we only have a symmetric power of a curve.\nThis fact will play a crucial role in subsequent sections.\n\\end{Rem} \n\n\n\n\n\n\n\n\n\\sec{ch6}{Rank $1$ sheaves and the Calogero--Moser space}\n\nIn this section we study the Gieseker and the Uhlenbeck moduli spaces of rank 1 and degree 0\ntorsion free sheaves on $\\BPt$.\n\n\\ssec{calmos}{The compactifications}\n\n\n\n\n\n\nTo unburden the notation we write\n$$\n\\obMt^n=\\GMt(1,0,n),\\quad \\hbMt^n=\\UMt(1,0,n), \\quad\n\\obMt^{n,k}=\\GMt^k(1,0,n), \\quad\n\\hbMt^{n,k}=\\UMt^k(1,0,n).\n$$\nIt is well known, cf. \\cite[Prop.~8.13]{ns}, that the open strata $\\GMt^{n,0} \\subset \\GMt^n$ and\n$\\UMt^{n,0} \\subset \\UMt$ can be identified with the Calogero--Moser\nspace.\nThus, the varieties $\\obMt^n$ and $\\hbMt^n$ provide\ntwo {\\em different}  compactifications  of the Calogero--Moser\nspace,\nto be called\n{\\sf the Gieseker and the Uhlenbeck compactifications}, respectively.\nFurthermore, the variety $\\obMt^n$ being smooth, the morphism $\\gamma_\\tau \\colon \\obMt^n \\to \\hbMt^n$ is a resolution of singularities.\nLater on, we will use this resolution to compute the stalks of the IC sheaf of the Uhlenbeck compactification.\n\n\n\n\\th{isom}\nFor $\\tau \\ne 0$ we have $\\SL(H)$-equivariant isomorphisms\n\\begin{equation*}\n\\GMt^{n,0} = \\UMt^{n,0} \\cong \\sM_\\tau^n.\n\\end{equation*}\n\\eth\n \\begin{proof} \nThe first isomorphism follows from~\\refp{nevins} and~\\cite[Prop.~8.13]{ns}. The second\nis a consequence of the first and~\\reft{str}(2).\n \\end{proof} \n\n\n\n\nIn \\refss{moduli-spaces}, we have introduced a contraction map $\\gamma_\\tau\\colon\\obMt^n\\to\\hbMt^n$.\nBy~\\reft{str} it sends a torsion free sheaf $E$ to $(E^{**},\\supp(E^{**}\/E))$,\nand for any $0\\leq k\\leq n$, we have\n\\begin{equation*}\n\\gamma_\\tau(\\obMt^{n,k})\\ =\\ \\hbMt^{n,k}\\ =\\ \\sM_\\tau^{n-k}\\x S^k{\\mathbb P}(H).\n\\end{equation*}\nBelow, we are going to describe the fibers of the map  $\\gamma_\\tau$.\nChoose $0 \\ne h \\in H$ and let\n\\begin{equation*}\n{\\mathbb A}^1_h := {\\mathbb P}(H) \\smallsetminus \\{h\\}.\n\\end{equation*}\nThus, ${\\mathbb A}^1_h$ is an affine line and, for any $n\\geq 0$, the set\n$S^n{\\mathbb A}^1_h$ is  Zariski open and dense in $S^n{\\mathbb P}(H)$.\nIt is clear that these sets for all $h \\in H$ form an open covering of $S^n\\PP(H)$.\n\nConsider the zeroth Calogero--Moser space $\\sM_\\tau^0$. Clearly, this is just a point,\nand under the isomorphism of~\\reft{isom} it corresponds to the trivial line bundle $\\CO_{\\BPt}$.\nFor each nonzero vector $h \\in H$ consider the open subset\n$\\{\\CO\\} \\times S^n{\\mathbb A}^1_h = \\sM^0_\\tau \\times S^n{\\mathbb A}^1_h \\subset \\sM^0_\\tau \\times S^n\\PP(H) = \\hbMt^{n,n}$\nand its preimage under the map $\\gamma_\\tau:\\obMt^{n,n} \\to \\hbMt^{n,n}$:\n\\begin{equation}\\label{equation-bnh}\nB^n_h =\n\\gamma_\\tau^{-1}(\\{\\calO\\}\\x S^n{\\mathbb A}^1_h).\n\\end{equation}\nAnalogously, we can take arbitrary locally free sheaf $\\CE$ of rank 1 and degree 0, consider the locally closed subset\n$\\{\\CE\\} \\times S^k{\\mathbb A}^1_h \\subset \\sM^m_\\tau \\times S^k{\\mathbb A}^1_h \\subset \\sM^m_\\tau \\times S^k\\PP(H) = \\hbMt^{m+k,k}$\nand its preimage under the map $\\gamma_\\tau:\\obMt^{m+k,k} \\to \\hbMt^{m+k,k}$:\n\n\n\\prop{dep}\nFor any locally free sheaf $\\calE$ of rank $1$, $\\calE\\in\\sM_\\tau^{m}$,\nthere is an isomorphism\n\\begin{equation*}\n\\gamma_\\tau^{-1}(\\{\\calE\\}\\x S^k{\\mathbb A}^1_h)\\cong B^k_h.\n\\end{equation*}\n\\end{propos}\n \\begin{proof} \nThere is an integer $p \\in \\ZZ$ and two maps\n\\begin{equation*}\n\\phi:\\CO(-p) \\to \\CE\n\\qquad\\text{and}\\qquad\n\\phi':\\CE \\to \\CO(p)\n\\end{equation*}\nsuch that $i^*(\\phi) = h^p$ and $i^*(\\phi') = h^p$. Indeed, take $p$ sufficiently large to have\n\\begin{equation*}\n\\Ext^1(\\CE,\\CO(p-1)) = \\Ext^1(\\CO(-p),\\CE(-1)) = 0\n\\end{equation*}\nand define $\\phi$ and $\\phi'$ as lifts of the compositions in the next two diagrams\n\\begin{equation*}\n\\xymatrix@C=1em{\n&& \\CO(-p) \\ar[r] \\ar@{..>}[d]_\\phi & i_*i^*\\CO(-p) \\ar[d]^{h^p} \\\\\n0 \\ar[r] & \\CE(-1) \\ar[r] & \\CE \\ar[r] & i_*i^*\\CE \\ar[r] & 0\n}\n\\qquad\n\\xymatrix@C=1.5em{\n&& \\CE \\ar[r] \\ar@{..>}[d]_\\phi' & i_*i^*\\CE \\ar[d]^{h^p} \\\\\n0 \\ar[r] & \\CO(p-1) \\ar[r] & \\CO(p) \\ar[r] & i_*i^*\\CO(p) \\ar[r] & 0\n}\n\\end{equation*}\n\n\nThe maps $\\phi$ and $\\phi'$ give morphisms between the moduli spaces of surjections\n$\\CE \\twoheadrightarrow F$ and $\\CO \\twoheadrightarrow F$ onto Artin sheaves $F$\nof length $k$ with $\\supp(F) \\subset {\\mathbb A}^1_h$:\n\\begin{equation*}\n(f:\\CE \\twoheadrightarrow F) \\mapsto (f \\circ \\phi(p):\\CO \\twoheadrightarrow F)\n\\qquad\\text{and}\\qquad\n(g:\\CO \\twoheadrightarrow F) \\mapsto (g(p) \\circ \\phi':\\CE \\twoheadrightarrow F).\n\\end{equation*}\nwhere in both cases we identify $F$ with $F(p)$ via $h^p$. It is straightforward to check\nthat these maps are mutually inverse. As the preimages $\\gamma_\\tau^{-1}(\\CE\\times S^k{\\mathbb A}^1_h)$ and\n$B^k_h = \\gamma_\\tau^{-1}(\\CO\\times S^k{\\mathbb A}^1_h)$ are identified by~\\reft{str}(3) with the reduced schemes\nunderlying these moduli spaces, the constructed maps provide an isomorphism between them as well.\n \\end{proof} \n\n\n\\newcommand{{{\\on{disj}}}}{{{\\on{disj}}}}\n\n\nThe spaces $B^n_h$ come with a natural map $\\gamma_\\tau:B^n_h \\to S^n{\\mathbb A}^1_h$.\nIn fact they enjoy the following factorization property. Define\nthe open subset $(S^{k_1}{\\mathbb A}^1\\times S^{k_2}{\\mathbb A}^1)_{{\\on{disj}}} \\subset S^{k_1}{\\mathbb A}^1\\times S^{k_2}{\\mathbb A}^1$ as\n\\begin{equation}\\label{disjoint-divisors}\n(S^{k_1}{\\mathbb A}^1\\times S^{k_2}{\\mathbb A}^1)_{{\\on{disj}}} = \\{ (D_1,D_2) \\in S^{k_1}{\\mathbb A}^1 \\times S^{k_2}{\\mathbb A}^1 \\ |\\ \\supp(D_1) \\cap \\supp(D_2) = \\emptyset \\}\n\\end{equation}\nand\n\\begin{equation*}\n(B_h^{k_1} \\times B_h^{k_2})_{{\\on{disj}}} := (\\gamma_\\tau \\times \\gamma_\\tau)^{-1}((S^{k_1}\\AA_h^1 \\times S^{k_2}\\AA_h^1)_{{\\on{disj}}}) \\subset B_h^{k_1} \\times B_h^{k_2}.\n\\end{equation*}\n\n\\prop{factorization-b}\nThe collection of spaces $B^n_h$ has a factorization property, i.e.\\ there is a collection of maps\n\\begin{equation*}\n\\psi_{k_1,k_2}:(B_h^{k_1}\\times B_h^{k_2})_{{\\on{disj}}} \\to B_h^{k_1+k_2}\n\\end{equation*}\nfor all positive integers $k_1,k_2$ which has the following properties: \n\\begin{itemize}\n\\item (associativity) $\\psi_{k_1+k_2,k_3} \\circ (\\psi_{k_1,k_2} \\times \\id) = \\psi_{k_1,k_2+k_3} \\circ (\\id \\times \\psi_{k_1,k_2})$ for all $k_1,k_2,k_3$;\n\\item (commutativity) the maps $\\psi_{k,k} \\colon (B_h^k\\times B_h^k)_{{\\on{disj}}} \\to B_h^{2k}$ commute with the transposition of the factors on the source for all $k$;\n\\item (compatibility with the addition) the following diagram is Cartesian\n\\begin{equation*}\n\\xymatrix{\n(B_h^{k_1}\\times B_h^{k_2})_{{\\on{disj}}} \\ar[rr]^-{\\psi_{k_1,k_2}} \\ar[d]_{\\gamma_\\tau\\times\\gamma_\\tau} && B_h^{k_1+k_2} \\ar[d]^{\\gamma_\\tau} \\\\\n(S^{k_1}{\\mathbb A}^1\\times S^{k_2}{\\mathbb A}^1)_{{\\on{disj}}} \\ar[rr]^-{a_{k_1,k_2}} \t&& S^{k_1+k_2}{\\mathbb A}^1\n}\n\\end{equation*}\nwhere the bottom arrow is the addition morphism: $(D_1,D_2) \\mapsto D_1 + D_2$.\n\\end{itemize}\n\n\\end{propos}\n \\begin{proof} \nA point of $(B^{k_1}_h \\times B^{k_2}_h)_{{\\on{disj}}}$ can be represented by a pair of Artin sheaves $F_1,F_2$ of length $k_1$ and $k_2$ respectively\nwith epimorphisms $\\CO \\twoheadrightarrow F_1$ and $\\CO \\twoheadrightarrow F_2$. \nConsider the sum $F = F_1 \\oplus F_2$ and the map $\\CO \\to F$ given by the sum of the two above maps. Let us show it is surjective.\nBy~\\refp{morph2}(2) it is enough to check that the map $\\CO_{\\PP(H)} \\to i^*F = i^*F_1 \\oplus i^*F_2$ is surjective.\nBut as the supports of the sheaves $i^*F_1$ and $i^*F_2$ are disjoint, this is equivalent to the surjectivity\nof each of the maps $\\CO_{\\PP(H)} \\to i^*F_1$ and $\\CO_{\\PP(H)} \\to i^*F_2$ which we have again by~\\refp{morph2}(2).\nThis means that the sheaf $F$ with the constructed epimorphism $\\CO \\twoheadrightarrow F$ give a point of $B^{k_1+k_2}_h$, and thus a morphism\n\\begin{equation*}\n\\psi^B_{k_1,k_2} \\colon (B^{k_1}_h \\times B^{k_2}_h)_{{\\on{disj}}} \\to B^{k_1+k_2}_h\n\\end{equation*}\nis defined. Let us show it is a factorization. Indeed, the associativity and the commutativity properties\nare evident, so it remains to check the compatibility with the addition, i.e.\\ that the corresponding diagram is Cartesian.\nThe commutativity of the diagram follows from~\\refl{artin-support}, so it remains to note that if $F$ is an Artin sheaf \nof length $k_1+k_2$ such that $\\supp(F) = D_1 + D_2$ with disjoint divisors $D_1 \\in S^{k_1}\\AA^1_h$ and $D_2 \\in S^{k_2}\\AA^1_h$, \nthen $F$ has a unique representation as a direct sum $F = F_1 \\oplus F_2$ with $\\supp(F_1) = D_1$\nand $\\supp(F_2) = D_2$ (this follows easily from~\\refp{Art0}). \n \\end{proof} \n\n\nThe variety $B_h^k$ has a nice linear algebra description.\nFix a vector space $V$ of dimension~$k$. Let\n\\eq{rel}\n\\widetilde{B}{}^k_h = \\{ (Y,Z,v)\\in \\End(V)\\times \\End(V)\\times V\\  |\\  [Y,Z]=\\tau Z^3\\ \\text{and $v$ is cyclic} \\}.\n \\end{equation} \nHere we say that a vector $v$ is {\\sf cyclic} for a pair of matrices $(Y,Z)$ if\nthere is no proper vector subspace $V' \\subset V$ that contains $v$ and is both $Y$-stable and $Z$-stable.\nOne has a natural  $\\GL(V)$-action on $\\widetilde{B}{}^k_h$\ngiven by $g:\\ (Y,Z,v)\\mapsto (gYg^{-1}, gZg^{-1}, gv)$.\n\n\n\\th{26}\nThe action of $\\GL(V)$ on $\\widetilde{B}{}^k_h$ is free and\n\\begin{equation*}\nB^k_h \\cong (\\widetilde{B}{}^k_h\/\\GL(V))_\\reduced.\n\\end{equation*}\nUnder this isomorphism the map $\\gamma_\\tau:B^k_h \\to S^k{\\mathbb A}^1_h$ is induced by the map $\\widetilde{B}^k_h \\to S^k{\\mathbb A}^1_h$\nwhich takes $(Y,Z,v)$ to $\\Spec(Y)$.\n\\eth\n\n \\begin{proof} \nAssume that $g\\in \\GL(V)$ acts trivially on a triple $(Y,Z,v)$. Let\n$V^g\\subset V$ be the space of invariants of $g$. Then $v\\in V^g$ and\n$Y(V^g)\\subset V^g,\\ Z(V^g)\\subset V^g$, hence $V^g=V$ as $v$ is cyclic, and so $g=1$.\n\n\nNow consider the moduli space of surjections $\\CO \\twoheadrightarrow F$ with $F$ an Artin sheaf\nof length $k$ with $\\supp(F) \\subset {\\mathbb A}^1_h$. Let us show it is isomorphic to the quotient\n$\\widetilde{B}{}^k_h\/\\GL(V)$. Then passing to the underlying reduced schemes will prove the Theorem.\n\n\nChoose symplectic coordinates $x,y$ in $H$ such that the point $h \\in \\PP(H)$ is given by the equation $x = 0$.\nLet $(Y,Z,v)$ be a point of $\\widetilde{B}{}^k_h$.\nConsider a graded vector space $V[x] := V \\otimes {\\mathbb C}[x]$ with $\\deg x = 1$, with $x$ acting by multiplications, and with the action of $y$ and $z$ defined by\n\\begin{equation*}\ny = xY -\\tau x^2Z^2\\partial_x,\n\\qquad\nz = xZ.\n\\end{equation*}\nThe commutation $[x,z] = 0$ is clear. Moreover, we have\n\\begin{equation*}\n[y,z] = [xY -\\tau x^2Z^2\\partial_x, xZ] = x^2[Y,Z] - \\tau x^2Z^2[\\partial_x,x]Z = \\tau x^2Z^3 - \\tau x^2Z^3 = 0\n\\end{equation*}\nand\n\\begin{equation*}\n[x,y] = [x, xY -\\tau x^2Z^2\\partial_x] = - \\tau x^2Z^2[x,\\partial_x] = \\tau x^2Z^2 = \\tau z^2.\n\\end{equation*}\nThis shows that $V[x]$ is a graded $A^\\tau$-module. Let $F$ be the corresponding coherent sheaf on $\\BPt$.\nBy definition the map $x:V[x] \\to V[x]$ is injective with finite dimensional cokernel, hence the map $x:F \\to F(1)$\nis an isomorphism. In particular, the Hilbert polynomial $h_F(t)$ is constant, hence $F$ is an Artin sheaf\nwith $\\supp(F) \\subset {\\mathbb A}^1_h$ by~\\refl{artin-support}. Moreover, the length of $F$ is equal to $\\dim V = k$\nand the vector $v \\in V \\subset V[x]$ gives a morphism of $A^\\tau$-modules $A^\\tau \\to V[x]$. By cyclicity\nassumption the map is surjective in all components of sufficiently large degree, hence the corresponding\nmorphism of sheaves $\\CO \\to F$ is surjective.\nNote that the construction is $\\GL(V)$-invariant.\n\nVice versa, let $\\CO \\twoheadrightarrow F$ be a surjection with $F$ an Artin sheaf of length $k$\nand $\\supp(F) \\subset {\\mathbb A}^1_h$. Choose an isomorphism $V \\cong H^0(\\BPt,F)$. Note that\nby~\\refl{artin-support} the map $x:F \\to F(1)$ is an isomorphism, hence it also\ninduces an isomorphism on the spaces of global sections $x \\colon H^0(\\BPt,F) \\xrightarrow{\\ \\sim\\ } H^0(\\BPt,F(1))$.\nWe denote by $x^{-1}$ the inverse isomorphism. We put $Y = x^{-1}y$, $Z = x^{-1}z$\nconsidered as endomorphisms of $V = H^0(\\BPt,F)$. Finally, we take $v$ to be the image\nof $1 \\in H^0(\\BPt,\\CO)$ in $V$ under the map $\\CO \\twoheadrightarrow F$.\n\nLet us show that~\\refe{rel} holds. First, we have $y = xY$, $z = xZ$ which gives relations\n\\begin{equation*}\nx^2Z = xZx,\n\\qquad\nxYxZ = xZxY,\n\\qquad\\text{and}\\qquad\nx^2Y - xYx = \\tau xZxZ.\n\\end{equation*}\nIt follows that\n\\begin{equation*}\nx^2(YZ - ZY) =\nx^2YZ - x^2ZY =\nxYxZ + \\tau xZxZ^2 - xZxY =\n\\tau x^2Z^3.\n\\end{equation*}\nIn the second equality we used the third relation,\nand in the third equality we used the first two relations.\nAs $x$ is an isomorphism, we deduce $[Y,Z] = \\tau Z^3$. So, it remains to show that $v$ is cyclic. For this\ntake an arbitrary subspace $V' \\subset V$ containing $v$ and closed under the action of $Y$ and $Z$.\nThe triple $(Y,Z,v)$ in the vector space $V'$ then gives an Artin sheaf $F'$ and a surjection $\\CO \\twoheadrightarrow F'$.\nThe embedding $V' \\subset V$ gives an embedding of sheaves $F' \\hookrightarrow F$ and it is clear that the original map\n$\\CO \\twoheadrightarrow F$ factors as $\\CO \\twoheadrightarrow F' \\hookrightarrow F$. It follows that $F' = F$\nand hence $V' = H^0(\\BPt,F') = H^0(\\BPt,F) = V$.\n\nThe two constructions are clearly mutually inverse and thus prove an isomorphism of moduli spaces\nand hence the first part of the Theorem.\nFor the second part it remains to show that $\\supp(F) = \\Spec(Y)$. But this is clear since\nthe action of the coordinate $y$ from $H^0(\\BPt,F) = V$ to $H^0(\\BPt,F(1)) = V$ is given by the operator $Y$.\n \\end{proof} \n\nWe use the identification $B^k_h \\cong\\ (\\widetilde{B}{}^k_h\/\\GL(V))_\\reduced$\nto investigate the properties of $B^k_h$.\n\n\\lem{nil}\nIf a pair $(Y,Z)$ satisfies \\refe{rel} then $Z$ is nilpotent.\n\\end{lemma}\n \\begin{proof} \nNote that $[Y,Z^p]=p\\tau Z^{p+2}$ for $p \\ge 3$. Since $\\tau\\ne0$, it follows that\n$\\Tr Z^{p+2}=0$ for any $p \\ge 3$. Hence $Z$ is nilpotent.\n \\end{proof} \n\n\\lem{ex}\nFor any nilpotent $Z$ there exist $Y$ and $v$ such that~\\refe{rel} holds.\n\\end{lemma}\n \\begin{proof} \nFirst, for any $u \\in {\\mathbb C}$ take\n\\begin{equation*}\nV = \\C[t]\/t^k,\\qquad\nY = u + \\tau t^3\\partial_t,\\quad\nZ = t,\\quad\nv = 1.\n\\end{equation*}\nClearly, \\refe{rel} holds,\nso we have an example in case when $Z$ is just one Jordan block.\nNote that $\\Spec(Y) = ku \\in S^k{\\mathbb A}^1_h$.\n\nFor arbitrary nilpotent $Z$ the Jordan decomposition of $Z$ is a direct sum\ndecomposition $Z = Z_1 \\oplus \\dots \\oplus Z_m$ with blocks of size $k_1, \\dots, k_m$.\nChoosing $m$ distinct complex numbers $u_1,\\dots,u_m$ we construct triples $(Y_i,Z_i,v_i)$\non $V_i = {\\mathbb C}[t]\/t^{k_i}$ such that $\\supp(Y_i,Z_i,v_i) = k_iu_i$. Factorization property of~\\refp{factorization-b}\nthen shows that the direct sum $(\\oplus Y_i,\\oplus Z_i,\\oplus v_i)$ is a point of $B^{k_1+\\dots+k_m}_h$.\n \\end{proof} \n\n\n\nWe will use a natural  one-to-one correspondence $\\lam\\mapsto O_\\lam$,\nbetween partitions of $k$ and the nilpotent conjugacy classes in\n$\\operatorname{End}(V)$, provided by Jordan normal form.\nLet $B_h^\\lam\\subset B^k_h$ denote the set of all triples $(Y,Z,v)$ satisfying \\refe{rel} with $Z\\in O_\\lam$.\n\n\\th{b-components}\nWe have a decomposition into a union of connected components\n\\begin{equation*}\nB^k_h=\\coprod_{\\lam \\in \\frP(k)} B^\\lam_h.\n\\end{equation*}\nThe component $B^\\lam_h$ is smooth, connected and $k$-dimensional.\n\\eth\n \\begin{proof} \nLet $O=O_\\lam$ be a nilpotent orbit of the group $\\GL(V)$. Let\n\\begin{equation*}\nN^*_O\\End(V)=\\{(Y,Z)\\ |\\ [Y,Z]=0\\}\\subset\\End(V) \\times \\End(V)\n\\end{equation*}\nbe the conormal bundle of $O$. Let\n\\begin{equation*}\n_\\tau\\!N^*_O\\End(V)=\\{(Y,Z)\\ |\\ Z\\in O,\\ [Y,Z]=\\tau Z^3\\}.\n\\end{equation*}\nThen according to~\\refl{ex} the space $(_\\tau\\!N^*_O\\End(V))_\\reduced$ is a\n$\\GL(V)$-equivariant $N^*_O\\End(V)$-torsor over~$O$.\nIn particular $(_\\tau\\!N^*_O\\End(V))_\\reduced$ is smooth and $k^2$-dimensional.\n\n\nIt is clear that for any pair $(Y,Z)$ the set of cyclic $v\\in V$ is open.\nTherefore,\n\\begin{equation*}\n\\widetilde{B}^k_h \\subset \\coprod_{\\lam \\in \\frP(k)} \\left(_\\tau\\!N^*_{O_\\lam}\\End(V)\n\\times V \\right)\n\\end{equation*}\nis an open subset.\nMoreover, by~\\refl{ex} it has a nonempty intersection with every component above.\nThe theorem now follows from~\\reft{26}.\n \\end{proof} \n\n\n\n\\cor{dim}\nConsider the map $\\gamma_\\tau:B^k_h \\to S^k{\\mathbb A}^1_h$.\nThe set of points $D \\in S^k{\\mathbb A}^1_h$ such that $\\dim \\gamma_\\tau^{-1}(D) \\ge m$\nhas codimension at least $m$ in $S^k{\\mathbb A}^1_h$, and is empty for $m \\ge k$.\n\\end{corol}\n \\begin{proof} \nAs $B^k_h$ is equidimensional of dimension $k$ by~\\reft{b-components}, it is enough to show that no component of $B^k_h$\nis contained in the fiber of $\\gamma_\\tau:B^k_h \\to S^k{\\mathbb A}^1_h$. For this note that the map $\\gamma_\\tau$\nis equivariant with respect to the action of the group $\\GG_a \\subset \\SL(H)$, the unipotent radical of the parabolic\nwhich fixes $h \\in H$, and that its action on ${\\mathbb A}^1_h$\nis free.\n \\end{proof} \n\n\n\\th{sml}\nThe map $\\gamma_\\tau\\colon\\GMt^n\\to\\UMt^n$ is small.\n\\eth\n \\begin{proof} \nLet $(\\UMt^n)_m \\subset \\UMt^n$ be the set of points over which the fiber of $\\gamma_\\tau$ has dimension~$m$.\nTake any $0 \\ne h \\in H$. By~\\refp{dep} for any $(\\CE,D) \\in \\sM_\\tau^{n-k} \\times S^k{\\mathbb A}^1_h$ the fiber $\\gamma_\\tau^{-1}(\\CE,D)$\nis isomorphic to the fiber of the map $\\gamma_\\tau:B^k_h \\to S^k{\\mathbb A}^1_h$ over $D$. In particular, by~\\refc{dim}\nthe codimension of the set $(\\UMt^n)_m \\cap (\\sM_\\tau^{n-k} \\times S^k{\\mathbb A}^1_h)$ in $\\sM_\\tau^{n-k} \\times S^k{\\mathbb A}^1_h$\nis at least $m$, and moreover $k > m$. Therefore\n\\begin{multline*}\n\\dim ((\\UMt^n)_m \\cap (\\sM_\\tau^{n-k} \\times S^k{\\mathbb A}^1_h)) \\le\n\\dim (\\sM_\\tau^{n-k} \\times S^k{\\mathbb A}^1_h) - m = \\\\\n2(n-k) + k - m = 2n - k - m < 2n - 2m.\n\\end{multline*}\nSince the sets $\\sM_\\tau^{n-k} \\times S^k{\\mathbb A}^1_h$ form an open covering of the stratum $\\sM_\\tau^{n-k} \\times S^k{\\mathbb A}^1 = \\UMt^{n-k,k}$\nof a stratification of $\\UMt^{n}$, the result follows.\n \\end{proof} \n\n\\ssec{defo}{Deformation of $\\GMt^n$ and $\\UMt^n$}\n\nThe goal of this section is to show that the Gieseker and the Uhlenbeck compactifications\nform a family over $\\AA^1$ (with coordinate $\\tau$) and check that the former is smooth.\nTo be more precise, consider the following graded algebra:\n\\begin{align*}\n\\sA =& {\\mathbb C}\\langle x,y,z,\\btau\\rangle\\Bigl.\\Bigr\/\n\\Bigl\\langle[x,z]=[y,z]=[\\btau,x]=[\\btau,y]=[\\btau,z]=0,[x,y]=\\btau z ^2\\Bigr\\rangle, \\\\\n& \\deg x=\\deg y=\\deg z=1,\\ \\deg\\btau=0.\n\\end{align*}\nAs $\\btau$ is central of degree 0, this is an algebra over ${\\mathbb C}[\\btau]$. In particular, we can specialize $\\btau$\nto any complex number $\\tau$, which gives back the algebra $A^\\tau$ we considered before.\n\nAnalogously, we consider the Koszul dual of $\\sA$ over ${\\mathbb C}[\\btau]$:\n\\begin{align*}\n\\sA^! =& {\\mathbb C}\\langle \\xi,\\eta,\\zeta,\\btau \\rangle\/\\langle \\xi^2=\\eta^2=\\eta\\xi+\\xi\\eta=\\zeta\\xi + \\xi\\zeta = \\eta\\zeta + \\zeta\\eta = \\zeta^2 + \\btau(\\xi\\eta - \\eta\\xi) = 0 \\rangle,\\\\\n& \\deg \\xi=\\deg \\eta = \\deg \\zeta = 1,\\ \\deg\\btau=0.\n\\end{align*}\nThis is a graded ${\\mathbb C}[\\btau]$-algebra. Note that each of its graded components $\\sA^!_0$, $\\sA^!_1$, $\\sA^!_2$, $\\sA^!_3$\nis a free ${\\mathbb C}[\\btau]$-module of finite rank (equal to $1$, $3$, $3$, and $1$ respectively).\n\n\nFurther, we consider the quiver $\\bQ$ over ${\\mathbb C}[\\btau]$ defined as\n\\begin{equation*}\n\\xymatrix{\n*+[o][F]{1} \\ar[rrr]_{\\sA^!_1}\\ar@\/^1pc\/[rrrrrr]^{\\sA^!_2} &&&\n*+[o][F]{2} \\ar[rrr]_{\\sA^!_1} &&&\n*+[o][F]{3} }\n\\end{equation*}\n(analogously to the quiver $\\bpt$), and its representations in the category of ${\\mathbb C}[\\btau]$-modules.\nBy definition such a representation is the data of three ${\\mathbb C}[\\btau]$-modules $(\\bV_1,\\bV_2,\\bV_3)$\nand two morphisms of ${\\mathbb C}[\\btau]$-modules $\\bV_1 \\otimes_{{\\mathbb C}[\\btau]} \\sA^!_1 \\to \\bV_2$ and\n$\\bV_2 \\otimes_{{\\mathbb C}[\\btau]} \\sA^!_2 \\to \\bV_3$ such that the composition\n$\\bV_1 \\otimes_{{\\mathbb C}[\\btau]} \\sA^!_1 \\otimes_{{\\mathbb C}[\\btau]} \\sA^!_1 \\to \\bV_2 \\otimes_{{\\mathbb C}[\\btau]} \\sA^!_1 \\to \\bV_3$\nfactors through $\\bV_1 \\otimes_{{\\mathbb C}[\\btau]} \\sA^!_2 \\to \\bV_3$.\n\nAssuming each of $\\bV_i$ is a free ${\\mathbb C}[\\btau]$-module of finite rank, the space\n\\begin{equation*}\n\\Hom_{{\\mathbb C}[\\btau]}(\\bV_1 \\otimes_{{\\mathbb C}[\\btau]} \\sA^!_1, \\bV_2) \\oplus \\Hom_{{\\mathbb C}[\\btau]}(\\bV_2 \\otimes_{{\\mathbb C}[\\btau]} \\sA^!_1, \\bV_3)\n\\end{equation*}\nis also a free ${\\mathbb C}[\\btau]$-module of finite rank. We consider the associated vector bundle over $\\Spec({\\mathbb C}[\\btau]) = \\AA^1$\nand its total space $\\Tot(\\Hom_{{\\mathbb C}[\\btau]}(\\bV_1 \\otimes_{{\\mathbb C}[\\btau]} \\sA^!_1, \\bV_2) \\oplus \\Hom_{{\\mathbb C}[\\btau]}(\\bV_2 \\otimes_{{\\mathbb C}[\\btau]} \\sA^!_1, \\bV_3))$\nwhich is fibered over $\\AA^1$ with fiber an affine space. The above factorization condition defines a Zarisky closed subspace\n\\begin{equation*}\n\\Rep_\\bQ(\\bV_\\bu) \\subset \\Tot(\\Hom_{{\\mathbb C}[\\btau]}(\\bV_1 \\otimes_{{\\mathbb C}[\\btau]} \\sA^!_1, \\bV_2) \\oplus \\Hom_{{\\mathbb C}[\\btau]}(\\bV_2 \\otimes_{{\\mathbb C}[\\btau]} \\sA^!_1, \\bV_3))\n\\end{equation*}\nparameterizing all representations of the quiver $\\bQ$ in $\\bV_\\bu$. By definition this is an affine variety over $\\AA^1$.\n\n\n\n\nNow we take a relatively prime triple $(r,d,n)$ such that~\\eqref{assumptions-r-d-n} hold,\nconsider a triple of free ${\\mathbb C}[t]$-modules $(\\bV_1,\\bV_2,\\bV_3)$ of ranks given\nby the dimension vector $\\alpha(r,d,n)$ of~\\eqref{alpha}, and put\n\\begin{equation*}\n\\Rep_\\bQ(\\alp(r,d,n)) = \\Rep_\\bQ(\\bV_\\bu).\n\\end{equation*}\nThe group $\\GL(\\alp(r,d,n))$ acts naturally on the space $\\Rep_\\bQ(\\alp(r,d,n))$\nalong the fibers of the projection $\\Rep_\\bQ(\\alp(r,d,n)) \\to \\AA^1$.\nAny rational polarization $\\tet$ (in the sense ~\\refss{stabilo}) linearizes this action and thus\ngives rise to the GIT quotient\n\\begin{equation*}\n\\CM^\\tet_\\bQ(\\alpha(r,d,n)) = \\Rep_\\bQ(\\alp(r,d,n)) \/\\!\/_\\tet \\GL(\\alp(r,d,n)).\n\\end{equation*}\nBy construction it comes with a map $\\CM^\\tet_\\bQ(\\alpha(r,d,n)) \\to \\AA^1$,\nand clearly its fiber over a point $\\tau \\in \\AA^1$ identifies with the moduli space\n$\\CM^\\tet_\\tau(\\alpha(r,d,n))$.\n\nUsing this construction for $\\tet = \\tet^0$ and for $\\tet = \\tet^0 + \\varepsilon\\tet^1$ we construct\nthe relative versions of the Gieseker and the Uhlenbeck compactifications\n\\begin{equation*}\n\\GM(r,d,n) = \\CM^{\\tnt}_\\bQ(\\alpha(r,d,n))\n\\qquad\\text{and}\\qquad\n\\UM(r,d,n) = \\CM^{\\tet^0}_\\bQ(\\alpha(r,d,n)).\n\\end{equation*}\nBy standard GIT we have a contraction $\\gamma:\\GM(r,d,n) \\to \\UM(r,d,n)$ commuting with the morphisms to $\\AA^1$.\n\n\n\n\n\n\\prop{smd}\nIf $\\operatorname{gcd}(r,d,n)=1$ then the map $\\GM(r,d,n) \\to \\AA^1$ is smooth and projective.\nIn particular, $\\GM(r,d,n)$ is a smooth variety.\n\\end{propos}\n \\begin{proof} \nBy~\\refl{kronecker-stability} the moduli space $\\GM(r,d,n)$ coincides with the moduli space of Gieseker semistable\nsheaves of rank $r$, degree $d$ and second Chern class $n$ for the family $\\sA$ of Artin--Schelter algebras over ${\\mathbb C}[\\btau]$\nconstructed in~\\cite{ns}. The smoothness and the projectivity of the latter is proved in Theorem~8.1 of {\\em loc.\\ cit.}\n \\end{proof} \n\n\nWhen considering the case $r = 1$, $d = 0$ we abbreviate $\\GM(1,0,n)$ to just $\\GM^n$ and $\\UM(1,0,n)$ to $\\UM^n$.\n\n\n\n\\ssec{fixed}{Fixed points}\n\nWe choose a torus $T \\subset \\SL(H)$ and consider its action on the Calogero--Moser space\nand its Gieseker and Uhlenbeck compactifications. The stratifications and the map $\\gam$\nare $\\SL(H)$-equivariant and hence $T$-equivariant as well. We aim at a description\nof the set of $T$-fixed points on $\\UMt^n$. Recall first what is known about the $T$-fixed locus of $\\sM^n_\\tau$.\n\n\\lem{fixed-cm}(\\cite[Proposition~6.11]{W})\nFor $\\tau \\ne 0$ the set of $T$-fixed points in $\\sM^n_\\tau$ is in a natural bijection\nwith the set $\\frP(n)$ of partitions of $n$.\n\\end{lemma}\n\n\nWe denote by $\\mathbf{c}^n_\\lambda \\in \\sM^n_\\tau$ the $T$-fixed point corresponding to a partition $\\lambda \\in \\frP(n)$.\nIn particular, $\\mathbf{c}^0 \\in \\sM^0_\\tau$ is the unique point (it is automatically $T$-fixed).\nLet also $P_0,P_\\infty \\in \\PP(H)$ be the $T$-fixed points on the line $\\PP(H)$.\n\n\\lem{fixed-uhlenbeck}\nFor any $\\tau \\ne 0$ the set of $T$-fixed points in $\\UMt^n$ is finite. Moreover,\n\\begin{equation*}\n(\\UMt^n)^T = \\{ (\\mathbf{c}^m_\\lambda, k_0 P_0 + k_\\infty P_\\infty) \\in \\sM^m_\\tau \\times S^{n-m}{\\mathbb P}^1 \\mid \\lambda\\in\\frP(m),\\ k_0+k_\\infty=n-m\\}.\n\\end{equation*}\n\\end{lemma}\n \\begin{proof} \nIt is enough to describe $T$-fixed points on each of the strata $\\sM^m_\\tau \\times S^{n-m}\\PP(H)$ of the stratification of $\\UMt^n$.\nAs the product decomposition is $\\SL(H)$-invariant, it is enough to describe fixed points on each factor.\nOn first factor we use~\\refl{fixed-cm}, and on $S^k\\PP(H)$ a description of fixed points is evident.\n \\end{proof} \n\nRecall that a $T$-fixed point $P$ is called {\\sf attracting} if all the weights of the $T$-action\non the tangent space at point $P$ are positive.\n\n\\lem{attracting-uhlenbeck}\nThe Uhlenbeck compactification $\\UMt^n$ of the Calogero--Moser space has a unique attracting $T$-fixed\npoint $(\\mathbf{c}^0,nP_0) \\in \\sM^0_\\tau\\times S^n\\PP(H) \\subset \\UMt^n$.\n\\end{lemma}\n \\begin{proof} \nSince $\\UMt^n$ is a projective variety, the $T$-action on it should have at least one attracting point.\nOn the other hand, $\\sM^n_\\tau$ is a sympletic manifold, and the $T$-action preserves\nthe sympletic structure~\\cite{kks}, hence for $m > 0$ the weights of $T$ on the tangent\nspaces at points $\\mathbf{c}^m_\\lambda$ are pairwise opposite, and thus for $m > 0$ the $T$-fixed\npoints $(\\mathbf{c}^m_\\lambda,k_0 P_0 + k_\\infty P_\\infty)$ are not attracting. Therefore, each\nattracting point of the $T$-action on $\\UMt^n$ lies on $\\sM^0\\times S^n\\PP(H) = S^n\\PP(H)$.\nAs it also should be an attracting point for the $T$-action on $S^n\\PP(H)$, it should\ncoincide with $(\\mathbf{c}^0,nP_0)$.\n \\end{proof} \n\n\n\n\n\n\\ssec{comp}{The IC sheaf of the Uhlenbeck compactification}\n\n\nIn this section we will prove~\\reft{main}. \nThe statement of the Theorem and the arguments we use are purely topological.\nWe refer to~\\cite{bbd} for the notion of IC sheaf and the general machinery.\n\n\n\n\nWe start with computing the stalks of the IC-sheaf at the deepest stratum of \nthe Uhlenbeck stratification.\nSince for $n=0$ the Calogero--Moser space $\\sM^0_\\tau$ is just a point, by~\\reft{str}(2) we have\n$S^n\\PP(H) = S^n\\PP(H) = \\sM^0_\\tau \\times S^n\\PP(H) \\subset \\UMt^n$.\nRecall also the diagonal stratification~\\eqref{diagonal-stratification} of $S^n\\PP(H)$\nand its deepest stratum $S_{(n)}\\PP(H) \\subset S^n\\PP(H)$.\n\n\n\n\n\n\n\\prop{one stalk}\nFor any $P \\in \\PP(H)$ the stalk of the sheaf $\\ic(\\UMt^n)$ at the point $(\\CO,nP)$ of the stratum $\\sM^0_\\tau \\times S_{(n)}\\PP(H) \\subset \\UMt^n$ is\nisomorphic to\n\\eq{chow}\n\\ic(\\UMt^n)_{(\\CO,nP)} = \\bigoplus_{\\mu\\in\\frP(n)}{\\mathbb C}[2l(\\mu)].\n\\end{equation}\n\\end{propos}\n \\begin{proof} \nLet $T \\subset \\SL(H)$ be a torus such that $P = P_0$ is the attracting point for the action of~$T$ on $\\PP(H)$.\nThe computation\nis based on the following ``deformation diagram'':\n\\begin{equation*}\n\\xymatrix{\n\\GM^n_0\\ar[d]^<>(0.5){\\gamma_0}\\ar@{^{(}->}[rr]^<>(0.5){\\tilde\\varsigma} &&\n\\GM^n\\ar[d]^<>(0.5){\\gamma} &&\n\\GM^n_\\bseta\\ar[d]^<>(0.5){\\gamma_\\bseta}\\ar@{_{(}->}[ll]\n\\\\\n\\UM^n_0\\ar[d]\\ar@{^{(}->}[rr]^<>(0.5){\\varsigma} &&\n\\UM^n\\ar[d]^<>(0.5){p} &&\n\\UM^n_\\bseta\\ar[d]\\ar@{_{(}->}[ll]\n\\\\\n\\{0\\}\\ar@{^{(}->}[rr] &&\n{\\mathbb A}^1\\ar@\/^\/[u]^<>(0.5){\\sigma} &&\n{\\mathbb A}^1\\sminus\\{0\\}\\ar@{_{(}->}[ll]\n}\n\\end{equation*}\nHere the middle column is the deformation family over $\\AA^1$ of~\\refp{smd}\nwith $p$ being the structure map. The left column is the fiber over the point $0 \\in \\AA^1$,\nwhile the right column is the base change to $\\AA^1 \\setminus \\{0\\} \\subset \\AA^1$. Finally,\nthe map $\\sigma:\\AA^1 \\to \\UM^n$ is defined as follows.\n\nFor any $\\tau \\ne 0$ we put $\\sigma(\\tau) = n\\cdot P \\in \\UMt^n \\subset \\UM^n$.\nClearly, this is a regular map $\\AA^1 \\setminus \\{0\\} \\to \\UMt$. By properness\nof $\\UM^n$ over $\\AA^1$ it extends to a map $\\sigma:\\AA^1 \\to \\UM^n$. Note that\n$\\sigma$ is a section of the map $p$. Indeed, this is clear over $\\AA^1 \\setminus\\{0\\}$\nby definition of $\\sigma$, and over $0$ this s true by continuity.\n\n\n\n\nLet $\\C^\\times_{\\leq1} \\subset {\\mathbb C}^\\times = T$ be the sub-semigroup of\nformed by the complex numbers with absolute value $\\leq1$, and\nlet $F = \\sigma({\\mathbb A}^1) \\sset \\UM^n$ be the image  of the section $\\sigma$.\nFurther, let $\\CU\\subset \\UM^n$ be a small open neighborhood (in the analytic topology)\nof the point $\\sigma(0)$. Without loss of generality,\nwe may choose the set $\\CU$ to be  $\\C^\\times_{\\leq1}$-stable.\nNote that $F$ is the attracting connected component\nof $(\\UM^n)^T$, by \\refl{attracting-uhlenbeck}.\nTherefore, shrinking  $\\CU$ further, if necessary,\none may assume  in addition that we have $\\CU^T=F\\cap \\CU$.\nThe action of $\\C^\\times_{\\leq1}$ preserves the fibers of $p:\\ \\CU\\to{\\mathbb A}^1$,\nand contracts $\\CU$ to the section $F=\\sigma({\\mathbb A}^1)$. According\nto~\\cite[Lemma~6]{Br},\nfor any $\\C^\\times$-equivariant complex ${\\mathcal F}$ of constructible sheaves on~$\\UMt^n$,\nthe natural morphism $\\sigma^*{\\mathcal F}\\to p_*({\\mathcal F}|_\\CU)$ is an isomorphism.\nIn other words, for any $\\tau\\in{\\mathbb A}^1$, there is a natural isomorphism\n\\eq{is0}\nH^\\bullet(\\CU\\cap \\UMt^n, \\ {\\mathcal F})\\cong {\\mathcal F}|_{\\sigma(\\tau)}.\n \\end{equation} \n\n\n\n\nNext, let $\\psi_p$, resp. $\\psi_\\pg$,  denote the nearby cycles\nfunctor~\\cite[8.6]{KS}\nwith respect to the function $p$, resp. $\\pg$. Note  that the morphism $\\pg$\nbeing smooth, we have\n$\\psi_\\pg(\\unl\\C{}_{\\GM^n_\\bseta})=\\unl\\C{}_{\\GM^n_0}$.\nTherefore, using the proper base change for\nnearby cycles (see e.g.~\\cite[Exercise~VIII.15]{KS}) we obtain\n\\begin{equation*}\n(\\gamma_0)_*\\unl\\C{}_{\\GM^n_0}=(\\gamma_0)_*(\\psi_\\pg(\\unl\\C{}_{\\GM^n_\\bseta}))\n=\\psi_p((\\gamma_\\bseta)_*\\unl\\C{}_{\\GM^n_\\bseta}).\n\\end{equation*}\nThe map  $\\gamma_\\bseta: \\GM^n_\\bseta\\to \\UM^n_\\bseta$\n is  a small and proper morphism.\nHence, we have an isomorphism ${\\operatorname{IC}}(\\UM^n_\\bseta) \\cong (\\gamma_\\bseta)_*\\unl\\C{}_{\\GM^n_\\bseta}[2n]$.\nCombining the above isomorphisms and\ntaking stalks at the point $\\sigma(0)$ yields\n\\eq{is1}\n\\big((\\gamma_0)_*\\unl\\C{}_{\\GM_0^n}\\big)|_{\\sigma(0)}\\cong\n\\big(\\psi_p((\\gamma_\\bseta)_*\\unl\\C{}_{\\GM^n_\\bseta})\\big)|_{\\sigma(0)}\n\\cong\\big(\\psi_p({\\operatorname{IC}}(\\UM^n_\\bseta))\\big)|_{\\sigma(0)}[-2n].\n \\end{equation} \n\nFurther, by definition\nof the functor $\\psi_p$, for a sufficiently small open set $\\CU$\nas above and for any $\\tau\\neq 0$\nwith a sufficiently small absolute value\none has $\n\\big(\\psi_p({\\operatorname{IC}}(\\UM^n_\\bseta))\\big)|_{\\sigma(0)}\n\\cong H^\\bullet\\big(\\CU\\cap p^{-1}(\\tau),\\ {\\operatorname{IC}}(\\UM^n_\\bseta)\\big)\n\\cong H^\\bullet\\big(\\CU\\cap\\UMt^n,\\ {\\operatorname{IC}}(\\UMt^n)\\big).\n$$\n\n\n\nThus, comparing the LHS and the RHS in \\refe{is1},\nwe obtain\n\\begin{align}H^\\bullet(\\gamma_0^{-1}(\\sigma(0)))[2n]\\cong\n\\big((\\gamma_0)_*\\unl\\C{}_{\\GM_0^n}\\big)|_{\\sigma(0)}[2n]&\\cong\n\\big(\\psi_p({\\operatorname{IC}}(\\UM^n_\\bseta))\\big)|_{\\sigma(0)}\\label{is4}\\\\\n&\\cong\nH^\\bullet\\big(\\CU\\cap\\UMt^n,\\ {\\operatorname{IC}}(\\UMt^n)\\big)\n\\cong{\\operatorname{IC}}(\\UMt^n)|_{\\sigma(\\tau)},\\nonumber\n\\end{align}\nwhere the last isomorphism is a special case of  \\refe{is0}\nfor ${\\mathcal F}={\\operatorname{IC}}(\\UMt^n)$.\n\nTo complete the proof, we observe that the\nfiber $\\gamma_0^{-1}(\\sigma(0)))$ is the ``central fiber''\nof the Hilbert--Chow morphism $\\Hilb^n({\\mathbb P}^2)\\to S^n{\\mathbb P}^2$.\nIn other words, the variety $\\gamma_0^{-1}(\\sigma(0)))$\nis nothing but $\\Hilb^n_0({\\mathbb A}^2)$, the punctual Hilbert scheme of\n$n$ infinitesimally close points in ${\\mathbb A}^2$.\nThe Betti numbers of the punctual Hilbert scheme\nare well known, cf. e.g.~\\cite{N}. Specifically,\nall odd Betti numbers vanish and one\nhas the formula\n\\[\\dim H^{2k-2}(\\Hilb^n_0({\\mathbb A}^2))=\\#\\{\\mu\\in \\frP(n)\\mid l(\\mu)=k\\}.\\]\nIt follows from (\\ref{is4}) that, for\n$\\tau$ sufficiently small, the dimensions of the cohomology\ngroups of the stalk ${\\operatorname{IC}}(\\UMt^n)|_{\\sigma(\\tau)}$\nare given by the same formula. This is equivalent to the\nstatement of the proposition.\n \\end{proof} \n\n\n\n\nNow~\\reft{main} follows from~\\refp{one stalk} and the factorization\nproperty of~\\refp{factorization-b}. In effect, due to $\\SL(H)$-equivariance,\nit suffices to find the stalks of ${\\operatorname{IC}}(\\UMt^n)$ at $S_\\lambda{\\mathbb A}^1_h\\subset S_\\lambda\\PP(H)$.\nGiven a point $\\CE \\in \\sM^m$ and $D\\in S_\\lambda{\\mathbb A}^1_h$, due to the smallness of $\\gamma_\\tau$,\n\\begin{equation*}\n{\\operatorname{IC}}(\\UMt^n)_{(\\CE,D)} = H^\\bullet(\\gamma_\\tau^{-1}(\\{\\CE\\}\\times D),{\\mathbb C}).\n\\end{equation*}\nFurther, by~\\refp{dep} we have\n\\begin{equation*}\nH^\\bullet(\\gamma_\\tau^{-1}(\\{\\CE\\}\\times D),{\\mathbb C}) \\cong H^\\bullet(\\gamma_\\tau^{-1}(\\{\\CO\\}\\times D),{\\mathbb C}).\n\\end{equation*}\nNow if $D = \\sum k_iP_i$ with pairwise distinct points $P_i \\in {\\mathbb A}^1_h$,\nthen according to~\\refp{factorization-b},\n$\\gamma_\\tau^{-1}(\\{\\CO\\}\\times D)\\cong \\prod_i\\gamma_\\tau^{-1}(\\{\\CO\\}\\times k_iP_i)$,\nand hence\n\\begin{equation*}\nH^\\bullet(\\gamma_\\tau^{-1}(\\{\\CO\\}\\times D),{\\mathbb C})\\cong\n\\bigotimes_i H^\\bullet(\\gamma_\\tau^{-1}(\\{\\CO\\}\\times k_iP_i),{\\mathbb C}).\n\\end{equation*}\nDue to the smallness of $\\gamma_\\tau$,\n$H^\\bullet(\\gamma_\\tau^{-1}(\\{\\CO\\}\\times k_iP_i),{\\mathbb C})={\\operatorname{IC}}(\\UMt^i)_{(\\CO,k_iP_i)}$,\nand the latter stalk is known from~\\refp{one stalk}. This completes the proof of~\\reft{main}. \\qed\n\n\n\n\n\n\n\n\n\n\n\\sec{app}{Appendix}\n\nIn this Appendix we collect the proofs of some results from section~\\refs{ch5}.\nThroughout, we assume that $\\tau\\neq 0$.\n\\ssec{artin-proofs}{Moduli spaces of Artin sheaves}\n\nDenote by $\\PP^1_3$ the third infinitesimal\nneighborhood of the line at infinity $\\PP(H)$ in $\\BPt$,\ni.e.\\ the projective spectrum of a commutative graded algebra $\\C[x,y,z]\/z^3$.\n\n\n\\lem{artin-moduli}\nThe moduli space ${}^A\\mathsf{M}_\\tau(1,2,1)$ is a fine moduli space.\nIt is isomorphic to the third infinitesimal neighborhood of a line on a plane:\n${}^A\\mathsf{M}_\\tau(1,2,1) \\cong \\PP^1_3$.\n\\end{lemma}\n \\begin{proof} \nThe data of a $(1,2,1)$-dimensional representation of $\\bpt$ amounts to two maps\n\\begin{equation*}\n\\C \\xrightarrow{f} \\C^2\\otimes A^\\tau_1\n\\qquad\\text{and}\\qquad\n\\C^2 \\xrightarrow{g} \\C\\otimes A^\\tau_1\n\\end{equation*}\nwith the condition saying that the composition\n\\begin{equation*}\n\\C \\xrightarrow{f} \\C^2\\otimes A^\\tau_1 \\xrightarrow{g\\otimes 1} \\C \\otimes A^\\tau_1\\otimes A^\\tau_1 \\to \\C \\otimes A^\\tau_2\n\\end{equation*}\nis zero. In other words, it can be rewritten as saying that\n\\begin{equation*}\n\\sigma := (g\\otimes 1)(f(1)) \\in K := \\Ker(A^\\tau_1\\otimes A^\\tau_1 \\to A^\\tau_2).\n\\end{equation*}\nThe $\\tet^0$-semistability is equivalent to the injectivity of the maps $f^T,g \\colon {\\mathbb C}^2 \\to {\\mathbb C}\\otimes A^\\tau_1$. This means\nthat the element $\\sigma$ considered as an element of $A^\\tau_1\\otimes A^\\tau_1 = \\Hom((A^\\tau_1)^*,A^\\tau_1)$ has rank~2\n(then $\\C^2$-component of the representation is just the image of $\\sigma$). Thus the moduli space is nothing\nbut the degeneration scheme of the morphism\n\\begin{equation*}\n(A^\\tau_1)^* \\otimes \\CO_{\\PP(K)}(-1) \\to A^\\tau_1 \\otimes \\CO_{\\PP(K)}\n\\end{equation*}\non $\\PP(K)$. As $K \\subset A^\\tau_1 \\otimes A^\\tau_1$ can be written as\n\\begin{equation*}\nK = \\{ u(y\\otimes z - z\\otimes y) + v(x\\otimes z - z\\otimes x) + w(x\\otimes y - y\\otimes x - \\tau z\\otimes z) \\mid u,v,w \\in {\\mathbb C} \\},\n\\end{equation*}\nthe above morphism is given by the matrix\n\\begin{equation}\\label{matrix-u-v-w}\n\\left(\n\\begin{matrix}\n0 & w & v \\\\\n-w & 0 & u \\\\\n-v & -u & -\\tau w\n\\end{matrix}\n\\right)\n\\end{equation}\nand the degeneration condition is given by its determinant which is equal to\n\\begin{equation*}\n\\det\\left(\n\\begin{matrix}\n0 & w & v \\\\\n-w & 0 & u \\\\\n-v & -u & -\\tau w\n\\end{matrix}\n\\right) = -\\tau w^3.\n\\end{equation*}\nThis means that the moduli space is the subscheme of $\\PP(K)$ given by the equation $w^3 = 0$, i.e.\\\nthe third infinitesimal neighborhood $\\PP^1_3$ of the line $\\PP^1 = \\{w = 0\\}$ in the plane $\\PP^2$.\n\nTo show that the moduli space is fine we should construct a universal family. For this we restrict the map\n$(A^\\tau_1)^* \\otimes \\CO_{\\PP(K)}(-1) \\to A^\\tau_1 \\otimes \\CO_{\\PP(K)}$\nto $M := \\PP^1_3$. This is a morphism of constant rank $2$ (the rank does not drop to 1 since among the 2-by-2 minors of the matrix~\\eqref{matrix-u-v-w} one easily finds $u^2$, $v^2$, and $w^2$),\nhence its image is a rank $2$ vector bundle~$\\CV_2$. It comes equipped\nwith a surjective map $(A^\\tau_1)^* \\otimes \\CO_{M}(-1) \\to \\CV_2$ and an injective map\n$\\CV_2 \\to A^\\tau_1\\otimes \\CO_M$. Clearly these two maps provide $(\\CO_M(-1),\\CV_2,\\CO_M)$ with a structure of a family of representations of the quiver~$\\bpt$.\nThe above arguments show it is a universal family.\n \\end{proof} \n\n\n\n\n\n\nNow we give a description of the reduced structure of the space ${}^A\\mathsf{M}_\\tau(k,2k,k)$ for $k > 1$.\n\n \\begin{proof} [Proof of~\\refp{artin-moduli-k}]\nConsider the subset ${}^A\\mathsf{R}_\\tau^{\\tet^0}(k,2k,k) \\subset {}^A\\mathsf{R}_\\tau(k,2k,k)$ of all $\\tet^0$-semistable $(k,2k,k)$-dimensional representations of $\\bpt$\nand let $\\CW_\\bullet$ be the universal representation of the quiver over $\\BPt$. Let ${\\mathcal F}$ be the universal\nsheaf on the product ${}^A\\mathsf{R}_\\tau^{\\tet^0}(k,2k,k) \\times \\BPt$, i.e., the sheaf defined by exact sequence\n\\begin{equation*}\n0 \\to \\CW_1 \\boxtimes \\CO(-1) \\to \\CW_0 \\boxtimes \\CO \\to \\CW_1 \\boxtimes \\CO(1) \\to {\\mathcal F} \\to 0.\n\\end{equation*}\nThen the support map defined in~\\refl{artin-support} gives a map $\\supp:{}^A\\mathsf{R}_\\tau^{\\tet^0}(k,2k,k) \\to S^k\\PP^1$.\nThe map is clearly $\\GL(k,2k,k)$-equivariant, hence descends to a map from the moduli space\n\\begin{equation*}\n\\supp:{}^A\\mathsf{M}_\\tau(k,2k,k) \\to S^k\\PP^1.\n\\end{equation*}\nOn the other hand, we clearly have an embedding which takes a $k$-tuple of $(1,2,1)$-dimensional Artin representations\n$W^1_\\bullet$, $W^2_\\bullet$, \\dots, $W^k_\\bullet$ to their direct sum\n\\begin{equation*}\n({}^A\\mathsf{R}_\\tau(1,2,1))^k \\to {}^A\\mathsf{R}_\\tau(k,2k,k),\n\\qquad\n(W^1_\\bullet, W^2_\\bullet, \\dots, W^k_\\bullet) \\mapsto W^1_\\bullet \\oplus W^2_\\bullet \\oplus \\dots \\oplus W^k_\\bullet.\n\\end{equation*}\nThis map is equivariant with respect to the action of the group $\\GL(1,2,1)^k \\rtimes \\fS_k$ on the source,\nsuch that the $i$-th factor $\\GL(1,2,1)$ acts naturally on the $i$-th factor of $({}^A\\mathsf{R}_\\tau(1,2,1))^k$ and $\\fS_k$ permutes the factors,\nand the action on the target is given by\na natural embedding $\\GL(1,2,1)^k \\rtimes \\fS_k \\subset \\GL(k,2k,k)$. The $\\Proj$ construction of the GIT quotient implies that the map\ninduces a morphism of the GIT quotients\n\\begin{equation*}\n({}^A\\mathsf{R}_\\tau(1,2,1))^k \/\\!\/_{\\tet^0} (\\GL(1,2,1)^k \\rtimes \\fS_k) \\to {}^A\\mathsf{R}_\\tau(k,2k,k) \/\\!\/_{\\tet^0} \\GL(k,2k,k).\n\\end{equation*}\nThe quotient on the right is just the moduli space ${}^A\\mathsf{M}_\\tau(k,2k,k)$. The quotient on the left\ncan be identified with $({}^A\\mathsf{M}_\\tau(1,2,1))^k \/ \\fS_k$,\nso it is isomorphic to $S^k(\\PP^1_3)$ by~\\refl{artin-moduli}.\nRestricting to the reduced subscheme we obtain a map\n\\begin{equation*}\n\\Sigma \\colon S^k\\PP^1 = S^k(\\PP^1_3)_\\reduced \\to {}^A\\mathsf{M}_\\tau(k,2k,k).\n\\end{equation*}\nWe are going to show that the constructed maps $\\supp$ and $\\Sigma$ induce isomorphisms between $S^k\\PP^1$ and\nthe reduced moduli space ${}^A\\mathsf{M}_\\tau(k,2k,k)_{\\mathrm{red}}$.\n\nFor this we note that the maps give bijections between the sets of closed points of $S^k\\PP^1$ and ${}^A\\mathsf{M}_\\tau(k,2k,k)$,\nsince by~\\refp{Art0}(3) any Artin sheaf is S-equivalent to a direct sum of structure sheaves for a unique collection\nof points (which are given back by the support map). Note also that both $S^k\\PP^1$ and ${}^A\\mathsf{M}_\\tau(k,2k,k)$ are projective\nvarieties, hence the map $\\Sigma$ is proper. Finally, $S^k\\PP^1 \\cong \\PP^k$ is normal.\n\nSo, it is enough to show that any proper regular map from a reduced normal scheme to a reduced scheme inducing\na bijection on the sets of closed points is an isomorphism. Locally, we just have an integral (due to properness)\nextension of rings with the bottom ring being integrally closed (by normality), hence it is an isomorphism.\n \\end{proof} \n\n\n\n\\newcommand{{}^G\\Rt}{{}^G\\mathsf{R}_\\tau}\n\\newcommand{{}^U\\Rt}{{}^U\\mathsf{R}_\\tau}\n\n\\ssec{statifications-proofs}{Stratifications}\n\nHere we construct the required stratifications of the Gieseker and Uhlenbeck moduli spaces.\n\n \\begin{proof} [Proof of~\\refl{stratification}]\nLet ${}^G\\Rt := \\osRt^{\\tnt}(\\alpha(r,d,n)) \\subset \\osRt(\\alpha(r,d,n))$ be the open subset of\n$\\tnt$-semistable $\\alpha(r,d,n)$-dimensional representations of $\\bpt$.\nLet $\\CV_\\bu$ be the universal family of representations over ${}^G\\Rt$. Consider the universal monad\n\\begin{equation*}\n\\CV_1 \\boxtimes \\CO(-1) \\to \\CV_2 \\boxtimes \\CO \\to \\CV_3 \\boxtimes \\CO(1)\n\\end{equation*}\non ${}^G\\Rt \\times \\BPt$ and denote its cohomology sheaf by $E$. For each point $s \\in {}^G\\Rt$\nwe denote by $E_s$ the restriction of $E$ to $\\{s\\} \\times \\BPt$. Note that this is just\nthe cohomology sheaf of the monad $\\CV_{1s}\\otimes \\CO(-1) \\to \\CV_{2s} \\otimes \\CO \\to \\CV_{3s}\\otimes\\CO(1)$.\nIn particular, the sheaf $E$ is flat over ${}^G\\Rt$.\n\nConsider also the dual monad on $S\\times\\BPt$\n\\begin{equation*}\n\\CV_3^\\vee\\boxtimes\\CO(-1) \\to \\CV_2^\\vee \\boxtimes\\CO \\to \\CV_1^\\vee \\boxtimes\\CO(1)\n\\end{equation*}\nand let ${\\mathcal F}$ be the cokernel of the last map\n\\begin{equation*}\n{\\mathcal F} := \\Coker( \\CV_2^\\vee \\boxtimes\\CO \\to \\CV_1^\\vee \\boxtimes\\CO(1) ).\n\\end{equation*}\nFor each point $s \\in S$ we have\n\\begin{equation*}\n{\\mathcal F}_s \\cong \\Coker( \\CV_{2s}^\\vee \\boxtimes\\CO \\to \\CV_{1s}^\\vee \\boxtimes\\CO(1) ) \\cong \\underline{\\Ext}^1(E_s,\\CO) \\cong \\underline{\\Ext}^2(E_s^{**}\/E_s,\\CO).\n\\end{equation*}\nThus it is an Artin sheaf, but its length may vary from point to point.\nConsider the flattening stratification of $S$ for ${\\mathcal F}$:\n\\begin{equation*}\n{}^G\\Rt = {}^G\\Rt^{\\ge 0} \\supset {}^G\\Rt^{\\ge 1} \\supset {}^G\\Rt^{\\ge 2} \\supset \\dots \\supset {}^G\\Rt^{\\ge n} \\supset {}^G\\Rt^{\\ge n+1} = \\emptyset,\n\\end{equation*}\nwhere ${}^G\\Rt^{\\ge k}$ is the subscheme of points $s \\in {}^G\\Rt$ where the length of ${\\mathcal F}_s$ is at least $k$.\nThis stratification is $\\GL(\\alpha(r,d,n))$-invariant, so it gives a stratification\nof the GIT quotient ${}^G\\Rt\/\\!\/_{\\tnt}\\GL(\\alpha(r,d,n))$, i.e., of the Gieseker moduli space $\\GMt(r,d,n)$.\nFinally, we replace each stratum by its underlying reduced subscheme.\n \\end{proof} \n\n\nBelow we will need the following result on universal families.\n\n\\prop{family-on-strata}\nLet $\\CV_\\bu$ be the universal family of $\\tnt$-semistable $\\alpha(r,d,n)$-dimensional representations\nof $\\bpt$ over ${}^G\\Rt^k := {}^G\\Rt^{\\ge k} \\setminus {}^G\\Rt^{\\ge k+1}$.\nThen there is a natural exact sequence\n\\begin{equation*}\n0 \\to \\CW_\\bu \\to \\CV_\\bu \\to \\CU_\\bu \\to 0\n\\end{equation*}\nof families of representations over ${}^G\\Rt^k$ where $\\CW_\\bu$ is a family of Artin representations\nof dimension $(k,2k,k)$ and $\\CU_\\bu$ is a family of supermonadic $\\tnt$-semistable representations.\n\\end{propos}\n \\begin{proof} \nWe use freely the notation introduced in the proof of~\\refl{stratification}.\nBy assumption ${\\mathcal F}$ is a flat (over~${}^G\\Rt^k$)\nfamily of Artin sheaves of length $k$.\nConsider its Beilinson resolution\n\\begin{equation*}\n\\CW'_1 \\boxtimes \\CO(-1) \\to \\CW'_2 \\boxtimes \\CO \\to \\CW'_3 \\boxtimes \\CO(1),\n\\end{equation*}\nBy flatness of ${\\mathcal F}$ we know that $\\CW'_1$, $\\CW'_2$, $\\CW'_3$ are vector bundles of ranks $k$, $2k$, and $k$ respectively.\nBy functoriality of the Beilinson resolution there is a morphism of resolutions\n\\begin{equation*}\n\\xymatrix{\n\\CV_3^\\vee \\boxtimes\\CO(-1) \\ar[r] \\ar[d] & \\CV_2^\\vee \\boxtimes\\CO \\ar[r] \\ar[d] & \\CV_1^\\vee \\boxtimes\\CO(1) \\ar[d] \\\\\n\\CW'_1 \\boxtimes\\CO(-1) \\ar[r] & \\CW'_2 \\boxtimes\\CO \\ar[r] & \\CW'_3 \\boxtimes\\CO(1)\n}\n\\end{equation*}\nNote that the induced morphisms $\\CV_i^\\vee \\to \\CW'_{4-i}$ of vector bundles on ${}^G\\Rt^k$ are surjective.\nIndeed, this can be verified pointwise, i.e.\\ just for one representation instead of a family. In this case note that both\n$\\CV^\\vee_\\bu$ and $\\CW'_\\bu$ are $\\tet^0$-semistable, hence so is the image of the map. But any $\\tet^0$-semistable\nsubrepresentation of an Artin representation is also Artin. If ${\\mathcal F}' \\subset {\\mathcal F}$ is the corresponding Artin sheaf then it follows\nthat the map $\\CV^\\vee_1 \\to {\\mathcal F}$ factors through ${\\mathcal F}'$ which by definition of ${\\mathcal F}$ implies ${\\mathcal F}' = {\\mathcal F}$.\n\n\nLet $\\CW_i = (\\CW'_{4-i})^\\vee$ and $\\CU_i = \\Ker(\\CV_i^\\vee \\to \\CW'_{4-i})^\\vee$, so that we have an exact sequence of monads\n\\begin{equation*}\n\\xymatrix{\n\\CW_1 \\boxtimes\\CO(-1) \\ar[r] \\ar[d] & \\CW_2 \\boxtimes\\CO \\ar[r] \\ar[d] & \\CW_3 \\boxtimes\\CO(1) \\ar[d] \\\\\n\\CV_1 \\boxtimes\\CO(-1) \\ar[r] \\ar[d] & \\CV_2 \\boxtimes\\CO \\ar[r] \\ar[d] & \\CV_3 \\boxtimes\\CO(1) \\ar[d] \\\\\n\\CU_1 \\boxtimes\\CO(-1) \\ar[r] & \\CU_2 \\boxtimes\\CO \\ar[r] & \\CU_3 \\boxtimes\\CO(1)\n}\n\\end{equation*}\nBy construction, the family $\\CU_\\bu$ is $\\tnt$-semistable and supermonadic, so this exact sequence\nis the one we need.\n \\end{proof} \n\nAnd now we are ready to construct the stratification of the Uhlenbeck moduli space.\nThe situation here is a bit more complicated than in the Gieseker case, since Artin\nrepresentations can appear both as subrepresentations and as quotient representations\nof a $\\tet^0$-semistable representation. So, we perform a two-step construction, first\ndealing with the latter, and then with the former.\n\n\n \\begin{proof} [Proof of~\\refl{stratification-uhlenbeck}]\nLet ${}^U\\Rt := \\hsRt^{\\tet^0}(\\alpha(r,d,n)) \\subset \\hsRt(\\alpha(r,d,n))$ be the open subset of\n$\\tet^0$-semistable $\\alpha(r,d,n)$-dimensional representations of $\\bpt$. Let $\\CV_\\bu$ be\nthe universal family of representations over ${}^U\\Rt$.\nConsider the family of sheaves ${\\mathcal F}' := \\Coker(\\CV_2 \\boxtimes \\CO \\to \\CV_3 \\boxtimes \\CO(1))$ over ${}^U\\Rt \\times \\BPt$.\nNote that these are Artin sheaves of length at most $n$. Let\n\\begin{equation*}\n{}^U\\Rt = {}^U\\Rt^{\\ge 0,\\bu} \\supset {}^U\\Rt^{\\ge 1,\\bu} \\supset {}^U\\Rt^{\\ge 2,\\bu} \\supset \\dots \\supset {}^U\\Rt^{\\ge n,\\bu} \\supset {}^U\\Rt^{\\ge n+1,\\bu} = \\emptyset,\n\\end{equation*}\nbe the flattening stratification for the sheaf ${\\mathcal F}'$. Restricting the family $\\CV_\\bu$ to each stratum ${}^U\\Rt^{k,\\bu} = {}^U\\Rt^{\\ge k,\\bu} \\setminus {}^U\\Rt^{\\ge k+1,\\bu}$\nand repeating the arguments of~\\refp{family-on-strata} (without dualization) we construct on ${}^U\\Rt^k$ a natural exact sequence\n\\begin{equation*}\n0 \\to \\CV'_\\bu \\to \\CV_\\bu \\to \\CW_\\bu \\to 0\n\\end{equation*}\nof representations with $\\CW_\\bu$ being a $(k,2k,k)$-dimensional family of Artin representations, and\n$\\CV'_\\bu$ being a $\\tnt$-semistable family of $\\alpha(r,d,n-k)$-dimensional representations of the quiver $\\bpt$.\nApplying to the latter family the arguments of the proof of~\\refl{stratification} we obtain a natural stratification\n\\begin{equation*}\n{}^U\\Rt^{k,,\\bu} = {}^U\\Rt^{k,\\ge 0} \\supset {}^U\\Rt^{k,\\ge 1} \\supset {}^U\\Rt^{k,\\ge 2} \\supset \\dots \\supset {}^U\\Rt^{k,\\ge n-k} \\supset {}^U\\Rt^{k,\\ge n+1-k} = \\emptyset,\n\\end{equation*}\nby the length of the Artin sheaf $\\Coker((\\CV'_2)^\\vee \\boxtimes\\CO \\to (\\CV'_1)^\\vee \\boxtimes\\CO(1))$, and moreover,\non the stratum ${}^U\\Rt^{k,l} := {}^U\\Rt^{k,\\ge l} \\setminus {}^U\\Rt^{k,\\ge l+1}$ an exact sequence of representations\n\\begin{equation*}\n0 \\to \\CW'_\\bu \\to \\CV'_\\bu \\to \\CU_\\bu \\to 0\n\\end{equation*}\nwith $\\CW'_\\bu$ being a $(l,2l,l)$-dimensional family of Artin representations, and\n$\\CU_\\bu$ being a $\\tnt$-semistable family of $\\alpha(r,d,n-k-l)$-dimensional supermonadic representations.\n\nIt is clear that the subsets\n\\begin{equation*}\n{}^U\\Rt^{\\ge m} := \\bigsqcup_{k + l \\ge m} {}^U\\Rt^{k,l} \\subset {}^U\\Rt\n\\end{equation*}\nare $\\GL(\\alpha(r,d,n))$-invariant closed subsets,\nhence they induce a stratification of the GIT quotient ${}^U\\Rt\/\\!\/_{\\tet^0}\\GL(\\alpha(r,d,n))$, i.e., of the Uhlenbeck moduli space $\\UMt(r,d,n)$.\nAgain, we finish by replacing each stratum with its reduced underlying scheme.\n \\end{proof} \n\n\n\\rem{another-stratification}\nWe could start with splitting of Artin subrepresentations first (and define in this way closed subsets ${}^U\\Rt^{\\bullet,\\ge l} \\subset {}^U\\Rt$)\nand then continue with splitting Artin quotient representations. Note that this will give {\\em different}\\\/ two-index\nstratification of the space ${}^U\\Rt$, but the resulting total stratification will be the same.\n\\end{remark}\n\n\\rem{urt-grt}\nWe always have an embedding ${}^G\\Rt^k \\subset {}^U\\Rt^{0,k}$. Moreover, if $r$ and $d$ are coprime then this inclusion becomes an equality\n\\begin{equation*}\n{}^G\\Rt^k = {}^U\\Rt^{0,k}.\n\\end{equation*}\nIndeed, the union of ${}^U\\Rt^{0,k}$ over all $k$ parameterizes all monadic $\\tet^0$-semistable representations,\nand by~\\refc{coprime-stable} these are precisely all $\\tnt$-semistable representations.\n\\end{remark}\n\nBefore we go to the proof of the main Theorem we need one more result.\nWe will use the notation of the proof of~\\refl{stratification-uhlenbeck}.\nConsider the stratum ${}^U\\Rt^{k,0}$ of the space ${}^U\\Rt$. We checked in the proof\nof~\\refl{stratification-uhlenbeck} that the universal representation over it\nfits into an exact sequence\n\\begin{equation}\\label{uvw-sequence}\n0 \\to \\CU_\\bu \\to \\CV_\\bu \\to \\CW_\\bu \\to 0\n\\end{equation}\nwith $\\CU_\\bu$ being supermonadic and $\\CW_\\bu$ being Artin.\n\n\\lem{sequence-splits}\nThe subset ${}^U\\Rt^{k,\\mathrm{split}} \\subset {}^U\\Rt^k$ of all points over which the exact sequence~\\eqref{uvw-sequence} splits is closed.\n\\end{lemma}\n \\begin{proof} \nIndeed, ${}^U\\Rt^{k,\\mathrm{split}} = {}^U\\Rt^k \\cap {}^U\\Rt^{\\ge k,\\bullet} \\cap {}^U\\Rt^{\\bullet,\\ge k}$\nand both ${}^U\\Rt^{\\ge k,\\bullet}$ and ${}^U\\Rt^{\\bullet,\\ge k}$ are closed subsets in ${}^U\\Rt$ by their construction.\n \\end{proof} \n\n\n\\ssec{}{Proof of~\\reft{str}}\n\n\n(1) Follows immediately from the inclusion ${}^G\\Rt^k \\subset {}^U\\Rt^{0,k}$.\n\n(2) Equality $\\UMt^0(r,d,n) = \\GMt^0(r,d,n)$ for $r$ and $d$ coprime follows immediately\nfrom the equality ${}^G\\Rt^0 = {}^U\\Rt^{0,0}$ which is the only component of the stratum of ${}^U\\Rt$ giving $\\UMt^0(r,d,n)$.\n\n\nFurther, consider the subset ${}^U\\Rt^{k,\\mathrm{split}} \\subset {}^U\\Rt^k$.\nBy~\\refl{sequence-splits} it is closed.\nIt is also $\\GL(\\alpha(r,d,n))$-invariant. Finally, each point in ${}^U\\Rt^k$ is S-equivalent to a point in ${}^U\\Rt^{k,\\mathrm{split}}$.\nTherefore\n\\begin{equation*}\n\\UMt^k(r,d,n) = {}^U\\Rt^k\/\\!\/_{\\tet^0} \\GL(\\alpha(r,d,n)) = {}^U\\Rt^{k,\\mathrm{split}}\/\\!\/_{\\tet^0} \\GL(\\alpha(r,d,n)).\n\\end{equation*}\nSo, it is enough to find a direct product decomposition for the right hand side of the equality.\nFor this recall that the universal family of representations restricted to ${}^U\\Rt^{k,\\mathrm{split}}$\nsplits canonically as a direct sum\n\\begin{equation*}\n\\CV_\\bu = \\CU_\\bu \\oplus \\CW_\\bu\n\\end{equation*}\nof a supermonadic $\\tnt$-semistable $\\alpha(r,d,n-k)$-dimensional representation and of an Artin representation of dimension $(k,2k,k)$.\nThis decomposition gives a $\\GL(\\alpha(r,d,n))$-equivariant morphism to the homogeneous space\n\\begin{equation*}\n{}^U\\Rt^{k,\\mathrm{split}} \\to \\GL(\\alpha(r,d,n)) \/ (\\GL(\\alpha(r,d,n-k) \\times \\GL(k,2k,k))\n\\end{equation*}\nsuch that the fiber over a point is the product $\\hsRt^0(r,d,n-k) \\times {}^A\\mathsf{R}_\\tau(k,2k,k)$. It follows that\n\\begin{multline*}\n{}^U\\Rt^{k,\\mathrm{split}} \/\\!\/_{\\tet^0}  \\GL(\\alpha(r,d,n))\n\\\\ \\cong (\\hsRt^0(r,d,n-k) \\times {}^A\\mathsf{R}_\\tau(k,2k,k)) \/\\!\/_{\\tet^0}  (\\GL(\\alpha(r,d,n-k) \\times \\GL(k,2k,k))\n\\\\ \\cong (\\hsRt^0(r,d,n-k) \/\\!\/_{\\tet^0}  \\GL(\\alpha(r,d,n-k)) \\times ({}^A\\mathsf{R}_\\tau(k,2k,k)) \/\\!\/_{\\tet^0}  \\GL(k,2k,k))\n\\\\ \\cong \\UMt^0(r,d,n-k) \\times {}^A\\mathsf{M}_\\tau(k,2k,k).\n\\end{multline*}\nThis together with~\\refp{artin-moduli-k} proves part (2) of the Theorem.\n\n(3) The split representation in a closure of the $\\GL(\\alpha(r,d,n))$-orbit of a $\\tnt$-semistable representation $V_\\bu(E)$\nis just the direct sum of the supermonadic quotient and the Artin subrepresentation of $V_\\bu$. By~\\refp{tnt-ss}\nthese correspond to the sheaves $E^{**}$ and $E^{**}\/E$ respectively, hence the claim.\n\\qed\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\n\nSingle electron control in semiconductor quantum dots  would be valuable for spintronics, quantum information, and spin qubits\\cite{Aws}.\nAdiabatic time evolution of degenerate eigenstates of a quantum system provides a means\nfor controlling individual quantum states\\cite{Sha,Wil,Zee}. \nThey exhibit  non-Abelian gauge structures\nand often  give finite  non-Abelian Berry phases (they are also called matrix Berry phase  or holonomic phase).\nThese phases often depend on the geometry of the path traversed in the parameter space of the Hamiltonian.\nNuclei\\cite{jones2}, superconducting nanocircuits\\cite{fal}, optical systems\\cite{pach2}, and \natomic systems\\cite{Mead,duan,Un} \nhave such degenerate quantum states.\nIt has been shown that universal quantum computation is possible by means of non-Abelian unitary\noperations\\cite{Zan,Pach}.\nManipulation is  expected to be stable\nsince symmetries of the Hamiltonian that give rise to degeneracy are not broken\nduring the adiabatic transformations.  \nHowever, the degree of its stability is under investigation\\cite{Sol1}.\n\nRecently several interesting possibilities for electronic manipulation in\nsemiconductors have been proposed.\nSpin manipulation of quasi-two-dimensional electrons by time-dependent gate voltage is possible\nthrough the Dresselhaus and Rashba spin-orbit coupling mechanisms \\cite{Rashba1}.\nA geometric spin manipulation technique based on acceptor states in p-type semiconductors with spin-orbit coupling\nwas  proposed\\cite{Bern}. \nSpin-orbit coupling and a revolving external electric field\nmay generate holonomic qubit operations  in CdSe nanocrystals \\cite{Sere2}.\nHolonomic quantum computation using excitons in semiconductor nanostructures has been also proposed\\cite{Sol2}.\n\nA matrix Berry  phase is experimentally interesting because it represents mixing between degenerate levels.\nLet us explain briefly the basic ideas using a\nsimple  system possessing  Kramers' double degeneracy\\cite{Mead}.  The Hamiltonian depends on some\nexternal parameters $\\lambda_p$.\nIf the system is  in a superposition  state $|\\psi'(0)\\rangle =c_1(0)|\\psi_1(0)\\rangle +c_2(0)|\\psi_2(0)\\rangle$ \nat time t=0  an adiabatic  evolution  of the parameters\n$\\lambda_p$ can  transform this state into another state $|\\psi'(t)\\rangle =c_1(t)|\\psi_1(t)\\rangle +c_2(t)|\\psi_2(t)\\rangle$ \nafter some time $t$. Here the orthonormal basis states $|\\psi_i(t)\\rangle$ are the instantaneous eigenstates of the Hamiltonian. \nFor a cyclic change with the  period  $T$, represented by a closed contour $C$ in the parameter space,\nthe states $|\\psi_1(T)\\rangle$ and  $|\\psi_2(T)\\rangle$ return to the initial states $|\\psi_1(0)\\rangle$ and \n$|\\psi_2(0)\\rangle$, but the coefficients\n$c_1(T)$ and $c_2(T)$ may not return the initial values.  In such a case a $2\\times2$ matrix Berry phase\n(non-Abelian Berry phase) $\\Phi_C$ is generated\n\\begin{eqnarray}\n\\left(\n\\begin{array}{c}\nc_1(T) \\\\\nc_2(T)\n\\end{array}\n\\right)=\\Phi_C\n\\left(\n\\begin{array}{c}\nc_1(0) \\\\\nc_2(0)\n\\end{array}\\right).\n\\end{eqnarray}\nThis   non-Abelian geometric phase (holonomy)\nis connected to non-Abelian gauge potentials, as we discuss below.\nDuring the adiabatic cycle  the degenerate energy  $E(t)$ varies with time. But   hereafter we will \nset $E(t)=0$ since  it can be easily restored when needed.\nThe expansion coefficients satisfy the time-dependent Schr$\\ddot{o}$dinger equation\n\\begin{eqnarray}\ni \\hbar \\dot{c}_i=-\\sum_j A_{i j} c_j \\qquad i=1,2.\n\\label{eq:time_Schrod}\n\\end{eqnarray}\nThe matrix elements $A_{ij}$ are given by \n$A_{ij}= \\hbar \\sum_p(A_p)_{i,j}\\frac{d\\lambda_p}{dt}$, where the \nsum over $p$ in $A_{ij}$ is meant to be the sum  over $\\lambda_p$.\n$A_p$ are  $2\\times2$ matrices and are called  the non-Abelian vector potentials.\nThey  are given by\n\\begin{eqnarray}\n(A_p)_{i,j}=\ni \\langle\\psi_i|\\frac{\\partial\\psi_j}{\\partial\\lambda_p}\\rangle .\n\\end{eqnarray}\nThe formal solution of the  time-dependent Schr$\\ddot{o}$dinger equation gives the matrix Berry phase $\\Phi_C=Pe^{ -\\oint_C\\sum_p A_p d\\lambda_p}$,\nwhere $P$ represents a path ordering.\nUnder a unitary transformation\n$| \\psi'_i \\rangle =\\sum_j U^*_{i j}| \\psi_j \\rangle$\nthe non-Abelian gauge structure emerges\n\\begin{eqnarray}\nA'_k=UA_kU^\\dagger+i U \\frac{\\partial U^\\dagger}{\\partial \\lambda_k}.\n\\label{eq:vec_pot}\n\\end{eqnarray}\n\n\nHere we propose that quantum dots in n-type semiconductors with spin-orbit interaction can have matrix Berry phases.\nAll the   discrete energy levels of  quantum dots possess  a double degeneracy \nbecause  the Rashba and\/or Dresselhaus spin orbit coupling terms have time reversal symmetry.\nIn II-VI semiconductors\nthe Rashba term is expected to be larger than the\nthe Dresselhaus coupling. In III-V semiconductors, such as GaAs, the\nopposite is true\\cite{Rashba2}. In our work both the Rashba and Dresselhaus terms are included.\nThe adiabatic transformation\ncan be performed electrically by changing the confinement potentials of the quantum dot, see Fig.\\ref{fig:setup}(d). \nWe find that, although the spin orbit terms formally give  rise to a\nnon-Abelian structure for the  matrix vector potentials, \ndouble degeneracy does not necessarily lead to \nfinite non-Abelian Berry phases.\nWe show  that when  the circular symmetry of two-dimensional harmonic potentials is broken  \nmatrix Berry phases can be  produced (such a distortion is shown in Fig.\\ref{fig:setup}(b)).\nWe propose that the presence of a matrix Berry phase may be detected by measuring the  \nthe optical spectrum.\n\n\nIn Sec. II we describe the   Hamiltonian of the system  in detail, and in Sec. III we discuss \na  model calculation of the matrix Berry phase. A possible experimental\ndetection of the the matrix Berry phase is suggested in Sec. IV.  Conclusions  are given in Sec. V.\n\n\n\n\\section{Hamiltonian}\nAn electric field $E$ is applied \nalong the z-axis and electrons are confined in a triangular potential $V(z)$, see Fig.\\ref{fig:setup}(a).\nWhen the the width of the quantum well along the z-axis is sufficiently {\\it small} we may include only the lowest\nsubband state along  the z-axis. We denote this  wavefunction by $f(z)$, see Fig.\\ref{fig:setup}(a).\nFrom the expectation value\n$\\langle f(z)|k_z^2|f(z)\\rangle =0.8(2m^*eE\/\\hbar^2)^{2\/3}$\nwe estimate the characteristic length scale along the z-axis: $R_z=1\/\\sqrt{0.8(2m^*eE\/\\hbar^2)^{2\/3}}$.\nThe Hamiltonian in the absence of the spin orbit coupling is \n$H_K=-\\frac{\\hbar^2\\nabla^2}{2m^*}+U(x,y)+V(z)$.\nWe take the two-dimensional potential to be $U(x,y)=\\frac{1}{2}m^*\\omega^2_x x^2+\\frac{1}{2}m^*\\omega^2_y y^2+\nV_p(x,y)$, see Fig.\\ref{fig:setup}(b).\nThe strengths of the harmonic potentials are denoted by $\\omega_x$ and $\\omega_y $.   They \nmay be varied by {\\it changing gate potentials} of the quantum dot system. \nThe characteristic lengths scales along x- and y-axis are $R_{x,y}=\\sqrt{\\frac{\\hbar}{m^*\\omega_{x,y}}}$.\nThe potential  $V_p(x,y)=\\epsilon' y$ represents a  distortion of the \nharmonic potentials and its strength $\\epsilon'$ may be {\\it varied electrically}. \n\nIn a periodic crystal potential of a semiconductor \nthe spin orbit interaction has two contributions.\nThe Rashba spin orbit term is \n\\begin{eqnarray}\nH_\\mathrm{R}=c_\\mathrm{R} \\left( \\sigma_x k_y -\\sigma_y k_x \\right).\n\\end{eqnarray}\nHere $\\sigma_{x,y}$ are Pauli spin matrices and $k_{x,y}$ are momentum operators ($k_x=\\frac{1}{i}\\frac{d}{dx}$\nand similarly with $k_y$.).  \nThe constant $c_R$ {\\it depends on the external electric field} $E$ applied along the z-axis.\nThe Dresselhaus spin orbit term is\n\\begin{eqnarray}\nH_\\mathrm{D}=c_\\mathrm{D}\\left( \n\\left( \\sigma_x k_x \\left(k_y^2-k_z^2 \\right) \\right)\n+\\left( \\sigma_y k_y \\left(k_z^2-k_x^2 \\right) \\right)\n\\right).\n\\label{Dresselhaus}\n\\end{eqnarray}\nThere is another term of the form  $ \\sigma_z \\langle k_z\\rangle \\left(k_x^2-k_y^2 \\right) $ in the  Dresselhaus spin orbit term\nbut it vanishes  since the expectation value\n$\\langle k_z\\rangle =\\langle f(z)|k_z|f(z)\\rangle =0$ for the first subband  along  z-axis.\nThe  constant $c_D$  represents breaking of inversion symmetry by the crystal  in zinc blende structures. \n\n\nThe total Hamiltonian of an electron in  a semiconductor quantum dot\nis\n$H=H_\\mathrm{K}+H_\\mathrm{R}+H_\\mathrm{D}$.\nAn  eigenstate of the Hamiltonian $H$ may be expanded as a linear combination of the eigenstates of $H_K$ \n\\begin{eqnarray}\n| \\psi \\rangle&=&\\sum_{m n}c_{m n\\uparrow }|m n \\uparrow \\rangle \n+\\sum_{m n}c_{m n\\downarrow }|m n \\downarrow \\rangle .\n\\end{eqnarray}\nThe expansion coefficients \nsatisfy a matrix equation \n$\\sum_{m'n'\\sigma'}\\langle mn\\sigma|H|m'n'\\sigma'\\rangle c_{m'n'\\sigma'}=Ec_{mn\\sigma}$.\nIn the basis states $|m n \\sigma \\rangle $ the quantum number $m (n)$ and $\\sigma$ label \nthe harmonic oscillator levels along  the x-axis (y-axis) and\nthe component of electron spin.  The subband wavefunction $f(z)$ is suppressed in the notation $|m n \\sigma \\rangle$.\n\nIn the absence of the Zeeman term the total Hamiltonian is invariant under time reversal symmetry:\n$\\vec{k}\\rightarrow -\\vec{k}$ and $\\vec{\\sigma}\\rightarrow -\\vec{\\sigma}$.\nThe time reversal operator is \n$K=-i \\sigma_y C$,\nwhere the operator $C$ stands for complex conjugation. The time reversed state of $ | \\psi \\rangle$ is\n\\begin{eqnarray}\n| \\overline{\\psi} \\rangle=K | \\psi \\rangle\n=-\\sum_{m n}c_{m n \\downarrow}^* |m n \\uparrow \\rangle\n+\\sum_{m n} c_{m n\\uparrow}^* |m n \\downarrow \\rangle.\n\\end{eqnarray}\nNote that $K^2| \\psi \\rangle=-| \\psi \\rangle$.\nThese two states are degenerate and   orthonormal.  \nWe have suppressed the Bloch wavefunction of the conduction band in applying the time reversal\noperator since it is unaffected by the operator $K$. Our wavefunctions are \nall effective mass wavefunctions  and only the  conduction band Bloch\nwavefunction at $\\vec{k}=0$ is relevant.\n\nAn  adiabatic change implemented \nby changing the energy parameters  $E_R$ and $E_p$ that characterize the Rashba term and the distortion potential $V_p(x,y)$:\n\\begin{eqnarray}\nE_R=\\frac{c_R}{\\sqrt{2}R_y} \\ \\textrm{and}\\ E_p=\\epsilon R_y,\n\\label{parameters}\n\\end{eqnarray}\nwhere\n$\\epsilon=\\epsilon'\/\\sqrt{2}$.\nThe first parameter  $E_R$ \ndepends on the electric field through the constant $c_R$.  \nThe typical value of  the energy scale associated with \nthe Rashba constant  depends on the electric field applied along the z-axis and the semiconductor material: it\nis of  order  $E_R=c_R\/ R\\sim 0.01-10meV$, where the   length\nscale $R\\sim 100 \\AA$ is the lateral dimension of the quantum dot.\nThe second adiabatic parameter $E_p$ represents the strength of the distortion potential $\\epsilon' y$:\nthe expectation value of the distortion potential is $E_p=\\langle 0|\\epsilon' y|1\\rangle$.  Its \nmagnitude is of order $1-10meV$, depending on the electric field applied along the y-axis and the\nwidth of the triangular potential $V(z)$.\n\n\n\n\n\n\n\n\n\n\n\\section{Model calculation of non-Abelian Berry phase}    \n\\subsection{A truncated Hamiltonian matrix}\n\nWe work out a  model that can be solved analytically.\nThis model is simple but much can be learned from it.\nLet us take $\\omega_x=2\\omega_y$ (Other values of $\\omega_x$ can also be chosen). Then the lowest eigenenergy state\nof $H_K$ is $|mn\\rangle=|00\\rangle$ with the energy $E_0=\\frac{3}{2}\\hbar\\omega_y$ and \nthe next lowest eigenenergy state  is $|01\\rangle$ with the energy $E_1=\\frac{5}{3}E_0$.\nThe typical value of the energy spacing between the quantum dot levels, $E_0$, is of  order $1-10 meV$.\nLet us write down the eigenstates of the total Hamiltonian  as a  linear combination\nof four basis states made out of these\nstates and spin degree of freedom:\n\\begin{eqnarray}\n| \\psi \\rangle&=&\nc_{0,0,\\uparrow}|0,0, \\uparrow \\rangle+c_{0,1,\\uparrow}|0,1, \\uparrow \\rangle  \\nonumber\\\\\n&+&c_{0,0,\\downarrow}|0,0, \\downarrow \\rangle+c_{0,1,\\downarrow}|0,1, \\downarrow \\rangle.\n\\label{eq:lin_comb1}\n\\end{eqnarray}\nThe  $4\\times4$ truncated Hamiltonian matrix is\n\\begin{eqnarray}\n\\left(\n\\begin{array}{cccc}\nE_0 & E_p & 0 & -i E_R - E_D \\\\\nE_p & E_1 & i E_R +E_D & 0 \\\\\n0 & -i E_R + E_D & E_0 & E_p \\\\\ni E_R -E_D & 0 & E_p & E_1\n\\end{array}\n\\right),\\nonumber\\\\\n\\label{matrix_Hamil}\n\\end{eqnarray}\nor\n\\begin{eqnarray}\n&&\\frac{1}{2}(E_0+E_1)I\n+\\frac{1}{2}(E_0-E_1)\\left( \\begin{array}{cc} \\sigma_z & 0 \\\\ 0 & \\sigma_z \\end{array} \\right) \\nonumber\\\\\n&&+E_p \\left( \\begin{array}{cc} \\sigma_x & 0 \\\\ 0 & \\sigma_x \\end{array} \\right)\n+E_R \\left( \\begin{array}{cc} 0 & \\sigma_y \\\\ \\sigma_y & 0 \\end{array} \\right)\\nonumber\\\\\n&&+E_D \\left( \\begin{array}{cc} 0 & -i \\sigma_y \\\\ i \\sigma_y & 0 \\end{array} \\right).\n\\end{eqnarray}\nThe matrix elements of the distortion potential are\n$\\langle m|y|n\\rangle=\\sqrt{\\frac{\\hbar}{2 m^* \\omega_y}} (\\sqrt{n+1} \\delta_{m,n+1}+\\sqrt{n} \\delta_{m,n-1})$. \nNote that the distortion potential couples even and odd parity\nstates of the one-dimensional harmonic potential.  Other functional form of $V_p(x,y)$ may be also\nused to generate the non-Abelian Berry phase as long as it couples even and odd parity\nstates.\nThe first term in the Dresselhaus spin orbit term, Eq. (\\ref{Dresselhaus}), is zero since $\\langle 0|k_x|0\\rangle=0$. \nFrom the second term of the Dresselhaus term we get\n$-ic_D\\langle 0|k_y|1\\rangle(\\langle f(z)|k_z^2|f(z)\\rangle-\\langle 0|k_x^2|0\\rangle)=-E_D$,\nwhere the constant $E_D=f(E_R)-g(E_0)$ with\n$f(E_R)=c_D\/R_yR_z^2$\nand\n$g(E_0)=c_D \/R_yR_x^2$.\nHere we have used\nthe momentum matrix elements \n$\\langle m|\\hbar k_{x,y}|n\\rangle=i \\sqrt{\\frac{m^* \\hbar \\omega_{x,y}}{2}} (\\sqrt{n+1} \\delta_{m,n+1}-\\sqrt{n} \\delta_{m,n-1})$.\nThe magnitude of the energy scale associated with the Dresselhaus term is of order  $E_D=c_D\/ R^3$, and\nit can be larger or smaller than the Rashba term, depending on the material\\cite{Rashba2}.\nThe function $f(E_R)$ depends on  $E_R$ because\nthe  electric field $E$ enters through $R_z$. \nThere is no simple way to calculate $c_R$ because it depends both on the electric field inside the \nsemiconductor heterostructure and on the detailed boundary conditions at the interface.\nFor simplicity we take $f(E_R)=aE_R$  and $g(E_0)=bE_0$, where $a$ and $b$ are numerical constants.\nMore complicated functions, for example, $f(E_R)=E_R^2\/E_0$ and $g(E_0)=E_0$, could be used, but our calculation indicates that \nthe essential physics does not change.\n\n\n\\subsection{Finite matrix Berry phase }\nDiagonalization of  this $4\\times4$ Hamiltonian matrix\ngives the eigenenergies\n$\\lambda=\\frac{1}{3}(4E_0\\pm \\sqrt{E_0^2+9E_D^2+9E_p^2 +9E_R^2})$.\nLet us choose the following  state\n\\begin{eqnarray}\n|\\psi\\rangle=\n\\frac{1}{N}\n\\left(\n\\begin{array}{c}\n3 E_p \\\\\nE_0-\\sqrt{E_0^2+9(E_D^2+E_p^2+E_R^2)} \\\\\n3(E_D-i E_R) \\\\\n0\n\\end{array}\n\\right),\\nonumber\\\\\n\\label{eq:lin_comb2}\n\\end{eqnarray}\nand its time reversal state  \n\\begin{eqnarray}\n|\\overline{\\psi}\\rangle=\n\\frac{1}{N^*}\n\\left(\n\\begin{array}{c}\n-3(E_D+i E_R) \\\\\n0 \\\\\n3 E_p \\\\\nE_0-\\sqrt{E_0^2+9(E_D^2+E_p^2+E_R^2)}\n\\end{array}\n\\right).\\nonumber\\\\\n\\end{eqnarray}\nas the basis states in the lowest energy degenerate Hilbert space.\n\n\n\n\n\nIn the evaluation of the matrix $A_p$ we use \n$2\\int \\phi_k(r)^*\\frac{\\partial\\phi_k(r)}{\\partial \\lambda_p}=\\frac{\\partial}{\\partial \\lambda_p}\n\\int|\\phi_k(r)|^2dr=0$,\nwhere  $k=mn\\sigma$.  Note that\n$\\phi_k(r)$ is the two-dimensional harmonic oscillator wavefunction and that it is a real function. \nOne can show that\n$(A_p)_{i,j}=i \\sum_k \\alpha_k^*\\frac{\\partial \\beta_k}{\\partial\\lambda_p}$,\nwhere the pair of degenerate states are \n$| \\psi_i \\rangle=\\sum_k \\alpha_k|k\\rangle$   and   $| \\psi_j \\rangle=\\sum_k \\beta_k|k\\rangle$.\nThe orthonormalization  $\\langle\\psi_i|\\psi_j\\rangle=\\delta_{ij}$    gives that the diagonal matrix elements\n$(A_p)_{i,i}$ are real   and that the off-diagonal elements satisfy $(A_p)_{i,j}=(A_p)^*_{j,i}$.\nThe adiabatic parameters are  $\\lambda_1=E_R$, $\\lambda_2=E_p$.\nWe calculate the matrix Berry phase for\n$E_D=E_R-E_0$. \nThe matrix vector potentials with respect to $|\\psi\\rangle$   and $|\\overline{\\psi}\\rangle$   have the following structures \n\\begin{eqnarray}\nA_{E_R}\n&=&\n\\left(\n\\begin{array}{cc}\nc_1 & a_1+i b_1 \\\\\na_1-i b_1 & -c_1\n\\end{array}\n\\right) \\nonumber \\\\\n&=&a_1 \\sigma_x -b_1 \\sigma_y +c_1 \\sigma_z,  \n\\end{eqnarray}\nand \n\\begin{eqnarray}\nA_{E_p}\n&=&\n\\left(\n\\begin{array}{cc}\n0 & a_2-i b_2 \\\\\na_2+i b_2 & 0\n\\end{array}\n\\right)\\nonumber \\\\\n&=&a_2 \\sigma_x +b_2 \\sigma_y. \n\\end{eqnarray}\nThe functions  $a_i$, $b_i$, and $c_i$ depend on $E_R$ and $E_p$ and are real. \nUnder an adiabatic time evolution a state in this degenerate Hilbert space changes as \n$|\\psi'(t)\\rangle=c_1(t) |\\psi(t)\\rangle + c_2(t) |\\overline{\\psi}(t)\\rangle$.\nSuppose initially  $c_1(0)=1$ and $c_2(0)=0$, i.e., $|\\psi'(0)\\rangle=|\\psi\\rangle$.\nThe cyclic adiabatic path with the period $T=2\\pi\/\\omega$ is shown in Fig.\\ref{fig:setup}(d) and is given by \n\\begin{eqnarray}\n&(&E_R(t),E_p(t))= \\nonumber \\\\\n&(&E_{R,c}+\\Delta E_R \\cos(\\omega t),E_{p,c}+\\Delta E_p \\sin(\\omega t)).\n\\label{closed_loop}\n\\end{eqnarray}\nThe frequency $\\omega$ can be taken to be a fraction of $E_0$.\nWe solve Eq.(\\ref{eq:time_Schrod})   numerically  and find that $c_2(T)$ is non-zero.\nFor the parameters $E_{R,c}= 2 E_0$, $E_{p,c}=E_0$, $\\Delta E_R= 1.9 E_0$, $\\Delta E_p= 0.9 E_0 $, and $\\omega=E_0 \/10$\nwe find $c_1(T)=0.8884-i 0.0897$  and $c_2(T)=-0.4429 +i 0.0874$.\n\n\n\n\\subsection{Absence of matrix  Berry phase}\nThe matrix Berry phase is absent when the distortion potential $V_p(x,y)$ is zero.\nWhen only Rashba and\/or Dresselhaus terms are present \nin  Eq.(\\ref{matrix_Hamil})\nthe adiabatic transformation can be performed\n by varying the parameters $\\lambda_1=E_R$  and $\\lambda_2=E_0$.\nIn this case  {\\it an orthonormal basis set exists} in the degenerate Hilbert space such that\nthe matrix vector potentials are diagonal and the non-Abelian \nBerry phase is zero. This degenerate basis set,  $|\\psi_1\\rangle$ and $|\\psi_2\\rangle$, has the property that\nfor each $k$  either $c_k^{(1)}=0$\nor  $c_k^{(2)}=0$, where $|\\psi_i\\rangle=\\sum_{k=1}^{4}c_k^i|k\\rangle$\n(see, for example, Eqs.(\\ref{eq:finite_B_1}) and (\\ref{eq:finite_B_2})).\nThe off-diagonal matrix elements of the vector potential $(A_p)_{1,2}\n=i \\langle\\psi_1| \\frac{d\\psi_2}{d\\lambda_p}\\rangle=i \\sum_{k=1}^{4}c_k^{(1)*}\n\\frac{dc^{(2)}_{k}}{d\\lambda_p}=0$\nbecause either $c_k^{(1)}=0$\nor  $c_k^{(2)}=0$.\nTherefore the matrix vector potentials are diagonal.\nIn this case  the matrix Berry phase will be absent.\nLet us construct  explicitly  $| \\psi_1 \\rangle$ and $| \\psi_2 \\rangle$  when only the  Rashba term is present, i.e., when $E_D=E_p=0$.\nThey are given  by\n\\begin{eqnarray}\n| \\psi_1 \\rangle&=&\\frac{1}{\\sqrt{2} \\sqrt{E_0^2+9 E_R^2+E_0 \\sqrt{E_0^2+ 9 E_R^2}}}  \\nonumber\\\\\n&\\times&\n\\left(\n\\begin{array}{c}\ni(E_0+\\sqrt{E_0^2+9 E_R^2})\\\\\n0 \\\\\n0 \\\\\n3 E_R\n\\end{array}\n\\right),\\nonumber\\\\\n\\label{eq:finite_B_1}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n| \\psi_2 \\rangle&=&\\frac{1}{\\sqrt{9 E_R^2+\\left(E_0 -\\sqrt{E_0^2+ 9 E_R^2}\\right)^2}} \\nonumber\\\\\n&\\times&\n\\left(\n\\begin{array}{c}\n0 \\\\\ni (E_0 -\\sqrt{E_0^2+9 E_R^2}) \\\\\n3 E_R \\\\\n0\n\\end{array}\n\\right).\\nonumber\\\\\n\\label{eq:finite_B_2}\n\\end{eqnarray}\nWith these new  eigenstates it is possible to show that the non-Abelian gauge potentials \nare not only diagonal but also zero: $A_{E_0}=0$ and $A_{E_R}=0$.\n\n\\subsection{Non-Abelian gauge structure}\nLet us also test the non-Abelian gauge structure given by   Eq. (\\ref{eq:vec_pot}).\nWe make a unitary transformation from ($| \\psi_1 \\rangle$, $| \\psi_2 \\rangle$), given in Eqs.(\\ref{eq:finite_B_1})\nand (\\ref{eq:finite_B_2}),\nto a  pair of time reversed degenerate eigenstates \n$(| \\psi'\\rangle,| \\overline{\\psi}'\\rangle)$,\n\\begin{eqnarray}\n| \\psi' \\rangle=\\frac{3}{2 \\sqrt{E_0^2+9E_R^2}}\n\\left(\n\\begin{array}{c}\nE_R \\\\\nE_R \\\\\n\\frac{1}{3} i \\left(E_0+\\sqrt{E_0^2+9E_R^2} \\right) \\\\\n-\\frac{1}{3} i \\left(-E_0+\\sqrt{E_0^2+9E_R^2} \\right)\n\\end{array}\n\\right),\\nonumber\\\\\n\\end{eqnarray}\nand \n\\begin{eqnarray}\n| \\overline{\\psi}' \\rangle=\\frac{3}{2 \\sqrt{E_0^2+9E_R^2}}\n\\left(\n\\begin{array}{c}\n\\frac{1}{3} i \\left( E_0+\\sqrt{E_0^2+9E_R^2} \\right) \\\\\n-\\frac{1}{3} i \\left( -E_0+\\sqrt{E_0^2+9E_R^2} \\right) \\\\\nE_R \\\\\nE_R \\\\\n\\end{array}\n\\right). \\nonumber\\\\\n\\end{eqnarray}\nUsing these new basis states  one can show that    the non-Abelian gauge potentials are  \n\\begin{eqnarray}\nA'_{E_R}\n&=&\\frac{1}{E_0^2+ 9 E_R^2}\n\\left(\n\\begin{array}{cc}\n0 & 3E_0\/2 \\\\\n3E_0\/2 & 0\n\\end{array}\n\\right) \\nonumber \\\\\n&=&\\frac{3 E_0}{2(E_0^2+ 9 E_R^2)}\\sigma_x,\n\\end{eqnarray}\nand \n\\begin{eqnarray}\nA'_{E_0}\n&=&\\frac{1}{E_0^2+ 9 E_R^2}\n\\left(\n\\begin{array}{cc}\n0 & -3E_R\/2 \\\\\n-3E_R\/2 & 0\n\\end{array}\n\\right) \\nonumber \\\\\n&=&-\\frac{3 E_R}{2(E_0^2+ 9 E_R^2)}\\sigma_x.\n\\end{eqnarray}\nSince the old vector potentials are $A_{E_R}=0$ and $A_{E_0}=0$ it  follows from  Eq. (\\ref{eq:vec_pot}) \nthat $A'_k=i U \\frac{\\partial U^\\dagger}{\\partial \\lambda_k}$.\nWe have explicitly verified that this relation holds by computing the unitary matrix $U$.\nWe have also verified independently the absence of  the matrix Berry phase\nby solving the time-dependent Schr$\\ddot{o}$dinger Eq.(\\ref{eq:time_Schrod}) \nwith $A'_{E_R}$  and $A'_{E_0}$.  This provides a check on our numerical method of \nsolving the time-dependent Schr$\\ddot{o}$dinger equation.\n\n\n\n\n\\section{Detection of matrix Berry phase} \n\nAfter an adiabatic cycle the electron acquires a matrix Berry phase.\nThe presence of such a phase may be detected by measuring the strength of dipole optical transitions before\nand after the cycle.  In this section we calculate the optical strengths using\nthe truncated Hamiltonian.  This calculation is not quantitatively accurate but\nit will give us an estimate  of the   magnitude of the effect.\n\nFirst we need to prepare physically some particular pair of  degenerate \nstates.  In the presence of a magnetic field  \nalong the z-axis any degenerate pair of states at zero magnetic field  will be split into two\nstates. \nWe  define  the lowest energy  pair of  degenerate eigenstates in the zero magnetic field limit of $B_z$ \nas $|\\psi_1\\rangle=\\lim_{B_z\\rightarrow 0}|\\psi_1(B_z)\\rangle$\nand  $|\\psi_2\\rangle=\\lim_{B_z\\rightarrow 0} |\\psi_2(B_z)\\rangle$ , where $|\\psi_1(B_z)\\rangle$ and \n$|\\psi_2(B_z)\\rangle$ are the split lowest and next lowest energy states in a finite magnetic field, see Fig.(\\ref{fig:mag_field_limit}a).\nIn a similar way we define degenerate eigenstates in the zero magnetic field limit of $B_x$:\n$|\\phi_1\\rangle=\\lim_{B_x\\rightarrow 0}|\\phi_1(B_x)\\rangle$\nand  $|\\phi_2\\rangle=\\lim_{B_x\\rightarrow 0} |\\phi_2(B_x)\\rangle$, see Fig.(\\ref{fig:mag_field_limit}b).\nThe states $|\\psi_{1,2}(B_z)\\rangle$ and $|\\phi_{1,2}(B_x)\\rangle$ are calculated from the Hamiltonian by replacing\n$\\vec{k}$ with $\\vec{k}+\\frac{e}{c}\\vec{A}$ and adding the Zeeman term ($\\vec{A}$ is the vector potential and $e>0$).\n\n\\begin{figure}[hbt]\n\\begin{center}\n\\includegraphics[width = 0.44 \\textwidth]{limit.eps}\n\\caption{(a)$|\\psi_1\\rangle$ and $|\\psi_2\\rangle$ are generated in the zero limit of $B_z$.\n(b)$|\\phi_1\\rangle$ and $|\\phi_2\\rangle$ are generated in the zero limit of $B_x$.\n(c)Dipole transition from the first to third lowest energy states in a finite  $B_z$.  The initial\nstate in this transition is   $|\\phi_1(0) \\rangle$ or  $|\\phi_1(T) \\rangle$.  These states represent, respectively,\nthe states before and after the adiabatic cycle.  The incident photon is polarized along the y-axis.\n}\n\\label{fig:mag_field_limit}\n\\end{center}\n\\end{figure}\nSuppose that the electron is in the lowest energy state.  In order to measure the matrix Berry phase we  perform the following set of procedures:\n\\begin{enumerate}\n\\item  \nWe apply a  magnetic field along the x-axis and take the zero field limit.   The resulting state $|\\phi_1 \\rangle$\n(Fig.(\\ref{fig:mag_field_limit}b))\ncan be written as a linear combination of  $|\\psi_1 \\rangle$\nand  $|\\psi_2 \\rangle$ (Fig.(\\ref{fig:mag_field_limit}a)): $|\\phi_1 \\rangle$=$c_1(0)|\\psi_1 \\rangle+c_2(0)|\\psi_2 \\rangle$ with  \n$(|c_1(0)|^2,|c_2(0)|^2)=(1\/2,1\/2)$.  \n\n\\item We apply  adiabatically a magnetic field along the z-axis.  \nWe increase it to a value $B_z$. Then the probability for  the electron to be  in the state $|\\psi_1(B_z)\\rangle$ \nis given by $|c_1(0)|^2$.\n\n\\item We then measure the intensity of the dipole transition  from the lowest energy state to the third lowest\nenergy state, see Fig.(\\ref{fig:mag_field_limit}c).  This intensity is proportional to $|c_1(0)|^2$, i.e, proportional\nto the probability that   $|\\psi_1(B_z)\\rangle$\nis occupied.\nThe lowest and third lowest energy  states can be written as  a \nlinear combination of the basis states $|mn\\sigma \\rangle$ : both states can be written in the form \n$ c_{0,0,\\uparrow}|0,0, \\uparrow \\rangle+c_{0,1,\\uparrow}|0,1, \\uparrow \\rangle\n+c_{0,0,\\downarrow}|0,0, \\downarrow \\rangle+c_{0,1,\\downarrow}|0,1, \\downarrow \\rangle$.\nWhen the photon is polarized along the y-axis  \nonly the basis states with different parities are coupled in   the dipole approximation,  for example,\n$\\langle 00\\sigma|k_y|01\\sigma\\rangle=- \\langle 01\\sigma|k_y|00\\sigma\\rangle$ is non-zero.\n\\end{enumerate}\nAgain we assume that the electron is in the lowest energy state.  We  perform the second  set of procedures:\n\\begin{enumerate}\n\\item  We apply a  magnetic field along the x-axis and take the zero field limit.\n\n\\item We then perform an adiabatic cycle following a closed loop \nin the parameter space of $(E_R,E_p)$, given by Eq.(\\ref{closed_loop}).\nAfter\nthe  adiabatic cycle the electron will be in the state $|\\phi_1(T) \\rangle=c_1(T)|\\psi_1 \\rangle+c_2(T)|\\psi_2 \\rangle$ \nwith $(|c_1(T)|^2,|c_2(T)|^2)=(0.559,0.441)$\n(the parameters are $E_{R,c}=E_0$, $E_{p,c}=3E_0$, $\\Delta E_R= 0.8E_0 $, $\\Delta E_p= 2.5E_0$, and $\\omega=0.2 E_0 $).\n\n\\item We apply adiabatically a magnetic field along the z-axis, see Fig.(\\ref{fig:mag_field_limit}c).   Then the probability for  the electron \nto be in the first  lowest energy state, $|\\psi_1(B_z)\\rangle$, is $|c_1(T)|^2$\nwhile  the probability that the electron\nwill be in the second lowest energy state, $|\\psi_2(B_z)\\rangle$, is $|c_2(T)|^2$.\nHere the value of $B_z$ is the same as that of step 2 in the first set of procedures.\n\n\\item We then measure the intensity of the dipole transition  from the lowest energy state to the third lowest\nenergy state.  This intensity is proportional to $|c_1(T)|^2$,  i.e, proportional\nto the probability that   $|\\psi_1(B_z)\\rangle$\nis occupied.\n\\end{enumerate}\n\nThese two sets of measurements are repeated many times.   Then the ratio \n$\\frac{|c_1(T)|^2 }{|c_1(0)|^2}=1.12$ gives the intensity ratio of the dipole transitions  in the first and second \nsets of procedures (the dipole matrix elements cancel each other).   It is the direct measure of \nthe matrix Berry phase.   The analysis of the intensity of optical transitions in zero magnetic field is \ncomplicated due to the presence of the degeneracy in the final states of the transition. \nIn a finite magnetic field this degeneracy is lifted.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Discussions}\n\n\n\n\nEach discrete eigenstate  of a semiconductor quantum dot with the Rashba and\/or Dresselhaus spin orbit coupling terms\npossesses  a double degeneracy due to time reversal symmetry.\nWe have investigated the matrix Berry phase of such a quantum dot \nin a simple truncated model Hamiltonian that can be solved analytically.\nWe have found that the double degeneracy  does not necessarily  lead to  a finite \nnon-Abelian Berry phase.  \nThe addition of a parity breaking distortion potential to the Hamiltonian when both the  Rashba and Dresselhaus spin orbit coupling terms\nare present  gives rise to\na finite matrix Berry phase.\nWe have proposed that this phase may be detected in the dipole  transitions\nbetween the ground and first excited states in a magnetic field.\n\nFor an accurate modeling of possible experiments\nthe number of basis vectors in the truncated Hamiltonian matrix must be chosen \nsufficiently large.  This number  will be determined by the ratios  $E_0\/E_{R,D,p}$.\nCalculation of the matrix Berry phase in such a case requires a heavy numerical computation.\nAccurate experimental determination of the functional dependence of the Rashba  constant on the electric field \nwould be also valuable\nin determining the magnitude of matrix Berry phases.\nThe quantum dot studied in this work contains  a single electron.\nIt may be worthwhile to investigate the effect of many body interactions.\n\n\n\n\n\\begin{acknowledgments}\nThis work was  supported by grant No. R01-2005-000-10352-0 from the Basic Research Program\nof the Korea Science and Engineering\nFoundation and by Quantum Functional Semiconductor Research Center (QSRC) at Dongguk University\nof the Korea Science and Engineering\nFoundation. \n\\end{acknowledgments}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\par The concept of continuity is natural when it comes to service, otherwise its counterpart, i.e. discontinuity (or fragmentation)  can hurt the system performance. Fragmentation is an age old problem in storage devices, and recently it also was shown to affect the network performance in elastic optical networks (EON). In EON, spectrum can be sliced to a smaller units (e.g., 6.25 GHz) to suit the diverse applications. While serving heterogeneous application demands for bandwidth, setting up and termination of connection requests can cause spectrum fragmentation of an elastic optical link (EOL), which results in higher blocking probability. To alleviate this problem, network defragmentation, - similar to the disk defragmentation, can be used to reconfigure some of the connections such that largest possible free resource block are created for the future requests. However, reconfiguration of a connection causes  disruption of its traffic for the duration of retuning time of transceivers   and  reconfiguration time of the optical switches. One can avoid reconfiguration time of switches  by making the connection over new frequency slots (FS) before breaking it from its existing path, however migration (retuning) will still cause some interruption.\n\n\\par In this paper, we coin the term Defragmentation-as-a-Service (DaaS) in elastic optical networks, and analyze the  positive effect of defragmentation (DF) process as a function of the rate at which it is performed, load (or, call arrival rates) and the available resources. We also analyze the impact of DF rate on the blocking probability for different spectrum allocation (SA) policies, such as first-fit (FF) and random-fit (RF) \\cite{yu2014exact,beyranvand2014analytical,Rosa2015}.\nTo understand the effect of defragmentation, we model an EOL using a multi-class continuous-time Markov chain (CTMC), and include DaaS states in addition to the normal service\\footnote{We refer to \\emph{normal service} for the case of service which does not employ defragmentation} states. We have the following assumptions for our model: i) the DF process is called when a request is blocked due to the fragmentation of a link; ii) all existing requests are interrupted during DF period, iii) all incoming requests are blocked during the DF period, and iv) system returns to normality, i.e. services of existing requests are resumed when DF is completed.\nAlthough assumptions, particularly (ii) and (iv), are stronger, traffic interruption of some (but not all) existing connections does happen during reconfiguration in the real system. Note that the DaaS states are blocking states for all types of connections, hence if DF is performed at lower rate then we would experience an increase in overall blocking probability (BP). Using our model, we show that DF reduces the overall call blocking probability only if: a) it is performed at the higher rates as compare to the normal service rate of the calls, b) load on system is moderate (i.e., call arrival rates are not high), and c) resources (bandwidth) are sufficient.\n\\vspace{-2mm}\n\\section{Proposed Model: Defragmentation-As-a-Service}\n\\begin{figure}[t]\n \\centering\n\\includegraphics[width= 1\\textwidth]{system.pdf}\n\\vspace{-8mm}\n  \\caption{DaaS: its effect, and state transitions leading to defragmentation for RF and FF SA policies.}\n\\label{fig:system}\n\\vspace{-2mm}\n\\end{figure}\n\\vspace{-1mm}\nWe adopt a multi-class CTMC model to analyze the blocking probability of EOL.\nHere, we assume that a DaaS process is triggered when a request is blocked due to fragmentation of an EOL, and not due to the unavailability of resources. To model the reconfiguration of the occupied FS, we introduce DaaS states. We assume that both inter-arrival  and call holding times of class-k requests are independent and  exponentially distributed with average rates $\\lambda_k$ and  $\\mu_k$, respectively. In Fig. \\ref{fig:system}(a), let us consider an example of an EOL link with $C=6$ FS to illustrate the idea.  In Fig. \\ref{fig:system}(b) and (c), we consider two classes of requests with demands $2$ and $3$ consecutive slots for RF and FF SA policies, respectively and show some of the transitions, due to the arrivals and departures, leading to a DaaS state. In contrast to FF, which allocates first available consecutive slots to a request, RF randomly allocates one of the possible assignments. In Fig. \\ref{fig:system}(b), under the RF method, consider a case where a new request (say $R_1$) of class $k=1$ arrives in state $S_1$ with demand $d_1=2$. There are $a(S_1,k)=2$ different ways to occupy the $3$ free slots of state $S_1$, thus transitions ($S_1 \\rightarrow S_2$ and $S_1 \\rightarrow S_6$) occurs with rates $\\lambda_1\/2$.\nFurthermore, when a class-1 request (two slots) arrives, blocking of the request due to fragmentation occurs in the set of states $FB(1)=\\{S_2, S_6, S_7\\}$. \nThese blocking events will trigger the transition from states $S_i \\in FB(1)$  to the DaaS state $S_{d1}$ (red arrows). During the exponentially distributed DaaS time $T_{DF}$ with mean $1\/\\mu_d$, arriving calls are blocked. Furthermore, services of existing connections effected by spectrum reconfiguration are interrupted during $T_{DF}$. Finally, system returns to a non-fragmented state (green transition to $S_5$), where services of the existing requests are resumed. Due to the memoryless property of the exponential distribution, the remaining call holding time is again exponentially distributed with mean $1\/\\mu_k$. Thus, transitions due to call terminations are modeled with rates $\\mu_k$.\n\\vspace{-3mm}\n\\begin{figure}[ht!]\n \\centering\n\\includegraphics[width=1\\textwidth, height =3.8in]{formulae.pdf}\n\\label{table:formulae}%\n\\vspace{-7mm}\n\\end{figure}\nThe global balance equations (GBE) for the RF policy are presented in Table 1. Eq.(1a) gives the GBE for normal occupancy states $S_i, i=1,...,N_{SA}$, whereas (1b) gives the GBE for DaaS states $Sd_\\nu, \\nu=1,...,N_D$. The stationary state probabilities  $\\pi = [ \\pi(S_1),...,\\pi(S_{N_{SA}}), \\pi(Sd_1),...,\\pi(Sd_{N_D})]$ can be calculate by solving $\\pi$ $Q=0$ subject to $\\sum_i \\pi(S_i)+\\sum_\\nu \\pi(Sd_\\nu)=1 $, where $Q$ is the transition rate matrix. For example, the GBE for state $S_1$ is given by $(\\lambda_1 + \\lambda_2 + \\mu_1)\\pi(S_1) = \\lambda_1\/5 \\pi(S_3)+ \\mu_1(\\pi(S_2)+\\pi(S_6)) +\\mu_2\\pi(S_4)$. The left (right) side characterizes the probability flow out of (into) $S_1$. The input rate $\\lambda_1\/a(S_3,1)=\\lambda_1\/5$ takes into account that 6 free slots of $S_3$ results into 5 different spectrum pattern, one of which is $S_1$. For the output flow we have $n_1(S_1)=1$, since there is only one existing class-1 call. The DaaS-state $Sd_1$ is reached from $S_i \\in FB(1)=\\{S_2,S_6,S_7\\}$, which have different spectrum pattern, but results in same pattern after shifting, i.e. $\\overleftarrow{S_i}\\equiv  Sd_1, i = 2,6,7$. Thus, the function $\\delta(S_i,Sd_1)=1, i=2,6,7$ assigns to the right DaaS-state, and $\\delta(S_5,Sd_1)=1$ links to the final de-fragmented state. Using (1b), the GBE of state $Sd_1$ is given by $\\mu_d \\pi(Sd_1)=\\lambda_1 (\\pi(S_2)+\\pi(S_6)+\\pi(S_7))$. The FF policy is also modeled by Eq.(1), taking into account that some transitions are not allowed due to the assignment strategy. Particularly, we have $a(S_i,k) = 1$. A more detailed presentation of the model is omitted for sake of brevity. The blocking probability of a class-k request is given in (2a), and the overall blocking in (2b). In our model, blocking occurs due to the fragmentation ($1^{st}$ term in (2a)), resource unavailability ($2^{nd}$ term), and also in DaaS states (last term).\n \n\\vspace{-2mm}\n\n\\section{Numerical Results}\n\\begin{figure}[t]\n \\centering\n\\includegraphics[width=1\\textwidth, height=2.4in]{BlockingandGain.pdf}\n\\vspace{-8mm}\n  \\caption{Effect of DaaS when applied to RF and FF SA policies.}\n\\label{fig:defrag}\n\\vspace{-2mm}\n\\end{figure}\nWe show the analytical and verifying Monte-Carlo simulation results for an EOL having 20 FS, and three classes of requests with demand $d_1=4, d_2=6$ and $d_3=8$ slots. Arrival rates are uniformly distributed and mean holding time is one unit for all classes of requests. Load on the link is calculated as $\\sum_{k =1 }^ K\\frac{\\lambda_k d_k}{\\mu_k}$. DaaS rates of $\\mu_d=1$ and $\\mu_d=10$ are used. A regular system without DaaS is shown by $\\mu_d=0$. In Fig. \\ref{fig:defrag}(left), we compare the blocking parts under RF policy (Eq.(2a)) due to the effects of lack of resources, fragmentation and defragmentation. In contrast to other work, our analytical model allows us to show accurate results for the important practical range of blocking probabilities below $10\\%$. As expected, without DaaS and for a suitable traffic load, the blocking is dominated by fragmentation and not by resource unavailability. For $\\mu_d=10$, the DaaS  reduces the fragmentation blocking at cost of the resource blocking, which becomes dominant for higher load ($>6$ Erl.). The additional blocking due to the DaaS is insignificantly lower (about two magnitudes lower). As expected (in Fig. \\ref{fig:defrag}(center)), when average defragmentation service time ($t_{DF}=1\/\\mu_d$) is same as the average service time of the class-k connection (i.e., $t_{Sk}=1\/\\mu_k$), then blocking probabilities are higher as compared to the system without DaaS in both FF and RF methods. To get the advantage of DaaS, $t_{DF}$ must be lower than the average service time $t_{Sk}$. As shown, all parts of the blocking probability reduces significantly for RF and marginally for FF methods when we increase the DaaS rate from 1 to 10, as corresponding service time decreases proportionally. We show the change in blocking probabilities in Fig. \\ref{fig:defrag}(right) as the difference in blocking probabilities of with and without DaaS. First we notice, that the FF policy has overall no significant advantage of defragmentation, also for small $t_{DF}$. For RF, the achieved gain (change in BP) is positive for a smaller $t_{DF}$ (= 1\/10) for the load range shown here, however, the gain decreases at higher loads due to the negative effect of DaaS. The reason is that when arrival rates are higher, DaaS states would be called very frequently, which increases blocking. \n\\vspace{-2mm}\n\n\n\\section{Conclusion}\nWe model for the first time the defragmentation-as-a-service and showed the correlation among DF service rate, load and the available resources. The results showed that over exploitation of DF can hurt  the system performance. \n\\vspace{-2mm}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:int}\n\nMagnetic fields are thought to play a key role in the formation of stars and their planets \\citep[e.g.,][]{Andre09, Baruteau14}, \nand for their subsequent evolution into maturity.  For instance, large-scale fields of low-mass pre-main-sequence (PMS) stars, the \nso-called T~Tauri stars (TTSs), are known to control and even trigger physical processes such as accretion, outflows and angular \nmomentum transport, through which they mostly dictate the rotational evolution of TTSs \\citep[e.g.,][]{Bouvier07, Frank14}.  \nLarge-scale fields of TTSs may also help newborn close-in giant planets to avoid falling into their host stars and survive the \nfast migration that accretion discs efficiently trigger, thanks to the magnetospheric gaps that they carve at the disc centre  \n\\citep[e.g.,][]{Lin96, Romanova06}.  The recent discoveries (or candidate detections) of newborn close-in giant planets around T~Tauri stars \n\\citep{vanEyken12, Mann16, JohnsKrull16, Donati16, David16} render the study of the latter topic particularly attractive and timely.  \n\nAlthough first detected long ago \\citep[e.g.,][]{Johns99b, Johns07}, magnetic fields of TTSs are not yet fully characterized, \nneither for those still surrounded by their accretion discs (the classical T~Tauri stars \/ cTTSs) nor for those whose discs \nhave dissipated already (the weak-line T~Tauri stars \/ wTTSs).  Only recently were the field topologies of a dozen cTTSs unveiled \n\\citep[e.g.,][]{Donati07, Hussain09, Donati10b, Donati13} thanks to the MaPP (Magnetic Protostars and Planets) Large Observing \nProgramme on the 3.6~m Canada-France-Hawaii Telescope (CFHT) with the ESPaDOnS high-resolution spectropolarimeter (550~hr of clear \ntime over semester 2008b to 2012b).  This first exploration revealed for instance that large-scale fields of cTTSs can be either \nrelatively simple or quite complex depending on whether the host star is largely convective or mostly radiative \\citep{Gregory12, Donati13}; \nit also showed that these fields vary with time \\citep[e.g.,][]{Donati11, Donati12, Donati13} and mimic those of mature stars with similar \ninternal structures \\citep{Morin08b}, suggesting a dynamo origin.  \n\nThe ongoing MaTYSSE (Magnetic Topologies of Young Stars and the Survival of close-in giant Exoplanets) Large Programme, allocated at CFHT \nover semesters 2013a-2016b (510~hr) with complementary observations with the Narval spectropolarimeter on the 2-m T\\'elescope Bernard Lyot (TBL) \nat Pic du Midi in France (450~hr, allocated) and with the HARPS spectropolarimeter at the 3.6-m ESO Telescope at La Silla in Chile (135~hr, \nallocated), is carrying out the same kind of magnetic exploration on a few tens of wTTSs \\citep[][hereafter D14, D15]{Donati14, Donati15}.  MaTYSSE  \nalso aims at probing the potential presence of newborn close-in giant exoplanets (hot Jupiters \/ hJs) at an early stage of star \/ planet formation;  \nit recently succeeded at detecting the youngest such body orbiting only 0.057~au (or 6.1 stellar radii) away from the 2~Myr wTTS V830~Tau \n\\citep[][hereafter D16]{Donati16}, strongly suggesting that disc migration is a viable and likely efficient mechanism for generating hJs.  \n\nIn this new paper, we revisit the latest MaTYSSE data set collected on V830~Tau, including extended observations from early 2016 that follow \nthe late 2015 ones from which V830~Tau~b was detected, as well as contemporaneous photometry secured at the Crimean Astrophysical Observatory (CrAO).  \nAfter briefly documenting these additional data (Sec.~\\ref{sec:obs}), we apply Zeeman-Doppler Imaging (ZDI) to both subsets to accurately model \nthe surface features and large-scale magnetic fields generating the observed activity (Sec.~\\ref{sec:mod}).  This modelling is then used to predict \nthe activity jitter\\footnote{Throughout the paper, we call ``activity jitter'' or ``jitter'' the RV signal that activity generates, and not \nan ``independent, identically distributed Gaussian noise'' as in, e.g., \\citet{Aigrain12}.} \nand retrieve the planet signature using two complementary methods, yielding results in agreement with a third completely \nindependent technique based on Gaussian-process regression \\citep[e.g.,][]{Haywood14, Rajpaul15} and with those of D16 (Sec.~\\ref{sec:fil}).  We finally \nsummarize our results and stress how MaTYSSE-like explorations can unlock current limitations in our understanding of how giant planets and \nplanetary systems form (Sec.~\\ref{sec:dis}).  \n\n\n\\section{Spectropolarimetric and photometric observations of V830~Tau}\n\\label{sec:obs}\n\nFollowing our intensive campaign in late 2015 (D16), V830~Tau was re-observed from 2016 Jan~14 to Feb~10, using again ESPaDOnS at the CFHT, \nits clone Narval at the TBL, and ESPaDOnS coupled to Gemini-North \nthrough the GRACES fiber link \\citep{Chene14}.  ESPaDOnS and Narval collect spectra covering 370 to 1,000~nm at a resolving power of \n65,000 \\citep{Donati03}.  A total of 15, 6 and 6 spectra were respectively collected with ESPaDOnS, Narval and ESPaDOnS\/GRACES, at a \ndaily rate from Jan~14 to 30 and more sparsely afterwards.  ESPaDOnS and NARVAL were used in spectropolarimetric modes, with all collected \nspectra consisting of a sequence of 4 individual subexposures (of duration 690 and 1200~s each for ESPaDOnS and Narval respectively) \nrecorded in different polarimeter configurations to allow the removal of all spurious polarisation signatures at first order.  \nESPaDOnS\/GRACES spectra were collected in spectroscopic ``star only'' mode, with a resolution similar to that of all other spectra, \nand consist of single 300~s observations.  All raw frames are processed with the reference pipeline {\\sc Libre ESpRIT} implementing optimal \nextraction and radial velocity (RV) correction from telluric lines, yielding a typical rms RV precision of 20--30~\\hbox{m\\,s$^{-1}$}\\ \\citep{Moutou07, Donati08b}.  \nLeast-Squares Deconvolution \\citep[LSD,][]{Donati97b} was applied to all spectra, using the same line list as in our previous studies (D15, D16).  \nThe full journal of observations is presented in Table~\\ref{tab:log}.\n\nRotational and orbital cycles of V830~Tau (denoted $r$ and $o$ in the following equations) are computed from Barycentric Julian Dates (BJDs)\naccording to the ephemerides:\n\\begin{eqnarray}\n\\mbox{BJD} \\hbox{\\rm ~(d)} & = & 2,457,011.80 + 2.741 r \\\\\n\\mbox{BJD} \\hbox{\\rm ~(d)} & = & 2,457,360.52 + 4.93  o\n\\label{eq:eph}\n\\end{eqnarray}\nin which the photometrically-determined rotation periods \\hbox{$P_{\\rm rot}$}\\ and the orbital period \\hbox{$P_{\\rm orb}$}\\ of the hJ are set to \n2.741~d and 4.93~d respectively \\citep[][D16]{Grankin13}.  Whereas the initial Julian date of the first ephemeris \nis chosen arbitrarily, that of the second one coincides with the inferior conjunction (with the hJ in front).  \n\nAs in our late-2015 data (D16), a few spectra (8 altogether, corresponding to cycles 1.347, 2.090, 2.692, 2.820, 3.068, 3.185, \n3.914 and 4.135) were weakly affected by moonlight in the far blue wing of the spectral lines, due to the proximity of the moon (passing \nwithin Taurus in Dec and Jan) and \/ or to non-photometric conditions.  To filter this contamination from our Stokes $I$ LSD profiles, we \napplied the dual-step method described in D16, specifically designed for this purpose and shown to be quite efficient at restoring the \noriginal RVs down to noise level (50~\\hbox{m\\,s$^{-1}$}\\ rms in our case, see Table~\\ref{tab:log}).  \n\n\\begin{table*}\n\\caption[]{Journal of ESPaDOnS observations of V830~Tau collected in from 2016 Jan~14 to Feb~10.  \nESPaDOnS and Narval spectropolarimetric observations consist of sequences of 4 subexposures (each \nlasting 690~s and 1200~s respectively) whereas ESPaDOnS\/GRACES exposures correspond to single (unpolarized) \nobservations lasting 300~s each.  Columns $1-5$ respectively list (i)~the UT date of the observation, (ii)~the \ninstrument used, (iii)~the corresponding UT time (at mid-exposure), (iv)~the Barycentric Julian Date (BJD), \nand (v)~the peak signal to noise ratio \\hbox{S\/N}\\ (per 2.6~\\hbox{km\\,s$^{-1}$}\\ velocity bin) of each observation.  \nColumns~6 and 7 respectively list the \\hbox{S\/N}\\ in Stokes $I$ LSD profiles (per 1.8~\\hbox{km\\,s$^{-1}$}\\ velocity bin), and the rms \nnoise level (relative to the unpolarized continuum level $I_{\\rm c}$) in Stokes $V$ LSD profiles (whenever relevant).  \nColumns~8 and 9 indicate the rotational $r$ and orbital $o$ cycles associated with each exposure (using the \nephemerides given by Eq.~\\ref{eq:eph}).  Columns 10--12 respectively give the raw and ZDI-filtered RVs \\hbox{$v_{\\rm raw}$}\\ and \n\\hbox{$v_{\\rm fil}$}, as well as the corresponding 1$\\sigma$ error bars \\hbox{$\\sigma_{\\rm RV}$}.  No \\hbox{$v_{\\rm fil}$}\\ estimates are available for Jan~30 \nand Feb~10 spectra, affected by strong flares.  The observation log of our late 2015 data can be found in D16 \n(Extended Data Table 1).  }\n\\begin{tabular}{cccccccccccc}\n\\hline\nDate   & Instrument & UT      & BJD      & \\hbox{S\/N}\\ & $\\hbox{S\/N}_{\\rm LSD}$ & $\\sigma_{\\rm LSD}$ & $r$ & $o$ & \\hbox{$v_{\\rm raw}$}\\ & \\hbox{$v_{\\rm fil}$}\\ & \\hbox{$\\sigma_{\\rm RV}$}\\ \\\\\n(2016) &            & (hh:mm:ss) & (2,457,400+) &      &     &   (0.01\\%)  & (142+)      & (8+)        & (km\/s) & (km\/s)    & (km\/s)  \\\\\n\\hline\nJan 14 & ESPaDOnS  & 08:19:57 &  1.85135 & 150 & 1460 & 3.3 &  0.303 & 0.384 &   0.254  & $-0.017$ & 0.049 \\\\\nJan 15 & ESPaDOnS  & 08:16:30 &  2.84889 & 150 & 1400 & 3.3 &  0.667 & 0.586 &   0.789  &   0.020  & 0.051 \\\\\nJan 16 & ESPaDOnS  & 08:34:49 &  3.86153 & 160 & 1480 & 2.9 &  1.036 & 0.791 & $-0.287$ &   0.005  & 0.048 \\\\\nJan 17 & ESPaDOnS  & 05:02:34 &  4.71408 & 170 & 1470 & 2.9 &  1.347 & 0.964 & $-0.008$ &   0.000  & 0.049 \\\\\nJan 18 & ESPaDOnS  & 07:32:40 &  5.81823 & 160 & 1420 & 3.1 &  1.750 & 1.188 & $-0.380$ & $-0.016$ & 0.050 \\\\\nJan 19 & ESPaDOnS  & 05:55:30 &  6.75069 & 170 & 1470 & 2.9 &  2.090 & 1.377 & $-0.123$ & $-0.092$ & 0.049 \\\\\nJan 20 & Narval    & 21:30:38 &  8.39998 &  90 & 1130 & 5.1 &  2.692 & 1.712 &   0.546  &   0.089  & 0.063 \\\\\nJan 21 & ESPaDOnS  & 05:55:40 &  8.75065 & 150 & 1440 & 3.3 &  2.820 & 1.783 & $-1.013$ &   0.034  & 0.050 \\\\\nJan 21 & Narval    & 22:16:00 &  9.43141 & 100 & 1240 & 4.6 &  3.068 & 1.921 & $-0.099$ &   0.035  & 0.058 \\\\\nJan 22 & ESPaDOnS  & 05:56:39 &  9.75126 & 140 & 1450 & 3.5 &  3.185 & 1.986 &   0.386  & $-0.006$ & 0.049 \\\\\nJan 23 & ESPaDOnS  & 07:00:55 & 10.79581 & 160 & 1450 & 3.0 &  3.566 & 2.198 &   1.170  &   0.015  & 0.050 \\\\\nJan 24 & ESPaDOnS  & 05:57:09 & 11.75144 & 170 & 1450 & 3.0 &  3.914 & 2.392 & $-1.258$ & $-0.069$ & 0.049 \\\\\nJan 24 & Narval    & 20:25:57 & 12.35475 &  70 &  970 & 6.5 &  4.135 & 2.514 &   0.180  &   0.025  & 0.074 \\\\\nJan 25 & ESPaDOnS  & 07:23:59 & 12.81166 & 150 & 1470 & 3.4 &  4.301 & 2.607 &   0.284  & $-0.011$ & 0.049 \\\\\nJan 26 & ESPaDOnS  & 06:59:05 & 13.79429 & 150 & 1420 & 3.5 &  4.660 & 2.806 &   0.840  & $-0.011$ & 0.051 \\\\\nJan 26 & Narval    & 19:34:05 & 14.31857 &  90 & 1130 & 5.4 &  4.851 & 2.912 & $-1.160$ &   0.039  & 0.064 \\\\\nJan 27 & ESPaDOnS  & 06:05:23 & 14.75691 & 170 & 1470 & 2.9 &  5.011 & 3.001 & $-0.482$ &   0.009  & 0.049 \\\\\nJan 28 & ESPaDOnS  & 06:05:42 & 15.75705 & 160 & 1440 & 3.1 &  5.376 & 3.204 & $-0.176$ & $-0.031$ & 0.050 \\\\\nJan 29 & ESPaDOnS  & 06:58:43 & 16.79378 & 150 & 1410 & 3.3 &  5.754 & 3.415 & $-0.444$ & $-0.036$ & 0.051 \\\\\nJan 29 & Narval    & 20:01:53 & 17.33762 &  80 & 1210 & 5.6 &  5.952 & 3.525 & $-0.956$ &   0.011  & 0.059 \\\\\nJan 30 & Narval    & 20:15:57 & 18.34730 &  80 & 1190 & 6.1 &  6.321 & 3.730 &   0.000  &          & 0.060 \\\\\nFeb 04 & GRACES    & 07:12:12 & 22.80262 & 140 & 1560 &     &  7.946 & 4.633 & $-1.017$ & $-0.005$ & 0.046 \\\\\nFeb 04 & GRACES    & 07:18:12 & 22.80678 & 150 & 1560 &     &  7.948 & 4.634 & $-0.978$ &   0.023  & 0.046 \\\\\nFeb 09 & GRACES    & 07:16:45 & 27.80532 & 150 & 1470 &     &  9.771 & 5.648 & $-0.621$ & $-0.019$ & 0.049 \\\\\nFeb 09 & GRACES    & 07:22:39 & 27.80941 & 150 & 1470 &     &  9.773 & 5.649 & $-0.628$ & $-0.010$ & 0.049 \\\\\nFeb 10 & GRACES    & 05:21:11 & 28.72497 & 160 & 1600 &     & 10.107 & 5.835 & $-0.653$ &          & 0.045 \\\\\nFeb 10 & GRACES    & 05:27:05 & 28.72907 & 160 & 1590 &     & 10.108 & 5.836 & $-0.647$ &          & 0.045 \\\\\n\\hline\n\\end{tabular}\n\\label{tab:log}\n\\end{table*}\n\nContemporaneous BVR$_{\\rm J}$I$_{\\rm J}$ photometric observations were also collected from the CrAO 1.25~m telescope \n(see Table~\\ref{tab:pho}), showing that V830~Tau exhibited significantly larger brightness fluctuations than a year \nbefore (D15), with a full amplitude of 0.28~mag and a period of $2.7424\\pm0.0014$~d \\citep[compatible within \nerror bars with the average periods of][used to phase our spectroscopic data, see Eq.~\\ref{eq:eph}]{Grankin13}.\n\n\\begin{table}\n\\caption[]{Journal of contemporaneous CrAO multicolour photometric observations of V830~Tau collected from 2015 Oct~30 to \n2016 Mar~15, respectively listing the Heliocentric Julian Date (HJD) of the observation, the measured \\hbox{${\\rm V}$}\\ magnitude,\n\\hbox{${\\rm B-V}$}, \\hbox{${\\rm V-R_{\\rm J}}$}\\ and \\hbox{${\\rm V-I_{\\rm J}}$}\\ Johnson photometric colours, and the corresponding rotational cycle (using again the ephemerides of \nEq.~\\ref{eq:eph}).  The middle line separates observations collected in 2015 and 2016.  The typical 1$\\sigma$ error bar \non \\hbox{${\\rm V}$}\\ is 20~mmag.  }\n\\begin{tabular}{cccccc}\n\\hline\nHJD          & \\hbox{${\\rm V}$}    & \\hbox{${\\rm B-V}$}  & \\hbox{${\\rm V-R_{\\rm J}}$} & \\hbox{${\\rm V-I_{\\rm J}}$} & $r$ \\\\\n(2,457,300+) & (mag) & (mag) & (mag) & (mag) & (114+) \\\\\n\\hline\n 26.4574 & 12.410 &       & 1.339 & 2.182 &  0.797 \\\\\n 28.5085 & 12.378 &       &       & 2.242 &  1.545 \\\\\n 30.6098 & 12.322 & 1.375 & 1.325 & 2.151 &  2.311 \\\\\n 31.5960 & 12.474 & 1.413 & 1.341 & 2.231 &  2.671 \\\\\n 32.5964 & 12.267 & 1.349 &       & 2.143 &  3.036 \\\\\n 40.5358 & 12.307 &       & 1.340 & 2.165 &  5.933 \\\\\n 44.4530 & 12.321 &       & 1.321 & 2.146 &  7.362 \\\\\n 47.5027 & 12.342 &       & 1.307 & 2.162 &  8.475 \\\\\n 47.5524 & 12.367 &       & 1.335 & 2.180 &  8.493 \\\\\n 73.3208 & 12.317 &       & 1.319 & 2.161 & 17.894 \\\\\n 73.5082 & 12.261 &       & 1.305 & 2.130 & 17.962 \\\\\n 74.2657 & 12.268 &       & 1.310 & 2.128 & 18.238 \\\\\n\\hline\n 91.3173 & 12.281 &       & 1.307 & 2.122 & 24.459 \\\\\n101.2599 & 12.199 &       & 1.282 & 2.094 & 28.087 \\\\\n105.2842 & 12.356 &       & 1.319 & 2.155 & 29.555 \\\\\n112.2820 & 12.244 &       & 1.284 & 2.113 & 32.108 \\\\\n118.2602 & 12.306 &       & 1.339 & 2.160 & 34.289 \\\\\n127.2569 & 12.362 &       & 1.325 & 2.165 & 37.571 \\\\\n129.2079 & 12.252 &       & 1.296 & 2.105 & 38.283 \\\\\n141.2201 & 12.413 &       & 1.343 & 2.204 & 42.665 \\\\\n142.2211 & 12.210 &       & 1.301 & 2.106 & 43.031 \\\\\n153.2096 & 12.219 &       & 1.297 & 2.106 & 47.040 \\\\\n156.2255 & 12.196 &       & 1.288 & 2.094 & 48.140 \\\\\n158.2696 & 12.329 &       & 1.332 & 2.148 & 48.886 \\\\\n163.2378 & 12.425 &       & 1.322 & 2.217 & 50.698 \\\\\n\\hline\n\\end{tabular}\n\\label{tab:pho}\n\\end{table}\n\nWe note that V830~Tau features emission in various spectral activity proxies, as expected from its youth and fast rotation.  \nMore specifically, Balmer lines and in particular \\hbox{H${\\alpha}$}, are in emission, as well as the central core of the \\hbox{Ca$\\;${\\sc ii}}\\ \ninfrared triplet (IRT) lines, with typical equivalent widths of 85 and 16~\\hbox{km\\,s$^{-1}$}\\ for \\hbox{H${\\alpha}$}\\ and the \\hbox{Ca$\\;${\\sc ii}}\\ IRT emission \ncore respectively.  The \\hbox{He$\\;${\\sc i}}\\ $D_3$ line is most of the time quite shallow, with an average equivalent width of 5~\\hbox{km\\,s$^{-1}$}.  \n\nIn 2016 however, we detected several flares of V830~Tau, showing up as enhanced red-shifted emission in all activity \nproxies including \\hbox{He$\\;${\\sc i}}, a reliable proxy whose high excitation potential makes it possible to separate flares from phases \nof enhanced chromospheric activity \\citep[e.g.,][]{Montes97}.  \nThe most intense flare occurred on Feb~10 during our last pair of observations (cycles 10.107 and 10.108), when \\hbox{H${\\alpha}$}, \n\\hbox{Ca$\\;${\\sc ii}}\\ IRT and \\hbox{He$\\;${\\sc i}}\\ emission reach equivalent widths of 280, 32 and 25~\\hbox{km\\,s$^{-1}$}\\ and feature large red-shifts of 15--35~\\hbox{km\\,s$^{-1}$}\\ \n(with respect to the stellar rest frame, shifted from the Barycentric rest frame by $\\simeq$17~\\hbox{km\\,s$^{-1}$}) and asymmetric profiles \n(with a conspicuous red tail for \\hbox{H${\\alpha}$}, see Fig.~\\ref{fig:hal}, and \\hbox{He$\\;${\\sc i}}).  \nWe note that one of our photometric measurements was secured just after this large flare (at rotation cycle 152.283, or \n10.283 in the reference frame of Table~\\ref{tab:log}).  At this time, the star was observed to be 54~mmag (i.e., 2.7$\\sigma$) \nbrighter than 4 rotation cycles earlier at almost the same phase (cycle 148.289, see Table~\\ref{tab:pho}).  This shows \nthat even the largest flare of our run was barely detectable in the light curve, to the point that it is not even clear \nwhich of the two photometric measurements at this phase deviates most from the bulk of our data points (see Sec.~\\ref{sec:mod}).  \n\nA weaker flare was detected 10.3~d earlier on Jan~30 (cycle 6.321), with activity proxies exhibiting similar albeit less \ndrastic characteristics, e.g., \\hbox{He$\\;${\\sc i}}\\ emission with an equivalent width of 11~\\hbox{km\\,s$^{-1}$}\\ and a redshift of $\\simeq$20~\\hbox{km\\,s$^{-1}$}\\ \n(with respect to the stellar rest frame, or $\\simeq$10 \\hbox{km\\,s$^{-1}$}\\ with respect to the average velocity of the \\hbox{He$\\;${\\sc i}}\\ line).  \nA third flare was recorded on Jan~26 (cycle 4.851), mostly in \\hbox{H${\\alpha}$}\\ (with an equivalent width reaching 122~\\hbox{km\\,s$^{-1}$}), but \nshort enough to be seen only with Narval, but neither a few hours before (cycle 4.660) nor later (cycle 5.011) with ESPaDOnS;  \nthis flare has only mild \\hbox{He$\\;${\\sc i}}\\ characteristics however, with an equivalent width only slightly above average and no \nsignificant redshift (with respect to the average line velocity).  \n\nThe 3 Stokes $I$ spectra corresponding to the 2 first flares turned out to yield discrepant RV estimates (with excess \nblue-shifts of order 0.3~\\hbox{km\\,s$^{-1}$}), most likely as \na result of flaring, and were removed from the subsequent modelling (see Secs.~\\ref{sec:mod} and \\ref{sec:fil}).  \nThe Stokes $V$ spectrum associated with the second flare compares well with those collected at similar phases but previous\ncycles (0.303, 4.301), suggesting that it was largely unaffected by the flare and thus used for magnetic imaging (see Sec.~\\ref{sec:mod}).  \nThe Stokes $I$ (and $V$) spectra corresponding to the third, milder, flare, yielding an RV estimate consistent with those \nfrom the two unperturbed ESPaDOnS spectra bracketing the flare, were also kept in the sample.  \n\n\\begin{figure}\n\\includegraphics[scale=0.35,angle=-90]{fig\/v830_hal.ps} \n\\caption[]{\\hbox{H${\\alpha}$}\\ profiles of V830~Tau on 2016 Jan~17 (cycle 1.347, red line), Feb~10 (cycle 10.107, blue) and Jan.~30 \n(cycle 6.321, green).  A red component \/ tail is clearly present in the latter two profiles (recorded during a flare) \nwhile absent in the first one (more typical of V830~Tau). } \n\\label{fig:hal}\n\\end{figure}\n\n\\begin{figure*}\n\\center{\\hbox{\\hspace{10mm}\n\\includegraphics[scale=0.6,angle=-90]{fig\/v830_fiti16.ps}\\hspace{3mm}\n\\includegraphics[scale=0.6,angle=-90]{fig\/v830_fitv16.ps}}}\n\\caption[]{Maximum-entropy fit (thin red line) to the observed (thick black line) Stokes $I$ (left panel) \nand Stokes $V$ (right panel) LSD photospheric profiles of V830~Tau in early 2016.  (The red and black lines \nalmost perfectly overlap for Stokes $I$ LSD profiles.)  Stokes $I$ LSD profiles \nprior to their filtering from lunar contamination (in the far blue wing) are also shown (cyan line).  \nRotational cycles and 3$\\sigma$ error bars (for Stokes $V$ profiles) are also shown next to each profile. }\n\\label{fig:fit16}\n\\end{figure*}\n\n\n\\section{Tomographic modelling of surface features, magnetic fields and activity}\n\\label{sec:mod}\n\nWe applied ZDI to both our late-2015 and early-2016 sets of phase-resolved Stokes $I$ and $V$ LSD profiles, keeping \nthem separate from each other in a first step.  ZDI is a tomographic technique inspired from medical imaging, with \nwhich distributions of brightness features and magnetic fields at the surfaces of rotating stars can be reconstructed from \ntime-series of high-resolution spectropolarimetric observations \\citep{Brown91, Donati97c, Donati01, Donati06b}.  Technically \nspeaking, ZDI follows the principles of maximum-entropy image reconstruction, and iteratively looks for the image with lowest \ninformation content that fits the data at a given \\hbox{$\\chi^2$}\\ level.  By working out the amount of latitudinal shearing that surface \nmaps are subject to as a function of time, ZDI can also infer an estimate of differential rotation at photospheric level \n\\citep{Donati97a, Donati03b}.  \n\nFor this study, we used the latest implementation of ZDI, where the large-scale field is decomposed into its poloidal and \ntoroidal components, both expressed as spherical harmonics expansions \\citep{Donati06b}, and where the brightness distribution \nincorporates both cool spots and warm plages\\footnote{In this paper, the term ``plage'' refers to a photospheric region \nbrighter than the quiet photosphere, and not to a bright region at chromospheric level (as in solar physics). } \n(D14, D15, D16).  The local Stokes $I$ and $V$ profiles are computed \nusing Unno-Rachkovsky's analytical solution to the polarized radiative transfer equations in a Milne-Eddington model atmosphere, \ntaking into account the local brightness and magnetic field;  these local profiles are then integrated over the visible hemisphere \nto derive the synthetic profiles of the rotating star, to be compared with our observations.  This computation scheme provides \na reliable description of how line profiles are distorted in the presence of magnetic fields \\citep[including magneto-optical\neffects, e.g.,][]{Landi04}.\n\nIn this new paper, we assume for V830~Tau the same parameters as in our previous studies in particular an inclination of the rotation \naxis to the line of sight $i$ equal to $55\\pm10$\\hbox{$^\\circ$}\\ and a line-of-sight-projected equatorial rotation velocity \\hbox{$v \\sin i$}\\ equal to \n$30.5\\pm0.5$~\\hbox{km\\,s$^{-1}$}\\ (D15, D16)\\footnote{The distance assumed for V830~Tau in D15 and D16, i.e., $131\\pm3$~pc, is likely underestimated, \nsince V830~Tau is located in L1529 rather than L1495, and thus close to DG~Tau for which the adopted distance is $150\\pm5$~pc \\citep{Rodriguez12}.  \nGiven that this difference in distance is comparable flux-wise to the uncertainty on the unspotted magnitude of V830~Tau, we still assume \nfor V830~Tau the same stellar parameters as in D15 and D16 (see Table~\\ref{tab:spar}).  Assuming instead that V830~Tau is 30\\% brighter \nwould mostly imply that it is younger, with an age of $\\simeq$1.5~Myr (using the evolutionary models of \\citealt{Siess00} as in D15).}.  \nWe recall that the inclination angle $i$ is derived both from the measured stellar parameters (see Sec.~3 of D15) and by minimizing \nthe information content of reconstructed images, with a typical error bar of order 10\\hbox{$^\\circ$}.  \nWe further assume that the (weak) surface differential rotation of V830~Tau is as derived by D16 from our late 2015 data alone, before \nrevisiting the subject using the whole data set in Sec.~\\ref{sec:dr}.  The parameters of V830~Tau used in our study are summarized \nin Table~\\ref{tab:spar}.  \n\n\\begin{table}\n\\caption[]{Summary of the main parameters of V830~Tau, with references as mentioned whenever appropriate (G13 and R12 stand for \n\\citealt{Grankin13} and \\citealt{Rodriguez12}).}\n\\center{\n\\begin{tabular}{ccc}\n\\hline\nParameter            & Value            & Reference \\\\ \n\\hline\n\\hbox{$M_{\\star}$}\\ (\\hbox{${\\rm M}_{\\odot}$})      & $1.00\\pm0.05$       & D15 \\\\ \n\\hbox{$R_{\\star}$}\\ (\\hbox{${\\rm R}_{\\odot}$})      & $2.0\\pm0.2$         & D15 \\\\ \nage (Myr)            & $\\simeq$2.2         & D15 \\\\ \n\\hbox{$P_{\\rm rot}$}\\ (d)           & 2.741               & G13 \\\\ \nBJD$_0$              & 2,457,011.80        & D15 \\\\ \n\\hbox{$\\Omega_{\\rm eq}$}\\ (\\hbox{rad\\,d$^{-1}$})        & $2.29525\\pm0.00020$ & D16 \\\\\n\\hbox{$d\\Omega$}\\ (\\hbox{rad\\,d$^{-1}$})         & $0.0172\\pm0.0014$   & D16 \\\\\n$i$ (\\hbox{$^\\circ$})          & $55\\pm10$           & D15 \\\\ \n\\hbox{$v \\sin i$}\\ (\\hbox{km\\,s$^{-1}$})       & $30.5\\pm0.5$        & D15 \\\\ \ndistance (pc)        & $150\\pm5$           & R12 \\\\ \n\\hbox{$T_{\\rm eff}$} (K)            & $4250\\pm50$         & D15 \\\\ \n\\hline\n\\end{tabular}} \n\\label{tab:spar}\n\\end{table}\n\n\\subsection{Brightness and magnetic imaging}\n\n\\begin{figure*}\n\\hbox{\\hspace{15mm}\n\\includegraphics[scale=0.45,angle=-90]{fig\/v830_mapi16_vsmall.ps}\\hspace{20mm}\n\\includegraphics[scale=0.45,angle=-90]{fig\/v830_mapi15_vsmall.ps}} \n\\caption[]{Maps of the logarithmic brightness (relative to the quiet photosphere), at the surface of V830~Tau in \nearly 2016 (left) and late 2015 (right).  Cool spots \/ bright plages show up as brown \/ blue features. \nThe star is shown in flattened polar projection down to latitudes of $-30\\hbox{$^\\circ$}$,\nwith the equator depicted as a bold circle and parallels as dashed circles.  Radial ticks around\neach plot indicate phases of observations. }\n\\label{fig:mapi}\n\\end{figure*}\n\n\\begin{figure*}\n\\hbox{\\hspace{4mm}\\includegraphics[scale=0.35,angle=-90]{fig\/v830_pho16_2.ps}\\hspace{6mm}\\includegraphics[scale=0.35,angle=-90]{fig\/v830_pho15.ps}} \n\\caption[]{Brightness variations of V830~Tau in early 2016 (left) and late 2015 (right) as predicted from the tomographic modelling \nof our spectropolarimetric data (see Fig.~\\ref{fig:mapi}, green line), compared with contemporaneous photometric observations \nin the \\hbox{${\\rm V}$}\\ band (open symbols and 1$\\sigma$ error bars of 20~mmag) at the 1.25-m CrAO telescope (see Table~\\ref{tab:pho}).  \nThe photometric measurement collected immediately after the large flare detected in our early 2016 spectroscopy data (see Sec.~\\ref{sec:obs}) \nis shown as an open blue square in the left panel.   }\n\\label{fig:pho}\n\\end{figure*}\n\nIn Fig.~\\ref{fig:fit16}, we show our sets of Stokes $I$ and $V$ LSD profiles of V830~Tau from early 2016, along with the fit to the \ndata.  A similar plot is provided in Appendix~\\ref{sec:fig} for our late-2015 data set (see Fig.~\\ref{fig:fit15}, repeating Fig.~1 \nof D16 for Stokes $I$ profiles, and including Stokes $V$ profiles not previously shown in D16).  \nThe fit we obtain in both cases corresponds to a \\hbox{$\\chi^2$}\\ equal to the number of data points, i.e., to a unit \\hbox{$\\chi^2_{\\rm r}$}\\ \nlevel (where \\hbox{$\\chi^2_{\\rm r}$}\\ is simply taken here as \\hbox{$\\chi^2$}\\ divided by the number of data points\\footnote{This is the usual convention in \nregularized tomographic imaging techniques where the number of model parameters, reflecting the (ill-defined) number of resolution \nelements in the reconstructed image, is much smaller than the number of fitted data points and not taken into account in the expression \nof \\hbox{$\\chi^2_{\\rm r}$}. }, respectively equal to 1104 and 2208 for \nthe early-2016 and late-2015 Stokes $I$ data sets, and to 966 and 1472 for the corresponding Stokes $V$ data sets).  \nThe initial \\hbox{$\\chi^2_{\\rm r}$}\\ values, corresponding to input maps with null fields and no brightness features, are equal to 27 and 19 for the \nearly-2016 and late-2015 data sets respectively, clearly demonstrating the overall success of ZDI at modelling the observed modulation of \nboth Stokes $I$ and $V$ LSD profiles. \n\nThe reconstructed brightness maps of V830~Tau at both epochs are shown in Fig.~\\ref{fig:mapi}.  The two maps share obvious similarities \nand exhibit similar spottedness levels, i.e., $\\simeq$13\\%\\ of the stellar \nsurface\\footnote{We stress that ZDI is only sensitive to large brightness features, and not to small ones evenly distributed at \nthe surface of the star;  for this reason, the value we quote here for the spot coverage of V830~Tau is likely to be a lower limit, \nin agreement with photometric monitoring suggesting a typical spot coverage in the range 30--50\\% for V830~Tau \\citep{Grankin08}.}  \n(7\\%\\ and 6\\%\\ for cool and warm features respectively). \nIn particular, most cool spots and warm plages present in either maps are recovered at both epochs.  \nOne can also notice differential rotation shearing the brightness distribution between late 2015 and early 2016 (a time gap corresponding to \n49~d or 18 rotation cycles), with equatorial and polar features being both shifted by a few \\% of a rotation cycle to smaller and larger phases \nrespectively\\footnote{For instance, the equatorial plage at phase 0.46 in early 2016 is found at phase 0.48 in late 2015, while the cool polar \ncap rotated by $\\simeq$0.1~cycle in the other direction at the same time.  Note that the latest map is shown first in Fig.~\\ref{fig:mapi} and \nfollowing plots.} (implying a fast equator and a slow pole, in good quantitative agreement \nwith D16).  Some intrinsic temporal evolution beyond differential rotation may be visible in our images as well, with, e.g., the appearance of a \nwarm equatorial plage at phase 0.08 in early 2016 that was not visible (or not as strong) in late 2015;  however, even though phase \ncoverage is fairly good in our case at both epochs, quantifying spot evolution by visually comparing images derived from differently \nsampled data sets is notoriously ambiguous and misleading.  We come back on this point in Sec.~\\ref{sec:dr}. \n\nWe stress that the derived brightness images predict light curves that are in good agreement with our observations (see \nFig.~\\ref{fig:pho}), even though these images were produced from our sets of LSD profiles only.  Note the small temporal evolution \nin the predicted light curves between both epochs, that our photometric observations cannot confirm due to their limited sampling \nand precision.  This further demonstrates that LSD profiles contain enough information to accurately predict the surface distribution \nof brightness features, and in particular those responsible of the RV activity jitter (see Sec.~\\ref{sec:fil});  on the opposite, it \nis quite obvious that photometric information is way too limited (even when better sampled and more precise) to infer complex spot \ndistributions such as those we reconstruct for V830~Tau.  It implies that jitter-filtering techniques based solely on photometry \n\\citep[e.g.,][]{Aigrain12} are likely to yield poorer results, especially for moderate to fast rotators whose optical RV curves are \nmuch more sensitive than photometry to small features in surface brightness distributions.  \n\nThe large-scale magnetic topologies we retrieve for V830~Tau at both epochs (see Fig.~\\ref{fig:mapv}) are again very similar, with \nrms surface magnetic fluxes of 350~G, and resemble that found previously for this star (D15).  As for the brightness maps, the main \nmagnetic regions that we recover are visible at both epochs.  More specifically, the field is found to be 90\\%\\ poloidal, featuring a \n340~G dipole field tilted at $22\\pm5$\\hbox{$^\\circ$}\\ to the rotation axis towards phase $0.79\\pm0.03$ (in late 2015) and $0.88\\pm0.03$ (in early 2016), \nand that gathers 60\\%\\ of the poloidal field energy.  \nWeaker quadrupolar and octupolar components (of strength 100--150~G) and smaller-scale features are also present on V830~Tau, \ngiving the field close to the stellar surface a more complex appearance than that of the dominating dipole.  With a rms flux of $\\simeq$110~G, the toroidal field \nis weak and of rather complex topology.  The extrapolated large-scale magnetic topology (in the assumption of a potential field) is \nshown in Fig.~\\ref{fig:mag} at both epochs.  \n\n\\begin{figure*}\n\\vspace{0mm}\n\\hbox{\\hspace{5mm}\\includegraphics[scale=0.65,angle=-90]{fig\/v830_mapv16_vsmall.ps}\\vspace{-2mm}} \n\\hbox{\\hspace{5mm}\\includegraphics[scale=0.65,angle=-90]{fig\/v830_mapv15_vsmall.ps}\\vspace{-2mm}}\n\\vspace{-2mm}\n\\caption[]{Maps of the radial (left), azimuthal (middle) and meridional (right) components of the \nmagnetic field $\\bf B$ at the surface of V830~Tau in early 2016 (top) and late 2015 (bottom).  \nMagnetic fluxes in the color lookup table are expressed in G.  The star is shown in flattened polar \nprojection as in Fig.~\\ref{fig:mapi}. }\n\\label{fig:mapv}\n\\end{figure*}\n\n\\begin{figure*}\n\\hbox{\\includegraphics[scale=0.55,angle=-90]{fig\/v830_mag16_vsmall.ps}\\hspace{3mm}\n      \\includegraphics[scale=0.55,angle=-90]{fig\/v830_mag15_vsmall.ps}}\n\\caption[]{Potential extrapolations of the magnetic field reconstructed for V830~Tau in early 2016 (left) and late 2015 (right),\nas seen by an Earth-based observer at phase 0.10.  Open and closed field lines are shown in blue and white respectively, whereas colors \nat the stellar surface depict the local values (in G) of the radial field (see left panels of Fig.~\\ref{fig:mapv}).\nThe source surface at which the field becomes radial is set at a distance of 4~\\hbox{$R_{\\star}$}, close to the corotation radius of V830~Tau \n(at which the Keplerian orbital period equals the stellar rotation period and beyond which field lines tend to open under the effect \nof centrifugal forces, \\citealt{Jardine04}) but smaller than the Alfv\\'en radius expected for a T~Tauri star like V830~Tau \n\\citep[$>$6~\\hbox{$R_{\\star}$}, see][]{Vidotto16}.  Note how the high-latitude open-field regions slightly lag behind rotation between both epochs \nas a result of differential rotation.  }\n\\label{fig:mag}\n\\end{figure*}\n\nAs for the brightness maps, the magnetic images show evidence of a global differential rotation shear similar to that reported by D15, \nwith equatorial regions (e.g., the strong negative azimuthal feature at phase 0.17) moving to slightly earlier phases from late 2015 \nto early 2016, and higher latitude regions (e.g., the positive radial field region at phase 0.05 and latitude 60\\hbox{$^\\circ$}) moving to later \nphases at the same time.  The increase in the phase towards which the dipole is tilted (0.79 and 0.88 in late 2015 and early 2016 respectively) \ncomes as additional evidence that high latitudes (at which the dipole poles are anchored) are rotating more slowly than average, by \ntypically 1 part in 200;  this is further confirmed by the fact that the line-of-sight projected (longitudinal) magnetic fields \n\\citep[proportional to the first moment of the Stokes $V$ profiles, e.g.,][and most sensitive to the low-order components of the \nlarge-scale field]{Donati97b} exhibit a recurrence timescale of $1.004\\pm0.003$~\\hbox{$P_{\\rm rot}$}, i.e.\\ slightly longer than \\hbox{$P_{\\rm rot}$}\\ by a \nsimilar amount.  \n\nWe also report that the phase of maximum \\hbox{H${\\alpha}$}\\ emission of V830~Tau coincides, in both late 2015 and early 2016, with that of the \nhigh-latitude regions at which the dipole field is anchored;  this is obvious from the dynamic spectra of the \\hbox{H${\\alpha}$}\\ residuals that \nwe provide as an additional figure in the Appendix (see Fig.~\\ref{fig:haldyn}).  A logical by-product is that \\hbox{H${\\alpha}$}\\ emission of \nV830~Tau, like its longitudinal field, is modulated by a period slightly longer than \\hbox{$P_{\\rm rot}$}, and equal to $1.004\\pm0.002$~\\hbox{$P_{\\rm rot}$}.  \nFrom the solar analogy, one would have expected chromospheric emission to be minimum when open field lines point towards the observer, \ni.e., at phase 0.8--0.9 (see Fig.~\\ref{fig:mag});  this is however not what we observe, suggesting that the \\hbox{H${\\alpha}$}\\ emission we detect \ncomes from regions close to (but not coinciding with) the strongest radial field regions that we reconstruct at high latitudes (see Fig.~\\ref{fig:mapv}). \n\nWe also note apparent temporal evolution of the magnetic topology, with, e.g., the positive radial field \nregion close to the equator at phase 0.33 growing much stronger between late 2015 and early 2016, though we caution again that a \nsimple visual image comparison of individual features can be misleading.  \n\n\\begin{figure*}\n\\center{\\hbox{\\hspace{-2mm}\n\\includegraphics[scale=0.3,angle=-90]{fig\/v830_drv2.ps}\\hspace{2mm}\n\\includegraphics[scale=0.328,angle=-90]{fig\/v830_dr.ps}\\hspace{2mm}\n\\includegraphics[scale=0.3,angle=-90]{fig\/v830_dri2_vsmall.ps}}}\n\\caption[]{Variations of \\hbox{$\\Delta \\chi^2$}\\ as a function of the differential rotation parameters \\hbox{$\\Omega_{\\rm eq}$}\\ and \\hbox{$d\\Omega$}, derived from modelling our \nfull set of Stokes $I$ (right) and $V$ (left) LSD profiles of V830~Tau at constant information content.  \nA well defined paraboloid is observed in both cases, with the outer \ncolor contour tracing the 99.99\\% confidence interval (corresponding to a \\hbox{$\\Delta \\chi^2$}\\  of 18.4 for the 2438 Stokes $V$ and 3312 \nStokes $I$ data points).  In both cases, the minimum \\hbox{$\\chi^2_{\\rm r}$}\\ achieved (equal to 1.62 and 1.18 for Stokes $I$ and Stokes $V$ data \nrespectively) is significantly larger than 1;  this value is used to normalise \\hbox{$\\chi^2$}\\ before computing \\hbox{$\\Delta \\chi^2$}\\ so as to account \nfor intrinsic variability (affecting the brightness distribution and magnetic field of V830~Tau over the course of our 91-d run, see \nSec~\\ref{sec:dr}) when estimating errors bars on differential rotation parameters.  The values we obtain for these parameters are equal to \n$\\hbox{$\\Omega_{\\rm eq}$}=2.29455\\pm0.00014$~\\hbox{rad\\,d$^{-1}$}\\ and $\\hbox{$d\\Omega$}=0.0156\\pm0.0009$~\\hbox{rad\\,d$^{-1}$}\\ for the Stokes $I$ data, and \n$\\hbox{$\\Omega_{\\rm eq}$}=2.29360\\pm0.00025$~\\hbox{rad\\,d$^{-1}$}\\ and $\\hbox{$d\\Omega$}=0.0131\\pm0.0010$~\\hbox{rad\\,d$^{-1}$}\\ for the Stokes $V$ data.  \nThe middle plot emphasizes how the confidence intervals from both measurements compare with each other, \nand with that derived from the late-2015 Stokes $I$ LSD profiles only (D16).  The 68.3\\% and 99.99\\% confidence intervals \n(dashed and full lines) are shown in green, red and blue for the full Stokes $V$, the full Stokes $I$ and the late-2015 Stokes $I$ \ndata sets respectively. }\n\\label{fig:dr}\n\\end{figure*}\n\n\n\\subsection{Intrinsic variability and surface differential rotation}\n\\label{sec:dr}\n\nThe most reliable way to assess whether intrinsic variability occurred at the surface of V830~Tau between late 2015 and early 2016 is to \nattempt modelling both data sets simultaneously with a unique brightness and magnetic topology, and see whether one can fit the full set to the \nsame \\hbox{$\\chi^2_{\\rm r}$}\\ level as that achieved for the individual sets (i.e., 1.0, see Sec.~\\ref{sec:mod}).  We find that this is not possible, \nwith a minimum achievable \\hbox{$\\chi^2_{\\rm r}$}\\ of 1.62 and 1.18 for Stokes $I$ and Stokes $V$ data respectively (starting from initial \\hbox{$\\chi^2_{\\rm r}$}\\ of 35 \nand 5);  this confirms our previous suspicion that intrinsic variability occurred at the surface of V830~Tau throughout the 91~d (33 \nrotation cycles) of our observing campaign, and in particular over the 49~d shift between our two data sets.  \nThe global fit to the full data set we obtain nonetheless captures most of the observed line profile \nfluctuations, indicating that the intrinsic variability at work at the surface of V830~Tau remained moderate and local without altering \nthe brightness and magnetic surface distributions too drastically;  this further confirms our visual impression that images from both \nepochs shared obvious similarities.  \n\nDespite this intrinsic variability, we attempted to estimate differential rotation from our full data set.  As in previous papers, we \nachieve this by assuming that the rotation rate at the surface of V830~Tau $\\Omega(\\theta)$ varies with latitude $\\theta$ as $\\sin^2 \\theta$ \nand depends on 2 main parameters, the rotation rate at the equator \\hbox{$\\Omega_{\\rm eq}$}\\ and the difference in rotation rate \\hbox{$d\\Omega$}\\ between the equator and \nthe pole (so that $\\Omega(\\theta)=\\hbox{$\\Omega_{\\rm eq}$}-\\hbox{$d\\Omega$}\\sin^2\\theta$).  \nBoth parameters are derived by looking for the pair that minimizes the \\hbox{$\\chi^2_{\\rm r}$}\\ of the fit to the data (at constant \ninformation content in the reconstructed image), whereas the corresponding error bars are computed from the curvature of the \\hbox{$\\Delta \\chi^2$}\\ \nparaboloid at its minimum \\citep{Donati03b}.  (\\hbox{$\\Delta \\chi^2$}\\ is defined as the \\hbox{$\\chi^2$}\\ increase with respect to the minimum \\hbox{$\\chi^2$}\\ in the \nmap.)  Results are shown in Fig.~\\ref{fig:dr}.  The differential rotation we derive from our \ncomplete data set is slightly smaller (though still compatible at a $\\simeq$3$\\sigma$ level) than that inferred from the late-2015 \nStokes $I$ LSD profiles only (D16).  Despite the fact that this weakening is observed in both Stokes $I$ and $V$ data, \nwe think that this small change likely results from intrinsic variability at the surface of V830~Tau\\footnote{For this reason, \nthe differential rotation parameters of D16 were used as reference throughout this paper, their impact on most results being however \nquite small given how weakly the photosphere of V830~Tau is sheared.}.  \n\nFurther evidence that high latitudes of V830~Tau are rotating more slowly than average (in agreement with the differential rotation pattern \nwe recover) comes from the drift to later phases of the polar regions at which the large-scale dipole field component is anchored and \nwhere \\hbox{H${\\alpha}$}\\ emission is strongest. \n\n\n\\section{Filtering the activity jitter and modelling the planet signal}\n\\label{sec:fil}\n\nWe describe below the results of 3 independent techniques aimed at characterizing the RV signature of V830~Tau~b from our data.  \nThe first 2 methods are those already outlined in D16 and used to detect V830~Tau~b from the late-2015 data alone, that we now apply \nto both late-2015 and early-2016 data sets, with some modifications to account for the intrinsic variability between the 2 epochs (see Sec.~\\ref{sec:dr}).  \nThe third one follows the approach of \\citet{Haywood14} and \\citet{Rajpaul15}, and uses Gaussian-process regression (GPR) to model \nactivity directly from the raw RVs.  The results obtained with each technique are described and compared in the following sections.  \n\n\\subsection{Modeling the planet signal from filtered RVs (ZDI~\\#1)}\n\nThe first technique consists in using the ZDI brightness images of Fig.~\\ref{fig:mapi} to predict the RV curves expected for \nV830~Tau at each epoch, and compare them with observed raw RVs.  Modeled and raw RVs are both computed as the first order moment of \nStokes $I$ LSD profiles (i.e., $\\int (1-I(v)) v dv \/ \\int (1-I(v)) dv$ where $v$ is the radial velocity across the line profile) \nwhile error bars on raw RVs are derived from those propagated from the observed spectra to the Stokes $I$ LSD profiles \n(and checked for consistency through simulated data sets as in D16);  \nactivity-filtered RVs are then derived by simply subtracting the modelled RVs from the observed ones \n(D16, see Table~\\ref{tab:log}).  Even though the intrinsic variability observed at the surface of V830~Tau is only moderate (see \nSec.~\\ref{sec:dr}), using a specific ZDI map for each data subset (i.e., late 2015 and early 2016) is essential to obtain precise \nfiltered RVs;  using a single image for both subsets and ignoring the temporal evolution of the surface brightness distribution \nbetween the two epochs (beyond that caused by differential rotation) significantly degrades the quality of the modelling and therefore \nthe precision of the filtered RVs.  \n\n\\begin{figure*}\n\\center{\n\\includegraphics[scale=0.6,angle=-90]{fig\/v830_sinfit.ps}\\vspace{3mm}\n\\includegraphics[scale=0.4,angle=-90]{fig\/v830_kepfit.ps}}\n\\caption[]{{\\bf Top panel:} Raw (top), filtered (middle) and residual (bottom) RVs of V830~Tau (open symbols and 1$\\sigma$ error bars, \nwith circles, squares and triangles depicting ESPaDOnS, NARVAL and ESPaDOnS\/GRACES data, and colors coding rotation cycles).  The raw \nRVs exhibit a semi-amplitude of 1.2~\\hbox{km\\,s$^{-1}$}\\ and a rms dispersion of 0.65~\\hbox{km\\,s$^{-1}$}\\ throughout the whole run.  (Rotation cycles \nof the 2016 data are shifted by $+24$ in this plot with respect to their values in Table~\\ref{tab:log} and Fig.~\\ref{fig:fit16}).  \nThe RV jitter predicted by ZDI at both epochs, as well as the best sine fit to the filtered RVs, are added in the top and middle plots \n(cyan lines).  Note how the jitter model changes between late 2015 and early 2016, and how both of them slowly evolve with time as a \nresult of differential rotation.  \nThe rms dispersion of the residual RVs is 44~\\hbox{m\\,s$^{-1}$}, in agreement with our measurement errors (see Table~\\ref{tab:log}). \n{\\bf Bottom panel:} Activity-filtered RVs phase-folded on the planet orbital period.  The fit \nto the data is only marginally better with an eccentric orbit (dashed line) than with a circular one (solid line).} \n\\label{fig:rvs}\n\\end{figure*}\n\n\\begin{figure*}\n\\center{\n\\includegraphics[scale=0.6,angle=-90]{fig\/v830_mlsper.ps}\\vspace{3mm}\n\\includegraphics[scale=0.6,angle=-90]{fig\/v830_bloper.ps}\\vspace{3mm}\n\\includegraphics[scale=0.6,angle=-90]{fig\/v830_halper.ps}}\n\\caption[]{{\\bf Top panel:} Lomb-Scargle periodograms of the raw (top), filtered (middle) and residual RVs (bottom) shown in Fig~\\ref{fig:rvs}. \nThe black line is for the full set, while the dashed red, green, blue and pink lines are for the late-2015, the early-2016, \nthe even, and the odd points only.  The stellar rotation period, its first harmonic and the planet orbital period are depicted with \nvertical dashed lines. The horizontal dotted and dashed lines trace the 33\\%, 10\\%, 3\\% and 1\\% FAP levels.  The planet signal in the \nfiltered RVs is detected in the full set with a FAP level $<10^{-5}$. {\\bf Middle panel:} Periodogram of the longitudinal magnetic field, \na reliable activity proxy \\citep{Haywood16}, featuring a clear peak at the stellar rotation period but no power at the planet \norbital period. {\\bf Bottom panel:} Same as middle panel for the \\hbox{H${\\alpha}$}\\ emission.  }\n\\label{fig:pdg}\n\\end{figure*}\n\nThe results we obtain are shown in Fig.~\\ref{fig:rvs} for the raw, filtered and residual RVs, and in Fig.~\\ref{fig:pdg} for the \ncorresponding periodograms.  The planet RV signal is very clearly detected in the filtered RVs, with a false-alarm probability (FAP) \nlower than $10^{-5}$.  The \\hbox{$\\chi^2$}\\ decrease that we obtain with our fit to the filtered RVs (with respect to a case with no planet) \nis about 36 (for 72 RV points and 4 degrees of freedom), suggesting a similarly-low FAP value of $<$10$^{-6}$.  \nThe corresponding curve features a semi-amplitude equal to $K=60\\pm10$~\\hbox{m\\,s$^{-1}$}\\ and an orbital period of $\\hbox{$P_{\\rm orb}$}=4.97\\pm0.03$~d, in agreement \nwith the estimates of D16 ($K=75\\pm12$~\\hbox{m\\,s$^{-1}$}\\ and $\\hbox{$P_{\\rm orb}$}=4.93\\pm0.05$~d).  Fitting a Keplerian orbit through the data marginally improves the fit, \nbut the derived eccentricity ($0.21\\pm0.15$) is not measured with enough precision to be reliable \\citep{Lucy71};  it confirms at least \nthat V830~Tau~b is close to circular or only weakly eccentric.  The residual RVs show a rms dispersion of 44~\\hbox{m\\,s$^{-1}$}, fully compatible with \nthe errors of our RV estimates (see Table~\\ref{tab:log}) that mostly reflect the photon noise in our LSD profiles \n(and to a lesser extent the intrinsic RV precision of ESPaDOnS, equal to 20--30~\\hbox{m\\,s$^{-1}$}, \\citealt{Moutou07, Donati08b}).  Residual RVs in \nthe first part of the run (late 2015) exhibit a larger-than-average dispersion (of 50~\\hbox{m\\,s$^{-1}$}\\ rms, i.e., close to the value of 48~\\hbox{m\\,s$^{-1}$}\\ \nfound by D16 from modelling the late 2015 data only) that mostly reflects the limits in our assumption of a constant brightness \ndistribution at the surface of the star (sheared by differential rotation) on a relatively long data set (15 rotation cycles) and to a \nsmall extent potential residual pollution by the moon between rotational cycles 6.0 and 7.2 (see Fig.~\\ref{fig:fit15}).  \n\nLomb-Scargle periodograms of the longitudinal fields and the \\hbox{H${\\alpha}$}\\ emission fluxes of V830~Tau (see Fig.~\\ref{fig:pdg}, middle and bottom panels) \nboth show that activity concentrates mostly at the rotation period (with a recurrence period slightly longer than \\hbox{$P_{\\rm rot}$}, see \nSec.~\\ref{sec:mod}) and first harmonic, but not in a significant way at the planet orbital period.  This further confirms that the \nRV signal from V830~Tau~b cannot be attributed to activity.  \n\n\\subsection{Deriving planet parameters from LSD Stokes $I$ profiles (ZDI~\\#2)}\n\nThe second method, proposed by \\citet{Petit15} and inspired from our differential rotation measurement technique, directly works \nwith Stokes $I$ LSD profiles, and consists in finding out the planet characteristics and brightness distribution that best explain the \nobserved profile modulation.  More specifically, we assume the presence of a close-in planet in circular orbit with given parameters \n($K$, \\hbox{$P_{\\rm orb}$}\\ and phase of inferior conjunction), correct our LSD profiles from the reflex motion induced by the planet, reconstruct with ZDI the \nbrightness image associated with the corrected LSD profiles at given information content (i.e., image spottedness) and iteratively derive \nwhich planet parameters allow the best fit to the data.  This technique was found to yield results in agreement with those our first \ndirect method gave when previously applied to our V830~Tau data (D15, D16).  \n\nThe method was slightly modified to handle 2 different subsets of data at the same time, following \\citet{Yu16}.  The main \ndifference is that, for each set of planet parameters, we now reconstruct 2 different brightness images (one for each subset) with ZDI, \nboth at constant information content;  we then compute a global \\hbox{$\\chi^2_{\\rm r}$}\\ for this dual image reconstruction as a weighted mean of the \n\\hbox{$\\chi^2_{\\rm r}$}'s associated with the 2 ZDI images (with weights equal to the number of data points in the subsets).  This allows us in particular \nto handle different brightness distributions for different epochs, without which data cannot be optimally fitted as a result of the \nintrinsic variability that the spot configuration is subject to (see Sec.~\\ref{sec:dr}).  \n\nThe planet parameters we derive with this second technique are equal to $K=62\\pm9$~\\hbox{m\\,s$^{-1}$}\\ and $\\hbox{$P_{\\rm orb}$}=4.97\\pm0.03$~d, very similar to those \nobtained with our first method and again in agreement with those of D16.  The corresponding \\hbox{$\\chi^2$}\\ map (projected onto the $K$ vs $\\hbox{$P_{\\rm orb}$}$ \nplane that passes through the global minimum), shown in Fig.~\\ref{fig:ppar}, features a clear minimum.  With respect to our best model \nincorporating a planet, a model with no planet corresponds to a \\hbox{$\\Delta \\chi^2$}\\ of 75, indicating that the planet is detected with a \nFAP level $<10^{-15}$;  the much lower FAP directly reflects the larger \\hbox{$\\Delta \\chi^2$}\\ obtained with this method, reflecting that \nline profiles of rapid rotators contain more (or less-noisy) information than their first moments (the raw and filtered RVs).  \n\n\\begin{figure}\n\\includegraphics[scale=0.35,angle=-90]{fig\/v830_ppar2.ps}\n\\caption[]{Variations of \\hbox{$\\Delta \\chi^2$}\\ of the ZDI fits to the late-2015 and early-2016 LSD profiles of V830~Tau (for a fixed spottedness \nlevel at both epochs), after removing the reflex motion of a close-in a planet and for a range of orbital periods \\hbox{$P_{\\rm orb}$}\\ (actually the \nratio of the orbital to rotation period $\\hbox{$P_{\\rm orb}$}\/\\hbox{$P_{\\rm rot}$}$) and semi-amplitudes $K$ of the planet RV signature. (This is a 2D cut from a 3D map, \nwith the phase of the RV signal also included as a search parameter).  A clear minimum is obtained in the \\hbox{$\\Delta \\chi^2$}\\ landscape, whose \nprojection in a $K$ vs $\\hbox{$P_{\\rm orb}$}\/\\hbox{$P_{\\rm rot}$}$ plane passing through the minimum is shown here.  The outer color contour traces the projected 99.99\\% \nconfidence interval, corresponding to a \\hbox{$\\Delta \\chi^2$}\\ of 21.1 for a 3-parameter fit to the 3312 data points of the LSD profiles. } \n\\label{fig:ppar}\n\\end{figure}\n\n\\subsection{Deriving planet parameters from raw RVs using Gaussian-process regression (GPR)}\n\nThe third method we applied to our data works directly from raw RVs and uses GPR to model the activity jitter as well \nas its temporal evolution, given its covariance function \\citep[e.g.,][]{Haywood14, Rajpaul15}.  Assuming again the presence of a \nclose-in planet of given characteristics, we correct the raw RVs from the reflex motion induced by the planet and fit the \ncorrected RVs with a Gaussian process (GP) based on a pseudo-periodic covariance function $c(t,t')$ of the form: \n\\begin{eqnarray}\nc(t,t') & = & \\theta_1^2 \\exp \\left( -\\frac{(t-t')^2}{\\theta_3^2} -\\frac{\\sin^2 \\left( \\frac{\\pi (t-t')}{\\theta_2} \\right)}{\\theta_4^2} \\right) \n\\label{eq:covar}\n\\end{eqnarray}\nwhere $\\theta_1$ is the amplitude of the GP (in \\hbox{km\\,s$^{-1}$}), $\\theta_2$ the recurrence timescale (i.e., close to 1 here, in units of \\hbox{$P_{\\rm rot}$}), \n$\\theta_3$ the decay timescale (i.e., the typical spot lifetime here, in units of \\hbox{$P_{\\rm rot}$}) and $\\theta_4$ a smoothing parameter (within [0,1]) \nsetting the amount of high frequency structure that we allow the fit to include.  For a given set of planet parameters and of the 4 GP hyper \nparameters $\\theta_1$ to $\\theta_4$, we can compute the GP that best fits the corrected raw RVs (denoted $y$) and estimate the log likelihood \n$\\log \\mathcal{L}$ of the corresponding parameter set from: \n\\begin{eqnarray}\n2 \\log \\mathcal{L} & = & -n \\log(2\\pi) - \\log|C+\\Sigma| - y^T (C+\\Sigma)^{-1} y\n\\label{eq:llik}\n\\end{eqnarray}\nwhere $C$ is the covariance matrix for all observing epochs, $\\Sigma$ the diagonal variance matrix of the raw RVs and $n$ the number of data points.  Coupling this with \na Markov Chain Monte-Carlo (MCMC) simulation to explore the parameter domain, we can determine the optimal set of planet and GP hyper parameters \nthat maximizes likelihood, as well as the relative probability of this optimal model with respect to one with no planet (and only the GP modelling activity).  \n\nWe start by carrying out an initial MCMC run with input priors, whose results (the posterior distributions) are used to infer refined priors and \nproposal distributions capable of ensuring both an efficient mixing and convergence of the chain as well as a thorough exploration of the domain \nof interest (through a standard Metropolis-Hastings jumping scheme);  these refined priors are found to be weakly dependent on the input priors, \nalready suggesting that our data contain enough information to reliably characterize the GP and planet parameters.  The main MCMC run uses \nour refined priors, listed in Table~\\ref{tab:gpp} for the various parameters;  we usually carry out two successive main runs, a first one with all \n4 GP hyper parameters and 3 planet parameters free to vary, then a second one with both $\\theta_3$ and $\\theta_4$ fixed to their best values and \nthe remaining 5 parameters left free to vary.  The goal of this sequential approach is to incorporate as much prior information about the stellar \nactivity as possible into our model (hence the stronger refined priors) so that the GP yields a robust estimation of the uncertainties on the \nfinal parameters (particularly the planet mass) given these priors (see \\citealt{Haywood14} and \\citealt{Lopez16} for a similar approach).  \n\n\\begin{table}\n\\caption[]{Priors used in our MCMC simulation for the planet and GP hyper parameters.  \nWe mention the mean and standard deviation used for the refined Gaussian priors (plus the standard \ndeviation assumed for our initial MCMC run), the minimum and maximum \nvalues allowed for the uniform and Jeffreys priors, as well as the knee value for the modified \nJeffreys priors (following \\citealt{Haywood14}, with $\\sigma_{\\rm RV}$ noting the $\\hbox{S\/N}^2$-weighted \naverage RV error of our measurements, equal to 53~\\hbox{m\\,s$^{-1}$}, see Table~\\ref{tab:log}).  The planet phase \n$\\phi_0$ relates to the epoch of inferior conjunction \\hbox{BJD$_{\\rm t}$}\\ through $\\hbox{BJD$_{\\rm t}$}=\\phi_0\\hbox{$P_{\\rm orb}$} + t_0$ \nwhere $t_0=2,457,359.9069$~d (corresponding to rotation cycle 127.0) is the reference zero time we \nused for our observations.  } \n\\center{\n\\begin{tabular}{cc}\n\\hline\nParameter   & Prior   \\\\ \n\\hline\n\\hbox{$P_{\\rm orb}$}\/\\hbox{$P_{\\rm rot}$} & Gaussian (1.80, 0.012, initial 0.10) \\\\\n$K$ (\\hbox{km\\,s$^{-1}$})  & modified Jeffreys ($\\sigma_{\\rm RV}$)\\\\\n$\\phi_0$    & Gaussian (0.13, 0.04, initial 0.10) \\\\\n\\hline\nGP amplitude        $\\theta_1$ (\\hbox{km\\,s$^{-1}$})  & modified Jeffreys ($\\sigma_{\\rm RV}$) \\\\\nRecurrence period   $\\theta_2$ (\\hbox{$P_{\\rm rot}$}) & Gaussian (1.0, 0.001, initial 0.010) \\\\\nSpot lifetime       $\\theta_3$ (\\hbox{$P_{\\rm rot}$}) & Jeffreys (0.1, 500.0) \\\\\nSmoothing parameter $\\theta_4$         & Uniform  (0, 1) \\\\\n\\hline\n\\end{tabular}}\n\\label{tab:gpp}\n\\end{table}\n\n\\begin{figure*}\n\\includegraphics[scale=0.6,angle=-90]{fig\/v830_gpfit2.ps}\n\\caption[]{Same as Fig.~\\ref{fig:rvs} but now using a planet+GP fit to the data (pink line), where the GP is modelling the activity jitter (cyan line) \nwhile the planet and GP parameters are determined using a MCMC simulation.  Note how the GP succeeds at modelling the activity and its temporal evolution \nthroughout the whole observing window, and not just for the two separate subsets. The rms dispersion of the residual RVs is 37~\\hbox{m\\,s$^{-1}$}.  } \n\\label{fig:gpfit}\n\\end{figure*}\n\nWe find that $\\theta_3$, the hyper parameter describing spot lifetime, gives the best result for a value of $\\theta_3=44\\pm11~\\hbox{$P_{\\rm rot}$} = 120\\pm30$~d, \nonly slightly longer than the full duration of our observing run (91~d).  This further confirms the importance of taking into account \nthe temporal evolution of brightness maps in activity filtering studies, even in the case of wTTSs like V830~Tau whose spot distributions \nare known to be fairly stable on long timescales;  whereas this is true for the largest surface features, this is no longer the case for \nthe smaller ones whose effect on RV curves is significant.  Similarly, we get that $\\theta_4=0.6\\pm0.1$ yields the most likely fit to the data;  \nthis reflects the lack of fine structure in the RV curves, as expected from the fact that RVs are the first-order moment of Stokes $I$ LSD \nprofiles that acts as a low-pass filter on surface brightness distributions.  \nWith the final MCMC run, we obtain that the recurrence timescale $\\theta_2$ is equal to $\\theta_2=0.9986\\pm0.0007~\\hbox{$P_{\\rm rot}$}$, i.e., only very \nslightly shorter than the average rotation period $\\hbox{$P_{\\rm rot}$}$ on which our data were phased (see Eq.~\\ref{eq:eph});  we note that this period matches well \nthe equatorial rotation period of V830~Tau (see Table~\\ref{tab:spar} and Sec.~\\ref{sec:dr}), suggesting that RVs are primarily affected by \nequatorial features at the stellar surface.  For the GP amplitude \n$\\theta_1$, we find that $\\theta_1=0.878\\pm0.135$~\\hbox{km\\,s$^{-1}$}, $\\simeq$30\\% larger than the rms dispersion of our raw RVs (equal to 0.65~\\hbox{km\\,s$^{-1}$}\\ prior to \nany activity filtering, or removal of planetary-induced reflex motions).  \n\nFor the planet parameters, we find that $K=68\\pm11$~\\hbox{m\\,s$^{-1}$}\\ and $\\hbox{$P_{\\rm orb}$}=4.93\\pm0.03$~d, whereas the most accurate epoch of inferior conjunction \n(assuming a circular orbit) is found to be $\\hbox{BJD$_{\\rm t}$}=2,457,360.51\\pm0.14$. The corresponding fit to the data, shown in \nFig.~\\ref{fig:gpfit}, \ndemonstrates that the GP is doing a very nice job at modelling not only the activity, but also its evolution with time.  Comparing with \nthe results of our first method (see Fig.~\\ref{fig:rvs}), we can see that both the GP and ZDI predict similar RV curves.  However, \nthanks to its higher flexibility, the GP does a better job at matching the data, not only for our second data set where temporal \nvariability is higher (given the faster evolution of the predicted RV curve, see Fig.~\\ref{fig:gpfit}) and where the planet signal is \nclearly better recovered, but also for our first data set where the slower spot evolution is enhanced by the longer time span (of 15 rotation \ncycles).  As a result, the rms dispersion of the RV residuals has further decreased to 37~\\hbox{m\\,s$^{-1}$}, 16\\% smaller than with our first method, \nincluding in the first part of our run (late 2015) where the fit to the data is now tighter (rms dispersion of RV residuals of 40~\\hbox{m\\,s$^{-1}$}\\ \ninstead of 50~\\hbox{m\\,s$^{-1}$}, and close to that of the full run).  \nGiven this, we consider that the planet parameters derived with this third method, and in particular $K$ and \\hbox{$P_{\\rm orb}$}, \nare likely more accurate than the estimates obtained with the two previous techniques;  they also agree better with the initial estimates \nof D16 inferred from the late 2015 data only.  The phase plots of our\nfinal 5-parameter MCMC run are provided in Appendix~\\ref{sec:fig} (see Fig.~\\ref{fig:mcmc}, left panel), showing little correlation \nbetween the various parameters and thus minimum bias in the derived values.  \n\nWhen applying this technique to the full series of raw RVs collected to date on V830~Tau, including our original set secured in \nlate 2014 and early 2015 (D15, D16), we further enhance the precision \non the derived parameters, in particular on the orbital period that we can now pin down to $\\hbox{$P_{\\rm orb}$}=4.927\\pm0.008$~d.  The derived \nsemi-amplitude of the RV curve is the same as in the previous fit ($K=68\\pm11$~\\hbox{m\\,s$^{-1}$}) whereas the epoch of inferior conjunction\n(assuming a circular orbit) is only slightly improved ($\\hbox{BJD$_{\\rm t}$}=2,457,360.522\\pm0.124$).  The phase plots of this MCMC run are \nalso provided in Appendix~\\ref{sec:fig} (see Fig.~\\ref{fig:mcmc}, right panel).  \n\nApplying the method of \\citet{Chib01} to the MCMC posterior samples, we obtain that the marginal likelihood of the model including \nthe planet is higher than that of a model with no planet by a Bayes' factor of $10^8$ ($10^9$ when also including our raw RVs \nfrom late 2014 and early 2015), providing a strong and independent confirmation that V830~Tau hosts a close-in giant planet in a 4.93~d orbit.  \nAssuming now a planet on an elliptical orbit (and using $\\sqrt{e} \\cos \\omega$ and $\\sqrt{e} \\sin \\omega$ as search parameters where $e$ and $\\omega$ \nrespectively denote the eccentricity and argument of periapsis of the orbit, \\citealt{Ford06}) yields a low eccentricity of $0.16\\pm0.20$;  \nthe marginal likelihood of this latter model is however larger than that of the circular planet model by a Bayes' factor of $<3$, \nimplying that there is no evidence yet that the planet is eccentric.  We provide the MCMC phase plots of the eccentric orbit model in Fig.~\\ref{fig:mcmc2}.  \n\nThe planet parameters derived with all 3 methods are summarized in Table~\\ref{tab:tpar}, with those derived in D16 (from our \nlate 2015 data only) listed as well for an easy comparison.  \n\n\\begin{table*}\n\\caption[]{Summary of our results using the three filtering techniques described in Sec.~\\ref{sec:fil};  columns~2 to 4 are \nfor the 2 ZDI-based and GPR methods described in Secs.~4.1 to 4.3 and applied to our new data from late 2015 to early 2016, \nwhereas column~5 is for GPR applied to the entire data set (including that of D15).  \nColumn~6 recalls the results derived in D16 from the late~2015 data only as a comparison.  \nThe first table section lists the derived planet parameters (with $M_{\\rm p}$ denoting the planet mass), the second one \nmentions the inferred GP hyper parameters in the GPR case, and the last one recalls the achieved \\hbox{$\\chi^2_{\\rm r}$}\\ (to the filtered \nRVs for ZDI \\#1, to the Stokes $I$ profiles for ZDI \\#2 and to the raw RVs for GPR) and the rms dispersion of the RV residuals \n(whenever relevant). } \n\\center{\n\\begin{tabular}{cccccc}\n\\hline\nParameter                  & ZDI \\#1          & ZDI \\#2            & GPR                & GPR (all data)   & D16 \\\\ \n\\hline\n\\hbox{$P_{\\rm orb}$}\\ (d)                 & $4.97\\pm0.03$    & $4.97\\pm0.03$      & $4.93\\pm0.03$      & $4.927\\pm0.008$  & $4.93\\pm0.05$  \\\\ \n$K$ (\\hbox{m\\,s$^{-1}$})                  & $60\\pm10$        & $62\\pm9$           & $68\\pm11$          & $68\\pm11$        & $75\\pm12$      \\\\ \n$\\phi_0$                   & $0.128\\pm0.025$  & $0.142\\pm0.024$    & $0.122\\pm0.028$    & $0.125\\pm0.025$  & $0.123\\pm0.025$ \\\\ \n\\hbox{BJD$_{\\rm t}$}\\ (2,457,300+)       & $60.54\\pm0.13$   & $60.61\\pm0.12$     & $60.51\\pm0.14$     & $60.523\\pm0.124$ & $60.52\\pm0.13$ \\\\ \n$M_{\\rm p} \\sin i$ (\\hbox{${\\rm M}_{\\rm Jup}$}) & $0.50\\pm0.09$    & $0.52\\pm0.09$      & $0.57\\pm0.10$      & $0.57\\pm0.10$    & $0.63\\pm0.10$  \\\\ \n$M_{\\rm p}$ (\\hbox{${\\rm M}_{\\rm Jup}$}) assuming $i=55\\hbox{$^\\circ$}$ & $0.61\\pm0.11$ & $0.63\\pm0.11$ & $0.70\\pm0.12$ & $0.70\\pm0.12$   & $0.77\\pm0.12$  \\\\ \n$a$ (au)                   &                  &                    &                    & $0.057\\pm0.001$  &     \\\\\n$a\/\\hbox{$R_{\\star}$}$                 &                  &                    &                    & $6.1\\pm0.6$      &     \\\\ \n\\hline\nGP amplitude $\\theta_1$ (\\hbox{km\\,s$^{-1}$})       &        &                    & $0.878\\pm0.135$    & $0.842\\pm0.105$  &     \\\\ \nRecurrence period $\\theta_2$ (\\hbox{$P_{\\rm rot}$}) &        &                    & $0.9986\\pm0.0007$  & $0.9985\\pm0.0006$&     \\\\ \nSpot lifetime $\\theta_3$ (d)         &        &                    & $120\\pm30$         & & \\\\ \nSmoothing parameter $\\theta_4$       &        &                    & $0.6\\pm0.1$        & & \\\\ \n\\hline\n\\hbox{$\\chi^2_{\\rm r}$}\\                   & 0.68             & 1.0                & 0.48               & 0.42             & 0.75 \\\\ \nrms RV residuals (\\hbox{m\\,s$^{-1}$})     & 44               &                    & 37                 & 35               & 48  \\\\ \n\\hline\n\\end{tabular}} \n\\label{tab:tpar}\n\\end{table*}\n\n\n\\section{Summary \\& discussion}\n\\label{sec:dis}\n\nThis paper reports the results of an extended spectropolarimetric run on the wTTS V830~Tau, carried out in the framework of the \ninternational MaTYSSE Large Programme, using ESPaDOnS on the CFHT, Narval on the TBL and GRACES\/ESPaDOnS on Gemini-North, spanning from \n2015 Nov~11 to Dec~22, then from 2016 Jan~14 to Feb~10, and complemented by contemporaneous photometric observations from the \n\\hbox{1.25-m} telescope at CrAO.  This new study is an in-depth follow-up of a previous one, based only on the first part of \nthis data set and focussed on the detection of the young close-in hJ orbiting V830~Tau in 4.93~d (D16), and of an older one \nthat suspected the presence of V830~Tau~b, but from too sparse a data set to firmly demonstrate the existence of the planet (D15).  \n\nApplying ZDI to our two new data subsets, we derived the surface brightness and magnetic maps of V830~Tau.  Cool spots and warm \nplages are again present on V830~Tau, totalling 13\\%\\ of the overall stellar surface for those to which ZDI is sensitive.  The \nbrightness maps from late 2015 and early 2016 are similar, except for differential rotation slightly shearing the photosphere of \nV830~Tau and for small local changes in the spot distribution, reflecting their temporal evolution on a timescale of only a few weeks.  \nThe magnetic maps of V830~Tau are also quite similar at both epochs and to that reconstructed from our previous data set (D15), \nfeaturing a mainly poloidal field whose dominant component is a 340~G dipole tilted at 22\\hbox{$^\\circ$}\\ from the rotation axis.  As for \nthe brightness distribution, the magnetic field is also sheared by a weak surface differential rotation, and is evolving with \ntime over the duration of our observing run.  \n\nWe detected several flares of V830~Tau during the second part of our run, where one major event and a weaker precursor \nwere strong enough to impact RVs at a level of about 0.3~\\hbox{km\\,s$^{-1}$}.  In addition to generate intense emission in the usual spectral \nactivity proxies including the \\hbox{H${\\alpha}$}, \\hbox{Ca$\\;${\\sc ii}}\\ IRT and \\hbox{He$\\;${\\sc i}}\\ $D_3$ lines, these flares triggered large redshifts of the emission \ncomponent, especially for the \\hbox{He$\\;${\\sc i}}\\ $D_3$ line whose redshift reaches up to 35~\\hbox{km\\,s$^{-1}$}\\ with respect to the stellar rest frame, and \n25~\\hbox{km\\,s$^{-1}$}\\ with respect to the average line position in a quiet state.  By analogy with the Sun and young active stars \n\\citep[e.g.,][]{Cameron89, Cameron89b}, we propose that the flares we detect on V830~Tau relate to coronal mass ejections and \nreflect the presence of massive prominences in the magnetosphere of V830~Tau, likely confined by magnetic fields in the equatorial \nbelt of closed-field loops encircling the star (see Fig.~\\ref{fig:mag}), and whose stability is perturbed by the photospheric shear \nstressing the field or by the hot Jupiter itself in the case of large magnetic loops extending as far as the giant planet orbit \n(at 6.1~\\hbox{$R_{\\star}$}).  High-cadence spectral monitoring in various activity proxies is required to investigate such flares in more detail, \nwork out the fate of associated prominences once no longer magnetically confined, and diagnose the main triggering mechanism behind \nthem.  \n\nWe applied 3 different methods to our full data set to further confirm the existence of its hJ, and better characterize \nits orbital parameters.  The first two methods, using ZDI to model and predict the RV activity jitter, are those with which \nV830~Tau~b was originally detected, in a slightly modified version allowing them to handle two different ZDI images (corresponding \nto the late-2015 and early-2016 subsets) at the same time and account for the potential evolution of brightness distributions between \nthe 2 epochs.  Our third technique is fully independent from the 2 others and directly \nworks from raw RVs, using GPR to model the RV activity jitter and MCMC to infer the optimal planet and GP parameters and error \nbars in a Bayesian formalism, following \\citet{Haywood14}.  All 3 methods unambiguously confirm the existence of V830~Tau~b and \nyield consistent results for the planet parameters when applied to our new data;  in particular, all are able to reliably recover \nthe RV planet signal (of semi-amplitude $68\\pm11$~\\hbox{m\\,s$^{-1}$}) hiding behind the activity jitter (of semi-amplitude 1.2~\\hbox{km\\,s$^{-1}$}\\ and rms dispersion \n0.65~\\hbox{km\\,s$^{-1}$}) that the brightness distribution of V830~Tau is inducing.  \nThe third method is found to perform best, thanks to its higher flexibility and better performances at modelling the temporal \nevolution of the RV activity jitter.  Applying this third method to all raw RVs collected to date on V830~Tau (including those of \nD15) allows us to significantly improve the precision on the planet orbital period.  We also confirm that the planet orbit is more or less \ncircular, with no evidence for a non-zero eccentricity at a 1$\\sigma$ precision of 0.15--0.20. \nFurther work is needed to enable ZDI reconstructing time-variable features and make it as efficient as GPR for filtering activity \nfrom RV curves of young active stars.  \n\nSpectropolarimetry is found to be essential for retrieving the large-scale topology of the magnetic field that fuels all activity phenomena, \nbut not critical for modelling and filtering the activity jitter at optical wavelengths, largely dominated by the impact of surface \nbrightness features;  however, spectropolarimetry is expected to become crucial at nIR wavelengths where brightness features contribute \nless jitter and Zeeman distortions are much larger than in the optical \\citep[e.g.,][]{Reiners13, Hebrard14}.  \n\nAlong with the latest reports of similar detections (or candidate detections) of young close-in giants around TTSs \n\\citep[e.g.,][]{vanEyken12, Mann16, JohnsKrull16, David16, Yu16}, our result suggests that newborn hJs may be frequent, possibly \nmore so than their mature equivalents around Sun-like stars \\citep{Wright12}.  The orbital fate of young hJs like V830~Tau~b under \ntidal forces and strong winds as the host star progresses on its evolutionary track, contracts and spins up to the main sequence, and \nat the same time loses angular momentum to its magnetic wind and planet, is still unclear \\citep[e.g.,][]{Vidotto10, Bolmont16}.  \nOne can expect V830~Tau~b, whose orbital period is currently longer than the stellar spin period, to be spiralling outwards, at \nleast until V830~Tau is old enough to rotate more slowly than its close-in giant;  investigating whether tidal \nforces will still be strong enough by then to successfully drag V830~Tau~b back and kick it into its host star in the next few \nhundred Myrs, may tell whether and how frequent newborn close-in giants can be reconciled with the observed sparse population \nof mature hJs.  \n\nAlternatively, the MaTYSSE sample may be somehow biased towards wTTSs hosting hJs \\citep[e.g.,][]{Yu16}.  In particular, our sample is \nlikely biased towards wTTSs whose discs have dissipated early, i.e., at a time where the star, still fully convective, hosted a magnetic \nfield strong enough to carve a large magnetospheric gap \\citep{Gregory12, Donati13} and trigger stable accretion \\citep{Blinova16}.  \nThis may come as favorable conditions for hJs to survive type-II migration, when compared to more evolved cTTSs featuring weaker \nfields, smaller magnetospheric gaps and chaotic accretion.  \n\nLast but not least, we stress that V830~Tau is the first known non-solar planet host that exhibits radio emission \\citep{Bower16}, which \nopens very exciting perspectives for in-depth studies of star-planet interactions, and possibly even of exoplanetary magnetic fields \n\\citep{Vidotto10, Vidotto16}.  \n\nApplying the complementary detection techniques outlined in this paper to extended spectropolarimetric data sets such as those \ngathered within MaTYSSE, or forthcoming ones to be collected with SPIRou, the nIR spectropolarimeter \/ high-precision velocimeter \ncurrently in construction for CFHT (first light planned in 2017), should turn out extremely fruitful and enlightening for our \nunderstanding of star \/ planet formation, about which little observational constraints yet exist.  \n\n\n\\section*{Acknowledgements} \nThis paper is based on observations obtained at the CFHT (operated by the National Research Council of Canada \/ CNRC, the Institut \nNational des Sciences de l'Univers \/ INSU of the Centre National de la Recherche Scientifique \/ CNRS of France and the University of \nHawaii), at the TBL (operated by Observatoire Midi-Pyr\\'en\\'ees and by INSU \/ CNRS), and at the Gemini Observatory (operated by the \nAssociation of Universities for Research in Astronomy, Inc., under a cooperative agreement with the National Science Foundation \/ NSF \nof the United States of America on behalf of the Gemini partnership: the NSF, the CNRC, CONICYT of Chile, Ministerio de Ciencia, \nTecnolog\\'\\i a e Innovaci\\'on Productiva of Argentina, and Minist\\'erio da Ci\\^encia, Tecnologia e Inova\\c c\\~ao of Brazil). \nThis research also uses data obtained through the Telescope Access Program (TAP), which has been funded by the National Astronomical \nObservatories of China, the Chinese Academy of Sciences (the Strategic Priority Research Program ``The Emergence of Cosmological Structures''\nGrant \\#XDB09000000), and the Special Fund for Astronomy from the Ministry of Finance.\n\nWe thank the QSO teams of CFHT, TBL and Gemini for their great work and efforts at collecting the high-quality MaTYSSE data presented \nhere, without which this study would not have been possible.  MaTYSSE is an international collaborative research programme involving \nexperts from more than 10 different countries.  \n\nWe also warmly thank the IDEX initiative at Universit\\'e F\\'ed\\'erale Toulouse Midi-Pyr\\'en\\'ees (UFTMiP) for funding the STEPS \ncollaboration program between IRAP\/OMP and ESO and for allocating a ``Chaire d'Attractivit\\'e'' to GAJH allowing her regularly visiting \nToulouse to work on MaTYSSE data.  We acknowledge funding from the LabEx OSUG@2020 that allowed purchasing the ProLine PL230 CCD imaging \nsystem installed on the 1.25-m telescope at CrAO.  SGG acknowledges support from the Science \\& Technology Facilities Council (STFC) \nvia an Ernest Rutherford Fellowship [ST\/J003255\/1].  SHPA acknowledges financial support from CNPq, CAPES and Fapemig.  \n\nWe finally thank the referee, Teruyuki Hirano, for his valuable comments that helped us improve the paper.  \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\nMachine Learning (ML) has shown to be extremely successful in solving tedious problems across many domains. Due to its innate ability to capture complex properties and extract related information, processes developed manually by a human can not even compete with an ML-based system. \nLike other domains, the access control field has many issues where a manual human-driven solution could achieve at best a sub-optimal solution. Consequently, there are huge avenues for improvement in different sub-areas of access control domain. Over time, access control researchers took the initiative to utilize the power of ML to obtain more efficient solutions. For example, several research works proposed ML-based solutions to design access control policies that are more robust than the traditional approaches~\\cite{cotrini2018mining, karimi2021automatic, bui2020learning, frank2008class, bui2020decision, zhou2019automatic, jabal2020polisma}.\nBesides, process automation is another important application of machine learning. Instead of using ML to solve a whole problem, ML can also assist in solving parts of the domain. Researchers exploited the advancements of ML to automate underlying laborious tasks in access control such as attributes extraction from the information or text that exists in plain natural language~\\cite{alohaly2018deep, alohaly2019automated, alohaly2019towards, narouei2017towards, heaps2021access}, mapping between roles and permissions~\\cite{zhou2019automatic, ni2009automating}, security rules extraction from access logs~\\cite{cotrini2018mining, karimi2021automatic, mocanu2015towards}, or even extracting access control policies from the user stories~\\cite{heaps2021access}.\nRecent years have also witnessed the use of machine learning for access policy verification that could enhance the quality of implemented access control policy and ease the underlying burdens by automating the processes~\\cite{heaps2019toward, hu2021verificationNIST}. It is also researched how machine learning-based method can automatically monitor the existing deployed access control policy and warns system administrators if it finds any suspicious activities~\\cite{xiang2019towards}.\nMoreover, researchers have proposed utilizing machine learning for access control decision making where a trained ML model will decide whether an access request is to be granted or denied~\\cite{cappelletti2019quality, liu2021efficient, karimi2021adaptive, chang2006access, nobi2022toward}. \n\nAs evident, the effect of ML in access control is found to be positive and has huge potential to achieve even more. Since the use of ML for access control purposes is rapidly emerging, there are several challenges encountered in the current ML oriented access control research practice. We observe that researchers applied ML methods on a case-by-case basis~\\cite{frank2008class, karimi2018unsupervised, alohaly2019towards, cotrini2018mining, bui2019efficient, xiang2019towards}, thereby, there is no common strategy for using ML in the access control domain. The lack of an exhaustive overview of the application of ML in access control makes it even harder to gain in-depth insights into it and plan accordingly.\nIn addition, limited research efforts to determine the proper ML method for a distinct access control problem makes it challenging to select the best model for a novel problem.\nBesides, there are several other limitations, such as a lack of quality datasets from the real-world organization~\\cite{heaps2019toward, el2017abac} or the anonymous publicly available datasets may not contain relevant access control information capable of expressing a complete access control state of a system~\\cite{molloy2011adversaries}. Putting all these into perspective, it is essential to have a holistic view of how researchers use machine learning for access control. Also, such a cohesive picture will help learn under-developed areas and determine future research in the domain.\n\nIn this paper, we perform a detailed review and summarize existing literature that uses ML to solve different access control problems, including attribute\/roles or security policies extraction to access decisions and policy verification. To the best of our knowledge, this is the first comprehensive work which offers an encyclopedic view towards outlining the application of machine learning for access control system. We summarize our contributions to the field of access control as follows.\n\\begin{itemize}\n    \\item We perform a comprehensive review of existing access control literature that uses machine learning and discuss various research works done in different sub-domains of access control.\n    \\item We propose a novel taxonomy of machine learning in access control, and highlight research at each stage as the domain evolved chronologically.\n    \\item We summarize the publicly available real-world datasets used for machine learning based access control researches.\n    \\item We highlight open challenges and limitations faced by the research community as well as provide future research directions to thrive this critical security domain.\n\\end{itemize}\n\n\\subsection{Survey Methodology}\nThis survey is based on published research from 2006 to the first quarter of 2022. Before 2006, we did not find any ML-based access control solutions. We review papers from various search sources, including Google Scholar, ACM Digital Library, IEEEXplore, Springer, and preprints from arXiv. We use different keywords to filter related articles that use machine learning to solve \\emph{any} access control related problem. \nThis comprehensive review summarizes different approaches utilizing machine learning methods to solve access control problems.\nTo that end, we ask the following research questions while reviewing other works.\n\\begin{enumerate}\n    \\item What are the target access control models? In particular, is the proposed ML approach applicable for the access control domain as a whole or only suitable to any particular model?\n    \\item What are the ML methods that the respective approach uses?\n    \\item Why does the respective approach uses ML, and to what extent ML method contributed?\n    \\item What are the input and output of the ML model? Especially, can the trained machine learning model make access control decisions, or does the corresponding method only use an ML to improve or automate some sub-processes? \n    \\item What kind of data was used for training the ML algorithm? What kind of data pre-processing is required for a related approach?\n    \\item How well an ML method performed towards solving the desired access control problem?\n\\end{enumerate}\n\n\\subsection{Organization of the Survey}\n\nThe paper is organized according to our proposed taxonomy which is presented in Section~\\ref{sec:access-control-model}. This section provides a brief overview of access control. In Section~\\ref{sec:ml-methods-overview}, we summarize ML methods that are widely used in the access control domain. Section~\\ref{sec:categorization} demonstrates a timeline of how the application of ML evolved in access control context. In the same section, we review different approaches using machine learning to solve access control problems and briefly discuss each method following the proposed taxonomy. In Section~\\ref{sec:open-challenges}, we review open challenges and future research directions in the context of access control and machine learning. Finally, Section~\\ref{sec:conclusion} concludes our discussion.\n\n\n\\subsection{ML for Access Control}\n\n\\subsubsection{Access Decision}\nContemporary researches manifest the advantages of using an ML model for more accurate access control decision-making~\\cite{karimi2021adaptive,cappelletti2019quality,chang2006access,liu2021efficient,nobi2022toward,srivastava2020machine}. These systems decide accesses based on a trained ML model instead of a written access control policy.\nGenerally, these models make access control decisions (grant or deny) using user and resource metadata and attributes. Metadata and attributes are the user\/resource features that an ML model learns for subsequent access decisions. We briefly discuss these approaches below and summarize them in Table~\\ref{tab:access-decision}.\n\nThere are situations in various systems where access control policies need to be restricted with access hours and established policies.\nFor example, a user has access to a resource, but there is a policy that the user can not access the respective resource outside of office hours.\nChang et al.~\\cite{chang2006access} proposed a novel access control system, called time-constraint access control, that can be applied to access policies constrained with time.\nThe authors apply SVM to the proposed scheme and divide the processes into three phases: (1) the input pattern transforming, (2) the training phase for SVMs, and (3) the authority decision phase. The authors implemented SVMs using the Library of Support Vector Machines (LIBSVM)~\\cite{chang2011libsvm}, which is an efficient and easy-to-use software implementing support vector learning. As part of training data, the system administrator determines a login time and a password for each user. The trained SVMs can classify the users into their groups and give them corresponding security access using their passwords and system login time. Consequently, instead of written access control policies, trained SVMs are used for access decisions. The authors evaluate the performance of access decisions with the same data used for the training. The experimental results show that the system can authenticate users' access rights, which justifies the practicality of the proposed approach.\n\nIn general, centralized access control architectures with static security policies are limited in the IoT domain. The IoT devices don't have intensive and computational capabilities or storage to deal with a complete distributed access control process. However, outsourcing access control management to non-constrained nodes is not practical as it might introduce severe security and privacy concerns. As a result, IoT needs an access control framework suitable to its distributed nature, where users may control their privacy, and at the same time, the need arise for centralized entity handling. The author in~\\cite{outchakoucht2017dynamic} proposed a blockchain and machine learning-based access control model for the IoT domain. As presented, the distributed model will have distributed policy handled with blockchain, which is consistent with the decentralized aspect of IoT. Also, the authors proposed to utilize the power of AI, especially those of machine learning, to facilitate the adjustments in the access control policy dynamically. In particular, they proposed applying Reinforcement Learning due to its ability to adjust the knowledge incrementally while accesses are made to resources, and the security policy is executed. The authors didn't present any implementation of their work or case study as they kept it for future work. \n\nLuca Cappelletti et al~\\cite{cappelletti2019quality} investigate different ML techniques from both symbolic (Decision Tree (DT), Random Forest (RF)) and non-symbolic (Support Vector Classifier (SVC) and Multi-Layer Perceptron (MLP)) families for inferring ABAC policies from access logs. The experimentation is performed on two Amazon Datasets~\\cite{AmazonUCI2011, AmazonKaggle2013} and the Incident dataset~\\cite{amaral2018enhancing}. The details of the datasets are reported in SL:~\\ref{row:amazon-uci}, \\ref{row:amazon-kaggle}, and \\ref{row:incident} of Table~\\ref{tab:real-world-dataset}.\nThe Amazon datasets consist of access logs that indicate the decisions (grant or deny) for numerous accesses requested by thousands of Amazon users. Hence, the data is sparse and imbalanced in the sense that it has extensive logs with \\emph{grant} access. The Incident dataset, on the other hand, contains more balanced data from the different classes. \nThe authors further justify these characteristics using PCA and t-SNE~\\cite{xyntarakis2019data} data visualization of some instances of all these datasets. Both PCA and t-SNE demonstrate that the classes in the Incident log form well-separated clusters. In contrast, the negative class in Amazon dataset does not form an easily separable group. That makes it challenging to create straightforward and interpretable rules for Amazon datasets. \n\nAs part of Amazon datasets preprocessing, the columns with either many distinct classes or a single class are removed. Also, the method applies one-hot encoding to the original data before applying them to the ML algorithms as input.\nFor the experimentation on Amazon datasets, a variant of the balanced Monte Carlo cross-validation method was adopted for training and evaluation. As far as performance is concerned, the accuracy for Amazon datasets of all the techniques is about 50\\%, which is expected due to the inadequacy of typical ML techniques to handle imbalance and sparse data.\nOn the contrary, the accuracy is exceptionally high for the balanced and non-sparse Incident logs. However, in the case of sparse and imbalanced logs, among all the techniques, MLP performed better to demonstrate the ability of neural networks to capture complicated relationships in the data. Also, the policies obtained from symbolic ML techniques provide a better understanding of a decision than the non-symbolic peers. As a result, the authors raised the concern of \\textit{incompatibility} of such \\emph{black-box} ML techniques for the domains such as the ABAC system, where \\emph{explainability} and \\emph{verificability} are important.\n\n\\begin{figure}[t]\n    \\vspace{-2ex}\n\t\\includegraphics[width=0.8\\linewidth]{graphics\/epde_ml_permission_decision_user.png}\n\t\\caption{Permission Decision Method of EPDE-ML.}\n\t\\label{fig:epde_ml_permission_decision}\n\t\\vspace{-2ex}\n\\end{figure}\n\nOver time, various access control models such as cryptographic-based, identity-based, and trust-based access control models have been proposed for secure access in the cloud environment. However, such models overlook the user behavior and scalability of the trust management system, which are essential factors for cloud-based systems. Khilar et al.~\\cite{khilar2019trust} propose a trust-based approach that allows access to the cloud resources based on the accesses history and access behavior of the user. Also, the approach considers other factors, including user behavior, bogus request, unauthorized request, forbidden request, and range specifications.\nTo justify the efficacy of their work, the authors experiment with several ML techniques such as KNN, decision tree, logistic regression, naive Bayes, neural network, etc. They also consider different ensemble algorithms for the evaluation. They evaluate the performance of users and resources separately. The ensemble model of random forest and K-nearest neighbor performs better than other models. Overall, the neural network-based models achieve the highest performance, demonstrating the neural network's superior ability to capture access control relationships.\n\nThere are many real-world applications such as defense, airport surveillance, and hospital management systems. The legacy access control systems are ineffective as the policy decided initially cannot be changed dynamically. A situation may arise when a user needs to access some resources even though the current policy doesn't support the access. However, one could consider granting access considering the nature of applications, the authenticity, and the requesters' need. Srivastava et al.~\\cite{srivastava2020machine} propose a novel access control framework named risk adaptive access control (RAdAC), which understands the genuineness of the requester, calculates the risk, and then acts accordingly. The framework considers many real-world attributes in its design, such as time of access, location of access, frequency of the requests, and the sensitivity of requested resources. The proposed approach depends on multiple parameters such as a sensitivity score and relevance, and the authors followed some unique strategies to engineer those parameters.\nThe authors develop a prototype of their proposed method for a Hospital Management System (HMS). They experiment with a neural network with two hidden layers and an RF algorithm. For RF, they examine two cases. In one case, they use the input data without engineered parameters (e.g., sensitivity score) and apply the autoencoder~\\cite{farahnakian2018deep} to extract features from the data. In another case, they use both input data and the engineered parameters. The RF achieves the best results among all these approaches using both input data and parameters. However, based on evaluation, the authors confirm that domain expertise is required for determining the optimal values of different parameters.\n\n\n\nThe authors in ~\\cite{liu2021efficient} propose an Efficient Permission Decision Engine scheme based on Machine Learning (EPDE-ML) that improves the policy decision point (PDP) of the ABAC model.\nInternally, the EPDE-ML includes an RF algorithm trained based on user attributes and prior access control information. For any access control request from a user, the decision engine results either permit or deny indicating the user has access or not to the corresponding resource. \nThe EPDE-ML splits the traditional permission decision process into two phases (Phase 1 and Phase 2), shown in Figure~\\ref{fig:epde_ml_permission_decision}. \n\nIn Phase 2, to ensure the privacy protection of the policy information, the RF model is trained based on the current access control policy information. The trained model is used to predict access control decisions in Phase 1. In the case of policy change, the method updates the RF model in Phase 2 and replaces the model in Phase 1 after that.\nThe authors perform experiments on Amazon's real access control policy set~\\cite{AmazonKaggle2013} and compare the results with some other ML algorithms. The authors evaluate the performance of different approaches based on several metrics that include accuracy, precision, recall, and F1-score. The author also analyzes the receiver operating characteristic (ROC) curve and area under the curve (AUC) among all the methods. Overall, the EPDE-ML demonstrates superior performance with an AUC value of 0.975 and an accuracy of 92.6\\%. Furthermore, the authors also compare the decision time that shows the EPDE-ML requires a consistent decision time of around 0.115 seconds regardless of the policy size.\n\n\nThere are some challenges with supervised learning approaches in the real-world system, such as limited labeled data, sparse logs holding partial access information, etc. To tackle these issues, Karimi et al.~\\cite{karimi2021adaptive} propose an adaptive access control approach that learns from the feedback provided by the user.\nThe authors propose a reinforcement learning system in which an authorization engine adapts an ABAC model, as shown in Figure~\\ref{fig:adaptive_rl_policy_learning}. The model depends on interacting with the administrator of the system to receive the feedback that helps the model make an authorization decision. The authors suggested four methods for initializing the learning model based on attribute value hierarchy to speed up the learning process. The paper focus on developing an adaptive ABAC policy learning model for a home IoT environment.\n\nThe authors experiment on multiple data sets, including synthesized and real ones, to evaluate the proposed method properly. The synthesized authorization records are produced based on ABAC policies: a set of ABAC policies with manually written policy rules and one with randomly generated policy rules. The real dataset is built from the records provided by Amazon~\\cite{AmazonKaggle2013} (SL:~\\ref{row:amazon-kaggle} of Table~\\ref{tab:real-world-dataset}). Each access tuple in this dataset corresponds to an employee's request for a resource and shows whether the access was permitted or not. The access log consists of the employees' attribute values and the resources' identifier.\n\n\\begin{figure}[t]\n    \\vspace{-3ex}\n\t\\includegraphics[width=0.5\\textwidth]{graphics\/Adaptive_RL_ABAC_Policy.png}\n\t\\caption{Overview of Reinforcement Learning Based ABAC Policy Learning.}\n\t\\label{fig:adaptive_rl_policy_learning}\n\t\\vspace{-3ex}\n\\end{figure}\n\nGenerally, traditional access control systems are imitated in the context of modern systems that are dynamic, complex, and large-scale, where it is difficult to maintain an accurate access control state in the system for a human administrator. As a potential solution to this problem, Nobi et al.~\\cite{nobi2022toward} propose Deep Learning Based Access Control (DLBAC) utilizing advancements in deep neural networks.\nThe DLBAC works directly with the users and resources metadata and obviates the need for attribute and policy engineering. The output of the DLBAC model is a trained neural network, which takes user and resource metadata as input and makes access control decisions.\n\nThe authors create a prototype of DLBAC using ResNet~\\cite{he2016deep} neural network. The authors simulate complex real-world scenarios for the performance evaluation and develop eight synthetic datasets. The datasets and related source code are publicly available\\footnote{\\url{https:\/\/github.com\/dlbac\/DlbacAlpha}}. The authors also experiment with their proposed method with two Amazon datasets~\\cite{AmazonUCI2011, AmazonKaggle2013}. The empirical evaluation reveals that the DLBAC can make more accurate access control decisions and generalize better than traditional policy mining approaches and classical ML-based systems. The evaluation results demonstrate a trade-off while managing the over-provision (permitting unauthorized accesses) and under-provision (denying desired accesses). The proposed experiment suggests that traditional policy-based systems are highly inefficient in denying access, whereas DLBAC can properly balance under- and over-provision.\n\nPrior researches~\\cite{cappelletti2019quality} highlight that a neural network-based system, such as DLBAC, might not be feasible for access control due to the lack of decision interpretation. As DLBAC makes access decisions based on its underlying black-box neural network, it is crucial to understand the decisions in human terms. The authors propose Integrated Gradient and Knowledge Transferring based understanding methods to overcome this challenge of comprehending DLBAC decisions. The first method ranks user and resource metadata with respect to their influence on an access decision in question using an attribute score. The latter approach extracts a decision tree to understand a decision in the form of traditional rules. \nThe authors also suggest that their proposed understanding techniques could be utilized in other aspects of access control, such as access modification.\n\\subsection{Policy Administration and Monitoring}\nTable~\\ref{tab:admin-monitoring} outlines related methods proposed for the access control administration and monitoring using machine learning.\n\n\\subsubsection{Policy Administration}\n\nManual access policy updates are a laborious and error-prone task~\\cite{argento2018towards} and the system becomes vulnerable to different cyber threats.\nAuthors in~\\cite{argento2018towards} present an ML-based approach, named ML-AC, that can update policies at run-time automatically and prevent such threats. The ML-AC monitor and learn the \\textit{access behavioral features} (e.g., frequency of access, amount of data, location, etc.) of a user and adjust existing access control rules based on learned \\textit{contextual knowledge}. The authors apply this method by adding a novel \\textit{Contextual Behaviour Learning} component in the Policy Administration Point (PAP) of the access control system. This new component builds user profiles based on the monitored access pattern and adjusts access control policies by utilizing those profiles.\nThe authors introduce the idea of \\textit{user behavior} that indicates how users are utilizing resources and build a behavioral model to identify anomalous accesses. The behaviors are formally defined in access request attributes (e.g., user, resource, and action) and different contextual features (i.e., working time, location, types of activities, etc.). Next, ML-AC clusters such behaviors together based on uniformity and create various behavioral profiles of normal accesses. \n\nThe ML-AC uses RF to classify a given access control behavior as the \\textit{normal} or \\textit{anomalous}. To refine the existing policy, ML-AC obtains the output of the decision tree, defined as \\textit{ML-rules}. These ML rules are encoded in existing policy in additional conditions that transfer learned contextual knowledge of user behavior. Such processed rules have more control in determining abnormal accesses in the system. To keep the RF models accurate and updated, the ML-AC also monitors the evolution of user behaviors to detect the appearance of new clusters using a method Olindda~\\cite{spinosa2009novelty}.\nThe authors implement ML-AC based on a synthetic dataset having 3000 unique behaviors from normal and anomalous categories to evaluate the effectiveness and practicality. The method presents the behaviors as numerical features based on three-dimensional points. The source code and dataset are publicly available in GitHub\\footnote{https:\/\/github.com\/cybersoton\/ml-ac}. The authors compare ML-AC with a method name BBNAC~\\cite{frias2009network} which is similar to ML-AC. The authors also compare the performance with a variant of ML-AC named ML-AC$_{nok}$ where they did not use any contextual knowledge. The experimentation shows ML-AC performs significantly better in all the cases.\n\n\nThe authors in~\\cite{alkhresheh2020adaptive} propose an adaptive access control policy framework for IoT deployments. The framework dynamically refines the access policies based on the access behaviors of the IoT devices. In parallel to the traditional ABAC authorization server, the authors propose incorporating a policy management module that implements the access policy adaptation functionalities, including the access behavior classifier and the policy refinement components. For both servers, a context monitor component is added to access the contextual information.\nThe authors apply RF, and Recurrent Neural Networks (RNNs)~\\cite{zaremba2014recurrent} on three years of access history data of a door locking system of a university to justify the effectiveness of the proposed method. The authors did not publish their dataset. For RNN, the authors use Long Short-Term Memory (LSTM)~\\cite{hochreiter1997long} due to its efficiency for large data sets. The authors create different instances of the dataset based on the size. The empirical results imply that both RF and LSTM show similar performance for smaller-sized datasets. For larger instances, the LSTM outperforms RF because of the increased chances for the LSTM classifier to learn from longer access sequences. Therefore, the authors conclude that the LSTM classifier can scale better to IoT environments.\n\n\nGumma et al.~\\cite{gumma2021pammela} propose PAMMELA which is an ML-based ABAC policy administration method. The PAMMELA creates new rules for the proposed changes and extends existing policy by adjusting the newly developed rules with the current policy. PAMMELA can also develop new policies for a system by learning existing policy rules in a similar system.\nPAMMELA is a supervised ML-based approach that mainly works in two phases. In the first phase, PAMMELA trains an ML classifier on ABAC policy rules. The authors define a rule as the combination of subject and object attribute-value pairs indicating whether the subject has specific access or not to the associated object. In the second phase, PAMMELA provides a set of access requests to the trained classifier and creates a set of rules based on the classifier's access decision for respective requests. These generated rules capture the access control state of the given ABAC policy rules. The authors experiment with their proposed method using three manually created datasets with various ABAC rules, subject attributes values, object attribute values, and access requests. To evaluate the performance of PAMMELA, the authors apply different kinds of machine learning methods, including neural network, decision tree, RF, Extra Trees (ET)~\\cite{geurts2006extremely}, Gradient Boosting (GB)~\\cite{friedman2001greedy} and Extreme Gradient Boosting (XGB)~\\cite{chen2015xgboost}. For a proper justification of the proposed method, the authors follow multiple evaluation techniques. The authors highlight some insights on managing ABAC policies using PAMMELA based on empirical results.\n\n\n\\subsubsection{Policy Monitoring}\nDue to the lack of proper tools, standardized policy specifications such as XACML and their implementation can be error-prone. It demands rigorous verification and validation to ensure the implemented policy complies with the desired one. Martin et al.~\\cite{martin2006inferring} discover that the ML algorithms could summarize the basic properties of such a policy and help to identify specific bug-exposing requests. Especially, their proposed method can efficiently identify the discrepancies between the policy specification and its intended functionalities.\nThe authors first generate a set of access requests and apply them to the existing access control system to observe the behavior of the policy. These observations are structured as request-response pairs and given as input to the ML algorithm to summarize the policy in the form of inferred properties. Essentially, these inferred properties capture the general policy behavior to determine cases that might be bug-exposing and necessitate a manual inspection. The authors integrate Sun's XACML implementation~\\cite{SunXacml2005} and Weka~\\cite{ian2005data}, a collection of ML algorithms for data mining tasks, into their proposed tool that implements the entire approach through request generation, request evaluation, and policy property inference. Remarkably, the authors use the Prism classification algorithm~\\cite{cendrowska1987prism} for the rule sets generation. For policy verification, the tool is applied to an access control policy of a central grades repository system for a university~\\cite{fisler2005verification}. Eventually, the generated rules are translated into an XACML policy to evaluate the performance. Evaluation results show that the inferred properties can indeed summarize the given policy that supports the ability of ML in providing crucial insight into basic policy properties, and help identify specific bug-exposing requests.\n\n\\begin{figure}[t]\n    \\vspace{-2ex}\n\t\\includegraphics[width=0.6\\linewidth]{graphics\/time_changing_decision_tree.png}\n\t\\caption{Time-Changing Decision Tree (TCDT).}\n\t\\label{fig:time_changing_decision_tree}\n\t\\vspace{-3mm}\n\\end{figure}\n\nA regular access control policy update is required to accommodate intermediate access changes. The task is very challenging for system admins and error-prone (e.g., over-granting access privilege) due to the lack of proper tools~\\cite{xiang2019towards}. However, failing to detect such requests (or misconfigurations) in the early stage may cause severe security incidents.\nAuthors in~\\cite{xiang2019towards} develop a tool named P-DIFF to backing system admins to monitor access control policy changes. P-DIFF also supports investigating any malicious access by backtracking related changes that help system admins identify the reason behind unwanted access. The authors adopt a decision tree representation to convert heterogeneous access control behaviors to a standard rule-based format. However, a regular decision tree is limited in handling time-series information to consider access concerning time. To compensate for such a challenge, the authors extend the decision tree algorithm and propose a novel Time-Changing Decision Tree (TCDT). Unlike single binary value outcome (allow or deny), each rule of a TCDT is expressed as a time series as depicted in Figure~\\ref{fig:time_changing_decision_tree}. The TCDT learning algorithm is designed to infer the tree by considering access logs as a sequence of access events ordered by access time. Then, the TCDT allows modeling access control behavior at any given time along with monitoring changes in the access control rules. \n\nThe P-DIFF takes access logs as the input to generate a TCDT as the output that records the entire change history to aid both access policy change validation and forensics analysis.\nThe authors experiment effectiveness of P-DIFF across datasets from five real-world systems, including the access logs from a public dataset of request traces to Wikipedia~\\cite{urdaneta2009wikipedia}. Experiments demonstrate that P-DIFF can detect 86\\%$-$100\\% of access control policy changes with an average precision of 89\\%. In forensic analysis, P-DIFF shows 85\\%$-$98\\% efficacy in determining the root cause of several cases.\n\n\n\\subsection{Policy Mining}\nDue to the flexible policies and fewer management burdens, high-level access control models such as ABAC and ReBAC are adopted to support dynamic and complex security policies. However, adopting such policies from an existing lower-level policy like ACLs is challenging.\nGenerally, the policy mining techniques take attributes of users\/resources in the system and the current access control state of the system as input. For ABAC and ReBAC, the algorithm output a set of rules (policy) that grants the same permissions~\\cite{bui2019greedy, bui2017mining}. In contrast, RBAC mining algorithms output permission to roles (PA) and user to roles (UA) assignments~\\cite{zhou2019automatic, ni2009automating}. Table~\\ref{tab:ACPolicyMining} summarizes\naccess control policy mining approaches using machine learning.\n\n\\subsubsection{Attribute Based Access Control (ABAC)}\n\n\\textbf{\\\\Attributes and Policy Extraction from Natural Language.}\nNatural language policies, being the preferred expression of policy~\\cite{alohaly2018deep}, need to be transformed into a machine-readable form. Several researchers attempted to process such policies to extract access control-related information, including identifying policy sentences, triples of subject-object-action, etc. While manual extraction of such information is inefficient as the task becomes repetitive, requires more time, and is error-prone, several other approaches have been proposed to automate the process~\\cite{narouei2017towards, narouei2015towards, slankas2013access, slankas2014relation, xiao2012automated}.\n\nNarouei et al. \\cite{narouei2017towards} introduce a top-down policy engineering framework for ABAC that automatically extracts security policies from natural language (NL) documents using deep recurrent neural networks (RNN). The paper uses high-level requirement specification documents referred to as natural language access control policies (NLACPs) which define statements governing management and access of critical objects in an enterprise. These NLACPs are simple human-readable expressions translated into machine enforceable access control policies. This work creates a deep recurrent neural network that uses pre-trained word embeddings to identify sentences that contain access control policy content. The authors develop a dataset from real-world policy documents from where a randomly selected subset of 2660 sentences is used and annotated for the presence of access control policy (ACP) content. The RNN models' input is word embeddings, where each `word' from the sentence is represented as a vector, and each of the `sentences' is represented as a vector of these word vectors. The output of the model is ACP sentences from large requirements documents. Further, the authors compare the proposed approach to other methods, outperforming the SVM model results and improving 5.58\\% over the state-of-the-art. A prime limitation of this work is the involvement of the human factor in determining the correct identifiers of ACP sentences from NL documents. \n\nAlohaly et al.~\\cite{alohaly2018deep} propose a deep learning-based automated approach for extracting ABAC attributes from NL policy. The authors develop a framework to automate the attribute extraction and related information leveraging natural language processing (NLP), relation extraction (RE), and CNN.\nThe authors first locate the modifiers of each policy element by analyzing the NL requirements. It extracts the attribute value and corresponding policy element, then identifies the attribute's category, short name, and data type.\nAuthors apply CNN to capture subject-attribute and object-attribute element-specific relations. They first generate candidate instances involving common patterns and feed them into the CNN classifier to determine whether a candidate instance holds a relation or not. They utilize two separate binary classifiers to determine subject and object attribute relation. Authors apply word embedding to encode words to real-valued vectors using GloVe pre-trained word embedding~\\cite{pennington2014glove}. Finally, a sigmoid layer determines the binary classification.\nThe authors evaluate the framework based on a set of NLACPs by augmenting some existing datasets, including iTrust~\\cite{meneely2012appendix}, IBM Course Management App~\\cite{ibmcourse2004}, CyberChair~\\cite{van2012cyberchair}, and Collected ACP~\\cite{xiao2012automated}. As the policies in the existing datasets are more role-based focused and do not have enough attributes information, the method expands either subjects or object elements to inject some additional ABAC context.\nThe authors apply the F1 score as the evaluation metric. Based on their evaluation, the F1 score of the subject's attributes extraction is 0.96, while the performance is lowered by 0.05 for the object's attributes.\n\n\\begin{figure}[t]\n    \\vspace{-2ex}\n\t\\includegraphics[width=.6\\linewidth]{graphics\/hierarchical_abac_policy.png}\n\t\\caption{Attributes Identification from a Natural Language Hierarchical ABAC Policy.}\n\t\\label{fig:hierarchical_abac_policy}\n\t\\vspace{-2ex}\n\\end{figure}\n\nThe primary goal of the Alohaly et al.~\\cite{alohaly2018deep} is to detect values of attributes (attribute extraction) in ABAC policies with no hierarchy among subject or object elements (\\emph{flat ABAC}). From a practical perspective, extraction of authorization attributes of \\emph{hierarchical ABAC} system from natural language artifacts is more needed than the flat counter-part.\nThe authors in~\\cite{alohaly2019automated} extend the framework proposed in~\\cite{alohaly2018deep}, built upon recent advancements in NLP and ML techniques, with automatic attributes extraction ability from NL hierarchical ABAC policies. \nThe entire proposed framework works in multiple distinct phases. \n\nFirst, it extracts attribute value and related policy element. As shown in Figure~\\ref{fig:hierarchical_abac_policy}, \\emph{nurse} and \\emph{senior} are the attribute values of two policy elements \\emph{subject\\_type} and \\emph{rank}, respectively. \nThen, the algorithm extracts policy elements and attribute values capturing subject-attribute and object-attribute relations by identifying the grammatical patterns that encode each relation.\nThe next phase generates two lists of subject-attribute and object-attribute relations candidate instances utilizing commonly available patterns. The authors build two separate CNN-based classifiers for two relations based on respective candidate instances and determine whether a candidate instance encodes the corresponding relationship. The framework uses the GloVe pre-trained word embedding~\\cite{pennington2014glove} to embed candidate instances.\nLater on, the framework executes the second phase to suggest attribute's short names leveraging a density-based clustering method known as DBSCAN~\\cite{ester1996density}. The authors utilize traditional algorithms to perform later steps that include hierarchically assigning the attributes to namespaces and identifying category and data type of attributes. \nFor performance evaluation of the proposed technique, similar to Alohaly et al.~\\cite{alohaly2018deep}, the authors construct a set of realistic synthetic NLACPs using some real-world datasets~\\cite{meneely2012appendix, ibmcourse2004, van2012cyberchair, xiao2012automated}. Based on the experiments, the proposed framework shows a promising result with an average F1 score of 0.96 when extracting the subject's attribute values and 0.91 for object attribute values extracting.\n\nIn ABAC, the attributes often need to satisfy some constraints, which is critical for complying with organizational security policy. It requires extraordinary skills and efforts to define the constraints formally. The process is tedious and error-prone as security architects have to analyze several documents to develop them manually. \nThe Alohaly et al.~\\cite{alohaly2019towards} proposed a reliable and automated constraints extraction process exploiting tools in NLP. This automation also enables trace-ability formal constraints expressions and related policies that will also help to avoid errors while assigning unauthorized attributes.\nFirst, the proposed method extracts the \\emph{conflicting} factors, as the constraints are defined using the notion of \\emph{conflict}~\\cite{bijon2013constraints, bijon2013towards}. To determine the conflicts in NLACPs, the authors initially annotate the NLACPs by specifying different levels of conflict ranging from 0 to 3, which is expected to be done manually by security architects for each policy sentence. In the next phase, the method identifies the constraints expressions by determining the right \\emph{boundaries} (inside, beginning, or outside) of conflicting values within the policy sentence. \n\nTo this end, the authors apply the state-of-the-art Bidirectional Long-Short Term Memory (BiLSTM) model of Recurrent Neural Networks (RNNs)~\\cite{huang2015bidirectional}. As input, the BiLSTM takes each word's real-valued vector representation in the NLACP sentence using GloVe pre-trained word embedding. Internally, the BiLSTM layer trains two distinct LSTMs. One is trained to acquire the sequential information of given input, while the other learns information encoded in the reverse order~\\cite{alohaly2019towards}. Finally, the model assigns a \\emph{label} for each word indicating its boundary, whether the word is in inside, at the beginning, or in the outside. The extraction process ends with the normalization of the constraints.\nTo evaluate the effectiveness of the proposed approach, the authors created a dataset (not publicly available) from real-world policy documents with authorization rules for various departments, including Human Resources, Information Technology, etc., from educational institutions. The dataset comprises 801 occurrences of constraints in 747 NLACP sentences, where each of the sentences contains at least one constraint. The experimental results imply that the proposed approach holds promise for enabling constraint automation with an F1 score of 0.91 in detecting at least 75\\% of each constraint expression.\n\nThe user stories written by software developers better represent the actual code than high-level product documentation. Such stories can contain access control-related information, and extracting that information can be used to construct access control documentation. Also, stakeholders can use this information for access control engineering, development, and review. For example, as shown in Figure~\\ref{fig:user_story}, the user story contains access control information where `camp administrator' is the actor and `campers' is a data object that the camp administrator should be able to perform a `remove' operation. Authors in~\\cite{heaps2021access} developed an automated process using ML to extract access control information from a set of user stories that present the behavior of the software product in question. The proposed work takes a collection of user stories as input to a transformers-based deep learning model~\\cite{devlin2018bert} with three components: access control classification, named entity recognition, and access type classification. First, it classifies whether a user story contains access control information or not. Then, it identifies the actors, data objects, and operations in the user story as part of a named entity recognition task.\nEventually, the proposed technique determines the type of access between the identified actors, data objects, and operations through a classification prediction. The authors apply three distinct transformers-based learning models for the above three tasks. \n\nThe authors experiment and evaluate the proposed method with Dalpiaz dataset~\\cite{dalpiaz2018requirements,dalpiaz2018pinpointing} consisting of 21 web applications from various domains, each with 50\u2013130 user stories for a total of over 1600 user stories. SL.~\\ref{row:dalpiaz} of Table~\\ref{tab:real-world-dataset} outlines details of this dataset. The authors compare their results with CNN and SVM-based implementation for all three tasks. The overall results suggest that the transformers perform the best in all categories. However, CNN performs as good as Transformers for the named entity recognition task. Though the Transformers model performs better than the CNN and SVM, the difference is not that notable. The authors suggest that the first step would be to have a larger and more robust dataset for performance enhancement.\n\n\\begin{figure}[t]\n    \\vspace{-2ex}\n\t\\includegraphics[width=0.7\\linewidth]{graphics\/user_stories.png}\n\t\\caption{A sample user story.}\n\t\\label{fig:user_story}\n\t\\vspace{-2ex}\n\\end{figure}\n\n\\noindent\n\\textbf{Policy from Access Logs.}\nWork by Mocanu et al.~\\cite{mocanu2015towards} proposed a deep learning-based approach to infer policies from logs, which also supports negative authorization (i.e., a subject cannot access a resource). The proposed approach has two phases. The first phase generalizes the knowledge from logs providing a set of candidates rules in a binary vector format. In the second phase, this set of candidate rules is transformed to the format acceptable by Xu-Stoller~\\cite{xu2014mining} and compared. The work uses Restricted Boltzmann Machines (RBMs) as density estimators to propose a technique capable of producing a set of suitable candidate rules based on the knowledge extracted by the RBMs from the processed logs. The authors use a healthcare dataset to evaluate the effectiveness of their approach. A significant advantage of their approach is the support for negative authorization. Still, their proposed work is limited to the first phase, and more implementation results and evaluations are required for real-world policies containing various types of expressions.\n\nHowever, such kinds of methods~\\cite{xu2014mining, mocanu2015towards} are limited where the logs have only a tiny subset of all possible access requests. Evidently, the access logs in real-world systems only contain about 10\\% of all possible access requests~\\cite{cotrini2018mining}. Over time, the developed rules also become highly complex due to its adoptions to different changes in the system. Hence, the shorter and less-convoluted rules are better from a maintainability perspective.\nCotrini et al.~\\cite{cotrini2018mining} proposed an approach, known as Rhapsody, for mining ABAC rules from sparse access logs identifying patterns among the authorized requests. Rhapsody also solves various other issues of the existing state-of-the-art, including the rules size and over-permissiveness nature.\nThe Rhapsody builds upon APRIORI-SD~\\cite{kavvsek2006apriori},which the authors modify to overcome its weaknesses of mining overly permissive or unnecessarily large rules. Rhapsody takes an ABAC instance as input and provides a set of rules as the output. It performs the overall computation in three different stages. At first, it determines a set of candidate rules that cover a certain amount of requests available in the logs. Next, it filters out overly permissive rules. The algorithm opt-out any rule from the rules set if there is an equivalent shorter rule.\n\nThe authors evaluate the performance of Rhapsody using two Amazon access logs~\\cite{AmazonKaggle2013, AmazonUCI2011} and a log of students accessing a computer lab at ETH Zurich. Both organizations' logs are sparse where Amazons' logs show about 7-10\\% of all the employees have requested access, whereas the computer lab logs have access of less than 1\\% of the users. Besides, the authors also experiment with some synthetic datasets that contain approximately 50\\% of all possible requests. \nCotrini et al. also introduce a novel validation method named \\emph{universal cross-validation} that evaluates mined rules with requests not present in the dataset. Experiments show that the proposed evaluation approach validates policies with higher F1 scores (more accurate in deciding requests outside the log) than the standard cross-validation technique. The authors compare the performance of Rhapsody with Classification Tree~\\cite{breiman2017classification}, CN2~\\cite{clark1989cn2}, and two state-of-the-art ABAC policy mining algorithms (Xu and Stoller~\\cite{xu2014mining} and APRIORI-SD~\\cite{kavvsek2006apriori}). The results signify that Rhapsody can mine concise rules from sparse logs with higher accuracy than state-of-the-art mining algorithms. It also ensures better generalization and avoids mining overly permissive rules.\n\nJabal et al.~\\cite{jabal2020polisma} propose a novel framework for learning ABAC policies, named Polisma, combining data mining, statistical, and ML techniques. The methodology of the Polisma approach works mainly in four different stages. In the first step, a data mining technique generates a set of rules inferring associations between the users and resources. Next, the generated rules are generalized based on statistically significant attributes and context information. In the third step, the generalized rules are further augmented with `restriction rules' for limiting users' access to resources concerning the respective authorization domain. Policies learned by those three stages are `safe generalizations' (do not have unintended consequences) with limited \\emph{over-fitting}. However, there might be other access requests for which the generated policy set cannot make an access decision. To overcome this issue, Polisma applies Random Forest (RF) and KNN as the machine learning classifiers on requests not covered by the learned set of policies and uses the classification result to label these data and generate additional rules. The approach is evaluated empirically using both the real-world~\\cite{AmazonUCI2011} and synthetic datasets. Experimental results show that Polisma can develop ABAC policies that accurately control access requests.\n\nMany large-scale businesses need to grant authorizations to their users that are \\emph{distributed} across heterogeneous computing environments. In such a case, the manual development of a single access control policy framework for an entire organization is cumbersome and error-prone. Karimi et al.~\\cite{karimi2021automatic} proposed an automating ABAC policy extraction based on access logs following their other work on an unsupervised learning-based approach for ABAC policy mining~\\cite{karimi2018unsupervised}. The proposed method extracts policy rules containing positive and negative attributes and relationship filters. The negative filters are missing in existing policy mining approaches, which are essential in scenarios when an exception is expressed. The authors apply the K-modes algorithm~\\cite{cao2009new} to divide the access log into clusters based on finding similar patterns between features (i.e., attribute values) of\nthe records (i.e., access control tuples). The records in each cluster correspond to one access control rule in the system. Next, the authors extend rules extraction algorithms proposed in~\\cite{karimi2018unsupervised} to extract both positive and negative attribute filters. Rules extracted at this stage might not be accurate and concise desirably. To enhance the quality and accuracy of mined policy, the authors apply multiple iterations of the rule pruning and policy refinement process.\n\nThe empirical study based on both real-world and synthetic datasets demonstrates the ability of the proposed approach to mine high-quality policies with minimal impact on the size of\nattributes and attribute values. The proposed algorithm also shows effectiveness with an incomplete set of access logs and handles a reasonable amount of noise applicable in real-world applications. However, the proposed method is based on K-modes clustering, where finding the correct number of clusters (K) is challenging. Also, the technique has dependencies on tuning multiple parameters that equally affect the performance. As a result, the authors do not claim that the proposed method will extract the policy with the highest quality in every scenario. However, by trial and error, more randomization in cluster initialization and a more comprehensive range of parameters will help to achieve optimal performance.\n\n\n\\noindent\n\\textbf{Policy Optimization.}\nBenkaouz et al.~\\cite{benkaouz2016work} presents an approach for ABAC policies clustering and classification. The proposed approach uses KNN algorithms that help to reduce dimensionality and achieve higher flexibility for ABAC policies in high-scale systems. The core idea is to create a cluster of similar policies given a set of ABAC policies. The value of the parameter \\textit{k} (i.e., as defined for KNN algorithms, \\textit{k} represents the number of nearest neighbors) is determined based on the criticality level of the application. This adds more flexibility to the ABAC model since the granularity of the ABAC policies could be adjusted based on the value of the parameter \\textit{k}. A small \\textit{k} implies fine-grained ABAC model, and a bigger \\textit{k} implies a coarse-grained ABAC model. This is a work in progress, and several key questions remain unanswered, such as the default value of \\textit{k}, the most suitable KNN algorithms for policies clustering, and also if the same approach has any uses in different kinds of applications. \n\nEl Hadj et al.~\\cite{el2017abac} propose ABAC-PC (ABAC Policy Clustering) that groups the policy rules according to the decision effects, such as to permit or deny rules. The method also creates cluster rules based on similarity scores and produces the minimum set of rules representing each cluster. The proposed work is an extension of the Benkaouz et al.~\\cite{benkaouz2016work} method. The authors evaluate the proposed work suggesting that the reduction rate of policy rules can reach up to 10\\% for an ABAC policy with over 9000 rules. In addition, the authors claim that the approach can be extended with other policy analysis tools to detect and resolve anomalies among XACML policies. \n\n\n\\subsubsection{Role Based Access Control (RBAC)}\n\\hfill\\\\\n\n\\textbf{Role Mining.}\nThe work in \\cite{frank2008class} focuses on bottom-up approaches for RBAC role mining where the goal is to approximate the user-permission assignments by finding a minimal set of roles, user-role and role-permission assignments. Combinatorial algorithms can be used by identifying the differences from the original user-permission assignments but this often causes (1) presence of errors in the user-permission assignments and (2) identification of unmeaningful roles (e.g., roles should be meaningful to represent a particular job function within an organization). Instead, the authors propose a probabilistic framework to overcome the drawbacks of combinatorial approaches. The proposed framework works by generalizing the observations in the existing user-permission assignments. The authors use Gibbs sampler~\\cite{neal2000markov} which is a Markov chain Monte Carlo algorithm for their proposed Disjoint Decomposition Model.\n\nThe authors use synthetic and real-world datasets for evaluation and compare the performance with a combinatorial algorithm. The synthetic dataset includes 200 users, 200 permissions, ten business roles, and five technical roles, and the user-permission assignments are randomly generated. The real-world (not publicly available) dataset includes 5000 users and 1323 permissions. Since there is no way to know the errors present in the real-world dataset, the authors apply a metric of role meaningfulness to compare models. The result demonstrates that the proposed approach creates meaningful roles and identifies erroneous user-permission assignments in given data.\nOne limitation of this work is that the authors uniformly introduce user-permission errors in the synthetic dataset, whereas the number of errors in real-world scenarios is unknown.\n\n\\input{tables\/table_realworld_dataset}\n\n\\textbf{Role\/Permission Assignments.}\nThe authors in~\\cite{ni2009automating} propose a machine learning-based automated role maintenance system that provisions existing roles with entitlements from newly deployed applications and provisions new users with existing roles. The proposed technique utilizes the attributes of entitlements to simplify the management of entitlements assignments to roles. The authors solve the problem in the context of machine learning by defining a \\emph{function} that maps between the various attributes of the entitlements and business roles. The goal is to learn the function to automatically provision business roles with entitlements by running a classifier based on that function. \nThe authors accomplish the overall provisioning process in four different phases. First, it is required to collect existing role-entitlement mappings data from human experts. The collected data needs to be filtered for selecting essential attributes (features). It impacts the performance as all the features are irrelevant to the provisioning mapping. The filtered entitlements and users can belong to several roles (\\textit{many-to-many}), or there might have some missing values. Hence, as the next step, it needs to pre-process the provisioning data to replace all missing values with a new value \\textit{not applicable} and transform each \\textit{many-to-many} mapping into several \\textit{many-to-one} mappings. The classifier's training and selection of the best classifiers are performed in the third phase. The authors train multiple classifiers and compare their performance based on standard estimation methods (e.g., 10-fold cross-validation) to select the fittest classifiers for the next step. The provisioning process automatically classifies the given data to assign entitlements to roles in the final phase. \n\nTo justify the efficacy of the proposed technique, the authors used both provisioning data (with 35 business roles, 659 mappings between the UI entitlements and the business roles) from a real-world organization and synthetic provisioning data. The real-world dataset is not available publicly. The authors experimented with 18 candidate classification algorithms, including support vector machines (SVM), RF, Decision Tree, etc., before choosing SVM as the final classifier. The authors used multiple evaluation metrics such as accuracy, false positive, and false negative. The performance results of SVM for a majority of the roles demonstrate that the FP rates are between 0\\-5\\%, and most FN rates are between 0 to 30\\%. Also, 70\\% or more of the assignments can be executed automatically, and no more than 30\\% of the assignments will need some assistance.\n\nLu Zhou et al.~\\cite{zhou2019automatic} propose two machine learning-based approaches to automate the role assignment process in the context of the real-time Supervisory Control and Data Acquisition (SCADA) system. The authors first demonstrate how SVM can be applied by developing a context-aware RBAC with automated role engineering. They use different static (job function, job position, etc.) and dynamic attributes (time, location, etc.) of the SCADA system's users and devices as input to the SVM and obtain user-role or permission-role assignments as the output. The authors did not explain any detailed evaluation results for the SVM-based approach. However, the authors further experimented Adaboost algorithm for the role assignment automation process. Similar to SVM experimentation, the author used the same input (user and device attributes) to the Adaboost approach to obtain assigned roles or permission for the corresponding user. The authors use a dataset~\\cite{turnipseed2015new} designed for intrusion detection in a SCADA system. The authors compared the performance of both the real and discrete-valued Adaboost algorithms in terms of accuracy, where they demonstrated results varied significantly for different experiments.\n\n\\subsubsection{Relationship Based Access Control (ReBAC)}\nLike ABAC policy mining approaches, ReBAC policy mining algorithms can also potentially reduce the effort to obtain a high-level policy from lower-level access control data. \nThe author in~\\cite{bui2019efficient} propose an efficient and scalable ReBAC policy mining algorithm which is an extension over the existing evolutionary algorithm~\\cite{bui2019greedy} for the same problem. This enhancement simplifies the current approach, facilitating the easier adoption of new policy language features. However, the search space of candidate policies explored by the existing evolutionary algorithm increases with the extension of policy language. That makes the algorithm's overall computation inefficient as it gives either the worst results or longer running time. The authors overcome this problem by adding a \\emph{feature selection} phase to identify useful or essential features that helps to reduce the evolutionary algorithm's search space. \nDue to the flexibility and scalability of high-dimensional data and large datasets, the authors choose a neural network over other classification methods for the feature selection algorithm.\nThe algorithm maps different \\emph{feature vectors} with Boolean values to indicate the authorization for users to act an action on resources.\nEach \\textit{feature} in the vector is defined as `a subject atomic condition, resource atomic condition, or atomic constraint satisfying the user-specified limits on lengths of paths in conditions and constraints'.\n\nThe authors train a 2-layer neural network that learns the classification of \\emph{feature vectors}. The input layer contains various nodes where a node represents a feature in the feature vector. In contrast, the output layer has two different output nodes that determine the probabilities of classifying the input as deny or permit.\nA feature vector is considered as \\emph{unclassified} if there is an equal permit and deny probabilities. \nThe training is continued for a fixed time limit, or terminates early if all inputs are correctly classified and no inputs are left unclassified.\nThe weights learned by the network are then analyzed to quantify \\emph{how useful or necessary each feature is in determining the neural network's output}. Finally, the features are ordered using the score of each feature's contribution (usefulness) towards an output, and top-placed features are classified as the \\emph{useful features}.\nTo demonstrate the effectiveness of the proposed method, the authors used the two ReBAC case studies~\\cite{bui2019greedy} based on policies of real organizations~\\cite{decat2014workforce, decat2014document} and some synthetic policies.\nBased on the evaluation, the algorithms that include \\textit{feature selection} are significantly faster and achieve substantially better results than the algorithm without a feature selection stage.\n\nThe authors in~\\cite{bui2020decision} proposed DTRM (Decision Tree ReBAC Miner) and DTRM$^-$, with the ability to mine policies in any ReBAC language.\nThe decision tree-based DTRM algorithm mines policies in the recent version of Object-oriented Relationship-based Access-control Language (ORAL) that supports two set comparison operators on top of existing operators in the earlier version. The DTRM$^-$ algorithm is an extension of DTRM with the support of negative conditions and negative constraints.\nIn general, the DTRM algorithm works in two main phases. First, it learns an authorization policy in a decision tree and then extracts a set of candidate authorization rules from the tree. \nSimilar to~\\cite{bui2019efficient}, the decision tree is trained to learn the classification of feature vectors, which are also mapped with Boolean values indicating the authorization for users to act on resources. The definition of a feature is the same as defined in~\\cite{bui2019efficient}.\nAfter building the tree, the DTRM algorithm converts the trained decision tree into an equivalent set of rules and includes them in the candidate policy.\nIn the second phase, first, the algorithm determines the mined policy by optionally eliminating negative conditions and constraints from the candidate rules.\nThis process is only applicable for the DTRM$^-$ algorithm. Next, the algorithm merges and simplifies the candidate rules.\nThe authors evaluate their proposed method using datasets from real organizations~\\cite{decat2014workforce, decat2014document} and some synthetic policies~\\cite{bui2019greedy}. \nEvaluation results show that the proposed method is effective at discovering the desired rules, producing smaller policies, and significantly faster than state-of-the-art approaches~\\cite{iyer2019generalized, bui2019efficient}.\n\n\\begin{figure}[t]\n\\vspace{-2ex}\n\t\\includegraphics[width=0.6\\linewidth]{graphics\/ReBAC_Multiway_DT.png}\n\t\\caption{A multi-way decision tree with three truth values.}\n\t\\label{fig:multiway_decision_tree}\n\t\\vspace{-2ex}\n\\end{figure}\n\nIn most real-world data, information about permissions can be incomplete, or some attribute values can be missing (or unknown).\nAuthors in~\\cite{bui2018mining, cotrini2018mining, law2020fastlas, iyer2020active} solved different variants of the ABAC and ReBAC policy mining problem considering \\textit{incomplete permissions information}. However, all these works assume the attribute (and relationship in the case of ReBAC) information is complete (or known). The authors in~\\cite{bui2020learning} proposed decision tree-based algorithms, Decision-Tree ReBAC Miner with Unknown values and negation (DTRMU$^-$ and DTRMU), for mining ABAC and ReBAC policies from ACLs. The first of its kind, the algorithm can deal with `incomplete' information about entities, where the values of some attributes of some entities are unknown.\nTo handle the `unknown' values, the authors proposed to reduce the core part of the ReBAC policy mining problem to the general problem of Kleene's three-valued logic learning formula. The authors assigned another truth value, `U,' to conditions and constraints with `unknown' values along with true (T) and false (F) binary logic values. The author also discussed that avoiding a three-valued logic by regarding them as false instead is unsafe.\n\nTo deal with a concise three-valued logic formula from a set of labeled feature vectors involving unknowns, the authors develop an algorithm based on multi-way decision trees (shown in Figure~\\ref{fig:multiway_decision_tree}) as opposed to a binary decision tree. Identical to~\\cite{bui2019efficient, bui2020decision}, the decision tree learns based on a given set of \\emph{feature vectors} labeled with a permit or deny. The definition of feature is also same as~\\cite{bui2019efficient} and~\\cite{bui2020decision}.\nThe sets of features and feature vectors are extracted from an input ACL policy.\nThe learned decision tree is applied to classify authorization requests as permitted or denied and eventually create the tree's ReBAC rules (policy). \nThe authors present their algorithm in the ReBAC mining context. However, to apply in the ABAC domain, one must limit the length of considered path expressions.\nThe authors experiment with their algorithm using four sample policies developed by Bui et al.~\\cite{bui2019greedy} and two extensive case studies developed by Decat et al.~\\cite{decat2014workforce, decat2014document} based on real-world applications. They didn't report any separate performance for the decision tree. Instead, the authors compare the algorithm's overall performance by comparing mined rules with simplified original policy rules.\n\n\n\n\\subsection{Policy Verification and Testing}\n\n\\subsubsection{Policy Verification and Testing}\n\n\\begin{figure}[t]\n  \n    \\centering\n\t\\fbox{\\includegraphics[width= 0.5\\linewidth, scale = 0.6]{graphics\/code_snippet_function_no_border.png}}\n\t\\caption{The source code of a function that verifies a user as Admin.}\n\t\\label{fig:codeSnippetFunction}\n\t\\vspace{-2ex}\n\\end{figure}\n\nTraditional policy verification methods are infeasible for many systems as they are error-prone and time-consuming processes. These methods perform static or dynamic analysis on the access control system to determine the policy behavior. Such analysis techniques can not specify roles (as shown in Figure~\\ref{fig:codeSnippetFunction}) or permissions-specific code elements, find relations among these codes and associated policy elements, or tackle the case when a mapping is required for a new code and policy element. \nThe authors in~\\cite{heaps2019toward} discuss these issues and pretend a potentiality of developing a more robust and efficient system by leveraging recent advancements in deep learning. The authors propose to train a deep learning model based on the links among the code and policy elements. However, the model can not learn anything based on the input code elements since they do not have a numerical meaning. Therefore, embedding code elements and mapping the word into a high-dimensional space are required to acquire meaningful numerical information to expedite learning. The authors utilize the popular Skip-gram~\\cite{mikolov2013efficient} word embedding algorithm Word2Vec and adjust the algorithm to obtain word embedding for code elements in the source code. The authors experiment proposed technique on the Java Development Kit 8 (JDK8)\\footnote{www.oracle.com\/technetwork\/java\/javase\/overview\/index.html} and Apache Shiro\\footnote{https:\/\/shiro.apache.org\/} access control library. The evaluation results signify that the intended technique can produce high-quality word embedding and deliver state-of-the-art performance.\n\nAccess control policies are verified based on model proof, data structure, system simulation, and test oracle to ensure the policy works desirably. By default, it is assumed that the policy under verification is accurate unless the policy failed to hold for test cases. However, this comprehensive test case generation is challenging and somewhat impractical. To make the entire verification process more efficient and straightforward, in a NIST Internal Report (IR), Vincent C. Hu~\\cite{hu2021verificationNIST} propose a technique exploiting machine learning algorithm. Compared to traditional methods, the proposed method, which is feasible and efficient, can check the policy logic directly. For the verification task, a machine learning model is trained based on the values of the assigned attributes of policy rules to generate a classification model. The trained model will be used to predict the access permissions (e.g., grant, deny, etc.) assigned to various rules that allow for the detection of inconsistencies, indicating faults found among policy rules. The author applies RF as the ML method in the verification process. As part of training data preparation, it is required to encode access control policy rules in a data table. Each column contains an attribute, action, or permission value, and the row represents a policy rule. The access permission is determined based on the attribute and action values in the row. The data table contains enforceable rules only that ensure the \\textit{syntactical correctness} of a policy.\n\nBesides, any rule with more than one action or object attribute value is split into separate sub-rules to hold only one action and one object attribute value to comply with the RF tree node representing an attribute or action. After training an RF model, the evaluation is done to detect permission conflict rules and ensure that the model can recognize different policy rules semantic that include condition property, separation of duty property, and exclusion property. The analysis of the accuracy function provides the percentage of \\textit{semantic correctness} of the original policy versus the RF model. Correctness of less than 100\\% indicates that conflict (error) rules may exist in the policy. Overall, the algorithm can efficiently verify the policy and detect inconsistencies (i.e., faults) in the policy rules.\nTable~\\ref{tab:verification} summarizes\naccess control policy verification tools and methods using machine learning.\n\n\n\n\\section{Overview of Access Control}\n\\label{sec:access-control-model}\n\\subsection{Access Control Preliminaries}\n\n\nAs the cornerstone of any information system's security, access control restricts the authorization of subjects\/users to perform operations on system's resources\/objects. Different abstraction, models and mechanisms have been proposed to govern the access and operations on resources to better represent the access control state of a system at various granularity levels with better maintainability. We briefly discuss some mainstream access control approaches and models in this section.\n\n\\subsubsection{Access Control List (ACL)} \nThe ACL is the low-level representation of the access control state of a system. An ACL informs a system of a user's access privileges to system objects\/resources (e.g., a file or a directory) \\cite{sandhu1994access}. Each object has a security property that connects it to its access control list. The list has an entry for every user with access rights to the system.\nHowever, as each user and resource is managed separately based on an identifier, ACLs make it very difficult to administer.\n\n\\subsubsection{Discretionary Access Control (DAC)}\nIt is an access control model where the \\textit{Owner} or \\textit{Security Administrator} of a system sets policies defining who can access the system resources ~\\cite{sandhu1994access,6585778}. DAC is efficiently implemented using ACL or \\emph{capabilities} \\cite{thomas1993discretionary}. The security administrator defines a resource profile for each object and updates the access control list for the profile. In DAC, it is difficult to enforce the principle of least privilege and separation of duties~\\cite{downs1985issues}\n\n\\subsubsection{Mandatory Access Control (MAC)}\nIn the MAC model, the administrator makes the policy decision, which is based on the security labels of subjects (clearance) and objects (classification) \\cite{sandhu1993lattice}. The user does not control, define or change access rights. A common example of MAC is in military security, where the data owner does not decide on who has a top-secret clearance, or change the classification from top-secret to secret ~\\cite{hu2021verificationNIST}.\nIn MAC, configuration and maintenance of the system is a challenging task for the administrator, along with covert channel attacks.\n\n\n\\subsubsection{Role Based Access Control (RBAC)} In RBAC, the user's access permissions will depend on what role he\/she has in the organization~\\cite{485845,sandhu1998role,gupta2017object}. An administrator designs the role defining what permissions a role should have and which users will be assigned what roles. For example, some users may be assigned to a role which allows them to read and write a file, whereas other user may be assigned a different role, which constrains them to read the file only.\nA user can be assigned many roles, and at the same time, a role can be assigned to many users. Roles in RBAC offer easy administration of permissions, as role \\textit{clusters} the abilities a user can hold. RBAC is useful for such scenarios because, with RBAC, there is no need to change the policies whenever an employee leaves the organization or changes jobs. The administrator can remove the employee from the role or add him to the new role. Depending on his job profile, a new employee can be assigned to his role promptly. RBAC also offers static and dynamic separation of duties, with the ability to define other constraints such as mutually exclusive roles, permissions, and cardinality constraints. However, RBAC is vulnerable to role explosion problem, which can lead to leakage of permissions as the numbers of users and roles in an organization expands.  \n\n\n\\subsubsection{Attribute Based Access Control (ABAC)}\nIn ABAC, a user's access to a resource is determined based on the attributes of the user, attributes of the resource, environmental conditions, and a set of pre-defined policies \\cite{hu2015attribute,servos2017current,gupta2020secure}.\nAn ABAC policy defines the combinations of the user, resource, or environmental attributes and eventually these attributes are required to allow the user to operate on a resource.  \nWhen an access request is intercepted by the ABAC system, the system matches the request with the underlying ABAC policies, where all conditions and restrictions are defined. The access request will get the desired access if the request satisfy the policy. \nFor example, to restrict a department's resource to its `sales\\_manager' only, the ABAC policy could limit the rule so that users with `job\\_role' attribute has value `sales\\_manager' can access the respective resource. As the access policy is designed with respect to rules, ABAC is one of the most generalized, scalable, and reliable access control models~\\cite{gupta2018attribute,gupta2019dynamic}. However, there are challenges while implementing ABAC~\\cite{brossard2017systematic} as it involves multiple laborious sub-processes such as \\emph{attribute engineering}, \\emph{attributes assignment} to the user and objects, \\emph{policy engineering}, etc. Therefore, a skilled human administrator must design ABAC rules to satisfy overall security policy requirements.\n\n\\subsubsection{Relationship Based Access Control (ReBAC)}\nRelationship-based access control defines access decisions based on the \\emph{relationships} between subjects \\cite{fong2011relationship, bui2020learning, bui2019efficient} or objects\nThe most well-known examples of relationship-based access control are social networks. On Facebook, for example, the user gives view access to the photos or videos to friends or friends of friends. But friends of those friends cannot view the photos. Thus, ReBAC allows access when the user has a certain relationship with other entities in the system. \nReBAC offers more than other access control models because it grants access based on multi-relationship between entities and takes decisions for certain entities, not entity types. For example, the user can access a specific photo from the directory, not the total directory.\n\n\\subsection{ML in Access Control Domain}\n\n\\begin{figure*}\n    \\scriptsize\n    \\vspace{-2ex}\n    \\centering\n    \\input{graphics\/taxonomy}\n    \\caption{\n    A Taxonomy of Machine Learning in Access Control Domain. NL indicates Natural Language.}\n    \\label{fig:taxonomy-access-control-pipeline}\n    \\vspace{-2ex}\n\\end{figure*}\n\n\\subsubsection{Policy Mining}\nPolicy Mining is the automation process of access control policy extraction from an existing access control state. Migration to a new model manually can be an error-prone and time-consuming task. To reduce the effort and cost of this task, extracting policies from the given access control information can be partly or completely automated~\\cite{iyer2018mining}.\nFor example, if an organization had already an access control model implementation such as ACLs, and wants to migrate to more flexible, fine-grained access control model such as RBAC, ABAC, or ReBAC. In this case, an ABAC or ReBAC policy mining approach can be applied to obtain desired policies in automated and efficient manner~\\cite{xu2014mining}. In general, the output of ABAC or ReBAC mining is a set of rules~\\cite{bui2020learning}, whereas, the RBAC policy mining approaches produce permission to roles (PA) and user to roles (UA) assignments~\\cite{zhou2019automatic, ni2009automating}.\nThere are several approaches that use advancements in machine learning to automate the ABAC~\\cite{alohaly2019automated, alohaly2019towards, karimi2018unsupervised, jabal2020polisma, cotrini2018mining}, RBAC~\\cite{frank2008class, heaps2021access, zhou2019automatic}, and ReBAC~\\cite{bui2019efficient, bui2020decision, bui2020learning} policy mining process.\n\n\n\\subsubsection{Policy Verification and Testing}\nAccess control policy verification confirms no flaws within the policy that leaks or blocks access privileges. As a software test, access control policy verification depends on model proof, data structure, system simulation, and test oracle to validate that the policy logic functions are working properly. However, these methods have capability and performance concerns related to inaccuracy and complexity limited by applied technologies. For instance, model proof, test oracle, and data structure methods assume that the policy under verification is flawless unless the policy model cannot hold for test cases. Thus, the challenge of the method is to compose test cases that can systematically learn all faults. Moreover, a system simulation method needs to convert the policy to a simulated system. But, if the policy logic is complex or the number of policy rules is large, the translation between the system may be difficult or impractical \\cite{hu2021verificationNIST}.\nTo overcome these challenges machine learning approach is used for policy verification which does not require comprehensive test cases, oracle, or system translation. Rather, it checks the logic of the policy rule directly, making it more efficient and feasible compared to the traditional method~\\cite{heaps2019toward, hu2021verificationNIST}.\n\n\\subsubsection{Policy Administration and Monitoring}\n\nDesigning the policies is not an easy task and requires substantial administrative effort to modify the policy to accommodate changes. There are multiple ML-assisted methods to support additional adjustments in the existing access control policy and help detect improper behavior in the access control system.\nML based techniques can help system administrators to reduce the cost of generating the policies and adjust them dynamically~\\cite{gumma2021pammela, alkhresheh2020adaptive}.\nBesides, to accommodate changes in the system, the system admin may introduce errors and misconfiguration to the system due to the dependencies on user, data, functionality, and domain~\\cite{argento2018towards, xiang2019towards}. Despite efforts on verification and testing~\\cite{hu2021verificationNIST} to avoid access control misconfiguration, it is still difficult to avoid it in the real world system. An automated real-time monitoring system could help to observe the changes in access control behavior better and act accordingly~\\cite{xiang2019towards, martin2006inferring}.\n\n\\iffalse\n\\begin{figure}[!t]\n    \\vspace{-2ex}\n\t\\includegraphics[width=0.90\\linewidth]{graphics\/AccessControlFunctionalPoints.png}\n\t\\caption{Access Control Functional Points}\n\t\\label{fig:access_control_functionalities}\n\t\\vspace{-2ex}\n\\end{figure}\n\\todo[inline]{we did not refer Fig. 2 anywhere}\n\\fi\n\n\\subsubsection{ML for Access Control}\nThe massive policy scale and number of access control entities in open distributed information systems, such as Big Data, Internet of Things, and cloud computing, makes it impossible to list all possible rules and make access decision evaluating them. To ease the burden of implementing and maintaining access-control aspects in a system, there are multiple efforts to automate the access decision process using ML methods~\\cite{liu2021efficient, karimi2021adaptive, cappelletti2019quality, nobi2022toward}. In all such cases, the system trains an ML model based on available logs, access history information, or existing access control state. Later on, the trained ML model decides accesses.\n\\section{Summary}\n\\label{sec:conclusion}\n\nThis survey paper comprehensively explores the access control literature that uses machine learning approaches to solve various access control-related problems. We summarize that the benefits of machine learning are emerging in various areas of access control, including but not limited to attribute engineering, policy mining, access control policy verification, etc. We also discovered multiple efforts to utilize a trained machine learning model to decide access control requests as a substitute for a written access policy. In addition, we outline public real-world datasets used for machine learning-based access control research. We also propose a taxonomy of machine learning for access control and briefly discuss each approach following the proposed classification. Finally, we portray our observations regarding open challenges in the domain and provide potential guidelines to overcome them.\n\\section{Open Challenges and Future Research Directions}\n\\label{sec:open-challenges}\n\n\n\\subsection{Understanding Access Control Decisions}\nIn general, in complex situations where access control system has much overlapping, the non-symbolic ML models make better access control decisions than the traditional access control policies or symbolic ML models~\\cite{cappelletti2019quality}. A neural network or any other non-symbolic method has an innate ability to learn subtle differences among users\/resources and their relationships. However, to obtain such superior performance in these non-symbolic models, we have to compromise with \\emph{understandability}--- that is, understanding the reason for a specific access decision.\n\nEven though the use of ML models for access control decision is still in its inception, a lack of understandability could hinder the growth of this potential direction. This kind of understanding is straightforward for a written policy and not that complicated for the symbolic ML models. For example, the policy itself is written with humanly understandable logical rules, which is also applicable for the policies if an ML model generates such policies.\nIn the case of symbolic ML methods, one can easily dig down its underlying tree to extract logical rules and learn the underlying reasoning for a decision. In contrast, in non-symbolic ML model based systems, e.g., DLBAC~\\cite{nobi2022toward} or Karimi et al.~\\cite{karimi2021adaptive}, a `black box' function makes access control decision and is not in a human-understandable form which is crucial for security sensitive domains.\n\nAs illustrated in Figure~\\ref{fig:understanding}, as we go from the written policy to the non-symbolic approach, the performance goes up, though with the cost of a lack of understanding of decisions. This is one of the major limitations when choosing a non-symbolic ML model for access control. This issue is also applicable to other domains, including computer vision, malware analysis, financial systems, etc. The understanding issue, also known as interpretation or explainability in the typical ML arena, is a very active research area~\\cite{islam2022systematic}. \nIn many cases, it is observed that the solutions are domain-specific. For example, a \\emph{computer vision} specific interpretation method would not directly work in the access control domain. \nRecently, Nobi et al.~\\cite{nobi2022toward} explore this issue for the first time from access control perspectives and provide two methods for understanding neural network-based access control decisions in human terms to a large degree. \nHowever, as the proposed method does not guarantee the understanding of decisions with 100\\% accuracy, it needs to study the area further to reach a better interpretation.\n\n\n\\subsection{Administration}\nWith the development of traditional access control or ML-based approaches, maintenance becomes crucial for regular access and policy management. In the access control domain, this problem is known as \\textit{access control administration} and is crucial to keeping a system secured for the long run. In general, the administration is the task of accommodating the system's access authorization changes by modifying existing policy configurations or access control-related attributes. The administration's requirements could vary from model to model and system to system. In the case of RBAC, the administration could assign\/ remove permission to the role, create a new role, manage role hierarchy, etc.~\\cite{sandhu1999arbac99, bhatti2005x, singh2021role}. To accommodate changes in ABAC, one needs to adjust users\/ resources attributes, create new rules, update existing rules, etc.~\\cite{servos2017current, jha2016administrative}.\nSimilarly, ReBAC has relationship-related management and modifications to the policies to incorporate changes~\\cite{stoller2015administrative, cheng2016extended}. Researchers thoroughly investigated administration issues for respective models in traditional access control models. \nAs the application of machine learning for Access Control policy decisions is emerging, there are also related administration problems such as updating the corresponding ML model, adjusting access control-related information, etc. Currently, there is no approach to address administration issues from ML and access control perspectives. Therefore, it needs to explore the domain to formally define administration requirements for ML-based systems and find solutions for tackling underlying issues.\n\n\n\n\\subsection{Adversarial Attacks} The adversarial attack is a common concern for any ML-based system. An adversary can obtain unwarranted decisions~\\cite{ballet2019imperceptible} by fooling the network with adversarial samples that are indistinguishable from natural ones by human~\\cite{zheng2019distributionally}. It is an active research area in machine learning and studies this issue to design novel attacks that help determine loopholes in an ML model. There are several adversarial attacks available~\\cite{kurakin2016adversarial, carlini2017towards}, though most of them are applicable for computer vision-oriented applications. Also, there are efforts to develop prevention methods to tackle these attacks while the model is in deployment. Adversarial training is one of the most successful defense methods~\\cite{madry2017towards, hosseini2017blocking} where a model is trained with a combination of both natural and adversarial samples. \nWith the application of ML in access control, this issue is also inevitable in the domain. In the context of access control, an adversarial attack could force an ML-assisted access control system to obtain unauthorized access. The attacker could \\emph{trick} the system deliberately providing manipulated information of users and resources to prevail the desired access. Moreover, an ML-based access control system could be susceptible to the \\emph{attribute-hiding attack} where an attacker may attempt to secure access by hiding (or removing) a portion of their information. Therefore, it is crucial to investigate adversarial attacks from access control perspectives and establish solutions to protect a system against such vulnerabilities.  \n\n\n\n\n\\begin{figure}[t]\n   \n\t\\includegraphics[width=0.4\\linewidth]{graphics\/performance-decision-tradeoff.png}\n\t\\caption{Trade-off Between Performance and Understandability in ML-based Systems.}\n\t\\label{fig:understanding}\n\t\\vspace{-3ex}\n\\end{figure}\n\n\n\\subsection{Lack of `Good' Datasets}\nFor access control-related research and development, it is necessary to have a ground truth to determine the efficacy of many applications, including role mining, attribute engineering, policy mining, etc. The performance of developed methods in these areas highly depends on the availability of high-quality real-world data~\\cite{molloy2011adversaries}. This requirement is critical in the case of ML-based methods. Until now, only a few access control-related datasets are publicly available for research and experiments (Table~\\ref{tab:real-world-dataset} reports publicly available real-world datasets). Also, the quality of these available datasets is not up to the mark as the datasets are highly imbalanced~\\cite{cotrini2018mining, nobi2022toward} and contain less access control related information~\\cite{heaps2021access}, etc. Therefore, it is not feasible to design an appropriate system based on those available datasets in many cases. For proper balancing of these available real-world datasets, often, the authors rely on extensive data preprocessing~\\cite{liu2021efficient}, augmenting available datasets~\\cite{alohaly2018deep, alohaly2019towards, zhou2019automatic}, amalgamating data from various sources~\\cite{heaps2021access}, etc. Also, it is ubiquitous of generating synthetic access control datasets~\\cite{xu2014mining} and applying them for different researches~\\cite{argento2018towards, mocanu2015towards} to evaluate proposed approaches. In some cases, only real-world datasets or synthetically developed datasets are not sufficient due to the underlying issues in the datasets. To get rid of that, many researchers follow a hybrid approach by applying datasets from both publicly available real-world datasets and simulated datasets~\\cite{bui2020learning, jabal2020polisma, karimi2021automatic, nobi2022toward}.\n\nEven though we see the growing trend of the application of ML in access control, the lack of quality datasets from real-world organizations may be a critical blocking point. Many publicly available datasets are either highly anonymous or do not contain complete access control information. Besides, those real-world and simulated datasets often lack semantics and granularity of permissions~\\cite{molloy2011adversaries} and are incapable of expressing an entire access control state of a system. Therefore, to advance in ML-based access control, the availability of a high-quality dataset is essential.\n\n\\subsection{Bias and Fairness}\nVarious methods apply synthetic and real-world datasets to train respective machine learning models. The synthetic datasets are well crafted, considering different aspects of the data and access control characteristics in mind. Also, the datasets are designed to keep an equal portion of samples from all the classes (e.g., permit and deny are two classes while training a model for access decisions). However, this is not practical nor feasible for real-world datasets. For instance, the Amazon dataset~\\cite{AmazonKaggle2013} holds historical access data of two consecutive years where employees were manually allowed or denied access to resources over time. The ratio between access grant and deny is highly imbalanced, where more than 90\\% of the requests got access to the desired resources~\\cite{cotrini2018mining, nobi2022toward}. In other words, the majority of the training data are from \\emph{grant} class. Such disproportionate training data will force the trained model, irrespective of the type of machine learning model and architecture, to make \\emph{biased} decision towards the grant class~\\cite{nobi2022toward}. Therefore, to obtain a fair and reliable system, it is critical to understand the training data characteristics by auditing them before initiating the training phase~\\cite{mehrabi2019survey}. Moreover, besides evaluating a system for access control decisions, it is equally important to evaluate the \\emph{fairness} performance and establish a feedback loop~\\cite{cutillo2020machine, nobi2022toward}.\n\n\n\n\\subsection{Insufficient Tools for Verification}\nA reliable system design is usually ensured through a rigorous \\emph{testing} and \\emph{verification}. Testing is the evaluation of a system in several conditions to observe underlying behavior and detect errors. In contrast, the verification ensures that the system will not demonstrate any misbehavior under more general circumstances~\\cite{goodfellow2017challenge}. Likewise, in access control, the implemented policy is verified and tested akin to verifying the correctness of software functionality (i.e., software testing)~\\cite{hu2016ACPolicyVerification}. \nIt is required to ensure that the engineered access control rules make correct access control decisions, which is complex and needs expensive effort. Unlike many domains, failing to verify the access control system's correctness to a large extent may have serious consequences, including but not limited to over-provision, under-provision, adversarial attacks, etc. However, this area is well studied for traditional access control systems, and there are methods for proper access control verification~\\cite{kuhn2016testingKuhn, hu2021verificationNIST}. \n\nWhen the machine learning method is applied to any system, the system's performance is measured based on an unseen \\emph{test dataset} to test the learned models' correctness to some degree. Evidently, this testing method cannot find all possible cases that may be misclassified. Also, such misclassification does not have an equal impact (or cost) on every application, and it could vary from domain to domain and application to application.\nFor example, granting an access request for an unauthorized resource could be more expensive than denying a legit request in an access control context. Therefore, a comprehensive verification is crucial before deploying a machine learning-assisted access control system. There are methods to verify a machine learning model automatically~\\cite{leofante2018automated}. Also, each method has its own merits and demerits, and no one method will work for all the domains. Consequently, further research is needed to design a systematic verification and testing framework from machine learning and access control perspectives. \n\\section{Machine Learning in Access Control}\n\\label{sec:categorization}\nThis section briefly describes literature using machine learning to solve access control-related problems. We organize the whole section according to the proposed taxonomy in Figure~\\ref{fig:taxonomy-access-control-pipeline}. Before going into further details, we first present the timeline of seminal works using machine learning for access control in Figure~\\ref{fig:timeline-seminal-works}. The timeline, sorted according to the published year, clearly illustrates how the application of machine learning evolves in the access control domain. As we see in Figure, machine learning in access control is emerging fast, and researchers published most of the work in recent years. Also, we summarize all the publicly available datasets in Table~\\ref{tab:real-world-dataset}. The next section describes approaches developed for facilitating policy mining areas. The following sections discuss access control policy verification-related methods followed by policy enforcement-related works. \n\n\\begin{figure*}\n\\centering\n\\vspace{-2ex}\n\\input{graphics\/timelines}\n\\caption{\nA Timeline of Seminal Works Towards ML in Access Control. In each work, the first grey highlight indicates the access control model (`Common' implies the method is applicable for any access control model), and the second highlight denotes ML algorithms applied in the corresponding method.\n}\n\\label{fig:timeline-seminal-works}\n\\vspace{-2ex}\n\\end{figure*}\n\n\n\\input{contents\/PolicyMining}\n\\input{tables\/table_PolicyMining}\n\n\\input{contents\/PolicyVerification}\n\\input{tables\/table_PolicyVerfication}\n\n\\input{contents\/PolicyEnforcement}\n\\input{tables\/table_PolicyEnforcement}\n\n\\input{contents\/MLForAC}\n\\input{tables\/table_MLForAC}\n\\section{Overview of Machine Learning}\n\\label{sec:ml-methods-overview}\nML is a sub-domain of Artificial Intelligence that automatically improves with data and experience study.\nSupervised machine learning is a method in which we put the labeled data into an ML model~\\cite{zhao2007spectral}. The model is trained with known data to predict future outputs appropriately. Before feeding to the machine learning model, we must prepare data to enhance its quality. There are various supervised machine learning models such as Decision Tree, Random Forest, Support Vector Machine, Logistic Regression, etc. It requires a lot of training time to prepare the data. Sometimes it is difficult to label the real-time data and fill the gaps.\n\nUnsupervised learning is an ML method that uses neither classified nor labeled information~\\cite{sathya2013comparison}. The job of the ML model is to group unsorted information corresponding to similarities, patterns, and differences without any earlier training of data. There are various types of unsupervised ML, such as K-means clustering, Principal Component Analysis, and Independent Component Analysis. The unsupervised ML is computationally complex and less accurate than supervised Methods.\n\nThis section briefly introduces the common ML methods widely used in access control-related research. Based on the structure of the ML models, we discuss them splitting into two classes--- (1) symbolic approaches: which represent the learned information with highly-structured knowledge, e.g., rules, trees, etc. (2) non-symbolic approaches: a highly complex function represents the underlying ML model. In the former case, the model and its underlying rules are easily intelligible due to the structure of data in human understandable form, hence, also referred to as the ``white-box'' model. In the latter case, the models are often referred to as ``black-box'' due to the inability to understand a logic of the function or the model. We further discuss them below.\n\n\\subsection{Symbolic ML Methods}\n\\label{sec:symbolic-ml}\n\n\\subsubsection{Decision Tree (DT)}\nDecision trees are one of the powerful methods normally applied in several fields, such as machine learning, image processing, and identification of patterns \\cite{charbuty2021classification, geurts2006extremely}. A decision tree is a tree-based method in which each path beginning from the root indicates a sequence of data splitting until a Boolean outcome is reached at the leaf node. Each path in the decision tree is a decision rule that reasons the underlying decision of input. In contrast, the entire tree corresponds to a compound Boolean expression which involves conjunctions and disjunctions to get Boolean decisions~\\cite{yang2019extended}.\nThough the decision tree is simple to comprehend, there are a few limitations such as lower accuracy, less generalization, etc.\n\n\n\\subsubsection{Random Forest (RF)}\nRF is a supervised form of learning that can deal with classification and regression problems. It is an effective decision tree ensemble used for large-scale and multivariate pattern recognition~\\cite{breiman2001random}. This collective learning is determined based on the idea of the random subspace method and the stochastic discrimination method of classification. \nRandom Forests struggle with multi-valued attributes in several dimensions. One of the common issues in RF is over-fitting, mainly in regression tasks. \n\n\\subsubsection{K-Nearest Neighbours (KNN)}\nK-nearest neighbors (KNN) algorithm is a supervised ML algorithm that can be applied for both classification and regression predictive problems~\\cite{djenouri2019adapted,xin2018machine}. K-nearest neighbors (KNN) algorithm utilizes `feature similarity' to estimate the values of new data points, which shows that the new data point will be assigned a value based on how closely it matches the points in the training set. \nThough the KNN algorithm is easy to implement, it has some disadvantages. In a large dataset, the cost of calculating the distance between the two-point is huge, which diminishes the performance.\n\n\\subsubsection{K-modes and K-means}\nK-Means clustering is an unsupervised learning algorithm. K-Means construct the clusters the objects that share similarities and dissimilar objects that belong to another cluster. The K-means algorithm represents the number of clusters represented by ``K'' in the K-means algorithm. In K-Means clustering, the data points are allocated to a cluster to minimize the sum of the square distance between the data points and the centroid. That is, a small variation within the clusters indicates more similar data points within the same cluster.\nK-means algorithm requires prespecify the number of clusters. K means performing well for spherical shape. The algorithm is ineffective in identifying clusters with curving or complex geometric shapes. Because the K means algorithm does not allow the data points that are far from each other to share a similar cluster, even though they belong to one cluster~\\cite{likas2003global}.\n\nK-mode is an extension of k-means algorithm~\\cite{huang1998extensions}. The k-means algorithm is\nsuitable for numerical data and does not perform well for categorical data because of the inadequate spatial representation. But the K-mode algorithm performs well in this situation \\cite{huang1999fuzzy}.\n\n\\subsubsection{APRIORI-SD (Apriori Subgroup Discovery)}\nThe \\emph{rule learning} algorithms~\\cite{cohen1995fast} are designed to create classification and prediction rules. A \\emph{subgroup} could be considered as most interesting if the group is as large as possible and\nhave the most unusual statistical (distributional) characteristics with respect\nto a property of interest.\nThe APRIORI-SD ~\\cite{kavvsek2006apriori} method adapts the rule learning to subgroup discovery~\\cite{wrobel1997algorithm}. The APRIORI-SD finds population subgroups that are statistically \\emph{most interesting}.\nThe algorithm works in three different phases to classify a set of samples. First, it computes a set of rules covering a certain number of samples in each rule. Then, the algorithm measures the \\emph{confidence} for each rule and opt-out any rule if the measured confidence is below a pre-determined threshold. Finally, the algorithm determines a subset of filtered rules in the second phase based on the highest \\emph{weighted relative accuracy}. Overall, the APRIORI-SD produces rules that are well generalized and smaller in size~\\cite{cotrini2018mining, kavvsek2006apriori}.\n\n\\subsubsection{Gradient Boosting Method (GB)}\nThe Gradient Boosting technique is used for Regression and classification problems. This technique predicts in the form of a collection of weak prediction models \\cite{chen2015xgboost}. Like any other supervised method, the GB method utilizes a feature vector to predict an output variable through a probability distribution. Thus a labeled data set is required, and the goal is to find an approximation function in the form of a weighted sum of functions from various classes of weak learners (the decision tree with fewer branches)~\\cite{friedman2001greedy}. Though it is an important method to solve many classifications and regression problems, it is vulnerable to over-fitting issues.\n\n\n\\subsection{Non-symbolic ML Methods}\n\\label{sec:non-symbolic-ml}\n\n\\subsubsection{Support Vector Machine (SVM)}\nSVM algorithm is one of the supervised machine learning algorithms supported by statistical learning theory~\\cite{abdullah2021machine}. It chooses from the training samples a group of characteristic subsets, so the classification of the character subset is adequate for the division of the whole dataset~\\cite{chang2011libsvm}. The SVM has been applied to resolve various classification problems effectively in many applications. Examples include intrusion detection, classification of face expression, prediction of your time series, speech recognition, image recognition, signal processing, detection of genes, text classification, recognition of fonts, diagnosis of faults, qualitative analysis, recognition of images, and other fields. However, for large-scale data processing in SVM, the time complexity and space complexity increase linearly with the rise in data. Thus, SVM is more capable of solving smaller samples, like nonlinearity and high dimensional problems.\n\n\\subsubsection{Restricted Boltzmann Machine (RBM)}\nA restricted Boltzmann machine is a two-layer undirected graphical model where the first layer contains observed data variables (or visible units), and the second layer contains latent variables (or hidden units)~\\cite{zhang2018overview}. The visible layer is fully connected to the hidden layer via pair-wise potentials, while the visible and hidden layers are confined to having no within-layer connections.\nRBM is utilized for feature selection and extraction to solve many other tasks like dimensionality reduction, classification, and regression.\n\n\n\\subsubsection{Neural Network and Multi-Layer Perceptron (MLP)}\nA neural network consists of neurons, input layers, hidden layers, and output layers \\cite{carlini2017towards}. The neurons connect and have an associated weight and threshold. If the calculated result of any individual layer exceeds the threshold value, it can pass the value to the next layer. Otherwise, no data is passed to the next layer \\cite{leofante2018automated}.\n\nMLP is a neural network where the mapping between input and output is non-linear ~\\cite{verma2020machine,hasan2020diabetes}. A multi-neuron perceptron consists of input and output layers and one or more hidden layers stacked together. MLP is useful for studying complex non-linear problems and large data input. However, the functional advantages of the model highly correlated with the quality of the model training.\n\n\\subsubsection{Convolutional Neural Network (CNN)}\nIn deep learning, a convolutional neural network (CNN\/ConvNet) is a class of deep neural networks~\\cite{o2015introduction}. In CNN, mostly the inputs are images, which allows us to determine specific properties in the architecture to distinguish certain patterns in the images. The CNNs utilize the spatial nature of the data to identify the objects, like the human perception of several objects in nature. A CNN consists of three layers: convolutional layers, pooling layers, and fully connected layers. When these layers are assembled, a CNN architecture has been formed.\n\\subsubsection{Recurrent Neural Network (RNN)}\nRNN is a type of neural network that remember the data sequence and utilize data patterns for the prediction \\cite{sherstinsky2020fundamentals}. \nRNN utilizes feedback loops that are different than other neural networks. With these loops, RNN process the sequence of the data. This loop collects the data to be shared from various nodes and give predictions. Recurrent neural networks are used to solve the machine learning problems such as speech recognition, language modeling, and image classification.\n\n\n\\subsubsection{Reinforcement Learning (RL)}\nRL indicates a set of algorithms that train an agent to make a sequence of decisions through an interaction with an unknown environment to attain a goal (i.e., to increase the estimated collective discounted reward) \\cite{karimi2021adaptive}. At each time, the agent examines the recent state. The agent takes action according to a policy. Following the action, the agent receives a reward signal. The agent's goal is to attain a policy that maximizes the expected reward, which is the sum of future discounted rewards.\nThough Reinforcement learning is a useful approach to solving many real-world problems, too much reinforcement can overload states which can diminish the results.\n\n\\subsubsection{Residual Neural Network (ResNet)}\nOne of the common challenges of deeper neural networks is the training difficulty. ResNet overcomes this challenge of training for substantially deeper networks by explicitly reformulating the layers~\\cite{he2016deep}. It allows for training extremely deep neural networks with more than 150 layers.\nA typical ResNet network is formed with stacks of multiple residual blocks followed by a Global Average Pooling layer. Each block is built on multiple convolutions whose output is added to the residual block's input and fed to the next layer. \nResNet uses a skip connection to add the output from an earlier layer to a later layer.\n\n\\subsubsection{Bidirectional LSTM (BiLSTM)}\nA BiLSTM is a structural processing model that contains two LSTMs: one getting the input in a forward direction and the other in a backward direction\\cite{liu2016learning}. BiLSTMs efficiently enhance the quantity of information available to the network, increasing the context available to the algorithm \\cite{huang2015bidirectional}.\nBi-LSTM is usually needed where the sequence-to-sequence tasks are required. A Bi-LSTM method is also effective in text classification, speech recognition, and forecasting models.\n\n\\subsubsection{Transformers}\nA transformer is a process of transforming one sequence into another with the help of an encoder and decoder. BERT (Bidirectional Encoder Representation From Transformer), a notable NLP model, consists of several transformers encode layers. With more encode layers, it performs better in several language understanding task~\\cite{devlin2018bert}.\n\n\\subsection{ML Evaluation Metrics}\nIt is the way to evaluate the performance of a classification problem where the output can be one of two possible classes, that is grant or deny. The access control element may have real and predicted results and both the elements have cases like True Positive (TP), True Negative (TN), False Positive (FP), False Negative (FN). From access control decision-making perspective, we can define these metrics as follows:\n\\begin{itemize}\n    \\item TP: The number of access requests that are correctly authorized by ML model\/ policy.\n    \\item TN: The number of access requests that are correctly denied by ML model\/ policy.\n    \\item FP: The number of access requests that are incorrectly authorized by ML model\/ policy.\n    \\item FN: The number of access requests that are incorrectly denied by ML model\/ policy.\n\\end{itemize}\n\\subsubsection{Accuracy}\nThe accuracy is defined as the number of correct predictions made by ML model. The accuracy can be calculated by the following formula:\\[\nAccuracy = \\frac {TP+TN} {TP+FP+TN+FN}\n\\]\n\\subsubsection{Precision}\nPrecision represents the number of correctly permitted samples to the total number of predicted permitted samples. It is calculated by the formula:\\[\nPrecision = \\frac{TP} {TP+FP}\n\\]\n\\subsubsection{Recall or True Positive Rate (TPR)}\nTPR is the number of correctly permitted sample to the total number of real permitted sample. Recall is calculated by the following formula:\\[\nTPR = \\frac{TP} {TP+FN}\n\\]\n\\subsubsection{False Positive Rate (FPR)}\nFPR is the number of wrongly permitted sample to the total number of wrongly permitted sample. FPR is calculated by the formula: \\[\nFPR = \\frac{FP} {FP+TN}\n\\]\n\\subsubsection{F1 score}\nF1 score is the weighted average of precision and Recall.\n\\[\nF1=\\frac {2\\ast precision \\ast Recall} {Precision+Recall}\n\\]\n\n\\noindent Policies (or models) with a higher F1 score lead to better generalization~\\cite{cotrini2018mining}. Also, the higher TPR and Precision means how correctly and efficiently the policies (or models) can grant access. On the other hand, the policies (or models) with a lower FPR are better as they are less likely to give access to requests, which should be denied according to the ground truth policy.\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\\setcounter{equation}{0}\n\\setcounter{footnote}{0}\n\n\nThe Standard Model (SM) of electroweak interactions has withstood every recent experimental attempt\nat  directly or indirectly detecting new physics.\nOn the other hand, it is widely believed that it is not the final theory but rather an effective theory valid up to a cutoff scale. Ultimately one would like to be able to find a theory where the cutoff scale is \ngiven by the Planck mass $M_{Pl} \\sim 10^{19}$ GeV. The reason why the cutoff in the SM can not be very high is \nthat loop corrections to the Higgs mass are quadratically sensitive to the cutoff of the theory. Thus new physics is expected to enter at a scale not too far from the weak scale to shield the Higgs mass \nfrom  this sensitivity to the physics at the Planck scale. This is usually referred to as the (big) hierarchy problem.\nA qualitatively different tension arises when one takes into account the precision measurements that suggest a light Higgs boson, below $280$ GeV:\ndue to the hierarchy problem discussed above this would imply a low scale for the new physics around $1$ TeV.\nThis limit comes mainly from the top loop contributions to the Higgs mass, as the top Yukawa is the largest coupling to the Higgs. On the other hand, if one includes higher dimensional operators (which would presumably be generated\nin a theory beyond the SM) in the analysis\nof electroweak precisions observables~\\cite{littlehierarchy}, it turns out that the scale suppressing such operators, that is the new physics scale, has to be larger than $5-10$ TeV.\nThis factor of $5-10$ mismatch is the little hierarchy problem: it requires a few \\% fine tuning in the new physics contribution to the Higgs mass.\nThus, the precision measurements, mainly performed at LEP~\\cite{LEP}, are the main challenge for the realization of realistic models of New Physics beyond the SM~\\cite{LEP2}.\n\n\nThe most acknowledged paradigm addressing these problems is supersymmetry: due to the presence of partners of any SM particle with different spin, the dangerous quadratic dependence on the cutoff is tamed to a mild logarithmic dependence.\nHowever, a natural realization of this paradigm would require superpartners at the weak scale, more precisely around the mass of the weak gauge bosons.\nThe fact that superpartners have not been discovered at LEP and the Tevatron \nhas pushed up the viable scale of supersymmetry breaking.\nIn a minimal realization (the MSSM), this reintroduces the fine tuning problem, especially in the stop mass term\nwhich is  responsible both for EWSB and the taming of the top divergences.\n\nThe Large Hadron Collider (LHC), which is expected to turn on in just two years, is expected  to shed some\nexperimental light on the problem. This adds a great motivation to look for alternative models that naturally\nprotect the Higgs mass.\nIndeed a plethora of new mechanisms have been proposed, among them gauge extensions of the MSSM~\\cite{gaugeext},\nLittle Higgses~\\cite{littlehiggs}, fat Higgses~\\cite{fathiggs}, and many more.\nThe recent realization~\\cite{antoniadis} that extra dimensions could play a role in low energy physics and are\nnot necessarily only relevant at the Planck scale, has opened a Pandora box of new possibilities.\nAn incomplete list of such models (relevant to electroweak symmetry breaking)\nincludes the Randall-Sundrum model~\\cite{RS},\nextra dimensional supersymmetry~\\cite{BHN}, composite Higgs in warped space~\\cite{contino1,contino2}, and Higgsless models~\\cite{higgsless}.\n\n\nA very attractive idea utilizing extra dimensions is called\nGauge-Higgs unification, and was first discussed in~\\cite{gaugehiggs}, and then\ndeveloped by several authors both in 5~\\cite{5D,SSS,twisted} and 6\ndimensions~\\cite{ABQ,CGM,6D-1,tadpoles}. In a nutshell, the idea is\nto identify the Higgs as the component along the extra dimensions of\na gauge field. The residual gauge invariance after the orbifold\nbreaking will impose a shift symmetry on the Higgs. The potential\nyielding electroweak symmetry breaking is then radiatively generated\nand gauge invariance itself, embedded in the extra dimensional\nbackground, ensures the insensitivity of the Higgs mass and the\nelectroweak scale on the cutoff. In 6D, however, possible tadpoles\ncan be generated on the orbifold fixed points~\\cite{CGM,tadpoles},\nreintroducing the cutoff dependence on the Higgs mass. While this\nmechanism offers great simplicity and elegance, any effort to build\na realistic model up to date has been unfruitful. The main problems\nare the lightness of the Higgs and of the top quark. Regarding the\ntop, the Yukawas are generated via the gauge coupling itself, so it\nis generically hard to engineer a Yukawa of order 1 from a small\ngauge coupling. Regarding the Higgs mass, it turns out to be too\nsmall, below the value currently excluded by LEP, because the\nquartic scalar interaction term is generated at one loop. Since the\nentire potential (mass and quartic) is loop generated, the potential\nwill also generically prefer large values of the Higgs vacuum expectation value (VEV) relative\nto the compactification scale so that the scale of new physics stays\ndangerously low.\nIt is interesting to note that a deconstructed version of this mechanism~\\cite{dec} led to the idea of Little Higgs models.\nThe symmetry protecting the Higgs mass is now a discrete shift symmetry, and the construction is much less constrained by the absence of 5D Lorentz invariance.\nIn Little Higgs models, this idea has been pushed further: in this case the symmetry is protecting the Higgs mass at one loop, but allows a quartic coupling at tree level~\\cite{littlehiggs}.\n\n\nThe simplest possibility is to extend the weak gauge symmetry to SU(3)$_w$~\\cite{ABQ}, the smallest group\nthat allows to embed the Higgs in an adjoint representation together\nwith the SM gauge bosons. The SU(3)$_w$ is broken to the SM gauge\ngroup SU(2)$_L \\times$U(1)$_Y$ by an orbifold projection. Such model, in one flat extra dimension,\nhas been extensively studied in~\\cite{SSS}: in their numerical study\nthe authors indeed found typical values $m_H < 20 $ GeV and $m_t <\n100$ GeV. An interesting possibility to improve the situation is to\nwarp the extra dimension~\\cite{contino2}: this allows to raise both\nthe Higgs and top mass. However, in such background, the Higgs VEV\ndistorts the W and Z wave functions and generates corrections to the\ncouplings with fermions. Thus, one has to worry about tree level\noblique corrections, in particular protecting the $\\rho$ parameter\nrequires the inclusion of a custodial SU(2)$_R$ symmetry in the\nbulk~\\cite{custodial}. Another crucial parameter is the coupling of\nthe left-handed bottom to the Z: forcing a large top mass will\ninduce large corrections, due to the fact that the b$_l$ is part of\nthe same weak doublet as the top~\\cite{Zbb}. \nFrom this point of view, the flat case offers a great advantage.\nIndeed, the flatness of the Higgs VEV does not induce any tree level mixing between the KK modes, due to the orthogonality of the wave functions.\nThe zero modes will also generically have flat wave functions, thus forbidding couplings with one KK mode alone: this property is similar to the T-parity introduced in Little Higgs models~\\cite{Tparity}.\n\n\nIn this paper we will focus on a flat model, in particular the toy model analyzed in~\\cite{SSS}, where such tree level corrections are avoided, thanks to the flatness of the $W$ and $Z$ wave functions.\nWe show two mechanisms that allow to get a heavy top and a heavy Higgs mass in a minimal scenario.\nThe fermion masses are generated by the mixing of localized degrees of freedom that couple to a massive bulk\nfield~\\cite{CGM}.\nA minimal model would require at least one bulk field for each SM fermion.\nWe have re-analyzed the spectrum of the top quark tower, taking into account the\neffects of the localized mixings exactly.\nOur numerical analysis confirms the results in~\\cite{SSS}: the effective Yukawas are exponentially suppressed by the\nbulk masses of the bulk fermions or by small localized mixings.\nHowever, in the limit of vanishing bulk mass the light mode develops a mass of the order of the $W$ mass.\nThis limit corresponds to the case when the SM top is mostly a bulk degree of freedom.\nAn enhancement factor can be added if one considers larger representations: as a drawback adding large representations will also lower the scale where the theory enters a strong coupling regime, endangering the stability of the calculation.\nWe find that a relatively small representation may explain a factor of two between the top and the $W$ and still keep the theory under control. \nHowever, a more careful analysis of this issue is necessary.\nIn order to get rid of unwanted extra light modes, one can either add localized degrees of freedom on the brane and couple them with the zero modes via localized mass terms, or twist the fermionic boundary conditions and then introduce the SM fermions on the brane.\n\nThe second problem is the Higgs mass.\nIn~\\cite{SSS}, the Higgs is found to be too light, however only the contribution of the top quark tower is taken into account in calculating the Higgs potential.\nThe rationale behind this is the following: since all the other SM fermions are much lighter than the top,\none expects the corresponding bulk fermions to be heavier in order to naturally generate the hierarchies in the\nfermion spectrum.\nHowever, if one gives up this motivation and simply assumes that the hierarchies are generated by smaller mixings,\nother bulk fermions, that couple to the bottom, tau or light generations, can give a non-negligible contribution to\nthe Higgs potential.\nWe identify a set of scenarios where these contributions show cancellations in the potential, achieving two main goals: first of all the Higgs VEV in units of 1\/$R$ is much smaller, thus allowing the resonance scale to be much heavier than the W mass, and the Higgs mass receives additional contributions that push its value above the experimental bound.\nIt is important to notice that these cancellations are not to be considered a fine tuning.\nIndeed, they are generated by the contribution of different representations of the SU(3)$_w$ group and will not be spoiled by a continuous variation of the parameters involved.\nMoreover, it is important for us that the model is still minimal, in the sense that we do not add any bulk ``spectator'' field with the sole purpose of raising the Higgs mass: each fermion we take into account has to be introduced in the theory anyway in order to generate the masses of light fermions.\nAn important ingredient for the model to work is the introduction of antiperiodic bulk fermions (twisted boundary conditions): the twisting flips the sign of the fermion contribution to the Higgs mass~\\cite{twisted}, that in 4D is always negative, and this flipped sign is the main source for the cancellations in the potential.\nHowever, in a fully realistic theory there will still be contributions to the $\\rho$ parameter and to the coupling of the $Z$ boson with the bottom quark.\n$\\Delta \\rho$ is due to the fact that we need to introduce an extra U(1) to obtain the correct $\\sin^2 \\theta_w$ and to get the correct quark hypercharges.\nOnce the extra U(1) is broken by boundary terms, the $Z$ will become a mixture of the $A^3$ and $A^8$ fields from SU(3)$_w$ and the extra U(1) field, with a non-flat wave function.\nThis distortion also generates a correction to $Z b \\bar b$.\nAnother correction to the couplings of the bottom arises due to the presence of triplets in the representation containing the left-handed top (and bottom).\nThe orbifold projection will leave zero modes for those triplets, which have to be removed via localized mixing terms.\nThese triplet zero modes will couple to the quark doublet via the Higgs VEV.\nIn the end, a fine tuning comparable to that of the MSSM (few \\% level) will still be required.\n\n\n\nThe paper is organized as follows: after a brief introduction of the\ntoy model in Sec.~\\ref{sec:model}, we exhaustively discuss all the\npossible ways of generating fermion masses in\nSec.~\\ref{sec:fermmass}. In Sec.~\\ref{sec:5Dmass} we illustrate how\none can avoid a light Higgs and large VEV in the toy model\nof~\\cite{SSS}. Finally, in Sec.~\\ref{sec:topmass} we analyze how to\ngenerate a realistic top mass using a large representation, and how\nthe same mechanism in the previous section can raise the Higgs mass. \nFinally we briefly discuss the bounds from precision measurements in Sec.~\\ref{sec:EWPT}, before the conclusions and outlook in Sec.~\\ref{sec:concl}.\n\n\n\n\\section{A toy SU(3)$_w$ model}\n\\label{sec:model}\n\\setcounter{equation}{0}\n\\setcounter{footnote}{0}\n\n\nIn this section we will briefly review the toy model studied in~\\cite{SSS}.\nThe gauge group is SU(3)$_c \\times$SU(3)$_w$ on an $S^1\/Z_2$ orbifold: the enhanced weak symmetry allows the\nunification of the SM gauge bosons and the Higgs doublet.\nIn fact, the adjoint of SU(3) decomposes into ({\\bf 3}, 0) + ({\\bf 2}, 1\/2) + ({\\bf 2}, -1\/2) + ({\\bf 1}, 0).\nThe orbifold breaks SU(3)$_w$ to SU(2)$_L \\times$U(1)$_w$ via the projection matrix:\n\n\\begin{equation}\nP = \\left( \\begin{array}{ccc}\n-1 & 0 & 0\\\\\n0 & -1 & 0\\\\\n0 & 0 & 1\n\\end{array} \\right)\\,,\n\\end{equation}\nwhere the gauge fields transform as $A_\\mu \\rightarrow P A_\\mu P^\\dagger$ and $A_5 \\rightarrow - P A_5 P^\\dagger$.\nWith this choice, only the SM gauge fields have a zero mode.\nIn the scalar sector, the zero mode is a single complex SU(2)\ndoublet with the correct quantum numbers to play the role of a Higgs:\n\n\\begin{equation}\nA_5 = \\frac{1}{\\sqrt{2}} \\left( \\begin{array}{cc}\n- &  H_5 \\\\\n H_5^\\dagger & -\n\\end{array} \\right)\\,.\n\\end{equation}\nNote also that all the massive modes in $A_5$ are eaten by the massive KK modes of the gauge bosons, and play the role of the longitudinal degrees of freedom, like in the usual Higgs mechanism: the only physical scalar left in the spectrum is the zero mode.\nThe linearized gauge transformations in the bulk are:\n\n\\begin{equation}\n\\begin{array}{lcr}\nA_\\mu & \\rightarrow & A_\\mu + \\partial_\\mu \\lambda (x, x_5) + i [ \\lambda (x,x_5), A_\\mu ]\\,, \\\\\nA_5 & \\rightarrow & A_5 + \\partial_5 \\lambda (x, x_5) + i [ \\lambda (x,x_5), A_5 ]\\,.\n\\end{array} \\end{equation}\nOn the branes, $\\lambda = 0$ for the broken generators, however the gauge transformation will still impose on $A_5$ a shift coming from $\\partial_5 \\lambda$.\nThis is enough to forbid a tree level potential for $A_5$, also on the fixed points, and only loop contributions will generate a potential for the Higgs, that will be non-local from the 5D point of view, and finite.\nWe will assume for the moment that the potential does induce a VEV for the Higgs: we can use SU(2) transformations to align the VEV, analogously to the SM case, and parametrize it\n\n\\begin{equation}\n\\langle H_5 \\rangle = \\sqrt{2}  \\left( \\begin{array}{c}\n0\\\\\n\\alpha\/R \\end{array} \\right)\\,.\n\\end{equation}\nIt is now straightforward to compute the spectrum of the gauge bosons: we find\n\n\\begin{equation} \\label{spectrum}\nM_{W n} = \\frac{n+\\alpha}{R}\\,, \\quad M_{Z n} = \\frac{n + 2 \\alpha}{R}\\,, \\quad M_{\\gamma n} = \\frac{n}{R}\\,,\n\\end{equation}\nwhere $n \\in Z$, and we want to identify the lightest state in each tower with the SM gauge bosons, the photon, the $W$ and the $Z$.\nLet us first point out that the spectrum is invariant if we shift $\\alpha$ by an integer, and if we change its sign.\nIn other words, the physical range for $\\alpha$ is $[0,1\/2]$ and all other vacua outside this range are equivalent, as the radiatively induced potential will respect the same symmetries.\nAnother important feature is that $M_Z$ turns out to be twice the $W$ mass: this is a consequence of the gauge group SU(3) that predicts $\\theta_W = \\pi\/3$.\nOne possible way to fix it is to add localized gauge kinetic terms: SU(3) being broken on the boundaries, such terms can be different for the SU(2) and U(1) and, if large enough, can dominate and fix the correct value of $\\sin \\theta_W$.\nHowever, this scenario is equivalent to a warped extra dimension: integrating out a slice of the warped space near the Planck brane, where the warping is small, will mimic the localized kinetic terms, while the remaining space will be almost flat.\nWe will not pursue this\ndirection, as it has been already discussed in the literature~\\cite{contino2}. \nMoreover, it suffers from tree level corrections to the precision observables~\\cite{Zbb} from the mixing of the zero modes with the KK modes as a consequence of the non-flat profile of the Higgs.\nAnother possibility is to extend the gauge group with an extra U(1)$_X$.\nIn this case, if the bulk fermions are charged, only the combination of the two U(1)'s proportional to the hypercharge is anomaly free, and the orthogonal gauge boson will develop a mass~\\cite{ABQ}.\nAlternatively, one can use boundary conditions to break U(1)$_w \\times$U(1)$_X \\to$ U(1)$_Y$,\nfor instance by twisting the BC on one of the two branes, such that no zero mode is left in the scalar sector.\nIn both cases, the breaking is due to localized terms: as a result the wave function of the $Z$ is distorted, introducing corrections to the $\\rho$ parameter and $Z b \\bar b$.\nFinally, it might be possible to achieve the correct weak mixing angle starting from a different gauge group and using more complicated orbifold projections, thus without introducing distortions in the zero mode wave functions: this possibility has not been exaustively explored yet.\nNevertheless, the details of this mechanism will not affect the main results of this paper, so in the following we will assume the presence of the extra U(1).\n\n\n\\subsection{Bulk fermions}\n\n\nThe next problem is how to generate a mass for the SM matter fields.\nIf we added bulk fermions, with chiral zero modes thanks to the orbifold projection, the Higgs VEV would generate a\nspectrum similar to that in (\\ref{spectrum}): all the light modes would have masses larger than the $W$ mass, where the exact relation depends on group theory factors arising from the fermion representations.\nIndeed, gauge invariance forces the Higgs to couple to bulk fields and with strength determined by the 5D gauge coupling $g_5$.\nThere are two possible solutions: one is to include odd masses for these fermions, that will localize the zero modes towards the two fixed points.\nAs modes with different chirality will be localized towards different points, this mechanism will reduce the overlaps between the wave functions, and generate hierarchies between the various Yukawa couplings.\nAnother possibility, adopted in~\\cite{CGM,SSS} is to localize the SM fermions on the fixed points, and then mix them with massive bulk fields that will induce an effective Yukawa coupling {\\it a la} Froggatt-Nielsen.\nIn the following we will focus on the latter possibility.\n\nThe most general conditions on a bulk fermion in a representation $\\mathcal{R}$ are:\n\n\\begin{equation}\n\\psi (-y) = \\eta \\mathcal{R} (P) \\psi (y)\\,, \\qquad \\psi (2 \\pi R+y) = \\eta' \\psi (y)\\,.\n\\end{equation}\nThe presence of an extra parity $\\eta'$ only means that we will allow for antiperiodic fermions: let us first discuss the case of periodic fermions, the case considered in detail in~\\cite{SSS}.\nIn this case, the orbifold projection will leave chiral bulk zero modes: in order to get rid of them, for each fermion $\\Psi$, we add a second bulk fermion $\\tilde \\Psi$ with the same quantum numbers but opposite parity, so that we can write down an invariant bulk mass $M$ for them.\nNow, the localized fermions can mix with the even components that do not have vanishing wave functions on the fixed points.\nIn Table~\\ref{tab:bulkf}, we listed the bulk fields, with their parities $\\eta$ and SU(3)$_c \\times$SU(2)$_L \\times$U(1)$_Y$ decomposition, that contain components with the same quantum numbers as the SM fields.\nOf course, in the presence of an extra U(1)$_X$, the extra charge can be adjusted to fit the hypercharge, and the choice of representations is much less constrained.\n\n\\begin{table}[t]\n\\centering\n\\begin{tabular}{|c|c|c|l|}\n\\hline\nSM particle & SU(3)$_c \\times$SU(3)$_w$ & $\\eta$ & SU(3)$_c \\times$SU(2)$_L \\times$U(1)$_Y$ \\\\\n\\hline\ndown & ({\\bf 3},{\\bf 3}) & + & ({\\bf 3},{\\bf 1},-1\/3) + ({\\bf 3},{\\bf 2},1\/6) \\\\\nup & ({\\bf 3}, $\\bf \\bar 6$) & + & ({\\bf 3},{\\bf 1},2\/3) + ({\\bf 3},{\\bf 2},1\/6) + ({\\bf 3},{\\bf 3},-1\/3)  \\\\\nlepton & ({\\bf 1},{\\bf 10}) & + & ({\\bf 1},{\\bf 1},-1) + ({\\bf 1},{\\bf 2},-1\/2) + ({\\bf 1},{\\bf 3},0) + ({\\bf 1},{\\bf 4},1\/2)  \\\\\n\\hline\n\\end{tabular}\n\\caption{List of the bulk fermions that mix with the localized SM fermions.}\n\\label{tab:bulkf}\n\\end{table}\n\nAntiperiodic fermions are equivalent to fermions with different parities on the two fixed points, that we will call ``twisted'', there is no massless zero mode and the KK masses are given by $m_n = (n+1\/2)\/R$.\nIn general, we can also add a partner $\\tilde \\Psi$ and a bulk mass, as in the previous case: such term could be of phenomenological interest, as we will see later.\nSuch twisted fermions can also account for the SM field masses.\nIndeed, there will be components that do not vanish on the fixed points and can mix with localized fields.\n\n\n\\subsection{Higgs potential from the bulk fields}\n\n\nWe now discuss the contribution to the Higgs potential from these bulk fields.\nTheir spectrum, as a function of the Higgs VEV, generically takes the form:\n\n\\begin{equation}\nm_n^2 = \\frac{(n + \\beta)^2}{R^2}\\,, \\quad n  \\in Z\\,,\n\\end{equation}\nwhere the parameter $\\beta$ is proportional to the Higgs VEV $\\alpha$ via an integer, that is determined by the representation of the field.\nWe can use the Higgs-dependent spectrum to compute the full one-loop potential, using the Coleman-Weinberg formula: after summing over the KK modes~\\cite{ABQ}, we find\n\n\\begin{equation} \\label{eq:potgen}\nV_{eff} (\\beta) =  \\frac{\\mp 1}{32 \\pi^2} \\frac{1}{(\\pi R)^4} \\mathcal{F} (\\beta)\\,,\n\\end{equation}\nwhere the signs stand for bosons\/fermions and\n\n\\begin{equation}\n\\mathcal{F} (\\beta) = \\frac{3}{2} \\sum_{n = 1}^{\\infty} \\frac{\\cos (2 \\pi \\beta n)}{n^5} = \\frac{3}{2} \\mathcal{R}e \\left[ \\mbox{Li}_5 (e^{2\\pi \\beta i}) \\right]\\,,\n\\end{equation}\nwhere Li$_k$ is the polylogarithmic function of order $k$.\n\nIn the presence of a bulk mass, as for example for the fermions described above, the spectrum is shifted to\n\n\\begin{equation}\nm_n^2 = M^2 + \\frac{(n + \\beta)^2}{R^2}\\,, \\quad n \\in Z\\,,\n\\end{equation}\nand the effective potential becomes\n\n\\begin{multline}\n\\mathcal{F}_\\kappa (\\beta) = \\frac{3}{2} \\sum_{n=1}^{\\infty} \\frac{e^{- \\kappa n} \\cos (2 \\pi \\beta n)}{n^{3}} \\left( \\frac{\\kappa^2}{3} + \\frac{\\kappa}{n} + \\frac{1}{n^2} \\right) = \\\\\n\\frac{3}{2} \\mathcal{R}e \\left[ \\frac{\\kappa^2}{3} \\mbox{Li}_3 (e^{-\\kappa+2 \\pi \\beta i }) + \\kappa \\mbox{Li}_4\n(e^{-\\kappa+2 \\pi \\beta i }) + \\mbox{Li}_5 (e^{-\\kappa+2 \\pi \\beta i}) \\right]\\,,\n\\end{multline}\nwhere $\\kappa = 2 \\pi M R$.\nIn the limit of vanishing bulk mass $\\kappa \\rightarrow 0$ we obtain the previous result, on the other hand for large $\\kappa$ the contribution to the effective potential is exponentially suppressed.\nAs a consequence, the bulk fields with large bulk mass will contribute the less to the potential.\nAnother important feature is that the most important term in the series is a $\\cos 2 \\pi \\beta$: the minimum of such term is in $0$ for bosons and in $\\beta = 1\/2$ for fermions.\nThus, the value of the integers relating $\\beta$ to $\\alpha$ will roughly speaking fix the value of the minimum.\n\nIf the fermion is antiperiodic or ``twisted'', the spectrum is:\n\n\\begin{equation}\nm_n^2 = M^2 + \\frac{(n + 1\/2 + \\beta)^2}{R^2}\\,, \\quad n  \\in Z\\,.\n\\end{equation}\nThe contribution to the effective potential is given by the previous formulas, with $\\beta \\rightarrow \\beta+1\/2$.\nAs\n$$\\cos (2 \\pi n (\\beta+1\/2)) = (-1)^n \\cos (2 \\pi n \\beta)\\,,$$\nthe twisted parity approximately flips the overall sign of the contribution.\nIn this way, we can get positive contributions to the Higgs mass arising from fermions.\n\n\n\\begin{table}[t]\n\\centering\n\\begin{tabular}{|c|c|l|}\n\\hline\nbulk field & multiplicity &  \\\\\n\\hline\ngauge (adj.) & $- 3$ & $2 \\mathcal{F} (\\alpha) + \\mathcal{F} (2 \\alpha)$ \\\\\ndown (3) & $3\\times 8$ & $\\mathcal{F}_{\\kappa_d} (\\alpha)$ \\\\\nup (6) & $3\\times 8$ & $\\mathcal{F}_{\\kappa_u} (\\alpha) + \\mathcal{F}_{\\kappa_u} (2 \\alpha)$ \\\\\nlepton (10) & $8$ & $2 \\mathcal{F}_{\\kappa_l} (\\alpha) + \\mathcal{F}_{\\kappa_l} (2 \\alpha)+ \\mathcal{F}_{\\kappa_l} (3 \\alpha)$\\\\\n\\hline\n\\end{tabular}\n\\caption{Contribution to the Higgs potential from the bulk fields in the theory. The multiplicity counts the spin and color factors. Recall that for each SM fermion, there are 2 4-component bulk fermions.}\n\\label{tab:pot}\n\\end{table}\n\nThe contribution of each bulk field is now easily computed if we decompose the SU(3)$_w$ multiplets and compute the couplings to the Higgs.\nThe results are summarized in Table~\\ref{tab:pot}.\n\n\n\\section{Fermion masses}\n\\label{sec:fermmass}\n\\setcounter{equation}{0}\n\\setcounter{footnote}{0}\n\n\nIn this section we will discuss how the SM fermions develop a mass\nvia the localized mixings. The bulk fields in Table~\\ref{tab:bulkf}\ncontain a component with the same quantum numbers as the SM fields,\nwhich thus they can mix with at the fixed points. The localized\nLagrangians generically are\n\n\\begin{multline} \\label{eq:lloc}\n\\mathcal{L}_{loc} = \\left[ -i {\\bar Q}_L {\\bar \\sigma}^\\mu \\partial_\\mu Q_L + \\frac{\\epsilon_L}{\\sqrt{\\pi R}} \\psi^d Q_L + h.c. \\right] \\delta (y - y_L) +\\\\\n\\left[ -i q_R \\sigma^\\mu \\partial_\\mu {\\bar q}_R + \\frac{\\epsilon_R}{\\sqrt{\\pi R}} q_R \\chi^s  + h.c. \\right] \\delta (y - y_R)\\,,\n\\end{multline}\nwhere $\\psi^d$ and $\\chi^s$ are the doublet and singlet components of the bulk fermion and the mixing parameters $\\epsilon$ are dimensionless (the factor of $\\pi R$ has been chosen for future convenience).\nA similar Lagrangian needs to be added for all the SM quarks and leptons.\nThe two points $y_L$ and $y_R$ can be either one of the fixed points: for each choice of twisted or untwisted periodicity, there is a unique component on both branes that can be identified with $\\psi^d$ and $\\chi^s$.\nAt the end of the day, we will have four inequivalent possibilities, depending on the twisting and if the mixings are on the same brane or on different brane.\nIn the following we will discuss all the cases, pointing out some differences.\n\n\nIn order to find how the spectrum is affected by the localized terms, we can as usual solve the bulk equations of motion in the bulk, and convert the localized terms into boundary conditions on the bulk fields~\\cite{CGHST}.\nFollowing this procedure, it is possible to write a master equation whose zeros are the mass eigenstates.\nThis procedure takes the effect of the localized mixings into account exactly.\nFor small mixings, the light mode mass is proportional to the $\\epsilon$'s, so it is always possible to reproduce masses much smaller than the Higgs VEV.\nThe challenge is given by the top mass: the coupling of the Higgs is generated via gauge interactions, so it is not easy to achieve a Yukawa of order one from a much smaller gauge coupling.\n\nIn the following we will discuss the case of a fundamental for simplicity: in this case there are no extra fields except a doublet and singlet that can be identified with the SM fields.\nWe will consider all the four possible cases  with regards to the twisting and localization of the mixings, and we will discuss some interesting limits for the light mass eigenstate.\nFor untwisted boundary conditions ($\\eta' = 1$), the master equations are:\n\n\\begin{multline} \\label{Y3ds}\n\\mathcal{Y}_{3} (w) = \\left( \\cos w - \\cos (2 \\pi \\alpha) \\right)^2 + 2 \\frac{\\epsilon_L^2 + \\epsilon_R^2}{w} \\sin w \\left( \\cos w - \\cos (2 \\pi \\alpha) \\right) +\\\\\n- \\frac{4 \\epsilon_L^2 \\epsilon_R^2}{w^2}\\, \\cdot \\left\\{ \\begin{array}{ll}\n\\left( \\cos w + 1 \\right) \\left( \\cos w - 1 + 2 \\frac{w^2}{w^2+\\kappa^2} \\sin^2 (\\pi \\alpha) \\right) & \\mbox{different branes}\\,,\\\\\n\\frac{1}{2} \\left( \\cos 2 w - 1 + 2 \\frac{w^2}{w^2+\\kappa^2} \\sin^2 (2 \\pi \\alpha) \\right)  & \\mbox{same brane}\\,.\n\\end{array} \\right.\n\\end{multline}\nwhere $w^2 = (2 \\pi R m)^2 - \\kappa^2$, and $\\kappa = 2 \\pi R M$.\nAs expected the only difference between the different branes and same brane cases is in the ``interference'' term.\nFor twisted boundary conditions ($\\eta' = - 1$), the equation becomes:\n\n\\begin{multline} \\label{tildeY3ds}\n\\mathcal{\\tilde Y}_{3} (w) = \\left( \\cos w + \\cos (2 \\pi \\alpha) \\right)^2 + 2 \\frac{\\epsilon_L^2 + \\epsilon_R^2}{w} \\sin w \\left( \\cos w + \\cos (2 \\pi \\alpha) \\right) +\\\\\n- \\frac{4 \\epsilon_L^2 \\epsilon_R^2}{w^2}\\,\n \\cdot \\left\\{ \\begin{array}{ll}\n\\left( \\cos w - 1 \\right) \\left( \\cos w + 1 - 2 \\frac{\\kappa^2}{w^2+\\kappa^2} \\sin^2 (\\pi \\alpha) \\right) & \\mbox{different branes}\\,,\\\\\n\\frac{1}{2} \\left( \\cos 2 w - 1 + 2 \\frac{w^2}{w^2+\\kappa^2} \\sin^2 (2 \\pi \\alpha) \\right)  & \\mbox{same brane}\\,.\n\\end{array} \\right.\n\\end{multline}\nIn the case of vanishing $\\epsilon$'s, eqs~(\\ref{Y3ds}) and~(\\ref{tildeY3ds}) simplify to\n\n\\begin{equation}\n\\cos w \\mp \\cos (2 \\pi \\alpha)=0\\,,\n\\end{equation}\nwhere the signs refer to the untwisted\/twisted case.\nThe solutions are\n\n\\begin{equation}\nm_n^2 = M^2 + \\left\\{ \\begin{array}{l}\n\\frac{(n + \\alpha)^2}{R^2} \\quad \\mbox{untwisted}\\,,\\\\\n\\frac{(n + 1\/2 + \\alpha)^2}{R^2} \\quad \\mbox{twisted}\\,,\n\\end{array} \\right. \\end{equation}\nas we expect.\n\nWe can solve Eqs.~(\\ref{Y3ds}) and~(\\ref{tildeY3ds}) approximately\n in the limit of small Higgs VEV $\\alpha$: this limit is useful to understand how the effective Yukawa coupling depends on the parameters in the model.\nIn the four different cases we find that the ratio between the fermion mass and the $W$ mass ($\\alpha$) is\n\n\\begin{eqnarray}\n\\mbox{untw. d.:}\\; \\frac{m_q}{m_W}& = &\\epsilon_L \\epsilon_R \\frac{\\kappa \\coth \\frac{\\kappa}{2}}{\\sqrt{\\left( 2\\epsilon_L^2 \\cosh \\frac{\\kappa}{2} + \\kappa \\sinh \\frac{\\kappa}{2} \\right) \\left( 2\\epsilon_R^2 \\cosh \\frac{\\kappa}{2} + \\kappa \\sinh \\frac{\\kappa}{2} \\right)}}\\,, \\label{eq:untwd}\\\\\n\\mbox{untw. s.:}\\; \\frac{m_q}{m_W}& = &\\epsilon_L \\epsilon_R \\frac{\\kappa\\, \\mbox{cosech}\\, \\frac{\\kappa}{2}}{\\sqrt{\\left( 2\\epsilon_L^2 \\cosh \\frac{\\kappa}{2} + \\kappa \\sinh \\frac{\\kappa}{2} \\right) \\left(2 \\epsilon_R^2 \\cosh \\frac{\\kappa}{2} + \\kappa \\sinh \\frac{\\kappa}{2} \\right)}}\\,, \\label{eq:untws}\\\\\n\\mbox{tw. d.:}\\; \\frac{m_q}{m_W}& = &\\epsilon_L \\epsilon_R \\frac{\\kappa \\mbox{tanh} \\frac{\\kappa}{2}}{\\sqrt{\\left(2 \\epsilon_L^2 \\sinh \\frac{\\kappa}{2} + \\kappa \\cosh \\frac{\\kappa}{2} \\right) \\left( 2\\epsilon_R^2 \\sinh \\frac{\\kappa}{2} + \\kappa \\cosh \\frac{\\kappa}{2} \\right)}}\\,, \\label{eq:twd}\\\\\n\\mbox{tw. s.:}\\; \\frac{m_q}{m_W}& = &\\epsilon_L \\epsilon_R \\frac{\\kappa\\, \\mbox{sech} \\frac{\\kappa}{2}}{\\sqrt{\\left( 2\\epsilon_L^2 \\sinh \\frac{\\kappa}{2} + \\kappa \\cosh \\frac{\\kappa}{2} \\right) \\left( 2\\epsilon_R^2 \\sinh \\frac{\\kappa}{2} + \\kappa \\cosh \\frac{\\kappa}{2} \\right)}}\\,. \\label{eq:tws}\n\\end{eqnarray}\n\nIt is interesting to study such results in different limits in the bulk mass $\\kappa$.\nFor large $\\kappa$, the bulk fermion twisting becomes irrelevant: the masses of the KK modes are dominated by the bulk mass $\\kappa$.\nWe find a different limit depending on whether the localized masses are on the same or opposite branes:\n\n\\begin{eqnarray}\n\\mbox{diff. branes} & \\rightarrow & \\frac{4 \\epsilon_L \\epsilon_R}{\\sqrt{(2 \\epsilon_L^2+1)(2 \\epsilon_R^2+1)}}\\; \\frac{\\kappa}{2}\\,  e^{-\\kappa\/2}\\,, \\\\\n\\mbox{same brane} & \\rightarrow & \\frac{4 \\epsilon_L \\epsilon_R}{\\sqrt{(2 \\epsilon_L^2+1)(2 \\epsilon_R^2+1)}}\\; \\kappa\\, e^{-\\kappa}\\,.\n\\end{eqnarray}\nIn both cases the effective Yukawa is exponentially suppressed: the two different powers are easily understood.\nIf the interactions are on different branes, in order to feel both the mixings, the massive fermion has to propagate from one brane to the other, thus developing a suppression of order $\\exp (- \\pi R M)$.\nIn the case of same brane localization, the fermion has to propagate to the other brane and back, therefore accumulating a double suppression.\nFrom these limits, it is clear that we can fit the light fermions very easily, either with a large bulk mass or with small localized mixings.\n\nThe limit for small $\\kappa$ is more interesting.\nIn the untwisted case, we find that $m_q \\rightarrow m_W$.\nThe rationale is again simple: when the bulk mass vanishes, the two bulk fermions decouple.\nThe localized fields mix with one of the two bulk fermions, $\\Psi$, giving a mass to the zero modes of order $\\epsilon$.\nThe other fermion $\\tilde \\Psi$, on the other hand, has a light mode whose mass is exactly $\\alpha\/R$, as discussed in\nSection \\ref{sec:model}.\nWe can confirm these result if we expand for small $\\kappa$, without any assumptions on the Higgs VEV.\nThe result is that for both same and different brane we find a light mode with mass $m_q R \\pi \\sim \\sin \\pi \\alpha$, and a mode:\n\n\\begin{eqnarray}\n\\mbox{diff. branes} & \\rightarrow & m_q \\pi R = \\frac{\\epsilon_L \\epsilon_R}{\\sqrt{(1+\\epsilon_L^2)(1+\\epsilon_R^2) - \\cos^2 \\pi \\alpha}}\\,,\\\\\n\\mbox{same brane} & \\rightarrow & m_q \\pi R = \\frac{\\epsilon_L \\epsilon_R \\cos \\pi \\alpha}{\\sqrt{(1+\\epsilon_L^2)(1+\\epsilon_R^2) - \\cos^2 \\pi \\alpha}}\\,.\n\\end{eqnarray}\nIn the twisted case the situation is more complicated: in the case of mixings on the same brane, the bulk fields that enter the mixing terms are coming from the same bulk field $\\Psi$.\nThe other bulk field decouples, while the localized zero modes develop a mass via the massive modes in $\\Psi$.\nIndeed, we find only one light mode, with mass:\n\n\\begin{equation}\nm_q \\pi R = \\frac{\\epsilon_L \\epsilon_R \\sin \\pi \\alpha}{\\sqrt{(1+\\epsilon_L^2)(1+\\epsilon_R^2)-\\sin^2 \\pi \\alpha}} \\rightarrow \\frac{\\epsilon_L \\epsilon_R}{\\sqrt{(1+\\epsilon_L^2)(1+\\epsilon_R^2)}}\\, \\alpha\\,,\n\\end{equation}\nwhere the limit for small $\\alpha$ agrees with the limit of Eq.~(\\ref{eq:tws}).\nEven if we do not have a bulk mass, we can achieve again small values with small mixings.\nIn the case of twisted bulk fermions on different brane, expanding Eq.~(\\ref{eq:twd}), we find:\n\n\\begin{equation}\n\\frac{m_q}{m_W} = \\frac{\\epsilon_L \\epsilon_R}{\\sqrt{(1+\\epsilon_L^2)(1+\\epsilon_R^2)}} \\frac{\\kappa}{2}\\,.\n\\end{equation}\nThe mass vanishes when $\\kappa \\rightarrow 0$: the reason is that the localized fermions couple to different bulk fields, so they will not get a mass in the absence of a bulk mass.\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=10cm]{mass_top}\n\\end{center}\n\\caption{Plot of the ratio $m_q\/m_W$ as a function of the bulk mass\n$\\kappa$ with fixed mixing parameters $\\epsilon_L=5=\\epsilon_R$. The\nfour curves correspond to untwisted boundary conditions and mixings\non different branes (d) or on the same one (s), and to twisted\nboundary conditions (t).} \\label{fig:top}\n\\end{figure}\n\n\nFrom this analysis it is clear that the effective Yukawa coupling cannot be larger than the gauge coupling.\nMoreover, the only way to avoid an exponential suppression is to have a vanishing bulk mass.\nIn other words, we need bulk zero modes to get directly their mass via the Higgs mechanism.\nIf we consider larger representations, the only novelty is the presence of more states that will mix via the Higgs and complicate the equation.\nHowever the same qualitative behavior appears.\nAnother interesting feature is the appearance of group theory factors that may increase the effective Yukawa coupling.\nWe will discuss in a later section if it is possible to use large representations to enhance the top mass.\nIn the rest of this section, we will focus our interest on the Higgs mass, thus the next step is to compute the one loop potential in presence of localized mixings.\nRegarding the top mass, following~\\cite{SSS}, for the moment we will assume it is generated by a symmetric representation.\nIn this case we only get a group theory factor enhancement of $\\sqrt{2}$: numerically the top mass will be lower that 100 GeV.\n\n\nSince\nthe spectrum of KK modes is largely modified in the presence of localized mixings,\nwe also expect large corrections in the Higgs potential.\nIf the mixings are small, as it may be for light fermions, we can neglect such effect and only consider the bulk contribution computed in the previous section, however a new calculation of the potential is necessary for the top quark.\nFollowing Goldberger and Rothstein in~\\cite{GR}, we can relate the Coleman-Weinberg potential to the master equation in eqs.~(\\ref{Y3ds})-(\\ref{tildeY3ds}).\nIn general, if $\\mathcal{Y} (m)=0$ determines the spectrum, the contribution of the states in such a spectrum is given by:\n\n\\begin{equation}\nV_{eff} = \\frac{1}{2} \\int_0^\\infty \\frac{d^4 p}{(2 \\pi)^4} \\ln \\mathcal{Y} (i p)\\,.\n\\end{equation}\n\nIn our case, the contribution can be written in the form of eq.~\\ref{eq:potgen}, with:\n\n\\begin{equation} \\label{eq:potloc}\n\\mathcal{F}_\\epsilon (\\kappa, \\alpha) = \\frac{1}{8} \\int_\\kappa^\\infty d \\zeta \\zeta (\\zeta^2-\\kappa^2) \\ln \\frac{\\mathcal{Y} (i \\zeta)}{K (\\zeta)}\\,,\n\\end{equation}\nwhere the function $\\mathcal{K}$ has been added to regularize the\ndivergence of the integral for large $\\zeta$. This function is\nsomewhat arbitrary, but will not affect the Higgs physics as long as\nit does not depend on the Higgs itself: it will simply regularize\nthe divergent contribution to the vacuum energy. For instance, in\nthe case of eq.~\\ref{Y3ds} and~\\ref{tildeY3ds},\n\n\\begin{equation}\nK = \\cosh^2 \\zeta \\left( 1+\\frac{2 \\epsilon_L^2}{\\zeta} \\right) \\left( 1+\\frac{2 \\epsilon_R^2}{\\zeta} \\right)\\,,\n\\end{equation}\nwill ensure the exponential convergence of the integral.\n\n\n\n\\section{Higgs mass}\n\\label{sec:5Dmass}\n\\setcounter{equation}{0}\n\\setcounter{footnote}{0}\n\n\nIn this section we will finally study the dynamical determination of the Higgs VEV via the radiative potential.\nThe $W$ mass is given by:\n\n\\begin{equation}\nm_W = \\frac{\\alpha}{R}\\,.\n\\end{equation}\nThe value of the Higgs VEV $\\alpha$ determines the ratio between the $W$ mass and the scale of the gauge boson resonances $1\/R$.\nThus, the smaller $\\alpha$, the heavier the resonances and the scale of new physics.\n\nThe Higgs mass is also given by the radiative potential after we expand around the VEV\n\\begin{equation}\n\\alpha = \\alpha_{min} + \\frac{h R}{2}\\,,\n\\end{equation}\nthrough the formula\n\n\\begin{equation}\nm_h^2 = \\frac{g_4^2}{4} R^2 V'' (\\alpha_{min}) = \\frac{g_4^2}{128 \\pi^6} \\frac{1}{R^2} \\sum \\mathcal{F}'' (\\alpha_{min})\\,.\n\\end{equation}\nFor the moment we will focus on the model in~\\cite{SSS}, and only consider the bulk fermions in Table~\\ref{tab:pot}.\nThe gauge fields alone will preserve the gauge symmetries and their potential will keep $\\alpha=0$.\nOn the other hand, the fermion content plays a very important role, as they will generate a non trivial minimum in the potential, thus driving EWSB.\nIt is particularly relevant that the larger the representation the more minima we have, in particular at small values of $\\alpha$.\nFor example, a fundamental {\\bf 3} will have a minimum at $\\alpha = 1\/2$: this is a bad value, where the $Z$ is massless so there is an extra unwanted unbroken U(1).\nThe symmetric {\\bf 6} will have a local minimum at $\\alpha \\sim 1\/4$, and so on.\nAnother interesting point is that, twisting the boundary conditions for the fermions, we can reverse the sign of\nthe potential.\nThis feature is useful if we want to move the minimum towards small values.\nThe game we want to play here is to combine the contribution from different bulk fermions in Table~\\ref{tab:bulkf}, the ones that are responsible for the SM fermion masses, and obtain cancellations that ensure a small value of the minimum and a heavy Higgs mass~\\cite{contino2}.\n\nIn~\\cite{SSS} the authors considered a well motivated scenario: the mixing terms are assumed to be all of the same order, and the hierarchy in the SM fermion masses is generated by different bulk masses.\nAs a consequence, the only bulk fermion giving a sizable contribution to the Higgs potential is the top.\nAs already mentioned, this leads to a minimum at large values of $\\alpha$ and small Higgs masses.\nWe typically find $\\alpha \\sim 0.3$ and $m_h \\sim 0.2 \\div 0.3\\, m_W$, confirming the results in~\\cite{SSS}.\nAnother drawback of a large VEV is that it predicts a low scale for the new physics, $1\/R \\sim 3\\div 5\\, m_W = 250 \\div 400$ GeV.\nIt is interesting to note that the only way to get a realistic value for $\\alpha$ with only periodic fermions is to use a huge representation~\\cite{SSS} that will certainly spoil the perturbative stability of the theory.\n\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=10cm]{potential6}\n\\end{center}\n\\caption{Plot of the Higgs potential (in arbitrary units) from the gauge bosons (\\textcolor{red}{dashed-red}), tau and top (\\textcolor{blue}{blue}), twisted fermions (\\textcolor{green}{dashed-green}), and the total ({\\bf thick black}), for one light generation with $\\kappa_l = 3$. The other parameters are like described in the text.} \\label{fig:plotpot}\n\\end{figure}\n\nIn order to lower the value of $\\alpha_{min}$ and enhance the Higgs mass,\nwe consider a different scenario: giving up the motivation to explain the fermion mass hierarchies with bulk masses, we assume\nthat light fermions are suppressed by small mixings.\nIn this case the bulk fermions responsible for their masses can contribute to the Higgs potential.\nMoreover, we can also twist the boundary conditions for some of them, in order to achieve cancellations in the potential.\nThe potential will now depend on a throng of new parameters: we will concentrate on a particularly successful example.\nWe assume that the top {$\\bf \\bar 6$} has large mixing terms, say $\\epsilon_L = \\epsilon_R = 3$, and $\\kappa_t \\sim 1$.\nWith these numbers, we have a top around $1.3\\, m_W \\sim 100$ GeV.\nFor the bottom we add a twisted {\\bf 3} with $\\kappa_b = 0$.\nFor the tau, we include a {\\bf 10} with $\\kappa_\\tau = 1$.\nFinally, we add twisted fermions for the light generations (namely a {\\bf 3}, a {$\\bf \\bar 6$} and a {\\bf 10}), with a common bulk mass $\\kappa_l$ for simplicity, that we keep as a free parameter.\nIn Fig.~\\ref{fig:plotpot}, we plotted the single contributions to the potential for $\\kappa_l = 3$.\nThe role played by the individual  terms is quite clear: the top and tau contributions will drive EWSB.\nOn the other hand, the twisted fermions, those introduced for the bottom and light generations, tend to move the minimum back to $\\alpha=0$: the cancellation between these two terms allows us to get a low minimum.\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=6cm]{vev6} \\hspace{1cm} \\includegraphics[width=6cm]{higgsmass}\n\\end{center}\n\\caption{Plots of the minimum  of the Higgs potential $\\alpha$ (left) and the Higgs mass (right), as a function of $\\kappa_l$.} \\label{fig:higgsmass}\n\\end{figure}\n\nIn Fig.~\\ref{fig:higgsmass} we show the Higgs mass and the Higgs VEV as a function of $\\kappa_l$ in the case of one or two light generations taken into account.\nIt is important to notice that for small values of $\\kappa_l$, the contribution of the twisted fermions will dominate and force the Higgs in the symmetric phase.\nIn other words, there is a continuous transition to the value $\\alpha = 0$: however, when $\\alpha$ is too small the dependence on the parameter $\\kappa_l$ is very strong, signaling a fine tuning in the potential.\nValues of $\\alpha \\gtrsim 0.1$ can be achieved without fine tuning as the cancellation only depends on the representations included in the calculation. \nWe have checked that the results are also insensitive to variations of the other continuous parameters.\nThe LEP bound on the Higgs mass pushes $\\alpha \\lesssim 0.05$, in a region that shows a mild fine tuning.\nWe will be more quantitative in the next section, where we analyze a more realistic scenario including the top mass.\nMoreover, in this region of the parameter space, the scale of the KK resonances is $1\/R \\gtrsim 20\\, m_W \\sim 2$ TeV.\n\n\n\\section{Top mass from large representations of SU(3)}\n\\label{sec:topmass}\n\\setcounter{equation}{0}\n\\setcounter{footnote}{0}\n\n\nThe model presented in the last sections seems to have a realistic spectrum, except for the top that is too light.\nThe main reason is that the Yukawa is generated by the gauge interactions themselves, so the fermion masses\nhave an upper bound\n\\begin{equation}\nm_q \\leq k\\, m_W\\,,\n\\end{equation}\nwhere the bound is saturated when the bulk mass vanishes, or in\nother words the fermion is a bulk field zero mode, and the\nproportionality factor $k$ depends on the representation the top is\nembedded in. In order to fit the top mass, one should include in the\ntheory a large representation. But how large does it have to be in\norder to generate a realistic mass? It turns out that the number $k$\nis given by the square root of the rank of the\nrepresentation~\\cite{SSSflavor}, i.e. the number of indices. So, the\nsmallest useful representation would be a tensor with 4 indices. It\nwould lead to the nice prediction that at tree level $m_t = 2\\, m_W$,\nthen corrections from QCD loops could account for the\nextra enhancement. The first worry about the use of large reps is\nthat the cutoff of the theory may enter a strong coupling regime at low scales, due to large group theory factors\nin the fermion loops. We will comment on this issue later, and for the\nmoment just assume that the theory is under perturbative control.\n\nThe representations of rank 4 of $SU(3)_w$, with their decomposition under SU(2)$\\times$U(1), are:\n\n\\begin{equation} \\begin{array}{lcl} \\label{reps4}\n({\\bf \\bar {15}})_{-2\/3} &  \\rightarrow & ({\\bf 1}, 2\/3) + ({\\bf 2}, 1\/6) + ({\\bf 3}, - 1\/3) + ({\\bf 4}, -5\/6 ) + ({\\bf 5}, -4\/3)\\,, \\\\\n({\\bf \\bar {24}})_{0} &  \\rightarrow & ({\\bf 1}, 2\/3) + ({\\bf 2}, 1\/6) + ({\\bf 2}, 7\/6) + ({\\bf 3}, - 1\/3) + ({\\bf 3}, 2\/3) + ({\\bf 4}, -5\/6 ) \\\\\n & & + ({\\bf 4}, 1\/6 ) + ({\\bf 5}, -1\/3)\\,, \\\\\n({\\bf 27})_{2\/3} &  \\rightarrow & ({\\bf 1}, 2\/3) + ({\\bf 2}, 1\/6) + ({\\bf 2}, 7\/6) + ({\\bf 3}, - 1\/3) + ({\\bf 3}, 2\/3) + ({\\bf 3}, 5\/3) \\\\\n & &  + ({\\bf 4}, -5\/6 ) + ({\\bf 4}, 1\/6 ) + ({\\bf 5}, 2\/3)\\,,\n\\end{array} \\end{equation}\nwhere we have added a charge under the extra U(1)$_X$ in order to fix the hypercharges.\nIn the following we will use the smallest of these representations, the symmetric ${\\bf \\bar {15}}$.\nIn order to fully exploit the factor of two, we need the top to be a bulk fermion zero mode.\nSo, we only add one bulk field, and as usual\nuse the orbifold parity to obtain a chiral spectrum of zero modes.\nThe orbifold projection will leave unwanted zero modes in the large representations of SU(2)$_L$ as well: in order to get rid of them we add localized fermions, and a mass like in Eq.~(\\ref{eq:lloc}).\nFor simplicity, we will assume in the following that all three brane localized mass\nparameters are equal (and denote them by $\\epsilon$).\n\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=10cm]{potential15}\n\\end{center}\n\\caption{Plot of the Higgs potential (in arbitrary units) from the gauge bosons (\\textcolor{red}{red-dashed}), top (\\textcolor{blue}{blue}), bottom ({\\bf 3}) and tau ({\\bf 10}) (\\textcolor{green}{green-dashed}), and the total ({\\bf thick black}), for $\\epsilon=1.25$.} \\label{fig:plotpot15}\n\\end{figure}\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=15cm]{combo15}\n\\end{center}\n\\caption{Top left: Higgs VEV $\\alpha$ as a function of the localized masses $\\epsilon$.\nTop right: fine tuning as a function of $\\alpha$. The vertical lines are bounds on $\\alpha$ coming from corrections to the $Z b \\bar b$ vertex, discussed in Section~\\ref{sec:EWPT}.\nBottom left: Higgs mass as a function of $\\alpha$. The horizontal line is the LEP bound of $115$ GeV.\nBottom right: top and first massive mode (dashed) masses as function of $\\alpha$. In all the plots, we show the results for model a (\\textcolor{blue}{blue}) and model b (\\textcolor{red}{red}).} \\label{fig:top15}\n\\end{figure}\n\n\nIn order to determine if the spectrum is realistic, we need again to\nminimize the Higgs potential. We analyze a minimal scenario, where\nonly the third generation fermions contribute. The masses for bottom\nand tau are generated in the usual way, with localized degrees of freedom that feel EWSB via the mixing to massive bulk fields.\nThe simplest choice is to add two fermions with twisted boundary condition: they will both give them small masses and induce cancellations in the Higgs potential as in the previous section.\nDue to the presence of the extra U(1)$_X$ we can always adjust the overall hypercharge, so we are free to use any representation: however, we will concentrate on small representations, like the {\\bf 3}, {\\bf 6} and {\\bf 10} in Table~\\ref{tab:bulkf}, so that they will not worsen the problem with the low strong coupling scale.\nFor simplicity we will present results from two cases, that summarize the qualitative properties of this scenario:\n\n\\begin{center}\n\\begin{tabular}{||l||c|c||}\n\\hline\n\\hline\n & bottom & tau  \\\\\n\\hline\nmodel \\textcolor{blue}{a} & ({\\bf 3}, {\\bf 3})$_0$ & ({\\bf 1}, {\\bf 10})$_0$ \\\\\nmodel \\textcolor{red}{b} & ({\\bf 3}, {\\bf 6})$_{1\/3}$ &  ({\\bf 1}, {\\bf 3})$_{-2\/3}$\\\\\n\\hline\n\\hline\n\\end{tabular} \\end{center}\nwhere we have eventually assigned U(1)$_X$ charges to fix the hypercharge of the doublet and singlet components.\nA typical shape for the\nHiggs potential is given in Fig.~\\ref{fig:plotpot15}, for model a: EWSB is again\ninduced by the contribution of the top (${\\bf \\bar {15}}$), while the twisted\nfermions push the VEV towards small values. The only free parameter\nin this case is $\\epsilon$: in the top--left panel of\nFig.~\\ref{fig:top15} we show the value of the Higgs VEV as a function of $\\epsilon$ in the two cases. \nYou can see that EW symmetry is broken for natural values of $\\epsilon \\sim 1$.\nFor large\nlocalized masses, $\\epsilon \\gtrsim 1.5$, there is no EWSB. What happens is\nthe following: the zero modes from the unwanted states are removed,\nand this is equivalent to twisted boundary conditions for such\ncomponents of the bulk fermion. The spectrum of the ($\\bf \\bar {15}$) in this limit is\nequivalent to a twisted fermion with large mixing with a localized\ndoublet and singlet. However, the induced potential will resemble\nthe one of a twisted bulk fermion as well, thus it will not drive\nEWSB. For this reason we did not consider this possibility. \nIn the plot, we also see that the sensitivity of the minimum to $\\epsilon$ becomes large when it approaches the EW preserving values.\nThis signals that in that region a parametric fine tuning is at work in the potential.\nA simple way to quantify the amount of fine tuning $f$ is via the logarithmic derivative:\n\n\\begin{equation}\nf = \\frac{d \\log \\alpha (\\epsilon)}{d \\log \\epsilon}\\,.\n\\end{equation}\nwhich we plotted in the top--right panel of Fig.~\\ref{fig:top15}.\nValues of $\\alpha \\gtrsim 0.05$ require a fine tuning milder than 10\\%: this shows that we can naturally obtain values of $1\/R \\lesssim 2$ TeV.\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{||l||c|c|c|c|c||}\n\\hline\n\\hline\n$\\alpha$ & $1\/R$ & $f$ & $m_H$ & $m_t$ & $m'_t$ \\\\\n\\hline\n\\hline\n0.08 & 1 TeV & $\\tabarr{31\\%}{42\\%}$ & $\\tabarr{110}{125}$ GeV  & $\\tabarr{113}{110}$ GeV & $\\tabarr{189}{186}$ GeV \\\\\n\\hline\n0.05 & 1.6 TeV & $\\tabarr{11\\%}{14\\%}$ & $\\tabarr{120}{133}$ GeV &  $\\tabarr{149}{149}$ GeV & $\\tabarr{381}{375}$ GeV \\\\\n\\hline\n0.04 & 2 TeV  & $\\tabarr{7\\%}{9\\%}$ & $\\tabarr{124}{136}$ GeV & $\\tabarr{154}{154}$ GeV & $\\tabarr{519}{514}$ Gev\\\\\n\\hline\n0.03 & 2.7 TeV  & $\\tabarr{4\\%}{5\\%}$ & $\\tabarr{128}{140}$ GeV & $\\tabarr{157}{157}$ GeV & $\\tabarr{753}{746}$ Gev\\\\\n\\hline\n0.02 & 4 TeV  & $\\tabarr{2\\%}{2\\%}$ & $\\tabarr{134}{144}$ GeV & $\\tabarr{159}{159}$ GeV & $\\tabarr{1224}{1213}$ Gev\\\\\n\\hline\n\\hline\n\\end{tabular} \\end{center}\n\\caption{Higgs ($m_H$), top ($m_t$) and first massive fermion ($m_t'$) masses for different values of the Higgs VEV $\\alpha$, in the two models \\textcolor{blue}{a}\\, (top row) and \\textcolor{red}{b}\\, (bottom row). We also list an estimate of the fine tuning $f$ required to obtain these minima.}\n\\label{tab:res15}\n\\end{table}\n\n\n\n\nIn the bottom--left panel of Fig.~\\ref{fig:top15}, we show the mass of the\nHiggs as a function of $\\alpha$ in the two cases.\nIn model a, the LEP bound of $115$ GeV requires that $\\alpha \\lesssim 0.06$: this means that in order to push the Higgs mass above the direct bound, we need to allow a moderate fine tuning in the potential.\nOn the other hand, in model b no such fine tuning is needed: the reason is that the colored bulk fermion is in a larger representation, so it will enhance the loop induced quartic term.\nIn general, $m_h \\sim 120 \\div 150$ GeV can be obtained, where the precise value depends on the choice of bulk fermions.\nFinally, in the bottom--right panel we plotted the top mass and the mass of the first massive mode.\nThe results are the same in the two cases, as they only depend on the bulk $\\bf \\bar {15}$.\nThe top, in order to saturate the value $2\\, m_W$, also prefers small values for $\\alpha \\lesssim 0.04$, where the first massive mode is heavier than $\\sim 500$ GeV.\nThese modes are bound by direct searches of a fourth generation of quarks to be heavier than $\\sim 200\\div 300$ GeV (for a $b'$ the bound is $\\sim 200$ GeV~\\cite{PDG}, and we only expect group theory factors coming from the different representations).\n\nIn Table~\\ref{tab:res15} we list some numerical results for different choices of the Higgs VEV.\nIn the next section, we will analyze possible constraints on the parameter space.\n\n\n\n\\section{Tree level corrections to electroweak precision observables}\n\\label{sec:EWPT}\n\\setcounter{equation}{0}\n\\setcounter{footnote}{0}\n\n\nModels of EWSB in extra dimensions generically obtain tree level corrections to electroweak precision observables, generated by the wave function overlap that affects the couplings between particles.\nOne possible source is the mixing between zero modes and KK modes of gauge bosons or fermions, generated by the Higgs VEV responsible for EWSB.\nThis is the case in models of Gauge-Higgs unification in warped background~\\cite{contino1,Zbb}.\nIn a flat background, the flatness of the $W$ and $Z$ wave functions and of the Higgs profile ensures the\nabsence of such corrections.\nThe reason behind this is that the Higgs VEV does not mix the zero modes with the KK resonances, thanks to the\northogonality of their wave functions.\nThe KK modes will generate non vanishing corrections at loop level, but we expect them to be small due to the heaviness of the scale of new physics.\nThe only tree level corrections in the flat case can be generated by the presence of exotic zero modes that mix with the SM particles and pick up a mass via boundary terms.\n\n\nA source of such deviations is the presence of large SU(2)$_L$ representations among the bulk fermions needed to generate masses for the SM fermions.\nTheir eventual zero modes will mix with the SM fermions via the Higgs VEV.\nFor the light fermions such corrections are highly suppressed by the masses, so they are negligible.\nThe only worrisome coupling is $Z b_l \\bar b_l$, because the left handed component of the b is in the same multiplet as the top, and is mixed with the large reps present in the $\\bf \\bar {15}$.\nThe leading contribution will arise at order $\\alpha^2$.\nAt this order, the only relevant representations are the ones linked to the quark doublet by one Higgs insertion.\nGauge invariance allows only couplings with triplets, with hypercharge -1\/3 or 2\/3:\n\n\\begin{equation}\n\\mathcal{Y}_{-1\/3}\\, Q_L H^\\dagger {\\bf \\bar 3}_{-1\/3} +  \\mathcal{Y}_{2\/3}\\, Q_L H {\\bf \\bar 3}_{2\/3}\\,.\n\\end{equation}\nWe can compute the corrections to the vertex in two simple limits.\nIf the localized mass is small, then only the zero mode contributes, as the KK modes do not couple and the mixing induced by the localized term is small.\nIn this limit, namely $\\alpha \\ll \\epsilon \\ll 1$, the contribution of the two triplets is:\n\n\\begin{equation} \\label{eq:dzbb}\n\\Delta = \\frac{\\delta g}{g} = \\frac{1}{1-\\frac{2}{3} \\sin^2 \\theta_W} (\\mathcal{Y}_{2\/3}^2 - \\mathcal{Y}_{-1\/3}^2) \\left( \\frac{m_W}{m_3} \\right)^2\\,,\n\\end{equation}\nwhere we have assumed that the two triplets have the same mass $m_3$.\nIn the limit of large localized mass, the BCs of the triplets are effectively twisted: we can then compute the contribution of a tower of twisted states.\nThe contribution is the same as (\\ref{eq:dzbb}), with\n\n\\begin{equation}\n\\frac{1}{m_3^2} \\rightarrow \\frac{\\pi^2}{3} R^2\\,.\n\\end{equation}\nIn this limit, we will have a direct bound on $R$, or equivalently on $\\alpha = m_W R$.\n\nIn the case of a {\\bf 15}, discussed in the previous section, there is only a triplet with hypercharge $Y=-1\/3$.\nThe effective Yukawa is $\\mathcal{Y}_{-1\/3} = \\sqrt{3}$, and the correction is\n\n\\begin{equation} \\label{eq:zbb3}\n\\Delta = - \\frac{\\pi^2}{1-\\frac{2}{3} \\sin^2 \\theta_W} \\alpha^2\\, \\approx - 11\\, \\alpha^2\\,.\n\\end{equation}\nThe LEP experiments have constrained the deviation $\\Delta$\nto be less than about one percent~\\cite{LEP}.\nA one percent deviation would imply the bound\n$\\alpha < 0.03$ or $1\/R > 2.7$ TeV (while a 0.5\\% bound would imply\n$\\alpha < 0.021$ or $1\/R > 3.9$ TeV). This bound would start pushing $\\alpha$ in the\nregion where a few percent fine tuning in the Higgs potential is required. It is\ninteresting to note that the triplet with hypercharge $Y = 2\/3$ will\ngive a positive contribution to $\\Delta$, thus relaxing the\nbound a little. This might be the case of larger representations like the\n{\\bf 24} or {\\bf 27}, see Eq.~\\ref{reps4}, that contain both the triplets. Note that this\ncorrection is present also in the case of a small representation,\nlike the {\\bf 6} considered in Sec.~\\ref{sec:5Dmass}: the main\ndifference is that the effective Yukawa $\\mathcal{Y}_{-1\/3} =\n1$ is smaller significantly reducing the correction to $Zb\\bar{b}$.\n\nAnother source of deviations is the presence of the extra U(1)$_X$ needed to fit the weak mixing angle.\nThe orbifold projection breaks SU(3)$_w \\times$U(1)$_X \\to$ SU(2)$_L \\times$U(1)$_w \\times$U(1)$_X$, leaving an unwanted zero mode.\nThe two U(1)'s can be broken to the hypercharge by a localized Higgs mechanism or the presence of a (localized) anomaly.\nIn both cases, the net effect is the presence of a localized mass term for the combination\n\n\\begin{equation}\nX_\\mu = \\frac{1}{\\sqrt{3 g^2 + g_x^2}} \\left( \\sqrt{3} g A^8_\\mu - g_x A^x_\\mu \\right)\\,,\n\\end{equation}\nwhere $A^8$ is the gauge boson of the U(1)$_w$, while the orthogonal combination can be identified with the hypercharge gauge boson\n\n\\begin{equation}\nB_\\mu =  \\frac{1}{\\sqrt{3 g^2 + g_x^2}} \\left( g_x A^8_\\mu + \\sqrt{3} g A^x_\\mu \\right)\\,,\n\\end{equation}\nwith gauge coupling\n\n\\begin{equation}\ng' = \\frac{\\sqrt{3} g_x g}{\\sqrt{3 g^2 + g_x^2}}\\,.\n\\end{equation}\nAt leading order, $g_x$ can be tuned to fit the SM coupling of the hypercharge.\nHowever, the presence of the $X$ boson will generate corrections to EWP observables.\nThe localized mass term is not protected by any symmetry, so it will generically be as large as the cutoff of the theory: in this limit it will distort the wave functions of the KK modes and introduce mixings of such massive states with the zero modes.\nIf the light fermions are localized on one fixed point (this is the same point where the anomaly is localized) they will not couple to the $X$ boson, thus there will not be any correction to the couplings at order $\\alpha^2$: the corrections will be of order $\\alpha^4$, or induced by the mixing with the bulk fermions that generate the mass, thus being of order $(m_f R)^2 \\alpha^2$.\nIn both cases, they are safely small: this implies that the $S$ parameter is negligibly small.\nHowever, the $X$ boson will also mix with the $Z$, correcting its mass and generating a deviation in the $\\rho$ parameter (or alternatively $T$) given by\n\n\\begin{equation}\nT =  \\frac{4 \\pi}{e^2} \\Delta \\rho = \\frac{4 \\pi}{e^2} \\frac{\\pi^2}{3} \\frac{3 - 4 \\sin^2 \\theta_W}{\\cos^2 \\theta_W}\\, \\alpha^2\\, \\approx 1.2\\cdot 10^3\\, \\alpha^2.\n\\end{equation}\nThe experimental bound $|T| \\lesssim 0.3$ poses a bound $\\alpha < 0.015$ ($1\/R > 5$ TeV).\n\nAnother correction arises in the third generation sector: the $b_l$ is a bulk field, so it couples directly to the $X$ bosons.\nThis coupling induces a correction to the coupling with the $Z$ given by\n\n\\begin{equation}\n\\Delta = - \\frac{2}{3} \\left[ \\frac{3 - 4 \\sin^2 \\theta_W}{\\cos^2 \\theta_W} \\frac{1}{6} - 3 Q_x \\right]\\, \\frac{\\pi^2}{1-\\frac{2}{3} \\sin^2 \\theta_W} \\alpha^2\\,,\n\\end{equation} \nwhere the U(1)$_X$ charge of the $\\bf \\bar {15}$ is $Q_x = -2\/3$.\nThis contribution is negative, and when added to the correction from the triplet in Eq.~(\\ref{eq:zbb3}), it gives a bound $\\alpha < 0.018$ or $1\/R > 4.5$ TeV for $|\\Delta| < 1\\%$ ($\\alpha < 0.013$ or $1\/R > 6$ TeV for $|\\Delta| < 0.5\\%$).\n\nA potentially tight bound on the scale $1\/R$ comes from the couplings of gauge resonances of SU(2)$_L \\times$U(1)$_w$ with the light fermions, that are localized on the orbifold fixed points.\nTheir coupling is generically $\\sqrt{2}\\, g$, and they will induce four fermion operators at tree level.\nThe bound from precision electroweak observables~\\cite{KKgb,LEP2} would require $1\/R > 4$ TeV.\nAn even tighter bound, around $7$ TeV, would emerge from the analysis of LEP2 data off the $Z$ peak~\\cite{cheung}.\nHowever, the light fermions do not play any active role in the extra dimensional construction.\nMoreover, their effect on the little hierarchy problem is negligible, due to the smallness of the Yukawa couplings.\nWe can thus couple them to the zero modes of the gauge bosons only, and add an explicit Yukawa with the Higgs without spoiling the good features of the model.\nA consistent inclusion of them is postponed to a UV completion of this model.\nThe only fermions that play an active role in the model under discussion are the top and bottom, and the constraints in this sector are milder.\nIt is anyway very easy to think of a scenario where four fermion operators involving top and bottom are absent or suppressed.\nFor instance, we can have a bulk $\\bf \\bar {15}$, that contains the doublet $Q_L$ and right handed top $t_R$. After EWSB their wave functions remain flat and the top gets its mass.\nThe right handed bottom $b_R$ can also be identified with the zero mode of a bulk singlet of SU(3)$_w$.\nIn order to give a mass to the bottom, we can introduce a twisted bulk fermion (like the {\\bf 6} in model b of \nSection~\\ref{sec:topmass}), and mix it with the bulk $Q_L$ and $b_R$ on the branes.\nFor the bottom mass, it is enough to have mixings $\\epsilon \\sim 0.1$.\nThe flatness of the wave function of the bulk fields will forbid couplings of two zero modes with one KK mode, thus removing the four fermi operators at tree level.\nThis mechanism is precisely the extra dimensional version of the $T$ parity advocated for Little Higgs models.\nThe presence of localized mixings will distort the wave functions, and generate very small contributions suppressed by $\\epsilon^4 \\sim (m_b\/m_W)^2 \\sim 10^{-3}$. \nNote that the tau (the bulk {\\bf 3}) does not play an important role in model b, so it can be treated as a light fermion.\n\n\nAnother effect of the large representations that we need to fit the top mass is that they tend to lower the scale where the theory is strongly coupled, due to their large contributions to loops.\nNaive dimensional analysis would suggest that the strong coupling scale is:\n\n\\begin{equation}\n\\Lambda_{strong} R \\sim \\frac{24 \\pi^3 R}{g_5^2} = \\frac{24 \\pi^2}{g^2} \\sim 10^3\\,,\n\\end{equation}\nhowever this estimation does not take into account large group\ntheory factors. For example, a fermionic representation will\ncontribute to a one loop result with a factor $4 C(r)$, $C(r)$ being\nthe Dynkin index of the representation. \nNotice that this same factor will enhance the contribution to the Higgs mass term~\\cite{matchev}.\nFor the {\\bf 15}, the\nsmallest representation needed to generate a top mass, this would\nmean $4 \\cdot 3 \\cdot 35\/2 \\sim 200$, where we also took into\naccount a color factor of $3$. Thus the naive strong scale is only\nfew times the resonance scale $1\/R$: it is then very important to\nverify if the results are stable under further radiative\ncorrections. For example, a calculation of the two loop corrections\nto the Higgs mass would be useful. In the case of a {\\bf 24}, {\\bf\n27} or higher rank representations, the Dynkin index is much larger,\nthus spoiling the predictive power of the theory. \n\n\n\n\\section{Conclusions and Outlook}\n\\label{sec:concl}\n\\setcounter{equation}{0}\n\\setcounter{footnote}{0}\n\n\n\nWe have shown that in a minimal model of Gauge-Higgs unification in a flat extra dimension,\nit is possible to accommodate a large Higgs mass and a heavy top. We\ndo not add extra fields just for the sake of the Higgs potential,\nbut all the bulk fields are also used to generate masses for the SM\nfields: in this sense we preserve minimality. \nTree level corrections to electroweak precision measurements generated by the mixing between KK modes are avoided thanks to the flatness of the Higgs profile.\nThe most serious bound on the size of the extra dimension comes from the presence of zero modes that are not projected away by the orbifold: extra zero modes in the large representation needed to fit the top mass correct the $Z b \\bar b$ vertex, while the extra U(1)$_X$ needed to fit the weak mixing angle introduces a $\\Delta \\rho$ (or $T$).\nThe top representation will also lower the scale where the theory becomes strongly coupled to few $\\times\\,1\/R$.\n\nThe model we analyzed\nconsists of a SU(3)$_c \\times$SU(3)$_w \\times$U(1)$_X$ gauge\ngroup, where the extra gauged U(1)$_X$ fixes the value of $\\sin^2\n\\theta_W$. The orbifold projection breaks SU(3)$_w \\to$ SU(2)$_L\n\\times$U(1)$_w$, and U(1)$_X \\times$U(1)$_w$ can be broken to\nU(1)$_Y$ either by twisted boundary conditions or by a localized anomaly. The\ntop mass is generated via gauge interaction, so one is forced to\nintroduce a $\\bf \\bar {15}$ of SU(3)$_w$ in order to\ngenerate an enhancement of the fermion mass with respect to the $W$\nmass.\nRegarding the Higgs potential, introducing twisted bulk fermions, that also\ngive mass to the bottom and tau, one can lower the value of the\nHiggs VEV and enhance the Higgs mass itself above the experimental\nbounds via cancellations in the potential.\nPart of these cancellations are due to the presence of twisted fermions: they allow to achieve naturally, without fine tuning, values $\\alpha \\gtrsim 0.05$ and $m_h \\sim 120 \\div 140$ GeV.\nIn this region we estimate a fine tuning milder than 10 \\%.\nThe scale of new physics is also large, naturally around few TeV.\nThe only tree level corrections to the electroweak precision measurements are corrections to the $Z b \\bar b$ coupling, arising via the mixing of the left-handed bottom with triplets of SU(2)$_L$ contained in the top bulk $\\bf \\bar {15}$, and $\\Delta \\rho$ induced by the extra U(1)$_X$.\nThey push the Higgs VEV in a region of the parameter space where a few \\% fine tuning is needed, and require $1\/R \\gtrsim 5 \\div 6$ TeV.\nThis bound is removed if the breaking of the gauge symmetry only comes from the orbifold projection, as it may be possible if we consider different gauge groups and more complicated orbifold projections.\nThe deviation in $Z b \\bar b$ given by the large top representation alone would require $1\/R \\gtrsim 4$ TeV (at $0.5 \\%$).\n\nAnother consequence of the large top representation is that the theory becomes strongly coupled at a relatively small scale, of order few $\\times\\,1\/R$.\nThis is a borderline situation, and a calculation or estimation of the two loop effects is needed to decide if the calculation of the Higgs mass that we presented in this paper is reliable.\nThis scale is nevertheless large enough, so that higher order operators generated by the non-perturbative physics will give a negligible contribution to the precision measurements.\nAbove the strong coupling scale, the theory is no longer under perturbative control.\nIf the gauge symmetry, responsible for the protection of the Higgs mass, is not broken by non-perturbative effects, the Higgs mass is protected up to the Planck scale, thus addressing the Big Hierarchy problem.\nThe only quantity that may be sensitive to the UV physics, and reintroduce a UV sensitivity in the Higgs mass is the compactification scale $1\/R$.\nHowever, we can imagine to add a stabilization mechanism~\\cite{stabilize}, that only couples to the gauge sector via gravitational interactions, thus without spoiling the stability of the Higgs mass.\nMoreover, this stabilization mechanism will induce small distortions of the background and small 5D Lorentz violating effects in the SM sector.\n\n\n\n\\section*{Note added}\n\nWhile this work was completed, we became aware of a related work by G.~Panico, M.~Serone and A.~Wulzer~\\cite{PSW}, where the authors also address the top and Higgs mass problem in this context.\nThe main difference between the two models is that they enhance the top mass via large explicit violation of 5 dimensional Lorentz invariance in the bulk, without the introduction of large representations.\n\n\n\\section*{Acknowledgments}\nWe thank Roberto Contino, Marco Serone and Andrea Wulzer for useful discussions and comments.\nWe thank Giuliano Panico, Marco Serone and Andrea Wulzer also for sharing with us a draft of their forthcoming paper~\\cite{PSW}, and for useful comments on our manuscript.\nG.C. and C.C. thank the Aspen Center for Physics for its hospitality while part of this work was completed.\nThe research of G.C. and C.C.\nis supported in part by the DOE OJI grant DE-FG02-01ER41206 and in part\nby the NSF grants PHY-0139738  and PHY-0098631. S.C.P. was supported by the Korea Research Foundation Grant funded by the\nKorean Government (MOEHRD), grant KRF-2005-214-C00147.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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D}{{\\mathcal D}}\n\\def{\\mathcal E}{{\\mathcal E}}\n\\def{\\mathcal F}{{\\mathcal F}}\n\\def{\\mathcal G}{{\\mathcal G}}\n\\def{\\mathcal H}{{\\mathcal H}}\n\\def{\\mathcal I}{{\\mathcal I}}\n\\def{\\mathcal J}{{\\mathcal J}}\n\\def{\\mathcal K}{{\\mathcal K}}\n\\def{\\mathcal L}{{\\mathcal L}}\n\\def{\\mathcal M}{{\\mathcal M}}\n\\def{\\mathcal N}{{\\mathcal N}}\n\\def{\\mathcal O}{{\\mathcal O}}\n\\def{\\mathcal P}{{\\mathcal P}}\n\\def{\\mathcal Q}{{\\mathcal Q}}\n\\def{\\mathcal R}{{\\mathcal R}}\n\\def{\\mathcal S}{{\\mathcal S}}\n\\def{\\mathcal T}{{\\mathcal T}}\n\\def{\\mathcal U}{{\\mathcal U}}\n\\def{\\mathcal V}{{\\mathcal V}}\n\\def{\\mathcal W}{{\\mathcal W}}\n\\def{\\mathcal X}{{\\mathcal X}}\n\\def{\\mathcal Y}{{\\mathcal Y}}\n\\def{\\mathcal Z}{{\\mathcal Z}}\n\n\n\\def\\Sc#1{{\\hbox{\\sc #1}}}     \n\\def\\Sf#1{{\\hbox{\\sf #1}}}     \n\n\n\\def\\slpa{\\slash{\\pa}}                           \n\\def\\slin{\\SLLash{\\in}}                                  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\n\n\n\\def\\stackrel{<}{{}_\\sim}{\\stackrel{<}{{}_\\sim}}\n\\def\\stackrel{>}{{}_\\sim}{\\stackrel{>}{{}_\\sim}}\n\n\n\\def\\sp#1{{}^{#1}}                             \n\\def\\sb#1{{}_{#1}}                             \n\\def\\oldsl#1{\\rlap\/#1}                         \n\\def\\slash#1{\\rlap{\\hbox{$\\mskip 1 mu \/$}}#1}     \n\\def\\Slash#1{\\rlap{\\hbox{$\\mskip 3 mu \/$}}#1}     \n\\def\\SLash#1{\\rlap{\\hbox{$\\mskip 4.5 mu \/$}}#1}   \n\\def\\SLLash#1{\\rlap{\\hbox{$\\mskip 6 mu \/$}}#1}     \n\\def\\wt#1{\\widetilde{#1}}                   \n\\def\\Hat#1{\\widehat{#1}}                       \n\\def\\Bar#1{\\bar{#1}}                      \n\\def\\bra#1{\\left\\langle #1\\right|}             \n\\def\\ket#1{\\left| #1\\right\\rangle}             \n\\def\\VEV#1{\\left\\langle #1\\right\\rangle}       \n\\def\\abs#1{\\left| #1\\right|}                   \n\\def\\leftrightarrowfill{$\\mathsurround=0pt \\mathord\\leftarrow \\mkern-6mu\n        \\cleaders\\hbox{$\\mkern-2mu \\mathord- \\mkern-2mu$}\\hfill\n        \\mkern-6mu \\mathord\\rightarrow$}      \n\\def\\dvec#1{\\vbox{\\ialign{##\\crcr\n        \\leftrightarrowfill\\crcr\\noalign{\\kern-1pt\\nointerlineskip}\n        $\\hfil\\displaystyle{#1}\\hfil$\\crcr}}}          \n\\def\\dt#1{{\\buildrel {\\hbox{\\LARGE .}} \\over {#1}}}    \n\\def\\dtt#1{{\\buildrel \\bullet \\over {#1}}}             \n\\def\\der#1{{\\pa \\over \\pa {#1}}}               \n\\def\\fder#1{{\\delta \\over \\delta {#1}}}                \n\n\n\\def\\partder#1#2{{\\partial #1\\over\\partial #2}}  \n\\def\\parvar#1#2{{\\delta #1\\over \\delta #2}}              \n\\def\\secder#1#2#3{{\\partial^2 #1\\over\\partial #2 \\partial #3}} \n\\def\\on#1#2{\\mathop{\\null#2}\\limits^{#1}}              \n\\def\\bvec#1{\\on\\leftarrow{#1}}                 \n\\def\\oover#1{\\on\\circ{#1}}                             \n\n\n\n\\def\\begin{equation}{\\begin{equation}}\n\\def\\end{equation}{\\end{equation}}\n\n\n\\def\\begin{displaymath}{\\begin{displaymath}}\n\\def\\end{displaymath}{\\end{displaymath}}\n\n\n\\def\\begin{eqnarray}{\\begin{eqnarray}}\n\\def\\end{eqnarray}{\\end{eqnarray}}\n\\def\\nonumber{\\nonumber}\n\n\n\\begin{document}\n\\DeclareGraphicsExtensions{.jpg,.pdf,.mps,.eps,.png}\n\\begin{flushright}\n\\baselineskip=12pt\nEFI-09-30 \\\\\nANL-HEP-PR-09-102\n\\end{flushright}\n\n\\begin{center}\n\\vglue 1.5cm\n\n\n\n\n{\\Large\\bf Prospects for Higgs Searches at the Tevatron and LHC in the MSSM with Explicit $CP$-violation } \\vglue 2.0cm {\\Large Patrick Draper$^{a}$, Tao Liu$^{a}$, and Carlos E.M. Wagner$^{a,b,c}$}\n\\vglue 1cm {\n$^a$ Enrico Fermi Institute and\n$^b$ Kavli Institute for Cosmological Physics, \\\\\nUniversity of Chicago, 5640 S. Ellis Ave., Chicago, IL\n60637\\\\\\vglue 0.2cm\n$^c$ HEP Division, Argonne National Laboratory,\n9700 Cass Ave., Argonne, IL 60439\n}\n\\end{center}\n\n\n\n\n\\vglue 1.0cm\n\\begin{abstract}\nWe analyze the Tevatron and Large Hadron Collider (LHC) reach for the Higgs sector\nof the Minimal Supersymmetric Standard Model (MSSM) in the\npresence of explicit $CP$-violation. Using the most recent studies\nfrom the Tevatron and LHC collaborations, we examine the CPX\nbenchmark scenario for a range of $CP$-violating phases in the soft\ntrilinear and gluino mass terms and compute the exclusion\/discovery\npotentials for each collider on the $(M_{H^+}, \\tan\\beta)$ plane.\nProjected results from Standard Model (SM)-like, non-standard, and\ncharged Higgs searches are combined to maximize the statistical\nsignificance. We exhibit complementarity between the SM-like Higgs searches at the LHC with low luminosity and the Tevatron, and estimate the combined reach of the two colliders in the early phase of LHC running.\n\\end{abstract}\n\n\\newpage\n\n\\section{Introduction}\n\nThe origin of electroweak symmetry breaking remains a principal open question in high-energy physics. In the Standard Model (SM), the breakdown of the electroweak symmetry is induced by the vacuum\nexpectation value of a scalar field, which transforms non-trivially\nunder the $SU(2)_L\\times U(1)_Y$ symmetry. A consequence of this\nmechanism of spontaneous symmetry breaking is the presence of a physical\nscalar Higgs particle, with well-defined couplings to fermions and\ngauge bosons. The search for such a particle at lepton and hadron\ncolliders is therefore a paramount goal in particle\nphysics. The LEP experiments have already excluded at 95\\% C.L. the presence of a SM-like\nHiggs with mass below 114.4~GeV~\\cite{Barate:2003sz}.\nIn the coming years, Higgs searches will be performed at\nhadron colliders.\n\nThe Tevatron collider at Fermilab has an active Higgs search program and\nhas already excluded a SM-like Higgs at 95\\% C.L. in the mass range 160--170~GeV~\\cite{:2009pt}.\nThe Tevatron is expected to operate until the end of 2011. It is likely that by this time\nthe CDF and D0 experiments will collect 10--12~$\\mbox{ fb}^{-1}$\napiece and achieve\nsome improvements in the analysis, yielding a significant chance that they\nwill be able to probe the entire SM Higgs mass range\n$110-190\\mbox{ GeV}$. A similar conclusion is reached for the parameter\nspace of the Higgs sector in the $CP$-conserving MSSM, provided that\nthe limits derived from SM-like Higgs searches are statistically combined\nwith limits from direct searches for the non-standard Higgs\nbosons~\\cite{Draper:2009fh}.\n\nThe Large Hadron Collider (LHC) at CERN will begin collisions at\nthe end of 2009\nwith an expected center-of-mass energy of several TeV, and the ATLAS and\nCMS experiments will collect on the order\nof a few hundred pb$^{-1}$ of data during 2010.\nA higher center of mass energy of $14\\mbox{ TeV}$ is expected to be\nachieved after this run, once the necessary upgrades are completed.\nThe anticipated rate of data acquisition at $14 \\mbox{ TeV}$ in the early years of the LHC is expected to be\na few to 10~fb$^{-1}$\/year. Once the LHC acquires a few fb$^{-1}$ of data\nat $14\\mbox{ TeV}$, the Tevatron\nand LHC reaches for a light Higgs boson may be comparable and even\ncomplementary in some searches.\nEventually, once\nthe LHC experiments collect 10 to 30~fb$^{-1}$ of $14\\mbox{ TeV}$ data,\nthe LHC will probe the SM and MSSM Higgs sectors at high statistical\nsignificance, far superior to what is attainable at the Tevatron~\\cite{Aad:2009wy,Ball:2007zza}.\n\nIn this study, we analyze the Tevatron and LHC reach for the MSSM Higgs\nsector~\\cite{ERZ}--\\cite{mhiggsFD2} in the presence of explicit $CP$-violation~\\cite{Pilaftsis:1998dd}--\\cite{Ibrahim:2002zk}.\nAt the Tevatron we consider 10 fb$^{-1}$ in all channels\nand provide projections for a set of\npossible improvements in signal efficiencies. For the LHC, to study\nthe possible initial complementarity with the Tevatron results, we\nconsider the case of 3 fb$^{-1}$ at $14\\mbox{ TeV}$. To display the\nlong-term LHC\ncapabilities, we also show the results for\n30 fb$^{-1}$ at $14\\mbox{ TeV}$ in the most challenging scenario.\nLet us mention that\nthere are alternative\npossibilities for the LHC timeline where the collision energy is kept below $14\\mbox{ TeV}$ for several years, and the upgrade to $14\\mbox{ TeV}$ is\nonly completed later. We do not attempt an analysis of these scenarios\nbecause we base our study on the\nHiggs reach projections presented by the LHC collaborations, which\nare only fully complete for a center of mass energy of $14\\mbox{ TeV}$.\nFor reference, preliminary results indicate that to obtain the same Higgs reach, the\nluminosity at $10\\mbox{ TeV}$ should be twice as large as that\nwhich is required at 14~TeV~\\cite{Delmastro:2009wc}.\n\nOur work differs in three significant ways from previous analyses of\nHiggs searches in the MSSM with\nexplicit $CP$-violation\nperformed in\nRefs.~\\cite{Carena:2002bb,Accomando:2006ga}.\nFirst, for the Tevatron we use the 2009 limits from the CDF\nand D0 experiments given in\nRefs.~\\cite{CDFSM}--\\cite{cdfcharged}. For the LHC\nwe incorporate the projections presented by the experimental collaborations\nin the most recent technical design and expected physics performance\ndocuments~\\cite{Aad:2009wy,Ball:2007zza}. These projections show\nmarked differences from the earlier TDRs, and as a result the priority\nfor some channels has been reduced, while others have been elevated.\nSecondly, for both colliders we analyze the potential for the non-standard MSSM Higgs searches, and provide the combination with the SM-like Higgs reach. These two types of searches offer\nconsiderable complementarity and together can be used to cover most of the\nanalyzed parameter space. For the Tevatron, we also include the reach for the charged Higgs. Finally, we present the combination of the Tevatron discovery reach with the LHC reach at 3~fb$^{-1}$. At this low LHC integrated luminosity, the statistical significances offered by the two colliders may be comparable, and so it may be of interest to perform the combination. We present this analysis only for the SM-like Higgs search channels, which offer greater complementarity than the non-standard channels.\n\nFor the LHC reach we compute and combine discovery significances using Poisson statistics and the profile likelihood ratio, evaluated on data fixed to the expected values for the signals and backgrounds. For the Tevatron we work with 95\\% C.L. upper bounds on the signal presented by CDF and D0, combining them in inverse quadrature. This combination method is strictly valid only in the Gaussian limit; however, it was tested in Ref.~\\cite{Draper:2009fh} and found to match well with the combination derived from a full analysis performed by the collaborations. A further discussion of statistical methods is presented in Appendix A.\n\nIn the MSSM Higgs sector, $CP$-violation can occur via the incorporation of explicit phases in the supersymmetry breaking parameters.\n$CP$-violating phases can be removed from the tree-level Higgs\npotential by field redefinitions. However, phases in the soft trilinear couplings and the gaugino mass\nterms influence the effective Higgs Lagrangian through loop corrections~\\cite{Pilaftsis:1998pe,Pilaftsis:1999qt}.\nWe shall work in the CPX benchmark scenario, defined by the following\nparameter values at the soft scale~\\cite{Carena:2000yi}:\n\\begin{eqnarray*}\nM_S=500\\mbox{ GeV, }& &|A_t|=1\\mbox{ TeV,}\\nonumber\\\\\n\\mu=2\\mbox{ TeV, }& &M_{1,2}=200\\mbox{ GeV,}\\nonumber\\\\\nA_{b,\\tau}=A_t\\mbox{, }& &|M_{\\tilde{g}}|=1\\mbox{ TeV}.\n\\end{eqnarray*}\nIn the above, $A_f$ are the trilinear Higgs sfermion couplings,\n$M_S$ is the characteristic scale of soft supersymmetry breaking\nscalar masses, and $M_{\\tilde{g}}$ is the gluino mass.\nWe set the top quark mass to $m_t=173.1\\mbox{ GeV}$. We shall perform our\nanalysis scanning over the charged Higgs\nmass $M_{H^+}$ and $\\tan\\beta$ over the ranges\n$(100\\mbox{ GeV},400\\mbox{ GeV})$ and $(2,60)$, respectively, for a\nvariety of complex phases of $A_{t,b,\\tau}$ and $M_{\\tilde{g}}$\\footnote{If we began with phases for these parameters at a higher scale, phases for first and second generation trilinear parameters would be generated by RG running, which are highly constrained by EDM measurements~\\cite{Garisto:1996dj}. This can be avoided either by fixing these phases to zero at the weak scale, or by increasing the soft masses of the first and second generation sfermions. However, note that the first and second generation parameters do not have a significant influence on the results presented in this work.}. Masses, mixings, and branching ratios are computed with CPsuperH~\\cite{Lee:2007gn} and HDECAY~\\cite{Djouadi:1997yw}; SM Higgs cross sections are taken from Ref.~\\cite{maltoni} and rescaled to obtain cross sections in the MSSM. The dominant effects of the phases are twofold. First, they cause the neutral Higgs\nmass eigenstates to become admixtures of $CP$-even and $CP$-odd components, modifying\nthe couplings to gauge bosons relative to those in the case with no\n$CP$-violation. For example, the lightest neutral Higgs can now have a\nsignificant $CP$-odd component, strongly suppressing its couplings to the\n$W$ and $Z$ bosons. Secondly, the Yukawa couplings are altered, leading\nin particular to modifications of the neutral Higgs decay branching ratios to $b\\bar{b}$ and\n$\\tau^+\\tau^-$, and different production cross sections through the\nbottom quark fusion and gluon fusion mechanisms. We examine separately\nthe reach in those channels designed to search for a Higgs with SM-like\ngauge couplings (hereafter referred to as an ``SM-like Higgs\"), and in those\nchannels which probe either neutral scalars with negligible gauge couplings\n(hereafter, ``non-standard Higgs\") or charged Higgs states, in order to understand the\ncomplementarity of their coverages. Afterwards we combine the statistical significances of all channels to obtain an overall reach in the MSSM Higgs parameter space.\n\nOur presentation is organized as follows. In section 2 we review the\ncouplings of the MSSM effective Lagrangian that are of particular\nrelevance for understanding the CPX reach. In section 3 we present\nand analyze the results for the Tevatron and make conservative estimates\nof the improvements in signal efficiency necessary to cover large regions\nof parameter space. Section 4 contains the projections for the LHC, and we\noffer conclusions in section 5. Appendix A offers a brief review of statistical methods, followed by a discussion of the\napproximations used in the text to compute and combine exclusion limits\nand discovery significances for multiple channels at the Tevatron and LHC. Appendix B extends some of the discussion in the text to the case of $CP$-conserving benchmark scenarios.\n\n\\section{Effective Yukawa Couplings}\n\nThe radiative corrections to the Yukawa couplings of Higgs states to\ndown-type fermions~\\cite{Carena:1998gk}--\\cite{Carena:2001bg} play a significant role in inducing $CP$-violating\neffects in the Higgs sector and can strongly affect the SM-like Higgs search\nchannels at colliders.\nThe scalar and pseudoscalar neutral Higgs couplings to bottom quarks in the\neffective Lagrangian are given by~\\cite{Carena:2002bb}\n\\begin{equation}\n\\mathcal{L}=-g_fH_i\\bar{b}(g^S_{H_ib\\bar{b}}+\\imath\\gamma_5g^P_{H_ib\\bar{b}})b\n\\end{equation}\nwhere $g_f$ is the SM scalar coupling given by the bottom quark mass over the vacuum expectation value of the Higgs, and $g^{S,P}$ are given by\n\\begin{eqnarray}\ng^S_{H_ib\\bar{b}}&=&\\mbox{Re}\\left(\\frac{1}{1+\\kappa_b\\tan\\beta}\\right)\n\\frac{\\mathcal{O}_{1i}}{\\cos\\beta}+\\mbox{Re}\n\\left(\\frac{\\kappa_b}{1+\\kappa_b\\tan\\beta}\\right)\n\\frac{\\mathcal{O}_{2i}}{\\cos\\beta} \\nonumber\\\\\n&+&\\mbox{Im}\\left(\\frac{\\kappa_b(\\tan^2\\beta+1)}{1+\\kappa_b\\tan\\beta}\\right)\n\\mathcal{O}_{3i}\\nonumber\\\\\ng^P_{H_ib\\bar{b}}&=&-\\mbox{Re}\\left(\\frac{\\tan\\beta-\\kappa_b}\n{1+\\kappa_b\\tan\\beta}\\right)\\mathcal{O}_{3i}\n+\\mbox{Im}\\left(\\frac{\\kappa_b\\tan\\beta}{1+\\kappa_b\\tan\\beta}\\right)\n\\frac{\\mathcal{O}_{1i}}{\\cos\\beta} \\nonumber\\\\\n&-&\\mbox{Im}\\left(\\frac{\\kappa_b}{1+\\kappa_b\\tan\\beta}\\right)\n\\frac{\\mathcal{O}_{2i}}{\\cos\\beta}.\n\\label{effcoup}\n\\end{eqnarray}\nHere $\\mathcal{O}_{jk}$ is the neutral Higgs mixing matrix, where $j$\nis associated with the gauge eigenstate $\\{H^0_u,H^0_d,A\\}$ and\n$k$ runs over the\nmass states $\\{H_1,H_2,H_3\\}$ which are ordered so that $(M_{H_3}\\geq M_{H_2}\\geq M_{H_1})$, and $\\kappa_b$ parameterizes the\nradiative contributions from sbottom-gluino and stop-chargino loops,\n\\begin{eqnarray}\n\\kappa_b&=&\\frac{(\\Delta h_b\/h_b)}{1+(\\delta h_b\/h_b)}\\nonumber\\\\\n\\Delta h_b\/h_b&=&\\frac{2\\alpha_s}{3\\pi}\nM_{\\tilde{g}}^*\\mu^*I(m^2_{\\tilde{d}_1},m^2_{\\tilde{d}_2},|M_{\\tilde{g}}|^2)\n+\\frac{|h_u|^2}{16\\pi^2}A_u^*\\mu^*\nI(m^2_{\\tilde{u}_1},m^2_{\\tilde{u}_2},|M_{\\tilde{g}}|^2)\\nonumber\\\\\n\\delta h_b\/h_b&=&-\\frac{2\\alpha_s}{3\\pi}M_{\\tilde{g}}^*A_b\nI(m^2_{\\tilde{d}_1},m^2_{\\tilde{d}_2},|M_{\\tilde{g}}|^2)\n-\\frac{|h_u|^2}{16\\pi^2}|\\mu|^2\nI(m^2_{\\tilde{u}_1},m^2_{\\tilde{u}_2},|M_{\\tilde{g}}|^2)\n\\end{eqnarray}\nwhere $I(a,b,c)$ is a function that behaves as $1\/\\max(a^2,b^2,c^2)$~\\cite{deltamb1}--\\cite{deltamb2b}.\nThe size of the dominant loop corrections to the Higgs sector is controlled\nby $\\mu$, so a large value of $|\\mu|$ is taken in CPX to accentuate the\n$CP$-violating effects.\n\nFor illustration, let us consider the behavior of the effective couplings\nin the simplest case of vanishing phases. In this scenario $\\Delta h_b\/h_b$\nand $\\delta h_b\/h_b$ take the approximate numerical values $1\/20$ and $-1\/20$,\nrespectively, and $\\kappa_b\\approx\\Delta h_b\/h_b\\approx 1\/20$\\footnote{For\ncomparison, in the $CP$-conserving Maximal Mixing\nscenario $\\kappa_b\\simeq 1\/200$, and in the Minimal Mixing scenario\n$\\kappa_b\\simeq 1\/400$. In both of these scenarios, $|\\mu|= 200$~GeV.}. We denote the $CP$-even mass eigenstates by $h$ and $H$, where $M_h\\leq M_H$. We can always identify $h$ with $H_1$, but due to strong radiative corrections $H$ can either be $H_2$ or $H_3$ depending on $M_{H^+}$ and $\\tan\\beta$.\nThe mass states are related to the gauge eigenstates by a mixing angle $\\alpha$, which satisfies $(-\\sin\\alpha)=\\mathcal{O}_{1h}$ and $\\cos\\alpha=\\mathcal{O}_{1H}$. Furthermore, the pseudoscalar\neffective couplings for these states vanish, and the scalar couplings\nare rescaled relative to their tree level values. At tree level $g^S_{hb\\bar{b}}$ and $g^S_{Hb\\bar{b}}$ are given by $(-\\sin\\alpha\/\\cos\\beta)$ and $(\\cos\\alpha\/\\cos\\beta)$, respectively, and the rescaling factors are given by\n\\begin{eqnarray}\n\\frac{g^S_{hb\\bar{b}}}{-\\sin\\alpha\/\\cos\\beta}&=&\n\\frac{1-\\kappa_b\\cot\\alpha}{1+\\kappa_b\\tan\\beta}\\nonumber\\\\\n\\frac{g^S_{Hb\\bar{b}}}{\\cos\\alpha\/\\cos\\beta}&=&\n\\frac{1+\\kappa_b\\tan\\alpha}{1+\\kappa_b\\tan\\beta}.\n\\label{reltree}\n\\end{eqnarray}\nIn Fig.~\\ref{effhbb} we plot the squares\nof $g^S_{hb\\bar{b}}$ and $g^S_{Hb\\bar{b}}$.\nIn the large $M_{H^+}$ limit, $(-\\sin\\alpha)\\rightarrow\\cos\\beta$\nand $\\cos\\alpha\\rightarrow\\sin\\beta$, so the $h$ scalar coupling converges\nto the SM value. Similarly, in the small $M_{H^+}$ limit, the\n$H_3$ scalar coupling becomes that of the SM.\nFinally, in the non-standard Higgs limit (large $M_{H^+}$ for $H$, or\nsmall $M_{H^+}$ for $h$) and for large $\\tan\\beta$, the effective coupling\nof the non-standard Higgs approaches $\\tan\\beta\/(1+\\kappa_b\\tan\\beta)$, which\nshould be contrasted with the tree level behavior proportional to $\\tan\\beta$.\nAlthough the coupling is still enhanced relative to the SM value, it is\nsuppressed relative to the tree level MSSM value.\n\nIt is important to be precise with terminology: as defined in the\nIntroduction, an SM-like Higgs is one with significant couplings to vector\nbosons, not necessarily one with SM-like fermionic couplings. The\ndistinction is relevant, for example, at moderate values of $M_{H^+}$,\nwhere the first two terms of the $h$ scalar coupling are comparable because of the factor of $\\kappa_b$ in the\nsecond term. For these values of $M_{H^+}$, $h$ can be simultaneously\nSM-like and have an altered coupling to $b\\bar{b}$. Of course, this is true\neven for the tree level coupling, but the effective coupling is modified\nrelative to tree level by the factor in Eq.~\\ref{reltree} which can generate\na significant suppression when $(-\\cot\\alpha) < \\tan\\beta$.\n\nIt is also important to observe that the $\\tau^+\\tau^-H_i$ effective coupling must take a similar form to that of\n$b\\bar{b}H_i$, but with radiative terms proportional to $\\alpha_{1,2}$ instead of $\\alpha_s$. The effects of the threshold corrections are therefore much smaller for the $\\tau^+\\tau^-$ coupling and can be qualitatively neglected.\n\nWhen phases are introduced, $\\mathcal{O}_{3i}$ can become nonzero for all\nstates. Then the third term of $g^S_{H_ib\\bar{b}}$ in\nEq.~\\ref{effcoup} is nonzero and proportional to the imaginary part\nof $\\kappa_b$, which is dominated by the imaginary part of $\\Delta h_b\/h_b$.\nThis term is $\\tan\\beta$-enhanced, so significant phases for $A_t$\nand $M_{\\tilde{g}}$ can affect the conclusions drawn above for vanishing\nphases, potentially even countering the suppression effect in\n$g^S_{H_ib\\bar{b}}$.\n\n\\begin{figure}[!htbp]\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=0.45\\textwidth]{.\/plots\/CPX_0000_mhptb_gSH1bb_nonorm.png} &\n\\includegraphics[width=0.45\\textwidth]{.\/plots\/CPX_0000_mhptb_gSH3bb_nonorm.png}\n\\end{tabular}\n\\caption{Effective bottom Yukawa couplings squared for the $CP$-even Higgs states $h$ and $H$ in the $CP$-conserving\nlimit of CPX.}\n\\label{effhbb}\n\\end{center}\n\\end{figure}\n\n\\section{Tevatron Results}\n\nNeutral Higgs states with SM-like couplings to gauge bosons are sought at\nthe Tevatron in associated production with a $W$ or $Z$ boson and\nin gluon fusion channels, with the Higgs decay to $b\\bar{b}$\nproviding the dominant decay channel in the former\ncase and Higgs decay to $W^+W^-$ providing the dominant channel in the\nlatter~\\cite{Abazov:2008eb}--\\cite{Aaltonen:2008ec}.\nWe compute projections for the expected upper limits on the signal from the combination of these channels at CDF and\nD0 with 10 fb$^{-1}$ per channel and 0\\%, 25\\%, and 50\\%\nimprovements in signal efficiencies. Neutral states with non-standard gauge\ncouplings are probed mainly in the inclusive $\\phi\\rightarrow\\tau^+\\tau^-$\nchannel~\\cite{:2008hu,Abulencia:2005kq}, but also exclusively in associated production with bottom quarks,\nwith the Higgs decaying into either bottom quark or $\\tau$ lepton pairs~\\cite{Abazov:2008zz,CDF3b}.\nFor these channels we consider 7 and 10 fb$^{-1}$, but without\nimprovement in efficiency. The extension of the reach by efficiency\nimprovements can be easily estimated: since the signal scales approximately\nwith $\\tan^2\\beta$, any eventual improvement in efficiency will produce a further\nextension of the reach in $\\tan\\beta$ by the square root of the efficiency\nimprovement.\nFurthermore, the 95\\% C.L. expected upper bound on the signal with\n25\\% improvements is essentially equivalent to the 90\\% limit with no\nimprovements\\footnote{This can be understood from the approximate formula for\nthe expected n$\\sigma$ upper limit on the signal,\n$s_{n\\sigma}\\approx n\\sqrt{b}\/(\\epsilon\\sqrt{L})$, where $L$ is the\nluminosity, $\\epsilon$ is the signal efficiency, and $b$ is the expected\nbackground.}. The charged Higgs is sought in decays of the top quark~\\cite{Collaboration:2009zh,Abulencia:2005jd}, $t\\rightarrow H^+b$, with $H^+\\rightarrow\\tau^+\\nu$ providing the dominant $H^+$ decay channel for $\\tan\\beta > 1$. We present the non-standard Higgs results in combination with those from the charged Higgs, since both particles can be classified together as strictly beyond-the-SM scalars. Finally, we combine the two classes of searches, SM-like and non-standard~+~charged, to derive\nthe strongest possible constraint on the MSSM Higgs parameter space. In all figures the shaded gray regions denote exclusion limits from\nLEP~\\cite{Schael:2006cr}, and solid black indicates theoretically\ndisallowed regions.\n\nNote that the expected 95\\% C.L. limits are obtained under the assumption that the data reflects only the average number of background events. If signal is also present and the data reflects the average value of signal+background for some point in the MSSM parameter space, the observed limit will be somewhat weaker, with an average value given by $R_{obs}\\approx R_{exp}+1$, where $R$ is the upper bound on the signal normalized to the expected signal in the MSSM\\footnote{Interestingly enough, the present combined Tevatron bounds on the SM Higgs given in Ref.~\\cite{Collaboration:2009je} show an observed bound that differs by about 1 from the expected bound in the low mass range. At $M_h=115\\mbox{ GeV}$, $R_{exp}=1.78$ and $R_{obs}=2.7$.}. For further discussion, see Appendix A.\n\n\\begin{figure}[!htbp]\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=0.40\\textwidth]{.\/plots\/CPX_0000_mhptb_notau_1.png} &\n\\includegraphics[width=0.40\\textwidth]{.\/plots\/CPX_9090_mhptb_notau_1.png} \\\\\n\\includegraphics[width=0.40\\textwidth]{.\/plots\/CPX_0000_mhptb_onlytau_1.png} &\n\\includegraphics[width=0.40\\textwidth]{.\/plots\/CPX_9090_mhptb_onlytau_1.png} \\\\\n\\includegraphics[width=0.40\\textwidth]{.\/plots\/CPX_0000_mhptb_1.png} &\n\\includegraphics[width=0.40\\textwidth]{.\/plots\/CPX_9090_mhptb_1.png}\n\\end{tabular}\n\\caption{Projected Tevatron exclusion contours at 90\\% and 95\\% C.L. in the\nCPX scenario with $CP$-violating phases $\\arg(A_{t,b,\\tau})=0^{\\circ}$,\n$\\arg(M_{\\tilde{g}})=0^{\\circ}$ (\\textit{left}) and\n$\\arg(A_{t,b,\\tau})=90^{\\circ}$, $\\arg(M_{\\tilde{g}})=90^{\\circ}$\n(\\textit{right}). Row 1 gives results for SM-like Higgs searches,\nrow 2 includes only the non-standard Higgs searches, and row 3\ngives the combined constraints.}\n\\label{CPX0000}\n\\end{center}\n\\end{figure}\n\nIn this work we will consider three sets of phases for\n$A_{t,b,\\tau}$ and $M_{\\tilde{g}}$: $(0^{\\circ},0^{\\circ})$,\n$(90^{\\circ},90^{\\circ})$, and $(140^{\\circ},140^{\\circ})$. In the course\nof our study we examined other values of the $CP$-violating phases, including\nthe departure from setting common phases for $A_{t,b,\\tau}$ and\n$M_{\\tilde{g}}$. However, the results were qualitatively similar, and\nin particular, all unique features of interest also appeared in one or more\ncases discussed here.\n\n\nWe consider first the case without explicit $CP$-violation, in order\nto understand features which are independent of the phases.  The results\nfrom SM-like searches, non-standard + charged Higgs search channels, and the combination are\ngiven in the first column of Fig.~\\ref{CPX0000}. The decoupling limit\nis probed at 95\\% C.L. with a 50\\% improvement in signal efficiency for the\nSM-like search channels. In this limit $M_{H_1}\\approx 121\\mbox{ GeV}$ with\nthe top mass set to $173.1\\mbox{ GeV}$. Smaller efficiency improvements more\nreadily probe the region of lower $\\tan\\beta$,\nwhere $M_{H_1}\\lesssim 121\\mbox{ GeV}$ and the SM Higgs constraint is stronger.\n\nHere a word about experimental mass resolution is in order. If the mass\ndifference between two Higgs states is under a certain finite threshold,\nthe detectors cannot resolve the particles. Consequently, their statistical\nsignificances should be added directly rather than in quadrature,\nleading to a stronger limit.  We take a representative experimental mass\nresolution of $10\\mbox{ GeV}$. The main effect of this approximate treatment appears as a discontinuous\nspike in the range $M_{H^+}\\approx 135-155\\mbox{ GeV}$ of the non-standard\nHiggs search constraint. The reason for this behavior is that in this range of $M_{H^+}$ and for moderate\nto large $\\tan\\beta$, radiative corrections drive a $CP$-even Higgs mass to within $10\\mbox{ GeV}$ of the $CP$-odd Higgs mass. For slightly lower or higher values of $M_{H^+}$ the approximate degeneracy is lifted. In all subsequent figures of this work, we include the finite mass resolution effects.\n\n\nContrary to the $CP$-conserving cases analyzed in Ref.~\\cite{Draper:2009fh},\nthe SM-like Higgs search constraints become weaker with moderate,\ndecreasing $M_{H^+}$. At tree level, the increase in $\\mathcal{O}_{11}$\nsignificantly enhances the coupling of the SM-like Higgs to down-type fermions\nfor smaller $M_{H^+}$, typically leading to a large region just above the\nintense coupling regime where the constraint is stronger than in the\ndecoupling limit (see, for example, the Maximal Mixing scenario examined in\nRef.~\\cite{Draper:2009fh}). This feature does not appear in the CPX scenario\nin the absence of phases. The reason is that, as discussed before and\nshown in Fig.~\\ref{effhbb},\nthe $\\kappa_b$ threshold corrections in Eq.~\\ref{effcoup} are significant due\nto the large value of $\\mu$ taken in CPX. These corrections suppress the\n$b\\bar{b}H_1$ effective coupling relative to the tree level value, and therefore $Br(H_1\\rightarrow b\\bar{b})$ is\ndecreased while $Br(H_1\\rightarrow \\tau^+\\tau^-)$ is increased in this region.\nThis can be seen from the upper left plot of Fig.~\\ref{h1tth1bb}, where we\nshow the ratio of the $\\tau^+\\tau^-$ to $b\\bar{b}$ branching ratios, each\nnormalized to their SM values.\n\n\\begin{figure}[!htbp]\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=0.40\\textwidth]\n{.\/plots\/CPX_0000_mhptb_BrH1ttH1bb.png} &\n\\includegraphics[width=0.40\\textwidth]\n{.\/plots\/CPX_9090_mhptb_BrH1ttH1bb.png} \\\\\n\\end{tabular}\n\\includegraphics[width=0.40\\textwidth]\n{.\/plots\/CPX_140140_mhptb_BrH1ttH1bb.png} \\\\\n\\caption{The ratio of $H_1$ branching ratios into $\\tau^+\\tau^-$\nand $b\\bar{b}$, normalized to their SM values, for vanishing\nphases (\\textit{upper left}), for $\\arg(A_{t,b,\\tau})=90^{\\circ}$,\n$\\arg(M_{\\tilde{g}})=90^{\\circ}$ (\\textit{upper right}), and for $\\arg(A_{t,b,\\tau})=140^{\\circ}$,\n$\\arg(M_{\\tilde{g}})=140^{\\circ}$ (\\textit{bottom}).}\n\\label{h1tth1bb}\n\\end{center}\n\\end{figure}\n\nThe constraint from the non-standard Higgs search in the $\\tau^+\\tau^-$\ninclusive channel is similar to what is obtained in other benchmark scenarios\ngeared towards the $CP$-conserving MSSM. At tree level the non-standard Higgs\nhas a $\\tan\\beta$ enhanced coupling to $\\tau^+\\tau^-$ that is not subject to\nlarge radiative corrections, and it is light enough to be produced for low to\nmoderate $M_{H^+}$. As a result the constraint is significant in this region.\nThe radiative increase in the $\\tau^+\\tau^-$ branching fraction due\nto the suppression of $b\\bar{b}$ is mostly compensated by the threshold\nsuppression of the non-standard Higgs coupling to bottom quarks, which enters\nin both of the dominant production mechanisms of $b\\bar{b}$ fusion and\ngluon fusion through a bottom loop~\\cite{Carena:2005ek}. $H_1$ becomes highly non-standard for moderate to large $\\tan\\beta$ and $M_{H^+}\\lesssim 150\\mbox{ GeV}$.\nTherefore the limit comes mostly from $H_1$ and the $CP$-odd Higgs for $M_{H^+}\\lesssim 150\\mbox{ GeV}$, and\nfrom the $CP$-odd and heavy $CP$-even Higgs for larger $M_{H^+}$. The non-standard constraint\nhas the virtue of mostly filling the dip in the LEP constraint\nat $M_{H^+}\\approx 140\\mbox{ GeV}$, which was due to a marginal excess\nin the LEP data around $M_{H_1}\\approx 90\\mbox{ GeV}$. The charged Higgs searches from top decays become also\nrelevant in this region of parameters, although in the absence of phases they do not independently reach 95\\% C.L. due to the large bottom coupling suppression in this scenario~\\cite{Carena:1999py}.\n\nWe stress the fact that neither the SM-like nor the non-standard~+~charged Higgs\nsearch is sufficient to reach the entire plane even with significant improvement.\nHowever, each search is most effective in the region where the other is\nweakest, providing excellent complementarity and strongly motivating a\nstatistical combination. In the bottom left plot of Fig.~\\ref{CPX0000},\nwe demonstrate that the combination of SM-like searches with 50\\%\nimprovements in signal efficiency and non-standard~+~charged search channels is\nsufficient to cover the entire region previously unprobed by LEP.\n\n\nNow we consider the effects of $CP$-violation, setting\n$\\arg(A_{t,b,\\tau})=\\arg(M_{\\tilde{g}})=90^{\\circ}$. The results are\npresented in the right-hand column of Fig.~\\ref{CPX0000}. In the\ndecoupling limit, $H_1$ is still SM-like; however,\nfor $M_{H^+}\\lesssim 160\\mbox{ GeV}$, $H_1$ becomes mostly $CP$-odd.\n$H_2$ is $CP$-odd in the decoupling limit, but transitions rapidly to become\nSM-like around $M_{H^+}\\approx 150\\mbox{ GeV}$, and\nfinally acquires non-standard couplings to gauge bosons for\n$M_{H^+}\\lesssim 135\\mbox{ GeV}$.\nThe fast transitions create a region of increased sensitivity compared to the $CP$-conserving case centered around\n$M_{H^+}\\approx 150\\mbox{ GeV}$ and stretching from low to moderate\n$\\tan\\beta$. Although $g_{ZZH_2}$ increases with $\\tan\\beta$, the $H_2\\rightarrow b\\bar{b}$ branching ratio suppression limits the height of the region. The region ends sharply at $M_{H^+}\\approx 130\\mbox{ GeV}$, where $H_3$ has become SM-like but the opening of the $H_3\\rightarrow H_1H_1$ channel heavily reduces the $H_3\\rightarrow b\\bar{b}$ branching ratio. As before the LEP constraint from $e^+e^-\\rightarrow H_1H_2\\rightarrow4b,2b2\\tau$ takes over for lower values of $M_{H^+}$.\n\nThe phase for $M_{\\tilde{g}}$ influences the Higgs masses more mildly than phases\nfor the trilinear couplings because it enters the mass matrix only at\nthe 2-loop level. Nonetheless, it strengthens the SM-like constraint around\n$M_{H^+}\\approx 200\\mbox{ GeV}$, primarily by counteracting the threshold\nsuppression of $g^S_{H_ib\\bar{b}}$ as discussed in Section 2 and leading to\na $g^S_{H_ib\\bar{b}}$ that is enhanced over the tree level value in this\nregion. Correspondingly the feature familiar from the Maximal Mixing scenario,\nwhich we noted earlier was absent in CPX with vanishing phases, has begun to\nreemerge in the small $M_{H^+}$, moderate $\\tan\\beta$ region. For comparison with the case of vanishing phases,\nin the upper right plot of Fig.~\\ref{h1tth1bb} we again present the ratio of\nthe $b\\bar{b}$ to $\\tau^+\\tau^-$ branching ratios, each normalized to their\nSM values, now in the presence of the $90^{\\circ}$ phases.\n\nThe non-standard~+~charged Higgs reach is similar to the case without phases. The two\nspikes, most visible on the combined plot, are the result of the mass\nresolution prescription discussed above. Below\n$M_{H^+}=150\\mbox{ GeV}$, $H_3$ and $H_2$ are within $10\\mbox{ GeV}$;\nabove, $H_2$ and $H_1$ share this property. From the final figure, it is\nagain evident that the combination of channels is essential to cover nearly the entire plane at 95\\% C.L.\n\nWe note that with $90^{\\circ}$ phases there appears a small hole in the\nLEP coverage at low $M_{H^+}$ and low $\\tan\\beta$ adjacent to the\ntheoretically disallowed region, which is unprobed by any of the\nTevatron search channels.\nThe presence of this hole was\nfirst discussed in Ref.~\\cite{Carena:2002bb} and is generated by the possible\ndecay of the SM-like Higgs boson into a pair of $H_1$'s, which acquire a\nsignificant $CP$-odd Higgs component. The hole is discussed in detail in Ref.~\\cite{Williams:2007dc}\nand channels which may help to cover it at hadron colliders are studied in\nRef.~\\cite{Kaplan:2009qt}--\\cite{Chang:2005ht}.\nOur results are qualitatively consistent with the LEP experiment plots\nfrom Ref.~\\cite{Schael:2006cr}\\footnote{In our plots the hole is somewhat\nsmaller than in Ref.~\\cite{Schael:2006cr} due to the finite grid size in our scan and\nthe approximations we used to implement the LEP constraints.}.\nIn the $M_{H^+}$ coordinates this hole appears as a very small region;\nhowever, it covers a significant portion of the range\n$M_{H_1}\\lesssim 45\\mbox{ GeV}$, and indicates that a light Higgs scenario\nphenomenologically similar to what has been proposed in the context of the\nNMSSM~\\cite{Dermisek:2005gg} has still not been fully ruled out in the\nMSSM. In this hole the decays $H_2\\rightarrow H_1H_1\\rightarrow 4b,4\\tau$\nconsidered by LEP are significant; however, since $\\tan\\beta$ is small,\n$H_2$ is of mixed composition and its dominant production mechanisms are\nsuppressed. Recently, the authors of Ref.~\\cite{Cranmer} reanalyzed the ALEPH data, extending the reach of the $4\\tau$ channel to higher values of $M_{H_2}$. However, this channel still only covers a small subset of this hole, because it is only efficient for $2m_{\\tau}<M_{H_1}<2m_b$, and the $ZZH_2$ coupling is typically suppressed in this region.\n\n\\begin{figure}[!htbp]\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=0.40\\textwidth]\n{.\/plots\/CPX_140140_mhptb_notau_1.png} \\\\\n\\includegraphics[width=0.40\\textwidth]\n{.\/plots\/CPX_140140_mhptb_onlytau_1.png} \\\\\n\\includegraphics[width=0.40\\textwidth]\n{.\/plots\/CPX_140140_mhptb_1.png} \\\\\n\\end{tabular}\n\\caption{Projected Tevatron exclusion contours at 90\\% and 95\\% C.L.\nin the CPX scenario with $CP$-violating phases\n$\\arg(A_{t,b,\\tau})=140^{\\circ}$, $\\arg(M_{\\tilde{g}})=140^{\\circ}$.\nRow 1 gives results for SM-like Higgs searches, row 2 includes only the\nnon-standard Higgs searches, and row 3 gives the combined constraints.}\n\\label{CPX140140}\n\\end{center}\n\\end{figure}\n\nFinally, we consider a case with larger values for the phases, with\nprojections given in Fig.~\\ref{CPX140140}. For\n$\\arg(A_{t,b,\\tau})=140^{\\circ}$ and $\\arg(M_{\\tilde{g}})=140^{\\circ}$,\nthe coverage appears more unusual than in previous cases.\nNear $M_{H^+}\\approx 160\\mbox{ GeV}$ and $\\tan\\beta\\approx 45$, $M_{H_1}$\nand $M_{H_2}$ become degenerate. Around this point there is sufficient variation in the mixing\nmatrix so that in one direction the SM Higgs becomes strongly up-type, suppressing the $\\tau^+\\tau^-$ width, while in another direction the different terms in Eq.~\\ref{effcoup}\ninterfere destructively and suppress the $b\\bar{b}$ width. The result is demonstrated in the bottom plot of Fig.~\\ref{h1tth1bb}: there exist both large areas of parameter space\nwhere the branching ratio to $b\\bar{b}$ is suppressed while $\\tau^+\\tau^-$ is\nenhanced, and large areas where the reverse occurs. The former leads to the\nunprobed red stripes in the SM-like searches, centered on the degeneracy point\nat moderate to large $\\tan\\beta$, but the latter causes the coverage to be\nextended to a 95\\% C.L. limit with only a 25\\% efficiency improvement for\nlarge $\\tan\\beta$ and moderate to large $M_{H^+}$.\n\nThe LEP coverage traces\nout a region of moderate $\\tan\\beta$ where sbottom corrections to\n$M_{H_1}$ are maximized~\\cite{Carena:2002bb}. These corrections are negative and proportional\nto $|h_b|^4$, where $h_b$ is the effective bottom quark Yukawa coupling.\nThe modulus depends on the $CP$-violating phases in such a way that\nif $\\cos\\phi<0$ (assuming a common phase $\\phi$ for $A_t$\nand $M_{\\tilde{g}}$), then $M_{H_1}$ has a local minimum\nat $\\cot\\beta=-|\\Delta h_b\/h_b|\\cos\\phi$. As we noted earlier,\n$|\\Delta h_b\/h_b|\\approx -1\/20$, so around\n$\\tan\\beta\\approx 25$ the LEP constraints are stronger than\nfor either lower or higher values of $\\tan\\beta$.\n\nIn the $(140^{\\circ},140^{\\circ})$ case the utility of the non-standard\nand SM-like combination is particularly manifest. Most of parameter space\ncan be covered by SM-like searches, with the exception of the stripes where\nthe $b\\bar{b}$ branching ratio of the SM-like Higgs is suppressed. The\nstripes occur precisely in the low $M_{H^+}$, large $\\tan\\beta$ region where\nthe non-standard search is most effective. This effect is not unique to the\nMSSM with $CP$-violation; a similar suppression of the $b\\bar{b}$ channels\nin this region and the corresponding complementarity of coverage exists in\nthe small-$\\alpha_{eff}$ scenario studied in Refs.~\\cite{Draper:2009fh},\\cite{Carena:2002qg}--\\cite{Hahn:2006my}.\n\n\\section{LHC Results}\n\nFor the LHC we examine the discovery reach for an integrated luminosity of 3 fb$^{-1}$ in all channels. For the case of $140^{\\circ}$ phases, which is the most difficult to probe, we also present the reach for 30 fb$^{-1}$. Table~\\ref{Tch} lists the search channels we use from CMS, taken from Refs.~\\cite{Ball:2007zza,CMSWW1,CMSWW2}, and the channels from ATLAS, given in Ref.~\\cite{Aad:2009wy}. The searches for $\\phi\\rightarrow\\tau^+\\tau^-,\\mu^+\\mu^-$ in association with bottom quarks are probes of the non-standard Higgs; the rest are SM-like Higgs channels. The experimental studies of the non-standard channels present the expected $5\\sigma$ contour in the $(M_{A},\\tan\\beta)$ plane in $CP$-conserving benchmark scenarios. However, to a good approximation the production cross section of the non-standard Higgs via bottom quark fusion is independent of the benchmark values, and scales as $\\tan^2\\beta$ in most of parameter space. Therefore we employ this scaling relation to extend the experimental analyses to expected significances on the full $(M_{H^+},\\tan\\beta)$ plane in CPX. Furthermore, for simplicity we omit channels which probe the charged Higgs, where the expected discovery region is mostly\\footnote{Small regions at low $\\tan\\beta\\lesssim 5$ that are not reached by the non-standard Higgs searches may be probed by the charged Higgs channels. However, these regions tend to be either theoretically disallowed or already excluded by LEP.} a subset of the region probed by the non-standard neutral Higgs at high significance.\n\n\\begin{table}[!htbp]\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n  \\hline\n \n  Experiment & Production & Decay \\\\\n  \\hline\n  \\hline\n  & Weak Boson Fusion & $\\phi\\rightarrow\\tau^+\\tau^-$ \\\\\n  \\cline{2-3}\n  & Inclusive & $\\phi\\rightarrow\\gamma\\gamma,W^+W^-,ZZ$ + tagged\\\\\n  & & leptons from each $W$ and $Z$\\\\\n  \\cline{2-3}\n  CMS & Weak Boson Fusion & $\\phi\\rightarrow W^+W^-$ with one tagged\\\\\n  & & lepton and two jets \\\\\n  \\cline{2-3}\n  & $t\\bar{t}\\phi$ & $\\phi\\rightarrow b\\bar{b}$ \\\\\n  \\cline{2-3}\n  & $b\\bar{b}\\phi$ & $\\phi\\rightarrow\\tau^+\\tau^-,\\mu^+\\mu^-$ \\\\\n  \\hline\n  & Weak Boson Fusion + Gluon Fusion & $\\phi\\rightarrow W^+W^-$ \\\\\n  \\cline{2-3}\n  & Inclusive & $\\phi\\rightarrow\\gamma\\gamma,ZZ$ \\\\\n  \\cline{2-3}\n  ATLAS & Weak Boson Fusion & $\\phi\\rightarrow\\tau^+\\tau^-$ \\\\\n  \\cline{2-3}\n  & $t\\bar{t}\\phi$ & $\\phi\\rightarrow b\\bar{b},W^+W^-$ \\\\\n  \\cline{2-3}\n  & W$\\phi$ & $\\phi\\rightarrow b\\bar{b}$ \\\\\n  \\cline{2-3}\n  & $b\\bar{b}\\phi$ & $\\phi\\rightarrow\\tau^+\\tau^-,\\mu^+\\mu^-$ \\\\\n  \\hline\n\\end{tabular}\n  \\caption{Search channels from the LHC employed in this study.}\n    \\label{Tch}\n\\end{center}\n\\end{table}\n\nThe left-hand column of Fig.~\\ref{LHC0000} gives the projections\nfor $(\\arg(A_{t,b,\\tau}),\\arg(M_{\\tilde{g}}))=(0^{\\circ},0^{\\circ})$, and\nthe right-hand column contains the case with phases set to $(90^{\\circ},90^{\\circ})$.\nFig.~\\ref{LHC140140} examines the reach with $(140^{\\circ},140^{\\circ})$, where\nthe left-hand column assumes 3~fb$^{-1}$ and the right-hand column uses 30~fb$^{-1}$.\n\nThe dominant SM-like search channels in all cases are\n$\\phi\\rightarrow\\tau^+\\tau^-$ with weak boson fusion (WBF)\nproduction~\\cite{Rainwater:1999sd,Plehn:1999nw,Plehn:1999xi} and the inclusive search for $\\phi\\rightarrow\\gamma\\gamma$~\\cite{Carena:1999bh}.\nSince the $\\tau^+\\tau^-$ channel relies on the coupling of the Higgs state\nto vector bosons, it is indeed an SM-like search; however, the strong\nenhancement of the $\\phi\\rightarrow\\tau^+\\tau^-$ branching ratio evident in Fig.~\\ref{h1tth1bb}\nimplies that this search is strongest in the intense\ncoupling regime, where multiple states may have simultaneously moderate\ngauge couplings and enhanced branchings to $\\tau^+\\tau^-$.\nHowever, as shown in Fig.~\\ref{h1tth1bb}, the introduction of $90^{\\circ}$\nphases reduces the\n$\\tan\\beta$-enhanced suppression of the coupling to bottom quarks,\npreventing the $\\tau^+\\tau^-$ branching fraction from becoming as large as\nin the case with vanishing phases. This makes the $\\tau^+\\tau^-$ channel\nrelatively weaker than in the case without $CP$-violation, and in particular\ncauses the low $\\tan\\beta$, large $M_{H^+}$ region to be unprobed by this\nchannel. Instead, the region is covered by the $\\phi\\rightarrow\\gamma\\gamma$\nsearch, which can probe the decoupling limit\nat $2-3\\sigma$ with 3 fb$^{-1}$.\n\nAs discussed previously for the Tevatron, in the $(140^{\\circ},140^{\\circ})$\nscenario there appears a region $M_{H^+}\\gtrsim 200\\mbox{ GeV}$\nand $\\tan\\beta\\gtrsim 45$ where the $H_1\\rightarrow\\tau^+\\tau^-$ branching ratio is significantly suppressed. The individual limits from the $\\tau^+\\tau^-$ and $\\gamma\\gamma$ channels are given for the case of 3~fb$^{-1}$ in Fig.~\\ref{LHC140indiv}. The $H_1\\rightarrow \\gamma\\gamma$ channel\nis strong enough to probe this region at the $2-3\\sigma$ level\nwith 3 fb$^{-1}$, or the $6-8\\sigma$ level with 30 fb$^{-1}$.\nHowever, there are also significant holes in the LEP coverage for lower values of $\\tan\\beta$ across most of the range of $M_{H^+}$, and in these regions, the branching fractions of the SM-like Higgs to $\\tau^+\\tau^-$ and $\\gamma\\gamma$ are always suppressed. Even after combining the channels there remain regions centered at $M_{H^+}\\approx 150\\mbox{ GeV}$ and $M_{H^+}\\approx 200\\mbox{ GeV}$ that may be probed at less than $2\\sigma$ with 3 fb$^{-1}$. Furthermore, when we increase the luminosity to 30 fb$^{-1}$, $5\\sigma$ discovery reach contour still does not cover the region around $150\\mbox{ GeV}$ and $\\tan\\beta\\approx 10-15$.\n\nIn all sets of phases, the combination of SM-like and non-standard channels considerably strengthens the statistical significance of the LHC results with 3 fb$^{-1}$. Note that in the context of this combination, a $5\\sigma$ significance does not necessarily imply a resonance peak in the data with this significance corresponding to any Higgs state; for example, it could be generated by only a $3\\sigma$ excess coming from the SM-like Higgs combined with a $4\\sigma$ excess from the nonstandard Higgs. The correct interpretation is that taken together, these excesses indicate an exclusion of the background-only hypothesis at $5\\sigma$. Of course, it is desirable to eventually discover the particles individually and measure their properties. This will take longer, but can be achieved with a higher integrated luminosity: we have checked that when the data set approaches 30 fb$^{-1}$, it is no longer necessary to go beyond the SM-like searches in order to probe the whole plane at 5$\\sigma$ in\nthe cases of vanishing and $90^{\\circ}$ phases. However, as mentioned previously, for $140^{\\circ}$ phases there is still a small region at low $\\tan\\beta$ and low $M_{H^+}$ which is unconstrained by LEP and is not probed at 5$\\sigma$ by the SM-like\nchannels with 30 fb$^{-1}$. This demonstrates that even with a considerable amount of data the combined analysis of standard and non-standard Higgs searches may still remain relevant.\n\nIt is interesting to note that whereas the strongest LHC SM-like Higgs searches are in the $\\tau^+\\tau^-$ and $\\gamma\\gamma$ channels, the Tevatron searches are mainly sensitive to the $b\\bar{b}$ branching ratio. This generates complementarity in the coverage offered by the two colliders.\nIn Fig.~\\ref{clashofthetitans} we give the discovery significance plots for each set of phases obtained by combining the projected reach of the SM-like search channels with 3~fb$^{-1}$ of data from the LHC and 10~fb$^{-1}$ + 50\\% efficiency\nimprovements from the Tevatron\\footnote{In our analysis of the Tevatron in Section 3, we test the signal~+~background hypothesis for the\npurpose of limit-setting, whereas for the LHC we calculate discovery significance, which is a test of the background-only hypothesis.\nHowever, in the limit of large backgrounds, the discovery significance at the Tevatron is inversely proportional to the\nthe upper bound derived on the signal. Further discussion of this approximation and the combination of channels is given in Appendix A.}. Particularly in the cases with nonzero phases, there are large regions where the LHC coverage is at the $2-3\\sigma$ level or lower. In combination with the Tevatron the potential reach achieves $3\\sigma$ on most of the plane. As discussed in the Introduction, depending on the data-taking rate of the LHC when it reaches a few fb$^{-1}$, this complementarity suggests that it may be worthwhile to perform the combination. Further examples of Tevatron-LHC combinations are presented in Appendix B for standard $CP$-conserving benchmark scenarios.\n\n\n\n\\begin{figure}[!htbp]\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=0.40\\textwidth]{.\/plots\/LHC_CPX_SM_0000_mhptb_3ifb_1.png} &\n\\includegraphics[width=0.40\\textwidth]{.\/plots\/LHC_CPX_SM_9090_mhptb_3ifb_1.png} \\\\\n\\includegraphics[width=0.40\\textwidth]{.\/plots\/LHC_CPX_NS_0000_mhptb_3ifb_1.png} &\n\\includegraphics[width=0.40\\textwidth]{.\/plots\/LHC_CPX_NS_9090_mhptb_3ifb_1.png} \\\\\n\\includegraphics[width=0.40\\textwidth]{.\/plots\/LHC_CPX_SMNS_0000_mhptb_3ifb_1.png} &\n\\includegraphics[width=0.40\\textwidth]{.\/plots\/LHC_CPX_SMNS_9090_mhptb_3ifb_1.png} \\\\\n\\end{tabular}\n\\caption{Projected LHC significance contours for 3 fb$^{-1}$ in the CPX scenario with phases $\\arg(A_{t,b,\\tau})=0^{\\circ}$, $\\arg(M_{\\tilde{g}})=0^{\\circ}$ (\\textit{left}), and $\\arg(A_{t,b,\\tau})=90^{\\circ}$, $\\arg(M_{\\tilde{g}})=90^{\\circ}$ (\\textit{right}). Row 1 gives results for SM-like Higgs searches, row 2 includes only the non-standard Higgs searches, and row 3 gives the combined constraints.}\n\\label{LHC0000}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[!htbp]\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=0.40\\textwidth]{.\/plots\/LHC_CPX_SM_140140_mhptb_3ifb_1.png} &\n\\includegraphics[width=0.40\\textwidth]{.\/plots\/LHC_CPX_SM_140140_mhptb_30ifb_1.png} \\\\\n\\includegraphics[width=0.40\\textwidth]{.\/plots\/LHC_CPX_NS_140140_mhptb_3ifb_1.png} &\n\\includegraphics[width=0.40\\textwidth]{.\/plots\/LHC_CPX_NS_140140_mhptb_30ifb_1.png} \\\\\n\\includegraphics[width=0.40\\textwidth]{.\/plots\/LHC_CPX_SMNS_140140_mhptb_3ifb_1.png} &\n\\includegraphics[width=0.40\\textwidth]{.\/plots\/LHC_CPX_SMNS_140140_mhptb_30ifb_1.png} \\\\\n\\end{tabular}\n\\caption{Projected LHC significance contours in the CPX scenario with phases $\\arg(A_{t,b,\\tau})=140^{\\circ}$, $\\arg(M_{\\tilde{g}})=140^{\\circ}$, for 3~fb$^{-1}$ (\\textit{left}) and 30~fb$^{-1}$ (\\textit{right}). Row 1 gives results for SM-like Higgs searches, row 2 includes only the non-standard Higgs searches, and row 3 gives the combined constraints.}\n\\label{LHC140140}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[!htbp]\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=0.40\\textwidth]{.\/plots\/LHC_CPX_VBFtautau_140140_mhptb_3ifb_1.png} &\n\\includegraphics[width=0.40\\textwidth]{.\/plots\/LHC_CPX_Inclgaga_140140_mhptb_3ifb_1.png} \\\\\n\\end{tabular}\n\\caption{Projected LHC significance contours in the CPX scenario with phases $\\arg(A_{t,b,\\tau})=140^{\\circ}$, $\\arg(M_{\\tilde{g}})=140^{\\circ}$ and 3~fb$^{-1}$, derived from the WBF $\\phi\\rightarrow\\tau^+\\tau^-$ channel  (\\textit{left}) and the inclusive $\\phi\\rightarrow\\gamma\\gamma$ channel (\\textit{right}).}\n\\label{LHC140indiv}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[!htbp]\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=0.40\\textwidth]{.\/plots\/TEVLHC_CPX_SM_0000_mhptb.png} &\n\\includegraphics[width=0.40\\textwidth]{.\/plots\/TEVLHC_CPX_SM_9090_mhptb.png} \\\\\n\\end{tabular}\n\\includegraphics[width=0.40\\textwidth]{.\/plots\/TEVLHC_CPX_SM_140140_mhptb.png}\n\\caption{Projected discovery significance contours from the combination of 10 fb$^{-1}$ + 50\\% efficiency-improved Tevatron data and 3 fb$^{-1}$ LHC data, using SM-like search channels in the CPX scenario for each of the previously considered sets of phases.}\n\\label{clashofthetitans}\n\\end{center}\n\\end{figure}\n\n\\section{Conclusions}\n\nIn this work we have analyzed in detail the Tevatron and LHC reach for the\nHiggs sector in the MSSM with explicit $CP$-violating phases,\nproviding the most up-to-date projections for the exclusion and\ndiscovery potential of these machines in this scenario.\nOur primary goals\nregarding the Tevatron search analyses\nwere to outline improvement factors necessary to probe\nmost of the parameter space, and to exhibit the considerable complementarity\noffered by search channels applicable for Higgs states with SM-like gauge\ncouplings, and channels relevant for charged Higgs bosons and\nneutral Higgs states with negligible gauge couplings\nbut enhanced couplings to fermions.\nWe find that taken in statistical\ncombination these two classes of Higgs searches can probe the\nentire parameter plane in a benchmark scenario that exhibits strong\n$CP$-violating  influence on the MSSM Higgs sector.\n\n\nThe low luminosity LHC offers complementary capabilities to the Tevatron.\nThe SM-like Higgs searches tend to be more efficient in the regions\nwhere the Higgs decay branching ratio into bottom quark pairs is suppressed,\na region that becomes quite difficult for searches at the Tevatron. With this in mind,\nwe have examined the possibility of combining SM-like Higgs search\nresults from the Tevatron and LHC, and found that it can serve to\nprobe nearly the whole parameter space at more than 2$\\sigma$ in regions in\nwhich neither the Tevatron nor a low-luminosity LHC can do so by\nstandard Higgs channels alone. Therefore,\nin these cases the combination may be a worthwhile exercise before the LHC has run for the\nfew years necessary to collect a large data sample. Furthermore, as\nhappens with the Tevatron, the combination of standard and\nnon-standard Higgs searches at the LHC allows essentially full coverage of the parameter space\nat more than 2$\\sigma$ with low integrated luminosity.  Finally, we showed\nthat this combination remains a powerful tool to disclose the full LHC discovery potential at\nhigher luminosities.\n\nIn our treatment of the CP-violating MSSM in this study, and previously in the CP-conserving case examined in Ref.~\\cite{Draper:2009fh}, we took specific representative values of the soft supersymmetry breaking parameters and the Higgsino mass parameter at the electroweak scale in order to study the reach of collider experiments for certain generic or interesting features of the MSSM Higgs sector. Constraints on the Higgs sector that can be derived from flavor physics or dark matter considerations are very sensitive to the supersymmetry breaking scale and to precise relations between the soft supersymmetry breaking parameters of the different generations as well as the gaugino masses. For this reason,\nwe restricted our analysis to the direct collider constraints on the Higgs sector alone. However, considering the renormalization group evolution of these parameters, our procedure can be applied to study particular UV-embeddings of the MSSM, including specific models of the mediation of supersymmetry breaking into the MSSM. The potential collider reach for these models is interesting in its own right, and furthermore requires the application of flavor and dark matter constraints. We leave the investigation of these interesting possibilities to future work.\n\n\n\n\\section*{Acknowledgments}\n\nWe would like to thank S. Farkas, T. Junk, B. Kilminster, and M. Oreglia for helpful discussions on statistical methods. Work at ANL is supported in part by the U.S. Department of Energy (DOE), Div.~of HEP, Contract DE-AC02-06CH11357. Work at EFI is supported in part by the DOE through Grant No. DE-FG02- 90ER40560.  T.L. is also supported by the\nFermi-McCormick Fellowship. This work was supported in part by the DOE under Task TeV of contract DE-FGO2-96-ER40956.\n\n\n\\newpage\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\nThe research area of performance modeling can be considered as the generalization of physiological processes into  mathematical  models  with  the  purpose  of  approximating  a  body's  response to exercise. Created models represent an objective understanding of body responses and provide opportunities to serve as reasoning tools to be applied in performance prediction, training simulation or exercise prescription~\\cite{clarke_rationale_2013}. Jones and Vanhatalo~\\cite{jones_critical_2017} as well as Sreedhara et al.~\\cite{sreedhara_survey_2019} agree, that current approaches require refinement but highlight their potential for future research opportunities.  \n\nWith their in 2019 published findings, Caen et al.~\\cite{caen_reconstitution_2019} propose that recovery kinetics depend on previous work rate in addition to time and exercise intensity during recovery. Established performance modeling approaches separate energy expenditure and recovery into two separate models \\cite{clarke_rationale_2013, jones_critical_2017, sreedhara_survey_2019} and are too restricted to address these findings. \n\nThis work proposes that evolutionary computation allows to revisit an alternative concept: The so-called ``three component hydraulic model'' is investigated by Morton~\\cite{morton_three_1986, morton_critical_2006} and represents human energy systems as three interconnected tanks. Energy is represented by liquids, the flow out of a tap corresponds to energy expenditure, and the refilling of tanks can be understood as recovery. Its conceptualization allows to combine expenditure and recovery in one model and recent findings by Caen et al.~\\cite{caen_reconstitution_2019} can be addressed. However, Morton still highlights that the model is not absolute realistic in its assumptions and that more---and more precise---physiological measurements are needed to see how many predictions of this model conform to reality~\\cite{morton_critical_2006}. \n\nWe introduce a generalized form of the three component hydraulic model that removes ties to such concrete metabolic measures and opens up the opportunity to use evolutionary computation as a new pathway to apply it to individual athletes. \n\n\n\\section{A Pathway to Apply The Hydraulic Performance Model}\n\\label{sec:pathway_to_apply_model}\n\nTo enable the application of evolutionary computation, the three component hydraulic system is further generalized. In short, concrete relations to lactate, carbohydrate or phosphocreatine are removed. The three interacting components of the model are instead understood as more abstract entities and referred to as the anaerobic fast component ($AnF$), the anaerobic slow component ($AnS$), and the aerobic contribution ($Ae$). Model equations are developed in accordance to approaches by~\\cite{morton_three_1986}. Detailed equations and more information about the generalization are provided in Appendix A.\n\nA configuration of our generalized three component hydraulic model consists of eight parameters, which affect how the model's three components ($AnF$, $AnS$, $Ae$) interact. To fit such a configuration of eight parameters to an athlete, two objectives are defined: One for energy expenditure and one for energy recovery.\n\nFor the energy expenditure objective, the so-called ``critical power concept'' is employed as the ground truth. It is the established model to estimate times to exhaustion and used in most of the currently applied performance models~\\cite{clarke_rationale_2013, jones_critical_2017, sreedhara_survey_2019}. A total of 12 performance tests at a constant exercise intensity until exhaustion are simulated by the generalized three component hydraulic model. The normalized root mean squared error of all differences between simulated and expected times to exhaustion is to be minimized as the expenditure objective.\n\nThe energy recovery objective uses the exercise protocol that Caen et al.~\\cite{caen_reconstitution_2019} employed to obtain their published recovery ratios. The generalized hydraulic model simulates all trials that were conducted by Caen et al., which results in 12 differences between simulated and expected recovery ratios. Also here the normalized root mean squared error of all differences is used to determine the energy recovery fitness measure to be minimized. More information on the exercise protocols and corresponding simulations is provided in Appendix B.\n\nTo find a configuration for the hydraulic model that optimizes both objectives, the established Multi-Objective Evolutionary Algorithm with Decomposition (MOEA\/D) approach coupled with the asynchronous islands functionality of Pygmo~\\cite{biscani_parallel_2020} is used. Since both objective functions result in a normalized root mean squared error of 12 measurements, they are directly comparable. Knowledge of the evolved Pareto front allows to derive the best trade-off between both dynamics as the configuration that has the smallest Euclidean distance to the minimal error, i.e., point $(0,0)$. If more details on this rationale are needed, please see Appendix C.\n\nWe want to emphasize that the introduced approach serves as a proof of concept and much more room for parameter optimization and exploration of problem-specific algorithms is left for future work. Nearly all parameters are at the default that Pygmo provides and only four parameters were investigated by grid search. More details are given in Appendix C. \n\n\\section{Results And Discussion}\n\\label{sec:results}\n\nTen independently estimated evolutionary fittings with the described hydraulic model and algorithm parameters are investigated and compared regarding consistency and quality.\n\nAll ten results behave similar to each other. As observable in \\Cref{fig:hyd_exp_compare}, the---by the critical power concept suggested---hyperbolic relationship of exhaustive exercise intensity and time to exhaustion is closely recreated with only slight deviations in the high intensities. Also simulated recovery ratios after various conditions are similar to the ones observed by Caen et al.~\\cite{caen_reconstitution_2019}. The values P4 and P8 represent preceding exhaustive work bout intensities, i.e., P4 is the intensity that is predicted to lead to exhaustion after 4 min. Published means and standard deviations for P4 (2 min : $51.8\\% \\pm 2.8\\%$, 4 min: $57.7\\% \\pm 4.3\\%$, 6 min: $64\\% \\pm 5.8\\%$) and P8 ($40.1\\% \\pm 3.9\\%$, $ 44.8\\% \\pm 3\\%$, $54.8\\% \\pm 3.8\\%$) are denoted in \\Cref{fig:hyd_rec_compare}. \n\nOverall, evolved configurations make the hydraulic model successfully resemble the hyperbolic intensity to time to exhaustion relationship (\\Cref{fig:hyd_exp_compare}) as well as recovery ratios that are affected by previous energy expenditure characteristics (\\Cref{fig:hyd_rec_compare}). The outlined evolutionary computation approach successfully allows to apply three component hydraulic model as a performance model that addresses findings by Caen et al.~\\cite{caen_reconstitution_2019} and combines expenditure and recovery in one concept. \n\n\\begin{figure}\n    \\centering\n    \\captionsetup{width=0.65\\linewidth}\n    \\includegraphics[width=0.65\\linewidth]{figures\/fitting_rec.png}\n    \\caption{Recovery dynamics of fitted hydraulic models in comparison to published observations by Caen et al.~\\cite{caen_reconstitution_2019}.\n    }\n    \\label{fig:hyd_rec_compare}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\captionsetup{width=0.65\\linewidth}\n    \\includegraphics[width=0.65\\linewidth]{figures\/fitting_exp.png}\n    \\caption{Exercise to time to exhaustion relationship of fitted hydraulic models in comparison to the critical power model.}\n    \\label{fig:hyd_exp_compare}\n\\end{figure}\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nThe proposed evolutionary computation pathway and generalized understanding of the three component hydraulic model add a beneficial new perspective to research in performance modeling. Results clearly motivate further investigations as a validation strategy for the hydraulic concept and to bring the model closer to the application on individual athletes. \n\nThe implemented hydraulic model and evolutionary approach are available at \\url{https:\/\/github.com\/faweigend\/three_comp_hyd}.\n\n\\section*{ACKNOWLEDGMENTS}\nWe thank Markus Wagner (University of Adelaide) for his support and feedback.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nReinforcement learning (RL) addresses the problem of learning an optimal control policy that maximizes the expected cumulative reward while interacting with an unknown environment  \\cite{sutton2018reinforcement}. Standard RL algorithms typically focus only on maximizing a single objective. However, in many real-world applications, the control policy learned by an RL algorithm has to additionally satisfy stringent safety constraints \\cite{dulac2019challenges, amodei2016concrete}. For example, an autonomous vehicle may need to reach its destination in the minimum possible time without violating safety constraints such as crossing the middle of the road. The Constrained Markov Decision Process (CMDP) \\cite{altman1999constrained, garcia2015comprehensive} formalism, where one seeks to maximize a reward while satisfying safety constraints, is a standard approach for modeling the necessary safety criteria of a control problem via constraints on cumulative costs. \n\nSeveral policy-gradient-based algorithms have been proposed to solve CMDPs. Lagrangian-based methods \\cite{tessler2018reward,stooke2020responsive, paternain2019safe,liang2018accelerated} formulate the CMDP problem as a saddle-point problem and optimize it via primal-dual methods, while Constrained Policy Optimization \\cite{achiam2017constrained, yang2019projection} (inspired by the trust region policy optimization \\cite{schulman2015trust}) computes new dual variables from scratch at each update to maintain constraints during learning. Although these algorithms provide ways to learn an optimal policy, performance guarantees about reward regret, safety violation or sample complexity are rare. \n\nOne class of RL algorithms for which performance guarantees are available follow the principle of Optimism in the Face of Uncertainty (OFU) \\cite{ding2021provably,efroni2020exploration,qiu2020upper}, and provide an $\\tilde{\\mathcal{O}}(\\sqrt{K})$ guarantee for the reward regret, where $K$ is the number of episodes. However, these algorithms also have $\\tilde{\\mathcal{O}}(\\sqrt{K})$ safety violations. Such significant violation of the safety constraints during learning may be unacceptable in many safety-critical real-world applications such as the control of autonomous vehicles or power systems. These applications demand a class of safe RL algorithms that can provably guarantee safety during learning. With this goal in mind,  we aim to answer the following open theoretical question in this paper:\n\n\\textbf{Can we design safe RL algorithms that can achieve an $\\tilde{\\mathcal{O}}(\\sqrt{K})$ regret with respect to the performance objective, while guaranteeing zero or bounded safety constraint violation with arbitrarily high probability?}\n\n\\begin{table}[t]\n  \\caption{Regret and constraint violation comparisons for algorithms on episodic CMDPs} \n  \\label{overall}\n  \\centering \n  \\begin{tabular}{lll} \n    \\toprule\n    Algorithm & Regret\\tablefootnote{This table is presented for $K \\ge \\text{poly}(|\\mathcal{S}|, |\\mathcal{A}|, H)$, with polynomial terms independent of $K$ omitted.\\label{omitpoly}} & Constraint violation \\footref{omitpoly}\\\\\n    \\midrule\n    OPDOP \\cite{ding2021provably} & $\\tilde{\\mathcal{O}}(H^3 \\sqrt{|\\mathcal{S}|^2 |\\mathcal{A}| K})$  & $\\tilde{\\mathcal{O}}(H^3 \\sqrt{|\\mathcal{S}|^2 |\\mathcal{A}| K})$ \\\\\n    OptCMDP \\cite{efroni2020exploration} \\tablefootnote{Efroni et al. \\cite{efroni2020exploration} use $\\mathcal{N}$, the maximum number of non-zero transition probabilities across the entire state-action space, in their regret and constraint violation analysis. For consistency, we use $|\\mathcal{S}|^2 |\\mathcal{A}|$ to bound $\\mathcal{N}$. \\label{UBN}} & $\\tilde{\\mathcal{O}}(H^2 \\sqrt{|\\mathcal{S}|^3 |\\mathcal{A}| K})$ & $\\tilde{\\mathcal{O}}(H^2 \\sqrt{|\\mathcal{S}|^3 |\\mathcal{A}| K})$ \\\\\n    OptCMDP-bonus \\cite{efroni2020exploration} \\footref{UBN} & $\\tilde{\\mathcal{O}}(H^2 \\sqrt{|\\mathcal{S}|^3 |\\mathcal{A}| K})$ & $\\tilde{\\mathcal{O}}(H^2 \\sqrt{|\\mathcal{S}|^3 |\\mathcal{A}| K})$ \\\\\n    OptDual-CMDP \\cite{efroni2020exploration} \\footref{UBN} & $\\tilde{\\mathcal{O}}(H^2 \\sqrt{|\\mathcal{S}|^3 |\\mathcal{A}| K})$ & $\\tilde{\\mathcal{O}}(H^2 \\sqrt{|\\mathcal{S}|^3 |\\mathcal{A}| K})$ \\\\\n    OptPrimalDual-CMDP \\cite{efroni2020exploration} \\footref{UBN} & $\\tilde{\\mathcal{O}}(H^2 \\sqrt{|\\mathcal{S}|^3 |\\mathcal{A}| K})$ & $\\tilde{\\mathcal{O}}(H^2 \\sqrt{|\\mathcal{S}|^3 |\\mathcal{A}| K})$ \\\\\n    C-UCRL \\cite{zheng2020constrained} \\tablefootnote{Zheng et al. \\cite{zheng2020constrained} assumes a known transition kernel and analyzes regret in the long-term average setting.} & $\\tilde{\\mathcal{O}}(T^{\\frac{3}{4}})$ & $0$ \\\\\n    \\midrule\n    \\textbf{OptPess-LP} & $\\tilde{\\mathcal{O}}(\\frac{H^3}{\\tau - c^0} \\sqrt{|\\mathcal{S}|^3 |\\mathcal{A}| K})$ & $0$ \\\\\n    \\textbf{OptPess-PrimalDual} & $\\tilde{\\mathcal{O}}(\\frac{H^3}{\\tau - c^0} \\sqrt{|\\mathcal{S}|^3 |\\mathcal{A}| K})$ & $\\mathcal{O}(1)$ \\tablefootnote{The detailed constraint violation is $\\mathcal{O}\\left(C''H + H^2 \\sqrt{|\\mathcal{S}|^3 |\\mathcal{A}| C'' \\log(C''\/\\delta')} \\right)$, which is independent of $K$. Here, $\\delta' = \\delta \/ (16 |\\mathcal{S}|^2 |\\mathcal{A}| H)$ \\mbox{ and } $C'' = \\mathcal{O} (\\frac{H^4 |\\mathcal{S}|^3 |\\mathcal{A}|}{(\\tau - c^0)^2} \\log\\frac{H^4 |\\mathcal{S}|^3 |\\mathcal{A}|}{(\\tau - c^0)^2\\delta'})$.}\\\\\n    \\bottomrule\n  \\end{tabular}\n\\end{table}    \n\nWe answer the above question affirmatively by proposing two algorithms and establishing their stringent safety performance during learning. Our focus is on the tabular episodic constrained RL setting (unknown transition probabilities, rewards, and costs). The key idea behind both algorithms is a concept used earlier for safe exploration in constrained bandits \\cite{pacchiano2021stochastic,liu2021efficient}, which we call ``Optimistic Pessimism in the Face of Uncertainty (OPFU)'' here. The optimistic aspect incentivizes the algorithm for using exploration policies that can visit new state-action pairs, while the pessimistic aspect disincentivizes the algorithm from using exploration policies that can violate safety constraints. By carefully balancing optimism and pessimism, the proposed algorithms guarantee zero or bounded safety constraint violation during learning while achieving an $\\tilde{\\mathcal{O}}(\\sqrt{K})$ regret with respect to the reward objective.\n\nThe two algorithms address two different classes of the safe learning problem: whether a strictly safe policy is known a priori or not. The resulting exploration strategies are very different in the two cases.\n\n\\begin{enumerate}\n    \\item \\textbf{OptPess-LP Algorithm:} This algorithm assumes the prior knowledge of a strictly safe policy. It ensures zero safety constraint violation during learning with high probability and utilizes the linear programming (LP) approach for solving a CMDP problem. The algorithm achieves a reward regret of $\\tilde{\\mathcal{O}}(\\frac{H^3}{\\tau - c^0} \\sqrt{|\\mathcal{S}|^3 |\\mathcal{A}| K})$ with respect to the performance objective, where $H$ is the number of steps per episode, $\\tau$ is the given constraint on safety violation, $c^0$ is the known safety constraint value of a strictly safe policy $\\pi^0$, and $|\\mathcal{S}|$ and $|\\mathcal{A}|$ are the number of states and actions respectively.\n    \\item \\textbf{OptPess-PrimalDual Algorithm:} This algorithm addresses the case where no strictly safe policy, but a feasible strictly safe cost is known. By allowing a bounded (in $K$) safety cost, it opens up space for exploration. The OptPess-PrimalDual algorithm avoids linear programming and its attendant complexity and exploits the primal-dual approach for solving a CMDP problem. The proposed approach improves the computational tractability, while ensuring a bounded safety constraint violation during learning and a reward regret of $\\tilde{\\mathcal{O}}(\\frac{H^3}{\\tau - c^0} \\sqrt{|\\mathcal{S}|^3 |\\mathcal{A}| K})$ with respect to the objective.\n\\end{enumerate}\n\nCompared with the other methods listed in Table \\ref{overall}, though the proposed algorithms have an additional $H\/(\\tau - c^0)$ or $\\sqrt{|\\mathcal{S}|}\/(\\tau-c^0)$ factor in the regret bounds, they are able to reduce the constraint violation to zero or constant with high probability. This improvement in safety can be extremely important for many mission-critical applications. \n\n\n\n\\subsection{Related work}\nThe problem of learning an optimal control policy that satisfies safety constraints has been studied both in the RL setting and the multi-armed bandits setting. \n\n\\paragraph{Constrained RL:}\nSeveral policy-gradient algorithms have seen success in practice \\cite{tessler2018reward,stooke2020responsive,paternain2019safe,liang2018accelerated,achiam2017constrained,yang2019projection}. Also of interest are works which utilize Gaussian processes to model the transition probabilities and value functions \\cite{berkenkamp2017safe,wachi2018safe,koller2018learning,cheng2019end}.  \n\nSeveral algorithms with provable guarantees are closely related to our work. Zheng et al. \\cite{zheng2020constrained} considered the constrained RL problem in an infinite horizon setting with unknown reward and cost functions. The approach is similar to the UCRL2 algorithm \\cite{jaksch2010near} and achieves a sub-linear reward regret $\\tilde{\\mathcal{O}}(T^{\\frac{3}{4}})$ while satisfying constraints with high probability during learning. In contrast to this work, we consider the setting of unknown transition probabilities. Efroni et al \\cite{efroni2020exploration} focused on the episodic setting of unknown non-stationary transitions over a finite horizon, attaining both a reward regret and a constraint violation of $\\tilde{\\mathcal{O}}(H^2 \\sqrt{|\\mathcal{S}|^3 |\\mathcal{A}| K})$. Ding et al. \\cite{ding2021provably} studied an episodic setting with linear function approximation (suitable for large state space cases), and proposed algorithms  that can achieve $\\tilde{\\mathcal{O}}(d H^3 \\sqrt{K})$ for both the regret and the constraint violation (where $d$ is the dimension of the feature mapping). The regret analysis in \\cite{ding2021provably} can be easily extended to the tabular case, yielding $\\tilde{\\mathcal{O}}(H^3 \\sqrt{|\\mathcal{S}|^2 |\\mathcal{A}| K})$.\n\n\\paragraph{Constrained Multi-Armed Bandits:}\nMulti-armed bandit problems are special cases of MDPs, with both the number of states as well as the episode length being one. Linear bandits with constraints (satisfied with high probability) have been investigated in different settings. One setting, referred to as conservative bandits \\cite{wu2016conservative, kazerouni2017conservative, garcelon2020improved}, requires the cumulative reward to remain above a fixed percentage of the cumulative reward of a given baseline policy. Another setting is where each arm is associated with two unknown distributions (similar to our setting), generating reward and cost signals respectively \\cite{amani2019linear,pacchiano2021stochastic,liu2021efficient,moradipari2019safe}.\n\n\n\n\\section{Problem formulation}\n\\label{sec:prob_formula}\nA finite-horizon constrained non-stationary MDP model is defined as a tuple $M = (\\mathcal{S}, \\mathcal{A}, H, P, r, c, \\tau, \\mu)$, where $\\mathcal{S}$ is the state space, $\\mathcal{A}$ is the action space, $H$ is the number of steps in each episode, $r: \\mathcal{S} \\times \\mathcal{A} \\rightarrow [0, 1]$ is the unknown reward function of interest, $c: \\mathcal{S} \\times \\mathcal{A} \\rightarrow [0, 1]$ is the unknown safety cost function used to model the constraint violation, $\\tau \\in (0, H]$ is the given constant used to define the safety constraint, and $\\mu$ is the known initial distribution of the state. $P_{\\cdot} (\\cdot | s, a) \\in \\Delta_{\\mathcal{S}}^H, \\forall s \\in \\mathcal{S}, \\forall a \\in \\mathcal{A}$, where $\\Delta_{\\mathcal{S}}$ is the $|\\mathcal{S}|$-dimensional probability simplex, and $P_{h} (s' | s, a)$ is the unknown transition probability that the next state is $s'$ when action $a$ is taken for state $s$ at step $h$. \n\nSome further notation is necessary to define the problem. A (randomized Markov) policy is defined by a map $\\pi: \\mathcal{S} \\times [H] \\rightarrow \\Delta_{\\mathcal{A}}$, with $\\pi_h(a | s)$ being the probability of taking action $a$ in state $s$ at time step $h$. With $S_t$ and $A_t$ representing the state and the action at time $t$ respectively, let   \n\\begin{align*}\nV^{\\pi}_{h}(s; g, P) := \\mathbb{E}_{P, \\pi}\\left[ \\sum_{t = h}^{H} g(S_t, A_t) | S_h = s\\right], \\quad \\forall s \\in \\mathcal{S}\n\\end{align*}\ndenote the expected cumulative value with respect to a function $g: \\mathcal{S} \\times \\mathcal{A} \\rightarrow \\mathbb{R}_+$ under $P$ for a policy $\\pi$ over a time interval $[h,h+1,\\ldots,H]$. With slightly abuse of notation, we use $V^{\\pi}_1(\\mu;g, P)$ to denote $\\mathbb{E}_{S_1 \\sim \\mu}[V^{\\pi}_1(S_1; g, P)]$.\n\nIn the formulation below there are a total of $K$ episodes with $H$ steps each.\nEach episode $k \\in [K]$ begins with an initial probability distribution $\\mu$ for $\\mathcal{S}_1$.\nThen, the agent determines a randomized Markov policy $\\pi^k$ for that episode based on the information gathered from the previous episodes, and executes it. At time step $h$ during the execution of the $k$-th episode, after taking action $A_h^k$ at state $S_h^k$, the agent receives\na noisy reward and cost of $R_h^k(S_h^k, A_h^k) = r_h(S_h^k, A_h^k) + \\xi^k_{h}(S_h^k, A_h^k; r)$ and $C_h^k(S_h^k, A_h^k) = c_{h}(S_h^k, A_h^k) + \\xi^k_h(S_h^k, A_h^k; c)$, respectively.\n\\begin{assumption}[Sub-Gaussian noise]\n    \\label{assump:noise}\n    For all $h \\in [H], k \\in [K]$, the reward and cost noise random variables are conditionally independent zero-mean 1\/2-sub-Gaussian, i.e., $\\mathbb{E}[\\xi_h^k|\\mathcal{F}_{k-1}] = 0$, $\\mathbb{E}[\\exp(\\lambda \\xi^k_h) | \\mathcal{F}_{k-1}] \\le \\exp(\\lambda^2\/4)$, $\\forall \\lambda \\in \\mathbb{R}$. Here $\\mathcal{F}_{k}$ is the $\\sigma$-algebra generated by the random variables up to episode $k$.\n\\end{assumption}\n\nLet $\\pi^*$ denote the optimal policy of the following CMDP model: \n\\begin{align}\n\\max_{\\pi} & \\quad V_{1}^{\\pi}(\\mu; r, P)  \\quad\\quad \\text{s.t.} \\quad V_{1}^{\\pi}(\\mu; c, P) \\leq \\tau.\n\\label{eqn:cmdp}\n\\end{align}\n\nA policy $\\pi$ is said to be \\emph{strictly safe} if $V_1^{\\pi}(\\mu;c, P) < \\tau$.\n\n\\begin{assumption}\n    There exists a strictly safe policy $\\pi^0$ with $V^{\\pi^0}_{1}(\\mu; c, P) = c^0 < \\tau$.\n    \\label{assumption_safe}\n\\end{assumption}\n\nThere are two important cases of the safe learning problem.\n\n\\noindent\\textbf{Zero constraint violation case}: \nThe agent has prior knowledge of a strictly safe policy $\\pi^0$ and its safety cost value $c^0 := V_1^{\\pi^0}(\\mu; c, P)$. The agent wishes to attain a sublinear (in $K$) \\emph{cumulative regret},\n\\begin{align}\n    Reg(K; r) := \\sum_{k = 1}^K \\left( V_{1}^{\\pi^*}(\\mu; r, P) - V^{\\pi^k}_{1}(\\mu; r, P) \\right),\n    \\label{eqn::regret}\n\\end{align}\nwhile incurring zero constraint violation with at least a specified high probability $(1-\\delta)$, i.e.,\n\\begin{align*}\n    \\mathbb{P}\\left( V^{\\pi^k}_1(\\mu;c , P) \\leq \\tau, \\forall k \\in [K] \\right) \\geq 1- \\delta.\n\\end{align*}\n\n\\noindent\\textbf{Bounded constraint violation case}:\nThe agent knows that there exists a strictly safe policy with a known safety cost value $c^0$, but does not know any strictly safe policy. It aims to achieve a cumulative regret (\\ref{eqn::regret}) that grows sublinearly with $K$, while ensuring that the \\emph{regret of constraint violation},\n\\begin{align*}\n        Reg(K; c) := \\left( \\sum_{k = 1}^{K} \\left( V^{\\pi^k}_{1}(\\mu; c, P) -  \\tau \\right) \\right)_{+} \\mbox{  (where $(a)_{+} := \\max\\{a, 0\\}$) },\n\\end{align*}\nsatisfies $\\sup_{K} Reg(K;c) < +\\infty$ with at least a specified high probability $(1 - \\delta)$. \n\n\\noindent\\textbf{Remarks.} \nFor the zero constraint violation case, the assumption of knowing $c^0$ can be relaxed, as shown in Appendix E.\n    \n\n\\section{Zero constraint violation case} \n\\label{sec:zero}\nWe start by considering the zero constraint violation case. On one hand, to balance the exploration-exploitation trade-off, we employ an optimistic estimate of the reward function, as embodied in the OFU principle. On the other hand, to maintain absolute safety with high probability during the exploration, we employ a pessimistic estimate of the safety cost. Such an OPFU principle was previously discussed in constrained bandits \\cite{amani2019linear} and we adapt it to the unknown CMDP setting.\n\nAt each episode $k$, we begin by forming empirical estimates of the transition probabilities, the reward function, and the cost function from step $h$ of all previous episodes:\n\\begin{align}\n    \\hat{P}_h^k(s'|s, a) &:= \\frac{\\sum_{k'=1}^{k-1} \\mathbbm{1}(S_h^{k'}=s, A_h^{k'}=a, S_{h+1}^{k'}=s')}{N^k_h(s, a) \\vee 1}, \\label{eqn:hatP}\\\\\n    \\hat{g}_h^k(s, a) &:= \\frac{\\sum_{k'=1}^{k-1} \\mathbbm{1}(S_h^{k'}=s, A_h^{k'}=a) (g_h(s, a) + \\xi_h^k(s, a; g))}{N^k_h(s, a) \\vee 1}, \\quad \\mbox{ for } g = r, c,\n    \\label{estimate2}\n\\end{align}\nwhere $a \\vee b := \\max\\{a, b\\}$, and\n\\begin{align}\n    N^k_h(s, a) := \\mbox{\\# of  visits to state-action pair $(s, a)$ at step $h$ in episodes $[1,2,\\ldots, k-1]$.}\n    \\label{eqn:bigN}\n\\end{align}\n\nNext we fix some $\\delta \\in (0, 1)$ and form a common (for notational simplicity) confidence radius $\\beta^k_h(s, a)$ for the transition probabilities, the rewards, and the costs, \n\\begin{align*}\n    \\beta^k_h(s, a) &:= \\sqrt{\\frac{1}{N^k_h(s, a) \\vee 1} Z}, \\quad \\text{where $Z := \\log(16|\\mathcal{S}|^2 |\\mathcal{A}|HK\/\\delta)$.}\n\\end{align*}\n\nAt each episode $k$, we define the \\emph{optimistically biased reward estimate} as\n\\begin{align}\n    \\bar{r}^k_h(s, a) &:= \\hat{r}^k_h(s, a) + \\alpha_r \\beta^k_h(s, a), \\quad \\forall (s,a, h) \\in \\mathcal{S} \\times \\mathcal{A} \\times [H], \\label{eqn:tilde_r1}\n\\end{align}\nwhere the scaling factor is\n\\begin{align}\n  &\\alpha_r := 1 + |\\mathcal{S}| H + \\frac{4 H(1+|\\mathcal{S}|H)}{\\tau - c^0}. \\label{alphar}\n\\end{align}\n\nTo guarantee safe exploration, define the \\emph{pessimistically biased safety cost estimate} at episode $k$ as\n\\begin{align}\n    \\underbar{c}{}_h^k(s, a) & := \\hat{c}_h^k(s,a) + (1 + H|\\mathcal{S}|) \\beta^k_h(s, a), \\quad \\forall (s, a, h) \\in \\mathcal{S} \\times \\mathcal{A} \\times [H].\n\\end{align}\n\nThe policy we execute at episode $k$ is chosen from a ``pessimistically safe\" policy set $\\Pi^k$, defined as\n\\begin{equation}\n  \\Pi^k := \n    \\begin{cases}\n      \\{\\pi^0\\} & \\text{if } V_1^{\\pi^0}(\\mu; \\underbar{c}^k, \\hat{P}^k) \\geq (\\tau + c^0)\/2,\\\\\n      \\{\\pi: V_1^{\\pi}(\\mu; \\underbar{c}^k, \\hat{P}^k) \\leq \\tau\\} & \\text{otherwise.} \n    \\end{cases}\n    \\label{eqn:PiPC}\n\\end{equation}\n\nWe simply use the strictly safe policy $\\pi^0$ until $V_1^{\\pi^0}(\\mu; \\underbar{c}^k, \\hat{P}^k) < (\\tau + c^0)\/2$, which is a sufficient condition to guarantee that the set $\\{\\pi: V_1^{\\pi}(\\mu; \\underbar{c}^k, \\hat{P}^k) \\leq \\tau\\}$ is non-empty. Within this set, we choose the optimistically best reward earning policy $\\pi^k$, which can be solved by linear programming with $\\Theta(|\\mathcal{S}||\\mathcal{A}|H)$ decision variables and constraints \\cite{efroni2020exploration}. The resulting Optimistic Pessimism-based Linear Programming (OptPess-LP) algorithm is presented below:\n\\begin{algorithm}\n    \\label{OptPess}\n    \\SetAlgoLined\n    \\KwInput{$K, \\delta, \\pi^0, c^0, \\tau$;}\n    \\noindent\\textbf{Initialization:} $N_h^1(s, a) = 0, \\ \\forall (s, a, h) \\in \\mathcal{S} \\times \\mathcal{A} \\times [H]$;\\\\\n    \\For{$k = 1,2, \\ldots, K$}{\n        Update empirical model (i.e., $\\hat{P}^k, \\hat{r}^k, \\hat{c}^k$) as in Equations (\\ref{eqn:hatP})-(\\ref{eqn:bigN});\\\\\n        Update $\\bar{r}^k, \\underbar{c}^k$, and $\\Pi^k$ as in Equations (\\ref{eqn:tilde_r1})-(\\ref{eqn:PiPC});\\\\\n        Calculate $\\pi^k \\in \\arg\\max_{\\pi \\in \\Pi^k} V^{\\pi}_1(\\mu; \\bar{r}^k, \\hat{P}^k)$;\\\\\n        Execute $\\pi^k$ and collect a trajectory $(S_h^k, A_h^k, R_h^k, C_h^k), \\ \\forall h \\in [H]$;\\\\\n        Update counters $N_h^{k+1}(S_h^k, A_h^k), \\ \\forall h \\in [H]$; \n    }\n    \\caption{\\textbf{OptPess-LP}}\n\\end{algorithm}\n \n\\begin{theorem}[Regret and constraint violation bounds for OptPess-LP]\n\\label{thm:safe-PC}\n    Fix any $\\delta \\in (0, 1)$. With probability at least ($1 - \\delta$), OptPess-LP has zero constraint violation with\n    \\begin{align*}\n        Reg^{\\textbf{OPLP}}(K; r) = \\tilde{\\mathcal{O}}\\left(\\frac{H^3}{\\tau - c^0} \\sqrt{|\\mathcal{S}|^3 |\\mathcal{A}| K} +  \\frac{H^5 |\\mathcal{S}|^3 |\\mathcal{A}|}{(\\tau - c^0)^2 \\land (\\tau - c^0)}  \\right), \\mbox{ where $a \\land b := \\min\\{a, b\\}$.}\n    \\end{align*}\n    \\label{thm1} \n\\end{theorem}\n\nTheorem \\ref{thm1} shows that it is possible to achieve sublinear regret in $K$, while simultaneously incurring no constraint violation with arbitrarily high probability. The proof is sketched in Section \\ref{zero_sketch}, with detailed proofs presented in Appendix B. \n\n\n\n\\section{Bounded constraint violation case}\n\\label{sec:bounded}\n\nWithout prior knowledge of a strictly safe policy $\\pi^0$, we may not be able to guarantee zero constraint violation with high probability. However, by relaxing the requirement to bounded (in $K$) safety constraint violation, we can incorporate more exploration and design a more efficient algorithm by a primal-dual approach. It is inspired by the design of the pessimistic term in constrained bandits \\cite{liu2021efficient}.\n\nDifferent from traditional optimistic dual analysis methods in \\cite{efroni2020exploration}, we introduce an additive pessimistic term $\\epsilon_k$ at each episode $k$ in the original optimization problem (\\ref{eqn:cmdp}), i.e., \n\\begin{align}\n    \\max_{\\pi} & \\quad V_1^{\\pi}(\\mu; r, P) \\quad\\quad \\text{s.t.}\\quad V_1^{\\pi}(\\mu; c, P) + \\epsilon_k \\le \\tau.\n    \\label{pertubed_problem}\n\\end{align}\n \nThe pessimistic term restrains the constraint violation and will be progressively decreased as learning proceeds.\n \nThe CMDP problem in (\\ref{pertubed_problem}) may however not have any feasible solution. To overcome this, we consider the Lagrangian,\n\\begin{align*}\n    L^k(\\pi, \\lambda) := V^{\\pi}_1(\\mu;r, P) + \\lambda \\left( \\tau - \\epsilon_k - V_1^{\\pi}(\\mu;c, P) \\right),\n\\end{align*}\nfor which, given any Lagrange multiplier $\\lambda$, we can always solve for the optimizer $\\max_{\\pi} L^k(\\pi, \\lambda)$ by dynamic programming. \n\nWith the introduction of the additive term $\\epsilon_k$, the dual variable (denoted by $\\lambda^k$) governed by the subgradient algorithm grows faster, which enhances safety constraints in the next episode. \n\nIn order to guarantee sufficient optimism of rewards and costs, we integrate the uncertainty of transitions into rewards and costs, i.e., \n\\begin{align}\n    \\label{empirical}\n    \\tilde{r}_h^k(s, a) &:= \\hat{r}_h^k(s, a) + \\beta(n_k(s, a, h)) + H |\\mathcal{S}| \\beta(n_k(s, a, h)), \\quad \\forall (s, a, h) \\in \\mathcal{S} \\times \\mathcal{A} \\times [H], \\notag\\\\\n    \\tilde{c}_h^k(s, a) &:= \\hat{c}_h^k(s, a) - \\beta(n_k(s, a, h)) - H |\\mathcal{S}| \\beta(n_k(s, a, h)), \\quad \\forall (s, a, h) \\in \\mathcal{S} \\times \\mathcal{A} \\times [H].\n\\end{align}\n\nNote that in contrast to the zero constraint violation case, we \\emph{optimistically} estimate the safety cost function, with the pessimism only governed by $\\epsilon_k$. \n\nWe employ truncated value functions due to the additional uncertainties from transitions, i.e.,\n\\begin{align}\n    \\hat{V}_1^{\\pi}(\\mu; \\tilde{r}^k, \\hat{P}^k) := \\min\\{H, V_1^{\\pi}(\\mu; \\tilde{r}^k, \\hat{P}^k)\\}, ~~\n    \\hat{V}_1^{\\pi}(\\mu; \\tilde{c}^k, \\hat{P}^k) := \\max\\{0, V_1^{\\pi}(\\mu; \\tilde{c}^k, \\hat{P}^k)\\}.\n    \\label{truncated}\n\\end{align}\n\nFor the policy update of the primal variable (denoted by $\\pi^k$), we can apply standard dynamic programming by viewing $\\tilde{r}_h^k(s, a) - \\frac{\\lambda^k}{\\eta^k}(\\tilde{c}_h^k(s, a) - \\tau)$ as the reward function. Specifically, we apply backward induction to solve for the optimal policy:\n\\begin{align*}\n    Q_h^k(s, a) = \\tilde{r}_h^k(s, a) - \\frac{\\lambda^k}{\\eta^k}(\\tilde{c}_h^k(s, a) - \\tau) + \\sum_{s' \\in \\mathcal{S}} \\hat{P}_h^k(s'|s, a) \\max_{a'} Q_{h+1}^k(s', a'), \\quad \\forall h \\in [H]\n\\end{align*}\nwith $Q_{H+1}^k (s, a) = 0, \\forall (s, a) \\in \\mathcal{S} \\times \\mathcal{A}$. Then, $\\pi_h^k \\in \\arg\\max_{a} Q_h^k(s, a)$, which is computationally efficient (as efficient as policy-gradient-based algorithms in the tabular case). \n\nThe resulting Optimistic Pessimism-based Primal-Dual (OptPess-PrimalDual) algorithm is shown in Algorithm \\ref{alg:dual}. It chooses $\\epsilon_k := 5 H^2 \\sqrt{|\\mathcal{S}|^3 |\\mathcal{A}|} (\\log\\frac{k}{\\delta'} + 1) \/ \\sqrt{k \\log\\frac{k}{\\delta'}}$, $\\delta' = \\delta \/ (16 |\\mathcal{S}|^2 |\\mathcal{A}| H)$, the scaling parameter $\\eta^k := (\\tau - c^0) H \\sqrt{k}$ in the primal policy update, and, for convenience, a step size of 1 for the dual update.\n\\begin{algorithm}\n    \\label{alg:dual}\n    \\SetAlgoLined\n    \\KwInput{$K, \\delta, c^0, \\tau$;\n    \\noindent\\textbf{Initialization:} $N_h^1(s, a) = 0, \\ \\forall (s, a, h) \\in \\mathcal{S} \\times \\mathcal{A} \\times [H]$, $\\lambda^1 = 0$;\\\\\n    \\For{$k = 1,2, \\ldots, K$}{\n        Set $\\epsilon_k = 5 H^2 \\sqrt{|\\mathcal{S}|^3 |\\mathcal{A}|} (\\log\\frac{k}{\\delta'} + 1) \/ \\sqrt{k \\log\\frac{k}{\\delta'}}$, $\\delta' = \\delta \/ (16 |\\mathcal{S}|^2 |\\mathcal{A}| H)$, $\\eta^k = (\\tau - c^0) H \\sqrt{k}$;\\\\\n        Update empirical model (i.e., $\\hat{P}^k, \\hat{r}^k, \\hat{c}^k$) as in Equations (\\ref{eqn:hatP})-(\\ref{eqn:bigN});\\\\\n        Update $\\tilde{r}^k$, $\\tilde{c}^k$ as in Equation (\\ref{empirical});\\\\\n        \\emph{(Policy Update)} $\\pi^k \\in \\arg\\max_{\\pi \\in \\Pi} \\hat{V}_1^{\\pi}(\\mu; \\tilde{r}^k, \\hat{P}^k) - \\frac{\\lambda^k}{\\eta^k} \\left(\\hat{V}_1^{\\pi}(\\mu; \\tilde{c}^k, \\hat{P}^k) - \\tau \\right)$;\\\\\n        \\emph{(Dual Update)} $\\lambda^{k+1} = \\left(\\lambda^k + \\hat{V}_1^{\\pi^k}(\\mu; \\tilde{c}^k, \\hat{P}^k) + \\epsilon_{k} - \\tau\\right)_{+}$;\\\\\n        Execute $\\pi^k$ and collect a trajectory $(S_h^k, A_h^k, R_h^k, C_h^k), \\ \\forall h \\in [H]$;\\\\\n        Update counters $N_h^{k+1}(S_h^k, A_h^k), \\ \\forall h \\in [H]$;\n    }\n    \\caption{\\textbf{OptPess-PrimalDual}}\n\\end{algorithm}\n\n\\begin{theorem}[Regret and constraint violation bounds for OptPess-PrimalDual]\n    Fix any $\\delta \\in (0, 1)$. Then,\n    \\begin{align*}\n        Reg^{\\textbf{OPPD}}(K; r) &= \\tilde{\\mathcal{O}}\\left(\\frac{H^3}{\\tau - c^0} \\sqrt{|\\mathcal{S}|^3 |\\mathcal{A}| K} + \\frac{H^5 |\\mathcal{S}|^3 |\\mathcal{A}|}{(\\tau - c^0)^2}\\right),\\\\\n        Reg^{\\textbf{OPPD}}(K; c) &= \\mathcal{O}\\left(C''(H-\\tau) + H^2 \\sqrt{|\\mathcal{S}|^3 |\\mathcal{A}| C''} \\right) = \\mathcal{O}(1),\n    \\end{align*}\n    where $C'' = \\mathcal{O} (\\frac{H^4 |\\mathcal{S}|^3 |\\mathcal{A}|}{(\\tau - c^0)^2} \\log\\frac{H^4 |\\mathcal{S}|^3 |\\mathcal{A}|}{(\\tau - c^0)^2\\delta'})$ is a coefficient independent of $K$.\n    \\label{thm2}\n\\end{theorem}\n\nTheorem \\ref{thm2} shows that it is possible to achieve an $\\tilde{\\mathcal{O}}(\\sqrt{K})$ reward regret,\nwhile only allowing bounded constraint violation with arbitrarily high probability. Detailed proofs are presented in Section \\ref{bounded_sketch} and Appendix C. \n\n\n\n\\section{Regret and constraint violation analysis}\n\\label{proof sketch}\n\nWe sketch the key steps in the proofs of Theorems \\ref{thm1} and \\ref{thm2}. Full details are relegated to Appendices B and C, respectively. Note that our analysis and results are conditioned on the same high probability event (specifically defined in Appendix A), which occurs with probability at least $(1 - \\delta)$. \n\n\\subsection{Analysis of OptPess-LP (Algorithm \\ref{OptPess})}\n\\label{zero_sketch}\n \n\\paragraph{Constraint violation analysis}\nThe zero constraint violation of OptPess-LP follows from the following property of the pessimistic policy set $\\Pi^k$:\n\\begin{lemma} \\label{lem:abs-safe}\nWith probability at least $(1 - \\delta)$, for any $k \\in [K]$ and policy $\\pi \\in \\Pi^k$, $V^{\\pi}_1(\\mu;c, P) \\leq \\tau$. \n\\end{lemma}\n\n\\paragraph{Regret of reward analysis} When the parameters $c, P$ are not well estimated, there may not exist any policy $\\pi$ such that $V^{\\pi}_1(\\mu;\\underbar{c}^k, \\hat{P}^k) \\leq \\tau$. Hence, as defined in (\\ref{eqn:PiPC}), $\\Pi^k$ is a singleton set $\\{\\pi^0\\}$, and accordingly $\\pi^0$ is executed, even though $\\pi^0$ is not safe for $(\\underbar{c}^k, \\hat{P}^k)$. It subsequently takes several episodes of exploration using $\\pi^0$ until it becomes strictly safe for $(\\underbar{c}^k, \\hat{P}^k)$. At that time, policies close enough to $\\pi^0$, of which there are infinitely many, are also safe for $(\\underbar{c}^k,\\hat{P}^k)$, and so $|\\Pi^k| = +\\infty$. At this point the learning algorithm can proceed to enhance reward while maintaining safety with respect to $(\\underbar{c}^k, \\hat{P}^k)$.\n\nTo analyze the algorithm, we decompose the reward regret as follows:\n\\begin{align}\n    Reg^{\\textbf{OPLP}}(K; r) =& \\sum_{k = 1}^K \\mathbbm{1}(|\\Pi^k| =1) \\left(V^{\\pi^*}_1(\\mu; r, P) - V^{\\pi^0}_1(\\mu; r, P)\\right) \\notag \\\\\n    +& \\sum_{k = 1}^K \\mathbbm{1}(|\\Pi^k| > 1) \\left( V^{\\pi^*}_1(\\mu; r, P) - V^{\\pi^k}_1(\\mu; \\bar{r}^k, \\hat{P}^k) \\right) \\notag \\\\\n    +& \\sum_{k = 1}^K  \\mathbbm{1}(|\\Pi^k| >1) \\left( V^{\\pi^k}_1(\\mu; \\bar{r}^k, \\hat{P}^k) - V^{\\pi^k}_1(\\mu; r, P)\\right) .\n    \\label{eqn:no_violation_decompose} \n\\end{align}\n\nTo bound the first term on the right-hand side (RHS) of (\\ref{eqn:no_violation_decompose}), we have the\nfollowing lemma which gives an upper bound on the number of episodes for exploration by policy $\\pi^0$: \n\\begin{lemma}\\label{lem:burn-in}\nWith probability at least $(1 - \\delta)$, $\\sum_{k=1}^K \\mathbbm{1}(|\\Pi^k| = 1) \\leq C'$, where $C' = \\tilde{\\mathcal{O}}(H^4 |\\mathcal{S}|^3 |\\mathcal{A}| \/ ((\\tau - c^0)^2 \\land (\\tau - c^0)))$.\n\\end{lemma}\n \nTurning to the second term on the RHS of (\\ref{eqn:no_violation_decompose}), we first note that $\\pi^*$ may not be in $\\Pi^k$. To ensure that the term $V^{\\pi^*}_1(\\mu; r, P) - V^{\\pi^k}_1(\\mu; \\bar{r}^k, \\hat{P}^k)$ is nevertheless non-positive even when $\\pi^* \\not\\in \\Pi^k$, we set $\\alpha_r$ to the large value shown in (\\ref{alphar}). This increases $\\bar{r}^k$, and hence also $V^{\\pi^k}_1(\\mu; \\bar{r}^k, \\hat{P}^k)$. In addition, we show that there is a policy $\\hat{\\pi}^k$ that attains the same reward as the (non-Markov) probabilistic mixed policy, $\\tilde{\\pi}^k := B_{\\gamma_k} \\pi^* + (1 - B_{\\gamma_k}) \\pi^0$, where $B_{\\gamma_k}$ is a Bernoulli distributed random variable with mean $\\gamma_k$ for $\\gamma_k \\in [0, 1]$. $\\gamma_k$ will be chosen as the largest coefficient such that $V_1^{\\tilde{\\pi}^k}(\\mu; \\underbar{c}^k, \\hat{P}^k) \\le \\tau$. This latter policy in turn has a larger reward than $\\pi^0$ since it is a mixture with $\\pi^*$, yielding the following:\n\\begin{lemma}\n    With probability at least $(1 - \\delta)$,\n    \\begin{align*}\n        \\sum_{k = 1}^K \\mathbbm{1}(|\\Pi^k| > 1) \\left( V^{\\pi^*}_1(\\mu; r, P) - V^{\\pi^k}_1(\\mu; \\bar{r}^k, \\hat{P}^k)\\right) \\le 0.\n    \\end{align*}\n    \\label{lem1}\n\\end{lemma}\n\nFinally, concerning the third term\non the RHS of (\\ref{eqn:no_violation_decompose}), akin to the closed-loop identifiability property \\cite{borkar1979adaptive,kumar1982new},\nwhile $\\hat{P}^k$ may not converge to $P$, the difference in the rewards $V^{\\pi^k}_1(\\mu; \\bar{r}^k, \\hat{P}^k) - V^{\\pi^k}_1(\\mu; r, P)$ grows sublinearly in $k$\nsince the same policy $\\pi^k$ is used in both values:\n\\begin{lemma}\n    With probability at least $(1 - \\delta)$,\n    \\begin{align*}\n        \\sum_{k = 1}^K \\mathbbm{1}(|\\Pi^k| > 1) \\left( V^{\\pi^k}_1(\\mu; \\bar{r}^k, \\hat{P}^k) - V^{\\pi^k}_1(\\mu; r, P)\\right) = \\tilde{\\mathcal{O}}\\left(\\frac{H^3}{\\tau - c^0} \\sqrt{|\\mathcal{S}|^3 |\\mathcal{A}| K} + \\frac{H^5 |\\mathcal{S}|^3 |\\mathcal{A}|}{\\tau-c^0} \\right).\n    \\end{align*}\n    \\label{lem2}\n\\end{lemma}\n\nCombining Lemmas \\ref{lem:burn-in}, \\ref{lem1}, and \\ref{lem2} yields Theorem \\ref{thm1}.\n\n\n\\subsection{Analysis of OptPess-PrimalDual (Algorithm \\ref{alg:dual})}\n\\label{bounded_sketch}\nIn this section, we outline the steps in the proof of Theorem \\ref{thm2} by analyzing regret and constraint violation of OptPess-PrimalDual respectively.\n\n\nRecall $\\epsilon_k = 5 H^2 \\sqrt{|\\mathcal{S}|^3 |\\mathcal{A}|} (\\log\\frac{k}{\\delta'} + 1) \/ \\sqrt{k \\log\\frac{k}{\\delta'}}$, where $\\delta' = \\delta \/ (16 |\\mathcal{S}|^2 |\\mathcal{A}| H)$. The existence of a feasible solution to (\\ref{pertubed_problem}) can be guaranteed if $\\epsilon_k \\le \\tau - c^0$. Let $C''$ be the smallest value such that $\\forall k \\geq C''$, $\\epsilon_k \\le (\\tau - c^0)\/2$. Then the perturbed optimization problem (\\ref{pertubed_problem}) has at least one feasible solution for any $k \\ge C''$. By simple calculation, one can verify that $C'' = \\mathcal{O} (\\frac{H^4 |\\mathcal{S}|^3 |\\mathcal{A}|}{(\\tau - c^0)^2} \\log\\frac{H^4 |\\mathcal{S}|^3 |\\mathcal{A}|}{(\\tau - c^0)^2\\delta'})$. Notice that $\\epsilon_k$ is a function not depending on $K$, and so is the coefficient $C''$.\n\n\\paragraph{Constraint violation analysis}\nThe bounded constraint violation of OptPess-PrimalDual is established as follows. We first decompose the constraint violation as\n\\begin{align}\n    Reg^{\\textbf{OPPD}}(K; c) &= \\left(\\sum_{k=1}^K \\left(V_1^{\\pi^k}(\\mu; c, P) - \\hat{V}_1^{\\pi^k}(\\mu; \\tilde{c}^k, \\hat{P}^k)\\right) + \\sum_{k=1}^K \\left(\\hat{V}_1^{\\pi^k}(\\mu; \\tilde{c}^k, \\hat{P}^k) - \\tau\\right)\\right)_{+} \\notag\\\\\n    & \\le \\left(\\sum_{k=1}^K \\left(V_1^{\\pi^k}(\\mu; c, P) - \\hat{V}_1^{\\pi^k}(\\mu; \\tilde{c}^k, \\hat{P}^k)\\right) + \\lambda^{K+1} - \\sum_{k=1}^K \\epsilon_{k} \\right)_{+}.\n    \\label{decompose_bounded_violation}\n\\end{align}\nThe first summation term and $\\lambda^{K+1}$ in (\\ref{decompose_bounded_violation}) can be bounded as follows:\n\\begin{lemma}\n    Recall $\\delta' = \\delta \/ (16 |\\mathcal{S}|^2 |\\mathcal{A}| H)$, with probability at least $(1 - \\delta)$,\n    \\begin{align*}\n        \\sum_{k=1}^K \\left(V_1^{\\pi^k}(\\mu;c, P) - \\hat{V}_1^{\\pi^k}(\\mu; \\tilde{c}^k, \\hat{P}^k)\\right) \\le 8 H^2 \\sqrt{|\\mathcal{S}|^3 |\\mathcal{A}| K \\log \\frac{K}{\\delta'}} + \\mathcal{O}(PolyLog(K)).\n    \\end{align*}\n    \\label{lem:violation_first_term}\n\\end{lemma}\n\n\\begin{lemma}\n    For any $k \\ge C''$, with probability at least $(1 - \\delta)$,\n    \\begin{align*}\n        \\lambda^k \\le \\frac{1}{\\zeta} \\ln \\frac{11 \\nu_{\\max}^2}{3 \\rho^2} + C''(H-\\tau) + \\sum_{u=1}^{C''} \\epsilon_u + H + \\frac{4(H^2 + \\epsilon_k^2 + \\eta^k H)}{\\tau - c^0},\n    \\end{align*}\n    where $\\rho = (\\tau - c^0)\/4$, $\\nu_{\\max} = H$, $\\zeta = \\rho \/ (\\nu_{\\max }^{2}+\\nu_{\\max} \\rho \/ 3)$.\n    \\label{lem7}\n\\end{lemma}\nTo guarantee bounded violation, we ensure that $\\sum_{k=1}^K \\epsilon_k$ in (\\ref{decompose_bounded_violation}) can cancel the dominant terms in the two lemmas above. According to Lemmas \\ref{lem:violation_first_term} and \\ref{lem7}, with probability at least $(1 - \\delta)$, the violation is bounded as\n\\begin{align*}\n    Reg^{\\textbf{OPPD}}(K; c) = \\mathcal{O}\\left(C''H + H^2 \\sqrt{|\\mathcal{S}|^3 |\\mathcal{A}| C'' \\log{(C'' \/ \\delta')}} \\right). \n\\end{align*}\n\n\n\\paragraph{Regret of reward analysis}\nFor episode $k$ with $k \\geq C''$, let $\\pi^{\\epsilon_k, *}$ be the optimal policy for (\\ref{pertubed_problem}), which is well-defined by the definition of $C''$. We decompose the reward regret as\n\\begin{align}\n    Reg^{\\textbf{OPPD}}&(K; r) = \\sum_{k=1}^{C''} \\left(V_1^{\\pi^*}(\\mu; r, P) - V_1^{\\pi^k}(\\mu; r, P)\\right) \\label{bound_regret_decompose}\\\\\n    + &\\sum_{k=C''}^K \\left(V_1^{\\pi^*}(\\mu; r, P) - V_1^{\\pi^{\\epsilon_k, *}}(\\mu; r, P)\\right) + \\sum_{k=C''}^K \\left(V_1^{\\pi^{\\epsilon_k, *}}(\\mu; r, P) - \\hat{V}_1^{\\pi^{\\epsilon_k, *}}(\\mu; \\tilde{r}^k, \\hat{P}^k)\\right) \\notag\\\\\n    + &\\sum_{k=C''}^K \\left(\\hat{V}_1^{\\pi^{\\epsilon_k, *}}(\\mu; \\tilde{r}^k, \\hat{P}^k) - \\hat{V}_1^{\\pi^k}(\\mu; \\tilde{r}^k, \\hat{P}^k)\\right) + \\sum_{k=C''}^K \\left(\\hat{V}_1^{\\pi^k}(\\mu; \\tilde{r}^k, \\hat{P}^k) - V_1^{\\pi^k}(\\mu; r, P)\\right). \\notag\n\\end{align}\n\nWe upper bound each term on the RHS of (\\ref{bound_regret_decompose}). Since $V_1^{\\pi}(\\mu;r, P) \\in [0, H]$ for any policy $\\pi$, the first term is upper bounded by $HC''$. The second and third terms can be bounded by the following two lemmas:\n\\begin{lemma}\n     With probability at least $(1 - \\delta)$,\n     \\begin{align*}\n         \\sum_{k=C''}^K \\left(V_1^{\\pi^*}(\\mu; r, P) - V_1^{\\pi^{\\epsilon_k, *}}(\\mu; r, P)\\right) \\le \\sum_{k=C''}^K \\frac{\\epsilon_k H}{\\tau - c^0} = \\tilde{\\mathcal{O}}\\left(\\frac{H^3}{\\tau - c^0} \\sqrt{|\\mathcal{S}|^3 |\\mathcal{A}| K}\\right).\n     \\end{align*}\n    \\label{lem3}\n\\end{lemma}\n\n\\begin{lemma}\n    With probability at least $(1 - \\delta)$,\n    $\\sum_{k=C''}^K \\left(V_1^{\\pi^{\\epsilon_k, *}}(\\mu; r, P) - \\hat{V}_1^{\\pi^{\\epsilon_k, *}}(\\mu; \\tilde{r}^k, \\hat{P}^k)\\right) \\le 0$.\n    \\label{lem4}\n\\end{lemma}\n\nThe pivotal step is to leverage optimism of $\\pi^k$ to further decompose the fourth term on the RHS of (\\ref{bound_regret_decompose}), and utilize the projected dual update to transfer it into the form of $\\lambda^k (\\lambda^k - \\lambda^{k+1})$. The following lemmas provide high probability bounds for the remaining two terms on the RHS of (\\ref{bound_regret_decompose}):\n\n\\begin{lemma}\n    With probability at least $(1 - \\delta)$,\n    \\begin{align*}\n        \\sum_{k=C''}^K \\left(\\hat{V}_1^{\\pi^{\\epsilon_k, *}}(\\mu; \\tilde{r}^k, \\hat{P}^k) - \\hat{V}_1^{\\pi^k}(\\mu; \\tilde{r}^k, \\hat{P}^k)\\right) = \\tilde{\\mathcal{O}}\\left( \\frac{H}{\\tau - c^0} \\sqrt{K}\\right).\n    \\end{align*}\n    \\label{lem5}\n\\end{lemma}\n\n\\begin{lemma}\n    With probability at least $(1 - \\delta)$, \n    \\begin{align*}\n        \\sum_{k=C''}^K \\left(\\hat{V}_1^{\\pi^k}(\\mu; \\tilde{r}^k, \\hat{P}^k) - V_1^{\\pi^k}(\\mu; r, P)\\right) = \\tilde{\\mathcal{O}}\\left(H^2 \\sqrt{|\\mathcal{S}|^3 |\\mathcal{A}| K} + H^4 |\\mathcal{S}|^3 |\\mathcal{A}|\\right).\n    \\end{align*}\n    \\label{lem6}\n\\end{lemma} \n\nApplying Lemmas \\ref{lem3}, \\ref{lem4}, \\ref{lem5}, and \\ref{lem6} yields Theorem \\ref{thm2}. \n\n\n\n\\section{Concluding remarks}\n\\label{conclusion}\nWe present two optimistic pessimism-based algorithms that maintain stringent safety constraints (either zero or bounded safety constraint violation) for unknown CMDPs with high probability, while still attaining an $\\tilde{\\mathcal{O}}(\\sqrt{K})$ regret of reward. The algorithms employ, respectively, a pessimistically safe policy set $\\Pi^k$ or an additional pessimistic term $\\epsilon_k$ into the safety constraint. \nThe first algorithm, OptPess-LP, guarantees zero violation with high probability by solving a linear programming with $\\Theta(|\\mathcal{S}||\\mathcal{A}|H)$ decision variables, while the second algorithm, OptPess-PrimalDual, is as efficient as policy-gradient-based algorithms in the tabular case, but violates constraints during initial episodes. A possible future direction for exploration is the application of the above OPFU principle in model-free algorithms.  \n\n\n\n\\section*{Acknowledgement}\nP. R. Kumar's work is partially supported by US National Science Foundation under CMMI-2038625, HDR Tripods CCF-1934904; US Office of Naval Research under N00014-21-1-2385; US ARO under W911NF1810331, W911NF2120064; and U.S. Department of Energy's Office of Energy Efficiency and Renewable Energy (EERE) under the Solar Energy Technologies Office Award Number DE-EE0009031. The views expressed herein and conclusions contained in this document are those of the authors and should not be interpreted as representing the views or official policies, either expressed or implied, of the U.S. NSF, ONR, ARO, Department of Energy  or the United States Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.\n\nDileep Kalathil gratefully acknowledges funding from the U.S. National Science Foundation (NSF) grants NSF-CRII- CPS-1850206 and NSF-CAREER-EPCN-2045783.\n\n\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\IEEEPARstart{A}s has been known, in conventional wireless communications, sending one information symbol from the transmitter to the receiver constitutes a transmission process over air interface.  Apart from the desired receiver, the transmitted symbol can be interference to the other receivers, except for broadcasting services. \n\nThis work proposes a new method that enables one information symbol to serve, over air interface, two users located in geometrically orthogonal directions as shown in Figure~1.  This scenario provides our research motivation to double transmission efficiency in the orthogonal geometry with the parallel- and perpendicular channel. The physical mechanism, allowing one symbol to appear with two different phases in the two directions, can be essentially attributed to the use of the spatial anisotropy of Doppler effect, which has been categorized into several sorts in general \\cite{Jackson1998}[2][3].   \n\nOur research starts from considering the Doppler effect on its emitted electromagnetic waves in the plane wave model and the observations from different angles with respect to the moving direction of the emission source.  \n\nWhen a harmonic signal is emitted, the Doppler effect can be expressed mathematically by \n\\begin{eqnarray}\n\t\\begin{array}{l}\\label{1}\n\t\\Delta f(\\theta) = f^{'}-f=\\frac{v_x}{\\lambda}\\cos\\theta,   \\ \\ \\\n\t\\end{array}\n\\end{eqnarray}\nwhere $\\Delta f$ is the Doppler frequency shift, $f^{'}$ is the frequency observed, $f$ is the frequency of the emission, $v_x$ is the speed of the source, $\\theta$ is the angle between directions of the observation and the source's moving, $\\lambda$ is the wavelength. \n\nIn a two-dimensional Cartesian coordinate model,  we use $\\Delta f_x$ and $\\Delta f_y$ to express $\\Delta f(0)$ and $\\Delta f(\\pi\/2)$ for showing that $\\Delta{f}_y = 0 $ holds no matter what value $\\Delta{f}_x $ is. This indicates that any effects resulted from $\\Delta f_x$ will be nullified in the direction of y-axis. \n\nNow, we work on $\\Delta f_x$ with Maxwell equation\\cite{Jackson1998} by\n\\begin{eqnarray}\n\\begin{array}{l}\\label{2}\nR_x(t)= Ae^{j2\\pi f t-j(kx_0-kv_xt)}, \\\\ \n\\end{array}\n\\end{eqnarray}\nwhere $j=\\sqrt{-1}$, $R_x(t)$ is the received wave of a receiver located at x-axis for $x>0$, $A$ is the received amplitude, $x_0$ is the initial position of the receiver, and $k=2\\pi\/\\lambda$ is a constant.   \n\n\\begin{figure}[!t]\n\t\\centering\n\t\\includegraphics[width=0.35\\textwidth]{Fig1}\n\t\\caption{ Multiple antennas at the crossroads working through  the parallel- and perpendicular channel.}\n\t\\label{Application Scenario}\n\\end{figure}\n\nThe Doppler frequency shift is found at $v_xk\/2\\pi = v_x\/\\lambda$ again in \\eqref{2}.  However, because $v_xt$ is actually the position the emission source arrives at time $t$, the frequency shift can be regarded as a continuous phase modulation with respect to the position of the emission source.  This can be more clearly seen at the baseband level by removing the carrier frequency $f$, i.e.,\n\\begin{eqnarray}\n\\begin{array}{l}\\label{3}\n\\hat{R}_x(x)=Ae^{jkx(t)-j\\phi_0},  \\ \\ \\ \n\\end{array}\n\\end{eqnarray}\nwhere  $\\hat{R}_x(x)$ is the received baseband signal, $x(t)=v_x t$ denotes the position of the emission source at time $t$, $\\phi_0= kx_0$ is the initial phase.  \n\nIt is noted that $kx(t)$ is a modulated phase that does not appear in direction of y-axis.  \n     \n\\section{Signal Modulation}\nIn this section, we propose a method in which the transmitter consists of a signal source and multiple antennas to transmit information symbols to the two receivers $\\hat{R}_x$ and $\\hat{R}_y$ located in two geometrically orthogonal directions, where the parallel and perpendicular channels are defined respectively, as shown in Fig.2. \n\n\\begin{figure}[!t]\n\t\\centering\n\t\\includegraphics[width=0.35\\textwidth]{Fig2.pdf}\n\t\\caption{The transmitter architecture of the QD method and the ideal geometric model.}\n\t\\label{PhD}\n\\end{figure}\n\n\\subsection{Quasi-Doppler Effect}\nLet us use a harmonic signal, in the plane wave model, to explain the mechanism of generating Doppler effect by using multiple antennas to replace the moving of the emission source.  These antennas are uniformly distributed along x-axis at $x_0 = 0,~x_1 = d,\\cdots,x_q=qd,\\cdots$, till $x_{Q-1} = (Q-1)d$ as shown in  Fig.2, where $d$ is the distance between two adjacent antennas and $Qd= \\lambda$ is required for limiting the geometric range of the multi antennas.  \n\nTo emulate the harmonic signal transmission of the Doppler source moving at speed $v_x$, we switch the signal source sequentially onto each antenna $x_0,~x_1,\\cdots,x_q,\\cdots, x_{Q-1}$, where the time of connecting the signal source and each antenna is set to $T=d\/v_x$ and the switching time, i.e., the time of hopping between any two antennas, is assumed to be zero. The received signal through the parallel channel can be written as\n\\begin{eqnarray}\n\\begin{array}{l}\\label{4}\n\\hat{R}_x(t)= h_xe^{jkv_x T \\lfloor t\/T\\rfloor+j\\phi_0} + n_0, \\ \\ \\\n\\end{array}\n\\end{eqnarray}\nwhere $\\hat{R}_x(t)$ is the received baseband signal, $v_x = d\/T$ is the speed of emulated moving, $\\lfloor \\cdot \\rfloor$ is the floor operator, $h_x$ is a complex channel response, $\\phi_0$ is an initial phase and $n_0$ is the Gaussian noise.\n\nThis equation shows the Doppler frequency shift at $kv_x\/2\\pi$, which is referred to as the Quasi Doppler (QD) effect, because the time variable behaves in a step manner with $\\lfloor t\/T \\rfloor$.  \n\nSimilar with the transformation from \\eqref{2} to \\eqref{3}, assuming $\\phi_0=0$, we can transfer the QD frequency shift into desiccate phase modulations, according to the positions of those antennas, as expressed by\n\\begin{eqnarray}\n\\begin{array}{l}\\label{5}\n\\hat{R}_x(x_q)=h_xe^{jk x_q+j\\phi_0} + n_0, \n\\end{array}\n\\end{eqnarray}\nwhere $x_q$ represents the position of $q^{th}$ antenna, $q = 0,1,\\cdots,Q-1$.  \n\nIn the following discussions, ``Antenna q'' stands for the antenna's position at position $x=x_q$.  It is found that the QD effect yields the discrete phase modulation instead of the continuous one in \\eqref{3}.  \n\nFurther, switching the signal source from antennas $q'$ to antenna $q$ can provide a phase modulation at $\\Delta {\\phi} = k(x_q-x_{q'}) = 2\\pi( x_q-x_{q'})\/\\lambda$.   Thus, by assuming $\\phi_0=0$, the transmission at antenna $q$ can be regarded as a QD phase of \n\\begin{eqnarray}\n\\begin{array}{l}\\label{6}\n\\phi_q =2\\pi x_q  \/\\lambda,  \n\\end{array}\n\\end{eqnarray}\nwhere $\\phi_q$ is defined as the QD phase, and $x_q$ is the position of the transmit antenna.  \n\nFor preparing a signal constellation for the transmission, $x_q$ should be pre-designed by\n\\begin{eqnarray}\n\\begin{array}{l}\\label{6-05}\nx_q = \\phi_q \\lambda\/2\\pi    \n\\end{array}\n\\end{eqnarray}\nfor $q=1,2,....,Q-1$, according to the required $\\phi_q$ in the signal constellation.  \n\nIn the communication scheme, the assumption of $\\phi_0 = 0$ is reasonable, because the channel characterization permits.  \n\n\\subsection{Joint Symbol Modulation}\nThe transmission issue arises with the use of one information symbol to provide two desired phases that appear through the parallel channel and the perpendicular channel.  \nTo be specific, $\\phi_x$ and $\\phi_y$ denote these two desired phases. The joint signal modulation is explained in the following two steps.  \n\nAt the first step, the QD transmitter uses the conventional modulation at its signal source to produce an information symbol as  \n\\begin{eqnarray}\n\\begin{array}{l}\\label{7}\nS_y = \\sqrt{E_s}e^{j\\phi_y},  \n\\end{array}\n\\end{eqnarray}\nwhere $S_y$ , $E_s$ and $\\phi_y$ are the modulated symbol, the symbol energy and the desired phase for the perpendicular channel, respectively. \n\nSecondly, for providing the desired phase, i.e., $\\phi_x$, through the parallel channel, the produced symbol in \\eqref{7} should be switched onto an appropriate antenna to complete the symbol transmission.  The QD phase adds its contribution by        \n\\begin{eqnarray}\n\\begin{array}{l}\\label{8}\nS_x = \\sqrt{E_s}e^{j(\\phi_y+ \\phi_q)}, \n\\end{array}\n\\end{eqnarray}\nand letting $\\phi_y+ \\phi_q = \\phi_x$ leads to \n\\begin{eqnarray}\\label{9}    \n\\phi_q = \\begin{cases}\n\\begin{split}\n&\\phi_x - \\phi_y,  \\quad &\\phi_x \\ge \\phi_y,\\\\\n&\\phi_x +2\\pi - \\phi_y, \\quad &\\phi_x <\\phi_y,\n\\end{split}\n\\end{cases}\n\\end{eqnarray}\nwhere $\\phi_q$ is the QD phase.  Further, by using \\eqref{6-05}, we find the transmit antenna $q$ for $x=x_q$.\n  \nBecause the QD phase. i.e., $\\phi_q$, does not show up through the perpendicular channel, the symbol transmission can be summarized in form of \n\\begin{eqnarray}\n\\begin{array}{l}\\label{10}\n\\hat{R}_\\beta = h_\\beta S_\\beta + n_0 =  h_\\beta\\sqrt{E_s}e^{j\\phi_\\beta} + n_0, \n\\end{array}\n\\end{eqnarray}\nwhere $\\hat{R}_\\beta$ and $h_\\beta$ are the received signal and the channel gain factor, respectively, with $\\beta=x$ and $y$ indicating the signal received through the parallel- and perpendicular channel, respectively.   \n  \nBased on the above derivations, we conclude that the QD method provides a possibility to double the transmission efficiency in the orthogonal geometry model, because one jointly modulated symbol can satisfy the needs of two receivers.   \n\n\\section{On the Performance}\nThe purpose of this section is to explore the application issues in the scenarios of crossroads, whereat the ideally orthogonal- and deviated geometric model are taken into the theoretical analysis and the simulations as well.  \n\n\\subsection {Theoretical Analysis}\nIn addition to the ideal geometric model, we define the deviated geometric model for the scenarios, where the orthogonality is slightly deviated, as shown in Fig. \\ref{fig3}, and work on the cases of the parallel- and the perpendicular channel separately for showing the discrepancies. \n\nAs can be expected, in comparison with the ideal geometry, the deviated angles can cause some phase offsets as analysed below. \n\nAssuming that there is a small angle deviation $\\theta_x$ between x-axis and the line of sight from the transmit antenna to the receiver $\\hat{R}_x$ as shown in Fig. 3(a),  we can find that the received $\\phi_y$ remains unchanged in \\eqref{8} and, however, the QD phase can change to (see the Appendix) \n\\begin{equation}\\label{11}\n{\\phi}_{q}' = k x_q\\cos\\theta_x =  2\\pi x_q \\cos\\theta_x\/\\lambda,\n\\end{equation}\nwhere ${\\phi}_q'$ is the QD phase received at the receiver $\\hat{R}_x$.  Consequently, the received signal can be expressed as \n\\begin{eqnarray}\n\\begin{array}{l}\\label{12}\n\\hat{R}_{\\theta_x} = h_{\\theta_x} e^{j(\\phi_y+\\phi_q')}\n\t + n_0 = h_{\\theta_x} e^{j\\phi_x+ j\\Delta{\\phi}_{x}} + n_0,\n\\end{array}\n\\end{eqnarray}\nwhere $h_{\\theta_x}$ and $\\Delta{\\phi_x} = 2\\pi x_q (\\cos\\theta_x-1)\/\\lambda$ are the channel gain factor and the phase-offset owing to the geometric deviation angle $\\theta_x$, respectively. \n\nSecondly, for the deviated perpendicular channel as shown in Fig.3(b), the received signal can be expressed as (see the Appendix)\n\\begin{eqnarray}\n\\begin{array}{l}\\label{13}\n\\hat{R}_{\\theta_y} = h_{\\theta_y} e^{j(\\phi_y+ kx_qsin\\theta_y)} + n_0 = h_{\\theta y}, e^{j(\\phi_y+\\Delta{\\phi_y})}+n_0\n\\end{array}\n\\end{eqnarray}\nwhere $h_{\\theta_y}$ and $\\Delta \\phi_y= kx_q\\sin\\theta_y \/ \\lambda$ are the channel gain factor and the phase-offset owing to $\\theta_y$, respectively.  \n\nApplying the Tailor expansion to \\eqref{12} and \\eqref{13}, we can find that the phase offsets are limited by the geometry deviation as $|\\Delta \\phi_x|\\le  \\frac{\\pi\\theta_x^2}{2} $ and $ |\\Delta \\phi_y|\\le \\pi \\theta_y $.   By comparison, the perpendicular channel is more sensitive to the deviation angle than the parallel channel, because $|\\Delta \\phi_y|$ converges in a linear manner, while $|\\Delta \\phi_x|$ with square, when the deviation angles approach zero.  \n    \nAs to the transmission performance, the QD method does not degrade much in principle, when the maximum values of $|\\Delta \\phi_x| $ and $|\\Delta \\phi_y|$ are much smaller than the phase difference between two adjacent points in the signal constellation used in the communication. To solidify this part of the work, simulations will be performed in the next subsection.\n\nFinally, we note that the assumption of zero switching-time in the theoretical model can be removed, because the phase detection is done by comparing the RF oscillations between the arriving signal and the local reference, both of which experience the same switching-time.  Thus, there is no relative phase change as compared to the case with zero switching-time. \n\\begin{figure}[!t]\n\t\\centering\n\t\\subfigure[]{\n\t\t\\centering\n\t\t\\includegraphics[width=0.38\\textwidth]{Fig3a.pdf}\n\t\t\\label{fig3a}\n\t}\n\t\\subfigure[]{\n\t\t\\centering\n\t\t\\includegraphics[width=0.38\\textwidth]{Fig3b.pdf}\n\t\t\\label{fig3b}\n\t}\n\t\\caption{Deviated geometric model:  an deviated angel with the parallel channel in (a) and that with perpendicular channel in (b). }\n\t\\label{fig3}\n\\end{figure}\n\n\\subsection{Simulations}\nIn this subsection, simulation results are reported for investigating the performances of one symbol through two directions in the ideally orthogonal- and deviated geometric model. \n\nFor simplicity, we use BPSK modulation at the signal source and two omni-directional antennas located at positions $x_0 = 0 $ and $x_1 =\\lambda\/2$ to construct the QD transmitter.  In addition, considering the crossroad scenarios where line of sight must exist from either the cars on the road or from people along the roadside to the traffic light, we neglect the multipath signals, because our antennas can be placed near the position of the traffic light.  Actually, we select the AWGN channel model to the simulations for testing the bit-error-rate (BER) performances on various SNR-values, \n\nThe random phases $\\phi_x$ and $\\phi_y$ are generated at the QD transmitter by using Matlab 2019, and the Monte-Carlo method is used to simulate the receiver with perfect channel catheterization, where the noise power is set to zero in the pilots.  Then, hard decision is used in the symbol demodulation.   \n\nTo confirm the QD method, the ideal geometric model is taken to the simulations and the results are plotted in Fig. 4(a)(b) for the parallel- and perpendicular channel, respectively.  Then, by setting a tolerable SNR loss at 0.8dB with $BER=10^{-8}$ in comparison with the ideal geometric model, we make a large number of simulations and found the corresponding deviation angles to $\\theta_x = 30^o$ and $\\theta_y = 8^o$, for which the BERs are also shown in Fig. \\ref{BER_Performance}(a)(b), respectively. The relativity of the two deviation angles, i.e., $\\theta_y < \\theta_x$, agrees with the theoretical conclusion in the analysis, i.e. the perpendicular channel is more sensitive compared to the parallel channel.  \n\n\\begin{figure}[!t]\n\t\\centering\n\t\\subfigure[]{\n\t\t\\centering\n\t\t\\includegraphics[width=0.42\\textwidth]{BER_Performance_thetax.pdf}\n\t\t\\label{fig4a}\n\t}\n\t\\subfigure[]{\n\t\t\\centering\n\t\t\\includegraphics[width=0.42\\textwidth]{BER_Performance_thetay.pdf}\n\t\t\\label{fig4b}\n\t}\n\t\\caption{The BER performance of the ideal channel and the deviated channel: (a) the parallel channel and (b) the perpendicular channel. }\n\t\\label{BER_Performance}\n\\end{figure}\n\nFinally, the author would get rid of any possible confusion between the QD method and the spatial modulation (SM) by the following short statement: the QD method can double the transmission efficiency and work in the one-to-two system, while the SM can increase the transmission efficiency slightly \\cite{Yang2008} in the point-to-point manner \\cite{Mesleh2008,Rajashekar2013,Wen2014}. \n\n\\section{Conclusion}\nThe concept of the QD effect has been created for realizing the discrete phase modulation.  By combining the anisotropy of electromagnetic wave propagation, we developed the QD method that can double the transmission efficiency.  The application issues are addressed in the scenarios of crossroads with the ideal and deviated channel models, wherein the feasibility is confirmed by simulation results. \n\n\\section*{Appendix}\nBy considering the plane wave model, we use Fig. A to show the angle deviations between the ideal- and deviated geometric model by $\\theta_x$ and $\\theta_y$. \n\nExamining the angle deviation from the parallel channel, one can find that the wave propagation path-difference $x_q -x_0$ in the ideal  geometric model should be projected onto the line in the direction of $\\hat{R}_\\theta$.  The projected result is equal to $(x_q-x_0) cos\\theta_x$ that leads to the QD phase $kx_q cos\\theta_x$ at the receiver. \n \nFor the deviated perpendicular channel, the propagation path-difference is described by the deviation angle $\\theta_y$.  Consequently, the QD phase can be changed to $kx_q sin\\theta_y$ because $ cos\\theta_x = sin\\theta_y$.     \n\\renewcommand{\\thefigure}{\\Alph{figure}}\n\\setcounter{figure}{0}\n\\begin{figure}[!t]\n\t\\centering\n\t\\includegraphics[width=0.4\\textwidth]{appendix.pdf}\n\t\\caption{Illustration of the QD phase difference. }\n\t\\label{appendix}\n\\end{figure}\n\\section*{Acknowledgment}\n\nThe author expresses his thanks to Dongsheng Zheng, a Ph.D candidate, for his work on the simulations. \n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Algebra}\\label{subsection algebra}\n\nIn the following, $\\mathbf{F}$ denotes the field $\\mathbf{Z} \/ 2 \\mathbf{Z}$. Vector spaces are always over $\\mathbf{F}$.\n\n\\begin{defin}\\label{definition category}\n\t\n\tAn $A_{\\infty}$-category $\\mcA$ is the data of\n\t\\begin{enumerate}\n\t\t\n\t\t\\item a collection of objects $\\mathrm{ob} \\, \\mcA$,\n\t\t\n\t\t\\item for every objects $X,Y$, a graded vector space of morphisms $\\mcA \\left( X,Y \\right)$,\n\t\t\n\t\t\\item a family of degree $\\left( 2-d \\right)$ linear maps\n\t\t\\[\\mu^d : \\mcA \\left( X_0, X_1 \\right) \\otimes \\dots \\otimes \\mcA \\left( X_{d-1}, X_d \\right) \\to \\mcA \\left( X_0, X_d \\right) \\]\n\t\tindexed by the sequences of objects $\\left( X_0, \\dots, X_d \\right)$, $d \\geq 1$, such that\n\t\t\\[\\sum\\limits_{0 \\leq i < j \\leq d} \\mu^{d-(j-i)+1} \\circ \\left( \\mathbf{1}^i \\otimes \\mu^{j-i} \\otimes \\mathbf{1}^{d-j} \\right) = 0, \\]\n\t\tfor all $d \\geq 1$.\n\t\n\t\\end{enumerate}\n\t\n\\end{defin}\n\n\\begin{defin}\\label{definition coalgebra}\n\t\n\tAn $A_{\\infty}$-cocategory $\\mcC$ is the data of \n\t\t\\begin{enumerate}\n\t\t\n\t\t\\item a collection of objects $\\mathrm{ob} \\, \\mcC$,\n\t\t\n\t\t\\item for every objects $X,Y$, a graded vector space of morphisms $\\mcC \\left( X,Y \\right)$,\n\t\t\n\t\t\\item a family of degree $\\left( 2-d \\right)$ linear maps\n\t\t\\[\\delta^d : \\mcC \\left( X_0, X_d \\right) \\to \\bigoplus \\limits_{d \\geq 1} \\bigoplus_{X_1, \\dots, X_{d-1}} \\mcC (X_0, X_1) \\otimes \\dots \\otimes \\mcC (X_{d-1}, X_d) \\]\n\t\tindexed by the sequences of objects $\\left( X_0, \\dots, X_d \\right)$, $d \\geq 1$, such that\n\t\t\\begin{itemize}\n\t\t\t\n\t\t\t\\item for all $d \\geq 1$,\n\t\t\t\\[\\sum\\limits_{0 \\leq i < j \\leq d} \\left( \\mathbf{1}^i \\otimes \\delta^{j-i} \\otimes \\mathbf{1}^{d-j} \\right) \\circ \\delta^{d-(j-i)+1} = 0, \\]\n\t\t\t\n\t\t\t\\item the map \n\t\t\t\\[C \\to \\prod_{d \\geq 1} C^{\\otimes d}, \\quad x \\mapsto \\left( \\delta^d \\left( x \\right) \\right)_{d \\geq 1} \\]\n\t\t\tfactors through the inclusion $\\bigoplus\\limits_{d \\geq 1} C^{\\otimes d} \\to \\prod\\limits_{d \\geq 1} C^{\\otimes d}$.\n\t\t\t\n\t\t\\end{itemize}\n\t\t\n\t\\end{enumerate}\n\t\n\\end{defin}\n\n\\begin{rmks}\n\t\n\t\\begin{enumerate}\n\t\t\n\t\t\\item If $\\mcE$ is some set, denote by $\\mathbf{F}_{\\mcE}$ the semi-simple algebra over $\\mathbf{F}$ generated by elements $e_X$, $X \\in \\mcE$, such that\n\t\t\\[e_X \\cdot e_Y = \n\t\t\\left\\{   \n\t\t\\begin{array}{ll}\n\t\t\te_X & \\text{if } X = Y \\\\\n\t\t\t0 & \\text{if } X \\ne Y.\n\t\t\\end{array}\n\t\t\\right.  \\]\n\t\tTo any $A_{\\infty}$-category $\\mcA$ with $\\mathrm{ob} (\\mcA) = \\mcE$, we can associate an $A_{\\infty}$-algebra over $\\mathbf{F}_{\\mcE}$ where \n\t\t\\begin{itemize}\n\n\t\t\t\\item the underlying graded vector space is $\\bigoplus_{X,Y \\in \\mcE} \\mcA (X, Y)$,\n\t\t\t\n\t\t\t\\item given $x \\in \\mcA (X_0, Y_0)$, \n\t\t\t\\[e_X \\cdot x = \n\t\t\t\\left\\{   \n\t\t\t\\begin{array}{ll}\n\t\t\t\tx & \\text{if } X = X_0 \\\\\n\t\t\t\t0 & \\text{if } X \\ne X_0\n\t\t\t\\end{array}\n\t\t\t\\right. \\text{ and } \n\t\t\tx \\cdot e_Y = \n\t\t\t\\left\\{   \n\t\t\t\\begin{array}{ll}\n\t\t\t\tx & \\text{if } Y = Y_0 \\\\\n\t\t\t\t0 & \\text{if } Y \\ne Y_0,\n\t\t\t\\end{array}\n\t\t\t\\right. \\]\t\n\t\t\t\n\t\t\t\\item operations are the same as on $\\mcA$.\n\n\t\t\\end{itemize}\n\t\tConversely, to any $A_{\\infty}$-algebra over $\\mathbf{F}_{\\mcE}$, one can associate an $A_{\\infty}$-category with $\\mathrm{ob} (\\mcA) = \\mcE$. \n\t\tNote that the above discussion also applies to $A_{\\infty}$-cocategories. As a result, the theory of $A_{\\infty}$-(co)categories with $\\mcE$ as set of objects is equivalent to the theory of $A_{\\infty}$-(co)algebras over $\\mathbf{F}_{\\mcE}$.\n\t\n\t\t\\item In this paper, we will appeal to several standard notions in the theory of $A_{\\infty}$-(co)categories that we choose not to recall: instead, we list them and give corresponding references.\n\t\t\\begin{itemize}\n\t\t\t\n\t\t\t\\item For $A_{\\infty}$-(co)maps, (co)augmentations and (co)bar, graded dual, Koszul dual constructions, see \\cite[section 2]{EL21} (where everything is written in the language of $A_{\\infty}$-(co)algebras over $\\mathbf{F}_{\\mcE}$).\n\t\t\t\n\t\t\t\\item For general definitions and results about $A_{\\infty}$-categories (in particular about homotopy between $A_{\\infty}$-functors, homological perturbation theory, directed (sub)categories and twisted complexes), see \\cite[chapter 1]{Sei08}.\n\t\t\t\n\t\t\t\\item For quotient of $A_{\\infty}$-categories, see \\cite{LO06}, and for localization of $A_{\\infty}$-categories, see \\cite[section 3.1.3]{GPS20}.\n\n\t\t\\end{itemize}\n\t\n\t\t\\item An Adams-graded vector space is a $\\mathbf{Z} \\times \\mathbf{Z}$-graded vector space: if $x$ is an element in the $(i,j)$ component, we say that $i$ is the cohomological degree of $x$, and $j$ is the Adams degree of $x$.\n\t\tAn Adams-graded $A_{\\infty}$-(co)category is an $A_{\\infty}$-(co)category enriched over Adams-graded vector spaces, where the operations are required to be of degree $0$ with respect to the Adams grading. \n\t\tSee \\cite{LPWZ08} for a treatment of Koszul duality in the context of Adams-graded $A_{\\infty}$-algebras.\n\t\t\n\t\\end{enumerate}\n\n\\end{rmks}\n\n\\subsection{Modules over $A_{\\infty}$-categories}\n\nLet $\\mcC, \\mcD$ be two $A_\\infty$-categories, and let $\\mcA, \\mcB$ be two full subcategories of $\\mcC, \\mcD$ respectively.\n\n\\begin{defin}\\label{definition bimodule}\n\t\n\tA $\\left( \\mcC, \\mcD \\right)$-bimodule $\\mcM$ consists of the following data: \n\t\\begin{enumerate}\n\t\t\n\t\t\\item for every pair $\\left( X, Y \\right) \\in \\mathrm{ob} \\left( \\mcC \\right) \\times \\mathrm{ob} \\left( \\mcD \\right)$, a vector space $\\mcM \\left( X, Y \\right)$, \n\t\t\n\t\t\\item a family of degree $\\left( 1-p-q \\right)$ linear maps \n\t\t\\begin{align*}\n\t\t\\mu_{\\mcM} : \\mcC \\left( X_0, X_1 \\right) \\otimes & \\dots \\otimes \\mcC \\left( X_{p-1}, X_p \\right) \\otimes \\mcM \\left( X_p, Y_q \\right) \\\\ \n\t\t& \\otimes \\mcD \\left( Y_q, Y_{q-1} \\right) \\otimes \\dots \\otimes \\mcD \\left( Y_1, Y_0 \\right) \\to \\mcM \\left( X_0, Y_0 \\right)\n\t\t\\end{align*}\n\t\tindexed by the sequences \n\t\t\\[\\left( X_0, \\dots, X_p, Y_0, \\dots, Y_q \\right) \\in \\mathrm{ob} \\left( \\mcC \\right)^{p+1} \\times \\mathrm{ob} \\left( \\mcD \\right)^{q+1}, \\]\n\t\twhich satisfy the relations \n\t\t\\begin{align*}\n\t\t\\sum \\mu_{\\mcM} \\left( \\dots , \\mu_{\\mcC} \\left( \\dots \\right) , \\dots, u \\dots \\right) & + \\sum \\mu_{\\mcM} \\left( \\dots , \\mu_{\\mcM} \\left( \\dots, u, \\dots \\right), \\dots \\right) \\\\ \n\t\t& + \\sum \\mu_{\\mcM} \\left( \\dots , u, \\dots, \\mu_{\\mcD} \\left( \\dots \\right) , \\dots \\right) = 0 .\n\t\t\\end{align*}\n\t\t\n\t\\end{enumerate}\n\t\n\tA degree $s$ morphism $t : \\mcM_1 \\to \\mcM_2$ between two $\\left( \\mcC, \\mcD \\right)$-bimodules consists of a family of degree $\\left( s-p-q \\right)$ linear maps\n\t\\begin{align*}\n\tt : \\mcC \\left( X_0, X_1 \\right) \\otimes & \\dots \\otimes \\mcC \\left( X_{p-1}, X_p \\right) \\otimes \\mcM_1 \\left( X_p, Y_q \\right) \\\\ \n\t& \\otimes \\mcD \\left( Y_q, Y_{q-1} \\right) \\otimes \\dots \\otimes \\mcD \\left( Y_1, Y_0 \\right) \\to \\mcM_2 \\left( X_0, Y_0 \\right)\n\t\\end{align*}\n\tindexed by the sequences \n\t\\[\\left( X_0, \\dots, X_p, Y_0, \\dots, Y_q \\right) \\in \\mathrm{ob} \\left( \\mcC \\right)^{p+1} \\times \\mathrm{ob} \\left( \\mcD \\right)^{q+1}. \\]\n\tThe differential of such a morphism is defined by\n\t\\begin{align*}\n\t\\mu_{\\mathrm{Mod}_{\\mcC, \\mcD}}^1 & \\left( t \\right) \\left( \\dots, u, \\dots \\right)  \\\\\n\t& = \\sum t \\left( \\dots , \\mu_{\\mcC} \\left( \\dots \\right) , \\dots, u, \\dots \\right) + \\sum t \\left( \\dots , \\mu_{\\mcM_1} \\left( \\dots, u, \\dots \\right), \\dots \\right) \\\\ \n\t& +\\sum t \\left( \\dots , u, \\dots, \\mu_{\\mcD} \\left( \\dots \\right), \\dots \\right) + \\sum \\mu_{\\mcM_2} \\left( \\dots , t \\left( \\dots, u, \\dots \\right), \\dots \\right) .\n\t\\end{align*} \n\tFinally, the composition of $t_1 : \\mcM_1 \\to \\mcM_2$ and $t_2 : \\mcM_2 \\to \\mcM_3$ is such that\n\t\\[\t\\mu_{\\mathrm{Mod}_{\\mcC}}^2 \\left( t_1 , t_2 \\right) \\left( \\dots, u, \\dots \\right)  = \\sum t_2 \\left( \\dots , t_1 \\left( \\dots, u, \\dots \\right), \\dots \\right) .\\]\n\tWe denote by $\\mathrm{Mod}_{\\mcC, \\mcD}$ the DG-category of $\\left( \\mcC, \\mcD \\right)$-bimodules.\n\t\n\\end{defin}\n\n\\begin{defin}\\label{definition pullback bimodule}\n\t\n\tLet $\\Phi_1, \\Phi_2 : \\mcC \\to \\mcD$ be two $A_{\\infty}$-functors. Then there is a $\\mcC$-bimodule $\\mcD \\left( \\Phi_1 (-), \\Phi_2 (-) \\right)$ defined as follows:\n\t\\begin{enumerate}\n\t\t\n\t\t\\item on objects, it sends $(X_1, X_2)$ to $\\mcD \\left( \\Phi_1 X_1, \\Phi_2 X_2 \\right)$,\n\t\t\n\t\t\\item on morphisms, it sends a sequence $\\left( \\dots, y, \\dots \\right)$ in\n\t\t\\begin{align*}\n\t\t\\mcC \\left( X_0, X_1 \\right) \\times & \\dots \\times \\mcC \\left( X_{p-1}, X_p \\right)  \\times \\mcD \\left( \\Phi_1 X_p, \\Phi_2 X_{p+1} \\right) \\\\ \n\t\t& \\times \\mcC \\left( X_{p+1}, X_{p+2} \\right) \\times \\dots \\times \\mcC \\left( X_{p+q}, X_{p+q+1} \\right)\n\t\t\\end{align*}\n\t\tto \n\t\t\\begin{align*}\n\t\t\\mu_{\\mcD \\left( \\Phi_1 (-), \\Phi_2 (-) \\right)} & \\left( \\dots, y, \\dots \\right) \\\\ \n\t\t& = \\sum \\mu_{\\mcD} \\left( \\Phi_1 \\left(\\dots \\right) , \\dots , \\Phi_1 \\left( \\dots \\right) , y, \\Phi_2 \\left( \\dots \\right) , \\dots , \\Phi_2 \\left( \\dots \\right) \\right) .\n\t\t\\end{align*}\n\t\t\n\t\\end{enumerate}\n\t\n\\end{defin}\n\nIn the following, we will focus on \\emph{left} $\\mcC$-modules, which correspond to $\\left( \\mcC, \\mathbf{F} \\right)$-bimodules.\nWe denote by $\\mathrm{Mod}_{\\mcC}$ the DG-category of (left) $\\mcC$-modules.\n\n\\begin{defin}\\label{definition quasi-isomorphism between modules}\n\t\n\tLet $t : \\mcM_1 \\to \\mcM_2$ be a degree $0$ closed $\\mcC$-module map. We say that $t$ is a quasi-isomorphism if the induced chain map $t : \\mcM_1 \\left( X \\right) \\to \\mcM_2 \\left( X \\right)$ is a quasi-isomorphism for every object $X$ in $\\mcC$. (See \\cite[section A.2]{GPS19} for a discussion on quasi-isomorphisms between $A_{\\infty}$-modules).\n\t\n\\end{defin}\n\n\\begin{defin}\\label{definition homotopy between morphisms of modules}\n\t\n\tLet $t, t' : \\mcM_1 \\to \\mcM_2$ be two degree $0$ closed morphisms of $\\mcC$-modules. A homotopy between $t$ and $t'$ is a $\\mcC$-module map $h : \\mcM_1 \\to \\mcM_2$ such that\n\t\\[ t + t' = \\mu^1_{\\mathrm{Mod}_{\\mcC}} \\left( h \\right) . \\]\n\t\n\\end{defin}\n\n\\begin{defin}[See \\cite{Sei08} paragraph (1l) and \\cite{GPS19} section A.1]\\label{definition Yoneda functor}\n\t \n\tThere is an $A_{\\infty}$-functor \n\t\\[ \\mcC \\to \\mathrm{Mod}_{\\mcC}, \\quad Y \\mapsto \\mcC \\left( - , Y \\right), \\]\n\tcalled the Yoneda $A_{\\infty}$-functor, defined as follows.\n\tFor every object $X$, \n\t\\[ \\mcC \\left( -,Y \\right) \\left( X \\right) = \\mcC \\left( X,Y \\right). \\]\n\tBesides, a sequence \n\t\\[ \\left( x_0 , \\dots , x_{d-1} \\right) \\in \\mcC \\left( X_0 , X_1 \\right) \\times \\dots \\times \\mcC \\left( X_{d-1} , X_d \\right) \\]\n\tacts on an element $u$ in $\\mcC \\left( X_d , Y \\right)$ via the operations \n\t\\[ \\mu_{\\mcC \\left( -,Y \\right)} \\left( x_0 , \\dots , x_{d-1} , u \\right) = \\mu_{\\mcC} \\left( x_0 , \\dots , x_{d-1} , u \\right) . \\]\n\tFinally, let \n\t\\[\\mathbf{y} = \\left( y_0 , \\dots , y_{p-1} \\right) \\in \\mcC \\left( Y_0 , Y_1 \\right) \\times \\dots \\times \\mcC \\left( Y_{p-1} , Y_p \\right) \\]\n\tbe a sequence of morphisms in $\\mcC$. Then the Yoneda functor gives a morphism of $\\mcC$-modules $t_{\\mathbf{y}} : \\mcC \\left( -,Y_0 \\right) \\to \\mcC \\left( -,Y_p \\right)$ which sends every sequence $\\left( x_0 , \\dots , x_{d-1} , u \\right)$ as above to \n\t\\[ \\mu_{\\mcC} \\left( x_0 , \\dots , x_{d-1} , u , y_0, \\dots, y_{p-1} \\right) \\in \\mcC \\left( X_0,Y_p \\right) . \\]\n\t\n\\end{defin}\n\nWe have the following important result.\n\n\\begin{prop}[Yoneda lemma]\\label{prop Yoneda lemma}\n\t\n\tThe Yoneda $A_{\\infty}$-functor \n\t\\[ \\mcC \\to \\mathrm{Mod}_{\\mcC}, \\quad Y \\mapsto \\mcC \\left( - , Y \\right) \\]\n\tis cohomologically full and faithful.\n\t\n\\end{prop}\n\n\\begin{proof}\n\t\n\tThis is Lemma 2.12 in \\cite{Sei08}, and also Lemma A.1 in \\cite{GPS19}.\n\t\n\\end{proof}\n\nThe Yoneda lemma has the following easy consequence. We state it for future reference.\n\n\\begin{coro}\\label{coro closed module map homotopic to Yoneda module map}\n\t\n\tEvery closed $\\mcC$-module map $f : \\mcC \\left( -, X \\right) \\to \\mcC \\left( -, Y \\right)$ is homotopic to the $\\mcC$-module map $t_{f \\left( e_X \\right)}$ induced by $f \\left( e_X \\right) \\in \\mcC \\left( X, Y \\right)$. (see Definition \\ref{definition Yoneda functor}).\n\t\n\\end{coro}\n\n\\begin{proof}\n\t\n\tAccording to the Yoneda lemma, $f$ is homotopic to $t_x$ for some closed $x$ in $\\mcC \\left( X , Y \\right)$. Thus, there exists a $\\mcC$-module map $h : \\mcC \\left( -, X \\right) \\to \\mcC \\left( -, Y \\right)$ such that \n\t\\[f = t_x + \\mu_{\\mathrm{Mod}_{\\mcC}}^1 \\left( h \\right) . \\]\n\tEvaluating the latter relation at the unit $e_X \\in \\mcC \\left( X, X \\right)$ gives \n\t\\[f \\left( e_X \\right) = x + \\mu_{\\mcC}^1 \\left( h e_X \\right) . \\]\n\tTherefore, $x$ is homotopic to $f \\left( e_X \\right)$, and this implies that $t_x$ is homotopic to $t_{f \\left( e_X \\right)}$ by the Yoneda lemma. Finally, we have that $f$ is homotopic to $t_{f \\left( e_X \\right)}$.\n\t\n\\end{proof}\n\n\\paragraph{Pullback of $A_{\\infty}$-modules.}\n\n\\begin{defin}[See \\cite{Sei08} paragraph (1k)]\\label{definition pullback functor}\n\t \n\tLet $\\Phi : \\mcC \\to \\mcD$ be an $A_{\\infty}$-functor. Then there is a DG-functor \n\t\\[\\Phi^* : \\mathrm{Mod}_{\\mcD} \\to \\mathrm{Mod}_{\\mcC}, \\quad \\mcN \\mapsto \\Phi^* \\mcN \\]\n\tdefined as follows.\n\tLet $\\mcN$ be a $\\mcD$-module. For every object $X$, \n\t\\[ \\Phi^* \\mcN \\left( X \\right) = \\mcN \\left( \\Phi X \\right) . \\]\n\tBesides, a sequence \n\t\\[ \\left( x_0 , \\dots , x_{d-1} \\right) \\in \\mcC \\left( X_0 , X_1 \\right) \\times \\dots \\times \\mcC \\left( X_{d-1} , X_d \\right) \\]\n\tacts on an element $u \\in \\Phi^* \\mcN \\left( X_d \\right)$ via the operations \n\t\\[ \\mu_{\\Phi^* \\mcN} \\left( x_0 , \\dots , x_{d-1} , u \\right) = \\sum \\mu_{\\mcN} \\left( \\Phi \\left( x_0 , \\dots , x_{i_1-1} \\right) , \\dots , \\Phi \\left( x_{d-i_r} , \\dots , x_{d-1} \\right) , u \\right) . \\]\n\tFinally, let $t : \\mcN_1 \\to \\mcN_2$ be a $\\mcD$-module map. Then the above functor gives a $\\mcC$-module map $\\Phi^* t : \\Phi^* \\mcN_1 \\to \\Phi^* \\mcN_2$ which sends every sequence $\\left( x_0 , \\dots , x_{d-1} , u \\right)$ as above to \n\t\\[ \\Phi^* t \\left( x_0 , \\dots , x_{d-1} , u \\right) = \\sum t \\left( \\Phi \\left( x_0 , \\dots , x_{i_1-1} \\right) , \\dots , \\Phi \\left( x_{d-i_r} , \\dots , x_{d-1} \\right) , u \\right) . \\]\n\t\n\\end{defin}\n\n\\begin{rmk}\\label{rmk composition of pullback functors}\n\t\n\tLet $\\Phi : \\mcC \\to \\mcD$ be an $A_{\\infty}$-functor, and let $\\Psi : \\mcD \\to \\mcE$ be another $A_{\\infty}$-functor towards a third $A_{\\infty}$-category $\\mcE$. Then $\\Phi^* \\circ \\Psi^* = \\left( \\Psi \\circ \\Phi \\right)^*$ as DG-functors.\n\t\n\t\n\\end{rmk}\n\n\\begin{defin}\\label{definition modules morphism induced by functor}\n\t\n\tLet $Y$ be an object of $\\mcC$, and let $\\Phi : \\mcC \\to \\mcD$ be an $A_{\\infty}$-functor.\n\tThen there is a degree $0$ closed $\\mcC$-module map $t_{\\Phi} : \\mcC \\left( - , Y \\right) \\to \\Phi^* \\mcD \\left( - , \\Phi \\left( Y \\right) \\right)$ which sends any sequence \n\t\\[ \\left( x_0 , \\dots , x_{d-1} , u \\right) \\in \\mcC \\left( X_0 , X_1 \\right) \\times \\dots \\times \\mcC \\left( X_{d-1} , X_d \\right) \\times \\mcC \\left( X_d , Y \\right) \\]\n\tto \n\t\\[t_{\\Phi} \\left( x_0 , \\dots , x_{d-1} , u \\right) = \\Phi \\left( x_0 , \\dots , x_{d-1} , u \\right) \\in \\mcD \\left( \\Phi X_0 , \\Phi Y \\right) . \\]\n\t\n\\end{defin}\n\n\\paragraph{Quotient of $A_{\\infty}$-modules.}\n\n\\begin{defin}[See \\cite{GPS20} section 3.1.3]\\label{definition localization of modules}\n\t \n\tThere is a DG-functor \n\t\\[ \\mathrm{Mod}_{\\mcC} \\to \\mathrm{Mod}_{\\mcC \/ \\mcA}, \\quad \\mcM \\, \\mapsto \\, _{\\mcA \\backslash} \\mcM \\]\n\tdefined as follows. Let $\\mcM$ be a $\\mcC$-module. For every object $X$, \n\t\\begin{align*}\n\t_{\\mcA \\backslash} \\mcM \\left( X \\right) & = \\\\\n\t& \\mcM \\left( X \\right) \\bigoplus \\left( \\bigoplus\\limits_{\\substack{ p \\geq 1 \\\\ A_1,\\dots,A_p \\in \\mcA}} \\mcC \\left( X , A_1 \\right) [1] \\otimes \\dots \\otimes \\mcC \\left( A_{p-1} , A_p \\right) [1] \\otimes \\mcM \\left( A_p \\right) \\right) .\n\t\\end{align*}\n\tBesides, a sequence\n\t\\[ \\mathbf{x}_i = \\left( x_i^0 , \\dots , x_i^{p_i-1} \\right) \\in \\mcC \/ \\mcA \\left( X_i , X_{i+1} \\right) \\quad \\left( 0 \\leq i \\leq d-1 \\right) \\]\n\tacts on an element\n\t\\[ \\mathbf{u} = \\left( x_d^0 , \\dots , x_d^{p_d-1} , u \\right) \\in \\, _{\\mcA \\backslash} \\mcM \\left( X_d \\right) \\]\n\tvia the operations \n\t\\begin{align*}\n\t\\mu_{_{\\mcA \\backslash} \\mcM} & \\left( \\mathbf{x}_0 , \\dots , \\mathbf{x}_{d-1} , \\mathbf{u} \\right) = \\\\ \n\t& \\sum\\limits_{ \\substack{ 0 \\leq i \\leq p_0, 1 \\leq j \\leq p_d \\\\ i<j \\text{ if } d=0 } } x_0^0 \\otimes \\dots \\otimes x_0^{i-1} \\otimes \\mu_{\\mcC} \\left( x_0^i, \\dots , x_d^{j-1} \\right) \\otimes x_d^j \\otimes \\dots \\otimes x_d^{p_d-1} \\otimes u \\\\\n\t& + \\sum\\limits_{ 0 \\leq i \\leq p_0 } x_0^0 \\otimes \\dots \\otimes x_0^{i-1} \\otimes \\mu_{\\mcM} \\left( x_0^i , \\dots , x_d^{p_d-1} , u \\right) .\n\t\\end{align*}\n\tFinally, let $t : \\mcM_1 \\to \\mcM_2$ be a $\\mcC$-module map. Then the above functor gives a $\\mcC \/ \\mcA$-module map $_{\\mcA \\backslash} t : \\, _{\\mcA \\backslash} \\mcM_1 \\to \\, _{\\mcA \\backslash} \\mcM_2$ which sends every sequence $\\left( \\mathbf{x}_0 , \\dots , \\mathbf{x}_{d-1} , \\mathbf{u} \\right)$ as above to \n\t\\[ _{\\mcA \\backslash} t \\left( \\mathbf{x}_0 , \\dots , \\mathbf{x}_{d-1} , \\mathbf{u} \\right) = \\sum\\limits_{0 \\leq i \\leq p_0} x_0^0 \\otimes \\dots \\otimes x_0^{i-1} \\otimes t \\left( x_0^i , \\dots , x_d^{p_d-1} , u \\right) . \\] \n\t\n\\end{defin}\n\n\\paragraph{Relations between pullback and quotient of $A_{\\infty}$-modules.}\n\n\\begin{defin}\\label{definition natural transformation}\n\t\n\tLet $\\Phi : \\mcC \\to \\mcD$ be an $A_{\\infty}$-functor such that $\\Phi \\left( \\mcA \\right)$ is contained in $\\mcB$, and let $X$ be a fixed object of $\\mcC$. Then, for each $\\mcD$-module $\\mcN$, there is a chain map $_{\\mcA \\backslash} \\left(\\Phi^* \\mcN \\right) \\left( X \\right) \\to \\, _{\\mcB \\backslash} \\mcN \\left( \\Phi X \\right)$ which sends an element \n\t\\[ \\mathbf{u} = \\left( x^0 , \\dots , x^{p-1} , u \\right) \\in \\, _{\\mcA \\backslash} \\left( \\Phi^* \\mcN \\right) \\left( X \\right) \\]\n\tto\n\t\\[ \\sum \\Phi \\left( x^0 , \\dots , x^{i_1-1} \\right) \\otimes \\dots \\otimes \\Phi \\left( x^{i_r} , \\dots , x^{p-1} \\right) \\otimes u \\in \\, _{\\mcB \\backslash} \\mcN \\left( \\Phi X \\right) . \\]\n\t\n\tThis defines a natural transformation between the functors $\\mcN \\mapsto \\, _{\\mcA \\backslash} \\left(\\Phi^* \\mcN \\right) \\left( X \\right)$ and $\\mcN \\mapsto \\, _{\\mcB \\backslash} \\mcN \\left( \\Phi X \\right)$ from $\\mathrm{Mod}_{\\mcD}$ to $\\mathrm{Ch}$.\n\tIn other words, for every $\\mcD$-module map $t : \\mcN_1 \\to \\mcN_2$, the following diagram of chain complexes commutes \n\t\\[\\begin{tikzcd}\n\t_{\\mcA \\backslash} \\left(\\Phi^* \\mcN_1 \\right) \\left( X \\right) \\ar[r] \\ar[d, \"_{\\mcA \\backslash} \\left(\\Phi^* t \\right)\"] & \\, _{\\mcB \\backslash} \\mcN_1 \\left( \\Phi X \\right) \\ar[d, \"_{\\mcB \\backslash} t\"] \\\\\n\t_{\\mcA \\backslash} \\left(\\Phi^* \\mcN_2 \\right) \\left( X \\right) \\ar[r] & \\, _{\\mcB \\backslash} \\mcN_2 \\left( \\Phi X \\right) \n\t\\end{tikzcd} . \\]\n\t\n\\end{defin}\n\n\\begin{rmk}\\label{rmk composition}\n\t\n\tLet $Y$ be an object of $\\mcC$, and let $\\Phi : \\mcC \\to \\mcD$ be an $A_{\\infty}$-functor such that $\\Phi \\left( \\mcA \\right)$ is contained in $\\mcB$.\n\tLet $\\widetilde{\\Phi} : \\mcC \/ \\mcA \\to \\mcD \/ \\mcB$ be the $A_{\\infty}$-functor induced by $\\Phi$ (see \\cite[section 3]{LO06}).\n\tLocalize the morphism $t_{\\Phi} : \\mcC \\left( - , Y \\right) \\to \\Phi^* \\mcD \\left( - , \\Phi Y \\right)$ of Definition \\ref{definition modules morphism induced by functor} at $\\mcA$ and evaluate at $X$ to get a chain map\n\t\\[\\mcC \/ \\mcA \\left( X,Y \\right) = \\, _{\\mcA \\backslash} \\mcC \\left( - , Y \\right) \\left( X \\right) \\xrightarrow{_{\\mcA \\backslash} t_{\\Phi}} \\, _{\\mcA \\backslash} \\left( \\Phi^* \\mcD \\left( - , \\Phi Y \\right) \\right) \\left( X \\right) . \\]\n\tThen the composition of this map with the chain map \n\t\\[ _{\\mcA \\backslash} \\left( \\Phi^* \\mcD \\left( - , \\Phi Y \\right) \\right) \\left( X \\right) \\to \\, _{\\mcB \\backslash} \\mcD \\left( - , \\Phi Y \\right) \\left( \\Phi X \\right) = \\, \\mcD \/ \\mcB \\left( \\Phi X, \\Phi Y \\right) \\]\n\tof Definition \\ref{definition natural transformation} is the chain map $\\widetilde{\\Phi} : \\mcC \/ \\mcA \\left( X,Y \\right) \\to \\mcD \/ \\mcB \\left( \\Phi X, \\Phi Y \\right)$.\n\t\n\\end{rmk}\n\n\\begin{prop}\\label{prop quasi-isomorphism between localizations}\n\t\n\tLet $\\Phi : \\mcC_1 \\to \\mcC_2$ be an $A_{\\infty}$-functor such that $\\Phi \\left( \\mcA_1 \\right)$ is contained in $\\mcA_2$, and let $\\widetilde{\\Phi} : \\mcC_1 \/ \\mcA_1 \\to \\mcC_2 \/ \\mcA_2$ be the $A_{\\infty}$-functor induced by $\\Phi$.\n\t\n\tLet $Y_1$ be an object of $\\mcC_1$ and set $Y_2 := \\Phi (Y_1)$.  Assume that there exists a $\\mcC_i$-module $\\mcM_{\\mcC_i}$, a degree $0$ closed $\\mcC_i$-module map $t_{\\mcC_i} : \\mcC_i \\left( - , Y_i \\right) \\to \\mcM_{\\mcC_i}$ and a degree $0$ closed $\\mcC_1$-module map $t_0 : \\mcM_{\\mcC_1} \\to \\Phi^* \\mcM_{\\mcC_2}$ such that the following diagram of $\\mcC_1$-modules commutes  \n\t\\[\\begin{tikzcd}\n\t\\mcC_1 \\left( - , Y_1 \\right) \\ar[d, \"t_{\\mcC_1}\"] \\ar[r, \"t_{\\Phi}\"] & \\Phi^* \\mcC_2 \\left( - , Y_2  \\right) \\ar[d, \"\\Phi^* t_{\\mcC_2}\"] \\\\\n\t\\mcM_{\\mcC_1} \\ar[r, \"t_0\"] & \\Phi^* \\mcM_{\\mcC_2}\n\t\\end{tikzcd} \\]\n\t(see Definition \\ref{definition modules morphism induced by functor} for the map $t_{\\Phi}$). Then for every object $X$ in $\\mcC_1$, there is a chain map $u : \\, _{\\mcA_1 \\backslash} \\mcM_{\\mcC_1} \\left( X \\right) \\to \\, _{\\mcA_2 \\backslash} \\mcM_{\\mcC_2} \\left( \\Phi X \\right)$ such that the following diagram of chain complexes commutes\n\t\\[\\begin{tikzcd}\n\t\\mcC_1 \/ \\mcA_1 \\left( X , Y_1 \\right) \\ar[d, \"_{\\mcA_1 \\backslash} t_{\\mcC_1}\"] \\ar[r, \"\\widetilde{\\Phi}\"] & \\mcC_2 \/ \\mcA_2 \\left( \\Phi X , Y_2 \\right) \\ar[d, \"_{\\mcA_2 \\backslash} t_{\\mcC_2}\"] \\\\\n\t_{\\mcA_1 \\backslash} \\mcM_{\\mcC_1} \\left( X \\right) \\ar[r, \"u\"] & _{\\mcA_2 \\backslash} \\mcM_{\\mcC_2} \\left( \\Phi X \\right) \\\\\n\t\\mcM_{\\mcC_1} \\left( X \\right) \\ar[u, hook] \\ar[r, \"t_0\"] & \\mcM_{\\mcC_2} \\left( \\Phi X \\right) \\ar[u, hook]\n\t\\end{tikzcd} \\]\n\t(the two lowest vertical maps are the inclusions). \n\tIf moreover the following holds:\n\t\\begin{enumerate}\n\t\t\n\t\t\\item for every objects $A$ in $\\mcA_i$, the complexes $\\mcM_{\\mcC_i} \\left( A \\right)$ are acyclic,\n\t\t\n\t\t\\item the maps $_{\\mcA_i \\backslash} t_{\\mcC_i} : \\, _{\\mcA_i \\backslash} \\mcC_i \\left( X , Y_i \\right) \\to \\, _{\\mcA_i \\backslash} \\mcM_{\\mcC_i} (X)$ are quasi-isomorphisms, and\n\t\t\n\t\t\\item the map $t_0 : \\mcM_{\\mcC_1} (X) \\to \\Phi^* \\mcM_{\\mcC_2} (X)$  is a quasi-isomorphism,\n\t\t\n\t\\end{enumerate}\n\tthen the map $\\widetilde{\\Phi} : \\mcC_1 \/ \\mcA_1 \\left( X , Y_1 \\right) \\to \\mcC_2 \/ \\mcA_2 \\left( \\Phi X , Y_2 \\right)$ is a quasi-isomorphism.\n\t\n\\end{prop}\n\n\\begin{proof}\n\t\n\tWe apply the functor $\\mcP \\mapsto \\, _{\\mcA_1 \\backslash} \\mcP$ to the first diagram, we evaluate at $X$ and we use the natural map of Definition \\ref{definition natural transformation} to get the following commutative diagram of chain complexes \n\t\\[\\begin{tikzcd}\n\t_{\\mcA_1 \\backslash} \\mcC_1 \\left( - , Y_1 \\right) \\left( X \\right) \\ar[d, \"_{\\mcA_1 \\backslash} t_{\\mcC_1}\"] \\ar[r, \"_{\\mcA_1 \\backslash} t_{\\Phi}\"] & _{\\mcA_1 \\backslash} \\left( \\Phi^* \\mcC_2 \\left( - , Y_2 \\right) \\right) \\left( X \\right) \\ar[d, \"_{\\mcA_1 \\backslash} \\left( \\Phi^* t_{\\mcC_2} \\right)\"] \\ar[r] & \\, _{\\mcA_2 \\backslash} \\mcC_2 \\left( - , Y_2 \\right) \\left( \\Phi X \\right) \\ar[d, \"_{\\mcA_2 \\backslash} t_{\\mcC_2}\"] \\\\\n\t_{\\mcA_1 \\backslash} \\mcM_{\\mcC_1} \\left( X \\right) \\ar[r, \"_{\\mcA_1 \\backslash} t_0\"] & _{\\mcA_1 \\backslash} \\left( \\Phi^* \\mcM_{\\mcC_2} \\right) \\left( X \\right) \\ar[r] & \\, _{\\mcA_2 \\backslash} \\mcM_{\\mcC_2} \\left( \\Phi X \\right)\n\t\\end{tikzcd} . \\]\n\tThen we compose the horizontal maps and we use Remark \\ref{rmk composition} to get a commutative diagram of chain complexes\n\t\\[\\begin{tikzcd}\n\t\\mcC_1 \/ \\mcA_1 \\left( X , Y_1 \\right) \\ar[d, \"_{\\mcA_1 \\backslash} t_{\\mcC_1}\"] \\ar[r, \"\\widetilde{\\Phi}\"] & \\mcC_2 \/ \\mcA_2 \\left( \\Phi X , Y_2 \\right) \\ar[d, \"_{\\mcA_2 \\backslash} t_{\\mcC_2}\"] \\\\\n\t_{\\mcA_1 \\backslash} \\mcM_{\\mcC_1} \\left( X \\right) \\ar[r, \"u\"] & _{\\mcA_2 \\backslash} \\mcM_{\\mcC_2} \\left( \\Phi X \\right) \\\\\n\t\\end{tikzcd} . \\]\n\tThis proves the first part of the Proposition because the following diagram of chain complexes commutes  \n\t\\[\\begin{tikzcd}\n\t_{\\mcA_1 \\backslash} \\mcM_{\\mcC_1} \\left( X \\right) \\ar[r, \"u\"] & _{\\mcA_2 \\backslash} \\mcM_{\\mcC_2} \\left( \\Phi X \\right) \\\\\n\t\\mcM_{\\mcC_1} \\left( X \\right) \\ar[u, hook] \\ar[r, \"t_0\"] & \\mcM_{\\mcC_2} \\left( \\Phi X \\right) \\ar[u, hook] \\\\\n\t\\end{tikzcd} . \\]\n\tThe second part of the Proposition follows directly with \\cite[Lemma 3.13]{GPS20}.\n\t\n\\end{proof}\n\n\\paragraph{Cone of module maps.}\n\n\\begin{defin}\\label{definition cone of a morphism between modules}\n\t\n\tLet $t : \\mcM_1 \\to \\mcM_2$ be a degree $0$ closed morphism of $\\mcC$-modules. \n\tWe denote by \n\t\\[\\mathrm{Cone} \\left( \\mcM_1 \\xrightarrow{t} \\mcM_2 \\right) = \\left[\n\t\\begin{tikzcd}\n\t\t\\mcM_1 \\ar[d, \"t\"] \\\\\n\t\t\\mcM_2\n\t\\end{tikzcd}\n\t\\right] \\]\n\tthe $\\mcC$-module $\\mcM$ defined as follows. For every object $X$ in $\\mcC$, \n\t\\[\\mcM \\left( X \\right) = \\mcM_1 \\left( X \\right) \\left[ 1 \\right] \\oplus \\mcM_2 \\left( X \\right) \\]\n\tas graded vector space, and any sequence \n\t\\[ \\left( x_0 , \\dots , x_{d-1} \\right) \\in \\mcC \\left( X_0 , X_1 \\right) \\times \\dots \\times \\mcC \\left( X_{d-1} , X_d \\right) \\]\n\tacts on an element $u_1 \\oplus u_2 $ in $\\mcM \\left( X_d \\right)$ via the operations \n\t\\begin{align*}\n\t\t\\mu_{\\mcM} \\left( x_0 , \\dots , x_{d-1} , u_1 \\oplus u_2 \\right) & = \\\\ \\mu_{\\mcM_1} \\left( x_0, \\dots, x_{d-1}, u_1 \\right) \n\t\t& \\oplus \\left( \\mu_{\\mcM_2} \\left( x_0, \\dots, x_{d-1}, u_2 \\right) + t \\left( x_0, \\dots, x_{d-1}, u_1 \\right) \\right) .\n\t\\end{align*}\n\t\n\\end{defin}\n\nIf we have two $\\mcC$-module maps $t : \\mcM_1 \\to \\mcM_2$ and $t' : \\mcM_1 \\to \\mcM_2'$, then we set \n\\[\\left[\n\\begin{tikzcd}\n\t& \\mcM_1 \\ar[ld, \"t\"] \\ar[rd, \"t'\"] & \\\\\n\t\\mcM_2 & & \\mcM_2'\n\\end{tikzcd}\n\\right] := \n\\left[\n\\begin{tikzcd}\n\t\\mcM_1 \\ar[d, \"{\\left( t, t' \\right)}\"] \\\\\n\t\\mcM_2 \\oplus \\mcM_2'\n\\end{tikzcd}\n\\right] . \\]\n\n\n\\begin{prop}\\label{prop morphism induced by homotopy}\n\t\n\tConsider a diagram of $\\mcC$-modules \n\t\\[ \n\t\\begin{tikzcd}\n\t\t\\mcM_1 \\ar[r, \"t_1\"] \\ar[d, \"t_1'\"] & \\mcM_2 \\ar[d, \"t_2\"] \\\\\n\t\t\\mcM_2' \\ar[r, \"t_2'\"] & \\mcM_3\n\t\\end{tikzcd}\n\t\\]\n\twhere all the morphisms are of degree $0$ and closed. Then any homotopy $h : \\mcM_1 \\to \\mcM_3$ between\n\t\\[t := \\mu_{\\mathrm{Mod}_{\\mcC}}^2 \\left( t_1, t_2 \\right) \\text{ and } t' := \\mu_{\\mathrm{Mod}_{\\mcC}}^2 \\left( t_1', t_2' \\right) \\]\n\tinduces a degree $0$ closed $\\mcC$-module map \n\t\\[t_h : \\left[\n\t\\begin{tikzcd}\n\t\t& \\mcM_1 \\ar[ld, \"t_1\"] \\ar[rd, \"t_1'\"] & \\\\\n\t\t\\mcM_2 & & \\mcM_2'\n\t\\end{tikzcd}\n\t\\right] \\to \\mcM_3 \\]\n\tdefined by \n\t\\begin{align*}\n\t\tt_h \\left( x_0 , \\dots , x_{d-1} , u_1 \\oplus u_2 \\oplus u_2' \\right) & = \\\\\n\t\th \\left( x_0 , \\dots , x_{d-1} , u_1 \\right) & + t_2 \\left( x_0 , \\dots , x_{d-1} , u_2 \\right) + t_2' \\left( x_0 , \\dots , x_{d-1} , u_2' \\right) .\n\t\\end{align*}\n\t\n\\end{prop}\n\n\\begin{proof}\n\t\n\tThe only thing to check is that $\\mu_{\\mathrm{Mod}_{\\mcC}}^1 \\left( t_h \\right) = 0$, which is straightforward.\n\t\n\\end{proof}\n\n\\begin{rmk}\\label{rmk cone and localization commute}\n\t\n\tIf $t : \\mcM_1 \\to \\mcM_2$ is a degree $0$ closed $\\mcC$-module map, then \n\t\\[_{\\mcA \\backslash} \\mathrm{Cone} \\left( \\mcM_1 \\xrightarrow{t} \\mcM_2 \\right) = \\mathrm{Cone} \\left( _{\\mcA \\backslash} \\mcM_1 \\xrightarrow{_{\\mcA \\backslash} t} \\, _{\\mcA \\backslash} \\mcM_2 \\right) . \\]\n\t\n\\end{rmk}\n\n\\subsection{Grothendieck construction and homotopy pushout}\\label{Grothendieck subsection}\n\nAn exposition on Grothendieck constructions and homotopy colimits in the context of $A_{\\infty}$-categories can be found in \\cite[appendix A]{GPS19}. We recall here definitions and basic facts that will serve us.  In this section, $A_{\\infty}$-categories are always assumed to be \\emph{strictly unital} (see \\cite[paragraph (2a)]{Sei08}).\n\n\\begin{defin}\\label{definition Grothendieck construction}\n\t\n\tConsider a diagram of $A_{\\infty}$-categories \n\t\\[\\begin{tikzcd}\n\t\\mcC \\ar[r, \"\\Phi_1\"] \\ar[d, \"\\Phi_2\" left] & \\mcD_1 \\\\\n\t\\mcD_2 \n\t\\end{tikzcd}. \\]\n\tThe Grothendieck construction of this diagram is the $A_{\\infty}$-category $\\mcG$ such that \n\t\\begin{enumerate}\n\t\t\n\t\t\\item the set of objects is $\\mathrm{ob} \\left( \\mcC \\right) \\sqcup \\mathrm{ob} \\left( \\mcD_1 \\right) \\sqcup \\mathrm{ob} \\left( \\mcD_2 \\right)$,\n\t\t\n\t\t\\item the space of morphisms between two objects $X$ and $Y$ is given by\n\t\t\\[\\mcG \\left( X, Y \\right) = \\left\\{\n\t\t\\begin{array}{ll}\n\t\t\\mcC \\left( X, Y \\right) & \\text{if } X, Y \\in \\mathrm{ob} \\left( \\mcC \\right) \\\\\n\t\t\\mcD_i \\left( X, Y \\right) & \\text{if } X, Y \\in \\mathrm{ob} \\left( \\mcD_i \\right) \\\\\n\t\t\\mcD_i \\left( \\Phi_i X, Y \\right) & \\text{if } X \\in \\mathrm{ob} \\left( \\mcC \\right) \\text{ and } Y \\in \\mathrm{ob} \\left( \\mcD_i \\right) \\\\\n\t\t0 & \\text{otherwise} .\n\t\t\\end{array}\n\t\t\\right. \\]\n\t\t\n\t\t\\item the operations involving only objects of $\\mcC$, respectively of $\\mcD_i$, are the same as in $\\mcC$, respectively in $\\mcD_i$, and for every sequence \n\t\t\\begin{align*}\n\t\t\\left( x_0 , \\dots , x_{p-1} , y , z_0 , \\dots, z_{q-1} \\right) \\in \\mcC \\left( X_0 , X_1 \\right) \\otimes \\cdots \\otimes \\mcC \\left( X_{p-1} , X_p \\right) \\\\ \\otimes \\mcG \\left( X_p , Y_0 \\right) \\otimes \\mcD_i \\left( Y_0 , Y_1 \\right) \\otimes \\cdots \\otimes \\mcD_i \\left( Y_{q-1} , Y_q \\right), \n\t\t\\end{align*}\n\t\twe have \n\t\t\\begin{align*}\n\t\t\\mu_{\\mcG} & \\left( x_0 , \\dots , x_{p-1} , y , z_0 , \\dots, z_{q-1} \\right) = \\\\\n\t\t& \\sum \\mu_{\\mcD_i} \\left( \\Phi_i \\left( x_0 , \\dots , x_{i_1-1} \\right) , \\dots , \\Phi_i \\left( x_{p-i_r} , \\dots , x_{p-1} \\right) , y , z_0 , \\dots, z_{q-1} \\right) .\n\t\t\\end{align*}\n\t\t\n\t\\end{enumerate}\n\t\n\tWe will call \\emph{adjacent unit} of $\\mcG$ any morphism in $\\mcG \\left( X , \\Phi_i \\left( X \\right) \\right)$ which corresponds to the unit in $\\mcD_i \\left( \\Phi_i \\left( X \\right) , \\Phi_i \\left( X \\right) \\right)$.\n\tThe \\emph{homotopy colimit} $\\mcH$ of the above diagram is the localization of $\\mcG$ at its adjacent units.\n\t\n\\end{defin}\n\n\\begin{prop}\\label{prop induced functor from Grothendieck construction}\n\t\n\tLet $\\mcG$ be the Grothendieck construction of a diagram \n\t\\[\\begin{tikzcd}\n\t\\mcC \\ar[r, \"\\Phi_1\"] \\ar[d, \"\\Phi_2\" left] & \\mcD_1 \\\\\n\t\\mcD_2 .\n\t\\end{tikzcd} \\] \n\tThen any strictly commutative square \n\t\\[\\begin{tikzcd}\n\t\\mcC \\ar[r, \"\\Phi_1\"] \\ar[d, \"\\Phi_2\" left] & \\mcD_1 \\ar[d, \"\\Psi_1\"] \\\\\n\t\\mcD_2 \\ar[r, \"\\Psi_2\"] & \\mcE \n\t\\end{tikzcd} \\]\n\tinduces a functor $\\sigma : \\mcG \\to \\mcE$ defined as follows. On the objects, $\\sigma$ acts on $\\mcD_i$ as $\\Psi_i$, and on $\\mcC$ as $\\Psi_1 \\circ \\Phi_1 = \\Psi_2 \\circ \\Phi_2$ ; on the morphisms, $\\sigma$ acts on $\\mcD_i$ as $\\Psi_i$, on $\\mcC$ as $\\Psi_1 \\circ \\Phi_1 = \\Psi_2 \\circ \\Phi_2$, and it sends any sequence  \n\t\\begin{align*}\n\t\\left( x_0 , \\dots , x_{p-1} , y , z_0 , \\dots, z_{q-1} \\right) \\in \\mcC \\left( X_0 , X_1 \\right) \\otimes \\cdots \\otimes \\mcC \\left( X_{p-1} , X_p \\right) \\\\ \\otimes \\mcG \\left( X_p , Y_0 \\right) \\otimes \\mcD_i \\left( Y_0 , Y_1 \\right) \\otimes \\cdots \\otimes \\mcD_i \\left( Y_{q-1} , Y_q \\right), \n\t\\end{align*} \n\tto \n\t\\begin{align*}\n\t\\sigma & \\left( x_0 , \\dots , x_{p-1} , y , z_0 , \\dots, z_{q-1} \\right) = \\\\\n\t& \\sum \\Psi_i \\left( \\Phi_i \\left( x_0 , \\dots , x_{i_1-1} \\right) , \\dots , \\Phi_i \\left( x_{p-i_r} , \\dots , x_{p-1} \\right) , y , z_0 , \\dots, z_{q-1} \\right) .\n\t\\end{align*}  \n\t\n\\end{prop}\n\n\\begin{proof}\n\t\n\tThis is a straightforward verification. \n\t\n\\end{proof}\n\n\\begin{prop}[\\cite{GPS19} Lemma A.5]\\label{prop invariance of homotopy colimits}\n\t\n\tA strictly commutative diagram of $A_{\\infty}$-categories\n\t\\[\\begin{tikzcd}\n\t\\mcB_1 \\ar[d] & \\mcA \\ar[l] \\ar[r] \\ar[d] & \\mcB_2 \\ar[d] \\\\\n\t\\mcD_1 & \\mcC \\ar[l] \\ar[r] & \\mcD_2\n\t\\end{tikzcd} \\] \n\tinduces an $A_{\\infty}$-functor from the Grothendieck construction of the top line to the Grothendieck construction of the bottom line which preserves adjacent units. \n\tIf moreover each vertical arrow is a quasi-equivalence, then the induced functor\n\t\\[\\mathrm{hocolim} \\left( \n\t\\begin{tikzcd}\n\t\\mcA \\ar[r] \\ar[d] & \\mcB_1 \\\\\n\t\\mcB_2 &\n\t\\end{tikzcd} \\right)\n\t\\to \n\t\\mathrm{hocolim} \\left( \n\t\\begin{tikzcd}\n\t\\mcC \\ar[r] \\ar[d] & \\mcD_1 \\\\\n\t\\mcD_2 &\n\t\\end{tikzcd} \\right) \\]\n\tis a quasi-equivalence.\n\t\n\\end{prop}\n\n\\begin{prop}\\label{prop homotopic functors induce quasi-isomorphic homotopy colimits}\n\t\n\tConsider two diagrams of $A_{\\infty}$-categories \n\t\\[\\begin{tikzcd}\n\t\\mcC \\ar[r, \"\\Phi_1\"] \\ar[d, \"\\Phi_2\" left] & \\mcD_1 \\\\\n\t\\mcD_2 &\n\t\\end{tikzcd}\n\t\\text{ and } \n\t\\begin{tikzcd}\n\t\\mcC \\ar[r, \"\\Psi_1\"] \\ar[d, \"\\Psi_2\" left] & \\mcD_1 \\\\\n\t\\mcD_2 . &\n\t\\end{tikzcd} \\]\n\tIf $\\Phi_i$ and $\\Psi_i$ (for $i \\in \\{1,2\\}$) are homotopic (see \\cite[paragraph (1h)]{Sei08}), then the homotopy colimits of the diagrams above are quasi-equivalent.\n\t\n\\end{prop}\n\n\\begin{proof}\n\t\n\tLet $\\mcG_0$ and $\\mcG_1$ the Grothendieck constructions of the above diagrams.\n\t\n\tLet $T_i$ be an homotopy from $\\Phi_i$ to $\\Psi_i$. This means that \n\t\\begin{align*}\n\t\\Phi_i + \\Psi_i & = \\sum T_i \\left( \\dots , \\mu_{\\mcC} \\left( \\dots \\right) , \\dots \\right) \\\\ \n\t& + \\sum \\mu_{\\mcD_i} \\left( \\Psi_i \\left( \\dots \\right) , \\dots , \\Psi_i \\left( \\dots \\right) , T_i \\left( \\dots \\right) , \\Phi_i \\left( \\dots \\right), \\dots, \\Phi_i \\left( \\dots \\right) \\right) .\n\t\\end{align*}\n\t\n\tWe consider the functor $\\kappa : \\mcG_0 \\to \\mcG_1$ such that \n\t\\[ \\kappa_{|\\mcC} = \\mathrm{id}_{\\mcC}, \\quad \\kappa_{|\\mcD_i} = \\mathrm{id}_{\\mcD_i}, \\]\n\tand which sends every sequence \n\t\\begin{align*}\n\t\\left( \\dots , y , \\dots \\right) & \\in \\mcC \\left( X_0,X_1 \\right) \\times \\cdots \\times \\mcC \\left( X_{p-1},X_p \\right) \\times \\mcG_0 \\left( X_p , Y_0 \\right) \\\\\n\t& \\times \\mcD_i \\left( Y_0,Y_1 \\right) \\times \\cdots \\times \\mcD_i \\left( Y_{q-1},Y_q \\right),\n\t\\end{align*}\n\tto\n\t\\begin{align*}\n\t& \\kappa \\left( \\dots , y , \\dots \\right) = \\\\ \n\t& \\sum \\mu_{\\mcD_i} \\left( \\Psi_i \\left( \\dots \\right) , \\dots , \\Psi_i \\left( \\dots \\right) , T_i \\left( \\dots \\right), \\Phi_i \\left( \\dots \\right) , \\dots , \\Phi_i \\left( \\dots \\right) , y , \\dots \\right)\n\t\\end{align*}\n\tif $p$ is positive, and to\n\t\\[\\Phi \\left( y , \\dots \\right) = \\mathrm{id}_{\\mcD_i} \\left( y , \\dots \\right) \\]\n\totherwise.\n\tUsing the facts that $\\Phi_i , \\Psi_i$ are $A_{\\infty}$-functors, that $T_i$ is an homotopy from $\\Phi_i$ to $\\Psi_i$, and gathering the terms depending on if they contain $T_i^k \\left( \\dots \\right)$ or $y$, we conclude that $\\kappa$ satisfies the $A_{\\infty}$-relations. This proves the result because $\\kappa$ is a quasi-equivalence sending the adjacent units of $\\mcG_0$ onto those of $\\mcG_1$. \n\t\n\\end{proof}\n\n\\subsection{Cylinder object and homotopy}\n\nLet $\\mcA_{\\bot}$, $\\mcA_I$ and $\\mcA_{\\top}$ be three copies of an $A_{\\infty}$-category $\\mcA$. \nWe denote by $\\mcC$ the Grothendieck construction of the following diagram\n\\[\\begin{tikzcd}\n\\mcA_I \\ar[r, \"\\mathrm{id}\"] \\ar[d, \"\\mathrm{id}\" left] & \\mcA_{\\top} \\\\\n\\mcA_{\\bot} &\n\\end{tikzcd}, \\]\nand we let $\\iota_{\\bot}, \\iota_I, \\iota_{\\top} : \\mcA \\to \\mcC$ be the strict inclusions with images $\\mcA_{\\bot}$, $\\mcA_I$, $\\mcA_{\\top}$ respectively. Finally, we denote by $W_{\\mcC}$ the set of adjacent units in $\\mcC$, and we let $\\mcC yl_{\\mcA} = \\mcC \\left[ W_{\\mcC}^{-1} \\right]$ be the homotopy colimit of the diagram above. We say that $\\mcC yl_{\\mcA}$ is a cylinder object for $\\mcA$. \n\nWe denote by $\\pi : \\mcC \\to \\mcA$ the $A_{\\infty}$-functor induced by the following commutative square \n\\[\\begin{tikzcd}\n\\mcA \\ar[r, \"\\mathrm{id}\"] \\ar[d, \"\\mathrm{id}\" left] & \\mcA \\ar[d, \"\\mathrm{id}\"] \\\\\n\\mcA \\ar[r, \"\\mathrm{id}\"] & \\mcA \n\\end{tikzcd} \\]\n(see Proposition \\ref{prop induced functor from Grothendieck construction}). \n\n\\begin{prop}\\label{prop cylinder object}\n\t\n\tThe following diagram of $A_{\\infty}$-categories commutes \n\t\\[\\begin{tikzcd}\n\t\\mcA \\sqcup \\mcA \\ar[r, \"\\iota_{\\bot} \\sqcup \\iota_{\\top}\"] \\ar[rr, \"\\mathrm{id} \\sqcup \\mathrm{id}\" below, bend right] & \\mcC \\ar[r, \"\\pi\"] & \\mcA .\n\t\\end{tikzcd} \\]\n\tMoreover, $\\pi$ sends $W_{\\mcC}$ to the set of units in $\\mcA$, and the induced $A_{\\infty}$-functor $\\widetilde{\\pi} : \\mcC yl_{\\mcA} \\to \\mcA \\left[ \\mathrm{units}^{-1} \\right]$ is a quasi-equivalence.  \n\t\n\\end{prop}\n\n\\begin{proof}\n\t\n\tThe facts that $\\pi \\circ \\left( \\iota_{\\bot} \\sqcup \\iota_{\\top} \\right) = \\mathrm{id} \\sqcup \\mathrm{id}$ and that $\\pi$ sends $W_{\\mcC}$ to the set of units in $\\mcA$ are clear. We now show that $\\widetilde{\\pi} : \\mcC yl_{\\mcA} \\to \\mcA \\left[ \\mathrm{units}^{-1} \\right]$ is a quasi-equivalence. \n\t\n\tFirst observe that it is enough to show that the map  \n\t\\[\\widetilde{\\pi} : \\mcC yl_{\\mcA} \\left( X, Y \\right) \\to \\mcA \\left[ \\{ \\mathrm{units} \\}^{-1} \\right] \\left( \\pi X, \\pi Y \\right) \\]\n\tis a quasi-isomorphism for every objects $X$, $Y$ in $\\mcA_{\\bot}$  because every object of $\\mcC$ can be related to one of $\\mcA_{\\bot}$ by a zigzag of morphisms in $W_{\\mcC}$, which are quasi-isomorphisms in $\\mcC yl_{\\mcA}$ (see \\cite[Lemma 3.12]{GPS20}). \n\tOur strategy is to apply Proposition \\ref{prop quasi-isomorphism between localizations}.\n\tLet $Y = \\iota_{\\bot} (Z)$ be an object in $\\mcA_{\\bot}$.\n\tFor the $\\mcC$-module we take \n\t\\[\\mcM_{\\mcC} = \\left[ \n\t\\begin{tikzcd}\n\t& \\mcC \\left( -, \\iota_I (Z) \\right) \\ar[ld, \"t_{I \\bot}\"] \\ar[rd, \"t_{I \\top}\"] & \\\\\n\t\\mcC \\left( -, \\iota_{\\bot} (Z) \\right) & & \\mcC \\left( -, \\iota_{\\top} (Z) \\right)\n\t\\end{tikzcd} \n\t\\right], \\]\n\twhere \n\t\\[t_{I \\triangle} : \\mcC \\left( -, \\iota_I (Z) \\right) \\to \\mcC \\left( -, \\iota_{\\triangle} (Z) \\right), \\quad \\triangle \\in \\{ \\bot, \\top \\}, \\]\n\tis the $\\mcC$-module map induced by the adjacent unit in $\\mcC \\left( \\iota_I (Z), \\iota_{\\triangle} (Z) \\right)$ (see Definition \\ref{definition Yoneda functor})).\n\tFor the $\\mcA$-module we simply take $\\mcA \\left( -, Z \\right)$. Besides, we let $t_{\\mcC} : \\mcC \\left( -, \\iota_{\\bot} (Z) \\right) \\to \\mcM_{\\mcC}$ be the $\\mcC$-module inclusion, and we let $t_{\\mcA} : \\mcA \\left( -, Z \\right) \\to \\mcA \\left( -, Z \\right)$ be the identity map. \n\tWe now define the morphism $t_0 : \\mcM_{\\mcC} \\to \\pi^* \\mcA \\left( -, Z \\right)$. Consider the following diagram of $\\mcC$-modules\n\t\\[\\begin{tikzcd}\n\t\\mcC \\left( -, \\iota_I (Z) \\right) \\ar[r, \"t_{I \\bot}\"] \\ar[d, \"t_{I \\top}\"] & \\mcC \\left( -, \\iota_{\\top} (Z) \\right) \\ar[d, \"t_{\\pi}\"] \\\\\n\t\\mcC \\left( -, \\iota_{\\bot} (Z) \\right) \\ar[r, \"t_{\\pi}\"] & \\pi^* \\mcA \\left( -, Z \\right) . \n\t\\end{tikzcd} \\]\n\tObserve that this diagram is commutative, and thus it induces a strict $\\mcC$-module map $t_0 : \\mcM_{\\mcC} \\to \\pi^* \\mcA \\left( -, Z \\right)$ according to Proposition \\ref{prop morphism induced by homotopy}. It is then easy to see that the following diagram commutes \n\t\\[\\begin{tikzcd}\n\t\\mcC \\left( - , \\iota_{\\triangle} (Z) \\right) \\ar[d, \"t_{\\mcC}\"] \\ar[r, \"t_{\\pi}\"] & \\pi^* \\mcA \\left( - , Z \\right) \\ar[d, \"\\pi^* t_{\\mcA}\"] \\\\\n\t\\mcM_{\\mcC} \\ar[r, \"t_0\"] & \\pi^* \\mcA \\left( -, Z \\right) .\n\t\\end{tikzcd} \\]\n\tTo conclude the proof, it suffices to check the three items of Proposition \\ref{prop quasi-isomorphism between localizations}. \n\tObserve that the pair $\\left( \\mcA \\left( -, Z \\right), t_{\\mcA} \\right)$ trivially satisfies the two first items.\n\t\n\tWe check that $\\mcM_{\\mcC}$ satisfies the first item of Proposition \\ref{prop quasi-isomorphism between localizations}. Let $Z'$ be an object in $\\mcA$ and let $w$ be the adjacent unit in $\\mcC \\left( \\iota_{I} (Z'), \\iota_{\\bot} (Z') \\right)$ (the proof is the same for the adjacent unit in $\\mcC \\left( \\iota_{I} (Z'), \\iota_{\\top} (Z') \\right) \\cap W_{\\mcC}$). Then \n\t\\begin{align*}\n\t\\mcM_{\\mcC} \\left( \\mathrm{Cone} \\, w \\right) & = \\mathrm{Cone} \\left( \\mcM_{\\mcC} \\left( \\iota_{\\bot} (Z') \\right) \\xrightarrow{\\mu_{\\mcC}^2 \\left( w , - \\right)} \\mcM_{\\mcC} \\left( \\iota_{I} (Z') \\right) \\right) \\\\\n\t& = \\mathrm{Cone} \\left( \\mcC \\left( \\iota_{\\bot} (Z') , \\iota_{\\bot} (Z) \\right) \\xrightarrow{\\mu_{\\mcC}^2 \\left( w , - \\right)} K \\right).\n\t\\end{align*}\n\twhere \n\t\\[K = \\left[ \n\t\\begin{tikzcd}\n\t& \\mcC \\left( \\iota_{I} (Z') , \\iota_I (Z) \\right) \\ar[ld, \"t_{I \\bot}\"] \\ar[rd, \"t_{I \\top}\"] \\\\\n\t\\mcC \\left( \\iota_{I} (Z') , \\iota_{\\bot} (Z) \\right) & & \\mcC \\left( \\iota_{I} (Z') , \\iota_{\\top} (Z) \\right)\n\t\\end{tikzcd}\n\t\\right] . \\]\n\tObserve that $\\mu_{\\mcC}^2 \\left( w , - \\right) : \\mcC \\left( \\iota_{\\bot} (Z') , \\iota_{\\bot} (Z) \\right) \\to K$ is injective so its cone is quasi-isomorphic to its cokernel, which is the cone of $t_{I \\top} : \\mcC \\left( \\iota_{I} (Z') , \\iota_I (Z) \\right) \\to \\mcC \\left( \\iota_{I} (Z') , \\iota_{\\top} (Z) \\right)$. The latter map is a quasi-isomorphism, so $\\mcM_{\\mcC} \\left( \\mathrm{Cone} \\, w \\right)$ is acyclic.\n\t\n\tWe now check that $\\left( \\mcM_{\\mcC}, t_{\\mcC} \\right)$ satisfies the second item of Proposition \\ref{prop quasi-isomorphism between localizations}. Observe that \n\t\\[_{W_{\\mcC}^{-1}} \\mcM_{\\mcC} = \\left[ \n\t\\begin{tikzcd}\n\t& _{W_{\\mcC}^{-1}} \\mcC \\left( -, \\iota_I (Z) \\right) \\ar[ld, \"_{W_{\\mcC}^{-1}} t_{I \\bot}\"] \\ar[rd, \"_{W_{\\mcC}^{-1}} t_{I \\top}\"] & \\\\\n\t_{W_{\\mcC}^{-1}} \\mcC \\left( -, \\iota_{\\bot} (Z) \\right) & & _{W_{\\mcC}^{-1}} \\mcC \\left( -, \\iota_{\\top} (Z) \\right)\n\t\\end{tikzcd} \n\t\\right] \\]\n\tand $_{W_{\\mcC}^{-1}} t_{\\mcC} : \\, _{W_{\\mcC}^{-1}} \\mcC \\left( -, \\iota_{\\bot} (Z) \\right) \\to \\, _{W_{\\mcC}^{-1}} \\mcM_{\\mcC}$ is the inclusion. Thus if $X$ is some object of $\\mcC$, the cone of $_{W_{\\mcC}^{-1}} t_{\\mcC} : \\mcC yl_{\\mcA} \\left( X, \\iota_{\\bot} (Z) \\right) \\to \\, _{W_{\\mcC}^{-1}} \\mcM_{\\mcC} \\left( X \\right)$ is quasi-isomorphic to the cone of the multiplication in $\\mcC yl_{\\mcA}$ by an element of $W_{\\mcC}$, which is a quasi-isomorphism. Thus the map $_{W_{\\mcC}^{-1}} t_{\\mcC} : \\, _{W_{\\mcC}^{-1}} \\mcC \\left( -, \\iota_{\\bot} (Z) \\right) \\to \\, _{W_{\\mcC}^{-1}} \\mcM_{\\mcC}$ indeed is a quasi-isomorphism. \n\t\n\tIt remains to check the third item of Proposition \\ref{prop quasi-isomorphism between localizations}, which is that the map $t_0 : \\mcM_{\\mcC} \\left( X \\right) \\to \\pi^* \\mcA \\left( -, Z \\right) \\left( X \\right)$ is a quasi-isomorphism when $X$ is in $\\mcA_{\\bot}$. This is true because $\\mcM_{\\mcC} \\left( \\iota_{\\bot} (Z') \\right) = \\mcC ( \\iota_{\\bot} (Z'), \\iota_{\\bot} (Z) ) = \\mcA ( Z', Z )$, and \n\t\\[t_0 : \\mcA ( Z', Z ) = \\mcM_{\\mcC} \\left( \\iota_{\\bot} (Z') \\right) \\to \\pi^* \\mcA \\left( -, Z \\right) \\left( \\iota_{\\bot} (Z') \\right) = \\mcA ( Z', Z ) \\]\n\tis the identity.\n\tThis concludes the proof. \n\t\n\\end{proof}\n\n\\begin{rmk}\n\t\n\tProposition \\ref{prop cylinder object} can be thought as saying that $\\mcC yl_{\\mcA}$ is a cylinder object for $\\mcA$.\n\t\n\\end{rmk}\n\n\\begin{prop}\\label{prop homotopy is generalized homotopy}\n\t\n\tIf two $A_{\\infty}$-functors $\\Phi, \\Psi : \\mcA \\to \\mcB$ are homotopic (see \\cite[paragraph (1h)]{Sei08}), then there is an $A_{\\infty}$-functor $\\eta : \\mcC \\to \\mcB$ which sends the adjacent units of $\\mcC$ to the units in $\\mcB$ and such that the following diagram commutes\n\t\\[\\begin{tikzcd}\n\t\\mcA \\ar[d, \"\\iota_{\\top}\" left] \\ar[rd, bend left, \"\\Phi\"] & \\\\\n\t\\mcC \\ar[r, \"\\eta\"] & \\mcB \\\\\n\t\\mcA \\ar[u, \"\\iota_{\\bot}\"] \\ar[ru, bend right, \"\\Psi\"] . & \n\t\\end{tikzcd} \\]\n\t\n\\end{prop}\n\n\\begin{proof}\n\t\n\tOn the objects, we set $\\eta \\left( X_{\\triangle} \\right) = \\Phi \\left( X \\right) = \\Psi \\left( X \\right)$ for every object $X$ of $\\mcA$ and $\\triangle \\in \\{ \\bot, I, \\top \\}$. \n\tOn the morphisms, we set \n\t\\[\\eta_{| \\mcA_{\\bot}} = \\eta_{| \\mcA_I} = \\Psi, \\, \\eta_{| \\mcA_{\\top}} = \\Phi \\]\n\tand ask for the restriction of $\\eta$ to\n\t\\[\\mcA_I \\left( X_0 , X_1 \\right) \\otimes \\cdots \\otimes \\mcA_I \\left( X_{p-1} , X_p \\right) \\otimes \\mcC \\left( X_p , X_{p+1} \\right) \\otimes \\mcA_{\\bot} \\left( X_{p+1} , X_{p+2} \\right) \\otimes \\cdots \\otimes \\mcA_{\\bot} \\left( X_{p+q} , X_{p+q+1} \\right) \\] \n\tto be $\\Psi$.\n\tIt remains to define $\\eta$ for \n\t\\begin{align*}\n\t\\left( \\dots, x, \\dots \\right) \\in & \\mcA_I \\left( X_0 , X_1 \\right) \\otimes \\cdots \\otimes \\mcA_I \\left( X_{p-1} , X_p \\right) \\\\ \n\t& \\otimes \\mcC \\left( X_p , X_{p+1} \\right) \\otimes \\mcA_{\\top} \\left( X_{p+1} , X_{p+2} \\right) \\otimes \\cdots \\otimes \\mcA_{\\top} \\left( X_{p+q} , X_{p+q+1} \\right) . \n\t\\end{align*}\n\tFor this we take a homotopy $T$ between $\\Phi$ and $\\Psi$, which means that\n\t\\begin{align*}\n\t\\Phi + \\Psi & = \\sum T \\left( \\dots , \\mu_{\\mcA} \\left( \\dots \\right) , \\dots \\right) \\\\ \n\t& + \\sum \\mu_{\\mcB} \\left( \\Phi \\left( \\dots \\right) , \\dots , \\Phi \\left( \\dots \\right) , T \\left( \\dots \\right) , \\Psi \\left( \\dots \\right), \\dots, \\Psi \\left( \\dots \\right) \\right) .\n\t\\end{align*}\n\tThen we let\n\t\\begin{align*}\n\t\\eta & \\left( \\dots, x, \\dots \\right) = \\\\\n\t& \\sum \\mu_{\\mcB} \\left( \\Phi (\\dots), \\dots, \\Phi (\\dots), T (\\dots), \\Psi (\\dots), \\dots, \\Psi (\\dots), \\Psi ( \\dots, x, \\dots ) \\Psi (\\dots), \\dots, \\Psi (\\dots) \\right)\n\t\\end{align*}\n\tif $p$ is positive, and $\\eta \\left( x, \\dots \\right) = \\Psi \\left( x, \\dots \\right)$ otherwise.\n\t\n\\end{proof}\n\n\\subsection{Adjunctions between Adams-graded and non Adams-graded}\n\nWe end this this section by describing specific adjunctions between the category of Adams-graded $A_{\\infty}$-categories concentrated in non-negative Adams-degree and the category of (non Adams-graded) $A_{\\infty}$-categories.  \n\n\\begin{defin}\\label{definition change of grading}\n\t\n\tIf $V$ is an Adams-graded vector space and $m$ is an integer, we denote by $V_m$ the graded vector space whose degree $n$ component is the direct sum of the bidegree $\\left( p,k \\right)$ components of $V$, where the sum is over the set of couples $\\left( p,k \\right) \\in \\mathbf{Z} \\times \\mathbf{Z}$ such that $p-mk=n$.\n\t\n\\end{defin}\n\n\\begin{defin}\\label{definition forgetful functor}\n\t\n\tIf $\\mcC$ is an Adams-graded $A_{\\infty}$-category, we denote by $\\mcC_m$ the (non Adams-graded) $A_{\\infty}$-category obtained from $\\mcC$ by changing the grading so that\n\t\\[\\mcC_m \\left(X_0, X_1 \\right) = \\mcC \\left( X_0, X_1 \\right)_m \\] \t\n\tObserve that any $A_{\\infty}$-functor $\\Phi : \\mcC_1 \\to \\mcC_2$ between two Adams-graded\u00a0$A_{\\infty}$-categories induces an $A_{\\infty}$-functor from $(\\mcC_1)_m$ to $(\\mcC_2)_m$ (that we still denote by $\\Phi$) which acts exactly as $\\Phi$ on objects and morphisms. \n\tThis defines a functor $\\mcC \\mapsto \\mcC_m$ from the category of Adams-graded $A_{\\infty}$-categories to the category of (non Adams-graded) $A_{\\infty}$-categories. \n\t\n\\end{defin}\n\nWe denote by $\\mathbf{F} \\left[ t_m \\right]$ the augmented Adams-graded associative algebra generated by a variable $t_m$ of bidegree $(m, 1)$, and by $t_m \\mathbf{F} \\left[ t_m \\right]$ its augmentation ideal (or equivalently, the ideal generated by $t_m$).\n\n\\begin{defin}\\label{definition cofree functor}\n\t\n\tIf $\\mcD$ is a (non Adams-graded) $A_{\\infty}$-category, we denote by $\\mathbf{F} \\left[ t_m \\right] \\otimes \\mcD$ the Adams-graded $A_{\\infty}$-category such that \n\t\\begin{enumerate}\n\t\t\n\t\t\\item the objects of $\\mathbf{F} \\left[ t_m \\right] \\otimes \\mcD$ are those of $\\mcD$,\n\t\t\n\t\t\\item the space of morphisms from $Y_1$ to $Y_2$ is $\\mathbf{F} \\left[ t_m \\right] \\otimes \\mcD \\left( Y_1, Y_2 \\right)$, and if $y \\in \\mcD \\left( Y_1, Y_2 \\right)$ is of degree $j$,  $t_m^k \\otimes y$ is of bidegree $\\left( j + mk, k \\right)$,\n\t\t\n\t\t\\item the operations send any sequence $\\left( t_m^{k_0} \\otimes y_0, \\dots, t_m^{k_{d-1}} \\otimes y_{d-1} \\right)$ of morphisms to \n\t\t\\[\\mu_{\\mathbf{F} \\left[ t_m\\right] \\otimes \\mcD} \\left( t_m^{k_0} \\otimes y_0, \\dots, t_m^{k_{d-1}} \\otimes y_{d-1} \\right) = t_m^{k_0 + \\dots + k_{d-1}} \\otimes \\mu_{\\mcD} \\left( y_0, \\dots, y_{d-1} \\right) . \\]\n\t\t\n\t\\end{enumerate}\n\tObserve that any $A_{\\infty}$-functor $\\Psi : \\mcD_1 \\to \\mcD_2$ between (non Adams-graded) $A_{\\infty}$-categories induces an $A_{\\infty}$-functor $\\mathbf{F} \\left[ t_m \\right] \\otimes \\mcD_1 \\to \\mathbf{F} \\left[ t_m \\right] \\otimes \\mcD_2$ which acts as $\\Psi$ on objects, and which sends any sequence $\\left( t_m^{k_0} \\otimes y_0, \\dots, t_m^{k_{d-1}} \\otimes y_{d-1} \\right)$ of morphisms to $t_m^{k_0 + \\dots k_{d-1}} \\otimes \\Psi \\left( y_0, \\dots, y_{d-1} \\right)$.\n\tThis defines a functor $\\mcD \\mapsto \\mathbf{F} \\left[ t_m \\right] \\otimes \\mcD$ from the category of (non Adams-graded) $A_{\\infty}$-categories to the category of Adams-graded $A_{\\infty}$-categories. \n\t\n\\end{defin}\n\n\\begin{defin}\\label{definition adjunction}\n\t\n\tLet $\\mcC$ be an Adams-graded $A_{\\infty}$-category concentrated in non-negative Adams-degree, and let $\\mcD$ be a (non Adams-graded) $A_{\\infty}$-category. To any $A_{\\infty}$-functor $\\Psi_m : \\mcC_m \\to \\mcD$, we associate an $A_{\\infty}$-functor $\\Psi : \\mcC \\to \\mathbf{F} \\left[ t_m \\right] \\otimes \\mcD$ which sends a sequence $\\left( x_0, \\dots, x_{d-1} \\right)$, where $x_j$ is of bidegree $\\left( i_j, k_j \\right)$, to\n\t\\[\\Psi \\left( x_0, \\dots, x_{d-1} \\right) = t_m^{k_0 + \\dots + k_{d-1}} \\otimes \\Psi_m \\left( x_0, \\dots, x_{d-1} \\right) . \\]\n\tThis defines an adjunction between the category of Adams-graded $A_{\\infty}$-categories concentrated in non-negative Adams-degree and the category of (non Adams-graded) $A_{\\infty}$-categories.  \n\t\n\\end{defin}\n\\chapter{Ekholm-Lekili revisited}\\label{section EL}\n\nIn this appendix, we revisit some of the results of \\cite{EL21} using localization of $A_{\\infty}$-categories. \n\n\\section{Setting}\n\nLet $X$ be a Weinstein manifold. \nNow let $L$ be a connected cylindrical exact Lagrangian submanifold of $X$. We denote by $\\Lambda = \\partial_{\\infty} L \\subset Y$ the corresponding Legendrian submanifold. We will assume that $\\Lambda$ is connected and chord generic with respect to $\\alpha$. Finally, we denote by $\\varepsilon_L : CE_{-*} \\left( \\Lambda \\right) \\to \\mathbf{F}$ the augmentation induced by $L$. \n\nWe fix two positive Morse functions $F_1 : L \\to \\mathbf{R}$ and $f_1 : \\Lambda \\to \\mathbf{R}$ with a unique minimum and a unique maximum, and such that in the ends \n\\[F_1 \\left( t,q \\right) = a_1 \\left( t + f_1 \\left( q \\right) \\right) + b_1 \\text{ for } 0 < a_1 \\ll 1 \\text{ and } b_1 > 0. \\]\nWe denote by $\\mcU$ a small open neighborhood of $\\left( F_1, f_1 \\right)$ in $C^{\\infty} \\left( L \\right) \\times C^{\\infty} \\left( \\Lambda \\right)$ with the following property: for every pair $\\left( F, f \\right)$ in $\\mcU$, the functions $F$, $f$ are positive, Morse, and their critical points and gradient trajectories are in bijection with those of $F_1$, $f_1$ respectively. \nThen we choose a family $\\left( F_n, f_n \\right)_{n \\geq 2}$ of pairs in $\\mcU$ such that in the ends,\n\\[F_n \\left( t,q \\right) = a_n \\left( t + f_n \\left( q \\right) \\right) + b_n \\text{ for } 0 < a_n \\ll 1 \\text{ and } b_n > 0. \\]\nWe also choose a decreasing sequence $\\left( \\delta_k \\right)_{k \\geq 1}$ of small enough positive numbers so that the family\n\\[\\left( H_n = \\sum_{k=1}^{n} \\delta_k F_k, h_n = \\sum_{k=1}^{n} \\delta_k f_k \\right)_{n \\in \\mathbf{N}} \\]\nhas the following property: for every $i<j$, the pair $\\left( H_j - H_i, h_j - h_i \\right)$ lies in the cone over $\\mcU$ (this can be easily achieved by induction on $n$).\nFinally, we let $L_n \\subset X$ be the Lagrangian corresponding to the graph of $d H_n$ in a Weinstein neighborhood of $L = L_0$, and we let $\\Lambda_n \\subset Y$ be the Legendrian corresponding to the one-jet of $h_n$ in a Weinstein neighborhood of $\\Lambda = \\Lambda_0$. Observe that we \\emph{do not} require the copies to be parallel in the sense of the third item of Lemma 31 (or Theorem 76) in \\cite{EL21}.\n\n\\section{Some $A_{\\infty}$-categories}\n\n\n\n\n\n\n\nWe denote by $\\mcA_+$, $\\mcA_{++}$, $\\mcA_0$ the $A_{\\infty}$-categories associated to $\\Lambda$ and $\\varepsilon_L$ in Definition \\ref{definition several categories}. We now associate an $A_{\\infty}$-category to $L$.\n\n\\begin{defin}\n\t\n\tLet \n\t\\[\\mathbf{L} := \\left( L_n \\right)_{n \\in \\mathbf{Z}}. \\]\n\tWe denote by $\\mcO$ the $A_{\\infty}$-category defined as follows:\n\t\\begin{enumerate}\n\t\t\n\t\t\\item the objects of $\\mcO$ are the Lagrangians $L_n$, $n \\in \\mathbf{Z}$, \n\t\t\n\t\t\\item the space of morphisms from $L_i$ to $L_j$ is either generated by $L_i \\cap L_j$ if $i<j$, or $\\mathbf{F}$ if $i=j$, or $0$ if $i>j$ ; the bigrading of an intersection point $c \\in \\mcO \\left( \\Lambda^i, \\Lambda^j \\right)$ corresponding to $q_0 \\in \\mathrm{Crit} \\, h_0$ is $\\left( 2 \\left( j-i \\right) + \\mathrm{ind} \\left( q_0 \\right) \\right)$, and \n\t\t\n\t\t\\item the operations are such that $e_{L_n} \\in \\mcO \\left( L_n, L_n \\right)$ is a strict unit, and for every sequence of integers $i_0 < \\dots < i_d$, for every sequence of intersection points \n\t\t\\[\\left( x_1, \\dots, x_d \\right) \\in \\left( L_{i_0} \\cap L_{i_1} \\right) \\times \\dots \\times \\left( L_{i_{d-1}} \\cap L_{i_d} \\right) \\]\n\t\twe have \n\t\t\\[\\mu_{\\mcO} \\left( x_1, \\dots, x_d \\right) = \\sum \\limits_{x_0 \\in L_{i_0} \\cap L_{i_d}} \\# \\mcM_{x_0, x_d \\dots x_1} \\left( \\mathbf{L}, j \\right) x_0 \\]\n\t\t(see Definition \\ref{definition moduli spaces for Lagrangians}).\n\t\tBesides, let $W_L$ be the set made of the morphisms $u^n \\in \\mcO \\left( L_n, L_{n+1} \\right)$ corresponding to the minimum of $\\left( H_{n+1} - H_n \\right)$ for $n \\in \\mathbf{N}$.\n\t\tWe define\n\t\t\\[CF^* \\left( L \\right) := \\mathrm{End}_{\\mcO \\left[ W_L^{-1} \\right]} \\left( L \\right) . \\]\n\t\t\n\t\\end{enumerate} \t\n\t\n\\end{defin}\n\n\nThere are natural $A_{\\infty}$-functors $\\psi : \\mcO \\to \\mcA_{++}$ (see \\cite{EL21} section 5), $\\iota : \\mcA_{++} \\to \\mcA_+$ (the inclusion) and $\\pi : \\mcA_+ \\to \\mcA_0$ (the projection).  \nThe composition $\\phi = \\pi \\circ \\iota \\circ \\psi : \\mcO \\to \\mcA_0$ sends $W_L$ to $W_0$, and thus it induces an $A_{\\infty}$-map $\\widetilde{\\phi} : CF^* \\left( L \\right) \\to LA_0^* \\left( \\Lambda \\right)$.\n\n\\begin{defin}\\label{definition relative cohomology} \n\t\n\tThe Floer complex of $L$ relative to its boundary $\\Lambda$ is the non unital $A_{\\infty}$-algebra\n\t\\[CF^* \\left( L, \\Lambda \\right) := \\mathrm{Cone} \\left( CF^* \\left( L \\right) \\xrightarrow{\\widetilde{\\phi}} LA_0^* \\left( \\Lambda \\right) \\right) \\]\n\t(see Definition \\ref{definition cone of an algebra map}). \n\t\n\\end{defin}\n\n\\section{Results}\n\nRecall that we are considering $A_{\\infty}$ functors $\\psi : \\mcO \\to \\mcA_{++}$ (as in \\cite{EL21} section 5) and $\\iota : \\mcA_{++} \\to \\mcA_+$ (the inclusion). Moreover, the composition $\\iota \\circ \\psi : \\mcO \\to \\mcA_+$ sends $W_L$ to $W_+$.\n\n\\begin{thm}\\label{thm relation Lagrangians Legendrians}\n\t(Part of Theorem 63 in \\cite{EL21})\n\t\n\tIf $HW^* \\left( L \\right) = 0$, then the $A_{\\infty}$-map\n\t\\[\\widetilde{\\iota \\circ \\psi} : CF^* \\left( L \\right) \\to LA_+^* \\left( \\Lambda \\right) \\]\n\tinduced by $\\iota \\circ \\psi$ is a quasi-isomorphism.  \n\t\n\\end{thm}\n\n\\begin{proof}\n\t\n\tOn the objects, $\\psi \\left( L_n \\right) = \\Lambda_n$ and, on the morphisms, $\\psi$ is defined using the exact same count of pseudo-holomorphic curves than in \\cite{EL21} Theorem 63 (the corresponding $A_{\\infty}$-map is denoted $\\mathfrak{e}$ there). In \\cite{EL21} Theorem 63, the authors prove that for every $i < j$,\n\t\\[\\mathrm{Cone} \\left( \\mcO \\left( L_i, L_j \\right) \\xrightarrow{\\psi} \\mcA_{++} \\left( \\Lambda_i, \\Lambda_j \\right) \\right)^{\\#} \\simeq CW^* \\left( L_i, \\widehat{L_j} \\right) \\]\n\twhere $\\widehat{L_j}$ is a negative wrapping of $L_j$ across $L_i$.\n\tBy hypothesis, $CW^* \\left( L \\right)$ is acyclic, which implies that $\\psi^1$ is a quasi-isomorphism. Thus, $\\psi$ is a quasi-equivalence, and since it sends $W_L$ into $W_+^{dir}$, the induced functor $\\widetilde{\\psi} : \\mcO \\left[ W_L^{-1} \\right] \\to \\mcA_{++} \\left[ \\left( W_+^{dir} \\right)^{-1} \\right]$ is also a quasi-equivalence (see Lemma 2.4 in \\cite{GPS19}). This concludes the proof according to Lemma \\ref{lemma directed and non-directed positive augmentation categories}.\n\n\n\n\n\n\t\n\\end{proof}\n\n\\begin{thm}\n\t(Part of Theorem 63 in \\cite{EL21})\n\t\n\tIf $HW^* \\left( L \\right) = 0$, then there is a quasi-isomorphism of $A_{\\infty}$-algebras\n\t\\[\\mathbf{F} \\oplus CF^* \\left( L, \\Lambda \\right) \\xrightarrow{\\sim} LA^* \\left( \\Lambda \\right) . \\] \n\t\n\\end{thm}\n\n\\begin{proof}\n\t\n\tFirst recall that the projection to the space of short Reeb chords induces an $A_{\\infty}$-functor $\\pi : \\mcA_+ \\to \\mcA_0$ which sends $W_+$ to $W_0$, and Proposition \\ref{prop Lagrangian algebra as a cone} gives\n\t\\[\\overline{LA^*} \\left( \\Lambda \\right) \\simeq \\mathrm{Cone} \\left( LA_+^* \\left( \\Lambda \\right) \\xrightarrow{\\widetilde{\\pi}} LA_0^* \\left( \\Lambda \\right) \\right) . \\] \n\tThen, recall that the composition $\\phi = \\pi \\circ \\iota \\circ \\psi : \\mcO \\to \\mcA_0$ sends $W_L$ to $W_0$, and thus it induces an $A_{\\infty}$-map $\\widetilde{\\phi} : CF^* \\left( L \\right) \\to LA_0^* \\left( \\Lambda \\right)$. \n\tNow observe that we have the following commutative diagram\n\t\\[\\begin{tikzcd}\n\tCF^* \\left( L \\right) \\ar[r, \"\\widetilde{\\iota \\circ \\psi}\"] \\ar[d, \"\\widetilde{\\phi}\"] & LA_+^* \\left( \\Lambda, \\varepsilon_L \\right) \\ar[d, \"\\widetilde{\\pi}\"] \\\\\n\tLA_0^* \\left( \\Lambda, \\varepsilon_L \\right) \\ar[r, equal] &  LA_0^* \\left( \\Lambda, \\varepsilon_L \\right) .\n\t\\end{tikzcd} \\]\n\tThe top horizontal arrow is a quasi-isomorphism by Theorem \\ref{thm relation Lagrangians Legendrians}. According to Proposition \\ref{prop invariance cone of an algebra map}, we get a quasi-isomorphism of non unital $A_{\\infty}$-algebras  \n\t\\[CF^* \\left( L, \\Lambda \\right) \\xrightarrow{\\sim} \\overline{LA^*} \\left( \\Lambda \\right). \\]\n\tThis concludes the proof because \n\t\\[LA^* \\left( \\Lambda \\right) = \\mathbf{F} \\oplus \\overline{LA^*} \\left( \\Lambda \\right). \\]\n\t\n\\end{proof}\n\n\n\\begin{thm}\\label{thm dga with coefficients in chains of based loop space}\n\t(Theorem 51 in \\cite{EL21})\n\t\n\tIf $\\Lambda$ is simply connected, then \n\t\\[E \\left( CE_{-*}^+ \\left( \\Lambda \\right) \\right) \\simeq LA_+^* \\left( \\Lambda \\right) . \\]\n\t\n\\end{thm}\n\n\\begin{proof}[Idea of proof]\n\t\n\tAssume that we can choose the copies to be parallel. In \\cite{EL21} section 3.5, the authors define a quasi-isomorphism $\\phi : CE_{-*}^+ \\left( \\Lambda\\right) \\to CE_{-*}^{\\parallel} \\left( \\Lambda \\right)$. Applying first bar and then graded dual, we get a quasi-isomorphism $B \\left( CE_{-*}^{\\parallel} \\left( \\Lambda \\right) \\right)^{\\#} \\to B \\left( CE_{-*}^+ \\left( \\Lambda \\right) \\right)^{\\#}$. Besides, recall that \n\t\\[CE_{-*}^{\\parallel} \\left( \\Lambda \\right) = \\Omega LC_*^{\\parallel} \\left( \\Lambda \\right). \\]\n\tThere is a natural quasi-isomorphism $B \\left( \\Omega LC_*^{\\parallel} \\left( \\Lambda \\right) \\right) \\to LC_*^{\\parallel} \\left( \\Lambda \\right)$ (see section 2.2.2 in \\cite{EL21}), which induces a quasi-isomorphism $LA^*_{\\parallel} \\left( \\Lambda \\right) \\to B \\left( \\Omega LC_*^{\\parallel} \\left( \\Lambda \\right) \\right)^{\\#}$. As a result, we get a quasi-isomorphism \n\t\\[LA^*_{\\parallel} \\left( \\Lambda \\right) \\to B \\left( CE_{-*} \\left( \\Lambda \\right) \\right)^{\\#} . \\] \n\t\n\tNow going back to the case where the copies are not parallel, we should still be able to define an $A_{\\infty}$-functor $\\Phi : \\mcA_+ \\to B \\left( CE_{-*} \\left( \\Lambda, C_{-*} \\left( \\Omega \\Lambda \\right) \\right) \\right)^{\\#}$ which mimic the map we just described. This functor should look like this. Take sequences \n\t\\[\\left\\{\n\t\\begin{array}{ccl}\n\t\\left( c_0, \\dots, c_{d-1} \\right) & \\in & \\mcR \\left( \\Lambda_{n_0}, \\Lambda_{n_1} \\right) \\times \\dots \\times \\mcR \\left( \\Lambda_{n_{d-1}}, \\Lambda_{n_d} \\right) \\\\\n\t\\left( x_0, \\dots, x_{r-1} \\right) & \\in & CE_{-*}^+ \\left( \\Lambda \\right)^{r}\n\t\\end{array}\n\t\\right. \\]\n\tsuch that none of the Reeb chords $c_i$ correspond to the minimum of the Morse function $f_1$, and each $x_i$ is a word in elements of $C_{-*} \\left( \\Omega \\Lambda \\right) \\cup \\mcR \\left( \\Lambda \\right)$.\n\tIf $r$ is positive, then \n\t\\[\\langle \\Phi \\left( c_0, \\dots, c_{d-1} \\right), s x_0 \\otimes \\dots \\otimes s x_{r-1} \\rangle = 0 . \\]\n\tAssume now that $r=0$. Let $\\left( \\mathbf{y}_1, \\dots, \\mathbf{y}_p \\right)$ be the family of short Reeb chords appearing in $c_0 \\cdots c_{d-1}$.\n\tThen \n\t\\[\t\\langle \\Phi \\left( c_0, \\dots, c_{d-1} \\right), s x_0 \\rangle = \\prod_{i=1}^{p} \\vert \\mcT \\left( \\sigma_i, \\mathbf{y}_i \\right) \\vert \\]\n\t(the moduli spaces $\\mcT \\left( \\sigma_i, \\mathbf{y}_i \\right)$ are defined in \\cite{EL21} section 3.5) if $x_0$ is obtained from $c_0 \\cdots c_{d-1}$ by replacing each $\\mathbf{y}_i$ by $\\sigma_i$, and $0$ otherwise. \n\t\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\t\n\tIt remains to show that $\\Phi$ sends $W_+$ to the unit (this should follow from Lemma 38 in \\cite{EL21}), and that the induced functor $\\widetilde{\\Phi} : \\mcA_+ \\left[ W_+^{-1} \\right] \\to B \\left( CE_{-*}^+ \\left( \\Lambda \\right) \\right)^{\\#} \\left[ \\mathbf{1}^{-1} \\right]$ is a quasi-equivalence. \n\t\n\\end{proof}\n\n\n\n\\section{Extended Legendrian $A_{\\infty}$-algebra}\\label{subsection positive augmentation category}\n\nIn this appendix, we study the endomorphism $A_{\\infty}$-algebra $LA_+^* \\left( \\Lambda \\right)$ of an augmentation $\\varepsilon : CE_{-*} \\left( \\Lambda \\right) \\to \\mathbf{F}$ in the positive augmentation category $\\mcA ug_+ \\left( \\Lambda \\right)$ defined by Chantraine in \\cite{Cha19} (it was first introduced by Ng, Rutherford, Shende, Sivek and Zaslow in \\cite{NRS+20} for Legendrians in $\\mathbf{R}^3$).\nThe goal is to prove Proposition \\ref{prop Lagrangian algebra as a cone}, which shows that there is a quasi-isomorphism of non-unital $A_{\\infty}$-algebras between $\\overline{LA^* \\left( \\Lambda \\right)}$ (which is the endomorphism algebra of $\\varepsilon$ in the augmentation category $\\mcA ug_- \\left( \\Lambda \\right)$ defined in \\cite{BC14}) and the cone of an $A_{\\infty}$-map from $LA_+^* \\left( \\Lambda \\right)$ to some $A_{\\infty}$-algebra $LA_0^* \\left( \\Lambda \\right)$ generated by Morse type generators.\n\n\\subsection{Setting}\n\nLet $(V, \\xi)$ be a contact manifold of dimension $(2n+1)$ equipped with a hypertight contact form $\\alpha$, and let $\\Lambda$ be a connected compact Legendrian submanifold in $\\left( V, \\xi \\right)$ which is chord generic with respect to $\\alpha$ (see section \\ref{subsection Legendrian invariants}).\nIn the following we fix an augmentation $\\varepsilon : CE_{-*} \\left( \\Lambda \\right) \\to \\mathbf{F}$. \n\nChoose a positive Morse function $f_1 : \\Lambda \\to \\mathbf{R}$ with a unique minimum.\nWe denote by $\\mcU$ a small open neighborhood of $f_1$ in $C^{\\infty} \\left( \\Lambda \\right)$ with the following property: for every $f$ in $\\mcU$, $f$ is positive, Morse, and its critical points and gradient trajectories are in bijection with those of $f_1$. \nThen we choose a family $\\left( f_n \\right)_{n \\geq 2}$ of functions in $\\mcU$.\nWe also choose a decreasing sequence $\\left( \\delta_k \\right)_{k \\geq 1}$ of small enough positive numbers so that the family\n\\[h_n = \\sum_{k=1}^{n} \\delta_k f_k, \\quad n \\in \\mathbf{N}, \\]\nhas the following property: for every $i<j$, $\\left( h_j - h_i \\right)$ lies in the cone over $\\mcU$ (this can be easily achieved by induction on $n$).\nFinally, we let $\\Lambda_n \\subset V$ be the Legendrian corresponding to the one-jet of $h_n$ in a Weinstein neighborhood of $\\Lambda = \\Lambda_0$. \nObserve that if the numbers $\\delta_k$ are small enough, the Chekanov-Eliashberg DG-algebra of $\\Lambda_n$ is naturally identified with $CE_{-*} \\left( \\Lambda \\right)$ for every $n$. In particular, $\\varepsilon = \\varepsilon_0$ induces an augmentation $\\varepsilon_n : CE_{-*} \\left( \\Lambda_n \\right) \\to \\mathbf{F}$ for every $n$. \n\n\\begin{rmk}\n\t\n\tLet $\\eta$ be the length of the smallest Reeb chord of $\\Lambda$. The set $\\mcR \\left( \\Lambda_i, \\Lambda_j \\right)$ of Reeb chords starting on $\\Lambda_i$ and ending on $\\Lambda_j$ splits as \n\t\\[\\mcR \\left( \\Lambda_i, \\Lambda_j \\right) = \\mcR_{long} \\left( \\Lambda_i, \\Lambda_j \\right) \\sqcup \\mcR_{short} \\left( \\Lambda_i, \\Lambda_j \\right), \\]\n\twhere $\\mcR_{long} \\left( \\Lambda_i, \\Lambda_j \\right)$ is the set of chords with length greater than $\\eta \/ 2$ (and $\\mcR_{short} \\left( \\Lambda_i, \\Lambda_j \\right)$ is the complementary set). In the following, we choose the numbers $\\delta_k$ small enough so that the maximum of $h_n$ is very small compare to $\\eta$. In particular, the set $\\mcR_{short} \\left( \\Lambda_i, \\Lambda_j \\right)$ is empty when $i \\geq j$, and it is in bijection with $\\mathrm{Crit} \\left( h_j - h_i \\right)$ when $i < j$. \n\t\n\\end{rmk}\n\n\\subsection{Some $A_{\\infty}$-categories and $A_{\\infty}$-algebras}\n\nIn the following, we fix a regular almost complex structure $J$ on $\\xi$ (see subsection \\ref{subsection moduli spaces}).  \nWe define three $A_{\\infty}$-categories $\\mcA_{\\triangle}$, for $\\triangle \\in \\{ ++, +, 0 \\}$.\n$\\mcA_{++}$ corresponds to the directed $A_{\\infty}$-category used in \\cite{Cha19} subsection 3.3 in order to define the endomorphism algebra of $\\varepsilon$ in $\\mcA ug_+ \\left( \\Lambda \\right)$, $\\mcA_+$ is a non-directed version of $\\mcA_{++}$, and $\\mcA_0$ is an $A_{\\infty}$-category whose spaces of morphisms are generated by short Reeb chords.\n\n\\begin{defin}\\label{definition several categories}\n\t\n\tLet\n\t\\[\\mathbf{\\Lambda} := \\bigsqcup_{n \\in \\mathbf{N}} \\Lambda_n.  \\]\n\tWe consider the $A_{\\infty}$-categories $\\mcA_{\\triangle}$, $\\triangle \\in \\{ ++, +, 0 \\}$, defined as follows:\n\t\\begin{enumerate}\n\t\t\n\t\t\\item the set of objects of $\\mcA_{\\triangle}$ is $\\{ \\Lambda_n \\mid n \\in \\mathbf{N} \\}$,\n\t\t\n\t\t\\item the space of morphisms from $\\Lambda_i$ to $\\Lambda_j$ is\n\t\t\\[\\mcA_{\\triangle} \\left( \\Lambda_i, \\Lambda_j \\right) = \\left\\{\n\t\t\\begin{array}{ll}\n\t\t\\langle \\mcR \\left( \\Lambda_i, \\Lambda_j \\right) \\rangle & \\text{if } \\triangle \\in \\{ +, ++ \\} \\\\\n\t\t\\langle \\mcR_{short} \\left( \\Lambda_i, \\Lambda_j \\right) \\rangle & \\text{if } \\triangle = 0 \n\t\t\\end{array}\n\t\t\\right. \\]\n\t\tif $i < j$,\n\t\t\\[\\mcA_{\\triangle} \\left( \\Lambda_i, \\Lambda_i \\right) = \\left\\{\n\t\t\\begin{array}{ll}\n\t\t\\mathbf{F} & \\text{if } \\triangle \\in \\{ ++, 0 \\} \\\\\n\t\t\\mathbf{F} \\oplus \\langle \\mcR \\left( \\Lambda_i, \\Lambda_i \\right) \\rangle & \\text{if } \\triangle = + ,\n\t\t\\end{array}\n\t\t\\right. \\] \n\t\tand $\\mcA_{\\triangle} \\left( \\Lambda_i, \\Lambda_j \\right) = 0$ if $i > j$. \n\t\t\n\t\t\\item the operations are such that $e_{\\Lambda_n} \\in \\mcA_{\\triangle} \\left( \\Lambda_n, \\Lambda_n \\right)$ is a strict unit, and for every sequence of generators \n\t\t\\[\\left( x_1, \\dots, x_d \\right) \\in \\mcA_{\\triangle} \\left( \\Lambda_{i_0}, \\Lambda_{i_1} \\right) \\times \\dots \\times \\mcA_{\\triangle} \\left( \\Lambda_{i_{d-1}}, \\Lambda_{i_d} \\right), \\]\n\t\twe have \n\t\t\\begin{align*}\n\t\t\\mu_{\\mcA_{\\triangle}} & \\left( x_1, \\dots, x_d \\right) \\\\ \n\t\t& = \\sum \\limits_{\\substack{ a, \\boldsymbol{\\gamma_0}, \\dots, \\boldsymbol{\\gamma_d} }} \\# \\mcM_{a, \\boldsymbol{\\gamma_d} x_d \\boldsymbol{\\gamma_{d-1}} x_{d-1} \\dots x_1 \\boldsymbol{\\gamma_0}} \\left( \\mathbf{R} \\times \\mathbf{\\Lambda}, J \\right) \\varepsilon_{i_0} \\left( \\boldsymbol{\\gamma_0} \\right) \\dots \\varepsilon_{i_d} \\left( \\boldsymbol{\\gamma_d} \\right)  a ,\n\t\t\\end{align*}\n\t\twhere in the sum, $a$ is a generator of $\\mcA_{\\triangle} \\left( \\Lambda_{i_0}, \\Lambda_{i_d} \\right)$ and $\\boldsymbol{\\gamma_j}$ is a word of Reeb chords in $\\mcR \\left( \\Lambda_{i_j}, \\Lambda_{i_j} \\right)$\n\t\t(see Definition \\ref{definition moduli spaces} for the moduli spaces).\n\t\t\n\t\\end{enumerate}\n\tMoreover, we let $W_{\\triangle}$ be the set made of the generators $c_n \\in \\mcA_{\\triangle} \\left( \\Lambda_n, \\Lambda_{n+1} \\right)$ corresponding to the minimum of $\\left( h_{n+1} - h_n \\right)$ for $n \\in \\mathbf{N}$.\n\tWe define\n\t\\[LA_{\\triangle}^* \\left( \\Lambda \\right) := \\mathrm{End}_{\\mcA_{\\triangle} \\left[ W_{\\triangle}^{-1} \\right]} \\left( \\Lambda \\right) . \\]\n\t\n\\end{defin}\n\n\\begin{lemma}\\label{lemma multiplication by the minimum in the different versions}\n\t\n\tFor every integer $j$, for every $\\triangle \\in \\{ ++, +, 0 \\}$ the morphism $c_j \\in \\mcA_{\\triangle} \\left( \\Lambda_j, \\Lambda_{j+1} \\right) \\cap W_{\\triangle}$ satisfies the following. For every $i < j < k$, the chain maps\n\t\\[\\left\\{\n\t\\begin{array}{ccccl}\n\t\\mu_{\\mcA_{\\triangle}}^2 \\left( - , c_j \\right) & : & \\mcA_{\\triangle} \\left( \\Lambda_i , \\Lambda_j \\right) & \\to & \\mcA_{\\triangle} \\left( \\Lambda_i , \\Lambda_{j+1} \\right) \\\\\n\t\\mu_{\\mcA_{\\triangle}}^2 \\left( c_j , - \\right) & : & \\mcA_{\\triangle} \\left( \\Lambda_{j+1}, \\Lambda_{k+1} \\right) & \\to & \\mcA_{\\triangle} \\left( \\Lambda_j, \\Lambda_{k+1} \\right)\n\t\\end{array}\n\t\\right. \\]\n\tare quasi-isomorphisms. \n\tMoreover, if we denote by $\\mcA_{long} \\left( \\Lambda_m, \\Lambda_n \\right)$ the subcomplex of $\\mcA_+ \\left( \\Lambda_m, \\Lambda_n \\right)$ generated by $\\mcR_{long} \\left( \\Lambda_m, \\Lambda_n \\right)$, then the induced chain maps \n\t\\[\\left\\{\n\t\\begin{array}{ccccl}\n\t\\mu_{\\mcA_+}^2 \\left( - , c_j \\right) & : & \\mcA_{long} \\left( \\Lambda_i , \\Lambda_j \\right) & \\to & \\mcA_{long} \\left( \\Lambda_i , \\Lambda_{j+1} \\right) \\\\\n\t\\mu_{\\mcA_+}^2 \\left( c_j , - \\right) & : & \\mcA_{long} \\left( \\Lambda_{j+1}, \\Lambda_{k+1} \\right) & \\to & \\mcA_{long} \\left( \\Lambda_j, \\Lambda_{k+1} \\right)\n\t\\end{array}\n\t\\right. \\]\n\talso are quasi-isomorphisms. \n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tWhen the contact manifold is $V = \\mathbf{R} \\times P$, the result follows from the main analytic theorem of \\cite{EES09} (Theorem 3.6). Indeed, this result implies that (when the perturbation functions $h_n$ are small enough) the 3-punctured pseudo-holomorphic discs with boundary on $\\mathbf{R} \\times \\bigsqcup_n \\Lambda_n$, asymptotic to $c_j$ at one of their punctures, correspond to trivial cylinders with boundary on $\\mathbf{R} \\times \\Lambda$.\n\t\n\tThe general case is part of a work in progress of Chantraine, Dimitroglou-Rizell and Ghiggini. \n\t\n\\end{proof}\n\n\n\\subsection{Relations between $A_{\\infty}$-categories and $A_{\\infty}$-algebras}\n\nObserve that the inclusion $\\iota : \\mcA_{++} \\to \\mcA_+$ defines a strict $A_{\\infty}$-functor which sends $W_{++}$ to $W_+$.\n\n\n\\begin{lemma}\\label{lemma directed and non-directed positive augmentation categories}\n\t\n\tThe $A_{\\infty}$-functor\n\t\\[\\widetilde{\\iota} : \\mcA_{++} \\left[ W_{++}^{-1} \\right] \\xrightarrow{\\sim} \\mcA_+ \\left[ W_+^{-1} \\right] \\]\n\tinduced by $\\iota$ (see Definition \\ref{definition induced functor between quotients}) is a quasi-equivalence.\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tFix an object $Y = \\Lambda_m$ in $\\mcA_{++}$. We want to prove that the map \n\t\\[\\widetilde{\\iota} : \\mcA_{++} \\left[ W_{++}^{-1} \\right] \\left( X, Y \\right) \\to \\mcA_+ \\left[ W_+ ^{-1} \\right] \\left( \\widetilde{\\iota} X, \\widetilde{\\iota} Y \\right) \\]\n\tis a quasi-isomorphism for every object $X$. \n\tOur strategy is to apply Proposition \\ref{prop quasi-isomorphism between localizations}. We consider the modules $\\mcM_{\\mcA_+}$, $\\mcM_{\\mcA_{++}}$ and module maps $t_{\\mcA_{++}} : \\mcA_{++} \\left( -, \\Lambda_m \\right) \\to \\mcM_{\\mcA_{++}}$, $t_{\\mcA_+} : \\mcA_+ \\left( -, \\Lambda_m \\right) \\to \\mcM_{\\mcA_+}$ of Definition \\ref{definition specific module}. \n\tObserve that the diagram of $\\mcA_{++}$-modules \n\t\\[\\begin{tikzcd}\n\t\\bigoplus \\limits_{n \\geq 0} \\mcA_{++} \\left( -, \\Lambda_n \\right) \\ar[d, \"\\mathrm{id}\"] \\ar[r, \"\\bigoplus \\limits_{n \\geq 0} t_{c_n}\"] & \\bigoplus \\limits_{n \\geq 0} \\mcA_{++} \\left( -, \\Lambda_{n+1} \\right) \\ar[d, \"t_{\\iota}\"] \\\\\n\t\\bigoplus \\limits_{n \\geq 0} \\mcA_{++} \\left( -, \\Lambda_n \\right) \\ar[r, \"t_{\\iota}\"] & \\iota^* \\mcM_{\\mcA_+}\n\t\\end{tikzcd} \\]\n\tcommutes up to the homotopy $h : \\bigoplus \\limits_{n \\geq 0} \\mcA_{++} \\left( -, \\Lambda_n \\right) \\to \\iota^* \\mcM_{\\mcA_+}$ which is the strict module map sending $x \\in \\mcA_{++} \\left( \\Lambda_k, \\Lambda_n \\right)$ to \n\t\\[h(x) = sx \\in \\mcA_+ \\left( \\Lambda_k, \\Lambda_n \\right) \\left[ 1 \\right] \\subset \\iota^* \\mcM_{\\mcA_+} \\left( \\Lambda_k \\right) \\]\n\t($s : \\mcA_+ \\left( \\Lambda_k, \\Lambda_n \\right) \\to \\mcA_+ \\left( \\Lambda_k, \\Lambda_n \\right) \\left[ 1 \\right]$ is the usual degree $(-1)$ map).\n\tAccording to Proposition \\ref{prop morphism induced by homotopy}, this induces a closed $\\mcA_{++}$-module map $t_0 : \\mcM_{\\mcA_{++}} \\to \\iota^* \\mcM_{\\mcA_+}$. Moreover, the following diagram of $\\mcA_{++}$-modules is commutative\n\t\\[\\begin{tikzcd}\n\t\\mcA_{++} \\left( - , Y \\right) \\ar[d, \"t_{\\mcA_{++}}\"] \\ar[r, \"t_{\\iota}\"] & \\iota^* \\mcA_+ \\left( - , \\iota Y  \\right) \\ar[d, \"\\iota^* t_{\\mcA_+}\"] \\\\\n\t\\mcM_{\\mcA_{++}} \\ar[r, \"t_0\"] & \\iota^* \\mcM_{\\mcA_+} .\n\t\\end{tikzcd} \\] \n\t\n\tIt remains to check the three items of Proposition \\ref{prop quasi-isomorphism between localizations}. The first two items are satisfied according to Lemmas \\ref{lemma multiplication by the minimum in the different versions} and \\ref{lemma second result for specific module}. We now have to prove that $t_0 : \\mcM_{\\mcA_{++}} \\to \\iota^* \\mcM_{\\mcA_+}$ is a quasi-isomorphism. Let $X = \\Lambda_k$ be some object in $\\mcA_{++}$. Observe that the following diagram of chain complexes is commutative \n\t\\[\\begin{tikzcd}\n\t\\mcM_{\\mcA_{++}} \\left( \\Lambda_k \\right) \\ar[r, \"t_0\"] & \\iota^* \\mcM_{\\mcA_+} \\left( \\Lambda_k \\right) = \\mcM_{\\mcA_+} \\left( \\Lambda_k \\right) \\\\\n\t\\mcA_{++} \\left( \\Lambda_k, \\Lambda_{k+1} \\right) \\ar[u, hook] \\ar[r, \"\\iota\"] & \\mcA_+ \\left( \\Lambda_k, \\Lambda_{k+1} \\right) \\ar[u, hook] .\n\t\\end{tikzcd} \\]\n\tEach vertical map is a quasi-isomorphism according to Lemma \\ref{lemma first result for specific module}, and the bottom horizontal map is the identity map. This implies that $t_0$ is a quasi-isomorphism, which is what we needed to prove.\n\t\n\\end{proof}\n\nObserve that the projection $\\pi : \\mcA_+ \\to \\mcA_0$ defines a strict $A_{\\infty}$-functor which sends $W_+$ to $W_0$. Thus it induces a strict $A_{\\infty}$-functor $\\widetilde{\\pi} : \\mcA_+ \\left[ W_+^{-1} \\right] \\to \\mcA_0 \\left[ W_0^{-1} \\right]$ (see Definition \\ref{definition induced functor between quotients}). \nBesides, let $\\overline{LA^*} \\left( \\Lambda \\right)$ be the augmented ideal of \n\\[LA^* \\left( \\Lambda \\right) = \\mathrm{End}_{\\mcA_+} \\left( \\Lambda \\right) \\]\n($\\overline{LA^*} \\left( \\Lambda \\right)$ is the endomorphism algebra of $\\varepsilon$ in the $A_{\\infty}$-category $\\mcA ug_- \\left( \\Lambda \\right)$ defined in \\cite{BC14}, see subsection \\ref{subsection Chekanov-Eliashberg algebra}). The inclusion $\\overline{LA^*} \\left( \\Lambda \\right) \\hookrightarrow \\mcA_+ \\left( \\Lambda, \\Lambda \\right)$ induces a strict $A_{\\infty}$-map $\\overline{LA^*} \\left( \\Lambda \\right) \\to LA_+^* \\left( \\Lambda \\right)$ whose image is contained in $\\ker \\left( \\widetilde{\\pi} \\right)$. Since $\\ker \\left( \\widetilde{\\pi} \\right)$ is an $A_{\\infty}$-subalgebra of $\\mathrm{Cone} \\left( LA_+^* \\left( \\Lambda \\right) \\xrightarrow{\\widetilde{\\pi}} LA_0^* \\left( \\Lambda \\right) \\right)$ (see Definition \\ref{definition cone of an algebra map}), we get a strict $A_{\\infty}$-map \n\\[\\phi : \\overline{LA^*} \\left( \\Lambda \\right) \\to \\mathrm{Cone} \\left( LA_+^* \\left( \\Lambda \\right) \\xrightarrow{\\widetilde{\\pi}} LA_0^* \\left( \\Lambda \\right) \\right) . \\]\n\n\\begin{prop}\\label{prop Lagrangian algebra as a cone}\n\t\n\tThe $A_{\\infty}$-map\n\t\\[\\phi : \\overline{LA^*} \\left( \\Lambda \\right) \\to \\mathrm{Cone} \\left( LA_+^* \\left( \\Lambda \\right) \\xrightarrow{\\widetilde{\\pi}} LA_0^* \\left( \\Lambda \\right) \\right) \\]\n\tis a quasi-isomorphism. \n\t\n\\end{prop}\n\n\\begin{proof}\n\t\n\tWe will apply Proposition \\ref{prop quasi-isomorphism between localizations} to the $A_{\\infty}$-functor $\\pi : \\mcA_+ \\to \\mcA_0$ and the object $\\Lambda_0 = \\Lambda$ of $\\mcA_+$. \n\tWe consider the modules $\\mcM_{\\mcA_0}$, $\\mcM_{\\mcA_+}$ and module maps $t_{\\mcA_+} : \\mcA_+ \\left( -, \\Lambda \\right) \\to \\mcM_{\\mcA_+}$, $t_{\\mcA_0} : \\mcA_0 \\left( -, \\Lambda \\right) \\to \\mcM_{\\mcA_0}$ of Definition \\ref{definition specific module}. \n\tObserve that the diagram of $\\mcA_+$-modules \n\t\\[\\begin{tikzcd}\n\t\\bigoplus \\limits_{n \\geq 0} \\mcA_+ \\left( -, \\Lambda_n \\right) \\ar[d, \"\\mathrm{id}\"] \\ar[r, \"\\bigoplus \\limits_{n \\geq 0} t_{c_n}\"] & \\bigoplus \\limits_{n \\geq 0} \\mcA_+ \\left( -, \\Lambda_{n+1} \\right) \\ar[d, \"t_{\\pi}\"] \\\\\n\t\\bigoplus \\limits_{n \\geq 0} \\mcA_+ \\left( -, \\Lambda_n \\right) \\ar[r, \"t_{\\pi}\"] & \\pi^* \\mcM_{\\mcA_0}\n\t\\end{tikzcd} \\]\n\tcommutes up to the homotopy $h : \\bigoplus \\limits_{n \\geq 0} \\mcA_+ \\left( -, \\Lambda_n \\right) \\to \\pi^* \\mcM_{\\mcA_0}$ which is the strict module map sending $x \\in \\mcA_+ \\left( \\Lambda_k, \\Lambda_n \\right)$ to \n\t\\[h(x) = s \\pi \\left( x \\right) \\in \\mcA_0 \\left( \\Lambda_k, \\Lambda_n \\right) \\left[ 1 \\right] \\subset \\pi^* \\mcM_{\\mcA_0} \\left( \\Lambda_k \\right) \\]\n\t($s : \\mcA_0 \\left( \\Lambda_k, \\Lambda_n \\right) \\to \\mcA_0 \\left( \\Lambda_k, \\Lambda_n \\right) \\left[ 1 \\right]$ is the usual degree $(-1)$ map).\n\tAccording to Proposition \\ref{prop morphism induced by homotopy}, this induces a closed $\\mcA_+$-module map $t_0 : \\mcM_{\\mcA_+} \\to \\pi^* \\mcM_{\\mcA_0}$. Moreover, the following diagram of $\\mcA_+$-modules is commutative\n\t\\[\\begin{tikzcd}\n\t\\mcA_+ \\left( - , \\Lambda \\right) \\ar[d, \"t_{\\mcA_+}\"] \\ar[r, \"t_{\\pi}\"] & \\pi^* \\mcA_0 \\left( - , \\Lambda  \\right) \\ar[d, \"\\pi^* t_{\\mcA_0}\"] \\\\\n\t\\mcM_{\\mcA_+} \\ar[r, \"t_0\"] & \\pi^* \\mcM_{\\mcA_0} .\n\t\\end{tikzcd} \\] \n\t\n\tNow according to Proposition \\ref{prop quasi-isomorphism between localizations}, there is a chain map $u : \\, _{W_+^{-1}} \\mcM_{\\mcA_+} \\left( \\Lambda \\right) \\to \\, _{W_0^{-1}} \\mcM_{\\mcA_0} \\left( \\Lambda \\right)$ such that the following diagram of chain complexes commutes\n\t\\[\\begin{tikzcd}\n\t\\mcA_+ \\left[ W_+^{-1} \\right] \\left( \\Lambda , \\Lambda \\right) \\ar[d, \"_{W_+^{-1}} t_{\\mcA_+}\"] \\ar[r, \"\\widetilde{\\pi}\"] & \\mcA_0 \\left[ W_0^{-1} \\right] \\left( \\Lambda , \\Lambda \\right) \\ar[d, \"_{W_0^{-1}} t_{\\mcA_0}\"] \\\\\n\t_{W_+^{-1}} \\mcM_{\\mcA_+} \\left( \\Lambda \\right) \\ar[r, \"u\"] & _{W_0^{-1}} \\mcM_{\\mcA_0} \\left( \\Lambda \\right) \\\\\n\t\\mcM_{\\mcA_+} \\left( \\Lambda \\right) \\ar[u, hook] \\ar[r, \"t_0\"] & \\mcM_{\\mcA_0} \\left( \\Lambda \\right) \\ar[u, hook]\n\t\\end{tikzcd} \\]\n\t(the two vertical maps on the bottom are the inclusions). \n\tMoreover, the inclusions \n\t\\[\\overline{LA^*} \\left( \\Lambda \\right) \\hookrightarrow LA_+^* \\left( \\Lambda \\right), \\, \\overline{LA^*} \\left( \\Lambda \\right) \\hookrightarrow \\, _{W_+^{-1}} \\mcM_{\\mcA_+} \\left( \\Lambda \\right), \\, \\overline{LA^*} \\left( \\Lambda \\right) \\hookrightarrow \\mcM_{\\mcA_+} \\left( \\Lambda \\right) \\]\n\tmake the following diagram commutes \n\t\\[\\begin{tikzcd}\n\t\\overline{LA^*} \\left( \\Lambda \\right) \\ar[r, hook] \\ar[d, equal] & LA_+^* \\left( \\Lambda \\right) \\ar[d, \"_{W_+^{-1}} t_{\\mcA_+}\"] \\ar[r, \"\\widetilde{\\pi}\"] & LA_0^* \\left( \\Lambda \\right) \\ar[d, \"_{W_0^{-1}} t_{\\mcA_0}\"] \\\\\n\t\\overline{LA^*} \\left( \\Lambda \\right) \\ar[r, hook] \\ar[d, equal] & _{W_+^{-1}} \\mcM_{\\mcA_+} \\left( \\Lambda \\right) \\ar[r, \"u\"] & _{W_0^{-1}} \\mcM_{\\mcA_0} \\left( \\Lambda \\right) \\\\\n\t\\overline{LA^*} \\left( \\Lambda \\right) \\ar[r, hook] & \\mcM_{\\mcA_+} \\left( \\Lambda \\right) \\ar[u, hook] \\ar[r, \"t_0\"] & \\mcM_{\\mcA_0} \\left( \\Lambda \\right) \\ar[u, hook] .\n\t\\end{tikzcd} \\]\n\tAccording to Lemma \\ref{lemma multiplication by the minimum in the different versions}, $\\mcA_+$ and $\\mcA_0$ satisfy the hypotheses of Lemma \\ref{lemma second result for specific module}. Thus, using Lemma \\ref{lemma second result for specific module} and Proposition \\ref{prop quasi-iso between a module and its localization}, we know that the vertical maps are quasi-isomorphisms. Therefore, we get the following commutative diagram of chain complexes\n\t\\[\\begin{tikzcd}\n\t\\overline{LA^*} \\left( \\Lambda \\right) \\ar[r, \"\\phi\"] \\ar[d, equal] & \\mathrm{Cone} \\left( LA_+^* \\left( \\Lambda \\right) \\xrightarrow{\\widetilde{\\pi}} LA_0^* \\left( \\Lambda \\right) \\right) \\ar[d, \"\\sim\"] \\\\\n\t\\overline{LA^*} \\left( \\Lambda \\right) \\ar[r] \\ar[d, equal] & \\mathrm{Cone} \\left( _{W_+^{-1}} \\mcM_{\\mcA_+} \\left( \\Lambda \\right) \\xrightarrow{u} \\, _{W_0^{-1}} \\mcM_{\\mcA_0} \\left( \\Lambda \\right) \\right) \\\\\n\t\\overline{LA^*} \\left( \\Lambda \\right) \\ar[r, \"\\psi\"] & \\mathrm{Cone} \\left( \\mcM_{\\mcA_+} \\left( \\Lambda \\right) \\xrightarrow{t_0} \\mcM_{\\mcA_0} \\left( \\Lambda \\right) \\right) \\ar[u, \"\\sim\" right] .\n\t\\end{tikzcd} \\]\n\tTherefore, in order to show that \n\t\\[\\phi : \\overline{LA^*} \\left( \\Lambda \\right) \\to \\mathrm{Cone} \\left( LA_+^* \\left( \\Lambda \\right) \\xrightarrow{\\widetilde{\\pi}} LA_0^* \\left( \\Lambda \\right) \\right) \\]\n\tis a quasi-isomorphism, it suffices to show that \n\t\\[\\psi : \\overline{LA^*} \\left( \\Lambda \\right) \\to \\mathrm{Cone} \\left( \\mcM_{\\mcA_+} \\left( \\Lambda \\right) \\xrightarrow{t_0} \\mcM_{\\mcA_0} \\left( \\Lambda \\right) \\right) \\]\n\tis a quasi-isomorphism. Observe that the inclusion $\\ker \\left( t_0 \\right) \\hookrightarrow \\mathrm{Cone} (t_0)$ is a quasi-isomorphism because $t_0$ is surjective. Thus it suffices to show that the inclusion $\\overline{LA^*} \\left( \\Lambda \\right) \\hookrightarrow \\ker \\left( t_0 \\right)$ is a quasi-isomorphism.\n\tRecall that we denote by $\\mcA_{long} \\left( \\Lambda_m, \\Lambda_n \\right)$ the subcomplex of $\\mcA_+ \\left( \\Lambda_m, \\Lambda_n \\right)$ generated by $\\mcR_{long} \\left( \\Lambda_m, \\Lambda_n \\right)$. Then observe that $\\overline{LA^*} \\left( \\Lambda \\right) = \\mcA_{long} \\left( \\Lambda, \\Lambda \\right)$ and \n\t\\[\\ker \\left( t_0 \\right) = \\left[\n\t\\begin{tikzcd}\n\t\\mcA_{long} \\left( \\Lambda , \\Lambda_0 \\right) \\ar[d, \"\\mathrm{id}\"] \\ar[rd, \"t_{c_0}\"] & \\mcA_{long} \\left( \\Lambda , \\Lambda_1 \\right) \\ar[d, \"\\mathrm{id}\"] \\ar[rd, \"t_{c_1}\"] & \\dots \\\\\n\t\\mcA_{long} \\left( \\Lambda , \\Lambda_0 \\right) & \\mcA_{long} \\left( \\Lambda , \\Lambda_1 \\right) & \\dots\n\t\\end{tikzcd} \n\t\\right] . \\]\n\tThus the result follows from Lemmas \\ref{lemma multiplication by the minimum in the different versions} and \\ref{lemma first result for specific module}. \n\t\n\n\\end{proof}\n\\section*{Introduction}\n\\addcontentsline{toc}{section}{Introduction} \n\nLegendrian contact homology was introduced by Chekanov \\cite{Che02} and Eliashberg \\cite{Eli98}, and it fits into the Symplectic Field Theory introduced in \\cite{EGH00}. It has been rigorously defined in the contactization of a Liouville manifold in \\cite{EES07}, following \\cite{EES05}. The importance of Legendrian contact homology goes beyond its applications to the Legendrian isotopy problem: for example, it was used by Bourgeois, Ekholm and Eliashberg in \\cite{BEE12} to compute symplectic invariants of Weinstein manifolds, and in a different way by Chantraine, Dimitroglou Rizell, Ghiggini and Golovko in \\cite{CDGG17} to prove a generation result for the wrapped Fukaya category of Weinstein manifolds. \n\nThe motivation for this paper is the study of Legendrian contact homology in subcritically fillable and Boothby-Wang contact manifolds, the latter being named after \\cite{BW58}. This has been done combinatorially in dimension three by Ekholm and Ng in \\cite{EN15} for the subcritically fillable case, and by Sabloff in \\cite{Sab03} for the Boothby-Wang case. The importance of the first kind of manifolds comes from the fact that every Weinstein manifold is obtained from a subcritical Weinstein manifold (of the form $\\mathbf{C} \\times P$ for some Weinstein manifold $P$) by attaching handles along Legendrian submanifolds in its boundary at infinity. The importance of the second kind of manifolds comes from a theorem of Donaldson in \\cite{Don96}, which states that any integral symplectic manifold $\\left( X, \\omega \\right)$ admits a symplectic submanifold $D \\subset X$ of codimension $2$, such that $X \\setminus D$ is a Liouville manifold whose boundary at infinity is a Boothby-Wang contact manifold. \nThe first step before attacking both cases presented above is to study Legendrian contact homology in the circular contactization of a Liouville manifold. In fact, both subcritically fillable and Boothby-Wang contact manifolds can be seen as compactifications of such spaces. This paper links together the Fukaya $A_{\\infty}$-category of a family of connected compact exact Lagrangians in a Liouville manifold $(P, \\lambda)$, and the Chekanov-Eliashberg DG-category of Legendrian lifts in the circular contactization $\\left( S^1 \\times P, \\ker ( d \\theta - \\lambda ) \\right)$.\n\nThe strategy we follow is to lift the situation to the usual contactization $\\mathbf{R} \\times P$ which has been much more studied. This naturally leads to consider an $A_{\\infty}$-category whose objects are the lifts in $\\mathbf{R} \\times P$ of our starting Legendrians,  and morphisms spaces are generated by Reeb chords. Moreover, the deck transformations of the cover $\\mathbf{R} \\to S^1$ induce an $A_{\\infty}$-autoequivalence of this category. \nThe rest of the proof has two main ingredients: \n\\begin{enumerate}\n\t\n\t\\item Functorial properties of the Legendrian invariants, which are used to bring us in a situation where we can apply the correspondence result of \\cite{DR16} between discs in the symplectization $\\mathbf{R} \\times \\mathbf{R} \\times P$ and polygons in $P$.\n\t\n\t\\item Two algebraic results of independent interest about mapping tori of $A_{\\infty}$-autoequivalences, that allow us to bridge the gaps between the algebraic invariants we are interested in.  \n\t\n\\end{enumerate} \nWe now proceed to describe the organization of the paper and state our main results.\n\n\\paragraph{Algebra.} \n\nIn section \\ref{subsection algebra}, we briefly recall the definitions of $A_{\\infty}$-(co)categories and give references for standard notions that we do not recall, such as (co)bar, graded dual and Koszul dual constructions.\nOn the other hand, we discuss in some details the notions of modules over $A_{\\infty}$-categories, as well as Grothendieck construction and homotopy pushout associated to a diagram of $A_{\\infty}$-categories following \\cite[section A.4]{GPS19}.\nWe use it to introduce the notion of ``cylinder object for an $A_{\\infty}$-category'', which is supposed to mimic the corresponding notion in homotopy theory. \n\n\\paragraph{Mapping torus of an $A_{\\infty}$-autoequivalence.} \n\nIn section \\ref{subsection mapping torus}\\footnote{In this section, $A_{\\infty}$-categories are always assumed to be \\emph{strictly unital} (see \\cite[paragraph (2a)]{Sei08}).}, we define the mapping torus associated to a quasi-autoequivalence $\\tau$ of an $A_{\\infty}$-category $\\mcA$ as the $A_{\\infty}$-category\n\\[\\mathrm{MT} (\\tau) := \\mathrm{hocolim} \\left( \n\\begin{tikzcd}\n\t\\mcA \\sqcup \\mcA \\ar[r, \"\\mathrm{id} \\sqcup \\tau\"] \\ar[d, \"\\mathrm{id} \\sqcup \\mathrm{id}\" left] & \\mcA \\\\\n\t\\mcA .\n\\end{tikzcd} \\right). \\]\nObserve that this terminology was also used in \\cite{Kar21}, but we do not know if the two notions coincide. When considering an $A_{\\infty}$-autoequivalence $\\tau : \\mcA \\to \\mcA$, we always assume that $\\mcA$ is equipped with a $\\mathbf{Z}$-splitting of $\\mathrm{ob} (\\mcA)$ compatible with $\\tau$, which is a bijection \n\\[\\mathbf{Z} \\times \\mcE \\xrightarrow{\\sim} \\mathrm{ob} \\left( \\mcA \\right), \\quad \\left( n, E \\right) \\mapsto X^n \\left( E \\right) \\]\nsuch that $\\tau \\left( X^n \\left( E \\right) \\right) = X^{n+1} \\left( E \\right)$ for every $n \\in \\mathbf{Z}$ and $E \\in \\mcE$ (see Definition \\ref{definition group-action}).\nThis naturally turns $\\mcA$ into an Adams-graded $A_{\\infty}$-category, where the Adams-degree of a morphism in $\\mcA (X^i (E), X^j (E'))$ is defined to be $(j-i)$.\nIt then follows that the mapping torus of $\\tau$ is also Adams-graded.\n\nSection \\ref{subsection mapping torus} contains two results about mapping tori of $A_{\\infty}$-autoequivalences: we choose to state only the most important in this introduction.\nWe denote by $\\mathbf{F} \\left[ t_m \\right]$ the augmented Adams-graded associative algebra generated by a variable $t_m$ of bidegree $(m, 1)$.\nObserve that if $\\mcC$ is a subcategory of an $A_{\\infty}$-category $\\mcD$ with $\\mathrm{ob} (\\mcC) = \\mathrm{ob} (\\mcD)$, then $\\mcC \\oplus (t_m \\mathbf{F} [t_m] \\otimes \\mcD)$ is naturally an Adams-graded $A_{\\infty}$-category, where the Adams degree of $t_m^k \\otimes x$ equals $k$.\nBesides, if $\\mcC$ is an $A_{\\infty}$-category equipped with a $\\mathbf{Z}$-splitting of $\\mathrm{ob} \\left( \\mcC \\right)$, we denote by $\\mcC^0$ the full $A_{\\infty}$-subcategory of $\\mcC$ whose set of objects corresponds to $\\{ 0 \\} \\times \\mcE$.\nFinally, we use the functor $\\mcC \\mapsto \\mcC_m$ of Definition \\ref{definition forgetful functor}. \n\n\\begin{thmintro}\\label{thm mapping torus in weak situation introduction}\n\t\n\tLet $\\tau$ be a quasi-autoequivalence of an $A_{\\infty}$-category $\\mcA$, weakly directed with respect to some compatible $\\mathbf{Z}$-splitting of $\\mathrm{ob} \\left( \\mcA \\right)$.\n\tAssume that there exists a closed degree $0$ bimodule map $f : \\mcA_m \\left( -, - \\right) \\to \\mcA_m \\left( -, \\tau(-) \\right)$ such that $f : \\mcA_m \\left( X^i(E) , X^j(E') \\right) \\to \\mcA_m \\left( X^i(E), X^{j+1} (E') \\right)$ is a quasi-isomorphism for every $i < j$ and $E, E' \\in \\mcE$.\n\tThen there is a quasi-equivalence of Adams-graded $A_{\\infty}$-categories \n\t\\[\\mathrm{MT} (\\tau) \\simeq \\mcA_m^0 \\oplus \\left( t_m \\mathbf{F} \\left[ t_m \\right] \\otimes \\mcA_m \\left[ f \\left( \\mathrm{units} \\right)^{-1} \\right]^0 \\right) . \\]\n\t\n\\end{thmintro}\n\n\\begin{rmksintro}\n\t\n\t\\begin{enumerate}\n\t\t\n\t\t\\item In \\cite{Gan13}, the chain complex of $\\mcA$-bimodule maps from the diagonal bimodule $\\mcA \\left( -, - \\right)$ to some $\\mcA$-bimodule $\\mcB$ is called the two-pointed complex for Hochschild cohomology of $\\mcA$ with coefficients in $\\mcB$. According to \\cite[Proposition 2.5]{Gan13}, this complex is quasi-isomorphic to the (ordinary) Hochschild cochain complex of $\\mcA$ with coefficients in $\\mcB$. In particular, the bimodule map $f$ in Theorem \\ref{thm mapping torus in weak situation introduction} defines a class in the Hochschild cohomology of $\\mcA_m$ with coefficients in $\\mcA_m \\left( -, \\tau (-) \\right)$.\n\t\t\n\t\t\\item The $A_{\\infty}$-category which computes the mapping torus in Theorem \\ref{thm mapping torus in weak situation introduction} is very similar to the categories studied in \\cite{Sei08bis}, with main difference the presence of curvature in Seidel's setting.\n\t\t\n\t\t\\item The use of the functor $\\mcC \\mapsto \\mcC_m$ in Theorem \\ref{thm mapping torus in weak situation introduction} is not of any deep importance. It was convenient for us to introduce it here for our application to Legendrian contact homology (see Theorem \\ref{thm mainthm} below).\n\t\t\n\t\\end{enumerate}\n\t\n\\end{rmksintro}\n\n\\paragraph{Chekanov-Eliashberg DG-algebra.} \n\nIn section \\ref{section Legendrian invariants}, we recall the definition and functorial properties of the Chekanov-Eliashberg DG-category associated to a family of Legendrians in a hypertight contact manifold.\n\n\\paragraph{Legendrian lifts of exact Lagrangians in the circular contactization.} \n\nIn section \\ref{subsection proof of main thm}, we start with a family \n\\[\\mathbf{L} = \\left( L(E) \\right)_{E \\in \\mcE}, \\quad \\mcE = \\left\\{ 1, \\dots, N \\right\\}, \\]\nof mutually transverse compact connected exact Lagrangian submanifolds in a Liouville manifold $\\left( P, \\lambda \\right)$, and we study a Legendrian lift of $\\mathbf{L}$ in the circular contactization $\\left( S^1 \\times P, \\ker (d \\theta - \\lambda) \\right)$.\nMore precisely, we assume\\footnote{This can always be achieved by applying the Liouville flow in backwards time.} that there are primitives $f_E : L(E) \\to \\mathbf{R}$ of $\\lambda_{| L(E)}$ such that $0 \\leq f_1 < \\dots < f_N \\leq 1\/2$,\nand we consider the family of Legendrians\n\\[\\mathbf{\\Lambda^{\\circ}} := (\\Lambda^{\\circ}(E))_{E \\in \\mcE}, \\text{ where } \\Lambda^{\\circ}(E) = \\left\\{ (f_E(x), x) \\in (\\mathbf{R} \/ \\mathbf{Z}) \\times P \\mid x \\in L(E) \\right\\} . \\]\nWe denote by $CE (\\mathbf{\\Lambda^{\\circ}})$ the Chekanov-Eliashberg category of $\\mathbf{\\Lambda^{\\circ}}$, by $\\mcF uk (\\mathbf{L})$ the Fukaya category generated by the Lagrangians $L(E)$ (see for example \\cite[chapter 2]{Sei08}), and by $\\overrightarrow{\\mcF uk} (\\mathbf{L})$ its directed subcategory (see \\cite[paragraph (5n)]{Sei08}).\n\nIn order for the latter algebraic objects to be $\\mathbf{Z}$-graded, we assume that $H_1(P)$ is free, that the first Chern class of $P$ (equipped with any almost complex structure compatible with $(- d \\lambda)$) is 2-torsion, and that the Maslov class of the Lagrangians $L(E)$ vanish.  \nAs explained in section \\ref{subsection Conley-Zehnder index}, the grading on $CE (\\mathbf{\\Lambda^{\\circ}})$ depends on the choice of a symplectic trivialization of the contact structure along a fiber $h_0 = S^1 \\times \\{ a_0 \\}$.\nWe denote by $CE_{-*}^r \\left( \\mathbf{\\Lambda^{\\circ}} \\right)$ the Chekanov-Eliashberg DG-category of $\\mathbf{\\Lambda^{\\circ}}$ with grading induced by the trivialization\n\\[\\left( \\xi^{\\circ}_{| h_0}, d \\alpha^{\\circ} \\right) \\xrightarrow{\\sim} \\left( h_0 \\times \\mathbf{C}^n, dx \\wedge dy \\right), \\quad \\left( \\left( \\theta, a_0  \\right), \\left( \\lambda_{a_0} (v), v \\right) \\right) \\mapsto \\left( \\left( \\theta, a_0  \\right) , e^{2 i \\pi r \\theta} \\psi (v) \\right), \\]\nwhere  $\\psi : \\left( T_{a_0} P, - d \\lambda_{a_0} \\right) \\xrightarrow{\\sim} \\left( \\mathbf{C}^n, dx \\wedge dy \\right)$ is a symplectic isomorphism.\n\nIn this setting, $CE_{-*}^r \\left( \\mathbf{\\Lambda^{\\circ}} \\right)$ is augmented (with the trivial augmentation) and Adams-graded (by the number of times a Reeb chord winds around the fiber).\nAs above, we denote by $\\mathbf{F} \\left[ t_m \\right]$ the augmented Adams-graded associative algebra generated by a variable $t_m$ of bidegree $(m, 1)$.\nMoreover, we denote by $E(-) = B(-)^{\\#}$ (graded dual of bar construction) the Koszul dual functor (see  \\cite[section 2]{LPWZ08} or \\cite[section 2.3]{EL21}).\n\n\\begin{thmintro}\\label{thm mainthm}\n\t\n\tKoszul duality holds for $CE_{-*}^r \\left( \\mathbf{\\Lambda^{\\circ}} \\right)$, and there is a quasi-equivalence of augmented Adams-graded $A_{\\infty}$-categories\n\t\\[E \\left( CE_{-*}^r \\left( \\mathbf{\\Lambda^{\\circ}} \\right) \\right) \\simeq \\overrightarrow{\\mcF uk} (\\mathbf{L}) \\oplus \\left( t_{2r} \\mathbf{F} \\left[ t_{2r} \\right] \\otimes \\mcF uk (\\mathbf{L}) \\right). \\]\n\t\n\\end{thmintro}\n\n\\begin{rmkintro}\n\t\t\n\tKoszul duality has many important consequences, see for example \\cite{LPWZ08} or \\cite{EL21}. In particular, by definition of Koszul duality (see \\cite[Theorem 2.4]{LPWZ08} or \\cite[Definition 17]{EL21}), Theorem \\ref{thm mainthm} implies that there is a quasi-equivalence of augmented Adams-graded DG-categories\n\t\\[CE_{-*}^r \\left( \\mathbf{\\Lambda^{\\circ}} \\right) \\simeq E \\left( \\overrightarrow{\\mcF uk} (\\mathbf{L}) \\oplus \\left( t_{2r} \\mathbf{F} \\left[ t_{2r} \\right] \\otimes \\mcF uk (\\mathbf{L}) \\right) \\right). \\]\n\tObserve that this formula is closely related to Conjecture 6.3 in \\cite{Sei08bis}, which was also discussed by Ganatra and Maydanskiy in the appendix of \\cite{BEE12}.\n\t\n\\end{rmkintro}\n\nWe now give a corollary of the latter result.\nIf $B$ is a (unpointed) space, we consider its one-point compactification $B^*$ and view it as a pointed space (with base point the point at infinity). If moreover $X$ is a pointed space, we consider the half-smash product of $B$ and $X$,\n\\[X \\rtimes B := X \\wedge B^* \\]\n(where $\\wedge$ denotes the smash product of pointed spaces).  \nFinally, if $Y$ is a pointed space, we denote by $\\Omega Y$ its loop space.\n\n\\begin{corointro}\n\t\n\tIf $L$ is a connected compact exact Lagrangian and $\\Lambda^{\\circ}$ is a Legendrian lift of $L$ in the circular contactization, then there is a quasi-equivalence of augmented DG-algebras \n\t\\[ CE_{-*}^1 \\left( \\Lambda^{\\circ} \\right) \\simeq C_{-*} \\left( \\Omega \\left( \\mathbf{CP}^{\\infty} \\rtimes L \\right) \\right). \\]\n\n\t\n\\end{corointro}\n\n\\paragraph{Acknowledgments.}\n\nThis work is part of my PhD thesis that I did at Nantes Universit\u00e9 under the supervision of Paolo Ghiggini and Vincent Colin, who I thank for their guidance and support. I also thank Baptiste Chantraine, Georgios Dimitroglou Rizell and Tobias Ekholm for helpful discussions.\n\\section{Legendrian lifts of exact Lagrangians in the circular contactization}\\label{subsection proof of main thm}\n\nIn this section, we start with a family $\\mathbf{L}$ of mutually transverse compact connected exact Lagrangian submanifolds in a Liouville manifold, and we study a Legendrian lift $\\mathbf{\\Lambda^{\\circ}}$ of $\\mathbf{L}$ in the circular contactization.  \nFor the standard contact form, each point on a Legendrian gives rise to a (countable) infinite set of Reeb chords, and thus the situation is degenerate. In section \\ref{subsection setting and Legendrian invariants}, we explain how we perturb the contact form and we state our central result, which relates the Chekanov-Eliashberg DG-category of $\\mathbf{\\Lambda^{\\circ}}$ and the Fukaya $A_{\\infty}$-category of $\\mathbf{L}$. \n\n\\subsection{Setting}\\label{subsection setting and Legendrian invariants}\n\nLet $\\left( P , \\lambda \\right)$ be a Liouville manifold, and let \n\\[\\mathbf{L} = \\left( L(E) \\right)_{E \\in \\mcE}, \\quad \\mcE = \\left\\{ 1, \\dots, N \\right\\}, \\]\nbe a family of mutually transverse compact connected exact Lagrangian submanifolds in $(P, \\lambda)$ such that there are primitives $f_E : L(E) \\to \\mathbf{R}$ of $\\lambda_{| L(E)}$ satisfying $0 \\leq f_1 < \\dots < f_N \\leq 1\/2$.\nWe consider the contact manifold \n\\[ \\left( V^{\\circ}, \\xi^{\\circ} \\right) = \\left( S^1 \\times P, \\mathrm{ker} \\, \\alpha^{\\circ} \\right), \\text{ where } S^1 = \\mathbf{R}_{\\theta} \/ \\mathbf{Z}, \\, \\alpha^{\\circ} = d \\theta - \\lambda, \\]\nand the family of Legendrian submanifolds \n\\[\\mathbf{\\Lambda^{\\circ}} := (\\Lambda^{\\circ}(E))_{E \\in \\mcE}, \\text{ where } \\Lambda^{\\circ}(E) = \\left\\{ (f_E(x), x) \\in (\\mathbf{R} \/ \\mathbf{Z}) \\times P \\mid x \\in L(E) \\right\\} . \\]\nIn order for the Chekanov-Eliashberg category of $\\mathbf{\\Lambda^{\\circ}}$ and the Fukaya category of $\\mathbf{L}$ to be $\\mathbf{Z}$-graded, we assume that $H_1(P)$ is free, that the first Chern class of $P$ (equipped with any almost complex structure compatible with $(- d \\lambda)$) is 2-torsion, and that the Maslov classes of the Lagrangians $L(E)$ vanish.\n\n\\subsubsection{Reeb chords}\\label{subsection perturbed Reeb chords}\n\nObserve that $\\Lambda^{\\circ} = \\bigcup_{E \\in \\mcE} \\Lambda (E)$ is not chord generic for $\\alpha^{\\circ}$ (see Definition \\ref{definition chord generic}). We will choose a compactly supported function $H : P \\to \\mathbf{R}$, and consider the perturbed contact form \n\\[ \\alpha_H^{\\circ} = e^H \\alpha^{\\circ} . \\]\nThe Reeb vector field of $\\alpha_H^{\\circ}$ is then \n\\[ R_{\\alpha_H^{\\circ}} = e^{- H} \\left( \n\\begin{array}{c}\n1 + \\lambda \\left( X_H \\right) \\\\\nX_H\n\\end{array} \\right), \\]\nwhere $X_H$ is the unique vector field on $P$ satisfying $\\iota_{X_H} d \\lambda = -d H$.\n\nWe fix a compact neighborhood $K$ of $L$ which is contained in a Weinstein neighborhood of $L$ in $P$. \nIt is not hard to see that for every positive integer $N$, the space of smooth functions $H$ on $P$ supported in $K$, such that the $R_{\\alpha_H^{\\circ}}$-chords of $\\Lambda^{\\circ}$ with action less than $N$ are generic, is open and dense in $C_K^{\\infty} \\left( P \\right)$. Therefore, the space of functions $H \\in C_K^{\\infty} \\left( P \\right)$ such that $\\Lambda^{\\circ}$ is chord generic with respect to $\\alpha_H^{\\circ}$ is a Baire subset of $C_K^{\\infty} \\left( P \\right)$. In particular, the latter is dense in $C_K^{\\infty} \\left( P \\right)$. In the following, we choose $H \\in C_K^{\\infty} \\left( P \\right)$ such that \n\\begin{enumerate}\n\t\n\t\\item $\\Lambda^{\\circ}$ is chord generic with respect to $\\alpha_H^{\\circ}$,\n\t\n\t\\item $H$ is sufficiently close to $0$ so that\n\t\\[d \\theta \\left( R_{\\alpha_H^{\\circ}} \\right) = e^{-H} \\left( 1 + \\lambda \\left( X_H \\right) \\right) \\geq 1\/2 . \\]\n\t\n\\end{enumerate}\n\n\\begin{exa}\\label{example cotangent case}\n\t\n\tAssume that we are in the case\n\t\\[\\left( P, \\lambda \\right) = \\left( T^* M, p dq \\right), \\, L = 0_M, \\text{ and } H \\left( q, p \\right) = h(q), \\]\n\twhere $h : M \\to \\mathbf{R}$ is a Morse function (we present this example in order to see what happens, even if $H$ is not compactly supported in $T^*M$).\n\tThe Reeb vector field of $\\alpha_H^{\\circ}$ is \n\t\\[ R_{\\alpha_H^{\\circ}} = e^{- h} \\left( \n\t\\begin{array}{c}\n\t1 \\\\\n\t0 \\\\\n\t-d h\n\t\\end{array} \\right), \\]\n\tand therefore the Reeb flow satisfies\n\t\\[ \\varphi_{R_{\\alpha_H^{\\circ}}}^t \\left( \\theta , \\left( q,p \\right) \\right) = \\left( \\theta + t e^{- h (q)}, \\left( q , p - t e^{-h (q)} dh \\left( q \\right) \\right) \\right) . \\]\n\tThus, the $R_{\\alpha_H^{\\circ}}$-chords of $\\Lambda^{\\circ}$ are the paths $c : \\left[ 0,T \\right] \\to S^1 \\times T^* M$ of the form \n\t\\[ c(t) = \\left( t e^{- h (q_0)} , \\left( q_0 , 0 \\right) \\right), \\text{ with } T e^{- h (q_0)} \\in \\mathbf{Z}_{\\geq 1} \\text{ and } q_0 \\in \\mathrm{Crit} \\, h. \\]\n\tObserve that these Reeb chords are transverse but lie on top of each others. See Figure \\ref{figure perturbed Reeb chords}, where we illustrate this perturbation when $M = S^1$.\n\t\n\\end{exa}\n\n\\begin{figure}\n\t\\def1\\textwidth{1.2\\textwidth}\n\t\\input{Invariants_lift\/Figures\/dessin10.pdf_tex}\t\t\n\t\\caption{Reeb chords (in blue) of $\\Lambda^{\\circ} = \\{ 0 \\} \\times 0_{S^1}$ for $\\alpha^{\\circ}$ (on the left) and for $\\alpha_H^{\\circ}$ (on the right)}\t\n\t\\label{figure perturbed Reeb chords}\n\\end{figure}\n\n\\subsubsection{Conley-Zehnder index}\n\nIn order to define the Conley-Zehnder index (see section \\ref{subsection Conley-Zehnder index}), we need to choose a family $(h_0, h_1, \\dots, h_s)$ of embedded circles in $V^{\\circ} = S^1 \\times P$ which represent a basis of $H_1 ( V^{\\circ} )$, and a symplectic trivialization of $\\xi^{\\circ}$ over each $h_i$. We let $h_0 = S^1 \\times \\{ a_0 \\}$ be some fiber of $S^1 \\times P \\to P$, and we fix $\\left( h_1, \\dots, h_s \\right)$ to be any family of embedded circles in $P$ which represent a basis of $H_1 ( P )$. \nWe choose a symplectic isomorphism $\\psi : \\left( T_{a_0} P, - d \\lambda_{a_0} \\right) \\xrightarrow{\\sim} \\left( \\mathbf{C}^n, dx \\wedge dy \\right)$, and then we choose the symplectic trivialization\n\\[\\left( \\xi^{\\circ}_{| h_0}, d \\alpha^{\\circ} \\right) \\xrightarrow{\\sim} \\left( h_0 \\times \\mathbf{C}^n, dx \\wedge dy \\right), \\quad \\left( \\left( \\theta, a_0  \\right), \\left( \\lambda_{a_0} (v), v \\right) \\right) \\mapsto \\left( \\left( \\theta, a_0  \\right) , e^{2 i \\pi r \\theta} \\psi (v) \\right) \\]\nwhere $r$ is some integer, that we call \\emph{$r$-trivialization of $\\xi^{\\circ}$ over the fiber}.\nFinally, we choose some trivialization of $\\xi^{\\circ}$ over each $h_i$, $1 \\leq i \\leq s$.\n\n\\begin{exa}\\label{example CZ index in cotangent case}\n\t\n\tWe compute the Conley-Zehnder index of a Reeb chord in the case of Example \\ref{example cotangent case}, i.e. when \n\t\\[\\left( P, \\lambda \\right) = \\left( T^* M, p dq \\right), \\, L = 0_M, \\text{ and } H \\left( q, p \\right) = h(q), \\]\n\twhere $h : M \\to \\mathbf{R}$ is a Morse function .\n\tIn this case, the Reeb flow is given by\n\t\\[ \\varphi_{R_{\\alpha_H^{\\circ}}}^t \\left( \\theta , \\left( q,p \\right) \\right) = \\left( \\theta + t e^{-  h (q)}, \\left( q , p - t  e^{-  h (q)} dh \\left( q \\right) \\right) \\right) . \\]\n\tLet $c : \\left[ 0,T \\right] \\to V^{\\circ}$ be a Reeb chord of $\\Lambda^{\\circ}$. Then there exists a positive integer $k$ and a critical point $q_0$ of $h$ such that \n\t\\[c(t) = \\left( t e^{-  h (q_0)} , \\left( q_0 , 0 \\right) \\right) \\text{ and } T e^{-  h (q_0)} = k. \\]\n\tObserve that $c(0)  = c(T)$, and thus there is no need to choose a capping path for $c$. \n\tBesides, for every $u$ in $T_{q_0} M$, we have  \n\t\\[D \\varphi_{R_{\\alpha_H^{\\circ}}}^t \\left( c(0) \\right) \\left( 0, u, 0 \\right) = \\left( 0, u, -  t e^{-  h (q_0)} D^2 h \\left( q_0 \\right) u \\right) . \\]\n\tIn order to compute the index of $c$, we first choose coordinates $\\left( x_1, \\dots, x_n \\right)$ around $q_0 \\in M$ in which  \n\t\\[h = h \\left( q_0 \\right) + \\frac{1}{2} \\sum_{j=1}^{\\mathrm{dim}(M)} \\sigma_j x_j^2, \\text{ where } \\sigma_j = \\pm 1, \\]\n\tand we extend it to symplectic coordinates $\\left( x_1, \\dots, x_n, y_1, \\dots, y_n \\right)$ around $\\left( q_0, 0 \\right) \\in T^*M$ by setting \n\t\\[y_j \\left( q,p \\right) = \\langle p, \\frac{\\partial}{\\partial x_j} (q) \\rangle . \\] \n\tOur choice of trivialization for a fiber of $S^1 \\times \n\tP \\to P$ induces the trivialization \n\t\\[e^{2 i \\pi r k t \/ T} \\left( dx + i dy \\right) : c^{-1} \\xi^{\\circ} \\xrightarrow{\\sim} \\left( \\mathbf{R} \/ T \\mathbf{Z} \\right) \\times \\mathbf{C}^n \\]\n\t(observe that $\\xi^{\\circ}_{c(t)} = \\{ 0 \\} \\times T_{\\left( q_0, 0 \\right)} \\left( T^*M \\right)$).\n\tAccordingly, the path $t \\mapsto D \\varphi_{R_{\\alpha_H^{\\circ}}}^t \\left( T_{c(0)} \\Lambda^{\\circ} \\right)$ induces a path of Lagrangians \n\t\\[\\Gamma_c : t \\in \\left[ 0, T \\right] \\mapsto \\left\\{ \\left( e^{2 i \\pi m k t \/ T} \\left( u_j - i  t e^{- h(q_0)} \\sigma_j u_j \\right) \\right)_{1 \\leq j \\leq n} \\mid u \\in \\mathbf{R}^n \\right\\} \\subset \\mathbf{C}^n . \\]\n\tWe close this path using a counterclockwise rotation $\\Gamma$, and call the resulting loop $\\overline{\\Gamma_c}$. In order to compute the Conley-Zehnder index of $c$, we have to look at how $\\overline{\\Gamma_c}$ intersects the Lagrangian $i \\mathbf{R}^n$ (as explained in \\cite[section 2.2]{EES05}).\n\tObserve that $\\Gamma_c$ intersects $i \\mathbf{R}^n$ positively $2rk$ times, so that $\\Gamma_c$ contributes $2rk$ to the Conley-Zehnder index of $c$. Moreover, since $\\Gamma$ is a counterclockwise rotation bringing\n\t\\[\\left\\{ \\left( u_j - i  T e^{- h(q_0)} \\sigma_j u_j \\right)_{1 \\leq j \\leq n} \\mid u \\in \\mathbf{R}^n \\right\\} \\text{ to } \\mathbf{R}^n, \\]\n\tthe contributions to the intersection between $\\Gamma$ and $i \\mathbf{R}^n$ come from the negative eigenvalues $\\sigma_j$. The computation done in \\cite[Lemma 3.4]{EES05bisbis} implies that $\\Gamma$ contributes $\\mathrm{ind} (q_0)$ to the Conley-Zehnder index of $c$. We conclude that the Conley-Zehnder index of $c$ is\n\t\\[CZ \\left( c \\right) = \\mu \\left( \\overline{\\Gamma_c} \\right) = 2rk + \\mathrm{ind} (q_0) . \\] \n\n\\end{exa}\n\n\\subsubsection{Main result and strategy of proof}\\label{subsection invariants and statement of the result}\n\nLet $j$ be an almost complex structure on $P$ compatible with $\\left( -d\\lambda \\right)$, and let $J^{\\circ}$ be its lift to a complex structure on $\\xi^{\\circ}$.\nRecall from section \\ref{subsection Chekanov-Eliashberg algebra} the definition of the Chekanov-Eliashberg DG-category of a family of Legendrians. In our situation, $CE_{-*}^r \\left( \\mathbf{\\Lambda^{\\circ}} \\right) = CE_{-*} \\left( \\mathbf{\\Lambda^{\\circ}}, J^{\\circ}, \\alpha_{H}^{\\circ} \\right)$ (with grading induced by the $r$-trivialization of $\\xi^{\\circ}$ over the fiber) is an Adams-graded DG-algebra, where the Adams-degree of a Reeb chord $c$ is the number of times $c$ winds around the fiber. Besides, the map $CE_{-*}^r \\left( \\mathbf{\\Lambda^{\\circ}} \\right) \\to \\mathbf{F}$ which sends every Reeb chord to zero (and preserves units) defines an augmentation of $CE_{-*}^r \\left( \\mathbf{\\Lambda^{\\circ}} \\right)$. \n\n\\begin{rmk}\n\t\n\tIn the case of Example \\ref{example cotangent case}, the cohomological degree of a Reeb chord $c$ in $CE_{-*}^r \\left( \\mathbf{\\Lambda^{\\circ}} \\right)$ corresponding to a positive integer $k$ and a critical point $q_0$ is \n\t\\[1 - CZ (c) = 1 -2rk - \\mathrm{ind} (q_0) \\]\n\t(see Example \\ref{example CZ index in cotangent case}).\n\t\n\\end{rmk}\n\nBesides, we denote by $\\mcF uk (\\mathbf{L})$ the Fukaya category generated by the Lagrangians $L(E)$ (see for example \\cite[chapter 2]{Sei08}), and by $\\overrightarrow{\\mcF uk} (\\mathbf{L})$ its directed subcategory:\n\\[\\hom_{\\overrightarrow{\\mcF uk} (\\mathbf{L})} (L(E), L(E')) =\n\\left\\{\n\\begin{array}{cl}\n\t\\langle L(E) \\cap L(E') \\rangle & \\text{if } E < E' \\\\\n\t\\mathbf{F} & \\text{if } E = E' \\\\\n\t0 & \\text{if } E > E' \\\\\n\\end{array}\n\\right. \\]\n(see \\cite[paragraph (5n)]{Sei08}). \n\nLet $\\mathbf{F} \\left[ t_m \\right]$ be the augmented Adams-graded associative algebra generated by a variable $t_m$ of bidegree $\\left( m,1 \\right)$. \nObserve that if $\\mcC$ is a subcategory of an $A_{\\infty}$-category $\\mcD$ with $\\mathrm{ob} (\\mcC) = \\mathrm{ob} (\\mcD)$, then $\\mcC \\oplus (t_m \\mathbf{F} [t_m] \\otimes \\mcD)$ is naturally an Adams-graded $A_{\\infty}$-category, where the Adams degree of $t_m^k \\otimes x$ equals $k$.\nMoreover, we denote by $E(-) = B(-)^{\\#}$ (graded dual of bar construction) the Koszul dual functor (see \\cite[section 2.3]{EL21} or \\cite[section 2]{LPWZ08}).\n\n\\begin{thm}[Theorem \\ref{thm mainthm} in the Introduction]\\label{thm mainthm restatement}\n\t\n\tKoszul duality holds for $CE_{-*}^r \\left( \\mathbf{\\Lambda^{\\circ}} \\right)$, and there is a quasi-equivalence of augmented Adams-graded $A_{\\infty}$-categories\n\t\\[E \\left( CE_{-*}^r \\left( \\mathbf{\\Lambda^{\\circ}} \\right) \\right) \\simeq \\overrightarrow{\\mcF uk} (\\mathbf{L}) \\oplus \\left( t_{2r} \\mathbf{F} \\left[ t_{2r} \\right] \\otimes \\mcF uk (\\mathbf{L}) \\right). \\]\n\t\n\\end{thm}\n\t\n\\begin{coro}\\label{coro topological expression}\n\t\n\tIf $L$ is a connected compact exact Lagrangian and $\\Lambda^{\\circ}$ is a Legendrian lift of $L$ in the circular contactization, then there is a quasi-equivalence of augmented DG-algebras \n\t\\[ CE_{-*}^1 \\left( \\Lambda^{\\circ} \\right) \\simeq C_{-*} \\left( \\Omega \\left( \\mathbf{CP}^{\\infty} \\rtimes L \\right) \\right). \\]\n\t\n\\end{coro}\n\n\\begin{proof}\n\t\n\tLet $x_0$ be the base point of $\\mathbf{CP}^{\\infty}$, and set $P := \\mathbf{CP}^{\\infty} \\setminus \\{ x_0 \\}$. \n\tObserve that \n\t\\[\\left( P \\times L \\right)^* = P^* \\wedge L^* = \\mathbf{CP}^{\\infty} \\wedge L^* = \\mathbf{CP}^{\\infty} \\rtimes L . \\]\n\tWe have  \n\t\\[\\mathbf{F} \\oplus \\left( t_2 \\mathbf{F} \\left[ t_2 \\right] \\otimes CF^* \\left( L \\right) \\right) \\simeq \\mathbf{F} \\oplus \\left( t_2 \\mathbf{F} \\left[ t_2 \\right] \\otimes C^* \\left( L \\right) \\right) \\simeq C^* \\left( \\left( P \\times L \\right)^* \\right) \\simeq C^* \\left( \\mathbf{CP}^{\\infty} \\rtimes L \\right). \\]\n\tThus, it follows from Theorem \\ref{thm mainthm restatement} that \n\t\\[E \\left( CE_{-*}^1 \\left( \\Lambda^{\\circ} \\right) \\right) \\simeq C^* \\left( \\mathbf{CP}^{\\infty} \\rtimes L \\right) . \\]\n\tSince Koszul duality holds for $CE_{-*}^1 \\left( \\Lambda^{\\circ} \\right)$, \n\t\\[CE_{-*}^1 \\left( \\Lambda^{\\circ} \\right) \\simeq E \\left( C^* \\left( \\mathbf{CP}^{\\infty} \\rtimes L \\right)  \\right) . \\]\n\tObserve that the graded algebra $H^*\\left( \\mathbf{CP}^{\\infty} \\rtimes L \\right)$ is locally finite (i.e. each degree component is finitely generated) and simply connected (i.e. its augmentation ideal is concentrated in components of degree strictly greater than 1). Thus, according to the homological perturbation lemma (see \\cite[Proposition 1.12]{Sei08}), we can assume that $C^*\\left( \\mathbf{CP}^{\\infty} \\rtimes L \\right)$ is a locally finite  and simply connected $A_{\\infty}$ model for the DG-algebra of cochains on $\\mathbf{CP}^{\\infty} \\rtimes L$. \n\tTherefore, \\cite[Lemma 10]{EL21} implies that\n\t\\[CE_{-*}^1 \\left( \\Lambda^{\\circ} \\right) \\simeq \\Omega \\left( C_{-*} \\left( \\mathbf{CP}^{\\infty} \\rtimes L \\right) \\right). \\]\n\tNow, since $\\mathbf{CP}^{\\infty} \\rtimes L$ is simply connected, Adams result (see \\cite{Ada56}, \\cite{AH56} and also \\cite{EL21}) yields\n\t\\[\\Omega \\left( C_{-*} \\left( \\mathbf{CP}^{\\infty} \\rtimes L \\right) \\right) \\simeq C_{-*} \\left( \\Omega \\left( \\mathbf{CP}^{\\infty} \\rtimes L \\right) \\right) . \\]\n\tThis concludes the proof.\n\t\n\\end{proof}\n\n\\paragraph{Strategy of proof.}\n\nWe explain the startegy to compute $E ( CE_{-*}^r \\left( \\mathbf{\\Lambda^{\\circ}} \\right) )$.\nRecall from the last paragraph of section \\ref{subsection Chekanov-Eliashberg algebra} that there is a coaugmented $A_{\\infty}$-cocategory $LC_* \\left( \\mathbf{\\Lambda^{\\circ}} \\right)$ such that \n\\[CE_{-*}^r \\left( \\mathbf{\\Lambda^{\\circ}} \\right) = \\Omega ( LC_* \\left( \\mathbf{\\Lambda^{\\circ}} \\right) ). \\]\n$LC_* \\left( \\mathbf{\\Lambda^{\\circ}} \\right)$ inherits an Adams-grading from $CE_{-*}^r \\left( \\mathbf{\\Lambda^{\\circ}} \\right)$ (the same), and we denote by $LA^* \\left( \\mathbf{\\Lambda^{\\circ}} \\right)$ its graded dual (see \\cite[section 2.1.3]{EL21}). In our situation, $LA^* \\left( \\mathbf{\\Lambda^{\\circ}} \\right)$ is an augmented Adams-graded $A_{\\infty}$-category whose augmentation ideal is generated by the Reeb chords of $\\mathbf{\\Lambda^{\\circ}}$ (and the Adams-degree of a Reeb chord $c$ is the number of times $c$ winds around the fiber).\nSince there is a quasi-isomorphism $B \\left( \\Omega C \\right) \\simeq C$ for every $A_{\\infty}$-cocategory $C$ (see \\cite[section 2.2.2]{EL21}), it follows that \n\\[E \\left( CE_{-*}^r \\left( \\mathbf{\\Lambda^{\\circ}} \\right) \\right) = B \\left( CE_{-*}^r \\left( \\mathbf{\\Lambda^{\\circ}} \\right) \\right)^{\\#} \\simeq LC_* \\left( \\mathbf{\\Lambda^{\\circ}} \\right)^{\\#} = LA^* \\left( \\mathbf{\\Lambda^{\\circ}} \\right) \\]\n(graded dual preserves quasi-isomorphisms).\n\n\\begin{rmk}\n\t\n\tIn the case of Example \\ref{example cotangent case}, the cohomological degree of a Reeb chord $c$ in $LA^* \\left( \\mathbf{\\Lambda^{\\circ}} \\right)$ corresponding to a positive integer $k$ and a critical point $q_0$ is \n\t\\[CZ (c) = 2rk + \\mathrm{ind} (q_0) \\]\n\t(see Example \\ref{example CZ index in cotangent case}).\n\t\n\\end{rmk}\n\nIn order to compute $LA^* \\left( \\mathbf{\\Lambda^{\\circ}} \\right)$, we apply a sequence of geometric modifications to the situation. Each of the section numbered from \\ref{subsection lift} to \\ref{subsection projection on P} explains one of these modifications and describes how the algebraic invariants change accordingly. \nThe main ingredients are respectively: Theorem \\ref{thm mapping torus in strict situation} in section \\ref{subsection lift}, Lemma \\ref{lemma rectify contact form} in section \\ref{subsection rectification of the contact form}, Theorem \\ref{thm invariance} in section \\ref{subsection back to the original acs}, and \\cite[Theorem 2.1]{DR16} in section \\ref{subsection projection on P}.\nFinally, we end the proof in section \\ref{subsection proof of the main result} using Theorem \\ref{thm mapping torus in weak situation} (Theorem \\ref{thm mapping torus in weak situation introduction} in the introduction).\n\n\\subsection{Lift to $\\mathbf{R} \\times P$}\\label{subsection lift}\n\nIn the following we will consider the contact manifold \n\\[ \\left( V, \\xi \\right) = \\left( \\mathbf{R}_{\\theta} \\times P, \\ker \\, \\alpha \\right) \\text{ where } \\alpha = d \\theta - \\lambda, \\]\nand the family of Legendrian submanifolds\n\\[\\mathbf{\\Lambda} := (\\Lambda^n(E))_{(n, E) \\in \\mathbf{Z} \\times \\mcE}, \\text{ where } \\Lambda^{\\theta}(E) = \\left\\{ (f_E(x) + \\theta, x) \\in \\mathbf{R} \\times P \\mid x \\in L(E) \\right\\} . \\]\nMoreover we set $\\Lambda^n := \\bigcup_{E \\in \\mcE} \\Lambda^n (E)$ and $\\Lambda := \\bigcup_{n \\in \\mathbf{Z}} \\Lambda^n$.\n\nRecall from section \\ref{subsection perturbed Reeb chords} that we chose a compactly supported function $H : P \\to \\mathbf{R}$ such that \n\\begin{enumerate}\n\t\n\t\\item $\\Lambda^{\\circ}$ is chord generic with respect to $\\alpha_H^{\\circ}$,\n\t\n\t\\item $H$ is sufficiently close to $0$ so that\n\t\\[d \\theta \\left( R_{\\alpha_H^{\\circ}} \\right) = e^{-H} \\left( 1 + \\lambda \\left( X_H \\right) \\right) \\geq 1\/2 . \\]\n\t\n\\end{enumerate}\nWe consider the contact form \n\\[ \\alpha_{H} = e^{ H} \\alpha, \\]\nwith Reeb vector field \n\\[ R_{\\alpha_H} = e^{-  H} \\left( \n\\begin{array}{c}\n1 +  \\lambda \\left( X_H \\right) \\\\\n X_H\n\\end{array} \\right). \\]\nMoreover, we denote by $J$ the lift of $J^{\\circ}$ to an almost complex structure on $\\xi$. \n\n\\subsubsection{The $A_{\\infty}$-category $\\mcA$}\n\n\\begin{defin}\\label{definition category A}\n\t\n\tWe consider the $A_{\\infty}$-category $\\mcA$ defined as follows\n\t\\begin{enumerate}\n\t\t\n\t\t\\item the objects of $\\mcA$ are the Legendrians $\\Lambda^n (E)$, $(n, E) \\in \\mathbf{Z} \\times \\mcE$,\n\t\t\n\t\t\\item the space of morphisms from $\\Lambda^i (E)$ to $\\Lambda^j (E')$ is either generated by the $R_{\\alpha_{H}}$-chords from $\\Lambda^i (E)$ to $\\Lambda^j (E')$ if $(i,E)<(j,E')$, or $\\mathbf{F}$ if $(i,E)=(j,E')$, or $0$ otherwise,\n\t\t\n\t\t\\item the operations are such that $1 \\in \\mcA \\left( \\Lambda^n (E), \\Lambda^n (E) \\right)$ is a strict unit, and for every sequence $(i_0, E_0) < \\dots < (i_d, E_d)$, for every sequence of Reeb chords \n\t\t\\[\\left( c_1, \\dots, c_d \\right) \\in \\mcR \\left( \\Lambda^{i_0} (E_0), \\Lambda^{i_1} (E_1) \\right) \\times \\dots \\times \\mcR \\left( \\Lambda^{i_{d-1}} (E_{d-1}), \\Lambda^{i_d} (E_d) \\right), \\]\n\t\twe have \n\t\t\\[\\mu_{\\mcA} \\left( c_1, \\dots, c_d \\right) = \\sum \\limits_{c_0 \\in \\mcR \\left( \\Lambda^{i_0} (E_0), \\Lambda^{i_d} (E_d) \\right)} \\# \\mcM_{c_d, \\dots, c_1, c_0} \\left( \\mathbf{R} \\times \\Lambda, J, \\alpha_{H} \\right) c_0 \\]\n\t\t(see Definition \\ref{definition moduli spaces} for the moduli spaces).\n\t\t\n\t\\end{enumerate} \n\t\n\\end{defin}\n\n\\subsubsection{The quasi-autoequivalence $\\tau$} \n\nWe introduce an $A_{\\infty}$-endofunctor of $\\mcA$ that will be important in the following.\n\n\\begin{defin}\\label{definition functor tau}\n\t\n\tWe denote by $\\tau : \\mcA \\to \\mcA$ the $A_{\\infty}$-functor defined as follows:\n\t\\begin{enumerate}\n\t\t\n\t\t\\item on objects, $\\tau$ sends $\\Lambda^n (E)$ to $\\Lambda^{n+1} (E)$,\n\t\t\n\t\t\\item on morphisms, the map \n\t\t\\[\\tau : \\mcA \\left( \\Lambda^{i_0} (E_0), \\Lambda^{i_1} (E_1) \\right) \\otimes \\dots \\otimes \\mcA \\left( \\Lambda^{i_{d-1}} (E_{d-1}), \\Lambda^{i_d} (E_d) \\right) \\to \\mcA \\left( \\Lambda^{i_0+1} (E_0), \\Lambda^{i_d+1} (E_d) \\right)  \\]\n\t\tis obtained by dualizing the components of the DG-isomorphism\n\t\t\\[CE_{-*} \\left( \\left( \\Lambda^{n+1} \\right)_{i_0 \\leq n \\leq i_d}, J, \\alpha_{H} \\right) \\to CE_{-*} \\left( \\left( \\Lambda^n \\right)_{i_0 \\leq n \\leq i_d}, J, \\alpha_{H} \\right) \\]\n\t\tinduced by the path $ \\left( \\left( \\Lambda^{n+1-t} \\right)_{i_0 \\leq n \\leq i_d}, J \\right)_{0 \\leq t \\leq 1}$ (see Theorem \\ref{thm invariance}). \n\t\t\n\t\\end{enumerate}\n\t\n\\end{defin}\n\n\\begin{lemma}\\label{lemma tau is a quasi-equivalence}\n\t\n\tThe $A_{\\infty}$-functor $\\tau : \\mcA \\to \\mcA$ is a quasi-equivalence.\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tConsider the $A_{\\infty}$-functor $\\overline{\\tau} : \\mcA \\to \\mcA$ defined as follows:\n\t\\begin{enumerate}\n\t\t\n\t\t\\item on objects, $\\overline{\\tau}$ sends $\\Lambda^n (E)$ to $\\Lambda^{n-1} (E)$,\n\n\t\t\\item on morphisms, the map \n\t\t\\[\\overline{\\tau} : \\mcA \\left( \\Lambda^{i_0} (E_0), \\Lambda^{i_1} (E_1) \\right) \\otimes \\dots \\otimes \\mcA \\left( \\Lambda^{i_{d-1}} (E_{d-1}), \\Lambda^{i_d} (E_d) \\right) \\to \\mcA \\left( \\Lambda^{i_0-1} (E_0), \\Lambda^{i_d-1} (E_d) \\right)  \\]\n\t\tis obtained by dualizing the components of the \\emph{inverse} of the DG-isomorphism\n\t\t\\[CE_{-*} \\left( \\left( \\Lambda^{n+1} \\right)_{i_0 \\leq n \\leq i_d}, J, \\alpha_{H} \\right) \\to CE_{-*} \\left( \\left( \\Lambda^n \\right)_{i_0 \\leq n \\leq i_d}, J, \\alpha_{H} \\right) \\]\n\t\tinduced by the path $ \\left( \\left( \\Lambda^{n+1-t} \\right)_{i_0 \\leq n \\leq i_d}, J \\right)_{0 \\leq t \\leq 1}$. \n\t\t\n\t\\end{enumerate} \n\tThen $\\tau \\circ \\overline{\\tau} = \\overline{\\tau} \\circ \\tau = \\mathrm{id}_{\\mcA}$. \n\t\n\\end{proof}\n\nHere the $\\mathbf{Z}$-splitting \n\\[\\mathbf{Z} \\times \\mcE \\xrightarrow{\\sim} \\mathrm{ob} \\left( \\mcA \\right), \\quad (n, E) \\mapsto \\Lambda^n (E) \\]\nis compatible with the quasi-autoequivalence $\\tau$ in the sense of Definition \\ref{definition group-action}. As explained there, this turns $\\mcA$ into an Adams-graded $A_{\\infty}$-category: the Adams-degree of a morphism $c \\in \\mcA \\left( \\Lambda^i (E), \\Lambda^j (E') \\right)$ is $\\left( j-i \\right)$. \n\n\\subsubsection{Relation between $LA^* \\left( \\Lambda^{\\circ} \\right)$ and $\\left( \\mcA, \\tau \\right)$}\n\nWe now explain how $LA^* \\left( \\Lambda^{\\circ} \\right)$ and $\\left( \\mcA, \\tau \\right)$ are related.\nSee Figure \\ref{figure relation usual and circular contactization}, where we illustrate the action of the projection $\\Pi_{S^1 \\times P}$ in the case \n\\[\\left( P, \\lambda \\right) = \\left( T^* S^1, p dq \\right), \\, L = 0_{S^1}, \\text{ and } H \\left( q, p \\right) = h(q), \\]\nwhere $h : S^1 \\to \\mathbf{R}$ is a Morse function. \n\n\\begin{figure}\n\t\\def1\\textwidth{1.3\\textwidth}\n\t\\input{Invariants_lift\/Figures\/dessin12.pdf_tex}\t\t\n\t\\caption{Action of the projection $\\Pi_{S^1 \\times T^* S^1}$}\t\n\t\\label{figure relation usual and circular contactization}\n\\end{figure}\n\n\\begin{lemma}\\label{lemma tau is strict}\n\t\n\tThe $A_{\\infty}$-functor $\\tau$ is strict, and it sends a Reeb chord $t \\mapsto \\left( \\theta \\left( t \\right) , x \\left( t \\right) \\right)$ in $\\mcA \\left( \\Lambda^i (E) , \\Lambda^j (E') \\right)$ to the Reeb chord $t \\mapsto \\left( \\theta \\left( t \\right) + 1 , x \\left( t \\right) \\right)$ in $\\mcA \\left( \\Lambda^{i+1} (E) , \\Lambda^{j+1} (E') \\right)$.\n\tIn particular, $\\tau$ acts bijectively on hom-sets. \n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tRecall that $\\alpha_H = e^H \\alpha$, with $H$ a function defined on the base manifold $P$.\n\tIn particular, the flow $\\varphi_{\\partial_{\\theta}}^t$ of $\\partial_{\\theta}$ is a strict contactomorphism of $\\left( V, \\alpha_{H} \\right)$.\n\tMoreover, since $J$ is the lift of an almost complex structure $j$ on $P$, we have\n\t\\[\\left( \\left( \\Lambda^{n+1-t} \\right)_{i_0 \\leq n \\leq i_d}, J \\right) = \\left( \\left( (\\varphi_{\\partial_{\\theta}}^t)^{-1} \\Lambda^{n+1} \\right)_{i_0 \\leq n \\leq i_d}, (\\varphi_{\\partial_{\\theta}}^t)^* J \\right) . \\]\n\tThe result follows from Theorem \\ref{thm invariance}.\n\t\n\\end{proof}\n\nWe denote by $\\mcA_{\\tau}$ the Adams-graded $A_{\\infty}$-category associated to $\\tau$ as in Definition \\ref{definition category of coinvariants}. \n\n\\begin{lemma}\\label{lemma A^rond = A_Z}\n\t\n\tThere is a quasi-isomorphism of Adams-graded $A_{\\infty}$-categories\n\t\\[LA^* \\left( \\Lambda^{\\circ} \\right) \\simeq \\mcA_{\\tau} . \\]\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tConsider the map which sends a Reeb chord $c \\in \\mcR \\left( \\Lambda^i (E), \\Lambda^j (E') \\right)$ to the corresponding chord $\\Pi_{S^1 \\times P} \\left( c \\right) \\in \\mcR \\left( \\Lambda^{\\circ} (E), \\Lambda^{\\circ} (E') \\right)$ (where $\\Pi_{S^1 \\times P} : \\mathbf{R} \\times P \\to S^1 \\times P$ is the projection). According to Lemma \\ref{lemma tau is strict}, $\\Pi_{S^1 \\times P} \\left( \\tau c \\right) = \\Pi_{S^1 \\times P} \\left( c \\right)$, and thus the map $c \\mapsto \\Pi_{S^1 \\times P} \\left( c \\right)$ induces a map $\\psi : \\mcA_{\\tau} \\to LA^* \\left( \\Lambda^{\\circ} \\right)$. Moreover, observe that $\\psi$ is a bijection on hom-spaces. It remains to prove that $\\psi$ is an $A_{\\infty}$-map. This follows from the fact that the map\n\t\\[ u = \\left( \\sigma , v \\right) \\mapsto \\left( \\sigma , \\Pi_{S^1 \\times P} \\circ v \\right) \\]\n\tinduces a bijection\n\t\\[\\mcM_{c_d, \\dots, c_1, c_0} \\left( \\mathbf{R} \\times \\Lambda, J, \\alpha_{H} \\right) \\xrightarrow{\\sim} \\mcM_{\\psi (c_d), \\dots, \\psi (c_1), \\psi (c_0)} \\left( \\mathbf{R} \\times \\Lambda^{\\circ}, J^{\\circ}, \\alpha_{H}^{\\circ} \\right) . \\] \n\t\n\\end{proof}\n\n\\begin{lemma}\\label{lemma A^rond = mapping torus of tau}\n\t\n\tThe Adams-graded $A_{\\infty}$-category $LA^* \\left( \\Lambda^{\\circ} \\right)$ is quasi-equivalent to the mapping torus of $\\tau : \\mcA \\to \\mcA$ (see Definition \\ref{definition mapping torus}). \n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tThis follows directly from Theorem \\ref{thm mapping torus in strict situation} using Lemmas \\ref{lemma tau is strict} and \\ref{lemma A^rond = A_Z}. \n\t\n\\end{proof}\n\n\\subsection{Rectification of the contact form}\\label{subsection rectification of the contact form}\n\nNow that we are in the usual contactization, we have the following result. \n\n\\begin{lemma}\\label{lemma rectify contact form} \n\t\n\tThere exists a contactomorphism $\\phi_H$ of $\\left( V,\\xi \\right)$ such that \n\t\\[ \\phi_H^* \\alpha_{H} = \\alpha . \\]\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tRecall that $\\alpha_{H} = e^{ H} \\alpha$, with $H$ a compactly supported function on the base manifold $P$ such that $e^{-H} \\left( 1 + \\lambda \\left( X_H \\right) \\right) \\geq 1\/2$.\n\t\n\tAssume that there is a contact isotopy $\\left( \\phi_{t} \\right)_{0 \\leq t \\leq 1}$ such that $\\phi_0 = \\mathrm{id}$ and \n\t\\begin{equation}\\label{eq1}\n\t\\phi_{t}^* \\alpha_{t H} = \\alpha\n\t\\end{equation}\n\tfor every $t$. Let $\\left( F_{t} \\right)_{t}$ be the family of functions on $V$ such that \n\t\\[ \\frac{d}{d t} \\phi_{t} = Y_{F_{t}} \\circ \\phi_{t}, \\]\n\twhere $Y_F$ is the contact vector field on $V$ satisfying \n\t\\begin{align*}\n\t\\left\\{\n\t\\begin{array}{ccl}\n\t\\alpha \\left( Y_F \\right) & = & F \\\\\n\t\\iota_{Y_F} d \\alpha & = & d F \\left( R_{\\alpha} \\right) \\alpha  - d F . \n\t\\end{array}\n\t\\right.\n\t\\end{align*}\n\tTaking the derivative of equation \\eqref{eq1} with respect to $t$, we get \n\t\\begin{equation}\\label{eq2}\n\tH + d \\left( e^{t H} F_{t} \\right) \\left( R_{\\alpha_{t H}} \\right) = 0\n\t\\end{equation}\n\tbecause $Y_F$ satisfies \n\t\\begin{align*}\n\t\\left\\{\n\t\\begin{array}{ccl}\n\t\\alpha_{t H} \\left( Y_F \\right) & = & e^{t H} F \\\\\n\t\\iota_{Y_F} d \\alpha_{t H} & = & d \\left( e^{t H} F \\right) \\left( R_{\\alpha_{t H}} \\right) \\alpha_{t H}  - d \\left( e^{t H} F \\right) .\n\t\\end{array}\n\t\\right.\n\t\\end{align*}\n\tBesides, we deduce from \n\t\\[ R_{\\alpha_{t H}} = e^{- t H} \\left( \n\t\\begin{array}{c}\n\t1 + t \\lambda \\left( X_H \\right) \\\\\n\tt X_H\n\t\\end{array} \\right), \\quad \\iota_{X_H} d \\lambda = - dH, \\]\n\tthat \n\t\\[ d H \\left( R_{\\alpha_{t H}} \\right) = 0 . \\]\n\tThen equation \\eqref{eq2} gives \n\t\\begin{equation}\\label{eq3}\n\td F_{t} \\left( R_{\\alpha_{t H}} \\right) = - H e^{- t H} .\n\t\\end{equation} \n\t\n\tConversely, if $\\left( F_{t} \\right)_{t}$ is a family of functions on $V$ satisfying equation \\eqref{eq3}, then the contact isotopy $\\left( \\phi_{t} \\right)_{t}$ defined by \n\t\\[ \\phi_0 = \\mathrm{id} \\text{ and } \\frac{d}{d t} \\phi_{t} = Y_{F_{t}} \\circ \\phi_{t}, \\]\n\tsatisfies\n\t\\[ \\frac{d}{d t} \\left( \\phi_{t}^* \\alpha_{t H} \\right) = 0 , \\]\n\tand thus $\\phi_H := \\phi_1$ gives the desired result.\n\t\n\tTherefore, it remains to find a family $\\left( F_{t} \\right)_{t}$ satisfying equation \\eqref{eq3}. First recall that \n\t\\[ R_{\\alpha_{t H}} = e^{- t H} \\left( \n\t\\begin{array}{c}\n\t1 + t \\lambda \\left( X_H \\right) \\\\\n\tt X_H\n\t\\end{array} \\right) . \\]\n\tBy assumption on $H$, the function $d \\theta \\left( R_{\\alpha_{t H}} \\right)$ is greater than $1\/2$ for every $t \\in \\left[ 0,1 \\right]$. Thus, for every $t \\in \\left[ 0,1 \\right]$ and every $\\left( \\theta , x \\right)$ in $V$, there exists a unique real number $\\rho_{t} \\left( \\theta , x \\right)$ such that \n\t\\[ \\varphi_{R_{\\alpha_{t H}}}^{- \\rho_{t} \\left( \\theta , x \\right) } \\left( \\theta , x \\right) \\in \\{ 0 \\} \\times P . \\]\n\tThen we let \n\t\\[ F_{t}:= - \\rho_{t} H e^{- t H} . \\]\n\tFor every real number $t$, we have \n\t\\[ F_{t} \\circ \\varphi_{R_{\\alpha_{t H}}}^t = - \\left( \\rho_{t} \\circ \\varphi_{R_{\\alpha_{t H}}}^t \\right) H e^{- t H} \\text{ because } dH \\left( R_{\\alpha_{t H}} \\right) = 0 . \\]\n\tBut the map $\\varphi_{R_{\\alpha_{t H}}}^{- \\rho_{t} \\circ \\varphi_{R_{\\alpha_{t H}}}^t + t}$ takes its values in $\\{ 0 \\} \\times P$ by definition of $\\rho_{t}$, so by uniqueness we have\n\t\\[ \\rho_{t} \\circ \\varphi_{R_{\\alpha_{t H}}}^t = \\rho_{t} + t . \\] \n\tThen we have \n\t\\[ F_{t} \\circ \\varphi_{R_{\\alpha_{t H}}}^t = - \\left( \\rho_{t} + t \\right) H e^{- t H}, \\]\n\tand thus \n\t\\[ d F_{t} \\left( R_{\\alpha_{t H}} \\right) = - H e^{- t H} . \\]\n\tThis concludes the proof.\n\t\n\\end{proof}\n\n\\begin{exa}\\label{example rectify contact form in cotangent case}\n\t\n\tAssume that we are in the case \n\t\\[\\left( P, \\lambda \\right) = \\left( T^* M, p dq \\right), \\, L = 0_M, \\text{ and } H \\left( q, p \\right) = h(q), \\]\n\twhere $h : M \\to \\mathbf{R}$ is a Morse function. \n\tThen the diffeomorphism $\\phi_H$ defined by\n\t\\[\\phi_H^{-1} \\left( \\theta, \\left( q,p \\right) \\right) = \\left( \\theta e^{h(q)}, \\left( q, e^{h(q)} p + \\theta e^{h(q)} dh (q) \\right) \\right) \\]\n\tsatisfies $\\phi_H^* \\alpha_H = \\alpha$.\n\tWith this choice of $\\phi_H$, we have in particular\n\t\\[\\phi_H^{-1} \\left( \\{ \\theta \\} \\times 0_M \\right) = j^1 \\left( \\theta e^h \\right) \\subset \\mathbf{R} \\times T^*M . \\] \n\t\n\\end{exa}\n\n\\subsubsection{The $A_{\\infty}$-category $\\mcA_1$}\n\nIn the following, we fix a contactomorphism $\\phi_H$ as in Lemma \\ref{lemma rectify contact form}.\nWe define an $A_{\\infty}$-category $\\mcA_1$, which is roughly obtained by pulling back the data of $\\mcA$ by $\\phi_H$. \n\n\\begin{defin}\n\t\n\tLet \n\t\\[\\Lambda_H^{\\theta} (E) := \\phi_H^{-1} \\left( \\Lambda^{\\theta} (E) \\right), \\, \\Lambda_H^n := \\bigcup\\limits_{E \\in \\mcE} \\Lambda_H^n (E), \\, \\Lambda_H := \\bigcup\\limits_{n \\in \\mathbf{Z}} \\Lambda_H^n \\text{ and } J_H := \\phi_H^* J. \\]\n\tWe consider the $A_{\\infty}$-category $\\mcA_1$ defined as follows\n\t\\begin{enumerate}\n\t\t\n\t\t\\item the objects of $\\mcA_1$ are the Legendrians $\\Lambda_H^n (E)$, $(n, E) \\in \\mathbf{Z} \\times E$,\n\t\t\n\t\t\\item the vector space $\\mcA_1 \\left( \\Lambda_H^i (E), \\Lambda_H^j (E') \\right)$ is either generated by the $R_{\\alpha}$-chords from $\\Lambda_H^i (E)$ to $\\Lambda_H^j (E')$ if $(i,E)<(j,E')$, or $\\mathbf{F}$ if $(i,E)=(j,E')$, or $0$ otherwise, and\n\t\t\n\t\t\\item the operations are such that $1 \\in \\mcA_1 \\left( \\Lambda_H^n (E), \\Lambda_H^n (E) \\right)$ is a strict unit, and for every sequence $(i_0, E_0) < \\dots < (i_d, E_d)$, for every sequence of Reeb chords \n\t\t\\[\\left( c_1, \\dots, c_d \\right) \\in \\mcR \\left( \\Lambda_H^{i_0} (E_0), \\Lambda_H^{i_1} (E_1) \\right) \\times \\dots \\times \\mcR \\left( \\Lambda_H^{i_{d-1}} (E_{d-1}), \\Lambda_H^{i_d} (E_d) \\right) \\]\n\t\twe have \n\t\t\\[\\mu_{\\mcA_1} \\left( c_1, \\dots, c_d \\right) = \\sum \\limits_{c_0 \\in \\mcR \\left( \\Lambda_H^{i_0} (E_0), \\Lambda_H^{i_d} (E_d) \\right)} \\# \\mcM_{c_d, \\dots, c_1, c_0} \\left( \\mathbf{R} \\times \\Lambda_H, J_H, \\alpha \\right) c_0 \\]\n\t\t(see Definition \\ref{definition moduli spaces} for the moduli spaces).\n\t\t\n\t\\end{enumerate}\n\t\n\\end{defin}\n\n\\subsubsection{The quasi-autoequivalence $\\tau_1$}\n\n\\begin{defin}\\label{definition functor tau_1}\n\t\n\tWe denote by $\\tau_1 : \\mcA_1 \\to \\mcA_1$ the $A_{\\infty}$-functor defined as follows:\n\t\\begin{enumerate}\n\t\t\n\t\t\\item on objects, $\\tau_1 \\left( \\Lambda_H^n (E) \\right) = \\Lambda_H^{n+1} (E)$,\n\t\t\n\t\t\\item on morphisms, the map \n\t\t\\[\\tau_1^d : \\mcA_1 \\left( \\Lambda_H^{i_0} (E_0), \\Lambda_H^{i_1} (E_1) \\right) \\otimes \\dots \\otimes \\mcA_1 \\left( \\Lambda_H^{i_{d-1}} (E_{d-1}), \\Lambda_H^{i_d} (E_d) \\right) \\to \\mcA_1 \\left( \\Lambda_H^{i_0+1} (E_0), \\Lambda_H^{i_d+1} (E_d) \\right)  \\]\n\t\tis obtained by dualizing the components of the DG-isomorphism\n\t\t\\[CE_{-*} \\left( \\left( \\Lambda_H^{n+1} \\right)_{i_0 \\leq n \\leq i_d}, J_H, \\alpha \\right) \\to CE_{-*} \\left( \\left( \\Lambda_H^n \\right)_{i_0 \\leq n \\leq i_d}, J_H, \\alpha \\right) \\]\n\t\tinduced by the path $ \\left( \\left( \\Lambda_H^{n+1-t} \\right)_{i_0 \\leq n \\leq i_d}, J_H \\right)_{0 \\leq t \\leq 1}$ (see Theorem \\ref{thm invariance}). \n\t\t\n\t\\end{enumerate}\n\t\n\\end{defin}\n\n\\begin{lemma}\\label{lemma tau_1 is a quasi-equivalence}\n\t\n\tThe $A_{\\infty}$-functor $\\tau_1 : \\mcA_1 \\to \\mcA_1$ is a quasi-equivalence.\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tThis follows because $\\tau_1$ is defined by dualizing the components of a DG-isomorphism (see the proof of Lemma \\ref{lemma tau is a quasi-equivalence}). \n\t\n\\end{proof}\n\nHere the $\\mathbf{Z}$-splitting \n\\[\\mathbf{Z} \\times \\mcE \\xrightarrow{\\sim} \\mathrm{ob} \\left( \\mcA_1 \\right), \\quad (n, E) \\mapsto \\Lambda_H^n (E), \\]\nis compatible with the quasi-autoequivalence $\\tau_1$ in the sense of Definition \\ref{definition group-action}. As explained there, this turns $\\mcA_1$ into an Adams-graded $A_{\\infty}$-category. \n\n\\subsubsection{Relation between $\\left( \\mcA, \\tau \\right)$ and $\\left( \\mcA_1, \\tau_1 \\right)$}\n\nWe now explain how the pairs $\\left( \\mcA, \\tau \\right)$ and $\\left( \\mcA_1, \\tau_1 \\right)$ are related.\nSee Figure \\ref{figure action of the contactomorphism} where we illustrate the action of the contactomorphism $\\phi_H^{-1}$ in the case \n\\[\\left( P, \\lambda \\right) = \\left( T^* S^1, p dq \\right), \\, L = 0_{S^1}, \\text{ and } H \\left( q, p \\right) = h(q), \\]\nwhere $h : S^1 \\to \\mathbf{R}$ is a Morse function.\n\n\\begin{figure}\n\t\\def1\\textwidth{1.4\\textwidth}\n\t\\input{Invariants_lift\/Figures\/dessin13.pdf_tex}\t\t\n\t\\caption{Action of the contactomorphism $\\phi_H^{-1}$}\t\n\t\\label{figure action of the contactomorphism}\n\\end{figure}\n\n\\begin{lemma}\\label{lemma relation A - A_1}\n\t\n\tThere is a strict $A_{\\infty}$-isomorphism $\\zeta_1 : \\mcA \\to \\mcA_1$ defined as follows:\n\t\\begin{enumerate}\n\t\t\n\t\t\\item on objects, $\\zeta_1 \\left( \\Lambda^n (E) \\right) = \\Lambda_H^n (E)$,\n\t\t\n\t\t\\item on morphisms, $\\zeta_1$ sends a Reeb chord $c$ in $\\mcA \\left( \\Lambda^i (E), \\Lambda^j (E') \\right)$ to the Reeb chord \n\t\t\\[\\zeta_1 \\left( c \\right) = \\phi_H^{-1} \\circ c \\]\n\t\tin $\\mcA_1 \\left( \\Lambda_H^i (E), \\Lambda_H^j (E') \\right)$.  \n\t\t\n\t\\end{enumerate} \n\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tWe have to show that $\\zeta_1$ is an $A_{\\infty}$-map.\n\tThis follows from the fact that the map \n\t\\[ u = \\left( \\sigma , v \\right) \\mapsto \\left( \\sigma , \\phi_H^{-1} \\circ v \\right) \\]\n\tinduces a bijection\n\t\\[\\mcM_{c_d, \\dots, c_1, c_0} \\left( \\mathbf{R} \\times \\Lambda, J, \\alpha_H \\right) \\xrightarrow{\\sim} \\mcM_{\\phi_H^{-1} (c_d) \\dots \\phi_H^{-1} (c_1), \\phi_H^{-1} (c_0)} \\left( \\mathbf{R} \\times \\Lambda_H, J_H, \\alpha \\right) . \\]\n\t\n\\end{proof}\n\n\\begin{lemma}\\label{lemma relation tau - tau_1}\n\t\n\tWe have \n\t\\[\\tau_1 = \\zeta_1 \\circ \\tau \\circ \\zeta_1^{-1} . \\]\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tThis follows from Theorem \\ref{thm invariance} using that $\\phi_H^* \\alpha_H = \\alpha$ and\n\t\\[\\left( \\left( \\Lambda_H^{n+1-t} \\right)_{i_0 \\leq n \\leq i_d}, J_H \\right) = \\left( \\left( \\phi_H^{-1} \\Lambda^{n+1-t} \\right)_{i_0 \\leq n \\leq i_d}, \\phi_H^* J \\right). \\]\n\t\n\\end{proof}\n\n\\begin{lemma}\\label{lemma mapping torus of tau = mapping torus of tau_1}\n\t\n\tThe mapping torus of $\\tau : \\mcA \\to \\mcA$ is quasi-equivalent to the mapping torus of $\\tau_1 : \\mcA_1 \\to \\mcA_1$ (see Definition \\ref{definition mapping torus}).\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tAccording to Lemma \\ref{lemma relation tau - tau_1} the following diagram of Adams-graded $A_{\\infty}$-categories is commutative \n\t\\[\\begin{tikzcd}\n\t\\mcA \\ar[d, \"\\zeta_1\"] & \\mcA \\sqcup \\mcA \\ar[l, \"\\mathrm{id} \\sqcup \\mathrm{id}\" above] \\ar[r, \"\\mathrm{id} \\sqcup \\tau\"] \\ar[d, \"\\zeta_1 \\sqcup \\zeta_1\"] & \\mcA \\ar[d, \"\\zeta_1\"] \\\\\n\t\\mcA_1 & \\mcA_1 \\sqcup \\mcA_1 \\ar[l, \"\\mathrm{id} \\sqcup \\mathrm{id}\" above] \\ar[r, \"\\mathrm{id} \\sqcup \\tau_1\"] & \\mcA_1 .\n\t\\end{tikzcd} \\]\n\tMoreover, each vertical arrow is a quasi-equivalence according to Lemma \\ref{lemma relation A - A_1}. Thus the result follows from Proposition \\ref{prop invariance of homotopy colimits}.\n\t\n\\end{proof}\n\n\\subsection{Back to the original almost complex structure}\\label{subsection back to the original acs}\n\nIn this section, we introduce an $A_{\\infty}$-category $\\mcA_2$ which has same objects and morphisms as $\\mcA_1$, but whose operations count punctured discs in $\\mathbf{R} \\times V$ which are pseudo-holomorphic for the almost complex structure induced by $\\alpha$ and $J$ (instead of $J_H$). \n\n\\subsubsection{The $A_{\\infty}$-category $\\mcA_2$}\n\nRecall that we chose a contactomorphism $\\phi_H$ as in Lemma \\ref{lemma rectify contact form}, and recall that\n\\[\\Lambda_H^{\\theta} (E) := \\phi_H^{-1} \\left( \\Lambda^{\\theta} (E) \\right), \\, \\Lambda_H^n := \\bigcup\\limits_{E \\in \\mcE} \\Lambda_H^n (E), \\, \\Lambda_H := \\bigcup\\limits_{n \\in \\mathbf{Z}} \\Lambda_H^n \\text{ and } J_H := \\phi_H^* J. \\]\n\n\\begin{defin}\n\t\n\tWe consider the $A_{\\infty}$-category $\\mcA_2$ defined as follows\n\t\\begin{enumerate}\n\t\t\n\t\t\\item the objects of $\\mcA_2$ are the Legendrians $\\Lambda_H^n (E)$, $(n, E) \\in \\mathbf{Z} \\times \\mcE$,\n\t\t\n\t\t\\item the vector space $\\mcA_2 \\left( \\Lambda_H^i (E), \\Lambda_H^j (E') \\right)$ is either generated by the $R_{\\alpha}$-chords from $\\Lambda_H^i (E)$ to $\\Lambda_H^j (E')$ if $(i,E)<(j,E')$, or $\\mathbf{F}$ if $(i,E)=(j,E')$, or $0$ otherwise, and\n\t\t\n\t\t\\item the operations are such that $1 \\in \\mcA_2 \\left( \\Lambda_H^n, \\Lambda_H^n \\right)$ is a strict unit, and for every sequence $(i_0, E_0) < \\dots < (i_d, E_d)$, for every sequence of Reeb chords \n\t\t\\[\\left( c_1, \\dots, c_d \\right) \\in \\mcR \\left( \\Lambda_H^{i_0} (E_0), \\Lambda_H^{i_1} (E_1) \\right) \\times \\dots \\times \\mcR \\left( \\Lambda_H^{i_{d-1}} (E_{d-1}), \\Lambda_H^{i_d} (E_d) \\right) \\]\n\t\twe have \n\t\t\\[\\mu_{\\mcA_2} \\left( c_1, \\dots, c_d \\right) = \\sum \\limits_{c_0 \\in \\mcR \\left( \\Lambda_H^{i_0} (E_0), \\Lambda_H^{i_d} (E_d) \\right)} \\# \\mcM_{c_d, \\dots, c_1, c_0} \\left( \\mathbf{R} \\times \\Lambda_H, J, \\alpha \\right) c_0 . \\]\n\t\t\n\t\\end{enumerate}\n\t\n\\end{defin}\n\n\\subsubsection{The quasi-autoequivalence $\\tau_2$}\n\n\\begin{defin}\\label{definition functor tau_2}\n\t\n\tWe denote by $\\tau_2 : \\mcA_1 \\to \\mcA_1$ the $A_{\\infty}$-functor defined as follows:\n\t\\begin{enumerate}\n\t\t\n\t\t\\item on objects, $\\tau_2 \\left( \\Lambda_H^n (E) \\right) = \\Lambda_H^{n+1} (E)$,\n\t\t\n\t\t\\item on morphisms, the map \n\t\t\\[\\tau_2^d : \\mcA_1 \\left( \\Lambda_H^{i_0} (E_0), \\Lambda_H^{i_1} (E_1) \\right) \\otimes \\dots \\otimes \\mcA_1 \\left( \\Lambda_H^{i_{d-1}} (E_{d-1}), \\Lambda_H^{i_d} (E_d) \\right) \\to \\mcA_1 \\left( \\Lambda_H^{i_0+1} (E_0), \\Lambda_H^{i_d+1} (E_d) \\right)  \\]\n\t\tis obtained by dualizing the components of the DG-isomorphism\n\t\t\\[CE_{-*} \\left( \\left( \\Lambda_H^{n+1} \\right)_{i_0 \\leq n \\leq i_d}, J, \\alpha \\right) \\to CE_{-*} \\left( \\left( \\Lambda_H^n \\right)_{i_0 \\leq n \\leq i_d}, J, \\alpha \\right) \\]\n\t\tinduced by the path $ \\left( \\left( \\Lambda_H^{n+1-t} \\right)_{i_0 \\leq n \\leq i_d}, J \\right)_{0 \\leq t \\leq 1}$ (see Theorem \\ref{thm invariance} or \\cite[Proposition 2.6]{EES07}). \n\t\t\n\t\\end{enumerate}\n\t\n\\end{defin}\n\n\\begin{lemma}\\label{lemma tau_2 is a quasi-equivalence}\n\t\n\tThe $A_{\\infty}$-functor $\\tau_2 : \\mcA_2 \\to \\mcA_2$ is a quasi-equivalence.\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tThis follows from the fact that $\\tau_2$ is defined by dualizing the components of a DG-isomorphism (see the proof of Lemma \\ref{lemma tau is a quasi-equivalence}). \n\t\n\\end{proof}\n\nHere the $\\mathbf{Z}$-splitting \n\\[\\mathbf{Z} \\times \\mcE \\xrightarrow{\\sim} \\mathrm{ob} \\left( \\mcA_2 \\right), \\quad (n, E) \\mapsto \\Lambda_H^n (E) \\]\nis compatible with the quasi-autoequivalence $\\tau_2$ in the sense of Definition \\ref{definition group-action}. As explained there, this turns $\\mcA_2$ into an Adams-graded $A_{\\infty}$-category. \n\n\\subsubsection{Relation between $\\left( \\mcA_1, \\tau_1 \\right)$ and $\\left( \\mcA_2, \\tau_2 \\right)$}\n\n\\begin{lemma}\\label{lemma relation A_1 - A_2}\n\t\n\tChoose a generic path $(J_t^{12})_{0 \\leq t \\leq 1}$ such that $J_0^{12} = J$ and $J_1^{12} = J_H$. \n\tThere is an $A_{\\infty}$-isomorphism $\\zeta_{12} : \\mcA_1 \\to \\mcA_2$ defined as follows\n\t\\begin{enumerate}\n\t\t\n\t\t\\item on objects, $\\zeta_{12} \\left( \\Lambda_H^n (E) \\right) = \\Lambda_H^n (E)$, \n\t\t\n\t\t\\item on morphisms, the map\n\t\t\\[\\zeta_{12} : \\mcA_1 \\left( \\Lambda_H^{i_0} (E_0), \\Lambda_H^{i_1} (E_1) \\right) \\otimes \\dots \\otimes \\mcA_1 \\left( \\Lambda_H^{i_{d-1}} (E_{d-1}), \\Lambda_H^{i_d} (E_d) \\right) \\to \\mcA_2 \\left( \\Lambda_H^{i_0} (E_0), \\Lambda_H^{i_d} (E_d) \\right) \\]\n\t\tis obtained by dualizing the components of the DG-isomorphism \n\t\t\\[CE_{-*} \\left( \\left( \\Lambda_H^n \\right)_{i_0 \\leq n \\leq i_d}, J, \\alpha \\right) \\to CE_{-*} \\left( \\left( \\Lambda_H^n \\right)_{i_0 \\leq n \\leq i_d}, J_H, \\alpha \\right) \\]\n\t\tinduced by the path $(\\left( \\Lambda_H^n \\right)_{i_0 \\leq n \\leq i_d}, J_t^{12})_{0 \\leq t \\leq 1}$ (see Theorem \\ref{thm invariance}).\n\t\t\n\t\\end{enumerate}\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tWe have to prove that $\\zeta_{12}$ is an isomorphism. This follows from the fact that it is defined by dualizing the components of a DG-isomorphism (see the proof of Lemma \\ref{lemma tau is a quasi-equivalence}). \n\t\n\\end{proof}\n\n\\begin{lemma}\\label{lemma relation tau_1 - tau_2}\n\t\n\tThe $A_{\\infty}$-functor $\\tau_2$ is homotopic to $\\zeta_{12} \\circ \\tau_1 \\circ \\zeta_{12}^{-1}$ (see \\cite[paragraph (1h)]{Sei08}). \n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tFirst recall that $\\tau_1$ is obtained by dualizing the components of the DG-map\n\t\\[CE_{-*} \\left( \\left( \\Lambda_H^{n+1} \\right)_{i_0 \\leq n \\leq i_d}, J_H, \\alpha \\right) \\to CE_{-*} \\left( \\left( \\Lambda_H^n \\right)_{i_0 \\leq n \\leq i_d}, J_H, \\alpha \\right) \\]\n\tinduced by the path $\\left( \\left( \\Lambda_H^{n+1-t} \\right)_{i_0 \\leq n \\leq i_d}, J_H \\right)_{0 \\leq t \\leq 1}$. Thus, $\\zeta_{12} \\circ \\tau_1$ is obtained by dualizing the components of the composition\n\t\\[CE_{-*} \\left( \\left( \\Lambda_H^{n+1} \\right)_{i_0 \\leq n \\leq i_d}, J, \\alpha \\right) \\to CE_{-*} \\left( \\left( \\Lambda_H^{n+1} \\right)_{i_0 \\leq n \\leq i_d}, J_H, \\alpha \\right) \\to CE_{-*} \\left( \\left( \\Lambda_H^n \\right)_{i_0 \\leq n \\leq i_d}, J_H, \\alpha \\right) . \\]\n\tOn the other hand, $\\tau_2$ is obtained by dualizing the components of the DG-map \n\t\\[CE_{-*} \\left(\\left( \\Lambda_H^{n+1} \\right)_{i_0 \\leq n \\leq i_d}, J, \\alpha \\right) \\to CE_{-*} \\left( \\left( \\Lambda_H^n \\right)_{i_0 \\leq n \\leq i_d}, J, \\alpha \\right) \\]\n\tinduced by the path $\\left( \\left( \\Lambda_H^{n+1-t} \\right)_{i_0 \\leq n \\leq i_d}, J \\right)_{0 \\leq t \\leq 1}$. Thus, $\\tau_2 \\circ \\zeta_{12}$ is obtained by dualizing the components of the composition\n\t\\[CE_{-*} \\left( \\left( \\Lambda_H^{n+1} \\right)_{i_0 \\leq n \\leq i_d}, J, \\alpha \\right) \\to CE_{-*} \\left( \\left( \\Lambda_H^n \\right)_{i_0 \\leq n \\leq i_d}, J, \\alpha \\right) \\to CE_{-*} \\left( \\left( \\Lambda_H^n \\right)_{i_0 \\leq n \\leq i_d}, J_H, \\alpha \\right) . \\]\n\tAccording to Theorem \\ref{thm invariance}, the DG-maps used to define $\\zeta_{12} \\circ \\tau_1$ and $\\tau_2 \\circ \\zeta_{12}$ are DG-homotopic. Therefore the $A_{\\infty}$-functors $\\zeta_{12} \\circ \\tau_1$ and $\\tau_2 \\circ \\zeta_{12}$ are homotopic. This concludes the proof.\n\t\n\\end{proof}\n\n\\begin{lemma}\\label{lemma mapping torus of tau_1 = mapping torus of tau_2}\n\t\n\tThe mapping torus of $\\tau_1 : \\mcA_1 \\to \\mcA_1$ is quasi-equivalent to the mapping torus of $\\tau_2 : \\mcA_2 \\to \\mcA_2$ (see Definition \\ref{definition mapping torus}).\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tLet $\\tau_{12} := \\zeta_{12} \\circ \\tau_1 \\circ \\zeta_{12}^{-1}$.\n\tConsider the following commutative diagram of Adams-graded $A_{\\infty}$-categories\n\t\\[\\begin{tikzcd}\n\t\\mcA_1 \\ar[d, \"\\zeta_{12}\"] & \\mcA_1 \\sqcup \\mcA_1 \\ar[l, \"\\mathrm{id} \\sqcup \\mathrm{id}\" above] \\ar[r, \"\\mathrm{id} \\sqcup \\tau_1\"] \\ar[d, \"\\zeta_{12} \\sqcup \\zeta_{12}\"] & \\mcA_1 \\ar[d, \"\\zeta_{12}\"] \\\\\n\t\\mcA_2 & \\mcA_2 \\sqcup \\mcA_2 \\ar[l, \"\\mathrm{id} \\sqcup \\mathrm{id}\" above] \\ar[r, \"\\mathrm{id} \\sqcup \\tau_{12}\"] & \\mcA_2 .\n\t\\end{tikzcd} \\]\n\tEach vertical arrow is a quasi-equivalence according to Lemma \\ref{lemma relation A_1 - A_2}, so it follows from Proposition \\ref{prop invariance of homotopy colimits} that the mapping torus of $\\tau_1$ is quasi-equivalent to the mapping torus of $\\tau_{12}$. \n\tNow according to Lemma \\ref{lemma relation tau_1 - tau_2}, $\\tau_{12}$ is homotopic to $\\tau_2$. Thus the result follows from Proposition \\ref{prop homotopic functors induce quasi-isomorphic homotopy colimits}.\n\t\n\\end{proof}\n\n\\subsection{Projection to $P$}\\label{subsection projection on P}\n\n\\subsubsection{The $A_{\\infty}$-category $\\mcO$}\n\nIn order to define the $A_{\\infty}$-category $\\mcO$, we need to introduce moduli spaces of pseudo-holomorphic discs in $P$. \n\n\\begin{defin}\\label{definition moduli spaces for Lagrangians}\n\t\n\tLet $\\mathbf{L} = (L^n (E))_{(n, E) \\in \\mathbf{Z} \\times \\mcE}$ be a family of mutually transverse connected compact exact Lagrangians in $(P, \\lambda)$. \n\tConsider a sequence of integers $i_0 < \\dots < i_d$, and a family of intersection points $\\left( x_0, x_1, \\dots, x_d \\right)$, where\n\t\\[x_0 \\in L^{i_0} (E_0) \\cap L^{i_d} (E_d) \\text{ and } x_k \\in L^{i_{k-1}} (E_{k-1}) \\cap L^{i_k} (E_k), \\, 1 \\leq k \\leq d. \\] \n\t\\begin{enumerate}\n\t\t\n\t\t\\item If $d=1$, we denote by $\\mcM_{x_1, x_0} \\left( \\mathbf{L}, j \\right)$ the set of equivalence classes of maps $u : \\mathbf{R} \\times \\left[ 0, 1 \\right] \\to P$ such that\n\t\t\\begin{itemize}                  \n\t\t\t\\item $u$ maps $\\mathbf{R} \\times \\{ 0 \\}$ to $L^{i_0} (E_0)$ and $\\mathbf{R} \\times \\{ 1 \\}$ to $L^{i_1 (E_1)}$,\n\t\t\t\\item $u$ satisfies the asymptotic conditions\n\t\t\t\\[u \\left( s, t \\right) \\underset{s \\to - \\infty}{\\longrightarrow} x_1 \\text{ and } u \\left( s, t \\right) \\underset{s \\to + \\infty}{\\longrightarrow} x_0, \\]\n\t\t\t\\item $u$ is $(i, j)$-holomorphic,\n\t\t\\end{itemize}\n\t\twhere two maps $u$ and $u'$ are identified if there exists $s_0 \\in \\mathbf{R}$ such that $u' (\\cdot, \\cdot) = u (\\cdot + s_0, \\cdot)$.\n\t\t\n\t\t\\item If $d \\geq 2$, we denote by $\\mcM_{x_d, \\dots, x_1, x_0} \\left( \\mathbf{L}, j \\right)$ the set of of pairs $\\left( r, u \\right)$ such that\n\t\t\\begin{itemize}\n\t\t\t\\item $r \\in \\mcR^{d+1}$ and $u : \\Delta_r \\to P$ maps the boundary arc $\\left( \\zeta_{k+1}, \\zeta_k \\right)$ of $\\Delta_r$ to $L^{i_k} (E_k)$,\n\t\t\t\\item $u$ satisfies the asymptotic conditions\n\t\t\t\\[\\left( u \\circ \\epsilon_k (r) \\right) \\left( s, t \\right) \\underset{s \\to - \\infty}{\\longrightarrow} x_k \\text{ and } \\left( u \\circ \\epsilon_0 (r) \\right) \\left( s, t \\right) \\underset{s \\to + \\infty}{\\longrightarrow} x_0, \\]\n\t\t\t\\item $u$ is $(i, j)$-holomorphic.\n\t\t\\end{itemize}\n\t\t\n\t\\end{enumerate}\n\t\n\\end{defin}\n\nRecall that we chose a contactomorphism $\\phi_H$ as in Lemma \\ref{lemma rectify contact form}.\nWe set \n\\[L_H^n := \\Pi_{P} \\left( \\Lambda_H^n (E) \\right) \\subset P \\text{ and } \\mathbf{L}_H := \\left( L_H^n (E) \\right)_{(n, E) \\in \\mathbf{Z} \\times \\mcE}. \\]\n\n\\begin{defin}\\label{definition category O}\n\t\n\tWe denote by $\\mcO$ the $A_{\\infty}$-category defined as follows:\n\t\\begin{enumerate}\n\t\t\n\t\t\\item the objects of $\\mcO$ are the Lagrangians $L_H^n (E)$, $(n, E) \\in \\mathbf{Z} \\times \\mcE$, \n\t\t\n\t\t\\item the space of morphisms from $L_H^i (E)$ to $L_H^j (E')$ is either generated by $L_H^i (E) \\cap L_H^j (E')$ if $(i, E) < (j, E')$, or $\\mathbf{F}$ if $(i, E) = (j, E')$, or $0$ otherwise, and\n\t\t\n\t\t\\item the operations are such that $1 \\in \\mcO \\left( L_H^n (E), L_H^n (E) \\right)$ is a strict unit, and for every sequence $(i_0, E_0) < \\dots < (i_d, E_d)$, for every sequence of intersection points \n\t\t\\[\\left( x_1, \\dots, x_d \\right) \\in \\left( L_H^{i_0} (E_0) \\cap L_H^{i_1} (E_1) \\right) \\times \\dots \\times \\left( L_H^{i_{d-1}} (E_{d-1}) \\cap L_H^{i_d} (E_d) \\right), \\]\n\t\twe have \n\t\t\\[\\mu_{\\mcO} \\left( x_1, \\dots, x_d \\right) = \\sum \\limits_{x_0 \\in L_H^{i_0} (E_0) \\cap L_H^{i_d} (E_d)} \\# \\mcM_{x_d, \\dots, x_1, x_0} \\left( \\mathbf{L}_H, j \\right) x_0 . \\]\n\t\t\n\t\\end{enumerate} \t\n\t\n\\end{defin}\n\n\\subsubsection{The quasi-autoequivalence $\\gamma$}\n\nBefore defining the $A_{\\infty}$-functor $\\gamma : \\mcO \\to \\mcO$, we recall Legendrian contact homology as defined in \\cite{EES07}. To each generic Legendrian $\\Lambda$ in $\\mathbf{R} \\times P$, the authors associate a semi-free DG-algebra $A = A \\left( \\Lambda, j \\right)$ generated by the self-intersection points of $\\Pi_P \\left( \\Lambda \\right)$, with a differential $\\partial : A \\to A$ defined using $j$-holomorphic discs in $P$. In our case, the differential of $A \\left( \\bigsqcup_k \\Lambda_H^k (E), j \\right)$ on a generator $x_0 \\in L_H^{i_0} (E_0) \\cap L_H^{i_d} (E_d)$ is given by \n\\[\\partial x_0 = \\sum_{\\left( x_1, \\dots, x_d \\right)} \\# \\mcM_{x_d, \\dots, x_1, x_0} \\left( \\mathbf{L}_H, j \\right) x_d \\cdots x_1  \\]\nwhere the sum is over the sequences \n\\[\\left( x_1, \\dots, x_d \\right) \\in \\left( L_H^{i_0} (E_0) \\cap L_H^{i_1} (E_1) \\right) \\times \\dots \\times \\left( L_H^{i_{d-1}} (E_{d-1}) \\cap L_H^{i_d} (E_d) \\right) . \\]\nAccording to \\cite[Theorem 2.1]{DR16}, Legendrian contact homology as defined in \\cite{EES07} coincides with the version exposed in section \\ref{section Legendrian invariants}: \n\\[A \\left( \\Lambda, j \\right) = CE_* \\left( \\Lambda, \\left( D \\Pi_P \\right)_{|\\xi}^* j, \\alpha \\right) . \\]\nWe introduced this version only because it makes clearer the fact that some operations are defined using pseudo-holomorphics polygons in the base $P$.\n\n\\begin{defin}\\label{definition functor gamma}\n\t\n\tWe denote by $\\gamma : \\mcO \\to \\mcO$ the $A_{\\infty}$-functor defined as follows\n\t\\begin{enumerate}\n\t\t\n\t\t\\item on objects, $\\gamma \\left( L_H^n (E) \\right) = L_H^{n+1} (E)$,\n\t\t\n\t\t\\item on morphisms, the map\n\t\t\\[\\gamma : \\mcO \\left( L_H^{i_0} (E_0), L_H^{i_1} (E_1) \\right) \\otimes \\dots \\otimes \\mcO \\left( L_H^{i_{d-1}} (E_{d-1}), L_H^{i_d} (E_d) \\right) \\to \\mcO \\left( L_H^{i_0 + 1} (E_0), L_H^{i_d+1} (E_d) \\right) \\]\n\t\tis obtained by dualizing the components of the DG-isomorphism\n\t\t\\begin{align*}\n\t\tA \\left( \\bigsqcup \\limits_{k=i_0}^{i_d} \\Lambda_H^{k+1}, j \\right) & = CE_{-*} \\left( \\mathbf{R} \\times \\bigsqcup \\limits_{k=i_0}^{i_d} \\Lambda_H^{k+1},  \\left( D \\Pi_P \\right)_{|\\xi}^* j, \\alpha \\right) \\\\\n\t\t & \\to CE_{-*} \\left( \\mathbf{R} \\times \\bigsqcup \\limits_{k=i_0}^{i_d} \\Lambda_H^k,  \\left( D \\Pi_P \\right)_{|\\xi}^* j, \\alpha \\right) = A \\left( \\bigsqcup \\limits_{k=i_0}^{i_d} \\Lambda_H^k, j \\right)\n\t\t\\end{align*}\n\t\tinduced by the Legendrian isotopy $\\left( \\bigsqcup \\limits_{k=i_0}^{i_d} \\Lambda_H^{k+1-t} \\right)_{0 \\leq t \\leq 1}$ (see Theorem \\ref{thm invariance}).\n\t\t\n\t\\end{enumerate}\n\t\n\\end{defin}\n\n\\begin{lemma}\\label{lemma gamma is a quasi-equivalence}\n\t\n\tThe $A_{\\infty}$-functor $\\gamma : \\mcO \\to \\mcO$ is a quasi-equivalence.\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tThis follows from the fact that $\\gamma$ is defined by dualizing the components of a DG-isomorphism (see the proof of Lemma \\ref{lemma tau is a quasi-equivalence}). \n\t\n\\end{proof}\n\nHere the $\\mathbf{Z}$-splitting \n\\[\\mathbf{Z} \\times \\mcE \\xrightarrow{\\sim} \\mathrm{ob} \\left( \\mcO \\right), \\quad (n, E) \\mapsto L_H^n (E) \\]\nis compatible with the quasi-autoequivalence $\\gamma$ in the sense of Definition \\ref{definition group-action}. As explained there, this turns $\\mcO$ into an Adams-graded $A_{\\infty}$-category. \n\n\\subsubsection{Relation between $\\left( \\mcA_2, \\tau_2 \\right)$ and $\\left( \\mcO, \\gamma \\right)$}\n\nWe now explain how the pairs $\\left( \\mcA_2, \\tau_2 \\right)$ and $\\left( \\mcO, \\gamma \\right)$ are related.\nSee Figure \\ref{figure action of the projection to P}, where we illustrate the action of the projection $\\Pi_P$ in the case \n\\[\\left( P, \\lambda \\right) = \\left( T^* S^1, p dq \\right), \\, L = 0_{S^1}, \\text{ and } H \\left( q, p \\right) = h(q), \\]\nwhere $h : S^1 \\to \\mathbf{R}$ is a Morse function.\n\n\\begin{figure}\n\t\\def1\\textwidth{1\\textwidth}\n\t\\input{Invariants_lift\/Figures\/dessin14.pdf_tex}\t\t\n\t\\caption{Action of the projection $\\Pi_{T^*S^1}$}\t\n\t\\label{figure action of the projection to P}\n\\end{figure}\n\t\n\\begin{lemma}\\label{lemma relation A_2 - O}\n\t\n\tThere is a strict $A_{\\infty}$-isomorphism $\\zeta_2 : \\mcA_2 \\to \\mcO$ defined as follows:\n\t\\begin{enumerate}\n\t\t\n\t\t\\item on objects, $\\zeta_2 \\left( \\Lambda_H^n (E) \\right) = L_H^n (E)$,\n\t\t\n\t\t\\item on morphisms, $\\zeta_2$ sends a Reeb chord $c$ in $\\mcA_2 \\left( \\Lambda_H^i (E), \\Lambda_H^j (E') \\right)$ to the intersection point \n\t\t\\[\\zeta_2 \\left( c \\right) = \\Pi_P \\left( c \\right) \\]\n\t\tin $\\mcO \\left( L_H^i (E), L_H^j (E') \\right)$.\n\t\t\n\t\\end{enumerate}\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tWe have to show that $\\zeta_2$ is an $A_{\\infty}$-map.\n\tSince $J = \\left( D \\Pi_P \\right)_{|\\xi}^* j$, it follows from \\cite[Theorem 2.1]{DR16} that the map \n\t\\[ u = \\left( \\sigma , v \\right) \\mapsto \\Pi_P \\circ v \\]\n\tinduces a bijection\n\t\\[\\mcM_{c_d, \\dots, c_1, c_0} \\left( \\mathbf{R} \\times \\Lambda_H, J, \\alpha \\right) \\xrightarrow{\\sim} \\mcM_{\\Pi_P (c_d) \\dots \\Pi_P (c_1), \\Pi_P (c_0)} \\left( \\mathbf{L}_H, j \\right) . \\]\n\tThis implies the result.\n\t\n\\end{proof}\n\n\\begin{lemma}\\label{lemma relation tau_2 - gamma}\n\t\n\tWe have\n\t\\[\\gamma = \\zeta_2 \\circ \\tau_2 \\circ \\zeta_2^{-1} . \\]\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tThis follows from the definitions of $\\tau_2$, $\\gamma$, $\\zeta_2$ and the fact that $J = \\left( D \\Pi_P \\right)_{|\\xi}^* j$.\n\t\n\\end{proof}\n\n\\begin{lemma}\\label{lemma mapping torus of tau_2 = mapping torus of gamma}\n\t\n\tThe mapping torus of $\\tau_2 : \\mcA_2 \\to \\mcA_2$ is quasi-equivalent to the mapping torus of $\\gamma : \\mcO \\to \\mcO$ (see Definition \\ref{definition mapping torus}).\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tAccording to Lemma \\ref{lemma relation tau_2 - gamma} the following diagram of Adams-graded $A_{\\infty}$-categories is commutative \n\t\\[\\begin{tikzcd}\n\t\\mcA_2 \\ar[d, \"\\zeta_2\"] & \\mcA_2 \\sqcup \\mcA_2 \\ar[l, \"\\mathrm{id} \\sqcup \\mathrm{id}\" above] \\ar[r, \"\\mathrm{id} \\sqcup \\tau_2\"] \\ar[d, \"\\zeta_2 \\sqcup \\zeta_2\"] & \\mcA \\ar[d, \"\\zeta_2\"] \\\\\n\t\\mcO & \\mcO \\sqcup \\mcO \\ar[l, \"\\mathrm{id} \\sqcup \\mathrm{id}\" above] \\ar[r, \"\\mathrm{id} \\sqcup \\gamma\"] & \\mcO .\n\t\\end{tikzcd} \\]\n\tMoreover, each vertical arrow is a quasi-equivalence according to Lemma \\ref{lemma relation A_2 - O}. Thus, the result follows from Proposition \\ref{prop invariance of homotopy colimits}.\n\t\n\\end{proof}\n\n\\subsection{Mapping torus of $\\gamma$}\\label{subsection proof of the main result}\n\nIn this section, we show that we can apply Theorem \\ref{thm mapping torus in weak situation} (Theorem \\ref{thm mapping torus in weak situation introduction} in the introduction) in order to compute the mapping torus of $\\gamma : \\mcO \\to \\mcO$. This allows us to finish the proof of Theorem \\ref{thm mainthm restatement}.  \n\nRecall that we fixed a contactomorphism $\\phi_H$ of $V$ such that $\\phi_H^* \\alpha_H = \\alpha$. Also recall that if $\\theta$ is some real number, then \n\\[\\Lambda^{\\theta} (E) = \\left\\{ \\left( f_E(x) + \\theta, x \\right) \\mid x \\in L \\right\\}, \\, \\Lambda_H^{\\theta} (E) = \\phi_H^{-1} \\left( \\Lambda^{\\theta} (E) \\right), \\text{ and } L_H^{\\theta} (E) = \\Pi_P \\left( \\Lambda_H^{\\theta} (E) \\right) . \\]\n\n\\subsubsection{Continuation elements}\n\nWe denote by $\\mcO_{2r}$ the $A_{\\infty}$-category obtained from $\\mcO$ by applying the functor of Definition \\ref{definition forgetful functor}.   \nBesides, we denote by\n\\[\\Gamma = \\left\\{ c_n (E) \\in \\mcO_{2r} (L^n (E), L^{n+1} (E)) \\mid (n,E) \\in \\mathbf{Z} \\times \\mcE \\right\\} \\]\nthe set of continuation elements in $\\mcO_{2r}$ induced by the exact Lagrangian isotopies $(L_H^{n+t})_{0 \\leq t leq 1}$ (see for example \\cite[section 3.3]{GPS20}).\n\nRecall that if $\\mcC$ is an $A_{\\infty}$-category equipped with a $\\mathbf{Z}$-splitting of $\\mathrm{ob} \\left( \\mcC \\right)$, we denote by $\\mcC^0$ the full $A_{\\infty}$-subcategory of $\\mcC$ whose set of objects corresponds to $\\{ 0 \\} \\times \\mcE$.\n\n\\begin{lemma}\\label{lemma localization of O equals CF}\n\t\n\tThere are quasi-equivalences of $A_{\\infty}$-categories\n\t\\[\\mcO_{2r}^0 \\simeq \\overrightarrow{\\mcF uk} (\\mathbf{L}_H) \\text{ and } \\mcO_{2r} \\left[ \\Gamma^{-1} \\right]^0 \\simeq \\mcF uk (\\mathbf{L}_H) . \\]\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tThis is explained in \\cite[Lecture 10]{Sei13}. See also \\cite{GPS20}, where they define Fukaya categories by localization.\n\t\n\\end{proof}\n\n\\subsubsection{The $\\mcO_{2r}$-bimodule map}\\label{subsection bimodule map}\n\nIn order to apply Theorem \\ref{thm mapping torus in weak situation}, we need a degree $0$ closed $\\mcO_{2r}$-module map $f : \\mcO_{2r} \\left( -, - \\right) \\to \\mcO_{2r} \\left( -, \\gamma \\left( - \\right) \\right)$ such that the elements in $f \\left( \\text{units} \\right)$ satisfy certain hypotheses. \nAs usual, we would like to find such an $f$ ``geometrically'', i.e. using some Lagrangian (or Legendrian) isotopy. However, here the unit $1 = e_{L_H^k (E)} \\in \\mcO \\left( L_H^k (E), L_H^k (E) \\right)$, which is not a ``geometric'' morphism, is supposed to be sent by $f$ to something in $\\mcO \\left( L_H^k (E), L_H^{k+1} (E) \\right)$, which is generated by the ``geometric'' elements of $L_H^k (E) \\cap L_H^{k+1} (E)$.\nTherefore, we need to somehow replace this unit by some intersection point between Lagrangians. The idea is that we will slightly perturb $L_H^k (E)$ to $L_H^{k+\\delta} (E)$, and then replace $e_{L_H^k (E)}$ by the continuation element in the vector space generated by $L_H^k (E) \\cap L_H^{k+\\delta} (E)$. \n\nObserve that if $\\delta$ is small enough, $L_H^{k+\\delta} (E)$ is a small perturbation of $L_H^k (E)$. Therefore, in a Weinstein neighborhood of $L_H^k (E)$, the Lagrangian $L_H^{k+\\delta} (E)$ is the graph of $d h_{\\delta, k, E}$, where $h_{\\delta, k, E}$ is some Morse function on $L (E)$.\nIn particular, the intersection points between $L_H^k (E)$ and $L_H^{k+\\delta} (E)$ correspond to the critical points of $h_{\\delta, k, E}$.\nMoreover, the continuation element in the vector space generated by $L_H^k (E) \\cap L_H^{k + \\delta} (E)$ corresponds to the sum of the minima of $h_{\\delta, k, E}$. \n\n\\begin{exa}\n\t\n\tAssume that we are in the case\n\t\\[\\left( P, \\lambda \\right) = \\left( T^* M, p dq \\right), \\, L = 0_M, \\text{ and } H \\left( q, p \\right) = h(q), \\]\n\twhere $h : M \\to \\mathbf{R}$ is a Morse function. \n\tAs explained in example \\ref{example rectify contact form in cotangent case}, in this case we have \n\t\\[L_H^{\\theta} = \\Pi_{T^*M} \\left( j^1 \\left( \\theta e^h \\right) \\right) = \\mathrm{graph} \\left( d \\left( \\theta e^h \\right) \\right) . \\]\n\tThus, $L_H^{k+\\delta}$ is the graph of $d \\left( \\delta e^h \\right)$ over $L_H^k$. \n\t\n\\end{exa}\n\nThe following result will allow us to ``replace'' the units by geometric morphisms as explained above. We denote by $g = - d\\lambda (-, j -)$ the metric on $P$ induced by $j$ and $\\left( -d \\lambda \\right)$.\n\n\\begin{lemma}\\label{lemma discs with boundary on small deformation}\n\t\n\tFor every positive integer $n$, there exists $\\delta_n > 0$ such that the following holds for every $\\delta \\in \\left] 0, \\delta_n \\right]$. For every sequence of integers\n\t\\[(-n, E_0) \\leq (j_0, E_0) < \\dots < (j_p, E_p) \\leq (\\ell_0, E_0') < \\dots < (\\ell_q, E_q') \\leq (n, E_q'), \\quad p,q \\geq 0, \\]\n\tthe rigid $j$-holomorphic discs in $P$ with boundary on \n\t\\[L_H^{j_0} (E_0) \\cup \\dots \\cup L_H^{j_p} (E_p) \\cup L_H^{\\ell_0 + \\delta} (E_0') \\cup \\dots \\cup L_H^{\\ell_q+\\delta} (E_q') \\]\n\tare \n\t\\begin{enumerate}\n\t\t\n\t\t\\item in bijection with the rigid $j$-holomorphic discs in $P$ with boundary on \n\t\t\\[L_H^{j_0} (E_0) \\cup \\dots \\cup L_H^{j_p} (E_p) \\cup L_H^{\\ell_0} (E_0') \\cup \\dots \\cup L_H^{\\ell_q} (E_q') \\]\n\t\tif $(j_p, E_p) < (\\ell_0, E_0')$, or\n\t\t\n\t\t\\item in bijection with the rigid $j$-holomorphic discs in $P$ with boundary on \n\t\t\\[L_H^{j_0} (E_0) \\cup \\dots \\cup L_H^{j_{p-1}} (E_{p-1}) \\cup L_H^{\\ell_0} (E_0') \\cup L_H^{\\ell_1} (E_1') \\cup \\dots \\cup L_H^{\\ell_q} (E_q') \\]\n\t\twith a flow line of $\\left( - \\nabla_g h_{\\delta, k, E_0'} \\right)$ attached on the component in $L_H^{\\ell_0} (E_0')$ if $(j_p, E_p) = (\\ell_0, E_0')$.\n\t\t\n\t\\end{enumerate}\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tThe case $j_p < \\ell_0$ follows from transversality of the moduli spaces in consideration. The case $j_p = \\ell_0$ follows from the main analytic theorem of \\cite{EES09} (Theorem 3.6). \n\t\n\\end{proof}\n\nIn order to define the $\\mcO_{2r}$-bimodule map $f$ properly, we will use Lemma \\ref{lemma discs with boundary on small deformation} to modify the $A_{\\infty}$-category $\\mcO_{2r}$.\nIn the following, we fix a \\emph{decreasing} sequence of positive numbers $\\left( \\delta_n \\right)_{n \\geq 1}$ such that, for every $n$,\n\\begin{enumerate}\n\t\n\t\\item Lemma \\ref{lemma discs with boundary on small deformation} holds with $\\delta_n$, and\n\t\n\t\\item $\\delta_n$ is small enough so that there is no handle slide instant in the Legendrian isotopy\n\t\\[\\bigcup_{\\ell = -n}^n \\Lambda_H^{\\ell + \\delta_n t} = \\bigcup_{\\ell = -n}^n \\bigcup_{E \\in \\mcE} \\Lambda_H^{\\ell + \\delta_n t} (E), \\quad t \\in \\left[ 0, 1 \\right]. \\]\n\t\n\\end{enumerate}\n\nWe define two families of $A_{\\infty}$-categories $\\left( \\mcO_{n, k} \\right)_{n,k}$ and $\\left( \\widetilde{\\mcO}_{n,k} \\right)_{n,k}$ indexed by the couples $(n,k)$, where $n \\geq 1$ and $-n \\leq k \\leq n$. \nThe $A_{\\infty}$-category $\\mcO_{n, k}$ is basically obtained from $\\mcO_{2r}$ by restricting to objects $L_H^i (E)$, $-n \\leq i \\leq n$, and adding a copy of the object $L_H^k (E)$.\n\n\\begin{defin}\\label{definition category O n,k}\n\t\n\tFor every $(j,E) \\in \\mathbf{Z} \\times \\mcE$, let $\\overline{L_H^j} (E)$ be a copy of $L_H^j (E)$.\n\tWe denote by $\\mcO_{n,k}$ the $A_{\\infty}$-category defined as follows:\n\t\\begin{enumerate}\n\t\t\n\t\t\\item the set of objects of $\\mcO_{n,k}$ is  \n\t\t\\[\\mathrm{ob} \\left( \\mcO_{n,k} \\right) = \\left\\{ L_H^j (E) \\mid -n \\leq j \\leq k, E \\in \\mcE \\right\\} \\cup \\left\\{ \\overline{L_H^{\\ell}} (E) \\mid k \\leq \\ell \\leq n, E \\in \\mcE \\right\\}, \\]\n\t\t\n\t\t\\item the spaces of morphisms in $\\mcO_{n,k}$ are the corresponding spaces of morphisms in $\\mcO_{2r}$ when we replace $\\overline{L_H^{\\ell}} (E)$, $k \\leq \\ell \\leq n$, by $L_H^{\\ell} (E)$, except that \n\t\t\\[\\mcO_{n,k} \\left( \\overline{L_H^k} (E), L_H^k (E) \\right) = \\{ 0 \\}, \\]\n\t\t\n\t\t\\item the operations are the same as in $\\mcO_{2r}$.\n\t\t\n\t\\end{enumerate} \t\n\t\n\\end{defin}\n\nThe $A_{\\infty}$-category $\\widetilde{\\mcO}_{n,k}$ is obtained from $\\mcO_{n,k}$ by perturbing the objects $\\overline{L_H^{\\ell}} (E)$, $k \\leq \\ell \\leq n$, to $L_H^{\\ell + \\delta_n} (E)$.\n\n\\begin{defin}\\label{definition category O n,k tilde}\n\t\n\tLet \n\t\\[\\Theta_{n,k} := \\left\\{ -n, \\dots, k \\right\\} \\cup \\left\\{ \\ell + \\delta_n \\mid k \\leq \\ell \\leq n \\right\\} \\subset \\mathbf{R}, \\text{ and } \\widetilde{\\mathbf{L}_H} := \\left( L_H^{\\theta} (E) \\right)_{(\\theta, E) \\in \\Theta_{n,k} \\times \\mcE}. \\]\n\tWe denote by $\\widetilde{\\mcO}_{n,k}$ the $A_{\\infty}$-category defined as follows:\n\t\\begin{enumerate}\n\t\t\n\t\t\\item the objects of $\\widetilde{\\mcO}_{n,k}$ are the Lagrangians $L_H^{\\theta} (E)$, $(\\theta, E) \\in \\Theta_{n,k} \\times \\mcE$,\n\t\t\n\t\t\\item the space of morphisms from $L_H^{\\theta} (E)$ to $L_H^{\\theta'} (E')$ is either generated by $L_H^{\\theta} (E) \\cap L_H^{\\theta'} (E')$ if $(\\theta, E) < (\\theta', E')$, or $\\mathbf{F}$ if $(\\theta, E) = (\\theta', E')$, or $0$ otherwise,\n\n\t\t\\item the operations are such that $e_{L_H^{\\theta} (E)} = 1 \\in \\widetilde{\\mcO}_{n,k} \\left( L_H^{\\theta} (E), L_H^{\\theta} (E) \\right)$ is a strict unit, and for every sequence $(\\theta_0, E_0) < \\dots < (\\theta_d, E_d)$, for every sequence of intersection points \n\t\t\\[\\left( x_1, \\dots, x_d \\right) \\in \\left( L_H^{\\theta_0} (E_0) \\cap L_H^{\\theta_1} (E_1) \\right) \\times \\dots \\times \\left( L_H^{\\theta_{d-1}} (E_{d-1}) \\cap L_H^{\\theta_d} (E_d) \\right), \\]\n\t\twe have \n\t\t\\[\\mu_{\\widetilde{\\mcO}_{n,k}} \\left( x_1, \\dots, x_d \\right) = \\sum \\limits_{x_0 \\in L_H^{\\theta_0} (E_0) \\cap L_H^{\\theta_d} (E_d)} \\# \\mcM_{x_d, \\dots, x_1, x_0} \\left( \\widetilde{\\mathbf{L}_H}, j \\right) x_0 . \\]\n\t\t\n\t\\end{enumerate} \t\n\t\n\\end{defin}\n\nThese $A_{\\infty}$-categories being defined, Lemma \\ref{lemma discs with boundary on small deformation} implies the following result.\n\n\\begin{lemma}\\label{lemma rho is a functor}\n\t\n\tThere is a strict $A_{\\infty}$-functor $\\rho_{n,k} : \\mcO_{n,k} \\to \\widetilde{\\mcO}_{n,k}$  defined as follows:\n\t\\begin{enumerate}\n\t\t\n\t\t\\item on objects, $\\rho_{n,k} \\left( L_H^j (E) \\right) = L_H^j (E)$ if $-n \\leq j \\leq k$ and $\\rho_{n,k} \\left( \\overline{L_H^{\\ell}} (E) \\right) = L_H^{\\ell + \\delta_n} (E)$ if $k \\leq \\ell \\leq n$,\n\t\t\n\t\t\\item on morphisms, $\\rho_{n,k}$ sends the unit of $\\mcO_{n,k} \\left( L_H^k (E), \\overline{L_H^k} (E) \\right) = \\mathbf{F}$ to the continuation element in $\\widetilde{\\mcO}_{n,k} \\left( L_H^k (E), L_H^{k + \\delta_n} (E) \\right)$, and it sends any other morphism of $\\mcO_{n,k}$ to the corresponding one in $\\widetilde{\\mcO}_{n,k}$. \n\t\t\n\t\\end{enumerate}\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tConsider a sequence $\\left( x_0, \\dots, x_{d-1} \\right)$ of morphisms in $\\mcO_{n,k}$.\n\tIf in this sequence there is no morphism from $L_H^k (E)$ to $\\overline{L_H^k} (E)$, then the relation \n\t\\[\\mu_{\\widetilde{\\mcO}_{n,k}} \\left( \\rho_{n,k} x_0, \\dots, \\rho_{n,k} x_d \\right) = \\rho_{n,k} \\left( \\mu_{\\mcO_{n,k}} \\left( x_0, \\dots, x_d \\right) \\right) \\]\n\tfollows directly from the first item of Lemma \\ref{lemma discs with boundary on small deformation}.\n\tNow assume that there is $p \\in \\{ 0, \\dots, d-1 \\}$ such that $x_p = e_{L_H^k (E)} \\in \\mcO_{n,k} \\left( L_H^k (E), \\overline{L_H^k} (E) \\right)$. Recall that the continuation element in $\\widetilde{\\mcO}_{n,k} \\left( L_H^k (E), L_H^{k + \\delta_n} (E) \\right)$ corresponds to the sum of the minima of $h_{\\delta_n, k, E}$. \n\tThen the second item of Lemma \\ref{lemma discs with boundary on small deformation} implies that \n\t\\[\\mu_{\\widetilde{\\mcO}_{n,k}} \\left( \\rho_{n,k} x_0, \\dots, \\rho_{n,k} x_d \\right) = \\left\\{\n\t\\begin{array}{ll}\n\t\\rho_{n,k} x_1 & \\text{if } d=1 \\text{ and } p=0 \\\\\n\t\\rho_{n,k} x_0 & \\text{if } d=1 \\text{ and } p=1 \\\\\n\t0 & \\text{otherwise} .\n\t\\end{array}\n\t\\right. \\]\n\tThus, the $A_{\\infty}$-relation for $\\rho_{n,k}$ is still satisfied according to the behavior of the operations $\\mu_{\\mcO_{n,k}}$ with respect to the unit $e_{L_H^k (E)}$.\n\t\n\\end{proof}\n\nNow that we have in some sense ``replace'' the unit $e_{L_H^k (E)} \\in \\mcO_{n,k} \\left( L_H^k (E), \\overline{L_H^k} (E) \\right)$ by the continuation element in $\\widetilde{\\mcO}_{n,k} \\left( L_H^k (E), L_H^{k + \\delta_n} (E) \\right)$, we can define geometrically an $A_{\\infty}$-functor that will finally allow us to define the $\\mcO_{2r}$-bimodule map $f$.\n\n\\begin{defin}\\label{definition functor nu}\n\t\n\tWe denote by $\\nu_{n,k} : \\widetilde{\\mcO}_{n, k} \\to \\mcO_{2r}$ the $A_{\\infty}$-functor defined as follows:\n\t\\begin{enumerate}\n\t\t\n\t\t\\item on objects, $\\nu_{n,k} \\left( L_H^j (E) \\right) = L_H^j (E)$ if $-n \\leq j \\leq k$, and $\\nu_{n,k} \\left( L_H^{\\ell + \\delta_n} (E) \\right) = L_H^{\\ell + 1} (E)$ if $k \\leq \\ell \\leq n$,\n\t\t\n\t\t\\item on morphisms, $\\nu_{n,k}$ is obtained by dualizing the components of the DG-isomorphism\n\t\t\\[A \\left( \\bigsqcup\\limits_{i=-n}^{n+1} \\Lambda_H^i \\right) \\xrightarrow{\\sim} A \\left( \\bigsqcup\\limits_{j=-n}^{k} \\Lambda_H^j \\sqcup \\bigsqcup\\limits_{\\ell=k}^{n} \\Lambda_H^{\\ell+\\delta_n} \\right) . \\]\n\t\tinduced by the Legendrian isotopy  \n\t\t\\[\\left( \\bigsqcup\\limits_{j=-n}^{k} \\Lambda_H^j \\right) \\sqcup \\left( \\bigsqcup\\limits_{\\ell=k}^{n} \\Lambda_H^{\\ell+1-t(1-\\delta_n)} \\right), \\quad t \\in \\left[ 0,1 \\right] \\]  \n\t\t(see Theorem \\ref{thm invariance} or \\cite[Proposition 2.6]{EES07}).\n\t\t\n\t\\end{enumerate}\n\t\n\\end{defin}\n\n\\begin{rmks}\\label{rmk properties of sigma}\n\t\n\tWe point out some properties of the $A_{\\infty}$-functors\n\t\\[\\sigma_{n,k} := \\nu_{n,k} \\circ \\rho_{n,k} : \\mcO_{n,k} \\to \\mcO_{2r} . \\]\n\t\\begin{enumerate}\n\t\t\n\t\t\\item Let $n \\leq p$ be two positive integers, and let $k \\in \\{ -n, \\dots, n \\}$. Recall that we chose $\\delta_n$ small enough so that there is no handle slide instant in the Legendrian isotopy \n\t\t\\[\\bigsqcup_{\\ell=-n}^n \\Lambda_H^{\\ell + \\delta_n t}, 0 \\leq t \\leq 1. \\]\n\t\tSince $\\delta_p \\leq \\delta_n$, neither is there any handle slide instant in the Legendrian isotopy \n\t\t\\[\\bigsqcup_{\\ell=-n}^n \\Lambda_H^{\\ell + \\delta_p t}, 0 \\leq t \\leq 1. \\]\n\t\tTherefore, $\\sigma_{p,k}$ agrees with $\\sigma_{n,k}$ on $\\mcO_{n,k} \\subset \\mcO_{p,k}$.\n\t\t\n\t\t\\item \tConsider a sequence of integers \n\t\t\\[-n \\leq j_0 < \\dots < j_p \\leq k_1 < k_2 \\leq \\ell_0 < \\dots < \\ell_q \\leq n , \\]\n\t\tand a sequence of morphisms \n\t\t\\begin{align*}\n\t\t\t& \\left( x_0, \\dots, x_{p-1}, u, y_0, \\dots, y_{q-1} \\right) \\in \\mcO_{n,k_i} \\left( L_H^{j_0} (E_0), L_H^{j_1} (E_1) \\right) \\times \\dots \\times \\mcO_{n,k_i} \\left( L_H^{j_{p-1}} (E_{p-1}), L_H^{j_p} (E_p) \\right) \\\\\n\t\t\t& \\times \\mcO_{n,k_i} \\left( L_H^{j_p} (E_p), \\overline{L_H^{\\ell_0}} (E_0') \\right) \\times \\mcO_{n,k_i} \\left( \\overline{L_H^{\\ell_0}} (E_0'), \\overline{L_H^{\\ell_1}} (E_1') \\right) \\times \\dots \\times \\mcO_{n,k_i} \\left( \\overline{L_H^{\\ell_{q-1}}} (E_{q-1}'), \\overline{L_H^{\\ell_q}} (E_q') \\right) .\n\t\t\\end{align*}\n\t\tSince the Legendrian isotopy defining $\\nu_{n,k_i}$ is \n\t\t\\[\\left( \\bigsqcup\\limits_{j=-n}^{k_i} \\Lambda_H^j \\right) \\sqcup \\left( \\bigsqcup\\limits_{\\ell=k_i}^{n} \\Lambda_H^{\\ell+1-t(1-\\delta_n)} \\right), \\quad t \\in \\left[ 0,1 \\right], \\] \n\t\twe have\n\t\t\\[\\left\\{\n\t\t\\begin{array}{l}\n\t\t\\sigma_{n, k_1} \\left( x_0, \\dots, x_{p-1} \\right) = \\delta_{1 p} x_0 \\\\\n\t\t\\sigma_{n, k_2} \\left( y_0, \\dots, y_{q-1} \\right) = \\gamma \\left( y_0, \\dots, y_{q-1} \\right) \\\\\n\t\t\\sigma_{n, k_2} \\left( x_0, \\dots, x_{p-1}, u, y_0, \\dots, y_{q-1} \\right) = \\sigma_{n, k_1} \\left( x_0, \\dots, x_{p-1}, u, y_0, \\dots, y_{q-1} \\right) .\n\t\t\\end{array}\n\t\t\\right. \\]\n\t\t\n\t\t\\item By construction, the $A_{\\infty}$-functor $\\nu_{n,k}$ sends the continuation element in $\\widetilde{\\mcO}_{n,k} \\left( L_H^k (E), L_H^{k + \\delta_n} (E) \\right)$ (corresponding to the sum of the minima of $h_{\\delta_n, k, E}$) to the continuation element $c_k (E)$ in $\\mcO_{2r} \\left( L_H^k (E), L_H^{k+1} (E) \\right)$.\n\t\tIn other words, $\\sigma_{n,k}$ sends the unit $e_{L_H^k (E)} \\in \\mcO_{n,k} \\left( L_H^k (E), \\overline{L_H^k} (E) \\right)$ to $c_k (E)$.\n\t\t\n\t\t\\item The map $\\sigma_{n,k} : \\mcO_{n,k} \\left( L_H^j(E) , \\overline{L_H^k}(E') \\right) \\to \\mcO_{2r} \\left( L_H^j(E), L_H^{k+1} (E') \\right)$ is a quasi-somorphism for every $j < k$ and $E, E' \\in \\mcE$\n\t\t\n\t\\end{enumerate}\n\t\n\\end{rmks}\n\nWe can now state and prove the desired result. \n\n\\begin{lemma}\\label{lemma bimodule map}\n\t\n\tThere exists a degree $0$ closed $\\mcO_{2r}$-bimodule map $f : \\mcO_{2r} \\left( -, - \\right) \\to \\mcO_{2r} \\left( -, \\gamma (-) \\right)$ which sends the unit $e_{L_H^k (E)} \\in \\mcO_{2r} \\left( L_H^k (E), L_H^k (E) \\right)$ to the continuation element $c_k (E) \\in \\mcO_{2r} \\left( L_H^k (E), L_H^{k+1} (E) \\right) \\cap \\Gamma$, and such that $f : \\mcO_{2r} \\left( L_H^j(E) , L_H^k(E') \\right) \\to \\mcO_{2r} \\left( L_H^j(E), L_H^{k+1} (E') \\right)$ is a quasi-somorphism for every $j< k$ and $E, E' \\in \\mcE$.\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tConsider a sequence  \n\t\\[(j_0, E_0) < \\dots < (j_p, E_p) \\leq (k ,E) = (\\ell_0, E_0') < \\dots < (\\ell_q, E_q') , \\]\n\tand a sequence of morphisms \n\t\\begin{align*}\n\t& \\left( x_0, \\dots, x_{p-1}, u, y_0, \\dots, y_{q-1} \\right) \\in \\mcO_{2r} \\left( L_H^{j_0} (E_0), L_H^{j_1} (E_1) \\right) \\times \\dots \\times \\mcO_{2r} \\left( L_H^{j_{p-1}} (E_{p-1}), L_H^{j_p} (E_p) \\right) \\\\\n\t& \\times \\mcO_{2r} \\left( L_H^{j_p} (E_p), L_H^{k} (E_0') \\right) \\times \\mcO_{2r} \\left( L_H^{k} (E_0'), L_H^{\\ell_1} (E_1') \\right) \\times \\dots \\times \\mcO_{2r} \\left( L_H^{\\ell_{q-1}} (E_{q-1}'), L_H^{\\ell_q} (E_q') \\right) .\n\t\\end{align*}\n\tWe choose $n \\geq 1$ such that $-n \\leq j_0 \\leq \\ell_q \\leq n$, and we set \n\t\\[f \\left( x_0, \\dots, x_{p-1}, u, y_0, \\dots, y_{q-1} \\right) := \\sigma_{n, k} \\left( x_0, \\dots, x_{p-1}, u, y_0, \\dots, y_{q-1} \\right) \\in \\mcO_{2r} \\left( L_H^{j_0} (E_0), \\gamma L_H^{\\ell_q} (E_q') \\right), \\]\n\twhere on the right hand side we consider that \n\t\\begin{align*}\n\t& \\left( x_0, \\dots, x_{p-1}, u, y_0, \\dots, y_{q-1} \\right) \\in \\mcO_{n,k} \\left( L_H^{j_0} (E_0), L_H^{j_1} (E_1) \\right) \\times \\dots \\times \\mcO_{n,k} \\left( L_H^{j_{p-1}} (E_{p-1}), L_H^{j_p} (E_p) \\right) \\\\\n\t& \\times \\mcO_{n,k} \\left( L_H^{j_p} (E_p), \\overline{L_H^k} (E_0') \\right) \\times \\mcO_{n,k} \\left( \\overline{L_H^k} (E_0'), \\overline{L_H^{\\ell_1}} (E_1') \\right) \\times \\dots \\times \\mcO_{n,k} \\left( \\overline{L_H^{\\ell_{q-1}}} (E_{q-1}'), \\overline{L_H^{\\ell_q}} (E_q') \\right) .\n\t\\end{align*}\n\tObserve that $f$ is well defined (it does not depend on the choice of $n$) according to the first item of Remark \\ref{rmk properties of sigma}.\n\n\tWe now verify that $f$ is closed. \n\tAccording to Definition \\ref{definition bimodule}, we have \n\t\\begin{align*}\n\t\\mu_{\\mathrm{Mod}_{\\mcC, \\mcC}}^1 & \\left( f \\right) \\left( x_0, \\dots, x_{p-1}, u, y_0, \\dots, y_{q-1} \\right) \\\\\n\t& = \\sum \\sigma_{n, k} \\left( \\dots, \\mu_{\\mcO_{2r}} \\left( \\dots \\right), \\dots, u, \\dots \\right) \\\\ \n\t& + \\sum \\sigma_{n, \\ell_s} \\left( \\dots, \\mu_{\\mcO_{2r}} \\left( x_r, \\dots, x_{p-1}, u, y_0, \\dots, y_{s-1} \\right), \\dots \\right) \\\\\n\t& + \\sum \\sigma_{n, k} \\left( \\dots, u, \\dots, \\mu_{\\mcO_{2r}} \\left( \\dots \\right), \\dots \\right) \\\\ \n\t& + \\sum \\mu_{\\mcO_{2r}} \\left( \\dots, \\sigma_{n, k} \\left( \\dots, u, \\dots \\right), \\gamma \\left( \\dots \\right), \\dots, \\gamma \\left( \\dots \\right) \\right) .\n\t\\end{align*}\n\tNow according to the second item of Remark \\ref{rmk properties of sigma}, we have \n\t\\begin{align*}\n\t\\sum \\sigma_{n, \\ell_s} & \\left( \\dots, \\mu_{\\mcO_{2r}} \\left( x_r, \\dots, x_{p-1}, u, y_0, \\dots, y_{s-1} \\right), \\dots \\right) \\\\\n\t& = \\sum \\sigma_{n, k} \\left( \\dots, \\mu_{\\mcO_{2r}} \\left( x_r, \\dots, x_{p-1}, u, y_0, \\dots, y_{s-1} \\right), \\dots \\right)\n\t\\end{align*}\n\tand \n\t\\begin{align*}\n\t\\sum & \\mu_{\\mcO_{2r}} \\left( \\dots, \\sigma_{n, k} \\left( \\dots, u, \\dots \\right), \\gamma \\left( \\dots \\right), \\dots, \\gamma \\left( \\dots \\right) \\right) \\\\\n\t& = \\sum \\mu_{\\mcO_{2r}} \\left( \\sigma_{n, k} \\left( \\dots \\right), \\dots, \\sigma_{n, k} \\left( \\dots \\right), \\sigma_{n, k} \\left( \\dots, u, \\dots \\right), \\sigma_{n, k} \\left( \\dots \\right), \\dots, \\sigma_{n, k} \\left( \\dots \\right) \\right) .\n\t\\end{align*}\n\tTherefore, we get \n\t\\[\\mu_{\\mathrm{Mod}_{\\mcC, \\mcC}}^1 \\left( f \\right) \\left( x_0, \\dots, x_{p-1}, u, y_0, \\dots, y_{q-1} \\right) = 0 \\]\n\tfrom the fact that $\\sigma_{n, k}$ is an $A_{\\infty}$-functor. \n\t\n\tNow $f$ sends the unit $e_{L_H^k (E)} \\in \\mcO_{2r} \\left( L_H^k (E), L_H^k (E) \\right)$ to the continuation element $c_k (E) \\in \\mcO_{2r} \\left( L_H^k (E), L_H^{k+1} (E) \\right) \\cap \\Gamma$ according to the third item of Remark \\ref{rmk properties of sigma}. \n\tFinally, the map $f : \\mcO_{2r} \\left( L_H^j(E) , L_H^k(E') \\right) \\to \\mcO_{2r} \\left( L_H^j(E), L_H^{k+1} (E') \\right)$ is a quasi-somorphism for every $j< k$ and $E, E' \\in \\mcE$ according to the last item of Remark \\ref{rmk properties of sigma}.\n\t\n\\end{proof}\n\n\\subsubsection{Proof of the main result}\n\nWe end the section with the proof of Theorem \\ref{thm mainthm restatement} (Theorem \\ref{thm mainthm} in the Introduction). \n\nRecall that we denote by $\\mathbf{F} \\left[ t_m \\right]$ the augmented Adams-graded associative algebra generated by a variable $t_m$ of bidegree $(m, 1)$, and by $t_m \\mathbf{F} \\left[ t_m \\right]$ its augmentation ideal (or equivalently, the ideal generated by $t_m$).\nThe key result is the following.\n\n\\begin{lemma}\\label{lemma mapping torus of gamma}\n\t\n\tThe mapping torus of $\\gamma$ is quasi-equivalent to the Adams-graded $A_{\\infty}$-category $\\overrightarrow{\\mcF uk} (\\mathbf{L}) \\oplus \\left( t_{2r} \\mathbf{F} \\left[ t_{2r} \\right] \\otimes \\mcF uk (\\mathbf{L}) \\right)$.\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tLet $f : \\mcO_{2r} \\left( -, - \\right) \\to \\mcO_{2r} \\left( -, \\gamma (-) \\right)$ be the degree $0$ closed bimodule map of Lemma \\ref{lemma bimodule map}. According to the latter, the hypotheses of Theorem \\ref{thm mapping torus in weak situation} are satisfied, and $f(\\mathrm{units}) = \\Gamma$. \n\tThus the mapping torus of $\\gamma$ is quasi-equivalent to the Adams-graded $A_{\\infty}$-algebra $\\mcO_{2r}^0 \\oplus \\left( t_{2r} \\mathbf{F} \\left[ t_{2r} \\right] \\otimes \\mcO_{2r} \\left[ \\Gamma^{-1} \\right]^0 \\right)$ (recall that if $\\mcC$ is an $A_{\\infty}$-category equipped with a $\\mathbf{Z}$-splitting $\\mathbf{Z} \\times \\mcE \\simeq \\mathrm{ob} \\left( \\mcC \\right)$, we denote by $\\mcC^0$ the full $A_{\\infty}$-subcategory of $\\mcC$ whose set of objects corresponds to $\\{ 0 \\} \\times \\mcE$).\n\tAccording to Lemma \\ref{lemma localization of O equals CF} we have\n\t\\[\\mcO_{2r}^0 \\simeq \\overrightarrow{\\mcF uk} (\\mathbf{L}_H) \\text{ and } \\mcO_{2r} \\left[ \\Gamma^{-1} \\right]^0 \\simeq \\mcF uk (\\mathbf{L}_H) . \\]\n\tThe result follows from invariance of the Fukaya category\n\t\\[\\overrightarrow{\\mcF uk} (\\mathbf{L}_H) \\simeq \\overrightarrow{\\mcF uk} (\\mathbf{L}) \\text{ and } \\mcF uk (\\mathbf{L}_H) \\simeq \\mcF uk (\\mathbf{L}) . \\]\n\t\n\\end{proof}\n\nWe now give the proof of Theorem \\ref{thm mainthm restatement} (Theorem \\ref{thm mainthm} in the introduction).\nAccording to \\cite[Theorem 2.4]{LPWZ08}, Koszul duality holds for the augmented Adams-graded DG-algebra $CE_{-*}^r \\left( \\mathbf{\\Lambda^{\\circ}} \\right)$ because it is \\emph{Adams connected} (see \\cite[Definition 2.1]{LPWZ08}). Indeed, recall from section \\ref{subsection invariants and statement of the result} that the Adams-degree in $CE_{-*}^r \\left( \\mathbf{\\Lambda^{\\circ}} \\right)$ of a Reeb chord $c$ is the number of times $c$ winds around the fiber. \nBesides, recall from section \\ref{subsection invariants and statement of the result} that there is a coaugmented Adams-graded $A_{\\infty}$-cocategory $LC_* \\left( \\mathbf{\\Lambda^{\\circ}} \\right)$ such that \n\\[CE_{-*}^r \\left( \\mathbf{\\Lambda^{\\circ}} \\right) = \\Omega \\left( LC_* \\left( \\mathbf{\\Lambda^{\\circ}} \\right) \\right) \\text{ and } LA^* \\left( \\mathbf{\\Lambda^{\\circ}} \\right) = LC_* \\left( \\mathbf{\\Lambda^{\\circ}} \\right)^{\\#}. \\]\nSince there is a quasi-isomorphism $B \\left( \\Omega C \\right) \\simeq C$ for every $A_{\\infty}$-cocategory $C$ (see \\cite[section 2.2.2]{EL21}), it follows that \n\\[E \\left( CE_{-*}^r \\left( \\mathbf{\\Lambda^{\\circ}} \\right) \\right) = B \\left( CE_{-*}^r \\left( \\mathbf{\\Lambda^{\\circ}} \\right) \\right)^{\\#} \\simeq LC_* \\left( \\mathbf{\\Lambda^{\\circ}} \\right)^{\\#} = LA^* \\left( \\mathbf{\\Lambda^{\\circ}} \\right) \\]\n(graded dual preserves quasi-isomorphisms).\nNow the quasi-equivalence \n\\[LA^* \\left( \\mathbf{\\Lambda^{\\circ}} \\right) \\simeq \\overrightarrow{\\mcF uk} (\\mathbf{L}) \\oplus \\left( t_{2r} \\mathbf{F} \\left[ t_{2r} \\right] \\otimes \\mcF uk (\\mathbf{L}) \\right) \\]\nfollows from Lemmas \\ref{lemma A^rond = mapping torus of tau}, \\ref{lemma mapping torus of tau = mapping torus of tau_1}, \\ref{lemma mapping torus of tau_1 = mapping torus of tau_2}, \\ref{lemma mapping torus of tau_2 = mapping torus of gamma} and \\ref{lemma mapping torus of gamma}.\nThis concludes the proof.\n\\section{Chekanov-Eliashberg DG-category}\\label{section Legendrian invariants}\n\nIn this section, we recall the definition of the Chekanov-Eliashberg DG-category associated to a family of Legendrians in a contact manifold equipped with a hypertight contact form $\\alpha$, which means that the Reeb vector field of $\\alpha$ has no contractible periodic orbits. . We also describe the behavior of the Chekanov-Eliashberg DG-category under change of data. \n\nIn the following, $(V, \\xi)$ is a contact manifold of dimension $(2n+1)$.\nIn order to have well-defined gradings in $\\mathbf{Z}$, we assume that $H_1 \\left( V \\right)$ is free and that the first Chern class of $\\xi$ (equipped with any compatible almost complex structure) is $2$-torsion.\nWe will need the following definition.\n\n\\begin{defin}\\label{definition chord generic}\n\t\n\tWe say that a Legendrian submanifold $\\Lambda$ in $\\left( V, \\xi \\right)$ is \\emph{chord generic} with respect to a contact form $\\alpha$ if the following holds:\n\t\\begin{enumerate}\n\t\t\n\t\t\\item for every Reeb chord $c : \\left[ 0,T \\right] \\to V$ of $\\Lambda$, the space $D \\varphi_{R_{\\alpha}}^T \\left( T_{c(0)} \\Lambda \\right)$ is transverse to $T_{c(T)} \\Lambda$ in $\\xi$,\n\t\t\n\t\t\\item different Reeb chords belong to different Reeb trajectories. \n\t\t\n\t\\end{enumerate}\n\t\n\\end{defin}\n\n\\subsection{Conley-Zehnder index}\\label{subsection Conley-Zehnder index}\n\nLet $\\alpha$ be a hypertight contact form on $(V, \\xi)$ and let $\\Lambda$ be a chord generic Legendrian submanifold of $(V, \\alpha)$.\nIn the following, we define the Conley-Zehnder index of a Reeb chord of $\\Lambda$ starting and ending on the same connected component (such chords are called \\emph{pure}).\n\nWe briefly recall what is the Maslov index of a loop in the Grassmanian of Lagrangian subspaces in $\\mathbf{C}^n$. We refer to \\cite{RS93} for a precise exposition. Fix a Lagrangian subspace $K$, and denote by $\\Sigma_k \\left( K \\right)$ the set of Lagrangian subspaces in $\\mathbf{C}^n$ whose intersection with $K$ is $k$ dimensional. Consider the \\emph{Maslov cycle} \n\\[\\Sigma = \\Sigma_1 \\left( K \\right) \\cup \\dots \\cup \\Sigma_n \\left( K \\right) . \\]\nThis is an algebraic variety of codimension one in the Lagrangian Grassmanian. Now if $\\Gamma$ is a loop in the Lagrangian Grassmanian, its Maslov index $\\mu \\left( \\Gamma \\right) \\in \\mathbf{Z}$ is the intersection number of $\\Gamma$ with $\\Sigma$. The contribution of an intersection instant $t_0$ is computed as follows. Choose a Lagrangian complement $W$ of $K$ in $\\mathbf{C}^n$. Then for each $v$ in $\\Gamma (t_0) \\cap K$, there exists a vector $w(t)$ in $W$ such that $v + w(t)$ is in $\\Gamma (t)$ for every $t$ near $t_0$. Consider the quadratic form\n\\[Q(v) = \\frac{d}{dt} \\omega \\left( v, w(t) \\right)_{|t=t_0} \\]\non $\\Gamma (t_0) \\cap K$. Without loss of generality, $Q$ can be assumed non-singular and the contribution of $t_0$ to $\\mu (\\Gamma)$ is the signature of $Q$.\n\nRecall that $H_1 \\left( V \\right)$ is assumed to be free. We choose a family $(h_1, \\dots, h_r)$ of embedded circles in $V$ which represent a basis of $H_1 ( V )$, and a symplectic trivialization of $\\xi$ over each $h_i$. If $\\gamma$ is some loop in $\\Lambda$, there is a unique family $(a_1, \\dots, a_r)$ of integers such that $\\left[ \\gamma_c - \\sum_i a_i h_i \\right]$ is zero in $H_1 (V)$. Choose a surface $\\Sigma_{\\gamma}$ in $V$ such that \n\\[ \\partial \\Sigma_{\\gamma} = \\gamma - \\sum_i a_i h_i . \\]\nThere is a unique trivialization of $\\xi$ over $\\Sigma_{\\gamma}$ which extends the chosen trivializations over $h_i$.\nThus we get a trivialization $\\gamma^{-1} \\xi \\simeq S^1 \\times \\mathbf{C}^n$ (where $n$ is the dimension of $\\Lambda$). We denote by $\\Gamma$ the loop of Lagrangian planes in $\\mathbf{C}^n$ corresponding, via the latter trivialization, to the loop $t \\mapsto T_{\\gamma (t)} \\Lambda$. The Maslov index of $\\Gamma$ does not depend on the choice of the surface $\\Sigma_{\\gamma}$ because we assumed $2 c_1 (\\xi) = 0$. This construction defines a morphism $H_1 \\left( \\Lambda, \\mathbf{Z} \\right) \\to \\mathbf{Z}$, and the \\emph{Maslov number} $m(\\Lambda)$ of $\\Lambda$ is the generator of its image. \nIn the following, we assume that the Maslov number of $\\Lambda$ is zero.  \n\nNow, let $c$ be a pure Reeb chord of $\\Lambda$. We choose a path $\\gamma_c : \\left[ 0,1 \\right] \\to \\Lambda$ which starts at the endpoint of $c$, and ends at its starting point ($\\gamma_c$ is called a \\emph{capping path} of $c$). We denote by $\\overline{\\gamma}_c$ the loop obtained by concatenating $\\gamma$ and $c$. Let $(a_1, \\dots, a_r)$ be the unique family of integers such that $\\left[ \\overline{\\gamma}_c - \\sum_i a_i h_i \\right]$ is zero in $H_1 (V)$, and choose a surface $\\Sigma_c$ in $V$ such that \n\\[ \\partial \\Sigma_c = \\overline{\\gamma}_c - \\sum_i a_i h_i . \\]\nThere is a unique trivialization of $\\xi$ over $\\Sigma_c$ which extends the chosen trivializations over $h_i$.\nThus we get a trivialization $\\overline{\\gamma}_c^{-1} \\xi \\simeq S^1 \\times \\mathbf{C}^n$ (where $n$ is the dimension of $\\Lambda$). We denote by $\\Gamma_c$ the path of Lagrangian planes in $\\mathbf{C}^n$ corresponding, via the latter trivialization, to the concatenation of $t \\mapsto T_{\\gamma (t)} \\Lambda$ and $t \\mapsto D \\varphi_{R_{\\alpha}}^t \\left( T_{c(0)} \\Lambda \\right)$. Since $\\Lambda$ is chord generic, $\\Gamma_c$ is not a loop: we close it in the following way. Let $I$ be a complex structure on $\\mathbf{C}^n$ which is compatible with the standard symplectic form on $\\mathbf{C}^n$ and such that $I \\left( \\Gamma_c (1) \\right) = \\Gamma_c (0)$. Then we let $\\overline{\\Gamma}_c$ be the loop of Lagrangian subspaces obtained by concatenating $\\Gamma_c$ and the path $t \\mapsto e^{tI} \\Gamma_c (1)$. The Conley-Zehnder index of $c$ is the Maslov index of $\\overline{\\Gamma}_c$: \n\\[CZ (c) := \\mu \\left( \\overline{\\Gamma}_c \\right). \\]\nThe Conley-Zehnder index of a Reeb chord does not depend on the choice of $\\Sigma_c$ because the first Chern class of $\\xi$ is $2$-torsion, and it does not depend on the choice of $\\gamma_c$ because the Maslov number of $\\Lambda$ vanishes.  \n\n\\subsection{Moduli spaces}\\label{subsection moduli spaces}\n\nRecall that $(V, \\xi)$ is a contact manifold such that $H_1 (V)$ is free and the first Chern class of $\\xi$ (equipped with any compatible almost complex structure) is 2-torsion. \nLet $\\alpha$ be a hypertight contact form on $(V, \\xi)$ and let $\\Lambda$ be a chord generic Legendrian submanifold of $(V, \\alpha)$ with vanishing Maslov number.\nIn the following, we introduce the moduli spaces needed to define the Chekanov-Eliashberg category of $\\Lambda$. \n\n\\begin{defin}\\label{definition Riemann disk}\n\t\n\tA Riemann $(d+1)$-pointed disk is a triple $\\left( D, \\boldsymbol{\\zeta}, j \\right)$ such that \n\t\\begin{enumerate}\n\t\t\n\t\t\\item $D$ is a smooth oriented manifold-with-boundary diffeomorphic to the closed unit disk in $\\mathbf{C}$,\n\t\t\n\t\t\\item a family $\\boldsymbol{\\zeta} = \\left( \\zeta_d, \\dots, \\zeta_1, \\zeta_0 \\right)$ is a cyclically ordered family of distinct points on $\\partial D$,\n\t\t\n\t\t\\item $j$ is an integrable almost complex structure on $D$ which induces the given orientation on $D$.\n\t\t\n\t\\end{enumerate}\n\tIf $\\left( D, \\boldsymbol{\\zeta}, j \\right)$ is a Riemann pointed disk, we denote by $\\Delta := D \\setminus \\{ \\zeta_d, \\dots, \\zeta_1, \\zeta_0 \\}$ the corresponding punctured disk.\n\t\n\\end{defin}\n\n\\begin{defin}\n\t\n\tA family of Riemann $(d+1)$-pointed discs is a bundle $\\mcS \\to \\mcR$ with \n\t\\begin{enumerate}\n\t\t\n\t\t\\item a family $\\boldsymbol{\\zeta} = \\left( \\zeta_d, \\dots, \\zeta_1, \\zeta_0 \\right)$ of non-intersecting sections $\\zeta_k : \\mcR \\to \\mcS$ and\n\t\t\n\t\t\\item a section $j : \\mcR \\to \\mathrm{End} (T \\mcS )$\n\t\t\n\t\\end{enumerate}\n\tsuch that $(\\mcS_r, \\boldsymbol{\\zeta} (r), j(r))$ is a Riemann $(d+1)$-pointed disk for every $r \\in \\mcR$.\n\t\n\\end{defin}\n\n\\begin{defin}\\label{definition choice of strip-like ends}\n\t\n\tLet $\\mcS \\to \\mcR$ be a family of Riemann $(d+1)$-pointed discs.\n\tA choice of strip-like ends for $\\mcS \\to \\mcR$ is a family of sections\n\t\\[\\epsilon_d, \\dots, \\epsilon_1 : \\mcR \\times \\mathbf{R}_{\\leq 0} \\times \\left[ 0,1 \\right] \\to \\Delta_r, \\quad \\epsilon_0 : \\mcR \\times \\mathbf{R}_{\\geq 0} \\times \\left[ 0,1 \\right] \\to \\Delta_r \\]\n\tsuch that\n\t\\begin{enumerate}\n\t\t\n\t\t\\item $\\epsilon_d (r), \\dots, \\epsilon_1 (r), \\epsilon_0 (r)$ are proper embeddings with\n\t\t\\[\\epsilon_k (r) \\left( \\mathbf{R}_{\\leq 0} \\times \\{ 0, 1 \\} \\right) \\subset \\partial \\Delta_r \\text{ and } \\epsilon_0 (r) \\left( \\mathbf{R}_{\\geq 0} \\times \\{ 0, 1 \\} \\right) \\subset \\partial \\Delta_r,  \\]\n\t\t\n\t\t\\item $\\epsilon_d (r), \\dots, \\epsilon_1 (r), \\epsilon_0 (r)$ satisfy the asymptotic conditions\n\t\t\\[\\epsilon_k (r) \\left( s, t \\right) \\underset{s \\to - \\infty}{\\longrightarrow} \\zeta_k (r) \\text{ and } \\epsilon_0 (r) \\left( s, t \\right) \\underset{s \\to + \\infty}{\\longrightarrow} \\zeta_0 (r) \\]\n\t\t\n\t\t\\item $\\epsilon_d (r), \\dots, \\epsilon_1 (r), \\epsilon_0 (r)$ are $\\left( i, j(r) \\right)$-holomorphic, where $i$ is the standard complex structure on $\\mathbf{C}$.\n\t\t\n\t\\end{enumerate}\n\t\n\\end{defin}\n\nAs explained in \\cite[section (9c)]{Sei08}, there is a universal family $\\mcS^{d+1} \\to \\mcR^{d+1}$ of Riemann $(d+1)$-pointed discs when $d \\geq 2$, which means that any other family $\\mcS \\to \\mcR$ is isomorphic to the pullback of $\\mcS^{d+1} \\to \\mcR^{d+1}$ by a map $\\mcR \\to \\mcR^{d+1}$.\nIn the following, we fix a choice of strip-like ends for the unversal family $\\mcS^{d+1} \\to \\mcR^{d+1}$.\n\n\\begin{defin}\\label{definition moduli spaces}\n\t\n\tLet $J$ be an almost complex structure on $\\xi$ compatible with $\\left( d \\alpha \\right)_{| \\xi}$. We denote by $J^{\\alpha}$ the unique almost complex structure on $\\mathbf{R}_{\\sigma} \\times V$ which sends $\\partial_{\\sigma}$ to $R_{\\alpha}$ and which restricts to $J$ on $\\xi$.\n\tLet $c_d, \\dots, c_1, c_0$ be a Reeb chords of $\\Lambda$, where $c_k : [0, T_k] \\to V$. \n\t\\begin{enumerate}\n\t\t\n\t\t\\item If $d=1$, we denote by $\\widetilde{\\mcM}_{c_1, c_0} \\left( \\mathbf{R} \\times \\Lambda, J, \\alpha \\right)$ the set of equivalence classes of maps $u : \\mathbf{R} \\times \\left[ 0, 1 \\right] \\to \\mathbf{R} \\times V$ such that\n\t\t\\begin{itemize}                  \n\t\t\t\\item $u$ maps the boundary of $\\mathbf{R} \\times \\left[ 0, 1 \\right]$ to $\\mathbf{R} \\times \\Lambda$,\n\t\t\t\\item $u$ satisfies the asymptotic conditions\n\t\t\t\\[u \\left( s, t \\right) \\underset{s \\to - \\infty}{\\longrightarrow} \\left( - \\infty, c_1 (T_1 t) \\right) \\text{ and } u \\left( s, t \\right) \\underset{s \\to + \\infty}{\\longrightarrow} \\left( + \\infty, c_0 (T_0 t) \\right), \\]\n\t\t\t\\item $u$ is $(i, J^{\\alpha})$-holomorphic,\n\t\t\\end{itemize}\n\t\twhere two maps $u$ and $u'$ are identified if there exists $s_0 \\in \\mathbf{R}$ such that $u' (\\cdot, \\cdot) = u (\\cdot + s_0, \\cdot)$.\n\t\t\n\t\t\\item If $d \\geq 2$, we denote by $\\widetilde{\\mcM}_{c_d, \\dots, c_1, c_0} \\left( \\mathbf{R} \\times \\Lambda, J, \\alpha \\right)$ the set of of pairs $\\left( r, u \\right)$ such that\n\t\t\\begin{itemize}\n\t\t\t\\item $r \\in \\mcR^{d+1}$ and $u : \\Delta_r \\to \\mathbf{R} \\times V$ maps the boundary of $\\Delta_r$ to $\\mathbf{R} \\times \\Lambda$,\n\t\t\t\\item $u$ satisfies the asymptotic conditions\n\t\t\t\\[\\left( u \\circ \\epsilon_k (r) \\right) \\left( s, t \\right) \\underset{s \\to - \\infty}{\\longrightarrow} \\left( -\\infty, c_k (T_k t) \\right) \\text{ and } \\left( u \\circ \\epsilon_0 (r) \\right) \\left( s, t \\right) \\underset{s \\to + \\infty}{\\longrightarrow} \\left( + \\infty, c_0 (T_0 t) \\right), \\]\n\t\t\t\\item $u$ is $(i, J^{\\alpha})$-holomorphic.\n\t\t\\end{itemize}\n\t\t\n\t\\end{enumerate}\n\tObserve that $\\mathbf{R}$ acts on $\\widetilde{\\mcM}_{c_d, \\dots, c_1, c_0} \\left( \\mathbf{R} \\times \\Lambda, J, \\alpha \\right)$ by translation in the $\\mathbf{R}_{\\sigma}$-coordinate. \n\tWe set\n\t\\[\\mcM_{c_d, \\dots, c_1, c_0} \\left( \\mathbf{R} \\times \\Lambda, J, \\alpha \\right) := \\widetilde{\\mcM}_{c_d, \\dots, c_1, c_0} \\left( \\mathbf{R} \\times \\Lambda, J, \\alpha \\right) \/ \\mathbf{R} . \\] \n\t\n\\end{defin}\n\nThe moduli space $\\widetilde{\\mcM}_{c_d, \\dots, c_1, c_0} \\left( \\mathbf{R} \\times \\Lambda, J, \\alpha \\right)$ can be realized as the zero-set of a section $\\overline{\\partial} : \\mcB \\to \\mcE$ of a Banach bundle $\\mcE \\to \\mcB$ (see for example \\cite{EES07}). We say that $\\widetilde{\\mcM}_{c_d, \\dots, c_1, c_0} \\left( \\mathbf{R} \\times \\Lambda, J \\right)$ is transversely cut out if $\\overline{\\partial}$ is transverse to the 0-section. \n\n\\begin{defin}\\label{definition regular almost complex structure}\n\t\n\tWe say that $J$ is \\emph{regular} (with repect to $\\alpha$ and $\\Lambda$) if the moduli spaces $\\widetilde{\\mcM}_{c_d, \\dots, c_1, c_0} \\left( \\mathbf{R} \\times \\Lambda, J, \\alpha \\right)$ are all transversely cut out.\n\t\n\\end{defin}\n\n\\begin{prop}[\\cite{DR16bis} Proposition 3.13]\\label{prop transverslity moduli spaces}\n\n\tThe set of regular almost complex structures on $\\xi$ is Baire.  \n\tMoreover, the dimension of a transversely cut out moduli space is \n\t\\[\\mathrm{dim} \\, \\widetilde{\\mcM}_{c_d, \\dots, c_1, c_0} \\left( \\mathbf{R} \\times \\Lambda, J, \\alpha \\right) = CZ \\left( a \\right) - \\left( \\sum_{k=1}^{d} CZ \\left( b_k \\right) \\right) + d-1 . \\]\n\n\\end{prop}\n\n\\subsection{Chekanov-Eliashberg DG-category}\\label{subsection Chekanov-Eliashberg algebra}\n\nRecall that $(V, \\xi)$ is a contact manifold such that $H_1 (V)$ is free and the first Chern class of $\\xi$ (equipped with any compatible almost complex structure) is 2-torsion. \nLet $\\alpha$ be a hypertight contact form on $(V, \\xi)$ and let $\\mathbf{\\Lambda} = (\\Lambda (E))_{E \\in \\mcE}$ be a family of Legendrian submanifolds of $(V, \\xi)$. We set $\\Lambda := \\bigcup_{E \\in \\mcE} \\Lambda(E)$ and we assume that $\\Lambda$ is chord generic with vanishing Maslov number. \nMoreover, we denote by $\\mcC (\\Lambda(E), \\Lambda(E'))$ the graded vector space generated by the words of Reeb chords $c_1 \\cdots c_d$ where $c_1$ starts on $\\Lambda(E)$, $c_d$ ends on $\\Lambda(E')$, and the ending component of $c_i$ is the starting component of $c_{i+1}$ for every $1 \\leq i \\leq d-1$, with grading \n\\[\\vert c_1 \\cdots c_d \\vert := \\sum_{i=1}^{d} \\left( CZ \\left( c_i \\right) - 1 \\right). \\]\nFinally, let $J$ be a regular almost complex structure on $\\xi$.\n\n\\begin{defin}\\label{definition Chekanov-Eliashberg}\n\t\n\tWe denote by $CE_* \\left( \\mathbf{\\Lambda} \\right) = CE_* \\left( \\mathbf{\\Lambda}, J, \\alpha \\right)$ the graded category defined as follows \n\t\\begin{enumerate}\n\t\t\n\t\t\\item the objects are the Legendrians $\\Lambda (E)$, $E \\in \\mcE$,\n\t\t\n\t\t\\item the space of morphisms from $\\Lambda (E)$ to $\\Lambda (E')$ is either $\\mcC (\\Lambda(E), \\Lambda(E'))$ if $E \\ne E'$, or $\\mathbf{F} \\oplus \\mcC (\\Lambda(E), \\Lambda(E'))$ if $E = E'$,\n\t\t\n\t\t\\item the composition is given by concatenation of words.\n\t\t\n\t\\end{enumerate}\n\tIf $c_0$ is a Reeb chord in $CE_* \\left( \\mathbf{\\Lambda} \\right)$, we set \n\t\\[\\partial (c_0) := \\sum \\limits_{c_d, \\dots,c_1} \\# \\mcM_{c_d, \\dots, c_1, c_0} \\left( \\mathbf{R} \\times \\Lambda, J, \\alpha \\right) c_d \\cdots c_1, \\]\n\twhere $\\# \\mcM \\in \\mathbf{F}$ denotes the number of elements modulo $2$ in $\\mcM$ if $\\mcM$ is finite, and $0$ otherwise. Finally, we extend $\\partial$ to $CE_* \\left( \\mathbf{\\Lambda} \\right)$ so that it is linear and satisfies the Leibniz rule with respect to the concatenation product.\n\t\n\\end{defin}\n\n\\begin{thm}\\label{thm d rond d egal 0}\n\t\n\t$\\partial : CE_* \\left( \\mathbf{\\Lambda} \\right) \\to CE_* \\left( \\mathbf{\\Lambda} \\right)$ decreases the grading by $1$ and satisfies $\\partial \\circ \\partial = 0$.\n\tAs a result, $(CE_{-*} \\left( \\mathbf{\\Lambda} \\right), \\partial)$ is a DG-category. \n\t\n\\end{thm}\n\n\\begin{proof}\n\t\n\tThis follows from Proposition  \\ref{prop transverslity moduli spaces}, SFT compactness (see \\cite{BEHW03} and \\cite{Abb14}, in particular \\cite[Theorem 3.20]{Abb14}) and pseudo-holomorphic gluing. See \\cite{Ekh08}, \\cite{EES05} and \\cite{EES07} for details.\n\t\n\\end{proof}\n\n\\paragraph{Augmentations and Legendrian $A_{\\infty}$-(co)category.} \n\nLet $\\mathbf{F}_{\\mcE}$ be the category with $\\mcE$ as set of objects, and morphism space from $E$ to $E'$ equal to $\\mathbf{F}$ if $E = E'$, or $0$ if $E \\ne E'$.\nAssume that we have an augmentation of $CE_{-*} \\left( \\mathbf{\\Lambda} \\right)$, i.e. a DG-functor $\\varepsilon : CE_{-*} \\left( \\mathbf{\\Lambda} \\right) \\to \\mathbf{F}_{\\mcE}$. Denote by $\\phi_{\\varepsilon}$ the automorphism of $CE_{-*} \\left( \\mathbf{\\Lambda} \\right)$ defined by \n\\[\\phi_{\\varepsilon} (c) = c + \\varepsilon (c) \\]\nfor every Reeb chord $c$ of $\\Lambda$.  We denote by $CE_{-*}^{\\varepsilon} \\left( \\mathbf{\\Lambda} \\right)$ the DG-category whose underlying graded category is the same as for $CE_{-*} \\left( \\mathbf{\\Lambda} \\right)$, but the differential is $\\partial_{\\varepsilon} = \\phi_{\\varepsilon} \\circ \\partial \\circ \\phi_{\\varepsilon}^{-1}$. Now let $\\overline{LC_*^{\\varepsilon} \\left( \\mathbf{\\Lambda} \\right)}$ be the graded pre-category (no composition) with\n\\begin{enumerate}\n\t\n\t\\item objects the Legendrians $\\Lambda (E)$, $E \\in \\mcE$,\n\t\n\t\\item space of morphisms from $\\Lambda (E)$ to $\\Lambda (E')$ generated by Reeb chords $c$ which starts on $\\Lambda(E)$ and ends on $\\Lambda(E')$, with grading \n\t\\[\\vert c \\vert := -CZ(c). \\]\n\\end{enumerate} \nObserve that, as a graded pre-category, we have\n\\[CE_{-*}^{\\varepsilon} \\left( \\mathbf{\\Lambda} \\right) = \\mathbf{F}_{\\mcE} \\oplus \\left( \\bigoplus \\limits_{d \\geq 1} \\overline{LC_*^{\\varepsilon} \\left( \\mathbf{\\Lambda} \\right)} \\left[ -1 \\right]^{\\otimes d} \\right) . \\] \nIf we write\n\\[\\left( \\partial_{\\varepsilon} \\right)_{| \\overline{LC_*^{\\varepsilon} \\left( \\mathbf{\\Lambda} \\right)} } = \\sum_{d \\geq 0} \\partial_{\\varepsilon}^d \\text{ with } \\partial_{\\varepsilon}^d : \\overline{LC_*^{\\varepsilon} \\left( \\mathbf{\\Lambda} \\right)} \\to \\overline{LC_*^{\\varepsilon} \\left( \\mathbf{\\Lambda} \\right)}^{\\otimes d}, \\]\nthen $\\partial_{\\varepsilon}^0 = \\varepsilon \\circ \\partial = 0$. Moreover, the operations $\\left( \\partial_{\\varepsilon}^d \\right)_{d \\geq 1}$ make $\\overline{LC_*^{\\varepsilon} \\left( \\mathbf{\\Lambda} \\right)}$ a (non-counital) $A_{\\infty}$-cocategory (see Definition \\ref{definition coalgebra}). We define the coaugmented $A_{\\infty}$-cocategory of $\\left( \\mathbf{\\Lambda}, \\varepsilon \\right)$ to be\n\\[LC_*^{\\varepsilon} \\left( \\mathbf{\\Lambda} \\right) := \\mathbf{F}_{\\mcE} \\oplus \\overline{LC_*^{\\varepsilon} \\left( \\mathbf{\\Lambda} \\right)} \\]\n(the $A_{\\infty}$-cooperations are naturally extended so that $1 \\in \\mathbf{F}_{\\mcE} (E, E)$, $E \\in \\mcE$ are counits).\nNow observe that, as a DG-category, \n\\[CE_{-*}^{\\varepsilon} \\left( \\mathbf{\\Lambda} \\right) = \\Omega \\left( LC_*^{\\varepsilon} \\left( \\mathbf{\\Lambda} \\right) \\right) \\]\n(see \\cite[section 2.2]{EL21} for the cobar construction).\nFinally, we define the augmented $A_{\\infty}$-category of $\\left( \\mathbf{\\Lambda}, \\varepsilon \\right)$ to be the graded dual (see \\cite[section 2.1.3]{EL21}) of $LC_*^{\\varepsilon} \\left( \\mathbf{\\Lambda} \\right)$:\n\\[LA^*_{\\varepsilon} \\left( \\mathbf{\\Lambda} \\right) = LC_*^{\\varepsilon} \\left( \\mathbf{\\Lambda} \\right)^{\\#} . \\]\n\n\\subsection{Functoriality}\\label{subsection invariance}\n\nRecall that $(V, \\xi)$ is a contact manifold such that $H_1 (V)$ is free and the first Chern class of $\\xi$ (equipped with any compatible almost complex structure) is 2-torsion. \nLet $\\mathbf{M}=(M(E))_{E \\in \\mcE}$ be a family of $n$-dimensional manifolds.\nWhen we write a map $\\mathbf{\\Lambda} : \\mathbf{M} \\to V$, we mean that $\\mathbf{\\Lambda}$ is a family of maps $\\Lambda (E) : M(E) \\to V$ indexed by $\\mcE$, and we set\n\\[\\Lambda = \\bigsqcup_{E \\in \\mcE} \\Lambda (E) : \\bigsqcup_{E \\in \\mcE} M (E) \\to V . \\]\n\n\\begin{defin}\\label{definition geometric category}\n\t\n\tLet $\\alpha$ be a hypertight contact form on $(V, \\xi)$.\n\tWe denote by $\\mcL_\\mathbf{M} (\\alpha)$ the bicategory where\n\t\\begin{enumerate}\n\t\t\n\t\t\\item objects are the pairs $(\\mathbf{\\Lambda}, J)$, where $\\mathbf{\\Lambda} : \\mathbf{M} \\to V$ is a family of Legendrian embedding such that $\\Lambda$ is chord generic with vanishing Maslov number, and $J$ is a regular almost complex structure on $\\xi$,\n\t\t\\item morphisms from $(\\mathbf{\\Lambda}_0, J_0)$ to $(\\mathbf{\\Lambda}_1, J_1)$ are the smooth paths $\\Phi = (\\mathbf{\\Lambda}_t, J_t)_{0 \\leq t \\leq 1}$ going from $(\\mathbf{\\Lambda}_0, J_0)$ to $(\\mathbf{\\Lambda}_1, J_1)$, where $\\mathbf{\\Lambda}_t : \\mathbf{M} \\to V$ is a family of Legendrian embeddings and $J_t$ is an almost complex structure on $\\xi$, \n\t\t\\item homotopies from a morphism $\\Phi = (\\mathbf{\\Lambda}_t, J_t)_{0 \\leq t \\leq 1} : (\\mathbf{\\Lambda}_0, J_0) \\to (\\mathbf{\\Lambda}_1, J_1)$ to another morphism $\\Phi' = (\\mathbf{\\Lambda}_t', J_t')_{0 \\leq t \\leq 1} : (\\mathbf{\\Lambda}_0, J_0) \\to (\\mathbf{\\Lambda}_1, J_1)$ are the smooth families $(\\mathbf{\\Lambda}_{s,t}, J_{s,t})_{0 \\leq s \\leq S, 0 \\leq t \\leq 1}$, where $\\mathbf{\\Lambda}_{s,t} : \\mathbf{M} \\to V$ is a family of Legendrian embeddings, $J_{s,t}$ is an almost complex structure on $\\xi$, and\n\t\t\\[(\\mathbf{\\Lambda}_{s,0}, J_{s,0}) = (\\mathbf{\\Lambda}_0, J_0), \\; (\\mathbf{\\Lambda}_{s,1}, J_{s,1}) = (\\mathbf{\\Lambda}_1, J_1), \\quad (\\mathbf{\\Lambda}_{0,t}, J_{0,t}) = (\\mathbf{\\Lambda}_t, J_t), \\; (\\mathbf{\\Lambda}_{S,t}, J_{S,t}) = (\\mathbf{\\Lambda}_t', J_t'). \\]\n\t\t\n\t\\end{enumerate}\n\t\t\n\\end{defin}\n\n\\begin{defin}\\label{definition action of contact isotopy}\n\t\n\tLet $\\alpha$, $\\alpha'$ be hypertight contact forms on $(V, \\xi)$, and let $\\varphi$ be a contactomorphism of $(V, \\xi)$ such that $\\varphi^* \\alpha  = \\alpha'$.\n\tIf $\\Phi = (\\mathbf{\\Lambda}_t, J_t)_{0 \\leq t \\leq 1}$ is a morphism in $\\mcL_\\mathbf{M} (\\alpha)$, we denote by \n\t\\[\\varphi^* \\Phi = (\\varphi^{-1} (\\mathbf{\\Lambda}_t), \\varphi^* J_t)_{0 \\leq t \\leq 1} \\]\n\tthe corresponding morphism in $\\mcL_\\mathbf{M} (\\alpha')$, and by \n\t\\[f_{(\\mathbf{\\Lambda}_t, J_t)}^{\\varphi} : CE_{-*} (\\mathbf{\\Lambda}_t, J_t, \\alpha) \\to CE_{-*} (\\varphi^{-1} (\\mathbf{\\Lambda}_t), \\varphi^* J_t, \\alpha') \\]\n\tthe DG-functor which sends a Reeb chord $c$ to $\\varphi^{-1} (c)$.\n\t\n\\end{defin}\n\n\\begin{defin}\\label{definition handle slide instant}\n\t\n\tLet $\\alpha$ be a hypertight contact form on $(V, \\xi)$, and let $\\Phi = (\\mathbf{\\Lambda}_t, J_t)_{0 \\leq t \\leq 1}$ be a morphism in $\\mcL_\\mathbf{M} (\\alpha)$.\n\tA \\emph{handle slide instant} in $\\Phi$ is a time $t_0$ where $\\mathbf{\\Lambda}_{t_0}$ is chord generic and has Reeb chords $c_d, \\dots, c_1, c_0$ such that the moduli space $\\widetilde{\\mcM}_{c_d, \\dots, c_1, c_0} \\left( \\mathbf{R} \\times \\Lambda_{t_0}, J_{t_0}, \\alpha \\right)$ is not transversely cut out.\n\t\n\\end{defin}\n\n\\begin{conj}\\label{conjecture functoriality}\n\t\n\tThere exist functors $\\mcF_{\\alpha}$ from $\\mcL_\\mathbf{M} (\\alpha)$ to the bicategory\\footnote{Homotopies between DG-maps are DG-homotopies, see for example \\cite[section 2.1]{PR21}.} of DG-categories such that \n\t\\begin{enumerate}\n\t\t\n\t\t\\item $\\mcF_{\\alpha}$ sends an object $( \\mathbf{\\Lambda}, J )$ to $CE_{-*} ( \\mathbf{\\Lambda}, J, \\alpha)$,\n\t\t\n\t\t\\item $\\mcF_{\\alpha}$ sends a morphism to a homotopy equivalence,\n\t\t\n\t\t\\item if $\\varphi$ is a contactomorphism of $(V, \\xi)$ such that $\\varphi^* \\alpha  = \\alpha'$ and if $\\Phi = (\\mathbf{\\Lambda}_t, J_t)_{0 \\leq t \\leq 1}$ is a morphism in $\\mcL_\\mathbf{M} (\\alpha)$, then \n\t\t\\[\\mcF_{\\alpha'} (\\varphi^* \\Phi) = f_{(\\mathbf{\\Lambda}_1, J_1)}^{\\varphi} \\circ \\mcF_{\\alpha} (\\Phi) \\circ (f_{(\\mathbf{\\Lambda}_0, J_0)}^{\\varphi})^{-1} \\]\n\t\t\n\t\t\\item if $(\\varphi_t)_{0 \\leq t \\leq 1}$ is a contact isotopy of $(V, \\xi)$ satisfying $\\varphi_t^* \\alpha  = \\alpha'$ for every $t$, and if $(\\mathbf{\\Lambda}, J)$ is an object of $\\mcL_\\mathbf{M} (\\alpha)$ such that there is neither birth\/death of Reeb chords nor handle slide instants in the path $\\Phi' = (\\varphi_t^{-1} (\\mathbf{\\Lambda}), \\varphi_t^* J)$, then \n\t\t\\[\\mcF_{\\alpha'} (\\Phi') = f_{(\\mathbf{\\Lambda}, J)}^{\\varphi_1} \\circ (f_{(\\mathbf{\\Lambda}, J)}^{\\varphi_0})^{-1} . \\]\n\t\t\t\t\n\t\\end{enumerate}\n\t\n\\end{conj}\n\nWe chose to state the latter result as a conjecture out of caution, even if some parts of it have already been proved, or at least detailed strategies of proofs have been given. The existence of such functors at the category level (without homotopies) has been completely established in the case $(V, \\alpha) = (\\mathbf{R} \\times P, dz - \\lambda)$, see \\cite[section 2.4]{EES07}. Detailed strategies of proofs for the general case can be found in \\cite[section 4]{Ekh08} and \\cite[section 5]{EO17}.\n \nNote that I proved a weaker version of this result in my thesis by generalizing methods of \\cite{EES05} and \\cite{PR21}.\nThe following is the only version of Conjecture \\ref{conjecture functoriality} that we will use in this paper. \n\n\\begin{thm}[\\cite{Pet22} Theorem 3.8]\\label{thm invariance}\n\t\n\tConjecture \\ref{conjecture functoriality} is true if we replace the categories $\\mcL_\\mathbf{M} (\\alpha)$ by the subcategories $\\mcL_\\mathbf{M}^0 (\\alpha)$ where\n\t\\begin{enumerate}\n\t\t\n\t\t\\item objects are the pairs $(\\mathbf{\\Lambda}, J)$ such that $\\Lambda$ has finitely many Reeb chords,\n\t\t\n\t\t\\item morphisms from $(\\mathbf{\\Lambda}_0, J_0)$ to $(\\mathbf{\\Lambda}_1, J_1)$ are the families $\\Phi = (\\mathbf{\\Lambda}_t, J_t)_{0 \\leq t \\leq 1}$ such that $\\Lambda_t$ is chord generic and has finitely many Reeb chords for every $t$,\n\t\t\n\t\t\\item homotopies from a morphism $\\Phi = (\\mathbf{\\Lambda}_t, J_t)_{0 \\leq t \\leq 1} : (\\mathbf{\\Lambda}_0, J_0) \\to (\\mathbf{\\Lambda}_1, J_1)$ to another morphism $\\Phi' = (\\mathbf{\\Lambda}_t', J_t')_{0 \\leq t \\leq 1} : (\\mathbf{\\Lambda}_0, J_0) \\to (\\mathbf{\\Lambda}_1, J_1)$ are the families $(\\mathbf{\\Lambda}_{s,t}, J_{s,t})_{0 \\leq s \\leq S, 0 \\leq t \\leq 1}$ such that $\\Lambda_{s,t}$ is chord generic and has finitely many Reeb chords for every $s, t$.\n\t\t\n\t\\end{enumerate}\n\t\n\\end{thm}\n\n\\begin{rmk}\n\n\tWe expect that the finiteness of Reeb chords condition in Theorem \\ref{thm invariance} (which is very restrictive) can be easily dropped using (homotopy) colimits of DG-categories diagrams. \n\tOn the other hand, studying birth\/death of Reeb chords phenomena is a more serious issue that we will address in a future work.\n\n\\end{rmk}\n\\section{Mapping torus of an $A_{\\infty}$-autoequivalence}\\label{subsection mapping torus}\n\nIn this section, we introduce the notion of mapping torus for a quasi-autoequivalence of an $A_{\\infty}$-category, by analogy with the mapping torus associated to an automorphism of a topological space. This terminology was also used in \\cite{Kar21}, but we do not know if the two notions coincide. The two main theorems of this section allow us to compute this mapping torus under different hypotheses.  \n\n\\begin{rmk}\n\t\n\tIn this section, $A_{\\infty}$-categories are always assumed to be \\emph{strictly unital} (see \\cite[paragraph (2a)]{Sei08}).\n\t\n\\end{rmk}\n\n\\subsection{Definitions and main results}\n\n\\subsubsection{Definitions}\n\n\\begin{defin}\\label{definition mapping torus}\n\t\n\tLet $\\tau$ be a quasi-autoequivalence of an Adams-graded $A_{\\infty}$-category $\\mcA$. \n\tThe mapping torus of $\\tau$ is the $A_{\\infty}$-category\n\t\\[\\mathrm{MT} (\\tau) := \\mathrm{hocolim} \\left( \n\t\\begin{tikzcd}\n\t\\mcA \\sqcup \\mcA \\ar[r, \"\\mathrm{id} \\sqcup \\tau\"] \\ar[d, \"\\mathrm{id} \\sqcup \\mathrm{id}\" left] & \\mcA \\\\\n\t\\mcA\n\t\\end{tikzcd} \\right) \\]\n\t(see Definition \\ref{definition Grothendieck construction}).\n\t\n\\end{defin}\n\n\\begin{rmks}\n\t\n\t\\begin{enumerate}\n\t\t\n\t\t\\item We use the terminology ``mapping torus'' by analogy with the analogous situation in the category of topological spaces. Indeed, if $f$ is an automorphism of some topological space $X$, then the mapping torus of $f$\n\t\t\\[M_f = \\left( X \\times \\left[ 0,1 \\right] \\right) \/ \\left( \\left( x, 0 \\right) \\sim \\left( f(x), 1 \\right) \\right) \\]\n\t\tis the homotopy colimit of the following diagram\n\t\t\\[\\begin{tikzcd}\n\t\t\tX \\sqcup X \\ar[r, \"\\mathrm{id} \\sqcup f\"] \\ar[d, \"\\mathrm{id} \\sqcup \\mathrm{id}\" left] & X \\\\\n\t\t\tX .\n\t\t\\end{tikzcd} \\]\n\t\n\t\t\\item The terminology ``mapping torus of an autoequivalence of $A_{\\infty}$-categories'' also appears in \\cite{Kar21}, and it is used in \\cite{Kar21bis} in order to distinguish open symplectic mapping tori. We do not know if the two notions (the one of \\cite{Kar21} and the one of Definition \\ref{definition mapping torus}) coincide.\n\t\t\n\t\t\\item The mapping torus of a quasi-autoequivalence is also Adams-graded, because it is the localization of an Adams-graded $A_{\\infty}$-category at morphisms of Adams-degree $0$.  \n\t\t\n\t\\end{enumerate}\n\t\n\\end{rmks}\n\n\\begin{defin}\\label{definition group-action}\n\t\n\tLet $\\mcA$ be an $A_{\\infty}$-category. A $\\mathbf{Z}$-splitting of $\\mathrm{ob} \\left( \\mcA \\right)$ is a bijection \n\t\\[\\mathbf{Z} \\times \\mcE \\xrightarrow{\\sim} \\mathrm{ob} \\left( \\mcA \\right), \\quad \\left( n, E \\right) \\mapsto X^n \\left( E \\right) . \\]\n\tIf such a splitting has been chosen, we define the Adams-grading of an homogeneous element $x \\in \\mcA \\left( X^i (E), X^j (E) \\right)$ to be $(j-i)$. This turns $\\mcA$ into an Adams-graded $A_{\\infty}$-category.\n\t\n\tLet $\\tau$ be a quasi-autoequivalence of $\\mcA$.\n\tWe say that a $\\mathbf{Z}$-splitting of $\\mathrm{ob} \\left( \\mcA \\right)$ is compatible with $\\tau$ if \n\t\\[\\tau \\left( X^n \\left( E \\right) \\right) = X^{n+1} \\left( E \\right) \\]\n\tfor every $n \\in \\mathbf{Z}$ and $E \\in \\mcE$. \n\t\n\tWe say that $\\mcA$ is weakly directed with respect to a $\\mathbf{Z}$-splitting of $\\mathrm{ob} \\left( \\mcA \\right)$ if \n\t\\[\\mcA \\left( X^i (E), X^j (E') \\right) = 0 \\]\n\tfor every $i > j$ and $E, E' \\in \\mcE$ (we use the term ``weakly directed'' $A_{\\infty}$-category because the notion is slightly more general than that of directed $A_{\\infty}$-category defined by Seidel in \\cite[paragraph (5m)]{Sei08}).\n\t\n\\end{defin}\n\n\\begin{rmk}\n\t\n\tCompatible $\\mathbf{Z}$-splittings naturally arise in the context of $\\mathbf{Z}$-actions. \n\tA strict $\\mathbf{Z}$-action on an $A_{\\infty}$-category $\\mcA$ is a family of $A_{\\infty}$-endofunctors $\\left( \\tau_n \\right)_{n \\in \\mathbf{Z}}$ such that $\\tau_0 = \\mathrm{id}_{\\mcA}$ and $\\tau_{i+j} = \\tau_i \\circ \\tau_j$ (see \\cite[paragraph (10b)]{Sei08}). \n\tIf the induced $\\mathbf{Z}$-action on $\\mathrm{ob} \\left( \\mcA \\right)$ is free, then any section $\\sigma$ of the projection $\\mathrm{ob} ( \\mcA ) \\to \\mcE$, where $\\mcE$ is the set of equivalence classes of objects in $\\mcA$ under the $\\mathbf{Z}$-action, gives a $\\mathbf{Z}$-splitting \n\t\\[\\mathbf{Z} \\times \\mcE \\xrightarrow{\\sim} \\mathrm{ob} \\left( \\mcA \\right), \\quad \\left( n, E \\right) \\mapsto \\tau_n \\left( \\sigma \\left( E \\right) \\right) \\]\n\twhich is compatible with the automorphism $\\tau_1$. \n\t\n\\end{rmk}\n\n\\subsubsection{Main results}\n\n\\paragraph{First result.}\n\n\\begin{defin}\\label{definition category of coinvariants}\n\t\n\tLet $\\tau$ be a quasi-autoequivalence of an $A_{\\infty}$-category $\\mcA$ equipped with a compatible $\\mathbf{Z}$-splitting of $\\mathrm{ob} \\left( \\mcA \\right)$.\n\tAssume that $\\tau$ is strict, i.e $\\tau^d=0$ for $d \\geq 2$, and acts bijectively on hom-sets. \n\tIn this case, we define an Adams-graded $A_{\\infty}$-category $\\mcA_{\\tau}$ as follows:\n\t\\begin{enumerate}\n\t\t\n\t\t\\item the set of objects of $\\mcA_{\\tau}$ is $\\mcE$,\n\t\t\n\t\t\\item the space of morphisms $\\mcA_{\\tau} \\left( E, E' \\right)$ is the Adams-graded vector space given by\n\t\t\\[\\mcA_{\\tau} \\left( E, E' \\right) = \\left( \\bigoplus \\limits_{i,j \\in \\mathbf{Z}} \\mcA \\left( X^i \\left( E \\right), X^j \\left( E' \\right) \\right) \\right) \/ \\left( \\tau (x) \\sim x \\right) \\]\n\t\t\n\t\t\\item the operations are the unique linear maps such that for every sequence \n\t\t\\[ \\left( x_0, \\dots, x_{d-1} \\right) \\in \\mcA \\left( X^{i_0} \\left( E_0 \\right), X^{i_1} \\left( E_1 \\right) \\right) \\times \\dots \\times \\mcA \\left( X^{i_{d-1}} \\left( E_{d-1} \\right), X^{i_d} \\left( E_d \\right) \\right), \\]\n\t\twe have \n\t\t\\[\\mu_{\\mcA_{\\tau}} \\left( [ x_0 ], \\dots, [ x_{d-1} ] \\right) = \\left[ \\mu_{\\mcA} \\left( x_0, \\dots, x_{d-1} \\right) \\right], \\]\n\t\twhere $[\\cdot] : \\mcA \\left( X^i \\left( E \\right), X^j \\left( E' \\right) \\right) \\to \\mcA_{\\tau} \\left( E, E' \\right)$ denotes the projection. (It is not hard to see that such operations exist and satisfy the $A_{\\infty}$-relations.) \n\t\t\n\t\\end{enumerate}\n\t\n\\end{defin}\n\n\\begin{thm}\\label{thm mapping torus in strict situation}\n\n\tLet $\\tau$ be a quasi-autoequivalence of an $A_{\\infty}$-category $\\mcA$ equipped with a compatible $\\mathbf{Z}$-splitting of $\\mathrm{ob} \\left( \\mcA \\right)$.\n\tAssume that $\\tau$ is strict and acts bijectively on hom-sets.\n\tThen there is a quasi-equivalence of Adams-graded $A_{\\infty}$-categories \n\t\\[\\mathrm{MT} (\\tau) \\simeq \\mcA_{\\tau}. \\] \n\t\n\\end{thm}\n\n\\begin{rmk}\n\t\n\tThe $A_{\\infty}$-category $\\mcA_{\\tau}$ is the (ordinary) colimit of the diagram used to define $\\mathrm{MT} (\\tau)$. Thus, Theorem \\ref{thm mapping torus in strict situation} can be thought of as a ``homotopy colimit equals colimit\" result.\n\t\n\\end{rmk}\n\n\\paragraph{Second result.}\n\nWe denote by $\\mathbf{F} \\left[ t_m \\right]$ the augmented Adams-graded associative algebra generated by a variable $t_m$ of bidegree $(m, 1)$.\nObserve that if $\\mcC$ is a subcategory of an $A_{\\infty}$-category $\\mcD$ with $\\mathrm{ob} (\\mcC) = \\mathrm{ob} (\\mcD)$, then $\\mcC \\oplus (t \\mathbf{F} [t] \\otimes \\mcD)$ is naturally an Adams-graded $A_{\\infty}$-category, where the Adams degree of $t^k \\otimes x$ equals $k$.\nBesides, if $\\mcC$ is an $A_{\\infty}$-category equipped with a $\\mathbf{Z}$-splitting of $\\mathrm{ob} \\left( \\mcC \\right)$, we denote by $\\mcC^0$ the full $A_{\\infty}$-subcategory of $\\mcC$ whose set of objects corresponds to $\\{ 0 \\} \\times \\mcE$.\nFinally, we use the functor $\\mcC \\mapsto \\mcC_m$ of Definition \\ref{definition forgetful functor}. \n\n\\begin{thm}[Theorem \\ref{thm mapping torus in weak situation introduction} in the Introduction]\\label{thm mapping torus in weak situation}\n\t \n\tLet $\\tau$ be a quasi-autoequivalence of an $A_{\\infty}$-category $\\mcA$, weakly directed with respect to some compatible $\\mathbf{Z}$-splitting of $\\mathrm{ob} \\left( \\mcA \\right)$.\n\tAssume that there exists a closed degree $0$ bimodule map $f : \\mcA_m \\left( -, - \\right) \\to \\mcA_m \\left( -, \\tau(-) \\right)$ such that $f : \\mcA_m \\left( X^i(E) , X^j(E') \\right) \\to \\mcA_m \\left( X^i(E), X^{j+1} (E') \\right)$ is a quasi-somorphism for every $i < j$ and $E, E' \\in \\mcE$.\n\tThen there is a quasi-equivalence of Adams-graded $A_{\\infty}$-categories \n\t\\[\\mathrm{MT} (\\tau) \\simeq \\mcA_m^0 \\oplus \\left( t_m \\mathbf{F} \\left[ t_m \\right] \\otimes \\mcA_m \\left[ f \\left( \\mathrm{units} \\right)^{-1} \\right]^0 \\right) . \\]\n\t\n\\end{thm}\n\n\\paragraph{Outline of the section} \n\nIn section \\ref{subsection resuts about a specific module}, we consider an $A_{\\infty}$-category $\\mcA$ equipped with a $\\mathbf{Z}$-splitting of $\\mathrm{ob} \\left( \\mcA \\right)$ and a choice of a closed degree $0$ morphism $c_n (E) \\in \\mcA \\left( X^n (E), X^{n+1} (E) \\right)$ for every $n \\in \\mathbf{Z}$ and every $E \\in \\mcE$. We give technical results about specific modules associated to this data. This will be used in the proof of Theorem \\ref{thm mapping torus in weak situation} with $c_n \\left( E \\right) = f \\left( e_{X^n(E)} \\right)$.\n\nIn section \\ref{subsection category and modules for the mapping torus}, we consider the Grothendieck construction $\\mcG$ of a slightly different diagram than the one in Definition \\ref{definition mapping torus}, together with a set $W_{\\mcG}$ of closed degree 0 morphisms. The idea is that the localization $\\mcH = \\mcG \\left[ W_{\\mcG}^{-1} \\right]$ is the homotopy colimit of a diagram obtained from the one in Definition \\ref{definition mapping torus} by a cofibrant replacement of the diagonal functor $\\mcA \\sqcup \\mcA \\to \\mcA$. Thus it is not surprising that $\\mcH$ is quasi-equivalent to the mapping torus of $\\tau$.\nMoreover, we prove technical results about specific modules over $\\mcG$ that will be used in the proofs of Theorems \\ref{thm mapping torus in strict situation} and \\ref{thm mapping torus in weak situation}. \n  \nIn section \\ref{subsection proof of the first result}, we prove Theorem \\ref{thm mapping torus in strict situation}. We first define an $A_{\\infty}$-functor $\\Phi : \\mcG \\to \\mcA_{\\tau}$ which sends $W_{\\mcG}$ to the set of units in $\\mcA_{\\tau}$. Then we prove that the induced $A_{\\infty}$-functor $\\widetilde{\\Phi} : \\mcH \\to \\mcA_{\\tau} \\left[ \\{ \\text{units} \\}^{-1} \\right]$ is a quasi-equivalence. To do that, our strategy is to apply Proposition \\ref{prop quasi-isomorphism between localizations} using the results of section \\ref{subsection category and modules for the mapping torus} about the specific $\\mcG$-modules. \n\nIn section \\ref{subsection proof of the second result}, we prove Theorem \\ref{thm mapping torus in weak situation}. We use the fact that $\\mcG$ is ``big enough'' (there are more objects and morphisms than in the Grothendieck construction of the diagram in Definition \\ref{definition mapping torus}) in order to define an $A_{\\infty}$-functor $\\Psi_m : \\mcG_m \\to \\mcA_m$ (see Definition \\ref{definition forgetful functor}). This induces an $A_{\\infty}$-functor $\\widetilde{\\Psi} : \\mcH \\to \\mathbf{F} \\left[ t_m \\right] \\otimes \\mcA_m \\left[ f \\left( \\{ \\text{units} \\} \\right)^{-1} \\right]$. Then we prove that for every Adams degree $j \\geq 1$, and for every objects $X,Y$ in $\\mcH$, the map\n\\[\\widetilde{\\Psi} : \\mcH \\left( X,Y \\right)^{*,j} \\to \\left( \\mathbf{F} \\left[ t_m \\right] \\otimes \\mcA_m \\left[ f \\left( \\{ \\text{units} \\} \\right)^{-1} \\right] \\right) \\left( \\Psi X , \\Psi Y \\right)^{*,j} \\]\nis a quasi-isomorphism (if $V$ is an Adams-graded vector space, $V^{*,j}$ denotes the subspace of Adams degree $j$ elements).\nTo do that, we apply once again Proposition \\ref{prop quasi-isomorphism between localizations} using the results of sections \\ref{subsection resuts about a specific module} and \\ref{subsection category and modules for the mapping torus} about the specific modules over $\\mcA_m$ and $\\mcG$ respectively.\nThis allows us to finish the proof of Theorem \\ref{thm mapping torus in weak situation}.\n\n\\subsection{Results about specific modules}\\label{subsection resuts about a specific module}\n\nIn this section, we give technical results that will allow us to apply Proposition \\ref{prop quasi-isomorphism between localizations} in the proof of Theorem \\ref{thm mapping torus in weak situation}.\n\nLet $\\mcA$ be an $A_{\\infty}$-category equipped with a $\\mathbf{Z}$-splitting of $\\mathrm{ob} \\left( \\mcA \\right)$.\nAssume that we chose, for every $n \\in \\mathbf{Z}$ and every $E \\in \\mcE$, a closed degree $0$ morphism $c_n (E) \\in \\mcA \\left( X^n (E), X^{n+1} (E) \\right)$. Moreover, assume that we chose a set $W_{\\mcA}$ of closed degree $0$ morphisms which contains the morphisms $c_n(E)$.\n\n\\begin{rmk}\n\t\n\tAccording to Definition \\ref{definition group-action}, the $\\mathbf{Z}$-splitting of $\\mathrm{ob} \\left( \\mcA \\right)$ naturally induces an Adams-grading on $\\mcA$. However in this section, we do not consider $\\mcA$ as being Adams-graded. \n\t\n\\end{rmk}\n\nIn the following, we fix some element $E_{\\diamond} \\in \\mcE$. When we write an object $X^n$ or a morphism $c_n$ without specifying the element of $\\mcE$, we mean $X^n (E_{\\diamond})$ or $c_n (E_{\\diamond})$ respectively. \nRecall that $t_{c_n} : \\mcA \\left( -, X^n \\right) \\to \\mcA \\left( -, X^{n+1} \\right)$ denotes the $\\mcA$-module map induced by $c_n \\in \\mcA \\left( X^n, X^{n+1} \\right)$ (see Definition \\ref{definition Yoneda functor}). \n\n\\begin{defin}\\label{definition specific module}\n\t\n\tWe set $\\mcM_{\\mcA}$ to be the $\\mcA$-module\n\t\\[ \\mcM_{\\mcA} := \\left[\n\t\\begin{tikzcd}\n\t\\dots \\ar[rd, \"t_{c_{-1}}\"] & \\mcA \\left( - , X^0 \\right) \\ar[d, \"\\mathrm{id}\"] \\ar[rd, \"t_{c_0}\"] & \\mcA \\left( - , X^1 \\right) \\ar[d, \"\\mathrm{id}\"] \\ar[rd, \"t_{c_1}\"] & \\dots \\\\\n\t\\dots & \\mcA \\left( - , X^0 \\right) & \\mcA \\left( - , X^1 \\right) & \\dots\n\t\\end{tikzcd} \n\t\\right] \\]\n\t(see Definition \\ref{definition cone of a morphism between modules}).\n\tBesides, we set $t_{\\mcA}^n : \\mcA \\left( - , X^n \\right) \\to \\mcM_{\\mcA}$ to be the $\\mcA$-module inclusion for every $n \\in \\mathbf{Z}$.\n\t\n\\end{defin}\n\nThe first result highlights a key property of the module $\\mcM_{\\mcA}$.\n\n\\begin{lemma}\\label{lemma multiplication by continuation element becomes homotopic to inclusion}\n\t\n\tFor every $n \\in \\mathbf{Z}$, the closed $\\mcA$-module map $t_{\\mcA}^{n+1} \\circ t_{c_n} : \\mcA \\left( -, X^n \\right) \\to \\mcM_{\\mcA}$ is homotopic to $t_{\\mcA}^n : \\mcA \\left( -, X^n \\right) \\to \\mcM_{\\mcA}$. \n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tConsider the degree $(-1)$ strict $\\mcA$-module map $s : \\mcA \\left( -, X^n \\right) \\to \\mcM_{\\mcA}$ which sends a morphism in $\\mcA \\left( X, X^n \\right)$ to the corresponding shifted element in $\\mcA \\left( X, X^n \\right) \\left[ 1 \\right]$.\n\tThen an easy computation gives\n\t\\[\\mu_{\\mathrm{Mod}_{\\mcA}}^1 \\left( s \\right) = t_{\\mcA}^{n+1} \\circ t_{c_n} + t_{\\mcA}^n . \\]\n\tThis concludes the proof. \n\t\n\\end{proof}\n\nIn the proof of the two results below, we will use specific $\\mcA$-modules. If $p$ is a fixed non negative integer, we set \n\\[K_p = \\left[\n\\begin{tikzcd}\n\\dots \\ar[rd, \"t_{c_{p-2}}\"] & \\mcA \\left( - , X^{p-1} \\right) \\ar[d, \"\\mathrm{id}\"] \\ar[rd, \"t_{c_{p-1}}\"] & \\mcA \\left( -  , X^p \\right) \\ar[d, \"\\mathrm{id}\"] \\\\\n\\dots & \\mcA \\left( - , X^{p-1} \\right) & \\mcA \\left( -  , X^p \\right)\n\\end{tikzcd} \n\\right] \\]\nand\n\\[\\widetilde{K}_p = \\left[\n\\begin{tikzcd}\n\\mcA \\left( -  , X^p \\right) \\ar[rd, \"t_{c_p}\"] & \\mcA \\left( -  , X^{p+1} \\right) \\ar[d, \"\\mathrm{id}\"] \\ar[rd, \"t_{c_{p+1}}\"] & \\dots \\\\\n& \\mcA \\left( -  , X^{p+1} \\right) & \\dots \n\\end{tikzcd} \n\\right] . \\]\nMoreover, we will consider the sequences of $\\mcA$-modules $\\left( F_p^q \\right)_{q \\geq 0}$, $\\left( \\widetilde{F}_p^q \\right)_{q \\geq 0}$ starting at $F_p^0 = \\widetilde{F}_p^0 = 0$ and with\n\\[ F_p^q = \\left[\n\\begin{tikzcd}\n\\mcA \\left( - , X^{p-q+1} \\right) \\ar[d, \"\\mathrm{id}\"] \\ar[rd, \"t_{c_{p-q+1}}\"] & \\dots \\ar[rd, \"t_{c_{p-1}}\"] & \\mcA \\left( - , X^p \\right) \\ar[d, \"\\mathrm{id}\"] \\\\\n\\mcA \\left( - , X^{p-q+1} \\right) & \\dots & \\mcA \\left( - , X^p \\right)\n\\end{tikzcd}\n\\right] \\]\nand \n\\[\\widetilde{F}_p^q = \\left[\n\\begin{tikzcd}\n\\mcA \\left( - , X^p \\right) \\ar[rd, \"t_{c_p}\"] & \\dots  \\ar[rd, \"t_{c_{p+q-1}}\"] & \\\\\n& \\dots & \\mcA \\left( - , X^{p+q} \\right)\n\\end{tikzcd} \n\\right] \\]\nfor $q \\geq 1$.\n\nThe following Lemma is mostly technical. It will be used in the proofs of Lemmas \\ref{lemma second result for specific module} and \\ref{lemma G-module map for relation G-A}.\n\n\\begin{lemma}\\label{lemma first result for specific module}\n\t\n\tAssume that for every $i < j$, for every $E \\in \\mcE$, the chain map\n\t\\[\\mu_{\\mcA}^2 \\left( - , c_j \\right) : \\mcA \\left( X^i (E) , X^j \\right) \\to \\mcA \\left( X^i (E) , X^{j+1} \\right) \\]\n\tis a quasi-isomorphism. \n\tThen for every $k < n$, for every $E \\in \\mcE$, the inclusion $\\mcA \\left( X^k (E), X^n \\right) \\hookrightarrow \\mcM_{\\mcA} \\left( X^k (E) \\right)$ is a quasi-isomorphism.\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tThe cone of the inclusion $\\mcA \\left( X^k (E)  , X^n \\right) \\hookrightarrow  \\mcM_{\\mcA} \\left( X^k (E)  \\right)$ is quasi-isomorphic to its cokernel, which is $K_{n-1} \\left( X^k (E) \\right) \\oplus \\widetilde{K}_n \\left( X^k (E) \\right)$.\n\t\n\tWe have to show that these complexes are acyclic. \n\tObserve that $\\left( F_{n-1}^q \\left( X^k (E) \\right) \\right)_{q \\geq 0}$ and $\\left( \\widetilde{F}_n^q \\left( X^k (E) \\right) \\right)_{q \\geq 0}$ are increasing, exhaustive, and bounded from below filtrations of $K_{n-1} \\left( X^k (E) \\right)$ and $\\widetilde{K}_n \\left( X^k (E) \\right)$ respectively.\n\tFor every $q \\geq 1$, we have \n\t\\[F_{n-1}^q \\left( X^k (E) \\right) \/ F_{n-1}^{q-1} \\left( X^k (E) \\right) = \\left[\n\t\\begin{tikzcd}\n\t\\mcA \\left( X^k (E) , X^{n-q} \\right) \\ar[d, \"\\mathrm{id}\"] \\\\\n\t\\mcA \\left( X^k (E) , X^{n-q} \\right)\n\t\\end{tikzcd} \\right] \\]\n\tand\n\t\\[\\widetilde{F}_n^q \\left( X^k (E) \\right) \/ \\widetilde{F}_n^{q-1} \\left( X^k (E) \\right) = \\left[\n\t\\begin{tikzcd}\n\t\\mcA \\left( X^k (E) , X^{n+q-1} \\right) \\ar[rd, \"t_{c_{n+q-1}}\"] & \\\\\n\t& \\mcA \\left( X^k (E) , X^{n+q} \\right) \n\t\\end{tikzcd} \\right] . \\]\n\tThe first of the two latter complexes is clearly acyclic, and the second one is acyclic by assumption on the morphisms $c_j$. Thus the entire complex $K_{n-1} \\left( X^k (E) \\right) \\oplus \\widetilde{K}_n \\left( X^k (E) \\right)$ is acyclic, which is what we needed to prove. \n\t\n\\end{proof}\n\nThe following two lemmas will be used later in order to apply Proposition \\ref{prop quasi-isomorphism between localizations}. \n\n\\begin{lemma}\\label{lemma second result for specific module}\n\t\n\tAssume that for every $i < j < k$, for every $E \\in \\mcE$, the chain maps\n\t\\[\\left\\{\n\t\\begin{array}{ccccl}\n\t\\mu_{\\mcA}^2 \\left( - , c_j \\right) & : & \\mcA \\left( X^i (E) , X^j \\right) & \\to & \\mcA \\left( X^i (E) , X^{j+1} \\right) \\\\\n\t\\mu_{\\mcA}^2 \\left( c_j (E) , - \\right) & : & \\mcA \\left( X^{j+1} (E), X^{k+1} \\right) & \\to & \\mcA \\left( X^j (E), X^{k+1} \\right)\n\t\\end{array}\n\t\\right. \\]\n\tare quasi-isomorphisms. \n\tThen for every $(n, E) \\in \\mathbf{Z} \\times \\mcE$, the complex $\\mcM_{\\mcA} \\left( \\mathrm{Cone} \\, c_n (E) \\right)$ is acyclic.\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tWe have\n\t\\[\\mcM_{\\mcA} \\left( \\mathrm{Cone} \\, c_n (E) \\right) = \\mathrm{Cone} \\left( \\mcM_{\\mcA} \\left( X^{n+1} (E)  \\right) \\xrightarrow{\\mu_{\\mcM_{\\mcA}}^2 \\left( c_n (E) , - \\right)} \\mcM_{\\mcA} \\left( X^n (E)  \\right) \\right), \\]\n\tso we have to prove that $\\mu_{\\mcM_{\\mcA}}^2 \\left( c_n (E) , - \\right) : \\mcM_{\\mcA} \\left( X^{n+1} (E)  \\right) \\to \\mcM_{\\mcA} \\left( X^n (E)  \\right)$ is a quasi-isomorphism. \n\tObserve that we have the following commutative diagram\n\t\\[\\begin{tikzcd}[column sep = 2.5cm]\n\t\\mcM_{\\mcA} \\left( X^{n+1} (E)  \\right) \\ar[r, \"\\mu_{\\mcM_{\\mcA}}^2 {(c_n (E),-)}\"] & \\mcM_{\\mcA} \\left( X^n (E)  \\right) \\\\\n\t\\mcA \\left( X^{n+1} (E)  , X^{n+2} \\right) \\ar[u, hook] \\ar[r, \"\\mu_{\\mcA}^2 {\\left( c_n (E) , - \\right)}\"] & \\mcA \\left( X^n (E)  , X^{n+2} \\right) \\ar[u, hook] .\n\t\\end{tikzcd} \\]\n\tThe bottom horizontal map is a quasi-isomorphism by assumption on the morphisms $c_j (E)$. Moreover, the vertical maps are quasi-isomorphisms according to Lemma \\ref{lemma first result for specific module}. This implies that $\\mu_{\\mcM_{\\mcA}}^2 \\left( c_n (E) , - \\right)$ is indeed a quasi-isomorphism. \n\t\n\\end{proof}\n\n\\begin{lemma}\\label{lemma third result for specific module}\n\t\n\tThe $\\mcA$-module map $_{W_{\\mcA}^{-1}} t_{\\mcA}^n : \\, _{W_{\\mcA}^{-1}} \\mcA \\left( - , X^n \\right) \\to \\, _{W_{\\mcA}^{-1}} \\mcM_{\\mcA}$ is a quasi-isomorphism for every $n \\in \\mathbf{Z}$.\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tLet $X$ be some object of $\\mcA$. We want to prove that the chain map $_{W_{\\mcA}^{-1}} t_{\\mcA}^n : \\, _{W_{\\mcA}^{-1}} \\mcA \\left( X , X^n \\right) \\to \\, _{W_{\\mcA}^{-1}} \\mcM_{\\mcA} \\left( X \\right)$ is a quasi-isomorphism. Observe that \n\t\\[_{W_{\\mcA}^{-1}} \\mcM_{\\mcA} \\left( X \\right) = \\left[\n\t\\begin{tikzcd}\n\t\t\\dots \\ar[rd] & \\mcA \\left[ W_{\\mcA}^{-1} \\right] \\left( X , X^0 \\right) \\ar[d, \"\\mathrm{id}\"] \\ar[rd, \"_{W_{\\mcA}^{-1}} t_{c_0}\" below = 2.5] & \\mcA \\left[ W_{\\mcA}^{-1} \\right] \\left( X , X^1 \\right) \\ar[d, \"\\mathrm{id}\"] \\ar[rd, \"_{W_{\\mcA}^{-1}} t_{c_1}\"] & \\dots \\\\\n\t\t\\dots & \\mcA \\left[ W_{\\mcA}^{-1} \\right] \\left( X , X^0 \\right) & \\mcA \\left[ W_{\\mcA}^{-1} \\right] \\left( X , X^1 \\right) & \\dots\n\t\\end{tikzcd} \n\t\\right] \\]\n\tand the chain map $_{W_{\\mcA}^{-1}} t_{\\mcA}^n : \\, _{W_{\\mcA}^{-1}} \\mcA \\left( X , X^n \\right) \\to \\, _{W_{\\mcA}^{-1}} \\mcM_{\\mcA} \\left( X \\right)$ is the inclusion. The cone of the latter is then quasi-isomorphic to its cokernel, which is $_{W_{\\mcA}^{-1}} K_{n-1} \\left( X \\right) \\oplus \\, _{W_{\\mcA}^{-1}} \\widetilde{K}_n \\left( X \\right)$.\n\tObserve that $\\left( _{W_{\\mcA}^{-1}} F_{n-1}^q \\left( X \\right) \\right)_{q \\geq 0}$, $\\left( _{W_{\\mcA}^{-1}} \\widetilde{F}_n^q \\left( X \\right) \\right)_{q \\geq 0}$ are increasing, exhaustive, and bounded from below filtrations of $_{W_{\\mcA}^{-1}} K_{n-1} \\left( X \\right)$, $_{W_{\\mcA}^{-1}} \\widetilde{K}_n \\left( X \\right)$ respectively. For every $q \\geq 1$, we have \n\t\\[_{W_{\\mcA}^{-1}} F_{n-1}^q \\left( X \\right) \/ \\, _{W_{\\mcA}^{-1}} F_{n-1}^{q-1} \\left( X \\right) = \\left[\n\t\\begin{tikzcd}\n\t\t\\mcA \\left[ W_{\\mcA}^{-1} \\right] \\left( X , X^{n-q} \\right) \\ar[d, \"\\mathrm{id}\"] \\\\\n\t\t\\mcA \\left[ W_{\\mcA}^{-1} \\right] \\left( X , X^{n-q} \\right)\n\t\\end{tikzcd} \\right] \\]\n\tand\n\t\\[_{W_{\\mcA}^{-1}} \\widetilde{F}_n^q \\left( X \\right) \/ \\, _{W_{\\mcA}^{-1}} \\widetilde{F}_n^{q-1} \\left( X \\right) =\n\t\\left[\n\t\\begin{tikzcd}\n\t\t\\mcA \\left[ W_{\\mcA}^{-1} \\right] \\left( X , X^{n-1+q} \\right) \\ar[rd, \"_{W_{\\mcA}^{-1}} t_{c_{n-1+q}}\"] & \\\\\n\t\t& \\mcA \\left[ W_{\\mcA}^{-1} \\right] \\left( X , X^{n+q} \\right) \n\t\\end{tikzcd} \\right] . \\]\n\tThe first of the two latter complexes is clearly acyclic, and the second one is acyclic because $c_{n-1+q}$ belongs to the set $W_{\\mcA}$ by which we localized (see \\cite[Lemma 3.12]{GPS20}). Thus the entire complex $_{W_{\\mcA}^{-1}} K_{n-1} \\left( X \\right) \\oplus \\, _{W_{\\mcA}^{-1}} \\widetilde{K}_n \\left( X \\right)$ is acyclic, which is what we needed to prove.\n\t\n\\end{proof}\n\n\\subsection{The $A_{\\infty}$-category and modules for the mapping torus}\\label{subsection category and modules for the mapping torus}\n\nIn this section, we consider an $A_{\\infty}$-category $\\mcG$, together with a set $W_{\\mcG}$ of closed degree 0 morphisms. We prove that $\\mcH = \\mcG \\left[ W_{\\mcG}^{-1} \\right]$ is quasi-equivalent to the mapping torus of $\\tau$, and we prove technical results about specific $\\mcG$-modules that will allow us to apply Proposition \\ref{prop quasi-isomorphism between localizations} in the proofs of Theorems \\ref{thm mapping torus in strict situation} and \\ref{thm mapping torus in weak situation}. \n\nLet $\\tau$ be a quasi-autoequivalence of an $A_{\\infty}$-category $\\mcA$ equipped with a compatible $\\mathbf{Z}$-splitting of $\\mathrm{ob} \\left( \\mcA \\right)$.\nIf $\\mcA_{\\triangle}$ is a copy of $\\mcA$, we denote by $X_{\\triangle}^n \\left( E \\right)$ the object of $\\mcA_{\\triangle}$ corresponding to $\\left( n, E \\right) \\in \\mathbf{Z} \\times \\mcE$.\n\n\\subsubsection{The Grothendieck construction $\\mcG$}\n\nThe $A_{\\infty}$-category $\\mcG$ will be the Grothendieck construction of a slightly different diagram than the one in Definition \\ref{definition mapping torus}. The idea is to introduce an $A_{\\infty}$-category $\\mcC$ together with a set of closed degree $0$ morphisms $W_{\\mcC}$ such that the localization $\\mcC \\left[ W_{\\mcC}^{-1} \\right]$ is a cylinder object for $\\mcA$. \nObserve that this kind of cofibrant replacement is common in homotopy colimits computation, and indeed we need it to prove Theorem \\ref{thm mapping torus in weak situation}.\n\n\\begin{defin}\\label{definition cylinder object}\n\t\n\tLet $\\mcA_{\\bot}$, $\\mcA_I$ and $\\mcA_{\\top}$ be three copies of $\\mcA$. \n\tWe denote by $\\mcC$ the Grothendieck construction (see Definition \\ref{definition Grothendieck construction}) of the following diagram\n\t\\[\\begin{tikzcd}\n\t\\mcA_I \\ar[r, \"\\mathrm{id}\"] \\ar[d, \"\\mathrm{id}\" left] & \\mcA_{\\top} \\\\\n\t\\mcA_{\\bot} &\n\t\\end{tikzcd} \\]\n\tand we let $\\iota_{\\bot}, \\iota_I, \\iota_{\\top} : \\mcA \\to \\mcC$ be the strict inclusions with images $\\mcA_{\\bot}$, $\\mcA_I$, $\\mcA_{\\top}$ respectively. Finally, we denote by $W_{\\mcC}$ the set of adjacent units in $\\mcC$, and we let $\\mcC yl_{\\mcA} = \\mcC \\left[ W_{\\mcC}^{-1} \\right]$ be the homotopy colimit of the diagram above.\n\t\n\\end{defin}\n \n\\begin{defin}\\label{definition category G}\n\t\n\tLet $\\mcA_-$, $\\mcA_{+}$, $\\mcA_{\\bullet}$ be three copies of $\\mcA$.\n\tWe denote by $\\mcG$ the Grothendieck construction of the following diagram\n\t\\[\\begin{tikzcd}\n\t\\mcA_- \\sqcup \\mcA_+ \\ar[r, \"\\mathrm{id} \\sqcup \\tau\"] \\ar[d, \"\\iota_{\\bot} \\sqcup \\iota_{\\top}\" left] & \\mcA_{\\bullet} \\\\\n\t\\mcC .\n\t\\end{tikzcd} \\]\n\tBesides, we denote by $W_{\\mcG}$ the union of $W_{\\mcC}$ and the set of adjacent units in $\\mcG$, and we set\n\t\\[\\mcH := \\mcG \\left[ W_{\\mcG}^{-1} \\right]. \\]\n\t\n\\end{defin}\n\nAccording to Proposition \\ref{prop cylinder object}, $\\mcC yl_{\\mcA}$ can be thought as a cylinder object for $\\mcA$.\nTherefore, the following result should not be surprising. \n\n\\begin{lemma}\\label{lemma relation G - mapping torus}\n\t\n\tThe mapping torus of $\\tau$ is quasi-equivalent to $\\mcH$. \n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tLet $\\pi : \\mcC \\to \\mcA$ be the $A_{\\infty}$-functor induced by the following commutative diagram \n\t\\[\\begin{tikzcd}\n\t\\mcA_I \\ar[r, \"\\mathrm{id}\"] \\ar[d, \"\\mathrm{id}\"] & \\mcA_{\\top} \\ar[d, \"\\mathrm{id}\"] \\\\\n\t\\mcA_{\\bot} \\ar[r, \"\\mathrm{id}\"] & \\mcA\n\t\\end{tikzcd} \\]\n\t(see Proposition \\ref{prop induced functor from Grothendieck construction}).\n\tWe get a commutative diagram \n\t\\[\\begin{tikzcd}\n\t\\mcC \\ar[d, \"\\pi\"] & \\mcA_- \\sqcup \\mcA_+ \\ar[l, \"\\iota_{\\bot} \\sqcup \\iota_{\\top}\" above] \\ar[r, \"\\mathrm{id} \\sqcup \\tau\"] \\ar[d, \"\\mathrm{id}\"] & \\mcA_{\\bullet} \\ar[d, \"\\mathrm{id}\"] \\\\\n\t\\mcA & \\mcA_- \\sqcup \\mcA_+ \\ar[l, \"\\mathrm{id} \\sqcup \\mathrm{id}\"] \\ar[r, \"\\mathrm{id} \\sqcup \\tau\" below] & \\mcA_{\\bullet} \n\t\\end{tikzcd} \\]\n\twhich induces an $A_{\\infty}$-functor $\\chi$ from $\\mcG$ to the Grothendieck construction of the bottom line (see Proposition \\ref{prop invariance of homotopy colimits}).\n\tObserve that $\\chi$ sends $W_{\\mcC}$ to the set $U$ of units in $\\mcA$. \n\tNow, according to Proposition \\ref{prop cylinder object}, the $A_{\\infty}$-functor $\\widetilde{\\pi} : \\mcC yl_{\\mcA} = \\mcC \\left[ W_{\\mcC}^{-1} \\right] \\to \\mcA \\left[ U^{-1} \\right]$ is a quasi-equivalence. According to Lemma A.6 in \\cite{GPS19} (called ``localization and homotopy colimits commute''), this implies that the $A_{\\infty}$-functor induced by $\\chi$\n\t\\[\\mathrm{hocolim} \\left(  \n\t\\begin{tikzcd}[column sep = 0.5cm]\n\t\\mcA_- \\sqcup \\mcA_+ \\ar[d, \"\\iota_{\\bot} \\sqcup \\iota_{\\top}\"] \\ar[r, \"\\mathrm{id} \\sqcup \\tau\"] & \\mcA_{\\bullet} \\\\\n\t\\mcC\n\t\\end{tikzcd}\n\t\\right) \\left[ W_{\\mcC}^{-1} \\right] \\xrightarrow{\\widetilde{\\chi}}\n\t\\mathrm{hocolim} \\left(  \n\t\\begin{tikzcd}[column sep = 0.5cm]\n\t\\mcA_- \\sqcup \\mcA_+ \\ar[d, \"\\mathrm{id} \\sqcup \\mathrm{id}\"] \\ar[r, \"\\mathrm{id} \\sqcup \\tau\"] & \\mcA_{\\bullet} \\\\\n\t\\mcA\n\t\\end{tikzcd}\n\t\\right) \\left[ U^{-1} \\right] \\]\n\tis a quasi-equivalence.\n\tThis completes the proof because the source of $\\widetilde{\\chi}$ is exactly $\\mcH$, and its target is quasi-equivalent to the mapping torus of $\\tau$. \n\t\n\\end{proof}\n\n\\subsubsection{Modules over $\\mcG$}\n\nIn the following, we fix some element $E_{\\diamond} \\in \\mcE$. When we write an object $X_{\\triangle}^n$ without specifying the element of $\\mcE$, we mean $X_{\\triangle}^n (E_{\\diamond})$.\nMoreover, we denote by \n\\[t_{\\triangle \\square}^n : \\mcG \\left( -, X_{\\triangle}^n \\right) \\to \\mcG \\left( -, X_{\\square}^{n+\\delta_{\\triangle \\square}} \\right) \\]\nthe $\\mcG$-module map induced by the adjacent unit in $\\mcG \\left( X_{\\triangle}^n, X_{\\square}^{n+\\delta_{\\triangle \\square}} \\right)$ (see Definition \\ref{definition Yoneda functor}), where \n\\[\\delta_{\\triangle \\square} = \\left\\{\n\\begin{array}{ll}\n1 & \\text{if } \\left( \\triangle, \\square \\right) = \\left( +, \\bullet \\right) \\\\\n0 & \\text{otherwise} .\n\\end{array}\n\\right. \\]\n\n\\begin{defin}\\label{definition module for G}\n\t\n\tWe denote by $\\mcM_{\\mcG}$ the $\\mcG$-module defined by\n\t\\[\\mcM_{\\mcG} = \n\t\\left[ \n\t\\begin{tikzcd}[column sep = scriptsize]\n\t\\dots \\ar[rd, \"t_{+ \\bullet}^{-1}\"] & \\mcG \\left( - , X_-^{0} \\right) \\ar[d, \"t_{- \\bullet}^0\"] \\ar[rd, \"t_{- \\bot}^0\"] & \\mcG \\left( - , X_I^{0} \\right) \\ar[d, \"t_{I \\bot}^0\"] \\ar[rd, \"t_{I \\top}^0\"] & \\mcG \\left( - , X_+^{0} \\right) \\ar[d, \"t_{+ \\top}^0\"] \\ar[rd, \"t_{+ \\bullet}^0\"] & \\mcG \\left( - , X_{-}^{1} \\right) \\ar[d, \"t_{- \\bullet}^1\"] \\ar[rd, \"t_{- \\bot}^1\"] & \\dots \\\\\n\t\\dots & \\mcG \\left( - , X_{\\bullet}^{0} \\right) & \\mcG \\left( - , X_{\\bot}^0 \\right) & \\mcG \\left( - , X_{\\top}^0 \\right) & \\mcG \\left( - , X_{\\bullet}^{1} \\right) & \\dots \n\t\\end{tikzcd}\n\t\\right] \\]\n\t(see Definition \\ref{definition cone of a morphism between modules}). \n\tFor practical reasons, we also consider the $\\mcG$-modules\n\t\\[ \\mcM_{\\star}^n := \n\t\\left[\n\t\\begin{tikzcd}\n\t& \\mcG \\left( - , X_I^n \\right) \\ar[ld, \"t_{I \\bot}^n\"] \\ar[rd, \"t_{I \\top}^n\"] & \\\\\n\t\\mcG \\left( - , X_{\\bot}^n \\right) & & \\mcG \\left( - , X_{\\top}^n \\right)\n\t\\end{tikzcd}\n\t\\right], \\quad n \\in \\mathbf{Z}. \\]\n\tBesides, we denote by $t_{\\mcG} : \\mcG \\left( - , X_{\\bullet}^0 \\right) \\to \\mcM_{\\mcG}$ the $\\mcG$-module inclusion.\n\t\n\\end{defin}\n\n\\begin{rmk}\n\t\n\tWe can write \n\t\\[ \\mcM_{\\mcG} = \n\t\\left[\n\t\\begin{tikzcd}\n\t& \\bigoplus \\limits_{n \\in \\mathbf{Z}} \\left( \\mcG \\left( - , X_-^n \\right) \\oplus \\mcG \\left( - , X_+^n \\right) \\right) \\ar[ld, \"\\bigoplus \\limits_{n \\in \\mathbf{Z}} \\left( t_{- \\bot}^n \\oplus t_{+ \\top}^n \\right)\"] \\ar[rd, \"\\bigoplus \\limits_{n \\in \\mathbf{Z}} \\left( t_{- \\bullet}^n \\oplus t_{+ \\bullet}^n \\right)\"] & \\\\\n\t\\bigoplus \\limits_{n \\in \\mathbf{Z}} \\mcM_{\\star}^n & & \\bigoplus \\limits_{n \\in \\mathbf{Z}} \\mcG \\left( - , X_{\\bullet}^n \\right)\n\t\\end{tikzcd}\n\t\\right] . \\]\n\t\n\\end{rmk}\n\nThe following two lemmas are analogs of Lemmas \\ref{lemma second result for specific module} and \\ref{lemma third result for specific module} respectively. They will be used later in order to apply Proposition \\ref{prop quasi-isomorphism between localizations}.\n\n\\begin{lemma}\\label{lemma properties of modules}\n\t\n\tFor every $w$ in $W_{\\mcG}$, the complex $\\mcM_{\\mcG} \\left( \\mathrm{Cone} \\, w \\right)$ is acyclic.\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tLet $w$ be the morphism in $W_{\\mcG} \\cap \\mcG \\left( X_I^k (E) , X_{\\top}^k (E) \\right)$ (the proof is analogous for the morphism in $W_{\\mcG} \\cap \\mcG \\left( X_I^k (E) , X_{\\bot}^k (E) \\right)$). Then \n\t\\begin{align*}\n\t\\mcM_{\\mcG} \\left( \\mathrm{Cone} \\, w \\right) & = \\mathrm{Cone} \\left( \\mcM_{\\mcG} \\left( X_{\\top}^k (E) \\right) \\xrightarrow{\\mu_{\\mcM_{\\mcG}}^2 \\left( w , - \\right)} \\mcM_{\\mcG} \\left( X_I^k (E) \\right) \\right) \\\\\n\t& = \\bigoplus\\limits_{n} \\mathrm{Cone} \\left( \\mcG \\left( X_{\\top}^k (E) , X_{\\top}^n \\right) \\xrightarrow{\\mu_{\\mcM_{\\mcG}}^2 \\left( w , - \\right)} \\mcM_{\\star}^n \\left( X_I^k (E) \\right) \\right) .\n\t\\end{align*}\n\tWe want to prove that $\\mu_{\\mcM_{\\mcG}}^2 \\left( w , - \\right) : \\mcG \\left( X_{\\top}^k (E) , X_{\\top}^n \\right) \\to \\mcM_{\\star}^n \\left( X_I^k (E) \\right)$ is a quasi-isomorphism for every $n$. Observe that the following diagram of chain complexes is commutative \n\t\\[\\begin{tikzcd}\n\t\\mcG \\left( X_{\\top}^k (E) , X_{\\top}^n \\right) \\ar[rr, \"{ \\mu_{\\mcM_{\\mcG}}^2 \\left( w , - \\right) }\"] & & \\mcM_{\\star}^n \\left( X_I^k (E) \\right) \\\\\n\t\\mcG \\left( X_{\\top}^k (E) , X_{\\top}^n \\right) \\ar[u, equal] \\ar[rr, \"{\\mu_{\\mcG}^2 \\left( w, - \\right)}\"] & & \\mcG \\left( X_I^k (E) , X_{\\top}^n \\right) \\ar[u, hook] .\n\t\\end{tikzcd} \\]\n\tThe rightmost vertical arrow is injective, so its cone is quasi-isomorphic to its cokernel, which is the cone of $t_{I \\bot}^n : \\mcG \\left( X_I^k (E), X_I^n \\right) \\to \\mcG \\left( X_I^k (E), X_{\\bot}^n \\right)$. Since the latter map is a quasi-isomorphism, the cone of $\\mu_{\\mcM_{\\mcG}}^2 \\left( w , - \\right) : \\mcG \\left( X_{\\top}^k (E) , X_{\\top}^n \\right) \\to \\mcM_{\\star}^n \\left( X_I^k (E) \\right)$ is quasi-isomorphic to the cone of $\\mu_{\\mcG}^2 \\left( w, - \\right) : \\mcG \\left( X_{\\top}^k (E) , X_{\\top}^n \\right) \\to \\mcG \\left( X_I^k (E) , X_{\\top}^n \\right)$. The latter map is a quasi-isomorphism, so we conclude that $\\mu_{\\mcM_{\\mcG}}^2 \\left( w , - \\right) : \\mcG \\left( X_{\\top}^k (E) , X_{\\top}^n \\right) \\to \\mcM_{\\star}^n \\left( X_I^k (E) \\right)$ is a quasi-isomorphism for every $n$, and thus $\\mcM_{\\mcG} \\left( \\mathrm{Cone} \\, w \\right)$ is acyclic. \n\t\n\tNow let $w$ be the morphism in $W_{\\mcG} \\cap \\mcG \\left( X_+^k (E) , X_{\\top}^k (E)  \\right)$ (the proof is analogous for the morphism in $W_{\\mcG} \\cap \\mcG \\left( X_-^k (E) , X_{\\bot}^k (E) \\right)$). Then \n\t\\begin{align*}\n\t\\mcM_{\\mcG} \\left( \\mathrm{Cone} \\, w \\right) & = \\mathrm{Cone} \\left( \\mcM_{\\mcG} \\left( X_{\\top}^k (E) \\right) \\xrightarrow{\\mu_{\\mcM_{\\mcG}}^2 \\left( w , - \\right)} \\mcM_{\\mcG} \\left( X_+^k (E) \\right) \\right) \\\\\n\t& = \\bigoplus\\limits_{n} \\mathrm{Cone} \\left( \\mcG \\left( X_{\\top}^k (E) , X_{\\top}^n \\right) \\xrightarrow{\\mu_{\\mcM_{\\mcG}}^2 \\left( w , - \\right)} K^n \\right).\n\t\\end{align*}\n\twhere \n\t\\[K^n = \\left[ \n\t\\begin{tikzcd}\n\t& \\mcG \\left( X_+^k (E) , X_+^n \\right) \\ar[ld, \"t_{+ \\top}^n\"] \\ar[rd, \"t_{+ \\bullet}^n\"] \\\\\n\t\\mcG \\left( X_+^k (E) , X_{\\top}^n \\right) & & \\mcG \\left( X_+^k (E) , X_{\\bullet}^{n+1} \\right)\n\t\\end{tikzcd}\n\t\\right] . \\]\n\tObserve that $\\mu_{\\mcM_{\\mcG}}^2 \\left( w , - \\right) : \\mcG \\left( X_{\\top}^k (E) , X_{\\top}^n \\right) \\to K^n$ is injective, so its cone is quasi-isomorphic to its cokernel, which is the cone of $t_{+ \\bullet}^n : \\mcG \\left( X_+^k (E) , X_+^n \\right) \\to \\mcG \\left( X_+^k (E) , X_{\\bullet}^{n+1} \\right)$. The latter map is a quasi-isomorphism because $\\tau$ is a quasi-equivalence. This implies that the cone of $\\mu_{\\mcM_{\\mcG}}^2 \\left( w , - \\right) : \\mcG \\left( X_{\\top}^k (E) , X_{\\top}^n \\right) \\to K^n$ is acyclic for every $n$, and thus $\\mcM_{\\mcG} \\left( \\mathrm{Cone} \\, w \\right)$ is acyclic.\n\t\n\tIt remains to consider a morphism $w$ in $W_{\\mcG} \\cap \\mcG \\left( X_+^k (E) , X_{\\bullet}^{k+1} (E) \\right)$ (the proof is analogous for the morphism in $W_{\\mcG} \\cap \\mcG \\left( X_-^k (E) , X_{\\bullet}^k (E) \\right)$). Then \n\t\\begin{align*}\n\t\\mcM_{\\mcG} \\left( \\mathrm{Cone} \\, w \\right) & = \\mathrm{Cone} \\left( \\mcM_{\\mcG} \\left( X_{\\bullet}^{k+1} (E) \\right) \\xrightarrow{\\mu_{\\mcM_{\\mcG}}^2 \\left( w , - \\right)} \\mcM_{\\mcG} \\left( X_+^k (E) \\right) \\right) \\\\\n\t& = \\bigoplus\\limits_{n} \\mathrm{Cone} \\left( \\mcG \\left( X_{\\bullet}^{k+1} (E) , X_{\\bullet}^{n} \\right) \\xrightarrow{\\mu_{\\mcM_{\\mcG}}^2 \\left( w , - \\right)} K^n \\right).\n\t\\end{align*}\n\twhere \n\t\\[K^n = \\left[ \n\t\\begin{tikzcd}\n\t& \\mcG \\left( X_+^k (E) , X_+^{n-1} \\right) \\ar[ld, \"t_{+ \\top}^{n-1}\"] \\ar[rd, \"t_{+ \\bullet}^{n-1}\"] \\\\\n\t\\mcG \\left( X_+^k (E) , X_{\\top}^{n-1} \\right) & & \\mcG \\left( X_+^k (E) , X_{\\bullet}^{n} \\right)\n\t\\end{tikzcd}\n\t\\right] . \\]\n\tObserve that $\\mu_{\\mcM_{\\mcG}}^2 \\left( w , - \\right) : \\mcG \\left( X_{\\bullet}^{k+1} (E) , X_{\\bullet}^{n} \\right) \\to K^n$ is injective, so its cone is quasi-isomorphic to its cokernel, which is the cone of $t_{+ \\top}^{n-1} : \\mcG \\left( X_+^k (E) , X_+^{n-1} \\right) \\to \\mcG \\left( X_+^k (E) , X_{\\top}^{n-1} \\right)$. The latter map is a quasi-isomorphism, so we conclude that the cone of $\\mu_{\\mcM_{\\mcG}}^2 \\left( w , - \\right) : \\mcG \\left( X_{\\bullet}^{k+1} (E) , X_{\\bullet}^{n} \\right) \\to K^n$ is acyclic for every $n$, and thus $\\mcM_{\\mcG} \\left( \\mathrm{Cone} \\, w \\right)$ is acyclic.\n\t\n\\end{proof}\n\n\\begin{lemma}\\label{lemma properties of modules bis}\n\t\n\tThe $\\mcH$-module map $_{W_{\\mcG}^{-1}} t_{\\mcG} : \\, _{W_{\\mcG}^{-1}} \\mcG \\left( - , X_{\\bullet}^0 \\right) \\to \\, _{W_{\\mcG}^{-1}} \\mcM_{\\mcG}$ is a quasi-isomorphism.\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tWe fix an object $X$ in $\\mcG$, and we want to prove that $_{W_{\\mcG}^{-1}} t_{\\mcG} : \\, _{W_{\\mcG}^{-1}} \\mcG \\left( X , X_{\\bullet}^0 \\right) \\to \\, _{W_{\\mcG}^{-1}} \\mcM_{\\mcG} \\left( X \\right)$ is a quasi-isomorphism. \n\tObserve that\n\t\\[_{W_{\\mcG}^{-1}} \\mcM_{\\mcG} := \\left[ \n\t\\begin{tikzcd}\n\t\t\\dots \\ar[rd] & \\mcG \\left[ W_{\\mcG}^{-1} \\right] \\left( - , X_+^{-1} \\right) \\ar[d, \"_{W_{\\mcG}^{-1}} t_{+ \\top}^{-1}\"] \\ar[rd, \"_{W_{\\mcG}^{-1}} t_{+ \\bullet}^{-1}\" below = 2] & \\mcG \\left[ W_{\\mcG}^{-1} \\right] \\left( - , X_{-}^{0} \\right) \\ar[d, \"_{W_{\\mcG}^{-1}} t_{- \\bullet}^0\"] \\ar[rd, \"_{W_{\\mcG}^{-1}} t_{- \\bot}^0\"] & \\dots \\\\\n\t\t\\dots & \\mcG \\left[ W_{\\mcG}^{-1} \\right] \\left( - , X_{\\top}^{-1} \\right) & \\mcG \\left[ W_{\\mcG}^{-1} \\right] \\left( - , X_{\\bullet}^0 \\right) & \\dots \n\t\\end{tikzcd}\n\t\\right] \\]\n\tand that the chain map $_{W_{\\mcG}^{-1}} t_{\\mcG} : \\, _{W_{\\mcG}^{-1}} \\mcG \\left( X , X_{\\bullet}^0 \\right) \\to \\, _{W_{\\mcG}^{-1}} \\mcM_{\\mcG} \\left( X \\right)$ is the inclusion. The cone of the latter is then quasi-isomorphic to its cokernel, which can be written $K' \\oplus K''$ with\n\t\\[K' = \\left[ \n\t\\begin{tikzcd}\n\t\t\\dots \\ar[rd] & \\mcG \\left[ W_{\\mcG}^{-1} \\right] \\left( X , X_I^{-1} \\right) \\ar[d, \"_{W_{\\mcG}^{-1}} t_{I \\bot}^{-1}\"] \\ar[rd, \"_{W_{\\mcG}^{-1}} t_{I \\top}^{-1}\" below = 2] & \\mcG \\left[ W_{\\mcG}^{-1} \\right] \\left( X , X_+^{-1} \\right) \\ar[d, \"_{W_{\\mcG}^{-1}} t_{+ \\top}^{-1}\"] \\\\\n\t\t\\dots & \\mcG \\left[ W_{\\mcG}^{-1} \\right] \\left( X , X_{\\bot}^{-1} \\right) & \\mcG \\left[ W_{\\mcG}^{-1} \\right] \\left( X , X_{\\top}^{-1} \\right) & \n\t\\end{tikzcd}\n\t\\right] \\]\n\tand \n\t\\[K'' = \\left[ \n\t\\begin{tikzcd}\t\n\t\t\\mcG \\left[ W_{\\mcG}^{-1} \\right] \\left( X , X_-^0 \\right) \\ar[rd, \"_{W_{\\mcG}^{-1}} t_{- \\bot}^0\" below=2] & \\mcG \\left[ W_{\\mcG}^{-1} \\right] \\left( X , X_I^0 \\right) \\ar[d, \"_{W_{\\mcG}^{-1}} t_{I \\bot}^0\" left=1] \\ar[rd, \"_{W_{\\mcG}^{-1}} t_{I \\top}^0\"] & \\dots \\\\\n\t\t& \\mcG \\left[ W_{\\mcG}^{-1} \\right] \\left( X , X_{\\bot}^0 \\right) & \\dots\n\t\\end{tikzcd}\n\t\\right] . \\]\n\tObserve that the maps defining the chain complexes structures in $K'$ and $K''$ are all quasi-isomorphisms (see \\cite[Lemma 3.12]{GPS20}). Thus it is not difficult to show using an increasing exhaustive and bounded from below filtration of $K'$ and $K''$ that these complexes are acyclic (compare the proof of Lemma \\ref{lemma third result for specific module}). This implies that the map $_{W_{\\mcG}^{-1}} t_{\\mcG} : \\, _{W_{\\mcG}^{-1}} \\mcG \\left( X , X_{\\bullet}^0 \\right) \\to \\, _{W_{\\mcG}^{-1}} \\mcM_{\\mcG} \\left( X \\right)$ is a quasi-isomorphism. \n\t\n\\end{proof}\n\n\\subsection{Proof of the first result}\\label{subsection proof of the first result}\n\nLet $\\tau$ be a quasi-autoequivalence of an $A_{\\infty}$-category $\\mcA$ equipped with a compatible $\\mathbf{Z}$-splitting of $\\mathrm{ob} \\left( \\mcA \\right)$.\nAssume that $\\tau$ is strict and acts bijectively on hom-sets.\n\nObserve that there is a strict $A_{\\infty}$-functor $\\sigma : \\mcA \\to \\mcA_{\\tau}$ which sends $X^n (E)$ to $E$, and which sends $x \\in \\mcA \\left( X^i (E_1) , X^j (E_2) \\right)$ to $[x] \\in \\mcA_{\\tau} \\left( E_1, E_2 \\right)$. \nBesides, let $\\pi : \\mcC \\to \\mcA$ be the $A_{\\infty}$-functor induced by the following commutative diagram \n\\[\\begin{tikzcd}\n\\mcA \\ar[r, \"\\mathrm{id}\"] \\ar[d, \"\\mathrm{id}\"] & \\mcA \\ar[d, \"\\mathrm{id}\"] \\\\\n\\mcA \\ar[r, \"\\mathrm{id}\"] & \\mcA\n\\end{tikzcd} \\]\n(see Proposition \\ref{prop induced functor from Grothendieck construction}).\nThen the diagram of Adams-graded $A_{\\infty}$-categories\n\\[\\begin{tikzcd}\n\\mcA_- \\sqcup \\mcA_+ \\ar[r, \"\\mathrm{id} \\sqcup \\tau\"] \\ar[d, \"\\iota_{\\bot} \\sqcup \\iota_{\\top}\" left] & \\mcA_{\\bullet} \\ar[d, \"\\sigma\"] \\\\\n\\mcC \\ar[r, \"\\sigma \\circ \\pi\"] & \\mcA_{\\tau}\n\\end{tikzcd} \\]\nis commutative because $\\sigma \\circ \\tau = \\sigma$. Moreover, the induced $A_{\\infty}$-functor $\\Phi : \\mcG \\to \\mcA_{\\tau}$ is strict, and it sends $W_{\\mcG}$ to the set of units in $\\mcA_{\\tau}$.\nLet \n\\[ \\widetilde{\\Phi} : \\mcH \\to \\mcA_{\\tau} \\left[ \\{ \\mathrm{units} \\}^{-1} \\right] \\]\nbe the $A_{\\infty}$-functor induced by $\\Phi$.\n\nAccording to Lemma \\ref{lemma relation G - mapping torus}, $\\mcH$ is quasi-equivalent to the mapping torus of $\\tau$. Moreover, $\\mcA_{\\tau} \\left[ \\{ \\mathrm{units} \\}^{-1} \\right]$ is quasi-equivalent to $\\mcA_{\\tau}$. Thus, Theorem \\ref{thm mapping torus in strict situation} will follow if we prove that $\\widetilde{\\Phi}$ is a quasi-equivalence. \nOur strategy is to apply Proposition \\ref{prop quasi-isomorphism between localizations}. \nObserve that it suffices to prove that \n\\[ \\widetilde{\\Phi} : \\mcH (X, Y) \\to \\mcA_{\\tau} \\left[ \\{ \\mathrm{units} \\}^{-1} \\right] (\\Phi X, \\Phi Y) \\]\nis a quasi-isomorphism for every object $X$, $Y$ in $\\mcA_{\\bullet}^0$ (recall that $\\mcA_{\\bullet}^0$ denotes the subcategory of $\\mcA_{\\bullet}$ generated by the objects $X_{\\bullet}^0 (E)$, $E \\in \\mcE$) because every object of $\\mcG$ can be related to one of $\\mcA_{\\bullet}^0$ by a zigzag of morphisms in $W_{\\mcG}$, which are quasi-isomorphisms in $\\mcH$ (see \\cite[Lemma 3.12]{GPS20}).  \n\nIn the following, we fix some element $E_{\\diamond} \\in \\mcE$. When we write an object $X_{\\triangle}^n$ without specifying the element of $\\mcE$, we mean $X_{\\triangle}^n (E_{\\diamond})$.\nWe consider the corresponding $\\mcG$-module $\\mcM_{\\mcG}$ and the $\\mcG$-module map $t_{\\mcG} : \\mcG \\left( -, X_{\\bullet}^0 \\right) \\to \\mcG$ of Definition \\ref{definition module for G}. \nMoreover, we set \n\\[\\mcM_{\\mcA_{\\tau}} := \\mcA_{\\tau} \\left( -, E_{\\diamond} \\right) \\text{ and } t_{\\mcA_{\\tau}} := \\mathrm{id} : \\mcA_{\\tau} \\left( -, E_{\\diamond} \\right) \\to \\mcM_{\\mcA_{\\tau}}. \\]\n\n\\begin{lemma}\\label{lemma G-module map for relation G - coinvariants}\n\t\n\tThere exists a $\\mcG$-module map $t_0 : \\mcM_{\\mcG} \\to \\Phi^* \\mcM_{\\mcA_{\\tau}}$ (see Definition \\ref{definition pullback functor} for the pullback functor) such that\n\t\\begin{enumerate}\n\t\t\n\t\t\\item the following diagram of $\\mcG$-modules commutes\n\t\t\\[\\begin{tikzcd}\n\t\t\t\\mcG \\left( - , X_{\\bullet}^0 \\right) \\ar[d, \"t_{\\mcG}\"] \\ar[r, \"t_{\\Phi}\"] & \\Phi^* \\mcA_{\\tau} \\left( - , E_{\\diamond} \\right) \\ar[d, \"\\Phi^* t_{\\mcA_{\\tau}} = \\mathrm{id}\"] \\\\\n\t\t\t\\mcM_{\\mcG} \\ar[r, \"t_0\"] & \\Phi^* \\mcM_{\\mcA_{\\tau}}\n\t\t\\end{tikzcd} \\]\n\t\t(see Definition \\ref{definition modules morphism induced by functor} for the map $t_{\\Phi}$),\n\t\t\n\t\t\\item for every $E \\in \\mcE$, the map $t_0 : \\mcM_{\\mcG} (X^0_{\\bullet} (E)) \\to \\Phi^* \\mcM_{\\mcA_{\\tau}} (X^0_{\\bullet} (E))$ is a quasi-isomorphism.\n\t\t\n\t\\end{enumerate}\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tObserve that the diagram of $\\mcG$-modules \n\t\\[\\begin{tikzcd}\n\t\\mcG \\left( - , X_I^n \\right) \\ar[r, \"t_{I \\top}^n\"] \\ar[d, \"t_{I \\bot}^n\"] & \\mcG \\left( - , X_{\\top}^n \\right) \\ar[d, \"t_{\\Phi}\"] \\\\\n\t\\mcG \\left( - , X_{\\bot}^n \\right) \\ar[r, \"t_{\\Phi}\"] & \\Phi^* \\mcM_{\\mcA_{\\tau}}\n\t\\end{tikzcd} \\]\n\tis commutative, so that it induces a $\\mcG$-module map $\\mcM_{\\star}^n \\to \\Phi^* \\mcM_{\\mcA_{\\tau}}$ (see Proposition \\ref{prop morphism induced by homotopy}). \n\tNow observe that the following diagram of $\\mcG$-modules commutes\n\t\\[\\begin{tikzcd}[column sep = huge]\n\t\\bigoplus \\limits_{n \\in \\mathbf{Z}} \\left( \\mcG \\left( - , X_-^n \\right) \\oplus \\mcG \\left( - , X_+^n \\right) \\right) \\ar[r, \"\\bigoplus \\limits_{n \\in \\mathbf{Z}} \\left( t_{- \\bullet}^n \\oplus t_{+ \\bullet}^n \\right)\"] \\ar[d, \"\\bigoplus \\limits_{n \\in \\mathbf{Z}} \\left( t_{- \\bot}^n \\oplus t_{+ \\top}^n \\right)\"] & \\bigoplus \\limits_{n \\in \\mathbf{Z}} \\mcG \\left( - , X_{\\bullet}^n \\right) \\ar[d, \"t_{\\Phi}\"] \\\\\n\t\\bigoplus \\limits_{n \\in \\mathbf{Z}} \\mcM_{\\star}^n \\ar[r] & \\Phi^* \\mcM_{\\mcA_{\\tau}}\n\t\\end{tikzcd} . \\]\n\tWe let $t_0 : \\mcM_{\\mcG} \\to \\Phi^* \\mcM_{\\mcA_{\\tau}}$ be the induced $\\mcG$-module map. It is then easy to verify that the following diagram of $\\mcG$-modules is commutative\n\t\\[\\begin{tikzcd}\n\t\\mcG \\left( - , X_{\\bullet}^0 \\right) \\ar[d, \"t_{\\mcG}\"] \\ar[r, \"t_{\\Phi}\"] & \\Phi^* \\mcA_{\\tau} \\left( - , E_{\\diamond} \\right) \\ar[d, \"\\Phi^* t_{\\mcA_{\\tau}} = \\mathrm{id}\"] \\\\\n\t\\mcM_{\\mcG} \\ar[r, \"t_0\"] & \\Phi^* \\mcM_{\\mcA_{\\tau}} .\n\t\\end{tikzcd} \\]\n\t\n\tWe now prove the second part of the lemma.\n\tWe have \n\t\\[\\mcM_{\\mcG} \\left( X_{\\bullet}^0 (E) \\right) = \\bigoplus\\limits_n \\mcG \\left( X_{\\bullet}^0 (E), X_{\\bullet}^n \\right) = \\bigoplus\\limits_n \\mcA \\left( X^k (E) , X^n \\right) \\]\t\n\tand \n\t\\[t_0 : \\bigoplus\\limits_n \\mcA \\left( X^k (E) , X^n \\right) = \\mcM_{\\mcG} \\left( X_{\\bullet}^0 (E) \\right) \\to \\Phi^* \\mcM_{\\mcA_{\\tau}} \\left( X_{\\bullet}^0 (E) \\right) = \\mcA_{\\tau} \\left( E, E_{\\diamond} \\right) \\]\n\tis the sum of the projections, which is an isomorphism. This concludes the proof\n\t\n\\end{proof}\n\n\\begin{lemma}\\label{lemma relation G - coinvariants}\n\t\n\tFor every $E \\in \\mcE$, the chain map \n\t\\[ \\widetilde{\\Phi} : \\mcH \\left( X_{\\bullet}^0 (E), X_{\\bullet}^0 \\right) \\to \\mcA_{\\tau} \\left[ \\{ \\mathrm{units} \\}^{-1} \\right] \\left( E, E_{\\diamond} \\right) \\]\n\tis a quasi-isomorphism.\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tAccording to Lemmas \\ref{lemma properties of modules}, \\ref{lemma properties of modules bis} and \\ref{lemma G-module map for relation G - coinvariants}, the assumptions of Proposition \\ref{prop quasi-isomorphism between localizations} are satisfied. This concludes the proof. \n\t\n\\end{proof}\n\nAs explained above, Theorem \\ref{thm mapping torus in strict situation} follows from Lemma \\ref{lemma relation G - coinvariants} since $\\mcH$ is quasi-equivalent to the mapping torus of $\\tau$ (see Lemma \\ref{lemma relation G - mapping torus}) and $\\mcA_{\\tau} \\left[ \\{ \\mathrm{units} \\}^{-1} \\right]$ is quasi-equivalent to $\\mcA_{\\tau}$.\n\n\\subsection{Proof of the second result}\\label{subsection proof of the second result}\n\nLet $\\tau$ be a quasi-autoequivalence of an $A_{\\infty}$-category $\\mcA$ equipped with a compatible $\\mathbf{Z}$-splitting of $\\mathrm{ob} \\left( \\mcA \\right)$.\nAssume that the following holds: \n\\begin{enumerate}\n\t\n\t\\item $\\mcA$ is weakly directed with respect to the $\\mathbf{Z}$-splitting of $\\mathrm{ob} \\left( \\mcA \\right)$ (see Definition \\ref{definition group-action}),\n\t\n\t\\item there exists a closed degree $0$ bimodule map $f : \\mcA_m \\left( -, - \\right) \\to \\mcA_m \\left( -, \\tau(-) \\right)$ (see Definitions \\ref{definition bimodule} and \\ref{definition pullback bimodule}) such that $f : \\mcA_m \\left( X^i(E) , X^j(E') \\right) \\to \\mcA_m \\left( X^i(E), X^{j+1} (E') \\right)$ is a quasi-somorphism for every $i < j$ and $E, E' \\in \\mcE$.\n\t\n\\end{enumerate}\n\n\\begin{rmk}\\label{rmk multiplication by f(unit) is quasi-iso}\n\t\n\tIt follows from Lemma \\ref{coro closed module map homotopic to Yoneda module map} and $\\tau$ being a quasi-equivalence that the chain maps\n\t\\[\\left\\{\n\t\\begin{array}{ccccl}\n\t\t\\mu_{\\mcA_m}^2 \\left( - , f(e_{X^j(E)}) \\right) & : & \\mcA_m \\left( X^i(E') , X^j(E) \\right) & \\to & \\mcA_m \\left( X^i(E') , X^{j+1} (E) \\right) \\\\\n\t\t\\mu_{\\mcA_m}^2 \\left( f(e_{X^j (E)}) , - \\right) & : & \\mcA_m \\left( X^{j+1} (E) , X^{k+1} (E') \\right) & \\to & \\mcA_m \\left( X^j (E) , X^{k+1} (E') \\right)\n\t\\end{array}\n\t\\right. \\]\n\tare quasi-isomorphisms for every $i < j < k$ and $E, E' \\in \\mcE$. \n\t\n\\end{rmk}\n\nIn the following, we set \n\\[c_n (E) := f \\left( e_{X^n (E)} \\right) \\]\nfor every $n \\in \\mathbf{Z}$, $E \\in \\mcE$, and\n\\[W_{\\mcA_m} := \\left\\{ c_n (E) \\mid n \\in \\mathbf{Z}, E \\in \\mcE \\right\\} \\cup \\left\\{ \\text{units of } \\mcA_m \\right\\} . \\]\n\n\\subsubsection{Generalized homotopy}\\label{subsection generalized homotopy}\n\nRecall that we introduced a functor $\\mcB \\mapsto \\mcB_m$ from the category of Adams-graded $A_{\\infty}$-categories to the category of (non Adams-graded) $A_{\\infty}$-categories. \nAlso, recall that we introduced Adams-graded $A_{\\infty}$-categories $\\mcC$ and $\\mcG$ in Definitions \\ref{definition cylinder object} and \\ref{definition category G} respectively.\nObserve that $\\mcC_m$ and $\\mcG_m$ are the Grothendieck constructions of the diagrams\n\\[\\begin{tikzcd}\n(\\mcA_I)_m \\ar[r, \"\\mathrm{id}\"] \\ar[d, \"\\mathrm{id}\" left] & (\\mcA_{\\top})_m \\\\\n(\\mcA_{\\bot})_m &\n\\end{tikzcd} \\text{ and }\n\\begin{tikzcd}\n(\\mcA_-)_m \\sqcup (\\mcA_+)_m \\ar[r, \"\\mathrm{id} \\sqcup \\tau\"] \\ar[d, \"\\iota_{\\bot} \\sqcup \\iota_{\\top}\" left] & (\\mcA_{\\bullet})_m \\\\\n\\mcC_m\n\\end{tikzcd}\n\\]\nrespectively. \nWe denote by $W_{\\mcC_m}$ the set of adjacent units in $\\mcC_m$, and by $W_{\\mcG_m}$ the union of $W_{\\mcC_m}$ and the set of adjacent units in $\\mcG_m$. \n\nWe would like to define an $A_{\\infty}$-functor $\\Psi_m : \\mcG_m \\to \\mcA_m$ which sends $W_{\\mcG_m}$ to $W_{\\mcA_m}$. According to Proposition \\ref{prop induced functor from Grothendieck construction}, it is enough to prove the following result.\n\n\\begin{lemma}\\label{lemma relation tau continuation}\n\t\n\tThere exists an $A_{\\infty}$-functor $\\eta : \\mcC_m \\to \\mcA_m$ which sends $W_{\\mcC_m}$ to $W_{\\mcA_m}$, and such that\n\t\\[\\eta \\circ \\iota_I = \\eta \\circ \\iota_{\\bot} = \\mathrm{id}, \\quad \\eta \\circ \\iota_{\\top} = \\tau . \\]\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tWe first define $\\eta$ to be $\\mathrm{id}$ on $(\\mcA_{\\bot})_m$, $(\\mcA_I)_m$. and to be $\\tau$ on $(\\mcA_{\\top})_m$. Observe that this completely defines $\\eta$ on the objects.\n\tBesides, we ask for $\\eta$ to act as the identity on the sequences involving an adjacent morphism from $(\\mcA_I)_m$ to $(\\mcA_{\\bot})_m$.\n\t\n\tIt remains to define $\\eta$ on the sequences involving an adjacent morphism from $(\\mcA_I)_m$ to $(\\mcA_{\\top})_m$.\n\tConsider a sequence of morphisms\n\t\\begin{align*}\n\t\\left( x_0, \\dots, x_{p+q} \\right) & \\in \\mcC_m \\left( X_I^{i_0} (E_0), X_I^{i_1} (E_1) \\right) \\times \\dots \\times \\mcC_m \\left( X_I^{i_{p-1}} (E_{p-1}), X_I^{i_p} (E_p) \\right) \\\\ \n\t& \\times \\mcC_m \\left( X_I^{i_p} (E_p), X_{\\top}^{i_{p+1}} (E_{p+1}) \\right) \\\\\n\t& \\times \\mcC_m \\left( X_{\\top}^{i_{p+1}} (E_{p+1}), X_{\\top}^{i_{p+2}} (E_{p+2}) \\right) \\times \\dots \\times \\mcC_m \\left( X_{\\top}^{i_{p+q}} (E_{p+q}), X_{\\top}^{i_{p+q+1}} (E_{p+q+1}) \\right) .\n\t\\end{align*} \n\tObserve that \n\t\\[\\mcC_m \\left( X_I^i (E), X_I^j (E') \\right) = \\mcC_m \\left( X_I^i (E), X_ {\\top}^j (E') \\right) = \\mcC_m \\left( X_{\\top}^i (E), X_{\\top}^j (E') \\right) = \\mcA_m \\left( X^i (E), X^j (E) \\right) . \\]\n\tThen we set \n\t\\[\\eta \\left( x_0, \\dots, x_{p+q} \\right) := f \\left( x_0, \\dots, x_{p-1}, x_p, x_{p+1}, \\dots, x_{p+q+1} \\right) \\in \\mcA_m \\left( X^{i_0} (E_0), \\tau X^{i_{p+q+1}} (E_{p+q+1}) \\right) . \\]\n\tThe functor $\\eta$ we defined satisfies the $A_{\\infty}$-relations because $f : \\mcA_m \\left( -, - \\right) \\to \\mcA_m \\left( -, \\tau (-) \\right)$ is a closed degree $0$ bimodule map. Moreover, $\\eta$ sends $W_{\\mcC_m}$ to $W_{\\mcA_m}$ by construction. \n\t\n\\end{proof}\n\n\\begin{rmk}\n\t\n\tFirst observe that\n\t\\[\\mcC yl_{\\mcA_m} = \\mcC_m \\left[ W_{\\mcC_m}^{-1} \\right] = \\left( \\mcC \\left[ W_{\\mcC}^{-1} \\right] \\right)_m = \\left( \\mcC yl_{\\mcA} \\right)_m . \\]\n\tAccording to Lemma \\ref{lemma relation tau continuation}, the functor $\\eta$ induces an $A_{\\infty}$-functor $\\widetilde{\\eta} : \\mcC yl_{\\mcA_m} \\to \\mcA_m \\left[ W_{\\mcA_m}^{-1} \\right]$. Moreover, Lemma \\ref{lemma relation tau continuation} implies that the following diagram commutes\n\t\\[\\begin{tikzcd}\n\t(\\mcA_+)_m \\ar[d, \"\\lambda_{\\mcC_m} \\circ \\iota_{\\top}\" left] \\ar[rd, bend left, \"\\lambda_{\\mcA_m} \\circ \\tau\"] & \\\\\n\t\\mcC yl_{\\mcA_m} \\ar[r, \"\\widetilde{\\eta}\"] & \\mcA_m \\left[ W_{\\mcA_m}^{-1} \\right] \\\\\n\t(\\mcA_-)_m \\ar[u, \"\\lambda_{\\mcC_m} \\circ \\iota_{\\bot}\"] \\ar[ru, bend right, \"\\lambda_{\\mcA_m}\"] & \n\t\\end{tikzcd}\n\t\\] \n\t($\\lambda_{\\mcA_m} : \\mcA_m \\to \\mcA_m \\left[ W_{\\mcA_m}^{-1} \\right]$ and $\\lambda_{\\mcC_m} : \\mcC_m \\to \\mcC_m \\left[ W_{\\mcC_m}^{-1} \\right]$ denote the localization functors). Since $\\mcC yl_{\\mcA_m}$ should be thought as a cylinder object for $\\mcA_m$ (see Proposition \\ref{prop cylinder object}), we should think that the functors $\\lambda_{\\mcA_m}$ and $\\lambda_{\\mcA_m} \\circ \\tau$ are homotopic (even if they do not act the same way on objects) and that $\\widetilde{\\eta}$ is a generalized homotopy between them (see Proposition \\ref{prop homotopy is generalized homotopy} for a justification of this terminology).\n\t\n\\end{rmk}\n\n\\subsubsection{Relation between $\\mcG$ and $\\mcA_m$}\n\nUsing the $A_{\\infty}$-functor $\\eta : \\mcC_m \\to \\mcA_m$ of Lemma \\ref{lemma relation tau continuation}, we get a commutative diagram of (non Adams-graded) $A_{\\infty}$-categories \n\\[\\begin{tikzcd}\n(\\mcA_-)_m \\sqcup (\\mcA_+)_m \\ar[r, \"\\mathrm{id} \\sqcup \\tau\"] \\ar[d, \"\\iota_{\\bot} \\sqcup \\iota_{\\top}\" left] & (\\mcA_{\\bullet})_m \\ar[d, \"\\mathrm{id}\"] \\\\\n\\mcC_m \\ar[r, \"\\eta\"] & \\mcA_m\n\\end{tikzcd} \\]\nand the induced $A_{\\infty}$-functor $\\Psi_m : \\mcG_m \\to \\mcA_m$ (see Proposition \\ref{prop induced functor from Grothendieck construction}) sends $W_{\\mcG_m}$ to $W_{\\mcA_m}$ (see Lemma \\ref{lemma relation tau continuation}). Let \n\\[ \\widetilde{\\Psi_m} : \\mcH_m = \\mcG_m \\left[ W_{\\mcG_m}^{-1} \\right] \\to \\mcA_m \\left[ W_{\\mcA_m}^{-1} \\right] \\]\nbe the $A_{\\infty}$-functor induced by $\\Psi_m$. \nObserve that, since $\\mcA$ is assumed to be weakly directed and since the Adams-degree of $\\mcA$ comes from the $\\mathbf{Z}$-splitting of $\\mathrm{ob} \\left( \\mcA \\right)$ (see Definition \\ref{definition group-action}), $\\mcH$ is concentrated in non-negative Adams-degree.\nIn particular, we can apply the adjunction of Definition \\ref{definition adjunction} to $\\widetilde{\\Psi_m}$, which gives an $A_{\\infty}$-functor\n\\[ \\widetilde{\\Psi} : \\mcH = \\mcG \\left[ W_{\\mcG}^{-1} \\right] \\to \\mathbf{F} \\left[ t_m \\right] \\otimes \\mcA_m \\left[ W_{\\mcA_m}^{-1} \\right] . \\]\nWe would like to prove that for every Adams degree $j \\geq 1$, and for every objects $X,Y$ in $\\mcA_{\\bullet}^0$ (recall that $\\mcA_{\\bullet}^0$ denotes the subcategory of $\\mcA_{\\bullet}$ generated by the objects $X_{\\bullet}^0 (E)$, $E \\in \\mcE$), the map\n\\[\\widetilde{\\Psi} : \\mcH \\left( X,Y \\right)^{*,j} \\to \\left( \\mathbf{F} \\left[ t_m \\right] \\otimes \\mcA_m \\left[ W_{\\mcA_m}^{-1} \\right] \\right) \\left( \\Psi X , \\Psi Y \\right)^{*,j} = \\mathbf{F}_m^j \\otimes \\mcA_m \\left[ W_{\\mcA_m}^{-1} \\right] \\left( \\Psi X , \\Psi Y \\right) \\]\nis a quasi-isomorphism, ($\\mathbf{F}_m^j$ is the vector space generated by $t_m^j$ ; also recall that if $V$ is an Adams-graded vector space, $V^{*,j}$ denotes the subspace of Adams degree $j$ elements). Our strategy is once again to apply Proposition \\ref{prop quasi-isomorphism between localizations}.\n\nIn the following we fix some element $E_{\\diamond} \\in \\mcE$. When we write $X_{\\triangle}^n$ or $c_n$ without specifying the element of $\\mcE$, we mean $X_{\\triangle}^n (E_{\\diamond})$ or $c_n (E_{\\diamond})$ respectively.\nWe consider the corresponding $\\mcG$-module $\\mcM_{\\mcG}$, and the $\\mcG$-module map $t_{\\mcG} : \\mcG \\left( -, X_{\\triangle}^n \\right) \\to \\mcG$ of Definition \\ref{definition module for G}. \nMoreover, we consider the $\\mcA_m$-module $\\mcM_{\\mcA_m}$ and the $\\mcA_m$-module maps\n\\[t_{\\mcA_m}^n : \\mcA_m \\left( - , X^n \\right) \\to \\mcM_{\\mcA_m}, \\quad n \\in \\mathbf{Z}, \\]\nassociated to the morphisms $\\left( c_n \\right)_{n \\in \\mathbf{Z}}$ as in Definition \\ref{definition specific module}.\n\nThe following result is a first step in order to define a $\\mcG_m$-module map $(t_0)_m : (\\mcM_{\\mcG})_m \\to \\Psi_m^* \\mcM_{\\mcA_m}$ as in Proposition \\ref{prop quasi-isomorphism between localizations}.\n\n\\begin{lemma}\\label{lemma first step to define the module map}\n\t\n\tFor every $n \\in \\mathbf{Z}$, the diagram of $\\mcG_m$-modules \n\t\\[\\begin{tikzcd}[column sep = huge]\n\t\\mcG_m \\left( - , X_I^n \\right) \\ar[r, \"t_{I \\top}^n\"] \\ar[d, \"t_{I \\bot}^n\"] & \\mcG_m \\left( - , X_{\\top}^n \\right) \\ar[d, \"\\Psi_m^* t_{\\mcA_m}^{n+1} \\circ t_{\\Psi_m}\"] \\\\\n\t\\mcG_m \\left( - , X_{\\bot}^n \\right) \\ar[r, \"\\Psi_m^* t_{\\mcA_m}^n \\circ t_{\\Psi_m}\"] & \\Psi_m^* \\mcM_{\\mcA_m}\n\t\\end{tikzcd} \\]\n\tcommutes up to homotopy.\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tFirst observe that $\\iota_I^* \\mcG_m \\left( - , X_I^n \\right) = \\mcA_m \\left( -, X^n \\right)$, and $\\iota_I^* \\Psi_m^* \\mcM_{\\mcA_m} = \\mcM_{\\mcA_m}$\tbecause $\\Psi_m \\circ \\iota_I = \\mathrm{id}$ (see Remark \\ref{rmk composition of pullback functors}).\n\tMoreover, it suffices to show that the $\\mcA_m$-module maps \n\t\\[\\iota_I^* \\left( \\Psi_m^* t_{\\mcA_m}^n \\circ t_{\\Psi_m} \\circ t_{I \\bot}^n \\right) = t_{\\mcA_m}^n \\circ \\iota_I^* \\left( t_{\\Psi_m} \\circ t_{I \\bot}^n \\right) : \\mcA_m \\left( -, X^n \\right) \\to \\mcM_{\\mcA_m} \\]\n\tand \n\t\\[\\iota_I^* \\left( \\Psi_m^* t_{\\mcA_m}^{n+1} \\circ t_{\\Psi_m} \\circ t_{I \\top}^n \\right) = t_{\\mcA_m}^{n+1} \\circ \\iota_I^* \\left( t_{\\Psi_m} \\circ t_{I \\top}^n \\right) : \\mcA_m \\left( -, X^n \\right) \\to \\mcM_{\\mcA_m} \\]\n\tare homotopic because \n\t\\[\\mcG_m \\left( X_{\\triangle}^k, X_I^n \\right) = 0 \\text{ if } \\triangle \\ne I . \\]\n\tOn the one hand, \n\t\\[t_{\\mcA_m}^n \\circ \\iota_I^* \\left( t_{\\Psi_m} \\circ t_{I \\bot}^n \\right) = t_{\\mcA_m}^n .\\]\n\tOn the other hand, $\\iota_I^* \\left( t_{\\Psi_m} \\circ t_{I \\top}^n \\right) : \\mcA_m \\left( -, X^n \\right) \\to \\mcA_m \\left( -, X^{n+1} \\right)$ is closed (as composition and pullback of closed module maps), and \n\t\\[\\iota_I^* \\left( t_{\\Psi_m} \\circ t_{I \\top}^n \\right) \\left( e_{X^n} \\right) = \\eta \\left( e_{X^n} \\right) = c_n \\]\n\taccording to Lemma \\ref{lemma relation tau continuation}. \n\tTherefore, $\\iota_I^* \\left( t_{\\Psi_m} \\circ t_{I \\top}^n \\right)$ is homotopic to $t_{c_n}$ according to Corollary \\ref{coro closed module map homotopic to Yoneda module map}, and thus $t_{\\mcA_m}^{n+1} \\circ \\iota_I^* \\left( t_{\\Psi_m} \\circ t_{I \\top}^n \\right)$ is homotopic to $t_{\\mcA_m}^{n+1} \\circ t_{c_n}$. Now according to Lemma \\ref{lemma multiplication by continuation element becomes homotopic to inclusion}, $t_{\\mcA_m}^{n+1} \\circ t_{c_n}$ is homotopic to $t_{\\mcA_m}^n$. This concludes the proof.\n\t\n\\end{proof}\n\nWe can now state the result establishing the existence of a $\\mcG_m$-module map $(t_0)_m : (\\mcM_{\\mcG})_m \\to \\Psi_m^* \\mcM_{\\mcA_m}$ as in Proposition \\ref{prop quasi-isomorphism between localizations}.\n\n\\begin{lemma}\\label{lemma G-module map for relation G-A}\n\t\n\tThere exists a $\\mcG_m$-module map $(t_0)_m : (\\mcM_{\\mcG})_m \\to \\Psi_m^* \\mcM_{\\mcA_m}$ such that the following holds. \n\t\\begin{enumerate}\n\t\t\n\t\t\\item The following diagram of $\\mcG_m$-modules commutes\n\t\t\\[\\begin{tikzcd}\n\t\t\t\\mcG_m \\left( - , X_{\\bullet}^0 \\right) \\ar[d, \"t_{\\mcG_m}\"] \\ar[r, \"t_{\\Psi_m}\"] & \\Psi_m^* \\mcA_m \\left( - , X^0 \\right) \\ar[d, \"\\Psi_m^* t_{\\mcA_m}^0\"] \\\\\n\t\t\t(\\mcM_{\\mcG})_m \\ar[r, \"(t_0)_m\"] & \\Psi_m^* \\mcM_{\\mcA_m}\n\t\t\\end{tikzcd} \\]\n\t\n\t\t\\item for every $E \\in \\mcE$ and $j \\geq 1$, the induced map $t_0 : \\mcM_{\\mcG} (X_{\\bullet}^0 (E)) \\to \\mathbf{F} \\left[ t_m \\right] \\otimes \\Psi_m^* \\mcM_{\\mcA_m} (X_{\\bullet}^0 (E))$ (see Definition \\ref{definition adjunction}) is a quasi-isomorphism in each positive Adams-degree.\n\t\t\n\t\\end{enumerate}\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tUsing Lemma \\ref{lemma first step to define the module map} and Proposition \\ref{prop morphism induced by homotopy}, we get a $\\mcG_m$-module map $t_{\\star}^n : \\left( \\mcM_{\\star}^n \\right)_m \\to \\Psi_m^* \\mcM_{\\mcA_m}$ for every $n \\in \\mathbf{Z}$ (see Definition \\ref{definition module for G} for the $\\mcG$-modules $\\mcM_{\\star}^n$). \n\tObserve that the diagram of $\\mcG_m$-modules \n\t\\[\\begin{tikzcd}[column sep = huge]\n\t\\bigoplus \\limits_{n \\in \\mathbf{Z}} \\left( \\mcG_m \\left( - , X_-^n \\right) \\oplus \\mcG_m \\left( - , X_+^n \\right) \\right) \\ar[r, \"\\bigoplus \\limits_{n \\in \\mathbf{Z}} \\left( t_{- \\bullet}^n \\oplus t_{+ \\bullet}^n \\right)\"] \\ar[d, \"\\bigoplus \\limits_{n \\in \\mathbf{Z}} \\left( t_{- \\bot}^n \\oplus t_{+ \\top}^n \\right)\"] & \\bigoplus \\limits_{n \\in \\mathbf{Z}} \\mcG_m \\left( - , X_{\\bullet}^n \\right) \\ar[d, \"\\bigoplus \\limits_{n \\in \\mathbf{Z}} \\Psi_m^* t_{\\mcA_m}^n \\circ t_{\\Psi_m}\"] \\\\\n\t\\bigoplus \\limits_{n \\in \\mathbf{Z}} \\left( \\mcM_{\\star}^n \\right)_m \\ar[r, \"\\bigoplus \\limits_{n \\in \\mathbf{Z}} t_{\\star}^n\"] & \\Psi_m^* \\mcM_{\\mcA_m}\n\t\\end{tikzcd} \\]\n\tis commutative (the composition is $\\mathrm{id}$ for $-$-terms and $\\tau$ for $+$-terms), so that it induces a $\\mcG_m$-module map $(t_0)_m : (\\mcM_{\\mcG})_m \\to \\Psi_m^* \\mcM_{\\mcA_m}$. It is then easy to verify that the following diagram of $\\mcG_m$-modules is commutative\n\t\\[\\begin{tikzcd}\n\t\\mcG_m \\left( - , X_{\\bullet}^0 \\right) \\ar[d, \"t_{\\mcG_m}\"] \\ar[r, \"t_{\\Psi_m}\"] & \\Psi_m^* \\mcA_m \\left( - , X^0 \\right) \\ar[d, \"\\Psi_m^* t_{\\mcA_m}^0\"] \\\\\n\t(\\mcM_{\\mcG})_m \\ar[r, \"(t_0)_m\"] & \\Psi_m^* \\mcM_{\\mcA_m} .\n\t\\end{tikzcd} \\]\n\t\n\tIt remains to show that the map $t_0 : \\mcM_{\\mcG} \\left( X_{\\bullet}^0 (E) \\right)^{*,j} \\to \\mathbf{F}_m^j \\otimes \\mcM_{\\mcA_m} \\left( X^0 (E) \\right)$ is a quasi-isomorphism for every $E \\in \\mcE$ and $j \\geq 1$.\n\tNote that \n\t\\[\\mcM_{\\mcG} \\left( X_{\\bullet}^0 (E) \\right)^{*,j} = \\mcG \\left( X_{\\bullet}^0 (E), X_{\\bullet}^j \\right) = \\mcA \\left( X^0 , X^j \\right) \\\\  \\]\n\tand the map\n\t\\[\\mcA \\left( X^0 (E) , X^j \\right) = \\mcM_{\\mcG} \\left( X_{\\bullet}^0 (E) \\right)^{*,j} \\xrightarrow{t_0} \\mathbf{F}_m^j \\otimes \\mcM_{\\mcA_m} \\left( X^0 (E) \\right) \\]\n\tis the inclusion. \n\tNow observe that the following diagram of chain complexes commutes \n\t\\[\\begin{tikzcd}\n\t\\mcA \\left( X^0 (E), X^j \\right) \\ar[r, \"t_0\"] & \\mathbf{F}_m^j \\otimes \\mcM_{\\mcA_m} \\left( X^0 (E) \\right) \\\\\n\t\\mathbf{F}_m^j \\otimes \\mcA_m \\left( X^0 (E), X^j \\right) \\ar[u, equal] \\ar[r, equal] & \\mathbf{F}_m^j \\otimes \\mcA_m \\left( X^0 (E), X^j \\right) \\ar[u, hook] .\n\t\\end{tikzcd} \\]\n\tThe inclusion $\\mcA_m \\left( X^0 (E) , X^j \\right) \\hookrightarrow \\mcM_{\\mcA_m} \\left( X^0 (E) \\right)$ is a quasi-isomorphism according to Lemma \\ref{lemma first result for specific module} (observe that it is important here that $j$ is \\emph{strictly} greater than $0$). Therefore the map $t_0 : \\mcA \\left( X^0 (E), X^j \\right) \\to \\mathbf{F}_m^j \\otimes \\mcM_{\\mcA_m} \\left( X^0 (E) \\right)$ is a quasi-isomorphism, which is what we needed to prove. \n\t\n\\end{proof}\n\n\\begin{lemma}\\label{lemma relation G-A}\n\t\n\tFor every $E \\in \\mcE$ and $j \\geq 1$, the map\n\t\\[\\widetilde{\\Psi} : \\mcH \\left( X_{\\bullet}^0 (E), X_{\\bullet}^0 \\right)^{*,j} \\to \\left( \\mathbf{F} \\left[ t_m \\right] \\otimes \\mcA_m \\left[ W_{\\mcA_m}^{-1} \\right] \\right) \\left( X^0 (E) , X^0 \\right)^{*,j} = \\mathbf{F}_m^j \\otimes \\mcA_m \\left[ W_{\\mcA_m}^{-1} \\right] \\left( X^0 (E) , X^0 \\right) \\]\n\tis a quasi-isomorphism.\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tUsing the first part of Lemma \\ref{lemma G-module map for relation G-A} and Proposition \\ref{prop quasi-isomorphism between localizations}, we know that there exists a chain map $u_m : \\, _{W_{\\mcG_m}^{-1}} (\\mcM_{\\mcG})_m \\left( X_{\\bullet}^0 (E) \\right) \\to \\, _{W_{\\mcA_m}^{-1}} \\mcM_{\\mcA_m} \\left( X^0 \\right)$ such that the following diagram of chain complexes commutes\n\t\\[\\begin{tikzcd}\n\t\\mcH_m \\left( X_{\\bullet}^0 (E) , X_{\\bullet}^0 \\right) \\ar[d, \"_{W_{\\mcG_m}^{-1}} t_{\\mcG_m}\"] \\ar[r, \"\\widetilde{\\Psi}_m\"] & \\mcA_m \\left[ W_{\\mcA_m}^{-1} \\right] \\left( X^0 (E) , X^0 \\right) \\ar[d, \"_{W_{\\mcA_m}^{-1}} t_{\\mcA_m}^0\"] \\\\\n\t_{W_{\\mcG_m}^{-1}} (\\mcM_{\\mcG})_m \\left( X_{\\bullet}^0 (E) \\right) \\ar[r, \"u_m\"] & _{W_{\\mcA_m}^{-1}} \\mcM_{\\mcA_m} \\left( \\Psi_m X \\right) \\\\\n\t(\\mcM_{\\mcG})_m \\left( X_{\\bullet}^0 (E) \\right) \\ar[u, hook] \\ar[r, \"(t_0)_m\"] & \\mcM_{\\mcA_m} \\left( \\Psi_m X \\right) \\ar[u, hook] .\n\t\\end{tikzcd}  \\]\n\tObserve that \n\t\\[\\left\\{\n\t\\begin{array}{lll}\n\t\\mcH_m \\left( X_{\\bullet}^0 (E) , X_{\\bullet}^0 \\right) & = & \\mcH \\left( X_{\\bullet}^0 (E) , X_{\\bullet}^0 \\right)_m \\\\\n\t_{W_{\\mcG_m}^{-1}} (\\mcM_{\\mcG})_m \\left( X_{\\bullet}^0 (E) \\right) & = & _{W_{\\mcG}^{-1}} \\mcM_{\\mcG} \\left( X_{\\bullet}^0 (E) \\right)_m \\\\\n\t(\\mcM_{\\mcG})_m \\left( X_{\\bullet}^0 (E) \\right) & = & \\mcM_{\\mcG} \\left( X_{\\bullet}^0 (E) \\right)_m .\n\t\\end{array}\n\t\\right. \\]\n\tApplying the adjunction of Definition \\ref{definition adjunction} to the last diagram, we get the following commutative diagram of Adams-graded chain complexes\n\t\\[\\begin{tikzcd}\n\t\\mcH \\left( X_{\\bullet}^0 (E) , X_{\\bullet}^0 \\right) \\ar[d, \"_{W_{\\mcG}^{-1}} t_{\\mcG}\"] \\ar[r, \"\\widetilde{\\Psi}\"] & \\mathbf{F} \\left[ t_m \\right] \\otimes \\mcA_m \\left[ W_{\\mcA_m}^{-1} \\right] \\left( X^0 (E) , X^0 \\right) \\ar[d, \"\\mathrm{id} \\otimes _{W_{\\mcA_m}^{-1}} t_{\\mcA_m}^0\"] \\\\\n\t_{W_{\\mcG}^{-1}} \\mcM_{\\mcG} \\left( X_{\\bullet}^0 (E) \\right) \\ar[r, \"u\"] & \\mathbf{F} \\left[ t_m \\right] \\otimes _{W_{\\mcA_m}^{-1}} \\mcM_{\\mcA_m} \\left( \\Psi_m X \\right) \\\\\n\t\\mcM_{\\mcG} \\left( X_{\\bullet}^0 (E) \\right) \\ar[u, hook] \\ar[r, \"t_0\"] & \\mathbf{F} \\left[ t_m \\right] \\otimes \\mcM_{\\mcA_m} \\left( \\Psi_m X \\right) \\ar[u, hook] .\n\t\\end{tikzcd} \\]\n\tSpecializing to the components of fixed Adams degree $j \\geq 1$, we get the following commutative diagram of chain complexes \n\t\\[\\begin{tikzcd}\n\t\\mcH \\left( X_{\\bullet}^0 (E) , X_{\\bullet}^0 \\right)^{*,j} \\ar[d, \"_{W_{\\mcG}^{-1}} t_{\\mcG}\"] \\ar[r, \"\\widetilde{\\Psi}\"] & \\mathbf{F}_m^j \\otimes \\mcA_m \\left[ W_{\\mcA_m}^{-1} \\right] \\left( X^0 (E) , X^0 \\right) \\ar[d, \"\\mathrm{id} \\otimes _{W_{\\mcA_m}^{-1}} t_{\\mcA_m}^0\"] \\\\\n\t_{W_{\\mcG}^{-1}} \\mcM_{\\mcG} \\left( X_{\\bullet}^0 (E) \\right)^{*,j} \\ar[r, \"u\"] & \\mathbf{F}_m^j \\otimes _{W_{\\mcA_m}^{-1}} \\mcM_{\\mcA_m} \\left( \\Psi_m X \\right) \\\\\n\t\\mcM_{\\mcG} \\left( X_{\\bullet}^0 (E) \\right)^{*,j} \\ar[u, hook] \\ar[r, \"t_0\"] & \\mathbf{F}_m^j \\otimes \\mcM_{\\mcA_m} \\left( \\Psi_m X \\right) \\ar[u, hook] .\n\t\\end{tikzcd} \\]\n\tUsing Lemmas \\ref{lemma properties of modules}, \\ref{lemma properties of modules bis}, and \\cite[Lemma 3.13]{GPS20}, we know that all the vertical maps on the left are quasi-isomorphisms. \n\tSimilarly, using Lemmas \\ref{lemma second result for specific module}, \\ref{lemma third result for specific module}, and \\cite[Lemma 3.13]{GPS20}, we know that all the vertical maps on the right are quasi-isomorphisms. \n\tMoreover, the second part of Lemma \\ref{lemma G-module map for relation G-A} states that the lowest horizontal map is a quasi-isomorphism. \n\tThus, the chain map $\\widetilde{\\Psi} : \\mcH \\left( X_{\\bullet}^0 (E) , X_{\\bullet}^0  \\right)^{*,j} \\to \\mathbf{F}_m^j \\otimes \\mcA_m [ W_{\\mcA_m}^{-1} ] \\left( X^0 (E) , X^0 \\right)$ is a quasi-isomorphism. \n\t\n\\end{proof}\n\n\\subsubsection{End of the proof}\n\nWe end the section with the proof of Theorem \\ref{thm mapping torus in weak situation}.\nNow that we have proved Lemma \\ref{lemma relation G-A} which takes care of the \\emph{positive} Adams-degrees, we have to treat the zero Adams-degree part (recall that $\\mcH$ is concentrated in non-negative Adams-degree because $\\mcA$ is assumed to be weakly directed).\n \nLet $\\mcI$ be the (non full) $A_{\\infty}$-subcategory of $\\mcH$ with \n\\[\\mathrm{ob} \\left( \\mcI \\right) = \\left\\{ X_{\\bullet}^0 (E) \\mid E \\in \\mcE \\right\\} \\text{ and } \\mcI \\left( X, Y \\right) = \\mcG \\left( X, Y \\right) \\oplus \\left( \\bigoplus \\limits_{j \\geq 1} \\mcH \\left( X, Y \\right)^{*,j} \\right) \\]\n(recall that if $V$ is an Adams-graded vector space, we denote by $V^{*,j}$ its component of Adams-degree $j$). \n\n\\begin{lemma}\n\t\n\tThe inclusion $\\mcI \\hookrightarrow \\mcH$ is a quasi-equivalence.\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tObserve that the inclusion $\\mcI \\hookrightarrow \\mcH$ is cohomologically essentially surjective because every object of $\\mcH$ can be related to one of $\\mcI$ by a zigzag of morphisms in $W_{\\mcG}$, which are quasi-isomorphisms in $\\mcH$ (see \\cite[Lemma 3.12]{GPS20}). \n\tTherefore, it suffices to show that the inclusion \n\t\\[\\mcG \\left( X_{\\bullet}^0 \\left( E \\right), X_{\\bullet}^0 \\left( E_{\\diamond} \\right) \\right) \\hookrightarrow \\mcH \\left( X_{\\bullet}^0 \\left( E \\right), X_{\\bullet}^0 \\left( E_{\\diamond} \\right) \\right)^{*,0} \\]\n\tis a quasi-isomorphism for every $E, E_{\\diamond} \\in \\mcE$.\n\t\n\tLet $E_{\\diamond}$ be an element of $\\mcE$. When we write an object $X_{\\bullet}^n$ without specifying the element of $\\mcE$, we mean $X_{\\bullet}^n (E_{\\diamond})$.\n\tRecall that we introduced a pair $\\left( \\mcM_{\\mcG}, t_{\\mcG} \\right)$ in Definition \\ref{definition module for G}.\n\tAccording to Lemmas \\ref{lemma properties of modules}, \\ref{lemma properties of modules bis} and \\cite[Lemma 3.13]{GPS20}, the inclusion $\\mcM_{\\mcG} \\left( X^0_{\\bullet} (E) \\right) \\hookrightarrow \\, _{W_{\\mcG}^{-1}} \\mcM_{\\mcG} \\left( X^0_{\\bullet} (E) \\right)$ and the map $_{W_{\\mcG}^{-1}} t_{\\mcG} : \\, \\mcH \\left( X^0_{\\bullet} (E) , X^0_{\\bullet} \\right) \\to \\, _{W_{\\mcG}^{-1}} \\mcM_{\\mcG} \\left( X^0_{\\bullet} (E) \\right)$ are quasi-isomorphisms for every $E \\in \\mcE$. Besides, observe that \n\t\\[\\mcM_{\\mcG} \\left( X^0_{\\bullet} (E) \\right)^{*,0} = \\mcG \\left( X^0_{\\bullet} (E),X^0_{\\bullet} \\right) . \\]\n\tThe result then follows from the commutativity of the following diagram\n\t\\[\\begin{tikzcd}\n\t\\mcG \\left( X_{\\bullet}^0 \\left( E \\right), X_{\\bullet}^0 \\right) \\ar[d, equal] \\ar[r, equal] & \\mcG \\left( X_{\\bullet}^0 \\left( E \\right), X_{\\bullet}^0 \\right) \\ar[d, hook] \\ar[r, equal] & \\mcG \\left( X_{\\bullet}^0 \\left( E \\right), X_{\\bullet}^0 \\right) \\ar[d, hook] \\\\\n\t\\mcM_{\\mcG} \\left( X^0_{\\bullet} (E) \\right)^{*,0} \\ar[r, hook, \"\\sim\"] & _{W_{\\mcG}^{-1}} \\mcM_{\\mcG} \\left( X^0_{\\bullet} (E) \\right)^{*,0} & \\mcH \\left( X^0_{\\bullet} (E) , X^0_{\\bullet} \\right)^{*,0} \\ar[l, \"\\sim\" above, \"_{W_{\\mcG}^{-1}} t_{\\mcG}\" below] .\n\t\\end{tikzcd} \\]  \n\t\n\\end{proof}\n\nThe following diagram of Adams-graded $A_{\\infty}$-categories is commutative\n\\[ \\begin{tikzcd}\n\\mcH \\ar[r, \"\\widetilde{\\Psi}\"] & \\mathbf{F} \\left[ t_m \\right] \\otimes \\mcA_m \\left[ W_{\\mcA_m}^{-1} \\right] \\\\\n\\mcI \\ar[u, hook, \"\\sim\" right] \\ar[r, \"\\widetilde{\\Psi}\"] & \\mcA_m^0 \\oplus \\left( t \\mathbf{F} \\left[ t_m \\right] \\otimes \\mcA_m \\left[ W_{\\mcA_m}^{-1} \\right]^0 \\right) \\ar[u, hook]\n\\end{tikzcd} \\]\n(recall that if $\\mcC$ is an $A_{\\infty}$-category equipped with a splitting $\\mathrm{ob} \\left( \\mcC \\right) \\simeq \\mathbf{Z} \\times \\mcE$, then we denote by $\\mcC^0$ the full $A_{\\infty}$-subcategory of $\\mcC$ whose set of objects corresponds to $\\{ 0 \\} \\times \\mcE$).\nMoreover, since $\\mcA$ is assumed to be weakly directed with respect to the $\\mathbf{Z}$-splitting of $\\mathrm{ob} \\left( \\mcA \\right)$, Lemma \\ref{lemma relation G-A} implies that the bottom horizontal $A_{\\infty}$-functor is a quasi-equivalence. \nTherefore we have\n\\[\\mcH \\simeq \\mcA_m^0 \\oplus \\left( t \\mathbf{F} \\left[ t_m \\right] \\otimes \\mcA_m \\left[ W_{\\mcA_m}^{-1} \\right]^0 \\right) . \\]\nRecall that $W_{\\mcA_m} = f \\left( \\mathrm{units} \\right) \\cup \\{ \\mathrm{units} \\}$, so that \n\\[\\mcA_m \\left[ W_{\\mcA_m}^{-1} \\right] \\simeq \\mcA_m \\left[ f \\left( \\mathrm{units} \\right)^{-1} \\right] . \\]\nThis concludes the proof of Theorem \\ref{thm mapping torus in weak situation}, since $\\mcH$ is quasi-equivalent to the mapping torus of $\\tau$ (see Lemma \\ref{lemma relation G - mapping torus}).\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{INTRODUCTION}\n\nEarly-type galaxies have been studied in great detail in rich clusters\nup to redshift one and above (see Stanford, Eisenhard \\&\nDickinson, 1998), with the basic\nresult that passive evolution in luminosity and no evolution in mass is\nruling them to at least $z\\simeq 1$. Old ages and an early coeval epoch of\nformation are then implied by these observations.  It is still unclear, \nhowever, how much the rich-cluster environment is representative\nof the general population. Indeed, contradictory results have been\nreported about the evolutionary properties of early-type field galaxies.\n\nThe analysis by Im et al. (1996) of a sample of elliptical galaxies in the HST\nMedium Deep Survey seems to indicate that these galaxies share similar\nproperties with the cluster objects, a result based however on a sample of\n376 galaxies with only 24 spectroscopic redshifts and photometric \nredshifts based on (V--I) colors. Similarly, Lilly et al. (1996) find that the luminosity function of red galaxies in the Canada-France\nRedshift Survey (CFRS) does not exhibit significant changes in the redshift\ninterval $0.2\\leq z \\leq1$. These observations imply an early\nepoch of formation for the elliptical galaxies.\n\nThese conclusions have been questioned by Kauffmann, Charlot, \\& White (1996)\n(see also Baugh, Cole, \\& Frenk, 1996), who claim evidence for a decrease \nin the mass \ndensity already at $z\\leq 1$ for early-type galaxies in the CFRS, \nthese too selected from V--I colors.\n\nAn exceedingly deep and clean view of the field galaxy populations to high\nredshifts is provided by a long integration of HST in the so-called {\\it\nHubble Deep Field} (HDF, Williams et al., 1996).  Though having provided\nrelevant constraints on galaxy formation and evolution and about the\nepoch of production of metals (e.g. Madau et al.\\\n1996; Connolly et al.\\ 1997; Sawicki, Lin, \\& Yee, 1997), \nthe interpretation of these data alone is made difficult by the\nrelatively short selection wavelengths (essentially those\nof the U, B, V, I bands), which\nimply strong evolutionary and K-corrections as a function of redshift\n(Giavalisco, Steidel, \\& Macchetto, 1996). \nThese may be particularly severe for\nthe early-type galaxies, because of their quickly evolving optical\nspectra and extreme K-corrections (e.g. Maoz, 1997).\n\nAnother unsolved problem specifically affecting short wavelength observations\nconcerns the possible effects of dust present\nin the line-of-sight to the object: even small amounts may seriously affect\nspectra corresponding to rest-frame far-UV wavelengths.\nVarious analyses attempting to solve this issue have compared the observed\nUV spectra with templates of young dusty galaxies (e.g. Meurer et al.\\ 1997).\nIt is evident, however,\nthat even small variations in the dust properties and in the assumed spectral\ntemplates imply widely discrepant predictions for the extinction corrections\n(published estimates range from less than 1 to more than 3 magnitudes of \nextinction in high-z galaxies). Any inferences\nabout the history of star-formation are correspondingly uncertain. \n\nWe try in this paper an alternative approach to the past history of galaxies\nbased on a thorough photometric and morphological analysis of a complete\nK-band selected sample of elliptical and S0 galaxies in the HDF.\nThe selection in the K infrared band helps to overcome most of\nthe above problems of optical selection, in particular minimizes the effects\nof K- and evolutionary corrections and of extinction by any residual\nintervening dust. Another crucial advantage of using near-infrared data\nis that the integrated fluxes at these wavelengths are contributed by\nall (low-mass) stars dominating the baryonic content of a galaxy.\n\nVarious deep integrations have been performed in the HDF at near-IR\n(Cowie et al.\\ 1996; Dickinson et al. 1997) and mid-IR wavelengths\n(Rowan-Robinson et al. 1997; see also \nAussel et al. 1998), which, combined with\nthe extreme quality of the morphological information and the very good\nspectroscopic coverage, make this area unique, in particular for the\ninvestigation of early-type galaxies outside rich clusters.\n\nWe concentrate here on a (morphologically selected)\nsample of 35 early-type galaxies. Our interest for this sub-population rests\non its homogeneous morphological properties, indicative of a\npurely stellar emission (hence requiring a relatively simple\nmodelling), and on the expected old ages (by comparison with their rich cluster\nanalogues). Altogether, this class of objects is expected to inform about\nthe high-redshift side of the star-formation history, which has perhaps been \nonly partly sampled by direct optical selection via the {\\it drop-out} \ntechnique. In our approach, \nthe inferred evolutionary history of stellar populations is essentially\nunaffected by the mentioned problems related to dust extinction.\n\nSection 2 of this paper provides details about the selection of the sample,\nthe adopted procedures in the photometric and morphological analyses, and\nstatistical tests of completeness carefully dealing with \nobservational limits on the total flux and surface brightness. \nIn Section 3 we analyze physical properties of galaxies as\ninferred from the morphology and from fits to the broad-band UV-optical-IR\nspectra. Section 4 addresses the statistical properties of the sample\n(counts, redshift distributions, identification statistics), and compare them\nwith those of local samples of early-type galaxies. \nWe discuss, in particular, evidence that the redshift distribution\nbreaks above $z\\gtrsim 1.3$.\nVarious interpretations of these results are discussed in Section 5, \nwhere the effects of repeated merging events, probably in the presence \nof a dust-polluted medium, are considered. \nThis information is used in Section 6 to constrain the history of star \nformation of stellar populations, and to compare it with other published \nestimates.\nOur main results are finally summarized in Section 7.\n\nWe adopt $H_0=50\\ Km\/s\/Mpc$ throughout the paper. The analysis is made for\ntwo values of the cosmological deceleration parameter, $q_0=0.5$ and 0.15, assuming\nzero cosmological constant $\\Lambda$. Note that the effects of increased time-scales \nand volumes of an open universe might be also obtained in a closure world model\nwith a non-zero $\\Lambda$. \n\n\n\\section{SAMPLE SELECTION AND PHOTOMETRY}\n\nDickinson et al. (1997) obtained deep near-infrared images of the $HDF$ \nwith the IRIM camera mounted at the KPNO-4m Telescope. The\ncamera employs a 256x256 NICMOS-3 array with $0''.16$\/pixel, but the\nreleased images are geometrically transformed and rebinned (with\nappropriate pixel weighting) into a 1024x1024 format. \nIRIM has observed the same area in the J, H and K filters, for a\ntotal of 12, 11.5 and 23 hours, respectively. Formal\n5-$\\sigma$ limiting magnitudes for the HDF\/IRIM images, computed\nfrom the measured sky noise within a $2''$ diameter circular aperture,\nare $23^m.45$ at J, $22^m.29$ at H, and $21^m.92$ at K, whereas the\nimage quality is $\\sim 1''.0$ $FWHM$.\n\nOur galaxy sample has been extracted from the HDF\/IRIM K--band image \nthrough a preliminar selection based on\nthe automatic photometry provided by SExtractor (Bertin and Arnouts 1996).\nIt is flux--limited in the K--band and it includes only galaxies whose\nmorphology strongly suggests an {\\it early--type} classification. The measure of\nmagnitude is based on fixed  2.5$''$ aperture photometry (which best accounts for\nthe seeing of the K image) and applying a stellar correction for the outer part.\n\n\n\\subsection{Monte-Carlo tests of completeness}\n\nIn order to set the appropriate value of the magnitude limit $K_L$ for\ninclusion in our sample, we have produced a synthetic frame containing\na 10$\\times$10 grid of toy galaxies with $r^{1\/4}$ luminosity profiles,\nhaving total $K$ magnitudes and effective radii $r_e$ spanning the\nranges $19^m.00\\div 21^m.25$ (step $0^m.25$) and\n$0^{\\prime\\prime}.08\\div 0^{\\prime\\prime}.80$ (step\n$0^{\\prime\\prime}.08$), respectively. These limits were chosen to\nprovide a full characterization of the performances of SExtractor\n(i.e. identification capability and magnitude estimate of galaxies) in\nthe critical range of $K$ magnitudes and for values of the true\neffective radius typical of the $HDF$ ellipticals (Fasano and\nFilippi~1998, hereafter FF98; see also Fasano et al.~1998, hereafter\nFA98).\n\nWe are particularly interested in evaluating the probability of\ndetection, as well as possible biases in the magnitude estimation, as\na function of the magnitude itself and of $r_e$ (i.e. of the `true'\naverage surface brightness $<\\mu_e^K>$, and, after all, of the\nredshift).  We have convolved this synthetic frame using a $PSF$\nderived by multi--gaussian fitting of the few stars included in the\n$K$--band image. Then, a bootstrapping procedure has been carried out\nin order to produce 10 different images mimicking the 'true' image\nnoise. These images have been processed with SExtractor to get,\nfor each toy galaxy, ten different SExtractor estimates of the total \n$K$ magnitude, and then the average magnitude $<K_{SEx}>$, the standard\ndeviation $\\sigma_{K_{SEx}}$ (or the difference $\\Delta K =\n<K_{SEx}>-K_{true}$).\n\nFrom these simulations of galaxies having $r^{1\/4}$ luminosity profiles\nwe conclude the following. {\\it a)} For the\nwhole range of tested magnitudes and radii, the fraction of detections\nis equal to unity up to $<K_{SEx}>\\simeq 21^m.4$ and $<\\mu_e^K>\\simeq\n22^m.0$. {\\it b)} \nThe magnitudes estimated by SExtractor are systematically fainter\nthan the true magnitudes, the bias mainly depending on the\ngalaxy surface brightness.\n\nFigure 1a shows that, if we consider only galaxies with\n$<K_{SEx}>\\lesssim 20^m.7$ ($K_{lim}$ hereafter), the standard\ndeviation of the SExtractor magnitude estimates is less than $0^m.05$\n($\\sigma_{max}$ hereafter).  Figure 1b reports $\\Delta K$ as a\nfunction of $<\\mu_e^K(SEx)>$ for the sub--sample of toy galaxies with\n$<K_{SEx}>\\lesssim K_{lim}$. It shows that $\\Delta K$ systematically\nincreases at increasing $<\\mu_e^K(SEx)>$, approaching the maximum value\n$\\Delta K_{max}\\sim 0^m.5$ for $<\\mu_e^K(SEx)>\\simeq 22^m.0$. The solid\nline in Figure 1b is a polynomial fit to the data:\n\n\\begin{equation}\n\\log(\\Delta K) = -0.830 + 0.147\\times<\\mu_*^K> + 0.019\\times<\\mu_*^K>^2\n\\end{equation}\n\n\\noindent where $<\\mu_*^K> = <\\mu_e^K(SEx)> - 19$.\n\nThe plots in Figure 1 provide a straightforward indication of the\nstrategy to be used in order to obtain a complete flux limited sample\nof early--type galaxies from the $IRIM$ $K$--band image using\nSExtractor. The procedure consists of the following steps: 1) to\nproduce a catalog of objects with $K_{SEx}\\lesssim K_{lim}=20^m.7$;\n2) to check the morphology of each object on the WFPC2 frames,\nremoving stars and late--type (or irregular) galaxies from the sample;\n3) to exploit the high resolution and depth of the $V_{606}$ and\/or\n$I_{814}$ images to derive the effective radii of the galaxies and use\nthese values of $r_e$ to compute $<\\mu_e^K>$, assuming a roughly\nconstant color profile; 4) to apply to the $K_{SEx}$ magnitudes the\nstatistical corrections given in the previous equation; 5) to include\nin the final sample only galaxies with corrected magnitude less than\nor equal to $K_L = K_{lim} - \\Delta K_{max} - \\sigma_{max} = 20^m.15$.\n\n\\subsection{The morphological filter}\n\nThe most delicate steps in the previous scheme are those concerning\nthe morphological analysis and the evaluation of the effective radii\n(points 2 and 3 of the previous list). They are widely discussed in\nFF98 and FA98 and we refer to these papers for details. Here we wish\nto summarize the 'flow--chart' of the morphological filtering for the\npresent sample. The SExtractor automatic photometry of the $IRIM$\n$K$--band image produced a preliminar sample of 109 objects with\n$K_{SEx}\\leq 20^m.7$ (first point of the previous\nselection scheme). All these objects were examined with the {\\it\nIMEXAM--IRAF} tool to produce a first (conservative) screening against\nstars or late-type and irregular galaxies, resulting in a temporary\nlist of 47 {\\it early--type} candidates. \nOur quantitative morphological classification filter assumes a dominant\n$r^{-1\/4}$ profile for the bulk of the light distribution in the galaxy.\n\nFor 29 galaxies in this list\nthe surface photometry in the ST--$V_{606}$ band is available in FF98\nand the total AB magnitudes in the four $WFPC2-HDF$ bands are given in\nFA98, together with the equivalent effective radii derived after\ndeconvolution of the luminosity profiles. Four more galaxies included\nin both our and FF98 samples, were not analyzed by FA98. For these\ngalaxies we computed optical AB magnitudes and effective radii according \nto the prescriptions given in FA98. The remaining 14 galaxies in the\ntemporary list are not in the FF98 sample, since in the $V_{606}$ \nband they did not satisfy the selection criteria adopted in order to \nsecure a reliable morphological analysis. We performed the detailed\nsurface photometry of these galaxies in the $I_{814}$ band (where they \nshow a better S\/N ratio), producing luminosity and geometric profiles \nof each galaxy. From this analysis, 3 objects were recognized to\nbe '{\\it disk-dominated}' galaxies (likely $Sa$), whereas two more objects \nshowed peculiar or unclassifiable profiles. These galaxies were excluded\nfrom the temporary list. \n\nThe remaining 9 galaxies with surface photometry \nin the $I_{814}$ band were analyzed following the same procedure described \nin FA98 to derive the total AB magnitudes in the optical bands, \nas well as the '{\\it true}' (deconvolved) equivalent effective radii.\nFigure 2 shows the luminosity profiles of these galaxies derived from \nthe surface photometry in the $I_{814}$ band.\n\n\\subsection{The final sample}\n\nAltogether, of the 47 candidate E\/S0 galaxies with $K_{SEx}\\leq 20.7$, 33 were\nalready classified as E\/S0 by FA98 and FF98, while 9 are confirmed as E\/S0\nby the present paper, for a total of 42 objects in the incomplete sample at\n$K_{SEx}\\leq 20.7$. To ensure a highly reliable completeness limit, following\nsteps 4 and 5 of the selection scheme discussed  in Sect. 2.1, \nwe used equation (1) to correct the $K_{SEx}$ magnitudes of the 42\ngalaxies in the temporary sample. In this way we \nobtain our final sample of 35 galaxies with corrected magnitude $K\\leq 20^m.15$ over the\nHDF (3 WF + 1 Planetary Camera) area of 5.7 square arcmin. Some basic data \non the sample are listed in Table 1. \n\nThe $H$ and $J$ magnitudes listed in the Table have been obtained\nrunning SExtractor on the corresponding $IRIM$ images and\naccounting for the expected biases with the same procedure adopted for\nthe $K$ magnitudes (simulations of toy galaxies, convolution with proper\n$PSF$, noise bootstrapping and correlation between bias and average\nsurface brightness). The corresponding equations are:\n\n\\begin{equation}\n\\log(\\Delta H) = -0.903 + 0.288\\times<\\mu_*^H> - 0.0158\\times<\\mu_*^H>^2\n\\end{equation}\n\n\\begin{equation}\n\\log(\\Delta J) = -1.014 + 0.162\\times<\\mu_*^J> + 0.0140\\times<\\mu_*^J>^2\n\\end{equation}\n\n\\noindent \nwhere $<\\mu_*^H> = <\\mu_e^H(SEx)> - 20$ and $<\\mu_*^J> = <\\mu_e^J(SEx)> \n- 19$.\n\nFor 15 galaxies in Table 1 spectroscopic redshifts are available in the\nliterature or in the WEB (see notes to Table 1), while for the remaining \n20 objects a\nreliable estimate of $z$ is obtained from fits of the optical-IR\nbroad-band spectra (see Sect. 3.2).\n\nThe last column in Table 1 gives some information on the surface photometry.\nIn particular, the symbol (814 Ph.) indicates galaxies whose surface \nphotometry has been obtained in the $I_{814}$ band, while the symbols \n(T1), (T2) and (T3) indicate galaxies which, according to FF98, belong \nto the `{\\it Normal}', '{\\it Flat}' and '{\\it Merger}' class, respectively. \nThese attributes refers to the classification by FF98 \n(see also Fasano et al. 1996),\nbased on the luminosity profiles, of {\\it early--type} galaxies \nin the $HDF$: {\\it 1)} the '{\\it Normal}' class, for which\nthe de~Vaucouleurs law is closely followed down to the innermost isophote \nnot significantly affected by the $PSF$; {\\it 2)} the '{\\it Flat}' class, \ncharacterized by an inward flattening of the luminosity \nprofiles (with respect to the de~Vaucouleurs' law) which cannot be \nascribed to the effect of the $PSF$ ; {\\it 3)} the '{\\it Merger}' class, \nin which isophotal contours show the existence of complex \ninner structures (two or more nuclei) embedded inside a common envelope, \nwhich roughly obeys the $r^{1\/4}$ law.\n\nThe completeness of the sample has been evaluated from numerical simulations \nas described in Sect. 2.1. \nWe have not used the classical $V\/V_{max}$ test to this purpose,\nsince it is equally sensitive to departures from spatial homogeneity\nthan to completeness. The question of the distribution of objects in the\nspace-time will be addressed in Sect. 4 below.\n\n\n\n\n\n\\section{ANALYSIS OF THE BROAD-BAND SPECTRA AND OF THE SURFACE BRIGHTNESS\nDISTRIBUTIONS}  \n\n\nThe surface photometry of the images of our sample galaxies has shown that \nfor the large majority of them \nthere are no morphological signatures of the presence of dust,\ne.g. obscured lanes across the galaxy, asymmetric brightness distributions\nor profiles deviating from the $r^{-1\/4}$ over the bulk of the galaxy\n(see however Witt, Thronson, \\& Capuano, 1992, for a cautionary remark about the\nlatter point).\nOnly three galaxies belong to the morphological class T3, where on-going\nstar-formation, possibly enshrouded by \ndust, is likely to have a role. Four more objects \nbelong to the class T2, for which any dust, whenever present, should most probably \nbe confined to the inner core of the galaxy, hence should not affect the \nglobal emission. Five out of 7 of the morphologically peculiar objects have blue (class [a], as discussed in Sect. 3.2) spectra.\n\nAn obvious way to test for the presence of dust would be to look for its\nre-radiation at infrared wavelengths. The Infrared Space Observatory has\nobserved the HDF in two broad-band filters centered at 6.7 and 15 $\\mu m$.\nWhile the former, for redshifted objects, is dominated by photospheric \nemission of old stars, the latter is contributed by very hot dust and\ntransiently heated molecules or very small dust particles present in the ISM.\nMann et al. (1997) and Goldschmidt et al. (1997) report catalogues of sources\ndown to rather conservative flux limits. At 15 $\\mu m$ only one ISO source is\nin common with our list (ID 2-251-0), a galaxy with a point-source evidencing\nin the U and B, probably an AGN. Though a conclusion has to wait for a \nrefined analysis of the ISO observations, aimed at deeper flux limits \n(Aussel et al. 1998, Desert et al. 1988), \nthe available information seems to confirm       \nthat there is little room for dust emission by our E\/S0 galaxies. \n\nThe present analysis will take advantage of this simplified behavior for the \nbulk of our objects,\nby allowing modelling of galaxy spectra as the integrated emission of purely \nstellar populations. The inclusion of the additional effect of dust extinction,\nwhenever present, would have substantially weakened our conclusions, since\ndegenerate sets of solutions could have been possible, implying\nuncertain estimates of the photometric redshift and of the galaxy age.\n\n\n\\subsection{The interpretative tool: a model for the evolutionary\nspectral energy distributions of the integrated starlight}  \n\nThe UV-optical-near\/IR Spectral Energy Distributions (SED) for all\ngalaxies in the sample have been fitted with synthetic spectra based on\na model briefly summarized here. A general framework for this modelization \ncan be\nfound in Bressan, Chiosi, \\& Fagotto (1994), while a critical assessment of problems\nand limitations is discussed by Charlot, Worthey, \\& Bressan (1996).\n\nThe code follows two paths, one describing the chemical evolution \nof the galaxy's ISM, the other\nassociating to any galactic time a generation of stars with the\nappropriate metallicity, and adding up the contributions to the integrated\nlight of all previous stellar populations.\n\nThe chemical path adopts a Salpeter IMF with a lower limit $M_l= 0.15\\ \nM_\\odot$, and a Schmidt-type law for the\nStar Formation Rate (SFR): $\\Psi(t)=\\nu \\, M_{g}(t)^{k}$, where $\\nu$ is the\nSF efficiency. A further parameter is the time-scale $t_{infall}$ for the \ninfall of the primordial gas. The library of isochrones is \nbased on the Padova models, spanning a\nrange in metallicities from $Z=0.004$ to 0.05 (i.e. from 0.2 to 2.5 solar).\nThis range is appropriate for stellar populations in local early-type \ngalaxies, which have solar metallicity on average \n(e.g. Grillmair et al. 1996).\n\nThe isochrones are modified to account for the contribution by dust-embedded \nMira and OH stars during the AGB phase (Bressan, Granato \\& Silva 1998), when\ncircum-stellar dust reddens the optical emission and produces an IR bump\nat 5-20 $\\mu m$. This IR emission from AGB stars is important for\ngalactic ages of 0.1 to a few Gyrs.\n\nA number of evolutionary patterns for the time-dependent SFR $\\Psi(t)$\nhave been tried to reproduce the galaxy SEDs, to estimate the photometric\nredshifts for galaxies lacking the spectroscopic identification, and to\nfit the global statistical properties of the sample.\nHowever, we will mostly refer in the following to two paradigmatic evolution \ncases.\n\nThe first model (henceforth Model 1) reproduces a classical scheme for the formation\nof ellipticals (e.g. Larson 1974), i.e. a huge starburst on short\ntimescales, expected to occur at high redshifts.\nIn our approach the star-formation has a maximum at a galactic age of\n0.3 Gyr and continues at substantial rates up to 0.8 Gyr, after which it\nis assumed that the input of energy by stellar winds and supernovae\nproduced a sudden outflow of the ISM through a galactic wind, stopping\nthe bulk of the SF. The evolution at later epochs is mostly due to\npassive aging of already formed stellar populations. \nThis evolution pattern is achieved with the following choice of the\nparameters: $t_{infall}=0.1$ Gyr, $k=1$, $\\nu=2\\ Gyr^{-1}$. \nThe precise time dependence of the SF rate is reproduced in Figure 3 (dotted line).\nThe redshift for the onset of star-formation $z_F$ in the galaxy\nis a free parameter.\n\nOur second model (henceforth Model 2) \ngives up the concept that the stellar populations in field\nearly-type galaxies are almost coeval, and assumes instead that the\nstar-formation lasts for a significant fraction of the Hubble time.\nThis is obtained with the following parameters:\n$t_{infall}=1$ Gyr, $k=1$, $\\nu=1.3\\ Gyr^{-1}$. \nThe corresponding SF law has a broad peak at 1.4 Gyrs but goes on at\nsubstantial rates for a couple Gyrs more. For a value of the\nbaryonic mass of $M=10^{11}M_\\odot$ (as typical for our sample galaxies,\nsee below), this brings to an energetic unbalance in the ISM followed by\na galactic wind at 3.1 Gyrs (see continuous line in Fig. 3 \nfor the detailed evolution of the SF). \n\nThis adopted SF law is clearly an over-simplified\npicture of a process which has been likely more complex. In particular,\na protracted SF, as implied by the second model, is likely to have been occurred \nthrough a set of successive starbursts (e.g. due to mergers or strong dynamical interactions). Since we are dealing with the integrated emission of all\nstellar generations, there is virtually no difference, as for the broad-band\nspectral appearance in the after-burst phase, between a set\nof starbursts occurring over 3 Gyrs and a continuous SF during this period.\n\nTo provide these simplified schemes with more flexibility, we have allowed\na residual star-formation $\\Psi_0$ at a basal level to be added to the flux\nemitted by the passively evolving populations. \nThis allows to fit the spectra of the four bluest objects, and to improve\nthe fits at the shortest (U, B) wavelengths for additional galaxies.\nSince typical values for  $\\Psi_0$ are much less than 1 $M_{\\odot}\/yr$\nand this residual SF does not contribute significantly to the mass and\nenergetics,\nthis parameter is used only in the spectral fits of Fig. 4 below and\nnever more in the subsequent analyses.\n\n$\\Psi_0$ is then a second free \nparameter. The baryonic mass $M$ and the photometric redshift (for cases\nwhere it is needed) are further parameters in the spectral fitting procedure.\n\n\n\\subsection{Fitting the galaxy broad-band spectra: evaluation of the\nphotometric redshifts and of the galactic ages} \n\n\\subsubsection{The photometric redshifts } \n\nFigure 4 is a collection of the observed spectra for the 35 sample galaxies. \nThe figure displays essentially two kinds of spectral behaviors: \nclass {\\it (a)} spectra, which are rather blue at all \nwavelengths, dominated by young stellar populations (10 objects\nout of 35); class {\\it (b)} spectra, with overall redder properties, \ndisplaying a typical two power-law behavior, with a break at $\\lambda\n\\simeq 0.4\\ \\mu m$ in the rest-frame spectrum in correspondence of the \nBalmer decrement (25 sources). \nNo objects, even those at the lowest redshifts, are found to\ndisplay very red colors, as would be expected for a very old stellar\npopulation dominating the spectrum. This result is consistent with \na statistical study by Zepf (1997) revealing a lack of very red objects in \nthe field.\n\nA crucial step in our analysis is the evaluation of redshifts\nfrom broad-band spectral fitting\nfor sources lacking a spectroscopic identification. The uncertainties \nrelated with this estimate have been discussed by many authors (see\nin particular Connolly et al. 1997; Hogg et al. 1998),\nwith the general outcome that the inclusion of near-infrared data\nin the analysis (added to the 4 HST bands) makes the redshift\nestimate quite more reliable.\n\nWith respect to these analyses we benefit here of various advantages.\nThe first one is due to the lack of evidence for dust in our objects,\nwhich breaks down at least one possible degeneracy in the parameter space\n(that is estimating a lower photometric z from a dust-reddened template).\nThe second one is that most of the observed spectra have rather \nhomogeneous and monotonic behaviors as typical of early-type galaxies,\nwith an easily discernible Balmer feature.\nA third potential advantage, as verified \"a posteriori\", \nis given by our relatively bright K-band selection, which\ntends to select objects only up to moderate redshifts ($z<1.5$, see Sect.\n4-5) and to exclude very high-z galaxies, whose redshift estimate would be\nquite more uncertain because of the strong evolutionary corrections in the \ntemplate spectra.\n\nMore specifically, \nfor class {\\it (b)} sources lacking spectroscopic redshifts (i.e. 15 galaxies\nin total)\nthe estimate of the redshift from spectral fitting is quite robust, thanks\nto a well-defined spectral break corresponding to the Balmer decrement,\ntypically occurring at $\\lambda \\sim$ 0.6 to 1 $\\mu m$. \nThe combined use of the\n3 near-IR and the 4 HST bands, with very small photometric errors, allows\nan accurate determination of the 4000 A break from a model-assisted spectral \ninterpolation.\nFor class {\\it (a)} spectra without spectroscopic redshifts (5 more galaxies),\nthere is still clear evidence for a 4000 A break, which is however less\nprecisely characterized in some instances. In these cases the photometric \nredshift may turn out to be significantly more uncertain.\n\nFigure 5 summarizes a comparison of our redshift estimate based on broad-band\nspectral fitting with the actual measurement from mid-resolution spectroscopy, for\nthe 15 galaxies in our sample with this information. The errorbars associated with\nthe photometric estimate are conservative $\\sim$95\\% confidence intervals based on\n$\\chi^2$ fitting using Model 2 as spectral template. In only one case \n(object 2-251-0) the photometric\nmeasure is significantly discrepant with respect to the spectroscopic one, but\nthis happens for the galaxy containing a nuclear point-like source, \nwhich affects the match of HST and near-IR photometry.\nFig. 5 shows that our process exploiting 7-band spectral data is \noverall quite reliable and that no systematic effects are present.\nOn the other hand, it is clear that the 15 galaxies with optical spectroscopy \nare not randomply sampled from our source list, as shown in particular by Fig. \n7 below: they tend to be lower-redshift, bluer objects, with SF protracted\nto recent cosmic epochs. Such spectral behaviour has encouraged a spectroscopic\nfollow-up, while the redder higher-z ones will require a substantial dedicated\nfuture effort.\n\nAltogether, we estimate that the errors associated with redshifts\nevaluated from class {\\it (b)} spectra should not typically exceed $10\\%$\nin $z$,\nwhile a conservative estimate may be closer to $20\\%$ for class {\\it (a)}\nspectra (see also Connolly et al. and Hogg et al. for similar conclusions).\nNone of our results will be significantly affected by these uncertainties.\n\n\n\\subsubsection{Evaluation of the galactic ages } \n\nFor all sample galaxies, but for object 4-727-0, we found acceptable fits \nwith Model 2. These solutions were found \nby optimizing the 3 (4) free parameters discussed in the previous Section.\nThe spectral solutions illustrated in Fig. 4 (whose parameters are reported\nin Table 2) refer to the case $q_0=0.15$. Consistent solutions were also \nfound for Model 2 in a closure world model \n(see again Table 2 for the corresponding best-fitting parameters).\n\nThe failure of Model 1 to reproduce the data is illustrated in Figure 6,\nwhich compares the broad-band spectrum of a typical galaxy in the sample\nwith two \npredictions of the model at varying $z_F$. The continuous curve is\nthe spectrum predicted by Model 1 with $z_F=5$ and $q_0=0.15$. With this choice \nof the parameters, this corresponds to a stellar population beginning \nformation at z=5 and ending it at z=3.6 (hence observed 5.5 Gyrs after the \nend of the SF in the source frame): the prediction is clearly far too \nred both in the optical and even in the near-IR. The dashed curve\ncorresponds to the same model with $z_F=1.3$, and the additional contribution\nof ongoing SF by 0.2 $M_{\\odot}\/yr$ to better reproduce the U band flux.\nIf the overall fit is better, there is still a significant excess flux\nin the observed B band with respect to the model spectrum. We have found this\nexcess, as well as sometimes one in the V band, to be a general\ncharacteristic of the observed spectra: \neven for the reddest galaxies in the sample it seemed difficult to reproduce the spectra  with stellar populations\nformed during a relatively brief time interval (as in Model 1), even including the additional contribution of ongoing star-formation.\n\nThe better performance of Model 2 is interpreted by us as indicating a rather\nprotracted star formation activity within each galaxy (typically 3 Gyrs in the \nmodel), rather than a single short-lived starburst (even one occurring at low\nredshift). A qualitatively similar result is achieved \nby FA98 analyzing a sample of HDF early-type galaxies \nselected in the $V_{606}$ band.\n\nThe second conclusion concerns the epoch when the stellar \nformation has taken place in the typical sample galaxy. Figure 7 is a plot of\nthe redshift corresponding to the peak of the star-formation versus the\nmass in baryons, according to our best-fits obtained with Model 2. \nPanels (a) and (b) refer to the two solutions with $q_0=0.15$ and $q_0=0.5$.\nThere are clearly two zones of avoidance in the figure: low-mass objects\n($M<2\\ 10^{10}\\ M_{\\odot}$) have peak SF confined to $z\\lesssim 1$, while \nthe high-mass ones \nappear to form stars mostly at $z\\gtrsim 1$. This is partly an artifact\nof the magnitude limits (in the former case) and of the small sampled\nvolume combined with the low space density of very massive galaxies \n(in the latter). While only a careful examination of all selection effects\nwill allow to obtain unbiased estimates of the cosmic SF history\nfrom this dataset (see Sect. 6 below), {\\it it is clear from Fig. 7 that the \nredshift interval of\nz=1 to 2 (1 to 3 for $q_0=0.5$) corresponds to a very active phase of SF\nfor the sample galaxies}.\n\nTo translate this into a constraint on the age for the typical\nearly-type galaxy in the field, we report in Figure 8 the rest-frame V--K\nand B--J colors as a function of redshift, compared with the predictions of \nsingle stellar populations with solar metal abundances. The rest-frame B--J\ncolors are computed by interpolating the galaxy observed spectra using the \nbest-fit models of Table 2, while the V--K's require substantial extrapolation\nat the longer wavelengths, hence are to be taken with some care.\nThe figure shows that the typical ages range from 1.5 to 3 Gyrs, rather\nindependently from the redshift (if any, there is quite a marginal tendency \nfor the higher-z galaxies to display bluer colors).\nFor comparison, the average colors of local galaxies are V--K$\\simeq$ \n3.2--3.3, quite significantly redder. \nThis difference is probably enhanced by a bias induced by the flux limit \nin the K-selected sample, emphasizing relatively bluer objects observed\nclose to their main event of star-formation. Again this will be subject of\nfurther inspection in the next chapters.\n\nAnother check of this is reported in Fig. 6, where the broad-band\nspectrum of the typical galaxy is compared with that of the local\nearly-type M32 (F. Bertola, private communication). The present\ninterpretation of M32's blue light is that it may be dominated by the\nHR turn-off population\nof a $\\sim 4$ Gyrs old stellar population of about solar\nmetallicity (O'Connell 1986), whose bright RGB and AGB stars have been\nrecently resolved (Freedman 1992, Grillmair et al. 1996). Older ages\nmight fit only for a metal-deficient dominant population (Renzini \\& Buzzoni\n1986). Our typical galaxy is bluer than  M32 and, assuming  a solar \nmetallicity, it turns out to be younger than 4 Gyrs.\n\nThe previous considerations have impinged upon a well-known problem when \ndating stellar populations from broad-band spectral analyses, i.e. the \ndegeneracy between age and metallicity. An estimate of the age needs \na guess on the metallicity.\nAs already discussed, the very regular morphologies and the lack of evidence\nfor the presence of dust in the large majority of our sample galaxies\nsuggest that they have already mostly completed their bulk of the SF \nand have metallicities comparable to those of the \nlocal counterparts (i.e. solar on average, see Carollo \\& Danziger 1994). \nIf true, then a conclusion seems unavoidable, that their typical age is close to \n1.5 to 2 Gyrs on average, at the observed redshift. \n\nFigure 9 illustrates the effects of drastic changes in the metallic \ncontent of the stellar populations on the rest-frame colors. \nThe two horizontal lines in both panels bracket\nthe typical colors for the sample galaxies. \nLarge age differences are found as a function of the metallicity for \na given color and for old stellar populations. \nOn the other hand, {\\it since our observed colors are rather blue, \nthe uncertainty due to the unknown metallicity is moderate in absolute terms}.\nFor example, with reference to Fig. 9b, the typical observed color\nB--J$\\simeq 2.5$ would correspond to an age of 1 Gyr for 2 times solar,\nto 2 Gyr for a 0.4 solar and 3 Gyr for a 0.2 solar metallicity.\nThe conclusion is that, {\\it unless we accept that the observed galaxies \nare very significantly metal-deficient (which would be difficult to \nreconcile with observations of local objects), our age determinations should\nnot be seriously in error}.\n\n\n\\subsection{Constraints from the size vs. surface brightness distributions\n}  \n\nFurther significant constraints onto the nature and evolutionary status\nof our sample galaxies may be gained from a detailed analysis of their\nmorphological properties.\n\nThe study of the galaxy spectral shapes in the previous Sect. has \nindicated that intermediate (rather than old) ages, with large spreads, \nare typical. If so, then one would expect quite appreciable luminosity \nevolution due to the -- mostly passive -- aging of the stellar populations\nfrom the redshift of the observation to the present time.\nA way to check it, as discussed in FA98, would be to compare\nthe sizes and the average surface brightness (within the effective radius)\nof distant galaxies with those of the local ones. \nTable 1 (columns 6 and 15) reports these data for all galaxies in our sample.\n\nFigure 10 compares the same data with the mean locus ((thick continuos \nline) of the\nrelation between the effective radius $r_e$ and the corresponding average \nsurface brightness in the V band $<\\mu_e>_V$ (the Kormendy relation). \nIn this figure the effective radii are expressed in Kpc,\nwhile the surface brightness has been corrected (K- and evolutionary\ncorrections) according to Model 1.\n\nThere is a clear offset in Fig. 10, by $\\sim 1.5$ magnitudes, with respect to\nthe locus representing local galaxies (see for more details on the latter\nFA98 and Jorgensen et al., 1995): \nthe surface brightness of distant galaxies is\nmore luminous on average than implied by Model 1 -- one in which, we remind,\nthe bulk of SF was completed by $z=3.5$.\n\nFigure 11 shows that much better consistency is achieved with Model 2,\nboth in the V (panel [a]) and in the K bands (panel [b]).\nThe local Kormendy relation in the band plotted in Fig. 11b is taken from \nPahre et al. (1995). \nIn this case the evolution in luminosity between the redshift of the\nobservation and the present time is stronger (stars are younger on average),\nwith local and distant galaxies consistently tracing the same population,\nas expected.\n\n\n\\section{STATISTICAL PROPERTIES OF THE COMPLETE SAMPLE: THE REDSHIFT \nDISTRIBUTION}\n\nThough small, our K-selected sample has been tested with great care\nin Sect. 2 for completeness and reliability in source selection,\nand it is then suitable for a detailed statistical analysis.\nIn consideration of the high-quality morphological and photometric data\navailable, this study may provide an accurate characterization of the \ndistribution of elliptical\/S0 galaxies in the space-time. \n\nWe consider here two statistical observables, the counts as a function of\nmorphological type and of limiting K magnitude, and the redshift distribution \nD(z).\n\nFigure 12 is a collection of galaxy counts in the K and HK bands, splitted \ninto two morphological components: the early-type galaxies (open squares), \nselected according to the criteria described in Section  2 (see Table 1), \nand late-type systems -- including spirals, irregulars, and starbursts \n(open triangles). \nThe K band counts in the left-hand panel\n are derived from our sample discussed in \nSect. 2 down to $K=20.1$ . The counts in the H+K band in the right-hand panel \nare derived from a sample by Cowie et al. (1996), including complete \nmorphological information for HK brighter than 21.5.\n\n\nThe number counts for early- and late-type galaxies display rather different \nslopes at the faint end, with the early-types converging very fast at\n$K>19$ and the late-types showing steadily increasing counts.\nSimilar results about the morphological number counts are reported by \nDriver et al. (1988), who find in particular a deficit of E\/S0 at\n$I_{814}>22$ compared with passively evolving models.\n\nThe distribution of the -- either spectroscopic or photometrically estimated -- \nredshifts for the 35 sample galaxies is reproduced as a thick line in Figure\n13. The observational distribution shows monotonic increasing values\nfrom $z$=0 to $z\\simeq 1.3$, with a marked peak at $z\\sim 1.1$. \n\nA remarkable feature in the distribution is \na sudden disappearance of objects at $z\\gtrsim 1.3$.\nThis absence may look surprising at first sight, taking into account that\nthe dominant stellar populations \nfor objects observed at $z\\sim 1$ are 1 to 3 Gyr old, whose luminosity\nhas then to increase, if any, at $z>1$.\n\nLet us first discuss the significance of this apparent redshift cutoff, in the \nlight of all selection effects operating in the sample. The first obvious\nconcern is our ability to identify in the K-band image\nextended emissions corresponding to high redshift elliptical\/S0's when going \nfrom the typical observed redshift $z\\sim 1$ to larger cosmic distances.\n\nFigure 14 illustrates the effect of increasing redshift on the average surface \nbrightness and the effective radius, for two evolutionary paths\n(both adopting $q_0=0.15$). \nThe top line corresponds to a typical galaxy in our sample having \n$<\\mu_e^K>=18$, $r_e=2$ Kpc (cfr. the distribution of values scaled to \nzero-redshift in Fig. 11b), and observed at $z$ ranging from 0 to 2.5.\nThe scaling with $z$ of the surface brightness has been calculated taking into\naccount all cosmological, K- and evolutionary effects. \nTo be conservative, these were computed according\nto Model 1 with a low formation redshift $z_F=3$. It turns out \nthat the brightening of the stellar populations, while approaching the SF \nphase at $z>1.5$, counter-balances any cosmological and K-correction\ndimming. In this case there is no significant dependence of the surface\nbrightness on redshift. A limiting situation is provided by the bottom curve\nin Fig. 14, representing the lowest surface brightness galaxy in the sample\nand luminosity evolution after Model 1 with a high formation redshift $z_F=6$. \nIn this case the brightness dimming effects dominate and there is an appreciable\nexcursion in  $<\\mu_e^K>$ as a function of $z$. \nIn spite of this, in this case as well as\nin all other more favorable ones, the faint object would be still be\ndetectable up to z=2.5\nabove the completeness limit, estimated by simulations in Sect. 2 to be  \n$<\\mu_e^K>=22$.\nNote that the same conclusions hold for any more actively evolving models\n(e.g. Model 2).\n\nAltogether, the K-band image is sensitive enough to allow easy detection of \neven moderately or non evolving galaxies up to at least $z\\sim 2.5$. \nThen the \nturnover in D(z) apparently occurring at much lower $z$ cannot be due\nto limitations in detecting faint extended structures.\n\nWe investigate in the following how the other fundamental selection condition, that \nin the total flux ($K<20.15$), operates for different \ncosmological and evolutionary models.\nTo perform this exercise, a reliable local luminosity function (LLF)\nof galaxies in the K band is required. Our adopted LLF is taken from\nGardner et al. (1997), which significantly updates previous\ndeterminations. The separate contributions of early- (E\/S0) and late-type \n(Sp\/Ir) galaxies have been calculated using the optical LLFs \nsplitted into various morphological types by Franceschini et al.\n(1988) and translated to the K band with type-dependent B-K colours.\n\nThe K-band luminosity function of galaxies has been combined with \nvarious evolutionary models to predict sample statistics,\nunder the simplifying assumption of a galaxy mass function\nconstant with cosmic time.\n\nWe find that the shape of the LLF estimated by Gardner et al. \ncan be more naturally\nreconciled with the total faint K-band counts of Fig. 12 in an open universe with\n$\\Lambda=0$, \nwhile a closure one would require some \"ad hoc\" evolutionary prescriptions. \nThis is because the latter has not enough volume at $z\\sim 1$ to fill in the \ncounts, given the space density of galaxies implied by the K LLF.\nThen Model 1 for E\/S0 galaxies (see Sect. 2.2), \nsupplemented with a moderate evolution for Sp\/Ir galaxies as in Mazzei\net al. (1992), could in principle provide a fair fit to the total\ncounts for $q_0=0.15$.\n\nBut a further crucial constraint is set by the observed $z$-distribution\nD(z) of early-type galaxies in the K-HDF (Fig. 13).  From $z=0$ to $z=1.2$, \nD(z) is consistent with a (mostly passive) \nevolution in an open universe (with $q_0$=0.1-0.2) with zero cosmological \nconstant. Again, a closure world \nmodel with $\\Lambda=0$ would require a much larger comoving \nluminosity density of massive E\/S0s at $z$=1 than locally.\n\nHence, there is no evidence, up to this epoch, of an evolution of the baryon\nmass function. This is difficult to reconcile with a decrease of the mass \nand luminosity due\nto progressive disappearance of big galaxies in favor of smaller mass \nunits, as implied by some specifications of the hierarchical clustering \nscheme (e.g. Baugh, Cole, \\& Frenk, 1996). \nThe case for a strong negative evolution of the early-type population\n(by a factor 2-3 less in number density at $z$=1; e.g. \nKauffmann, Charlot \\& White 1996)\ndoes not seem supported by our analysis of the HDF.\n\nAbove z=1.2, however, both Models 1 and 2 (whichever formation redshift $z_F$ \nis assumed) have difficulties to\nreproduce the redshift distribution reported in Fig. 13 and the identification\nstatistics in Figure 12. \nIn particular it is not able to explain the large number of sources  \nat $z=1$ followed by the rapid convergence above: the\nprediction would be of a much more gentle distribution, with less a\npronounced peak and a large tail of galaxies (including typically half of\nsample objects) observable above $z=1.2$. \nLower values of $q_0$ would allow better fits of the observed D(z) up \nto $z=1.3$, but would also worsen the mis-fit at higher $z$.\n\nA similar effect is observed in the\ncounts as a function of the morphological class (Fig. 12). Here again\nsources identified as E\/S0s show steep counts to K=19 and a sudden\nconvergence thereafter, while Model 1 would predict a much less rapid\nchange in slope. These problems remain forcing in various ways the \nmodel parameters. \n\n\n\\section{DUST EXTINCTION AND MERGING EFFECTS DURING A PROTRACTED SF PHASE}\n\nWe briefly discuss here possible explanations of the results obtained \nin Sections 3 and 4.\nWe look for solutions explaining the global\nstatistical properties of the sample (see Sect. 4) as well as the \nvariety of the single galactic spectra (Sect. 3).\n\n\\subsection{Merging}\n\nIn the previous Sect. we have found that\nat redshifts larger than 1.3 early-type galaxies suddenly disappear \nfrom the K-selected sample, while we would expect to observe them to much\nhigher $z$, given the predicted luminosity enhancement when approaching\nthe most intense phase of star formation. \nA comparison of Figures 7 and 13 suggests that there may be a relation \nbetween the redshift interval where such a disappearance occurs and the one\nbracketing the major episodes of star formation, \nas evaluated from the analysis of the stellar populations in distant galaxies.\nIt is clear from Fig. 7 that the interval $1<z<3$ is where the peak \nof star formation occurs for the bulk of our sample galaxies, and this \nroughly coincides with the redshift interval where virtually no objects \nare found in our sample.\n\nAs discussed in Sect 3.,\nthere is not only a substantial spread in the formation epochs $z_F$\nof the sample galaxies (see Table 2), but also\nin single galaxy spectra there is evidence for the presence of stellar \npopulations with widely different ages. A model with SF distributed\nover a 3 Gyr time-interval was able to fit such spectra. \n\nSo, {\\it during this relatively long-lasting period of formation of stars,\nour sample galaxies escape detection by our selection procedure. }\nTwo filters may operate in producing this:\neither the morphological filter, or the K-band limiting flux, or possibly \nboth simultaneously.\n\nAn obvious morphological transformation accompanying the formation of a sizable\nfraction of stars is predicted by the merging picture for the formation\nof early-type galaxies: two gas-rich systems undergo a deep dynamical\ninteraction, eventually bringing to a single merge product. During this process\nthere is a full reshuffling of both the gas component, with dynamically  \ntriggered SF, and also of the old stellar population, reflecting the \nrapidly changing gravitational field. Numerical simulations show\nthat an $r^{-1\/4}$-law stellar distribution may originate in this way (\nBarnes \\& Hernquist 1996).\nGiven the spread of ages in our observed objects, it is likely that a few\nto several of these merging events have taken place per single massive galaxy\nduring the first few Gyrs of the galaxy's lifetime.\nMerging, mostly occurring at $z>1$, is then a very appealing interpretation\nfor our results.\n\nAnother effect of merging would be to lower the flux detectability during\nthe SF phase, due to the mass function rapidly evolving at $z>1$.\nWe did not attempt to quantify the possible effects of merging on the\nvisibility of the early evolution phases of E\/S0 galaxies. We defer for an\nexercise of this kind to dedicated treatments (e.g. Baugh, Cole, \\& Frenk \n1996).\n\nAnother plausible reason of flux leakage during SF is related to\nthe extinction by dust in the medium where stars are forming. This dust\nis very likely present, left over by preexisting generations of massive \nstars.  We spend the rest of this Section\nin a more quantitative evaluation of the possible effects of dust.\nAn argument strongly suggestive of its occurrence will be given in Section 6.\n\n\n\\subsection{Dust effects during a prolonged SF phase in field ellipticals}\n\nDust is observed in significant, or even large, amounts in a wide variety\nof objects at any redshifts. The most distant astrophysical sites\nexplored so far, the quasars at redshift 4 to 5, have shown to contain,\nat that early epoch, dust amounts comparable with those of the most\nmassive galaxies today (Omont et al. 1997; Andreani, Franceschini, \\& Granato, 1997).\nBut also distant radio galaxies, Lyman \"drop-outs\",\nand damped $Ly\\alpha$ galaxies (Pettini et al. 1997)\nhave invariably shown the presence of dust.\nIn several galaxies, among the infrared selected, IRAS has found \nthat the dust re-processes a dominant fraction of the stellar UV radiation\ninto the infrared (the ultra- and hyper-luminous IR galaxies, see e.g. Sanders \nand Mirabel, 1996; Rowan-Robinson 1997).\n\nIndeed, a single short-lived starburst may produce enough metals via\nmassive stellar outflows during the AGB and supernova phase. Recent ISO\nobservations (Lagage et al. 1996) have proven that dust grains are synthesized\nalmost simultaneously with the metals, as soon as they are made available \nto the medium.\n\nA common interpretation of ultra-luminous IR galaxies is that, during a\nmerger or a close interaction, even moderate amounts of dust are spread\naround to make an extended dusty core in the starburst. In these conditions\nit is very likely that any major merger happens in a dust enshrouded medium,\nseveral examples of which have been found at both low- (e.g. Arp 220,\nNGC 6090, M82, see Silva et al. 1998) and high-redshifts (e.g. IRAS F10214;\nHR10; see also Ivison et al. 1998).\n\nWe provide here a simplified description of the effects of dust during the\nstar-formation phase and test it on the statistical properties of our\nsample as discussed in the previous Section.\n\nThe simplification we adopt is to assume, instead of a discrete set of \nsuccessive merging-driven starbursts as would be a realistic physical\nsituation, a more continuous process of star-formation with a temporal\nevolution following that of Model 2 in Figure 3 and occurring in a \ndust-enriched medium (we remind that in Model 2 the SF has a peak \nat 1.4 Gyr after beginning and keeps on until 3.1 Gyrs. We do not consider the \neffects of dust in Model 1, as it would only remove the $z>3$ sources and because\nit is in any case inconsistent, see Sect. 3.2.2). \nDust associated with the residual ISM \nextinguishes the light emitted by high redshift objects.\nThe idea is that the SF efficiency in field galaxies with deep dark-matter \npotentials is not so high to produce a galactic wind on\nshort timescales after the onset of SF. Hence an ISM is present in the \ngalaxy for an appreciable fraction of the Hubble time, during which the ISM \nis progressively enriched of metals and dust. While the gas fraction\ndiminishes as stars are continuously formed, \nits metallicity increases and keeps the dust optical depth\nroughly constant with time.\n\nWe have then complemented the Model 2 of Sect. 3.1 to account for\nthe effect of extinction and re-radiation by dust in star-forming\nmolecular regions and more diffused in the galaxy body during the active\nphase of SF using a model by Silva et al. (1988). \nThe model is precisely as described in Sect. 3.1 for the whole passively\nevolving phase after the end of SF.\n\nA detailed description of the code is given in Silva et al.\nHere we summarize its basic features.\nThe amount of dust in the galaxy is assumed proportional to the residual gas\nfraction and chemical abundance of C and Si, while the\nrelative amount of molecular to diffuse gas ($M_m\/M_g$) is a model parameter.\nThe molecular gas is divided into spherical clouds of assigned mass\n($5\\ 10^{5}\\; \\rm{M}_{\\odot}$) and radius ($16\\ pc$).  \nEach generation of stars born within the cloud progressively\nescapes it on timescales of several tens of Myrs. The two parameters \nregulating this process (see eq. [8] in Silva et al. 1998) are the age \n$t_0$ of \nthe oldest stars still embedded and the escape timescale $\\alpha$.\nThe emerging spectrum is obtained by\nsolving the radiative transfer through the cloud.\nBefore escaping the galaxy, the light arising from young stars\/molecular\ncomplexes, as well as from older stellar generations, further interacts\nwith dust present in the diffuse gas component (the latter dominates the \nglobal galactic extinction at late epochs, when the SF rate is low).\nThe dimming of starlight and consequent dust emission are\ncomputed by describing the galaxy as a spherically symmetric system\nsubdivided into volume elements, with radial dependencies of stars and gas \ndescribed by a King profile with a core radius $r_c=200\\ pc$,\nas reasonable for early-type systems. \n\nIn the general framework for the time evolution of the SF rate \ngiven by Model 2 (see Sect. 3.1 and the continuous line in Fig. 3), \nwe found a self-consistent \nsolution for the dust extinction and emission with the following \nparameter values: baryonic mass $M=10^{11}M_\\odot$, (typical for our sample\ngalaxies), $M_m\/M_g=0.3$, $t_0=0.1\\ Gyr$ and $\\alpha=100\\ Gyr^{-1}$. After \n3.1 Gyr,\nthe balance of gas thermal energy vs. input from supernovae breaks, and\nformally most of the small residual gas is then lost by the galaxy. \nAfter this event, a very low-level SF activity may keep on, due to either\npartial efficiency of the wind, or to stellar re-cycling.\nThe average stellar metallicity of the remnant is roughly solar.\n\nThis scheme naturally accounts for the evidence  previously discussed \nin Sect. 3 of a substantial spread in the ages of distant field E\/S0,\nshowing a combination of massive\namounts of old stellar populations and younger stars.\n\nThe same scheme, when convolved with a K-band LLF, \nsuccessfully accounts for the statistical \ndistributions in Figs. 12 and 13, in addition to other galaxy counts in the\nK-band. Now the cutoffs of both D(z) at $z>1.3$ and of the counts for E\/S0s \nat $K>19$ are reproduced as the effect of dust extinction during the SF phase.\n\nWe stress again that, though not physically unplausible, our treatment of\nthe SF as a continuous process during a substantial fraction of the\nHubble time at high-z is likely an over-semplification of a more complex\nprocess characterized by a discrete set of SF episodes. However, there is    \nvirtually no difference between the two cases as far as the \"observable\"\nlater, passively evolving, phase is considered.\n\n\n\\section{THE STAR-FORMATION HISTORY OF EARLY-TYPE FIELD GALAXIES}\n\n\nThis Section is devoted to a quantitative evaluation  \nof the  star-formation history of early-type galaxies in the field. \nThough it is not certainly the first time this is attempted,\nour approach is substantially different and complementary, and brings some\nsignificant advantages with respect to previous efforts\n(see e.g. Lilly et al. Madau et al. 1996; Connolly et al. 1997; Madau, \nPozzetti, \\& Dickinson 1998).\n\nMost of the previous analyses have concentrated on very large \n(e.g. thousands of objects for analyses dealing with the HDF itself or the \nCFRS) samples of galaxies without morphological differentiation and with \nsometimes very limited fractions (typically $<10\\%$ if we exclude CFRS) of\nspectroscopic identifications, directly\nobserved throughout the whole redshift range sampled by the very\nsensitive optical images.\n\nIn addition to the admittedly uncertain redshift estimation via the Lyman\ndrop-out technique, a recognized difficulty inherent in these \"direct\"\nevaluations of the cosmological star-formation rate is due to the\nessentially unknown effect of dust. Given the poor knowledge of dust\nproperties, an extinction correction is subject to tremendous uncertainties\neven for high-z galaxies with good optical spectra, which are in any\ncase a minority.\n\nOur present approach attempts to by-pass all these kinds of problems by\nconfining the analysis to a relatively small sample (35 objects)\nwith extremely accurate information per object.\nSuch an information concerns high-quality photometric data over a large\nwavelength interval ($0.3<\\lambda <2.2\\ \\mu m$), a good fraction (43\\%)\nof spectroscopically confirmed redshifts, a very accurate morphological\ninformation allowing to select a sample (that of early-types)\nwith likely homogeneous properties of star-formation.\nThe sample galaxies have been processed through a grid of synthetic\ngalaxy spectra, with the essential aim to date the stellar populations\npresent in the galaxy at the time of observation.\n\nThe selection waveband, the K band, has been chosen not only to minimize\nthe selection effects, due to the evolutionary and K-corrections, but\nalso to allow testing a wide range of stellar ages and masses, not\npossible if only optical data were considered.\nThe basic uncertain factor when dating stellar populations, i.e. \nthe metallicity of stars, was discussed in Sect. 3.2: our\nconclusion was that, given the relatively blue colors of galaxies\nat the time of observation, our results on the ages cannot be\ndrastically in error unless we consider highly non- (either super- or\nsub-) solar metallicities. In particular our results should be correct\nif our objects, having mostly completed their star formation, are \ncharacterized by the same solar metallicities which are typically observed \nin local galaxies.\nThe results of this best-fitting process are summarized in Table 2.\n\nOur task here is to account for all reasonable selection effects operating\nin the sample, to correct for incompleteness (that is for the part of the LF\nunobserved because of the flux limit) and to build up a \"population\" \nhistory from those of the single galaxies.\n\nThis derivation of the SF history, though subject to the mentioned\nmodellistic uncertainties, should be minimally or not affected at all by the\ndust extinction problem, as it deals with stellar populations during\na relatively late dust-free phase.\n\nOur approach is otherwise the same as attempted by Lilly et al. (1996) and\nMadau et al. (1996), among others.\nWe consider the contribution of all galaxies in our sampled volume\nto the star-formation rate per unit comoving volume. The time-dependent\nstar-formation rate per any single galaxy is given by that of the  \nbest-fitting model\n(i.e. Model 2 in the vast majority of galaxies, Model 1 for a \ncouple of them), and by the formation redshift\n$z_F$, everything scaled according to the best-fit baryonic mass $M$.\n\nThe contribution of each galaxy to the global comoving SF rate has been\nestimated by dividing the time-dependent SF by the maximum\ncomoving volume $V_{max}$ within which the object would still be visible\nabove the sample flux limit.\nWe defer to Lilly et al. (1995, 1996) for a detailed treatment of how \n$V_{max}$ is computed.\n\nA further correction factor to apply to the comoving SF rate takes into \naccount the portion of the\nluminosity function lost by the flux-limited sample within any\nredshift interval (at any $z$ only galaxies brighter than K=20.1\ncan be detected). The correction is simply computed as the \nratio of the total luminosity density to the luminosity-weighted integral \nof the LF above the luminosity corresponding to the flux limit at that $z$. \nThis correction for completeness is not\nvery large (of the order of a few tens percent), because our adopted\nK-band LLF is relatively flat at the faint end (see Connolly et al., 1997,\nfor further details on this point).\n\nThe global rate of SF $\\Psi (z)$, which is the mass in stars formed per\nyear and unit comoving volume (expressed in solar masses per year and \nper cubic Mpc), is then the summed contribution by all \ngalaxies in our sample. Figure 15 reports different estimates of  \n$\\Psi (z)$ based on the two assumptions of $q_0=0.15$ and $q_0=0.5$.  \nThe latter, in particular, is compared with estimates\nby Lilly et al. (1996), Madau et al. (1996) and Connolly et al. (1997),\nwhile a similar match is not possible for our best guess solution\nwith $q_0=0.15$. \n\nThe calculation of $V_{max}$ and of the completeness correction depend\non the assumed rate of luminosity evolution.\nThe shaded regions in Fig. 15 mark two boundaries estimated with two  \ndifferent evolutionary \ncorrections for luminosity evolution, to provide an idea of\nthe uncertainties. \nThe lower limit is computed from Model 1 (moderate evolution at $z<3.5$, no\nz-cutoff). As such, it provides a strict lower boundary to $\\Psi (z)$.\nFor the upper curve, $V_{max}$ is computed after Model 2, i.e. assuming\nstronger luminosity evolution and a cutoff in the available volume at\n$z\\simeq 1.5$. This upper curve corresponds to our best-guess $\\Psi (z)$.\nUnfortunately, a precise evaluation of the uncertainties is not\npossible at this stage.\n\nFigures 16 and 17 transform the information on the global SF rate\n$\\Psi (z)$ of Fig. 15 into one on the actual mass density in stars \nsynthesized within any redshift interval (in solar masses\nper unit comoving volume and per redshift interval). \nThis is simply the time integral of the function $\\Psi [z]$, and better \nquantifies when the various stellar populations are actually formed.\nFigs. 16 and 17 correspond to the usual two choices for $q_0$, with \n$V_{max}$ computed from Model 2.\nIn each figure this information is also differentiated into various \ngalactic mass intervals.\n\nAs it is apparent, there is a tendency for low-mass galaxies to have \na star-formation activity protracted to lower redshifts. Indeed, several \nmoderate to low-mass galaxies show young ages at low redshifts (see Fig. 7).\nHowever,  although we have been as careful as possible in correcting for all\nvarious selection and incompleteness effects, we cannot be conclusive about this,\nin particular because of the limited size of our sample. Deeper and richer\nsamples are needed to settle the issue.\n\nThe dashed lines in the various panels of Figs. 16 and 17 provide the\nintegral distributions (reported on a linear vertical scale ranging from 0 to\n100\\% in each panel)\nof the stellar mass synthesized as a function of $z$. For example, for $q_0=0.15$,\nthe dashed line in the top panel of Fig. 16 shows that in massive \ngalaxies the SF is already over at $z=1$, while the third panel \nindicates that, for $M<5\\ 10^{10}\\ M_\\odot$, 40\\% of the stars are generated \nat $z<1$.\n\nIt is evident from Figs. 15 to 17 that the two cases considered, of an\nEinstein-de Sitter or an open universe, entail somehow different\nsolutions for the evolutionary star-formation rate. In the closure world model\nwith $\\Lambda=0$\nthere is a limited amount of time between the redshift of the observation\n(typically $z\\sim 1$) and the Big-Bang: this implies relatively higher\nvalues of $z_F$ and a function $\\Psi (z)$ keeping flat to the highest $z$.\nAn open universe, instead, with more cosmic time available, entails typically\nlower values of $z_F$ and a $\\Psi (z)$ peaked at slightly lower redshifts.\n\nThis reflects on the mass fractions of stars generated at the various\nredshifts. For $q_0=0.15$, 80\\% of the mass in stars is formed between\nz=1 and z=3, while the remaining 20\\% is made at higher or lower $z$.\n For $q_0=0.5$, the same 80\\% of stars are formed between z=1.2 and z=4.2.\nThe median redshifts in the stellar formation process turn out to be\nclose to z=1.8 and z=2.5 in the two cases respectively.\n\nThe SF rate $\\Psi (z)$ for our early-type field galaxy population\nis compared in Fig. 15 with the average rate\n(shaded horizontal regions) expected for early-type galaxies in rich clusters,\nassumed that these synthesize between z=5 and 1.5 all metals observed \nin the intra-cluster medium (Mushotzky \\& Loewenstein 1997). \nThe comparison is mediated by a crucial\nassumption we made on the stellar IMF to have a Salpeter's form. \nAn IMF more  weighted in favor of massive stars would decrease the \nSF requirement for cluster ellipticals. \nThere is in any case an indication that field ellipticals\nbehave as less efficient metal producers than the rich cluster counterparts.\nIt is quite conceivable that the large number of dynamical interactions\nunderwent by galaxies in the denser cluster environment might have triggered\na more intense SF during the early Gyrs of the cluster lifetime, with a\nglobally more efficient metal yield.\n\nThis evidence also matches a likely difference in the ages of cluster \nversus field ellipticals: while the properties of cluster galaxies imply\nthat the SF was completed already before z=2 (Stanford et al. 1998), \ntypically 50\\% of stars in the field are produced at more recent epochs.\n\nFinally, it is interesting to compare in Figure 15 (lower panel) the \nindications on the SF history $\\Psi (z)$, as inferred from direct \ninspection of high-redshift galaxies (see data points in the figure), \nwith our predictions based on \nmodelling of the stellar populations. The latter refer to only a sub-class \nof galaxies, those\nclassified as early-types. Yet, this prediction already exceeds the global\namount inferred by high-z \"drop-outs\".\nThere is a suggestion, in particular, that for $z>1.5$ the direct estimate\nbased on optical-UV measurements misses some light, if a sub-population alone\nalready exceeds the global estimated amount. This may be taken as an evidence\nthat dust indeed played a role in at least partly obscuring the young\nstars during the active phase of star formation. If moderately \ntrue for the average\ngas-rich spiral, this might have been particularly relevant during the\nstarburst events bringing to the formation of early-type galaxies.\n\n\n\\section{CONCLUSIONS AND PERSPECTIVES}\n\nEarly-type galaxies are commonly believed to contain among the oldest stellar \nsystems in the universe. Analyses of galaxies in rich clusters,\nwhich found weakly evolving spectra and non-evolving dynamical conditions\nup to z=1 and slightly above, have confirmed this intuition.\n\nIn spite of a morphological similarity which may be taken as indicative of a \ncommon origin, no conclusive results were achieved so far about the early-type \ngalaxies in the field. Our contribution here has been devoted to exploit\na near-IR sample \nof 35 distant early-types in the HDF with $K<20.15$, with optimal \nmorphological information and spectro-photometric coverage, to study\nproperties of distant early-type galaxies outside rich clusters. \nNear-IR observations provide a view of galaxy evolution minimally biased\nby the effects of the evolutionary K-correction, dust extinction and\nchanges in the $M\/L$ ratio due to the aging of stellar populations. \nThe basic conclusions of our analysis may be summarized as follows.\n\n\n\\begin{itemize}\n\n\n\\item \nThe broad-band spectral energy distributions of the sample galaxies, \ntogether with the assumption of a Salpeter's initial mass function\nwithin $M_l=0.15$ and $M_u=100\\ M_\\odot$, allow us to date their dominant \nstellar populations. The majority of\nbright early-type galaxies in this field are found at redshifts $z\n\\lesssim 1.3$ to display colors indicative of a fairly wide range of\nages (typically 1.5 to 3 Gyrs).\nThis evidence adds to that of a substantially broad distribution of the\nformation redshifts $z_F$ from galaxy to galaxy in the sample. So, there is\nno coeval event of star-formation for early-types in this field, \nprobably at variance with what happens for cluster galaxies.\nThere is a tendency for lower-mass systems to be typically younger than the\nmassive counterparts, but a firm conclusion has to wait for deeper and richer \nsamples.\nA spread in the ages and a somewhat protracted SF activity in field, as \nopposed to cluster, ellipticals may be consistent with their observed\nnarrow-band spectral indices ($H_\\beta$, $Mg2$, etc.; see\nBressan, Chiosi \\& Tantalo, 1996).\n\nA protracted SF activity in field ellipticals may also bear some relationship\nwith their observed\npreferentially \"disky\" morphology (Shioya and Taniguchi, 1993).\nUnfortunately, our sample is too faint to allow testing this. Of the 15\ngalaxies (out of 35) bright enough to have the $c_4$ \nmorphological parameter measured, 11 show rather clearly a \"disky\" morphology, \nand the other 4 a \"boxy\" shape. \n\n\n\\item\nThe basic uncertainty when dating stellar populations -- the metallicity of \nstars -- has been the subject of a careful discussion. Given the relatively \nblue colors of the detected galaxies at the typical redshift of one, \nthe age uncertainty cannot be very large, as it would when considering\nold populations in local galaxies. Our result should then be fairly\nrobust.\n\n\\item\nBecause of the different cosmological timescales, the redshift-dependent\nstar-formation history inferred from these data depends to some extent on the\nassumed value for the cosmological deceleration parameter. \nWe find that the major episodes of star-formation\nbuilding up 80\\% of the mass in stars for typical $M^{\\star}$ galaxies \nhave taken place during the wide\nredshift interval between $z=1$ and $z=4$ for $q_0$=0.5, which becomes $z=1$ \nto $z=3$ for $q_0$=0.15.\nLower-mass ($M<5\\ 10^{10}\\ M_{\\odot}$) systems tend to have their bulk of SF \nprotracted to lower redshifts (down to almost the present time).\n\n\\item\nThe previous items dealt with the ages of the dominant stellar populations\nin distant field galaxies. The evolution history for the dynamical \nassembly of these stars into a galactic body might\nhave been entirely different. However, our estimated galactic\nmasses, for a Salpeter IMF, are found in the range from a few $\\sim 10^{9}\\\nM_\\odot$ to a few $10^{11}\\ M_\\odot$ already at $z\\simeq 1$.  So the\nmassive end of the E\/S0 population appears to be mostly in place by that \ncosmic epoch, with space densities, masses and luminosities consistent with\nthose of the local field population. \nIt is to be investigated how this result compares with published findings\n(see e.g. Kauffmann \\& Charlot 1998)\nof a strong decrease of the comoving mass density of early-type\ngalaxies already by $z\\simeq 1$. We suggest that a question to keep under\nscrutiny in these analyses concerns the color classification, as we find,\nafter a detailed morphological and photometric analysis, that these objects \nusually display blue young populations mixed with old red stars.\n\n\\item\nThe present sample is characterized by a remarkable absence of objects at \n$z>1.3$, which should be \ndetectable during the luminous star-formation phase expected to\nhappen at these redshifts. This conclusion seems robust:\nif optical spectroscopy has difficulties to enter this redshift\ndomain, our analysis, using broad-band galaxy SEDs when spectroscopic\nredshifts are not available, has no obvious biases.\nThe uncertainty in the photometric estimate of $z$ for this kind\nof spectra is small. We discuss solutions for this sudden disappearance\nin terms of: \n{\\it a)} merging events, triggering the SF, which imply strongly perturbed \nmorphologies and which may prevent selecting them by our\nmorphological classification filter, and {\\it b)} a dust-polluted\nISM obscuring the (either continuous or episodic) events of star-formation, \nafter which gas consumption (or a galactic wind) cleans up the galaxy.\nWe conclude that the likely solution is a combination thereof, \ni.e. a set of dust-enshrouded merging-driven starbursts occurred during\nthe first few Gyrs of the galaxy's lifetime.\n\n\\item\nA comparison between the observed SF rates with the level needed to synthesize\nthe metals in the ICM indicates that field ellipticals could have been \nslightly less efficient\nmetal producers than the cluster galaxies. This difference adds to the\none in the ages and age spread.\n\n\\item\nThe comparison in Fig. 15b of our estimated SF rate $\\Psi (z)$ \nat $z>1.5$ with those inferred\nfrom the optical colors of high-redshift galaxies (via the Lyman \"drop-out\" \nor other photometric techniques) provides a {\\it direct indication that a \nfraction of light\nemitted during the starburst episodes (in particular those concerning the\nformation of early-type galaxies) has been lost, probably obscured by dust}. \nThe recently detected IR\/sub-millimetric\nextragalactic background (Puget et al. 1996; Hauser et al. \n1997) may be a trace\nof this phenomenon. Our results on the cosmological SF rate reported \nin Fig. 15b (note that our best-guess coincides with the upper limit\nof the shaded region) are very close to the level predicted by Burigana\net al. (1997) to reproduce the spectral intensity of the cosmic \ninfrared background.\n\nIt will obviously be essential to have a direct test of such a \ndust-extinguished SF, but this will not be easy until powerful\ndedicated instrumentation will be available. \nThe difficulty is illustrated in Figure 18, \nshowing possible spectra corresponding to various ages of\na massive ($M=3\\ 10^{11}\\ M_\\odot$) galaxy at $z=2$, according to Model 2, \nduring the starburst and post-starburst phases. \nAs indicated by the figure, various planned missions\n(in launch-time order, SIRTF, ESA's FIRST,\nNASA's NGST, and some ground-based observatories in\nexceptionally dry sites, like the South Pole) could discover \nintense activity of star-formation at $z=1.5$ to 3 or 4.\nAlso interesting tests of these ideas could be soon achieved with SCUBA \non JCMT, and perhaps with ISO.\nIn any case, observations at long wavelengths will be needed to completely\ncharacterize the early evolutionary phases of (spheroidal) galaxies.\n\n\\item\nWhile our main conclusions are moderately dependent on the\nassumed value of $q_0$, an open universe or one with non-zero $\\Lambda$ are \nfavored in our analysis by \nthe match of the K-band local luminosity functions with the observed \nnumbers of faint distant galaxies.  \n\n\\item\nThere are two main sources of uncertainty in our analysis. \nThe redshift distribution in Figure 12 (and in particular the\npronounced peak at $z=1.1$) may be somehow affected\nby the possible presence of a background cluster or group at $z\\sim 1$.\nA few to several of the galaxies with photometric redshifts in the interval \n$1<z<1.2$ have a non-random distribution in the HDF, being clustered\nin one region (at RA$\\simeq$87.98, DEC$\\simeq$62.218). \nUntil we will not have a complete\nspectroscopic identification of most of the galaxies with precisely\nmeasured redshifts, it will be impossible to check this. However the\neffect should be limited to a small fraction of the sample galaxies\nand should not drastically influence our statistics.\nThe second basic uncertainty is due to the photometric estimate of the\nredshift for half of the sample. In this case too, however, this is not likely\nto affect our main conclusions. In any case, these uncertainties will be \nreduced soon by \nnew observations in the Southern HDF and deep spectroscopic surveys from \nground with large telescopes.\n\n\\end{itemize}\n\n\n\n\n\n\\acknowledgments\nThis work has been supported by the European Community with the TMR Grants\nERBFMRX-CT96-0086 and ERBFMRX-CT96-0068.\nWe also acknowledge partial funding by the Italian Space Agency under \nContract ASI-ARS-9689. We are indebted to an anonymous referee for very helpful \ncomments on the paper.\n\n\n\n\\newpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\\vspace{-0.05cm}\n\n\nWith the increasing use of smart-systems ubiquitous availability of sensitive data has become a matter of serious concern. \\textit{Privacy-Preserving Data-Aggregation} (PPDA) has gained a lot of research attention in recent days \\cite{survey1,survey2} as a potential solution to this issue. However, most of the existing PPDA solutions rely on highly computation-intensive \\textit{Homomorphic Encryption} (HE). Hence, they mostly do not fit with resource-constrained IoT systems. \\textit{Secure-Multi Party Computation} (SMPC) based strategies, in contrast, approach a collaborative solution for PPDA and hence, depend comparatively less on computation, while heavily using communication\/data sharing among the entities. However, the communication hardware being the most energy-hungry unit, the IoT devices always try minimization of their communication requirements too in order to have sustained life. Thus, none of the existing solutions for PPDA is directly applicable to resource-constrained IoT systems.\n\n\n\nEfficient communication in low-power IoT systems has been an active research area. In recent works, \\textit{Concurrent-Transmission} (CT) based communication in IoT\/WSN \\cite{sync_survey} has got quite good popularity because of its ability to achieve high reliability and simultaneously consume very less time. Different standard communication patterns including one-to-many, many-to-one as well as many-to-many have been quite successfully realized using CT. In the current work, we approach taking the help of CT to efficiently realize the communication-intensive component of SMPC. In particular, in this work, we make an endeavor to efficiently realize the well-known \\textit{Shamir Secret Sharing} (SSS) \\cite{sss} strategy with the help of a CT-based data sharing protocol MiniCast \\cite{minicast}. \n\n\\section{Background \\& Design}\n\\label{sec:background}\n\\vspace{-0.045cm}\nIn the following, we first briefly explain SSS and MiniCast, and subsequently, we explain the main design considerations to optimally integrate the two with each other.\n\n\\textbf{Shamir-Secret Sharing (SSS)}: SSS achieves PPDA using polynomial interpolation over finite fields in a semi-honest adversarial setting. Every node $n_i$, assumes a $k$-degree polynomial $P_{i}$ with coefficients $c_{1,i}$, $c_{2,i}$,\\ldots, $c_{k-1,i}$, where $c_{k-1,i} = S_i (= P_{i}(0))$, i.e., the secret value of node $n_i$. The aggregation process happens in two steps. During aggregation, every node evaluates its own polynomial at a set of $n$ number of public-points and subsequently shares these values to specific nodes through secure channels. This is referred to as the \\textit{sharing phase}. Every node is designated for a specific public-point based on the ID of the node. Each node then locally sums up the share values received from different nodes. Finally, the nodes re-share these sum values with each other. This is referred to as \\textit{reconstruction-phase}. At this stage, a node can use any set of $k+1$ values to reconstruct a final polynomial $(P_{s}(x)$ (using \\textit{Lagrange Interpolation} technique) which is the sum of the polynomials hold by all the nodes, i.e., $P_{s}(x) = \\sum_{i=1}^{n} P_{i}(x)$. The aggregation of the secret value is obtained as the constant term of $P_{s}(x)$, i.e., $P_{s}(0) = \\sum_{i=1}^{n} S_{i}$.\n\n\n\n\\textbf{MiniCast}: In SSS, a very frequent and important job is the dissemination of data from many source nodes to all the nodes which we carry out using the CT-based protocol MiniCast\\cite{minicast}. MiniCast extends the functionality of the CT-based pioneering protocol Glossy\\cite{glossy}. Fundamentally, it enables multiple instances of the Glossy-based flood to run together in an interspersed manner to realize all-to-all\/many-to-many data sharing. To accommodate multiple floods within the same time frame, the protocol arranges all the transmissions in the chain of packets, i.e., as a sequence of packets, based on a TDMA schedule. The process starts from a designated initiator node. Subsequently, the first-hop neighbors of the initiator transmit their packets which in turn trigger the transmission from the second hop. The parameter NTX in MiniCast defines the number of times a node transmits a full chain of packets. \n\n\\textbf{MiniCast hosting SSS}: The two rounds of SSS directly maps to two rounds of MiniCast. In sharing phase, in order to enable a node $n_i$ to share $n$ number of evaluated values destined for distinct nodes, the chain size is extended to contain $n^2$ sub-slots (packets) where the packet transmitted in each sub-slot is encrypted by a key which is assumed to be already shared with the destination node during the bootstrapping phase. In the reconstruction phase, the nodes share the sum values for different public points, hence, the chain size of $n$ is enough. However, chain size in the sharing phase is $O(n^2)$. \n\n\\section{Scalable Shamir Secret Sharing}\n\n \nThe degree of the polynomial being used in SSS decides the collusion threshold and overall protection of privacy. In the best case, the degree can be set as the number of nodes. However, for many practical purposes setting a lower degree is also enough. Therefore, we observe that the use of low-degree polynomial can bring multi-fold optimization. First, for a low-degree polynomial, the overall chain size in the sharing phase will decrease substantially. Second, to support a low-degree polynomial the sharing phase may run for a duration that would be enough to reach out only to the necessary number of neighbors, instead of attaining full network coverage. Moreover, having a low-degree polynomial also introduces an advantage of a higher degree of fault tolerance. In particular, when a degree $k$ polynomial is used where $k<n$, in the reconstruction phase even the final polynomial can be formed by combining any $k+1$ sum values. This alleviates the need for strict all-to-all sharing of data in the reconstruction phase also making the protocol fault-tolerant. \n\nThe parameter NTX in MiniCast mainly controls the degree of coverage\/reliability the dissemination process achieves. For sufficiently large NTX, a node gets the data from all the other nodes in the network. However, MiniCast shows a very stable and consistent behavior even for low NTX, although fails to receive the full network coverage. A node successfully receives the data from its neighbor within a certain perimeter depending on the exact value of NTX. The behavior is quite non-linear, i.e., with a short increase in NTX, a large amount of data becomes available in a node, while it takes a comparatively higher time (NTX) to have the full network coverage. \n\nWe exploit the above-mentioned behavior of MiniCast to optimize the sharing phase of SSS. In summary, we assume that the degree of the polynomial ($p$) is low. To improve the performance, not only the chain size in sharing phase is trimmed but also, the data sharing process is executed for quite a low NTX value. However to achieve this, in the bootstrapping phase every node is assumed to take note of which neighbor is reachable at what NTX value. Using this information, the chain in the sharing phase is constructed in a way where a node shares evaluation values and encrypts them for a few known pre-determined neighbors. The process completes fast with low NTX and enters the reconstruction phase to complete the process. \n\n\n\\vspace{-0.2cm}\n\\section{Evaluation}\n\\label{sec:results}\n\nThe proposed strategy is implemented in Contiki OS for nRF52840. The naive implementation of SSS is referred to as S3 while the scalable version is referred to as S4. We execute S3 and S4 in two publicly available testbeds FlockLab \\cite{flocklab} and DCube\\cite{dcube} having 26 and 45 nRF devices. In the sharing phase, each packet is encrypted using AES-128 while the reconstruction phase runs in plane text. We experiment with the different number of source nodes. We assume a polynomial of degree nearly equal to \\(\\ \\lfloor \\frac{n}{3} \\rfloor\\), $n$ being the total number of nodes in both S3 and S4. Each experiment is repeated for 2000 iterations. The value of NTX as 6 and 5 are found to be enough for sharing the data within the necessary number of neighbors in the sharing phase in FlockLab and DCube respectively.\n\nTwo standard metrics \\textbf{Latency} (time required to obtain the final aggregation in each node) and \\textbf{Radio-on time} (time necessary to complete the communication process in a round) are used for comparing the performance of S3 and S4 as depicted in Fig. \\ref{fig:result}. For complete network, it can be seen that S4 achieves private aggregation at least 6\\(\\times\\) faster and consuming 7\\(\\times\\) lesser radio-on time in FlockLab and 9\\(\\times\\) faster and consuming 10\\(\\times\\) lesser radio-on time in DCube compared to S3. Note that further improvement in the latency and radio-on time would be visible in S4 compared to S3 for an even lesser degree of the polynomial used.\n\n\n\\begin{figure}\n\\vspace{-0.2cm}\n    \\centering\n    \\includegraphics[scale=0.8]{Figures\/flocklab_dcube_final_icdcs_1.pdf}\n    \\vspace{-0.9cm}\n    \\caption{Performance comparison between S3 and S4 in FlockLab (26 nodes) and DCube (45 nodes).}\n    \\label{fig:result}\n    \\vspace{-0.7cm}\n\\end{figure}\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nIn this work, we show a possible lightweight and efficient realization of the well-known Multi-Party Computation-based strategy SSS for resource-constrained IoT systems. To make it time and energy-efficient, we introduced a couple of optimizations over the naive strategy based on the observation that for many practical cases we do not need the highest degree of privacy protection or collision resistance.\n\\section{Conclusion}\n\n\n\n\n\n\n\n\n\n\n\n\n\\end{document}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe transient X-ray source X0726$-$260\\ (=4U0728-25, 3A0726-260) has a Be star\noptical counterpart that was optically identified by Steiner et al.\n(1984). This star was classified as B0Ve by Corbet \\& Mason (1984) with\na corresponding distance of 4.6$\\pm$1.3 kpc. Negueruela et al. (1996),\nhowever, derive a slightly earlier spectral type of O8-9Ve with a\ndistance of 6.1$\\pm$0.3 kpc.  The X-ray source itself has not been\nextensively observed, although X0726$-$260\\ was also detected with Uhuru\n(Forman et al.  1978), HEAO-1 (Wood et al. 1984) and ROSAT (Haberl,\ncited in Negueruela et al 1996), the only published X-ray light curve\ncomes from Steiner et al. (1984) who present a rather sparse long term\nlight curve from the Ariel V Sky Survey Instrument (SSI). Pointed\nEXOSAT observations on 1984 Oct. 29 and 1984 Nov. 29 failed to detect\nany X-ray emission, even though simultaneous optical observations from\nthe Isaac Newton Telescope showed the presence of H$\\alpha$\\ emission\nindicating that a circumstellar envelope was still present.\n\nIn this paper we present the results of observations of X0726$-$260\\ made with\ntwo instruments on board the Rossi X-ray Timing Explorer (RXTE): the\nAll Sky Monitor (ASM) and the Proportional Counter Array (PCA). These\nobservations provide information on the orbital period and spin period\nof the neutron star in the system.\n\n\\section{Observations}\n\\subsection{All Sky Monitor}\nThe ASM detector on board RXTE (Bradt, Rothschild, \\& Swank 1983) is\ndescribed by Levine et al. (1996). The ASM consists of three similar\nScanning Shadow Cameras, sensitive to X-rays in an energy band of\napproximately 2-12 keV, which perform sets of 90 second pointed\nobservations (``dwells'') so as to cover \\sqig80\\% of the sky every\n\\sqig90 minutes.  The analysis presented here makes use of both ``daily\naveraged\" light curves and light curves from individual dwell data.\nThe light curve for X0726$-$260\\ in two day bins (derived from the daily\naveraged light curve) is shown in Figure 1 and this covers a period of\napproximately 1.5 years.  Note that, for comparison, the Crab produces\na mean count rate of 75 counts\/s in the ASM.  The mean count rate of\nX0726$-$260\\ is 0.14 counts\/s or \\sqig1.9 mCrab.  However, ASM observations\nof blank field regions away from the Galactic center suggest\nthat background\nsubtraction may yield a systematic uncertainty of about 0.1 counts\/s\n(A. Levine, private communication, Remillard 1997).\n\n\\subsection{Proportional Counter Array}\n\nThe RXTE Proportional Counter Array (PCA )is described by Jahoda et\nal.  (1996).  The PCA consists of five individual, nearly identical,\nProportional Counter Units (PCUs) sensitive to X-rays with energies\nbetween 2 - 60 keV with a total effective area of \\sqig6500cm$^2$. The\nPCUs each have a multi-anode xenon filled volume, with a front propane\nvolume which is primarily used for background rejection.  For the\nentire PCA across the complete energy band the Crab produces a count\nrate of 13,000 counts\/s. The high-energy HEXTE instrument is coaligned\nwith the PCA but the results from this instrument are not considered\nhere due to the faintness of the source and HEXTE's smaller effective\narea.\n\n\nThe PCA serendipitously scanned across the position of X0726$-$260\\ on 1997\nMay 5 (MJD 50573) at a rate of 6 degrees\/minute with a minimum distance\nof the center of the field of view from the source of approximately\n0.2$^{\\circ}$. The source was clearly detected at a background subtracted\npeak rate of 105 counts\/s which yields a collimator-response corrected\nflux of approximately 10 mCrab (also reported in Corbet \\& Peele\n1997).  Because of this serendipitous source detection, and a possible\nperiodicity in the ASM light curve (see Section 3), we undertook a\npointed observation at a subsequent time when the source was predicted\nto be bright.\n\n\nA pointed RXTE observation of X0726$-$260\\ was undertaken on 1997 June 7 (MJD\n50606) from 02:54 to 04:54 with an on-source observing time of 5 ks.\nData collection was obtained with a ``Good Xenon'' mode which records\nthe time and energy of each detected photon with maximum resolution.\nDuring the observation, one of the five PCUs experienced a breakdown\nand, to simplify data analysis, we therefore only use data from the\nother four detectors.  In addition, we primarily only use data from\nxenon layer 1; layer 1 gives better signal to noise than the other\nlayers but with a reduced high energy efficiency. A comparison of layer\n1 data only with data extracted from all three layers in fact reveals\nlittle difference for this observation.  Our analysis presented here\nmakes use of processed ``real-time'' data which is available more\nrapidly than the standard ``level zero'' data products.  The light\ncurve from this observation is shown in Figure 2.  The source was\nclearly detected and also showed significant variability during the\nobservation.\n\n\n\\section{Results}\n\nA Fourier Transform of the ASM data shows the strongest peak at a\nperiod of approximately 35 days (Figure 3). In addition, the second\nstrongest peak is at a harmonic of this at \\sqig17.5 days.  Using the\nLomb-Scargle method (Lomb 1976, Scargle, 1982) the nominal false alarm\nprobability (FAP) of the 35 day period is approximately\n2$\\times$10$^{-3}$ although this does not take into account, for\nexample, any systematic effects.  A fit of two sine waves (harmonic\nand fundamental) to the ASM light\ncurve yields a period of 34.46 $\\pm$ 0.12 days with epoch of maximum\nflux at MJD 50365.5 $\\pm$ 0.8. \nThe serendipitous scan detection of X0726$-$260\\ thus occurred at\na phase of \\sqig0.02 and the pointed observation took place at\na phase of \\sqig0.97. The light curve folded on this period is\nshown in Figure 4 and this shows that, on average, the\nsource brightens to about 0.4 counts\/s ($\\sim$ 5mCrab) for a few days\nand is close to 0 counts\/s for a substantial fraction of the time.\n\nThe Fourier Transform of the PCA light curve (Figure 5) shows a maximum\nat a period of \\sqig103 seconds with an associated FAP of\n$<$10$^{-6}$. A sine wave fit yields a period of 103.2 $\\pm$ 0.1 s.\nThe pulse profile as a function of energy is shown in Figure 6. There\nis evidence for a change in profile with energy with a secondary peak\n0.5 out of phase from the main peak appearing at higher energies.\n\n\nThe ``classic'' accreting X-ray pulsar model of an absorbed power law\n(e.g.  White, Swank, \\& Holt 1983) provides an acceptable fit to the\nobserved 2-20 keV spectrum. While an iron line at $\\sim$ 6.6 keV with an\nequivalent width of $\\sim$ 150 eV apparently improves the fit, the\nstatistical significance of this is difficult to estimate due to the\neffects of the Xe L edge in the response matrix. To allow for errors in\nthe response matrix in a simple way, additional 2\\% errors were\nincluded together with the statistical errors when doing the fit.  When\nthese systematic errors are included, a satisfactory fit is obtained\nwithout the need to include an iron line, and we derive a photon index\nof 1.58 $\\pm$ 0.03 and a hydrogen column density of 1.1$\\pm$ 0.3\n$\\times$10$^{22}$cm$^{-2}$ ($\\chi^2_{\\nu}$=1.01). The unabsorbed 2-20\nkeV flux with this model is 6.9$\\times$10$^{-11}$ ergs cm$^{-2}$\ns$^{-1}$ which corresponds to a luminosity of 2.8$\\times$10$^{35}\n(d\/6$kpc)$^2$ ergs s$^{-1}$\n\n\n\n\\section{Discussion}\n\nThe natural interpretation of the 35 day and 103 second periods found\nfrom our observations is that they represent the orbital period of\na neutron star in a 35 day period eccentric orbit, rotating with\na 103 second period.\nWe note that Reig et al. (1997) have proposed a correlation between\nH$\\alpha$\\ equivalent width and orbital period in Be\/neutron star\nbinaries. This is postulated to occur because the orbit of the neutron\nstar acts in some way to truncate the Be star circumstellar\nenvelope.  The 34.5 day orbital period for X0726$-$260\\ derived \nfrom the ASM data is consistent with this\ncorrelation (Reig, private communication). We have also examined the Ariel V SSI data presented\nby Steiner et al. (1984) for any evidence of a $\\sim$ 35 day period.\nHowever, these data are too sparse to enable any strong\nconclusions to be drawn. In addition, the source is not present in\nthe archival Ariel V or Vela 5B ASM light curves available at the\nHEASARC at GSFC.\n\nThe parameters of X0726$-$260, derived from our RXTE observations, when\nplotted on a orbital period\/spin period digram (cf. Corbet 1984, 1986),\nshow a longer pulse period (or shorter orbital period) than would be\nexpected from the general correlation between orbital and pulse periods\nobserved for Be star systems. We note the the 195.6s pulsar GROJ2058$+$42\\\nwhich is suspected to be a Be star system also\nappears to share this property. In the case of GROJ2058$+$42\\ the discrepancy\narises if\nthe \\sqig55 day period derived from RXTE ASM observations is used (Corbet,\nPeele, \\& Remillard 1997). If, however, the BATSE period of twice this\nvalue is taken (Wilson, Strohmayer, \\& Chakrabarty 1996, Bildsten et al.\n1997) then GROJ2058$+$42\\ {\\em does} lie on the correlation curve. We note\nthat, in the case of the {\\em supergiant} system 4U1223-624, a small\nflare at {\\em apastron} can sometimes also be seen (e.g. Pravdo et al.\n1995) as well as a large flare near periastron. The mechanism that\nproduces this effect could perhaps be invoked to explain the difference\nin orbital periods obtained for GROJ2058$+$42\\ from the RXTE ASM and BATSE data\nsets if, for these sources, an apastron flare could be of the same\nlevel as a periastron flare at some times. For Be\/neutron star\nit has sometimes been suggested that these systems might\ndisplay two outbursts per cycle if the plane of the Be star\nenvelope and the orbital plane are offset\n(Shibazaki 1982, Priedhorsky \\& Holt 1987). Such a system could\nthen exhibit flares when the neutron star passed through the\ncircumstellar envelope, in contrast to the single flare per\norbit at periastron that has so far actually been observed\nfor Be\/neutron star binaries.\nIf X0726$-$260\\ were to display two approximately equal flares per orbit,\nthen its orbital period would this be $\\sim$ 70 days and would be\nin better\nagreement with the overall Be star orbital\/pulse period correlation.\nThe ASM observations do not show any compelling evidence for\nan underlying \\sqig70 day period. Although there is a small peak \nin the Fourier Transform at the corresponding frequency (Figure 3),\nthe data folded on this period (Figure 7) do not show a strong\ndifference between ``odd'' and ``even'' cycles of the \\sqig35 day\ncycle.\nThe simplest interpretation may be that, in fact,\nX0726$-$260\\ does not lie on the general relationship between orbital and\npulse periods.  This correlation is believed to arise due to a\nbalance, on average, between spin-up and down torques (e.g. Corbet 1984,\nWaters \\& van Kerkwijk 1989). The equilibrium period is then dependent\non the mass transfer rate which itself should depend on the orbital period.\nThe strength of the correlation would thus result from the other\nproperties\nof most of the Be star systems (e.g. Be star\ncircumstellar envelopes and neutron star magnetic fields)\nbeing rather similar. X0726$-$260, and\nperhaps GROJ2058$+$42, may simply have different properties from the average\nsystem.  Indeed, Negueruela et al. (1996) propose that the spectral\ntype of the optical counterpart of X0726$-$260\\ is unusually early. This could,\nfor example,\nperhaps result in a different circumstellar environment in X0726$-$260\\ compared to\nother Be star systems.\nFor GROJ2058$+$42\\ there is no secure optical counterpart although\ntwo candidates have been suggested by Castro-Tirado (1996).\n\nTo unequivocally determine the orbital periods of both X0726$-$260\\ and\nGROJ2058$+$42\\ it appears that Doppler curves for these sources must be\nobtained which would also give measures of system eccentricities. If\nthe orbital periods of these systems are indeed relatively short, then\ntheir deviations from the overall correlations will make these systems\nof interest for further study.\n\nThe spectrum that we find from the PCA is typical of X-ray pulsars. The\nhydrogen column density at 1.1$\\pm$ 0.3 $\\times$10$^{22}$cm$^{-2}$,\nwhile not strongly constrained, is somewhat higher than but broadly\nconsistent with, the estimates of 0.35$\\pm$0.02 and 0.44$\\pm$0.12\n$\\times$ 10$^{22}$ cm$^{-2}$ derived by Negueruela et al. (1996) and\nCorbet \\& Mason (1984) respectively from measurements of optical\nreddening.\n\n\\section{Conclusion}\nRXTE observations of X0726$-$260\\ indicate a pulse period of 103.2 seconds and a\nmost likely orbital period of \\sqig35 days. Additional more extensive\npulse timing observations would be valuable to determine the orbital\nparameters of this system.\n\n\n\\acknowledgments\nThis paper made use of quick look data provided by the RXTE ASM team at\nMIT and GSFC. We thank many colleagues in the RXTE team for useful\ndiscussions and, in particular, T. Takeshima for help with the slew\ndata, and A. Levine and R. Remillard  for advice on ASM data.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe theory of regularity of nonlinear \\emph{scalar} elliptic and parabolic\nequations is by now classical. It goes back to the ground breaking work on\nequations in divergence form by De Giorgi, Moser and Nash in the late fifties.\nSince then, H\\\"older estimates for general nonlinear elliptic and parabolic\nequations were derived by Krylov and Safonov. However, it became clear quite early on, after discovery of counter examples, that nonlinear elliptic and parabolic \\emph{systems} do not, in general, possess \\emph{everywhere regularity}. Instead, only \\emph{partial regularity} results are available\n~\\cite{Giaquinta83}.\n\nDespite the lack of everywhere H\\\"older continuity of weak solutions of general elliptic and parabolic systems, there are several nontrivial examples that do possess everywhere regularity due to their special structure. One of the earliest such examples of a fully nonlinear elliptic system whose weak solutions are everywhere H\\\"older continuous is due to Uhlenbeck ~\\cite{Uhlenbeck77}. She considered elliptic systems of the form\n$$\n-(g (|\\mathbf{\\nabla} \\mathbf{u}|^2)u^i_{x_\\alpha})_{x_\\alpha} = 0 \\quad \\textrm{for }\ni\\in \\set{1\\ldots N},\n$$\nwith some additional ellipticity and grows conditions on $g$. Uhlenbeck's\nproof of H\\\"older continuity of weak solutions of these systems relies crucially on the existence of an auxiliary function which is subharmonic. This auxiliary function (\\emph{entropy}) is tied to $|\\mathbf{\\nabla}^2 \\mathbf{u}|^2$ by an inequality, and this allows local control on the second derivatives of $\\mathbf{u}$.\n\nThe parabolic examples of fully nonlinear systems possessing everywhere regularity followed somewhat later. Wiegner in ~\\cite{Wiegner92} provided an original example of such a system. He showed everywhere H\\\"older continuity of weak solutions of \\textbf{strongly coupled systems}\n\\begin{equation}\\label{eq:strongly_coupled_begin}\nu^i_t - (a_{\\alpha\\beta}u^i_{x_\\beta} +\nc^i_{\\alpha\\beta}H_{x_\\beta})_{x_\\alpha} = 0,\n\\end{equation}\nwith $H:= H(\\mathbf{u})$ a strictly convex function of $\\mathbf{u}$ and strict ellipticity conditions on the coefficients. His work has been followed by several others,\nincluding Dung ~\\cite{Dung99}, on strongly coupled system, and  Bae and\nChoe ~\\cite{BaeChoe04} on a parabolic analog of the example of Uhlenbeck. Unlike Uhlenbeck's work, however, none of the proofs of everywhere regularity for aforementioned parabolic examples rely explicitly on the existence of entropy.\n\nIn this paper I provide new examples of parabolic systems whose weak\nsolutions are everywhere H\\\"older continuous. These systems are a type of \\textbf{nonlinear diffusion systems}, and are of the form\n\\begin{equation}\\label{eq:diffusion_system}\n\\mathbf{u}_t -\\Delta\\left( \\mathbf{\\nabla}\\Phi(\\mathbf{u})\\right) = 0\n\\end{equation}\nwhere $\\Phi(z)$ is a strictly convex function. These systems are interesting since they generalize \\textbf{scalar nonlinear diffusion} equations of the form\n$$\nu_t - \\Delta ( \\gamma(u)) = 0,\n$$\nwhere $\\gamma$ is strictly increasing. I will show that if $\\Phi(z)$\ndepends only on the norm of $z$, then, together with some smoothness and\ngrows conditions on $\\Phi$, weak solutions of diffusion systems\n\\eqref{eq:diffusion_system} are everywhere H\\\"older continuous. I will do this by exhibiting an entropy, and showing that existence of entropy together with properties of general parabolic systems suffices to prove everywhere regularity. In the conclusion, I will show that regularity of bounded weak solutions to the strongly coupled systems \\eqref{eq:strongly_coupled_begin} can also be obtained with the help of an entropy.\n\n\n Let me introduce notation that I will use throughout this\npaper. I consider parabolic space to consist of space and time with space\nbeing of $n$ dimensions and time of one dimension. In this paper I will deal\nwith cylindrical domains for simplicity. By elliptic domain $\\Omega\\subset \\Real^n_x$ I\nwill mean space domain. Cylindrical parabolic domain $\\Omega\\times (0,\nT)\\subset \\Real_x^n\\times\\Real_t$, where $T>0$, will be denoted by $ {\\Omega_T} $. The time\ninterval of the cylindrical domain $ {\\Omega_T} $, that is the interval $(0,T)$ in\nthe case of $ {\\Omega_T}  = \\Omega\\times (0,T)$, will be denoted by $I( {\\Omega_T} )$.\nBy $\\Ball{x}{r}$ I will mean an $n$ dimensional ball as a subset of $\\Real^n_x$ with center at $x$ and\nof radius $r$. $Q(x,t,r)$ will denote the cylinder\n$$\nQ(x,t,r):=\\Ball{x}{r}\\times (t-r^2,t).\n$$\nI will write $\\textrm{B}_r$ or $\\textrm{B}$ instead of\n$\\Ball{x}{r}$ when $x$ or $r$ are clear from the context. Similarly,\nsometimes I will write $Q_r$ or $Q$ instead of $Q(x,t,r)$. For a\nLebesgue measurable set $S$ by $|S|$ I will mean Lebesgue measure of\n$S$. Finally $V( {\\Omega_T} ; \\Real^n)$ will denote the closure of $C^1( {\\Omega_T} ; \\Real^n)$\nfunctions under the norm\n\\begin{equation*}\n\\norm{v}_{V( {\\Omega_T} )}^2 = \\sup_{t\\in I( {\\Omega_T} )} \\int_{\\Omega}|v(x,t)|^2\\,\\textrm{d}x +\n\\iint_{ {\\Omega_T} } |\\mathbf{\\nabla} v(x,t)|^2\\,\\textrm{d}x\\,\\textrm{d}t.\n\\end{equation*}\n\n\\section{Partial regularity of nonlinear systems}\nIn this section we recall partial regularity results for weak\nsolutions of quasi-linear parabolic systems of the form\n\\begin{equation}\\label{eq:nonlinear_system}\nu^i_t - (A^{\\alpha\\beta}_{ij}(x, \\mathbf{u}) u^j_{x_\\beta})_{x_\\alpha}=0 \\quad\n\\forall i \\in \\set{1\\ldots N},\n\\end{equation}\nwhere coefficients $A^{\\alpha\\beta}_{ij}$ satisfy  the following condition of\n\\textbf{strong ellipticity}:\n\\begin{equation}\\label{eq:strong_ellipticity}\nA^{\\alpha\\beta}_{ij}\\xi^i_\\alpha \\xi^j_\\beta \\ge \\lambda|\\xi|^2\n\\textrm{ for all }\\xi\\in  \\mathds{R} ^{nN}, \\textrm{ for some }\\lambda>0.\n\\end{equation}\nBy a weak solution in this case I mean a function $\\mathbf{u}\\in\nV( {\\Omega_T} ; \\Real^N)$ that satisfies\n$$\n\\iint_ {\\Omega_T}  - u^iv^i_t + A^{\\alpha\\beta}_{ij} u^j_{x_\\beta}\nv^i_{x_\\alpha}\\,\\textrm{d}x\\textrm{d}t = 0, \\quad \\textrm{for all }\\mathbf{v}\\in \\Hkz{1}( {\\Omega_T} ;\\Real^N).\n$$\nIt is well known that weak solutions of systems of this type possess partial\nregularity under some appropriate continuity conditions on\n$A^{\\alpha\\beta}_{ij}$. The important result in this area is the following\nlocal regularity result due to Giaquinta and Struwe:\n~\\cite{GiaquintaStruwe82}\n\\begin{theorem}[Local regularity condition]\\label{thm:local_regularity}\nSuppose coefficients $A^{\\alpha\\beta}_{ij}$ satisfy condition\n\\eqref{eq:strong_ellipticity}, are continuous and bounded. Also\nsuppose $\\mathbf{u}\\in V( {\\Omega_T} )$ is a weak solution of\n\\eqref{eq:nonlinear_system}. Then, if for some $(x_0, t_0)\\in  {\\Omega_T} $\n\\begin{equation}\\label{eq:local_regularity_condition}\n\\liminf_{R\\rightarrow 0}\\frac{1}{R^n}\\iint_{Q(x_0, t_0 ,R)} |\\mathbf{\\nabla} \\mathbf{u}|^2 \\,\\textrm{d}x\\textrm{d}t= 0,\n\\end{equation}\nthen $\\mathbf{u}$ is H\\\"older continuous in the neighborhood of $(x_0, t_0)$.\n\\end{theorem}\nCondition \\eqref{eq:local_regularity_condition} is the basis for the proofs of\neverywhere regularity that we will discuss in the rest of this paper. We will, however, need one more result due to Giaquinta and Struwe ~\\cite{GiaquintaStruwe82}.\n\\begin{lemma}[$\\Lp{p}$ estimate]\\label{lem:lp_estimate}\nLet $\\mathbf{u}$ be a weak solution of the system \\eqref{eq:nonlinear_system}. Then there exists an exponent $p>2$ such that $|\\mathbf{\\nabla} \\mathbf{u}|\\in \\Lp{p}_{loc}( {\\Omega_T} )$; moreover for all $Q_R\\subset Q_{4R} \\subset  {\\Omega_T} $ we have\n\\begin{equation}\\label{eq:Lp_estimate}\n\\left(\\Xiint{\\rule{1.2em}{0.5pt}}_{Q_R} |\\mathbf{\\nabla} \\mathbf{u}|^p\\,\\textrm{d}x\\textrm{d}t\\right)^{1\/p}\\leq C \\left(\\Xiint{\\rule{1.2em}{0.5pt}}_{Q_{4R}} |\\mathbf{\\nabla} \\mathbf{u}|^2\\,\\textrm{d}x\\textrm{d}t\\right)^{1\/2}.\n\\end{equation}\n\\end{lemma}\n\n\\section{Generalized diffusion equations}\nIn this section I discuss the type of parabolic systems I will refer to as\n\\textbf{diffusion system}. Let $\\Phi:\\Real^N\\rightarrow \\mathds{R} $ be a strictly convex, twice continuously differentiable function with\n\\begin{equation}\\label{eq:strict_convexity}\n\\lambda |\\xi|^2 \\le \\Phi_{z_i z_j} \\xi^i \\xi^j \\leq \\Lambda |\\xi|^2.\n\\end{equation}\nThen we say that $\\mathbf{u}$ is a weak solution of\n\\begin{equation}\\label{eq:gen_diff_eq}\n\\mathbf{u}_t - \\Delta\\left[\\mathbf{\\nabla}\\Phi(\\mathbf{u})\\right] = 0\n\\end{equation}\nif $\\mathbf{u}\\in V( {\\Omega_T} ; \\Real^N)$ and for\nall $\\mathbf{w} \\in \\Hkz{1}( {\\Omega_T} ; \\Real^N)$\n\\begin{equation}\n\\iint_ {\\Omega_T}  -u^iw^i_t + (\\Phi_{z_i}(\\mathbf{u}))_{x_\\alpha}\nw^i_{x_\\alpha} \\,\\textrm{d}x\\textrm{d}t = 0.\n\\end{equation}\nThis is a standard quasi-linear elliptic system of the type in\n(\\ref{eq:nonlinear_system}), since we can rewrite it as\n\\begin{equation}\nu^i_t + \\left(\\Phi_{z_i z_j}(\\mathbf{u})u^j_{x_\\alpha}\\right)_{x_\\alpha} = 0\n\\quad \\textrm{for all }i\\in \\set{1\\ldots N}.\n\\end{equation}\nIf in addition $\\mathbf{u}$ is bounded I say $\\mathbf{u}$ is bounded weak solution.\n\nAs I have mentioned before this equation is a generalization of scalar nonlinear diffusion equation. It has several nice properties. First, its flow is a contraction in $\\Hk{-1}(\\Omega)$ and its solutions are unique. Second, weak solutions of \\eqref{eq:gen_diff_eq} are in $\\Hk{1}_{loc}(0,T; \\Lp{2}_{loc}(\\Omega))$, that is their weak derivatives in time are in $\\Lp{2}_{loc}( {\\Omega_T} )$. Furthermore, if the solution is bounded, the gradient in $x$ is actually in $\\Lp{4}_{loc}( {\\Omega_T} )$.\n\nFirst I show that flow is a contraction in $\\Hk{-1}(\\Omega)$.\n\\begin{theorem}[Uniqueness]\\label{thm:flow_contraction}\nLet $u_0, u_1\\in V( {\\Omega_T} ; \\Real^N)$ be two weak solutions of \\eqref{eq:gen_diff_eq} with the same boundary conditions, that is $u_0(\\cdot,t)\\equiv u_1(\\cdot,t)$ on $\\partial \\Omega$ for almost all $t\\in [0,T]$. Denote by $i:\\Lp{2}(\\Omega) \\rightarrow \\Hk{-1}(\\Omega)$ the natural embedding of square integrable functions in $\\Hk{-1}$ defined by\n$$\ni(f)(\\phi) := \\int_\\Omega f\\phi \\,\\textrm{d}x.\n$$\nThen we have for $T \\geq t_1 \\geq t_0 \\geq 0$\n\\begin{equation}\\label{eq:contraction}\n\\norm{i(u_0(t_1)-u_1(t_1))}_{\\Hk{-1}(\\Omega)}\\leq e^{\\lambda(t_1-t_0)}\\norm{i(u_0(t_0)-u_1(t_0))}_{\\Hk{-1}(\\Omega)},\n\\end{equation}\nwhere $\\lambda$ as in \\eqref{eq:strict_convexity}.\n\\end{theorem}\n\\begin{proof}\nLet us denote by $f_h$ the Steklov average of $f$ defined as\n$$\nf_h(x,t):=\\int_t^{t+h} f(x,s)\\,\\textrm{d}s.\n$$\nAlso for simplicity we will write $\\mathbf{v}_k$ for $\\mathbf{\\nabla}\\Phi(\\mathbf{u}_k)$ with $k=1,2$. It is not very hard to show that $(\\mathbf{u}_k)_h$ and $(\\mathbf{v}_k)_h$ weakly satisfy\n\\begin{equation}\\label{eq:milified_eq}\n((\\mathbf{u}_k)_h)_t - \\Delta((\\mathbf{v}_k)_h)=0.\n\\end{equation}\nLet us denote the solution of\n$$\n\\Delta w = f, \\quad w\\equiv 0\\textrm{ on }\\Omega\n$$\nby $\\Delta^{-1}f$. Then the $\\Hk{-1}(\\Omega)$ norm of $i(f)$ is given by\n$$\n\\norm{i(f)}_{\\Hk{-1}(\\Omega)}^2 = \\int_\\Omega |\\mathbf{\\nabla} (\\Delta^{-1} f)|^2\\,\\textrm{d}x.\n$$\nFor simplicity for an $f\\in \\Lp{2}(\\Omega)$ let us write $\\norm{f}_{\\Hk{-1}}$ instead of $\\norm{i(f)}_{\\Hk{-1}(\\Omega)}$. Now fix $h$. For $t_0, t_1 \\in (h, T-h)$ we compute\n\\begin{equation}\\label{eq:uniquenes_mollify}\n\\left(e^{2\\lambda t}\\norm{(\\mathbf{u}_1)_h - (\\mathbf{u}_0)_h}^2_{\\Hk{-1}}\\right)\\Big|_{t_0}^{t_1}   = I_1 + I_2,\n\\end{equation}\nwhere\n\\begin{equation*}\nI_1 = \\int_{t_0}^{t_1}2\\lambda e^{2\\lambda t}\\norm{(\\mathbf{u}_1)_h - (\\mathbf{u}_0)_h}^2_{\\Hk{-1}}\\Big|_{t}dt,\n\\end{equation*}\nand\n\\begin{equation*}\nI_2 = \\int_{t_0}^{t_1} \\int_\\Omega 2 e^{2\\lambda t}\\mathbf{\\nabla}\\Delta^{-1}( (\\mathbf{u}_1)_h - (\\mathbf{u}_0)_h)\\cdot \\mathbf{\\nabla}\\Delta^{-1}( (\\mathbf{u}_1)_h - (\\mathbf{u}_0)_h)_t \\,\\textrm{d}x\\textrm{d}t.\n\\end{equation*}\nUsing the equation and the fact that $\\mathbf{u}_0$ and $\\mathbf{u}_1$ have the same trace, for $I_2$ we obtain\n\\begin{equation*}\nI_2  =  -\\int_{t_0}^{t_1} \\int_\\Omega 2 e^{2\\lambda t} ( (\\mathbf{u}_1)_h - (\\mathbf{u}_0)_h)\\cdot ( (\\mathbf{v}_1)_h - (\\mathbf{v}_0)_h)\\,\\textrm{d}x\\textrm{d}t\n\\end{equation*}\nTaking the limit of both sides of the equation \\eqref{eq:uniquenes_mollify} as $h\\rightarrow 0$ we get\n\\begin{equation}\n\\left(e^{2\\lambda t}\\norm{(\\mathbf{u}_1 - \\mathbf{u}_0}^2_{\\Hk{-1}}\\right)\\Big|_{t_0}^{t_1} = I_3 + I_4,\n\\end{equation}\nwhere\n\\begin{equation*}\nI_3 = \\int_{t_0}^{t_1}2\\lambda e^{2\\lambda t}\\norm{\\mathbf{u}_1 - \\mathbf{u}_0}^2_{\\Hk{-1}}\\Big|_{t}dt,\n\\end{equation*}\nand\n\\begin{eqnarray*}\nI_4 & = & -\\int_{t_0}^{t_1} \\int_\\Omega 2 e^{2\\lambda t} ( \\mathbf{u}_1 - \\mathbf{u}_0)\\cdot ( \\mathbf{v}_1 - \\mathbf{v}_0)\\,\\textrm{d}x\\textrm{d}t\\\\\n& \\leq & -\\int_{t_0}^{t_1} \\int_\\Omega 2\\lambda e^{2\\lambda t} |\\mathbf{u}_1 - \\mathbf{u}_0|^2\\,\\textrm{d}x\\textrm{d}t\\\\\n& \\leq & -\\int_{t_0}^{t_1} 2 \\lambda e^{2\\lambda t} \\norm{\\mathbf{u}_1 - \\mathbf{u}_0}^2_{\\Hk{-1}}\\Big|_{t}dt.\n\\end{eqnarray*}\nThus we see that $I_3+I_4 \\leq 0$ and we establish the result for $T > t_1 \\geq t_0 > 0$. We establish the Theorem by taking the limit as $t_0\\rightarrow 0$ and $t_1\\rightarrow T$.\n\\qed\\end{proof}\nLet us now derive the $\\Hk{2}$ estimates for the solutions of \\eqref{eq:gen_diff_eq}. In general, quasi-linear parabolic systems do not have $\\Hk{2}$ estimates and the fact that solutions of generalized diffusion equations do have them is the consequence of the special structure that equation \\eqref{eq:gen_diff_eq} possesses.\n\\begin{theorem}[$\\Hk{2}$ estimates]\\label{H2_estimate}\nLet $u\\in V( {\\Omega_T} ; \\Real^N)$ be a weak solution of \\eqref{eq:gen_diff_eq}. Then $u_t\\in \\Lp{2}_{loc}( {\\Omega_T} )$ and for $Q(x,t,r)\\subsetneq Q(x,t,R) \\subset  {\\Omega_T} $ we have the following estimates\n\\begin{equation}\\label{eq:H2_estimate}\n\\iint_{Q(x,t,r)} |\\mathbf{u}_t|^2\\,\\textrm{d}x\\textrm{d}t\n\\leq \\frac{C}{(R-r)^2} \\iint_{Q(x,t,R)} |\\mathbf{\\nabla}\\mathbf{u}|^2 \\,\\textrm{d}x\\textrm{d}t,\n\\end{equation}\n\\begin{equation}\\label{eq:Hessian_estimate}\n\\iint_{Q(x,t,r)} |\\mathbf{\\nabla}^2 (\\mathbf{\\nabla}_z\\Phi(\\mathbf{u}))|^2\\,\\textrm{d}x\\textrm{d}t \\leq \\frac{C}{(R-r)^2} \\iint_{Q(x,t,R)} |\\mathbf{\\nabla}\\mathbf{u}|^2 \\,\\textrm{d}x\\textrm{d}t.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nAs before we will write $\\mathbf{v}$ for $\\mathbf{\\nabla}\\Phi(\\mathbf{u})$ and $f_h$ for Steklov average of $f$. Denote by $\\xi\\in C^\\infty_0(\\Omega)$ a smooth bump function supported in $\\Ball{x}{R}\\subset\\Omega$, which is identically one on $\\Ball{x}{r}\\subset\\Omega$ with $\\norm{\\mathbf{\\nabla}\\xi}_\\infty \\leq C\/(R-r)$. Also denote by $\\eta\\in C^\\infty_0((t-R^2,t])$ a function that is identically one on $[t-r^2,t]$ and supported in $[t-R^2,t]$ with $|\\eta'|\\leq C\/(R^2-r^2)$. Then multiplying equation\n$$\n(\\mathbf{u}_h)_t - \\Delta (\\mathbf{v}_h) = 0\n$$\nby $(\\mathbf{v}_h)_t\\xi^2\\eta$ we obtain\n$$\n\\int_{t-R^2}^t \\int_\\Omega (\\mathbf{u}_h)_t(\\mathbf{v}_h)_t\\xi^2\\eta + \\mathbf{\\nabla} (\\mathbf{v}_h)\\mathbf{\\nabla}(\\mathbf{v}_h)_t\\xi^2\\eta\n+ \\mathbf{\\nabla} (\\mathbf{v}_h)(\\mathbf{v}_h)_t 2\\xi\\mathbf{\\nabla}\\xi\\eta \\,\\textrm{d}x\\textrm{d}t = 0.\n$$\nUsing strict convexity on the first term, integrating the second term in time by parts and using H\\\"older inequality twice we obtain\n\\begin{eqnarray*}\n\\int_{t-R^2}^t \\int_\\Omega \\frac{\\lambda}{2}|(\\mathbf{u}_h)_t|^2\\xi^2\\eta \\,\\textrm{d}x\\textrm{d}t & + & \\int_\\Omega |\\mathbf{\\nabla}(\\mathbf{v}_h)|^2\\xi^2\\eta \\Big|_t \\,\\textrm{d}x \\\\\n&\\leq & C (\\norm{\\eta'}_\\infty + \\norm{\\mathbf{\\nabla}\\xi}_\\infty^2)\\int_{t-R^2}^t \\int_\\Omega |\\mathbf{\\nabla}(\\mathbf{v}_h)|^2\\xi^2\\,\\textrm{d}x\n\\end{eqnarray*}\nThus taking a limit as $h\\rightarrow 0$ we deduce that $u_t\\in L^2_{loc}(\\Omega)$ and derive estimate \\eqref{eq:H2_estimate} as claimed.\nTo prove \\eqref{eq:Hessian_estimate}, we use $\\Hk{2}$ estimates for the Laplacian to get the following for a fixed $t$:\n$$\n\\int_{\\Ball{x}{r}\\times\\set{t}} |\\mathbf{\\nabla}^2 \\mathbf{v}|^2\\,\\textrm{d}x \\leq \\frac{C}{(R-r)^2} \\int_{\\Ball{x}{R}\\times\\set{t}} |\\mathbf{\\nabla}\\mathbf{v}|^2 \\,\\textrm{d}x+ \\int_{\\Ball{x}{R}\\times\\set{t}} |u_t|^2\\,\\textrm{d}x.\n$$\nIntegrating in time and using estimate \\eqref{eq:H2_estimate} we get\n\\begin{equation*}\n\\iint_{Q(x,t,r)} |\\mathbf{\\nabla}^2 \\mathbf{v}|^2\\,\\textrm{d}x\\textrm{d}t \\leq \\frac{C}{(R-r)^2} \\iint_{Q(x,t,R)} |\\mathbf{\\nabla}\\mathbf{v}|^2 \\,\\textrm{d}x\\textrm{d}t.\n\\end{equation*}\n\\qed\\end{proof}\n\nIn addition to $\\Hk{2}$ estimates for weak solutions, bounded weak solutions of equation \\eqref{eq:gen_diff_eq} are actually in $\\Lp{4}_{loc}$. This is rather unusual since most quasi-linear equations do not have $\\Lp{4}$ estimates.\n\\begin{theorem}[$\\Lp{4}$ estimate for bounded solutions]\\label{thm:lp_bound_est}\nLet $\\mathbf{u}$ be a weak bounded solution of the equation \\eqref{eq:gen_diff_eq}. Then  $\\mathbf{\\nabla} \\mathbf{u}$ is locally in $\\Lp{4}$ and for $Q(x,t,r)\\subsetneq Q(x,t,R)\\subset  {\\Omega_T} $ we have the following estimate:\n\\begin{equation}\\label{eq:L4_estimate}\n\\iint_{Q(x,t,r)} |\\mathbf{\\nabla}\\mathbf{u}|^4\\,\\textrm{d}x\\textrm{d}t \\leq\n\\frac{C\\norm{\\mathbf{u}}_\\infty^2}{(R-r)^2}\\iint_{Q(x,t,R)} |\\mathbf{\\nabla}\\mathbf{u}|^2\\,\\textrm{d}x\\textrm{d}t.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nLet us again denote $\\mathbf{\\nabla} \\Phi(\\mathbf{u})$ by $\\mathbf{v}$ and let $\\tau: \\mathds{R}  \\rightarrow \\mathds{R} $ be a smooth increasing function that is linear on $(-\\infty, 1]$ and constant on $[2, \\infty)$. For some large enough constant $C_1$, $C_1\\tau(z)\\geq z (\\tau'(z))^2$. Define $\\tau_\\epsilon$ as $\\tau_\\epsilon(x):= \\tau(\\epsilon x)\/\\epsilon$. Notice that $\\norm{\\tau_\\epsilon'}_\\infty \\leq C_0$ and $C_1\\tau_\\epsilon(z)\\geq z (\\tau_\\epsilon'(z))^2$ with constants independent of $\\epsilon$. Letting $\\xi$ be a smooth bump function as in the proof of Theorem above, multiply the equation \\eqref{eq:gen_diff_eq} by $\\mathbf{v}\\tau_\\epsilon(|\\mathbf{\\nabla}\\mathbf{v}|^2)\\xi^2$ and integrate by parts to obtain\n\\begin{multline*}\n\\iint_{Q(x,t,R)} u^i_tv^i\\tau_\\epsilon(|\\mathbf{\\nabla}\\mathbf{v}|^2)\\xi^2 + |\\mathbf{\\nabla}\\mathbf{v}|^2\\tau_\\epsilon(|\\mathbf{\\nabla}\\mathbf{v}|^2)\\xi^2 \\\\\n+ 2v^iv^i_{x_\\alpha}v^j_{x_\\beta}\\tau_\\epsilon'(|\\mathbf{\\nabla}\\mathbf{v}|^2)v^j_{x_\\alpha x_\\beta}\\xi^2 + 2v^iv^i_{x_\\alpha}\\tau_\\epsilon(|\\mathbf{\\nabla} \\mathbf{v}|^2)\\xi\\xi_{x_\\alpha}\\,\\textrm{d}x\\textrm{d}t=0.\n\\end{multline*}\nHence we compute\n\\begin{eqnarray*}\n\\iint_{Q(x,t,R)} |\\mathbf{\\nabla}\\mathbf{v}|^2\\tau_\\epsilon(|\\mathbf{\\nabla}\\mathbf{v}|^2)\\xi^2\\,\\textrm{d}x\\textrm{d}t  \\leq  \\iint_{Q(x,t,R)} C_2\\norm{\\mathbf{v}}_\\infty^2|\\mathbf{u}_t|^2\\xi^2 + \\frac{1}{2}\\tau^2_\\epsilon(|\\mathbf{\\nabla}\\mathbf{v}|^2)\\xi^2\\\\\n+ C_3\\norm{\\mathbf{\\nabla}\\xi}^2_\\infty \\norm{\\mathbf{v}}_\\infty^2 |\\mathbf{\\nabla}\\mathbf{v}|^2\n+ C_1\\norm{\\mathbf{v}}^2_\\infty|\\mathbf{\\nabla}^2\\mathbf{v}|^2 \\xi^2 + \\frac{1}{4C_1}|\\mathbf{\\nabla}\\mathbf{v}|^4 \\tau_\\epsilon'(|\\mathbf{\\nabla}\\mathbf{v}|^2)^2\\xi^2 \\,\\textrm{d}x\\textrm{d}t.\n\\end{eqnarray*}\nSimplifying and using estimates \\eqref{eq:H2_estimate}, \\eqref{eq:Hessian_estimate} and $C_1\\tau_\\epsilon(z)\\geq z (\\tau_\\epsilon'(z))^2$ we deduce\n$$\n\\iint_{Q(x,t,r)} |\\mathbf{\\nabla}\\mathbf{v}|^2\\tau_\\epsilon(|\\mathbf{\\nabla}\\mathbf{v}|^2)\\,\\textrm{d}x\\textrm{d}t \\leq\n\\frac{C\\norm{\\mathbf{u}}_\\infty^2}{(R-r)^2}\\iint_{Q(x,t,R)} |\\mathbf{\\nabla}\\mathbf{u}|^2\\,\\textrm{d}x\\textrm{d}t.\n$$\nTaking a limit as $\\epsilon\\rightarrow 0$ we deduce by monotone convergence theorem that $\\mathbf{\\nabla}\\mathbf{u}$ is locally in $\\Lp{4}$ and the estimate as claimed in the statement of the Theorem.\n\\qed\\end{proof}\n\\begin{remark}\nThe fact that the gradient is actually locally in $\\Lp{4}$ implies that the singular set of a bounded solution has parabolic Hausdorff dimension smaller than $n-2$. As you may recall from Theorem \\ref{thm:local_regularity} the singular set is contained in the set\n$$\n\\set{(x,t)\\in  {\\Omega_T}  \\ \\Big| \\ \\liminf_{R\\rightarrow 0}\\frac{1}{R^n}\\iint_{Q(x, t ,R)} |\\mathbf{\\nabla} \\mathbf{u}|^2\\,\\textrm{d}x\\textrm{d}t > 0 }.\n$$\nNow using the H\\\"older inequality and the fact that $\\mathbf{u}$ locally in $\\Lp{4}$ we compute\n\\begin{eqnarray*}\n\\frac{1}{R^n}\\iint_{Q(x,t,R)} |\\mathbf{\\nabla}\\mathbf{u}|^2 \\,\\textrm{d}x\\textrm{d}t & \\leq &\n\\frac{1}{R^n}\\left( \\iint_{Q(x,t,R)} |\\mathbf{\\nabla}\\mathbf{u}|^4\\,\\textrm{d}x\\textrm{d}t\\right)^{1\/2}|Q(x,t,R)|^{1\/2}\\\\\n&\\leq& C R^{\\frac{2-n}{2}}\\left(\\iint_{Q(x,t,R)} |\\mathbf{\\nabla}\\mathbf{u}|^4 \\,\\textrm{d}x\\textrm{d}t\\right)^{1\/2}\\\\\n&\\leq& C \\left(R^{2-n}\\iint_{Q(x,t,R)} |\\mathbf{\\nabla}\\mathbf{u}|^4 \\,\\textrm{d}x\\textrm{d}t\\right)^{1\/2}.\n\\end{eqnarray*}\nHence the singular set must be contained in the set\n$$\n\\set{(x,t)\\in  {\\Omega_T}  \\ \\Big| \\ \\liminf_{R\\rightarrow 0}\\frac{1}{R^{n-2}}\\iint_{Q(x, t ,R)} |\\mathbf{\\nabla} \\mathbf{u}|^4 \\,\\textrm{d}x\\textrm{d}t > 0 },\n$$\nwhich has $n-2$ parabolic Hausdorff measure zero.\n\\end{remark}\n\n\n\\section{The key lemma}\nIn this section I discuss the parabolic version of the lemma that seems to be\nkey in the proof of everywhere regularity of some elliptic systems. Not\nsurprisingly, it will turn out that the parabolic lemma is crucial to proving\neverywhere regularity of solutions to some types of parabolic systems. The\nelliptic lemma, to which I refer, is well known and the proof of it can be found\nin ~\\cite{Giaquinta83} in Chapter 7 as part of Theorem 1.1.\n\nBefore we proceed with the discussion of this elliptic lemma let us recall that\ncoefficients $a_{\\alpha\\beta}$ are called \\textbf{strictly elliptic} if there\nexists $\\lambda >0$ such that\n$$\na_{\\alpha\\beta}\\xi^\\alpha\\xi^\\beta \\ge \\lambda |\\xi|^2\\quad\\textrm{for all }\\xi\\in \\Real^n.\n$$\n\\begin{lemma}\\label{lem:elliptic_key_lemma}\nSuppose coefficients $a_{\\alpha\\beta}(x)$ are strictly elliptic, bounded and measurable. Let $u\\in V(\\Omega)$, $f\\in \\Lp{1}(\\Omega)$ be nonnegative functions satisfying\n\\begin{equation}\\label{eq:elliptic_key_lemma_key_pdi}\n- (a_{\\alpha\\beta}u_{x_\\beta})_{x_\\alpha} + f \\le 0\n\\end{equation}\non $\\Omega$. For any $x_0\\in\\Omega$ for which\n$\\Ball{x_0}{R_0}\\subset \\Omega$ for some $R_0$, we have the\nfollowing:\n\\begin{equation}\\label{eq:elliptic_key_lemma_Morrey_decay}\n\\liminf_{R\\rightarrow 0} \\frac{1}{R^{n-2}}\\int_{\\Ball{x_0}{R}} f\\,\\textrm{d}x  = 0.\n\\end{equation}\n\\end{lemma}\n\nThe proof of this lemma is rather simple and follows easily from the elliptic\nHarnack inequality. Because of the peculiar geometry of the parabolic\nHarnack inequality, the elliptic proof does not translate directly into the parabolic case. Instead, by adopting proof of elliptic lemma to parabolic equations, one is able to control $f$ on the cylinders whose top centers are slightly shifted back in time. In fact it is not true that $f$ can be controlled on the cylinders whose top centers are not fixed, without additional assumptions on $f$. It turns out, however, that the assumption that one needs to impose to prove the parabolic version of the lemma is satisfied in applications to everywhere regularity of parabolic systems. What we need to assume is that for some $\\alpha > 1$ the $\\Lp{\\alpha}$ norm of $f$ on a cylinder is controlled by the $\\Lp{1}$ norm of $f$, perhaps on a larger cylinder.\n\n\\begin{lemma}[Key Lemma]\\label{lem:key_lemma}\nSuppose coefficients $a_{\\alpha\\beta}(x,t)$ are strictly\nelliptic, bounded and measurable.\nLet $u\\in V( {\\Omega_T} )$, $f\\in \\Lp{1}( {\\Omega_T} )$ be nonnegative\nfunctions weakly satisfying\n\\begin{equation}\\label{eq:key_lemma_key_pdi}\nu_t - (a_{\\alpha\\beta}u_{x_\\beta})_{x_\\alpha} + f \\le 0\n\\end{equation}\non $ {\\Omega_T} $.\nFurther suppose that for some $\\alpha >1$ our $f$ satisfies the following:\n\\begin{equation}\\label{eq:higher_Lp}\n\\left(\\Xiint{\\rule{1.2em}{0.5pt}}_{Q_R} f^\\alpha \\,\\textrm{d}x\\textrm{d}t\\right)^{1\/\\alpha}\\leq C \\ \\Xiint{\\rule{1.2em}{0.5pt}}_{Q_{4R}} f \\,\\textrm{d}x\\textrm{d}t,\n\\end{equation}\nfor all $Q_R\\subset Q_{4R}\\subset {\\Omega_T} $. Then for any $(x_0, t_0)\\in {\\Omega_T} $ for which\n$$\n\\textrm{B}_{R_0}(x_0)\\times (t_0 - R_0^2, t_0+R_0^2)\\subset  {\\Omega_T} , \\textrm{ for some } R_0,\n$$\nwe have the following:\n\\begin{equation}\\label{eq:key_lemma_Morrey_decay}\n\\liminf_{R\\rightarrow 0} \\frac{1}{R^{n}}\\iint_{Q(x_0,t_0,R)} f \\,\\textrm{d}x\\textrm{d}t  = 0.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nFix $0 <\\sigma \\le 1\/4$. Set\n$$\nR_i:= \\sigma^i R_0, \\quad Q_i:= Q(x_0, t_0, R_i), \\quad M_i:= \\sup_{Q_i} u\n$$\nand\n$$\nQ'_i:= Q(x_0,t_0 - 4\\sigma^2 R_i^2, (1-8\\sigma^2)^{1\/2}R_i),\n$$\n$$\nQ''_i:= Q(x_0,t_0 - 2\\sigma^2 R_i^2, (1-4\\sigma^2)^{1\/2}R_i).\n$$\nWe divide the proof in three steps. In the first step we show that for any $\\sigma \\in (0, 1\/4]$ we have\n$$\n\\lim_{i\\rightarrow \\infty}\\frac{1}{R_i^n}\\iint_{Q'_i} f \\,\\textrm{d}x\\textrm{d}t = 0.\n$$\nOnce we have done that, we show that we can control $f$ on $Q_i\\backslash Q'_i$ with the help of the assumption on $f$. Finally we will put it all together to conclude the lemma. \\\\\n\\noindent \\emph{Step 1}. \nFix $i$, and set $z := M_i - u$. We see that $z \\geq 0$ on $Q_i$ and $z$ satisfies\n\\begin{equation}\\label{eq:kl_z}\nz_t - (a_{\\alpha\\beta} z_{x_\\beta})_{x_\\alpha} \\ge f.\n\\end{equation}\nIn particular, due to parabolic Harnack inequality (see Theorem 6.24 in ~\\cite{Lieberman96})\n\\begin{equation}\\label{eq:kl_harnack_z}\n\\Xiint{\\rule{1.2em}{0.5pt}}_{Q''_i} z\\,\\textrm{d}x\\textrm{d}t \\leq C \\inf_{Q_{i+1}} z.\n\\end{equation}\nLet $w$ solve backward time parabolic equation\n\\begin{equation}\\label{eq:kl_w}\n-w_t - (a_{\\alpha\\beta}w_{x_\\alpha})_{x_\\beta} = \\frac{1}{R^{2}_i}\\chi_{Q''_{i}}\n\\end{equation}\non $Q_i$ with $w\\equiv 0$ on the backward time parabolic\nboundary, that is\n$$\nw \\equiv 0 \\textrm{ on }(\\partial \\textrm{B}_{R_i}(x_0) \\times [t_0-R^2_{i}, t_0])\\cup\n(B_{R_i}(x_0)\\times \\set{t_0}).\n$$\nAt this point if $zw$ were actually differentiable in time, we\nwould multiply equation \\eqref{eq:kl_w} by $zw$ and\nintegrate by parts to obtain\n$$\n\\iint_{Q_i} -z\\left(\\frac{w^2}{2}\\right)_t + a_{\\alpha\\beta}\nz_\\beta\\left(\\frac{w^2}{2}\\right)_{x_\\alpha}  + a_{\\alpha\\beta}\nw_{x_\\alpha}w_{x_\\beta}z \\,\\textrm{d}x\\textrm{d}t = \\frac{1}{R_i^2}\\iint_{Q''_{i}} zw\\,\\textrm{d}x\\textrm{d}t,\n$$\nand since $z$ satisfies equation \\eqref{eq:kl_z} we\nwould conclude that\n\\begin{equation}\\label{eq:key_lemma_sec_to_last}\n\\frac{1}{R_i^n}\\iint_{Q'_i} f\\left(\\frac{w^2}{2}\\right)\\,\\textrm{d}x\\textrm{d}t \\leq\n\\frac{1}{R_i^{n+2}}\\iint_{Q''_{i}} zw\\,\\textrm{d}x\\textrm{d}t.\n\\end{equation}\nIn general we cannot expect $zw$ to be differentiable in time.\nTo obtain equation \\eqref{eq:key_lemma_sec_to_last} rigorously\none would need to use Steklov average\n$$\n(zw)^h(x,t):=\\frac{1}{h}\\int_{t-h}^t z(x,\\tau)w(x,\\tau)\\,\\textrm{d} \\tau\n$$\nas a test function in \\eqref{eq:kl_w}. However, we will not do this\nhere, instead I refer the reader to Lemma 6.1 in\n~\\cite{Lieberman96}, where similar computation has been carried out.\n\nNow, since $w$ solves \\eqref{eq:kl_w}, by strong maximal\nprinciple $w \\ge \\theta >0$ on $Q'_i$ and also $w \\le C$\non $Q_i$, with bounds independent of $i$ (one can see this by\nscaling for example). Therefore, combining this observation with\ninequality \\eqref{eq:kl_harnack_z}, we obtain\n$$\n\\frac{1}{R_i^n}\\iint_{Q'_i} f \\,\\textrm{d}x\\textrm{d}t \\le C\\Xiint{\\rule{1.2em}{0.5pt}}_{Q''_{i}} z \\,\\textrm{d}x\\textrm{d}t \\le C \\inf_{Q_{i+1}} z = C (M_i - M_{i+1}).\n$$\nHowever, since\n$$\n\\sum_{i=0}^\\infty M_i - M_{i+1} \\leq \\sup_{Q_0} u,\n$$\nwe conclude that\n$$\n\\frac{1}{R_i^n}\\iint_{Q'_i} f \\,\\textrm{d}x\\textrm{d}t\\rightarrow 0 \\quad \\textrm{as}\\quad i \\rightarrow \\infty.\n$$\n\\\\\n\\noindent\\emph{ Step 2}.\nLet $A$ be some measurable set. We will show that for all $Q_R\\subset Q_{8R} \\subset Q_{R_0}$ and for all $\\epsilon$, there exists $\\delta$ such that if $|A\\cap Q_R|\\leq \\delta |Q_R|$, then\n$$\n\\frac{1}{R^n}\\iint_{A\\cap Q_R}f \\,\\textrm{d}x\\textrm{d}t \\leq \\epsilon.\n$$\nFirst, we can easily deduce by the argument similar to the one in part 1, that\n\\begin{equation}\\label{eq:kl_bound_f}\n\\frac{1}{(4R)^n}\\iint_{Q_{4R}}f \\,\\textrm{d}x\\textrm{d}t \\le C,\n\\end{equation}\nwhere constant is independent of $R$.\n\nWe now use \\eqref{eq:higher_Lp} to conclude that\n\\begin{eqnarray*}\n\\iint_{A\\cap Q_R} f\\,\\textrm{d}x\\textrm{d}t & \\le & \\left(\\frac{1}{|Q_R|}\\iint_{A\\cap Q_R} f^\\alpha \\,\\textrm{d}x\\textrm{d}t \\right)^{1\/\\alpha}|Q_R|^{1\/\\alpha}|A\\cap Q_R|^{1-1\/\\alpha} \\\\\n& = & C|Q_R|^{1\/\\alpha}|A\\cap Q_R|^{1- 1\/\\alpha} \\Xiint{\\rule{1.2em}{0.5pt}}_{Q_{4R}} f \\,\\textrm{d}x\\textrm{d}t \\\\\n& = & C\\left(\\frac{|A\\cap Q_R|}{|Q_R|}\\right)^{1-1\/\\alpha} \\iint_{Q_{4R}}f \\,\\textrm{d}x\\textrm{d}t \\leq C_1R^n\\delta^{1-1\/\\alpha}.\n\\end{eqnarray*}\nAbove, the last inequality follows by \\eqref{eq:kl_bound_f}.\n\n\\noindent \\emph{Step 3}.\\\\\nFinally we put everything together. Fix $\\epsilon >0$. First notice that by choosing $\\sigma$ small enough we can make\n$$\n|Q_i\\backslash Q'_i| \\leq \\delta |Q_i|,\n$$\nwhere $C_1\\delta^{1-1\/\\alpha} < \\epsilon \/2$. Then by first step we can find $i >2$ such that\n$$\n\\frac{1}{R_i^n}\\iint_{Q'_i} f \\,\\textrm{d}x\\textrm{d}t \\leq \\epsilon\/2.\n$$\nFinally, the above together with conclusion of second step gives us\n$$\n\\frac{1}{R_i^n}\\iint_{Q_i} f \\,\\textrm{d}x\\textrm{d}t = \\frac{1}{R_i^n}\\iint_{Q'_i} f \\,\\textrm{d}x\\textrm{d}t + \\frac{1}{R_i^n}\\iint_{Q_i\\backslash Q'_i} f \\,\\textrm{d}x\\textrm{d}t < \\epsilon.\n$$\n\\qed\\end{proof}\n\nNow we are in position to apply our key Lemma to deduce crucial importance of entropy in questions of everywhere regularity for parabolic systems.\n\\begin{theorem}[Entropy condition]\\label{thm:entropy_condition}\nLet $\\mathbf{u}$ be weak solution of \\eqref{eq:nonlinear_system}. Suppose there exists $\\phi\\in V( {\\Omega_T} )$ that together with $\\mathbf{u}$ weakly satisfy the following inequality:\n$$\n\\phi_t - (a_{\\alpha\\beta}\\phi_{x_\\beta})_{x_\\alpha} + \\lambda|\\mathbf{\\nabla} \\mathbf{u}|^2 \\leq 0,\n$$\nwhere $a_{\\alpha\\beta}$ are bounded and strictly elliptic. Then $\\mathbf{u}$ is everywhere H\\\"older continuous on the interior of $ {\\Omega_T} $.\n\\end{theorem}\n\\begin{proof}\nSince $\\mathbf{u}$ satisfies condition \\eqref{eq:Lp_estimate}, we see immediately that conditions of the key Lemma \\ref{lem:key_lemma} are satisfied. Therefore, we conclude that\n$$\n\\liminf_{R\\rightarrow 0}\\frac{1}{R^n}\\iint_{Q(X_0,R)} |\\mathbf{\\nabla} \\mathbf{u}|^2 \\,\\textrm{d}x\\textrm{d}t= 0.\n$$\nHowever, this is precisely the condition \\eqref{eq:local_regularity_condition} of Theorem \\ref{thm:local_regularity}. Hence we conclude that $\\mathbf{u}$ is everywhere H\\\"older continuous on the interior of $ {\\Omega_T} $.\n\\qed\\end{proof}\n\n\\section{Everywhere regularity of certain diffusion systems}\nIn this section we come back to the discussion of weak solutions of the diffusion system \\eqref{eq:gen_diff_eq}, imposing additional requirement that $\\Phi(z)$ is a function of only the norm of $z$. As we will see this allows us to conclude much more about solutions of this equation. In particular solutions are bounded if they are bounded initially and at the boundary. More importantly, solutions are actually everywhere H\\\"older continuous and thus smooth if $\\Phi$ is.\n\nFirst I will show that in the case when $\\Phi$ is a functions of only the norm, weak solutions of equation \\eqref{eq:gen_diff_eq} that are bounded initially and bounded at the boundary will remain bounded for all time.\n\\begin{theorem}[Boundedness]\\label{thm:boundedness}\nLet $\\mathbf{u}$ be a weak solution of the generalized diffusion equation \\eqref{eq:gen_diff_eq}. Also suppose that $\\norm{\\mathbf{u}(\\cdot,0)}_{\\Lp{\\infty}(\\Omega)}< \\infty$ and $\\norm{\\mathbf{u}(\\cdot,t)}_{\\Lp{\\infty}(\\partial\\Omega)} < \\infty$ for all $t\\in [0,T]$. Then $\\norm{\\mathbf{u}}_{\\Lp{\\infty}(\\Omega)} < \\infty$ and we have\n\\begin{equation}\\label{eq:sup_bound}\n\\norm{\\mathbf{u}(\\cdot, t)}_{\\Lp{\\infty}(\\Omega)}\\leq \\max\\set{\\norm{\\mathbf{u}(\\cdot,0)}_{\\Lp{\\infty}(\\Omega)}, \\sup_{s\\in [0,t]}\\norm{\\mathbf{u}(\\cdot,s)}_{\\Lp{\\infty}(\\partial\\Omega)} }.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nSet $B$ to\n$$\nB = \\max\\set{\\norm{\\mathbf{u}(\\cdot,0)}_{\\Lp{\\infty}(\\Omega)}, \\sup_{s\\in [0,t]}\\norm{\\mathbf{u}(\\cdot,s)}_{\\Lp{\\infty}(\\partial\\Omega)}}.\n$$\nFix $\\epsilon > 0$ that is less than one. Let $\\gamma: \\mathds{R} _+\\rightarrow \\mathds{R} $ be a smooth convex function which is identically zero on $[0,B+\\epsilon]$, positive and increasing otherwise, and linear on $[B+1, \\infty)$. Also set $\\Gamma(z)$ to $\\gamma(|z|)$. Then we compute\n\\begin{eqnarray*}\n\\frac{d}{dt}\\int_\\Omega \\Gamma(\\mathbf{u}(x,t))\\,\\textrm{d}x & = & \\int_\\Omega \\Gamma_{z_i}u^i_t\\,\\textrm{d}x \\\\\n& = & \\int_\\Omega \\left(\\Gamma_{z_i}\\Phi_{z_i z_j} u^j_{x_\\alpha}\\right)_{x_\\alpha}\n- \\Gamma_{z_i z_k}\\Phi_{z_k z_j} u^i_{x_\\alpha}u^j_{x_\\alpha} \\,\\textrm{d}x \\\\\n& \\leq & \\int_{\\partial\\Omega} \\Gamma_{z_i}\\Phi_{z_i z_j} u^j_{x_\\alpha} \\nu_\\alpha \\,\\textrm{d}S \\\\\n& = & 0.\n\\end{eqnarray*}\nThe inequality is true because Hessians of two functions of only the norm commute and both $\\Gamma$ and $\\Phi$ are convex. The last equality is true because $\\gamma$ is identically zero on $[0,B+\\epsilon]$ and $|\\mathbf{u}|$ is less than or equal to $B$ on the boundary. Since $\\Gamma(\\mathbf{u})$ is positive and initially zero we conclude that $\\Gamma$ is zero up to time $t$ and thus $\\norm{\\mathbf{u}(\\cdot, t)}_\\infty \\leq B +\\epsilon$. Since the inequality is true for all $\\epsilon >0$ the Theorem follows.\n\\qed\\end{proof}\n\nThe next Lemma will show that if we suppose that $\\Phi(z)$ is a function only of the norm of $z$, then there exists an entropy that satisfies conditions of Theorem \\ref{thm:entropy_condition}.\n\\begin{lemma}\\label{lem:gen_diff_scalar}\nLet $\\mathbf{u}$ be a weak bounded solution of \\eqref{eq:gen_diff_eq} and suppose\n$\\Phi$ is of the form\n$$\n\\Phi(\\mathbf{u}) = \\phi(|\\mathbf{u}|).\n$$\nThen there is a continuously differentiable, strictly increasing function\n$\\gamma: \\mathds{R} \\rightarrow \\mathds{R} $ such that $\\phi=\\phi(|\\mathbf{u}|)$ weakly satisfies the following\ninequality:\n\\begin{equation}\n\\phi_t - \\Delta(\\gamma(\\phi)) + \\lambda^2 |\\mathbf{\\nabla}\\mathbf{u}|^2 \\leq 0.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nFirst of all, without loss of generality we can suppose that $\\Phi(0)=0$.\nNotice that $\\phi$ satisfies the following equation weakly:\n$$\n\\phi_t - (\\Phi_{z_i}\\Phi_{z_i z_j}u^j_{x_\\alpha})_{x_\\alpha} +\n|\\mathbf{\\nabla}_x(\\mathbf{\\nabla}\\Phi(\\mathbf{u}))|^2=0.\n$$\nLooking at the quantity inside the divergence term we see that it is equal to\n$$\n\\Phi_{z_i}\\Phi_{z_i z_j}u^j_{x_\\alpha} =\n\\left(\\frac{1}{2}|\\mathbf{\\nabla}\\Phi(\\mathbf{u})|^2\\right)_{x_\\alpha}.\n$$\nSince $\\Phi(z) = \\phi(|z|)$, we observe that\n$$\n\\frac{1}{2}|\\mathbf{\\nabla} \\Phi(\\mathbf{u})|^2 = \\frac{1}{2} \\phi'(|\\mathbf{u}|)^2.\n$$\nLet $\\psi$ be the continuous inverse of $\\phi$. Set\n$$\n\\gamma(z) = \\int_0^z \\phi''(\\psi(t)) \\,\\textrm{d}t.\n$$\nThen multiplying $\\gamma'(\\phi(z))$ by $\\phi'(z)$ and integrating, we see that $\\gamma$ and $\\phi$ satisfy\n$$\n\\gamma(\\phi(z)) = \\frac{1}{2}\\phi'(z)^2.\n$$\nThis $\\gamma$ is continuously differentiable and strictly\nincreasing, since $\\phi$ is strictly convex. Therefore we\nconclude that $\\phi$ satisfies\n$$\n\\phi_t - \\Delta(\\gamma(\\phi))+|\\mathbf{\\nabla}_x(\\mathbf{\\nabla}\\Phi(\\mathbf{u}))|^2=0,\n$$\nand due to strict convexity the last term on the left hand side is greater or\nequal to $\\lambda^2 |\\mathbf{\\nabla} \\mathbf{u}|^2$.\n\\qed\\end{proof}\n\nNow we are in position to use the above Lemma \\ref{lem:gen_diff_scalar}\ntogether with Theorem \\ref{thm:entropy_condition} to deduce\n\\begin{theorem}\nWeak solutions of equation (\\ref{eq:gen_diff_eq}) are H\\\"older\ncontinuous in $ {\\Omega_T} $.\n\\end{theorem}\n\\begin{proof}\nLemma \\ref{lem:gen_diff_scalar} tells us, that for $\\mathbf{u}$ a weak bounded solution of \\eqref{eq:gen_diff_eq}, there exists $\\phi$ satisfying\n$$\n\\phi_t - (a\\phi_{x_\\alpha})_{x_\\alpha} + \\lambda^2 |\\mathbf{\\nabla}\\mathbf{u}|^2 \\leq 0,\n$$\nwhere $a(x,t) := \\gamma'(\\phi(x,t))$. However, this is precisely the condition of Theorem \\ref{thm:entropy_condition}. Therefore, we conclude the proof.\n\\qed\\end{proof}\n\n\\section{Strongly coupled parabolic systems}\nOne of the earliest nontrivial examples of quasi-linear\nparabolic systems whose solutions have interior everywhere\nregularity was due to Wiegner ~\\cite{Wiegner92}. These are the\nso-called strongly coupled parabolic systems of the following\nform:\n\\begin{equation}\\label{eq:strongly_coupled}\nu^i_t - (a_{\\alpha\\beta}u^i_{x_\\beta} +\nc^i_{\\alpha\\beta}H_{x_\\beta})_{x_\\alpha} = 0,\n\\end{equation}\nwhere\n\\begin{enumerate}\n\\item $H = H(\\mathbf{u})$ is a function of $\\mathbf{u}$;\n\\item $A^{ij}_{\\alpha\\beta}:= a_{\\alpha\\beta}\\delta^{ij} + c^i_{\\alpha\\beta}H_{z_j}$ and $a_{\\alpha\\beta}$\nare strictly elliptic in the sense that\n$$\n \\lambda |\\xi|^2\n\\le A^{ij}_{\\alpha\\beta}\\xi^i_\\alpha\\xi^j_\\beta,\\textrm{ and } \\lambda |\\zeta|^2 \\le a_{\\alpha\\beta} \\zeta^i \\zeta^j, \\quad \\textrm{for all }\\xi\\in  \\mathds{R} ^{nN}, \\zeta\\in \\Real^n;\n$$\n\\item $H(z)$ is twice continuously differentiable and\n$\\lambda |\\zeta|^2 \\le  H_{z_i z_j} \\zeta^i \\zeta^j;$\n\\item $a_{\\alpha \\beta}$ and $c^i_{\\alpha\\beta}$ are bounded.\n\\end{enumerate}\n\nAfter Wiegner, Dung ~\\cite{Dung99} also worked on these types of system. However, neither Wiegner's nor Dung's proofs of everywhere regularity for solutions of strongly coupled parabolic systems reduce to something analogous to the key Lemma \\ref{lem:key_lemma}. It is instructive to prove everywhere regularity of weak solutions to \\eqref{eq:strongly_coupled} using the key Lemma to illustrate an underlying similarity between strongly coupled parabolic systems and generalized diffusion equations \\eqref{eq:gen_diff_eq}. It appears that for both systems discussed in this paper the existence of an \\emph{entropy} is crucial for everywhere regularity of their solutions.\n\n\\begin{remark}\nWhen $\\Phi$ only depends on the norm of the gradient, diffusion system \\eqref{eq:gen_diff_eq} actually has the form of a strongly coupled system, except with possibly \\emph{non-convex} $H$. Indeed, if $\\Phi(z)= \\phi(|z|)$, then\n$$\n\\Phi_{z_i z_j}(z) = \\frac{\\phi'(|z|)}{|z|}\\delta_{ij} +\n\\frac{z_i}{|z|}\\left(\\phi''(|z|) - \\frac{\\phi'(|z|)}{|z|}\\right)\\frac{z_j}{|z|},\n$$\ntherefore, with\n$$\na_{\\alpha\\beta} = \\frac{\\phi'(|\\mathbf{u}|)}{|\\mathbf{u}|}\\delta_{\\alpha\\beta}, \\quad\nc^i_{\\alpha\\beta} =\\frac{u^i}{|\\mathbf{u}|}\\delta_{\\alpha\\beta},\n$$\nand\n$$\nH(z) = \\phi'(|z|) - \\int_0^{|z|} \\frac{\\phi'(s)}{s} \\,\\textrm{d}s,\n$$\nsystem \\eqref{eq:gen_diff_eq} has the form \\eqref{eq:strongly_coupled}.\n\\end{remark}\n\nWe can prove an interior everywhere regularity result for strongly coupled\nparabolic systems rather easily using Theorem \\ref{thm:entropy_condition}.\nAs in the previous section I will show existence of an entropy.\n\\begin{lemma}\nLet $\\mathbf{u} \\in V( {\\Omega_T} ; \\Real^N)$ be a weak bounded solution of\n(\\ref{eq:strongly_coupled}), then for some large enough $s$ there is a positive\nconstant $c$ such that $v := e^{sH}$ is a subsolution of the following\nequation:\n\\begin{equation}\\label{eq:strongly_coupled_scalar}\nv_t - (A_{\\alpha\\beta} v_{x_\\beta})_{x_\\alpha} + c|\\mathbf{\\nabla}\\mathbf{u}|^2 \\leq 0.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nWe compute\n\\begin{eqnarray*}\n(e^{sH})_t  &=& se^{sH}H_{z_i}(a_{\\alpha\\beta}u^i_{x_\\beta} +\nc^i_{\\alpha\\beta}H_{x_\\beta})_{x_\\alpha} \\\\\n& = & (A_{\\alpha\\beta}(e^{sH})_{x_\\beta})_{x_\\alpha} -\n(se^{sH}H_{z_i})_{x_\\alpha}(a_{\\alpha\\beta}u^i_{x_\\beta} +\nc^i_{\\alpha\\beta}H_{x_\\beta}).\n\\end{eqnarray*}\nThe last term on the right becomes\n\\begin{multline*}\n(se^{sH}H_{z_i})_{x_\\alpha}(a_{\\alpha\\beta}u^i_{x_\\beta}\n+c^i_{\\alpha\\beta}\nH_{x_\\beta})  =  s^2 e^{sH}A_{\\alpha\\beta}H_{x_\\alpha} H_{x_\\beta}  \\\\\n+ se^{sH}H_{z_i z_j}a_{\\alpha\\beta}u^i_{x_\\alpha}u^j_{x_\\beta} +\nse^{sH}H_{z_i z_j}c^i_{\\alpha\\beta}H_{x_\\beta}u^i_{x_\\alpha} \\\\\n\\ge se^{sH}\\left( \\lambda s|\\mathbf{\\nabla} H|^2 + \\lambda |\\mathbf{\\nabla} \\mathbf{u}|^2\n-C(\\epsilon)|\\mathbf{\\nabla} H|^2 - \\epsilon|\\mathbf{\\nabla} \\mathbf{u}|^2\\right) \\ge\n\\frac{\\lambda}{2} se^{sH}|\\mathbf{\\nabla} \\mathbf{u}|^2.\n\\end{multline*}\nThe last inequality follows by first making $\\epsilon$ small and then $s$ large. Since\n$\\mathbf{u}$ is bounded, we have $H$ is bounded from below. Therefore, for some\n$c$, $v$ satisfies equation (\\ref{eq:strongly_coupled_scalar}) as claimed.\n\\qed\\end{proof}\nAt this point we immediately conclude that conditions of Theorem \\ref{thm:entropy_condition} are satisfied. Therefore we have established\n\\begin{theorem}\nBounded weak solutions of (\\ref{eq:strongly_coupled}) are\nH\\\"older continuous in $ {\\Omega_T} $.\n\\end{theorem}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\nConsider the following coin-tossing game. To play the game, we will begin with a graph that has one active node with infinitely many inactive nodes on either side of it. In this paper, we will also refer to active nodes as occupied sites and inactive nodes as unoccupied sites. We will label the initial occupied site the origin. At the origin, there is a queue of particles that will perform random walks on the graph. This is seen in Figure \\ref{fig:begstate}, where occupied sites are black, unoccupied sites are white, the origin is labeled with ``$\\star$'' and the particles in the queue are indicated with ``$\\circ$''.\n\n\\begin{figure}\n\\[\n\\begin{tikzpicture}\n\\tikzstyle{wn}=[inner sep =3, fill=white, draw=black, circle];\n\\tikzstyle{bn}=[inner sep =3, fill=black, draw=black, circle];\n\\draw (0,0)--(1,0) node[wn] {} --(2,0) node[wn] {}--(3,0) node[wn] {}--(4,0) node[bn] {}--(5,0) node[wn] {}--(6,0) node[wn] {}--(7,0);\n\\draw (4,.2) node[above] {$\\circ$}; \\draw (4,.45) node[above] {$\\circ$}; \\draw (4,.7) node[above] {$\\circ$}; \n\\draw (4,1) node[above] {$\\vdots$};\n\\draw (4,-.2) node[below] {$\\star$};\n\\draw (0,0) node[left] {$\\cdots$};\n\\draw (7,0) node[right] {$\\cdots$};\n\\end{tikzpicture}\n\\]\n\\caption{The starting graph for our coin-tossing game.}\\label{fig:begstate}\n\\end{figure}\n\nWe imagine dropping the first particle in the queue onto the origin. Flipping heads will move the particle one position to the right and flipping tails will move it one position to the left. The game consists of flipping a coin repeatedly until the particle reaches an unoccupied site. Once an unoccupied site is reached, the particle settles there and occupies it. Now, the next particle in the queue is dropped onto the origin and the game begins again.\n\nThe illustration in Figure \\ref{fig:game} shows an example of the game being played for the first two particles in the queue. \nIn the example, we first flip heads. Thus, the first particle moves to the unoccupied site immediately to the right of the origin, settles there, and occupies it. Now, the graph has two occupied sites, each with infinitely many unoccupied sites on each side. Since a new site was occupied, the second particle in the queue is dropped onto the origin and the game begins again. With the next toss, there are two unoccupied sites that could become occupied: the site to the left of the origin or the site two to the right of the origin. Suppose the next coin toss is heads, followed by tails, then tails again. These three tosses would move the new particle to the occupied site, back to the origin, and then to the unoccupied site to the left of the origin. As a result, the particle settles at this site and occupies it. Next, the third next particle in the queue is dropped on the origin and the process continues.\n\n\\begin{figure}\n\\[\n\\begin{tikzpicture}\n\\draw (0,0) node[scale=.75] (a) {\n\\begin{tikzpicture}\n\\draw (2.5,0) -- (3,0) node[inner sep =3, fill=white, draw=black, circle] {}--(4,0) node[inner sep =3, fill=black, draw=black, circle] {}--(5,0) node[inner sep =3, fill=white, draw=black, circle] {} -- (5.5,0);\n\\draw (4,-.2) node[below] {$\\star$}; \\draw (4,.2) node[above] {$\\circ$}; \\draw (2.5,0) node[left] {$\\dots$}; \\draw (5.5,0) node[right] {$\\dots$};\n\n\\end{tikzpicture}\n};\n\\draw (0,-2) node[scale=.75] (a2) {\n\\begin{tikzpicture}\n\\draw (2.5,0) -- (3,0) node[inner sep =3, fill=white, draw=black, circle] {}--(4,0) node[inner sep =3, fill=black, draw=black, circle] {}--(5,0) node[inner sep =3, fill=white, draw=black, circle] {} -- (5.5,0);\n\\draw (4,-.2) node[below] {$\\star$}; \\draw (5,.2) node[above] {$\\circ$}; \\draw (2.5,0) node[left] {$\\dots$}; \\draw (5.5,0) node[right] {$\\dots$};\n\\end{tikzpicture}\n};\n\\draw (5.5,-2) node[scale=.75] (b) {\n\\begin{tikzpicture}\n\\draw (2.5,0) -- (3,0) node[inner sep =3, fill=white, draw=black, circle] {}--(4,0) node[inner sep =3, fill=black, draw=black, circle] {}--(5,0) node[inner sep =3, fill=black, draw=black, circle] {}--(6,0) node[inner sep =3, fill=white, draw=black, circle] {} -- (6.5,0);\n\\draw (4,-.2) node[below] {$\\star$}; \\draw (4,.2) node[above] {$\\circ$}; \\draw (2.5,0) node[left] {$\\dots$}; \\draw (6.5,0) node[right] {$\\cdots$};\n\\end{tikzpicture}\n};\n\\draw (5.5,-4) node[scale=.75] (b2) {\n\\begin{tikzpicture}\n\\draw (2.5,0) -- (3,0) node[inner sep =3, fill=white, draw=black, circle] {}--(4,0) node[inner sep =3, fill=black, draw=black, circle] {}--(5,0) node[inner sep =3, fill=black, draw=black, circle] {}--(6,0) node[inner sep =3, fill=white, draw=black, circle] {} -- (6.5,0);\n\\draw (4,-.2) node[below] {$\\star$}; \\draw (5,.2) node[above] {$\\circ$}; \\draw (2.5,0) node[left] {$\\dots$}; \\draw (6.5,0) node[right] {$\\cdots$};\n\\end{tikzpicture}\n};\n\\draw (5.5,-6) node[scale=.75] (b3) {\n\\begin{tikzpicture}\n\\draw (2.5,0) -- (3,0) node[inner sep =3, fill=white, draw=black, circle] {}--(4,0) node[inner sep =3, fill=black, draw=black, circle] {}--(5,0) node[inner sep =3, fill=black, draw=black, circle] {}--(6,0) node[inner sep =3, fill=white, draw=black, circle] {} -- (6.5,0);\n\\draw (4,-.2) node[below] {$\\star$}; \\draw (4,.2) node[above] {$\\circ$}; \\draw (2.5,0) node[left] {$\\dots$}; \\draw (6.5,0) node[right] {$\\cdots$};\n\\end{tikzpicture}\n};\n\\draw (5.5,-8) node[scale=.75] (b4) {\n\\begin{tikzpicture}\n\\draw (2.5,0) -- (3,0) node[inner sep =3, fill=white, draw=black, circle] {}--(4,0) node[inner sep =3, fill=black, draw=black, circle] {}--(5,0) node[inner sep =3, fill=black, draw=black, circle] {}--(6,0) node[inner sep =3, fill=white, draw=black, circle] {} -- (6.5,0);\n\\draw (4,-.2) node[below] {$\\star$}; \\draw (3,.2) node[above] {$\\circ$}; \\draw (2.5,0) node[left] {$\\dots$}; \\draw (6.5,0) node[right] {$\\cdots$};\n\\end{tikzpicture}\n};\n\\draw (11.5,-8) node[scale=.75] (c) {\n\\begin{tikzpicture}\n\\draw (1.5,0) -- (2,0) node[inner sep =3, fill=white, draw=black, circle] {}--(3,0) node[inner sep =3, fill=black, draw=black, circle] {}--(4,0) node[inner sep =3, fill=black, draw=black, circle] {}--(5,0) node[inner sep =3, fill=black, draw=black, circle] {}--(6,0) node[inner sep =3, fill=white, draw=black, circle] {} -- (6.5,0);\n\\draw (4,-.2) node[below] {$\\star$}; \\draw (4,.2) node[above] {$\\circ$}; \\draw (1.5,0) node[left] {$\\dots$}; \\draw (6.5,0) node[right] {$\\cdots$};\n\\end{tikzpicture}\n};\n\\draw[->] (a) -- node[midway,right] {H} (a2);\n\\draw[->] (b) -- node[midway,right] {H} (b2);\n\\draw[->] (b2) -- node[midway,right] {T} (b3);\n\\draw[->] (b3) -- node[midway,right] {T} (b4);\n\\draw[->] (a2) -- node[midway,above,sloped] {new particle} (b);\n\\draw[->] (b4) -- node[midway,above,sloped] {new particle} (c);\n\\end{tikzpicture}\n\\] \n\\caption{Playing the coin-tossing game.}\\label{fig:game}\n\\end{figure}\n\nIn our game, each particle on its own is performing a simple one-dimensional random walk, which is a well-studied stochastic model. See \\cite{Sauer}. This game we have described, of sending particles to take random walks in succession, is known as Internal Diffusion Limited Aggregation (Internal DLA). This process was defined by Meakin and Deutch in \\cite{MeakinDeutch} to serve as a model for ``chemical etching.'' In a more mathematical work by Diaconis and Fulton \\cite{DiaconisFulton}, Internal DLA was studied using the language of Markov chains, with interesting algebraic connections. The limiting behavior of this model has been frequently studied in dimensions two and three (e.g., in a square grid, the limiting shape is a ball), but here we will consider the one-dimensional case.     \n\nSome potential questions we can ask in the one-dimensional case for Internal DLA are:\n\\begin{enumerate}\n  \\item What is the probability that, after releasing $n$ particles, $k$ of them settle to the right of the origin? \n  \\item On average, how many coin tosses will it take for $n$ particles to settle? \n  \\item Given a large queue of particles, if the coin is tossed $N$ times, what is the probability that there are $n$ occupied sites? \n  \\item What if the particles do not perform a simple random walk and instead there is a probability assigned to each site? \n\\end{enumerate}\nIn this paper, we prove results for questions $(1)$ and $(2)$. Diaconis and Fulton addressed question $(1)$ and found the limiting distribution (as $n \\to \\infty$) is normal \\cite[Proposition 3.2]{DiaconisFulton}. We give a more precise characterization here. While we have investigated questions $(3)$ and $(4)$, we do not yet have any major results. Some comments on further pursuit of these questions are included at the end of the paper. \n\nIn Section 1, we consider question $(1)$ as a probability distribution for our coin-tossing game. That is, given that $n$ particles have left the origin and settled, we determine the probability that $k$ are to the right of the origin. We first used a Monte Carlo simulation to predict a distribution. Each simulation ran $100\\,000$ trials of the game and returned the minimum and maximum positions reached on the graph (assuming the origin is at position 0). Table \\ref{tab:Monte1} shows the observed probability of the game reaching maximum ending position $k$ for $1 \\leq n \\leq 7$.\n\n\\begin{table}\n\\[\n\\begin{array}{c | c c c c c c c | c}\nn \\backslash k & 0 & 1 & 2 & 3 & 4 & 5 & 6 & SSE\\\\\n\\hline\n1 & 1.00000 &&&&&&& 0.00000 \\\\\n2 & 0.50266 & 0.49734 &&&&&& 1.41512\\mathrm{e}{-10} \\\\\n3 & 0.16730 & 0.66561 & 0.16709 &&&&& 1.69687\\mathrm{e}{-10} \\\\\n4 & 0.04235 & 0.45694 & 0.45763 & 0.04308 &&&& 4.90051\\mathrm{e}{-10}\\\\\n5 & 0.00833 & 0.21704 & 0.54711 & 0.21906 & 0.00846 &&& 1.42356\\mathrm{e}{-10}\\\\\n6 & 0.00131 & 0.08109 & 0.41734 & 0.41990 & 0.07903 & 0.00133 && 8.36380\\mathrm{e}{-10}\\\\\n7 & 0.00018 & 0.02418 & 0.23675 & 0.47821 & 0.23671 & 0.02381 & 0.00016 & 1.82768\\mathrm{e}{-10}\n\\end{array}\n\\]\n\\caption{Monte Carlo results for reaching ending position $k$ for $1 \\leq n \\leq 7$. }\\label{tab:Monte1}\n\\end{table}\n\nWe know that when $n=1, k=0$ with probability $1$ because the only occupied site is the origin. Furthermore, we know that when $n=2,$ both $k=0$ and $k=1$ occur with probability $\\frac{1}{2}$ because $50\\%$ of the time we will flip heads and settle at the site to the right of the origin, and $50\\%$ of the time we will flip tails and settle at the site to the left of the origin. The data in Table \\ref{tab:Monte1} supports this fact. \n\nWhen $n=3$, our data suggests that there is a common denominator of $6$, and when $n=4$, our data suggests the common denominator may be $24$. For example, when $n=3$ and $k=0$, we see a probability of $0.16730$ which is very close to $0.1\\overline{666} = \\frac{1}{6}$. Similarly, when $n=4$ and $k=0$, we see $0.04235$ which is very close to $0.041\\overline{6} = \\frac{1}{24}$. If we consider these observations with our known probabilities for $n=1$ and $n=2$, we see that our denominators for $1 \\leq n \\leq 4$ may be $n!$. \n\nUpon recognizing this, we multiplied each row of Table \\ref{tab:Monte1} by a common denominator of $n!$ to try and find a nice distribution for the numerators. For example, multiplying row $5$ by $5!$ yields\n\\[\n[0.9996 \\quad 26.0448 \\quad 65.6532 \\quad 26.2872 \\quad 1.0152].\n\\]\nAfter multiplication by $n!$, we found that our data looked very close to the triangle of Eulerian numbers. See Table \\ref{table:eulnum}. Upon comparing our data in Table \\ref{tab:Monte1} to the corresponding Eulerian number divided by $n!$, we got a very close fit. The sum of squared errors, denoted $SSE$, for each $1 \\leq n \\leq 7$ is shown in the last column in Table \\ref{tab:Monte1}, comparing our data to the corresponding Eulerian number divided by $n!$. Thus, our Monte Carlo simulation suggests that our coin-tossing game follows an Eulerian probability distribution and we will prove this result in Theorem \\ref{thm:eulprob}.\\footnote{As communicated by Lionel Levine, it seems Jim Propp made this connection in unpublished work. In fact it was Levine, in a visit to DePaul University, who suggested we investigate this question in the first place.}\n\nRecall that for any $0 \\leq k \\leq n-1,$ the \\textit{Eulerian number $\\eulerian{n}{k}$} is the number of permutations of $\\{1,2,\\dots,n\\}$ with $k$ descents. See \\cite{Petersen}. The Eulerian numbers satisfy the recurrence\n\\begin{equation}\\label{eq:eulrec}\n\\eulerian{n}{k} = (n - k)\\eulerian{n - 1}{k - 1} + (k + 1)\\eulerian{n - 1}{k},\n\\end{equation}\nfor any $n \\geq 1$ and $0 \\leq k \\leq n - 1$, with boundary values $\\eulerian{n}{0} = \\eulerian{n}{n-1} = 1$ for all $n\\geq 0$. Table \\ref{table:eulnum} shows the triangle of Eulerian numbers for $1 \\leq n \\leq 6$. The Eulerian numbers are a finite combinatorial distribution and our coin-tossing game is a stochastic process. How these mathematical concepts are related is very puzzling at first glance. \n\\begin{table}\n\\begin{tabular}{c | c c c c c c}\n$n \\backslash k$ & 0 & 1 & 2 & 3 & 4 & 5\\\\\n\\hline\n1 & 1 \\\\\n2 & 1 & 1 \\\\\n3 & 1 & 4 & 1 \\\\\n4 & 1 & 11 & 11 & 1 \\\\\n5 & 1 & 26 & 66 & 26 & 1 \\\\\n6 & 1 & 57 & 302 & 302 & 57 & 1\n\\end{tabular}\n\\caption{Triangle of the Eulerian numbers for $1 \\leq n \\leq 6$.}\n\\label{table:eulnum}\n\\end{table}\n\nIn Section 2, we consider question $(2)$, the expected run time of the game. From our Monte Carlo simulations we recorded the average number of coin tosses needed for $n$ particles to settle. Let $E_n$ denote this number of expected tosses. The data from our simulation is shown in Table \\ref{tab:Monte2}. \n\n\\begin{table}\n\\[\n\\begin{array}{c | c ccccccccccccc}\nn& 1 & 2 & 3 & 4& 5& 6& 7& 8& 9 & 10  & 15 & 20 \\\\\n\\hline\nE_n & 0 & 1 & 3.00 & 6.66 & 12.49 & 20.98 & 32.64 & 47.96 & 67.54 & 91.68 & 300.24 & 699.79\n\\end{array}\n\\]\n\\caption{Monte Carlo results for $E_n$ for $1 \\leq n \\leq 10$, $n=15,$ and $n=20$.}\\label{tab:Monte2}\n\\end{table}\n\nUsing Lagrange interpolation, we find the number of tosses needed for the game to have $n$ occupied sites is roughly\n\\[\n\\frac{1}{12} n^3 + \\frac{1}{12} n^2.\n\\]\nWe will prove this result for expected run time in Theorem \\ref{thm:tosstosn}. \n\n\\section{Probability Distribution}\n\nTo begin, we define the possible states of the game and a two-parameter family of probabilities relating these states. \n\nFor any $n \\geq 1$ and $0 \\leq k \\leq n - 1$, let $s(n,k)$ denote the graph with $n$ occupied sites, $k$ of which are to the right of the origin. State $s(n,k)$ is shown in Figure \\ref{fig:snk}. We think of these graphs as possible states of our coin-tossing game. For example, \n\\[\ns(7,4) = \\begin{tikzpicture}[baseline=0]\n\\tikzstyle{wn}=[inner sep =3, fill=white, draw=black, circle];\n\\tikzstyle{bn}=[inner sep =3, fill=black, draw=black, circle];\n\\draw (0,0)--(1,0) node[wn] {} -- (2,0) node[bn] {}--(3,0) node[bn] {}--(4,0) node[bn] {}--(5,0) node[bn] {}--(6,0) node[bn] {}--(7,0) node[bn] {}--(8,0) node[bn] {}--(9,0) node[wn] {}--(10,0);\n\\draw (4,-.2) node[below] {$\\star$}; \\draw (0,0) node[left] {$\\cdots$}; \\draw (10,0) node[right] {$\\cdots$};\n\\end{tikzpicture}.\n\\]\n\n\\begin{figure}\n\\[\n\\begin{tikzpicture}[baseline=0]\n\\tikzstyle{wn}=[inner sep =3, fill=white, draw=black, circle];\n\\tikzstyle{bn}=[inner sep =3, fill=black, draw=black, circle];\n\\draw (0,0)--(1,0) node[wn] {} -- (2,0) node[bn] {}--(3,0) node[bn] {}--(4,0);\n\\draw (4.75,0)--(5.75,0) node[bn] {}--(6.75,0) node[bn] {}--(7.75,0);\n\\draw (8.5,0)--(9.5,0) node[bn] {}--(10.5,0) node[bn] {}--(11.5,0) node[wn] {}-- (12.5,0);\n\\draw (0,0) node[left] {$\\cdots$}; \\draw (4,0) node[right] {$\\cdots$}; \\draw (7.75,0) node[right] {$\\cdots$}; \\draw (12.5,0) node[right] {$\\cdots$};\n\\draw (5.75,-.2) node[below] {$\\star$};\n\\draw [decoration={brace,amplitude=6pt},decorate] (6.75,0.3) -- (10.5,0.3) node[midway,above=8pt] {$k$};\n\\draw [decoration={brace,amplitude=6pt},decorate] (2,0.3) -- (5.75,0.3) node[midway,above=8pt] {$n-k$};\n\\end{tikzpicture}\n\\]\n\\caption{State $s(n,k)$.}\\label{fig:snk}\n\\end{figure}\n\nThe graph $s(n,k)$ can only be reached from $s(n - 1,k - 1)$ or $s(n - 1,k)$ for any $n \\geq 1$ and $1 \\leq k \\leq n-2$. For example, state $s(4,2)$ can be reached only from state $s(3,1)$ or state $s(3,2)$. The boundary cases, states $s(n,0)$ and $s(n, n - 1)$, are exceptions: $s(n,0)$ can only be reached from $s(n-1,0)$ and $s(n,n-1)$ can only be reached from $s(n-1,n-2)$. The possible states of the game for $0 \\leq n \\leq 5$ are shown in Figure \\ref{fig:states}, with edges to indicate which states can be obtained from another.\n\\begin{figure}\n\\begin{center}\n\\[\n\\begin{tikzpicture}[scale=.7]\n  \\node (oz) at (0,4) {$s(1,0)$}; \n  \\node (tz) at (-2,2) {$s(2,0)$}; \\node (two) at (2,2) {$s(2,1)$};\n  \\node (thz) at (-4,0) {$s(3,0)$}; \\node (tho) at (0,0) {$s(3,1)$}; \\node (thtw) at (4,0) {$s(3,2)$};\n  \\node (fz) at (-6,-2) {$s(4,0)$}; \\node (fo) at (-2,-2) {$s(4,1)$}; \\node (ftw) at (2,-2) {$s(4,2)$}; \\node (fth) at (6,-2) {$s(4,3)$};\n  \\node (ffz) at (-8,-4) {$s(5,0)$}; \\node (ffo) at (-4,-4) {$s(5,1)$}; \\node (fftw) at (0,-4) {$s(5,2)$}; \\node (ffth) at (4,-4) {$s(5,3)$}; \\node (fff) at (8,-4) {$s(5,4)$};\n  \\draw (oz) -- (tz); \\draw (oz) -- (two); \n  \\draw (tz) -- (thz); \\draw (tz) -- (tho); \\draw (two) -- (tho); \\draw (two) -- (thtw);\n  \\draw (thz) -- (fz); \\draw (thz) -- (fo); \\draw (tho) -- (fo); \\draw (tho) -- (ftw); \\draw (thtw) -- (ftw); \\draw (thtw) -- (fth);\n  \\draw (fz) -- (ffz); \\draw (fz) -- (ffo); \\draw (fo) -- (ffo); \\draw (fo) -- (fftw); \\draw (ftw) -- (fftw); \\draw (ftw) -- (ffth); \\draw (fth) -- (ffth); \\draw (fth) -- (fff);\n  \\draw (ffz) node[below] {\\vdots}; \\draw (ffo) node[below] {\\vdots}; \\draw (fftw) node[below] {\\vdots}; \\draw (ffth) node[below] {\\vdots}; \\draw (fff) node[below] {\\vdots};\n\\end{tikzpicture}\n\\]\n\\end{center}\n\\caption{Possible states of the game for $0 \\leq n \\leq 5$.}\\label{fig:states}\n\\end{figure}\n\nWhile we know that the game will transition from one state to the next by moving down the lattice in Figure \\ref{fig:states}, we would like to know how likely each transition is. For example, we can ask how likely it is that the game will go from state $s(2,1)$ to state $s(3,1)$.\n\nFor any $n \\geq 1$ and $0 \\leq k \\leq n - 1$, let $p_{n,k}$ denote the probability of transitioning from $s(n - 1,k)$ to $s(n,k)$. Similarly, let $q_{n,k}$ denote the probability of transitioning from $s(n - 1,k - 1)$ to $s(n,k)$. We will call $p_{n,k}$ and $q_{n,k}$ transition probabilities. The transition probabilities in and out of state $s(n,k)$ are shown in Figure \\ref{fig:snktrans}. \n\nTo help us calculate the transition probabilities, we consider the classic Gambler's Ruin problem. \n\n\\subsection{Gambler's Ruin}\\label{gambruin}\n\nIn the Gambler's Ruin problem, we consider two gamblers: Gambler A and Gambler B. The gamblers are playing a game where they toss a coin. If the coin flip is tails, Gambler A takes a dollar from Gambler B. Similarly, if heads, Gambler B takes a dollar from Gambler A. Each player starts with some money, say $k$ dollars for Gambler A and $l$ dollars for Gambler B. The game ends when either Gambler A wins all $k+l$ dollars (Gambler B is ruined), or when Gambler B wins all $k+l$ dollars (Gambler A is ruined). The result says that the probability that Gambler A wins is $\\frac{k}{k+l}$ and the probability Gambler B wins is $\\frac{l}{k+l}$. See \\cite{GrinsteadSnell}.\n\nThis relates to our model in a straightforward way. Let's assume Gambler A has $4$ dollars and Gambler B has $3$ dollars. This corresponds to the following graph: \n\n\\[\n\\begin{tikzpicture}[baseline=0]\n\\tikzstyle{wn}=[inner sep =2.5, fill=white, draw=black, circle];\n\\tikzstyle{bn}=[inner sep =2.5, fill=black, draw=black, circle];\n\\draw (0,0)--(.75,0) node[wn] {} -- (1.5,0) node[bn] {}--(2.25,0) node[bn] {}--(3,0) node[bn] {}--(3.75,0) node[bn] {}--(4.5,0) node[bn] {}--(5.25,0) node[bn] {}--(6,0) node[wn] {}--(6.75,0);\n\\draw (3,-.2) node[below] {$\\star$}; \\draw (3,.2) node[above] {$\\circ$}; \\draw (0,0) node[left] {$\\cdots$}; \\draw (6.75,0) node[right] {$\\cdots$};\n\\draw [->] (.75,-1) node[below] {B is ruined} -- (.75,-.2); \\draw [->] (6,-1) node[below] {A is ruined} -- (6,-.2);\n\\draw [decoration={brace,amplitude=6pt},decorate] (1.5,0.15) -- (3,0.15) node[midway,above=12pt] {B};\n\\draw [decoration={brace,amplitude=6pt},decorate] (3,0.15) -- (5.25,0.15) node[midway,above=12pt] {A};\n\\end{tikzpicture}.\n\\]\nWe see that the location of the particle on the graph keeps track of the fortune for each player. That is, the money Gambler A has is equal to the number of active nodes to the right of and including the particle, while Gambler B's fortune equals the number of active nodes to the left of and including the particle. \n\nIf the coin toss is heads, Gambler B will take a dollar from Gambler A. So, continuing with our example, Gambler B will have $4$ dollars and Gambler A will have $3$ dollars. Thus, the game moves to the following state:\n\\[\n\\begin{tikzpicture}[baseline=0]\n\\tikzstyle{wn}=[inner sep =2.5, fill=white, draw=black, circle];\n\\tikzstyle{bn}=[inner sep =2.5, fill=black, draw=black, circle];\n\\draw (0,0)--(.75,0) node[wn] {} -- (1.5,0) node[bn] {}--(2.25,0) node[bn] {}--(3,0) node[bn] {}--(3.75,0) node[bn] {}--(4.5,0) node[bn] {}--(5.25,0) node[bn] {}--(6,0) node[wn] {}--(6.75,0);\n\\draw (3,-.2) node[below] {$\\star$}; \\draw (3.75,.2) node[above] {$\\circ$}; \\draw (0,0) node[left] {$\\cdots$}; \\draw (6.75,0) node[right] {$\\cdots$};\n\\draw [->] (.75,-1) node[below] {B is ruined} -- (.75,-.2); \\draw [->] (6,-1) node[below] {A is ruined} -- (6,-.2);\n\\draw [decoration={brace,amplitude=6pt},decorate] (1.5,0.15) -- (3.75,0.15) node[midway,above=12pt] {B};\n\\draw [decoration={brace,amplitude=6pt},decorate] (3.75,0.15) -- (5.25,0.15) node[midway,above=12pt] {A};\n\\end{tikzpicture}\n\\]\n\n\nIn general, state $s(n-1,k-1)$ corresponds to Gambler A having $k$ dollars and Gambler B having $l$ dollars such that $k+l=n$. Thus we see that Gambler A wins if the game goes to state $s(n,k-1)$ and Gambler B wins if the game goes to state $s(n,k)$. The Gambler's Ruin problem tells us that Gambler A will win with probability $\\frac{k}{n}$ and Gambler B will win with probability $\\frac{n-k}{n}$. Thus, we conclude for any $n \\geq 1$ and $1 \\leq k \\leq n - 1$, \n\\begin{equation}\\label{eq:pqtransprobs}\np_{n,k-1} = \\frac{k}{n} \\quad \\mbox{ and } \\quad q_{n,k} = \\frac{n-k}{n}.\n\\end{equation} \n\n\\begin{figure}\n\\[\n\\begin{tikzpicture}[scale=.7]\n \\node (nk) at (2,0) {$s(n,k)$};\n \\node (qnk) at (0,2) {}; \n \\node (pnk) at (4,2) {};\n \\node (pn1k) at (0,-2) {};\n \\node (qn1k1) at (4,-2) {};\n \\draw (qnk) -- node[left]{$q_{n,k}$} (nk); \\draw (pnk) -- node[right]{$p_{n,k}$} (nk); \\draw (pn1k) -- node[left]{$p_{n+1,k}$}(nk); \\draw (qn1k1) -- node[right]{$q_{n+1,k+1}$}(nk);\n \\end{tikzpicture}\n \\]\n \\caption{Transition probabilities for state $s(n,k)$.}\\label{fig:snktrans}\n \\end{figure}\n \n \\subsection{Eulerian distribution}\n\nWe use these equations for our transition probabilities to obtain our first main result, which is the probability of starting the game at $s(1,0)$ and arriving at state $s(n,k)$. Let $P(n,k)$ denote this probability. For example, $P(1,0) = 1,$ and since we move immediately to $s(2,0)$ or $s(2,1)$ after our first coin toss, we see $P(2,0) = \\frac{1}{2} = P(2,1)$. For $n \\geq 3,$ things are more interesting.  \n\nFirst, we will consider the transition to the boundary states (when $k=0$ or $k=n-1$). We know that state $s(n,0)$ can only be reached from $s(n-1,0)$, so $P(n,0)$ will only depend on $P(n-1,0)$. Similarly, state $s(n,n-1)$ can only be reached from $s(n-1,n-2)$, so $P(n,n-1)$ depends only on $P(n-1,n-2)$. Using our formulas for the transition probabilities in Equation \\eqref{eq:pqtransprobs}, we have \n\\[\nP(n,0) = p_{n,0} P(n-1,0) \\quad \\mbox{and} \\quad P(n,n-1) = q_{n,n-1} P(n-1,n-2).\n\\]\nSince $p_{n,0}=q_{n,n-1}= \\frac{1}{n}$ and $P(1,0)=1$, we find \n\\begin{equation}\\label{eq:boundary}\nP(n,0) = \\frac{P(n-1,0)}{n} = \\frac{1}{n!} \\quad \\mbox{ and } \\quad P(n,n-1) = \\frac{P(n-1,n-2)}{n} = \\frac{1}{n!}.\n\\end{equation}\n\nNext, we will look at the non-boundary cases. Because $s(n,k)$ can only be reached from states $s(n-1,k-1)$ and $s(n-1,k)$, we know that $P(n,k)$ will only depend on $P(n-1,k-1)$ and $P(n-1,k)$. Using our formulas for the transition probabilities in Equation \\eqref{eq:pqtransprobs}, we find for $1 \\leq k \\leq n-2$, the following recurrence for $P(n,k)$:\n\\begin{align}\nP(n,k) &= q_{n,k} P(n-1,k-1) + p_{n,k} P(n-1,k), \\nonumber \\\\\n&= \\frac{n - k}{n} \\cdot P(n - 1,k - 1) + \\frac{k + 1}{n} \\cdot P(n - 1,k) \\label{eq:recursiveP}.\n\\end{align}\nThe recurrence relation of $P(n,k)$ in Equation \\eqref{eq:recursiveP} looks similar to that of the Eulerian number $\\eulerian{n}{k}$ from Equation \\eqref{eq:eulrec}. \n\nIf we assume \n\\[\nP(n-1,k) = \\frac{\\eulerian{n-1}{k}}{(n-1)!},\n\\] \nfor $1 \\leq k \\leq n-2$, then,\n\\begin{align*}\nP(n,k) &= q_{n,k}P(n-1,k-1) + p_{n,k}P(n-1,k) \\\\\n &= \\frac{n-k}{n} \\cdot \\frac{\\eulerian{n - 1}{k - 1}}{(n-1)!} + \\frac{k+1}{n} \\cdot \\frac{\\eulerian{n - 1}{k}}{(n-1)!},\\\\\n  &=\\frac{(n - k)\\eulerian{n - 1}{k - 1} + (k + 1)\\eulerian{n - 1}{k}}{n!},\\\\\n  &= \\frac{\\eulerian{n}{k}}{n!},\n\\end{align*}\nwhere the final equality follows from the recurrence for Eulerian numbers in Equation \\eqref{eq:recursiveP}.\nThis proves our first theorem.\n\n\\begin{thm}\\label{thm:eulprob}\nFor any $n \\geq 1$ and $0 \\leq k \\leq n - 1$,\n\\[\nP(n,k) = \\frac{\\eulerian{n}{k}}{n!}.\n\\]\n\\end{thm} \n\nOur first main result is that $P(n,k)$, the probability of arriving at state $s(n,k)$, is modeled by the Eulerian probability distribution. In other words, we found a nice result for the probability that after emptying a queue of $n$ particles, $k$ of them settle to the right of the origin. With question (1) answered, we now move onto question (2): the expected amount of time (i.e., the number of coin tosses) it takes for $n$ particles to settle. \n\n\\section{Expected run time}\n\nWe next turn to the expected run time of our coin-tossing game, i.e., we want to know how many tosses it will take for $n$ particles to settle. Recall from the introduction that $E_n$ denotes the expected number of tosses needed to reach a state with $n$ occupied sites. Notice that $E_1 = 0$, and $E_2 = 1$ because flipping heads or tails will cause a site to become occupied immediately. It gets more complicated once we start looking at cases for $n \\geq 3$. In the introduction, we gave an estimate for $E_n$ based on our Monte Carlo simulation. We now turn this into our next theorem.\n\n\\begin{thm}\\label{thm:tosstosn}\nFor any $n \\geq 1$, the expected number of tosses needed to reach a state with $n$ occupied sites is \n\\begin{equation}\\label{eq:actualtoss}\nE_n = \\frac{1}{12} n^3 + \\frac{1}{12} n^2.\n\\end{equation}\n\\end{thm}\n\nIf we compute the first differences of this sequence ($E_{n} - E_{n-1}$), we find the change in expected tosses. Let $\\Delta E_n$ denote the expected number of tosses it takes for the $n^{th}$ particle to perform a random walk and settle, given $n-1$ sites are already occupied.  \nAssuming Equation \\eqref{eq:actualtoss}, we find\n\\[\n\\Delta E_n = \\frac{1}{12} n^3 + \\frac{1}{12} n^2 - \\left(\\frac{1}{12} (n-1)^3 + \\frac{1}{12} (n-1)^2\\right) = \\frac{1}{4}n^2 - \\frac{1}{12}n.\n\\]\nThus to obtain our run time result in Theorem \\ref{thm:tosstosn}, it suffices to show that the number of expected tosses for the $n^{th}$ particle, once dropped, to settle in an unoccupied site is\n\\begin{equation}\\label{eq:newtossquad}\n\\Delta E_n = \\frac{1}{4}n^2 - \\frac{1}{12}n.\n\\end{equation}\n\nBut by definition, the expected number of new steps is \n\\[\n\\Delta E_n = \\sum_{k=1}^{n-1}P(n-1,k-1) \\cdot E(n-1,k-1),\n\\]\nwhere $E(n-1,k-1)$ denotes the expected number of tosses to go from state $s(n-1,k-1)$ to $s(n,k)$ or $s(n,k-1)$. By Theorem \\ref{thm:eulprob}, we can write\n\\[\n \\Delta E_n = \\sum_{k=1}^{n-1} \\frac{\\eulerian{n-1}{k-1}}{(n-1)!} \\cdot E(n-1,k-1) = \\frac{1}{(n-1)!} \\sum_{k=1}^{n-1} \\eulerian{n-1}{k-1} E(n-1,k-1),\n\\]\ni.e., $\\Delta E_n$ is a weighted sum of Eulerian numbers. We will show $E(n-1,k-1)$ is simply $k(n-k)$, which will help us to evaluate the sum and verify \\eqref{eq:newtossquad}, and hence Theorem \\ref{thm:tosstosn}.\n\n\\subsection{Escape time for a random walk}\n\nConsider that the game is in state $s(n-1,k-1)$ pictured below, \n\\[\n\\begin{tikzpicture}\n\\tikzstyle{wn}=[inner sep =3, fill=white, draw=black, circle];\n\\tikzstyle{bn}=[inner sep =3, fill=black, draw=black, circle];\n\\draw (0,0)--(1,0) node[wn] {} --(2,0) node[bn] {}--(3,0) node[bn]{} -- (4,0);\n\\draw (4,0) node[right] {$\\cdots$};\n\\draw (4.75,0)--(5.75,0) node[bn] {} --(6.75,0) node[bn] {} -- (7.75,0) node[bn] {} -- (8.75,0); \n\\draw (8.75,0) node[right]{$\\cdots$};\n\\draw (9.5,0)--(10.5,0) node[bn] {}--(11.5,0) node[bn]{} -- (12.5,0) node[wn]{} -- (13.5,0);\n\\draw (0,0) node[left] {$\\cdots$};\n\\draw (13.5,0) node[right] {$\\cdots$};\n\\draw (6.75,-.2) node[below] {$\\star$}; \\draw (12.5,.2) node[above] {$k$}; \\draw (1,.2) node[above] {$l$};\n\\draw [decoration={brace,amplitude=6pt},decorate] (7.75,0.3) -- (11.5,0.3) node[midway,above=8pt] {$k-1$};\n\\draw [decoration={brace,amplitude=6pt},decorate] (2,0.3) -- (5.75,0.3) node[midway,above=8pt] {$l-1$};\n\\end{tikzpicture}\n\\]\nwhere $k+l = n$. We would like to know how to compute $E(n-1,k-1),$ i.e., how many coin tosses it will take for the game to reach either state $s(n,k)$ (activating on the right) or state $s(n,k-1)$ (activating on the left). To help us, recall that for the simple random walk, the expected time to reach the boundary of an interval $[-b,a]$ is $ab$. See \\cite{Sauer}. We translate this escape time result into the language of our game with the following claim. \n\\begin{cl}\\label{cl:esctime}\nFor any $k,l \\geq 1$ such that $k+l=n$, the expected number of tosses to go from state $s(n-1,k-1)$ to $s(n,k)$ or $s(n,k-1)$ is $E(n-1,k-1) = kl = k(n-k)$.\n\\end{cl}\n\nFor example, consider state $s(5,3)$ below.\n\\[\n\\begin{tikzpicture}\n\\tikzstyle{wn}=[inner sep =3, fill=white, draw=black, circle];\n\\tikzstyle{bn}=[inner sep =3, fill=black, draw=black, circle];\n\n\\draw (9,-2)--(9.5,-2) node[wn] {} -- (10.25,-2) node[bn] {} -- (11,-2) node[bn] {} -- (11.75,-2) node[bn] {} -- (12.5,-2) node[bn]{} -- (13.25,-2) node[bn] {} -- (14,-2) node[wn] {} -- (14.5,-2); \n\\draw (9,-2) node[left] {$\\cdots$}; \\draw (14.5,-2) node[right] {$\\cdots$};\n\\draw (11,-2.2) node[below] {$\\star$}; \\draw (9.5,-2.2) node[below] {\\scriptsize $l=2$}; \\draw (14,-2.2) node[below] {\\scriptsize $k=4$};\n\\draw (11.75,-2.65) node[below] {$s(5,3)$};\n\\end{tikzpicture}\n\\]\n\nAs shown, $l=2$ and $k=4$ in $s(5,3)$. We can calculate, using Claim \\ref{cl:esctime}, the expected number of tosses it takes to go from each state with $n=5$ to a state with $n=6$ occupied sites. Consider the chart below:\n\\[\n\\begin{centering}\n\\begin{tabular}{c | c | c | c} \nState & $k$ & $l$ & $E(5,k-1) = kl$ \\\\\n\\hline\n$s(5,0)$ & $1$ & $5$ & 5 \\\\\n$s(5,1)$ & $2$ & $4$ & 8 \\\\\n$s(5,2)$ & $3$ & $3$ & 9 \\\\\n$s(5,3)$ & $4$ & $2$ & 8 \\\\\n$s(5,4)$ & $5$ & $1$ & 5 \\\\\n\\end{tabular}\n\\end{centering}\n\\]\nIf we know there are five occupied sites (but not which 5), and we want to know how many tosses it would take for the sixth particle to settle, we calculate a weighted sum using our Eulerian probabilities. This is precisely our definition for $\\Delta E_6$. \n\\[\n\\Delta E_6 = \\left(\\frac{1}{120}\\right)(5) + \\left(\\frac{26}{120}\\right)(8) + \\left(\\frac{66}{120}\\right)(9) + \\left(\\frac{26}{120}\\right)(8) + \\left(\\frac{1}{120}\\right)(5) = \\frac{17}{2}. \n\\]\nThis agrees with Equation \\eqref{eq:newtossquad}, \n\\[ \n\\frac{1}{4} \\left(6^2\\right) - \\frac{1}{12} \\left(6\\right) = \\frac{17}{2}.\n\\]\n\nIn general, since $P(n,k) = \\frac{\\eulerian{n}{k}}{n!}$ by Theorem \\ref{thm:eulprob} and $E(n-1,k-1) = kl$ by Claim \\ref{cl:esctime}, we have the following Corollary of Theorem \\ref{thm:eulprob} and Claim \\ref{cl:esctime}.\n\n\\begin{cor}\\label{cor:deltaen}\nFor any $n \\geq 3$, the expected number of tosses for the $n^{th}$ particle to settle, given $n-1$ sites are occupied is \n\\begin{align*}\n\\Delta E_n &= \\sum_{k=1}^{n-1} \\frac{\\eulerian{n-1}{k-1}}{(n-1)!} \\cdot k(n-k) \\\\\n&= \\frac{1}{(n-1)!} \\sum_{k=1}^{n-1} \\eulerian{n-1}{k-1} \\cdot k(n-k).\n\\end{align*}\n\\end{cor}\n\nIt remains to see that this also equals the formula in Equation \\eqref{eq:newtossquad}. We will do so using generating function techniques.\n\n\\subsection{Eulerian polynomials and generating functions}\n\nThroughout this section we use the technique of \\emph{generating functions} which is explored in \\cite{Wilf}, and in particular \\cite{Petersen} for Eulerian numbers. The basic idea is to study power series whose coefficients are the numbers we are interested in, and then to use properties of these functions to deduce properties of the coefficients. \n\nTo this end, we define the $n^{th}$ bivariate Eulerian polynomial, denoted $A_{n}(s,t)$, as \n\\[\nA_{n}(s,t) = \\sum_{k=1}^{n} \\eulerian{n}{k-1} s^{k} t^{n+1-k}, \n\\]\ne.g., $A_5 (s,t) = st^5 + 26s^2t^4 + 66s^3t^3 + 11s^4t^2 + s^5t$. \n\nWe can get $\\Delta E_n$ from $A_{n-1} (s,t)$ as follows. First, we differentiate with respect to $s$ and $t$,\n\\begin{align*}\n\\frac{\\partial}{\\partial s} \\frac{\\partial}{\\partial t} \\left[A_{n-1}(s,t)\\right] &= \\frac{\\partial}{\\partial s} \\frac{\\partial}{\\partial t} \\left[\\sum_{k=1}^{n-1} \\eulerian{n-1}{k-1}s^kt^{n-k} \\right], \\\\\n&= \\sum_{k=1}^{n-1} \\eulerian{n-1}{k-1}k(n-k)s^{k-1}t^{n-k-1},\n\\end{align*}\nand when $s=1$ and $t=1$, this is \n\\begin{equation}\\label{eq:wgtsum}\n\\sum_{k=1}^{n-1} \\eulerian{n-1}{k-1}k(n-k) = (n-1)! \\cdot \\Delta E_n.\n\\end{equation}\n\nFor example, consider the case for $n=6$. We differentiate $A_5 (s,t)$ with respect to $s$ and $t$ and then set $s=t=1$,\n\\[\n\\frac{\\partial}{\\partial s} \\frac{\\partial}{\\partial t} \\left[A_{5}(s,t)\\right]_{s=t=1} = (1\\cdot 5) + 26(2 \\cdot 4) + 66(3\\cdot 3) + 26(4 \\cdot 2) + (5\\cdot 1).\n\\]\nNotice that this is precisely the weighted sum for $n=6$ from Equation \\eqref{eq:wgtsum},\n\\[\n\\sum_{k=0}^{4} \\eulerian{5}{k-1} k(6-k) = 5! \\cdot \\Delta E_6.\n\\]\n\nBefore we can use this idea to prove Equation \\eqref{eq:newtossquad} holds in general, we consider the exponential generating function for the Eulerian polynomial $A_n(s,t)$, denoted $F(s,t,z)$. By definition, we have   \n\\begin{equation}\\label{eq:expgenf}\nF(s,t,z) = \\sum_{n \\geq 1}A_{n}(s,t) \\frac{z^n}{n!}.\n\\end{equation}\nThat is, expanding in $z$, we find \n\\[\nF(s,t,z) =  stz + (s^2t + st^2 ) \\frac{z^2}{2} + (s^3t + 4s^2t^2 + st^3) \\frac{z^3}{3!} + \\cdots .\n\\]\n\nTo find a useful formula for $F(s,t,z),$ we work to the bivariate case from the univariate case, which is better known. \nNotice that \n\\begin{equation}\\label{eq:uni2bi}\nA_n (s,t) = \\sum_{k=1}^n \\eulerian{n}{k-1}s^kt^{n+1-k} = t^{n+1}\\sum_{k=1}^n \\eulerian{n}{k-1} \\left(\\frac{s}{t}\\right)^k= t^{n+1} A_n \\left(\\frac{s}{t}\\right),\n\\end{equation}\nwhere $A_n(x) = \\sum_{k=1}^{n} \\eulerian{n}{k-1} x^k$ is the single-variable Eulerian polynomial.\n\nThe exponential generating function for $A_n(x)$, denoted $G(x,z)$, is \n\\[\nG(x,z) = \\sum_{n\\geq 1} A_n (x) \\frac{z^n}{n!} = \\frac{x\\left(1-e^{z(x-1)}\\right)}{e^{z(x-1)} -x}.\n\\]\nSee \\cite{Petersen}. Thus, we see that \n\\begin{align}\nF(s,t,z) &= \\sum_{n \\geq 1} A_n (s,t) \\frac{z^n}{n!} \\nonumber \\\\\n&= t \\cdot \\sum_{n \\geq 1} A_n \\left(\\frac{s}{t}\\right) \\frac{(tz)^n}{n!} \\nonumber \\\\\n&= t \\cdot G\\left(\\frac{s}{t},tz\\right) \\nonumber \\\\\n&= \\frac{st\\left(1- e^{z(s-t)}\\right)}{te^{z(s-t)} -s} \\label{eq:fstz}. \n\\end{align}\n\nWith the function $F(s,t,z)$ in Equation \\eqref{eq:fstz}, we can calculate $\\Delta E_n$ simultaneously for all $n$ through the same steps of differentiation and substitution, i.e., \n\\begin{align*}\n\\frac{\\partial}{\\partial s} \\frac{\\partial}{\\partial t} [F(s,t,z)]_{s=t=1} &= \\sum_{n \\geq 1} \\left(\\frac{\\partial}{\\partial s} \\frac{\\partial}{\\partial t} [A_n(s,t)]_{s=t=1}\\right) \\frac{z^n}{n!}, \\\\\n&= \\sum_{n \\geq 1} n!\\cdot \\Delta E_{n+1} \\frac{z^n}{n!} = \\sum_{n \\geq 1} \\Delta E_{n+1} z^n.\n\\end{align*}\nTaking the deriviative at $s=t=1$ of the formula for $F(s,t,z)$ given in \\eqref{eq:fstz}, we find\n\\[\n\\frac{z^4 - 4z^3 + 6z^2 -6z}{6(z-1)^3} = \\sum_{n \\geq 1} \\Delta E_{n+1} \\cdot z^n. \n\\]\n\nNote that from Equation \\eqref{eq:newtossquad}, we would like to see\n\\begin{equation}\\label{eq:en1}\n\\Delta E_{n+1} = \\frac{1}{4} \\left(n+1\\right)^2 - \\frac{1}{12} \\left(n+1\\right) = \\frac{1}{4} n^2 + \\frac{5}{12} n + \\frac{1}{6}. \n\\end{equation}\n\nWe will now use generating function techniques to show that \n\\[\n\\frac{z^4 - 4z^3 + 6z^2 -6z}{6(z-1)^3} = \\sum_{n\\geq 1} \\left(\\frac{1}{4} n^2 + \\frac{5}{12} n + \\frac{1}{6}\\right) z^n.\n\\]\n \nBy differentiating the geometric series $1\/(1-z) = 1+z+z^2 + \\cdots$ twice (and dividing by $2$), we know that \n\\begin{equation}\\label{eq:genfunc}\n\\frac{1}{(1-z)^3} = \\sum_{n \\geq 0} \\binom{n+2}{2} z^n = \\binom{2}{2} + \\binom{3}{2}z + \\binom{4}{2} z^2 + \\cdots.\n\\end{equation}\n\nWe split our generating function for expectation into four pieces:\n\\[\n\\frac{z^4 - 4z^3 + 6z^2 -6z}{6(z-1)^3} = \\frac{6z}{6(1-z)^3} - \\frac{6z^2}{6(1-z)^3} + \\frac{4z^3}{6(1-z)^3} - \\frac{z^4}{6(1-z)^3},\n\\]\nand using the formula from Equation \\eqref{eq:genfunc}, we find \n\\begin{align*}\n\\frac{6z}{6(1-z)^3} &= z \\cdot \\sum_{n \\geq 0} \\binom{n+2}{2} z^n = \\sum_{n\\geq 1} \\binom{n+1}{2} z^n, \\\\\n-\\frac{6z^2}{6(1-z)^3} &=- z^2 \\cdot \\sum_{n \\geq 0} \\binom{n+2}{2} z^n = -\\sum_{n\\geq 2} \\binom{n}{2} z^n, \\\\\n\\frac{4z^3}{6(1-z)^3} &=\\frac{2z^3}{3} \\cdot \\sum_{n \\geq 0} \\binom{n+2}{2} z^n = \\sum_{n\\geq 3} \\frac{2}{3} \\binom{n-1}{2}z^n,\\\\\n- \\frac{z^4}{6(1-z)^3} &= - \\frac{z^4}{6} \\cdot \\sum_{n \\geq 0} \\binom{n+2}{2} z^n = \\sum_{n \\geq 4} \\frac{1}{6} \\binom{n-2}{2} z^n.\n\\end{align*}\n\nSumming these, we get\n\\begin{align*}\n \\frac{z^4 - 4z^3 + 6z^2 -6z}{6(z-1)^3} &= z + 2z^2 + \\frac{11}{3}z^3 \\\\\n  & \\quad + \\sum_{n\\geq 4}\\left[\\binom{n+1}{2} -\\binom{n}{2} + \\frac{2}{3}\\binom{n-1}{2} + \\frac{1}{6}\\binom{n-2}{2} \\right] z^n.\n\\end{align*}\n\nUsing the binomial formula to simplify the general term for $n\\geq 4$, we find\n\\[\n\\binom{n+1}{2} - \\binom{n}{2} + \\frac{2}{3}\\binom{n-1}{2} - \\frac{1}{6}\\binom{n-1}{2} = \\frac{1}{4}n^2 + \\frac{5}{12}n + \\frac{1}{6},\n\\]\nwhich establishes Equation \\eqref{eq:en1}, and thus Theorem \\ref{thm:tosstosn}.\n\n\\section{Further remarks}\n\nWe also have Monte Carlo results about question (3), the probability that $n$ sites are occupied given a coin is tossed $N$ times (with a large queue of particles). From the data,\nit appears that the probabilities have a common denominator of $2^{n-1}$. After clearing denominators and rounding the values to the nearest integer, the results for $1\\leq n \\leq 7$ are as follows:\n\\[\n\\begin{array}{c | c c c c c}\nN \\backslash n & 2 & 3 & 4 & 5 \\\\\n\\hline\n1 & 1 \\\\\n2 & 1 & 1 \\\\\n3 & 1 & 3 \\\\\n4 & 1 & 4 & 3 \\\\\n5 & 1 & 9 & 6 \\\\\n6 & 1 & 10 & 18 & 3 \\\\\n7 & 1 & 21 & 32 & 10 \n\\end{array}\n\\]\nWe did not find this array in the On-Line Encyclopedia of Integer Sequences \\cite{Sloane}, though the first two columns have simple explanations. Also, we remark that the first nonzero entry in column $n$ occurs when $N = \\lceil \\frac{n}{2} \\rceil \\cdot \\lfloor \\frac{n}{2} \\rfloor$.\n \nWe finish with some thoughts and observations about question (4) about what happens when we no longer have a simple random walk. Using linear algebra, one can analyze the one-dimensional random walk for any fixed choice of probabilities of a particle moving left or right at a given site. The coin toss model we used is just the case where the probability is $\\frac{1}{2}$ at each site. We have experimented with other choices of probabilities, but without interesting results, except in the case of a biased coin toss. Here each site has the same fixed probability, but that probability is not necessarily $\\frac{1}{2}$. In this case we can use analysis similar to that in this paper to obtain similar results, which we summarize now. \n\nLet $p$ be the probability of a particle going to the right and $q$ the probability of going to the left, and let $\\rho=p\/q$. The biased Gambler's ruin gives transition probabilities of $[k]\/[n]$ where\n\\[\n [n] = \\frac{1-\\rho^n}{1-\\rho} = 1+\\rho+\\rho^2 + \\cdots + \\rho^{n-1}.\n\\]\nThen the same arguments we used in the fair coin case will show that \n\\begin{equation}\n P(n,k) = \\frac{ \\eulerian{n}{k}^{\\maj}}{[n]!},\n\\end{equation}\nwhere $[n]! = [n][n-1]\\cdots[2][1]$, and $\\eulerian{n}{k}^{\\maj}$ counts permutations of length $n$ with $k$ descents, weighted according to a power of $\\rho$ given by a statistic called \\emph{major index}. That is, \n\\[\n\\eulerian{n}{k}^{\\maj} = \\sum_{\\substack{w \\in S_n, \\\\ \\des(w) =k }} \\rho^{\\maj(w)}.\n\\]\nThis gives a refinement of the Eulerian distribution in terms of $\\rho$, as explained in Chapter 6 of \\cite{Petersen}. Studying questions like the limiting distribution and run time when $\\rho\\neq 1$ is the starting point for future work on this problem.\n\n\\section*{Acknowledgements}\n\nThe author is grateful for support from the Undergraduate Summer Research Program from the College of Science and Health at DePaul University, and the guidance of her advisor Kyle Petersen. Thanks also to Lionel Levine for suggesting the problem, and thanks to Jim Propp for comments on an earlier draft.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Phenotype Sparsity}\\label{sec:spars_app}\n\nAs suggested in Section~\\ref{sec:exp-sparsity}, there is an inherent tradeoff between fit to the cost function and desired sparsity. The trade-off is made explicit for $\\lambda$--CNMF in Figure~\\ref{fig:sp_acc}. The sparsity of LLDA is controlled by tuning the hyperparameter ($\\beta$) of the word-topic multinomial parameters~\\citep{blei2003latent} and for MLC ~via the $\\ell_1$ regularization parameter $\\eta$. A smaller value of $\\beta$ ensures that the word-topic probabilities are sparse. As the value of $\\beta$ is increased, sparsity decreases (i.e. number of non-zero elements increases). For logistic regression (used by MLC), as the $\\ell_1$ regularization parameter increases, sparsity increases. Figure~\\ref{fig:sp_llda} demonstrates the sparsity of the estimated phenotypes for LLDA and Figure~\\ref{fig:sp_sip} shows that of logistic regression. We choose phenotypes obtained at $\\beta =  1 \\times 10^{-8}$ and $\\eta = 100$ for qualitative annotation. The parameters were chosen to achieve the lowest median sparsity while ensuring that for each chronic condition, the corresponding phenotype candidate is represented by at least $5$ non-zero clinical terms. Our fourth baseline (NMF + support) did not estimate sparse phenotypes and does not have a tuneable sparsity parameter (but were nevertheless annotated for qualitative evaluation). The proposed model provides the best sparsity among all baselines.\n\\begin{figure}[htbp]\n  \\centering \n  \\small\n  \\includegraphics[width=\\textwidth]{cnmf_48_full_sparsity.pdf} \n  \\caption{Sparsity--Accuracy Trade--off. Sparsity of the model is measured as the median of the number of non-zero entries in columns of the phenotype matrix $\\mathbf{A}$. (a) shows a box plots of the median sparsity across the $30$ chronic conditions for varying $\\lambda$ values. The median and third--quartile values are explicitly noted on the plots. (b) divergence function value of the estimate from Algorithm~\\ref{alg:alg1} plotted against $\\lambda$ parameter.  \\label{fig:sp_acc} }\n\\end{figure}\n\n\n\\begin{figure}[b]\n\\centering\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n  \\includegraphics[width=\\linewidth]{llda_48_full_pheno_sparsity.pdf}\n  \\caption{LLDA}\n  \\label{fig:sp_llda}\n\\end{subfigure}%\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n  \\includegraphics[width=\\linewidth]{logistic_48_full_sparsity.pdf}\n  \\caption{MLC}\n  \\label{fig:sp_sip}\n\\end{subfigure}\n\\caption{Phenotype sparsity for baseline models}\n\\end{figure}\n\\section{Sample Phenotypes for Baseline Models}\\label{sec:samples}\nFigures~\\ref{fig:liver_disease}--\\ref{fig:blood_loss_anemia} show the top $15$ terms learned for all target chronic conditions for the proposed model and baselines.\nThe sparsity level chosen is based on the criterion described in Section~\\ref{sec:exp-sparsity}. For all conditions, the terms are ordered in decreasing order of importance as learned by the models.\n\n\\begin{figure}[!htbp]\n\\small\n  \\centering\n  \\includegraphics[width=\\textwidth]{liver_disease.pdf}\n  \\caption{Learned Phenotypes for Liver Disease}\n  \\label{fig:liver_disease}\n\\end{figure}\n\n\\begin{figure}[!htbp]\n\\small\n  \\centering\n  \\includegraphics[width=\\textwidth]{solid_tumor.pdf}\n  \\caption{Learned Phenotypes for Solid Tumor}\n  \\label{fig:solid_tumor}\n\\end{figure}\n\n\\begin{figure}[!htbp]\n\\small\n  \\centering\n  \\includegraphics[width=\\textwidth]{metastatic_cancer.pdf}\n  \\caption{Learned Phenotypes for Metastatic Cancer}\n  \\label{fig:metastatic_cancer}\n\\end{figure}\n\n\\begin{figure}[!htbp]\n\\small\n  \\centering\n  \\includegraphics[width=\\textwidth]{chronic_pulmonary.pdf}\n  \\caption{Learned Phenotypes for Chronic Pulmonary Disorder}\n  \\label{fig:chronic_pulmonary}\n\\end{figure}\n\n\\begin{figure}[!htbp]\n\\small\n  \\centering\n  \\includegraphics[width=\\textwidth]{alcohol_abuse.pdf}\n  \\caption{Learned Phenotypes for Alcohol Abuse}\n  \\label{fig:alcohol_abuse}\n\\end{figure}\n\n\n\\begin{figure}[!htbp]\n\\small\n  \\centering\n  \\includegraphics[width=\\textwidth]{diabetes_uncomplicated.pdf}\n  \\caption{Learned Phenotypes for Diabetes Uncomplicated}\n  \\label{fig:diabetes_uncomplicated}\n\\end{figure}\n\n\\begin{figure}[!htbp]\n\\small\n  \\centering\n  \\includegraphics[width=\\textwidth]{diabetes_complicated.pdf}\n  \\caption{Learned Phenotypes for Diabetes Complicated}\n  \\label{fig:diabetes_complicated}\n\\end{figure}\n\\begin{figure}[!htbp]\n\\small\n  \\centering\n  \\includegraphics[width=\\textwidth]{peripheral_vascular.pdf}\n  \\caption{Learned Phenotypes for Peripheral Vascular Disorder}\n  \\label{fig:peripheral_vascular}\n\\end{figure}\n\\clearpage\n\\begin{figure}[!htbp]\n\\small\n  \\centering\n  \\includegraphics[width=\\textwidth]{renal_failure.pdf}\n  \\caption{Learned Phenotypes for Renal Failure}\n  \\label{fig:renal_failure}\n\\end{figure}\n\n\\begin{figure}[!htbp]\n\\small\n  \\centering\n  \\includegraphics[width=\\textwidth]{other_neurological.pdf}\n  \\caption{Learned Phenotypes for Other Neurological Disorders}\n  \\label{fig:other_neurological}\n\\end{figure}\n\n\\begin{figure}[!htbp]\n\\small\n  \\centering\n  \\includegraphics[width=\\textwidth]{cardiac_arrhythmias.pdf}\n  \\caption{Learned Phenotypes for Cardiac Arrhythmias}\n  \\label{fig:cardiac_arrhythmias}\n\\end{figure}\n\n\\begin{figure}[!htbp]\n\\small\n  \\centering\n  \\includegraphics[width=\\textwidth]{drug_abuse.pdf}\n  \\caption{Learned Phenotypes for Drug Abuse}\n  \\label{fig:drug_abuse}\n\\end{figure}\n\n\\begin{figure}[!htbp]\n\\small\n  \\centering\n  \\includegraphics[width=\\textwidth]{paralysis.pdf}\n  \\caption{Learned Phenotypes for Paralysis}\n  \\label{fig:paralysis}\n\\end{figure}\n\n\\begin{figure}[!htbp]\n\\small\n  \\centering\n  \\includegraphics[width=\\textwidth]{aids.pdf}\n  \\caption{Learned Phenotypes for AIDS}\n  \\label{fig:aids}\n\\end{figure}\n\n\\begin{figure}[!htbp]\n\\small\n  \\centering\n  \\includegraphics[width=\\textwidth]{fluid_electrolyte.pdf}\n  \\caption{Learned Phenotypes for Fluid Electrolyte Disorders}\n  \\label{fig:fluid_electrolyte}\n\\end{figure}\n\n\\begin{figure}[!htbp]\n\\small\n  \\centering\n  \\includegraphics[width=\\textwidth]{rheumatoid_arthritis.pdf}\n  \\caption{Learned Phenotypes for Rheumatoid Arthritis}\n  \\label{fig:rheumatoid_arthritis}\n\\end{figure}\n\n\\begin{figure}[!htbp]\n\\small\n  \\centering\n  \\includegraphics[width=\\textwidth]{lymphoma.pdf}\n  \\caption{Learned Phenotypes for Lymphoma}\n  \\label{fig:lymphoma}\n\\end{figure}\n\n\\begin{figure}[!htbp]\n\\small\n  \\centering\n  \\includegraphics[width=\\textwidth]{coagulopathy.pdf}\n  \\caption{Learned Phenotypes for Coagulopathy}\n  \\label{fig:coagulopathy}\n\\end{figure}\n\n\\begin{figure}[!htbp]\n\\small\n  \\centering\n  \\includegraphics[width=\\textwidth]{obesity.pdf}\n  \\caption{Learned Phenotypes for Obesity}\n  \\label{fig:obsesity}\n\\end{figure}\n\n\\begin{figure}[!htbp]\n\\small\n  \\centering\n  \\includegraphics[width=\\textwidth]{pulmonary_circulation.pdf}\n  \\caption{Learned Phenotypes for Pulmonary Circulation Disorder}\n  \\label{fig:pulmonary_circulation}\n\\end{figure}\n\\clearpage\n\\begin{figure}[!htbp]\n\\small\n  \\centering\n  \\includegraphics[width=\\textwidth]{valvular_disease.pdf}\n  \\caption{Learned Phenotypes for Valvular Disease}\n  \\label{fig:valvular_disease}\n\\end{figure}\n\n\\begin{figure}[!htbp]\n\\small\n  \\centering\n  \\includegraphics[width=\\textwidth]{peptic_ulcer.pdf}\n  \\caption{Learned Phenotypes for Peptic Ulcer}\n  \\label{fig:peptic_ulcer}\n\\end{figure}\n\n\\begin{figure}[!htbp]\n\\small\n  \\centering\n  \\includegraphics[width=\\textwidth]{congestive_heart_failure.pdf}\n  \\caption{Learned Phenotypes for Congestive Heart Failure}\n  \\label{fig:congestive_heart_failure}\n\\end{figure}\n\n\\begin{figure}[!htbp]\n\\small\n  \\centering\n  \\includegraphics[width=\\textwidth]{hypothyroidism.pdf}\n  \\caption{Learned Phenotypes for Hypothyroidism}\n  \\label{fig:hypothyroidism}\n\\end{figure}\n\n\\begin{figure}[!htbp]\n\\small\n  \\centering\n  \\includegraphics[width=\\textwidth]{weight_loss.pdf}\n  \\caption{Learned Phenotypes for Weight loss}\n  \\label{fig:weight_loss}\n\\end{figure}\n\n\\begin{figure}[!htbp]\n\\small\n  \\centering\n  \\includegraphics[width=\\textwidth]{deficiency_anemias.pdf}\n  \\caption{Learned Phenotypes for Deficiency Anemias}\n  \\label{fig:deficiency_anemias}\n\\end{figure}\n\n\n\\begin{figure}[!htbp]\n\\small\n  \\centering\n  \\includegraphics[width=\\textwidth]{blood_loss_anemia.pdf}\n  \\caption{Learned Phenotypes for Blood Loss Anemia}\n  \\label{fig:blood_loss_anemia}\n\\end{figure}\n\n\\begin{figure}[!htbp]\n\\small\n  \\centering\n  \\includegraphics[width=\\textwidth]{depression.pdf}\n  \\caption{Learned Phenotypes for Depresssion}\n  \\label{fig:depression}\n\\end{figure}\n\\newpage\n\\section{Augmented Mortality Prediction}\\label{sec:mort-pred}\n\nFigure~\\ref{fig:aug_cnmf_feat1} shows weights learned by the classifier for all features. The weights shaded red correspond to phenotypes and are relatively high compared to raw notes based features (shaded blue), indicating that comorbidities capture significant amount of predictive information on mortality and achieve comparable performance to full EHR model when augmented with additional raw clinical terms.\n\n\\begin{figure}[htb]\n\\small\n\\centering\n\\includegraphics[width=0.65\\textwidth]{sparsity.pdf}\n\\caption{Weights learned by the CNMF+Full EHR classifier for all features. The weights shaded red correspond to phenotypes.\n\\label{fig:aug_cnmf_feat1}\n\\end{figure}\n\\subsection{Interpretability--accuracy trade--off}\\label{sec:exp-sparsity}\nSparsity of the latent factors is used as a proxy for interpretability of phenotypes. Sparsity is measured as the median of the number of non--zero entries in columns of the phenotype matrix $\\mathbf{A}$ (lower is better). The $\\lambda$ parameter in $\\lambda$--CNMF controls the sparsity by imposing scaled simplex constraints on $\\mathbf{A}$. CNMF was trained on multiple $\\lambda$ in the range of  $0.1$ to $1$.  Stronger sparsity-inducing constraints results in worse fit to the cost function. This trade--off is indeed observed in all models (see \\ref{sec:spars_app} for details).\n\n \nFor all models, we pick estimates with lowest median sparsity while ensuring that the phenotype candidate for every condition is represented by at least $5$ non-zero clinical terms.\n\n\\subsection{Clinical relevance of phenotypes}\\label{sec:exp-phenotyping}\n\\begin{figure}[htb]\n\\small \n\\centering\n\\includegraphics[width=0.65\\textwidth]{phenotype_ratings.pdf}\n\\vspace{-4mm}\n\\caption{Qualitative Ratings from Annotation: The two bars represent the  ratings  provided by the two annotators. Each bar is a  histogram of the scores for the $30$ comorbidities sorted by scores. }\n\\label{fig:ratings_bargraph}\n\\end{figure}\n\nWe requested two clinicians to evaluate the candidate phenotypes based on the top $15$ terms learned by each model. The ratings were requested on a scale of $1$ (poor) to $4$ (excellent). The experts were asked to rate based on whether \\textit{the terms are relevant towards the corresponding condition and whether the terms are jointly discriminative of the  condition}. \nFigure~\\ref{fig:ratings_bargraph} shows the summary of qualitative ratings obtained for all models. For each model, we show two columns (corresponding to two experts). The stacked bars show the histogram of the ratings for the models. Nearly $50\\%$ of the phenotypes learned from our model were rated `good' or better by both annotators. In contrast, NMF with support constraints but \\emph{without} sparsity inducing constraints hardly learns clinically relevant phenotypes.\n\\begin{table}[htb]{\n\\small\n\\centering\n\\begin{tabular}{|l|l|l|l|l|}\n\\hline\n &  \\bf 0.4--CNMF &  LLDA  &   MLC  &  NMF \\\\ \\hline\n\\bf 0.4--CNMF &  \\bf 0 & \\bf 28  & \\bf 20 &  \\bf 44 \\\\ \\hline\n LLDA  & \\bf 7 & 0  & 12 & 35 \\\\ \\hline\n MLC   & \\bf 6 & 21 &0  & 42 \\\\ \\hline\n NMF+support & \\bf 1 &0  & 1 & 0 \\\\ \\hline\n\\end{tabular}\n\\caption{Relative Rankings Matrix: Each row of the table is the number of times the model along the row was rated \\textit{strictly} better than the model along the column by clinical experts, e.g., column $3$ in row $2$ implies that LLDA was rated better than MLC~$12$ times over all conditions by all experts.}\n\\label{tab:ratings}}\n\\end{table}\nThe proposed model $0.4$--CNMF also received significantly higher number of `excellent' and `good' ratings from both experts. Although LLDA and MLC~estimate sparse phenotypes, they are not at par with $\\lambda$--CNMF.\nTable~\\ref{tab:ratings} shows a summary of relative rankings for all models. Each cell entry shows the number of times the model along the corresponding row was rated \\textit{strictly better} than that along the column. $0.4$--CNMF is better than all three baselines.  The supervised baseline MLC outperforms LLDA even though LLDA learns comorbidities jointly suggesting that the simplex constraint imposed by LLDA may be restrictive.\n\n\nFigure~\\ref{fig:psychoses}\nis an example of a phenotype (top 15 terms) learned by all models for psychoses. For this condition, the proposed model was rated ``excellent\" and strictly better than both LLDA and MLC by both annotators while LLDA and MLC ratings were tied. However, the phenotype for Hypertension (in Figure~\\ref{fig:hypertension}) learned by $0.4$--CNMF has more terms related to `Renal Failure' or `End Stage Renal Disease' rather than hypertension. One of our annotators pointed out that  ``Candidate 1 is a fairly good description of renal disease, which is an end organ complication of hypertension'', where the anonymized Candidate 1 refers to $0.4$--CNMF. Exploratory analysis suggests that hypertension and renal failure are the most commonly co-occurring set of conditions. Over 93\\% of patients that have hypertension (according to ECI) also suffer from Renal Failure. Thus, our model is unable to distinguish between highly co-occurring conditions. Other baselines were also rated poorly for hypertension, while LLDA was rated only slightly better. More examples of phenotypes are provided in \\ref{sec:samples}.\n\\begin{figure}[tb]\n\\small\n  \\centering\n  \\includegraphics[width=0.8\\textwidth]{psychoses.pdf}\n  \\caption{Phenotypes learned for `Psychoses' (words are listed in order of importance)}\n  \\label{fig:psychoses}\n\\end{figure}\n\\begin{figure}[tb]\n\\small\n  \\centering\n  \\includegraphics[width=0.8\\textwidth]{hypertension.pdf}\n  \\caption{Phenotypes learned for `Hypertension'}\n  \\label{fig:hypertension}\n\\end{figure}\n\\subsection{Mortality prediction}\\label{sec:exp-pred}\n\nTo quantitatively evaluate the utility of the learned phenotypes, we consider the $30$ day mortality prediction task. We divide the EHR into $5$ cross-validation folds of $80\\%$ training and $20\\%$ test patients. As this is an imbalanced class problem, the training--test splits are stratified by mortality labels.\nFor each split, all models were applied on the training data to obtain phenotype candidates $\\tilde{\\mathbf{A}}$ and feature biases $\\tilde{b}$. For each model, the patient loadings $\\tilde{\\mathbf{W}}$ along the respective phenotype space $(\\tilde{\\mathbf{A}},\\tilde{b})$ are used as features to train a logistic regression classifier for mortality prediction.\n For CNMF and NMF+support, these are obtained as  $\\mathbf{W}_\\text{train\/test}={\\text{argmin}}_{\\small \\mathbf{W}\\in[0,1]^{K\\times N}} \\mathcal{D}( \\tilde{\\mathbf{A}}\\mathbf{W}+\\tilde{b}\\mathbbm{1}^\\top,\\mathbf{X}_\\text{train\/test})$ for fixed $(\\tilde{\\mathbf{A}},\\tilde{b})$. For LLDA, these are obtained using Gibbs sampling with fixed topic--word distributions. For MLC, the predicted class probabilities of the comorbidities are used as features. Additionally, we train a logistic regression classifier using the full EHR matrix as features. \n \nWe clarify the following points on the methodology:\n\\begin{inparaenum}[(1)]\n\\item $\\tilde{\\mathbf{A}}$ is learned on the patients in the training dataset only, hence there is no information leak from test patients into  training. \n \\item Test patients' comorbidities from ECI are \\textit{not} used as support constraints on their loadings. \n \\item Regularized logistic regression classifiers are used to learn models for  mortality prediction. The regularization parameters are chosen via grid-search.\n\\end{inparaenum}\n\\begin{table}[t]\n\\centering\n\\small\n\\begin{tabular}{|l|l|l|l|l|}\n\\hline\n& Model       & AUROC           & Sensitivity    & Specificity    \\\\ \\hline\n1. &{$0.4$--CNMF}          &  0.63(0.02)  & 0.59(0.04) & 0.62(0.03)  \\\\\n2.  & NMF+support          & 0.52(0.02)  & 0.56(0.13) & 0.51(0.14)  \\\\\n3.  & LLDA &  0.64(0.02) & 0.62(0.03) & 0.61(0.05)  \\\\\n4.  & {MLC} &  0.66(0.01) & 0.62(0.06) & 0.62(0.05) \\\\\n\\bf 5.  & \\bf Full EHR & \\bf 0.72(0.02) & \\bf 0.69(0.02) & \\bf 0.63(0.04) \\\\ \\hline\n\\bf 6.  & \\bf CNMF+Full EHR ($\\ell_1$, $C=0.1$)& \\bf 0.72(0.02) & \\bf 0.61(0.09) & \\bf 0.71(0.07) \\\\\\hline\n\\end{tabular}\n\\caption{$30$ day mortality prediction: $5$--fold cross-validation performance of logistic regression classifiers. Classifiers for $0.4$--CNMF and competing baselines (NMF+support, LLDA, MLC) were trained on the $30$ dimensional phenotype loadings as features. Full EHR denotes the baseline classifier ($\\ell_1$-regularized logistic regression) using full $\\sim3500$ dimensional EHR as features. CNMF+Full EHR denotes the  performance of the $\\ell_1$-regularized classifier learned on Full EHR augmented with CNMF features (hyperparameter was manually tuned to match performance of the Full EHR model). \n\\label{tab:mortality_pred}}\n\\end{table}\n\nThe performance of the above baselines trained on $\\ell_2$ regularized logistic regression over a $5$-fold cross-validation is reported in  Table~\\ref{tab:mortality_pred}: rows $1$--$5$. The classifier trained on the full EHR unsurprisingly outperforms all baselines as it uses richer high dimensional information.\nAll phenotyping baselines, except NMF+support, show comparable performance on mortality prediction which in spite of learning on a small number of $30$ features, is only slightly worse than predictive performance of full EHR with $\\sim 3500$ features.\n\n\n\n\n\\paragraph{Augmented features for mortality prediction (CNMF+Full EHR)}\nUnsurprisingly, Table~\\ref{tab:mortality_pred} suggests that the high dimensional EHR data has additional information towards mortality prediction which are lacking in the $30$ dimensional features generated via phenotyping. To evaluate whether this additional information can be captured by CNMF if augmented with a small number of raw EHR features, we train a mortality prediction classifier using $\\ell_1$ regularized logistic regression on CNMF features\/loadings combined with raw {bag--of--words} features, with parameters tuned to match the performance of the full EHR model. The results are reported in the final row of  Table~\\ref{tab:mortality_pred}. \n\nIn exploring the weights learned by the classifier for all features, we observe that only $8.3\\%$ of the features corresponding to raw EHR based \\emph{bag-of-words} features have non--zero weights.\nThis suggests that comorbidities capture significant amount of predictive information on mortality and achieve comparable performance to full EHR model with a small number of additional terms. See Figure~\\ref{fig:aug_cnmf_feat1} in Appendix showing the weights learned by the classifier for all features. \nFigure~\\ref{fig:aug_cnmf_feat2} shows comorbidities and EHR terms with top magnitude weights learned by the CNMF+full EHR classifier. For example, it is interesting to note that the top weighted EHR term -- dnr or `Do Not Resuscitate' is not indicative of any comorbidity but is predicitive of patient mortality.\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=0.35\\textwidth,angle=0]{ehr.png}\n~\n\\includegraphics[width=0.33\\textwidth,angle=0]{cnmf.png}\n\\caption{Top magnitude weights  on (a) EHR and (b) CNMF features in CNMF+Full EHR classifier}\n\\label{fig:aug_cnmf_feat2}\n\\end{figure}\n\n\n\n\n\\section{Introduction}\\label{sec:intro}\n\\input{intro}\n\\input{problem_description}\n\\input{model}\\label{sec:model}\n\\section{Empirical Evaluation}\\label{sec:expts}\n\\input{experiments}\n\\section{Discussion and Related Work}\\label{sec:discussion}\n\\input{relatedwork}\n\n\\clearpage\n\\section*{Acknowledgements}\nWe thank Dr. Saul Blecker and Dr. Stephanie Kreml for their qualitative evaluation of the computational phenotypes. SJ, SG and JG were supported by NSF: SCH \\#1418511. DS was supported by NSF CAREER award \\#1350965. We also thank Yacine Jernite for sharing a code used in preprocessing clinical notes.\n\n\n\n\\section{Data Extraction}\\label{sec:data}\nThe MIMIC-III dataset consists of de-identified EHRs for $\\sim38,000$ adult ICU patients at the Beth Isreal Deaconess Medical Center, Boston, Massachusetts from $2001$--$2012$. For all ICU stays within each admission, clinical notes including nursing progress reports, physician notes, discharge summaries, ECG, etc. are available. We analyze patients who have stayed in the ICU for at least $48$ hours ($\\sim17000$ patients). We derive phenotypes using clinical notes collected within the first $48$ hours of patients' ICU stay to evaluate the quality of phenotypes when limited patient data is available. Further, we evaluate the phenotypes on a $30$ day mortality prediction problem. To avoid obvious indicators of mortality and comorbidities, apart from restricting to 48 hour data, we exclude discharge summaries as they explicitly mention patient outcomes (including mortality).\n\\begin{table}[t]\n\\centering\n\\caption{Target comorbidities}\\label{tab:conds1}\n\\resizebox{\\textwidth}{!}{\n\\begin{tabular}{lllll}\n\\hline\nCongestive Heart Failure & Cardiac Arrhythmias & Valvular Disease   & Pulmonary Circulation Disorder &  Peripheral Vascular  Disorder  \\\\\nHypertension             & Paralysis          & Other Neurological Disorders &   Chronic Pulmonary Diseases    & Diabetes Uncomplicated \\\\ \nDiabetes Complicated     & Hypothyroidism  & Renal Failure      & Liver Disease   (excluding bleeding)      & Peptic Ulcer \\\\\nAIDS &  Lymphoma           & Metastatic Cancer  & Solid Tumor          (without metastasis) & Rheumatoid Arthritis \\\\\n Coagulopathy             & Obesity            & Weight loss        & Fluid Electrolyte Disorder   &  Blood Loss Anemia    \\\\\n Deficiency Anemia       & Alcohol abuse      & Drug abuse      &  Psychoses             & Depression    \\\\\n\\hline\n\\end{tabular}}\n\\end{table}\n\\begin{asparaenum}\n\\item {\\bf{Clinically relevant bag-of-words features:}}\\label{sec:data-vocab}\n Aggregated clinical notes from all sources are represented as a single \\emph{bag-of-words} features. To enhance clinical relevance, we create a custom vocabulary containing clinical terms from two sources (a) the Systematized Nomenclature of Medicine-Clinical Terms (SNOMED CT), and (b) the level-0 terms provided by the Unified Medical Language System (UMLS), consolidated into a standard vocabulary format using Metamorphosys --- an application provided by UMLS for custom vocabulary creation.\\footnote{See \\url{https:\/\/www.nlm.nih.gov\/healthit\/snomedct\/} and \\url{https:\/\/www.nlm.nih.gov\/research\/umls\/}} To extract clinical terms from the raw text, the notes were tagged for chunking using a conditional random field tagger\\footnote{\\url{https:\/\/taku910.github.io\/crfpp\/}}. The tags are looked up against the custom vocabulary (generated from Metamorphosys) to obtain the \\emph{bag-of-words} representation. Our final vocabulary has $\\sim$3600 clinical terms.\n\\item {\\bf{Computable weak diagnosis:}}\\label{sec:data-supp}\nWe incorporate domain constraints from weak supervision to ground ~the latent factors to have a one-to-one mapping with the conditions of interest. In the model described in Section~\\ref{sec:model}, this is enforced by constraining the non-zero entries on patient loading along the latent factors using a weak diagnosis for comorbidities. The weak diagnoses of target comorbidities in Table~\\ref{tab:conds1} are obtained using  ECI\\footnote{\\url{https:\/\/git.io\/v6e7q}}, computed solely from patient administrative data without human annotation. We refer to this index as \\textit{weak diagnoses} as it is not a physician's exact diagnosis and is subject to noise and misspecification.\nNote that ECI ignores diagnoses code related to the primary diagnoses of admission. Thus, ECI models presence and absence of conditions other than the primary reason for admission (comorbidities). The phenotype candidates from the proposed model can be considered as concise representations of such comorbidities.\n\\end{asparaenum}\n\n\n\n\n\n\n\\section{Identifiable High--Throughput Phenotyping}\\label{sec:setup}\nThe notation used in the paper are enumerated in Table~\\ref{tab:notation}. \n\\begin{table}[t]\n\\small\n\\centering\n\\begin{tabular}{lp{11cm}}\n\\hline\nNotation                         & Description                                                                   \\\\ \\hline\n$[m]$ for integer $m$ & Set of indices $[m]=\\{1,2,\\dots,m\\}$. \\\\\n$\\Delta^{d-1}$ & Simplex in dimension $d$, ${\\Delta^{d-1}=\\{x\\in\\mathbb{R}^d_+:\\sum x_i=1\\}}$. \\\\\n$x^{(j)} $     & Column $j$  of a matrix $\\mathbf{X}$.  \\\\\n$\\text{supp}(x)$ & Support of a vector $x$, $\\text{supp}(x)=\\{i:x_i\\neq 0\\}$. \\\\\n\\bf Observations &\\\\\n$N$, $d$                             & Number of patients  ($\\sim17000$) and  features  ($\\sim3600$), respectively.   \\\\\n$\\mathbf{X}\\in\\mathbb{R}_+^{d\\times N}$ & EHR matrix from MIMIC III: Clinically relevant bag-of-words features from notes in first $48$ hours of ICU stay for $N$ patients. \\\\\n$k=1,2,\\ldots,K$               & Indices for  $K=30$ comorbidities in Table \\ref{tab:conds1}. \\\\\n$C_j\\subseteq [K]$ for $j\\in[N]$        & Set of comorbidities  patient $j$ is diagnosed with using  ECI . \\\\\n\\bf Factor matrices &\\\\\n$\\tilde{\\mathbf{W}} \\in [0,1]^{K\\times N}$ & Estimate of \\textit{patients' risk} for the $K$ conditions.\\\\%: $\\mathbf{W}^*$ is unknown ground truth, and $\\tilde{\\mathbf{W}}$ is the estimate form Algorithm~\\ref{alg:alg1}\\\\\n$\\tilde{\\mathbf{A}} \\in \\mathbb{R}_+^{d\\times K}$, $\\tilde{b}\\in\\mathbb{R}_+^{d}$ & Estimate of \\textit{phenotype factor matrix} and \\textit{feature bias vector}. \\\\%The columns represent phenotype definitions of $K$ chronic conditions\\\\%,   unknown ground truth, and  estimate form Algorithm~\\ref{alg:alg1}, respectively\\\\\n\\hline\n\\end{tabular}\n\\caption{Notation used in the paper}\n\\label{tab:notation}\n\\end{table}\nIn summary, for each patient $j\\in[N]$, \n\\begin{inparaenum}[(a)]\n\\item the {bag-of-words} features from clinical notes is represented as column $x^{(j)}$ of EHR matrix $\\mathbf{X}\\in\\mathbb{R}_{+}^{d \\times N}$, and \n\\item  the list of comorbidities diagnosed  using ECI is denoted as $C_j\\subseteq [K]$.\n\\end{inparaenum}\n Let an unknown  $\\mathbf{W}^*\\in[0,1]^{K\\times N}$ represent the risk of $N$ patients for $K$ comorbidities of interest; each entry $w^*_{kj}$ lies in the interval $[0,1]$, with $0$ and $1$ indicating no-risk and maximum-risk, respectively, of patient $j$ being afflicted with condition $k$. If $C_j^*\\subseteq[K]$ denotes an accurate diagnosis  for patient  $j$, then $w^{*(j)}$ satisfies $\\text{supp}(w^{*(j)})\\subseteq C_j^*$\n\\begin{definition}[EHR driven phenotype]\\label{def:phenotype} \\normalfont  \\textit{EHR driven phenotypes} for  $K$ co--occurring  conditions are a set of vectors $\\{a^{*(k)}\\in\\mathbb{R}^d_+: k\\in[K]\\}$, such that for a patient $j$ afflicted with conditions $C^*_j\\subseteq[K]$,\n\\begin{equation}\n\\b{E}[x^{(j)}|w^{*(j)}] = \\textstyle \\sum_{k\\in C^*_j}\\;w^*_{kj} a^{*(k)} + b^*,\n\\end{equation}\nwhere $b^*$ is a  bias representing the feature component observed  independent of the $K$ target conditions. $\\mathbf{A}^*\\in\\mathbb{R}^{d\\times K}$ with $a^{*(k)}$ as columns is referred as  the \\textit{phenotype factor matrix}.\n\\end{definition}\nNote that we explicitly model a feature bias $b^*$ to capture frequently occurring terms that are not discriminative of the target  conditions, e.g., temperature, pain, etc\n\n\\section{Related Work}\n\n\nOur approach achieves identifiability using support constraints to ground~the latent factors and interpretability using sparsity constraints.\nThe phenotypes learned are clinically interpretable and predictive of mortality when augmented with a sparse set of raw \\emph{bag-of-words} features on unseen patient population.\nThe model outperforms baselines in terms of clinical relevance according to experts and significantly better than the model which includes supervision but no sparsity constraints. The proposed method can be easily extended to other non--negative data to obtain more comprehensive phenotypes. However, it was observed that the algorithm does not discriminate between frequently co--occurring conditions, e.g. renal failure and hypertension. Further, the weak supervision (using ECI) does not account for the primary diagnoses of admission. Additional model flexibility to account for a primary condition in explaining the observations could potentially improve performance.\nAddressing the above limitations along with quantitative evaluation of risk for disease prediction, and understanding conditions for uniqueness of phenotyping solutions are interesting areas of follow-up work.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:1}\n\n\\subsection{General picture}\n\nAccording to current models the entire post-supernova system is tightly coupled. \nThe central engine -- a rapidly spinning neutron star -- is responsible for creation of particles, their acceleration and finally outflow in the form of a relativistic wind. The energy in the wind is transported by Poynting flux and released to the plasma at large distances from the star, in the nebula, where the energized particles radiate away the acquired energy and perform $p$d$V$ work on the surroundings. \nThe global evolution of the system is usually modelled using relativistic MHD.\nAlthough it allows us to get a glimpse on how the system is globally interconnected, we still do not understand the details of the coupling mechanisms and the processes that lead to the release of the magnetic energy.\n\nThe abundant pair plasma, transported by the wind, is thought to be created in electromagnetic cascades in the closest neighbourhood of the star, in the magnetosphere. The conventional boundary of this region is defined by the light cylinder of a cylindrical radius $r_{\\rm L}=cP\/(2\\pi)$ (where $P$ is the rotational period of a pulsar), at which the corotation velocity of the particles with the star would be equal to the speed of light. The corotating plasma is coupled to the magnetic field lines, which in the dipole-like geometry close within the light cylinder.  \nThe field lines which cross the light cylinder carry the currents that brake the central rotator via Lorentz forces. \n\nThis rotational energy of the central neutron star is transported outwards by the wind, a mixture of plasma and electromagnetic fields. The outflow is terminated at the roughly standing shock, where its ram pressure becomes balanced by the pressure of the confining external medium. The shock is located at a large distance from the central star -- in the case of the Crab $r_{\\rm shock}\\sim 10^8 r_{\\rm L}$. At the shock the magnetic energy of the wind is randomised and the particles carried by the wind become efficiently accelerated via the Fermi I-order process. They are further injected into the nebula downstream of the shock, where they radiate in a synchrotron process, observed as diffuse emission from radio to X-rays; in gamma-rays a dominant process is the inverse-Compton (IC) scattering on the cosmic microwave background radiation and on the synchrotron photons produced in the nebula itself (synchrotron self-Compton (SSC) emission). \n\nBoth the diffuse radiation from the nebula, as well as the high-energy pulses of emission observed from the central point-like source, provide the observational evidence that efficient mechanisms of particle acceleration are at work, challenging our understanding of the physics of relativistic flows. \nIn the following we will concentrate on one particular underlying question: how and where the enormous magnetic energy carried by the wind is dissipated and transferred to the energised population of radiating particles?  \n\n\n\\subsection{Problems}\n \nThe energy of the central rotator is transported over long distances, and, therefore, the wind zone proves to be an important ingredient in the global description of the system. On the other hand, very little direct information is provided by observations, in which the wind region appears underluminous \n\\cite{2000ApJ...536L..81W}. This is related to the fact that the plasma in the wind is expected to be cold and, therefore, not producing any emission. However, if due to some internal dissipation the plasma were heated, the relativistic beaming and the radial dependence of the emissivity would cause the emission to appear as point-like \\cite{2009ASSL..357..421K}.\n\nThe point-like emission from the relativistic, MHD pulsar wind would be seen by a distant observer as pulsed \\cite{2002A&A...388L..29K}, because the internal dissipation in the wind, due to e.g. reconnection, is expected to occur abruptly in a thin, corrugated layer -- a current sheet (see Sect.~\\ref{section:sw}). \nReconnection regions are known to be powerful particle accelerators, and, therefore, this scenario provides a promising possibility to explain puzzling very high-energy $\\gamma$-ray observations from the Crab \\cite{2016A&A...585A.133A} and Vela \\cite{velahess2, 2014ApJ...797L..13L} pulsars, as well as the millisecond pulsar J0614--3329 \\cite{2016arXiv160408710X}. \nThis is an alternative to the magnetospheric models, which associate the $\\gamma$-ray pulses with cyclotron-self Compton scattering \\cite{2013MNRAS.431.2580L}, synchro-curvature radiation \\cite{2015MNRAS.449.3755V} and SSC emission inside the light cylinder \\cite{2015ApJ...811...63H}.   \n\nThe pulsed emission, however, constitutes only a fraction of the total energy budget.  \nOn the global scale, the dissipation of the magnetic energy is believed to take place somewhere close to the termination shock. The properties of a shock, in particular its ability to dissipate the energy, strongly depend on the magnetisation of the upstream plasma. This magnetisation is quantified by the parameter $\\sigma$. In a cold plasma it measures the ratio of the magnetic energy flux to the kinetic energy flux in the flow:\n\\eqb \\sigma=\\frac{B^2}{8\\pi N mc^2 \\Gamma}, \\eqe\nwhere $B$ is the strength of the magnetic field, $N$ is the plasma density in the laboratory frame and $\\Gamma$ is the Lorentz factor of the flow. \nThe magnetisation is thought to be large close to the light cylinder, where the wind is launched. \nHowever, Kennel \\& Coroniti \\cite{1984ApJ...283..694K} have shown that the observational limits on the magnetisation can be put downstream of the termination shock, in the nebula, where it must be small. It seems, therefore, that the transition between large- and small-$\\sigma$ regimes requires some magnetic dissipation in the wind zone or at the shock itself. The mechanism responsible for this dissipation is unknown, with the magnetic reconnection being one of the most important candidates. \n\n\nIn the following we review the physics of relativistic pulsar winds and the dissipation processes that can lead to emission observed from those outflows. We start by considering their launching from the magnetospheres in Sect. \\ref{section:launch}, and we discuss their \\lq\\lq striped\\rq\\rq\\ structure in Sect. \\ref{section:sw}. Further, in Sect. \\ref{section:why}, we summarise the arguments in favour of the wind model of pulsed high-energy emission. \nIn Sect. \\ref{problem1} we consider a dissipation mechanism -- relativistic reconnection, which is likely to set in in the current sheet of the wind; a population of particles accelerated in the course of reconnection can be responsible for the pulses of emission, which would be, therefore, a direct probe of the physics in the current sheet. \nIn Sect. \\ref{section:implic} we discuss the implications of this model for the gamma-ray spectra of pulsars and its predictions for future observations. Finally, in Sect. \\ref{section:shocks}, we describe the structure of shocks of striped winds and the $\\sigma$-problem -- a failure to describe the conversion of the Poynting flux into the particle energy flux before the wind reaches the termination shock. A solution for this problem is still not clear, but we discuss two scenarios that have been proposed, namely the driven reconnection of wind stripes at the shock and the generation of an electromagnetic dissipative precursor to the shock. \n\n\n\\section{MHD picture: striped winds}\n\n\\subsection{Launching from a magnetosphere}\n\\label{section:launch}\n\nUnderstanding of pulsar wind launching requires the knowledge\nof (1) the magnetic field topology, (2) the location of particle acceleration regions, and (3) the mechanisms\nof supplying charges to the outflow. The latter are closely linked to radiation processes,\nwhich in the strong-field regime often change their character or become dominated by exotic phenomena, resulting from energy quantization in the direction perpendicular to the field (Landau states) or the\nnonconservation of transverse photon momenta which can be absorbed by the field \\cite{1991Sci...251.1033H}.\n\nThe magnetic field is usually modelled as a rotating dipole (strongly distorted close to the light cylinder), but higher multipoles may also contribute \\cite{1982ApJ...254..713B, 2001ApJ...550..383G, 2015MNRAS.450..714P}. The strength of the magnetic field can be measured in two ways: (1) from the pulsar spin down, assuming that it results from the dipole emission, (2) directly from the\nenergies of cyclotron absorption lines in the X-ray spectra of neutron stars (for discussion see e.g. \\cite{1991Sci...251.1033H}). \nThese measurements imply that the magnetic fields at the stellar surface are very strong $B_*\\gtrsim 10^{12}$~G.\n\nA rotating neutron star, endowed with a dipole magnetic field, generates a quad\\-ru\\-po\\-le electric field, in a\ndirect analogy to a unipolar inductor: like a conductor rotating in a magnetic field, charges\non the pulsar's surface are redistributed by the Lorentz force, which induces an electric potential\ndifference between the poles and the equator.\nThe generated electric field is thought to be strong enough to rip the charges off from the stellar surface and accelerate them along the magnetic field lines, because any transverse motion would be immediately suppressed by radiation of synchrotron photons. \nThe accelerated charges emit curvature gamma-rays, which in the strong magnetic field can materialise into secondary $e^{\\pm}$ pairs, further accelerated and radiating. \n\nThe number of pairs created in these electromagnetic cascades is not clear. \nThe critical Goldreich-Julian density $N_{\\rm GJ}$ defines the number of charges that are necessary to screen out the component of the electric field parallel to the magnetic field:\n\\eqb N_{\\rm GJ}=\\frac{\\mathbf{B}\\cdot\\boldsymbol{\\rm \\omega}}{2\\pi e  c}\\frac{1}{1-(r\/r_{\\rm L})^2\\sin^2\\theta}, \\label{nGJ}\\eqe\nwhere $\\mathbf{\\rm \\omega}$ is the angular velocity of the pulsar. \nThe density of pairs $N$, created in the magnetosphere, is usually expressed in terms of the pair multiplicity \n\\eqb \\kappa=N\/N_{\\rm GJ}. \\eqe \nIf the pair injection in the magnetosphere is somewhat below $N_{\\rm GJ}$, the pulsar magnetosphere is described as the \\lq\\lq electrosphere\\rq\\rq, in which a charge-separated plasma is confined to domes above the poles and a differentially rotating equatorial disc \\cite{1985MNRAS.213P..43K,1985A&A...144...72K, 2002A&A...384..414P, 2004IAUS..218..357S, 2015MNRAS.448..606C}. \nIf the beam of primary particles ripped off from the stellar surface initiates prolific cascades, a pair plasma in the magnetosphere is abundant $\\kappa\\gg 1$. \nSimulations of electromagnetic cascades suggest multiplicities in the range $\\kappa \\sim 10^3-10^5$ (see \\cite{2015ApJ...810..144T} and references therein). \nA constraint on the multiplicity $\\kappa \\sim10^6$ is implied by the models of emission from the Crab Nebula, under the assumption that all the radiating particles are being constantly supplied by the pulsar and that these particles are in energy  equipartition with the magnetic field \\cite{1974MNRAS.167....1R,1996ApJ...457..253D,2002ASPC..271...99G} (for discussion of the radio brightness see also \\cite{2003MNRAS.345..153L,2009ASSL..357..421K}). Modelling of other nebulae in general suggests $\\kappa\\sim10^5$ \\cite{2011MNRAS.410..381B}.\n \nSuch large values of multiplicity imply that the magnetospheric plasma easily screens the parallel component of the electric field. However, to maintain the activity, a pulsar has to sustain the constant or at least intermittent replenishment of pairs that outflow with the wind. Current theories postulate that the pair creation processes take place in limited regions of an unscreened component of the electric field -- so called vacuum gaps.   \nTheir location is model-dependent: they are associated with \nthe polar caps (located near the stellar surface close to the magnetic axis of the pulsar) \\cite{1971ApJ...164..529S, 1975ApJ...196...51R, 1978ApJ...225..226H, 1982ApJ...252..337D}, outer magnetosphere \\cite{1986ApJ...300..500C, 1995ApJ...438..314R}, and the region along the last open field line \\cite{1979ApJ...231..854A,2003ApJ...588..430M, 2004ApJ...606.1143M,2003ApJ...598.1201D}. \nLyubarsky \\cite{1996A&A...311..172L} suggested that the pairs can be produced in the current sheet just outside the light cylinder, where the magnetic field in the sheet is strongest. This scenario is observed in kinetic Particle-In-Cell (PIC) simulations of pulsar magnetospheres with low obliquity, whereas the pulsars with high obliquity tend to create pairs rather in polar cap regions \\cite{2015ApJ...801L..19P}. \n\nThe existence of gaps, although important for understanding of how pulsars really work, is usually neglected in the modelling of the global plasma dynamics. In particular, the MHD, fluid description of the plasma is thought to be valid.  \nSince the energetics is dominated by electromagnetic fields, the particle \ninertia is negligible in the first approximation, but the particles are still assumed to carry charge and currents. \nThis zero-inertia limit of MHD is called force-free electrodynamics (FFE). \nThe equations for a force-free magnetosphere of an aligned rotator were obtained by Michel\n\\cite{1973ApJ...180L.133M}, as well as Scharlemann \\& Wagoner \\cite{1973ApJ...182..951S}. \nThe plasma, rigidly corotating with the star, is assumed to have infinite conductivity, such that \nin the comoving frame the electric field vanishes; after transformation into the laboratory frame, this implies the frozen-in condition between the plasma and the magnetic field \n\\eqb \\mathbf{E}+(1\/c)\\mathbf{v}\\times \\mathbf{B}=0. \\label{frozen-in} \\eqe  \nCorotation of a plasma with the star is possible only within the light cylinder. \nSince the particles cannot corotate with superluminal speed, the magnetic field lines that would cross the light cylinder are bent backwards and they open beyond $r_{\\rm L}$,  \neven in the case of an aligned rotator (and in contrast with an aligned rotator in vacuum). \nAs explained by Michel \\cite{1973ApJ...180..207M}, in vacuum, without the plasma which ensures the frozen-in condition, the magnetic field of such an aligned rotator is static, whereas in the presence of a  plasma, the magnetic field is \\lq\\lq rotating\\rq\\rq. As a consequence, there is a nonvanishing Poynting vector $\\mathbf{S}=\\mathbf{E}\\times \\mathbf{B}$, which describes the transport of energy; the closer to the light cylinder, the more the corotation speed increases, and so does the magnitude of the Poynting vector. Ultimately, beyond $r_{\\rm L}$, it would exceed the energy density that can be transported with the speed of light. Thus, the field lines become open by the centrifugal force exerted by the corotating energy density of the electromagnetic fields. \n\nFor an oblique rotator, with the angle $\\chi$ between the magnetic and rotational axes, the presence of the plasma increases the torque when compared to a rotator in vacuum. The latter emits dipole radiation at the expense of the rotational energy, and the luminosity is given by:\n\\eqb L_{\\rm vac}=L_{0}\\sin^2\\chi, \\eqe\nwhere $L_0=B_*^2\\omega^4r_*^6\/(6c^3)$, $B_*$ is the strength of the magnetic field at the stellar equator, $r_*$ is the stellar radius. \nThe plasma-filled magnetosphere, on the other hand, spin downs according to\n\\eqb L_{\\rm FFE}\\approx \\frac{3}{2}L_{0}(1 +\\sin^2\\chi) \\eqe\n\\cite{2006ApJ...648L..51S,2012MNRAS.424..605P}. \nIn comparison to the vacuum case, the topology of the magnetic field lines in the force-free description is similar inside the light cylinder, but exhibits the formation of a current sheet outside this distance \\cite{2012MNRAS.424..605P}. \n\nThe analytical studies of pulsar magnetospheres of oblique rotators are nowadays extended to the simulations: 3D force-free \\cite{2006ApJ...648L..51S, 2012MNRAS.424..605P}, resistive force-free \\cite{2012ApJ...749....2K, 2012ApJ...746...60L}, relativistic MHD \\cite{2006MNRAS.367...19K, 2013MNRAS.435L...1T}, and also PIC simulations \\cite{2015ApJ...801L..19P}.  \nIn general relativity force-free simulations have been performed recently by P\\'etri \\cite{2016MNRAS.455.3779P}. \n\nAn analytical two-fluid approach to describe the pulsar magnetosphere has been proposed by Petrova \\cite{2015MNRAS.446.2243P}. In this solution, electron and positron fluids exhibit a velocity shift with respect to each other, which can underlie the instabilities and production of waves, convertible into the radio emission from pulsars. Interestingly, in this model the plasma conductivity in the magnetosphere is finite due to inertial effects, which provides a physical justification for the resistivity, considered only phenomenologically as a free parameter in the resistive force-free simulations \\cite{2012ApJ...749....2K,2012ApJ...746...60L}.    \n\n\\subsection{The striped wind structure}\n\\label{section:sw}\n\nAn exact force-free monopole solution for an outflow of an aligned rotator was found by Michel \\cite{1973ApJ...180L.133M} and it was generalized for an oblique rotator by Bogovalov \\cite{1999A&A...349.1017B}. \nIn the pulsar models, the magnetic field is believed to change smoothly from the dipolar geometry inside the light cylinder to the monopolar, radial geometry at infinity \\cite{1973ApJ...180..207M}. However, this change of magnetic field topology, captured numerically, has not been described analytically.  \nRecently though, the exact dipole solution for an aligned rotator has been reported and discussed by Petrova \\cite{2016arXiv160807998P}.\n\nIn the limit of negligible\nparticle inertia, a monopole configuration is radial in the meridional plane, \nand\nin the equatorial plane these lines develop a perfect Archimedean spiral\n$r_s = r_{\\rm L} \\phi$ \nregardless of a field topology inside the light cylinder. Globally, more appropriate is a split-monopole configuration, where two half magnetic monopoles of opposite polarity are joined together \nin the equatorial plane. This change in magnetic field direction must be, according to Amp\\`ere's law,\naccompanied by formation of a current sheet, within which the magnetic field vanishes and the\npressure, necessary to keep the equilibrium, is supported by a hot plasma. \nIn the oblique case, a current sheet oscillates around the equatorial plane\nas the pulsar rotates, connecting the equator with field lines of opposite polarity every half\na period (see Fig.~\\ref{fig_cs}). This corrugated current sheet far from the light cylinder can be approximated by spherical shells, separating the stripes of magnetized plasma with opposite magnetic polarity -- this is the structure of a so-called striped wind \\cite{1990ApJ...349..538C, 1994ApJ...431..397M}. \nA solution for the wind with analytically prescribed current sheet and a radial velocity, has been recently reported \\cite{2013MNRAS.434.2636P}.\n \n\n\\figuresheet \n\nIn the split-monopole solution the Poynting flux has a maximum value at the equator \\cite{1973ApJ...180L.133M}. Most of the energy, therefore, is carried within the wedge of the striped wind, defined by the spherical coordinate $\\theta$, $\\pi\/2-\\chi<\\theta<\\pi\/2+\\chi$, where $\\chi$ is the angle between the magnetic and rotational axes of the pulsar (obliquity).\nIn the equatorial plane the stripes have the equal width, hence the phase-averaged magnetic field vanishes. This is not the case for the higher latitudes, for which \na phase-averaged magnetic field increases with the latitude. It reaches the maximum value at the surface defined in spherical coordinates by $\\theta=\\pi\/2\\pm\\chi$. This surface passes through the cusps of the current sheet. Above it, in the polar region of the wind, the total magnetic field has only one polarity.\n\n\nThe radial dependence of the magnetic field components can be obtained from the magnetic flux conservation. \nIntegrating over the volume of a sphere of a radius $r$, one gets from the Gauss\ntheorem\n\\eqb \nB_r\\propto r^{-2}. \\label{eqbr}\n\\eqe\nSimilarly, as the number of field lines in the sectional area is constant,\n\\eqb\nB_{\\phi}\\propto r^{-1} \\label{eqbphi}\n\\eqe \n(for a detailed discussion see e.g. \\cite{1994ApJ...431..397M, 2009ASSL..357..421K}). \nClose to the light cylinder the radial and poloidal component of the magnetic field are comparable, \nbut the poloidal component, which in this solution is purely radial, decreases much faster than the\ntoroidal one. Observations of synchrotron radiation close to the termination shock, in particular its uniform\nlinear polarization, are consistent with the toroidal structure of the magnetic field, expected in the wind at large distances from the pulsar.\n\nThe radial wind is believed to emerge from the magnetosphere as submagnetosonic. \nIn the force-free solution the fast magnetosonic (FMS) point is located at infinity, and the outflow accelerates with the distance $\\Gamma\\propto r$ \\cite{1977MNRAS.180..125B} at the expense of the magnetic energy. As a result, the wind described by the FFE arrives at the termination shock as kinetically dominated \\cite{2002ApJ...566..336C}. However, the FFE model neglects the particle inertia, which, by definition at the FMS point become important (propagation of the FMS wave is an interplay between particle inertia and magnetic\ntension, implying that the former cannot be neglected). The FMS speed in a highly magnetized plasma is relativistic, defined by the Lorentz factor $\\Gamma_{\\rm fms}\\approx\\sqrt{\\sigma} \\gg 1$. For large-$\\sigma$ flow, the FMS point should be crossed somewhere close to the light cylinder. Further out the wind becomes supermagnetosonic, described by MHD rather than FFE equations. An MHD relativistic wind is moving ballistically, with almost a constant, relativistic speed, and it stays strongly magnetically dominated up to the large distances (for a discussion of this topic see \\cite{2009ASSL..357..421K}). \n \n\n\\section{Reconnection and pulsed emission from the wind}\n\n\\subsection{Why the wind model?}\n\\label{section:why}\n\nThe striped wind is a promising site of emission of high-energy (optical to gamma-ray)\nnonthermal photons observed from pulsars, an idea proposed by \\cite{2002A&A...388L..29K}.  \nThe difference between the wind and the magnetospheric scenarios lies in assumptions about the physics of production of the pulsed emission.\nThe wind model offers an analytic description of the magnetic field structure, and the peaks in a lightcurve appear as a result of the strong beaming (since the wind is relativistic) of the radiation produced in a thin layer -- a current sheet. This layer is naturally prone to dissipative processes, in particular to reconnection.  \nOn the other hand, in the magnetospheric models the peaks in a light curve\nare caused by the overlap of emission from different\nregions in the magnetosphere (caustics) (see e.g. \\cite{2010ApJ...715.1270B}); this overlap is sensitive\nto both the definition of gaps and the exact geometry of the magnetic field -- firstly, because the\ngeometry of the field defines how the effects of aberration of light and light\ntravel delay add together to form the caustics; \nsecondly, because the geometry of the field close to the light cylinder determines\nthe shape of the polar cap on the star, which in turn controls the shape of the radiation region \\cite{2010ApJ...715.1270B}. This requires a global modelling of the pulsar magnetospheres, and presumably can be done only via numerical simulations.\n\nMoreover, the reconnection in the wind is expected to be a very powerful particle accelerator, whereas details of acceleration processes (and accelerating fields) that lead to magnetospheric emission are strongly model-dependent. \nExtreme conditions of particle acceleration are indeed required to explain \nrecent observations of pulsars above a hundred of GeV \\cite{2016A&A...585A.133A,velahess2, 2016arXiv160408710X}.\n\n\nTo understand how the pulses from the wind appear in the lightcurves, we assume that the corrugated current sheet can be described as subsequent shells, which radiate after crossing a sphere of a radius $r_0$ (see Fig.~\\ref{figfive}). A distant observer will notice a peak of emission from the shell that has crossed the radius $r_0$ along his line of sight, followed by a quick decline of the emissivity as the sheet propagates outwards, due to adiabatic cooling and radial dependence of magnetic field (Eq. \\ref{eqbr} and \\ref{eqbphi}). \nDepending on the viewing angle, the observer can detect up to two pulses per rotational period of a pulsar -- \n(1) two pulses with changing separation if his line of sight lies in the wedge $\\pi\/2-\\chi<\\theta_{\\rm obs}<\\pi\/2+\\chi$; (2) one pulse, if the line of sight is close to $\\theta_{\\rm obs}\\approx\\pi\/2\\pm\\chi$; (3) no pulse if $\\theta_{\\rm obs}<\\pi\/2-\\chi$ or $\\theta_{\\rm obs}>\\pi\/2+\\chi$.  \n\\figurefive\n\n\\figurefour\nThe pulses of radiation, if emitted too far from the pulsar, would be, however, smeared out. Let the observer be located is at the distance $D$ from the pulsar. In Fig.~\\ref{figfour} photons 1 and 2 are emitted at the same time from the shell passing the radius $r_0$ (for simplicity we assume that the shell is spherical, which is true in the first approximation far from the light cylinder). The first photon arrives at the observer after time: $t_1=(D-r_0)\/c$, whereas the second photon arrives after time $t_2=(D-r_0\\cos\\theta)\/c$. Time time difference between arrival of the photons is therefore \n$\\Delta t=r_0\\left(1-\\cos\\theta\\right)\/c\\approx r_0\/(2\\Gamma^2c)$,  \nwhere for ultrarelativistc flow with the Lorentz factor $\\Gamma\\gg1$, $\\theta\\approx 1\/\\Gamma$. \nIf this time difference is smaller than the time between two subsequent current sheets crossing the sphere $r_0$, the emission from these sheets will appear at the observer as pulsed. At the equator the shells are separated by half of the wavelength of the wind $\\lambda\/2=\\pi r_{\\rm L}$. Thus, the condition for the maximal distance at which the emission emitted still will appear as pulsed can be estimated as:\n\\eqb r_0 < 2\\pi r_{\\rm L} \\Gamma^2 \\eqe\nFor a detailed discussion and the calculations taking into account an exact form of the Archimidean spiral, see \\cite{2016arXiv160804895P}.\n\nIt is worth mentioning that the pulse and off-pulse polarization features, modelled in the wind scenario for optical emission of the Crub pulsar, fit very well with this picture \\cite{2005ApJ...627L..37P}. For more details see also \\cite{2009ASSL..357..421K}. \n\n\nIf the pulsed emission originates in the wind, one expects that the spectrum of the received radiation should directly probe the physics of particle acceleration in the current sheet. \n\n\n\\subsection{Relativistic reconnection and particle acceleration in the wind}\n\\label{problem1}\n\nIn highly conducting plasmas, the magnetic energy can be released by the magnetic field reconfiguration near the singular regions, where the magnetic field changes direction and the localized currents introduce some non-ideal effects, breaking the frozen-in condition (\\ref{frozen-in}). In the collisionless plasmas, these non-ideal effects include the electron kinetic\/inertial terms in the Ohm's law, and microinstabilities, which on the macroscopic level play a role similar to the collisional resistivity. Rearrangement of the field topology occurs locally at so-called X-points. \n\nIn the pulsar wind, magnetic reconnection is expected to set in in the current sheet, which separates the regions of oppositely polarized magnetic field. In highly magnetized plasma in the wind it proceeds in the relativistic regime, because initially the magnetic energy per particle is significantly greater than the particle energy. Thus, when the particles enter the reconnecting layer, they become accelerated to relativistic energies.  \nWhile the steady-state description of relativistic reconnection has been studied analytically\n\\cite{2003ApJ...589..893L,2005MNRAS.358..113L}, numerical simulations (see \\cite{2015SSRv..191..545K} and references therein) demonstrate that this process is essentially time-dependent; current layers are violently unstable to\nthe plasmoid (tearing) instability which leads to the continuous formation and ejection of multiple\nsecondary islands -- plasmoids (see Fig.~\\ref{figcs}). \n\\figuresironi\n\nThe whole process of particle acceleration depends on how the particles penetrate and interact with plasmons, which in turn merge and grow. In the full time-dependent picture the process of particle acceleration can only be captured by fully kinetic PIC simulations \\cite{2001ApJ...562L..63Z,2004PhPl...11.1151J, 2007PhPl...14e6503B, 2014ApJ...783L..21S, 2014PhRvL.113o5005G, 2014ApJ...782..104C}.\nThe results suggest that the particles gain most of the energy when they are accelerated linearly by the electric field at X-points. \nWhen the tension force of the reconnected magnetic field advects the particles away from the X-point, they become trapped in magnetic islands, with a roughly isotropic energy distribution. \nThe larger the size of a plasmoid, the more energetic particles it can confine. This provides a Hillas criterion for the Larmor radius of particles accelerated in the course of relativistic reconnection \\cite{2016MNRAS.462...48S}. It has been reported \\cite{2016MNRAS.462...48S} that the observed highest energy particles have Larmor radius of about 3\\% of the system length (along the reconnection layer).\n\nIntegrated over the whole reconnection region, the energy spectrum of particles is generally observed to be a power law, with a very hard index $s<2$ for magnetisation $\\sigma>10$ \\cite{2014ApJ...783L..21S}, and being the harder the larger is the magnetisation parameter (see Fig.~\\ref{figspectr}). \n\n\\figuredistrib\n\nThe most energetic particles are the ones that interact closer to the centre of the X-point, because they are less prone to be advected away by the reconnected field and, therefore, they stay longer in the acceleration region \\cite{2014ApJ...783L..21S}. \nThe maximum energy to which a power-law particle distribution extends, as well as the shape of the high-energy cutoff in this distribution, both depend on the interplay between the acceleration process and the losses that the particles experience. \nThe particle spectra obtained from PIC simulations by Werner et al. \\cite{2016ApJ...816L...8W} exhibit either exponential or super-exponential behaviour. The exponential cutoff is more appropriate if the maximum energy is limited by the total available magnetic energy per particle, whereas the super-exponential shape better fits the results if the acceleration is limited by the particle escape from the acceleration region. \n\nIn pulsars there are three important timescales that play a role in shaping a particle distribution during the acceleration and radiation processes. The first one is the timescale of acceleration of a particle to a Lorentz factor $\\gamma'$ in the reconnection electric fields $E'=\\tau B'$, where $\\tau$ is the reconnection rate, and the prime denotes the quantities considered in the wind comoving frame:\n\\eqb t'_{\\rm acc}=\\gamma' mc\/(eE'). \\eqe\nThe second timescale is defined by radiative cooling, given by usual synchrotron losses of a particle with a Lorentz factor $\\gamma'$: \n\\eqb t'_{\\rm rad}=6\\pi mc^2\/(\\sigma_{\\rm T}c\\gamma' B'^2). \\eqe \nThe third timescale can be related to the particle escape from the acceleration region, when its gyroradius $r_{\\rm g}=\\gamma'mc\/(eB')$ becomes comparable to the size of the confining region. In this case the highest energy particles are defined by the Hillas criterion \\cite{2016MNRAS.462...48S}.  \n\n\nA particle is able to reach the radiation reaction limit when the timescale of the acceleration is equal to the timescale of radiative cooling in the wind comoving frame; in this case:\n\\eqb \\gamma'_{\\rm rad}=\\left(\\frac{6\\pi e\\tau}{\\sigma_{\\rm T}B'}\\right)^{1\/2}. \\label{gamma_rad} \\eqe\nIn fact, at the X-points the particles can reach even higher energies \\cite{2004PhRvL..92r1101K,2012ApJ...746..148C,2014ApJ...782..104C}, because the radiation reaction force vanishes in the direct field acceleration. As a consequence, the particle distribution is not cut-off sharply at (\\ref{gamma_rad}), but this energy defines where the cut-off starts in the distribution. \n\nThe escape from the acceleration region will occur when the particle gyroradius becomes a fraction $\\zeta$ of a current sheet thickness $\\Delta_{\\rm cs}=2\\pi \\beta c\/\\omega$ (the confining region could be similar to the size of the largest plasmoids, according to the results of simulations, it is a few percent of the system length -- here $\\sim r_{\\rm L}$ \\cite{2016MNRAS.462...48S}), which implies the limiting energy:\n\\eqb \\gamma'_{\\rm esc}=\\frac{\\Gamma \\zeta (\\Delta_{\\rm sc} 2\\pi c) eB'}{mc^2\\omega}. \\label{gamma_esc} \\eqe\n\nIt can be expected, therefore, that in large current sheets with large plasmoids the particles will stay confined and accelerated until they reach the radiation reaction limit. On the other hand, in smaller systems, they will be mainly escaping, before reaching the radiation reaction limit.\n\n\nIt is interesting to consider two cases in which the limiting energy is defined by:\n\\begin{itemize}\n\\item the radiative cooling, and the particle distribution takes a form of a power-law with an exponential cutoff:\n\\eqb n(\\gamma') \\propto \\gamma'^{-s}e^{-\\gamma'\/\\gamma'_{\\rm rad}}, \\label{distr_rad} \\eqe\n\\item the escape from accelerating\/confining region. In this case the particle distribution is cut-off super-exponentially:\n\\eqb n(\\gamma') \\propto \\gamma'^{-s}e^{-\\gamma'^2\/\\gamma'^2_{\\rm esc}}. \\label{distr_esc} \\eqe\n\\end{itemize}\nThis approach is, in fact, phenomenological, because in reality one should solve the continuity equation with a given acceleration process and radiative losses, in order to obtain the exact form of the particle distribution function. This, however, is a difficult problem, given, in particular, the complexity of the particle acceleration at reconnection sites.\n\nThe reconnection is assumed to set in in the wind at some distance $r=\\hat{r}r_{\\rm L}$ from a pulsar. Only a small fraction $\\varepsilon_{\\rm d}$ of the magnetic energy goes into acceleration of the plasma particles so that the process has some efficiency, but it does not alter the dynamics of the wind itself, and the rest of the energy stays either in the magnetic field -- for instance plasmons -- or in a thermal population of particles. \nWith these assumptions the regime of the particle acceleration (radiative cooling\/escape) for a given pulsar depends only on its parameters -- the spin down power and the period, given that $\\gamma'_{\\rm rad}\\sim\\gamma'_{\\rm esc}$ when \n\\eqb \\left(\\frac{\\dot{E}}{10^{38} {\\rm erg\/s}}\\right)^{3\/2}\\left(\\frac{P}{0.01 {\\rm s}}\\right)^{-1}\\sim 0.005 \\left(\\frac{r}{10 r_{\\rm L}}\\right) \\left(\\frac{0.1}{\\zeta}\\right)\\left(\\frac{0.1}{\\Delta_{\\rm cs}}\\right) \\left(\\frac{\\Gamma}{100}\\right)^{3\/2} \\left(\\frac{\\varepsilon_{\\rm d}}{0.01}\\right)^{1\/2}. \\label{line} \\eqe\nThis suggests that only the most energetic and the fastest pulsars exhibit acceleration occurring in the radiative cooling regime \\cite{2015MNRAS.449L..51M}. When plotted on the $P\\dot{P}$-diagram (see Fig.~\\ref{figpp}) the dividing line separates two pulsar populations.\n\n\\figurethree\n\n\n\\subsection{Implications for gamma-ray spectra of pulsars}\n\\label{section:implic}\n\nRadiative signatures of each acceleration regime are determined by the power-law index of the particle distribution, as well as the shape of the cutoff at the highest energies. \nThe most important emission process in the current sheet is the synchrotron radiation, produced by the particles moving in the leftover and\/or adjacent magnetic field. \nIn addition, the SSC spectral component may become significant in the very high energy band, because the same population of leptons that emits synchrotron photons in the sheet, can also upscatter them to higher energies in the IC mechanism. This scattering proceeds in the Thomson regime when the electron Lorentz factor $\\gamma$ and the photon energy normalized to the electron rest mass $\\epsilon$ satisfy the condition $\\gamma \\epsilon\\lesssim 1$. Thus, the soft photon field in the current sheet has to be sufficiently dense in order to make the scattering by the most energetic particles observable. \n\nThe behaviour of the synchrotron spectrum at the highest energies can be calculated analytically from the synchrotron flux integral using the steepest descent method \\cite{2015MNRAS.449L..51M}. The summary of the two considered cases is shown below in Table~\\ref{tableregimes}.\n\n\\begin{table}\n\\centering\n\\normalsize\n\\begin{tabular}{c c c c c }\n\\hline \\vspace{-5pt} \\\\ \nacceleration limit & \\hspace{10pt} & radiation & \\hspace{10pt} & escape \\vspace{7pt}  \\\\ \nparticle distribution & \\hspace{10pt} & $n(\\gamma)=n_0\\gamma^{-s}e^{-\\gamma\/\\gamma_{\\rm rad}}$ & \\hspace{10pt} & $n(\\gamma)=n_0\\gamma^{-s}e^{-\\gamma^2\/\\gamma^2_{\\rm esc}}$ \\vspace{7pt} \\\\ \nasymptote of & \\hspace{10pt} &\n\\multirow{2}{*}{$x^{1.3-(s+0.6)\/3}e^{-1.9\\,x^{1\/3}}$}  & \\hspace{10pt}  &\n\\multirow{2}{*}{$x^{1.3-(s+0.6)\/4}e^{-2\\,x^{1\/2}}$}  \\\\\nsynchrotron spectrum & \\hspace{10pt}  & &  \\hspace{10pt} &\\vspace{7pt} \\\\\nSSC component & & detectable & & weak (not shown) \\vspace{7pt} \\\\ \\hline  \n\\end{tabular}\n\\caption{Summary of two regimes of particle acceleration in the wind.}\n\\label{tableregimes}\n\\end{table}\n\nIn order to model the emission from the wind the following procedure is adopted: (1) calculation of the synchrotron and SSC spectrum of an accelerated particle population (\\ref{distr_rad}) and (\\ref{distr_esc}) in the wind comoving frame, (2) transformation of the emission to the observer frame, (3) comparing the obtained spectrum of emission with the observed pulsed emission from a given pulsar, and finally (4) fitting the model to the data points in order to constrain $\\Gamma$, $\\hat{r}$, $s$ and $\\varepsilon_{\\rm d}$. \n\nTwo scenarios of acceleration, limited either by radiation reaction or the particle escape, are considered for an example of the Crab and Vela pulsars, respectively. The results are shown in Fig.~\\ref{fig1} and Fig.~\\ref{fig2}.\n\n\\figureone\n\\figuretwo\n\nThe emphasis should be put on the fact that fitting the spectrum to the data is not unique. \nIf the dissipation efficiency $\\varepsilon_{\\rm d}$ is small, the density of radiating particles is low, and, therefore, to match the data points, their emission must be generated close to the light cylinder, where the magnetic field is stronger, and, in addition, the radiation has to be boosted by a large Lorentz factor of the wind. \nIf $\\epsilon_{\\rm d}$ is larger, there are more accelerated particles, and, therefore, in order to not overproduce the synchrotron flux, the emission should be produced further from the light cylinder, where the magnetic field is weaker.\n\nInterestingly, the difference between power-law indices $s$ in the two acceleration scenarios is close to 1, as expected for the radiatively cooling population with $s\\sim2$ in comparison to the injected population with a very hard index $s\\sim1$. That would be consistent with the model, in which  the cooling population is observed above the cooling break, and the escaping population is a real injection distribution.\n\nMoreover, each of two considered regimes predicts different SSC component at TeV energies (compare Fig. \\ref{fig1} with Fig. \\ref{fig2}).\nThe maximum energy to which the synchrotron photons can be upscattered is given by\n\\eqb \\epsilon_{\\rm max}\\approx 2\\Gamma \\gamma'_{\\rm rad}mc^2\\approx 3.6\\,\\, {\\rm TeV} \\left(\\frac{\\Gamma}{10}\\right)^{5\/2}\\left(\\frac{P}{0.01 {\\rm s}}\\right)^{1\/2}\\left(\\frac{\\varepsilon_{\\rm d}}{0.01}\\right)^{1\/2}\\left(\\frac{\\dot{E}}{10^{38} {\\rm erg\/s}}\\right)^{-1\/4} \\eqe\n\nThe particle index changes the brightness of the SSC component, because it controls how many soft photons, available for upscattering (in the Thomson regime), are produced. For the steep particle index $s>2$ (like for the Crab), there are more low energy particles that can produce low-energy synchrotron photons; in this case SSC component is brighter. From the currently available data we can constrain the upper limit on this index to be $\\sim2.4$, because steeper indices would overproduce the SSC emission. On the other hand, a fit to the Vela spectrum requires a much harder index $s<2$. In this case the SSC component is\nweaker by several orders of magnitude (not shown in the\nplot), because the energy density resides mainly in the highest energy particles, which do not produce enough low-energy synchrotron photons that can be upscattered to very\nhigh energies.\n\n\nFor the Crab, the key to constrain the model and the pulsar wind parameters is partly given by the recent MAGIC results \\cite{2016A&A...585A.133A}, which seems to prefer emission produced at smaller distances from the light cylinder. More accurate limits will be given by observations from the experiments HESS II and CTA. \n\n\n\\section{MHD and beyond: $\\sigma$-problem and the structure of termination shocks of striped winds}\n\\label{section:shocks}\n\nAccording to MHD models, the energy transported by the wind is released in the nebula downstream of the wind termination shock, where a broad-band  diffuse emission is produced. \nThe Rankine-Hugoniot jump conditions at the shock have been solved by Kennel \\& Coroniti \\cite{1984ApJ...283..694K,1984ApJ...283..710K}. With the assumption that the particles are efficiently accelerated at the shock, they calculated the synchrotron emissivity downstream of the shock as a function of the upstream wind parameters. They found that MHD shocks in highly magnetized flows are very weak, implying that the magnetisation and the flow velocity practically do not change across them. If the flow is initially highly magnetized, it stays so after passing a shock without any significant energy dissipation.\nThus, in order to satisfy the observational constraints on the synchrotron emission, as well as the nonrelativistic expansion of the outer edge of the nebula, the plasma downstream, as well as upstream of the shock, has to be weakly magnetised.   \nHowever,\naccording to theoretical pulsar models, the wind is highly\nmagnetised at its launching point and it stays so up to the termination shock. The reason is that a radial, large-$\\sigma$ MHD flow\ndoes not collimate as it propagates, and therefore it does not\nconvert the energy from the electromagnetic (EM) fields to the kinetic form, arriving at the shock still Poynting dominated.\nThus, it is not clear how and where the wind dissipates its EM energy to\nthe plasma, a puzzle known as the \\lq\\lq$\\sigma$-problem\\rq\\rq. \nIt has been shown that the phase-averaged, non-oscillating component of the fields in the\nwind can dissipate in the bulk of the nebula\n\\cite{2013MNRAS.431L..48P}. Therefore the\n$\\sigma$-problem concerns mainly the dissipation of the wave-like\noscillating component of the fields. \n\nLyubarsky \\cite{2003MNRAS.345..153L} has\nproposed a solution, in which the striped wind, due to interaction with the shock, becomes compressed and the stripes\ndissipate the energy by the driven magnetic reconnection. This scenario provides not only the solution to the $\\sigma$-paradox, but also explains the very hard spectrum of the nebula in the radio band as the synchrotron emission of the particles accelerated at the shock in the course of reconnection. \nPIC simulations \\cite{2007A&A...473..683P,2011ApJ...741...39S} show that this mechanism operates also in 2D and 3D. \nThe only caveat is that in order to reproduce the observed particle spectra,\na high plasma density has to be assumed $\\kappa\\sim10^8$, much higher than usually invoked in pulsar studies. \nIn fact, in\nplasmas of the assumed densities, reconnection in the wind\nwould start much earlier, before it arrives at the shock \\cite{2003ApJ...591..366K}. \n\n\nThe question arises if there is a mechanism dissipating the magnetic energy in the regime of low plasma density.  \nAs pointed out by Usov \\cite{1975Ap&SS..32..375U} and Michel \\cite{1994ApJ...431..397M}, according to the mass continuity\n$r^2 N v_r ={\\rm const}$, the density of current carriers $N$, as measured in the laboratory frame, drops as $N\\propto r^{-2}$,\n faster than the magnetic field $B_{\\phi}\\propto r^{-1}$ that they are required to maintain. \nAs discussed by Coroniti \\cite{1990ApJ...349..538C}, Michel \\cite{1994ApJ...431..397M}, and also Lyubarsky \\&\nKirk \\cite{2001ApJ...547..437L}, \nthe particles whose density decreases are forced to stream with higher\nand higher drift-speed in order to satisfy the Amp\\`ere's law in the flow, but, since the drift speed cannot reach $c$, the anomalous resistivity arises, which in\nturn will trigger the magnetic reconnection. As a result, magnetic energy would be released into\nheating of a plasma, which, however, will perform work\non the flow, leading to its acceleration \\cite{2001ApJ...547..437L}.  \nIn a generic case, therefore, no significant dissipation will occur before the MHD wind arrives at the\nshock, which must be then itself responsible for the wave dissipation.\nOn the other hand, if beyond a certain distance from the pulsar the flow becomes charge-starved, and therefore non-stationary, the displacement current in Amp\\`ere's law cannot be neglected anymore. Usov \\cite{1975Ap&SS..32..375U} and Melrose \\& Melatos \\cite{1996MNRAS.279.1168M} suggested that the MHD framework is not able to describe the whole physics of a diluted flow, which resembles more an EM wave in a plasma than an MHD wave.  \nIn this case one must refer to a more general, two fluid approach. Self consistent, analytic solutions of Maxwell and two-fluid equations have been discussed in numerous studies \\cite{1971PhRvL..27.1342M,1974JPlPh..12..297C,1976JPlPh..15..335K,1984A&A...139..417A,2012ApJ...745..108A,2013ApJ...771...53M}. \n\nThe propagation of EM waves in a plasma is possible only when their frequency $\\omega_{\\rm w}$ exceeds the proper plasma frequency $\\omega_{\\rm pl}$. If the amplitude of an EM is large $a=eB\/(mc\\omega)\\gg1$, these waves can drive particles to relativistic Lorentz factors $\\gamma\\approx a\\gg1$, and the condition of propagation is less restrictive. In the vicinity of the pulsar the propagation condition of a wave with a frequency of the rotating pulsar,\n\\eqb \\omega_{\\rm w}=\\omega_{\\rm pulsar} > \\omega_{\\rm pl}=\\left(\\frac{8\\pi e^2 N}{\\gamma m}\\right)^{1\/2}, \\eqe\nwhere $N=\\kappa N_{\\rm GJ}r_{\\rm L}^2\/r^2$ and $N_{\\rm GJ}$ is given by Eq.~\\ref{nGJ}, translates to\n\\eqb r > r_{\\rm crit}, \\qquad r_{\\rm crit} = \\kappa r_{\\rm L} \\label{propcond} \\eqe\n\nFor most isolated pulsars the shock is located at a sufficiently large distance to allow the existence of the underdense region. \nFor the Crab pulsar $\\kappa\\sim10^6$, and thus the shock located at $r_{\\rm shock}\\sim10^9 r_{\\rm L}$, satisfies the condition (\\ref{propcond}). \nSince the EM waves become new eigenmodes in the system, they can be generated when the rarified wind interacts with the shock, forming a dissipative precursor. This precursor accelerates the particles to relativistic energies and \ntransfers the energy from the fields into the plasma. \n\n\\figureamano\nThe two-fluid simulations \\cite{2013ApJ...770...18A} of this process have indeed demonstrated the formation of such precursors in diluted plasmas (see Fig.~\\ref{figamano}). The authors simulated a circularly polarized MHD wave (magnetic shear), mimicking an MHD wind, launched against a shock in a low density plasma, i.e. whose plasma frequency is smaller than the frequency of the wave. When the wave hits the shock, it is  observed that EM waves are generated in front of the shock, and the system relaxes to an equilibrium in which an electromagnetic precursor is formed, and the Poynting flux of the initial wave is dissipated by the EM modes. As a consequence, a shock becomes essentially unmagnetized and thus potentially a very efficient Fermi accelerator. \nRecent studies \\cite{giacchekirk} of the particle dynamics in the precursor indeed demonstrate its ability to energise particles via wave-particle interactions. In particular, they prove that a significant fraction of such a pre-accelerated particles is reflected upstream and further picked up by the Fermi I-order mechanism. \n\n\nAn interesting possibility arises in pulsar binary systems, where\nthe shocks may switch between different regimes -- when the\nbinary members are close, such that the distance between\nthe pulsar and the shock is smaller than the critical radius $r_{\\rm shock}<r_{\\rm crit}$ (\\ref{propcond}),\nthe shock is in the MHD regime and the energy is dissipated via driven reconnection of stripes. \nWhen the separation between binary members becomes larger, such that the shock is located at a radius $r_{\\rm shock}>r_{\\rm crit}$, a shock can acquire an EM precursor. At this shock regime transition, a sudden appearance of a precursor ahead of the shock should be\naccompanied by enhanced emission in the synchrotron and IC (on the photons from the stellar companion) processes, because the\nprecursor emission is the most efficient close to the cut-off \\cite{2013ApJ...776...40M}.\nThis scenario has been invoked in the eccentric gamma-ray binary B1259-63 \\cite{2013ApJ...776...40M}, where a mysterious flare, observed by the Fermi-LAT \\cite{2011ApJ...736L..11A,2011ApJ...736L..10T,2015ApJ...798L..26T}, challenges the models \\cite{2012ApJ...753..127K,2012ApJ...752L..17K,2013A&A...557A.127D}. \n\n\\section{Summary}\n\nPulsar winds play an important role in the coupling between the central engine -- a rotating magnetized star -- and the nebula, downstream of the pulsar wind shock, where the electromagnetic energy carried by the wind is released into the plasma.  \nThe mechanism leading to the energy release is not clear, however. It seems plausible that it is related to the particular structure of the equatorial component of the wind, which has a form of stripes of opposite magnetic polarity, separated by the current sheet that performs a wavy pattern in space as the pulsar rotates. This structure is believed to be prone to the relativistic magnetic reconnection. \n\nAlthough it has been shown that the reconnection in the wind is not able to dissipate the electromagnetic energy before it reaches the termination shock, it provides a promising scenario for the production of the high-energy pulsed emission. The particles accelerated in the course of reconnection emit synchrotron and SSC photons, whose spectrum can reach TeV energies. Remarkably, recent observations of \nthe Crab \\cite{2016A&A...585A.133A}, as well as the millisecond pulsar J0614--3329 \\cite{2016arXiv160408710X} confirm the detection of the very high energy spectral components, favouring the scenario, in which the gamma-rays originate from outside the light cylinder.\n\nMost of the wind electromagnetic energy must be, however, released somewhere close to the termination shock of the wind. In this case, the stripes become compressed when the wind impacts on the shock, and the driven reconnection annihilates the alternating field component. The nonthermal spectrum of the particles accelerated in the course of this process can be observed, if the plasma carried by the wind is sufficiently dense. \nIn the low plasma density, on the other hand, the electromagnetic fields of the wind behave more like\nan electromagnetic wave instead of a familiar MHD wind. Recent studies show that the shocks in diluted plasmas exhibit dissipative, electromagnetic precursors, which mediate the Poynting flux dissipation and pre-accelerate particles, which are further a subject to the Fermi process. This scenario is expected to be realised in shocks of isolated pulsars, located at large stand-off distances, $r_{\\rm shock}>\\kappar_{\\rm L}$.\n\n\n\\begin{acknowledgement}\nI thank Tomasz Rembiasz and J\\'er\\^{o}me  P\\'etri for helpful comments on the manuscript.\n\\end{acknowledgement} \n\n\\input{author1\/referenc}\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Explicit Expanders}\n Many constructions of expander graphs have been explored in the past (see \\cite{expsurvey}). We will be using a construction that is comparatively easy to describe and implement. We will be considering Cayley graphs, which are obtained from the theory of finite Fields\/Groups. The vertex set of such graphs is a group and the edges are defined by addition operation on the vertices. Also we will be describing expanders that are simple undirected graphs. Given an undirected graph $G=(V,E)$, we can obtain  a bipartite expander $G=(V,V',E)$, by making a copy of the vertices $V$ on the other side and adding edges according to $E$.\n\n\\subsection{Cayley Expander}\nLet $V$ be a group under an operation $+$ and let $H \\subset V$ be a set of generators of the group $V$. The Cayley graph defined by $V,H$ is the graph with vertex set $V$ and the edges $E =\\{(x,x+h) : h \\in H\\}$. For our construction, we will consider the group $\\{0,1\\}^n$ under coordinate-wise XOR operations. For $H \\subset \\{0,1\\}^n$, the Cayley graph defined by $H$ is $|H|$-regular. It is a well-known result in spectral graph theory that there are set of generators $H$, for which the Cayley graph defined by $\\{0,1\\}^n, H$ forms an expander with spectral gap $\\gamma$.\n\\begin{theorem*}[Alon, Roichman, Proposition 4 \\cite{cayley}]\nFor every $\\epsilon$,  there exists explicit $H \\subset \\{0,1\\}^n$ of size $\\leq O(n^2\/\\epsilon^2)$ such that the Cayley graph defined by $\\{0,1\\}^n, H$ is an expander with spectral gap $\\gamma = 1-\\epsilon$.\n\\end{theorem*}\n\n\n\\section{Sensitivity in Expanders}\nIn this section, we will give details about some properties of expander graphs that are required for proving Theorems 1 and 2. Expander graphs are characterized by the spectral gap, which in turn implies the connectivity properties listed in Section 3.4.\n\nThe properties given in Section 3.4 follows from the fact that expanders have a spectral gap (for proofs see \\cite{pseudo}). For the sensitivity proofs, we need the following lemmas. The first lemma states that having a spectral gap, implies that all set of vertices of size $\\leq n\/2$ expands.\n\\begin{lemma}[Theorem 4.6 \\cite{pseudo}] \n\\label{lab:spec}\nIf $G=(V,E)$ is an expander with spectral gap of $\\gamma$ then for every subset $S\\subset V$ of size $\\leq |V|\/2$, the size of the set of neighbors $|N(S)| \\geq (1+\\gamma) |S|$.\n\\end{lemma}\nThe next lemma is known as the expander mixing lemma, deals with the uniform connectivity properties of the expander.\n\\begin{lemma}[Lemma 4.15 \\cite{pseudo}]\n\\label{lab:exp-mix-lem}\nIf $G=(V,E)$ is an $D$-regular expander with spectral gap of $\\gamma$ then for every subset $S,T\\subset V$,\n$$\\left| E(S,T) - D \\cdot |S| \\cdot |T| \/ n  \\right| \\leq (1-\\gamma)\\sqrt{|S| \\cdot |T|}$$\nwhere $E(S,T)$ is the set of edges from $S$ to $T$.\n\\end{lemma}\n\n\n\nIn this section, we prove Theorem 1 and Theorem 2 using the properties described above along with the connectivity properties defined in Section 3.1 of the paper.\n\\begin{theorem}[Sensitivity of X-Nets]\\label{thm:conn}\nLet $n$ be the number of input as well as output nodes in the network and $G_1,G_2,\\cdots, G_t$ be $D$ regular bipartite expander graphs with $n$ nodes on both sides. Then  \nevery output neuron is sensitive to every input in a Deep X-Linear Network defined by $G_i$'s with depth $t = O( \\log n)$.\n\\end{theorem}\n\\begin{proof}\nFor showing sensitivity, we show that for every pair of input and output $(u,v)$,  there is a path in the X-Net with the $i^{th}$ edge from the $i^{th}$ graph. \nWe use the expansion property of the expander graphs. Let $N_1(u)$ be the set of neighbors of $u$ in $G_1$ and $N_i(u)$ be the set of neighbors of $N_{i-1}(u)$ in the graph $G_i$. Since each of the graphs are expanding $|N_{i}(u)| \\geq (1+\\gamma) \\times |N_{i-1}(u)|$, using Lemma \\ref{lab:spec}. Since $(1+\\gamma) > 1$, we can obtain that for $i = O(\\log n)$, $|N_i(u)| \\geq n\/2$. Similarly we can start from $v$ and define $N_1(v)$ as the set of neighbors of $v$ in $G_t$ and $N_i(v)$ be the set of neighbors of $N_{i-1}(v)$ in the graph $G_{n-i -1}$. Due to the expansion property, for $j = O(\\log n)$, $|N_j(v)| \\geq n\/2$. Now we choose $t = i + j = O(\\log n)$ so that the $i^{th}$ graph from $1$ is the same as $j+1^{th}$ graph from $t$. Then we will have that $N_i(u) \\cap N_j(v) \\neq \\phi$. That is there is some vertex in the $i^{th}$ graph that is both connected to $u$ and $v$, which implies that there is a path from $u$ to $v$.\n\\end{proof}\n\n\\begin{theorem}[Mixing in Deep Expander Networks]\nLet $n$ be the number of input as well as output nodes in the network and $G_1,G_2,\\cdots, G_t$ be $D$ regular bipartite expander graphs with $n$ nodes on both sides. Let $S,T$ be subsets of input and output nodes in the X-Linear Network defined by the $G_i$'s. The number of paths between $S$ and $T$ is $\\approx D|S||T|\/n$\n\\end{theorem}\n\\begin{proof}\nFirst we prove the theorem for the case when $G_1 = G_2 \\cdots = G_t = G$. Note that for $t=1$, the theorem is same as the expander mixing lemma (Lemma \\ref{lab:exp-mix-lem}). For $t>1$, consider the graph of $t$ length paths denoted by $G^t$. An edge in this graph denotes that there is a path of length $t$ in $G$. Observe that the adjacency matrix of $G^t$ is given by the $t$th power of the adjacency matrix of $G$ and hence the spectral gap of $G^t$, $\\gamma_t = \\gamma^t$. Now applying the expander mixing lemma on $G^t$ (Lemma \\ref{lab:exp-mix-lem}), proves the theorem.\n\nFor the case when the graph $G_i$'s are different, let $\\gamma_{\\min}$ be the minimal spectral gap among the graphs. Since all the graphs are $D$-regular, the largest eigenvector is the all ones vector with eigenvalue $D$ for all the graphs. The eigenvector corresponding to second largest eigenvalue can be different for each graph, but they are orthogonal to the all ones vector. Hence the spectral gap of the $t$ length path graph $G$ is at least $\\gamma_{\\min}^t$. Finally we apply the expander mixing lemma on $G^t$ to prove the theorem.\n\\end{proof}\n\n\n\n\\section{Model Details}\n\nWe discuss the model structures in detail, along with tabulated  values of size and flops of various architectures presented as graphs in the main paper. \n\n\\begin{table}[!tbh]\n\\centering\n\\resizebox{0.6\\columnwidth}{!}{\n\\begin{tabular}{|l|c|c|c|}\n\\hline\n{\\bf AlexNet} & {\\bf Filter Shape} & {\\bf Filter Shape} & {\\bf Filter Shape} \\\\\n\\hline\n &  & ({\\bf X-AlexNet-1}) & ({\\bf X-AlexNet-2})\\\\\n \\hline\nConv2d & 64 x 3 x 11 x 11 & 64 x 3 x 11 x 11 & 64 x 3 x 11 x 11\\\\\nConv2d & 192 x 64 x 5 x 5 & 192 x 64 x 5 x 5 & 192 x 64 x 5 x 5\\\\\nConv2d & 384 x 192 x 3 x 3 & 384 x 192 x 3 x 3 & 384 x 192 x 3 x 3\\\\\nConv2d & 256 x 384 x 3 x 3 & 256 x 384 x 3 x 3 & 256 x 384 x 3 x 3\\\\\nConv2d & 256 x 256 x 3 x 3 & 256 x 256 x 3 x 3 & 256 x 256 x 3 x 3\\\\\n\\hline\nLinear & 9216 x 4096 & 1024 x 4096 & 512 x 4096\\\\\nLinear & 4096 x 4096 & 512 x 4096 & 512 x 4096\\\\\nLinear & 4096 x 1000 & 1024 x 1000 & 1024 x 1000\\\\\n\\hline\n\\end{tabular}}\n\\caption{Filter sizes for the AlexNet model. Notice the filter sizes of the linear layers of the original model has $|V|\\times |U|$ parameters, whereas X-AlexNet models have $|V|\\times D$ parameters. Note that $D << |U|$ as stated in Section 3.2. Hence, expander graphs model connections in linear layers (X-Linear) effectively.}\n\\label{tab:vgg}\n\\end{table}\n\\begin{table}[!tbh]\n\\centering\n\\resizebox{0.6\\columnwidth}{!}{\n\\begin{tabular}{|l|c|c|c|}\n\\hline\n{\\bf VGG} & {\\bf Filter Shape} & {\\bf Filter Shape} & {\\bf Filter Shape}\\\\\n\\hline\n &  & ({\\bf X-VGG16-1}) & ({\\bf X-VGG16-2})\\\\\n \\hline\nConv2d & 64 x 3 x 3 x 3 & 64 x 3 x 3 x 3 & 64 x 3 x 3 x 3\\\\\nConv2d & 64 x 64 x 3 x 3 & 64 x 64 x 3 x 3 & 64 x 64 x 3 x 3\\\\\nConv2d & 128 x 64 x 3 x 3 & 128 x 64 x 3 x 3 & 128 x 64 x 3 x 3\\\\\nConv2d & 128 x 128 x 3 x 3 & 128 x 64 x 3 x 3 & 128 x 64 x 3 x 3\\\\\nConv2d & 256 x 128 x 3 x 3 & 256 x 32 x 3 x 3 & 256 x 16 x 3 x 3\\\\\nConv2d & 256 x 256 x 3 x 3 & 256 x 32 x 3 x 3 & 256 x 16 x 3 x 3\\\\\nConv2d & 256 x 256 x 3 x 3 & 256 x 32 x 3 x 3 & 256 x 16 x 3 x 3\\\\\nConv2d & 512 x 256 x 3 x 3 & 512 x 32 x 3 x 3 & 512 x 16 x 3 x 3\\\\\nConv2d & 512 x 512 x 3 x 3 & 512 x 32 x 3 x 3 & 512 x 16 x 3 x 3\\\\\nConv2d & 512 x 512 x 3 x 3 & 512 x 32 x 3 x 3 & 512 x 16 x 3 x 3\\\\\nConv2d & 512 x 512 x 3 x 3 & 512 x 32 x 3 x 3 & 512 x 16 x 3 x 3\\\\\nConv2d & 512 x 512 x 3 x 3 & 512 x 32 x 3 x 3 & 512 x 16 x 3 x 3\\\\\nConv2d & 512 x 512 x 3 x 3 & 512 x 32 x 3 x 3 & 512 x 16 x 3 x 3\\\\\n\\hline\nLinear & 512 x 512 & 128 x 512 & 128 x 512\\\\\nLinear & 512 x 10 & 512 x 10 & 512 x 10\\\\\n\\hline\n\\end{tabular}}\n\\caption{Filter sizes for the VGG-16 model on CIFAR-10 dataset. The filter sizes given are $|V|\\times |U| \\times c \\times c$ in original VGG network, $|V|\\times D \\times c \\times c$ in our X-VGG16 models. Note that $D << |U|$ as stated in Section 3.2. Hence, expander graphs model connections in Convolutional layers (X-Conv) effectively.}\n\\label{tab:alexnet}\n\\end{table}\n\n\\begin{table}[!tbh]\n\\centering\n\\resizebox{0.6\\columnwidth}{!}{\n\\begin{tabular}{|l|c|c|c|}\n\\hline\n{\\bf Model} & {\\bf Accuracy} & {\\bf \\#Params } & {\\bf \\#FLOPs}\\\\\n\\hline\nCIFAR10 &  & {\\bf(in M)} & {\\bf(in 100M)}\\\\\n\\hline\nX-DenseNetBC-2-40-24 & {\\bf 94.83\\%} & {\\bf 0.4M} & {\\bf 1.44}\\\\\nDenseNetBC-40-24 & 94.79\\% & 0.7M & 2.88\\\\\nX-DenseNetBC-2-40-36 & 94.98\\% & 0.75M & 3.24\\\\\nX-DenseNetBC-2-40-48 & {\\bf 95.48\\%} & {\\bf 1.4M} & {\\bf 5.75}\\\\\nDenseNetBC-40-36 & 95.26\\% & 1.5M & 6.47\\\\\nX-DenseNetBC-2-40-60 & {\\bf 95.71\\%} & {\\bf 2.15M} & {\\bf 8.98}\\\\\nDenseNetBC-40-48 & 95.64\\% & 2.8M & 11.50\\\\\nDenseNetBC-40-60 & 95.91\\% & 4.3M & 17.96\\\\\n\\hline\nCIFAR100 &  &  &\\\\\n\\hline\nX-DenseNetBC-2-40-24 & 74.37\\% & 0.4M & 1.44\\\\\nDenseNetBC-40-24 & 76.05\\% & 0.7M & 2.88\\\\\nX-DenseNetBC-2-40-36 & {\\bf 76.69\\%} & {\\bf 0.75M} & {\\bf 3.24}\\\\\nDenseNetBC-40-36 & 77.84\\% & 1.5M & 6.47\\\\\nX-DenseNetBC-2-40-60 & 78.53\\% & 2.15M & 8.98\\\\\nX-DenseNetBC-4-70-60 & {\\bf 79.56\\%} & {\\bf 2.6M} & {\\bf 10.26}\\\\\nDenseNetBC-40-48 & 79.03\\% & 2.8M & 11.50\\\\\nDenseNetBC-40-60 & 79.87\\% & 4.3M & 17.96\\\\\nX-DenseNetBC-2-70-60 & 80.89\\% & 5.18M & 20.52\\\\\nDenseNetBC-70-60 & 81.28\\% & 10.36M & 41.05\\\\\n\\hline\n\\end{tabular}}\n\\caption{Results obtained on the state-of-the-art models on CIFAR-10 and CIFAR-100 datasets, ordered by FLOPs per model. X-Nets give significantly better accuracies with corresponding DenseNet models in the same limited computational budget and correspondingly significant parameter and FLOP reduction for models with similar accuracy.}\n\\label{tab:cifar}\n\\end{table}\n\\subsection{Filter structure of AlexNet and VGG}\n\nIn Tables \\ref{tab:vgg} and \\ref{tab:alexnet}, the detailed layer-wise filter structure is tabulated as stated in Section 5.3 of the paper. We compare the sizes of input channels between the filters, and show that with X-Conv and X-Linear layers, we can train models effectively even with upto 32x and 18x times smaller filters in input dimension in VGG16 and AlexNet models respectively. Hence, modeling connections as weighted adjacency matrix of an Expander graph is an effective method to model connections between neurons, producing highly efficient X-Nets.\n\n\\subsection{Results}\nAs stated in Section 5.2, Tables \\ref{tab:cifar} and \\ref{tab:imagenet} presented below give the detailed accuracy, parameters and FLOPs of models displayed in the Figure 3 in the paper. Details of other models are also provided in the same table, which could not be displayed in the paper due to lack of space.\n\nTable \\ref{tab:cifar} displays the performance of X-DenseNet models on the CIFAR-10 and CIFAR-100 datasets. If we compare models that have similar number of parameters, we achieve around 0.2\\% and 0.6\\% increase in accuracy over DenseNet-BC models on CIFAR-10 and CIFAR-100 datasets respectively. In the same manner, we can achieve upto using only two-thirds of the parameter and runtime cost cost respectively, keeping accuracy constant on CIFAR-10 and CIFAR-100 datasets as stated in the paper.\n\n\\begin{table}[!tbh]\n\\centering\n\\resizebox{0.6\\columnwidth}{!}{\n\\begin{tabular}{|l|c|c|c|}\n\\hline\n{\\bf Model} & {\\bf Accuracy} & {\\bf \\#Params } & {\\bf \\#FLOPs}\\\\\n\\hline\nResNet &  & {\\bf(in M)} & {\\bf(in 100M)}\\\\\n\\hline\nX-ResNet-2-34 & 69.23\\% & 11M & 35\\\\\nX-ResNet-2-50 & {\\bf 72.85\\%} & {\\bf 13M} & {\\bf 40}\\\\\nResNet-34 & 71.66\\% & 22M & 70\\\\\nX-ResNet-2-101 & {\\bf 74.87\\%} & {\\bf 22.5M} & {\\bf 80}\\\\\nResNet-50 & 74.46\\% & 26M & 80\\\\\nResNet-101 & 75.87\\% & 45M & 160\\\\\n\\hline\nDenseNetBC &  &  & \\\\\n\\hline\nMobileNet \\cite{howard2017mobilenets} & 70.6\\% & 4.2M & 5.7 \\footnote{These are reported as mult-add operations} \\\\\nShuffleNet \\cite{zhang2017shufflenet} & 70.9\\% & 5M & 5.3 \\footnote{These are reported as mult-add operations} \\\\\nX-DenseNetBC-2-121 & {\\bf 70.5\\%} & {\\bf 4M} & 28\\\\\nX-DenseNetBC-2-169 & 71.7\\% & 7M & 33\\\\\nX-DenseNetBC-2-201 & 72.5\\% & 10M & 43\\\\\nX-DenseNetBC-2-161 & {\\bf 74.3\\%} & 14.3M & {\\bf 55}\\\\\nDenseNetBC-121 & 73.3\\% & 8M & 55\\\\\nDenseNetBC-169 & 74.8\\% & 14M & 65\\\\\nDenseNetBC-201 & 75.6\\% & 20M & 85\\\\\nDenseNetBC-161 & 76.3\\% & 28.5M & 110\\\\\n\\hline\n\\end{tabular}}\n\\caption{Results obtained on the state-of-the-art models on ImageNet dataset, ordered by FLOPs. We also observe that X-DenseNetBC models outperform ResNet and X-ResNet models in both compression, parameters and FLOPs and achieve comparable accuracies with the highly efficient MobileNets and ShuffleNets in the same parameter budget, albeit with much higher FLOPs due to architectural constraints.}\n\\label{tab:imagenet}\n\\end{table}\n\nSimilarly, Table \\ref{tab:imagenet} displays the performance of X-ResNet and X-DenseNet models on the ImageNet datasets. We can observe that we achieve around 3.2\\% and 1\\% increase in accuracy over ResNet and DenseNet-BC models in the same computational budget. Also, we can observe that we require approximately 15\\% less FLOPs for achieving similar accuracies over DenseNet models and 15\\% less parameters for achieving similar accuracies over the ResNet model respectively as stated in the paper. We also observe that X-DenseNetBC models outperform ResNet and X-ResNet models in both compression, parameters and FLOPs and achieve comparable accuracies with MobileNets \\cite{howard2017mobilenets} and ShuffleNets \\cite{zhang2017shufflenet} in the same parameter budget, albeit with much higher computational cost due to architectural constraints. \n\n\\begin{table}[!tbh]\n\\centering\n\\resizebox{0.6\\columnwidth}{!}{\n\\begin{tabular}{|l|c|c|c|}\n\\hline\n{\\bf Model} & {\\bf Accuracy} & {\\bf \\#Params } & {\\bf \\#FLOPs}\\\\\n\\hline\nWider & & {\\bf(in M)} & {\\bf(in 100M)} \\\\\n\\hline\nDenseNetBC-40-60 & 79.87\\% & 4.3M & 17.96\\\\\nX-DenseNetBC-2-40-60 & 78.53\\% & 2.15M & 8.98\\\\\nX-DenseNetBC-4-40-60 & 77.54\\% & 1.08M & 4.49\\\\\nX-DenseNetBC-8-40-60 & 75.29\\% & 0.54M & 2.24\\\\\nX-DenseNetBC-16-40-60 & 74.44\\% & 0.27M & 1.12\\\\\n\\hline\nDenseNetBC-40-100 & 80.9\\% & 11.85M & 49.85\\\\\nX-DenseNetBC-4-40-100 & 78.87\\% & 2.9M & 12.46\\\\\nX-DenseNetBC-8-40-100 & 77.75\\% & 1.48M & 6.23\\\\\nX-DenseNetBC-16-40-100 & 76.2\\% & 0.74M & 3.12\\\\\n\\hline\nDenseNetBC-40-200 & 81.62\\% & 47.19M & 199.28\\\\\nX-DenseNetBC-4-40-200 & 80.66\\% & 11.79M & 49.82\\\\\nX-DenseNetBC-10-40-200 & 79.46\\% & 4.71M & 19.93\\\\\nX-DenseNetBC-20-40-200 & 78.33\\% & 2.3M & 9.96\\\\\nX-DenseNetBC-30-40-200 & 77.29\\% & 1.6M & 6.64\\\\\nX-DenseNetBC-50-40-200 & 75.7\\% & 0.9M & 3.99\\\\\nX-DenseNetBC-80-40-200 & 73.26\\% & 0.5M & 2.49\\\\\n\\hline\nDeeper &  &  &\\\\\n\\hline\nDenseNetBC-40-60 & 79.87\\% & 4.3M & 17.96\\\\\nX-DenseNetBC-2-40-60 & 78.53\\% & 2.15M & 8.98\\\\\nX-DenseNetBC-8-40-60 & 77.54\\% & 0.54M & 2.24\\\\\nX-DenseNetBC-16-40-60 & 75.29\\% & 0.27M & 1.12\\\\\n\\hline\nDenseNetBC-58-60 & 80.79\\% & 7.66M & 30.96\\\\\nX-DenseNetBC-2-58-60 & 80.29\\% & 3.83M & 15.48\\\\\nX-DenseNetBC-4-58-60 & 78.74\\% & 1.9M & 7.74\\\\\nX-DenseNetBC-8-58-60 & 77.98\\% & 0.95M & 3.87\\\\\nX-DenseNetBC-16-58-60 & 75.87\\% & 0.47M & 1.93\\\\\n\\hline\nDenseNetBC-70-60 & 81.28\\% & 10.36M & 41.05\\\\\nX-DenseNetBC-2-70-60 & 80.89\\% & 5.18M & 20.52\\\\\nX-DenseNetBC-4-70-60 & 79.56\\% & 2.6M & 10.26\\\\\nX-DenseNetBC-8-70-60 & 77.48\\% & 1.3M & 5.13\\\\\nX-DenseNetBC-16-70-60 & 77.23\\% & 0.65M & 2.57\\\\\n\\hline\n\\end{tabular}}\n\\caption{We display accuracies, parameters and FLOPs of all the wider and deeper networks on CIFAR-100 listed in increasing compression order. This proves that efficiently designing layers like X-Conv and X-Linear allows us to train wider and deeper networks frugally.}\n\\label{tab:ultranets}\n\\end{table}\n\nTable \\ref{tab:ultranets} displays accuracies, parameters and FLOPs of all the wider and deeper networks trained on CIFAR-100 dataset as discussed in Section 5.4. They are listed in increasing compression order from DensenetBC (1x) to the highest compressed X-DenseNet. The figure indicates that Expander Graphs modeling can scale up to high compression ratios without drastic drops in accuracies, enabling us to train deeper and wider networks retaining similar FLOPs and parameters effectively. We believe this modeling can open up a interesting exploration of training significantly deeper and wider range of models, unlike the current compression techniques as X-Nets are highly compressed networks since definition. \n\n\\subsection{Experimental Details}\n\\label{sec:expdetails}\nTo ensure better reproducibility, we used the same hyper-parameters, models, training schedules and dataset structures from the official \\href{https:\/\/github.com\/pytorch\/examples\/blob\/master\/imagenet\/main.py}{PyTorch repository} for ResNet and DenseNet-BC ImageNet experiments. Similarly, we followed the \\href{https:\/\/github.com\/marvis\/pytorch-mobilenet}{pytorch-mobilenet repository} including all hyperparameters for all the mobilenet experiments. Likewise, we followed the \\href{https:\/\/github.com\/andreasveit\/densenet-pytorch\/}{densenet-pytorch repository} by Andreas Veit for all DenseNet-BC experiments on CIFAR-10\/100 datasets, and the \\href{https:\/\/github.com\/chengyangfu\/pytorch-vgg-cifar10}{pytorch-vgg-cifar10} repository by Cheng-Yang Fu for VGG16 experiments on the CIFAR-10 dataset. We deleted a linear layer and remove dropouts from the VGG16 model. There were two differences between our code and the PyTorch ImageNet repository. Our ImageNet training code used a compressed version of Imagenet with images resized to 256x256, we used a batchsize of 128 for all experiments, hence typically results in slightly lower accuracies. Our AlexNet model used BatchNorm layers to stabilize training, with a batchsize of 384.\n\nWe trained all the models on a setup consisting of 10 Intel Xeon E5-2640 cores and 2 GeForce GTX 1080 Ti GPUs. All networks are trained from scratch. We modeled connections as  random expanders for all experiments. The X-Conv layers did not have a bias. No dropouts were used. Additional details regarding all models, accuracies, and numbers of parameters and FLOPs is tabulated in the supplementary material for reference due to lack of space.  \n\n\\par\nAll models in the plot were trained from the ImageNet code available on Pytorch Official repository including the original DenseNet and ResNet models. Note that this training code is common for all models in the repository and not fine-tuned to any specific model like ResNet or DenseNet. Hence the accuracies we report is slightly lower that those reported using model-specific training code. However, note that we have used the same code for training both the original models and the expander versions of these models. Hence the comparison is a fair one. Same is true with the  AlexNet model. Note that we use the original AlexNet architecture, and not CaffeNet. \nIn contrast, training with fine-tuned code is expected to give a 1-2\\% improvement over MobileNets in Table \\ref{tab:imagenet}. Training schedules suited to X-Nets, hyper-parameter tuning, Activation layers tailored to X-Nets could further improve the accuracies. Overall, we believe that investigating training methods for X-Nets has a lot of potential for improving their performance.\n\n\\section{Efficient Algorithms}\\label{sec:implementation}\n \nIn this section, we present efficient algorithms of X-Conv layers. Our algorithms achieve speedups and save memory in the training as well as the inference phase. This enables one to experiment with significantly wider and deeper networks given  memory and runtime constraints.\nWe exploit the structured sparsity of expander graphs to design fast algorithms.  We propose two methods of training X-Nets, both requiring substantially less memory and computational cost than their vanilla counterparts: \\\\1) Using Sparse Representations \\\\2) Expander-Specific Fast Algorithms.\n\n\\subsection{Using Sparse Representation}\n\nThe adjacency matrices of expander graphs are highly sparse for $D << n$. Hence, we can initialize a sparse matrix with non-zero entries corresponding to the edges of the expander graphs. Unlike most pruning techniques, the sparse connections are determined before training phase, and stay fixed. Dense-Sparse convolutions are easy to implement, and are supported by most deep learning libraries. CNN libraries like Cuda-convnet\\cite{cudaconvnet} support such random sparse convolution algorithms.\n\n \\begin{algorithm}[t]\n \\textbf{Algorithm 1: } Fast Algorithm for Convolutions in X-Conv Layer\\\\\n \\begin{algorithmic}[1]\n \\State For every vertex $v \\in \\{1,\\cdots, n\\}$, let $N(v,i)$ denote the $i$th neighbor of $v$ ($i \\in \\{1,\\cdots, D\\}$).\n \\State Let $K_v$ be the $c\\times c \\times D \\times 1$ sized kernel associated with the $v$th output channel.\n \\State Let $O_v[x,y]$ be the output value of the $v$th channel at the position $x,y$.\n \\For{$v$= 1 to n}\n     \\State $O_v[x,y] = K_v * \\textit{Mask}_{N(v,1),\\cdots N(v,D)}(I)[x,y]$.\n \\EndFor\n \\end{algorithmic}\n \\label{alg:cnnalgo}\n \\end{algorithm}\n \n \n\\subsection{X-Net based Fast Dense Convolution}\nNext, we present fast algorithms that exploit the sparsity of expander graphs.\\\\\n\n{\\bf X-Conv:} In an X-Conv layer, every output channel is only sensitive to $out$ rom input channels. We propose to use a mask to select $D$ channels of the input, and then convolve with a $c \\times c \\times D \\times 1$ kernel, obtaining a single channel per filter in the output. The mask is obtained by choosing D samples uniformly (without replacement) from the set $\\{1,\\cdots N\\}$, where $N$ is the number of input channels. The mask value is $1$ for each of the selected $D$ channels and $0$ for others (see Algorithm \\ref{alg:cnnalgo}). This is illustrated in Figure \\ref{fig:efficientmatrix}. There has been recent work about fast CUDA implementations called Block-Sparse GPU Kernels\\cite{blocksparse}, which can implement this algorithm efficiently.\n\n\\section{Conclusion}\nWe proposed a new network layer architecture for deep networks using expander graphs that give strong theoretical guarantees on connectivity. The resulting architecture (X-Net) is shown to be highly efficient in terms of both computational requirements and model size. In addition to being compact and computationally efficient, the connectivity properties of the network allow us to achieve significant improvements over the state-of-the-art architectures in performance on a parameter or run-time budget. In short, we show that the use of principled approaches that sparsify a model while maintaining global information flows can help in developing efficient deep networks.\n\nTo the best of our knowledge, this is the first attempt at using theoretical results from graph theory in modeling connectivity to improve deep network architectures. We believe that the field of deep networks can gain significantly from other similar explorations.\n\n\\section{Approach}\\label{sec:approach}\nRecent breakthroughs in CNN architectures like ResNet\\cite{he2015convolutional} and DenseNet-BC\\cite{huang2016densely} are ideas  based on increasing connectivity, which resulted in better performance trade-offs. These works suggest that connectivity is an important property for improving the performance of deep CNNs. In that vein, we investigate ways of preserving connectivity between neurons while significantly sparsifying the connections between them. Such networks are expected to preserve accuracy (due to connectivity) while being runtime efficient (due to the sparsity). We empirically demonstrate this in the later sections.\n\n\\subsection{Graphs and Deep CNNs}\n\nWe model the connections between neurons as graphs. This enables us to leverage well-studied concepts from Graph Theory like Expander Graphs. Now, we proceed to formally describe the connection between graphs and Deep CNNs. \n\n{\\bf Linear Layer defined by a Graph:} Given a bipartite graph $G$ with vertices $U, V$, the Linear layer defined by $G$, is a layer with $|U|$ input neurons, $|V|$ output neurons and each output neuron $v \\in V$ is only connected to the neighbors given by $G$. Let the graph $G$ be sparse, having only $M$ edges. Then this layer has only $M$ parameters as compared to $|V|\\times |U|$, which is the size of typical linear layers. \n{\\bf Convolutional Layer defined by a Graph:} Let a Convolutional layer be defined as a bipartite graph $G$ with vertices $U,V$ and a window size of $c\\times c$. This layer takes a 3D input with $|U|$ channels and produces a 3D output with $|V|$ channels. The output channel corresponding to a vertex $v \\in V$ is computed only using the input channels corresponding the the neighbors of $v$. Let $G$ be sparse, having only $M$ edges. Hence the kernel of this convolutional layer has $M \\times c \\times c$ parameters as compared to $|V|\\times |U| \\times c \\times c$, which is the number of parameters in a vanilla CNN layer.\n\n\\subsection{Sparse Random Graphs}\n\nWe want to constrain  our convolutional layers to form a sparse graph $G$. Without any prior knowledge of the data distribution, we take inspiration from randomized algorithms and propose choosing the neighbours of every output neuron\/channel uniformly and independently at random from the set of all its input channels. It is known that a graph $G$ obtained in this way belongs to a well-studied category of graphs called Expander Graphs, known to be sparse but well connected. \n\n{\\bf Expander Graph:} A bipartite expander with degree $D$ and spectral gap $\\gamma$, is a bipartite graph $G=(U,V,E)$ ($E$ is the set of edges,  $ E \\subseteq U\\times V$) in which: \n\n{\\bf1.) Sparsity:} Every vertex in $V$ has only $D$ neighbors in $U$. We  will be using constructions with $D << |U|$. Hence the number of edges is only $D \\times |V|$ as compared to $|U| \\times |V|$ in a dense graph.\n\n {\\bf 2.) Spectral Gap:} The eigenvalue with  the second largest absolute value $\\lambda$ of the adjacency matrix is bounded away from D (the largest eigenvalue). Formally $1-\\lambda\/D \\geq \\gamma.$\n\n{\\bf Random expanders:} A random bipartite expander of degree $D$ on the two vertex sets $U, V$, is a graph in which for every vertex $v \\in V$, the $D$ neighbors are chosen independently and uniformly from $U$. It is a well-known result in graph theory that such graphs have a large spectral gap (\\cite{pseudo}).\nSimilar to random expanders, there exist several explicit expander constructions. More details about explicit expanders can be found in the supplementary section.\nWe now proceed to give constructions of deep networks that have connections defined by an expander graph.\n\n{\\bf Expander Linear Layer (X-Linear):} The Expander Linear (X-Linear) layer is a layer defined by a random bipartite expander $G$ with degree $D$. The expander graphs that we use have values of $D << |U|$, while having an expansion factor of $K \\approx D$, which ensures that the layer still has good expressive power.\n\n{\\bf Expander Convolutional Layer (X-Conv):} The Expander Convolutional (X-Conv) layer is a convolutional layer defined by a random bipartite expander graph $G$ with degree $D$, where $D << |U|$. \n\n{\\bf Deep Expander Networks (X-Nets):} Given expander graphs \n$$G_1 = (V_0,V_1,E_1), G_2 = (V_1,V_2,E_2), \\cdots, G_t = (V_{t-1},V_t,E_t)$$, we define the Deep Expander Convolutional Network (Convolutional X-Net or simply X-Net) as a $t$ layer deep network in which the convolutional layers are replaced by X-Conv layers and linear layers are replaced by X-Linear layers defined by the corresponding graphs. \n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[scale=0.15]{images\/Expander.png}\n\n\\caption{The proposed fast convolution algorithm for X-Conv layer. We represent all the non-zero filters in the weight matrix of the X-Conv layer as a compressed dense matrix of $D$ channels. The algorithm starts by selecting $D$ channels from input (with replacement) using a mask created while initializing the model. The output is computed by convolving these selected channels with the compressed weight matrices.}\n\\label{fig:efficientmatrix}\n\n\\end{figure*}\n\n\\subsection{Measures of Connectivity}\n\nIn this subsection, we describe some connectivity properties of Expander graphs (see \\cite{pseudo}, for the proofs). These will be used to prove the properties of sensitivity and mixing of random walks in X-Nets .\n\n{\\bf Expansion:} For every subset $S \\subseteq V$ of size $\\leq \\alpha |V|$ ($\\alpha \\in (0,1)$ depends on the construction), let $N(S)$ be the set of neighbors. Then $|N(S)| \\geq K |S|$ for $K\\approx D$. That is, the neighbors of the vertices in $S$ are almost distinct. It is known that random expanders have expansion factor $K \\approx D$ (see Theorem 4.4 in \\cite{pseudo}).\n\n{\\bf Small Diameter:} The diameter of a graph is the length of the longest path among all shortest paths. If $G(U,V,E)$ is a $D$-regular expander with expansion factor $K > 1$ and diameter $d$, then  $d\\leq O(\\log n )$. This bound on the diameter implies that for any pair of vertices, there is a path of length $O(\\log n)$ in the graph.\n\n{\\bf Mixing of Random Walks:} Random walks in the graph quickly converge to the uniform distribution over nodes of the graph. If we start from any vertex and keep moving to a random neighbor, in $O(\\log n)$ steps the distribution will be close to uniform over the set of vertices.\n\n\\subsection{Sensitivity of X-Nets}\n\nX-Nets have multiple layers, each of which have connections derived from an expander graph. We can guarantee that the output nodes in such a network are sensitive to all the input nodes.\n\n\\begin{theorem}[Sensitivity of X-Nets]\\label{thm:conn}\nLet $n$ be the number of input as well as output nodes in the network and $G_1,G_2,\\cdots, G_t$ be $D$ regular bipartite expander graphs with $n$ nodes on both sides. Then  \nevery output neuron is sensitive to every input in a Deep X-Net defined by $G_i$'s with depth $t = O( \\log n)$.\n\\end{theorem}\n\\begin{proof}\nFor every pair of input and output $(u,v)$, we show that there is a path in the X-Net. The proof is essentially related to the the fact that expander graphs have diameter $O(\\log n)$. A detailed proof can be found in the supplementary material.\n\\end{proof}\n\nNext, we show a much stronger connectivity property known as mixing for the X-Nets. The theorem essentially says that the number of edges between subsets of input and output nodes is proportional to the product of their sizes. This result implies that the connectivity properties are uniform and rich across all nodes as well as subsets of nodes of the same size. Simply put, all nodes tend to have equally rich representational power. \n\n\\begin{theorem}[Mixing in X-Nets]\nLet $n$ be the number of input as well as output nodes in the network and $G$ be $D$ regular bipartite expander graph with $n$ nodes on both sides. Let $S,T$ be subsets of input and output nodes in the X-Net layer defined by $G$. The number of edges between $S$ and $T$ is $\\approx D|S||T|\/n$\n\\end{theorem}\n\\begin{proof}\nA detailed proof is provided in the supplementary material.\n\\end{proof}\n\n\n\\section{Experiments and Results}\\label{sec:experiments}\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.3]{images\/mobilenet.png}\n\\caption{Comparison between Grouped convolutions and X-Conv using MobileNet architecture trained on ImageNet. X-$d$ or G-$d$ represents the 1x1 conv layers are compressed by $d$ times using X-Conv or Groups. We observe X-MobileNets beat Group-MobileNet by 4\\% in accuracy on increasing sparsity.}\n\\label{fig:mobilenet}\n\\end{figure}\nIn this section, we benchmark and empirically demonstrate the effectiveness of X-Nets on a variety of CNN architectures. Our code is available at: \\url{https:\/\/github.com\/DrImpossible\/Deep-Expander-Networks}. \n\\subsection{Comparison with Grouped Convolution}\n\\label{sec:group}\n\n\nFirst, we compare our Expander Convolutions (X-Conv) against Grouped Convolutions (G-Conv). We choose G-Conv as it is a popular approach, on which a lot of concurrent works \\cite{zhang2017shufflenet} have developed their ideas. G-Conv networks have the same sparsity as X-Conv networks but lack only the connectivity property. This will test whether increasing connectivity increases accuracy, i.e does a graph without good connectivity properties provides worse accuracy? We choose MobileNet as the base model for this experiment, since it is the state-of-the-art in efficient CNN architectures.\n\\begin{figure*}[t] \n\\begin{tabular}{cc}\n \\includegraphics[scale=0.28]{images\/cifar10flops.png}  & \\includegraphics[scale=0.28] {images\/cifar100flops.png} \\\\\n\\\\\n(a) CIFAR10 & (b) CIFAR100 \\\\\n\\end{tabular}\n\n\\caption{We show the error as a function of \\#FLOPs during test-time (below) for DenseNet-BC with X-DenseNet-BCs on CIFAR10 and CIFAR100 datasets. We observe X-DenseNet-BCs achieve better performance tradeoffs over DenseNet-BC models. For each datapoint, we mention the X-C-D-G notation (see Section \\ref{sec:denres}) along with the accuracy.\n}\n\n\\label{fig:cifar}\n\\end{figure*}\nWe compare X-Conv against grouped convolutions using MobileNet-0.5 on the ImageNet classification task. We replace the $1\\times 1$ convolutional layers in MobileNet-0.5 with X-Conv layers forming X-MobileNet-0.5. Similarly, we replace them with G-Conv layers to form Group-MobileNet-0.5. Note that we perform this only in layers with most number of parameters (after the 8th layer as given in Table 1 of \\cite{howard2017mobilenets}). We present our results in Figure \\ref{fig:mobilenet}. The reference original MobileNet-0.5 has an error of 36.6\\% with a cost of 150M FLOPs. Additional implementation details are given in the supplementary material.\n\nWe can observe that X-MobileNets beat Group-MobileNets by over 4\\% in terms of accuracy when we increase sparsity. This also demonstrates that X-Conv can be used to further improve the efficiency of even the most efficient architectures like MobileNet.\n\n\\subsection{Comparison with Efficient CNN Architectures}\n\\label{sec:denres}\n\nIn this section, we test whether Expander Graphs can improve the performance trade-offs even in state-of-the-art architectures such as DenseNet-BCs \\cite{huang2016densely} and ResNets\\cite{he2015convolutional} on the ImageNet \\cite{deng2009imagenet} dataset. We additionally train DenseNet-BCs on CIFAR-10 and CIFAR-100 \\cite{krizhevsky2009learning} datasets to demonstrate the robustness of our approach across datasets. \n\nOur X-ResNet-C-D is a $D$ layered ResNet that has every layer except the first and last replaced by an X-Conv layer that compresses connections between it and the previous layer by a factor of $C$. We compare across various models like ResNets-34,50,101. Similarly, our X-DenseNet-BC-C-D-G architecture has depth $D$, and growth rate $G$. We use DenseNet-BC-121-32,169-32,161-48,201-32 as base models. These networks have every layer except the first and last replaced by an X-Conv layer that compresses connections between it and the previous layer by a factor of $C$. More details are provided in the supplementary material.\n\\begin{minipage}{\\textwidth}\n  \\begin{minipage}[b]{0.45\\textwidth}\n    \\centering\n    \\includegraphics[scale=0.25]{images\/resnet.png}\n\\captionof{figure}{We show the error as a function of \\#FLOPs to compare between ResNet and X-ResNet on the ImageNet dataset. We observe X-ResNets achieve better performance tradeoffs over original ResNet models.}\n\\label{fig:resnet}\n \n  \\end{minipage}\n  \\hfill\n  \\begin{minipage}[b]{0.5\\textwidth}\n    \n    \\resizebox{0.9\\columnwidth}{!}{\n    \\begin{tabular}{|l|c|c|}\n\\hline\n{\\bf Model} & {\\bf Accuracy}  & {\\bf \\#FLOPs}\\\\\n\\hline\nResNet &   & {\\bf(in 100M)}\\\\\n\\hline\nX-ResNet-2-34 & 69.23\\%  & 35\\\\\nX-ResNet-2-50 & {\\bf 72.85\\%}  & {\\bf 40}\\\\\nResNet-34 & 71.66\\%  & 70\\\\\nX-ResNet-2-101 & {\\bf 74.87\\%}  & {\\bf 80}\\\\\nResNet-50 & 74.46\\%  & 80\\\\\nResNet-101 & 75.87\\%  & 160\\\\\n\\hline\nDenseNet-BC &  &   \\\\\n\\hline\nX-DenseNet-BC-2-121 & 70.5\\% & 28\\\\\nX-DenseNet-BC-2-169 & 71.7\\% & 33\\\\\nX-DenseNet-BC-2-201 & 72.5\\% & 43\\\\\nX-DenseNet-BC-2-161 & {\\bf 74.3\\%} & {\\bf 55}\\\\\nDenseNet-BC-121 & 73.3\\% & 55\\\\\nDenseNet-BC-169 & 74.8\\% & 65\\\\\nDenseNet-BC-201 & 75.6\\% & 85\\\\\nDenseNet-BC-161 & 76.3\\% & 110\\\\\n\\hline\n\\end{tabular}}\n      \\captionof{table}{Results obtained by ResNet and DenseNet-BC models on ImageNet dataset, ordered by \\#FLOPs. or each datapoint, we use the X-C-D-G notation (see Section \\ref{sec:denres}) along with the accuracy.}\n\\label{tab:imagenet}\n    \\end{minipage}\n    \n  \\end{minipage}\n\\subsection{Comparison with Pruning Techniques}\n\\label{sec:prun}\nWe plot the performance tradeoff of X-ResNets against ResNets in Figure \\ref{fig:resnet} . We achieve significantly better performance tradeoffs compared to the original model. More specifically, we can reduce the \\#FLOPs in ResNets by half while incurring only 1-1.5\\% decrease in accuracy. Also, we can compare models with similar \\#FLOPs or accuracy with the help of Table \\ref{tab:imagenet}. We observe that  X-ResNet-2-50 has 43\\% fewer FLOPs than ResNet-34, but achieves a 1\\% improvement in accuracy against it. Similarly, X-DenseNet-BC-2-161 has similar \\#FLOPs as DenseNet-BC-121, but achieves a 1\\% improvement in accuracy.  \n\nTo further prove the robustness of our approach on DenseNet-BC, we test the same on CIFAR10 and CIFAR100, and plot the tradeoff curve in Figure \\ref{fig:cifar}.\nWe observe that we can achieve upto 33\\% compression keeping accuracy constant on CIFAR-10 and CIFAR-100 datasets. \\\\\n\n\nWe compare our approach with methods which prune the weights during or after training. Our method can be thought of as constraining the weight matrices with a well studied sparse connectivity pattern even before the training starts. This results in fast training for the compact X-Conv models, while the trained pruning techniques face the following challenges: \\\\\n\n\\noindent 1) Slow initial training due to full dense model.\\\\\n2) Several additional phases of pruning and retraining.\\\\\n\nHence they achieve the compactness and runtime efficiency only in test time. Nevertheless we show similar sparsity can be achieved by our approach without explicitly pruning. We benchmark on VGG16 and AlexNet architectures since most previous results in the pruning literature have been reported on these architectures. In Table \\ref{tab:cifar_fullcomp}, we compare two X-VGG-16 models against existing pruning techniques. We achieve comparable accuracies to the previous state-of-the-art model with 50\\% fewer parameters and \\#FLOPs. Similarly, in Table \\ref{tab:imagenet_fullcomp} we compare X-AlexNet with trained pruning techniques on the Imagenet dataset. Despite having poor connectivity due to parameters being concentrated only in the last three fully connected layers, \nwe achieve similar accuracy to AlexNet model using only 7.6M-9.7M parameters out of 61M, comparable to the state-of-the-art pruning techniques which have upto 3.4M-5.9M parameters. Additionally, it is possible to improve compression by applying pruning methods on our compact architectures, but pruning X-Nets is out of the scope of our current work.\n\\begin{table}[t\n  \\centering\n\\begin{tabular}{|l|c|c|c|}\n\\hline\n{\\bf Method} & {\\sc { \\bf Accuracy}} & {\\bf \\#Params} & {\\bf Training }  \\\\\n\n\\hline\nLi et al. \\cite{li2016pruning} & 93.4\\%  & 5.4M &  \\ding{55} \\\\\nLiu et al. \\cite{liu2017learning} &  93.8\\% & 2.3M &  \\ding{55} \\\\\n\\hline\nX-VGG16-1 & 93.4\\%  & 1.65M (9x) &  \\ding{51} \\\\\nX-VGG16-2 & 93.0\\%  & 1.15M (13x) &  \\ding{51} \\\\\n\\hline\nVGG16-Orig &  94.0\\% & 15.0M & - \\\\\n\\hline\n\\end{tabular}\n\\captionof{table}{Comparison with other methods on CIFAR-10 dataset using VGG16 as the base model. We significantly outperform popular compression techniques, achieving similar accuracies with upto 13x compression rate.}\n\\label{tab:cifar_fullcomp}\n  \\end{table}\n  \n\\begin{table}[t\n    \\centering\n\\begin{tabular}{|l|c|c|c|}\n\\hline\n{\\bf Method} & {\\bf Accuracy} & {\\bf \\#Params} & {\\bf Training} \\\\\n & {\\bf } &  & {\\bf  Speedup}?\\\\\n\\hline\nNetwork Pruning & & & \\\\\n\\hline\nCollins et al.\\cite{collins2014memory} & 55.1\\% & 15.2M &  \\ding{55} \\\\\nZhou et al. \\cite{zhou2016less} & 54.4\\% & 14.1M & \\ding{55} \\\\\nHan et al. \\cite{han2015deep} &  57.2\\% & 6.7M & \\ding{55}  \\\\\nHan et al. \\cite{han2015deep} & 57.2\\% & 6.7M & \\ding{55} \\\\\n Srinivas et al. \\cite{srinivas2017training} & 56.9\\% & 5.9M & \\ding{55} \\\\\nGuo et al. \\cite{guo2016dynamic} & 56.9\\% & 3.4M & \\ding{55} \\\\\nX-AlexNet-1 & 55.2\\% & 7.6M & \\ding{51} \\\\\nX-AlexNet-2 & 56.2\\% & 9.7M & \\ding{51} \\\\\n\\hline\nAlexNet-Orig & 57.2\\% & 61M & - \\\\\n\\hline\n\\end{tabular}\n\\captionof{table}{Comparison with other methods on ImageNet-2012 using AlexNet as the base model. We are able to achieve comparable accuracies using only 9.7M parameters.}\n\\label{tab:imagenet_fullcomp}\n    \\end{table}\n  \n\n\\subsection{Stability of Models}\nWe give empirical evidence as well as a theoretical argument regarding the stability of our method. For the vanilla DNN training, the weights are randomly initialized, and randomized techniques like dropouts, augmentation are used. Hence there is some randomness present and is well accepted in DNN literature prior to our method. We repeat experiments on different datasets (Imagenet and CIFAR10) and architectures (VGG, DenseNet and MobileNet0.5) to empirically show that the accuracy of expander based models has variance similar to vanilla DNN training over multiple runs.\n\nWe repeated the experiments with independent sampling of random expanders on the VGG and DenseNet baselines on the CIFAR10 dataset. The results can be seen in Table~\\ref{tab:cifar}. It is noted that the accuracy values changes only by less than $0.3$\\% across runs and the standard deviation of expander method is also comparable to the vanilla DNN training.\n\nWe also repeated experiments of our main result, which is the comparison with grouped convolutions on ImageNet dataset. We rerun the experiment with MobileNet0.5 feature extractor twice with Groups and the expander method. As can be seen from Table~\\ref{tab:imgnet}, the accuracy variations are comparable between the two models, and it is less than 1\\%.\n\n\\begin{table}[!thb]\n\\begin{minipage}{.5\\linewidth}\n\\centering\n\\scalebox{0.9}{\n\\begin{tabular}{|l|c|c|c|c|}\n\\hline\n{\\bf Model} & {\\bf Accuracy\\%} & {\\bf Max\\%} & {\\bf Min\\%}\\\\\n\\hline\nVGG & 93.96$\\pm0.12$ & 94.17 & 93.67 \\\\\n\\hline\nX-VGG-1 & 93.31$\\pm0.18$ & 93.66 & 93.06 \\\\\nX-VGG-2 & 92.91$\\pm0.19$ & 93.26 & 92.69 \\\\\n\\hline\nXDNetBC-40-24 & 94.41$\\pm$0.19 & 94.63 & 94.18 \\\\\nXDNetBC-40-36 & 94.98$\\pm$0.14 & 95.21 & 94.84 \\\\\nXDNetBC-40-48 & 95.49$\\pm$0.15 & 95.65 & 95.28 \\\\\nXDNetBC-40-60 & 95.75$\\pm$0.07 & 95.81 & 95.68 \\\\\n\n\\hline\n\\end{tabular}}\n\\caption{The accuracies (mean $\\pm$ stddev) of various models over 10 training runs on CIFAR-10 dataset.} \\label{tab:cifar}\n\\end{minipage}%\n\\hfill\n \\begin{minipage}{.45\\linewidth}\n \\centering\n \\scalebox{0.9}{\n\\begin{tabular}{|c|c|c|}\n\\hline\n{\\bf MobileNet} & {\\bf Mean} & {\\bf Range}\\\\\n{\\bf Variant}  & {\\bf Accuracy} & {\\bf (Max-Min)}\\\\\n\\hline\nBase & $63.39\\%$ & $0.11\\%$\\\\\n\n\\hline\nG2 & $57.45~\\%$ & $0.06~\\%$\\\\\nX2 & \\bf{$58.22~\\%$} & $ 0.14~\\%$\\\\\n\\hline\nG4 & $49.41~\\%$ & $0.55~\\%$\\\\\nX4 & \\bf{$54.00~\\%$} & $0.53~\\%$\\\\\n\\hline\nG8 & $45.13~\\%$ & $0.03~\\%$\\\\\nX8 & \\bf{$49.23~\\%$} & $0.60~\\%$\\\\\n\\hline\nG16 & $39.03~\\%$ & $0.64~\\%$\\\\\nX16 & \\bf{$44.63~\\%$} & $0.18~\\%$\\\\\n\\hline\n\\end{tabular}}\n\\caption{The mean accuracy and range of variation over  2 runs of MobileNet0.5 variants on ImageNet dataset.} \\label{tab:imgnet}\n\\end{minipage}%\n\\end{table}\nA theoretical argument also concludes that choosing random graphs doesn't degrade stability. It is a well known result (See Theorem 4.4 in  \\cite{pseudo}) in random graph theory, that graphs chosen randomly are well connected with overwhelmingly high probability (with only inverse exponentially small error, due to the Chernoff's Tail bounds) and satisfies the Expander properties. Hence the chance that for a specific run, the accuracy gets affected due to the selection of a particularly badly connected graph is insignificant.\n\n\\subsection{Training Wider and Deeper  networks}\\label{sec:ultrawide}\n\n\\begin{figure*}[t]\n\\begin{tabular}{cc}\n\n\\includegraphics[scale=0.25]{images\/ultrawide.png}  & \n\\includegraphics[scale=0.25] {images\/ultradeep.png}\\\\\n(a) Effect of Width  & (b) Effect of Depth \\\\\n\\end{tabular}\n\\caption{We show the performance tradeoff obtained on training significantly wider and deeper networks on CIFAR-100 dataset. Every datapoint is X-$C$ specified along with the number of parameters, $C$ being the compression factor. We show that training wider or deeper networks along with more  compression using X-Nets achieve better accuracies with upto two-thirds of the total parameter and FLOPs on CIFAR-100 dataset. }\n\\label{fig:deepnet}\n\\end{figure*}\n\nSince X-Nets involve constraining the weight matrices to sparse connectivity patterns before training, the fast algorithms can make it possible to utilize memory and runtime efficiently in training phase. This makes it possible to train significantly deeper and wider networks. Note the contrast with pruning techniques, where it is necessary to train the full, bulky model, inherently limiting the range of models that can be compressed.\n\nWide-DenseNets\\footnote{https:\/\/github.com\/liuzhuang13\/DenseNet\\#wide-densenet-for-better-timeaccuracy-and-memoryaccuracy-tradeoff} offered a better accuracy-memory-time trade-off. We increase the width and depth of these networks to train significantly wider and deeper networks. The aim is to study whether leveraging the effectiveness of X-Nets in this fashion can lead to better accuracies. \n  \nWe widen and deepen the DenseNet-BC-40-60 architecture, increasing the growth rate from 60 to 100 and 200 respectively and compare the effect of increasing width on these new models. Similarly, we increase the depth from 40 to 58 and 70 to obtain deeper networks. We benchmark these approaches using CIFAR-100 dataset and present the results in Figure \\ref{fig:deepnet}. \n\nWe have two interesting observations. First, the deeper X-DenseNet-BC-70-60 significantly outperforms X-DenseNet-BC-58-60 and wider X-DenseNet-40-200 outperforms X-DenseNet-BC-40-100 with fewer parameters for a wide range of $C$ values (Expander degree). \n\nThe second interesting observation is the decreasing slope of the curves. This  indicates that expander graph modeling seems to be effective on wider and deeper X-Nets i.e X-DenseNet-BC models suffer lesser penalty with increasing depth and width compression. This enables X-Nets to work at high compression rates of 30x, compressing DenseNet-BC-40-200 model from 19.9B FLOPs to 0.6B FLOPs with only $4.3\\%$ drop in accuracy. We hope this preliminary investigation holds significant value in alleviating the constraint of GPU memory and resources.\n\n\n\\section{Related Work}\\label{sec:rel-works}\nOur approach lies at the intersection of trained pruning techniques and efficient layer design techniques. We present a literature survey regarding both the directions in detail. \n\n\\subsection{Efficient Layer Designs}\n\nCurrently there is extensive interest in developing novel convolutional layers\/blocks and effectively leveraging them to improve architectures like \\cite{squeezenet,howard2017mobilenets,hu2017squeeze}. Such micro-architecture design is in a similar direction as our work. In contrast, approaches like \\cite{googlenet} try to design the macro-architectures by connecting pre-existing blocks. Recent concurrent work has been on performing architecture searches effectively \\cite{liu2017hierarchical,zoph2017learning,zhong2017practical,liu2017progressive}. Our work is complementary to architecture search techniques as we can leverage their optimized macro-architectures. \n\nAnother line of efficient architecture design is Grouped Convolutions: which was first proposed in AlexNet\\cite{krizhevsky2012imagenet}, recently popularized by MobileNets\\cite{howard2017mobilenets} and XCeption\\cite{xception} architectures . This is currently a very active area of current research, with a lot of new concurrent work being proposed \\cite{zhang2017shufflenet,sandler2018inverted,huang2017condensenet}. \n\nIt is interesting to note that recent breakthroughs in designing accurate deep networks \\cite{resnet,huang2016densely,xie2017aggregated} were mainly by introducing additional connectivity to enable the efficient flow of information through deep networks. This enables the training of compact, accurate deep networks. These approaches, along with Grouped Convolutions are closely related to our approach. \n\\subsection{Network Compression} \nSeveral methods have been introduced to compress pre-trained networks as well as train-time compression. Models typically range from low-rank decomposition  \\cite{sainath2013low,novikov2015tensorizing,masana2017domain} to network pruning  \\cite{blundell2015weight,liu2015sparse,he2017channel,molchanov2016pruning}.\n\nThere is also a major body of work that quantizes the networks at train-time to achieve efficiency \\cite{rastegari2016xnor,courbariaux2016bnn,han2015deep,wu2016quantized,bagherinezhad2016lcnn,zhu2017trained,zhou2016dorefa}. The problem of pruning weights in train-time have been extensively explored  \\cite{wen2016structured,li2016pruning} primarily from weight-level \\cite{lebedev2016fast,scardapane2017group,srinivas2015data,guo2016dynamic} to channel-level pruning \\cite{liu2017learning,li2016pruning,wen2016structured}. Weight-level pruning has the highest compression rate while channel-level pruning is easier to practically exploit and has compression rates almost on par with the former. Hence, channel-level pruning is currently considered superior \\cite{liu2017learning}. Channel-level pruning approaches started out with no guidance for sparsity \\cite{chen2015compressing} and eventually added constraints \\cite{srinivas2017training,zhou2016less,yoon2017combined}, tending towards more structured pruning.\n\nHowever, to the best of our knowledge, this is the first attempt at constraining neural network connections by graph-theoretic approaches to improve deep network architecture designs. Note that we do not prune weights during training.  \n\n\\section{Introduction}\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=0.175]{images\/Expander2.png}\n\n\\caption{Popular sparse approximations are agnostic to the global information flow in a network, possibly creating disconnected components. In contrast, expander graph-based models produce sparse yet highly connected networks.}\n\\label{fig:intro}\n\n\\end{figure}\n\nConvolutional Neural Networks (CNNs) achieve state-of-the-art results in a variety of machine learning applications\\cite{krizhevsky2012imagenet,he2016identity,huang2016densely,googlenet}. However, they are also computationally intensive and consume a large amount of computing power and runtime memory. After the success of VGG Networks \\cite{vgg}, there has been significant interest in designing compact neural network architectures due to the wide range of applications valuing mobile and embedded devices based use cases.\n \nResNet\\cite{resnet} and DenseNet-BC\\cite{huang2016densely} directed the focus of efficient designs of convolutional layers on increasing connectivity. Additional connectivity with residual connections to previous layers provided efficient information flow through the network, enabling them to achieve an order of magnitude reduction in storage and computational requirements. We take inspiration from these approaches, to focus on designing highly connected networks. We explore making networks efficient by designing sparse networks that preserve connectivity properties. \nRecent architectures like MobileNet\\cite{howard2017mobilenets} improves the efficiency by an order of magnitude over a ResNet. However, in order to achieve this, they sparsify a network by removing several connections from a trained network, reducing their accuracies in the process. We ask a basic question: If we try to maximize the connectivity properties and information flow, can we achieve the same efficiency gains with minimal loss in accuracy? \nIt is essential that the connections allow information to flow through the network easily. That is, each output node must at least have the capacity to be sensitive to features of previous layers. As we can see from Fig.\\ref{fig:intro}, traditional model compression techniques such as pruning can aggravate the problem, since they can prune the neuron connections of a layer, while being agnostic of global connectivity of the network. A necessary condition for having good representational power is efficient information flow through the network, which is particularly suited to be modeled by graphs. \nWe propose to make the connections between neurons (filters in the case of CNNs) according to specific graph constructions known as expander graphs. They have been widely studied in spectral graph theory \\cite{sgt} and pseudorandomness \\cite{pseudo}, and are known to be sparse but highly connected graphs. Expander graphs have a long history in theoretical computer science, also being used in practice in computer networks, constructing error correcting codes, and in cryptography (for a survey, see \\cite{expsurvey}). \n\n\n{\\bf Main Contributions:} i.) We propose to represent neuronal connections in deep networks using expander graphs (see Section \\ref{sec:approach}). We further prove that X-Nets have strong connectivity properties (see Theorem \\ref{thm:conn}). ii.) We provide memory-efficient implementations of  Convolutional (X-Conv) layers using sparse matrices and propose a fast expander-specific algorithm (see Section \\ref{sec:implementation}). iii.) We empirically compare X-Conv layers with grouped convolutions that have the same level of sparsity but worse connectivity. X-Conv layers obtain a 4\\% improvement in accuracy when both the techniques are applied to the MobileNet architecture trained on Imagenet (see Section \\ref{sec:group}).  iv.) We also demonstrate the robustness of our approach by applying the technique to some of the state of the art models like DenseNet-BC and ResNet, obtaining better performance trade-offs (see Section \\ref{sec:denres}). v.) Additionally, our simple design achieves comparable compression rates to even the state-of-the-art trained pruning techniques. (see Section \\ref{sec:prun}). vi.) Since we enforce the sparsity before the training phase itself, our models are inherently compact and faster to train compared to pruning techniques. We leverage this and showcase the performance of wider and deeper X-Nets (see Section \\ref{sec:ultrawide}). \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nLet $\\mathbb{D}:=\\{z\\in\\mathbb{C}:|z|<1\\}$ be the open unit disc and $\\mathcal{H}$ be the class of functions, analytic on $\\mathbb{D}.$ For any complex number $a$ and a positive integer $n,$ let $\\mathcal{H}[a,n]$ be the subclass of $\\mathcal{H}$ containing functions of the form $f(z)=a+a_nz^n+a_{n+1}z^{n+1}+\\cdots.$ Functions in $\\mathcal{H}$ with the normalization $f(0)=0,$ $f'(0)=1$ constitutes a subclass of $\\mathcal{H},$ which we denote by $\\mathcal{A}.$ Further, $\\mathcal{S}$ denotes the subclass of $\\mathcal{A}$ consisting of univalent functions. The class of functions $p$ of the form $p(z)=1+c_1z+c_2z^2+\\cdots$ with the property $\\RE p(z)>0$ on $\\mathbb{D},$ is known as the Carath\\'eodory class and is denoted by $\\mathcal{P}.$ For any two analytic functions $f$ and $F,$ it is said that $f$ is subordinate to $F,$ written as $f\\prec F$ if there exists an analytic function $\\omega(z),$ satisfying $\\omega(0)=0$ and $|\\omega(z)|<1,$ such that $f(z)=F(\\omega(z)).$ In particular, if $F$ is univalent then $f\\prec F$ if and only if $f(0)=F(0)$ and $f(\\mathbb{D})\\subset F(\\mathbb{D}).$ Let $\\mathcal{R}$ denotes the class of functions $f\\in\\mathcal{A}$ satisfying $\\RE f'(z)>0$ on $\\mathbb{D},$ then clearly $\\mathcal{R}\\subset\\mathcal{S}.$ A subclass $\\mathcal{S}^*$ of $\\mathcal{S},$ known as the class of starlike functions, consists of the functions $f$ satisfying $\\RE(zf'(z)\/f(z))>0$ in $\\mathbb{D}.$ The class of convex functions, denoted by $\\mathcal{C},$ is another subclass of $\\mathcal{S}$ consisting of the functions $f$ with $\\RE(1+zf''(z)\/f'(z))>0$ in $\\mathbb{D}.$ An analytic function $f$ is said to be close-to-convex if $\\RE(zf'(z)\/g(z))>0\\;(z\\in\\mathbb{D}),$ where $g$ is a starlike function. The class of close-to-convex functions is denoted by $\\mathcal{K}.$ Let $\\mathcal{S}^*(\\alpha)$ $(0\\leq\\alpha<1)$ be the subclass of $\\mathcal{S}^*$ consisting of the functions $f$ satisfying $\\RE(zf'(z)\/f(z))>\\alpha,$ known as the class of starlike functions of order $\\alpha.$ The class of $\\alpha$-convex  functions of order $\\beta$ is defined as\n\\begin{equation*}\n\t\\mathcal{M}_{\\alpha}(\\beta):=\\left\\{f\\in\\mathcal{A}:\t\\RE\\left((1-\\alpha)\\dfrac{zf'(z)}{f(z)}+\\alpha\\left(1+\\dfrac{zf''(z)}{zf'(z)}\\right)\\right)>\\beta\\right\\}\n\\end{equation*}\nand the class of $\\alpha$-starlike functions of order $\\beta$ is defined as\n\\begin{equation*}\n\t\\mathcal{L}_{\\alpha}(\\beta): =\\left\\{f\\in\\mathcal{A}:\\RE\\left(\\dfrac{zf'(z)}{f(z)}\\right)^{1-\\alpha}\\left(1+\\dfrac{zf''(z)}{f'(z)}\\right)^{\\alpha}>\\beta\\right\\},\t\n\\end{equation*}\nwhere $z\\in\\mathbb{D},$ $\\alpha$ is any real number and $0\\leq\\beta<1.$ For $0\\leq\\alpha\\leq1,$ the three classical means namely arithmetic, geometric and harmonic mean of two numbers $x$ and $y,$ in their special form, known as convex weighted means are defined respectively as follows:\n\\begin{eqnarray*}\n\tA_{\\alpha}(x,y)= (1-\\alpha)x+\\alpha y;\\;\n\tG_{\\alpha}(x,y)= x^{\\alpha}y^{1-\\alpha};\\;\n\tH_{\\alpha}(x,y)= \\dfrac{xy}{\\alpha y+(1-\\alpha)x}.\n\\end{eqnarray*}\nNote that the classes $\\mathcal{M}_{\\alpha}(\\beta)$ and $\\mathcal{L}_{\\alpha}(\\beta)$ are defined by using the convex arithmetic mean and convex geometric mean, of the quantities $zf'(z)\/f(z)$ and $1+zf''(z)\/f'(z)$ respectively. Let $h$ be univalent on $\\mathbb{D}$ and $\\psi:\\mathbb{C}^2\\times\\mathbb{D}\\rightarrow\\mathbb{C}$ be a complex function, then an analytic function $p,$ which satisfies\n\\begin{equation}\\label{h15}\n\t\\psi(p(z),zp'(z))\\prec h(z)\n\\end{equation}\nis called the solution of the differential subordination~\\eqref{h15}. The theory of differential subordination introduced by Miller and Mocanu~\\cite{ds} is developed on the basis of the following implication\n\\begin{equation*}\n\t\\psi(p(z),zp'(z))\\prec h(z)\\Rightarrow p(z)\\prec q(z),\\qquad z\\in\\mathbb{D},\n\\end{equation*}\nwhere $h$ and $q$ are univalent in $\\mathbb{D}$ and $q$ is a dominant of the solutions of the differential subordination~\\eqref{h15}. In this direction, a good amount of work has been carried out by many authors(see~\\cite{pri3,first,ssp,adiba,vibha,shagun4}). In 1996, Kanas et. al~\\cite{kanas1996} introduced and studied differential subordinations involving geometric mean of $p(z)$ and $p(z)+zp'(z).$ Later in 2011, the authors in~\\cite{lecko2} proved several differential subordination results associated with arithmetic as well as geometric mean of certain analytic functions. In~\\cite{kanasels}, Cri\\c{s}an and Kanas considered a combination of arithmetic and geometric mean for which they established the following implication:\n\\begin{equation}\\label{h16}\n\t\\gamma (p(z))^{\\delta}+(1-\\gamma)(p(z))^{\\mu}\\left(p(z)+\\dfrac{zp'(z)}{p(z)}\\right)^{1-\\mu}\\prec h(z)\\Rightarrow p(z)\\prec h(z).\n\\end{equation}\nThis result is applied to prove some univalence and starlikeness criteria. A particular form of this expression, for different choices of $h(z)$ has been worked upon by Kanas~\\cite{kanasbmms}. Recently, Gavri\\c{s}~\\cite{gavris} melded all the three pythagorean means in one expression and proved an implication similar to~\\eqref{h16} for a specific choice of $h(z).$ In the present investigation, we extend the results of~\\cite{gavris} by proving a similar implication for another choice of $h(z)$ and generalize many existing results. In 2014, Cho et. al~\\cite{geo} established conditions on an analytic function $\\Phi(z)$ so that the geometric mean of $p(z)$ and $p(z)+zp'(z)\\Phi(z)$ is subordinate to $h(z)$ leads $p(z)$ to be subordinate to $h(z),$ where $h$ is a univalent function. Later in the same year, Chojnacka and Lecko~\\cite{harmonic} proved a similar result for harmonic mean. The arithmetic mean is already covered by Miller and Mocanu in~\\cite{ds}. In our study, we prove some general differential subordination results involving the harmonic mean of the quantities $p(z)$ and $p(z)\\Theta(z)+zp'(z)\\Phi(z),$ where $\\Theta(z)$ and $\\Phi(z)$ are analytic functions. As an application to our main results, we establish various differential subordination results pertaining to several subclasses of $\\mathcal{S}^*.$ Further, we obtain sufficient conditions for starlikeness and univalence, which generalize some earlier known results. Here are a few prerequisites for our main findings:\n\n\\begin{definition}\\label{defH}\n\tLet $t\\in[0,1]$ and $\\Theta, \\Phi\\in\\mathcal{H}$ with $\\Theta(0)=1.$ By $\\mathcal{H}(t; \\Theta, \\Phi),$ we mean the subclass of $\\mathcal{H}$ of all functions $f$ such that\n\t\\begin{equation}\\label{5}\n\t\tH_{t;\\Theta,\\Phi,f}(z):=\\begin{cases}\n\t\t\t\\dfrac{P_{0;\\Theta,\\Phi,f}(z)P_{1;\\Theta,\\Phi,f}(z)}{P_{1-t;\\Theta,\\Phi,f}(z)} & \\quad P_{1-t;\\Theta,\\Phi,f}(z)\\neq0, \\\\\n\t\t\t\\lim_{\\mathbb{D}\\backin\\zeta\\rightarrow z}\\dfrac{P_{0;\\Theta,\\Phi,f}(\\zeta)P_{1;\\Theta,\\Phi,f}(\\zeta)}{P_{1-t;\\Theta,\\Phi,f}(\\zeta)} & \\quad P_{1-t;\\Theta,\\Phi,f}(z)=0,\n\t\t\\end{cases}\n\t\\end{equation}\n\tis an analytic function in $\\mathbb{D},$\n\twhere\n\t\t\\begin{equation*}\n\t\tP_{t;\\Theta,\\Phi,f}(z):=(1-t+t\\Theta(z))f(z)+t\\Phi(z)zf'(z),\\quad z\\in\\mathbb{D}\n\t\\end{equation*}\nand define $H_{t;\\Theta,\\Phi,0}\\equiv 0.$\n\\end{definition}\n\\begin{definition}\nLet $\\mathcal{Q}$ denotes the class of all convex functions $h$ with the following properties:\n\t\\begin{enumerate}\n\t\t\\item $h(\\mathbb{D})$ is bounded by finitely many smooth arcs which form corners at their end points (including corners at infinity),\n\t\t\\item $E(h)$ is the set of all points $\\zeta\\in\\partial\\mathbb{D}$ which corresponds to corners $h(\\zeta)$ of $\\partial h(\\mathbb{D}),$\n\t\t\\item $h'(\\zeta)\\neq 0$ exists at every $\\zeta\\in\\partial\\mathbb{D}\\backslash E(h).$\n\\end{enumerate}\n\\end{definition}\n\\begin{lemma}[Lemma~2.2a]\\cite{ds}\\label{jack}\n  Let $z_0\\in\\mathbb{D}$ and $r_0=|z_0|.$ Let $f(z)=a_nz^n+a_{n+1}z^{n+1}+\\cdots$ be continuous on $\\overline{\\mathbb{D}}_{r_0}$ and analytic on $\\mathbb{D}\\cup\\{z_0\\}$ with $f(z)\\neq 0$ and $n\\geq 1.$ If $$|f(z_0)|=\\max\\{|f(z)|:z\\in\\overline{\\mathbb{D}}_{r_0}\\}$$ then there exists an $m\\geq n$ such that $$\\dfrac{z_0f'(z_0)}{f(z_0)}=m\\;\\text{and}\\;\\RE\\left(\\dfrac{z_0f''(z_0)}{f'(z_0)}+1\\right)\\geq m.$$\n\\end{lemma}\n\n\\section{Main Results}\nWe begin with the following result.\n  \\begin{lemma}\\label{mainn}\n\tLet $\\delta\\in[0,1],$ $h\\in\\mathcal{Q}$ with $0\\in\\overline{h(\\mathbb{D})}$ and $\\Theta,\\Phi\\in\\mathcal{H}$ be such that $\\Theta(0)=1,$ $\\RE \\Phi(z)>0\\;(z\\in\\mathbb{D})$ and\n\t\\begin{equation}\\label{hypo}\n\t\t\\RE\\left( \\Phi(z)+\\dfrac{h(\\zeta)}{\\zeta h'(\\zeta)}(\\Theta(z)-1)\\right)>0,\\quad z\\in\\mathbb{D},\\;\\zeta\\in\\partial\\mathbb{D}.\n\t\\end{equation}\n\tIf $p\\in\\mathcal{H}(\\delta;\\Theta,\\Phi),$ $p(0)=h(0)$ and $H_{\\delta;\\Theta,\\Phi,p}\\prec h,$ then $p\\prec h.$\n\\end{lemma}\n\\begin{proof}\n\tFrom~\\eqref{5} it is easy to see that $H_{0;\\Theta,\\Phi,p}= p,$ therefore the implication given in the hypothesis holds when $\\delta=0$. Now let us take $\\delta\\in(0,1].$ If $p= p(0)\\in\\mathcal{H}(\\delta;\\Theta,\\Phi),$ then using the fact that $\\Theta(0)=1$ and from~\\eqref{5}, we have $p(0)\\in h(\\mathbb{D})$ and thus in this case, the implication holds obviously. Now let $p\\in\\mathcal{H}(\\delta;\\Theta,\\Phi)$ be a nonconstant function and define\n\t\\begin{equation}\\label{6}\n\t\tx:=p(z_0)\\quad \\text{and}\\quad y:=p(z_0)\\Theta(z_0)+z_0p'(z_0)\\Phi(z_0).\n\t\\end{equation}\n\tSince $h\\in\\mathcal{Q},$ therefore $h'(\\zeta_0)\\neq 0$ exists. Let $p\\nprec h,$ then from~\\cite[Lemma~2.2]{geo} and~\\cite[Lemma~2.3]{harmonic}, we have $z_0\\in\\mathbb{D}\\backslash\\{0\\}$ and $\\zeta_0\\in\\partial\\mathbb{D}\\backslash E(h)$ such that\n\t\\begin{equation}\\label{7}\n\t\tp(\\mathbb{D}_{|z_0|})\\subset h(\\mathbb{D}),\\quad p(z_0)=h(\\zeta_0)\n\t\\end{equation}\n\tand\n\t\\begin{equation}\\label{8}\n\t\tz_0p'(z_0)=m\\zeta_0h'(\\zeta_0)\\quad\\text{for some}\\;m\\geq 1.\n\t\\end{equation}\n\tUsing~\\eqref{7} and~\\eqref{8} in~\\eqref{6}, we obtain\n\t\\begin{equation*}\n\t\tx=h(\\zeta_0)\\quad\\text{and}\\quad y=h(\\zeta_0)\\Theta(z_0)+m\\zeta_0h'(\\zeta_0)\\Phi(z_0).\n\t\\end{equation*}\n\tLet $\\mathbb{P}$ be an open half plane, which supports the convex domain $h(\\mathbb{D})$ at $h(\\zeta_0).$ So\n\t\\begin{equation}\\label{x}\n\t\tx=h(\\zeta_0)\\in\\overline{\\mathbb{P}}\\quad\\text{and}\\quad h(\\mathbb{D})\\cap\\mathbb{P}=\\phi.\n\t\\end{equation}\n\tWe observe that\n\t\\begin{eqnarray}\\label{y}\n\t\n\t\ty = h(\\zeta_0)\\Theta(z_0)+m\\zeta_0h'(\\zeta_0)\\Phi(z_0) =h(\\zeta_0)+\\zeta_0h'(\\zeta_0)\\Psi(z_0),\n\t\\end{eqnarray}\n\twhere\n\t\\begin{equation}\\label{9}\n\t\t\\Psi(z):=m\\Phi(z_0)+\\dfrac{h(\\zeta_0)}{\\zeta_0h'(\\zeta_0)}(\\Theta(z_0)-1).\n\t\\end{equation}\n\tClearly~\\eqref{hypo} together with the fact that $m\\geq 1$ and $\\RE\\Phi(z)>0$ implies $\\RE\\Psi(z)>0.$ Using this along with~\\eqref{y}, we can say that $y\\in\\mathbb{P}.$ For $x,y\\in\\mathbb{P}$ and $\\delta\\in(0,1],$ it follows from~\\cite[Lemma~2.1]{harmonic} that the harmonic mean of $x$ and $y,$ $H_{\\delta;\\Theta,\\Phi,p}(z_0)\\in\\overline{\\mathbb{P}}$ provided $y+\\delta(x-y)\\neq 0.$ Taking into account~\\eqref{x}, it follows that $H_{\\delta;\\Theta,\\Phi,p}(z_0)\\notin h(\\mathbb{D}),$ which contradicts the hypothesis and thus the result holds in this case.\\\\\n\t\\indent For the case when $y+\\delta(x-y)=0,$ we have $P_{1-\\delta;\\Theta,\\Phi,p}(z_0)=0.$ By Definition~\\ref{defH}, we know that the limit\n\t\\begin{equation*}\n\t\tH_{\\delta;\\Theta,\\Phi,p}(z_0)= \\lim_{\\mathbb{D}\\backin\\zeta\\rightarrow z_0}\\dfrac{P_{0;\\Theta,\\Phi,p}(\\zeta)P_{1;\\Theta,\\Phi,p}(\\zeta)}{P_{1-\\delta;\\Theta,\\Phi,p}(\\zeta)}\n\t\\end{equation*}\n\tis finite and so $P_{0;\\Theta,\\Phi,p}(z_0)P_{1;\\Theta,\\Phi,p}(z_0)=xy=0.$ First let us suppose that $x=0,$ which implies $(1-\\delta)y=0.$ Since $x=0,$ $y$ reduces to $y=m\\zeta_0h'(\\zeta_0)\\Phi(z_0).$ Clearly $y$ can not be zero as $h'(\\zeta_0)\\neq 0$ and $\\RE \\Phi(z)>0.$ Hence $x=0$ if and only if $\\delta=1.$ Now we observe from Definition~\\ref{defH},\n\t\\begin{equation*}\n\t\tH_{1;\\Theta,\\Phi,p}(z_0)=P_{1;\\Theta,\\Phi,p}(z_0)=y\\in\\mathbb{P},\n\t\\end{equation*}\n\twhich further implies $H_{1;\\Theta,\\Phi,p}(z_0)\\notin h(\\mathbb{D}).$ This is a contradiction to hypothesis and the result follows. Next let us suppose that $y=0,$ then we have $\\delta x=0.$ Since $\\delta\\in(0,1],$ it follows that $x=0$ and thus $y$ becomes $y=m\\zeta_0h'(\\zeta_0)\\Phi(z_0),$ which can never be equal to 0. Therefore such a case is never possible. This completes the proof.\n\\end{proof}\n\\begin{remark}\n\tIf we take $t=1\/2,$ $\\Theta(z)=1$ and $\\Phi(z)=1$ in $H_{t;\\Theta,\\Phi,p}(z),$ from Definition~\\ref{defH} we can say that it reduces to\n\t\\begin{equation*}\n\t\tH_{1\/2;1,1,p}(z)=\\dfrac{2p(z)(p(z)+zp'(z))}{2p(z)+zp'(z)}=:P(z).\n\t\\end{equation*}\n\tFurther if we take $h(z)=(1+z)\/(1-z)$ in Lemma~\\ref{mainn}, it reduces to a result of Kanas and Tudor~\\cite[Theorem~2.1]{kanasbmms}.\n\\end{remark}\n\\begin{remark}\n\tIf $h(z)=((1+z)\/(1-z))^{\\gamma},$ where $\\gamma\\in(0,1],$ then for $t=1\/2,$ $\\Theta(z)=1$ and $\\Phi(z)=1,$ Lemma~\\ref{mainn} reduces to~~\\cite[Theorem~2.6]{kanasbmms}.\n\\end{remark}\nSome applications of the above theorem are discussed in the next section. So far in this direction, authors have proved differential subordination implications of the form\n\\begin{equation*}\n\t\\psi(p(z),zp'(z))\\prec h(z)\\Rightarrow p(z)\\prec h(z)\n\\end{equation*}\nand the above theorem generalizes many such results in case of harmonic mean. Henceforth, we consider differential subordinations of the form\n\\begin{equation*}\n\t\\psi(p(z),zp'(z))\\prec h(z)\\Rightarrow p(z)\\prec q(z)\n\\end{equation*}\nfor different choices of $h,$ $q$ and $\\psi.$ We enlist below a few examples.\n\\begin{example}\nLet $p(z)=1+a_1z+a_2z^2+\\cdots$ be analytic in $\\mathbb{D}$ with $p(z)\\not\\equiv1.$ Then $p(z)\\prec e^z,$ whenever\n\\begin{equation*}\n  \\dfrac{2p(z)(p(z)+zp'(z))}{2p(z)+zp'(z)}\\prec \\phi_i(z)\\;(i=1,2,..,5),\n\\end{equation*}\nwhere $\\phi_1(z)=\\sqrt{1+z},\\;\\phi_2(z)=2\/(1+e^{-z}),\\;\\phi_3(z)=z+\\sqrt{1+z^2},\\;\\phi_4(z)=1+\\sin{z}$ and $\\phi_5(z)=1+4z\/3+2z^2\/3.$\n\\end{example}\n\\begin{proof}\nLet $q(z)=e^z$ and $\\Omega_i=\\phi_i(\\mathbb{D})\\;(i=1,2..,5)$. Suppose that  $\\psi:\\mathbb{C}^2\\times\\mathbb{D}\\rightarrow\\mathbb{C}$ be a function defined by $\\psi(a,b;z)=2a(a+b)\/(2a+b).$ Using the admissibility conditions for $e^z$ given by Naz et al.~\\cite{adiba}, it is sufficient to prove that $\\psi\\in\\Psi[\\Omega_i,e^z],$ or equivalently, $\\psi(r,s;z)\\notin\\Omega,$ where $r=e^{e^{i\\theta}}$ and $s=me^{i\\theta}e^{e^{i\\theta}}=mre^{i\\theta}\\;(-\\pi\\leq\\theta\\leq\\pi).$ We observe that\n \\begin{equation*}\n  \\psi(r,s;z)=\\dfrac{2r(r+s)}{2r+s}=\\dfrac{2r(r+mre^{i\\theta})}{2r+mre^{i\\theta}}=2e^{e^{i\\theta}}\\left(1-\\dfrac{1}{2+me^{i\\theta}}\\right).\n\\end{equation*}\nTherefore\n\\begin{eqnarray*}\n  \\RE\\psi(r,s;z)&=&\\dfrac{2e^{\\cos{\\theta}} \\left(\\left(m^2+3 m \\cos{\\theta}+2\\right) \\cos{(\\sin{\\theta})}-m \\sin {\\theta} \\sin{(\\sin{\\theta})}\\right)}{m^2+4 m \\cos (\\theta)+4}\\\\\n  &=:&g(\\theta),\n\\end{eqnarray*}\nwhich at $\\theta=0$ becomes $g(0)=2e(m^2+3m+2)\/(5+4m).$ Since $m\\geq 1,$ we have $g(0)\\geq 4e\/3.$ Thus it is easy to conclude that $\\psi(r,s,z)\\notin\\Omega_i\\;(i=1,2,..,5)$ and the result follows.\n\\end{proof}\n\\begin{example}\nLet $p(z)=1+a_1z+a_2z^2+\\cdots$ be analytic $\\mathbb{D}$ with $p(z)\\not\\equiv1.$ Then $p(z)\\prec \\sqrt{1+z},$ whenever\n\\begin{equation*}\n  \\dfrac{2p(z)(p(z)+zp'(z))}{2p(z)+zp'(z)}\\prec\\dfrac{2}{1+e^{-z}}.\n\\end{equation*}\n\\end{example}\n\\begin{proof}\nFor $\\phi(z)=2\/(1+e^{-z}),$ let us take $\\Omega=\\phi(\\mathbb{D})$ and $q(z)=\\sqrt{1+z}$. Now let  $\\psi:\\mathbb{C}^2\\times\\mathbb{D}\\rightarrow\\mathbb{C}$ be defined as $\\psi(a,b;z)=2a(a+b)\/(2a+b).$ Then by using the admissibility conditions for $\\sqrt{1+z}$ given by Madaan et al.~\\cite{vibha}, it suffices to show that $\\psi\\in\\Psi[\\Omega,\\sqrt{1+z}],$ which is equivalent to, $\\psi(r,s;z)\\notin\\Omega,$ where $r=\\sqrt{2\\cos{2\\theta}}e^{i\\theta}$ and $s=me^{3i\\theta}\/(2\\sqrt{2\\cos{2\\theta}})=me^{2i\\theta}\/(2r)\\;(-\\pi\/4\\leq\\theta\\leq\\pi\/4).$ We observe that\n \\begin{equation*}\n  \\psi(r,s;z)=\\dfrac{2r(r+s)}{2r+s}=\\dfrac{2r\\left(r+\\dfrac{me^{2i\\theta}}{2r}\\right)}{2r+\\dfrac{me^{2i\\theta}}{2r}}=2\\sqrt{2\\cos{2\\theta}}e^{i\\theta}\\left(\\dfrac{m+4\\cos{2\\theta}}{m+8\\cos{2\\theta}}\\right).\n\\end{equation*}\nTherefore\n\\begin{equation*}\n  \\RE\\psi(r,s;z)=2\\sqrt{2\\cos{2\\theta}}\\cos{\\theta}\\left(\\dfrac{m+4\\cos{2\\theta}}{m+8\\cos{2\\theta}}\\right),\n\\end{equation*}\nwhich is an increasing function of $m.$ So for $m\\geq 1,$ we have\n\\begin{equation*}\n \\RE\\psi(r,s;z)\\geq2\\sqrt{2\\cos{2\\theta}}\\cos{\\theta}\\left(\\dfrac{1+4\\cos{2\\theta}}{1+8\\cos{2\\theta}}\\right)=:g(\\theta),\n\\end{equation*}\nwhich at $\\theta=0$ becomes $g(0)=10\\sqrt{2}\/9\\approx 1.57.$ Since $\\max\\RE(2\/(1+e^{-z}))\\leq 2e\/(1+e)\\approx 1.46,$ it is easy to conclude that $\\psi(r,s,z)\\notin\\Omega$ and the result follows.\n\\end{proof}\n\\begin{example}\nLet $p(z)=1+a_1z+a_2z^2+\\cdots$ be analytic $\\mathbb{D}$ with $p(z)\\not\\equiv1.$ Then $p(z)\\prec 2\/(1+e^{-z}),$ whenever\n\\begin{equation*}\n  \\dfrac{2p(z)(p(z)+zp'(z))}{2p(z)+zp'(z)}\\prec\\sqrt{1+z}.\n\\end{equation*}\n\\end{example}\n\\begin{proof}\nLet us suppose $q(z)=2\/(1+e^{-z})$ and $\\Omega=\\phi(\\mathbb{D}),$ with $\\phi(z)=\\sqrt{1+z}$. Also, let  $\\psi:\\mathbb{C}^2\\times\\mathbb{D}\\rightarrow\\mathbb{C}$ is a function given by $\\psi(a,b;z)=2a(a+b)\/(2a+b).$ By applying the admissibility conditions for $2\/(1+e^{-z})$ given by Kumar and Goel~\\cite{pri3}, it is sufficient to prove that $\\psi\\in\\Psi[\\Omega,2\/(1+e^{-z})],$ which means $\\psi(r,s;z)\\notin\\Omega,$ with $r=2\/(1+e^{-e^{i\\theta}})$ and $s=(2me^{i\\theta}e^{-e^{i\\theta}})\/(1+e^{-e^{i\\theta}})^2=(mre^{i\\theta}e^{-e^{i\\theta}})\/(1+e^{-e^{i\\theta}})\\;(-\\pi\\leq\\theta\\leq\\pi).$ We observe that\n \\begin{equation*}\n  \\psi(r,s;z)=\\dfrac{2r(r+s)}{2r+s}=\\dfrac{4}{1+e^{-e^{i\\theta}}}\\left(\\dfrac{1+e^{-e^{i\\theta}}+me^{i\\theta}e^{-e^{i\\theta}}}{2+2e^{-e^{i\\theta}}+me^{i\\theta}e^{-e^{i\\theta}}}\\right).\n\\end{equation*}\nThen\n\\begin{equation*}\n  \\RE\\psi(r,s;z)=\\dfrac{N(\\theta)}{D(\\theta)}=:g(\\theta),\n\\end{equation*}\nwhere\n\\begin{eqnarray*}\nN(\\theta)&=&m^2 e^{\\cos \\theta}+m^2 \\cos (\\sin \\theta)+5 m e^{\\cos \\theta} \\cos \\theta+3 m e^{2 \\cos \\theta} \\cos (\\theta-\\sin \\theta)\\\\\n& &+m \\cos (\\theta-\\sin \\theta)+2 m \\cos (\\theta+\\sin \\theta)+m e^{\\cos \\theta} \\cos (\\theta-2 \\sin \\theta)+4 e^{\\cos \\theta}\\\\\n& &+2 e^{3 \\cos \\theta}+6 e^{2 \\cos \\theta} \\cos (\\sin \\theta)+2 \\cos (\\sin \\theta)+2 e^{\\cos \\theta} \\cos (2 \\sin \\theta)\n\\end{eqnarray*} and \\begin{eqnarray*}\nD(\\theta)&=&(1+e^{2\\cos{\\theta}}+2e^{\\cos{\\theta}}\\cos{(\\sin{\\theta})})(4+m^2+4e^{2\\cos{\\theta}}+4m\\cos{\\theta}\\\\\n& &+8e^{\\cos{\\theta}}\\cos{(\\sin{\\theta})}+4me^{\\cos{\\theta}}\\cos{(\\theta-\\sin{\\theta})}).\n\\end{eqnarray*}\nWe observe that $g(0)=4e(1+e+m)\/(1+e)(2+2e+m)$ and $m\\geq 1,$ so we have $g(0)\\geq 4e(2+e)\/(1+e)(3+2e)\\approx 1.635.$ Thus it is easy to conclude that $\\psi(r,s;z)\\notin\\Omega$ and the result follows.\n\\end{proof}\n \\begin{theorem}\n Let $-1\\leq B<A\\leq 1,$ $-1\\leq E<D\\leq 1,$ $L:=k+2,$ $M:=2(A+B)+k(2A-B),$ $N:=2AB,$ $G:=2(E-D),$ $H:=2AE(k+2)-2BE(k-1)-AD(k+2)+BD(k-4),$ $I:=2A^2E(k+2)-2ABE(k-2)-ABD(k+4)+B^2D(k-2)$ and $J:=2A^2BE-2AB^2D$ with\n\\begin{equation}\\label{3}\n  2E(1+A)-D(1+B))>0.\n\\end{equation}\n In addition, assume\n\\begin{equation}\\label{2}\nGH+HI-3GJ+IJ+12GJ\\geq 4|GI+HJ|\\quad (k\\geq 1)\n\\end{equation}\nand\n\\begin{equation}\\label{4}\n  3+2AB+D(B+1)(A(2B+3)+B+2)\\geq 2 E(A+1)(A(B+2)+1)+|4A+B|.\n\\end{equation}\n   Let $p(z)=1+a_1z+a_2z^2+\\cdots$ be analytic in $\\mathbb{D}$ with $p(z)\\not\\equiv 1.$ Then\n   \\begin{equation*}\n     \\dfrac{2p(z)(p(z)+zp'(z))}{2p(z)+zp'(z)}\\prec\\dfrac{1+Dz}{1+Ez} \\quad\\Rightarrow\\quad p(z)\\prec\\dfrac{1+Az}{1+Bz}.\n   \\end{equation*}\n \\end{theorem}\n \\begin{proof}\n  Define $P(z)$ by\n  \\begin{equation*}\n    P(z):=\\dfrac{2p(z)(p(z)+zp'(z))}{2p(z)+zp'(z)}\\quad\\text{and}\\quad\\omega(z):=\\dfrac{p(z)-1}{A-Bp(z)},\n  \\end{equation*}\n  or equivalently $p(z)=(1+A\\omega(z))\/(1+B\\omega(z)).$ Then $\\omega(z)$ is clearly meromorphic in $\\mathbb{D}$ and $\\omega(0)=0.$ We need to show that $|\\omega(z)|<1$ in $\\mathbb{D}.$ We have\n \\begin{equation*}\n   P(z)=\\dfrac{2\\left(\\dfrac{1+A\\omega(z)}{1+B\\omega(z)}\\right)\\left(\\dfrac{1+A\\omega(z)}{1+B\\omega(z)}+\\dfrac{(A-B)z\\omega'(z)}{(1+B\\omega(z))^2}\\right)}{2\\left(\\dfrac{1+A\\omega(z)}{1+B\\omega(z)}\\right)+\\dfrac{(A-B)z\\omega'(z)}{(1+B\\omega(z))^2}}.\n \\end{equation*}\n Therefore\n{\\small \\begin{equation*}\n   \\dfrac{P(z)-1}{D-E P(z)}=-\\dfrac{(A-B)(2\\omega(z)(1+A\\omega(z))(1+B\\omega(z))+(1+2A\\omega(z)-B\\omega(z))z\\omega'(z))}{2(1+A\\omega(z))(1+B\\omega(z))\\Phi_1(z)+(A-B)\\Phi_2(z)z\\omega'(z)},\n \\end{equation*}}\n where\n \\begin{equation*}\n \\Phi_1(z)=E(1+A\\omega(z))-D(1+B\\omega(z))\\;\\text{and}\\;\\Phi_2(z)=2E(1+A\\omega(z))-D(1+B\\omega(z)).\n \\end{equation*}\nOn the contrary if there exists a point $z_0\\in\\mathbb{D}$ such that $\\max_{|z|\\leq|z_0|}|\\omega(z)|=|\\omega(z_0)|=1,$ then by~\\cite[Lemma~1.3]{rush}, there exists $k\\geq1$ such that $z_0\\omega'(z_0)=k\\omega(z_0).$ Let $\\omega(z_0)=e^{i\\theta},$ then we have\n \\begin{eqnarray*}\n   \\left|\\dfrac{P(z_0)-1}{D-EP(z_0)}\\right|=(A-B)\\left|\\dfrac{L+Me^{i\\theta}+Ne^{2i\\theta}}{G+He^{i\\theta}+Ie^{2i\\theta}+Je^{3i\\theta}}\\right|.\n \n \\end{eqnarray*}\n We observe that\n \\begin{equation*}\n   |L+Me^{i\\theta}+Ne^{2i\\theta}|^2=L^2+M^2+N^2-2LN+2(L+N)M\\cos{\\theta}+4LN\\cos^2{\\theta}.\n \\end{equation*}\nChoose $t:=\\cos{\\theta}\\in[-1,1].$ Since\n \\begin{equation*}\n   \\min\\{at^2+bt+c:-1\\leq t\\leq1\\}=\\begin{cases}\n                                      \\dfrac{4ac-b^2}{4a}, & \\quad\\text{if}\\;a>0\\; \\text{and}\\;|b|<2a \\\\\n                                      a-|b|+c, & \\quad\\text{otherwise,}\n                                    \\end{cases}\n \\end{equation*}\nwe have $|L+Me^{i\\theta}+Ne^{2i\\theta}|^2\\geq (L-|M|+N)^2.$ Next we consider\n\\begin{eqnarray*}\n  |G+He^{i\\theta}+Ie^{2i\\theta}+Je^{3i\\theta}|^2&=&G^2+H^2+I^2+J^2-2GI-2HJ+(2GH\\\\\n& &+2HI-6GJ+2IJ)\\cos{\\theta}+(4GI+4HJ)\\cos^2{\\theta}\\\\\n& &+8GJ\\cos^3{\\theta},\n\\end{eqnarray*}\nwhich is an increasing function of $t=\\cos{\\theta}\\in[-1,1]$ in view of~\\eqref{2}. Thus we have $|G+He^{i\\theta}+Ie^{2i\\theta}+Je^{3i\\theta}|^2\\leq (G+H+I+J)^2.$ Therefore\n\\begin{equation*}\n  \\left|\\dfrac{P(z_0)-1}{D-EP(z_0)}\\right|^2\\geq \\left(\\dfrac{L-|M|+N}{G+H+I+J}\\right)^2=:\\psi(k) ,\\end{equation*}\nwhich in view of~\\eqref{3} is an increasing function of $k.$ So we have $\\psi(k)\\geq\\psi(1)$ and therefore\n\\begin{equation*}\n\\left|\\dfrac{P(z_0)-1}{D-EP(z_0)}\\right|\\geq \\dfrac{3+2AB-|4A+B|}{2E(A+1)(A(B+2)+1)-D(B+1)(A(2 B+3)+B+2)},\n\\end{equation*}\nwhich in view of~\\eqref{4} is greater than or equal to $1.$ This contradicts that $P(z)\\prec(1+Dz)(1+Ez)$ and that completes the proof.\n\\end{proof}\n\\begin{note}\nThe fact that the equations~\\eqref{3},~\\eqref{2} and~\\eqref{4} hold simultaneously is validated by the following set of values: $A=3\/8,\\;B=0,\\;D=1,\\;E=123\/128$.\n\\end{note}\nIn the next two results, we consider a combination of harmonic mean, geometric mean and arithmetic mean of $p(z)$ and $p(z)+zp'(z)\/p(z)$. Note that these results involve some constants given by $x=\\IM p(z_0)$ and $y=z_0p'(z_0)\/m,$ where $z_0$ and $m$ are as defined in Lemma~\\ref{jack} for $f(z)=(p(z)-1)\/(1-2\\alpha+p(z)).$\n\\begin{theorem}\\label{htheo1}\n\tLet $\\gamma\\in[0,1],$ $\\alpha\\in[0,1),$ $\\delta\\in[1,2],$ $\\beta\\in[0,1)$ and $\\rho\\in[0,1]$ be such that $\\rho\\geq \\alpha(1+2\\alpha),$ whenever $\\alpha\\in[0,1\/2].$ Also, let $p$ be an analytic function with $p(0)=1$ and\n\\begin{equation*}\n\\RE\\left(\\gamma(p(z))^{\\delta}+(1-\\gamma)\\dfrac{p(z)+\\dfrac{zp'(z)}{p(z)}}{1+\\rho\\dfrac{zp'(z)}{p^2(z)}}\\right)>\\beta,\n\\end{equation*}\nfor $\\beta\\geq \\gamma\\alpha+(1-\\gamma)\\beta_0,$ where $\\beta_0$ is given as follows:\\\\\n\\begin{equation}\\label{h4}\n\t\\beta_0=\\begin{cases}\\alpha\\dfrac{(1+\\alpha)(1-2\\alpha)}{\\rho(1-\\alpha)-2\\alpha^2},\\quad &\\text{if}\\;I_1\\;\\text{holds}\\\\\n\t\t\\alpha\\quad &\\text{if}\\;(\\sim I_1)\\wedge I_2\\;\\text{holds}\\\\\n\t\t\\alpha+\\dfrac{\\rho(1-\\rho)(1-\\alpha)(2(1-\\alpha)+\\rho)}{16\\alpha(2\\alpha^2-\\rho(1-\\alpha))} & \\text{if}\\;(\\sim I_1)\\wedge(\\sim I_2)\\wedge I_3\\; \\text{holds}\\\\\n\t\t\\alpha+\\dfrac{\\rho(1-\\rho)}{16\\alpha(1-\\alpha)}\\left(\\dfrac{2\\alpha^2-\\rho(1-\\alpha)}{2(1-\\alpha)+\\rho}\\right)&\\text{if}\\;(\\sim I_1)\\wedge(\\sim I_2)\\wedge(\\sim I_3)\\;\\text{holds},\n\t\\end{cases}\n\\end{equation}\nwith\n\\begin{equation}\\label{h14}\nI_1: 0\\leq\\alpha\\leq 1\/2,\\;\\;I_2:\\alpha^2-x^2+\\rho my>0\\;\\;\\text{and}\\;\\;I_3:x^2\\geq \\tfrac{(2\\alpha^2-\\rho(1-\\alpha))(1-\\alpha)}{2(1-\\alpha)+\\rho},\n\\end{equation}\nprovided $x>0$ and $my\\leq-\\tfrac{(1-\\alpha)^2+x^2}{2(1-\\alpha)}.$ Then $\\RE p(z)>\\alpha.$\n\\end{theorem}\n\\begin{proof}\nIf we let $q(z)=(1+(1-2\\alpha)z)\/(1-z),$ then it suffices to show that $p\\prec q.$ Let us suppose that $p$ is not subordinate to $q.$ Then by~\\cite[Lemma~2.2d,p~24]{ds} and~\\cite[Lemma~2.2f,p~26]{ds}, there exist $z_0\\in\\mathbb{D},$ $\\zeta_0\\in\\partial\\mathbb{D}\\backslash\\{1\\}$ and $m\\geq 1$ such that\n\\begin{equation}\\label{h1}\n\tp(z_0)=q(\\zeta_0)=\\alpha+ix\\quad\\text{and}\\quad z_0p'(z_0)=m\\zeta_0 q'(\\zeta_0)=:my\\leq -\\dfrac{(1-\\alpha)^2+x^2}{2(1-\\alpha)}.\n\\end{equation}\nConsequently, we have\n\\begin{equation}\\label{h6}\n\\gamma(p(z_0))^{\\delta}+(1-\\gamma)\\dfrac{p(z_0)+\\dfrac{z_0p'(z_0)}{p(z_0)}}{1+\\rho\\dfrac{z_0p'(z_0)}{p^2(z_0)}}=\\gamma(q(\\zeta_0))^{\\delta}+(1-\\gamma)\\dfrac{q(\\zeta_0)+\\dfrac{m\\zeta_0q'(\\zeta_0)}{q(\\zeta_0)}}{1+\\rho\\dfrac{m\\zeta_0q'(\\zeta_0)}{q^2(\\zeta_0)}}.\n\\end{equation}\nNow let $E(\\delta)=(q(\\zeta_0))^{\\delta}$ and $L=\\{E(\\delta):\\delta\\in[1,2]\\}.$ Geometrically, L represents an arc of the logarithmic spiral joining the points $E(1)=q(\\zeta_0)$ and $E(2)=(q(\\zeta_0))^2.$ We know that L cuts each radial halfline at an angle, which is constant. Clearly, $\\arg(q(\\zeta_0))^{\\delta}$ is an increasing function of $\\delta$ and thus, L is in the closed halfplane containing the origin and bounded by the line $\\RE z=\\RE E(1).$ As a result, we have\n\\begin{equation}\\label{h7}\n\t\\RE (q(\\zeta_0))^{\\delta}\\leq \\alpha,\\quad\\delta\\in[1,2].\n\\end{equation}\nWe observe that\n\\begin{equation}\\label{h5}\n\\dfrac{q(\\zeta_0)+\\dfrac{m\\zeta_0q'(\\zeta_0)}{q(\\zeta_0)}}{1+\\rho\\dfrac{m\\zeta_0q'(\\zeta_0)}{q^2(\\zeta_0)}}=q(\\zeta_0)+(1-\\rho)\\dfrac{m\\zeta_0q'(\\zeta_0)}{q(\\zeta_0)+\\rho m\\zeta_0 q'(\\zeta_0)\/q(\\zeta_0)}.\n\\end{equation}\nUsing~\\eqref{h1} and~\\eqref{h5}, we have\n\\begin{eqnarray}\\nonumber\\label{hh2}\n\t\\RE\\left(\\dfrac{q(\\zeta_0)+\\dfrac{m\\zeta_0q'(\\zeta_0)}{q(\\zeta_0)}}{1+\\rho\\dfrac{m\\zeta_0q'(\\zeta_0)}{q^2(\\zeta_0)}}\\right)&=&\\RE \\left(\\alpha+ix+(1-\\rho)\\dfrac{my}{\\alpha+ix+\\tfrac{\\rho my}{\\alpha+ix}}\\right)\\\\ \\nonumber\n\t& =& \\alpha+(1-\\rho)\\dfrac{my\\alpha(\\alpha^2+x^2+\\rho m y)}{(\\alpha^2-x^2+\\rho m y)^2+4\\alpha^2x^2}.\\\\\n\t\\end{eqnarray}\nSince $my<0,$ we may observe that\n\\begin{equation}\\label{h2}\n\\RE\\left(\\dfrac{q(\\zeta_0)+\\dfrac{m\\zeta_0q'(\\zeta_0)}{q(\\zeta_0)}}{1+\\rho\\dfrac{m\\zeta_0q'(\\zeta_0)}{q^2(\\zeta_0)}}\\right)\\leq  \\alpha+(1-\\rho)\\dfrac{my\\alpha(\\alpha^2-x^2+\\rho m y)}{(\\alpha^2-x^2+\\rho m y)^2+4\\alpha^2x^2}.\n\\end{equation}\nCase~(i): $0\\leq \\alpha\\leq 1\/2.$\\\\\nFrom hypothesis, we have $\\rho\\geq\\alpha(1+2\\alpha)\\geq 2\\alpha^2\/(1-\\alpha)$ and from~\\eqref{h1}, we have $my\\leq -\\tfrac{(1-\\alpha)^2+x^2}{2(1-\\alpha)}.$ Therefore\n\\begin{equation*}\n\t\\alpha^2-x^2+\\rho my\\leq \\dfrac{2\\alpha^2-\\rho(1-\\alpha)}{2}-x^2\\dfrac{2(1-\\alpha)+\\rho}{2(1-\\alpha)}\\leq 0.\n\\end{equation*}\nNow using the fact that $\\alpha^2x^2>0$ and that $my\\leq -\\tfrac{(1-\\alpha)^2+x^2}{2(1-\\alpha)}\\leq -\\tfrac{1-\\alpha}{2},$ we obtain from~\\eqref{h2},\n\\begin{eqnarray*}\n\t\\RE\\left(\\dfrac{q(\\zeta_0)+\\dfrac{m\\zeta_0q'(\\zeta_0)}{q(\\zeta_0)}}{1+\\rho\\dfrac{m\\zeta_0q'(\\zeta_0)}{q^2(\\zeta_0)}}\\right)&\\leq&  \\alpha+(1-\\rho)\\dfrac{my\\alpha(\\alpha^2-x^2+\\rho m y)}{(\\alpha^2-x^2+\\rho m y)^2}\\\\\n\t&=&\\alpha+(1-\\rho)\\dfrac{\\alpha}{\\tfrac{\\alpha^2-x^2}{my}+\\rho}\\\\\n\t&\\leq& \\alpha +(1-\\rho)\\dfrac{\\alpha}{\\rho-\\tfrac{2\\alpha^2}{1-\\alpha}}\\\\\n\t&=&\\alpha\\dfrac{(1+\\alpha)(1-2\\alpha)}{\\rho (1-\\alpha)-2\\alpha^2}.\n\\end{eqnarray*}\nCase~(ii): $1\/2<\\alpha<1$ and $\\alpha^2-x^2+\\rho my>0.$\\\\\nSince $\\alpha>1\/2,$ we have $2\\alpha^2\/(1-\\alpha)>1$ and therefore, $\\rho<2\\alpha^2\/(1-\\alpha).$ Also, since $my<0$ and $\\alpha^2-x^2+\\rho my>0,$ we have\n\\begin{eqnarray*}\n\t\t\\RE\\left(\\dfrac{q(\\zeta_0)+\\dfrac{m\\zeta_0q'(\\zeta_0)}{q(\\zeta_0)}}{1+\\rho\\dfrac{m\\zeta_0q'(\\zeta_0)}{q^2(\\zeta_0)}}\\right)&\\leq&\\alpha+(1-\\rho)\\dfrac{my\\alpha(\\alpha^2-x^2+\\rho m y)}{(\\alpha^2-x^2+\\rho m y)^2+4\\alpha^2x^2}\\leq \\alpha.\n\\end{eqnarray*}\nCase~(iii): $1\/2<\\alpha<1,$ $\\alpha^2-x^2+\\rho my\\leq0$ and $ x^2\\geq \\tfrac{(2\\alpha^2-\\rho(1-\\alpha))(1-\\alpha)}{2(1-\\alpha)+\\rho}.$\nFrom~\\eqref{hh2}, we have\n\\begin{eqnarray}\\nonumber\\label{h3}\n\t\\RE\\left(\\dfrac{q(\\zeta_0)+\\dfrac{m\\zeta_0q'(\\zeta_0)}{q(\\zeta_0)}}{1+\\rho\\dfrac{m\\zeta_0q'(\\zeta_0)}{q^2(\\zeta_0)}}\\right)&\\leq&\\alpha+(1-\\rho)my\\dfrac{\\alpha(\\alpha^2+x^2+\\rho my)}{(\\alpha^2-x^2+\\rho my)^2+4\\alpha^2 x^2}\\\\ \\nonumber\n\t&\\leq& \\alpha+(1-\\rho)\\dfrac{\\alpha\\rho m^2y^2}{(\\alpha^2-x^2+\\rho my)^2+4\\alpha^2 x^2}\\\\ \\nonumber\n\t&\\leq&\\alpha+(1-\\rho)\\dfrac{\\alpha\\rho m^2y^2}{4\\alpha^2 x^2}\\\\\n\t&=&\\alpha+(1-\\rho)\\dfrac{\\rho m^2y^2}{4\\alpha x^2}.\n\\end{eqnarray}\nUsing the inequality $my\\leq-\\tfrac{(1-\\alpha)^2+x^2}{2(1-\\alpha)}\\leq -\\tfrac{1-\\alpha}{2}$ and the condition on $x^2$ in~\\eqref{h3}, we obtain\n\\begin{eqnarray*}\n\t\\RE\\left(\\dfrac{q(\\zeta_0)+\\dfrac{m\\zeta_0q'(\\zeta_0)}{q(\\zeta_0)}}{1+\\rho\\dfrac{m\\zeta_0q'(\\zeta_0)}{q^2(\\zeta_0)}}\\right)&\\leq& \\alpha+(1-\\rho)\\dfrac{\\rho}{4\\alpha}\\left(\\dfrac{1-\\alpha}{2}\\right)^2\\left(\\dfrac{2(1-\\alpha)+\\rho}{(2\\alpha^2-\\rho(1-\\alpha))(1-\\alpha)}\\right)\\\\\n&=&\\alpha+\\dfrac{\\rho(1-\\rho)(1-\\alpha)(2(1-\\alpha)+\\rho)}{16\\alpha(2\\alpha^2-\\rho(1-\\alpha))}.\n\\end{eqnarray*}\nCase~(iv): $1\/2<\\alpha<1,$ $\\alpha^2-x^2+\\rho my\\leq0$ and $ x^2< \\tfrac{(2\\alpha^2-\\rho(1-\\alpha))(1-\\alpha)}{2(1-\\alpha)+\\rho}.$\\\\\nConsidering $my\\leq-\\tfrac{(1-\\alpha)^2+x^2}{2(1-\\alpha)}\\leq -\\tfrac{x^2}{2(1-\\alpha)}$ and using it in~\\eqref{h3} along with the condition on $x^2$, we get\n\\begin{eqnarray*}\n\t\t\\RE\\left(\\dfrac{q(\\zeta_0)+\\dfrac{m\\zeta_0q'(\\zeta_0)}{q(\\zeta_0)}}{1+\\rho\\dfrac{m\\zeta_0q'(\\zeta_0)}{q^2(\\zeta_0)}}\\right)&\\leq& \\alpha+(1-\\rho)\\dfrac{\\rho}{4\\alpha x^2}\\left(\\dfrac{-x^2}{2(1-\\alpha)}\\right)^2\\\\\n\t&=&\\alpha+\\dfrac{\\rho(1-\\rho)x^2}{16\\alpha(1-\\alpha)^2}\\\\\n\t&\\leq& \\alpha+\\dfrac{\\rho(1-\\rho)}{16\\alpha(1-\\alpha)}\\left(\\dfrac{2\\alpha^2-\\rho(1-\\alpha)}{2(1-\\alpha)+\\rho}\\right).\n\\end{eqnarray*}\nCombining all the cases we obtain\n\\begin{equation}\\label{ha7}\n\t\\RE\\left(\\dfrac{q(\\zeta_0)+\\dfrac{m\\zeta_0q'(\\zeta_0)}{q(\\zeta_0)}}{1+\\rho\\dfrac{m\\zeta_0q'(\\zeta_0)}{q^2(\\zeta_0)}}\\right)\\leq\\beta_0,\t\n\\end{equation}\nwhere $\\beta_0$ is given by~\\eqref{h4}. From~\\eqref{h6},~\\eqref{h7} and~\\eqref{ha7}, we have\n\\begin{equation*}\n\t\\RE\\left(\\gamma(p(z_0))^{\\delta}+(1-\\gamma)\\dfrac{p(z_0)+\\dfrac{z_0p'(z_0)}{p(z_0)}}{1+\\rho\\dfrac{z_0p'(z_0)}{p^2(z_0)}}\\right)\\leq \\gamma\\alpha+(1-\\gamma)\\beta_0,\n\\end{equation*}\nwhich contradicts the hypothesis. Hence the result follows.\n\\end{proof}\n\\begin{remark}\n\tIf we take $\\gamma=0$ and $\\alpha\\in[0,1\/2],$ the above result reduces to a result of Kanas~\\cite[Theorem~2.4]{kanas2017}. Infact, we extended this result for $\\alpha\\in[0,1).$\n\\end{remark}\n\\begin{remark}\n\tBy taking $\\gamma=0,\\;\\rho=1\/2$ and $\\alpha=0,$ we obtain a result of Kanas and Tudor~\\cite[Theorem~2.3]{kanasbmms}\n\\end{remark}\n\\begin{theorem}\\label{genels}\n\tLet $\\gamma\\in[0,1],$ $\\alpha\\in[0,1),$ $\\mu\\in[0,1],$ $\\delta\\in[1,2],$ $\\beta\\in[0,1)$ and $\\rho\\in[0,1]$ be such that $\\rho\\geq \\alpha(1+2\\alpha),$ whenever $\\alpha\\in[0,1\/2].$ Also, let $p$ be an analytic function with $p(0)=1$ and\n\\begin{equation*}\n\t\\RE\\left(\\gamma(p(z))^{\\delta}+(1-\\gamma)\\dfrac{(p(z))^{\\mu}\\left(p(z)+\\dfrac{zp'(z)}{p(z)}\\right)^{1-\\mu}}{1+\\rho\\dfrac{zp'(z)}{p^2(z)}}\\right)>\\beta,\n\\end{equation*}\nfor $\\beta\\geq \\gamma\\alpha+(1-\\gamma)\\beta_1,$ where $\\beta_1$ is defined as follows:\\\\\n{\\small \\begin{equation*}\n\t\t\\beta_1=\\begin{cases}\\alpha&I_4\\;\\text{holds}\\\\\n\t\t\t\\alpha+\\dfrac{2\\alpha\\rho^2(1-2\\alpha)}{(\\rho-2(1-\\alpha))^2(\\rho(1-\\alpha)-2\\alpha^2)},\\quad &I_1\\wedge (\\sim I_4)\\;\\text{holds}\\\\\n\t\t\\alpha+\\dfrac{\\alpha\\rho(1-\\alpha)(4\\alpha^2-\\rho(1-\\alpha))}{4(2\\alpha^2-\\rho(1-\\alpha))^2}\\quad &(\\sim I_1)\\wedge I_2\\wedge(\\sim I_4)\\;\\text{holds}\\\\\n\t\t\\alpha+\\dfrac{\\alpha\\rho(1-\\alpha)(2(1-\\alpha)+\\rho)}{2(4\\alpha^2(1-\\alpha)+\\rho(2\\alpha-1))} & (\\sim I_1)\\wedge(\\sim I_2)\\wedge I_3\\wedge({\\sim I_4})\\; \\text{holds}\\\\\n\t\t\\alpha+\\dfrac{\\alpha\\rho(2\\alpha^2-\\rho(1-\\alpha))}{2(4\\alpha^2(1-\\alpha)+\\rho(2\\alpha-1))}&(\\sim I_1)\\wedge(\\sim I_2)\\wedge(\\sim I_3)\\wedge(\\sim I_4)\\;\\text{holds},\n\\end{cases}\n\\end{equation*}}\nwith $I_1,\\;I_2$ and $I_3$ given by~\\eqref{h14} and $I_4:\\alpha^2+x^2+\\rho my\\leq 0,$ provided $x>0$ and $my\\leq-\\tfrac{(1-\\alpha)^2+x^2}{2(1-\\alpha)}.$ Then $\\RE p(z)>\\alpha.$\n\\end{theorem}\n\\begin{proof}\nWe proceed as done in Theorem~\\ref{htheo1} to show that $p\\prec q,$ where $q(z)=(1+(1-2\\alpha)z)\/(1-z).$ For if $p\\nprec q,$ then there exists $z_0\\in\\mathbb{D},$ $\\zeta_0\\in\\partial\\mathbb{D}$ and $m\\geq1$ such that\n\\begin{align*}\n\t\\gamma(p(z_0))^{\\delta}&+(1-\\gamma)\\dfrac{(p(z_0))^{\\mu}\\left(p(z_0)+\\dfrac{z_0p'(z_0)}{p(z_0)}\\right)^{1-\\mu}}{1+\\rho\\dfrac{z_0p'(z_0)}{p^2(z_0)}}\\\\\n\t&=\\gamma(q(\\zeta_0))^{\\delta}+(1-\\gamma)\\dfrac{(q(\\zeta_0))^{\\mu}\\left(q(\\zeta_0)+\\dfrac{m\\zeta_0q'(\\zeta_0)}{q(\\zeta_0)}\\right)^{1-\\mu}}{1+\\rho\\dfrac{m\\zeta_0q'(\\zeta_0)}{q^2(\\zeta_0)}}.\n\\end{align*}\nFrom the proof of Theorem~\\ref{htheo1}, we have\n\\begin{equation}\\label{h99}\n\\RE (q(\\zeta_0))^{\\delta}\\leq \\alpha.\n\\end{equation}\nNow we set\n\\begin{equation*}\n\tE(\\mu)=\\dfrac{(q(\\zeta_0))^{\\mu}\\left(q(\\zeta_0)+\\dfrac{m\\zeta_0q'(\\zeta_0)}{q(\\zeta_0)}\\right)^{1-\\mu}}{1+\\rho\\dfrac{m\\zeta_0q'(\\zeta_0)}{q^2(\\zeta_0)}}=\\dfrac{q(\\zeta_0)\\left(1+\\dfrac{m\\zeta_0q'(\\zeta_0)}{q^2(\\zeta_0)}\\right)^{1-\\mu}}{1+\\rho\\dfrac{m\\zeta_0q'(\\zeta_0)}{q^2(\\zeta_0)}}\n\\end{equation*}\nand let $L=\\{E(\\mu):\\mu\\in[0,1]\\}.$ Note that\n\\begin{equation*}\n\tE(0)=\\dfrac{q(\\zeta_0)+\\dfrac{m\\zeta_0q'(\\zeta_0)}{q(\\zeta_0)}}{1+\\rho\\dfrac{m\\zeta_0q'(\\zeta_0)}{q^2(\\zeta_0)}}\\quad\\text{and}\\quad E(1)=\\dfrac{q(\\zeta_0)}{1+\\rho\\dfrac{m\\zeta_0q'(\\zeta_0)}{q^2(\\zeta_0)}}.\n\\end{equation*}\nIt was shown in the proof of Theorem~\\ref{htheo1} that $\\RE E(0)\\leq \\beta_0,$ where $\\beta_0$ is given by~\\eqref{h4}. Now we consider $\\RE E(1),$ which by using the conditions~\\eqref{h1} becomes\n\\begin{eqnarray*}\n\\RE E(1)&=&\\RE\\left(\\dfrac{\\alpha+ix}{1+\\dfrac{\\rho my}{(\\alpha+ix)^2}}\\right)\\\\\n&=&\\dfrac{\\alpha(\\alpha^4+x^4+2\\alpha^2x^2+\\alpha^2\\rho my-3x^2\\rho my)}{(\\alpha^2-x^2+\\rho my)^2+4\\alpha^2x^2}\\\\\n&=&\\alpha-\\dfrac{\\alpha\\rho my(\\alpha^2+x^2+\\rho my)}{(\\alpha^2-x^2+\\rho my)^2+4\\alpha^2x^2}.\\\\\n\\end{eqnarray*}\nIt is easy to note that $\\RE E(1)\\leq\\alpha,$ whenever $\\alpha^2+x^2+\\rho my\\leq 0.$ So now let us assume that $\\alpha^2+x^2+\\rho my>0$ for the rest of the cases. Before we start our first case, we observe that\n\\begin{eqnarray}\\label{h9}\\nonumber\n-\\alpha\\rho my\\left(\\dfrac{\\alpha^2+x^2+\\rho my}{(\\alpha^2-x^2+\\rho my)^2+4\\alpha^2x^2}\\right)&\\leq&-\\alpha\\rho my\\left(\\dfrac{\\alpha^2+x^2}{(\\alpha^2-x^2+\\rho my)^2+4\\alpha^2x^2}\\right)\\\\ \\nonumber\n&\\leq&\\dfrac{-\\alpha\\rho my(\\alpha^2+x^2)}{(\\alpha^2+x^2+\\rho my)^2}\\\\\n&=&\\dfrac{-\\alpha\\rho (\\alpha^2+x^2)}{my\\left(\\tfrac{\\alpha^2+x^2}{my}+\\rho\\right)^2},\n\\end{eqnarray}\n since $my<0$ and $\\alpha^2+x^2+\\rho my>0.$\\\\\nCase~(i): $0\\leq\\alpha\\leq 1\/2.$\\\\\nFrom~\\eqref{h1}, we have $my\\leq -\\tfrac{(1-\\alpha)^2+x^2}{2(1-\\alpha)},$ which further implies\n\\begin{equation}\\label{h8}\n\\alpha^2+x^2+\\rho my\\leq \\alpha^2-\\dfrac{\\rho(1-\\alpha)}{2}+x^2\\left(1-\\dfrac{\\rho}{2(1-\\alpha)}\\right).\n\\end{equation}\nThe inequality $\\alpha\\leq1\/2$ implies $1-\\rho\/2(1-\\alpha)>0.$ Also from hypothesis, we have $\\rho\\geq\\alpha(1+2\\alpha)\\geq 2\\alpha^2\/(1-\\alpha),$ which is sufficient to conclude that\n\\begin{equation*}\n\t\\alpha^2+x^2+\\rho my\\leq x^2\\left(1-\\dfrac{\\rho}{2(1-\\alpha)}\\right).\n\\end{equation*}\nThus\n\\begin{equation}\\label{h10}\n\t\\dfrac{\\alpha^2+x^2}{my}+\\rho\\geq \\rho-2(1-\\alpha).\n\\end{equation}\nWe know that $\\alpha^2+x^2+\\rho my>0,$ so from~\\eqref{h8} we obtain\n\\begin{equation}\\label{h11}\n\tx^2\\geq \\dfrac{(\\rho(1-\\alpha)-2\\alpha^2)(1-\\alpha)}{2(1-\\alpha)-\\rho}.\n\\end{equation}\nUsing~\\eqref{h10} and~\\eqref{h11} in~\\eqref{h9}, we get\n\\begin{eqnarray*}\n-\\alpha\\rho my\\left(\\dfrac{\\alpha^2+x^2+\\rho my}{(\\alpha^2-x^2+\\rho my)^2+4\\alpha^2x^2}\\right)&\\leq& \\dfrac{2\\alpha\\rho(1-\\alpha)(\\alpha^2+x^2)}{x^2(\\rho-2(1-\\alpha))^2}\t\\\\\n&=&\\dfrac{2\\alpha\\rho(1-\\alpha)\\left(\\tfrac{\\alpha^2}{x^2}+1\\right)}{(\\rho-2(1-\\alpha))^2}\\\\\n&\\leq&\\dfrac{2\\alpha(1-2\\alpha)\\rho^2}{(\\rho-2(1-\\alpha))^2(\\rho(1-\\alpha)-2\\alpha^2)}.\n\\end{eqnarray*}\nCase~(ii): $1\/2<\\alpha<1$ and $\\alpha^2-x^2+\\rho my>0$\\\\\nUsing the above condition, we obtain\n\\begin{eqnarray*}\n\\alpha^2+x^2+\\rho my&<&2(\\alpha^2+\\rho my)\\leq 2\\left(\\alpha^2-\\dfrac{\\rho(1-\\alpha)}{2}\\right).\n\\end{eqnarray*}\nThus we have\n\\begin{equation}\\label{h12}\n\t\\dfrac{\\alpha^2+x^2}{my}+\\rho\\geq \\dfrac{-2\\left(2\\alpha^2-\\rho(1-\\alpha)\\right)}{1-\\alpha}.\n\\end{equation}\nNext we observe that\n\\begin{eqnarray}\\label{ha13}\n\\alpha^2+x^2&\\leq& 2\\alpha^2+\\rho my\\leq 2\\alpha^2-\\dfrac{\\rho(1-\\alpha)}{2}.\n\\end{eqnarray}\nUsing~\\eqref{h12} and~\\eqref{ha13} in~\\eqref{h9}, we get\n\\begin{eqnarray*}\n-\\alpha\\rho my\\left(\\dfrac{\\alpha^2+x^2+\\rho my}{(\\alpha^2-x^2+\\rho my)^2+4\\alpha^2x^2}\\right)&\\leq& \\dfrac{2\\alpha\\rho\\left(2\\alpha^2-\\dfrac{\\rho(1-\\alpha)}{2}\\right)}{4(1-\\alpha)\\left(\\dfrac{2\\alpha^2-\\rho(1-\\alpha)}{1-\\alpha}\\right)^2}\\\\\n&=&\\dfrac{\\alpha\\rho(1-\\alpha)(4\\alpha^2-\\rho(1-\\alpha))}{4(2\\alpha^2-\\rho(1-\\alpha))^2}.\n\\end{eqnarray*}\nCase~(iii): $1\/2<\\alpha<1,$ $\\alpha^2-x^2+\\rho my\\leq0$ and $ x^2\\geq \\tfrac{(2\\alpha^2-\\rho(1-\\alpha))(1-\\alpha)}{2(1-\\alpha)+\\rho}.$\\\\\nSince $my<0,$ $\\alpha^2+x^2+\\rho my\\leq \\alpha^2+x^2,$ which further implies\n\\begin{equation}\\label{h13}\n\t\\dfrac{\\alpha^2+x^2}{my}+\\rho\\geq \\dfrac{-2(\\alpha^2+x^2)}{1-\\alpha}.\n\\end{equation}\nNow by using~\\eqref{h13} and the condition $my\\leq -(1-\\alpha)\/2$ in~\\eqref{h9}, we obtain\n\\begin{equation*}\n-\\alpha\\rho my\\left(\\dfrac{\\alpha^2+x^2+\\rho my}{(\\alpha^2-x^2+\\rho my)^2+4\\alpha^2x^2}\\right)\\leq \\dfrac{\\alpha\\rho(1-\\alpha)}{2(\\alpha^2+x^2)},\n\\end{equation*}\nwhich after applying the condition on $x^2$ becomes\n\\begin{equation*}\n-\\alpha\\rho my\\left(\\dfrac{\\alpha^2+x^2+\\rho my}{(\\alpha^2-x^2+\\rho my)^2+4\\alpha^2x^2}\\right)\\leq\\dfrac{\\alpha\\rho(1-\\alpha)(2(1-\\alpha)+\\rho)}{2(4\\alpha^2(1-\\alpha)+\\rho(2\\alpha-1))}.\n\\end{equation*}\nCase~(iv): $1\/2<\\alpha<1,$ $\\alpha^2-x^2+\\rho my\\leq0$ and $ x^2< \\tfrac{(2\\alpha^2-\\rho(1-\\alpha))(1-\\alpha)}{2(1-\\alpha)+\\rho}.$\\\\\nProceeding similarly as done in case~(iii) and replacing the condition $my\\leq -(1-\\alpha)\/2$ by $my\\leq -x^2\/2(1-\\alpha),$ we get\n\\begin{equation*}\n-\\alpha\\rho my\\left(\\dfrac{\\alpha^2+x^2+\\rho my}{(\\alpha^2-x^2+\\rho my)^2+4\\alpha^2x^2}\\right)\\leq \\dfrac{\\alpha\\rho}{2\\left(\\dfrac{\\alpha^2}{x^2}+1\\right)(1-\\alpha)},\t\n\\end{equation*}\nwhich by using the condition on $x^2$ becomes\n\\begin{eqnarray*}\n\t-\\alpha\\rho my\\left(\\dfrac{\\alpha^2+x^2+\\rho my}{(\\alpha^2-x^2+\\rho my)^2+4\\alpha^2x^2}\\right)&\\leq& \\dfrac{\\alpha\\rho}{2\\left(\\dfrac{\\alpha^2}{x^2}+1\\right)(1-\\alpha)}\\\\\n\t&\\leq & \\dfrac{\\alpha\\rho(2\\alpha^2-\\rho(1-\\alpha))}{2(4\\alpha^2(1-\\alpha)+\\rho(2\\alpha-1))}.\t\n\\end{eqnarray*}\nCombining all the cases, we obtain that $\\RE E(1)\\leq \\beta_1.$ We observe that $L$ represents an arc of logarithmic spiral with end points $E(0)$ and $E(1),$ such that it cuts every radial halfline at a constant angle. Also\n\\begin{equation*}\n\\arg E(\\mu)=\\arg(q(\\zeta_0))+(1-\\mu)\\left(1+\\dfrac{m\\zeta_0 q'(\\zeta_0)}{q^2(\\zeta_0)}\\right)-\\arg\\left(1+\\rho\\dfrac{m\\zeta_0 q'(\\zeta_0)}{q^2(\\zeta_0)}\\right)\n\\end{equation*}\nis a decreasing function of $\\mu$ and thus, $\\arg E(1)\\leq \\arg E(\\mu)\\leq \\arg E(0),\\; \\mu\\in[0,1].$ So we may conclude that $L$ lies in the closed halfplane containing the origin and determined by the line $\\RE z=\\RE E(1),$ which means\n\\begin{equation}\\label{h98}\n  \\RE E(\\mu)\\leq \\beta_1.\n\\end{equation}\nFrom~\\eqref{h99} and~\\eqref{h98}, we obtain\n\\begin{equation*}\n\t\\RE\\left(\\gamma(p(z))^{\\delta}+(1-\\gamma)\\dfrac{(p(z))^{\\mu}\\left(p(z)+\\dfrac{zp'(z)}{p(z)}\\right)^{1-\\mu}}{1+\\rho\\dfrac{zp'(z)}{p^2(z)}}\\right)\\leq\\gamma\\alpha+(1-\\gamma)\\beta_1\\leq\\beta,\n\\end{equation*}\nwhich contradicts the hypothesis and hence the result follows.\n\\end{proof}\n\\begin{remark}\n\tTaking $\\rho=0,$ we obtain the result \\cite[Theorem~2.3]{kanasels}\n\\end{remark}\n\\begin{remark}\n\tBy taking $\\mu=0,\\;\\gamma=0,\\;\\rho=1\/2$ and $\\alpha=0,$ we obtain the result~\\cite[Theorem~2.3]{kanasbmms}.\n\\end{remark}\n\\begin{remark}\n\tFor $\\gamma=0,\\;\\rho=0$ and $\\alpha=0,$ we obtain a result of Lewandowski et al.~\\cite{lewa}.\n\\end{remark}\n\\begin{remark}\n\tFor $\\delta=1,\\;\\rho=0,\\;\\mu=0$ and $\\alpha=0,$ we obtain a result of Sakaguchi~\\cite{sakaguchi}.\n\\end{remark}\n\\section{Applications}\nWe begin with the applications of Lemma~\\ref{mainn}:\n\\begin{theorem}\\label{kcorr}\n\tLet $\\delta\\in[0,1],$ $h\\in\\mathcal{Q}$ with $0\\in\\overline{h(\\mathbb{D})}$ and $\\Theta,\\Phi\\in\\mathcal{H}$ are such that $\\Theta(0)=1$ and\n\t\\begin{equation}\\label{hypo2}\n\t\t\\RE \\Phi(z)\\geq 5|\\Theta(z)-1|-\\RE(\\Theta(z)-1),\\quad z\\in\\mathbb{D}.\n\t\\end{equation}\n\tIf $p\\in\\mathcal{H}(\\delta;\\Theta,\\Phi),$ $p(0)=h(0)$ and $H_{\\delta;\\Theta,\\Phi,p}\\prec h,$ then $p\\prec h.$\n\\end{theorem}\n\\begin{proof}\n\tIn view of Lemma~\\ref{mainn}, it is sufficient to show that $\\Theta,\\;\\Phi$ and $h$ satisfies~\\eqref{hypo}. Since $h$ is convex and $h(0)=1,$ we can say that $h_1:=h-1\\in\\mathcal{C}.$ Using Marx Strohh\\\"{a}cker theorem~\\cite{ds}, we have $\\RE(\\zeta h_1'(\\zeta)\/h_1(\\zeta))>1\/2,$ which is equivalent to\n\t\\begin{equation*}\n\t\t\\left|\\dfrac{h_1(\\zeta)}{\\zeta h_1'(\\zeta)}-1\\right|\\leq 1,\n\t\\end{equation*}\n\twhich means\n\t\\begin{equation}\\label{10}\n\t\t\\left|\\dfrac{h(\\zeta)}{\\zeta h'(\\zeta)}-1\\right|\\leq 1+\\dfrac{1}{|h'(\\zeta)|}.\n\t\\end{equation}\n\tSince $h_1\\in\\mathcal{C},$ we have $|h_1'(z)|\\geq 1\/(1+r)^2$ on $|z|=r$~\\cite[Theorem.9, pp~118]{goodman}. We know that $\\zeta\\in\\partial\\mathbb{D},$ so we have $|h'(\\zeta)|=|h_1'(\\zeta)|\\geq 1\/4.$ Thus~\\eqref{10} reduces to\n\t\\begin{equation}\\label{11}\n\t\t\\left|\\dfrac{h(\\zeta)}{\\zeta h'(\\zeta)}-1\\right|\\leq 5.\n\t\\end{equation}\n\tNote that if $X,\\;Y\\in\\mathbb{C}$ and $|X-1|\\leq K,$ then\n\t\\begin{equation*}\n\t\t\\RE(X.Y)=\\RE Y+\\RE Y(X-1))\\geq \\RE Y-|Y|K.\n\t\\end{equation*}\n\tApplying this inequality to~\\eqref{hypo} and using~\\eqref{11}, we get\n\t\\begin{equation*}\n\t\t\\RE\\left( \\Phi(z)+\\dfrac{h(\\zeta)}{\\zeta h'(\\zeta)}(\\Theta(z)-1)\\right)\\geq \\RE \\Phi(z)+\\RE(\\Theta(z)-1)- 5|\\Theta(z)-1|.\n\t\\end{equation*}\n\tFrom~\\eqref{9} and~\\eqref{hypo2}, we conclude that $\\RE \\Psi(z)>0$ and~\\eqref{hypo} holds. Therefore the result follows by Lemma~\\ref{mainn}.\n\\end{proof}\n\\begin{corollary}\n\tLet $\\delta\\in[0,1],$ $h\\in\\mathcal{Q}$ with $0\\in\\overline{h(\\mathbb{D})}$ and $\\Theta\\in\\mathcal{H}$ be a bounded function such that $\\Theta(0)=1$ and $|\\Theta(z)|\\leq M\\;(z\\in\\mathbb{D})$ for some $M>0.$ Suppose $\\Phi\\in\\mathcal{H}$ be such that\n\t\\begin{equation}\\label{hypo3}\n\t\t\\RE \\Phi(z)\\geq 6(M+1).\n\t\\end{equation}\n\tIf $p\\in\\mathcal{H}(\\delta;\\Theta,\\Phi),$ $p(0)=h(0)$ and $H_{\\delta;\\Theta,\\Phi,p}\\prec h,$ then $p\\prec h.$\n\\end{corollary}\n\\begin{proof}\n\tWe know that $-\\RE(\\Theta(z)-1)\\leq |\\Theta(z)-1|,$ which gives\n\t\\begin{equation*}\n\t\t5|\\Theta(z)-1|-\\RE(\\Theta(z)-1)\\leq 6|\\Theta(z)-1|\\leq 6(|\\Theta(z)|+1)\\leq 6(M+1).\n\t\\end{equation*}\n\tClearly~\\eqref{hypo3} is sufficient for~\\eqref{hypo2} to hold true. Thus the result follows as an application of Theorem~\\ref{kcorr}.\n\\end{proof}\n\\begin{theorem}\n\tLet $\\delta\\in[0,1],$ and $\\Theta,\\Phi\\in\\mathcal{H}$ be such that $\\Theta(0)=1$ and\n\t\\begin{equation}\\label{z1}\n\t\t\\RE \\Phi(z)>2(|\\Theta(z)-1|-\\RE (\\Theta(z)-1)),\\quad z\\in\\mathbb{D}.\n\t\\end{equation}\n\tIf $p\\in\\mathcal{H}(\\delta;\\Theta,\\Phi),\\;p(0)=1$ and $H_{\\delta;\\Theta,\\Phi,p}\\prec \\sqrt{1+z},$ then $p\\prec \\sqrt{1+z}.$\n\\end{theorem}\n\\begin{proof}\n\tIn view of Lemma~\\ref{mainn}, if we take $h(z)=\\sqrt{1+z},$ then~\\eqref{hypo} reduces to\n\t\\begin{equation*}\n\t\t\\RE\\left(2(\\Theta(z)-1)\\left(1+\\dfrac{1}{\\zeta}\\right)+\\Phi(z)\\right)\\geq 2\\RE(\\Theta(z)-1)-2|\\Theta(z)-1|+\\RE \\Phi(z),\n\t\\end{equation*}\n\twhich by~\\eqref{z1} is greater than 0. Thus the result follows as an application of Lemma~\\ref{mainn}.\n\\end{proof}\n\\begin{theorem}\n\tLet $\\delta\\in[0,1],$ $\\gamma\\in(0,1]$ and $\\Theta,\\Phi\\in\\mathcal{H}$ be such that $\\Theta(0)=1,\\Theta'(0)>0$ and $\\RE \\Phi(z)>0.$ Suppose $p\\in\\mathcal{H}(\\delta;\\Theta,\\Phi\n\t),\\;p(0)=1$ and $p'(0)>0,$ then\n\t\\begin{equation}\\label{z2}\n\t\tH_{\\delta;\\Theta,\\Phi,p}\\prec\\left(\\dfrac{1+z}{1-z}\\right)^{\\gamma}\\quad\\Rightarrow\\quad p(z)\\prec \\left(\\dfrac{1+z}{1-z}\\right)^{\\gamma}.\n\t\\end{equation}\n\\end{theorem}\n\\begin{proof}\n\tLet $h(z)=((1+z)\/(1-z))^{\\gamma}.$ From the proof of~\\cite[Corollary 6]{pri4}, it is easy to observe that\n\t\\begin{equation*}\n\t\t\\RE (\\Theta(z)-1)\\dfrac{h(\\zeta)}{\\zeta h'(\\zeta)}=\\RE (\\Theta(z)-1)\\dfrac{1-\\zeta^2}{2\\gamma\\zeta}>0,\n\t\\end{equation*}\n\tprovided $\\Theta'(0)>0$ and $p'(0)>0$ which is true by hypothesis. Using the fact that $\\RE \\Phi(z)>0$ and Lemma~\\ref{mainn}, the implication~\\eqref{z2} follows and hence the result.\n\\end{proof}\n\\begin{theorem}\n\tLet $f\\in\\mathcal{A}$ and $g\\in\\mathcal{S}^*$ with $g(z)\\neq zf'(z)$ and\n\t\\begin{equation}\\label{0i}\n\t\t\\RE\\left(\\dfrac{2zf'(z)}{g(z)}-\\dfrac{2z(f'(z))^2}{3g(z)f'(z)+zf''(z)g(z)-zg'(z)f'(z)}\\right)>0,\n\t\\end{equation}\n\tthen $f\\in\\mathcal{K}.$\n\t\\end{theorem}\n\t\\begin{proof}\n\t\tLet $p(z)=zf'(z)\/g(z),$ then\n\t\t\\begin{equation*}\n\t\t\t\\dfrac{2zf'(z)}{g(z)}-\\dfrac{2z(f'(z))^2}{3g(z)f'(z)+zf''(z)g(z)-zg'(z)f'(z)}=\\dfrac{2p(z)(p(z)+zp'(z))}{2p(z)+zp'(z)}.\n\t\t\\end{equation*}\n\t\tFrom~\\eqref{0i}, we have\n\t\t\\begin{equation*}\n\t\t\t\\RE\\left(\\dfrac{2p(z)(p(z)+zp'(z))}{2p(z)+zp'(z)}\\right)>0,\n\t\t\\end{equation*}\n\t\tor equivalently,\n\t\t\\begin{equation*}\n\t\t\t\\dfrac{2p(z)(p(z)+zp'(z))}{2p(z)+zp'(z)}\\prec\\dfrac{1+z}{1-z}.\n\t\t\\end{equation*}\n\t\tBy applying Lemma~\\ref{mainn} with $\\Theta(z)=\\Phi(z)=1,\\;t=1\/2$ and $h(z)=(1+z)\/(1-z),$ we get $p(z)\\prec(1+z)\/(1-z),$ which is equivalent to\n\t\t\\begin{equation*}\n\t\t\t\\RE p(z)=\\RE\\dfrac{zf'(z)}{g(z)}>0.\n\t\t\\end{equation*}\n\t\tHence the result.\n\t\\end{proof}\nBy taking $p(z)=zf'(z)\/f(z)$ in Theorem~\\ref{genels}, we obtain the following result.\n\\begin{corollary}\n\tLet $\\gamma\\in[0,1],$ $\\alpha\\in[0,1),$ $\\mu\\in[0,1],$ $\\delta\\in[1,2],$ $\\beta\\in[0,1)$ and $\\rho\\in[0,1]$ be such that $\\rho\\geq \\alpha(1+2\\alpha),$ whenever $\\alpha\\in[0,1\/2].$ Also, let $f\\in\\mathcal{A}$ satisfies\n\t\\begin{equation*}\n\t\\RE\\left(\\gamma\\left(\\dfrac{zf'(z)}{f(z)}\\right)^{\\delta}+(1-\\gamma)\\dfrac{\\left(\\dfrac{zf'(z)}{f(z)}\\right)^{1+\\mu}\\left(1+\\dfrac{zf''(z)}{f'(z)}\\right)^{1-\\mu}}{\\rho\\left(1+\\dfrac{zf''(z)}{f'(z)}\\right)+(1-\\rho)\\dfrac{zf'(z)}{f(z)}}\\right)>\\beta,\n\t\\end{equation*}\nwhere $\\beta$ is as defined in Theorem~\\ref{genels}. Then $f$ is starlike of order $\\alpha.$\n\\end{corollary}\n\\begin{remark}\n\tIf we take $\\delta=1,\\;\\mu=0$ and $\\rho=0$ in the above result, then it implies that the class of $\\gamma$-convex functions of order $\\beta$ is contained in the class of starlike functions of order $\\alpha.$ Further by taking $\\alpha=0,$ it reduces to a well known result which states that every $\\gamma$-convex functions is starlike(see~\\cite{mocanu1969,mocanu1972,mocanu1973}).\n\\end{remark}\n\\begin{remark}\nBy taking $\\delta=1,\\rho=0$ and $\\alpha=0$ in the above result, we obtain a well known result which states that a $\\mu$-starlike function is starlike~\\cite{lewa,lewa2}.\n\\end{remark}\nBy taking $p(z)=f'(z)$ in Theorem~\\ref{genels}, we obtain the following result.\n\\begin{corollary}\n\tLet $\\gamma\\in[0,1],$ $\\alpha\\in[0,1),$ $\\mu\\in[0,1],$ $\\delta\\in[1,2],$ $\\beta\\in[0,1)$ and $\\rho\\in[0,1]$ be such that $\\rho\\geq \\alpha(1+2\\alpha),$ whenever $\\alpha\\in[0,1\/2].$ Also, let $f\\in\\mathcal{A}$ satisfies\n\t\\begin{equation*}\n\t\\RE\\left(\\gamma(f'(z))^{\\delta}+(1-\\gamma)\\dfrac{(f'(z))^{1+\\mu}\\left(f'(z)+\\dfrac{zf''(z)}{f'(z)}\\right)^{1-\\mu}}{f'(z)+\\rho\\dfrac{zf''(z)}{f'(z)}}\\right)>\\beta,\n\t\\end{equation*}\n\twhere $\\beta$ is as defined in Theorem~\\ref{genels}. Then $\\RE(f'(z))>\\alpha$ and therefore $f$ is univalent in $\\mathbb{D}$.\n\\end{corollary}\nBy taking $p(z)=f(z)\/z$ in Theorem~\\ref{genels}, we obtain the following result.\n\\begin{corollary}\n\t\tLet $\\gamma\\in[0,1],$ $\\alpha\\in[0,1),$ $\\mu\\in[0,1],$ $\\delta\\in[1,2],$ $\\beta\\in[0,1)$ and $\\rho\\in[0,1]$ be such that $\\rho\\geq \\alpha(1+2\\alpha),$ whenever $\\alpha\\in[0,1\/2].$ Also, let $f\\in\\mathcal{A}$ satisfies\n\t\\begin{equation*}\n\t\t\\RE\\left(\\gamma\\left(\\dfrac{f(z)}{z}\\right)^{\\delta}+(1-\\gamma)\\dfrac{\\left(\\dfrac{f(z)}{z}\\right)^{1+\\mu}\\left(\\dfrac{f(z)}{z}+\\dfrac{zf'(z)}{f(z)}-1\\right)^{1-\\mu}}{\\dfrac{f(z)}{z}+\\rho\\dfrac{zf'(z)}{f(z)}}-1\\right)>\\beta,\n\t\\end{equation*}\n\twhere $\\beta$ is as defined in Theorem~\\ref{genels}. Then $\\RE(f(z)\/z)>\\alpha.$\n\\end{corollary}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe infrared singularities of gauge theory scattering amplitudes\nplay a fundamental role in particle physics for both phenomenological\nand theoretical reasons. Knowing the structure of long-distance\nsingularities is necessary for combining the real and virtual contributions\nto the cross section, as the divergences of the separate contributions\nonly cancel once they are added. In addition, infrared singularities\ndictate the structure of large logarithmic contributions\nto the cross section, allowing such terms to be resummed.\nLong-distance singularities, moreover, have several highly interesting properties.\nThey have a universal structure among different gauge theories;\ntheir exponentiation properties \\cite{Yennie:1961ad,Gatheral:1983cz,Frenkel:1984pz,Gardi:2010rn,Mitov:2010rp,Gardi:2013ita}\nand their relation to the renormalization of Wilson line correlators\n\\cite{Korchemsky:1985xj,Korchemsky:1987wg}\nallow the exploration of the all-order structure of their perturbative expansion,\na feat currently unattainable for complete scattering amplitudes.\n\nThe key tool for computing the infrared singularities of\nscattering amplitudes is provided by the eikonal approximation\nin which each parton $i$ emerging from the hard scattering\nacts as a source of soft gluon radiation and\nis replaced by a semi-infinite path ordered Wilson line\n\\begin{equation}\n\\Phi_{v_i} \\equiv \\mathcal{P}\n\\exp\\left( ig \\int_0^\\infty \\hspace{-0.6mm} {\\rm d} t \\hspace{0.9mm}\nv_i \\cdot A(tv_i) \\right) \\,,\n\\label{eq:def_of_Wilson_line}\n\\end{equation}\nwhich extends from time $t=0$, when the hard scattering takes place,\nto infinity along the classical trajectory of the hard parton,\nspanned by its four-velocity $v_i^\\mu$. The long-distance singularities\nof the scattering amplitude of the hard partons are then given by\nthe eikonal amplitude\n\\begin{equation}\n\\mathcal{S}(v_i \\cdot v_j, \\epsilon) \\equiv\n\\langle 0 \\hspace{0.1mm} | \\hspace{0.1mm} \\Phi_{v_1} \\otimes \\Phi_{v_2}\n\\otimes \\cdots \\otimes \\Phi_{v_n} \\hspace{0.1mm} | \\hspace{0.1mm} 0 \\rangle \\,,\n\\label{eq:Wilson_line_correlator_def}\n\\end{equation}\nwhich has the exact same soft singularities as the original full\namplitude, but is much simpler to compute. Owing to the scale\ninvariance of the Wilson line correlator (\\ref{eq:Wilson_line_correlator_def}),\nits infrared singularities can be computed equivalently\nby studying its ultraviolet renormalization factor\n\\cite{Korchemsky:1985xj,Korchemsky:1987wg}.\nThis renormalization factor forms a matrix in the space of available\ncolor configurations, called the soft anomalous dimension matrix. This matrix has been\ncomputed through two loops for massless \\cite{Aybat:2006wq,Aybat:2006mz}\nas well as massive \\cite{Becher:2009kw,Ferroglia:2009ep,Ferroglia:2009ii,Mitov:2010xw}\nWilson lines, and there has been recent progress toward the\nthree-loop result in Refs.~\\cite{Henn:2013wfa,Gardi:2013saa,Falcioni:2014pka}.\nIn processes involving only two Wilson lines, the soft matrix\nreduces to the cusp anomalous dimension, which has been computed\nin QCD up to three loops \\cite{Korchemsky:1987wg,Kidonakis:2009ev,Grozin:2014hna}.\n\nIn this Letter we introduce a notion of cuts of eikonal diagrams---i.e.,\nthe diagrams contributing to the eikonal amplitude. Applied to\nany eikonal diagram, the cuts produce its discontinuities,\nin analogy with the Cutkosky rules for standard Feynman\ndiagrams. The discontinui\\-ties are in turn readily combined\nto yield the imaginary part of the diagram. A direct computational\nmethod of the latter is desirable in a variety of contexts,\ne.g. rapidity gaps \\cite{Forshaw:2006fk,Forshaw:2008cq},\ncross section calculations \\cite{Korchemsky:1992xv,Gardi:2005yi},\nand the breaking of collinear factorization theorems caused by exchanges\nof Glauber-region (i.e., transverse) gluons \\cite{Catani:2011st,Forshaw:2012bi}.\nRegarding the latter, the resulting factorization-breaking\nterms are purely imaginary and take the form of the non-Abelian analog\nof the QED Coulomb phase. By utilizing the all-order exponentiation\nproperty of the eikonal amplitude, the latter could be obtained\nby computing the imaginary part of the exponent.\nIntroducing cuts of eikonal diagrams can also be viewed as\nthe first step toward extending the modern unitarity\nmethod~\\cite{Bern:1994zx,Bern:1994cg,Bern:1996je,Britto:2004nc,Britto:2005ha,Forde:2007mi,Mastrolia:2009dr,Kosower:2011ty,CaronHuot:2012ab}\nto Wilson line correlators. In unitarity,\nloop-level (non-eikonal) amplitudes are computed by expanding\nthe amplitude in a basis of integrals and determining the basis\ncoefficients by taking cuts which measure the discontinuity of\nthe amplitude in its various kinematic channels.\n\nWe emphasize that a cutting prescription acting on the\nmomentum-space representation of eikonal diagrams was defined in Ref.~\\cite{Korchemsky:1987wg}.\nIn contrast, the cuts introduced here\nare applied to the position-space representation of the diagrams.\nAs we shall see, position-space cuts offer a substantial simplification\nover momentum-space cuts in the computation of imaginary parts\nof eikonal diagrams.\n\n\n\\section{Imaginary parts and their physical origin}\n\nIn this section we discuss the origin of the imaginary part of\nWilson line correlators from the point of view of causality\nas well as unitarity.\n\nWe will adopt the convention that all velocities are outgoing.\nUltraviolet divergences are regulated by computing\nall diagrams in $D=4-2\\epsilon$ dimensions with $\\epsilon > 0$.\nTo avoid complications arising from regulating collinear singularities,\nwe take all velocities to be timelike, $v_i^2 = 1$.\nFor notational convenience, we will drop color factors,\ncoupling constants and factors of $\\frac{\\Gamma(D\/2 -1)}{4\\pi^{D\/2}}$.\n\nWe will start our investigations by examining\nthe simplest eikonal diagram, the one-loop exchange.\nThe position-space representation of this diagram\nis obtained by direct perturbative expansion of\nEq.~(\\ref{eq:Wilson_line_correlator_def}), yielding\n\\begin{equation}\nF^{(1)} = \\mu^{2\\epsilon} \\int_0^\\infty \\hspace{-0.8mm}\n\\int_0^\\infty \\frac{{\\rm d} t_1 \\hspace{0.3mm} {\\rm d} t_2 \\hspace{0.7mm} v_1 \\cdot v_2}\n{\\big[ {-}(t_1 v_1 - t_2 v_2)^2 + i\\eta \\big]^{1-\\epsilon}} \\,,\n\\label{eq:one-loop_diagram_x-space}\n\\end{equation}\nwhere $t_1, t_2$ have the dimension of time and denote the positions of\nthe attachment points of the soft-gluon propagator on the\nWilson lines spanned by the four-velocities $v_1$ and $v_2$.\nThe integrations in Eq.~(\\ref{eq:one-loop_diagram_x-space}) yield\nan infrared divergence which can be extracted via the change of variables\n$(t_1, t_2)= (\\lambda x, \\lambda(1-x))$ with $0\\leq x \\leq 1$,\nwhere $\\lambda$ has the dimension of length,\n\\begin{equation}\nF^{(1)}\\hspace{-0.8mm}= \\mu^{2\\epsilon} \\hspace{-0.9mm} \\int_0^\\infty\n\\hspace{-1.3mm} \\frac{{\\rm d} \\lambda \\hspace{0.7mm} e^{-\\Lambda \\lambda}}{\\lambda^{1-2\\epsilon}}\n\\hspace{-0.7mm} \\int_0^1 \\frac{{\\rm d} x \\hspace{0.8mm} v_1 \\cdot v_2}\n{\\big[{-}(x v_1 \\hspace{-0.3mm}-\\hspace{-0.3mm} (1\\hspace{-0.3mm}-\\hspace{-0.3mm}x) v_2)^2\n\\hspace{-0.3mm}+\\hspace{-0.3mm} i\\eta \\big]^{1-\\epsilon}} \\,,\n\\label{eq:one-loop_diagram_x-space_lambda_s-channel}\n\\end{equation}\nwhere the infrared divergence arising from the exchange of\ngluons of increasingly longer wavelength is regularized\nby the exponential damping factor $e^{-\\Lambda \\lambda}$\nwith $\\Lambda \\ll 1$.\n\nThe diagram is then readily evaluated, yielding\nin, respectively, time- and spacelike kinematics to $\\mathcal{O}(\\epsilon^{-1})$,\n\\begin{equation}\nF^{(1)} \\hspace{-0.5mm}=\\hspace{-0.5mm}\n\\frac{1}{2\\epsilon} \\left( \\frac{\\mu}{\\Lambda} \\right)^{2\\epsilon}\n\\hspace{-0.5mm} \\times \\hspace{0.5mm} \\left\\{ \\hspace{-0.7mm}\n\\begin{array}{cllr}\n(\\gamma - \\pi i) \\coth \\gamma \\hspace{2mm} &\\mathrm{for} & v_1 \\cdot v_2 &\\hspace{-1mm}> 0 \\,\\phantom{.} \\\\[2.5mm]\n\\gamma \\coth \\gamma \\hspace{2mm} &\\mathrm{for} & v_1 \\cdot v_2 &\\hspace{-1mm}< 0 \\,,\n\\end{array} \\right.\n\\label{eq:one-loop_diagram_result}\n\\end{equation}\nwhere the angle $\\gamma$ is defined through\n$\\cosh \\gamma \\equiv |v_1 \\cdot v_2|$, and where, e.g.,\nthe timelike result can be obtained from the spacelike\none by the analytic continuation $\\gamma \\to \\pi i - \\gamma$.\n\nWe observe that the imaginary part of the one-loop diagram\nin Eq.~(\\ref{eq:one-loop_diagram_result}) is, respectively,\nnon-vanishing and vanishing. From the position-space representation\n(\\ref{eq:one-loop_diagram_x-space}) of the diagram,\nthe origin of the imaginary part can be understood from a\nsimple causality consideration as follows. (As our focus is on\ncomputing the imaginary part to the leading order in $\\epsilon$,\nthe $\\epsilon$ in the propagator exponent can be dropped once\nthe infrared divergence has been extracted.)\nFor timelike kinematics $v_1 \\cdot v_2 >0$,\nthere are regions $\\frac{t_1}{t_2} = e^{\\pm \\gamma}$ within the\nintegration domain where $(t_1 v_1 - t_2 v_2)^2 =0$,\nso that the $-i\\eta$ term becomes relevant and generates\nan imaginary part. Physically, what is happening at\nsuch times $t_1, t_2$ is that the two partons traveling\nalong $v_1$ and $v_2$ become lightlike separated. As a result, the\nphases of their states can be changed through\nthe exchange of lightlike gluons (or photons).\nIn contrast, for spacelike kinematics $v_1 \\cdot v_2 <0$,\nthe integral in Eq.~(\\ref{eq:one-loop_diagram_x-space})\nhas a vanishing imaginary part: the denominator\n$(t_1 v_1 - t_2 v_2)^2$ is strictly positive within the region\nof integration, and the $-i\\eta$ can thus be dropped.\nIn this case, the partons are never lightlike separated,\nand the phases of their states cannot be changed\nthrough the exchange of lightlike massless gauge bosons.\n\nThese observations on the evolution of the phases of the hard-parton\nstates, related in the interaction picture through time\nevolution by $|f\\rangle_I = e^{i \\int_0^{1\/\\Lambda}\n{\\rm d} t \\hspace{0.6mm} V_I t} |i\\rangle_I$, suggest that\nthe imaginary part of the anomalous dimension of the correlator of\ntwo Wilson lines defines an interparton potential.\nThis is indeed the case in conformal gauge theories\nowing to the state-operator correspondence \\cite{Chien:2011wz};\nin QCD, the relation holds up to terms proportional\nto the beta function \\cite{Grozin:2014hna}.\n\nLet us turn to the complementary question of\nhow the imaginary part of the one-loop diagram\nmay be obtained from its momentum-space representation,\n\\begin{equation}\nF^{(1)} \\hspace{-0.6mm}=\\hspace{-0.2mm} i \\mu^{2\\epsilon}\n\\hspace{-1.2mm} \\int \\hspace{-0.9mm} \\frac{{\\rm d}^D k}{(2\\pi)^D}\n\\frac{v_1 \\cdot v_2}{(k^2 \\hspace{-0.3mm}+\\hspace{-0.3mm} i\\eta)\n(v_1 \\hspace{-0.2mm} \\cdot \\hspace{-0.2mm} k \\hspace{-0.3mm}+\\hspace{-0.3mm} i\\eta)\n(v_2 \\hspace{-0.2mm} \\cdot \\hspace{-0.2mm} k \\hspace{-0.3mm}-\\hspace{-0.3mm} i\\eta)} \\,.\n\\label{eq:one-loop_diagram_p-space}\n\\end{equation}\nSuch a cutting prescription was provided in Ref.~\\cite{Korchemsky:1987wg}\nwhere it was shown that the imaginary part of the one-loop diagram\nin Eq.~(\\ref{eq:one-loop_diagram_p-space}) is obtained by\nreplacing the two eikonal propagators by delta functions,\n\\begin{align}\n2i \\hspace{0.4mm} \\mathrm{Im} \\hspace{0.6mm} F^{(1)} &= (2\\pi)^2 i \\hspace{0.6mm}\n\\theta (v_1^0) \\theta (v_2^0) (v_1 \\cdot v_2) \\mu^{2\\epsilon} \\nonumber\\\\\n&\\hspace{7mm}\\times \\int \\hspace{-0.5mm} \\frac{{\\rm d}^D k}{(2\\pi)^D}\n\\frac{\\delta(v_1 \\cdot k) \\delta(v_2 \\cdot k)}{k^2 + i\\eta} \\,.\n\\label{eq:map_p-space_cut_to_x-space_1}\n\\end{align}\nFrom this representation we observe that the support of the delta\nfunctions in Eq.~(\\ref{eq:map_p-space_cut_to_x-space_1})\nis the region where the momentum of the exchanged gluon\nis maximally transverse, $v_i \\cdot k \\approx 0$,\nwhich was identified in Ref.~\\cite{Korchemsky:1987wg} as\nthe Glauber region \\cite{Collins:1983ju}. Furthermore,\nthe delta functions have the effect of putting the\nhard partons on shell, making them asymptotic states.\nThus, in momentum space, the imaginary part\narises from the two hard partons going on shell and exchanging\nGlauber gluons.\n\nWe conclude that the position- and momentum-space representations\nof eikonal diagrams offer complementary points of view\non the origin of their imaginary part, based, respectively,\non causality and unitarity considerations.\n\nThe momentum-space cuts applied in\nEq.~(\\ref{eq:map_p-space_cut_to_x-space_1}) have the conceptual advantage\nof factoring eikonal diagrams into on-shell lower-loop and tree diagrams\nwhich can be computed as independent objects. However, at $L$ loops,\nthe resulting cut diagrams involve integrals over up to $(L+1)$-particle phase space.\nThe evaluation of these integrals poses a substantial\ncomputational challenge which limits the applicability\nof the momentum-space cutting prescription\nfor obtaining imaginary parts.\n\n\n\\section{Cuts of eikonal diagrams without internal vertices}\n\nIn this section we will derive a formula for the imaginary part\nof $L$-loop eikonal diagrams containing no internal (i.e., three-\nor four-gluon) vertices to\nthe leading order in the dimensional regulator $\\epsilon$.\nWe will interchangeably refer to such diagrams as ladder-type diagrams.\nThe basic observation is that in position space\nthese diagrams are iterated integrals, and thus their imaginary part\ncan be obtained by decomposing the real-line integrations into\nprincipal-value and delta function contributions.\n\nIn position space, an arbitrary $L$-loop ladder-type eikonal diagram\nconsists of $L$ soft-gluon propagators, or \\emph{rungs}.\nEach rung extends between the eikonal lines\nspanned by any two (possibly identical) external four-velocities\n$v_1, \\ldots, v_n$ where $1 \\leq n \\leq L+1$.\nFor the $j$th rung we will denote these four-velocities\nby $v_{\\ell_j}$ and $v_{r_j}$. Let $t_{i,k}$ denote the position\nof the $k$th attachment on the eikonal line spanned by $v_i$, counting\nfrom the hard interaction vertex and outwards so that\n$0\\leq t_{i,1} < \\cdots < t_{i,N_i}$ where\n$N_i$ denotes the total number of soft-gluon attachments on the eikonal\nline. Furthermore, for the $j$th rung, we will let the variables\n$m_j$ and $n_j$ record the soft-gluon attachment numbers on\nthe eikonal lines spanned by $v_{\\ell_j}$ and $v_{r_j}$, respectively.\nThe $L$-loop eikonal diagram is then defined as the $2L$-fold integral\n\\begin{align}\nF^{(L)} &= \\mu^{2L\\epsilon} \\prod_{j=1}^L \\int_0^\\infty\n\\frac{{\\rm d} t_{\\ell_j, m_j} {\\rm d} t_{r_j, n_j} (v_{\\ell_j} \\cdot v_{r_j})}\n{\\big[ {-}(t_{\\ell_j, m_j} v_{\\ell_j} - t_{r_j, n_j} v_{r_j})^2\n+ i\\eta \\big]^{1-\\epsilon}} \\nonumber \\\\\n&\\hspace{18mm}\\times \\prod_{i=1}^n \\prod_{k=0}^{N_i}\n\\theta (t_{i,k+1} - t_{i,k}) \\,,\n\\label{eq:L-loop_ladder_diagram_def}\n\\end{align}\nwhere it is implied that $t_{i,N_i +1} \\equiv \\infty$ and $t_{i,0} \\equiv 0$.\nWithout loss of generality, we assume that no rungs attach with\nboth end points to the same Wilson line. In such diagrams these\nrungs can be integrated out, each producing to leading order in\n$\\epsilon$ a factor of $1\/\\epsilon$ times the diagram without\nthese rungs~\\footnote{More explicitly, these integrations will\nproduce a factor of $1\/\\epsilon$ times epsilonic powers of the\nattachment points of rungs attached in between the end points of\nthe rung being integrated out.}. For the latter we can then use our formalism.\n\n\n\\begin{figure*}[!t]\n\\begin{center}\n\\includegraphics[width=0.90\\textwidth]{Fig_ThreeLoopImPart.eps}\n{\\vskip 1mm}\n\\caption{Schematic illustration of the formula~(\\ref{eq:Master Im F})\nfor the imaginary part of an eikonal diagram with no internal vertices.\nThe black dots at the end points of a soft-gluon propagator indicate\nthat the propagator has been cut; i.e., replaced by a delta function.\nPhysically, the black dots represent the emission and absorption\nof a lightlike gauge boson. It is implied that the integrals over\nthe attachment points of uncut soft propagators are principal-value integrals.}\n\\label{fig:Im_of_three-loop_ladder}\n\\end{center}\n\\end{figure*}\n\nIn order to extract the imaginary part of $F^{(L)}$ from the integral\nrepresentation in Eq.~(\\ref{eq:L-loop_ladder_diagram_def})\nit will be useful to perform a change of variables which leaves\neach soft propagator dependent on a single variable.\nTo this end, we adopt the change of variables introduced\nin Ref.~\\cite{Gardi:2011yz},\n\\begin{equation}\n\\left( \\hspace{-1.1mm} \\begin{array}{c} t_{\\ell_j, m_j} \\\\\nt_{r_j, n_j} \\end{array} \\hspace{-1.1mm} \\right)\n\\hspace{0.2mm}=\\hspace{0.2mm} \\rho_j \\left( \\hspace{-1.1mm}\n\\begin{array}{c} x_j \\\\ 1 - x_j \\end{array} \\hspace{-1.1mm} \\right)\n\\hspace{3mm} \\mathrm{where} \\hspace{3mm}\n\\left\\{ \\hspace{-0.6mm} \\begin{array}{l}\n0 \\leq \\rho_j < \\infty \\\\[0.8mm]\n0 \\leq x_j \\leq 1 \\,.\n\\end{array} \\right.\n\\label{eq:radial_coordinates}\n\\end{equation}\nFor notational convenience, we define the nesting function\nin terms of the new variables as follows,\n\\begin{equation}\n\\Theta (\\boldsymbol{\\rho},\\boldsymbol{x})\n\\hspace{-0.3mm}\\equiv\\hspace{-0.3mm} \\prod_{i=1}^n \\prod_{k=0}^{N_i}\n\\theta (t_{i,k+1} - t_{i,k}) \\bigg|_{\\tiny \\left( \\hspace{-1.5mm} \\begin{array}{c} t_{\\ell_j, m_j} \\\\\nt_{r_j, n_j} \\end{array} \\hspace{-1.9mm} \\right)\n\\hspace{0.7mm}=\\hspace{0.7mm} \\rho_j \\left( \\hspace{-1.3mm}\n\\begin{array}{c} x_j \\\\ 1 - x_j \\end{array} \\hspace{-1.7mm} \\right)} \\,,\n\\label{eq:nesting_function_def}\n\\end{equation}\nand the soft propagators through\n\\begin{equation}\nP_{ij}^{[\\epsilon]} (x) \\equiv \\frac{v_i \\cdot v_j}\n{\\big[ {-}\\big(x v_i - (1-x)v_j\\big)^2 + i\\eta \\big]^{1-\\epsilon}} \\,.\n\\label{eq:propagator_notation}\n\\end{equation}\nThe diagram then takes the form\n\\begin{equation}\nF^{(L)} \\hspace{-0.8mm}=\\hspace{-0.3mm} \\mu^{2L\\epsilon} \\prod_{j=1}^L\n\\int_0^\\infty \\hspace{-1.0mm} \\frac{{\\rm d} \\rho_j}{\\rho_j^{1-2\\epsilon}} \\hspace{-0.5mm}\n\\int_0^1 \\hspace{-0.4mm} {\\rm d} x_j \\hspace{0.4mm} P_{\\ell_j r_j}^{[\\epsilon]} (x_j)\n\\Theta (\\boldsymbol{\\rho},\\boldsymbol{x}) \\,.\n\\label{eq:L-loop_ladder_diagram_rep_1}\n\\end{equation}\nWe observe that the dependence of the soft propagators on the\nradial coordinates $\\rho_j$ has scaled out, and that\neach propagator now depends only on a single variable $x_j$.\n\nNext, we extract the overall infrared divergence of the\ndiagram by setting $\\tau_1 \\equiv \\rho_1$ and applying\nthe following sequence of $L-1$ substitutions\n\\begin{equation}\n\\left( \\hspace{-1.1mm} \\begin{array}{c} \\tau_j \\\\\n\\rho_{j+1} \\end{array} \\hspace{-1.1mm} \\right) =\n\\tau_{j+1} \\left( \\hspace{-1.1mm} \\begin{array}{c} y_j \\\\\n1 - y_j \\end{array} \\hspace{-1.1mm} \\right) \\hspace{4mm} \\mathrm{with} \\hspace{3mm}\n\\left\\{ \\hspace{-0.7mm} \\begin{array}{l}\n0\\leq \\tau_j < \\infty \\\\[0.1mm]\n0\\leq y_j \\leq 1 \\,,\n\\end{array} \\right.\n\\label{eq:tau_y_variables}\n\\end{equation}\nwhere $j=1,\\ldots,L-1$, the variables $\\tau_j$\nhave the dimension of length, and the $y_j$ are dimensionless.\n\nThe $L$-loop eikonal diagram then becomes\n\\begin{equation}\nF^{(L)} = \\prod_{j=1}^L \\int_0^1 {\\rm d} x_j \\hspace{0.5mm}\nP_{\\ell_j r_j}^{[\\epsilon]} (x_j) \\hspace{0.4mm} \\mathcal{I} (\\boldsymbol{x}) \\,.\n\\label{eq:L-loop_ladder_diagram_rep_2}\n\\end{equation}\nThe infrared divergence of the diagram has been absorbed into the kernel\n\\begin{equation}\n\\mathcal{I} (\\boldsymbol{x}) \\hspace{-0.6mm}=\\hspace{-0.2mm} \\Gamma(2L\\epsilon) \\hspace{-0.5mm}\n\\left( \\frac{\\mu}{\\Lambda} \\right)^{2L\\epsilon}\n\\prod_{j=1}^{L-1} \\hspace{-0.4mm} \\int_0^1 \\hspace{-0.1mm}\n\\frac{{\\rm d} y_j \\hspace{0.8mm} y_j^{-1+2j\\epsilon}}\n{(1 \\hspace{-0.2mm}-\\hspace{-0.2mm} y_j)^{1-2\\epsilon}}\n\\Theta \\big(\\{\\boldsymbol{y}, \\boldsymbol{x} \\}\\big),\n\\label{eq:kernel}\n\\end{equation}\nwhere the notation $\\Theta \\big(\\{\\boldsymbol{y}, \\boldsymbol{x} \\}\\big)$\nrefers to the result of applying the substitutions\n(\\ref{eq:tau_y_variables}) to Eq.~(\\ref{eq:nesting_function_def}).\nHere we have regulated the infrared divergence through\nthe damping factor $e^{-\\Lambda \\tau_L}$\nwith $\\Lambda \\ll 1$. In addition, Eq.~(\\ref{eq:kernel}) contains any potential\nultraviolet subdivergences of the diagram.\n\nHaving written the $L$-loop eikonal diagram in the form\n(\\ref{eq:L-loop_ladder_diagram_rep_2}), we now turn to\nthe question of extracting its imaginary part. We will\nrestrict our attention to the leading order in the dimensional\nregulator $\\epsilon$ and drop the dependence of\nthe soft propagators on $\\epsilon$,\n\\begin{equation}\nF^{(L)} \\hspace{-0.3mm}=\\hspace{-0.3mm}\n\\prod_{j=1}^L \\int_0^1 {\\rm d} x_j \\hspace{0.5mm}\nP_{\\ell_j r_j}^{[0]} (x_j) \\hspace{0.4mm} \\mathcal{I} (\\boldsymbol{x})\n\\hspace{0.4mm} + \\hspace{0.4mm} \\mathcal{O}(\\epsilon^{-d+1}) \\,,\n\\label{eq:L-loop_ladder_diagram_rep_3}\n\\end{equation}\nwith $d$ denoting the degree of divergence of the diagram,\n$F^{(L)} \\sim \\frac{1}{\\epsilon^d} \\times \\mathrm{(finite)}$.\n\nTo compute the imaginary part of Eq.~(\\ref{eq:L-loop_ladder_diagram_rep_3})\nwe start by observing that Eq.~(\\ref{eq:kernel}) is\nmanifestly real. The Feynman $i\\eta$'s are thus\nthe only source of imaginary parts of\nEq.~(\\ref{eq:L-loop_ladder_diagram_rep_3}). We can therefore decompose each\nof the $x_j$-integration paths into a principal-value part and\nsmall semicircles around the propagator poles.\nAs the integrand takes purely imaginary values in the regions\nclose to the poles and is real-valued on the remaining domain of\nintegration, the resulting $2^L$ terms (each involving\n$L$ integrations) will be either purely real or purely imaginary.\n\nIn order to collect the imaginary contributions, we define\nthe cut propagator\n\\begin{align}\n\\label{eq:Dij}\n\\Delta_{ij}(x) \\equiv\n- \\pi \\, v_i \\cdot v_j \\, \\delta\\big( (x v_i-(1-x) v_j)^2 \\big) \\,,\n\\end{align}\nand the $p$-fold cutting operator\n\\begin{align}\n\\label{eq:Cut_x F}\n\\Cut_{x_{i_1},\\dotsc,x_{i_p}} F^{(L)} &=\n\\prod_{j=1 \\atop \\!\\!\\! j \\neq i_1,\\dotsc,i_p \\!\\!\\!\\!\\!\\!}^{n} \\PV \\int_0^1 {\\rm d} x_j \\, P(x_j)\\nonumber\\\\\n&\\hspace{6mm} \\times \\prod_{k=1}^{p} \\int_0^1 {\\rm d} x_{i_k} \\,\n\\Delta(x_{i_k}) \\hspace{0.3mm} \\mathcal{I}(\\boldsymbol{x})\\,.\n\\end{align}\nFor notational brevity we omitted the indices on the (cut)\npropagators: $P(x_j) \\equiv P_{\\ell_j r_j}^{[0]} (x_j)$\nand $\\Delta (x_j) \\equiv \\Delta_{\\ell_j r_j} (x_j)$.\n\nThe imaginary part of any $L$-loop eikonal diagram with no\ninternal vertices is then, to the leading order in $\\epsilon$,\n\\begin{equation}\n\\Im F^{(L)}=\n\\sum_{p=1 \\atop p \\hspace{0.7mm} \\text{odd}}^L \\hspace{0.0mm}\n\\sum_{i_1,\\dotsc,i_p = 1 \\atop i_1 < \\dotsm < i_p}^L i^{\\,p-1} \\Cut_{x_{i_1}, \\ldots, x_{i_p}} F^{(L)} \\,.\n\\label{eq:Master Im F}\n\\end{equation}\nThe formula~(\\ref{eq:Master Im F})\nis illustrated for a generic ladder diagram in\nFig.~\\ref{fig:Im_of_three-loop_ladder}.\nNote that Eq.~(\\ref{eq:Master Im F})\nshows that the imaginary part of\nthe integrated result for the diagram\nwill have transcendentality weight one less than\nthe real part.\n\n\\begin{figure*}[!t]\n\\begin{center}\n\\includegraphics[width=0.90\\textwidth]{Fig_ThreeGluonVertexImPart.eps}\n{\\vskip 1mm}\n\\caption{Schematic illustration of the formula~(\\ref{eq:Master Im F})\nfor the imaginary part of an eikonal diagram with an internal vertex.}\n\\label{fig:Im_of_3g_vertex_diagram}\n\\end{center}\n\\end{figure*}\n\nWe have verified the formula~(\\ref{eq:Master Im F})\nfor ladder-type diagrams with up to three loops,\nfinding agreement with results in the literature%\n~\\footnote{The diagram $F^{(3)}$ can be obtained from Ref.~\\cite{Falcioni:2014pka}\nas an appropriate linear combination of eqs.~(4.29) and (4.33) there.\nSubsequent analytic continuation from spacelike to timelike kinematics\nyields an imaginary part in agreement with Eq.~(\\ref{eq:ImF3-v2}).}.\nFor example, for the diagram in Fig.~\\ref{fig:Im_of_three-loop_ladder},\nEq.~(\\ref{eq:Master Im F}) yields the imaginary part\n($\\gamma\\equiv\\gamma_{12}$, $\\chi \\equiv e^{-\\gamma}$)\n\\begin{align}\n\\label{eq:ImF3-v2}\n\\Im F^{(3)} \\hspace{-0.6mm}&=\\hspace{-0.6mm}\n\\frac{\\pi}{6\\epsilon} \\left( \\frac{\\mu}{\\Lambda}\n\\right)^{6\\epsilon} \\hspace{-1.0mm} \\coth^3 \\hspace{-0.9mm} \\gamma \\hspace{0.3mm}\n\\Big[H_{3,1}(\\chi^2) \\hspace{-0.2mm}+\\hspace{-0.2mm}H_{2,2}(\\chi^2)\n\\hspace{-0.2mm}+\\hspace{-0.2mm}H_4(\\chi^2) \\nonumber \\\\\n&\\hspace{-8mm} {-}\\tfrac{1}{3}\\log^4 \\chi +\\log^2 \\hspace{-0.4mm} \\chi\n\\hspace{0.4mm} \\big(H_2(\\chi^2) \\hspace{-0.2mm}+\\hspace{-0.2mm} 3\\zeta_2\\big)\n-\\log \\chi \\big(H_3(\\chi^2) \\hspace{-0.2mm} - \\hspace{-0.2mm} \\zeta_3\\big) \\nonumber\\\\\n&\\hspace{-8mm}\n-\\zeta_2 H_2(\\chi^2)\n+\\tfrac{1}{2}\\zeta_4\n\\Big] + \\mathcal{O}(\\epsilon^0)\n\\,.\n\\end{align}\nThe $H_{i,j}$ and $H_i$ denote harmonic polylogarithms\naccording to the conventions of Ref.~\\cite{Remiddi:1999ew}.\n\nA natural question concerns the relation of the imaginary part\nof the eikonal diagram to the discontinuities in its various\nkinematic channels. Expressed in terms of the \\emph{exponentials}\nof the cusp angles, $\\chi_{ij} \\equiv e^{-\\gamma_{ij}}$,\nrather than the cusp angles $\\cosh \\gamma_{ij} = |v_i \\cdot v_j|$\nthemselves, the eikonal diagram has branch cuts located on the real line and\nsatisfies Schwarz reflection, $F^{(L)}(\\overline{\\chi_{ij}})\n= \\overline{F^{(L)}(\\chi_{ij})}$. As a result, the discontinuities\ngive rise to the imaginary part through the relation\n\\begin{equation}\n2i \\hspace{0.7mm} \\mathrm{Im} \\hspace{0.6mm}\nF^{(L)} (\\boldsymbol{\\chi})\n\\hspace{0.7mm}=\\hspace{0.7mm} \\sum_{j=1}^L \\theta(v_{\\ell_j} \\cdot v_{r_j}) \\hspace{0.2mm}\n\\mathop{\\mathrm{Disc}}_{\\chi_{\\ell_j r_j}} \\hspace{0.3mm}\nF^{(L)} (\\boldsymbol{\\chi}) \\,.\n\\label{eq:Im_and_discontinuities_relation}\n\\end{equation}\nThe step functions account for the fact that the\nimaginary part has non-vanishing and vanishing contributions\nfrom channels with, respectively, timelike ($v_{\\ell_j} \\cdot v_{r_j} > 0$)\nand spacelike ($v_{\\ell_j} \\cdot v_{r_j} < 0$) kinematics,\nin agreement with the causality considerations of the previous\nsection.\n\n\\section{Cuts of eikonal diagrams with internal vertices}\n\nEikonal diagrams with internal vertices have a more\ncomplicated structure than ladder-type diagrams. Nonetheless,\nwe expect that our cutting prescription applies to such\ndiagrams as well. As an explicit example, we consider\nthe two-loop three-line diagram involving\na three-gluon vertex, illustrated in Fig.~\\ref{fig:Im_of_3g_vertex_diagram}.\nIn order to apply our prescription,\nthe overall infrared divergence of the diagram is extracted\nby integrating out the radial distance of the three-gluon\nvertex, leaving three remaining integrations over real\nprojective space $u \\in \\mathds{R}\\mathds{P}^{1,2}$.\nAfter some algebraic manipulations, the three-gluon vertex\ndiagram has the following position-space\nrepresentation, to the leading order in $\\epsilon$ \\cite{Mitov:2009sv},\n\\begin{align}\nF_{3g}\n\\propto \\frac{1}{\\epsilon}\n\\int {\\rm d}^3 u\n~{\\cal V}(\\{v_i\\},u)\n\\hspace{0.4mm} \\prod_{\\ell=1}^{3} \\int_{0}^{\\infty} {\\rm d} x_\\ell \\,\nP_{\\ell}^{[0]}(\\zeta_\\ell, x_\\ell) \\,,\n\\label{eq:Three_gluon_vertex_diagram}\n\\end{align}\nwith the three-gluon vertex related differential operator\n${\\cal V}(\\{v_i\\},u)=\\sum_{i,j,k=1}^{3} \\varepsilon_{ijk}\n\\, v_i \\cdot v_j \\, \\zeta_i \\, \\zeta_k \\, \\frac{\\partial}{\\partial \\zeta_i}$,\nin terms of the scalar products $\\zeta_i = v_i \\cdot u$.\nThe position-space propagators are given by\n$P_{i}^{[\\epsilon]}(\\zeta_i,x_i) = (-u^2 + 2 x_i \\zeta_i - x_i^2 + i\\eta)^{-1+\\epsilon}$.\nThe imaginary part of this diagram is given by the formula~(\\ref{eq:Master Im F})\nwith the cut propagators,\n\\begin{align}\n\\Delta_{i}(x) \\equiv\n- \\pi \\, \\delta\\big(u^2 - 2 x_i \\zeta_i + x_i^2 \\big) \\,.\n\\end{align}\nThe formula is illustrated in Fig.~\\ref{fig:Im_of_3g_vertex_diagram}.\nWe have checked the formula by performing the integral over the\ndirection of the three-gluon vertex $u$ numerically.\nFor small cusp angles, where the numerics proves to behave well,\nwe find agreement with the analytic result in Ref.~\\cite{Ferroglia:2009ii}.\nThis in turn suggests the applicability of our cutting\nprescription to any eikonal diagram.\n\n\n{\\bf Acknowledgments}\n\nWe thank S.~Abreu, S.~Caron-Huot, E.~Gardi, J.~Henn, P.~Hoyer,\nL.~Magnea, G.~Sterman, I.~Stewart, W.~Vleeshouwers, A.~Waelkens, C.~White\nand especially G.~Korchemsky for useful discussions.\nKJL is grateful for the hospitality of\nthe Institute for Advanced Study in Princeton and\nthe Institut de Physique Th{\\'e}orique, CEA Saclay,\nwhere part of this work was carried out.\nThe research leading to these results has received\nfunding from the European Union Seventh Framework\nProgramme (FP7\/2007-2013) under grant agreement no.~627521.\nThis work was supported by the Foundation for\nFundamental Research of Matter (FOM),\nprogram 104 ``Theoretical Particle Physics in the Era of the LHC''\nand by the Research Executive Agency (REA)\nof the European Union under the Grant Agreement number\nPITN-GA-2010-264564 (LHCPhenoNet).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\n\nConsider a hydrogen atom, initially in a high Rydberg state with the\nprincipal quantum number $n_0$ illuminated by the microwave field\nof a frequency $\\omega$ close to the frequency of the unperturbed Kepler\nmotion $\\omega_K=1\/n_0^3$. Quantum mechanically speaking, a resonant\nperiodic field couples strongly several $n$ states due to\n almost resonant transitions\n$n\\rightarrow n'=n\\pm 1$ for $n$ close to $n_0$. Since the driving field\nis periodic, applying the Floquet theorem \\cite{s65} one may find eigenstates\nof the atom-field system (the so called Floquet\nor dressed \\cite{cct} states). The eigenenergies are then referred to as\nquasienergies of the system and are defined modulo $\\hbar\\omega$.\nThe Floquet states, periodic in time, may be viewed as linear\ncombinations of unperturbed system eigenstates. While the construction\nof Floquet states is possible both for nonresonant and resonant\ndriving,  in the latter case they may have quite unusual properties,\nespecially in a semiclassical limit.\n\nClassically a resonance between the driving frequency and the frequency of the\nunperturbed motion leads to a strong perturbation of the system and a creation\nof a stable island in the phase space centered on a periodic orbit\nof the frequency $\\omega$. The motion in the island is locked to the microwave\nfrequency due to the nonlinear resonance \\cite{lil}. Semiclassically one then\nexpects that the corresponding Floquet time-periodic state will follow\na classical trajectory (in the vicinity of the periodic orbit) i.e.\nform a wavepacket which will not disperse in time.\n\nStates localized in the resonance island for such a periodic perturbation\nhave been first considered more than 20 years ago \\cite{beza77} and details\nof their semiclassical construction for some one-dimensional (1D) model systems\nhave been analyzed \\cite{holt,sk95}. The wavepacket character of the time \nevolution\nof individual Floquet states has been realized only quite recently \\cite{ab3}\nfor hydrogen atom driven by linearly polarized microwaves. Independently,\nit has been shown that Gaussian wave packets may propagate almost\nwithout dispersion along circular periodic orbits\nin hydrogen atoms driven by circularly polarized fields \\cite{ibb}.\nThe fact that the harmonic approximation implied in \\cite{ibb} and resulting\nthe Gaussian wave packet form is not a necessary condition for non-spreading\nproperties have been discussed in \\cite{dzb95,zdb95} where it was shown\nthat the harmonic expansion provides a good approximation for exact\nFloquet states of the system (which by definition, being time periodic, do\nnot spread on a long time scale).\n\nThe fact that one may construct in a nonlinear system wavepackets that\ndo not spread induced a flurry of activity in the field. Some \n\\cite{far_c,ibb_r,fluA95,bufL96,lbf,clfu97}\nconcentrated on modifying the potential so as to make the harmonic\napproximation as good as possible aiming at the construction of Gaussian\nnonspreading wavepackets. Claiming that the anharmonic corrections are\nbig for a hydrogen atom driven by circularly polarized fields those authors\nadded a magnetic field perpendicular to the microwave polarization plane\n\\cite{far_c,ibb_r,fluA95,bufL96,lbf,clfu97}. \nThis helps to minimize the unharmonicities {\\it in this plane}\nleaving, however, unaffected the terms along the magnetic field axis.\nThus the Gaussian wave packets remain even then an approximation to the\nreal dynamics and must disperse (although very slowly in time). Another\napproach aimed at optimizing the coordinate system to the symmetry of the\nproblem \\cite{keb}.\n\nAs discussed by us elsewhere \\cite{dzb95,zdb95,szd98,zdb98,jasz98,dz98} much more\nfruitful is another approach, already outlined above. Namely we define\nthe nonspreading wavepacket as a single Floquet state (for which the\nGaussian packet may be merely an approximation). Then a localization of\nthe wavepacket in the vicinity of a stable fixed point is assured by the\ncorrespondence principle provided the size of the surrounding island\nin the phase space is comparable to $\\hbar$. One may construct\nan approximate resonance Hamiltonian in the vicinity of the island\nwhose eigenstates will approximate well those Floquet states which\nare localized in the vicinity of the island (see below).\n Simultaneously, time-periodicity\nof Floquet eigenstates assures that the exact Floquet states will not disperse.\n\nThe existence of such wave packet Floquet eigenstates has been proven by\nan exact numerical diagonalization of the problem both for linear \npolarization (LP) \\cite{ab3}\nand circular polarization (CP) \\cite{dzb95,zdb95} of the microwaves. To\nallow their detection one should consider the ways of populating\neffectively such states. For a CP, where the direct optical\nexcitation from a weakly perturbed low lying state is impossible (since\nthe wavepacket is built from predominantly circular atomic states\nunaccessible from low lying states due to dipole selection rules) one\nshould first prepare the atomic circular state \\cite{kl,dd} and then switch on\nthe microwaves sufficiently fast \\cite{dzb95}. It has been shown that\nthe wavepacket states may be populated in this way with about 90\\% efficiency\n\\cite{zd97}. For a LP case, an addition of a static electric\nfield allows to control the trajectory of the wavepacket \\cite{szd98}, in\nparticular wavepackets moving along elongated, \nlow angular momentum trajectories may be created.\n Such wavepackets may be accessible to a direct\nexcitation from low lying states.\n\nThe next important question is the possible mechanism of the detection of these\nstates. At least  three possible ways suggest themselves. Two of them\nutilize the residual decay of wavepackets, \neither via the spontaneous emission (treated both\nfor LP \\cite{hb98} and CP \\cite{zibb97,dz98}) or the ionization. The\nformer may not be efficient, since, at least for wavepackets moving on circular\ntrajectories, the corresponding spontaneous emission rates are quite small\n\\cite{dz98}. In the ionization experiment, \nthe population of the wavepacket state may be detected by a strong decrease in\nthe ionization yield (since ionization rates of wavepacket states\nare typically very small \\cite{dzb95,zdb95}). On the other hand, \nthese rates  fluctuate strongly  (the mechanism of\ntheir ionization, via ``chaos assisted tunneling'' is discussed in detail\nelsewhere \\cite{zdb98,jasz98}) -- this may make their detection \nin the ionization yield\nquite ambiguous.\n\nBy far the most promising method is the Floquet spectroscopy \\cite{ab4}\ni.e. probing, by a second weak microwave field, the structure of Floquet\n(dressed \\cite{cct} by microwaves) states. To this end a precise estimation\nof the quasienergies of wavepacket states is necessary. An exact\ndiagonalization of the problem gives all the Floquet states and a time\nconsuming inspection of individual eigenvectors is necessary to identify\nthe wave packet states.\nThis process may be optimized by calculating properties of\nmatrix elements of appropriately chosen operators but certainly it is desirable\nto have good semiclassical predictions for the quasienergies. For CP case\nthose are given, to a very good accuracy from the harmonic approximation\nHamiltonian \\cite{ibb,dzb95,zdb95}, this approach being, however, restricted\nto this particular system.\n\nFor a general case of periodically driven system there is no simple unitary\ntransformation which removes the time-dependence (as it is in the CP case)\nand the correct approach is to use approximate resonant Hamiltonians. The\nsemiclassical quantization of such a Hamiltonian gives not only  the\ngood estimate for wavepacket states but allows for the classification of\nresonant states for systems of more than one dimensions. Recently using\nsuch an approach we could discuss the resonant dynamics in a realistic \nthree dimensional (3D)\nmodel of a hydrogen atom in the LP case \\cite{bsdz98}. Similarly, we\nhave discussed the control of wavepacket trajectories using an\nadditional static electric field \\cite{szd98}.\n\nUp till now the discussion of nonspreading wavepacket states has been\nrestricted to linear and circular polarization cases only. The aim of this paper\nis to treat a resonant dynamics of a hydrogen atom, both semiclassically and\nquantum-mechanically in a general case of an elliptical polarization (EP).\nApart from generalizing the notion of nonspreading wavepackets to an\narbitrary EP, we discuss the full dynamics\nof quasienergies as a function of the ellipticity of the microwaves for\na resonant case, being\n stimulated by recent\nionization experiments \\cite{koch96}. Unfortunately, the experiments do not\nallow for a full selection of the initial state of an atom (states with\nall possible angular momenta are simultaneously excited) but this situation\nmay improve in the near future.\n\nThe EP case is highly nontrivial.\nFor LP microwaves the conservation of\nthe angular momentum projection onto the polarization axis, $L_z$, makes\nthe dynamics effectively two-dimensional (2D). For the CP case\nwhile $L_z$ is not conserved, the transformation to the frame rotating\nwith the microwave frequency removes the explicit oscillatory\ntime-dependence (see e.g. \\cite{nauen2,How92,zdgr93,ZGD96}).\n Both these simplifications are no longer\npossible in the general EP microwave field. In effect, the exact\nquantum diagonalization approach for the part\nof the spectrum corresponding to the strongly perturbed atomic Rydberg\nspectrum would require very big computer memory.\n  Let us mention also that in the effective\n2D LP case, the state of the art computations \\cite{ab3} consider\ninitial atomic states with the principal quantum number of the order of\n20.\n\nFor that reason we shall consider not the realistic fully 3D model\nof an atom but rather the restricted 2D model in which the electronic\nmotion is restricted to the polarization plane. Study of such\nsimplified models have been most successful in the past both for\nLP (where one dimensional model has been a main source\n of quantum results for\na long time \\cite{JS91,KoLe95,sk95}) and in CP \\cite{nauen2,How92,zdgr93,ZGD96}\n where also the 2D, polarization plane restricted model has been\nutilized. Additional argument favoring the 2D model comes from our\nclassical study of dynamics in EP microwaves \\cite{sz97,sz98}\nwhere comparison of 2D \\cite{sz97} and 3D \\cite{sz98} analysis\nshows the similarity of physical phenomena in both cases. Simply\nput the perturbation is most effective if the polarization plane\ncoincides with the plane of Kepler motion.\n\n\n\\section{The semiclassical versus quantum approaches}\n\nThe Hamiltonian of the hydrogen 2D model atom\ndriven by an elliptically polarized electromagnetic field reads\nin the dipole approximation and\nin the length gauge (in atomic units)\n\\begin{equation}\n        H=\\frac{p_{x}^{2}+p_{y}^{2}}{2}\n        -\\frac{1}{r}+ F(\n        x\\cos{\\omega t}+\\alpha y\\sin{\\omega t}),\n\\label{ham}\n\\end{equation}\nwhere $r= \\sqrt{x^2+y^2}$ while $F$ and $\\omega$ denote\n the amplitude and the frequency of the microwave\nfield, respectively.\n$\\alpha$ defines the ellipticity of the microwaves with\n$\\alpha=0$ ($\\alpha=1$) corresponding to a LP (CP) limiting case.\n\nUsing the Floquet theorem the solution of the quantum problem\nis equivalent to diagonalizing the Floquet Hamiltonian\n\\begin{equation}\n\\left ( H-i\\frac{\\partial}{\\partial t}\\right ) \\psi_n =H_F\\psi_n=\n E_n\\psi_n\n\\label{floq}\n\\end{equation}\nwith $E_n$ being the quasienergies while $\\psi_n$ time-periodic\nFloquet eigenstates.\n\nThe details of the numerical method are described in the Appendix.\nIn short, the calculations proceed by expressing the\nFloquet eigenvalue\nequation in the scaled semi-parabolic variables\n\\begin{eqnarray}\nx=\\Lambda\\frac{u^2-v^2}{2}, & y=\\Lambda uv.\n\\label{scaluv}\n\\end{eqnarray}\nwhere $\\Lambda$ is an arbitrary scaling factor.\nThis allows to remove the Coulomb singularity and cast the Schr\\\"odinger\nequation into the generalized eigenvalue problem for\npolynomial like operators. Standard harmonic oscillator\ncreation and annihilation operators\nallow for simple evaluation of matrix elements.\n The approach closely\nresembles that for the 2D atom in the CP field \\cite{ZGD96} except that\nthe explicit time dependence is treated by the Fourier expansion. It\nis worth stressing that using the complex scaling parameter $\\Lambda$\none effectively realizes the complex rotation in the system which\nenables the exact treatment of the coupling to the continuum (ionization),\nfor details see \\cite{ab3,ZGD96} for LP and CP cases, respectively.\nIn this paper, however, we shall concentrate on the Floquet level dynamics only\nfor the sake of the comparison with the semiclassics. \nThe analysis of the ionization\nphenomenon is left for a future publication.\n\nThe semiclassical quantization of resonant dynamics closely resembles\nthe similar procedure applied by us recently for the LP case \\cite{szd98,bsdz98}.\nThat in turn originates from a general prescription for EBK quantization\nof Floquet spectra \\cite{BrHo91}.\n\nStarting with the Hamiltonian (\\ref{ham}) we remove the explicit time dependence\nby going to the extended phase space \\cite{lil}.\n Defining the momentum $p_t$\nconjugate to $t$ (time) variable we get the new Hamiltonian\n\\begin{equation}\n{\\cal H}=H+p_t\n\\label{ext}\n\\end{equation}\nwhich is conserved during the motion.\nThe quasi-energies of the system will be then\n the quantized values of ${\\cal H}$.\n\nAs the next step we express the Hamiltonian\n in action-angle variables of the unperturbed Coulomb problem \\cite{sz97}.\n For the 2D model atom those are e.g.\n the canonically conjugate\npairs $(J,\\theta)$ and $(L,\\phi)$. $J$\nis the principal action\n (corresponding to the principal quantum number, $n_0$).\nThe corresponding angle, $\\theta$, determines the position of the\nelectron on its elliptic trajectory.\n$L$ is the angular\nmomentum (equal to $L_z$ for the 2D motion in the $x-y$ plane)\n while $\\phi$\nis the conjugate angle (the angle between the Runge-Lenz vector\nand the $x$-axis, i.e. the main axis of the polarization ellipse).\n\nWe shall consider below the case of the resonant driving,\n i.e. when the frequency\nof the Kepler motion $\\omega_K=1\/J^3$ is close to the microwave driving frequency\n$\\omega$. Applying the secular perturbation theory \\cite{lil}\nto average over the nonresonant\nterms one obtains an approximate resonant Hamiltonian (in the frame\nrotating together with the electron) of the form\n\\begin{equation}\n{\\cal H}_r=-\\frac{1}{2J^2}-\\omega J+\nF\\Gamma(L,\\phi;\\alpha)\\cos(\\hat\\theta-\\delta)+\\hat p_t\n\\label{EPr}\n\\end{equation}\nwhere $\\hat\\theta=\\theta-\\omega t$ while $\\hat p_t=p_t+\\omega J$.\n${\\cal H}_r$ yields the pendulum like principal action motion\nwith the strength and the equilibrium position determined by \n$\\Gamma(L,\\phi;\\alpha)$ and $\\delta=\\delta(L,\\phi;\\alpha)$ respectively.\nThe pendulum Hamiltonian is obtained by additionally expanding ${\\cal H}_r$\naround the center of the resonance island given by $J=\\omega^{-1\/3}$\nup to the second order in $\\Delta J=J-\\omega^{-1\/3}$ but we do not apply this\nexpansion.\nBoth $\\Gamma$ and $\\delta$  \ndepend on the initial shape and orientation of the electronic ellipse\n(via $L,\\phi$) as well as on the ellipticity of microwaves, $\\alpha$,\nand are given by \\cite{sz97,sth}\n\\begin{equation}\n\\Gamma(L,\\phi;\\alpha)\n=\\left[\\left(\\frac{1+\\alpha}{2}V_{1}\n\\right)^{2}+2\\cos 2\\phi \\frac{1+\\alpha}{2}V_{1}\\frac{1-\\alpha}{2}V_{-1}\n+\\left(\\frac{1-\\alpha}{2}V_{-1}\\right)^{2}\\right]^{1\/2},\n\\label{gam}\n\\end{equation} \n\\begin{equation}\n\\tan\\delta=\\frac{(1-\\alpha)V_{-1}-(1+\\alpha)V_1}{(1-\\alpha)V_{-1}+(1+\\alpha)V_1}\n\\tan\\phi,\n\\end{equation}\nwhere\n\\begin{equation}\n V_{\\pm 1}(J,L) =\\omega^{-2\/3}[{\\cal J}_{1}^{'}(e) \\pm\n \\frac{\\sqrt{1-e^2}}{e}{\\cal J}_{1}(e)].\n\\end{equation}\n${\\cal J}_{1}(x)$ and ${\\cal J}_{1}^{'}(x)$ denote the ordinary \nBessel function and its derivative, respectively, while \n$e=\\sqrt{1-L^2\/J^2}=\\sqrt{1-L^2\\omega^{2\/3}}$ stands for an eccentricity of \nan electronic ellipse.\nFor completness let\nus mention that a similar approximation on the purely quantum level,\nleading to Mathieu equation\nhas been performed for the CP case only in \\cite{ke96}.\n\n\nThe semiclasical quantization of (\\ref{EPr}) is straightforward and\nfollows closely the procedure described in detail elsewhere\n\\cite{bsdz98} for arbitrary $m:1$ resonance. For $1:1$ resonance\nconsidered here, the trivial\nquantization of\n $\\hat p_t$,  exploring the time periodicity of the system,\n yields additive terms\n$k\\omega$ to quasi-energies (different values of $k$\ncorrespond to  different Floquet zones).\nAs discussed first in \\cite{lr86}, see also \\cite{bsdz98}, the\norbital motion\n in $(J,\\hat \\theta)$ variables\n (along the perturbed Kepler ellipse)\n  is much     faster than the modification of the ellipse shape and\nits movement (precession) as described by the motion in $(L,\\phi)$.\nThus the slow and fast motions may be adiabatically decoupled.\nMaking Born-Oppenheimer approximation and using standard WKB rules,\nthe fast $(J,\\hat\\theta)$ motion is quantized taking the\nMaslov index $\\nu=2$ (corresponding to librations, i.e.,\n we quantize states inside the resonance island).\n  Being interested in resonantly\nlocalized states we shall consider later the\nground\nstate of the radial motion, only.\nFor the slow angular $(L,\\phi)$ motion\nwe take the Maslov index  $\\mu=0$ or 2 for a rotational\nor librational motion, respectively.\n\n\n\n\nSimilarly, as in the LP microwaves \\cite{bsdz98},\n it is easier to quantize first the\nslow motion\ngenerated by constant values of $\\Gamma(L,\\phi;\\alpha)$ and later\ntreat the fast motion.  Such a procedure is justified since \nquantizing the fast motion one takes $\\Gamma(L,\\phi;\\alpha)$ as a constant\nquantity and thus  the order of the quantization\ndoes not matter.\n\n\nThe existence of the resonance island in $(J,\\hat\\theta)$ space\nensures that the radial motion is localized.\nSo the remaining analysis should\nconcern the angular $(L,\\phi)$ motion which reflects the slow evolution\nof the Kepler ellipse. The structure of the $(L,\\phi)$ space\ninfluences values of quasi-energies as well as the structure of corresponding\nsemiclassical eigenstates. Trajectories in the $(L,\\phi)$ space are determined\nby constant values of $\\Gamma$\nalso responsible for the size of the resonance island\nin $(J,\\hat\\theta)$ space (recall that the island's size is determined \n approximately  by $\\sqrt{F\\Gamma}$).\n\nBefore presenting the results let us define scaled variables, typically used\nas a convenient parameterization of the microwave ionization problem (since\nthe dynamics scales classically \\cite{perc}). The common choice is to link\nthe scaling to the initial energy of the electron \\cite{KoLe95}.\n For an unperturbed 2D hydrogen atom the eigenenergies are given by\n  $E_n=-1\/2(n+1\/2)^2$ with $n$ being non-negative\ninteger. This allows us to define scaled variables via the principal quantum\nnumber of the initial state $n_0$ via\n\\begin{eqnarray}\n\\omega_0&=&\\omega(n_0+1\/2)^3\\\\\nF_0&=& F(n_0+1\/2)^4\\\\\nL_0&=&L\/(n_0+1\/2).\n\\end{eqnarray}\nIn particular note that the scaled angular momentum $L_0$ may take values\nfrom $[-1,1]$ interval with extremal values corresponding to circular\norbits on which the electron moves in two opposite directions.\n\n\\section{Results and discussion}\n\nTo compare the semiclassical predictions to the exact quantum calculations,\nwe consider the $n_0=21$ manifold of our 2D model atom. \nThis value is a compromise\nbetween the requirement to be in the semiclassical, large $n_0$ regime\nand computer memory limitations (size of the Floquet matrix to be diagonalized).\nFor resonant driving we take the microwave frequency to be\n $\\omega=\\omega_K=1\/(n_0+1\/2)^3$, ($\\omega_0=1$).\n  \nIn Fig.~\\ref{qep1} we present values of $\\Gamma$ as a function of\nthe scaled angular momentum, $L_0$\n and the $\\phi$ angle for two different values of\nthe degree of the field ellipticity given by $\\alpha$. Equipotential curves of\n$\\Gamma$ corresponding to semiclassical states originating from\n$n_0=21$ hydrogenic manifold are also shown in the Figure. Due to the\nspecific form of the resonance Hamiltonian (\\ref{EPr}) those lines are\n{\\it independent} of the microwave amplitude, $F$.\n\n\nFor $\\alpha=0$ the $(L,\\phi)$ space is symmetric with respect to the\n$L_0=0$ axis since, in the LP case, dynamics is not\naffected by the direction of the rotation of an electron. The left column\n in Fig.~\\ref{qep1} presents the results for $\\alpha=0.1$, i.e.,\nthe case very close to the LP problem. Note the presence of\nfour stable fixed points: two of them,\n$L_0\\approx 0$ and $\\phi=\\pi\/2,\\ 3\\pi\/2$,\ncorrespond to almost straight line orbits oriented perpendicularly to the\nmain axis of the polarization ellipse,\n the other two fixed points correspond to\ncircular orbits, $|L_0|\\approx 1$\n(for such orbits $\\phi$ is a dummy variable), with an electron\nrotating in the same or opposite direction to the direction of\nthe rotation of the field\nvector. In the vicinity of these fixed points the electronic motion\nis ``shape'' localized (since the motion remains in the vicinity of a given\n$(L_0,\\phi)$ point the eccentricity of the orbit, $e=\\sqrt{1-L_0^2}$, as well\nas its orientation with respect to the polarization ellipse, given by $\\phi$ \nis approximately preserved). Whether the motion is well localized in\nits radial motion in $(J,\\hat\\theta)$ variables, i.e. along the ellipse \ndepends on the size of the resonance island, given by $\\sqrt{F\\Gamma}$\nas mentioned above. If the island in $(J,\\hat\\theta)$ space is sufficiently\nlarge (comparable in size to $\\hbar$ or larger)\n the quantum state will be localized in the island. On the contrary,\n too small resonance island cannot lead to localization in $(J,\\hat\\theta)$\n space, possible quantum states will spread over the whole ellipse\n (then also the semiclassical approach used becomes obviously non\n adequate).\n Thus to observe interesting, nonspreading wavepackets the resonance island\n must be large enough.\n \n In this respect, two fixed points corresponding for small $\\alpha$ \n to almost circular orbits, $|L_0|\\approx 1$ lay at the maxima of\n $\\Gamma$ (compare the left top panel in the figure) and for sufficiently large\n $F$ may enable\n  a strong radial localization. The other two stable fixed points, corresponding\n  to elongated orbits lay at local minima of $\\Gamma$. \n There exist also two unstable fixed points around $L_0\\approx 0$ and\n$\\phi=0,\\ \\pi$, they form the origin of the separatrix dividing\nthe space.\n\nWhen $\\alpha$ increases the fixed points situated initially around\n$L_0\\approx 0$ and $\\phi=\\pi\/2,\\ 3\\pi\/2$ move in the direction of greater\nnegative values of $L_0$ (see the right column in Fig.~\\ref{qep1}),\nthus, the eccentricity of the corresponding orbits decreases.\nThe circular orbits do not change the shape but the resonance island\nin $(J,\\hat\\theta)$ space\nassociated with  $L_0\\approx 1$ becomes larger while that\nassociated with  $L_0\\approx -1$ becomes smaller as reflected\nby the greater and smaller values of $\\Gamma$, respectively.\nNote also that the islands in the $(L,\\phi)$\nspace containing\nlibrational states  shrink with the increase of $\\alpha$, hence,\nthey can support\nless and less semiclassical states -- while $\\alpha$ increases\nlibrational states vault over the separatrices and become rotational.\nIn the limit of $\\alpha=1$ the islands disappear and there exist only\nrotational states.\n\n\nConsider the level dynamics corresponding to the change of $\\alpha$.\nFig.~\\ref{levd} shows the level dynamics for the group of states\noriginating from the $n_0=21$ hydrogenic manifold. We take the scaled\nfield amplitude\n$F_0=0.03$, which for the CP case corresponds already to a significant\nionization yield \\cite{ZGD96} and may be, therefore, considered as a \nquite large value.\n For the LP case,\ni.e. $\\alpha=0$, all states are doubly degenerate. Those of them which\ncorrespond to rotational states in the $(L,\\phi)$ space are\ndegenerate because the change of the rotation of an electron does\nnot affect dynamics in the LP problem [the $(L,\\phi)$ space is symmetric\nwith respect to the $L_0=0$ axis]. The remaining librational states\nare degenerate as the two islands, around $L_0=0$, $\\phi=\\pi\/2$ and\n$L_0=0$, $\\phi=3\\pi\/2$, are identical and\nsupport identical states. Of course\nthe degeneracies may be removed due to tunneling\nprocesses, for instance, the wavepackets localized on circular orbits\n(the highest levels) rotate in the opposite directions\nand belong to distinct semiclassical states but quantum mechanical\nstates are the symmetric or antisymmetric linear combination of them.\nThus two wavepackets\npropagating on a circular orbit in the opposite directions correspond to\na single eigenstate\n(these wavepackets were already discussed in \\cite{ke95} for the limiting\nLP case only).\nSimilarly tunneling effects affect the librational states, i.e.\nquantum states are the linear combinations of\nsolutions in each island.\n\nAn increase of $\\alpha$ removes the semiclassical\ndegeneracy of rotational states as\none can see in Fig.~\\ref{levd} -- observe splittings of upper levels in the\nmanifold. It corresponds to the broken symmetry with respect to\nchange of a direction of rotation of an electron. Quasi-energies of\nstates corresponding to the motion in the opposite sense to the rotation of\nthe field vectors, i.e. corresponding to negative values of $L_0$, move\ndown while those with positive $L_0$ go up. It is a consequence of\nthe behavior of $\\Gamma$, i.e. the greater\n$\\Gamma$ the greater the quasi-energy value, see Figs.~\\ref{qep1}.\nObserve also that the degeneracy of all librational states (lower levels\nin Fig.~\\ref{levd}) is not immediately removed after a slight\nchange of the field ellipticity from $\\alpha=0$ but is removed successively\nduring an increase of $\\alpha$. This reflects the shrinking of the\ncorresponding\nislands (the situation mentioned above) which causes the librational states\nto vault over the separatrix and become rotational.\nThe levels with the smallest energy difference\ncorrespond to the librational and\nrotational orbits closest to the separatrix.\nThe narrowing of the level spacing in their vicinity\nis just a consequence of the slowing down of the classical motion \\cite{dd2}.\nIn the limit of $\\alpha=1$ there is no degeneracy in the manifold.\n\nNote that the energy splitting of the manifold\nis the largest\nin CP case while the smallest for LP.\nIt is associated with the corresponding strength\nof the perturbation and simply expresses the dependence of $\\Gamma$ on\n$\\alpha$. \n\n Exact quantum results coming from the\nnumerical diagonalization of the Floquet Hamiltonian are presented also\nin Fig.~\\ref{levd}. Among the  multitude\nof Floquet states appearing in the same energy range only those with the\nlargest overlap on the initial manifold are plotted for clarity.\nOne can see that\nthe agreement with the semiclassical predictions is very good except\nin the region of broad avoided\ncrossings (with other levels - partners in the crossing eliminated by the\noverlap selection) which appear in the upper part of the Figure.\n The semiclassical\nmethod does not take into account the interaction of the considered\n$n_0=21$ manifold with other manifolds (the method describes only\na single resonant\nmanifold), thus such avoided crossings have no chance to appear\nin our semiclassical calculations.\nFor a more accurate comparison we plotted, in Fig.~\\ref{energ}, quasi-energies\nfor two different values of $\\alpha$ separately.\nThe agreement between the semiclassical\nand quantum results is very good for high lying levels\n(greater values of $\\Gamma$) while for lower levels some differences\nappear as expected due to small size of the resonance island.\n\nIn the limiting case $\\alpha=1$, for CP case, another analysis of resonant\ndynamics is possible, namely a harmonic approximation around a stable\nequilibrium point in the rotating frame \\cite{ibb} (corresponding to the\nexpansion about a stable fixed point $L_0=1$, $\\phi$ arbitrary in our picture).\nIt is interesting to compare both approaches. The present semiclassical\nquantization of the resonance hamiltonian is certainly a more general approach\nvalid for a general EP case and for the whole manifold. The harmonic\nexpansion is limited to the CP case  and valid for highest lying states\nin the manifold only. On the other hand this expansion is quadratic in\ndeviations from the equilibrium point but nonlinear in the microwave field $F$\nstrength while the resonance Hamiltonian approach is a first order in $F$\nexpansion. Clearly in the deep semiclassical regime (large $n_0$) and for\nsufficiently large $F_0$ the harmonic expansion approach yields a better\napproximation for wavepacket states in the CP than the present resonant\nHamiltonian analysis (yielding, however, little information on the energies\nof other states in the resonant manifold). On the other hand, quite\nsurprisingly, we have found, by a direct comparison of numerical values,\nthat for $F_0$ around 0.01 or 0.06 and $n_0$ up to 30-40 the resonant\nHamiltonian yields semiclassical values closer to exact quantum data than\nthe harmonic expansion. Thus for intermediate range of $n_0$ and $F_0$ values\nthe resonant Hamiltonian quantization is surprisingly good in accuracy,\nyielding, at the same time, the predictions for the whole resonant manifold and\nfor an arbitrary polarization.\n\nTo complete the picture we show\nin Fig.~\\ref{levd1} the level dynamics of the same $n_0=21$ manifold\nversus scaled field amplitude, $F_0$, for $\\alpha=0.6$.\nQuantum results are presented\ntogether with the semiclassical ones. Again, for the\nresonance island in orbital $(J,\\hat\\theta)$\nmotion sufficiently large to capture\nquantum states, semiclassical results reproduce quantum ones\nquite well even beyond a classical chaos border (of course\nexcept avoided crossings).\n\nExact quantum Floquet matrix diagonalizations, performed using the Lanczos code\nyields not only the eigenvalues but also the corresponding eigenvectors.\nTheir time evolution may be visualized \\cite{abgre} to confirm \ndirectly that\nindeed the Floquet states localized both in $(J,\\hat\\theta)$ and in\n$(L,\\phi)$ spaces correspond to the nonspreading localized wavepackets.\nConsider again the level dynamics presented in Fig.~\\ref{levd}. As\ndiscussed above the higher lying state in the manifold corresponds to the\nwavepacket motion of the electron on the circular orbit in the direction \ncoinciding with the direction of rotation of the electric field vector.\nAnd indeed such a motion is revealed by the plots of the corresponding\nFloquet state (see Fig.~\\ref{packets}, upper row). This Floquet state\nchanges little with change of the ellipticity of the microwaves, $\\alpha$,\nand in the limiting case of CP becomes a well known CP nonspreading wavepacket\n\\cite{ibb,dzb95}. In the opposite limit of $\\alpha=0$ i.e. the LP microwave case\nthis state becomes almost degenerate with another one corresponding to\nthe different direction of the electron rotation. The two, almost\ndegenerate exact Floquet states are linear combinations of the two wavepackets\n(at least in 2D) as discussed in \\cite{ke95}.\n\nIt seems interesting to see what happens to the second member of the pair\nas polarization is changed from being linear to elliptical\n($\\alpha$ increases). The two states separate fast in\n$\\alpha$ and {\\it each} of them represents a distinct motion. As mentioned\nbefore the state going down in energy corresponds to wavepacket moving\non the circular orbit in the direction opposite to the field. It undergoes \na series of avoided crossings with other states of the manifold loosing\nprogressively its localized character. Still, for $\\alpha$ not too large,\nand far from avoided crossings its wavepacket character is clearly visible\n(compare lower row in Fig.~\\ref{packets}). This state looses its\nwavepacket character when the librational islands (compare Fig.~\\ref{qep1})\nmove sufficiently far down so as  rotational states with large negative\n$L_0$ disappear since there is no ``space left'' for them due to the finite\nvalue of $\\hbar$.  \nAs may be seen by comparison of Fig.~\\ref{qep1} and Fig.~\\ref{levd} this\ntransition occurs around $\\alpha=0.45$ for $n_0=21$, manifesting itself\nin the quantum data (Fig.~\\ref{levd}) \nby relatively broad avoided crossings encountered by\nthe wavepacket state. The transition point is, of course, $\\hbar$ dependent, for\nsmaller $\\hbar$ (larger $n_0$) the rotational, wavepacket-like, state with\nlarge negative $L_0$ exists for larger $\\alpha$. \nIt is worth stressing that for $\\alpha$ below this critical, $n_0$\n(and thus driving frequency) dependent value there are two distinct Floquet\nstates corresponding to two wavepackets moving in opposite directions.\n \n\\section{Conclusions}\n\nTo conclude we have shown, by the comparison with the quantum numerical\nresults, that the proposed semiclassical method, based on the first order\nresonant Hamiltonian  gives good quantitative\npredictions. Numerical calculations produce nothing but\nnumbers and an advantage\nof a semiclassical analysis is an understanding what physics is hidden\nbehind them. Apparently the behavior of H atom placed in EP microwave field,\nin the range of parameters used above, is determined by the underlying\nclassical dynamics, especially, one may build wavepackets which follow\nclassical trajectories without spreading.\nWe have shown that the wavepacket discovered in CP microwaves\n\\cite{ibb,dzb95} corresponding\nto the motion of the electron on a circular orbit in the same direction\nas the rotation of the microwave field exists also in EP case. For \na sufficiently small ellipticity of the microwave polarization, $\\alpha$,\nthere exist also another wavepacket state, corresponding to electron \nrotating in the opposite direction to the microwave field.  \nIn the limiting LP case both wavepackets coalesce, the exact Floquet states\ncorrespond semiclassically to linear combinations of  two wavepackets\npropagating in the opposite direction on the circular orbit \n\\cite{ke95}.\n\nThe analysis presented is restricted to the two-dimensional model atom\n(similar assumption is implicit in \\cite{ke95}), its validity for the\nreal three-dimensional atom is an open question. Certainly, in the limiting\nLP case, due to rotational symmetry with respect to the field axis, \nthe two wavepackets moving in the\nopposite directions in 2D lead in 3D to a doughnut shaped localized\nfunction oscillating between north and south pole of the sphere (assuming\na vertical polarization of LP microwaves) \\cite{bsdz98}.\n For CP, on the other hand,\nthe wavepacket motion remains essentially two-dimensional \\cite{zdb95}.\nIt will be most interesting to see how the third dimension affects the\ndynamics also for a general case of EP microwave field since already\nclassical studies \\cite{sz97,sz98} indicate that some qualitative\ndifferences may appear. This subject is left for future studies. \n\nStill\nthe present 2D analysis shows that the nonspreading wavepacket states\nlocalized on circular orbits are not restricted to circular polarization\nof microwaves only. Thus the perfect circular polarization is not essential\nfor a possible experimental observation of nonspreading wavepackets. \n\nInterestingly, in the EP case,\nthere appears a possibility of the angular localization in minima of\nthe effective potential $\\Gamma$ with the position of the minima being\n$\\alpha$ dependent (compare Fig.~\\ref{qep1}).\n If the creation of localized wavepackets in such minima\nwere possible it would allow to control the shape of the trajectories\non which the wavepackets propagate by a change of microwave polarization,\nand not by an additional static field as proposed in \\cite{szd98}.\nStudy of the corresponding wavefunctions indicates, however, that\ndue to the small resonance island width the localization {\\it along}\nthe ellipse [i.e. in ($J,\\hat\\theta$) space] is not very effective\nat least in the range of $n_0$ and $F_0$ studied by us. \n\n\n\\section{Acknowledgments}\nWe are grateful to Dominique Delande for numerous discussions\nand the possibility to use his Lanczos diagonalization routines.\nK.S. is a fellow of the Foundation for Polish Science.\nSupport of KBN\nunder project No.~2P302B-00915 (K.S. and J.Z.) is acknowledged.\nNumerical calculations were performed at Academic Computer Center Cyfronet\nin Krak\\'ow with the help of grants KBN\/S2000\/UJ\/067\/1998 and\nKBN\/S2000\/UJ\/068\/1998.\n\\section{Appendix}\n\nConsider the Floquet Hamiltonian, $H_F$, as defined in (\\ref{floq}) with\nthe hamiltonian given by (\\ref{ham}).\nFor the sake of efficiency of the numerical diagonalization it is\nconvenient\nto rewrite the Floquet Hamiltonian in the velocity gauge by applying\na standard unitary transformation \\cite{cct,sth}\n\\begin{equation}\nH_F=-i\\frac{\\partial}{\\partial t}+\n\\frac{p_x^2+p_y^2}{2}-\\frac{1}{r}+\\frac{F}{\\omega}[p_x\\sin\\omega t-\n   p_y\\alpha\\cos\\omega t]\n   +\\frac{F^2}{4\\omega^2}(\\alpha^2+1).\n\\label{velo}\n\\end{equation}\nNext we pass to scaled semiparabolic coordinates (\\ref{scaluv}) which\nallows us to express the\nFloquet eigenvalue equation\n\\begin{equation}\nH_F\\mid u_n(t)\\rangle=E_n\\mid u_n(t)\\rangle\n\\label{eigen}\n\\end{equation}\nas a generalized eigenvalue problem\n\\begin{equation}\n({\\cal A}-E_n{\\cal B})\\mid u_n(t)\\rangle=0\n\\label{eigen1}\n\\end{equation}\nwhere\n\\begin{equation}\n{\\cal A}=(u^2+v^2)H_F \\ \\ \\  \\mbox{and} \\ \\ \\ {\\cal B}=u^2+v^2\n\\end{equation}\nwith\n\\begin{equation}\nH_F=\\frac{p_u^2+p_v^2}{2\\Lambda^2(u^2+v^2)}-\\frac{2}{\\Lambda(u^2+v^2)}\n-i\\frac{\\partial}{\\partial t} +\\frac{F}{\\Lambda\\omega (u^2+v^2)}[\n(up_u-vp_v)\\sin\\omega t-\\alpha (vp_u+up_v)\\cos\\omega t].\n\\end{equation}\n(with the additive pondoromotive term omitted).\n\n\nFloquet states are periodic in time\n(with the period $T=2\\pi\/\\omega$) thus we may expand $\\mid u_n(t)\\rangle$\nin the Fourier series\n\\begin{equation}\n\\mid u_n(t)\\rangle=\\sum_{K=-\\infty}^{+\\infty}e^{-iK\\omega t}\\mid u_n^K\\rangle\n\\end{equation}\nThis allows to cast Eq.~(\\ref{eigen1}) into the equivalent set of coupled\nequations\n\n\\begin{equation}\n\\left[\\frac{p_u^2+p_v^2}{2\\Lambda^2}-\\frac{2}{\\Lambda}-K\\omega\n(u^2+v^2)\\right]\\mid u_n^K\\rangle \n+(W_s-W_c)\\mid u_n^{K-1}\\rangle\n-(W_s+W_c)\\mid u_n^{K+1}\\rangle  =\nE_n(u^2+v^2)\\mid u_n^K\\rangle\n\\label{seteq}\n\\end{equation}\nwhere\n\\begin{eqnarray}\nW_s&=&i\\frac{F}{2\\omega\\Lambda}(up_u-vp_v) \\cr\nW_c&=&\\alpha\\frac{F}{2\\omega\\Lambda}(vp_u+up_v).\n\\end{eqnarray}\nAll terms in the above equation have the polynomial form in coordinates\nand momenta. This suggests to use the harmonic oscillator basis\nfor an efficient evaluation of matrix elements \\cite{sth}. The method becomes\nthen analogous to the treatment of the circular polarization case discussed\nin detail elsewhere \\cite{ZGD96}. The resulting generalized eigenvalue\nequation is diagonalized using the Lanczos code which allows for\nan extraction of eigenvalues in a selected energy range and the corresponding\neigenvectors. For completeness let us mention only that\nthe Floquet Hamiltonian is invariant under the generalized parity\ntransformation, i.e. the parity transformation combined with the\ntranslation in time by $\\pi\/\\omega$.\n Thus ${\\cal A}$ and ${\\cal B}$\nmatrices may be split into uncoupled matrices that are two times\nsmaller. This  makes\nthe numerical calculations more efficient.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Background}\n\n\\subsection{Binary Neural Networks}\\label{s:bnn_networks}\nLike full-precision DNNs, Binary Neural Networks (BNNs) are composed of many layers (i.e., convolution, dense, etc) stacked on top of each other. BNNs use one-bit weights and inputs constrained to $+1$ and $-1$, but they usually are trained using full-precision weights. Then, the weights and inputs are binarized using the following equation:      \n\n\\begin{equation}\nx^b = \\begin{cases}\n+1    &\\text{if }  x>=0, \\\\\n-1         &\\text{otherwise, }  \n\\end{cases}\n\\label{e:x_binarization}\n\\end{equation}\n\nwhere $x$ is a full-precision weight or input and $x^b$ is the binarized value.\n\nConvolution operations with binary inputs and weights are computed as their full-precision counterpart. However, to calculate an output feature Equation~\\ref{e:binary_mac} is employed. In this equation, $o$ is a feature of the output feature map, whereas $w^b$ and $x^b$ are the binarized kernel (weights) and input vectors. Note that the kernel and the input are composed of $1$ and $-1$, but they are stored as $1$ and $0$, respectively.\n\n\\begin{equation}\no = popcount(xnor(w^b,x^b))\n\\label{e:binary_mac}\n\\end{equation}\n\nEquation~\\ref{e:binary_mac} is essentially computing a multiply-accumulate operation (MAC) which is the main computation on DNNs. Hence, as mentioned earlier, BNNs are very efficient in computations and memory requirements since MAC operations can be computed using xnor and popcount functions, and weights are stored using 1 bit. \n\n\\subsection{BNN model}\\label{s:cnn_model}\nRecently, on image classification tasks, BNN models can achieve top-1 accuracy, similar to the accuracy of full-precision models tailored to edge devices. In this regard, ReActNet~\\cite{reactnet} is a BNN model based on MobileNet~\\cite{mobilenetv2} with a top-1 accuracy of 69.4\\% on the ImageNet dataset. Note that the accuracy of Mobilenet V2 for the same task is 72\\%. Since the accuracy of ReActNet is very close to the full-precision counterpart, we employ this model as the baseline in this work.    \n\nThe main building block of ReActNet is shown in Figure~\\ref{f:reacnet_arch}. This block comprises two convolutional operations that employ binarized input and kernels. Also, it performs other operations such as batch-norm and Prelu activation functions which are computed using full-precision. One of the critical contributions of this model is that the Prelu activation is biased by shifting and reshaping its input. By performing this transformation, the accuracy of the network is substantially improved~\\cite{reactnet}.    \n\nReActNet is composed of 15 layers which include one input, one output layer, and 13 layers that are constructed using the basic block shown in Figure~\\ref{f:reacnet_arch}. The input layer is convolutional, whereas the output layer is fully-connected. Both layers are computed using full-precision values, and in this work, we quantize them using 8 bits. \n\nTable~\\ref{t:reacnet_storage} shows the storage requirements and execution time for the main operations in ReActNet. As can be seen, the storage requirement due to the 3x3 convolutions accounts for 68\\% of the total. Furthermore, these 3x3 convolutions represent 67\\% of the execution time; hence in this work, we focus on improving these convolutions.    \n  \n\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=3.375in]{figures\/reacnet_arch.pdf}\n    \t\\vspace*{-2mm}\n    \\caption{ReActNet basic block and Architecture.}\n    \\label{f:reacnet_arch}\n\\end{figure}\n\n\n\n\n \\begin{table}[t!]\n\t\\caption{ReActNet storage and execution time breakdown. Others refers to activation and normalization layers,etc.}\n\t\\label{t:reacnet_storage}\n\t\\centering\n\t\\begin{tabular}{cccc}\n\t\t\\cellcolor[gray]{0.9}\\small\\textbf{Operation}&\\cellcolor[gray]{0.9}\\cellcolor[gray]{0.9}\\small\\textbf{Storage (\\%)} &\\cellcolor[gray]{0.9}\\small\\textbf{Precision (bits)}&\\cellcolor[gray]{0.9}\\small\\textbf{Execution time(\\%)}   \\\\\n\t\t\\small Input Layer &\\small 0.02&\\small 8&\\small 4.0     \\\\\n\t\t\\cellcolor[gray]{0.9}\\small Output Layer &\\cellcolor[gray]{0.9}\\small 22.17\\ &\\cellcolor[gray]{0.9}\\small 8&\\cellcolor[gray]{0.9}\\small 18.7   \\\\\n\t\t\\small Conv 1x1 &\\small 8.5&\\small 1&\\small 6.9    \\\\\n\t\t\\cellcolor[gray]{0.9}\\small Conv 3x3 &\\cellcolor[gray]{0.9}\\small 68.0&\\cellcolor[gray]{0.9}\\small 1&\\cellcolor[gray]{0.9}\\small 66.8 \\\\\n\t\t\t\\small Others &\\small 1.31& \\small 32& \\small 3.6   \\\\\t\n\t\t\n\t\\end{tabular}\n\\end{table}\n\\section{Kernel Compression}\n\nWe observe that for the 3x3 binarized kernels found in ReActNet, each channel is composed of nine values that are either one or zero; thus,  there are only 512 unique $bit~sequences$. However, some $bit~sequences$ are rarely used, while other sequences tend to appear very frequently. Therefore, we could exploit this observation by employing a variable encoding that assigns fewer bits to the most common $bit~sequences$. Moreover, we experimentally found that some rarely used $bit~sequences$ can be substituted by some others that are more common without negatively affecting the overall network accuracy.   \n\n\nFigure~\\ref{f:unique_sequences_ex} shows a binarized 3x3 kernel with 2 channels. As mentioned earlier, each of these channels is composed of nine values which are either $1$ or $-1$, and to represent one channel a least 9 bits are needed. For clarity, to refer to a given $bit~sequence$,  we employ a Natural mapping, as shown in Figure~\\ref{f:unique_sequences_ex}, that assigns a Natural number to each $bit~sequence$. In this regard, we assign the value in position 0,0 to the most significant bit and the value in position 2,2 to the least significant bit. For example, a channel where all the values are $zeros$ (i.e., -1) will be referred to as 0. Similarly, a channel where all the values are $ones$ will be mapped to 511. Note that this is a conceptual representation, and the actual layout of these bits in memory can be different.  \n\nThe rest of this section presents an analysis of the use frequency for each $bit~sequence$ in ReActNet. Then, we describe our methodology to replace uncommon $bit~sequences$. Finally, we detail our compressing scheme. \n\n\n\\subsection{Frequency of use for $bit~sequences$}\\label{s:sequence_frequency}\n\nFigure~\\ref{f:frequency_sequence_l12} shows the frequency of use for the 16 more common $bit~sequences$ in all the 3x3 binary kernels in one of the basic blocks of ReActNet.\nAs shown in the plot, $bit~sequences$ containing only zeros or ones account for 25\\%. In contrast,  the most common 16 and 64 $bit~sequences$ account for 46\\% and 75\\% of all of them, respectively. Regarding the kernels in other basic blocks, their behavior is similar to the one in Figure~\\ref{f:frequency_sequence_l12}. As shown in Table~\\ref{t:distribution_by_layer}, in  all the blocks, the top 64 $bit~sequences$ account for more than\n50\\% of the sequences. Moreover, for all the blocks, the 256 most frequently used $bit~sequences$ account for up to 92\\%. Henceforth, instead of employing nine bits to store each channel, we could use 5 bits to store half of them and nine or less for the rest.\n   \n\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=3.375in]{figures\/unique_sequences_mapping.pdf}\n    \t\\vspace*{-2mm}\n    \\caption{3x3 binary kernel with 2 channels. Conceptually the values of a channel are mapped to an integer number. }\n    \\label{f:unique_sequences_ex}\n\\end{figure}\n\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=3.375in]{figures\/frequency_sequence_l12.pdf}\n    \t\\vspace*{-2mm}\n    \\caption{Frequency of use for the top 16 $bit~sequences$ for the 3x3 binary kernels in one of  ReActNet's basic blocks.}\n    \\label{f:frequency_sequence_l12}\n\\end{figure}\n\n \\begin{table}[t!]\n\t\\caption{ Distribution of $bit~sequences$ for the 3x3 kernel in each basic block.}\n\t\\label{t:distribution_by_layer}\n\t\\centering\n\t\\begin{tabular}{ccc}\n\t\t\\cellcolor[gray]{0.9}\\small\\textbf{Layer}&\\cellcolor[gray]{0.9}\\small\\textbf{Top 64 (\\%)} &\\cellcolor[gray]{0.9}\\small\\textbf{Top 256 (\\%)}\\\\\n\t\t\\small Block 1 &\\small 53.4&\\small 90.6\\\\\n\t\t\\cellcolor[gray]{0.9}\\small Block 2 &\\cellcolor[gray]{0.9}\\small 64.5 &\\cellcolor[gray]{0.9}\\small 95.1\\\\\n\t\t\\small Block 3 &\\small 56.3&\\small~87.11\\\\\n\t\t\\cellcolor[gray]{0.9}\\small Block 4 &\\cellcolor[gray]{0.9}\\small 64.8&\\cellcolor[gray]{0.9}\\small 92.7\\\\\n\t\t\t\\small Block 5 &\\small 63.2& \\small 88.3\\\\\n\t\t\\cellcolor[gray]{0.9}\\small Block 6 &\\cellcolor[gray]{0.9}\\small 63.1 &\\cellcolor[gray]{0.9}\\small 90.86 \\\\\n\t\t\\small Block 7 &\\small 62.4&\\small 91.64\\\\\n\t\t\\cellcolor[gray]{0.9}\\small Block 8 &\\cellcolor[gray]{0.9}\\small 60.8&\\cellcolor[gray]{0.9}\\small 90.24\\\\\n\t\t\t\\small Block 9 &\\small 55.2& \\small 92.9\\\\\t\n\t\t\\cellcolor[gray]{0.9}\\small Block 10 &\\cellcolor[gray]{0.9}\\small 62.2 &\\cellcolor[gray]{0.9}\\small 89.9 \\\\\n\t\t\\small Block 11  &\\small 67.97&\\small 92.\\\\\n\t\t\\cellcolor[gray]{0.9}\\small Block 12 &\\cellcolor[gray]{0.9}\\small 75.3&\\cellcolor[gray]{0.9}\\small 93.4\\\\\n\t\t\\small Block 13  &\\small 58.3&\\small 86.9    \n\t\t\t\t\n\t\\end{tabular}\n\\end{table}\n\n\\subsection{Compressing $bit~sequences$ }\\label{s:kernel_compression}\nAiming to increase storage efficiency and to exploit the variability in the frequency of use of the $bit~sequences$, we propose to use Huffman Encoding to encode them. When using Huffman encoding to compress the kernels, we must first create a Huffman tree, where each $bit~sequence$ is assigned to a tree node. Then, before using a $bit~sequence$, we must decode it with the help of lookup tables that contain the uncompressed value. Note that the tree is created offline, whereas the decoding is done at runtime.      \n\nA significant issue when decoding a Huffman encoded sequence of bits is that it requires optimized lookup tables or complex hardware if high throughput is needed~\\cite{phuffman}. In this regard, we proposed to use a simplified Huffman tree where we limit the tree size to a small number of nodes (i.e., four), as shown in Figure~\\ref{f:simplified_tree}. In this tree,  several encoded $bit~sequences$  are assigned to each node. Note that we could employ a table to store the encoded $bit~sequences$ belonging to a given node. Hence, during the decoding process, we use the first bits of the encoded $bit~sequence$ to find out to which table (node) it belongs and then employ the remaining bits to address the table and get the uncompressed $bit~sequence$. This simplified version provides a good trade-off between simplicity and compression rate, as shown in Section~\\ref{s:results}.\n\n    \\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=3.375in]{figures\/simplified_tree.pdf}\n    \t\\vspace*{-2mm}\n    \\caption{Simplified Huffman tree with four nodes. The prefix of an encoded bit sequence is used as a pointer to a table with the uncompressed value.}\n    \\label{f:simplified_tree}\n\\end{figure}\n\n\\subsection{Removing less frequent $bit~sequences$}\\label{s:removing_less_common}\n\nWe observe that in some cases, a rarely used $bit~sequence$ ($sa$) can be replaced by one employed more frequently ($sb$) without negatively affecting the model accuracy. For this, we constrain the hamming distance between $sa$ and $sb$ to 1 (i.e., only a one-bit difference ) to keep the error introduced during the evaluation low. Hence, we can further improve the compression ratio by increasing the frequency of use of the most common $bit~sequences$ and removing some of the less common ones. To exploit this observation, we employ the following algorithm.  \n\n\nFirst, for all the 3x3 binary kernels in a given basic block, we create a set $st$ containing the $M$ most commonly used $bit~sequences$. Then, we create a second set $su$ containing the $N$ less commonly employed. After this, we try to replace the $bit~sequences$ in $su$ with $bit~sequences$ from $st$. This process is done by first taking a $bit~sequence$ ($sa$) from $su$. Then comparing its hamming distance with the hamming distance of the $bit~sequences$ in $st$. When a $bit~sequence$ ($sb$) is found, such that the hamming distance between $sa$ and $sb$ is one, we replace $sa$ by $sb$. Note that if many $bit~sequences$ match this criterion, we employ the $bit~sequence$ with the highest frequency to replace $sa$. Finally, we repeat this process for all the $bit~sequences$ in $su$. We empirically searched for some combinations of $M$ and $N$; while this search is not exhaustive, we found some combinations that significantly improve the compression ratio, shown in Section~\\ref{s:results}.     \n\n\n\n\n\n\n\\section{Related Work}\nHardware acceleration for BNNs has been an active area of research in recent years.\nIn this regards, there have been several works targeting CPUs~\\cite{trusov2022fast}, FPGAs~\\cite{zhang2021fracbnn,du2022bnn,fiscaletti}, and ASICs~\\cite{hosseini2021binary,cho2021reconfigurable}. FracBnn~\\cite{zhang2021fracbnn} proposes a FPGA accelerator for ReActNet. In addition, it improves performance by employing up to two bits for activation and increasing the level of sparsity in the kernels. Also, they binarize the first layer of the model, wheres it is quantized in this work. The work in~\\cite{du2022bnn} proposes an FPGA accelerator that combines the xnor and accumulation to improve performance. BNNsplit~\\cite{fiscaletti} is another accelerator that divides the evaluation of the BNN model across multiple FPGAs. Our proposal is different from these works, since we focus on compressing the kernels and we target mobile CPUs.\n\nHosseini et al.~\\cite{hosseini2021binary} propose a programmable many-core accelerator targeting the primary operations found in BNN networks. The work in~\\cite{cho2021reconfigurable} is another BNN accelerator that analyzes each layer and, based on the given parameters, reconfigures the hardware to increase parallelism. Both works target ASICs, whereas our work focuses on mobile CPUs.\n\nThe work in~\\cite{trusov2022fast} proposes a new blocking algorithm to partition the binary kernels using a small block size such that the utilization of CPU vector registers is more efficient. In Deep Compression~\\cite{han2015deep} Huffman encoding is applied to full-precision networks after pruning them. This scheme is proposed for GPUs, and the optimizations are in software. Our proposal is different since it is a hybrid software\/hardware approach.     \n\n\n\\section{Conclusion}\nThis work presents a novel compressing scheme for binary neural networks. We show that \nin the binary kernels, some \\textit{bit sequences} occur more frequently than others. Aiming to exploit this observation, we encode each sequence with a simplified Huffman Tree. We propose a simple hardware extension to conventional CPUs to support this scheme, since implementing it on software degrades performance. After applying our compression scheme, the binary kernels are compressed by 1.32x on average, whereas the BNN model is compressed by 1.2x. Also, compared to the software baseline, our proposal achieves a speedup of 1.35x.\n\n\n\\section{Hardware and Software Implementation}\n\nThis section describes our proposal to improve BNN performance using compressed kernels. First, we present an overview of our proposal. Next, we detail the software baseline used to evaluate the model. Then, we motivate the need for some hardware optimizations. Finally, we discuss the hardware implementation of our scheme.\n\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=3.375in]{figures\/packing_ex.pdf}\n    \t\\vspace*{-2mm}\n    \\caption{Channel packing a 2-channel 3x3 binary kernel. In this case, a 2-bit register is used to store a bit from each channel. }\n    \\label{f:packing_ex}\n\\end{figure}\n\n\\subsection{Overview}\\label{s:overview}\nOne of the goals of this work is to employ the compressed 3x3 binary kernels during the evaluation of the \nBNN model. Our overall proposal work as follows. First, we compute the frequency of use for each $bit~sequence$ in all the 3x3 binary kernels. Then, based on those frequencies, we create a Huffman Tree and assign an encoding to each of the $bit~sequences$. Note that the previous steps are done offline and only once. \nNext, when evaluating a 3x3 kernel, we store the Huffman tree corresponding to that kernel in memory. After this, we start fetching the compressed $bit~sequences$ and inputs from memory. Then, the $bit~sequences$ are decompressed, and the kernel\nevaluation is performed. This process is repeated for all $bit~sequences$ in the kernel. Nonetheless, compressing the kernels decreases the storage requirements; it adds an overhead to decode the $bit~sequences$ resulting in a slowdown when implemented in software. However, the overhead can be mitigated with additional hardware support. We detail these issues in the following sections.  \n\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=3.375in]{figures\/decoding_unit.pdf}\n    \t\\vspace*{-2mm}\n    \\caption{Decoding Unit}\n    \\label{f:decoding_unit}\n\\end{figure}\n\n\\subsection{Software Baseline}\\label{s:software_baseline}\n\nWhen evaluating BNNs in modern CPUs, one of the most critical components is laying out the kernels in memory such that the number of bits loaded into the internal register of the CPU~\\cite{zhang2019dabnn,trusov2022fast} is maximized. In this regard, one of the most efficient methods is to pack bits from different channels together so that the number of bits packed together fills an entire register when loaded from memory~\\cite{zhang2019dabnn}. For simplicity, we refer to this method as \\textit{channel packing}.       \n\nIn \\textit{channel packing}, the group of bits packed from each channel must belong to the same position (i.e., 0,0), as shown in Figure~\\ref{f:packing_ex}. Also, when the number of channels is larger than the CPU register's size, the channels are split into several sub-groups such that they can be loaded independently into the CPU registers.\nNote that \\textit{channel packing} works best when the number of channels is divisible by the size of CPU registers; otherwise, padding is needed. Padding BNN kernels is challenging since the zero value represents a -1. Subsequently, when padding is done, the kernel computation must account for the extra -1. In this work, we employ \\textit{channel packing} to store the uncompressed 3x3 binary kernels, and it is done offline, as in~\\cite{zhang2019dabnn}. Also, since the number of channels for all the kernels is a power of two, we do not employ padding.\n\n\n \nRegarding the layout in memory of the compressed kernels, note that each of the encoded $bit~sequence$ has a variable length, and thus we cannot perform channel packing offline. In this case, we store them consecutively in memory as a sequence of encoded words. As a result, during the kernel evaluation, once an encoded $bit~sequence$ is decoded, we need to pack it into the CPU registers. Note that the size of a decoded $bit~sequence$ is 9 bits, whereas the typical size of a CPU register is 32 bits or more; hence if packing is not done, the CPU registers are underutilized.\n\n\n\n\nWe implemented our compression scheme in software to evaluate its feasibility. Then, we compare the performance of this implementation against a baseline using uncompressed kernels. For the compression scheme, we implemented the simplified Huffman Tree described in Section~\\ref{s:kernel_compression}. In this case, we limit the size of each tree to four nodes and use four tables to store the decoded $bit~sequence$. Then, we employ the first bits of each encoded $bit~sequence$ to indicate the table where it belongs. Moreover, we use the remaining bits in the uncompressed $bit~sequence$ to get the original from its corresponding table.\nAfter comparing these two implementations, the compressed kernel implementation is 1.47x slower than the baseline. \n\n\n    \n\n\\subsection{Hardware Support}\\label{s:hardware_compression}\n\n\nThe main reason for the slowdown of the software implementation of our scheme is the overhead of decoding and packing the $bit~sequences$ at runtime. Hence, to mitigate this issue, we propose performing it in hardware. For this, we add a \\textit{decoding unit} to the load-store unit (LSU) of the CPU. Furthermore, we add two new instructions to manage this unit. The \\textit{decoding unit}, shown in  \nFigure~\\ref{f:decoding_unit}, is tailored to streaming and decoding the encoded \\textit{bit sequences}. As can be seen, it is composed of a \\textit{streaming unit} and a \\textit{packing unit}.\n\n \n\n\n \n\nThe \\textit{streaming unit} is employed to load a stream of encoded \\textit{bit sequences} from the main memory and decompress them. It works as follows, first, it uses the \\textit{stream address register} to store the base address of the \\textit{bit sequences} that will be fetched. Then, a request to load $T$ bytes from the address in the \\textit{stream address register} is sent to the LSU decoder. Once the $T$ bytes are fetched from memory, they are stored in the \\textit{input buffer}. After this, the decoding process starts by sending $m$-bits of data from the \\textit{input buffer} to the \\textit{stream parser}. Note that, we also send a new request to fetch more bytes while doing the decoding. In the \\textit{stream parser}, from this $m$-bits the first $n$ bits are used to create the \\textit{node address} which identifies the node of the Huffman tree belonging to the current \\textit{bit sequence}. Note that since the length of the encoded \\textit{bit sequence} is variable, after finding the node, we need to find its length. It is done by reading it from the \\textit{length table}, which is addressed with the \\textit{node address}. Then, with the length information, we can get the remaining bits of the encoded \\textit{bit sequence} and store them in the \\textit{decoded address register}. Next, in the \\textit{decoder unit}, the uncompressed \\textit{bit sequences} are stored in a scratchpad memory (\\textit{uncompressed table}), which is partitioned into multiple banks. Then, we employ the \\textit{node address} and the value in the \\textit{decoded address register} to address the \\textit{uncompressed table} and get the uncompressed \\textit{bit sequence}. Finally, the uncompressed sequence is passed to the $packing~unit$. We repeat the previous process until the whole stream of bits in the \\textit{input buffer} have been processed.   \n\n\n\n\n\nThe $packing~unit$ is in charge of \\textit{channel packing} each one of the \\textit{bit sequences} received from the $streaming~unit$ into the \\textit{packing registers}. In this unit, after receiving a \\textit{bit sequence}, it is distributed into $k$ register (i.e., 9 ) each of $R$ bits (i.e., 128 bits). Note that $R$ \\textit{bit sequences} are channel packed sequentially into \\textit{k} registers.\nAlso, once $R$ \\textit{bit sequences} are packed, a new set of $k$ register is used to store the incoming \\textit{bit sequences}.\n\n\nTo use the \\textit{decoding unit}, the programmer first must configure it. For this, we add the instruction \\textit{lddu} (i.e., load decoder unit configuration). This instruction uses a pointer to a  configuration structure to load its values in the \\textit{decoding unit}. Table~\\ref{t:configuration_structure} shows the main configuration values for the \\textit{decoding unit}  which are stored in the main memory. Once the configuration values are loaded, the \\textit{decoding unit} is reset and it starts decoding the \\textit{bit sequences}.    \n\nFor reading the packed and uncompressed \\textit{bit sequences} from the \\textit{decoding unit}, we add the instruction \\textit{ldps} (i.e., load packed bit sequence). To use this instruction, the programmer must specify the destination register to store the \\textit{bit sequence} being read. Note that this instruction will read the oldest \\textit{bit sequence} that was decoded in the \\textit{decoding unit}. Hence, the programmer is responsible for setting this unit before using \\textit{ldps}.      \n\nTo summarize, the \\textit{decoding unit} works as follows. First, a \\textit{lddu} instruction is executed to start its configuration and the decoding process. Then, in the background, the \\textit{decoding unit} fetches and decodes the compressed \\textit{bit sequences}. Later, when evaluating a 3x3 convolution, the programmer uses the \\textit{lddu} instruction to load the uncompressed and channel-packed \\textit{bit sequences} into the CPU registers. This process is repeated during the computation of a kernel until all the \\textit{bit sequences} are evaluated.  \n\n\n\n\n\n\\begin{table}[t!]\n\t\\caption{Configuration Structure.}\n\t\\label{t:configuration_structure}\n\t\\centering\n\t\\begin{tabular}{cc}\n\t\t\\hline\n\t\t\\multicolumn{2}{c}{Number of bit sequences }\\\\\n\t\t\\multicolumn{2}{c}{\\cellcolor[gray]{0.9}\\small{Compressed sequences pointer}}\\\\\n\t\t\t\\multicolumn{2}{c}{\\small Compressed sequences length }\\\\\n\t\t\t\\multicolumn{2}{c}{\\cellcolor[gray]{0.9}\\small Huffman tree nodes}\\\\\n\t\n\t\\end{tabular}\n\\end{table}\n\t\\vspace*{-2mm}\n\n\n\n\n\n\n\n\n\n\\section{Introduction}\n\n\nDeep Neural Networks (DNNs) have recently achieved tremendous success in various tasks such as image classification and speech recognition. Not surprisingly, they have become the most critical component for Machine Learning Applications. \nNonetheless, the cost of storing and evaluating DNNs is high since they usually are composed of millions of parameters. Mainly, evaluating DNNs on edge devices is quite challenging due to the constrained resources on edge devices. In this regard, Binary neural networks (BNNs) are an attractive solution to reduce the cost of evaluating DNNs on edge devices. In BNNs, the weights and input features can only take the value of $1$ and $-1$; thus, one bit is used to store them. Therefore, the cost of storing a BNN layer can be reduced dramatically compared to those that use integer or floating point values. Moreover, BNNs can be executed very efficiently with modest hardware since the matrix multiplications required to evaluate a DNN, can be performed using a series of xnor and popcount operations. \n\n\n\nWhile BNNs are very computationally efficient, their accuracy tends to be relatively low compared to a similar state-of-the-art full-precision network. However, recent BNN models, such as ReactNet~\\cite{reactnet} and FracBNN~\\cite{zhang2021fracbnn}, achieve a top-1 accuracy comparable to state-of-the-art full-precision networks. For instance, ReActNet obtains a top-1 accuracy of 69\\% on the Imagenet dataset while MobileNet~V2~\\cite{mobilenetv2} (i.e., CNN model tailored to edge devices) has a top-1 accuracy of 72\\%. Nevertheless, the number of bits required to store ReactNet is 29 million, whereas it is 27 million for MobileNet~V2. Hence, evaluating these models on edge devices is still challenging due to their limited storage and computational resources. Most of the storage and computational requirements of BNNs are due to convolutional operations.  \n\n\n\nIn this work, we focus on improving the performance and storage requirement of BNN models using convolutional operations. In this regard, in a binary kernel (i.e., weights are 1 or -1), for any given \\textit{n by n} channel stored as a sequence of bits, there are $2^{n*n}$ possible sequences of bits.\nHowever, we observe that during the evaluation of a convolution operation for a given kernel, some sequences of bits are used more frequently than others. For instance, in some layers of ReActNet, 32 individual sequences of bits account for more than 50\\% of all the possible sequences. To exploit this observation, we propose to employ a variable length compression algorithm to encode each sequence of bits. Then, we assign an encoding with fewer bits to the most common sequences since they are more likely to be used when evaluating a kernel. For simplicity in the rest of the paper, we refer to the sequence of bits formed with the values of a given channel as \\textit{bit~sequence}.  \n\n\n\n\n\n A significant challenge in implementing a variable-length encoding is providing a software- and hardware-friendly solution while also being efficient~\\cite{phuffman}. A software-only implementation is insufficient since the performance is severely affected by the overhead of decoding the compressed bit sequences. We address this issue by providing hardware support to decompress the \\textit{bit sequences}. Moreover, when evaluating BNNs on mobile CPUs, the matrix multiplications are done very fast since the latency of computing a xnor and a popcount is very low. Therefore, the loads to fetch the weights are in the critical path. Hence, we also provide hardware support to stream the weights automatically to mitigate this issue. \n\n\nAnother important observation is that for some layers, a less frequently used \\textit{bit sequence} can be replaced by one that is employed more often without negatively impacting the accuracy of the network. Accordingly, we propose a clustering algorithm that substitutes some of the less common \\textit{bit sequences} with others that are similar to them but also very common. By reducing the less common \\textit{bit sequences}, we can compress the binary kernels more efficiently since the number of bits to store the most common ones is smaller.\n\n\n \n\n\n\n\n\n\nIn summary, the main focus of this paper is improving the performance and memory requirements of BNNs on mobile CPUs. We claim the following contributions:\n\n\\begin{itemize}\n    \\item \nWe analyze the frequency distribution of all the \\textit{bit sequences} in the network and show that some of them are used more frequently than the rest.\n\n    \\item  We propose to use a simple variable-length encoding to compress the binary kernels. Also, we describe a novel clustering algorithm that increases the frequency of the most common \\textit{bit sequences}. Our scheme compresses the BNN layers by 1.32x on average and improves the model storage requirements by 1.2x.\n    \n    \\item We propose simple hardware extensions and evaluate them on top of a modern ARM CPU. Our proposals improve the performance by 1.35x.\n\n    \n\\end{itemize}\n\n\n\n\n\n\\section{Evaluation Methodology}\\label{s:methodology}\nWe employ the Pytorch implementation of ReActNet and the ImageNet dataset as our baseline to evaluate the accuracy. Also, we use Python to compress and analyze the frequency of use for each bit sequence. For the performance evaluation, we use the daBnn~\\cite{zhang2019dabnn} C++ framework to implement the BNN model. DaBnn employs \\textit{channel packing} for the kernels and inputs. Moreover, it provides several C++ functions to compute the most common layers found on BNNs. This library is highly optimized for the ARMv8 architecture and uses the NEON extensions.  \n\n\nAs a hardware baseline, we use an ARM A53 CPU. We use Gem5 to implement and evaluate our hardware optimizations. To assess the software baseline with the proposed hardware extensions, we rewrote the assembly functions used by daBnn, such that we exploit them when computing the binary kernels. Also, to estimate the latencies of our hardware structures, we implement them in Verilog and use the Synopsys Design Compiler for synthesis. We modeled a 4GB DDR4 DRAM for the main memory. The main configuration parameters for the CPU and the \\textit{decoding unit} are shown in Table~\\ref{t:hardware_params}.\n\n\\begin{table}[t!]\n\t\\caption{Configuration Parameters.}\n\t\\label{t:hardware_params}\n\t\\centering\n\t\\begin{tabular}{cc}\n\t\t\\hline\n\t\t\\cellcolor[gray]{0.9}\\small\\textbf{Parameter}&\\cellcolor[gray]{0.9}\\small\\textbf{Value}\\\\\n\t\t\\small CPU&\\small ARM A53\\\\\n\t\t\\cellcolor[gray]{0.9}\\small Main Memory&\\cellcolor[gray]{0.9}\\small 4 GB DDR4\\\\\n\t\t\\small Vector Registers&\\small 32 (128 bits)\\\\\n\t\t\\cellcolor[gray]{0.9}\\small Frequency&\\cellcolor[gray]{0.9}\\small 1 GHz\\\\\n\t\t\\small CPU L1 Cache&\\small 32KB\\\\\n\t\t\\cellcolor[gray]{0.9}\\small CPU L2 Cache&\\cellcolor[gray]{0.9}\\small 256 KB\\\\\n\t\t\\hline\n\t\t\\multicolumn{2}{c}{\\cellcolor[gray]{0.9}\\textbf{Decoding Unit }}\\\\\n\t\t\n\t\t\\small Max number of Nodes&\\small 4\\\\\n\t\t\\cellcolor[gray]{0.9}\\small Uncompressed table &\\cellcolor[gray]{0.9}\\small 1 KB\\\\\n\t\t\\small Register file&\\small 256 bytes\\\\\n\t\t\\cellcolor[gray]{0.9}\\small Input Buffer&\\cellcolor[gray]{0.9}\\small 256 bytes \\\\\n\t\t\\hline\n\t\t\n\t\\end{tabular}\n\\end{table}\n\n\\section{Results}\\label{s:results}\n\nThis section presents the evaluation of the proposed scheme implemented on top of an ARM CPU. We limit the Huffman tree to four nodes. Also, we restrict the number of \\textit{bit sequences} stored on each node to 32, 64, 64, and 256. Hence, we use 6 bits to store the 32 most common sequences. Similarly, we use 8,9 and 12 bits for the other nodes, respectively. \n\n\nTable~\\ref{t:compression_ratio_by_layer} shows the compression ratio by layer. In this table, the Encoding column refers to encoding the 3x3 kernels without removing the less frequent $bit~sequences$, whereas the 256 most uncommon are removed for the Clustering column. For the Encoding, the compression ratio is between 1.18x and 1.25. The improvements came from employing only 6 bits to store the 32 most common \\textit{bit sequences}, which have an average frequency of use of 46\\%. Moreover, the rest of the \\textit{bit sequences} are stored using 8, 9, and 12 bits with a frequency of use of 24\\%, 23\\%, and 5\\%, respectively. After removing some of the least used \\textit{bit sequences}, the compression ratio is improved by 1.32x on average. These improvements are due to increasing the frequency of the top 32 \\textit{bit sequences} to 65\\% on average. In this case, the frequency of use of the stored sequences using 12 bits is 0.6\\%, whereas, for 8 and 9, it is 25\\% and 8\\%, respectively. Finally, the overall BNN model is compressed by 1.2x.\n\nRegarding the performance, we compare our compression scheme with hardware support with the software implementation of ReAcNet. In this case, we obtain a speedup of 1.35x. The primary source for this improvement is the reduced latency for loading the 3x3 kernels. Note that since the \\textit{decoding unit} fetches some of the \\textit{bit sequences} while decoding and evaluating others, the load's latency overlaps with the computations.\n\n\n\n\n\\begin{table}[t!]\n\t\\caption{ Compression ratio of $bit~sequences$ for the 3x3 kernel in each basic block.}\n\t\\label{t:compression_ratio_by_layer}\n\t\\centering\n\t\\begin{tabular}{cccc}\n\t\t\\cellcolor[gray]{0.9}\\small\\textbf{Layer}&\\cellcolor[gray]{0.9}\\small\\textbf{Encoding} &\\cellcolor[gray]{0.9}\\small\\textbf{Clustering}\\\\\n\t\t\\small Block 1 &\\small 1.18&\\small 1.30\\\\\n\t\t\\cellcolor[gray]{0.9}\\small Block 2 &\\cellcolor[gray]{0.9}\\small 1.22 &\\cellcolor[gray]{0.9}\\small 1.30\\\\\n\t\t\\small Block 3 &\\small 1.21&\\small1.31\\\\\n\t\t\\cellcolor[gray]{0.9}\\small Block 4 &\\cellcolor[gray]{0.9}\\small 1.21&\\cellcolor[gray]{0.9}\\small 1.32\\\\\n\t\t\t\\small Block 5 &\\small 1.19& \\small 1.30\\\\\n\t\t\\cellcolor[gray]{0.9}\\small Block 6 &\\cellcolor[gray]{0.9}\\small 1.20 &\\cellcolor[gray]{0.9}\\small 1.33 \\\\\n\t\t\\small Block 7 &\\small 1.18&\\small 1.33\\\\\n\t\t\\cellcolor[gray]{0.9}\\small Block 8 &\\cellcolor[gray]{0.9}\\small 1.20&\\cellcolor[gray]{0.9}\\small 1.32\\\\\n\t\t\t\\small Block 9 &\\small 1.20& \\small 1.31\\\\\t\n\t\t\\cellcolor[gray]{0.9}\\small Block 10 &\\cellcolor[gray]{0.9}\\small 1.18 &\\cellcolor[gray]{0.9}\\small 1.32 \\\\\n\t\t\\small Block 11  &\\small 1.19&\\small 1.33\\\\\n\t\t\\cellcolor[gray]{0.9}\\small Block 12 &\\cellcolor[gray]{0.9}\\small 1.25&\\cellcolor[gray]{0.9}\\small 1.36\\\\\n\t\t\\small Block 13  &\\small 1.22&\\small 1.35    \n\t\t\t\t\n\t\\end{tabular}\n\\end{table}\n\n\n\n\t\t\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\nUnsupervised domain adaptation (UDA) aims to leverage an annotated ``source'' dataset in designing learning schemes for  a ``target'' dataset for which no ground-truth is available. \nThis problem arises when annotations are easy to obtain in one domain (\\eg, synthetic images) but expensive in another (\\eg, real images), and is exacerbated in tasks where the annotation is laborious, as in semantic segmentation where each pixel in an image is assigned one of $K$ labels.\nIf the two datasets are sampled from the same distribution, this is a standard semi-supervised learning problem. \nThe twist in UDA is that the distributions from which source and target data are drawn differ enough that a model trained on the former performs poorly, out-of-the-box, on the latter.\n\n\n\\begin{figure}[!t]\n\\begin{center}\n  \\includegraphics[width=0.4\\textwidth]{figures\/phase_constraint.png}\n\\end{center}\n\\vspace{-0.2cm}\n\\caption{\\sl\\small {\\bf Semantic content is mainly carried by the phase component of the Fourier Transform.} Replacing the amplitude component of the image depicting a panda by the amplitude from other images within a wide range, will not prevent us from recognizing a panda in the images generated by the inverse Fourier Transform.}\n\\vspace{-0.2cm}\n\\label{fig:phase-constraint}\n\\end{figure}\n\n\nTypical domain adaptation work employing deep neural networks (DNNs) proceeds by either learning a map that aligns the source and target (marginal) distributions, or by training a backbone to be insensitive to the domain change through an auxiliary discrimination loss for the domain variable. \nEither way, these approaches operate on the marginal distributions, since the labels are not available in the target domain. \nHowever, the marginals could be perfectly aligned, yet the labels could be scrambled: Trees in one domain could map to houses in another, and vice-versa. \nSince we want to transfer information about the classes, ideally we would want to align the class-conditional distributions, which we do not have.\nRecent improvements in UDA, for instance cycle-consistency, only enforce the invertibility of the map, but not preservation of semantic information such as the class identity, see Fig. \\ref{fig:phase_result}. Since the problem is ill-posed, constraints or prior have to be enforced in UDA. \n\nWe introduce two priors or constraints, one on the map between the domains, the other on the classifier in the target domain, both unknown at the outset. \n\nFor the map between domains, we know from visual psychophysics that semantic information in images tends to be associated with the phase of its Fourier transform. Changes in the amplitude of the Fourier transform can significantly alter the appearance but not the interpretation of the image. This suggests placing an incentive for the transformation between domains to be phase-preserving. Indeed, we show from ablation studies that imposing phase consistency improves the performance of current UDA models.\n\nFor the classifier in the target domain, even in the absence of annotations, a target image informs the set of possible hypotheses (segmentations), due to the statistical regularities of natural scenes (ecological statistics, \\cite{brunswik1953ecological,elder2002ecological}). \nSemantic segments are unlikely to straddle many boundaries in the image plane, and their shape is unlikely to be highly irregular due to the regularity of the shape of objects in the scene. \nSuch generic priors, informed by each single un-annotated images, could be learned from other (annotated) images and transfer across image domains, since they arise from properties of the scene they portray. \nWe use a Conditional Prior Network \\cite{yang2018conditional} to learn a data-dependent prior on segmentations that can be imposed in an end-to-end framework when learning a classifier in the target domain in UDA.\n\nThese two priors yield improvement in UDA benchmarks. We conduct ablation studies to quantify the effect of each prior on the overall performance of learned classifiers (segmentation networks).\n\nIn the next section, we describe current approaches to UDA and then describe our method, which is summarized in Sect. \\ref{sec:final}, before testing it empirically in Sect. \\ref{sec:experiments}.\n\n\n\\subsection{Related Work}\n\n\nEarly works on UDA mainly focus on image classification \\cite{glorot2011domain,fernando2013unsupervised,baktashmotlagh2013unsupervised}, by minimizing a discrepancy measure between two domains \\cite{geng2011daml}. Recent methods apply adversarial learning \\cite{ganin2014unsupervised,tzeng2017adversarial} for classification, by instantiating a discriminator that encourages the alignment in feature space \\cite{sener2016learning,kumar2018co,shu2018dirt}. Unfortunately, none of these methods achieves the same success on semantic segmentation tasks.\n\nRecent progress in image-to-image transformation techniques \\cite{CycleGAN2017,liu2017unsupervised} aligns domains in image space, with some benefit to semantic segmentation \\cite{hoffman2016fcns,hoffman2017cycada}. \\cite{hoffman2016fcns} is the first UDA semantic segmentation method utilizing both global and categorical adaptation techniques. CyCADA \\cite{hoffman2017cycada} adapts representations in both image and feature space while enforcing cycle-consistency to regularize the image transformation network. \\cite{sankaranarayanan2018learning} also applies image alignment by projecting the learned intermediate features into the image space. \\cite{zhang2017curriculum} proposes curriculum learning to gradually minimize the domain gap using anchor points. \\cite{wu2018dcan} reduces domain shift at both image and feature levels by aligning statistics in each channel of CNN feature maps in order to preserve spatial structures. \\cite{gong2019dlow} generates a sequence of intermediate shifted domains from source to target to further improve the transferability by providing multi-style translations. \\cite{luo2019taking} introduces a category-level adversarial network to prevent the degeneration of well-aligned categories during global alignment. \\cite{hong2018conditional} conditions on both source images and random noise to produce samples that appear similar to the target. Despite the difficulty in training the domain discriminators, generally, the alignment criteria provided by the domain discriminators do not guarantee consistency of the semantic content between the original and transformed images. In addition to cycle-consistency, \\cite{li2019bidirectional,Chen_2019_CVPR} propose using the segmentation network on the target domain to encourage better semantic consistency. However, this will make the performance depend highly on the employed surrogate network.\n\nIn psychophysics, \\cite{piotrowski1982demonstration} demonstrates that certain phase modifications can hinder or prevent the recognition of visual scenes. \\cite{oppenheim1981importance} shows that many important features of a signal can be preserved by the phase component of the Fourier Transform, and under some conditions a signal can be completely reconstructed with only the phase. Moreover, \\cite{hansen2007structural} shows psychophysically that the Fourier phase spectrum plays a critical role in human vision. \nThe concurrent work \\cite{yang2020fda} shows that swapping the amplitude component of an image with one from the other domain preserves the semantic content while aligning the two domains.\nWith all these observations, we propose to use phase to provide an effective semantic consistency constraint that does not depend on any surrogate networks.\n\nBesides discriminators applied to the image or in feature space, \\cite{tsai2018learning, tsai2019domain} find that adaptation on the structured output space is also beneficial for semantic segmentation. \\cite{chen2018road} proposes spatially-aware adaptation along with target guided distillation using activation supervision with a pretrained classification network. Further, \\cite{chen2019learning} proposes a geometrically guided adaptation aided with depth in a multi-task learning framework. \\cite{chang2019all} extracts the domain invariant structure from the image to disentangle images into domain invariant structure and domain-specific variations. \\cite{zou2018unsupervised} performs iterative class-balanced self-training as well as refinement of the generated pseudo-labels using a spatial prior. A similar strategy is also applied in \\cite{li2019bidirectional,tsai2019domain}. \\cite{vu2019advent} approaches UDA for semantic segmentation by entropy minimization of the pixel-wise predictions. An adversarial loss on the entropy map is also used to introduce regularity in the output space. However, none of them explicitly models the scene compatibility that regularizes the training of the target domain segmentation network.\n\n\n\n\n\\vspace{-0.1cm}\n\\section{Method}\n\\vspace{-0.1cm}\n\n\n\\deffigures\/gta{figures\/phase}\n\\def0.142\\textwidth{0.15\\textwidth}\n\\begin{figure}[!ht]\n\\centering\n{\\footnotesize\n\\begin{tabular}{c@{\\hspace{0.01in}}c@{\\hspace{0.01in}}c}\\\\\nImage&Cycle Consistency&Phase Consistency\\\\\n\\includegraphics[width=0.142\\textwidth]{figures\/gta\/ori1.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/cyc1.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/bda1.png}\n\\vspace{-0.07cm}\\\\\n\\includegraphics[width=0.142\\textwidth]{figures\/gta\/ori2.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/cyc2.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/bda2.png}\n\\vspace{-0.07cm}\\\\\n\\includegraphics[width=0.142\\textwidth]{figures\/gta\/ori3.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/cyc3.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/bda3.png}\n\\vspace{-0.07cm}\\\\\n\\end{tabular}\n}\n\\caption{\\sl\\small {\\bf Cycle consistency does not guarantee semantic consistency, but phase does.} Note the sky is transformed to trees (1st row), the cloud is transformed to a mountain (2nd row), and the buildings are also transformed to trees (3rd row) even if cycle consistency is enforced. Phase consistency enforces the semantic information to be preserved and provides enough flexibility to align the two domains.}\n\\vspace{-0.3cm}\n\\label{fig:phase_result}\n\\end{figure}\n\nWe first describe general image translation for unsupervised domain adaptation (UDA) and how it is used in semantic segmentation. We point to some drawbacks as inspiration for the two complementary constraints, which we introduce in Sect. \\ref{sec:phase-constraint} and \\ref{sect:cpn-on-phi}, and incorporate into a model of UDA for semantic segmentation in Sect. \\ref{sec:final}, which we validate empirically in Sect. \\ref{sec:experiments}.\n\n\n\\subsection{Preliminaries: Image Translation for UDA}\n\n\nWe consider two probabilities, a source $P^s$ and a target $P^\\tau$, which are generally different (covariate shift), as measured by the Kullbach-Liebler divergence $KL(P^s || P^\\tau)$. In UDA, we are given ground-truth annotation in the source domain only. So, if $x \\in \\mathbb{R}^{H\\times W \\times 3}$ are color images, and $y \\in [1, \\dots, K]^{H\\times W}$ are segmentation masks where each pixel has an associated label, we have images and segmentations in the source domain, $D^s = \\{ (x^s_i, y^s_i) \\sim P^s(x,y) \\}_{i=1}^{N_s} $ but only images in the target domain, $\\{x_i^\\tau \\sim P^\\tau(x)\\}_{i=1}^{N_\\tau}$. \nThe goal of UDA for semantic segmentation is to train a model $\\phi^\\tau$, for instance a deep neural network (DNN), that maps target images to estimated segmentations, $x^\\tau \\mapsto \\hat y^\\tau = \\arg\\max_y \\phi^\\tau(x^\\tau)_y$, leveraging source domain annotations. \nBecause of the covariate shift, simply applying to the target data a model trained on the source generally yields disappointing results. As observed in \\cite{ben2010theory}, the upper bound on the target domain risk can be minimized by reducing the gap between two distributions. \nAny invertible map $T$ between samples in the source and target domains, for instance $x^s \\mapsto T(x^s)$ induces a (pushforward) map between their distributions $P^s \\mapsto T_* P^s$ where $T_*P^s(x^\\tau) = P^s(T^{-1}(x^\\tau))$. The map can be implemented by a ``transformer'' network, and the target domain risk is minimized by the cross-entropy loss, whose empirical approximation is:\n\\begin{equation}\n    \\mathcal{L}_{ce}(\\phi^\\tau, T; D^s) = \\sum_{(x_i, y_i) \\in D^s} - \\log [\\phi^\\tau(T(x_i))]_{y_i} \n\\label{eq:segmentation-loss}\n\\end{equation}\nwhere $T$ maps data sampled from the source distribution to the target domain. The gap is measured by $KL(P^\\tau || T_*P^s)$, and can be minimized by (adversarially) maximizing the domain confusion, as measured by a domain discriminator $\\theta$ that maps each image into the probability of it coming from the source or target domains:\n\\begin{equation}\n    \\mathcal{L}_D(\\theta, T; x^s_i) = -\\log[ \\theta(T(x^s_i)) ].\n\\label{eq:discriminator-loss-T}\n\\end{equation}\nIdeally, $\\theta$ returns $1$ for images drawn from the target $P^\\tau$, and $0$ otherwise.\n\n\n\\subsection{Limitations and Challenges} \n\n\nIdeally, jointly minimizing the two previous equations would yield a segmentation model that operates in the target domain, producing estimated segmentations $y^\\tau = \\phi(x^\\tau)$.\nUnfortunately, a transformation network $T$ trained by minimizing Eq. \\eqref{eq:discriminator-loss-T} does not yield a good target domain classifier, as $T$ is only asked by Eq. \\eqref{eq:discriminator-loss-T} to match the marginals, which it could do while scrambling all labels (images of class $i$ in the source can be mapped to images of class $j$ in the target). In other words, the transformation network can match the image statistics, but there is nothing that encourages it to match semantics. Cycle-consistency \\cite{CycleGAN2017,hoffman2017cycada} does not address this issue, as it only enforces the invertibility of $T$:\n\\begin{equation}\n    \\mathcal{L}_{cyc}(T, T^{-1}; x^s_i) = \\| x^s_i - T^{-1}\\circ T(x^s_i) \\|_{1}.\n\\label{eq:cycle-consistency-T}\n\\end{equation}\nEven after imposing this constraint, buildings in the source domain could be mapped to trees in the target domain, and vice-versa (Fig. \\ref{fig:phase_result}). Ideally, if $\\phi^s$ is a model trained on the source, and $\\phi^\\tau$ the one operating on the target, we would like:\n\\begin{equation}\n    \\phi^s(x^s_i) = {\\phi^\\tau}( T(x^s_i)), \\forall i.\n    \\label{eq:ideal-semantic-consistency}\n\\end{equation}\nUnfortunately, training ${\\phi^\\tau}$ would require ground-truth in the target domain, which is unknown. We could use $\\phi^s$ as a surrogate, apply $\\phi^s$ on the target domain, and penalize the discrepancy between the two sides in Eq. \\eqref{eq:ideal-semantic-consistency} with respect to the unknowns. Absent any regularization, this yields the trivial result where $T(x) = x$ and $\\phi^\\tau = \\phi^s$. While Eq. \\eqref{eq:ideal-semantic-consistency} is useless in providing information on $T$ {\\em and} $\\phi^\\tau$, it can be seen as a vehicle to transfer prior information {\\em from one} (\\eg, $T$) {\\em onto the other} (\\eg., $\\phi^\\tau$). In the next two sections we discuss additional constraints and priors that can be imposed on $T$ (Sect. \\ref{sec:phase-constraint}) and $\\phi^\\tau$ (Sect. \\ref{sect:cpn-on-phi}) that make the above constraint non-trivial, and usable in the context of UDA.\n\n\n\\subsection{Phase Consistency}\n\\label{sec:phase-constraint}\n\n\nIt is well known in perceptual psychology that manipulating the spectrum of an image can lead to different effects: Changes in the amplitude of the Fourier transform alters the image but does not affect its interpretation, whereas altering the phase produces uninterpretable images \\cite{kermisch1970image,piotrowski1982demonstration,oppenheim1981importance,hansen2007structural}. This is illustrated in Fig. \\ref{fig:phase-constraint}, where the amplitude of the Fourier transform of an image of a panda is replaced with the amplitude from an image of a bear, a tourist landmark and a landscape, yet the reconstructed images portray a panda. In other words, it appears that semantic information is included in the phase, not the amplitude, of the spectrum. This motivates us to hypothesize that the transformation $T$ should be phase-preserving.\n\nTo this end, let $\\mathcal{F}: \\mathbb{R}^{H\\times W} \\rightarrow \\mathbb{R}^{H\\times W \\times 2}$ be the Fourier Transform. Phase consistency, for a transformation $T$, for a single channel image $x$, is obtained by minimizing:\n\\begin{equation}\n    \\mathcal{L}_{ph}(T; x) = - \\sum_{j} \\dfrac{\\langle\\mathcal{F}(x)_j, \\mathcal{F}(T(x))_j\\rangle}{\\|\\mathcal{F}(x)_j\\|_{2}\\cdot\\|\\mathcal{F}(T(x))_j\\|_{2}}\n    \\label{eq:phase-constraint}\n\\end{equation}\nwhere $\\langle , \\rangle$ is the dot-product, and $\\| \\cdot \\|_{2}$ is the $L_2$ norm. Note that Eq. \\eqref{eq:phase-constraint} is the negative cosine of the difference between the original and the transformed phases, thus, by minimizing Eq. \\eqref{eq:phase-constraint} we can directly minimize their difference and increase phase consistency. We demonstrate the effectiveness of phase consistency in the ablation studies in Sect. \\ref{sec:experiments}.\n\n\n\\subsection{Prior on Scene Compatibility} \\label{sect:cpn-on-phi}\n\n\nWhile target images have no ground-truth labeling, not all semantic segmentations are equally likely at the outset.\nGiven an unlabeled image, we may not know what classes $\\{1, \\dots, K\\}$ may interest a user, but we do know that objects in the scene have certain regularities, so it is unlikely that photometrically homogeneous regions are segmented into many pieces, or that a class segment straddles many image boundaries. \nIt is also unlikely that the segmented map is highly irregular. These characteristics inform the probability of a segmentation given the image in the target domain, $Q(\\phi(x)| x)$. $Q$ can be thought of as a function that scores each hypothesis $\\phi(x)$ based on the plausibility of the resulting segmentation given the input image $x$. \nThe function can be learned using images for which the ground-truth segmentation is given, for instance the source dataset $D^s$, and then used at inference time as a scoring function. Such a scoring function can be implemented by a Conditional Prior Network (CPN) \\cite{yang2018conditional}. However, note that $D^s=\\{ (x^s_i,y^s_i) \\}$ is sampled from $P^s(x,y)$. Simply training a CPN with $D^s$ will make $Q(y| x)$ approximate $P^s(y|x)$\\footnote{We abuse the notation and use $y$ to indicate both the class and the soft-max (log-likelihood) vector that approximates its indicator function.}, making the exercise moot. The CPN would capture both the domain-related unary prediction term and the domain irrelevant pairwise term that depends on the image structure. To make this point explicit, we can decompose $P^s(y|x)$ as follows:\n\n\\begin{equation}\n    P^s(y|x) \\approx \\prod_{j}P^s(y_j|x) \\prod_{m < n} P(y_m= y_n|x)\n\\label{eq:ps-dcomp}\n\\end{equation}\nwhere we omit higher-order terms for simplicity. The unary terms $P^s(y_j|x)$ measure the likelihood of the semantic label of a single pixel given the image; \\eg, pixels in a white region indicate sky in the source domain, which depends highly on the domains. The pairwise terms $P(y_m=y_n|x)$ measure the labeling compatibility between pixels, which would depend much less on the domain; \\eg, pixels in a white region may not be sky in the target domain, but they should be labeled the same. Absent at least binary terms, the unary terms would lead to overfitting the source domain. To prevent this, we randomly permute the labels in $y^s$ according to a uniform distribution:\n\\begin{equation}\n    y^s\\mid_{y^s=i} \\ = \\mathrm{PM}^{K}(i)\n    \\label{eq:permutate}\n\\end{equation}\n\n\n\\begin{figure}[!t]\n\\begin{center}\n  \\includegraphics[width=0.3\\textwidth]{figures\/CPN_schematic.png}\n\\end{center}\n\\vspace{-0.1cm}\n\\caption{\\sl\\small {\\bf CPN Architecture.} To reconstruct the segmentation map that is encoded into a narrow bottleneck, the decoder needs to leverage structural information from the image. The CPN reconstructs better the prediction $\\phi(x)$ with higher compatibility with $x$. Labels are randomly permuted during training to avoid overfitting to the domain dependent unary prediction terms in Eq. \\eqref{eq:ps-dcomp}.}\n\\vspace{-0.05cm}\n\\label{fig:CPN}\n\\end{figure}\n\n\\noindent where $\\mathrm{PM}^K$ is a random permutation of the class ID's for $K$ classes, and we denote the permuted semantic segmentation masks as $\\hat{y^s}$, which scales the original dataset up in size by a factor of $K!$. We denote the new source dataset with permuted ground-truth masks as $\\hat{D^s}$, which will render the conditional distribution invariant to the domain-dependent unaries, \\ie:\n\\begin{equation}\n    \\hat{P^s}(y|x) \\approx \\prod_{m<n}P(y_m=y_n|x)\n\\label{eq:ps-permute}\n\\end{equation}\nNote, $\\hat{P^s}(y|x)$ only evaluates the compatibility based on the segmentation layout but not the semantic meanings. Thus, we train a CPN $Q$ using the following training loss \\cite{yang2018conditional} with an information capacity constraint:\n\\begin{equation}\n    \\min_{Q} \\mathbb{E}_{x} {KL}( \\hat{P^s}(y|x), Q(y|x)) + \\beta \\mathbb{I}( y, Q^e(y))\n    \\label{eq:loss-cpn}\n\\end{equation}\nwhere $\\mathbb{I}$ denotes the mutual information between $y$ and its CPN encoding $Q^e(y)$. Then, we obtain a compatibility function\n\\begin{equation}\n    Q(y|x) \\approx \\prod_{m<n}P(y_m=y_n|x)\n\\end{equation}\n\n\nThe proposed CPN architecture is illustrated in Fig. \\ref{fig:CPN} and the training details, including the encoding metric, is described in Sect. \\ref{sec:experiments}. We now summarize the overall training loss, that exploits regularities implied by each constraint.\n\n\n\n\\begin{table*}[!ht]\n\\footnotesize\n\\begin{tabular}{|c|c|c|p{0.3cm}<{\\centering}p{0.3cm}<{\\centering}p{0.3cm}<{\\centering}p{0.3cm}<{\\centering}p{0.3cm}<{\\centering}p{0.3cm}<{\\centering}p{0.3cm}<{\\centering}p{0.3cm}<{\\centering}p{0.3cm}<{\\centering}p{0.3cm}<{\\centering}p{0.3cm}<{\\centering}p{0.3cm}<{\\centering}p{0.3cm}<{\\centering}p{0.3cm}<{\\centering}p{0.3cm}<{\\centering}p{0.3cm}<{\\centering}p{0.2cm}<{\\centering}p{0.3cm}<{\\centering}p{0.4cm}<{\\centering}|c|}\n\\hline\n\\rot{Architecture}&SSL&CPN&\\rot{road}&\\rot{sidewalk}&\\rot{building}&\\rot{wall}&\\rot{fence}&\\rot{pole}&\\rot{light}&\\rot{sign}&\\rot{vegetation}&\\rot{terrain}&\\rot{sky}&\\rot{person}&\\rot{rider}&\\rot{car}&\\rot{truck}&\\rot{bus}&\\rot{train}&\\rot{motorcycle}&\\rot{bicycle}&mIoU\n\\\\\n\\hline\nA&&&88.2&41.3&\\bf{83.2}&28.8&21.9&\\bf{31.7}&\\bf{35.2}&28.2&\\bf{83.0}&26.2&\\bf{83.2}&57.6&27.0&77.1&27.5&34.6&2.5&28.3&\\bf{36.1}&44.3\\\\\nA&&\\checkmark&\\bf{91.4}&\\bf{47.2}&82.9&\\bf{29.2}&\\bf{22.9}&31.4&33.3&\\bf{30.2}&80.8&\\bf{27.8}&81.3&\\bf{59.1}&\\bf{27.7}&\\bf{84.4}&\\bf{31.5}&\\bf{40.9}&\\bf{3.2}&\\bf{30.2}&24.5&\\bf{45.3}\\\\\n\\cline{2-23}\nA&\\checkmark&&91.2&46.1&83.9&31.6&20.6&29.9&36.4&31.9&\\bf{85.0}&39.7&\\bf{84.7}&57.5&29.6&83.1&38.8&\\bf{46.9}&\\bf{2.5}&27.5&38.2&47.6\\\\\nA&\\checkmark&\\checkmark&\\bf{91.3}&\\bf{48.2}&\\bf{85.0}&\\bf{39.4}&\\bf{26.1}&\\bf{32.4}&\\bf{37.4}&\\bf{40.7}&84.9&\\bf{41.9}&83.0&\\bf{59.8}&\\bf{30.2}&\\bf{83.6}&\\bf{40.0}&46.1&0.1&\\bf{31.7}&\\bf{43.3}&\\bf{49.7}\\\\\n\\hline\nB&&&\\bf{86.4}&39.5&79.2&27.4&\\bf{24.3}&\\bf{23.4}&29.0&18.0&80.5&33.2&70.1&47.2&18.1&75.4&20.6&23.3&0.0&\\bf{16.1}&5.4&37.7\\\\\nB&&\\checkmark&86.0&\\bf{39.9}&\\bf{80.6}&\\bf{32.3}&21.9&21.6&\\bf{29.5}&\\bf{23.9}&\\bf{83.1}&\\bf{37.5}&\\bf{75.9}&\\bf{53.2}&\\bf{24.4}&\\bf{79.3}&\\bf{22.8}&\\bf{32.4}&\\bf{0.9}&13.9&\\bf{18.9}&\\bf{40.9}\\\\\n\\cline{2-23}\nB&\\checkmark&&89.2&40.9&\\bf{81.2}&29.1&19.2&14.2&29.0&19.6&\\bf{83.7}&35.9&\\bf{80.7}&\\bf{54.7}&23.3&\\bf{82.7}&\\bf{25.8}&28.0&\\bf{2.3}&\\bf{25.7}&19.9&41.3\\\\\nB&\\checkmark&\\checkmark&\\bf{90.1}&\\bf{44.7}&81.0&\\bf{29.3}&\\bf{26.4}&\\bf{20.9}&\\bf{33.7}&\\bf{34.3}&83.4&\\bf{37.4}&71.2&54.0&\\bf{27.4}&79.9&23.7&\\bf{39.6}&1.1&18.5&\\bf{22.6}&\\bf{43.1}\\\\\n\\hline\nC&&&79.1&33.1&77.9&\\bf{23.4}&\\bf{17.3}&32.1&33.3&31.8&81.5&26.7&69.0&\\bf{62.8}&\\bf{14.7}&74.5&\\bf{20.9}&\\bf{25.6}&6.9&\\bf{18.8}&\\bf{20.4}&39.5\\\\\nC&&\\checkmark&\\bf{89.1}&\\bf{41.4}&\\bf{81.2}&22.2&15.3&\\bf{34.0}&\\bf{35.0}&\\bf{37.1}&\\bf{84.8}&\\bf{32.1}&\\bf{76.2}&61.7&12.5&\\bf{82.1}&20.8&25.2&\\bf{7.3}&15.6&18.9&\\bf{41.7}\\\\\n\\hline\n\\end{tabular}\n\\vspace{0.15cm}\n\\caption{\\sl\\small {\\bf The learned Scene Compatibility improves segmentation accuracy.} Training the segmentation model with the learned scene compatibility $Q$ improves segmentation accuracy under all experimental settings, with different network backbones: A: ResNet-101, B: VGG-16, C: DRN-26. SSL: Self-supervised Learning. Note that whenever $Q$ is added in the training loss while the other terms are fixed, the overall semantic segmentation performance gets improved.}\n\\vspace{-0.1cm}\n\\label{tab:cpn_result}\n\\end{table*}\n\n\\subsection{Overall Training Loss}\n\\label{sec:final}\n\nCombining the adversarial losses and our novel constraints for both phase consistency and scene compatibility, we have the overall training loss for the proposed domain adaptation method for training the image transformation networks $T, T^{-1}$ and the target domain segmentation network $\\phi^\\tau$:\n\n\\begin{multline}\n    \\mathcal{L}(\\phi^\\tau, T, T^{-1}; \\theta^s, \\theta^\\tau, x^s_i, y^s_i, x^\\tau_i) = \\\\\n    \\lambda_D( \\mathcal{L}_D(\\theta^\\tau, T; x^s_i) + \\mathcal{L}_D(\\theta^s, T^{-1}; x^\\tau_i) )\\\\\n    + \\lambda_{cyc}( \\mathcal{L}_{cyc}(T, T^{-1}; x^s_i) + \\mathcal{L}_{cyc}(T^{-1}, T; x^\\tau_i) )\\\\\n    + \\lambda_{ph}( \\mathcal{L}_{ph}(T; x^s_i) + \\mathcal{L}_{ph}(T^{-1}; x^\\tau_i) ) \\\\\n    \\mathcal{L}_{ce}(\\phi^\\tau, T; x^s_i, y^s_i) - \\lambda_{cpn}\\log [Q(\\phi^\\tau(x^\\tau_i)|x^\\tau_i)]\n\\label{eq:final-training-loss}\n\\end{multline}\nwith $\\lambda$'s the corresponding weights on each term (hyperparameters), whose values will be reported in Sect.\\ref{sec:experiments}. Note when training $\\phi^\\tau$ using Eq. \\eqref{eq:final-training-loss}, we do not permute its output to evaluate the scene compatibility term. And the scene compatibility $Q$ is fixed after it is trained using Eq. \\eqref{eq:loss-cpn}. We follow the standard procedure in \\cite{hoffman2017cycada,li2019bidirectional} to train the domain discriminators.\n\n\n\n\n\\section{Experiments}\n\\label{sec:experiments}\n\nWe evaluate the proposed UDA method on synthetic-to-real semantic segmentation tasks, where the source images (GTA5 \\cite{Cordts2016Cityscapes} and Synthia \\cite{RosCVPR16synthia}) and corresponding annotations are generated using graphics engines, and the adapted segmentation models are tested on real-world images. We use average intersection-over-union score (mIoU) across semantic classes as the evaluation metric in all experiments. Moreover, the frequency weighted IoU (fwIoU), which is the sum of the IoUs of different classes but weighted by how frequent a certain class appears in the dataset, is calculated and compared in the GTA5-to-Cityscapes experiments.\n\nWe first describe the data used for training and the implementation details, followed by a comprehensive ablation study demonstrating the effectiveness of each proposed component in our method. Then we show quantitative and qualitative comparisons against the state-of-the-art methods, using networks with different backbones, on the GTA5-to-Cityscapes and Synthia-to-Cityscapes benchmarks.\n\n\\subsection{Datasets}\n\n\\textbf{Cityscapes} \\cite{Cordts2016Cityscapes} is a real-world semantic segmentation dataset containing 2975 street view training images and 500 validation images with original resolution $2048 \\times 1024$, which is resized to $1024 \\times 512$ for training. The images are collected during the day in multiple European cities and densely annotated. We train the image transformation network and the adapted segmentation network using the training set, and report the result on the validation set.\n\n\\textbf{GTA5} \\cite{Richter_2016gta} contains 24966 synthesized images from the Grand Theft Auto game with resolution $1914 \\times 1052$. It exhibits a wide range of variations including weather and illumination. We resize the images to $1280 \\times 720$ and use the 19 compatible classes for the  training and evaluation.\n\n\\textbf{Synthia} \\cite{RosCVPR16synthia} is a synthetic dataset focusing on driving scenarios rendered from a virtual city. We use the SYNTHIA-RAND-CITYSCAPES subset as source data, which contains 9400 images with the resolution of $1280 \\times 760$ for training the 16 common classes with Cityscapes, and we evaluate the trained network using both the 16 classes or a subset of 13 classes following previous works \\cite{tsai2018learning,li2019bidirectional,Du_2019_ICCV}.\n\n\\subsection{Implementation Details}\n\n\\textbf{Image Transformation Network:} We adapt the public CycleGAN \\cite{CycleGAN2017} framework, and use the ``cycle$\\_$gan'' model therein. We set $\\lambda_D=1.0$, $\\lambda_{cyc}=10.0$ and $\\lambda_{ph}=5.0$ for training the image transformation networks $T, T^{-1}$. Images from source and target domains are resized to $1024 \\times 512$ and then cropped to $452\\times 452$ before feeding into the network. We set the batch-size to $1.0$ and use ``resnet$\\_$9blocks'' as the backbone.\n\n\n\n\\begin{table}[!t]\n\\footnotesize \\centering\n\\begin{tabular}{|c|c|c|c|}\n\\hline\nMethod & Surrogate & Output Space & mIoU \\\\\n\\hline\n\\multirow{2}*{CyCADA \\cite{hoffman2017cycada}} & \\checkmark & & 43.5 \\\\\n    & \\checkmark & \\checkmark & 43.1 \\\\\n    \\hline\n\\multirow{3}*{AdaptSegNet \\cite{tsai2018learning}} & & & 36.6 \\\\\n    & \\checkmark & & 39.3 \\\\\n    & \\checkmark & \\checkmark & 42.4 \\\\\n    \\hline\n\\multirow{3}*{BDL \\cite{li2019bidirectional}} & & \\checkmark & 41.1 \\\\\n    & \\checkmark* & \\checkmark & 42.7 \\\\\n    & \\checkmark$^\\dagger$ & \\checkmark & 44.4 \\\\\n    \\hline\n\\multirow{2}*{Ours}& & \\checkmark & 44.8 \\\\\n    & & & \\bf{45.3} \\\\\n\\hline\n\\end{tabular}\n\\vspace{0.15cm}\n\\caption{\\sl\\small {\\bf Phase consistency (ours) achieves better performance.} Note our model trained only with the phase consistency outperforms other methods that utilize a surrogate network to impose semantic consistency (Surrogate), or employ output space regularization (Output Space). * and $^\\dagger$: first and second round of improved image transformation using a self-trained surrogate network.}\n\\vspace{-0.1cm}\n\\label{tab:phase_result}\n\\end{table}\n\n\\textbf{Conditional Prior Network:} We adopt the standard UNet \\cite{RFB15a} architecture, and add the segmentation encoder branch. We instantiate 6 convolutional layers, whose channel numbers are \\{16, 32, 64, 128, 256, 256\\}, to encode the image. Each of the first five layers is followed by $2 \\times 2$ max pooling, similarly, for semantic segmentation maps. Encoded image and segmentation are stacked at the bottleneck, then passed through a 6-layer decoder with channel numbers \\{512, 256, 128, 64, 32, 16\\}, followed by a fully connected layer for class prediction. Skip connections are enabled between the image encoder and the decoder. The network is trained with batch size four by ADAM optimizer with initial learning rate 1e-4. The learning rate is reduced by a factor of 10 after every 30000 iterations.\n\nDuring training, the network aims at reconstructing the encoded $\\hat{y}^s$, which is the randomly permutated ground-truth segmentation, by utilizing image information, leading to the training loss:\n\\begin{equation}\n    \\mathcal{L}_{cpn}(Q; \\hat{y}^s, x^s) = \\mathrm{NLL}( Q(\\hat{y}^s|x^s), \\hat{y}^s )\n    \\label{eq:cpn_loss}\n\\end{equation}\nwhere $\\mathrm{NLL}$ denotes the negative log likelihood loss derived from the KL-divergence term in the CPN training loss Eq. \\eqref{eq:loss-cpn}. Lower indicates better scene compatibility \\ie higher $Q(y|x)$. Note the information capacity constraint in Eq. \\eqref{eq:loss-cpn} is implemented by a structural bottleneck as in \\cite{yang2018conditional}.\n\n\\textbf{Semantic Segmentation Network:} We experiment with different segmentation network backbones. Due to memory constraint, we choose to train the segmentation network after the transformation networks are trained. We first train from scratch the segmentation network using transformed source images and the corresponding annotations using Eq. \\eqref{eq:final-training-loss}. We fix $\\lambda_{cpn} = 0.5$ for all the experiments. Finally, we apply the self-supervised training technique as in \\cite{li2019bidirectional,tsai2019domain} to further improve the performance on the target domain. We accept the high confidence ($>0.9$) predictions as the pseudo labels. All networks are trained using the ADAM optimizer, with learning rate 2.5e-4, 1e-5, and 1e-4 for ResNet-101, VGG-16, and DRN-26, respectively.\n\n\n\n\\subsection{Ablation Study}\n\nHere we carry out an ablation study to investigate the effectiveness and robustness of the proposed priors.\n\n\\deffigures\/gta{figures\/cpn}\n\\def0.142\\textwidth{0.12\\textwidth}\n\\begin{figure}[t]\n\\hspace*{-2.5mm}\\centering\n{\\footnotesize\n\\begin{tabular}{c@{\\hspace{0.01in}}c@{\\hspace{0.01in}}c@{\\hspace{0.01in}}c}\\\\\n\\vspace{-1cm}&&&\\\\\nImage&w\/o CPN&w\/ CPN&Ground-truth\\\\\n\\vspace{-0.07cm}\\includegraphics[width=0.142\\textwidth]{figures\/gta\/50_org.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/50_0.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/50_1.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/50_gt.png}\\\\\n\\vspace{-0.07cm}\\includegraphics[width=0.142\\textwidth]{figures\/gta\/54_org.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/54_0.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/54_1.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/54_gt.png}\\\\\n\\vspace{-0.07cm}\\includegraphics[width=0.142\\textwidth]{figures\/gta\/64_org.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/64_0.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/64_1.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/64_gt.png}\\\\\n\\includegraphics[width=0.142\\textwidth]{figures\/gta\/70_org.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/70_0.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/70_1.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/70_gt.png}\n\\end{tabular}\n}\n\\vspace{-0.1cm}\n\\caption{\\sl\\small {\\bf The learned scene compatibility prior imposes regularity on the predictions.} When the scene compatibility is added, the segmentation network yields predictions better aligned with object boundaries, and are more consistent within the objects.}\n\\label{fig:cpn_result}\n\\vspace{-0.0cm}\n\\end{figure}\n\n\n\n\n\\begin{table*}[!ht]\n\\footnotesize \\centering\n\\begin{tabular}{|c|c|p{0.23cm}<{\\centering}p{0.23cm}<{\\centering}p{0.23cm}<{\\centering}p{0.23cm}<{\\centering}p{0.23cm}<{\\centering}p{0.23cm}<{\\centering}p{0.23cm}<{\\centering}p{0.23cm}<{\\centering}p{0.23cm}<{\\centering}p{0.23cm}<{\\centering}p{0.23cm}<{\\centering}p{0.23cm}<{\\centering}p{0.23cm}<{\\centering}p{0.23cm}<{\\centering}p{0.23cm}<{\\centering}p{0.23cm}<{\\centering}p{0.1cm}<{\\centering}p{0.23cm}<{\\centering}p{0.4cm}<{\\centering}|p{0.4cm}<{\\centering}|p{0.4cm}<{\\centering}|}\n\\hline\nMethod&\\rot{Architecture}&\\rot{road}&\\rot{sidewalk}&\\rot{building}&\\rot{wall}&\\rot{fence}&\\rot{pole}&\\rot{light}&\\rot{sign}&\\rot{vegetation}&\\rot{terrain}&\\rot{sky}&\\rot{person}&\\rot{rider}&\\rot{car}&\\rot{truck}&\\rot{bus}&\\rot{train}&\\rot{motorcycle}&\\rot{bicycle}&\\rot{mIoU}&\\rot{fwIoU}\n\\\\\n\\hline\nAdaptSegNet \\cite{tsai2018learning}&A&86.5&25.9&79.8&22.1&20.0&23.6&33.1&21.8&81.8&25.9&75.9&57.3&26.2&76.3&29.8&32.1&\\bf{7.2}&29.5&32.5&41.4&75.5\\\\\nDCAN \\cite{wu2018dcan}&A&85.0&30.8&81.3&25.8&21.2&22.2&25.4&26.6&83.4&36.7&76.2&58.9&24.9&80.7&29.5&42.9&2.5&26.9&11.6&41.7&76.2\\\\\nCyCADA \\cite{hoffman2017cycada}&A&88.3&40.9&81.4&26.9&19.7&31.3&31.8&31.9&81.6&22.3&77.1&56.3&25.1&80.8&33.4&38.6&0.0&24.6&35.5&43.6&77.9\\\\\nSSF-DAN \\cite{Du_2019_ICCV}&A&90.3&38.9&81.7&24.8&22.9&30.5&37.0&21.2&84.8&38.8&76.9&58.8&30.7&\\bf{85.7}&30.6&38.1&5.9&28.3&36.9&45.4&79.6\\\\\nBDL \\cite{li2019bidirectional}&A&\\bf{91.0}&44.7&84.2&34.6&27.6&30.2&36.0&36.0&85.0&\\bf{43.6}&83.0&58.6&31.6&83.3&35.3&\\bf{49.7}&3.3&28.8&35.6&48.5&81.1\\\\\nOurs&A&\\bf{91.0}&\\bf{49.2}&\\bf{85.6}&\\bf{37.2}&\\bf{29.7}&\\bf{33.7}&\\bf{38.1}&\\bf{39.2}&\\bf{85.4}&35.4&\\bf{85.1}&\\bf{61.1}&\\bf{32.8}&84.1&\\bf{45.6}&46.9&0.0&\\bf{34.2}&\\bf{44.5}&\\bf{50.5}&\\bf{82.0}\\\\\n\\hline\nAdaptSegNet \\cite{tsai2018learning}&B&87.3&29.8&78.6&21.1&18.2&22.5&21.5&11.0&79.7&29.6&71.3&46.8&6.5&80.1&23.0&26.9&0.0&10.6&0.3&35.0&74.9\\\\\nCyCADA \\cite{hoffman2017cycada}&B&85.2&37.2&76.5&21.8&15.0&23.8&22.9&21.5&80.5&31.3&60.7&50.5&9.0&76.9&17.1&28.2&\\bf{4.5}&9.8&0.0&35.4&73.8\\\\\nDCAN \\cite{wu2018dcan}&B&82.3&26.7&77.4&23.7&20.5&20.4&30.3&15.9&80.9&25.4&69.5&52.6&11.1&79.6&24.9&21.2&1.3&17.0&6.7&36.2&72.9\\\\\nSSF-DAN \\cite{Du_2019_ICCV}&B&88.7&32.1&79.5&\\bf{29.9}&22.0&23.8&21.7&10.7&80.8&29.8&72.5&49.5&16.1&82.1&23.2&18.1&3.5&24.4&8.1&37.7&76.3\\\\\nBDL \\cite{li2019bidirectional}&B&89.2&40.9&81.2&29.1&19.2&14.2&29.0&19.6&\\bf{83.7}&35.9&\\bf{80.7}&\\bf{54.7}&23.3&\\bf{82.7}&\\bf{25.8}&28.0&2.3&\\bf{25.7}&19.9&41.3&78.4\\\\\nOurs&B&\\bf{90.2}&\\bf{44.7}&\\bf{82.0}&28.4&\\bf{28.4}&\\bf{24.4}&\\bf{33.7}&\\bf{35.6}&\\bf{83.7}&\\bf{40.5}&75.1&54.4&\\bf{28.2}&80.3&23.8&\\bf{39.4}&0.0&22.8&\\bf{30.8}&\\bf{44.6}&\\bf{79.3}\\\\\n\\hline\nCyCADA \\cite{hoffman2017cycada}&C&79.1&33.1&77.9&23.4&17.3&32.1&33.3&31.8&81.5&26.7&69.0&\\bf{62.8}&\\bf{14.7}&74.5&\\bf{20.9}&25.6&6.9&\\bf{18.8}&20.4&39.5&72.7\\\\\nOurs&C&\\bf{90.7}&\\bf{49.8}&\\bf{81.9}&\\bf{23.4}&\\bf{18.5}&\\bf{37.3}&\\bf{35.5}&\\bf{34.3}&\\bf{82.9}&\\bf{36.5}&\\bf{75.8}&61.8&12.4&\\bf{83.2}&19.2&\\bf{26.1}&4.0&14.3&\\bf{21.8}&\\bf{42.6}&\\bf{79.7}\\\\\n\\hline\n\\end{tabular}\n\\vspace{0.15cm}\n\\caption{\\sl\\small {\\bf Quantitative Evaluation on the GTA5-to-Cityscapes benchmark.} Our method achieves the best mIoU and fwIoU using different segmentation architectures: A (ResNet-101), B (VGG-16), C (DRN-26). }\n\\vspace{-0.1cm}\n\\label{tab:gta5_result}\n\\end{table*}\n\n\n\n\\begin{table*}[ht]\n\\footnotesize \\centering\n\\begin{tabular}{|c|c|p{0.31cm}<{\\centering}p{0.31cm}<{\\centering}p{0.31cm}<{\\centering}p{0.31cm}<{\\centering}p{0.31cm}<{\\centering}p{0.31cm}<{\\centering}p{0.31cm}<{\\centering}p{0.31cm}<{\\centering}p{0.31cm}<{\\centering}p{0.31cm}<{\\centering}p{0.31cm}<{\\centering}p{0.31cm}<{\\centering}p{0.31cm}<{\\centering}p{0.31cm}<{\\centering}p{0.31cm}<{\\centering}p{0.4cm}<{\\centering}|c|c|}\n\\hline\nMethod&\\rot{Architecture}&\\rot{road}&\\rot{sidewalk}&\\rot{building}&\\rot{wall*}&\\rot{fence*}&\\rot{pole*}&\\rot{light}&\\rot{sign}&\\rot{vegetation}&\\rot{sky}&\\rot{person}&\\rot{rider}&\\rot{car}&\\rot{bus}&\\rot{motorcycle}&\\rot{bicycle}&mIoU&mIoU*\n\\\\\n\\hline\nAdaptPatch \\cite{tsai2019domain}&A&82.4&38.0&78.6&8.7&0.6&26.0&3.9&11.1&75.5&84.6&53.5&21.6&71.4&32.6&19.3&31.7&40.0&46.5\\\\\nAdaptSegNet \\cite{tsai2018learning}&A&84.3&42.7&77.5&-&-&-&4.7&7.0&77.9&82.5&54.3&21.0&72.3&32.2&18.9&32.3&-&46.7\\\\\nSSF-DAN \\cite{Du_2019_ICCV}&A&84.6&41.7&80.8&-&-&-&11.5&14.7&\\bf{80.8}&\\bf{85.3}&\\bf{57.5}&21.6&\\bf{82.0}&36.0&19.3&34.5&-&50.0\\\\\nBDL \\cite{li2019bidirectional}&A&\\bf{86.0}&\\bf{46.7}&80.3&-&-&-&14.1&11.6&79.2&81.3&54.1&27.9&73.7&\\bf{42.2}&25.7&45.3&-&51.4\\\\\nOurs&A&85.9&44.6&\\bf{80.8}&\\bf{9.0}&\\bf{0.8}&\\bf{32.1}&\\bf{24.8}&\\bf{23.1}&79.5&83.1&57.2&\\bf{29.3}&73.5&34.8&\\bf{32.4}&\\bf{48.2}&\\bf{46.2}&\\bf{53.6}\\\\\n\\hline\nAdaptSegNet \\cite{tsai2018learning}&B&78.9&29.2&75.5&-&-&-&0.1&4.8&72.6&76.7&43.4&8.8&71.1&16.0&3.6&8.4&-&37.6\\\\\nAdaptPatch \\cite{tsai2019domain}&B&72.6&29.5&77.2&\\bf{3.5}&0.4&21.0&1.4&7.9&73.3&79.0&45.7&14.5&69.4&19.6&7.4&16.5&33.7&39.6\\\\\nDCAN \\cite{wu2018dcan}&B&\\bf{79.9}&30.4&70.8&1.6&\\bf{0.6}&22.3&6.7&23.0&76.9&73.9&41.9&16.7&61.7&11.5&10.3&38.6&35.4&41.7\\\\\nBDL \\cite{li2019bidirectional}&B&72.0&30.3&74.5&0.1&0.3&\\bf{24.6}&\\bf{10.2}&25.2&\\bf{80.5}&80.0&\\bf{54.7}&23.2&\\bf{72.7}&24.0&7.5&44.9&39.0&46.1\\\\\nOurs&B&79.7&\\bf{35.2}&\\bf{78.7}&1.4&\\bf{0.6}&23.1&10.0&\\bf{28.9}&79.6&\\bf{81.2}&51.2&\\bf{25.1}&72.2&\\bf{24.1}&\\bf{16.7}&\\bf{50.4}&\\bf{41.1}&\\bf{48.7}\\\\\n\\hline\n\\end{tabular}\n\\vspace{0.15cm}\n\\caption{\\sl\\small {\\bf Quantitative Evaluation on the Synthia-to-Cityscapes Benchmark.} mIoU and mIoU* are the mean IoU computed on the 16 classes and the 13 subclasses respectively (* excluded). Our method achieves the best performance using different segmentation network backbones: A (ResNet-101), B(VGG-16). }\n\\vspace{-0.1cm}\n\\label{tab:synthia_result}\n\\end{table*}\n\n\n\\textbf{Phase Consistency:} Here we train the segmentation network Deeplab-V2 \\cite{CP2016Deeplab} on the transformed source dataset with phase consistency. To make the comparison fair, all competing methods also use the same segmentation network as ours. The results of \\cite{tsai2018learning} and \\cite{li2019bidirectional} are reported by the original papers. We retrain \\cite{hoffman2017cycada} and report its best performance with hyperparameter tuning. The result is presented in Tab. \\ref{tab:phase_result}. Without any surrogate semantic consistency provided by a surrogate semantic segmentation network, our segmentation model achieves higher accuracy. Note that introducing surrogate semantic consistency for regularizing the transformation networks will also incur more memory cost. Moreover, several rounds of training to improve the performance of the surrogate network can also be time-consuming. However, our phase consistency can be implemented at low computational overhead (see Sect. \\ref{sec:computational_cost}).\n\nInterestingly, output space regularization, which aligns the marginal distributions of the segmentations, occasionally leads to worse performance in some settings, including \\cite{hoffman2017cycada} and ours. This is somewhat reasonable since aligning the marginal distributions does not guarantee the conditional alignment given the observations.\n\n\n\\textbf{Scene Compatibility:}\nTo better understand the performance gain from the scene compatibility prior, we compare to competing methods on the same transformed source images. We collect the scores for all the other methods using the same setting as ours, if needed, we retrain their model.\n\nIn Tab. \\ref{tab:cpn_result}, we show that under all experimental settings, scene compatibility prior improves accuracy for most of the semantic classes as well as the overall average. The performance gain is preserved during self-supervised learning. We present qualitative comparisons in Fig. \\ref{fig:cpn_result}, showing that the scene compatibility prior provides strong spatial regularity to align segmentation to the object boundaries. Incorporating the scene compatibility prior into the training process significantly improves the overall segmentation smoothness and integrity, resulting in more consistent label prediction within each object. \n\n\\subsection{Benchmark Results}\n\n\\deffigures\/gta{figures\/gta}\n\\def0.142\\textwidth{0.142\\textwidth}\n\\begin{figure*}[!t]\n\\hspace*{-5mm}\\centering\n{\\footnotesize\n\\begin{tabular}{c@{\\hspace{0.01in}}c@{\\hspace{0.01in}}c@{\\hspace{0.01in}}c@{\\hspace{0.01in}}c@{\\hspace{0.01in}}c@{\\hspace{0.01in}}c@{\\hspace{0.01in}}c}\\\\\n\n\\vspace{-0.15cm}\\rot{\\quad Image}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/1_img.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/2_img.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/3_img.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/4_img.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/5_img.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/6_img.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/9_img.png}\\\\\n\\vspace{-0.07cm} \\rot{\\scriptsize \\!\\!\\! CyCADA \\cite{hoffman2017cycada}}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/1_cycada.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/2_cycada.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/3_cycada.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/4_cycada.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/5_cycada.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/6_cycada.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/9_cycada.png}\\\\\n\\vspace{-0.07cm}\\rot{BDL \\cite{li2019bidirectional}}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/1_bdl.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/2_bdl.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/3_bdl.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/4_bdl.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/5_bdl.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/6_bdl.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/9_bdl.png}\\\\\n\\vspace{-0.07cm}\\rot{\\quad Phase}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/1_phase.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/2_phase.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/3_phase.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/4_phase.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/5_phase.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/6_phase.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/9_phase.png}\\\\\n\\vspace{-0.07cm}\\rot{\\quad\\,\\,Full}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/1_full.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/2_full.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/3_full.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/4_full.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/5_full.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/6_full.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/9_full.png}\\\\\n\\rot{\\quad \\,\\,\\,GT}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/1_gt.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/2_gt.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/3_gt.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/4_gt.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/5_gt.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/6_gt.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/9_gt.png}\n\\end{tabular}\n}\n\\caption{\\sl\\small {\\bf Qualitative comparison with state-of-the-art methods.} Our method outputs more spatially regularized segmentations align well with the underlying scene structure. All visualized models are based on DeepLab-V2 with ResNet-101 under the same setting. Phase: trained with phase consistency only; Full: our full model.}\n\\vspace{-0.3cm}\n\\label{fig:gta_result}\n\\end{figure*}\n\nTo recall, feature space alignment has been explored by \\textbf{DCAN} \\cite{wu2018dcan} and \\textbf{CyCADA} \\cite{hoffman2017cycada}. CyCADA also applies image level domain alignment by training cross-domain cycle consistent image transformation. Output space alignment methods include \\textbf{AdaptSegNet} \\cite{tsai2018learning}, \\textbf{AdaptPatch} \\cite{tsai2019domain} and \\textbf{SSF-DAN} \\cite{Du_2019_ICCV}, in which various ways of adversarial learning to the segmentation output are applied for better domain confusion. \\textbf{BDL} \\cite{li2019bidirectional} propagates information from semantic segmentation back to the image transformation network as semantic consistent regularization. \n\nWe apply ResNet-101 \\cite{He2015DeepRL} based Deeplab-V2 \\cite{CP2016Deeplab} and VGG-16 \\cite{Simonyan2014VeryDC} based FCN-8s \\cite{Shelhamer2014FullyCN} for the segmentation network to compare with \\cite{tsai2018learning,wu2018dcan,li2019bidirectional,vu2019advent,Du_2019_ICCV} under the same experimental setting. To better understand the robustness to different neural network settings, We also apply our method to the retrain the DRN-26 \\cite{Yu2017} model from \\cite{hoffman2017cycada}. \n\n\nThe result on the GTA5-to-Cityscapes benchmark is summarized in Tab. \\ref{tab:gta5_result}. Our method achieves state-of-the-art performance with all network backbones in terms of mIoU and fwIoU. Moreover, across different settings, our method achieves the best score for most of the classes, indicating that the proposed priors improve the segmentation accuracy consistently across different semantic categories. We also present a qualitative comparison in Fig. \\ref{fig:gta_result}. Our proposed method outputs more spatially regularized predictions, which are also consistent with the scene structures. We relatively achieve 4.1\\% and 8.0\\% improvement over the second-best method with the backbone ResNet-101 and VGG-16, respectively.\n\n\nThe result on the Synthia-to-Cityscapes benchmark can be found in Tab. \\ref{tab:synthia_result}. The mIoUs of either 13 or 16 classes are evaluated according to the evaluation protocol in the literature. Our method outperforms competing methods on both sets. It also achieves the best result on most of the semantic categories. Again, we relatively achieve 4.3\\% and 5.4\\% improvement over the second-best using different backbones.\n\n\n\n\\subsection{Computational Cost} \\label{sec:computational_cost}\nAll networks are trained using a single Nvidia Titan Xp GPU. Enforcing the phase consistency will incur a $<$0.001s overhead for a $1024 \\times 512$ image, which is negligible. Training the CPN for scene compatibility takes 2.5 seconds to process a batch of 4 images, given the images are cropped to $1280 \\times 768$. Incorporating CPN into segmentation training adds 1.5 seconds overhead to each iteration. Note that CPN is not required at the time of inference to segment target images.\n\n\n\\section{Discussion}\n\nIt is empirically shown in Sect. \\ref{sec:experiments} that the proposed priors improve UDA semantic segmentation accuracy under different settings, however, how to impose semantic consistency and ecological statistics priors to general UDA tasks besides semantic segmentation remains an open problem. \n\nAnalysis of the CPN is another unsolved task. Currently, the capacity of the CPN bottleneck is chosen empirically. In order to estimate the optimal bottleneck capacity for specific tasks, quantitative measurement of the information that CPN leverages from the image is necessary, which requires future exploration.\n\n\nUnsupervised domain adaptation is key for semantic segmentation, where dense annotation in real images is costly and rare, but comes automatically in rendered images. UDA is a form of transfer learning that hinges on regularities and assumptions or priors on the relationship between the distributions from which the source and target data are sampled. We introduce two assumptions, and the corresponding priors and variational renditions that are integrated into end-to-end differential learning. One is that the transformations mapping one domain to another only affect the magnitude, but not the phase, of their spectrum. This is motivated by empirical evidence that image semantics, as perceived by the human visual system, go with the phase but not the magnitude of the spectrum. The other is a prior meant to capture the ecological statistics, that are characteristics of the images induced by regularities in the scene, and therefore shared across different imaging modalities and domains. We show that the resulting priors improve performance in UDA benchmarks, and quantify their impact through ablation studies.\n\n\n\n\\section*{Acknowledgements}\n\nResearch supported by ARO W911NF-17-1-0304 and ONR N00014-19-1-2066. Dong Lao is supported by KAUST through the VCC Center Competitive Funding.\n\n\n\n\n{\\small\n\\bibliographystyle{ieee_fullname}\n\n\\section{Introduction}\n\n\n\nUnsupervised domain adaptation (UDA) aims to leverage an annotated ``source'' dataset in designing learning schemes for  a ``target'' dataset for which no ground-truth is available. \nThis problem arises when annotations are easy to obtain in one domain (\\eg, synthetic images) but expensive in another (\\eg, real images), and is exacerbated in tasks where the annotation is laborious, as in semantic segmentation where each pixel in an image is assigned one of $K$ labels.\nIf the two datasets are sampled from the same distribution, this is a standard semi-supervised learning problem. \nThe twist in UDA is that the distributions from which source and target data are drawn differ enough that a model trained on the former performs poorly, out-of-the-box, on the latter.\n\n\n\\begin{figure}[!t]\n\\begin{center}\n  \\includegraphics[width=0.4\\textwidth]{figures\/phase_constraint.png}\n\\end{center}\n\\vspace{-0.2cm}\n\\caption{\\sl\\small {\\bf Semantic content is mainly carried by the phase component of the Fourier Transform.} Replacing the amplitude component of the image depicting a panda by the amplitude from other images within a wide range, will not prevent us from recognizing a panda in the images generated by the inverse Fourier Transform.}\n\\vspace{-0.2cm}\n\\label{fig:phase-constraint}\n\\end{figure}\n\n\nTypical domain adaptation work employing deep neural networks (DNNs) proceeds by either learning a map that aligns the source and target (marginal) distributions, or by training a backbone to be insensitive to the domain change through an auxiliary discrimination loss for the domain variable. \nEither way, these approaches operate on the marginal distributions, since the labels are not available in the target domain. \nHowever, the marginals could be perfectly aligned, yet the labels could be scrambled: Trees in one domain could map to houses in another, and vice-versa. \nSince we want to transfer information about the classes, ideally we would want to align the class-conditional distributions, which we do not have.\nRecent improvements in UDA, for instance cycle-consistency, only enforce the invertibility of the map, but not preservation of semantic information such as the class identity, see Fig. \\ref{fig:phase_result}. Since the problem is ill-posed, constraints or prior have to be enforced in UDA. \n\nWe introduce two priors or constraints, one on the map between the domains, the other on the classifier in the target domain, both unknown at the outset. \n\nFor the map between domains, we know from visual psychophysics that semantic information in images tends to be associated with the phase of its Fourier transform. Changes in the amplitude of the Fourier transform can significantly alter the appearance but not the interpretation of the image. This suggests placing an incentive for the transformation between domains to be phase-preserving. Indeed, we show from ablation studies that imposing phase consistency improves the performance of current UDA models.\n\nFor the classifier in the target domain, even in the absence of annotations, a target image informs the set of possible hypotheses (segmentations), due to the statistical regularities of natural scenes (ecological statistics, \\cite{brunswik1953ecological,elder2002ecological}). \nSemantic segments are unlikely to straddle many boundaries in the image plane, and their shape is unlikely to be highly irregular due to the regularity of the shape of objects in the scene. \nSuch generic priors, informed by each single un-annotated images, could be learned from other (annotated) images and transfer across image domains, since they arise from properties of the scene they portray. \nWe use a Conditional Prior Network \\cite{yang2018conditional} to learn a data-dependent prior on segmentations that can be imposed in an end-to-end framework when learning a classifier in the target domain in UDA.\n\nThese two priors yield improvement in UDA benchmarks. We conduct ablation studies to quantify the effect of each prior on the overall performance of learned classifiers (segmentation networks).\n\nIn the next section, we describe current approaches to UDA and then describe our method, which is summarized in Sect. \\ref{sec:final}, before testing it empirically in Sect. \\ref{sec:experiments}.\n\n\n\\subsection{Related Work}\n\n\nEarly works on UDA mainly focus on image classification \\cite{glorot2011domain,fernando2013unsupervised,baktashmotlagh2013unsupervised}, by minimizing a discrepancy measure between two domains \\cite{geng2011daml}. Recent methods apply adversarial learning \\cite{ganin2014unsupervised,tzeng2017adversarial} for classification, by instantiating a discriminator that encourages the alignment in feature space \\cite{sener2016learning,kumar2018co,shu2018dirt}. Unfortunately, none of these methods achieves the same success on semantic segmentation tasks.\n\nRecent progress in image-to-image transformation techniques \\cite{CycleGAN2017,liu2017unsupervised} aligns domains in image space, with some benefit to semantic segmentation \\cite{hoffman2016fcns,hoffman2017cycada}. \\cite{hoffman2016fcns} is the first UDA semantic segmentation method utilizing both global and categorical adaptation techniques. CyCADA \\cite{hoffman2017cycada} adapts representations in both image and feature space while enforcing cycle-consistency to regularize the image transformation network. \\cite{sankaranarayanan2018learning} also applies image alignment by projecting the learned intermediate features into the image space. \\cite{zhang2017curriculum} proposes curriculum learning to gradually minimize the domain gap using anchor points. \\cite{wu2018dcan} reduces domain shift at both image and feature levels by aligning statistics in each channel of CNN feature maps in order to preserve spatial structures. \\cite{gong2019dlow} generates a sequence of intermediate shifted domains from source to target to further improve the transferability by providing multi-style translations. \\cite{luo2019taking} introduces a category-level adversarial network to prevent the degeneration of well-aligned categories during global alignment. \\cite{hong2018conditional} conditions on both source images and random noise to produce samples that appear similar to the target. Despite the difficulty in training the domain discriminators, generally, the alignment criteria provided by the domain discriminators do not guarantee consistency of the semantic content between the original and transformed images. In addition to cycle-consistency, \\cite{li2019bidirectional,Chen_2019_CVPR} propose using the segmentation network on the target domain to encourage better semantic consistency. However, this will make the performance depend highly on the employed surrogate network.\n\nIn psychophysics, \\cite{piotrowski1982demonstration} demonstrates that certain phase modifications can hinder or prevent the recognition of visual scenes. \\cite{oppenheim1981importance} shows that many important features of a signal can be preserved by the phase component of the Fourier Transform, and under some conditions a signal can be completely reconstructed with only the phase. Moreover, \\cite{hansen2007structural} shows psychophysically that the Fourier phase spectrum plays a critical role in human vision. \nThe concurrent work \\cite{yang2020fda} shows that swapping the amplitude component of an image with one from the other domain preserves the semantic content while aligning the two domains.\nWith all these observations, we propose to use phase to provide an effective semantic consistency constraint that does not depend on any surrogate networks.\n\nBesides discriminators applied to the image or in feature space, \\cite{tsai2018learning, tsai2019domain} find that adaptation on the structured output space is also beneficial for semantic segmentation. \\cite{chen2018road} proposes spatially-aware adaptation along with target guided distillation using activation supervision with a pretrained classification network. Further, \\cite{chen2019learning} proposes a geometrically guided adaptation aided with depth in a multi-task learning framework. \\cite{chang2019all} extracts the domain invariant structure from the image to disentangle images into domain invariant structure and domain-specific variations. \\cite{zou2018unsupervised} performs iterative class-balanced self-training as well as refinement of the generated pseudo-labels using a spatial prior. A similar strategy is also applied in \\cite{li2019bidirectional,tsai2019domain}. \\cite{vu2019advent} approaches UDA for semantic segmentation by entropy minimization of the pixel-wise predictions. An adversarial loss on the entropy map is also used to introduce regularity in the output space. However, none of them explicitly models the scene compatibility that regularizes the training of the target domain segmentation network.\n\n\n\n\n\\vspace{-0.1cm}\n\\section{Method}\n\\vspace{-0.1cm}\n\n\n\\deffigures\/gta{figures\/phase}\n\\def0.142\\textwidth{0.15\\textwidth}\n\\begin{figure}[!ht]\n\\centering\n{\\footnotesize\n\\begin{tabular}{c@{\\hspace{0.01in}}c@{\\hspace{0.01in}}c}\\\\\nImage&Cycle Consistency&Phase Consistency\\\\\n\\includegraphics[width=0.142\\textwidth]{figures\/gta\/ori1.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/cyc1.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/bda1.png}\n\\vspace{-0.07cm}\\\\\n\\includegraphics[width=0.142\\textwidth]{figures\/gta\/ori2.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/cyc2.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/bda2.png}\n\\vspace{-0.07cm}\\\\\n\\includegraphics[width=0.142\\textwidth]{figures\/gta\/ori3.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/cyc3.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/bda3.png}\n\\vspace{-0.07cm}\\\\\n\\end{tabular}\n}\n\\caption{\\sl\\small {\\bf Cycle consistency does not guarantee semantic consistency, but phase does.} Note the sky is transformed to trees (1st row), the cloud is transformed to a mountain (2nd row), and the buildings are also transformed to trees (3rd row) even if cycle consistency is enforced. Phase consistency enforces the semantic information to be preserved and provides enough flexibility to align the two domains.}\n\\vspace{-0.3cm}\n\\label{fig:phase_result}\n\\end{figure}\n\nWe first describe general image translation for unsupervised domain adaptation (UDA) and how it is used in semantic segmentation. We point to some drawbacks as inspiration for the two complementary constraints, which we introduce in Sect. \\ref{sec:phase-constraint} and \\ref{sect:cpn-on-phi}, and incorporate into a model of UDA for semantic segmentation in Sect. \\ref{sec:final}, which we validate empirically in Sect. \\ref{sec:experiments}.\n\n\n\\subsection{Preliminaries: Image Translation for UDA}\n\n\nWe consider two probabilities, a source $P^s$ and a target $P^\\tau$, which are generally different (covariate shift), as measured by the Kullbach-Liebler divergence $KL(P^s || P^\\tau)$. In UDA, we are given ground-truth annotation in the source domain only. So, if $x \\in \\mathbb{R}^{H\\times W \\times 3}$ are color images, and $y \\in [1, \\dots, K]^{H\\times W}$ are segmentation masks where each pixel has an associated label, we have images and segmentations in the source domain, $D^s = \\{ (x^s_i, y^s_i) \\sim P^s(x,y) \\}_{i=1}^{N_s} $ but only images in the target domain, $\\{x_i^\\tau \\sim P^\\tau(x)\\}_{i=1}^{N_\\tau}$. \nThe goal of UDA for semantic segmentation is to train a model $\\phi^\\tau$, for instance a deep neural network (DNN), that maps target images to estimated segmentations, $x^\\tau \\mapsto \\hat y^\\tau = \\arg\\max_y \\phi^\\tau(x^\\tau)_y$, leveraging source domain annotations. \nBecause of the covariate shift, simply applying to the target data a model trained on the source generally yields disappointing results. As observed in \\cite{ben2010theory}, the upper bound on the target domain risk can be minimized by reducing the gap between two distributions. \nAny invertible map $T$ between samples in the source and target domains, for instance $x^s \\mapsto T(x^s)$ induces a (pushforward) map between their distributions $P^s \\mapsto T_* P^s$ where $T_*P^s(x^\\tau) = P^s(T^{-1}(x^\\tau))$. The map can be implemented by a ``transformer'' network, and the target domain risk is minimized by the cross-entropy loss, whose empirical approximation is:\n\\begin{equation}\n    \\mathcal{L}_{ce}(\\phi^\\tau, T; D^s) = \\sum_{(x_i, y_i) \\in D^s} - \\log [\\phi^\\tau(T(x_i))]_{y_i} \n\\label{eq:segmentation-loss}\n\\end{equation}\nwhere $T$ maps data sampled from the source distribution to the target domain. The gap is measured by $KL(P^\\tau || T_*P^s)$, and can be minimized by (adversarially) maximizing the domain confusion, as measured by a domain discriminator $\\theta$ that maps each image into the probability of it coming from the source or target domains:\n\\begin{equation}\n    \\mathcal{L}_D(\\theta, T; x^s_i) = -\\log[ \\theta(T(x^s_i)) ].\n\\label{eq:discriminator-loss-T}\n\\end{equation}\nIdeally, $\\theta$ returns $1$ for images drawn from the target $P^\\tau$, and $0$ otherwise.\n\n\n\\subsection{Limitations and Challenges} \n\n\nIdeally, jointly minimizing the two previous equations would yield a segmentation model that operates in the target domain, producing estimated segmentations $y^\\tau = \\phi(x^\\tau)$.\nUnfortunately, a transformation network $T$ trained by minimizing Eq. \\eqref{eq:discriminator-loss-T} does not yield a good target domain classifier, as $T$ is only asked by Eq. \\eqref{eq:discriminator-loss-T} to match the marginals, which it could do while scrambling all labels (images of class $i$ in the source can be mapped to images of class $j$ in the target). In other words, the transformation network can match the image statistics, but there is nothing that encourages it to match semantics. Cycle-consistency \\cite{CycleGAN2017,hoffman2017cycada} does not address this issue, as it only enforces the invertibility of $T$:\n\\begin{equation}\n    \\mathcal{L}_{cyc}(T, T^{-1}; x^s_i) = \\| x^s_i - T^{-1}\\circ T(x^s_i) \\|_{1}.\n\\label{eq:cycle-consistency-T}\n\\end{equation}\nEven after imposing this constraint, buildings in the source domain could be mapped to trees in the target domain, and vice-versa (Fig. \\ref{fig:phase_result}). Ideally, if $\\phi^s$ is a model trained on the source, and $\\phi^\\tau$ the one operating on the target, we would like:\n\\begin{equation}\n    \\phi^s(x^s_i) = {\\phi^\\tau}( T(x^s_i)), \\forall i.\n    \\label{eq:ideal-semantic-consistency}\n\\end{equation}\nUnfortunately, training ${\\phi^\\tau}$ would require ground-truth in the target domain, which is unknown. We could use $\\phi^s$ as a surrogate, apply $\\phi^s$ on the target domain, and penalize the discrepancy between the two sides in Eq. \\eqref{eq:ideal-semantic-consistency} with respect to the unknowns. Absent any regularization, this yields the trivial result where $T(x) = x$ and $\\phi^\\tau = \\phi^s$. While Eq. \\eqref{eq:ideal-semantic-consistency} is useless in providing information on $T$ {\\em and} $\\phi^\\tau$, it can be seen as a vehicle to transfer prior information {\\em from one} (\\eg, $T$) {\\em onto the other} (\\eg., $\\phi^\\tau$). In the next two sections we discuss additional constraints and priors that can be imposed on $T$ (Sect. \\ref{sec:phase-constraint}) and $\\phi^\\tau$ (Sect. \\ref{sect:cpn-on-phi}) that make the above constraint non-trivial, and usable in the context of UDA.\n\n\n\\subsection{Phase Consistency}\n\\label{sec:phase-constraint}\n\n\nIt is well known in perceptual psychology that manipulating the spectrum of an image can lead to different effects: Changes in the amplitude of the Fourier transform alters the image but does not affect its interpretation, whereas altering the phase produces uninterpretable images \\cite{kermisch1970image,piotrowski1982demonstration,oppenheim1981importance,hansen2007structural}. This is illustrated in Fig. \\ref{fig:phase-constraint}, where the amplitude of the Fourier transform of an image of a panda is replaced with the amplitude from an image of a bear, a tourist landmark and a landscape, yet the reconstructed images portray a panda. In other words, it appears that semantic information is included in the phase, not the amplitude, of the spectrum. This motivates us to hypothesize that the transformation $T$ should be phase-preserving.\n\nTo this end, let $\\mathcal{F}: \\mathbb{R}^{H\\times W} \\rightarrow \\mathbb{R}^{H\\times W \\times 2}$ be the Fourier Transform. Phase consistency, for a transformation $T$, for a single channel image $x$, is obtained by minimizing:\n\\begin{equation}\n    \\mathcal{L}_{ph}(T; x) = - \\sum_{j} \\dfrac{\\langle\\mathcal{F}(x)_j, \\mathcal{F}(T(x))_j\\rangle}{\\|\\mathcal{F}(x)_j\\|_{2}\\cdot\\|\\mathcal{F}(T(x))_j\\|_{2}}\n    \\label{eq:phase-constraint}\n\\end{equation}\nwhere $\\langle , \\rangle$ is the dot-product, and $\\| \\cdot \\|_{2}$ is the $L_2$ norm. Note that Eq. \\eqref{eq:phase-constraint} is the negative cosine of the difference between the original and the transformed phases, thus, by minimizing Eq. \\eqref{eq:phase-constraint} we can directly minimize their difference and increase phase consistency. We demonstrate the effectiveness of phase consistency in the ablation studies in Sect. \\ref{sec:experiments}.\n\n\n\\subsection{Prior on Scene Compatibility} \\label{sect:cpn-on-phi}\n\n\nWhile target images have no ground-truth labeling, not all semantic segmentations are equally likely at the outset.\nGiven an unlabeled image, we may not know what classes $\\{1, \\dots, K\\}$ may interest a user, but we do know that objects in the scene have certain regularities, so it is unlikely that photometrically homogeneous regions are segmented into many pieces, or that a class segment straddles many image boundaries. \nIt is also unlikely that the segmented map is highly irregular. These characteristics inform the probability of a segmentation given the image in the target domain, $Q(\\phi(x)| x)$. $Q$ can be thought of as a function that scores each hypothesis $\\phi(x)$ based on the plausibility of the resulting segmentation given the input image $x$. \nThe function can be learned using images for which the ground-truth segmentation is given, for instance the source dataset $D^s$, and then used at inference time as a scoring function. Such a scoring function can be implemented by a Conditional Prior Network (CPN) \\cite{yang2018conditional}. However, note that $D^s=\\{ (x^s_i,y^s_i) \\}$ is sampled from $P^s(x,y)$. Simply training a CPN with $D^s$ will make $Q(y| x)$ approximate $P^s(y|x)$\\footnote{We abuse the notation and use $y$ to indicate both the class and the soft-max (log-likelihood) vector that approximates its indicator function.}, making the exercise moot. The CPN would capture both the domain-related unary prediction term and the domain irrelevant pairwise term that depends on the image structure. To make this point explicit, we can decompose $P^s(y|x)$ as follows:\n\n\\begin{equation}\n    P^s(y|x) \\approx \\prod_{j}P^s(y_j|x) \\prod_{m < n} P(y_m= y_n|x)\n\\label{eq:ps-dcomp}\n\\end{equation}\nwhere we omit higher-order terms for simplicity. The unary terms $P^s(y_j|x)$ measure the likelihood of the semantic label of a single pixel given the image; \\eg, pixels in a white region indicate sky in the source domain, which depends highly on the domains. The pairwise terms $P(y_m=y_n|x)$ measure the labeling compatibility between pixels, which would depend much less on the domain; \\eg, pixels in a white region may not be sky in the target domain, but they should be labeled the same. Absent at least binary terms, the unary terms would lead to overfitting the source domain. To prevent this, we randomly permute the labels in $y^s$ according to a uniform distribution:\n\\begin{equation}\n    y^s\\mid_{y^s=i} \\ = \\mathrm{PM}^{K}(i)\n    \\label{eq:permutate}\n\\end{equation}\n\n\n\\begin{figure}[!t]\n\\begin{center}\n  \\includegraphics[width=0.3\\textwidth]{figures\/CPN_schematic.png}\n\\end{center}\n\\vspace{-0.1cm}\n\\caption{\\sl\\small {\\bf CPN Architecture.} To reconstruct the segmentation map that is encoded into a narrow bottleneck, the decoder needs to leverage structural information from the image. The CPN reconstructs better the prediction $\\phi(x)$ with higher compatibility with $x$. Labels are randomly permuted during training to avoid overfitting to the domain dependent unary prediction terms in Eq. \\eqref{eq:ps-dcomp}.}\n\\vspace{-0.05cm}\n\\label{fig:CPN}\n\\end{figure}\n\n\\noindent where $\\mathrm{PM}^K$ is a random permutation of the class ID's for $K$ classes, and we denote the permuted semantic segmentation masks as $\\hat{y^s}$, which scales the original dataset up in size by a factor of $K!$. We denote the new source dataset with permuted ground-truth masks as $\\hat{D^s}$, which will render the conditional distribution invariant to the domain-dependent unaries, \\ie:\n\\begin{equation}\n    \\hat{P^s}(y|x) \\approx \\prod_{m<n}P(y_m=y_n|x)\n\\label{eq:ps-permute}\n\\end{equation}\nNote, $\\hat{P^s}(y|x)$ only evaluates the compatibility based on the segmentation layout but not the semantic meanings. Thus, we train a CPN $Q$ using the following training loss \\cite{yang2018conditional} with an information capacity constraint:\n\\begin{equation}\n    \\min_{Q} \\mathbb{E}_{x} {KL}( \\hat{P^s}(y|x), Q(y|x)) + \\beta \\mathbb{I}( y, Q^e(y))\n    \\label{eq:loss-cpn}\n\\end{equation}\nwhere $\\mathbb{I}$ denotes the mutual information between $y$ and its CPN encoding $Q^e(y)$. Then, we obtain a compatibility function\n\\begin{equation}\n    Q(y|x) \\approx \\prod_{m<n}P(y_m=y_n|x)\n\\end{equation}\n\n\nThe proposed CPN architecture is illustrated in Fig. \\ref{fig:CPN} and the training details, including the encoding metric, is described in Sect. \\ref{sec:experiments}. We now summarize the overall training loss, that exploits regularities implied by each constraint.\n\n\n\n\\begin{table*}[!ht]\n\\footnotesize\n\\begin{tabular}{|c|c|c|p{0.3cm}<{\\centering}p{0.3cm}<{\\centering}p{0.3cm}<{\\centering}p{0.3cm}<{\\centering}p{0.3cm}<{\\centering}p{0.3cm}<{\\centering}p{0.3cm}<{\\centering}p{0.3cm}<{\\centering}p{0.3cm}<{\\centering}p{0.3cm}<{\\centering}p{0.3cm}<{\\centering}p{0.3cm}<{\\centering}p{0.3cm}<{\\centering}p{0.3cm}<{\\centering}p{0.3cm}<{\\centering}p{0.3cm}<{\\centering}p{0.2cm}<{\\centering}p{0.3cm}<{\\centering}p{0.4cm}<{\\centering}|c|}\n\\hline\n\\rot{Architecture}&SSL&CPN&\\rot{road}&\\rot{sidewalk}&\\rot{building}&\\rot{wall}&\\rot{fence}&\\rot{pole}&\\rot{light}&\\rot{sign}&\\rot{vegetation}&\\rot{terrain}&\\rot{sky}&\\rot{person}&\\rot{rider}&\\rot{car}&\\rot{truck}&\\rot{bus}&\\rot{train}&\\rot{motorcycle}&\\rot{bicycle}&mIoU\n\\\\\n\\hline\nA&&&88.2&41.3&\\bf{83.2}&28.8&21.9&\\bf{31.7}&\\bf{35.2}&28.2&\\bf{83.0}&26.2&\\bf{83.2}&57.6&27.0&77.1&27.5&34.6&2.5&28.3&\\bf{36.1}&44.3\\\\\nA&&\\checkmark&\\bf{91.4}&\\bf{47.2}&82.9&\\bf{29.2}&\\bf{22.9}&31.4&33.3&\\bf{30.2}&80.8&\\bf{27.8}&81.3&\\bf{59.1}&\\bf{27.7}&\\bf{84.4}&\\bf{31.5}&\\bf{40.9}&\\bf{3.2}&\\bf{30.2}&24.5&\\bf{45.3}\\\\\n\\cline{2-23}\nA&\\checkmark&&91.2&46.1&83.9&31.6&20.6&29.9&36.4&31.9&\\bf{85.0}&39.7&\\bf{84.7}&57.5&29.6&83.1&38.8&\\bf{46.9}&\\bf{2.5}&27.5&38.2&47.6\\\\\nA&\\checkmark&\\checkmark&\\bf{91.3}&\\bf{48.2}&\\bf{85.0}&\\bf{39.4}&\\bf{26.1}&\\bf{32.4}&\\bf{37.4}&\\bf{40.7}&84.9&\\bf{41.9}&83.0&\\bf{59.8}&\\bf{30.2}&\\bf{83.6}&\\bf{40.0}&46.1&0.1&\\bf{31.7}&\\bf{43.3}&\\bf{49.7}\\\\\n\\hline\nB&&&\\bf{86.4}&39.5&79.2&27.4&\\bf{24.3}&\\bf{23.4}&29.0&18.0&80.5&33.2&70.1&47.2&18.1&75.4&20.6&23.3&0.0&\\bf{16.1}&5.4&37.7\\\\\nB&&\\checkmark&86.0&\\bf{39.9}&\\bf{80.6}&\\bf{32.3}&21.9&21.6&\\bf{29.5}&\\bf{23.9}&\\bf{83.1}&\\bf{37.5}&\\bf{75.9}&\\bf{53.2}&\\bf{24.4}&\\bf{79.3}&\\bf{22.8}&\\bf{32.4}&\\bf{0.9}&13.9&\\bf{18.9}&\\bf{40.9}\\\\\n\\cline{2-23}\nB&\\checkmark&&89.2&40.9&\\bf{81.2}&29.1&19.2&14.2&29.0&19.6&\\bf{83.7}&35.9&\\bf{80.7}&\\bf{54.7}&23.3&\\bf{82.7}&\\bf{25.8}&28.0&\\bf{2.3}&\\bf{25.7}&19.9&41.3\\\\\nB&\\checkmark&\\checkmark&\\bf{90.1}&\\bf{44.7}&81.0&\\bf{29.3}&\\bf{26.4}&\\bf{20.9}&\\bf{33.7}&\\bf{34.3}&83.4&\\bf{37.4}&71.2&54.0&\\bf{27.4}&79.9&23.7&\\bf{39.6}&1.1&18.5&\\bf{22.6}&\\bf{43.1}\\\\\n\\hline\nC&&&79.1&33.1&77.9&\\bf{23.4}&\\bf{17.3}&32.1&33.3&31.8&81.5&26.7&69.0&\\bf{62.8}&\\bf{14.7}&74.5&\\bf{20.9}&\\bf{25.6}&6.9&\\bf{18.8}&\\bf{20.4}&39.5\\\\\nC&&\\checkmark&\\bf{89.1}&\\bf{41.4}&\\bf{81.2}&22.2&15.3&\\bf{34.0}&\\bf{35.0}&\\bf{37.1}&\\bf{84.8}&\\bf{32.1}&\\bf{76.2}&61.7&12.5&\\bf{82.1}&20.8&25.2&\\bf{7.3}&15.6&18.9&\\bf{41.7}\\\\\n\\hline\n\\end{tabular}\n\\vspace{0.15cm}\n\\caption{\\sl\\small {\\bf The learned Scene Compatibility improves segmentation accuracy.} Training the segmentation model with the learned scene compatibility $Q$ improves segmentation accuracy under all experimental settings, with different network backbones: A: ResNet-101, B: VGG-16, C: DRN-26. SSL: Self-supervised Learning. Note that whenever $Q$ is added in the training loss while the other terms are fixed, the overall semantic segmentation performance gets improved.}\n\\vspace{-0.1cm}\n\\label{tab:cpn_result}\n\\end{table*}\n\n\\subsection{Overall Training Loss}\n\\label{sec:final}\n\nCombining the adversarial losses and our novel constraints for both phase consistency and scene compatibility, we have the overall training loss for the proposed domain adaptation method for training the image transformation networks $T, T^{-1}$ and the target domain segmentation network $\\phi^\\tau$:\n\n\\begin{multline}\n    \\mathcal{L}(\\phi^\\tau, T, T^{-1}; \\theta^s, \\theta^\\tau, x^s_i, y^s_i, x^\\tau_i) = \\\\\n    \\lambda_D( \\mathcal{L}_D(\\theta^\\tau, T; x^s_i) + \\mathcal{L}_D(\\theta^s, T^{-1}; x^\\tau_i) )\\\\\n    + \\lambda_{cyc}( \\mathcal{L}_{cyc}(T, T^{-1}; x^s_i) + \\mathcal{L}_{cyc}(T^{-1}, T; x^\\tau_i) )\\\\\n    + \\lambda_{ph}( \\mathcal{L}_{ph}(T; x^s_i) + \\mathcal{L}_{ph}(T^{-1}; x^\\tau_i) ) \\\\\n    \\mathcal{L}_{ce}(\\phi^\\tau, T; x^s_i, y^s_i) - \\lambda_{cpn}\\log [Q(\\phi^\\tau(x^\\tau_i)|x^\\tau_i)]\n\\label{eq:final-training-loss}\n\\end{multline}\nwith $\\lambda$'s the corresponding weights on each term (hyperparameters), whose values will be reported in Sect.\\ref{sec:experiments}. Note when training $\\phi^\\tau$ using Eq. \\eqref{eq:final-training-loss}, we do not permute its output to evaluate the scene compatibility term. And the scene compatibility $Q$ is fixed after it is trained using Eq. \\eqref{eq:loss-cpn}. We follow the standard procedure in \\cite{hoffman2017cycada,li2019bidirectional} to train the domain discriminators.\n\n\n\n\n\\section{Experiments}\n\\label{sec:experiments}\n\nWe evaluate the proposed UDA method on synthetic-to-real semantic segmentation tasks, where the source images (GTA5 \\cite{Cordts2016Cityscapes} and Synthia \\cite{RosCVPR16synthia}) and corresponding annotations are generated using graphics engines, and the adapted segmentation models are tested on real-world images. We use average intersection-over-union score (mIoU) across semantic classes as the evaluation metric in all experiments. Moreover, the frequency weighted IoU (fwIoU), which is the sum of the IoUs of different classes but weighted by how frequent a certain class appears in the dataset, is calculated and compared in the GTA5-to-Cityscapes experiments.\n\nWe first describe the data used for training and the implementation details, followed by a comprehensive ablation study demonstrating the effectiveness of each proposed component in our method. Then we show quantitative and qualitative comparisons against the state-of-the-art methods, using networks with different backbones, on the GTA5-to-Cityscapes and Synthia-to-Cityscapes benchmarks.\n\n\\subsection{Datasets}\n\n\\textbf{Cityscapes} \\cite{Cordts2016Cityscapes} is a real-world semantic segmentation dataset containing 2975 street view training images and 500 validation images with original resolution $2048 \\times 1024$, which is resized to $1024 \\times 512$ for training. The images are collected during the day in multiple European cities and densely annotated. We train the image transformation network and the adapted segmentation network using the training set, and report the result on the validation set.\n\n\\textbf{GTA5} \\cite{Richter_2016gta} contains 24966 synthesized images from the Grand Theft Auto game with resolution $1914 \\times 1052$. It exhibits a wide range of variations including weather and illumination. We resize the images to $1280 \\times 720$ and use the 19 compatible classes for the  training and evaluation.\n\n\\textbf{Synthia} \\cite{RosCVPR16synthia} is a synthetic dataset focusing on driving scenarios rendered from a virtual city. We use the SYNTHIA-RAND-CITYSCAPES subset as source data, which contains 9400 images with the resolution of $1280 \\times 760$ for training the 16 common classes with Cityscapes, and we evaluate the trained network using both the 16 classes or a subset of 13 classes following previous works \\cite{tsai2018learning,li2019bidirectional,Du_2019_ICCV}.\n\n\\subsection{Implementation Details}\n\n\\textbf{Image Transformation Network:} We adapt the public CycleGAN \\cite{CycleGAN2017} framework, and use the ``cycle$\\_$gan'' model therein. We set $\\lambda_D=1.0$, $\\lambda_{cyc}=10.0$ and $\\lambda_{ph}=5.0$ for training the image transformation networks $T, T^{-1}$. Images from source and target domains are resized to $1024 \\times 512$ and then cropped to $452\\times 452$ before feeding into the network. We set the batch-size to $1.0$ and use ``resnet$\\_$9blocks'' as the backbone.\n\n\n\n\\begin{table}[!t]\n\\footnotesize \\centering\n\\begin{tabular}{|c|c|c|c|}\n\\hline\nMethod & Surrogate & Output Space & mIoU \\\\\n\\hline\n\\multirow{2}*{CyCADA \\cite{hoffman2017cycada}} & \\checkmark & & 43.5 \\\\\n    & \\checkmark & \\checkmark & 43.1 \\\\\n    \\hline\n\\multirow{3}*{AdaptSegNet \\cite{tsai2018learning}} & & & 36.6 \\\\\n    & \\checkmark & & 39.3 \\\\\n    & \\checkmark & \\checkmark & 42.4 \\\\\n    \\hline\n\\multirow{3}*{BDL \\cite{li2019bidirectional}} & & \\checkmark & 41.1 \\\\\n    & \\checkmark* & \\checkmark & 42.7 \\\\\n    & \\checkmark$^\\dagger$ & \\checkmark & 44.4 \\\\\n    \\hline\n\\multirow{2}*{Ours}& & \\checkmark & 44.8 \\\\\n    & & & \\bf{45.3} \\\\\n\\hline\n\\end{tabular}\n\\vspace{0.15cm}\n\\caption{\\sl\\small {\\bf Phase consistency (ours) achieves better performance.} Note our model trained only with the phase consistency outperforms other methods that utilize a surrogate network to impose semantic consistency (Surrogate), or employ output space regularization (Output Space). * and $^\\dagger$: first and second round of improved image transformation using a self-trained surrogate network.}\n\\vspace{-0.1cm}\n\\label{tab:phase_result}\n\\end{table}\n\n\\textbf{Conditional Prior Network:} We adopt the standard UNet \\cite{RFB15a} architecture, and add the segmentation encoder branch. We instantiate 6 convolutional layers, whose channel numbers are \\{16, 32, 64, 128, 256, 256\\}, to encode the image. Each of the first five layers is followed by $2 \\times 2$ max pooling, similarly, for semantic segmentation maps. Encoded image and segmentation are stacked at the bottleneck, then passed through a 6-layer decoder with channel numbers \\{512, 256, 128, 64, 32, 16\\}, followed by a fully connected layer for class prediction. Skip connections are enabled between the image encoder and the decoder. The network is trained with batch size four by ADAM optimizer with initial learning rate 1e-4. The learning rate is reduced by a factor of 10 after every 30000 iterations.\n\nDuring training, the network aims at reconstructing the encoded $\\hat{y}^s$, which is the randomly permutated ground-truth segmentation, by utilizing image information, leading to the training loss:\n\\begin{equation}\n    \\mathcal{L}_{cpn}(Q; \\hat{y}^s, x^s) = \\mathrm{NLL}( Q(\\hat{y}^s|x^s), \\hat{y}^s )\n    \\label{eq:cpn_loss}\n\\end{equation}\nwhere $\\mathrm{NLL}$ denotes the negative log likelihood loss derived from the KL-divergence term in the CPN training loss Eq. \\eqref{eq:loss-cpn}. Lower indicates better scene compatibility \\ie higher $Q(y|x)$. Note the information capacity constraint in Eq. \\eqref{eq:loss-cpn} is implemented by a structural bottleneck as in \\cite{yang2018conditional}.\n\n\\textbf{Semantic Segmentation Network:} We experiment with different segmentation network backbones. Due to memory constraint, we choose to train the segmentation network after the transformation networks are trained. We first train from scratch the segmentation network using transformed source images and the corresponding annotations using Eq. \\eqref{eq:final-training-loss}. We fix $\\lambda_{cpn} = 0.5$ for all the experiments. Finally, we apply the self-supervised training technique as in \\cite{li2019bidirectional,tsai2019domain} to further improve the performance on the target domain. We accept the high confidence ($>0.9$) predictions as the pseudo labels. All networks are trained using the ADAM optimizer, with learning rate 2.5e-4, 1e-5, and 1e-4 for ResNet-101, VGG-16, and DRN-26, respectively.\n\n\n\n\\subsection{Ablation Study}\n\nHere we carry out an ablation study to investigate the effectiveness and robustness of the proposed priors.\n\n\\deffigures\/gta{figures\/cpn}\n\\def0.142\\textwidth{0.12\\textwidth}\n\\begin{figure}[t]\n\\hspace*{-2.5mm}\\centering\n{\\footnotesize\n\\begin{tabular}{c@{\\hspace{0.01in}}c@{\\hspace{0.01in}}c@{\\hspace{0.01in}}c}\\\\\n\\vspace{-1cm}&&&\\\\\nImage&w\/o CPN&w\/ CPN&Ground-truth\\\\\n\\vspace{-0.07cm}\\includegraphics[width=0.142\\textwidth]{figures\/gta\/50_org.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/50_0.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/50_1.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/50_gt.png}\\\\\n\\vspace{-0.07cm}\\includegraphics[width=0.142\\textwidth]{figures\/gta\/54_org.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/54_0.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/54_1.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/54_gt.png}\\\\\n\\vspace{-0.07cm}\\includegraphics[width=0.142\\textwidth]{figures\/gta\/64_org.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/64_0.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/64_1.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/64_gt.png}\\\\\n\\includegraphics[width=0.142\\textwidth]{figures\/gta\/70_org.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/70_0.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/70_1.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/70_gt.png}\n\\end{tabular}\n}\n\\vspace{-0.1cm}\n\\caption{\\sl\\small {\\bf The learned scene compatibility prior imposes regularity on the predictions.} When the scene compatibility is added, the segmentation network yields predictions better aligned with object boundaries, and are more consistent within the objects.}\n\\label{fig:cpn_result}\n\\vspace{-0.0cm}\n\\end{figure}\n\n\n\n\n\\begin{table*}[!ht]\n\\footnotesize \\centering\n\\begin{tabular}{|c|c|p{0.23cm}<{\\centering}p{0.23cm}<{\\centering}p{0.23cm}<{\\centering}p{0.23cm}<{\\centering}p{0.23cm}<{\\centering}p{0.23cm}<{\\centering}p{0.23cm}<{\\centering}p{0.23cm}<{\\centering}p{0.23cm}<{\\centering}p{0.23cm}<{\\centering}p{0.23cm}<{\\centering}p{0.23cm}<{\\centering}p{0.23cm}<{\\centering}p{0.23cm}<{\\centering}p{0.23cm}<{\\centering}p{0.23cm}<{\\centering}p{0.1cm}<{\\centering}p{0.23cm}<{\\centering}p{0.4cm}<{\\centering}|p{0.4cm}<{\\centering}|p{0.4cm}<{\\centering}|}\n\\hline\nMethod&\\rot{Architecture}&\\rot{road}&\\rot{sidewalk}&\\rot{building}&\\rot{wall}&\\rot{fence}&\\rot{pole}&\\rot{light}&\\rot{sign}&\\rot{vegetation}&\\rot{terrain}&\\rot{sky}&\\rot{person}&\\rot{rider}&\\rot{car}&\\rot{truck}&\\rot{bus}&\\rot{train}&\\rot{motorcycle}&\\rot{bicycle}&\\rot{mIoU}&\\rot{fwIoU}\n\\\\\n\\hline\nAdaptSegNet \\cite{tsai2018learning}&A&86.5&25.9&79.8&22.1&20.0&23.6&33.1&21.8&81.8&25.9&75.9&57.3&26.2&76.3&29.8&32.1&\\bf{7.2}&29.5&32.5&41.4&75.5\\\\\nDCAN \\cite{wu2018dcan}&A&85.0&30.8&81.3&25.8&21.2&22.2&25.4&26.6&83.4&36.7&76.2&58.9&24.9&80.7&29.5&42.9&2.5&26.9&11.6&41.7&76.2\\\\\nCyCADA \\cite{hoffman2017cycada}&A&88.3&40.9&81.4&26.9&19.7&31.3&31.8&31.9&81.6&22.3&77.1&56.3&25.1&80.8&33.4&38.6&0.0&24.6&35.5&43.6&77.9\\\\\nSSF-DAN \\cite{Du_2019_ICCV}&A&90.3&38.9&81.7&24.8&22.9&30.5&37.0&21.2&84.8&38.8&76.9&58.8&30.7&\\bf{85.7}&30.6&38.1&5.9&28.3&36.9&45.4&79.6\\\\\nBDL \\cite{li2019bidirectional}&A&\\bf{91.0}&44.7&84.2&34.6&27.6&30.2&36.0&36.0&85.0&\\bf{43.6}&83.0&58.6&31.6&83.3&35.3&\\bf{49.7}&3.3&28.8&35.6&48.5&81.1\\\\\nOurs&A&\\bf{91.0}&\\bf{49.2}&\\bf{85.6}&\\bf{37.2}&\\bf{29.7}&\\bf{33.7}&\\bf{38.1}&\\bf{39.2}&\\bf{85.4}&35.4&\\bf{85.1}&\\bf{61.1}&\\bf{32.8}&84.1&\\bf{45.6}&46.9&0.0&\\bf{34.2}&\\bf{44.5}&\\bf{50.5}&\\bf{82.0}\\\\\n\\hline\nAdaptSegNet \\cite{tsai2018learning}&B&87.3&29.8&78.6&21.1&18.2&22.5&21.5&11.0&79.7&29.6&71.3&46.8&6.5&80.1&23.0&26.9&0.0&10.6&0.3&35.0&74.9\\\\\nCyCADA \\cite{hoffman2017cycada}&B&85.2&37.2&76.5&21.8&15.0&23.8&22.9&21.5&80.5&31.3&60.7&50.5&9.0&76.9&17.1&28.2&\\bf{4.5}&9.8&0.0&35.4&73.8\\\\\nDCAN \\cite{wu2018dcan}&B&82.3&26.7&77.4&23.7&20.5&20.4&30.3&15.9&80.9&25.4&69.5&52.6&11.1&79.6&24.9&21.2&1.3&17.0&6.7&36.2&72.9\\\\\nSSF-DAN \\cite{Du_2019_ICCV}&B&88.7&32.1&79.5&\\bf{29.9}&22.0&23.8&21.7&10.7&80.8&29.8&72.5&49.5&16.1&82.1&23.2&18.1&3.5&24.4&8.1&37.7&76.3\\\\\nBDL \\cite{li2019bidirectional}&B&89.2&40.9&81.2&29.1&19.2&14.2&29.0&19.6&\\bf{83.7}&35.9&\\bf{80.7}&\\bf{54.7}&23.3&\\bf{82.7}&\\bf{25.8}&28.0&2.3&\\bf{25.7}&19.9&41.3&78.4\\\\\nOurs&B&\\bf{90.2}&\\bf{44.7}&\\bf{82.0}&28.4&\\bf{28.4}&\\bf{24.4}&\\bf{33.7}&\\bf{35.6}&\\bf{83.7}&\\bf{40.5}&75.1&54.4&\\bf{28.2}&80.3&23.8&\\bf{39.4}&0.0&22.8&\\bf{30.8}&\\bf{44.6}&\\bf{79.3}\\\\\n\\hline\nCyCADA \\cite{hoffman2017cycada}&C&79.1&33.1&77.9&23.4&17.3&32.1&33.3&31.8&81.5&26.7&69.0&\\bf{62.8}&\\bf{14.7}&74.5&\\bf{20.9}&25.6&6.9&\\bf{18.8}&20.4&39.5&72.7\\\\\nOurs&C&\\bf{90.7}&\\bf{49.8}&\\bf{81.9}&\\bf{23.4}&\\bf{18.5}&\\bf{37.3}&\\bf{35.5}&\\bf{34.3}&\\bf{82.9}&\\bf{36.5}&\\bf{75.8}&61.8&12.4&\\bf{83.2}&19.2&\\bf{26.1}&4.0&14.3&\\bf{21.8}&\\bf{42.6}&\\bf{79.7}\\\\\n\\hline\n\\end{tabular}\n\\vspace{0.15cm}\n\\caption{\\sl\\small {\\bf Quantitative Evaluation on the GTA5-to-Cityscapes benchmark.} Our method achieves the best mIoU and fwIoU using different segmentation architectures: A (ResNet-101), B (VGG-16), C (DRN-26). }\n\\vspace{-0.1cm}\n\\label{tab:gta5_result}\n\\end{table*}\n\n\n\n\\begin{table*}[ht]\n\\footnotesize \\centering\n\\begin{tabular}{|c|c|p{0.31cm}<{\\centering}p{0.31cm}<{\\centering}p{0.31cm}<{\\centering}p{0.31cm}<{\\centering}p{0.31cm}<{\\centering}p{0.31cm}<{\\centering}p{0.31cm}<{\\centering}p{0.31cm}<{\\centering}p{0.31cm}<{\\centering}p{0.31cm}<{\\centering}p{0.31cm}<{\\centering}p{0.31cm}<{\\centering}p{0.31cm}<{\\centering}p{0.31cm}<{\\centering}p{0.31cm}<{\\centering}p{0.4cm}<{\\centering}|c|c|}\n\\hline\nMethod&\\rot{Architecture}&\\rot{road}&\\rot{sidewalk}&\\rot{building}&\\rot{wall*}&\\rot{fence*}&\\rot{pole*}&\\rot{light}&\\rot{sign}&\\rot{vegetation}&\\rot{sky}&\\rot{person}&\\rot{rider}&\\rot{car}&\\rot{bus}&\\rot{motorcycle}&\\rot{bicycle}&mIoU&mIoU*\n\\\\\n\\hline\nAdaptPatch \\cite{tsai2019domain}&A&82.4&38.0&78.6&8.7&0.6&26.0&3.9&11.1&75.5&84.6&53.5&21.6&71.4&32.6&19.3&31.7&40.0&46.5\\\\\nAdaptSegNet \\cite{tsai2018learning}&A&84.3&42.7&77.5&-&-&-&4.7&7.0&77.9&82.5&54.3&21.0&72.3&32.2&18.9&32.3&-&46.7\\\\\nSSF-DAN \\cite{Du_2019_ICCV}&A&84.6&41.7&80.8&-&-&-&11.5&14.7&\\bf{80.8}&\\bf{85.3}&\\bf{57.5}&21.6&\\bf{82.0}&36.0&19.3&34.5&-&50.0\\\\\nBDL \\cite{li2019bidirectional}&A&\\bf{86.0}&\\bf{46.7}&80.3&-&-&-&14.1&11.6&79.2&81.3&54.1&27.9&73.7&\\bf{42.2}&25.7&45.3&-&51.4\\\\\nOurs&A&85.9&44.6&\\bf{80.8}&\\bf{9.0}&\\bf{0.8}&\\bf{32.1}&\\bf{24.8}&\\bf{23.1}&79.5&83.1&57.2&\\bf{29.3}&73.5&34.8&\\bf{32.4}&\\bf{48.2}&\\bf{46.2}&\\bf{53.6}\\\\\n\\hline\nAdaptSegNet \\cite{tsai2018learning}&B&78.9&29.2&75.5&-&-&-&0.1&4.8&72.6&76.7&43.4&8.8&71.1&16.0&3.6&8.4&-&37.6\\\\\nAdaptPatch \\cite{tsai2019domain}&B&72.6&29.5&77.2&\\bf{3.5}&0.4&21.0&1.4&7.9&73.3&79.0&45.7&14.5&69.4&19.6&7.4&16.5&33.7&39.6\\\\\nDCAN \\cite{wu2018dcan}&B&\\bf{79.9}&30.4&70.8&1.6&\\bf{0.6}&22.3&6.7&23.0&76.9&73.9&41.9&16.7&61.7&11.5&10.3&38.6&35.4&41.7\\\\\nBDL \\cite{li2019bidirectional}&B&72.0&30.3&74.5&0.1&0.3&\\bf{24.6}&\\bf{10.2}&25.2&\\bf{80.5}&80.0&\\bf{54.7}&23.2&\\bf{72.7}&24.0&7.5&44.9&39.0&46.1\\\\\nOurs&B&79.7&\\bf{35.2}&\\bf{78.7}&1.4&\\bf{0.6}&23.1&10.0&\\bf{28.9}&79.6&\\bf{81.2}&51.2&\\bf{25.1}&72.2&\\bf{24.1}&\\bf{16.7}&\\bf{50.4}&\\bf{41.1}&\\bf{48.7}\\\\\n\\hline\n\\end{tabular}\n\\vspace{0.15cm}\n\\caption{\\sl\\small {\\bf Quantitative Evaluation on the Synthia-to-Cityscapes Benchmark.} mIoU and mIoU* are the mean IoU computed on the 16 classes and the 13 subclasses respectively (* excluded). Our method achieves the best performance using different segmentation network backbones: A (ResNet-101), B(VGG-16). }\n\\vspace{-0.1cm}\n\\label{tab:synthia_result}\n\\end{table*}\n\n\n\\textbf{Phase Consistency:} Here we train the segmentation network Deeplab-V2 \\cite{CP2016Deeplab} on the transformed source dataset with phase consistency. To make the comparison fair, all competing methods also use the same segmentation network as ours. The results of \\cite{tsai2018learning} and \\cite{li2019bidirectional} are reported by the original papers. We retrain \\cite{hoffman2017cycada} and report its best performance with hyperparameter tuning. The result is presented in Tab. \\ref{tab:phase_result}. Without any surrogate semantic consistency provided by a surrogate semantic segmentation network, our segmentation model achieves higher accuracy. Note that introducing surrogate semantic consistency for regularizing the transformation networks will also incur more memory cost. Moreover, several rounds of training to improve the performance of the surrogate network can also be time-consuming. However, our phase consistency can be implemented at low computational overhead (see Sect. \\ref{sec:computational_cost}).\n\nInterestingly, output space regularization, which aligns the marginal distributions of the segmentations, occasionally leads to worse performance in some settings, including \\cite{hoffman2017cycada} and ours. This is somewhat reasonable since aligning the marginal distributions does not guarantee the conditional alignment given the observations.\n\n\n\\textbf{Scene Compatibility:}\nTo better understand the performance gain from the scene compatibility prior, we compare to competing methods on the same transformed source images. We collect the scores for all the other methods using the same setting as ours, if needed, we retrain their model.\n\nIn Tab. \\ref{tab:cpn_result}, we show that under all experimental settings, scene compatibility prior improves accuracy for most of the semantic classes as well as the overall average. The performance gain is preserved during self-supervised learning. We present qualitative comparisons in Fig. \\ref{fig:cpn_result}, showing that the scene compatibility prior provides strong spatial regularity to align segmentation to the object boundaries. Incorporating the scene compatibility prior into the training process significantly improves the overall segmentation smoothness and integrity, resulting in more consistent label prediction within each object. \n\n\\subsection{Benchmark Results}\n\n\\deffigures\/gta{figures\/gta}\n\\def0.142\\textwidth{0.142\\textwidth}\n\\begin{figure*}[!t]\n\\hspace*{-5mm}\\centering\n{\\footnotesize\n\\begin{tabular}{c@{\\hspace{0.01in}}c@{\\hspace{0.01in}}c@{\\hspace{0.01in}}c@{\\hspace{0.01in}}c@{\\hspace{0.01in}}c@{\\hspace{0.01in}}c@{\\hspace{0.01in}}c}\\\\\n\n\\vspace{-0.15cm}\\rot{\\quad Image}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/1_img.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/2_img.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/3_img.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/4_img.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/5_img.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/6_img.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/9_img.png}\\\\\n\\vspace{-0.07cm} \\rot{\\scriptsize \\!\\!\\! CyCADA \\cite{hoffman2017cycada}}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/1_cycada.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/2_cycada.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/3_cycada.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/4_cycada.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/5_cycada.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/6_cycada.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/9_cycada.png}\\\\\n\\vspace{-0.07cm}\\rot{BDL \\cite{li2019bidirectional}}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/1_bdl.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/2_bdl.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/3_bdl.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/4_bdl.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/5_bdl.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/6_bdl.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/9_bdl.png}\\\\\n\\vspace{-0.07cm}\\rot{\\quad Phase}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/1_phase.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/2_phase.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/3_phase.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/4_phase.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/5_phase.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/6_phase.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/9_phase.png}\\\\\n\\vspace{-0.07cm}\\rot{\\quad\\,\\,Full}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/1_full.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/2_full.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/3_full.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/4_full.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/5_full.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/6_full.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/9_full.png}\\\\\n\\rot{\\quad \\,\\,\\,GT}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/1_gt.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/2_gt.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/3_gt.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/4_gt.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/5_gt.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/6_gt.png}&\\includegraphics[width=0.142\\textwidth]{figures\/gta\/9_gt.png}\n\\end{tabular}\n}\n\\caption{\\sl\\small {\\bf Qualitative comparison with state-of-the-art methods.} Our method outputs more spatially regularized segmentations align well with the underlying scene structure. All visualized models are based on DeepLab-V2 with ResNet-101 under the same setting. Phase: trained with phase consistency only; Full: our full model.}\n\\vspace{-0.3cm}\n\\label{fig:gta_result}\n\\end{figure*}\n\nTo recall, feature space alignment has been explored by \\textbf{DCAN} \\cite{wu2018dcan} and \\textbf{CyCADA} \\cite{hoffman2017cycada}. CyCADA also applies image level domain alignment by training cross-domain cycle consistent image transformation. Output space alignment methods include \\textbf{AdaptSegNet} \\cite{tsai2018learning}, \\textbf{AdaptPatch} \\cite{tsai2019domain} and \\textbf{SSF-DAN} \\cite{Du_2019_ICCV}, in which various ways of adversarial learning to the segmentation output are applied for better domain confusion. \\textbf{BDL} \\cite{li2019bidirectional} propagates information from semantic segmentation back to the image transformation network as semantic consistent regularization. \n\nWe apply ResNet-101 \\cite{He2015DeepRL} based Deeplab-V2 \\cite{CP2016Deeplab} and VGG-16 \\cite{Simonyan2014VeryDC} based FCN-8s \\cite{Shelhamer2014FullyCN} for the segmentation network to compare with \\cite{tsai2018learning,wu2018dcan,li2019bidirectional,vu2019advent,Du_2019_ICCV} under the same experimental setting. To better understand the robustness to different neural network settings, We also apply our method to the retrain the DRN-26 \\cite{Yu2017} model from \\cite{hoffman2017cycada}. \n\n\nThe result on the GTA5-to-Cityscapes benchmark is summarized in Tab. \\ref{tab:gta5_result}. Our method achieves state-of-the-art performance with all network backbones in terms of mIoU and fwIoU. Moreover, across different settings, our method achieves the best score for most of the classes, indicating that the proposed priors improve the segmentation accuracy consistently across different semantic categories. We also present a qualitative comparison in Fig. \\ref{fig:gta_result}. Our proposed method outputs more spatially regularized predictions, which are also consistent with the scene structures. We relatively achieve 4.1\\% and 8.0\\% improvement over the second-best method with the backbone ResNet-101 and VGG-16, respectively.\n\n\nThe result on the Synthia-to-Cityscapes benchmark can be found in Tab. \\ref{tab:synthia_result}. The mIoUs of either 13 or 16 classes are evaluated according to the evaluation protocol in the literature. Our method outperforms competing methods on both sets. It also achieves the best result on most of the semantic categories. Again, we relatively achieve 4.3\\% and 5.4\\% improvement over the second-best using different backbones.\n\n\n\n\\subsection{Computational Cost} \\label{sec:computational_cost}\nAll networks are trained using a single Nvidia Titan Xp GPU. Enforcing the phase consistency will incur a $<$0.001s overhead for a $1024 \\times 512$ image, which is negligible. Training the CPN for scene compatibility takes 2.5 seconds to process a batch of 4 images, given the images are cropped to $1280 \\times 768$. Incorporating CPN into segmentation training adds 1.5 seconds overhead to each iteration. Note that CPN is not required at the time of inference to segment target images.\n\n\n\\section{Discussion}\n\nIt is empirically shown in Sect. \\ref{sec:experiments} that the proposed priors improve UDA semantic segmentation accuracy under different settings, however, how to impose semantic consistency and ecological statistics priors to general UDA tasks besides semantic segmentation remains an open problem. \n\nAnalysis of the CPN is another unsolved task. Currently, the capacity of the CPN bottleneck is chosen empirically. In order to estimate the optimal bottleneck capacity for specific tasks, quantitative measurement of the information that CPN leverages from the image is necessary, which requires future exploration.\n\n\nUnsupervised domain adaptation is key for semantic segmentation, where dense annotation in real images is costly and rare, but comes automatically in rendered images. UDA is a form of transfer learning that hinges on regularities and assumptions or priors on the relationship between the distributions from which the source and target data are sampled. We introduce two assumptions, and the corresponding priors and variational renditions that are integrated into end-to-end differential learning. One is that the transformations mapping one domain to another only affect the magnitude, but not the phase, of their spectrum. This is motivated by empirical evidence that image semantics, as perceived by the human visual system, go with the phase but not the magnitude of the spectrum. The other is a prior meant to capture the ecological statistics, that are characteristics of the images induced by regularities in the scene, and therefore shared across different imaging modalities and domains. We show that the resulting priors improve performance in UDA benchmarks, and quantify their impact through ablation studies.\n\n\n\n\\section*{Acknowledgements}\n\nResearch supported by ARO W911NF-17-1-0304 and ONR N00014-19-1-2066. Dong Lao is supported by KAUST through the VCC Center Competitive Funding.\n\n\n\n\n{\\small\n\\bibliographystyle{ieee_fullname}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nWe have introduced Deep Sketches, which are compact representations of databases that allow us to estimate the result sizes of SQL queries.\nOur demonstration allows users to experience the end-to-end training and querying process of these sketches.\nThe goal of this work is to show that a learned cardinality model can compete with and even outperform traditional cardinality estimators, especially for highly correlated data.\nOur audience can specify ad-hoc queries and thereby observe that ML might indeed be the right hammer for the decades-old cardinality estimation job.\nClearly, more research is needed to automate the training and utilization of Deep Sketches in query optimizers.\nOne question---that we currently outsource to our users---is for which schema parts we should build such sketches.\nBesides further improving cardinality estimation for query optimization, another avenue we could take in future work is to deploy Deep Sketches---which offer a small footprint size---on the client to preview query results.\n\\balance\n\n\\section{Demonstration}\n\\label{sec:demonstration}\n\nAs stated earlier, a Deep Sketch is essentially a wrapper for a trained MSCN model and a set of materialized samples.\nIn our demonstration, users can experience the end-to-end process of defining, training, and using trained sketches to estimate the result sizes of ad-hoc SQL queries.\nWe support the TPC-H and IMDb datasets.\n\nFigure~\\ref{fig:training} shows the four steps involved to create a new sketch.\nFirst (\\tikz[baseline=-0.7ex]{\\node[alexNumber] {1}}), users need to select a subset of tables from either schema and define a few parameters, including the number of materialized base table samples, the number of training queries, and the number of training epochs.\nNext (\\tikz[baseline=-0.7ex]{\\node[alexNumber] {2}}), we generate uniformly distributed training queries on the specified tables in our backend, and (\\tikz[baseline=-0.7ex]{\\node[alexNumber] {3}}) execute them with HyPer to obtain true cardinalities and to extract bitmaps indicating qualifying samples.\nTo accelerate this process during our demonstration, we plan to execute the training queries (in parallel) on multiple HyPer instances.\nFinally (\\tikz[baseline=-0.7ex]{\\node[alexNumber] {4}}), we featurize the training queries and train the MSCN model for the specified number of epochs.\n\nTo give a point of reference on the training costs, training the model with 90,000 queries over 100 epochs takes almost 39 minutes on an Amazon Web Services (AWS) ml.p2.xlarge instance using the PyTorch framework~\\cite{pytorch} with CUDA.\nSince this number is too high for an interactive user experience, we address this problem in three ways.\n\nFirst, we allow users to control the number of training queries and epochs.\nFor a small number of tables, 10,000 queries will already be sufficient to achieve good results.\nNote that the training time decreases linearly with fewer epochs.\nFrom our experience, 25 epochs are usually enough to achieve a reasonable mean q-error on a separate validation set.\nSecond, we offer pre-built (high quality) models that can be queried right away.\nThird, we allow users to train new models \\emph{while} querying existing ones.\n\nFigure~\\ref{fig:inference} illustrates a Deep Sketch on two tables A and B.\nThe interface of a sketch is very simple, it consumes a SQL query and returns a cardinality estimate.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.55\\linewidth]{deepsketches}\n\\caption{Web interface for Deep Sketches.}\n\\label{fig:deepsketches}\n\\end{figure*}\n\nFigure~\\ref{fig:deepsketches} shows our web interface for Deep Sketches.\nOn the left, we allow users to specify SQL queries, and on the right, we display query results (estimated and true cardinalities).\nOn the top (\\texttt{SHOW SKETCHES}), users can select existing and create new sketches as described above.\nTo query a sketch, users do not need enter SQL directly (the SQL string shown in the web interface is only displayed for information purposes).\nInstead, we provide them with a simple graphical query interface.\nBy clicking on a table, it is added to the query.\nWhen a user selects multiple tables, we automatically add the corresponding join predicates to the query (based on the single PK\/FK relationships that exist between tables).\n\nUsers can also define selections on base tables by clicking the respective columns in the schema.\nWe support both equality and range predicates.\nIn addition, we allow users to specify a placeholder for a certain column.\nSince our Deep Sketch implementation can only estimate single queries, we automatically instantiate such query templates and---in the background---execute each instance separately against the sketch.\nTo create such an instance, we draw a value from the column sample that is part of the sketch.\nOptionally, users can select a \\emph{function} to be applied to these values.\nFor example, for columns with many distinct values---such as \\texttt{Date} columns, users may want to ``group'' the results by year (e.g.,\\@\\xspace \\texttt{EXTRACT(YEAR FROM date)}).\nTo serve such queries, we generate multiple range queries (one for each year found in the sample) to be issued against the sketch.\nWe also support grouping the output into equally sized buckets based on the minimum and maximum values from the sample.\n\nWhen a user hits the \\texttt{EXECUTE} button, we issue the query against HyPer to compute its true cardinality as well as against the Deep Sketch and the cardinality estimators of HyPer and PostgreSQL to obtain estimates.\nThe query results are displayed with different overlays as they arrive.\nOn the X-axis we denote values from the placeholder column and on the Y-axis we plot the estimated and true cardinalities.\nWe support both bar and line charts and allow users to hide the results of individual systems.\n\nWe also use TensorBoard~\\cite{tensorboard} to visualize the neural network architecture of our model and the training phase.\n\n\n\\section{Deep Sketches}\n\\label{sec:deepsketches}\n\n\\begin{figure*}\n    \\centering\n    \\begin{subfigure}[b]{0.54\\textwidth}\n        \\includegraphics[width=\\textwidth]{training}\n        \\caption{Creation of a sketch.}\n        \\label{fig:training}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.44\\textwidth}\n        \\includegraphics[width=\\textwidth]{inference}\n        \\caption{Result size estimation with a sketch.}\n        \\label{fig:inference}\n    \\end{subfigure}\n    \\caption{Creation and usage of a Deep Sketch. Depending on the number of training queries, training can be expensive. However, once a sketch is trained, it allows for an efficient result size estimation of SQL queries.}\n    \\label{fig:training-inference}\n\\end{figure*}\n\nDeep Sketches are powered by a new (supervised) deep learning approach to cardinality estimation~\\cite{DBLP:conf\/cidr\/KipfKRLBK19} that has recently been adopted in~\\cite{DBLP:journals\/corr\/abs-1903-09999}.\nThe idea of this approach is to execute generated queries against a database to obtain true cardinalities (\\emph{labels} in ML), featurize these queries, and feed them into a neural network.\nOf course, instead of generating queries and uniformly sampling the search space, one could also use past user queries.\n\nBesides static query features---such as selections or joins, we featurize information about qualifying base table samples.\nIn other words, in addition to executing a training query against the full database, we execute each base table selection against a set of materialized samples (e.g.,\\@\\xspace 1000 tuples per base table).\nThus, we derive \\emph{bitmaps} indicating qualifying samples for each base table.\nThese bitmaps are then used as an additional input to the deep learning model.\nBesides this integration of (runtime) sampling, another differentiating factor from other learning-based approaches to cardinality estimation~\\cite{DBLP:conf\/sigmod\/OrtizBGK18, DBLP:journals\/pvldb\/WuJAPLQR18} is the use of a model that employs \\emph{set semantics}, inspired by recent work on \\textit{Deep Sets}~\\cite{zaheer2017deep}, a neural network module that operates on sets.\nThis decision rests on the fact that the cardinality of a query is independent of its query plan---e.g.,\\@\\xspace both $(A \\Join B) \\Join C$ and $A \\Join (B \\Join C)$ can be represented as $\\{A, B, C\\}$.\nWhile the Deep Sets model only addresses single sets, our model---called multi-set convolutional network (MSCN)---represents three sets (tables, joins, and predicates) and can capture correlations between sets.\n\nOn a high level, the MSCN model can be described as follows:\nFor each set, it has a separate module, comprised of one fully-connected multi-layer perceptron (MLP) per set element with shared parameters.\nWe average module outputs, concatenate them, and feed them into a final output MLP, which captures correlations between sets and outputs a cardinality estimate.\nThe featurization of a query is very straightforward.\nBased on the training data, we enumerate tables, columns, joins, and predicate types ($=$, $<$, and $>$) and represent them as unique one-hot vectors.\nWe represent each literal in a query as a value $val$ ($val \\in[0, 1]$), normalized using the minimum and maximum values of the respective column.\nSimilarly, we logarithmize and then normalize cardinalities (labels) using the maximum cardinality present in the training data.\nFor a detailed description of the model and the featurization, we refer the reader to~\\cite{DBLP:conf\/cidr\/KipfKRLBK19}.\nFinally, we train our model with the objective of minimizing the mean \\emph{q-error}~\\cite{DBLP:journals\/pvldb\/MoerkotteNS09} $q$ ($q \\geq 1$).\nThe q-error is the factor between the true and the estimated cardinality.\n\nOne advantage of our approach over pure sampling-based cardinality estimators is that it addresses 0-tuple situations, which is when no sampled tuples qualify.\nIn such situations, sampling-based approaches usually fall back to an ``educated'' guess---causing large estimation errors.\nOur approach, in contrast, handles such situations reasonably well \\cite{DBLP:conf\/cidr\/KipfKRLBK19} as it can use the signal of individual query features (e.g.,\\@\\xspace predicate types) to provide a more precise estimate.\nIn addition, it can---to some degree---capture correlations across joins and can thus estimate joins without assuming independence.\n\n\\begin{table}\n  \\small\n\\begin{tabular}{@{}lllllll@{}}\n\\toprule\n                 & median        & 90th          & 95th          & 99th          & max           & mean          \\\\ \\midrule\nDeep Sketch      & \\textbf{3.82}          & \\textbf{78.4} & \\textbf{362} & \\textbf{927} & \\textbf{1110} & \\textbf{57.9}   \\\\\nHyPer            & 14.6          & 454           & 1208          & 2764          & 4228          & 224           \\\\\nPostgreSQL       & 7.93          & 164           & 1104          & 2912          & 3477          & 174           \\\\ \\bottomrule\n\\end{tabular}\n\\caption{Estimation errors on the JOB-light workload.}\n\\label{tab:job-qerror}\n\\vspace{-1.0em}\n\\end{table}\n\nTable~\\ref{tab:job-qerror} shows the estimation errors (q-errors) of our approach (Deep Sketch) compared to the cardinality estimators of HyPer and PostgreSQL version 10.3.\nThe results are on JOB-light~\\cite{learnedpy}, which is a workload derived from the Join Order Benchmark (JOB)~\\cite{qoleis} containing 70 of the original 113 queries.\nIn contrast to JOB, JOB-light does not contain any predicates on strings nor disjunctions and only contains queries with one to four joins.\nMost queries in JOB-light have equality predicates on dimension table attributes. The only range predicate is on \\texttt{production\\_year}.\nConsidering that MSCN was trained with a uniform distribution between $=$, $<$, and $>$ predicates, it performs reasonably well.\nThis experiment shows that MSCN can generalize to workloads with distributions different from the training data.\nBy allowing users of our demonstration to issue ad-hoc queries, we want to enable them to experience this generalizability.\n\nWe believe that Deep Sketches are an important step towards a learned database system~\\cite{DBLP:conf\/cidr\/KraskaABCKLMMN19}, and can be used in conjunction with other recently proposed ML-powered components for join enumeration~\\cite{DBLP:conf\/sigmod\/MarcusP18, DBLP:journals\/corr\/abs-1808-03196}, adaptive query processing~\\cite{DBLP:journals\/pvldb\/TrummerMMJA18}, indexing~\\cite{DBLP:conf\/sigmod\/KraskaBCDP18, DBLP:journals\/corr\/abs-1903-11203}, view materialization~\\cite{DBLP:journals\/corr\/abs-1903-01363}, workload management~\\cite{DBLP:conf\/cidr\/JainYCH19}, and query performance prediction~\\cite{DBLP:journals\/corr\/abs-1902-00132}.\n\n\n\n\\section{Introduction}\n\nWe introduce Deep Sketches, compact model-based representations of databases that allow us to estimate the result sizes of SQL queries.\nDeep Sketches are powered by a new deep learning approach to cardinality estimation~\\cite{DBLP:conf\/cidr\/KipfKRLBK19} (code:~\\cite{learnedpy}).\nThis approach builds on sampling-based estimation and addresses its weaknesses when no sampled tuples qualify a predicate.\nA Deep Sketch is essentially a wrapper for a (serialized) neural network and a set of materialized samples.\n\nEstimates of intermediate query result sizes are the core ingredient to cost-based query optimizers~\\cite{qoleis,leis2018query}.\nWhile the focus of this work is \\emph{not} to show the effect of better cardinality estimates on the quality of resulting query plans---which is orthogonal to having better estimates in the first place, we demonstrate that the estimates produced by Deep Sketches are superior to estimates of traditional optimizers and often close to the ground truth.\nThe estimates produced by Deep Sketches can directly be leveraged by existing, sophisticated join enumeration algorithms and cost models.\nThis is a more gradual approach than the one taken by machine learning (ML)-based end-to-end query optimizers~\\cite{DBLP:conf\/cidr\/MarcusP19}.\n\nOur demonstration allows users to define Deep Sketches on the TPC-H and Internet Movie Database (IMDb) datasets.\nThe latter is a real-world dataset that contains many correlations and therefore proves to be very challenging for cardinality estimators.\nTo create a Deep Sketch, users select a subset of tables and define a few parameters such as the number of training queries.\nUsers can then monitor the training progress, including the execution of training queries and the training of the deep learning model.\nOnce the model has been trained, users can issue ad-hoc queries against the resulting Deep Sketch.\nOur user interface makes it easy to create such queries graphically.\nUsers can optionally specify a placeholder for a certain column to define a query template.\nFor example, a movie producer might be interested in the popularity of a certain keyword over time:\n\\begin{verbatim}\nSELECT COUNT(*)\nFROM title t, movie_keyword mk, keyword k\nWHERE mk.movie_id=t.id AND mk.keyword_id=k.id\nAND k.keyword='artificial-intelligence'\nAND t.production_year=?\n\\end{verbatim}\nA placeholder has a similar effect as a group-by operation, except that it does not operate on all distinct values of the group-by column but instead only on the values present in the column sample that comes with the sketch.\nIn other words, we instantiate the query template with values (literals) from the column sample.\nBesides this being an interesting feature for data analysts, it serves the purpose of visualizing the robustness of our deep learning approach to cardinality estimation\\footnote{Note that the deep learning model is not necessarily trained with literals present in the column sample. In fact, it can happen (and is even likely for columns with many distinct values) that a literal from the column sample has never been seen by the model.}.\nThe result of a query template can be displayed as a bar or as a line plot with one data point per template instance.\nUsing overlays, we show the difference to the cardinality estimates produced by HyPer\\footnote{We are referring to the research version of HyPer developed at the Technical University of Munich.}~\\cite{hyper} and PostgreSQL as well as to the true cardinalities---obtained by executing the queries with HyPer.\n\nDeep Sketches feature a small footprint size (a few MiBs) and are fast to query (within milliseconds).\nDue to these facts, another application of Deep Sketches lies in the area of previewing query result sizes.\nOften, rough estimates are sufficient to inform users whether executing a certain query would be worthwhile, and sometimes even all they need---e.g.,\\@\\xspace to get an idea about the data distribution of a certain column given some selections and\/or joins.\nFor example, Deep Sketches could be deployed in a web browser or within a cell phone to preview query results.\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nWe have introduced Deep Sketches, which are compact representations of databases that allow us to estimate the result sizes of SQL queries.\nOur demonstration allows users to experience the end-to-end training and querying process of these sketches.\nThe goal of this work is to show that a learned cardinality model can compete with and even outperform traditional cardinality estimators, especially for highly correlated data.\nOur audience can specify ad-hoc queries and thereby observe that ML might indeed be the right hammer for the decades-old cardinality estimation job.\nClearly, more research is needed to automate the training and utilization of Deep Sketches in query optimizers.\nOne question---that we currently outsource to our users---is for which schema parts we should build such sketches.\nBesides further improving cardinality estimation for query optimization, another avenue we could take in future work is to deploy Deep Sketches---which offer a small footprint size---on the client to preview query results.\n\\balance\n\n\\section{Deep Sketches}\n\\label{sec:deepsketches}\n\n\\begin{figure*}\n    \\centering\n    \\begin{subfigure}[b]{0.54\\textwidth}\n        \\includegraphics[width=\\textwidth]{training}\n        \\caption{Creation of a sketch.}\n        \\label{fig:training}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.44\\textwidth}\n        \\includegraphics[width=\\textwidth]{inference}\n        \\caption{Result size estimation with a sketch.}\n        \\label{fig:inference}\n    \\end{subfigure}\n    \\caption{Creation and usage of a Deep Sketch. Depending on the number of training queries, training can be expensive. However, once a sketch is trained, it allows for an efficient result size estimation of SQL queries.}\n    \\label{fig:training-inference}\n\\end{figure*}\n\nDeep Sketches are powered by a new (supervised) deep learning approach to cardinality estimation~\\cite{DBLP:conf\/cidr\/KipfKRLBK19} that has recently been adopted in~\\cite{DBLP:journals\/corr\/abs-1903-09999}.\nThe idea of this approach is to execute generated queries against a database to obtain true cardinalities (\\emph{labels} in ML), featurize these queries, and feed them into a neural network.\nOf course, instead of generating queries and uniformly sampling the search space, one could also use past user queries.\n\nBesides static query features---such as selections or joins, we featurize information about qualifying base table samples.\nIn other words, in addition to executing a training query against the full database, we execute each base table selection against a set of materialized samples (e.g.,\\@\\xspace 1000 tuples per base table).\nThus, we derive \\emph{bitmaps} indicating qualifying samples for each base table.\nThese bitmaps are then used as an additional input to the deep learning model.\nBesides this integration of (runtime) sampling, another differentiating factor from other learning-based approaches to cardinality estimation~\\cite{DBLP:conf\/sigmod\/OrtizBGK18, DBLP:journals\/pvldb\/WuJAPLQR18} is the use of a model that employs \\emph{set semantics}, inspired by recent work on \\textit{Deep Sets}~\\cite{zaheer2017deep}, a neural network module that operates on sets.\nThis decision rests on the fact that the cardinality of a query is independent of its query plan---e.g.,\\@\\xspace both $(A \\Join B) \\Join C$ and $A \\Join (B \\Join C)$ can be represented as $\\{A, B, C\\}$.\nWhile the Deep Sets model only addresses single sets, our model---called multi-set convolutional network (MSCN)---represents three sets (tables, joins, and predicates) and can capture correlations between sets.\n\nOn a high level, the MSCN model can be described as follows:\nFor each set, it has a separate module, comprised of one fully-connected multi-layer perceptron (MLP) per set element with shared parameters.\nWe average module outputs, concatenate them, and feed them into a final output MLP, which captures correlations between sets and outputs a cardinality estimate.\nThe featurization of a query is very straightforward.\nBased on the training data, we enumerate tables, columns, joins, and predicate types ($=$, $<$, and $>$) and represent them as unique one-hot vectors.\nWe represent each literal in a query as a value $val$ ($val \\in[0, 1]$), normalized using the minimum and maximum values of the respective column.\nSimilarly, we logarithmize and then normalize cardinalities (labels) using the maximum cardinality present in the training data.\nFor a detailed description of the model and the featurization, we refer the reader to~\\cite{DBLP:conf\/cidr\/KipfKRLBK19}.\nFinally, we train our model with the objective of minimizing the mean \\emph{q-error}~\\cite{DBLP:journals\/pvldb\/MoerkotteNS09} $q$ ($q \\geq 1$).\nThe q-error is the factor between the true and the estimated cardinality.\n\nOne advantage of our approach over pure sampling-based cardinality estimators is that it addresses 0-tuple situations, which is when no sampled tuples qualify.\nIn such situations, sampling-based approaches usually fall back to an ``educated'' guess---causing large estimation errors.\nOur approach, in contrast, handles such situations reasonably well \\cite{DBLP:conf\/cidr\/KipfKRLBK19} as it can use the signal of individual query features (e.g.,\\@\\xspace predicate types) to provide a more precise estimate.\nIn addition, it can---to some degree---capture correlations across joins and can thus estimate joins without assuming independence.\n\n\\begin{table}\n  \\small\n\\begin{tabular}{@{}lllllll@{}}\n\\toprule\n                 & median        & 90th          & 95th          & 99th          & max           & mean          \\\\ \\midrule\nDeep Sketch      & \\textbf{3.82}          & \\textbf{78.4} & \\textbf{362} & \\textbf{927} & \\textbf{1110} & \\textbf{57.9}   \\\\\nHyPer            & 14.6          & 454           & 1208          & 2764          & 4228          & 224           \\\\\nPostgreSQL       & 7.93          & 164           & 1104          & 2912          & 3477          & 174           \\\\ \\bottomrule\n\\end{tabular}\n\\caption{Estimation errors on the JOB-light workload.}\n\\label{tab:job-qerror}\n\\vspace{-1.0em}\n\\end{table}\n\nTable~\\ref{tab:job-qerror} shows the estimation errors (q-errors) of our approach (Deep Sketch) compared to the cardinality estimators of HyPer and PostgreSQL version 10.3.\nThe results are on JOB-light~\\cite{learnedpy}, which is a workload derived from the Join Order Benchmark (JOB)~\\cite{qoleis} containing 70 of the original 113 queries.\nIn contrast to JOB, JOB-light does not contain any predicates on strings nor disjunctions and only contains queries with one to four joins.\nMost queries in JOB-light have equality predicates on dimension table attributes. The only range predicate is on \\texttt{production\\_year}.\nConsidering that MSCN was trained with a uniform distribution between $=$, $<$, and $>$ predicates, it performs reasonably well.\nThis experiment shows that MSCN can generalize to workloads with distributions different from the training data.\nBy allowing users of our demonstration to issue ad-hoc queries, we want to enable them to experience this generalizability.\n\nWe believe that Deep Sketches are an important step towards a learned database system~\\cite{DBLP:conf\/cidr\/KraskaABCKLMMN19}, and can be used in conjunction with other recently proposed ML-powered components for join enumeration~\\cite{DBLP:conf\/sigmod\/MarcusP18, DBLP:journals\/corr\/abs-1808-03196}, adaptive query processing~\\cite{DBLP:journals\/pvldb\/TrummerMMJA18}, indexing~\\cite{DBLP:conf\/sigmod\/KraskaBCDP18, DBLP:journals\/corr\/abs-1903-11203}, view materialization~\\cite{DBLP:journals\/corr\/abs-1903-01363}, workload management~\\cite{DBLP:conf\/cidr\/JainYCH19}, and query performance prediction~\\cite{DBLP:journals\/corr\/abs-1902-00132}.\n\n\\section{Demonstration}\n\\label{sec:demonstration}\n\nAs stated earlier, a Deep Sketch is essentially a wrapper for a trained MSCN model and a set of materialized samples.\nIn our demonstration, users can experience the end-to-end process of defining, training, and using trained sketches to estimate the result sizes of ad-hoc SQL queries.\nWe support the TPC-H and IMDb datasets.\n\nFigure~\\ref{fig:training} shows the four steps involved to create a new sketch.\nFirst (\\tikz[baseline=-0.7ex]{\\node[alexNumber] {1}}), users need to select a subset of tables from either schema and define a few parameters, including the number of materialized base table samples, the number of training queries, and the number of training epochs.\nNext (\\tikz[baseline=-0.7ex]{\\node[alexNumber] {2}}), we generate uniformly distributed training queries on the specified tables in our backend, and (\\tikz[baseline=-0.7ex]{\\node[alexNumber] {3}}) execute them with HyPer to obtain true cardinalities and to extract bitmaps indicating qualifying samples.\nTo accelerate this process during our demonstration, we plan to execute the training queries (in parallel) on multiple HyPer instances.\nFinally (\\tikz[baseline=-0.7ex]{\\node[alexNumber] {4}}), we featurize the training queries and train the MSCN model for the specified number of epochs.\n\nTo give a point of reference on the training costs, training the model with 90,000 queries over 100 epochs takes almost 39 minutes on an Amazon Web Services (AWS) ml.p2.xlarge instance using the PyTorch framework~\\cite{pytorch} with CUDA.\nSince this number is too high for an interactive user experience, we address this problem in three ways.\n\nFirst, we allow users to control the number of training queries and epochs.\nFor a small number of tables, 10,000 queries will already be sufficient to achieve good results.\nNote that the training time decreases linearly with fewer epochs.\nFrom our experience, 25 epochs are usually enough to achieve a reasonable mean q-error on a separate validation set.\nSecond, we offer pre-built (high quality) models that can be queried right away.\nThird, we allow users to train new models \\emph{while} querying existing ones.\n\nFigure~\\ref{fig:inference} illustrates a Deep Sketch on two tables A and B.\nThe interface of a sketch is very simple, it consumes a SQL query and returns a cardinality estimate.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.55\\linewidth]{deepsketches}\n\\caption{Web interface for Deep Sketches.}\n\\label{fig:deepsketches}\n\\end{figure*}\n\nFigure~\\ref{fig:deepsketches} shows our web interface for Deep Sketches.\nOn the left, we allow users to specify SQL queries, and on the right, we display query results (estimated and true cardinalities).\nOn the top (\\texttt{SHOW SKETCHES}), users can select existing and create new sketches as described above.\nTo query a sketch, users do not need enter SQL directly (the SQL string shown in the web interface is only displayed for information purposes).\nInstead, we provide them with a simple graphical query interface.\nBy clicking on a table, it is added to the query.\nWhen a user selects multiple tables, we automatically add the corresponding join predicates to the query (based on the single PK\/FK relationships that exist between tables).\n\nUsers can also define selections on base tables by clicking the respective columns in the schema.\nWe support both equality and range predicates.\nIn addition, we allow users to specify a placeholder for a certain column.\nSince our Deep Sketch implementation can only estimate single queries, we automatically instantiate such query templates and---in the background---execute each instance separately against the sketch.\nTo create such an instance, we draw a value from the column sample that is part of the sketch.\nOptionally, users can select a \\emph{function} to be applied to these values.\nFor example, for columns with many distinct values---such as \\texttt{Date} columns, users may want to ``group'' the results by year (e.g.,\\@\\xspace \\texttt{EXTRACT(YEAR FROM date)}).\nTo serve such queries, we generate multiple range queries (one for each year found in the sample) to be issued against the sketch.\nWe also support grouping the output into equally sized buckets based on the minimum and maximum values from the sample.\n\nWhen a user hits the \\texttt{EXECUTE} button, we issue the query against HyPer to compute its true cardinality as well as against the Deep Sketch and the cardinality estimators of HyPer and PostgreSQL to obtain estimates.\nThe query results are displayed with different overlays as they arrive.\nOn the X-axis we denote values from the placeholder column and on the Y-axis we plot the estimated and true cardinalities.\nWe support both bar and line charts and allow users to hide the results of individual systems.\n\nWe also use TensorBoard~\\cite{tensorboard} to visualize the neural network architecture of our model and the training phase.\n\n\n\\section{Introduction}\n\nWe introduce Deep Sketches, compact model-based representations of databases that allow us to estimate the result sizes of SQL queries.\nDeep Sketches are powered by a new deep learning approach to cardinality estimation~\\cite{DBLP:conf\/cidr\/KipfKRLBK19} (code:~\\cite{learnedpy}).\nThis approach builds on sampling-based estimation and addresses its weaknesses when no sampled tuples qualify a predicate.\nA Deep Sketch is essentially a wrapper for a (serialized) neural network and a set of materialized samples.\n\nEstimates of intermediate query result sizes are the core ingredient to cost-based query optimizers~\\cite{qoleis,leis2018query}.\nWhile the focus of this work is \\emph{not} to show the effect of better cardinality estimates on the quality of resulting query plans---which is orthogonal to having better estimates in the first place, we demonstrate that the estimates produced by Deep Sketches are superior to estimates of traditional optimizers and often close to the ground truth.\nThe estimates produced by Deep Sketches can directly be leveraged by existing, sophisticated join enumeration algorithms and cost models.\nThis is a more gradual approach than the one taken by machine learning (ML)-based end-to-end query optimizers~\\cite{DBLP:conf\/cidr\/MarcusP19}.\n\nOur demonstration allows users to define Deep Sketches on the TPC-H and Internet Movie Database (IMDb) datasets.\nThe latter is a real-world dataset that contains many correlations and therefore proves to be very challenging for cardinality estimators.\nTo create a Deep Sketch, users select a subset of tables and define a few parameters such as the number of training queries.\nUsers can then monitor the training progress, including the execution of training queries and the training of the deep learning model.\nOnce the model has been trained, users can issue ad-hoc queries against the resulting Deep Sketch.\nOur user interface makes it easy to create such queries graphically.\nUsers can optionally specify a placeholder for a certain column to define a query template.\nFor example, a movie producer might be interested in the popularity of a certain keyword over time:\n\\begin{verbatim}\nSELECT COUNT(*)\nFROM title t, movie_keyword mk, keyword k\nWHERE mk.movie_id=t.id AND mk.keyword_id=k.id\nAND k.keyword='artificial-intelligence'\nAND t.production_year=?\n\\end{verbatim}\nA placeholder has a similar effect as a group-by operation, except that it does not operate on all distinct values of the group-by column but instead only on the values present in the column sample that comes with the sketch.\nIn other words, we instantiate the query template with values (literals) from the column sample.\nBesides this being an interesting feature for data analysts, it serves the purpose of visualizing the robustness of our deep learning approach to cardinality estimation\\footnote{Note that the deep learning model is not necessarily trained with literals present in the column sample. In fact, it can happen (and is even likely for columns with many distinct values) that a literal from the column sample has never been seen by the model.}.\nThe result of a query template can be displayed as a bar or as a line plot with one data point per template instance.\nUsing overlays, we show the difference to the cardinality estimates produced by HyPer\\footnote{We are referring to the research version of HyPer developed at the Technical University of Munich.}~\\cite{hyper} and PostgreSQL as well as to the true cardinalities---obtained by executing the queries with HyPer.\n\nDeep Sketches feature a small footprint size (a few MiBs) and are fast to query (within milliseconds).\nDue to these facts, another application of Deep Sketches lies in the area of previewing query result sizes.\nOften, rough estimates are sufficient to inform users whether executing a certain query would be worthwhile, and sometimes even all they need---e.g.,\\@\\xspace to get an idea about the data distribution of a certain column given some selections and\/or joins.\nFor example, Deep Sketches could be deployed in a web browser or within a cell phone to preview query results.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSeveral effects derived from the twisting of the polarization and wavefronts of electromagnetic fields, specifically the spin and orbital angular momenta, are  a subject of increasing study in  recent years  \\cite{allenlibro,allenlibro1,babiker,yao,cameron2,molina1,molina2}. This is accompanied by a steady improvement in particle manipulation techniques and theories  \\cite{garces,grier,dunlop,chaumet,brasse,chen,MNV2015}, and by  the use of spatially structured waves \\cite{barron2} with enhanced helicity \\cite{tang1} to increase the signal in circular dichroism \\cite{schellmann,barron1}  for enantiomeric discrimination \\cite{tang2, tang3,choi}. In addition, recent studies \\cite{muka} in fluorescence resonance energy transfer (FRET) \\cite{fret1,fret2}, (see also \\cite{craig,salam}),  show an electromagnetic force between excited molecules, different from the Van der Waals force when they  are in their ground-state.\n\n A consequence of this research was the derivation of a  conservation law for the helicity of electromagnetic fields \\cite{lipkin,cameron1} that appears as fundamental as that for the energy.\n\nIn this paper we discuss the physical significance of this helicity conservation   law.  Dealing  with quasimonochromatic electromagnetic fields, we establish the optical theorem which constitutes the main consequence of this law concerning  optical, or electromagnetic, scattering. In this way, we show that this new theorem  provides an expression for the helicity excitation rate of a particle, (dipolar in the wide sense, i.e. such that its scattering may be fully described by its first electric and magnetic partial waves), by extinction of  the helicity of the irradiating field. In particular for magnetodielectric bi-isotropic objects,  this leads to a relationship  between polarizabilities, complementary and compatible with that of the optical theorem for energies. For circularly polarized light this also  establishes a necessary and  sufficient condition between their  duality and  scattering characteristics. \n\n In this respect, we do not address here quadrupoles or other multipolar excitations. Although extensions of dipolar models have been carried out in studies of  the energy conveyed by those higher order terms, showing the observable signal due to the electric dipole-quadrupole polarizability for chiral configurations \\cite{Yang}, (see also \\cite{choi} remarking the similarity in magnitude of the electric quadrupole and magnetic dipole moments according to  quantum electrodynamical calculations in \\cite{craig, salam}),\nas regards the purpose of our study which deals  with a  different quantity: the helicity, we show  that the (broad sense) dipolar formulation already leads to new physical phenomena that should be observed in future novel experiments, even though of course this theory  is amenable of further generalizations to account for effects due to higher order excitations.\n\nMore importantly, this novel equation opens a new landscape for:\n\n1.The emission and absorption of helicity in complex environments, also  in particular  at the nanoscale, e.g. in FRET between molecules, or other nanoscructures, where rather than anlysing the transference of energy, one establishes and addresses the behavior of  the helicity  lifetimes, taking the  bi-isotropy, and chirality in particular, into account.\n\n2. Enantiomeric discrimination, where chiral molecules, or other nanoparticles, are studied by circular dichroism. This is done  by means of a new dissymmetry  factor introduced in this  work stemming from this novel  optical theorem. This factor has higher sensitivity than the standard one \\cite{barron1}  based  on the extinction of incident energy  and its transfer to the object by  measuring its intensity excitation, since it involves a new experimental procedure  which detects the total scattered helicity and its flow by means of  an ellipsometry set-up \\cite{marston1}.\n\n\\section{The helicity}\n\nWe consider  fields, currents and potentials  with a time-harmonic dependence, so that the electric and magnetic vectors are   ${\\bf \\cal E}({\\bf r},t)$ and  ${\\bf \\cal B}({\\bf r},t)$:  ${\\bf \\cal E}({\\bf r},t)=\\Re [{\\bf E}({\\bf r}) \\exp(-i\\omega t)]$ and  ${\\bf \\cal B}({\\bf r},t)=\\Re [{\\bf B}({\\bf r}) \\exp(-i \\omega t)]$.  $\\Re$ denotes real part.\n\nWe introduce the helicity density $\\mathscr{ H}$ and the density of flow of helicity  $\\mathscr{ F}$  of this  field in a non-absorbing dielectric medium of  refractive index $n=\\sqrt{\\epsilon \\mu}$, ($\\epsilon$ and $\\mu$ represent the dielectric permittivity and the magnetic permeability),   as:\n\\begin{equation}\n\\hel=\\frac{1}{2}(\\frac{1}{\\mu}{\\bf \\cal A}\\cdot\\bf{\\cal B}-\\epsilon {\\bf \\cal C} \\cdot \\bf \\cal E), \\label{h}\n\\end{equation}\n\\begin{equation}\n\\flow=\\frac{c}{2 \\mu}({\\bf \\cal E}\\times\\bf{\\cal A} +  {\\bf \\cal B} \\times \\bf \\cal C). \\label{f}\n\\end{equation}\n ${\\bf \\cal A}$ and ${\\bf \\cal C}$ are vector potentials  such that: ${\\bf \\cal B}=\\nabla \\times {\\bf \\cal A}$ and  ${\\bf \\cal E}=-\\nabla \\times {\\bf \\cal C}$\n\\cite{cameron1}, so that  working in a Coulomb gauge: $\\nabla\\cdot {\\bf \\cal A}=\\nabla\\cdot {\\bf \\cal C}= 0$, and  one has from Maxwell's equations: \n\\be\n\\dot{\\bf \\cal A}= -c {\\bf\\cal  E},\n\\dot{\\bf \\cal C}= - \\frac{c}{\\epsilon\\mu} \\nabla\\times {\\bf \\cal A } + \\frac{4 \\pi}{\\epsilon}  {\\bf \\cal K };\n {\\bf \\cal J}= \\nabla \\times {\\bf \\cal K }. \\label{pots}\n\\ee\n\nThe upper dot stands for $\\partial_t$, $c$ is the light speed in vacuum, and $\\bf \\cal J$ denotes the electric current density which is transversal since the existence of ${\\bf \\cal A}$ and the law $\\nabla\\cdot \\epsilon {\\bf \\cal E}= 4 \\pi \\rho$ imply that the electric charge density $\\rho$ is zero . From the above equations one obtains the conservation law \\cite{cameron1}\n\\begin{equation}\n\\dot{\\hel}+ \\nabla \\cdot \\flow = - \\perd . \\label{contH}\n\\end{equation}\nWhere dissipation in the  interaction of the fields with matter is represented by $\\perd= 2\\pi ({\\bf  \\cal E}\\cdot {\\bf  \\cal K}- {\\bf \\cal J} \\cdot {\\bf \\cal  C})$.\n\nSince the fields and potentials  are time-harmonic, we convert the quantities holding Eqs. (\\ref{pots})  and  (\\ref{contH}) into:\n\\be\n{\\bf  A}= - \\frac{i}{k} {\\bf E}, \\,\\,\\,\\,\n{\\bf  C}= - \\frac{i}{\\epsilon} [\\frac{{\\bf B}}{k \\mu} - \\frac{4 \\pi}{\\omega} {\\bf K}],  \\label{potsr}\n\\ee\nand\n\\begin{equation}\n\\hel= <\\hel>=\\frac{1}{2k}\\sqrt{\\frac{\\epsilon}{\\mu}} \\Im ({\\bf E}\\cdot {\\bf B}^*), \\label{hh}\n\\end{equation}\n\\begin{equation}\n\\flow= <\\flow> =  \\frac{c}{4n k } \\Im ( \\epsilon {\\bf E}^* \\times {\\bf  E}  + \\frac{1}{\\mu}{\\bf  B}^*\\times \\bf B)  . \\label{ff}\n\\end{equation}\nWhere $<\\cdot>$ denotes time-average, $\\Im$  means imaginary part and $k=n \\omega\/c$, ${\\bf \\cal A}= \\Re[ {\\bf A}({\\bf r}) \\exp (-i \\omega t)]$, \n${\\bf \\cal C}= \\Re[{\\bf C}({\\bf r})  \\exp (-i \\omega t)]$,   ${\\bf \\cal J}= \\Re[{\\bf J}({\\bf r})  \\exp(-i \\omega t)]$,  ${\\bf \\cal K}=\\Re[ {\\bf K}({\\bf r})  \\exp(-i \\omega t)]$.  Now $\\flow$ coincides with the spin angular momentum density. Eq.(\\ref{contH}) is then fulfilled by these time-averaged quantities  with $\\perd$  replaced by:\n\\begin{eqnarray}\n< \\perd >=\\pi [\\frac{2}{ck}\\sqrt{\\frac{\\mu}{\\epsilon}} \\nabla\\cdot \\Im ({\\bf K}\\times {\\bf B}^*)  \\nonumber \\\\ -\\frac{1}{kn} \\Im ( {\\bf J} \\cdot {\\bf  B}^*)  +\\frac{4\\pi}{ck}\\sqrt{\\frac{\\mu}{\\epsilon}}\\Im  ({\\bf J}\\cdot {\\bf K}^*)  \n +\\Re  ({\\bf E}\\cdot {\\bf K}^*) ]  . \\label{pp}\n\\end{eqnarray}\nFor these monochromatic fields, Maxwell's  equations, and the above relations, show   that (\\ref{hh}) and (\\ref{ff}) are proportional to Lipkin's zilches \\cite{lipkin,cameron1}, used in recent works as chirality $\\qui$  and flow of chirality $\\s$ \\cite{tang1,tang2}:\n\\begin{equation}\n\\qui= <\\qui>=k^2 \\hel= k^2 <\\hel>\\label{kk}\n\\end{equation}\n\\begin{equation}\n\\s= <\\s> = k^2 \\flow= k^2 <\\flow>\\label{ss}\n\\end{equation}\nThe dissipative terms are however different. \nWe follow the criterion  of \\cite{cameron1} according to which $\\flow$ is the quantity with dimensions of angular momentum, so that we deal with  the helicity and its flow; although  (\\ref{kk}) and (\\ref{ss}) show that both pairs yield equivalent  mesurements   for monochromatic fields.\n\\section{The optical theorem for the helicity}\nLet a  monochromatic, elliptically polarized, plane wave be incident on a scattering body, e.g. a polarizable particle, (cf. Fig.1). The field at any point of the exterior medium may be represented as the sum of the incident  and the scattered vectors  as:  ${\\bf E}({\\bf r})={\\bf E}_i({\\bf r})+{\\bf E}_s({\\bf r})$,   ${\\bf B}({\\bf r})={\\bf B}_i({\\bf r})+ {\\bf B}_s({\\bf r})$. \n\n\\begin{figure}[htbp]\n\\centerline{\\includegraphics[width=1.0\\columnwidth]{fig1.eps}}\n\\caption{(Color online). An elliptically polarized plane wave incides on a  polarizable particle. The fields are evaluated at the point {\\bf P}: ${\\bf r}=R \\bf s$ , of  coordinates   $(R, \\theta, \\phi)$, of  a sphere of integration   of radius $R$, centered at some point  ${\\bf r}_ {0}$  of the particle.  ${\\bf r}_ {0}$ acts as the framework center {\\bf 0}. The point {\\bf Q} is the projection of {\\bf P} on the plane $ OXY$; the scattering plane being $OPQ$.  We  show the three orthonormal vectors:  ${\\bf s}$,  ${\\bm \\epsilon}_{\\parallel}$ (in the plane $OPQ$ and in the sense of rotation of $\\theta$), and ${\\bm \\epsilon}_{\\perp}$ (normal to $OPQ$).}\n\\end{figure}\n\nThe incident fields being  ${\\bf E}_i={\\bf e}_{i} e^{ik({\\bf s}_{i}\\cdot {\\bf r})}$,  ${\\bf B}_i={\\bf b}_{i} e^{ik({\\bf s}_{i}\\cdot {\\bf r})}$; whereas in  the far zone the scattered fields are:  ${\\bf E}_s={\\bf e}({\\bf s}) \\exp(ikr)\/r$, ${\\bf B}_s={\\bf b}({\\bf s}) \\exp(ikr)\/r$ . Also  ${\\bf b}_{i}=n {\\bf s}_{i} \\times  {\\bf e}_{i}$, ${\\bf e}_{i} \\cdot {\\bf s}_{i}= {\\bf b}_{i} \\cdot {\\bf s}_{i}=0$;  ${\\bf b}=n {\\bf s} \\times  {\\bf e}$, ${\\bf e} \\cdot {\\bf s}= {\\bf b} \\cdot {\\bf s}=0$.\n\n The flow (or time-averaged flow) density of helicity is:  $\\flow= <\\flow> =  \\flow^i + \\flow^s +\n\\flow' $  . Where\n\\begin{eqnarray}\n\\flow^i=<\\flow^i>=\\frac{c}{4n k } \\Im ( \\epsilon {\\bf E}_{i}^{ *} \\times {\\bf  E}_{i }  + \\frac{1}{\\mu}{\\bf  B}_{i}^{ *}\\times {\\bf B}_{i })  .   (10a)  \\nonumber\\\\\n\\flow^s=<\\flow^s>=\\frac{c}{4n k } \\Im ( \\epsilon {\\bf E}_{s}^{ *} \\times {\\bf  E}_{s }  + \\frac{1}{\\mu}{\\bf  B}_{s}^{ *}\\times {\\bf B}_{s}).  \\nonumber (10b) \\\\\n\\flow'=<\\flow'>=\\frac{c}{4n k } \\Im ( \\epsilon {\\bf E}_{i}^{ *} \\times {\\bf  E}_{s }  + \\frac{1}{\\mu}{\\bf  B}_{i}^{ *}\\times {\\bf B}_{s }).  \\nonumber (10c)\n\\end{eqnarray}\n\nFrom Eq.(\\ref{contH}) the rate ${\\cal W}_{\\hel}^{a}$ at which the helicity is dissipated on interaction with the body is given by the $\\Sigma$-integral that gives the outward flow of helicity: $\\int_{\\Sigma}  d\\Omega R^2 \\flow  \\cdot {\\bf s}$   through the surface of a large sphere $\\Sigma$ of radius $R$ with center at some point  ${\\bf r}_0$ of the object. $d\\Omega$ is the element of solid angle  and ${\\bf s}$ denotes the outward normal.  I.e., according to Eq.(\\ref{contH}):\n\\begin{equation}\n- {\\cal W}_{\\hel}^{a}= {\\cal W}_{\\hel}^{i}+{\\cal W}_{\\hel}^{s}+{\\cal W}_{\\hel}'. \\label{who}\n\\end{equation}\nWhere ${\\cal W}_{\\hel}^{i}$, ${\\cal W}_{\\hel}^{s}$ and ${\\cal W}_{\\hel}'$ are respectively the  $\\Sigma$-integrals of the projections on $\\bf s$ of  $\\flow^i$, $\\flow^s$ and $\\flow' $.  On the other hand,  ${\\cal W}_{\\hel}^{a}=\\int_{\\Sigma} d R d\\Omega R^2 <\\perd >$. \n\n\nFrom these equations we have that  $ {\\cal W}_{\\hel}^{i}=0$, so that  (\\ref{who}) becomes\n\\be\n {\\cal W}_{\\hel}^{a}+ {\\cal W}_{\\hel}^{s}=-{\\cal W}_{\\hel}'. \\label{wh}\n\\ee\nWhereas the integrals  of $\\flow^s \\cdot {\\bf s}$ and $\\flow' \\cdot {\\bf s}$ across $\\Sigma$  are \n\\be\n{\\cal W}_{\\hel}^{s}=\\int_{\\Sigma}  d\\Omega R^2   \\flow^{s} \\cdot {\\bf s} =\\nonumber \\\\\n\\frac{c}{4n k } \\Im \\int_{\\Sigma}  d\\Omega {\\bf s} \\cdot [ \\epsilon {\\bf e}^{*}({\\bf s}) \\times  {\\bf e}({\\bf s}) + \\frac{1}{\\mu}{\\bf b}^{*}({\\bf s}) \\times  {\\bf b}({\\bf s})] , \\label{fss}\n\\ee\nand\n\\be\n{\\cal W}_{\\hel}'=\\int_{\\Sigma}  d\\Omega R^2   \\flow'    \\cdot {\\bf s}=\\nonumber \\\\\n\\frac{2 \\pi c }{n  k^2 }\\sqrt{\\frac{\\epsilon}{\\mu}} \\Re[{\\bf B}_{i}^{*}(\\bf r_0)   \\cdot  {\\bf e}({\\bf s}_{i})] = \\nonumber \\\\\n-\\frac{2 \\pi c }{n  k^2 }\\sqrt{\\frac{\\epsilon}{\\mu}} \\Re[{\\bf E}_{i}^{*}(\\bf r_0)  \\cdot  {\\bf b}({\\bf s}_{i})] . \\label{fis}\n\\ee\nIn deriving (\\ref{fis}) we have used Jones' lemma based on the principle of the stationary phase \\cite{jones, born}:\n\n\\be\n\\frac{1}{R}\\int  d\\Omega R^2  F({\\bf s}) e^{-ik ({\\bf s}_{i} \\cdot {\\bf s})R} \\sim \\frac{2\\pi i}{k}[F({\\bf s}_{i})e^{-ik R} \\nonumber \\\\\n - F(-{\\bf s}_{i})e^{ik R} ] . \\label{psp}\n\\ee\nEqs.(\\ref{fss}) and  (\\ref{fis})  together with (\\ref{wh})  constitute the {\\it optical theorem} that represents the {\\it conservation of helicity} on scattering by an arbitrary body. They state that the rate at which the helicity is dissipated from the incident wave, in the form of losses in the obstacle,  ${\\cal W}_{\\hel}^{a}$, and  of  helicity of the scattered field integrated in all directions, ${\\cal W}_{\\hel}^{s}$, is proportional to a certain helicity component of the scattered field, $-{\\cal W}_{\\hel}'$,  on interference with the incident wave in the forward direction ${\\bf s}_i$. This suggests to  introduce a {\\it helicity extinction cross-section} of the body ${\\cal Q}_{\\hel}$  dividing the terms of (\\ref{wh}) by  the rate $|\\flow^{i}|=\\frac{c}{2nk}\\sqrt{\\frac{\\epsilon}{\\mu}}\\Im [{\\bf E}_{i} ({\\bf r}_0) \\cdot {\\bf B}_{i}^{*}({\\bf r}_{0}) ]=\\frac{c}{2nk}\\sqrt{\\frac{\\epsilon}{\\mu}}{\\Im [{\\bf e}_{i} \\cdot {\\bf b}_{i}^{*}]} $ at which the helicity is incident on a unit cross-sectional area of the object, so that (\\ref{wh})  and (\\ref{fis})   give:\n\\be\n{\\cal Q}_{\\hel}=\\frac{{\\cal W}_{\\hel}^{a}+{\\cal W}_{\\hel}^{s}}{|\\flow^{i}|}=\\nonumber\\\\\n-\\frac{4 \\pi }{k}\\frac{\\Re[{\\bf B}_{i}^{*}(\\bf r_0)   \\cdot  {\\bf e}({\\bf s}_{i})]} {\\Im [{\\bf e}_{i } \\cdot {\\bf b}_{i}^{*} ]}= \\frac{4\\pi }{k}\\frac{\\Re[{\\bf E}_{i}^{*}(\\bf r_0)   \\cdot  {\\bf b}({\\bf s}_{i})]}{\\Im [{\\bf e}_{i}  \\cdot {\\bf b}_{i}^{*} ]}. \\label{ot}\n\\ee\nWhere $\\Im [{\\bf E}_{i} \\cdot {\\bf B}_{i}^{*}]=\\Im [{\\bf e}_{i} \\cdot {\\bf b}_{i}^{*}]=2k \\sqrt{ \\mu\/ \\epsilon} {\\hel^{i}}$ .   $ {\\hel^{i}}$ being  the helicity of the incident wave.    For this field  $\\hel^{i}=(n\/c)\\flow^{i} \\cdot {\\bf s}_{i}$.\n\nEq.(\\ref{ot})  is the {\\it optical theorem} for the {\\it helicity cross-section}. Likewise,  the {\\it  absorption} and {\\it scattering helicity cross-sections} ${\\cal Q}_{\\hel}^{a}$ and ${\\cal Q}_{\\hel}^{s}$ are  introduced as:\n\\be\n{\\cal Q}_{\\hel}^{a}=\\frac{{\\cal W}_{\\hel}^{a}}{|\\flow^{i}|} , \\,\\,\\,\\,\\,\n{\\cal Q}_{\\hel}^{s}=\\frac{{\\cal W}_{\\hel}^{s}}{|\\flow^{i}|}. \\label{qs}\n\\ee\nAnd of course ${\\cal Q}_{\\hel}={\\cal Q}_{\\hel}^{a}+{\\cal Q}_{\\hel}^{s}$.\n\nThese laws embody a close analogy with those of the optical theorem for the energy \\cite{born}, and suggest the determination of these magnitudes in scattering experiments.\n\n\\section{Magnetodielectric bi-isotropic dipolar particle}\nLet us consider a  magnetodielectric bi-isotropic particle \\cite{kong}, dipolar in the wide sense, i.e.  if for example we consider it a sphere, its  magnetodielectric response, is characterized by its electric,  magnetic, and magnetoelectric polarizabilities $\\alpha_{e}$,  $\\alpha_{m}$,  $\\alpha_{em}$,  $\\alpha_{me}$, given by the first order Mie coefficients as: $\n\\alpha_{e}=i\\frac{3}{2k^{3}}a_{1}$,  $\\alpha_{m}=i\\frac{3}{2k^{3}}b_{1}$,  $\\alpha_{em}=i\\frac{3}{2k^{3}}c_{1}$,  $\\alpha_{me}=i\\frac{3}{2k^{3}}d_{1}=-\\alpha_{em}$.  $a_{1}$, $b_{1}$ and  $c_{1}=-d_{1}$ standing for the electric, magnetic, and magnetoelectric first Mie coefficients, respectively \\cite{MNV2010,Nieto2011,chanlat}. Notice that such sphere is chiral since   $\\alpha_{em}=-\\alpha_{me}$. \n\nThe electric and magnetic dipole moments, $ {\\bf p}$ and  ${\\bf m}$, induced on the particle by the incident field are:\n\\be\n{\\bf p}=\\alpha_{e} {\\bf E}_i+\\alpha_{em}{\\bf  B}_i, \\,\\,\\,\\,\\,\n{\\bf m}=\\alpha_{me}{\\bf E}_i+\\alpha_{m}{\\bf B}_i. \\label{consti}\n\\ee\nAnd the fields scattered by this particle  in the far-zone read:\n\\begin{equation}\n{\\bf e}({\\bf s})=k^2 \\frac{e^{ikr}}{r} [\\epsilon^{-1}({\\bf s} \\times {\\bf\np})\\times {\\bf s}- \\sqrt{\\frac{\\mu}{\\epsilon}}({\\bf s}\\times\n{\\bf m})], \\label{dipe}\n\\end{equation}\n\\begin{equation}\n{\\bf b}({\\bf s})=k^2 \\frac{e^{ikr}}{r}[\\mu({\\bf s} \\times {\\bf m})\\times\n{\\bf s}+\\sqrt{\\frac{\\mu}{\\epsilon}} ({\\bf s}\\times {\\bf p})]. \\label{dipm}\n\\end{equation}\nIntroducing (\\ref{dipe}) and (\\ref{dipm}) into (\\ref{fss}) and (\\ref{fis}), evaluating the angular integrals, and substituting the results in (\\ref{wh}) we obtain:\n\\be\n {\\cal W}_{\\hel}^{a}+\\frac{8\\pi c k^3}{3 \\epsilon} \\Im[{\\bf p} \\cdot {\\bf m}^{*}]= \\nonumber \\\\\n\\frac{2\\pi c}{n} \\{ \\Re[ ({\\bf p}  \\times {\\bf E}_{i}^{*})    \\cdot{\\bf s}_{i}] + \\Re[({\\bf m}\\times  {\\bf B}_{i}^{*}) \\cdot{\\bf s}_{i} ] \\}. \\label{todip1}\n\\ee\nWhich normalizing to ${|\\flow^{i}|}$, becomes {\\it  the optical theorem for the helicity}  expressed as:\n\\be\n {\\cal Q}_{\\hel}^{a}+\\frac{4\\pi}{3}  \\frac{k^3}{\\hel^{i}} \\sqrt{\\frac{\\mu}{\\epsilon} } \\Im[{\\bf p} \\cdot {\\bf m}^{*}]= \\nonumber \\\\\n\\frac{\\pi }{\\hel^{i}} \\{ \\Re[({\\bf p} \\times {\\bf E}_{i}^{*}) \\cdot{\\bf s}_{i} ]+ \\Re[({\\bf m}  \\times {\\bf B}_{i}^{*})    \\cdot{\\bf s}_{i}] \\}. \\label{todip}\n\\ee\n\nThe second term of  the left side of  (\\ref {todip}) is the \"total scattered helicity cross section\"  or  helicity  scattering-cross section defined above, [cf. (\\ref{qs})],\n\\be\n{\\cal Q}_{\\hel}^{s}=\\frac{4\\pi}{3}  \\frac{k^3}{\\hel^{i}} \\sqrt{\\frac{\\mu}{\\epsilon} } \\Im[{\\bf p} \\cdot {\\bf m}^{*}],   \\label{qs1}\n\\ee\nassociated to the rate of helicity excitation;  as such it accounts for optical rotation effects like e.g. {\\it circular dichroism}  \\cite{schellmann,barron1}. \n\nOn the other hand, the right side of  (\\ref {todip1}) or (\\ref {todip}) is proportional to the projection on ${\\bf s}_i$ of the extinction optical torque  $\\bm \\Gamma$ felt by the particle \\cite{MNV2015}:\n\\be\n{\\bm \\Gamma}=\n\\frac{1 }{2}  \\Re \\{[ {\\bf p}\\times {\\bf E}_{i}^{*} ] + [ {\\bf m}\\times{\\bf B}_{i}^{*} ] \\} , \\label{torque}\n\\ee\n exerted by the spin of the incident wave.  Notice also that since $({\\bf p} \\times {\\bf E}_{i}^{*}) \\cdot{\\bf s}_{i}= -(1\/n) {\\bf p}  \\cdot {\\bf B}_{i}^{*}$ and  $({\\bf m} \\times {\\bf B}_{i}^{*}) \\cdot{\\bf s}_{i}= n {\\bf m}  \\cdot {\\bf E}_{i}^{*}$, this term may also be expressed as $\n\\frac{2\\pi c}{\\mu} \\Re \\{ -\\frac{1}{\\epsilon} {\\bf p}  \\cdot {\\bf B}_{i}^{*} +\\mu  {\\bf m}\\cdot  {\\bf E}_{i}^{*}  \\}$, therefore the {\\it conservation of helicity}  (\\ref{todip1}) may be written as\n\\be\n{\\cal W}_{\\hel}^{a}+\\frac{8\\pi c k^3}{3 \\epsilon} \\Im[{\\bf p} \\cdot {\\bf m}^{*}]= \n\\frac{2\\pi c}{\\mu} \\Re \\{ -\\frac{1}{\\epsilon} {\\bf p}  \\cdot {\\bf B}_{i}^{*} +\\mu  {\\bf m}\\cdot  {\\bf E}_{i}^{*}  \\}. \\,\\,\\,\\,\\label{todip4}\n\\ee\n\nThe condition (\\ref{todip4}) must be compatible with the optical theorem for energies \\cite{born}\n\\be\n {\\cal W}^{a}+\\frac{c k^4}{3 n} [\\epsilon^{-1} |{\\bf p}|^2 + \\mu |{\\bf m}|^{2}]= \\nonumber \\\\\n\\frac{\\omega}{2} \\Im[ {\\bf p}  \\cdot {\\bf E}_{i}^{*}   +{\\bf m}\\cdot  {\\bf B}_{i}^{*}] . \\label{top}\n\\ee\n${\\cal W}^{a}$ being the rate of energy absorption, the second term of the left side constituting the total energy scattered by the dipolar object, and the right side representing the energy rate dissipated  from the illuminting field.  In this connection, notice the interesting formal analogy in the two conservation laws (\\ref{todip4}) and (\\ref{top}) where we observe a duality of  ${\\bf E}$  and ${\\bf B}$.  Also  the comparison between the second terms of their respective left sides is intriguing. We shall  discuss these points in the  next section. \n\nNotwithstanding let us remark that for circularly polarized light,  when we add  the two scattering  cross sections, namely, that of helicity: ${\\cal Q}_{\\hel}^{s}$ given by Eq. (\\ref{qs1}), and that of  energy:  ${\\cal Q}=\\frac{c k^4}{3 n}[\\epsilon^{-1} |{\\bf p}|^2 + \\mu |{\\bf m}|^{2}]\/|<{\\bf S}_i>|$, (the denominator is the incident energy flow magnitude), then {\\it  the product}: $[{\\cal Q}_{\\hel}^{s}+{\\cal Q}]{\\cal W}_i$, (where ${\\cal W}_i$ is the  incident illuminating energy density), {\\it represents the  rate  of excitation} of a chiral molecule or particle. An expression  usually derived from  quantum mechanics \\cite{craig} and that here we have obtained on the basis of Maxwell's equations.\n\n\nThe conservation of helicity is generalized to an arbitrary illuminating wavefield, which we express as a decomposition of plane wave components \\cite{mandel, nietolib}: \n\\be\n{\\bf E}^{(i)}({\\bf r})=\\int_{\\cal D} {\\bf e}_{i}({\\bf s}) e^{ik({\\bf s}\\cdot {\\bf r})} d\\Omega, \\nonumber \\\\\n{\\bf B}^{(i)}({\\bf r})=\\int_{\\cal D} {\\bf b}_{i}({\\bf s}) e^{ik({\\bf s}\\cdot {\\bf r})} d\\Omega.  \\label{ang1}\n\\ee\nThe integration being done in the  contour $\\cal D$ that contains both propagating and evanescent waves \\cite{mandel,nietolib},   and to include them  both, ${\\bf s}_i$ in  (\\ref{todip1})  and (\\ref{todip}) must be replaced by  ${\\bf s}_{i}^{*}$, complex conjugated  of  ${\\bf s}_{i}=(s_{i}^{x},\ns_{i}^{y}, s_{i}^{z}) $, where  $s_{i}^{z}= \\sqrt{ 1-(s_{i}^{x 2}+s_{i}^{y 2})}$  if  $ s_{i}^{x 2}+s_{i}^{y 2} \\leq 1$, (propagating components); and   $s_{i}^{z}= i \\sqrt{ (s_{i}^{x 2}+s_{i}^{y 2})-1}$ if   $s_{i}^{x 2}+s_{i}^{y 2} > 1$, (evanescent  components). Then by the same procedure as before and summing up for all plane wave components,  one sees that in Eqs.(\\ref{todip1})-(\\ref{todip4}) now ${\\bf  E}_{i}$ and   ${\\bf  B}_{i}$  must replaced by  ${\\bf  E}^{(i)}$ and  ${\\bf  B}^{(i)}$;  therefore instead of  (\\ref{todip4}) we now obtain the  fundamental  {\\it conservation relation for the helicity}:\n\\be\n {\\cal W}_{\\hel}^{a}+\\frac{8\\pi c k^3}{3 \\epsilon} \\Im[{\\bf p} \\cdot {\\bf m}^{*}]= \\nonumber \\\\\n2\\pi c \\Re \\{ -\\frac{1}{n^2} {\\bf p}  \\cdot {\\bf B}^{(i) *} + {\\bf m}\\cdot  {\\bf E}^{(i) *}  \\}. \\label{todip3}\n\\ee\nWe remark that in this general case the incident fields in the right side of  (\\ref{top}) should also be ${\\bf E}^{(i)}$ and  $ {\\bf B}^{(i)}$.\n\nEquations (\\ref{todip1}),  (\\ref{todip}) or (\\ref{todip4}), as well as Eq. (\\ref{todip3}), express the extinction of helicity from the incident field by interaction with the dipolar particle. The right side is the helicity dissipated by the dipole from the illuminating wave, and plays for this magnitude a role analogous to that of  $\\frac{\\omega}{2} \\Im[{\\bf p} \\cdot  {\\bf E}^{(i) *} +  {\\bf m}    \\cdot {\\bf B}^{(i)  *}]  $ for the dissipated energy. As such, the term $2\\pi c \\Re \\{ -\\frac{1}{n^2} {\\bf p}  \\cdot {\\bf B}^{(i) *} + {\\bf m}\\cdot  {\\bf E}^{(i) *}  \\}$  has a potential for determining both dissipated and radiated, or scattered, helicity by a bi-isotropic dipolar particle, (e.g. in particular a chiral one) in an arbitrary, homogeneous or inhomogeneous, embedding medium. Also in FRET observations on transmission of energy and helicity between chiral molecules, and in its consequences for the torque exerted on each other \\cite{muka,MNV2015}. In addition we shall see below that Eq.(\\ref{todip3}) constitutes the basis for introducing  a new dissymmetry factor in circular dichroism and enantiomeric discrimination. Hence this new law  gives rise to  avenues worthy of further research.\n\n\\section{Consequences for the polarizabilities}\nIt will be  is useful to consider a Cartesian framework where  the elliptically polarized incident plane wave  has ${\\bf s}_i$ along $OZ$, expressing  its electric vector in an helicity basis ${\\bm \\epsilon}^{\\pm}=(1\/\\sqrt{2})(1, \\pm i, 0)$ as the sum of a left-hand (LCP)  and a right-hand  (RCP) circularly polarized plane wave,  so that ${\\bf e}_{i}=( e_{i x},e_{i y},0)=e_{i}^{+}{\\bm \\epsilon}^{+}+ e_{i}^{-}{\\bm \\epsilon}^{-}$ and  ${\\bf  b}_{i}= ( b_{i x},b_{i y},0)=n ( -e_{i y},e_{i x},0)=b_{i}^{+}{\\bm \\epsilon}^{+}+ b_{i}^{-}{\\bm \\epsilon}^{-}=-ni(e_{i}^{+}{\\bm \\epsilon}^{+}- e_{i}^{-}{\\bm \\epsilon}^{-})$. The upper and lower sign of  $\\pm$ standing for LCP (+) and RCP (-), respectively. In this representation, the incident helicity density  reads:    $ {\\hel^{i}}=(\\epsilon\/k)\\Im[e_{i x}^{*}e_{i y}]= (\\epsilon\/2k) S_{3}=(\\epsilon\/2k)[|e_{i}^{+}|^2-|e_{i}^{-}|^2]$, namely, it is the difference between the LCP and RCP intensities of the field.   $S_3 = 2 \\Im[e_{i x}^{*}e_{i y}]= |e_{i}^{+}|^2-|e_{i}^{-}|^2$ is the 4th Stokes parameter \\cite{marston1,born}.\n\nUsing Eq.(\\ref{consti})  in the above geometry, we obtain from the helicity conservation theorem (\\ref{todip1}):\n\\be\n{\\cal W}_{\\hel}^{a}+\\frac{8\\pi c k^3}{3 \\epsilon}\\{ \\Im[\\alpha_{e}^{*}   \\alpha_{me} + n^2 \\alpha_{m} \\alpha_{em}^{*}] |e_{i}|^2  \\nonumber \\\\\n -2k \\sqrt{\\frac{\\mu}{\\epsilon}} \\Re[\\alpha_{e}^{*}   \\alpha_{m} -\\alpha_{me} \\alpha_{em}^{*}]{\\hel^{i}}\\}=\n    \\nonumber \\\\\n\\frac{2\\pi c}{n}\\{n  [\\alpha_{em}^{R} -  \\alpha_{me}^{R}] |e_i|^2 - 2 (\\alpha_{e}^{I} +n^2  \\alpha_{m}^{I})  \\frac{k}{\\epsilon} {\\hel^{i}}\\}  . \\label{todip2}\n\\ee\nThe superscripts $R$ and $I$ denote the real and imaginary parts of the polarizabilities, respectively.  $|e_{i}|^2= | e_{i x}|^2+|e_{i y}|^2=\\frac{8\\pi}{c}\\sqrt{\\frac{\\mu}{\\epsilon}} <S>=\\frac{8\\pi}{\\epsilon} <w>$. $<S>$ and $<w>$ representing the incident field  time-averaged Poynting vector magnitude  and electromagnetic  energy density, respectively. $<w>=<w_e>+<w_m>$ .  $<w_e>=(\\epsilon\/16 \\pi)|{\\bf E}_{i}|^{2}$, $<w_m>=(1\/16 \\pi  \\mu)|{\\bf B}_{i}|^{2}$.  \n\nIn addition, in this reference frame,  the extinction torque (\\ref{torque})  is: ${\\bm \\Gamma}= (0,0,\\Gamma)=\\Gamma {\\bf s}_{i}$ so that the right side of Eq.(\\ref{todip1}) obviously is $(4\\pi c\/n)\\Gamma$ which is given by the right side of Eq.(\\ref{todip2}), \n\nOn the other hand, we should recall that the optical theorem for energies, Eq.(\\ref{top}), leads to\n\\be\n{\\cal W}^{a}+\\frac{2 k^3}{3 }\\{[\\epsilon^{-1} (|\\alpha_{e}|^{2}    \n+ n^{2}|\\alpha_{em}|^2) + \\mu (|\\alpha_{me}|^{2}   \\nonumber \\,\\,\\,\\,\\,\\, \\\\\n+ n^{2}|\\alpha_{m}|^2)]|e_{i}|^2  \n -4k \\sqrt{\\frac{\\mu}{\\epsilon}}\\Im [ \\epsilon^{-1} \\alpha_{em}^{*}   \\alpha_{e}  + \\mu \\alpha_{me} \\alpha_{m}^{*}]{\\hel^{i}}\\}=\n    \\nonumber \\\\\n2k \\sqrt{\\frac{\\mu}{\\epsilon}}(\\alpha_{me}^{R} -  \\alpha_{em}^{R}) {\\hel^{i}}+ (\\alpha_{e}^{I} +n^2  \\alpha_{m}^{I}) |e_i|^2   .  \\,\\,\\,\\,\\,\\,\\, \\,\\,\\,\\,\\, \\label{top2}\n\\ee\n Considering from now on absence of absorption from electric currents, $  {\\cal W}_{\\hel}^{a}=0$ and  ${\\cal W}^{a}=0$. The compatibility of  the new equation (\\ref{todip2})  with (\\ref{top2}) implies that their combination yields\n\\be\n\\sqrt{\\frac{\\mu}{\\epsilon}}(\\alpha_{me}^{R} -  \\alpha_{em}^{R})(\\frac{4k^{2}}{\\epsilon^{2}}{\\hel^{i}}^{2}-|e_{i}|^{4})=\\nonumber \\\\\n\\frac{4 k^{3}}{3 }\\{(|\\epsilon^{-1}\\alpha_{e}-\\mu \\alpha_{m}|^{2}+\\frac{\\mu}{\\epsilon}|\\alpha_{em}+\\alpha_{me}|^{2})\\frac{k}{\\epsilon}\\hel^{i}|e_{i}|^{2} - \\nonumber \\\\\n\\sqrt{\\frac{\\mu}{\\epsilon}}\\Im[  \\alpha_{em}^{*}(\\frac{4 k^{2}}{\\epsilon^{2}}{\\hel^{i}}^{2} \\,\\,\\frac{\\alpha_{e}}{\\epsilon}-|e_{i}|^{4 } \\mu \\alpha_{m})+   \\nonumber\\\\\n \\alpha_{me}^{*}(|e_{i}|^{4} \\,\\, \\frac{\\alpha_{e}}{\\epsilon}-\\frac{4k^{2}}{\\epsilon^{2}}{\\hel^{i}}^{2}  \\mu\\alpha_{m})] \\}. \\,\\, \\, \\label{cond1}\n\\ee\nEq.(\\ref{cond1}) constitutes the constraint between the four polarizabilities $\\alpha_e$, $\\alpha_m$, $\\alpha_{em}$ and  $\\alpha_{me}$ imposed by  the conservation of the two quantities: energy and helicity.\n\nIn particular,  if the particle is not bi-isotropic, ($\\alpha_{em}= \\alpha_{me}=0$) and $\\hel^{i}\\neq 0$, the conservation of both helicity and energy, Eq.(\\ref{cond1}), states that the particle is dual \\cite{molina1,molina2},  i.e.  $\\epsilon^{-1}\\alpha_{e}=\\mu \\alpha_{m}$ and thus  fulfills the well-known {\\it first Kerker condition} (K1) \\cite{kerk,greffin} according to which it produces  zero angular distribution of scattered intensity in the backscattering direction. However, as seen next, this  also occurs for chiral particles.\n\nSeveral other cases are in order, as shown next.\n\n\\subsection{Circular polarization of the incident wave}\nIn this case  $e_{ix}=e$ and  $e_{iy}=\\pm i e$, $e$ being real, depending on whether the incident wave is LCP or RCP. Then   $|e_{i}|^2=2e^2$, and   $|e_{i}|^{4}-\\frac{4k^{2}}{\\epsilon^{2}}{\\hel^{i}}^{2}  =0$, i.e.   $\\pm |e_{i}|^{2}= \\frac{2k}{\\epsilon}{\\hel^{i}}$ with  the sign + and - applying when  the wave is LCP and RCP, respectively. Then (\\ref{cond1}) becomes\n\\be\n\\pm \\{(|\\epsilon^{-1}\\alpha_{e}-\\mu \\alpha_{m}|^{2}+\\frac{\\mu}{\\epsilon}|\\alpha_{em}+\\alpha_{me}|^{2}) \\} =\\nonumber \\\\\n2\\sqrt{\\frac{\\mu}{\\epsilon}}\\Im[ ( \\alpha_{em}^{*} +  \\alpha_{me}^{*}) (\\frac{\\alpha_{e}}{\\epsilon}- \\mu \\alpha_{m})]. \\label{condcircul}\n\\ee\nWhich yields:\n\\be\n(\\epsilon^{-1}\\alpha_{e}-\\mu \\alpha_{m}) \\pm i\\frac{\\mu}{\\epsilon} (\\alpha_{em}+\\alpha_{me})=0. \\label{condcircul1}\n\\ee\n If the particle is chiral, then \\cite{tang1}   $\\alpha_{em}= - \\alpha_{me}$ and either (\\ref{condcircul}) or  (\\ref{condcircul1}) imply that  $\\epsilon^{-1}\\alpha_{e}=\\mu \\alpha_{m}$, i.e.  the particle is dual and thus holds K1.\n\nReciprocally, if the particle is such that $\\epsilon^{-1}\\alpha_{e}=\\mu \\alpha_{m}$, then (\\ref{condcircul}) or (\\ref{condcircul1}) imply that  $\\alpha_{em}= - \\alpha_{me}$, namely the particle is chiral and hence dual.\n\n\nIn addition, in this case  ${\\bf p}=\\pm i n {\\bf m}$ and  ${\\bf b}({\\bf s})=\\mp n i{\\bf e}({\\bf s})$ [cf.  Eqs.(\\ref{consti}),   (\\ref{dipe}) and (\\ref{dipm})] i.e. the scattered field is circularly polarized (CP) with respect to the Cartesian system of orthogonal  axes defined by the unit vectors: $({\\bm \\epsilon}_{\\perp},{\\bm \\epsilon}_{\\parallel}, {\\bf s})$, (see Fig.1). ${\\bm \\epsilon}_{\\perp} $ and  ${\\bm \\epsilon}_{\\parallel}$ being respectively  perpendicular and parallel to the scattering plane $OPQ$.  I.e.: ${\\bf e}({\\bf s})=({\\bf e}({\\bf s}) \\cdot {\\bm \\epsilon}_{\\perp})(1,\\pm i, 0)$ and ${\\bf b}({\\bf s})=( n {\\bf e}({\\bf s}) \\cdot {\\bm \\epsilon}_{\\perp})  (\\mp i, 1,0)$.  The helicity density of the scattered field being proportional to its intensity density: ${\\hel^{s}}=\\pm \\frac{\\epsilon}{2k} |{\\bf e}({\\bf s})|^2$; and the flow of helicity density  (spin) being proportional to that of energy density (Poynting vector).  Then, in this case the optical theorem for the helicity (\\ref{todip1}) and that for the energy (\\ref{top})  are equivalent.  In fact, it is known \\cite{cameron1} that for circularly polarized waves there is a mapping of the helicity to the energy. Thus  both conservation laws coincide when both the  incident wave and the scattered field (like under K1)  have circular polarization.\n\n\nWe then conclude that  {\\it the necessary and sufficient condition for a non-absorbing bi-isotropic particle to be chiral, $ \\alpha_{em}=  - \\alpha_{me}$, is that $\\epsilon^{-1}\\alpha_{e}=\\mu \\alpha_{m}$, i.e. it is dual. Its scattering by a circularly polarized plane wave, which must satisfy both energy and  helicity conservation, produces a circularly polarized scattered field with zero  differential scattering cross section  in the backscattering direction. Namely, the particle satisfies the first Kerker condition}.\n\nAs stated before, if $\\alpha_{em}= \\alpha_{me}=0$, the conservation of helicity and energy also implies duality, namely K1 and hence zero backscattering. Even though, of course, circular polarization of the scattered field will occur when the incident plane wave is circularly polarized.\n\n\\section{Significance for helicity emission\/absorption from dipolar objects.  Helicity transfer  in F.R.E.T.}\nThe new  law (\\ref{todip3}) stating the conservation of electromagnetic helicity shows us how to determine the rate of helicity dissipation from a dipolar particle or molecule in an arbitrary environment, whether homogeneous or inhomogeneous:\n\\be\n \\frac{d{\\cal W}_{\\hel}}{dt}=\n2\\pi c \\Re \\{ -\\frac{1}{n^2} {\\bf p}  \\cdot {\\bf B}^{*}({\\bf r}_0) + {\\bf m}\\cdot  {\\bf E}^{*}({\\bf r}_0)  \\}. \\label{todip5}\n\\ee\n${\\bf r}_0$ being a point of the  particle, (which is  usually convenient to consider its center if it is e.g. a sphere).  We write: ${\\bf E}({\\bf r}_0)= {\\bf E}_{i}({\\bf r}_0)+{\\bf E }_{s}({\\bf r}_0)$,     ${\\bf B}({\\bf r}_0)= {\\bf B}_{i}({\\bf r}_0)+{\\bf B }_{s}({\\bf r}_0)$. Where now the index $i$  denote the dipole field that the particle would emit in isolation; whereas $s$ stands for the field resulting from multiple scattering with  surrounding particles  or  near objects.\n\nEq.(\\ref{todip5})  constitutes the starting point  for future studies on  the {\\it helicity decay rate}  $\\gamma_{\\hel}$, either between the dipole and arbitrary near bodies, or between dipolar objects; in particular in the phenomenon of {\\it fluorescence resonant energy transfer} (FRET) between molecules. For the latter, one does not have to be limited to the transfer of  energy, but likewise it is possible to analyse the {\\it  flow of helicity between nearby particles} with  interesting effects to disclose from the additional degrees of freedom introduced by the helicity and its flow. Such technique based on Eq.(\\ref{todip5}), [recall also (\\ref{todip1})], that we shall call {\\it resonant helicity transfer} (RHELT), or {\\it fluorescence resonant helicity transfer}  (FRHELT) when fluorescence is involved, will use the concept that we herewith coin as {\\it helicity transfer rate} $\\gamma_{\\hel}^{DA}$ between donor  $D$ and acceptor $A$, also taking their possible  bi-isotropy  (and chirality, in particular) into account, which in analogy with energy transfer  (see e.g. \\cite{novotny}), we express by\n\\be\n\\frac{\\gamma_{\\hel}^{DA}}{\\gamma_{\\hel}^{0}}=\\frac{{\\cal W}_{\\hel}^{DA}}{{\\cal W}_{\\hel}^{0}}.\n\\ee\n$\\gamma_{\\hel}^{0}$ and ${\\cal W}_{\\hel}^{0}=\\frac{4\\pi}{3}  \\frac{k^3}{\\hel^{i}} \\sqrt{\\frac{\\mu}{\\epsilon} } \\Im[{\\bf p} \\cdot {\\bf m}^{*}]$,  [cf. Eq.(\\ref{qs1})], representing the helicity decay rate  and  helicity yield from the donor in absence of acceptor, respectively.  And \n\\be\n{\\cal W}_{\\hel}^{DA}=2\\pi c \\Re \\{ -\\frac{1}{n^2} {\\bf p}_{A}  \\cdot {\\bf B}_{D}^{*}({\\bf r}_A) + {\\bf m}_{A}\\cdot  {\\bf E}_{D}^{*}({\\bf r}_A) \n\\ee\nThe subindex $D$ in the fields means that they are generated by the donor, whereas the subindices $A$ stand for the excited dipole moments and position points in the acceptor.\n\nIncreasingly investigated   structures with electric and magnetic dipoles \\cite{Nieto2011,luki,peng,kiv}, and their mutual interaction  \\cite{MNV2010,greffin,kiv,evly},  make (\\ref{todip5}) of appealing and intriguing consequences in such future studies. The details on these quantities will be the subject of a future study. \n\n\\section{Effects on helicity enantiomeric discrimination by circular dichroism}\n\nThe weakness of the signal in enantiomeric discrimination is well known \\cite{tang1,tang2,tang3,choi}. Proposals to enhance it by acting on the helicity of the illuminating wave have been  studied, \\cite{tang1,tang2}. Such enhancement comes from the use of the right side of  the optical theorem of energy conservation, Eq.(\\ref{top}), which for a chiral molecule or dipolar object, leads to the so-called dissymmetry factor: \\cite{barron1,schellmann,tang1}\n\\be\ng=2\\frac{{\\cal W}^{+}-{\\cal W}^{-}}{{\\cal W}^{+}+{\\cal W}^{-}}. \\label{gfactor1}\n\\ee\nWhere ${\\cal W}^{\\pm}$ is the energy  excitation  of the object, (i.e. the molecule or particle), which  equals the dissipated energy from the illuminating wave:  $\\frac{\\omega}{2} \\Im[{\\bf p} \\cdot  {\\bf E}_{i}^{*} +  {\\bf m}    \\cdot {\\bf B}_{i}^{*}]  $.  Considering for example the pair of fields: $\\pm {\\bf \\cal E}_{i}({\\bf r},t)$ and  ${\\bf \\cal H}_{i}({\\bf r},t)$:  ${\\bf \\cal E}_{i}({\\bf r},t)=\\Re [{\\bf E}_{i}({\\bf r}) \\exp(-i\\omega t)]$ and  ${\\bf \\cal H}_{i}({\\bf r},t)=\\Re [{\\bf H}_{i}({\\bf r}) \\exp(-i \\omega t)]$, whose respective helicites are: ${\\hel}^{+}$ and  ${\\hel}^{-}$, with ${\\hel}^{+}={\\hel}= -{\\hel}^{-}$,   this leads to:\n\\be\ng=- \\frac{2 k n}{\\pi}\\frac{\\alpha_{em}^{R}}{\\alpha_{e}^{I} + n^{2} \\alpha_{m}^{I}} \\frac{{\\hel}}{<w>}. \\label{gfactor2}\n\\ee\nThe term $n^2\\alpha_{m}^{I}$  being negligible in cases in which  $|\\alpha_{m}|<<|\\alpha_{e}|$ \\cite{schellmann,tang1}. It should be remarked, however, that this is not always the case, see \\cite{choi,Nieto2011}. Also, if the illumination is with a CP plane wave, we have seen above  that if the particle is dipolar in the wide sense and chiral then $ \\alpha_e= n^2 \\alpha_m$, wich would pose a difficulty to neglect $\\alpha_m$ while retaining $\\alpha_e$. Thus in this case one should replace the denominator of  (\\ref{gfactor2}) by $2 \\pi \\alpha_e  <w>$ . \n\nIt is known that the quantity $g$  may be small for the usually employed circularly polarized illumination,  for which $\\pm {\\bf E}_{i}$ represents the two polarization states: LCP and RCP, respectively. Then  ${\\hel}^{\\pm}=\\pm (4\\pi n\/k) <w> $    and, if one can neglect the    $ \\alpha_{m}^{I}$ term,  $g$ becomes the classical dissymmetry factor:  $- 8 n^2 \\alpha_{em}^{R}\/ \\alpha_{e}^{I}$, which may be as small as $10^{-2}$ or $10^{-6}$, depending on whether there is  electronic or vibrational excitation. By contrast  the  factor  ${\\hel}\/<w>$  may be large enough  to overcome the above limitation of $g$ by devising differently spatially structured illumination \\cite{tang1}. \n\nNow on account of the optical theorem of helicity established here, Eq.(\\ref{todip3}), we suggest  to use the rate of helicity excitation of the object, [cf. the right side of  (\\ref{todip3})]:\n\\be\n -{\\cal W}_{\\hel}'= 2\\pi c \\Re \\{ -\\frac{1}{n^2} {\\bf p}  \\cdot {\\bf B}_{i}^{*} + {\\bf m}\\cdot  {\\bf E}_{i}^{*}  \\}=  \\nonumber \\\\\n -32 \\pi^2 c \\alpha_{em}^{R}<w> + 2 \\pi c\\{(\\alpha_{m}^{R}- \\frac{ \\alpha_{e}^{R}}{n^2} ) \\leh +\\nonumber \\\\\n2k \\sqrt{\\frac{\\mu}{\\epsilon} } (\\alpha_{m}^{I}+\\frac{\\alpha_{e}^{I}}{n^2}) \\hel \\}. \\label{wgfactor}\n\\ee\n Where $\\pm  \\leh= \\Re ( \\pm {\\bf E}_{i}\\cdot {\\bf B}_{i}^{*})$.\n\nHence, we propose a new method with  measurements based on a  novel  {\\it dissymmetry factor} $g_{\\hel}$ that we introduce as: \n\\be\ng_{\\hel}=2\\frac{(-{\\cal W}_{\\hel}'^{+}) - (-{\\cal W}_{\\hel}'^{-})}{(-{\\cal W}_{\\hel}'^{+}) + (-{\\cal W}_{\\hel}'^{-})}. \\label{dfactor1}\n\\ee\nWhich, rather than (\\ref{gfactor2}), yields  [cf.  (\\ref{wgfactor})]:\n\\be\ng_{\\hel}=-\\frac{(\\alpha_{m}^{R}- \\frac{ \\alpha_{e}^{R}}{n^2} ) \\leh +\n2k \\sqrt{\\frac{\\mu}{\\epsilon} } (\\alpha_{m}^{I}+\\frac{\\alpha_{e}^{I}}{n^2}) \\hel }{8 \\pi \\alpha_{em}^{R}<w>}.  \\label{ghfactor}\n\\ee\n{\\it In contrast with $g$}, Eq.(\\ref{gfactor2}), {\\it  that has    the usually small factor $\\alpha_{em}^{R}\/(\\alpha_{e}^{I} + n^{2} \\alpha_{m}^{I})$,  $g_{\\hel}$ is a large quantity since  it contains a term with the inverse factor:  $(\\alpha_{e}^{I} + n^{2} \\alpha_{m}^{I})\/\\alpha_{em}^{R}$.}\n\nIn common situations in which  $ \\leh=0$,  Eq.(\\ref{ghfactor}) becomes:\n\\be\ng_{\\hel}=-\\frac{k}{4\\pi n \\epsilon }\\frac{\\alpha_{e}^{I}+ n^2{\\alpha_{m}^{I}}}{\\alpha_{em}^{R}}\n\\frac{ \\hel }{<w>}.  \\label{ghfactornl}\n\\ee\n(Again the term $n^2\\alpha_{m}^{I}$ may be  negligible only  in cases in which  $|\\alpha_{m}|<<|\\alpha_{e}|$ ).\n\nIn fact for a  circularly polarized plane wave,  $ \\leh=0$ and  Eq.(\\ref{ghfactor}) becomes:\n\\be\ng_{\\hel}=-\\frac{k}{4\\pi n  }\\frac{\\alpha_{e}^{I}+ n^2{\\alpha_{m}^{I}}}{\\alpha_{em}^{R}}\\frac{ \\hel }{<w>} = \n-\\frac{\\alpha_{e}^{I}+ n^2{\\alpha_{m}^{I}}}{\\alpha_{em}^{R}}. \\label{ghfactornl}\n\\ee\nThus being of the order of the inverse of the usual dissymmetry factor, i.e. of  $g^{-1}$. In addition, the helicity factor  $\\frac{ \\hel }{<w>}$ common to Eq.(\\ref {gfactor2}) and (\\ref{ghfactornl}),  enhances  not only $g$ with \"superchiral light\" as shown in \\cite{tang1}, but then also  $g_{\\hel}$.\n\nBecause of  (\\ref{who}),  (\\ref{todip1}) and (\\ref {todip3}) one may equally determine  a helicity dissymmetry factor   on using in (\\ref{dfactor1}) the quantity ${\\cal W}_{\\hel}^{s}=(8\\pi c k^3\/3 \\epsilon) \\Im[{\\bf p} \\cdot {\\bf m}^{*}]$ instead of $-{\\cal W}_{\\hel}'$. For particles with small imaginary parts of the polarizabilities, both dissymmetry factors are equivalent, and specially when one employs  CP illumination, both   $-{\\cal W}_{\\hel}'$  and ${\\cal W}_{\\hel}^{s}$ lead to: $g_{\\hel}=\\frac{2 }{n} \\frac{\\alpha_{e}^{I}}{\\alpha_{em}^{R}}$. \n\nTo take advantage of $g_{\\hel}$,  detection should be carried out  by measuring the total scattered or radiated helicity,  rather than the energy excitation, through an experiment that involves determination of Stokes parameters, including $S_3$, when LCP is used. This, according to the above helicity optical theorem,  equals the  helicity dissipated from the incident field. In the case of an incident LCP plane wave, one may also perform the measurement by  determining the projection of the extinction optical torque on the particle according to [cf. Eqs.(\\ref{todip1}), (\\ref{todip}) and (\\ref{torque})]\n\\be  -{\\cal W}_{\\hel}'=\\frac{2\\pi c}{n} \\{ \\Re[( {\\bf p}  \\times {\\bf e}_{i}^{*}   ) \\cdot{\\bf s}_{i}] + \\Re[({\\bf m}\\times  {\\bf b}_{i}^{*}) \\cdot{\\bf s}_{i} \\}]= \\nonumber \\\\\n\\frac{4 \\pi c }{n}\\{n \\alpha_{em}^{R}|e|^2 - (  \\alpha_{e}^{I} + n^2 \\alpha_{m}^{I})\\frac{k}{\\epsilon} \\hel \\} ,  \n\\ee\neither directly through an optical force experiment, or equivalently again by  the optical theorem (\\ref{todip}), to measure it through a determination of the total scattered helicity.\n\nOne should  remark that determining $ g_{\\hel}$ is  an approach different to ellipsometric chiroptical spectroscopy, (see e.g. \\cite{choi}), in which one does not exclusively employ helicity flows ${\\cal W}_{\\hel}$ as proposed here, but instead one uses  the ratio \n${\\cal W}_{\\hel}^{s}\/{\\cal W}^{s} \\sim {\\cal Q}_{\\hel}^{s}\/{\\cal Q}^{s}$ of the scattering helicity cross section, Eq. (\\ref {qs1}),\n\n to the total scattering cross section, [or scattered energy flow (\\ref{ot}), normalized to the incident one: $ |<S^{i}>|=\\sqrt{\\frac{\\epsilon}{\\mu}}(c|e_i|^2\/8\\pi)$], for two LCP and RCP waves: \n\\be  {\\cal Q}^s= \\sqrt{\\frac{\\epsilon}{\\mu}}(c\\|e_i|^2\/8\\pi) \\frac{c k^4}{3 n} [\\epsilon^{-1} |{\\bf p}|^2 + \\mu |{\\bf m}|^{2}] ,  \\ee\n\n\nHowever the  signal obtained by this latter procedure is weaker than the one proposed here based on the factor of Eq.(\\ref{dfactor1}).\n\n\\section{Conclusions}\nThe optical theorem that expresses the conservation of electromagnetic helicity has been put forward, from which one can define a  helicity cross section for extinction, scattering and absorption. This equation suggests intriguing consequences both  for FRET and circular dichroism. As for the former a new technique: RHELT,  (or FRHELT if emission is due to fluorescence), based on the helicity transfer rate, is proposed, whereas for the  latter we suggest a new  procedure  employing ellipsometric measurements,  from   which a dissymmetry factor based on the emitted helicity rate, here introduced and larger than the standard one, yields  greater sensitivity.\n\nIn the emerging field of silicon photonics with magnetodielectric structures, the behavior of $g$ and $g_{\\hel}$ drastically changes since then the resonant $\\alpha_{m}^{I}$ associated to the excitation of the  magnetic dipole  \\cite{Nieto2011,luki,greffin,peng,kiv,evly}, would dominate   $\\alpha_{e}^{I}$ and  $\\alpha_{em}^{R}$; thus $g$ may  diminish while $g_{\\hel}$  would be enhanced. Further advances in the study of  magnetodielectric chiral objects should manifest such effects.\n\n\\section{Acknowledgments}\nWork  supported by the MINECO through grants FIS2012-36113-C03-03 and FIS2014-55563-REDC. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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\\end{list}}\n\n\\newlength{\\myStackWidth}\n\n\n\\newlength{\\aligntop}\n\\setlength{\\aligntop}{-0.53em}\n\\newlength{\\alignbot}\n\\setlength{\\alignbot}{-0.85\\baselineskip}\n\\addtolength{\\alignbot}{-0.1em} \\makeatletter\n\\renewenvironment{align}{%\n  \\vspace{\\aligntop}\n  \\start@align\\@ne\\st@rredfalse\\m@ne\n}{%\n  \\math@cr \\black@\\totwidth@\n  \\egroup\n  \\ifingather@\n    \\restorealignstate@\n    \\egroup\n    \\nonumber\n    \\ifnum0=`{\\fi\\iffalse}\\fi\n  \\else\n    %\n  \\fi\n  \\ignorespacesafterend%\n  \\vspace{\\alignbot}\\par\\noindent\n} \\makeatother\n\n\n\\IEEEoverridecommandlockouts\n\n\n\n\n\\topmargin = 0em\n\n\\begin{document}\n\n\\title{Integer-Forcing Message Recovering in Interference Channels}\n\n\\author{\\authorblockN{Seyed Mohammad Azimi-Abarghouyi, Mohsen Hejazi, Behrooz Makki, Masoumeh Nasiri-Kenari, \\emph{Senior Member, IEEE}, and Tommy Svensson, \\emph{Senior Member, IEEE}}\\\\\n   \\thanks{\n  S.M. Azimi-Abarghouyi and M. Nasiri-Kenari are with Electrical Engineering Department, Sharif University of Technology, Tehran, Iran. Emails:\n{\\textit{azimi$\\_$sm@ee.sharif.edu, mnasiri@sharif.edu}}. M. Hejazi is with Electrical and Computer Engineering Department, University of Kashan, Kashan, Iran. Email: {\\textit{mhejazi@ee.sharif.edu}}.\nB. Makki and T. Svensson are with Department of Signals and Systems, Chalmers University\nof Technology, Gothenburg, Sweden, Emails: {\\textit{behrooz.makki@chalmers.se,\ntommy.svensson@chalmers.se}}. This work has been supported in part by VR research link project \"Green Communication via Multi-relaying\".} }\n\n\n\\maketitle\n\n\n\\begin{abstract}\nIn this paper, we propose a scheme referred to as integer-forcing message recovering (IFMR) to enable receivers to recover their desirable messages in interference channels. Compared to the state-of-the-art integer-forcing linear receiver (IFLR), our proposed IFMR approach needs to decode considerably less number of messages. In our method, each receiver recovers independent linear integer combinations of the desirable messages each from two independent equations. We propose an efficient algorithm to sequentially find the equations and integer combinations with maximum rates. We evaluate the performance of our scheme and compare the results with the minimum mean-square error (MMSE) and zero-forcing (ZF), as well as the IFLR schemes. The results indicate that our IFMR scheme outperforms the MMSE and ZF schemes, in terms of achievable rate, considerably. Also, compared to IFLR, the IFMR scheme achieves slightly less rates in moderate signal-to-noise ratios, with significantly less implementation complexity.\n\n\\end{abstract}\n\n\\vspace{-10pt}\n\\section{Introduction}\nVarious wireless communication setups can be modeled as interference channels consisting of multiple coexisting transmitter-receiver pairs. To reduce the interference in such systems, there are mainly two categories of receiver structures [1]-[2]. The first category are maximum likelihood (ML)-based receivers achieving the highest possible rates [1]. However, the ML-based estimation may be practically infeasible, as the size of the search space grows exponentially with the codeword length, the number of antennas, and the number of transmitters [1]. The second category are linear receivers (LR) which have low complexity in filtering the received signals through a linear structure for decoding. LRs are often proposed based on the criteria of zero-forcing (ZF) and minimum mean-square error (MMSE) [1]-[4]. \n\nRecently, a novel linear receiver referred to as integer-forcing linear receiver (IFLR) has been designed to simultaneously recover the transmitted messages in point-to-point multiple-input multiple-output (MIMO) systems [5]. This idea was derived from the compute-and-forward scheme [6]. Based on noisy linear combinations of the transmitted messages, IFLR recovers independent equations of messages through a linear receiver structure. In this way, in contrast to MMSE and ZF schemes, instead of combating, IFLR exploits the interference for a higher throughput. Application of the IFLR scheme in MIMO multi-pair two-way relaying is proposed in [7]. It is shown in [8] and [9] that precoding in IFLR can achieve the full diversity and the capacity of Gaussian MIMO channels up to a gap, respectively. Also, [10] applies successive decoding in IFLR and proves its sum rate optimality.\n\nIFLR recovers all desirable and undesirable transmitted messages by decoding sufficient number of the best independent equations in terms of achievable rate. Hence, considering IFLR in interference networks, the complexity of the lattice decoding and also the best equation selection process grows considerably with the number of transmitters and data streams. The combination of IFLR and interference alignment [11], referred to as integer-forcing interference alignment (IFIA), is proposed in [12] to decode sufficient equations to recover the desirable messages. However, IFIA requires channel state information at the transmitter (CSIT). This is the motivation for our paper in which we design an efficient low-complexity receiver for interference channels with no need for CSIT. \n\nHere, we propose a linear receiver scheme, referred to as integer-forcing message recovering (IFMR), for interference networks. Benefiting from a special equation structure of IFLR, we propose a novel receiver model in which the required number of decodings is limited to twice the number of desirable messages. In our IFMR, independent integer combinations of the desirable messages are recovered in each receiver. Each integer combination, referred to as desirable combined message (DCM), is recovered by decoding two independent equations. Here, with a new formulation, the equations can be optimized such that a DCM is recovered with maximum achievable rate. Despite of its much less complexity, we prove that our sequential approach in optimizing DCMs achieves the same rate as the optimal approach when we can jointly optimize DCMs (Theorem 1). \n\nInstead of NP hard exhaustive search in optimizing the equations of IFMR, we present a practical and efficient suboptimal algorithm to maximize the achievable rate in polynomial time. The proposed algorithm iterates in three steps, one for the coefficient factors of the two equations and the others for the coefficient vectors of an undesirable combined message (UCM) and DCM. The associated problem with each step is solved in polynomial time. The convergence of the proposed algorithm is also proved (Theorem 3). Hence, our IFMR scheme provides a low-complexity scheme in recovering the desirable messages through a few decodings of near-optimal integer combinations in interference channels.  \n\nOur scheme is different and much less complex compared to the IFLR scheme that uses a large number of equations for message recovery. Particularly, the complexity of IFMR does not depend on the number of transmitters and the data streams of the interfering transmitters. Also, as opposed to IFIA, our scheme requires no CSIT. \n\nWe evaluate the performance of our scheme and compare the results with the minimum mean-square error (MMSE) and zero-forcing (ZF), as well as the IFLR schemes. The results indicate that, in all signal-to-noise ratios (SNRs), our IFMR scheme outperforms the MMSE and ZF schemes, in terms of achievable rate, substantially. Also, the IFMR scheme achieves slightly less rates in moderate SNRs, compared to IFLR, with significantly less implementation complexity. In addition, our proposed algorithm provides a tight lower bound for the results obtained via the NP hard exhaustive search. For instance, consider a three-pair interference channel with single antenna at the transmitters\/receivers. Then, the achievable rate of the exhaustive search is only 1 dB better than our proposed algorithm in 1 bit\/channel use. \n\nThe remainder of this paper is organized as follows. In Section II, the system model and IFLR are briefly described. Section III presents the IFMR scheme. Numerical results are given in Section IV. Finally, Section V concludes this paper.\n\n\\textbf{Notations:} The operators ${(\\mathbf{A})^*}$, $\\text{det}(\\mathbf{A})$, $\\text{Tr}(\\mathbf{A})$, $||\\mathbf{A}||$, and $\\text{span}\\left\\{\\mathbf{A}\\right\\}$ stand for conjugate transpose, determinant, trace, frobenius norm, and the space spanned by the column vectors of matrix $\\mathbf{A}$, respectively. The $\\mathbf{Z}^{n \\times 1}$ and $\\mathbf{R}^{n \\times 1}$ are the $n$ dimensional integer field and $n$ dimensional real field, respectively. Moreover, ${\\rm{lo}}{{\\rm{g}}^ + }\\left( x \\right)$ denotes ${\\rm{max}}\\left\\{ {\\log \\left( x \\right),0} \\right\\}$. The operator $\\succcurlyeq$ refers to the generalized inequality associated with the positive semidefinite cone. Also, ${\\nabla _{{\\mathbf{a}}}}{f}$ represents the partial derivative of function $f$ with respect to vector $\\mathbf{a}$. Finally, $\\mathbf{I}$ and $\\mathbf{1}$ stand for the identity matrix and the vector with all elements equal to one, respectively.\n\n\n\\section{System Model and Integer-Forcing Linear Receiver (IFLR)} \t\n\\subsection{System Model} We consider $K$-pair interference channels where $K$ transmitters are transmitting independent data streams to $K$ receivers simultaneously, as shown in Fig 1. It is assumed that there is no coordination among the transmitters and receivers. We assume no CSIT and, as a result, we do not use beamforming. This is an acceptable assumption in simple setups with no coordinations and central processing units in which channel state information (CSI) feedback and beamforming is infeasible. Incorporating partial CSIT is left for future work. In this system, the $k$-th transmitter and receiver are equipped with $N_{{t_k}}$ and $N_{{r_k}}$ antennas, respectively. The matrix $\\mathbf{H}_{kj}$ denotes the channel matrix from transmitter $k$ to receiver $j$, with dimension ${N_{{r_j}}} \\times {N_{{t_k}}}$. The elements of $\\mathbf{H}_{kj}$ are assumed to be independent identically distributed (IID) Gaussian variables with variance $\\rho _{kj}^2$. We focus on real-valued channels. However, our scheme and results are directly applicable to complex-valued channels via a real-valued decomposition, as in [5]-[6]. Transmitter $k$ exploits a lattice encoder with power constraint $P$ to map $N_{{t_k}}$ message streams $\\mathbf{w}_k$ to a real-valued codeword matrix $\\mathbf{x}_k$ with dimension ${N_{{t_k}}} \\times n$, where $n$ is the codeword length. \n\nAccording to Fig. 1, the received signal at receiver $k$ is given by\n\\begin{eqnarray}\n{\\mathbf{Y}_k} = {\\mathbf{H}_{kk}}{\\mathbf{x}_k} +\\mathop \\sum \\limits_{j = 1,j \\ne k}^K {\\mathbf{H}_{jk}}{\\mathbf{x}_j} + {\\mathbf{n}_k},\n\\end{eqnarray}\nwhere $\\mathbf{n}_k$ is IID additive white Gaussian noise with the variance $\\sigma^2, \\forall k$. \n\\subsection{Integer-Forcing Linear Receiver (IFLR)} Since the objective of our proposed approach is to limit the complexity of the IFLR scheme [5] for interference channels, it is interesting to briefly review this scheme as follows. The readers familiar with the IFLR scheme can skip this part.\n\\begin{figure}[tb!]\n\\centering\n\n\\includegraphics[width =5in]{inferifmr.png}\n\n\\caption{$K$-pair interference channel.}\n\n\\end{figure}\n\nLet us rewrite (1) as\n\\begin{eqnarray}\n{\\mathbf{Y}_k} = {\\mathbf{\\hat H}_k}\\mathbf{X} + {\\mathbf{n}_k},\n\\end{eqnarray}\nwhere ${\\mathbf{\\hat H}_k} \\buildrel \\Delta \\over = \\left[ {{\\mathbf{H}_{1k}}, \\ldots ,{\\mathbf{H}_{Kk}}} \\right]$ and $\\mathbf{X} \\buildrel \\Delta \\over = \\left[ {\\begin{array}{*{20}{c}}\n{{\\mathbf{x}_1^*}},\n \\ldots,\n{{\\mathbf{x}_K^*}}\n\\end{array}} \\right]^*$. Since $\\mathbf{X}$ is of size $L \\buildrel \\Delta \\over = \\mathop \\sum \\limits_{k = 1}^K {N_{{t_k}}}$, the IFLR scheme recovers $L$ independent equations from $\\mathbf{Y}_k$. The $L$ independent equations with equation coefficient vectors (ECVs) $\\mathbf{a}_l^k$, $l=1, \\ldots ,L$, totally shown by matrix $\\mathbf{A}^k \\buildrel \\Delta \\over = \\left[ {\\begin{array}{*{20}{c}}\n{\\mathbf{a}_1^k},\n \\ldots, \n{\\mathbf{a}_L^k}\n\\end{array}} \\right]^*$, are then solved to recover the desirable messages of the receiver $k$. \nQuantizing $\\mathbf{Y}_k$, the answer of the equation with ECV $\\mathbf{a}_l^k$ can be recovered as [6, Eq. (68)]\n\\begin{eqnarray}\n{t_l^k} = \\mathbf{a}{_l^k}^*\\mathbf{X} = Q\\left( {\\mathbf{b}{_l^k}^*{\\mathbf{Y}_k}} \\right),\n\\end{eqnarray}\nwhere $Q(\\cdot)$ denotes lattice equation quantizer, and vector $\\mathbf{b}_l^k$, of length $N_{{r_k}}$, is the projection vector. Also, $\\mathbf{b}_l^k$ is given by [5, Eq. (28)]\n\\begin{eqnarray}\n\\mathbf{b}{_l^k}^* = \\mathbf{a}{_l^k}^* \\mathbf{\\hat H}_k^*{\\left( {\\frac{1}{{\\text{SNR}}}\\mathbf{I} + {{\\mathbf{\\hat H}}_k} \\mathbf{\\hat H}_k^*} \\right)^{ - 1}},\n\\end{eqnarray}\nwhere $\\text{SNR} = \\frac{P}{{{\\sigma ^2}}}$. Finally, the rate of the equation with ECV $\\mathbf{a}_l^k$ is obtained by [5, Eq. (30)]\n\\begin{eqnarray}\nR\\left( {{\\mathbf{a}_l^k}} \\right) = {\\log ^ + }\\left( {{{\\left( {\\mathbf{a}{_l^k}^*\\left( {\\mathbf{I} - \\mathbf{\\hat H}_k^*{{\\left( {\\frac{1}{{\\text{SNR}}}\\mathbf{I} + {{\\mathbf{\\hat H}}_k} \\mathbf{\\hat H}_k^*} \\right)}^{ - 1}}{{\\mathbf{\\hat H}}_k}} \\right){\\mathbf{a}_l^k}} \\right)}^{ - 1}}} \\right).\n\\end{eqnarray}\nHence, the optimal value of $\\mathbf{A}^k$, in terms of (5), is obtained by solving the following problem\n\\begin{eqnarray}\n\\mathbf{A}_\\text{opt}^k= \\arg \\mathop {\\min}\\limits_{\\mathbf{A}^k \\in {\\mathbf{Z}^{L \\times L}}} {\\max _{l = 1, \\ldots ,L}} \\mathbf{a}{_l^k}^*\\left( {\\mathbf{I} -  \\mathbf{\\hat H}_k^*{{\\left( {\\frac{1}{{\\text{SNR}}}\\mathbf{I} + {{\\mathbf{\\hat H}}_k} \\mathbf{\\hat H}_k^*} \\right)}^{ - 1}}{{ \\mathbf{\\hat H}}_k}} \\right){\\mathbf{a}_l^k},\\nonumber\\\\\n\\text{subject to} \\hspace{+10pt}\n\\text{det}(\\mathbf{A}^k) \\ne 0. \\hspace{+200pt}\n\\end{eqnarray}\nThe problem (6) is an NP hard integer programming and its complexity grows with $L$ significantly. \n\nNote that the IFLR scheme does the lattice equation quantization (3) $L$ times, which increases the implementation complexity with $L$ significantly. Hence, the IFLR scheme leads to significantly higher complexity compared to the MMSE and ZF schemes [1]-[2], i.e., $L_c \\buildrel \\Delta \\over = L - N_{{t_k}}$ more decoding for each receiver $k$.\n\nIn Section III, we propose our IFMR scheme where, independently of $K$ and ${N_{{t_i}}},\\forall i \\ne k$, each receiver $k$ only requires lattice equation decoding twice the number of the desirable messages, i.e., $2 \\times {N_{{t_k}}}$, with a low complexity best equation selection process.\n\n\n\n\n\n\\section{Integer-Forcing Message Recovering (IFMR)}\nIn summary, our proposed IFMR scheme is based on the following procedure. From the received signals ${\\mathbf{Y}_k}$ in (1), independent DCMs are recovered. For each DCM, the observed interfered signal is integer-forced to an UCM. Then, two independent equations of the DCM and UCM are decoded by the lattice quantizer as in (3) which lead to recovering the DCM. Finally, solving the recovered DCMs results in the desirable messages. \n\nIn Subsection III.A, the structure of an equation in IFMR is proposed, and accordingly its receiver model is presented. Then, in Subsection III.B, we develop a sequential three-step algorithm to efficiently find the coefficient factors of the required equations in the first step and their associated coefficient vectors of UCMs and DCMs in the second and third steps, respectively, with maximum rates in polynomial time. Theorem 1 proves that our scheme with sequential selection of DCMs achieves the same rate as the optimal scheme jointly selecting DCMs. Theorem 2 proves that Lenstra-Lenstra-Lovasz (LLL) algorithm [13] is qualified to be used for the optimization problem of the first step, and Theorem 3 proves the convergence of the proposed algorithm. Simulation results are presented in Section IV where we compare the performance of our proposed scheme with those in the literature.\n\n\\subsection{Receiver Structure} We consider an equation in the general form $t^k=d^k x_k^{\\text{DCM}}+e^k x_k^{\\text{UCM}}$ for receiver $k$, which is an integer combination of two messages $x_k^{\\text{DCM}}$ and $x_k^{\\text{UCM}}$. Here, $x_k^{{\\rm{DCM}}} \\buildrel \\Delta \\over = {\\mathbf{a}^k}^*{\\mathbf{x}_k}$ and $x_k^{{\\rm{UCM}}} \\buildrel \\Delta \\over = \\mathop \\sum \\limits_{j = 1,j \\ne k}^K \\mathbf{c}{{_j^k}^*}{\\mathbf{x}_j}$ are referred to as DCM and UCM, respectively. In other words, according to the IFLR receiver structure, $t^k$ has ECV equal to $\\left[ {\\begin{array}{*{20}{c}}\n{{d^k}{\\mathbf{a}^k}}\\\\\n{{e^k}{\\mathbf{c}^k}}\n\\end{array}} \\right]$, where ${\\mathbf{c}^k} \\buildrel \\Delta \\over = {\\left[ {\\mathbf{c}{{_1^k}^*}, \\ldots ,\\mathbf{c}{{_{k - 1}^k}^*},\\mathbf{c}{{_{k + 1}^k}^*}, \\ldots ,\\mathbf{c}{{_K^k}^*}} \\right]^*}$. $d^k$ and $e^k$ are integer coefficient factors in $\\mathbf{Z}$ space. Also, $\\mathbf{a}^k$ and\n$\\mathbf{c}_j^k, \\forall j,$ are integer coefficient vectors in ${\\mathbf{Z}^{{N_{{t_k}}} \\times 1}}$ and ${\\mathbf{Z}^{{N_{{t_j}}} \\times 1}}$, respectively. \n\nIt is straightforward to show that two equations with independent set of coefficient factors $(d_1^k,e_1^k)$ and $(d_2^k,e_2^k)$, and same $\\mathbf{a}^k$ and $\\mathbf{c}^k$ for the combined messages can obtain $x_k^{{\\rm{DCM}}} = {\\mathbf{a}^k}^*{\\mathbf{x}_k}$. According to (5) and for given coefficient vector of $x_k^{\\text{UCM}}$ and coefficient factors of the two equations, the rate of recovering $x_k^{\\text{DCM}}$ is obtained by\n\\begin{eqnarray}\n{R_{\\text{DCM}}}\\left({\\mathbf{a}^k}|{\\mathbf{c}^k},d_1^k,e_1^k,d_2^k,e_2^k\\right) = {\\rm{min}}\\left\\{ {R\\left( {\\left[ {\\begin{array}{*{20}{c}}\n{d_1^k{\\mathbf{a}^k}}\\\\\n{e_1^k{\\mathbf{c}^k}}\n\\end{array}} \\right]} \\right),R\\left( {\\left[ {\\begin{array}{*{20}{c}}\n{d_2^k{\\mathbf{a}^k}}\\\\\n{e_2^k{\\mathbf{c}^k}}\n\\end{array}} \\right]} \\right)} \\right\\},\n\\end{eqnarray}\nwith $R(\\cdot)$ given in (5). Hence, the unconditional achievable rate of $x_k^{\\text{DCM}}$ is determined by\n\\begin{eqnarray}\n{R_{\\text{DCM}}}\\left({\\mathbf{a}^k}\\right) = \\mathop {\\max }\\limits_{d_1^k,e_1^k,d_2^k,e_2^k\\in \\mathbf{Z},{\\mathbf{c}^k} \\in {\\mathbf{Z}^{{L_c} \\times 1}}} {R_{\\text{DCM}}}\\left({\\mathbf{a}^k}|{\\mathbf{c}^k},d_1^k,e_1^k,d_2^k,e_2^k\\right).\n\\end{eqnarray}\n\nDue to the size of $\\mathbf{x}_k$, it is sufficient to recover $N_{{t_k}}$ independent DCMs. An illustration of the receiver structure is given in Fig. 2. \n\\begin{figure}[tb!]\n\\centering\n\n\\includegraphics[width =5in]{recifmradv.png}\n\n\\caption{The proposed structure of receiver $k$. In each branch $i=1,...,N_{t_k}$, $\\mathbf{B}_i^k$ includes the projection vectors in (4) related to the two equations of the branch with integer coefficients $e_1^{k,i}$, $d_1^{k,i}$, $e_2^{k,i}$, $d_2^{k,i}$, $\\mathbf{a}^{k,i}$, and $\\mathbf{c}^{k,i}$. Note $\\mathbf{c}^{k,i}$, related to undesirable recovered messages, is not shown in the figure.}\n\n\\end{figure}\n\\subsection{Best Integer Coefficients Selection} From (7) and (8), it is clear that the coefficients of the optimal independent DCMs with maximum rates are jointly selected from the following optimization\n\\begin{eqnarray}\n{\\max _{d{_l^{k,m}},e{_l^{k,m}}\\in \\mathbf{Z},{\\mathbf{a}^{k,m}} \\in {\\mathbf{Z}^{{N_{{t_k}}} \\times 1}},{\\mathbf{c}^{k,m}} \\in {\\mathbf{Z}^{{L_c} \\times 1}}}}\\mathop {\\min }\\limits_{m = 1, \\ldots ,{N_{t_k}}} \\mathop {\\min }\\limits_{l = 1,2} R\\left( {\\left[ {\\begin{array}{*{20}{c}}\n{d{_l^{k,m}}\\mathbf{a}^{k,m}}\\\\\n{e{_l^{k,m}}\\mathbf{c}^{k,m}}\n\\end{array}} \\right]} \\right),\\nonumber\n\\end{eqnarray}\nsubject to\n\\begin{eqnarray}\n\\left\\{ {\\begin{array}{*{20}{c}}\n{\\text{det}\\left(\\left[ {\\begin{array}{*{20}{c}}\n{d{_1^{k,m}}}&{e{_1^{k,m}}}\\\\\n{d{_2^{k,m}}}&{e{_2^{k,m}}}\n\\end{array}} \\right]\\right) \\ne 0}, \\forall m=1,...,N_{t_k}\\\\\n{\\text{det}\\left( {\\left[ {\\mathbf{a}^{k,1}, \\ldots ,\\mathbf{a}^{k,{N_{t_k}}}} \\right]} \\right) \\ne 0}\n\\end{array}} \\right. . \n\\end{eqnarray}\nThe problem (9) is complex, because it requires searches over space $\\mathbf{Z}^{(L+2)^{N_{t_k}}\\times 1}$. For this reason, we propose a sequential selection in $N_{{t_k}}$ stages which only requires a search over space $\\mathbf{Z}^{{N_{t_k}}(L+2)\\times 1}$. Hence, the sequential scheme is of interest because it simplifies the search process, compared to (9), significantly. In the sequential selection, each stage $t$ is to recover the best DCM $x_k^{\\text{DCM},t}$ with maximum rate independently of the previously recovered messages $x_k^{\\text{DCM},j}, \\forall j < t$. To be more specific, in each stage $t$, it is required to solve\n\\begin{eqnarray}\n{\\max _{d_l^k,e_l^k \\in \\mathbf{Z},{\\mathbf{a}^k} \\in {\\mathbf{Z}^{{N_{{t_k}}} \\times 1}},{\\mathbf{c}^k} \\in {\\mathbf{Z}^{{L_c} \\times 1}}}}}\\min _{l=1,2} {R\\left( {\\left[ {\\begin{array}{*{20}{c}}\n{d_l^k{\\mathbf{a}^k}}\\\\\n{e_l^k{\\mathbf{c}^k}}\n\\end{array}} \\right]} \\right),\\nonumber\n\\end{eqnarray}\nsubject to\n\\begin{eqnarray}\n\\left\\{ {\\begin{array}{*{20}{c}}\n{\\text{det}\\left(\\left[ {\\begin{array}{*{20}{c}}\n{d_1^k}&{e_1^k}\\\\\n{d_2^k}&{e_2^k}\n\\end{array}} \\right]\\right) \\ne 0}\\\\\n{\\text{det}\\left( {\\left[ {{\\mathbf{a}^k},\\mathbf{g}_1^k, \\ldots ,\\mathbf{g}_{t - 1}^k} \\right]} \\right) \\ne 0}\n\\end{array}} \\right. , \n\\end{eqnarray}\nwhere $\\mathbf{g}_j^k$ is the integer coefficient vector associated with $x_k^{{\\text{DCM},j}}$ obtained in the stage $j<t$. \nIn Theorem 1, we prove that the sequential selection (10) is optimal, in the sense that it achieves the same rate as optimal search (9), with considerably less implementation complexity.\n\\begin{theorem}\nThe sequential selection (10) achieves the same rate as the optimal selection (9).\n\\end{theorem}\n\\begin{proof}\nSee Appendix I.\n\\end{proof}\nNote that (10) is still an NP hard integer programming problem, requiring an exhaustive search over integer coefficients. For this reason, we propose a suboptimal scheme presented in Algorithm 1 to efficiently solve (10) in polynomial time and iteratively in three steps. In words, the algorithm is based on the following procedure. In Step I, the coefficient factors of the equations are optimized to maximize the rate of recovering given DCM and UCM. Then, in Step II, using equation factors obtained in Step I and given coefficient vector of DCM, we find the optimal coefficient vector of UCM. Finally, in Step III, for the obtained coefficient vector of UCM in Step II and the equation factors obtained in Step I, the coefficient vector of DCM is optimized. The convergence of the algorithm is proved in Theorem 3. \n\n\\textit{Step I:} For given $\\mathbf{c}^k$ and $\\mathbf{a}^k$, solve \n\\begin{eqnarray}\n&\\mathop {\\min }\\limits_{d_1^k,e_1^k,d_2^k,e_2^k \\in \\mathbf{Z}} \\mathop {\\max }\\limits_{l = 1,2} f_l (\\mathbf{a}^k,\\mathbf{c}^k),\\nonumber\\\\\n&\\text{subject to}\\hspace{+10pt}\n{\\rm{det}}\\left(\\left[ {\\begin{array}{*{20}{c}}\n{d_1^k}&{e_1^k}\\\\\n{d_2^k}&{e_2^k}\n\\end{array}} \\right]\\right) \\ne 0 .\n\\end{eqnarray}\nDefining $f_l (\\mathbf{a}^k,\\mathbf{c}^k)\\buildrel \\Delta \\over = R\\left( {\\left[ {\\begin{array}{*{20}{c}}\n{d_l^k{\\mathbf{a}^k}}\\\\\n{e_l^k{\\mathbf{c}^k}}\n\\end{array}} \\right]} \\right)$ and ${\\mathbf{H}_k} \\buildrel \\Delta \\over = \\left[ {{\\mathbf{H}_{1k}}, \\ldots ,{\\mathbf{H}_{\\left( {k - 1} \\right)k}},{\\mathbf{H}_{\\left( {k + 1} \\right)k}}, \\ldots ,{\\mathbf{H}_{Kk}}} \\right]$, we use (5) to expand $f_l (\\mathbf{a}^k,\\mathbf{c}^k)$ as\n\\begin{multline}\nf_l (\\mathbf{a}^k,\\mathbf{c}^k)=d{{_l^k}^2}{\\mathbf{a}^k}^*{\\mathbf{a}^k} + e{{_l^k}^2}{\\mathbf{c}^k}^*{\\mathbf{c}^k} - \\left( {d_l^k{\\mathbf{a}^k}^*\\mathbf{H}_{kk}^* + e_l^k{\\mathbf{c}^k}^*\\mathbf{H}_k^*} \\right) {\\left( {\\frac{1}{{\\text{SNR}}}\\mathbf{I} + {\\mathbf{H}_{kk}}\\mathbf{H}_{kk}^* + {\\mathbf{H}_k}\\mathbf{H}_k^*} \\right)^{ - 1}}\\times \\\\ \\left( {d_l^k{\\mathbf{H}_{kk}}{\\mathbf{a}^k} + e_l^k{\\mathbf{H}_k}{\\mathbf{c}^k}} \\right).\n\\end{multline}\nHence, with some simplifications, the optimization (11) can be written as\n\\begin{eqnarray}\n&\\mathop {\\min }\\limits_{d_1^k,e_1^k,d_2^k,e_2^k \\in \\mathbf{Z}} \\mathop {\\max }\\limits_{l = 1,2} \\left[ \\begin{array}{*{20}{c}}\n{d_l^k}&{e_l^k}\n\\end{array} \\right]\\mathbf{U}\\left[ \\begin{array}{*{20}{c}}\n{d_l^k}\\\\\n{e_l^k}\n\\end{array} \\right],\\nonumber\\\\\n&\\text{subject to} \\hspace{+10pt}\n{\\rm{det}}\\left(\\left[ {\\begin{array}{*{20}{c}}\n{d_1^k}&{e_1^k}\\\\\n{d_2^k}&{e_2^k}\n\\end{array}} \\right]\\right) \\ne 0 ,\n\\end{eqnarray}\nwhere \n\\begin{multline}\n\\mathbf{U} \\buildrel \\Delta \\over = \\left[ {\\begin{array}{*{20}{c}}\n{{\\mathbf{a}^k}^*{\\mathbf{a}^k}}&0\\\\\n0&{{\\mathbf{c}^k}^*{\\mathbf{c}^k}}\n\\end{array}} \\right] - \\left[ {\\begin{array}{*{20}{c}}\n{{\\mathbf{a}^k}^*\\mathbf{H}_{kk}^*}\\\\\n{{\\mathbf{c}^k}^*\\mathbf{H}_k^*}\n\\end{array}} \\right]{\\left( {\\frac{1}{{\\text{SNR}}}{\\mathbf{I}} + {\\mathbf{H}_{kk}}\\mathbf{H}_{kk}^* + {\\mathbf{H}_k}\\mathbf{H}_k^*} \\right)^{ - 1}}\\left[ {\\begin{array}{*{20}{c}}\n{{\\mathbf{H}_{kk}}{\\mathbf{a}^k}}&{{\\mathbf{H}_k}{\\mathbf{c}^k}}\n\\end{array}} \\right].\n\\end{multline}\nIn Theorem 2 we prove $\\mathbf{U}$ to be positive definite, the proof of which uses Lemma 1 as follows.\n\\begin{lemma}\nMatrix ${\\mathbf{I}} - \\frac{{\\mathbf{x}{\\mathbf{x}^*}}}{{{\\mathbf{x}^*}\\mathbf{x}}}$ is semi-definite, where $\\mathbf{x} \\ne 0$ is a vector in $\\mathbf{R}^{L\\times 1}$.\n\\end{lemma}\n\\begin{proof}\nSee Appendix II.\n\\end{proof}\n\\begin{theorem}\n$\\mathbf{U}$ is a positive definite matrix.\n\\end{theorem}\n\\begin{proof}\nSee Appendix III.\n\\end{proof}\nAccording to Theorem 2, $\\mathbf{U}$ admits a unique Cholesky decomposition. Hence, (13) can be solved efficiently in polynomial time with the LLL method [13]. \n\\begin{small}\n\\begin{table}[t\n   \n  \\centering\n  \\caption\n    \"Algorithm 1\"}\n    \\vspace*{-1em}\n  \\begin{tabular}{l}\n      \\hline\n\\underline{For} $t = 1, \\ldots ,{N_t}$\n\\\\\n\\hspace*{+10pt} \\underline{Initialize} ${\\mathbf{a}^{k,0}}$ and ${\\mathbf{c}^{k,0}}$\n\\\\\n\\hspace*{+10pt} \\underline{Iterate}\n\\\\\n\\hspace*{+20pt}1. Step I: Update $\\left[ {\\begin{array}{*{20}{c}}\n{d_1^{k,j + 1}}&{e_1^{k,j + 1}}\\\\\n{d_2^{k,j + 1}}&{e_2^{k,j + 1}}\n\\end{array}} \\right]$ by solving (13) with the \\\\\\hspace*{+22pt}assumption of given $\\mathbf{a}^{k,j}$ and $\\mathbf{c}^{k,j}$ .\n\\\\\n\\hspace*{+20pt}2. Step II: Update $\\mathbf{c}^{k,j+1}$ by solving (21) with the assumption of \\\\\\hspace*{+22pt}given $\\mathbf{a}^{k,j}$ and $\\left[ {\\begin{array}{*{20}{c}}\n{d_1^{k,j + 1}}&{e_1^{k,j + 1}}\\\\\n{d_2^{k,j + 1}}&{e_2^{k,j + 1}}\n\\end{array}} \\right]$. \n\\\\\n\n\\hspace*{+20pt}3. Step III: Check which of the cases 1-3 are valid and update \\\\\\hspace*{+22pt}$\\mathbf{a}^{k,j+1}$ accordingly with the assumption of given $\\mathbf{c}^{k,j+1}$ and\\\\\\hspace*{+22pt}$\\left[ {\\begin{array}{*{20}{c}}\n{d_1^{k,j + 1}}&{e_1^{k,j + 1}}\\\\\n{d_2^{k,j + 1}}&{e_2^{k,j + 1}}\n\\end{array}} \\right]$.\n\\\\\n\\hspace*{+10pt}\\underline{Until} \n$\\min \\left\\{ {R\\left( {\\left[ {\\begin{array}{*{20}{c}}\n{d_1^k{\\mathbf{a}^k}}\\\\\n{e_1^k{\\mathbf{c}^k}}\n\\end{array}} \\right]} \\right),R\\left( {\\left[ {\\begin{array}{*{20}{c}}\n{d_2^k{\\mathbf{a}^k}}\\\\\n{e_2^k{\\mathbf{c}^k}}\n\\end{array}} \\right]} \\right)} \\right\\}$ converges with \\\\ \\hspace*{+22pt}convergence threshold $\\delta$. \n\\\\\n\\hspace*{+10pt}\\underline{Then} $\\mathbf{g}_t^k \\leftarrow {\\mathbf{a}^{k,{\\rm{end}}}}$\n\\\\\n\\underline{End}\n    \\vspace*{.5em}\n\\\\\n\n      \\hline\n    \\end{tabular}\n\n\\end{table}\n\\end{small}\n\n\\textit{Step II:} For given $\\mathbf{a}^k$ and coefficient factors $(e_1^k, d_1^k)$ and $(e_2^k, d_2^k)$, solve\n\\begin{eqnarray}\n\\mathop {\\min }\\limits_{{\\mathbf{c}^k} \\in \\mathbf{Z}^{{L_{{c}}} \\times 1}} \\mathop {\\max }\\limits_{l = 1,2} f_l (\\mathbf{a}^k,\\mathbf{c}^k),\n\\end{eqnarray}\nwhich, according to (12), can be rewritten by\n\\begin{eqnarray}\n\\mathop {\\min }\\limits_{{\\mathbf{c}^k} \\in \\mathbf{Z}^{{L_{{c}}} \\times 1}} \\mathop {\\max }\\limits_{l = 1,2}{\\mathbf{c}^k}^*\\mathbf{Q}_l{\\mathbf{c}^k}-2\\mathbf{q}_l^*\\mathbf{c}^k,\n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\n\\mathbf{Q}_l \\buildrel \\Delta \\over= e{{_l^k}^2}\\mathbf{I} -e{{_l^k}^2}\\mathbf{H}_k^* {\\left( {\\frac{1}{{\\text{SNR}}}\\mathbf{I} + {\\mathbf{H}_{kk}}\\mathbf{H}_{kk}^* + {\\mathbf{H}_k}\\mathbf{H}_k^*} \\right)^{ - 1}}{\\mathbf{H}_k},\n\\end{eqnarray}\n\\begin{eqnarray}\n\\mathbf{q}_l \\buildrel \\Delta \\over= e{_l^k} d{_l^k}\\mathbf{H}_{k}^*{\\left( {\\frac{1}{{\\text{SNR}}}\\mathbf{I} + {\\mathbf{H}_{kk}}\\mathbf{H}_{kk}^* + {\\mathbf{H}_k}\\mathbf{H}_k^*} \\right)^{ - 1}}\\mathbf{H}_{kk}\\mathbf{a}_{k}.\n\\end{eqnarray}\nThe min-max quadratic problem (16) is an NP-hard integer programming. Therefore, we propose an efficient suboptimal solution for (16) which is obtained in polynomial time as follows.\nFirst, we relax the constraint ${\\mathbf{c}^k} \\in {\\mathbf{Z}^{{L_{c}} \\times 1}}$, and let the optimal value of $\\mathbf{c}^k$ to be continuous, i.e., ${\\mathbf{c}^k} \\in {\\mathbf{R}^{{L_{c}} \\times 1}}$, with the constraint $\\text{diag}\\left\\{{\\mathbf{c}^k}\\right\\}({\\mathbf{c}^k}-\\mathbf{1}) \\geq \\mathbf{0}$. Then, the obtained real-valued solution is rounded to its closest integer point. It is shown in [14] that the constraint $x_i(x_i-1)\\geq 0$ for each element $i$ of a real-valued vector $\\mathbf{x}$ can achieve a tight lower bound on the optimal value of the integer quadratic minimization of $\\mathbf{x}$. \n\nFollowing the same approach as in [14], we relax (16) as\n\\begin{eqnarray}\n&\\mathop {\\min }\\limits_{{\\mathbf{c}^k} \\in \\mathbf{R}^{{L_{{c}}} \\times 1}} \\mathop {\\max }\\limits_{l = 1,2}{\\mathbf{c}^k}^*\\mathbf{Q}_l{\\mathbf{c}^k}-2\\mathbf{q}_l^*\\mathbf{c}^k, \\nonumber\\\\\n&\\text{subject to} \\hspace{10pt}\n\\text{diag}\\left\\{{\\mathbf{c}^k}\\right\\}({\\mathbf{c}^k}-\\mathbf{1}) \\geq \\mathbf{0}.\n\\end{eqnarray}\n\nWith the definition of ${\\mathbf{C}^k} \\buildrel \\Delta \\over = {\\mathbf{c}^k}{\\mathbf{c}^k}^*$, the problem (19) is reformulated as\n\\begin{eqnarray}\n&\\mathop {\\min }\\limits_{{\\mathbf{c}^k} \\in \\mathbf{R}^{{L_{{c}}} \\times 1}} \\mathop {\\max }\\limits_{l = 1,2}{\\rm{\\text{Tr}}}\\left\\{ {{\\mathbf{Q}_l}{\\mathbf{C}^k}} \\right\\} - 2\\mathbf{q}_l^*{\\mathbf{c}^k},\\nonumber\\\\\n&\\text{subject to} \\hspace{10pt}\\left\\{ {\\begin{array}{*{20}{c}}\n\\text{diag}\\left\\{ {{\\mathbf{C}^k}} \\right\\} \\ge {\\mathbf{c}^k}\\\\\n{\\mathbf{C}^k} = {\\mathbf{c}^k}{\\mathbf{c}^k}^*\n\\end{array}} \\right. .\n\\end{eqnarray}\nThen, relaxing the nonconvex constraint ${\\mathbf{C}^k} = {\\mathbf{c}^k}{\\mathbf{c}^k}^*$ into a convex constraint ${\\mathbf{C}^k} \\succcurlyeq {\\mathbf{c}^k}{\\mathbf{c}^k}^*$, the non-convex problem (20) with the help of a Schur complement is relaxed to a convex problem as\n\\begin{eqnarray}\n&\\mathop {\\min }\\limits_{\\varepsilon, {\\mathbf{c}^k} \\in \\mathbf{R}^{{L_{{c}}} \\times 1}} \\varepsilon , \\nonumber\\\\\n&\\text{subject to} \\hspace{10pt}\\left\\{ {\\begin{array}{*{20}{c}}\n{\\begin{array}{*{20}{c}}\n{\\rm{\\text{Tr}}}\\left\\{ {{\\mathbf{Q}_1}{\\mathbf{C}^k}} \\right\\} - 2\\mathbf{q}_1^*{\\mathbf{c}^k} \\le \\varepsilon \\\\\n{\\rm{\\text{Tr}}}\\left\\{ {{\\mathbf{Q}_2}{\\mathbf{C}^k}} \\right\\} - 2\\mathbf{q}_2^*{\\mathbf{c}^k} \\le \\varepsilon \n\\end{array}}\\\\\n{\\begin{array}{*{20}{c}}\n\\text{diag}\\left\\{ {{\\mathbf{C}^k}} \\right\\} \\ge {\\mathbf{c}^k}\\\\\n\\left[ {\\begin{array}{*{20}{c}}\n{{\\mathbf{C}^k}}&{{\\mathbf{c}^k}}\\\\\n{{\\mathbf{c}^k}^*}&1\n\\end{array}} \\right] \\succcurlyeq \\mathbf{0}\n\\end{array}}\n\\end{array}} \\right. .\n\\end{eqnarray}\nThe problem (21) is a semidefinite programming (SDP) and can be efficiently solved by CVX [15], which is a software package developed for convex optimization problems.\n\n\\textit{Step III:} For given $\\mathbf{c}^k$ and coefficient factors $(e_1^k, d_1^k)$ and $(e_2^k, d_2^k)$, solve\n\\begin{eqnarray}\n&\\mathop {\\min }\\limits_{{\\mathbf{a}^k} \\in \\mathbf{Z}^{{N_{{t_k}}} \\times 1}} \\mathop {\\max }\\limits_{l = 1,2} f_l (\\mathbf{a}^k,\\mathbf{c}^k),\\nonumber\\\\\n&\\text{subject to} \\hspace{10pt}\n{\\rm{det}}\\left( {\\left[ {{\\mathbf{a}^k},\\mathbf{g}_1^k, \\ldots ,\\mathbf{g}_{t - 1}^k} \\right]} \\right) \\ne 0, \n\\end{eqnarray}\nwhich is an NP-hard integer programming. Here, because of the constraint ${\\rm{det}}\\left( {\\left[ {{\\mathbf{a}^k},\\mathbf{g}_1^k, \\ldots ,\\mathbf{g}_{t - 1}^k} \\right]} \\right) \\ne 0$, we cannot achieve a tight bound with an approach similar to Step II. Therefore, we propose a search over integer space $\\mathbf{Z}^{{N_{{t_k}}}\\times 1}$ which can obtain an efficient suboptimal solution of (22) as follows. First, we optimize (22) with a relaxation on the constraint ${\\mathbf{a}^k} \\in {\\mathbf{Z}^{{N_{{t_k}}} \\times 1}}$ as ${\\mathbf{a}^k} \\in {\\mathbf{R}^{{N_{{t_k}}} \\times 1}}$. Then, we search over a $N_{t_k}$-dimensional quantization sphere which has the obtained real valued solution ${\\mathbf{a}^k} \\in {\\mathbf{R}^{{N_{{t_k}}} \\times 1}}$ as its center, and find the best candidate according to (22). Since $f_l (\\mathbf{a}^k,\\mathbf{c}^k)$ is a convex quadratic function, the proposed search can achieve a tight suboptimal solution of (22) when the quantization sphere has sufficiently large radius. The quantization scheme will be further discussed in the sequel.\n\nHere, the problem (22) is relaxed as\n\\begin{eqnarray}\n\\mathop {\\min}\\limits_{{\\mathbf{a}^k} \\in {\\mathbf{R}^{{N_{t_k}} \\times 1}}} \\mathop {\\max }\\limits_{l = 1,2} f_l (\\mathbf{a}^k,\\mathbf{c}^{k}).\n\\end{eqnarray}\nTo obtain the solution of (23) in closed form, we use the same procedure as in [16, Subsection III.A] to convert (23) to an equivalent problem as \n\\begin{eqnarray}\n\\mathop {{\\rm{max}}}\\limits_{0 \\le \\alpha  \\le 1} \\mathop {{\\rm{min}}}\\limits_{{\\mathbf{a}^k} \\in {\\mathbf{R}^{{N_{{t_k}}} \\times 1}}} V\\left( {\\alpha ,{\\mathbf{a}^k}} \\right),\n\\end{eqnarray}\nwhere $V(\\alpha ,{\\mathbf{a}^k}) = \\alpha {f_1}({\\mathbf{a}^k},{\\mathbf{c}^{k}}) + (1 - \\alpha ){f_2}({\\mathbf{a}^k},{\\mathbf{c}^{k}})$, and $0 \\le \\alpha  \\le 1$ is an auxiliary parameter. As details are given in Appendix IV, the solution of (24) is \n\\begin{multline}\n{\\mathbf{a}^k} = u^k({\\alpha ^*}){\\left( {v^k({\\alpha ^*})\\mathbf{I} - \\mathbf{H}_{kk}^*{{\\left( {\\frac{1}{{\\text{SNR}}}\\mathbf{I} + {\\mathbf{H}_{kk}}\\mathbf{H}_{kk}^* + {\\mathbf{H}_k}\\mathbf{H}_k^*} \\right)}^{ - 1}}{\\mathbf{H}_{kk}}} \\right)^{ - 1}}\\\\ \\times \\mathbf{H}_{kk}^*{\\left( {\\frac{1}{{\\text{SNR}}}\\mathbf{I} + {\\mathbf{H}_{kk}}\\mathbf{H}_{kk}^* + {\\mathbf{H}_k}\\mathbf{H}_k^*} \\right)^{ - 1}}{\\mathbf{H}_k}{\\mathbf{c}^{k}},\n\\end{multline}\nwhere $\\alpha^*$ is obtained according to the considered three cases in Appendix IV. Also, functions $u^k(p)$ and $v^k(p)$ are defined as follows\n\\begin{eqnarray}\nu^k(p) \\buildrel \\Delta \\over = p{e_1^k}{d_1^k} + \\left( {1 - p} \\right){e_2^k}{d_2^k} ,\\nonumber\n\\end{eqnarray}\n\\vspace{-20pt}\n\\begin{eqnarray}\nv^k(p)\\buildrel \\Delta \\over = pd{_1^k}^2 + \\left( {1 - p} \\right)d{_2^k}^2 .\n\\end{eqnarray}\n\nAs a polynomial-time approach to search over the quantization sphere, we can consider slowest descent lines with directions of the eignevectors of the hessien of the cost function $f_l (\\mathbf{a}^k,\\mathbf{c}^k)$ in (12), i.e., $d{{_l^k}^2}\\mathbf{I} - d{{_l^k}^2}\\mathbf{H}_{kk}^*{\\left( {\\frac{1}{{\\text{SNR}}}\\mathbf{I} + {\\mathbf{H}_{kk}}\\mathbf{H}_{kk}^* + {\\mathbf{H}_k}\\mathbf{H}_k^*} \\right)^{ - 1}} {\\mathbf{H}_{kk}}$, which crosses the center ${\\mathbf{a}^k} \\in {\\mathbf{R}^{{N_{{t_k}}} \\times 1}}$ in (25). Then, the closest integer points to the lines and independent of $\\mathbf{g}_1^k, \\ldots ,\\mathbf{g}_{t-1}^k$ are checked to find the best candidate. This approach is based on the slowest descent method which can efficiently search over discrete points [17]. \n\nAssume that the quantization radius is $R$ and the number of the slowest descent lines is $W$. It is straightforward to show that our approach needs to search over at most $W\\times (2R+1)\\times N_{t_k}$ integer points. Through the following lemma, we can exclude those ${\\mathbf{a}^k}$ from the quantization sphere for which the rate (7) are zero. It also determines the maximum required radius for the quantization sphere, which guarantees to include the optimal solution of (22). The Lemma is of interest because it reduces the complexity for searching in the quantization sphere.\n\\begin{lemma}\nAssume $e_1$, $e_2$, $d_1$, $d_2$, and $\\mathbf{c}^k$ are given. The search space $\\mathbf{a}^k$ with the following norm leads to rate $0$ in (7).\n\\begin{eqnarray}\n{\\left| {\\left| {{\\mathbf{a}^k}} \\right|} \\right|^2} \\geq \\mathop {\\min }\\limits_{l = 1,2} \\frac{1}{{e_l^2}}\\left( {1 + {\\rm{\\text{SNR}}}{\\lambda _{{\\rm{max}}}^2}\\left( {{{\\mathbf{\\hat H}}_k}} \\right) - d_l^2{\\left| {\\left| {{\\mathbf{c}^k}} \\right|} \\right|^2}} \\right),\n\\end{eqnarray}\nwhere ${\\lambda _{\\rm{max}}}\\left( {{\\mathbf{\\hat H}}_k} \\right)$ is the maximum singular value of $\\mathbf{\\hat H}_k$.\n\\end{lemma}\n\\begin{proof}\nSee Appendix V.\n\\end{proof}\n\n\\vspace{+50pt}\nAlgorithm 1, summarized in Table 1, is iterated until a convergence threshold $\\delta$, considered by the algorithm designer, is reached. In the simulation results, we will show the performance of our polynomial time suboptimal algorithm in comparison with the NP-hard optimal exhaustive search of the equations and UCM and DCM coefficients over the cost function of (10). The following theorem proves the convergence of Algorithm 1.\n\n\\begin{theorem}\nAlgorithm 1 is convergent.\n\\end{theorem}\n\\begin{proof}\nSee Appendix VI.\n\\end{proof}\n\n\\section{Simulation Results}\nIn this section, we provide simulation results that demonstrate the performance of the proposed IFMR scheme. Consider a three pair interference channel in which each node is equipped with $N$ antennas, unless otherwise stated. The elements of the channel matrices are assumed to have Gaussian distribution with variance 1, i.e., $\\rho _{kj}^2 = 1$, $\\forall k, j$. The additive white Gaussian noise has $\\sigma^2 = 1$. The convergence threshold parameter $\\delta$ in Algorithm 1 is set to $10^{-3}$. We average over 10000 randomly generated channel realizations.\n\nIn Figs. 3 and 4, we evaluate the achievable rates of our proposed IFMR scheme and compare the results with the state-of-the-art works, i.e., MMSE and ZF [2], and IFLR scheme [5], for $N=1$ and $N=2$, respectively. As observed, Algorithm 1 can achieve almost the same performance as in the optimal exhaustive  search-based scheme. For instance, in the cases with $N = 1$ and $N = 2$, the performance degradation, compared to the optimal exhaustive search-based approach, is less than 1 dB in 1 bit\/channel use and 1 dB in 2 bit\/channel use, respectively. It is also observed that the IFMR scheme outperforms the conventional MMSE and ZF receivers at all SNRs, and the performance gap increases with SNR which is because of the increase in interference. Also, the IFMR scheme achieves slightly higher rates compared to the IFLR scheme at low SNRs. It is due to the fact that the optimal equations recovered from (6) may have zero elements with high probability at low SNRs [18], whereby a subset of the equations would be enough for recovering the desirable messages. Note that the IFLR scheme leads to better achievable rates compared to the IFMR scheme at high SNRs, at the expense of much higher complexity. For example, the IFLR scheme has 2 dB improvement compared to the IFMR scheme at 1.15 bit\/channel use in the one antenna case (Fig. 3) and 2.5 dB improvement at 2.5 bit\/channel use in two antennas case (Fig. 4). That is because, in comparison with IFMR, the IFLR scheme has more flexibility in decoding the interference as equations. \n\nIn Fig. 5, we investigate the average number of required iterations as a function of SNR for the cases with $N = 2$. It is observed that for all considered SNRs less than 5 iterations are required for the algorithm convergence. Thus, our algorithm can be effectively applied in delay-constrained applications.\n\nFig. 6 shows the throughput versus the target rate $R_\\text{t}$ for the case with $N = 2$. The throughput is defined as, e.g., [19, Eq. (4)]\n\\begin{eqnarray} \n\\eta = R_\\text{t}\\times\\left(1-\\Pr(R_\\text{achievable}<R_\\text{t})\\right).  \\nonumber\n\\end{eqnarray}\nAs observed, for small values of $R_\\text{t}$, the throughput increases with the rate almost linearly, because with high probability the data is correctly decoded. On the other hand, the outage probability increases and the throughput goes to zero for large values of $R_\\text{t}$. Moreover, depending on the SNR, there may be a finite optimum for the target rate in terms of throughput.\n\nIn Figs. 7 and 8, the effect of the number of receiving antennas $N$ is assessed on the achievable rate and the outage probability of the proposed algorithm when each transmitter has one antenna. The outage probability is defined as $\\Pr(R_\\text{achievable}<R_\\text{t})$. Here, $R_\\text{t}$ = 1 bit\/channel use is considered. As can be observed from Fig. 7, the achievable rate increases with the number of antennas $N$. For example, in sum rate of 2 bit\/channel use, the system with $N = 4$ improves the power efficiency by 4 dB and 10 dB compared to the cases with $N = 3$ and $N = 2$, respectively. Also, from Fig. 8, the IFMR scheme results in diversity, i.e., the slope of the outage probability curves at high SNRs, approximately equal to $N$.\n\n\n\n \n\\begin{figure}[t!]\n\\centering\n\\center\n\\vspace{-1ex}\n\\includegraphics[width =4.5in]{oneantenna.eps}\n\\caption{Achievable rate of IFMR vs conventional MMSE, ZF, and IFLR for SISO, i.e., $1\\times1$ MIMO, three pair interference channel.}\n\\end{figure}\n\n\n\n\\begin{figure}[t!]\n\\centering\n\\center\n\\vspace{-1ex}\n\\includegraphics[width =4.5in]{twonew.eps}\n\\caption{Achievable rate of IFMR vs conventional MMSE, ZF, and IFLR for $2\\times2$ MIMO three pair interference channel.}\n\n\\end{figure}\n\n\\begin{figure}[t!]\n\\centering\n\\center\n\\vspace{-1ex}\n\\includegraphics[width =4.5in]{iteration2.eps}\n\\caption{The average number of required iterations of IFMR for $2\\times2$ MIMO three pair interference channel.}\n\n\\end{figure}\n\\begin{figure}[t!]\n\\centering\n\\center\n\\vspace{-1ex}\n\\includegraphics[width =4.5in]{throput.eps}\n\\caption{Throughput versus the target rate $R_\\text{t}$ for $2\\times2$ MIMO three pair interference channel.}\n\n\\end{figure}\n\\begin{figure}[t!]\n\\centering\n\\center\n\\vspace{-1ex}\n\\includegraphics[width =4.5in]{multiantenna.eps}\n\\caption{Achievable rate of IFMR for SIMO, i.e., $1\\times N$ MIMO, three pair interference channel with $N$ receiving antennas.}\n\n\\end{figure}\n\n\\begin{figure}[t!]\n\\centering\n\\center\n\\vspace{-1ex}\n\\includegraphics[width =4.5in]{outage.eps}\n\\caption{Outage probability of IFMR for SIMO, i.e., $1\\times N$ MIMO, three pair interference channel with $N$ receiving antennas and $R_\\text{t}$ = 1 bit\/channel use.}\n\n\\end{figure}\n\n\n\n\\section{Conclusion}\nIn this paper, we proposed a low-complexity linear receiver scheme, referred to as IFMR, for interference channels. In IFMR, an integer combination of the desirable messages of each receiver can be recovered with the help of only two equations independently of the number of transmitters and data streams. We first proved that the sequential selection of the integer combinations can achieve the same rate as in the optimally joint selection. Then, we proposed a suboptimal algorithm to optimize the required equations and integer combinations in polynomial time and proved its convergence. Despite of its much less complexity for IFMR, our proposed algorithm can achieve almost the same performance as in the exhaustive search scheme. The IFMR scheme also shows a significantly better performance, in terms of the achievable rate, in comparison with the MMSE and ZF schemes.\n\\appendices\n\\section{Proof of Theorem 1}\nLet the independent DCM coefficient vectors $\\left\\{ {{\\mathbf{g}_1^k}, \\ldots ,{\\mathbf{g}_{N_t}^k}} \\right\\}$ be selected by the sequential method in (10). According to the constraint in (10), we have $R_{\\text{DCM}}\\left( {{\\mathbf{g}_1^k}} \\right) \\ge R_{\\text{DCM}}\\left( {{\\mathbf{g}_2^k}} \\right) \\ge  \\ldots  \\ge R_{\\text{DCM}}\\left( {{\\mathbf{g}_{N_t}^k}} \\right)$. Hence, the achievable rate of the sequential technique is ${R_{\\text{seq}}} = {N_{{t}}} \\times {R_{\\text{DCM}}}\\left( {{\\mathbf{g}_{N_{{t}}}^k}} \\right)$. Suppose also that the independent set $\\left\\{ {\\mathbf{d}_1^k}, \\ldots ,{\\mathbf{d}_{N_t}^k} \\right\\}$, i.e., $\\text{rank}\\left\\{ {\\mathbf{d}_1^k}, \\ldots ,{\\mathbf{d}_{N_t}^k} \\right\\} = {N_t}$, are the optimum solution of (9). Without loss of generality, assume that $R_{\\text{DCM}}\\left( {\\mathbf{d}_1^k} \\right) \\ge R_{\\text{DCM}}\\left( {\\mathbf{d}_2^k} \\right) \\ge  \\ldots  \\ge R_{\\text{DCM}}\\left( {\\mathbf{d}_{N_t}^k} \\right)$. Thus, the achievable rate of the optimal technique is ${R_{\\text{opt}}} = {N_{t}} \\times {R_{\\text{DCM}}}\\left( {\\mathbf{d}_{N_{t}}^k} \\right)$. \n\nUsing contradiction, assume $R_{\\text{opt}} > R_{\\text{seq}}$. Hence, ${R_{\\text{DCM}}}\\left( {{\\mathbf{d}_{N_{t}}^k}} \\right) > {R_{\\text{DCM}}}\\left( {{\\mathbf{g}_{N_{t}}^k}} \\right)$. From (10), $\\mathbf{g}_{N_{{t}}}^k$ is obtained from two equations which have the maximum achievable rate among all set of two equations whose associated DCM coefficient vectors are linearly independent of $\\left\\{ {\\mathbf{g}_1^k}, \\ldots ,{\\mathbf{g}_{{N_t}-1}^k} \\right\\}$. This implies that every DCM coefficient vector with a rate higher than ${R_{\\text{DCM}}}\\left( \\mathbf{g}_{N_{t}}^k \\right)$ is linearly dependent to the set $\\left\\{ {{\\mathbf{g}_1^k}, \\ldots ,{\\mathbf{g}_{{N_t}-1}^k}} \\right\\}$. Thus, we conclude $\\mathbf{d}_{N_{t}}^k$ exists in the $\\text{span}\\left\\{ {\\mathbf{g}_1^k}, \\ldots ,{\\mathbf{g}_{{N_t}-1}^k} \\right\\}$. As a result, for all ${\\mathbf{d}_{{i}}^k}, \\forall i \\le N_{t}$, we have \n\\begin{eqnarray}\n\\left\\{ {{\\mathbf{d}_1^k}, \\ldots ,{\\mathbf{d}_{{N_t}}^k}} \\right\\} \\in \\text{span}\\left\\{ {{\\mathbf{g}_1^k}, \\ldots ,{\\mathbf{g}_{{N_t}-1}^k}} \\right\\},\n\\end{eqnarray}\nwhich indicates that $\\text{rank}\\left\\{ {{\\mathbf{d}_1^k}, \\ldots ,{\\mathbf{d}_{{N_t}}^k}} \\right\\} \\le {N_t}-1$. However, this contradicts the assumption of linear-independency of these equations. Hence, ${R_{\\text{opt}}} = {R_{\\text{seq}}}$.\n\n\\section{Proof of Lemma 1}\nFor every vector $\\mathbf{y} \\ne 0$ in $\\mathbf{R}^{L\\times 1}$, we can write\n\\begin{eqnarray}\n{\\bf{\\Gamma }} \\buildrel \\Delta \\over = {\\mathbf{y}^*}\\left( {{\\mathbf{I}} - \\frac{{\\mathbf{x}{\\mathbf{x}^*}}}{{{\\mathbf{x}^*}\\mathbf{x}}}} \\right)\\mathbf{y} = {\\mathbf{y}^*}\\mathbf{y} - {\\mathbf{y}^*}\\frac{{\\mathbf{x}{\\mathbf{x}^*}}}{{{\\mathbf{x}^*}\\mathbf{x}}}\\mathbf{y} = {\\mathbf{y}^*}\\mathbf{y} - \\frac{1}{{{\\mathbf{x}^*}\\mathbf{x}}}{\\mathbf{y}^*}\\mathbf{x}{\\mathbf{x}^*}\\mathbf{y}.\n\\end{eqnarray}\nThen, from the Cauchy-Schwarz inequality ${\\mathbf{y}^*}\\mathbf{x}{\\mathbf{x}^*}\\mathbf{y} \\le \\left( {{\\mathbf{y}^*}\\mathbf{y}} \\right)\\left( {{\\mathbf{x}^*}\\mathbf{x}} \\right)$, we conclude ${\\bf{\\Gamma }} \\ge 0$. Thus, ${\\mathbf{I}} - \\frac{{\\mathbf{x}{\\mathbf{x}^*}}}{{{\\mathbf{x}^*}\\mathbf{x}}}$ is semi-definite. \n\n\n\\section{Proof of Theorem 2}\nFrom the definition of $\\mathbf{U}$ in (14) and adding then subtracting a term, we can write\n\\begin{multline}\n\\mathbf{U} = \\left[ {\\begin{array}{*{20}{c}}\n{{\\mathbf{a}^k}^*{\\mathbf{a}^k}}&0\\\\\n0&{{\\mathbf{c}^k}^*{\\mathbf{c}^k}}\n\\end{array}} \\right] - \\left[ {\\begin{array}{*{20}{c}}\n{{\\mathbf{a}^k}^*\\mathbf{H}_{kk}^*}\\\\\n{{\\mathbf{c}^k}^*\\mathbf{H}_k^*}\n\\end{array}} \\right]{\\left( {\\frac{1}{{\\text{SNR}}}{\\mathbf{I}} + {\\mathbf{H}_{kk}}\\frac{{{\\mathbf{a}^k}{\\mathbf{a}^k}^*}}{{{\\mathbf{a}^k}^*{\\mathbf{a}^k}}}\\mathbf{H}_{kk}^* + {\\mathbf{H}_k}\\frac{{{\\mathbf{c}^k}{\\mathbf{c}^k}^*}}{{{\\mathbf{c}^k}^*{\\mathbf{c}^k}}}\\mathbf{H}_k^*} \\right)^{ - 1}}\\\\ \\times \\left[ {\\begin{array}{*{20}{c}}\n{{\\mathbf{H}_{kk}}{\\mathbf{a}^k}}&{{\\mathbf{H}_k}{\\mathbf{c}^k}}\n\\end{array}} \\right]+ \\left[ {\\begin{array}{*{20}{c}}\n{{\\mathbf{a}^k}^*\\mathbf{H}_{kk}^*}\\\\\n{{\\mathbf{c}^k}^*\\mathbf{H}_k^*}\n\\end{array}} \\right]{\\left( {\\frac{1}{{\\text{SNR}}}{\\mathbf{I}} + {\\mathbf{H}_{kk}}\\frac{{{\\mathbf{a}^k}{\\mathbf{a}^k}^*}}{{{\\mathbf{a}^k}^*{\\mathbf{a}^k}}}\\mathbf{H}_{kk}^* + {\\mathbf{H}_k}\\frac{{{\\mathbf{c}^k}{\\mathbf{c}^k}^*}}{{{\\mathbf{c}^k}^*{\\mathbf{c}^k}}}\\mathbf{H}_k^*} \\right)^{ - 1}} \\left[ {\\begin{array}{*{20}{c}}\n{{\\mathbf{H}_{kk}}{\\mathbf{a}^k}}&{{\\mathbf{H}_k}{\\mathbf{c}^k}}\n\\end{array}} \\right] \\\\- \\left[ {\\begin{array}{*{20}{c}}\n{{\\mathbf{a}^k}^*\\mathbf{H}_{kk}^*}\\\\\n{{\\mathbf{c}^k}^*\\mathbf{H}_k^*}\n\\end{array}} \\right]{\\left( {\\frac{1}{{\\text{SNR}}}{\\mathbf{I}} + {\\mathbf{H}_{kk}}\\mathbf{H}_{kk}^* + {\\mathbf{H}_k}\\mathbf{H}_k^*} \\right)^{ - 1}}\\left[ {\\begin{array}{*{20}{c}}\n{{\\mathbf{H}_{kk}}{\\mathbf{a}^k}}&{{\\mathbf{H}_k}{\\mathbf{c}^k}}\n\\end{array}} \\right].\n\\end{multline}\nAccording to matrix inverses identities in [20, Eqs. (159) and (165)], we can rewrite (30) as\n\\begin{multline}\n\\mathbf{U}=\\biggl( \\left[ {\\begin{array}{*{20}{c}}\n{\\frac{1}{{{\\mathbf{a}^k}^*{\\mathbf{a}^k}}}}&0\\\\\n0&{\\frac{1}{{{\\mathbf{c}^k}^*{\\mathbf{c}^k}}}}\n\\end{array}} \\right] + \\text{SNR}\\left[ {\\begin{array}{*{20}{c}}\n{\\frac{1}{{{\\mathbf{a}^k}^*{\\mathbf{a}^k}}}{\\mathbf{a}^k}^*\\mathbf{H}_{kk}^*}\\\\\n{\\frac{1}{{{\\mathbf{c}^k}^*{\\mathbf{c}^k}}}{\\mathbf{c}^k}^*\\mathbf{H}_k^*}\n\\end{array}} \\right] \\left[ {\\begin{array}{*{20}{c}}\n{\\frac{1}{{{\\mathbf{a}^k}^*{\\mathbf{a}^k}}}{\\mathbf{H}_{kk}}{\\mathbf{a}^k}}&{\\frac{1}{{{\\mathbf{c}^k}^*{\\mathbf{c}^k}}}{\\mathbf{H}_k}{\\mathbf{c}_k}}\n\\end{array}} \\right] \\biggr)^{ - 1} \\\\+ \\left[ {\\begin{array}{*{20}{c}}\n{{\\mathbf{a}^k}^*\\mathbf{H}_{kk}^*}\\\\\n{{\\mathbf{c}^k}^*\\mathbf{H}_k^*}\n\\end{array}} \\right] \\biggl\\{\\left( {\\frac{1}{{\\text{SNR}}}{\\mathbf{I}} + {\\mathbf{H}_{kk}}\\frac{{{\\mathbf{a}_k}{\\mathbf{a}_k}^*}}{{{\\mathbf{a}^k}^*{\\mathbf{a}^k}}}\\mathbf{H}_{kk}^* + {\\mathbf{H}_k}\\frac{{{\\mathbf{c}_k}{\\mathbf{c}_k}^*}}{{{\\mathbf{c}^k}^*{\\mathbf{c}^k}}}\\mathbf{H}_k^*} \\right)^{ - 1}\\\\ \\times \\left( {{\\mathbf{H}_{kk}}\\mathbf{H}_{kk}^* - {\\mathbf{H}_{kk}}\\frac{{{\\mathbf{a}_k}{\\mathbf{a}_k}^*}}{{{\\mathbf{a}^k}^*{\\mathbf{a}^k}}}\\mathbf{H}_{kk}^* + {\\mathbf{H}_k}\\mathbf{H}_k^* - {\\mathbf{H}_k}\\frac{{{\\mathbf{c}_k}{\\mathbf{c}_k}^*}}{{{\\mathbf{c}^k}^*{\\mathbf{c}^k}}}\\mathbf{H}_k^*} \\right)\n \\left( {\\frac{1}{{\\text{SNR}}}{\\mathbf{I}} + {\\mathbf{H}_{kk}}\\mathbf{H}_{kk}^* + {\\mathbf{H}_k}\\mathbf{H}_k^*} \\right)^{ - 1}\\biggr\\} \\\\ \\times \\left[ {\\begin{array}{*{20}{c}}\n{{\\mathbf{H}_{kk}}{\\mathbf{a}^k}}&{{\\mathbf{H}_k}{\\mathbf{c}_k}}\n\\end{array}} \\right].\n\\end{multline}\nIt is straightforward to show that matrices $\\mathbf{F}$, $\\mathbf{G}$, and $\\mathbf{T}$ with\n\\begin{eqnarray}\n\\mathbf{F} = \\biggl( \\left[ \\begin{array}{*{20}{c}}\n{\\frac{1}{{{\\mathbf{a}^k}^*{\\mathbf{a}^k}}}}&0\\\\\n0&{\\frac{1}{{{\\mathbf{c}^k}^*{\\mathbf{c}^k}}}}\n\\end{array} \\right] + \\text{SNR}\\left[ \\begin{array}{*{20}{c}}\n{\\frac{1}{{{\\mathbf{a}^k}^*{\\mathbf{a}^k}}}{\\mathbf{a}^k}^*\\mathbf{H}_{kk}^*}\\\\\n{\\frac{1}{{{\\mathbf{c}^k}^*{\\mathbf{c}^k}}}{\\mathbf{c}^k}^*\\mathbf{H}_k^*}\n\\end{array} \\right] \\left[ {\\begin{array}{*{20}{c}}\n{\\frac{1}{{{\\mathbf{a}^k}^*{\\mathbf{a}^k}}}{\\mathbf{H}_{kk}}{\\mathbf{a}^k}}&{\\frac{1}{{{\\mathbf{c}^k}^*{\\mathbf{c}^k}}}{\\mathbf{H}_k}{\\mathbf{c}^k}}\n\\end{array}} \\right] \\biggr)^{ - 1},\\nonumber\n\\end{eqnarray}\n\\begin{eqnarray}\n\\mathbf{G} = \\left( {\\frac{1}{{\\text{SNR}}}{\\mathbf{I}} + {\\mathbf{H}_{kk}}\\frac{{{\\mathbf{a}^k}{\\mathbf{a}^k}^*}}{{{\\mathbf{a}^k}^*{\\mathbf{a}^k}}}\\mathbf{H}_{kk}^* + {\\mathbf{H}_k}\\frac{{{\\mathbf{c}^k}{\\mathbf{c}^k}^*}}{{{\\mathbf{c}^k}^*{\\mathbf{c}^k}}}\\mathbf{H}_k^*} \\right)^{ - 1},\\nonumber\n\\end{eqnarray}\n\\begin{eqnarray}\n\\mathbf{T} = \\left( {\\frac{1}{{\\text{SNR}}}{\\mathbf{I}} + {\\mathbf{H}_{kk}}\\mathbf{H}_{kk}^* + {\\mathbf{H}_k}\\mathbf{H}_k^*} \\right)^{ - 1},\n\\end{eqnarray}\nare positive definite. \n\nAccording to Lemma 1, since ${\\mathbf{H}_{kk}}\\mathbf{H}_{kk}^* - {\\mathbf{H}_{kk}}\\frac{{{\\mathbf{a}^k}{\\mathbf{a}^k}^*}}{{{\\mathbf{a}^k}^*{\\mathbf{a}^k}}}\\mathbf{H}_{kk}^*$ and ${\\mathbf{H}_k}\\mathbf{H}_k^* - {\\mathbf{H}_k}\\frac{{{\\mathbf{c}^k}{\\mathbf{c}^k}^*}}{{{\\mathbf{c}^k}^*{\\mathbf{c}^k}}}\\mathbf{H}_k^*$ are positive semi-definite matrices, the matrix $\\mathbf{X}$ with\n\\begin{multline}\n\\mathbf{X}={\\left( {\\frac{1}{{\\text{SNR}}}{\\mathbf{I}} + {\\mathbf{H}_{kk}}\\frac{{{\\mathbf{a}^k}{\\mathbf{a}^k}^*}}{{{\\mathbf{a}^k}^*{\\mathbf{a}^k}}}\\mathbf{H}_{kk}^* + {\\mathbf{H}_k}\\frac{{{\\mathbf{c}^k}{\\mathbf{c}^k}^*}}{{{\\mathbf{c}^k}^*{\\mathbf{c}^k}}}\\mathbf{H}_k^*} \\right)^{- 1}}\\\\ \\times \\biggl( {{\\mathbf{H}_{kk}}\\mathbf{H}_{kk}^* - {\\mathbf{H}_{kk}}\\frac{{{\\mathbf{a}^k}{\\mathbf{a}^k}^*}}{{{\\mathbf{a}^k}^*{\\mathbf{a}^k}}}\\mathbf{H}_{kk}^* + {\\mathbf{H}_k}\\mathbf{H}_k^*- {\\mathbf{H}_k}\\frac{{{\\mathbf{c}^k}{\\mathbf{c}^k}^*}}{{{\\mathbf{c}^k}^*{\\mathbf{c}^k}}}\\mathbf{H}_k^*} \\biggr){\\left( {\\frac{1}{{\\text{SNR}}}{\\mathbf{I}} + {\\mathbf{H}_{kk}}\\mathbf{H}_{kk}^* + {\\mathbf{H}_k}\\mathbf{H}_k^*} \\right)^{ - 1}},\n\\end{multline}\n is also semi-definite. Hence, the overall matrix $\\mathbf{U}$, which is sum of a positive definite matrix and a semi-definite matrix, is positive definite.\n \n \n\\section{Details for the Solution of (24)}\nFor (24), we further define a function\n\\begin{eqnarray}\nV\\left( \\alpha  \\right) \\buildrel \\Delta \\over = {\\rm{mi}}{{\\rm{n}}_{{\\mathbf{a}^k} \\in {\\mathbf{R}^{{N_{{t_k}}} \\times 1}}}}V\\left( {\\alpha ,{\\mathbf{a}^k}} \\right) = V\\left( {\\alpha ,{\\mathbf{a}^k}^{\\left( \\alpha  \\right)}} \\right),\n\\end{eqnarray}\nwhere ${\\mathbf{a}^k}^{\\left( \\alpha  \\right)}$ minimizes $V(\\alpha,{\\mathbf{a}^k})$ for given $\\alpha$. \nLet ${\\alpha ^*}$ denote the solution of ${\\rm{ma}}{{\\rm{x}}_{0 \\le \\alpha  \\le 1}}V\\left( \\alpha  \\right)$. There are three cases according to the relationship of ${f_1}\\left( {{\\mathbf{a}^k}^{\\left( {{\\alpha ^*}} \\right)}},{\\mathbf{c}^{{k}}} \\right)$ and ${f_2}\\left( {{\\mathbf{a}^k}^{\\left( {{\\alpha ^*}} \\right)}},{\\mathbf{c}^{{k}}} \\right)$, one of which includes the solution of (24).\n\n\\underline{Case 1}: If ${\\alpha ^*} = 0$, we have \n\\begin{eqnarray}\n{f_1}\\left( {{\\mathbf{a}^k}^{\\left( 0 \\right)},{\\mathbf{c}^{k}}} \\right) \\le {f_2}\\left( {{\\mathbf{a}^k}^{\\left( 0 \\right)},{\\mathbf{c}^{k}}} \\right).\n\\end{eqnarray}\nHence, (24) is changed to\n\\begin{eqnarray}\n\\mathop {{\\rm{min}}}\\limits_{{\\mathbf{a}^k}^{\\left( 0 \\right)} \\in {\\mathbf{R}^{{N_{{t_k}}} \\times 1}}} {f_2}\\left( {{\\mathbf{a}^k}^{\\left( 0 \\right)},{\\mathbf{c}^{k}}} \\right),\n\\end{eqnarray}\nwhich can be effectively solved by setting the derivative of ${f_2}\\left( {{\\mathbf{a}^k}^{\\left( 0 \\right)},{\\mathbf{c}^{k}}} \\right)$ with respect to ${\\mathbf{a}^k}^{\\left( 0 \\right)}$ equal to zero. Hence, according to (12), the optimal ${\\mathbf{a}^k}^{\\left( 0 \\right)}$ is given by \n\\begin{multline}\n{\\nabla _{{\\mathbf{a}^k}}}{f_2}\\left( {{\\mathbf{a}^k},{\\mathbf{c}^{k}}} \\right) = \\biggl( {d{{_2^k}^2}\\mathbf{I} - \\mathbf{H}_{kk}^*{{\\biggl( {\\frac{1}{{\\text{SNR}}}\\mathbf{I} + {\\mathbf{H}_{kk}}\\mathbf{H}_{kk}^* + {\\mathbf{H}_k}\\mathbf{H}_k^*} \\biggr)}^{ - 1}} {\\mathbf{H}_{kk}}} \\biggr){\\mathbf{a}^k}\\\\ - e_2^kd_2^k\\mathbf{H}_{kk}^*{\\left( {\\frac{1}{{\\text{SNR}}}\\mathbf{I} + {\\mathbf{H}_{kk}}\\mathbf{H}_{kk}^* + {\\mathbf{H}_k}\\mathbf{H}_k^*} \\right)^{ - 1}}{\\mathbf{H}_k}{\\mathbf{c}^{k}} = 0,\n\\end{multline}\nwhich respectively leads to\n\\begin{multline}\n{\\mathbf{a}^k}^{\\left( 0 \\right)} = {e_2^k}{d_2^k}\\biggl( d{{_2^k}^2}\\mathbf{I} - \\mathbf{H}_{kk}^*{\\left( {\\frac{1}{{\\text{SNR}}}\\mathbf{I} + {\\mathbf{H}_{kk}}\\mathbf{H}_{kk}^* + {\\mathbf{H}_k}\\mathbf{H}_k^*} \\right)}^{ - 1} {\\mathbf{H}_{kk}} \\biggr)^{ - 1} \\\\ \\times \\mathbf{H}_{kk}^*\\left( \\frac{1}{{\\text{SNR}}}\\mathbf{I} + {\\mathbf{H}_{kk}}\\mathbf{H}_{kk}^*+ {\\mathbf{H}_k}\\mathbf{H}_k^* \\right)^{ - 1}{\\mathbf{H}_k}{\\mathbf{c}^{{k}}}.\n\\end{multline}\n\\underline{Case 2}: If ${\\alpha ^*} = 1$, then we have \n\\begin{eqnarray}\n{f_1}\\left( {{\\mathbf{a}^k}^{\\left( 1 \\right)},{\\mathbf{c}^{k}}} \\right) \\ge {f_2}\\left( {{\\mathbf{a}^k}^{\\left( 1 \\right)},{\\mathbf{c}^{k}}} \\right).\n\\end{eqnarray}\nThus, similar to Case 1, we can find the ${\\mathbf{a}^k}^{\\left( 1 \\right)}$ as\n\\begin{multline}\n{\\mathbf{a}^k}^{\\left( 1 \\right)} = {e_1^k}{d_1^k}\\biggl( d{{_1^k}^2}\\mathbf{I} - \\mathbf{H}_{kk}^*{{\\left( {\\frac{1}{{\\text{SNR}}}\\mathbf{I} + {\\mathbf{H}_{kk}}\\mathbf{H}_{kk}^* + {\\mathbf{H}_k}\\mathbf{H}_k^*} \\right)}^{ - 1}}{\\mathbf{H}_{kk}} \\biggr)^{ - 1} \\\\ \\times\\mathbf{H}_{kk}^*{\\left( {\\frac{1}{{\\text{SNR}}}\\mathbf{I} + {\\mathbf{H}_{kk}}\\mathbf{H}_{kk}^* + {\\mathbf{H}_k}\\mathbf{H}_k^*} \\right)^{ - 1}}{\\mathbf{H}_k}{\\mathbf{c}^{k}}.\n\\end{multline}\n\\underline{Case 3}: If $0 < {\\alpha ^*} < 1$, then we have \n\\begin{eqnarray}\n{f_1}\\left( {{\\mathbf{a}^k}^{\\left( {{\\alpha ^*}} \\right)},{\\mathbf{c}^{k}}} \\right) = {f_2}\\left( {{\\mathbf{a}^k}^{\\left( {{\\alpha ^*}} \\right)},{\\mathbf{c}^{k}}} \\right),\n\\end{eqnarray}\nin which ${\\alpha ^*}$ can be found by the Bisection method. In this case, (24) is rephrased as\n\\begin{eqnarray}\n{\\rm{mi}}{{\\rm{n}}_{{\\mathbf{a}^k}^{\\left( {{\\alpha ^*}} \\right)} \\in {\\mathbf{R}^{{N_{{t_k}}} \\times 1}}}}V\\left( {{\\alpha ^*},{\\mathbf{a}^k}^{\\left( {{\\alpha ^*}} \\right)}} \\right) ={\\alpha ^*}{f_1}\\left( {{\\mathbf{a}^k}^{\\left( {{\\alpha ^*}} \\right)},{\\mathbf{c}^{k}}} \\right) + \\left( {1 - {\\alpha ^*}} \\right){f_2}\\left( {{\\mathbf{a}^k}^{\\left( {{\\alpha ^*}} \\right)},{\\mathbf{c}^{k}}} \\right),\n\\end{eqnarray}\nwhich can be solved by setting the derivative of $V\\left( {{\\alpha ^*},{\\mathbf{a}^k}^{\\left( {{\\alpha ^*}} \\right)}} \\right)$ with respect to ${\\mathbf{a}^k}^{\\left( {{\\alpha ^*}} \\right)}$ equal to zero. With the same arguments and using some manipulations, ${\\mathbf{a}^k}^{\\left( {{\\alpha ^*}} \\right)} $ is obtained by\n\\begin{multline}\n{\\mathbf{a}^k}^{\\left( {{\\alpha ^*}} \\right)} = u^k({\\alpha ^*}){\\left( {v^k({\\alpha ^*})\\mathbf{I} - \\mathbf{H}_{kk}^*{{\\left( {\\frac{1}{{\\text{SNR}}}\\mathbf{I} + {\\mathbf{H}_{kk}}\\mathbf{H}_{kk}^* + {\\mathbf{H}_k}\\mathbf{H}_k^*} \\right)}^{ - 1}}{\\mathbf{H}_{kk}}} \\right)^{ - 1}}\\\\ \\times \\mathbf{H}_{kk}^*{\\left( {\\frac{1}{{\\text{SNR}}}\\mathbf{I} + {\\mathbf{H}_{kk}}\\mathbf{H}_{kk}^* + {\\mathbf{H}_k}\\mathbf{H}_k^*} \\right)^{ - 1}}{\\mathbf{H}_k}{\\mathbf{c}^{k}},\n\\end{multline}\nwhere \n\\begin{eqnarray}\nu^k({\\alpha ^*}) \\buildrel \\Delta \\over = {\\alpha ^*}{e_1^k}{d_1^k} + \\left( {1 - {\\alpha ^*}} \\right){e_2^k}{d_2^k} ,\\nonumber\n\\end{eqnarray}\n\\vspace{-20pt}\n\\begin{eqnarray}\nv^k({\\alpha ^*})\\buildrel \\Delta \\over =  {\\alpha ^*}d{_1^k}^2 + \\left( {1 - {\\alpha ^*}} \\right)d{_2^k}^2 .\n\\end{eqnarray}\n\n\n\n\\section{Proof of Lemma 2}\nAn equation with ECV $\\mathbf{a}_l^k$ has rate (5) equal to zero if ${\\left| {\\left| {{\\mathbf{a}_l^k}} \\right|} \\right|^2} \\geq 1+{\\rm{\\text{SNR}}}{\\lambda _{{\\rm{max}}}^2}\\left( {{{\\mathbf{\\hat H}}_k}} \\right)$ [5]. Thus, the rate (7) is zero if \n\\begin{eqnarray}\n{e_1^2\\left| {\\left| {{\\mathbf{a}^k}} \\right|} \\right|^2+d_1^2\\left| {\\left| {{\\mathbf{c}^k}} \\right|} \\right|^2} \\geq 1+{\\rm{\\text{SNR}}}{\\lambda _{{\\rm{max}}}^2}\\left( {{{\\mathbf{\\hat H}}_k}} \\right),\n\\end{eqnarray}\nor\n\\begin{eqnarray}\n {e_2^2\\left| {\\left| {{\\mathbf{a}^k}} \\right|} \\right|^2+d_2^2\\left| {\\left| {{\\mathbf{c}^k}} \\right|} \\right|^2} \\geq 1+{\\rm{\\text{SNR}}}{\\lambda _{{\\rm{max}}}^2}\\left( {{{\\mathbf{\\hat H}}_k}} \\right).\n \\end{eqnarray}\n Accordingly, we should have\\\\ \n${\\left| {\\left| {{\\mathbf{a}^k}} \\right|} \\right|^2} \\geq \\frac{1}{{e_1^2}}\\left( {1 + {\\rm{\\text{SNR}}}{\\lambda _{{\\rm{max}}}^2}\\left( {{{\\mathbf{\\hat H}}_k}} \\right) - d_1^2{\\left| {\\left| {{\\mathbf{c}^k}} \\right|} \\right|^2}} \\right)$ or $\\frac{1}{{e_2^2}}\\left( {1 + {\\rm{\\text{SNR}}}{\\lambda _{{\\rm{max}}}^2}\\left( {{{\\mathbf{\\hat H}}_k}} \\right) - d_2^2{\\left| {\\left| {{\\mathbf{c}^k}} \\right|} \\right|^2}} \\right)$, which completes the proof.\n\\section{Proof of Theorem 3}\nFor each $t$, assume $\\epsilon^j (\\mathbf{a}^{k,j},\\mathbf{c}^{k,j}) = \\mathop {{\\rm{max}}}\\limits_{l = 1,2} {f_l}^j\\left( {{\\mathbf{a}^{k,j}},{\\mathbf{c}^{k,j}}} \\right)$, where ${f_l}^j\\left( {{\\mathbf{a}^{k,j}},{\\mathbf{c}^{k,j}}} \\right)$ corresponds to the $j$-th iteration. For the iteration $j+1$ of Step I, we have $\\epsilon^{j+1} (\\mathbf{a}^{k,j},\\mathbf{c}^{k,j}) \\leq \\epsilon^j (\\mathbf{a}^{k,j},\\mathbf{c}^{k,j})$, in Step II, $\\epsilon^{j+1} (\\mathbf{a}^{k,j},\\mathbf{c}^{k,j+1} ) \\leq \\epsilon^{j+1} (\\mathbf{a}^{k,j},\\mathbf{c}^{k,j})$, and in Step III, $\\epsilon^{j+1} (\\mathbf{a}^{k,j+1},\\mathbf{c}^{k,j+1} ) \\leq \\epsilon^{j+1} (\\mathbf{a}^{k,j},\\mathbf{c}^{k,j+1})$. According to $f_l (\\mathbf{a}^k,\\mathbf{c}^k)$, the latter is guaranteed when we assume the quantization sphere has sufficiently large radius to find a suitable $\\mathbf{a}^{k,j+1}$. Even for a small quantization sphere with no candidate, we can update as $\\mathbf{a}^{k,j+1} = \\mathbf{a}^{k,j}$ which in the worst case of $\\mathbf{c}^{k,j+1} = \\mathbf{c}^{k,j}$ leads to $\\epsilon^{j+1} (\\mathbf{a}^{k,j+1},\\mathbf{c}^{k,j+1} ) = \\epsilon^{j+1} (\\mathbf{a}^{k,j},\\mathbf{c}^{k,j})$. Hence, $\\epsilon^{j+1} (\\mathbf{a}^{k,j+1},\\mathbf{c}^{k,j+1}) \\leq \\epsilon^j (\\mathbf{a}^{k,j},\\mathbf{c}^{k,j})$ at the end of iteration $j+1$. In this way, in each iteration, the function $\\epsilon = \\mathop {{\\rm{max}}}\\limits_{l = 1,2} f_l (\\mathbf{a}^k,\\mathbf{c}^k )$ either decreases or remains unchanged, and is lower bounded by zero. Thus, the proposed algorithm is convergent. \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIt is accepted that the energy source that sustains the high temperature of solar corona is in the magnetic field \\citep{2006SoPh..234...41K}. Regarding the way in which this energy is converted to thermal energy, we can broadly distinguish between two scenarios that depend on the timescale of the energy release: one is continuous heating, the other is in the form of discrete, rapid pulses. The latter is consistent with the nanoflare model proposed by  \\cite{1988ApJ...330..474P}. According to this model, the magnetic field tubes are displaced by the photospheric motions, and can approach and interact. When two flux tubes are almost in contact they will form a current sheet, where the field lines can reconnect. The reconnection can release a large quantity of energy in impulsive events called nanoflares. Signatures of these events are difficult to observe for several reasons, one of which is the fine structuring of the magnetic loops, that is hardly resolved with present-day instruments (e.g., \\citealt{2013ApJ...770L...1T}). \n\nOne feature to discriminate the heating release is the presence or absence of very hot plasma. In active regions, the mean coronal temperature is typically 2-3 MK. If heat pulses occur, we expect that a small amount of plasma hotter than the average (6-10 MK) will be ever-present. \n\nRecently, a number of studies, mostly based on data from the Hinode and the Solar Dynamics Observatory (SDO) missions, have shown increasing evidence for such small very hot components in active regions (\\citealt{2009ApJ...704L..58R}, \\citealt{2009ApJ...698..756R}, \\citealt{2009ApJ...697...94M}, \\citealt{2009ApJ...693L.131S}, \\citealt{2009ApJ...697.1956K}, \\citealt{2009ApJ...696..760P}, \\citealt{2010AstL...36...44S}, \\citealt{2010A&A...514A..82S}), but the issue is still debated (\\citealt{2012ApJ...754L..40T}, \\citealt{2011ApJ...734...90W}). In SDO images taken with a channel sensitive also to emission of plasma at 6 MK (94 \\AA), cores of active regions contain bright strands \\citep{2011ApJ...736L..16R}, as predicted by models of nanoflaring loops \\citep{2010ApJ...719..576G}. It remains to be proven whether this plasma is really at such high temperatures or not. Spectroscopic observations should help greatly. In this work, we analyse an active region that shows evidence for this very hot component in SDO data, but for which spectroscopic data are also available from the EUV spectrometer EIS on-board the Hinode mission. In a previous work  (\\citealt{2012ApJ...750L..10T}, hereafter Paper I), emission in the CaXVII line, which forms around temperature of 6-8MK, was detected in the hot structures identified with SDO\/AIA data. In that work they built a 3 color image, to highlight the presence of a very hot component of emitting plasma inside the active region. The AIA 94~\\AA\\ band  is known to be multi-thermal.\n It is sensitive to hot plasma, due to the presence of an \nFe XVIII line, formed around 6~MK, but is also sensitive to plasma at 1 MK, because of the presence of \na Fe X line and  cooler Fe IX and Fe VIII (see, e.g., \\citealt{2011A&A...535A..46D}, \\citealt{2012ApJ...745..111T}, \\citealt{2011ApJ...743...23M}, \\citealt{2011ApJ...740L..52F}, \\citealt{2012A&A...537A..22O}, \\citealt{2012A&A...546A..97D}).\nRecently, also a Fe XIV line was identified in  \\cite{2012A&A...546A..97D}. This line is normally stronger than the other cool components in \nactive region cores, as shown in \\cite{2013A&A...558A..73D}. It is therefore not simple to assess if the \nhot emission seen in the AIA 94~\\AA\\ band is really due to Fe XVIII. To clarify this point, \\cite{2012ApJ...750L..10T}\n compared the AIA 94~\\AA\\ image with the Ca XVII image obtained from the EIS spectrometer. They showed a strong correlation between the hot CaXVII and the emission in the 94\\AA\\ AIA band, so concluded that the hot emission seen in the  AIA 94~\\AA\\ band \nis effectively due to very hot plasma (6-8MK). More direct evidence has been found from observations of another Fe XVIII line by the  SUMER spectrometer on board the SOHO mission \\citep{2012ApJ...754L..40T}. \n\n\nHowever, even if the indication is rather strong, it is still not enough to establish that the plasma is actually so hot, since in theory it is possible that plasma at a lower temperature, but with very high emission measure, can give the same line intensity, as suggested by \n\\cite{2012ApJ...754L..40T}. Indeed  \\cite{2013A&A...558A..73D} used simultaneous EIS and AIA observations of \nactive regions cores to show that a significant fraction of the Fe XVIII 94~\\AA\\ intensity can be due to plasma  at 3~MK and not\n6~MK. In fact,  \\cite{2013A&A...558A..73D} showed that often Fe XVIII 94~\\AA\\ emission is present in the cores of active regions, \nbut Ca XVII (which has a narrower formation temperature, hence is sensitive to hotter plasma) is not.\nThe only way to disentangle the various contributions to the AIA 94~\\AA\\ band  is therefore  to perform an emission measure modelling.\nTo this purpose, here we use the same observations from Hinode and SDO as in Paper I, that include both high-resolution spectroscopic data, over a wide spectral window, and images with high spatial resolution. \n\nAll this information provides simultaneous constraints on the plasma thermal structure along the line of sight. To further support the analysis we replicate the same analysis  on the same data set but taking a region outside of the core that shows no evidence of these very hot components.We also compare two different inversion methods.\nIn Section~\\ref{sec:obs} we describe the observation and the data analysis, and in Section~\\ref{sec:discuss} the results are discussed.\n\n\\section{Observation and data analysis}\n\\label{sec:obs}\n\nWe analyse the active region (AR 11289) observed on 2011-09-13, from 10.30 to 11.30 UT. Our analysis is focused on spectroscopic data of EIS on board of Hinode \\citep{2007SoPh..243...19C}. We analyse data from a study designed by one of us (GDZ), called ATLAS\\_60, where the entire full spectral range, 178-213 \\AA\\ and 245-290 \\AA, is extracted. \nThe observations are obtained by stepping the 2$\"$ slit from solar west to east, with a 120$\"$x160$\"$ field of view. The exposure time was 60 s. The EIS data were processed using eis\\_prep, available in SolarSoft. This routine removes CCD dark current, cosmic ray strikes and takes into account hot, warm and dusty pixels. After that, radiometric calibration was applied to convert digital data (related to photon counts) into physical units (erg s$^{-1}$ cm$^{-2}$ sr$^{-1}$ \\AA). \nWe then re-aligned the fields of view by correcting for the wavelength offset of the two CCDs (by using the EIS routine eis\\_ccd\\_offset), obtained a new cropped field of view 120$\" \\times 140\"$. \n\n\n\\begin{figure*}[t!]\n\t\\centering\n\t\\includegraphics[scale=0.8]{lines.ps}  \n\t\\caption{EIS, AIA and XRT images of the analysed region, i.e. 6 EIS spectral lines (Fe VIII, Fe IX, Fe XII, Fe XV, Fe XVI, Ca XVII, see Table~\\ref{tab:eislines}), 2 AIA channels (171 \\AA, 94 \\AA), and 1 XRT channel (Ti\\_poly). The position of the hot (H) and cold (C) regions analysed here are marked in all images.}\n\t\\label{fig:lines}\n\\end{figure*}\n\nIn the same temporal window we selected frames from two imagers, the X-ray telescope (XRT, \\citealt{2007SoPh..243...63G}) on board Hinode (512$\"$x512$\"$ in Ti\\_poly filter, exposure times between 0.7 and 1 s) and the Atmospheric Imaging Assembly (AIA, \\citealt{2012SoPh..275...17L}) on board SDO (full disk in 171 \\AA, 335 \\AA, 94 \\AA\\ channels, exposure times 2, 2.9, 2.9 s, respectively) . \n\nThe two sets of images were processed with the standard routines available in the Solar Soft package (xrt\\_prep and aia\\_prep). The images of the two instruments were co-aligned to each other (tr\\_get\\_disp.pro in SolarSoftware package) and to the EIS raster image in the He II 256 \\AA\\ line, to match the EIS field of view. To improve the homogeneity between the XRT or AIA images and the EIS images we have to consider that the latter are built from rastering that takes some time, while the former are \"instantaneous\". We then built-up new composite XRT and AIA images, that account for this different time spacing, as follows: each vertical strip is extracted from the images closest in time to the time when the EIS slit was at that location. XRT errors were computed taking into account the effect of the deterioration of the CCD response \\citep{2013arXiv1312.4850K}. Fig.\\ref{fig:lines} shows EIS, AIA and XRT representative images of the same field of view and with the same timing. The EIS images are obtained by integrating the spectrum in a narrow band (0.1 \\AA wide) centered at the position of each line. \n\n\nThe information available for very hot plasma from EIS data is mostly based on the Ca XVII line, the strongest EIS line formed around 6 MK \\citep{2008A&A...481L..69D}. This line is severely blended with Fe XI and O V lines, as discussed in \\cite{2007PASJ...59S.857Y}, \\cite{2009A&A...508..501D}, \\cite{2011A&A...526A...1D}, \\cite{2012ApJ...750L..10T}. As in Paper I in the case of the Ca XVII line we use the procedure developed by \\cite{2009ApJ...697.1956K} for de-blending the line from Fe XI and O V lines, and extract the Ca XVII emission.\n\nAIA and XRT images are composite images as explained above. The line images span emission from plasma in a broad temperature range $5.7 < \\log T < 6.7$. The \"cold\" ones ($\\log T < 6$) show a very bright feature consisting of \"fan\" loops in the top-left region, consistently with the image in the AIA 171 \\AA\\ channel. In the images around $\\log T = 6.4$ the morphology becomes very different: the whole region in the bottom part is quite bright and closed loops are visible, while where the bright \"cold\" features are present there is little emission.\nThat hot and cold emission is often not co-spatial has been known for a long time, and is often true even at the \nhigh spatial resolution of AIA \\citep{2013A&A...558A..73D}.\nThe Ca XVII image shows only two bright loops bifurcating southwards from a common point near the middle of the field of view. \nThese features clearly show similar morphology to the AIA 94 \\AA\\ emission, and are the same features found in Paper I. Here we also analyze XRT observations, which show that these loops are also bright in the X-rays.\nWe see also some other emission around that recalls the EIS Fe XV and Fe XVI images.\n\n\\begin{center}\n\\begin{table*}[htb!]\n\\begin{center}\n\\caption{EIS line fluxes measured after fitting each selected line with a Gaussian profile, in the two regions (cold, hot) marked in Fig.\\ref{fig:aia3col}. The ratio of the model to the observed fluxes are also reported for the DEM reconstructions with the MCMC and Del Zanna (DZ) methods (see Section~\\ref{subsec:DEMrec}).}\n\\begin{tabular}{lccccccccc} \n\\hline\\hline\n {\\bf Line}\t&   Wavelength \t&\tTemperature & Cold region &Ratio &Ratio &Hot region &Ratio &Ratio \\\\\n  \t&   (\\AA)\t&\t ($\\log (T)$)& (erg cm$^{-2}$ sr$^{-1}$ s$^{-1}$)&DZ&MCMC&(erg cm$^{-2}$ sr$^{-1}$ s$^{-1}$)&DZ&MCMC\\\\\n  \\hline\n\tFeVIII & $185.21$ & $5.7$ & $4250 \\pm 80$&1.03& 1.07 &  $380 \\pm 26$&1.33&0.99\\\\\n\tMgVII &\t$278.40$ & $5.8$&$3240 \\pm 90$&0.3& 0.35&  $65  \\pm   10$&0.25&0.95\\\\       \n\tFeIX & $197.862$\t& $5.9$&$380 \\pm 12$&1.00& 1.13& $27 \\pm    3$&0.97&1.09\\\\ \n\tFeX\t& $184.536$\t& $6.0$&$1186  \\pm    45$&0.89&0.81   &   $255   \\pm   23$&0.94&0.58\\\\\n\tFeXI\t& $180.401$\t& $6.15$&$2270    \\pm  140$&1.07&0.83  &  $ 1340   \\pm   110$&1.11&0.76\\\\\t\n\tFeXII\t& $195.119$\t& $6.20$&$1354   \\pm   20 $&1.02&1.04  & $  1546   \\pm   21$&1.09&1.06\\\\\n\tFeXIII\t&$ 202.044$\t& $6.25$& $1001  \\pm    34 $&0.49& 0.57 & $  1650   \\pm   40$&0.60&0.58\\\\\n\tFeXIV   &$ 211.318 $& $6.3$& $1120  \\pm    100 $&0.85&1.15 & $   3420   \\pm   170$&0.88&0.86\\\\\t  \n\tFeXV\t& $284.160\t$&$ 6.35$&$3700  \\pm    120$&1.13&1.08   &  $ 20070    \\pm  280$&1.17&1.07\\\\\n\tFeXVI\t& $262.984\t$&$ 6.45$&$180   \\pm   20   $&0.8&0.56& $  2870   \\pm   80$&0.79&0.81\\\\\t  \n\tSXIII\t&$ 256.685\t$& $6.45$&$399   \\pm   44  $&0.49&0.37 & $  2380   \\pm   100$&0.78&0.65\\\\\t   \n\tCaXIV  & $193.866\t$&$ 6.6$&$13   \\pm   2  $&1.13&0.49 &$   512  \\pm    13$&0.95&1.13\\\\      \n\tCaXV\t& $200.972$ &$ 6.65$&$46  \\pm    5  $&0.10& 0.05& $  650   \\pm   20$&0.56&0.73\\\\\t \n\tCaXVII\t& $192.858$ &$ 6.70$&$0  \\pm    2  $&1.05& 0.99&$   540  \\pm    10$&0.98&1.00\\\\     \n\\hline\n\\label{tab:eislines}\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\\end{center}\n\nInside this field of view, we selected two small regions for more detailed analysis. In Paper I a special three-color coding is devised to highlight immediately hot and cool regions. In the three-color image (where each color is the intensity in a different AIA channel, green 171 \\AA, blue 335 \\AA, red 94 \\AA), we selected a strip one-pixel wide and 7-pixels long, deep inside the hot region (where the hot part of the 94 \\AA\\ emission is high), and another (equal) one inside a colder region (high 171 \\AA\\ emission, Fig.\\ref{fig:aia3col}).\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[scale= 0.6]{aia_3_colors.ps}  \n\t\\caption{Image of the active region in three AIA channels: 171 \\AA\\ (green), 335 \\AA\\ (blue) and  94 \\AA\\ (red). We analyse the two small regions marked by the strips, one is hot (pink), the other is cold (green), in more detail.}\n\t\\label{fig:aia3col}\n\\end{figure}\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[scale= 0.7]{cold_spectrum.ps}  \n\t\\caption{Average Hinode\/EIS spectrum over the strip in the cold region (green in Fig.~\\ref{fig:aia3col}). The lines selected for further analysis are marked and labelled (with their temperature of maximum formation, $\\log T$). }\n\t\\label{fig:cold_spectrum}\n\\end{figure*}\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[scale= 0.7]{hot_spectrum.ps}  \n\t\\caption{As Fig.~\\ref{fig:cold_spectrum} for the strip in the Hot Region (pink in Fig.\\ref{fig:aia3col}).}\n\t\\label{fig:hot_spectrum}\n\\end{figure*}\n\nWe extracted EIS spectra in each pixel of the selected strips and, to increase the signal to noise ratio, we averaged the spectra over all the pixels in each strip. The resulting average spectra are shown in Fig.\\ref{fig:cold_spectrum} and \\ref{fig:hot_spectrum}. In these spectra the continuum is low and we see many emission lines for several elements, e.g. Fe, Ca, Mg.  We selected a subset of the emission lines with the following criterion. We conceptually divided the temperature range $5.5 < \\log T < 7$ into bins $\\Delta \\log T \\sim 0.1$ and choose approximately only one line that has the maximum formation temperature in a given bin, possibly the most intense one in at least one of the two spectra. We ended up with 14 lines, listed in Table~\\ref{tab:eislines}, 9 from Fe ions, 3 from Ca, 1 from S and Mg, that provide a reasonably uniform coverage of the temperature range, as shown in Fig.~\\ref{fig:eisemis}.  We use data from CHIANTI v 7.1 \\citep{2013ApJ...763...86L}. \nWe then fitted each line profile with a Gaussian (or multi-Gaussian for blended lines) and then we measured the flux by integrating the area below each Gaussian (Table~\\ref{tab:eislines}). To each flux we applied the new EIS radiometric calibration, which also includes a \ncorrection for the long-term degradation of the EIS effective area \\citep{2013A&A...555A..47D}. This leads to significantly higher (by a factor of about two) radiances of the EIS lines in the LW channel.\nOur DEM modelling is mostly constrained by strong iron lines, for which the atomic data are reliable.\nTo a first approximation, we can therefore neglect any uncertainty due to atomic data and chemical abundances.\nIt is difficult to assess the accuracy of the new EIS calibration, so we decided to \nassociate to each flux only the uncertainty due to photon statistics, and then use the results of the DEM modelling to discuss systematic errors (see Section~\\ref{sec:discuss}). We measure zero flux for the Ca XVII line in the cold region, and the value in the table is the corresponding upper limit.\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[scale= 0.7]{emis_eis_c7.1.ps}  \n\t\\caption{Emissivity of the selected lines per unit emission measure.}\n\t\\label{fig:eisemis}\n\\end{figure*}\n\n\\subsection{DEM reconstruction}\n\\label{subsec:DEMrec}\n \n We use the radiances in the selected lines (Table~\\ref{tab:eislines}) to derive the distribution of the emission measure vs temperature, DEM(T).  As a first step of the analysis we use the EM loci method \\citep{1978PhDT.......298S}, by which each line intensity is divided by its emissivity. The loci of these curves represent, at each temperature, the maximum value of the  emission measure at that temperature, if all the plasma were isothermal.  Therefore, if the plasma were isothermal along the line of sight, all the curves would \ncross at a single point, exactly at the plasma temperature (see \\citealt{2002A&A...385..968D} for more details). There are many inversion methods to obtain the DEM.\n\nWe applied the widely-used Markov-Chain Monte Carlo method (MCMC,  \\citealt{1998ApJ...503..450K}). This technique is based on a Bayesian statistical formalism to determine the most probable DEM curves that reproduce the observed line intensities by Monte Carlo simulations. A very useful feature of this technique is the possibility to obtain an estimate of the uncertainty of the DEM in each temperature bin in which the DEM is computed (see e.g., \\citealt{2011ApJ...728...30T}, \\citealt{2012ApJ...758...54T} for more detail).\n\nFor the application of the MCMC method, we assumed an electron density of $3 \\times 10^{9}$ $cm^{-3}$, \\cite{1992PhyS...46..202F} coronal abundances and CHIANTI 7.1  ionization equilibrium to compute the emissivity (per unit emission measure) of each line. The temperature range was divided into equal bins $\\Delta \\log T\\sim 0.1$.  Fig.\\ref{fig:dem14} shows the result of the reconstruction in which we added the EM-loci curves. We mark the most probable solution and the cloud of solutions in each temperature bin. The broader the cloud, the less constrained is the value of the best fit curve in that bin.\n\n\\begin{figure*}[h!]\n\t\\centering\n\t\\subfigure[Cold Region]{\\includegraphics[scale= 0.6]{cold_dem.ps}}\n\t\\subfigure[Hot Region]{\\includegraphics[scale= 0.6]{hot_dem.ps}}\n\t\\caption{Results of DEM reconstruction using the MCMC technique on 14 Hinode\/EIS line fluxes (see text, and Table 1) for (a) the cold region, (b) the hot region, shown in Fig. \\ref{fig:aia3col}. The red (histogram) curve is the solution that better reproduces the observed fluxes, the histogram cloud contains the solutions. The EM-loci curves are also shown for reference, each color marks a different element.}\n\t\\label{fig:dem14}\n\\end{figure*}\n\nThere is no small region in the plot where EM-loci curves all intersect; the plasma is therefore clearly multi-thermal both in the cold and in the hot segments. The information about the very hot plasma comes mostly from the Ca lines. The very low flux of the Ca XVII line in the cold region puts a strong constraint to reduce the emission measure for $\\log T > 6.5$, while the flux is much higher in the hot region. The Fe and Ca lines show a good overall agreement while the Mg line leads to a higher emission measure in the cold region.\n\nThe MCMC method applied to the EIS data provides very different distributions in the cold and hot regions. In the cold region, the distribution is mostly flat for $\\log T < 6.4$, then it decreases rapidly. The local peak at $\\log T = 6.3$ might be significant, because of the small uncertainty. Overall the impression of a cool distribution is confirmed.  In the hot region, we see a different trend: the DEM increases monotonically for $\\log T > 6$, with a slope between 2 and 3, it has a peak at $\\log T = 6.6$ and then decreases gradually at higher temperature. This shape is not new for active region core \\citep{2011ApJ...734...90W}, but our treatment of uncertainties makes the solutions better constrained. In particular, the components on the hot side of the DEM peak look quite well defined even with respect to the cooler side, where several lines have maximum formation temperatures. These hot emission measure components are at a level of $\\sim 20$\\% of the emission measure peak, therefore a significant fraction.\n\nTable~\\ref{tab:eislines} contains also the ratio of the line flux computed from the DEM reconstruction with the MCMC method to the observed one. Except for some well-known lines (Mg VII, Fe XIII) and for hot lines ($log T \\geq 6.45$) in the cold region, not relevant for our analysis, good agreement (within 20\\%) is found, comparable to that obtained in other similar analyses (e.g. \\citealt{2011ApJ...734...90W}).\n\nTo try to better constrain the high-temperature part of the DEM,  first we add the information provided by the flux measured in the 94\\AA\\ AIA channel. As we briefly discussed previously, this band contains lines sensitive to emission from plasma at $\\log T \\sim 6.8$ (Fe XVIII),  but also lines sensitive to lower temperatures, so the intensity in the AIA band can be written as \n\n\\begin{equation}\n\tI_{94} = EM_{cold}G_{94}(T_{cold})+EM_{hot}G_{94}(T_{hot})\n\t\\label{eq:I94comps}\n\\end{equation}\nwhere $EM_{cold}$ and $EM_{hot}$ are the emission measures of the cold and hot components, $G_{94}(T_{cold})$ and $G_{94}(T_{hot})$ are the values of the channel response functions of the cold and hot peaks respectively. \n\nWe use the technique devised by \\cite{2011ApJ...736L..16R} \n to separate the two contributions and pick up the hot one. The technique uses the flux measured in the 171 \\AA\\ channel: \n\n\\begin{equation}\n\tI_{171} = EM_{cold}G_{171}(T_{cold})\n\t\\label{eq:I171}\n\\end{equation}\n\nto constrain the cold component. Substituting Eq.\\ref{eq:I171} in Eq.\\ref{eq:I94comps} we obtain \n\n\\begin{equation}\n\tI_{94} = \\frac{I_{171}}{G_{171}(T_{cold})}G_{94}(T_{cold})+EM_{hot}G_{94}(T_{hot})\n\\end{equation}\n\nWe can then subtract the flux extrapolated from the 171 \\AA\\ channel from the 94 \\AA\\ flux. \nWhat is left is the hot part of the 94 \\AA\\ emission:\n\n\\begin{equation}\n\tI_{hot} = I_{94} - \\frac{I_{171}}{G_{171}(T_{cold})}G_{94}(T_{cold})\n\\end{equation}\n\nWe note that a similar procedure, using the 171 and 193~\\AA\\ AIA bands was devised by \n\\cite{2012ApJ...759..141W}. \\cite{2013A&A...558A..73D}\nsuggested the use of the  171 and 211~\\AA\\ AIA bands instead, the latter one being \nused to estimate the contribution of the Fe XIV line to the 94~\\AA\\ band.\nAs shown in  \\cite{2013A&A...558A..73D}, this method produces \nsimilar results as the method of \\cite{2012ApJ...759..141W}.\nThe 94 \\AA\\ flux in the cold region is compatible with zero flux, so we put an upper limit as we did for the Ca XVII flux in the same region in Table~\\ref{tab:eislines}.\n\nAfter including the information from the AIA 94 \\AA\\ channel, the EM-loci AIA curves are in good agreement with the Ca XVII curves. In the hot region, the curves intersect both at $\\log T \\sim 6.6$ and at $\\log T \\sim 6.8$ but they are very similar in between. The DEM solutions derived with MCMC method are very similar to those obtained without the AIA flux, thus confirming coherent information.\n\nAdditional information about the hot components is independently available from the X-ray observation with Hinode\/X-Ray Telescope (XRT). However, we expect looser constraints from the XRT filters because they are broadband, and have a broader temperature response.  We can simply plug in our analysis the flux measured in one XRT filter; in particular, we consider the Ti\\_poly filter, which has the highest sensitivity to emission from plasma at $\\log T \\sim 6.9$. The result after including both AIA and XRT fluxes is shown in Fig.\\ref{fig:dem14xrt}. We do not see qualitative difference from the DEM shown in Fig.~\\ref{fig:dem14}, as expected from the broader AIA and XRT temperature responses (see EM-loci curve). A quantitative difference is that the DEM in the hot side of the DEM peak of the hot region is reduced by a factor $\\sim 2$, i.e. to about 10\\% of the peak, while maintaining a narrow cloud distribution for $\\log T < 6.9$. \n\n\\begin{center}\n\\begin{table*}[htb!]\n\\begin{center}\n\\caption{Fluxes measured in the AIA-94 \\AA\\ channel and in the XRT-Ti\\_poly bands. \nFor the 94\\AA\\ band we subtracted the cold component, as explained in subsection \\ref{subsec:DEMrec}.}\n\\begin{tabular}{lcccc}\n\\hline\\hline\nInstrument&Cold region flux&Ratio&Hot region flux&Ratio\\\\\n& (DN s$^{-1}$ pixel$^{-1}$)&MCMC & (DN s$^{-1}$ pixel$^{-1}$)&MCMC\\\\\n\\hline\nAIA-94 \\AA&$<0.4$&0.58 &$66 \\pm 3$& 0.69 \\\\\nXRT-TI\\_poly&$53 \\pm 5$& 0.94 &$2030\t\\pm 40$& 1.28\\\\ \n\\hline\n\\label{tab:aiaxrtflx}\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\\end{center}\n\n\\begin{figure*}[h]\n\t\\centering\n\t\\subfigure[Cold Region]{\\includegraphics[scale= 0.6]{cold_demaiaxrt.ps}}\n\t\\subfigure[Hot Region]{\\includegraphics[scale= 0.6]{hot_demaiaxrt.ps}}\n\t\\caption{Same as Fig.\\ref{fig:dem14} adding the contribution of the fluxes measured in the \"hot part\" (see text) of the AIA 94 \\AA\\ channel (black line) and in the XRT Ty\\_poly filter (red line).}\n\t\\label{fig:dem14xrt}\n\\end{figure*}\n\nWe further supported our analysis by comparing the best solutions of the DEM reconstruction on EIS data from the MCMC method with those from another method, devised by \\cite{1999PhDT.........8D}. \nThe method assumes that a smooth DEM distribution exists, and models it with a spline function.\nThe choice of the nodes of the spline is somewhat subjective, but the actual inversion \nis carried out following the maximum entropy method described in \\cite{1991AdSpR..11..281M}.\nWe note that the same atomic data and input parameters were used for both inversions.  \nThe comparison is shown in Fig.~\\ref{fig:petdzdem14}.\n\n\t\\begin{figure*}[h!]\n\t\\centering\n\t\\subfigure[Cold Region]{\\includegraphics[scale= 0.6]{petdz_cold_dem.ps}}\n\t\\subfigure[Hot Region]{\\includegraphics[scale= 0.6]{petdz_hot_dem.ps}}\n\t\\caption{Best DEM solutions from MCMC method (histogram, red in Fig.\\ref{fig:dem14}) and from Del Zanna method (solid line $+$ symbols) for (a) the cold and (b) the hot region.  The Emission Measure values with MCMC are divided by the temperature to match the output from Del Zanna method.}\n\t\\label{fig:petdzdem14}\n\t\\end{figure*}\n\nThe results with the spline method are obviously smoother, but overall there is very good agreement between the two methods for both regions, especially in the temperature ranges of interest, and in particular in the hot part of the hot region. \nGood agreement (within 20\\%) between observed and predicted intensities is found also with this method.\n\n\\section{Discussion and conclusions}\n\\label{sec:discuss}\n\nWe analysed the thermal distribution of the plasma along the line of sight in two different regions where the narrow-band imagers diagnosed very different dominant temperatures, a hotter and a cooler one. The analysis was mostly based on the spectral data from Hinode\/EIS. Our major attention was to the hottest components that may be signature of impulsive heating at work, and here our aim was to reconstruct the whole EM distribution and check the coherence of the overall scenario, including the hot component. At variance from many other similar analyses, our approach was to consider a limited number of spectral lines ($\\sim 1$ per temperature bin), and we picked the most intense lines uniformly covering the temperature range. In this way we simplified our control and interpretation of the reconstruction results, and all measured fluxes had the same weight in determining the global emission measure distribution. We also had a particular approach regarding the assessment of the uncertainties. The typical choice is to associate the same percentage error to all measured fluxes; 20\\% is the most commonly assumed value (e.g. \\citealt{2011ApJ...734...90W}). This uncertainty safely includes possible systematic unknown effects due to the instrument, atomic data and chemical abundances, but weights the same all measured values, independently of whether a line is strong or weak. Our choice is different. The measured fluxes were assigned exclusively the poissonian error, dictated only by the photon statistics. This allowed us to weight more the intense lines and to minimise the uncertainties of the solutions of the DEM reconstructions.\n\nAs a consequence, the DEM reconstruction with the MCMC technique leads to solution clouds with a narrow distribution around the best solution for many temperature bins (factor 2-3), significantly narrower than in previous works. Only a minority of temperature bins show a spread solution cloud. The constraint on the hot components is tighter. \n\nThe error estimate is important in this analysis, and we are aware that the real uncertainties are surely larger than those that we assume. The uncertainties can influence considerably the error on the DEM solution, and sometimes even the solution itself, as thoroughly discussed in \\cite{2012ApJ...745..111T}. We should also consider that the cloud spread may underestimate the real error on the DEM solution. The small uncertainties typically lead to a larger number of iterations with the MCMC method for better convergence \\citep{2012ApJ...745..111T}, but our 400 iterations is certainly an appropriate number for our cases. Tests show that DEM structures are reliably recovered on scales larger than $\\Delta \\log T = 0.2$, so we do not discuss narrower features. \n\nThe good agreement between predicted and observed line intensities confirms this, although we note that \nsome significant discrepancies are present, in particular with the  Ca XV line, as also \nfound previously \\citep{2013A&A...558A..73D}. The good agreement between the two inversion methods,\nalready found in \\cite{2011A&A...535A..46D}, is confirmed. This suggests that the main source of uncertainty resides in the \nchoice of parameters, atomic data and instrument calibration.\n\nAlthough our assumptions probably underestimate the errors, the coherent support from other instruments (AIA, XRT), the agreement with the results from another reconstruction method and the coherence with the morphology seen in the images spanning the different temperature regimes makes us confident that other errors should not affect our results considerably.\n\nOverall, the analysis has confirmed the two general characteristics anticipated by the imagers. The comparison of the DEM distributions of the cold and hot regions has, on the one hand, revealed substantial thermal components for $logT < 6.3$ in the cold region, without showing prominent features. The reconstruction of the cold region also shows  minor components for $logT > 6.3$. Since the images show no bright features in the hot channels and lines, we may use the values of these emission measure components as sensitivity limits of our analysis, i.e. we may not trust emission measure components below $10^{26}$ cm$^{-5}$. \n\nThe hot region shows a much more peaked thermal structure, with the positive gradient typically found in previous studies in coronal loops of active region cores \\citep{2011ApJ...734...90W}. The peak is at a rather high temperature ($\\log T = 6.6$) and beyond that the emission measure declines, but not very steeply and still showing a significant fraction of emission measure at temperature $\\log T \\leq 6.8$. Some components might be present at even higher temperature, although with higher uncertainties, up to the limit of the thermal sensitivity of our analysis ($\\log T \\sim 7$). We believe that the joint use of hot spectral lines, AIA 94 \\AA\\ channel and XRT filterbands, helps to partially remove the so-called \"blind spot\" for $\\log T > 6.8$ \\citep{2012ApJ...746L..17W}. Although the presence of the very hot component looks confirmed, still it is based on a very limited amount of information, and therefore some care should still be used. Some further feedback is provided by the comparison with the images, and between the images. The morphology of the region in the Ca XVII line only partially overlaps the morphology in immediately cooler lines (Fe XVI), while it matches well the X-ray and AIA 94 \\AA\\ morphology. \nThis confirms that in the hot region selected in the present analysis there is indeed significant \nemission at 6~MK, resulting in strong  Ca XVII, so that the Fe XVIII emission in the  AIA 94 \\AA\\ band is also \ndue to this component, unlike other cases in the cores of active regions where Ca XVII is not observed,\nand the  Fe XVIII emission is due to a large emission measure at 3~MK \\citep{2013A&A...558A..73D}.\n\nOur analysis has remarked how spectral data are by far more constraining than the data from imagers, because the spectral lines are much more sensitive to temperature variations that the broader bands of the imagers (in agreement with the findings of \\cite{2012ApJ...758...54T}. Still it stresses that a quantum leap in the diagnostics of the hottest DEM components needs constraints from more lines sensitive to emission from high temperature plasma. These might be, for instance, easily accessible to broad-band X-ray spectrometers, to which we look forward in future space missions.\n\n\n\\acknowledgements\n{\nAP and FR acknowledge support from Italian Ministero dell'Universit\\`a e Ricerca (MIUR). GDZ acknowledges support from STFC (UK).\nPT was supported by contract SP02H1701R from Lockheed-Martin, NASA contract NNM07AB07C to SAO, and NASA grant NNX11AC20G. \\emph{CHIANTI} is a collaborative project involving the \\emph{NRL (USA)}, the \\emph{Universities of Florence (Italy)} and\n\\emph{Cambridge (UK)}, and \\emph{George Mason University (USA)}. Hinode is a Japanese mission developed and launched by ISAS\/JAXA, with NAOJ, NASA, and STFC (UK) as partners, and operated by these agencies in cooperation with ESA and NSC (Norway). SDO data were supplied courtesy of the SDO\/AIA consortium. SDO is the first mission to be launched for NASA's Living With a Star Program. \nWe thank the International Space Science Institute (ISSI) for hosting the International Team of S. Bradshaw and H. Mason: Coronal Heating \u2013 Using Observables to Settle the Question of Steady vs. Impulsive Heating.\n\n}\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn the theory of quantum entanglement the familiar `distant labs' scenario motivates the restriction of allowed operations to those that can be carried out by local operations and classical communication (LOCC). Since no entangled state can be created from an unentangled one by LOCC, we regard entanglement as a resource in this context. In the case of the bipartite pure states, the picture of how this resource can be quantified and how it can be transformed (by LOCC) is quite well developed: For bipartite pure states the entanglement of distillation, $E_d$, and the entanglement cost, $E_c$, have been shown to both be equal to the entropy of entanglement (for a review see Plenio and Virmani \\cite{emeas}). Since $E_d$ and $E_c$ are equal, LOCC transformation of bipartite pure state entanglement is asymptotically reversible, and the entropy of entanglement is a unique measure of the amount of entanglement available in the many copy limit.\n\nFor mixed bipartite states, entanglement transformations are not in general asymptotically reversible, and consequently, LOCC no longer induces a unique measure of the amount of entanglement (in the asymptotic regime). For instance, there are bound entangled states (which by definition have zero entanglement of distillation) which have been shown to have non-zero entanglement cost \\cite{irrev}. This has motivated the search for non-trivial extensions of LOCC, with respect to which entanglement transformations are asymptotically reversible for all bipartite states. Recently, Plenio and Brand\\~{a}o \\cite{pb} proved that the set of operations which asymptotically cannot generate entanglement is such a set, with the regularised relative entropy of entanglement as the corresponding measure of entanglement. An open conjecture \\cite{ape} is that the set of PPT operations \\cite{rains} (defined later in this section) also renders bipartite entanglement transformations asymptotically reversible (in \\cite{ape} it is shown that it does so for the anti-symmetric Werner state).\n\nFor exact, finite transformations, the problem of determining when a given pure bipartite state can be deterministically converted into another by LOCC is completely solved by Nielsen's majorization theorem \\cite{nielsen}: The process is possible if and only if the Schmidt coefficient vector of the initial state is majorized by that of the final state.\n\nIn the present paper we investigate exact, finite transformations of pure states by PPT operations. This topic was first treated by Ishizaka and Plenio \\cite{IshizakaPlenio}), where the emphasis is on conversion of LOCC-inequivalent forms of multi-partite entanglement (e.g. GHZ and W states for three parties) by PPT operations. In the present work we are only concerned with bipartite states.\n\nHere, we introduce some conventions that will be used throughout the rest of the paper. Logarithms are always taken to base two. If the deterministic transformation of a state $\\rho$ into $\\rho'$ can be accomplished by the class of operations $OP$ then we write $\\rho \\stackrel{OP}{\\rightarrow} \\rho'$; if it cannot be then we write $\\rho \\stackrel{OP}{\\nrightarrow} \\rho'$.  In discussing the transformation of bipartite pure states by any class of operations which contains LOCC, we need only consider the Schmidt coefficients of the states since states with the same Schmidt coefficients are equivalent up to local unitary transformations (which are obviously contained in LOCC). Since we are only concerned here with classes that include LOCC, we will use the state\n\n\\begin{equation}\n\t\\rho_{\\lambda} = \\sum_{i=1}^d\\sum_{j=1}^d \\sqrt{\\lambda_{i}\\lambda_{j}} \\ket{ii}\\bra{jj}\n\\end{equation}\n\n(where $\\ket{ij} = \\ket{i}_A \\otimes \\ket{j}_B$ for orthonormal bases $\\{\\ket{i}_A\\}$, $\\{\\ket{i}_B\\}$ for Alice and Bob's quotients of the bipartite Hilbert space) as a representative of all states with Schmidt coefficient vector $\\lambda$ without loss of generality. We use $\\lambda^{\\uparrow}$ ($\\lambda^{\\downarrow}$) to denote the vector obtained by putting the components of $\\lambda$ in non-decreasing (non-increasing) order.\n\nUsing this notation, Nielsen's theorem is\n\\begin{equation}\n\t\\rho_{\\lambda} \\stackrel{LOCC}{\\rightarrow} \\rho_{\\mu} \\iff \\lambda \\prec \\mu,\n\\end{equation}\nwhere the majorization relation is defined on vectors in $\\mathbb{R}^d$ whose components sum to one by\n\\begin{equation}\n\t\\lambda \\prec \\mu \\iff \\sum_{i=1}^{j}\\lambda_i^{\\downarrow} \\leq \\sum_{i=1}^{j}\\mu_i^{\\downarrow}, \\textrm{ for all } j \\in \\{1,...,d\\}.\n\\end{equation}\nWe shall use $\\Phi_K$ to denote a maximally entangled state of rank $K$ (where we assume that $K \\geq 2$).\n\nWe will use the symbol $X^{\\Gamma}$ to denote the partial transpose an operator $X$. We define the (linear) partial transpose map by\n\\begin{equation}\n\t\\ketbra{ij}{kl}^{\\Gamma} = \\ketbra{il}{kj}.\n\\end{equation}\nClearly this is basis dependent, but the eigenvalues (and hence the positivity) of the partial transpose of an operator does not depend on this basis choice. For convenience we will use the same basis that we are using as our representative Schmidt basis.\n\nThe set of PPT operations on a bipartite system is the set of completely positive trace-preserving (CPTP) maps $\\mathcal{L}$ such that the composition $\\Gamma \\circ \\mathcal{L} \\circ\\Gamma$ is also completely positive, where $\\Gamma$ denotes the partial transpose map $\\rho \\to \\rho^{\\Gamma}$. Equivalently, PPT maps are CPTP maps that \\emph{completely} preserve the PPT property of states in the same sense that completely positive maps \\emph{completely} preserve non-negativity of states: Any extension of a PPT map on a system Q onto a larger system QR where we apply the original map on Q and the identity map on R is PPT preserving (it maps PPT states to PPT states).\n\nWe make frequent use of the R\\'{e}nyi entropies: For $t \\in [0,\\infty]$ the \\emph{R\\'{e}nyi entropy at $t$} is defined by\n\n\\begin{equation}\n\t\tS_t\\left(\\lambda\\right) = \\left\\{ \\begin{array}{ll}\n\t\t\t\\frac{1}{1-t}\\log \\left( \\sum_{i=1}^{d} \\lambda_i^t \\right) &\\textrm{ for } t \\in (0,1)\\cup(1,\\infty), \\\\\n\t\t\t\\log |\\lambda| &\\textrm{ for } t = 0,\\\\\n\t\t\tH(\\lambda) &\\textrm{ for } t = 1,\\\\\n\t\t\t-\\log \\lambda^{\\downarrow}_1 &\\textrm{ for } t = \\infty,\n\t\t\\end{array} \\right.\n\\end{equation}\nwhere $|\\lambda|$ denotes the number of non-zero components of $\\lambda$ (i.e. the Schmidt rank) and $H(\\lambda) = -\\sum_i \\lambda_i\\log\\lambda_i$ is the Shannon entropy of $\\lambda$.\n\nIn the next section we discuss some necessary conditions for general bipartite pure state transformations by PPT. After that we provide part of the theory analogous to Nielsen's theory for deterministic transformations of pure bipartite states by PPT operations. In particular, we look at the case where the initial state is maximally entangled. In section \\ref{sec:frommax} we provide a necessary condition for transformations of this type, which we conjecture is also sufficient. We make use of this result in section \\ref{sec:cat} to show that the phenomenon of catalysis \\cite{catalysis} can occur for PPT operations and give a necessary and sufficient condition for when this can occur if both the initial state and catalyst state are maximally entangled. We conclude with some suggestions for future work, including a conjecture for extending this condition to arbitrary initial and catalyst states.\n\n\\section{General PPT pure state transformations}\\label{sec:general}\n\nDetermining whether a particular pure state transformation can be carried out by a PPT operation can in fact be formulated as a semidefinite programming problem \\cite{IshizakaPlenio}, the difficulty is in phrasing the relevant constraints in terms of conditions on the Schmidt coefficients of the pure states. Here we show that certain R\\'{e}nyi entropies of the Schmidt coefficient vectors correspond to operationally motivated PPT monotones (allowing us to give some necessary conditions for transformations).\n\nFor completeness let us define these operational monotones for any class of operations $X$: Let $\\rho$ be a density operator in $\\mathcal{B}(\\mathcal{H}_A\\otimes\\mathcal{H}_B)$, where $\\mathcal{B}(\\mathcal{H})$ denotes the set of hermitian operators on the Hilbert space $\\mathcal{H}$.\n\\begin{definition}\n\tThe distillable entanglement of $\\rho$ under the class of operations $X$ is defined by\n\t\\begin{equation}\n\t\tE_{d}^{{\\rm X}}( \\rho ) := \\sup_{(\\mathcal{L}_j)_{j\\in\\mathbb{N}}}\\left\\{ \\lim_{j\\to\\infty} \\frac{\\log K_j}{n_j} \\bigg| \\lim_{j\\to\\infty} \\| \\Phi_{K_j} - \\mathcal{L}_j ( \\rho^{\\otimes n_j} ) \\|_1 = 0 \\right\\},\n\t\\end{equation}\n\tand the exact distillable entanglement by\n\t\\begin{equation}\n\t\tE_{dx}^{{\\rm X}}( \\rho ) := \\lim_{j\\to\\infty} \\max_{\\mathcal{L}_j}\\left\\{ \\frac{\\log K_j}{n_j} \\bigg| \\| \\Phi_{K_j} - \\mathcal{L}_j ( \\rho^{\\otimes n_j} ) \\|_1 = 0 \\right\\},\n\t\\end{equation}\n\twhere in both cases $(\\mathcal{L}_j)_{j\\in\\mathbb{N}}$ is a sequence of completely positive trace preserving (CPTP) maps such that each map, $\\mathcal{L}_j$, takes $\\mathcal{B}\\left(\\left(\\mathcal{H}_A\\otimes\\mathcal{H}_B\\right)^{\\otimes n_j}\\right)$ to $\\mathcal{B}\\left(\\mathbb{C}^{\\otimes K_j}\\otimes\\mathbb{C}^{\\otimes K_j}\\right)$ and belongs to the class $X$.\n\\end{definition}\n\n\\begin{definition}\n\tThe entanglement cost of a state $\\rho$, we define by\n\t\\begin{equation}\n\t\tE_{c}^{{\\rm X}}( \\rho ) := \\inf_{(\\mathcal{L}_j)_{j\\in\\mathbb{N}}}\\left\\{ \\lim_{j\\to\\infty} \\frac{\\log K_j}{n_j} \\bigg| \\lim_{j\\to\\infty} \\| \\mathcal{L}_j (\\Phi_{K_j} ) -  \\rho^{\\otimes n_j} \\|_1 = 0 \\right\\},\n\t\\end{equation}\n\tand the exact entanglement cost by\n\t\\begin{equation}\n\t\tE_{cx}^{{\\rm X}}( \\rho ) := \\lim_{j\\to\\infty} \\min_{\\mathcal{L}_j}\\left\\{ \\frac{\\log K_j}{n_j} \\bigg| \\| \\mathcal{L}_j (\\Phi_{K_j} ) -  \\rho^{\\otimes n_j} \\|_1 = 0 \\right\\},\n\t\\end{equation}\n\twhere in both cases $(\\mathcal{L}_j)_{j\\in\\mathbb{N}}$ is a sequence of CPTP maps such that each map, $\\mathcal{L}_j$, takes $\\mathcal{B}\\left(\\mathbb{C}^{\\otimes K_j}\\otimes\\mathbb{C}^{\\otimes K_j}\\right)$ to $\\mathcal{B}\\left(\\left(\\mathcal{H}_A\\otimes\\mathcal{H}_B)\\right)^{\\otimes n_j}\\right)$ and belongs to the class $X$.\n\\end{definition}\n\n\\begin{proposition}\n\tThe entanglement cost $E_{c}^{\\mathinner{\\mathrm{PPT}}}(\\rho_{\\lambda})$ and distillable entanglement $E_{d}^{\\mathinner{\\mathrm{PPT}}}(\\rho_{\\lambda})$ of $\\rho_{\\lambda}$ are both equal to $S_{1}(\\lambda)$, the entropy of entanglement of the state.\n\\end{proposition}\n\\begin{proof}\n\t\nIt is clear and well-known that by LOCC operations, both the entanglement cost and distillable entanglement of $\\rho_{\\lambda}$ is $S_{1}(\\lambda)$. By elementary\nresults of Rains' theory of PPT distillation \\cite{rains}, PPT operations\ncannot asymptotically increase the number of EPR pairs. Hence\n$$E_{c}^{\\mathinner{\\mathrm{PPT}}}(\\rho_{\\lambda}) \\geq S_1(\\lambda),\\quad\n  E_{d}^{\\mathinner{\\mathrm{PPT}}}(\\rho_{\\lambda}) \\leq S_1(\\lambda).$$\nSince the opposite inequalities are trivial by LOCC $\\subset$ PPT,\nwe conclude that\n$$E_{c}^{\\mathinner{\\mathrm{PPT}}}(\\rho_{\\lambda}) = E_{d}^{\\mathinner{\\mathrm{PPT}}}(\\rho_{\\lambda}) = S_{1}(\\lambda),$$\nthus coinciding with the corresponding LOCC entanglement cost and distillable entanglement: the entropy\nof entanglement.\n\\end{proof}\n\n\\begin{proposition}\n\tThe exact distillable entanglement $E_{xd}^{\\mathinner{\\mathrm{PPT}}}(\\rho_{\\lambda})$ of $\\rho_{\\lambda}$ is given by $S_{\\infty}(\\lambda)$.\n\\end{proposition}\n\\begin{proof}\nWe use a result of Rains~\\cite{rains} on the maximum fidelity\nobtainable from $\\ket{\\psi}$ via PPT operations, to a maximally entangled\nstate of Schmidt rank $K$: by this result, exact transformation (fidelity $1$)\nis possible if there exists an operator $F$ with\n$$\\rho_{\\lambda} \\leq F\\leq\\openone,\\quad -\\frac{1}{K}\\openone\\leq F^\\Gamma\\leq \\frac{1}{K}\\openone.$$\nThus, an upper bound to $K$ is given by\n$$\\max\\left\\{ \\left\\|F^\\Gamma\\right\\|_\\infty^{-1}\\ \\big|\\  \\rho_{\\lambda}\\leq F\\leq\\openone \\right\\}.$$\nWe claim that this is $1\/\\lambda^{\\downarrow}_1$: for assume any\n$F$ as above, and write it $F=\\rho_{\\lambda^{\\downarrow}}+A$, $A\\geq 0$.\n\\begin{equation}\\begin{split}\n  \\left\\|F^\\Gamma\\right\\|_\\infty &\\geq    \\mathinner{\\mathrm{Tr}}(\\ketbra{11}{11}(\\rho_{\\lambda^{\\downarrow}}^{\\Gamma} + A^{\\Gamma})) \\\\\n                               &=    \\lambda^{\\downarrow}_1 + \\mathinner{\\mathrm{Tr}}(\\ketbra{11}{11}^{\\Gamma} A) \\\\\n& = \\lambda^{\\downarrow}_1 + \\mathinner{\\mathrm{Tr}}(\\ketbra{11}{11} A) \\geq \\lambda^{\\downarrow}_1.\n\\end{split}\\end{equation}\nHence $\\left\\|F^\\Gamma\\right\\|_\\infty\\geq \\lambda^{\\downarrow}_1$, with equality\nachieved for $F=\\rho_{\\lambda}$.\n\\par\nThis gives $E_{xd}^{\\mathinner{\\mathrm{PPT}}}(\\rho_{\\lambda}) \\leq S_\\infty(\\lambda)$.\nThe opposite inequality comes from LOCC operations asymptotically\nachieving this bound~\\cite{morikoshi}.\n\\end{proof}\n\n\\begin{proposition}\\label{exactPPTcost}\nThe entanglement cost $E_{xc}^{\\mathinner{\\mathrm{PPT}}}(\\rho_{\\lambda})$ of $\\rho_{\\lambda}$ is $S_{1\/2}(\\lambda)$.\n\\end{proposition}\n\\begin{proof}\n\tThis is merely an application of the more general result of Audenaert \\emph{et al.} \\cite{ape} to pure states.\n\\end{proof}\n\nAll of these quantities are clearly PPT monotones so they provide a necessary condition for the possibility of a PPT pure state transformation:\n\\begin{proposition}\n\tIf $\\rho_{\\mu} \\stackrel{PPT}{\\rightarrow} \\rho_{\\lambda}$ then\n\t\\begin{align}\n\t\tS_{1\/2}(\\mu) \\geq S_{1\/2}(\\lambda), \\\\\n\t\tS_{1}(\\mu) \\geq S_{1}(\\lambda), \\\\\n\t\tS_{\\infty}(\\mu) \\geq S_{\\infty}(\\lambda).\n\t\\end{align}\n\\end{proposition}\n\nAs Schur concave functions, the R\\'{e}nyi entropies of the Schmidt coefficients at all values of $t \\in [0,\\infty]$ are monotones for LOCC state transformations. Under PPT however, the R\\'{e}nyi entropies for $0 \\leq t < 1\/2$ are \\emph{not} monotones.\n\n\\begin{example}\n  \\label{expl:small-alpha}\n  Consider a pure state $\\rho_{\\lambda}$ with Schmidt spectrum\n  $$\\lambda^{\\downarrow} = \\left( \\frac{1}{20},\\frac{1}{20},\\frac{1}{20},\\frac{4}{20},\\frac{4}{20},\\frac{9}{20} \\right).$$\n \\par\n  It is easily verified that $S_{1\/2}(\\lambda)=\\log 5$, and\n  indeed, in accordance with the Proposition \\ref{exactPPTcost} the transformation\n  $$\\Phi_5^{\\otimes(n+o(n))}\\longrightarrow\\rho_{\\lambda}^{\\otimes n}$$\n  is possible by PPT operations for all sufficiently large $n$.\n  \\par\n  But for $0\\leq \\alpha<1\/2$,\n  $$\\log 5 < S_{t}(\\lambda).$$\n  Since there are trivial examples of initial states such that the opposite\n  inequality is true (e.g.~$\\Phi_6$), the R\\'{e}nyi entropies\n  at $0\\leq t<1\/2$ are not PPT monotones.\n  \\par\n  $S_0(\\lambda)$ is just the logarithm of the Schmidt rank, so, as was noted in \\cite{IshizakaPlenio}, PPT operations can increase the Schmidt rank of pure states, a thing LOCC transformations cannot even do with nonzero probability!\n\\end{example}\n\nIt should be noted that a necessary condition for pure bipartite state transformations by \\emph{separable} operations was recently given by Gheorghiou and Griffiths \\cite{sepTrans}.\n\n\\section{PPT transformations from maximally entangled states.}\\label{sec:frommax}\n\nUnless otherwise stated, the final state is $\\rho_{\\lambda}$, where the Schmidt coefficient vector $\\lambda \\in \\mathbb{R}^d$ is assumed without loss of generality to have no vanishing components (so $d$ is the Schmidt rank of the final state). Since any state with Schmidt rank not greater than $K$ can be produced from $\\Phi_K$ by LOCC, the interesting case for PPT transformations is when the Schmidt rank is increased.\n\n\\begin{lemma}\n\tFor any (pure or mixed) final state $\\rho$, $\\Phi_K \\stackrel{PPT}{\\rightarrow} \\rho$ if and only if the solution to the semidefinite program\n\\begin{equation}\\label{mixedPrimalSDP}\n\t\\min \\{ \\mathinner{\\mathrm{Tr}} \\left(P\\right) | P \\ge 0, -\\left(K-1\\right)P^{\\Gamma} \\leq \\rho^{\\Gamma} \\leq \\left(K+1\\right)P^{\\Gamma} \\},\n\\end{equation}\n\twhere $P$ is an hermitian operator on the same Hilbert space as $\\rho$, is less than or equal to one.\n\\end{lemma}\n\\begin{proof}\n\tThe argument is almost the same as the one given in \\cite{ape}, but we present it here for convenience.\n\tIf there is a PPT map $\\mathcal{L}$ such that $\\mathcal{L}\\left(\\Phi_K\\right) = \\rho$ then the map $\\mathcal{L'}$ made by preceding $\\mathcal{L}$ with the twirl operation \\cite{rains} $\\mathcal{T}$ (which can be implemented by LOCC) is also PPT, and does the same transformation since $\\mathcal{T}\\left(\\Phi_K\\right) = \\Phi_K$. By symmetry it is always possible to write the new map in the form\n\t\\begin{equation}\n\t\t\\mathcal{L'}\\left(\\tau\\right) = F \\mathinner{\\mathrm{Tr}}\\left(\\Phi_K \\tau\\right) + G \\mathinner{\\mathrm{Tr}}\\left(\\left(\\openone - \\Phi_K\\right) \\tau\\right)\n\t\\end{equation}\n\twhere, in order for the map to be CPTP, $F$ and $G$ must be density operators. In order that $\\mathcal{L}'\\left(\\tau\\right) = \\rho$, we require $F = \\rho$, so the desired PPT transformation is possible if and only if there is a density operator $G$ such that the resulting $\\mathcal{L'}$ is PPT. $\\mathcal{L'}$ is PPT preserving if and only if the operator\n\t\\begin{align}\n\t\t\\mathcal{L'}\\left(\\tau ^{\\Gamma}\\right)^{\\Gamma} &= \\rho^{\\Gamma} \\mathinner{\\mathrm{Tr}}\\left(\\Phi_K^{\\Gamma} \\tau\\right) + G^{\\Gamma} \\mathinner{\\mathrm{Tr}}\\left(\\left(\\openone - \\Phi_K\\right)^{\\Gamma} \\tau\\right)\\\\\n\t\t&= \\rho^{\\Gamma} \\mathinner{\\mathrm{Tr}}\\left(\\left(\\mathcal{S} - \\mathcal{A}\\right) \\tau\\right)\/K + G^{\\Gamma} \\left(\\left(\\left(1-1\/K\\right)\\mathcal{S} + \\left(1+1\/K\\right)\\mathcal{A}\\right) \\tau\\right) \\\\\n\t\t&= \\mathinner{\\mathrm{Tr}}\\left(\\mathcal{S} \\tau\\right) \\left(\\rho^{\\Gamma}\/K + \\left(1-1\/K\\right)G^{\\Gamma}\\right) + \\mathinner{\\mathrm{Tr}}\\left(\\mathcal{A} \\tau\\right) \\left(\\left(1+1\/K\\right)G^{\\Gamma} - \\rho^{\\Gamma}\/K\\right)\n\t\\end{align}\n\tis positive semidefinite for all positive semidefinite $\\tau$, where $\\mathcal{S}$ and $\\mathcal{A}$ are the projectors onto the symmetric and anti-symmetric subspaces of the bipartite space. Since $\\mathcal{S}\\mathcal{A} = 0$, this condition holds if and only if $\\rho^{\\Gamma}\/K + \\left(1-1\/K\\right)G^{\\Gamma} \\geq 0$ and $\\left(1+1\/K\\right)G^{\\Gamma} - \\rho^{\\Gamma}\/K \\geq 0$. We also require $G \\geq 0$ and $\\mathinner{\\mathrm{Tr}} G = 1$. The set of operators which satisfy these constraints is precisely the set of feasible points of the SDP \\eref{mixedPrimalSDP} which have trace 1. If $P_0$ is feasible, then points $P_0 + t \\openone$ are also feasible for all $t \\geq 0$, so if an optimal point has trace not greater than one, it ensures the existence of a feasible point which satisfies all the constraints on $G$ (so the transformation is possible); if an optimal point has trace greater than one, then clearly no such point exists and hence the transformation is not possible.\n\t\n\\end{proof}\n\t\n\\begin{proposition}\\label{prop:primal}\n\t\\begin{equation}\n\t\t\\Phi_K \\stackrel{PPT}{\\rightarrow} \\rho_{\\lambda} \\iff T\\left(K;\\lambda\\right) \\leq 1,\n\t\\end{equation}\n\twhere we define $T\\left(K;\\lambda\\right)$ to be the solution to the semidefinite program\n\t\\begin{equation}\\label{primalSDP}\n\t\tT\\left(K;\\lambda\\right) := \\min \\left\\{ \\displaystyle\\sum_{i\\geq j}^{d} s_{ij} + \\displaystyle\\sum_{i > j}^{d} a_{ij} \\bigg| s_{ij} \\geq \\frac{\\sqrt{\\lambda_i \\lambda_j}}{K+1}, a_{ij} \\geq \\frac{\\sqrt{\\lambda_i \\lambda_j}}{K-1}, \\displaystyle\\sum_{i\\geq j} s_{ij} \\sigma_{ij}^{\\Gamma} + \\displaystyle\\sum_{i > j} a_{ij} \\alpha_{ij}^{\\Gamma} \\geq 0 \\right\\},\n\t\\end{equation}\n\twith\n\t\\begin{eqnarray*}\n\t&\\sigma_{i j}& = \\left\\{ \\begin{array}{l}\n\t\t(\\ket{ij} + \\ket{ji}) (\\bra{ij} + \\bra{ji})\/2 \\\\\n\t\t\\ketbra{ii}{ii}\n\t\t  \\end{array} \\right. \\begin{array}{c}\n\t\t    \\textrm{ when } i \\neq j, \\\\\n\t\t    \\textrm{ when } i = j,\n\t\t  \\end{array}\\\\\n\t\t  &\\alpha_{i j}& = \\left(\\ket{ij} - \\ket{ji}\\right) \\left(\\bra{ij} - \\bra{ji}\\right)\/2,\n\t\\end{eqnarray*}\n\tand $\\{s_{ij} | 1 \\leq j \\leq i \\leq d \\}$, $\\{a_{ij} | 1 \\leq j < i \\leq d \\}$ together constitute $d^2$ real variables.\n\\end{proposition}\n\\begin{proof}\n\\begin{equation}\n\t\\rho_{\\lambda}^{\\Gamma} = \\displaystyle\\sum_{i\\geq j}\\sqrt{\\lambda_i \\lambda_j} \\sigma_{ij} - \\displaystyle\\sum_{i > j}\\sqrt{\\lambda_i \\lambda_j} \\alpha_{ij}.\n\\end{equation}\nLet $\\Pi$ be the projection map on the space of hermitian operators given by\n\\begin{equation}\n\t\\Pi\\left(\\tau\\right) = \\displaystyle\\sum_{i\\geq j}\\sigma_{ij}\\tau\\sigma_{ij} + \\displaystyle\\sum_{i > j}\\alpha_{ij}\\tau\\alpha_{ij}.\n\\end{equation}\nTo show that this map preserves positivity of partial-transpose, we note that\n\\begin{equation}\n\t\\Pi\\left(\\tau^{\\Gamma}\\right)^{\\Gamma} = \\frac{1}{2}\\displaystyle\\sum_{i\\neq j}\\bra{ij}\\tau\\ket{ij}\\left(\\ketbra{ij}{ij} + \\ketbra{ji}{ji}\\right) + \\frac{1}{2}\\displaystyle\\left(\\sum_{i}\\ketbra{ii}{ii}\\right)\\left(\\tau+\\tau^{\\ast}\\right)\\left(\\sum_{j}\\ketbra{jj}{jj}\\right).\n\\end{equation}\nClearly then, $\\Pi$ is positive and preserves positivity of partial transpose. Since $\\rho^{\\Gamma}$ lies in the image of $\\Pi$, if $P$ is a feasible point of the semidefinite program \\eref{mixedPrimalSDP} then $\\Pi\\left(P^{\\Gamma}\\right)^{\\Gamma}$ is also a feasible point. Since $\\Pi$ is also trace reducing, it will not change the optimal value of \\eref{mixedPrimalSDP} if we impose the additional restriction that $P = \\Pi\\left(P^{\\Gamma}\\right)^{\\Gamma}$. This restriction is obeyed if and only if we can write $P$ in the form\n\\begin{equation}\n\tP = \\displaystyle\\sum_{i\\geq j} s_{ij} \\sigma_{ij}^{\\Gamma} + \\displaystyle\\sum_{i > j} a_{ij} \\alpha_{ij}^{\\Gamma}.\n\\end{equation}\nSubstituting this into \\eref{mixedPrimalSDP}, we obtain \\eref{primalSDP}.\n\\end{proof}\n\n\\begin{lemma}\n\t\\begin{equation}\\label{T-upperBound}\n\t\tT\\left(K; \\lambda\\right) \\leq \\left(2^{S_{1\/2}\\left(\\lambda\\right)}-1\\right)\/\\left(K-1\\right).\n\t\\end{equation}\n\\end{lemma}\n\\begin{proof}\n\tThe point\n\t\\begin{equation}\n\t\ts_{ij} = \\sqrt{\\lambda_i \\lambda_j}\/\\left(K-1\\right), a_{ij} = \\sqrt{\\lambda_i\\lambda_j}\/\\left(K-1\\right)\n\t\\end{equation}\n\tis primal feasible since\n\t\\begin{equation}\n\t\t\\displaystyle\\sum_{i} \\lambda_i \\sigma_{ii}^{\\Gamma} + \\displaystyle\\sum_{i > j} \\sqrt{\\lambda_i\\lambda_j} \\left(\\sigma_{ij} + \\alpha_{ij}\\right)^{\\Gamma} = \\displaystyle\\sum_{i} \\lambda_i \\ketbra{ii}{ii} + \\displaystyle\\sum_{i > j} \\sqrt{\\lambda_i\\lambda_j} \\left(\\ketbra{ij}{ij} + \\ketbra{ji}{ji}\\right)\\geq 0\n\t\\end{equation}\n\tand the other inequalities are obviously satisfied. The primal objective function at this point is $2^{S_{1\/2}\\left(\\lambda\\right)}\/\\left(K-1\\right)$.\n\\end{proof}\n\nSince the semidefinite program \\eref{primalSDP} is strictly feasible (take the point $s_{ij} = 2, a_{ij} = 2$, for example), its solution is equal to the solution of the dual SDP \\cite{vbSDP} so we have\n\n\\begin{proposition}\n\t\\begin{align}\n\t\tT\\left(K;\\lambda\\right) = \\max \\left\\{ \\frac{K 2^{S_{1\/2}\\left(\\lambda\\right)} - 1}{K^2 - 1} - \\left(\\sum_{i\\geq j} \\frac{\\mu_{ij}\\sqrt{\\lambda_i \\lambda_j}}{K+1} + \\sum_{i > j} \\frac{t_{ij}\\sqrt{\\lambda_i \\lambda_j}}{K-1} \\right) \\bigg| \\left(\\mu,t\\right) \\in R \\right\\}\\label{dualSDP}\\\\\n\t\tR := \\left\\{\\left(\\mu,t\\right) \\bigg| \\mu_{ij} \\leq 1, t_{ij} \\leq 1, \\sum_{i\\geq j} \\mu_{ij} \\sigma_{ij}^{\\Gamma} + \\sum_{i > j} t_{ij} \\alpha_{ij}^{\\Gamma} \\geq 0 \\right\\}.\n\t\\end{align}\n\tHere $\\{\\mu_{ij} | 1 \\leq j \\leq i \\leq d \\}$, $\\{t_{ij} | 1 \\leq j < i \\leq d \\}$ together constitute $d^2$ real variables.\n\\end{proposition}\n\\begin{proof}\n\tThe semidefinite program dual to \\eref{primalSDP} is\n\t\\begin{align}\n\t\t\\textrm{maximize } \\frac{1}{K+1}\\sum_{i\\geq j}\\sqrt{\\lambda_i \\lambda_j} \\left(1-\\mathinner{\\mathrm{Tr}}\\left(Z \\sigma_{ij}^{\\Gamma}\\right)\\right) + \\frac{1}{K-1}\\sum_{i > j}\\sqrt{\\lambda_i \\lambda_j} \\left(1-\\mathinner{\\mathrm{Tr}}\\left(Z \\alpha_{ij}^{\\Gamma}\\right)\\right)\n\t\\end{align}\n\t\\begin{align}\t\n\t\t\\textrm{subject to } Z &\\geq 0\\\\\n\t\t\\mathinner{\\mathrm{Tr}}\\left(Z \\sigma_{ij}^{\\Gamma}\\right) &\\leq 1\\\\\n\t\t\\mathinner{\\mathrm{Tr}}\\left(Z \\alpha_{ij}^{\\Gamma}\\right) &\\leq 1.\n\t\\end{align}\n\tIf $Z$ is feasible point of this program then so is $\\Pi\\left(Z^{\\Gamma}\\right)^{\\Gamma}$ (where $\\Pi$ is the map defined in the proof of Proposition \\ref{prop:primal}). So the substitutions\n\t\\begin{equation}\n\t\tZ = \\sum_{i\\geq j} \\mu_{ij} \\sigma_{ij}^{\\Gamma} + \\sum_{i > j} t_{ij} \\alpha_{ij}^{\\Gamma}, x_{ij} = 1 - \\mu_{ij}, y_{ij} = 1 - t_{ij}\n\t\\end{equation}\n\tresult in an SDP with the same solution and this is the one given in the proposition.\n\\end{proof}\n\nThe dual objective at any feasible point of the dual semidefinite program \\eref{dualSDP} is a lower bound on $T\\left(K;\\lambda\\right)$, and therefore\n\n\\begin{corollary}\n\t\\begin{equation}\\label{T-lowerBound}\n\t\tT\\left(K; \\lambda\\right) \\geq \\left(K 2^{S_{1\/2}\\left(\\lambda\\right)} - 1\\right)\/\\left(K^2 - 1\\right).\n\t\\end{equation}\n\\end{corollary}\n\\begin{proof}\n\tThe point $\\mu_{ij} = 0, t_{ij} = 0$ is clearly dual feasible.\n\\end{proof}\n\n\\begin{theorem}\\label{t:rank1opt}\n\tThe optimal value of the dual objective that can be attained by a dual feasible point which satisfies the additional constraint that\n\t\\begin{equation}\\label{rankconstraint}\n\t\t\\mathinner{\\mathrm{rank}}\\left(\\displaystyle\\sum_{i\\geq j} \\mu_{ij} \\sigma_{ij}^{\\Gamma} + \\displaystyle\\sum_{i > j} t_{ij} \\alpha_{ij}^{\\Gamma}\\right) = 1,\n\t\\end{equation}\n \tis given by\n\t\\begin{equation}\\label{rankConstrainedOptimum}\n\t\tT_1\\left(K; \\lambda\\right) = \\frac{K 2^{S_{1\/2}\\left(\\lambda\\right)} - 1}{K^2 - 1}+\\frac{K}{\\left(K^2-1\\right)\\left(K+c^{\\ast}-d\\right)}\\left( \\left(\\displaystyle\\sum_{i=1}^{c^{\\ast}}\\sqrt{\\lambda^{\\uparrow}_i}\\right)^2-\\left(K+c^{\\ast}-d\\right)\\displaystyle\\sum_{i=1}^{c^{\\ast}}\\lambda^{\\uparrow}_i\\right),\n\t\\end{equation}\n\twhere $c^{\\ast}$ is the smallest number $c \\in \\{1 + d - K,...,d-1 \\}$ satisfying\n\t\\begin{equation}\\label{c-constraint}\n\t\t\\frac{\\sum_{i=1}^{c}\\sqrt{\\lambda^{\\uparrow}_i}}{K+c-d} \\leq \\sqrt{\\lambda^{\\uparrow}_{c+1}}.\n\t\\end{equation}\n\tIf none of the integers in the range satisfy this relation then $c^{\\ast} = d$.\n\\end{theorem}\n(The proof of this theorem is given in the appendix.)\n\\begin{remark}\n\tClearly $T_1\\left(K; \\lambda\\right) \\leq T\\left(K; \\lambda\\right)$, so a necessary condition for the transformation $\\Phi_K \\stackrel{PPT}{\\rightarrow} \\rho_{\\lambda}$ is $T_1\\left(K; \\lambda\\right) \\leq 1$.\n\\end{remark}\n\n\\begin{corollary}\\label{c:tt1}\n\tIf $\\left(\\displaystyle\\sum_{i=1}^{d-1}\\sqrt{\\lambda^{\\uparrow}_i}\\right)\/\\left(K-1\\right) > \\sqrt{\\lambda^{\\uparrow}_{d}}$ then\n\t\\begin{equation}\n\t\tT\\left(K;\\lambda\\right) = T_1\\left(K;\\lambda\\right) = \\left(2^{S_{1\/2}\\left(\\lambda\\right)}-1\\right)\/\\left(K-1\\right).\n\t\\end{equation}\n\\end{corollary}\n\\begin{proof}\n\tIn this case $c^{\\ast} = d$ and so, $T_1\\left(K;\\lambda\\right) = \\left(2^{S_{1\/2}\\left(\\lambda\\right)}-1\\right)\/\\left(K-1\\right)$ which, by Lemma \\ref{T-upperBound}, is an upper bound on $T\\left(K;\\lambda\\right)$. Since $T_1$ is also a lower bound on $T$, the result follows.\n\\end{proof}\n\n\\begin{corollary}\\label{c:borderline}\n\tIf $S_{1\/2}\\left(\\lambda\\right) = \\log K$ and $d \\geq K$ then the transformation is possible only in the trivial case where the goal state is also a maximally entangled state of rank $K$.\n\\end{corollary}\n\\begin{proof}\n\tThe value $c = d - 1$ satisfies \\eref{c-constraint} provided that $\\lambda^{\\uparrow}_d \\geq 1\/K$ or equivalently if $S_{\\infty}\\left(\\lambda\\right) \\leq \\log K$. Since $S_{t}\\left(\\lambda\\right)$ is a non-increasing function of $t$, this condition is indeed satisfied. Using this value of $c$ in \\eref{rankConstrainedOptimum} yields the lower bound\n\t\\begin{equation}\n\t\tT_1\\left(K; \\lambda\\right) \\geq 1 + \\frac{K\\left(\\sqrt{K \\lambda^{\\uparrow}_d} - 1\\right)^2}{\\left(K^2-1\\right)\\left(K-1\\right)}\n\t\\end{equation}\n\twhich is clearly greater than $1$ (implying the impossibility of the transformation) except where $S_{\\infty}\\left(\\lambda\\right) = \\log K $ which (together with $S_{1\/2}\\left(\\lambda\\right) = \\log K$) implies $\\lambda$ is the uniform distribution of size $K$, so the goal state is a maximally entangled state of rank $K$.\n\\end{proof}\n\n\\begin{proposition}\\label{p:rank3}\n\tIn the case where the goal state has Schmidt rank three, $T\\left(2; \\lambda\\right) = T_1\\left(2; \\lambda\\right)$.\n\\end{proposition}\n\\begin{proof}\n\tTo simplify notation we shall here assume that $\\lambda_i = \\lambda^{\\uparrow}_i$. If $\\sqrt{\\lambda_1} + \\sqrt{\\lambda_2} > \\sqrt{\\lambda_3}$ then we can apply Corollary \\ref{c:tt1} and we're done, so we assume that $\\sqrt{\\lambda_1} + \\sqrt{\\lambda_2} \\leq \\sqrt{\\lambda_3}$. In this case\n\t\\begin{equation}\n\t\tT_1 = \\left(1 + 4\\left(\\sqrt{\\lambda_3 \\lambda_2} + \\sqrt{\\lambda_3 \\lambda_1}\\right) + 8\\sqrt{\\lambda_1 \\lambda_2 }\\right)\/3.\n\t\\end{equation}\n\tWe shall show that $T$ is the same by constructing a primal optimal solution. Let\n\t\\begin{eqnarray}\n\t\ts^{\\ast}_{11} = \\frac{1}{3}\\lambda_1 + \\frac{4}{9}\\sqrt{\\lambda_1 \\lambda_2}, s^{\\ast}_{22} = \\frac{1}{3}\\lambda_2 + \\frac{4}{9}\\sqrt{\\lambda_1 \\lambda_2}, s^{\\ast}_{33} = \\frac{1}{3}\\lambda_3, \\\\\n\t\ts^{\\ast}_{12} = \\left(\\frac{1}{3} + \\frac{4}{9}\\right)\\sqrt{\\lambda_1 \\lambda_2}, s^{\\ast}_{13} = \\frac{1}{3}\\sqrt{\\lambda_1 \\lambda_3}, s^{\\ast}_{23} = \\frac{1}{3}\\sqrt{\\lambda_2 \\lambda_3}, \\\\\n\t\ta^{\\ast}_{ij} = \\sqrt{\\lambda_i \\lambda_j}\n\t\\end{eqnarray}\n\t\\begin{equation}\n\t\\displaystyle\\sum_{i\\geq j} s^{\\ast}_{ij} \\sigma_{ij}^{\\Gamma} + \\displaystyle\\sum_{i > j} a^{\\ast}_{ij} \\alpha_{ij}^{\\Gamma} = P_1\\oplus P_2\n\t\\end{equation}\n\t\\begin{eqnarray*}\n\t  P_1 = \\frac{2}{3} \\sum_{i \\neq j} \\sqrt{\\lambda_i\n\t  \\lambda_j} \\ketbra{ij}{ij} + \\frac{2}{9} \\left(\\ketbra{23}{23} + \\ketbra{32}{32}\\right) \\geq 0\n\t\\end{eqnarray*}\n\tWritten as a matrix in the $\\{\\ketbra{ii}{jj}\\}$ basis,\n\t\\begin{eqnarray*}\n\t  P_2 = \\frac{1}{9} \\left(\\begin{array}{ccc}\n\t    3 \\lambda_1 + 4 \\sqrt{\\lambda_2 \\lambda_3} & - 3 \\sqrt{\\lambda_1 \\lambda_2} & - 3 \\sqrt{\\lambda_1\n\t    \\lambda_3}\\\\\n\t    - 3 \\sqrt{\\lambda_2 \\lambda_1} & 3 \\lambda_2 + 4 \\sqrt{\\lambda_2\n\t    \\lambda_3} & - \\sqrt{\\lambda_2 \\lambda_3}\\\\\n\t    - 3 \\sqrt{\\lambda_3 \\lambda_1} & - \\sqrt{\\lambda_3 \\lambda_2} & 3\n\t    \\lambda_3\n\t  \\end{array}\\right) &  & \n\t\\end{eqnarray*}\n\twhich can be seen to be positive semidefinite by the Sylvester criterion. Therefore $P_1 \\oplus P_2 \\geq 0$, and since the other primal constraints are clearly satisfied, the point $\\left(s^{\\ast},a^{\\ast}\\right)$ is primal feasible so\n\t\\begin{equation}\n\t\tT_1\\left(2;\\lambda\\right) \\leq T\\left(2;\\lambda\\right) \\leq \\displaystyle\\sum_{i\\geq j} s^{\\ast}_{ij} + \\displaystyle\\sum_{i > j} a^{\\ast}_{ij} = \\left(1 + 4\\left(\\sqrt{\\lambda_3 \\lambda_2} + \\sqrt{\\lambda_3 \\lambda_1}\\right) + 8\\sqrt{\\lambda_1 \\lambda_2 }\\right)\/3 = T_1\\left(2;\\lambda\\right).\n\t\\end{equation}\n\tTherefore $T\\left(2;\\lambda\\right) = T_1\\left(2;\\lambda\\right)$.\n\\end{proof}\nIt follows directly from this proposition that:\n\\begin{theorem}\\label{t:rank3}\n\tA pure state of Schmidt rank three can be produced from $\\Phi_2$ (an EPR pair) by PPT operations if and only if its Schmidt coefficients obey\n\t\\begin{equation}\n\t\t2\\left( \\sqrt{\\lambda^{\\uparrow}_3 \\lambda^{\\uparrow}_2} + \\sqrt{\\lambda^{\\uparrow}_3 \\lambda^{\\uparrow}_1} \\right) + 4\\sqrt{\\lambda^{\\uparrow}_1\\lambda^{\\uparrow}_2} \\leq 1.\n\t\\end{equation}\n\\end{theorem}\n\nThis fact, and some suggestive numerical evidence, leads us to make the following conjecture\n\\begin{conjecture}\\label{conj:rankIs1}\n\tIn the dual program \\eref{dualSDP}, the optimal value of the dual objective function is always attained by a point satisfying the rank constraint \\eref{rankconstraint} and as a consequence $T\\left(K; \\lambda\\right) = T_1\\left(K; \\lambda\\right)$.\n\\end{conjecture}\n\n\\section{Catalysis}\\label{sec:cat}\n\nNielsen's theorem was used to show that a phenomenon analogous to chemical catalysis can occur in LOCC entanglement transformation \\cite{catalysis}. That is, there exist pairs of states $\\rho_{\\lambda}, \\rho_{\\mu}$ such that $\\rho_{\\lambda} \\stackrel{LOCC}{\\nleftrightarrow} \\rho_{\\mu}$ but where there is a third state $\\rho_{\\xi}$ such that $\\rho_{\\lambda}\\otimes\\rho_{\\xi} \\stackrel{LOCC}{\\rightarrow} \\rho_{\\mu}\\otimes\\rho_{\\xi}$.\n\nA necessary and sufficient condition for the existence of a deterministic catalytic transformation for bipartite pure states was recently given in papers by Turgut \\cite{turgut} and Klimesh \\cite{klimesh}:\n\n\\begin{theorem}\\label{t:locccatcond}\nGiven two pure bipartite states $\\rho_{\\lambda}$ and $\\rho_{\\mu}$, where $\\lambda^{\\uparrow} \\neq \\mu^{\\uparrow}$ and $\\lambda$ and $\\mu$ don't both have components equal to zero, there exists a pure state $\\rho_{\\xi}$ such that $\\rho_{\\lambda}\\otimes\\rho_{\\xi} \\stackrel{LOCC}{\\rightarrow} \\rho_{\\mu}\\otimes\\rho_{\\xi}$ if and only if the following conditions are satisfied\n\\begin{align}\n\tS_{t}\\left(\\lambda\\right) &> S_{t}\\left(\\mu\\right) \\textrm{ for } t \\in \\left(0, \\infty\\right), \\\\\t\n\tf_{t}\\left(\\lambda\\right) &> f_{t}\\left(\\mu\\right) \\textrm{ for } t \\in \\left(-\\infty, 0\\right],\n\\end{align}\nwhere\n\\begin{equation}\n\tf_t\\left(\\lambda\\right) := \\left\\{ \\begin{array}{l}\n\t\t\\frac{1}{t - 1}\\log \\left( \\sum_{i=1}^{d} \\lambda_i^t \\right)\\\\\n\t\t\\sum_i \\log \\lambda_i \\\\\n\t\t-\\infty\n\t\t  \\end{array} \\right. \\begin{array}{l}\n\t\t    \\textrm{ when } t > 0 \\textrm{ and } \\lambda_i \\neq 0 \\textrm{ for all } i \\in \\{1,...,d\\}, \\\\\n\t\t\t\\textrm{ when } t = 0 \\textrm{ and } \\lambda_i \\neq 0 \\textrm{ for all } i \\in \\{1,...,d\\}, \\\\\n\t\t    \\textrm{ otherwise. }\n\t\t  \\end{array}\n\\end{equation}\n\\end{theorem}\n\nUsing the results established in the last section, we can show that catalysis can also occur under PPT operations.\n\n\\begin{theorem}\\label{t:catcond}\n\t$\\Phi_C\\otimes\\Phi_K \\stackrel{PPT}{\\rightarrow} \\Phi_C\\otimes\\rho_{\\lambda}$ (where $\\rho_\\lambda \\neq \\Phi_K$) for some sufficiently large value of $C$ if and only if $S_{1\/2}\\left(\\lambda\\right) < \\log K$.\n\\end{theorem}\n\\begin{proof}\n\tFor a given value of $C$, the transformation is possible if and only if $T\\left(KC;\\lambda\\otimes U_C\\right) \\leq 1$, where $U_C$ denotes the uniform probability distribution vector with $C$ elements.\n\tAs $C$ tends to infinity, both the upper bound \\eref{T-upperBound} and lower bound \\eref{T-lowerBound} on $T\\left(KC;\\lambda\\otimes U_C\\right)$ tend to $2^{S_{1\/2}\\left(\\lambda\\right)}\/K$. Therefore, if $S_{1\/2}\\left(\\lambda\\right) < \\log K$ then for some sufficiently large $C$, $T\\left(KC;\\lambda\\otimes U_C\\right) \\leq 1$ so the corresponding transformation is possible. That the condition is necessary follows from Proposition \\ref{exactPPTcost} ($S_{1\/2}$ cannot increase in \\emph{any} PPT pure state transformation and the R\\'{e}nyi entropies are additive) and Corollary \\ref{c:borderline} (deals with the case where $S_{1\/2}$ stays the same).\n\\end{proof}\n\nIn the case where the goal state has Schmidt rank three, the states satisfying\n\\begin{equation}\n\t2\\left (\\sqrt{\\lambda^{\\uparrow}_3 \\lambda^{\\uparrow}_2} + \\sqrt{\\lambda^{\\uparrow}_3 \\lambda^{\\uparrow}_1} + \\sqrt{\\lambda^{\\uparrow}_1 \\lambda^{\\uparrow}_2 } \\right) \\leq 1\n\\end{equation}\nare exactly those that can be reached from $\\Phi_2$ by PPT operations when maximally entangled catalysts of arbitrarily high rank are available, according to Theorem \\ref{t:catcond}. This is a strict superset of those Schmidt rank three states which can be obtained from $\\Phi_2$ without a catalyst (see Theorem \\ref{t:rank3}). These regions are illustrated on one cell of the simplex of Schmidt coefficient vectors in Fig. \\ref{fig:allowed-regions}.\n\t\n\\begin{figure}\n\\includegraphics[scale=0.5]{regionsDiag.pdf}\n\\caption{Pure states of Schmidt rank three accessible deterministically from a single EPR pair by PPT and by PPT assisted by a maximally entangled catalyst.}\\label{fig:allowed-regions}\n\\end{figure}\n\n\\section{Conclusions}\\label{sec:conc}\n\nWe have provided a necessary condition for the exact preparation of a pure bipartite state from a maximally entangled state by PPT operations in terms of the Schmidt coefficients of the final state. We conjecture that this condition is also sufficient and have shown that this is true when the final state has Schmidt rank three. We have demonstrated that the phenomenon of catalysis occurs in the context of PPT operations. A notable difference from LOCC catalysis is that maximally entangled states can act as catalysts under PPT operations (this is impossible with LOCC). In the case where both the initial state and the catalyst are maximally entangled, we have given a necessary and sufficient condition for the production of a pure state: the R\\'{e}nyi entropy at $1\/2$ of the Scmidt coefficient vector must decrease, unless the final state is the same as the initial state. To give a direction for future work -- comparing Theorem \\ref{t:catcond} and the observations on PPT monotones given in Section \\ref{sec:general} with Theorem \\ref{t:locccatcond}, we make the following conjecture:\n\n\\begin{conjecture}\n\tWhen $\\lambda^{\\uparrow} \\neq \\mu^{\\uparrow}$, there exists a catalyst state $\\rho_{\\xi}$ such that $\\rho_{\\xi}\\otimes\\rho_{\\lambda} \\stackrel{PPT}{\\rightarrow} \\rho_{\\xi}\\otimes\\rho_{\\mu}$ if and only if\n\t\\begin{equation}\n\tS_{t}\\left(\\lambda\\right) > S_{t}\\left(\\mu\\right), \\forall t \\in [1\/2,\\infty).\n\t\\end{equation}\n\\end{conjecture}\n\nIt would also be desirable to determine the validity of Conjecture \\ref{conj:rankIs1}.\n\n\\begin{acknowledgments}\nWM acknowledges support from the U.K. EPSRC. \nAW is supported by the U.K. EPSRC (project ``QIP IRC'' and an Advanced\nResearch Fellowship), by a Royal Society Wolfson Merit Award, and\nthe EC, IP ``QAP''. The Centre for Quantum Technologies is funded by the\nSingapore Ministry of Education and the National Research Foundation as\npart of the Research Centres of Excellence programme.\nWe thank Armin Uhlmann and Jens Eisert for early discussions on the topic of this\npaper, and Richard Low for his comments on presentation.\n\\end{acknowledgments}\n\n\t\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{s-in}\n\nThe classical Pohozaev-Schoen identity \\cite{Schoen} lies within a larger set of geometric identities which rely on conservation principles arising from symmetries of specific variational problems, as has been noted and explained in \\cite{Gover} (see also \\cite{allan} for another variant, in manifolds with boundary, of this useful identity). In this work, the authors encapsulate several geometric identities which have had important impact in geometric analysis within a single identity derived from such symmetry principles. It is important to note that these type of geometric identities have been established for compact manifolds. Taking into account how useful these conservation principles have proved to be when analysing geometric problems, we intend to present a version of the Pohozaev-Schoen identity established in \\cite{allan} for asymptotically euclidean (AE) manifolds. These manifolds are complete non-compact manifolds with a particular simple structure at infinity. In general relativity, for example, we often need to consider initial data problems on non-compact manifolds, with natural restrictions on the asymptotic geometry (see, for instance, \\cite{CB2},\\cite{maxwell} and references therein). The positive mass theorem and the Penrose inequality are example of such situations (see \\cite{Bartnik},\\cite{Huisken},\\cite{Schoen-Yau1},\\cite{Schoen-Yau},\\cite{Witten}). In fact, geometric problems on this scenario have received plenty of attention since these structures play a central role in general relativity, serving as a model for initial data of isolated gravitational systems. Thus, plenty of analytic tools have been developed in this scenario. In this direction, our aim is to prove the validity of a version of the Pohozaev-Schoen identity in this context, which is sufficiently powerful to be useful when analysing a wide variety of geometric problems on AE manifolds. Explicitly, we will prove the following theorem.\\footnote{See Section 2 for the precise definition of the functional spaces involved.}\n\n\\begin{thm*}\nLet $(M,g)$ be a $H_{s,\\delta}$-asymptotically euclidean manifold, with $s>\\frac{n}{2}+1$ and $\\delta>-\\frac{n}{2}$. Also, let $B$ be a symmetric $(0,2)$-tensor field, satisfying $\\mathrm{div}_gB=0$, and let $X$ be a vector field on $M$. Suppose $X\\in C^1$ is bounded with $\\nabla X\\in L^2$. Furthermore, suppose that $X\\in L^2_{-(\\rho+1)}$ and $B\\in H_{s+1,\\rho}$, with $\\rho\\geq 0$. Then, the following equality holds:\n\\begin{align}\n\\int_M\\langle \\overset{\\circ}{B},\\pounds_X g \\rangle\\mu_g   = \\frac{2}{n} \\int_M X(\\mathrm{tr}_gB)\\mu_g + 2 \\int_{\\partial M}\\overset{\\circ}{B}(X,\\nu)\\mu_{\\partial M}\n\\end{align}\n\\end{thm*}\n\n\\bigskip\nWe will employ this version of the generalized Pohozaev-Schoen identity to analyse different geometric problems, such as generalized solitons on AE manifolds. These soliton-type equations arise as a generalization of Ricci almost solitons, as proposed by Pigola \\textit{et al.} \\cite{Pigola}. Also these type of equations were analysed in the \\textit{extrinsic} context by Al\\'ias \\textit{et al.} in \\cite{ALR}. During the analysis of these generalized solitons, we will prove the following rigidity theorem.\n\n\\begin{thm*}\nLet $(M^n,g)$ be an  asymptotically euclidean manifold without boundary, $n\\geq 3$, where $g$ satisfies the hypotheses of the above theorem, and suppose that the tensor field $B\\in H_{s+1,0}$, $s>\\frac{n}{2}$, is $g$-divergence-free. Furthermore, let $X\\in H_{s+2,-1}$ and $\\mathrm{tr}_gB=0$. Then, under these assumptions, any $B$-generalized soliton is $B$-flat.\n\\end{thm*}\n\nFrom this theorem, we will extract some interesting geometric consequences. For instance, we will show that any AE Ricci almost soliton with zero scalar curvature is trivial, \\textit{i.e}, is isometric to the euclidean flat space. \n\n\\bigskip\nAnother important result, where we will make use of the generalized Pohozaev-Schoen identity, is the following almost-Schur theorem for AE manifolds (see in Section \\ref{applications} a review and motivation for this type of inequalities): \n\n\\begin{thm*}\nLet $(M^n,g)$ is an $H_{s+3,\\delta}$-asymptotically euclidean manifold without boundary, with $n\\geq 3$, $s>\\frac{n}{2}$, $\\delta>-\\frac{n}{2}$ and $\\delta\\geq -2$. Now, let $B$ be a symmetric $(0,2)$-tensor field such that $\\mathrm{div}_gB=0$ and $B\\in H_{s+1,\\rho}$, with $\\rho>-\\frac{n}{2} + 2$, furthermore, suppose that $\\rho\\geq 0$, and denote its trace by  $b\\doteq \\mathrm{tr}_gB$.  Then:\\\\\nI)  there is a constant $C$ independent of the specific choice of such tensor $B$, such that the following inequality holds\n\\begin{align}\n||b||_{L^2}\\leq n\\left( \\frac{n-1}{n} + C||\\mathrm{Ric}_g||_{C^0_2}\\right)^{\\frac{1}{2}} ||\\overset{\\circ}{B}||_{L^2},\n\\end{align}\nII) If $\\mathrm{Ric}_g$ is non-negative, \\textit{i.e}, $\\mathrm{Ric}_g(X,X)\\geq 0$ $\\forall$ $X\\in\\Gamma(TM)$, then\n\\begin{align}\n||b||_{L^2}\\leq \\sqrt{n(n-1)} ||\\overset{\\circ}{B}||_{L^2}.\n\\end{align}\n\\end{thm*}\n\n\\bigskip\nNotice that the hypotheses of the theorem split depending on whether $n=3,4$ or $n\\geq 5$. In the first case, the hypotheses on the weights $\\delta$ and $\\rho$ are $\\delta>-\\frac{n}{2}$ and $\\rho>-\\frac{n}{2}+2$, while, if $n\\geq 5$, we get $\\delta\\geq -2$ and $\\rho\\geq 0$. One reason for these seemingly odd behaviour is that we need, as an underlying hypotheses, both $\\mathrm{Ric}_g$ and $B$ in $L^2$. In dimensions $3$ and $4$ this behaviour of the Ricci tensor is guaranteed by the condition $\\delta>-\\frac{n}{2}$. In contrast, for $n\\geq 5$, this is not the case, and the condition $\\delta\\geq -2$ guarantees the desired asymptotic behaviour. The weight $\\rho$ is subtlety tied to $\\delta$ via the solutions of a PDE (see the proof of Theorem \\ref{B-AS}).\n\n\nFrom this general result, we will extract as corollaries an almost-Schur inequality for the scalar curvature, analogous to the one proved by De Lellis-Topping in \\cite{Topping}; for $Q$-curvature, analogous to \\cite{chinos}, and for the mean curvature of AE-hypersurfaces in Ricci-flat spaces, analogous to \\cite{Cheng1}. Some of these results, in the compact case, where summarized and put together in \\cite{Cheng2}.\n\n\\bigskip\nFinally, we will analyse static potentials on AE manifolds. These structures have recieved plenty of interest and some important results, together with applications, can be consulted in J. Corvino's influential paper \\cite{Corvino}. Strong rigidity of these structures has been observed as a consequence of integral geometric identities, for instance, in \\cite{allan} by Barbosa \\textit{et al.} and in \\cite{Miao} by Miao-Tam for compact and AE manifolds respectively. In this context, we will show the validity of the following result.\n\n\\begin{thm*}\nSuppose that $(M^n,g)$ is a $H_{s+3,\\delta}$-AE manifold with $n\\geq 3$, $s>\\frac{n}{2}$ and $\\delta>-1$, which admits a non-negative static potential $f$. Furthermore, suppose that $\\mathrm{Ric}_g\\in H_{s+1,\\rho}$, for some $\\rho>\\frac{n}{2}-1$, and that $\\partial M=f^{-1}(0)$ consists of $N$-closed connected components, labelled by $\\{\\Sigma_i \\}_{i=1}^{N}$. Then, it follows that \n\\begin{align}\n\\int_Mf|\\mathrm{Ric}_g|^2_g\\mu_g = \\frac{1}{2}\\sum_{i=i}^Nc_i\\int_{\\Sigma_i}R_{h_i}\\mu_{\\partial M},\n\\end{align}\nwhere the constants $c_i=|\\nabla f|_{\\Sigma_i}$. In particular, if $\\partial M=\\emptyset$, then $(M^n,g)$ is isometric to $(\\mathbb{R}^n,e)$, where $e$ is the euclidean metric.\n\\end{thm*}\n\nThis theorem, in dimension three, was proved in \\cite{Miao}. We will show how their result has a natural (although non-trivial) extension to general dimensions. Furthermore, we will show how this result can be used to give very simple characterizations for the admissible topologies for the event horizon of static black holes. The characterization we will provide is not new, although it is derived using much simpler techniques than the ones used to prove more general results, such as the ones appearing in \\cite{EGP} and \\cite{Hawking}.  \n\n\\bigskip\nWith the above in mind, the organization of the paper will be as follows. We will first review some of the analytic tools needed for the core of the paper. Then, under sufficient conditions, we will establish the validity of the Pohozaev-Schoen identity on asymptotically euclidean manifolds. Finally, we will go into the applications, where we will show how some rigidity results of generalized solitons can be derived as a straightforward consequence of this identity. Furthermore, a generalized almost-Schur-type inequality, in this non-compact setting, can also be obtained using these tools. As our last application, we will show how some identities which are related with rigidity of static potentials on asymptotically euclidean manifolds, also follow directly from these principles.\n\n\\section{AE Manifolds}\n\nIn this section, the idea will be to present the main definitions and properties of AE manifolds that we will use in the subsequent sections. The analysis of these structures has been developed along the years, including the seminal works \\cite{Bartnik}, \\cite{Cantor}, \\cite{CB1}, \\cite{Maxwell-Dilts}, \\cite{maxwell}, \\cite{maxwell1}. We are following the classical notations and results established in \\cite{CB1}, where the detailed proofs can be consulted.\n\n\\begin{defn}\nA $n$-dimensional smooth Riemannian manifold $(M,e)$ is called euclidean at infinity if there exits a compact set $K$ such that $M\\backslash K$ is the disjoint union of a finite number of open sets $U_i$, such that each $(U_i,e)$ is isometric to the exterior of an open ball in the Euclidean space.   \n\\end{defn}\n\nOn manifolds euclidean at infinity, we define $d=d(x,p)$ the distance in the Riemannian metric $e$ of an arbitrary point $x$ to a fixed point $p$. We will typically omit the dependence on $(x,p)$. \n\n\\begin{defn}\nA weighted Sobolev space $H_{s,\\delta}$, with $s$ a nonnegative integer and $\\delta\\in\\mathbb{R}$, is a space of tensor fields $u$ of some given type on the manifold $(M,e)$ Euclidean at infinity with generalized derivatives of order up to $s$ in the metric $e$ such that $D^mu(1 + d^2)^{\\frac{1}{2}(m+\\delta)}\\in L^2$, for $0\\leq m\\leq s$. It is a Banach space with norm\n\\begin{align}\\label{norm}\n||u||^{2}_{H_{s,\\delta}}\\doteq \\sum_{0\\leq m\\leq s}\\int_M|D^mu|_e^2(1 + d^2)^{(m+\\delta)}d\\mu_{e}\n\\end{align}\nwhere $D$ represents the $e$-covariant derivative and $\\mu_e$ the Riemannian volume form associated with $e$.\n\\end{defn}\n\n\\begin{remark}\nIt is important to note that the space $C^{\\infty}_0(M,E)$ of compactly supported tensor fields of some given type is dense in $H_{s,\\delta}$ for all $s\\in\\mathbb{N}$ and $\\delta\\in\\mathbb{R}$. \n\\end{remark}\n\n\n\\begin{defn}\\label{AE-def}\nWe will say that a Riemannian metric $g$ on $M$ is asymptotically euclidean if $g-e\\in H_{s,\\delta}$, $s>\\frac{n}{2}$ and $\\delta>-\\frac{n}{2}$.\n\\end{defn}\n\nThese weighted spaces share several properties of the usual Sobolev spaces. For instance, we have the continuous embedding:\n\n\\begin{lemma}\\label{sobolevembedding}\nLet $(M, e)$ be a manifold euclidean at infinity. The following inclusion holds and is continuous:\n\\begin{align*}\nH_{s,\\delta}\\subset C^{s'}_{\\delta'}\n\\end{align*}\nif $s' <s -n\/2$ and $\\delta' <\\delta +n\/2$.\n\\end{lemma}\n\nIn the above lemma, the weighted-$C^k$ spaces are equipped with the following norm:\n\\begin{align*}\n||u||_{C^k_{\\beta}}\\doteq \\sup_{x\\in M}\\sum_{0\\leq l\\leq k}|D^lu|_e(1+d^2_e)^{\\frac{1}{2}(\\beta+l)}.\n\\end{align*}\n\nAlso, the following multiplication property is very useful to analyse the range of differential operators acting between these weighted spaces. \n\n\\begin{lemma}\\label{sobolevmultiplication}\nIf $(M,e)$ is an euclidean at infinity, then the following continuous multiplication property holds\n\\begin{align*}\nH_{s_1,\\delta_1}\\times H_{s_2,\\delta_2}&\\mapsto H_{s,\\delta},\\\\\n(f_1,f_2)&\\mapsto f_1\\otimes f_2,\n\\end{align*}\nif $s_1,s_2\\geq s$, $s<s_1+s_2-\\frac{n}{2}$ and $\\delta<\\delta_1+\\delta_2+\\frac{n}{2}$.\n\\end{lemma}\n\n\\subsection*{AE manifolds with boundary}\n\nThe idea in this subsection is to extend to above ideas to AE manifolds with boundary. We will consider $(M,e)$ to be a manifold euclidean at infinity, according to the definition presented above, but allowing that within the compact region $K$ we have a boundary. Then, we have a finite number of end charts, say $\\{U_i,\\varphi_i \\}_{i=1}^{k}$, with $\\varphi(U_i)\\simeq \\mathbb{R}^n\\backslash B_1(0)$, and a finite number of coordinate charts covering the compact region $K$, say $\\{U_i,\\varphi_i\\}_{i=k+1}^N$. We can consider a partition of unity $\\{\\eta_i\\}_{i=1}^N$ subordinate to the coordinate  cover $\\{U_i,\\varphi_i\\}_{i=1}^N$, and let $V_i$ be equal to either $\\mathbb{R}^n$ or $\\mathbb{R}^n_{+}$, depending on whether $U_i$, $i\\geq k+1$, is an interior of boundary chart respectively. Then, given a vector bundle $E\\xrightarrow{\\pi} M$, we can define $H_{s,\\delta}(M,E)$ to be the subset of $H_{s,loc}(M,E)$ such that \n\\begin{align}\\label{locsob}\n||u||_{H_{s,\\delta}}&=\\sum_{i=1}^k||{\\varphi^{-1}_i}^{*}(\\eta_i u)||_{H_{s,\\delta}(\\mathbb{R}^n)} + \\sum_{i=k+1}^N||{\\varphi^{-1}_i}^{*}(\\eta_i u)||_{H_{s}(V_i)}<\\infty. \n\\end{align}\n\nIt should be noted that Lemmas \\ref{sobolevembedding}-\\ref{sobolevmultiplication} remain true, since they hold for both compact manifolds with boundary and for the asymptotic region.\n\nWe should now comment on the properties of the \\textit{traces} of fields in $H_{s,\\delta}(M)$. From the theory of Sobolev spaces on compact manifolds with boundary, we know that if $\\Omega$ is a compact smooth manifold with smooth boundary $\\partial\\Omega$, then, for any function $u\\in H_{s}(M)$, with $s>\\frac{1}{2}$, we have a continuous trace map\\footnote{Here we are appealing to Sobolev spaces defined for non-integer $s\\in\\mathbb{R}$, which we regard as a standard tool. See, for instance, \\cite{Taylor} for the details.}\n\\begin{align*}\n\\tau: H_{s}(\\Omega) \\mapsto H_{s-\\frac{1}{2}}(\\partial\\Omega).\n\\end{align*}\n\nIn the case that $u$ is actually a section of a vector bundle $E\\xrightarrow{\\pi} \\Omega$, we get an analogous result. To see this, notice that the use of boundary charts trivializing $E$ naturally gives us an induced vector bundle $\\tilde{E}$ over $\\partial\\Omega$ with the same typical fibre as $E$, whose vector bundle structure comes from the boundary coordinate charts, which come from the coordinate systems of $\\Omega$ adapted  to the boundary that trivialize $E$. Clearly, we can both extend the sections of $\\tilde{E}$ to section of $E$, and, vice versa, restrict sections of $E$ to sections of $\\tilde{E}$. Hence, we will drop the tilde from $\\tilde{E}$, and take this setting as implicit. Now, notice that the restriction of a section of $E$ to $\\partial\\Omega$ will produce a map $\\tau u: \\partial\\Omega\\mapsto E$. Furthermore, by localizing $u$ in a fashion analogous to what was done prior to introducing the norm (\\ref{locsob}), we get that $u\\in H_{s}(\\Omega,E)$ iff given a partition of unity $\\{\\alpha_i \\}$ subordinate to a coordinate cover $\\{U_i,\\varphi_i \\}$, the localized fields $\\alpha_i u \\in H_s(U_i,E)$. We can then use boundary charts to trace these localized fields, and get, $\\forall$ $s>\\frac{1}{2}$, a continuous trace map \\footnote{The interested reader can find the details on \\cite{Palais}}\n\\begin{align*}\n\\tau:H_{s}(\\Omega,E) \\mapsto H_{s-\\frac{1}{2}}(\\partial\\Omega,E).\n\\end{align*}\n\nIn the case of $AE$-manifolds, since the trace map only involves the compact core $K\\subset\\subset M$, from the above and a partition of unity argument, we get that, for $s>\\frac{1}{2}$, we have continuous trace map\na continuous trace map\n\\begin{align}\n\\tau:H_{s,\\delta}(M,E) \\mapsto H_{s-\\frac{1}{2}}(\\partial M,E).\n\\end{align}\n\n\\begin{remark}\nIn the subsequent sections, in the notation $H_{s,\\delta}(M,E)$, whenever the vector bundle is implicit from the context, we will simply write $H_{s,\\delta}$. The same will be done with other functional spaces.\n\\end{remark}\n\n\\begin{remark}\nWe would like to point out that the definition of AE Riemannian manifold provided in Definition \\ref{AE-def} follows the same lines as the definition given in \\cite{CB1}-\\cite{CB2} and \\cite{Bartnik}, among others. In particular, the definition adopted by Bartnik in \\cite{Bartnik} is related to ours by a simple shift in the weight parameter. Explicitly, $\\delta_{b}=-(\\delta + \\frac{n}{2})$, where $\\delta_b$ is the weighting parameter used in the definition of the functional spaces in \\cite{Bartnik}. Nevertheless, analogous definitions can be given without invoking weighted Sobolev spaces. For instance, it is quite usual to define an AE metric on a manifold euclidean at infinity by its behaviour in each end in coordinate systems. That is, demanding $g$, near infinity, to satisfy a condition of the form\n\\begin{align}\\label{AE-def2}\ng_{ij}=\\delta_{ij} + o_{k}(|x|^{-\\tau}),\n\\end{align}\nwhere $o_{k}(|x|^{-\\tau})$ implies that $\\partial^{l}g_{ij}=o(|x|^{-\\tau-l})$ for all $1\\leq l \\leq k$ and $\\tau>0$. It should be completely clear that the condition expressed in Definition \\ref{AE-def} controls the behaviour of $g$ and its derivatives at infinity and implies that the metric satisfies a conditions of the form of (\\ref{AE-def2}). Also, similar definitions could be given by means of weighted $C^k$-spaces and H\\\"older spaces. All of these definitions are very easily related by means of embedding theorems and analysis of the behaviour of the field at infinity, and the precise choice relies on the specific problems to be treated and which of them seems to be more transparently tailored to deal with such problems.\n\\end{remark}\n\n\\section{Pohozaev-Schoen identity on AE manifolds}\n\nThe idea now will be to try to deduce the integral identity in the context of AE manifolds. We will consider that such manifolds $M$ may have compact boundary $\\partial M$ as described in the previous section.\n\n\n\n\\begin{thm}\\label{thm1}\nLet $(M,g)$ be a $H_{s,\\delta}$-asymptotically euclidean manifold, with $s>\\frac{n}{2}+1$ and $\\delta>-\\frac{n}{2}$. Also, let $B$ be a symmetric $(0,2)$-tensor field, satisfying $\\mathrm{div}_gB=0$, and let $X$ be a vector field on $M$. Suppose $X\\in C^1$ is bounded with $\\nabla X\\in L^2$. Furthermore, suppose that $X\\in L^2_{-(\\rho+1)}$ and $B\\in H_{s+1,\\rho}$, with $\\rho\\geq 0$. Then, the following equality holds:\n\\begin{align}\\label{PS}\n\\int_M\\langle \\overset{\\circ}{B},\\pounds_X g \\rangle\\mu_g   = \\frac{2}{n} \\int_M X(\\mathrm{tr}_gB)\\mu_g + 2 \\int_{\\partial M}\\overset{\\circ}{B}(X,\\nu)\\mu_{\\partial M}\n\\end{align}\n\\end{thm}\n\n\n\\begin{proof}\n\nWith a slight abuse of notation, making contact with standard notation for differential forms, we will denote the contraction of $B$ and $X$ by $i_XB\\doteq B(X,\\cdot)$.  Now, if we consider $B\\in C^{\\infty}_0$ and $X$ satisfying the hypotheses described above, then the following is standard:\n\\begin{align}\\label{PS1}\n\\frac{1}{2}\\int_M\\langle B,\\pounds_{X}g\\rangle\\mu_g=\\int_M\\mathrm{div}_g(i_XB)\\mu_g=\\int_{\\partial M}B(X,\\nu)\\mu_{\\partial M},\n\\end{align}\nwhere $\\nu$ denotes the unit normal of $\\partial M$. Now, consider $\\{B_n\\}_{n=1}^{\\infty}\\subset C^{\\infty}_0(M)$, such that \n\\begin{align*}\nB_n&\\xrightarrow{H_{s+1,\\rho}} B.\n\\end{align*}\nThen, we have the following:\n\\begin{align*}\n\\Big\\vert \\int_M(\\langle B,\\pounds_{X}g\\rangle-\\langle B_n,\\pounds_{X}g\\rangle)\\mu_g\\Big\\vert&\\leq \\int_M|\\langle B-B_n,\\pounds_{X}g\\rangle|\\mu_g,\\\\\n&\\leq \\int_M|B-B_n|_g|\\pounds_{X}g|_g\\mu_g,\\\\\n&\\leq ||B-B_n||_{L^2}||\\pounds_{X}g||_{L^2}\\xrightarrow[n\\rightarrow\\infty]{} 0,\n\\end{align*}\nwhere we have made use of the fact that, since $\\nabla X\\in L^2$, so does $\\pounds_{X}g$. All this gives us \n\\begin{align}\\label{PS2}\n\\frac{1}{2}\\int_M\\langle B,\\pounds_{X}g\\rangle\\mu_g=\\lim_{n\\rightarrow\\infty}\\int_{\\partial M}B_n(X,\\nu)\\mu_{\\partial M}.\n\\end{align}\nNow, a similar argument gives us\n\\begin{align*}\n\\Big\\vert\\int_{\\partial M}(B-B_n)(X,\\nu)\\mu_{\\partial M}\\Big\\vert\\lesssim \\int_{\\partial M} |B-B_n|_g|X|_g\\mu_{\\partial M}\n\\end{align*}\nwhere  we have used that, given two tensor fields $A,B$ on $M$ then the following pointwise estimate holds $|C(A\\otimes B)|_g\\leq K(n,d_1,d_2)|A|_g|B|_g$, where $C$ denotes an arbitrary contraction and $K$ is a constant depending only on the dimension $n$ of $M$ and the dimensions $d_1$ and $d_2$ of the fibres. Notice that, since we have a continuous trace $\\tau: H_{s+1,\\rho}(M)\\mapsto H_{s}(\\partial M)$, then $\\tau(B-B_n)\\xrightarrow[n\\rightarrow \\infty]{} 0$ in $L^2(\\partial M)$. Hence, from the above, we get\n\\begin{align*}\n\\Big\\vert\\int_{\\partial M}(B-B_n)(X,\\nu)\\mu_{\\partial M}\\Big\\vert\\lesssim ||B-B_n||_{L^2(\\partial M)} ||X||_{L^2(\\partial M)}\\xrightarrow[n\\rightarrow \\infty]{} 0\n\\end{align*}\nPutting this together with (\\ref{PS2}), we get\n\\begin{align}\\label{PS3}\n\\frac{1}{2}\\int_M\\langle B,\\pounds_{X}g\\rangle\\mu_g=\\int_{\\partial M}B(X,\\nu)\\mu_{\\partial M}.\n\\end{align}\n\n\nNow, if we write $B=\\overset{\\circ}{B}+\\frac{1}{n}\\mathrm{tr}_gB \\: g$, where $\\overset{\\circ}{B}$ is the traceless part of $B$, we get the following\n\\begin{align}\\label{PS4}\n\\langle B, \\pounds_Xg \\rangle &= \\langle \\overset{\\circ}{B},\\pounds_X g \\rangle + \\frac{2}{n}\\mathrm{tr}_gB\\mathrm{div}_gX.\n\\end{align}\nNotice that $\\mathrm{tr}_gB\\mathrm{div}_gX=\\mathrm{div}_g(\\mathrm{tr}_gB X) - X(\\mathrm{tr}_gB)$, which gives us that \n\\begin{align*}\n\\int_M\\mathrm{tr}_gB\\mathrm{div}_gX\\mu_g=\\int_{\\partial M}\\mathrm{tr}_gB\\langle X,\\nu\\rangle \\mu_{\\partial M} -\\int_M X(\\mathrm{tr}_gB)\\mu_g \\;\\; \\forall \\;\\; B\\in C^{\\infty}_{0}.\n\\end{align*}\nDoing as above and approximating $B$ by compactly supported fields converging to $B$ in $H_{s+1,\\rho}$, we get\n\\begin{align*}\n\\Big\\vert\\int_M(\\mathrm{tr}_gB\\mathrm{div}_gX-\\mathrm{tr}_gB_n\\mathrm{div}_gX)\\mu_g\\Big\\vert&\\leq \\int_M|\\mathrm{tr}_g(B-B_n)||\\mathrm{div}_gX| \\mu_g,\\\\\n&\\lesssim \\int_M|B-B_n|_g|\\nabla X|_g\\mu_g,\\\\\n&\\leq ||B-B_n||_{L^2}||\\nabla X||_{L^2}\\xrightarrow[n\\rightarrow\\infty]{} 0,\n\\end{align*}\nsince, by hypothesis, $\\nabla X\\in L^2(M)$. The above shows that \n\\begin{align}\n\\int_M\\mathrm{tr}_gB\\mathrm{div}_gX\\mu_g=\\lim_{n\\rightarrow\\infty}\\Big\\{ \\int_{\\partial M}\\mathrm{tr}_gB_n\\langle X,\\nu\\rangle \\mu_{\\partial M} -\\int_M X(\\mathrm{tr}_gB_n)\\mu_g \\Big\\}.\n\\end{align}\nFurthermore, notice that \n\\begin{align*}\n\\Big\\vert\\int_M X(\\mathrm{tr}_g(B-B_n))\\mu_g\\Big\\vert&\\leq \\int_M|\\langle \\nabla\\mathrm{tr}_g(B-B_n),X\\rangle|\\mu_g,\\\\\n&\\lesssim \\int_M|\\nabla(B-B_n)|_g|X|_g\\mu_g,\\\\\n&=\\int_M|\\nabla(B-B_n)|_g(1+d^2_e)^{\\frac{1}{2}(\\rho+1)}(1+d^2_e)^{-\\frac{1}{2}(\\rho+1)}|X|_g\\mu_g,,\\\\\n&\\leq ||X||_{L^2_{-(\\rho+1)}}||\\nabla(B-B_n)||_{L^2_{\\rho+1}}\\xrightarrow[n\\rightarrow\\infty]{} 0\n\\end{align*}\n\nSimilarly, we also get that\n\\begin{align*}\n\\Big\\vert\\int_{\\partial M}\\mathrm{tr}_g(B - B_n)\\langle X,\\nu\\rangle \\mu_{\\partial M}\\Big\\vert&\\lesssim \\int_{\\partial M}|B - B_n|_g |X|_g,\\\\\n&\\leq ||B-B_n||_{L^2(\\partial M)} ||X||_{{L^2(\\partial M)}} \\xrightarrow[n\\rightarrow\\infty]{} 0,\n\\end{align*}\nwhere we have again used the continuity of the trace map. All of the above, gives us that\n\\begin{align}\n\\int_M\\mathrm{tr}_gB\\mathrm{div}_gX\\mu_g=\\int_{\\partial M}\\mathrm{tr}_gB\\langle X,\\nu\\rangle \\mu_{\\partial M} -\\int_M X(\\mathrm{tr}_gB)\\mu_g.\n\\end{align}\nThus, integrating (\\ref{PS4}) and applying the above identity, we see that the following holds\n\\begin{align*}\n\\int_M\\langle B, \\pounds_Xg \\rangle \\mu_g = \\int_M\\langle \\overset{\\circ}{B},\\pounds_X g \\rangle\\mu_g  +  \\frac{2}{n}\\int_{\\partial M}\\mathrm{tr}_gB\\langle X,\\nu\\rangle \\mu_{\\partial M} - \\frac{2}{n} \\int_M X(\\mathrm{tr}_gB)\\mu_g.\n\\end{align*}\nPutting this together with (\\ref{PS3}), we get that the Pohozaev-Schone equality is valid in the present context:\n\\begin{align}\n\\int_M\\langle \\overset{\\circ}{B},\\pounds_X g \\rangle\\mu_g   = \\frac{2}{n} \\int_M X(\\mathrm{tr}_gB)\\mu_g + 2 \\int_{\\partial M}\\overset{\\circ}{B}(X,\\nu)\\mu_{\\partial M}\n\\end{align}\n\n\\end{proof}\n\n\n\\section{Applications} \\label{applications}\n\nBefore going into the applications of the above identity, we would like to state and give an independent proof of the following rigidity statement linked with AE-manifolds. It should be stressed that similar statements, with slightly different hypotheses, can be found in several papers, such as \\cite{Lee-Parker},\\cite{Bartnik},\\cite{volcomp}.\n\n\\begin{thm}\\label{Ricciflatrigidity}\nLet $(M,e)$ be a, $n$-dimensional manifold euclidean at infinity, with $n\\geq 3$, and consider a $H_{s,\\delta}$-AE metric $g$, with $s> \\frac{n}{2}+2$ and $\\delta>-\\frac{n}{2}$. Then, if $g$ Ricci-flat, it holds that $(M,g)$ is isometric to $\\mathbb{R}^n$ with its standard flat metric.  \n\\end{thm} \n\n\\begin{proof}\nThis proof will be carried out via a number of well-known results. First, notice that under our hypotheses on $g$, its Ricci tensor can be written, on any particular end, in harmonic coordinates giving us an expression of the form\n\\begin{align}\n{\\mathrm{Ric}_g}_{ij}=-g^{ab}\\partial_a\\partial_bg_{ij} + Q_{ij}(g,\\partial g),\n\\end{align}\nwhere $Q(g,\\partial g)$ is a quadratic polynomial in $\\partial g$. Thus, in these coordinates, the operator appearing in the right hand side of the above identity is an elliptic second order operator acting on the metric component functions written in such coordinates. Nevertheless, since the left-hand side is a tensor, we get that if $\\mathrm{Ric}_g\\in L^2_{\\tau}$, so is the left hand side on any given coordinate system. Thus, via elliptic theory we can improve the decay of the metric. This can be achieved by appealing to Theorem 5.1 in \\cite{CB1} plus an inductive argument of the type of Lemma 3.6 in Appendix II of \\cite{CB2}. In any case, the final result is that we can guarantee that $g-e\\in H_{2,\\tau-2}$, as long as $\\tau-2<\\frac{n}{2}-2$. In particular, if $g$ is Ricci-flat, then $g-e\\in H_{2,\\rho}\\cap C^2$ for any $\\rho<\\frac{n}{2}-2$. This implies that, near infinity, $g_{ij}=\\delta_{ij}+o_2(|x|^{-(\\rho+\\frac{n}{2})})$. Thus, we get that\n\\begin{align*}\ng_{ij}=\\delta_{ij}+o_2(|x|^{-\\rho'}), \\;\\; \\forall \\;\\; \\rho'<n-2.\n\\end{align*}\nThe above, plus Ricci-flatness, imply that the ADM mass of $g$ is actually well-defined, and, in particular, we are under the decay assumptions of Theorem 2.1 of \\cite{Miao-ADM}. Hence, we can rewrite the ADM mass of $g$, say $m(g)$, in terms of its Einstein tensor \\textit{at infinity}. Thus, Ricci-flatness implies that $m(g)=0$. From this condition we get that $(M,g)$ is isometric to $(\\mathbb{R}^n,\\cdot)$ via the well-known rigidity of the positive mass theorem, which can be obtained in this setting via Proposition 10.2 in \\cite{Lee-Parker}.\n\\end{proof}\n\n\n\\subsection*{Generalized Solitons}\n\nIn this section, we will consider $B$-\\textit{generalzed solitons} and use the Pohozaev-Schoen identity (\\ref{PS2}) to obtain some rigidity results. To begin with, we will consider an asymptotically euclidean manifold $(M,g)$, a symmetric tensor field $B$ and a vector field $X$. Then, we say that $X$ defines a $B$-generalized soliton structure on the AE-manifold $M$ if the following equation is satisfied\n\\begin{align}\\label{Bgensol}\nB+\\frac{1}{2}\\pounds_Xg=\\lambda g,\n\\end{align}\nfor some function $\\lambda$. Then, we have the following immediate lemma.\n\\begin{thm}\\label{Bgensol}\nLet $(M^n,g)$ be an  asymptotically euclidean manifold without boundary, $n\\geq 3$, where $g$ satisfies the hypotheses of Theorem \\ref{thm1}, and suppose that the tensor field $B\\in H_{s+1,0}$, $s>\\frac{n}{2}$, is $g$-divergence-free. Furthermore, let $X\\in H_{s+2,-1}$ and $\\mathrm{tr}_gB=0$. Then, under these assumptions, any $B$-generalized soliton is $B$-flat.\n\\end{thm}\n\\begin{proof}\nUnder our working hypotheses, since $H_{s+2,-1}\\hookrightarrow C^1_{-1}$, we have both that $X\\in H_{1,-1}\\cap C^1$. Thus, we know that the identity (\\ref{PS2}) holds. Hence, if $\\mathrm{tr}_gB=0$, we have that\n\\begin{align*}\n\\int_M\\langle \\overset{\\circ}{B},\\pounds_X g \\rangle\\mu_g=0.\n\\end{align*}\nFurthermore, since $X$ satisfies (\\ref{Bgensol}), the following identities holds\n\\begin{align*}\n\\langle \\overset{\\circ}{B},\\pounds_X g \\rangle &= \\langle B,\\pounds_{X,conf} g \\rangle,\\\\\n&=- \\frac{1}{2}\\langle \\pounds_Xg,\\pounds_{X,conf} g \\rangle,\\\\\n&=- \\frac{1}{2}\\langle \\pounds_{X,conf}g,\\pounds_{X,conf} g \\rangle,\n\\end{align*}\nwhere in the first line we have introduced the conformal Killing Laplacian, defined by $\\pounds_{X,\\text{conf}}g\\doteq \\pounds_Xg-\\frac{2}{n}\\mathrm{div}_gX g$, which defines a traceless tensor; in the second line we used the soliton equation (\\ref{Bgensol}); and in the last line we used this traceless property of the conformal Killing Laplacian. With the above, we see that (\\ref{Bgensol}) implies that \n\\begin{align*}\n\\int_M|\\pounds_{X,conf}g|_g^2\\mu_g=0,\n\\end{align*}\nwhich is equivalent to $X\\in H_{s+1,-1}$ being a conformal Killing field of $g$. But it is known that there are no non-trivial Killing fields with this asymptotic behaviour \\cite{Chris}-\\cite{maxwell}, thus $X=0$, and thus $B$-flat.\n\\end{proof}\n\nFrom this theorem, we get the following corollaries.\n\n\\begin{coro}\nConsider an $n$-dimensional AE-Riemannian manifold $(M,g)$ without boundary, with $g-e\\in H_{s+3,\\delta}$, with $s>\\frac{n}{2}$ and $\\delta>-\\min\\{\\frac{n}{2},2\\}$ and $n\\geq 3$. Then, if $R_g=0$, any Ricci almost soliton $X\\in H_{s+2,\\delta+1}$ on $(M,g)$ is isometric to $(\\mathbb{R}^n,e)$, with $e$ the euclidean metric.\n\\end{coro}\n\\begin{proof}\nFirst, recall a Ricci almost soliton structure on a Riemannian manifold $(M,g)$ is given by a vector field $X$ satisfying the equation\n\\begin{align}\\label{Riccisol1}\nRic_g+\\frac{1}{2}\\pounds_Xg=\\lambda g,\n\\end{align}\nfor some function $\\lambda$. Notice that the above equation is equivalent to the $G$-generalized soliton equation given by\n\\begin{align}\\label{Riccisol2}\nG_g +\\frac{1}{2}\\pounds_Xg=(\\lambda - \\frac{1}{2}R_g)g.\n\\end{align}\nwhere $G_g\\doteq \\mathrm{Ric}_g-\\frac{1}{2}R_gg$ stands for the Einstein tensor associated to the metric $g$. Notice that under our hypotheses for $g$, we have that $G_g\\in H_{s+1,0}$. Thus, from this fact and the hypotheses on $X$, we are under the hypotheses of Theorem \\ref{Bgensol}. Hence, if $\\mathrm{tr}_gG_g=\\frac{2-n}{2}R_g=0$, we have that $Ric_g=0$ as a consequence of the Theorem. Thus, from Theorem \\ref{Ricciflatrigidity}, we get that $(M,g)$ is isometric to $(\\mathbb{R}^n,e)$.\n\n\n\\end{proof}\n\nSimilarly, we can consider \\textit{Codazzi-generalized solitons} for some AE-hypersurface $(M^n,g)$ isometrically immersed in a Ricci-flat space $(\\tilde{M}^{n+1},\\tilde{g})$. In such a case, from Codazzi's equation, we know that the tensor $\\pi\\doteq K - \\mathrm{tr}_gK g$ satisfies the conservation equation\n\\begin{align}\\label{piconservation}\n\\mathrm{div}_g\\pi=0,\n\\end{align}\nwhere $K$ stands for the \\textit{extrinsic curvature} of the hypersurface $M\\hookrightarrow \\tilde{M}$, which is defined as $K\\doteq \\langle \\RN{2},n \\rangle$, where $\\RN{2}$ stands for the second fundamental form associated to the immersion, and $n$ for the unit normal field. Now, we define a Codazzi-generalized soliton structure on such an immersion as the choice of a vector field $X$ on $M$ satisfying the following soliton-type equation\n\\begin{align}\\label{codsol}\nK + \\frac{1}{2}\\pounds_Xg = \\lambda g,\n\\end{align} \nfor some function $\\lambda$ on $M$. \n\n\\begin{coro}\nConsider the setting described above, suppose that $M$ does not have a boundary, and suppose that $(M,g,X)$ provides a Codazzi-generalized soliton structure in a Ricci-flat space. Furthermore, suppose that $K\\in H_{s+1,0}$ and $X\\in H_{s+2,-1}$. Then, if $M$ is a minimal or maximal hypersurface\\footnote{We are enabling $\\tilde{g}$ to be either Riemannian or Lorentzian.}, that is, $\\mathrm{tr}_gK=0$, then $(M,g)$ is a totally geodesic scalar-flat hypersurface. In particular, if the ambient space is actually flat, then $(M,g)$ is isometric to $(\\mathbb{R}^n,e)$, with $e$ the euclidean metric.\n\\end{coro}\n\\begin{proof}\nSince our ambient manifold is Ricci-flat, we know that the Codazzi equation for hypersurfaces implies that (\\ref{piconservation}) holds. Furthermore, the hypothesis of $\\mathrm{tr}_gK=0$ implies that $\\pi=K$, and thus $K$ is a \\textit{conserved} symmetric tensor, which satisfies all the hypotheses demanded for the tensor field $B$ in Theorem \\ref{thm1}. Furthermore, following the same arguments as in the previous corollary, we have that $X$ also satisfies all the hypotheses imposed in Theorem \\ref{thm1}. Thus, using Theorem \\ref{Bgensol}, we immediately get that $M\\hookrightarrow\\tilde{M}$ is totally geodesic. Now, notice that the Gauss equation can be written en the following way. Given $X,Y,Z,W\\in\\Gamma(TM)$, we have that\n\\begin{align}\\label{Gauss}\n\\tilde{g}(\\tilde{R}(X,Y)Z,W)=g(R(X,Y)Z,W) + \\epsilon\\big(  K(X,Z)K(Y,W) - K(Y,Z)K(X,W) \\big),\n\\end{align}\nwhere $\\epsilon=\\tilde{g}(n,n)=\\pm 1$, depending on whether the normal direction is \\textit{space-like} or \\textit{time-like}. Now, we have just shown that $M$ is totally geodesic, thus $K=0$. Thus, consider an orthonormal frame $\\{E_{\\alpha}\\}_{\\alpha=0}^n$, where $E_{0}=n$ and thus $E_i$ is tangent to $M$ $\\forall$ $i=1,\\cdots,n$, so that we can write\n\\begin{align*}\n\\mathrm{Ric}_g(X,Y)&=\\sum_{i=1}^n\\tilde{g}(\\tilde{R}(E_i,Y)Z,E_i),\\\\\n&=\\mathrm{Ric}_{\\tilde{g}}(X,Y)-\\epsilon\\tilde{g}(\\tilde{R}(n,Y)Z,n),\\\\\n&=-\\epsilon\\tilde{g}(\\tilde{R}(n,Y)Z,n),\n\\end{align*} \nwhere in the third line we have used the hypotheses that $\\tilde{g}$ is Ricci-flat. From this, we get\n\\begin{align*}\nR_g=-\\epsilon\\sum_{j=1}^n\\tilde{g}(\\tilde{R}(n,E_j)E_j,n)=-\\epsilon\\sum_{j=1}^n\\tilde{g}(\\tilde{R}(E_j,n)n,E_j)=-\\epsilon\\mathrm{Ric}_{\\tilde{g}}(n,n)=0,\n\\end{align*}\nwhich proves that $(M,g)$ is scalar-flat. Finally, since $K=0$, it is a trivial consequence of (\\ref{Gauss}) that if $\\mathrm{Riem}_{\\tilde{g}}=0$, then $\\mathrm{Riem}_{g}=0$ and thus $(M^n,g)$ is isometric to $(\\mathbb{R}^n,e)$.\n\n\\end{proof}\n\\begin{remark}\nIn order to gain some insight about the kind of geometric picture that a Codazzi-soliton represents, it is an interesting observation to note that, following closely Al\\'ias-Lira-Rigoli \\cite{ALR}, we can understand (\\ref{codsol}) following the kind of definition provided in \\cite{ALR} for mean curvature flow solitons. That is, if, given an immersion $\\psi:M^n\\mapsto \\tilde{M}^{n+1}$, we say it defines a Codazzi-soliton with respect to a conformal Killing field $X\\in\\Gamma(T\\tilde{M})$ if it satisfies\n\\begin{align*}\nX^{\\perp}=-n,\n\\end{align*}\nalong $\\psi$, where $X^{\\perp}$ denotes the orthogonal component of $X$ to $\\psi(M)$ and $n$ denotes the unit normal of $\\psi(M)$. Then, following the same line of argument presented in \\cite{ALR}, equation (\\ref{codsol}) is an equation that any Codazzi-soliton must satisfy. Thus, in this sense, Codazzi solitons stand in perfect analogy with interesting soliton-like equations related to interesting geometric flows.  \n\n\\end{remark}\n\n\\begin{remark}\nTaking into account the above remark and corollary, it is also interesting to note that, in the case of manifolds euclidean at infinity, admitting a scalar-flat metric is not a trivial statement. This can be viewed as a consequence of some important results, concerning for instance the Yamabe classification for AE manifolds \\cite{Maxwell-Dilts} and topological obstructions related with the positive mass theorem \\cite{Schoen-Yau}. More explicitly, using the Yamabe classification provided in \\cite{Maxwell-Dilts}, we know that AE manifolds produced by removing points from closed manifolds which do not admit Yamabe positive metric do not carry any metric of zero scalar curvature. Using the results established in \\cite{Schoen-Yau}, it is possible to produce a wide variety of closed manifolds which do not admit a Yamabe positive metric. For instance, their main claim is that manifolds of the form $M^n\\# T^n$, with $M^n$ closed, lie in this category. Now, all this implies that AE manifolds obtained by removal of a finite number of points from such closed manifolds do not admit any Codazzi-soliton structure in any Ricci-flat space. \n\\end{remark}\n\nSimilarly to what we have done above, we could easily define new soliton-type equation related with other conserved second rank tensor fields, which we could relate to other generalized geometric flows. \n\n\\subsection*{Generalized Almost-Schur type inequality}\n\nThe classical Schur's lemma says that every connected Einstein manifold of dimension $n\\geq 3$ has constant scalar curvature. In \\cite{Topping}, De Lellis and Topping studied a quantitative version of the classical Schur Lemma, that allows to infer that, if a closed Riemannian manifold is close to being Einstein in the $L^{2}$ sense, then its scalar curvature is close to a constant in $L^{2}$. In this scope, they proved the so called ``Almost Schur Lemma\" that claims if $(M^{n},g)$ is a closed Riemannian manifold of dimension $n\\geq 3$, with nonnegative Ricci curvature, then\n\n$$\\displaystyle\\int_{M}(R-\\bar{R})^{2}\\mu_{g}\\leq \\frac{4n(n-1)}{(n-2)^{2}}\\displaystyle\\int_{M}\\left|Ric-\\frac{R}{n}g\\right|^{2}\\mu_{g},$$\nwhere we denote $\\bar{R}=\\frac{1}{Vol M}\\int_{M}R\\mu_{g}$. Furthermore, the equality occurs if, and only if, $M$ is an Einstein manifold.\n\nWe observe that a central step in the proof is the following integral identity (see \\cite{Topping})\n\n$$-\\displaystyle\\int_{M}\\langle dR, df\\rangle \\mu_{g}=\\frac{2n}{n-2}\\displaystyle\\int_{M}\\langle{\\stackrel{\\circ}{Ric}},D^{2}f\\rangle,$$\nwhere $f$ is a solution of a PDE. In fact, this integral formula is the Pohoz\\v{a}ev-Sch\\\"oen identity, if $X=\\nabla f$ and $\\partial M=\\emptyset$. In fact, this idea can be used to obtain these type of inequalities for more general tensors and under others suitable conditions on the Ricci curvature (\\cite{Barbosa},\\cite{Cheng1},\\cite{Cheng2}). The idea of this section is to generalize these type of inequalities obtained in \\cite{Topping} to non-compact manifolds in our case of interest. Thus, suppose that $(M^n,g)$ is an $H_{s+3,\\delta}$-asymptotically euclidean manifold, with $n\\geq 3$, $\\delta>-\\frac{n}{2}$, $\\delta\\geq -2$ and $s>\\frac{n}{2}$. Recall from \\cite{CB1} that $\\Delta_g:H_{s+2,\\delta_0}\\mapsto H_{s,\\delta_0+2}$ is an isomorphism for $-\\frac{n}{2}<\\delta_0<\\frac{n}{2}-2$. It will be an interesting observation to note that this is not true for closed manifolds, where what we have is that the operators of the form $\\Delta_g-a:H_{s+2}\\mapsto H_s$ with $a>0$ and $a\\in H_{s}$  have this isomorphism property. This difference can be traced back to the fact that on closed manifolds constant functions are in the Kernel of $\\Delta_g$, while on AE-manifolds we have a Poincar\\'e inequality \\cite{Bartnik}-\\cite{Maxwell-Dilts}, which guarantees that $\\Delta_g$ has the isomorphism property. In fact, a stronger statement that can be made. That is, given a $H_{s+1,\\delta}$-AE metric, with $s>\\frac{n}{2}$ and $\\delta>-\\frac{n}{2}$, then the Laplacian operator associated the $g$, that is $\\Delta_g$, is injective on $H_{2,\\rho}$ for any $\\rho>-\\frac{n}{2}$. This result is a consequence of Corallary 4.3 presented in Appendix 2 of \\cite{CB2} and an application of the Sobolev embedding theorems. These points are key features that will allow us to produce an almost-Schur-type inequality without any restrictions on the Ricci-curvature. It is known that, for $n\\geq 5$, being bounded from below is a necessary condition in the case of closed manifolds \\cite{Topping}. \n\n\\begin{thm}\\label{B-AS}\nLet $(M^n,g)$ is an $H_{s+3,\\delta}$-asymptotically euclidean manifold without boundary, with $n\\geq 3$, $s>\\frac{n}{2}$, $\\delta>-\\frac{n}{2}$ and $\\delta\\geq -2$. Now, let $B$ be a symmetric $(0,2)$-tensor field satisfying the hypotheses of Theorem \\ref{thm1}, that is, $\\mathrm{div}_gB=0$ and $B\\in H_{s+1,\\rho}$, with $\\rho\\geq 0$, furthermore, suppose that $\\rho>-\\frac{n}{2} + 2$, and denote its trace by  $b\\doteq \\mathrm{tr}_gB$.  Then:\\\\\nI)  there is a constant $C$ independent of the specific choice of such tensor $B$, such that the following inequality holds\n\\begin{align}\\label{AS}\n||b||_{L^2}\\leq n\\left( \\frac{n-1}{n} + C||\\mathrm{Ric}_g||_{C^0_2}\\right)^{\\frac{1}{2}} ||\\overset{\\circ}{B}||_{L^2},\n\\end{align}\nII) If $\\mathrm{Ric}_g$ is non-negative, \\textit{i.e}, $\\mathrm{Ric}_g(X,X)\\geq 0$ $\\forall$ $X\\in\\Gamma(TM)$, then\n\\begin{align}\\label{AS.01}\n||b||_{L^2}\\leq \\sqrt{n(n-1)} ||\\overset{\\circ}{B}||_{L^2}.\n\\end{align}\n\\end{thm}\n\n\\begin{proof}\nUnder the above hypotheses, since $\\delta>-\\frac{n}{2}$, we have that $b\\in H_{s+1,\\rho}$. Now, assume that $\\rho<\\frac{n}{2}$. If actually $\\rho\\geq \\frac{n}{2}$, then we would have a continuous embedding $H_{s+1,\\rho}\\hookrightarrow H_{s+1,\\sigma}$, for any $\\sigma<\\frac{n}{2}$. Thus, this assumption posses no actual restrictions, and from now on, so as to avoid introducing one further weighting parameter, we will assume it. Also, from the hypotheses bounding $\\rho$ from below, we have that $-\\frac{n}{2}+2<\\rho<\\frac{n}{2}$, for $n=3,4$; and $0\\leq\\rho<\\frac{n}{2}$, for $n\\geq 5$. Notice that these conditions can always be satisfied, and that, as stated before, we are not posing any further restrictions. \n\nNow, consider $\\Delta_g:H_{s+3,\\rho-2}\\mapsto H_{s+1,\\rho}$, and consider the equation\n\\begin{align}\\label{AS1}\n\\Delta_gf=b.\n\\end{align}\nSince $-\\frac{n}{2}<\\rho-2<\\frac{n}{2}-2$ for any $n\\geq 3$, then, from the discussion preceding the theorem, we know that there is a solution $f\\in H_{s+3,\\rho-2}$. Hence, we get the following:\n\\begin{align}\\label{AS2}\n\\begin{split}\n\\int_M b^2\\mu_g&=\\int_Mb\\Delta_gf\\mu_g,\\\\\n&=-\\int_M\\nabla f(b)\\mu_g + \\int_M\\mathrm{div}_g(b\\nabla f)\\mu_g.\n\\end{split}\n\\end{align} \nNow, to see that the last term actually does not produce any contribution, the procedure is standard: Consider sequences $\\{b_n\\}\\subset C^{\\infty}_0$ of compactly supported functions converging to $b$ in $H_{s+1,\\rho}$ and $\\{\\nabla f_n\\}\\subset C^{\\infty}_0$ converging to $\\nabla f$ in $H_{s+2,\\rho-1}$. Then, consider the difference\n\\begin{align}\\label{AS3}\n\\begin{split}\n\\Big\\vert\\int_M\\{\\mathrm{div}_g(b\\nabla f)-\\mathrm{div}_g(b_n\\nabla f_n)\\}\\mu_g\\Big\\vert\\leq &\\int_M|\\mathrm{div}_g(b(\\nabla f-\\nabla f_n)|\\mu_g \\\\\n&+ \\int_M|\\mathrm{div}_g((b-b_n)\\nabla f_n)|\\mu_g.\n\\end{split}\n\\end{align}\nWe also have that $\\mathrm{div}_g(b(\\nabla f-\\nabla f_n))=\\langle\\nabla b,\\nabla f-\\nabla f_n\\rangle + b\\:\\mathrm{div}_g(\\nabla f-\\nabla f_n)$, which gives point wise estimates of the form:\n\\begin{align*}\n|\\mathrm{div}_g(b(\\nabla f-\\nabla f_n))|\\lesssim |\\nabla f-\\nabla f_n|_g|\\nabla b|_g + |b||\\nabla(\\nabla f-\\nabla f_n)|_g.\n\\end{align*}\nNotice that, since $\\rho\\geq 0$, then $H_{s+3,\\rho-1}\\hookrightarrow L^2_{-1}$ and $H_{s,\\rho+1}\\hookrightarrow L^2_{1}$. Thus, after integration, we get the following estimate:\n\\begin{align*}\n\\int_M|\\mathrm{div}_g(b(\\nabla f-\\nabla f_n)|\\mu_g&\\lesssim ||\\nabla f-\\nabla f_n||_{L^2_{-1}}||\\nabla b||_{L^2_1} \\\\\n&+ ||b||_{L^2}|||\\nabla(\\nabla f - \\nabla f_n)||_{L^2}\\xrightarrow[n\\rightarrow\\infty]{} 0.\n\\end{align*}\nA similar argument can be applied to the second term in the right-hand side of (\\ref{AS3}), showing that\n\\begin{align*}\n\\int_M\\mathrm{div}_g(b\\nabla f)\\mu_g = \\lim_{n\\rightarrow\\infty} \\int_M\\mathrm{div}_g(b_n\\nabla f_n)\\mu_g=0.\n\\end{align*} \nThus, going back to (\\ref{AS2}), we get the following:\n\\begin{align}\\label{AS4}\n\\begin{split}\n\\int_M b^2\\mu_g&=-\\int_M\\nabla f(b)\\mu_g,\\\\\n&=-\\frac{n}{2}\\int_M\\langle \\overset{\\circ}{B},\\pounds_{\\nabla f}g \\rangle\\mu_g,\\\\\n&=-n\\int_M\\langle \\overset{\\circ}{B},\\nabla^2f \\rangle\\mu_g,\\\\\n&=-n\\int_M\\langle \\overset{\\circ}{B},\\nabla^2f - \\frac{1}{n}g\\Delta_gf \\rangle\\mu_g\n\\end{split}\n\\end{align}\nwhere in the second line we have used (\\ref{PS}). Clearly, since the left hand side is non-negative, then we have that\n\\begin{align}\\label{AS5}\n\\begin{split}\n0\\leq -n\\int_M\\langle \\overset{\\circ}{B},\\nabla^2f - \\frac{1}{n}g\\Delta_gf \\rangle\\mu_g &= n\\Big\\vert\\int_M\\langle \\overset{\\circ}{B},\\nabla^2f - \\frac{1}{n}g\\Delta_gf \\rangle\\mu_g\\Big\\vert,\\\\\n&\\leq n\\int_M |\\overset{\\circ}{B}|_g |\\nabla^2f - \\frac{1}{n}g\\Delta_gf|_g \\mu_g,\\\\\n&\\leq n||\\overset{\\circ}{B}||_{L^2}||\\nabla^2f - \\frac{1}{n}g\\Delta_gf||_{L^2}.\n\\end{split}\n\\end{align}\nNotice that due to the choice of our functional spaces the right-hand side is well-defined, since both $|\\overset{\\circ}{B}|_g,|\\nabla^2f|_g\\in L^2$. Also, since $H_{s+3,\\rho-2}\\subset C^{3}$, we get the usual Ricci identity\n\\begin{align*}\n{\\mathrm{Ric}_g}_{uj}\\nabla^uf=\\nabla_i\\nabla_j\\nabla^if - \\nabla_j(\\Delta_gf),\n\\end{align*}\nwhich implies \n\\begin{align}\\label{ricciid}\n{\\mathrm{Ric}_g}(\\nabla f,\\nabla f)=\\langle \\nabla f,\\mathrm{div}_g\\left(\\nabla^2 f\\right)\\rangle_g - \\langle \\nabla f,\\nabla(\\Delta_gf)\\rangle_g.\n\\end{align}\nSince $\\nabla f\\in H_{s+2,\\rho-1}$ and $\\mathrm{Ric}_g\\in H_{s+1,\\delta+2}$, with $\\delta> -\\frac{n}{2}$ and $\\rho\\geq 0$, then ${\\mathrm{Ric}_g}(\\nabla f,\\nabla f)\\in H_{s+1,\\sigma}$ for any $\\sigma< \\delta + \\rho + \\frac{n}{2}$, which implies ${\\mathrm{Ric}_g}(\\nabla f,\\nabla f)\\in L^2$. Furthermore, since $\\nabla^2f\\in H_{s+1,\\rho}$, then both $\\nabla^3f$ and $\\nabla(\\Delta_gf)$ lie in $H_{s,\\rho+1}\\hookrightarrow L^{2}_1$. Thus, since $\\nabla f\\in L^2_{-1}$, then, both terms in the right hand side of (\\ref{ricciid}) are actually in $L^1$, since \n\\begin{align*}\n\\Big\\vert \\int_M\\langle \\nabla f,\\mathrm{div}_g\\nabla^2 f\\rangle_g\\mu_g\\Big\\vert&\\leq \\int_M |\\nabla f| |\\mathrm{div}_g\\nabla^2 f| \\mu_g \\lesssim ||\\nabla f||_{L^2_{-1}}||\\nabla^3f||_{L^2_{1}}<\\infty,\\\\\n\\Big\\vert \\int_M\\langle \\nabla f,\\nabla(\\Delta_gf)\\rangle_g\\mu_g\\Big\\vert&\\lesssim ||\\nabla f||_{L^2_{-1}}||\\nabla (\\Delta_gf)||_{L^2_{1}}<\\infty.\n\\end{align*}\nThis implies that we can integrate (\\ref{ricciid}) over $M$. Then, we can integrate by parts, so as to get\n\\begin{align*}\n\\int_M\\langle \\nabla f,\\mathrm{div}_g\\nabla^2 f\\rangle_g\\mu_g= - \\int_M|\\nabla^2f|^2_g\\mu_g + \\int_M\\mathrm{div}_g\\left(i_{\\nabla f}\\nabla^2f \\right)\\mu_g.\n\\end{align*}\nNotice that from the above estimates, approximating $f$ in $H_{s+3,\\rho-2}$ by $\\{f_n \\}_{n=1}^{\\infty}\\subset C^{\\infty}_0$, we get that\n\\begin{align*}\n\\Big\\vert \\int_M\\left(\\langle \\nabla f,\\mathrm{div}_g\\nabla^2 f \\rangle_g - \\langle \\nabla f_n,\\mathrm{div}_g\\nabla^2 f_n\\rangle_g\\right)\\mu_g\\Big\\vert&\\lesssim ||\\nabla (f-f_n)||_{L^2_{-1}}||\\nabla^3f||_{L^2_{1}} \\\\\n&+ ||\\nabla f_n||_{L^2_{-1}}||\\nabla^3(f-f_n)||_{L^2_{1}}\\xrightarrow[n\\rightarrow\\infty]{}0,\n\\end{align*}\nwhich implies that\n\\begin{align*}\n\\int_M\\langle \\nabla f,\\mathrm{div}_g\\nabla^2 f \\rangle_g\\mu_g=-\\lim_{n\\rightarrow\\infty}\\int_M|\\nabla^2f_n|^2_g\\mu_g=-\\int_M|\\nabla^2f|^2_g\\mu_g,\n\\end{align*}\nwhere the last equality comes from the same kind of approximation argument, plus the fact that $\\nabla^2f\\in L^2$. Following the lines as described above, we also get that\n\\begin{align*}\n\\int_M\\langle \\nabla f,\\nabla(\\Delta_gf)\\rangle_g\\mu_g=-\\lim_{n\\rightarrow\\infty}\\int_{M}|\\Delta_gf_n|^2\\mu_g=-\\int_{M}|\\Delta_gf|^2\\mu_g.\n\\end{align*}\nThus, finally, we get that the following expression, familiar for compact manifolds, holds under our hypotheses\n\\begin{align}\n\\int_M{\\mathrm{Ric}_g}(\\nabla f,\\nabla f)\\mu_g= \\int_{M}|\\Delta_gf|^2\\mu_g - \\int_M|\\nabla^2f|^2_g\\mu_g.\n\\end{align}\nNoticing that $|\\nabla^2f-\\frac{1}{n}g\\Delta_gf|^2_g=|\\nabla^2f|^2_g - \\frac{1}{n} |\\Delta_gf|^2$, we get\n\\begin{align*}\n||\\nabla^2f-\\frac{1}{n}g\\Delta_gf||^2_{L^2}&=||\\nabla^2f||^2_{L^2} - \\frac{1}{n} ||\\Delta_gf||^2_{L^2}\\\\\n&=\\frac{n-1}{n} ||\\Delta_gf||^2_{L^2} - \\int_M{\\mathrm{Ric}_g}(\\nabla f,\\nabla f)\\mu_g.\n\\end{align*}\nThus, notice that, parallel to what happens in the compact scenario, if $\\mathrm{Ric}_g$ is non-negative, then we can get an estimate of the $L^2$-norm of $\\nabla^2f-\\frac{1}{n}g\\Delta_gf$ in terms of the $L^2$ norm of $\\Delta_gf$. Doing this, we can prove the second statement in the Lemma. That is, if $\\mathrm{Ric}_g$ is non-negative, then putting the above identity together with (\\ref{AS4})-(\\ref{AS5}), we get\n\\begin{align}\n||b||^2_{L^2}\\leq \\sqrt{n(n-1)}||\\overset{\\circ}B||_{L^2}||b||_{L^2}.\n\\end{align}\n\nNevertheless, we can actually get rid of this limitation on the Ricci tensor with the following argument. Notice that, from the point wise continuity of contractions, we get\n\\begin{align*}\n\\Big\\vert \\int_M\\mathrm{Ric}_g(\\nabla f,\\nabla f)\\mu_g \\Big\\vert \\lesssim \\int_{M}|\\mathrm{Ric}_g|_g|\\nabla f|^2_g\\mu_g. \n\\end{align*}\nAlso, since $\\mathrm{Ric}_g\\in H_{s+1,\\delta+2}$, with $s>\\frac{n}{2}$ and $\\delta>-\\frac{n}{2}$, then $H_{s+1,\\delta+2}\\hookrightarrow C^0_{2}$, where \n\\begin{align*}\n||\\mathrm{Ric}_g||_{C^0_2}=\\sup_{M}|\\mathrm{Ric}_g|_g(1+d^2_e),\n\\end{align*}\nthus, we get\n\\begin{align*}\n\\Big\\vert \\int_M\\mathrm{Ric}_g(\\nabla f,\\nabla f)\\mu_g \\Big\\vert &\\lesssim ||\\mathrm{Ric}_g||_{C^0_2}\\int_M|\\nabla f|^2_g(1+d^2)^{-1}\\mu_g\\\\\n&=||\\mathrm{Ric}_g||_{C^0_2}\\int_M|\\nabla f|^2_g(1+d^2)^{\\rho-1}(1+d^2)^{-\\rho}\\mu_g\\\\\n&\\leq ||\\mathrm{Ric}_g||_{C^0_2}||\\nabla f||^2_{L^2_{\\rho-1}}.\n\\end{align*}\nNotice, again, that since $f\\in H_{s+3,\\rho-2}$, with $\\rho\\geq 0$, then $\\nabla f\\in L^2_{\\rho-1}$. Thus, the above expression is well-defined. We then get the following estimate, where $C_n$ stands for some constant depending only on $n$,\n\\begin{align*}\n||\\nabla^2f-\\frac{1}{n}g\\Delta_gf||^2_{L^2}&\\leq \\frac{n-1}{n} ||\\Delta_gf||^2_{L^2} + C_{n}||\\mathrm{Ric}_g||_{C^0_2}||\\nabla f||^2_{L^2_{\\rho-1}},\\\\\n&\\leq \\frac{n-1}{n} ||\\Delta_gf||^2_{L^2} + C_n||\\mathrm{Ric}_g||_{C^0_2}||f||^2_{H_{2,\\rho-2}}\n\\end{align*}\nNow, notice that $\\rho-2>-\\frac{n}{2}$, and since $\\Delta_g:H_{2,\\rho-2}\\mapsto H_{0,\\rho}$ is injective for $\\rho-2>-\\frac{n}{2}$, we have the following elliptic estimate (see \\cite{CB1}):\n\\begin{align*}\n||f||_{H_{2,\\rho-2}}\\leq C ||\\Delta_gf||_{H_{0,\\rho}}\\leq C ||b||_{L^2},\n\\end{align*}\nfor some constant $C$, which does not depend on the particular $f\\in H_{2,\\rho-2}$, where the last inequality is a consequence of the embedding $L^2_{\\rho}\\subset L^2$ for $\\rho\\geq 0$. This implies that\n\\begin{align*}\n||\\nabla^2f-\\frac{1}{n}g\\Delta_gf||_{L^2}&\\leq \\left( \\frac{n-1}{n} + C||\\mathrm{Ric}_g||_{C^0_2}\\right)^{\\frac{1}{2}}||b||_{L^2},\n\\end{align*}\nFor some other constant $C$. Putting this together with (\\ref{AS4})-(\\ref{AS5}), we get\n\\begin{align*}\n||b||^2_{L^2}\\leq n\\left( \\frac{n-1}{n} + C||\\mathrm{Ric}_g||_{C^0_2}\\right)^{\\frac{1}{2}}||\\overset{\\circ}{B}||_{L^2}||b||_{L^2} , \n\\end{align*}\n\\end{proof}\n\n\\begin{coro}\nUnder the same hypotheses of the previous lemma, if $n\\geq 5$, then (\\ref{AS.01}) can be simplified by the following estimate\n\\begin{align}\n||b||^2_{L^2}\\leq nC_g ||\\overset{\\circ}B||_{L^2},\n\\end{align}\nwhere the constant $C_g$ is the best constant satisfying the elliptic estimate $||h||_{H_{2,-2}}\\leq C_g||\\Delta_gh||$ for all $h\\in H_{2,-2}$.\n\\end{coro}\n\\begin{proof}\nUnder these assumptions we can look at $b\\in H_{s+1,0}$, and consider the map $\\Delta_g:H_{s+3,-2}\\mapsto H_{s+1,0}$, which will give us a solution $f\\in H_{s+3,-2}$ to (\\ref{AS1}). Then, from (\\ref{AS4})-(\\ref{AS5}), we can actually deduce that\n\\begin{align*}\n||b||^2_{L^2}\\leq n ||\\overset{\\circ}B||_{L^2}||\\nabla^2f||_{L^2}.\n\\end{align*}\nThus, since $||\\nabla^2f||_{L^2}\\leq ||\\nabla^2f||_{H_{2,-2}}$, and, again, for $n\\geq 5$ we have that $\\Delta_g$ is injective on $H_{2,-2}$, we get the elliptic estimate  $||\\nabla^2f||_{H_{2,-2}}\\leq C_g||b||_{L^2}$. In this case, we would directly get\n\\begin{align}\n||b||^2_{L^2}\\leq nC_g ||\\overset{\\circ}B||_{L^2}||b||_{L^2},\n\\end{align}\nwhich is a simpler estimate than the one presented in the main proof.\n\\end{proof}\n\nFrom this general theorem, we get a version of the usual almost-Schur inequality for the scalar-curvature \\cite{Topping}-\\cite{Barbosa} in the context of AE manifolds. In our case, we are able to state such inequality without restriction on the Ricci tensor, but we do not provide an explicit value for the constant $C$. \n\n\\begin{coro}\\label{R-AS}\nLet $(M^n,g_0)$ is an $H_{s+3,\\delta}$-asymptotically euclidean manifold without boundary, with $n\\geq 3$, $\\delta>-\\frac{n}{2}$ and $\\delta\\geq-2$ and $s>\\frac{n}{2}$. Then, there is a constant $C$, such that for any $H_{s+3,\\delta}$-AE metric $g$ on $M$ the following inequality holds\n\\begin{align}\n||R_g||_{L^2}\\leq C ||\\overset{\\circ}{\\mathrm{Ric}_g}||_{L^2}\n\\end{align}\n\\end{coro}\n\\begin{proof}\nNotice that since $g$ is $H_{s+3,\\delta}$-AE, with $\\delta>-\\frac{n}{2}$ and $\\delta\\geq -2$, then we have $\\mathrm{Ric}_g,R_g\\in H_{s+1,\\rho}$, for $\\rho>-\\frac{n}{2}+2$ and $\\rho\\geq 0$, which also implies $G_g\\in H_{s+1,\\rho}$ for the same $\\rho$. Hence, since $\\mathrm{div}_gG_g=0$, we get as an immediate corollary of the above Lemma that\n\\begin{align*}\n||R_g||_{L^2}\\leq C ||\\overset{\\circ}{\\mathrm{G}_g}||_{L^2}.\n\\end{align*}\nFinally, since $\\overset{\\circ}{\\mathrm{G}_g}=\\overset{\\circ}{\\mathrm{Ric}_g}$ our claim holds.\n\\end{proof}\n\nClearly, the following also holds as a direct consequence of the above Lemma.\n\\begin{coro}\nLet $(M^n,g)$ is an $H_{s+3,\\delta}$-asymptotically euclidean manifold without boundary, with $n\\geq 3$, $\\delta> - \\frac{n}{2}$, $\\delta\\geq -2$ and $s>\\frac{n}{2}$. Suppose that there is an isometric immersion $(M^n,g)\\hookrightarrow (\\tilde{M}^n,\\tilde{g})$ into a Ricci-flat space, \\textit{i.e}, $\\mathrm{Ric}_{\\tilde{g}}=0$. Then, if the extrinsic curvature $K\\in H_{s+1,\\rho}$ for $\\rho>-\\frac{n}{2}+2$ and $\\rho\\geq 0$, then, there is a constant $C$, independent of the immersion, such that\n\\begin{align}\\label{AS-K}\n||\\tau||_{L^2}\\leq C ||\\overset{\\circ}{K}||_{L^2},\n\\end{align} \nwhere $\\tau\\doteq\\mathrm{tr}_gK$.\n\\end{coro}\n\\begin{proof}\nAs we have already commented, under our hypotheses, we have that the Codazzi equation for hypersurfaces guarantees that the tensor $\\pi=K - \\tau g\\in H_{s+1,0}$ and satisfies $\\mathrm{div}_g\\pi=0$. Furthermore, we have that $\\overset{\\circ}{\\pi}=\\overset{\\circ}{K}$. Thus (\\ref{AS}) proves (\\ref{AS-K}).\n\\end{proof}\n\nFinally, we will present an almost-Schur-type inequality for $Q$-curvature on AE-manifolds. Following \\cite{Topping}-\\cite{Barbosa}, it has been shown in \\cite{chinos} that such an inequality holds in the context of closed manifolds.  \nIt should be stress that problems related with $Q$-curvature have gained plenty of attention, see \\cite{Qcurv} for a survey. Before presenting this result, we should introduce some definitions. First of all, given a Riemannian manifold $(M^n,g)$, with $n\\geq 3$, we will define its $Q$-curvature in the following way\n\\begin{align}\\label{Qcurv}\nQ_g\\doteq A_n\\Delta_gR_g + B_n|\\mathrm{Ric}_g|^2_g + C_nR^2_g,\n\\end{align}\nwhere $A_n=-\\frac{1}{2(n-1)}$, $B_n=-\\frac{2}{(n-2)^2}$ and $C_n=\\frac{n^2(n-4)+16(n-1)}{8(n-1)^2(n-2)^2}$. \n\nNow, following \\cite{chinos}, we will explain how we can canonically associate to $Q$-curvature a divergence-free $(0,2)$-symmetric tensor field.\nNotice that we can view $Q$-curvature as a map on the set of Riemannian metrics on a given manifold $M$, given by\n\\begin{align*}\nQ:\\mathrm{Riem}^{k}(M)&\\mapsto C^{k-4}(M),\\\\\ng &\\mapsto Q_g,\n\\end{align*} \nwhere $\\mathrm{Riem}^{k}(M)$ denotes the set of $C^k$ Riemannian metrics on $M$ and we are considering $k\\geq 4$. Then, we can analyse its linearisation $L_g\\doteq D_gQ:S^2M\\mapsto C^{k-4}(M)$ which is given by an operator acting on the space of $C^k$-symmetric tensor fields on $M$. Let $L^{*}_g$ be its formal adjoint operator, which acts on functions on $M$ and produces a symmetric tensor-field on $M$. Now, consider any point $p\\in M$ and a bounded neighbourhood $U$ of $p$. Furthermore, let $V\\subset U$ be another neighbourhood of $p$. Let $f\\in C^{\\infty}_{0}(U)$ and $X$ be a compactly supported smooth vector field on $U$. Then, we have that\n\\begin{align*}\n\\int_U\\langle X,\\mathrm{div}_g(L^{*}_gf) \\rangle \\mu_g = -\\frac{1}{2}\\int_U\\langle \\pounds_Xg,L^{*}_gf \\rangle \\mu_g=-\\frac{1}{2}\\int_U f L_g(\\pounds_Xg) \\mu_g.\n\\end{align*}\nNow, let $\\psi_t$ be the flow associated to the vector field $X$, then, we have that\n\\begin{align*}\nX(Q_g)=\\frac{d}{dt}\\psi^{*}_t(Q_{g})|_{t=0}=\\frac{d}{dt}Q_{\\psi^{*}_tg}|_{t=0}=D_{\\psi^{*}_tg}Q\\cdot\\frac{d}{dt}\\psi^{*}_tg|_{t=0}=D_{g}Q\\cdot\\pounds_Xg=L_g(\\pounds_Xg),\n\\end{align*}\nthus, we have that\n\\begin{align*}\n\\int_U\\langle \\mathrm{div}_g(L^{*}_gf),X \\rangle \\mu_g = -\\frac{1}{2}\\int_U \\langle fdQ_g, X \\rangle \\mu_g, \n\\end{align*}\nwhich implies that $\\mathrm{div}_g(L^{*}_gf)+\\frac{1}{2}fdQ_g=0$ $\\forall$ $f\\in C^\\infty_0(U)$. In particular, if we consider $f|_{V}\\equiv 1$, then we get that, in a neighbourhood of $p$, it holds that\n\\begin{align*}\n\\mathrm{div}_g(L^{*}_g1)+\\frac{1}{2}dQ_g=\\mathrm{div}_g\\big( L^{*}_g1 + \\frac{1}{2}Q_gg \\big)  =0.\n\\end{align*}\nIn particular, this means that if we define the tensor field $J_g\\doteq -\\frac{1}{2}L^{*}_g1$ then,  $B_J\\doteq  J_g - \\frac{1}{4}Q_gg$ is a conserved $(0,2)$-symmetric tensor field. The explicit expression for such tensor field is given in terms of the $Q$-curvature, the Bach tensor and the Schoutten tensor, and, as a map on the metric, is a fourth order operator. Explicitly, it can be written as follows \\cite{chinos}.\n\\begin{align*}\n\\begin{split}\n&J_g=\\frac{1}{n}Q_gg-\\frac{1}{n-2}B_g-\\frac{n-4}{4(n-1)(n-2)}T_g,\\\\\n&{B_g}_{jk}\\doteq \\nabla^i{C_g}_{ijk} + {W_g}_{ijkl}S_g^{il},\\\\\n&{C_g}_{ijk}\\doteq \\nabla_i{S_g}_{jk} - \\nabla_j{S_g}_{ik},\\\\\n&S_g\\doteq \\frac{1}{n-2}(\\mathrm{Ric}_g - \\frac{1}{2(n-1)}R_gg),\\\\\n&W_g\\doteq \\mathrm{Riem}_g - \\frac{1}{n-2}\\overset{\\circ}{\\mathrm{Ric}}\\owedge g - \\frac{R_g}{2n(n-1)}g\\owedge g,\\\\\n&h\\owedge k(X,Y,Z,V)\\doteq h(X,Z)k(Y,W) + h(Y,V)k(X,Z) - h(X,V)k(Y,Z) - h(Y,Z)k(X,V),\\\\\n&T_g\\doteq (n-2)(\\nabla^2\\mathrm{tr}_gS_g - \\frac{g}{n}\\Delta_g\\mathrm{tr}_gS_g) + 4(n-1)(S\\times S - \\frac{1}{n}|S_g|^2_g g) - n\\mathrm{tr}_g\\mathrm{S}_g \\overset{\\circ}{S}_g,\\\\\n&(S\\times S)_{jk}\\doteq S^i_jS_{ik},\n\\end{split}\n\\end{align*}\nwhere in the above we have defined several useful tensor and operations, namely, the Bach tensor $B_g$; the Cotton tensor $C_g$; the Schouten tensor $S_g$; the Weyl curvature tensor $W_g$, which uses the $\\owedge$-operator, which for two $(0,2)$-tensor fields $h$ and $k$ produces a $(0,4)$-tensor field. Since the Bach tensor is known to be traceless, and from the above expressions we have that $\\mbox{tr}_gT_g=0$, then $\\mathrm{tr}_g J_g=Q_g$. Taking into consideration all this notation, we are now in a position to present the following result, whose proof runs in complete analogy to the previous ones.\n\n\\begin{coro}\nLet $(M^n,g_0)$ is an $H_{s+3,\\delta_0}$-asymptotically euclidean manifold without boundary, with $n\\geq 3$, $\\delta>-\\frac{n}{2}$, $\\delta_0\\geq-2$ and $s>\\frac{n}{2}$. Then, there is a constant $C$, such that for any $H_{s+5,\\delta}$-AE metric $g$, with $\\delta>-\\min\\{\\frac{n}{2}+2,4\\}$, the following inequality holds\n\\begin{align}\n||Q_g||_{L^2}\\leq C ||\\overset{\\circ}{J_g}||_{L^2}\n\\end{align}\n\\end{coro}\n\\begin{proof}\nWe basically only need to show that $J_g$ and $Q_g$ satisfy the hypotheses of  Lemma \\ref{B-AS}. Since $g$ is $H_{s+5,\\delta}$-AE with $\\delta>-\\frac{n}{2}$, then from the continuous multiplication property and the fact that  $\\mathrm{Riem}_g, \\mathrm{Ric}_g\\in H_{s+3,\\delta+2}$, we have that $W_g,S_g\\in H_{s+3,\\delta+2}$. Thus, since contractions define continuous operations on tensor fields, the multiplication property gives us that the contracted tensor ${W_g}_{\\cdot}S\\doteq {W_g}_{ijkl}S^{il}$ is in $H_{s+3,\\delta+4}$. Also, we get that $C_g\\in H_{s+2,\\delta+3}$, which, together with ${W_g}_{\\cdot}S\\in H_{s+3,\\delta+4}$, gives us $B_g\\in H_{s+1,\\delta+4}$. Similarly, we have that $\\Delta_gR_g\\in H_{s+1,\\delta+4}$ and $R^2_g,|\\mathrm{Ric}_g|^2_g\\in H_{s+3,\\delta+4}$, which gives us that $Q_g\\in H_{s+1,\\delta+4}$. Finally, the fact that $S\\in H_{s+3,\\delta+2}$ and the multiplication property give us that $T\\in H_{s+1,\\delta+4}$. All this implies that $Q_g,J_g\\in H_{s+1,\\delta+4}$, where $\\rho=\\delta+4\\geq 0$ and $\\rho>-\\frac{n}{2}+2$, since $\\delta>-\\min\\{\\frac{n}{2}+2,4\\}$. Thus, we are under the hypotheses of Lemma \\ref{B-AS}, and the result follows.\n\\end{proof}\nFinally, it is worth noticing that Lemma \\ref{B-AS} has been stated in a quite general form, that, as shown above, can be adapted to analyse different versions of almost-Schur type inequalities for different interesting geometric tensors related to conserved quantities. Obviously, we cannot exhaust all the examples here, but, surely, there are other rather obvious candidates for such examples, such us Lovelock curvatures, which have been in the center of plenty of resent research in geometric analysis.\n\n\\subsection*{Static potentials}\n\nRecall that a Riemannian metric $g$ on a manifold $M$ is called static if the linearised scalar curvature map has nontrivial cokernel. This is equivalent to stating that the equation\n\\begin{align}\n-g(\\Delta_gf)+\\nabla^2f-fRic_g=0\n\\end{align}\nadmits some nontrivial solution $f$. Such solution is referred to as a \\textit{static potential}. It has been shown in \\cite{Corvino} that any $C^3$-static metric must have constant scalar curvature, which implies that any $C^3$-asymptotically-flat static metrics must have zero scalar curvature. This means that a static potential of an asymptotically flat static metric must satisfy the system\\footnote{Notice that this means that no static potential can exist in $H_{s,\\delta}$ for $\\delta>-\\frac{n}{2}$, since $\\Delta_g$ is injective in this case.}\n\\begin{align}\\label{staticeq}\n\\begin{split}\n\\nabla^2f&=fRic_g,\\\\\n\\Delta_gf&=0.\n\\end{split}\n\\end{align}\n\nPositive static potentials are an interesting object of study, since they are intimately related with vacuum space-time solutions of the Einstein equations, as was shown in \\cite{Corvino}. In fact, there it is shown that a Riemannian manifold $(M^n,g)$ admits a static potential iff the warped product metric $\\tilde{g}=-f^2dt\\otimes dt + g$ is Einstein. Thus, in the case of $(M,g)$ being an AE manifold, since $R(g)=0$, then $\\tilde{g}$ must actually be Ricci-flat. Thus, we get a correspondence between \\textit{static} vacuum solutions of the space-time Einstein equations with static Riemannian manifolds. Furthermore, in \\cite{Corvino} it is also shown that the zero level set of a static potential (in case it exists), is given by a regular totally geodesic hypersurface in $M$. Notice that in case $f^{-1}(0)\\neq \\emptyset$, then, the space-time metric $\\tilde{g}$ will actually degenerate on such set, possibly signalling the existence of some \\textit{pathology} of the space-time structure. In fact, since the scalar curvature of a static Riemannian metric is zero, then it is trivially a solution of the \\textit{time-symmetric} vacuum Einstein constraint equations, which under evolution generate the static space-time. In this scenario, the hypersurface $f^{-1}(0)$ defines an \\textit{apparent horizon} in the initial data. Such structures are related with the formation of black holes under the space-time evolution. For some details concerning these ideas, see, for instance, \\cite{maxwell}-\\cite{Corvino}-\\cite{chrusciel-mazzeo} and references therein.  \n\nTaking into account the above ideas, it is not surprissing that static potentials in the context of AE manifolds have atractted quite a lot of attention, and some strong rigidity has been observed for instance in \\cite{Miao} for the $3$-dimensional case. In this section, we intend to show how some higher-dimensional analogues of their results follow naturally from the Pohozaev-Schoen identity (\\ref{PS}), plus some mild hypotheses on $f$. Before going to the main statements, we make explicit the fact that throughout this section, when we consider static AE manifolds with boundary, we will consider that $\\partial M=f^{-1}(0)$, this being motivated by the above discussion. That this, we have a model in mind where the boundary components would signal the existence of apparent horizons in time-symmetric vacuum initial data sets for the Einstein equations, which, under evolution, should evolve into black hole static space-times. The following lemma, which can be derived using some results that can be found \\cite{Miao}, is presented within our current notations and conventions. We include the proof for sake of completeness.\n\n\\begin{lemma}\nSuppose that $f$ is a static potential of an $H_{s+3,\\delta}$-asymptotically euclidean metric $g$, with $s>\\frac{n}{2}$, $\\delta>-1$, and $n\\geq 3$. Then $f\\in C^2$ has bounded gradient $\\nabla f$ and $\\nabla^2f\\in L^2$.\n\\end{lemma}\n\\begin{proof}\nFirst of all, concerning the regularity of the solution, since $H_{s+3,\\delta}$ embeds in $C^3$, appealing to Proposition 2.5 in \\cite{Corvino}, we get that the static potential is $C^2$. Now, consider a point $x\\in M$ which lies in one of the ends $E_i$. Let $B_{r_0}(p)$ be the $g$-geodesic ball of radius $r_0$ around the origin $p$, with $r_0$ chosen large enough such that its boundary $\\partial B_{r_0}$ lies in the ends of $M$, and, furthermore, suppose that $x$ has been chosen so that it lies outside this geodesic ball. Let $\\gamma(t)$ be a $g$-minimizing geodesic joining $p$ and $x$, parametrized by arc-length, so that $\\gamma(r_0)\\in\\partial B_{r_0}$, and suppose that $\\gamma(T)=x$. Finally, consider $f(t)\\doteq f(\\gamma(t))$, $r_0\\leq t\\leq T$. Then, since $f$ is a static potential of $g$, we have\n\\begin{align}\\label{miao1}\nf''(t)=\\mathrm{Ric}_g(\\gamma',\\gamma')f(t) \\;\\;\\ \\forall \\;\\; r_0\\leq t\\leq T\n\\end{align}\nSince $g-e\\in H_{s+2,\\delta}$, then $\\mathrm{Ric}_g\\in H_{s,\\delta+2}\\subset L^2_{\\delta+2}\\cap C^0$, which implies that $|\\mathrm{Ric}_g|_e=o(d_e(x)^{-(\\delta+2+\\frac{n}{2})})$ near infinity, in each end. The asymptotic condition on $g$ actually gives us that $|\\mathrm{Ric}_g|_e=o(d_g(x)^{-(\\delta+2+\\frac{n}{2})})$. This implies that, given an arbitrary $\\epsilon>0$, if $t$ is sufficiently large, then \n\\begin{align}\\label{miao2}\n|\\mathrm{Ric}_g(\\gamma'_t,\\gamma'_t)|(\\gamma_t)\\leq \\epsilon d_g(\\gamma_t)^{-(\\delta+\\frac{n}{2}+2)}\\leq \\epsilon t^{-2}.\n\\end{align}\n\nNow, define $\\alpha\\doteq \\frac{1}{2}(1+\\sqrt{1+4\\epsilon})$; $a\\doteq \\sup_{\\partial B_{r_0}}(|f|+|Df|_e)$ and $\\omega(t)\\doteq At^{\\alpha}$, with $r_0\\leq t\\leq T$, where $A$ is a constant chosen such that $Ar_0^{\\alpha}>a$ and $\\alpha Ar_0^{\\alpha-1}>a$ and $\\epsilon$ is some positive constant. Then, the function $\\omega(t)$ satisfies the following properties\n\\begin{align}\\label{miao3}\n\\omega''(t)=\\epsilon t^{-2}\\omega(t), \\;\\; |f(r_0)|\\leq \\omega(r_0), \\;\\; |f'(r_0)|\\leq \\omega'(r_0).\n\\end{align}\n\nSuppose that $|f(t)|>\\omega(t)$ for some $r_0\\leq t\\leq T$. Then, define $t_1\\doteq \\inf \\{t\\in [r_0,T] \\;\\; \\backslash \\;\\;  |f(t)|>\\omega(t)  \\}$. We know that $t_1$ satisfies that $t_1>r_0$ and $f(t_1)=\\omega(t_1)$. Using (\\ref{miao1})-(\\ref{miao2}), we have that \n\\begin{align*}\n|f''(t)|\\leq \\epsilon t^{-2}\\omega(t)=\\omega''(t) \\;\\; \\forall \\;\\; r_0\\leq t\\leq t_1.\n\\end{align*}\nNow, integrating he above inequality twice and using (\\ref{miao3}), gives $|f(t)|\\leq \\omega$ for all $r_0\\leq t\\leq t_1$, which is a contradiction, thus we get that $|f(t)|\\leq \\omega(t)$ for all $t\\in [r_0,T]$. Thus, since $\\delta>-\\frac{n}{2}$, we can choose an $\\epsilon>0$ such that $\\alpha<1+\\frac{1}{2}(\\delta+\\frac{n}{2})$, which implies that\n\\begin{align*}\n|f(t)|\\leq At^{1+\\frac{1}{2}(\\delta+\\frac{n}{2})}.\n\\end{align*}\nThe above inequality together with (\\ref{miao1})-(\\ref{miao2}), gives that\n\\begin{align*}\n|f''(t)|\\leq A\\epsilon t^{-(\\delta+2+\\frac{n}{2})}t^{1+\\frac{1}{2}(\\delta+\\frac{n}{2})}= A\\epsilon t^{-(1+\\frac{1}{2}(\\delta+\\frac{n}{2}))} \\;\\; \\forall \\;\\; r_0\\leq t \\leq T.\n\\end{align*}\nThis implies that there is a constant $C_1$, independent of the point $x\\in E_i\\backslash B_{r_0}$ and of $t$, such that \n\\begin{align}\\label{miao4}\n|f''(t)|\\leq C_1 t^{-(1+\\frac{1}{2}(\\delta+\\frac{n}{2}))} \\;\\; \\forall \\;\\; r_0\\leq t \\leq T.\n\\end{align}\nIntegrating this inequality between $r_0$ and $t$, we can show that $|f'(t)|\\leq C_2$ for some other constant $C_2$, which shows that for sufficiently large $|x|$ it holds that\n\\begin{align}\n|f|(x)\\leq C_3|x|.\n\\end{align}\n\nNow, using the equation $\\nabla^2f=f\\mathrm{Ric}_g$, we have that $|\\nabla^2f|_e(x)\\lesssim |x||x|^{-(\\delta+2+\\frac{n}{2})}=|x|^{-(1+\\delta+\\frac{n}{2})}\\in L^2(\\mathbb{R}^n\\backslash B_1(0))$ for $\\delta>-1$, which proves that $\\nabla^2f\\in L^2$. In order to prove that $\\nabla f$ is bounded, define $\\phi\\doteq|\\nabla f|^2_g$, and notice that \n\\begin{align*}\n|\\nabla\\phi|^2_g(x)\\leq 4|\\nabla^2f|^2_g\\phi \n\\end{align*}\nDoing as above, consider $\\phi(t)\\doteq \\phi(\\gamma(t))$, and notice that since $g$ is AE, then $g$ and $e$ define \\textit{equivalent} metrics, that is, for any tangent vector $v_p$ to $M$, there are positive constants $a$ and $b$, independent of the point $p\\in M$, such that $a|v_p|_e\\leq |v_p|_g \\leq b|v|_e$. Thus, the above inequality tells us that $|\\nabla\\phi|^2_e(x)\\lesssim |\\nabla^2f|^2_e\\phi$. Then, we have that\n\\begin{align*}\n|\\phi'(t)|=|g(\\nabla\\phi_{\\gamma(t)},\\gamma'_t)|\\leq |\\nabla\\phi|_g\\leq 2|\\nabla^2f|_g\\phi^{\\frac{1}{2}} \\lesssim |\\nabla^2f|_e\\phi^{\\frac{1}{2}} \\lesssim t^{-(1+\\delta+\\frac{n}{2})}\\phi^{\\frac{1}{2}}(t),\n\\end{align*}\nwhich, after integration, gives us that\n\\begin{align*}\n\\frac{1}{\\delta+\\frac{n}{2}}\\left( t^{-(\\delta+\\frac{n}{2})} - r^{-(\\delta+\\frac{n}{2})}_0\\right)\\lesssim \\phi^{\\frac{1}{2}}(t) -\\phi^{\\frac{1}{2}}(r_0) \\lesssim -\\frac{1}{\\delta+\\frac{n}{2}}\\left( t^{-(\\delta+\\frac{n}{2})} - r^{-(\\delta+\\frac{n}{2})}_0\\right).\n\\end{align*}\nSince $t> r_0$ and $\\delta+\\frac{n}{2}>0$, the above inequality implies that\n\\begin{align*}\n\\phi^{\\frac{1}{2}}(r_0)-\\frac{1}{\\delta+\\frac{n}{2}} r^{-(\\delta+\\frac{n}{2})}_0\\lesssim \\phi^{\\frac{1}{2}}(t) \\lesssim \\phi^{\\frac{1}{2}}(r_0)+ \\frac{1}{\\delta+\\frac{n}{2}} r^{-(\\delta+\\frac{n}{2})}_0 \\;\\;\\; \\forall \\;\\;\\; t> r_0.\n\\end{align*}\nThe above relation implies that $|\\nabla f|_g(x)=\\phi^{\\frac{1}{2}}(x)$ is bounded.\n\n\\end{proof}\n\n\n\\begin{thm}\nSuppose that $(M^n,g)$ is a $H_{s+3,\\delta}$-AE manifold with $n\\geq 3$, $s>\\frac{n}{2}$ and $\\delta>-1$, which admits a non-negative static potential $f$. Furthermore, suppose that $\\mathrm{Ric}_g\\in H_{s+1,\\rho}$, for some $\\rho>\\frac{n}{2}-1$, and that $\\partial M=f^{-1}(0)$ consists of $N$ closed connected components, labelled by $\\{\\Sigma_i \\}_{i=1}^{N}$. Then, it follows that \n\\begin{align}\n\\int_Mf|\\mathrm{Ric}_g|^2_g\\mu_g = \\frac{1}{2}\\sum_{i=i}^Nc_i\\int_{\\Sigma_i}R_{h_i}\\mu_{\\partial M},\n\\end{align}\nwhere the constants $c_i=|\\nabla f|_{\\Sigma_i}$. In particular, if $\\partial M=\\emptyset$, then $(M^n,g)$ is isometric to $(\\mathbb{R}^n,e)$, where $e$ is the euclidean metric.\n\\end{thm}\n\\begin{proof}\nFirst, notice that under our hypotheses the above lemma shows that $X=\\nabla f$ satisfies the general hypotheses of Theorem \\ref{thm1} if we pick $\\rho>\\frac{n}{2}-1$. Also, since a static AE metric has zero scalar curvature, we get that $\\mathrm{Ric}_g\\in H_{s+1,\\rho}$ is a conserved $(0,2)$-tensor field. Thus, our choice of $\\rho>\\frac{n}{2}-1$ shows that we are under the hypotheses of Theorem \\ref{thm1}. Then, we can apply (\\ref{PS}) choosing the vector field $X=\\nabla f$ so as to get\n\\begin{align}\\label{SP1}\n\\int_M\\langle \\mathrm{Ric}_g,\\nabla^2f \\rangle\\mu_g=\\int_{\\partial M}\\mathrm{Ric}(\\nabla f,\\nu)\\mu_{\\partial M}.\n\\end{align}\nAlso, from the static equation, we have that $\\langle \\mathrm{Ric}_g,\\nabla^2f \\rangle=f|\\mathrm{Ric}_g|^2_g$. Furthermore, since $\\partial M=f^{-1}(0)$, then, we get that $\\nu=-\\frac{\\nabla f}{|\\nabla f|}$. In addition, the static equation implies that on $\\partial M$ it holds that $\\nabla^2f(X,\\nabla f)=0$ for any $X$ tangent to $\\partial M$. Then, since $\\nabla^2f(X,\\nabla f)=\\frac{1}{2}X(|\\nabla f|^2_g)$, we get that $|\\nabla f|$ is constant along each connected component of $\\partial M$. Thus, we see that\n\\begin{align*}\n\\int_Mf|\\mathrm{Ric}_g|^2_g\\mu_g = -\\sum_{i=i}^Nc_i\\int_{\\Sigma_i}\\mathrm{Ric}(\\nu,\\nu)\\mu_{\\partial M},\n\\end{align*}\nwhere the sum is carried along the $N$ connected components $\\{\\Sigma_i\\}_{i=1}^N$ of $\\partial M$, and the constants $c_i=|\\nabla f|_{\\Sigma_i}$. Now, denote by $h_i$ the induced Riemannian metric on $\\Sigma_i$. Now, let $\\{E_i,\\nu\\}_{i=1}^{n-1}$ denote an orthonormal frame of $M$ along $\\Sigma_i$, where $\\{ E_i \\}_{i=1}^{n-1}$ gives an orthonormal frame on $\\Sigma_i$. Then, we get that for any pair of tangent vectors $X,Y$ to $\\Sigma_i$ it holds that\n\\begin{align*}\n\\mathrm{Ric}_g(X,Y)&=\\sum_{i=1}^{n-1}g(R_g(X,E_i)E_i,Y) + g(R_g(X,\\nu)\\nu,Y),\n\\end{align*}\nwhich, from the Gauss equation and the fact that all the components are totally geodesic, implies that\n\\begin{align*}\nR_g&=\\sum_{j=1}^{n-1}\\mathrm{Ric}_g(E_j,E_j) + \\mathrm{Ric}_g(\\nu,\\nu)=\\sum_{j=1}^{n-1}\\sum_{i=1}^{n-1}g(R_g(E_j,E_i)E_i,E_j)  + \\sum_{j=1}^{n-1}g(R_g(E_j,\\nu)\\nu,E_j) \\\\\n&+ \\mathrm{Ric}_g(\\nu,\\nu),\\\\\n&=\\sum_{j=1}^{n-1}\\sum_{i=1}^{n-1}h(R_{h_i}(E_j,E_i)E_i,E_j) + \\sum_{j=1}^{n-1}g(R_g(\\nu,E_j)E_j,\\nu) + \\mathrm{Ric}_g(\\nu,\\nu),\\\\\n&=R_{h_i} + 2\\mathrm{Ric}_g(\\nu,\\nu).\n\\end{align*}\nAgain, since $R_g=0$, we get $-\\mathrm{Ric}_g(\\nu,\\nu)|_{\\Sigma_i}=\\frac{1}{2}R_{h_i}$. Thus, going back, this gives us that\n\\begin{align}\\label{static1}\n\\int_Mf|\\mathrm{Ric}_g|^2_g\\mu_g = \\frac{1}{2}\\sum_{i=i}^Nc_i\\int_{\\Sigma_i}R_{h_i}\\mu_{\\partial M}.\n\\end{align}\n\nFrom the above equation it is clear that if $\\partial M=\\emptyset$, then $\\mathrm{Ric}_g=0$ on $M$. In such case, we have already shown this implies that $(M,g)$ has to be isometric to $(\\mathbb{R}^n,e)$.\n \n\\end{proof}\n\n\\begin{remark}\nNotice that the two conditions $g-e\\in H_{s+3,\\delta}$, with $\\delta>-1$, and $\\mathrm{Ric}_g\\in H_{s+1,\\rho}$, with $\\rho>\\frac{n}{2}-1$, in dimensions 3 and 4 are actually redundant, since, from $g-e\\in H_{s+3,\\delta}$, we get that $\\mathrm{Ric}_g\\in H_{s+1,\\delta+2}$, with $\\delta+2>1$. Thus, in dimension $n=3$, we get $\\frac{n}{2}-1=\\frac{1}{2}<\\delta+2$. Similarly, for $n=4$, we get $\\frac{n}{2}-1=1<\\delta+2$. Nevertheless, for $n\\geq 5$ the condition on the Ricci tensor becomes a necessary additional information. \n\nFrom the above discussion, we see that the identity (\\ref{static1}) is presented for $n=3$ in \\cite{Miao}, and that the same identity holds with the same hypotheses for $n=4$. Nevertheless, for $n\\geq 5$ the identity extends naturally, although non-trivially. \n\\end{remark}\n\n\n\\bigskip\nNow, consider a connected static manifold $(M^n,g,f)$, with $n\\geq 3$, where $g$ and $\\mathrm{Ric}_g$ satisfy the hypotheses of the above Lemma, and suppose that $\\partial M=\\emptyset$. Notice that if $f$ changes sign, then $f^{-1}(0)\\neq \\emptyset$ actually represents the boundary of both $f^{-1}(0,\\infty)$ and $f^{-1}(-\\infty,0)$, which are two $n$-dimensional submanifolds of $M$. We can then add $f^{-1}(0)$ to any of these submanifolds, generating two manifolds with boundary, say $M_{+}$ and $M_{-}$ respectively. Clearly, $(M_{+},g,f)$ is a static manifold with boundary which satisfies all the hypotheses of the above theorem provided suitable decaying conditions for $g$. Furthermore, $(M_{-},g,-f)$ is also a static manifold since the static equation is linear. Thus, $(M_{-},g,-f)$ also satisfies all the hypotheses of the above theorem. Thus, we get the following corollary.\n\n\\begin{coro}\nLet $(M^n,g,f)$ be a connected static manifold, with $n\\geq 3$, where $g$ and $\\mathrm{Ric}_g$ satisfy the hypotheses of the above theorem. Moreover, suppose that $f^{-1}(0)\\neq \\emptyset$ and $\\partial M=\\emptyset$. Then, we have that\n\\begin{align}\\label{balckholetopology.1}\n\\sum_{i=i}^Nc_i\\int_{\\Gamma_i}R_{h_i}\\mu_{\\partial M} \\geq 0,\n\\end{align}\nwhere $\\{\\Gamma_i\\}_{i=1}^N$ denote the connected components of $f^{-1}(0)$. Furthermore, the equality holds if and only if $(M,g)$ is isometric to $(\\mathbb{R}^n,e)$.\n\\end{coro}\n\\begin{proof}\nConsider the notation introduced above. Then, under our hypotheses, we have that both $(M_{+},g,f)$ and $(M_{-},g,-f)$ are static manifolds with boundary, for which the above theorem holds. Then, we get that\n\\begin{align*}\n\\sum_{i=i}^Nc_i\\int_{\\Gamma_i}R_{h_i}\\mu_{\\partial M}=2\\int_{M_{+}}f|\\mathrm{Ric}_g|^2_g\\mu_g\\geq 0,\\\\\n\\sum_{i=i}^Nc_i\\int_{\\Gamma_i}R_{h_i}\\mu_{\\partial M}=-2\\int_{M_{-}}f|\\mathrm{Ric}_g|^2_g\\mu_g \\geq 0,\n\\end{align*}\nwhere, in both cases, the equality holds iff $\\mathrm{Ric}_g=0$ on both $M_{+}$ and $M_{-}$. That is, the equality holds iff $\\mathrm{Ric}_g=0$ on $M$. But then, from Theorem \\ref{Ricciflatrigidity}, we get the rigidity statement in the equality case.\n\\end{proof}\n\nFrom the above corollary, we get the following.\n\n\\begin{coro}\nUnder the same hypotheses of the above corollary, if $n=3$ and $f^{-1}(0)$ consists of one connected component, that is $N=1$, then $f^{-1}(0)$ is homeomorphic to $\\mathbb{S}^2$. If $N\\geq 2$, then there must be at least one topological sphere within the components of $f^{-1}(0)$.\n\\end{coro}\n\n\\begin{proof}\nThis is straightforward, since, from (\\ref{balckholetopology.1}), we get that if $N=1$ the total scalar curvature of $\\partial M$ must be positive. Thus, from the Gauss-Bonnet theorem, we know that the Euler characteristic of $\\partial M$ must be positive, thus $\\partial M$ is a topological $2$-sphere.\n\nOn the other hand, in the case $N\\geq 2$, since the sum of the total scalar curvatures of the different components has to be positive, then we must have at least one sphere from the argument given above.\n\\end{proof}\n\n\\begin{remark}\nIt is important to stress that the above corollary is very closely related to well-known results related with the classification of allowable black hole topologies. Recall that the zero level set of a static potential models the intersection of the event horizon of a static black hole with a $t$-constant hypersurface in space-time. Thus, the characterization given above shows that such slices of the event horizon of an isolated static black hole have to be a topological 2-spheres. Once this connection between the two problems is clear, it also becomes clear that the above corollary is not a new result, but something that can be extracted from more general statements, such as Hawking's black hole uniqueness theorem \\cite{Hawking}, or even more generally, a recent result by Eichmair, Galloway and Pollack \\cite{EGP}. In fact, these results are much stronger than the above corollary, since the show that every component of $\\partial M$ should be a topological $2$-sphere and they are not restricted to static space-times. Nevertheless, we consider that the value of the above corollary relies on its simplicity, since this result is a straightforward application of the generalized Pohozaev-Schoen identity, which, in turn, relies only on a conservation identity derived from a symmetry principle. In contrast, the sharper characterizations presented in \\cite{Hawking} and \\cite{EGP} rely on more delicate techniques. In particular, for instance, the result presented in \\cite{EGP} relies on the positive resolution of the geometrization conjecture. \t\n\\end{remark}\n\n\\section*{Acknowledgements}\n\nThe first author would like to thank professor Paul Laurain for reading a previous version of this paper and making several relevant comments and suggestions.\n \nThe first author would also like to thank CAPES\/PNPD for financial support.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe axion\\cite{Peccei,Wilczek,Kim,DFS} is the Nambu-Goldstone boson\nassociated with the Peccei-Quinn symmetry breaking which was invented\nas the most natural solution to the strong CP problem\\cite{tHoot} of QCD.\nThe Peccei-Quinn symmetry breaking scale $F_a$ is stringently\nconstrained by the consideration of accelerator experiments, stellar\ncooling and cosmology. The allowed range of $F_a\/N$ ( axion window ) is\nbetween $10^{10}\\mbox{GeV}$ and $10^{12}\\mbox{GeV}$, where the integer $N$\nis the color anomaly of Peccei-Quinn symmetry. \n\nSince the Peccei-Quinn symmetry is a global U(1) symmetry,  global\nstrings (axionic strings) are produced during spontaneous symmetry\nbreaking. At this\nstage the potential for the axion field is flat, i.e., the axion is\nmassless.  However the axion has a mass at QCD scale\nthrough the instanton effect. \nThe potential is written by\n\\begin{equation}\n   V(A) = f_{\\pi}^2m_{\\pi}^2\\left( 1 - \\cos\\frac{NA}{F_a}\\right),\n   \\label{potential}\n\\end{equation}\nwhere $f_{\\pi}$ is the pion decay constant and $m_{\\pi}$ is its mass.\nThen the mass of the axion is given by $m_a \\simeq\nf_{\\pi}m_{\\pi}\/{F_a\/N}$. This potential(\\ref{potential}) has $Z_n$\nsymmetry and takes its minimum at $A = 0, (f_a\/N)\\pi, 2(f_a\/N)\\pi,\n\\ldots 2f_a\\pi$. $Z_n$ discrete symmetry is spontaneously broken to\nproduce two dimensional topological defects, i.e., axionic domain walls.\nThe property of the axionic domain wall is characterized by $N$.\nSince $N$ domain walls stretch out from each axionic string, the network\nof the string-wall systems is very complicated and survives long enough\nto dominate the universe for $N > 1$ axionic domain wall \\cite{RPS}.\nTherefore the $N>1$ domain wall is not accepted cosmologically.\n\nOne might expect that the density of the domain wall can be diluted in\nthe inflationary universe. However, for the dilution mechanism to\nwork, both the reheating temperature and the expansion rate during\ninflation should be smaller than $F_a$ \\cite{Linde}, which is rather\nunnatural requirement for many inflation models. Therefore the domain\nwall problem in the axion model is serious one.\n\nIn the case of $N=1$, the domain wall is a disk bounded by the axionic\nstring. The wall with the string boundary is no longer stable and it\nmight collapse by the surface tension.  In fact, since, as seen later,\nthe surface tension is stronger than the tension due to the string for\nthe domain wall with size much greater than the width of the wall, the\ndynamics of the wall is controlled by the wall tension. Therefore the\nstring that bounds the wall cannot prevent the wall from collapsing.\nTypically the wall has size of horizon length ($\\sim t_{QCD}$) when it\nis formed.  Therefore the time scale for collapse is about $t_{QCD}\n\\sim 10^{-4}$ sec and the walls disappear quickly without overclosing\nthe universe and hence the domain wall seems harmless in the case of\n$N=1$ axion model.  However it is expected that a number of axions are\nproduced when the axionic walls collapse. If produced axions are\nnon-relativistic or cold, their energy density decreases slowly\n($\\propto a^{-3}$ $a$: scale factor) compared with the radiation\ndensity ($ \\propto a^{-4}$). Hence the relative contribution of axions\nto the total density of the universe increases with $a$ until the\nuniverse becomes matter dominated.\n\nIn this paper we study the collapse of the axionic domain wall ($N=1$)\nby the numerical integration of equations of motion for the axion\nfield and estimate the energy density of axion field after the\ncollapse.  It is found that the energy carried by the walls is\nconverted to the axions, which gives significant contribution to the\nmass density of the present universe and might account for the dark\nmatter of the universe.\n\nThe axion emission is also expected from axionic string before\n$t_{QCD}$\\cite{Davis} or annihilation of axionic domain\nwalls\\cite{Lyth}.  However, since it is shown later that parallel two\ndomain walls go through each other without annihilation, it is\nunlikely that a large number of axions are produced by interaction of\ntwo domain walls.  The massless axion can be produced by axionic\nstrings and they acquire mass after the QCD phase transition, which\nmight give significant contribution to the total density of the\npresent universe. However there are two independent quantitative\nestimations by Davis\\cite{Davis} and by Harari and\nSikivie\\cite{Hagmann} and, unfortunately, they are quite different.\nThe contribution from axionic domain walls is expected to be at least\ncomparable to that from axionic strings if the estimation by Harari\nand Sikivie is correct.\n\n\\section{Dynamics of Domain Wall}\n\nThe dynamics of the Peccei-Quinn scalar field $\\phi$ is described by\nthe lagrangian:\n\\begin{eqnarray}\n    {\\cal L} & = & \\partial^{\\mu} \\phi^{*} \\partial_{\\mu} \\phi \n    + \\frac{\\lambda}{4} [|\\phi|^2 - F_a^2]^2\n    \\nonumber\\\\\n    & & = \\partial^{\\mu} \\phi^{*} \\partial_{\\mu} \\phi + V_s(|\\phi|)\n   \\label{eq:lagra}\n\\end{eqnarray}\nwhere $\\lambda$ is the coupling constant. As the universe cools down,\nPeccei-Quinn U(1) symmetry ($\\phi \\rightarrow e^{i\\theta}\\phi$) is\nspontaneously broken and the scalar field has vacuum expectation value\n$\\langle | \\phi | \\rangle = F_a$.  After global U(1) symmetry is\nbroken, axionic strings are formed. From causality argument, about one\naxionic string is produced within the horizon at $T\\simeq F_a$. Since\nthe line energy density of the string is $4\\pi F_a^2$, the string\ndensity $\\rho_{st}$ is about $4\\pi F_a^2 t\/t^3 \\sim 4\\pi\nF_a^4(F_a\/m_{pl})^2$ which is only $\\sim 10^{-13} - 10^{-17}$ of total\ndensity of the universe at the formation epoch.  Defining\n$\\phi = |\\phi|\\exp(iA\/F_a)$, the lagrangian for axion field\n$A$ is derived from eq.(\\ref{eq:lagra}) as\n\\begin{eqnarray}\n    {\\cal L}_a & = & \\partial^{\\mu} A \\partial_{\\mu} A \n    + f_{\\pi}^2m_{\\pi}^2\\left( 1 - \\cos\\frac{A}{F_a}\\right) ,\n    \\nonumber\\\\\n    & & = \\partial^{\\mu} A \\partial_{\\mu} A + V_w(A)\n   \\label{eq:alagra}\n\\end{eqnarray}\nwhere the second term comes from QCD instanton effect which gives\naxion mass $m_a \\simeq f_{\\pi}m_{\\pi}\/{F_a}$.  The potential $V_w(A)$\nhas a minimum at $A = 0, 2\\pi F_a$ and domain walls are produced\nbetween $A=0$ phase and $A = 2\\pi F_a$ phase. More precisely, the\naxion mass has temperature dependence and increase as \n\\begin{equation}\n    m_a(T) \\simeq 0.1m_a(T=0) (\\Lambda_{QCD}\/T)^{3.7},\n\\end{equation}\nwhere $\\Lambda_{QCD}$ is the QCD scale $\\sim 200\\mbox{MeV}$. The domain\nwalls are produced when the axion mass becomes greater than the\nexpansion rate of the universe, i.e., $m_a(T_1) \\simeq\n\\dot{a}(T_1)\/a$.  The axionic domain wall has size of about horizon\nlength at the formation time $t_1(T_1)$ and has the axionic string on\nits boundary. The surface tension of the axionic domain wall is\n$\\sigma \\simeq 16 m_a F_a^2$\\cite{Vilenkin}.  For wall with size $R$,\nthe ratio of surface energy to string energy is given by\n\\begin{equation}\n    \\frac{\\mu R}{\\sigma R^2} = \\frac{4\\pi F_a^2}{16m_a F_a^2 R} \\simeq\n    \\frac{1}{m_a R}.\n    \\label{walltostring}\n\\end{equation}\nTherefore the dynamics of the axionic wall with size greater than\n$R^{*} \\simeq 1\/(m_a) \\gg t_1$ is determined by the wall tension and\nthe string tension can be neglected.\n\nThe cosmological evolution of the wall is determined by the surface\ntension and interaction of walls; the former makes the wall shrink and\nthe latter cuts the wall into small pieces. In both cases the wall\nfinally collapses by the surface tension.  After collapse the domain\nwalls disappear and the energy that the wall had is converted into\nkinetic energy of axion fields.  If the energy of axion fields changes\nlike non-relativistic particles as the universe expands, the axion\nenergy density is \n\\begin{equation}\n    \\rho_a(t) = \\rho_{wall}(t_1)\\left(\\frac{a(t_1)}{a(t)}\\right)^3,\n    \\label{axiondensity}\n\\end{equation}\nwhere $\\rho_a(t)$ and $\\rho_{wall}$ are the energy density of the\naxion field and the axionic wall, respectively.  For relativistic\naxion, the axion density decreases as $\\sim a^{-4}$ until axion\nbecomes non-relativistic. Then eq.(\\ref{axiondensity}) is changed to\n\\begin{equation}\n    \\rho_a(t) = \\rho_{wall}(t_1)\\left(\\frac{a(t_1)}{a(t)}\\right)^3 \n    \\left(\\frac{\\langle E_a \\rangle}{m_a}\\right)^{-1},\n    \\label{axiondensity2}\n\\end{equation}\nwhere $\\langle E_a \\rangle$ is the average energy of emitted axion at\n$t_1$.  Assuming the mean distance and the mean radius of\nwalls are $\\alpha t_1$ and $\\beta t_1$, $\\rho_{wall}$ is given by\n\\begin{equation}\n    \\rho_{wall}(t_1) = \n    \\frac{\\sigma \\pi (\\beta t_1)^2}{(\\alpha t_1)^3}\n    = 16\\pi\\alpha^{-3}\\beta^2 m_a F_a^2\n    \\left(\\frac{16\\pi^3 {\\cal N}}{45}\\right)^{1\/2}\\frac{T_1}{m_{pl}}, \n    \\label{walldensity}\n\\end{equation}\nwhere ${\\cal N}$ is the relativistic degrees of freedom at $t_1\\simeq 1\n\\mbox{GeV}$. We expect that the numerical parameters $\\alpha$ and $\\beta$\nare O(1) from causality, although the precise values should be determined\nby the realistic simulation of cosmological formation of the wall-string\nsystem, which is beyond the scope of the present paper.\n \nUsing the entropy density of the universe $s (= 2\\pi^2{\\cal N}T^3\/45)$\nand its conservation  ( $s a^3 =$ const.),\nthe number density of the axion  can be written as\n\\begin{equation}\n    \\frac{n_a}{s} = 380 \\alpha^{-3}\\beta^2 {\\cal N}^{-1\/2} \n    \\left(\\frac{F_a^2}{T_1 m_{pl}}\\right)\n    \\left(\\frac{\\langle E_a \\rangle}{m_a}\\right)^{-1}.\n    \\label{axion-entropy}\n\\end{equation}\nThen the present density of the axion is \n\\begin{equation}\n    \\rho_a = 1.06 \\times 10^6 \\mbox{cm}^{-3} \\alpha^{-3}\\beta^2\n    {\\cal N}^{-1\/2} m_a \n    \\left(\\frac{F_a^2}{T_1 m_{pl}}\\right)\n    \\left(\\frac{\\langle E_a \\rangle}{m_a}\\right)^{-1}.\n    \\label{present-axion-density}\n\\end{equation}\nSince ${\\cal N}=289\/4$, $T_1 \\simeq 2\\mbox{GeV} (F_a\/10\\mbox{GeV})^{-0.18}$\nand $m_a = 6.2\\times 10^{-4}\\mbox{eV}(F_a\/10^{10}\\mbox{GeV})^{-1}$, \n\\begin{equation}\n    \\rho_a = 3.2 \\times 10^2 \\mbox{eV} \\mbox{cm}^{-3} \\alpha^{-3}\\beta^2\n    \\left(\\frac{F_a}{10^{10}\\mbox{GeV}}\\right)^{1.18}\n    \\left(\\frac{\\langle E_a \\rangle}{m_a}\\right)^{-1}.\n\\end{equation}\nThe contribution of the axion to the density parameter $\\Omega$ is given by\n\\begin{equation}\n    \\Omega_a h^2 = 0.030 \\alpha^{-3}\\beta^2\n    \\left(\\frac{F_a}{10^{10}\\mbox{GeV}}\\right)^{1.18}\n    \\left(\\frac{\\langle E_a \\rangle}{m_a}\\right)^{-1},\n    \\label{omega-axion}\n\\end{equation}\nwhere $h$ is the Hubble constant in units of $100\\mbox{km}\/\\sec\/\\mbox{Mpc}$.\nTherefore the axion density is large enough to account for the dark\nmatter in the universe unless the axion is ultra-relativistic when it\nis emitted.  We estimate the $\\langle E_a \\rangle\/m_a$ by numerical\nsimulation in the next section.\n\n\n\\section{Simulation of Collapse}\n\\label{sec:simulation}\n\nIn order to follow the motion of domain walls, we have solved the\nevolution equation of the axion field numerically. When a wall piece\nmuch smaller than the cosmological horizon is considered, the cosmic\nexpansion can be ignored.\nThen the field equation under the Minkowski background is written as\n\\begin{equation}\n    \\frac{\\partial^2 \\phi}{\\partial t^2}-\\nabla^2\\phi\n    =-\\frac{~\\partial}{\\partial \\phi}(V_s+V_w)~,\n    \\label{eq:eveq}\n\\end{equation}\nusing the potentials in the equations (\\ref{eq:lagra}) and\n(\\ref{eq:alagra}).  We have employed the staggered leapfrog method to\nsolve the differential equation. The model parameters are chosen such\nthat the width of the wall is equal to ten simulation meshes and the\nvacuum energy of the string is a hundred times larger than that of the\nwall. The variation of the numerical value in the latter condition\ndoes not alter our conclusion since only the wall tension governs the\nmotion of walls as we have mentioned above.  The boundary condition is\nperiodic, which is useful to check the accuracy of our calculation\nsince the total energy in the simulation box is conserved.  The\nsolution of a static infinite planar wall under $|\\phi|=v$ is used in\nthe initial configuration.\nWhen the wall lies in $yz-$ plane, it is expressed as\n\\begin{equation}\n    \\frac{A(x)}{F_a}=\\pi +2\\sin^{-1}(\\tanh m_a x) ~.\n\\end{equation}\nThus the scale of the inverse axion mass represents the characteristic\nthickness of the axionic wall. The basic numerical technique is\nthe same one in the previous paper.\nSee the reference \\cite{Naga} for more details.\n\nFirst we have confirmed that approaching walls that face in parallel\neach other pass through one another. Widrow showed this is true in the\ncase of a toy sine-Gordon potential\\cite{Widrow}. We have reproduced\nthe passing phenomenon by numerical simulations using the potential\n(\\ref{eq:eveq}).  Fig.\\ref{fig:pass} shows the result. The initial\ncondition is set so that the relative velocity is $0.05c$, where $c$\nis light velocity and the separation between walls is 200 meshes.\nNotice that the size of one mesh is equal to $F_a^{-1}$ in our\nsimulation and we take $m_a = 0.1 F_a$. As the time evolves, two\nwalls become close and go away without any crush. When the initial\nrelative velocity is much larger, for example, equal to $0.5c$, the\npair annihilation of walls occurs neither.  Therefore it is unlikely\nthat annihilation of domain walls occurs and produces axions\nduring the collision process. The possibility which is described in\nthe reference \\cite{Lyth} should be unfavorable.\n\nOn the other hand, in the case of the encounter of a wall with a wall\nedge, {i.e.}, a string cuts the wall and the process of disintegration\nadvances\\cite{Naga}.  Hence by repeated intercommutations, large walls\nare broken to small pieces. When their size becomes comparable with\nthe thickness scale of the wall, they collapse and radiate energy as\naxions. In order to see how fast and effective the wall collapse is,\nwe have performed the simulation of the evolution of a small wall piece.\nAs a simple example we followed the time evolution of a disk wall\nsurrounded by a circular string in the previous work \\cite{Naga}.\nThe result shows that the disk wall shrinks at the\nvelocity of light and the energy of the wall is converted to axions.\nWe can say that the wall collapse process is so rapid that the\nradiation of gravitational waves is hardly expected.\n\nIn order to see the wall collapse with higher accuracy, we have\nperformed the simulations of a strip wall. Figs.\\ref{fig:obi}(a)(b)(c)\ndemonstrate the two-dimensional distribution of $V_w$. The first one\nshows the initial configuration in which there is one strip wall of\ninfinite height and 20 meshes length. As the time evolves, the edges\nof the strip approach each other, that is, the wall size decreases.\nThus the wall collapse proceeds also at about the light velocity and\nthe result of the three-dimensional simulation is confirmed.\nIn the case of Fig.\\ref{fig:obi}, the initial $\\dot{\\phi}$\nis zero everywhere in the box. However, the initial motion may make\nthe wall move periodically so that the death of wall might be avoided.\nWe have found that this is not the case by performing the simulations\nwith various initial motions. Even in the most extreme case where the\nstrings attached to the wall edges go away each other with the light\nvelocity, such a motion could do nothing more than the slight\nextension of the wall life time.\n\nFinally we have estimated the $\\langle E_a \\rangle\/m_a$ in the\nsimulation. The time evolution of the potential energy $V_w$ and\nkinetic energy of the axion field $A$ is shown in\nFig.\\ref{fig:pot}(a). In these simulations, the wall edges are\nsmoothly connected to the true vacuum region which is different from\nthe disk wall case. Since the overestimation of the field gradient is\nremoved by this smoothing, the quantitative analysis of energy\ndistribution is enabled.  After the wall collapse there remains only\naxionic waves which we identify as axions and the values of the\npotential and kinetic energy of the axion become constant. Since the\naxion waves oscillate like $\\sim e^{-iE_at}$, the ratio of the kinetic\nenergy to potential energy is equal to $\\langle E_a \\rangle^2\/m_a^2$.\nThe simulation shows $\\langle E_a \\rangle\/m_a \\simeq 3$, which means\nthat the emitted axion is mildly relativistic and becomes\nnon-relativistic soon by cooling due to the cosmic expansion. As\nmentioned before we check the accuracy of our simulation by the\nconservation of the total energy in the simulation box. In the\nsimulation above the total energy is conserved within accuracy of 20\\%.\n\nIn order to confirm the estimated value of $\\langle E_a \\rangle\/m_a $\nwe have performed the simulation of higher resolution ( string\nwidth $= 3$ meshes, wall width $= 30$ meshes, wall length $=60$\nmeshes) . The result (Fig.\\ref{fig:pot}(b)) shows $\\langle E_a\n\\rangle\/m_a \\simeq 3$ which is quite consistent with the result of\nthe lower resolution run.\n\nOur model parameters in the simulations above correspond to $m_a\/F_a =\n0.1$ which is much larger than the actual axion model ( $ m_a\/F_a \\sim\n10^{-23}$ ). To check if the estimation for $\\langle E_a \\rangle\/m_a$\ndepends on $m_a\/F_a$, we have run the simulation of thicker wall\n( string width $= 1$ meshes wall width $= 50$ meshes, wall length $=100$\nmeshes ) in which $m_a\/F_a =0.02$. Although the conservation of the total\nenergy becomes as bad as 50\\% in this case, the result indicates that\nthe value of $\\langle E_a \\rangle\/m_a$ is affected only slightly, which\nmeans the generality of our estimation concerning the relativisity of\nemitted axions.\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nIn the previous section, it is found that the axionic wall collapses\nwith time scale $R\/c$ and the mildly-relativistic axions remain\nafter the collapse. The estimated $\\langle E_a \\rangle\/m_a$ is about 3,\nwhich leads to the relic density of the axion given by\n\\begin{equation}\n    \\Omega_a h^2 \\simeq  0.01 \\alpha^{-3}\\beta^2\n    \\left(\\frac{F_a}{10^{10}\\mbox{GeV}}\\right)^{1.18}\n    \\label{omega-axion2}\n\\end{equation}\nSince $\\alpha$ and $\\beta$ are expected to be $O(1)$, the contribution\nof the axion to the present universe is large for $F_a \\gtrsim\n10^{10}\\mbox{GeV}$ (compared with the baryon density $\\Omega_B h^2 \\simeq\n0.013$\\cite{Walker}) and might account for the dark matter of the\nuniverse. However eq.(\\ref{omega-axion2}) should be compared with the\ndensity of the coherent oscillation of axion field (cold\naxion)\\cite{Preskill,Turner} given by\n\\begin{equation}\n    \\Omega_a({\\rm cold}) = 6.5\\times 10^{-3\\pm 0.4}\n    \\left(\\frac{F_a}{10^{10}\\mbox{GeV}}\\right)^{1.18}.\n    \\label{cold}\n\\end{equation}\nTherefore the axion from the wall has comparable to or higher density\nthan the cold axion unless $\\alpha^{-3}\\beta^2$ is much less than 1.\nIn deriving (\\ref{omega-axion2}), we assume that the domain wall\ncollapses rapidly after its formation.  As shown in the simulations\nthis assumption is quite reasonable for walls whose size is much\nsmaller than the horizon.  However, a large wall does not shrink until\nits size becomes smaller than the horizon, which leads to the increase\nof the density of axionic domain wall and as a result the density of\nthe axion produced by the collapse of the wall. Therefore\neq.(\\ref{omega-axion2}) may underestimate the actual axion density and\nthe axion from axionic walls may be more important than the cold axion.\n\nAnother important source of axions is an oscillating axionic string.\nLet us compare our result with the axion emission from the axionic\nstrings. The density of the axion from the strings is given by\n\\begin{equation}\n    \\Omega_a({\\rm string}) \\simeq (1 - 0.01) (F_a\/10^{10}\\mbox{GeV})^{1.18},\n    \\label{string}\n\\end{equation}\nwhere uncertainty of a factor of 100 is due to two different\nestimations by Davis\\cite{Davis} and Harari and Sikivie\\cite{Hagmann}.\nThus the importance of the axion from the wall depends on which\nestimation is correct.  The difference comes from the different\nassumption for the energy spectrum of the emitted axion and it is hard to\njudge which assumption is better.  In the case of the axionic domain\nwall the spectrum or average energy of emitted axions can be estimated\nby numerical simulations more easily since axions are mostly produced\nat the final stage of the collapsing wall whose size is much smaller\nthan the cosmic scale.\n\nIn conclusion, the relic density of the axion produced by the collapse\nof axionic walls is, at least, larger than the baryon density for $F_a\n\\gtrsim 10^{10}$ GeV and accounts for a part or all of the dark matter in\nthe universe.  The axion from the axionic domain wall is more important\nthan the cold axion and comparable to that from strings if Harari and\nSikivie's estimation is correct.  To make a more precise prediction of\nthe relic density we have to know the numeric parameter $\\alpha$ and\n$\\beta$, which is beyond the scope of the present work and will be\nstudied in future work.\n\n\\vskip 0.5cm\nMN is grateful to Professor K. Sato for his continuous encouragement.\nThis work was in part supported by the Japanese Grant in Aid for\nScience Research Fund of the Ministry of Education, Science and\nCulture (No. 3253). Numerical computations were partially performed by\nSUN SPARC stations at Uji Research Center, Yukawa Institute for\nTheoretical Physics, Kyoto University.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nSets are widely used in a variety of control theory and applications including reachability analysis for system verification \\cite{Asarin2006,Girard2005,Girard2006,Kurzhanskiy2007}, robust Model Predictive Control (MPC) \\cite{Mayne2005,Langson2004,Bravo2006}, and state estimation \\cite{Chisci1996,Alamo2005,Le2013}. However, the sets used in control theory are not always practical to compute in application. For example, the minimal Robust Positively Invariant (mRPI) set \\cite{Rakovic2005} is widely used in robust MPC \\cite{Mayne2014,Richards2006,Limon2010}.  However, in general, mRPI sets are not finitely represented and must be approximated. Furthermore, existing techniques for determining finite approximations of the mRPI set do not scale well with the dimension of the state space. Such scalability issues are found in many set computations \\cite{Tiwary2008}, motivating the need for alternative set representations and efficient approximation algorithms. \n\nWhen computing a set, there is often a trade-off between accuracy, complexity, and computation time. The desired balance of these three aspects varies depending on if the set computations are performed off-line prior to controller execution or on-line in real-time. Certain applications permit iterative set computation methods while others require one-step methods that allow set computations to be embedded within an existing optimization problem \\cite{Trodden2016_OneStep}.\n\nAdditionally, this trade-off is highly dependent on the specific representation of the set. Widely used set representations include the halfspace representation (H-Rep) based on the intersection of a finite number of halfspace inequalities and the vertex representation (V-Rep) based on the convex hull of a finite number of vertices. As an alternative, zonotopes (G-Rep) \\cite{McMullen1971} and, more recently, constrained zonotopes (CG-Rep) \\cite{Scott2016} have enabled significant reductions in the cost and complexity associated with commonly used set computations in dynamic systems and control.\n\nA \\emph{zonotope} is the Minkowski sum of a finite set of line segments or, equivalently, the image of a hypercube under an affine transformation \\cite{Fukuda2004,Maler2008}. Due to their computational efficiency, zonotopes have been widely used in reach set calculations for hybrid system verification, estimation, and MPC \\cite{Maler2008,Althoff2010,Scott2016,Bravo2006}. As with the iterative algorithm in \\cite{Scibilia2011}, computing these reach sets utilizes linear transformation and Minkowski sum operations. Zonotopes are closed under these operations (i.e. the Minkowski sum of two zonotopes is a zonotope) and the number of generators grows linearly with the number of Minkowski sum operations, compared to the potential exponential growth of the number of halfspaces in H-Rep. Unfortunately, zonotopes in general are not closed under intersection and the conversion from G-Rep to H-Rep for intersection operations is inefficient. \n\n\\emph{Constrained zonotopes} were developed in \\cite{Scott2016} to overcome the limitations caused by the inherent symmetry of zonotopes. Constrained zonotopes are closed under linear transformation, Minkowski sum, and generalized intersection and can be used to represent any convex polytope. Constrained zonotopes provide the computational advantages of zonotopes while enabling exact computations of a much wider class of sets. In \\cite{Koeln2019ACC}, reach set computations using constrained zonotopes were shown to be several orders-of-magnitude faster than the same set computations using H-Rep, enabling the on-line computation of these reach sets for use in a hierarchical MPC formulation.\n\nWhile zonotopes and constrained zonotopes provide a significant computational advantage, various set operations can increase the complexity of the resultant sets beyond a desired upper limit. Thus, there is a need for techniques that provide reduced-complexity approximations of the desired set. Currently there exist reduced-order outer-approximation techniques for zonotopes \\cite{Kopetzki2017a,Yang2018} and constrained zonotopes \\cite{Scott2016}.  Outer-approximations are widely used in the field of reachability analysis for system verification to determine if a system will always operate in a desired region of the state space \\cite{Girard2005,Girard2006}.  \n\nHowever, in many applications there is a need for computing reduced-order inner-approximations. In general computing inner-approximations of sets is considered a more difficult problem \\cite{Kurzhanski2000}. Inner-approximations are particularly important when computing backward reachable sets that define a set of initial states for which a system will enter a specified target region after some allotted time \\cite{Xue2017}. While there are existing techniques for zonotopes \\cite{Girard2006,Han2016a}, inner-approximation techniques for constrained zonotopes are lacking.\n\nThe goal of this paper is to further increase the practicality of applying set-based control techniques through the use of zonotopes and constrained zonotopes. Specifically, this paper provides improved methods for i) representing set intersections with halfspaces, ii) removing redundancy from set representations, and iii) computing reduced-order inner-approximations, convex hulls, RPI sets, and Pontryagin differences. Approaches for both zonotopes and constrained zonotopes are provided along with numerical examples that demonstrate the features and applicability of each approach.\\footnote{The source code for all of the constrained zonotope operations and numerical examples is provided at https:\/\/github.com\/ESCL-at-UTD\/ConZono.}\n\nThe remainder of the paper is organized as follows. Section~\\ref{Notation} provides some initial notation and preliminary background on set operations, zonotopes, and constrained zonotopes. Methods for checking and computing halfspace intersections for zonotopes and constrained zonotopes are presented in Section~\\ref{halfspaceIntersections}. Section~\\ref{Sec_Redundancy} addresses the issue of redundancy in set representations along with methods for redundancy removal. Techniques for computing reduced-complexity inner-approximations of zonotopes and constrained zonotopes are provided in Section~\\ref{Sec_InnerApprox}. Zonotope and constrained-zonotope based methods for computing the convex hull of two sets, the outer-approximation of the mRPI set, and the Pontryagin difference of two sets are presented in Sections \\ref{Sec_ConvexHull}, \\ref{Sec_RPI}, and \\ref{Sec_Pontryagin}, respectively. Section~\\ref{Sec_Hier} provides a practical application of these techniques for computing and approximating a backward reachable set within the context of hierarchical control. Finally, Section~\\ref{Conclusions} summarizes the conclusions of the paper.\n\\section{Notation and Preliminaries} \\label{Notation}\n\nFor sets $ Z, W \\subset \\mathbb{R}^n $, $ Y \\subset \\mathbb{R}^m $, and matrix $ \\mathbf{R} \\in \\mathbb{R}^{m \\times n} $, the linear transformation of $ Z $ under $ \\mathbf{R} $ is $ \\mathbf{R} Z = \\left\\{\\mathbf{R} \\mathbf{z} \\mid \\mathbf{z} \\in Z \\right\\} $, the Minkowski sum of $ Z $ and $ W $ is $ Z \\oplus W = \\left\\{\\mathbf{z}+\\mathbf{w} \\mid \\mathbf{z} \\in Z, \\mathbf{w} \\in W \\right\\} $, and the generalized intersection of $ Z $ and $ Y $ under $ \\mathbf{R} $ is $ Z \\cap_{\\mathbf{R}} Y = \\left\\{ \\mathbf{z} \\in Z \\mid \\mathbf{R}\\mathbf{z} \\in Y \\right\\} $. The standard intersection, corresponding to the identity matrix $ \\mathbf{R} = \\mathbf{I}_n $, is simply denoted as $ Z \\cap Y$.\n\nThe convex polytope $ H \\subset \\mathbb{R}^n $ in H-Rep is defined as $ H = \\{\\mathbf{x} \\in \\mathbb{R}^n \\mid \\mathbf{H} \\mathbf{x} \\leq \\mathbf{f} \\} $ where $ \\mathbf{H} \\in \\mathbb{R}^{n_h \\times n} $, $ \\mathbf{f} \\in \\mathbb{R}^{n_h} $, and $ n_h $ is the number of halfspaces. A centrally symmetric set $ Z \\subset \\mathbb{R}^n $ can be represented as a zonotope in G-Rep where $ Z = \\left\\{ \\mathbf{G} \\boldsymbol{\\xi} + \\mathbf{c} \\mid \\lVert \\boldsymbol{\\xi} \\rVert_\\infty \\leq 1 \\right\\} $. The vector $ \\mathbf{c} \\in \\mathbb{R}^n $ is the center and the $ n_g $ generators, denoted $ \\mathbf{g}_i $, form the columns of the generator matrix $ \\mathbf{G} \\in \\mathbb{R}^{n \\times n_g } $. Similarly, a constrained zonotope $ Z_c \\subset \\mathbb{R}^n $ is defined in CG-Rep as $ Z = \\left\\{ \\mathbf{G}\\boldsymbol{\\xi} + \\mathbf{c} \\mid \\lVert \\boldsymbol{\\xi} \\rVert_\\infty \\leq 1, \\mathbf{A} \\boldsymbol{\\xi} = \\mathbf{b} \\right\\} $. With $ \\mathbf{A} \\in \\mathbb{R}^{n_c \\times n_g} $ and $ \\mathbf{b} \\in \\mathbb{R}^{n_c} $, constrained zonotopes include $ n_c $ equality constraints that break the symmetry of zonotopes and allow any convex polytope to be written in CG-Rep. The complexity of a zonotope is captured by its order, $ o = \\frac{n_g}{n} $ while the complexity of a constrained zonotope is captured by the degrees-of-freedom order, $o_d = \\frac{n_g-n_c}{n} $. Zonotopes and constrained zonotopes are denoted as $ Z = \\left\\{\\mathbf{G},\\mathbf{c}\\right\\} $  and $ Z_c = \\left\\{\\mathbf{G},\\mathbf{c},\\mathbf{A},\\mathbf{b}\\right\\} $, respectively. \n\nAs shown in \\cite{Scott2016}, constrained zonotopes are closed under linear transformation, Minkowski sum, and generalized intersection where\n\n\\begin{equation} \\label{affineMap}\n\\mathbf{R} Z = \\left\\{\\mathbf{R} \\mathbf{G}_z, \\mathbf{R} \\mathbf{c}_z, \\mathbf{A}_z, \\mathbf{b}_z\\right\\},\n\\end{equation}\n\\begin{equation} \\label{MinkowskiSum}\nZ \\oplus W = \\left\\{\\left[\\mathbf{G}_z \\; \\mathbf{G}_w\\right], \\mathbf{c}_z+\\mathbf{c}_w, \\begin{bmatrix} \\mathbf{A}_z & \\mathbf{0} \\\\ \\mathbf{0} & \\mathbf{A}_w \\end{bmatrix}, \\begin{bmatrix} \\mathbf{b}_z \\\\ \\textbf{b}_w \\end{bmatrix} \\right\\},\n\\end{equation}\n\\begin{equation} \\label{generalized_Intersection}\nZ \\cap_{\\mathbf{R}} Y = \\left\\{\\left[\\mathbf{G}_z \\; \\mathbf{0}\\right], \\mathbf{c}_z,  \\begin{bmatrix} \\mathbf{A}_z & \\mathbf{0} \\\\ \\mathbf{0} & \\mathbf{A}_y \\\\ {\\scriptstyle \\mathbf{R} \\mathbf{G}_z} & {\\scriptstyle -\\mathbf{G}_y} \\end{bmatrix}, \\begin{bmatrix} \\mathbf{b}_z \\\\ \\mathbf{b}_y \\\\ {\\scriptstyle \\mathbf{c}_y - \\mathbf{R} \\mathbf{c}_z }\\end{bmatrix} \\right\\}.\n\\end{equation}\nAdditional notation is defined as follows. The set of non-negative real numbers is denoted as $ \\mathbb{R}_+ $. The matrix $ \\mathbf{T} \\in \\mathbb{R}^{n \\times m} $ with values $ t_{i,j} $ in the $i^{th}$ row and $ j^{th} $ column is denoted as $ \\mathbf{T} = [t_{i,j}] $. A $ n \\times m $ matrix of zeros is denoted as $ \\mathbf{0}_{n \\times m} $ or simply $ \\mathbf{0} $ if the dimension can be readily determined from context. Similarly, a vector of ones is denoted as $ \\mathbf{1} $. For a matrix $ \\mathbf{A} $, the null space is denoted $ \\mathcal{N}(\\mathbf{A}) $ and the pseudoinverse is denoted $ \\mathbf{A}^\\dagger $. Parallel vectors $ \\mathbf{v}_1 $ and $ \\mathbf{v}_2 $ are denoted as $ \\mathbf{v}_1 \\parallel  \\mathbf{v}_2 $. The unit hypercube in $ \\mathbb{R}^n $ is defined as $ B_\\infty = \\left\\{ \\boldsymbol{\\xi} \\mid \\| \\boldsymbol{\\xi} \\|_\\infty \\leq 1 \\right\\}$ while $ B_\\infty(\\mathbf{A},\\mathbf{b}) = \\left\\{ \\boldsymbol{\\xi} \\in  B_\\infty \\mid \\mathbf{A} \\boldsymbol{\\xi} = \\mathbf{b} \\right\\} $. With the volume of a set $ X $ denoted as $ V(X) $, the volume ratio for sets $ X, Y \\in \\mathbb{R}^n $ is defined as $ V_r = \\left( \\frac{V(X)}{V(Y)} \\right)^{1\/n} $. All numerical examples were generated using MATLAB on a desktop computer with a 3.6 GHz i7 processor and 16 GB of RAM.\nAll optimization problems were formulated and solved with YALMIP \\cite{Lofberg2004} and Gurobi \\cite{Gurobi2019}.\n\n\\section{Halfspace Intersections} \\label{halfspaceIntersections}\n\nThis section presents methods for determining if a zonotope or constrained zonotope intersects a given halfspace along with the exact representation of this intersection in CG-Rep. The need for computing this intersection arises in reachability analysis \\cite{Althoff2012} and in MPC when determining the set of feasible initial conditions \\cite{Scibilia2011}. The use of CG-Rep enables exact representations unlike existing techniques that rely on zonotopic approximations of the intersection \\cite{Girard2008}.\n\n\\subsection{Zonotope-Halfspace Intersection}\nFor a zonotope in $ \\mathbb{R}^n $ with $ n_g $ generators, the intersection between a zonotope and a hyperplane can be tested algebraically with complexity $O(nn_g)$.\n\\begin{lem} (Section 5.1 of \\emph{\\cite{Girard2005}}) \\label{Zono_halfspace_int_check}\n\tThe zonotope $ Z = \\{\\mathbf{G},\\mathbf{c}\\} \\subset \\mathbb{R}^n $ intersects the hyperplane $ H = \\{\\mathbf{x} \\in \\mathbb{R}^n \\mid \\mathbf{h}^T \\mathbf{x} = f \\} $ if and only if \n\t\\begin{equation} \\label{hyperplane_int}\n\t| f - \\mathbf{h}^T \\mathbf{c} | \\leq \\sum_{i = 1}^{n_g} | \\mathbf{h}^T \\mathbf{g}_i |.\n\t\\end{equation} \n\\end{lem}\nIf a zonotope intersects a hyperplane, the intersection between the zonotope and the corresponding halfspace can be represented in CG-Rep by the addition of exactly one generator and one equality constraint.\n\n\\begin{thm} \\label{zono_halfspace_int}\n\tIf the zonotope $ Z = \\{\\mathbf{G},\\mathbf{c}\\} \\subset \\mathbb{R}^n $ intersects the hyperplane $ H = \\{\\mathbf{x} \\in \\mathbb{R}^n \\mid \\mathbf{h}^T \\mathbf{x} = f \\} $ corresponding to the halfspace $ H_- = \\{\\mathbf{x} \\in \\mathbb{R}^n \\mid \\mathbf{h}^T \\mathbf{x} \\leq f \\} $, then the intersection $ Z_h = Z \\cap H_- $ is a constrained zonotope where\n\t\\begin{equation}\\label{zono_halfspace_int_def}\n\tZ_h = \\{[\\mathbf{G} \\; \\mathbf{0}], \\mathbf{c}, \\begin{bmatrix} \\mathbf{h}^T\\mathbf{G} \\; \\frac{d_m}{2}\t\\end{bmatrix}, \\begin{matrix} f - \\mathbf{h}^T\\mathbf{c}-\\frac{d_m}{2}\\end{matrix}\\}, \n\t\\end{equation} \n\tand $ d_m = f - \\mathbf{h}^T\\mathbf{c} + \\sum_{i = 1}^{n_g} | \\mathbf{h}^T \\mathbf{g}_i | $.\n\\end{thm}\n\\begin{pf}\nConsidering any element $\\mathbf{x} \\in Z_h$, it is to be proven that $\\mathbf{x} \\in Z \\cap H_{-}$. From the definition of $Z_h$ in \\eqref{zono_halfspace_int_def}, $\\exists \\; \\boldsymbol{\\xi} \\in \\mathbb{R}^{n_g}$ and $\\xi_{n_g + 1} \\in \\mathbb{R}$ such that\n\\begin{equation*}\n    \\mathbf{x} = \\mathbf{G}\\boldsymbol{\\xi} + \\mathbf{0}\\xi_{n_g + 1} + \\mathbf{c}, \\quad ||\\boldsymbol{\\xi}||_{\\infty} \\leq 1, \\quad |\\xi_{n_g + 1}| \\leq 1,\n\\end{equation*}\n\\begin{equation}\\label{conzono_const}\n    \\mathbf{h}^T\\mathbf{G}\\boldsymbol{\\xi} + \\frac{d_m}{2}\\xi_{n_g+1} = f - \\mathbf{h}^T\\mathbf{c} - \\frac{d_m}{2}.\n\\end{equation}\n\n\nBy the assumption that $Z \\cap H \\neq \\emptyset$, the definition of $d_m$ and \\eqref{hyperplane_int} ensure $d_m \\geq 0$. If $d_m = 0$, then \\eqref{conzono_const} results in $\\mathbf{h}^T\\mathbf{G}\\boldsymbol{\\xi} = f- \\mathbf{h}^T\\mathbf{c}$, which can be rewritten as $\\mathbf{h}^T(\\mathbf{G}\\boldsymbol{\\xi} + \\mathbf{c}) = f$. Therefore, $\\mathbf{x} \\in Z_h \\subset Z$ and $\\mathbf{x} \\in H \\subset H_{-}$. If $d_m > 0$, \\eqref{conzono_const} can be solved for $\\xi_{n_g + 1}$ as\n\\begin{equation} \\label{eq_xi_ngplus1}\n    \\xi_{n_g + 1} = \\frac{2}{d_m}(f - \\mathbf{h}^T\\mathbf{c}- \\frac{d_m}{2} - \\mathbf{h}^T\\mathbf{G}\\boldsymbol{\\xi}).\n\\end{equation}\nCombining \\eqref{eq_xi_ngplus1} and the inequality constraint $-1 \\leq \\xi_{n_g + 1}$ results in \\begin{subequations}\n    \\begin{align*}\n      -1 \\leq & \\; \\xi_{n_g + 1} = \\frac{2}{d_m}(f - \\mathbf{h}^T\\mathbf{c}- \\frac{d_m}{2} - \\mathbf{h}^T\\mathbf{G}\\boldsymbol{\\xi}),  \\\\ \n        -\\frac{d_m}{2} \\leq & \\; f - \\mathbf{h}^T\\mathbf{c} - \\frac{d_m}{2} - \\mathbf{h}^T\\mathbf{G}\\boldsymbol{\\xi},  \\\\ \n        \\mathbf{h}^T(\\mathbf{c} + \\mathbf{G}\\boldsymbol{\\xi}) \\leq & \\; f. \n    \\end{align*}\n\\end{subequations}\n\\setcounter{equation}{\\value{equation}-1}\nTherefore, $\\mathbf{x} \\in Z$ and $\\mathbf{x} \\in H_{-}$.\nNext, considering any $\\mathbf{x} \\in Z \\cap H_{-}$, it is to be proven that $\\mathbf{x} \\in Z_h$. For all $\\mathbf{x} \\in Z \\cap H_{-}$, $\\exists \\; \\boldsymbol{\\xi} \\in \\mathbb{R}^{n_g}$ such that \n\\begin{equation}\\label{eq_x_Z_intsct_H}\n\\mathbf{x} = \\mathbf{G}\\boldsymbol{\\xi} + \\mathbf{c}, \\; ||\\boldsymbol{\\xi}||_{\\infty} \\leq 1, \\; \\mathbf{h}^T\\mathbf{x} \\leq f\n\\end{equation}\nTo show that $\\mathbf{x} \\in Z_h$ requires proving the existence of $\\xi_{n_g + 1} \\in \\mathbb{R}$ such that\n\\begin{equation*}\n\\mathbf{x} = \\mathbf{G}\\boldsymbol{\\xi} + \\mathbf{0}\\xi_{n_g + 1} + \\mathbf{c}, \\quad |\\xi_{n_g + 1}| \\leq 1,\n\\end{equation*}\nand \\eqref{conzono_const} holds for all $\\mathbf{x}$ satisfying \\eqref{eq_x_Z_intsct_H}. If $d_m = 0$, then \\eqref{conzono_const} is independent of $\\xi_{n_g + 1}$ and holds $\\forall \\; \\mathbf{x} \\in Z \\cap H_{-}$. Thus, $\\xi_{n_g + 1}$ can be arbitrarily chosen such that $|\\xi_{n_g + 1}| \\leq 1$. If $d_m > 0$, let $\\xi_{n_g+1}$ be chosen as in \\eqref{eq_xi_ngplus1}, which satisfies \\eqref{conzono_const}.\nTo prove $|\\xi_{n_g + 1}| \\leq 1$, consider $\\mathbf{x}$ as in \\eqref{eq_x_Z_intsct_H}.\nSince, $f - \\mathbf{h}^T\\mathbf{x} \\geq 0$, $\\xi_{n_g +1}$ satisfies\n\\begin{equation}\\label{xi_ngplus1_exp2}\n    \\xi_{n_g + 1} = \\frac{2}{d_m}(f - \\mathbf{h}^T\\mathbf{x} - \\frac{d_m}{2}) \\geq -1.\n\\end{equation}\nFinally, using \\eqref{eq_xi_ngplus1}, the fact that $-\\mathbf{h}^T\\mathbf{G}\\boldsymbol{\\xi} \\leq \\sum \\limits_{i = 1}^{n_g} |\\mathbf{h}^T \\mathbf{g}_i |$, and the definition of $d_m$ results in\n\\begin{subequations}\n\\begin{align*}\n    & \\xi_{n_g + 1} \\leq \\frac{2}{d_m}(f - \\mathbf{h}^T\\mathbf{c} + \\sum\\limits_{i = 1}^{n_g}|\\mathbf{h}^T \\mathbf{g}_i | - \\frac{d_m}{2}), \\nonumber \\\\ \n    & \\xi_{n_g + 1} \\leq \\frac{2}{d_m}(d_m - \\frac{d_m}{2}) = 1. \\nonumber\n\\end{align*}    \n\\end{subequations}\n\\setcounter{equation}{\\value{equation}-1}\nThus, $\\forall \\; \\mathbf{x} \\in Z \\cap H_{-}$, $\\mathbf{x} \\in Z_h$. \\hfill \\hfill \\qed\n \\end{pf}\n\n\t\n\n\n\\begin{exmp}\\label{exmp_1}\n\tThe left subplot in Fig. \\ref{Fig_Halfspace_Intersection} shows the zonotope $ Z $ and halfspace $ H_- $ where\n\t\\begin{equation*}\n\t\\setlength\\arraycolsep{2pt}\n\tZ = \\left\\{ \\begin{bmatrix} 1 & 1 \\\\ 0 & 2 \\end{bmatrix}, \\begin{bmatrix} 0 \\\\ 0 \\end{bmatrix} \\right\\}, \\; H_- = \\{\\mathbf{x} \\in \\mathbb{R}^2 \\mid \\left[ 3 \\enspace 1 \\right] \\mathbf{x} \\leq 3 \\}.\n\t\\end{equation*}\n\tFrom \\emph{\\textbf{Lemma \\ref{Zono_halfspace_int_check}}}, $ Z $ intersects the associated hyperplane $ H $ since \\eqref{hyperplane_int} evaluates to $ 3 \\leq 8 $. From \\emph{\\textbf{Theorem \\ref{zono_halfspace_int}}}, the intersection $ Z \\cap H_- $ is a constrained zonotope and \\eqref{zono_halfspace_int_def} evaluates to\n\t\\begin{equation*}\n\tZ_h = \\left\\{\\begin{bmatrix} 1 & 1 & 0 \\\\ 0 & 2 & 0 \\end{bmatrix}, \\begin{bmatrix} 0 \\\\ 0 \\end{bmatrix}, \\left[ 3 \\enspace 5 \\enspace 5.5\t\\right], -2.5\\right\\}.\n\t\\end{equation*}\n\n\tThe left subplot in Fig. \\ref{Fig_Halfspace_Intersection} also shows the physical interpretation of $ d_m $ where $ d_m = d_1 + d_2 $.  With $ d_1 = f - \\mathbf{h}^T\\mathbf{c} $, $ d_1 $ captures the orthogonal distance from the hyperplane $ H $ to the center, $ \\mathbf{c} $, of the zonotope. With $ d_2 = \\sum_{i = 1}^{n_g} | \\mathbf{h}^T \\mathbf{g}_i | $, $ d_2 $ captures the orthogonal distance from center of the zonotope to the point in $ Z $ farthest from $ H $.\n\\end{exmp}\n\n\\begin{figure}\n\t\\begin{center}\n\t\\includegraphics[width=8.6cm]{Halfspace_Int.pdf}\n\t\t\\vspace{-10pt}\n\t\t\\caption{Left: The intersection of the zonotope $ Z $ and the halfspace $ H_- $ corresponding to the hyperplane $ H $ results in the constrained zonotope $ Z_h $. The distances $ d_1 $ and $ d_2 $, measured orthogonally to $ H $, are shown to provide a geometric interpretation of the equality constraints in \\eqref{zono_halfspace_int_def}. Right: An example where the constrained zonotope $  Z_c = \\{\\mathbf{G},\\mathbf{c},\\mathbf{A},\\mathbf{b}\\} $, with corresponding unconstrained zonotope  $ Z = \\{\\mathbf{G},\\mathbf{c}\\} $, where $ Z $ intersects the hyperplane $ H $ but $ Z_c $ does not.}\n\t\t\\label{Fig_Halfspace_Intersection}                      \n\t\\end{center}                         \n\\end{figure}\n\n\\subsection{Constrained Zonotope-Halfspace Intersection} \\label{Sec_conZonoHalfspace}\n\nFor the intersection $ Z_h = Z_c \\cap H_- $ of a constrained zonotope $ Z_c = \\{\\mathbf{G},\\mathbf{c},\\mathbf{A},\\mathbf{b}\\} $ and a halfspace $ H_- $, \\textbf{Theorem~\\ref{zono_halfspace_int}} is readily modified where \n\\begin{equation} \\label{conzono_halfspace_int}\n \\scriptsize{Z_h} = \\left\\{[\\mathbf{G} \\; \\mathbf{0}], \\mathbf{c}, \\begin{bmatrix} \\mathbf{A} & \\mathbf{0} \\\\ \\mathbf{h}^T\\mathbf{G} & \\frac{d_m}{2}\t\\end{bmatrix}, \\begin{bmatrix} \\mathbf{b} \\\\ f - \\mathbf{h}^T\\mathbf{c}-\\frac{d_m}{2}\\end{bmatrix}\\right\\}.\n\\end{equation} \n\nHowever, if the constrained zonotope is completely contained in the halfspace, $ Z_c \\subset H_- $, and does not intersect the corresponding hyperplane $ H $, then $ Z_h = Z_c $ and the addition of the $ n_g + 1 $ generator and $ n_c + 1 $ constraint is redundant and increases the order of $ Z_h $ unnecessarily.\n\nHowever, when determining if a constrained zonotope $ Z_c $ intersects a hyperplane $ H $, the inequality \\eqref{hyperplane_int} is necessary but not sufficient. The equality constraints $ \\mathbf{A} \\boldsymbol{\\xi} = \\mathbf{b} $ impose restrictions such that $ Z_c \\subset Z =  \\{\\mathbf{G},\\mathbf{c}\\} $. Thus, the parent zonotope $ Z $ may intersect $ H $ while $ Z_c $ does not (as shown in right subplot of Fig.~\\ref{Fig_Halfspace_Intersection}). The intersection of a constrained zonotope with a hyperplane can be checked by solving two Linear Programs (LPs), each with $n_g$ decision variables.\n\t\n\\begin{lem}\\label{conzono_hyplane_intersection}\n\tThe constrained zonotope $ Z_c = \\{\\mathbf{G},\\mathbf{c},\\mathbf{A},\\mathbf{b}\\} \\subset \\mathbb{R}^n $ intersects the hyperplane $ H = \\{\\mathbf{x} \\in \\mathbb{R}^n \\mid \\mathbf{h}^T \\mathbf{x} = f \\} $ if $ f_{min} \\leq f \\leq f_{max} $, where\n\t\\begin{subequations}\n\t\t\\begin{align*}\n\t\tf_{min} \\triangleq \\text{\\emph{min}} \\{\\mathbf{h}^T (\\mathbf{c} + \\mathbf{G} \\boldsymbol{\\xi}) \\mid \\| \\boldsymbol{\\xi} \\|_\\infty \\leq 1, \\mathbf{A} \\boldsymbol{\\xi} = \\mathbf{b} \\}, \\\\\n\t\tf_{max} \\triangleq \\text{\\emph{max}} \\{\\mathbf{h}^T (\\mathbf{c} + \\mathbf{G} \\boldsymbol{\\xi}) \\mid \\| \\boldsymbol{\\xi} \\|_\\infty \\leq 1, \\mathbf{A} \\boldsymbol{\\xi} = \\mathbf{b} \\}.\n\t\t\\end{align*}\n\t\\end{subequations}\n\\end{lem}\n\\setcounter{equation}{\\value{equation}-1}\n\\begin{pf}\n\tFrom the definition of $ f_{min} $ and $ f_{max} $, if  $ f_{min} \\leq f \\leq f_{max} $, then there exists $ \\mathbf{x}_{min}, \\mathbf{x}_{max} \\in Z_c $ such that $ \\mathbf{h}^T \\mathbf{x}_{min} \\leq f \\leq \\mathbf{h}^T \\mathbf{x}_{max} $.\n\tBy the convexity of constrained zonotopes \\cite{Scott2016}, there exists $ \\mathbf{x}_\\lambda \\in Z_c $ such that $ \\mathbf{x}_\\lambda = \\lambda \\mathbf{x}_{min} + (1-\\lambda) \\mathbf{x}_{max} $, $ \\lambda \\in \\left[0,1\\right] $. For the case where $f_{min} = f_{max} = f $, any choice of $\\lambda \\in [0,1]$ results in $\\mathbf{h}^T\\mathbf{x}_{\\lambda} = f$. Otherwise, if $f_{min} \\neq f_{max}$, choosing $ \\lambda = \\frac{f-f_{max}}{f_{min}-f_{max}} \\in \\left[0,1\\right] $ results in $ \\mathbf{h}^T \\mathbf{x}_\\lambda = f $. Thus $ \\mathbf{x}_\\lambda \\in H $ and $ \\mathbf{x}_\\lambda \\in Z_c $, proving $ Z_c \\cap H \\neq \\emptyset $. \\hfill \\hfill \\qed\n\\end{pf}\nNote that $f_{min}$ and $f_{max}$ obtained using \\textbf{Lemma \\ref{conzono_hyplane_intersection}} represent the largest orthogonal distance between a point in $Z_c$ and either side of the hyperplane  providing additional insight to the location of constrained zonotope with respect to the hyperplane.\n\n\n\\begin{rem}\\label{rem_conzono_hyp_intersection}\n\tWhile the knowledge of $f_{min}$ and $f_{max}$ can be useful, checking for the non-empty intersection of a constrained zonotope and a hyperplane can be achieved by assessing the feasibility of a single LP with constraints\n\t\\begin{equation*}\n\t    \\mathbf{h}^{T}(\\mathbf{c} + \\mathbf{G}\\boldsymbol{\\xi}) \\leq f, \\quad \\mathbf{A}\\boldsymbol{\\xi}= \\mathbf{b}, \\quad ||\\boldsymbol{\\xi}||_{\\infty} \\leq 1.\n\t\\end{equation*}\n\\end{rem}\nWhen solving these LPs is undesirable, an iterative method based on interval arithmetic from \\cite{Scott2016} provides an approach for checking constrained zonotope-halfspace intersection with complexity $O(n_cn_g^2)$. Reproduced from \\cite{Scott2016}, \\textbf{Algorithm \\ref{gen_Bounds}} computes the interval set $ E = [\\boldsymbol{\\xi}^L, \\boldsymbol{\\xi}^U] $ such that $ B_\\infty(\\mathbf{A},\\mathbf{b}) \\subset E \\subset [-\\mathbf{1},\\mathbf{1}] $ and $ R = [\\boldsymbol{\\rho}^L, \\boldsymbol{\\rho}^U] \\subset \\mathbb{R}^{n_g} $ where\n\\begin{equation*}\nR_j \\supset \\{\\xi_j \\mid \\mathbf{A} \\boldsymbol{\\xi} = \\mathbf{b}, |\\xi_i| \\leq 1, \\forall i \\neq j \\}, \\quad \\forall j \\in [1,n_g].\n\\end{equation*}\nAs discussed in \\cite{Scott2016}, this iterative method has the potential to detect empty constrained zonotopes without solving a LP. Specifically, if $ E \\cap R = \\emptyset $, then $ Z_c = \\emptyset $. Since $ E, R $ are intervals, $ E \\cap R = \\emptyset $ if $ \\xi_j^U < \\rho_j^L $ or $ \\xi_j^L > \\rho_j^U $ for any $ j \\in [0,n_g] $.\n\n\\IncMargin{1.5em}\n\\begin{algorithm2e}\n\t\\SetAlgoLined\n\t\\SetKwInOut{Input}{Input}\\SetKwInOut{Output}{Output}\n\t\\Input{$ Z_c = \\{\\mathbf{G},\\mathbf{c},\\mathbf{A},\\mathbf{b}\\} $}\n\t\\Output{$ E_j, R_j, \\forall j \\in [1,n_g] $}\n\t\\BlankLine\n\tInitialize $ E_j \\leftarrow [-1, 1], R_j \\leftarrow [-\\infty, \\infty], \\; i,j \\leftarrow 1 $\n\t\\While{$ i \\leq n_c $}{\n\t\t\\While{$ j \\leq n_g $}{\n\t\t\t\\If{$ a_{ij} \\neq 0 $}{\n\t\t\t\t$ R_j \\leftarrow R_j \\cap ( a_{ij}^{-1}b_i - \\sum_{k \\neq j} a_{ij}^{-1} a_{ik} E_{k} ) $\\;\n\t\t\t\t$ E_j \\leftarrow E_j \\cap R_j $\\;}\n\t\t\t$ j \\leftarrow j+1 $\\;}\n\t\t$ i \\leftarrow i + 1, j \\leftarrow 1 $\\;}\n\t\\caption{\\cite{Scott2016} Constrained zonotope intervals.}\n\t\\label{gen_Bounds}\n\\end{algorithm2e}\n\\DecMargin{1.5em}\nThe goal is to detect if $ Z_c \\subset H_- $, resulting in $ Z_h = Z_c $ and thus avoiding the unnecessary addition of generators and constraints from the application of \\eqref{conzono_halfspace_int}. The proposed approach uses the fact that $ Z_c \\subset H_- $ if and only if $ Z_c \\cap H_+ = \\emptyset $, where $ H_+ = \\{\\mathbf{x} \\in \\mathbb{R}^n \\mid \\mathbf{h}^T \\mathbf{x} \\geq f \\} $ is the complement of $ H_- $. By modifying \\eqref{conzono_halfspace_int} such that $ Z_{h^{+}} = Z_c \\cap H_+ $, \\textbf{Algorithm \\ref{gen_Bounds}} can then be applied to $ Z_{h^{+}} $ to check if $ Z_{h^{+}} = \\emptyset $. Specifically, if $ E \\cap R = \\emptyset $, then $ Z_{h^{+}} = \\emptyset $ and $ Z_c \\subset H_- $. Note that applying \\textbf{Algorithm 1} does not guarantee the detection of $ Z_{h^{+}} = \\emptyset $.\nAs discussed in \\cite{Scott2016}, \\textbf{Algorithm~\\ref{gen_Bounds}} can be applied iteratively to refine the interval set $ E $. In fact, two iterations of \\textbf{Algorithm~\\ref{gen_Bounds}} were required to detect that $ Z_c \\subset H_- $ for the example shown on the right subplot of Fig. \\ref{Fig_Halfspace_Intersection}.\n\n\\begin{rem}\nTo provide an unbiased evaluation of constrained-zonotope hyperplane intersection using \\textbf{\\emph{Algorithm \\ref{gen_Bounds}}}, the intersection of $Z_h$ (from \\textbf{\\emph{Example \\ref{exmp_1}}})\nwith 100 randomly chosen hyperplanes is checked. Note that for all instances, the parent zonotope $Z$ satisfying $Z \\supset Z_h$ intersected the random hyperplanes. The constrained zonotope $Z_h$ intersected these random hyperplanes $61$ times and did not intersect for the remaining $39$ times. In all cases, \\textbf{\\emph{Algorithm \\ref{gen_Bounds}}} accurately detected the intersection\/non-intersection of the constrained zonotope and randomly generated hyperplanes. Iteration of \\textbf{\\emph{Algorithm \\ref{gen_Bounds}}} to further refine $E$ was only required in $13$ of these $100$ cases.\n\\end{rem}\n\n\\section{Redundancy Removal} \\label{Sec_Redundancy}\n\nIt is important to recognize that certain set operations can create redundancy in the set representation. For example, the Minkowski sum can create redundancy in the resultant zonotope if the two operands have parallel generators.  Additionally, the generalized intersection can create redundancy within the generators and constraints of a constrained zonotope. Detecting and removing this redundancy can provide order reduction without reducing the volume of the set.\nFirst, if a zonotope $ Z = \\{\\mathbf{G},\\mathbf{c}\\} $ has parallel generators, $ \\mathbf{g}_i \\parallel \\mathbf{g}_j $, then the same set can be represented using one less generator by simply combining parallel generators through addition $\\mathbf{g}_i + \\mathbf{g}_j$. For a zonotope in $ \\mathbb{R}^n $ with $ n_g $ generators,  parallel generators can be detected and combined using a typical sorting algorithm with complexity $O(n n_g^2)$. To set a desired numerical precision, two generators are considered parallel if $ \\frac{|\\mathbf{g}_i^T \\mathbf{g}_j|}{\\|\\mathbf{g}_i\\|_2 \\|\\mathbf{g}_j\\|_2} \\geq 1 - \\epsilon $, where $ \\epsilon > 0 $ is a small number.\n\nThe same is true for a constrained zonotope $ Z_c = \\{\\mathbf{G},\\mathbf{c},\\mathbf{A},\\mathbf{b}\\} $ if the lifted zonotope \\cite{Scott2016}\n\\begin{equation*\nZ^{+} = \\left\\{ \\begin{bmatrix} \\mathbf{G} \\\\ \\mathbf{A} \\end{bmatrix}, \\begin{bmatrix} \\phantom{-}\\mathbf{c} \\\\ -\\mathbf{b} \\end{bmatrix} \\right\\} = \\{\\mathbf{G}^+,\\mathbf{c}^+\\},\n\\end{equation*}\nhas parallel generators, $ \\mathbf{g}_i^+ \\parallel \\mathbf{g}_j^+ $. In this case, the parallel generators can be similarly reduced but with higher complexity $O(n+n_c)n_g^2$ due to the $ n_c $ constraints added to the rows of the lifted zonotope structure. Once the reduced lifted zonotope is obtained, it\nis transformed back to a reduced constrained zonotope with fewer generators\n\n\nFor constrained zonotopes, redundancy can also come from the combination of constraints $ \\mathbf{A} \\mathbf{\\xi} = \\mathbf{b} $ and $ \\lVert \\boldsymbol{\\xi} \\rVert_\\infty \\leq~1 $. By representing these constraints as\n\\begin{equation} \\label{redund_cons}\n\\mathbf{A} \\mathbf{\\xi} = \\mathbf{b} \\Longleftrightarrow \\mkern -20mu \\sum_{j\\in \\{ 1, \\cdots, n_g \\}} \\mkern -20mu a_{i,j} \\xi_j = b_i, \\forall i \\in \\{ 1, \\cdots, n_c\\},\n\\end{equation} \nand $ \\lVert \\boldsymbol{\\xi} \\rVert_\\infty \\leq 1 \\Leftrightarrow |\\xi_j | \\leq 1, \\forall j \\in \\{ 1, \\cdots, n_g \\}$, the following theorem provides a condition for detecting redundancy and a method for removing one generator and one constraint with complexity $O(n_cn_g^2)$.\n\n\\begin{thm} \\label{Redundant_ConZono}\n\tFor $ Z_c = \\{ \\mathbf{G},\\mathbf{c},\\mathbf{A},\\mathbf{b} \\} \\subset \\mathbb{R}^n $ with $ n_g $ generators and $ n_c $ constraints, if there exists indices $ r \\in \\{ 1, \\cdots, n_c \\} $ and $ c \\in \\{1, \\cdots, n_g \\} $ such that $ a_{r,c} \\neq 0 $ and \n\t\\begin{equation} \\label{Ranges_indexed}\n\tR_{r,c} \\triangleq a_{r,c}^{-1} b_r - a_{r,c}^{-1} \\sum_{k \\neq c} a_{r,k} E_k \\subseteq [-1,1],\n\t\\end{equation}\n\twith $E_k$ computed using \\textbf{\\emph{Algorithm 1}}, then $ Z_c $ \n\n\tcan be exactly represented by a constrained zonotope $ Z_r $ with $ n_g -1 $ generators and $ n_c -1 $ constraints.\n\\end{thm}\n\\begin{pf}\n\tFollowing the procedure in \\cite{Scott2016}, let  \n\t\\begin{equation*}\n\tZ_r = \\{\\mathbf{G}-\\mathbf{\\Lambda}_G\\mathbf{A}, \\mathbf{c}+\\mathbf{\\Lambda}_G\\mathbf{b}, \\mathbf{A}-\\mathbf{\\Lambda}_A\\mathbf{A}, \\mathbf{b}-\\mathbf{\\Lambda}_A\\mathbf{b}\\},\n\t\\end{equation*}\n\twhere $ \\mathbf{\\Lambda}_G = \\mathbf{G} \\mathbf{E}_{c,r} a_{r,c}^{-1} \\in \\mathbb{R}^{n \\times n_c} $, $ \\mathbf{\\Lambda}_A = \\mathbf{A} \\mathbf{E}_{c,r} a_{r,c}^{-1} \\in \\mathbb{R}^{n_c \\times n_c} $, and $ \\mathbf{E}_{c,r} \\in \\mathbb{R}^{n_g \\times n_c} $ is zero except for a one in the $ (c,r) $ position. With $ Z_r = \\{\\mathbf{G}_r,\\mathbf{c}_r,\\mathbf{A}_r,\\mathbf{b}_r\\} $, this transformation uses the $ r^{th} $ of row of \\eqref{redund_cons} to solve for $ \\xi_c $ in terms of $ \\xi_k, k\\neq c $. This results in the $ c^{th} $ column of $ \\mathbf{G}_r $ and $ \\mathbf{A}_r $ and the $ r^{th} $ row of $ \\mathbf{A}_r $ to equal zero. Removing these columns and rows of zeros results in a constrained zonotope with $ n_g -1 $ generators and $ n_c -1 $ constraints. Through this transformation, the $ r^{th} $ constraint is still imposed in $ Z_r $ but the ability to constraint $ | \\xi_c | \\leq 1 $ is lost. However, since $ R_{r,c} \\subseteq [-1,1] $, this constraint is imposed by the remaining equality and norm constraints, and thus $ Z_r = Z_c $. \\hfill \\hfill \\qed\n\\end{pf}\n\nAs in \\cite{Scott2016}, Gauss-Jordan elimination with full pivoting should be applied to $ Z_c $ prior to applying \\textbf{Algorithm \\ref{gen_Bounds}} to determine the intervals $  E_k $ required to compute \\eqref{Ranges_indexed}.\nThe procedure discussed in the proof of \\textbf{Theorem \\ref{Redundant_ConZono}} can be applied iteratively until $ R_{r,c} \\nsubseteq [-1,1] $ for any indices. However, there is no guarantee that the resulting constrained zonotope will be without redundancy since \\textbf{Theorem \\ref{Redundant_ConZono}} only provides a sufficient condition. \n\\begin{exmp}\n\tConsider the two zonotopes shown in Fig. \\emph{\\ref{Fig_Redundancy}}\n\t\\begin{equation*}\n\tZ_1 = \\left\\{ \\begin{bmatrix} 1 & \\phantom{-}1 \\\\ 1 & -1 \\end{bmatrix}, \\begin{bmatrix} 0 \\\\ 0 \\end{bmatrix} \\right\\}, \\quad\n\tZ_2 = \\left\\{ \\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\end{bmatrix}, \\begin{bmatrix} 0 \\\\ 0 \\end{bmatrix} \\right\\},\n\t\\end{equation*}\n and the constrained zonotope $ Z_c = Z_1 \\cap Z_2 $.  Applying \\eqref{generalized_Intersection} results in \n\t\\begin{equation} \\label{Redund_gen_int_example}\n\t\\setlength\\arraycolsep{1.7pt}\n\tZ_c = \\left\\{ \\begin{bmatrix} 1 & \\phantom{-}1 & 0 & 0 \\\\ 1 & -1 & 0 & 0 \\end{bmatrix}, \\begin{bmatrix} 0 \\\\ 0 \\end{bmatrix}, \\begin{bmatrix} 1 & \\phantom{-}1 & -1 & \\phantom{-}0 \\\\ 1 & -1 & \\phantom{-}0 & -1 \\end{bmatrix}, \\begin{bmatrix} 0 \\\\ 0 \\end{bmatrix} \\right\\},\n\t\\end{equation}\n\twith $ n_g = 4 $ generators and $ n_c = 2 $ constraints. However, since $ Z_2 \\subset Z_1 $, the intersection is also represented exactly by $ Z_2 $. By applying Gauss-Jordan elimination with full pivoting and two iterations of the procedure from \\emph{\\textbf{Theorem \\ref{Redundant_ConZono}}}, two constraints and two generators are removed to reduce $ Z_c $ from \\eqref{Redund_gen_int_example} to $ Z_c = Z_2 $ with $ n_g = 2 $ and $ n_c = 0 $. To provide an unbiased evaluation of \\emph{\\textbf{Theorem \\ref{Redundant_ConZono}}}, the axis-aligned generators of $Z_2$ above were replaced by randomly chosen generators. In each of the $45$ out of $100$ cases where $Z_2 \\subseteq Z_1$, $Z_c$ was successfully reduced to $Z_c = Z_2$ with $n_g = 2$ and $n_c = 0$. \n\t\\end{exmp}\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=4cm]{Redundancy_Removal.pdf}\n\t\t\\caption{Zonotopes $ Z_1 $ and $ Z_2 $, where $ Z_1 \\cap Z_2 = Z_2 $, used to demonstrate the ability to remove redundancy from constrained zonotopes that can arise from operations like the generalized intersection.}\n\t\t\\label{Fig_Redundancy}                 \t\\end{center}                        \n\\end{figure}\n\n\\begin{rem}\nFor a constrained zonotope $Z_c$ with $n_c$ constraints and $n_g$ generators and a set $H$ in H-Rep with with $n_h$ halfspaces, \\textbf{\\emph{Algorithm \\ref{gen_Bounds}}} can be applied in two different ways to either prevent or remove redundancy in the set representation of $ Z_c \\cap H $. The approach from Section \\ref{Sec_conZonoHalfspace} based on preventing the addition of unnecessary generators and constraints has a best-case complexity of $O(n_h n_c n_g^2)$ if $ Z_c \\subset H $ and a worst-case complexity of $O(n_h (n_c+n_h) (n_g+n_h)^2)$ if $ Z_c $ intersects each of the $ n_h $ halfspaces. Alternatively, $ n_h $ constraints and $n_h$ generators can be directly added to $ Z_c $ using \\eqref{conzono_halfspace_int} and then \\textbf{\\emph{Theorem \\ref{Redundant_ConZono}}} can be applied to reduce set complexity. This approach has a best-case complexity of $O((n_c+n_h)(n_g+n_h)^2)$ when no generators\/constraints can be removed and a worst-case complexity $O(n_h(n_c+n_h)(n_g+n_h)^2)$ when all of the added $ n_h $ constraints and $n_h$ generators can be removed. Thus, both approaches have the same worst-case complexity but the preventative approach has the potential to require fewer computations in practice.\n\\end{rem}\n\n\\section{Inner-Approximations} \\label{Sec_InnerApprox}\n\nOnce attempts have been made to remove redundancy from the representation of a zonotope or constrained zonotope, further complexity reduction may be required. As discussed in the Introduction, the majority of order reduction techniques have focused on outer-approximations. This section establishes inner-approximation order reduction for zonotopes and constrained zonotopes.\n\n\\subsection{Zonotopes}\n\nThe proposed reduced-order inner-approximation of a zonotope requires the following zonotope containment conditions.\n\\begin{lem} (Theorem 3 of\n\t\\emph{\\cite{Sadraddini2019}}) \\label{zonoContainment}\n\tGiven two zonotopes $ X = \\{\\mathbf{G}_x,\\mathbf{c}_x\\} \\subset \\mathbb{R}^n $ and $ Y = \\{\\mathbf{G}_y,\\mathbf{c}_y\\} \\subset \\mathbb{R}^n $, $ X \\subseteq Y $ if there exists $ \\mathbf{\\Gamma} \\in \\mathbb{R}^{n_y \\times n_x} $ and $ \\boldsymbol{\\beta} \\in \\mathbb{R}^{n_y} $ such that\n\t\\begin{equation} \\label{zonoContainment_Conditions}\n\t\\mathbf{G}_x = \\mathbf{G}_y \\mathbf{\\Gamma}, \\quad \\mathbf{c}_y - \\mathbf{c}_x = \\mathbf{G}_y \\boldsymbol{\\beta}, \\quad |\\mathbf{\\Gamma}|\\mathbf{1} + |\\boldsymbol{\\beta}| \\leq \\mathbf{1}.\n\t\\end{equation}\n\\end{lem}\n\\begin{thm} \\label{ZonoInnerApprox}\n\tThe zonotope $ Z_r = \\{\\mathbf{G}_r,\\mathbf{c}\\} \\subset \\mathbb{R}^n $ is a reduced-order inner-approximation of $ Z = \\{\\mathbf{G},\\mathbf{c}\\} \\subset \\mathbb{R}^n $ such that $ Z_r \\subseteq Z $ with $ \\mathbf{G}_r \\in \\mathbb{R}^{n \\times n_r} $, $ \\mathbf{G} \\in \\mathbb{R}^{n \\times n_g} $, and $ n_r < n_g $ if $ \\mathbf{G}_r = \\mathbf{G} \\mathbf{T} $ where $ \\mathbf{T} = [t_{i,j}] \\in \\mathbb{R}^{n_g \\times n_r} $, $ t_{i,j} \\in \\{-1,0,1\\} $, and $ \\sum_{j=1}^{n_r} |t_{i,j}| = 1 $, $\\forall i \\in \\{ 1, \\cdots, n_g\\}$.\n\\end{thm}\n\\begin{pf}\n\tFrom \\textbf{Lemma \\ref{zonoContainment}}, $ Z_r \\subseteq Z $ if there exist $ \\mathbf{\\Gamma} \\in \\mathbb{R}^{n_g \\times n_r} $ and $ \\boldsymbol{\\beta} \\in \\mathbb{R}^{n_g} $ such that\n\t\\begin{equation*}\n\t\\mathbf{G}\\mathbf{T} = \\mathbf{G} \\mathbf{\\Gamma}, \\quad \\mathbf{c} - \\mathbf{c} = \\mathbf{G} \\boldsymbol{\\beta}, \\quad |\\mathbf{\\Gamma}|\\mathbf{1} + |\\boldsymbol{\\beta}| \\leq \\mathbf{1}.\n\t\\end{equation*}\n\tThe first two equations hold by setting $ \\mathbf{\\Gamma} = \\mathbf{T} $ and $ \\boldsymbol{\\beta} = \\mathbf{0} $. The third equation holds since $ \\sum_{j=1}^{n_r} |t_{i,j}| = 1, \\; \\forall i \\in \\{1, \\cdots, n_g\\} $, if and only if $ |\\mathbf{T}|\\mathbf{1} = \\mathbf{1} $. \\hfill \\hfill \\qed\n\\end{pf}\nThe specific definition of $ \\mathbf{T} $ in \\textbf{Theorem \\ref{ZonoInnerApprox}} produces an inner-approximation of $ Z $ by forming the generators of $ Z_r $ through the addition of the generators in $ Z $. Typically, the largest inner-approximation of $ Z $ is desired. The proposed method for determining $ \\mathbf{T} $ is inspired by the methods for determining outer-approximations of zonotopes presented in \\cite{Kopetzki2017a}. First, let the generators $ \\mathbf{g}_i $ of $ Z $ be arranged such that $ \\| \\mathbf{g}_i \\|_2 \\geq \\| \\mathbf{g}_{i+1} \\|_2, \\; \\forall i \\in \\{1, \\cdots, n_g -1 \\} $. Then partition the generator matrix such that $ \\mathbf{G} = [\\mathbf{G}_1 \\; \\mathbf{G}_2] $ where $ \\mathbf{G}_1 \\in \\mathbb{R}^{n \\times n_r} $ and $ \\mathbf{G}_2 \\in \\mathbb{R}^{n \\times (n_g-n_r)} $. For each generator $ \\mathbf{g}_{2,j} $ in $ \\mathbf{G}_2 $, compute the magnitude of the dot product $ \\alpha_{i,j} = | \\mathbf{g}_{1,i}^T \\mathbf{g}_{2,j}| $ with all generators $ \\mathbf{g}_{1,i} $ in $ \\mathbf{G}_1 $. The goal is to add the generators $ \\mathbf{g}_{2,j} $ to the most aligned generator $ \\mathbf{g}_{1,i} $. Thus, let $ \\mathbf{T} = [t_{i,j}] $ where\n\\begin{equation} \\label{T_zonoapprox}\nt_{i,j} = \\begin{Bmatrix} 1 & \\text{if } i=j\\leq n_r \\\\\n\\frac{1}{\\alpha_{i,j}}\\mathbf{g}_{1,i}^T \\mathbf{g}_{2,j} & \\text{if } \\alpha_{i,j} > \\alpha_{i,k}, \\forall k \\neq j \\\\\n0 & \\text{otherwise}\n\\end{Bmatrix}.\n\\end{equation}\nNote that computing $Z_r$ using \\textbf{Theorem \\ref{ZonoInnerApprox}} and \\eqref{T_zonoapprox} has an overall complexity of $O(nn_g^2 + nn_gn_r)$, where the first term is associated with sorting the generators based on the $2$-norm and the second term is associated with computing the product $\\mathbf{G}_r = \\mathbf{G}\\mathbf{T}$ in \\textbf{Theorem \\ref{ZonoInnerApprox}}.\n\\begin{exmp}\\label{exmp_zon_innerapprox}\n\tConsider the zonotope \n\t\\begin{equation*}\n\tZ = \\left\\{ \\begin{bmatrix} 4 & 3 & -2 & 0.2 & 0.5 \\\\ 0 & 2 & 3 & 0.6 & -0.3 \\end{bmatrix}, \\boldsymbol{0} \\right\\} \\subset \\mathbb{R}^2. \n\t\\end{equation*}\n\tNote that the generators are already arranged in order of decreasing 2-norm. With $ n_g = 5 $, the goal is to determine $ Z_r \\subseteq Z $ such that $ n_r = 3 $. From \\textbf{\\emph{Theorem \\ref{ZonoInnerApprox}}} and \\eqref{T_zonoapprox}, the matrix $\\mathbf{T}$ and the reduced-order zonotope $Z_r$ are \n\t\\begin{equation*}\n\t    \\mathbf{T} = \\begin{bmatrix} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\\\ 0 & 1 & 0 \\\\ 1 & 0 & 0 \\end{bmatrix}, \\; Z_r = \\left\\{ \\begin{bmatrix} \\phantom{-}4.5 & 3.2 & -2 \\\\ -0.3 & 2.6 & \\phantom{-}3 \\end{bmatrix}, \\boldsymbol{0} \\right\\}.\n\t\\end{equation*}\n\n\tFig. \\emph{\\ref{Fig_Zono_Inapprox_Tmat}} confirms $ Z_r \\subseteq Z $ with volume ratio $V_r = 0.97$.\n\tWhile this numerical example resulted in relatively large volume ratio, the reduction in volume is highly dependent on the distribution of generator lengths and the number of generators removed. For 100 randomly generated zonotopes in $ \\mathbb{R}^2 $ with $n_g = 5$, applying \\textbf{\\emph{Theorem \\ref{ZonoInnerApprox}}} and \\eqref{T_zonoapprox}, resulted in all reduced zonotopes satisfying $Z_r \\subseteq Z$ with $n_r = 3$ and mean volume ratio $V_r = 0.84$.\n\\end{exmp}\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=4cm]{Zono_Inner_Approx_Tmat.pdf}\n\t\t\\caption{The inner-approximation of $ Z $ with $ n_g = 5 $ by the reduced-order zonotope $ Z_r $ with $ n_r = 3 $.}\n\t\t\\label{Fig_Zono_Inapprox_Tmat}              \n\t\\end{center}                         \n\\end{figure}\n\n\\subsection{Constrained Zonotopes}\\label{subsec_conzono_innerapprox}\n\nFor constrained zonotopes, a reduced-order inner-approximation $ Z_r $ of $ Z_c $ can be computed based on the set containment criteria for the affine transformation of polytopes in H-Rep (AH-polytopes) developed in \\cite{Sadraddini2019} since AH-polytopes and constrained zonotopes are equivalent.\n\n\\begin{defn} \\label{AHPolytope_def} \\emph{\\cite{Sadraddini2019}}\n\tAn AH-polytope $ X \\subset \\mathbb{R}^n$ is an affine transformation of a H-Rep polytope $ P \\subset \\mathbb{R}^m$ where\n\t\\begin{equation} \\label{AH_def}\n\tX = \\bar{\\mathbf{x}} + \\mathbf{X} P, \\quad \\mathbf{X} \\in \\mathbb{R}^{n \\times m}, \\quad \\bar{\\mathbf{x}} \\in \\mathbb{R}^n.\n\t\\end{equation}\n\\end{defn}\n\nThe following theorem proves the equivalency between constrained zonotopes and AH-polytopes in addition to providing a method to convert constrained zonotopes to AH-polytopes with complexity $O(nn_g^2+n_c^2n_g)$,\nwhere the first term is associated with computing an affine transformation and the second term is associated with computing the basis of $\\mathcal{N}(\\mathbf{A})$ for $Z_c = \\{\\mathbf{G}, \\mathbf{c}, \\mathbf{A}, \\mathbf{b} \\}$.\n\t\\begin{thm} \\label{ConZono_AHpolytope}\n\tA non-empty set $ Z_c \\subset \\mathbb{R}^n $ is a constrained zonotope if and only if it is an AH-polytope.\n\\end{thm}\n\\begin{pf}\n\tTo prove that every AH-polytope is a constrained zonotope, let $ P = \\{ \\mathbf{z} \\in \\mathbb{R}^m \\mid \\mathbf{H} \\mathbf{z} \\leq \\mathbf{k} \\} $. Per Theorem 1 in \\cite{Scott2016}, the set $ P $ can always be represented as a constrained zonotope $ P =  \\{\\mathbf{G}_p, \\mathbf{c}_p, \\mathbf{A}_p, \\mathbf{b}_p\\} $. Thus, from \\eqref{AH_def} and the properties of constrained zonotopes \\eqref{affineMap} and \\eqref{MinkowskiSum}, $ X $ is a constrained zonotope where $ X = \\{\\mathbf{X}\\mathbf{G}_p, \\bar{\\mathbf{x}}+\\mathbf{X}\\mathbf{c}_p, \\mathbf{A}_p, \\mathbf{b}_p\\} $. \n\tTo prove that every constrained zonotope is an AH-polytope, consider $ Z_c  =  \\{\\mathbf{G}, \\mathbf{c}, \\mathbf{A}, \\mathbf{b}\\} $ with $ n_g $ generators and $ n_c $ constraints. If $ n_c = 0 $, $ Z_c = Z = \\{\\mathbf{G},\\mathbf{c}\\} $ is a zonotope and can be represented in AH-polytope form of \\eqref{AH_def} with $ \\bar{\\mathbf{x}} = \\mathbf{c} $, $ \\mathbf{X} = \\mathbf{G} $, and $ P = B_\\infty $. For $ n_c > 0 $, assume that any rank deficiency in $ \\mathbf{A} $ has been detected as a row of zeros in the reduced row echelon form achieved through Gauss-Jordan elimination with full pivoting (see \\cite{Scott2016} for details). Thus, the rank of $ \\mathbf{A} $ is $ n_c $ and there exists $\\mathbf{s} = \\mathbf{A}^{\\dagger}\\mathbf{b} \\in \\mathbb{R}^{n_g}$ and the matrix $ \\mathbf{T} \\in \\mathbb{R}^{n_g \\times (n_g-n_c)} $ with columns that form a basis for $\\mathcal{N}(\\mathbf{A})$. Using the change of variables $ \\boldsymbol{\\xi} = \\mathbf{T} \\bar{\\boldsymbol{\\xi}} + \\mathbf{s} $, the equality constraint $\\mathbf{A}\\boldsymbol{\\xi} = \\mathbf{b}$\\\\ is satisfied for all $ \\bar{\\boldsymbol{\\xi}} \\in \\mathbb{R}^{n_g-n_c} $.\n\n    Hence, $Z_c$ can be expressed as \n\t\\begin{equation*}\n\tZ_c = \\left\\{ \\mathbf{c}+\\mathbf{G}\\mathbf{s} + \\mathbf{G}\\mathbf{T}\\bar{\\boldsymbol{\\xi}} \\mid \\| \\mathbf{T} \\bar{\\boldsymbol{\\xi}} + \\mathbf{s} \\|_\\infty \\leq 1 \\right\\}.\n\t\\end{equation*}\n\n\n\tFurthermore, the norm constraints $ \\| \\mathbf{T} \\bar{\\boldsymbol{\\xi}} + \\mathbf{s} \\|_\\infty \\leq 1 $ can be represented in H-Rep as $P = \\{ \\bar{\\boldsymbol{\\xi}} \\mid \\mathbf{H} \\bar{\\boldsymbol{\\xi}} \\leq \\mathbf{k} \\}$, where \n\t\\begin{equation*}\n\t\\mathbf{H} = \\begin{bmatrix} \\phantom{-}\\mathbf{T} \\\\ -\\mathbf{T} \\end{bmatrix},  \\quad \n\t\\mathbf{k} = \\begin{bmatrix} \\mathbf{1} - \\mathbf{s} \\\\ \\mathbf{1} + \\mathbf{s} \\end{bmatrix}.\n\t\\end{equation*}\n\tThus, with $ \\bar{\\mathbf{x}} = \\mathbf{c}+\\mathbf{G}\\mathbf{s} $ and $ \\mathbf{X} =  \\mathbf{G}\\mathbf{T} $,\n$ Z_c $ is an AH-polytope of the form \\eqref{AH_def}. \t\\hfill \\hfill \\qed\n\\end{pf}\n\n\n\\begin{rem}\n\t\tThe convexity of the constrained zonotope $Z_c = (\\mathbf{G}, \\mathbf{c}, \\mathbf{A}, \\mathbf{b})$ also facilitates representation as a polynomial zonotope $Z_p = (\\mathbf{c}, \\mathbf{G}, \\mathbf{E})$ in Z-Rep \\cite{Kochdumper2019}. However, the reverse is not true.\n\\end{rem}\n\\begin{lem}(Theorem 1 of \\emph{\\cite{Sadraddini2019}}) \\label{AHContainment}\n\tGiven AH-polytopes $ X,Y \\subset \\mathbb{R}^n $ where $ X = \\bar{\\mathbf{x}} + \\mathbf{X} P_x $, $ Y = \\bar{\\mathbf{y}} + \\mathbf{Y} P_y $, $ P_x = \\left\\{ \\mathbf{x} \\in \\mathbb{R}^{n_x} \\mid \\mathbf{H}_x \\mathbf{x} \\leq \\mathbf{f}_x \\right\\} $, and $ P_y = \\left\\{ \\mathbf{y} \\in \\mathbb{R}^{n_y} \\mid \\mathbf{H}_y \\mathbf{y} \\leq \\mathbf{f}_y \\right\\}$, $ X \\subseteq Y $ if there exists $\\mathbf{\\Gamma} \\in \\mathbb{R}^{n_y \\times n_x}, \\boldsymbol{\\beta} \\in \\mathbb{R}^{n_y}$ and $\\mathbf{\\Lambda} \\in \\mathbb{R}^{n_{hy} \\times n_{hx}}_{+}$ such that \n\t\\begin{subequations} \\label{AHContainment_Cons}\n\t\t\\begin{align}\n\t\t\\mathbf{X} &= \\mathbf{Y} \\mathbf{\\Gamma}, & \\bar{\\mathbf{y}} - \\bar{\\mathbf{x}} &= \\mathbf{Y} \\boldsymbol{\\beta}, \\\\ \n\t\t\\mathbf{\\Lambda} \\mathbf{H}_x &= \\mathbf{H}_y \\mathbf{\\Gamma}, & \\mathbf{\\Lambda} \\mathbf{f}_x &\\leq \\mathbf{f}_y + \\mathbf{H}_y \\boldsymbol{\\beta}. \n\t\t\\end{align}\n\t\\end{subequations}\n\\end{lem}\n\nTo achieve a reduced-order inner-approximation $ Z_r $ of constrained zonotope $ Z_c $, \\textbf{Theorem \\ref{ConZono_AHpolytope}} can be used to convert both $ Z_r $ and $ Z_c $ in to AH-polytopes while \\textbf{Lemma \\ref{AHContainment}} can be used to ensure $ Z_r \\subseteq Z_c $. Assuming $ Z_c $ is known, consider $ Z_r =  \\{\\mathbf{G}_r \\mathbf{\\Phi}, \\mathbf{c}_r, \\mathbf{A}_r, \\mathbf{b}_r\\} $ where $ \\mathbf{\\Phi} = diag(\\boldsymbol{\\phi}) $ is a scaling matrix with $ \\phi_i > 0, \\forall i \\in \\{1,\\cdots, n_{gr} \\} $. Assuming $ \\mathbf{G}_r $, $ \\mathbf{A}_r $, and $ \\mathbf{b}_r $ are known, the following optimization problem can be formulated with $4n_{gr}^2 + n_{gr} + (n_g - n_c)(1 + n_{gr} - n_{cr}) + n$ decision variables that maximizes the $p = 1$, $2$, or $\\infty$ norm of the diagonal elements $\\phi$ of the scaling matrix $\\mathbf{\\Phi}$ by solving\n\\begin{subequations}\\label{Conzono_containment}\n\t\\begin{align}\n\t& \\underset{\\mathbf{\\Phi}, \\mathbf{\\Gamma}, \\boldsymbol{\\beta}, \\mathbf{\\Lambda}, \\mathbf{c}_r}{\\text{max}} \\mkern 20mu ||\n\t\\mathbf{\\phi}\n\t||_p, \\\\\n\t& \\text{s.t.} \\nonumber \\\\\n\t& (\\mathbf{c} + \\mathbf{G}\\mathbf{s}) - (\\mathbf{c}_r + \\mathbf{G}_r\\mathbf{\\Phi}\\mathbf{s}_r) = \\mathbf{G}\\mathbf{T}\\boldsymbol{\\beta}, \\\\\n\t& \\mathbf{G}_r\\mathbf{\\Phi}\\mathbf{T}_r = \\mathbf{G}\\mathbf{T}\\mathbf{\\Gamma},\\quad  \\mathbf{\\Lambda}\\begin{bmatrix}\\phantom{-}\\mathbf{T}_r \\\\ \\mathbf{-T}_r\\end{bmatrix} = \\begin{bmatrix} \\phantom{-}\\mathbf{T} \\\\ -\\mathbf{T} \\end{bmatrix}\\mathbf{\\Gamma}, \\\\\n\t& \\mathbf{\\Lambda}\\begin{bmatrix} \\mathbf{1} - \\mathbf{s}_r \\\\ \\mathbf{1} + \\mathbf{s}_r \\end{bmatrix} \\leq \\begin{bmatrix} \\mathbf{1} - \\mathbf{s} \\\\ \\mathbf{1} + \\mathbf{s} \\end{bmatrix} + \\begin{bmatrix} \\phantom{-}\\mathbf{T} \\\\ -\\mathbf{T}\\end{bmatrix}\\boldsymbol{\\beta},\n\t\\end{align}\n\\end{subequations}\nwith parameters $\\mathbf{s} = \\mathbf{A}^{\\dagger}\\mathbf{b} \\in \\mathbb{R}^{n_g} $, $ \\mathbf{s}_r = \\mathbf{A}_r^{\\dagger}\\mathbf{b}_r \\in \\mathbb{R}^{n_{gr}} $, and matrices $ \\mathbf{T} \\in \\mathbb{R}^{n_g \\times (n_g - n_c)}, \\mathbf{T}_r \\in \\mathbb{R}^{n_{gr} \\times (n_{gr} - n_{cr})} $ with columns that form bases for $ \\mathcal{N}(\\mathbf{A}) $ and $ \\mathcal{N}(\\mathbf{A}_r) $, respectively.\nNote that the majority of the decision variables in \\eqref{Conzono_containment} come from the matrices $\\mathbf{\\Gamma} \\in \\mathbb{R}^{(n_g - n_c) \\times (n_{gr} - n_{cr})}$ and $\\mathbf{\\Lambda} \\in \\mathbb{R}^{2n_{gr} \\times 2n_{gr}}_{+}$.\nWhile this procedure applies to any $ Z_r $, the process discussed in Section \\ref{Sec_Redundancy} can be used to compute $ Z_r $ by removing exactly one constraint and one generator from $ Z_c $. For the case where $ Z_c $ satisfies the conditions in \\textbf{Theorem~\\ref{Redundant_ConZono}}, the $ r^{th} $ constraint and the $ c^{th} $ generators were chosen such that $ R_{r,c} \\subseteq [-1,1] $ and thus an exact reduced-order representation was achieved with $ Z_r = Z_c $. To achieve further reduction through the inner-approximation of $ Z_c $, the same procedure from Section \\ref{Sec_Redundancy} can be applied by choosing appropriate indices and scaling $ Z_r $ via optimization while enforcing $ Z_r \\subseteq Z_c $ using the constraints from \\eqref{Conzono_containment}.\nSince $ R_j = [\\rho_j^L, \\rho_j^U] $ represents the range of $ \\xi_j $ if the constraints $ | \\xi_j | \\leq 1 $ were omitted \\cite{Scott2016}, the $ c^{th} $ generator should be removed that minimizes $ \\max(|\\rho_j^L|, |\\rho_j^U|) $. Once $ c $ is chosen, $ r $ should be chosen such that the entry in the $ (r,c) $ position of $ \\mathbf{A}_r $ has the largest absolute value of all entries in the $ c^{th} $ column. \n\n\\begin{exmp} \\label{example_innerApprox}\n\tConsider the constrained zonotope $ Z_c $ shown in Fig. \\ref{Fig_ConZono_Inapprox} where\n\t\\begin{subequations}\n\t\t\\setlength\\arraycolsep{2pt}\n\t\t\\begin{align*}\n\t\tZ_c = \\Bigg \\{ &\\begin{bmatrix} -1 & \\phantom{-}3 & \\phantom{-}4 & 0 & 0 \\\\ \\phantom{-}4 & -2 & -5 & 0 & 0 \\end{bmatrix},  \\begin{bmatrix} 0 \\\\ 0 \\end{bmatrix}, \\\\\n\t\t&\\begin{bmatrix} -1 & \\phantom{-}3 & \\phantom{-}4 & 6.5 & 0 \\\\ \\phantom{-}4 & -2 & -5 & 0 & 8 \\end{bmatrix}, \\begin{bmatrix} -1.5 \\\\ -3 \\end{bmatrix} \\Bigg \\}.\n\t\t\\end{align*}\n\t\\end{subequations}\n\t\\setcounter{equation}{\\value{equation}-1}\n\tFirst, Gauss-Jordan elimination with full pivoting was applied to $ Z_c $, followed by the transformation in \\emph{\\textbf{Theorem \\ref{Redundant_ConZono}}} by picking the $ c^{th} $ generator that minimizes $ \\max(|\\rho_j^L|, |\\rho_j^U|) $ and the $ r^{th} $ row with the largest entry in $ c^{th} $ column of $ \\mathbf{A} $. Then an LP was formulated and solved using the constraints from \\eqref{AHContainment_Cons} and a cost function that maximized $ \\| \\boldsymbol{\\phi} \\|_\\infty $. The resulting reduced-order zonotope $ Z_r $ is shown in Fig. \\ref{Fig_ConZono_Inapprox} where \n\t\\begin{subequations}\n\t\t\\setlength\\arraycolsep{2pt}\n\t\t\\begin{align*}\n\t\tZ_r = \\Bigg \\{ &\\begin{bmatrix} 0 & \\phantom{-}3.17 & \\phantom{-}2.38 & -0.79 \\\\ 0 & -3.97 & -1.59 & \\phantom{-}3.17 \\end{bmatrix},  \\begin{bmatrix} -1.34 \\\\ \\phantom{-}1.03 \\end{bmatrix}, \\\\\n\t\t&\\begin{bmatrix} 1 & -0.63 & -0.25 & \\phantom{-}0.50 \\end{bmatrix}, \\begin{bmatrix} -0.38 \\end{bmatrix} \\Bigg \\}.\n\t\t\\end{align*}\n\t\\end{subequations}\n\t\\setcounter{equation}{\\value{equation}-1}\nUsing a similar approach, Fig, \\ref{Fig_ConZono_Inapprox} also shows the inner-approximations of $ Z_c $ by zonotope $ Z $ and interval set $ B $ where\n\t\\begin{subequations}\n\t\t\\setlength\\arraycolsep{2pt}\n\t\t\\begin{align*}\n\t\tZ &= \\left\\{ \\begin{bmatrix} \\phantom{-}2.31 & \\phantom{-}1.93 & -0.27 \\\\ -2.84 & -0.94 & \\phantom{-} 2.57\\end{bmatrix}, \\begin{bmatrix} \\phantom{-}0.49 \\\\ -1.35 \\end{bmatrix} \\right\\}, \\\\\n\t\tB &= \\left\\{ \\begin{bmatrix} 2 & 0 \\\\ 0 & 2 \\end{bmatrix}, \\begin{bmatrix} \\phantom{-}2.55 \\\\ -3.18 \\end{bmatrix} \\right\\}.\n\t\t\\end{align*}\n\t\\end{subequations}\n\t\\setcounter{equation}{\\value{equation}-1}\n\tTo compute $ Z $, the equality constraints from $ Z_c $ were removed via the same change of variables used in the proof of \\emph{\\textbf{Theorem~\\ref{ConZono_AHpolytope}}}\n\tTypically this would result in an outer-approximation of $ Z_c $, however the scaling matrix $ \\mathbf{\\Phi} $ is used to reduce the length of each generator such that $ Z \\subseteq Z_c $. For the interval set $ B $, the generator matrix is initialized as the identity matrix and then scaled by $ \\mathbf{\\Phi} $. The resulting volume ratios with respect to $ Z_c $ are $ V_r = 0.86 $, $ V_r = 0.83 $, $ V_r = 0.46 $ for $ Z_r $, $ Z $, and $ B $, respectively.  \n\tRepeating this process for 100 randomly generated constrained zonotopes with $ 4 \\leq n_g \\leq 20 $ and $ 1 \\leq n_c \\leq \\frac{1}{2}n_g $, Fig. \\ref{Fig_ConZono_Inapprox_Rand} shows the volume ratios for constrained zonotope, zonotope, and interval set inner-approximations. Both constrained zonotopes and zonotopes provide better approximations compared to interval sets while constrained zonotopes provide only a slightly higher mean volume ratio.\n\\end{exmp}\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=8.6cm]{ConZono_Inner_Approx.pdf}\n\t\t\\caption{Left: The inner-approximation of $ Z_c $ by a constrained zonotope $ Z_r $ with one less generator and constraint. Right: The inner-approximation of $ Z_c $ by a zonotope $ Z $ and an interval set $ B $.}\n\t\t\\label{Fig_ConZono_Inapprox}             \n\t\\end{center}      \n\\end{figure}\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=5cm]{ConZono_Inner_Approx_Rand.pdf}\n\t\t\\caption{The volume ratios for the inner-approximation of 100 randomly generated constrained zonotopes by a constrained zonotope $ Z_r $ with one less generator and constraint, a zonotope $ Z $, and an interval set $ B $. The red crosses denote outliers that do not fit the box plot distribution.}\n\t\t\\label{Fig_ConZono_Inapprox_Rand} \t\\end{center}               \n\\end{figure}\n\n\\section{Convex Hulls} \\label{Sec_ConvexHull}\nThis section computes the CG-Rep of the convex hull of two constrained zonotopes $Z_1, Z_2 \\subset \\mathbb{R}^n$ with complexity $O(n+n_{c1}+n_{c2})$ where $n_{c1}$ and $n_{c2}$ are the number of constraints in $Z_1$ and $Z_2$, respectively. Since zonotopes are a subset of constrained zonotopes with $ n_c = 0 $, the following result also applies to zonotopes.\n\n\\begin{defn} \\label{CH_def} \\emph{\\cite{Tiwary2008}} \n\tThe convex hull of the union of two polytopes $ P_1, P_2 \\subset \\mathbb{R}^n $ is defined as\n\t\\begin{equation*}\n\tCH(P_1 \\cup P_2) \\triangleq \\left\\{\\mathbf{x}_1 \\lambda + \\mathbf{x}_2 (1-\\lambda) \\mid \\begin{matrix} \\mathbf{x}_1 \\in P_1,\\\\ \\mathbf{x}_2 \\in P_2, \\\\ 0 \\leq \\lambda \\leq 1 \\end{matrix}\\right\\}.\n\t\\end{equation*}\n\\end{defn}\n\\begin{thm} \\label{convexHull}\n\tThe convex hull of the union of two constrained zonotopes $ Z_1 = \\{\\mathbf{G}_1,\\mathbf{c}_1,\\mathbf{A}_1,\\mathbf{b}_1\\} \\subset \\mathbb{R}^n $ and $ Z_2 = \\{\\mathbf{G}_2,\\mathbf{c}_2,\\mathbf{A}_2,\\mathbf{b}_2\\} \\subset \\mathbb{R}^n $ is a constrained zonotope $ Z_h = \\{\\mathbf{G}_h,\\mathbf{c}_h,\\mathbf{A}_h,\\mathbf{b}_h\\} $ where\n\t\\begin{equation*}\n\t\\begin{matrix*}[l]\n\t\\mathbf{G}_h = \\begin{bmatrix}\n\t\\mathbf{G}_1 & \\mathbf{G}_2 & \\frac{\\mathbf{c}_1-\\mathbf{c}_2}{2} & \\mathbf{0\n\t\\end{bmatrix}, & \n\t\n\t\\mathbf{c}_h = \\frac{\\mathbf{c}_1+\\mathbf{c}_2}{2}, \\\\\n\t\n\t\\mathbf{A}_h = \\begin{bmatrix}\n\t\\mathbf{A}_1 & \\mathbf{0} & -\\frac{\\mathbf{b}_1}{2} & \\mathbf{0} \\\\\n\t\\mathbf{0} & \\mathbf{A}_2 & \\phantom{-}\\frac{\\mathbf{b}_2}{2} & \\mathbf{0} \\\\\n\t\\mathbf{A}_{3,1} & \\mathbf{A}_{3,2} & \\mathbf{A}_{3,0} & \\mathbf{I}\n\t\\end{bmatrix}, &\n\t\n\t\\mathbf{b}_h = \\begin{bmatrix} \\phantom{-}\\frac{1}{2} \\mathbf{b}_1 \\\\ \\phantom{-}\\frac{1}{2} \\mathbf{b}_2 \\\\ -\\frac{1}{2} \\mathbf{1}\t\\end{bmatrix}, \\\\\n\t\n\t\\mathbf{A}_{3,1} = \\begin{bmatrix} \\phantom{-} \\mathbf{I} \\\\ -\\mathbf{I} \\\\ \\phantom{-} \\mathbf{0} \\\\ \\phantom{-} \\mathbf{0} \\end{bmatrix},\n\t\n\t\\mathbf{A}_{3,2} = \\begin{bmatrix} \\phantom{-} \\mathbf{0} \\\\ \\phantom{-}\\mathbf{0} \\\\ \\phantom{-}\\mathbf{I} \\\\ -\\mathbf{I} \\end{bmatrix}, &\n\t\n\t\\mathbf{A}_{3,0} = \\begin{bmatrix} -\\frac{1}{2}\\mathbf{1} \\\\ -\\frac{1}{2}\\mathbf{1} \\\\ \\phantom{-}\\frac{1}{2}\\mathbf{1} \\\\ \\phantom{-}\\frac{1}{2}\\mathbf{1} \\end{bmatrix}.\n\t\\end{matrix*}\n\t\\end{equation*}\n\\end{thm}\n\\begin{pf}\n    Considering any element $\\mathbf{x} \\in Z_h$, it is to be proven that $\\mathbf{x} \\in CH(Z_1 \\cup Z_2)$. By the definition of $Z_h$, $\\exists \\; \\boldsymbol{\\xi}_1 \\in \\mathbb{R}^{n_{g1}}, \\boldsymbol{\\xi}_2 \\in \\mathbb{R}^{n_{g2}}, \\xi_0 \\in \\mathbb{R}$, and $\\boldsymbol{\\xi}_s \\in \\mathbb{R}^{2(n_{g1} + n_{g2})}$ such that\n    \\begin{subequations}\\label{eq_Zh_exist_all}\n        \\begin{align}\n            & \\scriptsize{\\mathbf{x} = \\mathbf{G}_1\\boldsymbol{\\xi}_1+\\mathbf{G}_2\\boldsymbol{\\xi}_2+\\frac{\\mathbf{c}_1-\\mathbf{c}_2}{2}\\xi_0+\\mathbf{0}\\boldsymbol{\\xi}_s +\\frac{\\mathbf{c}_1+\\mathbf{c}_2}{2},}  \\label{eq_Zh_exist_element} \\\\\n            & ||\\boldsymbol{\\xi}_1||_{\\infty} \\leq 1, \\; ||\\boldsymbol{\\xi}_2||_{\\infty} \\leq 1, \\; |\\xi_0| \\leq 1, \\; ||\\boldsymbol{\\xi}_s||_{\\infty} \\leq 1, \\label{eq_Zh_inf_norm_const} \\\\\n            & \\mathbf{A}_h [\n            \\boldsymbol{\\xi}_1^T \\; \\boldsymbol{\\xi}_2^T \\; \\xi_0 \\; \\boldsymbol{\\xi}_s^T\n            ]^T = \\mathbf{b}_h. \\label{eq_Zh_exist_cons}\n        \\end{align}\n    \\end{subequations}\n   \n    To prove $\\mathbf{x} \\in CH(Z_1 \\cup Z_2)$ requires the existence of elements $\\mathbf{z}_1$, $\\mathbf{z}_2 \\in \\mathbb{R}^n$, $\\lambda \\in \\mathbb{R}, \\boldsymbol{\\xi}_1^{'} \\in \\mathbb{R}^{n_{g1}}$, and $\\boldsymbol{\\xi}_2^{'} \\in \\mathbb{R}^{n_{g2}}$ such that\n\\begin{subequations}\\label{eq_CH_Z1_Z2_exist_all}\n    \\begin{align}\n        & \\mathbf{x} = \\mathbf{z}_1\\lambda + \\mathbf{z}_2(1 - \\lambda), \\quad 0 \\leq \\lambda \\leq 1, \\label{eq_CH_Z1_Z2_exp_base} \\\\ \n        & \\mathbf{z}_1 = \\mathbf{c}_1 + \\mathbf{G}_1\\boldsymbol{\\xi}_1^{'}, \\quad ||\\boldsymbol{\\xi}_1^{'}||_{\\infty} \\leq 1, \\quad \\mathbf{A}_1\\boldsymbol{\\xi}_1^{'} = \\mathbf{b}_1, \\label{eq_Zh_exist_element_Z1}\\\\\n        & \\mathbf{z}_2 = \\mathbf{c}_2 + \\mathbf{G}_2\\boldsymbol{\\xi}_2^{'}, \\quad ||\\boldsymbol{\\xi}_2^{'}||_{\\infty} \\leq 1, \\quad \\mathbf{A}_2\\boldsymbol{\\xi}_2^{'} = \\mathbf{b}_2. \\label{eq_Zh_exist_element_Z2}\n    \\end{align}\n\\end{subequations}\nThis is shown by defining $\\lambda$, $\\boldsymbol{\\xi}_1^{'}$, and $\\boldsymbol{\\xi}_2^{'}$ as\n\\begin{equation}\\label{eq_cvxhull_lambda_xi_varsubs}\n    \\lambda = \\frac{1}{2}(1 + \\xi_0), \\quad \\boldsymbol{\\xi}_1 = \\boldsymbol{\\xi}_1^{'}\\lambda, \\quad \\boldsymbol{\\xi}_2 = \\boldsymbol{\\xi}_2^{'} (1 - \\lambda).\n\\end{equation}\nBy rearranging \\eqref{eq_Zh_exist_element}, substituting using the variable definitions in \\eqref{eq_cvxhull_lambda_xi_varsubs}, and then rearranging to simplify using the definitions for $\\mathbf{z}_1$ and $\\mathbf{z}_2$ from \\eqref{eq_Zh_exist_element_Z1} and \\eqref{eq_Zh_exist_element_Z2}, the expression for $\\mathbf{x}$ from \\eqref{eq_CH_Z1_Z2_exp_base} can be established as\n\\begin{subequations}\n    \\begin{align}\n        \\mathbf{x} &= \\frac{\\mathbf{c}_1}{2}(1 + \\xi_0) + \\mathbf{G}_1\\boldsymbol{\\xi}_1 + \\frac{\\mathbf{c}_2}{2}(1 - \\xi_0) + \\mathbf{G}_2\\boldsymbol{\\xi}_2, \\label{eq_Zh_exist_element_exp1}\\\\ \n        &= \\mathbf{c}_1\\lambda + \\mathbf{G}_1\\boldsymbol{\\xi}_1^{'}\\lambda + \\mathbf{c}_2(1 - \\lambda) + \\mathbf{G}_2\\boldsymbol{\\xi}_2^{'}(1 - \\lambda), \\label{eq_Zh_exist_element_exp2}\\\\ \n        &= \\mathbf{z}_1\\lambda + \\mathbf{z}_2(1 - \\lambda).\n    \\end{align}\n\\end{subequations}\nSince $|\\xi_0| \\leq 1$, the definition for $\\lambda$ in \\eqref{eq_cvxhull_lambda_xi_varsubs} results in $0 \\leq \\lambda \\leq 1$.\nFrom the definition of $\\mathbf{A}_h$ and $\\mathbf{b}_h$, the first two sets of equality constraints are\n\\begin{equation}\\label{eq_cvxhull_Zh_cons}\n    \\mathbf{A}_1\\boldsymbol{\\xi}_1 - \\frac{\\mathbf{b}_1}{2}\\xi_0 = \\frac{1}{2}\\mathbf{b}_1, \\quad \\mathbf{A}_2\\boldsymbol{\\xi}_2 - \\frac{\\mathbf{b}_2}{2}\\xi_0 = \\frac{1}{2}\\mathbf{b}_2.\n\\end{equation}\nUsing \\eqref{eq_cvxhull_lambda_xi_varsubs}, \\eqref{eq_cvxhull_Zh_cons} simplifies to\n\\begin{equation*}\n    \\mathbf{A}_1\\boldsymbol{\\xi}_1^{'} = \\mathbf{b}_1, \\; \\forall \\; \\lambda \\in (0, 1], \\quad \\mathbf{A}_2\\boldsymbol{\\xi}_2^{'} = \\mathbf{b}_2, \\; \\forall \\; \\lambda \\in [0, 1).\n\\end{equation*}\nNote that if $\\lambda = 0$, then $\\mathbf{z}_1$ does not affect $\\mathbf{x}$ or $\\boldsymbol{\\xi}_1$ and an arbitrary value of $\\boldsymbol{\\xi}_1^{'}$ can be chosen satisfying the infinity norm and equality constraints from \\eqref{eq_Zh_exist_element_Z1}. Similarly, if $\\lambda = 1$, an arbitrary value of $\\boldsymbol{\\xi}_2^{'}$ can be chosen satisfying constraints from \\eqref{eq_Zh_exist_element_Z2}. Otherwise, the norm constraints $||\\boldsymbol{\\xi}_1^{'}||_{\\infty} \\leq 1$ and $||\\boldsymbol{\\xi}_2^{'}||_{\\infty} \\leq 1$ are guaranteed since $||\\boldsymbol{\\xi}_1||_{\\infty} \\leq 1$, $||\\boldsymbol{\\xi}_2||_{\\infty} \\leq 1$, and $0 \\leq \\lambda \\leq 1$. Thus, $\\mathbf{x} \\in CH(Z_1 \\cup Z_2)$.\n\nNext, considering any $\\mathbf{x} \\in CH(Z_1 \\cup Z_2)$, it is to be proven that $\\mathbf{x} \\in Z_h$.\nBy \\textbf{Definition \\ref{CH_def}}, there exists elements $\\mathbf{z}_1$, $\\mathbf{z}_2 \\in \\mathbb{R}^n$, $\\lambda \\in \\mathbb{R}, \\boldsymbol{\\xi}_1^{'} \\in \\mathbb{R}^{n_{g1}}$, and $\\boldsymbol{\\xi}_2^{'} \\in \\mathbb{R}^{n_{g2}}$ such that \\eqref{eq_CH_Z1_Z2_exp_base}-\\eqref{eq_Zh_exist_element_Z2} hold.\nTo prove $\\mathbf{x} \\in Z_h$ requires the existence of variables $\\boldsymbol{\\xi}_1 \\in \\mathbb{R}^{n_{g1}}$, $\\boldsymbol{\\xi}_2 \\in \\mathbb{R}^{n_{g2}}$, $\\xi_0 \\in \\mathbb{R}$, $\\boldsymbol{\\xi}_s \\in \\mathbb{R}^{2(n_{g1} + n_{g2})}$ such that \\eqref{eq_Zh_exist_element}-\\eqref{eq_Zh_exist_cons} hold.\nConsider the following definitions for variables $\\boldsymbol{\\xi}_1$, $\\boldsymbol{\\xi}_2$, ${\\xi}_0$, and $\\boldsymbol{\\xi}_s$ with\n\\begin{subequations}\\label{eq_xi1_xi2_xi_0_xi_s_assum_all}\n    \\begin{align}\n        & \\boldsymbol{\\xi}_1 = \\boldsymbol{\\xi}_1^{'}\\lambda, \\quad \\boldsymbol{\\xi}_2 = \\boldsymbol{\\xi}_2{'}(1-\\lambda), \\quad \\xi_0 = 2\\lambda -1, \\label{eq_xi1_xi2_xi_0_assum}\\\\ \n        & \\boldsymbol{\\xi}_s = -\\frac{1}{2}\\mathbf{1} - (\\mathbf{A}_{31}\\boldsymbol{\\xi}_1 + A_{32}\\boldsymbol{\\xi}_2 + \\mathbf{A}_{30}\\xi_0). \\label{eq_xi_s_assum}\n    \\end{align}\n\\end{subequations} \nUsing \\eqref{eq_xi1_xi2_xi_0_assum} and \\eqref{eq_xi_s_assum}, it can be readily shown that the equality constraints in \\eqref{eq_CH_Z1_Z2_exp_base}-\\eqref{eq_Zh_exist_element_Z2} can be rewritten to achieve \\eqref{eq_Zh_exist_element} and \\eqref{eq_Zh_exist_cons}. Thus, all that remains is to show $\\big|\\big| \\; [\n\\boldsymbol{\\xi}_1^T \\; \\boldsymbol{\\xi}_2^T \\; \\xi_0 \\; \\boldsymbol{\\xi}_s^T ]^T\\; \\big|\\big|_{\\infty} \\leq 1$. Since $0 \\leq \\lambda \\leq 1$ holds, \n\\begin{equation*}\n||\\boldsymbol{\\xi}_1^{'}||_{\\infty} \\geq ||\\boldsymbol{\\xi}_1^{'}\\lambda||_{\\infty} = ||\\boldsymbol{\\xi}_1||_{\\infty},\n\\end{equation*}\nis satisfied. By \\eqref{eq_Zh_exist_element_Z1}, $||\\boldsymbol{\\xi}_1^{'}||_{\\infty} \\leq 1$ implies $||\\boldsymbol{\\xi}_1||_{\\infty} \\leq 1$. Similarly, it can be shown that $||\\boldsymbol{\\xi}_2||_{\\infty} \\leq 1$. Using the definition of $\\xi_0$ from \\eqref{eq_xi1_xi2_xi_0_assum} and $0 \\leq \\lambda \\leq 1$ proves that $|\\xi_0| \\leq 1$. Finally, using the definition of $\\boldsymbol{\\xi}_s$ from \\eqref{eq_xi_s_assum} and interval arithmetic, it can be shown that \n\\begin{equation*}\n||\\boldsymbol{\\xi}_1||_{\\infty} \\leq 1, \\; ||\\boldsymbol{\\xi}_2||_{\\infty} \\leq 1, \\; |\\boldsymbol{\\xi}_0|\\leq 1 \\implies ||\\boldsymbol{\\xi}_s||_{\\infty} \\leq 1.\n\\end{equation*} Thus, $\\forall \\; \\mathbf{x} \\in CH(Z_1 \\cup Z_2)$, $\\mathbf{x} \\in Z_h$. \\hfill \\hfill \\qed\n\\end{pf}\n\nThe resulting constrained zonotope $Z_h$ obtained using \\textbf{Theorem \\ref{convexHull}} has $n_{gh} = 3(n_{g1}+n_{g2})+1$ generators and $n_{ch} = n_{c1}+n_{c2}+2(n_{g1}+n_{g2})$ constraints.\n\n\n\\begin{exmp}\n\tFor the zonotopes\n\t\\begin{subequations} \n\t\t\\begin{align*}\n\t\tZ_1 &= \\left\\{ \\begin{bmatrix} 0 & 1 & 0 \\\\ 1 & 1 & 2 \\end{bmatrix}, \\begin{bmatrix} 0 \\\\ 0\t\\end{bmatrix} \\right\\}, \\\\ \n\t\tZ_2 &= \\left\\{ \\begin{bmatrix} -0.5 & 1 & -2\\\\ \\phantom{-}0.5 & 0.5 & 1.5\\end{bmatrix}, \\begin{bmatrix} -5 \\\\ \\phantom{-}0\t\\end{bmatrix} \\right\\},\n\t\t\\end{align*}\n\t\\end{subequations}\n\t\\setcounter{equation}{\\value{equation}-1}\n\tFig. \\ref{Fig_CVX_Hull} shows the convex hull $ Z_h = CH(Z_1 \\cup Z_2) $ with $n_g = 19 $ generators and $ n_c = 12 $ constraints, as computed using \\textbf{\\emph{Theorem \\ref{convexHull}}}. Fig. \\ref{Fig_CVX_Hull} also shows the convex hull $ Z_{ch} = CH(Z_{c1} \\cup Z_{c2}) $ with $n_g = 25 $ generators and $ n_c =~18 $ constraints, where $ Z_{c1} = Z_1 \\cap H_{1-} $, $ Z_{c2} = Z_2 \\cap H_{2-} $, $ H_{1-} = \\{\\mathbf{z} \\mid [1 \\; 1] \\mathbf{z} \\leq 0 \\} $, and $ H_{2-} = \\{\\mathbf{z}\\mid [-2.5 \\; 1] \\mathbf{z} \\leq 9.5 \\} $.\n\\end{exmp}\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=8.6cm]{Convex_Hull.pdf}\n\t\t\\caption{Left: The convex hull $ Z_h $ of zonotopes $ Z_1 $ and $ Z_2 $. \n\t\t\tRight: The convex hull $ Z_{ch} $ of constrained zonotopes $ Z_{c1} $ and $ Z_{c2} $, where each constrained zonotope is a zonotope-halfspace intersection corresponding to the shown hyperplanes.} \n\t\t\\label{Fig_CVX_Hull}                      \n\t\\end{center}                         \n\\end{figure}\n\n\n\\section{Robust Positively Invariant (RPI) Sets} \\label{Sec_RPI}\nThis section provides both iterative and one-step optimization based methods for computing approximations of the minimal robust positively invariant set using zonotopes. Consider the autonomous discrete-time linear time-invariant system\n\\begin{equation} \\label{autoSys}\n\\mathbf{x}_{k+1} = \\mathbf{A} \\mathbf{x}_k + \\mathbf{w}_k,\n\\end{equation}\nwhere $ \\mathbf{x}_k \\in \\mathbb{R}^n $, $ \\mathbf{A} \\in \\mathbb{R}^{n \\times n} $ is a strictly stable matrix, and $ \\mathbf{w}_k \\in W \\subset \\mathbb{R}^n $, where $ W $ is a convex and compact set containing the origin.  \n\n\\begin{defn} \\emph{\\cite{Blanchini1999}}\n\tThe set $ \\Omega \\subset \\mathbb{R}^n $ is a robust positively invariant (RPI) set of \\eqref{autoSys} if and only if $ \\mathbf{A} \\Omega \\oplus W \\subseteq \\Omega $.\n\\end{defn}\n\n\\begin{defn} \\emph{\\cite{Rakovic2005}}\n\tThe minimal RPI (mRPI) set $ F_\\infty $ of \\eqref{autoSys} is the RPI set that is contained in every closed RPI set of \\eqref{autoSys} and is given by\n\t\\begin{equation} \\label{RPI_infsum}\n\tF_\\infty = \\bigoplus_{i=0}^{\\infty} \\mathbf{A}^i W.\n\t\\end{equation}\n\\end{defn}\n\n\\subsection{Iterative Method}\nUnless specific conditions are met, such as $ \\mathbf{A} $ being nilpotent, the infinite sequence of Minkowski sums in \\eqref{RPI_infsum} makes it impossible to compute $ F_\\infty $ exactly. Thus, outer-approximations of the mRPI set are typically used. An iterative approach is developed in \\cite{Rakovic2005} that computes the RPI set $ F(\\alpha,s) $ such that $ F_\\infty \\subseteq F(\\alpha,s) \\subseteq F_\\infty \\oplus \\epsilon B_\\infty $, where $ \\epsilon $ is a user defined bound on the error of the approximation with $ s \\in \\mathbb{N}_+ $, $ \\alpha \\in [0,1) $ such that $ \\mathbf{A}^s W \\subseteq \\alpha W $. Starting at $ s = 0 $, the approach increments $ s $ until the approximation error is less than $ \\epsilon $, at which point $ F_s $ is computed as \n\\begin{equation} \\label{RPI_finitesum}\nF_s = \\bigoplus_{i=0}^{s} \\mathbf{A}^i W,\n\\end{equation}\nand $ F(\\alpha,s) = (1-\\alpha)^{-1} F_s $. The iterative algorithm in \\cite{Rakovic2005} requires use of multiple support functions at each iteration. When $ W $ is expressed in H-Rep, an LP must be solved for each support function calculation. As discussed in \\cite{Trodden2016_OneStep}, computing $ F(\\alpha,s) $ using this method may require the solution of thousands of LPs, even for a system with only two states. As briefly mentioned in Remark 3 in \\cite{Rakovic2005}, if $ W $ is expressed in G-Rep, then the support function can be evaluated algebraically without the use of an LP, significantly reducing the computational cost. Thus, the use of zonotopes for RPI set calculations provides both improved scalability and reduced computational cost for the Minkowski sums in \\eqref{RPI_finitesum} and by removing the need to solve LPs.\n\n\\subsection{One-step Optimization Method}\n\nAs an alternative for the iterative method in \\cite{Rakovic2005}, a one-step method for computing an outer-approximation of the mRPI set is presented in \\cite{Trodden2016_OneStep}. By expressing the RPI set in H-Rep, this method requires solving a single LP, assuming both the number and normal vectors of the hyperplanes associated with each halfspace inequality are provided \\emph{a priori}. Inspired by this approach, the following presents a similar one-step method\nfor computing an outer-approximation of the mRPI set using G-Rep, where the generator vectors are predetermined.\n\\begin{thm} \\label{RPI_OneStep}\n\tThe zonotope $ Z = \\{\\mathbf{G}\\mathbf{\\Phi},\\mathbf{c}\\} \\subset \\mathbb{R}^n $, with $ \\mathbf{\\Phi} = diag(\\boldsymbol{\\phi}), \\phi_i > 0, \\forall i \\in \\{1, \\cdots, n_g \\} $, is an RPI set of \\eqref{autoSys} if $ W = \\{\\mathbf{G}_w,\\mathbf{c}_w\\} $ and there exists $ \\mathbf{\\Gamma}_1 \\in \\mathbb{R}^{n_g \\times n_g} $, $ \\mathbf{\\Gamma}_2 \\in \\mathbb{R}^{n_g \\times n_w} $, and $ \\boldsymbol{\\beta} \\in \\mathbb{R}^{n_g} $ such that\n\t\\begin{subequations} \\label{RPI_conditions}\n\t\t\\begin{align} \n\t\t\\mathbf{A}\\mathbf{G}\\mathbf{\\Phi} &= \\mathbf{G}\\mathbf{\\Gamma}_1, \\label{RPI_condition1}\\\\\n\t\t\\mathbf{G}_w &= \\mathbf{G}\\mathbf{\\Gamma}_2, \\label{RPI_condition2}\\\\\n\t\t(\\mathbf{I} - \\mathbf{A})\\mathbf{c} - \\mathbf{c}_w &= \\mathbf{G}\\boldsymbol{\\beta}, \\label{RPI_condition3} \\\\\n\t\t|\\mathbf{\\Gamma}_1|\\mathbf{1} + |\\mathbf{\\Gamma}_2|\\mathbf{1} + |\\boldsymbol{\\beta}| &\\leq \\mathbf{\\Phi}\\mathbf{1}. \\label{RPI_condition4}\n\t\t\\end{align}\n\t\\end{subequations}\n\\end{thm}\n\\begin{pf}\n\tThe proof requires showing that \\eqref{RPI_conditions} enforces the zonotope containment conditions from \\textbf{Lemma \\ref{zonoContainment}} such that $ X \\subseteq Y $, where $ X = \\mathbf{A} Z \\oplus W $ and $ Y = Z $. Consider the change of variables $ \\mathbf{\\Gamma}_1 = \\mathbf{\\Phi}\\tilde{\\mathbf{\\Gamma}}_1 $, $ \\mathbf{\\Gamma}_2 = \\mathbf{\\Phi}\\tilde{\\mathbf{\\Gamma}}_2 $, $ \\boldsymbol{\\beta} = \\mathbf{\\Phi}\\tilde{\\boldsymbol{\\beta}} $ and define $ \\tilde{\\mathbf{\\Gamma}} = [\\tilde{\\mathbf{\\Gamma}}_1 \\; \\tilde{\\mathbf{\\Gamma}}_2] $. Then the zonotope containment conditions from \\eqref{zonoContainment_Conditions} are satisfied by 1) rearranging and combining \\eqref{RPI_condition1} and \\eqref{RPI_condition2} to get $ [\\mathbf{A}\\mathbf{G}\\mathbf{\\Phi} \\; \\mathbf{G}_w] = \\mathbf{G}\\mathbf{\\Phi} \\tilde{\\mathbf{\\Gamma}} $, 2) rearranging \\eqref{RPI_condition3} to get $ \\mathbf{c} - (\\mathbf{A}\\mathbf{c} + \\mathbf{c}_w) = \\mathbf{G}\\mathbf{\\Phi}\\tilde{\\boldsymbol{\\beta}} $, and 3) multiplying \\eqref{RPI_condition4} by $ \\mathbf{\\Phi}^{-1} $, since $ \\phi_i > 0 $, to get $ |\\tilde{\\mathbf{\\Gamma}}|\\mathbf{1} + |\\tilde{\\boldsymbol{\\beta}}| \\leq \\mathbf{1} $. \\hfill \\hfill \\qed\n\\end{pf}\n\nWhen using \\textbf{Theorem \\ref{RPI_OneStep}} to determine the RPI set $ Z $ in G-Rep, the generator matrix $ \\mathbf{G} $ is assumed to be known \\emph{a~priori} in the same way that the normal vectors are chosen \\emph{a~priori} in \\cite{Trodden2016_OneStep} for the one-step RPI set computation in H-Rep. Given a desired order of $ Z $, $ \\mathbf{G} $ can be computed using \\eqref{RPI_finitesum} where $ \\mathbf{G} = [\\mathbf{G}_w \\; \\mathbf{A}\\mathbf{G}_w \\; ... \\; \\mathbf{A}^s\\mathbf{G}_w] $, for some $ s \\in \\mathbb{N}_+ $ that provides the desired order. \nOnce $ \\mathbf{G} $ is determined, the diagonal matrix $ \\mathbf{\\Phi} $ provides the ability to scale the size of $ Z $ such that $ Z $ is an RPI set. Since the minimal RPI set is typically desired, an optimization problem can be formulated with the constraints from \\eqref{RPI_conditions} and a objective function that minimizes the scaling variables in $ \\mathbf{\\Phi} $. With $ \\mathbf{c} $, $ \\mathbf{\\Phi} $, $ \\mathbf{\\Gamma}_1 $, $ \\mathbf{\\Gamma}_2 $, and $ \\boldsymbol{\\beta} $ as decision variables in this optimization problem, \\eqref{RPI_conditions} consists of only linear constraints and thus an LP or QP can be formulated based on the norm used to minimize the vector $ \\boldsymbol{\\phi} $, where $ \\mathbf{\\Phi} = diag(\\boldsymbol{\\phi}) $. In the following example, an LP is formulated by minimizing $ \\| \\boldsymbol{\\phi} \\|_\\infty $ subject to \\eqref{RPI_conditions}. Computing RPI set $Z$ using \\textbf{Theorem \\ref{RPI_OneStep}} requires solving an LP with $n_g^2 + n_g(n_w + 2) + n$ decision variables.\n\\begin{exmp}\n\n\n\tConsider the system from \\emph{\\cite{Trodden2016_OneStep}}\n\t\\begin{equation}\\label{RPISet_Trodden_Sys}\n\t\\mathbf{x}_{k+1} = \\begin{bmatrix}\t1 & 1 \\\\ 0 & 1\t\\end{bmatrix}\\mathbf{x}_k + \\begin{bmatrix} 0.5 \\\\ 1\n\t\\end{bmatrix} u_k + \\mathbf{w}_k, \n\t\\end{equation} \n\twith $ \\mathbf{w}_k \\in W = \\{ \\mathbf{w} \\in \\mathbb{R}^2 \\mid \\| \\mathbf{w} \\|_{\\infty} \\leq 0.1\\} $. As in \\emph{\\cite{Trodden2016_OneStep}}, the state feedback control law $ u_k = \\mathbf{K} \\mathbf{x}_k $, where $\\mathbf{K} $ corresponds to the LQR solution with $ \\mathbf{Q} = \\mathbf{I} $ and $ \\mathbf{R} = 1 $, converts \\eqref{RPISet_Trodden_Sys} to an autonomous system of the form \\eqref{autoSys}. For this  system, four methods for computing outer-approximations of the mRPI set are compared in Fig. \\ref{Fig_RPI_Set} with respect to volume ratio $ V_r $ and computation time $ \\Delta t_{calc} $ as a function of set complexity ($ n_g $~for zonotopes in G-Rep, $ \\frac{1}{2}n_h $ for polytopes in H-Rep). The seminal work from \\emph{\\cite{Rakovic2005}}, denoted as $\\epsilon$-mRPI (H-Rep), is the most computationally expensive since evaluating support functions for polytopes in H-Rep requires the solution of an LP. Using zonotopes in G-Rep, computational cost of this $\\epsilon$-mRPI approach can be reduced by an order-of-magnitude since evaluating support functions for zonotopes is algebraic, as mentioned in Remark 3 of \\emph{\\cite{Rakovic2005}}. Alternatively, the 1-step approaches from \\emph{\\cite{Trodden2016_OneStep}} and \\emph{\\textbf{Theorem \\ref{RPI_OneStep}}}, provide similar computational advantages. However, the 1-step approach from \\emph{\\cite{Trodden2016_OneStep}} is sensitive to the choice of hyperplanes. Using the same choice of hyperplanes from \\emph{\\cite{Trodden2016_OneStep}}, Fig. \\ref{Fig_RPI_Set} shows that the volume ratio does not decrease with increasing set complexity as quickly as the zonotope-based approach. Note that volume ratio is defined with respect to an approximation of the true mRPI set volume computed using the $\\epsilon$-mRPI method with $\\epsilon = 10^{-9}$. \n\t\n\tTo assess the scalability of these methods with respect to system order, Fig. \\ref{Fig_RPI_Set_Scalability} shows a comparison of these methods based on set complexity and computation time as a function of system order $ n $. Note that the  $\\epsilon$-mRPI (H-Rep) method became impractical for higher system orders and is not included in Fig. \\ref{Fig_RPI_Set_Scalability}. Similarly, the 1-step (H-Rep) method became impractical for $ n > 6 $. These results are generated using a $n^{th}$-order integrator system similar to that of \\eqref{RPISet_Trodden_Sys}. While the $\\epsilon$-mRPI method in G-Rep provides the lowest computational cost, the complexity of the resulting set is roughly ten times larger than the set used for the 1-step approach. While scaling better than the 1-step H-Rep approach, the 1-step G-Rep approach requires solving a linear program with the constraints from \\eqref{RPI_conditions} which includes the large decision variable $ \\mathbf{\\Gamma}_1 \\in \\mathbb{R}^{n_g \\times n_g} $. To manage this computational cost for higher order systems, the number of steps $ s \\in \\mathbb{N}_+ $ in \\eqref{RPI_finitesum} can be chosen to balance set complexity and accuracy.\n\\end{exmp}\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=8.6cm]{RPI_Sets_K1_ave10.pdf}\n\t\t\\caption{Comparison of volume ratio and computation time as a function of set complexity for outer-approximations of the mRPI set using iterative and 1-step approaches based on H-Rep or G-Rep.}\n\t\t\\label{Fig_RPI_Set}                    \n\t\\end{center}                         \n\\end{figure}\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=8.6cm]{RPI_Sets_Scalability_K1_ave10.pdf}\n\t\t\\caption{Comparison of set complexity and computation time as a function of system order for outer-approximations of the mRPI set using iterative and 1-step approaches based on H-Rep or G-Rep.}\n\t\\label{Fig_RPI_Set_Scalability}         \n\t\\end{center}\n\\end{figure}\n\n\\section{Pontryagin Difference} \\label{Sec_Pontryagin}\n\nThis section provides an iterative method for computing the constrained zonotope representation of the Pontryagin difference of two zonotopes and a one-step optimization method for computing the zonotopic inner-approximation of the Pontryagin difference.\n\n\\begin{defn} \\emph{\\cite{Althoff2015a}}\n\tGiven two sets $ Z_1, Z_2 \\subset \\mathbb{R}^n $, the Pontryagin difference $ Z_d = Z_1 \\ominus Z_2 $ is defined as\n\t\\begin{equation} \\label{Pontryagin_Diff_Def}\n\tZ_d = \\{z \\in \\mathbf{R}^n \\mid z \\oplus Z_2 \\subseteq Z_1\\}.\n\t\\end{equation}\n\\end{defn}\nThe Pontryagin difference is also referred to as the Minkowski difference or the erosion of set $ Z_1 $ by $ Z_2 $.\n\n\\subsection{Iterative Method}\nIf $ Z_1 $ and $ Z_2 $ are zonotopes, then \\cite{Althoff2015a} provides the following iterative method for computing $Z_d$.\n\n\\begin{lem} (Theorem 1 of  \\emph{\\cite{Althoff2015a}}) \\label{Pontryagin_Diff}\n\tIf $ Z_1 = \\{\\mathbf{G}_1,\\mathbf{c}_1\\} $ and $ Z_2 = \\{\\mathbf{G}_2,\\mathbf{c}_2\\} $, then the Pontryagin difference $ Z_d = Z_1 \\ominus Z_2 $ is computed using the $ n_{g2} $ generators $ \\mathbf{g}_{2,i} $ of $ Z_2 $ by applying the following recursion:\n\t\\begin{subequations} \\label{Pontryagin_Diff_calc}\n\t\t\\begin{align}\n\t\tZ_{int}^{(0)} &= Z_1 - c_2, \\\\\n\t\tZ_{int}^{(i)} &= (Z_{int}^{(i-1)} + \\mathbf{g}_{2,i}) \\cap (Z_{int}^{(i-1)} - \\mathbf{g}_{2,i}), \\label{Pontryagin_intersection}\\\\\n\t\tZ_d &= Z_{int}^{(n_{g2})}.\n\t\t\\end{align}\n\t\\end{subequations}\n\\end{lem}\n\nAs shown in \\cite{Althoff2015a}, zonotopes are not closed under the Pontryagin difference. Thus, the methods in \\cite{Althoff2015a} require the use of a combination of G-Rep and H-Rep to compute approximations of $ Z_d $ in G-Rep. While this combination results in faster calculations than methods that solely use H-Rep, the majority of computation time comes from the conversion from G-Rep to H-Rep, which scales exponentially with the number of generators. \n\nHowever, since $ Z_d $ is computed via the intersection of zonotopes, $ Z_d $ can be exactly represented as a constrained zonotope. Thus, \\eqref{Pontryagin_intersection} can be directly computed using the generalized intersection from \\eqref{generalized_Intersection} without the need for H-Rep. Note that iterative method from \\textbf{Lemma \\ref{Pontryagin_Diff}} is also applicable if $ Z_1 = \\{\\mathbf{G}_1,\\mathbf{c}_1,\\mathbf{A}_1,\\mathbf{b}_1\\} $ is a constrained zonotope, since \\eqref{Pontryagin_Diff_calc} only requires $ Z_2 $ to be the Minkowski sum of generators $ \\mathbf{g}_{2,i} $. For a constrained zonotope $Z_1$ in $\\mathbb{R}^n$ with $n_{c1}$ constraints and $n_{g1}$ generators and a zonotope $Z_2$ in $\\mathbb{R}^n$ with $n_{g2}$ generators, $Z_d = Z_1 \\ominus Z_2$ is a constrained zonotope with $n_{gd} = 2^{n_{g2}}n_{g1}$ generators and $n_{cd} = 2^{n_{g2}}n_{c1} + n(2^{n_{g2}} - 1)$ constraints.\n\n\n\\subsection{One-step Optimization Inner-Approximation Method}\n\nAs an alternative to the iterative method from \\textbf{Lemma~\\ref{Pontryagin_Diff}}, the following theorem presents a one-step method for computing an zonotopic inner-approximation of the Pontryagin difference $ \\tilde{Z}_d \\subseteq Z_d = Z_1 \\ominus Z_2$ using a single LP.\n\n\\begin{thm} \\label{Pontryagin_OneStep}\n\tGiven $ Z_1 = \\{\\mathbf{G}_1,\\mathbf{c}_1\\} $ and $ Z_2 = \\{\\mathbf{G}_2,\\mathbf{c}_2\\} $, then $ \\tilde{Z}_d = \\{[\\mathbf{G}_1 \\; \\mathbf{G}_2]\\mathbf{\\Phi},\\mathbf{c}_d\\} $, with $ \\mathbf{\\Phi} = diag(\\boldsymbol{\\phi}), \\phi_i > 0, \\forall i \\in \\{1, \\cdots, n_{g1} + n_{g2}\\}$, is an inner-approximation of the Pontryagin difference such that $ \\tilde{Z}_d \\subseteq Z_1 \\ominus Z_2 $ if there exists $ \\mathbf{\\Gamma} \\in \\mathbb{R}^{n_{g1} \\times (n_{g1} + 2 n_{g2})} $ and $ \\boldsymbol{\\beta} \\in \\mathbb{R}^{n_{g1}} $, such that\n\t\\begin{subequations} \\label{Pontryagin_conditions}\n\t\t\\begin{align} \n\t\t\\begin{bmatrix}\n\t\t[\\mathbf{G}_1 \\; \\mathbf{G}_2]\\mathbf{\\Phi} & \\mathbf{G}_2\n\t\t\\end{bmatrix} &= \\mathbf{G}_1 \\mathbf{\\Gamma}, \\label{Pontryagin_condition1}\\\\\n\t\t\\mathbf{c}_1 - (\\mathbf{c}_d + \\mathbf{c}_2) &= \\mathbf{G}_1 \\boldsymbol{\\beta}, \\label{Pontryagin_condition2}\\\\\n\t\t|\\mathbf{\\Gamma}|\\mathbf{1} + |\\boldsymbol{\\beta}| &\\leq \\mathbf{1}. \\label{Pontryagin_condition3}\n\t\t\\end{align}\n\t\\end{subequations}\n\\end{thm}\n\\begin{pf}\n\tBy viewing \\eqref{Pontryagin_conditions} in the context of the zonotope containment conditions from \\textbf{Lemma \\ref{zonoContainment}}, it is clear that \\eqref{Pontryagin_conditions} enforces the  Pontryagin difference condition $ \\tilde{Z}_d \\oplus Z_2 \\subset Z_1 $ from \\eqref{Pontryagin_Diff_Def}. \\hfill \\hfill \\qed\n\\end{pf}\n\nWhen using \\textbf{Theorem \\ref{Pontryagin_OneStep}} to compute $ \\tilde{Z}_d \\subset Z_d $ in G-Rep, the generator matrix $ [\\mathbf{G}_1 \\; \\mathbf{G}_2]\\mathbf{\\Phi} $ is assumed to be comprised of the generators from both $ Z_1 $ and $ Z_2 $ scaled by the diagonal matrix $ \\mathbf{\\Phi} $. Since maximizing the size of $ \\tilde{Z}_d $ is typically desired, an optimization problem can be formulated with the constraints from \\eqref{Pontryagin_conditions} and an objective function that maximizes the scaling variables in $ \\mathbf{\\Phi} $. With $ \\mathbf{c}_d $, $ \\mathbf{\\Phi} $, $ \\mathbf{\\Gamma} $, and $ \\boldsymbol{\\beta} $ as decision variables in this optimization problem, \\eqref{Pontryagin_conditions} consists of only linear constraints and thus an LP or QP can be formulated based on the norm used to maximize the vector $ \\boldsymbol{\\phi} $, where $ \\mathbf{\\Phi} = diag(\\boldsymbol{\\phi})$. Computing $\\tilde{Z}_d$ using \\textbf{Theorem \\ref{Pontryagin_OneStep}} requires solving a LP with $n_{g1}^2 +2n_{g1}n_{g2} + 2n_{g1} + n_{g2} + n$ decision variables.\n\n\\begin{exmp}\n\tConsider the zonotopes from \\emph{\\cite{Althoff2015a}}\n\t\\begin{equation*}\n\t\\setlength\\arraycolsep{2pt}\n\tZ_1 = \\left\\{ \\begin{bmatrix} 1 & 1 & 0 & 0 \\\\ 1 & 0 & 1 & 0 \\\\ 1 & 0 & 0 & 1 \\end{bmatrix}, \\boldsymbol{0} \\right\\}, \\quad Z_2 = \\left\\{ \\frac{1}{3} \\begin{bmatrix} -1 & 1 & 0 & 0 \\\\ \\phantom{-}1 & 0 & 1 & 0 \\\\ \\phantom{-}1 & 0 & 0 & 1 \\end{bmatrix}, \\boldsymbol{0} \\right\\}.\n\t\\end{equation*}\n\tFig. \\ref{Fig_Pontryagin_Diff_3D} shows the Pontryagin difference $ Z_d = Z_1 \\ominus Z_2 $ with $ n_g = 64 $ and $ n_c = 45 $ computed using \\emph{\\textbf{Lemma~\\ref{Pontryagin_Diff}}}. As discussed in \\emph{\\cite{Althoff2015a}}, zonotopes are not closed under the Pontryagin difference, which can be seen in Fig. \\ref{Fig_Pontryagin_Diff_3D} by the asymmetric facets of $ Z_d $. Using \\emph{\\textbf{Theorem \\ref{Pontryagin_OneStep}}}, the inner-approximation of the Pontryagin difference $ \\tilde{Z}_d $ is also shown in Fig. \\ref{Fig_Pontryagin_Diff_3D}. Choosing to maximize $ \\| [\\mathbf{G}_1 \\; \\mathbf{G}_2]\\mathbf{\\Phi} \\|_\\infty $ subject to \\eqref{Pontryagin_conditions} produced $ \\tilde{Z}_d \\subset Z_d $ with a volume ratio of $ V_r = 0.924 $.\n\\end{exmp}\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=7cm]{Pontryagin_Diff_3D.pdf}\n\t\t\\caption{Left: The Pontryagin difference $ Z_d = Z_1 \\ominus Z_2 $ where $ Z_1 $ and $ Z_2 $ are zonotopes but $ Z_d $ is not \\cite{Althoff2015a}. Right: The inner-approximation of $ Z_d $ by a zonotope $ \\tilde{Z}_d \\subseteq Z_d $. }\n\t\t\\label{Fig_Pontryagin_Diff_3D}                   \n\t\\end{center}                     \n\\end{figure}\n\n\\begin{table*}[!t]\n\t\\renewcommand{\\arraystretch}{1.1}\n\t\\caption{Pontryagin difference set complexity and computation time (seconds)}\n\t\\label{Tab_Ptryagindiff}\n\t\\centering\n\t\\begin{tabular}{ M{0.3cm} M{0.4cm}  M{0.4cm}|  M{0.6cm}  M{0.6cm} | M{3.5cm} M{0.9cm} | M{0.6cm} M{0.6cm} M{0.6cm} }\n\t\t\\hline\n\t\t\\multicolumn{3}{c}{} & \\multicolumn{7}{c}{$Z_1 \\ominus Z_2$} \\\\\n\t\t\\cline{4-10}\n\t\t\\multicolumn{1}{c}{} & \\multicolumn{1}{c}{$Z_1$} & \\multicolumn{1}{c}{$Z_2$} & \\multicolumn{2}{c}{H-Rep} & \\multicolumn{2}{c}{CG-Rep} & \\multicolumn{3}{c}{1-Step (G-Rep)} \\\\ \\cline{2-3}  \\cline{4-10}\n\t\t\\multicolumn{1}{c}{$n$} & \\multicolumn{2}{c}{$n_g$} & \\multicolumn{1}{c}{$n_h$} & \\multicolumn{1}{c}{$t_h$} & \\multicolumn{1}{c}{$n_{c} \\times n_g$} & \\multicolumn{1}{c}{$t_h\/t_{cg}$} & \\multicolumn{1}{c}{$n_g$} & \\multicolumn{1}{c}{$V_r$} & \\multicolumn{1}{c}{$t_h\/t_{g}$} \\\\\n\t\t\\hline\n\t\t2 & 4  & 4  & 16    & 0.01 & 30 \t $\\times$ 64  \t\t& 33.2 \t& 2.5 & 0.64 & 3.3\\\\\n\t\t2 & 8  & 4  & 32    & 0.01 & 30 \t $\\times$ 128 \t\t& 48.2 \t& 3.0 & 0.54 & 3.5\\\\\n\t\t2 & 4  & 8  & 16    & 0.01 & 510\t $\\times$ 1024 \t\t& 17.6 \t& 2.6 & 0.67 & 3.3\\\\\n\t\t2 & 8  & 8  & 32    & 0.02 & 510 \t $\\times$ 2048  \t& 20.9 \t& 3.2 & 0.53 & 3.1\\\\ \\hline \n\t\t3 & 6  & 6  & 60    & 0.03 & 189  \t $\\times$ 384 \t\t& 68.8 \t& 3.7 & 0.55 & 7.5 \\\\\n\t\t3 & 12 & 6  & 264   & 0.16 & 189 \t $\\times$ 768 \t\t& 261 \t& 4.8 & 0.46 & 16.0 \\\\\n\t\t3 & 6  & 12 & 60    & 0.13 & 12,285  $\\times$ 24,576 \t& 18.6 \t& 3.8 & 0.54 & 21.5\\\\\n\t\t3 & 12 & 12 & 264   & 0.40 & 12,285  $\\times$ 49,152 \t& 27.2\t& 4.9 & 0.43 & 28.2\\\\ \\hline \n\t\t4 & 8  & 8  & 224   & 0.38 & 1,020   $\\times$ 2,048  \t& 359 \t& 4.9 & 0.50 & 46.2 \\\\ \n\t\t4 & 16 & 8  & 2,240 & 41.0 & 1,020   $\\times$ 4,096  \t& 2,370 & 6.2 & 0.45 & 1,890\\\\ \n\t\t4 & 8  & 16 & 224   & 59.0 & 262,140 $\\times$ 524,288   & 271 \t& 4.7 & 0.43 & 4,078\\\\ \n\t\t4 & 16 & 16 & 2,240 & 243  & 262,140 $\\times$ 1,048,576 & 556 \t& 6.2 & 0.48 & 6,510\\\\ \\hline \n\t\\end{tabular}\n\\end{table*}\n\n\\begin{exmp}\n\tSimilar to \\emph{\\cite{Althoff2015a}}, the scalability of exact constrained zonotope representations of the Pontryagin difference via \\emph{\\textbf{Lemma~\\ref{Pontryagin_Diff}}} and  zonotopic inner-approximations via \\emph{\\textbf{Theorem~\\ref{Pontryagin_OneStep}}} is compared with the standard H-Rep approach provided in the Multi-Parametric Toolbox \\cite{Herceg2013}. Table \\ref{Tab_Ptryagindiff} shows the complexity and computational time for computing the Pontryagin difference $ Z_d = Z_1 \\ominus Z_2 $ using each of the three methods for zonotopes in $ \\mathbb{R}^2 $, $ \\mathbb{R}^3 $, and $ \\mathbb{R}^4 $. Each entry in Table~\\ref{Tab_Ptryagindiff} represents an average of 100 computations using randomly generated zonotopes $ Z_1 $ and $ Z_2 $. These random zonotopes are generated using the procedure provided in \\emph{\\cite{Althoff2015a}} and the CORA toolbox \\cite{Althoff2018a}. Cases where $ Z_d = \\emptyset $ were disregarded and not considered in the set of 100 computations. For CG-Rep and G-Rep, the ratio of computation times relative to that of H-Rep is presented. Since the G-Rep approach is an inner-approximation, the average volume ratio is also provided. From these results, it is clear that both the set complexity $ n_h $ and the computation time $ t_h $ for the H-Rep approach increase by approximately an order-of-magnitude as the set dimension $ n $ increases. While the CG-Rep approach increases the computation speed by approximately two orders-of-magnitude, the set complexity increases exponentially. Sparse matrices were used to reduce the memory requirements for these computations.  The redundancy removal approach presented in Section \\ref{Sec_Redundancy} was not able to detect the high-degree of redundancy in these set representations.  Alternatively, the one-step G-Rep approximation approach also provided significant reductions in computational cost while maintaining a small number of generators. However, for these randomly generated zonotopes, the inner-approximation only captures approximately 50\\% of the volume of $ Z_d $.  While these methods will likely work well for many practical applications, future work is needed to improve redundancy detection and removal for the CG-Rep approach and improved optimization formulations are needed for the G-Rep approach to further maximize volume ratio.\n\\end{exmp}\n\\section{Application to Reachability Analysis} \\label{Sec_Hier}\n\nTo demonstrate the applicability of algorithms developed in this paper, this section considers the exact and approximate computations of backwards reachable sets of a constrained linear system in the context of the two-level hierarchical MPC framework developed in \\cite{Koeln2019ACC,Koeln2019_Aut}. The high-level goal is to compute a \\emph{wayset} $ Z_c(k) $ at discrete time step $ k $ that captures all of the initial states $ \\mathbf{x}(k) \\in Z_c(k) \\subset \\mathbb{R}^n $ for which there are state and input trajectories $ \\mathbf{x}(k+j) $ and $ \\mathbf{u}(k+j) $ that satisfy, for all $ j \\in \\{0,\\cdots,N-1\\} $, \\emph{i}) the dynamics $ \\mathbf{x}(k+j+1) = \\mathbf{A} \\mathbf{x}(k+j) + \\mathbf{B} \\mathbf{u}(k+j) $, \\emph{ii}) the state and input constraints $ \\mathbf{x}(k+j) \\in \\mathcal{X}$ and $ \\mathbf{u}(k+j) \\in \\mathcal{U}$, and \\emph{iii}) the terminal constraint $ \\mathbf{x}(k+N) = \\mathbf{x}^* $ for some predetermined target $ \\mathbf{x}^* \\in \\mathbb{R}^n $. In the context of the hierarchical MPC framework from \\cite{Koeln2019ACC,Koeln2019_Aut}, $ \\mathbf{x}^* $ is a future state on the optimal trajectory determined by an upper-level controller and $ Z_c(k) $ is a terminal constraint imposed on a lower-level controller. Since $ \\mathbf{x}^* $ is updated at every evaluation of the upper-level controller, $ Z_c(k) $ must be recomputed in real-time, which is enabled through the use of constrained zonotopes.\n\n\\textbf{Algorithm \\ref{waysetComputation}} shows a simplified version of the backward reachable wayset algorithms presented in \\cite{Koeln2019ACC,Koeln2019_Aut}. Fig. \\ref{Fig_HierMPC_Wayset_evol} shows the results of this algorithm when applied to the simplified vehicle system model from \\cite{Koeln2019ACC,Koeln2019_Aut} with\n\\begin{equation}\n    \\mathbf{x}(k+1) = \\begin{bmatrix} 1 & 1 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{bmatrix}\\mathbf{x}(k) + \\begin{bmatrix} \\phantom{-}0 & \\phantom{-}0 & \\phantom{-}0 \\\\ \\phantom{-}1 & -1 & \\phantom{-}0 \\\\ -1 & -1 & -1\n    \\end{bmatrix}\\mathbf{u}(k),\n\\end{equation}\nwhere the states represent position, velocity, and on-board energy storage and the inputs represent acceleration, deceleration, and power to an on-board load. The  discretization time step size is $\\Delta t = 1$ second and the state and input constraints defining $\\mathcal{X}$ and $\\mathcal{U}$ are\n\\begin{equation*}\n    \\begin{bmatrix} -1 \\\\ -20 \\\\ 0 \\end{bmatrix} \\leq \\mathbf{x}(k) \\leq \\begin{bmatrix}\n    105 \\\\ 20 \\\\ 100\n    \\end{bmatrix}, \\; \\begin{bmatrix} 0 \\\\ 0 \\\\ 0 \\end{bmatrix} \\leq \\mathbf{u}(k) \\leq \\begin{bmatrix} 1 \\\\ 1 \\\\ 1\\end{bmatrix}.\n\\end{equation*}\n\nTo demonstrate the halfspace intersection results from Section \\ref{halfspaceIntersections}, Table \\ref{Tab_wayset_cmplx_comptime} compares the set representation complexity and computation time of four different CG-Rep methods with those using H-Rep via the Multi-Parametric Toolbox \\cite{Herceg2013}. All computation times are averaged over 100 runs. Overall, the CG-Rep methods result in significantly less set complexity and computation time. The CG-Rep methods differ in the computation of $ \\hat{Z}_c(k + j-1) \\cap \\mathcal{X}$ in \\textbf{Algorithm \\ref{waysetComputation}}. Specifically, this intersection is computed using $1)$ the zonotope-hyperplane (ZH) method from \\textbf{Lemma \\ref{Zono_halfspace_int_check}} based on the parent zonotope $ \\hat{Z}(k + j-1) \\supset \\hat{Z}_c(k + j-1) $ and the H-Rep of $ \\mathcal{X} $, $2)$ the generalized intersection (GI) (from \\eqref{generalized_Intersection}) of the constrained zonotope wayset and the G-Rep of $ \\mathcal{X} $, $3)$ the linear program (LP) method from \\textbf{Lemma \\ref{conzono_hyplane_intersection}} for checking the intersection of a constrained zonotope and a hyperplane, and $4)$ the interval arithmetic (IA) approach using \\textbf{Algorithm \\ref{gen_Bounds}} to detect empty sets when $ \\hat{Z}_c(k + j-1) \\subset \\mathcal{X}$.\nIn the ZH, LP, and IA methods, if the wayset intersects the hyperplanes associated with the halfspaces of $ \\mathcal{X} $, generators and constraints are added using \\eqref{conzono_halfspace_int} to exactly compute $ \\hat{Z}_c(k + j-1) \\cap \\mathcal{X}$ in CG-Rep.\n\nAs expected, the GI approach resulted in the highest set complexity since generators and constraints are added even if $ \\hat{Z}_c(k + j-1) \\subset \\mathcal{X} $. The LP approach results in the lowest complexity by only adding generators and constraints when needed to exactly define the intersection. In this application, the ZH method also achieves this low set complexity and requires significantly less computation time. However, achieving this low complexity is not expected in general. Finally, the IA approach did not perform as well in this application, resulting in unnecessary generators and constraints and a large computation time. However, in practice, the zonotope-halfspace check from \\textbf{Theorem \\ref{zono_halfspace_int}} would be applied first so that \\textbf{Algorithm \\ref{gen_Bounds}} is only used in cases where the parent zonotope intersects the hyperplane. \n\n\\IncMargin{1.5em}\n\\begin{algorithm2e}[t]\n\t\\SetAlgoLined\n\t\\SetKwInOut{Input}{Input}\\SetKwInOut{Output}{Output}\n\t\\Input{$\\mathbf{x}^*$}\n\t\\Output{$ Z_c(k)$}\n\t\\BlankLine\n\t\\SetAlgoLined\n\tinitialize $ j \\leftarrow N $\\\\\n\t$Z_c(k + j) =  \\mathbf{x}^* $\\;\n\t\\While{$ j \\geq 1 $}{\n\t\t$\\hat{Z}_c(k + j-1) = A^{-1}Z_c(k + j) \\oplus (-A^{-1}B) \\mathcal{U}$\\;\n\t\t$ Z_c(k + j-1) = \\hat{Z}_c(k + j-1) \\cap \\mathcal{X}$\\;\n\t\t$j \\leftarrow j - 1 $\\;\n\t}\n\t$Z_c(k) = Z_c(k + j) $\n\t\\caption{Wayset $Z_c(k)$ for target $ \\mathbf{x}^*$.}\n\t\\label{waysetComputation}\t\n\\end{algorithm2e}\n\\DecMargin{1.5em}\n\n\\begin{figure}[t]\n\t\\begin{center}\t\t\\includegraphics[width=7cm]{Wayset_S13_evol_time.pdf}\n\t\\caption{The evolution of backward reachable wayset $Z_c(k)$ for $k = 40$ and $N = 10$ time steps starting from $\\mathbf{x}^*$ projected on the position and energy states. The sets $Z_c(k + j), \\forall \\; j \\in \\{ 7, 8, 9\\}$ are zonotopes (evident from symmetry) while the sets $Z_c(k+j), \\forall \\; j \\in \\{ 0, \\cdots, 6\\}$, are constrained zonotopes. The constrained zonotope wayset $Z_c(k)$ contains $\\mathbf{x}_{-}^*$ ensuring the control feasibility from \\cite{Koeln2019ACC,Koeln2019_Aut}.}\n\t\\label{Fig_HierMPC_Wayset_evol}   \n\t\\end{center}        \n\\end{figure}\n\n\\begin{table}\n\t\\renewcommand{\\arraystretch}{1.25}\n\t\\caption{Complexity and Computation Time of Waysets}\n\t\\label{Tab_wayset_cmplx_comptime}\n\t\\centering\n\t\\begin{tabular}{ M{1cm} | M{1.5cm} | M{0.8cm} | M{1.5cm} | M{0.8cm}}\n\t\t\\multicolumn{5}{c}{} \\\\\n\t\t\\hline\t\\multicolumn{1}{c}{} & \\multicolumn{1}{c}{$Z_c$} & \\multicolumn{1}{c}{$t_{calc}$} & \\multicolumn{1}{c}{$\\tilde{Z}_c$} & \\multicolumn{1}{c}{$t_{calc}$} \\\\\n\t\t\\cline{2-5}\t\\multicolumn{1}{c}{Method} & \\multicolumn{1}{c}{$n_c \\times n_g$} & \\multicolumn{1}{c}{sec}\n\t\t& \\multicolumn{1}{c}{$\\tilde{n}_c \\times \\tilde{n}_g$} & \\multicolumn{1}{c}{sec}\\\\\n\t\t\\hline\n\t\tZH & $ 7 \\times 37$ & $1e^{-3}$ & \n\t\t$7 \\times 37$ & $4e^{-3}$ \\\\\n\t\tGI & $30 \\times 60$ & $2e^{-3}$ & $7 \\times 37$ & $2e^{-1}$ \\\\\n\t\tLP & $7 \\times 37$ & $1e^{-1}$ & $7 \\times 37$ & $2e^{-3}$ \\\\\n\t\tIA & $15 \\times 45$ & $1e^{-1}$ & $7 \\times 37$ & $4e^{-2}$\\\\\n\t\tH-Rep & $n_h = 5047$ & $161$ & $n_h = 153$ & $333$\n\t\\end{tabular}\n\\end{table}\n\nTo demonstrate redundancy removal results from Section \\ref{Sec_Redundancy}, \\textbf{Algorithm \\ref{waysetComputation}} and \\textbf{Theorem \\ref{Redundant_ConZono}} were applied to successfully remove all unnecessary generators and constraints resulting in the irredundant constrained zonotope wayset $ \\tilde{Z}_c $ in Table \\ref{Tab_wayset_cmplx_comptime}. Overall, when compared to H-Rep, any of the four CG-Rep approaches are computationally efficient with less set complexity and the preferred CG-Rep approach is likely to be application dependent.\n\nWhen computing these waysets for complex systems, it is likely that inner-approximations are needed to restrict the complexity of the set to satisfy a predetermined upper bound on the number of generators and constraints. Demonstrating the inner-approximations from Section \\ref{Sec_InnerApprox} and the convex hull operation from Section \\ref{Sec_ConvexHull}, the top row of plots in Fig. \\ref{Fig_HierMPC_InApprox_cvxhull} shows the inner-approximating interval set $ B \\subset Z_c $ computed using the method described in \\textbf{Example \\ref{example_innerApprox}} with $ n_g = 3 $ and $ n_c = 0 $.  However, in the hierarchical MPC framework from \\cite{Koeln2019ACC,Koeln2019_Aut} the wayset must also include a key element denoted here as $ \\mathbf{x}^*_- $. Since $ \\mathbf{x}^*_- \\notin B $, the wayset can be computed as $ CH(B \\cup \\mathbf{x}^*_-) $ resulting in $ n_g = 10 $ and $ n_c = 6 $. If this increase in set complexity is undesirable for a particular application, the point containment $ \\mathbf{x}^*_- \\in B \\subseteq Z_c $ can be readily added to the LP defined in \\eqref{Conzono_containment}. The resulting inner-approximating interval set with this point containment is shown in the bottom row of plots in Fig. \\ref{Fig_HierMPC_InApprox_cvxhull}. The computation time for these inner-approximating interval sets are approximately $0.18$ and $0.25$ seconds for the top and bottom rows, respectively.\n\n\\begin{figure}\n\t\\begin{center}\n \t\t\\includegraphics[width=8cm]{HierMPC_Wayset_InApprox_cvxhull_nd_pt_containment.pdf}\n\t\t\\caption{Top: The wayset $Z_c$, inner-approximating interval set $ B $ with $ V_r = 0.35 $, and $CH(B \\cup \\mathbf{x}^*_-)$ with $V_r = 0.39$ are shown on the left and the projections on to the position and velocity states are shown on the right. Bottom: The wayset $Z_c$, inner-approximating interval set $ B $ containing $ \\mathbf{x}^*_- $ with $ V_r = 0.30 $ shown on the left with the projection shown on the right.}\n\t\t\\label{Fig_HierMPC_InApprox_cvxhull}\n\t\t\\end{center}      \n\\end{figure}\n\n\\section{Conclusions and Future Work} \\label{Conclusions}\nThe use of zonotopes and constrained zonotopes for set operations provides significant computational advantages that improve the practicality of set-based techniques commonly used in systems and control theory. Operations such as halfspace intersections, convex hulls, invariant sets, and Pontryagin differences have been shown to benefit from zonotope and constrained zonotope set representations. \nComplexity reduction techniques were developed based on redundancy removal and inner-approximations to further improve the practicality of these set representations. Future work will focus on improved redundancy detection algorithms and optimization formulations that more accurately capture the volume of the approximated set.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe interpretation presented here completely revives classical ontology: Reality is described completely by a classical trajectory $q(t)$, in the classical configuration space $Q$. \n\nThe wave function is also interpreted in a completely classical way -- as a particular description of a classical probability flow, defined by the probability distribution $\\rho(q)$ and average velocities $v^i(q)$. The formula which defines this connection has been proposed by Madelung 1926 \\cite{Madelung} and is therefore as old as quantum theory itself. It is the polar decomposition\n\\begin{equation}\\label{polar}\n\\psi(q)=\\sqrt{\\rho(q)} e^{\\frac{i}{\\hbar} S(q)},\n\\end{equation}\nwith the phase $S(q)$ being a potential of the velocity $v^i(q)$: \n\\begin{equation}\\label{guiding}\nv^i(q) = m^{ij} \\pd_j S(q),\n\\end{equation}\nso that the flow is a potential one.\\footnote{Here, $m^{ij}$ denotes a quadratic form -- a ``mass matrix'' -- on configuration space. I assume in this paper that the Hamiltonian is quadratic in the momentum variables, thus,\n\\begin{equation}\\label{HamiltonFunction}\nH(p,q) = \\frac{1}{2}m^{ij}p_ip_j + V(q).\n\\end{equation}\nThis is quite sufficient for relativistic field theory, see app. \\ref{relativity}.\n}\n\nThis puts the interpretation into the classical realist tradition of interpretation of quantum theory, the tradition of  de Broglie-Bohm (dBB) theory \\cite{deBroglie}, \\cite{Bohm} and Nelsonian stochastics \\cite{Nelson}. \n\nBut there is a difference -- the probability flow is interpreted as describing incomplete information about the true trajectory $q(t)$. This puts the interpretation into another tradition -- the interpretation of probability theory as the logic of plausible reasoning in situations with incomplete information, as proposed by Jaynes \\cite{Jaynes}. This objective variant of the Bayesian interpretation of probability follows the classical tradition of Laplace \\cite{Laplace} (to be distinguished from the subjective variant proposed by de Finetti \\cite{deFinetti}). \n\nSo this interpretation combines two classical traditions -- classical realism about trajectories and the classical interpretation of probability as the logic of plausible reasoning. \n\nThere is also some aspect of novelty -- a clear and certain program for development of subquantum theory. Quantum theory makes sense only as an approximation for potential probability flows. It has to be generalized to non-potential flows, described by the flow variables  $\\rho(q), v^i(q)$. Without a potential $S(q)$ there will be also no wave function $\\psi(q)$ in such a subquantum theory. And the flow variables are not fundamental fields themself, they also describe only incomplete information. The fundamental theory has to be one for the classical trajectory $q(t)$ alone. \n\nSo this interpretation is also a step toward the development of a subquantum theory. This is an aspect which invalidates prejudices against quantum interpretations as pure philosphy leading nowhere. Nonetheless even this program for subquantum theory has also classical character, making subquantum theory closer to a classical theory. \n\nThinking about how to name this interpretation of quantum theory, I have played around with ``neoclassical'', but rejected it, for a simple reason: There is nothing ``neo'' in it. Instead, it deserves to be named ``paleo''.\\footnote{Association with the ``paleolibertarian'' political direction is welcome and not misleading -- it also revives classical libertarian ideas considered to be out of date for a long time.} \n\nAnd so I have decided to name this interpretation of quantum theory ``paleoclassical''. \n\n\\subsection{The Bayesian character of the paleoclassical interpretation}\n\nThe interpretation of the wave function is essentially Bayesian. \n\nIt is the objective (information-dependent) variant of Bayesian probability, as proposed by Jaynes \\cite{Jaynes}, which is used here. It has to be distinguished from the subjective variant proposed by de Finetti \\cite{deFinetti}, which is embraced by the Bayesian interpretation of quantum theory proposed by Caves, Fuchs and Schack \\cite{Bayesian}. \n\nThe Bayesian interpretation of the wave function is in conflict with the objective character assigned to the wave function in dBB theory. Especially Bell has emphasized that the wave function has to be understood as real object. \n\nBut the arguments in favour of the reality of the wave function, even if strong, appear insufficient: The effective wave function of small subsystems depends on the configuration of the environment. This dependence is sufficient to explain everything which makes the wave function similar to a really existing object. It is only the wave function of a closed system, like the whole universe, which is completely interpreted in terms of incomplete information about the system itself. \n\nComplexity and time dependence, even if they are typical properties of real things, are characteristics of incomplete information as well. \n\nThe Bayesian aspects essentially change the character of the interpretation: The dBB ``guiding equation'' \\eqref{guiding}, instead of guiding the configuration, becomes part of the definition of the information about the configuration contained in the wave function. \n\nThis has the useful consequence that there is no longer any ``action without reaction'' asymmetry: While it is completely natural that the real configuration does not have an influence on the information available about it, there is also no longer any ``guiding'' of the configuration by the wave function. \n\n\\subsection{The realistic character of the paleoclassical interpretation}\n\nOn the other hand, there is also a strong classical realistic aspect of the paleoclassical interpretation: First of all, it is a realistic interpretation. But, even more, its ontology is completely classical -- the classical trajectory $q(t)$ is the only beable.  \n\nThis defines an important difference between the paleoclassical interpretation and other approaches to interprete the wave function in terms of information: It answers the question ``information about what'' by explicitly defining the ``what'' -- the classical trajectory -- and even the ``how'' -- by giving explicit formulas for the probability distribution $\\rho(q)$ and the average velocity $v^i(q)$. \n\nThe realistic character leads also to another important difference -- that of motivation. For the paleoclassical interpretation, there is no need to solve a measurement problem -- there is none already in dBB theory, and the dBB solution of this problem -- the explicit non-Schr\\\"{o}dinger\\\/  evolution of the conditional wave function of the measured system, defined by the wave function of system and measurement device and the actual trajectory of the measurement device -- can be used without modification.  \n\nAnd there is also no intention to save relativistic causality by getting rid of the non-local collapse -- the paleoclassical interpretation accepts a hidden preferred frame, which is yet another aspect of its paleo character. Anyway, because of Bell's theorem, there is no realistic alternative.\n\\footnote{Here I, of course, have in mind the precise meaning of ``realistic'' used in Bell's theorem, instead of the metaphorical one used in ``many worlds'', which is, in my opinion, not even a well-defined interpretation.}\n\n\\subsection{The justification}\n\nSo nor the realistic approach of dBB theory, which remains influenced by the frequentist interpretation of probability (an invention of the positivist Richard von Mises \\cite{Mises}) and therefore tends to objectivize the wave function, nor the Bayesian approach, which embraces the anti-realistic rejection of unobservables and therefore rejects the preferred frame, are sufficiently classical to see the possibility of a complete classical picture. \n\nBut it is one thing to see it, and possibly even to like it because of its simplicity, and another thing to consider it as justified.  \n\nThe justification of the paleoclassical interpretation presented here is based on a reformulation of classical mechanics in term of -- a wave function. It is a simple variant of Hamilton-Jacobi theory with a density added, but this funny reformulation of classical mechanics appears to be the key for the paleoclassical interpretation. The point is that its exact equivalence to classical mechanics (in the domain of its validity) and even the very fact that it shortly becomes invalid (because of caustics) almost force us to accept -- for this classical variant -- the interpretation in terms of insufficient information. And it also provides all the details we use. \n\nBut, then, why should we change the ontological interpretation if all what changes is the equation of motion?  Moreover, if (as it appears to be the case) the Schr\\\"{o}dinger\\\/  equation is simply the linear part of the classical equation, so that there is not a single new term which could invalidate the interpretation?  And where one equation is the classical limit of the other one?\n\nWe present even more evidence of the conceptual equivalence between the two equations: A shared explanation, in terms of information, that there should be a global $U(1)$ phase shift symmetry, a shared explanation of the product rule for independence, a shared explanation for homogenity of the equation. And there is, of course, the shared Born probability interpretation. All these things being equal, why should one even think about giving the two wave functions a different interpretation? \n\n\\subsection{Objections} \n\nThere are a lot of objections against this interpretation to care about. \n\nSome of them are quite irrelevant, because they are handled already appropriately from point of view of de Broglie-Bohm theory, so I have banned them into appendices: The objection of incompatibility with relativity (app. \\ref{relativity}), the Pauli objection that it destroys the symmetry between configuration and momentum variables (app. \\ref{Pauli}), some doubts about viability of field-theoretic variants related with overlaps (app. \\ref{fields}). \n\nThere are the arguments in favour of interpreting the wave function as describing external real beables. These are quite strong but nonetheless insufficient: The wave functions of small subsystems -- and we have no access to different ones -- \\emph{depend} on real beables external to the system itself, namely the trajectories of the environment. And complexity and dynamics are properties of incomplete information as well. \n\nAnd there is the Wallstrom objection \\cite{Wallstrom}, the in my opinion most serious one. How to justify, in terms of $\\rho(q)$ and $v^i(q)$, the ``quantization condition''\n\\begin{equation\n\\oint m_{ij} v^i(q) dq^j = \\oint \\pd_j S(q) dq^j = 2\\pi m\\hbar, \\qquad m\\in\\mbox{$\\mathbb{Z}$}.\n\\end{equation}\nfor closed trajectories around zeros of the wave function, which is, in quantum theory, a trivial consequence of the wave function being uniquely defined globally, but which is completely implausible if formulated in terms of the velocity field?\n\nFortunately, I have found a solution of this problem in \\cite{againstWallstrom}, based on an additional regularity postulate. All what has to be postulated is that \\emph{$0< \\Delta \\rho(q) < \\infty$ almost everywhere where $\\rho(q)=0$}. This gives the necessary quantization condition. Moreover, there are sufficiently strong arguments that it can be justified by a subquantum theory. \n\nThe idea that what we observe are localized wave packets instead of the configurations themself I reject in app. \\ref{wavepackets}. So all the objections I know about can be answered in a satisfactory way. Or at least I think so. \n\n\\subsection{Directions of future research} \n\nDifferent from many other interpretations of quantum theory, the paleoclassical interpretation suggests a quite definite program of development of a more fundamental, subquantum theory: It defines the ontology of subquantum theory as well as the equation which can hold only approximately -- the potentiality condition for the velocity $v^i(q)$. The consideration of the Wallstrom objection even identifies the domain where modifications of quantum theory are necessary -- the environment of the zeros of the wave function. \n\nOne can identify also another domain where quantum predictions will fail in subquantum theory -- the reliability of quantum computers, in particular their ability to reach exponential speedup in comparison with classical computers. \n\nAnother interesting question is what restrictions follow for $\\rho(q), v^i(q)$ from the interpretation as a probability flow for a more fundamental theory for the trajectory $q(t)$ alone. An answer may be interesting for finding answers to the ``why the quantum'' question. A question which cannot be answered by an interpretation, which is restricted to the ``what is the quantum'' question. \n\n\\section{A wave function variant of Hamilton-Jacobi theory}\n\nDo you know that one can reformulate classical theory in terms of a wave function?  With an equation for this wave function which is completely classical, but, nonetheless, quite close to the Schr\\\"{o}dinger\\\/  equation?  \n\nIn fact, this is a rather trivial consequence of the mathematics of Hamilton-Jacobi theory and the insights of Madelung \\cite{Madelung}, de Broglie \\cite{deBroglie}, and Bohm \\cite{Bohm}. All one has to do is to look at them from another point of view. Their aim was to understand quantum theory, by representing quantum theory in a known, more comprehensible, classical form -- a form remembering the classical Hamilton-Jacobi equation\n\\begin{equation}\\label{HamiltonJacobi}\n\\pd_t S(q) + \\frac12 m^{ij} \\pd_i S(q) \\pd_j S(q) + V(q) = 0.\n\\end{equation}\nSo, the real part of the Schr\\\"{o}dinger\\\/  equation, divided by the wave function, gives\n\\begin{equation}\\label{Bohm}\n\\pd_t S(q) + \\frac12 m^{ij} \\pd_i S(q) \\pd_j S(q) + V(q) + Q[\\rho] = 0,\n\\end{equation}\nwith only one additional term -- the quantum potential \n\\begin{equation}\\label{Qdef}\nQ[\\rho] = -\\frac{\\hbar^2}{2} \\frac{\\Delta \\sqrt{\\rho}}{\\sqrt{\\rho}}\n\\end{equation}\nThe imaginary part of the Schr\\\"{o}dinger\\\/  equation (also divided by $\\psi(q)$) is the continuity equation for $\\rho$\n\\begin{equation}\\label{continuity}\n\\pd_t\\rho(q,t) + \\pd_i(\\rho(q,t)v^i(q,t)) = 0\n\\end{equation}\n \nNow, all what we have to do is to revert the aim -- instead of presenting the Schr\\\"{o}dinger\\\/  equation like a classical equation, let's present the classical Hamilton-Jacobi equation like a Schr\\\"{o}dinger\\\/  equation. There is almost nothing to do -- to add a density $\\rho(q)$ together with a continuity equation \\eqref{continuity} is trivial.  We use the same polar decomposition formula \\eqref{polar} to define the classical wave function. It remains to do the same procedure in the other direction. The difference is the same -- the quantum potential. So we obtain, as the equation for the classical wave function, an equation I have named pre-Schr\\\"{o}dinger\\\/  equation:\n\\begin{equation}\\label{pre}\ni\\hbar \\pd_t \\psi(q,t) = -\\frac{\\hbar^2}{2} m^{ij}\\pd_i\\pd_j\\psi(q,t) + (V(q) - Q[\\rho]) \\psi(q,t) = \\hat{H} \\psi - Q[|\\psi|^2]\\psi.\n\\end{equation}\n\nOf course, the additional term is a nasty, nonlinear one. But that's the really funny point: We can obtain now quantum theory as the linear approximation of classical theory, in other words, as a simplification.\n\nBut there is more than this in this classical equation for the classical wave equation. The point is that there is an exact equivalence between the different formulations of classical theory, despite the fact that one is based on a classical trajectory $q(t)$ only, and the other has a wave function $\\psi(q,t)$ together with the trajectory $q(t)$. And it is this exact equivalence which can be used to identify the meaning of the classical wave function. \n\nBecause of its importance, let's formulate this in form of a theorem:\n\\begin{theorem}[equivalence]\nAssume $\\psi(q,t),q(t)$ fulfill the pre-Schr\\\"{o}dinger\\\/  equation \\eqref{pre} for some Hamilton function $H(p,q)$ of type \\eqref{HamiltonFunction}, together with the guiding equation \\eqref{guiding}. \n\nThen, whatever the initial values $\\psi_0(q),q_0$ for the wave function $\\psi_0(q)=\\psi(q,t_0)$ and the initial configuration $q_0=q(t_0)$, there exists initial values $q_0,p_0$ so that the Hamilton equation for $H(p,q)$ with these initial values gives $q(t)$.  \n\\end{theorem}\n\\begin{proof} \nThe difficult part of this theorem is classical Hamilton-Jacobi theory, so, presupposed here as known. The simple modifications of this theory used here -- adding the density $\\rho(q)$, with continuity equation \\eqref{continuity}, and rewriting the result in terms of the wave function $\\psi(q)$ defined by the polar decomposition \\eqref{polar}, copying what has been done by Bohm \\cite{Bohm} -- do not endanger the equivalence between Hamiltonian (or Lagrangian) and Hamilton-Jacobi theory. \n\\end{proof}\n\n\\section{The wave function describes incomplete information}\n\nSo let's evaluate now what follows from the equivalence theorem about the physical meaning of the (yet classical) wave function. \n\nFirst, it is the classical (Hamiltonian or Lagrangian) variant which is preferable as a fundamental theory. There are three arguments to justify this:\n\n\\begin{itemize}\n\n\\item Simplicity of the set of fundamental variables: We need only a single trajectory $q(t)$ instead of trajectory $q(t)$ together with a wave function $\\psi(q,t)$.  \n\n\\item Correspondence between fundamental variables and observables: It is only the trajectory $q(t)$ in configuration space which is observable. \n\n\\item Stability in time: The wave function develops caustics after a short period of time and becomes invalid. The classical equations of motion does not have such a problem. \n\n\\end{itemize}\n\nSo it is the classical variant which is preferable as a fundamental theory. Thus, we can identify the true beable of classical theory with the classical trajectory $q(t)$. \n\nThe next observation is that, once $q(t)$ is known and fixed, the wave function contains many degrees of freedom which are unobservable in principle: Many different wave functions define the same trajectory $q(t)$. So we can conclude that these different wave functions do \\emph{not} describe physically different states, containing additional beables. \n\nOn the other hand, this is true only if $q$ is known. What if $q$ is not known? In this case, the wave function defines simply a subset of all possible classical trajectories, and a probability measure on this subset.  \n\nTo illustrate this, a particular most important example of a Hamilton-Jacobi function is useful: It is the function $S(q_0,t_0,q_1,t_1)$ defined by\n\\begin{equation}\\label{Minimumproblem}\n S(q_0,t_0,q_1,t_1) = \\int_{t_0}^{t_1} L(q(t),\\dot{q}(t),t) dt,\n\\end{equation} \nwhere the integral is taken over the classical solutions $q(t)$ of this minimum problem with initial and final values $q(t_0)=q_0$, $q(t_1)=q_1$. This function fulfills the Hamilton-Jacobi equation in the variables $q_0,t_0$ as well as $q_1,t_1$. In above versions, it can be characterized as a Hamilton-Jacobi function $S(q,t)$ which is defined by a subset of trajectories: The function $S(q_0,t_0,q,t)$ describes the subset of trajectories going through $q_0$ at $t_0$, while the function $S(q,t,q_1,t_1)$ describes the subset of trajectories going through $q_1$ at $t_1$.  \n\nWe can generalize this. The phase $S(q,t)$ tells us the value of $\\dot{q}(t)$ given $q(t)$: \\emph{If} $q(t)=q_0$ \\emph{then} $\\dot{q}(t)=v_0$, with $v_0$  defined by the guiding equation \\eqref{guiding}. So, $S(q,t)$ always distinguishes a particular subset of classical trajectories. \n\nEven more specific, this subset described by $S(q,t)$ can be uniquely described in terms of a subset of the possible initial values at the moment of time $t$ -- the configuration $q(t_0)$ and the momentum $p(t_0)=\\nabla S(q(t_0),t_0)$, that means, as a subset of the possible values for the fundamental beables $q,p$ (or $q,\\dot{q}$). \n\nThe other part -- the density $\\rho(q)$ -- is nothing but a probability density on this particular subset. \n\nOf course, a subset is nothing but a special delta-like probability measure, so that the wave function simply defines a probability density on the set of possible initial values:\n\\begin{equation}\\label{probability}\n\\rho(p,q)dpdq = \\rho(q)\\delta(p-\\nabla S(q))dpdq\n\\end{equation}\nThe pre-Schr\\\"{o}dinger\\\/  equation is, therefore, nothing but the evolution equation for this particular probability distribution. \n\nSo our classical, Hamilton-Jacobi wave function is nothing mystical, but simply a very particular form of a standard classical probability density $\\rho(p,q)$ on the phase space. \n\nIn particular, the pre-Schr\\\"{o}dinger\\\/  equation for the wave function is nothing but a particular case of the Liouville equation, the standard law of evolution of standard classical probability distributions $\\rho(p,q)$, for the particular ansatz \\eqref{probability}, and follows from the fundamental law of evolution for the true, fundamental classical beables. \n\nMoreover, the Liouville equation also defines the domain of applicability of the equation for the wave function. This domain is, in fact, restricted. In terms of $\\rho(p,q)$, it will always remain a probability distribution on some Lagrangian submanifold. But this Lagrangian submanifold will be, after some time, no longer a graphic of a function $p=\\nabla S(q)$ on configuration space -- there may be caustics, and in this case there will be several values of momentum for the same configuration $q$. If this happens, the wave function is no longer an adequate description. \n\nSuch an effect -- restricted validity -- is quite natural for the evolution of information, but not for fundamental beables. \n\nSo the wave function variant of Hamilton-Jacobi theory almost forces us to accept an interpretation of the wave function in terms of incomplete information. Indeed,\n\n\\begin{itemize}\n\n\\item The parts of the wave function, $\\rho(q)$ as well as $S(q)$, make sense as describing a well-defined type of incomplete information about the classical configuration, namely the probability distribution $\\rho(p,q)$ defined by \\eqref{probability}.  \n\n\\item The alternative, to interpret the wave function as describing some other, external beables, does not make sense, given the observational equivalence of the theory with simple Lagrangian classical mechanics, with $q(t)$ as the only observable. Additional beables should influence, at least in some circumstances, the $q(t)$. They don't.  \n\n\\end{itemize}\n\nSo it looks like a incomplete information, it behaves like incomplete information, it becomes invalid like incomplete information -- it is incomplete information.\n\nAnd, given that we know, in the case of the pre-Schr\\\"{o}dinger\\\/  equation, the fundamental law of evolution of the beables themself, it also makes no sense to reify this particular probability distribution as objective.  A probability distribution defines a set of incomplete information about the real configuration, that's all. \n\n\\subsection{What about the more general case?}\n\nOf course, the considerations above have been based on the equivalence theorem between the classical evolution and the evolution defined by the pre-Schr\\\"{o}dinger\\\/  equation. It was this exact equivalence which was able to give us some certainty, to remove all doubts that there is something else, some information about other beables, hidden in the wave function. \n\nBut what about the more general case, the case where we do not have an exact equivalence between an equation in terms of $q,\\psi(q)$ and an equation purely in terms of the classical trajectory $q(t)$?  In such a situation, the case for an interpretation of $\\psi(q)$ in terms of incomplete information about the $q(t)$ is, of course, a little bit weaker. \n\nNonetheless, given such an ideal correspondence for the pre-Schr\\\"{o}dinger\\\/  equation, the interpretation remains certainly extremely plausible in a more general situation too. Indeed, why should one change the interpretation, the ontological meaning, of $\\psi(q)$, if all what is changed is that we have replaced the pre-Schr\\\"{o}dinger\\\/  equation by another equation? The evolution equation is different, that's all. In itself, a change in the evolution equation does not give even a bit of motivation to change the interpretation. \n\nAnd it should be noted that they have a hard job. The similarity between the two variants of the Schr\\\"{o}dinger\\\/  equation does not makes it easier: In fact, the funny observation that the Schr\\\"{o}dinger\\\/  equation is the linear part of the pre-Schr\\\"{o}dinger\\\/  equation becomes relevant here.  If the Schr\\\"{o}dinger\\\/  equation would contain some new terms, this would open the door for attempts to show that the new terms do not make sense in the original interpretation.  But there are no new terms in the linear part of the pre-Schr\\\"{o}dinger\\\/  equation.  All terms of the Schr\\\"{o}dinger\\\/  equation are already contained in the pre-Schr\\\"{o}dinger\\\/  equation. So they all make sense.\\footnote{It has to be mentioned in this connection that there is something present in the Schr\\\"{o}dinger\\\/  equation which is not present in the pre-Schr\\\"{o}dinger\\\/  equation -- the dependence on $\\hbar$. So one can at least try to base an argument on this additional dependence. But, as described below, if one incorporates a Nelsonian stochastic process, the dependence on $\\hbar$ appears in a natural way as connected with the stochastic process. So to use this difference to justify a modification of the ontology remains quite nontrivial.}\n\n\\subsection{The classical limit}\n\nThere is also another strong argument for interpreting above theories in the same way -- that classical theory appears as the classical limit of quantum theory.  \n\nThe immediate consequence of this is that above theories have the same intention -- the description, as accurate as possible, of the same reality.  \n\nSo this is not a situation where the same mathematical apparatus is applied to quite different phenomena, so that it is natural use a different interpretation even if the same mathematical apparatus is applied. In our case, we use the same mathematical formalism to describe the same thing. At least in the classical limit, quantum theory has to describe the same thing -- with the same interpretation -- as the classical interpretation. \n\n\\section{What follows from the interpretation}\n\nBut there is not only the similarity between the equations, and that the object is essentially the same, which suggests to use the same interpretation for above variants of the wave function. \n\nThere are also some interesting consequences of the interpretation. And these consequences, to be shared by all wave functions which follow this interpretation, are fulfilled by the quantum wave function.  \n\n\\subsection{The relevance of the information contained in the wave function} \n\nA first consequence of the interpretation can be found considering the \\emph{relevance} of the information contained in the wave function, as information about the real beables. In fact, assume that the wave function $\\psi(q)$ contains a lot of information which is irrelevant as information about the $q$. This would open the door for some speculation about the nature of this additional information. Once it cannot be interpreted as information about the $q$, it has to contain information about some other real beables. So let's consider which of the information contained in the wave function is really relevant, tells us something about the true trajectory $q(t)$.\n\nNow, knowledge about the probability $\\rho(q)$ is certainly relevant if we don't know the real value of $q$. And it is relevant in all of its parts. \n\nThen, as we have seen, $S(q)$ gives us, via the ``guiding equation'' \\eqref{guiding}, the clearly relevant information about the value of $\\dot{q}$ given the value of $q$. Given that we don't know the true value of $q$, this information is clearly relevant. But, different from $\\rho(q)$, the function $S(q)$ contains also a little bit more: Another function $S'=S+c$ would give \\emph{exactly} the same information about the real beables. \n\nAs a consequence, the wave function $\\psi(q)$ also contains a corresponding additional, irrelevant information. The polar decomposition \\eqref{polar} defines what it is  -- a global constant phase factor.  \n\nSo we find that all the information contained in $\\psi(q)$ -- \\emph{except for a global constant phase factor} -- is relevant information about the real beables $q$. \n\nAt a first look, this seems trivial, but I think it is a really remarkable property of the paleoclassical interpretation. \n\nThe irrelevance of the phase factor is a property of the interpretation of the meaning of the wave function. It follows from this interpretation: We have considered all the ways how to obtain information about the beables from $\\psi(q)$, and we have found that all the information is relevant, except this constant phase factor. \n\nAnd now let's consider the equations. Above equations considered up to now -- the pre-Schr\\\"{o}dinger\\\/  equation as well as the Schr\\\"{o}dinger\\\/  equation -- have the same symmetry regarding multiplication with a constant phase factor: Together with $\\psi(q,t)$, $c\\psi(q,t)$ is a solution of the equations too.  \n\nThis is, of course, how it should be if the wave function describes incomplete information about the $q$. If, instead, it would describe some different, external beables, then there would be no reason at all to assume such a symmetry. In fact, why should the values of some function, values at points far away from each other, be connected with each other by such a global symmetry?  \n\nSo every interpretation which assigns the status of reality to $\\psi(q)$ has, in comparison, a problem to explain this symmetry. A popular solution is to assign reality not to $\\psi(q)$, but, instead, to the even more complicate density matrix $|\\psi\\rangle\\langle\\psi|$. The flow variable are, obviously, preferable by simplicity.\n\n\\subsection{The independence condition}\n\nIt is one of the great advantages of Jaynes' information-based approach to plausible reasoning \\cite{Jaynes} that it contains common sense principles to be applied in situations with insufficient information. If we have no information which distinguishs the probability of the six possible outcomes of throwing a die, we \\emph{have} to assign equal probability to them. Everything else would be irrational.\\footnote{The purely subjectivist, de Finetti approach differs here. It does not make prescriptions about the initial probability distributions -- all what is required is that one updates the probabilities following the rules of Bayesian updating. This difference is one of the reasons for my preference for the objective approach proposed by Jaynes \\cite{Jaynes}}  \n\nSimilar principles work if we consider the question of independence. From a frequentist point of view, independence is a quite nontrivial physical assumption, which has to be tested. And, in fact, there is no justification for it, at least none coming from the frequentist interpretation. From point of view of the logic of plausible reasoning, the situation is much better: Once we have no \\emph{information} which justifies any hypothesis of a dependence between two statements $A$ and $B$, we \\emph{have} to assume their independence. There is no information which makes $A$ more or less plausible given $B$, so we have to assign equal plausibilities to them: $P(A|B)=P(A)$.  But this is the condition of independence $P(AB)=P(A)P(B)$. \n\nThe different status of plausibilities in comparison with physical hypotheses makes this obligatory character reasonable. Probabilities are not hypotheses about reality, but logical conclusions derived from the available information, derived using the logic of plausible reasoning.\n\nNow, all this is much better explained in \\cite{Jaynes}, so why I talk here about this?  The point is that I want to obtain here a formula which appropriately describes different independent subsystems. Subsystems are independent if we have no information suggesting their dependence.  A quite typical situation, so independence is quite typical too.  \n\nSo assume we have two subsystems, and we have no information suggesting any dependence between them. What do we have to assume based on the logic of plausible reasoning? \n\nFor the probability distribution itself, the answer is trivial:  \n\\begin{equation}\n\\rho(q_1,q_2)=\\rho_1(q_1)\\rho_2(q_2).\n\\end{equation}\nBut what about the phase function $S(q)$?  We have to assume that the velocity of one system is independent of the state of the other one. So the potential of that velocity also should not depend on the state of the other system. So we obtain \n\\begin{equation}\nS(q_1,q_2) = S_1(q_1) + S_2(q_2).\n\\end{equation}\nThis gives, combined using the polar decomposition \\eqref{polar}, the following rule for the wave function:\n\\begin{equation}\n\\psi(q_1,q_2) = \\psi_1(q_1) \\psi_2(q_2). \n\\end{equation}\n\nOf course, again, the rule is trivial. Nobody would have proposed any other rule for defining a wave function in case of independence.  Nonetheless, I consider it as remarkable that it has been derived from the very logic of plausible reasoning and the interpretation of the wave function. \n\nAnd, again, this property is shared by above variants, the quantum as well as the classical one. And if one thinks about the applicability of this rule to the classical variant of the wave function, one has to recognize that it is nontrivial: From a classical point of view, the rule of combining $\\rho(q)$ and $S(q)$ into a wave function $\\psi(q)$ is quite arbitrary, and there is no a priori reason to expect that such an arbitrary combination transforms the rule for defining independent classical states into a simple rule for independent wave functions, moreover, into the one used in quantum theory. \n\n\\subsection{Appropriate reduction to a subsystem}\n\nWhile the pre-Schr\\\"{o}dinger\\\/  equation is non-linear, it shares with the Schr\\\"{o}dinger\\\/  equation the weaker property of homogenity of degree one: The operator condition \n\\begin{equation}\\label{homogenity}\n\\Omega(c\\psi) = c \\Omega(\\psi)\n\\end{equation}\nholds not only for the Schr\\\"{o}dinger\\\/  operator, but also for the non-linear pre-Schr\\\"{o}dinger\\\/  operator, and not only for $|c|=1$, as required by the $U(1)$ symmetry, which we have already justified, but for arbitrary $c$, so that the $U(1)$ symmetry is, in fact, a $GL(1)$ symmetry of the equations. \n\nSo one ingredient of linearity is shared by the pre-Schr\\\"{o}dinger\\\/  equation. This suggests that for this property there should be a more fundamental explanation, one which is shared by above equations. And, indeed, such an explanation is possible. There should be a principle which allows an appropriate reduction of the equation to subsystems, something like the following\n\n\\begin{principle}[splitting principle]\nThere should a be simple ``no interaction'' conditions for the operators on a system consisting of two subsystems of type $\\Omega=\\Omega_1+\\Omega_2$ such that the equation $\\pd_t\\psi=\\Omega(\\psi)$ splits for independent product states $\\psi(q_1,q_2)=\\psi_1(q_1)\\psi_2(q_2)$ into independent equation's for the two subsystems. \n\\end{principle}\n\nNow, the time derivative splits nicely:\n\\begin{equation}\n\\pd_t \\psi(q_1,q_2)= \\pd_t\\psi_1(q_1)\\psi_2(q_2) + \\psi_1(q_1)\\pd_t\\psi_2(q_2).\n\\end{equation}\nIt remains to insert the equations for the whole system $\\pd_t \\psi = \\Omega(\\psi)$ as well as for the two subsystems into this equation. This gives:\n\\begin{equation}\n\\Omega(\\psi_1(q_1)\\psi_2(q_2)) = \\Omega_1(\\psi_1(q_1))\\psi_2(q_2) + \\psi_1(q_1)\\Omega_2(\\psi_2(q_2)),\n\\end{equation}\nwhere the $\\Omega_i$ are subsystem operators acting only on the $q_i$, so that functions of the other variable are simply constants for them. On the other hand, in the splitting principle we have assumed $\\Omega = \\Omega_1+\\Omega_2$. Comparison gives\n\\begin{equation}\n\\begin{split}\n\\Omega_1(\\psi_1(q_1)\\psi_2(q_2)) &= \\Omega_1(\\psi_1(q_1))\\psi_2(q_2),\\\\\n\\Omega_2(\\psi_1(q_1)\\psi_2(q_2)) &= \\psi_1(q_1)\\Omega_2(\\psi_2(q_2)).\n\\end{split}\n\\end{equation}\nThis follows from the homogenity condition \\eqref{homogenity}. The weaker $U(1)$ symmetry is not sufficient, because the values of $\\psi_2(q_2)$ may be arbitrary complex numbers. \n\nSomething similar to the splitting property is necessary for any equation relevant for us -- we have not enough information to consider the equation of the whole universe, thus, have to restrict ourself to small subsystems. And the equations of these subsystems should be at least approximately independent of the state of the environment. \n\nSo the homogenity of the equations -- in itself a quite nontrivial property of the equations, given the definition of the wave function by polar decomposition -- can be explained by this splitting property. \n\n\\subsection{Summary}\n\nSo we have found three points, each in itself quite trivial, but each containing some nontrivial element of explanation based on the paleoclassical principles: The global $U(1)$ phase shift symmetry of the wave function, explained by the irrelevance of the phase as information about $q(t)$, the product rule of independence, explained by the logic of plausible reasoning, applied to the set of information described by the wave function, and the homogenity of the equations, explained in terms of a splitting property for independent equations for subsystems. \n\nAll three points follow the same scheme -- the interpretation in terms of incomplete information allows to derive a rule, and this rule is shared by above theories, quantum theory as well as the wave function variant of classical theory.  \n\nSo all three points give additional evidence that the simple, straightforward proposal to use the same interpretation of the wave function for above theories is the correct one.\n\n\\section{Incorporating Nelsonian stochastics}\n\nThe analogy between pre-Schr\\\"{o}dinger\\\/  and Schr\\\"{o}dinger\\\/  equation is a useful one, but it is that useful especially because the equations are nonetheless very different. And one should not ignore these differences. \n\nOne should not, in particular, try to interpret the Schr\\\"{o}dinger\\\/  equation as the linear approximation of the pre-Schr\\\"{o}dinger\\\/  equation: The pre-Schr\\\"{o}dinger\\\/  equation does not depend on $\\hbar$. Real physical effects do depend. And so there should be also something in the fundamental theory, the theory in terms of the trajectory $q(t)$, which depends on $\\hbar$. \n\nHere, Nelsonian stochastics comes into mind. In Nelsonian stochastics the development of the configuration $q$ in time is described by a deterministic drift term $b^i(q(t),t)dt$ and a stochastic diffusion term $dB^i_t$\n\\begin{equation}\\label{process}\ndq^i(t) = b^i(q(t),t) dt + dB^i_t,\n\\end{equation}\nwith  is a classical Wiener process $B^i_t$ with expectation $0$ and variance\n\\begin{equation}\n\\langle dB^i_t m_{ij} dB^j_t \\rangle = \\hbar dt,\n\\end{equation}\nso that we have a $\\hbar$-dependence on the fundamental level. The probability distribution $\\rho(q(t),t)$ has, then, to fulfill the Fokker-Planck-equation:\n\\begin{equation}\\label{FokkerPlanck}\n\\partial_t \\rho + \\pd_i (\\rho b^i) - \\frac{\\hbar}{2}m^{ij}\\pd_i\\pd_j\\rho = 0\n\\end{equation}\nFor the average velocity $v^i$ one obtains\n\\begin{equation}\\label{vdef}\nv^i(q(t),t) = b^i(q(t),t) - \\frac{\\hbar}{2}\\frac{m^{ij}\\pd_j \\rho(q(t),t)}{\\rho(q(t),t)},\n\\end{equation}\nwhich fulfills the continuity equation. The difference between flow velocity $b^i$ and average velocity $v^i$, the osmotic velocity\n\\begin{equation}\\label{osmotic}\nu^i(q(t),t) = b^i(q(t),t)-v^i(q(t),t) = \\frac{\\hbar}{2}\\frac{m^{ij}\\pd_j\\rho(q(t),t)}{\\rho(q(t),t)} = \\frac{\\hbar}{2}m^{ij}\\pd_j \\ln\\rho(q(t),t)\n\\end{equation}\nhas a potential $\\ln\\rho(q)$. The average acceleration is given by\n\\begin{equation}\\label{acceleration}\na^i(q(t),t) = \\partial_t v^i + v^j \\pd_j v^i - \\frac{\\hbar^2}{2} m^{ij}\\pd_j\n        \\left(\\frac{m^{kl}\\pd_k\\pd_l \\sqrt{\\rho(q(t),t)}}{\\sqrt{\\rho(q(t),t)}}\\right)\n\\end{equation}\nFor this average acceleration the classical Newtonian law\n\\begin{equation}\\label{Newton}\na^i(q(t),t) = - m^{ij} \\pd_j V(q(t),t)\n\\end{equation}\nis postulated. Putting this into the equation \\eqref{acceleration} gives\n\\begin{equation}\\label{derivativeBohm}\n\\partial_t v^i + (v^j \\pd_j)v^i =\n - m^{ij} \\pd_j \\left(V -\\frac{\\hbar^2}{2}\n       \n        \\frac{\\Delta \\sqrt{\\rho}}{\\sqrt{\\rho}}\n\\right) = - m^{ij}\\pd_j(V + Q[\\rho]).\n\\end{equation}\n\nThe next postulate is that the average velocity $v^i(q)$ has, at places where $\\rho>0$, a potential, that means, a function $S(q)$, so that \\eqref{guiding} holds. Then equation \\eqref{derivativeBohm} can be integrated (putting the integration constant into $V(q)$), which gives \\eqref{Bohm}. Finally, one combines the equations for $S(q)$ and $\\rho(q)$ into the Schr\\\"{o}dinger\\\/  equation as in de Broglie-Bohm theory. \n\nSo far the basic formulas of Nelsonian stochastics. Now, the beables of Nelsonian stochastics are quite different from those of the paleoclassical interpretation. The external flow $b^i(q,t)$ does not define some state of information, but some objective flow which takes away the configuration if it is at $q$. The configuration is guided in a way close to a swimmer in the ocean: He can randomly swim into one or another direction, but whereever he is, he is always driven by the flow of the ocean, a flow which exists independent of the swimmer. \n\nWhat would be the picture suggested for a stochastic process by the paleoclassical interpretation? It would be different, more close to a spaceship flying in the cosmos. If it changes its place because of some stochastic process, there will be no predefined, independent velocity $b^i(q)$ at the new place which can guide it. Instead, it would have to follow its previous velocity. Indeed, the velocity field $v^i(q)$, and, therefore, also $b^i(q)$, of the new place describes only information, not a physical field which could possibly influence the spaceship in its new position. \n\nBut what follows for the information if there is some probability that the stochastic process causes the particle to change its location to the new one? It means that the old average velocity has to be recomputed. \n\nOf course, there is such a necessity to recompute only if the velocity at the new location is different from the old one.  So, if $\\pd_i v^k=m^{jk}\\pd_i p_j=0$, there would be no need for such a recomputation at all. This condition is clearly too strong, it would cause the whole velocity field to become constant. Now there is an interesting subcondition, namely that $\\pd_i p_j-\\pd_j p_i=0$, the condition that the velocity field is a potential one. So the re-averaging of the average velocity $v^i(q)$ caused by the stochastic process will decrease the curl.\n\nThis gives a first advantage in comparison with Nelsonian stochastics: The potentiality assumption does not have to be postulated, without any justification, for the external flow $b^i(q,t)$. (It is postulated for average velocity, but their difference -- the osmotic velocity -- has a potential anyway.) In the paleoclassical interpretation we have an independent motivation for postulating this. But, let's recognize, there is no fundamental reason to postulate potentiality: On the fundamental level, the curl may be nontrivial. All what is reached is that the assumption of potentiality is not completely unmotivated, but that it appears as a natural consequence of the necessary re-averaging.\n\nThere is another strangeness connected with the external flow picture of Nelsonian stochastics. The probability distribution $\\rho(q)$ already characterizes the information about the configuration itself -- it is a probability for the swimmer in the ocean, not for the ocean. Once they depend on $\\rho(q)$, as the average velocity $v^i(q)$, as the average acceleration $a^i(q)$ also describe information about the swimmer, not about the ocean. Then, it is postulated that the average acceleration of the swimmer has to be defined by the Newtonian law. But because this average velocity is, essentially, already defined by the very process -- the flow of the ocean -- together with the initial value for the swimmer, this condition for the swimmer becomes an equation for the ocean. This is conceptually unsound -- as if the ocean has to care that the swimmer is always correctly accelerated.  \n\nBut this conceptual inconsistency disappears in the paleoclassical interpretation. The drift field is now part of the incomplete information about the configuration itself, as defined by the average velocity and osmotic velocity. There is no longer any external drift field. And, so, it is quite natural that a condition for the average acceleration of the configuration gives an equation for the average velocity $v^i(q)$. So the paleoclassical picture is internally much more consistent.\n\nBut is it viable? This is a quite non-trivial question discussed below in sec. \\ref{viability}. \n\n\\section{The character of the wave function}\n\nLet's start with the consideration of the objections against the paleoclassical interpretation. Given that the basic formulas are not new at all, I do not have to wait for reactions to this paper -- some quite strong arguments are already well-known.\n\nThe first one is the evidence in favour of the thesis that the wave function describes the behaviour of real degrees of freedom, degrees of freedom which actually influence the things we can observe immediately. Here, at first Bell's argumentation comes into mind -- an argumentation for a double ontology which, I think, has impressed many of those who support today realistic interpretations: \n\\begin{quote}\nIs it not clear fro the smalleness of the scintillation on the screen that we have to do with a particle? And is it not clear, from the diffraction and interference patterns, that the motion of the particle is directed by a wave? \n(\\cite{Bell} p. 191).\\end{quote}\nBut what are the points which make this argument that impressive? What is it what motivates us to accept some things as real? Here I see no way to express this better than Brown and Wallace: \n\\begin{quote}\nFrom the corpuscles' perspective, the wave-function is just a (time-dependent) function on their configuration space, telling them how to behave; it superficially appears similar to the Newtonian or Coulomb potential field, which is again a function on configuration space. No-one was tempted to reify the Newtonian potential; why, then, reify the wave-function?\n\nBecause the wave-function is a very different sort of entity. It is contingent (equivalently, it has dynamical degrees of freedom independent of the corpuscles); it evolves over time; it is structurally overwhelmingly more complex (the Newtonian potential can be written in closed form in a line; there is not the slightest possibility of writing a closed form for the wave-function of the Universe.) Historically, it was exactly when the gravitational and electric fields began to be attributed independent dynamics and degrees of freedom that they were reified: the Coulomb or Newtonian `fields' may be convenient mathematical fictions, but the Maxwell field and the dynamical spacetime metric are almost universally accepted as part of the ontology of modern physics.\n\nWe don't pretend to offer a systematic theory of which mathematical entities in physical theories should be reified. But we do claim that the decision is not to be made by fiat, and that some combination of contingency, complexity and time evolution seems to be a requirement.\n(\\cite{BrownWallace} p. 12-13)\\end{quote}\n\nSo, let's consider the various points in favour of the reality of the wave function:\n\n\\subsection{Complexity of the wave function} \n\nThe argument of complexity seems powerful. But, in fact, for an interpretation in terms of incomplete information \\emph{this} is not a problem at all. \n\nComplexity is, in fact, a natural consequence of incompleteness of information.  The complete information about the truth of a statement is a single bit:  true or false.  The incomplete information is much more complex: it is a real number, the probability.  \n\nAs well, the complete information about reality in this interpretation is simple: a single trajectory $q(t)$.  But incomplete information requires much more information: Essentially, we need probabilities for all possible trajectories.  \n\n\\subsection{Time dependence of the wave function} \n\nTime dependence is, as well, a natural property of information -- complete or incomplete.  The information about where the particle has been yesterday transforms into some other information about where the particle is now. \n\nThis transformation is, moreover, quite nontrivial and complex. \n\nIt is also worth to note here that the law of transformation of information is a derivative of the real, physical law of the behaviour of the real beables.  \n\nSo it necessarily has all the properties of a physical law. \n\nWe can, in particular, use the standard Popperian scientific method (making hypotheses, deriving predictions from them, testing and falsifying them, and inventing better hypotheses) to find these laws. \n\nThis is, conceptually, a quite interesting point: The laws of probability themself are best understood, following Jaynes \\cite{Jaynes}, as laws of extended logic, of the logic of plausible reasoning. \n\nBut, instead, the laws of \\emph{transformation} of probabilities in time follow from the laws of the original beables in time, and, therefore, have the character of physical laws.\n\nOr, in other words, incomplete information develops in time in a way indistinguishable from the development in time of real beables. In particular, we use the same objective scientific methods to find and test them. \n\nSo, nor the simple fact that there is a nontrivial time evolution, nor the very physical character of the dynamical laws, in all details, up to the scientific method we use to find them, give any argument against an interpretation in terms of incomplete information.  All this is quite natural for the evolution of incomplete information too. \n\n\\subsection{The contingent character of the wave function}\n\nThere remains the most powerful argument in favour of the reality of the wave function: Its contingent character.  \n\nThere are different wave functions, and these different wave functions lead to objectively different probability distributions of the observable results.  \n\nIf we have, in particular, different preparation procedures, leading to different interference pictures, we really observe different interference pictures. It is completely implausible that these different interference pictures -- quite objective pictures -- could be the result of different sets of incomplete information about the same reality. The different interference pictures are, clearly, the result of different things happening in reality.  \n\nBut, fortunately, there is not even a reason to disagree with this. The very point is that one has to distinguish the wave function of a small subsystem -- we have no access to any other wave functions -- from the wave function of a closed system. The last is, in fact, only an object of purely theoretical speculation, because there is no closed system in nature except the whole universe, but we have no idea about the wave function of the whole universe. \n\nFor the wave function of a small subsystem, the situation is quite different. It does not contain only incomplete information about the subsystem. In fact, it is only an effective wave function, and there is a nice formula of dBB theory, which can be used in our paleoclassical interpretation too: The formula which defines the conditional wave function of a subsystem $\\psi^S(q_S,t)$ in terms of the wave function of the whole system (say the whole universe) $\\psi(q_S,q_E,t)$ and the configuration of the environment $q_E(t)$:\n\\begin{equation}\\label{conditional}\n\\psi^S(q_S,t) = \\psi(q_S,q_E(t),t)\n\\end{equation}\nThis is a remarkable formula of dBB theory which contains, in particular, the solution of the measurement problem: The evolution of $\\psi^S(q_S,t)$ is, in general, not described by a Schr\\\"{o}dinger\\\/  equation -- if there is interaction with the environment, the evolution of $\\psi^S(q_S,t)$ is different, but, nonetheless, completely well-defined.  And this different evolution is the collapse of the wave function caused by measurement. \n\nLet's note that the paleoclassical interpretation requires to justify this formula in terms of the information about the subsystem. But this is not a problem. Indeed, assume the trajectory of the environment $q_E(t)$ is known -- say by observation of a classical, macroscopic measurement device. Then the combination of the knowledge described by the wave function of the whole system with the knowledge of $q_E(t)$ gives exactly the same knowledge as that described by $\\psi^S(q_S,t)$. Indeed, the probability distribution gives\n\\begin{equation}\n\\rho^S(q_S,t) = \\rho(q_S,q_E(t),t),\n\\end{equation}\nand, similarly, the velocity field defined by $S(q)$ follows the same reduction principle:\n\\begin{equation}\n\\nabla S^S(q_S,t) = \\nabla S(q_S, q_E(t),t).\n\\end{equation}\nSo in the paleoclassical interpretation the dBB formula the conditional wave function of a subsystem is a logical necessity. This provides yet another consistency check for the interpretation. \n\nBut the reason for considering this formula here was a different one: The point is that the wave function of the subsystem in fact contains important information about other real beables -- the actual configuration of the whole environment $q_E(t)$.  So there are real degrees of freedom, different from the configuration of the system $q_S(t)$ itself, which are distinguished by different wave functions $\\psi^S(q_S,t)$. \n\nAnd we do not have to object at all if one argues that the wave function contains such additional degrees of freedom. That's fine, it really contains them. These degrees of freedom are those of the configuration of the environment $q_E(t)$.  And this is not an excuse, but a logical consequence of the interpretation itself, a consequence of the definition of the conditional wave function of the subsystem \\eqref{conditional}.\n\n\\subsection{Conclusion}\n\nSo the consideration of the arguments in favour of a beable status for the wave function, even if they seem to be strong and decisive at a first look, appear to be in no conflict at all with the interpretation of the wave function of a closed system in terms of incomplete information about this system.\n\nThe most important point to understand this is, of course, the very fact that the conditional wave function of the small subsystems of the universe we can consider really have a different character -- they depend on the configuration of the environment. And, so, the argumentation that these conditional wave functions describe real degrees of freedom, external to the system itself, is accepted and even derived from the interpretation.  \n\nNonetheless, the point that nor the complexity of the wave function of the whole universe, nor its evolution in time, nor the physical character of the laws of this evolution are in any conflict with an interpretation in terms of incomplete information is an important insight too. \n\n\\section{The Wallstrom objection}  \n\nWallstrom \\cite{Wallstrom} has made an objection against giving the fields of the polar decomposition $\\rho(q)$, $S(q)$ (instead of the wave function $\\psi(q)$ itself) a fundamental role.  \n\nThe first point is that around the zeros of the quantum mechanical wave function, the flow has no longer a potential. The quantum flow is a potential one only where $\\rho(q)>0$. But in general there will be submanifolds of dimension $n-2$ where the wave function is zero. And for a closed path $q(s)$ around such a zero submanifold one finds that\n\\begin{equation}\\label{curl}\n\\oint m_{ij} v^i(q) dq^j \\neq 0.\n\\end{equation}\n\nThis, in itself, is unproblematic for the interpretation: The condition of potentiality is not assumed to be a fundamental one -- the fundamental object is not $S(q)$ but the $v^i(q)$. There will be some mechanism in subquantum theory which locally reduces violations of potentiality, so we can assume that the flow is a potential one only as an approximation. \n\nIt is also quite natural to assume that such a mechanism works more efficient for higher densities and fails near the zeros of the density. \n\nSo having them localized at the zeros of the density is quite nice -- not for a really fundamental equation, which is not supposed to have any infinities, but if we consider quantum theory as being only an approximation.  \n\nThe really problematic part of the Wallstom objection is a different one: It is that the quantum flow has to fulfill a  nontrivial \\emph{quantization condition}, namely\n\\begin{equation}\\label{curlquantization}\n\\oint m_{ij} v^i(q) dq^j = \\oint \\pd_j S(q) dq^j = 2\\pi m\\hbar, \\qquad m\\in\\mbox{$\\mathbb{Z}$}.\n\\end{equation}\nThe point is, in particular, that the equations \\eqref{Bohm}, \\eqref{continuity} in flow variables are not sufficient to derive this quantization condition. So, in fact, this set of equations is \\emph{not} empirically equivalent to the Schr\\\"{o}dinger\\\/  equation. \n\nThis is, of course, no wonder, given the fact that the equivalence holds only for $\\rho(q)=|\\psi(q)|^2>0$. But, however natural, empirical inequivalence is empirical inequivalence. \n\nThen, this condition looks quite artificial in terms of the $v^i$. What is a triviality in terms of the wave function -- that it has to globally uniquely defined -- becomes extremely artificial and strange if formulated in terms of the $v^i(q)$. As Wallstrom \\cite{Wallstrom} writes, to ``the best of my knowledge, this condition [\\eqref{curlquantization}] has not yet found any convincing explanation outside the context of the Schr\\\"{o}dinger\\\/  equation''.  \n\n\\subsection{A solution for this problem} \n\nFortunately I have found a solution for this problem in \\cite{againstWallstrom}. I do not claim that it is a complete one -- there is a part which is beyond the scope of an interpretation, which has to be left to particular proposals for a subquantum theory. One has to check if the assumptions I have made about such a subquantum theory are really fulfilled in that particular theory.  \n\nThe first step of the solution is to recognize that, for empirical equivalence with quantum theory, it is sufficient to recover only solutions with simple zeros. Such simple zeros give $m=\\pm 1$ in the quantization condition \\eqref{curlquantization}. This is a consequence of the principles of general position: A small enough modification of the wave function cannot be excluded by observation, but leads to a wave function in general position, and in general position the zeros of the wave function are non-degenerated. \n\nThe next step is a look at the actual solutions. For the simple, two dimensional, rotational invariant, zero potential case these solutions are defined by $S(q)=m\\varphi$, $\\rho(q)=r^{2|m|}$. And this extends to the general situation, where $S(q)=m\\varphi+\\tilde{S}(q)$,  $\\rho(q)=r^{2|m|}\\tilde{\\rho}(q)$, such that $\\tilde{S}(q)$ is well-defined in a whole environment of the zero, and $\\tilde{\\rho}(0)>0$.\n\nBut that means we can replace the problem of justifying an integer $m$ in $S(q)=m\\varphi$, where all values of $m$ seem equally plausible, by the quite different problem of justifying $\\rho(q)=r^2$ (once we need only $m=\\pm 1$) in comparison with other $\\rho(q)=r^\\a$. This is already a quite different perspective. \n\nWe make the natural conclusion and invent a criterion which prefers $\\rho(q)=r^2$ in comparison with other $r^\\a$.  This is quite easy:  \n\n\\begin{postulate}[regularity of $\\Delta\\rho$]\\label{preg}\nIf $\\rho(q)=0$, then $0< \\Delta \\rho(q) < \\infty$ almost everywhere. \n\\end{postulate}\n\nThis postulate already solves the inequivalence argument. The equations \\eqref{Bohm}, \\eqref{continuity} for $\\rho(q)>0$, together with postulate \\ref{preg}, already defines a theory empirically equivalent to quantum theory (even if the equivalence is not exact, because only solutions in general position are recovered). \n\nIt remains to invent a justification for this postulate. \n\nThe next step is to rewrite equation \\eqref{Bohm} for stable states in form of a balance of energy densities. In particular, we can rewrite the densitized quantum potential as\n\\begin{equation}\\label{Q}\nQ[\\rho]\\rho = \\frac12\\rho u^2 - \\frac14 \\Delta \\rho,\n\\end{equation}\nwith the ``osmotic velocity'' $u(q) = \\frac12 \\nabla \\ln \\rho(q)$. Then the energy balance looks like\n\\begin{equation}\n\\frac12\\rho v^2 + \\frac12 \\rho u^2 + V(q) = \\frac14\\Delta\\rho.\n\\end{equation}\n\nSo, the operator we have used in the postulate is not an arbitrary expression, but a meaningful term, which appears in an important equations -- an energy balance.  This observation is already sufficient to justify the $\\Delta \\rho(q) < \\infty$ part of the condition. There may be, of course, subquantum theories which allow for infinities in energy densities, but it is hardly a problem for a subquantum theory to justify that expressions which appear like energy densities in energy balances have to be finite.  \n\nLast but not least, subquantum theory has to allow for nonzero curl $\\nabla\\times v$, but has to suppress it to obtain a quantum limit. One way to suppress it is to add a penalty term $U(\\rho,\\nabla\\times v)$ which increases with $|\\nabla\\times v|$. This would give\n\\begin{equation}\n\\frac12\\rho v^2 + \\frac12 \\rho u^2 + V(q) + U(\\rho,\\nabla\\times v) = \\frac14\\Delta\\rho.\n\\end{equation}\nMoreover subquantum theory has to regularize the infinities of $v$ and $u$ at the zeros of the density. One can plausibly expect that this gives finite but large values of $|\\nabla\\times v|$ at zero which decrease sufficiently fast with $r$. Now, a look at the energy balance shows that, if the classical potential term $V(q)$ is neglected (for example assuming that it changes only smoothly) the only term which can balance $U(\\rho,\\nabla\\times v)$ at zero is the $\\Delta \\rho$ terms, which, therefore, has to be finite but nonzero. Or, at least, it would not be difficult to modify the definition of $U(\\rho,\\nabla\\times v)$ in such a way that the extremal value $\\Delta\\rho=0$ (we have necessarily $\\Delta\\rho\\ge 0$ at the minima) has to be excluded. \n\nSo the postulate seems nicely justifiable. For some more details I refer to \\cite{againstWallstrom}. What remains is the particular job of particular proposals for subquantum theories -- they have to check if the way to justify the postulate really works in this theory, or if it may be justified in some other way. But this is beyond the scope of the interpretation. What has to be done by the interpretation -- in particular, to obtain empirical equivalence with quantum theory -- has been done.\n \n\\section{A Popperian argument for preference of an information-based interpretation}\n\nOne of Popper's basic ideas was that we should prefer -- as long as possible without conflict with experience -- theories which are more restrictive, make more certain predictions, depend on less parameters.  And, while this criterion has been formulated for theories, it should be applied, for the same reasons, to more general principles of constructing theories too. \n\nThis gives an argument in preference for an interpretation in terms of incomplete information. \n\nIndeed, let's consider, from this point of view, the difference between interpretations of fields $\\rho(q)$, $v^i(q)$ in terms of a probability for some real trajectories $q(t)$, and interpretations which reify them as describing some external reality, different from $q(t)$, which influences the trajectory $q(t)$.  \n\nIt is quite clear and obvious which of the two approaches is more restrictive. Designing theories of the first type, we are restricted, for the real physics, to theories for single trajectories $q(t)$. Then, given that we have identified the connection between the fields $\\rho(q)$, $v^i(q)$ and $q(t)$ as those of a probability flow, everything else follows. There is no longer any freedom of choice for the field equations. If we have fixed the Hamiltonian evolution for $p(t), q(t)$, the Liouville field equation for $\\rho(p,q)$ is simply a logical consequence. Similarly, the continuity equation \\eqref{continuity} is a law of logic, it cannot be modified, is no longer a subject of theoretical speculation. It is fixed by the interpretation of $\\rho(q)$, $v^i(q)$ as a probability flow, in the same way as $\\rho(q)\\ge 0$ is fixed. \n\nIn the second case, we have much more freedom -- the full freedom of speculation about field theories in general. In particular, the continuity equation can be modified, introducing, say, some creation and destruction processes, which are quite natural if $\\rho(q)$ describes a density of some external objects. \n\nThe derivation of the $U(1)$ global phase shift symmetry is another particular example of such an additional logical law following from the interpretation.  \n\nSo there are some consequences of the interpretation which have purely logical character, including the continuity equation, $\\rho(q)\\ge 0$, and the $U(1)$ global phase shift symmetry.  But these will not be the only consequences. The other equations will be restricted, in comparison with field theories, too, but in a less clear and obvious way. There is, last but not least, a large freedom of choice for the equations of the real beables $q(t)$, which correspondence to a similarly large freedom of choice of the resulting equations for $\\rho(q)$, $v^i(q)$. But this freedom of choice will be, nonetheless, much smaller than the completely arbitrariness of a general field theory.  \n\nThis consideration strongly indicates that we have to prefer the interpretation in terms of incomplete information until it has been falsified, until it appears incompatible with observation. \n\nThe immediate, sufficiently trivial logical consequences we have found yet are compatible with the Schr\\\"{o}dinger\\\/  equation and therefore with observation. So we should prefer this interpretation. \n\n\\section{Open problems}\n\nInstead of using such a Popperian argumentation, I could have, as well, used simply Ockham's razor: Don't multiply entities without necessity. Once, given this interpretation, there is no necessity for more than a single classical trajectory $q(t)$, one should not introduce other real entities like really existing wave functions. \n\nBut the Popperian consideration has the advantage that it implicitly defines an interesting research program.\n\n\\subsection{Other restrictions following from the interpretation}\\label{viability}\n\nIn fact, given the restrictive character of the interpretation, there may be other, additional, more subtle restrictions of the equations for probility flows $\\rho(q)$, $v^i(q)$, restrictions which we have not yet identified, but which plausibly exist. \n\nSo what are these additional restrictions for equations for probability flows $\\rho(q)$, $v^i(q)$ in comparison with four general, unspecific fields fulfilling a continuity equation? I have no answer. \n\nThis is clearly an interesting question for future research. It is certainly also a question interesting in itself, interesting from point of view of pure mathematics, for a better understanding of probability theory.  \n\nThe consequences may be fatal for this approach -- it may be that we find that the Schr\\\"{o}dinger\\\/  equation does not fit into this set of restrictions. This possibility of falsifiction is, of course, the very point of the Popperian consideration. I'm nonetheless not afraid that this happens, but this is only a personal opinion. \n\nThe situation may be, indeed, much better: That this subclass of theories contains the Schr\\\"{o}dinger\\\/  equation, but appears to be heavily restricted by some additional, yet unknown, conditions. Then, all of these additional restrictions give us partial answers to the ``why the quantum'' question. \n\n\\subsection{Why is the Schr\\\"{o}dinger\\\/  equation linear?}\n\nThe most interesting question which remains open is why the Schr\\\"{o}dinger\\\/  equation is linear. We have found only some part of the answer -- an explanation for the global $U(1)$ phase symmetry based on the informational content and of the homogeneity based on the reduction of the equation to subsystems. \n\nBut, given that the pre-Schr\\\"{o}dinger\\\/  equation is non-linear, but interpreted in the same way, the linearity of the Schr\\\"{o}dinger\\\/  equation cannot follow from the interpretation taken alone. Some other considerations are necessary to obtain linearity.\n\nOne idea is to justify linearity as an approximation. Last but not least, linearization is a standard way to obtain approximations. \n\nThe problem of stability in time may be also relevant here. The pre-Schr\\\"{o}dinger\\\/  equation becomes invalid after a short period of time, then the first caustic appears. There is no such problem in quantum theory, which has a lot of stable solutions. Then, there should be not only stable solutions, but also slowly changing solutions: It doesn't even matter if the fundamental time scale is Planck time or something much larger -- even if it is only the time scale of strong interactions, all the things changing around us are changing in an extremely slow way in comparison with this fundamental time scale. But the linear character of the Schr\\\"{o}dinger\\\/  equation gives us a way to obtain solutions slowly changing in time by combining different stable solutions with close energies. \n\n\\section{A theoretical possibility to test: The speedup of quantum computers}\\label{computer}\n\nThere is an idea which suggests, at least in principle, a way to distinguish observationally the paleoclassical interpretation (or, more accurate, the class of all more fundamental theories compatible with the paleoclassical interpretation) from the minimal interpretation. \n\nThe idea is connected with the theory of quantum computers. If quantum computers really work as predicted by quantum theory, these abilities will provide fascinating tests of the accuracy of quantum theory. In the case of Simon's algorithm, the speed-up is exponential over any classical algorithm. It may be a key for the explanation of this speed-up that the state space (phase space) of a composite classical system is the Cartesian product of the state spaces of its subsystems, while the state space of a composite quantum system is the tensor product of the state spaces of its subsystems. For $n$ qubits, the quantum state space has $2^n$ instead of $n$ dimensions. So the information required to represent a general state increases exponentially with $n$ (see, for example, \\cite{Bub}). There is also the idea ``that a quantum computation is something like a massively parallel classical computation, for all possible values of a function. This appears to be Deutsch's view, with the parallel computations taking place in parallel universes.'' \\cite{Bub}. \n\nIt is this \\emph{exponential} speedup which suggests that the predictions of standard QM may differ from those of the paleoclassical interpretation. An exact quantum computer would have all beables contained in the wave function as degrees of freedom.  A quantum computer in the paleoclassical interpretation has only the resources provided by its beables.  But these beables are, essentially, only the classical states of the universe. Given the exponential difference between them, $n$ vs. $2^n$ dimensions for qubits instead of classical bits, an exact quantum computer realizable at least in principle in a laboratory on Earth can have more computational resources than a the corresponding computer of the paleoclassical interpretation, which can use only the classical degrees of freedom, even if these are the classical degrees of freedom of the whole universe. \n\nBut if we distort a quantum computer, even slightly, the result will be fatal for the computation. In particular, if this distortion is of the type of the paleoclassical interpretation, which replaces an exact computer with a $2^n$-dimensional state space by an approximate one with only $N$ dimensions, then even for quite large $N\\gg n$ the approximate computer will be simply unable to do the exact computations, even in principle. There simply are no parallel universes in the paleoclassical interpretation to make the necessary parallel computations. \n\nSo, roughly speaking, the prediction of the paleoclassical interpretation is that a sufficiently large quantum computer will fail to give the promised exponential speedup. The exponential speedup will work only up to a certain limit, defined by the logarithm of the relation between the size of the whole universe and the size of the quantum computer. \n\nOf course, we do not know the size of the universe. It may be much larger than the size of the observable universe, or even infinite. Nonetheless, this argument, applied to any finite model of the universe, shows that the true theory, the theory in the configurational beables alone, cannot be exactly quantum theory. This is in my opinion the most interesting conclusion. \n\nBut let's see if we can, nonetheless, make even testable (at least in principle) predictions. So let's presuppose that the universe is finite, and, moreover, let's assume that this size is not too many orders larger than the its observable part. This would be already sufficient to obtain some numbers about the number of qubits such that the $2^n$ exponential speedup is no longer possible. This number will be sufficiently small, small enough that a quantum computer in a laboratory on Earth will be sufficient to reach this limit. \n\nAnd, given the logarithmic dependence on $N$, increasing $N$ does not help much. If it is possible to build a quantum computer with $n$ qubits, why not with $2n$? This would already move the size of the universe into completely implausible regions. \n\n\\subsection{The speed of quantum information as another boundary}\n\nInstead of caring about the size of the universe, it may be more reasonable to care about the size of the region which can causally influence us. Here I do not have in mind the limits given by of relativity, by the speed of light. Given the violation of Bell's inequality, there has to be (from a realist's point of view) a hidden preferred frame where some other sort of information -- quantum information -- is transferred with a speed much larger than the speed of light.  But if we assume that the true theory has some locality properties, even if only in terms of a much larger maximal speed, the region which may be used by a quantum computer for its computational speedup decreases in comparison with the size of the universe. \n\nSo if we assume that there is such a speed limit for quantum information, then we obtain in the paleoclassical interpretation even more restrictive limits for the speedup reachable by quantum computers, limits which depend logarithmically on the speed limit for quantum information. \n\nNonetheless, I personally don't believe that quantum computers will really reach large speedups. I think the general inaccuracy of human devices will prevent us from constructing quantum computers which can really use the full $2^n$ power for large enough $n$. I would guess that the accuracy requirements necessary to obtain a full $2^n$ speedup will also grow exponentially. So I guess that quantum computers will fail already on a much smaller scale. \n\n\\section{Conclusions}\n\nSo it's time to summarize:\n\n\\begin{itemize}\n\n\\item The unknown, true theory of the whole universe is a theory defined on the classical configuration space $Q$, with the configuration $q(t)$ evolving in absolute time $t$ as a complete description of all beables. \n\n\\item The wave function of the whole universe is interpreted as a consistent set of incomplete information about these fundamental beables. \n\n\\item In particular, $\\rho(q)$ defines not some ``objective'' probability, but an incomplete set of information about the real position $q$, described, as required by the logic of plausible reasoning, by a probability distribution $\\rho(q)dq$.  \n\n\\item The phase $S(q)$ describes, via the ``guiding equation'', the expectation value $\\langle\\dot{q}\\rangle$ of the velocity given the actual configuration $q$ itself. So the ``guiding equation'' is not a physical equation, but has to be interpreted as part of the definition of $S(q)$, which describes which information about $q$ is contained in $S(q)$.    \n\n\\item Only a constant phase factor of $\\psi(q)$ does not contain any relevant information about the trajectory $q(t)$. Therefore, the equations for $\\psi(q)$ should not depend on such a factor. \n\n\\item The Schr\\\"{o}dinger\\\/  equation is interpreted as an approximate equation. More is not to be expected, given that it describes the evolution of an incomplete set of information. \n\n\\item The linear character of the Schr\\\"{o}dinger\\\/  equation is interpreted as an additional hint that it is only an approximate equation. \n\n\\item The interpretation can be used to reinterpret Nelsonian stochastics. The resulting picture is conceptually more consistent than the original proposal. \n\n\\item The Wallstrom objection appears much less serious than expected. The quantization condition for simple zeros (which is sufficient because it is the general position) can be derived from the much simpler regularity postulate that $0<\\Delta \\rho(q)<\\infty$ if $\\rho(q)=0$. While a final justification of this condition has to be left to a more fundamental theory, it is, as shown in \\cite{againstWallstrom}, plausible that this is not a problem for such theories. \n\n\\item If the true theory of the universe is defined on classical configurations, and the whole universe is finite, quantum computers can give their promised exponential speedup only up to an upper bound for the number of qubits, which is much smaller than the available qubits of the universe. This argument shows that the Schr\\\"{o}dinger\\\/  equation has to be approximate. \n\n\\item The dBB problem with of the ``action without reaction'' asymmetry is solved: For effective wave function, the collapse defines the back-reaction, for the wave function of the whole universe there should be no such back-reaction -- it is only an equation about incomplete information about reality, not about reality itself. \n\n\\item The wave functions of small subsystems obtain a seemingly objective, physical character only because they, as conditional wave functions, depend on the physical beables of the environment. \n\n\\end{itemize}\n\nFrom point of view of simplicity, the paleoclassical interpretation is superior to all alternatives. The identification of the fundamental beables with the classical configuration space trajectory $q(t)$ is sufficient for this point. \n\nIt has also the additional advantage that it leads to strong restrictions of the properties of a more fundamental, sub-quantum theory:  It has to be a theory completely defined on the classical configuration space. Moreover, it has to be a theory which, in its statistical variant, leads to a Fokker-Planck-like equations for the probability flow defined by the classical flow variables $\\rho(q)$ and $v^i(q)$. \n\n\\begin{appendix}\n \n\\section{Compatibility with relativity}\\label{relativity}\n\nMost physicists consider the problem of compatibility with relativity as the major problem of dBB-like interpretations -- sufficient to reject them completely. But I have different, completely independent reasons for accepting a preferred frame, so that I don't worry about this. \n\nThere are two parts of this compatibility problem, a physical and a philosophical one, which should not be mingled: \n\nThe physical part is that we need a dBB version of relativistic quantum field theories, in particular of the standard model of particle physics -- versions which do not have to change the fundamental scheme of dBB, and, therefore, may have a hidden preferred frame. \n\nThe philosophical part is the incompatibility of a hidden preferred frame with relativistic metaphysics.  \n\nThe physical problem is heavily overestimated, in part because of the way dBB theory is often presented: As a theory of many particles. I think it should be forbidden to introduce dBB theory in such a way. The appropriate way is to present it as a general theory in terms of an abstract configuration space $Q$, and to recognize that field theories as well as their lattice regularizations fit into this scheme.  The fields are, of course, fields on three-dimensional space $\\mbox{$\\mathbb{R}$}^3$ changing in time,  and their lattice regularizations live on three-dimensional spatial lattices $\\mbox{$\\mathbb{Z}$}^3$, not four-dimensional space-time lattices. But this violation of manifest relativistic symmetry is already part of the second, philosophical problem. \n\nThe simple, seemingly non-relativistic Hamiltonian \\eqref{HamiltonFunction}, with $p^2$ instead of $\\sqrt{p^2 + m^2}$, is also misleading: For relativistic field theories the quadratic Hamiltonian is completely sufficient. Indeed, a relativistic field Lagrangian is of type\n\\begin{equation}\\label{field}\n\\mathscr{L} = \\frac{1}{2}((\\pd_t\\varphi)^2 - (\\pd_i\\varphi)^2)-V(\\varphi).\n\\end{equation}\nThis gives momentum fields $\\pi = \\pd_t\\varphi$ and the Hamiltonian\n\\begin{equation}\n\\mathscr{H} = \\frac{1}{2}(\\pi^2 + (\\pd_i\\varphi)^2)+V(\\varphi) = \\frac{1}{2}\\pi^2 + \\tilde{V}(\\varphi)\n\\end{equation}\nquadratic in $\\pi$, thus, the straightforward field generalization of the standard Hamiltonian \\eqref{HamiltonFunction}. And a for lattice regularization, the Hamiltonian is already exactly of the form \\eqref{HamiltonFunction}. So, whatever one thinks about the dBB problems with other relativistic fields, it is certainly not relativity itself which causes the problem. \n\nThe problem with fermions and gauge fields is certainly more subtle. Here, my proposal is described in \\cite{clm}. It heavily depends on a preferred frame, but for completely different reasons -- interpretations of quantum theory are not even mentioned. Nonetheless, fermion fields are obtained from field theories of type \\eqref{field}, and gauge-equivalent states  are interpreted as fundamentally different beables, so that no BRST factorization procedure is necessary. \n\nAnother part of the physical problem is compatibility with relativistic gravity. Here I argue that it is the general-relativistic concept of background-freedom which is incompatible with quantum theory and has to be given up. I use a quantum variant of the classical hole argument for this purpose \\cite{hole}. As a replacement, I propose a theory of gravity with background and preferred frame \\cite{glet}.  \n\nSo there remains only the philosophical part. But here the violation of Bell's inequality gives a strong argument in favour of a preferred frame: Every realistic interpretation needs it. Moreover, the notion of metaphysical realism presupposed by ``realistic interpretation'' is so weak that Norsen \\cite{Norsen} has titled a paper ``against realism'', arguing that one should not mention realism at all in this context. The metaphysical notion of realism used there is so weak that to give it up does not save Einstein locality at all -- it is presupposed in this notion too. \n\n\\section{Pauli's symmetry argument}\\label{Pauli}\n\nThere is also another symmetry argument against dBB theory, which goes back to Pauli \\cite{Pauli}, which deserves to be mentioned: \n\n\\begin{quote}\n\\ldots the artificial asymmetry introduced in the treatment of the two variables of a canonically conjugated pair\ncharacterizes this form of theory as artificial metaphysics. (\\cite{Pauli}, as quoted by \\cite{Freire}),\n\n``\\ldots the Bohmian corpuscle picks out by fiat a preferred basis (position) \\ldots'' \\cite{BrownWallace}\n\\end{quote}\n\nHere my counterargument is presented in \\cite{kdv}. I construct there an explicit counterexample, based on the KdV equation,  that the Hamilton operator alone, without a specification which operator measures position, is not sufficient to fix the physics. It follows that the canonical operators have to be part of the complete definition of a quantum theory and so have to be distinguished by the interpretation as something special, different from the other similar pairs of operators. \n\nThe Copenhagen interpretation makes such a difference -- this is one of the roles played by the classical part. But attempts to get rid of the classical part of the Copenhagen interpretation, without adding something else as a replacement, are not viable \\cite{pure}. One has to introduce a replacement. \n\nRecognizing that the configuration space has to be part of the definition of the physics gives more power to an old argument in favour of the pilot wave approach, made already by de Broglie at the Solvay conference 1927:\n\\begin{quote}\n``It seems a little paradoxical to construct a configuration space with the coordinates of points which do not exist.'' \\cite{deBroglie}.\n\\end{quote}\n\n\\section{Problems with field theories}\\label{fields}\n\nIt has been argued that fields are problematic as beables in general for dBB theory, a point which could be problematic for the paleoclassical interpretation too. \n\nIn particular, the equivalence proof between quantum theory and dBB theory depends on the fact that the overlap of the wave function for different macroscopic states is irrelevant. But it appeared in field theory that for one-particle states there is always a non-trivial overlap, even if these field states are localized far away from each other.  \n\nBut, as I have shown in \\cite{overlap}, the overlap decreases sufficiently fast (approximately exponentially) with greater particle numbers. \n\n\\section{Why we observe configurations, not wave packets}\\label{wavepackets}\n\nIn the many worlds community there is a quite popular argument against dBB theory -- that it is many worlds in denial (for example, see \\cite{BrownWallace}). But this argument depends on the property of dBB theory that the wave function is a beable, a really existing object. So it cannot be applied against the paleoclassical interpretation, where the wave function is no longer a beable. \n\nBut in fact it is invalid also as an argument against dBB theory. In fact, already in dBB theory it is the configuration $q(t)$ which is observable and not the wave function.  \n\nThis fact is sometimes not presented in a clear enough way, so that misrepresentations become possible. For example Brown and Wallace \\cite{BrownWallace} find support for another interpretation even in Bohm's original paper \\cite{Bohm}:\n\\begin{quote}\n\\ldots even in his hidden variables paper II of 1952, Bohm seems to associate the wavepacket chosen by the corpuscles as the representing outcome of the measurement -- the role of the corpuscles merely being to point to it.\n(\\cite{BrownWallace} p. 15)\\end{quote}\nand support their claim with the following quote from Bohm\n\\begin{quote}\nNow, the packet entered by the apparatus variable $y$ determines the actual result of the measurement, which the observer will obtain when he looks at the apparatus.\n(\\cite{Bohm} p. 118)\\end{quote}\nThis quote may, indeed, lead to misunderstandings about this issue. So, maybe we obserse only the wave packet containing the configuration, instead of configuration itself? \n\nMy answer is a clear no. I don't believe into the existence of sufficiently localized wave packets to construct some effective reality out of them, as assumed by many worlders. \n\nToday they use decoherence to justify their belief that wave packets will be sufficiently localized. But decoherence presupposes another structure -- a decomposition of the world into systems. Only from point of view of such a subdivision of $q$ into, say, $(x,y)$, a non-localized wave function like $e^{-(x-y)^2\/2}$ may look similar to a superposition, for different $a$, of product states localized in $x$ and $y$ like $e^{-(x-a)^2}\\cdot e^{-(y-a)^2}$. \n\n\nBut where does this subdivision into systems come from?  The systems around us -- observers, planets, measurement devices -- cannot be used for this purpose. They do not exist on the fundamental level, as a predefined structure on the configuration space. But the subdivision into systems has to, once we need it to construct localized objects. Else, the whole construction would be circular.\n\nSo one would have to postulate something else as a fundamental subdivision into systems. This something else is undefined in the interpretations considered here, so an interpretation based on it is simply another interpretation, with another, additional fundamental structure -- a fundamental subdivision into systems. \n\n\\end{appendix}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nInfrared maps of star forming Giant Molecular Clouds (GMCs) are an essential tool in the\nmodern study of star formation.  When radio and millimeter maps also exist, the relationships\nbetween the regions of infrared, millimeter and radio activity provide one of the key new tools\nfor clarifying the varieties of star formation that can occur. The sensitivity of infrared techniques\nmeans that even shallow surveys can in principle reveal the processes of both low and high mass\nstar formation in clouds that are not too far away.   There are,  however, very few nearby GMCs\nwith which to take full advantage of these techniques. Among these available targets, one is the\nVela Molecular Ridge (VMR), a complex of four adjoining GMCs (Murphy \\& May 1991;\nYamaguchi et al. 1999), located in the galactic plane ($b$=$\\pm$3$^{\\circ}$) outside the solar\ncircle ($l \\sim$ 260$^{\\circ}$ - 275$^{\\circ}$);  most of the gas (clouds named A, C and D) is\nlocated at a distance of about 700 pc (Liseau et al. 1992).\n\nThis team has studied the star formation activity in the VMR for many years: the concentration of\nred and young sources (Liseau et al. 1992, Lorenzetti et al. 1993); the presence of embedded\nclusters (Massi et al. 2000, 2003); the occurrence of protostellar jets (Lorenzetti et al. 2002;\nGiannini et al. 2001, 2005, De Luca et al. 2007, hereinafter D07). Recently we mapped with the SIMBA\nbolometer array at SEST a $\\sim$ 1 deg$^2$ area of the cloud D in the 1.2\\,mm continuum of dust emission, \nand in the $^{12}$CO(1--0) and $^{13}$CO(2--1) transitions (Massi et al. 2007, hereinafter M07; Elia et\nal. 2007, hereinafter E07).\n\nThe advent of the Spitzer Space Telescope (SST, Werner et al. 2004) and the imaging\nphotometric facilities on board, i.e. the Multiband Imaging Photometer for Spitzer (MIPS, 24, 70, \n160\\,$\\mu$m; Rieke et al. 2004) and the InfraRed Array Camera (IRAC, Fazio et al.\n2004)   has enabled us to obtain maps of the VMR from 3.8 to 70~$\\mu$m across the same area\nalready surveyed in the millimeter emission of dust and gas. The primary goal of this survey is to\nobtain a census of the embedded young stellar population of VMR-D and to correlate it with\nits gas and dust cores. \n\nThis paper describes our MIPS observations of the VMR-D; it is the first of a series of papers we\nare preparing dealing with VMR-D as seen by Spitzer; the IRAC data of the same region will be\npresented in a separate paper. Spitzer surveys of several other star\nforming regions have been already published, most of them in the framework of the cores-to-disk\n(c2d) legacy project (Evans et al. 2003). Most of these regions, however, are located outside the\nGalactic plane ($|b|$ $>$ 10$^{\\circ}$) in regions that were originally selected in part to avoid\nstrong confusion and extinction problems. A huge amount of observational material has been so\nfar accumulated and published on those clouds. For the VMR, our current multi-frequency\ndatabase, when combined with the increased sensitivity of current instruments, has allowed us to\novercome many of the problems associated with observations of GMCs in the galactic plane. \nSince the plane is  where most of the material currently forming stars is located, it is both a\nnatural and critical region to understand, and will also help with the comparison between the\nderived properties of our Galaxy with those of external galaxies, whose planes are the unique\nzones we are able to sample.\n\nSpitzer has surveyed many different types of star formation regions.  \nTherefore, legitimate comparisons between them all would benefit from a standard analysis and\npresentation, although some problems could arise because their numerous differing parameters, \nas well as the various details of the observations. In this paper we therefore adopt as much \nas possible the methods that have already been used successfully in the c2d program. \n\nOur paper is organized as follows: in Sect.\\,2 we give the details of the observations and data\nreduction procedure; the results are presented in Sect.\\,3 and discussed in Sect.\\,4 and 5.\nConcluding remarks are given in Sect.6.\n\n\n\\section{Observations} \n\nVMR-D cloud was observed with MIPS on board the Spitzer Space Telescope\nwithin the Guaranteed Time Observation program (PID 30335).\nThe observations covered $\\sim$ 1.15 (in R.A.) $\\times$ 1.6 (in dec.) degrees centered \nat $\\alpha$, $\\delta$ (J2000) = 8$^h$~47$^m$~50$^s$,\n-43$^{\\circ}$~42$^{\\prime}$~13$^{\\prime\\prime}$ \nand 8$^h$~48$^m$~20$^s$, -43$^{\\circ}$~31$^{\\prime}$~26$^{\\prime\\prime}$  at \n24~$\\mu$m and 70~$\\mu$m, respectively (orientation: 145$^{\\circ}$ W of N).\n\nData were collected on 14 Jun 2006 in scan mode, medium speed, with 5 scan legs \nand 160$^{\\prime\\prime}$ cross-scan step, resulting in a total integration time \nof 40 seconds (for both 24 and 70~$\\mu$m) per pixel.\nThe mapping parameters were optimized for the 24 and 70~$\\mu$m bands: as a consequence, \nthe 160~$\\mu$m map suffers from coverage gaps and saturation and will be not considered \nin the following.\n\nThe SSC-pipeline, version S14.4.0, produced basic calibrated data (BCDs) \nthat we have used to obtain mosaiced, pointing refined images by means of \nthe MOPEX package provided by the Spitzer Science Center (Makovoz \\& Marleau 2005).\n\nThe main instrumental artifacts have been removed from the mosaiced images \nby means of the MOPEX package. Minor problems of residuals jailbars \n(expecially at 70~$\\mu$m) and background matching between adjacent frames \n(at 24~$\\mu$m) are still visible close to the brightest objects, but they do \nnot affect significantly the point source photometry discussed in this paper.\n\nThe final 24\\,$\\mu$m map global properties can be summarized as follows: \npixel scale of 2.45$^{\\prime\\prime}$\/pixel, background r.m.s. of 0.3~$\\mu$Jy\/arcsec$^2$, \nwithin the regions of high level of diffuse emission. The brightest sources saturate at the\nemission peak: for these we estimate a lower limit to the integrated flux of 4 Jy.\nIn the 70\\,$\\mu$m map the pixel scale is 4.0\\,$^{\\prime\\prime}$\/pixel and the background r.m.s.\nranges between 23 and 94\\,$\\mu$Jy\/arcsec$^2$. None of the detected sources appears saturated\nat this wavelength.\n\n\n\n\\section{Results} \n\n\\begin{figure*}\n\\includegraphics[angle=0,scale=0.8]{f1.eps}\n\\caption{MIPS two-color map (24\\,$\\mu$m in blue, 70\\,$\\mu$m in red) of\nVMR-D.\n\\label{MIPS24-70}}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\includegraphics[angle=0,scale=0.8]{f2.eps}\n\\caption{Mosaic of VMR-D map at 24~$\\mu$m, with superposed \nthe $^{12}$CO intensity map (whose limits are depicted in yellow), where the contours (in green)\nare in the range -2\\,-\\,20 km s$^{-1}$\n(adapted by E07). Also overlaid is the 1.2$\\it{mm}$\ndust emission map (red contours, adapted from M07).\nCO contour levels start from 5 K km s$^{-1}$ and are in steps of\n25 K km s$^{-1}$, while dust contours start from 50 mJy\/beam and\nare in steps of 50 mJy\/beam.  \\label{MIPS24}}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[angle=0,scale=0.8]{f3.eps}\n\\caption{As Figure~\\ref{MIPS24} for the 70\\,$\\mu$m map.\\label{MIPS70}}\n\\end{figure*}\n\nFigure\\,\\ref{MIPS24-70} shows the two-color final mosaic of VMR-D\n(24\\,$\\mu$m in blue, 70\\,$\\mu$m in red), while on Figures \\ref{MIPS24} and \n\\ref{MIPS70}, images in each of the two filters are shown separately.\nIn these latter, the 1.2\\,mm dust map (adapted from M07) and the $^{12}$CO intensity map\nintegrated in the velocity range -2\\,$\\div$\\,20 km s$^{-1}$ (adapted from E07) \nhave been superimposed for comparison.  We define as `on-cloud' all objects \ninside these latter contours. Such a definition is perforce just the first level effort of delimiting\nthe sources belonging to the molecular cloud; indeed, it is clear from Figures \\ref{MIPS24} and\n\\ref{MIPS70} that \nthe $^{12}$CO emission remains well above the 3$\\sigma$ level at the north and west borders \nof the gas map, and thus sources belonging to VMR-D could exist toward these directions. \nConsidering such sources as 'off-cloud' will have the effect of reducing the distinctions between\nthe 'on' and 'off' cloud populations; these sources should then be considered on a \ncase-by-case basis (see sect.5.1). We have also considered as 'off-cloud' those regions where the CO peak velocity\nis faster than  20~km s$^{-1}$, since they are likely to be more distant and unassociated\nwith VMR-D (see Figure~1 in Lorenzetti et al. 1993). \n\n\nThe point-source extraction and photometry processes were performed by using\nthe {\\it DAOPHOT} task of the astronomical data analysis package {\\it\nIRAF}\\footnote{IRAF, the Image Reduction and Analysis Facility, is a general\npurpose software written and supported by the IRAF programming group at\nthe National Optical Astronomy Observatories (NOAO) in Tucson, Arizona \n(http:\/\/iraf.noao.edu).}.\nGiven the size of the MIPS mosaic it was impossible to apply any \nautomatic procedure for finding sources down to the sensitivity \nlimits without being affected by a locally varying background level; we \ntherefore applied a searching algorithm as deep as possible, \nbut still compatible with an automatic procedure. The search algorithm\nwas applied to a differential image we produced between the final mosaic and a 'sky' image, \nthe latter obtained by applying to the mosaic a median filter over boxes of 5$\\times$5 pixels.\nA threshold of 30\\,$\\sigma$ has been imposed on the sky-subtracted image, which\ncorresponds at least to 5\\,$\\sigma$ (depending on the local background) \nin the unsubtracted image.\n\n\n\\begin{figure}\n\\includegraphics[angle=-90,scale=.40]{f4.eps}\n\\caption{Histogram of the sources detected at 24~$\\mu$m. The completeness\nlimit is around 5 mJy, as indicated by the vertical line.\\label{compl24}}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[angle=-90,scale=.40]{f5.eps}\n\\caption{Histogram of the sources detected at 70~$\\mu$m. The completeness\nlimit is around 250 mJy, as indicated by the vertical line.\\label{compl70}}\n\\end{figure}\n\n\\begin{figure*}\n\\includegraphics[angle=0,scale=1]{f6.eps}\n\\caption{Left panel: differential number counts at 24~$\\mu$m. Thick and thin lines refer to\nsource in VMR-D within and outside the $^{12}$CO contours, respectively.\nExtragalactic background sources from the SWIRE ELAIS N1 field are shown for \ncomparison (these latter have been taken from Figs.6 and 7 in Rebull et al. 2007, hereinafter\nR07). Right panel: as \nin left panel at  70~$\\mu$m.  \n\\label{fig:counts24_70}}\n\\end{figure*}\n\nThe automated methods just described lead to the detection of 838 and 61 point sources \nat 24~$\\mu$m and 70~$\\mu$m, respectively. A further 12 detections have been added\nto the 24~$\\mu$m list by applying local sky values in selected areas (see the discussion below).\nThe source distribution as a function of the measured flux is depicted in Figures~\\ref{compl24}\nand \\ref{compl70}, where the completeness limits can be evaluated as the flux bin corresponding\nto the maximum counts before the decline at lower fluxes due to\nthe instrumental sensitivity.  We determine that our sample is complete down to 5 and 250 mJy at\n24 and 70~$\\mu$m, respectively. \n\nA statistical summary of the detected sources is presented in Table \\ref{tab:tab1}. \nAbout 45\\% of the 24~$\\mu$m sources are spatially located inside the region \ndelimited  by the $^{12}$CO contours ($\\sim$ 0.61 deg$^2$), even though this latter \nis about one half the size of the remaining mapped area ($\\sim$ 1.23 deg$^2$). \nThis result gives an initial indication of how the IR source density increases by \nabout 100\\% moving from outside to inside the CO contours,  and the pattern becomes even\nmore significant when considering the 70\\,$\\mu$m sources, whose source density increase\nis four times. \nThe 24~$\\mu$m counts per deg$^2$ are \nrepresented in Figure~\\ref{fig:counts24_70}, left panel, where different symbols indicate those\nsources located respectively within and outside the CO contours.\nAlso in this plot, where the differential number density is shown, there is a drop \nfor F$_{\\nu}$ $<$ 35\\,mJy. For greater F$_{\\nu}$  values the number of objects inside the gas\ncontours (i.e. those more likely associated to the cloud) systematically\nexceeds the number of objects outside the cloud, giving  reasonable support to the\nempirical significance of this crude classification. In addition,   Figure~\\ref{fig:counts24_70}  is\na comparison between the on- and off-cloud samples and the Spitzer Wide-area Infrared\nExtragalactic Survey (SWIRE, Lonsdale et al. 2003) legacy program. A\nsignificant amount of contamination from the extragalactic background is predicted (at\n24~$\\mu$m) for flux densities $<$10 mJy down to the completeness limit, so that 'on' and 'off'\nsource populations at this level become undistinguishable. \n\nThe counts per deg$^2$ at 70~$\\mu$m are depicted \nin Figure~\\ref{fig:counts24_70}, right panel: here again the possible extragalactic contamination\nappears just at (or even below) the completeness limit.\\\\\nIn the same Figure~\\ref{fig:counts24_70}, we also note that the 70~$\\mu$m counts confirm the\non- and off-cloud distributions already found at 24\\,$\\mu$m. \n\n\nThe complete catalog of the detected sources is given in electronic form (a short sample version is printed in Table~\\ref{tab:tab2}). In Table~\\ref{tab:tab3}\nwe show the list of the 70~$\\mu$m detections: of the  61 sources, 52 of them are \ncoincident with a 24~$\\mu$m source (i.e. the distance in both right ascension \nand declination is less than the 20$^{\\prime\\prime}$ PSF radius \nat 70 $\\mu$m, see the summary of Table~\\ref{tab:tab1}). In Table~\\ref{tab:tab3}, we list the \n24~$\\mu$m coordinates (which are more accurate than the 70~$\\mu$m ones because of the\nsmaller PSF at 24~$\\mu$m), the distance from the 70~$\\mu$m coordinates, \n($\\Delta\\alpha$\/$\\Delta\\delta$)$_{70}$, the measured flux at 24 and 70~$\\mu$m along with \nthe relative uncertainties, a flag indicating whether or not \nthe source is located inside the region delimited by CO emission contours, and\nthe association with a dust core, if any. This latter is based on the distance between the 24$\\mu$m\nand mm coordinates, ($\\Delta\\alpha$\/$\\Delta\\delta$)$_{mm}$, which must be within the SIMBA HPBW \nof 24$^{\\prime\\prime}$. All the dust cores associated with a 24~$\\mu$m source are also associated with\nits 70~$\\mu$m counterpart.\\\\\nNine 70~$\\mu$m sources have no 24~$\\mu$m\ncounterpart.  Four of these were not imaged at 24~$\\mu$m because of the shift between the two\nmaps, four appear as diffuse or with a filamentary structure at 24~$\\mu$m, and one \nhas F$_{24}$ $<$ 1.2 mJy (3$\\sigma$ upper limit): this source (\\#1 in Table~\\ref{tab:tab3})\ncould be (if not a galaxy) a very young protostar which deserves further attention. \n\nWe also provide in Table~\\ref{tab:tab3} the association of\nMIPS 24\/70~$\\mu$m sources with dust cores found in VMR-D by M07.\n The detailed study of these sources will be addressed in a future paper;  here we give some\npreliminary results and point out some statistical aspects. In the region mapped in the dust\nemission at 1.2\\,mm \n(see Figures\\,\\ref{MIPS24} and \\ref{MIPS70}), a robust sample of 29 cores has been revealed,\nalong with 26\ncores whose size is below the map spatial resolution (24$^{\\prime\\prime}$). \nD07 have associated 12 of these cores \n(8 resolved and 4 under-resolved) with an IRAS or MSX point source, while the remaining 43\ncores\nare not associated with any FIR counterpart, so that they appear to be either cold Class\n0 sources\/starless cores (in case of resolved cores) or possibly data artifacts (in case of \nunder-resolved cores). As stated in D07,  such a high fraction of starless cores as compared to\nprotostellar cores is most likely a result of the poor sensitivity of the IRAS\/MSX facilities. Our\nsignificantly more sensitive MIPS data offers the opportunity to check whether \nor not such a bias exists, and to eventually find weak counterparts of the dust cores.\nIn order to resolve this issue we closely reexamined our maps, performing photometry on the mm\npeaks coordinates using local rather than global thresholds for the background level.  This\ntechnique turned up 12 new objects at a flux density as low as 0.7 mJy at 24~$\\mu$m, fainter\nthan the completeness limit by more than a factor of 7.\n\nThis procedure, together with automatic finding described above, \nwhen applied overall led to the association of 23 resolved and 20 under-resolved cores with 58\nsources at 24 ~$\\mu$m, 19 sources at 70~$\\mu$m  (in some cases we found multiple\nassociations), thereby dramatically increasing the\npercentage of cores associated with an embedded protostar from 22\\% (D07) to 78\\%. This\nresult is in general agreement with recent MIPS findings in other GMCs that have substantially\nmodified the percentage of active vs. inactive cores in favor of the former (e.g. Young et al.\n2004).  We also note that the existence of a MIPS counterpart to 20 out of 26 under-resolved\ncores significantly reduces the possibility that these objects are simply data artifacts.  The lack,\neven at the MIPS sensitivity,  of a FIR counterpart to five\nresolved dust peaks (namely MMS 6,\\,13,\\,15,\\,20,\\,24 in the list by M07) makes these objects  a\nvery robust sample of genuine starless cores. \n\n\n\\section{Comparison with IRAS sources}\nThe similarity of the MIPS 24 and 70~$\\mu$m bandpasses to the \n25 and 60\\,$\\mu$m filters on-board IRAS offers us the opportunity to \nevaluate directly the reliability of the IRAS point source catalogue (IRAS-PSC) \nfluxes in crowded and diffuse clouds like VMR-D, objects that are commonly found \nin the galactic plane. A similar study has already been performed \nby R07 in the Perseus molecular cloud; although in this case\nthe geographic location makes extended emission and source \nconfusion less critical, only 61\\% (at 25$\\mu$m) and 32\\% (at 60$\\mu$m)\nof the objects of the IRAS-PSC are recovered by MIPS as point-like sources, while all\nthe others, although detected, remain confused by nebulosity. Higher rates of coincidence \nare found, at least at 25\\,$\\mu$m, if the Faint Source Catalogue (FSC) -\nproduced by point-source filtering the individual detector data streams - is used.\n\nUnfortunately, the FSC does not\ncover the galactic plane, so that we cannot confirm this result on VMR-D. \nHere, a total of 57 high (f$_{qual}$=3) or moderate (f$_{qual}$=2) quality \ndetections are listed in the IRAS-PSC catalogue at 25\\,$\\mu$m;  46 of them (80\\%)\nare also seen by MIPS and recovered with our algorithm, while the remaining  11 IRAS objects\nappear as diffuse \nemission at 24\\,$\\mu$m and are thus undetected as point-sources. \nThe matching rate for VMR-D is thus higher than in Perseus.  The same trend is seen \nat 60\\,$\\mu$m, where out of 48 IRAS-PSC items, the retrieval rate is of about 50\\%. \nIn Table~\\ref{tab:tab4} we give the list of the IRAS-PSC (with any f$_{qual}$) not \nrecovered by MIPS. \nThe IRAS sources in the table marked as `off-edge' in one MIPS bandpass are necessarily \n`on-edge' in the other, because of the spatial shift between the two focal plane\narrays.  Along with the f$_{qual}$ flag, we also give in Table~\\ref{tab:tab4}\nthe IRAS correlation coefficient flag (cc) which provides an indication of the \npoint-likeness confidence of the detected source. This flag is coded as alphabetical\ncharacter and subsequent letters correspond to decreasing accuracy (i.e. A$>$99\\%,\nE$>$96\\%). \nNoticeably, PSC sources retrieved by MIPS (with f$_{qual}$=2,3), show, on average, \n`cc' flag equal to \"A\" or \"B\": such an occurrence can thus be translated into a\nsuitable tool to broadly distinguish between genuine point-source and diffuse emissions, if \nMIPS maps (and FSC detections) are unavailable.\n\n\n\n\\section{Color-Magnitude diagrams}\n\n\\subsection{K$_s$ vs. K$_s$-[24]}\n\n\\begin{figure}\n\\includegraphics[angle=0,scale=.50]{f7.eps}\n\\caption{Color-magnitude diagram for the 2MASS K$_s$-band\nand the MIPS 24\\,$\\mu$m sources. Of the 849 24~$\\mu$m sources \nin the MIPS map, 401 have a K$_s$ detection within a radius of 5 arcsec.\nThese are shown by red dots if located inside the CO contour map (180 sources)\nand by black dots if outside (221 sources).\nLarge dots denote sources with 70~$\\mu$m detections, while arrows refer to\nsources saturated at 24\\,$\\mu$m. Hatched\nareas are the {\\it loci} of the sources in the SWIRE survey (taken from R07). \nThe thick line indicates the effect of the extinction for different values of A$_V$ (open squares\nrefer to A$_V$=10 and 50~mag).\\label{fig:k-24}}\n\\end{figure}\n\nAbout half of the 24~$\\mu$m detections have identifiable 2MASS counterparts \nat K$_s$ (limiting magnitude of 15.3) within a radius of 5$^{\\prime\\prime}$. \nThese 2MASS fluxes have been used to construct the K$_s$ vs. K$_s$-[24] \ncolor-magnitude diagram given in Figure~\\ref{fig:k-24}, where MIPS \nsources inside and outside the CO contours are shown with different colors. \nAlso reported as hatched areas are the {\\it loci} of the extragalactic sources in the SWIRE survey. \nAs expected for a molecular cloud in the galactic plane, there are very few \nextragalactic sources seen. A remarkable number of objects fall at K$_s$$<$8.5 mag and K$_s$-[24]$\\sim$0, \nwhich, given our completeness limit at 24~$\\mu$m of 5~mJy, delimits the region\nof normal photospheres in VMR-D.   \nNoticeably, in this part of the diagram, the number density (per deg$^2$)\nof the 'off-cloud' sources is larger than that of the 'on-cloud'\nones (100 vs. 69, see Table~\\ref{tab:tab4}): in principle, all the unreddened\nphotospheres detected in VMR-D could be indeed foreground\/background stars. More reasonably,\nwe can affirm that no increase of main-sequence stars (with respect to the\nadjacent field) is registered in VMR-D, as expected because of the youth of the region.\n\nThe thick squares indicate the effects of an extinction of A$_V$ =\n10 and 50\\,mag, respectively, on the data.\nThe quantitative A$_V$ map of the overall region by Dobashi et al. (2005) does not \nprovide values in excess to 5-10\\,mag (below the saturation limit of the catalog of 15\nmag), while toward the dust cores A$_V$ can increase up to\n$\\sim$ 20\\,mag (M07 and E07).  We thus conclude that sources \nwith K$_s$-[24] $>$ 5 are probably reasonably young objects, and not merely extinguished\nobjects. \nIndeed, it is not accidental that the large majority of these sources\nbelong to the molecular cloud, nor that the objects with normal photospheres are outside the\ncloud.\\\\\nFrom an evolutionary point of view, protostars can be characterized on the basis of the spectral \nclassification between 2\\,-\\,10\\,$\\mu$m  (Greene et al. 1994), according to which\ndifferent evolutionary stages, from the accretion phase (Class I) to the beginning of \nthe main-sequence (Class III), are manifested. \nThe same authors have found that the 2\\,-10\\,$\\mu$m spectral index  \ndoes not change substantially when computed using fluxes up to 20\\,$\\mu$m \n(by using photometry in the Q band);  this result allows one to extrapolate the 24\\,$\\mu$m \nflux for different spectral indexes and accordingly to compute \nthe expected value of the K$_s$-[24] color. \nThe result of this procedure is given in R07, who furthermore\nrequested that, to select Class III sources from  \nnormal photospheres and foreground\/background stars, K$_s$-[24]$>$2. The spectral classification\nderived for VMR-D is depicted in Fig.\\,\\ref{fig:k-24} and also reported in Table \\ref{tab:tab5}.\nThe ratio of `on-cloud' over `off-cloud' objects within the same class\nincreases with increasing K$_s$-[24] values; \nmoreover, most of the younger 'on-cloud' objects are also detected in the 70~$\\mu$m band\n(large, red dots in Fig.\\,\\ref{fig:k-24}).\nNoticeably, 70\\% of the `off-cloud' objects showing the characteristics of the youngest\nand coldest sources (black dots at K$_s$-[24]$\\simeq$7) are located just outside the North and \nWest borders of the CO gas map, therefore they are reasonably genuine members of VMR-D; the \nremaining sources (30\\%) with the same colours, if not belonging to VMR-D, could represent \nstar forming regions at larger distances.\n\n \nIn summary, we find a definitely many more young sources associated with the cloud, but\nconfirming that active star formation behind and\/or in the close \nneighbourhood of our cloud is going on as well.\\\\\nThe relative percentages of sources attributed to different evolutionary stages (see\nTable~\\ref{tab:tab4}) can be compared with those of other well studied star forming regions.\nSchmeja et al. (2005), in particular, have investigated number ratios of sources \nin different evolutionary classes in several star forming regions ($\\rho$ Ophiuchi, Serpens,\nTaurus, Chamaleon\\,I, IC348) basing on data obtained before than the Spitzer advent. They find,\non average, that Class\\,I sources are $\\sim$ 1-10\\%, while  Class\\,II\/III sources are about 80-95\\%\nof the total. \nWith the advent of Spitzer, these percentages have increased slightly in favor of younger\nprotostars: by combining both IRAC and MIPS data, Reach et al. (2004) classified as Class\\,I\n11\\%  of the sources in IC1396A and  Muzerolle et al. (2004) 20\\% of the sources in  NGC7129.\nFinally, in L1630 a ratio of 0.25 is found between Class\\,I and Class\\,II protostars (Muzerolle et\nal. 2005). A comparison of our statistics with these is not straightforward both because of the\ndifferent classification adopted there (e.g. flat spectrum objects are not included) and the fact that\nthis paper uses MIPS data only. A more meaningful and direct comparison is with the works of Harvey\net al. (2007) and of R07, who performed a very similar analysis as the one we do here, but on \nthe Serpens star forming region and on the IC348 and NGC1333 clouds in Perseus. The percentage\nof Class\\,I vs. Class\\,II objects is 6\\% vs. 63\\% (Serpens), \n6\\% vs. 85\\% (IC348) and 7\\% vs. 67\\% (NGC1333), thus strongly in favoring the latter.    \nIn contrast, we find percentages in VMR-D of 23\\% of Class\\,I and 28\\%\nof Class\\,II.\n\nTwo possible alternatives could explain such a difference: {\\it i)} VMR-D is\nsignificantly younger than either Perseus or Serpens. Such an hypothesis is supported by \nthe age estimates of 1-2 Myr derived in the Perseus cloud (Palla \\& Stahler 2000, R07) and of\n2 Myr derived in Serpens (Djupvik et al. 2006) as compared with an age of 10$^5$-10$^6$ yr \ntowards the clusters of VMR-D (Massi et al. 2000); {\\it ii)} our K$_s$ vs. K$_s$-[24] \ndiagram suffers from missing two important categories of sources. One category \nis represented by the $\\sim$ 450 objects detected at 24$\\mu$m, but without a\nK$_s$-2MASS counterpart (see Table~\\ref{tab:tab1}). The sensitivity limits of our\nsurvey in terms of power density at a given wavelength are \n$\\lambda$F$_{\\lambda}$(2MASS) $\\simeq$ $\\lambda$F$_{\\lambda}$(24-MIPS) $\\sim$ 6 10$^{-16}$ W m$^{-2}$. \nThis implies that these 450 sources are objects whose SED is rising with\nwavelength and thus they could be additional young objects that tend even to increase the already anomalous \npercentage of Class~I sources. The second category, however, is represented \nby the about 5 10$^4$ 2MASS objects not having a MIPS 24$\\mu$m counterpart.\nTheir SEDs are allowed to decrease with increasing wavelength, therefore,\nalthough many of them could be foreground or background objects unrelated with\nthe VMR population, they undoubtly represent a potential reservoir of Class~II\nand III objects. It should be sufficient that a very small fraction of them\n($\\sim$ 1-2\\%) were genuine Class~II\/III sources to reduce significantly the\nrelative excess of Class~I in VMR, and then to increase the apparent age of the\nregion. In this view the disagreement with other star forming regions \ncould be reconciled, by considering that in those cases the lower\nbackground level implied by their location outside the Galactic plane\nhas permitted to reach detection limits at 24$\\mu$m up to an order of magnitude\nfainter than in Vela, therefore allowing to trace the SED also for faint \nK$_s$ sources that decline going from the near- to the far-infrared. In any case, we expect\nto provide a more certain answer to this issue in the next future, by means of\nforthcoming IRAC images covering the relevant spectral bands at more adequate\nsensitivity.\n\n\\subsection{[24] vs. [24]-[70]}\n\\begin{figure}\n\\includegraphics[angle=0,scale=.50]{f8.eps}\n\\caption{Color-magnitude diagram [24] vs. [24]-[70], where only sources\nnot saturated at 24\\,$\\mu$m are plotted. Red\/black dots refer\nto sources inside\/outside the CO contour map (41\/12 sources). Large dots denote sources\nassociated with a dust core, while\nnumbered sources are the candidates exciting sources of the jets discussed\nin Sect.\\,6. The hatched area shows the {\\it locus} of \nthe SWIRE survey (taken from Fig.11 in R07).\n\\label{fig:24-70}}\n\\end{figure}\n\nIn Figure~\\ref{fig:24-70} the color-magnitude diagram based on MIPS fluxes alone is\nshown. Here the sources detected in both bands are plotted; the large majority\nof them are on-cloud, although there is no clear difference between sources associated \nor not associated with dust cores (large dots). Remarkably, all sources (except 3) are located\nto the right of [24]-[70]=2. This value pertains to SED's that increase with wavelength in such a\nway that ($\\lambda$F$_{\\lambda}$)$_{70}$ = 2$\\times$($\\lambda$F$_{\\lambda}$)$_{24}$.\nThese red objects are much more numerous than the 70~$\\mu$m detections depicted in\nFigure~\\ref{fig:k-24}, since the majority of them lack a 2MASS counterpart.\nForthcoming IRAC data will help us to reconstruct their SED's more adequately, giving\nconstraints on their luminosity and evolutionary stage. A few sources (6) lie \nin the locus corresponding to black-body temperatures ranging between 40 and 50 K. These\nare values theoretically predicted (Shu, Adams \\& Lizano, 1987) for a collapsing isothermal\nsphere, identified as Class 0 objects. These 6 sources are all located inside the CO cloud;  two\nof them lie within a mm core and one (\\#38) is also associated to a compact H$_2$ jet\n(see Sect.\\,6). Although at the moment a quantitative evaluation of their sub-mm vs.\nbolometric luminosity cannot be given, they still represent the best candidates for \nmembers of the youngest population within the cloud.\n\n\n\\section{MIPS associations with H$_2$ protostellar jets}\\label{sec:jets} \nMIPS offers a chance to identify, for the first time, very embedded compact exciting sources of\nmolecular (H$_2$) jets, found by D07, that have so far remained undetected even at the longest IRAS\nwavelengths. We recall once again that our completeness limits are \nsignificantly higher than the sensitivity limits, so that by\nscrutinizing our data-base for the weakest MIPS sources in selected areas, we have been able to\ndiscover new objects down to 0.7 mJy at 24~$\\mu$m.\nWe used this technique to search for the sources driving a number \nof molecular jets that were previously found by narrow band imaging centered at the H$_2$\n(1-0)S(1) line (2.12$\\mu$m, see D07 for details). \nAlthough our H$_2$ driving source survey is still incomplete, we here identify out some\ninteresting cases in the VMR-D cloud. \n\n\nThe results of the correlation between H$_2$ maps with our MIPS maps are given\nin Table~\\ref{tab:tab6}; here we list each jet,  the length and the \ndynamical time of the jet itself (having adopted d=700 pc); the total flux detected at 2.12 $\\mu$m and the corresponding\nH$_2$ luminosity; \nthe identification  of the exciting source in our MIPS\ncatalogue; the coincidence with a\ndust mm-peak,  and finally an estimate of the source bolometric luminosity\nobtained by summing up all the contributions from the near-IR (if\nany) to the 1.2\\,mm flux, as derived by M07 SIMBA map.\nWe identify several different morphologies among our sources, including discovering the driving\nsources for three of the jets (namely \\# 2, 4, 5), objects that were not detected in either near-IR (K\nband) or the far-IR (N-band, IRAS, MSX). \n\n{\\bf Jet 1 -} The jet center lies towards a millimeter peak (MMS2),\nwhere no infrared source is detected down to K=17 mag. In the MIPS\n24~$\\mu$m band, an emission peak is found, although not aligned with the jet\naxis. The lack of any aligned source suggests two possible alternative\nscenarios: (i) we are observing just one jet lobe or (ii), more\nreasonably, the exciting source is too faint to be detected even by\nMIPS (F(24) $<$ 0.7 mJy). Scheduled APEX observations will hopefully\nanswer this question (Figure~\\ref{fig:jet1}).\n\n{\\bf Jets 2 and 3 -} Two small jets have been detected that correspond to faint dust peaks. The\nexciting sources, although not detected in the infrared (NIR bands, IRAS, MSX), are clearly\nrecognizable in the MIPS images and one of them (\\#38) is a\ncandidate Class 0 protostar. The compactness of these jets\nimplies a very short dynamical age, if reasonable conditions for\nboth shock velocity (v$_{shock}$ = 50 km s$^{-1}$) and inclination angle ({\\it i} =\n45$^{\\circ}$) are assumed (Figures~\\ref{fig:jet2} and \\ref{fig:jet3}).\n\n\n{\\bf Jet 4 -} A parsec scale jet emerges from a young near-infrared\ncluster centered on IRAS08476-4306. The proposed exciting source,\ndetected in the near-IR bands is the IRS20-\\#98\naccording to the Massi et al. (1999) list (Figure~\\ref{fig:jet4}).\n\n{\\bf Jet 5 -} A point-like 24\/70~$\\mu$m source aligned with the jet\nand corresponding to a dust peak  (umms19) is found about 2 arcmin\naway towards the NE. If this source is indeed driving the\njet then we are observing just one jet lobe, being the\ncounter-jet located outside the H$_2$ investigated field. A NIR\ncluster is also found at the MIPS source position (Figure~\\ref{fig:jet5}).\n\n{\\bf Jet 6 -} A chain of H$_2$ knots emerges from a MIPS source (not\nvisible in the H$_2$ band) centered at the dust emission peak MMS16. \nWe do not observe a counter jet (Figure~\\ref{fig:jet6}). \n\n\\begin{figure}\n\\includegraphics[angle=0, width=14cm]{f9.eps}\n\\caption{H$_2$ contours of jet 1 (green) superposed on the MIPS 24~$\\mu$m (left)\nand 70~$\\mu$m (right) images. Dust contours (from a 3$\\sigma$ level in steps\nof 3$\\sigma$) are shown in yellow. Dust core MMS2 is located\nat the jet center. \\label{fig:jet1}}\n\\end{figure}\n\n\n\\begin{figure}\n\\includegraphics[angle=0, width=14cm]{f10.eps}\n\\caption{The same as Fig.\\ref{fig:jet1} for jet 2. Peak umms16 \n(under-resolved at the SIMBA spatial resolution) is found near \nthe jet center. The proposed exciting source is \\#38 in \nTable~\\ref{tab:tab3}.\\label{fig:jet2}}\n\\end{figure}\n\n\n\\begin{figure}\n\\includegraphics[angle=0, width=14cm]{f11.eps}\n\\caption{The same as Fig.\\ref{fig:jet1} for jet 3. A MIPS source (\\#21)\nis found at the jet center, corresponding to a location of weak dust emission. \n\\label{fig:jet3}}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[angle=0, width=14cm]{f12.eps}\n\\caption{The same as Fig.\\ref{fig:jet1} for jet 4. Dust core MMS22 crosses\nthe jet. The candidate driving source (IRS20-MGL99\\#98) is indicated.\\label{fig:jet4}}\n\\end{figure}\n\n\n\\begin{figure}\n\\includegraphics[angle=0, width=14cm]{f13.eps}\n\\caption{The same as Fig.\\ref{fig:jet1} for jet 5. The driving source is likely to be the \nMIPS source (\\#44) associated with the under-resolved peak umms19. The lack of a\ncounter jet is probably due to the driving source being located near the edge \nof the H$_2$ image.\\label{fig:jet5}}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[angle=0, width=7cm]{f14.eps}\n\\caption{The same as Fig.\\ref{fig:jet1} for jet 6. The candidate exciting source is the \n24 and 70~$\\mu$m source associated with mm peak MMS16 (\\#50). Just one lobe \nhas been detected. \\label{fig:jet6}}\n\\end{figure}\n\n\\clearpage\n\n\\section{Conclusions} \nMIPS maps covering 1.8 square degrees across the Vela Molecular Cloud D at 24 and\n70\\,$\\mu$m are presented.  The data allowed us to derive the following results:\n\\begin{itemize}\n\\item[-] A total of 849 and 61 point sources at 24 and 70\\,$\\mu$m, respectively, have been\ndetected at completeness limits of 5 and 250 mJy.\n\\item[-] About half of the 24\\,$\\mu$m sources and two thirds of the 70\\,$\\mu$m ones \nare spatially located inside a region delimited by the $^{12}$CO contours (0.6 deg$^2$). \nThe implication is that the IR source density doubles (and is four times when considering \nsources at 70\\,$\\mu$m) inside the CO contours as compared to outside the molecular cloud. A\nquantitative analysis of the 24 and 70\\,$\\mu$m counts per deg$^2$ confirms this result. \n\\item[-] The maps allow us to correlate MIPS sources with the distribution of the dust cores\nfound within VMR-D, and, when we extend the search of MIPS sources down to the\ninstrumental sensitivity limit, we find that most of these cores result associated with\nred and cold objects.\n\\item[-] The MIPS sensitivity has enabled us to identify many new starless cores; the result will\nprompt a revision of the relative percentages of young objects known so far in VMR-D. \n\\item[-] IRAS-PSC detections of good quality (f$_{qual}$=3,2) are also seen by\nMIPS, but only when the IRAS point-likeness confidence is high (correlation coefficient, cc,\nequal to A or B). This result may be adopted as a broad confidence prescription \nfor finding genuine point sources in the IRAS catalogue.\n\\item[-] About 400 MIPS sources have 2MASS K$_s$ counterparts.  Color-magnitude\nplots constructed with magnitudes at 2.2, 24 and 70\\,$\\mu$m in VMR-D\nshow an excess of Class I objects in comparison with other well studied star formation \nregions.  This excess could be biased by the \nsensitivity limits of the 2MASS and MIPS surveys, otherwise, it\ncould reflect the short time elapsed since the first collapse of the cloud.\nFrom the MIPS colors, 6 objects appear as potential candidates Class\\,0 objects.\n\\item[-] We have detected the driving source in five out of six H$_2$ protostellar jets in VMR-D,\nfour of them embedded in mm-cores.  Such circumstance, along with the very low dynamical\ntime estimated for the jets, indicates ages of 10$^4$-10$^5$ yr for these sources.  \n\\item[-] We note that, given the southern location of VMR-D, many of the newly detected MIPS\nsources will be excellent candidate targets for ALMA. \n\\end{itemize}  \n\n\n\n\\section{Acknowledgements}\nThis paper is based on observations\nmade with the Spitzer Space Telescope, which is operated by the Jet\nPropulsion Laboratory, California Institute of Techonology under\na contract with NASA.  HAS acknowledges partial support from NASA grant NAG5-10659.\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe theory of weight functions is fundamental in the evaluation of stress intensity factors for asymptotic representations near non-regular boundaries such as cusps, \nwedges, cracks. The classical work of Bueckner defined weight functions for several types of cracks, both in 2D and 3D, as the stress intensity factors corresponding to \nthe point force loads applied to the faces of the cracks placed in a homogeneous continuum (Bueckner, 1987, 1989). In particular, for a penny shaped crack in three \ndimensions, as well as a half-plane crack, Bueckner has introduced the weight functions corresponding to a general distribution of forces on the opposite faces of \nthe crack, which also included the case of forces acting in the same direction. \n\nIn the present paper, we use the term ``symmetric'' load for forces acting on opposite crack faces and in opposite directions, whereas the term ``skew-symmetric'' or \n``anti-symmetric'' load will be used for forces acting on opposite crack faces but in the same direction, see Fig.~\\ref{fig01a}. Consequently, symmetric loads are always \nself-balanced, whereas skew-symmetric loads are not, in general (however, we shall consider only the case of self-balanced loading). Note that this terminology is \ndifferent from that usually in use, where the words ``symmetric'' and ``anti-symmetric'' are intended in terms of the symmetry about the plane containing the crack, \nso that, for example, shear forces acting on opposite crack faces but in the same direction are considered to be symmetric (see for example Meade and Keer, 1984).\n\\begin{figure}[!htb]\n\\begin{center}\n\\vspace*{3mm}\n\\includegraphics[width=10cm]{fig01a.eps}\n\\caption{\\footnotesize Symmetric and skew-symmetric parts of the loading in the case of single point forces. In practical cases, the loading is \ngiven as a combination of point forces and\/or distributed forces in such a way it is self-balanced (see for example the problem analysed in \nSection \\ref{example}).}\n\\label{fig01a}\n\\end{center}\n\\end{figure}\n\nAlthough in two dimensions skew-symmetric loading does not contribute to the stress intensity factors, it becomes essential in three dimensions. Both symmetric and \nskew-symmetric loads on the faces of a half-plane crack in a three dimensional homogeneous elastic space are also considered by Meade and Keer (1984).\n\nThe situation when the crack is placed at an interface between two dissimilar elastic media is substantially different from the analogue corresponding to a homogeneous \nelastic space with a crack. Here, even in the case of two dimensions (plane strain or plane stress), the stress components oscillate near the crack tip and also the \nskew-symmetric loads generate non-zero stress intensity factors. Also for the Mode III, where the stress components do not oscillate, there is a non-vanishing  \nskew-symmetric component of the weight function. \n\nSymmetric twodimensional weight functions (i.e. stress intensity factors for symmetric opening loads applied on the crack faces) for interfacial cracks were analysed by \nHutchinson, Mear and Rice (1987). The problem of symmetric three dimensional Bueckner's weight functions for interfacial cracks has been addressed by Lazarus and Leblond \n(1998) and Piccolroaz {\\em et al.}\\ (2007). On the basis of these results, Pindra {\\em et al.}\\ (2008) studied the evolution in time of the deformation of the front of a \nsemi-infinite 3D interface crack propagating quasistatically in an infinite heterogeneous elastic body.\n\nHowever, to our best knowledge, the skew-symmetric weight function for interfacial cracks has never been constructed for the plane strain case nor even for the Mode III \ndeformation. On the other hand, in most applications, especially for the interfacial crack, the loading is not symmetric and consequently the effects of the skew-symmetric \nloading in the analysis of fracture propagation require a thoroughly investigation.\n\nThe aim of this paper is to construct the aforementioned skew-symmetric weight functions for the twodimensional interfacial crack problem. To this purpose, we develop a \ngeneral approach, which allows us to obtain the general weight functions, as used by Willis and Movchan (1995), defined as non-trivial singular solutions of a boundary \nvalue problem for interfacial cracks with zero tractions on the crack faces but unbounded elastic energy. By taking the trace of these general weight functions on the \nreal axis, one can arrive to the notion of Bueckner's weight function associated with the point force load on the crack faces (Piccolroaz {\\em et al.}, 2007). It is then \nshown that the skew-symmetric part of the loading contributes to the stress singularity at the crack edge and thus to the resulting stress intensity factors. These \nresults are presented in Section \\ref{wfunc} for the plane strain problem and in Section \\ref{antiplane} for the Mode III problem.\n\nWe summarize that the symmetric weight function matrix $\\jump{0.15}{\\mbox{\\boldmath $U$}}$ for a plane strain interfacial crack has the form (compare with \\eq{jumpu} in the main text):\n\\begin{equation}\n\\label{symm}\n\\begin{array}{ll}\n\\displaystyle \\jump{0.15}{\\mbox{\\boldmath $U$}}(x_1) = \\frac{1}{2 d_0 \\sqrt{2 \\pi x_1}} \\left\\{ \\frac{x_1^{-i \\epsilon}}{c_1^+} \\mbox{\\boldmath $\\mathcal{B}$} + \\frac{x_1^{i \\epsilon}}{c_1^-} \\mbox{\\boldmath $\\mathcal{B}$}^\\top \\right\\} \n& \\text{for } x_1 > 0, \\\\[5mm]\n\\displaystyle \\jump{0.15}{\\mbox{\\boldmath $U$}}(x_1) = 0 & \\text{for } x_1 < 0,\n\\end{array}\n\\end{equation}\nwhere $i$ is the imaginary unit, $\\mbox{\\boldmath $\\mathcal{B}$} = \\begin{pmatrix} 1 & -i \\\\ i & 1 \\end{pmatrix}$, the superscript $^\\top$ denote transposition, and the bimaterial parameters \n$\\epsilon$ and $d_0$ are defined in Appendix \\ref{app1}. The quantities $c_1^+$, $c_1^-$ appear as coefficients in the representation of oscillatory terms in the \nasymptotics of stress near the tip of the interfacial crack, as defined in the sequel (see \\eq{stress1}--\\eq{stress2}). \n\nThe result for the skew-symmetric weight function matrix $\\langle {\\bf U} \\rangle$, unpublished in the earlier literature, reads as follows (compare with \\eq{meanu} \nin the main text):\n\\begin{equation}\n\\label{skew}\n\\begin{array}{ll}\n\\displaystyle \\langle \\mbox{\\boldmath $U$} \\rangle(x_1) = \\frac{\\alpha}{2} \\jump{0.15}{\\mbox{\\boldmath $U$}}(x_1) & \\text{for } x_1 >0, \\\\[5mm]\n\\displaystyle \\langle \\mbox{\\boldmath $U$} \\rangle(x_1) = -i \\frac{\\alpha (d_* - \\gamma_*)}{4d_0^3 \\sqrt{-2 \\pi x_1}} \n\\left\\{ \\frac{(-x_1)^{-i \\epsilon}}{c_1^+} \\mbox{\\boldmath $\\mathcal{B}$} - \\frac{(-x_1)^{i \\epsilon}}{c_1^-} \\mbox{\\boldmath $\\mathcal{B}$}^\\top \\right\\} & \\text{for } x_1 < 0,\n\\end{array}\n\\end{equation}\nwhere the bimaterial constant $\\gamma_*$ and the Dundurs parameters $\\alpha$ and $d_*$ are defined in Appendix \\ref{app1}. The conventional notations \n$\\jump{0.15}{f}(x) = f(x,0^+) - f(x,0^-)$ and $\\langle f \\rangle(x) = \\frac{1}{2}(f(x,0^+) + f(x,0^-))$ are in use here to denote the symmetric and skew-symmetric \ncomponents, respectively.\n\nNote that the result for the symmetrical weight functions is consistent with the representation for the stress intensity factors derived by Hutchinson, Mear and Rice \n(1987).\n\nFor the Mode III case, the scalar symmetric and skew-symmetric weight functions are (compare with \\eq{jumpu3} and \\eq{meanu3} in the main text):\n\\begin{equation}\n\\jump{0.15}{U_3}(x_1) = \\left\\{\n\\begin{array}{ll}\n\\displaystyle \\frac{1-i}{\\sqrt{2\\pi}} x_1^{-1\/2}, & \\text{for } x_1 > 0, \\\\[3mm]\n0, & \\text{for } x_1 < 0,\n\\end{array} \\right., \\quad\n\\langle U_3\\rangle = \\frac{\\eta}{2} \\jump{0.15}{U_3},\n\\end{equation}\nwhere $\\eta$ is another bimaterial constant defined in Appendix \\ref{app1}.\n\nIn Section \\ref{identity} we use both symmetric and skew-symmetric weight function matrices together with the Betti identity in order to derive the stress intensity \nfactors for an interfacial crack subjected to a general load. We summarize here that the integral formula for the computation of the complex stress intensity factor \nin the plane strain problem, $K = K_\\text{I} + i K_\\text{II}$, has the form (compare with \\eq{SIF} in the main text):\n\\begin{equation}\n\\mbox{\\boldmath $K$} = -i \\mbox{\\boldmath $\\mathcal{M}$}_1^{-1} \\lim_{x_1'\\to 0} \\int_{-\\infty}^{0} \\left\\{\\jump{0.15}{\\mbox{\\boldmath $U$}}^\\top(x_1'-x_1) \\mbox{\\boldmath $R$} \\langle \\mbox{\\boldmath $p$} \\rangle(x_1) \n+ \\langle\\mbox{\\boldmath $U$}\\rangle^\\top(x_1'-x_1) \\mbox{\\boldmath $R$} \\jump{0.15}{\\mbox{\\boldmath $p$}}(x_1) \\right\\} dx_1,\n\\end{equation}\nwhere $\\mbox{\\boldmath $K$} = [K,K^*]^\\top$, the superscript $^*$ denotes conjugation, $\\mbox{\\boldmath $\\mathcal{M}$}_1$ is defined in \\eq{emme1} and \\eq{stress2}, $\\mbox{\\boldmath $R$}$ is defined in \\eq{erre} and \n$\\langle \\mbox{\\boldmath $p$} \\rangle(x_1)$, $\\jump{0.15}{\\mbox{\\boldmath $p$}}(x_1)$ stand for the symmetric and skew-symmetric parts of the loading, respectively.\n\nFor the computation of the Mode III stress intensity factor, $K_\\text{III}$, the integral formula has the form (compare with \\eq{k3} in the main text):\n\\begin{equation}\nK_\\text{III} = - \\sqrt{\\frac{2}{\\pi}} \\int_{-\\infty}^{0} \\left\\{ \\langle p_3 \\rangle(x_1) + \\frac{\\eta}{2} \\jump{0.15}{p_3}(x_1) \\right\\} (-x_1)^{-1\/2} dx_1.\n\\end{equation}\n\nNeedless to say, by replacing the loading with the Dirac delta function, these integral formulae immediately give the Bueckner's weight functions. Also, this approach \nallows us to derive the constants in the high-order terms, which are essential in the perturbation analysis (Lazarus and Leblond, 1998; Piccolroaz {\\em et al.}, 2007). \n\nSection \\ref{perturbation} is devoted to the perturbation model of a crack advancing quasi-statically along the interface between two dissimilar media. We develop and prove \nan alternative approach consistent with the idea originally presented in Willis and Movchan (1995). Both symmetric and skew-symmetric weight function matrices are \nessential in this perturbation analysis.\n\nA simple illustrative example is given in Section \\ref{example}, which shows quantitatively the influence of the skew-symmetric loading for \ninterfacial cracks.\n\nSection \\ref{prel_analysis} is introductory and contains auxiliary results concerning the physical fields around the interfacial crack tip, which are necessary for the analysis presented \nin Sections \\ref{wfunc}--\\ref{antiplane}.\n\nFinally, the Appendix \\ref{app1} collects all bimaterial parameters involved in the analysis of the interfacial crack, including the well-known parameters as well as \nthe new parameters appearing in the skew-symmetric problem. The parameters have been classified according to whether they are involved in the symmetrical or \nskew-symmetrical weight functions. The Appendix \\ref{app3} provides the theorems required for proving the approach presented in Section \\ref{perturbation}.\n\n\\section{Preliminary results for an interfacial crack}\n\\label{prel_analysis}\nTo introduce the main notations and the mathematical framework for our model, we start with the analysis of the displacement and stress fields around a semi-infinite \nplane crack located on an interface between two dissimilar isotropic elastic media, with elastic constants denoted by $\\nu_\\pm,\\mu_\\pm$, see Fig. \\ref{fig02a}.\n\\begin{figure}[!htb]\n\\begin{center}\n\\vspace*{3mm}\n\\includegraphics[width=9cm]{fig02a.eps}\n\\caption{\\footnotesize Geometry of the model.}\n\\label{fig02a}\n\\end{center}\n\\end{figure}\n\nThe loading is given by tractions acting upon the crack faces. In terms of a Cartesian coordinate system with the origin at the crack tip, traction components are \ndefined as follows \n\\begin{equation}\n\\label{load}\n\\sigma_{2j}^\\pm(x_1,0^\\pm) = p_j^\\pm(x_1), \\quad \\text{for } x_1 < 0, \\quad j = 1,2,\n\\end{equation}\nwhere $p_j^\\pm(x_1)$ are prescribed functions.\n\nThe load is assumed to be self-balanced, so that its principal force and moment vectors are equal to zero. We also assume that the forces are applied outside a \nneighbourhood of the crack tip and vanish at infinity (specific assumptions on the behaviour of the loading will be discussed in the sequel). The body forces are \nassumed to be zero. It is convenient to use the notion of the symmetric and skew-symmetric parts of the loading\n\\begin{equation}\n\\label{parts}\n\\langle p_j \\rangle(x_1) = \\frac{p_j^+(x_1) + p_j^-(x_1)}{2}, \\quad \\jump{0.15}{p_j}(x_1) = p_j^+(x_1) - p_j^-(x_1), \\quad j = 1,2.\n\\end{equation}\n\nThe solution to the physical problem is sought in the class of functions which vanish at infinity and have finite elastic energy.\n\nThe results presented in Sections \\ref{mellin} and \\ref{asymptotics} are known in the literature (see for example: Williams, 1959; Erdogan, 1963; Rice, 1965; \nWillis, 1971), at least for symmetric loading. These results, together with the results on higher order terms presented in Section \\ref{high}, which are less common \nin the literature, provide the formulae needed for the perturbation analysis developed in the sequel of the paper.\n\n\n\\subsection{Representation in terms of Mellin transforms}\n\\label{mellin}\nIn polar coordinates the Mellin transforms for the \ndisplacement vector and the stress tensor with locally bounded elastic energy are defined as follows \n$$\n\\tilde{\\mbox{\\boldmath $u$}}(s,\\theta) = \\int_0^\\infty \\mbox{\\boldmath $u$}(r,\\theta) r^{s-1} dr, \\quad\n\\tilde{\\mbox{\\boldmath $\\sigma$}}(s,\\theta) = \\int_0^\\infty \\mbox{\\boldmath $\\sigma$}(r,\\theta) r^s dr, \n$$\nand they are represented by analytic functions in the strips $-\\vartheta_0 < \\mathop{\\mathrm{Re}}(s) < \\vartheta_\\infty$ and $-\\gamma_0 < \\mathop{\\mathrm{Re}}(s) < \\gamma_\\infty$, respectively, where \n$\\vartheta_0,\\vartheta_\\infty \\ge 0$ ($\\vartheta_0 + \\vartheta_\\infty > 0$), $\\gamma_0,\\gamma_\\infty > 0$ are constants related to the behaviour of the solution at the \ncrack tip and at infinity, namely\n\\begin{equation}\n\\label{exponents}\n\\mbox{\\boldmath $u$}(r,\\theta) = \n\\left\\{\n\\begin{array}{ll}\nO(r^{\\vartheta_0}), & r \\to 0, \\\\\nO(r^{-\\vartheta_\\infty}), & r \\to \\infty, \n\\end{array}\n\\right.\n\\quad\n\\mbox{\\boldmath $\\sigma$}(r,\\theta) = \n\\left\\{\n\\begin{array}{ll}\nO(r^{\\gamma_0-1}), & r \\to 0, \\\\\nO(r^{-\\gamma_\\infty-1}), & r \\to \\infty, \n\\end{array}\n\\right.\n\\end{equation}\n\nCorrespondingly, the inverse transforms are \n\\begin{equation}\n\\label{inverse}\n\\mbox{\\boldmath $u$}(r,\\theta) = \n\\frac{1}{2\\pi i} \\int_{\\omega_1 - i\\infty}^{\\omega_1 +i\\infty} \\tilde{\\mbox{\\boldmath $u$}}(s,\\theta) \nr^{-s} ds, \\quad\n\\mbox{\\boldmath $\\sigma$}(r,\\theta) = \n\\frac{1}{2\\pi i} \\int_{\\omega_2 - i\\infty}^{\\omega_2 + i\\infty} \\tilde{\\mbox{\\boldmath $\\sigma$}}(s,\\theta) \nr^{-s - 1} ds, \n\\end{equation}\nwhere $-\\vartheta_0 < \\omega_1 < \\vartheta_\\infty$ and $-\\gamma_0 < \\omega_2 < \\gamma_\\infty$.\n\n\\subsubsection{General solution to the field equations of linear elasticity} \n\\label{general}\nIn polar coordinates, with the origin at the crack tip, the Mellin transforms of stresses and displacements satisfy the relations (the material constants involved in \nthese representations are reported in Appendix \\ref{app1})\n$$\n\\tilde{\\sigma}_{\\theta\\theta}^\\pm = \nC_1^\\pm \\cos[(s + 1)\\theta] + C_2^\\pm \\cos[(s - 1)\\theta] + C_3^\\pm \\sin[(s + 1)\\theta] \n+ C_4^\\pm \\sin[(s - 1)\\theta],\n$$\n$$\n\\tilde{\\sigma}_{rr}^\\pm = -\\frac{1}{s}\\left[\\tilde{\\sigma}_{\\theta\\theta}^\\pm \n- \\frac{1}{s-1}\\nderiv{}{\\theta}{2}\\tilde{\\sigma}_{\\theta\\theta}^\\pm\\right], \\quad\n\\tilde{\\sigma}_{r\\theta}^\\pm = \\frac{1}{s-1} \\deriv{}{\\theta}\\tilde{\\sigma}_{\\theta\\theta}^\\pm,\n$$\n$$\n\\tilde{u}_r^\\pm = \\frac{1}{2s \\mu_\\pm}[\\tilde{\\sigma}_{\\theta\\theta}^\\pm \n- (1 - \\nu_\\pm)\\tilde{\\sigma}_0^\\pm], \\quad\n\\tilde{u}_\\theta^\\pm = -\\frac{1}{2s \\mu_\\pm}\\left[\\tilde{\\sigma}_{r\\theta}^\\pm \n+ \\frac{1 - \\nu_\\pm}{s + 1}\\deriv{}{\\theta}\\tilde{\\sigma}_0^\\pm\\right], \\quad\n$$\nwhere $\\tilde{\\sigma}_0^\\pm = \\tilde{\\sigma}_{rr}^\\pm + \\tilde{\\sigma}_{\\theta\\theta}^\\pm$.\n\nHence, we deduce\n$$\n\\tilde{\\sigma}_{rr}^\\pm = \n-\\frac{s + 3}{s - 1} C_1^\\pm \\cos[(s + 1)\\theta] - C_2^\\pm \\cos[(s - 1)\\theta]\n- \\frac{s + 3}{s - 1} C_3^\\pm \\sin[(s + 1)\\theta] - C_4^\\pm \\sin[(s - 1)\\theta],\n$$\n$$\n\\tilde{\\sigma}_{r\\theta}^\\pm = \n-\\frac{s + 1}{s - 1} C_1^\\pm \\sin[(s + 1)\\theta] - C_2^\\pm \\sin[(s - 1)\\theta]\n+ \\frac{s + 1}{s - 1} C_3^\\pm \\cos[(s + 1)\\theta] + C_4^\\pm \\cos[(s - 1)\\theta],\n$$\n$$\\begin{array}{l}\n\\displaystyle\n\\tilde{u}_r^\\pm = \n\\frac{1}{2s \\mu_\\pm} \\left\\{ C_1^\\pm \\cos[(s + 1)\\theta] + C_2^\\pm \\cos[(s - 1)\\theta]\n+ C_3^\\pm \\sin[(s + 1)\\theta] + C_4^\\pm \\sin[(s - 1)\\theta] \\right. \\\\[3mm]\n\\displaystyle \\hspace{60mm}\n\\left. + \\frac{4(1 - \\nu_\\pm)}{s - 1}\n\\left[ C_1^\\pm \\cos[(s + 1)\\theta] + C_3^\\pm \\sin[(s + 1)\\theta] \\right] \\right\\},\n\\end{array}\n$$\n$$\n\\begin{array}{l}\n\\displaystyle\n\\tilde{u}_\\theta^\\pm = \n-\\frac{1}{2s \\mu_\\pm} \\left\\{ -C_1^\\pm \\frac{s + 1}{s - 1} \\sin[(s + 1)\\theta] \n- C_2^\\pm \\sin[(s - 1)\\theta]\n+ C_3^\\pm \\frac{s + 1}{s - 1} \\cos[(s + 1)\\theta] \\right. \\\\[3mm]\n\\displaystyle \\hspace{30mm}\n\\left. + C_4^\\pm \\cos[(s - 1)\\theta] + \\frac{4(1 - \\nu_\\pm)}{s - 1}\n\\left[ C_1^\\pm \\sin[(s + 1)\\theta] - C_3^\\pm \\cos[(s + 1)\\theta] \\right] \\right\\}.\n\\end{array}\n$$\n\nThe coefficients $C_j^\\pm$ depend on the loading and will be given in the next section.\n\n\\subsubsection{The boundary conditions and full field solution} \n\\label{full}\nThe coefficients $C_j^\\pm$ are obtained from the boundary conditions on the crack faces, namely \n$$\n\\sigma_{\\theta\\theta}^\\pm(r,\\pm\\pi) = p^\\pm(r), \\quad \\sigma_{r\\theta}^\\pm(r,\\pm\\pi) = q^\\pm(r),\n$$\nwhere $p^\\pm(r), q^\\pm(r)$ are prescribed functions given by $p^\\pm(r) = p_2^\\pm(-r)$, $q^\\pm(r) = p_1^\\pm(-r)$, see \\eq{load}. Mishuris and Kuhn (2001) give the result in a \ncompact form as the solution to the following system of algebraic equations\n\\begin{equation}\n\\label{system}\n\\frac{2\\sin \\pi s}{s - 1}\n\\left[\n\\begin{array}{l}\nC_1^\\pm(s) \\\\[3mm]\nC_3^\\pm(s) \n\\end{array}\n\\right] = \n\\left[\n\\begin{array}{ll}\n-\\sin \\pi s & \\pm \\cos \\pi s \\\\[3mm]\n\\pm \\cos \\pi s & \\sin \\pi s\n\\end{array}\n\\right]\n\\mbox{\\boldmath $D$}(s)\n\\pm \\left[\n\\begin{array}{l}\n\\displaystyle \\langle \\tilde{q} \\rangle \\pm \\frac{1}{2} \\jump{0.15}{\\tilde{q}} \\\\[3mm]\n\\displaystyle \\langle \\tilde{p} \\rangle \\pm \\frac{1}{2} \\jump{0.15}{\\tilde{p}} \n\\end{array}\n\\right],\n\\end{equation}\n$$\n\\left[\n\\begin{array}{l}\n\\displaystyle\nC_1^\\pm(s) + C_2^\\pm(s) \\\\[3mm]\n\\displaystyle\n\\frac{s + 1}{s - 1} C_3^\\pm(s) + C_4^\\pm(s)\n\\end{array}\n\\right] = \\mbox{\\boldmath $D$}(s),\n$$\nwhere $\\mbox{\\boldmath $D$}(s) = - \\mbox{\\boldmath $\\Phi$}^{-1}(s) \\mbox{\\boldmath $F$}(s)$ and \n$$\n\\mbox{\\boldmath $\\Phi$}(s) = \n\\left[\n\\begin{array}{ll}\n\\cos \\pi s & d_* \\sin \\pi s \\\\[3mm]\n-d_* \\sin \\pi s & \\cos \\pi s \n\\end{array}\n\\right],\n\\quad\n\\mbox{\\boldmath $F$}(s) = \n\\left[\n\\begin{array}{l}\n\\displaystyle \\langle \\tilde{p} \\rangle(s) + \\frac{\\alpha}{2} \\jump{0.15}{\\tilde{p}}(s) \\\\[3mm]\n\\displaystyle \\langle \\tilde{q} \\rangle(s) + \\frac{\\alpha}{2} \\jump{0.15}{\\tilde{q}}(s)\n\\end{array}\n\\right].\n$$\n\nNote that the prescribed boundary conditions appear in \\eq{system} and in the definition of the vector $\\mbox{\\boldmath $F$}(s)$, in terms of the symmetric and skew-symmetric parts of the \nloading. The material constants $d_*$ and $\\alpha$ are the Dundurs parameters, as described in the Appendix \\ref{app1}.\n\nDenoting the determinant of $\\mbox{\\boldmath $\\Phi$}(s)$ by $\\delta(s) = \\cos^2 \\pi s + d_*^2 \\sin^2 \\pi s$, we obtain the full field solution in terms of the Mellin transform as follows \n$$\\begin{array}{l}\n\\displaystyle \nC_1^\\pm(s) = \\frac{s - 1}{2 \\delta(s)}\n\\Bigg\\{\n\\langle \\tilde{p} \\rangle[(1 \\mp d_*) \\cos \\pi s] \n+ \\frac{1}{2} \\jump{0.15}{\\tilde{p}}[(1 \\mp d_*)\\alpha \\cos \\pi s] \\\\[3mm]\n\\displaystyle \\hspace{15mm} \n+ \\langle \\tilde{q} \\rangle\\left[-(1 \\mp d_*)d_* \\sin \\pi s \\right]\n+ \\frac{1}{2} \\jump{0.15}{\\tilde{q}}\\left[(d_* - \\alpha)d_* \\sin \\pi s \n+ (1 \\mp \\alpha)\\frac{\\cos^2 \\pi s}{\\sin \\pi s}\\right]\n\\Bigg\\},\n\\end{array}\n$$\n$$\n\\begin{array}{l}\n\\displaystyle \nC_3^\\pm(s) = \\frac{s - 1}{2 \\delta(s)}\n\\Bigg\\{\n\\langle \\tilde{p} \\rangle\\left[-(1 \\mp d_*)d_* \\sin \\pi s\\right]\n+ \\frac{1}{2} \\jump{0.15}{\\tilde{p}}\\left[(d_*-\\alpha)d_* \\sin \\pi s \n+ (1 \\mp \\alpha)\\frac{\\cos^2 \\pi s}{\\sin \\pi s}\\right] \\\\[3mm]\n\\displaystyle \\hspace{15mm}\n+ \\langle \\tilde{q} \\rangle[-(1 \\mp d_*) \\cos \\pi s] \n+ \\frac{1}{2} \\jump{0.15}{\\tilde{q}}[-(1 \\mp d_*)\\alpha \\cos \\pi s] \n\\Bigg\\},\n\\end{array}\n$$\n$$\n\\begin{array}{l}\n\\displaystyle \nC_2^\\pm(s) = -\\frac{1}{2 \\delta(s)}\n\\Bigg\\{\n\\langle \\tilde{p} \\rangle[(1 + s \\pm (1 - s)d_*) \\cos \\pi s] \n+ \\frac{1}{2} \\jump{0.15}{\\tilde{p}}[(1 + s \\pm (1 - s)d_*)\\alpha \\cos \\pi s] \\\\[3mm]\n\\displaystyle \\hspace{40mm} \n+ \\langle \\tilde{q} \\rangle\\left[(\\pm(1 + s) + (1 - s)d_*)d_* \\sin \\pi s\\right] \\\\[3mm]\n\\displaystyle \\hspace{40mm}\n+ \\frac{1}{2} \\jump{0.15}{\\tilde{q}}\\left[-(\\alpha(1 + s) + d_*(1 - s))d_* \\sin \\pi s \n- (1 \\mp \\alpha)(1 - s)\\frac{\\cos^2 \\pi s}{\\sin \\pi s}\\right]\n\\Bigg\\},\n\\end{array}\n$$\n$$\n\\begin{array}{l}\n\\displaystyle \nC_4^\\pm(s) = -\\frac{1}{2 \\delta(s)}\n\\Bigg\\{\n\\langle \\tilde{p} \\rangle\\left[(1 - s \\pm (1 + s)d_*)d_* \\sin \\pi s\\right] \\\\[3mm]\n\\displaystyle \\hspace{15mm}\n+ \\frac{1}{2} \\jump{0.15}{\\tilde{p}}\\left[(\\alpha(1 - s) + d_*(1 + s))d_* \\sin \\pi s \n+ (1 \\mp \\alpha)(1 + s)\\frac{\\cos^2 \\pi s}{\\sin \\pi s}\\right] \\\\[3mm]\n\\displaystyle \\hspace{15mm}\n+ \\langle \\tilde{q} \\rangle[(1 - s \\pm (1 + s)d_*) \\cos \\pi s] \n+ \\frac{1}{2} \\jump{0.15}{\\tilde{q}}[(1 - s \\pm (1 + s)d_*)\\alpha \\cos \\pi s] \n\\Bigg\\}.\n\\end{array}\n$$\n\nWe note that some of the poles of the functions $C_j^\\pm(s)$ are obtained from the solution of the equation $\\delta(s) = \\cos^2 \\pi s + d_*^2 \\sin^2 \\pi s = 0$, so that \nthe poles are given by \n\\begin{equation}\n\\label{poles}\ns_n^\\pm = \\frac{1 - 2n}{2} \\pm i\\epsilon,\n\\end{equation}\nwhere $n$ is an integer, and $\\epsilon$ is the bimaterial constant defined in Appendix \\ref{app1}.\n\nFor the purpose of evaluation of residues, it will be useful to have at hand the derivative of $\\delta(s)$. This is given by \n$\\delta'(s) = 2\\pi(-1+d_*^2) \\sin \\pi s \\cos \\pi s$, and its value at $s = s_n^\\pm$ is as follows\n\\begin{equation}\n\\label{deltap}\n\\delta'\\left(s_n^\\pm\\right) = \\pm 2\\pi i d_*.\n\\end{equation}\nThis formula will be used in the next section for the derivation of the asymptotic estimates of stress and displacement fields near the crack tip.\n\n\\subsection{Asymptotic representations near the crack tip and the stress intensity factors}\n\\label{asymptotics}\nThe solution outlined above leads to the asymptotic representations of stress and displacement near the crack tip. Analysis of the singular terms in the stress components \nyields the complex stress intensity factor for the interfacial crack, where the stress singularity is accompanied by the oscillatory behaviour of the physical fields.\n\n\\subsubsection{Asymptotics of stress and displacement near the crack tip}\nTaking into account the assumptions on the applied forces, the inspection of the results for $\\tilde{\\mbox{\\boldmath $\\sigma$}}(s,\\theta)$ shows that $\\tilde{\\mbox{\\boldmath $\\sigma$}}(s,\\theta)$ is \nanalytic in the strip $-1\/2 < \\mathop{\\mathrm{Re}}(s) < 1\/2$. Therefore, in \\eq{exponents} we have $\\gamma_0 = \\gamma_\\infty = 1\/2$, and choosing $\\omega_2 = 0$, the inverse transform \nis given by \n$$\n\\mbox{\\boldmath $\\sigma$}(r,\\theta) = \\frac{1}{2\\pi i} \\int_{0-i\\infty}^{0+i\\infty} \n\\tilde{\\mbox{\\boldmath $\\sigma$}}(s,\\theta) r^{-s-1} ds.\n$$\n\nBy means of the Cauchy's residue theorem, we can get the asymptotics of $\\mbox{\\boldmath $\\sigma$}(r,\\theta)$ as $r \\to 0$ as follows\n$$\n\\mbox{\\boldmath $\\sigma$}(r,\\theta) =  \\sum_{\\pm} \\mathop{\\mathrm{Res}}[\\tilde{\\mbox{\\boldmath $\\sigma$}}(s,\\theta) r^{-s-1},s=s_1^\\pm] + \n\\frac{1}{2\\pi i} \\int_{\\omega-i\\infty}^{\\omega+i\\infty} \\tilde{\\mbox{\\boldmath $\\sigma$}}(s,\\theta) r^{-s-1} ds, \n$$\nwhere $\\omega < -1\/2$. The first two terms are the leading term, of the order $O(r^{-1\/2})$, and the last term is a higher order term, of the order \n$O(r^{\\beta}), \\beta > -1\/2$. The notation $\\mathop{\\mathrm{Res}}[f(s),s=s^*]$ stands for the residue of $f$ at the pole $s = s^*$,\n$$\n\\mathop{\\mathrm{Res}}[f(s),s=s^*] = \\frac{1}{(m-1)!} \\left. \\frac{d^{m-1} f(s)(s-s^*)^m}{ds^{m-1}}\\right|_{s=s^*},\n$$\nwhere $m$ is the order of the pole.\n\nIt suffices now to compute the residues at the poles $s = s_1^\\pm = -1\/2 \\pm i\\epsilon$,\n$$\n\\begin{array}{l}\n\\mathop{\\mathrm{Res}}\\left[\\tilde{\\mbox{\\boldmath $\\sigma$}}(s,\\theta)\\, r^{-s - 1}, s = -1\/2 \\pm i\\epsilon\\right] = \\\\[3mm]\n\\hspace{30mm}\n\\displaystyle\n= \\left[\\tilde{\\mbox{\\boldmath $\\sigma$}}(s,\\theta) \\delta(s)\\, r^{-s - 1}\\right]_{s = -1\/2 \\pm i\\epsilon} \n\\cdot \\mathop{\\mathrm{Res}}\\left[\\frac{1}{\\delta(s)}, s = -1\/2 \\pm i\\epsilon\\right] \\\\[3mm]\n\\hspace{30mm}\n\\displaystyle\n= \\left[\\tilde{\\mbox{\\boldmath $\\sigma$}}(s,\\theta) \\delta(s)\\, r^{-s - 1}\\right]_{s = -1\/2 \\pm i\\epsilon} \n\\cdot \\lim_{s \\to -1\/2 \\pm i\\epsilon} \\frac{s-(-1\/2 \\pm i\\epsilon)}{\\delta(s)} \\\\[3mm]\n\\hspace{30mm}\n\\displaystyle\n= \\left[\\tilde{\\mbox{\\boldmath $\\sigma$}}(s,\\theta) \\delta(s)\\, r^{-s - 1}\\right]_{s = -1\/2 \\pm i\\epsilon} \n\\cdot \\frac{1}{\\delta'(-1\/2 \\pm i\\epsilon)} \\\\[3mm]\n\\hspace{30mm}\n\\displaystyle\n= \\mp \\frac{i}{2\\pi d_*} \n\\left[\\tilde{\\mbox{\\boldmath $\\sigma$}}(s,\\theta) \\delta(s)\\right]_{s = -1\/2 \\pm i\\epsilon} \\cdot \nr^{-1\/2 \\mp i\\epsilon},\n\\end{array}\n$$\nwhere in the last equality we used the formula \\eq{deltap}.\n\nIt can be shown that $\\beta = 0$, so that the leading term asymptotics of the stress field is \n\\begin{equation}\n\\label{sigma}\n\\mbox{\\boldmath $\\sigma$}(r,\\theta) =  \\beth_{-1\/2}(r,\\theta) + O(1),\n\\end{equation}\nwhere\n$$\n\\beth_{-1\/2}(r,\\theta) = \n\\frac{i}{2\\pi d_*}\n\\left\\{-\\left[\\tilde{\\mbox{\\boldmath $\\sigma$}}(s,\\theta) \\delta(s)\\right]_{s = -1\/2 + i\\epsilon} \nr^{-1\/2 - i\\epsilon}\n+ \\left[\\tilde{\\mbox{\\boldmath $\\sigma$}}(s,\\theta) \\delta(s)\\right]_{s = -1\/2 - i\\epsilon} \nr^{-1\/2 + i\\epsilon}\\right\\}.\n$$\n\nAn inspection of the results for $\\tilde{\\mbox{\\boldmath $u$}}(s,\\theta)$ shows that it is analytic in the same strip $-1\/2 < \\mathop{\\mathrm{Re}}(s) < 1\/2$, except for a simple pole in $s = 0$. Since \nwe seek the solution vanishing at infinity, the strip of analyticity for the function $\\tilde{\\mbox{\\boldmath $u$}}(s,\\theta)$ is $0 < \\mathop{\\mathrm{Re}}(s) < 1\/2$. Therefore, in \\eq{exponents} we \nhave $\\vartheta_0 = 0$, $\\vartheta_\\infty = \\gamma_\\infty = 1\/2$. Choosing $\\omega_1$ in this interval and applying the Cauchy's residue theorem, we can write the \nleading term asymptotics of $\\mbox{\\boldmath $u$}(r,\\theta)$ as $r \\to 0$ as follows \n$$\n\\mbox{\\boldmath $u$}(r,\\theta) = \\mathop{\\mathrm{Res}}\\left[\\tilde{\\mbox{\\boldmath $u$}}(s,\\theta)\\, r^{-s}, s = 0\\right] + \\sum_{\\pm} \\mathop{\\mathrm{Res}}\\left[\\tilde{\\mbox{\\boldmath $u$}}(s,\\theta)\\, r^{-s}, s = s_1^\\pm\\right] \n+ \\frac{1}{2 \\pi i} \\int_{\\omega-i\\infty}^{\\omega+i\\infty} \\tilde{\\mbox{\\boldmath $u$}}(s,\\theta) r^{-s} ds,\n$$\nwhere $\\omega < -1\/2$. The first term of the order $O(1)$ corresponds to the translation of the crack tip and reads\n\\begin{equation}\n\\label{qtheta}\n\\mbox{\\boldmath $u$}(0,\\theta) = \\mbox{\\boldmath $V$}_0(\\theta) =  \n\\mbox{\\boldmath $Q$}(\\theta)\\mbox{\\boldmath $w$}_0, \\quad \\mbox{\\boldmath $Q$}(\\theta) = \\left[\\begin{array}{ll} \\cos\\theta & \\sin\\theta \\\\[3mm] -\\sin\\theta & \\cos\\theta \\end{array}\\right],\n\\end{equation}\nwhere\n$$\n\\begin{array}{ll}\n\\displaystyle w_{01} & \\displaystyle = u_1(0,0) \\\\[3mm]\n& \\displaystyle = \\frac{1}{2\\pi\\mu_\\pm} \\left\\{ [1 - 2\\nu_\\pm \\pm 2d_*(\\nu_\\pm - 1)] \\pi \\int_{0}^{\\infty} \\langle p \\rangle(r) dr + \n(-1 \\pm \\alpha)(\\nu_\\pm - 1) \\int_{0}^{\\infty} \\jump{0.15}{q}(r) (\\log r) dr \\right\\},\n\\end{array}\n$$\n$$\n\\begin{array}{ll}\n\\displaystyle w_{02} & \\displaystyle = u_2(0,0) \\\\[3mm]\n& \\displaystyle = \\frac{1}{2\\pi\\mu_\\pm} \\left\\{ -[1 - 2\\nu_\\pm \\pm 2d_*(\\nu_\\pm - 1)] \\pi \\int_{0}^{\\infty} \\langle q \\rangle(r) dr + \n(-1 \\pm \\alpha)(\\nu_\\pm - 1) \\int_{0}^{\\infty} \\jump{0.15}{p}(r) (\\log r) dr \\right\\},\n\\end{array}\n$$\nare the Cartesian components of the translation of the crack tip.\nThe second and third terms are of the order $O(r^{1\/2})$, and the last term is a higher order term, which is of the order $O(r^{\\beta}), \\beta > 1\/2$. \n\nIt can be shown that $\\beta = 1$, so that the computation of the residues leads to the asymptotics of the displacement field as follows\n\\begin{equation}\n\\label{disp}\n\\mbox{\\boldmath $u$}(r,\\theta) \n= \\mbox{\\boldmath $V$}_0(\\theta) + \\mbox{\\boldmath $V$}_{1\/2}(r,\\theta) + O(r),\n\\end{equation}\nwhere\n$$\n\\mbox{\\boldmath $V$}_{1\/2}(r,\\theta) = \n\\frac{i}{2\\pi d_*}\n\\left\\{-\\left[\\tilde{\\mbox{\\boldmath $u$}}(s,\\theta) \\delta(s)\\right]_{s = -1\/2 + i\\epsilon} \nr^{1\/2 - i\\epsilon}\n+ \\left[\\tilde{\\mbox{\\boldmath $u$}}(s,\\theta) \\delta(s)\\right]_{s = -1\/2 - i\\epsilon} \nr^{1\/2 + i\\epsilon}\\right\\}.\n$$\n\nNote that the remainders $O(1)$ in \\eq{sigma} and $O(r)$ in \\eq{disp} are rough estimates and will be refined in the Section \\ref{high}. However, their physical meaning is \nalready evident: the first corresponds to the so-called T-stress and the second to a rigid body rotation superimposed to a uniform deformation near the crack tip. \n\n\\subsubsection{The complex stress intensity factor $K = K_\\text{I} + i K_\\text{II}$}\nFrom the full field solution outlined in Sections \\ref{general} and \\ref{full} we obtain\n\\begin{equation}\n\\label{eq46}\n\\tilde{\\sigma}_{\\theta\\theta}(s,0) + i \\tilde{\\sigma}_{r\\theta}(s,0) = \n-\\left\\{\\langle \\tilde{p} \\rangle(s) + i\\langle \\tilde{q} \\rangle(s) + \\frac{\\alpha}{2} \\jump{0.15}{\\tilde{p}}(s) + i \\frac{\\alpha}{2} \\jump{0.15}{\\tilde{q}}(s)\\right\\}\n\\frac{\\cos \\pi s + i d_* \\sin \\pi s}{\\delta(s)}.\n\\end{equation}\n\nApplying the formula (\\ref{sigma}), the leading term asymptotics may be written as\n$$\n\\sigma_{\\theta\\theta}(r,0) + i \\sigma_{r\\theta}(r,0) = \\frac{K}{\\sqrt{2\\pi}} r^{-1\/2 + i\\epsilon} + O(1),\n$$\nwhere\n$$\nK = - \\sqrt{\\frac{2}{\\pi}} \\cosh(\\pi\\epsilon) \\left\\{\\langle \\tilde{p} \\rangle(s) + i\\langle \\tilde{q} \\rangle(s) \n+ \\frac{\\alpha}{2} \\jump{0.15}{\\tilde{p}}(s) + i \\frac{\\alpha}{2} \\jump{0.15}{\\tilde{q}}(s)\\right\\}_{s = -1\/2 - i\\epsilon}\n$$\nis the complex stress intensity factor.\n\nCorrespondingly, by means of the formula (\\ref{disp}), we obtain\n$$\n\\jump{0.15}{u_\\theta}(r) + i\\jump{0.15}{u_r}(r) = \n-\\frac{(1 - \\nu_+) \/ \\mu_+ + (1 - \\nu_-) \/ \\mu_-}{(1\/2 + i\\epsilon)\\cosh(\\pi\\epsilon)} \\frac{K}{\\sqrt{2\\pi}} r^{1\/2 + i\\epsilon} + O(r), \n$$\nwhere we used the notation $\\jump{0.15}{f}(r) = f(r,\\pi) - f(r,-\\pi)$. Note that the zero-order term $\\mbox{\\boldmath $V$}_0(\\theta)$ present in \\eq{disp} disappears in the last formula.\n\nThe formula for the complex stress intensity factor is then\n$$\n\\begin{array}{l}\n\\displaystyle\nK = - \\sqrt{\\frac{2}{\\pi}} \\cosh(\\pi\\epsilon) \n\\int_{0}^{\\infty}\\left\\{\\langle p \\rangle(r) + i\\langle q \\rangle(r) + \\frac{\\alpha}{2} \\jump{0.15}{p}(r) + i \\frac{\\alpha}{2} \\jump{0.15}{q}(r)\\right\\} \nr^{-1\/2 - i\\epsilon} dr.\n\\end{array}\n$$\n\nAs expected, this representation is consistent with Hutchinson, Mear and Rice (1987) who considered a crack loaded by point forces a distance $a$ behind the tip:\n$$\np^+(r) = p^-(r) = -P \\delta(r-a), \\quad q^+(r) = q^-(r) = -Q \\delta(r-a),\n$$\nwhere $\\delta(\\cdot)$ is the Dirac delta function, so that\n$$\n\\langle p \\rangle(r) = -P \\delta(r-a), \\quad \\jump{0.15}{p}(r) = 0, \\quad \\langle q \\rangle(r) = -Q \\delta(r-a), \\quad \\jump{0.15}{q}(r) = 0.\n$$\nFor this loading we obtain\n$$\nK  = \\sqrt{\\frac{2}{\\pi}} \\cosh(\\pi\\epsilon) \\int_0^\\infty \\left\\{ P \\delta(r-a) + i Q \\delta(r-a) \\right\\} r^{-1\/2-i\\epsilon} dr \n= \\sqrt{\\frac{2}{\\pi}} \\cosh(\\pi\\epsilon) (P+iQ) a^{-1\/2-i\\epsilon},\n$$\nwhich fully agrees with Hutchinson, Mear and Rice (1987).\n\n\\subsection{High-order asymptotics}\n\\label{high}\nThe asymptotic procedure involving the evaluation of stress intensity factors for cracks with a slightly perturbed front requires the high-order asymptotics of the \ndisplacement and stress fields near the crack edge. The procedure of the previous section can be extended to the high-order terms, and hence the high-order asymptotics \nof the stress field is given by\n\\begin{equation}\n\\label{beth}\n\\begin{array}{l}\n\\mbox{\\boldmath $\\sigma$}(r,\\theta) = \\beth_{-1\/2}(r,\\theta) + \\mbox{\\boldmath $T$}(\\theta) + \\beth_{1\/2}(r,\\theta)\n+ \\mbox{\\boldmath $S$}(\\theta)r + \\beth_{3\/2}(r,\\theta) + O(r^2), \n\\end{array}\n\\end{equation}\nwhere (the superscript $^\\top$ denotes transposition, $x_1 = r \\cos\\theta$, $x_2 = r \\sin\\theta$, $\\mbox{\\boldmath $Q$}(\\theta)$ is given in \\eq{qtheta})\n$$\n\\mbox{\\boldmath $T$}(\\theta) = \\mathop{\\mathrm{Res}}\\left[\\tilde{\\mbox{\\boldmath $\\sigma$}}(s,\\theta), s = -1\\right] = T \\mbox{\\boldmath $Q$}(\\theta) \\left[\\begin{array}{ll} 1 & 0 \\\\[3mm] 0 & 0 \\end{array}\\right] \\mbox{\\boldmath $Q$}^\\top(\\theta), \n$$\n$$\n\\begin{array}{l}\n\\displaystyle \\mbox{\\boldmath $S$}(\\theta) = \\mathop{\\mathrm{Res}}\\left[\\tilde{\\mbox{\\boldmath $\\sigma$}}(s,\\theta), s = -2\\right] = \\\\[3mm]\n\\displaystyle = \\frac{1 \\mp \\alpha}{\\pi \\sqrt{x_1^2 + x_2^2}} \\mbox{\\boldmath $Q$}(\\theta) \n\\left[\n\\begin{array}{ll}\n\\displaystyle x_2 \\int_{0}^{\\infty} \\jump{0.15}{p}(r) \\frac{dr}{r^2} - x_1 \\int_{0}^{\\infty} \\jump{0.15}{q}(r) \\frac{dr}{r^2} \n& \\displaystyle x_2 \\int_{0}^{\\infty} \\jump{0.15}{q}(r) \\frac{dr}{r^2} \\\\[3mm]\n\\displaystyle x_2 \\int_{0}^{\\infty} \\jump{0.15}{q}(r) \\frac{dr}{r^2} \n& \\displaystyle 0 \n\\end{array}\n\\right] \n\\mbox{\\boldmath $Q$}^\\top(\\theta),\n\\end{array}\n$$\nand\n$$\n\\begin{array}{ll}\n\\displaystyle \\beth_{\\,l\/2}(r,\\theta) \n&\n\\displaystyle = \\sum_{\\pm} \\mathop{\\mathrm{Res}}\\left[\\tilde{\\mbox{\\boldmath $\\sigma$}}(s,\\theta)\\, r^{-s - 1}, s = -l\/2 -1 \\pm i\\epsilon\\right] \\\\[5mm]\n&\n\\displaystyle = \\frac{i}{2\\pi d_*}\n\\left\\{-\\left[\\tilde{\\mbox{\\boldmath $\\sigma$}}(s,\\theta) \\delta(s)\\right]_{s = -l\/2 -1 + i\\epsilon} \nr^{l\/2 - i\\epsilon}\n+ \\left[\\tilde{\\mbox{\\boldmath $\\sigma$}}(s,\\theta) \\delta(s)\\right]_{s = -l\/2 -1 - i\\epsilon} \nr^{l\/2 + i\\epsilon}\\right\\}, \\quad l = 1,3.\n\\end{array}\n$$\n\nNote that $T$ is the T-stress given by\n$$\nT = \\frac{1 \\mp \\alpha}{\\pi} \\int_{0}^{\\infty} \\jump{0.15}{q}(r) \\frac{dr}{r}.\n$$\n\nApplying the formula \\eq{beth} to \\eq{eq46}, we finally obtain\n$$\n\\sigma_{\\theta\\theta}(r,0) + i \\sigma_{r\\theta}(r,0) = \\frac{K}{\\sqrt{2\\pi}} r^{-1\/2 + i\\epsilon} \n+ \\frac{A}{\\sqrt{2\\pi}} r^{1\/2 + i\\epsilon} + \\frac{B}{\\sqrt{2\\pi}} r^{3\/2 + i\\epsilon} \n+ O(r^{2}),\n$$\nwhere the constants in the high-order terms are \n$$\nA = \\sqrt{\\frac{2}{\\pi}} \\cosh(\\pi\\epsilon) \n\\int_0^\\infty \\left\\{\\langle p \\rangle(r) + i\\langle q \\rangle(r) + \\frac{\\alpha}{2} \\jump{0.15}{p}(r) + i \\frac{\\alpha}{2} \\jump{0.15}{q}(r)\\right\\} \nr^{-3\/2-i\\epsilon} dr, \n$$\n$$\nB = -\\sqrt{\\frac{2}{\\pi}} \\cosh(\\pi\\epsilon) \n\\int_0^\\infty \\left\\{\\langle p \\rangle(r) + i\\langle q \\rangle(r) + \\frac{\\alpha}{2} \\jump{0.15}{p}(r) + i \\frac{\\alpha}{2} \\jump{0.15}{q}(r)\\right\\} \nr^{-5\/2-i\\epsilon} dr. \n$$\nIf the loading is smooth enough, then we can integrate by parts to obtain\n$$\nA = \\sqrt{\\frac{2}{\\pi}} \\frac{\\cosh(\\pi\\epsilon)}{1\/2+i\\epsilon} \n\\int_0^\\infty \\left\\{\\langle p \\rangle'(r) + i\\langle q \\rangle'(r) + \\frac{\\alpha}{2} \\jump{0.15}{p}'(r) + i \\frac{\\alpha}{2} \\jump{0.15}{q}'(r)\\right\\} \nr^{-1\/2-i\\epsilon} dr,\n$$\n$$\nB = -\\sqrt{\\frac{2}{\\pi}} \\frac{\\cosh(\\pi\\epsilon)}{(1\/2+i\\epsilon)(3\/2+i\\epsilon)} \n\\int_0^\\infty \\left\\{\\langle p \\rangle''(r) + i\\langle q \\rangle''(r) + \\frac{\\alpha}{2} \\jump{0.15}{p}''(r) + i \\frac{\\alpha}{2} \\jump{0.15}{q}''(r)\\right\\} \nr^{-1\/2-i\\epsilon} dr.\n$$\n\nCorrespondingly, we obtain for the high-order asymptotics of the displacement field the expression\n$$\n\\mbox{\\boldmath $u$}(r,\\theta) = \\mbox{\\boldmath $V$}_0(\\theta) + \\mbox{\\boldmath $V$}_{1\/2}(r,\\theta) + \\mbox{\\boldmath $V$}_1(\\theta)r + \\mbox{\\boldmath $V$}_{3\/2}(r,\\theta) \n+ \\mbox{\\boldmath $V$}_2(\\theta)r^2 + \\mbox{\\boldmath $V$}_{5\/2}(r,\\theta) + O(r^3), \n$$\nwhere\n$$\n\\mbox{\\boldmath $V$}_l(\\theta) = \\mathop{\\mathrm{Res}}\\left[\\tilde{\\mbox{\\boldmath $u$}}(s,\\theta), s = -l\\right] = \\mbox{\\boldmath $Q$}(\\theta) r^{-l} \\mbox{\\boldmath $w$}_l(r,\\theta), \\quad l = 1,2,\n$$\n\\vspace{3mm}\n$$\n\\begin{array}{ll}\n\\displaystyle \\mbox{\\boldmath $V$}_{l\/2}  \n&\n\\displaystyle = \\sum_{\\pm} \\mathop{\\mathrm{Res}}\\left[\\tilde{\\mbox{\\boldmath $u$}}(s,\\theta)\\, r^{-s}, s = -l\/2 \\pm i\\epsilon\\right] \\\\[5mm]\n&\n\\displaystyle = \\frac{i}{2\\pi d_*}\n\\left\\{-\\left[\\tilde{\\mbox{\\boldmath $u$}}(s,\\theta) \\delta(s)\\right]_{s = -l\/2 + i\\epsilon} \nr^{l\/2 - i\\epsilon}\n+ \\left[\\tilde{\\mbox{\\boldmath $u$}}(s,\\theta) \\delta(s)\\right]_{s = -l\/2 - i\\epsilon} \nr^{l\/2 + i\\epsilon}\\right\\}, \\quad l = 3,5,\n\\end{array}\n$$\nand\n$$\nw_{11} = \\frac{1 \\mp \\alpha}{2\\pi \\mu_\\pm} (1 - \\nu_\\pm) \\left\\{x_1\\int_{0}^{\\infty} \\jump{0.15}{q}(r) \\frac{dr}{r} - \nx_2\\int_{0}^{\\infty} \\jump{0.15}{p}(r) \\frac{dr}{r}\\right\\},\n$$\n$$\nw_{12} = \\frac{1 \\mp \\alpha}{2\\pi \\mu_\\pm} \\left\\{(1 - \\nu_\\pm) x_1\\int_{0}^{\\infty} \\jump{0.15}{p}(r) \\frac{dr}{r} - \n\\nu_\\pm x_2\\int_{0}^{\\infty} \\jump{0.15}{q}(r) \\frac{dr}{r}\\right\\},\n$$\n$$\nw_{21} = \\frac{1 \\mp \\alpha}{8\\pi \\mu_\\pm} \\left\\{4(1 - \\nu_\\pm)x_1x_2\\int_{0}^{\\infty} \\jump{0.15}{p}(r) \\frac{dr}{r^2} + \n[(x_1^2 + x_2^2) - (3 - 2\\nu_\\pm)(x_1^2 - x_2^2)]\\int_{0}^{\\infty} \\jump{0.15}{q}(r) \\frac{dr}{r^2}\\right\\},\n$$\n$$\nw_{22} = \\frac{1 \\mp \\alpha}{8\\pi \\mu_\\pm} \\left\\{-[(x_1^2 + x_2^2) + (1 - 2\\nu_\\pm)(x_1^2 - x_2^2)]\\int_{0}^{\\infty} \\jump{0.15}{p}(r) \\frac{dr}{r^2} + \n4\\nu_\\pm x_1x_2 \\int_{0}^{\\infty} \\jump{0.15}{q}(r) \\frac{dr}{r^2}\\right\\}.\n$$\nWe note that $\\mbox{\\boldmath $w$}_l(r,\\theta)$ is homogeneous of degree $l$ in the variable $r$ and that $w_{11}$ and $w_{12}$ are the Cartesian components of the local rigid body rotation superimposed to a uniform deformation near the crack tip mentioned above.\n\nHence\n$$\n\\begin{array}{l}\n\\displaystyle \\jump{0.15}{u_\\theta}(r) + i\\jump{0.15}{u_r}(r) = \n-\\frac{(1 - \\nu_+) \/ \\mu_+ + (1 - \\nu_-) \/ \\mu_-}{\\cosh(\\pi\\epsilon)} \\times \\\\[3mm]\n\\displaystyle \\left\\{ \\frac{1}{1\/2 + i\\epsilon} \\frac{K}{\\sqrt{2\\pi}} r^{1\/2 + i\\epsilon} \n- \\frac{1}{3\/2 + i\\epsilon} \\frac{A}{\\sqrt{2\\pi}} r^{3\/2 + i\\epsilon} + \\frac{1}{5\/2 + i\\epsilon} \\frac{B}{\\sqrt{2\\pi}} r^{5\/2 + i\\epsilon} \\right\\} + O(r^{3}). \n\\end{array}\n$$\n\n\\section{Symmetric and skew-symmetric weight functions for interfacial cracks}\n\\label{wfunc}\nIn this section, we introduce a special type of singular solutions of a homogeneous problem for an interfacial crack. The traces of these functions on the plane containing \nthe crack are known as the weight functions, and they are used in the evaluation of the stress intensity factors in models of linear fracture mechanics. The notations \n$\\jump{0.15}{\\mbox{\\boldmath $U$}}$ and $\\langle \\mbox{\\boldmath $U$} \\rangle$ will be used for the symmetric and skew-symmetric weight function matrices, as outlined in the sequel of the paper.\n\nWe refer to the earlier publications by Antipov (1999), Bercial {\\em et al.}\\ (2005) and Piccolroaz {\\em et al.}\\ (2007) for the detailed discussion of the theory of \nweight functions and related functional equations of the Wiener-Hopf type. Here we give an outline, required for our purpose of evaluation of the weight function matrices \nfor interfacial cracks.\n\nThe special singular solutions for the interfacial crack are defined as the solutions of the elasticity problem where the crack is placed along the positive semi-axis, \n$x_1 > 0$, the boundary conditions are homogeneous (traction-free crack faces) and satisfying special homogeneity properties. In particular, \n\\begin{itemize}\n\\item[(a)]\nthe singular solution $\\mbox{\\boldmath $U$} = [U_1,U_2]^\\top$ satisfies the equation of equilibrium;\n\\item[(b)]\n$\\jump{0.15}{\\mbox{\\boldmath $U$}} = 0$ when $x_1 < 0$;\n\\item[(c)]\nthe associated traction vector acting on the plane containing the crack, $\\mbox{\\boldmath $\\Sigma$} = [\\Sigma_{21},\\Sigma_{22}]^\\top$, is continuous and $\\mbox{\\boldmath $\\Sigma$} = 0$ when $x_2 = 0$ and \n$x_1 > 0$ (homogeneous boundary conditions); \n\\item[(d)]\n$\\mbox{\\boldmath $U$}$ is a linear combination of homogeneous functions of degree $-1\/2 + i\\epsilon$ and $-1\/2 - i\\epsilon$.\n\\end{itemize}\n\nWe enphasise here that the domain for weight functions (where the crack is placed along the positive semi-axis) is different from the domain for the physical solution \n(where the crack is placed along the negative semi-axis). The reason for this is to have the fundamental integral identity, introduced in the sequel, eq. \\eq{betti}, \nin convolution form (see Piccolroaz {\\em et al.}, 2007, for details).\n\n\\subsection{The Wiener-Hopf equation}\n\\subsubsection{Factorization and solution}\nLet us introduce the Fourier transforms of the crack-opening singular displacements and the corresponding traction components\n$$\n\\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^+(\\beta) = \\int_{0}^{\\infty} \\jump{0.15}{\\mbox{\\boldmath $U$}}(x_1) e^{i\\beta x_1} dx_1, \\quad\n\\overline{\\mbox{\\boldmath $\\Sigma$}}^-(\\beta) = \\int_{-\\infty}^{0} \\mbox{\\boldmath $\\Sigma$}(x_1) e^{i\\beta x_1} dx_1.\n$$\nThe superscript $^+$ indicates that the function $\\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^+(\\beta)$ is analytic in the upper half-plane \n$\\ensuremath{\\mathbb{C}}^+ = \\{\\beta \\in \\ensuremath{\\mathbb{C}} : \\mathop{\\mathrm{Re}}\\beta \\in (-\\infty,\\infty), \\mathop{\\mathrm{Im}}\\beta \\in (0,\\infty)\\}$, whereas the superscript $^-$ indicates that the function \n$\\overline{\\mbox{\\boldmath $\\Sigma$}}^-(\\beta)$ is analytic in the lower half-plane $\\ensuremath{\\mathbb{C}}^- = \\{\\beta \\in \\ensuremath{\\mathbb{C}} : \\mathop{\\mathrm{Re}}\\beta \\in (-\\infty,\\infty), \\mathop{\\mathrm{Im}}\\beta \\in (-\\infty,0)\\}$.\n\nThese functions are related via the functional equation of the Wiener-Hopf type (Antipov, 1999):\n\\begin{equation}\n\\label{wh}\n\\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^+(\\beta) = \n-\\frac{b}{|\\beta|} \\mbox{\\boldmath $G$}(\\beta) \\overline{\\mbox{\\boldmath $\\Sigma$}}^-(\\beta), \\quad \\beta \\in \\ensuremath{\\mathbb{R}},\n\\end{equation}\nwhere $b = (1 - \\nu_+)\/\\mu_+ + (1 - \\nu_-)\/\\mu_-$ and \n$$\n\\mbox{\\boldmath $G$}(\\beta) = \n\\left[\n\\begin{array}{cc}\n1 & -i\\mathop{\\mathrm{sign}}(\\beta) d_* \\\\[3mm]\ni\\mathop{\\mathrm{sign}}(\\beta) d_* & 1\n\\end{array}\n\\right].\n$$\n\nThe factorization of $-b\/|\\beta| \\mbox{\\boldmath $G$}$ is given by\n\\begin{equation}\n\\label{fact}\n-\\frac{b}{|\\beta|} \\mbox{\\boldmath $G$}(\\beta) = \n-\\frac{b}{\\beta_+^{1\/2} \\beta_-^{1\/2}} \\mbox{\\boldmath $X$}^+(\\beta) [\\mbox{\\boldmath $X$}^-(\\beta)]^{-1}, \\quad \\beta \\in \\ensuremath{\\mathbb{R}},\n\\end{equation}\nwhere\n$$\n\\mbox{\\boldmath $X$}^+(\\beta) = d_0\n\\left[\n\\begin{array}{cc}\n\\cos B^+ & -\\sin B^+ \\\\[3mm]\n\\sin B^+ & \\cos B^+\n\\end{array}\n\\right], \\quad \\beta \\in \\ensuremath{\\mathbb{C}}^+, \\quad\n\\mbox{\\boldmath $X$}^-(\\beta) = d_0^{-1}\n\\left[\n\\begin{array}{cc}\n\\cos B^- & -\\sin B^- \\\\[3mm]\n\\sin B^- & \\cos B^-\n\\end{array}\n\\right], \\quad \\beta \\in \\ensuremath{\\mathbb{C}}^-,\n$$\n$d_0 = (1 - d_*)^{1\/4}$, and the limit values of the functions $\\beta_\\pm^{1\/2}$, as $\\mathop{\\mathrm{Im}}(\\beta) \\to 0^\\pm$, are\n$$\n\\beta_+^{1\/2} =\n\\left\\{ \n\\begin{array}{ll}\n\\beta^{1\/2}, & \\beta>0 \\\\[3mm]\ni(-\\beta)^{1\/2}, & \\beta<0\n\\end{array}\n\\right., \\quad\n\\beta_-^{1\/2} =\n\\left\\{ \n\\begin{array}{ll}\n\\beta^{1\/2}, & \\beta>0 \\\\[3mm]\n-i(-\\beta)^{1\/2}, & \\beta<0\n\\end{array}\n\\right.,\n$$\nand\n$$\nB^\\pm = -\\epsilon \\log(\\mp i\\beta), \\quad \\beta \\in \\ensuremath{\\mathbb{C}}^\\pm.\n$$\nFrom the definition of singular solutions given in the introduction of this section, it follows that the asymptotic behaviour of the weight functions is given by\n$$\n\\begin{array}{ll}\n\\jump{0.15}{\\mbox{\\boldmath $U$}}(x_1) \\sim x_1^{-1\/2} \\mbox{\\boldmath $F$}_1(x_1), & x_1 \\to 0^+, \\\\[3mm]\n\\mbox{\\boldmath $\\Sigma$}(x_1) \\sim (-x_1)^{-3\/2} \\mbox{\\boldmath $F$}_2(x_1), & x_1 \\to 0^-,\n\\end{array}\n$$ \n$$\n\\begin{array}{ll}\n\\jump{0.15}{\\mbox{\\boldmath $U$}}(x_1) \\sim x_1^{-1\/2} \\mbox{\\boldmath $F$}_3(x_1), & x_1 \\to \\infty, \\\\[3mm]\n\\mbox{\\boldmath $\\Sigma$}(x_1) \\sim (-x_1)^{-3\/2} \\mbox{\\boldmath $F$}_4(x_1), & x_1 \\to -\\infty,\n\\end{array}\n$$\nwhere $\\mbox{\\boldmath $F$}_1,\\mbox{\\boldmath $F$}_2,\\mbox{\\boldmath $F$}_3,\\mbox{\\boldmath $F$}_4$ are bounded functions. The application of the Abelian type theorem \\ref{abel} and the Tauberian type theorem \\ref{tau} \n(see Appendix \\ref{app3}) gives\n$$\n\\begin{array}{ll}\n\\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^+(\\beta) \\sim \\beta_+^{-1\/2} \\tilde{\\mbox{\\boldmath $F$}}_1(\\beta), & \n\\beta \\in \\ensuremath{\\mathbb{C}}^+, \\quad \\beta \\to \\infty, \\\\[3mm]\n\\overline{\\mbox{\\boldmath $\\Sigma$}}^-(\\beta) \\sim \\beta_-^{1\/2} \\tilde{\\mbox{\\boldmath $F$}}_2(\\beta), & \n\\beta \\in \\ensuremath{\\mathbb{C}}^-, \\quad \\beta \\to \\infty,\n\\end{array}\n$$ \n$$\n\\begin{array}{ll}\n\\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^+(\\beta) \\sim \\beta_+^{-1\/2} \\tilde{\\mbox{\\boldmath $F$}}_3(\\beta), & \n\\beta \\in \\ensuremath{\\mathbb{C}}^+, \\quad \\beta \\to 0, \\\\[3mm]\n\\overline{\\mbox{\\boldmath $\\Sigma$}}(\\beta) \\sim \\beta_-^{1\/2} \\tilde{\\mbox{\\boldmath $F$}}_4(\\beta), & \n\\beta \\in \\ensuremath{\\mathbb{C}}^-, \\quad \\beta \\to 0,\n\\end{array}\n$$\nwhere $\\tilde{\\mbox{\\boldmath $F$}}_1,\\tilde{\\mbox{\\boldmath $F$}}_2,\\tilde{\\mbox{\\boldmath $F$}}_3,\\tilde{\\mbox{\\boldmath $F$}}_4$ are bounded functions. Substituting the representation \\eq{fact} for the matrix $-b\/|\\beta|\\mbox{\\boldmath $G$}(\\beta)$ \ninto the Wiener-Hopf equation \\eq{wh}, we obtain\n\\begin{equation}\n\\label{wiener}\n\\beta_+^{1\/2} [\\mbox{\\boldmath $X$}^+(\\beta)]^{-1} \\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^+(\\beta) =\n-\\frac{b}{\\beta_-^{1\/2}} [\\mbox{\\boldmath $X$}^-(\\beta)]^{-1} \\overline{\\mbox{\\boldmath $\\Sigma$}}^-(\\beta), \\quad \\beta \\in \\ensuremath{\\mathbb{R}}.\n\\end{equation}\nSince $[\\mbox{\\boldmath $X$}^+(\\beta)]^{-1}$ and $[\\mbox{\\boldmath $X$}^-(\\beta)]^{-1}$ are bounded functions and taking into account the asymptotic behaviour of the weight functions at infinity, it \nfollows that the LHS and RHS of \\eq{wiener} behave as $O(1)$, $\\beta \\to \\infty$, and from the Liouville theorem, they are equal to the same constant \n$\\mbox{\\boldmath $e$} = [e_1,e_2]^\\top$, where $e_1,e_2$ are arbitrary, so that\n\\begin{equation}\n\\label{wf}\n\\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^+(\\beta) = \\frac{1}{\\beta_+^{1\/2}} \\mbox{\\boldmath $X$}^+(\\beta)\n\\left[ \n\\begin{array}{l}\ne_1 \\\\[3mm]\ne_2\n\\end{array}\n\\right], \\quad \\beta \\in \\ensuremath{\\mathbb{C}}^+, \\quad\n\\overline{\\mbox{\\boldmath $\\Sigma$}}^-(\\beta) = -\\frac{\\beta_-^{1\/2}}{b} \\mbox{\\boldmath $X$}^-(\\beta)\n\\left[ \n\\begin{array}{l}\ne_1 \\\\[3mm]\ne_2\n\\end{array}\n\\right], \\quad \\beta \\in \\ensuremath{\\mathbb{C}}^-.\n\\end{equation}\n\n\\subsubsection{The basis in the space of singular solutions}\nFrom now on, we shall use the notion of weight functions, defined as traces of the singular displacement fields obtained in the above section. It is evident from the \nsolution obtained above that the space of singular solutions is a 2-dimensional linear space. Two linearly independent weight functions can be selected as follows. \nSetting $e_1 = 1$, $e_2 = 0$ in \\eq{wf}, we get the first weight function as:\n\\begin{equation}\n\\label{first}\n\\begin{array}{l}\n\\displaystyle \\jump{0.15}{\\overline{U}_1^1}^+ = \\frac{d_0}{\\beta_+^{1\/2}} \\cos B^+ =\n\\frac{d_0}{2} \\beta_+^{-1\/2} \\left(e_0 \\beta_+^{i\\epsilon} + \\frac{1}{e_0} \\beta_+^{-i\\epsilon}\\right), \\quad \n\\beta \\in \\ensuremath{\\mathbb{C}}^+, \\\\[3mm]\n\\displaystyle \\jump{0.15}{\\overline{U}_2^1}^+ = \\frac{d_0}{\\beta_+^{1\/2}} \\sin B^+ =\n\\frac{i d_0}{2} \\beta_+^{-1\/2} \\left(e_0 \\beta_+^{i\\epsilon} - \\frac{1}{e_0} \\beta_+^{-i\\epsilon}\\right), \\quad \n\\beta \\in \\ensuremath{\\mathbb{C}}^+, \\\\[3mm]\n\\displaystyle \\overline{\\mbox{\\boldmath $\\Sigma$}}_{21}^{1-} = -\\frac{\\beta_-^{1\/2}}{bd_0} \\cos B^- =\n- \\frac{1}{2bd_0} \\beta_-^{1\/2} \\left(\\frac{1}{e_0} \\beta_-^{i\\epsilon} + e_0 \\beta_-^{-i\\epsilon}\\right), \\quad \n\\beta \\in \\ensuremath{\\mathbb{C}}^-, \\\\[3mm]\n\\displaystyle \\overline{\\mbox{\\boldmath $\\Sigma$}}_{22}^{1-} = -\\frac{\\beta_-^{1\/2}}{bd_0} \\sin B^- =\n- \\frac{i}{2bd_0} \\beta_-^{1\/2} \\left(\\frac{1}{e_0} \\beta_-^{i\\epsilon} - e_0 \\beta_-^{-i\\epsilon}\\right), \\quad \n\\beta \\in \\ensuremath{\\mathbb{C}}^-,\n\\end{array}\n\\end{equation}\nwhere $e_0 = e^{\\epsilon\\pi\/2}$.\n\nSetting $e_1 = 0$, $e_2 = 1$ in \\eq{wf}, we get the second weight function as:\n\\begin{equation}\n\\label{second}\n\\begin{array}{l}\n\\displaystyle \\jump{0.15}{\\overline{U}_1^2}^+ = -\\frac{d_0}{\\beta_+^{1\/2}} \\sin B^+ =\n-\\frac{i d_0}{2} \\beta_+^{-1\/2} \\left(e_0 \\beta_+^{i\\epsilon} - \\frac{1}{e_0} \\beta_+^{-i\\epsilon}\\right), \\quad \n\\beta \\in \\ensuremath{\\mathbb{C}}^+, \\\\[3mm]\n\\displaystyle \\jump{0.15}{\\overline{U}_2^2}^+ = \\frac{d_0}{\\beta_+^{1\/2}} \\cos B^+ =\n\\frac{d_0}{2} \\beta_+^{-1\/2} \\left(e_0 \\beta_+^{i\\epsilon} + \\frac{1}{e_0} \\beta_+^{-i\\epsilon}\\right), \\quad \n\\beta \\in \\ensuremath{\\mathbb{C}}^+, \\\\[3mm]\n\\displaystyle \\overline{\\mbox{\\boldmath $\\Sigma$}}_{21}^{2-} = \\frac{\\beta_-^{1\/2}}{bd_0} \\sin B^- =\n\\frac{i}{2bd_0} \\beta_-^{1\/2} \\left(\\frac{1}{e_0} \\beta_-^{i\\epsilon} - e_0 \\beta_-^{-i\\epsilon}\\right), \\quad \n\\beta \\in \\ensuremath{\\mathbb{C}}^-, \\\\[3mm]\n\\displaystyle \\overline{\\mbox{\\boldmath $\\Sigma$}}_{22}^{2-} = -\\frac{\\beta_-^{1\/2}}{bd_0} \\cos B^- =\n- \\frac{1}{2bd_0} \\beta_-^{1\/2} \\left(\\frac{1}{e_0} \\beta_-^{i\\epsilon} + e_0 \\beta_-^{-i\\epsilon}\\right), \\quad \n\\beta \\in \\ensuremath{\\mathbb{C}}^-.\n\\end{array}\n\\end{equation}\nIn order to use a compact notation in the sequel of the paper, we collect the weight function components together with components of the corresponding tractions in \nmatrices as follows\n$$\n\\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^+ = \n\\left[\n\\begin{array}{cc}\n\\jump{0.15}{\\overline{U}_1^1}^+ & \\jump{0.15}{\\overline{U}_1^2}^+ \\\\[3mm]\n\\jump{0.15}{\\overline{U}_2^1}^+ & \\jump{0.15}{\\overline{U}_2^2}^+ \n\\end{array}\n\\right], \\quad\n\\overline{\\mbox{\\boldmath $\\Sigma$}}^- = \n\\left[\n\\begin{array}{cc}\n\\overline{\\Sigma}_{21}^{1-} & \\overline{\\Sigma}_{21}^{2-} \\\\[3mm]\n\\overline{\\Sigma}_{22}^{1-} & \\overline{\\Sigma}_{22}^{2-} \n\\end{array}\n\\right].\n$$\n\nNote that the limit values of the functions $\\beta_+^{\\pm i\\epsilon}$, as $\\mathop{\\mathrm{Im}}(\\beta) \\to 0^+$, are\n$$\n\\beta_+^{i\\epsilon} = \n\\left\\{ \n\\begin{array}{ll} \n\\displaystyle \\beta^{i\\epsilon}, & \\beta>0, \\\\[3mm]\n\\displaystyle \\frac{(-\\beta)^{i\\epsilon}}{e_0^2}, & \\beta<0,\n\\end{array} \n\\right. \\qquad\n\\beta_+^{-i\\epsilon} = \n\\left\\{ \n\\begin{array}{ll} \n\\displaystyle \\beta^{-i\\epsilon}, & \\beta>0, \\\\[3mm]\n\\displaystyle (-\\beta)^{-i\\epsilon} e_0^2, & \\beta<0,\n\\end{array} \n\\right.\n$$\nwhereas the limit values of the functions $\\beta_-^{\\pm i\\epsilon}$, as $\\mathop{\\mathrm{Im}}(\\beta) \\to 0^-$, are\n$$\n\\beta_-^{i\\epsilon} = \n\\left\\{ \n\\begin{array}{ll} \n\\displaystyle \\beta^{i\\epsilon}, & \\beta>0, \\\\[3mm]\n\\displaystyle (-\\beta)^{i\\epsilon} e_0^2, & \\beta<0,\n\\end{array} \n\\right. \\qquad\n\\beta_-^{-i\\epsilon} = \n\\left\\{ \n\\begin{array}{ll} \n\\displaystyle \\beta^{-i\\epsilon}, & \\beta>0, \\\\[3mm]\n\\displaystyle \\frac{(-\\beta)^{-i\\epsilon}}{e_0^2}, & \\beta<0.\n\\end{array} \n\\right.\n$$\n\nIt is clear that the choice of a basis of two linearly independent singular solutions is not unique. Bercial-Velez {\\em et al.}\\ (2005) provided another set \nof linearly independent weight functions, namely\n$$\n\\begin{array}{l}\n\\displaystyle \\jump{0.15}{\\overline{\\ensuremath{\\mathcal{U}}}_1^1}^+ =  \n\\frac{1}{2d_0^2} \\beta_+^{-1\/2} \n\\left(-\\frac{e_0 \\beta_+^{i\\epsilon}}{c_1^-} + \\frac{\\beta_+^{-i\\epsilon}}{e_0 c_1^+}\\right), \\quad \n\\beta \\in \\ensuremath{\\mathbb{C}}^+, \\\\[3mm]\n\\displaystyle \\jump{0.15}{\\overline{\\ensuremath{\\mathcal{U}}}_2^1}^+ =  \n-\\frac{i}{2d_0^2} \\beta_+^{-1\/2} \n\\left(\\frac{e_0 \\beta_+^{i\\epsilon}}{c_1^-} + \\frac{\\beta_+^{-i\\epsilon}}{e_0 c_1^+}\\right), \\quad \n\\beta \\in \\ensuremath{\\mathbb{C}}^+, \\\\[3mm]\n\\displaystyle \\jump{0.15}{\\overline{\\ensuremath{\\mathcal{U}}}_1^2}^+ = -\\jump{0.15}{\\overline{\\ensuremath{\\mathcal{U}}}_2^1}^+, \\quad \\beta \\in \\ensuremath{\\mathbb{C}}^+, \\\\[3mm]\n\\displaystyle \\jump{0.15}{\\overline{\\ensuremath{\\mathcal{U}}}_2^2}^+ = \\jump{0.15}{\\overline{\\ensuremath{\\mathcal{U}}}_1^1}^+, \\quad \\beta \\in \\ensuremath{\\mathbb{C}}^+.\n\\end{array}\n$$\nIt can be shown that these weight functions are linear combinations of (\\ref{first}) and (\\ref{second}),\n$$\n\\left[\n\\begin{array}{cc}\n\\jump{0.15}{\\overline{\\ensuremath{\\mathcal{U}}}_1^1}^+ & \\jump{0.15}{\\overline{\\ensuremath{\\mathcal{U}}}_1^2}^+ \\\\[3mm]\n\\jump{0.15}{\\overline{\\ensuremath{\\mathcal{U}}}_2^1}^+ & \\jump{0.15}{\\overline{\\ensuremath{\\mathcal{U}}}_2^2}^+ \n\\end{array}\n\\right] = \\frac{1}{2 c_1^+ c_1^- d_0^3}\n\\left[\n\\begin{array}{cc}\n\\jump{0.15}{\\overline{U}_1^1}^+ & \\jump{0.15}{\\overline{U}_1^2}^+ \\\\[3mm]\n\\jump{0.15}{\\overline{U}_2^1}^+ & \\jump{0.15}{\\overline{U}_2^2}^+ \n\\end{array}\n\\right] \n\\left[\n\\begin{array}{cc}\n\\displaystyle -c_1^+ + c_1^- & \\displaystyle i (c_1^+ + c_1^-) \\\\[3mm]\n\\displaystyle -i (c_1^+ + c_1^-) & \\displaystyle -c_1^+ + c_1^- \n\\end{array}\n\\right].\n$$\n\n\\subsection{The half-plane problem and full representation of weight functions}\nIn this section, we will construct the full representation of singular solutions for the whole plane. This is needed in order to compute the skew-symmetric weight \nfunction matrix $\\langle \\mbox{\\boldmath $U$} \\rangle$. The complete singular solutions can be constructed by solving a boundary value problem for a semi-infinite half-plane subjected \nto traction boundary conditions at its boundary. \n\nLet us consider first the lower half-plane. A plane strain elasticity problem is usually solved by means of the Airy function. Introducing the Fourier transform with \nrespect to the $x_1$ variable, the problem is most easily solved considering the stress component $\\overline{\\sigma}_{22}$ as the primary unknown function, so that the \nproblem reduces to the following ordinary differential equation\n$$\n\\overline{\\sigma}_{22}'''' - 2\\beta^2 \\overline{\\sigma}_{22}'' + \\beta^4 \\overline{\\sigma}_{22} = 0,\n$$\nwhere a prime denotes derivative with respect to $x_2$. The general solution is then \n$$\n\\overline{\\sigma}_{22}(\\beta,x_2) = (A_2 + x_2 B_2) e^{|\\beta| x_2}, \\quad\n\\overline{\\sigma}_{11}(\\beta,x_2) = -\\frac{1}{\\beta^2} \\overline{\\sigma}_{22}'', \\quad\n\\overline{\\sigma}_{21}(\\beta,x_2) = -\\frac{i}{\\beta} \\overline{\\sigma}_{22}', \\quad \\beta \\in \\ensuremath{\\mathbb{R}}.\n$$\nCorrespondingly, the Fourier transforms of the displacement components are \n$$\n\\overline{u}_{1}(\\beta,x_2) = -\\frac{i}{2\\mu_- \\beta} \n\\{ \\overline{\\sigma}_{22} - (1-\\nu_-)\\overline{\\sigma}_0 \\}, \\quad\n\\overline{u}_{2}(\\beta,x_2) = \\frac{1}{2\\mu_- \\beta^2} \n\\{ \\overline{\\sigma}_{22}' + (1-\\nu_-)\\overline{\\sigma}_0' \\}, \\quad \\beta \\in \\ensuremath{\\mathbb{R}},\n$$\nwhere $\\overline{\\sigma}_0 = \\overline{\\sigma}_{11} + \\overline{\\sigma}_{22}$. The boundary conditions along the boundary $x_2 = 0^-$ are defined by\n$$\n\\overline{\\sigma}_{22}(\\beta,x_2 = 0^-) = \\overline{\\Sigma}_{22}^-(\\beta), \\quad\n\\overline{\\sigma}_{21}(\\beta,x_2 = 0^-) = \\overline{\\Sigma}_{21}^-(\\beta), \\quad \\beta \\in \\ensuremath{\\mathbb{R}},\n$$\nwhere $\\Sigma_{22}^-,\\Sigma_{21}^-$ are the tractions along the interface derived in the previous section. It follows that\n$$\n\\overline{\\sigma}_{22}(\\beta,x_2 = 0^-) = A_2 = \\overline{\\Sigma}_{22}^-(\\beta), \\quad\n\\overline{\\sigma}_{21}(\\beta,x_2 = 0^-) = -\\frac{i}{\\beta} (A_2|\\beta| + B_2) = \n\\overline{\\Sigma}_{21}^-(\\beta), \\quad \\beta \\in \\ensuremath{\\mathbb{R}},\n$$\nand thus\n$$\nA_2 = \\overline{\\Sigma}_{22}^-, \\quad\nB_2 = i\\beta \\overline{\\Sigma}_{21}^- - |\\beta| \\overline{\\Sigma}_{22}^-, \\quad \\beta \\in \\ensuremath{\\mathbb{R}}.\n$$\nThe full representations of Fourier transforms (with respect to $x_1$)  of the required singular displacements and the corresponding components of stress are\n$$\n\\begin{array}{l}\n\\overline{\\sigma}_{22}(\\beta,x_2) = \\{ i\\beta x_2 \\overline{\\Sigma}_{21}^- \n+ (1 - |\\beta|x_2) \\overline{\\Sigma}_{22}^- \\} e^{|\\beta|x_2}, \\\\[3mm]\n\\overline{\\sigma}_{11}(\\beta,x_2) = \\{ -i(2\\mathop{\\mathrm{sign}}(\\beta) + \\beta x_2) \\overline{\\Sigma}_{21}^- \n+ (1 + |\\beta|x_2) \\overline{\\Sigma}_{22}^- \\} e^{|\\beta|x_2}, \\\\[3mm]\n\\overline{\\sigma}_{21}(\\beta,x_2) = \\{ (1 + |\\beta|x_2) \\overline{\\Sigma}_{21}^- \n+ i\\beta x_2 \\overline{\\Sigma}_{22}^- \\} e^{|\\beta|x_2}.\n\\end{array}\n$$\n$$\n\\begin{array}{l}\n\\displaystyle\n\\overline{u}_{1}(\\beta,x_2) = \\frac{1}{2\\mu_-} \\left\\{ \\left[x_2 + \\frac{2(1-\\nu_-)}{|\\beta|}\\right] \n\\overline{\\Sigma}_{21}^- \n+ i\\left[\\mathop{\\mathrm{sign}}(\\beta)x_2 + \\frac{1 - 2\\nu_-}{\\beta}\\right] \n\\overline{\\Sigma}_{22}^- \\right\\} e^{|\\beta|x_2}, \\\\[3mm]\n\\displaystyle\n\\overline{u}_{2}(\\beta,x_2) = \\frac{1}{2\\mu_-} \n\\left\\{ i\\left[\\mathop{\\mathrm{sign}}(\\beta) x_2 - \\frac{1-2\\nu_-}{\\beta} \\right] \n\\overline{\\Sigma}_{21}^- \n+ \\left[\\frac{2(1-\\nu_-)}{|\\beta|} - x_2\\right] \n\\overline{\\Sigma}_{22}^- \\right\\} e^{|\\beta|x_2}.\n\\end{array}\n$$\n\nFor the upper half-plane, we find the same equations, subject to replacing $|\\beta|$ with $-|\\beta|$, $\\mu_-$ with $\\mu_+$ and $\\nu_-$ with $\\nu_+$.\n\n\\subsection{New results for skew-symmetric weight functions}\n\\subsubsection{The Fourier transform representations}\nIt is possible now to derive the jump and average of Fourier transforms of the singular displacement functions across the plane containing the crack. The traces on the \nplane containing the crack are given by\n$$\n\\begin{array}{l}\n\\overline{U}_{1}(\\beta) = \\overline{u}_{1}(\\beta,x_2 = 0^+) = \n\\left[\n\\begin{array}{ll}\n\\displaystyle -\\frac{1-\\nu_+}{\\mu_+ |\\beta|}, & \\displaystyle \\frac{i(1-2\\nu_+)}{2\\mu_+ \\beta}\n\\end{array}\n\\right]\n\\left[\n\\begin{array}{l}\n\\overline{\\Sigma}_{21}^- \\\\[3mm]\n\\overline{\\Sigma}_{22}^-\n\\end{array}\n\\right], \\\\[3mm]\n\\overline{U}_{2}(\\beta) = \\overline{u}_{2}(\\beta,x_2 = 0^+) = \n\\left[\n\\begin{array}{ll}\n\\displaystyle -\\frac{i(1-2\\nu_+)}{2\\mu_+ \\beta}, & \\displaystyle -\\frac{1-\\nu_+}{\\mu_+ |\\beta|}\n\\end{array}\n\\right]\n\\left[\n\\begin{array}{l}\n\\overline{\\Sigma}_{21}^- \\\\[3mm]\n\\overline{\\Sigma}_{22}^-\n\\end{array}\n\\right], \\\\[3mm]\n\\overline{U}_{1}(\\beta) = \\overline{u}_{1}(\\beta,x_2 = 0^-) = \n\\left[\n\\begin{array}{ll}\n\\displaystyle \\frac{1-\\nu_-}{\\mu_- |\\beta|}, & \\displaystyle \\frac{i(1-2\\nu_-)}{2\\mu_- \\beta}\n\\end{array}\n\\right]\n\\left[\n\\begin{array}{l}\n\\overline{\\Sigma}_{21}^- \\\\[3mm]\n\\overline{\\Sigma}_{22}^-\n\\end{array}\n\\right], \\\\[3mm]\n\\overline{U}_{2}(\\beta) = \\overline{u}_{2}(\\beta,x_2 = 0^-) = \n\\left[\n\\begin{array}{ll}\n\\displaystyle -\\frac{i(1-2\\nu_-)}{2\\mu_- \\beta}, & \\displaystyle \\frac{1-\\nu_-}{\\mu_- |\\beta|}\n\\end{array}\n\\right]\n\\left[\n\\begin{array}{l}\n\\overline{\\Sigma}_{21}^- \\\\[3mm]\n\\overline{\\Sigma}_{22}^-\n\\end{array}\n\\right],\n\\end{array}\n$$\nso that, we obtain in matrix form\n$$\n\\left[\n\\begin{array}{cc}\n\\jump{0.15}{\\overline{U}_1^1}^+ & \\jump{0.15}{\\overline{U}_1^2}^+ \\\\[3mm]\n\\jump{0.15}{\\overline{U}_2^1}^+ & \\jump{0.15}{\\overline{U}_2^2}^+\n\\end{array}\n\\right] = -\\frac{b}{|\\beta|} \n\\left[\n\\begin{array}{cc}\n1 & -i\\mathop{\\mathrm{sign}}(\\beta) d_* \\\\[3mm]\ni\\mathop{\\mathrm{sign}}(\\beta) d_* & 1\n\\end{array}\n\\right]\n\\left[\n\\begin{array}{cc}\n\\overline{\\Sigma}_{21}^{1-} & \\overline{\\Sigma}_{21}^{2-} \\\\[3mm]\n\\overline{\\Sigma}_{22}^{1-} & \\overline{\\Sigma}_{22}^{2-}\n\\end{array}\n\\right], \\quad \\beta \\in \\ensuremath{\\mathbb{R}},\n$$\nand\n$$\n\\left[\n\\begin{array}{cc}\n\\langle\\overline{U}_1^1\\rangle & \\langle\\overline{U}_1^2\\rangle \\\\[3mm]\n\\langle\\overline{U}_2^1\\rangle & \\langle\\overline{U}_2^2\\rangle\n\\end{array}\n\\right] = -\\frac{b\\alpha}{2|\\beta|} \n\\left[\n\\begin{array}{cc}\n1 & -i\\mathop{\\mathrm{sign}}(\\beta) \\gamma_* \\\\[3mm]\ni\\mathop{\\mathrm{sign}}(\\beta) \\gamma_* & 1\n\\end{array}\n\\right]\n\\left[\n\\begin{array}{cc}\n\\overline{\\Sigma}_{21}^{1-} & \\overline{\\Sigma}_{21}^{2-} \\\\[3mm]\n\\overline{\\Sigma}_{22}^{1-} & \\overline{\\Sigma}_{22}^{2-}\n\\end{array}\n\\right], \\quad \\beta \\in \\ensuremath{\\mathbb{R}},\n$$\nwhere $\\gamma_*$ is a material parameter,\n$$\n\\gamma_* = \\frac{\\mu_-(1 - 2\\nu_+) + \\mu_+(1 - 2\\nu_-)}{2\\mu_-(1 - \\nu_+) - 2\\mu_+(1 - \\nu_-)}.\n$$\n\nNote that the above equations can be rewritten as\n$$\n\\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^+ = \n-\\frac{b}{|\\beta|} \\left[\\begin{array}{cc} 1 & 0 \\\\[3mm] 0 & 1 \\end{array}\\right] \\overline{\\mbox{\\boldmath $\\Sigma$}}^- - \n\\frac{i b d_*}{\\beta} \\left[\\begin{array}{cc} 0 & -1 \\\\[3mm] 1 & 0 \\end{array}\\right] \\overline{\\mbox{\\boldmath $\\Sigma$}}^-, \\quad\n\\beta \\in \\ensuremath{\\mathbb{R}},\n$$\n$$\n\\langle\\overline{\\mbox{\\boldmath $U$}}\\rangle = \n-\\frac{\\alpha b}{2|\\beta|} \\left[\\begin{array}{cc} 1 & 0 \\\\[3mm] 0 & 1 \\end{array}\\right] \\overline{\\mbox{\\boldmath $\\Sigma$}}^- - \n\\frac{i \\alpha b \\gamma_*}{2\\beta} \\left[\\begin{array}{cc} 0 & -1 \\\\[3mm] 1 & 0 \\end{array}\\right]\n\\overline{\\mbox{\\boldmath $\\Sigma$}}^-, \\quad \\beta \\in \\ensuremath{\\mathbb{R}},\n$$\nso that we can derive the decomposition of $\\langle\\overline{\\mbox{\\boldmath $U$}}\\rangle$ in the sum of \"$+$\" and \"$-$\" functions as follows\n\\begin{equation}\n\\label{average}\n\\langle\\overline{\\mbox{\\boldmath $U$}}\\rangle = \n\\frac{\\alpha}{2} \\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^+ + \\alpha(d_* - \\gamma_*) \n\\frac{i b}{2\\beta} \\left[\\begin{array}{cc} 0 & -1 \\\\[3mm] 1 & 0 \\end{array}\\right] \n\\overline{\\mbox{\\boldmath $\\Sigma$}}^-, \\quad \\beta \\in \\ensuremath{\\mathbb{R}},\n\\end{equation}\nwhere the last term on the right-hand side is a \"$-$\" function.\n\n\\subsubsection{The weight functions - Fourier inversion}\n\\label{weight_f}\nAfter the inversion of the corresponding Fourier transforms, the weight functions, which will be needed for the computation of stress intensity factors, are as follows. \nThe symmetric weight function matrix $\\jump{0.15}{\\mbox{\\boldmath $U$}}(x_1)$ is equal to 0 for $x_1 < 0$, whereas for $x_1 > 0$ it is given by \n\\begin{equation}\n\\label{jumpu}\n\\begin{array}{l}\n\\displaystyle\n\\jump{0.15}{U_1^1}(x_1) = \\frac{x_1^{-1\/2}}{2d_0 \\sqrt{2\\pi}} \n\\left( \\frac{x_1^{-i\\epsilon}}{c_1^+} + \\frac{x_1^{i\\epsilon}}{c_1^-} \\right), \\\\[5mm]\n\\displaystyle\n\\jump{0.15}{U_2^1}(x_1) = \\frac{i x_1^{-1\/2}}{2d_0 \\sqrt{2\\pi}} \n\\left( \\frac{x_1^{-i\\epsilon}}{c_1^+} - \\frac{x_1^{i\\epsilon}}{c_1^-} \\right), \\\\[5mm]\n\\displaystyle \n\\jump{0.15}{U_1^2}(x_1) = -\\jump{0.15}{U_2^1}(x_1), \\\\[5mm]\n\\jump{0.15}{U_2^2}(x_1) = \\jump{0.15}{U_1^1}(x_1).\n\\end{array}\n\\end{equation}\nThe skew-symmetric weight function matrix $\\langle\\mbox{\\boldmath $U$}\\rangle(x_1)$ is equal to $\\displaystyle \\frac{\\alpha}{2}\\jump{0.15}{\\mbox{\\boldmath $U$}}(x_1)$ for $x_1 > 0$, whereas for $x_1 < 0$ it is \ngiven by \n\\begin{equation}\n\\label{meanu}\n\\begin{array}{l}\n\\displaystyle\n\\langle U_1^1 \\rangle(x_1) = -\\frac{i\\alpha(d_*-\\gamma_*)(-x_1)^{-1\/2}}{4d_0^3 \\sqrt{2\\pi}}\n\\left[ \\frac{(-x_1)^{-i\\epsilon}}{c_1^+} - \\frac{(-x_1)^{i\\epsilon}}{c_1^-} \\right], \\\\[3mm]\n\\displaystyle\n\\langle U_2^1 \\rangle(x_1) = \\frac{\\alpha(d_*-\\gamma_*)(-x_1)^{-1\/2}}{4d_0^3 \\sqrt{2\\pi}}\n\\left[ \\frac{(-x_1)^{-i\\epsilon}}{c_1^+} + \\frac{(-x_1)^{i\\epsilon}}{c_1^-} \\right], \\\\[3mm]\n\\displaystyle \n\\langle U_1^2 \\rangle(x_1) = -\\langle U_2^1 \\rangle(x_1), \\\\[3mm]\n\\langle U_2^2 \\rangle(x_1) = \\langle U_1^1 \\rangle(x_1).\n\\end{array}\n\\end{equation}\nThe function $\\mbox{\\boldmath $\\Sigma$}(x_1)$ is equal to 0 for $x_1 > 0$, whereas for $x_1 < 0$ it is given by \n$$\n\\begin{array}{l}\n\\displaystyle\n\\Sigma_{21}^1(x_1) = \\frac{(-x_1)^{-3\/2}}{2bd_0^3 \\sqrt{2\\pi}} \n\\left[ \\frac{1\/2 + i\\epsilon}{c_1^+}(-x_1)^{-i\\epsilon} + \\frac{1\/2 - i\\epsilon}{c_1^-}(-x_1)^{i\\epsilon} \\right], \\\\[3mm]\n\\displaystyle\n\\Sigma_{22}^1(x_1) = \\frac{i(-x_1)^{-3\/2}}{2bd_0^3 \\sqrt{2\\pi}} \n\\left[ \\frac{1\/2 + i\\epsilon}{c_1^+}(-x_1)^{-i\\epsilon} - \\frac{1\/2 - i\\epsilon}{c_1^-}(-x_1)^{i\\epsilon} \\right], \\\\[3mm]\n\\displaystyle \n\\Sigma_{21}^2(x_1) = -\\Sigma_{22}^1(x_1), \\\\[3mm]\n\\Sigma_{22}^2(x_1) = \\Sigma_{21}^1(x_1).\n\\end{array}\n$$\n\nIt will be shown in the sequel of the text that we only need the inverse transform of $\\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^+$  (see (\\ref{average})) in \norder to compute the stress intensity factors for both symmetric and skew-symmetric loading.\n\n\\section{The Betti identity and evaluation of the stress intensity factors}\n\\label{identity}\nHere we develop a general procedure for the evaluation of coefficients in the asymptotics of the stress components near the crack tip. This includes the stress intensity \nfactors as well as high-order asymptotics. In particular, the coefficients near the higher order terms require appropriate weight functions which are shown to be derived \nvia differentiation of the weight functions $\\jump{0.15}{\\mbox{\\boldmath $U$}}$ and $\\langle \\mbox{\\boldmath $U$} \\rangle$ along the crack.\n\n\\subsection{The fundamental Betti identity and its equivalent representation in terms of Fourier transforms}\nIn this section, the notations $\\mbox{\\boldmath $u$} = [u_1, u_2]^\\top$ and $\\mbox{\\boldmath $\\sigma$} = [\\sigma_{21},\\sigma_{22}]^\\top$ will be used for the physical displacement and traction fields discussed \nin Section \\ref{prel_analysis}. The notations $\\mbox{\\boldmath $U$} = [U_1,U_2]^\\top$ and $\\mbox{\\boldmath $\\Sigma$} = [\\Sigma_{21},\\Sigma_{22}]^\\top$ will apply to the auxiliary singular displacements and the \ncorresponding tractions obtained in Section \\ref{weight_f}. We note that $\\mbox{\\boldmath $U$}$ is discontinuous along the positive semi-axis $x_1 > 0$, whereas $\\mbox{\\boldmath $u$}$ is discontinuous \nalong the negative semi-axis $x_1 < 0$.\n\nSimilar to Willis and Movchan (1995) and Piccolroaz {\\em et al.}\\ (2007), we can apply the Betti formula to the physical fields and to the weight functions in order to \nevaluate the coefficients in the asymptotics near the crack tip. In particular, applying the Betti identity to the upper half-plane and lower half-plane, we obtain\n$$\n\\int_{(x_2 = 0^+)} \\left\\{ \\mbox{\\boldmath $U$}^\\top(x_1'-x_1,0^+) \\mbox{\\boldmath $R$} \\mbox{\\boldmath $\\sigma$}(x_1,0^+) - \\mbox{\\boldmath $\\Sigma$}^\\top(x_1'-x_1,0^+) \\mbox{\\boldmath $R$} \\mbox{\\boldmath $u$}(x_1,0^+) \\right\\} dx_1 = 0,\n$$\nand\n$$\n\\int_{(x_2 = 0^-)} \\left\\{ \\mbox{\\boldmath $U$}^\\top(x_1'-x_1,0^-) \\mbox{\\boldmath $R$} \\mbox{\\boldmath $\\sigma$}(x_1,0^-) - \\mbox{\\boldmath $\\Sigma$}^\\top(x_1'-x_1,0^-) \\mbox{\\boldmath $R$} \\mbox{\\boldmath $u$}(x_1,0^-) \\right\\} dx_1 = 0,\n$$\nrespectively, where \n\\begin{equation}\n\\label{erre}\n\\mbox{\\boldmath $R$} = \\begin{pmatrix} -1 & 0 \\\\ 0 & 1 \\end{pmatrix}\n\\end{equation}\nis a rotation matrix. Then, subtracting one from the other, we arrive at\n$$\n\\begin{array}{l}\n\\displaystyle\n\\int_{(x_2 = 0)} \\left\\{ \\mbox{\\boldmath $U$}^\\top(x_1'-x_1,0^+) \\mbox{\\boldmath $R$} \\mbox{\\boldmath $\\sigma$}(x_1,0^+) - \\mbox{\\boldmath $U$}^\\top(x_1'-x_1,0^-) \\mbox{\\boldmath $R$} \\mbox{\\boldmath $\\sigma$}(x_1,0^-) \\right. \\\\[3mm]\n\\displaystyle \\hspace{30mm}\n\\left. - \\left[\\mbox{\\boldmath $\\Sigma$}^\\top(x_1'-x_1,0^+) \\mbox{\\boldmath $R$} \\mbox{\\boldmath $u$}(x_1,0^+) - \\mbox{\\boldmath $\\Sigma$}^\\top(x_1'-x_1,0^-) \\mbox{\\boldmath $R$} \\mbox{\\boldmath $u$}(x_1,0^-)\\right] \\right\\} dx_1 = 0.\n\\end{array}\n$$\nThe traction components acting on the $x_1$ axis can be written as\n$$\n\\mbox{\\boldmath $\\sigma$}(x_1,0^+) = \\mbox{\\boldmath $p$}^+(x_1) + \\mbox{\\boldmath $\\sigma$}^{(+)}(x_1,0), \\quad \\mbox{\\boldmath $\\sigma$}(x_1,0^-) = \\mbox{\\boldmath $p$}^-(x_1) + \\mbox{\\boldmath $\\sigma$}^{(+)}(x_1,0), \n$$\nwhere $\\mbox{\\boldmath $p$}^+(x_1) = \\mbox{\\boldmath $\\sigma$}(x_1,0^+) H(-x_1)$, $\\mbox{\\boldmath $p$}^-(x_1) = \\mbox{\\boldmath $\\sigma$}(x_1,0^-) H(-x_1)$ is the loading acting on the upper and lower crack faces, respectively, and \n$\\mbox{\\boldmath $\\sigma$}^{(+)}(x_1,0)$ is the traction field ahead of the crack tip, with $H(x_1)$ being the unit-step Heaviside function. Consequently, the integral identity can be \nwritten as\n$$\n\\begin{array}{l}\n\\displaystyle\n\\int_{(x_2 = 0)} \\Big\\{ \\mbox{\\boldmath $U$}^\\top(x_1'-x_1,0^+) \\mbox{\\boldmath $R$} \\mbox{\\boldmath $p$}^+(x_1) + \\mbox{\\boldmath $U$}^\\top(x_1'-x_1,0^+) \\mbox{\\boldmath $R$} \\mbox{\\boldmath $\\sigma$}^{(+)}(x_1,0) \\\\[3mm]\n\\displaystyle \\hspace{15mm}\n- \\mbox{\\boldmath $U$}^\\top(x_1'-x_1,0^-) \\mbox{\\boldmath $R$} \\mbox{\\boldmath $p$}^-(x_1) - \\mbox{\\boldmath $U$}^\\top(x_1'-x_1,0^-) \\mbox{\\boldmath $R$} \\mbox{\\boldmath $\\sigma$}^{(+)}(x_1,0) \\\\[3mm]\n\\displaystyle \\hspace{30mm}\n- \\left[\\mbox{\\boldmath $\\Sigma$}^\\top(x_1'-x_1,0^+) \\mbox{\\boldmath $R$} \\mbox{\\boldmath $u$}(x_1,0^+) - \\mbox{\\boldmath $\\Sigma$}^\\top(x_1'-x_1,0^-) \\mbox{\\boldmath $R$} \\mbox{\\boldmath $u$}(x_1,0^-)\\right] \\Big\\} dx_1 = 0.\n\\end{array}\n$$\nFrom the continuity of $\\mbox{\\boldmath $\\sigma$}^{(+)}$ and $\\mbox{\\boldmath $\\Sigma$}$ we get\n$$\n\\begin{array}{l}\n\\displaystyle\n\\int_{(x_2 = 0)} \\Big\\{ \\jump{0.15}{\\mbox{\\boldmath $U$}}^\\top(x_1'-x_1) \\mbox{\\boldmath $R$} \\mbox{\\boldmath $\\sigma$}^{(+)}(x_1,0) - \\mbox{\\boldmath $\\Sigma$}^\\top(x_1'-x_1,0) \\mbox{\\boldmath $R$} \\jump{0.15}{\\mbox{\\boldmath $u$}}(x_1) \\Big\\} dx_1 = \\\\[3mm]\n\\displaystyle \\hspace{30mm}\n- \\int_{(x_2 = 0)} \\Big\\{ \\mbox{\\boldmath $U$}^\\top(x_1'-x_1,0^+) \\mbox{\\boldmath $R$} \\mbox{\\boldmath $p$}^+(x_1) - \\mbox{\\boldmath $U$}^\\top(x_1'-x_1,0^-) \\mbox{\\boldmath $R$} \\mbox{\\boldmath $p$}^-(x_1) \\Big\\} dx_1.\n\\end{array}\n$$\nIntroducing the symmetric and skew-symmetric parts of the loading (\\ref{parts}), we finally obtain\n\\begin{equation}\n\\label{betti}\n\\begin{array}{l}\n\\displaystyle\n\\int_{(x_2 = 0)} \\Big\\{ \\jump{0.15}{\\mbox{\\boldmath $U$}}^\\top(x_1'-x_1) \\mbox{\\boldmath $R$} \\mbox{\\boldmath $\\sigma$}^{(+)}(x_1,0) - \\mbox{\\boldmath $\\Sigma$}^\\top(x_1'-x_1,0) \\mbox{\\boldmath $R$} \\jump{0.15}{\\mbox{\\boldmath $u$}}(x_1) \\Big\\} dx_1 = \\\\[3mm]\n\\displaystyle \\hspace{30mm}\n- \\int_{(x_2 = 0)} \\Big\\{ \\jump{0.15}{\\mbox{\\boldmath $U$}}^\\top(x_1'-x_1) \\mbox{\\boldmath $R$} \\langle \\mbox{\\boldmath $p$} \\rangle(x_1) + \\langle \\mbox{\\boldmath $U$} \\rangle^\\top(x_1'-x_1) \\mbox{\\boldmath $R$} \\jump{0.15}{\\mbox{\\boldmath $p$}}(x_1) \\Big\\} dx_1.\n\\end{array}\n\\end{equation}\nTaking the Fourier transform of (\\ref{betti}) we derive\n\\begin{equation}\n\\label{betti2}\n\\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^{+\\top} \\mbox{\\boldmath $R$} \\overline{\\mbox{\\boldmath $\\sigma$}}^+ - \\overline{\\mbox{\\boldmath $\\Sigma$}}^{-\\top} \\mbox{\\boldmath $R$} \\jump{0.15}{\\overline{\\mbox{\\boldmath $u$}}}^- = \n- \\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^{+\\top} \\mbox{\\boldmath $R$} \\langle \\overline{\\mbox{\\boldmath $p$}} \\rangle - \\langle \\overline{\\mbox{\\boldmath $U$}} \\rangle^\\top \\mbox{\\boldmath $R$} \\jump{0.15}{\\overline{\\mbox{\\boldmath $p$}}}, \\quad \\beta \\in \\ensuremath{\\mathbb{R}}.\n\\end{equation}\n\n\\subsection{Stress intensity factors and high-order terms coefficients}\nIn this section, we will derive the stress intensity factors as well as high-order terms from the Betti formula \\eq{betti2} obtained in the previous section. The complex \nstress intensity factor will be denoted by $K = K_\\text{I} + i K_\\text{II}$, and analogous notations will be used for high-order terms: $A = A_\\text{I} + i A_\\text{II}$, \n$B = B_\\text{I} + i B_\\text{II}$.\n\nThe asymptotics of the physical traction field as $x_1 \\to 0^+$ are as follows\n$$\n\\mbox{\\boldmath $\\sigma$}^{(+)}(x_1) = \\frac{x_1^{-1\/2}}{2\\sqrt{2\\pi}} \\mbox{\\boldmath $\\mathcal{S}$}(x_1) \\mbox{\\boldmath $K$} + \\frac{x_1^{1\/2}}{2\\sqrt{2\\pi}} \\mbox{\\boldmath $\\mathcal{S}$}(x_1) \\mbox{\\boldmath $A$} + \\frac{x_1^{3\/2}}{2\\sqrt{2\\pi}} \\mbox{\\boldmath $\\mathcal{S}$}(x_1) \\mbox{\\boldmath $B$} + O(x_1^{5\/2}),\n$$\nwhere \n$$\n\\mbox{\\boldmath $\\mathcal{S}$}(x_1) = \n\\left[\\begin{array}{ll}\n\\displaystyle\n-i x_1^{i\\epsilon} & i x_1^{-i\\epsilon} \\\\[3mm]\n\\displaystyle\nx_1^{i\\epsilon} & x_1^{-i\\epsilon}\n\\end{array}\\right],\n$$\nand $\\mbox{\\boldmath $K$} = [K, K^*]^\\top$, $\\mbox{\\boldmath $A$} = [A, A^*]^\\top$, $\\mbox{\\boldmath $B$} = [B, B^*]^\\top$, where the superscript $^*$ denotes conjugation. The corresponding Fourier transforms, as \n$\\beta \\to \\infty$, $\\beta \\in \\ensuremath{\\mathbb{C}}^+$, are (compare with Piccolroaz {\\em et al.}, 2007)\n\\begin{equation}\n\\label{stress1}\n\\overline{\\mbox{\\boldmath $\\sigma$}}^+(\\beta) = \\frac{\\beta_+^{-1\/2}}{4} \\mbox{\\boldmath $\\mathcal{T}$}_1(\\beta) \\mbox{\\boldmath $K$} + \\frac{\\beta_+^{-1\/2}}{4\\beta} \\mbox{\\boldmath $\\mathcal{T}$}_2(\\beta) \\mbox{\\boldmath $A$} + \n\\frac{\\beta_+^{-1\/2}}{4\\beta^2} \\mbox{\\boldmath $\\mathcal{T}$}_3(\\beta) \\mbox{\\boldmath $B$} + O(\\beta^{-7\/2}),\n\\end{equation}\nwhere\n\\begin{equation}\n\\mbox{\\boldmath $\\mathcal{T}$}_j(\\beta) = \n\\left[\\begin{array}{ll}\n\\displaystyle \\frac{\\beta_+^{-i\\epsilon}}{c_j^+ e_0} & \\displaystyle -\\frac{e_0 \\beta_+^{i\\epsilon}}{c_j^-} \\\\[5mm]\n\\displaystyle \\frac{i\\beta_+^{-i\\epsilon}}{c_j^+ e_0}  & \\displaystyle \\frac{ie_0 \\beta_+^{i\\epsilon}}{c_j^-} \n\\end{array}\\right], \\quad j=1,2,3,\n\\end{equation}\n\\begin{equation}\n\\label{stress2}\nc_1^\\pm = \\frac{(1+i)\\sqrt{\\pi}}{2\\Gamma(1\/2 \\pm i\\epsilon)}, \\quad\nc_2^\\pm = \\frac{(1-i)\\sqrt{\\pi}}{2\\Gamma(3\/2 \\pm i\\epsilon)}, \\quad\nc_3^\\pm = -\\frac{(1+i)\\sqrt{\\pi}}{2\\Gamma(5\/2 \\pm i\\epsilon)},\n\\end{equation}\nand $\\Gamma(\\cdot)$ denotes the Gamma function.\n\nThe components of the displacement physical field as $x_1 \\to 0^-$ have the form\n$$\n\\jump{0.15}{\\mbox{\\boldmath $u$}}(x_1) = \\frac{b d_0^2}{\\sqrt{2\\pi}} (-x_1)^{1\/2} \\mbox{\\boldmath $\\mathcal{U}$}_1(x_1) \\mbox{\\boldmath $K$} -\n\\frac{b d_0^2}{\\sqrt{2\\pi}} (-x_1)^{3\/2} \\mbox{\\boldmath $\\mathcal{U}$}_2(x_1) \\mbox{\\boldmath $A$} + \\frac{b d_0^2}{\\sqrt{2\\pi}} (-x_1)^{5\/2} \\mbox{\\boldmath $\\mathcal{U}$}_3(x_1) \\mbox{\\boldmath $B$} + O[(-x_1)^{7\/2}],\n$$\nwhere\n$$\n\\mbox{\\boldmath $\\mathcal{U}$}_j(x_1) = \n\\left[\\begin{array}{ll}\n\\displaystyle -\\frac{i (-x_1)^{i\\epsilon}}{2j-1 + 2i\\epsilon} & \n\\displaystyle \\frac{i (-x_1)^{-i\\epsilon}}{2j-1 - 2i\\epsilon} \\\\[3mm]\n\\displaystyle \\frac{(-x_1)^{i\\epsilon}}{2j-1 + 2i\\epsilon} & \n\\displaystyle \\frac{(-x_1)^{-i\\epsilon}}{2j-1 - 2i\\epsilon}\n\\end{array}\\right], \\quad j=1,2,3.\n$$\n\nThe corresponding Fourier transforms, as $\\beta \\to \\infty$, $\\beta \\in \\ensuremath{\\mathbb{C}}^-$ are (compare with Piccolroaz {\\em et al.}, 2007)\n\\begin{equation}\n\\label{displac}\n\\jump{0.15}{\\overline{\\mbox{\\boldmath $u$}}}^-(\\beta) = -\\frac{bd_0^2}{4\\beta} \\beta_-^{-1\/2} \\mbox{\\boldmath $\\mathcal{V}$}_1(\\beta) \\mbox{\\boldmath $K$} - \\frac{bd_0^2}{4\\beta^2} \\beta_-^{-1\/2} \\mbox{\\boldmath $\\mathcal{V}$}_2(\\beta) \\mbox{\\boldmath $A$} \n- \\frac{bd_0^2}{4\\beta^3} \\beta_-^{-1\/2} \\mbox{\\boldmath $\\mathcal{V}$}_3(\\beta) \\mbox{\\boldmath $B$} + O(\\beta^{-9\/2}),\n\\end{equation}\nwhere\n$$\n\\mbox{\\boldmath $\\mathcal{V}$}_j(\\beta) = \n\\left[\\begin{array}{ll}\n\\displaystyle \\frac{e_0 \\beta_-^{-i\\epsilon}}{c_j^+} & \\displaystyle - \\frac{\\beta_-^{i\\epsilon}}{c_j^-e_0} \\\\[5mm]\n\\displaystyle \\frac{i e_0 \\beta_-^{-i\\epsilon}}{c_j^+} & \\displaystyle \\frac{i \\beta_-^{i\\epsilon}}{c_j^-e_0} \n\\end{array}\\right], \\quad j=1,2,3.\n$$\n\nFrom these asymptotics we can derive estimates of the terms in the LHS of the Betti identity \n\\eq{betti2}, as $\\beta \\to \\infty$,\n\\begin{equation}\n\\label{lhs}\n\\begin{array}{l}\n\\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^{+\\top} \\mbox{\\boldmath $R$} \\overline{\\mbox{\\boldmath $\\sigma$}}^+  = \n\\beta^{-1} \\mbox{\\boldmath $\\mathcal{M}$}_1 \\mbox{\\boldmath $K$} + \\beta^{-2} \\mbox{\\boldmath $\\mathcal{M}$}_2 \\mbox{\\boldmath $A$} + \\beta^{-3} \\mbox{\\boldmath $\\mathcal{M}$}_3 \\mbox{\\boldmath $B$} + O(\\beta^{-4}), \\quad \\beta \\in \\ensuremath{\\mathbb{C}}^+ \\\\[3mm]\n\\overline{\\mbox{\\boldmath $\\Sigma$}}^{-\\top} \\mbox{\\boldmath $R$} \\jump{0.15}{\\overline{\\mbox{\\boldmath $u$}}}^- = \n\\beta^{-1} \\mbox{\\boldmath $\\mathcal{M}$}_1 \\mbox{\\boldmath $K$} + \\beta^{-2} \\mbox{\\boldmath $\\mathcal{M}$}_2 \\mbox{\\boldmath $A$} + \\beta^{-3} \\mbox{\\boldmath $\\mathcal{M}$}_3 \\mbox{\\boldmath $B$} + O(\\beta^{-4}), \\quad \\beta \\in \\ensuremath{\\mathbb{C}}^-\n\\end{array}\n\\end{equation}\nwhere\n\\begin{equation}\n\\label{emme1}\n\\mbox{\\boldmath $\\mathcal{M}$}_j = \\frac{d_0}{4c_j^+c_j^-} \\left[ \\begin{array}{cc} -c_j^- & c_j^+ \\\\[3mm] ic_j^- & ic_j^+ \\end{array} \\right], \\quad j=1,2,3.\n\\end{equation}\n\nLet us introduce the following notation for the RHS of the Betti identity \\eq{betti2}\n$$\n\\mbox{\\boldmath$\\Psi$}(\\beta) = - \\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^{+\\top}(\\beta) \\mbox{\\boldmath $R$} \\langle \\overline{\\mbox{\\boldmath $p$}} \\rangle(\\beta) \n- \\langle\\overline{\\mbox{\\boldmath $U$}}\\rangle^\\top(\\beta) \\mbox{\\boldmath $R$} \\jump{0.15}{\\overline{\\mbox{\\boldmath $p$}}}(\\beta), \\quad \\beta \\in \\ensuremath{\\mathbb{R}}.\n$$\nNote that for any possible loading, $\\mbox{\\boldmath$\\Psi$}(\\beta) \\to 0$, as $\\beta \\to \\pm \\infty$. In fact, if the loading is given by point forces in terms of the Dirac delta function, \nthen $\\mbox{\\boldmath$\\Psi$}(\\beta) = O(|\\beta|^{-1\/2})$. If, for example, $\\mbox{\\boldmath $p$}^\\pm \\in L_1(\\ensuremath{\\mathbb{R}})$ or $(\\mbox{\\boldmath $p$}^\\pm)' \\in L_1(\\ensuremath{\\mathbb{R}})$ then $\\mbox{\\boldmath$\\Psi$}(\\beta) = o(|\\beta|^{-1\/2})$ or \n$\\mbox{\\boldmath$\\Psi$}(\\beta) = o(|\\beta|^{-3\/2})$, respectively.\n\nConsequently, we can split $\\mbox{\\boldmath$\\Psi$}(\\beta)$ in the sum of a plus function and a minus function \n$$\n\\mbox{\\boldmath$\\Psi$}(\\beta) = \\mbox{\\boldmath$\\Psi$}^+(\\beta) - \\mbox{\\boldmath$\\Psi$}^-(\\beta), \\quad \\beta \\in \\ensuremath{\\mathbb{R}},\n$$\nwhere\n$$\n\\mbox{\\boldmath$\\Psi$}^\\pm(\\beta) = \\frac{1}{2\\pi i} \\int_{-\\infty}^{\\infty} \\frac{\\mbox{\\boldmath$\\Psi$}(t)}{t-\\beta} dt, \\quad \n\\beta \\in \\ensuremath{\\mathbb{C}}^\\pm.\n$$\n\nMoreover $\\mbox{\\boldmath$\\Psi$}^\\pm(\\beta) \\sim 1\/\\beta$, $\\beta \\to \\infty$, $\\beta \\in \\ensuremath{\\mathbb{C}}^\\pm$, and, from the estimates (\\ref{lhs}) we can conclude that the inhomogeneous \nWiener-Hopf equation (\\ref{betti2}) has the unique solution \n$$\n\\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^{+\\top} \\mbox{\\boldmath $R$} \\overline{\\mbox{\\boldmath $\\sigma$}}^+ = \\mbox{\\boldmath$\\Psi$}^+, \\quad \\beta \\in \\ensuremath{\\mathbb{C}}^+ \\quad \\text{and} \\quad \n\\overline{\\mbox{\\boldmath $\\Sigma$}}^{-\\top} \\mbox{\\boldmath $R$} \\jump{0.15}{\\overline{\\mbox{\\boldmath $u$}}}^- = \\mbox{\\boldmath$\\Psi$}^-, \\quad \\beta \\in \\ensuremath{\\mathbb{C}}^-.\n$$\n\nFrom these identities we can extract asymptotic estimates. For this reason, let us first assume that $\\mbox{\\boldmath $p$}^\\pm \\in \\ensuremath{\\mathcal{S}}(\\ensuremath{\\mathbb{R}})$, where $\\ensuremath{\\mathcal{S}}(\\ensuremath{\\mathbb{R}})$ is \nthe Schwartz space of rapidly decreasing functions (Zemanian, 1968), so that \n$\\mbox{\\boldmath$\\Psi$}(\\beta)$ decays exponentially as $\\beta \\to \\pm\\infty$. Then we have, for $\\beta \\to \\infty$, $\\beta \\in \\ensuremath{\\mathbb{C}}^\\pm$,\n\\begin{equation}\n\\label{rhs}\n\\mbox{\\boldmath$\\Psi$}^\\pm(\\beta) = \\frac{1}{2\\pi i} \\int_{-\\infty}^{\\infty} \\frac{\\mbox{\\boldmath$\\Psi$}(t)}{t-\\beta} dt = \n-\\beta^{-1} \\frac{1}{2\\pi i} \\int_{-\\infty}^{\\infty} \\mbox{\\boldmath$\\Psi$}(t) dt  \n-\\beta^{-2} \\frac{1}{2\\pi i} \\int_{-\\infty}^{\\infty} t \\mbox{\\boldmath$\\Psi$}(t) dt\n+ O(\\beta^{-3}). \n\\end{equation}\n\nComparing corresponding terms in (\\ref{rhs}) and (\\ref{lhs}), we obtain the following formulae for the complex stress intensity factor and the coefficient in the \nsecond-order term\n\\begin{equation}\n\\label{coeff}\n\\begin{array}{l}\n\\displaystyle\n\\mbox{\\boldmath $K$} = \\frac{1}{2\\pi i} \\mbox{\\boldmath $\\mathcal{M}$}_1^{-1}\n\\int_{-\\infty}^{\\infty} \\left\\{\\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^{+\\top}(t) \\mbox{\\boldmath $R$} \\langle \\overline{\\mbox{\\boldmath $p$}} \\rangle(t)  \n+ \\langle\\overline{\\mbox{\\boldmath $U$}}\\rangle^\\top(t) \\mbox{\\boldmath $R$} \\jump{0.15}{\\overline{\\mbox{\\boldmath $p$}}}(t) \\right\\} dt, \\\\[3mm]\n\\displaystyle\n\\mbox{\\boldmath $A$} = \\frac{1}{2\\pi i} \\mbox{\\boldmath $\\mathcal{M}$}_2^{-1}\n\\int_{-\\infty}^{\\infty} t \\left\\{\\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^{+\\top}(t) \\mbox{\\boldmath $R$} \\langle \\overline{\\mbox{\\boldmath $p$}} \\rangle(t) \n+ \\langle\\overline{\\mbox{\\boldmath $U$}}\\rangle^\\top(t) \\mbox{\\boldmath $R$} \\jump{0.15}{\\overline{\\mbox{\\boldmath $p$}}}(t) \\right\\} dt,\n\\end{array}\n\\end{equation}\nrespectively.\n\nNote that\n$$\n\\begin{array}{l}\n\\displaystyle\n\\frac{1}{2\\pi} \\int_{-\\infty}^{\\infty} \\left\\{\\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^{+\\top}(t) \\mbox{\\boldmath $R$} \\langle \\overline{\\mbox{\\boldmath $p$}} \\rangle(t)  \n+ \\langle\\overline{\\mbox{\\boldmath $U$}}\\rangle^\\top(t) \\mbox{\\boldmath $R$} \\jump{0.15}{\\overline{\\mbox{\\boldmath $p$}}}(t) \\right\\} dt = \\\\[3mm]\n\\displaystyle\n= \\lim_{x_1'\\to 0} \\frac{1}{2\\pi} \n\\int_{-\\infty}^{\\infty} \\left\\{\\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^{+\\top}(t) \\mbox{\\boldmath $R$} \\langle \\overline{\\mbox{\\boldmath $p$}} \\rangle(t) \n+ \\langle\\overline{\\mbox{\\boldmath $U$}}\\rangle^\\top(t) \\mbox{\\boldmath $R$} \\jump{0.15}{\\overline{\\mbox{\\boldmath $p$}}}(t) \\right\\} e^{-i x_1' t} dt \\\\[3mm]\n\\displaystyle\n= \\lim_{x_1'\\to 0} \\ensuremath{\\mathcal{F}}^{-1}_{x_1'} \\left\\{\\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^{+\\top} \\mbox{\\boldmath $R$} \\langle \\overline{\\mbox{\\boldmath $p$}} \\rangle \n+ \\langle\\overline{\\mbox{\\boldmath $U$}}\\rangle^\\top \\mbox{\\boldmath $R$} \\jump{0.15}{\\overline{\\mbox{\\boldmath $p$}}} \\right\\} \\\\[3mm]\n\\displaystyle\n= \\lim_{x_1'\\to 0} \\int_{-\\infty}^{0} \\left\\{\\jump{0.15}{\\mbox{\\boldmath $U$}}^\\top(x_1'-x_1) \\mbox{\\boldmath $R$} \\langle \\mbox{\\boldmath $p$} \\rangle(x_1) \n+ \\langle\\mbox{\\boldmath $U$}\\rangle^\\top(x_1'-x_1) \\mbox{\\boldmath $R$} \\jump{0.15}{\\mbox{\\boldmath $p$}}(x_1) \\right\\} dx_1.\n\\end{array}\n$$\n\nThe formula for the complex stress intensity factor becomes\n\\begin{equation}\n\\label{SIF}\n\\mbox{\\boldmath $K$} = -i \\mbox{\\boldmath $\\mathcal{M}$}_1^{-1} \\lim_{x_1'\\to 0} \\int_{-\\infty}^{0} \\left\\{\\jump{0.15}{\\mbox{\\boldmath $U$}}^\\top(x_1'-x_1) \\mbox{\\boldmath $R$} \\langle \\mbox{\\boldmath $p$} \\rangle(x_1) \n+ \\langle\\mbox{\\boldmath $U$}\\rangle^\\top(x_1'-x_1) \\mbox{\\boldmath $R$} \\jump{0.15}{\\mbox{\\boldmath $p$}}(x_1) \\right\\} dx_1.\n\\end{equation}\n\nSimilarly, we can develop the integral in \\eq{coeff}$_2$ to obtain\n$$\n\\begin{array}{l}\n\\displaystyle\n\\frac{1}{2\\pi} \\int_{-\\infty}^{\\infty} t \\left\\{\\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^{+\\top}(t) \\mbox{\\boldmath $R$} \\langle \\overline{\\mbox{\\boldmath $p$}} \\rangle(t) \n+ \\langle\\overline{\\mbox{\\boldmath $U$}}\\rangle^\\top(t) \\mbox{\\boldmath $R$} \\jump{0.15}{\\overline{\\mbox{\\boldmath $p$}}}(t) \\right\\} dt = \\\\[3mm]\n\\displaystyle\n= i \\lim_{x_1'\\to 0} \\frac{d}{dx_1'} \\frac{1}{2\\pi} \n\\int_{-\\infty}^{\\infty} \\left\\{\\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^{+\\top}(t) \\mbox{\\boldmath $R$} \\langle \\overline{\\mbox{\\boldmath $p$}} \\rangle(t) \n+ \\langle\\overline{\\mbox{\\boldmath $U$}}\\rangle^\\top(t) \\mbox{\\boldmath $R$} \\jump{0.15}{\\overline{\\mbox{\\boldmath $p$}}}(t) \\right\\} e^{-i x_1' t} dt \\\\[3mm]\n\\displaystyle\n= i \\lim_{x_1'\\to 0} \\int_{-\\infty}^{0} \\left\\{\\jump{0.15}{\\mbox{\\boldmath $U$}}^\\top(x_1'-x_1) \\mbox{\\boldmath $R$} \\frac{d\\langle \\mbox{\\boldmath $p$} \\rangle(x_1)}{dx_1} \n+ \\langle\\mbox{\\boldmath $U$}\\rangle^\\top(x_1'-x_1) \\mbox{\\boldmath $R$} \\frac{d\\jump{0.15}{\\mbox{\\boldmath $p$}}(x_1)}{dx_1} \\right\\} dx_1.\n\\end{array}\n$$\n\nThe formula for the coefficient in the second order term is\n\\begin{equation}\n\\label{sec}\n\\mbox{\\boldmath $A$} = \\mbox{\\boldmath $\\mathcal{M}$}_2^{-1} \\lim_{x_1'\\to 0} \\int_{-\\infty}^{0} \\left\\{\\jump{0.15}{\\mbox{\\boldmath $U$}}^\\top(x_1'-x_1) \\mbox{\\boldmath $R$} \\frac{d\\langle \\mbox{\\boldmath $p$} \\rangle(x_1)}{dx_1} \n+ \\langle\\mbox{\\boldmath $U$}\\rangle^\\top(x_1'-x_1) \\mbox{\\boldmath $R$} \\frac{d\\jump{0.15}{\\mbox{\\boldmath $p$}}(x_1)}{dx_1} \\right\\} dx_1.\n\\end{equation}\n\nIf the loading $\\mbox{\\boldmath $p$}^\\pm$ is not smooth enough, singular or it is a distribution (such as the Dirac delta function), then we can replace $\\mbox{\\boldmath $p$}^\\pm$ \nwith a sequence $\\{\\mbox{\\boldmath $p$}^\\pm_n\\}$ such that $\\mbox{\\boldmath $p$}^\\pm_n \\in \\ensuremath{\\mathcal{S}}(\\ensuremath{\\mathbb{R}})$ and $\\mbox{\\boldmath $p$}^\\pm_n \\to \\mbox{\\boldmath $p$}^\\pm$ in $\\bar{\\ensuremath{\\mathcal{S}}}(\\ensuremath{\\mathbb{R}})$, when $n \\to \\infty$. \nHere $\\bar{\\ensuremath{\\mathcal{S}}}(\\ensuremath{\\mathbb{R}})$ denotes the space of distributions (Zemanian, 1968). Therefore, the representations \\eq{coeff}, \\eq{SIF}, \\eq{sec} can \nbe interpreted in the sense of distributions and we can integrate formally \\eq{sec} by parts to obtain\n$$\n\\mbox{\\boldmath $A$} = \\mbox{\\boldmath $\\mathcal{M}$}_2^{-1} \\lim_{x_1'\\to 0} \\int_{-\\infty}^{0} \\left\\{\\frac{d\\jump{0.15}{\\mbox{\\boldmath $U$}}^\\top(x_1'-x_1)}{dx_1} \\mbox{\\boldmath $R$} \\langle \\mbox{\\boldmath $p$} \\rangle(x_1) \n+ \\frac{d\\langle\\mbox{\\boldmath $U$}\\rangle^\\top(x_1'-x_1)}{dx_1} \\mbox{\\boldmath $R$} \\jump{0.15}{\\mbox{\\boldmath $p$}}(x_1) \\right\\} dx_1.\n$$\n\n\\section{Perturbation of the crack front}\n\\label{perturbation}\nFor the 2D case, the perturbation considered here is related to an advance of the crack front by a distance $a$. We denote quantities relative to the original unperturbed \ncrack problem by subscript $0$, and quantities relative to the perturbed crack problem by subscript $\\star$.\n\n\\subsection{General settings}\nThe complex stress intensity factor for the original unperturbed problem is given by\n$$\n\\mbox{\\boldmath $K$}_0 = -i \\mbox{\\boldmath $\\mathcal{M}$}_1^{-1} \\lim_{x_1'\\to 0} \\int_{-\\infty}^{0} \\left\\{\\jump{0.15}{\\mbox{\\boldmath $U$}}^\\top(x_1'-x_1) \\mbox{\\boldmath $R$} \\langle \\mbox{\\boldmath $p$} \\rangle(x_1) \n+ \\langle\\mbox{\\boldmath $U$}\\rangle^\\top(x_1'-x_1) \\mbox{\\boldmath $R$} \\jump{0.15}{\\mbox{\\boldmath $p$}}(x_1) \\right\\} dx_1.\n$$\nFor the perturbed problem, if the coordinate system is shifted by the quantity $a$ to the new crack tip position, the complex stress intensity factor is given by the \nformula\n$$\n\\mbox{\\boldmath $K$}_\\star(a) = -i \\mbox{\\boldmath $\\mathcal{M}$}_1^{-1} \\lim_{x_1'\\to 0} \\int_{-\\infty}^{0} \\left\\{\\jump{0.15}{\\mbox{\\boldmath $U$}}^\\top(x_1'-x_1) \\mbox{\\boldmath $R$} \\langle \\mbox{\\boldmath $p$} \\rangle(x_1+a) \n+ \\langle\\mbox{\\boldmath $U$}\\rangle^\\top(x_1'-x_1) \\mbox{\\boldmath $R$} \\jump{0.15}{\\mbox{\\boldmath $p$}}(x_1+a) \\right\\} dx_1.\n$$\n\nThe computation of the first order variation of the complex stress intensity factor is now straightforward\n$$\n\\begin{array}{l}\n\\displaystyle\n\\left.\\frac{d\\mbox{\\boldmath $K$}_\\star(a)}{da}\\right|_{a=0} =  \\\\[3mm]\n\\displaystyle\n=-i \\mbox{\\boldmath $\\mathcal{M}$}_1^{-1} \\lim_{x_1'\\to 0} \\lim_{a \\to 0^+} \\int_{-\\infty}^{0} \n\\left\\{\\jump{0.15}{\\mbox{\\boldmath $U$}}^\\top(x_1'-x_1) \\mbox{\\boldmath $R$} \\frac{\\langle \\mbox{\\boldmath $p$} \\rangle(x_1+a) - \\langle \\mbox{\\boldmath $p$} \\rangle(x_1)}{a} \\right. \\\\[3mm] \n\\hspace{50mm}\n\\displaystyle\n\\left. + \\langle\\mbox{\\boldmath $U$}\\rangle^\\top(x_1'-x_1) \\mbox{\\boldmath $R$} \\frac{\\jump{0.15}{\\mbox{\\boldmath $p$}}(x_1+a) - \\jump{0.15}{\\mbox{\\boldmath $p$}}(x_1)}{a} \\right\\} \ndx_1 \\\\[3mm]\n\\displaystyle\n=-i \\mbox{\\boldmath $\\mathcal{M}$}_1^{-1} \\lim_{x_1'\\to 0} \\int_{-\\infty}^{0} \n\\left\\{\\jump{0.15}{\\mbox{\\boldmath $U$}}^\\top(x_1'-x_1) \\mbox{\\boldmath $R$} \\frac{d\\langle \\mbox{\\boldmath $p$} \\rangle(x_1)}{dx_1} \n+ \\langle\\mbox{\\boldmath $U$}\\rangle^\\top(x_1'-x_1) \\mbox{\\boldmath $R$} \\frac{d\\jump{0.15}{\\mbox{\\boldmath $p$}}(x_1)}{dx_1} \\right\\} \ndx_1.\n\\end{array}\n$$\n\nSimilarly, for the second order term, we obtain\n$$\n\\left.\\frac{d\\mbox{\\boldmath $A$}_\\star(a)}{da}\\right|_{a=0} = \\mbox{\\boldmath $\\mathcal{M}$}_2^{-1} \\lim_{x_1'\\to 0} \n\\int_{-\\infty}^{0} \\left\\{\\jump{0.15}{\\mbox{\\boldmath $U$}}^\\top(x_1'-x_1) \\mbox{\\boldmath $R$} \\frac{d^2\\langle \\mbox{\\boldmath $p$} \\rangle(x_1)}{dx_1^2} \n+ \\langle\\mbox{\\boldmath $U$}\\rangle^\\top(x_1'-x_1) \\mbox{\\boldmath $R$} \\frac{d^2\\jump{0.15}{\\mbox{\\boldmath $p$}}(x_1)}{dx_1^2} \\right\\} dx_1.\n$$\n\nWe assume here that all the derivatives exist, otherwise, as previously, the formula should be understood in the generalized sense.\n\n\\subsection{Computation of the perturbations of stress intensity factor according to the approach of Willis and Movchan (1995)} \nThe Betti formula for the original crack problem and the perturbed crack problem reads\n\\begin{equation}\n\\label{betti4}\n\\begin{array}{l}\n\\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^{+\\top} \\mbox{\\boldmath $R$} \\overline{\\mbox{\\boldmath $\\sigma$}}_0^{+} - \\overline{\\mbox{\\boldmath $\\Sigma$}}^{-\\top} \\mbox{\\boldmath $R$} \\jump{0.15}{\\overline{\\mbox{\\boldmath $u$}}_0}^- = \n- \\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^{+\\top} \\mbox{\\boldmath $R$} \\langle \\overline{\\mbox{\\boldmath $p$}} \\rangle - \\langle\\overline{\\mbox{\\boldmath $U$}}\\rangle^\\top \\mbox{\\boldmath $R$} \\jump{0.15}{\\overline{\\mbox{\\boldmath $p$}}}, \\quad \\beta \\in \\ensuremath{\\mathbb{R}}, \\\\[3mm]\n\\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^{+\\top} \\mbox{\\boldmath $R$} \\overline{\\mbox{\\boldmath $\\sigma$}}_\\star^{\\ddag} - \\overline{\\mbox{\\boldmath $\\Sigma$}}^{-\\top} \\mbox{\\boldmath $R$} \\jump{0.15}{\\overline{\\mbox{\\boldmath $u$}}_\\star^{\\ddag}} = \n- \\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^{+\\top} \\mbox{\\boldmath $R$} \\langle \\overline{\\mbox{\\boldmath $p$}} \\rangle - \\langle\\overline{\\mbox{\\boldmath $U$}}\\rangle^\\top \\mbox{\\boldmath $R$} \\jump{0.15}{\\overline{\\mbox{\\boldmath $p$}}}, \\quad \\beta \\in \\ensuremath{\\mathbb{R}},\n\\end{array}\n\\end{equation}\nrespectively. According to \\eq{stress1} and \\eq{displac}, the asymptotics of physical fields in the unperturbed problem are\n\\begin{equation}\n\\label{unpert}\n\\begin{array}{l}\n\\displaystyle\n\\overline{\\mbox{\\boldmath $\\sigma$}}_0^+(\\beta) = \\frac{\\beta_+^{-1\/2}}{4} \\mbox{\\boldmath $\\mathcal{T}$}_1(\\beta) \\mbox{\\boldmath $K$}_0 + \\frac{\\beta_+^{-1\/2}}{4\\beta} \\mbox{\\boldmath $\\mathcal{T}$}_2(\\beta) \\mbox{\\boldmath $A$}_0 + O(\\beta^{-5\/2}), \\quad \\beta \\in \\ensuremath{\\mathbb{C}}^+, \\\\[3mm]\n\\displaystyle\n\\jump{0.15}{\\overline{\\mbox{\\boldmath $u$}}_0}^-(\\beta) = -\\frac{bd_0^2}{4\\beta} \\beta_-^{-1\/2} \\mbox{\\boldmath $\\mathcal{V}$}_1(\\beta) \\mbox{\\boldmath $K$}_0 - \\frac{bd_0^2}{4\\beta^2} \\beta_-^{-1\/2} \\mbox{\\boldmath $\\mathcal{V}$}_2(\\beta) \\mbox{\\boldmath $A$}_0 + O(\\beta^{-7\/2}), \\quad \n\\beta \\in \\ensuremath{\\mathbb{C}}^-.\n\\end{array}\n\\end{equation}\n\nFor the physical fields in the perturbed problem, $\\mbox{\\boldmath $\\sigma$}_\\star^\\ddag$ and $\\mbox{\\boldmath $u$}_\\star^\\ddag$, we can write the respective transforms in the new \ncoordinate system related to the new position of the crack tip\n$$\n\\begin{array}{l}\n\\displaystyle\n\\overline{\\mbox{\\boldmath $\\sigma$}}_\\star^\\ddag(\\beta,a) =  \n\\int_a^{\\infty} \\mbox{\\boldmath $\\sigma$}_\\star^\\ddag(x_1) e^{i\\beta x_1} dx_1 = \ne^{i\\beta a} \\int_0^{\\infty} \\mbox{\\boldmath $\\sigma$}_\\star(y) e^{i\\beta y} dy = \ne^{i\\beta a} \\overline{\\mbox{\\boldmath $\\sigma$}}_\\star^+(\\beta,a), \\\\[3mm]\n\\displaystyle\n\\jump{0.15}{\\overline{\\mbox{\\boldmath $u$}}_\\star^\\ddag}(\\beta,a) =  \n\\int_{-\\infty}^a \\jump{0.15}{\\mbox{\\boldmath $u$}_\\star^\\ddag}(x_1) e^{i\\beta x_1} dx_1 = \ne^{i\\beta a} \\int_{-\\infty}^0 \\jump{0.15}{\\mbox{\\boldmath $u$}_\\star}(y) e^{i\\beta y} dy = \ne^{i\\beta a} \\jump{0.15}{\\overline{\\mbox{\\boldmath $u$}}_\\star}^-(\\beta,a).\n\\end{array}\n$$\nNote that $\\overline{\\mbox{\\boldmath $\\sigma$}}_\\star^+$ and $\\jump{0.15}{\\overline{\\mbox{\\boldmath $u$}}_\\star}^-$ are \"$+$\" and \"$-$\" functions, respectively, with the same asymptotics as in \\eq{unpert}, \nsubject to replacing $\\mbox{\\boldmath $K$}_0$ with $\\mbox{\\boldmath $K$}_\\star(a)$ and $\\mbox{\\boldmath $A$}_0$ with $\\mbox{\\boldmath $A$}_\\star(a)$.\n\nSubtracting (\\ref{betti4})$_2$ from (\\ref{betti4})$_1$, we obtain\n\\begin{equation}\n\\label{betti5}\n\\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^{+\\top} \\mbox{\\boldmath $R$} (\\overline{\\mbox{\\boldmath $\\sigma$}}_0^{+} - e^{i\\beta a}\\overline{\\mbox{\\boldmath $\\sigma$}}_\\star^{+}) \n- \\overline{\\mbox{\\boldmath $\\Sigma$}}^{-\\top} \\mbox{\\boldmath $R$} (\\jump{0.15}{\\overline{\\mbox{\\boldmath $u$}}_0}^- - e^{i\\beta a}\\jump{0.15}{\\overline{\\mbox{\\boldmath $u$}}_\\star}^-) = \\mbox{\\boldmath $0$}, \n\\quad \\beta \\in \\ensuremath{\\mathbb{R}}.\n\\end{equation} \nNote that for any $a > 0$, the function $e^{i\\beta a}\\overline{\\mbox{\\boldmath $\\sigma$}}_\\star^{+}$ is a \"$+$\" function which decays exponentially as $\\beta \\to \\infty$, \n$\\beta \\in \\ensuremath{\\mathbb{C}}^+$, whereas $e^{i\\beta a}[\\overline{\\mbox{\\boldmath $u$}}_\\star]^-$ is a \"$-$\" function which grows exponentially as $\\beta \\to \\infty$, $\\beta \\in \\ensuremath{\\mathbb{C}}^-$.\n\nFollowing Willis and Movchan (1995), we expand the exponential term as $a \\to 0^+$, so that\n$$\n\\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^{+\\top} \\mbox{\\boldmath $R$} (\\overline{\\mbox{\\boldmath $\\sigma$}}_0^{+} - (1 + ia\\beta)\\overline{\\mbox{\\boldmath $\\sigma$}}_\\star^{+}) \n- \\overline{\\mbox{\\boldmath $\\Sigma$}}^{-\\top} \\mbox{\\boldmath $R$} (\\jump{0.15}{\\overline{\\mbox{\\boldmath $u$}}_0}^- - (1 + ia\\beta)\\jump{0.15}{\\overline{\\mbox{\\boldmath $u$}}_\\star}^-) = \\mbox{\\boldmath $0$}, \n\\quad \\beta \\in \\ensuremath{\\mathbb{R}}.\n$$\n\nThe substitution of two-terms asymptotics for the unperturbed physical fields, $\\overline{\\mbox{\\boldmath $\\sigma$}}_0^{+}$, $\\jump{0.15}{\\overline{\\mbox{\\boldmath $u$}}_0}^-$, and for the perturbed \nphysical fields, $\\overline{\\mbox{\\boldmath $\\sigma$}}_\\star^{+}$, $\\jump{0.15}{\\overline{\\mbox{\\boldmath $u$}}_\\star}^-$, leads to\n$$\n\\begin{array}{l}\n\\displaystyle\n\\mbox{\\boldmath $\\mathcal{M}$}_1 (\\mbox{\\boldmath $K$}_0 - \\mbox{\\boldmath $K$}_\\star(a))\\beta_+^{-1} - ia \\mbox{\\boldmath $\\mathcal{M}$}_1 \\mbox{\\boldmath $K$}_\\star(a) -ia \\mbox{\\boldmath $\\mathcal{M}$}_2 \\mbox{\\boldmath $A$}_\\star(a) \\beta_+^{-1} \\\\[3mm]\n\\hspace{30mm}\n- \\{ \\mbox{\\boldmath $\\mathcal{M}$}_1(\\mbox{\\boldmath $K$}_0 - \\mbox{\\boldmath $K$}_\\star(a))\\beta_-^{-1} - ia \\mbox{\\boldmath $\\mathcal{M}$}_1 \\mbox{\\boldmath $K$}_\\star(a) -ia \\mbox{\\boldmath $\\mathcal{M}$}_2 \\mbox{\\boldmath $A$}_\\star(a) \\beta_-^{-1} \\} + O(\\beta^{-2}) = \\\\[3mm]\n\\displaystyle \\hspace{30mm}\n= \\{\\mbox{\\boldmath $\\mathcal{M}$}_1 (\\mbox{\\boldmath $K$}_0 - \\mbox{\\boldmath $K$}_\\star(a)) - ia \\mbox{\\boldmath $\\mathcal{M}$}_2 \\mbox{\\boldmath $A$}_\\star(a) \\} (\\beta_+^{-1} - \\beta_-^{-1}) + O(\\beta^{-2}) = \\mbox{\\boldmath $0$}.\n\\end{array}\n$$\nWe get \n\\begin{equation}\n\\label{SIF_perturb}\n\\Delta \\mbox{\\boldmath $K$} = \\mbox{\\boldmath $K$}_\\star(a) - \\mbox{\\boldmath $K$}_0 \\sim -ia \\mbox{\\boldmath $\\mathcal{M}$}_1^{-1} \\mbox{\\boldmath $\\mathcal{M}$}_2 \\mbox{\\boldmath $A$}_0 = \na \\left[\\begin{array}{ll} 1\/2 + i\\epsilon & 0 \\\\[3mm] 0 & 1\/2 - i\\epsilon \\end{array}\\right] \\mbox{\\boldmath $A$}_0, \\quad a \\to 0^+,\n\\end{equation}\nand finally\n$$\n\\Delta K \\sim a (1\/2 + i\\epsilon) A_0, \\quad \\Delta K^* \\sim a (1\/2 - i\\epsilon) A_0^*, \\quad a \\to 0^+.\n$$\n\nNote that it is assumed here that $\\mbox{\\boldmath $K$}_\\star(a)$ and $\\mbox{\\boldmath $A$}_\\star(a)$ are continuous in $a = 0$. This fact will be proven in the next section.\n\n\\subsection{An alternative computation of the perturbations of stress intensity factors and high-order terms}\n\\label{SIF_Abelian}\nHere we provide a rigorous asymptotic procedure leading to the computation of perturbations in the stress intensity factors as well as high-order terms. The Abelian \nand Tauberian type theorems outlined in Appendix \\ref{app3} will be used for this purpose.\n\nWe rewrite now the equation (\\ref{betti5}) as follows \n\\begin{equation}\n\\label{w-h}\n\\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^{+\\top} \\mbox{\\boldmath $R$} (e^{-i\\beta a}\\overline{\\mbox{\\boldmath $\\sigma$}}_0^{+} - \\overline{\\mbox{\\boldmath $\\sigma$}}_\\star^{+}) \n- \\overline{\\mbox{\\boldmath $\\Sigma$}}^{-\\top} \\mbox{\\boldmath $R$} (e^{-i\\beta a}\\jump{0.15}{\\overline{\\mbox{\\boldmath $u$}}_0}^- - \\jump{0.15}{\\overline{\\mbox{\\boldmath $u$}}_\\star}^-) = \\mbox{\\boldmath $0$}, \n\\quad \\beta \\in \\ensuremath{\\mathbb{R}}.\n\\end{equation} \nThe function $e^{-i\\beta a} \\overline{\\mbox{\\boldmath $\\Sigma$}}^{-\\top} \\mbox{\\boldmath $R$} \\jump{0.15}{\\overline{\\mbox{\\boldmath $u$}}_0}^{-}$ is a \"$-$\" function which decays exponentially as $\\beta \\to \\infty$, \n$\\beta \\in \\ensuremath{\\mathbb{C}}^-$, whereas $e^{-i\\beta a} \\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^{+\\top} \\mbox{\\boldmath $R$} \\overline{\\mbox{\\boldmath $\\sigma$}}_0^+ $ is not a proper \"$+$\" function, because of the exponential \ngrowth as $\\beta \\to \\infty$, $\\beta \\in \\ensuremath{\\mathbb{C}}^+$. However, for $\\beta \\in \\ensuremath{\\mathbb{R}}$, \n$e^{-i\\beta a} \\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^{+\\top} \\mbox{\\boldmath $R$} \\overline{\\mbox{\\boldmath $\\sigma$}}_0^+  \\sim 1\/\\beta$, as $\\beta \\to \\pm \\infty$, so that it can be decomposed as \n$$\ne^{-i\\beta a} \\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^{+\\top} \\mbox{\\boldmath $R$} \\overline{\\mbox{\\boldmath $\\sigma$}}_0^+ = \\mbox{\\boldmath $\\Xi$}^+ - \\mbox{\\boldmath $\\Xi$}^-,\n$$\nwhere\n$$\n\\mbox{\\boldmath $\\Xi$}^\\pm(\\beta) = \\frac{1}{2\\pi i} \\int_{-\\infty}^{\\infty} \n\\frac{e^{-ia t} \\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^{+\\top} \\mbox{\\boldmath $R$} \\overline{\\mbox{\\boldmath $\\sigma$}}_0^+}{t - \\beta} dt, \n\\quad \\beta \\in \\ensuremath{\\mathbb{C}}^\\pm.\n$$\nNow, from the solution of the Wiener-Hopf equation (\\ref{w-h}), we obtain the following identities\n\\begin{equation}\n\\label{id}\n\\begin{array}{l}\n\\displaystyle\n\\mbox{\\boldmath $\\Xi$}^+ - \\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^{+\\top} \\mbox{\\boldmath $R$} \\overline{\\mbox{\\boldmath $\\sigma$}}_\\star^{+} = \\mbox{\\boldmath $0$}, \\quad \\beta \\in \\ensuremath{\\mathbb{C}}^+, \\\\[3mm]\n\\displaystyle\n-\\mbox{\\boldmath $\\Xi$}^- - \\overline{\\mbox{\\boldmath $\\Sigma$}}^{-\\top} \\mbox{\\boldmath $R$} (e^{-i\\beta a}\\jump{0.15}{\\overline{\\mbox{\\boldmath $u$}}_0}^- - \\jump{0.15}{\\overline{\\mbox{\\boldmath $u$}}_\\star}^-) = \\mbox{\\boldmath $0$}, \\quad \\beta \\in \\ensuremath{\\mathbb{C}}^-.\n\\end{array}\n\\end{equation}\nNote that\n$$\n\\int_{-\\infty}^{\\infty} e^{-iat} \\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^{+\\top} \\mbox{\\boldmath $R$} \\overline{\\mbox{\\boldmath $\\sigma$}}_0^+ dt < \\infty,\n$$\nand\n$$\ne^{-i\\beta a} \\overline{\\mbox{\\boldmath $\\Sigma$}}^{-\\top} \\mbox{\\boldmath $R$} \\jump{0.15}{\\overline{\\mbox{\\boldmath $u$}}_0}^- = o\\left( \\frac{1}{\\beta^n} \\right), \n\\quad \\beta \\to \\infty, \\quad \\beta \\in \\ensuremath{\\mathbb{C}}^-,\n$$\nfor any $n \\in \\ensuremath{\\mathbb{N}}$. Consequently, we obtain from the leading term asymptotic of both identities (\\ref{id}) the same result, namely\n\\begin{equation}\n\\label{k2}\n\\mbox{\\boldmath $\\mathcal{M}$}_1 \\mbox{\\boldmath $K$}_\\star(a) = -\\frac{1}{2\\pi i} \\int_{-\\infty}^{\\infty} e^{-iat} \\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^{+\\top} \\mbox{\\boldmath $R$} \\overline{\\mbox{\\boldmath $\\sigma$}}_0^+ dt.\n\\end{equation}\nThe application of the Tauberian type Theorem \\ref{theorem} (see Appendix \\ref{app3}) leads to\n$$\n\\lim_{a \\to 0^+} \\mbox{\\boldmath $\\mathcal{M}$}_1 \\mbox{\\boldmath $K$}_\\star(a) = -\\frac{1}{i} (- i \\mbox{\\boldmath $\\mathcal{M}$}_1 \\mbox{\\boldmath $K$}_0),\n$$\nand, since the matrix $\\mbox{\\boldmath $\\mathcal{M}$}_1$ is not singular,\n$$\n\\lim_{a \\to 0^+} \\mbox{\\boldmath $K$}_\\star(a) = \\mbox{\\boldmath $K$}_0.\n$$\nThis proves that the complex stress intensity factor $K$ is continuous in $a = 0$.\n\nIn order to derive the first order variation of the complex stress intensity factor in the perturbation problem, we consider the derivative of (\\ref{k2}) with \nrespect to $a$ (see Theorem \\ref{theorem}),\n$$\n\\mbox{\\boldmath $\\mathcal{M}$}_1 \\mbox{\\boldmath $K$}_\\star'(a) = \\frac{1}{2\\pi} \\int_{-\\infty}^{\\infty} e^{-iat} \\{t \\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^{+\\top} \\mbox{\\boldmath $R$} \\overline{\\mbox{\\boldmath $\\sigma$}}_0^+ - \\mbox{\\boldmath $\\mathcal{M}$}_1 \\mbox{\\boldmath $K$}_0\\} dt,\n$$\nwhere\n$$\nt \\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^{+\\top} \\mbox{\\boldmath $R$} \\overline{\\mbox{\\boldmath $\\sigma$}}_0^+ - \\mbox{\\boldmath $\\mathcal{M}$}_1 \\mbox{\\boldmath $K$}_0 = t^{-1} \\mbox{\\boldmath $\\mathcal{M}$}_2 \\mbox{\\boldmath $A$}_0 + t^{-2} \\mbox{\\boldmath $\\mathcal{M}$}_2 \\mbox{\\boldmath $B$}_0 + O(t^{-3}), \\quad t \\to \\infty, \n\\quad t \\in \\ensuremath{\\mathbb{C}}^+.\n$$\nApplying again the Theorem \\ref{theorem} we can conclude that\n$$\n\\lim_{a \\to 0^+} \\mbox{\\boldmath $\\mathcal{M}$}_1 \\mbox{\\boldmath $K$}_\\star'(a) = -i \\mbox{\\boldmath $\\mathcal{M}$}_2 \\mbox{\\boldmath $A$}_0, \n$$\nand, finally, the first order variation of the complex stress intensity factor in the perturbation problem is given by \n$$\n\\mbox{\\boldmath $K$}_\\star'(0) = -i \\mbox{\\boldmath $\\mathcal{M}$}_1^{-1} \\mbox{\\boldmath $\\mathcal{M}$}_2 \\mbox{\\boldmath $A$}_0,\n$$\nwhich is consistent with the result \\eq{SIF_perturb}.\n\nThis asymptotic procedure can be extended to compute the perturbations of high-order terms as follows. From the second-term asymptotics of (\\ref{id}) we derive \nthe coefficient $\\mbox{\\boldmath $A$}_\\star(a)$, which reads\n\\begin{equation}\n\\label{A2}\n\\mbox{\\boldmath $\\mathcal{M}$}_2 \\mbox{\\boldmath $A$}_\\star(a) = \\lim_{\\beta \\to \\infty} \\frac{\\beta}{2\\pi i}\n\\int_{-\\infty}^{\\infty} \\frac{t e^{-iat} \\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^{+\\top} \\mbox{\\boldmath $R$} \\overline{\\mbox{\\boldmath $\\sigma$}}_0^+}{t - \\beta} dt,\n\\quad \\beta \\in \\ensuremath{\\mathbb{C}}^+.\n\\end{equation}\nNote that the integral on the right-hand side exists, but the behaviour of the integrand does not allow us to take the limit directly.\n\nLet us consider the function\n$$\n\\mbox{\\boldmath $f$}(\\beta,a) = \\frac{\\beta}{2\\pi i}\n\\int_{-\\infty}^{\\infty} \\frac{t e^{-iat} \\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^{+\\top} \\mbox{\\boldmath $R$} \\overline{\\mbox{\\boldmath $\\sigma$}}_0^+}{t - \\beta} dt.\n$$\n\nIntegration by parts leads to\n\\begin{equation}\n\\label{effe2}\n\\mbox{\\boldmath $f$}(\\beta,a) = \\frac{\\beta}{2\\pi i} \\left\\{ \n\\left[g(t,\\beta) t \\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^{+\\top} \\mbox{\\boldmath $R$} \\overline{\\mbox{\\boldmath $\\sigma$}}_0^+ \\right]_{-\\infty}^{\\infty}\n- \\int_{-\\infty}^{\\infty} g(t,\\beta) (t \\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^{+\\top} \\mbox{\\boldmath $R$} \\overline{\\mbox{\\boldmath $\\sigma$}}_0^+ )' dt\n\\right\\},\n\\end{equation}\nwhere \n\\begin{equation}\n\\label{gi}\ng(t,\\beta) = \\int_{-\\infty}^{t} \\frac{e^{-ia\\xi}}{\\xi - \\beta} d\\xi, \\quad \\beta \\in \\ensuremath{\\mathbb{C}}^+.\n\\end{equation}\nThe integrand in (\\ref{gi}) is analytic in $\\ensuremath{\\mathbb{C}}^-$ and decays exponentially as $\\xi \\to \\infty$, \n$\\xi \\in \\ensuremath{\\mathbb{C}}^-$. As a result\n$$\n\\lim_{t \\to \\infty} g(t,\\beta) = 0, \\quad \\beta \\in \\ensuremath{\\mathbb{C}}^+,\n$$\nand therefore the first term in brackets in \\eq{effe2} vanishes and we can then write\n$$\n\\begin{array}{ll}\n\\displaystyle \\mbox{\\boldmath $f$}(\\beta,a) & \\displaystyle = -\\frac{\\beta}{2\\pi i} \\int_{-\\infty}^{\\infty} g(t,\\beta) (t \\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^{+\\top} \\mbox{\\boldmath $R$} \\overline{\\mbox{\\boldmath $\\sigma$}}_0^+ )' dt \\\\[3mm]\n& \\displaystyle = -\\frac{\\beta}{2\\pi i} \\int_{-\\infty}^{\\infty} g(t,\\beta) (t \\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^{+\\top} \\mbox{\\boldmath $R$} \\overline{\\mbox{\\boldmath $\\sigma$}}_0^+ - \\mbox{\\boldmath $\\mathcal{M}$}_1 \\mbox{\\boldmath $K$}_0)' dt \\\\[3mm]\n& \\displaystyle = -\\frac{\\beta}{2\\pi i} \\left\\{ \n\\left[g(t,\\beta) (t \\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^{+\\top} \\mbox{\\boldmath $R$} \\overline{\\mbox{\\boldmath $\\sigma$}}_0^+ - \\mbox{\\boldmath $\\mathcal{M}$}_1 \\mbox{\\boldmath $K$}_0)\\right]_{-\\infty}^{\\infty} \n- \\int_{-\\infty}^{\\infty} g'(t,\\beta) (t \\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^{+\\top} \\mbox{\\boldmath $R$} \\overline{\\mbox{\\boldmath $\\sigma$}}_0^+ - \\mbox{\\boldmath $\\mathcal{M}$}_1 \\mbox{\\boldmath $K$}_0) dt \n\\right\\} \\\\[3mm]\n& \\displaystyle = \\frac{\\beta}{2\\pi i}  \n\\int_{-\\infty}^{\\infty} \\frac{e^{-iat}}{t-\\beta} (t \\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^{+\\top} \\mbox{\\boldmath $R$} \\overline{\\mbox{\\boldmath $\\sigma$}}_0^+ - \\mbox{\\boldmath $\\mathcal{M}$}_1 \\mbox{\\boldmath $K$}_0) dt.\n\\end{array}\n$$\nIt is now possible to take the limit as $\\beta \\to \\infty$ in (\\ref{A2}), thus we deduce\n$$\n\\begin{array}{ll}\n\\displaystyle \\mbox{\\boldmath $\\mathcal{M}$}_2 \\mbox{\\boldmath $A$}_\\star(a) & \\displaystyle = \\lim_{\\beta \\to \\infty} \n\\frac{\\beta}{2\\pi i} \\int_{-\\infty}^{\\infty} \\frac{e^{-iat} (t \\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^{+\\top} \\mbox{\\boldmath $R$} \\overline{\\mbox{\\boldmath $\\sigma$}}_0^+ - \\mbox{\\boldmath $\\mathcal{M}$}_1 \\mbox{\\boldmath $K$}_0)}{t - \\beta} dt \\\\[3mm]\n& \\displaystyle = -\\frac{1}{2\\pi i} \\int_{-\\infty}^{\\infty} e^{-iat} (t \\jump{0.15}{\\overline{\\mbox{\\boldmath $U$}}}^{+\\top} \\mbox{\\boldmath $R$} \\overline{\\mbox{\\boldmath $\\sigma$}}_0^+ - \\mbox{\\boldmath $\\mathcal{M}$}_1 \\mbox{\\boldmath $K$}_0) dt.\n\\end{array}\n$$\nWe are now in the position to use the Theorem \\ref{theorem}, so that\n$$\n\\lim_{a \\to 0^+} \\mbox{\\boldmath $\\mathcal{M}$}_2 \\mbox{\\boldmath $A$}_\\star(a) = -\\frac{1}{i} (-i \\mbox{\\boldmath $\\mathcal{M}$}_2 \\mbox{\\boldmath $A$}_0),\n$$\nand, since the matrix $\\mbox{\\boldmath $\\mathcal{M}$}_2$ is not singular,\n$$\n\\lim_{a \\to 0^+} \\mbox{\\boldmath $A$}_\\star(a) = \\mbox{\\boldmath $A$}_0,\n$$\nwhich proves the continuity of the function $\\mbox{\\boldmath $A$}_\\star(a)$ in $a = 0$.\n\nMoreover, the first order variation of $\\mbox{\\boldmath $A$}_\\star(a)$ can now be derived following the same asymptotic procedure presented above and we deduce\n$$\n\\mbox{\\boldmath $A$}_\\star'(0) = -i \\mbox{\\boldmath $\\mathcal{M}$}_2^{-1} \\mbox{\\boldmath $\\mathcal{M}$}_3 \\mbox{\\boldmath $B$}_0,\n$$\nso that\n$$\nA_\\star'(0) = (3\/2 + i\\epsilon) B_0, \\quad A_\\star^{*\\prime}(0) = (3\/2 - i\\epsilon) B_0^*.\n$$\n\n\\section{An illustrative example}\n\\label{example}\nIn this section, we present an example regarding the computation of the complex stress intensity factor $K$ for an interfacial crack loaded by a simple asymmetric force \nsystem, as shown in Fig. \\ref{three-point}. The loading is given by a point force $F$ acting upon the upper crack face at a distance $a$ behind the crack tip and two \npoint forces $F\/2$ acting upon the lower crack face at a distance $a - b$ and $a + b$ behind the crack tip. In terms of the Dirac delta function $\\delta(\\cdot)$, the \nloading is then given by\n$$\n\\begin{array}{l}\n\\displaystyle p^+(x_1) = - F \\delta(x_1 + a), \\\\[3mm]\n\\displaystyle p^-(x_1) = -\\frac{F}{2} \\delta(x_1 + a + b) - \\frac{F}{2} \\delta(x_1 + a - b).\n\\end{array}\n$$\n\n\\begin{figure}[!htb]\n\\begin{center}\n\\vspace*{3mm}\n\\includegraphics[width=10cm]{fig03.eps}\n\\caption{\\footnotesize \"Three-point\" loading of an interfacial crack.}\n\\label{three-point}\n\\end{center}\n\\end{figure}\n\nThe loading is self-balanced, in terms of both principal force and moment vectors, and can be divided into symmetric and skew-symmetric parts (see Fig. \n\\ref{three-point}), \n$$\n\\begin{array}{l}\n\\displaystyle \\langle p \\rangle(x_1) = -\\frac{F}{2} \\delta(x_1 + a) - \\frac{F}{4} \\delta(x_1 + a + b) - \\frac{F}{4} \\delta(x_1 + a - b), \\\\[3mm]\n\\displaystyle \\jump{0.15}{p}(x_1) = - F \\delta(x_1 + a) + \\frac{F}{2} \\delta(x_1 + a + b) + \\frac{F}{2} \\delta(x_1 + a - b).\n\\end{array}\n$$\n\nFor this loading system, the complex stress intensity factor $K$ has been evaluated by means of the integral formula \\eq{SIF} and the symmetric and skew-symmetric \nweight functions, \\eq{symm} and \\eq{skew} respectively, obtaining $K = K^S + K^A$, where\n$$\n\\begin{array}{l}\n\\displaystyle K^S = F \\sqrt{\\frac{2}{\\pi}} \\cosh(\\pi\\epsilon) a^{-1\/2 - i\\epsilon} \\left\\{ \\frac{1}{2} + \\frac{1}{4} (1 + b\/a)^{-1\/2 - i\\epsilon} + \n\\frac{1}{4} (1 - b\/a)^{-1\/2 - i\\epsilon} \\right\\}, \\\\[3mm]\n\\displaystyle K^A = \\alpha F \\sqrt{\\frac{2}{\\pi}} \\cosh(\\pi\\epsilon) a^{-1\/2 - i\\epsilon} \\left\\{ \\frac{1}{2} - \\frac{1}{4} (1 + b\/a)^{-1\/2 - i\\epsilon} -\n\\frac{1}{4} (1 - b\/a)^{-1\/2 - i\\epsilon} \\right\\}.\n\\end{array}\n$$\n\nThe complex stress intensity factor has been computed for the case $\\nu_+ = 0.2$ and $\\nu_- = 0.3$ and five values of the parameter $\\eta$, namely \n$\\eta = \\{ -0.99, -0.5, 0, 0.5, 0.99 \\}$. The values have been normalized and plotted in Fig. \\ref{fig04} as functions of the ratio $b\/a$ \n($a$ is fixed equal to 1). The symmetric (skew-symmetric) \nstress intensity factor is reported on the left (right) of the figure, the real (imaginary) part is reported on the top (bottom).\n\\begin{figure}[!htb]\n\\begin{center}\n\\vspace*{3mm}\n\\includegraphics[width=11cm]{fig04arx.eps}\n\\caption{\\footnotesize Symmetric and anti-symmetric stress intensity factors as a function of $b\/a$ ($a$ is fixed equal to 1) for the case \n$\\nu_+ = 0.2$, $\\nu_-=0.3$ and: $\\eta = -0.99$ (green), $\\eta = -0.5$ (orange), $\\eta = 0$ (red), \n$\\eta = 0.5$ (blue), $\\eta = 0.99$ (black).}\n\\label{fig04}\n\\end{center}\n\\end{figure}\n\nCommenting on the results, first, we note from the figure that both $K_\\text{II}^S$ and $K_\\text{II}^A$ are not identically zero, even though the loading corresponds to a \ntensile (or opening) mode. This is typical of the interfacial crack problem, where there is no symmetry with respect to the plane containing the crack, so that Mode I and \nMode II are coupled.\n\nSecond, results pertaining to the skew-symmetric stress intensity factor show that $K_\\text{I}^A$ and $K_\\text{II}^A$ are both equal to zero for $b\/a = 0$, as expected, since \nin this case the loading is symmetric. As we increase $b\/a$ and the skew-symmetric part of the loading becomes more and more relevant, the skew-symmetric stress intensity \nfactor increases correspondingly. Both the symmetric and skew-symmetric stress intensity factors are singular as $b\/a \\to 1$, because a point force is approaching the \ncrack tip.\n\nThe analytical method developed in the present paper allows us also to evaluate the coefficients in the higher order terms of asymptotics of physical \nfields. As an example, we computed the coefficient in the second order term, $A$, which is reported in Fig. \\ref{fig05}, in \nanalogy with the stress intensity factor in the figures above.\n\\begin{figure}[!htb]\n\\begin{center}\n\\vspace*{3mm}\n\\includegraphics[width=11cm]{fig05arx.eps}\n\\caption{\\footnotesize Symmetric and anti-symmetric parts of $A$ as functions of $b\/a$ ($a$ is fixed equal to 1) for the case $\\nu_+ = 0.2$, \n$\\nu_-=0.3$ and: $\\eta = -0.99$ (green), $\\eta = -0.5$ (orange), $\\eta = 0$ (red), $\\eta = 0.5$ (blue), $\\eta = 0.99$ (black).}\n\\label{fig05}\n\\end{center}\n\\end{figure}\n\nIn order to appreciate the magnitude of the skew-symmetric stress intensity factor with respect to the magnitude of the symmetric stress intensity factor, the ratio \n$K_\\text{I}^A\/K_\\text{I}^S$ is plotted in Fig. \\ref{fig06} (on the left) as a function of $b\/a$. We notice from the figure that the magnitude of $K_\\text{I}^A$ may \neasily reach 20\\% of the magnitude of $K_\\text{I}^S$, a value which is not negligible and has to be taken into account in the view of the application of a tensile fracture \ncriterion. Note also that the coefficient $A$ is even more sensitive to the anti-symmetric loading, $A_\\text{I}^A$ may \neasily reach the same magnitude as $A_\\text{I}^S$, see Fig. \\ref{fig06} on the right. The ratios $K_\\text{II}^A\/K_\\text{II}^S$ and \n$A_\\text{II}^A\/A_\\text{II}^S$ are not reported here since, in this case, they are both equal to the same constant, namely \n$K_\\text{II}^A\/K_\\text{II}^S = A_\\text{II}^A\/A_\\text{II}^S = \\alpha$.\n\\begin{figure}[!htb]\n\\begin{center}\n\\vspace*{3mm}\n\\includegraphics[width=11cm]{fig06arx.eps}\n\\caption{\\footnotesize Ratios $K_\\text{I}^A\/K_\\text{I}^S$ and $A_\\text{I}^A\/A_\\text{I}^S$ as functions of\n$b\/a$ ($a$ is fixed equal to 1) for the case \n$\\nu_+ = 0.2$, $\\nu_-=0.3$ and: $\\eta = -0.99$ (green), $\\eta = -0.5$ (orange), $\\eta = 0$ (red), \n$\\eta = 0.5$ (blue), $\\eta = 0.99$ (black).}\n\\label{fig06}\n\\end{center}\n\\end{figure}\n\n\\section{Mode III} \n\\label{antiplane}\nThe Wiener-Hopf equation relating the Fourier transforms of the singular displacement and the corresponding traction is given by \n$$\n\\jump{0.15}{\\overline{U}_3}^+ = - \\frac{b+e}{|\\beta|} \\overline{\\Sigma}_{32}^-, \\quad \\beta \\in \\ensuremath{\\mathbb{R}},\n$$\nwhere $e = \\nu_+ \/ \\mu_+ + \\nu_- \/ \\mu_-$. This equation can be immediately factorized as follows\n$$\n\\beta_+^{1\/2} \\jump{0.15}{\\overline{U}_3}^+ = -\\frac{b+e}{\\beta_-^{1\/2}} \\overline{\\Sigma}_{32}^-.\n$$\nAssuming that $\\jump{0.15}{\\overline{U}_3}^+ \\sim \\beta_+^{-1\/2}$ and $\\overline{\\Sigma}_{32}^- \\sim \\beta_-^{1\/2}$, as $\\beta \\to \\infty$, $\\beta \\in \\ensuremath{\\mathbb{C}}^\\pm$, \nwe have\n$$\n\\beta_+^{1\/2} [\\overline{U}_3]^+ = -\\frac{b+e}{\\beta_-^{1\/2}} \\overline{\\Sigma}_{32}^- = O(1), \\quad\n\\beta \\to \\pm\\infty , \\quad \\beta \\in \\ensuremath{\\mathbb{R}},\n$$\nso that, from the Liouville theorem, it follows that both sides of the equation are equal to the same constant.\n\nThe weight functions for the Mode III problem are then given by\n$$\n\\jump{0.15}{\\overline{U}_3}^+(\\beta) = \\beta_+^{-1\/2}, \\quad \\text{and} \\quad \n\\overline{\\Sigma}_{32}^-(\\beta) = -\\frac{\\beta_-^{1\/2}}{b+e},\n$$\nwith corresponding inverse transforms\n\\begin{equation}\n\\label{jumpu3}\n\\jump{0.15}{U_3}(x_1) = \\left\\{\n\\begin{array}{ll}\n\\displaystyle \\frac{1-i}{\\sqrt{2\\pi}} x_1^{-1\/2}, & \\displaystyle x_1 > 0, \\\\[3mm]\n0, & x_1 < 0,\n\\end{array} \\right. \\qquad\n\\Sigma_{32}(x_1) = \\left\\{\n\\begin{array}{ll}\n0, & \\displaystyle x_1 > 0, \\\\[3mm]\n\\displaystyle \\frac{1-i}{2\\sqrt{2\\pi}(b+e)} (-x_1)^{-3\/2}, & x_1 < 0,\n\\end{array} \\right.\n\\end{equation}\n\nThe Betti identity reduces to \n$$ \n\\jump{0.15}{\\overline{U}_3}^+ \\overline{\\sigma}_{32}^+ - \\overline{\\Sigma}_{32}^- \\jump{0.15}{\\overline{u}_3}^- = \n- \\jump{0.15}{\\overline{U}_3}^+ \\langle \\overline{p}_3 \\rangle - \\langle\\overline{U}_3\\rangle^+ \\jump{0.15}{\\overline{p}_3}.\n$$\n\nThe asymptotics of physical fields are\n$$\n\\begin{array}{l}\n\\displaystyle\n\\sigma_{32} = \\frac{K_\\text{III}}{\\sqrt{2\\pi}} x_1^{-1\/2} + \\frac{A_\\text{III}}{\\sqrt{2\\pi}} x_1^{1\/2} + O(x_1^{3\/2}), \\quad x_1 \\to 0^+, \\\\[5mm]\n\\displaystyle\n\\jump{0.15}{u_3} = \\frac{2(b+e)K_\\text{III}}{\\sqrt{2\\pi}} (-x_1)^{1\/2} - \\frac{2(b+e)A_\\text{III}}{3\\sqrt{2\\pi}} (-x_1)^{3\/2} + O[(-x_1)^{5\/2}], \\quad x_1 \\to 0^-,\n\\end{array}\n$$\nwith corresponding Fourier transforms\n$$\n\\begin{array}{l}\n\\displaystyle\n\\overline{\\sigma}_{32}^+ = \\frac{(1+i)K_\\text{III}}{2} \\beta_+^{-1\/2} - \\frac{(1-i)A_\\text{III}}{4} \\beta_+^{-3\/2} + O(\\beta_+^{-5\/2}), \n\\quad \\beta \\to \\infty, \\quad \\beta \\in \\ensuremath{\\mathbb{C}}^+ \\\\[5mm]\n\\displaystyle\n\\jump{0.15}{\\overline{u}_3}^- = -\\frac{(1+i)(b+e)K_\\text{III}}{2} \\beta_-^{-3\/2} + \\frac{(1-i)(b+e)A_\\text{III}}{4} \\beta_-^{-5\/2} + O(\\beta_-^{-7\/2}), \n\\quad \\beta \\to \\infty, \\quad \\beta \\in \\ensuremath{\\mathbb{C}}^-.\n\\end{array}\n$$\n\nBy the same arguments used for Mode I and II, we obtain the formulae\n$$\n\\begin{array}{l}\n\\displaystyle K_\\text{III} = -(1+i) \\lim_{x_1' \\to 0^+} \\int_{-\\infty}^{0} \n\\{ \\jump{0.15}{U_3}(x_1'-x_1) \\langle p_3 \\rangle(x_1) + \\langle U_3 \\rangle(x_1'-x_1) \\jump{0.15}{p_3}(x_1) \\} dx_1, \\\\[3mm]\n\\displaystyle A_\\text{III} = -2(1+i) \\lim_{x_1' \\to 0^+} \\int_{-\\infty}^{0} \n\\left\\{ \\jump{0.15}{U_3}(x_1'-x_1) \\frac{d\\langle p_3 \\rangle(x_1)}{dx_1} + \\langle U_3 \\rangle(x_1'-x_1) \\frac{d\\jump{0.15}{p_3}(x_1)}{dx_1} \\right\\} dx_1.\n\\end{array}\n$$\n\nFrom the solution of the half-plane problem, we get the full representation of weight functions. For the lower half-plane:\n$$\n\\begin{array}{l}\n\\displaystyle\\overline{u}_3(\\beta,x_2) = \\frac{1}{\\mu_-|\\beta|} \\overline{\\Sigma}_{32}^- e^{|\\beta|x_2}, \\\\[3mm]\n\\displaystyle \\overline{\\sigma}_{31}(\\beta,x_2) = -i \\mathop{\\mathrm{sign}}(\\beta) \\overline{\\Sigma}_{32}^- e^{|\\beta|x_2}, \\\\[3mm]\n\\displaystyle \\overline{\\sigma}_{32}(\\beta,x_2) = \\overline{\\Sigma}_{32}^- e^{|\\beta|x_2}.\n\\end{array}\n$$\n\nFor the upper half-plane the equations are the same subject to replacing $|\\beta|$ with $-|\\beta|$ and $\\mu_-$ with $\\mu_+$, so that we obtain\n\\begin{equation}\n\\label{meanu3}\n\\jump{0.15}{\\overline{U}_3}^+ = -\\frac{\\mu_+ + \\mu_-}{\\mu_ + \\mu_-} \\frac{1}{|\\beta|}\\overline{\\Sigma}_{32}^-, \\quad\n\\langle\\overline{U}_3\\rangle^+ = \\frac{\\mu_+ - \\mu_-}{2\\mu_+\\mu_-} \\frac{1}{|\\beta|}\\overline{\\Sigma}_{32}^- = \\frac{\\eta}{2} \\jump{0.15}{\\overline{U}_3}^+,\n\\end{equation} \nwhere\n$$\n\\eta = \\frac{\\mu_- - \\mu_+}{\\mu_- + \\mu_+}.\n$$\n\nThe formulae for the evaluation of $K_\\text{III}$ and $A_\\text{III}$ are \n\\begin{equation}\n\\label{k3}\n\\begin{array}{l}\n\\displaystyle K_\\text{III} = - \\sqrt{\\frac{2}{\\pi}} \\int_{-\\infty}^{0} \\left\\{ \\langle p_3 \\rangle(x_1) + \\frac{\\eta}{2} \\jump{0.15}{p_3}(x_1) \\right\\} (-x_1)^{-1\/2} dx_1, \\\\[3mm]\n\\displaystyle A_\\text{III} = -2 \\sqrt{\\frac{2}{\\pi}} \\int_{-\\infty}^{0} \n\\left\\{ \\frac{d\\langle p_3 \\rangle(x_1)}{dx_1} + \\frac{\\eta}{2} \\frac{d\\jump{0.15}{p_3}(x_1)}{dx_1} \\right\\} (-x_1)^{-1\/2} dx_1.\n\\end{array}\n\\end{equation}\n\nThe full representation of weight functions after Fourier inversion is, for the lower half-plane, $x_2 < 0$,\n$$\n\\begin{array}{l}\n\\displaystyle u_3(x_1,x_2) = -\\frac{\\mu_+}{2\\sqrt{\\pi}(\\mu_- + \\mu_+)}\\left\\{ (ix_1 - x_2)^{-1\/2} - i(-ix_1 - x_2)^{-1\/2} \\right\\}, \\\\[3mm]\n\\displaystyle \\sigma_{31}(x_1,x_2) = \\frac{\\mu_+\\mu_-}{4\\sqrt{\\pi}(\\mu_- + \\mu_+)}\\left\\{ i(ix_1 - x_2)^{-3\/2} - (-ix_1 - x_2)^{-3\/2} \\right\\}, \\\\[3mm]\n\\displaystyle \\sigma_{32}(x_1,x_2) = -\\frac{\\mu_+\\mu_-}{4\\sqrt{\\pi}(\\mu_- + \\mu_+)}\\left\\{ (ix_1 - x_2)^{-3\/2} - i(-ix_1 - x_2)^{-3\/2} \\right\\},\n\\end{array}\n$$\nand, for the upper half-plane, $x_2 > 0$,\n$$\n\\begin{array}{l}\n\\displaystyle u_3(x_1,x_2) = \\frac{\\mu_-}{2\\sqrt{\\pi}(\\mu_- + \\mu_+)}\\left\\{ (ix_1 + x_2)^{-1\/2} - i(-ix_1 + x_2)^{-1\/2} \\right\\}, \\\\[3mm]\n\\displaystyle \\sigma_{31}(x_1,x_2) = -\\frac{\\mu_+\\mu_-}{4\\sqrt{\\pi}(\\mu_- + \\mu_+)}\\left\\{ i(ix_1 + x_2)^{-3\/2} - (-ix_1 + x_2)^{-3\/2} \\right\\}, \\\\[3mm]\n\\displaystyle \\sigma_{32}(x_1,x_2) = -\\frac{\\mu_+\\mu_-}{4\\sqrt{\\pi}(\\mu_- + \\mu_+)}\\left\\{ (ix_1 + x_2)^{-3\/2} - i(-ix_1 + x_2)^{-3\/2} \\right\\}.\n\\end{array}\n$$\n\nWe compare now the result obtained for the Mode III with that of Mode I and II. In particular, we notice from \\eq{k3} that the skew-symmetric part of $K_\\text{III}$ \nis proportional to the material parameter $\\eta$, whereas the skew-symmetric part of the complex stress intensity factor $K$ is proportional to the material parameter \n$\\alpha$, see \\eq{SIF} and \\eq{skew}. A comparison of these two parameters is then useful to understand whether the skew-symmetric loading is more relevant for Mode III \n($\\alpha > \\eta$) or Mode I and II ($\\eta$ > $\\alpha$). This comparison is shown in Fig. \\ref{alpha}.\n\\begin{figure}[!htb]\n\\begin{center}\n\\vspace*{3mm}\n\\includegraphics[width=6cm]{fig08arx.eps}\n\\caption{\\footnotesize The material parameter $\\alpha$ as a function of the material parameter $\\eta$ for: $\\nu_+ = 0$, $\\nu_-=0.5$ (green); \n$\\nu_+ = 0.2$, $\\nu_-=0.3$ (orange); $\\nu_+ = \\nu_-=0.3$ (red); \n$\\nu_+ = 0.5$, $\\nu_-=0$ (blue).}\n\\label{alpha}\n\\end{center}\n\\end{figure}\n\n\\vspace{60mm}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn networking or telecommunications the search for the minimum-delay\npath (that is the shortest path between two points) is always on.\nThe cost on  each edge, that is the time taken for a signal to\ntravel between two  adjacent nodes of the network, is often a\nfunction of real time. Hence the shortest path between any two nodes\nchanges with time. Of course one can run a shortest path algorithm\nevery time a signal has to be sent, but usually some prior knowledge of\nthe network graph is given in advance, such as the structure of the\nnetwork graph and the cost functions on each edge (with time as a\nvariable).\n\nHow can one benefit from this extra information? One plausible way\nis to preprocess the initial information and store the\npreprocessed information. Every time the rest of the input is given,\nusing the preprocessed information, one can solve the optimization\nproblem  faster than solving the problem from scratch. Even if the\npreprocessing step is expensive one would benefit by saving precious\ntime whenever the optimal solution has to be computed. Also, if the\nsame preprocessed information is used multiple times then the total\namount of resources used will be less in the long run.\n\nSimilar phenomena can be observed in various other combinatorial\noptimization problems that arise in practice; that is, a part of the\ninput does not change with time and is known in advance. However, many\ntimes it is hard to make use of this extra information.\n\nIn this paper we consider only those problems where the whole input is\na weighted graph. We assume that the graph structure and some\nknowledge of how the weights on the edges are generated are known in\nadvance. We call this the {\\it function-weighted graph} -- it is a graph\nwhose edges are labeled with continuous real functions. When all the\nfunctions are univariate (and all have the same variable), the graph\nis called a {\\em parametric weighted graph}. In other words, the graph\nis $G=(V,E,W)$ where $W:E \\rightarrow \\mathcal{F}$ and $\\mathcal{F}$ is the\nspace of all real continuous functions with the variable $x$. If $G$\nis a parametric weighted graph, and $r \\in \\mathbb{R}$ is any real\nnumber, then $G(r)$ is the standard weighted graph where the weight of\nan edge $e$ is defined to be $(W(e))(r)$.  We say that $G(r)$ is an\n{\\em instantiation} of $G$, since the variable $x$ in each function is\ninstantiated by the value $r$.  Parametric weighted graphs are\ntherefore, a generic instance of infinitely many instances of weighted\ngraphs.\n\nThe idea is to use the generic instance $G$ to precompute some general\ngeneric information $I(G)$, such that for any given\ninstantiation $G(r)$, we will be able to\nuse the precomputed information $I(G)$ in order to speed up the time\nto solve the given problem on $G(r)$, faster than just solving the\nproblem on $G(r)$ from scratch.  Let us make this notion more\nprecise.\n\nA {\\em parametric weighted graph algorithm} (or, for brevity, a {\\em\nparametric algorithm}) consists of two phases. A {\\em preprocessing\nphase} whose input is a parametric weighted graph $G$, and whose\noutput is a data structure (the advice) that is later used by the {\\em\ninstantiation phase}, where a specific value $r$ for the variable is\ngiven. The instantiation phase outputs the solution to the (standard)\nweighted graph problem on the weighted graph $G(r)$. Naturally, the\ngoal is to have the running time of the instantiation phase\nsignificantly smaller than the running time of any algorithm that\nsolves the weighted graph problem from scratch, by taking advantage of\nthe advice constructed in the preprocessing phase.  Parametric\nalgorithms are therefore evaluated by a pair of running times, the\n{\\em preprocessing time} and the {\\em instantiation time}.\n\n\nIn this paper we show that parametric algorithms are beneficial for\none of the most natural combinatorial optimization problems: the {\\em\nshortest path} problem in directed graphs. Recall that given a\ndirected real-weighted graph $G$, and two vertices $u,v$ of $G$, the\ndistance from $u$ to $v$, denoted by $\\delta(u,v)$, is the length of a\nshortest path from $u$ to $v$. The {\\em single pair} shortest path\nproblem seeks to compute $\\delta(u,v)$ and construct a shortest path\nfrom $u$ to $v$. Likewise, the {\\em single source} shortest path\nproblem seeks to compute the  distances and shortest paths from a\ngiven vertex to all other vertices, and the {\\em all pairs} version\nseeks to compute distances and shortest paths between all ordered\npairs of vertices.  In some of our algorithms we forgo the calculation\nof the path itself to achieve a shorter instantiation time. In all\nthose cases the algorithms can be easily modified to also output a\nshortest path, in which case their instantiation time is the sum of\nthe time it takes to calculate the distance and a time linear in the\nsize of the path to be output.\n\nOur first algorithm is a parametric algorithm for single source\nshortest path, in the case where the weights are {\\em linear}\nfunctions. That is, each edge $e$ is labeled with a function\n$a_ex+b_e$ where $a_e$ and $b_e$ are reals. Such linear\nparametrization has practical importance.  Indeed, in many problems\nthe cost of an edge is composed from some constant term plus a term\nwhich is a factor of some commodity, whose cost varies (e.g. bank\ncommissions, taxi fares, vehicle maintenance costs, and so on).  Our\nparametric algorithm has preprocessing time $\\tilde{O}(n^4)$ and\ninstantiation time $O(m+n\\log n)$ (throughout this paper $n$ and $m$\ndenote the number of vertices and edges of a graph, respectively). We\nnote that the fastest algorithm for the single source shortest path in\nreal weighted directed graphs requires $O(nm)$ time; the Bellman-Ford\nalgorithm \\cite{Be-1958}.  The idea of our preprocessing stage is to\nprecompute some other linear functions, on the {\\em vertices}, so that\nfor every instantiation $r$, one can quickly determine whether $G(r)$\nhas a negative cycle and otherwise use these functions to quickly\nproduce a reweighing of the graph so as to obtain only nonnegative\nweights similar to the weights obtained by Johnson's algorithm\n\\cite{Jo-1977}. In other words, we {\\em avoid} the need to run the\nBellman-Ford algorithm in the instantiation phase. The\n$\\tilde{O}(n^4)$ time in the preprocessing phase comes from the use of Megiddo's\\cite{M-79} technique that we need in order to compute the linear vertex functions.\n\n\\begin{theorem}\\label{thm:neg} There exists a parametric algorithm for\nsingle source shortest path in graphs weighted by linear functions,\nwhose preprocessing time is $\\tilde{O}(n^4)$ and whose instantiation\ntime is $O(m + n\\log n)$.\n\\end{theorem}\n\nOur next algorithm applies to a more general setting where the weights\nare polynomials of degree at most $d$. Furthermore, in this case our\ngoal is to have the instantiation phase answering distance queries\nbetween any two vertices in {\\em sublinear} time. Notice first that if\nwe allow exponential preprocessing time, this goal can be easily\nachieved. This is not hard to see since the overall\npossible number of shortest paths (when $x$ varies over the reals) is\n$O(n!)$, or from Fredman's decision tree for shortest paths whose\nheight is $O(n^{2.5})$ \\cite{Fr-1976}. But can we settle for {\\em\nsub-exponential} preprocessing time and still be able to have\nsublinear instantiation time? Our next result achieves this goal.\n\n\\begin{theorem}\\label{thm:gen} There exists a parametric algorithm for\nthe single pair shortest path problem in graphs weighted by degree $d$\npolynomials, whose preprocessing time is $O(n^{(O(1) + \\log f(d))\\log\nn})$ and instantiation time $O(\\log^2 n)$, where $f(d)$ is the time\nrequired to compute the intersection points of two degree $d$\npolynomials. The size of the advice that the preprocessing algorithm\nproduces is $O(n^{(O(1) + \\log d)\\log n})$.\n\\end{theorem}\n\nThe above result falls in the subject of sensitivity analysis where\none is interested in studying the effect on the optimal solution as the\nvalue of the parameter changes. We give a linear-time (linear in the\noutput size) algorithm that computes the breaking points.\n\nThe practical and theoretical importance of shortest path problems\nlead several researchers to consider fast algorithms that settle for\nan approximate shortest path. For the general case (of real weighted\ndigraphs) most of the algorithms guarantee an {\\em $\\alpha$-stretch}\nfactor. Namely, they compute a path whose length is at most $\\alpha\n\\delta(u,v)$. We mention here the $(1+\\epsilon)$-stretch algorithm\nof Zwick for the all-pairs shortest path problem, that runs in\n$\\tilde{O}(n^\\omega)$ time when the weights are non-negative reals\n\\cite{Zw-2002}. Here $\\omega < 2.376$ is the matrix multiplication\nexponent \\cite{CoWi-1990}.\n\nHere we consider probabilistic additive-approximation algorithms, or\n{\\em surplus} algorithms, that work for linear weights which may have\npositive and negative values (as long as there is no negative weight\ncycle).  We say that a shortest path algorithm has an\n$\\epsilon$-surplus if it computes paths whose lengths are at most\n$\\delta(u,v)+\\epsilon$.  We are unaware of any truly subcubic algorithm\nthat guarantees an $\\epsilon$-surplus approximation, and which\noutperforms the fastest general all-pairs shortest path algorithm\n\\cite{Ch-2007}.\n\nIn the linear-parametric setting, it is easy to obtain\n$\\epsilon$-surplus parametric algorithms whose preprocessing time is\n$O(n^4)$ time, and whose instantiation time, for any ordered pair of\nqueried vertices $u,v$ is constant. It is assumed instantiations are\ntaken from some interval $I$ whose length is independent of $n$.\nIndeed, we can partition $I$ into $O(n)$ subintervals $I_1,I_2,\\ldots$\nof size $O(1\/n)$ each, and solve, in cubic time (say, using\n\\cite{Fl-1962}), the exact all-pairs solution for any instantiation\n$r$ that is an endpoint of two consecutive intervals.  Then, given any\n$r \\in I_j=(a_j,b_j)$, we simply look at the solution for $b_j$ and\nnotice that we are (additively) off from the right answer only by\n$O(1)$. Standard scaling arguments can make the surplus smaller than\n$\\epsilon$.  But do we really need to spend $O(n^4)$ time for\npreprocessing? In other words, can we invest (significantly) less than\n$O(n^4)$ time and still be able to answer instantiated distance\nqueries in $O(1)$ time? The following result gives a positive answer\nto this question.\n\n\\begin{theorem}\\label{thm:approx} Let $\\epsilon > 0$, let\n$[\\alpha,\\beta]$ be any fixed interval and let $\\gamma$ be a fixed\nconstant. Suppose $G$ is a linear-parametric graph that has no\nnegative weight cycles in the interval $[\\alpha,\\beta]$, and for which\nevery edge weight $a_e+xb_e$ satisfies $|a_e|\\leq\\gamma$. There is a\nparametric randomized algorithm for the $\\epsilon$-surplus shortest\npath problem, whose preprocessing time is $\\tilde{O}(n^{3.5})$ and\nwhose instantiation time is $O(1)$ for a single pair, and hence\n$O(n^2)$ for all pairs.\n\\end{theorem} We note that this algorithm works in the restricted\naddition-comparison model.  We also note that given an ordered pair\n$u,v$ and $r \\in [\\alpha,\\beta]$, the algorithm outputs, in $O(1)$\ntime, a weight of an actual path from $u$ to $v$ in $G(r)$, and points\nto a linked list representing that path.  Naturally, if one wants to\noutput the vertices of this path then the time for this is linear in\nthe length of the path.\n\nThe rest of this paper is organized as follows. The next subsection\nshortly surveys related research on parametric shortest path\nproblems. In the three sections following it we prove Theorems\n\\ref{thm:neg}, \\ref{thm:gen} and \\ref{thm:approx}.\nSection~\\ref{sec:conclusion} contains some concluding remarks and open\nproblems.\n\n\\subsection{Related research} Several researchers have considered parametric\nversions of combinatorial optimization problems. In particular\nfunction-weighted graphs (under different names) have been\nextensively studied in the subject of sensitivity analysis (see\n\\cite{vHKRW89}) where they study the effect on the optimal solution\nas the parameter value changes.\n\nMurty~\\cite{Mur80} showed that for parametric linear programming\nproblems the optimal solution can change exponentially many times\n(exponential in the number of variables). Subsequently, Carstensen\n\\cite{Ca-1983} has shown that there are constructions  for which\nthe number of shortest path changes while $x$ varies over the reals\nis $n^{\\Omega(\\log n)}$. In fact, in her example each linear\nfunction is of the form $a_e+xb_e$ and both $a_e$ and $b_e$ are\npositive, and $x$ varies in $[0,\\infty]$. Carstensen also proved\nthat this is tight. In other words, for any linear-parametric graph\nthe number of changes in the shortest paths is $n^{O(\\log n)}$. A\nsimpler proof was obtained by Nikolova et al. \\cite{Ni-Mi-2006},\nthat also supply an $n^{O(\\log n)}$ time algorithm to compute the\npath breakpoints. Their method, however, does not apply to the case\nwhere the functions are not linear, such as in the case of degree\n$d$ polynomials. Gusfield~\\cite{gus} also gave a proof for the upper bound\nof the number of breakpoints in the linear function version of the parametric shortest\npath problem, in addition to studying a number of other parametric\nproblems.\n\n\nKarp and Orlin \\cite{KaOr-1981}, and, later, Young, Tarjan, and\nOrlin~\\cite{Yo-Or-1991} considered a special case of the\nlinear-parametric shortest path problem.  In their case, each edge\nweight $e$ is either some fixed constant $b_e$ or is of the form\n$b_e-x$. It is not too difficult to prove that for any given vertex\n$v$, when $x$ varies from $-\\infty$ to the largest $x_0$ for which\n$G(x_0)$ has no negative weight cycle (possibly $x_0=\\infty$), then\nthere are at most $O(n^2)$ distinct shortest path trees from $v$ to\nall other vertices. Namely, for each $r \\in [-\\infty,x_0]$ one of\nthe trees in this family is a solution for single-source shortest\npath in $G(r)$. The results in \\cite{KaOr-1981,Yo-Or-1991} cleverly\nand compactly compute all these trees, and the latter does it in\n$O(nm+n^2\\log n)$ time.\n\n\n\\section{Proof of Theorem~\\ref{thm:neg}}\n\nThe proof of Theorem~\\ref{thm:neg} follows from the following two\nlemmas.\n\n\\begin{lemma}\\label{lem:negcyl} Given a linear-weighted graph\n$G=(V,E,W)$, there exist $\\alpha, \\beta \\in \\mathbb{R}\\cup \\{-\\infty\\}\n\\cup \\{+\\infty\\}$ such that $G(r)$ has no negative cycles if and only\nif $\\alpha \\leq r \\leq \\beta$.  Moreover $\\alpha$ and $\\beta$ can be\nfound in $\\tilde{O}(n^4)$ time.\n\\end{lemma}\n\n\\begin{lemma}\\label{lem:reweight} Let $G=(V,E,W)$ be a linear-weighted\ngraph. Also let $\\alpha, \\beta \\in \\mathbb{R}\\cup \\{-\\infty\\} \\cup\n\\{+\\infty\\}$ be such that at least one of them is finite and for all\n$\\alpha \\geq r \\geq \\beta$ the graph $G(r)$ has no negative cycle.\nThen for every vertex $v \\in V$ there exists a linear function\n$g^{[\\alpha,\\beta]}_v$ such that if the new weight function $W'$ is\ngiven by\n$$\nW'\\left((u,v)\\right) = W\\left((u,v)\\right) + g^{[\\alpha, \\beta]}_u -\ng^{[\\alpha, \\beta]}_v\n$$\nthen the new linear-weighted graph $G'=(V,E,W')$ has the property that\nfor any real $\\alpha \\leq r \\leq \\beta$ all the edges in $G'(r)$ are\nnon-negative.  Moreover the functions $g^{[\\alpha, \\beta]}_v$ for all\n$v \\in V$ can be found in $O(mn)$ time.\n\\end{lemma}\n\nSo given a linear-weighted graph $G$, we first use Lemma\n\\ref{lem:negcyl} to compute $\\alpha$ and $\\beta$.  If at least one of\n$\\alpha$ and $\\beta$ is finite then using Lemma~\\ref{lem:reweight} we\ncompute the $n$ linear functions $g^{[\\alpha,\\beta]}_v$, one for each\n$v \\in V$.  If $\\alpha = -\\infty$ and $\\beta = +\\infty$, then using\nLemma~\\ref{lem:reweight} we compute the $2n$ linear functions\n$g^{[\\alpha, 0]}_v$ and $g^{[0,\\beta]}_v$. These linear functions will\nbe the advice that the preprocessing algorithm produces.  The above\nlemmas guarantee us that the advice can be computed in time\n$\\tilde{O}(n^4)$, that is the preprocessing time is $\\tilde{O}(n^4)$.\n\nNow when computing the single source shortest path problem from vertex\n$v$ for the graph $G(r)$ our algorithm proceeds as follows:\n\\begin{enumerate}\n\\item If $r<\\alpha$ or $r>\\beta$ output ``$-\\infty$'' as there exists\na negative cycle (such instances are considered invalid).\n\\item If $\\alpha\\leq r\\leq \\beta$ and at least one of $\\alpha$ or\n$\\beta$ is finite then compute $g_u(r)$ for all $u \\in V$. Use these\nto re-weight the edges in the graph as in Johnson's algorithm\n\\cite{Jo-1977}.  If $\\alpha = -\\infty$ and $\\beta = +\\infty$ then if\n$r\\leq 0$ compute $g^{[\\alpha,0]}_u(r)$ for all $u \\in V$ and if\n$r\\geq 0$ compute $g^{[0,\\beta]}_u(r)$ for all $u \\in V$.  Notice that\nafter the reweighing we have an instance of $G'(r)$.\n\\item Use Dijkstra's algorithm \\cite{Di-1959} to solve the single\nsource shortest path problem in $G'(r)$. Dijkstra's algorithm applies\nsince $G'(r)$ has no negative weight edges.  The shortest paths tree\nreturned by Dijkstra's algorithms applied to $G'(r)$ is also the\nshortest paths tree in $G(r)$. As in Johnson's algorithm, we use the\nresults $d'(v,u)$ of $G'(r)$ to deduce $d(v,u)$ in $G(r)$ since, by\nLemma \\ref{lem:reweight} $d(v,u)=d'(v,u)-g_v(r)+g_u(r)$.\n\\end{enumerate} The running time of the instantiation phase is\ndominated by the running time of Dijkstra's algorithm which is $O(m +\nn\\log n)$ \\cite{FrTa-1987}.\n\n\\subsection{Proof of Lemma~\\ref{lem:negcyl}}\n\nSince the weight on the edges of the graph $G$ are linear functions,\nwe have that the weight of any directed cycle in the graph is also a\nlinear function.  Let $C_1, C_2, \\ldots, C_T$ be the set of all\ndirected cycles in the graph.  The linear weight function of a cycle\n$C_i$ will be denoted by $\\mbox{wt}(C_i)$.  If $\\mbox{wt}(C_i)$ is not the\nconstant function, then let $\\gamma_i$ be the real number for which\nthe linear equation $\\mbox{wt}(C_i)$ evaluates to $0$.\n\n\\noindent Let $\\alpha$ and $\\beta$ be defined as follows:\n$$\n\\alpha = \\max_i \\left\\{ \\gamma_i \\mid \\mbox{ $\\mbox{wt}(C_i)$ has a positive\nslope}\\right\\}.\n$$\n$$\n\\beta = \\min_i \\left\\{ \\gamma_i \\mid \\mbox{ $\\mbox{wt}(C_i)$ has a negative\nslope}\\right\\}.\n$$\nNote that if $\\mbox{wt}(C_i)$ has a positive slope then\n$\\gamma_i = \\min_x \\left\\{ \\mbox{wt}(C_i)(x) \\geq 0\\right\\}.$\nThus for all $x \\geq \\gamma_i$ the value of $\\mbox{wt}(C_i)$ evaluated at\n$x$ is non-negative. So by definition for all $x \\geq \\alpha$ the\nvalue of the $\\mbox{wt}(C_i)$ is non-negative if the slope of $\\mbox{wt}(C_i)$ is\npositive, and for any $x<\\alpha$ there exists a cycle $C_i$ such that\n$\\mbox{wt}(C_i)$ has positive slope and $\\mbox{wt}(C_i)(x)$ is negative.\nSimilarly, for all $x \\leq \\beta$ the value of the $\\mbox{wt}(C_i)$ is\nnon-negative if the slope of $\\mbox{wt}(C_i)$ is negative and for any\n$x>\\beta$ there exists a cycle $C_i$ such that $\\mbox{wt}(C_i)$ has negative\nslope and $\\mbox{wt}(C_i)(x)$ is negative.\n\nThis proves the existence of $\\alpha$ and $\\beta$.  There are,\nhowever, two bad cases that we wish to exclude.  Notice that if\n$\\alpha > \\beta$ this means that for any evaluation at $x$, the\nresulting graph has a negative weight cycle.  The same holds if there\nis some cycle for which $\\mbox{wt}(C_i)$ is constant and negative.  Let us\nnow show how $\\alpha$ and $\\beta$ can be efficiently computed whenever\nthese bad cases do not hold.  Indeed, $\\alpha$ is the solution to the\nfollowing Linear Program (LP), which has a feasible solution if and\nonly if the bad cases do not hold.\n\\begin{center} \\framebox{\n\\parbox[s]{3.0in}{\\\n\n{\\bf Minimize $x$} under the constraints\n\n\\bigskip {\\bf $\\forall i$,  $\\mbox{wt}(C_i)(x) \\geq 0$}.\n\n\\bigskip }}\n\\end{center} This is an LP on one variable, but the number of\nconstraints can be exponential.\nUsing Megiddo's\\cite{M-79}\ntechnique for finding the minimum ratio cycles we can solve the\nlinear-program in $O(n^4\\log n)$ steps.\n\n\\subsection{Proof of Lemma~\\ref{lem:reweight}}\n\nLet $\\alpha$ and $\\beta$ be the two numbers such that for all\n$\\alpha\\leq r \\leq \\beta$ the graph $G(r)$ has no negative cycles and\nat least one of $\\alpha$ and $\\beta$ is finite.\n\nFirst let us consider the case when both $\\alpha$ and $\\beta$ are\nfinite. Recall that, given any number $r$, Johnson's algorithm\nassociates a weight function $h^{r}:V\\rightarrow \\mathbb{R}$ such\nthat, for any edge $(u,v)\\in E$,\n$$\nW_{(u,v)}(r) + h^r(u) - h^r(v) \\geq 0.\n$$\n(Johnson's algorithm computes this weight function by running the\nBellman-Ford algorithm over $G(r)$).  Define the weight function\n$g^{[\\alpha,\\beta]}_v$ as\n$$\ng^{[\\alpha, \\beta]}_v(x) = \\left(\\frac{h^{\\beta}(v) -\nh^{\\alpha}(v)}{\\beta - \\alpha}\\right)x + h^{\\alpha}(v) -\n\\left(\\frac{h^{\\beta}(v) - h^{\\alpha}(v)}{\\beta -\n\\alpha}\\right)\\alpha~.\n$$\nThis is actually the equation of the line joining $(\\alpha,\nh^{\\alpha}(v))$ and $(\\beta, h^{\\beta}(v))$ in $\\mathbb{R}^2$.\n\n\\noindent Now we need to prove that for every $\\alpha \\leq r\\leq\n\\beta$ and for every $(u,v)\\in V$,\n$$\nW_{(u,v)}(r) + g^{[\\alpha, \\beta]}_{u}(r) - g^{[\\alpha,\\beta]}_{v}(r)\n\\geq 0~.\n$$\nSince $\\alpha \\leq r \\leq \\beta$, one can write $r = (1-\\delta)\\alpha\n+ \\delta\\beta$ where $1 \\geq \\delta \\geq 0$. Then for all $v\\in V$,\n$$\ng^{[\\alpha, \\beta]}_v(r) = (1 - \\delta) h^{\\alpha}(v) + \\delta\nh^{\\beta}(v)~.\n$$\nSince $W_{(u,v)}(r)$ is a linear function we can write\n$$\nW_{(u,v)}(r) = (1-\\delta)W_{(u,v)}(\\alpha) + \\delta W_{(u,v)}(\\beta)~.\n$$\nSo after re-weighting the weight of the edge $(u,v)$ is\n$$\n(1-\\delta)W_{(u,v)}(\\alpha) + \\delta W_{(u,v)}(\\beta) +\n(1-\\delta)h^{\\alpha}(u) + \\delta h^{\\beta}(u) - (1 - \\delta)\nh^{\\alpha}(v) - \\delta h^{\\beta}(v)~.\n$$\nNow this is non-negative as by the definition of $h^{\\beta}$ and\n$h^{\\alpha}$ we know that both $W_{(u,v)}(\\beta) + h^{\\beta}(u) -\nh^{\\beta}(v)$ and $W_{(u,v)}(\\alpha) + h^{\\alpha}(u) - h^{\\alpha}(v)$\nare non-negative.\n\nWe now consider the case when one of $\\alpha$ or $\\beta$ is not\nfinite.  We will prove it for the case where $\\beta = +\\infty$. The\ncase $\\alpha = -\\infty$ follows similarly.  Consider the simple\nweighted graph $G_{\\infty}=(V,E,W_{\\infty})$ where the weight function\n$W_{\\infty}$ is defined as: if the weight of the edge $e$ is $W(e) =\na_ex + b_e$ then $W_{\\infty}(e) = a_e$.\n\nWe run the Johnson's algorithm on the graph $G_{\\infty}$. Let\n$h^{\\infty}(v)$ denote the weight that Johnson's algorithm associates\nwith the vertex $v$. Then define the weight function\n$g^{[\\alpha,\\infty]}_v$ as\n$$\ng^{[\\alpha,\\infty]}_v(x) = h^{\\alpha}(v) + (x-\\alpha)h^{\\infty}(v)~.\n$$\nWe need to prove that for every $\\alpha \\leq r$ and for every\n$(u,v)\\in V$,\n$$\nW_{(u,v)}(r) + g^{[\\alpha,\\infty]}_{u}(r) - g^{[\\alpha,\n\\infty]}_{v}(r) = W_{(u,v)}(r) + h^{\\alpha}(u) + (r-\\alpha)\nh^{\\infty}(u) - h^{\\alpha}(v) - (r-\\alpha) h^{\\infty}(v) \\geq 0~.\n$$\nLet $r = \\alpha + \\delta$ where $\\delta \\geq 0$. By the linearity of\n$W$ we can write $W_{(u,v)}(r) = W_{(u,v)}(\\alpha) + \\delta\na_{(u,v)}$, where $W_{(u,v)}(r) = a_{(u,v)}r + b_{(u,v)}$. So the\nabove inequality can be restated as\n$$\nW_{(u,v)}(\\alpha) + \\delta a_{(u,v)} + h^{\\alpha}(u) + \\delta\nh^{\\infty}(u) - h^{\\alpha}(v) - \\delta h^{\\infty}(v) \\geq 0~.\n$$\nThis now follows from the fact that both $W_{(u,v)}(\\alpha) +\nh^{\\alpha}(u) - h^{\\alpha}(v)$ and $a_{(u,v)} + h^{\\infty}(u) -\nh^{\\infty}(v)$ are non-negative.\n\nSince the running time of the reweighing part of Johnson's algorithm\ntakes $O(mn)$ time, the overall running time of computing the\nfunctions $g^{[\\alpha,\\beta]}_v$ is $O(mn)$, as claimed.\n\n\\section{Proof of Theorem~\\ref{thm:gen}}\n\nIn this section we construct a parametric algorithm that computes the\ndistance $\\delta(u,v)$ between a given pair of vertices. If one is\ninterested in the actual path realizing this distance, then it can be\nfound with some extra book-keeping that we omit in the proof.\n\nThe processing algorithm will output the following advice: for any\npair $(u,v) \\in V\\times V$ the advice consists of a set of $t+2$ increasing\nreal numbers $-\\infty = b_0 < b_1 < \\dots < b_{t} < b_{t+1} = \\infty$\nand an ordered set of degree-$d$ polynomials $p_0, p_1, \\dots, p_t$,\nsuch that for all $b_i \\leq r \\leq b_{i+1}$ the weight of a shortest\npath in $G(r)$ from $u$ to $v$ is $p_i(r)$.  Note that each $p_i$\ncorresponds to the weight of a path from $u$ to $v$. Thus if we are\ninterested in computing the exact path then we need to keep track of\nthe path corresponding to each $p_i$.\n\nGiven $r$, the instantiation algorithm has to find the $i$ such that\n$b_i \\leq r \\leq b_{i+1}$ and then output $p_i(r)$. So the output\nalgorithm runs in time $O(\\log t)$. To prove our result we need to\nshow that for any $(u,v) \\in V\\times V$ we can find the advice in time\n$O(f(d) n)^{\\log n}$. In particular this will prove that $t =\nO(dn)^{\\log n}$ and hence the result will follow.\n\n\\begin{definition} A \\textit{minBase} is a sequence of increasing real\nnumbers $-\\infty = b_0 < b_1 < \\dots < b_t < b_{t+1} = \\infty$ and an\nordered set of degree-$d$ polynomials $p_0, p_1, \\dots, p_t$, such\nthat for all $b_i \\leq r \\leq b_{i+1}$ and all $j \\neq i$, $p_i(r)\n\\leq p_j(r)$.\n\\end{definition} We call the sequence of real numbers the {\\em\nbreaks}. We call each interval $[b_i, b_{i+1}]$ the $i$-th interval of\nthe minBase and the polynomial $p_i$ the $i$-th polynomial. The {\\em\nsize} of the minBase is $t$.\n\nThe final advice that the preprocessing algorithm produces is a\nminBase for every pair $(u,v)\\in V\\times V$ where the $i$-th\npolynomial has the property that $p_i(r)$ is the distance from $u$ to\n$v$ in $G(r)$ for each $b_i \\le r \\le b_{i+1}$.\n\n\\begin{definition} A $minBase^{\\ell}(u,v)$ is a minBase corresponding\nto the ordered pair $u,v$, where the $i$-th polynomial $p_i$ has the\nproperty that for $r \\in [b_i,b_{i+1}]$, $p_i(r)$ is the length of a\nshortest path from $u$ to $v$ in $G(r)$, that is taken among all paths\nthat use at most $2^{\\ell}$ edges.\n\n\\noindent A $minBase^{\\ell}(u,w,v)$ is a minBase corresponding to the\nordered triple $(u,w, v)$ where the $i$-th polynomial $p_i$ has the\nproperty that for each $r \\in [b_i,b_{i+1}]$, $p_i(r)$ is the sum of\nthe lengths of a shortest path from $u$ to $w$ in $G(r)$, among all\npaths that use at most $2^{\\ell}$ edges, and a shortest path from $w$\nto $v$ in $G(r)$, among all paths that use at most $2^{\\ell}$ edges.\n\\end{definition} Note that in both of the above definitions some of\nthe polynomials can be $+\\infty$ or $-\\infty$.\n\n\\begin{definition} If $B_1$ and $B_2$ are two minBases (not\nnecessarily of the same size), with polynomials $p^1_i$ and $p^2_j$,\nwe say that another minBase with breaks $b'_k$ and polynomials $p'_k$\nis $\\min(B_1 + B_2)$ if the following holds.\n\\begin{enumerate}\n\\item For all $k$ there exist $i,j$ such that $p'_k = p^1_i +p^2_j$,\nand\n\\item For $b'_k \\leq r \\leq b'_{k+1}$ and for all $i,j$ we have\n$p'_k(r) \\le p^1_i(r) + p^2_j(r)$.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{definition} If $B_1,B_2,\\ldots,B_s$ are $s$ minBases (not\nnecessarily of the same size), with polynomials $p^1_{i_1}, p^2_{i_2},\n\\ldots, p^s_{i_s}$, another minBase with breaks $b'_k$ and polynomials\n$p'_k$ is $\\min\\{B_1,B_2,\\ldots,B_s\\}$ if the following holds.\n\\begin{enumerate}\n\\item For all $k$ there exist $q$ such that $p'_k = p^q_{i_q}$, and\n\\item For $b'_k \\leq r \\leq b'_{k+1}$ and for all $1 \\le q \\le s$ and\nall $i_q$, we have $p'_k(r) \\le p^q_{i_q}(r)$.\n\\end{enumerate}\n\\end{definition}\n\nNote that using the above definition we can write the following two\nequations:\n\\begin{equation}\\label{eq:minuwv} minBase^{\\ell+1}(u,v) = \\min_{w\\in\nV}\\left\\{minBase^{\\ell}(u,w,v)\\right\\}~.\n\\end{equation}\n\n\\begin{equation}\\label{eq:minuwwv} minBase^{\\ell}(u,w,v) =\nmin\\left(minBase^{\\ell}(u,w) + minBase^{\\ell}(w,v)\\right)~.\n\\end{equation}\n\nThe following claim will prove the result. The proof of the claim is\nomitted due to lack of space.\n\n\\begin{claim}\\label{cl:sizesminBase} If $B_1$ and $B_2$ are two\nminBases of sizes $t_1$ and $t_2$ respectively, then\n\\begin{enumerate}\n\\item[(a)] $\\min(B_1+B_2)$ can be computed from $B_1$ and $B_2$ in\ntime $O(t_1 + t_2)$.\n\\item[(b)] $\\min\\{B_1, B_2\\}$ can be computed from $B_1$ and $B_2$ in\ntime $O(f(d)(t_1 + t_2))$, where $f(d)$ is the time required to\ncompute the intersection points of two degree-$d$ polynomials. The\nsize of $\\min\\{B_1, B_2\\}$ is $O(d(t_1 + t_2))$.\n\\end{enumerate}\n\\end{claim} In order to compute $\\min\\{B_1,\\ldots,B_s\\}$ one\nrecursively computes $X=\\min\\{B_1,\\ldots,B_{s\/2}\\}$ and\n$Y=\\min\\{B_{s\/2+1},\\ldots,B_s\\}$ and then takes $\\min\\{X,Y\\}$.\n\nIf there are no negative cycles, then the advice that the\ninstantiation algorithm needs from the preprocessing algorithm\nconsists of $minBase^{\\lceil\\log n\\rceil}(u,v)$. To deal with negative\ncycles, both $minBase^{\\lceil\\log n\\rceil}(u,v)$ and\n$minBase^{\\lceil\\log n\\rceil+1}(u,v)$ are produced, and the\ninstantiation algorithm compares them. if they are not equal, then the\ncorrect output is $-\\infty$.\n\nAlso note that $minBase^{0}(u,v)$ is the trivial minBase where the\nbreaks are $-\\infty$ and $+\\infty$ and the polynomial is weight\n$W((u,v))$ associated to the edge $(u,v)$ if $(u,v) \\in E$ and\n$+\\infty$ otherwise.\n\nIf the size of $minBase^{\\ell}(u,v)$ is $s_{\\ell}$, then by\n(\\ref{eq:minuwv}), (\\ref{eq:minuwwv}), and by\nClaim~\\ref{cl:sizesminBase} the time to compute\n$minBase^{\\ell+1}(u,v)$ is $O(f(d))^{\\log n}s_{\\ell}$ and the size of\n$minBase^{\\ell+1}(u,v)$ is $O(d)^{\\log n}s_{\\ell}$. Thus one can\ncompute the advice for $u$ and $v$ in time\n$$\n(O(f(d))^{\\log n})^{\\log n} = O(n^{(O(1) + \\log f(d))\\log n})~,\n$$\nand the length of the advice string is $O(n^{(O(1) + \\log d)\\log n})$.\n\n\n\n\\section{Proof of Theorem \\ref{thm:approx}}\n\nGiven the linear-weighted graph $G=(V,E,W)$, our preprocessing phase\nbegins by verifying that for all $r \\in [\\alpha,\\beta]$, $G(r)$ has no\nnegative weight cycles. From the proof of Lemma \\ref{lem:reweight} we\nknow that this holds if and only if both $G(\\alpha)$ and $G(\\beta)$\nhave no negative weight cycles. This, in turn, can be verified in\n$O(mn)$ time using the Bellman-Ford algorithm.  We may now assume that\n$G(r)$ has no negative cycles for any $r \\in [\\alpha,\\beta]$.\nMoreover, since our preprocessing algorithm will solve a large set of\nshortest path problems, each of them on a specific instantiation of\n$G$, we will first compute the reweighing functions\n$g^{[\\alpha,\\beta]}_v$ of Lemma \\ref{lem:reweight} which will enable\nus to apply, in some cases, algorithms that assume nonnegative edge\nweights.  Recall that by Lemma \\ref{lem:reweight}, the functions\n$g^{[\\alpha,\\beta]}_v$ for all $v \\in V$ are computed in $O(mn)$ time.\n\nThe advice constructed by the preprocessing phase is composed of two\ndistinct parts, which we respectively call the {\\em crude-short}\nadvice and the {\\em refined-long} advice. We now describe each of\nthem.\n\nFor each edge $e \\in E$, the weight is a linear function\n$w_e=a_e+xb_e$.  Set $K=8(\\beta-\\alpha)\\max_e|a_e|$. Let $N_0 = \\lceil\nK\\sqrt{n}\\ln n\/\\epsilon \\rceil$ and let $N_1= \\lceil Kn\/\\epsilon\n\\rceil$.  We define $N_0+1$ and $N_1+1$ points in\n$[\\alpha,\\beta]$ and solve certain variants of shortest path problems\ninstantiated in these points.\n\nConsider first the case of splitting $[\\alpha,\\beta]$ into $N_0$\nintervals. Let $\\rho_0 = (\\beta-\\alpha)\/N_0$ and consider the points\n$\\alpha +i\\rho_0$ for $i=0,\\ldots,N_0$. The crude-short part of the\npreprocessing algorithm solves $N_0+1$ {\\em limited} all-pairs\nshortest path problems in $G(\\alpha +i\\rho_0)$ for $i=0,\\ldots,N_0$.\nSet $t=4\\sqrt{n}\\ln n$, and let $d_i(u,v)$ denote the length of a\nshortest path from $u$ to $v$ in $G(\\alpha +i\\rho_0)$ that is chosen\namong all paths containing at most $t$ vertices (possibly\n$d_i(u,v)=\\infty$ if no such path exists).  Notice that $d_i(u,v)$ is\nnot necessarily the distance from $u$ to $v$ in $G(\\alpha +i\\rho_0)$,\nsince the latter may require more than $t$ vertices.  It is\nstraightforward to compute shortest paths limited to at most $k$\nvertices (for any $1 \\le k \\le n$) in a real-weighted directed graph\nwith $n$ vertices in time $O(n^3 \\log k)$ time, by the repeated\nsquaring technique. In fact, they can be computed in $O(n^3)$ time\n(saving the $\\log k$ factor) using the method from \\cite{AHU-1974},\npp. 204--206.  This algorithm also constructs the predecessor data\nstructure that represents the actual paths.  It follows that for each\nordered pair of vertices $u,v$ and for each $i=0,\\ldots,N_0$, we can\ncompute $d_i(u,v)$ and a path $p_i(u,v)$ yielding $d_i(u,v)$ in\n$G(\\alpha +i\\rho_0)$ in $O(n^3|N_0|)$ time which is\n$O(n^{3.5} \\ln n)~.$\nWe also maintain, at no additional cost, linear functions $f_i(u,v)$\nwhich sum the linear functions of the edges of $p_i(u,v)$. Note also\nthat if $d_i(u,v)=\\infty$ then $p_i(u,v)$ and $f_i(u,v)$ are\nundefined.\n\nConsider next the case of splitting $[\\alpha,\\beta]$ into $N_1$\nintervals. Let $\\rho_1 = (\\beta-\\alpha)\/N_1$ and consider the points\n$\\alpha +i\\rho_1$ for $i=0,\\ldots,N_1$. However, unlike the\ncrude-short part, the refined-long part of the preprocessing algorithm\ncannot afford to solve an all-pairs shortest path algorithm for each\n$G(\\alpha +i\\rho_1)$, as the overall running time will be too large.\nInstead, we randomly select a set $H \\subset V$ of (at most)\n$\\sqrt{n}$ vertices. $H$ is constructed by performing $\\sqrt{n}$\nindependent trials, where in each trial, one vertex of $V$ is chosen\nto $H$ uniformly at random (notice that since the same vertex can be\nselected to $H$ more than once $|H| \\le \\sqrt{n}$).  For each $h \\in\nH$ and for each $i=0,\\ldots,N_1$, we solve the single source shortest\npath problem in $G(\\alpha +i\\rho_1)$ from $h$, and also (by reversing\nthe edges) solve the single-destination shortest path {\\em toward}\n$h$.  Notice that by using the reweighing functions\n$g^{[\\alpha,\\beta]}_v$ we can solve all of these single source\nproblems using Dijkstra's algorithm.  So, for all $h \\in H$ and\n$i=0,\\ldots,N_1$ the overall running time is\n$$\nO(|N_1||H|(m+n \\log n)) = O(n^{1.5}m + n^{2.5}\\log n) = O(n^{3.5})~.\n$$\nWe therefore obtain, for each $h \\in H$ and for each $i=0,\\ldots,N_1$,\na shortest path tree $T_i(h)$, together with distances $d^*_i(h,v)$\nfrom $h$ to each other vertex $v \\in V$, which is the distance from\n$h$ to $v$ in $G(\\alpha +i\\rho_1)$. We also maintain the functions\n$f^*_i(h,v)$ that sum the linear equations on the path in $T^*_i(h)$\nfrom $h$ to $v$.  Likewise, we obtain a ``reversed'' shortest path\ntree $S^*_i(h)$, together with distances $d^*_i(v,h)$ from each $v \\in\nV$ to $h$, which is the distance from $v$ to $h$ in $G(\\alpha\n+i\\rho_1)$. Similarly, we maintain the functions $f^*_i(v,h)$ that sum\nthe linear equations on the path in $S^*_i(h)$ from $v$ to $h$.\n\nFinally, for each ordered pair of vertices $u,v$ and for each\n$i=0,\\ldots,N_1$ we compute a vertex $h_{u,v,i} \\in H$ which attains\n$\n\\min_{h \\in H} d^*_i(u,h)+d^*_i(h,u)~.\n$\nNotice that the time to construct the $h_{u,v,i}$ for all ordered\npairs $u,v$ and for all $i=0,\\ldots,N_1$ is $O(n^{3.5})$.  This\nconcludes the description of the preprocessing algorithm. Its overall\nruntime is thus $O(n^{3.5} \\ln n)$.\n\nWe now describe the instantiation phase. Given $u,v \\in V$ and $r \\in\n[\\alpha,\\beta]$ we proceed as follows.  Let $i$ be the index for which\nthe number of the form $\\alpha+i\\rho_0$ is closest to $r$.  As we have\nthe advice $f_i(u,v)$, we let $w_0 = f_i(u,v)(r)$ (recall that\n$f_i(u,v)$ is a function).  Likewise, let $j$ be the index for which\nthe number of the form $\\alpha+j\\rho_1$ is closest to $r$.  As we have\nthe advice $h=h_{u,v,j}$, we let $w_1 = f^*_j(u,h)(r)+f^*_j(h,u)(r)$.\nFinally, our answer is $z=\\min \\{w_0,w_1\\}$. Clearly, the\ninstantiation time is $O(1)$.  Notice that if we also wish to output a\npath of weight $z$ in $G(r)$ we can easily do so by using either\n$p_i(u,v)$, in the case where $z=w_0$ or using $S^*_j(h)$ and\n$T^*_j(h)$ (we take the path from $u$ to $h$ in $S^*_j(h)$ and\nconcatenate it with the path from $h$ to $v$ in $T^*_j(h)$) in the\ncase where $z=w_1$.\n\nIt remains to show that, with very high probability, the result $z$\nthat we obtain from the instantiation phase is at most $\\epsilon$\nlarger than the distance from $u$ to $v$ in $G(r)$.  For this purpose,\nwe first need to prove that the random set $H$ possesses some\n``hitting set'' properties, with very high probability.\n\n\nFor every pair of vertices $u$ and $v$ and parameter $r$, let\n$p_{u,v,r}$ be a shortest path in $G(r)$ among all simple paths from\n$u$ to $v$ containing at least $t=4\\sqrt{n}\\ln n$ vertices (if $G$ is\nstrongly connected then such a path always exist, and otherwise we can\njust put $+\\infty$ for all $u,v$ pairs for which no such path exists).\nThe following simple lemma is used in an argument similar to one used\nin \\cite{Zw-2002}.\n\\begin{lemma}\n\\label{lem:prob} For fixed $u$, $v$ and $r$, with probability at least\n$1-o(1\/n^3)$ the path $p_{u,v,r}$ contains a vertex from $H$.\n\\end{lemma}\n\\begin{proof} Indeed, the path from $p_{u,v,r}$ by its definition has\nat least $4\\sqrt{n}\\ln n$ vertices.  The probability that all of the\n${\\sqrt n}$ independent selections to $H$ failed to choose a vertex\nfrom this path is therefore at most\n$$\n\\left(1-\\frac{4\\sqrt{n} \\ln n }{n} \\right)^{\\sqrt{n}} < e^{-4\\ln n} <\n\\frac{1}{n^4} = o(1\/n^3)~.\n$$\n\\end{proof}\n\nLet us return to the proof of Theorem \\ref{thm:approx}.  Suppose that\nthe distance from $u$ to $v$ in $G(r)$ is $\\delta$. We will prove that\nwith probability $1-o(1)$, $H$ is such that for every $u$, $v$ and $r$\nwe have $z \\le \\delta+\\epsilon$ (clearly $z \\ge \\delta$ as it is the\nprecise length of some path in $G(r)$ from $u$ to $v$).\nAssume first that there is a path $p$ of length\n$\\delta$ in $G(r)$ that uses less than $4\\sqrt{n}\\ln n$ edges.\nConsider the length of $p$ in $G(\\alpha+i\\rho_0)$. When going from $r$\nto $\\alpha+i\\rho_0$, each edge $e$ with weight $a_ex+b_e$ changed its\nlength by at most $|a_e|\\rho_0$. By the definition of $K$, this is at\nmost $\\rho_0 K\/(8(\\beta-\\alpha))$.  Thus, $p$ changed its weight by at\nmost\n$$\n(4\\sqrt{n} \\ln n) \\cdot \\rho_0 \\frac{K}{8(\\beta-\\alpha)} = (4\\sqrt{n}\n\\ln n)\\frac{K}{8N_0} < \\frac{\\epsilon}{2}.\n$$\nIt follows that the length of $p$ in $G(\\alpha+i\\rho_0)$ is less than\n$\\delta+\\epsilon\/2$.  But $p_i(u,v)$ is a shortest path from $u$ to\n$v$ in $G(\\alpha+i\\rho_0)$ of all the paths that contain at most $t$\nvertices. In particular, $d_i(u,v) \\le \\delta+\\epsilon\/2$.  Consider\nthe length of $p_i(u,v)$ in $G(r)$. The same argument shows that the\nlength of $p_i(u,v)$ in $G(r)$ changed by at most $\\epsilon\/2$. But\n$w_0=f_i(u,v)(r)$ is that weight, and hence $w_0 \\le\n\\delta+\\epsilon$. In particular, $z \\le \\delta+\\epsilon$.\n\nAssume next that every path of length $\\delta$ in $G(r)$ uses at least\n$4\\sqrt{n}\\ln n$ edges.  Let $p$ be one such path.  When going from\n$r$ to $r'=\\alpha+j\\rho_1$, each edge $e$ with weight $a_ex+b_e$\nchanged its length by at most $|a_e|\\rho_1$. By the definition of $K$,\nthis is at most $\\rho_1 K\/(8(\\beta-\\alpha))$.  Thus, $p$ changed its\nweight by at most\n$$\nn \\cdot \\rho_1 \\frac{K}{8(\\beta-\\alpha)} = n \\frac{K}{8N_1} <\n\\frac{\\epsilon}{8}.\n$$\nIn particular, the length of $p_{u,v,r'}$ is not more than the length\nof $p$ in $G(r')$, which, in turn, is at most $\\delta+\\epsilon\/8$.  By\nLemma \\ref{lem:prob}, with probability $1-o(1\/n^3)$, some vertex of\n$h$ appears on $p_{u,v,r'}$.  Moreover, by the union bound, with\nprobability $1-o(1)$ {\\em all} paths of the type $p_{u,v,r'}$\n(remember that $r'$ can hold one of $O(n)$ possible values) are thus\ncovered by the set $H$.  Let $h'$ be a vertex of $H$ appearing in\n$p_{u,v,r'}$. We therefore have $d^*_j(u,h')+d^*_j(h',v) \\le\n\\delta+\\epsilon\/8$. Since $h=h_{u,v,j}$ is taken as the vertex which\nminimizes these sums, we have, in particular, $d^*_j(u,h)+d^*_j(h,v)\n\\le \\delta+\\epsilon\/8$.  Consider the path $q$ in $G(\\alpha+j\\rho_1)$\nrealizing $d^*_j(u,h)+d^*_j(h,v)$.  The same argument shows that the\nlength of $q$ in $G(r)$ changed by at most $\\epsilon\/8$. But\n$w_1=f^*_j(u,h)(r)+f^*_j(h,v)(r)$ is that weight, and hence $w_1 \\le\n\\delta+\\epsilon\/4$. In particular, $z \\le \\delta+\\epsilon\/4$.\n\n\\section{Concluding remarks}\\label{sec:conclusion} We have constructed\nseveral parametric shortest path algorithms, whose common feature is\nthat they preprocess the generic instance and produce an advice that\nenables particular instantiations to be solved faster than running the\nstandard weighted distance algorithm from scratch.  It would be of\ninterest to improve upon any of these algorithms, either in their\npreprocessing time or in their instantiation time, or both.\n\nPerhaps the most challenging open problem is to improve the\npreprocessing time of Theorem \\ref{thm:gen} to a polynomial one, or,\nalternatively, prove an hardness result for this task.  Perhaps less\nambitious is the preprocessing time in Theorem \\ref{thm:neg}.\n\nFinally, parametric algorithms are of practical importance for other\ncombinatorial optimization problems as well. It would be interesting\nto find applications where, indeed, a parametric algorithm can be\ntruly beneficial, as it is in the case of shortest path problems.\n\n\\section*{Acknowledgment} We thank Oren Weimann and Shay Mozes for\nuseful comments.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\\section{Conclusion}\n\\label{conclusion}\n\nIn this work,  a new data-driven approach to generating a Koopman  linear model based on the direct encoding of Koopman Operator (DDE) is presented as an alternative to dynamic mode decomposition (DMD) and other related methods using least squares estimate (LSE). The major contributions include: 1) The analytical formula of Direct Encoding is converted to a numerical formula for computing the inner product integrals from given data; 2) An efficient algorithm is developed and its convergence conditions to the true results are analyzed; 3) Numerical experiments demonstrate a) greater accuracy compared to EDMD, b) lower sensitivity to data distribution, and c) rapid convergence of inner product computation. Furthermore, the DDE method is incorporated to Deep Koopman, i.e. neural network based methods for construction of the Koopman Operator, for improving prediction accuracy.\n\n\n\n\\section{Experiments}\n\\label{experiments}\n\nIn this section, the DDE algorithm is implemented for the sake of evaluating its validity and comparing its modeling accuracy to EDMD.\nConsider a 2nd order nonlinear system consisting of a pendulum with a nonlinear damper. See Fig. \\ref{fig:penddiagram}. The pendulum also bounces against  walls with nonlinear compliance. The state variables for this system are $x = [ \\theta, \\dot{\\theta}]^T$, and the equation of motion can be written as:\n\\begin{equation}\n    \\ddot{\\theta} = - sin(\\theta) + F_k + F_c\n\\end{equation}\nwhere $F_k$ and $F_c$ are wall reaction moment and damping moment, respectively,\n\\begin{align}\n    F_k &= \\begin{cases}-\\text{sign}(\\theta) \\: k(|\\theta| - \\frac{\\pi}{4})^2 &\\text{if }|\\theta| \\geq \\frac{\\pi}{4}\\\\\n    0 &\\text{if }|\\theta| < \\frac{\\pi}{4} \n    \\end{cases}\\\\\n    F_c &= -\\text{sign}(\\dot{\\theta}) \\: c \\:\\dot{\\theta}^2\n\\end{align}\nwhere $k = 200$ and $c = 1$. \nWe present a two part numerical experiment for this system. The first experiment regards variations in dataset size and distribution, and the second experiment varies the usage of observable functions.\n\n\\begin{figure} \n    \\centering\n    \\includegraphics[width = 0.9\\linewidth]{Figures\/Diagrams\/pendulum.png}\n    \\caption{Diagram of pendulum with with walls. (a) depicts the range of the pendulum where the walls are equally angularly displaced from the vertical. (b) depicts the forces exerted on the pendulum due to contact with walls. (c) depicts the damping force exerted onto the pendulum. }\n    \\label{fig:penddiagram}\n\\end{figure}\n\n\n\\subsection{Dataset Variations}\n\nThe datasets tested are of three types:\n\\begin{enumerate}\n\\item \\textbf{Uniform}: These datasets are composed of a rectangular dynamic range which is sampled uniformly, like an evenly divided grid. The range varies from $\\theta = [-\n0.8,0.8]$ and $\\dot{\\theta} = [-2,2]$, where the mass can hit the walls and the damping can vary from 0 to a significant value. See Fig. \\ref{fig:penddiagram}-(b), (c).\n\\item \\textbf{Gaussian}: Data points are sampled with a finite-support Gaussian distribution. The data are distributed non-uniformly with their highest density at the peak of the Gaussian placed at diverse locations. In addition, each dataset contains 100 data points uniformly distributed along the boarder of the dynamic range to guarantee the same dynamic range as the uniform datasets.  Samples outside the dynamic range are excluded.\n\\item \\textbf{Trajectories}: These datasets are composed of trajectories, beginning from 100 initial conditions that are simulated forward the same number of time steps. The dynamic range of this dataset differs from the two other dataset types.\n\\end{enumerate}\n\nThe models constructed for DDE and EDMD use the same observable functions. The observable functions chosen are two dimensional radial basis functions (RBFs), uniformly distributed between the maximum and minimum values of each state variable in their respective dataset, and the state variables. The total order of the system is 27th order with 25 RBFs and 2 state variables. \n\nA trajectory dataset graph is generated using Delaunay Triangles in DDE, shown in Fig. \\ref{fig:pend_graph}. \n\\begin{figure} \n    \\centering\n    \\includegraphics[width= 0.9\\linewidth]{Figures\/pendulum\/graph_shaded.png}\n    \\caption{Graph of connections for a trajectory dataset composed of 10000 points formed when utilizing the data-driven direct encoding method. The lightly red shaded region denotes the dynamic range. In the zoomed-in image, the difference in volumes associated to different data points can be observed. Noting the difference in size of the purple triangle versus the green triangle.}\n    \\label{fig:pend_graph}\n\\end{figure}\n\\begin{table}  \n\\caption{Sum of Squared Errors over dynamic range with varying dataset sizes.}\n\\label{tab:pred_dynrange} \\begin{center} \\begin{tabular}{c c c}\n\\hline \\hline \n\n \\multirow{ 2}{*}{Dataset Size} & Total SSE & SSE Variance\\\\\n & EDMD \/ DDE  & EDMD \/ DDE\\\\ \n\\hline \n\\hline\n\\multicolumn{3}{c }{\\textit{Uniform Datasets}} \\\\\n\\hline\n\\hline\n900  & 19.470 \/ \\textbf{17.167}  &  \\textbf{0.0094} \/ 0.0097\\\\\n2500 & 17.995 \/ \\textbf{16.471} &  \\textbf{0.0095} \/ 0.0103\\\\\n10000 & 17.010 \/ \\textbf{16.184} & \\textbf{0.0099} \/ 0.0105\\\\\n22500 & 16.698 \/ \\textbf{16.133}  &  \\textbf{0.0101} \/ 0.0105\\\\\n\\hline\n\\hline\n\\multicolumn{3}{c}{\\textit{Trajectory Datasets}} \\\\\n\\hline\n\\hline\n 1000  & 56.532 \/ \\textbf{33.392} & 0.0349 \/ \\textbf{0.0193}\\\\\n2500 & 33.330 \/ \\textbf{25.064} &  0.0200 \/ \\textbf{0.0130}\\\\\n5000 & 31.690 \/ \\textbf{25.099} &  0.0195 \/ \\textbf{0.0129}\\\\\n10000 & 30.184 \/ \\textbf{25.101} &  0.0186 \/ \\textbf{0.0129}\\\\\n25000 & 29.380 \/ \\textbf{25.106} &  0.0181 \/ \\textbf{0.0129}\\\\\n\\hline\n\\hline\n\\end{tabular} \\end{center} \\end{table}\n\nThe accuracy of the models is tested through calculating sum of squared errors (SSE) for one-step ahead predictions over the dynamic range of the datasets. These error values are calculated for a uniform grid of points, similar to that used in the uniform datasets. A visualization of the SSE is plotted in Fig. \\ref{fig:sse_space}. The results of these calculations are shown in Table \\ref{tab:pred_dynrange} and \\ref{tab:pred_gaussian}. In the computation, the dynamic range is discretized, and the SSE value of each point is summed. For the Gaussian datasets, the test is run for eight iterations of each dataset to account for randomness and the average result is noted.\n\\begin{figure} \n    \\centering\n    \\includegraphics[width = 1\\linewidth]{Figures\/pendulum\/heatmapcomp.png}\n    \\caption{Sum of Squared Error plots for various datasets. (a) and (b) are EDMD models for Gaussian datasets; (a) uses a dataset that is centered at $[0,0]$ and (b) uses a dataset that is centered at $[0,2]$. (c) DDE and (d) EDMD models using the trajectory dataset composed of 2500 data points and using 25 RBFs. Circled in red are the regions with greatest variation in SSE between models. Only the dynamic range is shown in all plots.}\n    \\label{fig:sse_space}\n\\end{figure}\n\\begin{table}  \n\\caption{Sum of Squared Errors over dynamic range for Gaussian distributions with different centers. Centers of the distribution are noted above each column.}\n\\label{tab:pred_gaussian} \\begin{center} \\begin{tabular}{c c c c}\n\\hline \\hline\n \\multirow{ 3}{*}{Dataset Size} & \\multicolumn{3}{c }{Total SSE} \\\\\n & Center: $[0,0]$ & Center: $[0.8, 0]$ & Center: $[0,2]$ \\\\\n & EDMD \/ DDE  & EDMD \/ DDE & EDMD \/ DDE\\\\ \n\\hline \\hline \n\\multicolumn{4}{c}{\\textit{Gaussian Datasets}} \\\\\n\\hline\n\\hline\n1000  & 25.565 \/ \\textbf{24.377} &  25.161 \/ \\textbf{22.98} & 24.338 \/ \\textbf{22.445}\\\\\n2500 & 24.991 \/ \\textbf{23.739}&  25.421 \/ \\textbf{22.747} & 24.221 \/ \\textbf{21.590}\\\\\n 5000 & 24.662 \/ \\textbf{23.518} &  26.805 \/ \\textbf{22.415} & 23.914 \/ \\textbf{21.195}\\\\\n10000 & 24.395 \/ \\textbf{23.085} &  28.424 \/ \\textbf{21.825} & 25.770 \/ \\textbf{21.052}\\\\\n25000 & 25.219 \/ \\textbf{22.167} &  30.788 \/ \\textbf{21.687} & 26.941 \/ \\textbf{20.704}\\\\\n\\hline \\hline\n\\end{tabular} \\end{center} \\end{table}\n\n\\begin{table} [!ht]\n\\caption{Sum of Squared Errors over dynamic range with varying order of lifted linear models.}\n\\label{tab:pred_obs} \\begin{center} \\begin{tabular}{c c c}\n\\hline \\hline \n \\multirow{ 2}{*}{\\# Observables} & \\multicolumn{2}{c}{Total SSE}\\\\\n  & EDMD & DDE \\\\\n\\hline \n\\hline\n\\multicolumn{3}{c}{\\textit{Trajectory Dataset, 5000 points}} \\\\\n\\hline\n27  & 31.690 & \\textbf{25.099} \\\\\n51  & 36.657 & \\textbf{21.637}\\\\\n83 & 28.437 & \\textbf{13.613}  \\\\\n\\hline\n\\end{tabular} \\end{center} \\end{table}\n\n\n\n\n\\subsection{Observable Function Variation}\n\nThe second experiment varies the number of observable functions selected, thus increasing the order. In this experiment, the number of RBFs is varied through uniformly increasing the density of the centers of the function over the range of the dataset. The results are noted in Table \\ref{tab:pred_obs}. In the table, the number detailing number of observables is the number of functions including the state variables.\n\n\\subsection{Discussion}\nFrom these results we can make the following observations.\n\n\\begin{itemize}\n    \\item All the numerical experiments show that DDE outperforms EDMD in total SSE.\n    \\item For uniform datasets, both models are nearly equivalent, though DDE has slightly lower SSE in all cases, shown in Table \\ref{tab:pred_dynrange}. This result is expected as all data points are weighted equally in a uniform distribution.\n    \\item EDMD models exhibit significantly different distributions of prediction error, depending on dataset distribution. In Fig. \\ref{fig:sse_space}-(a), the EDMD model learned from a dataset with high density near the origin produces a prediction error distribution that is low in accuracy in the top-right and the bottom-left corners of the dynamic range. In contrast, Fig.\\ref{fig:sse_space}-(b)  illustrates that when using EDMD to learn from data with high density at $ \\theta = 0, \\dot{\\theta} = 2$, the model results in low accuracy in the lower half of the dynamic range. DDE does not exhibit this high distribution dependency and achieves lower total SSE, as shown in Table \\ref{tab:pred_gaussian}.\n    \\item In the trajectory datasets in Table \\ref{tab:pred_dynrange}, DDE is not only lower in total SSE than EDMD but is also smaller in variance. Fig. \\ref{fig:sse_space}-(c) shows that DDE has a uniformly low error distribution across the dynamic range, while EDMD in Fig.\\ref{fig:sse_space}-(d) has two regions, as circled in the figure, with significantly larger error.\n    These regions coincide with sparsity in the dataset, providing evidence of EDMD's bias towards regions of high data density and explaining the difference in performance between models for the non-uniform datasets.\n    \\item In the trajectory datasets, the total SSE converges for DDE beginning with small dataset sizes. This result implies that the elements in the $R$ and $Q$ matrices of DDE, that is, the inner product integral computations, converge as the data size and the data density increase. This convergence is confirmed in Fig. \\ref{fig:qplot}, where several elements of the $Q$ matrix are plotted against the data size. \n    \\item The second experiment, regarding variations in observable function numbers demonstrates the effect of DDE remains even for significant increases in observable functions, referring to Table \\ref{tab:pred_obs}. In all cases tested, DDE significantly outperforms EDMD over the dynamic range, as expected for a non-uniform dataset.\n\\end{itemize}\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = 0.85\\linewidth]{Figures\/pendulum\/q_plot.png}\n    \\caption{Change in values of diagonal elements in the $Q$ matrix of DDE with respect to trajectory dataset size. The specific elements shown correspond to a state variable, $Q_{0,0}$, and three RBF functions of varying distances away from the equilibrium.}\n    \\label{fig:qplot}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\\section{Introduction}\n\\label{intro}\n\n\nDynamic Mode Decomposition (DMD) was presented as a method to produce linear models from data generated through nonlinear dynamical processes by using Singular Value Decomposition (SVD) \\cite{schmid2010dynamic}. Later, this method was developed further to create Extended Dynamic Mode Decomposition (EDMD), which introduced the concept of using observable functions, nonlinear functions of state variables, as a method of augmenting the state space \\cite{williams2015data}. EDMD referenced the Koopman Operator as justification and a theoretical underpinning for lifting the state space. In his seminal work, Bernard Koopman showed the existence of this operator that transforms nonlinear systems into linear systems \\cite{koopman1931hamiltonian}. Another extension of DMD has shown the viability of using DMD for control on non-autonomous systems \\cite{proctor2016dynamic}. This enabled complex nonlinear Model Predictive Control (MPC) to be converted to linear MPC \\cite{Korda_2018}, leading to numerous studies  utilizing the methodology to real systems\\cite{ng2021model, cibulkaVehicle2020, calderon2021, bruder2021}.\n \nTo improve the accuracy of the models based on the Koopman Operator, two avenues of research have formed. The first avenue regards the selection of the observable functions used for constructing a lifted state space, as these functions are a key ingredient in creating an accurate linear model. Various methods have been developed, including deep neural networks for learning effective observable functions \\cite{Lusch_2018, yeung2017, Selby_2021} and optimization \\cite{korda2020optimal}. However, an efficient selection of observables does not solve all the issues that arise when attempting to construct an accurate linear model. For example, it is known that unstable modes are involved in Koopman-based DMD models and their extensions although the underlying nonlinear systems are known to be stable. The second avenue of research involves the formulation of the linear transition matrix. Extensive studies have been done to create stable linear models to remedy the situations where an outright use of DMD would lead to the creation of an unstable linear model \\cite{fanstabkoop2021, mamakoukas2020learning, bevanda2022diffeomorphically, han2021desko}. Recently, an extension of DMD, called Robust Dynamic Mode Decomposition (RDMD), utilizes statistical measures to suppress the effect of outliers on modeling the linear Koopman matrix\\cite{Hossein_Abolmasoumi_2022}.\n\nA fundamental difficulty in constructing a proper linear model is data dependency.\nLeast Squares Estimation (LSE), involved in all DMD based methods, often produces a significant bias due to the distribution of the dataset. To eliminate this dependency on data distributions, the current work takes an alternative approach to LSE.\n\nRecently, a new formulation of the Koopman Operator, termed Koopman Direct Encoding (DE), was produced \\cite{asada2022}. This method directly encodes the nonlinear function of state transition using observables as basis functions to obtain a Koopman linear model. Inner products of observable functions in composition with the nonlinear state transition function are used to construct the state transition matrix without use of LSE. While DE theoretically guarantees the exact linear model, it requires access to the nonlinear state equations, which are often not available in practical applications. The current work aims to fill the gap between DE and data-driven approaches.\n\n\nThere are four significant contributions presented in this work. The first is the conversion of the DE formula of the Koopman Operator to a data-driven formula. The second is a computational algorithm and proof of its convergence to the true inner products that constitute the DE formula. The third is numerical experiments that provide evidence that the proposed method, unlike EDMD, does not exhibit biases to data distribution, but can produce consistently higher accuracy compared to EDMD.\nFinally, the DDE algorithm is utilized in modeling a high order nonlinear system in combination with deep learning.\n\n\n\n\\section{Data-Driven Koopman Encoding}\n\\label{methods}\n\nThe prevailing method for construction of the Koopman Operator, EDMD, is based on LSE and SVD. This method, however, cannot provide a consistent estimate; the result is highly dependent on the distribution of data as the Koopman Operator is being approximated \\cite{ljung1998system}. This dependency on distribution occurs because a core assumption of LSE is that the model structure is correct; when this assumption is violated, LSE is unable to create an unbiased estimator \\cite{freedman2009statistical}. As the Koopman operator is truncated in practical use, this assumption does not hold.\n\nNon-uniform data distributions inevitably occur in practical applications. For a nonlinear dynamical system with a stable equilibrium, for example, data collected from experiments and\/or simulation of the system tend to be dense in the vicinity of the equilibrium, as all trajectories that begin within a region of attraction converge to the equilibrium.\nBecause LSE applies equal weighting to all data points, the model is heavily tuned to the behavior of the densely populated region.\n\n\nThe Direct Encoding method described previously enables us to obtain the exact linear state transition matrix $A$ through inner product computations. As the formulation is based on integration over the entire state space, there is no bias towards particular parts of the domain. \n\nHowever, the original form of the Direct Encoding method utilizes the nonlinear state equation, i.e. the self-map $f(x)$, to compute the inner products. In practical applications, such a nonlinear function is not always available; only data are available. The objective of this section is to establish a computational algorithm to obtain the $A$ matrix by  numerically computing the inner products, $\\langle g_i, g_j \\rangle, \\langle g_i \\circ f, g_j \\rangle$, from a given set of data. \n\nThe method presented consists of three operations. \n\\begin{itemize}\n    \\item The integral of the inner products is reduced in range from the entire state space to the dynamic range encapsulated by the data.\n    \\item The dynamic range is discretized with data points.\n    \\item The inner product integral is reduced to a weighted summation of the integrand evaluated at each data point multiplied by the volume $\\Delta v$ associated to each point.\n\\end{itemize}\n Naturally, if data are densely populated in a small region, the discretized integral interval is small and thereby the volume also becomes small. Similarly, the volume tends to be larger where the data are sparse. In the summation, the integrand evaluated at individual data points are \"weighted\" by the size of the volume. This numerical inner product calculation prevents overemphasis of clustered data.\n\n\n\n\n\n\n\n\n\n\\subsection{Inner Product Computation}\n\nWe present the data-driven encoding method (DDE) as an alternative data-driven method to DMD for calculating a finite order approximation of the Koopman Operator.\nThe objective of this method is to compute the matrices $R$ and $Q$ in (13) and (14) from data. This entails the computation of inner products:\n\\begin{align}\n    \\langle g_i, g_j \\rangle = \\int_X G_{ij}(\\xi)  d\\xi \\\\\n    \\langle g_i \\circ f, g_j \\rangle = \\int_X F_{ij}(\\xi)  d\\xi\n\\end{align}\nwhere\n\\begin{equation}\n    G_{ij} (x) = g_i(x) \\bar{g}_j(x),  F_{ij} (x) = g_i[f(x)] \\bar{g}_j(x)\n\\end{equation}\nare assumed to be Riemann Integrable; the functions are bounded and continuous \\cite{davis2007methods}.\n\nThere are two data sets used for the inner product computation. The first data set is\n\\begin{equation}\n    D_N = \\{ x_i \\mid i = 1, \\cdots, N; x_i \\in X \\} \n\\end{equation}\nNote that all the data values are finite, $|x_i| < \\infty$. As such, the integral interval of the inner products is finite in computing them from the data. To define the integral interval, we consider the dynamic range of the system, $X_D$, determined from the data set $D_N$. See Fig.\\ref{fig:convexhull}. The dynamic range $X_D$ is defined to be the minimum domain in the space $X$ that includes all the data points in $D_N$, $X_D \\supset D_N$, and that is convex. Namely, for any two states in $X_D$, $x, x' \\in X_D$,\n\\begin{equation}\n    \\xi = \\alpha x + (1-\\alpha) x' \\in X_D\n\\end{equation}\nwhere $0 \\leq \\alpha \\leq 1$. \n\\begin{figure} [!h]\n    \\centering\n    \\includegraphics[width = 0.8\\linewidth]{Figures\/Diagrams\/convexhull.PNG}\n    \\caption{Illustration of the dynamic range of a dataset defined by the convex hull containing all points in the set, partitioned using a triangulation method. The data points are in black and the convex hull that encapsulates all data points is in grey.}\n    \\label{fig:convexhull}\n\\end{figure}\nEach data point $x_i$ is mapped to $f(x_i)$, following the state transition law in eq.(\\ref{eq:gen_nonlinear}). We assume that the transferred state, too, stays within the same dynamic range $X_D$. Collecting all the transferred states yields the second data set.\n\\begin{align}\n    D_N^f = \\{ f(x_i) \\mid i = 1, \\cdots, N; x_i \\in D_N \\} \\\\\n    D_N^f \\subset X_D\n\\end{align}\nThis implies that the state space of the nonlinear system under consideration is closed within the dynamic range $X_D$.\n\nWith this dynamic range, we redefine our objective to compute the inner products over $X_D$.\n\\begin{align}\n    R_{ij} = \\int_{X_D} G_{ij}(x) dx \\label{eq:G_ij integral} \\\\\n    Q_{ij} = \\int_{X_D} F_{ij}(x) dx \\label{eq:F_ij integral}\n\\end{align}\nThe integrals can be computed by partitioning the domain $X_D$ into many segments $X_1, \\cdots X_P$, as shown in Fig. \\ref{fig:convexhull}.\n\\begin{equation}\n    X_D = \\bigcup_{p=1}^P X_p\n\\end{equation}\nWe generate these segments by applying a meshing technique to the data set $D_N$, where the $n$-dimensional coordinates of individual data points are treated as nodes of a mesh. Delaunay Triangulation, for example, generates a triangular mesh structure with desirable properties \\cite{cheng2013delaunay}. As illustrated in Fig. \\ref{fig:convexhull}, each triangular element is convex and has no internal node. The volume of the dynamic range $V(X_D)$ is the sum of the volumes of all the elements.\n\\begin{equation}\n    V(X_D) = \\sum_{p=1}^P \\Delta v_p\n\\end{equation}\nAccordingly, the integral $R_{ij}$ in eq.(\\ref{eq:G_ij integral}) can be segmented to\n\\begin{equation}\n    R_{ij} = \\sum_{p=1}^P \\int_{X_p} G_{ij} (x) dx\n\\end{equation}\nSuppose that the $p$-th element has $K_p$ nodes, as shown in Fig.\\ref{fig:partition}. Renumbering these nodes 1 through $K_p$,\n\\begin{equation}\n    \\{ x[k_p] \\Big| x[k_p] \\in X_p; k_p = 1, \\cdots, K_p \\},  p = 1, \\cdots, P\n\\end{equation}\nThe integrand $G_{ij}$ within the $p$-th element can be approximated to the mean of the $K_p$ nodes involved in the $p$-th element.\n\\begin{equation}\n    G_{ij} (x;p) \\approx \\bar{G}_{ij,p}= \\frac{1}{K_p} \\sum_{k_p = 1}^{K_p} G_{ij}(x[k_p]), x[k_p] \\in X_p\n    \\label{integrand_approx}\n\\end{equation}\nIf Delaunay Triangulation is used, $K_p = n+1$.\nSee Fig. \\ref{fig:partition}. Substituting this into (\\ref{eq:G_ij integral}) yields the approximate value of $R_{ij}$.\n\\begin{equation}\n    \\hat{R}_{ij} = \\sum_{p=1}^P \\frac{1}{K_p} \\sum_{k_p = 1}^{K_p} G_{ij}(x[k_p])  \\Delta v_p\n    \\label{eq:r_calc}\n\\end{equation}\nwhere \n\\begin{equation}\n     \\Delta v_p = \\int_{X_p} 1 \\cdot dx\n\\end{equation}\n\nSimilarly, each component of the matrix $Q$ can be computed by using the same meshing.\n\\begin{equation}\n    \\hat{Q}_{ij} = \\sum_{p=1}^P \\frac{1}{K_p} \\sum_{k_p = 1}^{K_p} F_{ij}(x[k_p]) \\Delta v_p\n    \\label{eq:q_calc}    \n\\end{equation}\nNote that $F_{ij}$ is evaluated by using the data points in both $D_N^f$ and $D_N$,\n\\begin{equation}\n    F_{ij}(x[k_p]) = g_i[f(x[k_p])] \\bar{g}_j(x[k_p])\n\\end{equation}\nwhere $f(x[k_p]) \\in D_N^f$.\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = 0.8\\linewidth]{Figures\/Diagrams\/partition.PNG}\n    \\caption{Visualization of the integrand calculation process. The volume of the partition is encapsulated by the data points is denoted as $\\Delta v_p$. With this grouping, the value of $G_{ij}$ is calculated for each point and the average among this group is computed, $\\bar{G}_{ij,p}$.  In turn, this value, weighted by the volume of this partition, is summed across other partitions (not shown) to approximate the value of the element $R_{ij}$.}\n    \\label{fig:partition}\n\\end{figure}\n\n\n\\subsection {Convergence}\n\nConsider the center of each partition, $\\bar{x}_p = \\int_{X_p} x dx \/ \\Delta v_p$, and the distance between $\\bar{x}_p$ and each point, $x[k_p]$:\n\\begin{equation}\n    \\Delta x[k_p] = \\bar{x}_p - x[k_p]\n\\end{equation}\nSee Fig.\\ref{fig:partition}. The maximum distance from the center of the partition to each point that makes up the partition is \n\\begin{equation}\n    |\\Delta x_p| = \\max\\{|\\Delta x[1]|,\\, ... \\, |\\Delta x[k_p-1]|, \\, |\\Delta x[k_p]| \\}\n\\end{equation}\nConsider a sequence of refining the approximate inner product integral $\\hat{R}_{ij}$ by increasing data points $N$.  We can show that, as the number of partition $P$ tends infinity and the maximum subintervals $|\\Delta x_p|$ approach zero, the approximate inner product integral $\\hat{R}_{ij}$ converges to its true integral.  \n\n\\begin{equation}\n    R_{ij} =  \\lim_{\\substack{ P \\to \\infty \\\\ |\\Delta x_p| \\to 0}} \\ \\sum_{p=1}^P \\frac{1}{K_p} \\sum_{k_p = 1}^{K_p} G_{ij}(x[k_p])  \\Delta v_p\n\\end{equation}\n    \n   \nThis formulation takes the form of weighted sums, specifically Riemann sums.\nGiven functions that are bounded and continuous over the subdomain of interest, sequences of this form are known to have a common limit and thus converge upon refinement to the Riemann integral value over that subdomain, according to Numerical Integration theory \\cite[Section 1.5]{davis2007methods}.\n\n\n\\subsection{Algorithm}\n\nIn the prior section, integrals (\\ref{eq:r_calc}) and (\\ref{eq:q_calc}) are presented as summations over partitions. This computation can be streamlined by converting the summations over partitions to the one over nodes. Consider node 3 associated to data point $x_3$ in Fig.\\ref{fig:convexhull}, for example. This node is an apex of the 5 surrounding triangles. This implies that integrand $G_{ij}(x)$ is calculated or recalled 5 times in computing (\\ref{eq:r_calc}) and (\\ref{eq:q_calc}). This repetition can be eliminated by computing volume $\\Delta v_k$ associated to node $k$, rather than partition $p: \\Delta v_p$. Namely, we compute \n\\begin{equation}\n    \\Delta v_k = \\sum_{p=1}^{P} \\frac{\\Delta v_p}{K_p}I(k,p)\n\\end{equation}\nwhere $I(k,p)$ is a membership function that takes value 1 when node $k$ is an apex of triangle $p$, that is, node $k$ is involved in partition $p$. Using this volume as a new weight we can rewrite (\\ref{eq:r_calc}) and (\\ref{eq:q_calc}) to be\n\\begin{align}\n    \\hat{R}_{ij} = \\sum_{k=1}^K G_{ij}(x[k])  \\Delta v_k  \\label{eq:r_calc2}\\\\\n    \\hat{Q}_{ij} = \\sum_{k=1}^K F_{ij}(x[k]) \\Delta v_k  \\label{eq:q_calc2}\n\\end{align}\n\nUsing this conversion, the computation can be streamlined and cleanly separated into three steps, as shown by pseudo-code in Algorithm \\ref{algo:dde}. \nThe steps are: (1) \\textit{Graph Creation:} data are connected to create partitions of the domain using a mesh generator: lines 3 to 8, (2) \\textit{Weighting Calculation:} calculation of the weights for each data point: lines 10 to 17, and (3) \\textit{Matrix Calculation:} the calculation of the $R$ and $Q$ matrices to find the  matrix $A$, lines 19 to 21.\n\n\n\n\n\\begin{algorithm} [!b]\n    \\caption{Algorithm for DDE in pseudocode}\n    \\label{algo:dde}\n    \\KwIn{}\n    $D_N$, $D_N^f$\\\\ \n    \\KwOut{}\n    $P$: $p$, $K$: $k$, $R$, $Q$ $A$\\\\\n    \\textit{Graph Creation:} \\\\\n    \\For{$x_i$ in $D_N$ and corresponding $f(x_i)$ in $D_N^f$}{\n    Create node, $k$\\\\\n    Assign node attributes for current data point $k.x_t = x_t$ and $k.x_{t+1} = x_{t+1}$  \\\\\n    Append node to node list $K$.\n    }\n    \\textbf{then}{ \\\\\n    \\textit{Weighting Calculation:} \\\\\n    Create list of triangles, $P$ using Delaunay Triangulation on node list $K$ using $k.{x_t}$}\\\\\n    \\For{$p$ in $P$}{\n    Calculate volume, $V$, of $p$\\\\\n    \\For{$k$ corresponding to $K_p$}{\n    Update volume of each node $k.\\Delta v_k = k.\\Delta v_k + \\frac{V}{n+1}$}\n    }\n    \\textbf{then}{\\\\\n    \\textit{Matrix Calculation:}\\\\\n    Find $R$ and $Q$ via (\\ref{eq:r_calc2}) and (\\ref{eq:q_calc2})\\\\\n    Find $A = Q R^{-1}$\n    }\n\\end{algorithm}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Application to Neural Net Koopman Modeling}\n\\label{nnkoop}\nThe use of deep neural networks for finding effective observable functions and constructing a Koopman linear model has been reported by several groups \\cite{Lusch_2018, yeung2017, Selby_2021}. This method, sometimes referred to as Deep Koopman, is effective for approximating the Koopman operator to a low-order model, compared to the use of locally activated functions, such as RBFs, which scale poorly for high-order nonlinear systems. The proposed DDE method can be incorporated into Deep Koopman, further improving approximation accuracy.\n\nFig. \\ref{fig:nn} shows the architecture of the neural network similar to the prior works \\cite{Lusch_2018, yeung2017, Selby_2021}. The input layer receives training data of independent state variables. The successive hidden layers produce observable functions; these functions feed into the output layer consisting of linear activation functions. This linear output layer corresponds to the $A$ matrix that maps the observables of the current time to those of the next time step, i.e. the state transition in the lifted space. In the Deep Koopman approach, the output layer, that is, the $A$ matrix, is trained together with observable functions in the hidden layers. Using the observable functions learned from deep learning, the output layer weights are replaced with the $A$ matrix obtained from DDE, captioned in Fig. \\ref{fig:nn}.\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = 0.8\\linewidth]{Figures\/Diagrams\/nn.PNG}\n    \\caption{Feedforward neural network model used to generate observable functions. The final layer of the neural network is a linear layer. In the standard Deep Koopman model this remains the same after the model is fully trained. However, with the DDE model, this final layer is recalculated using DDE by taking in the dataset used to train the model.}\n    \\label{fig:nn}\n\\end{figure}\n\n\nThe effectiveness of this Deep Koopman-DDE method is applied to a multi-cable manipulation system \\cite{ng2021model}. A simplified single cable version is utilized consisting of six independent state variables. The network is constructed using PyTorch, trained with an Adam optimizer, to generate 40 observable functions. These model concatenates the state variables of the nonlinear system, resulting in a 46th order model. Table \\ref{tab:nndef} shows parameters used for training the Deep Koopman model. The training dataset is composed of 3000 data points drawn from trajectories. Table \\ref{table:SSE of Neural Net} compares the Deep Koopman model to the proposed model that uses DDE. Results are in terms of sum of squared error over a set of test trajectories. A significant improvement is achieved by incorporating DDE into the deep learning method.\n\\begin{table} [!h]\n    \\caption{Neural network parameters and characteristics.}\n    \\small\n    \\centering\n    \\begin{tabular}{c|c}\n    \\hline\n        Parameters and Characteristics & Value \\\\\n        \\hline\n        Number of Hidden layers & 3 \\\\\n        Activation Functions, Both Hidden Layers & ReLU \\\\\n         Width of 1st Hidden Layer & 16 \\\\\n         Width of 2nd Hidden Layer & 16 \\\\\n         Width of 3rd Hidden Layer & 40 \\\\\n         Learning Rate, $\\alpha$ & 0.01 \\\\\n    \\end{tabular}\n    \\label{tab:nndef}\n\\end{table}\n\\begin{table} [!h]\n\\caption{Average SSE prediction error for trajectories in set of test data for single winch system.}\n\\label{table:SSE of Neural Net} \\begin{center} \\begin{tabular}{c c c}\n\\hline \\hline \nModeling Method & 1 Time Step & 20 Time Steps  \\\\\n\\hline\n\\hline\nDeep Koopman only  & 0.2471 & 9.8610  \\\\\nDeep Koopman + DDE  & \\textbf{0.2350} & \\textbf{4.1131}  \\\\\n\\hline\n\\end{tabular} \\end{center} \\end{table}\n\n\n\n\n\n\nPractical concerns arose from the use of Delaunay Triangulation in this experiment. Specifically, the method was found to be limiting and inconsistent for systems that are eighth order or higher, due to computational cost and numerical instability. Alternative numerical integration approaches or alternative partitioning methods may be necessary when applying the method to higher order nonlinear systems. \n\n\n\n\n\n\n\n\n\n\n\\section{Koopman Operator and the Direct Encoding Method}\n\\label{preliminaries}\n\nIn this section, we give a brief overview of the Koopman Operator and dynamic mode decomposition, and introduce the direct encoding method for obtaining a Koopman operator directly from nonlinear dynamics.\n\n\\subsection{Least Squares Estimation of the Koopman operator}\n\nConsider a discrete-time dynamical system, given by\n\n\\begin{equation}\n    x_{t+1} = f(x_t) \n    \\label{eq:gen_nonlinear}\n\\end{equation}\nwhere $x \\in X \\subset \\mathbb{R}^n$ is the independent state variable vector representing the dynamic state of the system, \n$f$ is a self-map, nonlinear function $f : X \\rightarrow X$, and $t$ is the current time step. Also consider a real-valued observable function of the state variables $g : X \\rightarrow \\mathbb{R}$. The Koopman Operator $\\mathcal{K}$ is an infinite-dimensional linear operator acting on the observable function $g$ :\n\\begin{equation}\n    \\mathcal{K} g  = g \\circ f\n\\end{equation}\nwhere $g \\circ f$ is the composition of function $g$ with function $f$: $(g \\circ f)(x) = g(f(x))$.\n\nA common data-driven method for constructing the operator is Extended Dynamic Mode Decomposition (EDMD) \\cite{williams2015data}, where observables that are experimentally obtained or simulated from the governing equation of the system are augmented by including real-valued observable functions of the independent state vector $x_t$. Collectively, a high-dimensional state vector is formed. \n\\begin{equation}\n    z_t = \\begin{bmatrix} g_1(x_t) \\\\ g_2(x_t) \\\\ \\vdots \\\\ g_m(x_t) \\end{bmatrix}\n\\end{equation}\nwhere $m$ is the order of the lifted state corresponding to the number of observable functions. Underpinned by the Koopman Operator theory, EDMD assumes the existence of a linear state transition matrix $A$ relating $z_{t+1}$ to $z_t$, and determine $A$ by solving a least squares regression that minimizes the Sum of Squared Error (SSE).\n\n\\begin{equation}\n   A  = \\arg \\min_A \\sum_{t}|| z_{t+1} - A z_t || ^2\n\\end{equation}\nSingular Value Decomposition (SVD) is used for the least squares optimization.\n\\subsection{Direct Encoding of the Koopman Operator}\n\\label{koop:de}\n\nAn alternative to the least squares estimate and EDMD is to obtain the exact $A$ matrix by directly encoding the self-map, nonlinear state transition function $f(x)$ with an independent and complete set of observable functions through inner product computations. This Direct Encoding method is introduced next, while the full proof can be found in \\cite{asada2022}. \n\n\n\n\n\nLet us first consider the case where $g_1, g_2, g_3, ...$ are orthonormal basis functions spanning a Hilbert space $\\mathcal{H}$.\nWe assume that the self-map nonlinear function $f(x)$ is continuous and that the composition of $g_j$ with $f$ is also involved in the Hilbert space.\n\\begin{equation}\n    g_j \\circ f \\in \\mathcal{H} \\quad\\forall j\n\\end{equation}\nThis implies that the function $g_j \\circ f$ can be expanded in $[g_1, g_2, g_3, ...]$.\n\\begin{equation}\n    g_j \\circ f = \\sum_{k=1}^\\infty \\langle g_j \\circ f, g_k \\rangle g_k\n    \\label{eq:orthof_exp}\n\\end{equation}\n\n\nConcatenating $g_1, g_2, g_3, ...$ and $g_1 \\circ f, g_2 \\circ f, g_3 \\circ f ...$ in infinite dimensional column vectors, respectively,\n\\begin{equation}\n   \\bar{z}_t = \\begin{bmatrix} g_1(x_t) \\\\ g_2(x_t) \\\\ \\vdots  \\end{bmatrix},\n   \\bar{z}_{t+1} = \\begin{bmatrix} g_1[f(x_t)] \\\\ g_2[f(x_t)] \\\\ \\vdots  \\end{bmatrix}\n\\end{equation}\neq. (\\ref{eq:orthof_exp}) can be written in matrix and vector form.\n\\begin{equation}\n    \\bar{z}_{t+1} = \\bar{A} \\bar{z}_t\n    \\label{eq:X_t+1 = A X_t}\n\\end{equation}\nwhere $\\bar{A}$ is an infinite dimensional matrix consisting of the inner products involved in eq. (\\ref{eq:orthof_exp}),\n\n\\begin{equation}\n    \\bar{A} = \\begin{bmatrix} \\langle g_1\\circ f, g_1 \\rangle & \\langle g_1 \\circ f, g_2 \\rangle & \\dots \\\\ \\langle g_2 \\circ f, g_1 \\rangle & \\langle g_2 \\circ f, g_2 \\rangle & \\dots \\\\ \\vdots & \\vdots & \\ddots \\end{bmatrix}\n\\end{equation}\nEq. (\\ref{eq:X_t+1 = A X_t}) manifests that the state lifted to the infinite dimensional space $\\bar{z_t}$ makes linear state transition with matrix $\\bar{A}$.\n\nThe observables $g_1, g_2, g_3, ...$ were assumed to be orthonormal basis functions in the above derivation. This assumption can be relaxed to an independent and complete set of basis functions spanning the Hilbert space. Hereafter, let $[g_1, g_2, g_3, ...]$ be an independent and complete set of basis functions spanning the Hilbert space. \n\n\nIt can be shown that the time evolution of lifted state $z_t$ is given by a constant matrix $A_f$ for the independent and complete set of basis functions $[g_1, g_2, g_3, ...]$.\n\\begin{equation}\n    z_{t+1} = A_f z_t\n    \\label{eq:A_f 1}\n\\end{equation}\n\n    \\label{A_f with C}\n\nThe matrix $A_f$ can be computed directly from the self-map, state transition function $f(x)$ and an independent and complete set of observables $[g_1, g_2, g_3, ...]$ through inner product computations. Post-multiplying the transpose of $z_t$ to both sides of eq. (\\ref{eq:A_f 1}) and integrating them over $X$ yield:\n\\begin{equation}\n    \\int_X z(f(x)) z^T(x)dx = A_f \\int_X z(x) z^T(x) dx\n\\end{equation}\nwhich can be written as\n\\begin{equation}\n    Q = A_f R\n\\end{equation}\nwhere\n\\begin{align}\n    Q &= \\begin{bmatrix} \\langle g_1\\circ f, g_1 \\rangle & \\langle g_1 \\circ f, g_2 \\rangle & \\dots \\\\ \\langle g_2 \\circ f, g_1 \\rangle & \\langle g_2 \\circ f, g_2 \\rangle & \\dots \\\\ \\vdots & \\vdots & \\ddots \\end{bmatrix}\\\\\n    R &= \\begin{bmatrix} \\langle g_1, g_1 \\rangle & \\langle g_1, g_2 \\rangle & \\dots \\\\ \\langle g_2, g_1 \\rangle & \\langle g_2, g_2 \\rangle & \\dots \\\\ \\vdots & \\vdots & \\ddots \\end{bmatrix}\n\\end{align}\nBecause the observables are independent, the matrix $R$ is non-singular. Therefore, the matrix $A_f$ is given by\n\\begin{equation}\n    A_f = QR^{-1}\n    \\label{eq:A_f = Q*R_inv}\n\\end{equation}\n\nThis formula for obtaining the matrix $A_f$ directly from the governing nonlinear state equation with the function $f(x)$ and the independent observables through inner products, which are guaranteed to exist in Hilbert space $\\mathcal{H}$, is the Direct Encoding method.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{introduction}\n\nNetwork monitoring have been deeply studied in P2P, DTN and others using gossip-based or hierarchical-based approaches.\nHowever, when it is applied to a wireless mobile ad hoc network (MANET), new problematics arise mainly due to the absence of a centralized administration, the inherent MANETs properties and the node mobility.\nSome approaches propose a coordinator, nevertheless, due to energy efficiency,\ninfrastructure or other parameters, these solutions are not always applicable.\n\nWhile studying the monitoring of a network, the most common and intuitive approach is to define a central node as a coordinator for storage and processing of the observations.\nThis is notably proposed by \\cite{cormode2013continuous}, where the author surveys the different communication mechanisms.\nThese centralized architectures might be efficient for certain type of topologies, but become critical when considering dynamic topologies.\nThis is why there has been a lot of efforts on decentralized monitoring.\nGossip-based approaches show an extraordinary robustness and stability in dynamic scenarios and changing topologies.\nNonetheless, depending on the scalability, the cost and performance can be impacted.\nOn the other side, hierarchical approaches show an efficient performance, cost and scalability, although the  robustness and stability may decrease in dynamic scenarios.\nThis shows that the two major categories perform very good under different characteristics, requirements and constraints of a network \\cite{stingl2012benchmarking}.\n\nThe main contribution of this paper is the proposal of a hybrid algorithm for decentralized monitoring of MANETs.\nWe define an architecture combining gossip-based and hierarchical-based algorithms for query dissemination and data aggregation.\nWe perform the gossip-based approach to disseminate the query and in the process to build a virtual hierarchical topology (VHT) for a time window.\nOnce the query is disseminated through all the network, with the support of the VHT, a hierarchical-based aggregation takes place.\nThe second contribution of this paper is the definition of a monitoring protocol that aims at helping a decentralized monitoring process.\nOur expectation is not just to provide a structure but also a mathematical background for further model checking and testing. \nOur protocol has been successfully assessed using NS3 and Docker. \n\n\nThe remaining of our paper is as it follows.\nIn Section \\ref{methods}, we present our hybrid algorithm.\nIn Section \\ref{results}, we present our implementation, with a semi-formal support for our protocol. \nNext, in Section \\ref{relatedworks}, we present some interesting related works from which we got inspired.\nFinally, we conclude and give some perspectives in Section \\ref{conclusions}.\n\n\n\\section{Preliminaries} \\label{preliminaries}\n\n\n\nNetwork monitoring can be described as \\say{A number of observers making observations, and wish to work together to compute a function of the combination of all their observations} \\cite{cormode2013continuous}.\nThe goal is that all the nodes in the network compute a value $t\\mapsto f(t)$ [$\\mathbb{R}^{+*}\\rightarrow X$, $X$ being the domain targeted by $f$] in a given instant of time $t$ in a collaborative way.\nFor our purposes, $f$ is a linear and non-complex function (e.g., the average CPU)\n\nThe monitoring process can be classified in centralized and decentralized \\cite{battat2014monitoring}.\nThe centralized approach \\cite{cormode2013continuous}\nis when all the nodes report their observations to a central entity (centralizer or coordinator).\nThe decentralized one considers networks with no centralizer. The main current decentralized approaches are gossip and hierarchical \\cite{stingl2012benchmarking}.\n\n\nThe gossip approaches are based on the gossip or epidemic algorithms.\nGossip algorithms rely on selecting, from a set of reachable nodes, a random or a specific node (depending on the algorithm) to forward the data packet \\cite{chakchouk2015survey}. \nEpidemic algorithms try to forward the packet not only to one but to multiple nodes. \nGossip based monitoring algorithms have the advantage of being highly stable and perform better in increasing dynamic networks. \nIn the hierarchical approaches, the leaf nodes communicate recursively the values with their parent node. \nFor a hierarchical algorithm to work over a network, it will need to build the topology before being able to monitor.\n\n\n\\section{Hybrid Monitoring Approach} \\label{methods}\n\nNetwork monitoring can be described as \\say{A number of observers making observations, and wish to work together to compute a function of the combination of all their observations} \\cite{cormode2013continuous}.\nThe goal is that all the nodes in the network compute a value $t\\mapsto f(t)$ [$\\mathbb{R}^{+*}\\rightarrow X$, $X$ being the domain targeted by $f$] in a given instant of time $t$ in a collaborative way.\nFor our purposes, $f$ is a linear and non-complex function (e.g., the average CPU)\n\n\nOur hybrid algorithm architecture consists in two network states, the \\say{query state} and the \\say{aggregate state}.\nThe idea is to combine a gossip approach and a hierarchical approach to achieve the monitoring of a property of the network.\nThe communication between the nodes to achieve the monitoring of the network will be achieved through a package previously defined.\nThe idea is that a start node will start the monitoring process by propagating a monitoring query in a gossip approach.\nThe approach chosen will be described as epidemic.\nEach hop, the nodes will exchange information creating a virtual hierarchical topology (VHT) which will be valid only during the monitoring process.\nThen based on this topology, the nodes will start aggregating the information by sending their results to the parent node.\nOnce the aggregation is done and has reached the start node, there will be a global view of the measured property and the VHT will no longer be usable.\nIf the process starts again, a new VHT will be derived.\nThe purpose is to establish a VHT during one time window, duration of the monitoring process, to ease the analysis and the global view of the property.\n\n\n\n\n\\subsection{Hybrid Architecture} \\label{hybridArchitecture}\n\n\n\\subsubsection{Query State} \\label{queryState}\n\nThe query state refers to the process of propagating in an epidemic way the monitoring packet.\nThis state goal is to disseminate the query and the VHT layout to allow the nodes in the network to do an accurate and efficient aggregation in the next state.\nThis query will be forwarded in an epidemic approach to the nodes in the relay set.\nThe packet is explained in depth in Section \\ref{results}, containing the query itself but also the information to generate the VHT.\nThis will communicate all the network information to create the VHT, which is the foundation of the following state of the network.\nThis process will go on until a node on the edge of the network is reached.\n\nAlong this state, there are some specific challenges to discuss.\n\\begin{enumerate*}[label=(\\roman*)]\n  \\item The first challenge is if a node receives more than one monitoring packet once it is already in a monitoring state.\n \n  For this, the node will take the first monitoring packet and will discard all the subsequent monitoring packets.\n  \\item The second challenge is, what if the propagation of the query is interrupted by a node that remains in a cyclic state. \n  For this, we introduce a timeout for the packet to avoid these problems. \n \n  The idea is to provide a mechanism to avoid loops in the communications.\n \n  For this problem, the timeout will be triggered and once reached, this node will start the aggregation process by sending its result to the parent node.\n \n  \\item The third challenge is the broadcast of the packet itself.\n  Due to the nature of the simple epidemic dissemination approach, a packet will be forwarded to the next hop of nodes but also to the parent node.\n \n  We decided that this will work as an acknowledgment of the child node to the parent node.\n  This way, the parent node will receive $n$\n  acknowledgments and he will know how many packets he should wait for before changing to the aggregate state.\n \n\\end{enumerate*}\n\n\\subsubsection{Aggregate State} \\label{aggregateState}\n\nOnce the data is disseminated up to the edge of the network, the edge nodes will change from query state to aggregate state and will start sending recursively their information to their parent up to the start node.\nThis process will be an aggregation of all the data of a node and his children in order to collect the monitored values.\nThe aggregation will be computed in a hierarchical manner with a combination when required of a gossip approach.\nA node will compute based on his own observations the result of the function $f(x)$ that received from the query state.\nThis information will be aggregated with the same child nodes information.\nIn the edge nodes cases, where this state starts, it will be done only with information from themselves.\n\nAlong this state, there are some specific challenges to discuss.\n\\begin{enumerate*}[label=(\\roman*)]\n  \\item The first challenge is when a parent node and a corresponding child node goes out of range from when they first met.\n \n  When the child node sends an aggregate type of message and receives no acknowledgment it will trigger a forward packet to the corresponding node.\n \n  For this, we will rely on the routing protocol of the network.\n  This makes our approach dependent on the routing layer of the network and we will consider our own opportunistic routing mechanism in forthcoming works.\n \n  \\item The second challenge is when a parent node is off line.\n \n  For this, we propose that in the query state, a set of nodes are communicated to every child node for them to have an alternative path.\n  Since the child node will have the relay set of the parent node, he will fall back into one of these nodes to send the information.\n \n  Since it is a hierarchical approach, the parent will send the information about his parents in the VHT.\n \n  \\item The third challenge is when a node receives a grandchild node aggregate information.\n  For this, the node will assume that the child node is off line and that he will be aggregating that information.\n  Given that the node does not know the information of how many grandchild will send information, he will also rely on the timeout before he sends his own aggregate information.\n  For every grandchild packet he receives, he will restart the timeout to give time for additional packages.\n  If the timeout is reached, he will continue with his aggregation process.\n \n\\end{enumerate*}\n\n\n\n\n\n\n\\section{Experiments} \\label {results}\n\n\n\\subsection{Protocol Definition} \\label{protocolAutomata}\n\n\\input{04_results\/automata}\n\nThe protocol definition, depicted in Figure \\ref{fig:automata}, shows the expected behavior of the protocol to support as base ground for the hybrid monitoring architecture.\nThe set of states is $Q = (Initial, Q1, Q2, A1, A3, A3)$.\nWhere $Initial$ is the initial state.\nThe states $Q1$ and $Q2$ refer to the query states of the network.\nAnd the states $A1$, $A2$ and $A3$ refer to the aggregate states of the network.\nThe internal operations of the automaton are startMonitoring(), acc(IP), timeout(), done() and error().\nThe startMonitoring() refers to the process of starting the monitoring.\nThe acc(IP) refers to the process of the node of accumulating the IP of the acknowledgment messages source.\nThis is used to identify while the query is propagating if there are child nodes available for a given node.\nIf a node does not receive any acknowledgment, he will continue the monitoring process by using a timeout.\nThe timeout() refers to the process of counting time since the last package received.\nThe done() refers to\nthe restart of the state of the node.\nMeanwhile, error() refers to the process of not being able to send a message, which if it happens, it means that the node itself is out of the network range or a major outage is happening with the network.\nThe input and output operations of the automaton are determined by sending (SND) and receiving (RCV) messages.\nThe possible messages to be sent or received are the query, query ack, aggregate, aggregate ack, agregate route and aggregate forward.\nThe query message refers to the query itself and the base ground of the query state.\nFor simplicity purposes, in the automaton, there is a distinction between the query and the query ack message.\nBut in reality, they are meant to be the same package but received by a different node. \nThis is discussed in Section \\ref{queryState}.\nThe aggregate messages refer to the aggregation process and the same principle applies as the query messages.\nThe aggregate ack message is an aggregate message but received by a different node.\nThen we also have two extra messages which are the aggregate route and aggregate forward.\nThe aggregate route message refers to the process of routing a message through the network to the corresponding parent node in the VHT.\nAs explained in Section \\ref{aggregateState}, the idea is to make the hierarchical aggregation more robust through the addition of a gossip routing approach to route the package to the corresponding root node on the fly.\nAnd finally, the aggregate forward message, which in the case that the parent node is not found, probably because the parent node went offline due to an outage or something similar.\nIn this case, the message will be forwarded to one of the nodes defined in the relay set, which will be populated by the grandparents and siblings.\n\n\n\n\\subsection{Packet Definition} \\label{packetDefinition}\n\nIn order for the communication to be successful, we need to define the monitoring packet.\nThe packet will work equally in both states of the network, query and aggregate states, but different information will be sent depending on the state containing a set of common properties.\nIt needs to contain some basic information in order to be useful for the following nodes and hops.\nThe definition of such packet will be done using json.\nFor each state of the nodes, there will be a set of properties transmitted.\nThere will be a set of global properties that will always be transmitted. \nThese global properties are:\n\\begin{enumerate*}[leftmargin=0.4cm]\n  \\item Type: the type of message being sent, the set of values is listed in \\ref{protocolAutomata}.\n  \\item Parent: the IP address of the parent node.\n  \\item Source: the IP address of the node sending the message.\n \n  \\item Timeout: the timeout value in milliseconds.\n\\end{enumerate*}\nFor the query state the properties transmitted are: \n\\begin{enumerate*}[leftmargin=0.4cm]\n  \\item Function: the function $f$ to compute.\n  \\item Relay Set: the list of IPs for alternative paths, with at most three items.\n\\end{enumerate*}\nFor the aggregate state the properties transmitted are:\n\\begin{enumerate*}[leftmargin=0.4cm]\n  \\item Result: the result of the aggregation of the function $f$. \n \n  \\item Destination: the destination IP that should be the parent IP for most of the cases, unless the parent node is off line, then it will be a relay set IP.\n  \\item Observations: the number of aggregated observations.\n\\end{enumerate*}\nThe json definition of the complete package is the following:\n\n\\input{04_results\/packetDefinition}\n\n\\subsection{Results} \\label{implementation}\n\n\\input{04_results\/parametersTable}\n\nWe evaluate our proposal using an emulator built in-house based on DOCKEMU \\cite{to2015dockemu}. \nThis emulator is a combination between Docker and NS3, which allows to conduct highly scalable, replicable and robust experiments.\nThe testbed consisted in an implementation of the protocol in the language Go, that was deployed on our emulator.\nThe idea was to determine the convergence time, by which we mean the time it took from the moment that the monitoring started by the root node, to the moment that the root node was able to return a verdict.\nWe defined two scenarios, one scenario with no mobility and another with low mobility.\nFor this study, we are testing the implementation without the mobility support. This means we do not consider states $A2$ and $A3$ of Figure \\ref{fig:automata}.\nScenario 1 and scenario 2 consider the parameters of Table \\ref{table:scenario1param}.\nFor both scenarios the Mac Protocol is 802.11a with a data rate of 54Mbps.\nEach node had a range of $\\approx$125m.\nScenario 1 was designed to test the convergence time, the scenario 2 to prove that due to the high performance of the algorithm, we may monitor in a mobile environment without the mobility support.\n\nThe emulator was running on top of an Amazon EC2 t2.large instance and Ubuntu 16.04.\nVersions in use were Docker 1.12.1, NS3.25 and Go 1.6.2.\nThe containers were running as a base Ubuntu 16.04 LTS and IPv4.\n\n\\subsubsection{Scenario 1}\n\n\n\\imageref[0.45]{Rplot1.png}{Scenario 1 convergence results}{scenario1results\n\n\nFor our first scenario, we collected the convergence time using a different root node as a starting point in each run.\nWe decided to use different root node selected randomly to prove that it will work independently of who the root node is.\nThe results are summarized in Figure \\ref{fig:scenario1results}.\nWe can point out that there is a clear relationship between the number of nodes and the time it takes to converge.\nWith 50 nodes to 100 nodes, the average value seems to stabilize in around $\\approx$2.15s.\nAbout the number of packets sent, we empirically assumed that for the static environments, the number would be twice the number of nodes. This can be deducted because everyone will send their query message one time and their aggregate message one time as well.\nThe variability of the nodes will occur when the mobility support is added.\nFor $N$ nodes, the messages sent are $2*N$ for static environments.\nThe average message size is $\\approx$151 bytes.\n\n\\subsubsection{Scenario 2}\n\nFor the second scenario, we collected the convergence time but also the amount of observations collected by the root node at the end of each run.\nThe results are summarized in Table \\ref{table:scenario2results}.\nWe can observe that the nodes converge about the same amount of time that they do in a static environment.\nWe observed that it would converge but without all the possible observations on the network.\nAnd on top of that, between the more the speed of the nodes the lower the success rate would be.\nBy success rate, we define if the monitoring process was able to converge.\nThe algorithm has proved in static environments that is capable of converging really fast, even though is not using the mobility support, suggesting promising future results.\n\n{\\small\n\\begin{table}\n  \\caption{Scenario 2 results}\\label{table:scenario2results}\n  \\vspace{-1em}\n  \\begin{center}\n      \\begin{tabular}{| l | l | l |}\n      \\hline\n       & \\textbf{Speed 2m\/s} & \\textbf{Speed 5m\/s} \\\\ \\hline\n      \\textbf{Average convergence time (ms)} & 2018.06 & 2123.63  \\\\ \\hline\n      \\textbf{Average observations (\\# nodes)} & 22.68 & 21.25  \\\\ \\hline\n      \\textbf{Success rate} & 0.8 & 0.4  \\\\ \\hline\n      \\end{tabular}\n  \\end{center}\n  \\vspace{-2em}\n\\end{table}\n}\n\n\\section{Future Work} \\label{futurework}\n\nWe intend to study the selection of the root node.\nIt could be based on location, energy, computing power and other parameters.\nIt could also be an autonomous process, proactively or reactively, or a manual process.\n\nBesides we plan to introduce the mobility support and enhance the testbeds to this specific cases.\nWe will target diverse distributed monitored environment such as urban networks. \nWe will be looking for specific metrics under different scenarios in a scalable way.\n\n\nLastly we intent to analyze and take into consideration more complex functions.\nWe are interested in monitoring the interoperability within a MANET.\nFor this we need to define an optimal solution to propagate a more complex function through our query mechanism.\nTo monitor interoperability we believe that it is needed to analyze not all observation points in the network, but more specifically a subset of observation points.\n\\section{Related Works} \\label{relatedworks}\n\nMANET monitoring has been studied\nfor many objectives like their performances \\cite{mehrotra2014performance}, to test them \\cite{merouane2007methodology}, their security \\cite{nadeem2013survey} and more recently their energetic efficiency \\cite{palaniappan2015energy}.\n\nGossipico \\cite{van2012gossip} is an algorithm to calculate the average, the sum or the count of node values in a large dynamic network.\nThe foundation of the algorithm is through two parts: count and beacon.\nThe combination of these two mechanisms\nprovides the advantage for counting the nodes inside a network in an efficient and quick way.\nMobi-G \\cite{stingl2014mobi} is designed for urban outdoor areas with a focus on pedestrian that moves around.\nThe idea is to create the global view of an attribute incorporating all the nodes in the network.\nNevertheless the accuracy decreases for an increasing spatial network size.\nOn the hierarchical categorization, we can mention BlockTree \\cite{stingl2013blocktree}, which is a fully decentralized location-aware monitoring mechanism for MANETs.\nThe idea is to divide the network in proximity-based clusters, which are arranged hierarchical.\nThis approach scales with the spatial network size and provides accurate results. \nIn \\cite{stingl2012benchmarking}, the main key points in architectural description for decentralized monitoring mechanisms are depicted.\nHowever, it is difficult to determine a better performer since both perform better in diverse scenarios and workloads.\n\\section{Conclusions} \\label{conclusions}\n\nWe have presented in this paper a hybrid algorithm for monitoring decentralized networks that consists on the combination of gossip-based and hierarchical-based algorithms.\nThe gossip-based approach is applied to disseminate the query and the hierarchical approach is applied to aggregate the data.\nBesides, with the help of a time-based hierarchical approach, the computation of a global property is achieved.\nWe designed a scalable and configurable testbed using NS3 and Docker, based on DOCKEMU \\cite{to2015dockemu}.\nOur methodology and results seem promising for a wide set of scenarios. \n\nAs future works, we intend to study the selection of the root node.\nIt could be based on location, energy, computing power and other parameters, or to be an autonomous process, proactively or reactively, or a manual process.\nBesides we plan to introduce the mobility support and enhance the testbeds to this specific cases.\nWe also intend to consider more complex functions in monitoring the MANETs interoperability. For this we need to define an optimal solution to propagate a more complex function through our query mechanism.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:Motivation}\nThe neighborhood grid data structure can be used to compute estimates of neighborhoods in point sets. It has been introduced by Joselli et al. \\cite{joselli2009neighborhood}, \\cite{joselli2015neighborhood}. In order to give a short introduction to the data structure, consider the example in Figure \\ref{fig:PipelineGrid}. It shows how points from a point cloud (Figure \\ref{fig:Pipeline1PointCloud}) are placed in a grid (Figure \\ref{fig:Pipeline2Grid}). The order in which the points are given is random, thus their initial placement in the grid is also. After the placement, only the coordinates of the points are considered in the grid (Figure \\ref{fig:Pipeline3GridCoordinates}).\n\n\\begin{figure}[htb!]\n\t\\begin{subfigure}[c]{0.3\\linewidth}\n\t\t\\resizebox{1.0\\linewidth}{!}{\n\t\t\t\\begin{tikzpicture}\n\t\t\t\t\\draw[line width=5pt] (0,0) -- (0,10) -- (10,10) -- (10,0) -- (0,0);\n\t\t\t\t\\filldraw[red]    (2.06,7.76) circle (7pt);\n\t\t\t\t\\filldraw[orange] (3.73,6.84) circle (7pt);\n\t\t\t\t\\filldraw[ProcessBlue] (9.18,9.05) circle (7pt);\n\t\t\t\t\\filldraw[lightgray]   (1.77,5.46) circle (7pt);\n\t\t\t\t\\filldraw[blue]   (4.07,5.13) circle (7pt);\n\t\t\t\t\\filldraw[ForestGreen]  (8.23,4.79) circle (7pt);\n\t\t\t\t\\filldraw[black]  (1.53,1.30) circle (7pt);\n\t\t\t\t\\filldraw[teal]   (4.27,1.45) circle (7pt);\n\t\t\t\t\\filldraw[purple] (8.41,1.96) circle (7pt);\t\t\t\n\t\t\t\\end{tikzpicture}\n\t\t}\n\t\t\\caption{Point cloud with nine points in $\\mathbb{R}^2$.}\n\t\t\\label{fig:Pipeline1PointCloud}\n\t\\end{subfigure}\n\t\\hfill\n\t\\begin{subfigure}[c]{0.3\\textwidth}\n\t\t\\resizebox{1.0\\linewidth}{!}{\n\t\t\t\\begin{tikzpicture}\n\t\t\t\t\\draw (0,0) grid (3,3);\n\t\t\t\t\\filldraw[blue] (0.5,0.5) circle (7pt);\n\t\t\t\t\\filldraw[red] (0.5,1.5) circle (7pt);\n\t\t\t\t\\filldraw[ForestGreen] (0.5,2.5) circle (7pt);\n\t\t\t\t\\filldraw[orange] (1.5,0.5) circle (7pt);\n\t\t\t\t\\filldraw[lightgray] (1.5,1.5) circle (7pt);\n\t\t\t\t\\filldraw[purple] (1.5,2.5) circle (7pt);\n\t\t\t\t\\filldraw[teal] (2.5,0.5) circle (7pt);\n\t\t\t\t\\filldraw[ProcessBlue] (2.5,1.5) circle (7pt);\n\t\t\t\t\\filldraw[black] (2.5,2.5) circle (7pt);\n\t\t\t\\end{tikzpicture}\n\t\t}\n\t\t\\caption{The points are initially placed randomly.}\n\t\t\\label{fig:Pipeline2Grid}\n\t\\end{subfigure}\n\t\\hfill\n\t\\begin{subfigure}[c]{0.3\\textwidth}\n\t\t\\resizebox{1.0\\linewidth}{!}{\n\t\t\t\\begin{tikzpicture}\n\t\t\t\t\\draw[step=2.0] (0,0) grid (6,6);\n\t\t\t\t\\filldraw[blue,align=center] (1,1) node () {$4.07,5.13$};\n\t\t\t\t\\draw[red,align=center] (1,3) node () {$2.06,7.76$};\n\t\t\t\t\\filldraw[ForestGreen,align=center] (1,5) node () {$8.23,4.79$};\n\t\t\t\t\\filldraw[orange,align=center] (3,1) node () {$3.73,6.84$};\n\t\t\t\t\\filldraw[lightgray,align=center] (3,3) node () {$1.77,5.46$};\n\t\t\t\t\\filldraw[purple,align=center] (3,5) node () {$8.41,1.96$};\n\t\t\t\t\\filldraw[teal,align=center] (5,1) node () {$4.27,1.45$};\n\t\t\t\t\\filldraw[ProcessBlue,align=center] (5,3) node () {$9.18,9.05$};\n\t\t\t\t\\filldraw[black,align=center] (5,5) node () {$1.53,1.30$};\n\t\t\t\\end{tikzpicture}\n\t\t}\n\t\t\\caption{The point coordinates are considered.}\n\t\t\\label{fig:Pipeline3GridCoordinates}\n\t\\end{subfigure}\n\t\\caption{First part of the neighborhood grid pipeline.}\n\t\\label{fig:PipelineGrid}\n\\end{figure}\n\nThe grid as obtained in Figure \\ref{fig:Pipeline3GridCoordinates} will now be sorted. Each row should grow in the first coordinates from left to right, each column should grow in the second coordinates from bottom to top. A corresponding sorted grid is given in Figure \\ref{fig:Pipeline4GridSorted}. Note how it, in this example, recovers the combinatorial neighborhood relation from the points.\\\\\n\nIn order to use the neighborhood grid to determine a neighborhood estimate for a given point, find that point in the sorted grid. Then, consider a small neighborhood around that point, e.g. the one-ring around it. The size of this neighborhood should not depend on the number of inserted points such that this lookup runs in constant time (asymptotically). From that neighborhood, find the closest point to the considered point and output it as estimated nearest neighbor.\n\n\\begin{figure}\n\t\\begin{subfigure}[c]{0.4\\textwidth}\n\t\t\\resizebox{1.0\\linewidth}{!}{\n\t\t\t\\begin{tikzpicture}\n\t\t\t\t\\draw[step=2.0] (0,0) grid (6,6);\n\t\t\t\t\\filldraw[blue,align=center] (3,3) node () {$4.07,5.13$};\n\t\t\t\t\\draw[red,align=center] (1,5) node () {$2.06,7.76$};\n\t\t\t\t\\filldraw[ForestGreen,align=center] (5,3) node () {$8.23,4.79$};\n\t\t\t\t\\filldraw[orange,align=center] (3,5) node () {$3.73,6.84$};\n\t\t\t\t\\filldraw[lightgray,align=center] (1,3) node () {$1.77,5.46$};\n\t\t\t\t\\filldraw[purple,align=center] (5,1) node () {$8.41,1.96$};\n\t\t\t\t\\filldraw[teal,align=center] (3,1) node () {$4.27,1.45$};\n\t\t\t\t\\filldraw[ProcessBlue,align=center] (5,5) node () {$9.18,9.05$};\n\t\t\t\t\\filldraw[black,align=center] (1,1) node () {$1.53,1.30$};\n\t\t\t\\end{tikzpicture}\n\t\t}\n\t\t\\caption{Coordinates are sorted to grow in columns and rows.}\n\t\t\\label{fig:Pipeline4GridSorted}\n\t\\end{subfigure}\n\t\\hfill\n\t\\begin{subfigure}[c]{0.4\\textwidth}\n\t\t\\resizebox{1.0\\linewidth}{!}{\n\t\t\t\\begin{tikzpicture}\n\t\t\t\\filldraw[Apricot!30] (0,2) rectangle (4,6);\n\t\t\t\\filldraw[Lavender!30] (0,4) rectangle (2,6);\n\t\t\t\\draw[step=2.0] (0,0) grid (6,6);\n\t\t\t\\filldraw[blue,align=center] (3,3) node () {$4.07,5.13$};\n\t\t\t\\draw[red,align=center] (1,5) node () {$2.06,7.76$};\n\t\t\t\\filldraw[ForestGreen,align=center] (5,3) node () {$8.23,4.79$};\n\t\t\t\\filldraw[orange,align=center] (3,5) node () {$3.73,6.84$};\n\t\t\t\\filldraw[lightgray,align=center] (1,3) node () {$1.77,5.46$};\n\t\t\t\\filldraw[purple,align=center] (5,1) node () {$8.41,1.96$};\n\t\t\t\\filldraw[teal,align=center] (3,1) node () {$4.27,1.45$};\n\t\t\t\\filldraw[ProcessBlue,align=center] (5,5) node () {$9.18,9.05$};\n\t\t\t\\filldraw[black,align=center] (1,1) node () {$1.53,1.30$};\n\t\t\t\\end{tikzpicture}\n\t\t}\n\t\t\\caption{Determining the neighbors of the \\textcolor{Lavender}{upper left point} by looking at its \\textcolor{Apricot}{neighbors} in the grid.}\n\t\t\\label{fig:Pipeline5GridNeighbors}\n\t\\end{subfigure}\n\t\\caption{Second part of the neighborhood grid pipeline.}\n\t\\label{fig:PipelineGrid2}\n\\end{figure}\n\nIn this report, we collect new results on the neighborhood grid data structure. Malheiros and Walter \\cite{malheiros2015simple} investigated several iterative building strategies for the data structure. Despite the evidences of practical relevance, as given in the publication cited above, neither Joselli nor Malheiros investigated the asymptotic building times of the grid or answered the question for a time-optimal building algorithm. Therefore, this paper contains:\n\\begin{itemize}\n\t\\item a polynomial-time algorithm to build a neighborhood grid (Theorem \\ref{the:StableStateExistence}, page \\pageref{the:StableStateExistence}),\n\t\\item combinatorial results on the number of possible sorted placements (Theorem \\ref{the:Countings}, \\pageref{the:Countings}),\n\t\\item a complete list of unique sorted placements for $n\\in\\{1,2,3\\}$ (Section \\ref{subsec:EnumerationOfAllUniqueStableStates}),\n\t\\item a proof of non-existence of unique sorted placements for $n\\geq 4$ (Section \\ref{subsec:EnumerationOfAllUniqueStableStates}).\n\\end{itemize}\nSo far, some questions remain unanswered:\n\\begin{itemize}\n\t\\item Is the presented algorithm time-optimal?\n\t\\item For a given $n\\in\\mathbb{N}$, $n\\geq4$, what is a point configuration with the least or largest number of stable states?\n\\end{itemize}\nFor the case of the latter question we present a conjecture, see page \\pageref{con:UpperBoundStableStates}.\n\n\n\\section{The Neighborhood Grid}\n\\label{sec:TheNeighborhoodGrid}\n\nIn this first section, we present the neighborhood data structure, fix corresponding notation, and prove a first theorem on a polynomial-time building algorithm.\n\n\\subsection{Definition of the Data Structure}\n\\label{subsec:DefinitionOfTheDataStructure}\n\nGiven a set of points $P=\\{p_1,\\ldots,p_N\\mid p_i\\in\\mathbb{R}^d\\}$. In the following we will assume that $N=n^2$ for some $n\\in\\mathbb{N}$ and $d=2$. Therefore each point is given by $p_i=(p_{i1},p_{i2})\\in\\mathbb{R}^2$, where $p_{i1}$ will be referred to as $x$- and $p_{i2}$ as $y$-value. Furthermore, we assume that $p_i\\neq p_j$ for all $i\\neq j$. Consider Section \\ref{sec:GeneralCase} for the general case without these restrictions. Finally, we can restrict w.l.o.g. to $p_i\\in\\mathbb{N}^2$ for all $i\\in[N]$, see Section \\ref{subsec:CountingStableStates} for an explanation of this restriction.\\\\\n\nThe points will be placed in an $n\\times n$ matrix, where each cell of the matrix contains a point $p_i$. That is, we consider a matrix $M\\in(\\mathbb{R}^2)^{n\\times n}$ which then has the form\n\n\\begin{align}\n\\label{equ:matrix}\n\tM=\n\t\\begin{array}{|c|c|c|}\n\t\\hline\n\t\t(a_{1n},b_{1n}) & \\ldots & (a_{nn},b_{nn})\\\\\n\t\\hline\n\t\t\\vdots & \\iddots & \\vdots\\\\\n\t\\hline\n\t\t(a_{11},b_{11}) & \\ldots & (a_{n1},b_{n1})\\\\\n\t\\hline\n\t\\end{array}.\n\\end{align}\n\nUltimately, we want to order the points in the matrix such that the following state is reached.\n\n\\begin{defi}\n\t\\label{def:StableState}\n\tThe matrix $M$ as given in (\\ref{equ:matrix}) is said to be in a \\textbf{stable state} if and only if the following two conditions are satisfied for any $i,j\\in[n]$, $i\\neq j$.\n\t\\begin{enumerate}\n\t\t\\item For all $k\\in [n]$ it is: $i<j \\Rightarrow a_{ki}<a_{kj}\\vee (a_{ki}=a_{kj}\\wedge b_{ki}<b_{kj})$.\n\t\t\\item  For all $\\ell \\in [n]$ it is: $i<j \\Rightarrow b_{i\\ell}<b_{j\\ell}\\vee (b_{i\\ell}=b_{j\\ell}\\wedge a_{i\\ell}<a_{j\\ell})$\n\t\\end{enumerate}\n\\end{defi}\n\nIn other words, a matrix $M$ is in a stable state, if the points in each row of $M$ are ordered lexicographically according to the first and then the second coordinate. Similarly, all columns of $M$ have to be ordered lexicographically according to the second and then the first coordinate. An illustration of Definition \\ref{def:StableState} is given in Figure \\ref{fig:IllustrateStableState}. We call a stable state \\textbf{unique}, if there exists no other stable state for the same point configuration $P$. \n\n\\begin{figure}[htb!]\n\t\\begin{tikzpicture}\n\t\t\\draw[fill=green,opacity=0.1] (0,4) rectangle (4,6);\n\t\t\\draw[fill=red,opacity=0.1]   (2,2) rectangle (6,4);\n\t\t\\draw[fill=green,opacity=0.1] (0,0) rectangle (4,2);\n\t\t\\draw[step=2.0] (0,0) grid (6,6);\n\t\t\\draw (1,5) node {$(55,42)$};\n\t\t\\draw (3,5) node {$(60,49)$};\n\t\t\\draw (3,3) node {$(26,61)$};\n\t\t\\draw (5,3) node {$(13,69)$};\n\t\t\\draw (1,1) node {$(95,13)$};\n\t\t\\draw (3,1) node {$(95,65)$};\n\t\\end{tikzpicture}\n\t\\hfill\n\t\\begin{tikzpicture}\n\t\\draw[step=2.0] (0,0) grid (6,6);\n\t\\draw (1,5) node {$(06,69)$};\n\t\\draw (3,5) node {$(26,61)$};\n\t\\draw (5,5) node {$(86,89)$};\n\t\\draw (1,3) node {$(02,55)$};\n\t\\draw (3,3) node {$(80,34)$};\n\t\\draw (5,3) node {$(86,41)$};\n\t\\draw (1,1) node {$(05,19)$};\n\t\\draw (3,1) node {$(47,11)$};\n\t\\draw (5,1) node {$(95,13)$};\n\t\\end{tikzpicture}\n\t\\caption{On the left three cases where two marked in green satisfy the first condition of Definition \\ref{def:StableState}, while the red case violates it. On the right a $3\\times 3$ matrix $M$ in stable state.}\n\t\\label{fig:IllustrateStableState}\n\\end{figure}\n\n\\subsection{Polynomial-Time building algorithm}\n\\label{subsec:PolynomialTimeBuildingAlgorithm}\n\nNow the following question arises naturally. For any set of points $P$ as specified above, is there a bijective placement $\\pi:[n^2]\\rightarrow[n]\\times[n]$, $i\\mapsto(k,\\ell)$ such that the matrix $M_\\pi(P)$ with \n\n\\begin{align}\n\\label{equ:placedMatrix}\nM=\n\\begin{array}{|c|c|c|}\n\\hline\n(p_{\\pi^{-1}(n,1)1},p_{\\pi^{-1}(n,1)2}) & \\ldots & (p_{\\pi^{-1}(n,n)1},p_{\\pi^{-1}(n,n)2})\\\\\n\\hline\n\\vdots & \\iddots & \\vdots\\\\\n\\hline\n(p_{\\pi^{-1}(1,1)1},p_{\\pi^{-1}(1,1)2}) & \\ldots & (p_{\\pi^{-1}(1,n)1},p_{\\pi^{-1}(1,n)2})\\\\\n\\hline\n\\end{array}.\n\\end{align}\n\nis in a stable state. In other words, given $n^2$ points, can these be written into an $n\\times n$ matrix s.t. it is in a stable state as defined in Definition \\ref{def:StableState}.\n\n\\begin{theo}\n\t\\label{the:StableStateExistence}\n\tFor every set of points $P=\\{p_1,\\ldots,p_N\\mid p_i\\in\\mathbb{R}^2\\}$ there is a bijective placement $\\pi$ such that $M_\\pi(P)$ is in a stable state. A placement $\\pi$ can be found in $\\mathcal{O}(N\\log(N))$.\n\\end{theo}\n\\begin{proof}\n\tConsider the points $p_1,\\ldots,p_N$ as a sequence. Sort this sequence according to the first condition given in Definition \\ref{def:StableState}. Obtain a sequence\n\t\\begin{align*}\n\t\t(q_{11},q_{12}),(q_{21},q_{22}),\\ldots,(q_{N1},q_{N2}),\n\t\\end{align*}\n\twhere for $i,j\\in[n]$, $i<j$ we have $q_{i1}<q_{j1}$ or $(q_{i1}=q_{j1} \\wedge q_{i2}<q_{j2})$. Now split this sequence into $n$ blocks as follows:\n\t\\begin{align*}\n\t\t\\underbrace{(q_{11},q_{12}),\\ldots,(q_{n1},q_{n2})}_{=:Q_1},\n\t\t\\underbrace{(q_{(n+1)1},q_{(n+1)2}),\\ldots,(q_{(2n)1},q_{(2n)2})}_{=:Q_2},\\\\\n\t\t\\ldots\n\t\t\\underbrace{(q_{(n^2-n+1)1},q_{(n^2-n+1)2}),\\ldots,(q_{N1},q_{N2}),}_{=:Q_n}.\n\t\\end{align*}\n\tNow consider each sequence $Q_i$ and sort it according to the second condition given in Definition \\ref{def:StableState}. Obtain a sequence\n\t\\begin{align*}\n\t\tR_k:=(r_{11},r_{12}),(r_{21},r_{22}),\\ldots,(r_{n1},r_{n2}),\\ k\\in[n],\n\t\\end{align*}\n\twhere for $i<j$ we have $r_{i2}<r_{j2}$ or $(r_{i2}=r_{j2}\\wedge r_{i1}<r_{j1})$. That is, the points in the sequence $R_k$ are sorted according to the second condition of Definition \\ref{def:StableState}. Furthermore, for $i<j$, any point from $R_i$ satisfies the first condition of Definition \\ref{def:StableState} when compared to any point from $R_j$, since the $R_k$ derive from the $Q_k$.\tTherefore, placing the sequence $R_k$ into the $k$th column of the matrix $M$ results in a stable state.\n\t\n\t\n\tConcerning the runtime, in the first step, $N$ points were sorted, which takes $\\mathcal{O}(N\\log(N))$. In the second step, $n$ sets of $n$ points each were sorted, which takes $n\\cdot\\mathcal{O}(n\\log(n))=\\mathcal{O}(n^2\\log(\\sqrt{N}))=\\mathcal{O}(N\\log(N))$. Hence, the stable state was computed in $\\mathcal{O}(N\\log(N))$.\n\\end{proof}\n\nAn illustration of the procedure presented by this theorem is given in \\mbox{Figure \\ref{fig:IllustrateSortingAlgorithm}}.\n\n\\begin{figure}[htb!]\n\t\\centering\n\t$\\{(26,61),\\ (06,69),\\ (95,13),\\ (86,41),\\ (05,19),\\ (02,55),\\ (86,89),\\ (47,11),\\ (80,34)\\}$\\\\[0.25cm]\n\t\n\t$\\downarrow$ Consider all points as one sequence. $\\downarrow$\\\\[0.25cm]\n\t\n\t\\begin{tikzpicture}[scale=0.6]\n\t\t\\draw[step=2.0] (0,0) grid (18,2);\n\t\t\\draw (1,1) node {\\footnotesize $(26,61)$};\n\t\t\\draw (3,1) node {\\footnotesize $(06,69)$};\n\t\t\\draw (5,1) node {\\footnotesize $(95,13)$};\n\t\t\\draw (7,1) node {\\footnotesize $(86,41)$};\n\t\t\\draw (9,1) node {\\footnotesize $(05,19)$};\n\t\t\\draw (11,1) node {\\footnotesize $(02,55)$};\n\t\t\\draw (13,1) node {\\footnotesize $(86,89)$};\n\t\t\\draw (15,1) node {\\footnotesize $(47,11)$};\n\t\t\\draw (17,1) node {\\footnotesize $(80,34)$};\n\t\\end{tikzpicture}\\\\[0.25cm]\n\t\n\t$\\downarrow$ Sort the sequence according to the first condition of Definition \\ref{def:StableState}. $\\downarrow$\\\\[0.25cm]\n\t\n\t\\begin{tikzpicture}[scale=0.6]\n\t\t\\draw[step=2.0] (0,0) grid (18,2);\n\t\t\\draw (1,1) node {\\footnotesize $(02,55)$};\n\t\t\\draw (3,1) node {\\footnotesize $(05,19)$};\n\t\t\\draw (5,1) node {\\footnotesize $(06,69)$};\n\t\t\\draw (7,1) node {\\footnotesize $(26,61)$};\n\t\t\\draw (9,1) node {\\footnotesize $(47,11)$};\n\t\t\\draw (11,1) node {\\footnotesize $(80,34)$};\n\t\t\\draw (13,1) node {\\footnotesize $(86,41)$};\n\t\t\\draw (15,1) node {\\footnotesize $(86,89)$};\n\t\t\\draw (17,1) node {\\footnotesize $(95,13)$};\n\t\\end{tikzpicture}\\\\[0.25cm]\n\t\n\t$\\downarrow$ Separate into chunks of size $n$ and sort them according to the second condition of Definition \\ref{def:StableState}. $\\downarrow$\\\\[0.25cm]\n\t\n\t\\begin{tikzpicture}[scale=0.6]\n\t\\draw[step=2.0] (0,0) grid (2,6);\n\t\\draw[step=2.0] (4,0) grid (6,6);\n\t\\draw[step=2.0] (8,0) grid (10,6);\n\t\\draw (1,1) node {\\footnotesize $(05,19)$};\n\t\\draw (1,3) node {\\footnotesize $(02,55)$};\n\t\\draw (1,5) node {\\footnotesize $(06,69)$};\n\t\\draw (5,1) node {\\footnotesize $(47,11)$};\n\t\\draw (5,3) node {\\footnotesize $(80,34)$};\n\t\\draw (5,5) node {\\footnotesize $(26,61)$};\n\t\\draw (9,1) node {\\footnotesize $(95,13)$};\n\t\\draw (9,3) node {\\footnotesize $(86,41)$};\n\t\\draw (9,5) node {\\footnotesize $(86,89)$};\n\t\\end{tikzpicture}\n\t\\caption{An illustration of the algorithm outlined in Theorem \\ref{the:StableStateExistence}. The last step gives the rows of the final matrix which is then in a stable state.}\n\t\\label{fig:IllustrateSortingAlgorithm}\n\\end{figure}\n\nTheorem \\ref{the:StableStateExistence} imposes an upper bound on the runtime of any time-optimal comparison-based algorithm that creates a stable state of a matrix $M$. The next question is then: What is a lower bound? A trivial lower bound is given by the number of points in the matrix, $N$. In the next section, we investigate whether this lower bound can be raised to $N\\log(N)$.\n\n\\section{Combinatorial Results}\n\\label{sec:CombinatorialResults}\n\nIn order to better understand the search space for any sorting algorithm that is to build a stable state of a matrix $M$, we start by counting the number of possible stable states and giving two necessary conditions on their uniqueness. Then, we proceed to give a list of all point configurations $P$ for $n\\in\\{1,2,3\\}$ that omit to a single, unique stable state. As a last part of this section, we investigate which point configuration has the largest number of stable states.\n\n\\subsection{Counting Stable States and a lower Bound}\n\\label{subsec:CountingStableStates}\n\nConcerning the stable state of a matrix, it is not important what actual $x$- or $y$-values the points have, but only how these values compare to each other. Therefore, the setup can be simplified by fixing the $x$ and $y$ values on the set $[N]$. In terms of the $p_i$ the restriction is\n\\begin{align}\n\t\\label{equ:restriction}\n\t\\{p_{11},\\ldots,p_{N1}\\}=\\{p_{12},\\ldots,p_{N2}\\}=\\{1,\\ldots,N\\}.\n\\end{align}\nIn this restricted set we can quantify several of the occurring states.\n\n\\begin{theo}\n\t\\label{the:Countings}\n\tGiven some $N,n\\in\\mathbb{N}$ with $N=n^2$, there are:\n\t\\begin{enumerate}\n\t\t\\item $(n^2)!$ possible point sets $P$ and $((n^2)!)^2$ ways to fill the matrix,\n\t\t\\item $((n^2)!\/(n!)^n)^2$ ways to fill the matrix with a stable state,\n\t\t\\item $1\/(n!)^{2n}$ of all fillings of the matrix are stable.\n\t\\end{enumerate}\n\\end{theo}\n\\begin{proof}\n\tConcerning the first statement, when building the points $p_i$, iterate through the $x$-values. For each $x$-value pick one of the $y$-values from $[N]$ not picked yet. Then for $x=1$, there are $N$ choices, for $x=2$, there are $N-1$ choices, etc. Hence there are $N!$ possible point sets. Each $p_i$ of the point set $P$ occupies one of the $N$ points in the matrix, therefore, there are $N!$ ways to write each point set into the matrix. Therefore, there are $(N!)^2$ possible ways to fill the matrix.\n\t\n\tConcerning the second statement, a matrix is in a stable state if both conditions of Definition \\ref{def:StableState} are met. Because of the restriction given by (\\ref{equ:restriction}), the equality case is never met in either condition of Definition \\ref{def:StableState}. Therefore, for each condition, it suffices to check the $x$- or $y$-values respectively, neglecting the other. When setting up the $x$-values for the first row, one can pick $n$ of the possible $N$ values, which then admit to a unique order. Therefore, for the $x$-values in the first row, there are ${n^2 \\choose n}$ possibilities. For the second row, there are ${n^2-n \\choose n}$ possibilities, until there is ${n^2-(n-1)n \\choose n}=1$ possibility for the last row. Overall, there are\n\t\\begin{align*}\n\t\t\\prod_{k=0}^{(n-1)} {n^2-kn \\choose n}=\\frac{n^2!}{(n^2-n)!n!}\\cdot\\frac{(n^2-n)!}{(n^2-2n)!n!}\\cdot\\ldots\\cdot\\frac{(2n)!}{n!n!}\\cdot\\frac{n!}{n!}=\\frac{(n^2)!}{(n!)^n}\t\t\n\t\\end{align*}\n\tpossibilities to put $x$-values into the matrix and obtain a stable state from them. Accordingly, there are $\\frac{(n^2)!}{(n!)^n}$ ways to write $y$-values into the matrix and obtain a stable state. Hence, overall, there are\n\t\\begin{align*}\n\t\t\\left(\\frac{(n^2)!}{(n!)^n}\\right)^2\n\t\\end{align*}\n\tstable states.\n\t\n\tFinally, because of the first two results, the fraction of stable fillings amongst all fillings of the matrix are\n\t\\begin{align*}\n\t\t\\frac{\\left(\\frac{(n^2)!}{(n!)^n}\\right)^2}{((n^2)!)^2}\t= \\frac{1}{(n!)^{2n}}.\n\t\\end{align*}\n\\end{proof}\n\n\\subsection{Uniqueness of Stable States}\n\\label{subsec:UniquenessOfStableStates}\n\nIn the previous section it was shown that the running time of the algorithm outlined in Theorem \\ref{the:StableStateExistence} gives an upper bound on the sorting problem. To prove a lower bound, we assume the following. \n\n\\begin{assump}\n\\label{ass:UniqueStableState}\n\tGiven some $n\\in\\mathbb{N}$, assume that there is a set $P$ of points $p_1,\\ldots,p_N$ such that there exists a unique stable state for $P$.\n\\end{assump}\n\nGiven that assumption, optimality of the Algorithm of Theorem \\ref{the:StableStateExistence} follows by the following argument: \n\\begin{proof}\n\tGiven an $n\\in\\mathbb{N}$ and the set of points $P$ assumed to exist. Then the unique stable state in particular satisfies the first condition of Definition \\ref{def:StableState} for every row. Therefore, the algorithm needs to sort every row, which takes $\\mathcal{O}(n\\log(n))$ time for each row, that is $n\\cdot\\mathcal{O}(n\\log(n))=\\mathcal{O}(N\\log(N))$ time in total. Hence, given Assumption \\ref{ass:UniqueStableState}, $\\mathcal{O}(N\\log(N))$ is a lower bound on the computation of a stable state.\n\\end{proof}\n\nNote that the assumption of the existence of a unique stable state is crucial in this argument. If the assumption was proven to be wrong, the algorithm can pick any stable state to create a sorting of the rows there, which could lead to a faster than $\\mathcal{O}(N\\log(N))$ algorithm because of the larger search space.\n\n\nConcerning Assumption \\ref{ass:UniqueStableState}, it is not at all obvious that for a given point configuration, there has to be a unique stable state. Three point configurations are given in Figure \\ref{fig:MultipleStableStates}, where two have a unique stable state, while the other one has two stable states. The uniqueness of the stable state of the $2\\times 2$ matrix shown follows by this reasoning: The element $(1,1)$ has to be placed in the upper left corner, since it has $x$-, as well as lowest $y$-value compared to all other elements. The element $(2,4)$ has largest $y$-value, therefore it needs to be placed in the lower row. But it also has second lowest $x$-value, therefore it has to be placed in the lower left or upper right corner. Taken these two arguments together, the element has to be placed in the lower left corner. Finally, the shown configuration is the only way to complete the matrix to a stable state.\n\n\\begin{figure}[htb!]\n\t\\begin{tikzpicture}[scale=0.6]\n\t\\draw[step=2.0] (0,0) grid (4,4);\n\t\\draw (1,3) node {$(2,4)$};\n\t\\draw (3,3) node {$(3,3)$};\n\t\\draw (1,1) node {$(1,1)$};\n\t\\draw (3,1) node {$(4,2)$};\n\t\\end{tikzpicture}\n\t\\hfill\n\t\\begin{tikzpicture}[scale=0.6]\n\t\\draw[step=2.0] (0,0) grid (4,4);\n\t\\draw (1,3) node {$(2,2)$};\n\t\\draw (3,3) node {$(4,4)$};\n\t\\draw (1,1) node {$(1,1)$};\n\t\\draw (3,1) node {$(3,3)$};\n\t\\end{tikzpicture}\n\t\\hspace{0.1cm}\n\t\\begin{tikzpicture}[scale=0.6]\n\t\\draw[step=2.0] (0,0) grid (4,4);\n\t\\draw (1,3) node {$(3,3)$};\n\t\\draw (3,3) node {$(4,4)$};\n\t\\draw (1,1) node {$(1,1)$};\n\t\\draw (3,1) node {$(2,2)$};\n\t\\end{tikzpicture}\n\t\\hfill\n\t\\begin{tikzpicture}[scale=0.6]\n\t\\draw[step=2.0] (0,0) grid (6,6);\n\t\\draw (1,1) node {$(1,1)$};\n\t\\draw (3,1) node {$(6,2)$};\n\t\\draw (5,1) node {$(7,3)$};\n\t\\draw (1,3) node {$(2,6)$};\n\t\\draw (3,3) node {$(5,5)$};\n\t\\draw (5,3) node {$(8,4)$};\n\t\\draw (1,5) node {$(3,7)$};\n\t\\draw (3,5) node {$(4,8)$};\n\t\\draw (5,5) node {$(9,9)$};\n\t\\end{tikzpicture}\n\t\\caption{On the left a point configuration with $n=2$ which has a unique stable state. In the middle a point configuration with $n=2$ with its two possible stable states. On the right a point configuration with $n=3$ which has a unique stable state.}\n\t\\label{fig:MultipleStableStates}\n\\end{figure}\n\nThe examples from Figure \\ref{fig:MultipleStableStates} pose two further questions:\n\\begin{enumerate}\n\t\\item Is there a unique stable state for every $n\\in\\mathbb{N}$?\n\t\\item Can we classify all unique stable states?\n\\end{enumerate}\n\n\n\\subsection{First Necessary Condition on Stable States}\n\\label{subsec:FirstNecessaryConditionOnStableStates}\n\nFor a necessary condition on the uniqueness of a stable state reconsider the algorithm outlined in Theorem \\ref{the:StableStateExistence}. Note that the $n$ smallest $x$-values are placed in the first column, the $n$ second-smallest $x$-values are placed in the second column, etc. Finally, a stable state is obtained from this procedure. Furthermore, the algorithm can be run the other way around, placing the $n$ smallest $y$-values in the first row, the $n$ second-smallest $y$ values in the second row, and so on to reach a stable state. In the following, we will call these two conditions \\textbf{$x$-bin and $y$-bin condition}. \n\nAssume now that a stable state is given, which violates the $x$-, the $y$-, or both bin conditions. Then applying the algorithm from Theorem \\ref{the:StableStateExistence} with sorting according to the violated bin condition gives another stable state for the same points. This observation establishes the following necessary condition for the uniqueness of a stable state.\n\n\\begin{theo}\n\t\\label{the:NecessaryConditionStableState}\n\tGiven a matrix $m$ in stable state. If the stable state is unique, then both the $x$- and the $y$-bin condition hold.\n\\end{theo}\n\nNote that Theorem \\ref{the:NecessaryConditionStableState} does not give an equivalence. Consider the point configuration $P=\\{(1,1),(2,3),(3,2),(4,4)\\}$. A stable state of this configuration satisfying both $x$- and $y$-bin condition is shown on the left side of Figure \\ref{fig:NonEquivalenceXYBin}. However, the point configuration admits to a second stable state, shown on the right side of Figure \\ref{fig:NonEquivalenceXYBin}. This showcases that $x$- and $y$-bin condition are only necessary, but not sufficient conditions.\n\n\\begin{figure}[htb!]\n\t\t\\centering\n\t\t\\begin{tikzpicture}[scale=0.6]\n\t\t\\draw[step=2.0] (0,0) grid (4,4);\n\t\t\\draw (1,3) node {$(2,3)$};\n\t\t\\draw (3,3) node {$(4,4)$};\n\t\t\\draw (1,1) node {$(1,1)$};\n\t\t\\draw (3,1) node {$(3,2)$};\n\t\t\\end{tikzpicture}\n\t\t\\hspace{2cm}\n\t\t\\begin{tikzpicture}[scale=0.6]\n\t\t\\draw[step=2.0] (0,0) grid (4,4);\n\t\t\\draw (1,3) node {$(3,2)$};\n\t\t\\draw (3,3) node {$(4,4)$};\n\t\t\\draw (1,1) node {$(1,1)$};\n\t\t\\draw (3,1) node {$(2,3)$};\n\t\t\\end{tikzpicture}\n\t\t\\caption{On the left hand side a configuration in stable state, satisfying both $x$- and $y$-bin condition. On the right a stable state of the same configuration $P$, showing that $x$- and $y$-bin condition do not imply uniqueness.}\n\t\t\\label{fig:NonEquivalenceXYBin}\n\\end{figure}\n\nTheorem \\ref{the:Countings} already tells us that there are $(n^2)!\/(n!)^n$ ways to put $x$-values into the matrix such that they form a stable state. Furthermore, there are $(n^2)!$ possible ways to fill the matrix with $x$-values. How many of these satisfy the condition of Theorem \\ref{the:NecessaryConditionStableState}.\n\n\\begin{theo}\n\t\\label{the:CountingNecessaryStableStates}\n\tThere are $(n!)^{2n}$ stable states that satisfy both the $x$- and the $y$-bin condition.\n\\end{theo}\n\\begin{proof}\n\tAssume the $x$- and $y$-bin condition holds, then the first row contains the $n$ smallest $y$-values, while the first column contains the $n$ smallest $x$-values. Either values can be permuted arbitrarily without destroying the stable state. This holds for every row and every column. Therefore, there are $(n!)^n$ ways to reorganize the rows and similarly $(n!)^n$ ways to reorganize the columns and still obtain a stable state. Hence, $(n!)^{2n}$ stable states satisfy both bin conditions.\n\\end{proof}\n\nThe situation established in Theorems \\ref{the:Countings} and \\ref{the:CountingNecessaryStableStates} can be visualized as in \\mbox{Figure \\ref{fig:DistributionOfStates}}.\n\n\\begin{figure}[htb!]\n\t{\\footnotesize\n\t\\begin{align*}\n\t\\bordermatrix{\n\t\t~ \t\t\t\t\t\t& (n!)^n \t\t\t& (n^2)!\/(n!)^n-(n!)^n & (n^2)!-(n^2)!\/(n!)^n \\cr\n\t\t(n!)^n \t\t\t\t\t& \\text{stable }x-y-\\text{bin}\t& \\text{stable }x-\\text{bin}\t& \\text{not stable} \\cr\n\t\t(n^2)!\/(n!)^n-(n!)^n \t& \\text{stable }y-\\text{bin} \t\t& \\text{stable} \t\t& \\text{not stable}\\cr\n\t\t(n^2)!-(n^2)!\/(n!)^n\t& \\text{not stable}\t\t& \\text{not stable}\t& \\text{not stable}\\cr\n\t}\n\t\\end{align*}}\n\t\\caption{Distribution of possibly unique stable states amongst all stable states and amongst all states.}\n\t\\label{fig:DistributionOfStates}\n\\end{figure}\n\nHowever, we can derive even more from the algorithm of Theorem \\ref{the:StableStateExistence}.\n\n\\subsection{Second Necessary Condition on Stable States}\n\\label{subsec:SecondNecessaryConditionOnStableStates}\n\nGiven the terminology and results from Section \\ref{subsec:FirstNecessaryConditionOnStableStates}, we can derive an even stronger necessary condition for unique stable states. Namely, the following holds:\n\n\\begin{theo}\n\tGiven an $n\\times n$ matrix $m$ in stable state. If the stable state is unique, then any $k\\times k$ submatrix of $m$ with $k\\in[n]$ is in a unique stable state.\n\\end{theo}\n\\begin{proof}\n\tGiven some $n\\times n$ matrix $m$ in a unique stable state. Assume there exists some $k\\times k$ submatrix $\\tilde{m}$ of $m$, $k\\in[n]$, such that $\\tilde{m}$ is not unique, but has a different stable state $\\bar{m}$. Assume that $\\tilde{m}$ occupies rows $r,\\ldots,r+(k-1)$  and columns $c,\\ldots,c+(k-1)$ in $m$.\n\t\n\tBecause of the first necessary condition on stable states (Section \\ref{subsec:FirstNecessaryConditionOnStableStates}) we know that the $x$-values of any elements in columns $1,\\ldots,c-1$ are smaller and the $x$-values of any elements in the columns $c+k,\\ldots,n$ are larger than all $x$-values in $\\tilde{m}$ respectively. This remains true independent of any reordering, in particular for $\\bar{m}$. The same argument holds for the $y$-values in rows $1,\\ldots,r-1$ and the $y$-values in rows $r+k,\\ldots,n$ when compared with the $y$-values of $\\tilde{m}$.\n\t\n\tTherefore, replacing $\\tilde{m}$ by $\\bar{m}$ in $m$ gives another stable state, which violates the uniqueness of $m$.\n\\end{proof}\n\n\\subsection{Enumeration of all unique Stable States}\n\\label{subsec:EnumerationOfAllUniqueStableStates}\n\nThe number of unique stable states for any $n\\in\\mathbb{N}$ is given in Table \\ref{tab:NumberOfUniqueStableStates}. All unique stable states for $n=2$ and $n=3$ are shown in Figures \\ref{fig:UniqueStableStates2} and \\ref{fig:UniqueStableStates3} respectively. Brute-Force computation shows that none of the $37,536$ possible $4\\times 4$ point configurations satisfying the second necessary condition from Section \\ref{subsec:SecondNecessaryConditionOnStableStates} is uniquely stable. That is, Assumption \\ref{ass:UniqueStableState} does not hold. Furthermore, from $n=4$ upward, there can never again be any uniquely stable point configuration, as if there was one, it would have to include a unique stable $4\\times 4$ state, which does not exist.\n\n\\begin{table}[htb!]\n\t\\centering\n\t\\begin{tabular}{r|r}\n\t\tn & \\# stable states\\\\\n\t\t\\hline\n\t\t$1$ & 1\\\\\n\t\t$2$ & 12\\\\\n\t\t$3$ & 966\\\\\n\t\t$\\geq4$ & 0\n\t\\end{tabular}\n\t\\caption{Number of unique stable states for $n\\times n$ matrices.}\n\t\\label{tab:NumberOfUniqueStableStates}\n\\end{table}\n\n\\begin{figure}[htb!]\n\t\\centering\n\t\\includegraphics[width=0.29\\textwidth]{allUnique_2by2-scale}\n\t\\caption{All $12$ unique $2\\times 2$ stable states. Red edges correspond to horizontal neighborhood relation in the grid, while blue edges indicate a vertical neighborhood relation in the grid.}\n\t\\label{fig:UniqueStableStates2}\n\\end{figure}\n\n\\begin{landscape}\n\\begin{figure}[htb!]\n\t\\includegraphics[height=1.0\\textwidth]{allUnique_3by3}\n\t\\caption{All $966$ unique $3\\times 3$ stable states.}\n\t\\label{fig:UniqueStableStates3}\n\\end{figure}\n\\end{landscape}\n\nThe fact that for $n\\geq4$ there is no point configuration with a single unique stable state, it raises the following question: \n\\begin{openQuest}\n\tGiven $n\\in\\mathbb{N}$, $n\\geq4$, what is a point configuration $P$ with the minimum number of stable states among any point configuration with $n^2$ points?\t\n\\end{openQuest}\n\n\\subsection{The worst Stable State}\n\\label{subsec:TheWorstStbaleState}\n\nWe proceed by turning the question from the last paragraph around. What is the maximal number of stable states a point configuration can obtain for some given $n\\in\\mathbb{N}$? In order to investigate this question, we first turn to a specific point configuration, for which we can count the number of stable states. Consider the \"identity\": $\\{(1,1),(2,2),\\ldots,(n^2,n^2)\\}$. Counting the number of stable states for the identity is equivalent to placing only one number in each field of the $n\\times n$ matrix, which then has to satisfy both conditions of Definition \\ref{def:StableState}.\n\nIn order to count the number of stable states for the identity, we introduce the concept of Ferrers Diagrams and Young tableaux. Taken from \\cite{sagan2001symmetric}.\n\n\\begin{defi}\n\tLet $\\lambda=(\\lambda_1,\\lambda_2,\\ldots,\\lambda_\\ell)$ be a partition of $N$. The Ferrers diagram of $\\lambda$ is an array of $N$ cells having $\\ell$ left-justified rows with row $i$ containing $\\lambda_i$ cells for $1\\leq i\\leq\\ell$.\n\\end{defi}\n\nConsider for example $N=4$. Then $N$ can be partitioned in five different ways: \n\\begin{align*}\n\t4&&=3+1&&=2+2&&=2+1+1&&=1+1+1+1,\n\\end{align*}\nas illustrated in Figure \\ref{fig:FerrersDiagram}.\n\n\\begin{figure}[htb!]\n\t\\centering\n\t\\begin{tikzpicture}[scale=0.6]\n\t\\draw (0,0) grid (4,1);\n\t\\end{tikzpicture}\n\t\\hfill\n\t\\begin{tikzpicture}[scale=0.6]\n\t\\draw (0,0) grid (3,1);\n\t\\draw (0,0) grid (1,-1);\n\t\\end{tikzpicture}\n\t\\hfill\n\t\\begin{tikzpicture}[scale=0.6]\n\t\\draw (0,0) grid (2,2);\n\t\\end{tikzpicture}\n\t\\hfill\n\t\\begin{tikzpicture}[scale=0.6]\n\t\\draw (0,0) grid (2,1);\n\t\\draw (0,0) grid (1,-2);\n\t\\end{tikzpicture}\n\t\\hfill\n\t\\begin{tikzpicture}[scale=0.6]\n\t\\draw (0,0) grid (1,4);\n\t\\end{tikzpicture}\n\t\\caption{The five possible Ferrers diagrams for the partitions of $N=4$ given by $\\lambda=(4)$, $\\lambda=(3,1)$, $\\lambda=(2,2)$, $\\lambda=(2,1,1)$ and $\\lambda=(1,1,1,1)$.}\n\t\\label{fig:FerrersDiagram}\n\\end{figure}\n\n\\begin{defi}\n\tLet $\\lambda=(\\lambda_1,\\lambda_2,\\ldots,\\lambda_\\ell)$ be a partition of $N$. A Young tableau of shape $\\lambda$ is an array $t$ obtained by filling the cells of the Ferrers diagram of $\\lambda$ with the numbers $1,2,\\ldots,N$ bijectively.\n\t\n\tA tableau $t$ is standard if the rows and columns of $t$ are increasing sequences.\n\\end{defi}\n\n\\begin{figure}[htb!]\n\t\\centering\n\t\\begin{tikzpicture}[scale=0.6]\n\t\\draw (0,0) grid (3,1);\n\t\\draw (0,0) grid (1,-1);\n\t\\draw (0.5,0.5) node {1};\n\t\\draw (1.5,0.5) node {2};\n\t\\draw (2.5,0.5) node {3};\n\t\\draw (0.5,-0.5) node {4};\n\t\\end{tikzpicture}\n\t\\hspace{2cm}\n\t\\begin{tikzpicture}[scale=0.6]\n\t\\draw (0,0) grid (3,1);\n\t\\draw (0,0) grid (1,-1);\n\t\\draw (0.5,0.5) node {4};\n\t\\draw (1.5,0.5) node {1};\n\t\\draw (2.5,0.5) node {3};\n\t\\draw (0.5,-0.5) node {2};\n\t\\end{tikzpicture}\n\t\\caption{Two Young tableaus corresponding to the partition $\\lambda=(3,1)$ of $N=4$. The left tableau is standard, while the right one is not.}\n\t\\label{fig:YoungTableau}\n\\end{figure}\n\nClearly, the number of standard Young tableaux of shape $\\lambda=(n,\\ldots,n)$ for $N=n^2$ is equal to the number of stable states for the identity. Denote by $f^{\\lambda}$ the number of standard $\\lambda$-tableaux. In order to compute it, we introduce the concept of hooks.\n\n\\begin{defi}\n\tIf $v=(i,j)$ is a node in the diagram of $\\lambda$, then it has hook\n\t\\begin{align*}\n\t\tH_v=H_{i,j}=\\{(i,j')\\mid j'>j\\}\\cup\\{(i',j)\\mid i'\\geq i\\}\n\t\\end{align*}\n\twith corresponding hooklength\n\t\\begin{align*}\n\t\th_v=h_{i,j}=|H_{i,j}|.\n\t\\end{align*}\n\\end{defi}\n\n\\begin{figure}[htb!]\n\t\\centering\n\t\\begin{tikzpicture}[scale=0.6]\n\t\\draw (0,0) grid (5,1);\n\t\\draw (0,0) grid (4,-1);\n\t\\draw (0,-1) grid (3,-2);\n\t\\draw (0,-2) grid (3,-3);\n\t\\draw (0,-3) grid (1,-4);\n\t\\filldraw (1.5,-0.5) circle (3pt);\n\t\\filldraw (2.5,-0.5) circle (3pt);\n\t\\filldraw (3.5,-0.5) circle (3pt);\n\t\\filldraw (1.5,-1.5) circle (3pt);\n\t\\filldraw (1.5,-2.5) circle (3pt);\n\t\\end{tikzpicture}\n\t\\caption{Given the partition $\\lambda=(5,4,3,3,1)$ of $N=16$, the dots show the hook $H_{2,2}$ with a hook length of $h_{2,2}=5$.}\n\t\\label{fig:Hook}\n\\end{figure}\n\nIt is now easy to state the hook formula of Frame, Robinson, and Thrall.\n\n\\begin{theo}\n\tLet $\\lambda=(\\lambda_1,\\ldots,\\lambda_\\ell)$ be a partition of $N$. Then\n\t\\begin{align*}\n\t\tf^\\lambda = \\frac{N!}{\\prod_{(i,j)\\in\\lambda} h_{i,j}}.\n\t\\end{align*}\n\\end{theo}\n\nIn the concrete case of the identity, where $N=n^2$, $\\lambda=(n,\\ldots,n)$, we have\n\\begin{align*}\n\th_{i,j} = n-i+n-j+1 = 2n-i-j+1.\n\\end{align*}\nTherefore, the number of stable states of the identity is given by\n\\begin{align}\n\\label{equ:NumStableStatesIdentity}\n\tf^{(n,\\ldots,n)} = \\frac{N!}{\\prod_{i=1}^n\\prod_{j=1}^n (2n-i-j+1)}.\n\\end{align}\nThe results for $n\\in\\{1,2,3\\}$ and computational experiments lead us to state the following conjecture.\n\n\\begin{conj}\n\\label{con:UpperBoundStableStates}\n\tGiven $n\\in\\mathbb{N}$, the number of stable states of any point configuration $P$ on $n^2$ points is less or equal to $f^{(n,\\ldots,n)}$.\n\\end{conj}\n\nIf the number of stable states of any point configuration is indeed bounded from above by $f^{(n,\\ldots,n)}$, then the running time of any decision-based algorithm is bounded from below\n\n\\begin{align*}\n\t\\log\\left(\\frac{n^2!}{f^{(n,\\ldots,n)}}\\right)\n\t\t&= \\log\\left(\\prod_{i=1}^n\\prod_{j=1}^n (2n-i-j+1)\\right)\\\\\n\t\t&= \\sum_{i=1}^n\\sum_{j=1}^n \\log(2n-i-j+1)\\\\\n\t\t&= n\\cdot\\log(n) + \\sum_{i=1}^n\\left(i\\cdot\\log(i)+i\\cdot\\log(2n-i)\\right).\n\\end{align*}\n\nTherefore, given Conjecture \\ref{con:UpperBoundStableStates}, the algorithm presented in Theorem \\ref{the:StableStateExistence} has optimal running time.\n\n\\subsection{Connection to Posets}\n\\label{subsec:ConnectionToPosets}\n\nThe number $f^{(n,\\ldots,n)}$ established in Equation \\ref{equ:NumStableStatesIdentity} does not only count the number of Young Tableaus of the shape $(n,\\ldots,n)$ and thus the number of stable states of the identity, but also has a connection to posets (partially ordered sets) which we will explain here shortly.\n\n\\begin{defi}\n\tGiven a set $A$ and a symmetric, anti-symmetric, and transitive relation $\\succeq$ on the elements of $A$. Then $(A,\\succeq)$ is called a partially ordered set, short: poset.\n\\end{defi}\n\nThe specific entities that we want to relate to in poset theory are the linear extensions of a poset.\n\n\\begin{defi}\n\tGiven a poset $P=(A,\\succeq)$ with $|A|=N$. A linear extension of $P$ is a function $f:P\\rightarrow[N]$ such that $f$ is bijective and\n\t\\begin{align*}\n\t\tx\\succeq y \\Rightarrow f(x)\\geq f(y),\n\t\\end{align*}\n\twith $\\geq$ the regular order relation on $\\mathbb{N}$.\n\\end{defi}\n\nOur aim is to count the number of linear extensions of a specific poset, the so-called $n\\times n$ lattice.\n\n\\begin{defi}\n\tThe poset $(X_n,\\gg)$ with $X_n=\\{(i,j)\\mid 1\\leq i\\leq j\\leq n\\}$ and $(i,j)\\gg (k,\\ell)$ if and only if $i\\geq k$ and $j\\geq\\ell$ is called the $n\\times n$ lattice.\n\\end{defi}\n\nThe following theorem now relates the number of linear extensions of the $n\\times n$ lattice to the number of standard Young Tableaus of the shape $(n,\\ldots,n)$.\n\n\\begin{theo}\n\tThe number of linear extensions of the $n\\times n$ lattice is exactly $f^{(n,\\ldots,n)}$ as given in Equation \\ref{equ:NumStableStatesIdentity}.\n\\end{theo}\n\\begin{proof}\n\tGiven a linear extension $f$ of the $n\\times n$ lattice, it induces an $(n,\\ldots,n)$ Young Tableau by $(A_{ij})_{i,j=1,\\ldots,n}=(f(i,j))_{i,j=1,\\ldots,n}$. Furthermore, the map $f\\mapsto (f(i,j))_{i,j=1,\\ldots,n}$ is injective.\n\\end{proof}\n\nThis relationship gives another possible leverage on Conjecture \\ref{con:UpperBoundStableStates}. See Figure \\ref{fig:LinearExtensions} for the two possible linear extensions of the $2\\times2$ lattice and the two corresponding Young Tableaus of shape $(2,2)$.\n\n\\begin{figure}[htb!]\n\t\\centering\n\t\\begin{tikzpicture}\n\t\t\\node (bottom) at (0,0) {$(1,1)$};\n\t\t\\node [xshift=0.325cm, left of = bottom, red] {$1$};\n\t\t\\node [above left of  = bottom, xshift=-0.5cm, yshift=0.5cm] (left)  {$(2,2)$};\n\t\t\\node [xshift=0.325cm, left of = left, red] {$2$};\n\t\t\\node [above right of = bottom, xshift= 0.5cm, yshift=0.5cm] (right) {$(3,3)$};\n\t\t\\node [xshift=0.325cm, left of = right, red] {$3$};\n\t\t\\node [above left of  = right,  xshift=-0.5cm, yshift=0.5cm]  (top)   {$(4,4)$};\n\t\t\\node [xshift=0.325cm, left of = top, red] {$4$};\n\t\t\\draw (bottom) -- (left);\n\t\t\\draw (bottom) -- (right);\n\t\t\\draw (left) -- (top);\n\t\t\\draw (right) -- (top);\n\t\\end{tikzpicture}\n\t\\begin{tikzpicture}\n\t\t\\draw[white] node at (0,0) {$1$};\n\t\t\\draw (0,0.5) rectangle (2,2.5);\n\t\t\\draw (0,1.5) -- (2,1.5);\n\t\t\\draw (1,0.5) -- (1,2.5);\n\t\t\\node[red] at (0.5,1) {$1$};\n\t\t\\node[red] at (0.5,2) {$2$};\n\t\t\\node[red] at (1.5,1) {$3$};\n\t\t\\node[red] at (1.5,2) {$4$};\n\t\\end{tikzpicture}\n\t\\hfill\n\t\\begin{tikzpicture}\n\t\\node (bottom) at (0,0) {$(1,1)$};\n\t\\node [xshift=0.325cm, left of = bottom, blue] {$1$};\n\t\\node [above left of  = bottom, xshift=-0.5cm, yshift=0.5cm] (left)  {$(2,2)$};\n\t\\node [xshift=0.325cm, left of = left, blue] {$3$};\n\t\\node [above right of = bottom, xshift= 0.5cm, yshift=0.5cm] (right) {$(3,3)$};\n\t\\node [xshift=0.325cm, left of = right, blue] {$2$};\n\t\\node [above left of  = right,  xshift=-0.5cm, yshift=0.5cm]  (top)   {$(4,4)$};\n\t\\node [xshift=0.325cm, left of = top, blue] {$4$};\n\t\\draw (bottom) -- (left);\n\t\\draw (bottom) -- (right);\n\t\\draw (left) -- (top);\n\t\\draw (right) -- (top);\n\t\\end{tikzpicture}\n\t\\begin{tikzpicture}\n\t\\draw[white] node at (0,0) {$1$};\n\t\\draw (0,0.5) rectangle (2,2.5);\n\t\\draw (0,1.5) -- (2,1.5);\n\t\\draw (1,0.5) -- (1,2.5);\n\t\\node[blue] at (0.5,1) {$1$};\n\t\\node[blue] at (0.5,2) {$3$};\n\t\\node[blue] at (1.5,1) {$2$};\n\t\\node[blue] at (1.5,2) {$4$};\n\t\\end{tikzpicture}\n\t\\caption{The two possible linear extensions of the $2\\times2$ lattice corresponding to the respective $(2,2)$ Young Tableaus.}\n\t\\label{fig:LinearExtensions}\n\\end{figure}\n\n\n\\section{General Case}\n\\label{sec:GeneralCase}\n\nIn Section \\ref{subsec:DefinitionOfTheDataStructure} we made several restrictions. In the following, we will show that the data structure, as well as the presented results and algorithms do not suffer from these restrictions.\\\\\n\n\\paragraph{Not a square number of points.} A first assumption was that we are given exactly $N=n^2$ points in the point set $P$. How can the structure handle a general number of points? Given $m\\in\\mathbb{N}$ points $P=\\{p_1,\\ldots,p_m\\}$ with $n^2\\geq m$ being the smallest square larger or equal to $m$. Let $x_{\\max}$ be the largest first coordinate and let $y_{\\max}$ be the largest second coordinate, i.e.\n\\begin{align*}\n\tx_{\\max} := \\max\\{p_{11},\\ldots,p_{m1}\\}, && y_{\\max} := \\max\\{p_{12},\\ldots,p_{m2}\\}.\n\\end{align*}\nAdd $n^2-m$ auxiliary points of the form $(x_{\\max}+1,y_{\\max}+1)$ to $P$. In the algorithm of Theorem \\ref{the:StableStateExistence}, these auxiliary points can be forced into the last two sequences $Q_{n-1},Q_n$ and thus be placed at the lowest rows or rightmost columns of the grid where they can be neglected in any neighborhood queries.\n\n\\paragraph{Larger dimension than $2$.} The second assumption was that the points $p_i\\in P$ are two-dimensional. For higher dimensions, $d\\geq2$, consider a $d$-dimensional grid with side length $n$ instead of a matrix, i.e. $M\\in(\\mathbb{R}^d)^{n\\times\\ldots\\times n}$. Denote by $p_i=(p_{i1},\\ldots,p_{id}), p_j=(p_{j1},\\ldots,p_{jd})\\in P$ two point from $P$ stored at cells $(c_1,\\ldots,c_d),(c'_1,\\ldots,c'_d)\\in[n]^d$ in $M$. There are now $d$ sorting conditions, for each $\\ell\\in[d]$ there is a condition:\\\\\nEither $c_\\ell\\neq c'_\\ell$ or w.l.o.g. $c_\\ell<c'_\\ell$ and \n\\begin{align}\n\\label{equ:sorting}\n\\begin{split}\n\t\t\t& p_{i\\ell}<p_{j\\ell} \\\\\n\t\\vee\t& (p_{i\\ell}=p_{j\\ell}\\wedge p_{i(\\ell+1)}<p_{j(\\ell+1)}) \\\\\n\t\\vee\t& \\ldots \\\\\n\t\\vee \t& (p_{i\\ell}=p_{j\\ell}\\wedge\\ldots\\wedge p_{i(d-1)}=p_{j(d-1)}\\wedge p_{id}<p_{jd})\\\\\n\t\\vee \t& (p_{i\\ell}=p_{j\\ell}\\wedge\\ldots\\wedge p_{id}=p_{jd}\\wedge p_{i1}<p_{j1}) \\\\\n\t\\vee\t& \\ldots\\\\\n\t\\vee \t& (p_{i\\ell}=p_{j\\ell}\\wedge\\ldots\\wedge p_{id}=p_{jd}\\wedge p_{i1}=p_{j1} \\wedge\\ldots\\wedge p_{i(\\ell-2)}=p_{j(\\ell-2)}\\\\\n\t\t\t& \\wedge p_{i(\\ell-1)}<p_{j(\\ell-1)}).\n\\end{split}\n\\end{align}\nThat is, in the $\\ell$th dimension of the grid, two points are compared starting from the $\\ell$th coordinate. The comparison is performed cyclically through all coordinates. $p_i$ is smaller than $p_j$ if starting from $\\ell$, the first coordinate where $p_i$ and $p_j$ differ is smaller in $p_i$.\n\n\\paragraph{Equal points.} If there are points $p_i,p_j\\in P$ such that $p_i=p_j$, no total ordering can be imposed on the points using Equation (\\ref{equ:sorting}). This tie can (in case of equality of the points) be easily broken by declaring $p_i$ to be \"smaller\" than $p_j$ if and only if $i<j$.\n\n\n\\section{Iterative Parallelized Procedure}\n\\label{sec:IterativeParallelizedProcedure}\n\nA benefit of the neighborhood grid data structure not discussed so far is the straight forward parallelization of an algorithm creating a stable state. In this section, we discuss the parallelization of a building algorithm for stable states, investigate its runtime and how iterative procedures can speed up the re-building of a stable state in case of addition or deletion of points.\n\n\\subsection{Parallelization of Stable State Creation}\n\\label{subsec:ParallelizationOfStableStateCreation}\n\nThe idea of iterative and parallel creation of stable states is discussed in \\cite{malheiros2015simple} at great detail and we are only going to state the basic ideas and results here. However, some questions are not investigated, which we are going to tackle here. The general scheme for a two-dimensional grid is quite simple and given in Algorithm 1 (see Figure \\ref{fig:Algorithm1}).\n\n\\begin{figure}[htb]\n\t\\textbf{Algorithm 1}\\\\[0.5cm]\n\t\\phantom{aaa}sorted := false\\\\\n\t\\phantom{aaa}\\textbf{while} $\\neg$sorted \\textbf{do}\\\\\n\t\\phantom{aaa}\\phantom{aaa}\\textbf{for} each row $r$ \\textbf{do}\\\\\n\t\\phantom{aaa}\\phantom{aaa}\\phantom{aaa}run a sorting algorithm on $r$, comparing $x$\\\\\n\t\\phantom{aaa}\\phantom{aaa}\\textbf{for} each row $r$ \\textbf{do}\\\\\n\t\\phantom{aaa}\\phantom{aaa}\\phantom{aaa}run a sorting algorithm on $r$, comparing $y$\\\\\n\t\\phantom{aaa}\\phantom{aaa}\\textbf{if} matrix is stable \\textbf{then}\\\\\n\t\\phantom{aaa}\\phantom{aaa}\\phantom{aaa}sorted := true\t\n\t\\caption{Algorithm 1}\n\t\\label{fig:Algorithm1}\n\\end{figure}\n\nAn immediate question following this algorithm concerns its convergence. Does it converge and if so, is the final state stable? Malheiros and Walter did not answer this question in \\cite{malheiros2015simple}, nor did Joselli et al in the original publications \\cite{joselli2009neighborhood,joselli2015neighborhood}, where the idea of Algorithm 1 is also used. We will prove the convergence here.\n\n\\begin{theo}\n\t\\label{the:ParallelConvergence}\n\tGiven any point set $P=\\{p_1,\\ldots,p_N\\mid p_i\\in\\mathbb{R}^2\\}$, placing the points in a matrix $M$ as described in Section \\ref{subsec:DefinitionOfTheDataStructure}, and running Algorithm 1 on the matrix, the algorithm converges and yields a stable state of $M$.\n\\end{theo}\n\\begin{proof}\n\tGiven a matrix $M$ as in (\\ref{equ:matrix}), consider the following expression:\n\t\\begin{align}\n\t\\label{equ:Energy}\n\tE(M)=\\sum_{i,j=1}^n i\\cdot a_{ij}+j\\cdot b_{ij},\n\t\\end{align}\n\tfor each sorting step of Algorithm 1, this expression grows, but it can at most attain $N!$ many different values.\n\\end{proof}\n\nIf the matrix is not in a stable state, i.e. there is a row or column violating the stable state, sorting this row or column lets expression (\\ref{equ:Energy}) grow and resolves the conflict in the given row or column, possibly creating a new conflict in another row or column. Therefore, a local maximum of this expression is equivalent to a stable state in the matrix.\n\nNote that the energy functional (\\ref{equ:Energy}) can easily be extended to higher-dimensional settings. Given a corresponding version of Algorithm 1 for the higher-dimensional case, Theorem \\ref{the:ParallelConvergence} holds true for arbitrary dimensional point sets.\n\nIn their work \\cite{malheiros2015simple}, Malheiros and Walter investigate a slight variation of Algorithm 1. Namely, they do not perform a full sorting of a row or column, but rather consider one step of the odd-even sort algorithm. Such step performs an exchange between all those cells in odd columns and their respective right neighboring cells, if this pair of cells violates the stable state conditions of Definition \\ref{def:StableState}. Then, all cells in even columns and their respective right neighboring cells are compared and exchanged if necessary. The same is performed on all odd rows and even rows, yielding a four step mechanism, see Figure \\ref{fig:OddEvenSort}.\n\n\\begin{figure}\n\t\\includegraphics[width=1.0\\textwidth]{odd_even_sort_Malheiros}\n\t\\caption{Step-based odd-even sort on a two-dimensional neighborhood grid, figure from \\cite{malheiros2015simple}.}\n\t\\label{fig:OddEvenSort}\n\\end{figure}\n\nNote that the argument in the proof of Theorem \\ref{the:ParallelConvergence} holds true also for this algorithm. Therefore, it also converges to a stable state. Question remains how fast the algorithm works asymptotically. Note that, defying intuition, elements can cycle using this procedure. An example is given in Figure \\ref{fig:IllustrateCycling}, where the algorithm depicted in Figure \\ref{fig:OddEvenSort} is used.\n\n\\begin{figure}[htb!]\n\t\\begin{tikzpicture}\n\t\t\\draw (0,0.5) rectangle (2,2.5);\n\t\t\\draw (0,1.5) -- (2,1.5);\n\t\t\\draw (1,0.5) -- (1,2.5);\n\t\t\\node at (0.5,1) {$(0,3)$};\n\t\t\\node at (0.5,2) {$(3,1)$};\n\t\t\\node at (1.5,1) {$(1,0)$};\n\t\t\\node[red] at (1.5,2) {$(2,2)$};\n\t\\end{tikzpicture}\n\t$\\stackrel{x}{\\rightarrow}$\n\t\\begin{tikzpicture}\n\t\t\\draw (0,0.5) rectangle (2,2.5);\n\t\t\\draw (0,1.5) -- (2,1.5);\n\t\t\\draw (1,0.5) -- (1,2.5);\n\t\t\\node at (0.5,1) {$(0,3)$};\n\t\t\\node[red] at (0.5,2) {$(2,2)$};\n\t\t\\node at (1.5,1) {$(1,0)$};\n\t\t\\node at (1.5,2) {$(3,1)$};\n\t\\end{tikzpicture}\n\t$\\stackrel{y}{\\rightarrow}$\n\t\\begin{tikzpicture}\n\t\t\\draw (0,0.5) rectangle (2,2.5);\n\t\t\\draw (0,1.5) -- (2,1.5);\n\t\t\\draw (1,0.5) -- (1,2.5);\n\t\t\\node[red] at (0.5,1) {$(2,2)$};\n\t\t\\node at (0.5,2) {$(0,3)$};\n\t\t\\node at (1.5,1) {$(1,0)$};\n\t\t\\node at (1.5,2) {$(3,1)$};\n\t\\end{tikzpicture}\n\t$\\stackrel{x}{\\rightarrow}$\n\t\\begin{tikzpicture}\n\t\t\\draw (0,0.5) rectangle (2,2.5);\n\t\t\\draw (0,1.5) -- (2,1.5);\n\t\t\\draw (1,0.5) -- (1,2.5);\n\t\t\\node at (0.5,1) {$(1,0)$};\n\t\t\\node at (0.5,2) {$(0,3)$};\n\t\t\\node[red] at (1.5,1) {$(2,2)$};\n\t\t\\node at (1.5,2) {$(3,1)$};\n\t\\end{tikzpicture}\n\t$\\stackrel{y}{\\rightarrow}$\n\t\\begin{tikzpicture}\n\t\t\\draw (0,0.5) rectangle (2,2.5);\n\t\t\\draw (0,1.5) -- (2,1.5);\n\t\t\\draw (1,0.5) -- (1,2.5);\n\t\t\\node at (0.5,1) {$(1,0)$};\n\t\t\\node at (0.5,2) {$(0,3)$};\n\t\t\\node at (1.5,1) {$(3,1)$};\n\t\t\\node[red] at (1.5,2) {$(2,2)$};\n\t\\end{tikzpicture}\n\t\\caption{Performing an odd-even sort alternating on all the rows and columns causes the red element $(2,2)$ to cycle through the matrix.}\n\t\\label{fig:IllustrateCycling}\n\\end{figure}\n\nGiven the possibility of cycling elements, the theoretical asymptotic bounds of the parallel iterative algorithm remain unclear and only experimental results are available, as presented in \\cite{malheiros2015simple}. \n\n\\begin{openQuest}\n\tGiven the step-wise odd-even algorithm depicted in Figure \\ref{fig:OddEvenSort}, what is its parallelized worst-case runtime aside from the upper bound of $\\mathcal{O}(N!)$ as established in the proof of Theorem \\ref{the:ParallelConvergence}?\n\\end{openQuest}\n\nFirst experiments and intuition lead us to state the following conjecture:\n\n\\begin{conj}\n\tThe step-wise odd-even algorithm depicted in Figure \\ref{fig:OddEvenSort} runs like Bubble-Sort in worst case time of $\\mathcal{O}(N^2)$, after parallelization in $\\mathcal{O}(N)$.\n\\end{conj}\n\n\\subsection{Adaptive Sorting Algorithms for Fast Modifications}\n\\label{subsec:AdaptiveSortingAlgorithmsForFastModifications}\n\nThe most wide-spread data structure for neighborhood computation, Kd-Trees, suffer from a severe problem. Namely, if the underlying point set is slightly altered, the Kd-Tree might become unbalanced. Although there are some heuristics how to modify Kd-Trees when adding or deleting points, at some point the Kd-Tree has to be rebuild. \n\n\\begin{openQuest}\n\tIs utilizing an adaptive sorting algorithm on a modified stable state faster than rebuilding a whole new stable state?\n\\end{openQuest}\n\n\n\\section{Quality of Neighborhood Approximation}\n\\label{sec:QualityOfNeighborhoodApproximation}\n\nAs stated above, the neighborhood estimates given by the neighborhood grid data structure are not necessarily precise. In this section, we will investigate the quality of the neighborhood approximation.\n\n\\subsection{Single Point Neighbor}\n\\label{subsec:SinglePointNeighbor}\n\nA first question question to answer in this subsection concerns the distance of two geometrical neighbors from $P$ in the stable state of $M$. In the following, we will create a point set $P$ with points $p,q\\in P$ that are respective nearest neighbors to each other, but that lie on the exact opposite sites of $M$.\n\nConsider the following points $p=(0,0)$, $q=(1,1)$, $p_i=(0,2+i\/n)$, $i=1,\\ldots,n-1$, $q_i=(1,-2-i\/n)$, $i=1,\\ldots,n-1$, and $r_{i,j}=(1-1\/i,2+j\/n)$, $i=2,\\ldots,n-1$, $j=1,\\ldots,n$. This yields a point set $P$ with $n^2$ points for $n\\geq2$, see Figure \\ref{fig:SingPointNeighborPointSet}. Given these points, the following matrix is in a stable state:\n\n\\begin{figure}[htb]\n\t\\centering\n\t\\begin{tikzpicture}\n\t\t\\draw[gray!50] (-2,-3) grid (2,4);\n\t\t\\draw (0,0) circle (1.4159);\n\t\t\\filldraw[black] (0,0) circle (2pt)  node[left] {$p$};\n\t\t\\filldraw[black] (1,1) circle (2pt) node[right] {$q$};\n\t\t\\filldraw[black] (0,2.25) circle (2pt)  node[left] {$p_1$};\n\t\t\\filldraw[black] (0,2.5) circle (2pt)  node[left] {$p_2$};\n\t\t\\filldraw[black] (0,2.75) circle (2pt)  node[left] {$p_3$};\n\t\t\\filldraw[black] (1,-2.25) circle (2pt)  node[right] {$q_1$};\n\t\t\\filldraw[black] (1,-2.5) circle (2pt)  node[right] {$q_2$};\n\t\t\\filldraw[black] (1,-2.75) circle (2pt)  node[right] {$q_3$};\n\t\t\\filldraw[black] (0.5,2.25) circle (2pt);\n\t\t\\filldraw[black] (0.5,2.5) circle (2pt);\n\t\t\\filldraw[black] (0.5,2.75) circle (2pt);\n\t\t\\filldraw[black] (0.5,3) circle (2pt);\n\t\t\\filldraw[black] (0.66,2.25) circle (2pt);\n\t\t\\filldraw[black] (0.66,2.5) circle (2pt);\n\t\t\\filldraw[black] (0.66,2.75) circle (2pt);\n\t\t\\filldraw[black] (0.66,3) circle (2pt) node[above] {$r_{i,j}$};\n\t\\end{tikzpicture}\n\t\\caption{Point set $P$ as given in Section \\ref{subsec:SinglePointNeighbor}.}\n\t\\label{fig:SingPointNeighborPointSet}\n\\end{figure}\n\n\\begin{align*}\nM(P)=\n\\begin{array}{|c|c|c|c|c|}\n\t\\hline\n\tp_{n-1} & r_{2,n} & \\ldots & r_{n-1,n} & q\\\\\n\t\\hline\n\t\\vdots & \\vdots & \\iddots & \\vdots & q_1\\\\\n\t\\hline\n\tp_1 & r_{2,2} & \\ldots & r_{n-1,2} & \\vdots\\\\\n\t\\hline\n\tp & r_{2,1} & \\ldots & r_{n-1,n} & q_{n-1}\\\\\n\t\\hline\n\\end{array}.\n\\end{align*}\n\nNote that the nearest neighbor to $p$ and $q$ in $P$ is $q$ and $p$ respectively. However, in $M(P)$, these points lie in the opposing corners of the matrix. That is, in order to find the geometrically closest neighbor to $p$ in $M(P)$, the $n$-ring around $p$ has to be checked - in other words, all points have to be checked, which takes $\\mathcal{O}(n^2)$ instead of $\\theta(\\log(n))$ as in Kd-Trees.\n\n\\subsection{All Point Nearest Neighbors}\n\\label{subsec:AllPointNearestNeighbors}\nIn the previous example we saw that for a single point, its unique nearest neighbor can be arbitrarily far away in the neighborhood grid. However, when considering all points, how is the overall estimate? Here, we provide an example, where no point has its corresponding neighbor within its one-ring in the neighborhood grid.\n\nFor $n\\in\\mathbb{N}$, $2\\mid n$, consider the following points $p_{i,j} = (i,j)$, $q_{i,j}=(i+n,j+0.5)$, $r_{i,j}=(i+0.5,j+n)$, and $s_{i,j}=(i+n+0.5,j+n+0.5)$ for $i,j\\in\\{0,\\ldots,\\frac{n}{2}-1\\}$. This yields a point set $P$ with $n^2$ points, see Figure \\ref{fig:AllPointNeighborPointSet}. Given these points, the following matrix is in a stable state:\n\n\\begin{figure}[htb]\n\t\\centering\n\t\\begin{tikzpicture}\n\t\\draw[gray!50] (-1,-1) grid (6,6);\n\t\\filldraw[black] (0,0) circle (2pt)  node[below left] {$p_{0,0}$};\n\t\\filldraw[black] (1,0) circle (2pt)  node[below left] {$p_{1,0}$};\n\t\\filldraw[black] (0,1) circle (2pt)  node[below left] {$p_{0,1}$};\n\t\\filldraw[black] (1,1) circle (2pt)  node[below left] {$p_{1,1}$};\n\n\t\\filldraw[red] (4,0.5) circle (2pt)  node[below left] {$q_{0,0}$};\n\t\\filldraw[red] (5,0.5) circle (2pt)  node[below left] {$q_{1,0}$};\n\t\\filldraw[red] (4,1.5) circle (2pt)  node[below left] {$q_{0,1}$};\n\t\\filldraw[red] (5,1.5) circle (2pt)  node[below left] {$q_{1,1}$};\n\t\n\t\\filldraw[blue] (0.5,4) circle (2pt)  node[below left] {$r_{0,0}$};\n\t\\filldraw[blue] (1.5,4) circle (2pt)  node[below left] {$r_{1,0}$};\n\t\\filldraw[blue] (0.5,5) circle (2pt)  node[below left] {$r_{0,1}$};\n\t\\filldraw[blue] (1.5,5) circle (2pt)  node[below left] {$r_{1,1}$};\n\t\n\t\\filldraw[ForestGreen] (4.5,4.5) circle (2pt)  node[below left] {$s_{0,0}$};\n\t\\filldraw[ForestGreen] (5.5,4.5) circle (2pt)  node[below left] {$s_{1,0}$};\n\t\\filldraw[ForestGreen] (4.5,5.5) circle (2pt)  node[below left] {$s_{0,1}$};\n\t\\filldraw[ForestGreen] (5.5,5.5) circle (2pt)  node[below left] {$s_{1,1}$};\n\t\\end{tikzpicture}\n\t\\caption{Point set $P$ as given in Section \\ref{subsec:AllPointNearestNeighbors} for $n=4$.}\n\t\\label{fig:AllPointNeighborPointSet}\n\\end{figure}\n\n\\begin{align*}\n\tM(P)=\n\t\\begin{array}{|c|c|c|c|c|c|c|}\n\t\\hline\n\t\\textcolor{red}{q_{0,\\frac{n}{2}-1}} & \\textcolor{ForestGreen}{s_{0,\\frac{n}{2}-1}} & \\textcolor{red}{q_{1,\\frac{n}{2}-1}} & \\textcolor{ForestGreen}{s_{1,\\frac{n}{2}-1}} & \\ldots & \\textcolor{red}{q_{\\frac{n}{2}-1,\\frac{n}{2}-1}} & \\textcolor{ForestGreen}{s_{\\frac{n}{2}-1,\\frac{n}{2}-1}}\\\\\n\t\\hline\n\tp_{0,\\frac{n}{2}-1} & \\textcolor{blue}{r_{0,\\frac{n}{2}-1}} & p_{1,\\frac{n}{2}-1} & \\textcolor{blue}{r_{1,\\frac{n}{2}-1}} & \\ldots & p_{\\frac{n}{2}-1,\\frac{n}{2}-1} & \\textcolor{blue}{r_{\\frac{n}{2}-1,\\frac{n}{2}-1}}\\\\\n\t\\hline\n\t\\vdots & \\vdots & \\vdots & \\vdots & \\iddots &\\vdots&\\vdots\\\\\n\t\\hline\n\t\\textcolor{red}{q_{0,1}} & \\textcolor{ForestGreen}{s_{0,1}} & \\textcolor{red}{q_{1,1}} & \\textcolor{ForestGreen}{s_{1,1}} & \\ldots & \\textcolor{red}{q_{\\frac{n}{2}-1,1}} & \\textcolor{ForestGreen}{s_{\\frac{n}{2}-1,1}}\\\\\n\t\\hline\n\tp_{0,1} & \\textcolor{blue}{r_{0,1}} & p_{1,1} & \\textcolor{blue}{r_{1,1}} & \\ldots & p_{\\frac{n}{2}-1,1} & \\textcolor{blue}{r_{\\frac{n}{2}-1,1}}\\\\\n\t\\hline\n\t\\textcolor{red}{q_{0,0}} & \\textcolor{ForestGreen}{s_{0,0}} & \\textcolor{red}{q_{1,0}} & \\textcolor{ForestGreen}{s_{1,0}} & \\ldots & \\textcolor{red}{q_{\\frac{n}{2}-1,0}} & \\textcolor{ForestGreen}{s_{\\frac{n}{2}-1,0}}\\\\\n\t\\hline\n\tp_{0,0} & \\textcolor{blue}{r_{0,0}} & p_{1,0} & \\textcolor{blue}{r_{1,0}} & \\ldots & p_{\\frac{n}{2}-1,0} & \\textcolor{blue}{r_{\\frac{n}{2}-1,0}}\\\\\n\t\\hline\n\t\\end{array}.\n\\end{align*}\n\nNote that the nearest neighbor to $p_{i,j}$ is some $p_{k,\\ell}$, the nearest neighbor to $q_{i,j}$ is some $q_{k,\\ell}$, etc. However, none of the points has its corresponding neighbor in its one-ring. Expanding this scheme by adding more four-point subsets, it is easily achievable to create point sets with immediate stable states where no respective nearest neighbor is in the $\\frac{n}{4}$ ring of all points.\n\n\\subsection{Open Questions}\nWe have given two constructions to create worst case behavior in the neighborhood estimates. However, these constructions do not account for the average estimate to be expected for a random point cloud. Therefore, we pose the following question:\n\n\\begin{openQuest}\n\tHow good is the average neighborhood estimate of the neighborhood grid?\n\\end{openQuest}\n\nSo far, this question can only be answered approximately by experiments. These are going to be performed.\n\nAnother open question concerns the procedure for creation of stable states. From observations, it seems that the sorting produced by the algorithm of Theorem \\ref{the:StableStateExistence} does not give a result that is as good as performing the following: As long as a state is not stable, find the two points in the grid such that exchanging them achieves a maximal grow in expression (\\ref{equ:Energy}).\n\n\\begin{openQuest}\n\tDoes exchanging points by maximal growth of (\\ref{equ:Energy}) give significantly better results in the grid than the algorithm of Theorem \\ref{the:StableStateExistence}?\n\\end{openQuest}\n\n\n\n\\section*{Acknowledgments}\nMartin Skrodzki and Konrad Polthier acknowledge the support by the DFG SFB\/TR 109 'Discretization in Geometry and Dynamics`. Martin Skrodzki also thanks ECMath, BMS, and the German National Academic Foundation for their support.\n\n\\newpage\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe electron mobility is a very important parameter of electronic devices. In heavily doped bulk semiconductors, the low temperature mobility is determined by the electron scattering on ionized donors and is relatively small. Larger low temperature mobilities can be achieved near the surface of a lightly doped say $n$-type semiconductor, where the electron accumulation layer is induced by the applied surface electric field. In such devices the  mobility becomes sensitive to the semiconductor surface roughness, which can be imagined as a collection of atomic-size steps of total height $\\Delta$ and characteristic size $d \\gg \\Delta$ along the surface. The roughness scattering dominates at high electric fields $E$ when electrons are squeezed closer to the surface. In this case~\\cite{Prange,Chaplik,*Entin,Ando_1977,Ando_1979,Ando_1982,Suris} the mobility $\\mu$ limited by the surface roughness scattering behaves as $\\mu \\propto 1\/E^{2}$. For a large enough field $E$ the two-dimensional (2D) concentration of electrons $n \\propto E$ so that $\\mu \\propto 1\/n^{2}$  and the surface conductivity $\\sigma = ne\\mu \\propto 1\/n $. This result holds for an inversion layer in a lightly doped $p$-type semiconductor when the electric field $E$ is larger than the electric field of the depletion layer. The low temperature mobility was also extensively studied in quantum wells, where it is limited by the surface roughness of both interfaces. This mobility strongly depends on the width of the quantum well~\\cite{Sakaki, Gold, Penner}.\n\nBecause of the interest in higher mobilities, the surface roughness scattering was studied theoretically only for relatively small concentrations $n$, when only the first energy quantization subband is filled at low temperatures~\\cite{Chaplik,*Entin,Ando_1977,Ando_1979,Ando_1982,Suris,Sakaki, Gold, Penner}. Also, it was difficult to induce large electron concentrations $n$ (higher than $10^{13} ~\\mathrm{cm}^{-2}$ in Si). So the the roughness scattering in the case of large concentrations $n$ when many subbands are filled at low temperatures has remained unexplored.\n\nThe last decade, however, witnessed growing interest in accumulation layers with large $n$ which allow to achieve qualitatively new properties of the electron gas, such as superconductivity or magnetism. New methods to create large electron concentrations were developed. One of them is based on ion gating with help of an electrolyte or a room temperature ionic liquid, which does not need an insulator layer and, therefore, makes a double layer with a very large capacitance. In Si concentrations $n \\sim 5 \\cdot 10^{13}~\\mathrm{cm}^{-2}$ were achieved using gating by an electrolyte~\\cite{Tardella} and by an ionic liquid~\\cite{JJ}. Even larger concentrations $\\sim 10^{14}~\\mathrm{cm}^{-2}$ were induced in ZnO~\\cite{Iwasa_2009}, $\\mathrm{MoS_2}$~\\cite{Iwasa_2012} and $\\mathrm{SrTiO_3}$ \\cite{Iwasa_2008,Gallagher_2014} with this method.\n\nAnother important method is based on heterojunctions of polar and nonpolar perovskites such as $\\text{GdTiO}_3$ and $\\text{SrTiO}_3$, which accumulate $3 \\cdot 10^{14}~\\mathrm{cm}^{-2}$ electrons \\cite{Stemmer}. Concentrations $n$ up to $10^{15}~\\mathrm{cm}^{-2}$ were obtained combining this effect with the electron spill-out for a special band alignment  \\cite{Bharat}. Similar physics takes place in GaN-based heterojunctions where concentrations up to $4.4\\times10^{13}$ cm$^{-2}$ were achieved \\cite{Shur, Speck_2009, Shur_2009}.\n\nAt such large concentrations $n$ the dimensionless parameter $na_{B}^{2} > 1$  and many electron subbands are filled. Here  $a_B=\\kappa\\hbar^2 \/m^*e^2$ is the effective Bohr radius, $\\kappa$ is the dielectric constant, $m^*$ is the effective electron mass. In the cases of ZnO and MoS$_2$ mentioned above $na_{B}^{2}$ reaches 5. In semiconductors with relatively large $a_{B}$ such as GaAs, InAs, InSb, and PbTe, it should be easy to reach  $na_{B}^{2} \\gg 1$.\n\n\nAs we said above the roughness scattering limited mobility for the multisubband case has not been theoretically studied. In this paper, we fill this gap and study the low temperature mobility limited by surface roughness in an accumulation layer with large $n$. Our result for the dimensionless conductivity $\\sigma(n) \/(2e^2\/h)$ at $d<a_B $ is shown on Fig. \\ref{fig:conductivity_the_result} as a function of the dimensionless concentration  $na_{B}^{2}$ for the exponential model of roughness \\cite{Goodnick} by the thick solid line (black). The conductivity first decreases with $n$ as $\\propto n^{-6\/5}$ and then saturates at the level\n\\begin{equation}\\label{eq:conductivity}\n\\frac{\\sigma}{2e^2\/h}\\simeq\\frac{da_B}{\\Delta^2},\n\\end{equation}\nwhich is much larger than unity assuming that both $d\\gg \\Delta$ and $a_B\\gg \\Delta$. The thin solid line (red) schematically shows the $1\/n$ dependence of the conductivity derived for a single subband by previous work~\\cite{Ando_1977}.\n\\begin{figure}[t]\n \\includegraphics[width=.45\\textwidth]{conductivity}\\\\\n \\caption{(Color online) The schematic log-log plot of the dimensionless conductivity of an accumulation layer $\\sigma \/(2e^2\/h)$ limited by the surface roughness scattering as a function of the dimensionless 2D electron concentration $na_{B}^{2}$ at $d<a_B$.  The thick solid line (black) shows the conductivity for the multisubband accumulation layer. It first decreases as $(na_B^2)^{-6\/5} (a_B^4\/d^2\\Delta^2)$ and then saturates at $na_B^2\\sim (a_B\/d)^{5\/2}$, where the wavelength $k_F^{-1}\\sim d$. The thin solid line (red) represents the $1\/n$ dependence derived for a single subband \\cite{Ando_1977}. Conjectured extrapolation \\cite{DasSarma} of this dependence to larger  concentrations is shown by the thin dashed thin (red).}\\label{fig:conductivity_the_result}\n \\end{figure}\n\nWe see that with growing $n$ our conductivity at first approximately continues the single subband dependence $1\/n$, but then saturates. The saturation happens when the electron wavelength $k_F^{-1}$ is equal to the size $d$ of the roughness. Before this point, the roughness felt by electrons is averaged over all irregularities within the region of size $k_F^{-1}$. As the concentration increases, $k_F$ increases and fewer irregularities are averaged over making the surface ``rougher\" for electrons. When $k_F^{-1}$ gets below $d$, the electron ``hits\" only a single irregularity and the level of ``roughness\" is fixed leading to the saturation of the conductivity.\n\nOur results contradict to the conjecture~\\cite{DasSarma} that the single subband result~\\cite{Ando_1977} can be extrapolated to the large-concentration multisubband case (see the thin dashed line (red) in Fig. \\ref{fig:conductivity_the_result}). This conjecture led to the dramatic prediction~\\cite{DasSarma} that the dimensionless conductivity could become smaller than unity implying the re-entrant metal-insulator transition with growing $n$. Our results show that at large $n$ the accumulation layer remains metallic. This agrees with decent mobilities observed in Refs. \\onlinecite{diamond,Iwasa_2009,Iwasa_2012,Tardella}.\n\nThe organization of our paper is as follows. In Sec. \\ref{sec:accu}, we explain the structure of the accumulation layer where electrons occupy many energy quantization subbands. In Sec. \\ref{sec:roughness}, we introduce two models of the surface roughness including the exponential one assumed in Fig. \\ref{fig:conductivity_the_result} and the Gaussian widely used in earlier studies. In Sec. \\ref{sec:class}, we present an intuitive quasi-classical interpretation of our mobility results for the exponential roughness. In Sec. \\ref{sec:sing}, we introduce the more formal quantum-mechanical approach starting from the case of a single subband connecting to previous studies. In Sec. \\ref{sec:inter}, we discuss the multisubband case, take into account the scattering of electrons between different subbands, and give the final scattering rate and mobility. We conclude in Sec. \\ref{sec:con}. In the main text of the paper we use the scaling approach and drop numerical coefficients. In Appendix \\ref{App:AppendixA} we estimate the coefficients of the conductivity for the exponential roughness.\n\n\n\\section{Electronic structure of an accumulation layer}\\label{sec:accu}\n\nThe accumulation layer is created near the surface of an $n$-type semiconductor when the orthogonal-to-surface electric field $E$ induces a large 2D concentration of excessive surface electrons $n=E\/4\\pi e$ in the layer of width $\\sim L$. We assume that $n\/L$ is much larger than the bulk concentration of electrons $N$ in the semiconductor. This means that the bulk Fermi level at low temperatures is either below the conduction band bottom or slightly above it. In Fig. \\ref{fig:subband} illustrating the accumulation layer we actually assumed that the bulk Fermi level coincides with the conduction band bottom, which in this paper serves as the reference point of the electron energy. Our description of the electron accumulation layer is applicable also to an inversion layer in the very lightly doped $p$-type semiconductor, where the 2D concentration $n_{depl}$ of ionized acceptors forming the depletion layer is much smaller than the 2D concentration $n$ of the electrons so that we can still use $n=E\/4\\pi e$.\n\\begin{figure}[h]\n\\includegraphics[width=.4\\textwidth]{subband}\\\\\n\\caption{(Color online) Schematic plot of the subbands electrons occupy at low temperatures in the accumulation layer. The subbands are represented by the black thick horizontal lines. The electron energy $\\varepsilon$ consists of the kinetic energy $\\hbar^2k^2\/2m^*$ where $k$ is the momentum and the potential energy $-e\\varphi(z)$. The surface potential well $-e\\varphi(z)$ is shown by the grey thin line (red) where $\\varphi(z)$ is given by Eq. \\eqref{eq:potential_linear}. The reference energy level $\\varepsilon=0$ is the conduction band bottom. In a lightly $n$-doped bulk semiconductor the electron Fermi level is close to zero. Most electrons are located within $z\\lesssim L$ where $L$ is the decay length given by Eq. \\eqref{eq:decay}.}\\label{fig:subband}\n\\end{figure}\n\nIn this paper we are focused on large $E$ and $n$ cases when the one-dimensional potential well created by the electric field $E$ along the $z$-axis (which is defined as normal to the surface towards the accumulation layer) is so deep that it has several quantized levels (see Fig. \\ref{fig:subband}). Each of such levels forms a subband with a two-dimensional (2D) Fermi gas moving freely parallel to the surface. At large $n$ all these 2D gases with the same Fermi level form a three-dimensional (3D) degenerate gas, which can be described in the Thomas-Fermi approximation neglecting the discreteness of subbands. The nonlinear screening of the electric field $E$ by such a gas was studied by solving the Poisson-Thomas-Fermi equation for the self-consistent potential $\\varphi(z)$ and the 3D electron density $N(z)$ by Frenkel~\\cite{Frenkel}. The result is~\\cite{Frenkel,Kostya}\n\\begin{equation}\n  \\label{eq:potential_linear}\n  \\varphi(z)=  C_1 \\frac{e}{\\kappa a_B}  \\left(\\frac{a_B}{z+L} \\right)^4,\n\\end{equation}\n\\begin{equation}\n\\label{eq:concentration_linear}\nN(z)= C_2 \\frac{1}{a_B^3} \\left(\\frac{a_B}{z+L}\\right)^{6},\n\\end{equation}\nwhere  $z$ is the distance from the interface, $a_B=\\kappa\\hbar^2\/m^*e^2$ is the effective Bohr radius, $\\kappa$ is the\ndielectric constant, $m^*$ is the isotropic effective electron mass, $C_1=225\\pi^2\/8 \\simeq 278$, $C_2=1125 \\pi \/8 \\simeq 442 $. $L$ is the characteristic decay length of the electron density\n\\begin{equation}\n\\label{eq:decay}\nL=C_3 \\frac{a_B}{(na_B^2)^{1\/5}},\n\\end{equation}\nwhere $C_3=(225\\pi\/8)^{1\/5} \\simeq 2.4$, $n$ is the total 2D concentration of electrons inside the accumulation layer. The width of the electron gas is $\\sim L$. This solution is valid for $na_B^2 \\gg 1$, when for $N(0)\\sim {n\/L} \\sim (na_B^2)^{6\/5}{a_B}^{-3}$ and for $k_F \\sim N(0)^{1\/3} = (na_B^2)^{2\/5}\/a_B$ the following inequalities hold: $N(0)a_B^{3} \\sim (na_B^2)^{6\/5} \\gg 1$ and $k_F L \\sim (na_B^2)^{1\/5} \\gg 1$. These inequalities confirm that we deal with a 3D degenerate gas and that the Thomas-Fermi approximation is valid and many subbands are filled.\nAt smaller concentrations when $na_B^2 < 1$ only one subband is filled so that we go back to the 2D case studied for the inversion layer or narrow quantum wells.\n\n\\section{Models of surface roughness}\\label{sec:roughness}\n\\begin{figure}[h]\n$\\begin{array}{c}\n\\includegraphics[width=0.47\\textwidth]{Gaussian_new}\\\\\n\\\\\n\\\\\n\\includegraphics[width=0.47\\textwidth]{exponential_new}\n\\end{array}$\n\\caption{Two types of surface roughness. (a) The Gaussian type of roughness. Here the lattice discreteness can be ignored where $\\Delta\\gg a$ and $a$ is the lattice constant. (b) The exponential kind of roughness. The size of the roughness is $d$ and the height fluctuates as $\\pm\\Delta$ with respect to the average interface ($z=0$) where $2\\Delta=a$. When the electron wavelength $k_F^{-1}\\gg d$, the felt height of roughness is averaged result $\\Delta\/\\sqrt{1\/k_F^2d^2}=\\Delta k_Fd$ on a length scale of $k_F^{-1}$ for both types of surface roughness. At $k_F^{-1}\\ll d$, the incident electron feels only a single hill\/valley or island. For Gaussian roughness, the electron scatters on the slope determined by the angle $\\alpha\\sim \\Delta\/d$ shown in (a). For the exponential roughness, however, the electron is scattered by the island edge which has a height $\\Delta$ and an effective length $k_F^{-1}$, and thus the effective angle is $\\alpha\\simeq\\Delta k_F$ as shown in (b).}\\label{fig:roughness}\n\\end{figure}\nThe surface roughness is a random shift of the interface $\\Delta(\\vec{r})$ from $z=0$ so that $<\\Delta(\\vec{r})>=0$, where $\\vec{r}=(x,y)$ is the coordinate in $z=0$ interface plane (see Fig. \\ref{fig:roughness}). The roughness is described by the height correlator and its Fourier transform\n\\begin{equation}\\label{eq:corre}\n\\begin{aligned}\n<\\Delta(\\vec{r})\\Delta(\\vec{r'})>=&W(\\vec{r}-\\vec{r'}),\\\\\n<|\\Delta(q)|^2>=&W(q).\n\\end{aligned}\n\\end{equation}\nTwo main models of roughness are used in literature. One is Gaussian\n\\begin{equation}\\label{eq:Gauss}\n\\begin{aligned}\nW(\\vec{r}-\\vec{r'})=&\\Delta^2e^{-(\\vec{r}-\\vec{r'})^2\/d^2},\\\\\nW(q)=&\\pi\\Delta^2d^2e^{-q^2d^2\/4},\n\\end{aligned}\n\\end{equation}\nwhere $d,\\,\\Delta$ are the characteristic size and height of the roughness, $\\Delta\\ll d, \\,L$. This is the model widely used in earlier studies such as for the single subband mobility \\cite{Entin,Ando_1977,Ando_1979,Ando_1982,Suris,Sakaki,Gold}.\n\nHowever, later experimental observations found that the spacial correlations are more likely to follow an exponential behavior \\cite{Goodnick,Feenstra_1994}\n\\begin{equation}\\label{eq:exp}\n\\begin{aligned}\nW(\\vec{r}-\\vec{r'})=&\\Delta^2e^{-\\sqrt{2}|\\vec{r}-\\vec{r'}|\/d},\\\\\nW(q)=&\\pi\\Delta^2d^2(1+q^2d^2\/2)^{-3\/2}.\n\\end{aligned}\n\\end{equation}\nThe important difference from the Gaussian case is that here $W(q)$ decays much slower as $q^{-3}$ at large $q$. This leads to a stronger scattering at large $n$. One way to envision this kind of roughness is to think about randomly distributed flat islands of an additional lattice layer with typical size $d$ on the top of the last complete layer of the crystal. These islands may, for example occupy half of the surface area, so that $\\Delta(\\vec{r})= \\pm \\Delta$ appear with equal probability where $2\\Delta = a$ is the lattice constant. Whenever two survey points $\\vec{r}$ and $\\vec{r'}$ fall within the same island, $\\Delta(\\vec{r})\\Delta(\\vec{r'})$ is $\\Delta^{2}$. This typically happens when points are close, i.e., $|\\vec{r}-\\vec{r'}|\\ll d$. When one of the points misses this island $\\Delta(\\vec{r})\\Delta(\\vec{r'})$ is $-\\Delta^{2}$. The probability of falling into different islands at $|\\vec{r}-\\vec{r'}|\\ll d$ is $\\propto |\\vec{r}-\\vec{r'}|\/d$. So $W(\\vec{r}-\\vec{r}')\\sim \\Delta^2(1-|\\vec{r}-\\vec{r'}|\/d)$. Such a behavior at small distances determines the large $q$ asymptote of the Fourier transform $W(q)$ as $\\sim 1\/q^3$, which is the result obtained at large $q$ from Eq. \\eqref{eq:exp}. Our calculations of the roughness-limited mobility for accumulation layers are focused on this type of surface roughness. However, to make a connection with earlier studies\\cite{Entin,Ando_1977,Ando_1979,Ando_1982,Suris,Sakaki,Gold}, we will also calculate the mobility for the Gaussian model and compare the results of these two models.\n\n\\section{Quasi-classical picture}\\label{sec:class}\nInspired by Ref. \\onlinecite{Suris}, in this section we start from an intuitive quasi-classical picture of the electron scattering by the surface roughness and get the scaling result shown in Fig. \\ref{fig:conductivity_the_result}.\n\nElectrons are scattered when they hit the rough ``hard wall\" surface. The time between two consecutive collisions of electrons with the surface is $\\sim L\/v_F\\sim m^*L\/\\hbar k_F$. For each bounce, the reflection is specular with respect to the tangential plane of the hitting point and therefore adds a random angle $\\alpha$ to the direction of the reflected momentum. Due to this angular diffusion, the total relaxation of the momentum direction requires $\\alpha^{-2}$ times collisions. Thus the relaxation time is\n\\begin{equation}\\label{eq:relaxation time}\n\\tau=\\frac{m^*L}{\\hbar k_F\\alpha^2}.\n\\end{equation}\nBelow we are going to investigate the deviation angle $\\alpha$ at different values of $k_F$ and thus find $\\tau$.\n\nFor the exponential surface roughness, one can imagine the irregularities as islands going up or down. Each island is flat on a scale $d$ and drops or rises abruptly by a height $\\Delta$ on the edges. An electron can be regarded as a particle only on length scales larger than the wavelength $ k_F^{-1}$. At $d\\ll k_F^{-1}$ or $1\/d\\gg k_F$, electrons can only feel an averaged roughness of all islands within the region of size $k_F^{-1}$ whose number is $(k_F^{-1})^2\/d^2=1\/k_F^2d^2$. Due to the randomness of the distribution of these islands, the resulting height or depth has a magnitude $\\sim \\Delta\/\\sqrt{1\/k_F^2d^2}=\\Delta k_Fd$. Such a height\/depth on a length scale $k_F^{-1}$ effectively results in $\\alpha=\\Delta k_Fd\/ k_F^{-1}=\\Delta dk_F^2$. Using Eq. \\eqref{eq:relaxation time}, we get for $k_Fd\\ll 1$\n\\begin{equation}\\label{eq:tau_small_d}\n\\tau\\sim\\frac{1\/(\\Delta d k_F^2)^2}{\\hbar k_F\/m^*L}=\\frac{m^*L}{\\hbar\\Delta^2d^2k_F^5}.\n\\end{equation}\nIn the opposite case at $k_F^{-1}\\ll d$, the electron hits a single island each time it bounces off the surface. However, when electrons hit the flat middle plane of the island, there is no momentum relaxation. Only when electrons happen to hit the sharp edges can they get a ``scattering\" reflection. Since only on a scale $k_F^{-1}$ can electrons be seen as quasi-classical particles, the scattering edge is then estimated to be of height $\\Delta$ and size $k_F^{-1}$ which gives rise to $\\alpha=\\Delta k_F$ (see Fig. \\ref{fig:roughness}b). The probability to hit one such edge is proportional to its area fraction $k_F^{-1}d\/d^2=1\/k_Fd$. This gives\n\\begin{equation}\\label{eq:tau_large d}\n\\tau\\sim\\frac{1\/(\\Delta k_F)^2}{1\/k_Fd}\\frac{m^*L}{\\hbar k_F}=\\frac{m^*Ld}{\\hbar\\Delta^2k_F^2}.\n\\end{equation}\nSince $L\\sim a_B\/(na_B^2)^{1\/5},$ $k_F\\sim (na_B^2)^{2\/5}\/a_B$ as we said in Sec. \\ref{sec:accu}, we get the mobility as\n\\begin{equation}\\label{eq:mobility}\n\\mu\\sim \\frac{e}{h}\\frac{1}{\\Delta^2}\\times\\left\\{ \\begin{array}{ll}\n\\displaystyle{\\frac{a_B^{8\/5}}{d^2 n^{11\/5} }}, &na_B^2\\ll(a_B\/d)^{5\/2},\\\\\n&\\\\\n\\displaystyle{\\frac{a_Bd}{n}} \\,\\,\\,,& na_B^2\\gg (a_B\/d)^{5\/2},\n\\end{array} \\right.\n\\end{equation}\nand the 2D conductivity $\\sigma=ne\\mu$ as shown in Fig. \\ref{fig:conductivity_the_result}, where $k_Fd\\sim 1$ at $na_B^2\\sim (a_B\/d)^{5\/2}$. When $d>a_B$, in the 3D regime where $na_B^2>1$ there is no range of $k_Fd\\ll 1$ and the mobility always decreases as $\\propto 1\/n$. One should note that the 2D conductivity saturation $\\sigma\/(2e^2\/h)\\sim da_B\/\\Delta^2$ at large concentration $n$ is usually much larger than unity as $d,\\,a_B\\gg \\Delta$ and implies that the accumulation layer remains metallic at large concentrations.\n\n\\section{Single subband case}\\label{sec:sing}\nIn previous section, we have employed a quasi-classical perspective to understand the electron scattering off the surface roughness. Now we turn to the more formal quantum-mechanical approach.\n\nLet us start from the single subband case where the scattering occurs within the same subband. The scattering rate $1\/\\tau$ of an electron at the Fermi level with the wave vector $\\vec{k'}$ can be found according to Fermi's golden rule:\n\\begin{equation}\n\\frac{1}{\\tau}=\\frac{2\\pi}{\\hbar}\\int \\frac{d^2\\vec{k}}{(2\\pi)^2} \\frac{|U(q)|^2}{\\epsilon(q)^2}\\delta(\\varepsilon-\\varepsilon_F)(1-\\cos\\theta')\n\\end{equation}\nwhere $\\varepsilon=\\hbar^2k^2\/2m^*$, $\\varepsilon_F=\\hbar^2k'^2\/2m^*=\\hbar^2k_F^2\/2m^*$ are the final and initial (Fermi level) energies of an electron, $\\vec{k}$ is the final electron momentum, $k_F$ is the Fermi wavenumber, $\\theta'$ is the angle between initial and final electron momenta and $q=2k_F\\sin(\\theta'\/2)$ is the magnitude of the transferred momentum $\\vec{q}=\\vec{k}-\\vec{k'}$. All the momenta here are in the $x$-$y$ plane. Due to the electronic screening inside a single subband, the Fourier transform of the scattering potential $U(q)$ is reduced by the dielectric function $\\epsilon(q)$ \\cite{Ando_1982}\n\\begin{equation}\\label{eq:2D_screening}\n\\epsilon(q)\\simeq 1+2\/a_Bq.\n\\end{equation}\nLet us first derive the scattering potential resulting from the surface roughness. We know that electrons are confined near the interface and thus have a quantization kinetic energy $E_z=\\hbar^2k_z^2\/2m^*$ where $k_z$ is a multiple of $\\pi\/L$ (for the first subband, $k_z=\\pi\/L$). Due to the surface roughness, the confinement width $L$ fluctuates by $\\Delta(\\vec{r})$ at position $r$. The kinetic energy then varies by $(dE_z\/dL)\\Delta(\\vec{r})\\sim E_z\\Delta(\\vec{r})\/L$. These fluctuations of the quantization kinetic energy act as a fluctuating potential $U(\\vec{r})$ for the 2-dimensional motion of confined electrons. Its scattering matrix element for 2D Bloch states $U(q)$ within a given subband then satisfies\n\\begin{equation}\\label{eq:scatter_matrix_element}\n|U(q)|^2= \\left(\\frac{E_z}{L}\\right)^2W(q) .\n\\end{equation}\nAs a result we get:\n\\begin{equation}\\label{eq:quantum_large_d}\n\\frac{1}{\\tau}\\sim\\frac{\\hbar}{m^*}\\frac{k_z^4}{L^2}\\int d\\theta' \\frac{W(q)}{\\epsilon(q)^2}(1-\\cos\\theta').\n\\end{equation}\n\nAt $k_Fd\\ll 1$, according to Eqs. \\eqref{eq:Gauss} and \\eqref{eq:exp},  two models of roughness give the same $W(q)\\sim \\Delta^2d^2$. For the one subband case, $k_Fa_B \\simeq 1$ and $\\epsilon(q) \\simeq 1$.\nThis gives the scattering rate as\n\\begin{equation}\\label{eq:quantum_small_d}\n\\frac{1}{\\tau}\\sim\\frac{\\hbar}{m^*}\\frac{k_z^4\\Delta^2d^2}{L^2}.\n\\end{equation}\nAs $k_z\\sim 1\/L$, the mobility is then\n\\begin{equation}\\label{eq:2D_mobility_small_d}\n\\mu\\sim\\frac{e}{\\hbar}\\frac{L^6}{\\Delta^2d^2}.\n\\end{equation}\nSince in the single subband case $na_B^2 \\leq 1$, the condition of validity of Eq. \\eqref{eq:2D_mobility_small_d} $k_Fd\\ll 1$ is fulfilled for the roughness with $d < a_B$. The case $d < a_B$ was studied for silicon inversion layers in Ref. \\onlinecite{Ando_1977} and discussed in Introduction above. For the 2D inversion layers, the width $L$ is determined by the applied electric field as $L\\propto E^{-1\/3}$ and one gets \\cite{Ando_1977, Entin,Suris} $\\mu\\propto 1\/E^2$. As the electron concentration increases, the interfacial electric field $E\\propto n$ and mobility $\\mu\\propto 1\/n^{2}$. Such a dependence was obtained in Ref. \\onlinecite{Ando_1977} and used in Ref.~\\onlinecite{DasSarma} for extrapolation to the multisubband case as shown in Fig. \\ref{fig:conductivity_the_result}.\nNote that for the 2D quantum wells, the width $L$ of the electron gas is the same as the well width so that Eq. \\eqref{eq:2D_mobility_small_d} agrees with the well known result of Ref. \\onlinecite{Sakaki}.\n\nHowever, if $d>a_B$, for the single subband case, there is also a range of concentrations that satisfies $k_Fd\\gg 1$. The results of two roughness models are different.\nFor the Gaussian case, the typical $q$ is $\\sim 1\/d$ and the typical $\\theta'$ is $\\sim q\/k_F\\sim 1\/k_Fd$, $W(q)\\sim \\Delta^2d^2$, $\\epsilon(q)\\sim 1+d\/a_B\\sim d\/a_B$.  The scattering rate is then\n\\begin{equation}\\label{eq:quantum_large_d_Gauss_screen}\n\\frac{1}{\\tau}\\sim \\frac{\\hbar}{m^*}\\frac{k_z^4\\Delta^2a_B^2}{L^2k_F^3d^3}.\n\\end{equation}\nPutting $k_z\\sim 1\/L$ for the single subband case, we get the mobility as\n\\begin{equation}\\label{eq:2D_mobility_large_d_Gauss}\n\\mu\\sim\\frac{e}{\\hbar}\\frac{L^6d^3k_F^3}{\\Delta^2a_B^2},\n\\end{equation}\n\\begin{comment}\n\\begin{numcases}\n{\\mu\\sim \\frac{e}{\\hbar}\\frac{L^6dk_F^3}{\\Delta^2}\\times}\n1, & $d\\ll a_B$\\label{eq:2D_mobility_large_d_Gauss_1}\\\\ \\nonumber\n&\\\\\n\\frac{d^2}{a_B^2}, ~~~~~~~& $d\\gg a_B$\\ \\label{eq:2D_mobility_large_d_Gauss_2}\n\\end{numcases}\n\\end{comment}\nwhich can also be obtained from results in Refs. \\onlinecite{Gold,Suris}.\n\nFor the exponential case, $W(q)$ decays in a much milder way as $\\propto 1\/q^3$ at large $q$. This leads to the large angle scattering. Indeed, let us consider the contribution to the integral in Eq. \\eqref{eq:quantum_large_d} from the small angles $\\theta' \\sim (k_Fd)^{-1}$ and large angles $\\theta' \\sim 1$. In the first case, $W(q\\sim 1\/d)\\sim\\Delta^2d^2$, $\\epsilon(q)\\sim d\/a_B$ and\n$$\\int_0^{(k_Fd)^{-1}}d\\theta'(1-\\cos\\theta') \\simeq \\theta'^3 \\simeq \\frac{1}{(k_Fd)^3}.$$\nAs a result the contribution from the small angles to the scattering rate is the same as Eq. \\eqref{eq:quantum_large_d_Gauss_screen}.\nFor the large angles  $W(q\\sim k_F) \\simeq \\Delta^2 d^2 \\left(k_Fd \\right)^{-3}$ is $(k_Fd)^3$ times smaller than that of the small-angle scattering, $\\epsilon(q)\\simeq 1$ and the integral over the angle is\n$$\\int \\limits_{(k_Fd)^{-1}}^{\\pi} d\\theta'(1-\\cos\\theta')  \\sim 1.$$ The angle integral is $(k_Fd)^3$ times larger than that from the small angles, and the scattering rate is then\n\\begin{equation}\\label{eq:quantum_large_d_exp}\n\\frac{1}{\\tau}\\sim\\frac{\\hbar}{m^*}\\frac{k_z^4\\Delta^2  }{L^2k_F^3 d},\n\\end{equation}\nwhich due to the absence of screening is $(d\/a_B)^2$ times larger than that from the small-angle scattering.\nIn this sense, the large-angle scattering is more effective and always gives the mobility as\n\\begin{equation}\\label{eq:2D_mobility_large_d_exp}\n\\mu \\sim \\frac{e}{\\hbar} \\frac{L^6dk_F^3}{\\Delta^2 }.\n\\end{equation}\nThe dominance of the large-angle scattering is a unique feature of the exponential roughness.\n\n\\section{Intersubband scattering in multisubband accumulation layers}\\label{sec:inter}\nIn Sec. \\ref{sec:sing} we have calculated the mobility limited by the surface roughness scattering of a single subband. We not only have recovered the results for the Gaussian type of roughness obtained by previous studies but also have got the results for the relatively unexplored exponential roughness. For multisubband accumulation layers, the situation is different from the single subband case. First, $k_z$ is not $\\sim1\/L$ but typically is $\\sim k_F$, where $k_F=(n\/L)^{1\/3}$ is the 3D Fermi wavenumber of the electron gas. Second, though the screening now is still two-dimensional \\footnote{One could worry about the use of 2D screening for the 3D accumulation layer with width $L$. Indeed, it is known that in a metallic film with width $L$ at the distance $q^{-1}\\gg L$ the screening is two-dimensional, while at $ q^{-1} \\ll L$ it becomes three-dimensional. In our case, the 3D screening radius inside the layer is $r_D=a_B\/(Na_B^3)^{1\/6}\\sim a_B\/(na_B^2)^{1\/5}\\sim L .$ Thus, for $q^{-1}< L$ the 3D dielectric function is $\\sim 1$. Therefore, the 3D screening may be ignored.}, the screening radius is $L$ instead of $a_B$ so that $\\epsilon(q)\\simeq 1+1\/Lq$ due to collective screening of multiple subbands \\footnote{The 2D screening radius $r_2$ here is not $a_B$ but $L$ because when deriving $r_2\\propto d(e\\varphi)\/dn$ where $e\\varphi$ characterizes the electron chemical potential for each subband, the total 2D concentration $n$ is the electron concentration within each subband times the number of subbands $k_FL$. Therefore, the 2D screening radius is reduced from $a_B$ to $a_B\/k_FL\\simeq a_B\/(na_B^2)^{1\/5}\\sim L$.}. Last, in addition to the intrasubband scattering, there is also intersubband scattering.\n\nTypically, the intersubband scattering rate is of the same order of the intrasubband scattering as shown in Appendix \\ref{App:AppendixA}. Therefore, the final scattering rate is approximately the product of the typical intrasubband scattering rate and the total number of subbands which the initial electron can be scattered into. It is easy to check that the typical transferred momentum in the $z$-direction is of the same order as the typical transferred momentum $q$ in the $x$-$y$ plane. At $1\/d\\gg k_F$ where $q\\sim k_F$, all subbands ``communicate\" with each other and the total number is $k_FL$. Multiplying by $k_FL$ the intrasubband result Eq. \\eqref{eq:quantum_small_d} with $k_z\\sim k_F$, we arrive at the final scattering rate given by Eq. \\eqref{eq:tau_small_d}.\nThis is a universal result for both Gaussian and exponential models.\nAt $1\/d\\ll k_F$, for the exponential roughness, the typical transferred momentum is $q\\sim 1\/d$ for the small-angle scattering and $q\\sim k_F$ for the large-angle one. So for the former, the number of subbands involved in the scattering process is $\\sim L\/d$, much smaller than the number $ k_FL$ for the latter. Considering also its immunity to screening as $k_FL\\gg1$, the large-angle scattering mechanism absolutely dominates. Using the intrasubband scattering rate as given by Eq. \\eqref{eq:quantum_large_d_exp} and $k_z\\sim k_F$, we arrive at the total scattering rate at $k_Fd\\gg 1$ for the exponential roughness given by Eq. \\eqref{eq:tau_large d}. Thereby, we get the same expressions for the mobility as in Eq. \\eqref{eq:mobility}. The corresponding 2D conductivity at large $n$ saturates as $\\sigma\/(2e^2\/h)\\sim da_B\/\\Delta^2$ (see Fig. \\ref{fig:conductivity_the_result}), which means that there is no re-entrant metal-insulator transition. In Appendix \\ref{App:AppendixA}, we do more careful estimation of the numerical coefficient in front of $da_B\/\\Delta^2$ and get that it is close to 1.\n\nCompared to the simple result in the exponential case, the mobility for the Gaussian roughness is more complicated. At $k_F^{-1}\\ll d<L$, the intrasubband scattering is unscreened for the Gaussian case and the rate is the same as Eq. \\eqref{eq:quantum_large_d_exp}. The typical momentum transfer is $q\\sim 1\/d$, so only a few subbands the number of which is $\\sim q\/L^{-1}=L\/d$ can participate in the intersubband scattering. The total scattering rate is then the result in Eq. \\eqref{eq:quantum_large_d_exp} times $L\/d$. At even larger $ d\\gg L\\gg k_F^{-1}$, the typical momentum transferred $1\/d$ is smaller than the $z$-direction momentum quantization $1\/L$. Therefore no intersubband scattering is possible. This situation resembles the single subband case and is natural since when $L\\ll d$ we actually are dealing with a 2D system. At $d\\gg L$, the screening of the potential adds the factor $(L\/a_B)^2$ to the total scattering rate given by Eq. \\eqref{eq:quantum_large_d_Gauss_screen} as the screening radius now is $L$ instead of $a_B$. The result is summarized as follows\n\\begin{numcases}\n{\\frac{1}{\\tau}\\sim }\n\\frac{\\hbar k_F \\Delta^2}{m^*Ld^2 }, & $k_F^{-1}\\ll d\\ll L\\,,$\\label{eq:3D_scattering_rate_large_d_Gauss_1}\\\\ \\nonumber\n&\\\\\n\\frac{\\hbar k_F\\Delta^2}{m^*d^3},& $L\\ll d\\,.$\\label{eq:3D_scattering_rate_large_d_Gauss_2}\\\\ \\nonumber\n\\end{numcases}\n$k_F^{-1}=d$ is reached at $na_B^2\\sim (a_B\/d)^{5\/2}$ and $d=L$ is achieved at $na_B^2\\sim (a_B\/d)^5$. By expressing $k_F$ and $L$ in terms of $n$, one can then get the mobility as a function of the 2D electron concentration. The corresponding results together with that for the exponential case are listed in Table. \\ref{tab:1}.\n\\begin{table\n\\caption{\\label{tab:1} Mobility $\\mu$ in units of $\\left(e\/\\hbar\\right)\\left(d^4\/\\Delta^2\\right)$ as a function of the 2D electron concentration $n$ at different values of $d$ for two types of surface roughness, the Gaussian model (G) and the exponential one (E). Since the Fermi wavenumber $k_F\\simeq (n\/L)^{1\/3}$, and the width of the 2D electron gas $L\\sim a_B\/(na_B^2)^{1\/5}$, we get $k_F^{-1}=d$ at $na_B^2\\sim (a_B\/d)^{5\/2}$ and $L=d$ at $na_B^2\\sim(a_B\/d)^5$. }\n\\begin{ruledtabular}\n\\renewcommand{\\arraystretch}{2}\n\\begin{tabular}{ c  | c| c| c}\n & $d<k_F^{-1}$ & $k_F^{-1}<d<L$& $L<d$ \\\\ \\hline\nG$\\quad$&$ (a_B\/d)^6\/(na_B^2)^{11\/5}$&$(a_B\/d)^{2}\/(na_B^2)^{3\/5}$&$(a_B\/d)\/(na_B^2)^{2\/5}$\\\\ \\hline\nE$\\quad$&$(a_B\/d)^6\/(na_B^2)^{11\/5}$&$(a_B\/d)^3\/(na_B^2)$&$(a_B\/d)^3\/(na_B^2)$\\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\nThe obtained $\\mu(n)$ dependence is presented in Fig. \\ref{fig:mobility}. For the exponential roughness, the corresponding 2D conductivity $\\sigma$ is shown in Fig. \\ref{fig:conductivity_the_result}. The smallest conductivity for the exponential case is larger than $2e^2\/h$ as mentioned above. For the Gaussian model, the smallest $\\sigma\/(2e^2\/h)$ is also $\\sim da_B\/\\Delta^2$. Since in reality, $d>\\Delta$, we get $\\sigma\/(2e^2\/h)\\gg1$. So, the smallest conductivity for the Gaussian roughness is also always above the critical value and no re-entrant metal-insulator transition will happen in realistic situations.\n\\begin{comment}\\begin{equation}\\label{eq:screen_total}\n\\mu\\sim \\frac{e}{h}\\times\\left\\{ \\begin{array}{ll}\n\\displaystyle{\\frac{d^3a_B}{\\Delta^2(na_B^2)^{4\/5}}}, & \\textrm{Gaussian-roughness}\\,,\\\\\n&\\\\\n\\displaystyle{\\frac{da_B^3}{\\Delta^2(na_B^2)}} \\,\\,\\,,& \\,\\textrm{island-roughness}\\,.\n\\end{array} \\right.\n\\end{equation}\n\\end{comment}\n\\begin{figure}[h]\n\\includegraphics[width=.45\\textwidth]{mobility_new}\\\\\n\\caption{The scaling behavior of the mobility $\\mu$ in units of $\\left(e\/\\hbar\\right)\\left(d^4\/\\Delta^2\\right)$ as a function of the scaled electron concentration $na_B^{2}$ plotted at $d<a_B$ in a double logarithmic scale. The thick solid line (black) denotes the mobility of the accumulation layer for the exponential roughness. The thin solid line (black) represents the mobility for the Gaussian roughness which also decreases. Here only powers of the $n$ dependence are shown while the complete scaling formulae are presented in Table. \\ref{tab:1}. The thin dashed line (red) represents the $1\/n^2$ dependence derived for a single subband \\cite{Ando_1977} and its conjectured extrapolation \\cite{DasSarma} to larger concentrations. }\\label{fig:mobility}\n\\end{figure}\n\n\n\n\nIn Sec. \\ref{sec:class}, we gave a quasi-classical explanation of the mobility limited by the exponential surface roughness. Inspired by Ref. \\onlinecite{Suris}, we can interpret the Gaussian roughness results quasi-classically as well. Below we again start from Eq. \\eqref{eq:relaxation time} and find $\\alpha$ for the Gaussian roughness in different situations.\nAt $k_Fd\\ll1$, the roughness relief is averaged over the electron wavelength and the resulting relaxation time is the same as in the exponential case given by Eq. \\eqref{eq:tau_small_d}.\nAt $k_F^{-1}\\ll d\\ll L$, the wavelength $k_F^{-1}$ is smaller than the size of the roughness hill. The electron then collides with a single hill (valley) each time it hits the surface and the deviation angle $\\alpha$ is the slope of each single hill (valley) $\\sim \\Delta\/d$ (see Fig. \\ref{fig:roughness}a). The relaxation time is given by Eq. \\eqref{eq:3D_scattering_rate_large_d_Gauss_1}.\nAt $L\\ll d$, the electronic screening changes the scattering potential by a factor $L\/d$, which can effectively be regarded as reducing the roughness height from $\\Delta$ to $\\Delta L\/d$. The size of each single hill is so large that the electron can hit the same hill consecutively for several times during which it has traveled back and forth for $\\sim d\/L$ times within the accumulation layer. Since these consecutive hits are on the same slope, the scattered angle is the same and $\\alpha$ accumulates, in contrast with uncorrelated random collisions on different hills (valleys). After the electron finishes colliding with the same slope, the accumulated angle is $(\\Delta L\/d^2)(d\/L)=(\\Delta\/d)\\ll1$ and the time of such a series of collisions is $\\sim(L\/k_F)( d\/L)=d\/k_F$. The resulting relaxation time is then given by Eq. \\eqref{eq:3D_scattering_rate_large_d_Gauss_2}.\nThus we obtained quasi-classically the same scaling behavior of the mobility limited by the Gaussian roughness as by the quantum-mechanical approach.\n\n\\section{Conclusion}\\label{sec:con}\nIn this paper, we have studied the surface-roughness limited mobility in inversion and multisubband accumulation layers as a function of the 2D electron concentration $n$\nfor two models of the surface roughness both quantum-mechanically and quasi-classically. For the more realistic exponential roughness, the mobility decreases as $\\propto 1\/n$ at large $n$ and results in a 2D conductivity saturation as $\\sigma\/(2e^2\/h)\\simeq d a_B\/\\Delta^2\\gg 1$ since the characteristic roughness size $d$ and the effective Bohr radius $a_B$ are larger than the characteristic roughness height $\\Delta\\simeq a\/2$ where $a$ is the lattice constant. For the Gaussian roughness which was widely used in earlier studies, the minimum conductivity is found to be larger than the critical value as well. One should note that the considerations here have not included the contribution from the tail electrons, which makes $\\sigma$ even larger in reality due to their larger distances from the surface and thus larger relaxation times \\cite{RAT}.\nSo there is no reason to expect the re-entrant metal-insulator transition~\\cite{DasSarma} at large concentrations. Indeed, decent conductivities were observed in large concentration accumulation layers in Refs. \\onlinecite{diamond,Iwasa_2009,Iwasa_2012,Tardella}.\n\n\n\n$\\phantom{}$\n\\vspace*{2ex} \\par \\noindent\n{\\em Acknowledgments.}\n\nWe are grateful to S. A. Campbell, A. V. Chaplik, M. V. Entin, B. Jalan, M. J. Manfra, M. Sammon, M. Shur, and R. A. Suris for helpful discussions.\nThis work was supported primarily by the National Science Foundation through the University of Minnesota MRSEC under Award No. DMR-1420013.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nGas cooling plays a key role in galaxy formation.  According to the \nhierarchical structure formation model, galaxies form in gravitationally \ncollapsed dark matter halos.  The dissipative baryonic matter cools by \nemitting radiation and condenses into halos to form galaxies \n\\citep{Rees.Ostriker:77, White.Rees:78}.\nSubsequently stars form through the radiative cooling of interstellar medium \n\\citep[ISM; e.g.,][]{McKee.Ostriker:07}.\n\nThe cooling rate of gas varies with its chemical abundance, because different \natoms and molecules have different cooling rates.  The chemical abundance of \ncosmic gas evolves as a function of time and environment.\nIn the early universe, the primordial gas consists of mainly hydrogen \n(mass fraction of $X\\simeq 0.74$) and helium (mass fraction of $Y\\simeq 0.25$) \nwith a negligible amount of light metals \\citep[e.g.][]{Burles.etal:01}.\nThroughout this paper, all the elements heavier than helium are collectively \ncalled `metals'.  Most metals are produced inside stars and spread out to the \nspace by galactic winds driven by supernova (SN) explosions.  \nIn the metal-enriched gas, the total cooling rate is significantly enhanced \nrelative to that of the primordial gas by the atomic emission lines from \nrecombination of ionised metals. \n\nMetals may influence galaxy formation in two different ways. \nFirstly, the SF efficiency is increased owing to the shorter gas cooling time. \nStar formation generally takes place in dusty metal-enriched ISM, therefore, \nthe enhanced cooling rate by metals would boost up the SF efficiency. \nHowever, the energy\/momentum feedback by SNe may suppress the subsequent star \nformation after the initial starburst by heating up the ambient gas.  \nThis complicates the situation, and it is not clear whether the net effect \nof metal cooling and SN feedback would be negative or positive.  \nThe feedback process is highly nonlinear, therefore, a direct numerical \nsimulation would be a useful tool to explore the effects of feedback and \nmetal cooling. \n\nSecondly, the metals dispersed into the IGM by SN feedback also enhance the \ncooling of IGM.  This can lead to the increase of IGM accretion onto \ngalaxies, because colder gas can sink into galaxies more easily.  \nBoth of the above two effects may affect the galaxy growth and \ncosmic SF history significantly.  Therefore it is essential to include the \neffects of metal cooling and chemical enrichment by feedback in the studies \nof galaxy formation and evolution, and capture the complex two-way \ninteractions between galaxies and IGM. \nIn addition, metal enrichment also affects the equation of state by \naltering the mean molecular weight of gas, which influences the hydrodynamic \ncalculation of gas thermal state. \n\nCosmological hydrodynamic simulations are widely used in the studies of galaxy \nformation and cosmic SF history \\citep[e.g.,][]{Cen:92, Katz.etal:92, \nCen.Ostriker:93, Katz.etal:96, Nagamine.etal:00, Pearce.etal:01, \nNagamine.etal:01a, Nagamine.etal:01b, \nAscasibar.etal:02, Murali.etal:02, Nagamine:02, Weinberg.etal:02, \nKawata.Gibson:03, Kravtsov:03, Marri.White:03, Springel.Hernquist:03:S, \nGovernato.etal:04, Robertson.etal:04, Weinberg.etal:04, \nNagamine.etal:04, Nagamine.etal:05a, Nagamine.etal:05b, Cen.etal:05, \nKeres.etal:05, Finlator.etal:06, Hoeft.etal:06, Nagamine.etal:06, \nScannapieco.etal:06, Tasker.Bryan:06, Dave.Oppenheimer:07, Finlator.etal:07, \nGovernato.etal:07, Kobayashi.etal:07, Tornatore.etal:07, Finlator.etal:08}.\nMost of these works included the treatments of radiative cooling by H \\& He, \nstar formation, and SN feedback. However, some of the simulations did not \ninclude the effects of metal cooling and\/or chemical enrichment by \ngalactic wind. \nFurthermore, the effects of metal cooling on galaxy growth and cosmic SF \nhistory has not been explored in detail using cosmological hydrodynamic \nsimulations and presented in the literature, as it is costly to run large \ncosmological simulations with and without the effect of metal cooling. \n\nIn this paper, we investigate the effects of metal cooling and metal \nenrichment on galaxy growth and cosmic SF history using a series of \ncosmological hydrodynamic simulations with and without metal cooling. \nThe aim of this paper is to single out the effects of metal cooling among \nour simulations.  This work can be regarded as our initial step towards the \nlong-term goal of developing more complete cosmological hydrodynamic code \nwith physically motivated models of star formation and feedback. \n\nThe paper is organised as follows. \nIn \\S~\\ref{sec:method}, we describe our simulation method focusing on how we \nimplement the metal cooling. In \\S~\\ref{sec:global}, we study the \nmetal cooling effects on the global properties, such as cosmic SFR, phase \nspace ($\\rho$ vs. $T$) distribution of gas, and the evolution of four \nphases (hot, warm-hot, diffuse, and condensed) of baryons.\nWe then study the galaxy mass functions (\\S~\\ref{sec:MF}) and \ngas mass fractions (\\S~\\ref{sec:GasFrac}). \nFinally, we discuss and summarise our findings in \n\\S~\\ref{sec:summary}.\n\n\n\n\\section{Numerical technique}\n\\label{sec:method}\n\n\\subsection{Simulation and Metal Cooling}\n\\label{sec:simulation}\n\nWe use the updated version of the Tree-particle-mesh (TreePM) smoothed \nparticle hydrodynamics (SPH) code GADGET-2 \\citep{Springel:05} for our \ncosmological simulations.  The gravitational dynamics is computed by a TreePM \nalgorithm, which uses a particle-mesh method \\citep{Hockney.Eastwood:88} for \nthe long-range gravitational force and a Tree method for the short-range \ngravitational force \\citep{Barnes.Hut:86}.  This hybrid algorithm makes the \ngravitational force calculation faster than a Tree method and allows better \nforce resolution than a PM method in dense regions.  The gas dynamics is \ncomputed by an SPH method.  The SPH is particularly useful if the simulation \nneeds to resolve large dynamical range, which is an inevitable requirement for\nthe study of galaxy formation in a cosmological context.  Therefore a \nTreePM-SPH simulation can provide a fast and high resolution calculation for \nboth gravitational dynamics and hydrodynamics.\n\nThe GADGET-2 code adopts the entropy-conservative formulation \\citep{Springel.Hernquist:02}, \nwhich alleviates the overcooling problem that previous SPH codes \nsuffered from.  Our basic simulations include radiative cooling and heating \nprocesses for hydrogen and helium using a method similar to \n\\citet*{Katz.etal:96}.  An external UV background radiation is treated as \na spatially uniform photoionising radiation \\citep{Haardt.Madau:96}, and \nmodified to match the Ly$\\alpha$ forest observations \\citep{Dave.etal:99}.  \nImplementing star formation and feedback \nfrom first principles is not feasible in current cosmological simulations, \nbecause the spatial \\& mass-scales of molecular clouds are not resolved. \nStar formation and SN feedback are represented by the subgrid multiphase ISM \nmodel developed by \\citet{Springel.Hernquist:03}. In the multiphase scheme, \na single gas particle represents both hot and cold gas.  The stars \nare formed in the cold portion when the density exceeds a given threshold, \n$\\rho_{\\rm th}$, which is derived self-consistently within the multiphase ISM model. \nThe related parameters, such as the normalisation of gas consumption time and \nevaporation efficiency of cold gas, are set to satisfy the empirical \nKennicutt-Schmidt law \\citep{Kennicutt:98a, Kennicutt:98b}.  \nThe SN feedback returns some fraction of the cold gas to the hot phase, \nand increases the thermal energy of the hot gas.  \n\nAs an extension to the multiphase ISM model, the simulation includes a \nphenomenological model for SN-driven galactic wind \\citep{Springel.Hernquist:03}.  \nThe galactic wind is particularly important for distributing the metals \nproduced by SNe into the IGM.  We use the strong kinematic wind with a \nvelocity of 484\\,km\\,s$^{-1}$.  It has been shown that this model produces\nfavourable results for the luminosity function of Lyman-break galaxies \nat the bright-end \\citep{Nagamine.etal:04} and the H\\,{\\sc i} column density \ndistribution function \\citep*{Nagamine.etal:04-dla} at $z=3$, when compared \nto the runs without the wind. \n\nHowever, \\citet{Dave.Oppenheimer:07} pointed out the problems of this \ngalactic wind model by comparing with the observations of C\\,{\\sc iv} \nabsorption lines in quasar spectra, and suggested that the momentum-driven \nwind model \\citep{Murray.etal:05} is a more viable model,\nwhich can carry more metals with lower wind velocities.  \nIn this paper we choose not to modify our galactic wind model in order to \nsingle out the effects of metal cooling, and to allow direct comparisons\nto the previous works \\citep{Nagamine.etal:04, Nagamine.etal:05a, \nNagamine.etal:05b}.  Here we focus on the effects of metal cooling on galaxy \ngrowth, while \\citet{Dave.Oppenheimer:07} focused on the C\\,{\\sc iv} \nstatistics of the IGM.  \n\n\n\\begin{figure*}\n\\centerline{\\includegraphics[width=2.0\\columnwidth,angle=0] {XYZ_table.eps}}\n\\caption{\nThe relationship between metallicity parameters in our implementation of \nmetal cooling. \nThe four panels show the following quantities as a function of metallicity\n$Z\/Z_\\odot$, where $Z_\\odot$ is the solar metallicity. \n[Fe\/H], hydrogen mass fraction $X$, helium mass fraction $Y$, and oxygen \nabundance $12 + \\log(O\/H)$.  Panel (d) represents how $\\alpha$-particles \ncontribute to the metal evolution.\n}\n\\label{fig:XYZ_table}\n\\end{figure*}\n\n\nThe metallicity of gas particles are also tracked by the code, assuming a \nclosed box model for each gas particle. The yield \n($y = \\Delta M_{\\rm metal} \/ \\Delta M_{\\rm gas}$) of 0.02 is assumed. \nIn principle, there could be a time delay between SN explosions and chemical \nenrichment of the ambient gas.  Unfortunately, current cosmological simulations\ndo not have sufficient resolution to track the detailed mixing process of \nmetals.  In our simulations, we ignore this time delay and assume an \ninstantaneous mixing within each gas particle.  This assumption is reasonable, \nbecause the overall time step of the simulations is longer than the \nmixing time-scale on small scales. \n\nIdeally we would like to track individual metals and compute the cooling \nrate of each element at each time-step of the simulation.  This approach has \nbeen used to study the evolution of individual galaxies\n\\citep {Recchi.etal:01,Lia.etal:02,Kawata.Gibson:03,Kobayashi:04}.\nFor cosmological simulations, it requires a large memory to track individual \nmetals and the computation becomes exceedingly expensive \n\\citep[but see][for such an attempts]{Scannapieco.etal:05,Martinez.etal:08}.\nOur simulations collectively track the total mass fraction of metals \n($Z\\equiv m_{\\rm metal}\/m_{\\rm gas}$) as the measure of metallicity. \nWe obtain the chemical abundance and cooling rates for a given $Z$ and\ntemperature by interpolating the table given in \n\\citet[][hereafter SD93]{Sutherland.Dopita:93} for the standard collisional \nionisation equilibrium model, and create a lookup table.\nMetal cooling is implemented in the full range of metallicity in the\ntable from SD93, from [Fe\/H]=-3 to [Fe\/H]=1. In addition, the metal cooling \noutside of this range is computed by extrapolating the table. \nWe require that the extrapolation does not go lower than the primordial cooling rate.  \nThe primordial abundance pattern is used where [Fe\/H] $\\le -1$ \n\\citep{Wheeler.etal:89,Bessell.etal:91}, and the solar abundance pattern is used where \n[Fe\/H] $\\ge 0$ \\citep{Anders.Grevesse:89}. \nFor $-1 <$ [Fe\/H] $ < 0$, the abundance pattern is computed by interpolating \nbetween the primordial and solar abundance patterns. \nNote that, [Fe\/H] = $\\log (n_{\\textup{Fe}}\/n_{\\textup{H}})$ - $\\log (n_{\\textup{Fe}_{\\sun}}\/n_{\\textup{H}_{\\sun}})$, where $n$ is the number density.\n\nFigure \\ref{fig:XYZ_table} shows the relationships between the metal \nparameters ([Fe\/H], $X$, $Y$, and $12 + \\log(O\/H)$) and the total \nmetallicity $Z$.\nBecause the abundance pattern is fixed for a given $Z$, \n[Fe\/H] is a monotonically increasing function of $Z$. The main difference \nbetween the solar and the primordial abundance pattern is the oxygen or \n$\\alpha$-particle enhancement.\nIn the early phase of metal enrichment, Type II SNe are the main source of \nmetals and they tend to generate more $\\alpha$-particles than SN Ia. \nTherefore, it is expected that a low metallicity gas to have more \n$\\alpha$-particles than the high metallicity one.\nAs mentioned above, we use two different chemical abundance pattern (at low- and \nhigh-end of the $Z$ values) to model this evolution in chemical composition.\nWe compute the oxygen abundance based on the Table 4 of SD93. \nIn our code, we incorporate the metal cooling effect by adding the additional \ncooling contribution from metals on top of the primordial cooling rate, \nwhich is computed in a similar fashion as described in \\citet{Katz.etal:96}.  \n\nMetal enrichment changes the mean molecular weight $\\mu$, which is needed to \ncompute the temperature and internal energy of gas.  In order to compute \n$\\mu$, we need to know the electron number density $n_e$, which varies with \nmetallicity.  We obtain the values of $n_e$ from SD93 in the same way as we \nobtained the metal cooling rates.  Our electron number density estimate \nignores the electrons from photoionised metals by the UV background \nradiation, but we expect that the number of electrons from ionised metals \nwill be small. \n\nVarying the mean molecular weight $\\mu$ also influences the SF threshold \ndensity $\\rho_{\\rm th}$ in the multiphase ISM model of \\citet{Springel.Hernquist:03}, \nas well as the cold and hot gas fractions of multiphase gas particles. \nIn this paper, we consider two different treatments of $\\rho_{\\rm th}$. \nIn the `constant $\\rho_{\\rm th}$' model, we take the value of $\\rho_{\\rm th}$ as in the \noriginal formulation by \\citet{Springel.Hernquist:03} and do not modulate\n$\\rho_{\\rm th}$ by the varying $\\mu$.  \nIn the `varying $\\rho_{\\rm th}$' model, we take into account of the modulation of \n$\\rho_{\\rm th}$ by the change in $\\mu$.  \n\n\n\\subsection{Simulation Setup}\n\\label{sec:IC}\n\nWe use three series of simulations with varying box size and resolution. \nThe effects of metal enrichment and metal cooling are tested by comparing \nthe runs listed in Table~\\ref{tbl:Simulation}.\nThe effects of galactic wind are also tested by turning on and off the wind.\nThe adopted cosmological parameters of all simulations are \n$(\\Omega_{m}\\, , \\Omega_{\\Lambda}\\, , \\Omega_{b}\\, , \\sigma_{8}\\, , h) \\: = \\: (0.3\\, ,0.7\\, ,0.04\\, ,0.85\\, ,0.7)$, which mostly concur with the results of \nthe Wilkinson Microwave Anisotropy Probe (WMAP1) \\citep{WMAP1}.\n\n\n\\begin{table*}\n\\centering\n\\begin{minipage}[t]{1.0\\textwidth}\n\\begin{tabular}{lcccccclc}\n\\hline\n  Run & \n  Box size \\footnote{Box size in units of $h^{-1}$\\,Mpc.} &  \n  $N_{p}$ \\footnote{Initial particle number of dark matter and gas particles.} & \n  $m_{\\rm DM}$ \\footnote{Dark matter particle mass in units of $h^{-1} \\textup{M}_{\\sun}$.}& \n  $m_{\\rm gas}$ \\footnote{Initial gas particle mass in units of $h^{-1} \\textup{M}_{\\sun}$. The star particle mass is set to $0.5 m_{\\rm gas}$ in our simulations.} & \n  $\\epsilon$ \\footnote{Comoving gravitational softening length in units of $h^{-1}$\\,kpc.  This is a proxy for the spatial resolution of the simulation, and the N216L10 series has a spatial resolution of physical $\\approx 0.5$\\,$h^{-1}$\\,kpc at $z=3$. }&  \n  $z_{\\rm end}$ & \n  Metal cooling \\footnote{`Primordial' means the cooling rate with primordial chemical composition.  See \\S~\\ref{sec:simulation} for the description of `constant $\\rho_{\\rm th}$' and `varying $\\rho_{\\rm th}$' models.} &\n  Wind \\footnote{If employed, the wind is a `strong' wind with $v_w = 484$\\,km\\,s$^{-1}$.}\\\\\n\\hline\n N144L10      & 10.00   & $2 \\times 144^{3}$ & $2.42 \\times 10^{7}$ & $3.72 \\times 10^{6}$ & 2.78 & 2.75 & No (primordial) & Yes \\\\\n N144L10mc    & 10.00   & $2 \\times 144^{3}$ & $2.42 \\times 10^{7}$ & $3.72 \\times 10^{6}$ & 2.78 &2.75 & Yes (const. $\\rho_{\\rm th}$) & Yes \\\\\n N216L10      & 10.00   & $2 \\times 216^{3}$ & $7.16 \\times 10^{6}$ & $1.10 \\times 10^{6}$ & 1.85 &2.75 & No (primordial) & Yes \\\\\n N216L10nw    & 10.00   & $2 \\times 216^{3}$ & $7.16 \\times 10^{6}$ & $1.10 \\times 10^{6}$ & 1.85 &2.75 & No (primordial) & No \\\\\n N216L10mc    & 10.00   & $2 \\times 216^{3}$ & $7.16 \\times 10^{6}$ & $1.10 \\times 10^{6}$ & 1.85 &2.75 & Yes (const. $\\rho_{\\rm th}$) & Yes \\\\\n N216L10mv    & 10.00   & $2 \\times 216^{3}$ & $7.16 \\times 10^{6}$ & $1.10 \\times 10^{6}$ & 1.85 &2.75 & Yes (varying $\\rho_{\\rm th}$)  & Yes \\\\\n N288L34      & 33.75   & $2 \\times 288^{3}$ & $1.16 \\times 10^{8}$ & $1.79 \\times 10^{7}$ & 4.69 &1.00 & No (primordial) & Yes \\\\\n N288L34mc    & 33.75   & $2 \\times 288^{3}$ & $1.16 \\times 10^{8}$ & $1.79 \\times 10^{7}$ & 4.69 &1.00 & Yes (const. $\\rho_{\\rm th}$) & Yes \\\\\n\\hline\n\\end{tabular}\n\\caption{The simulations employed in this paper}\n\\label{tbl:Simulation}\n\\end{minipage}\n\\end{table*}\n\nOur naming conventions for the simulations are as follows.\nThe first part of the run name denotes the particle number and box size\nused for the simulation series.  \nThe run with the extension of `mc' (for `metal cooling') adopts the \n`constant $\\rho_{\\rm th}$' model for metal cooling, the run with the extension `mv' \n(for `metal, varying') adopts the `varying $\\rho_{\\rm th}$' model, the run \nwith the extension `nw' has no galactic wind, and the run without any \nextension uses the cooling rates for the primordial chemical composition \n(i.e., only H and He). \n\nThe `N216L10' series is the fiducial simulation set in our study. \nWe implement one simulation without galactic wind (`N216L10nw' run). \nThe `N216L10' run has the same physical models as the `Q4' run used in \n\\citet{Springel.Hernquist:03:S} and \\citet{Nagamine.etal:04} with no metal \ncooling. \n\nTo examine the resolution effect, we implement lower resolution simulations, \nthe `N144L10' series.  Owing to the comparably small box size and missing \nlong wavelength perturbations, we evolve the N144L10 and N216L10 series \nonly down to $z=2.75$.  To evolve the simulation toward lower redshifts, \nwe need a simulation with a larger box size. Therefore we implement the \n`N288L34' series, which has a larger box size than other series, but a \nlower resolution.  This series is evolved down to $z=1$.\nThe runs within the same series use the identical initial condition,\nwhich makes the comparison more robust and enables halo-by-halo comparison \nif needed. \n\n\n\n\\subsection{Galaxy finding method}\n\\label{sec:Groups}\n\nIn cosmological simulations, dark matter, gas, and stars are represented \nby particles, and these particles are Monte Carlo representations of matter \ndistribution.  In order to study galaxy formation using a cosmological \nsimulation, we need a working definition for simulated galaxies.\nIn this paper the simulated galaxies are defined as isolated groups of star \nand gas particles, identified by a simplified variant of the \\textup{SUBFIND} \nalgorithm developed by \\citet{Springel.etal:01}.  In more detail, the code \nfirst computes a smoothed baryonic density field to identify candidate \ngalaxies with high density peaks.  The full extent of these groups are found \nby adding gas and star particles to the groups in the order of declining \ndensity.\nIf all $N_{\\rm min}$\\footnote{$N_{\\rm min}$ is the minimum number of gas and star \nparticles that constitute one isolated group.  In this paper we set \n$N_{\\rm min} = 32$.} nearest neighbour particles have lower densities, this group of \nparticles is considered as a new group.  If there is a denser neighbour, \nthe particle is attached to the group to which its nearest denser neighbour \nalready belongs to.  If two nearest neighbours belong to different groups and \none of them has less than $N_{\\rm min}$ particles, these the two groups are merged.\nIf two nearest neighbours belong to different groups and both of them has more \nthan $N_{\\rm min}$ particles, the particles are attached to the larger group, \nleaving the other group intact.  In addition, the gas particles in groups \nshould be denser than $0.01 \\rho_{\\rm th}$. \nIn this paper we are not concerned with groups without star particles. \nThe fraction of such groups are very small, therefore they do not affect \nour conclusions. \n\n\n\n\\section{Global properties}\n\\label{sec:global}\n\n\\subsection{Cosmic star formation history}\n\\label{sec:cos_sfr}\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=1.0\\columnwidth,angle=0] {sfr_N216L10.eps}}\n\\caption{\nCosmic star formation histories for the N216L10 series, comparing the runs \nwith and without metal cooling. \nIt also shows the run without a galactic wind (N216L10nw), \nand a lower resolution run (N144L10mc). \nOverall the SFR density is increased by $\\sim 20-30$\\% by metal cooling\nthroughout the entire redshift range. \n}\n\\label{fig:sfr_N216L10}\n\\end{figure}\n\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=1.0\\columnwidth,angle=0] {sfr_N288L34.eps}}\n\\caption{\nThe same as Figure~\\ref{fig:sfr_N216L10}, but for the N288L34 series, which \ngoes down to $z=1$. \nThe N216L10mc run is also shown for a resolution comparison. \nThe green dot-dashed line (HS03) is for the analytic model of \n\\citet[][Eq.\\,(2)]{Hernquist.Springel:03}. \n}\n\\label{fig:sfr_N288L34}\n\\end{figure}\n\nFigure~\\ref{fig:sfr_N216L10} shows the effect of metal cooling and galactic \nwind on the cosmic SF history for the N216L10 series.  \nThe comparison of N216L10mc and N216L10 runs clearly shows that metal cooling \nenhances the cosmic SFR throughout all redshifts by $20-30$\\%. \nPreviously, \\citet[][Fig.~13]{Hernquist.Springel:03} argued that\nthe SFR density is enhanced by metal cooling mostly at lower redshifts \nand hardly at high-$z$, however, our simulations suggest otherwise. \n(See \\S~\\ref{sec:summary} for more discussion on this point.)\n\nMetallicity needs to be higher than $\\sim 10^{-2} Z_\\odot$ to noticeably \nincrease the gas cooling rate, because the metal cooling rate is much \nsmaller than that by H and He at $Z \\leq 10^{-2} Z_\\odot$.\nThe metallicity of diffuse IGM is generally lower than $10^{-2} Z_\\odot$ at \n$z \\gtrsim 3$, and the metal cooling effect on IGM is small. \nHowever the local SFR could be enhanced significantly even at high-$z$ when\nthe ISM is enriched to $Z > 10^{-2} Z_\\odot$, which can be easily achieved \nafter a few events of SN II.  Our results suggest that star formation \nat high-$z$ can also be enhanced by metal cooling, and not just at low-$z$. \nWe also find that the peak of the cosmic SF history hardly shifts by the \nintroduction of metal cooling, consistently with the estimate by \n\\citet{Hernquist.Springel:03}.\n\nThe N216L10mv run has a lower SFR at $z\\gtrsim 9$ and a higher \nSFR at $z\\lesssim 7$ than the N216L10mc run. \nThis is because at high-$z$, the value of $\\rho_{\\rm th}$ is higher in the `mv' run \nthan in the `mc' run due to lower metallicity and mean molecular weight, \nleading to a larger fraction of gas not being able to satisfy the SF \ncriteria in the N216L10mv run than in the N216L10mc.\nThe opposite is true at $z\\lesssim 3$. \n\nFigure~\\ref{fig:sfr_N216L10} also shows that the SFR in the N216L10nw run \nis higher than other runs by an order of magnitude.  This is because the gas \ncan cool very efficiently without being ejected by the galactic wind. \nAlthough the current galactic wind models in cosmological simulations are not \nwell established yet and need to be improved, both theoretical and \nobservational studies evidently suggest the important role of galactic wind \nfeedback \\citep{Springel.Hernquist:03,Martin:05,Martin:06, Oppenheimer.Dave:06}.\n\nAs expected, the N144L10mc run has lower SFR densities at $z\\ge 7$ \ncompared to the N216L10 series.  This is because a lower\nresolution run is unable to resolve low-mass halos that would otherwise \nforms stars in them.  When the star formation at high-$z$ is underestimated, \nsome of the gas remain unconverted into stars until lower redshift, resulting\nin a higher SFR at $z<4$ in the N144L10mc run than in the N216L10mc run. \n\nFigure~\\ref{fig:sfr_N288L34} shows the cosmic SF history down to $z=1$ for \nthe N288L34 series.  Again, the SFR density in the N288L34mc run is \nsystematically higher than that in the N288L34 run at all redshifts.\nMoreover, the difference in SFR density between the two runs increases at \n$z\\lesssim 3$.  By $z \\sim 3$ the IGM metallicity becomes high enough to \nnoticeably increase the cooling rate, and the accretion of IGM onto galaxies \nbecomes more efficient, which later fuels the star formation. \nOur results suggest that the cosmic SFR density is enhanced by metal \ncooling at $z \\lesssim 3$ by both of the following two effects: \n1) more efficient IGM accretion onto galaxies due to higher metallicity of \nIGM, and 2) increased SF efficiency due to higher cooling rate\nwith the contribution from metal cooling. \n\nIn Figure~\\ref{fig:sfr_N288L34} the analytic model of \n\\citet[][Eq.\\,(2), hereafter H\\&S model]{Hernquist.Springel:03} is also \nshown in the green dot-dashed line.  This model is a result of many  \ncosmological SPH simulations with increasing resolution, and the authors \ngave a theoretical interpretation to the empirical fitting formula for the  \ncosmic SFR density. Since our runs adopt a lower value of $\\sigma_8=0.85$ \nthan the original value used to derive the H\\&S model ($\\sigma_8=0.90$), we \nmodify the H\\&S model by rescaling the $\\beta$ parameter in their Eq.\\,(2)\nand Eq.\\,(47).  This reduces the SFR density at $z\\gtrsim 5$ by \n$\\sim 10$\\%, but not much at $z\\lesssim 5$. \nThe results of the N288L34 and N216L10mc runs agree well with the H\\&S model \nat $z<4$ and $z<6$, respectively. \nThis is expected, because the N288L34 run adopts the same physical \nmodels as in the original simulations that were used to derive the \nH\\&S model. At high-$z$, both runs fall short compared to the H\\&S model \nowing to insufficient resolution. \n\n\n\n\\subsection{Phase space distribution of cosmic gas}\n\\label{sec:ps_gas}\n\n\\begin{figure*}\n\\centerline{\n  \\subfigure[N216L10]{\\includegraphics[width=0.475\\textwidth,angle=0] {phase.N216L10.eps}}\n  \\subfigure[N216L10nw]{\\includegraphics[width=0.475\\textwidth,angle=0] {phase.N216L10nw.eps}}\n}\\vspace{0.3cm}\n\\centerline{\n  \\subfigure[N216L10mc]{\\includegraphics[width=0.475\\textwidth,angle=0] {phase.N216L10mc.eps}}\n  \\subfigure[N216L10mv]{\\includegraphics[width=0.475\\textwidth,angle=0] {phase.N216L10mv.eps}}\n}\n\\caption{\nThe phase space and metallicity distribution of the cosmic gas at $z=3$ for \nthe four simulations of N216L10 series.  For each run, three panels are \nshown for $\\rho - T$, $Z - T$, and $\\rho - Z$ relationship.  \nFor plotting purpose, we plot only randomly selected 1\\% of the total gas \nparticles in the simulation.  The blue dashed line is based on the fitting \nfunction to the effective equation of state for the multiphase star-forming \ngas provided by \\citet{Robertson.etal:04}. \n}\n\\label{fig:phase.N216L10}\n\\end{figure*}\n\n\nThe metal enrichment and metal cooling also change the phase space \ndistribution ($\\rho - T$ diagram) of cosmic gas.  \nFigure~\\ref{fig:phase.N216L10} shows \nthe state of cosmic gas at $z=3$ for the N216L10 series.  As shown in \npanel ($a$), the baryons in the Universe can be broadly categorised \ninto four different phases according to their overdensity and temperature: \n`hot', `warm-hot', `diffuse', and `condensed' \\citep{Dave.etal:99, \nCen.Ostriker:99, Dave.etal:01}.  The first two phases are mostly the \nshock-heated gas in clusters and groups of galaxies.  \nThe `diffuse' phase is mostly the photoionised IGM with lower temperature, \nwhich can be observed as the Ly$\\alpha$ forest.  The tight power-law \nrelationship between $\\rho$ and $T$ at $\\rho\/\\bar{\\rho} \\lesssim 3$ and $T<10^4$\\,K \nis governed by the ionisation equilibrium \\citep{Hui.Gnedin:97}, where\n$\\bar{\\rho}$ is the mean density of baryons at $z=0$.\n\nThe `condensed' phase is the high density, cold gas in galaxies. \nBecause the primordial cooling curve has a sharp cutoff at $T\\simeq 10^4$\\,K, \nthe temperature of condensed phase decreases only down to this temperature\nin Figure~\\ref{fig:phase.N216L10}a,b.  The characteristic finger-like feature\nat $\\rho\/\\bar{\\rho} \\gtrsim 10^4$ is due to the pressurisation of the star-forming gas\nby SN feedback in the multiphase ISM model of \\citet{Springel.Hernquist:03}. \nA fitting formula for this effective equation of state was derived by \n\\citet{Robertson.etal:04}, which is shown by the blue dashed lines. \nAbove the SF threshold density $\\rho_{\\rm th}$, the gas particles in the simulation \ngo into the multiphase mode and are allowed to form stars.  \n\nThe most noticeable and interesting change in the $\\rho - T$ plot by the \nmetal cooling is in the distribution of `condensed' gas. Star formation \nand SN explosions occur in the multiphase star-forming gas, therefore \nthe immediate effect of metal cooling appears in the condensed phase. \nThe presence of gas at $T < 10^4$\\,K with metal cooling results \nfrom the effective temperature calculation for the multiphase medium,\nand not from the direct gas cooling down to $T < 10^4$\\,K.\nAccording to the multiphase scheme in \\citet{Springel.Hernquist:03}, \nthe star forming gas consists of two phases: hot gas \n($10^5$\\,K $< T <$ $10^8$\\,K) and cold gas ($T = 10^3$\\,K). \nThe effective temperature for the multiphase gas is computed from the \ninternal energy of hot and cold gases, which are derived using the cold gas\nfraction. The cold gas fraction is computed from the cooling rate of \nhot gas and the gas density. The original GADGET imposes \nthe effective temperature of $10^4$\\,K at the SF threshold density.\nIn our metal cooling scheme, however, the cooling rate of hot gas increases, \nand the cold gas fraction increases. Therefore, the multiphase gas above the SF \nthreshold density with higher metallicity could have lower effective \ntemperature of $T < 10^4$\\,K in our implementation.\n \nIn the N216L10mc run, the value of $\\rho_{\\rm th}$ was kept fixed as the original \nvalue, which causes a sharp edge at $\\rho\/\\bar{\\rho} \\simeq 10^4$ for the multiphase \ngas.  In the N216L10mv run, the value of $\\rho_{\\rm th}$ was varied \naccording to the change in mean molecular weight, therefore the condensed gas \nhas a spread in both temperature and density near $\\rho\/\\bar{\\rho} \\simeq 10^4$.\n\nGalactic wind returns the metal-enriched, star-forming gas to the hot and \nlow density IGM, as evidenced in the $\\rho - Z$ diagrams. This metal-enriched, \nlow-density ($\\rho\/\\bar{\\rho} \\lesssim 10$) gas is absent in the N216L10nw run, which \nclearly shows the necessity of galactic wind to explain the metallicity \nobserved in Ly$\\alpha$ forest.  \nThe phase distribution of IGM in the N216L10, N216L10mc, and N216L10mv runs \nare very similar.  This is presumably because the effects of metal cooling \nis not so significant yet in the IGM at $z>3$. \nAs we will discuss in the next section, this is not be the case at $z<3$. \n\n\n\n\\subsection{Evolution of four phases of baryons}\n\\label{sec:Evolv4}\n\n\\begin{figure*}\n\\centerline{\\includegraphics[width=2.0\\columnwidth,angle=0] {Evolv4_Frac.eps}}\n\\caption{\nThe evolution of mass fractions for the four phases of baryons: hot, warm-hot, \ndiffuse, and condensed.  {\\it Panel (b)} shows the addition of the two \nphases, `warm-hot + hot', and {\\it panel (c)} shows the total masses in the \nIGM, i.e., `diffuse + warm-hot + hot'. \nHere we compare the N216L10 and N288L34 series. \n}\n\\label{fig:Evolv4}\n\\end{figure*}\n\nThe connection between the four phases of cosmic gas is important in galaxy \nformation, and it is beneficial for us to study their evolution for a better \nunderstanding of metal cooling effects. \nA conventional view is that the diffuse gas is shock-heated to hot or warm-hot \nphase, which later cools down to the condensed phase \\citep{Dave.etal:99, \nCen.Ostriker:99, Dave.etal:01}.  \\citet{Keres.etal:05} also suggested that \nsome diffuse gas can migrate to the condensed phase without being \nshock-heated, which they called the `cold mode' accretion. \n\nFigure~\\ref{fig:Evolv4} shows the evolution of mass fractions of different phases.\nAll of our simulations show that the diffuse phase continuously decreases from \nhigh-$z$ to low-$z$, while the `warm-hot + hot' and `condensed' phases \ncontinuously increase, consistently with the general expectations. \nThe IGM (i.e., hot + warm-hot + diffuse) is accreted onto galaxies, \nto become the condensed phase. \nThe diffuse component (panel $a$) decreases from $\\approx 95$\\% \nat $z=6$ to $\\approx 55$\\% at $z=1$.  The `warm-hot + hot' component \n(panel $b$) increases from almost zero at $z=6$ to $\\approx 35$\\% at $z=1$. \nThe mass fraction of the total IGM (panel $c$) decreases from $96-98$\\% \nat $z=6$ to $92-93$\\% at $z=3$.  This reflects the increase of the  \ncondensed phase (panel $d$) from a few percent at $z=6$ to $\\sim 7$\\% at \n$z=3$.  These mass fractions are in general agreement with the results\nof \\cite{Dave.etal:01}.\n\nThe comparison between N288L34 and N288L34mc runs shows interesting differences. \nAt $z>3$, the mass fractions of all four phases are similar in the two \nruns. However, at $z<3$, the IGM mass fraction continues to decrease in the \nN288L10mc run, while it becomes almost constant in the N288L10 run.  \nA similar behaviour is seen for the condensed phase, and the N288L10mc \ncontinues to increase its condensed mass. \nThis suggests that the metal cooling enhances the IGM accretion onto \ngalaxies mostly at $z\\lesssim 3$, which supplies the fuel for star formation. \nThe cooling of warm+hot phase into diffuse phase is also enhanced, \nleading to a larger mass fraction of diffuse component in the N288L34mc run \nthan in the N288L34 run.  (The same is true for N216L10mc and N216L10 runs.)\n\nThe N216L10nw run shows stark differences from all other runs. \nIt has a significantly lower IGM mass fraction, and a very large mass fraction \nin the condensed phase.  This shows that the IGM is mostly enriched by \ngalactic wind, and that most of the masses are returned into the \n`warm-hot + hot' phase, but not to the diffuse phase.  \n\n\n\n\\section{Galaxy stellar \\& baryonic mass function}\n\\label{sec:MF}\n\n\\subsection{At $z\\ge 3$:}\n\n\\begin{figure*}\n\\centerline{\\includegraphics[width=2.0\\columnwidth,angle=0] {MF_star.eps}}\n\\caption\n{\nStellar mass functions of simulated galaxies.\n{\\it Panel (a):} For the runs without metal cooling at $z=3$. \nThe vertical dotted line indicates the galaxy masses with 32 star particles\nfor N144L10 run. \n{\\it Panel (b)} compares the N216L10 series at $z=3$. \nThe bottom two panels show the redshift evolution of GSMF in the N216L10 \n({\\it panel c}) and N216L10mc ({\\it panel d}) run.\n}\n\\label{fig:MF_Star}\n\\end{figure*}\n\n\nGalaxy stellar mass function (GSMF) tells us how the stellar mass is \ndistributed in different galaxies, and its redshift evolution reflects\nthe growth of structure in the hierarchical universe. \nFigure~\\ref{fig:MF_Star} compares the GSMF in different simulations. \nWe only show the range of $M_{\\rm star} \\ge 2.5\\times 10^7\\,M_\\odot$, which \ncorresponds to the limiting mass of 32 star particles in N216L10 runs.\nWe chose 32 star particles as our resolution limits because all mass \nfunction start to turn over around this mass range.\nPanel~($a$) shows that the run with no galactic wind (N216L10nw) completely \noverestimates the GSMF, and the run with lower resolution (N144L10) \nunderestimates the number of lower mass galaxies at \n$M_{\\rm star} \\lesssim 10^8\\,M_\\odot$ relative to the N216L10 run. \nPanel~($b$) shows that the metal cooling increases the masses of galaxies \nby $\\sim 20$\\% at $z=3$.  As a result, both the N216L10mc and N216L10mv runs \nshow a slight increase in the number of galaxies at the massive end, with a \nslightly stronger enhancement in the N216L10mv run. \n\nFigures~\\ref{fig:MF_Star}c,d demonstrate that the metal cooling also changes \nthe redshift evolution of GSMF.  In the N216L10mc run, the peak of \nmass function is shifting more toward the massive side than in the N216L10 run \nat $z=3$.  From $z=4$ to $z=3$, the low-mass galaxies merge to form more \nmassive galaxies, increasing the number of galaxies with \n$M_{\\rm star} > 10^8\\,M_\\odot$ furthermore. \n\n It also seems that the formation of low-mass galaxies ($M_{\\rm star} \n\\lesssim 10^8\\,M_\\odot$) is enhanced by metal cooling at early times ($z=5-6$). \nHowever we find that the behaviour of MF at the low-mass end is somewhat \ndependent on the threshold density for the grouping. \nAs mentioned in \\S~\\ref{sec:Groups}, our grouping code imposes a threshold \ngas density of $0.01 \\rho_{\\rm th}$ for a gas particle to be part of a galaxy.  \nWhen we lower this threshold density, we find that the two MFs for \nthe N216L10 and N216L10mc converged, as more particles are incorporated \ninto galaxies at the outskirts of galaxies.  Metal cooling increases the \ngas density in galaxies, enabling more low-mass galaxies to satisfy the \ngas density threshold and to be identified as simulated galaxies.   \nTherefore the behaviour of MF at $M_{\\rm star} \\lesssim 5\\times 10^7\\,M_\\odot$ is\nsomewhat dependent on the choice of the outer threshold density of galaxies. \n\n\\begin{figure*}\n\\centerline{\\includegraphics[width=2.0\\columnwidth,angle=0] {MF_comp.eps}}\n\\caption\n{\nThe stellar ({\\it panel a}), gas ({\\it panel b}), and the total baryonic \n({\\it panel c}) mass functions for the N216L10 series at $z=3$. \n}\n\\label{fig:MF_comp}\n\\end{figure*}\n\n\nFigure~\\ref{fig:MF_comp} compares the gas and baryonic (star + gas) \nMFs for the N216L10 series at $z=3$ (panels $a$ \\& $b$), and the \nredshift evolution of the baryonic MF from $z=5$ to $z=3$ in the N216L10\n(panel $c$) and N216L10mc (panel $d$) runs.\nHere we show only the range of $M_{\\rm star} \\ge 5.0\\times 10^7\\,M_\\odot$, which \ncorresponds to the limiting mass of 32 gas particles. \nPanel ($a$) shows that more gas is converted into stars in the runs with \nmetal cooling (N216L10mc and N216L10mv) compared to the N216L10 run, while\npanel ($b$) shows that the baryonic MF is similar in all the runs. \nAs we discussed in \\S~\\ref{sec:Evolv4}, the IGM accretion onto galaxies is \nnot so much enhanced yet before $z\\sim 3$ due to relatively low IGM \nmetallicity, which explains the similarity in the total baryonic MF.  \nIn the runs with metal cooling, the peak of GSMF is shifted \ntoward higher mass, whereas the peak of gas MF is shifted toward lower mass.  \nThis suggests that the enhanced GSMF at $z=3$ is due to the increased \nSF efficiency by metal cooling within the galaxies. \nAgain, the apparent enhancement in the number of low-mass galaxies\nin the N216L10mc run compared to the N216L10 run is somewhat dependent \non the threshold density of grouping, therefore it should be interpreted \nwith caution.\n \n\n\n\\subsection{At $z< 3$:}\n\n\\begin{figure*}\n\\centerline{\\includegraphics[width=2.0\\columnwidth,angle=0] {MF_comp_N288L34.eps}}\n\\caption\n{\nThe stellar, gas, and baryonic (star + gas) mass functions at $z=3$ (left) and $z=1$ (right) for the N288L34 series. \n}\n\\label{fig:MF_comp_N288L34}\n\\end{figure*}\n\n\nAs the accretion of IGM onto galaxies become more efficient at $z<3$ \ndue to the increased IGM cooling by the metals, we expect that the total \nbaryonic mass of galaxies would be more enhanced at $z=1$ than at $z=3$.\nFigure~\\ref{fig:MF_comp_N288L34} shows the MFs at $z=3$ and $z=1$ \nin the N288L34 series, and it clearly demonstrates that this expectation is \ntrue.  Here we show only the range of $M_{\\rm gas} \\ge 8.2\\times 10^8\\,M_\\odot$, \nwhich corresponds to the limiting mass of 32 gas particles. \nThe general trend in the three (star, gas, and total baryon) MFs is \nsimilar to that we saw in  Figure~\\ref{fig:MF_comp}, although the difference \nat $z=3$ between the N288L34 and N288L34mc runs is slightly smaller than \nin the N216L10 series due to poorer resolution.\nFigure~\\ref{fig:MF_comp_N288L34}f shows most prominently the enhancement \nof the total baryonic MF at $z=1$ in the N288L34mc run owing to the metal \ncooling. In panels ($e$) \\& ($f$), the N288L34 run actually has a longer\ntail at the most massive-end than the N288L34mc run.  The reason for this \nfeature is not fully clear, but it may be related to the balance between IGM \naccretion and feedback. Owing to metal cooling, IGM accretion rate increases in \nthe 'mc' run, and the SFR is also enhanced, leading to a stronger\nfeedback. The amount of mass loss is greater in low mass galaxies,\nbut the net heating of IGM is more significant for massive galaxies.\nThe 'mc' run has stronger feedback, therefore its feedback heating may \nbecome more significant than IGM accretion for very massive galaxies. \nThe significant IGM heating suppresses the growth of massive-end of \nmass function from $z=3$ to $z=1$ and results in shorter tail for the 'mc' run \nat $z=1$ as shown in Figure~\\ref{fig:MF_comp_N288L34}e,f.\n\n\n\n\\section{Gas fraction}\n\\label{sec:GasFrac}\n\n\\begin{figure*}\n\\centerline{\\includegraphics[width=2.0\\columnwidth,angle=0] {GFrac_evol_N216L10.eps}}\n\\caption{\nThe gas fraction of galaxies as a function of stellar mass.\nEach data point in {\\it panel (a)} is a simulated galaxy in the \nN216L10mc run at $z=3$.  Here we overplot two lines representing the \ndistribution: `average' and `median' (see text for details).\nPanels $(b)-(d)$ compare the gas fractions (`average' case) in the \nN216L10 series at $z=3, 4, \\&\\ 5$. \n}\n\\label{fig:GFrac_evol_N216L10}\n\\end{figure*}\n\nFor a fixed galaxy baryonic mass, the enhancement of star formation by metal \ncooling would decrease the gas mass fraction, \n$f_{\\rm gas} \\equiv M_{\\rm gas} \/ M_{\\rm baryon}$.\nFigure~\\ref{fig:GFrac_evol_N216L10} shows $f_{\\rm gas}$ as a function of galaxy \nstellar mass for the N216L10 series. \nPanel ($a$) shows the data only from N216L10mc run at $z=3$, and each data \npoint corresponds to a simulated galaxy. \nTo characterise the distribution, we compute the following two quantities \nin each logarithmic stellar mass bin: `average' and `median'.\nThe `average' is the ratio of total gas mass to total baryonic mass for \nall the galaxies in each mass bin, i.e., \n$\\sum_{i}^{} M_{\\rm gas, i} \/ \\sum_{i}^{} M_{\\rm baryon, i}$. \nThe `median' case is simply the median of $f_{\\rm gas}$ values in each mass bin.\nBoth quantities show a similar trend, however, there is a sharper drop-off at \n$M_{\\rm star} \\simeq 2.5 \\times 10^{7}\\,M_\\odot$ for the `median' case in \nFigure~\\ref{fig:GFrac_evol_N216L10}a.  This mass-scale corresponds to \n32 star particles in the N216L10 series.  \nWe find that there are many galaxies with $f_{\\rm gas} = 0$ above this \nmass-scale, which causes the sharp drop-off in the `median' line. \nBelow this limiting mass, galaxies are not resolved well, which results in \nan underestimate of star formation and an overestimate of $f_{\\rm gas}$. \nIf we had a higher resolution simulation with finer particle masses, this \nlimiting mass-scale would shift to a lower mass.  Therefore the location of \nthis sharp drop-off is currently determined by the resolution of our \nsimulation.  \nHowever, dark matter halos would stop forming stars at some \nlower limiting halo mass, if we had an infinitely high-resolution simulation, \nThis lower limit to the galaxy mass is presumably determined by the \nphotoevaporation of gas by the UV background radiation \\citep{Rees:86, \nEfstathiou:92, Quinn.etal:96,Gnedin:00, Nagamine.etal:04-dla, Pontzen.etal:08,\nOkamoto.etal:08}.  Recent works suggest that star formation \ncould be suppressed by the UV background in halos with \n$M_{\\rm halo} \\lesssim 10^9\\,M_\\odot$ at $z\\sim 3$.  Galaxies with \n$M_{\\rm star} \\simeq 2 \\times 10^{7}\\,M_\\odot$ would reside in halos with \n$M_{\\rm halo} \\approx 2\\times 10^9\\,M_\\odot$. \nThe N216L10 series would resolve such a halo with $\\sim 280$ dark matter \nparticles, and its mass resolution is actually close to the \nastrophysical limit for dwarf galaxy formation at $z\\sim 3$.  \nTherefore the sharp drop-off in $f_{\\rm gas}$ at $M_{\\rm star} \\simeq 2 \\times 10^{7}\\,M_\\odot$ \nmay not be so far from the true answer. \n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=1.0\\columnwidth,angle=0] {MGFrac.redshift.eps}\n\\includegraphics[width=1.0\\columnwidth,angle=0] {MGFrac.resolution.eps}\n\\caption{\n{\\it Panel (a)} highlights the redshift evolution of $f_{\\rm gas}$ in the \nN288L34 run from $z=5$ to $z=1$.  \n{\\it Panel (b)} shows the effect of numerical resolution on $f_{\\rm gas}$. \nThe vertical dotted lines indicate the galaxy masses with 32 star particles\nfor each run.  The simulation results shift to lower masses as the resolution \nis increased. We plot the `average' case in these figures. \n}\n\\label{fig:MGFrac_resolution}\n\\end{center}\n\\end{figure*}\n\n\nWe find that $f_{gas}$ increases with decreasing $M_{\\rm star}$ at all redshifts, \nregardless of metal cooling and wind effects.\nThis trend is qualitatively consistent with current observations.\n\\citet{Erb.etal:2006} estimate the gas fraction as a function of stellar mass\nusing the rest-frame UV-selected star-forming galaxies at $z \\sim 2$, and show\nthat the gas fraction decreases with increasing stellar mass. And using \nthe mean gas and stellar mass, they find the average $f_{gas} \\sim 0.35 $, \nwhich agrees with the predicted gas fraction for massive galaxies in our simulation.\nIn addition, \\citet{Geha.etal:06} reported that the average neutral \ngas fraction is $\\langle f_{\\rm gas} \\rangle = 0.6$ for the local dwarf\ngalaxies selected from the Sloan Digital Sky Survey.  In our N216L10 series\nwith metal cooling, the `average' $f_{\\rm gas}$ reaches 0.6 for galaxies with\n$M_{\\rm star} \\simeq 2\\times 10^7\\,M_\\odot$ at $z=3$ \\& 4. \n\nFigures\\,\\ref{fig:GFrac_evol_N216L10}b,c,d show the redshift evolution of \n$f_{\\rm gas}$ for the N216L10 series.  In the runs with metal cooling (N216L10mc and \nN216L10mv), $f_{\\rm gas}$ is lower than in the N216L10 run by $20-30$\\% at all \nredshifts for galaxies with $M_{\\rm star} = 10^{7.5} - 10^9\\,M_\\odot$.  \nThis result suggests that the metal cooling reduces $f_{\\rm gas}$ owing\nto more efficient star formation. \nThe values of $f_{\\rm gas}$ seem to be more convergent at the massive-end \n($M_{\\rm star} > 10^9\\,M_\\odot$). \nIn addition, we find that galaxies in the N216L10nw run are the most gas-rich, \nbecause almost no galactic gas is returned to the IGM.\n\nFigures~\\ref{fig:GFrac_evol_N216L10}b,c,d also show that $f_{\\rm gas}$ is higher \nat higher redshifts.  For example, the well-resolved galaxies \nwith $M_{\\rm star} = 10^9\\,M_\\odot$ have $f_{\\rm gas} \\simeq 0.45$ at $z=5$, but \n$f_{\\rm gas} = 0.25$ at $z=3$.  As we saw in Figure~\\ref{fig:MF_Star}d, there is a\nconsiderable growth in GSMF from $z=5$ to $z=3$.  Together with the decrease\nin the gas fraction, these results suggest that most of the gas accreted \nduring $z=3-5$ has been converted into stars. \n\nFigure~\\ref{fig:MGFrac_resolution}a highlights the redshift evolution of \n$f_{\\rm gas}$ from $z=5$ to $z=1$ in the N288L34mc run. The gas mass fraction \nclearly decreases with decreasing redshift as the gas is converted into \nstars.  The rate of decrease is greater at $z=5 \\rightarrow 3$ than at \n$z=3 \\rightarrow 1$, with $f_{\\rm gas} \\simeq 0.5, 0.25,$ \\& 0.2 for $z=5, 3$, \n\\& 1, respectively, for galaxies with $M_{\\rm star} \\gtrsim 3\\times 10^9\\,M_\\odot$.\n\nFigure~\\ref{fig:MGFrac_resolution}b shows the effect of resolution on $f_{\\rm gas}$. \nThe location of the drop-off in $f_{\\rm gas}$ \nshifts to lower masses as the resolution is increased from N144L10mc to \nN216L10mc run.  The location of the drop-off is at around our resolution limit --\nthe vertical lines in {\\it Panel (b)}. Therefore the values of $f_{\\rm gas}$ are \noverestimated in unresolved galaxies.\n\n\n\n\\section{Discussions and Conclusions}\n\\label{sec:summary}\n\nUsing cosmological hydrodynamic simulations with metal enrichment and metal \ncooling, we studied their effects on galaxy growth and cosmic SFR.  \nOwing to metal cooling, the SFR density increases about 20\\% \nat $z=3$ and about 50\\% at $z=1$.\nOur results suggest that metal cooling enhances the star formation \nthrough two different processes: \n1) more efficient conversion of local ISM into stars (i.e., increase of  \nlocal SF efficiency), and \n2) the increase of IGM accretion onto galaxies. \nThe former process is in effect essentially at all times at $z\\lesssim 15$, \nbecause the local ISM can be instantaneously enriched by SN explosions.\nThis process enhances the SFR, but does not noticeably change the total \nbaryonic mass of galaxies. \nThe latter process, on the other hand, can increase the total baryonic mass\nof galaxies, as well as enhancing the overall SFR density by supplying more \ngas for star formation.   This process becomes effective only at lower \nredshifts ($z\\lesssim 3$), because it takes some time to enrich the \nIGM up to $Z \\gtrsim 10^{-2}Z_\\odot$.  \n\nIn this paper, we used two different prescriptions for star formation \nin the runs with metal cooling:  \n1) the `constant $\\rho_{\\rm th}$' scheme and 2) the `varying $\\rho_{\\rm th}$' scheme. \nThe value of $\\rho_{\\rm th}$ was fixed in the former scheme, while \nit was modulated according to the metallicity in the latter scheme. \nBoth schemes show similar increase of cosmic SFR density (see \nFigure~\\ref{fig:sfr_N216L10}), which implies that the overall SF enhancement \nby metal cooling is not so sensitive to the choice of $\\rho_{\\rm th}$\n(but with some differences as we discussed in \\S~\\ref{sec:cos_sfr}).\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=1.9\\columnwidth,angle=0] {kenn_mcool_cut2.eps}\n\\caption{Comparison of the N216L10 series with the empirical Kennicutt law. \n\\label{fig:kenni}}\n\\end{center}\n\\end{figure*}\n\nThe multiphase ISM model for star formation by \\citet{Springel.Hernquist:03} \ncontains two free parameters: $\\rho_{\\rm th}$ and the normalisation of gas \nconsumption time-scale, $t_\\star^0$.  The values of these two parameters were \noriginally chosen to match the empirical Kennicutt-Schmidt law \n\\citep{Kennicutt:98a,Kennicutt:98b} using simulations of isolated disk \ngalaxies.  Since we did not change the value of $t_\\star^0$ in the runs\nwith metal cooling (as well as $\\rho_{\\rm th}$ in the `mc' runs), \nit is possible that our simulations may now violate the Kennicutt law.  \n\nHowever, as we show in Figure~\\ref{fig:kenni}, the plots of ${N_{\\rm HI}}$ vs. \n$\\Sigma_{\\rm SFR}$ for the N216L10 and N216L10mc runs are not so different \nfrom each other. \nThis can be understood as follows.  As Figure~\\ref{fig:Evolv4}c showed,  \nmetal cooling increases the density and lowers the temperature of \nstar-forming gas.  The fundamental scaling relationship between SFR and \ncold gas density does not need to change when the metal cooling is introduced; \nthe star-forming gas particles would simply slide upward along the \nKennicutt law, $\\Sigma_{\\rm SFR} \\propto {N_{\\rm HI}}^{1.4}$. \n\nIn the case of N216L10mv run, we varied $\\rho_{\\rm th}$ according to the gas \nmetallicity, primarily scattering it to lower values because higher metallicity\nincreases the cooling rate and lowers $\\rho_{\\rm th}$.  Therefore more gas particles \nbecome eligible to form stars and $\\Sigma_{\\rm SFR}$ is scattered upward \nabove the Kennicutt law for a given value of ${N_{\\rm HI}}$, resulting in \na broader deviation from the power-law relationship of the Kennicutt law near\n$\\log {N_{\\rm HI}} \\simeq 20$.  At higher values of ${N_{\\rm HI}}$, $\\Sigma_{\\rm SFR}$ is\nenhanced, but still within the range of the Kennicutt law. \nThis difference can be alleviated by adjusting the SF timescale \n\\citep[see][]{Springel.Hernquist:03}, however, we try to emphasise the effect\nof metal cooing on SFR by keeping this parameter the same in all the runs.\nFurther investigations of the H{\\sc i}\\,\\,\\ aspect of our simulations is beyond the \nscope of the present paper, and we will present the results elsewhere. \n\nUpon evaluating the effect of metal cooling, \\citet{Springel.Hernquist:03:S} \nconsidered two different gas phases where the cooling become important. \nOne is the diffuse gas in galactic halos, which must radiate its thermal \nenergy in order to collapse onto the high-density ISM, and the other is the \nmultiphase star-forming gas.  They argued in their \\S~5.2 that the cosmic SFR \ndensity at $z\\gtrsim 6$ would not be affected by the metal cooling very much, \nbecause at high-$z$ cooling is so efficient that the gas in diffuse \natmospheres of halos cools nearly instantly, even without any metal cooling.  \nThey expected their model results to be largely independent of metal \nenrichment, because in this regime the evolution of SFR density is driven by \nthe fast gravitational growth of the halo mass function. \nFor the star-forming multiphase ISM, they argued that the parameters of \ntheir model (e.g. cold gas evaporation efficiency and gas consumption \ntime-scale) can be adjusted such that the normalisation of the Kennicutt law \ncan be maintained, yielding to first order unaltered model predictions. \nHowever, our simulations, in particular the comparison between the N216L10 and \nN216L10mc run, suggest that the metal cooling can enhance the SFR density \neven at $z\\gtrsim 6$ by $\\sim 20-30$\\%.  This is because the gas density \nbecomes higher and temperature becomes lower (see Figure~\\ref{fig:Evolv4}c) \nby responding to the enhanced cooling rate by metals. \n\nIn summary, we consider that our study serves to demonstrate the importance \nof metal enrichment and metal cooling in galaxy formation, beyond \nour original motivation to perform a simple numerical study. \nOur study highlights the role of metals at low-$z$ and high-$z$ to \nenhance the star formation in the Universe, and demonstrates that the metals\nincrease the SF efficiency in the star-forming ISM, as well as \nenabling more IGM to accrete onto galaxies and fuel star formation.\nWe expect that metal cooling would also alter some of the galaxy properties \nsuch as the mass-metallicity relationship \\citep[e.g.,][]{Tremonti.etal:04} \nand specific SFR \\citep[e.g.,][]{Noeske.etal:07}.  We plan to investigate \nthese issues in the future, as well as alternative models for star formation\nand feedback. \n\n\n\n\\section*{Acknowledgements}\nWe thank V. Springel for allowing us to use the updated version \nof {\\small GADGET-2} code for our study, and for useful comments on \nthe manuscript. \nKN is grateful for the hospitality of Institute for the Physics and \nMathematics of the Universe, University of Tokyo, where part of this work was done.\nWe are also grateful to the anonymous referee for constructive comments which improved this paper.\nThis research was supported in part by the National Aeronautics and \nSpace Administration under Grant\/Cooperative Agreement No. NNX08AE57A \nissued by the Nevada NASA EPSCoR program, by the National Science \nFoundation through TeraGrid resources provided by the San Diego \nSupercomputer Center (SDSC), and  by the President's Infrastructure Award at UNLV. \nThe simulations were performed at the UNLV Cosmology Computing Cluster\nand the Datastar at SDSC.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nFixing bugs in programs, that is, program repair, is one of the core tasks in software maintenance,\nbut requires effort to\n analyze failed executions, locate the cause of the fault, synthesize a bug fix and validate that the fault has been corrected without introducing new ones \\cite{Mullerburg1983}.\n Automatic program repair \\cite{LeGoues2012, Long2016, Pradel2018, Monperrus:survey} attempts to alleviate most of the manual effort of locating and repairing faults.\nHowever, a major concern in industry is that linters and program repair methods approaches are required to have high precision without risking achieving high enough recall.\nAs an industrial example Google's Tricorder \\cite{Sadowski2015} enforces a false positive rate $< 10\\%$.\n\nOne way to find a ``sweet spot'' of maintaining high precision with adequate recall is to focus on\nrepairing types of simple bugs, such as one-line bugs,\nor  bugs that fall into a small set of templates,\nsuch as mutation operators\n\\cite{LeGoues2012} or other types of predefined templates \\cite{Long2015, Long2016, Pradel2018}.\nHowever, these have been evaluated on either a relatively small numbers of projects, e.g. 69 defects in 8 applications or on synthetic data.\nBecause of this lack of data, it has not previously been possible to estimate the \\emph{recall} of a set of repair templates,\nthat is, the percentage of real-world bugs that can be repaired by one of the templates.\nSimultaneously to the current work, a larger dataset of one-line bugs has been mined \\cite{Sequencer},\nbut even this dataset does not attempt to classify bugs into templates.\n\nAiming to fill this gap, we provide a dataset containing 25,539 single-statement bug-fix changes mined from 100 popular open-source Java Maven projects as well as a larger one containing 153,652 single-statement bug-fix changes mined from 1,000 popular open-source Java projects,\nannotated by whether they match any of a set of 16 bug templates, inspired by state-of-the-art program repair techniques.\nThe chosen templates aim at extracting bugs that compile both before and after repair as such can be quite tedious to manually spot, yet their fixes are so simple that many developers would call them ``stupid'' upon realization.\nWe will refer onwards to these bugs as ``simple stupid bugs'' (SStuBs)\\footnote{The\nacronym is intended to reflect the fact that, for the authors at least, finding such a bug can feel much like stubbing one's toe.} and the corresponding dataset as the Many\\sstubs4J\\ dataset. %\nAutomatic repair of SStuBs\\ is potentially an intermediate step toward more general program repair tools,\nwhile already being useful to developers.\nWe also think that SStuBs\\ might be a good start for the evaluation of machine learning based fault localization and repair methods.\n\n\nAn extra distinctive feature of our dataset is that the smaller version is restricted to projects that can be built automatically\nusing Maven.\nThose that contain a test suite can be built and used to evaluate test based techniques.\nIn an initial analysis, we find that 33.04\\% in the smaller version dataset and 33.47\\% in the larger version of all of the single-statement bugs that we mine match at least one of the SStuB\\ templates resulting in 10,231 and 63,923 SStuB\\ instances respectively.\nThis indicates that a remarkable number of singe-statement bugs can be repaired with a relatively small set of templates.\nIn further analysis we also estimated the frequency in lines of code with which these pattern based and general single-statement bugs appear.\nThis estimation is based on the size of the project's latest version and reveals that in the smaller dataset version SStuBs appear with a frequency of about 1 per 1,600 lines of code and 1 per 2,500 lines of code for the large version.\nWe hope that this dataset can serve as a valuable resource for both future work in program repair and studies in empirical software engineering.\n\n\\section{Methodology}\nWe next describe the methodology we employed to build the dataset.\nOur data generation tools along with documentation and detailed instructions for how to use them\nare available in a public GitHub repository\\footnote{https:\/\/github.com\/mast-group\/mineSStuBs} \nand the dataset is publicly available in Zenodo.\\footnote{DOI: https:\/\/doi.org\/10.5281\/zenodo.3653444}\n\n\\subsection{Selecting Appropriate Java Projects}\nIn order to mine a high quality dataset we opted to selecting high popularity projects.\nFor the small version of the dataset we selected the 100 most popular open source Java Maven \\cite{Miller2010} projects from GitHub up to 1\/4\/2017. \nTo allow evaluation of repair tools that might require building the projects, we selected only Maven ones because\nit is easy to automatically download the required dependencies for every project and build it.\nIn contrast, manual downloading of dependencies would require an immense amount of human effort.\nTo create a ranking for the projects we downloaded the MySQL dump of GHTorrent \\cite{Gousios2013} up to 1\/4\/2017. \nA project's popularity is determined by computing the sum of z-scores of its forks and stars \\cite{Allamanis2015a, Allamanis2016}.\nLastly, we pulled the projects' head commit by 28\/1\/2019 and considered commits until that date.\nThe same approach was used to rank projects for the larger version.\nHowever, the ranking was calculated using a later dump of GHTorrent from 1\/1\/2019.\nA download script along with the list of projects for both variants of the dataset are also available to ensure replicability.\n\n\\subsection{Classifying Commits as Bug-Fixing or not}\nFor every project our tool searches historically through all of its commits to locate bug-fixing ones. \nTo decide if a commit fixes a bug, we checked if its\ncommit message contains at least one of the\nkeywords: `error', `bug', `fix', `issue', `mistake', `incorrect', `fault', `defect', `flaw',\nand `type'. \nThis heuristic was previously used by Ray et al. \\cite{Ray2015} and was shown to achieve 96\\% accuracy on a set of 300 manually verified commits and 97.6\\% on a set of 384 manually verified commits \\cite{Tufano2018}.\nWe sampled 100 random commits containing SStuBs\\ from the small version of the dataset and found it to achieve 94\\% accuracy.\nThe above process produced a total of 115,929 and 883,982 bug-fixing  commits for the small and large dataset variants.\n\n\\subsection{Selecting Single Statement Changes}\nWe have opted to restrict the dataset to small bug fixes that do not require much code modification to fix.\nAdditionally, we are interested in bugs that are not just syntactic errors but cases where the code compiles both before and after the bug was located and repaired.\nAs we are interested in simple bugs that involve only a single statement, we filter out any commits that either add or delete a Java file.\nWe also filter out commits which make a multiple-statement change at any single position in the Java file.\nWe do \\emph{not} filter out commits that make single-line modifications at more than one position in the same file.\nSimilarly to the diff algorithm, we consider a modification as deleting the old lines\/statements and then adding the new ones.\nTo estimate whether a modification spans across multiple statements we calculate the diff for each modified Java file,\nand \nfor each modified chunk, we count how many statements were modified.\nIn the case of blocks each statement in the block's body is counted as a different statement. \nFor \\texttt{if} and \\texttt{while} statements, we count the condition as a separate statement\nfor this purpose.\nThis method allows to us include fixes to single simple statements that span across multiple lines (e.g. due to stylistic reasons)\nas a simple fix, unlike a line-based approach.\nAny commits that modify multiple statements in any single position returned by the diff are dropped while we still maintain commits for which a file's diff contains multiple positions with single statement modifications.\nIn the first case it is not trivial to align the deleted and added statements while it is in the latter.\nFor example, one or more of the deleted statements may have been replaced by multiple of the added ones while simultaneously one or more of the deleted statements may have simply been deleted.\nWe note that our tool ignores any changes to comments, blank lines as well as any formatting changes.\nOur methodology allows cases where the same expression containing a bug appeared multiple times in the file.\nThis filtering produces almost 13,000 and 86,769 commits for the two dataset versions.\nLastly, the employed methodology works in a similar way to the popular SZZ algorithm \\cite{Sliwerski2005} and its extensions \\cite{Kim2006, Williams2008} that have extensively been used to spot fix inducing changes.\n\n\n\\subsection{Creating Abstract Syntax Trees}\n\nEach file in the commit that contains one or more bugs is parsed, yielding an abstract syntax tree (AST) of the file before the repair.\nThen, for each repaired line in the file we extract the AST after applying the repair only on that line and leaving the rest of the lines as is.\nEach extracted pair of ASTs (original and single fix) only differ on the node(s) for the modified line.\nBy performing a simultaneous depth-first traversal on the two ASTs we locate the first node on which the two ASTs differ.\n\n\\subsection{Filtering out Clear Refactorings}\n\nAlthough we filter for bug-fixing changes in Step B,\nthere might still exist changes in the data that do not fix a bug or that do not even produce any behavioural changes.\nThis could happen because the commit-message filter had\na false positive, or because the change is tangled \\cite{herzig2013tangled}, and contains a bug-fixing modification\nalong with unrelated ones to other files.\nTo reduce the number of non-fixing changes in the dataset, we observe that\nthere is a class of refactorings that can produce small changes, namely renamings.\nThese are extracted via the diffs of the modified files.\nOur method spots variable, function, or class renaming as well as any uses of them across other modified files in the commit and excludes them.\n\n\\subsection{SStuB\\ Patterns}\n\nWe next describe the 16 SStuB\\ patterns.\nWe opted to choose patterns that appear often.\nMany of these have been used in pattern-based repair and mutation tools \\cite{LeGoues2012, Long2015, Long2016, Pradel2018}.\nHere we provide a brief description of each pattern. \nDue to page limitations we do not include examples here but in the README of the GitHub repository. %\n\\begin{itemize}\n  \n\\item \\emph{Change Identifier Used}\nChecks whether an identifier appearing in some expression in the statement was replaced with another one.\nIt is easy for developers to by accident utilize a different identifier than the intended one that has the same type.\nCopy pasting code is a potential source of such errors.\nSimilarly named identifiers may further contribute to the occurrence of such errors.\n\n\\item \\emph{Change Numeric Literal}\nChecks whether a numeric literal was replaced with another one.\nIt is easy for developers to mix two numeric values in their program.\n\n\\item \\emph{Change Boolean Literal}\nChecks whether a Boolean literal was replaced. True is replaced with False and vice-versa.\nIn many cases developers use the opposite Boolean value than the intended one.\n\n\\item \\emph{Change Modifier}\nChecks whether a variable, function, or class was declared with the wrong modifiers.\nFor example a developer can forget to declare one of the modifiers.\n\n\\item \\emph{Wrong Function Name}\nChecks if a function with the same parameter list but the wrong name was called. This is a usual pitfall.\n\n\\item \\emph{Same Function More Args}\nChecks whether an overloaded version of the function with more arguments was called.\nFunctions with multiple overload can often confuse developers.\n\n\\item \\emph{Same Function Less Args}\nChecks whether an overloaded version of the function with less arguments was called.\nFor instance, a developer can forget to specify one of the arguments and not realize it if the code still compiles due to function overloading.\n\n\\item \\emph{Same Function Change Caller}\nChecks whether in a function call expression the caller object for it was replaced with another one.\nWhen there are multiple variables with the same type a developer can accidentally perform an operation.\nCopy pasting code or mixing similar variables are common cases of such errors.\n\n\\item \\emph{Same Function Swap Args}\nChecks whether a function was called with two of its arguments swapped.\nWhen multiple function arguments are of the same type, developers %\ncan easily swap two of them without realizing.\nIt was also used in DeepBugs \\cite{Pradel2018}. %\n\n\\item \\emph{Change Binary Operator}\nChecks whether a binary operand was accidentally replaced with another one of the same type.\nFor example, developers very often mix comparison operators in expressions.\nA similar pattern was also used in DeepBugs \\cite{Pradel2018}.\n\n\\item \\emph{Change Unary Operator}\nChecks whether a unary operand was accidentally replaced with another one of the same type\n(e.g., developers often forget the ! operator in a boolean expression).\n\n\\item \\emph{Change Operand}\nChecks whether one of the operands in a binary operation was wrong.\nIt was also used in DeepBugs \\cite{Pradel2018}.\n\n\\item \\emph{More Specific If}\nChecks whether an extra condition (\\&\\& operand) was added  in an \\texttt{if} statement's condition.\n\n\\item \\emph{Less Specific If}\nChecks whether an extra condition which either this or the original one needs to hold ($\\|$ operand) was added in an \\texttt{if} statement's condition.\n\n\\item \\emph{Missing Throws Exception}\nChecks whether the fix added a \\texttt{throws} clause in a function declaration.\n\n\\item \\emph{Delete Throws Exception}\nChecks whether the fix deleted a \\texttt{throws} clause in a function declaration.\n\\end{itemize}\n\n\\subsection{SStuB\\ Pattern Matching}\n\nFinally, each pair of ASTs is automatically checked for fitting any of the SStuB\\ patterns.\nEach pattern is expressed as a mutation operation on the original AST that produces the new one.\nAll instances are added to the single-statement dataset, while only those that match SStuB\\ patterns are saved in the SStuBs\\ one.\n\n\n\\section{Many\\sstubs4J\\ Dataset Statistics}\n\n\n\\begin{table}[tpb]\n\\renewcommand{\\arraystretch}{1.05}\n\\caption{Statistics for each SStuB pattern.}\n\\begin{center}\n\\resizebox{0.935\\columnwidth}{!}{\n\\begin{tabular}{l r r r r}\n\\toprule\n\\textbf{Pattern Name} & \\textbf{\\textit{SStuBs}} & \\textbf{\\textit{Ratio}} & \\textbf{\\textit{SStuBs L}} & \\textbf{\\textit{Ratio L}} \\\\\n\\midrule\nChange Identifier Used \t\t& 3265 \t& 12.78\\%\t& 22668\t& 14.75\\% \\\\\nChange Numeric Literal \t\t& 1137 \t& 4.45\\%\t& 5447\t& 3.55\\% \\\\\nChange Modifier \t\t\t& 1852 \t& 7.25\\%\t& 5011\t& 3.26\\% \\\\\nChange Boolean Literal \t\t& 169 \t& 0.66\\%\t& 1842\t& 1.20\\% \\\\\n\\midrule\nWrong Function Name \t\t& 1486 \t& 5.82\\%\t& 10179\t& 6.62\\% \\\\\nSame Function More Args \t& 758 \t& 2.97\\%\t& 5100\t& 3.32\\% \\\\\nSame Function Less Args \t& 179 \t& 0.70\\%\t& 1588\t& 1.03\\% \\\\\nSame Function Wrong Caller\t& 187 \t& 0.73\\%\t& 1504\t& 0.98\\% \\\\\nSame Function Swap Args \t& 127 \t& 0.50\\%\t& 612\t& 0.39\\% \\\\\n\\midrule\nChange Binary Operator \t\t& 275 \t& 1.08\\%\t& 2241\t& 1.46\\% \\\\\nChange Unary Operator \t\t& 170 \t& 0.67\\%\t& 1016\t& 0.66\\% \\\\\nChange Operand \t\t\t\t& 120 \t& 0.47\\%\t& 807\t& 0.53\\% \\\\\n\\midrule\nLess Specific If \t\t\t& 215 \t& 0.84\\%\t& 2813\t& 1.83\\% \\\\\nMore Specific If\t\t\t& 175 \t& 0.69\\%\t& 2381\t& 1.55\\% \\\\\nMissing Throws Exception \t& 68 \t& 0.27\\%\t& 206\t& 0.13\\% \\\\\nDelete Throws Exception \t& 48 \t& 0.19\\%\t& 508\t& 0.33\\% \\\\\n\\midrule\nTOTAL NO DOUBLE COUNTS\t\t& 8438\t& 33.04\\%\t& 51433\t& 33.47\\% \\\\\nTOTAL \t\t\t\t\t\t& 10231\t& 40.06\\%\t& 63923\t& 41.59\\% \\\\\n\\bottomrule\n\\end{tabular}\n}\n\\label{tab:sstubPercentages}\n\\end{center}\n\\end{table}\n\n\nThe Many\\sstubs4J\\ dataset consists of 10,231 and 63,923 instances of single statement bugs mined from 12,598 and 86,771 bug-fix commits with only single-statement changes respectively for each version.\nConsequently, on average almost 2 single statement bugs and 0.75 SStuBs\\ were mined per valid commit.\nThe data is saved in JSON files and detailed information is available in the GitHub repository.\nEach SStuB\\ instance is also annotated with the SStuB\\ pattern satisfied, the project's name, the Java file's name, the hashes of the fix inducing commit and its parent, the line at which the bug starts, and the AST subtree's location. \nIn some cases a statement might fit more than one patterns.\nIn those cases it is counted as separate instances.\nHowever, in most cases the patterns are distinct.\nThe statistics for each of the 16 SStuB\\ patterns of the Many\\sstubs4J\\ dataset are shown in Table~\\ref{tab:sstubPercentages}.\nPatterns that are similar are grouped together (e.g. patterns that concern functions) and sorted in descending frequency order.\nThe three most common SStuB\\ patterns are \\emph{Change Identifier Used}, \\emph{Wrong Function Name}, and \\emph{Change Numeric Literal}.\n\nWe note that the mined bugs have not been annotated by severity and we expect that to vary.\nSome of the bugs appear in test code.\nAlthough bugs in test code will not reach a final product, they can have significant effect on it as they can potentially mask important bugs in it.\nTest oracle errors can bring confusion that slows down the debugging process while fixing them improves the performance of fault localization algorithms \\cite{Guo2015}.\nSuch bugs might also be quite tedious to locate as it is very rare to test a test suite and even if we follow that logic we would have to endlessly create tests for the tests.\nBy design we do not attempt to restrict the bugs to those that have a failing test case.\nThe goal is to reproduce the situtations that the bugs happen in the wild.\nLastly, as it was recently shown, unit tested code does not appear to be associated with fewer failures while increased coverage is associated with more failures \\cite{chioteli2019does}.\n\n\n\\section{Research Questions} \\label{sec:questions}\n\nAlthough the paper focuses on the dataset, we run a simple analysis to support\nour design decision to focus our new dataset on SStuBs.\nIn order to explore whether the SStuB\\ patterns are useful targets for program repair techniques,\nwe asked two research questions.\n\n\\subsection*{RQ1. Are SStuBs\\ common in open-source code?}%\n\\label{sec:questionPercentage}\nWe measured for each SStuB type the percentage of single statement modifications that are not clear refactorings and fit the pattern.\nThese are visualized in Table~\\ref{tab:sstubPercentages}.\nFor each project $P$ we also estimated the following two densities for the mined SStuBs:\n(a) the number of SStuBs\\ in project $P$ \/ total lines in $P$ at the final snapshot and\n(b) the number of SStuBs\\ in project $P$ \/ total lines added and deleted in $P$ by the final snapshot. \nThus, estimating the frequency per line of code modifications in the project's history. That is counting any line that was added or deleted to the project from the start to its latest version.\nA line modification is counted twice (once as a deletion and once as an addition). \nOnce for deleting the old and once for adding the new line.\nComments and empty lines were excluded from these estimations.\nWe found that in the smaller version of the dataset SStuBs appear with densities of about 2,400 and 30,000 lines of code (LOC) respectively.\n\nWe also estimated the same densities for the larger dataset variant.\nWe found that such bugs appear with a frequency of about 1,600 and 20,000 LOC respectively.\nAs a threat to validity, we acknowledge that the number of LOC in the final snapshot may not be the most informative denominator for a measure of bug density, but developing better ones is a thorny issue left for future work.\n\n\\subsection*{RQ2. Can SStuBs be spotted by existing tools such as static analyzers?} \\label{questionSpot}\n\nWe measure the proportion of bugs in our dataset that can be identified by\nthe popular static analysis tool SpotBugs.\\footnote{https:\/\/spotbugs.github.io\/}\nIf SpotBugs reports any bug for the line containing the SStuB\\ then we consider that SpotBugs successfully detected it.\nWe find that SpotBugs could only locate about $12\\%$ of SStuBs while also reporting more than $200$ million possible bugs when configured to report all warnings, even those with low confidence.\nIn fact, as explained the actual recall is even lower.\nThis is confirmed by a recent study where three static bug detectors including SpotBugs located only $4.5\\%$ of bugs \\cite{Habib2018}.\nThis means that a developer would have to look through hundreds of thousands of warnings produced by SpotBugs to locate a single SStuB.\nThis highlights the necessity for tools that are specifically built to detect SStuBs.\nThe scripts used to run and evaluate SpotBugs are also available in our repository.\n\n\n\n\\section{Related Work}\n\nSeveral previous data sets of real-world bugs have been curated.\nDefects4J \\cite{Just2014} is a popular dataset consisting 395 Java bugs.\nEach bug is fixed in a single commit but the fix may modify multiple source code lines.\nThe ManyBugs dataset \\cite{Goues2015} contains 185 C bugs, a subset of which were used by the GenProg \\cite{LeGoues2012}, Prophet \\cite{Long2016} and SPR \\cite{Long2015} papers.\nBugs.jar \\cite{Saha2018} is comprised of 1,158 Java bugs and their patches.\nThese datasets have the disadvantage of being relatively small.\nMore recently, a few larger-scale data sets of small bugs have been created.\nThe combined datasets are the CodRep dataset \\cite{CodRep} and the Bugs2Fix dataset \\cite{Bugs2Fix} resulting in 40,289 one-line bugs.\nThese datasets are combined into a single dataset of one line bugs in \\cite{Sequencer}.\nOur datasets are of similar size consisting of 25,539 and 153,652 single-statement bugs.\nIn contrast, our dataset focus on estimating the frequency of SStuB\\ templates, motivated by recent\nprogram repair tools and also operates on the statement level, which prevents falsely excluding instances due to formatting or stylistic reasons.\nAlso, the projects from which the small version of our dataset was generated can easily be built using Maven and we provide a list of projects containing tests and which tests fail for each instance (in GitHub repo).\nThus test based methods can be evaluated upon them.\nHowever, unlike Defects4J that aims in comparing test-based patch generation approaches, it aims in techniques that can accurately highlight SStuBs early in development allowing immediate patching since in many cases the fix might be trivial.\nLastly, unlike previous datasets, we take additional steps to filter out refactorings, although we acknowledge that such instances might be rare.\nIn our case however, we were able to filter out almost 5,000 and 35,000 refactored statements for the two dataset versions.\n\n\n\\section{Limitations - Threats to Validity}\n\nAlthough unlikely, it is possible for our SZZ like methodology to extract a pair of aligned statements that are unrelated (i.e. one line was deleted and one was added).\nWe do spot refactorings but there is no guarantee that we have detected 100\\% of them.\nThe heuristic used to spot bug fixing commits could introduce false positives, but this is mitigated by the fact that we focus on single line commits and as already discussed the false positive rate is low.\nOur dataset will not be useful for evaluating whether repair systems are good at fixing larger bugs.\nOur dataset is restricted to Java but could be replicated for other languages by using a parser and creating a module that checks if an AST pair fits any of the SStuB patterns.\nThe precise set of patterns might vary across languages and determining these might be an interesting direction for future work.\n\n\n\n\\section{Conclusions}\n\nWe introduce a new, large-scale dataset of real-world SStuBs, simple one-statement bugs,\nin Java for the evaluation of program repair techniques. The distinguishing feature of our dataset is\nthat where possible, the SStuBs\\ are categorized into one of 16 bug templates, which are inspired\nby those considered in state-of-the-art program repair methods.\nThese types of bugs often result in code that compiles, which means that they are particularly interesting for automated repair. \nWe find that SStuBs\\ occur relatively often --- one per 1,600 LOC in the projects\nwe study --- making them potentially a promising evaluation dataset for repair techniques that could be used to estimate their actual recall.\nThe data could also be used to answer other research questions, such as empirical questions about how and when simple\nbugs are introduced, or about evaluating program repair techniques for small bugs.\nAlso, it can aid in evaluating machine learning systems that learn to localize simple bugs via examples \\cite{Pradel2018} or a language model's entropy \\cite{Karampatsis2020big}.\nLast, coverage information for the maven projects with tests suits in the dataset could be used to estimate how often do tests cover SStuBs.\n\n\n\n\n\\bibliographystyle{ACM-Reference-Format}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\chapter{Accelerator Capabilities} \n\\label{chap:ceo}\n\n\n\\begin{center}\\begin{boldmath}\n\n\\input AccelCap\/authorlist.tex\n\n\n\\end{boldmath}\\end{center}\n\n\\section{Introduction}\n\nAccelerator-based experiments have been the mainstay and driving force of experimental particle physics ever since Rutherford recognized that radioactive sources provided an extremely limited capability to probe the nature of matter at the shortest time and distance scales.  From their earliest beginnings the history of invention in accelerator technology have enabled the history of particle physics discoveries especially on the energy frontier; see Fig.~\\ref{fig:Livingston}.  \n\nYet the figure of merit of energy reach in the constituent center-of-momentum frame is \nfar from the entire story of enabling capabilities. For experiments to be completed in a reasonable amount of time, the rate of interesting experimental events must be sufficiently high.  That is, the luminosity of the accelerator-target combination must be large enough to explore the physics of interest.\n\\begin{equation}\n\\mbox{Interesting\\ Events} = \\mbox{Cross\\ section} \\times \\mbox{Average\\ collision\\ rate} \\times \\mbox{Time} \\ . \n\\eeq{Interesting}\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{AccelCap\/Livingston.pdf}\n\\caption{The so-called Livingston plot illustrates how history of discovery on the energy frontier has been enabled by the history of invention (red arrows) in accelerator science and technology.}\n\\label{fig:Livingston}\n\\end{center}\n\\end{figure}\n\n\n\nGenerally high luminosity implies a large charge per bunch, a high collision frequency of beam bunches with the target, and focus of the beam to a small spot. The target can either be fixed or a counter-rotating beam bunch.\nEven energy-reach and accelerator-target luminosity is insufficient for discovery-level experiments. The accelerator should deliver beam particles with appropriate quantum numbers and with a timing structure which maximizes the signal-to-background noise of the experiment. \n\nAssessing the suite of characteristics of present and proposed accelerators in the light of current interests and priorities of particle physicists has been the task of the Accelerator Capabilities Study of the Snowmass process. For proposed or planned capabilities the study has recommended the critical research and development programs needed.\n\nThe study has been a synthesis of individual pre-Snowmass workshops conducted by six working groups culminating with the collective Snowmass meeting of all interested participants. Each working group addressed major challenges foreseen for their respective class of accelerators in their respective pre-Snowmass workshops. The areas of inquiry included but were not limited to the following specific questions:\n\n\\begin{itemize}\n\\item   How high a luminosity is possible for the LHC? \n\\item  \tHow high an energy is possible in the LHC tunnel? \n\\item  \tCould a Higgs factory be built in the LHC tunnel?\n\\item  \tCan the ILC and  CLIC designs be improved using new technologies?  \n\\item  \tCan one design a multi-TeV $\\mu^+\\mu^-$ collider?\n\\item  \tWhat secondary beams are needed for Intensity Frontier experiments?\n\\item  \tWhat proton beams are needed to generate these secondary beams, and can these be made by existing machines?\n\\item  \tWhat accelerator capabilities at heavy flavor factories are required to realize the full range of physics opportunities? \n\\item  \tWhat are new physics opportunities using high-power electron and positron beams?\n\\item  \tWhat is the broad range of test capabilities existing or needed for developing accelerator capabilities?\n\\end{itemize}\n\nThe working groups also considered a set of \"big questions\" regarding accelerator capabilities for the long-term future of high energy physics:\n \\begin{itemize}\n\\item  \tHow can one build a collider at the 10 -- 30 TeV constituent mass scale?\n\\item  \tWhat is the farthest practical energy reach of accelerator-based particle physics?\n\\item  \tHow would one generate ten or more megawatts of proton beam power? \n\\item  \tCan multi-megawatt targets survive and if so, for how long?\n\\item  \tCan plasma-based accelerators achieve energies and luminosities relevant to particle physics?\n\\item  \tCan accelerators be made an order of magnitude cheaper per GeV and\/or per MW?\n\\end{itemize}\n\nThis report is organized as a collection of the executive summaries of each of the six working groups: Energy Frontier proton colliders, Energy Frontier lepton colliders, Intensity Frontier proton sources, Intensity Frontier electron accelerators, electron-ion colliders, and accelerator research and test-beam facilities.\n\n\\section{Energy Frontier proton colliders}\n\nHigh-energy hadron colliders have been the tools for discovery at the highest mass scales of the Energy Frontier for more than a decade. They will remain so, unchallenged for the foreseeable future. The discovery of the Higgs boson at the LHC announced opened a new era for particle physics. Its measured properties are consistent, within the current uncertainties, with those of the Standard Model (SM) Higgs boson. \n\nAfter the discovery of the Higgs boson, understanding what is the real origin of electro-weak symmetry breaking becomes the next key challenge for collider physics. This challenge can be expressed in terms of two questions:  Up to what level of precision does the Higgs boson behave as predicted by the Standard Model? At what energy are the new particles that could offer some insight into the origin of dark matter, the matter-antimatter asymmetry, and neutrino masses? \n\n\\subsection{The LHC and its upgrades}\n\nThe approved LHC program, its future upgrade towards higher luminosities (HL-LHC), and the study of an LHC energy upgrade or of a new proton collider delivering collisions at center-of-mass energies up to 100~TeV, are all essential components of this endeavor.  High luminosity could increase the precision of several key measurements, uncover rare processes, and guide and validate the progress in theoretical modeling, thus reducing systematic uncertainties in the interpretation of the data.  To achieve this end,\n\n{\\it the full exploitation of the LHC is the highest priority of the energy-frontier hadron-collider program. }\n\nThe LHC is expected to restart in Spring 2015 at a center-of-mass energy of 13--14~TeV and reach its  design luminosity of $10^{34}$cm$^{-2}$s$^{-1}$  during 2015.  After 2020, some critical components of the accelerator will reach the radiation damage limit, and others will have lower reliability due to vulnerability to radiation. Furthermore, as the statistical gain in running the accelerator without substantially increased luminosity will become marginal, the LHC will need a substantial increase in luminosity~\\cite{ATLAS}. With a high luminosity upgrade (HL-LHC), the LHC should deliver 3000 fb$^{-1}$  during a decade of operation~\\cite{Zimmermann}. The HL-LHC project is the first priority of Europe, as stated by the Strategy Update for High Energy Physics approved by CERN council in the special session of May 30, 2013 in Brussels.\n\n\nSuperconducting magnets are the single most critical technology for the LHC and for proton colliders beyond the present LHC level of perormance.  Indeed, the present LHC is based on 30 years of development of magnets using NbTi wire. In the LHC NbTi-based magnets are pushed to their limits both in the collider arcs and in the interaction regions designed and built in collaboration with U.S. and Japanese national laboratories. \n\nDelivering the beam brightness required for HL-LHC will be a difficult challenge for the upgraded LHC injector chain; also challenging will be preserving it in the LHC storage ring. In light of this limitation, the preferred route towards increased luminosity is reducing the beta (amplitude) function at the interaction point, \n$\\beta^*$, by means of stronger triplet magnets with larger apertures in the interaction region. The present design of the HL-LHC interaction region requires quadrupoles with an aperture of 150~mm and with a peak field in excess of 12 T --- beyond the capabilities of NbTi conductor. It relies heavily on the success of the advanced Nb$_3$Sn technology developed by an integrated consortium of U.S. national laboratories under the LHC Accelerator Research Program (LARP). In addition to magnets, many other technologies will be involved in HL-LHC, such as crab cavities, advanced collimators, high-temperature superconducting links, and advanced remote handling. LARP plays a vital leadership role in this work to maximize the discovery potential of the extensive LHC infrastructure at CERN.\n\n{\\it A vigorous LARP leading to U.S. participation in the HL-LHC construction project is crucial to the full realization of the potential of LHC to deliver discovery physics to the thousands of U.S. researchers who are engaged at the LHC.}\n\nEnergy and luminosity play complementary roles in exploring the energy frontier with hadron collisions.  Production rates for signals of interest at the collider may be small either because the mass of the produced objects is large or because the coupling strength is small. In the former case, increasing the beam energy is clearly favored. In contrast, higher luminosity may be more effective in probing small couplings at smaller masses. In reaching higher energies, magnet technology will retain a pivotal role.\n\nGiven the progress in magnet technology and the maturity that Nb$_3$Sn is reaching, due largely to the LARP program for HL-LHC, Nb$_3$Sn magnets are expected to reach a limit of 15.5--16~T in operating field (that is with 15--20\\% margin with respect to quench) \nwithin the next decade, thereby opening a path towards a \ncollider with an energy significantly larger than the LHC. This would open the \nway towards very high energy hadron collisions with an exceptional potential for \nprobing the Energy Frontier.\n \n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=0.95\\textwidth]{AccelCap\/LHCsched.pdf}\n\\caption{LHC baseline plan for the next ten years in terms of energy of the collisions (upper line) \nand of luminosity (lower lines). The first long shutdown (LS) 2013-14 is to allow design parameters \nof beam energy and luminosity. The second LS, 2018, is to increase beam \nintensity and reliability as well as to upgrade the LHC Injectors.}\n\\label{fig:LHCsched}\n\\end{center}\n\\end{figure}\n\n\n\\subsection{Options for higher energy hadron colliders}\n\nThe first option is a machine housed in the LHC tunnel. The achievable center-of-mass energy depends on the dipole field strength. An energy of 26 TeV is within the reach of proven Nb$_3$Sn technology, although it requires engineering development. The energy reach would become 33 TeV, if 20 T magnets based on more futuristic high-temperature superconductors (HTS) were practical and affordable, and if they could fit in the limited space available in the LHC tunnel. In a new, longer tunnel, a  100~TeV proton collider is possible. Studies for a very large proton collider able to deliver center-of-mass energies of 100 to 200 TeV have been conducted over the past two decades. The multi-laboratory VLHC study led by Fermilab in 2001~\\cite{VLHC}\nwas noteworthy. The analysis remains valid and argues for a very large (233 km) tunnel that allows the use of NbTi magnet technology.\n\nContinuing an integrated, multi-laboratory program (LARP-like) towards the engineering development of 20~T magnets for HE-LHC or a new collider in a longer tunnel, needs to be an HEP priority. A program focused on engineering readiness, closely coordinated with CERN, would establish the limits of Nb$_3$Sn technology, investigate new conductor materials, and refine our present concepts for how to manage the enormous stresses produced by such high magnetic fields. The experience from RHIC, SSC, and LHC indicates that the dipoles account for about half of the total collider cost. Therefore, investing in magnet technology is critical and should represent a major focus of the research and development process.\n\nWith the renewed interest in the U.S. in a 100~TeV scale collider, the study group recommends participation in the international study for colliders in a large tunnel which CERN is now organizing.  The construction of a new tunnel reduces the demand for high dipole field strength, since there is a tradeoff between the tunnel circumference and required field strength (see Figure 3). The target collision energy is 100 TeV for 20 T dipoles in an 80 km tunnel ($R=13$ km). However, a slightly larger, 100 km circumference tunnel would provide the same collision energy of 100 TeV with field of 16 T, reachable with Nb$_3$Sn technology that relies on a far more mature and less expensive conductor than HTS. A large tunnel would also open up the possibility of a Higgs Factory based upon an $e^+e^-$ storage ring, if the International Linear Collider is not built~\\cite{Bhat}. \n\nThe 100~TeV study should address critical technology and cost feasibility issues and, thus, will provide input for developing a long-term roadmap of high-energy physics. Areas of particular interest to U.S. research groups include beam dynamics, magnets, vacuum systems, and machine protection.\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth]{AccelCap\/Super.pdf}\n\\caption{Role of the superconductor in the energy reach of hadron colliders as a function of field and bending radius, $R$, produced by the dipole magnets.}\n\\label{fig:Super}\n\\end{center}\n\\end{figure}\n\nWhether in the LHC tunnel or in a new, larger tunnel a collider with energy beyond the LHC will have to deal with an additional challenge, the emission by the beam of synchrotron radiation at least 20 times greater per meter than the LHC at its nominal parameters. The beam pipe and beam screen will have to absorb that radiation. Although synchrotron radiation is very beneficial for beam stabilization and will make the higher energy collider the first hadron machine dominated by synchrotron radiation damping, the power dissipated in synchrotron radiation must be removed at cryogenic temperature. In LHC it is removed at 5--10 K. One possible solution for the HE-LHC is based on a beam screen at 40--60 K. This approach has no major drawback in principle, although careful design, engineering and prototyping are needed to validate such a design. Merely relying on a beam screen at 10~K similar to the LHC would be a heavy burden for the cryogenics. Handling synchrotron radiation may be even more complex for a 100 TeV machine.\n\n{\\it Focused engineering development is no substitute for innovative R\\&D. }\n\nIn addition to the focused program of engineering development, the separate programs of long-range research in magnet materials and mechanical structures of magnets must be continued. These programs have provided the intellectual and infrastructure base for the success of LARP.  For example, the long-range future of high-energy physics using p-p collisions may need magnet operating fields beyond 16~T.  In practice, the decision will rest on a tradeoff between tunnel cost --- which scales roughly inversely with dipole field --- and the cost of the dipole magnets that scales roughly linearly with dipole field for long magnets.  The crossover in cost cannot be decided ab initio as the tunnel cost will depend strongly on the geology of the collider site. \nA new generation of magnets using high-temperature superconductors will require new engineering materials with small filament size that are available in multi-kilometer continuous pieces.  Advanced magnets may offer greater temperature margin against quenches due to stray radiation lost from the beam. Higher-field magnets will require innovative stress management techniques, exquisitely sensitive magnet protection schemes, and perhaps novel structural materials.\n\nAlthough synchrotron radiation damping is not dominant in proton colliders, as it is in electron colliders, it is still important to understand beam dynamics and other effects of marginal synchrotron radiation damping. As the energy stored in the beam becomes several gigajoules, control of tenuous beam halos becomes a pressing issue. Likewise machine protection from accidental beam loss and the design of special beam abort dumps become difficult challenges that demand innovation. An effective long-range research and development program must also consider beam physics of the injection chain, effects of noise and ground motion, and design and technology options for the configuration of the interaction regions.  \n\n\n\\section{Energy Frontier lepton colliders}\n\nIn its first three years of running experiments, the Large Hadron Collider (LHC) has begun to explore the energy region up to 1 TeV. The LHC experiments have discovered a Higgs boson which,  within current experimental precision,  is consistent with that of the Standard Model and have measured its mass as about 126~GeV. The discovery of the Higgs boson and the recent release of a technical design report for the International Linear Collider have energized high energy physicists to evaluate in depth the experiments that could be performed with a lepton collider. As described in the report of the Snowmass Energy Frontier Study Group, a future lepton or photon collider would provide a factory for measurements of the properties of the Higgs boson with ultimate precision.  A lepton collider would allow unambiguous searches for new particles that complement searches at the LHC. It would also provide opportunities to probe for and study new physics, both through the production of new particles predicted by models of physics beyond the Standard Model and through the study of indirect effects of new physics on the $W$ and $Z$ bosons, the top quark, and other systems.\n\n\\subsection{International Linear Collider }\n\nIn June 2013 the Global Design Effort (GDE) published a Technical Design Report (TDR)~\\cite{ILCTDR} of the International Linear Collider (ILC), an accelerator that will give these capabilities. The ILC is a superconducting linear $e^+e^-$  collider with a center-of-mass collision energy tunable between 200 and 500 GeV with a luminosity   exceeding 10$^{34}$ cm$^{-2}$s$^{-1}$\\  at 500 GeV, roughly scaling in proportion to the collision energy. The ILC is upgradable in luminosity by a factor of 2 and upgradable  in energy to 1 TeV.  The Japanese high-energy physics community has named the ILC as its first priority. An experienced cadre of U.S. accelerator physicists and engineers is capable and ready to work on this project. \n\n{\\it Our study welcomes the initiative for ILC in Japan. The U.S. accelerator community is capable of contributing as part of a balanced U.S. high-energy physics program.}\n\nThe key characteristics of the ILC accelerator are the relatively long interval between collisions of bunches, narrow beam energy spread, beam position and energy stability, and the ability to polarize both electrons and positrons. The design and technical details of the ILC have been developed over more than 20 years and incorporate extensive US leadership contributions in beam dynamics, damping ring design, electron and positron sources, superconducting RF (SRF) technology,  and beam delivery system design. As described in the TDR, the ILC represents a mature collection of technologies that are ready to proceed to construction.  \nThe 1.2 GeV VUV Free-Electron-Laser \"FLASH\" facility at DESY has provided a critical integrated system test experience. FLASH technology is quite similar to ILC, and ILC operational parameters are within the range of FLASH hardware. Tests at FLASH were done with the ILC bunch number, bunch repetition rate, bunch charge and peak beam current. These tests were successful and encountered no fundamental technology issues with operating a superconducting linac with the ILC design characteristics. The research program has successfully demonstrated the goal of 31.5 MV\/m in installed cryomodules with beam loading, using niobium cavities with no more than two surface-preparation processing cycles. With the specified accelerating gradient, the total length of the 500~GeV ILC is 31~km.  \n\nIndustrial and institutional partners of the GDE advised a strong industrialization program from the outset. A key ingredient was the development, within the GDE, of a project-governance and value-based cost-estimating strategy that was balanced regionally and that took advantage of the intrinsic modularity and relative maturity of the SRF technology. The most costly and time-consuming part of industrialization is the construction and commissioning of heavy infrastructure, notably institutional test facilities. Fabrication of superconducting cavities to the ILC design specifications has been industrialized, with qualified vendors in Europe, North America, and Asia.\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{l|l|c|c|c|c|c|c}\n$E_{CM}$ & GeV &   250   &  350 & 500 &  500 up &  250 up &  350 up \\\\ \\hline\nRep. rate  &\tHz  &\t5  &\t5  &\t5  &\t5  &\t10  &\t8   \\\\ \\hline\nBunches\/pulse   &\t &\t1312 &\t1312 &\t1312 &\t2625 &\t2625 &\t2625\\\\ \\hline\nTotal beam power &\tMW\t & 5.3 &\t7.4 &\t10.5 &\t21.0\t &21.0 &\t23.5\\\\ \\hline\nLuminosity  &\t$10^{34}$ cm$^{-2}$s$^{-1}$   &\t0.75\t & 1.0 &\t1.8 &\t3.2 &\t3.0\t &3.2  \\\\ \\hline\n\\end{tabular}\n\\caption{Summary of ILC performance in the range 250 to 500 GeV.   Columns 3--5 give\nthe TDR values. Columns 6--8  indicate scenarios beyond the parameters discussed in the TDR, but still within its technical scope. Units are specified  in the \nsecond column.}\n\\label{tab:ILCparams}\n\\end{center}\n\\end{table}\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{l|c|c}\nCenter-of-mass energy   & \\% of TDR cost &\tPower consumption (MW) \\\\ \\hline\n250 GeV \"Higgs factory\" &\t70\\%\t & 120  \\\\  \\hline\n500 GeV\t& 100\\%& \t163 - 204 \\\\ \\hline\n1 TeV upgrade &\t150\\%&\t240\\\\ \\hline \n1.5 TeV upgrade &\t200\\%&\t$> 300$\\\\ \\hline\n\\end{tabular}\n\\caption{Scaling of ILC cost and power consumption scaling as function of center-of-mass energy.}\n\\label{tab:ILCscaling}\n\\end{center}\n\\end{table}\n\n\n\nDuring the research phase, U.S. national laboratories and industry have contributed substantially to cavity performance, cryomodule design, high-level RF performance, demonstrations of beam dynamics, and specific components and systems. For the construction of ILC, it is reasonable to assume the U.S. contributions will continue along these lines. Specifically, construction and testing of completed cryomodules will contribute the greatest economic value to the project. The US community could also provide detailed accelerator design development efforts as an intellectual contribution. It is expected that the US would also contribute to design and construction of one or more linear collider subsystems. Given that the TDR estimates the required labor effort to be 13,000 person years, participation by America's highly experienced accelerator scientists and engineers will be crucial.\n\nExtension of the ILC to 1 TeV is straightforward. It requires lengthened linac tunnels and additional cryomodules, but would use the original ILC sources, damping rings, final focus and interaction regions and beam dumps. No new technological breakthroughs would be required, although research to develop higher-gradient cavities would permit shorter tunnel extensions and thus cost savings.\n\n\\subsection{Alternative $e^+e^-$  colliders}\n\nAlternative approaches to the ILC \u2013 albeit on a much longer time scale \u2013 include approaches with the potential to reach multi-TeV energies. One approach, the Compact Linear Collider (CLIC)~\\cite{CLICCDR}  two-beam accelerator, is based on a high-gradient X-band, warm linac powered by a high current, low energy \n(about 1~GeV) drive beam. CLIC would stretch for 50 km at an ultimate center-of-mass energy of 3 TeV. In contrast, the muon collider --- if proven feasible --- would fit a 3--5~TeV collider onto the present Fermilab site. Moreover, the muon approach has a very large overlap with capabilities needed for Intensity Frontier accelerators. Yet another approach would use wakefields driven either by beams or lasers to achieve accelerating fields of 10 to 100 GeV per meter. Many feasibility and practicality issues need to be solved in order to implement any of these programs.\n\nFor CLIC options at 350 GeV, 1.5 TeV and 3 TeV are currently being studied. The first and second option use a single drive-beam generation complex to feed both linacs, while in option three each linac would be fed by a separate complex. The CLIC design is based on three key technologies, which have been addressed experimentally: high gradient structures, drive beam generation, and power extraction. The normal-conducting accelerating structures in the main linac have a gradient of 100 MV\/m, to limit the length of the machine. The RF frequency of 12 GHz and detailed parameters of the structure have been derived from an overall cost optimization at 3 TeV. Experiments at KEK, SLAC, and CERN have verified the structure design and established its gradient and breakdown rate. The drive beams run parallel to the colliding beams through a sequence of power extraction and transfer structures, where they produce short, high-power RF pulses that are transferred into the accelerating structures. These drive beams are generated in a central complex. The drive-beam generation and power extraction have been demonstrated in a dedicated test facility at CERN.\n\nWakefields in plasma-based accelerators can potentially provide a thousand-fold or greater increase in acceleration gradient over standard technologies.  Two primary approaches are being investigated: beam-driven wakefields (plasma wake field acceleration, or PWFA) and laser driven wakefields (laser plasma acceleration, or LPA). Experiments at SLAC, LBNL, and at European laboratories have shown that plasmas can accelerate and focus high-energy beams at an accelerating gradient in excess of 50 GeV\/m. At present two large R\\&D facilities are spearheading this research in the U.S., FACET at SLAC (PWFA) and BELLA at LBNL (LPA). Smaller efforts are being pursued at university laboratories.\n\nFor any variant of wakefield accelerator to be practical as a linear collider, several feasibility and practicality issues must be resolved in the context of an integrated system test. Most importantly, wakefield accelerators, like standard accelerator modules, must be capable of being staged in a series of phase-locked segments.  Both PWFA and LPA approaches must demonstrate simultaneous positron acceleration and focusing in plasma densities consistent with preserving positron beam quality. Both must demonstrate timing, pointing, and focusing control consistent with the high luminosities required in a lepton collider.  Finally, both must demonstrate that multi-bunch plasma instabilities (such as zero-frequency, convective hose instability) can be overcome with operation at the tens of kHz repetition rate required for high luminosity.  Beyond the feasibility issues are questions of practicality related to overall cost, efficiency, and reliability. These issues may be more demanding for laser-driven concepts as cost and efficiency limitations of drive beam sources are better known.\n\n\nFor the study of the Higgs boson what matters are deliverable luminosity, maturity of design, risk, timescale, and cost. The type of $e^+e^-$collider, linear or circular, is of little importance.  A circular collider might offer a potential alternative to the linear collider. It can benefit from three characteristics of lower-energy circular machines: (1) high luminosity and reliability, (2) the availability of several interaction points, (3) excellent beam energy accuracy. For a given RF power, the luminosity of a storage ring collider rises linearly with its circumference. For a given tunnel size, and assuming the machine operates at the beam-beam tune-shift limit, luminosity rises linearly with the total dissipated synchrotron radiation (SR) power, which is approximately proportional to the total available power. Therefore, the analysis can be scaled to different machine sizes. The energy reach consistent with sufficient luminosity for Higgs studies depends strongly on the machine circumference; a machine of 27~km circumference might reach 240~GeV, while the limit is about \n350~GeV for a machine of 80~km. At a given circumference, the energy reach and luminosity are very strongly coupled; detailed characteristics of the design matter greatly.  \n\nCERN has begun a design study of the TLEP storage ring, a machine with a circumference of 80 to 100 km circumference and dissipated synchrotron radiation power set at 100 MW. The luminosity would be largest at the $Z$ peak, but fall rapidly as center-of-mass increases beyond that point.  The proposed TLEP storage ring would need ~ 1 km of superconducting RF cavities and low $\\beta^*$ insertions, would operate at fixed field and fed by a synchrotron situated in the same tunnel for continuous top-up injection. Multi-bunch operation would be necessary for high luminosity below the top energy, requiring separated beam pipes for electron and positron beams. Beamstrahlung  from the collision process can lead to a short luminosity-lifetime and large instantaneous energy spread in the beam of the order of several percent. Beam losses must be tightly controlled even in a lattice of very large dynamic aperture.  Maintaining vertical to horizontal emittance ratios considerably smaller than light sources needs further study. \n\nThe Chinese high-energy community has also started discussion and  first studies of a circular Higgs factory or a  proton collider in a very large tunnel~\\cite{IHEP}.  Should the ILC not go forward\n over the next decade and should the renewed interest in a very large circumference hadron collider be sustained, the possibility of a circular Higgs factory deserves extensive consideration.\n\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth]{AccelCap\/Muon.pdf}\n\\caption{ Muon accelerator timeline, including the MAP Feasibility Assessment period. It is anticipated that decision points for moving forward with a Neutrino Factory project supporting Intensity Frontier physics efforts could be reached by the end of this decade, and a decision point for moving forward with a Muon Collider physics effort, supporting a return to the Energy Frontier with a U.S. facility, could be reached by the middle of the next decade.}\n\\label{fig:Muon}\n\\end{center}\n\\end{figure}\n\n\n\\subsection{Muon accelerators}\nMuon accelerators~\\cite{Palmer} have the potential to provide world-leading experimental capabilities for physics at center-of-mass energies from the Higgs at 126 GeV up to the multi-TeV scale. A circular muon collider can potentially reach the higher energy range because the larger mass of the muons means they produce far less synchrotron radiation than electrons. For the same reason, beamstrahlung is suppressed leading to a very small spread of the center-of-mass energies in muon collisions. Muons, however, have a lifetime (at rest) of about 2.2 $\\mu$s and therefore decay in flight.  The unstable nature of the muon demands that the sequence of beam creation, manipulation, and acceleration be done rapidly; high-gradient acceleration is essential. Therefore, an Energy Frontier muon collider would necessarily be relatively compact. Even a 5~TeV collider would fit within the Fermilab site.\n\nResearch of critical importance to the performance of a muon collider includes: (1) development of a high-power target station including a high-field capture solenoid (nominally 20~T hybrid normal and superconducting magnet with about 3 GJ stored energy) that is ultimately capable of handling more than 4 MW of power; (2) control of collective effects in the high-intensity beams at the low energy end of the production, capture, and acceleration process; (3) cooling (reduction) of the muon six-dimensional phase space volume by six orders of magnitude to achieve the beam parameters required for the muon collider designs; (4) development of very large-aperture, superconducting dipole magnets with heavy inserts to protect the superconductors from the decay products of the stored muons in the collider. Once items (1) and (2) are resolved positively, a muon accelerator would provide an outstanding high-flux source of well-characterized neutrinos for a range of Intensity Frontier experiments. The potential for muon accelerators to address crucial questions on both the Intensity and Energy Frontiers argues for a robust development program.\n\n\n\n\\subsection{Photon colliders}\n\nIn a Higgs factory photon collider, two electron beams are accelerated to 80 GeV and converted to about 63 GeV photon beams by colliding with low-energy (3.5 eV photons), high-intensity (5 J per pulse) laser beams via the Inverse Compton Scattering (ICS) process. The two high-energy photon beams then collide and generate Higgs particles through the $s$-channel resonance $\\gamma\\gamma\\to H$. Among various options for a Higgs factory, a photon collider has the distinct advantage that the 80 GeV energy required for the electron beam is lower than for other colliders. Photon colliders have been discussed as options to accompany proposed linear or circular colliders or as a stand-alone facility. \nSince the overall acceleration efficiency of any facility is the product of the individual efficiencies of each of the systems involved in the transfer of power from the wall plug to the beam, each of the systems should be as efficient as possible. Innovative research on efficient RF generation would be extremely beneficial to all lepton collider designs. It should be strongly supported as a key technology to reduce the operating costs of future facilities for both the Energy and Intensity Frontiers.\n\n\\section{Capabilities requiring high-intensity proton beams}\n\nThe beam parameters needed for a variety of measurements of interest in the Intensity Frontier were presented at a community workshop held at Brookhaven National Laboratory in April, 2013~\\cite{Galambos}.\n Experiments included neutrino, kaon, muon, neutron, and proton EDM measurements. From the secondary beam specifications, the workshop identified the proton beam requirements for the different proposed Intensity Frontier measurements. Although the resulting sets of primary proton beam requirements were generally not based on detailed calculations but rather on general guidelines, they can be used as a guide for matching the desired measurements with existing or proposed proton facilities. The study compared 20 existing proton beamlines and 14 planned upgrades with the requirements of 19 secondary beam requests filled out by experiment advocates and drew the following conclusion:\n\n{\\it The next generation of Intensity Frontier experiments will require proton beam intensities and timing structures beyond the capabilities of any existing accelerators.}\n\nThe major proton accelerator facilities surveyed included U.S. capabilities at FNAL, BNL, LANL, and SNS. International representation included CERN, J-PARC, PSI, ISIS, and TRIUMF. New and proposed proton accelerator facilities included the European Spallation Source (ESS), the Chinese Neutron Spallation Source (CSNS), DAE$\\delta$ALUS, and Project X. In addition, existing facilities typically have upgrade proposals --- both near and long term --- which have been included in the study. High proton beam powers of 1 MW or more are desired for most proposed experiments. The beam energy requirement typically exceeds 1 GeV.  The required time structure of the beam varies significantly, depending on the nature of the measurements and the specifics of the measurement conditions; for example, underground detectors have special requirements. \n\nAccelerators with current and anticipated future programs of Intensity Frontier experiments include FNAL, J-PARC, and CERN. The LANL linac is undergoing a refurbishment and in a few years this beam would operate at 100 Hz and provide bunches with a 1 ms duration available for potential users. Cold neutron applications may be able to use this beam structure, as well as others.  Existing accelerators such as J-PARC and SNS might provide MW proton beams, with energy  less than 3 GeV with very low duty factor (less than 10$^{-4}$ to minimize background from cosmic sources) for lower-energy muon and electron neutrino studies.  ISIS (in the UK) has been used for this purpose in the past, and still has a cavern 10 m from their TS-1 target.  The ultimate technical potential of linac technology is suggested by the example of LEDA. The normal-conducting LEDA linac, which produced a 6.7 MeV, 100 mA CW beam, operated at Los Alamos from 1999 to 2001. \n\n\\subsection{Proposed new accelerators}\\label{sec:intenseaccel}\n\nProposed new accelerators for intensity frontier applications include Project X, DAE$\\delta$ALUS, and nuSTORM. Of the three, Project X~\\cite{Holmes}  has the broadest scope and, if required, could be structured as a multi-phase proposal. Project X is a high-intensity proton facility that will support a world-leading Intensity Frontier research program over the next few decades at Fermilab. When compared to other facilities in the planning stages elsewhere in the world, Project X would be unique in its ability to deliver, simultaneously, up to 6 MW of site-wide beam power to multiple experiments, at energies ranging from 233 MeV to 120 GeV, and with flexible beam formats. Project X is designed to support a wide range of experiments based on neutrinos, muons, kaons, nucleons, and nuclei. It will also lay the foundation for the long-term development of a neutrino factory and\/or a muon collider. \n\nProject X capitalizes on the very rapid development of superconducting RF technologies over the last 20 years, and their highly successful application to high-power H\u2212 acceleration at the Spallation Neutron Source (SNS) at Oak Ridge National Laboratory. As a result of the accelerator research leading to and supporting SNS construction and operation, excellent simulation and modeling tools exist for designing the Project X facility with high confidence that the performance goals can be achieved. The primary supporting technologies required to construct Project X exist today.\n\nThe Project X Collaboration is engaged in a comprehensive development program aimed at mitigating technical and cost risks associated with construction and operations. The major elements of the risk reduction program include an integrated, 25 MeV front-end system test, the Project X Injector Experiment (PXIE). To allow for beam accumulation in the Main Injector, the Project X linac will accelerate H$^-$ ions. Optimizing the superconducting RF cavities will improve the cost effectiveness of the final accelerator design. Achieving the performance required of high-power targets is expected to be challenging. Therefore, high-power target technology will be a key component of research in support of the project.\n\nThe overall scope and goals of the Project X development program are based on being prepared for a 2017 construction start. Essentially all elements described above are required to implement the first stage of Project X.\n\nDAE$\\delta$ALUS~\\cite{Conrad}  is a neutrino research program based on ``decay-at-rest'' sources.  Pions are produced by interaction of 800 MeV protons on a suitable target. This energy is sufficiently above threshold for good pion yield, and low enough that pions will stop in the target before decaying.  The DAE$\\delta$ALUS configuration consists of three nearly identical\n sources of neutrinos. The highest-power accelerator, located 20 km from the\n detector would provide 10 mA of protons at 800 MeV to the neutrino-generating targets. \nA current of 10 mA is approximately a factor of 5 times the highest achieved current at PSI, the world's leading high-power cyclotron today.  Accelerating H$_2^+$  ions rather than\n protons has the potential for reducing space-charge issues at injection. Extraction of \nthe beam at 800 MeV with a stripper foil minimizes the necessity for clean turn separation at the outer radii, only requiring a proton extraction channel with sufficiently large momentum \nacceptance to transport ions stripped from several overlapping turns. The use of H$_2^+$ acceleration represents a novel approach to reducing beam losses at extraction from the\n 800 MeV cyclotron.\n\nTo date, the DAE$\\delta$ALUS feasibility arguments are made by scaling from existing low-energy H$^-$ commercial cyclotrons as well as from the PSI high-power proton cyclotron.  Since the high-power H$_2^+$ concept is quite novel, a systematic research is being conducted to address challenges of meeting the required performance. The most critical elements are (1) ion-source development to achieve very bright, vibrationally cold  H$_2^+$ beams of at least 50 mA CW, (2) injection into cyclotrons with emphasis on bunching efficiency, space requirements and space-charge dynamics, (3) end-to-end simulations to evaluate beam stability and uncontrolled loss, and (4) atomic physics experiments to measure stripping and beam-gas scattering cross sections and possibly techniques for Lorentz dissociation of vibrational states in high-field (about 25~T) magnets in the transport line between the injection and the second-stage cyclotrons. \nAs is the case for Project~X, target systems are challenging and will require sustained research.\n\nThe first phase of DAE$\\delta$ALUS is IsoDAR~\\cite{Bungau}, a compact 600 kW cyclotron proposed to be located 15 m from the KamLAND detector for a definitive search for one or two sterile neutrinos.  The IsoDAR cyclotron would also be a prototypical injector for the superconducting ring cyclotron of DAE$\\delta$ALUS. At present space charge effects at injection are being studied experimentally in a collaboration with industrial partners.\n\nThe nuSTORM facility is designed to deliver beams of electron and muon neutrinos from the decay of a stored muon beam with a central momentum of 3.8 GeV\/c and a momentum acceptance of 10\\%. The facility will be able to (1) conduct highly sensitive searches for sterile neutrinos, (2) provide definitive measurements of neutrino-nucleon scattering cross sections with percent-level precision for the electron and muon neutrinos, and (3) take the crucial first step in developing of muon accelerators as a powerful new technique for particle physics.\n\nnuSTORM is the simplest implementation of the neutrino factory concept.  It utilizes protons from the Main Injector, has a conventional (NuMI-like) target station, pion collection with a NuMI-style horn, a pion transport line, and a 480 m circumference muon decay ring.  The nuSTORM facility can also provide an intense pulses of $10^{10}$  low-energy muons, appropriate for future 6-dimensional muon ionization cooling experiments.  This beam is available during operation of the neutrino physics program and operates in a completely parasitic mode.  Finally the muon decay ring provides the opportunity for accelerator R\\&D on beam instrumentation and potentially on muon acceleration technology.\n\n\\subsection{General accelerator research needs}\n\nIn addition to project-specific needs, the Snowmass workshop identified more general research needed to advance Intensity Frontier capabilities. That research includes but is not limited to the following~\\cite{Shiltsev}:\n\\begin{itemize}\n\\item \tUnderstanding and controlling beam loss: Very precise simulations of beam dynamics involving accurate predictions of beam halos are essential for enabling high-power multi-megawatt-scale machines to run without inducing excessive amounts of radioactivity. To benchmark simulation codes, reliable measurements of very weak beam halos will be required with a large (10$^4$--10$^5$) dynamic range.\n\\item \tSuperconducting RF technology: Most modern high-power proton facilities rely on linacs utilizing superconducting radio frequency (SRF) acceleration. The initial development of SRF technology emphasized maximizing the accelerating gradient to limit the size of the accelerators. Since  many Intensity Frontier applications require the delivery of high-duty-factor beams, the figure of merit is shifting from high-gradient to very high unloaded quality-factor, Q$_0$. Cavity fabrication and processing techniques must minimize cost, while reliably producing cavities with gradients of roughly 20~MV\/m at Q$_0$ in excess of 10$^{10}$ to support future facilities.\n\\item \tHigh-quality, high-current injection systems: Multi-MW linacs and cyclotrons will need high-current, low emittance sources of ion beams, effective beam chopping, and control of strong space-charge effects at low beam energies. While suitable ion sources are close to being available, effective beam chopping is beyond the present state of technology and is under active development. The challenge is to develop choppers to provide full beam deflection on a time scale less than the beam micro-bunch spacing, and\/or to cleanly transport partially deflected beam.\n\\item \tIsochronous ring cyclotrons: Isochronous ring cyclotrons (including fixed-field alternating gradient accelerators) are also good candidates for high continuous (CW) power. Clean extraction of the beam requires well-separated turns. This sets the scale of both the ring diameter and the energy gain per turn. The alternative is to use an ion such as H$_2^+$  that can be extracted with a foil. The large superconducting magnet sectors require some development beyond the present RIKEN magnets before costs can be confirmed. \n\\end{itemize} \n\n\\subsection{High-power targets}\n\nHigh-power target development and its implications for high-power proton accelerators should not be underestimated~\\cite{Hurh}. Three of the present high-power accelerator facilities (SNS, J-PARC-MLF, and FNAL-MINOS) have been limited in beam power during recent operational periods by target concerns, not by the accelerator capability. The cost of these systems is a non-negligible portion of high-power accelerator facilities, and the resources needed to operate and maintain them are also significant (remote maintenance, disposal of radioactive wastes, control of personnel exposure to radiation). Plans for new facilities should include the impacts of handling the proposed high beam powers.\n\nOf the many identified challenges of high-power targets, radiation damage is of special note because it is complex, difficult to evaluate, and time-consuming to investigate. Atom displacements and gas production are the main underlying damage mechanisms, the particulars of which depend on primary beam characteristics, target materials, and specifics of the mechanical design. In the regimes of beam energy and intensity foreseen for Intensity Frontier experiments, one cannot simply scale from experience with nuclear power reactors. Moreover, the formation and growth of defects in target materials in extreme conditions is a mesoscale physics problem that is presently beyond present modeling and computational capabilities.  In addition, techniques of accelerated testing are frequently not validated and\/or appropriate irradiation facilities may not be available.\n\n{\\it Sustained, focused research into target technology in the context of a broad international collaboration of interested laboratories will be essential.}\n\nExperiments on the Intensity Frontier offer a broad range of scientific opportunities accompanied by a similarly broad range of technical challenges that must be overcome to conduct these experiments in acceptable time frames. The technical challenges will require sustained, focused efforts. The time to begin is now.\n\n\\section{High-intensity electron and photon beams}\n\nTo offer new opportunities for physics discoveries heavy flavor factories must accelerate and store beams with characteristics that will be challenging to deliver. Two types of flavor factories are being constructed or proposed worldwide: super $B$ factories and super tau-charm factories. SuperKEKB, a super-high-luminosity $B$ factory, is an upgrade to the KEKB $B$-factory currently under construction in Japan, with commissioning scheduled to commence in January 2015. \nU.S. national labs and universities have made important contributions to the design. Two super tau-charm factories have been proposed: one at Frascati (Tor Vergata) in Italy and one at Novosibirsk in Russia. Both machines are two-ring, symmetric-energy machines, with provisions for longitudinally polarized beams.\n\nFor SuperKEKB to achieve its target luminosity of $8\\times 10^{35}$ cm$^{-2}$s$^{-1}$\\ --- a 40-fold increase over that of KEKB  --- it will require  beam currents approximately twice as high as used at KEKB and vertical bunch sizes at the collision point about 20 times smaller than those achieved at KEKB. Although it would be premature to contemplate increases in the design luminosity of a machine that has not yet started commissioning, there is performance headroom in machine subsystems for increasing luminosity beyond the its baseline \ndesign value~\\cite{Groupfour}. Continuing U.S. collaboration would strengthen the SuperKEKB project, and would support efforts to increase the luminosity to well above that targeted in the existing design. This result would mirror the operational experience with both KEKB and PEP-II.\n\nEnhanced collaboration could also enable polarized beams in the collider. Polarization in SuperKEKB will involve preparing and injecting a beam of polarized electrons in the correct spin orientation parallel to the local precession axis at the injection point. While polarized sources with the required intensity exist (e.g., the SLC polarized gun), they are not low-emittance polarized sources. BNL has developed a candidate gun which is ready for commissioning and testing. The alternative would be to use a polarized DC gun followed by a damping ring to lower the emittance. The physics case needs to be made for both higher luminosity and polarized beams.  \n\nThe technology development for super flavor factories exploits strong synergies with light sources and damping rings. Relevant areas of synergy include high top-up rate to compensate for low lifetimes, beam stability and control, control of electron cloud and fast ion instabilities, coherent synchrotron radiation issues with short bunches, timing jitter and the attendant energy jitter, and beam instrumentation. Many of these areas are important to the design of damping rings for a linear collider and would certainly apply to the design of a circular Higgs factory. Participation of U.S. accelerator personnel in commissioning and machine studies would help to maintain a high level of U.S. expertise with high-current, low-emittance electron storage rings, such as TLEP.\n\nThree plans or proposals address physics opportunities using high-power electron and positron beams. The DarkLight experiment plans to use the 100 MeV, 10 mA electron beam of the Jefferson Lab free electron laser (FEL), impinging on a hydrogen target, to search for gauge bosons associated with Dark Force theories. The gauge bosons are predicted to decay into $e^+e^-$, leading to the final state $e^- + p + e^+ + e^-$. DarkLight will be unique in its ability to detect all four particles in the final state. The leptons will be measured in a large high-rate time-projection chamber (TPC)\n and a silicon detector will measure the protons. A 0.5~T solenoidal magnetic field provides the momentum resolution and focuses the copious M\\o ller scattering background down the beam line, away from the detectors. A first beam test has shown the FEL beam is compatible with the target design and that the hall backgrounds are manageable. The DarkLight experiment has been approved by Jefferson Lab for first running in 2017.\nHigh-intensity electron beams may also be instrumental in fixed target searches for light, weakly-coupled particles including dark matter in the MeV to GeV range. One proposed experiment would require 10$^{20}$ to 10$^{23}$ electrons with energies of $\\geq$1 GeV striking stationary nuclei in a beam dump to produce new weakly interacting states, which pass through shielding material and scatter tens of meters downstream in a relatively small detector, one cubic meter or less.\nA more speculative proposal would use a high-intensity, low-emittance positron beam, impinging on a plasma target, to generate muon\/anti-muon pairs. The beam energy should be just above the $\\mu^+\\mu^-$ production threshold with maximal beam energy asymmetry, i.e., a $\\sim$45 GeV positron beam interacting on an electron target. A low-power experimental test of this muon production rate may be possible at the FACET facility at SLAC.\n\n\\section{Electron-ion colliders}\n\nSeveral designs of future electron-ion colliders have been under consideration in recent years. All of them are based at an existing accelerator facility. The collider configurations include both ring-ring and linac-ring options. Center-of-mass energies of the proposed colliders range from 14 GeV to 2000 GeV.  Most of the collider concepts share enabling technologies. In all designs superconducting RF cavities must be able to operate with high average and high peak beam currents, providing effective damping of high-order modes. The cryomodule design must be able to accommodate containing high beam power. The hadron beam transverse and longitudinal emittances must be small to achieve a high electron-ion collider luminosity. In the designs with medium hadron energy, this requirement calls for the application of powerful cooling techniques.\nThe low \u03b2\u2217 interaction region designs for all proposed colliders face the issues of strong focusing of beams at the collision point and fast separation of beams after the collision. The synchrotron radiation fan produced by electrons in the interaction region magnets must be kept from hitting the beam pipes whether inside or in the vicinity of the detectors or in superconducting magnets.  Requirements imposed by the detector integration must be satisfied and chromatic corrections to the beam optics have to be implemented while maintaining an acceptable dynamic aperture.  Approaches considered include the following:\n\\begin{itemize}\n\\item\tSpecial design of interaction region magnets, including Nb$_3$Sn superconductor technology.\n\\item\tCollimators, absorbers, and masks at appropriate locations to provide protection from synchrotron radiation.\n\\item\tIntegrated dipole field in the detector design.\n\\item\tModified crossing angle geometry (in some designs), including crab-crossings.\n\\end{itemize}\nThe linac-ring designs utilize a high-current polarized electron source, with the average current ranging from 6~mA to 50~mA. A Gatling gun arrangement or large-size cathode gun can likely produce up to 50~mA current. The linac-ring scheme introduces non-standard beam-beam effects, which must be explored to understand the limits on the luminosity and the beam parameters. The effects include the electron beam disruption, the hadron beam kink instability, and the effect of the electron beam parameter fluctuation on the hadron beam.\nOther shared technologies include polarized beam sources and techniques to preserve beam polarization. In ring-ring colliders spin matching and the harmonic correction techniques have to be investigated further with the goal of minimizing the beam depolarization due to synchrotron radiation, especially in the presence of spin rotators and solenoidal detector magnets.\n\n\\section{Accelerator R\\&D and test beam capabilities}\n\nThis study identified a broad range of test capabilities present in existing or needed for proposed Energy and Intensity Frontier accelerators. The first category of test facilities permits testing beam physics or accelerator components essential to manage technical risks in planned projects. A second category integrates accelerator systems to provide proof-of-practicality tests.  The third category provides tests of physics feasibility of concepts and\/or components.  The study identified 35 existing test facilities~\\cite{Groupsix} in the U.S. and overseas both with and without beam testing capability. Although these facilities provide substantial readiness to move forward with the highest priority accelerators for high-energy physics, the long-range future of high energy physics needs a few additional dedicated test capabilities in the near term.\n\n\\subsection{Energy Frontier accelerators}\n\nHadron colliders have dominated the high-energy physics landscape for several decades and are expected to continue in that status over many more years of the LHC operation. Several Energy Frontier hadron colliders are under consideration now, including upgrade of LHC luminosity to $5\\times 10^{34}$  cm$^{-2}$s$^{-1}$, upgrade of the LHC energy to more than 25 TeV, and an 80 to 100 TeV VLHC in a 80--100 km or longer tunnel.\nThe most crucial technology issues for hadron colliders are in the area of superconducting magnets which are the largest drivers of the cost and of the energy and luminosity reach of these colliders. Of greatest challenge are Nb$_3$Sn 15~T arc dipoles and possible HTS superconductors such as YBCO and Bi-2212 capable of supporting dipole fields well above 20 T. For the VLHC, cheaper lower-field magnets and innovative low-cost tunneling technologies are of utmost importance to establish the financial practicality of a collider. Superconducting crab-cavities and very high-power beam dumps need to be developed for beams of such high energy. \n\nMany engineering challenges will be associated with the expected few GJ of beam energy in colliders beyond the present LHC. Increased levels of synchrotron radiation will affect beam dynamics, machine design, and operating costs and complexity. These issues will require research to identify methods of efficient collimation and control, novel vacuum chamber designs, and development of the photon stops to protect the superconducting magnets and other equipment.  \n\nMany requisite technology developments for Energy Frontier hadron colliders \u2013 described previously  \u2013 can be carried out at the existing superconducting magnet and RF test facilities at the DOE national laboratories, as well as at similar facilities overseas, foremost of which is at CERN. Some of these facilities may need upgrades to accommodate the research needs of specific projects (cable tests, transmission line magnet tests, etc.). Many beam dynamics studies concerning collimation and coherent instabilities can be performed at the LHC and its injectors, at the Fermilab Main Injector and Booster, and at RHIC. Most Project X research and development activities and facilities will directly address many issues related to the injector chain for a hadron collider beyond the LHC.  Several promising, novel accelerator schemes \u2013 such as resonance-free, integrable optics, space charge compensation with electron devices, and optical stochastic cooling \u2013 require a dedicated test accelerator for evaluation. The Integrable Optics Test Accelerator (IOTA) at the Advanced Superconducting Test Accelerator (ASTA) facility at Fermilab is a suitable match for the task. \n\nAlthough a primary challenge of any Energy-Frontier $e^+e^-$ linear collider project is value engineering to reduce costs, specific technical challenges remain for design of the positron source and the beam delivery system. These issues are being addressed at facilities including ATF2 at KEK, FLASH at DESY, and ASTA at Fermilab. A CLIC-type system has additional challenges to develop and industrialize the accelerating structures and major components. These questions are being actively addressed at CesrTA, ATF2 and CTF3.\n\nAt energies less than 250 GeV per beam, in tunnels with sufficiently large circumference, circular colliders may potentially provide a high luminosity while drawing on a base of mature technology with rich experience. Primary challenges for rings are the large synchrotron power, beamstrahlung, and large momentum acceptance required. Many of these issues are being addressed in large part at flavor factories, diffraction-limited light sources, and beam test facilities.\n\nPhoton collider designs allow access to the CP property of the Higgs boson, and they can study it at a lower electron beam energy with a high degree of polarization without the need for positrons and ancillary systems. Primary challenges for photon colliders relate to generating sufficient photon flux with high-energy lasers as well as new interaction region design. Appropriate laser technology is being developed within the international consortiums and at LLNL in the U.S.\n\nA muon collider scales well to high energies because the relatively large mass of the muons allows for tight bends and compact accelerator geometry. The primary challenge for muons is creating and cooling them to sufficient phase space density to allow for high enough luminosity. The Muon Accelerator Program is coordinating the R\\&D into the major issues with experiments primarily at FNAL and RAL.\n\nThe main challenges for plasma accelerators include demonstrating a concept for positron acceleration and demonstrating staged acceleration with the required beam brightness and efficiency, and those are being addressed through multi-institution collaborations conducting experiments at dedicated and proposed facilities BELLA\/BELLA-II at LBNL and FACET\/FACET-II at SLAC. \n\nA major challenge in applying plasma wakefield accelerator concepts to high-energy colliders is staging successive drive beams to accelerate a sufficiently intense witness beam to the desired energy. Staging requires the generation of high-intensity, multiple-drive electron\/laser beams capable of high-repetition-rate operation. Tailoring the longitudinal characteristics of a drive beam may effectively increase the energy transfer efficiency from the drive beam to the witness beam, thus reducing the number of required stages. Issues of wakefield acceleration in dielectric structures include identifying materials with appropriate breakdown voltage characteristics, identifying optimal structure geometries, and demonstrating staging with the required efficiency. Relevant research is being conducted at the AWA at ANL, ATF at BNL and FACET at SLAC. The NLCTA facility at SLAC hosts a dedicated beamline studying the primary issues for laser-driven microstructures including damage limits, coupling geometry, staging, and maintaining phase tolerances.\n\nSeveral facilities in the U.S. are exploring wakefield accelerators. AWA at Argonne combines multiple 75 MeV drive beams with a 15 MeV witness beam. BELLA at LBL uses a 1 PW laser to conduct experiments that synchronize two laser beamlines to accelerate a single, sub-GeV electron beam. FACET at SLAC uses a 30~GeV beam from the SLAC linac to investigate positron acceleration, high-transformer ratio configurations and the ``afterburner'' concept. Multiple drive beam generation and pulse shaping is one of the major goals for the proposed FACET II upgrade.\n\n\\subsection{Intensity Frontier accelerators}\n\nThe short- and medium-term needs of the U.S. Intensity Frontier program will employ Fermilab proton accelerators --- power-upgraded Booster and Main Injector augmented\n by the ``muon campus'' rings ($g-2$, mu2e, nuSTORM, etc.). Longer-term aspirations require construction of the proposed Project X, described in Section~\\ref{sec:intenseaccel}.\n Project X could support a wide range of experiments based on neutrinos, muons, kaons, \nnucleons, and nuclei and also can form a basis of the front-end of a future Neutrino\n Factory. The proposed DAE$\\delta$ALUS experiment calls for construction of several \nvery high-power, 1 to 5~MW, \n800~MeV proton cyclotrons.    \nThe primary design constraints in high-power hadron accelerators for the Intensity Frontier are minimizing beam loss ($<$ 1 watt\/m, or $<$ 0.01\\% total beam loss), and flexible beam delivery in experiment-specific time formats ranging from quasi-CW for rare particle decays to single few-ns-long bunches. Additional challenges facing designers of Intensity Frontier accelerators include: (1) generating high-current, high-quality beams, (2) efficiently accelerating high beam currents to high energies, (3) controlling emittance and minimizing beam halo, (4) manipulating beams on a bunch-by-bunch basis for parallel experiments, (5) maintaining low loss during beam extraction, and (6) creating target systems for extreme power densities and extreme radiation environments.\n\nProject X R\\&D activities at the PXIE facility and beam studies with the SRF cryomodules at Fermilab's ASTA facility will directly address many issues listed above. Many beam experiments on efficient collimation, coherent instabilities, and novel extraction methods can be performed at Fermilab's Main Injector and Booster, at the ORNL's SNS, or at the LHC and its injectors. Studies of promising novel accelerator techniques --- like halo suppression in the integrable optics and space charge compensation with electron devices ---\n can be performed at the IOTA at the ASTA facility. A dedicated facility is needed for development and tests of high-power targets, though some target-related issues can be explored at existing accelerators. Many tests and technology demonstrations to support the DAE$\\delta$ALUS experiment design work can be performed at the existing facilities, such as INFN-Catania, RIKEN, ORNL, or in the private sector at medical cyclotron companies.\n\n\\section{Summary remarks}\n\nThe approved LHC program, its future upgrade towards higher luminosities (HL-LHC), and the study of an LHC energy upgrade or of a new proton collider delivering collisions at center-of-mass energies up to 100 TeV, are all essential components of exploring the Energy Frontier with proton colliders. Therefore the priority recommendation of our study is full exploitation of LHC.  Doing so requires a strong LHC Accelerator Research Program sponsored by the OHEP that transitions to a US-LHC high luminosity construction project. During the project period we recommend continuing a focused, integrated, laboratory program that emphasizes the engineering readiness of technologies suitable for a 26 TeV upgrade of the LHC or a machine of higher energy in a larger tunnel. The most critical technology development toward higher-energy hadron colliders is the next generation high-field Nb$_3$Sn magnets (limited to about 15 T) and adequate beam control technology to assure machine protection.\n\nOur study welcomes the initiative for ILC in Japan. The U.S. accelerator community is capable of contributing  as part of a balanced U.S. high-energy physics program. As described in its Technical Design Report (TDR), the ILC is technically ready to proceed to construction. The TDR incorporates leadership U.S. contributions to machine physics and technology in superconducting RF (SRF), high-power targets for positron production, beam delivery, damping ring design, and beam dynamics such as electron cloud effects. \nPlasma-based accelerators (beam or laser driven) providing GV\/m accelerating fields would open the way to linear colliders in the muli-TeV range with a relatively small footprint if the preservation of beam quality during multi-stage acceleration can be demonstrated. A circular muon collider, if feasible, could provide discovery-level experimental capabilities at energies from the Higgs $s$-channel threshold at 126 GeV up to the multi-TeV scale. Such a collider could probe narrow physics features with extreme precision, because the larger mass of the muon suppresses synchrotron radiation during the collision process that would spread the energy of the beams. \n\nRequired accelerator capabilities for Intensity Frontier  experiments are more diverse than for the energy frontier. The common characteristics required are (1) average beam power $>$ 1 MW delivered \nwith (2) a flexible, experiment-dependent time structure. The overarching conclusion of the study is that the next generation of intensity frontier experiments requires beam intensities and timing structures beyond the capabilities of any existing accelerator. Fermilab's proposed, multi-stage Project X would yield a world-leading facility that would simultaneously deliver a flexible ``on-demand'' beam structure that could serve multiple experiments over an energy range 0.25 -- 120 GeV. Our study heard exciting possibilities for\n DAE$\\delta$ALUS, a capability of narrower experimental scope than Project X. For both proposed \ncapabilities high-power targets are a difficult challenge that limits facility performance. One cannot \nsimply scale the performance with high-energy beams from experience with nuclear reactors. \nComputed radiation effects in inhomogeneous materials, subject to time-varying irradiation, \nneed validation with controlled, instrumented in-beam tests. \n\nThe relevant technologies of super flavor-factories exploit strong synergy with light sources and damping rings for lepton colliders. Doubtless the operation of such machines such as SuperKEKB, the super-high-luminosity $B$ factory,  will inform the final design of damping rings for a energy frontier lepton collider. \nAll electron-ion colliders studied in recent years would be based at an existing accelerator facility. The collider configurations include both ring-ring and linac-ring options; center-of-momentum energies \nrange from 14~GeV to 2000~GeV.  Most of the collider concepts share enabling technologies. \nSuperconducting RF cavities must be able to operate with high average and high peak beam \ncurrents, providing effective damping of high-order modes. \n\nAlthough existing accelerator test-bed facilities provide substantial readiness to advance the \nhighest priority accelerators for high-energy physics, the long range future of high energy physics \nneeds a few additional dedicated test capabilities in the near term, most notably, for high power \ntargetry, muon accelerators, and high-intensity accelerator systems.  The group did not examine \nthe process in basic accelerator physics and modeling that must accompany any long-range research \nprogram. Such research bears fruit beyond the confines of specific projects and is essential to the\n long-term health of the field.\n\nThe Snowmass Accelerator Capabilities Study offers the following summary findings and recommendations:\n\\begin{itemize}\n\\item \tTo maximize the exploration of the Energy Frontier, the full exploitation of the LHC is the highest priority of the hadron-collider program. \n\\item\tAs described in its Technical Design Report (TDR), the ILC is technically ready to proceed to construction.\n\\item \tA vigorous, integrated U.S. research program toward demonstrating feasibility of a muon collider is highly desirable. The current funding level is inadequate to assure timely progress.\n\\item \tThe overarching conclusion is that the next generation of Intensity Frontier experiments requires beam intensities and timing structures beyond the capabilities of any existing accelerator.\n\\item \tFermilab's proposed, multi-stage Project X would yield a world-leading facility that could serve multiple experiments over an energy range 0.25 -- 120 GeV.\n\\item \tSustained, focused research into high-power target technology in the context of a broad international collaboration of interested laboratories will be essential to frontier accelerator facilities.\n\\item \tThe relevant technologies of super flavor-factories exploit strong synergy with light sources and damping rings for lepton colliders.\n\\item \tAll electron-ion colliders studied in recent years would be based at an existing accelerator facility with center-of-momentum energies range from 14 GeV to 2000 GeV.\n\\item \tInnovations in acceleration and beam transport techniques such as plasma and dielectric wakefield acceleration have a significant potential to reduce the size of future accelerator facilities. \n\\item \tLong-term research including fundamental accelerator and beam physics theory and simulation will expand the technical options for any future accelerator-based facility. Focused engineering development is no substitute for innovative R\\&D. \n\\end{itemize}\n\n{\\bf Acknowledgements: }\nThis report has been partially supported by the Office of High Energy Physics and the Office of Nuclear Physics of the U.S. Department of Energy and the National Science Foundation. The conveners thank all those who have participated in the Snowmass 2013 process, both at workshops and by submitting white papers.  \n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe strain energy of a piece of homogeneous anisotropic elastic material depends on the orientation of the material relative to the directions of principal stress, although the orientation dependence vanishes trivially for isotropic solids.  This property is therefore an inherently anisotropic feature of elasticity, and it raises the question of how to find  the material orientation (if any) which minimizes the strain energy for a given state of stress or  strain. New results are presented in this paper on the determination of optimal orientations for both general and specific types of anisotropy. \n\nThe general  problem of determining optimal orientations in anisotropic elasticity has been the subject of several studies in the last two decades, beginning with the work of Seregin and Troitskii \\cite{Seregin81} in the context of  orthotropic solids. They determined the important {\\it coaxiality condition}: a minimum or maximum of strain energy requires that the stress and strain share common principal axes.  The coaxiality condition was subsequently and independently obtained by others: first, by  Rovati and Taliercio \\cite{Rovati91} who considered 3D  elastic materials with orthotropic and cubic symmetries (although their derivation is not restricted to these symmetries but is applicable to general anisotropy), and later by    \nCowin \\cite{Cowin94}.  Cowin  derived the coaxiality condition  independent of material symmetry considerations.  He showed that the commutativity of the stress and strain is a consequence of the stationarity condition of the strain energy with respect to rotations of the moduli.  Vianello \\cite{Vianello96a} provided a more formal derivation of coaxiality in linear elasticity.  He used the  tangent space of the rotation group to show that there are at least two orientations of the moduli that simultaneously make the energy stationary and stress and strain coaxial, a result later generalized to  hyperelasticity \\cite{Vianello96b} (it was subsequently shown that at least three such orientations exist for both linear elasticity \\cite{Sgarra97a} and for hyperelasticity \\cite{Sgarra97b}).  There is a slight distinction between the problems considered by Cowin and by Vianello that is important to note for our purposes \\cite{Cowin97}.  Thus, Cowin \\cite{Cowin94} considered  stress states with fixed principal directions but arbitrary amplitudes, whereas Vianello \\cite{Vianello96a} assumed a specific state of stress.  The former constraint  defines a smaller set of possible elastic moduli for which coaxility can be attained, because it requires that optimal condition be simultaneously satisfied by a family of coaxial stresses.  Not surprisingly, Cowin found that only materials with orthotropic symmetry meet these conditions.   In this paper the stress state is taken as given, in the same spirit as \\cite{Vianello96a} and also of Rovati and Taliercio \\cite{Rovati03}.   While the emphasis here is on three dimensional elasticity, the optimality problem has also been \n addressed within  the context of  two dimensional elasticity \\cite{Pedersen89,Pedersen90,Gea04}.\nCowin and Yang  \\cite{Cowin00} considered a related but more general question of optimality with respect to Kelvin modes, rather than simply the freedom to orient a given material.  \n  A more extensive review of the literature can be found in the recent paper of Rovati and Taliercio \\cite{Rovati03}.  \n\nIt is interesting to note that the coaxiality condition has been derived in a variety of  different ways: for particular symmetries (e.g. orthotropic) \\cite{Seregin81}, using Lagrange multipliers \\cite{Cowin94}, from general analytic considerations \\cite{Vianello96a}, and even using the 6-dimensional eigenvector properties of the elasticity tensor \\cite{Banichuk96}. The derivation of the coaxiality condition  presented here differs from all these previous methods.    Our starting point is a representation of the rotation matrix due to Mehrabadi, Cowin and Jaric \\cite{mcj}.  This formulation also enables derivation of conditions for minima or maxima, in a simpler  and more general form than that obtained by Cowin \\cite{Cowin94}.  Section \\ref{sec2} begins with the problem definition and notation.  The stationarity and min\/max conditions are discussed in Section \\ref{sec3}.  Specific conditions for particular material symmetries are derived in Section \\ref{sec4}, and we conclude in Section \\ref{sec5} by defining the {\\it strain deviation angle}, a concept which could have application in practical circumstances in anisotropic elasticity. \n\n\n\\section{Problem definition and notation}\\label{sec2}\n\n\\subsection{Optimal orientation of anisotropic solids}\n\nConsider a fixed coordinate system $\\{ \\bf{e}_1,\\, \\bf{e}_2,\\, \\bf{e}_3 \\}$ coincident with the principal axes of stress.  The stress tensor is therefore  \n\\beq{eq1}\n\\bosy{\\sigma} = \\sigma_{\\rm I}  \\,\\bf{e}_1 \\otimes  \\bf{e}_1 + \\sigma_{\\rm II} \\, \\bf{e}_2 \\otimes  \\bf{e}_2 + \\sigma_{\\rm III} \\, \\bf{e}_3 \\otimes  \\bf{e}_3, \n\\end{equation}\nwhere $\\sigma_{\\rm I}$, $\\sigma_{\\rm II}$ and $\\sigma_{\\rm III}$ are the principal stresses, in no particular order.  Alternative expressions for the stress include the $3\\times 3$ matrix representation, \n\\beqa{s1}\n \\bosy{\\sigma}    = \n\\left[ \\begin{array}{ccc}\n\\sigma_{\\rm I} & 0 & 0 \\\\\n0 & \\sigma_{\\rm II} &  0 \\\\\n0 & 0 &  \\sigma_{\\rm III}  \n\\end{array} \\right] , \n\\end{eqnarray} \nand indicial\\footnote{Lower case Latin suffices take on the values 1, 2, and 3, and the summation convention on repeated indices is assumed unless noted otherwise.} notation, \n\\beq{t2}\n\\sigma_{ij}=  \\sigma_{\\rm I}\\,  \\delta_{i1} \\delta_{j1} +  \\sigma_{\\rm II}\\,  \\delta_{i2} \\delta_{j2}+  \\sigma_{\\rm III}\\,  \\delta_{i3} \\delta_{j3}. \\quad\n\\end{equation}\nOur goal is to find the orientation or orientations which minimize the energy function for fixed stress\n\\beq{s4}\n{\\cal E}  \\equiv  \\sigma_{ij} \\sigma_{kl} \\, s_{ijkl}\\, . \n\\end{equation} \nHere, $s_{ijkl}$ are the components of the fourth order compliance tensor relative to $\\{ \\bf{e}_1,\\, \\bf{e}_2,\\, \\bf{e}_3 \\}$.  Think of the material as being free to be oriented in  such a way that the energy ${\\cal E}$ depends upon the orientation of the {\\it material}  with respect to the fixed principal axes of the stress. \nThe material moduli for stiffness and compliance are ${\\mathbb C}} \t\t\t\t\\newcommand{\\cS}{{\\mathbb S}^{(0)}$ and $\\cS^{(0)}$ when aligned with the fixed axes. It is not necessary  to  specify at this stage whether or not the moduli possess any symmetry with respect to these axes.  The main point is that  the material is free to orient in arbitrary directions with {\\it oriented} moduli  ${\\mathbb C}} \t\t\t\t\\newcommand{\\cS}{{\\mathbb S}$ and $\\cS$ while the stress  orientation remains  fixed. \n\n\\subsection{Notation and tensor rotation}\n Hooke's law  relating stress $\\sigma_{ij}$ and strain $\\varepsilon_{ij}$ is \n\\beq{00}\n\\sigma_{ij} = c_{ijkl}\\varepsilon_{kl}, \n\\qquad\n\\varepsilon_{ij} = s_{ijkl}\\sigma_{kl}. \n\\end{equation}\nHere $c_{ijkl}$ denote the components of the stiffness tensor, inverse to the compliance: \n$c_{ijkl}s_{klpq} = I_{ijpq}$, where $I_{ijpq} = (\\delta_{ip} \\delta_{jq} +\\delta_{iq}\\delta_{jp})\/2$ is the fourth order identity tensor.   \nThe rotated elasticity components could be expressed in terms of the unrotated components $c^{(0)}_{ijkl}$ and $s^{(0)}_{ijkl}$, using Euler angles, for instance.  The concise Voigt notation is used to represent the elements of the elasticity tensor in the fixed basis.     Thus, the compliance is  \n$\\cS = \\left[ S_{IJ} \\right],\\  I,J=1,2,\\ldots 6$, with $S_{12}= s_{1122}$, $S_{16}= s_{1112}$,  $S_{44}= s_{2323}$, etc.,   \n\\beqa{a1a}\n\\cS  = \\left[ \n\\ba{cccccc}\nS_{11} & S_{12} & S_{13} & \n S_{14} &  S_{15} & S_{16} \n\\\\ & & & & & \\\\\n & S_{22} & S_{23} & \n S_{24} &  S_{25} &  S_{26} \n\\\\ & & & & & \\\\\n & & S_{33} & \n  S_{34} &  S_{35} &  S_{36} \n\\\\ & & & & & \\\\\n &   &   \n & S_{44} & S_{45} & S_{46}\n\\\\ & & & & & \\\\\n   S &Y  &M & & S_{55} & S_{56}\n\\\\ & & & & & \\\\\n&&&& & S_{66}\n\\end{array} \\right] .  \n\\end{eqnarray}\n\nAn alternative representation for the elasticity tensor, closely related to \\rf{a1a}, is the $6\\times 6$ matrix   $\\hbS$ with elements $ \\left[ \\hS_{IJ}\\right]$  defined as \n\\beqa{defs}\n\\hbS = {\\bf T} \\, \\cS\\, {\\bf T},\n\\quad \\mbox{where  }\n{\\bf T} \\equiv \n\\left[ \n\\ba{cc}\n~ {\\bf I}~ & 0  \\\\ & \\\\\n0 & \\sqrt{2} {\\bf I}\n\\end{array} \\right] .\n\\end{eqnarray}\nExplicitly,   \n\\beqa{a1}\n\\hbS  \\equiv \\left[ \n\\ba{cccccc}\nS_{11} & S_{12} & S_{13} & \n 2^{\\frac12} S_{14} & 2^{\\frac12} S_{15} & 2^{\\frac12} S_{16} \n\\\\ & & & & & \\\\\nS_{12} & S_{22} & S_{23} & \n 2^{\\frac12} S_{24} & 2^{\\frac12} S_{25} & 2^{\\frac12} S_{26} \n\\\\ & & & & & \\\\\nS_{13} & S_{23} & S_{33} & \n 2^{\\frac12} S_{34} & 2^{\\frac12} S_{35} & 2^{\\frac12} S_{36} \n\\\\ & & & & & \\\\\n 2^{\\frac12} S_{14} & 2^{\\frac12} S_{24} & 2^{\\frac12} S_{34} \n & 2S_{44} & 2S_{45} & 2S_{46}\n\\\\ & & & & & \\\\\n 2^{\\frac12} S_{15} & 2^{\\frac12} S_{25} & 2^{\\frac12} S_{35} \n & 2S_{45} & 2S_{55} & 2S_{56}\n\\\\ & & & & & \\\\\n 2^{\\frac12} S_{16} & 2^{\\frac12} S_{26} & 2^{\\frac12} S_{36} \n & 2S_{46} & 2S_{56} & 2S_{66}\n\\end{array} \\right] , \n\\end{eqnarray}\nThis representation is useful in taking advantage of the fact that  fourth order elasticity tensors in 3 dimensions are \nequivalent to second order symmetric tensor of 6 dimensions \\cite{c3}.\nSimilar equations follow for ${\\mathbb C}} \t\t\t\t\\newcommand{\\cS}{{\\mathbb S}$ and  $\\widehat{\\bf C}}  \t\\newcommand{\\hbS}{\\widehat{\\bf S} = {\\bf T} \\, {\\mathbb C}} \t\t\t\t\\newcommand{\\cS}{{\\mathbb S}\\, {\\bf T}$.  Define \n\\beqa{eqs2}\n\\widehat{\\bosy{\\sigma}}  \\equiv \\left[ \n\\ba{c}\n\\sigma_{11} \\\\  \\sigma_{22} \\\\ \\sigma_{33} \\\\ \\sqrt{2} \\sigma_{23} \\\\  \\sqrt{2}\\sigma_{31} \\\\ \\sqrt{2}\\sigma_{12}\n \\end{array} \\right] , \n \\qquad\n \\widehat{\\bosy{\\varepsilon}}  \\equiv \\left[ \n\\ba{c}\n\\varepsilon_{11} \\\\  \\varepsilon_{22} \\\\ \\varepsilon_{33} \\\\ \\sqrt{2}\\varepsilon_{23} \\\\  \\sqrt{2}\\varepsilon_{31} \\\\ \\sqrt{2}\\varepsilon_{12}\n \\end{array} \\right] , \n\\end{eqnarray}\nthen the stress strain relations \\rf{00} become \n\\beqa{s3}\n\\widehat{\\bosy{\\sigma}}   = \\widehat{\\bf C}}  \t\\newcommand{\\hbS}{\\widehat{\\bf S} \\widehat{\\bosy{\\varepsilon}} , \n \\qquad\n \\widehat{\\bosy{\\varepsilon}}  = \\hbS \\widehat{\\bosy{\\sigma}}. \n\\end{eqnarray}\nNote that $\\hbS$ and $\\widehat{\\bf C}}  \t\\newcommand{\\hbS}{\\widehat{\\bf S}$ are the matrix inverse of each other; \n$\\hbS \\widehat{\\bf C}}  \t\\newcommand{\\hbS}{\\widehat{\\bf S} = \\widehat{\\bf C}}  \t\\newcommand{\\hbS}{\\widehat{\\bf S} \\hbS = \\hbI$ where $\\hbI=$diag$(1,1,1,1,1,1)$. \n\nThe rotation about ${\\bf n}$, $|{\\bf n} | = 1$  by angle $\\phi$ is defined as ${\\bf Q} ({\\bf n} , \\phi )\\in O(3)$, such that vectors (including the basis vectors) transform as ${\\bf v} \\rightarrow {\\bf v}' = {\\bf Q}{\\bf v}$. By considering small rotations, it may be readily seen that  ${\\bf Q} ({\\bf n} , \\phi )$ can be expressed in terms a skew symmetric tensor ${\\bf P}$ that is linear in $\\bf n$. Thus,  \n\\beq{qsat1}\n\\frac{d {\\bf Q}}{d\\phi} ({\\bf n} , \\phi )  = {\\bf P}\\, {\\bf Q} ({\\bf n} , \\phi ), \\quad \\mbox{where}\\quad P_{ij}({\\bf n})= e_{ijk}n_k\\, ,  \n\\end{equation}\nand hence \n\\beq{qeq}\n{\\bf Q} = e^{\\phi {\\bf P}},  \n\\end{equation}\nNote that    ${\\bf Q}$ possesses alternate well-known expressions\n\\beqa{a1a1} {\\bf Q}  ({\\bf n} , \\phi ) &=& {\\bf I} + \\sin\\phi\\, {\\bf P}+ (1 -  \\cos\\phi) {\\bf P}^2 \\nonumber \\\\\n&=& {\\bf n}\\otimes{\\bf n} + \\cos\\phi \\, ({\\bf I} -{\\bf n}\\otimes{\\bf n} )+\n\\sin\\phi\\, {\\bf P}\\, . \n\\end{eqnarray}\nIn particular for our needs here, the small angle expansion is \n\\beq{qsat2}\n{\\bf Q} ({\\bf n} , \\phi )  = {\\bf I} +  \\phi {\\bf P} + \\mbox{O}(\\phi^2)\\, .  \n\\end{equation}\n\nUnder the change of basis associated with ${\\bf Q} ({\\bf n} , \\phi )$, second order tensors (including stress and strain) transform as $\\bosy{\\sigma} \\rightarrow \\bosy{\\sigma} '$, where\n\\beq{qsat3}\n\\sigma_{ij} ' = Q_{ir}Q_{js} \\, \\sigma_{rs} \n\\quad \\Leftrightarrow \\quad \\sigma_{ij} ' = {\\cal Q}_{ijrs}\\, \\sigma_{rs}. \n\\end{equation}\nThe fourth order ``rotation\" tensor follows from \\rf{qsat3} as \n\\beq{qsat4}\n{\\cal Q}_{ijrs} = \\frac12 \\, \\big( Q_{ir}Q_{js} + Q_{is}Q_{jr}\\big),  \n\\end{equation}\nand eqs. \\rf{qsat1} and \\rf{qsat4} imply\n\\beq{qsat5}\n\\frac{d {\\cal Q}_{ijrs} }{d\\phi} ({\\bf n} , \\phi )  = {\\cal P}_{ijpq}{\\cal Q}_{pqrs}, \n\\quad \\mbox{with}\\quad {\\cal Q}_{ijrs}  ({\\bf n} , 0 ) = I_{ijpq}, \n\\end{equation}\nwhere\n\\beq{qsat6}\n{\\cal P}_{ijpq} = \n\\frac12 \\, \\big( \\delta_{ip}P_{jq}  +\\delta_{iq}P_{jp} +\\delta_{jp}P_{iq} + \\delta_{jq}P_{ip}\n\\big)\\, . \n\\end{equation}\nThe formal solution of \\rf{qsat5}, with meaning that should be clear, is \n\\beq{qsat7}\n{\\bf {\\cal Q}} = e^{\\phi {\\bf {\\cal P}}},   \n\\end{equation}\nand the small angle approximation is\n\\beq{qsat8}\n{\\cal Q}_{ijpq} = I_{ijpq}+\\phi  {\\cal P}_{ijpq} + \\mbox{O}(\\phi^2)\\, . \n\\end{equation}\n\nMehrabadi et al. \\cite{mcj}  derived an elegant expression for ${\\bf {\\cal Q}}$ analogous to the representation  for ${\\bf Q}({\\bf n} , \\phi ) $. The key is the characteristic equation of ${\\bf {\\cal P}}$, (${\\bf {\\cal P}}^5 +5{\\bf {\\cal P}}^3+4{\\bf {\\cal P}}=0$ where ${\\cal P}^2_{ijkl} = {\\cal P}_{ijpq}{\\cal P}_{pqkl}$ etc.) which permits the exponential expression \\rf{qsat7} to be simplified.  The result is most simply presented in terms of the $6\\times 6$ rotation matrix $\\hbQ $  introduced by Mehrabadi et al. \\cite{mcj}, and defined in the same manner as before.  Thus, \n$\\hbQ  = {\\bf T} {\\cal Q} {\\bf T} $  and $\\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q}  = {\\bf T} {\\cal P} {\\bf T} $  where ${\\bf T}$ is defined in \\rf{defs} and ${\\cal Q}$ and ${\\cal P}$ are the $6\\times 6$ Voigt matrices.   Explicitly,  $\\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q}$ is a skew symmetric six dimensional tensor  linear in  $\\bf n$, \n\\beqa{a1b}\n\\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q} ({\\bf n}) =  \\left[ \n\\ba{cccccc}\n0 & 0 & 0 & \n        0 & \\sqrt{2} n_2 & -\\sqrt{2} n_3  \n\\\\ & & & & & \\\\\n0 & 0 & 0 & \n        -\\sqrt{2} n_1 &0 &  \\sqrt{2} n_3\n\\\\ & & & & & \\\\\n0 & 0 & 0 & \n        \\sqrt{2} n_1 &  -\\sqrt{2} n_2 &0\n\\\\ & & & & & \\\\\n0 & \\sqrt{2} n_1 & -\\sqrt{2} n_1   & 0 & n_3 & -n_2\n\\\\ & & & & & \\\\\n-\\sqrt{2} n_2 &0 &  \\sqrt{2} n_2  & -n_3 & 0 & n_1\n\\\\ & & & & & \\\\\n\\sqrt{2} n_3 &  -\\sqrt{2} n_3 &0 &n_2 & -n_1 & 0 \n\\end{array} \\right] . \n\\end{eqnarray}\n $\\hbQ  ({\\bf n} , \\phi )$ is  an orthogonal second order tensor of six \ndimensions, satisfying $\\hbQ\\hbQ^T = \\hbQ^T\\hbQ = \\hbI$.  \nEquation \\rf{qsat7} becomes\n\\beq{they}    \n \\hbQ ({\\bf n} , \\phi ) =  e^{\\phi  \\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q}({\\bf n})} ,\n \\end{equation}\n and  has the explicit expansion  \\cite{mcj}\n\\beq{mc2}\n\\hbQ ({\\bf n} , \\phi )= \\hbI + \n\\sin\\phi\\, \\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q}\n+ (1 -  \\cos\\phi)\\, \\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q}^2 \n+\\frac13 \\sin\\phi (1 -  \\cos\\phi)\\,( \\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q} + \\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q}^3)\n+\\frac16  (1 -  \\cos\\phi)^2\\,( \\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q}^2 + \\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q}^4)\\, . \\quad\n\\end{equation}\nFinally, we note that  fourth order tensors ${\\mathbb C}} \t\t\t\t\\newcommand{\\cS}{{\\mathbb S}  $ transforms as $\\widehat{\\bf C}}  \t\\newcommand{\\hbS}{\\widehat{\\bf S} \\rightarrow \\widehat{\\bf C}}  \t\\newcommand{\\hbS}{\\widehat{\\bf S} ' = \\hbQ\\widehat{\\bf C}}  \t\\newcommand{\\hbS}{\\widehat{\\bf S}\\hbQ^T$.\n\n\\subsection{Orientation function revisited}\n \n\nDenote the matrix of rotation from the fixed to the ``current''  axes as $\\hbQ $.  Thus, \n\\beq{s3b} \n\\widehat{\\bf C}}  \t\\newcommand{\\hbS}{\\widehat{\\bf S}   = \\hbQ \\widehat{\\bf C}}  \t\\newcommand{\\hbS}{\\widehat{\\bf S}^{(0)} \\hbQ^T , \n\\qquad\n\\hbS   = \\hbQ \\hbS^{(0)} \\hbQ^T . \n\\end{equation}\nHence the objective function of   \\rf{s4} for the stress-based energy minimization becomes \n\\beq{s5}\n{\\cal E} = \\widehat{\\bosy{\\sigma}}^T\\,  \\hbS\\,  \\widehat{\\bosy{\\sigma}}\\, . \n\\end{equation}\nThis is the starting point in the next section to derive conditions necessary for a minimum.  It is important to emphasize the initial assumption that the stress is aligned with the fixed axes, \\rf{s1}, or in terms of $\\widehat{\\bosy{\\sigma}}$,     \n\\beqa{s1a}\n\\quad \n\\widehat{\\bosy{\\sigma}}  \\equiv \\left[ \n\\ba{c}\n\\sigma_{\\rm I} \\\\  \\sigma_{\\rm II} \\\\ \\sigma_{\\rm III} \\\\ 0 \\\\ 0 \\\\ 0\n \\end{array} \\right] .\n\\end{eqnarray} \nThis ensures that the energy varies as the material axes are rotated (if the stress were also rotated then the energy would be, trivially,  unchanged). \n\n\\section{Stationarity and min\/max conditions}\\label{sec3}\n\n\\subsection{Angular derivatives of the strain energy}\n\nConsider the energy  $\\cal E$ of \\rf{s5} as a function of the rotation $\\hbQ$.  A  stationary value is obtained if $\\cal E$ is unchanged  with respect to additional small rotations of $\\hbS$.  This requires calculating the first derivative with respect to rotation angle for arbitrary rotation. The second derivative is needed  to distinguish the stationary point as a minimum or maximum, or otherwise. \n\nThe strain energy obtained by arbitrary rotation  of the material about the axis $\\bf n$ is \n\\beq{s6}\n{\\cal E} ( {\\bf n}, \\phi) = \\widehat{\\bosy{\\sigma}}^T\\, \\hbQ({\\bf n} , \\phi ) \\hbS \\hbQ^T ({\\bf n} , \\phi ) \\,  \\widehat{\\bosy{\\sigma}}\\, .   \n\\end{equation}\nThe first derivative can be expressed, \n\\beqa{s8q}\n \\frac{\\partial \\, }{\\partial \\phi }{\\cal E} ( {\\bf n}, \\phi) &= &\n\\widehat{\\bosy{\\sigma}}^T\\, \\big[ \\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q}({\\bf n}) \\hbQ({\\bf n} , \\phi ) \\hbS \\hbQ^T ({\\bf n} , \\phi )  + \\hbQ({\\bf n} , \\phi ) \\hbS^T  \\hbQ^T ({\\bf n} , \\phi )\\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q}^T ({\\bf n}) \\big] \\,  \\widehat{\\bosy{\\sigma}}\n\\nonumber \\\\\n&= & 2\\widehat{\\bosy{\\sigma}}^T\\,  \\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q}({\\bf n}) \\hbQ({\\bf n} , \\phi ) \\hbS \\hbQ^T ({\\bf n} , \\phi )  \\,  \\widehat{\\bosy{\\sigma}}\n\\, .  \n\\end{eqnarray}\nThis follows from eqs. \\rf{qsat1}, \\rf{they} and \\rf{s6}, using the fact that $\\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q}$ and $\\hbQ$ commute.  Similarly, the second derivative follows as \n\\beq{s8qq}\n \\frac{\\partial^2 \\, }{\\partial \\phi^2 }{\\cal E} ( {\\bf n}, \\phi) \n= 2\\widehat{\\bosy{\\sigma}}^T\\, \\big[  \\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q}^2({\\bf n}) \\hbQ({\\bf n} , \\phi ) \\hbS \\hbQ^T ({\\bf n} , \\phi ) + \\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q}({\\bf n}) \\hbQ({\\bf n} , \\phi ) \\hbS \\hbQ^T ({\\bf n} , \\phi ) \\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q}^T ({\\bf n})\n\\big]  \\,  \\widehat{\\bosy{\\sigma}} \n\\, .  \\quad\n\\end{equation}\n\n\n\\subsection{Condition for stationary strain energy}\n \nAssume,  with no loss in generality, that the  stationary orientation is at $\\phi = 0$. \nIf  $\\hbS$ is at a stationary point, then the energy should be unchanged regardless of the axis $\\bf n$, or \n\\beq{s7}\n{\\cal E} \\mbox{ stationary} \\quad \\Leftrightarrow \\quad \n  \\left. \\frac{\\partial \\, }{\\partial \\phi } {\\cal E} ( {\\bf n}, \\phi)\\right|_{\\phi = 0} = 0, \\quad \\forall \\, |{\\bf n} |=1.\n\\end{equation}\nThis becomes, using \\rf{s8q} evaluated at  $\\phi = 0$,  \n\\beq{s9}\n  \\left. \\frac{\\partial \\, }{\\partial \\phi }{\\cal E} ( {\\bf n}, \\phi)\\right|_{\\phi = 0} = \n2\\widehat{\\bosy{\\sigma}}^T\\,   \\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q}({\\bf n}) \\hbS  \\,  \\widehat{\\bosy{\\sigma}}\\, .  \n\\end{equation}\nWe now take advantage of the fact that the stress is aligned with the fixed axes.  Thus, eqs.   \\rf{a1b} and \\rf{s1a} give \n\\beq{r1}\n\\widehat{\\bosy{\\sigma}}^T\\,   \\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q}({\\bf n})\n = \n \\bigg( 0,\\, 0,\\, 0,\\, \\sqrt{2} (\\sigma_{\\rm III} - \\sigma_{\\rm II})n_1 ,\\,  \\sqrt{2} (\\sigma_{\\rm I} - \\sigma_{\\rm III})n_2 ,\\, \n \\sqrt{2} (\\sigma_{\\rm II} - \\sigma_{\\rm I})n_3  \\bigg) . \n \\end{equation}\n Hence, \n\\beqa{r2}\n  \\widehat{\\bosy{\\sigma}}^T\\,   \\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q}({\\bf n}) \\hbS  \\,  \\widehat{\\bosy{\\sigma}}\\, \n   =   \n2  \n\\left[ \\begin{array}{c}\n(\\sigma_{\\rm III} - \\sigma_{\\rm II})n_1 \\\\ \\\\  (\\sigma_{\\rm I} - \\sigma_{\\rm III})n_2 \\\\ \\\\ (\\sigma_{\\rm II} - \\sigma_{\\rm I})n_3\n \\end{array} \\right]^T\\, \n \\left[ \\begin{array}{ccc}\n S_{14} &  S_{24} &  S_{34} \\\\\n && \\\\ \n S_{15} &  S_{25} &  S_{35} \\\\\n && \\\\\n S_{16} &  S_{26} &  S_{36} \n \\end{array} \\right] \n \\left[ \\begin{array}{c}\n \\sigma_{\\rm I} \\\\  \\\\ \n \\sigma_{\\rm II} \\\\  \\\\ \n \\sigma_{\\rm III} \\end{array} \\right] .\n   \\end{eqnarray}\nThis must vanish for arbitrary direction ${\\bf n}$, hence the energy $\\cal E$  is stationary if \n\\beqa{r3}\n   \\left[ \\begin{array}{ccc}\n\\sigma_{\\rm III} - \\sigma_{\\rm II} & 0 & 0 \\\\\n && \\\\\n 0 & \\sigma_{\\rm I} - \\sigma_{\\rm III} & 0 \\\\\n && \\\\\n 0 & 0 & \\sigma_{\\rm II} - \\sigma_{\\rm I} \\end{array} \\right]\n \\left[ \\begin{array}{ccc}\n S_{14} &  S_{24} &  S_{34} \\\\\n && \\\\ \n S_{15} &  S_{25} &  S_{35} \\\\\n && \\\\\n S_{16} &  S_{26} &  S_{36} \n \\end{array} \\right] \n \\left[ \\begin{array}{c}\n \\sigma_{\\rm I} \\\\  \\\\ \n \\sigma_{\\rm II} \\\\  \\\\ \n \\sigma_{\\rm III} \\end{array} \\right]  = \n \\left[ \\begin{array}{c}\n0 \\\\  \\\\ \n0 \\\\  \\\\ \n0 \\end{array} \\right] .\n   \\end{eqnarray}\n\nLet us assume for simplicity that the state of stress is triaxial, so that $\\sigma_{\\rm I},\\, \\sigma_{\\rm II},\\, \\sigma_{\\rm III}$ are distinct.  The left matrix in \\rf{r3} can be removed, implying a linear condition in the stress:\n\\beq{4r}\n{\\bf E} \\, \\big(\\sigma_{\\rm I},\\, \\sigma_{\\rm II},\\, \\sigma_{\\rm III}\\big)^T = \\big(0,\\, 0,\\, 0 \\big)^T, \n\\end{equation}\nwhere ${\\bf E}$ involves moduli (compliances) only\n\\beqa{r4}\n   {\\bf E}    = \n \\left[ \\begin{array}{ccc}\n S_{14} &  S_{24} &  S_{34} \\\\\n && \\\\ \n S_{15} &  S_{25} &  S_{35} \\\\\n && \\\\\n S_{16} &  S_{26} &  S_{36} \n \\end{array} \\right] .\n   \\end{eqnarray}\nThus, the energy function ${\\cal E}$ is stationary if \n $\\big(\\sigma_{\\rm I},\\, \\sigma_{\\rm II},\\, \\sigma_{\\rm III}\\big)$ is a right null vector of the $3\\times 3$ matrix $\\bf E$. \nBased on \\rf{s3} and \\rf{s1}, the condition \\rf{4r}  is equivalent to the requirement that the off-diagonal elements of the strain vanish: \n\\beq{e12}\n\\varepsilon_{23}=\\varepsilon_{31}=\\varepsilon_{12}=0\\, ,  \n\\end{equation}\nor, \n\\beqa{r6}\n{\\cal E} \\mbox{ stationary} \\quad \\Leftrightarrow \\quad \n\\widehat{\\bosy{\\varepsilon}}  = \\left[ \n\\ba{c}\n\\varepsilon_{1} \\\\  \\varepsilon_{2} \\\\ \\varepsilon_{3} \\\\ 0 \\\\ 0 \\\\ 0\n \\end{array} \\right] = \\left[ \n\\ba{c}\n\\varepsilon_{\\rm I} \\\\  \\varepsilon_{\\rm II} \\\\ \\varepsilon_{\\rm III} \\\\ 0 \\\\ 0 \\\\ 0\n \\end{array} \\right] .\n\\end{eqnarray} \nwhere $\\varepsilon_{\\rm I}, \\varepsilon_{\\rm II}\\varepsilon_{\\rm III}$ are the  principal strains.  \nWe have therefore derived the simple but important general result: \n\n\\medskip \n\\noindent\n{\\bf Result 1}: \\textcolor{blue}{The energy  \n$ \\textcolor{blue} {\\cal E}$  is stationary iff the stress and strain are coaxial. }\n\\medskip \n\nEquation \\rf{e12} states that the 3-vector $ (\\sigma_{\\rm I},\\, \\sigma_{\\rm II},\\,\\sigma_{\\rm III})$ is a right null-vector of ${\\bf E}$. \nThis  requires as a {\\it necessary but not sufficient condition} that \n\\beq{v1}\n\\mbox{det} \\, {\\bf E} = 0.  \n\\end{equation}\nConsequences of this condition were explored in detail by Rovati and Taliercio \\cite{Rovati03} for particular material symmetries: cubic, transversely isotropic and tetragonal.  A different approach is taken in Section \\ref{sec4} below, where the strain energy will be minimized directly.  \n\nWhile Result 1 is not new but has been derived by several authors \n\\cite{Seregin81,Rovati91,Cowin94,Vianello96a,Banichuk96,Rovati03}, \nthe present derivation is novel and explicit.  In particular, it allows us to go further and obtain the condition necessary for a minimum or maximum.  This is  explored next. \n\n\\subsection{Condition for an energy  minimum}\n\nThe second derivative of $\\cal E$ at the stationary point follows from \\rf{s8qq} evaluated at $\\phi = 0$, \n\\beq{cm1}\n  \\left. \\frac{\\partial^2 \\, }{\\partial \\phi^2 }{\\cal E}(  {\\bf n}, \\phi)\\right|_{\\phi = 0} = \n2\\widehat{\\bosy{\\sigma}}^T\\, \\big[ \\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q}^2 \\hbS  + \\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q} \\hbS \\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q}^T  \\big] \\,  \\widehat{\\bosy{\\sigma}}\\, .  \n\\end{equation}\nEach term on the right hand side will be examined in turn.  Using \\rf{r1}, it follows that \n\\beqa{cm4}\n  \\widehat{\\bosy{\\sigma}}^T\\, \\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q}^2 \\hbS  \\,  \\widehat{\\bosy{\\sigma}}\\, = 2\\, \n\n   \\left[ \\begin{array}{c}\n \\sigma_{\\rm I} \\\\ \\\\  \\sigma_{\\rm II} \\\\ \\\\ \\sigma_{\\rm III}\n \\end{array} \\right]^T\\, \n \\left[ \n\\ba{cccccc}\n S_{11} &  S_{12} &  S_{13} & \n    S_{14} &    S_{15} &    S_{16} \n\\\\ & & & & & \\\\\n S_{12} &  S_{22} &  S_{23} & \n    S_{24} &    S_{25} &    S_{26} \n\\\\ & & & & & \\\\\n S_{13} &  S_{23} &  S_{33} & \n    S_{34} &    S_{35} &    S_{36} \n\\end{array} \\right]\\, \n\\left[ \n\\ba{c}\n(\\sigma_{\\rm III} - \\sigma_{\\rm I})n_2^2 + (\\sigma_{\\rm II} - \\sigma_{\\rm I})n_3^2\\\\\n\\\\\n(\\sigma_{\\rm I} - \\sigma_{\\rm II})n_3^2 + (\\sigma_{\\rm III} - \\sigma_{\\rm II})n_1^2\\\\\n\\\\\n(\\sigma_{\\rm II} - \\sigma_{\\rm III})n_1^2 + (\\sigma_{\\rm I} - \\sigma_{\\rm III})n_2^2\\\\\n\\\\\n(\\sigma_{\\rm II} +\\sigma_{\\rm III} - 2\\sigma_{\\rm I})n_2n_3 \\\\\n\\\\\n(\\sigma_{\\rm III} +\\sigma_{\\rm I} - 2\\sigma_{\\rm II})n_3n_1 \\\\\n\\\\\n(\\sigma_{\\rm I} +\\sigma_{\\rm II} - 2\\sigma_{\\rm III})n_1n_2 \\\\\n\\end{array} \\right],\\quad \n\\end{eqnarray}\nand\n\\beqa{cm2}\n  \\widehat{\\bosy{\\sigma}}^T\\, \\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q} \\hbS \\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q}^T  \\,  \\widehat{\\bosy{\\sigma}}\\, =  \n   4\\,\n  \\left[ \\begin{array}{c}\n(\\sigma_{\\rm III} - \\sigma_{\\rm II})n_1 \\\\ \\\\  (\\sigma_{\\rm I} - \\sigma_{\\rm III})n_2 \\\\ \\\\ (\\sigma_{\\rm II} - \\sigma_{\\rm I})n_3\n \\end{array} \\right]^T\\, \n    \\left[ \\begin{array}{ccc}\n S_{44} &  S_{45} &  S_{46} \\\\\n && \\\\ \n S_{45} &  S_{55} &  S_{56} \\\\\n && \\\\\n S_{46} &  S_{56} &  S_{66} \n \\end{array} \\right]\\, \n \\left[ \\begin{array}{c}\n(\\sigma_{\\rm III} - \\sigma_{\\rm II})n_1 \\\\ \\\\  (\\sigma_{\\rm I} - \\sigma_{\\rm III})n_2 \\\\ \\\\ (\\sigma_{\\rm II} - \\sigma_{\\rm I})n_3\n \\end{array} \\right]\\, . \n\\end{eqnarray}\nThus, \n\\beq{cm6}\n   \\left. \\frac{\\partial^2 \\, }{\\partial \\phi^2 }{\\cal E} (  {\\bf n}, \\phi)\\right|_{\\phi = 0} =   4\\, {\\bf n}^T {\\bf F} {\\bf n} , \n\\end{equation}\nwhere the symmetric $3\\times 3$ matrix ${\\bf F}$ has elements  \n\\beqa{G}\nF_{11} &=& (\\sigma_{\\rm III} - \\sigma_{\\rm II})\\, \\big[ 2 S_{44} (\\sigma_{\\rm III} - \\sigma_{\\rm II}) +\n\t(S_{12}-S_{13})\\sigma_{\\rm I} + (S_{22}-S_{23})\\sigma_{\\rm II} + (S_{32}-S_{33})\\sigma_{\\rm III} \\big], \n\t\\nonumber \\\\\nF_{22} &=& (\\sigma_{\\rm I} - \\sigma_{\\rm III})\\, \\big[ 2 S_{55} (\\sigma_{\\rm I} - \\sigma_{\\rm III}) +\n\t(S_{13}-S_{11})\\sigma_{\\rm I} + (S_{23}-S_{21})\\sigma_{\\rm II} + (S_{33}-S_{31})\\sigma_{\\rm III} \\big], \n\t\\nonumber \\\\\nF_{33} &=& (\\sigma_{\\rm II} - \\sigma_{\\rm I})\\, \\big[ 2 S_{66} (\\sigma_{\\rm II} - \\sigma_{\\rm I}) +\n\t(S_{11}-S_{12})\\sigma_{\\rm I} + (S_{21}-S_{22})\\sigma_{\\rm II} + (S_{31}-S_{32})\\sigma_{\\rm III} \\big], \n\\nonumber \t \\\\\nF_{23} &=& 2 S_{56} (\\sigma_{\\rm I} - \\sigma_{\\rm III}) (\\sigma_{\\rm II} - \\sigma_{\\rm I}) + \\frac12 \\,\n(\\sigma_{\\rm II} +\\sigma_{\\rm III}- 2\\sigma_{\\rm I})\\, \\big(\n\tS_{14}\\sigma_{\\rm I} +S_{24}\\sigma_{\\rm II} +S_{34}\\sigma_{\\rm III}\\big) ,\n\\\\\nF_{31} &=& 2 S_{46} (\\sigma_{\\rm II} - \\sigma_{\\rm I}) (\\sigma_{\\rm III} - \\sigma_{\\rm II})  + \\frac12 \\,\n(\\sigma_{\\rm III} +\\sigma_{\\rm I}- 2\\sigma_{\\rm II})\\, \\big(\n\tS_{15}\\sigma_{\\rm I} +S_{25}\\sigma_{\\rm II} +S_{35}\\sigma_{\\rm III}\\big),\n\\nonumber \\\\\nF_{12} &=& 2 S_{45} (\\sigma_{\\rm III} - \\sigma_{\\rm II}) (\\sigma_{\\rm I} - \\sigma_{\\rm III}) +\\frac12 \\,\n(\\sigma_{\\rm I} +\\sigma_{\\rm II}- 2\\sigma_{\\rm III})\\, \\big(\n\tS_{16}\\sigma_{\\rm I} +S_{26}\\sigma_{\\rm II} +S_{36}\\sigma_{\\rm III}\\big)\\, . \\nonumber\n\\end{eqnarray}\n \nThe second derivative \\rf{cm6} must be positive for all directions ${\\bf n}$ at an orientation where $\\cal E$ is a local minimum.  Noting that \n\\beqa{G4}\nF_{23} &=& 2 S_{56} (\\sigma_{\\rm I} - \\sigma_{\\rm III}) (\\sigma_{\\rm II} - \\sigma_{\\rm I}) + \\frac12 \\,\n(\\sigma_{\\rm II} +\\sigma_{\\rm III}- 2\\sigma_{\\rm I})\\, \\varepsilon_{23} ,\n\\nonumber \\\\\nF_{31} &=& 2 S_{46} (\\sigma_{\\rm II} - \\sigma_{\\rm I}) (\\sigma_{\\rm III} - \\sigma_{\\rm II})  +\\frac12 \\,\n(\\sigma_{\\rm III} +\\sigma_{\\rm I}- 2\\sigma_{\\rm II})\\, \\varepsilon_{31},\n \\\\\nF_{12} &=& 2 S_{45} (\\sigma_{\\rm III} - \\sigma_{\\rm II}) (\\sigma_{\\rm I} - \\sigma_{\\rm III}) +\\frac12 \\,\n(\\sigma_{\\rm I} +\\sigma_{\\rm II}- 2\\sigma_{\\rm III})\\, \\varepsilon_{12} , \\nonumber\n\\end{eqnarray}\nit follows that at a stationary point the off-diagonal elements of $\\bf F$ become, \n\\beqa{G5}\nF_{23} &=& 2 S_{56} (\\sigma_{\\rm I} - \\sigma_{\\rm III}) (\\sigma_{\\rm II} - \\sigma_{\\rm I})  ,\n\\nonumber \\\\\nF_{31} &=& 2 S_{46} (\\sigma_{\\rm II} - \\sigma_{\\rm I}) (\\sigma_{\\rm III} - \\sigma_{\\rm II})  ,\n \\\\\nF_{12} &=& 2 S_{45} (\\sigma_{\\rm III} - \\sigma_{\\rm II}) (\\sigma_{\\rm I} - \\sigma_{\\rm III}) . \\nonumber\n\\end{eqnarray}\n Equivalently, by pre- and post-multiplication of $\\frac 12 \\bf F$ by the diagonal matrix\n diag$[(\\sigma_{\\rm III} - \\sigma_{\\rm II})^{-1},\\, (\\sigma_{\\rm I} - \\sigma_{\\rm III})^{-1},\\, (\\sigma_{\\rm II} - \\sigma_{\\rm I})^{-1}]$, it follows that $\\bf G$ must be  positive definite, where \n \\beqa{G6}\nG_{11} &=&  S_{44}+ \\frac12 (\\sigma_{\\rm III} - \\sigma_{\\rm II})^{-1}\\, \\big[  \n\t(S_{12}-S_{13})\\sigma_{\\rm I} + (S_{22}-S_{23})\\sigma_{\\rm II} + (S_{32}-S_{33})\\sigma_{\\rm III} \\big], \n\t\\nonumber \\\\\nG_{22} &=&  S_{55} + \\frac12 (\\sigma_{\\rm I} - \\sigma_{\\rm III})^{-1}\\, \\big[   \n\t(S_{13}-S_{11})\\sigma_{\\rm I} + (S_{23}-S_{21})\\sigma_{\\rm II} + (S_{33}-S_{31})\\sigma_{\\rm III} \\big], \n\t\\nonumber \\\\\nG_{33} &=&  S_{66}+ \\frac12 (\\sigma_{\\rm II} - \\sigma_{\\rm I})^{-1}\\, \\big[   \n\t(S_{11}-S_{12})\\sigma_{\\rm I} + (S_{21}-S_{22})\\sigma_{\\rm II} + (S_{31}-S_{32})\\sigma_{\\rm III} \\big], \n\\nonumber \t \\\\\nG_{23} &=&  S_{56}  \\,  ,\n\\\\\nG_{31} &=&  S_{46} \\,  ,\n\\nonumber \\\\\nG_{12} &=&  S_{45} \\, . \\nonumber\n\\end{eqnarray}\nNote that $G_{11} =  S_{44}- \\frac12 (\\sigma_{\\rm III} - \\sigma_{\\rm II})^{-1}\\, (\\varepsilon_3 - \\varepsilon_2)$, etc. or, using \\rf{r6},  \n\\beqa{G7}\n {\\bf G} = \\left[ \\begin{array}{ccc}\n   S_{44}- \\frac12 \\big( \\frac{\\varepsilon_{\\rm III} - \\varepsilon_{\\rm II} }{\\sigma_{\\rm III} - \\sigma_{\\rm II}}\\big)  &   S_{45} &   S_{46} \\\\\n && \\\\ \n   S_{45} &  S_{55} -\\frac12 \\big( \\frac{\\varepsilon_{\\rm I} - \\varepsilon_{\\rm III} }{\\sigma_{\\rm I} - \\sigma_{\\rm III}}\\big) &   S_{56}  \\\\\n && \\\\ \n   S_{46}&    S_{56} &   S_{66} - \\frac12 \\big(\\frac{\\varepsilon_{\\rm II} - \\varepsilon_{\\rm I} }{\\sigma_{\\rm II} - \\sigma_{\\rm I}} \\big)\n  \\end{array} \\right]\\, . \n  \\end{eqnarray}\nIn summary, \n\\beqa{cmsum}\n \\left. \\frac{\\partial^2 \\, }{\\partial \\phi^2 }{\\cal E} (  {\\bf n}, \\phi)\\right|_{\\phi = 0}\n =  \n   8\\,\n  \\left[ \\begin{array}{c}\n(\\sigma_{\\rm III} - \\sigma_{\\rm II})n_1 \\\\ \\\\  (\\sigma_{\\rm I} - \\sigma_{\\rm III})n_2 \\\\ \\\\ (\\sigma_{\\rm II} - \\sigma_{\\rm I})n_3\n \\end{array} \\right]^T\\, \n    {\\bf G}\\, \n \\left[ \\begin{array}{c}\n(\\sigma_{\\rm III} - \\sigma_{\\rm II})n_1 \\\\ \\\\  (\\sigma_{\\rm I} - \\sigma_{\\rm III})n_2 \\\\ \\\\ (\\sigma_{\\rm II} - \\sigma_{\\rm I})n_3\n \\end{array} \\right]\\, . \n\\end{eqnarray}\nThis must hold for arbitrary $\\bf n$, $|{\\bf n}|=1$, and  again assuming that the principal stresses are distinct, it follows that  $\\bf G$ must be positive definite.  Combined with Result 1 for the existence of a stationary point, this gives  \n\n\\medskip \n\\noindent\n{\\bf Result 2}:  \\textcolor{blue}{The energy\n${\\cal E}$  is a local minimum if the stress and strain are coaxial and the symmetric matrix ${\\bf G}$ of \\rf{G7} is positive definite.  }  \n \n This can be rewritten (with obvious notation)\n\\beqa{G7ex}\n  \\left[ \\begin{array}{ccc}\n  S_{44} &  S_{45} & S_{46} \\\\\n && \\\\ \nS_{45} &  S_{55}  &  S_{56}  \\\\\n && \\\\ \n  S_{46}&    S_{56} &   S_{66} \n  \\end{array} \\right]\\, > \\, \n  \\left[ \\begin{array}{ccc}\n   \\frac12 \\big( \\frac{\\varepsilon_{\\rm III} - \\varepsilon_{\\rm II} }{\\sigma_{\\rm III} - \\sigma_{\\rm II}}\\big)  &   0 &   0 \\\\\n && \\\\ \n   0 &  \\frac12 \\big( \\frac{\\varepsilon_{\\rm I} - \\varepsilon_{\\rm III} }{\\sigma_{\\rm I} - \\sigma_{\\rm III}}\\big) &   0  \\\\\n && \\\\ \n   0&    0 &   \\frac12 \\big(\\frac{\\varepsilon_{\\rm II} - \\varepsilon_{\\rm I} }{\\sigma_{\\rm II} - \\sigma_{\\rm I}} \\big)\n  \\end{array} \\right]  \\, . \n  \\end{eqnarray}\n The left matrix is positive definite because of the positive definite properties of the moduli.  The minimum condition therefore requires that this latter matrix be greater than\\footnote{The matrix $\\bf U$ is greater than the matrix $\\bf V$ if \n${\\bf r}^T {\\bf U} {\\bf r} > {\\bf r}^T {\\bf V} {\\bf r}$ for all nonzero  ${\\bf r} \\in R^3$.}  the right hand diagonal matrix defined by the principal stresses and strains.  The requirement that the full matrix  is positive definite can be relaxed if the stationarity is restricted in  orientation axis $\\bf n$.  Thus, only the single scalar quantity ${\\bf n}^T{\\bf G}{\\bf n}$ needs to be considered in the important special case of rotation about a single axis.  This situation is examined in detail in the Appendix. \n\n\n\n\n\\section{Optimal orientation of particular material symmetries}\\label{sec4}\n\n\n\\subsection{Partition of the energy}\nBefore considering specific material symmetries, several general results are presented which help focus attention on the anisotropic part of the strain energy.  Separate  contributions to the energy function ${\\cal E}$ of \\rf{s4} from  isotropic and anisotropic parts of the elastic moduli may be \ndistinguished as follows, \n\\beqa{fa}\n{\\cal E}&=& {\\cal E}^{\\rm (is)} + {\\cal E}^{\\rm (an)} \n\\nonumber \\\\\n &=&  \\sigma_{ij} \\sigma_{kl} \\, s^{\\rm (is)}_{ijkl} +\\sigma_{ij} \\sigma_{kl} \\, s^{\\rm (an)}_{ijkl}.  \n\\end{eqnarray}\nThe moduli are partitioned into isotropic and anisotropic parts \n\\beq{wr}\ns_{ijkl} =  s^{\\rm (is)}_{ijkl} + s^{\\rm (an)}_{ijkl},\n\\qquad\nc_{ijkl} =  c^{\\rm (is)}_{ijkl} + c^{\\rm (an)}_{ijkl},  \n\\end{equation}\nwith the  isotropic moduli defined by \n\\beq{wr2}\ns^{\\rm (is)}_{ijkl} = \\frac{1}{3\\kappa_s} \\, J_{ijkl} + \\frac{1}{2\\mu_s} \\, K_{ijkl},\n\\qquad\nc^{\\rm (is)}_{ijkl} = 3\\kappa_c \\, J_{ijkl} + 2\\mu_c \\, K_{ijkl}.  \n\\end{equation}\nHere,   \n\\beq{wr3}\n J_{ijkl}  = \\frac{1}{3}\\, \\delta_{ij} \\delta_{kl}, \n\\qquad\nK_{ijkl}= I_{ijkl}- J_{ijkl} \\, , \n\\end{equation}\nand $I_{ijkl}$ are the elements of the fourth order identity. \nThe effective ``bulk\" and ``shear\" moduli $\\kappa_s,\\, \\mu_s$ and $\\kappa_c,\\, \\mu_c$  are obtained as \n\\beqa{km}\n && \\frac{1}{\\kappa_s} =  3 s_{ijkl}J_{ijkl} = \nS_{11}+S_{22}+S_{33}+2S_{12}+2S_{13}+2S_{23}, \n\\\\\n&& \\frac{15}{4 \\mu_s} =  \\frac{3}{2}s_{ijkl}K_{ijkl} = S_{11}+S_{22}+S_{33} - S_{12}-S_{23}-S_{31} +3S_{44}+3S_{55}+3S_{66} ,\n\\qquad \n\\\\\n&& 9 \\kappa_c =  3 c_{ijkl}J_{ijkl} = \nC_{11}+C_{22}+C_{33}+2C_{12}+2C_{13}+2C_{23}, \n\\\\\n&& 15 \\mu_c =  \\frac{3}{2}c_{ijkl}K_{ijkl} = C_{11}+C_{22}+C_{33} - C_{12}-C_{23}-C_{31} +3C_{44}+3C_{55}+3C_{66} .\n\\qquad \n\\end{eqnarray}\nNote that in general $\\kappa_c \\ne \\kappa_s$ and $\\mu_c\\ne \\mu_s$.  The anisotropic parts of the moduli in \\rf{wr} are defined as the remainder after subtracting the isotropic parts, \n$s^{\\rm (an)}_{ijkl} = s_{ijkl} - s^{\\rm (is)}_{ijkl}$, etc. \n\nThe energy associated with the isotropic part of the moduli becomes \n\\beq{fis}\n{\\cal E}^{\\rm (is)} = \\frac{1}{\\kappa_s}\\overline{\\sigma}^2 + \\frac{1}{2\\mu_s} \\sigma_{ij}'\\sigma_{ij}',\n\\end{equation}\nwhere $\\overline{\\sigma}$ and ${\\bosy{\\sigma}}'$ are the hydrostatic and deviatoric stress, respectively, \n\\beq{fis2}\n\\overline{\\sigma}=  \\frac13 \\sigma_{kk}, \\qquad \\sigma_{ij}' = \\sigma_{ij} - \\overline{\\sigma}\\delta_{ij}.\n\\end{equation}\nThese may be written explicitly in terms of the principal stresses, from \\rf{s1}, as  \n\\beqa{sdev}\n\\overline{\\sigma} &=&  \\frac13\\, \\big( \\sigma_{\\rm I}+\\sigma_{\\rm II}+\\sigma_{\\rm III}\\big), \\\\\n \\bosy{\\sigma}'    &=& \\frac13\\, \n\\left[ \\begin{array}{ccc}\n2 \\sigma_{\\rm I}- \\sigma_{\\rm II}- \\sigma_{\\rm III} & 0 & 0 \\\\ && \\\\\n0 & 2 \\sigma_{\\rm II}- \\sigma_{\\rm III}- \\sigma_{\\rm I} &  0 \\\\ && \\\\\n0 & 0 &  2 \\sigma_{\\rm III}- \\sigma_{\\rm I}- \\sigma_{\\rm II}  \n\\end{array} \\right] \n\\equiv \\left[ \\begin{array}{ccc}\n\\sigma_{\\rm I} ' & 0 & 0 \\\\ && \\\\\n0 &  \\sigma_{\\rm II}' &  0 \\\\ && \\\\\n0 & 0 &   \\sigma_{\\rm III}'  \n\\end{array} \\right] .\\qquad \n\\end{eqnarray} \nThe energy associated with  the anisotropic part of the moduli is\n\\beq{fan}\n{\\cal E}^{\\rm (an)} = \\overline{\\sigma}^2 s^{\\rm (an)}_{jjkk} + 2\\overline{\\sigma} \\sigma_{ij}' \\, s^{\\rm (an)}_{ijkk} + \n\\sigma_{ij}' \\sigma_{kl}' \\, s^{\\rm (an)}_{ijkl} .\n\\end{equation}\nBy definition, the scalar quantity $s^{\\rm (an)}_{jjkk}$ is zero, and accordingly, the anisotropic energy simplifies to \n\\beq{fq}\n{\\cal E}^{\\rm (an)} =2\\overline{\\sigma} \\sigma_{ij}' \\, s^{\\rm (an)}_{ijkk} + \n\\sigma_{ij}' \\sigma_{kl}' \\, s^{\\rm (an)}_{ijkl} .\n\\end{equation}\n\nIt may be shown that the $3\\times 3$ matrices $\\bf E$ and $\\bf G$ of \\rf{r4} and \\rf{G7} vanish  for isotropic materials.  In general, they depend upon the anisotropic part of the material moduli. \n\n\n\n\\subsection{Materials with cubic symmetry}  \n\nIn the fixed coordinate system of the principal stress axes, the  elastic compliance for  a material with cubic symmetry is \n\\beqa{cub1}\n\\cS^{(0)}  = \\left[ \n\\ba{cccccc}\nS_{11}^{(0)} & S_{12}^{(0)} & S_{12}^{(0)} & \n 0&0&0 \n\\\\ & & & & & \\\\\n & S_{11}^{(0)} & S_{12}^{(0)} & \n 0&0&0\n\\\\ & & & & & \\\\\n & & S_{11}^{(0)} & \n 0&0&0\n\\\\ & & & & & \\\\\n &   &   \n & S_{44}^{(0)} & 0 & 0\n\\\\ & & & & & \\\\\n   S &Y  &M & & S_{44}^{(0)}&  0 \n\\\\ & & & & & \\\\\n&&&&& S_{44}^{(0)}\n\\end{array} \\right] . \n\\end{eqnarray}\nThere are three independent  moduli, $\\kappa$, $\\mu_1$ and $\\mu_2$, where\n\\beq{cub2}\n\\frac{1}{3\\kappa} =   S_{11}^{(0)}+2  S_{12}^{(0)},\\quad \n\\frac{1}{2\\mu_1} = 2S_{44}^{(0)},\\quad\n\\frac{1}{2\\mu_2} =   S_{11}^{(0)}-  S_{12}^{(0)}\\, . \n\\end{equation} \nThe associated fourth order tensors  can be expressed succinctly using the irreducible tensor  notation of Walpole \\cite{c1}, as\n\\beq{wr5}\nc^{(0)}_{ijkl} = 3\\kappa \\, J_{ijkl} + 2\\mu_1 \\, L^{(0)}_{ijkl}+ 2\\mu_2 \\, M^{(0)}_{ijkl}, \n\\qquad\ns^{(0)}_{ijkl} = \\frac{1}{3\\kappa} \\, J_{ijkl} + \\frac{1}{2\\mu_1} \\, L^{(0)}_{ijkl}+ \\frac{1}{2\\mu_2} \\, M^{(0)}_{ijkl}. \n\\end{equation}\nHere $L^{(0)}_{ijkl}= I_{ijkl}- D^{(0)}_{ijkl}$,  $M^{(0)}_{ijkl}= D^{(0)}_{ijkl}- J_{ijkl}$, and \n\\beq{Ddef}\nD^{(0)}_{ijkl} = \\delta_{i1} \\delta_{j1}\\delta_{k1} \\delta_{l1}\n+\\delta_{i2} \\delta_{j2}\\delta_{k2} \\delta_{l2} +\\delta_{i3} \\delta_{j3}\\delta_{k3} \\delta_{l3}\\, . \n\\end{equation}\nThis format makes it relatively straightforward to determine the effective isotropic moduli, \n\\beq{wr6}\n\\kappa_c = \\kappa_s = \\kappa, \\qquad 5\\mu_c = 3\\mu_1+2\\mu_2, \\qquad\n\\frac{5}{\\mu_s} = \\frac{3}{\\mu_1} + \\frac{2}{\\mu_2}\\, . \n \\end{equation}\nThus, \n\\beq{w8}\ns^{(an,0)}_{ijkl} = \\frac{1}{10}\\bigg(\\frac{1}{\\mu_1}- \\frac{1}{\\mu_2}\\bigg)\\, \\big(\n5J_{ijkl} + 2K_{ijkl} - 5D^{(0)}_{ijkl} \\big). \n\\end{equation}\n\nThe anisotropic  part of the energy, \\rf{fq}, depends only upon the deviatoric part of the stress,  \n\\beq{fq2}\n{\\cal E}^{\\rm (an)} =\\sigma_{ij}' \\sigma_{kl}' \\, s^{\\rm (an)}_{ijkl} .\n\\end{equation}\nThe reason is that the second order tensor $s^{(an,0)}_{ijkk}$ is identically zero for cubic symmetry, and hence  remains zero in the rotated material axes: $s^{\\rm (an)}_{ijkk} = 0$.  The first term in \\rf{fq} therefore vanishes, leaving the simpler expression \\rf{fq2}.  The isotropic tensors $J_{ijkl}$ and $K_{ijkl}$ are  unchanged under rotation, and consequently, from \\rf{fis}, \\rf{w8} and \\rf{fq2}, \n\\beq{fq3}\n{\\cal E} = \\frac{1}{\\kappa}\\overline{\\sigma}^2 + \\frac{1}{2\\mu_1} \\sigma_{ij}'\\sigma_{ij}' + {\\cal E}^{\\rm (ex)},\n\\qquad\n{\\cal E}^{\\rm (ex)} = \\frac{1}{2}\\bigg(\\frac{1}{\\mu_2}- \\frac{1}{\\mu_1}\\bigg)\\,\n\\sigma_{ij}' \\sigma_{kl}' \\, D_{ijkl} ,  \n\\end{equation}\nwhere $D_{ijkl}$ is the rotated version of $D^{(0)}_{ijkl}$.  In order to avoid ambiguity, let $\\sigma_{(rot)kl}$ denote the stress in the {\\it rotated} coordinate system, then it follows that\n\\beq{fq4}\n\\sigma_{ij}' \\sigma_{kl}' \\, D_{ijkl} = \\sigma_{(rot)11}^{'2} + \\sigma_{(rot)22}^{'2} +\\sigma_{(rot)33}^{'2}  \\ . \n\\end{equation}\nThe scalar second invariant of the deviatoric stress is \n\\beqa{in}\n\\sigma_{ij}'\\sigma_{ij}' &=& {\\sigma_{\\rm I}'}^2 + {\\sigma_{\\rm II}'}^2 + {\\sigma_{\\rm III}'}^2\n\\nonumber \\\\\n&=& \\sigma_{(rot)11}^{'2} + \\sigma_{(rot)22}^{'2} +\\sigma_{(rot)33}^{'2}\n+ 2\\sigma_{(rot)23}^{'2} + 2\\sigma_{(rot)31}^{'2} +2\\sigma_{(rot)12}^{'2} .  \n\\end{eqnarray}\nTherefore, the function ${\\cal E}^{\\rm (ex)}$ of \\rf{fq3} is stationary when either the right member of \\rf{fq4} or \n\\beq{fq6}\n\\Gamma = \\sigma_{(rot)23}^{'2} + \\sigma_{(rot)31}^{'2} +\\sigma_{(rot)12}^{'2},\n\\end{equation}\nare stationary.  Furthermore, \n\\beq{fq7}\n{\\cal E}^{\\rm (ex)} ={\\cal E}^{\\rm (ex,0)}  +  \\bigg(\\frac{1}{\\mu_1}- \\frac{1}{\\mu_2}\\bigg)\\,\n\\big[ \\sigma_{(rot)23}^{'2} + \\sigma_{(rot)31}^{'2} +\\sigma_{(rot)12}^{'2}\\big] \\, , \n\\end{equation}\nwhere ${\\cal E}^{\\rm (ex,0)}$ is the unrotated or fixed value, which follows from \\rf{sdev} as\n\\beq{q9}\n{\\cal E}^{\\rm (ex,0)} = \\frac{1}{2}\\bigg(\\frac{1}{\\mu_2}- \\frac{1}{\\mu_1}\\bigg)\\,  \n\\big( {\\sigma_{\\rm I}'}^2 + {\\sigma_{\\rm II}'}^2 + {\\sigma_{\\rm III}'}^2 \\big)\\, . \n\\end{equation}\n Hence, \n\\beq{wq3}\n \\mu_1 > \\mu_2 \\quad \\Rightarrow \\quad {\\cal E}^{\\rm (ex)}\\le  {\\cal E}^{\\rm (ex,0)}, \n \\end{equation}\nwith equality when the material and stress axes are aligned.  Thus, \\textcolor{blue}{a local minimum that is not aligned with the stress axes occurs iff $\\mu_1 > \\mu_2$ and occurs when $\\Gamma$ of \\rf{fq6} achieves a local maximum}.   It will be shown below that the maximum value is $\\frac12 \\sigma_{ij}'\\sigma_{ij}'$, or equivalently, that ${\\cal E}^{\\rm (ex)}$ is zero at the stationary point. \n\n\nAs  the material axes are rotated to transform $s^{(an,0)}_{ijkl} \\rightarrow s^{\\rm (an)}_{ijkl}$, the only part that contributes to the anisotropic strain energy  is $D_{ijkl}^{(0)} \\rightarrow D_{ijkl}$.  \nConditions for obtaining the stationary value of strain energy are next derived by focusing on the dependence upon $D_{ijkl}$.  The $6\\times 6$   matrix associated with the unrotated tensor $D^{(0)}_{ijkl}$ is\n\\beqa{D1}\n\\widehat{\\bf D}^{(0)}    = \n\\left[ \\begin{array}{ccc}\nI_{3\\times 3} & &0_{3\\times 3}  \\\\ && \\\\ \n0_{3\\times 3} && 0_{3\\times 3} \n\\end{array} \\right] .\n\\end{eqnarray} \nIt is convenient to split $ \\hbQ $ as follows into $3\\times 3$ matrices, \n\\beqa{D2}\n \\hbQ   = \n\\left[ \\begin{array}{ccc}\n\\hbQ_1 & &\\hbQ_2  \\\\ && \\\\ \n\\hbQ_3 && \\hbQ_4 \n\\end{array} \\right] ,   \n\\end{eqnarray} \nso that the rotated tensor    \n$\\widehat{{\\bf D}}  = \\hbQ \\widehat{{\\bf D}}^{(0)} \\hbQ^T$,  follows from \\rf{D1} and \\rf{D2} as \n\\beqa{D2a}\n\\widehat{{\\bf D}}  = \n\\left[ \\begin{array}{ccc}\n\\hbQ_1 \\hbQ_1^T & &\\hbQ_1 \\hbQ_3^T  \\\\ && \\\\ \n\\hbQ_3 \\hbQ_1^T && \\hbQ_3 \\hbQ_3^T \n\\end{array} \\right] .  \n\\end{eqnarray} \nThe term associated with the rotated energy  follows from \\rf{s1a} and \\rf{D2a} as  \n\\beq{dq2}\n\\sigma_{ij}' \\sigma_{kl}' \\, D_{ijkl} =   (\\sigma_{\\rm I}',\\, \\sigma_{\\rm II}',\\,\\sigma_{\\rm III}') \\, \\hbQ_1\\hbQ_1^T \\, (\\sigma_{\\rm I}',\\, \\sigma_{\\rm II}',\\,\\sigma_{\\rm III}')^T\\, .   \n\\end{equation}\n  Thus,  any  stress that is a null vector of $\\hbQ_1^T$\nalso  yields the minimum or maximum value for ${\\cal E}^{\\rm (ex)}$ of \\rf{fq3}, i.e. zero.  \nThis suggests $\\hbQ_1$ as the focus of attention, and implies    \n the important result: \\textcolor{blue}{Every stress state which is a null vector of $\\hbQ_1^T$ corresponds to a global minimum (maximum) of ${\\cal E}$ if $\\mu_1 >\\mu_2$ ($\\mu_2 >\\mu_1$)}.  We therefore search for null vectors of $\\hbQ_1^T$. \n\nBefore deriving two alternative methods to find null vectors of $\\hbQ_1^T$ in the next two subsections, note that the quantity \\rf{fq4} vanishes at a stationary orientation, and hence $\\sigma_{(rot)11} = \\sigma_{(rot)22}  = \\sigma_{(rot)22}$.  Thus, the  stress in each of the three rotated axial directions is equal, a result previously obtained by Rovati and Taliercio \\cite{Rovati91,Rovati03}.   Furthermore, at the stationary point it may be easily shown that the following identities hold:\n\\beq{idh}\n{\\cal D}\\bosy{\\sigma} = \\overline{\\sigma}\\, {\\bf I} , \\quad\n{\\cal L}\\bosy{\\sigma} = \\bosy{\\sigma}', \\quad \n{\\cal M}\\bosy{\\sigma} = 0\\, ,\n\\end{equation}\nwhere ${\\cal D}, {\\cal L}, {\\cal M}$ are the (rotated) tensors with components $D_{ijkl}, L_{ijkl}, M_{ijkl}$, respectively.  Hence, the strain at optimal orientation is simply\n\\beq{idh2}\n\\bosy{\\varepsilon} = \\frac{1 }{3\\kappa} \\, \\overline{\\sigma} \\, {\\bf I}   \\quad\n+ \\frac{ 1 }{2\\mu_1}\\, \\bosy{\\sigma}'\\, \\qquad (\\mbox{optimal orientation only}).\n\\end{equation}\nThis is clearly coaxial with the stress, which follows from the commutation property of coaxial second order tensors \n($\\bosy{\\sigma} \\bosy{\\varepsilon} - \\bosy{\\varepsilon}  \\bosy{\\sigma}=0$ in this case). \n\nIt is also worth remarking that we do not seek null vectors of  the matrix  $\\bf E$, although this approach is feasible and has been used to  advantage by Rovati and Taliercio \\cite{Rovati03}. Some comments on $\\bf E$ are in order though.  The $3\\times 3$ matrix follows from  eqs. \\rf{D1} through \\rf{D2a} as \n\\beq{D3}\n{\\bf E} = \\hbQ_3\\hbQ_1^T\\, ,  \n\\end{equation}\nand the condition \\rf{v1}  is satisfied if either  $\\mbox{det} \\, \\hbQ_1$ or $\\mbox{det} \\, \\hbQ_3$ vanish.  These can be made more explicit in terms of the elements of the rotation matrix.  \nLet \n\\beqa{qq}\n{\\bf Q} = \n\\left[ \\begin{array}{ccc}\nq_{11} & q_{12} & q_{13} \\\\\nq_{21} & q_{22} & q_{23} \\\\\nq_{31} & q_{32} & q_{33} \n\\end{array} \\right] ,  \n\\end{eqnarray}\nthen using the   the explicit representation of the $6\\times 6$ rotation matrix from Auld  \\cite{AuldI} or otherwise, the condition  \\rf{v1} implies \n\\beqa{D5}\n\\mbox{det} \\, \\hbQ_1 = \\left| \\begin{array}{cccccc}\nq_{11}^2 & q_{12}^2 & q_{13}^2    \\\\\n&& \\\\\nq_{21}^2 & q_{22}^2 & q_{23}^2    \\\\\n&& \\\\\nq_{31}^2 & q_{32}^2 & q_{33}^2  \n\\end{array} \\right| = 0,\n\\quad \n\\mbox{or}\\quad    \\mbox{det} \\, \\hbQ_3 = (2)^{3\/2}\\, \n\\left| \\begin{array}{cccccc}\n q_{21}q_{31} & q_{22}q_{32}& q_{23}q_{33} \\\\\n&& \\\\\n q_{31}q_{11} & q_{32}q_{12}& q_{33}q_{13} \\\\\n&& \\\\\n q_{11}q_{21} & q_{12}q_{22}& q_{13}q_{23}\n\\end{array} \\right| = 0.\\quad\n\\end{eqnarray}\nUsing the fact that the column vectors of a transformation matrix form an orthonormal triad, it follows that \n\\beq{D7a}\n \\hbQ_1^T\\, (1,\\, 1,\\, 1)^T =  (1,\\, 1,\\, 1)^T, \\quad  \\hbQ_3\\,(1,\\, 1,\\, 1)^T =  (0,\\, 0,\\, 0)^T.   \n\\end{equation}\nThat is, $(\\sigma_{\\rm I},\\, \\sigma_{\\rm II},\\,\\sigma_{\\rm III}) = \\lambda\\, (1,\\, 1,\\, 1)$ is a   null vector of $\\bf E$ for any $\\lambda$.  Hence, $\\bf E$ is not of full rank, implying that det$\\, {\\bf E}$ is always zero.  However, this   is not of interest as the null space corresponds to hydrostatic stress, for which the energy function is independent of material orientation.  The implications of \\rf{D5} are not considered further, and we return to the simpler task of finding null vectors of $\\hbQ_1^T$ alone. \n\n\nTwo  methods for achieving the minimum energy ${\\cal E}^{\\rm (ex)} = 0$ are  described, both using explicit forms of the rotation.  The first involves a single rotation about an arbitrary axis, and the second is in terms of standard Euler angles.   \n\n\\subsubsection{Minimum energy state with a single rotation}\n\nThe range of transformations which correspond to energy minima can be obtained using Euler's Theorem \\cite{Baruh} which states that any transformation matrix ${\\bf Q}$ can be represented in the form \\rf{qeq} for some axis ${\\bf n}$, $|{\\bf n}|=1$,  and angle $\\phi$.  Thus, \n\\beqa{qeq1}\n{\\bf Q}({\\bf n},\\, \\phi) = \n\\left[ \\begin{array}{ccc}\n \t\t1-2s^2(n_2^2+n_3^2) & 2s(sn_1n_2-cn_3) & 2s(sn_1n_3+cn_2)\n  \\\\ && \\\\\n \t 2s(sn_1n_2+cn_3)\t& ~ 1-2s^2(n_3^2+n_1^2) ~&  2s(sn_2n_3 -cn_1)\n \\\\ && \\\\\n \t 2s(sn_1n_3 -cn_2)   &  2s(sn_2n_3+cn_1) & 1-2s^2(n_1^2+n_2^2)  \n\\end{array} \\right] ,\\quad\n\\end{eqnarray}\nwhere $c=\\cos \\frac{\\phi}{2}$, $s=\\sin \\frac{\\phi}{2}$.\nand the elements of the associated $\\hbQ_1$ are determined by squaring each element in \\rf{qeq1}. It may be shown that \n\\beq{qeq2}\n\\mbox{det} \\, \\hbQ_1\\ ({\\bf n},\\, \\phi) = \\cos 2\\phi + 2(2+ 3 \\cos\\phi) (1-\\cos\\phi)^2 \\,  (\nn_1^2n_2^2 + n_2^2n_3^2 + n_3^2n_1^2 )+ 6(1-\\cos\\phi)^3 \\,  \nn_1^2n_2^2 n_3^2 \\, .  \\quad   \n\\end{equation}\nNote that $n_1^2n_2^2 + n_2^2n_3^2 + n_3^2n_1^2 \\le 1\/3$ and $n_1^2n_2^2 n_3^2 \\le 1\/27$ with equality when $n_1^2= n_2^2 =n_3^2 = 1\/3$.  \n\nFor a given $n_3^2$ and angle $\\phi$, \n\\beq{n1n2}\nn_1^2, n_2^2 =  \\frac{1}{2} (1-n_3^2) \\pm \\big[ \\frac{1}{4} (1-n_3^2)^2- g\\big]^{1\/2}, \n\\end{equation}\nwhere \n\\beq{geq}\ng(n_3^2, \\phi) = \n\\frac{-\\cos 2\\phi - 2(2+ 3 \\cos\\phi) (1-\\cos\\phi)^2 \\,  n_3^2(1-n_3^2 )}{ \n2 (1-\\cos\\phi)^2 [ (2+ 3 n_3^2 +3\\cos\\phi(1 - n_3^2)]}\\, .    \n\\end{equation}\n\n\nThe null vector of $\\hbQ_1$ is such that \n\\beqa{nv1}\n\\sigma_{\\rm I} ' Q_{11}^2 + \\sigma_{\\rm II} ' Q_{12}^2 + \\sigma_{\\rm III} ' Q_{13}^2 &=&0, \n\\nonumber \\\\\n\\sigma_{\\rm I} ' Q_{21}^2 + \\sigma_{\\rm II} ' Q_{22}^2 + \\sigma_{\\rm III} ' Q_{23}^2 &=&0,\n\\\\\n\\sigma_{\\rm I} ' Q_{31}^2 + \\sigma_{\\rm II} ' Q_{32}^2 + \\sigma_{\\rm III} ' Q_{33}^2 &=&0.\n\\nonumber\n\\end{eqnarray}\nUsing the fact that this is a deviatoric stress, we replace  $ \\sigma_{\\rm III} ' = -  \\sigma_{\\rm I} '- \\sigma_{\\rm II} '$ in the final equation of \\rf{nv1}, to get\n\\beq{nv2}\n\\sigma_{\\rm I} ' \\bigg(Q_{31}^2 - Q_{33}^2\\bigg) + \\sigma_{\\rm II} ' \\bigg(Q_{32}^2 - Q_{33}^2\\bigg) = 0.\n\\end{equation}\nHence, \n\\beq{nv3}\n\\sigma_{\\rm I} '  = a_0 \\bigg(Q_{32}^2 - Q_{33}^2\\bigg) ,\\quad  \\sigma_{\\rm II} ' = a_0\\bigg(Q_{33}^2 - Q_{31}^2\\bigg) ,\n\\end{equation}\nfor arbitrary  $a_0 \\ne 0$. Once again using the fact that $ \\sigma_{\\rm III} ' = -  \\sigma_{\\rm I} '- \\sigma_{\\rm II} '$ gives\n\\beq{nv3a}\n\\sigma_{\\rm III} '  = a_0 \\bigg(Q_{31}^2 - Q_{32}^2\\bigg) . \n\\end{equation}\nIn the same way, using the other equations in \\rf{nv1}, three alternative expressions for the null vector are found:\n\\beqa{3alt}\n\\big(\\sigma_{\\rm I} ',\\, \\sigma_{\\rm II} ',\\,  \\sigma_{\\rm III} '\\big)  \n&=& a_1 \\bigg(Q_{32}^2 - Q_{32}^2,\\,  Q_{33}^2 - Q_{31}^2,\\, Q_{31}^2 - Q_{32}^2 \\bigg) \\\\\n&=& a_2 \\bigg(Q_{12}^2 - Q_{12}^2,\\,  Q_{13}^2 - Q_{11}^2,\\, Q_{11}^2 - Q_{12}^2 \\bigg) \\\\\n&=& a_3 \\bigg(Q_{22}^2 - Q_{22}^2,\\,  Q_{23}^2 - Q_{21}^2,\\, Q_{21}^2 - Q_{22}^2 \\bigg) ,  \n\\end{eqnarray}\nfor some constants $a_1$, $a_2$, $a_3$.  Thus, from the first expression, with $a_1=1$, \n\\beqa{s3a}\n\\sigma_{\\rm I} '  &=&  \\big[   (1+n_1^2)(n_2^2-n_3^2)(1-\\cos \\phi) +n_3^2-n_2^2 - 4 n_1n_2n_3\\sin\\phi\\big](1-\\cos \\phi) , \\\\\n\\sigma_{\\rm II} '  &=&   \\big\\{   [(1-n_1^2)(n_1^2-n_3^2) -2n_2^2](1-\\cos \\phi) +n_2^2-n_1^2 +1 +2 n_1n_2n_3\\sin\\phi\\big\\}(1-\\cos \\phi) - 1, \\\\\n\\sigma_{\\rm III} '  &=&   \\big\\{   [(1-n_1^2)(n_2^2-n_1^2) +2n_3^2](1-\\cos \\phi) +n_1^2-n_3^2 -1 +2 n_1n_2n_3\\sin\\phi \\big\\}(1-\\cos \\phi) + 1, \\qquad \\label{s3c}\n\\end{eqnarray}\nThese equations provide us with a two parameter set of stress states, described by $0< n_3^2< 1 $ and $\\phi$.  The two are independent insofar as  $n_1^2$ and $n_2^2$ of \\rf{n1n2} lie in $(0,1)$.  This in turn requires that $g$ of \\rf{geq} satisfies \n\\beq{n1n2q}\n0 \\le g(n_3^2, \\phi) \\le  \\frac{1}{4} (1-n_3^2)^2, \n\\end{equation}\nwhich defines the range of $0< n_3^2< 1 $ and $\\phi$. \n\n\\subsubsection{Minimum energy using Euler angles}\n\nThe standard three Euler angles  $(\\theta_1 ,\\, \\theta_2 ,\\, \\theta_3)$ are used to transform from \n$\t\\{ {\\bf e}_1, {\\bf e}_2, {\\bf e}_3\\} \n\\rightarrow \n\t\\{ {\\bf e}_1', {\\bf e}_2', {\\bf e}_3' = {\\bf e}_3\\}\n\\rightarrow \n\t\\{ {\\bf e}_1''={\\bf e}_1', {\\bf e}_2'', {\\bf e}_3'' \\}\n\\rightarrow \n\t\\{ {\\bf e}_1''', {\\bf e}_2''', {\\bf e}_3'''={\\bf e}_3'' \\}\n$.  That is,  first rotate about the ${\\bf e}_3$ axis by $\\theta_1$, then about the ${\\bf e}_1'$ axis by $\\theta_2$, and finally about the ${\\bf e}_3''$ axis by $\\theta_3$.  The transformation matrix is \n\\beqa{eul}\n&&{\\bf Q} (\\theta_1,\\theta_2,\\theta_3) \\nonumber \\\\ && \\nonumber \\\\ && =  \n\\left[ \\begin{array}{ccc}\n \t\t\\cos \\theta_1 \\cos \\theta_3 - \\sin \\theta_1 \\cos \\theta_2 \\sin \\theta_3 \n & \t\\sin \\theta_1\\cos \\theta_3 + \\cos \\theta_1 \\cos \\theta_2  \\sin \\theta_3\n & \t\\sin \\theta_2 \\sin \\theta_3\n  \\\\ && \\\\\n \t\t-\\cos \\theta_1\\sin \\theta_3 - \\sin \\theta_1 \\cos \\theta_2 \\cos \\theta_3 \n & \t-\\sin \\theta_1\\sin \\theta_3 + \\cos \\theta_1 \\cos \\theta_2 \\cos \\theta_3 \n & \t\\sin \\theta_2 \\cos \\theta_3\n \\\\ && \\\\\n \t\\sin \\theta_1 \\sin \\theta_2 \n & \t-\\cos \\theta_1 \\sin \\theta_2  & \\cos \\theta_2\n\\end{array} \\right] , \\qquad\n\\end{eqnarray}\nand it follows from this and $\\rf{D5}_1$  that \n\\beq{detex}\n\\mbox{det} \\, \\hbQ_1\\ (\\theta_1,\\theta_2,\\theta_3) = \\cos 2\\theta_1\\, \\cos 2\\theta_2\\, \\cos 2\\theta_3\n-\\frac14\\, \\sin 2\\theta_1\\,  \\sin 2\\theta_3\\, \\cos \\theta_2\\, (3\\cos 2 \\theta_2 + 1)\\, .    \\quad\n\\end{equation}\nThe condition that this vanish  is equivalent to eq. (90) of Rovati and Taliercio \\cite{Rovati03}, although their result is obtained in a different manner.\n\n\nConsider, for instance,  $\\theta_3=0$, for which \n\\beq{D8}\n\\mbox{det} \\, \\hbQ_1\\ (\\theta_1,\\theta_2,0) = \\cos 2\\theta_1\\, \\cos 2\\theta_2,   \n\\end{equation}\nand hence there are null spaces associated with $\\hbQ_1(\\theta_1, \\pi\/4,0)$ and $\\hbQ_1(\\pi\/4,\\theta_2,0)$.  The  null spaces are lines in the stress space, \nwhich follow from the simplified form of $\\hbQ_1^T$, \n\\beqa{td}\n\\hbQ_1^T (\\theta_1,\\theta_2,0) = \n\\left[ \\begin{array}{ccc}\n \\cos^2 \\theta_1 ~&~  \\sin^2 \\theta_1 \\cos^2 \\theta_2  ~&~  \\sin^2 \\theta_1  \\sin^2 \\theta_2\n  \\\\ && \\\\\n \\sin^2 \\theta_1     & \\cos^2 \\theta_1\\cos^2 \\theta_2  \n & \\cos^2 \\theta_1 \\sin^2 \\theta_2\n \\\\ && \\\\\n0 &  \\sin^2 \\theta_2  & \\cos^2 \\theta_2\n\\end{array} \\right] . \\quad\n\\end{eqnarray}\nThe possible states of deviatoric stress are:  \n$(\\sigma_{\\rm I}',\\, \\sigma_{\\rm II}',\\,\\sigma_{\\rm III}') = \\lambda \\, (0, \\, -1,\\, 1)$ if $\\theta_2 = \\pi\/4,\\, \\theta_3 = 0$, and \n$(\\sigma_{\\rm I}',\\, \\sigma_{\\rm II}',\\,\\sigma_{\\rm III}') = \\lambda \\, (\\cos 2\\theta_2, \\, -\\cos^2\\theta_2,\\, \n\\sin^2\\theta_2)$, if $\\theta_1 = \\pi\/4,\\, \\theta_3 = 0$. The first family of stresses correspond to a 2D elasticity problem (see Appendix):  $\\sigma_{\\rm I}'=0,\\, \\sigma_{\\rm II}'+ \\sigma_{\\rm III}'=0$, and it is also a null vector of $\\hbQ_1 (0,  \\pi\/4,\\theta_3)$.  The second is also a null vector of $\\hbQ_1 (0, \\theta_2 , \\pi\/4)$.  Similarly, $\\lambda \\, ( 1,\\, -1,\\, 0)$ is a null vector of $\\hbQ_1( \\pi\/4,\\pi\/2, \\theta_3)$ and \n$\\lambda \\, ( -\\cos^2\\theta_1,\\, \\sin^2\\theta_1 , \\cos 2\\theta_1 )$ is a null vector of $\\hbQ_1( \\theta_1, \\pi\/2,\\pi\/4)$. \n\nConversely, an orientation which provides a minimum in energy can be found for a given state of stress.  Assume with no loss in generality that \n$\\sigma_{\\rm II}' < 0 < \\sigma_{\\rm III}'$.  Define the angle  $\\theta_2$ by \n\\beq{lp}\n\\tan^2 \\theta_2 = -\\frac{\\sigma_{\\rm III}'}{\\sigma_{\\rm II}'},\n\\end{equation}\nthen the deviatoric stress may be expressed\n\\beq{exp}\n(\\sigma_{\\rm I}',\\, \\sigma_{\\rm II}',\\,\\sigma_{\\rm III}') = \\big( \\sigma_{\\rm III}' - \\sigma_{\\rm II}'\\big)\\, (\\cos 2\\theta_2, \\, -\\cos^2\\theta_2,\\, \n\\sin^2\\theta_2). \n\\end{equation}\nIt follows from the above example  that this deviatoric stress is a null vector of $\\hbQ_1(\\pi\/4,\\theta_2,0)$. \n\n\nIt is instructive to  examine further the example of eq. \\rf{exp}.  The rotated material axes,  denoted  $\\{ {\\bf e}_1', {\\bf e}_2',{\\bf e}_3'\\}$, are given by the columns of ${\\bf  Q}(\\pi\/4,\\theta_2,0)$: \n\\beqa{inst1}\n{\\bf e}_1' = \\frac{1}{\\sqrt{2}}\n\\left[ \\begin{array}{r}\n 1 \\\\ \\\\\n -c  \\\\ \\\\\n s   \n\\end{array} \\right], \n\\quad\n{\\bf e}_2' = \\frac{1}{\\sqrt{2}}\n\\left[ \\begin{array}{r}\n 1  \\\\ \\\\\n c  \\\\ \\\\\n -s   \n\\end{array} \\right], \n\\quad\n{\\bf e}_3' = \n\\left[ \\begin{array}{c}\n0 \\\\ \\\\\n s \\\\ \\\\\n c  \n\\end{array} \\right], \n\\quad\n\\end{eqnarray}\nwhere $s=\\sin \\theta_2$, $c=\\cos \\theta_2$, or from \\rf{lp},\n\\beq{inst2}\ns = \\sqrt{ \\frac{\\sigma_{\\rm III}'}{\\sigma_{\\rm III}' - \\sigma_{\\rm II}'}}, \n\\quad\nc = \\sqrt{ \\frac{-\\sigma_{\\rm II}'}{\\sigma_{\\rm III}' - \\sigma_{\\rm II}'}}. \n\\end{equation}\nThe rotated tensor $D_{ijkl}$ is\n\\beq{inst3}\n{\\cal D} = {\\bf e}_1'\\otimes{\\bf e}_1'\\otimes{\\bf e}_1'\\otimes{\\bf e}_1'\n+ {\\bf e}_2'\\otimes{\\bf e}_2'\\otimes{\\bf e}_2'\\otimes{\\bf e}_2'\n+ {\\bf e}_3'\\otimes{\\bf e}_3'\\otimes{\\bf e}_3'\\otimes{\\bf e}_3'\\, , \n\\end{equation}\nand hence\n\\beq{inst4}\n{\\cal D} \\bosy{\\sigma}' = \\sum \\limits_{k=1}^3 {\\bf e}_k'\\otimes{\\bf e}_k'\\, \n ({\\bf e}_k'\\cdot \\bosy{\\sigma}' {\\bf e}_k' )\\, , \n\\end{equation}\nIt may be seen by direct calculation that the three scalars ${\\bf e}_k'\\cdot \\bosy{\\sigma}' {\\bf e}_k'$ (no sum) are identically zero by virtue of \\rf{inst1} and \\rf{inst2}.  This demonstrates  explicitly that \n\\beq{inst5}\n{\\cal D} \\bosy{\\sigma}' = 0 , \n\\end{equation}\nat the optimal orientation.  The identities \\rf{idh} follow accordingly. \n\n\\subsubsection{Summary for cubic symmetry}\n\n The extreme values of the energy for cubic materials are ${\\cal E}_1$ and ${\\cal E}_2$, where \n\\beq{ex12}\n{\\cal E}_j = \\frac{1}{\\kappa}\\overline{\\sigma}^2 + \\frac{1}{2\\mu_j}\n\\big( {\\sigma_{\\rm I}'}^2 + {\\sigma_{\\rm II}'}^2 + {\\sigma_{\\rm III}'}^2 \\big), \\qquad j=1,\\, 2. \n\\end{equation}  \nThe \nfixed axes are always one of the stationary orientations, since $\\bf E$ of \\rf{r4} vanishes.  The stationary value for the unrotated axes is ${\\cal E}_2$, which is the global minimum (maximum) if $\\mu_2 > \\mu_1$ ($\\mu_2 < \\mu_1$). The stationary value ${\\cal E}_1$ occurs at some rotated axes, \nthe existence of which is not in doubt for a material of cubic symmetry (or any material symmetry for that matter). The important point to note is that it is possible to explicitly determine  such orientations. Thus, we have shown  \nby direct construction the material orientation that achieves the stationary energy value ${\\cal E}_1$ for any state of stress.  This is a global minimum (maximum) if $\\mu_2 < \\mu_1$ ($\\mu_2 > \\mu_1$).   \n\nIt is interesting that the expressions for the extreme values in  \\rf{ex12} have the form of the energy for an isotropic solid, but with different shear moduli.  This is evident by writing ${\\cal E}_1$ and ${\\cal E}_2$ in terms of the invariants of the stress tensor: \n\\beq{ex122}\n{\\cal E}_j = \\frac{1}{9\\kappa}\\big({\\rm tr}\\, \\bosy{\\sigma}\\big)^2\n+ \\frac{1}{2\\mu_j}\\big[ {\\rm tr}\\, \\bosy{\\sigma}^2  - \\frac13 \\big({\\rm tr}\\, \\bosy{\\sigma}\\big)^2 \\big], \\qquad j=1,\\, 2. \n\\end{equation}\n\n\\subsubsection{Example materials}\nNoting that $\\mu_1 = c_{44}$ and $\\mu_2 = (c_{11}-c_{12})\/2$  allows us to determine the sign of $(\\mu_2 - \\mu_1)$. \nTable A.1 of Musgrave \\cite{Musgrave} provides data for $c^* = 2(\\mu_2 - \\mu_1)$ for a multitude of materials.  These show $c^*$ to be negative for most elemental and engineering materials with cubic symmetry and different lattice structures: aluminum, nickel, copper, silver, gold (all f.c.c structure), iron (b.c.c), brass (f.c.c. and b.c.c), diamond, silicon, germanium (all diamond structure), and GaSb, InSb (both zinc-blende).  Hence, for all of these cubic materials there exist optimal orientations of the axes that achieve the lowest energy state possible.  Materials with positive $c^*$ include crystalline compounds of potassium, sodium and silver with rock-salt structure: KF, KCL, KBr, KI, NaF, NaCl, NaBr, NaI, AgCl, AgBr; plus  caesium compounds with structure related to b.c.c.  For these, the orientation associated with \\rf{exp}, for instance, gives maximum strain energy.  The minimum energy is achieved by no rotation. \n\n\\subsection{Transverse isotropy} \nMaterials with hexagonal symmetry, or equivalently, transverse isotropy (TI), are characterized by five moduli.  In the coordinate system of the principal axes, the elements of the compliance are  \n\\beqa{TI1}\n\\cS^{(0)}  = \\left[ \n\\ba{cccccc}\nS_{11}^{(0)} & S_{12}^{(0)} & S_{13}^{(0)} & \n 0&0&0 \n\\\\ & & & & & \\\\\n & S_{11}^{(0)} & S_{13}^{(0)} & \n 0&0&0\n\\\\ & & & & & \\\\\n & & S_{33}^{(0)} & \n 0&0&0\n\\\\ & & & & & \\\\\n &   &   \n & S_{44}^{(0)} & 0 & 0\n\\\\ & & & & & \\\\\n   S &Y  &M & & S_{44}^{(0)}&  0 \n\\\\ & & & & & \\\\\n&&&&& S_{66}^{(0)}\n\\end{array} \\right] ,\n\\end{eqnarray}\nwith $S_{66}^{(0)} = \\frac12 ( S_{11}^{(0)} - S_{12}^{(0)} )$.  The TI material is characterized by an axis of symmetry, defined by the unit vector ${\\bf n}$, which is here chosen as the ${\\bf e}_3$ axis.  In general, the strain energy depends only upon the orientation of ${\\bf n}$ with respect to the stress axes, and  the problem is  formulated as one of selecting ${\\bf n}$ to minimize $\\cal E$. \n\nFirst note that two of the five moduli can be ascribed to the isotropic part of the elasticity; or conversely, an isotropic part   may be subtracted from the compliance  tensor $s_{ijkl}$ according to \\rf{wr}, \\rf{wr2} and \\rf{km}, where \n \\beq{km3}\n \\frac{1}{\\kappa_s} =  \n2S_{11}^{(0)}+S_{33}^{(0)}+2S_{12}^{(0)}+4S_{13}^{(0)}, \\qquad\n \\frac{15}{4 \\mu_s} =   2S_{11}^{(0)}+S_{33}^{(0)} - S_{12}^{(0)} - 2S_{13}^{(0)}+6S_{44}^{(0)}+3S_{66}^{(0)} ,\n \\end{equation}\n  leaving a tensor $s_{ijkl}^{\\rm (an)}$ with three constants.  The anisotropic compliance depends upon the orientation of the axis of symmetry as follows  \n\\beqa{t1}\ns_{ijkl}^{\\rm (an)} &=& a\\, n_in_jn_kn_l + b\\, (\\delta_{ij}n_kn_l+ \\delta_{kl}n_in_j) \n+ \\frac{c}{2}\\, (\\delta_{ik}n_jn_l+ \\delta_{il}n_jn_k + \\delta_{jk}n_in_l+ \\delta_{jl}n_in_k) \n\\nonumber \\\\ && - \\frac13 (a+6b+2c) J_{ijkl} - \\frac{2}{15}(a+5c) K_{ijkl} .\\quad\n\\end{eqnarray}\nThe tensors $J_{ijkl}$ and $K_{ijkl}$ are defined in \\rf{wr3}, and \n the elastic constants $a$, $b$ and $c$ follow from \\rf{TI1}-\\rf{t1} as \n\\beq{abc}\na=S_{11}^{(0)} + S_{33}^{(0)} - 2S_{13}^{(0)} - 4S_{44}^{(0)}, \\quad \nb=S_{13}^{(0)} - S_{12}^{(0)}, \\quad \nc=2S_{44}^{(0)} - 2S_{66}^{(0)}. \\quad \n\\end{equation}\n  The stress is, as usual, aligned with the fixed axes, so that the total strain energy follows from \\rf{fa}, \\rf{t1} and \\rf{t2}, as \n\\beq{t3a}\n{\\cal E} =  \\big[ \\frac{1}{\\kappa_s} - \\frac13 (a+6b+2c)\\big] \n\\overline{\\sigma}^2 + \\big[  \\frac{1}{2\\mu_s} - \\frac{2}{15}(a+5c)\\big]\\sigma_{ij}'\\sigma_{ij}' + \n{\\cal E}^{\\rm (ex)} , \n\\end{equation}\nwhere  $\\overline{\\sigma}$  is defined in \\rf{fis2} and the extra energy term is   \n\\beq{t4}\n{\\cal E}^{\\rm (ex)} =a\\, \\big( \\sigma_{\\rm I}n_1^2+ \\sigma_{\\rm II}n_2^2+ \\sigma_{\\rm III}n_3^2\\big)^2\n+ (3b\\overline{\\sigma}+ c\\sigma_{\\rm I})2\\sigma_{\\rm I}n_1^2\n+ (3b\\overline{\\sigma}+ c\\sigma_{\\rm II})2\\sigma_{\\rm II}n_2^2\n+ (3b\\overline{\\sigma}+ c\\sigma_{\\rm III})2\\sigma_{\\rm III}n_3^2\\, . \n\\end{equation}\nThe latter shows that the anisotropic part of the energy ${\\cal E}^{\\rm (ex)}$ depends upon the TI axis orientation through the three parameters \n$n_1^2$, $n_2^2$ and $n_3^2$, which satisfy $n_1^2+n_2^2+n_3^2=1$. Since each of  $n_1^2$, $n_2^2$ and $n_3^2$ must be non-negative, the permissible set is the equilateral  triangular area $A$ of the  plane $n_1^2+n_2^2+n_3^2-1=0$   bounded by the lines $L_1:\\, n_2^2+n_3^2=1$, $L_2:\\,n_3^2+n_1^2=1$ and $L_3:\\,n_1^2+n_2^2=1$. \n\n\nConsider first the possibility that the optimal orientation lies  \non one of the lines $L_i$, $I=1,2,3$.  Thus, along $L_3$, a simple calculation using $n_3^2 = 0$ shows that \n\\beq{t8}\n{\\cal E}^{\\rm (ex)}  = a (\\sigma_{\\rm II} - \\sigma_{\\rm I})^2\\, \\big( n_1^2 - N_1\\big)^2 + 2\\sigma_{\\rm II} \\big( \n3b\\overline{\\sigma} + c \\sigma_{\\rm II}\\big)  - a (\\sigma_{\\rm II} - \\sigma_{\\rm I})^2\\, N_1^2\\, \\quad \\mbox{on }L_3, \n\\end{equation}\nwhere\n\\beq{t7}\nN_1 \\equiv \\frac{a \\sigma_{\\rm II} +c (\\sigma_{\\rm I} + \\sigma_{\\rm II}) +3b\\overline{\\sigma} }{a(\\sigma_{\\rm II} - \\sigma_{\\rm I})}\\, . \n\\end{equation}\nThus, $n_1^2 = N_1$, is a {\\em possible} optimal orientation.  It must first be checked whether or not $N_1$  lies in $(0,1)$. If this is so, and if $a>0$, then an energy minimum occurs at the point $n_1^2=N_1$, $n_2^2=1-N_1$ on $L_3$.  \nSimilarly, a minimum occurs on $L_1$ at $n_2^2=N_2$, $n_3^2=1-N_2$ if $N_2 \\in (0,1)$ and $a>0$, and on $L_2$ at $n_3^2=N_3$, $n_1^2=1-N_3$ if $N_3 \\in (0,1)$ and $a>0$, where\n\\beqa{t9}\nN_2 &=& \\frac{a \\sigma_{\\rm III} +c(\\sigma_{\\rm II} + \\sigma_{\\rm III}) +3b\\overline{\\sigma} }{a(\\sigma_{\\rm III} - \\sigma_{\\rm II})},\n\\\\\\\nN_3 &=& \\frac{a \\sigma_{\\rm I} +c(\\sigma_{\\rm III} + \\sigma_{\\rm I}) +3b\\overline{\\sigma} }{a(\\sigma_{\\rm I} - \\sigma_{\\rm III})} \\, . \n\\end{eqnarray} \n\nNow consider the possibility of the minimum occurring in the interior of $A$.   Substitute $n_3^2=1 -n_1^2-n_2^2$ into \\rf{t4} and setting the partial derivatives with respect to $n_1^2$ and $n_2^2$ to zero, yields a pair of simultaneous conditions\n\\beqa{t5}\n\\big( \\sigma_{\\rm I} - \\sigma_{\\rm III}\\big)\\, \\big[ \na\\, \\big( \\sigma_{\\rm I}n_1^2+ \\sigma_{\\rm II}n_2^2+ \\sigma_{\\rm III}n_3^2\\big)\n+ 3b\\overline{\\sigma}+ c(\\sigma_{\\rm I}+ \\sigma_{\\rm III}) \\big] &=&0,\n\\\\\n\\big( \\sigma_{\\rm II} - \\sigma_{\\rm III}\\big)\\, \\big[ \na\\, \\big( \\sigma_{\\rm I}n_1^2+ \\sigma_{\\rm II}n_2^2+ \\sigma_{\\rm III}n_3^2\\big)\n+ 3b\\overline{\\sigma}+ c(\\sigma_{\\rm II}+\\sigma_{\\rm III})\\big] &=&0 \\, . \n\\end{eqnarray}\nThese cannot be satisfied in general if the three principle stresses are distinct.   We therefore conclude that \\textcolor{blue}{\\em the optimal $\\bf n$ will lie inside $A$ iff the stress is biaxial}.  \nFor instance, if $\\sigma_{\\rm II} $ and $ \\sigma_{\\rm III}$ are equal, then eq. \\rf{t5} combined with $n_3^2=1 -n_1^2-n_2^2$ imply that ${\\cal E}^{\\rm (ex)}$ of \\rf{t4} can be expressed as a function of   $n_1^2$ alone, and the expression is identical in form to that given in \\rf{t8}. Thus, the existence of a minimum inside $A$ requires biaxiality $(\\sigma_{\\rm II}=\\sigma_{\\rm III})$ and that $0 < N_1 < 1$.  The associated optimal direction is not unique, but is defined by the cone $n_1^2 = N_1$, $n_2^2+n_3^2 = 1-N_1$ (note that $N_3 = 1-N_1$ when $\\sigma_{\\rm II}=\\sigma_{\\rm III}$). \nAgain, \\rf{t8} indicates that the optimal orientation corresponds to a minimum (maximum) in energy if $a>0$ $(a<0)$.   Thus, the sign of the elastic compliance  $a$ is crucial in determining whether the stationary point  is a minimum or a maximum.  \n\nThese conclusions may also be confirmed by the coaxiality condition for the stress and strain. Thus, for arbitrary orientation\n\\beqa{od}\n\\varepsilon_{23} &=& n_2n_3\\, \\big[ \na\\big( \\sigma_{\\rm I}n_1^2+ \\sigma_{\\rm II}n_2^2+ \\sigma_{\\rm III}n_3^2\\big) \n+ 3b\\overline{\\sigma}+ c\\big( \\sigma_{\\rm II} + \\sigma_{\\rm III} \\big)\\big]\\, , \n\\nonumber \\\\\n\\varepsilon_{31} &=& n_3n_1\\, \\big[ \na\\big( \\sigma_{\\rm I}n_1^2+ \\sigma_{\\rm II}n_2^2+ \\sigma_{\\rm III}n_3^2\\big) \n+ 3b\\overline{\\sigma}+ c\\big( \\sigma_{\\rm III} + \\sigma_{\\rm I} \\big)\\big]\\, , \n\\\\\n\\varepsilon_{12} &=& n_1n_2\\, \\big[ \na\\big( \\sigma_{\\rm I}n_1^2+ \\sigma_{\\rm II}n_2^2+ \\sigma_{\\rm III}n_3^2\\big) \n+ 3b\\overline{\\sigma}+ c\\big( \\sigma_{\\rm I} + \\sigma_{\\rm II} \\big)\\big]\\, . \n\\nonumber \n\\end{eqnarray}\nThe requirement that these simultaneously  vanish is identical to the  above conditions for the existence of the minimum inside $A$ or along its perimeter. \n\n\nIn summary, $a>0$ is a necessary condition that an energy minimum occurs at points inside $A$ or along the lines $L_j$, $j=1,2,3$.  A minimum is achieved iff one or more of $N_1$, $N_2$ or $N_3$ lies in $(0,1)$.  The minimum occurs on the associated bounding line $L_j$ or on a cone of directions for biaxial states of stress.   Otherwise, the global energy minimum corresponds to one of the  vertices  of $A$, i.e. at $n_1^2 = 1$ or $n_2^2 = 1$ or $n_3^2 = 1$.  In this default case the TI axis of symmetry is aligned with one of the stress axes.  These findings are in agreement with those of Rovati and Taliercio \\cite{Rovati03}, who stated the condition as follows: At least one of the principal axes of stress must lie in a plane of transverse isotropy, or alternatively, the TI axis must lie in a plane defined a pair of principal axes of stress. \n\n\\subsection{Tetragonal symmetry} \n\nThe moduli have the same general form as in \\rf{TI1}, except that there is no relation between $S_{66}^{(0)}$, $S_{11}^{(0)}$ and $S_{12}^{(0)}$.  In this sense,  tetragonal symmetry is the same as TI but with  one additional elastic constant.  The isotropic moduli are given by eq. \\rf{km3}, and the anisotropic part of the compliance is \n\\beqa{t1tet}\ns_{ijkl}^{\\rm (an)} &=& a'\\, n_in_jn_kn_l + b'\\, (\\delta_{ij}n_kn_l+ \\delta_{kl}n_in_j) \n+ \\frac{c'}{2}\\, (\\delta_{ik}n_jn_l+ \\delta_{il}n_jn_k + \\delta_{jk}n_in_l+ \\delta_{jl}n_in_k) \n\\nonumber \\\\ && \n+ d\\, (p_ip_j- q_iq_j)(p_kp_l - q_kq_l) \n- \\frac13 (a'+6b'+2c') J_{ijkl} - \\frac{2}{15}(a'+5c' + 3d) K_{ijkl} .\\quad \\qquad\n\\end{eqnarray}\nThe additional fourth order tensor as compared to TI is  $ (p_ip_j- q_iq_j)(p_kp_l - q_kq_l)$, \nwhere $\\{ {\\bf n}, {\\bf p}, {\\bf q}\\}$ form an orthonormal triad.  \nThe elastic constants $a'$, $b'$, $c'$  and $d$ are\n\\beqa{abcd}\n&&a'=\\frac12 S_{11}^{(0)} +\\frac12 S_{12}^{(0)}+ S_{33}^{(0)} + S_{66}^{(0)}  - 2S_{13}^{(0)} - 4S_{44}^{(0)}, \n\\nonumber \\\\ \n&& b'=S_{13}^{(0)} - S_{12}^{(0)} - \\frac12 \\big( S_{11}^{(0)}- S_{12}^{(0)}\\big)  + S_{66}^{(0)}, \\quad \nc'=2S_{44}^{(0)} - 2S_{66}^{(0)},  \\\\\n&&\nd=\\frac12 \\big( S_{11}^{(0)}- S_{12}^{(0)}\\big)  - S_{66}^{(0)} . \\quad \\nonumber\n\\end{eqnarray}\nCompared to the TI constants $a$, $b$, $c$ of \\rf{abc}, \n\\beq{abcom}\na'=a-d,\\quad b'=b-d,\\quad c'=c. \n\\end{equation}\n\n\nThe strain energy of the tetragonal material can be split into a component similar in form to that for a TI material,  and an additional term proportional to the constant $d$.  The minimization of the TI part of the energy is as before (with $a',b',c'$ instead of $a,b,c$), and depends  upon the orientation of $\\bf n$ but not $\\bf p$ and $\\bf q$.   The additional energy term depends  on the deviatoric part of the stress and on these directions, \n\\beq{tet2}\n{\\cal E}^{\\rm tet} = d \\big[ (p_ip_j-q_iq_j)\\sigma_{ij}'\\big]^2 - \\frac25\\, d \\sigma_{ij}'\\sigma_{ij}', \n\\end{equation}\nor, in terms of the principal stresses,  \n\\beq{tet3}\n{\\cal E}^{\\rm tet} = d\\, \\big( \\sigma_{\\rm I} '\\Delta_1'  + \\sigma_{\\rm II} '\\Delta_2' + \\sigma_3 '\\Delta_{\\rm III}' \\big)^2 - \\frac25\\, d \\sigma_{ij}'\\sigma_{ij}' \\, , \n\\end{equation}\nwhere\n\\beq{di}\n \\Delta_i'  = p_i^2- q_i^2 \\, \\quad \\mbox{(no sum)} .\\quad \n\\end{equation}\nThe final term in \\rf{tet3} is independent of $\\{ {\\bf n}, {\\bf p}, {\\bf q}\\}$, and it is only  necessary to  consider the quantity\n\\beq{tq1}\n{\\cal E} '  = d\\, f^2,\\quad \\mbox{where } \nf = \\sigma_{\\rm I} '\\Delta_1'  + \\sigma_{\\rm II} '\\Delta_2' + \\sigma_3 '\\Delta_{\\rm III}'\\, . \n\\end{equation}\nThe orientation dependence is captured by the quantity $f$.  \n\nIt is now demonstrated that for any given $\\bf n$, there is at least one set  of $ {\\bf p}, {\\bf q}$ orthogonal to $\\bf n$  which make ${\\cal E} '$ vanish.  Let $ {\\bf p}^{(0)}, {\\bf q}^{(0)}$ be an arbitrary pair of unit vectors such that $\\{ {\\bf n}, {\\bf p}^{(0)}, {\\bf q}^{(0)}\\}$ form an orthonormal triad, then every possible   set $\\{ {\\bf n}, {\\bf p}, {\\bf q}\\}$ is defined by the pair $ {\\bf p}, {\\bf q}$ obtained by rotation about $\\bf n$ by angle $\\phi$:\n\\beq{tet5}\n{\\bf p}(\\phi) = \\cos \\phi \\, {\\bf p}^{(0)} - \\sin \\phi \\, {\\bf q}^{(0)}, \n\\qquad \n{\\bf q}(\\phi) = \\sin \\phi \\, {\\bf p}^{(0)} + \\cos \\phi \\, {\\bf q}^{(0)}. \n\\end{equation}\nIt may then be readily verified that \n \\beq{tet6}\n\\Delta_i'(\\phi) = \\Delta_i'(0)\\, \\cos 2\\phi - \\Delta_i'(\\frac{\\pi}{4})\\, \\sin 2\\phi , \\quad i=1,2,3. \n\\end{equation}\nEquation \\rf{tet6} implies that \n\\beq{tet7}\n{\\cal E} ' (\\phi) = d\\, \\big[ f(0)\\, \\cos 2\\phi - f(\\frac{\\pi}{4})\\, \\sin 2\\phi\\big]^2 \\, , \n\\end{equation}\nand hence \n\\beq{tet8}\n{\\cal E} ' (\\phi^*) = 0 , \\quad \\mbox{where } \\tan 2\\phi^* = f(0)\/f(\\frac{\\pi}{4}) \\, , \n\\end{equation}\nThus, if $d>0$, the situation for tetragonal symmetry is a simple addition to the TI situation: First find $\\bf n$ which minimizes  the TI part of the energy.  Then, select the pair  $  {\\bf p}, {\\bf q}$ such that they satisfy \\rf{tet8}.  The minimum energy is then exactly that achieved by the TI part of the moduli (although it depends upon $a',b',c'$ rather than $a,b$ and $c$). \n\nIf $d<0$ then the situation is more complicated, and the sequential minimization of first the  TI energy and then  the additional energy ${\\cal E} '$ does not work, although these do define stationary points for the strain energy.  The $d-$term must be taken into account when optimizing with respect to $\\bf n$, and a more complicated minimization problem is involved. \n\nTetragonal symmetry represents a demarcation between the simpler higher material symmetries for which explicit results can be obtained, and the  lower material symmetries which require numerical resolution, in general.  Exceptions may occur, however, it is useful and instructive to distinguish the cubic, TI and tetragonal symmetries from those of, for example, monoclinic symmetry with 13 independent moduli to consider. \n\n\\section{Strain deviation angle}\\label{sec5}\n\n\\subsection{Definition of the strain deviation angle}\n\nA necessary condition for an energy minimum is that the stress and strain are coaxial.  This is always the case in isotropic media,  whereas it will be the exception rather than the rule under conditions of  general anisotropy and arbitrary stress.  According to Euler's theorem \\cite{Baruh} the transformation from one set of principal axes to the other can be reduced to an axis of rotation $\\bf n$, $|{\\bf n}|=1$, and an angle of rotation $\\phi$. The stress axes  have been assumed to coincide with the fixed axes ${\\bf e}_j$, $j=1,2,3$.   Let  ${\\bf e}_j'$ be the orthonormal axes of the strain tensor, then it follows that the rotation matrix is simply the matrix composed of the three unit vectors as columns, \n\\beq{qr}\n{\\bf Q} = \\big[ {\\bf e}_1 '\\, {\\bf e}_2 '\\, {\\bf e}_3' \\big] \\, . \n\\end{equation}\nLet $\\bf Q$ be represented by \\rf{qeq}, then it follows from the latter that \n\\beq{qr2}\ne_{ijk}Q_{jk} = 2 \\sin \\phi\\, n_i.\n\\end{equation}\nThis provides a formula to determine both the angle $\\phi$ and the axis of rotation $\\bf n$.    \n\nThe strain deviation angle $\\phi$ is defined as the angle of rotation between the stress and strain axes. This angle is identically zero in isotropic materials for all stress states.  In anisotropic materials it depends on both the material constants and the state of stress.   However,  the above analysis tells us that $\\phi = 0$ is a necessary condition for energy minimization. Therefore, the magnitude of $\\phi$ provides, through a single parameter, the degree to which the given state of stress and material orientation are optimal.  It does so without requiring any calculation of the energy locally or globally.   It requires only that the principal strain axes are determined, and from those $\\phi$  can be immediately computed. \n\nFor a given material, stress and strain, the  strain deviation angle can be obtained from \\rf{qr2}.  A more explicit method is to use the  general identity for integer $m$: \n\\beq{ged}\n  \\cos m\\phi =  \\frac12\\, {\\rm tr} \\big({\\bf Q}^m\\big) - \\frac12. \n\\end{equation}\nThis follows from, for example, eqs. \\rf{qeq} and \\rf{a1a1}, which imply\n\\beq{ged2} {\\bf Q}^m  ({\\bf n} , \\phi ) = {\\bf n}\\otimes{\\bf n} + \\cos m\\phi \\, ({\\bf I} -{\\bf n}\\otimes{\\bf n} )+\n\\sin m\\phi\\, {\\bf P}\\, . \n\\end{equation}\nFor instance, $m=1$ gives the  strain deviation angle explicitly in terms of the first invariant of $\\bf Q$: \n\\beq{qr3}\n  \\phi = \\cos^{-1}\\big[ \\frac12 \\big({\\rm tr}{\\bf Q}  -1\\big)\\big].\n\\end{equation}\n\n\n\\subsection{Weak anisotropy}\nLet $\\varepsilon_{j}^{(0)}$, $j=1,2,3$ be the principal strains for the isotropic medium, i.e. the principal axes of \n$s_{ijkl}^{\\rm (is)}\\sigma_{kl}$, where $\\sigma_{ij}$ is given by \\rf{s1} and\/or \\rf{t2}.  In order to determine the strain deviation angle we first need to find the principal axes of strain.  \nIt is useful to express the strain as \n\\beq{usf}\n\\varepsilon_{ij} = s_{ijkl}^{\\rm (is)}\\sigma_{kl}+\\gamma_{ij},\n\\end{equation}\n where \n\\beqa{w0}\n\\gamma_{ij} &=& s_{ijkl}^{\\rm (an)}\\sigma_{kl}\n\\nonumber \\\\ &=&\ns_{ij11}^{\\rm (an)}\\sigma_{\\rm I} + s_{ij22}^{\\rm (an)}\\sigma_{\\rm II}+s_{ij33}^{\\rm (an)}\\sigma_{\\rm III}   \n\\end{eqnarray}\nIt is assumed that $\\bosy{\\gamma}$ is small, so that standard perturbation analysis may be used to find the first correction to the directions of principal strain, \n$\\{ {\\bf e}_1', {\\bf e}_2',{\\bf e}_3'\\}$ , which to leading order are coincident with the stress directions: \n\\beq{wa1}\n{\\bf e}_i' = {\\bf e}_i + \\sum\\limits_{j\\ne i}\\, \\big( \\varepsilon_{i}^{(0)} - \\varepsilon_{j}^{(0)}\\big)^{-1}\\, \n\\big( {\\bf e}_i\\cdot \\bosy{\\gamma}{\\bf e}_j\\big) \\, {\\bf e}_j, \\, \\quad \\mbox{no sum on }i. \n\\end{equation}\nLet $E$ and $\\nu$ be the isotropic Young's modulus and Poisson's ratio characterizing $s_{ijkl}^{\\rm (is)}$, then \\rf{wa1} implies that, to leading order, \n\\beq{wa2}\nQ_{ij} = - Q_{ji} \\approx \\frac{E}{1+\\nu}\\, \\frac{\\gamma_{ij}}{ (\\sigma_j - \\sigma_i)},\\quad i\\ne j.\n\\end{equation}\nHence, the strain deviation angle for weak anisotropy is \n\\beq{wa3}\n\\phi \\approx \\sin \\phi \\approx \n \\frac{E}{1+\\nu}\\, \\bigg[ \\frac{\\gamma_{12}^2}{ (\\sigma_{\\rm I} - \\sigma_{\\rm II})^2}\n + \\frac{\\gamma_{23}^2}{ (\\sigma_{\\rm II} - \\sigma_{\\rm III})^2} + \\frac{\\gamma_{31}^2}{ (\\sigma_{\\rm III} - \\sigma_{\\rm I})^2}\\bigg]^{1\/2}\\, . \n\\end{equation}\nIt is useful to write the stress dependence explicitly by eliminating $\\bosy{\\gamma}$, \n\\beqa{wa31}\n\\phi &\\approx &\n \\frac{E}{1+\\nu}\\, \\bigg[  \n \\bigg( \\frac{\\sigma_{\\rm I}s_{14}^{\\rm (an)}+\\sigma_{\\rm II}s_{24}^{\\rm (an)}+\\sigma_{\\rm III}s_{34}^{\\rm (an)}  }{ \\sigma_{\\rm II} - \\sigma_{\\rm III}}\\bigg)^2\n +\\bigg( \\frac{\\sigma_{\\rm I}s_{15}^{\\rm (an)}+\\sigma_{\\rm II}s_{25}^{\\rm (an)}+\\sigma_{\\rm III}s_{35}^{\\rm (an)}  }{ \\sigma_{\\rm III} - \\sigma_{\\rm I}}\\bigg)^2\n \\nonumber \\\\  && \\qquad \\qquad\n + \\bigg(\\frac{\\sigma_{\\rm I}s_{16}^{\\rm (an)}+\\sigma_{\\rm II}s_{26}^{\\rm (an)}+\\sigma_{\\rm III}s_{36}^{\\rm (an)}  }{ \\sigma_{\\rm I} - \\sigma_{\\rm II}}\\bigg)^2\n \\bigg]^{1\/2}\\, . \n\\end{eqnarray}\nThis shows that the strain deviation angle depends upon the same $9$ moduli that appear in the matrix $\\bf E$ of eq. \\rf{r4}. \n\nThe above formula breaks down for biaxial stress.  In this case the choice of fixed axes is arbitrary since any orthogonal pair in the the plane spanned by the equal principal stresses are valid.  However, the choice can be made {\\it a posteriori} such that the term that would otherwise be singular is zero.  For instance, if $\\sigma_{\\rm I} = \\sigma_{\\rm II}$, then the axes ${\\bf e}_1$ and ${\\bf e}_2$ can be selected such that $\\gamma_{12}= 0$.  \n\n\\section{Conclusions}\n\nThe 6-dimensional notation of Mehrabadi et al.  \\cite{mcj} is well suited to the problem of finding optimal orientations of anisotropic solids.  It leads quite naturally to the main results of the paper, which we recapitulate:\n\n\\medskip \n\\noindent\n{\\bf Result 1}: {The energy  \n$ \\textcolor{blue} {\\cal E}$  is stationary iff the stress and strain are coaxial. }\n\n\\noindent\n{\\bf Result 1a}: {A necessary (but not sufficient) condition for this to hold is that det ${\\bf E}=0$, where $\\bf E$ is defined in \\rf{r4}}\n\n\\noindent\n{\\bf Result 2}:  {The energy\n${\\cal E}$  is a local minimum if the stress and strain are coaxial and the symmetric matrix ${\\bf G}$ of \\rf{G7} is positive definite.  } \n\nResult 2 provides for the first time an explicit set of conditions that must be satisfied if  the stationary condition is a minimum or a maximum.  \n\nSpecific  results are given for materials of cubic, transversely isotropic  and tetragonal symmetries.  In each case the existence of a minimum or maximum depends on the sign of a single elastic constant.  For cubic symmetry we have several new findings.  For instance,  eqs. \\rf{s3a} - \\rf{s3c} provide a two parameter set of stress states which minimize or maximize the strain energy if a material of cubic symmetry is rotated  about an arbitrary axis $\\bf n$ by angle $\\phi$ (subject to the constraint \\rf{n1n2q}).  Alternatively, eqs. \\rf{lp} and \\rf{exp} provide a means to find the optimal orientation for a given state of stress.   In particular, the rotation of the material axes depends only upon the deviatoric stress.  This demonstrates that the stationary (minimum or maximum) value of energy can always be achieved for cubic materials.  Furthermore, it shows that \nthe optimal orientation of a solid with cubic material symmetry is not normally aligned with the symmetry directions.  \n\nThe remainder of the new results concern the optimal orientation of  TI and tetragonal materials,   and  are in general agreement with results of  Rovati and Taliercio \\cite{Rovati03} obtained by a different procedure.  However, the results obtained here are more direct and provide considerable insight into the nature of the optimal states for these material symmetries.  In particular,  the problem for tetragonal symmetry is very similar to that for TI, with an additional energy term that can be simply minimized or maximized (depending on the sign of the constant $d$ of \\rf{abcd}.   \n\nFinally, we have defined and  introduced the strain deviation angle.  The strain deviation angle is inherently anisotropic, and directly related to the problem of energy minimization since the angle   defines the degree to which a state of stress or strain is not optimal.  Future work will explore other consequences of this new concept.\n\n\n\\section*{Appendix: Two dimensional elasticity}\n\n\nOptimal orientation for  two dimensional elastic anisotropy  is an important special case of the general problem.  It was recently considered by  Gea and Luo \\cite{Gea04}, and is reconsidered here  in the context of the present theory.  We will see that some of the features Gea and Luo  obtained transfer to the 3D problem: in particular, the dependence of the minimization upon a single elastic constant. \n\nThe two dimensional strain energy function is \n\\beq{q1}\n{\\cal E}(\\theta ) = S_{11} \\sigma_{\\rm I}^2 + S_{22} \\sigma_{\\rm II}^2 + 2S_{12} \\sigma_{\\rm I} \\sigma_{\\rm II}\\, . \n\\end{equation}\nwhere $S_{11}$, $S_{22}$ and $S_{12}$ depend upon the angle $\\theta$ by which the material is rotated relative to the fixed $\\bf{e}_3$ axis.  Consider an orthotropic material with compliance elements \n$S^{(0)}_{11},\\,  S^{(0)}_{22},\\,  S^{(0)}_{12},\\,  S^{(0)}_{66}$ in the unrotated (fixed) axes.  \nUsing the standard relations \\cite{Christensen} for the transformation of the moduli, it may be shown that \n\\beq{q2}\n{\\cal E}(\\theta ) = \\frac{1 }{4} \\, d_0\\, ( \\sigma_{\\rm II} - \\sigma_{\\rm I})^2\\, \\big( \n\\cos 2\\theta -  \\Lambda \\big)^2\\,  + b_0 ,\n\\end{equation}\nwhere $\\Lambda$ is a combination of stress and moduli, \n\\beq{x6}\n \\Lambda  = \\bigg( \\frac{ \\sigma_{\\rm II} + \\sigma_{\\rm I} }{ \\sigma_{\\rm II} - \\sigma_{\\rm I} } \\bigg) \\,\n \\frac{ c_0 }{ d_0 }\\, , \n \\end{equation}\n $c_0$ and $d_0$ are   moduli, \n\\beq{x1a}\nc_0 = S^{(0)}_{11} - S^{(0)}_{22}, \\qquad \nd_0 = S^{(0)}_{11} + S^{(0)}_{22} - 2S^{(0)}_{12} - 4S^{(0)}_{66} , \n\\end{equation}\nand $b_0$ is a constant,\n\\beq{b0}\nb_0 = ( \\sigma_{\\rm II} - \\sigma_{\\rm I})^2\\, S^{(0)}_{66} + \\frac14 ( \\sigma_{\\rm II} + \\sigma_{\\rm I})^2\\, \\big(\nS^{(0)}_{11} + S^{(0)}_{22}+2 S^{(0)}_{12} -\\frac{c_0^2}{d_0}\\big) . \n\\end{equation}\n\nIt can be easily seen that the energy ${\\cal E}$ of \\rf{q2} is  stationary with respect to $\\theta$ when \n\\beq{st}\n\\cos 2\\theta = 1, \\mbox{ and  } \\cos 2\\theta = -1\n\\quad \\Leftrightarrow \\quad \\theta =0, \\mbox{ and  } \\theta = \\pi\/2, \n\\end{equation}\nrespectively.  Which of these yields the  smaller value for ${\\cal E}$ depends upon the sign of $d_0\\Lambda$, or equivalently, the sign of $ (\\sigma_{\\rm II}^2 - \\sigma_{\\rm I}^2)c_0$.  Specifically, the minimum is at $\\cos 2\\theta = $sgn$[(\\sigma_{\\rm II}^2 - \\sigma_{\\rm I}^2)c_0]$. A third stationary value is possible if $-1< \\Lambda<1 $, and occurs at \n\\beq{ste}\n\\cos 2\\theta = \\Lambda, \n\\quad \\Leftrightarrow \\quad \\theta  =\\pm \\theta^*, \n\\end{equation}\nwhere $\\theta^* = \\frac12\\, \\cos^{-1}\\Lambda$.  If this stationary point occurs, it follows from explicit evaluation that it corresponds to a global minimum or maximum of the energy.  \nThus, \n\\beqa{q3}\n\\left. \\ba{c}\n{\\cal E}(0)\\\\ \\\\\n{\\cal E}(\\frac{\\pi}{2})\\\\ \\\\\n{\\cal E}(\\theta^*)\n\\end{array} \\right\\} = b_0 + d_0\\, ( \\sigma_{\\rm II} - \\sigma_{\\rm I})^2\\, \\times \\, \n\\left\\{ \\ba{l}\n \\sin^4  \\theta^* \\, , \\\\ \\\\\n \\cos^4  \\theta^* \\, , \\\\ \\\\\n0\\, . \n\\end{array}\\right. \n\\end{eqnarray}\nIt is clear that $\\theta = \\pm \\theta^*$ is a repeated global minimum (maximum) if  $d_0 >0\\, (d_0 <0)$. This is the fundamental result of  Gea and Luo \\cite{Gea04} (although their conclusion is slightly different since they do not start with the stress in the principal axes frame): \n\\textcolor{blue}{ If  $-1< \\Lambda<1 $ and $d_0 >0$ then $ \\theta  =\\pm \\theta^*$ is a repeated global minimum of ${\\cal E}(\\theta)$.  Otherwise, the minimum occurs when $\\cos 2\\theta = $sgn$[(\\sigma_{\\rm II}^2 - \\sigma_{\\rm I}^2)c_0]$.\n}\n\nThe results of Gea and Luo are now reconsidered within the context of the general theory applied to 2D.   Based on the general theory for 3D, the 2D condition for a stationary orientation is \n\\beq{21}\n\\varepsilon_{12} = 0\\, , \n\\end{equation}\nor, in terms of the stress, assuming for simplicity that $\\sigma_{\\rm III} = 0$: \n\\beq{22}\nS_{16}\\sigma_{\\rm I} + S_{26}\\sigma_{\\rm II} = 0\\, . \n\\end{equation}\nThe latter is consistent with the general formulation, i.e. eq. \\rf{r3}, under the assumption that $\\sigma_{\\rm I}$ and $\\sigma_{\\rm II}$ are distinct (if they are not distinct, then the stress-based energy function is constant for all material orientations). \nThe additional condition that the stationary orientation is a local minimum follows from \n\\beq{cm8}\n\\left. \\frac{d^2 {\\cal E}(\\theta)}{d\\theta^2 } \\right|_{\\theta = 0} =  4 F_{33} , \n\\end{equation}\nas \n\\beq{h3}\nF_{33} = 2 (\\sigma_{\\rm II} - \\sigma_{\\rm I})^2 \\, G_{33} > 0 . \n\\end{equation}\nThe exact form of $G_{33}$  follows from  \\rf{G6}   with $\\sigma_{\\rm III} = 0$ as  \n\\beq{h4}\nG_{33} = S_{66}  + \\frac12 \n(\\sigma_{\\rm II} - \\sigma_{\\rm I})^{-1}\\, \\big[ \n\t(S_{11}-S_{12})\\sigma_{\\rm I} + (S_{21}-S_{22})\\sigma_{\\rm II}  \\big]\\, . \n\\end{equation}\nRearrangement gives \n\\beq{h5}\nG_{33} = \\frac14 \\,\n(\\sigma_{\\rm II} - \\sigma_{\\rm I})^{-1}\\, \\big[ (\\sigma_{\\rm I} + \\sigma_{\\rm II}) c + \n(\\sigma_{\\rm I} - \\sigma_{\\rm II}) d \\big]\\, , \n\\end{equation}\nwhere $c$ and $d$ are \n\\beq{x1}\nc = S_{11} - S_{22}, \\qquad \nd = S_{11} + S_{22} - 2S_{12} - 4S_{66} . \n\\end{equation}\n\nThe specific case of an orthotropic material is considered next.  \nIt may be shown by use of standard relations \\cite{Christensen} that the combinations of moduli in \\rf{x1} transform according to \n\\beq{x2}\nc (\\theta) = c_0 \\, \\cos 2\\theta , \\qquad \nd (\\theta) = d_0 \\, \\cos 4\\theta ,  \n\\end{equation}\nwhere \n$c_0 = c(0)$ and $d_0 = d(0)$ are the same moduli defined in \\rf{x1a}. \nAlso, \n\\beqa{x3}\nS_{16} &=&-\\frac{1}{4} \\big( c_0 +  d_0 \\cos 2\\theta \\big) \\, \\sin 2\\theta ,\n\\\\ \nS_{26} &=&-\\frac{1}{4} \\big( c_0 -  d_0 \\cos 2\\theta \\big) \\, \\sin 2\\theta . \n\\end{eqnarray}\nHence, \n\\beq{x4}\n\\varepsilon_{12} = -\\frac{1}{4}  \\sin 2\\theta \\, \n\\big[ (\\sigma_{\\rm I} + \\sigma_{\\rm II}) c_0 + (\\sigma_{\\rm I} - \\sigma_{\\rm II}) d_0 \\cos 2\\theta \\big]\\, . \n\\end{equation}\nThe strain $\\varepsilon_{12}$ vanishes if \n\\beq{x5}\n \\sin 2\\theta = 0,\\quad \\mbox{or}\\quad \n \\cos 2\\theta = \\Lambda .\n \\end{equation}\nThus, the stationary points are $\\theta = 0,\\, \\pi\/2$ and $\\pm \\theta^*$ where $\\cos 2\\theta^* = \\Lambda $, in agreement with the results of Gea and Luo \\cite{Gea04}. \n\nUsing the same notation, \\rf{h5} becomes\n\\beq{qwa}\nG_{33} (\\theta) =    \\frac14 \\,d_0\\,  \\big( \\Lambda\n\\cos 2\\theta -  \\cos 4\\theta \\big)\\, .\n\\end{equation}\nIn particular, if $- 1<\\Lambda <1$, then \n\\beq{qw}\nG_{33}(\\theta^*)  =   \\frac14 \\, d_0\\,  \\sin^2 2\\theta^*\\, .\n\\end{equation}\nThis implies that $\\theta= \\pm \\theta^*$ is a local minimum of ${\\cal E}$ iff $d_0$ is positive. The general analysis for 3D optimal orientation does not provide an explicit statement about global minima.   In order to show that it is a global maximum one must compare the value of the energy  ${\\cal E}$ at $\\theta= \\pm \\theta^*$ with its value at the other local minimum, as done in \\rf{q3}.  \n\n\\medskip\n\\noindent\n{\\bf Acknowledgment}\\,   I would like to acknowledge Hae Chang Gea for useful suggestions.  \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\n\nCavity opto-mechanics describes a paradigmatic system for quantum\nmetrology: a massive object with mechanical degrees of freedom is\ncoupled to and measured by a bosonic field. Interest in this generic\nsystem is motivated by several considerations.  For one, the system\nallows one to explore and address basic questions about quantum\nlimits to measurement. In this context, quantum limits to quadrature\nspecific and non-specific measurements, both for those performed\ndirectly on the mechanical object and also those performed through\nthe mediation of an amplifier \\cite{brag95qmbook}. Second, as a\ndetectors of weak forces, cavity opto-mechanical systems in the\nquantum regime may yield improvements in applications ranging from\nthe nanoscale (e.g.\\ for atomic or magnetic force microscopies) to\nthe macroscale (e.g.\\ in ground- or space-based gravity wave\nobservatories).  Finally, such systems, constructed with ever-larger\nmechanical objects, may allow one to test the validity of quantum\nmechanics for massive macroscopic objects.  Striking developments in\nthis field were presented at ICAP 2008 by Harris and Kippenberg.\n\nOur contribution to this developing field is the realization that a\ncavity opto-mechanical system can be constructed using a large gas\nof ultracold atoms as the mechanical object.  Having developed an\napparatus that allows quantum gases to be trapped within the optical\nmode of a high-finesse Fabry-Perot optical resonator, we are now\nable to investigate basic properties of opto-mechanical systems.\nSeveral of these investigations are described below.  The\natoms-based mechanical oscillator may be considered small by some,\nwith a mass ($\\simeq 10^{-17}$ g) lying geometrically halfway\nbetween the single-atom limit explored at the quantum regime in ion\nand atom traps ($10^{-22}$ g) \\cite[for example]{meek96,bouc99}, and\nthe small ($\\simeq 10^{-12}$ g) nanofabricated systems now\napproaching quantum limits \\cite{laha04approach,rega08}.\nNevertheless, our system offers the advantages of immediate access\nto the quantum mechanical regime, of the \\emph{ab initio}\ntheoretical basis derived directly from quantum optics and atomic\nphysics, and of the tunability and amenability to broad new probing\nmethods that are standard in ultracold atomic physics.  Our\nmotivation for probing cavity opto-mechanics with our setup is not\njust to poach the outstanding milestones of this field (e.g.\\\nreaching the motional ground state or observing measurement\nbackaction and quantum fluctuations of radiation pressure with a\nmacroscopic object \\cite{murc08backaction}).  Rather, we hope to\ncontribute to the development of macroscopic quantum devices by\nclarifying experimental requirements and the role of and limits to\ntechnical noise, developing optimal approaches to signal analysis\nand system control, exploring the operation and uses of multi-mode\nquantum devices, and defining different physical regimes for such\nsystems. Also, our opto-mechanical system may have direct\napplication as part of an atom-based precision (perhaps\ninterferometric) sensor.\n\n\\section{Collective modes of an intracavity atomic ensemble}\n\nThe theoretical reasoning for considering a trapped atomic gas\nwithin a high-finesse optical resonator as a macroscopic cavity\nopto-mechanical system is laid out in recent work\n\\cite{murc08backaction}. Recapping that discussion, we consider the\ndispersive coupling of an ensemble of $N$ identical two-level atoms\nto a single standing-wave mode of a Fabry-Perot cavity, obtaining\nthe spectrum of ``bright'' eigenstates of the atoms-cavity system\naccording to the following Hamiltonian:\n\\begin{equation}\n\\mathcal{H} = \\hbar \\omega_c \\hat{n} + \\sum_{i} \\frac{\\hbar g^2(z_i)}{\\Delta_{ca}}\n+ \\mathcal{H}_{a} + \\mathcal{H}_{in\/out}\n\\end{equation}\nHere $\\hat{n}$ is the cavity photon number operator, $\\Delta_{ca} = \\omega_c\n- \\omega_a$ is the difference between the empty-cavity and atomic\nresonance frequencies, and $g(z_i) = g_0 \\sin(k_p z)$ is the\nspatially dependent atom-cavity coupling frequency with $z_i$ being\nthe position of atom $i$ and $k_p$ being the wavevector at the\ncavity resonance.  The term $\\mathcal{H}_a$ describes the energetics of\natomic motion while $\\mathcal{H}_{in\/out}$ describes the electromagnetic\nmodes outside the cavity.  Note that this expression already treats\nthe atom-cavity coupling to second order in $g$. Repeating this\nanalysis starting from the first-order term does not change our\nconclusions substantially.\n\nNow, let us assume that all the atoms are trapped in harmonic\npotentials with ``mechanical'' trap frequency $\\omega_z$ and neglect\nmotion along directions other than the cavity axis.  Further, we\ntreat the atomic motion only to first order in atomic displacements,\n$\\delta z_i$, from their equilibrium positions, $\\bar{z}_i$;  i.e.\\\nwe assume atoms to be confined in the Lamb-Dicke regime with $k_p\n\\delta z_i \\ll 1$. We now obtain the canonical cavity\nopto-mechanical Hamitonian \\cite{kipp08sciencereview} as\n\\begin{equation}\n\\mathcal{H} = \\hbar \\omega_c^\\prime \\hat{n} + \\hbar \\omega_z \\hat{a}^\\dagger \\hat{a}\n- F \\hat{Z} \\hat{n} + \\mathcal{H}^\\prime_{a} + \\mathcal{H}_{in\/out} \\label{eq:hami}\n\\end{equation}\n\nWe make several steps to arrive at this expression. First, we allow\nthe cavity resonance frequency to be modified as $\\omega_c^\\prime =\n\\omega_c + \\sum_i g^2(\\bar{z}_i) \/ \\Delta_{ca}$, accounting for the cavity\nresonance shift due to the atoms at their equilibrium positions.\nSecond, we introduce the collective position variable $\\hat{Z} =\nN_{\\mbox{\\scriptsize{eff}}}^{-1} \\sum_i \\sin(2 k_p \\bar{z}_i) \\delta z_i$ that, along with\na weighted sum $\\hat{P} = \\sum_i \\sin(2 k_p \\bar{z}_i) p_i$ of the\natomic momenta $p_i$, describes the one collective motion within the\natomic ensemble that is coupled to the cavity-optical field. The\noperators $\\hat{a}$ and $\\hat{a}^\\dagger$ are defined conventionally for this\nmode. In our treatment, absent the presence of light within the\noptical cavity, this mode is harmonic, oscillating at the mechanical\nfrequency $\\omega_z$, and endowed with a mass $M$ equal to that of\n$N_{\\mbox{\\scriptsize{eff}}} = \\sum_i \\sin^2(2 k_p \\bar{z}_i)$ atoms. Third, we summarize\nthe opto-mechanical coupling by the per-photon force $F = N_{\\mbox{\\scriptsize{eff}}}\n\\hbar k g_0^2 \/ \\Delta_{ca}$ that acts on the collective mechanical mode.\nFinally, we lump all the remaining atomic degrees of freedom, and\nalso the neglected higher order atom-cavity couplings, into the term\n$\\mathcal{H}^\\prime_a$.\n\nWith this expression in hand, we may turn immediately to the\nliterature on cavity opto-mechanical systems to identify the\nphenomenology expected for our atoms-cavity system.  Several such\nphenomena are best described by referring to the opto-mechanical\nforce on the collective atomic mode, given as\n\\begin{equation}\n\\hat{\\mathcal{F}}_{\\mbox{\\scriptsize{opto}}} = - M \\omega_z^2 \\hat{Z} + F \\hat{n}. \\label{eq:optoforce}\n\\end{equation}\n\nWe consider the following effects:\n\\begin{itemize}\n\\item{} If we allow the state of the cavity to follow the atomic motion adiabatically ($\\omega_z \\ll \\kappa$),\nneglect quantum-optical fluctuations of the cavity field, and assume\nthe collective atomic displacement remains small, the linear\nvariation of $\\langle \\hat{n} \\rangle$ with $\\hat{Z}$ modifies the\nvibration frequency of the collective atomic motion.  Here, $\\kappa$\nis the cavity half-linewidth.  This modification, known as the\n``optical spring,'' has been observed in various opto-mechanical\nsystems and has been used to trap macroscopic objects optically\n\\cite{shea04spring,corb06spring,corb07gram}. We have made\npreliminary observations of the optical spring effect in our system\nas well.\n\n\\item{} For larger atomic displacements, the opto-mechanical force\nmay become notably anharmonic, and even, under suitable conditions,\nbistable \\cite{dors83bistability}.  Our observations of the\nresulting opto-mechanical bistability \\cite{gupt07nonlinear} are\ndiscussed in Sec.\\ \\ref{sec:bistab}.\n\n\\item{} When the cavity field no longer follows the atomic motion\nadiabatically, the opto-mechanical potential is no longer\nconservative.  The dramatic effects of such non-adiabaticity are the\ncavity-induced damping or coherent amplification of the mechanical\nmotion \\cite{brag67}.  Such effects of dynamical backaction have\nbeen detected in several micro-mechanical systems\n\\cite{kipp05anal,arci06,giga06cooling} and also for single\n\\cite{maun04cooling} or multiple atoms \\cite{chan03} trapped within\na cavity.\n\n\\item{} Finally, we consider also the effects of quantum-optical\nfluctuations of the intracavity photon number and, thereby, of the\noptical forces on the atomic ensemble.  It can be shown that these\nforce fluctuations represent the backaction of quantum measurements\nof the collective atomic position \\cite{murc08backaction}, as\ndescribed in Sec.\\ \\ref{sec:backaction}.\n\\end{itemize}\n\n\\section{Collective atomic modes in various regimes}\n\nThe theoretical treatment described above is suitable in the\nLamb-Dicke regime of atomic confinement and under the condition that\nthe linear opto-mechanical coupling term ($F \\hat{Z} \\hat{n}$) is\ndominant (i.e.\\ that the intracavity atomic gas is not tuned to\npositions of exclusively quadratic sensitivity).  These conditions\nare met in our experiments at Berkeley, where an ultracold gas of\nabout $10^5$ atoms of $^{87}$Rb is transported into the mode volume\nof a high-finesse Fabry-Perot optical resonator. The resonator\nlength is tuned so that the resonator supports one TEM$_{00}$ mode\nwith wavelength $\\lambda_T = 850$ nm (trapping light) and another\nwithin a given detuning $\\Delta_{ca}$ (in the range of 100's of GHz) of the\nD2 atomic resonance line (probe light). Laser light with wavelength\n$\\lambda_T$ is sent through the cavity to generate a 1D optical\nlattice potential in which the cold atomic gas is trapped (Fig.\\\n\\ref{fig:scheme}). The gas is strewn across over $> 100$ contiguous\nsites in this 1D optical lattice. Within each well, atoms are\nbrought by evaporative cooling to a temperature $T \\sim 700$ nK. At\nthis temperature, the atoms lie predominantly in the ground state of\nmotion along the cavity axis, with $\\hbar \\omega_z \/ k_B \\simeq 2 \\,\n\\mu\\mbox{K} \\gg T$, and the Lamb-Dicke condition is satisfied with\nrespect to the wavevector of probe light ($k_p \\simeq 2 \\pi \/ (780\n\\, \\mbox{nm})$) used to interrogate the atomic motion.\n\n\n\\begin{figure}\n\\begin{center}\n\\psfig{file=scheme.eps,width=4.5in}\n\\end{center}\n\\caption{Scheme for opto-mechanics with ultracold atoms in the\nLamb-Dicke confinement regime.  A high finesse cavity supports two\nlongitudinal modes -- one with wavelength of about 780 nm that is\nnear the D2 resonance of $^{87}$Rb atoms trapped within the\nresonator, and another with wavelength of about 850 nm.  Light at\nthe 850 nm resonance produces a one-dimensional optical lattice,\nwith trap minima indicated in orange, in which atoms are confined\nwithin the lowest vibrational band.  These atoms induce frequency\nshifts on the 780 nm cavity resonance.  The strength of this shift,\nand of its dependence on the atomic position, varies between the\ndifferent sites of the trapping optical lattice, as shown.\nNevertheless, in the Lamb-Dicke confinement regime, the complex\natoms-cavity interactions reduce to a simple opto-mechanical\nHamiltonian wherein a single collective mode of harmonic motion,\ncharacterized by position and momentum operators $\\hat{Z}$ and\n$\\hat{P}$, respectively, is measured, actuated, and subjected to\nbackaction by the cavity probe.} \\label{fig:scheme}\n\\end{figure}\n\nThe opto-mechanics picture of atomic motion in cavity QED has also\nbeen considered recently by the Esslinger group in Z\\\"{u}rich\n\\cite{bren08optonote}. There, a continuous Bose-Einstein condensate\nof $^{87}$Rb is trapped in a large-volume optical trap within the\ncavity volume. Yet, in spite of the stark differences in the\nexternal confinement and the motional response of the condensed gas,\na similar opto-mechanical Hamiltonian emerges.  In our prior\ndescription of the Lamb-Dicke regime, optical forces due to cavity\nprobe light are found to excite and, conversely, to make the cavity\nsensitive to a specific collective motion in the gas.  In the case\nof a continuous condensate, the cavity optical forces excite atoms\ninto a specific superposition of the $\\pm 2 \\hbar k_p$ momentum\nmodes. Interference between these momentum-excited atoms and the\nunderlying condensate creates a spatially (according to $k_p$) and\ntemporally (according to the excitation energy) periodic density\ngrating that is sensed via the cavity resonance frequency.  Thus,\nidentifying operators $\\hat{a}$ and $\\hat{a}^\\dagger$ with this\nmomentum-space excitation and the operator $\\hat{Z}$ with the density\nmodulation, we arrive again at the Hamiltonian of Eq.\\\n\\ref{eq:hami}.\n\nWe can attempt to bridge these two opto-mechanical treatments by\ntracking the response of an extended atomic gas to spatially\nperiodic optical forces (due to probe light at wavelength 780 nm) as\nwe gradually turn up the additional optical lattice potential (due\nto trapping light at wavelength 850 nm).  In the absence of the\nlattice potential, a zero-temperature Bose gas forms a uniform\nBose-Einstein condensate.  The excitations of this system are\ncharacterized by their momentum and possess an energy determined by\nthe Bogoliubov excitation spectrum; in Fig.\\ \\ref{fig:bands}(a), we\npresent this spectrum as a free-particle dispersion relation,\nneglecting the effects of weak interatomic interactions. The\nspatially periodic optical force of the cavity probe excites a\nsuperposition of momentum excitations as described above.\n\n\n\\begin{figure}\n\\begin{center}\n\\psfig{file=bands.eps,width=4.5in}\n\\end{center}\n\\caption{Influence of band structure on the opto-mechanical response\nof an ultracold atomic gas confined within a high-finesse\nFabry-Perot optical resonator.  We consider the relevant macroscopic\nexcitation produced by cavity probe light at wavevector $k_p = 2 \\pi\n\/ (780 \\, \\mbox{nm})$ within a Bose gas confined within a\none-dimensional optical lattice formed by light at wavevector $k_t =\n2 \\pi \/ (850 \\, \\mbox{nm})$ and with variable depth.  The gas is\ncooled to zero temperature, non-interacting, and extended evenly\nacross many lattice sites.  Energies are scaled by the recoil energy\n$E_r = \\hbar^2 k_t^2 \/ 2 m$ and wavevectors by $k_t$.  (a) With a\nweak lattice applied ($2 E_r$), the band structure for atomic\nexcitations (black lines) is slightly perturbed from the\nfree-particle excitations in the absence of a lattice (gray).  The\ncavity probe excites atoms primarily to states with quasi-momenta\n$\\pm 2 k_p$ within the second excited band, corresponding closely to\nmomentum eigenstates in the lattice-free regime.  (b) In a deep\nlattice ($15 E_r$), energy bands show little dispersion and are\nspaced by energies scaling as the square root of the lattice depth.\n(c) Lines show the energies of the three lowest energy states at\nquasi-momentum $2 k_p$ as a function of the lattice depth.  The\nrelative probability for excitation by cavity probe light to each of\nthese states, taken as the square of the appropriate matrix element,\nis shown by the width of the shaded regions around each line.  At\nzero lattice depth, cavity probe light excites the second excited\nband exclusively.  At large lattice depth, the excitation\nprobability to the first excited band grows while excitation to\nhigher bands is suppressed. At intermediate lattice depths, several\nexcited states are populated, indicating the onset of complex\nmulti-mode behaviour.} \\label{fig:bands}\n\\end{figure}\n\n\nAdding the lattice potential changes both the state of the\nBose-Einstein condensate, which now occupies the lowest Bloch state,\nand also the state of excitations, which are now characterized by\ntheir quasi-momentum and by the band index.  There are now many\nexcitations of the fluid that may be excited at the quasi-momentum\nselected by the spatially periodic cavity probe.  In the case that\nthe lattice is very shallow, shown in Fig.\\ \\ref{fig:bands}(a), the\ncavity probe will still populate only one excited state nearly\nexclusively.  Given the relation between the wavelengths of the\ntrapping (850 nm) and cavity-probe light (780 nm), this excited\nstate lies in the second excited band.  As the lattice is deepened,\nhowever, matrix elements connecting to quasi-momentum states on\nother bands will grow (shown in Fig.\\ \\ref{fig:bands}(c)).  Now our\nsimple opto-mechanical picture is made substantially more complex,\nwith multiple mechanical modes oscillating with differing mechanical\nfrequencies all influencing the optical properties of the cavity.\n\nContinuing to deepen the optical lattice this complexity will be\nalleviated when we reach the Lamb-Dicke regime, i.e.\\ as the\nLamb-Dicke parameter $k_p \\delta z$ becomes ever smaller, the\nprobabilities of excitation from the ground state via the cavity\nprobe hone in on the first excited band.  We calculate such\nprobabilities as $p_i \\propto \\left| <2 k_p; i| \\cos(2 k_p z) | g>\n\\right|^2$ where the ket is the $2 k_p$ quasi-momentum Bloch state\nin the $i$th band, and the bra is the ground state in the lattice\nconsidered. Here, we interpret excitations to higher bands as being\ncontrolled by terms of higher order in the Lamb-Dicke parameter,\ne.g.\\ excitations to the second excited band result from couplings\nthat are quadratic in the atomic positions.\n\nThus, we confirm that a simple opto-mechanics picture emerges for\nthe collective atomic motion within a cavity both in the shallow-\nand deep-lattice limits.  We note, however, that these limits differ\nin two important ways.  First, we see that the mechanical\noscillation frequency for the collective atomic motion is\nconstrained to lie near the bulk Bragg excitation frequency in the\nshallow-lattice limit, whereas it may be tuned to arbitrarily high\nfrequencies (scaling as the square root of the lattice depth) in the\ndeep-lattice limit.  The ready tunability of the mechanical\nfrequency in the latter limit may allow for explorations of quantum\nopto-mechanical systems in various regimes, e.g.\\ in the the\nresolved side-band regime where ground-state cavity cooling and also\nquantum-limited motional amplification are possible\n\\cite{marq07sideband,wils07groundstate}. Second, we see that the\nmechanical excitation frequency has a significant quasi-momentum\n(Doppler) dependence in the shallow-lattice limit. This dependence\nmakes it advantageous to use low-temperature Bose-Einstein\ncondensates for experiments of opto-mechanics, as indeed achieved in\nthe Z\\\"{u}rich experiments, so as to minimize the Doppler width of\nthe Bragg excitation frequency. In contrast, the excitation\nbandwidth is dramatically reduced (exponentially with the lattice\ndepth) in the deep-lattice limit. Thus, one can conduct\nopto-mechanics experiments with long-lived mechanical resonances in\nthe deep-lattice limit without bothering to condense the atomic gas.\nNevertheless, we note that variations in the mechanical frequency\ndue to the presence of significant radial motion (not considered in\nthis one-dimensional treatment) do indeed limit the mechanical\nquality factor in the Berkeley experiments.\n\n\\section{Effects of the conservative optomechanical potential: optomechanical bistability}\n\\label{sec:bistab}\n\nThe observation of cavity nonlinearity and bistability arising from\ncollective atomic motion is described in recent work\n\\cite{gupt07nonlinear}.  Briefly, we find that the optical force due\nto cavity probe light will displace the equilibrium collective\natomic position $\\langle \\hat{Z} \\rangle$, leading to a\nprobe-intensity-dependent shift of the cavity resonance frequency.\nBy recording the cavity transmission as cavity probe light was swept\nacross the cavity resonance, we observed asymmetric and shifted\ncavity resonance lines, and also hallmarks of optical bistability.\n\nRefractive optical bistability is well studied in a variety of\nexperimental systems \\cite{boyd03}.  One unique aspect of our\nexperiment is the observation of both branches of optical\nbistability at average cavity photon numbers as low as $0.02$.  The\nroot of such strong optical nonlinearities is the presence within\nthe cavity of a medium that (1) responds significantly to the\npresence of infrequent cavity photons (owing to the strong\ncollective cooperativity) and (2) recalls the presence of such\nphotons for long coherence times (here, the coherence is stored\nwithin the long-lived collective motion of the gas).  It is\ninteresting to consider utilizing such long-lived motional\ncoherence, rather than the shorter-lived internal state coherence\ntypically considered, for the various applications of cavity QED and\nnonlinear optics in quantum information science, e.g.\\ photon\nstorage and generation, single-photon detection, quantum logic\ngates, etc.\n\nSuch motion-induced cavity bistability can also be understood in the\ncontext of the opto-mechanical forces described by Eq.\\\n\\ref{eq:optoforce}.  Neglecting the non-adiabatic following of the\ncavity field to the collective motion (essentially taking $\\kappa \/\n\\omega_z \\rightarrow \\infty$ so that dynamical backaction effects\nare neglected) and also the quantum fluctuations of the cavity\nfield, we may regard atomic motion in an optically driven cavity to\nbe governed by an opto-mechanical potential of the form\n\\begin{equation}\nU(Z) = \\frac{1}{2} M \\omega_z^2 Z^2 + n_{\\mbox{\\scriptsize{max}}} \\hbar \\kappa \\,\n\\arctan\\left( \\frac{\\Delta_{pc} - F Z \/ \\hbar}{\\kappa}\\right)\n\\label{eq:ucoll}\n\\end{equation}\nHere $\\Delta_{pc}$ is the detuning of the constant frequency probe from the\nmodified cavity resonance frequency $\\omega_c^\\prime$, and $n_{\\mbox{\\scriptsize{max}}}$\nis the average number of cavity photons when the cavity is driven on\nresonance.\n\nThe form of this potential is sketched in Fig.\\\n\\ref{fig:bistablepotential} for different operating conditions of\nthe atoms-cavity system.  Cavity bistability \\cite{gupt07nonlinear}\nis now understood as reflecting an effective potential for the\ncollective atomic variable $Z$ that has two potential minima.\nRemarkably, these potential minima may be separated by just\nnanometer-scale displacements in $Z$.  Even though the inherent\nquantum position uncertainty of each individual atom (10's of nm) is\nmuch larger than this separation, the reduced uncertainty in the\ncollective variable $Z$ allows for these small displacements to\nyield robust and distinct experimental signatures in the cavity\ntransmission.\n\n\n\\begin{figure}\n\\begin{center}\n\\psfig{file = bistablepotential.eps, height =2in}\n\\end{center}\n\\caption{\\small For different stable regimes of cavity operation,\nthe cavity-relevant collective mode of the intracavity atomic\nensemble is trapped in a particular minimum of the effective\npotential $U(Z)$ (bottom).  In the regime of bistability, the two\nstable cavity states reflect the presence of two potential minima.}\n\\label{fig:bistablepotential}\n\\end{figure}\n\n\n\\section{Quantum fluctuations of the optomechanical potential: measurement backaction}\n\\label{sec:backaction}\n\nAside from the conservative forces described above, the intracavity\natomic medium is subject also to dipole force fluctuations arising\nfrom the quantum nature of the intracavity optical field. Indeed,\nshould these force fluctuations be especially large, the picture of\ncavity optical non-linearity and bistability described in the\nprevious section, in which we implicitly assume that the collective\natomic motion may follow adiabatically into a local minimum of an\nopto-mechanical potential, must be dramatically modified.  To assess\nthe strength of such force fluctuations, let us consider the impact\non the atomic ensemble of a single photon traversing the optical\ncavity.  During its residence time of $\\sim 1 \/ 2 \\kappa$, such a\nphoton would cause a dipole force that imparts an impulse of $\\Delta\nP = f\/(2 \\kappa)$ on the atomic medium, following which the\ncollective mode is displaced by a distance $\\Delta Z = \\Delta P \/ (M\n\\omega_z)$; in turn, this displacement will shift the cavity\nresonance frequency by $F \\Delta Z \/ \\hbar$.  Comparing this\nsingle-photon-induced, transient frequency shift with the cavity\nhalf-linewidth leads us to define a dimensionless ``granularity\nparameter'' as\n\\begin{equation}\n\\epsilon = \\sqrt{ \\frac{F \\Delta Z}{\\hbar \\kappa}} = \\frac{F\nZ_{\\mbox{\\scriptsize{ho}}}}{\\hbar \\kappa},\n\\end{equation}\nwhere $Z_{\\mbox{\\scriptsize{ho}}} = \\sqrt{\\hbar \/ 2 M \\omega_z}$ is the harmonic\noscillator length for the atomic collective mode.  The condition\n$\\epsilon > 1$ marks the granular (or strong) opto-mechanical\ncoupling regime in which the disturbance of the collective atomic\nmode by single photons is discernible both in direct quantum-limited\nmeasurements of the collective atomic motion and also in subsequent\nsingle-photon measurements of the cavity resonance frequency.  In\nour experiments, the granularity parameter is readily tuned by\nadjusting frequency difference between the cavity and atomic\nresonance $\\Delta_{ca}$.  Under conditions of our recent work, the\ngranular regime is reached at  $|\\Delta_{ca}|\/(2\\pi) < 27$ GHz.\n\n\nIn recent work, we have focused on effects of fluctuations of the\ndipole force in the non-granular regime, attained at atom-cavity\ndetunings in the 100 GHz range.  As described in our work\n\\cite{murc08backaction}, and also derived in earlier treatments\n\\cite{hora97,vule00,murr06,marq07sideband}, these fluctuations will\ncause the motional energy of the collective atomic mode to vary\naccording to the following relation:\n\\begin{equation}\n\\frac{d}{d t} \\langle a^\\dagger a \\rangle = \\kappa^{2} \\epsilon^2\n\\left[S_{nn}^{(-)} + \\left(S_{nn}^{(-)} - S_{nn}^{(+)}\\right) \\langle\na^\\dagger a \\rangle \\right] \\label{eq:adaga}\n\\end{equation}\nHere, the relevant dipole force fluctuations are derived from the\nspectral density of intracavity photon number fluctuations at the\nmechanical frequency $\\omega_z$, calculated for a\ncoherent-state-driven cavity as $S_{nn}^{(\\pm)} = 2 \\langle n \\rangle\n\\kappa (\\kappa^{2} + (\\Delta_{pc} \\pm \\omega_z)^{2})^{-1}$.  Eq.\\\n\\ref{eq:adaga}, which can be derived readily from a rate-equation\napproach \\cite{marq07sideband}, reveals two manners in which the\nmechanical oscillator responds to a cavity optical probe: momentum\ndiffusion, which raises the mechanical oscillator energy at a\nconstant rate, and the dynamic backaction effects of cavity-based\ncooling or amplification of the mechanical motion, described by an\nexponential gain or damping.\n\nThe mechanical momentum diffusion in an opto-mechanical system plays\nthe essential metrological role of providing the backaction\nnecessary in a quantum measurement, as discussed, for example, by\nCaves in the context of optical interferometry \\cite{cave81}.  For\n$\\omega_z \\ll \\kappa$, we see that, at constant circulating power in\nthe cavity, this diffusion is strongest for probe light at the\ncavity resonance and weaker away from resonance.  This dependence on\nthe intensity and detuning of the cavity probe light precisely\nmatches the rate of information carried by a cavity optical probe on\nthe state of the mechanical oscillator.  To elucidate this point, we\nrecall that, under constant drive by a monochromatic input field,\nthe intracavity electric field oscillates at the input field\nfrequency with complex amplitude $E_{cav} = \\eta \/ (\\kappa - i\n\\Delta_{pc})$.   A displacement by $\\Delta Z$ of the mechanical oscillator\nvaries the probe-cavity detuning by $F \\Delta Z \/ \\hbar$. In\nresponse, the electric field in the cavity varies as\n\\begin{equation}\nE_{cav} \\simeq E_0 \\left( 1 + \\frac{i}{\\kappa - i \\Delta_{pc}} \\frac{F\n\\Delta Z}{\\hbar} \\right) = E_0 + E_{sig},\n\\end{equation}\nwhere $E_0$ is the cavity field with the cantilever at its\nequilibrium position and we expand to first order in $\\Delta Z$. The\nsensitivity of the cavity field to the cantilever displacement, at\nconstant intracavity intensity (constant $E_0$), is determined by\nthe magnitude of $|E_{sig}\/E_0|^2 \\propto 1 \/ (1 + \\Delta_{pc}^2 \/\n\\kappa^2)$; this functional dependence matches that of the momentum\ndiffusion term, supporting its representing  measurement backaction.\n\nTo measure this backaction heating, we take advantage of several\nfeatures of our experiment.  First, by dint of the low temperature\nof our atomic ensemble, we ensure that the effects of dynamical\nbackaction (cooling and amplification) are negligible.  Second, the\nlow quality-factor of our mechanical oscillator ensures that the\nmomentum diffusion of the collective atomic motion leads to an\noverall heating of the atomic ensemble, allowing us to measure this\ndiffusion bolometrically.  Third, the large single-atom\ncooperativity in our cavity QED system implies that this backaction\nheating of the entire atomic ensemble dominates the single-atom\nheating due to atomic spontaneous emission.  And, fourth, owing to\nthe finite, measured depth of our intracavity optical trap,\nbackaction heating can be measured via the light-induced loss rate\nof atoms from the trap.  The measured light-induced heating rate was\nfound to be in good agreement with our predictions, providing the\nfirst quantification of measurement backaction on a macroscopic\nobject at a level consistent with quantum metrology limits.\n\n\\section{Future developments: cavity QED\/atom chips}\n\nWhile continuing explorations of quantum opto-mechanics in our\nexisting apparatus, we are also developing an experimental platform\nthat integrates the capabilities of single- and many-atom cavity QED\nonto microfabricated atom chips.  Similar platforms have been\ndeveloped recently by other groups \\cite{tepe06res,colo07}. Aside\nfrom enabling myriad applications in quantum atom optics and atom\ninterferometry, we anticipate the cavity QED\/atom chip to provide\nnew capabilities in cold-atoms-based opto-mechanics. For instance,\nthe tight confinement provided by microfabricated magnetic traps\nwill allow atomic ensembles to be confined into single sites of the\nintracavity optical lattice potential, providing a means of tuning\nthe opto-mechanical coupling between terms linear or quadratic in\n$\\hat{Z}$.  As emphasized by Harris and colleagues\n\\cite{thom08membrane}, a purely quadratic coupling may allow for\nquantum non-demolition measurements of the energy of the macroscopic\nmechanical oscillator.\n\n\\begin{figure}\n\\begin{center}\n\\psfig{file = atom_chip_figure.eps, height = 2 in}\n\\end{center}\n\\caption{An integrated cavity QED\/atom chip. (a) A top view of the\nmicrofabricated silicon chip shows etched trenches, later\nelectroplated with copper and used to tailor the magnetic field\nabove the chip surface.   The left portion of the image shows wire\npatterns used for producing the spherical-quadrupole field of a\nmagneto-optical trap and also the Ioffe-Pritchard fields for\nproducing stable magnetic traps.  Serpentine wires spanning the\nentire chip form a magnetic conveyor system to translate atoms to\nthe optical cavities that are located in the right half of the\nimage.  (b) A detailed view shows the serpentine wires and also a\ntwo-wire waveguide surrounding a central, rectangular hole that\npierces the atom chip.  (c) Fabry-Perot cavities are formed by\nmirrors straddling the atom chip. Between the mirrors, the chip is\nthinned to below 100 $\\mu$m -- outlines of the thinned areas are\nseen also in (a).  The cavity mode light passes unhindered through\nthe chip via the microfabricated holes shown in (b).}\n\\label{fig:atomchip}\n\\end{figure}\n\n\n\n\\bibliographystyle{ws-procs9x6}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} The CFHT Adaptive Optics Bonnette (AOB)\nprovides excellent correction using much fainter guide stars than\nprevious systems so that observations of many extragalactic targets are\nnow possible, using either a nearby reference star or the object\nitself. A wide variety of extragalactic objects have now been observed\nwith AOB and some of these are used as examples to discuss some of the\nproblems encountered and to demonstrate the potential of adaptive\noptics for studies of the high redshift universe.\n\n\\section{The Adaptive Optics Bonnette} \nThe CFHT AOB is based on a curvature wavefront system (Roddier et al. 1991)\nwith 19 subapertures.  A complete description of the system is given by\nArsenault et al. (1994), and a comprehensive discussion of its\nperformance by Rigaut et al. (1997).  If the target is to be observed in\nthe infrared, a dichroic is used to reflect the visible light to\nthe wavefront sensor (WFS) while transmitting the IR light to the\nscience detector, but if the target is to be observed in the visible, a\nbeamsplitter must be used to send a percentage of the light to the\nWFS.  The delivered image quality depends on the brightness of the\nreference star, its distance from the target, the seeing at the time of\nobservation and the wavelength of observation. To gain an appreciation\nof how these variables affect the images and what can be expected, the\nreader is encouraged to visit the ``performance meter'' at\nhttp:\/\/www.cfht.hawaii.edu\/manuals\/aob\/psf.html  To achieve FWHM\n$\\sim$0\\farcs12 imaging in the near-IR (diffraction-limited) and  at\n$I$ under median seeing conditions, the reference star must be brighter\nthan R $\\sim$14 and within $\\sim$30\\arcsec\\ of the target. However, the\nsystem will provide good correction on stars as faint as R = 17 under\ngood seeing conditions, and the reference star can be located anywhere\nwithin the 90\\arcsec\\ diameter field.  AOB offers considerable\nadvantages in terms of operational efficiency, since it is literally a\npush-button operation  that eliminates all the overhead of focussing\nand guiding associated with conventional observing.\n\nThe observations reported here were all made with the University of\nMontreal infrared camera MONICA (Nadeau et al. 1994) which was modified\nto give a pixel scale of 0\\farcs034 and hence a field of\n8\\farcs8$\\times$8\\farcs8.  This very small field is a significant\nhandicap for many observations, as is the fact that the detector\nsuffers from persistence problems, i.e., a bright source leaves a\nresidual image that slowly decays with time.  Since the site seeing is\nvariable, observations of relatively bright stars usually {\\it have} to\nbe carried out in order to monitor the PSF (point-spread function). The\nsequence of observations of very faint sources thus has to be carefully\nplanned to minimize such problems while adequately monitoring the PSF\nto attain the scientific goals.  V\\'eran et al. (1997) have recently\ndemonstrated that statistics of the wavefront sensor signals can be\nused to derive an excellent model of the average PSF. This is obviously\nideal in that  the synthetic PSF is simultaneous with the the target\nobservation and it obviates the requirement of directly observing the\nbright star.\n\n \n\\section{Examples of Self-referencing Targets} \nObjects such as the nuclei of nearby galaxies, seyferts and AGN can be\nusually guided on directly, i.e., the nucleus is often sufficiently\npoint-like to enable the WFS to function properly as long as there is\nsufficient flux.  For example guiding on the nucleus of M31 was\nstraightforward, resulting in diffraction-limited images (FWHM =\n0\\farcs12) of the stars in the nuclear bulge region, despite the fact\nthat it is double with a fainter component at a distance of\n$\\sim$0\\farcs5. The near-IR image quality is comparable to that of\nWFPC on $HST$, enabling accurate magnitudes and colors to be\ndetermined for a significant number of stars in the bulge (Davidge et\nal.  1997).  During the commissioning period, other nearby galaxies,\nAGN and Seyfert 1 type galaxies with point-like nuclei and magnitudes\nin the range m = 11--14 (e.g., bright ones like NGC 4151 to fainter\nones like NGC 6814) were observed with AOB, guiding on the bright\nnuclei themselves. However, guiding was  not successful on more diffuse\nobjects such as the V = 13 nucleus of the Seyfert 2 galaxy Mrk 266, or\nthe large bright elliptical galaxy which hosts 3C296.\n\nThe small MONICA  field  complicates observations of such targets since\nneither the sky nor PSF stars are normally included in the frames.  An\naccurate estimate of the PSF is essential for these sources in order to\nadequately subtract the effects of the bright point source from the\nnuclear region, so a nearby reference star of similar brightness must\nbe monitored throughout the sequence of observations.\n\n\\section{Targets with guide stars}\nNearby guide stars must be used for targets which are not sufficiently\nconcentrated or bright enough for the WFS to perform adequately. In\nthis case, not only are observations of the guide star itself required\nto monitor the PSF variations, but measurements of starfields (e.g.,\nglobular cluster fields) are also required to estimate the degradation\nof the PSF due to anisoplanatic effects.  Steinbring (1997) has developed\nsemi-empirical software that models this degradation over the field as\na function of the seeing and produces an ``off-axis PSF\" that can be\nused to help analyse the target data. For some projects this is not\nimportant, but for the observations of quasar host galaxies, for\nexample, it is vital.  Studies of the latter are notoriously difficult\ndue both to the faintness of the galaxy itself plus the superposition\nof the bright nuclear source. Some quasars have already been\nobserved with AOB, both in the visible region (at $I$) with a CCD detector and\nin the near infrared (mostly at $H$).  Analysis of the z = 1.1 quasar,\n1055.3+019, which is well-resolved in both $I$ and $H$, has recently\nbeen published by Hutchings et al. (1998) who suggest that the host\ngalaxy has been undergoing a close encounter or merger event.\n\nThe components of multiply-imaged gravitational lenses formed by\ngalaxies in the line of sight to distant quasars typically have\nsubarcsecond separations and consequently their study and monitoring\ncould significantly benefit from the improved resolution offered by\nadaptive optics.  At present, about $\\sim$35 such lenses are known\n(Kochanek 1997) but virtually none of them have\nnearby guide stars that are sufficiently bright to give good\ncorrection. One exception is SBS 1520+530, a\ndoubly-imaged BAL quasar with a separation of 1\\farcs6 which happens to\nbe only 13\\arcsec\\ from a m$\\sim$12 star. $H$ band images with FWHM =\n0\\farcs15 taken with AOB reveal the lensing galaxy 0\\farcs40 from the\nfainter component, offset 0\\farcs12 from the line joining components A\nand B (Crampton, Schecter and Beuzit 1997).\n\n\\section{Registering invisible targets}\n\nTargets which are very faint, particularly those that are diffuse,\noften present additional problems since the MONICA field is so small\nthat there aren't any objects or features that can be used to register\nthe images. In general, near-IR images have to be dithered to remove the\neffects of bad pixels and to improve the flat-field, offsets are often\nrequired to enlarge the area surveyed, and differential flexure between\nthe WFS and the detector may produce additional shifts. Consequently,\nregistering and superimposing the images is not trivial. The AOB WFS\ncoordinates are recorded in the FITS headers and observations of star\nfields have been used to calibrate these positions in terms of pixel\nlocation on the detector and\/or equatorial coordinates.  Repeated\nobservations of the same point source (e.g., a quasar) over several\nhours show that the WFS positions can be used to register frames with a\ndispersion of 1.4 pixels or 0\\farcs05. Since the delivered images in\nthe near-IR usually have FWHM $\\sim$0\\farcs12, this is clearly\ninadequate, so future systems should be designed with a low-order WFS\nincorporated close to the detector  to minimize flexure and allow for\nsub-pixel ``drizzling\" and registration.  Ideally, the coordinates of\nthis WFS should allow  diffraction-limited images to be registered to\nbetter than $\\sim$10\\% of their FWHM; for 8m telescopes this will be of\nthe order of a few mas.\n\nAlthough the 0\\farcs034 pixel scale is necessary to properly sample the\n$\\sim$0\\farcs1 resolution delivered by AOB, many extragalactic projects\nwould benefit from a larger field. KIR, a new 1K$\\times$1K camera with\nthree times higher sensitivity and a field of 36\\arcsec\\ has just been\ncommissioned, so registration will be easier. Even so, astrometry to\nthe precision that is possible with 0\\farcs1 images will be\nchallenging:  relative positions can be measured to an accuracy of a\nfew mas, but tying these to an absolute reference frame is difficult.\nSince many extragalactic targets are diffuse, a larger pixel scale\nwould not be a disadvantage for most investigations, and the larger\nfield would significantly improve the probability of including stars,\nproviding even better registration and astrometry.\n\n\nThe jet of 3C273 is an example of an object which is barely visible,\neven on background-subtracted 300s exposures at $H$ and $K$.  The jet\nis located between 11 -- 21\\arcsec\\ from 3C273 which, of magnitude R =\n13, was used as the guide star for AOB.  Given the small field of the\ndetector, the jet had to be observed at two different locations along\nthe jet and then all the frames were registered using WFS positions.\nUnfortunately the site seeing was poor on average and very variable\n(from 0\\farcs5 to 1\\farcs2) during the observations so, combined\nwith registration uncertainties, the resulting (preliminary)\nimage has a resolution of only $\\sim$0\\farcs3. Observations\nwith the new KIR camera should be significantly better.\n\n\n\n\\section{Summary}\nOur experience with AOB demonstrates that adaptive optic systems can be\nmade robust, reliable, efficient and user-friendly and will routinely\ndeliver 0\\farcs1 resolution images from $I$ to $K$ with substantial\ngains in the visible. AOB observations of many extragalactic targets\nare now feasible and competitive with $HST$.  Images of most of the\nobjects discussed in this article can be found on the web at\nhttp:\/\/www.hia.nrc.ca\/science\/instrumentation\/optical\/pueo2\/pueo2.html .\nSpectroscopic observations using instruments designed to exploit the\nAOB image quality are just beginning, although to obtain reasonable\nspectral resolution simultaneously with  high spatial resolution will\nrequire adaptive optic systems on 8--10m telescopes for the majority\nof extragalactic targets.\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\n\n\nProduced by Camilo Bernard and Marc D'Hooghe at\nhttp:\/\/www.freeliterature.org (Images at the Internet\nArchive, scanned by Google)\n\n\n\n\n\nTHE TRAIL HUNTER.\n\nA TALE OF THE FAR WEST.\n\nBY\n\nGUSTAVE AIMARD,\n\nAUTHOR OF \"THE PRAIRIE FLOWER,\" \"THE INDIAN CHIEF,\" ETC.\n\n\nLONDON:\n\nWARD AND LOCK,\n\n158, FLEET STREET.\n\n1861.\n\n\n\n\nPREFACE.\n\n\nThe present volume of Aimard's Indian Tales is devoted to the earlier\nadventures of those hunters, whose acquaintance the reader has formed,\nI trust with pleasure, in the preceding series. It does not become me to\nsay anything further in its favour, than that the sustained interest of\nthe narrative, which has been regarded as the charm of stories referring\nto life in the desert and prairie, has not been departed from in this\ninstance. The stories themselves supply an innate proof of the writer's\ncorrectness to Nature, and, in truth, many of the scenes are so\nstartling that they must be the result of personal observation.\n\nIn conclusion, I may be permitted to thank the Press generally for the\nkindly aid they have afforded me in making the English translation of\nAimard's volumes known to the British reading public, and the hearty way\nin which they have recognized the merits of the previous series. It\nwould be an easy task to collect paragraphs, expressing a belief that\nAimard is second to none of the writers who have hitherto described\nIndian life and scenery; but I prefer to rest my hopes of success on the\ninherent qualities of his stories.\n\n                                                      LASCELLES WRAXALL.\n\n\n\n\nCONTENTS.\n\n\n   PART I. RED CEDAR                       PART II. EL PRESIDIO DE SANTA FE\n\n\n   I.--The Virgin Forest                   I.--El Rancho de Coyote\n   II.--The Contest                        II.--The Cuchillada\n   III.--Don Miguel Zarate                 III.--The Hunters\n   IV.--The Peccaries                      IV.--Sunbeam\n   V.--The Wound                           V.--The Adoption\n   VI.--The Squatter's Shanty              VI.--The Missionary\n   VII.--The Rangers                       VII.--The Interview\n   VIII.--The Valley of the Buffalo        VIII.--The Prison\n   IX.--The Assassination                  IX.--The Embassy\n   X.--The Sachem of the Coras             X.--The Presentation\n   XI.--Conversation                       XI.--Psychological\n   XII.--El Meson                          XII.--Diamond Cut Diamond\n   XIII--Red Cedar                         XIII.--A Stormy Discussion\n   XIV.--The Two Hunters                   XIV.--The Mystery\n   XV.--Fray Ambrosio                      XV.--The Ambuscade\n   XVI.--Two Varieties of Villains         XVI.--A Friendly Discussion\n   XVII.--El Canyon Del Buitre             XVII.--Nathan\n   XVIII.--Father Seraphin                 XVIII.--The Wounded Man\n   XIX.--Unicorn                           XIX.--Indian Diplomacy\n   XX.--The Hunt of Wild Horses            XX--The Stranger\n   XXI.--The Surprise                      XXI.--General Ventura\n   XXII.--The Meeting                      XXII.--The Comanches\n   XXIII.--The Abduction                   XXIII.--Negotiations\n   XXIV.--The Revolt                       XXIV.--Free\n                                           XXV.--The Meeting\n                                           XXVI.--Dona Clara\n                                           XXVII.--El Vado del Toro\n\n\n\n\n\n\n\nTRAIL HUNTER BY GUSTAVE AIMARD.\n\n\n\nCHAPTER I.\n\nTHE VIRGIN FOREST.\n\n\nIn Mexico the population is only divided into two classes, the upper and\nthe lower. There is no intermediate rank to connect the two extremes,\nand this is the cause of the two hundred and thirty-nine revolutions\nwhich have overthrown this country since the declaration of its\nindependence. Why this is so is simple enough. The intellectual power is\nin the hands of a small number, and all the revolutions are effected by\nthis turbulent and ambitious minority; whence it results that the\ncountry is governed by the most complete military despotism, instead of\nbeing a free republic.\n\nStill the inhabitants of the States of Sonora, Chihuahua, and Texas have\nretained, even to the present day, that stern, savage, and energetic\nphysiognomy which may be sought in vain among the other States of the\nConfederation.\n\nBeneath a sky colder than that of Mexico, the winter, which frequently\ncovers the rivers of the region with a thick layer of ice, hardens the\nmuscles of the inhabitants, cleanses their blood, purifies their hearts,\nand renders them picked men, who are distinguished for their courage,\ntheir intelligence, and their profound love of liberty.\n\nThe Apaches, who originally inhabited the greater portion of New Mexico,\nhave gradually fallen back before the axe of the pioneers; and after\nretiring into the immense deserts that cover the triangle formed by the\nRio Gila, the Del Norte, and the Colorado, they ravage almost with\nimpunity the Mexican frontiers, plundering, firing, and devastating all\nthey meet with on their passage.\n\nThe inhabitants of the countries we alluded to above, held in respect by\nthese ever-shifting savages, are in a state of continual warfare with\nthem, always ready to fight, fortifying their haciendas, and only\ntravelling with weapons in their hands.\n\nEl Paso del Norte may be regarded as the outpost of the civilised\nportion of Mexico. Beyond that, to the north and north-west, extend the\nvast unfilled plains of Chihuahua, the _bolson_ of Mapimi, and the arid\ndeserts of the Rio Gila. These immense deserts, known by the name of\nApacheria, are still as little investigated as they were at the close of\nthe eighteenth century. El Paso del Norte owes its name to its situation\nnear a ford of the Rio Del Norte. It is the oldest of all the New\nMexican settlements, and its establishment dates back to the close of\nthe sixteenth century. The present settlement is scattered for a\ndistance of about ten miles along the banks of the Del Norte, and\ncontains four thousand inhabitants at the most. The _plaza_, or village\nof the Paso, is situated at the head of the valley: at the other\nextremity is the Presidio of San Elezario. All the interval is occupied\nby a continuous line of white, flat-roofed houses, buried in gardens,\nand surrounded by vineyards. About a mile above the ford the stream is\ndammed up, and led by a canal into the valley, which it waters.\nApacheria begins only a few miles from this settlement.\n\nIt is easily seen that the foot of civilised man has only trodden\ntimidly and at rare intervals this thoroughly primitive country, in\nwhich nature, free to develop herself under the omnipotent eye of the\ncreator, assumes an aspect of incredible beauty and fancifulness.\n\nOn a lovely morning in the month of May, which the Indians call \"the\nmoon of the flowers,\" a man of high stature, with harsh and marked\nfeatures, mounted on a tall, half-tamed steed, started at a canter from\nthe plaza, and after a few minutes of hesitation, employed in realising\nhis position, resolutely buried his spurs in the horse's flanks, crossed\nthe ford, and after leaving behind him the numerous cottonwood trees\nwhich at this spot cover the river banks, proceeded toward the dense\nforest that flashed on the horizon.\n\nThis horseman was dressed in the costume generally adopted on the\nfrontiers, and which was so picturesque that we will give a short\ndescription of it. The stranger wore a pelisse of green cloth,\nembroidered with silver, allowing a glimpse of an elegantly-worked\nshirt, the collar of which was fastened by a loosely-knotted black silk\nhandkerchief, the ends passed through a diamond ring. He wore green\ncloth breeches, trimmed with silver, and two rows of buttons of the same\nmetal, and fastened round the hips by a red silken scarf with gold\nfringe. The breeches, open on the side half way up the thigh, displayed\nhis fine linen drawers beneath: his legs were defended by a strip of\nbrown embossed and stamped leather, called _botas vaqueras_, attached\nbelow the knee by a silver garter. On his heels enormous spurs clanked.\nA _manga_, glistening with gold, and drawn up on the shoulder, protected\nthe upper part of his body, while his head was sheltered from the\nburning sunbeams by a broad-leafed hat of brown stamped felt, the crown\nof which was contracted by a large silver _toquilla_ passed twice or\nthrice round it.\n\nHis steed was caparisoned with graceful luxuriousness, which heightened\nall its beautiful points: a rich saddle of embossed leather, adorned\nwith massive silver, on the back of which the _zarape_ was fastened;\nwide Moorish silver stirrups, and handsome water bottles at the\nsaddle-bow; while an elegant _anquera_, made of openwork leather, and\ndecorated with small steel chains, entirely covered the horse's croup,\nand sparkled with its slightest movement.\n\nThe stranger appeared, judging from the luxury he displayed, to belong\nto the high class of society. A _machete_ hung down his right side,\ntwo pistols were passed through his girdle, the handle of a long knife\nprotruded from his right boot, and he held a superbly damascened rifle\nacross the saddle in front of him.\n\nBending over the neck of his galloping steed, he advanced rapidly\nwithout looking round him, although the landscape that lay extended\nbefore him was one of the most attractive and majestic in those regions.\n\nThe river formed the most capricious windings in the centre of a terrain\ndiversified in a thousand strange ways. Here and there on the sandy\nbanks enormous trees might be seen lying, which, dried up by the sun,\nevidenced, in their washed-out appearance, that they had been dead for\ncenturies. Near the shallow and marshy spots, caymans and alligators\nwandered about awkwardly. At other places, where the river ran almost\nstraight, its banks were uniform, and covered with tall trees, round\nwhich creepers had twined, and then struck root in the ground again,\nthus forming the most inextricable confusion. Here and there small\nclearings or marshy spots might be detected in the midst of the dense\nwood, often piled up with trees that had died of old age. Further on,\nother trees, which seemed still young, judging from their colour and the\nsolidity of their bark, fell into dust with the slightest breath of\nwind.\n\nAt times, the earth, entirely undermined beneath, drawn down by its own\nweight, dragged with it the wood which it bore, and produced a crashing,\nconfused sound, which was returned on all sides by the echo, and\npossessed a certain degree of grandeur in this desert, whose depths no\nman has ever yet ventured to scrutinise.\n\nStill the stranger galloped on, with his eye ardently fixed before him,\nand not appearing to see anything. Several hours passed thus: the\nhorseman buried himself deeper in the forest. He had left the banks of\nthe river, and only progressed with extreme difficulty, through the\nentanglement of branches, grass, and shrubs, which at every step\narrested his movements, and forced him to make innumerable turnings. He\nmerely reined in his horse now and then, took a glance at the sky, and\nthen started again, muttering to himself but one word:\n\n\"_Adelante_! (Forward!)\"\n\nAt length he stopped in a vast clearing, took a suspicious glance around\nhim, and probably reassured by the leaden silence which weighed on the\ndesert, he dismounted, hobbled his horse, and took off its bridle that\nit might browse on the young tree shoots. This duty accomplished, he\ncarelessly lay down on the ground, rolled a maize cigarette in his\nfingers, produced a gold _mechero_ from his waist belt, and struck a\nlight.\n\nThe clearing was of considerable extent. On one side the eye could\nsurvey with ease, through the trees, the widely extending prairie, on\nwhich deer were browsing with security. On the other side, the forest,\nwilder than ever, seemed, on the contrary, an impassable wall of\nverdure. All was abrupt and primitive at this spot, which the foot of\nman had so rarely trodden. Certain trees, either entirely or partially\ndried up, offered the vigorous remains of a rich and fertile soil;\nothers, equally ancient, were sustained by the twisted creepers, which\nin the course of time almost equalled their original support in size:\nthe diversity of the leaves produced the strangest possible mixture.\nOthers, containing in their hollow trunk a manure which, formed of the\nremains of their leaves and half-dead branches, had warmed the seeds\nthey had let fall, and offered, in the young shoots they contained, some\ncompensation for the loss of their father tree.\n\nIn the prairies, nature, ever provident, seems to have been desirous to\nshelter from the insults of time certain old trees, patriarchs of the\nforest which are crushed beneath the weight of ages, by forming them a\ncloak of greyish moss, which hangs in festoons from the highest branches\nto the ground, assuming the wildest and most fantastic shapes.\n\nThe stranger, lying on his back, with his head resting on his two\ncrossed hands, was smoking with that beatitude, full of ease and sloth,\nwhich is peculiar to the Hispano-Americans. He only interrupted this\ngentle occupation to roll a fresh cigarette and cast a glance around,\nwhile muttering:\n\n\"Hum! He keeps me waiting a long time.\"\n\nHe emitted a puff of bluish smoke, and resumed his first position.\nSeveral hours passed thus. Suddenly, a rather loud rustling was heard in\nthe thicket, some distance behind the stranger.\n\n\"Ah, ah!\" he said, \"I fancy my man is coming at last.\"\n\nIn the meanwhile, the sound became louder, and rapidly approached.\n\n\"Come on, hang it!\" the horseman shouted, as he rose. \"By our Lady of\nPilar! You have surely been keeping me waiting long enough.\"\n\nNothing appeared: the clearing was still deserted, although the sound\nhad attained a certain degree of intensity. The stranger, surprised at\nthe obstinate silence of the man he was addressing, and specially by his\ncontinuing not to show himself, at length rose to see for himself the\nreason. At this moment, his horse pricked up its ears, snorted\nviolently, and made a sudden effort to free itself from the lasso that\nheld it; but our new acquaintance rushed toward it and patted it. The\nhorse trembled all over, and made prodigious bounds in order to escape.\nThe stranger, more and more surprised, looked round for an explanation\nof these extraordinary movements, and was soon satisfied.\n\nScarce twenty yards from him a magnificent jaguar, with a\nsplendidly-spotted hide, was crouched on the main branch of an enormous\ncypress, and fixed on him two ferocious eyes, as it passed its\nblood-red, rugged tongue over its lips with a feline pleasure.\n\n\"Ah, ah!\" the stranger said to himself in a low voice, but displaying no\nfurther excitement, \"I did not expect you; but no matter, you are\nwelcome, comrade. _Caray_! We shall have a fight for it.\"\n\nWithout taking his eye off the jaguar, he convinced himself that his\nmachete quitted its scabbard readily, picked up his rifle, and, after\nthese precautions were taken, he advanced resolutely toward the\nferocious brute, which saw him coming without changing its position. On\narriving within ten yards of the jaguar, the stranger threw away the\ncigarette he had till now held between his lips, shouldered his rifle,\nand put his finger on the trigger. The jaguar drew itself together and\nprepared to leap forward. At the same moment a hoarse yell was heard\nfrom the opposite side of the clearing.\n\n\"Wait a minute,\" the stranger said to himself with a smile; \"it seems\nthere are two of them, and I fancied I had to do with a bachelor jaguar.\nThis is beginning to grow interesting.\"\n\nAnd he threw a glance on one side. He had not deceived himself: a second\njaguar, rather larger than the first, had fixed its flashing eyes upon\nhim.\n\n\n\n\nCHAPTER II.\n\nTHE CONTEST.\n\n\nThe dwellers on the Mexican frontiers are accustomed to fight\ncontinually with wild animals, both men and brutes, that continually\nattack them. Hence the stranger was but slightly affected by the\nunexpected visit of the two jaguars. Although his position between his\ntwo ferocious enemies was somewhat precarious, and he did not at all\nconceal from himself the danger he ran alone against two, he did not the\nless resolve to confront them bravely. Not taking his eye off the jaguar\nhe had first seen, he went back a few steps obliquely, so as to have his\nfoes nearly opposite him, instead of standing between them. This\nmanoeuvre, which demanded some little time, succeeded beyond his hopes.\nThe jaguars watched him, licking their lips, and passing their paws\nbehind their ears with those graceful movements peculiar to the feline\nrace. The two wild beasts, certain of their prey, seemed to be playing\nwith it and not over eager to pounce on it.\n\nWhile keeping his eye on the watch, the Mexican did not yield to any\ntreacherous feeling of security: he knew that the struggle he was about\nto undertake was a supreme one, and he took his precautions. Jaguars\nnever attack a man unless forced by necessity; and the latter tried,\nbefore all, to seize the horse. The noble animal, securely fastened by\nits master, exhausted itself in efforts to break the bonds that held it,\nand escape. It trembled with terror on scenting its ferocious enemies.\n\nThe stranger, when his precautions were completely taken, shouldered his\nrifle for the second time. At this moment the jaguars raised their\nheads, while laying back their ears and snuffing anxiously. An almost\nimperceptible sound was audible in the bushes.\n\n\"Who goes there?\" the Mexican asked in a loud voice.\n\n\"A friend, Don Miguel Zarate,\" was the reply.\n\n\"Ah! It is Don Valentine,\" the Mexican continued. \"You have arrived just\nin time to see some fine sport.\"\n\n\"Ah, ah!\" the man who had already spoken went on. \"Can I help you?\"\n\n\"It is useless; but make haste if you want to see.\"\n\nThe branches were sharply drawn aside, and two men appeared in the\nclearing. At the sight of the jaguars they stopped, not through alarm,\nfor they quietly placed the butts of their rifles on the ground, but in\norder to give the hunter every facility to emerge victoriously from his\nrash combat.\n\nThe jaguars seemed to comprehend that the moment for action had arrived.\nAs if by one accord, they drew themselves up and bounded on their enemy.\nThe first, struck in its leap by a bullet which passed through its right\neye, rolled on the ground, where it remained motionless. The second was\nreceived on the point of the hunter's machete, who after discharging his\nrifle, had fallen on his knee, with his left arm folded in his blanket\nin front, and the machete in the other hand. The man and the tiger\nwrithed together in a deadly embrace, and after a few seconds only one\nof the adversaries rose: it was the man. The tiger was dead: the\nhunter's machete, guided by a firm hand, had passed right through its\nheart.\n\nDuring this rapid fight the newcomers had not made a sign, but remained\nstoical spectators of all that was taking place. The Mexican rose,\nthrust his machete in the grass to clean the blade, and turning coldly\nto the strangers, said:\n\n\"What do you say to that?\"\n\n\"Splendidly played,\" the first answered; \"it is one of the best double\nstrokes I ever saw in my life.\"\n\nThe two men threw their rifles on their shoulders, and walked up to the\nMexican, who reloaded his piece with as much coolness and tranquillity\nas if he had not just escaped from a terrible danger by a miracle of\nskill.\n\nThe sun was sinking on the horizon, the shadow of the trees assumed a\nprodigious length, and the luminary appeared like a ball of fire amid\nthe limpid azure of the heavens. The night would soon arrive, and the\ndesert was awaking. On all sides could be heard, in the gloomy and\nmysterious depths of the virgin forest, the hoarse howling of the\ncoyotes and the other wild beasts, mingled with the song of the birds\nperched on all the branches. The desert, silent and gloomy during the\noppressive heat of the day, emerged from its unhealthy torpor on the\napproach of dark, and was preparing to resume its nocturnal sports.\n\nThe three men in the clearing collected dried branches, made a pile of\nthem and set fire to it. They doubtlessly intended to camp for a portion\nof the night at this spot. So soon as the flames rose joyously, skyward\nin long spirals, the two strangers produced from their game bags maize\ntortillas, jerked meat, and a gourd of pulque. These various comestibles\nwere complacently spread out on the grass, and the three men began a\nhunter's meal. When the gourd had gone the round several times, and the\ntortillas had disappeared, the newcomers lit their Indian pipes, and the\nMexican rolled a papelito.\n\nAlthough this meal had been short, it lasted, however, long enough for\nnight to have completely set in ere it was ended. Perfect darkness\nbrooded over the clearing, the ruddy reflections of the fire played on\nthe energetic faces of the three men, and gave them a fantastic\nappearance.\n\n\"And now,\" the Mexican said, after lighting his cigarette, \"I will, with\nyour permission, explain to you why I was so anxious to see you.\"\n\n\"One moment,\" one of the hunters answered. \"You know that in the deserts\nthe leaves have often eyes, and the trees ears. If I am not mistaken in\nyour hints, you invited us here that our interview might be secret.\"\n\n\"In truth, I have the greatest interest in nothing of what is said here\nbeing overheard, or even suspected.\"\n\n\"Very good. Curumilla, to work.\"\n\nThe second hunter rose, seized his rifle and disappeared noiselessly in\nthe gloom. His absence was rather long; but as long as it lasted, the\ntwo men left at the fire did not exchange a syllable. In about half an\nhour the hunter returned, however, and seated himself by his comrades'\nside.\n\n\"Well?\" the one who had sent him off asked him.\n\n\"My brother can speak,\" he replied laconically; \"the desert is quiet.\"\n\nOn this assurance the three men banished all anxiety. Still prudence did\nnot abandon them: they took up their pipes, and turned their backs to\nthe fire, so that they might watch the neighbourhood while conversing.\n\n\"We are ready to listen to you,\" the first hunter said.\n\n\"Listen to me with the greatest attention,\" the Mexican began; \"what you\nare about to hear is of the utmost importance.\"\n\nThe two men bowed silently, and the Mexican prepared to speak again.\n\nBefore going further we must introduce to the reader the two men we have\njust brought on the stage, and go back a few paces in order to make it\nperfectly understood why Don Miguel Zarate, in lieu of receiving them at\nhis own house, had given them the meeting in the heart of the virgin\nforest.\n\nThe two hunters seemed at the first glance to be Indians; but on\nexamining them more attentively, you could recognise that one of them\nbelonged to those white trappers whose boldness has become proverbial in\nMexico. Their appearance and equipment offered a singular medley of\nsavage and civilised life. Their hair was of a remarkable length; for in\nthose countries, where a man is frequently only fought for the glory of\nlifting his scalp, it is considered the thing to wear it long and easy\nto seize.\n\nThe hunters had their hair neatly plaited, and intertwined with beaver\nskins and bright  ribbons. The rest of their garb harmonised with\nthis specimen of their taste. A hunting shirt of bright red calico fell\ndown to their knees; gaiters decorated with woolen ribbons and bells\nsurrounded their legs; and their feet were shod with moccasins\nembroidered with beads which the squaws know so well how to make. A\nstriped blanket, fastened round the hips by a belt of tanned deer hide,\ncompleted their clothing, but was not so closely drawn that at their\nevery movement the butt of the pistols and the hilt of the machetes\nmight be seen glistening. As for their rifles, useless at this moment,\nand carelessly thrown on the ground by their side, if they had been\nstripped of the plume-worked elk skin that covered them, it would have\nbeen possible to see, with what care their owners had decorated them\nwith copper nails painted of various colours; for all about these two men\nbore the imprint of Indian habits.\n\nThe first of the two hunters was a man of thirty-eight at the most, tall\nand well-built; his muscular limbs denoted great bodily strength, allied\nto unequalled lightness. Although he affected all the manners of the\nredskins, it was an easy matter to perceive that he not only belonged to\nthe unmixed white race, but also to the Norman or Gaulish type. He was\nfair; his large, blue and pensive eyes, adorned with long lashes, had an\nexpression of undefinable sadness: his nose was slightly aquiline; his\nmouth large, and filled with teeth of dazzling whiteness; a thick\nchestnut beard covered the lower part of his face, which revealed\ngentleness, kindness, and courage without boasting, though the whole\nwere combined with a will of iron.\n\nHis companion evidently belonged to the Indian race, all the\ncharacteristic signs of which he displayed; but, strange to say, he was\nnot coppery like the American aborigines of Texas and North America; and\nhis skin was brown and slightly of an olive hue. He had a lofty brow, a\nbent nose, small but piercing eyes, a large mouth and square chin; in\nshort, he presented the complete type of the American race, which\ninhabits a limited territory in the South of Chili. This hunter had\nround his brow a purple- fillet, in which was thrust over the\nright ear a plume of the Andes Eagle, a sign which serves to distinguish\nthe chiefs of the Aucas.\n\nThese two men, whom the reader has doubtless already recognised, as they\nplayed an important part in our previously published works[1], were\nValentine Guillois, an ex-noncommissioned officer in the Spahis, and\nCurumilla, his friend--Ulmen of the Great Hare tribe.\n\nWe will introduce a parenthesis to explain their present position, and\nwhich is indispensable for a right understanding of what follows. The\nmoment is capitally selected, by the way, for opening this parenthesis;\nfor the three hunters are gaily talking round their fire, the night is\ngloomy, the forest quiet, and it does not appear likely that anything\nwill arise to disturb them.\n\n\n\n[1] \"The Chief of the Aucas,\" \"The Tiger Slayer,\" \"The Gold Finders,\"\n\"The Indian Chief.\"\n\n\n\n\nCHAPTER III.\n\nDON MIGUEL ZARATE.\n\n\nWere Mexico better governed, it would be, without contradiction, one of\nthe richest countries on the face of the globe. Indeed the largest\nprivate fortunes must still be sought in that country. Since the United\nStates Americans have revealed to the world, by seizing one-half of\nMexico, whither their ambition tends, the inhabitants of that fine\ncountry have slightly emerged from the torpor they enjoyed, and have\nmade great efforts to colonise their provinces, and summon to their\nsoil, which is so rich and fertile, intelligent and industrious\nlabourers, who might change the face of affairs, and cause abundance and\nwealth to abound at spots, where, prior to their arrival, there was\nnaught save ruin, desolation, carelessness, and misery.\n\nUnfortunately, the noble efforts made up to the present day have,\nthrough an inexplicable fatality, remained without result, either owing\nto the natural apathy of the inhabitants, or the fault of the Mexican\nGovernment itself. Still the large landowners, comprehending all the\nadvantages of the proposed measure, and how much it is to their interest\nto combat the deadly influence of the American invasions, have\ngenerously devoted themselves to the realization of this great question\nof social economy, which, unluckily is growing more and more\nunrealisable.\n\nIn fact, in Northern America two hostile races--the Anglo-Saxon and the\nSpanish--stand face to face. The Anglo-Saxons are devoured by an ardour\nfor conquest, and a rage for invasion, which nothing can arrest, or even\n<DW44>. It is impossible to see without amazement the expansive\ntendencies of this active and singular people, a heterogeneous composite\nof all the races which misery or evil instincts expelled from Europe\noriginally, and which feels restricted in the immense territory which\nits numerical weakness yet prevents it entirely occupying.\n\nImprisoned within its vast frontiers, making a right of strength, it is\ncontinually displacing its neighbours' landmarks, and encroaching on\nterritory of which it can make no use. Daily, bands of emigrants abandon\ntheir dwellings, and with their rifles on their shoulders, their axes in\ntheir hand, they proceed south, as if impelled by a will stronger than\nthemselves; and neither mountains, deserts, nor virgin forests are\nsufficient obstacles to make them halt even for an instant. The Yankees\nimagine themselves generally the instruments of Providence, and\nappointed by the decrees of the Omnipotent to people and civilise the\nNew World. They count with feverish impatience the hours which must\nelapse ere the day (close at hand in their ideas) arrive in which their\nrace and government system will occupy the entire space contained\nbetween Cape North and the Isthmus of Panama, to the exclusion of the\nSpanish republics on one side, and the English colonies on the other.\n\nThese projects, of which the Americans make no mystery, but, on the\ncontrary, openly boast, are perfectly well known to the Mexicans, who\ncordially detest their neighbours, and employ all the means in their\npower to create difficulties for them, and impede their successive\nencroachments.\n\nAmong the New Mexican landowners who resolved to make sacrifices in\norder to stop, or at least check, the imminent invasion from North\nAmerica, the richest, and possibly, first of all, through his\nintelligence and the influence he justly enjoyed in the country, was Don\nMiguel Acamarichtzin Zarate.\n\nWhatever may be asserted, the Indian population of Mexico is nearly\ndouble in number to the white men, and possesses an enormous influence.\nDon Miguel descended in a straight line from Acamarichtzin, first king\nof Mexico, whose name had been preserved in the family as a precious\nrelic. Possessed of an incalculable fortune, Don Miguel lived on his\nenormous estates like a king in his empire, beloved and respected by the\nIndians, whom he effectively protected whenever the occasion presented\nitself, and who felt for him a veneration carried almost to idolatry;\nfor they saw in him the descendant from one of their most celebrated\nkings, and the born defender of their race.\n\nIn New Mexico the Indian population has very largely increased during\nthe past fifty years. Some authors, indeed, assert that it is now more\nnumerous than prior to the conquest, which is very probable, through the\napathy of the Spaniards, and the carelessness they have ever displayed\nin their struggles against it. But the Indians have remained stationary\namid the incessant progress of civilization, and still retain intact the\nprincipal traits of their old manners. Scattered here and there in\nmiserable ranchos or villages, they live in separate tribes, governed by\ntheir caciques, and they have mingled but very few Spanish words with\ntheir idioms, which they speak as in the time of the Aztecs. The sole\napparent change in them is their conversion to Catholicism--a conversion\nmore than problematical, as they preserve with the utmost care all the\nrecollections of their ancient religion, follow its rites in secret, and\nkeep up all its superstitious practices.\n\nThe Indians--above all, in New Mexico--although called _Indios fideles_,\nare always ready on the first opportunity to ally themselves with their\ndesert congeners; and in the incursions of the Apaches and Comanches it\nis rare for the faithful Indians not to serve them as scouts, guides,\nand spies.\n\nThe family of Don Miguel Zarate had retired to New Mexico, which country\nit did not leave again--a few years after the conquests of the\nadventurer Cortez. Don Miguel had closely followed the policy of his\nfamily by maintaining the bonds of friendship and good neighbourhood\nwhich, from time immemorial, attached it to the Indians, believers or\nnot. This policy had borne its fruit. Annually, in September, when the\nterrible red warriors, preceded by murder and arson, rushed like a\ntorrent on the wretched inhabitants, whom they massacred in the farms\nthey plundered, without pity of age or sex, Don Miguel Zarate's estates\nwere respected; and not merely was no damage inflicted on them, but even\nif at times a field were unwittingly trampled by the horses' hoofs, or a\nfew trees destroyed by plunderers, the evil was immediately repaired ere\nthe owner had opportunity for complaint.\n\nThis conduct of the Indians had not failed to arouse against Don Miguel\nextreme jealousy on the part of the inhabitants, who saw themselves\nperiodically ruined by the _Indios Bravos._ Earnest complaints had been\nlaid against him before the Mexican Government; but whatever might be\nthe power of his enemies, and the means they employed to ruin him, the\nrich hacendero had never been seriously disturbed: in the first place,\nbecause New Mexico is too remote from the capital for the inhabitants to\nhave anything to fear from the governing classes; and secondly, Don\nMiguel was too rich not to render it easy for him to impose silence on\nthose who were most disposed to injure him.\n\nDon Miguel, whose portrait we drew in a previous chapter, was left a\nwidower after eight years' marriage, with two children, a boy and a\ngirl, the son being twenty-four, the daughter seventeen, at the period\nwhen our story opens. Dona Clara--such was the daughter's name--was one\nof the most delicious maidens that can be imagined. She had one of those\nMurillo's virgin heads, whose black eyes, fringed with long silky\nlashes, pure mouth, and dreamy brow seem to promise divine joys. Her\ncomplexion, slightly bronzed by the warm sunbeams, wore that gilded\nreflection which so well becomes the women of these intertropical\ncountries. She was short of stature, but exquisitely modelled. Gentle\nand simple, ignorant as a Creole, this delicious child was adored by her\nfather, who saw in her the wife he had so loved living once more. The\nIndians looked after her when she at times passed pensively, plucking a\nflower before their wretched huts, and scarce bending the slants on\nwhich she placed her delicate foot. In their hearts they compared this\nfrail maiden, with her soft and vaporous outline, to the \"virgin of the\nfirst loves,\" that sublime creation of the Indian religion which holds\nso great a place in the Aztec mythology.\n\nDon Pablo Zarate, the hacendero's son, was a powerfully built man, with\nharshly marked features, and a haughty glance, although at times it was\nimprinted with gentleness and kindness. Endowed with more than ordinary\nstrength, skilled in all bodily exercises, Don Pablo was renowned\nthrough the whole country for his talent in taming the most spirited\nhorses, and the correctness of his aim when on the chase. A determined\nhunter and daring wood ranger, this young man, when he had a good horse\nbetween his legs, and his rifle in his hand, knew none, man or animal,\ncapable of barring his passage. The Indians, in their simple faith,\nyielded to the son the same respect and veneration they entertained for\nthe father, and fancied they saw in him the personification of\n_Huitzilopochtli_, that terrible war god of the Aztecs, to whom 62,000\nhuman victims were sacrificed in one day, upon the inauguration of his\n_teocali_.\n\nThe Zarates, then, at the period when our story opens, were real kings\nof New Mexico. The felicity they enjoyed was suddenly troubled by one of\nthose vulgar incidents which, though unimportant in themselves, do not\nfail to cause a general perturbation, and a discomfort possessing no\napparent cause, from the fact that it is impossible to foresee or\nprevent them. The circumstance was as follows:--\n\nDon Miguel possessed, in the vicinity of the Paso, vast estates\nextending for a great distance, and consisting principally of haciendas,\nprairies, and forests. One day Don Miguel was returning from a visit to\nhis haciendas. It was late, and he pressed on his horse in order to\nreach ere night the ford, when, at about three or four leagues at the\nmost from the spot to which he was proceeding, and just as he was\nentering a dense forest of cottonwood trees, through which he must pass\nere reaching the ford, his attention was attracted by cries mingled with\ngrowls emerging from the wood he was about to enter. The hacendero\nstopped in order to account for the unusual sounds he heard, and bent\nhis head forward to detect what was happening. But it was impossible for\nhim to distinguish anything through the chaos of creepers and shrubs\nwhich intercepted vision. In the meanwhile, the noise grew louder, and\nthe shouts were redoubled, and mingled with oaths and passionate\nexclamations.\n\nThe Mexican's horse laid back its ears, neighed, and refused to advance.\nStill Don Miguel must make up his mind. Thinking that a man was probably\nattacked by wild beasts, he only consulted his heart; and, in spite of\nthe visible repugnance of his steed, he compelled it to go forward and\nenter the wood. He had scarce gone a few yards ere he stopped in\namazement at the strange spectacle that presented itself to him.\n\n\n\n\nCHAPTER IV.\n\nTHE PECCARIES.\n\n\nIn the middle of the clearing lay a ripped up horse, which six or eight\npeccaries were rending, while a dozen others were attacking with their\ntusks the stem of an enormous tree, in the topmost branches of which a\nman had sought shelter.\n\nLet us explain to our readers, who probably know little about them, what\nsort of animals the peccaries are. The peccaries hold the intermediate\ngrade between the domestic pig and the wild boar. Although this animal\ndoes not exceed two feet in height, and is not more than three feet long\nfrom the end of the snout to the beginning of the tail, it is\nindubitably one of the most dangerous animals in North America. The\nanimal's jaw is provided with tusks rather like those of the boar, but\nstraight and sharp, their length varying between four and six inches. In\nthe shape of the body it resembles a pig, but the bristles scattered\nover its warty hide are in  strips; the part nearest the skin is\nwhite, and the point of a chocolate tinge. So soon as the animal is\nenraged, these bristles stand out like the quills of a porcupine.\n\nThe movements of the peccaries are as quick and sharp as those of a\nsquirrel. They ordinarily live in herds of fifteen, thirty, and even\nfifty. The strength of the head, neck, and shoulders is so great when\nthey charge, that nothing can resist the impetuosity of their attacks. A\nremarkable peculiarity of this genus is the clumsy wart they have on\ntheir backs, whence a musty fluid evaporates when the animal is in a\nfury.\n\nThe peccary lives in preference on acorns, roots, wheat, sugar cane, and\nreptiles of every description. It is a proved fact that the most\nvenomous serpents are devoured by them without their feeling in the\nslightest degree incommoded.\n\nThe mode in which the peccary forms its lair is very singular. This lair\nis generally in the midst of tufted and impenetrable canes, found in\nmarshy spots round the monarchs of the forest, which still stand like\ncrushed giants, with their grappling lines of creepers and virgin vines.\nThe trunks of these trees, which at times measure forty feet in\ncircumference, are nearly all hollow, and thus afford a convenient\nshelter for the peccaries, which retire to them every night in herds of\ntwenty to twenty-five, entering the cavity one after the other\nbackwards; so that the last has the end of its snout placed just at the\nentrance of the hole, thus watching, as it were, over the rest of its\ncompanions.\n\nThe peccaries are unboundedly ferocious: they know not danger, or at\nleast despise it completely. They always attack in herds, and fight with\nunequalled rage until the last succumbs, no matter the nature of their\nfoe.\n\nHence men and animals all fly a meeting with these terrible beasts: the\njaguar, so strong and redoubtable, will become their prey if it be so\nimprudent as to attack them. This is the way they set about conquering\nthis wild beast:--\n\nWhen a jaguar has wounded a peccary, the latter collect, chase it, and\npursue until they can contrive to surround the common enemy. When every\nissue is closed, the jaguar, believing it can thus escape, seeks refuge\nup a tree. But the peccaries do not resign the vengeance; they establish\nthemselves at the foot of the tree, being incessantly recruited by fresh\nallies, and patiently waiting till the jaguar, driven to extremities by\nhunger and thirst, decides on descending from its improvised fortress.\nThis is almost always sure to happen at the end of two or three days at\nthe most. The jaguar bounds into the midst of its enemies, which boldly\nawait it, and attack it bravely; a terrible fight commences; and the\ntiger, after covering the ground with victims, at length succumbs\nbeneath the efforts of its assailants, and is ripped up by their tusks.\n\nAfter what we have said, it is easy to understand how precarious was the\nposition of the man perched on the top of the tree, and surrounded by\npeccaries. His enemies seemed determined not to leave their ground; they\ncraftily crept round the tree, attacked its base with their tusks, and\nthen recognising the inutility of their onsets, they quietly lay down by\nthe carcass of the horse, which they had already sacrificed to their\nfury. Don Miguel felt moved to pity for the poor fellow, whose position\ngrew momentarily more critical; but in vain did he rack his brains how\nto help the unhappy man whose destruction was assured.\n\nTo attack the peccaries would have been extreme imprudence, and have\nproduced no other result than that of turning on himself the fury of the\nanimals, while not saving the man he wished to help. Still time pressed.\nWhat was to be done? How, without sacrificing himself, save the man who\nran so great a risk?\n\nThe Mexican hesitated for a long period. It seemed to Don Miguel\nimpossible to leave, without help, this man whose death was certain.\nThis idea, which presented itself to his mind several times, he had\nenergetically repulsed, so monstrous did it appear to him. At length he\nresolved at all risks to attempt impossibilities in favour of this\nstranger, of whose death he would have eventually accused himself had he\nleft him to perish in the desert.\n\nThe stranger's position was the more critical because, in his haste to\ndefend himself from the attacks of his enemies, he had left his rifle\nfall at the foot of the tree, and was consequently unable to reduce the\nnumber of the peccaries. In spite of their fineness of scent, the latter\nhad not noticed Don Miguel's approach, who, by a providential accident,\nhad entered the wood on the side opposite the wind. The Mexican\ndismounted with a sigh, patted his horse, and then took off its\naccoutrements. The noble animal, habituated to its master's caresses,\nshook his head joyously, and fixed its large intelligent eyes on him.\nDon Miguel could not repress another sigh: a tear fell down on his\nbronzed cheeks. On the point of accomplishing the sacrifice, he\nhesitated.\n\nIt was a faithful companion, almost a friend, he was about to separate\nfrom; but the life of a man was at stake. The Mexican drove back the\nfeelings that agitated him, and his resolution was formed. He passed a\nlasso round his horse's neck, and, in spite of its obstinate resistance,\ncompelled it to advance to the entrance of the clearing in which the\npeccaries were assembled. A frail curtain of creepers and leaves alone\nhid it from their sight. On arriving here Don Miguel stopped: he had one\nmore moment's hesitation, but only one; for then seizing a piece of\ntinder, which he lighted, he thrust it into the poor animal's ear while\ncaressing it.\n\nThe effect was sudden and terrible. The horse uttered a snort of pain;\nand rendered mad by the burning, bounded forward into the clearing,\nstriving in vain to get rid of the tinder which caused it intolerable\nsuffering. Don Miguel had smartly leaped aside, and now followed with an\nanxious glance the result of the terrible tentative he had just made to\nsave the stranger. On seeing the horse appear suddenly in their midst,\nthe peccaries rose, formed a compact group and rushed with their heads\ndown in pursuit of the horse, thinking no longer of the man. The animal,\nspurred on still more by the sight of its ferocious enemies, shot ahead\nwith the speed of an arrow, breaking down with its chest all the\nobstacles in its way, and followed closely by the peccaries.\n\nThe man saved; but at what a price! Don Miguel repressed a last sigh of\nregret, and leaped into the clearing. The stranger had already descended\nfrom the tree; but the emotion he had undergone was so extreme, that he\nremained seated on the ground, almost in a state of unconsciousness.\n\n\"Quick, quick!\" Don Miguel said to him sharply. \"We have not a moment to\nlose: the peccaries may alter their minds and return.\"\n\n\"That is true,\" the stranger muttered in a hollow voice, as he cast a\nterrified glance around. \"Let us be off--off at once.\"\n\nHe made an effort over himself, seized his rifle, and rose. Through a\npresentiment for which he could not account to himself, Don Miguel\nexperienced at the sight of this man, whom he had hitherto scarce looked\nat, a feeling of invincible doubt and disgust. Owing to the life he was\nobliged to lead on these frontiers, frequented by people of every\ndescription, the hacendero had been often brought into relation with\ntrappers and hunters whose faces were no recommendation to them; but\nnever ere now had chance brought him in contact with an individual of\nsuch sinister appearance as this one.\n\nStill he did not allow his feelings to be seen through, and invited this\nman to follow him. The latter did not let the invitation be repeated;\nfor he was anxious to escape from the spot where he had been so near\ndeath. Thanks to the Mexican's acquaintance with the country, the wood\nwas speedily traversed, and the two men, after a walk of scarce an\nhour's duration, reached the banks of the Del Norte, just opposite the\nvillage. Their speed had been so great, their anxiety so serious, that\nthey had not exchanged a syllable, so terrified were they of seeing the\npeccaries appear at any moment. Fortunately this was not the case, and\nthey reached the ford without being again disturbed.\n\nDon Miguel was burdened with his horse's trappings, which he now threw\non the ground, and looked around him in the hope of finding someone who\nwould help him in crossing the river. His expectations were not\ndeceived; for just as they reached the ford an _arriero_ was preparing\nto cross to the other side of the river with his _recca_ of mules, and,\nwith the generosity innate in all Mexicans, he offered to carry them\nboth to the Paso. The two men eagerly accepted, each mounted a mule, and\nhalf an hour later they found themselves in safety at the village. After\ngiving the arriero a few reals to requite him for his services, Don\nMiguel took up his horse's trappings again, and prepared to start. The\nstranger stopped.\n\n\"We are about to part here, caballero,\" he said in a rough voice, with a\nvery marked English accent; \"but before leaving, let me express to you\nmy deep gratitude for the noble and generous manner in which you saved\nmy life at the peril of your own.\"\n\n\"Sir,\" the Mexican simply answered, \"I only did my duty in saving you.\nIn the desert all men are brothers, and owe each other protection. Hence\ndo not thank me, I beg, for a very simple action: any other in my place\nwould have acted as I have done.\"\n\n\"Perhaps so,\" the stranger continued; \"but be kind enough, pray, to tell\nme your name, so that I may know to whom I owe my life.\"\n\n\"That is needless,\" Don Miguel said with a smile. \"Still, as I fancy you\nare a stranger in these parts, let me give you a piece of advice.\"\n\n\"What is it, sir?\"\n\n\"Never in future to attack the peccaries. They are terrible enemies,\nonly to be conquered by a strong body of men; and an individual in\nattacking them commits an unpardonable folly, to which he must fall a\nvictim.\"\n\n\"Be assured, sir, that I shall profit by the lesson I have received this\nday, and shall never put myself in such a wasps' nest again. I was too\nnear paying dearly for my imprudence. But I beg you, sir, do not let us\nseparate ere I know the name of my preserver.\"\n\n\"As you insist, sir, you shall learn it. I am Don Miguel de Zarate.\"\n\nThe stranger took a peculiar glance at the speaker, while repressing a\nmovement of surprise.\n\n\"Ah!\" he said in a singular tone, \"Thanks, Don Miguel Zarate. Without\nknowing you personally, I was already acquainted with your name.\"\n\n\"That is possible,\" the hacendero answered; \"for I am well known in\nthis country, where my family has been established for many a long\nyear.\"\n\n\"I, sir, am the man whom the Indians call Witchasta Joute, the Maneater,\nand the hunters, my companions, Red Cedar.\"\n\nAnd after lifting his hand to his cap in salute, this man threw his\nrifle on his shoulder, turned on his heel, and went off at full speed.\nDon Miguel looked after him for a while, and then walked pensively\ntoward the house he inhabited at el Paso. The hacendero did not suspect\nthat he had sacrificed his favourite horse to save the life of his most\nimplacable enemy.\n\n\n\n\nCHAPTER V.\n\nTHE WOUND.\n\n\nAt sunrise, Don Miguel, mounted on an excellent horse, left the Paso,\nand proceeded toward the hacienda where he resided with his family. It\nwas situated a few miles from the Presidio of San Elezario, in a\ndelicious position, and was known as the _Hacienda de la Noria_ (the\nFarm of the Well). The estate inhabited by Don Miguel stood in the\ncentre of the vast delta formed by the Del Norte and the Rio San Pedro,\nor Devil's River. It was one of those strong and massive buildings which\nthe Spaniards alone knew how to erect when they were absolute masters of\nMexico.\n\nThe hacienda formed a vast parallelogram, supported at regular distances\nby enormous cross walls of carved stone. Like all the frontier\nhabitations, which are rather fortresses than houses, it was only\npierced on the side of the plain with a few narrow windows resembling\nloopholes, and protected by solid iron bars. This abode was begirt by a\nthick wall of circumvallation, defended on the top by that fretwork\ncalled _almenas_, which indicated the nobility of the owner. Within this\nwall, but separated from the chief apartments, were the stables,\nouthouses, barns and cabins for the peons.\n\nAt the extremity of the courtyard, in an angle of the hacienda, was the\ntall square belfry of the chapel, rising above its terraced roof. This\nchapel was served by a monk called Fray Ambrosio. A magnificent plain\nclosed in this splendid farm. At the end of a valley more than fifty\nmiles in length were cactus trees of a conical shape, loaded with fruit\nand flowers, and whose stems were as much as six feet in diameter.\n\nDon Miguel employed a considerable number of peons in the cultivation of\nthe sugar cane, which he carried on upon a very large scale. As\neverybody knows, the cane is planted by laying it horizontally in\nfurrows half a foot deep. From each knot springs a shoot which reaches a\nheight of about three yards, and which is cut at the end of a year to\nextract the juice.\n\nNothing can be more picturesque than the sight of a field of sugar\ncanes. It was one of those superb American mornings during which nature\nseems to be holding a festival. The _centzontle_ (American nightingale)\nfrequently poured forth its harmonious notes; the red throstled\ncardinals, the blue birds, the parakeets, chattered gaily beneath the\nfoliage; far away on the plain galloped flocks of light antelopes and\ntimid ashatas, while on the extreme verge of the horizon rushed startled\n_manadas_ of wild horses, which raised clouds of impalpable dust beneath\nthe vibration of their rapid hoofs. A few alligators, carelessly\nstretched out on the river mud, were drying their scales in the sun, and\nin mid air the grand eagles of the Sierra Madre hovered majestically\nabove the valley.\n\nDon Miguel advanced rapidly at the favourite pace of the Mexican\n_jinetes,_ and which consists in making the horse raise its front legs,\nwhile the hind ones almost graze the ground--a peculiar sort of amble\nwhich is very gentle and rapid. The hacendero only employed four hours\nin traversing the distance separating him from the hacienda, where he\narrived about nine in the morning. He was received on the threshold of\nthe house by his daughter, who, warned of his arrival, had hastened to\nmeet him.\n\nDon Miguel had been absent from home a fortnight; hence, he received his\ndaughter's caresses with the greatest pleasure. When he had embraced her\nseveral times, while continuing to hold her tightly clasped in his arms,\nhe regarded her attentively during several seconds.\n\n\"What is the matter, _mi querida_ Clara?\" he asked with sympathy. \"You\nseem very sad. Can you feel vexed at the sight of me?\" he added, with a\nsmile.\n\n\"Oh, you cannot believe that, father!\" she answered quickly; \"for you\nknow how happy your presence must render me.\"\n\n\"Thanks, my child! But whence, in that case, comes the sorrow I see\nspread over your features?\"\n\nThe maiden let her eyes sink, but made no reply.\n\nDon Miguel threw a searching glance around.\n\n\"Where is Don Pablo?\" he said. \"Why has he not come to greet me? Can he\nbe away from the hacienda?\"\n\n\"No, father, he is here.\"\n\n\"Well, then, what is the reason he is not by your side?\"\n\n\"Because--\" the girl said, with hesitation.\n\n\"Well?\"\n\n\"He is ill.\"\n\n\"My son ill!\" Don Miguel exclaimed.\n\n\"I am wrong,\" Dona Clara corrected herself.\n\n\"Explain yourself, in Heaven's name!\"\n\n\"My father, the fact is that Pablo is wounded.\"\n\n\"Wounded!\" the hacendero sharply said; and thrusting his daughter\naside, he rushed toward the house, bounded up the few steps leading to\nthe porch, crossed several rooms without stopping, and reached his son's\nchamber. The young man was lying, weak and faint, on his bed; but on\nperceiving his parent he smiled, and held his hand to him. Don Miguel\nwas fondly attached to his son, his sole heir, and walked up to him.\n\n\"What is this wound of which I have heard?\" he asked him in great\nagitation.\n\n\"Less than nothing, father,\" the young man replied, exchanging a meaning\nglance with his sister, who entered at the moment. \"Clara is a foolish\ngirl, who, in her tenderness, wrongly alarmed you.\"\n\n\"But, after all, you are wounded?\" the father continued.\n\n\"But I repeat that it is a mere nothing.\"\n\n\"Come, explain yourself. How and when did you receive this wound?\"\n\nThe young man blushed, and maintained silence.\n\n\"I insist on knowing,\" Don Miguel continued pressingly.\n\n\"Good heavens, father!\" Don Pablo replied with an air of ill-humour, \"I\ndo not understand why you are alarmed for so futile a cause. I am not a\nchild, whom a scratch should make frightened; and many times have I been\nwounded previously, and you have not disturbed yourself so much.\"\n\n\"That is possible; but the mode in which you answer me, the care you\nseem trying to take to keep me ignorant of the cause of this wound--in a\nword, everything tells me that this time you are trying to hide\nsomething grave from me.\"\n\n\"You are mistaken, father, and shall convince yourself.\"\n\n\"I wish nothing more: speak. Clara, my child, go and give orders to have\nbreakfast prepared, for I am dying of hunger.\"\n\nThe girl went out.\n\n\"Now it is our turn,\" Don Miguel continued. \"In the first place, where\nare you wounded?\"\n\n\"Oh! I have merely a slight scratch on my shoulder: if I went to bed it\nwas more through indolence than any other motive.\"\n\n\"Hum! and what scratched your shoulder?\"\n\n\"A bullet.\"\n\n\"What! A bullet! Then you must have fought a duel, unhappy boy!\" Don\nMiguel exclaimed with a shudder.\n\nThe young man smiled, pressed his father's hand, and bending toward him,\nsaid,--\n\n\"This is what has happened.\"\n\n\"I am listening to you,\" Don Miguel replied, making an effort to calm\nhimself.\n\n\"Two days after your departure, father,\" Don Pablo continued, \"I was\nsuperintending, as you wished me to do, the cutting of the cane crop,\nwhen a hunter whom you will probably remember having seen prowling about\nthe estate, a man of the name of Andres Garote, accosted me at the\nmoment I was about to return home after giving my orders to the\nmajordomo. After saluting me obsequiously as his wont, the scamp smiled\ncunningly, and lowering his voice so as not to be overheard by those\naround us, said, 'Don Pablo, I fancy you would give half an ounce to the\nman who brought you important news?' 'That depends,' I answered; for,\nhaving known the man a long time, I was aware much confidence could not\nbe placed in him. 'Bah! Your grace is so rich,' he continued\ninsidiously, 'that a miserable sum like that is less than nothing in his\npocket, while in mine it would do me a deal of good.'\n\n\"Apart from his defects, this scamp had at times done us a few small\nservices; and then, as he said, a half-ounce is but a trifle, so I gave\nit to him. He stowed it away in his pockets, and then bent down to my\near. 'Thanks, Don Pablo,' he said to me. 'I shall not cheat you of your\nmoney. Your horse is rested, and can stand a long journey. Proceed to\nBuffalo Valley, and there you will learn something to interest you.' It\nwas in vain that I urged him to explain himself more clearly; I could\ndraw no more from him. He merely added before parting from me, 'Don\nPablo, you have good weapons; so take them with you, for no man knoweth\nwhat may happen.' Somehow the scamp's veiled confidence aroused my\ncuriosity: hence I resolved to go to Buffalo Valley, and gain the clue\nof this riddle.\"\n\n\"Andres Garote is a villain, who laid a snare for you, into which you\nfell,\" Don Miguel interrupted.\n\n\"No, father, you are mistaken. Andres was honest towards me, and I have\nonly thanks to give him. Still he should have explained himself,\nperhaps, more distinctly.\"\n\nThe hacendero shook his head with a doubting air.\n\n\"Go on,\" he said.\n\n\"I entered my house, procured the weapons, and then, mounted on <DW64>,\nmy black charger, I proceeded toward Buffalo Valley. As you are aware,\nfather, the place we call so, and which belongs to us, is an immense\nforest of cedars and maples, nearly forty miles in circumference, and\ntraversed almost through its entire length by a wide confluent of the\nRio San Pedro.\"\n\n\"Of course I know it, and I intend next year to fell some of the wood\nthere.\"\n\n\"You need not take the trouble,\" the young man said with a smile, \"for\nsomeone has done it for you.\"\n\n\"What do you mean?\" the hacendero asked wrathfully. \"Who dared?\"\n\n\"Oh! One of those wretched heretic squatters, as they call themselves.\nThe villain found the spot to suit him, and has quietly settled there\nwith his three whelps--three big fellows with hang-dog faces, who\nlaughed at me when I told them the forest was mine, and answered, while\naiming at me, that they were North Americans, who cared as little for me\nas they did for a coyote; that the ground belonged to the first comer;\nand that I shall afford them lively pleasure by being off at full speed.\nWhat more shall I tell you, father? I take after you. I have hot blood,\nand I cordially hate that race of Yankee pirates, who, for some years\nback, have settled on our lovely country like a swarm of mosquitoes. I\nsaw our forest plundered, our finest trees cut down. I could not remain\nunmoved in the presence of these scoundrels' insolence, and the quarrel\nbecame so sharp that they fired at me.\"\n\n\"_Virgen Santisima_!\" Don Miguel exclaimed in fury, \"They shall pay\ndearly for the affront they have offered you I swear it! I will take\nexemplary vengeance.\"\n\n\"Why be so angry, father?\" the young man replied, visibly annoyed at the\neffect his story had produced. \"The harm these people do us is really\nvery trifling. I was in the wrong to let my passion carry me away.\"\n\n\"On the contrary, you were right. I will not have these Northern thieves\ncome and commit their plunder here. I will put a stop to it.\"\n\n\"I assure you that, if you will leave me to act, I feel certain of\narranging this affair to your entire satisfaction.\"\n\n\"I forbid you taking the slightest steps, for this matter concerns me\nnow. Whatever may occur, I do not wish you to interfere. Will you\npromise me this?\"\n\n\"As you insist, I do so, father.\"\n\n\"Very good. Get cured as speedily as possible, and keep your mind at\nrest. The Yankees shall pay me dearly for the blood they have shed.\"\n\nWith these words Don Miguel retired, and his son fell back on his bed\nstifling a sigh, and uttering a hoarse exclamation of passion.\n\n\n\n\nCHAPTER VI.\n\nTHE SQUATTER'S SHANTY.\n\n\nDon Pablo had not told his father the facts in all their truth or\ndetail. He had fallen into a perfect ambuscade. He was suddenly attacked\nby the three brothers, who would have mercilessly killed him, resolved\nto lay the blame of his death on the wild beasts, had not, at the moment\nwhen one of them lifted his knife on the young man, who was thrown down\nand rendered motionless by the others, a providential succour reached him\nin the person of a charming maid scarce sixteen years of age.\n\nThe courageous girl rushed from a copse with the rapidity of a fawn, and\nthrew herself resolutely into the midst of the assassins.\n\n\"What are you about, brother?\" she exclaimed in a melodious voice, whose\nharmonious notes echoed amorously in Don Pablo's ears. \"Why do you wish\nto kill this stranger?\"\n\nThe three squatters, surprised by this apparition, which they were far\nfrom expecting, fell back a few paces. Don Pablo profited by this truce\nto jump up and regain possession of his arms, which had fallen by his\nside.\n\n\"Was it not enough,\" the girl continued, \"to rob this man, that you must\nnow try to take his life? Fie, brothers! Do you not know that blood\nleaves on the hands of him who spills it stains which nothing can\nefface? Let this man retire in peace.\"\n\nThe young men hesitated. Although unconsciously yielding to their\nsister's influence, they were ashamed of thus executing her wishes.\nStill they did not dare express their thoughts, and merely bent on their\nenemy, who awaited them with a firm foot and pistols in hand, glances\nladen with hatred and anger.\n\n\"Ellen is right,\" the youngest of her brothers suddenly said. \"No, I\nwill not allow any harm to be done the stranger.\"\n\nThe others looked at him savagely.\n\n\"You would defend him, if necessary, I suppose, Shaw?\" Nathan said to\nhim ironically.\n\n\"Why should I not, were it required?\" the young man said boldly.\n\n\"Eh!\" Sutter remarked with a grin, \"He is thinking of the Wood\nEglantine.\"\n\nThis word had been scarce uttered ere Shaw, with purpled face,\ncontracted features, and eyes injected with blood, rushed with uplifted\nknife on his brother, who awaited him firmly. The girl dashed between\nthem.\n\n\"Peace, peace!\" she shrieked in a piercing voice, \"Do brothers dare\nthreaten one another?\"\n\nThe two young fellows remained motionless, but watching and ready to\nstrike in a moment. Don Pablo fixed an ardent glance on the girl, who\nwas really admirable at this moment. With her features animated by\nanger, her head erect, and her arms stretched out between the two men,\nshe bore a startling likeness to those Druidesses who in olden times\nsummoned the warriors to combat beneath the forests of Germany.\n\nIn her whole person she offered the complete type of the gentle Northern\nwoman. Her hair light and golden like ripe corn; her eyes of extreme\npurity, which reflected the azure of the sky; her earnest mouth, with\nrosy lips and pearly teeth; her flexible and small waist; the whiteness\nof her complexion, whose delicate and transparent skin still bore the\nflush of adolescence--all was combined in this charming maiden to render\nher the most seductive creature imaginable.\n\nDon Pablo, a stranger to this kind of beauty, felt himself involuntarily\nattracted toward the girl, and entirely subjugated by her. Forgetting\nthe reason that had brought him to this spot, the danger he had\nincurred, and that which still menaced him, he was fascinated and\ntrembling before this delicious apparition, fearing at each instant to\nsee it vanish like a vision, and not daring to turn his glance from her\nwhile he felt he had no strength left to admire her.\n\nThis young creature, so frail and delicate, formed a strange contrast\nwith the tall statures and marked features of her brothers, whose coarse\nand savage manners only served to heighten the elegance and charm\nexhaled by her whole person. Still this scene could not be prolonged,\nand must be ended at once. The maiden walked toward Don Pablo.\n\n\"Sir,\" she said to him with a soft smile, \"You have nothing more to fear\nfrom my brothers; you can mount your horse again, and set out, and no\none will oppose your departure.\"\n\nThe young man understood that he had no pretext to prolong his stay at\nthis spot; he therefore let his head sink, placed his pistols in his\nholsters, leaped on his horse, and set out with regret, and as slowly as\npossible.\n\nHe had scarce gone a league when he heard the hasty clatter of a horse\nbehind him. He turned back. The approaching horseman was Shaw, who soon\ncaught up with Don Pablo. The pair then proceeded some distance side by\nside without exchanging a syllable, and both seemed plunged in profound\nthought. On reaching the skirt of the forest, Shaw checked his horse,\nand softly laid his right hand on the Mexican's bridle. Don Pablo also\nstopped on this hint, and waited, while fixing an inquiring glance on\nhis strange comrade.\n\n\"Stranger,\" the young man said, \"my sister sends me. She implores you,\nif it be possible, to keep secret what occurred between us today. She\ndeeply regrets the attack to which you fell a victim, and the wound you\nhave received; and she will try to persuade Red Cedar, our father, to\nretire from your estates.\"\n\n\"Thank your sister for me,\" Don Pablo answered. \"Tell her that her\nslightest wish will ever be a command to me, and that I shall be happy\nto execute it.\"\n\n\"I will repeat your words to her.\"\n\n\"Thanks. Render me a parting service.\"\n\n\"Speak.\"\n\n\"What is your sister's name?\"\n\n\"Ellen. She is the guardian angel of our hearth. My name is Shaw.\"\n\n\"I am obliged to you for telling me your name, though I cannot guess the\nreason that induces you to do so.\"\n\n\"I will tell you. I love my sister Ellen before all: she urged me to\noffer you my friendship. I obey her. Remember, stranger, that Shaw is\nyours to the death.\"\n\n\"I shall not forget it, though I hope never to be under the necessity of\nreminding you of your words.\"\n\n\"All the worse,\" the American said, with a shake of his head; \"but if at\nany time the opportunity offers, I will prove to you that I am a man of\nmy word, so surely as I am a Kentuckian.\"\n\nAnd hurriedly turning his horse's head, the young man rapidly\ndisappeared in the windings of the forest.\n\nBuffalo Valley, illumined by the parting rays of the setting sun, seemed\na lake of verdure to which the golden mist of night imparted magical\ntones. A light breeze rustled through the lofty crests of the cedars,\ncatalpas, tulip and Peru trees, and agitated the grass on the banks of\nthe Rio San Pedro. Don Pablo let the reins float idly on his horse's\nneck, and advanced dreamily through the forest, where the birds were\nleaping from spray to spray, each saluting in its language the arrival\nof night.\n\nAn hour later, the young man reached the hacienda; but the wound he had\nreceived in his shoulder was more serious than was at first supposed. He\nwas obliged, to his great regret, to keep his bed, which prevented him\nseeking to meet again the maiden whose image was deeply engraved on his\nheart.\n\nSo soon as the Mexican had gone off, the squatters continued felling\ntrees and sawing planks, and did not abandon this work till the night\nhad grown quite black. Ellen had returned to the interior of the jacal,\nwhere she attended to the housekeeping duties with her mother. This\njacal was a wretched hut, hastily made with branches of intertwined\ntrees, which trembled with every breeze, and let the sun and rain\npenetrate to the interior.\n\nThis cabin was divided into three compartments: the one to the right\nserved as the bedroom of the two females, while the men slept in the one\nto the left. The central compartment, furnished with worm-eaten benches\nand a clumsily-planed table, was at once keeping room and kitchen.\n\nIt was late: the squatters, assembled round the fire, over which a huge\npot was boiling, were silently awaiting the return of Red Cedar, who had\nbeen absent since the morning. At length, a horse's hoofs sounded\nsharply on the detritus collected for years on the floor of the forest,\nthe noise grew gradually nearer, the horse stopped in front of the\njacal, and a man made his appearance. It was Red Cedar. The men slowly\nturned their heads toward him, but did not otherwise disturb themselves,\nor address a syllable to him.\n\nEllen alone rose and embraced her father affectionately. The giant\nseized the girl in his nervous arms, raised her from the ground, and\nkissed her several times, saying in his rough voice, which his\ntenderness sensibly softened,--\n\n\"Good evening, my dear.\"\n\nThen he put her down on the ground again, and not troubling himself\nfurther about her, fell heavily on a bench near the fire, and thrust his\nfeet toward the fire.\n\n\"Come, wife,\" he said, after the expiration of a moment, \"the supper, in\nthe fiend's name! I have a coyote's hunger.\"\n\nThe wife did not let this be repeated. A few moments later an immense\ndish of _frijoles_, with pimiento, smoked on the table, with large pots\nof pulque. The meal was short and silent, the four men eating with\nextreme rapacity. So soon as the beans had disappeared Red Cedar and his\nsons lit their pipes, and began smoking, while drinking large draughts\nof whiskey, though still not speaking. At length Red Cedar took his pipe\nfrom his lips, and hit the table sharply, while saying in a rough\nvoice,--\n\n\"Come, women, decamp! You have nothing more to do here. You are in our\nway, so go to the deuce!\"\n\nEllen and her mother immediately went out, and entered their separate\napartment. For a few minutes they could be heard moving about, and then\nall became silent again.\n\nRed Cedar made a sign, and Sutter rose and gently put his ear to the\nparting board. He listened for a few moments while holding his breath,\nand then returned to his seat, saying laconically,--\n\n\"They are asleep.\"\n\n\"Quick, my whelps!\" the old squatter said in a low voice. \"We have not a\nminute to lose: the others are expecting us.\"\n\nA strange scene then occurred in this mean room, which was merely\nillumined by the expiring light of the hearth. The four men arose,\nopened a large chest, and produced from it various objects of strange\nshapes--leggings, mittens, buffalo robes, collars of grizzly bear claws;\nin a word, the complete costumes of Apache Indians.\n\nThe squatters disguised themselves as redskins; and when they had put on\ntheir garments, which rendered it impossible to recognise them, they\ncompleted the metamorphosis by painting their faces of different colours.\n\nAssuredly the traveller whom accident had brought at this moment to the\njacal would have fancied it inhabited by Apaches or Comanches.\n\nThe garments which the squatters had taken off were locked up in the\nchest, of which Red Cedar took the key; and the four men, armed with\ntheir American rifles, left the cabin, mounted their horses, which were\nawaiting them ready saddled, and started at full gallop through the\nwinding forest paths.\n\nAt the moment they disappeared in the gloom Ellen stood in the doorway\nof the cabin, took a despairing glance in the direction where they had\ngone, and fell to the ground murmuring sadly,--\n\n\"Good Heaven! What diabolical work are they going to perform this\nnight?\"\n\n\n\n\nCHAPTER VII.\n\nTHE RANGERS.\n\n\nOn the banks of the Rio San Pedro, and on the side of a hill, stood a\n_rancheria_ composed of some ten cabins, inhabited by a population of\nsixty persons at the most, including men, women and children. These\npeople were Coras Indians, hunters and agriculturists, belonging to the\nTortoise tribe. These poor Indians lived there on terms of peace with\ntheir neighbours, under the protection of the Mexican laws. Quiet and\ninoffensive beings, during the nearly twenty years they had been\nestablished at this place they had never once offered a subject of\ncomplaint to their neighbours, who, on the contrary, were glad to see\nthem prosper, owing to their gentle and hospitable manners. Though\nMexican subjects, they governed themselves after their fashion, obeying\ntheir caciques, and regulating in the assembly of their elders all the\ndifficulties that arose in their village.\n\nOn the night when we saw the squatters leave the cabin in disguise, some\ntwenty individuals, armed to the teeth and clothed in strange costumes,\nwith their faces blackened so as to render them unrecognizable, were\nbivouacked at about two leagues from the rancheria, in a plain on the\nriver's bank. Seated or lying round huge fires, they were singing,\nlaughing, quarrelling or gambling with multitudinous yells and oaths.\nTwo men seated apart at the foot of an enormous cactus, were conversing\nin a low tone, while smoking their husk cigarettes. These two men, of\nwhom we have already spoken to the reader, were Fray Ambrosio, chaplain\nto the Hacienda de la Noria, and Andres Garote, the hunter.\n\nAndres was a tall, thin fellow, with a sickly and cunning face, who\ndraped himself defiantly in his sordid rags, but whose weapons were in a\nperfectly good condition.\n\nWho were the men causing this disturbance? They were \"rangers,\" but this\nrequires explanation.\n\nImmediately after each of the different revolutions which have\nperiodically overturned Mexico since that country so pompously declared\nits independence, the first care of the new president who reaches power\nis to dismiss the volunteers who had accidentally swollen the ranks of his\narmy, and supplied him the means of overthrowing his predecessor. These\nvolunteers, we must do them the justice of allowing, are the very scum\nof society, and the most degraded class human nature produces. These\nsanguinary men, without religion or law, who have no relations or\nfriends, are an utter leprosy to the country.\n\nRoughly driven back into society, the new life they are forced to adopt\nin no way suits their habits of murder and pillage. No longer able to\nwage war on their countrymen, they form free corps, and engage\nthemselves for a certain salary, to hunt the Indios Bravos--that is to\nsay, the Apaches and Comanches--who desolate the Mexican frontiers. In\naddition to this, the paternal government of North America in Texas, and\nof Mexico in the States of the Confederation, allots them a certain sum\nfor each Indian scalp they bring in.\n\nWe do not fancy we are saying anything new in asserting that they are\nthe scourge of the colonists and inhabitants, they plunder shamelessly\nin every way when they are not doing worse.\n\nThe men assembled at this moment on the banks of the Rio San Pedro were\npreparing for a war party--the name they give to the massacres they\norganise against the redskins.\n\nToward midnight Red Cedar and his three sons reached the rangers' camp.\nThey must have been impatiently expected, for the bandits received them\nwith marks of the greatest joy and the warmest enthusiasm. The dice, the\ncards, and botas of mezcal and whiskey were immediately deserted. The\nrangers mounted their horses, and grouped round the squatters, near whom\nstood Fray Ambrosio and his friend Andres Garote.\n\nRed Cedar took a glance round the mob, and could not repress a smile of\npride at the sight of the rich collection of bandits of every\ndescription whom he had around him, and who recognised him as chief. He\nextended his arm to command peace. When all were silent the giant took\nthe word.\n\n\"Senores caballeros,\" he said, in a powerful and marked voice, which\nmade all these scamps quiver with delight at being treated like honest\npeople, \"the audacity of the redskins is growing intolerable. If we let\nthem alone they would soon inundate the country, when they would end by\nexpelling us. This state of things must have an end. The government\ncomplains about the few scalps we supply; it says we do not carry out\nthe clauses of the agreement we have formed with it; it talks about\ndisbanding us, as our services are useless, and therefore burdensome to\nthe republic. It is our bounden duty to give a striking denial to these\nmalevolent assertions, and prove to those who have placed confidence in\nus that we are ever ready to devote ourselves to the cause of humanity\nand civilisation. I have assembled you here for a war party, which I\nhave been meditating for some time, and shall carry out this night. We\nare about to attack the rancheria of the Coras, who for some years past\nhave had the impudence to establish themselves near this spot. They are\npagans and thieves, who have one hundred times merited the severe\nchastisement we are about to inflict on them. But I implore you, senores\ncaballeros, display no mistaken pity. Crush this race of vipers--let not\none escape! The scalp of a child is worth as much as that of a man; so\ndo not let yourselves be moved by cries or tears, but scalp, scalp to\nthe end.\"\n\nThis harangue was greeted as it deserved to be; that is, by yells of\njoy.\n\n\"Senores,\" Red Cedar continued, \"the worthy monk who accompanies me will\ncall down the blessing of Heaven on our enterprise; so kneel down to\nreceive the absolution he is about to give you.\"\n\nThe bandits instantaneously dismounted, took off their hats, and knelt\non the sand. Fray Ambrosio then repeated a long prayer, to which they\nlistened with exemplary patience, repeating _amen_ after each occasion,\nand he ended by giving them absolution. The rangers rose, delighted at\nbeing thus freed from the burden of their sins, and got into their\nsaddles again.\n\nRed Cedar then whispered a few words in Fray Ambrosio's ears, who bowed\nhis head in assent, and immediately set out in the direction of the\nHacienda de la Noria, followed by Andres Garote. The squatter then\nturned to the rangers, who were awaiting his orders.\n\n\"You know where we are going, gentlemen,\" he said. \"Let us start, and,\nbefore all, be silent, if we wish to catch our game in its lair; for you\nknow that the Indians are as cunning as opossums.\"\n\nThe band started at a gallop, Red Cedar and his sons being at their\nhead. It was one of those calm nights which predispose the soul to\nreverie, such as America alone has the privilege of possessing. The dark\nblue sky was spangled with an infinite number of stars, in the centre of\nwhich shone the majestic Southern Cross, sparkling like a king's mantle;\nthe atmosphere was extraordinarily transparent, and allowed objects to\nbe noticed at a great distance; the moon profusely spread around her\nsilvery rays, which gave the scenery a fantastic appearance; a\nmysterious breeze sported through the tops of the great trees; and at\ntimes vague rumours traversed the space, and were lost in the distance.\n\nThe gloomy horsemen still went on, silent and frowning, like the\nphantoms of the ancient legends, which glide through the shadows to\naccomplish a deed without a name. At the end of scarce an hour the\nrancheria was reached. All were resting in the village--not a light\nflashed in the hut. The Indians, wearied with the hard toil of the day,\nwere reposing, full of confidence in the sworn faith, and apprehending\nno treason.\n\nRed Cedar halted twenty yards from the rancheria, and drew up his\nhorsemen so as to surround the village on all sides. When each had taken\nhis post, and the torches were lighted, Red Cedar uttered the terrible\nwar cry of the Apaches, and the rangers galloped at full speed on the\nvillage, uttering ferocious howls, and brandishing the torches, which\nthey threw on the cabins.\n\nA scene of carnage then took place which the human pen is powerless to\ndescribe. The unhappy Indians, surprised in their sleep, rushed\nterrified and half naked out of their poor abodes, and were pitilessly\nmassacred and scalped by the rangers, who waved with a demoniac laugh\ntheir smoking, blood-dripping scalps. Men, women, and children, all were\nkilled with refinements of barbarity. The village, fired by the rangers'\ntorches, soon became an immense funebral pile, in which victims and\nmurderers were huddled pell-mell.\n\nStill a few Indians had succeeded in collecting. Formed in a compact\ntroop of twenty men, they opposed a desperate resistance to their\nassassins, exasperated by the odour of blood and the intoxication of\ncarnage. At the head of this band was a half-nude, tall Indian of\nintelligent features, who, armed with a ploughshare, which he wielded\nwith extreme force and skill, felled all the assailants who came within\nreach of his terrible weapon. This man was the cacique of the Coras. At\nhis feet lay his mother, wife, and two children--dead. The unhappy man\nstruggled with the energy of despair. He knew his life would be\nsacrificed, but he wished to sell it as dearly as possible.\n\nIn vain had the rangers fired on the cacique--he seemed invulnerable:\nnot one of the bullets aimed at him had struck him. He still fought, and\nthe weight of his weapon did not seem to fatigue his arm. The rangers\nexcited each other to finish him; but not one dared to approach him.\n\nBut this combat of giants could not endure longer. Of the twenty\ncompanions he had round him on commencing the struggle, the cacique now\nonly saw two or three upright: the rest were dead. There must be an end.\nThe circle that inclosed the hapless Indian drew closer and closer.\nHenceforth it was only a question of time with him. The rangers,\nrecognising the impossibility of conquering this lion-hearted man, had\nchanged their tactics: they no longer attacked him, but contented\nthemselves with forming an impassable circle round him, waiting\nprudently for the moment when the strength of the prey, which could not\nescape them, was exhausted, in order to rush upon him.\n\nThe Coras understood the intention of his enemies. A contemptuous smile\ncontracted his haughty lips, and he rushed resolutely toward these men\nwho recoiled before him. Suddenly, with a movement quicker than thought,\nhe threw with extraordinary strength the ploughshare among the rangers,\nand bounding like a tiger, leaped on a horse, and clutched its rider\nwith superhuman vigour.\n\nEre the rangers had recovered from the surprise this unforeseen attack\noccasioned in them, by a desperate effort, and still holding the\nhorseman, the chieftain drew from his girdle a short sharp knife, which\nhe buried up to the hilt in the flanks of the horse. The animal uttered\na shriek of pain, rushed headlong into the crowd, and bore both away\nwith maddening speed.\n\nThe rangers, rendered furious at being played with by a single man, and\nseeing their most terrible enemy escape them, started in pursuit; but\nwith his liberty the Coras had regained all his energy: he felt himself\nsaved. In spite of the desperate efforts the rangers made to catch him\nup, he disappeared in the darkness.\n\nThe cacique continued to fly till he felt his horse tottering under him.\nHe had not loosed his hold of the horseman, who was half strangled by\nthe rude embrace, and both rolled on the ground. This man wore the\ncostume of the Apache Indians. The Coras regarded him for an instant\nattentively, and then a smile of contempt played round his lips.\n\n\"You are not a redskin,\" he said, in a hollow voice; \"you are only a\npaleface dog. Why put on the skin of the lion when you are a cowardly\ncoyote?\"\n\nThe ranger, still stunned by the fall he had suffered, and the hug he\nhad endured, made no reply.\n\n\"I could kill you,\" the Indian continued; \"but my vengeance would not be\ncomplete. You and yours must pay me for all the innocent blood you have\nshed like cowards this night. I will mark you, so that I may know you\nagain.\"\n\nThen, with fearful coolness, the Coras threw the ranger on his back, put\nhis knee on his chest, and burying his finger in the socket of his eye,\ngave it a sharp rotatory movement, and plucked out his eyeball. On this\nfrightful mutilation, the wretch uttered a cry of pain impossible to\ndescribe. The Indian got up.\n\n\"Go!\" he said to him. \"Now I am certain of finding you again whenever I\nwant you.\"\n\nAt this moment the sound of hoofs could be heard a short distance off:\nthe rangers had evidently heard their comrade's cry, and were hurrying\nto his aid. The Coras, rushed into the bushes and disappeared. A few\nmoments later the rangers came up.\n\n\"Nathan, my son!\" Red Cedar shouted as he leaped from his horse and\nthrew himself on the body of the wounded man. \"Nathan, my firstborn, is\ndead!\"\n\n\"No,\" one of the rangers answered; \"but he is very bad.\"\n\nIt was really the squatter's eldest son whom the cacique had mutilated.\nRed Cedar seized him in his arms, placed him before him on the saddle,\nand the band started again at a gallop. The rangers had accomplished\ntheir task: they had sixty human scalps hanging from their girdles. The\nrancheria of the Coras was no longer aught save a pile of ashes.\n\nOf all the inhabitants of this hapless village only the cacique\nsurvived; but he would suffice to avenge his brothers.\n\n\n\n\nCHAPTER VIII.\n\nTHE VALLEY OF THE BUFFALO.\n\n\nDon Miguel Zarate, on leaving his son, remounted his horse and rode\nstraight to Paso, to the house of Don Luciano Perez, the _juez de\nletras_ (police magistrate).\n\nThe hacendero was one of the richest landed proprietors in the country;\nand as he was thoroughly acquainted with the spirit of the depositaries\nof justice in those parts, he had consequently been careful to line his\npurse well. Here were two reasons, then, to interest the judge in his\nfavour, and this really happened.\n\nThe worthy Don Luciano shuddered on hearing the details of what had\noccurred between Don Pablo and the squatters. He swore that he would,\nwithout delay, take an exemplary vengeance for this starting felony on\nthe part of the heretic dogs, and that it was high time to bring them\nreason. Confirming himself more and more in his resolution, he buckled\non his sword, gave orders to twenty well-armed alguaciles to mount, and\nplacing himself at the head of this numerous escort, he proceeded toward\nBuffalo Valley.\n\nDon Miguel had witnessed with secret annoyance all these formidable\npreparations. He placed but slight confidence in the courage of the\npolicemen, and he would have preferred the judge leaving him master to\nact as he pleased. He had even adroitly attempted to obtain from Don\nLuciano a regular warrant, which he would have executed however he might\nthink proper; but the judge, burning with an unusual warlike ardor, and\nspurred on by the large sum he had received, would listen to nothing,\nbut insisted on himself taking the head of the expedition.\n\nDon Luciano Perez was a plump little man of about sixty years of age,\nround as a tub, with a jolly face, adorned with a rubicund nose and two\ncunning little eyes. This man cordially detested the North Americans;\nand, in the courageous deed he was committing at this moment, hatred was\nas much the instigation as avarice.\n\nThe little band set out at a canter, and proceeded rapidly toward the\nforest. The judge hurled fire and flames at the audacious usurpers, as\nhe called them; he spoke of nothing less than killing them without\nmercy, if they attempted even the slightest resistance to the orders he\nwas about to give them. Don Miguel, who was much calmer, and foreboded\nno good from this great wrath, sought in vain to pacify him by telling\nhim that he would in all probability have to do with men difficult to\nintimidate, against whom coolness would be the best weapon.\n\nThey gradually approached. The hacendero, in order to shorten the\njourney, had led the band by a cross road, which saved at least\none-third the distance; and the first trees of the forest already\nappeared about two miles off. The mischief produced by the squatters was\nmuch more considerable than Don Pablo had represented to his father;\nand, at the first glance, it seemed impossible that, in so short a time,\nfour men, even though working vigorously, could have accomplished it.\nThe finest trees lay on the ground; enormous piles of planks were\narranged at regular distances, and on the San Pedro an already completed\nraft only awaited a few more stems of trees to be thrust into the water.\n\nDon Miguel could not refrain from sighing at the sight of the\ndevastation committed in one of his best forests; but the nearer they\napproached the spot where they expected to meet the squatters, the more\nlukewarm grew the warlike zeal of the judge and his acolytes, and the\nhacendero soon found himself compelled to urge them on, instead of\nrestraining them as he had hitherto done. Suddenly the sound of an axe\nre-echoed a few paces ahead of the band. The judge impelled by the\nfeeling of his duty, and shame of appearing frightened, advanced boldly\nin the direction of the sound, followed by his escort.\n\n\"Stop!\" a rough voice shouted at the moment the policemen turned the\ncorner of a lane.\n\nWith that instinct of self-preservation which never abandons them, the\nalguaciles stopped as if their horses' feet had been suddenly welded to\nthe ground. Ten paces from them stood a man in the centre of the ride,\nleaning on an American rifle. The judge turned to Don Miguel with such\nan expression of hesitation and honest terror that the hacendero could\nnot refrain from laughing.\n\n\"Come, courage, Don Luciano,\" he said to him. \"This man is alone; he\ncannot venture to bar our passage.\"\n\n\"_Con mil diablos!_\" the judge exclaimed, ashamed of this impression\nwhich he could not master, and frowning portentously, \"forward, you\nfellows, and fire on that scoundrel if he make but a sign to resist\nyou.\"\n\nThe alguaciles set out again with prudential hesitation.\n\n\"Stop! I tell you again,\" the squatter repeated. \"Did you not hear the\norder I gave you!\"\n\nThe judge, reassured by the presence of the hacendero, then advanced,\nand said with a tone which he strove to render terrible, but which was\nonly ridiculous through the terror he revealed,--\n\n\"I, Don Luciano Perez, _juez de letras_ of the town of Paso, have come,\nby virtue of the powers delegated to me by the Government, to summon you\nand your adherents to quit within twenty-four hours this forest you have\nillegally entered, and which--\"\n\n\"Ta, ta!\" the stranger shouted, rudely interrupting the judge, and\nstamping his foot savagely. \"I care as much for all your words and laws\nas I do for an old moccasin. The ground belongs to the first comers. We\nare comfortable here, and mean to remain.\"\n\n\"Your language is very bold, young man,\" Don Miguel then said. \"You do\nnot consider that you are alone, and that, failing other rights, we have\nstrength on our side.\"\n\nThe squatter burst into a laugh.\n\n\"You believe that,\" he said. \"Learn, stranger, that I care as little for\nthe ten humbugs I now have before me as I do for a woodcock, and that\nthey will do well to leave me at peace, unless they want to learn the\nweight of my arm at their expense. However, here is my father; settle it\nwith him.\"\n\nAnd he began carelessly whistling \"Yankee Doodle.\" At the same instant\nthree men, at the head of whom was Red Cedar, appeared on the path. At\nthe sight of these unexpected reinforcements for their arrogant enemy\nthe alguaciles made a movement in retreat. The affair was becoming\nsingularly complicated, and threatened to assume proportions very grave\nfor them.\n\n\"Halloh! What's up?\" the old man asked roughly. \"Anything wrong,\nSutter?\"\n\n\"These people,\" the young man answered, shrugging his shoulders\ncontemptuously, \"are talking about driving us from the forest by virtue\nof some order.\"\n\n\"Halloh!\" Red Cedar said, his eyes flashing as he cast a savage glance\nat the Mexicans. \"The only law I recognise in the desert,\" he continued\nwith a gesture of terrible energy as he struck his rifle barrel, \"is\nthis. Withdraw, strangers, if you do not wish blood to be shed between\nus. I am a peaceful man, wishing to do no one hurt; but I warn you that\nI will not allow myself to be kicked out without striking a blow.\"\n\n\"You will not be turned out,\" the judge remarked timidly; \"on the\ncontrary, you have seized on what belongs to other people.\"\n\n\"I won't listen to your arguments, which I do not understand,\" the\nsquatter roughly exclaimed. \"God gave the ground to man that he might\nlabour on it. Every proprietor that does not fulfil this condition\ntacitly renounces his rights, and the earth then becomes the property of\nthe man who tills it with the sweat of his brow; so go to the devil! Be\noff at full speed, if you do not wish harm to happen to you!\"\n\n\"We will not suffer ourselves to be intimidated by your threats,\" the\njudge said, impelled by his anger, and forgetting for a moment his\nalarm; \"we will do our duty, whatever may happen.\"\n\n\"Try it,\" Red Cedar said with a grin.\n\nAnd he made a sign to his sons. The latter arranged themselves in a\nsingle line, and occupied the entire width of the path.\n\n\"In the name of the law,\" the judge said with energy, as he pointed out\nthe old man, \"alguaciles, seize that person.\"\n\nBut, as so frequently happens under similar circumstances, this order\nwas more easy to give than to execute. Red Cedar and his sons did not\nappear at all disposed to let themselves be collared. We must, however,\ndo the alguaciles the justice of stating that they did not hesitate for\na moment. They plainly refused to carry out the order they had received.\n\n\"For the last time, will you be off?\" the squatter shouted. \"Let them\nhave it.\"\n\nHis three sons raised their rifles. At this movement, which removed all\ndoubts that might still remain on their minds, and which proved to them\nthat the squatters would not hesitate to proceed to extremities, the\nalguaciles were seized with an invincible terror. They turned bridle and\ngalloped off at full speed, followed by the yells of the Americans.\n\nOne man alone remained motionless before the squatters--Don Miguel\nZarate. Red Cedar had not recognised him, either owing to the distance\nthat separated them, or because the hacendero had purposely pulled over\nhis eyes his broad-brimmed hat. Don Miguel dismounted, placed the\npistols from his holsters through his belt, fastened his horse to a\ntree, and coolly throwing his rifle across his shoulders, boldly\nadvanced toward the squatters. The latter, surprised by the courage of\nthis man, who alone attempted what his comrades had given up all hopes\nof achieving, let him come up to them without offering the slightest\nopposition. When Don Miguel was a couple of paces from the old squatter;\nhe stopped, put the butt of his rifle on the ground, and removing his\nhat, said,--\n\n\"Do you recognise me, Red Cedar?\"\n\n\"Don Miguel Zarate!\" the bandit shouted in surprise.\n\n\"As the judge deserts me,\" the hacendero continued, \"and fled like a\ncoward before your threats, I am obliged to take justice for myself,\nand, by heavens! I will do so! Red Cedar, I, as owner of this forest, in\nwhich you have settled without permission, order you to depart at once.\"\n\nThe young men exchanged a few muttered threats.\n\n\"Silence!\" Red Cedar commanded. \"Let the caballero speak.\"\n\n\"I have finished, and await your answer.\"\n\nThe squatter appeared to reflect deeply for a few minutes.\n\n\"The answer you demand is difficult to give,\" he at length said: \"my\nposition toward you is not a free one.\"\n\n\"Why so?\"\n\n\"Because I owe you my life.\"\n\n\"I dispense you from all gratitude.\"\n\n\"That is possible. You are at liberty to do so; but I cannot forget the\nservice you rendered me.\"\n\n\"It is of little consequence.\"\n\n\"Much more than you fancy, caballero. I may be, through my character,\nhabits, and the mode of life I lead, beyond the law of civilised beings;\nbut I am not the less a man, and if of the worst sort, perhaps, I no\nmore forget a kindness than I do an insult.\"\n\n\"Prove it, then, by going away as quickly as you can, and then we shall\nbe quits.\"\n\nThe squatter shook his head.\n\n\"Listen to me, Don Miguel,\" he said. \"You have in this country the\nreputation of being the providence of the unfortunate. I know from\nmyself the extent of your kindness and courage. It is said that you\npossess an immense fortune, of which you do not yourself know the\nextent.\"\n\n\"Well, what then?\" the hacendero impatiently interrupted him.\n\n\"The damage I can commit here, even if I cut down all the trees in the\nforest, would be but a trifle to you; then whence comes the fury you\ndisplay to drive me out?\"\n\n\"Your question is just, and I will answer it. I demand your departure\nfrom my estates, because, only a few days back, my son was grievously\nwounded by your lads, who led him into a cowardly snare; and if he\nescaped death, it was only through a miracle. That is the reason why we\ncannot live side by side, for blood severs us.\"\n\nRed Cedar frowned.\n\n\"Is this true?\" he said, addressing his sons.\n\nThe young men only hung their heads in reply.\n\n\"I am waiting,\" Don Miguel went on.\n\n\"Come, the question cannot be settled thus, so we will proceed to my\njacal.\"\n\n\"For what purpose? I ask you for a yes or no.\"\n\n\"I cannot answer you yet. We must have a conversation together, after\nwhich you shall decide to my future conduct. Follow me, then, without\nfear.\"\n\n\"I fear nothing, as I believe I have proved to you. Go on, as you demand\nit: I will follow you.\"\n\nRed Cedar made his sons a sign to remains here they were, and proceeded\nwith long strides toward his jacal, which was but a short distance off.\nDon Miguel walked carelessly after him. They entered the cabin. It was\ndeserted. The two females were doubtless also occupied in the forest.\nRed Cedar closed the door after him, sat down on a bench, made his guest\na sign to do the same, and began speaking in a low and measured voice,\nas if afraid what he had to say might be heard outside.\n\n\n\n\nCHAPTER IX.\n\nTHE ASSASSINATION.\n\n\n\"Listen to me, Don Miguel,\" Red Cedar said, \"and pray do not mistake my\nmeaning. I have not the slightest intention of intimidating you, nor do\nI think of attempting to gain your confidence by revelations which you\nmay fairly assume I have accidentally acquired.\"\n\nThe hacendero regarded with amazement the speaker, whose tone and\nmanner had so suddenly changed.\n\n\"I do not understand you,\" he said to him. \"Explain yourself more\nclearly, for the words you have just uttered are an enigma, the key to\nwhich I seek in vain.\"\n\n\"You shall be satisfied, caballero; and if you do not catch the meaning\nof my words this time it must be because you will not. Like all\nintelligent men, you are wearied of the incessant struggles in which the\nvital strength of your country is exhausted unprofitably. You have seen\nthat a land so rich, so fertile, so gloriously endowed as Mexico, could\nnot--I should say ought not--to remain longer the plaything of paltry\nambitions, and the arena on which all these transitory tyrannies sport\nin turn. For nearly thirty years you have dreamed of emancipation, not\nof your entire country, for that would be too rude a task, and\nunrealisable; but you said to yourself, 'Let us render New Mexico\nindependent; form it into a new State, governed by wise laws rigorously\nexecuted. By liberal institutions let us give an impetus to all the\nriches with which it is choked, give intellect all the liberty it\nrequires, and perhaps within a few years the entire Mexican\nConfederation, amazed by the magnificent results I shall obtain, will\nfollow my example. Then I shall die happy at what I have effected--my\nobject will be carried out. I shall have saved my country from the abyss\nover which it hangs, through the double pressure of the invasion of the\nAmerican Union and the exhaustion of the Spanish race.' Are not those\nideas yours, caballero? Do you consider that I have explained myself\nclearly this time?\"\n\n\"Perhaps so, though I do not yet see distinctly the point you wish to\nreach. The thoughts you attribute to me are such as naturally occur to\nall men who sincerely love their country, and I will not pretend that I\nhave not entertained them.\"\n\n\"You would be wrong in doing so, for they are great and noble, and\nbreathe the purest patriotism.\"\n\n\"A truce to compliments, and let us come to the point, for time\npresses.\"\n\n\"Patience: I have not yet ended. These ideas must occur to you sooner\nthan to another, as you are the descendant of the first Aztec kings, and\nborn defender of the Indians in this hapless country. You see that I am\nwell acquainted with you, Don Miguel Zarate.\"\n\n\"Too well, perhaps,\" the Mexican gentleman muttered.\n\nThe squatter smiled and went on:--\n\n\"It is not chance that led me to this country. I knew what I was doing,\nand why I came. Don Miguel, the hour is a solemn one. All your\npreparations are made: will you hesitate to give New Mexico the signal\nwhich must render it independent of the metropolis which has so long\nbeen fattening at its expense? Answer me.\"\n\nDon Miguel started. He fixed on the squatter a burning glance, in which\nadmiration at the man's language could be read. Red Cedar shrugged his\nshoulders.\n\n\"What! You still doubt?\" he said.\n\nHe rose, went to a box from which he took some papers, and threw them on\nthe table before the hacendero, saying,--\n\n\"Read.\"\n\nDon Miguel hurriedly seized the papers, and ran his eye over them.\n\n\"Well?\" he asked, looking fixedly at the strange speaker.\n\n\"You see,\" the squatter answered, \"that I am your accomplice. General\nIbanez, your agent in Mexico, is in correspondence with me, as is Mr.\nWood, your agent at New York.\"\n\n\"It is true,\" the Mexican said coldly, \"you have the secret of the\nconspiracy. The only point left is to what extent that goes.\"\n\n\"I possess it entirely. I have orders to enlist the volunteers who will\nform the nucleus of the insurrectionary army.\"\n\n\"Good!\"\n\n\"Now, you see, by these letters of General Ibanez and Mr. Wood, that I\nam commissioned by them to come to an understanding with you, and\nreceive your final orders.\"\n\n\"I see it.\"\n\n\"What do you purpose doing?\"\n\n\"Nothing.\"\n\n\"What, nothing!\" the squatter exclaimed, bounding with surprise. \"You\nare jesting, I suppose.\"\n\n\"Listen to me in your turn, and pay attention to my words, for they\nexpress my irrevocable resolution. I know not nor care to know, by what\nmeans, more or less honourable, you have succeeded in gaining the\nconfidence of my partners, and becoming master of our secrets. Still it\nis my firm conviction that a cause which employs such men as yourself is\ncompromised, if not lost; hence I renounce every combination in which\nyou are called to play a part. Your antecedents, and the life you lead,\nhave placed you without the pale of the law.\"\n\n\"I am a bandit--out with it! What matter so long as you succeed? Does\nnot the end justify the means?\"\n\n\"That may be your morality, but it will never be mine. I repudiate all\ncommunity of ideas with men of your stamp. I will not have you either as\naccomplice or partner.\"\n\nThe squatter darted a look at him laden with hatred and disappointment.\n\n\"In serving us,\" Don Miguel continued, \"you can only have an interested\nobject, which I will not take the trouble of guessing at. An\nAnglo-American will never frankly aid a Mexican to conquer his liberty;\nhe would lose too much by doing it.\"\n\n\"Then?\"\n\n\"I renounce forever the projects I had formed. I had, I grant, dreamed\nof restoring to my country the independence of which it was unjustly\nstripped: but it shall remain a dream.\"\n\n\"That is your last word?\"\n\n\"The last.\"\n\n\"You refuse?\"\n\n\"I do.\"\n\n\"Good; then I now know what is left me to do.\"\n\n\"Well, what is it? Let me hear,\" the hacendero said, as he crossed his\narms on his breast, and looked him boldly in the face.\n\n\"I will tell you.\"\n\n\"I am waiting for you to do so.\"\n\n\"I hold your secret.\"\n\n\"Entirely?\"\n\n\"Hence you are in my power.\"\n\n\"Perhaps.\"\n\n\"Who will prevent me going to the Governor of the State and denouncing\nyou?\"\n\n\"He will not believe you.\"\n\n\"You think so?\"\n\n\"I am sure of it.\"\n\n\"Perhaps, I will say in my turn.\"\n\n\"Why so?\"\n\n\"Oh! you shall easily see.\"\n\n\"I am curious to learn it.\"\n\n\"However rich you may be, Don Miguel Zarate, and perhaps because of\nthose very riches, and in spite of the kindness you sow broadcast, the\nnumber of your enemies is very considerable.\"\n\n\"I know it.\"\n\n\"Very good. Those enemies will joyfully seize the first opportunity that\npresents itself to destroy you.\"\n\n\"It is probable.\"\n\n\"You see, then. When I go to the governor and tell him you are\nconspiring, and, in support of my denunciation, hand him not only these\nletters, but, several others written and signed by you, lying in that\nchest, do you believe that the governor will treat me as an impostor,\nand refuse to arrest you?\"\n\n\"Then you have letters in my hand-writing?\"\n\n\"I have three, which will be enough to have you shot.\"\n\n\"Ah!\"\n\n\"Yes. Hang it all! you understand: that, in an affair so important as\nthis, it is wise to take one's precautions, for no one knows what may\nhappen; and men of my stamp,\" he added, with an ironical smile, \"have\nmore reasons than others for being prudent.\"\n\n\"Come, that is well played,\" the hacendero said, carelessly.\n\n\"Is it not?\"\n\n\"Yes, and I compliment you on it: you are a better player than I gave\nyou credit for.\"\n\n\"Oh! You do not know me yet.\"\n\n\"The little I do know suffices me.\"\n\n\"Then?\"\n\n\"We will remain as we are, if you will permit me.\"\n\n\"You still refuse?\"\n\n\"More than ever.\"\n\nThe squatter frowned.\n\n\"Take care, Don Miguel,\" he muttered, hoarsely. \"I will do what I told\nyou.\"\n\n\"Yes, if I allow you time.\"\n\n\"Eh?\"\n\n\"_Caspita!_ If you are a clever scamp, I am not altogether a fool. Do\nyou believe, in your turn, that I will let myself be intimidated by your\nthreats, and that I should not find means to keep you from acting, not\nfor my own sake, as I care little personally for what you can do, but\nfor my friends, who are men of honour, and whose lives I do not wish to\nbe compromised by your treachery?\"\n\n\"I am curious to know the means you will employ to obtain this result.\"\n\n\"You shall see,\" Don Miguel replied with perfect coolness.\n\n\"Well?\"\n\n\"I shall kill you.\"\n\n\"Oh, oh!\" the squatter said, as he looked complacently at his muscular\nlimbs, \"That is not easy.\"\n\n\"More so than you suppose, my master.\"\n\n\"Hum! and when do you reckon on killing me?\"\n\n\"At once!\"\n\nThe two men were seated in front of the hearth, each at the end of a\nbench: the table was between them, but a little back, so that while\ntalking they only leaned an elbow on it. While uttering the last word,\nDon Miguel bounded like a tiger on the squatter, who did not at all\nexpect the attack, seized him by the throat, and hurled him to the\nground. The two enemies rolled on the uneven flooring of the jacal.\n\nThe Mexican's attack had been so sudden and well directed that the\nhalf-strangled squatter, in spite of his Herculean strength, could not\nfree himself from his enemy's iron clutch, which pressed his throat like\na vice. Red Cedar could neither utter a cry nor offer the slightest\nresistance: the Mexican's knee crushed his chest, while his fingers\npressed into his throat.\n\nSo soon as he had reduced the wretch to utter impotence, Don Miguel drew\nfrom his vaquera boot a long sharp knife, and buried the entire blade in\nhis body. The bandit writhed convulsively for a few seconds; a livid\npallor suffused his face; his eyes closed, and he then remained\nmotionless. Don Miguel left the weapon in the wound, and slowly rose.\n\n\"Ah, ah!\" he muttered as he gazed at him with a sardonic air, \"I fancy\nthat rogue will not denounce me now.\"\n\nWithout loss of time he seized the letters lying on the table, took from\nthe box the few documents he found in it, hid them all in his bosom,\nopened the door of the cabin, which he carefully closed after him, and\nwent off with long strides.\n\nThe squatter's sons had not quitted their post; but, so soon as they\nperceived the Mexican, they went up to him.\n\n\"Well,\" Shaw asked him, \"have you come to an understanding with the old\nman?\"\n\n\"Perfectly so,\" the Mexican answered.\n\n\"Then the affair is settled?\"\n\n\"Yes, to our mutual satisfaction.\"\n\n\"All the better,\" the young men exclaimed joyously.\n\nThe hacendero unfastened his horse and mounted.\n\n\"Good-bye, gentlemen!\" he said to them.\n\n\"Good-bye!\" they replied, returning his bow.\n\nThe Mexican put his horse to a trot, but at the first turn in the road\nhe dug his spurs into its flanks, and started at full speed.\n\n\"Now,\" Sutter observed, \"I believe that we can proceed to the cabin\nwithout inconvenience.\"\n\nAnd they gently walked toward the jacal, pleasantly conversing together.\n\nDon Miguel, however, had not succeeded so fully as he imagined. Red\nCedar was not dead, for the old bandit kept a firm hold on life.\nAttacked unawares, the squatter had not attempted a resistance, which he\nsaw at the first glance was useless, and would only have exasperated his\nadversary. With marvellous sagacity, on feeling the knife blade enter\nhis body, he stiffened himself against the pain, and resolved on\n\"playing 'possum;\" that is to say, feigning death. The success of his\nstratagem was complete. Don Miguel, persuaded that he had killed him,\ndid not dream of repeating his thrust.\n\nSo long as his enemy remained in the jacal the squatter was careful not\nto make the slightest movement that might have betrayed him; but, so\nsoon as he was alone, he opened his eyes, rose with an effort, drew the\ndagger from the wound, which emitted a jet of black blood, and looking\nat the door, through which his assassin had departed, with a glance so\nfull of hatred that it is impossible to describe, he muttered,--\n\n\"Now we are quits, Don Miguel Zarate, since you have tried to take back\nthe life of him you saved. Pray God never to bring us face to face\nagain!\"\n\nHe uttered a deep sigh, and rolled heavily on the ground in a fainting\nfit. At this moment his sons entered the cabin.\n\n\n\n\nCHAPTER X.\n\nTHE SACHEM OF THE CORAS.\n\n\nA few days after the events we have described in the previous chapter\nthere was one of those lovely mornings which are not accorded to our\ncold climates to know. The sun poured down in profusion its warm beams,\nwhich caused the pebbles and sand to glisten in the walks of the garden\nof the Hacienda de la Noria. In a clump of flowering orange and lemon\ntrees, whose sweet exhalations perfumed the air, and beneath a copse of\ncactus, nopals, and aloes, a maiden was asleep, carelessly reclining in\na hammock made of the thread of the _Phormium tenax,_ which hung between\ntwo orange trees.\n\nWith her head thrown back, her long black hair unfastened, and falling\nin disorder on her neck and bosom; with her coral lips parted, and\ndisplaying the dazzling pearl of her teeth, Dona Clara (for it was she\nwho slept thus with an infantile slumber) was really charming. Her\nfeatures breathed happiness, for not a cloud had yet arisen to perturb\nthe azure horizon of her calm and tranquil life.\n\nIt was nearly midday: there was not a breath in the air. The sunbeams,\npouring down vertically, rendered the heat so stifling and\nunsupportable, that everyone in the hacienda had yielded to sleep, and\nwas enjoying what is generally called in hot countries the _siesta._\nStill, at a short distance from the spot where Dona Clara reposed, calm\nand smiling, a sound of footsteps, at first almost imperceptible, but\ngradually heightening, was heard, and a man made his appearance. It was\nShaw, the youngest of the squatter's sons. How was he at this spot?\n\nThe young man was panting, and the perspiration poured down his cheeks.\nOn reaching the entrance of the clump he bent an anxious glance on the\nhammock.\n\n\"She is there,\" he murmured with a passionate accent. \"She sleeps.\"\n\nThen he fell on his knees upon the sand, and began admiring the maiden,\ndumb and trembling. He remained thus a long time, with his glance fixed\non the slumberer with a strange expression. At length he uttered a sigh\nand tearing himself with an effort from this delicious contemplation, he\nrose sadly, muttering in a whisper,--\n\n\"I must go--if she were to wake--oh, she will never know how much I love\nher!\"\n\nHe plucked an orange flower, and softly laid it on the maiden; then he\nwalked a few steps from her, but almost immediately returning, he\nseized, with a nervous hand, Dona Clara's _rebozo,_ which hung down from\nthe hammock, and pressed it to his lips several times, saying, in a\nvoice broken by the emotion he felt,--\n\n\"It has touched her hair.\"\n\nAnd rushing from the thicket, he crossed the garden and disappeared. He\nhad heard footsteps approaching. In fact, a few seconds after his\ndeparture, Don Miguel, in his turn, entered the copse.\n\n\"Come, come,\" he said gaily, as he shook the hammock, \"sleeper, will you\nnot have finished your siesta soon?\"\n\nDona Clara opened her eyes, with a smile.\n\n\"I am no longer asleep, father,\" she said.\n\n\"Very good. That is the answer I like.\"\n\nAnd he stepped forward to kiss her; but, with sudden movement, the\nmaiden drew herself back as if she had seen some frightful vision, and\nher face was covered with a livid pallor.\n\n\"What is the matter with you?\" the hacendero exclaimed with terror.\n\nThe girl showed him the orange flower.\n\n\"Well,\" her father continued, \"what is there so terrific in that flower?\nIt must have fallen from the tree during your sleep.\"\n\nDona Clara shook her head sadly.\n\n\"No,\" she said: \"for some days past I have always noticed, on waking a\nsimilar flower thrown on me.\"\n\n\"You are absurd; chance alone is to blame for it all. Come, think no\nmore about it; you are pale as death, child. Why frighten yourself thus\nabout a trifle? Besides the remedy may be easily found. If so afraid of\nflowers now, why not take your siesta in your bedroom, instead of\nburying yourself in this thicket?\"\n\n\"That is true, father,\" the girl said, all joyous, and no longer\nthinking of the fear she had undergone. \"I will follow your advice.\"\n\n\"Come, that is settled, so say no more about it. Now give me a kiss.\"\n\nThe maiden threw herself into her father's arms, whom she stifled with\nkisses. Both sat down on a grassy mound, and commenced one of those\ndelicious chit-chats whose charm only those who are parents can properly\nappreciate. Presently a peon came up.\n\n\"What has brought you?\" Don Miguel asked.\n\n\"Excellency,\" the peon answered, \"a redskin warrior has just arrived at\nthe hacienda, who desires speech with you.\"\n\n\"Do you know him?\" Don Miguel asked.\n\n\"Yes, Excellency; it is Eagle-wing, the sachem of the Coras of the Rio\nSan Pedro.\"\n\n\"Mookapec! (Flying Eagle)\" the hacendero repeated with surprise. \"What\ncan have brought him to me? Lead him here.\"\n\nThe peon retired and in a few minutes returned, preceding Eagle-wing.\n\nThe chief had donned the great war-dress of the sachems of his nation.\nHis hair, plaited with the skin of a rattlesnake, was drawn up on the\ntop of his head; in the centre an eagle plume was affixed. A blouse of\nstriped calico, adorned with a profusion of bells, descended to his\nthighs, which were defended from the stings of mosquitoes by drawers of\nthe same stuff. He wore moccasins made of peccary skin, adorned with\nglass beads and porcupine quills. To his heels were fastened several\nwolves' tails, the distinguishing mark of renowned warriors. Round his\nloins was a belt of elk hide, through which passed his knife, his pipe\nand his medicine bag. His neck was adorned by a collar of grizzly bear\nclaws and buffalo teeth. Finally, a magnificent robe of a white female\nbuffalo hide, painted red inside, was fastened to his shoulders, and\nfell down behind him like a cloak. In his right hand he held a fan\nformed of a single eagle's wing, and in his left hand an American rifle.\nThere was something imposing and singularly martial in the appearance\nand demeanor of this savage child of the forest.\n\nOn entering the thicket, he bowed gracefully to Dona Clara, and then\nstood motionless and dumb before Don Miguel. The Mexican regarded him\nattentively, and saw an expression of gloomy melancholy spread over the\nIndian chief's features.\n\n\"My brother is welcome,\" the hacendero said to him. \"To what do I owe\nthe pleasure of seeing him?\"\n\nThe chief cast a side glance at the maiden. Don Miguel understood what\nhe desired, and made Dona Clara a sign to withdraw. They remained alone.\n\n\"My brother can speak,\" the hacendero then said; \"the ears of a friend\nare open.\"\n\n\"Yes, my father is good,\" the chief replied in his guttural voice. \"He\nloves the Indians: unhappily all the palefaces do not resemble him.\"\n\n\"What does my brother mean? Has he cause to complain of anyone?\"\n\nThe Indian smiled sadly.\n\n\"Where is there justice for the redskins?\" he said. \"The Indians are\nanimals: the Great Spirit has not given them a soul, as He has done for\nthe palefaces, and it is not a crime to kill them.\"\n\n\"Come, chief, pray do not speak longer in riddles, but explain why you\nhave quitted your tribe. It is far from Rio San Pedro to this place.\"\n\n\"Mookapec is alone: his tribe no longer exists.\"\n\n\"How?\"\n\n\"The palefaces came in the night, like jaguars without courage. They\nburned the village, and massacred all the inhabitants, even to the women\nand little children.\"\n\n\"Oh, that is frightful!\" the hacendero murmured, in horror.\n\n\"Ah!\" the chief continued with an accent full of terrible irony, \"The\nscalps of the redskins are sold dearly.\"\n\n\"And do you know the men who committed this atrocious crime?\"\n\n\"Mookapec knows them, and will avenge himself.\"\n\n\"Tell me their chief, if you know his name.\"\n\n\"I know it. The palefaces call him Red Cedar, the Indians the Maneater.\"\n\n\"Oh! As for him, chief, you are avenged, for he is dead.\"\n\n\"My father is mistaken.\"\n\n\"How so? Why, I killed him myself.\"\n\nThe Indian shook his head.\n\n\"Red Cedar has a hard life,\" he said: \"the blade of the knife my father\nused was too short. Red Cedar is wounded, but in a few days he will be\nabout again, ready to kill and scalp the Indians.\"\n\nThis news startled the hacendero: the enemy he fancied he had got rid\nof still lived, and he would have to begin a fresh struggle.\n\n\"My father must take care,\" the chief continued. \"Red Cedar has sworn to\nbe avenged.\"\n\n\"Oh! I will not leave him the time. This man is a demon, of whom the\nearth must be purged at all hazards, before his strength has returned,\nand he begins his assassinations again.\"\n\n\"I will aid my father in his vengeance.\"\n\n\"Thanks, chief. I do not refuse your offer: perhaps I shall soon need\nthe help of all my friends. And now, what do you purpose doing?\"\n\n\"Since the palefaces reject him, Eagle-wing will retire to the desert.\nHe has friends among the Comanches. They are redskins, and will welcome\nhim gladly.\"\n\n\"I will not strive to combat your determination, chief, for it is just;\nand if, at a later date, you take terrible reprisals on the white men,\nthey will have no cause of complaint, for they have brought it on\nthemselves. When does my brother start?\"\n\n\"At sunset.\"\n\n\"Rest here today: tomorrow will be soon enough to set out.\"\n\n\"Mookapec must depart this day.\"\n\n\"Act as you think proper. Have you a horse?\"\n\n\"No; but at the first manada I come to I will lasso one.\"\n\n\"I do not wish you to set out thus, but will give you a horse.\"\n\n\"Thanks; my father is good. The Indian chief will remember--\"\n\n\"Come, you shall choose for yourself.\"\n\n\"I have still a few words to say to my father.\"\n\n\"Speak, chief; I am listening to you.\"\n\n\"Koutonepi, the pale hunter, begged me to give my father an important\nwarning.\"\n\n\"What is it?\"\n\n\"A great danger threatens my father. Koutonepi wishes to see him as soon\nas possible, in order himself to tell him its nature.\"\n\n\"Good! My brother will tell the hunter that I shall be tomorrow at the\n'clearing of the shattered oak,' and await him there till night.\"\n\n\"I will faithfully repeat my father's words to the hunter.\"\n\nThe two men then quitted the garden, and hurriedly proceeded toward the\nhacienda. Don Miguel let the chief choose his own horse, and while the\nsachem was harnessing his steed in the Indian fashion, he withdrew to\nhis bedroom, and sent for his son to join him. The young man had\nperfectly recovered from his wound. His father told him that he was\nobliged to absent himself for some days: he intrusted to him the\nmanagement of the hacienda, while recommending him on no consideration\nto leave the farm, and to watch attentively over his sister. The young\nman promised him all he wished, happy at enjoying perfect liberty for a\nfew days.\n\nAfter embracing his son and daughter for the last time Don Miguel\nproceeded to the _patio_, where in the meanwhile, the chief had been\namusing himself by making the magnificent horse he had chosen curvet.\nDon Miguel admired for several moments the Indian's skill and grace, for\nhe managed a horse as well as the first Mexican _jinete;_ then mounted,\nand the two men proceeded together toward the Paso del Norte, which they\nmust cross in order to enter the desert, and reach the clearing of the\nshattered oak.\n\nThe journey passed in silence, for the two men were deeply reflecting.\nAt the moment they entered Paso the sun was setting on the horizon in a\nbed of red mist, which foreboded a storm for the night. At the entrance\nof the village they separated; and on the morrow, as we have seen in our\nfirst chapter, Don Miguel set out at daybreak, and galloped to the\nclearing.\n\nWe will now end this lengthy parenthesis, which was, however,\nindispensable for the due comprehension of the facts that are about to\nfollow, and take up our story again at the point where we left it.\n\n\n\n\nCHAPTER XI.\n\nCONVERSATION.\n\n\nValentine Guillois, whom we have already introduced to the reader in\nprevious works[1], had inhabited, or, to speak more correctly, traversed\nthe vast solitudes of Mexico and Texas during the past five or six\nyears. We saw him just now accompanied by the Araucano chief. These two\nmen were the boldest hunters on the frontier. At times, when they had\ncollected an ample harvest of furs, they went to sell them in the\nvillages, renewed their stock of powder and ball, purchased a few\nindispensable articles, and then returned to the desert.\n\nNow and then they engaged themselves for a week, or even a fortnight,\nwith the proprietors of the haciendas, to free them from the wild beasts\nthat desolated their herds; but so soon as the ferocious animals were\ndestroyed, and the reward obtained, no matter the brilliancy of the\noffers made them by the landowners, the two men threw their rifles on\ntheir shoulders and went off.\n\nNo one knew who they were, or whence they came. Valentine and his friend\nmaintained the most complete silence as to the events of their life\nwhich had preceded their appearance in these parts. Only one thing had\nbetrayed the nationality of Valentine, whom his comrade called\nKoutonepi, a word belonging to the language of the Aucas, and signifying\n\"The Valiant.\" On his chest the hunter wore the cross of the Legion of\nHonor. The deeds of every description performed by these hunters were\nincalculable, and their stories were the delight of the frontier\ndwellers during the winter night. The number of tigers they had killed\nwas no longer counted.\n\nChance had one day made them acquainted with Don Miguel Zarate under\nstrange circumstances, and since then an uninterrupted friendship had\nbeen maintained between them. Don Miguel, during a tempestuous night,\nnamely, had only owed his life to the accuracy of Valentine's aim, who\nsent a bullet through the head of the Mexican's horse at the moment\nwhen, mad with terror, and no longer obeying the bridle, it was on the\npoint of leaping into an abyss with its master. Don Miguel had sworn\neternal gratitude to his saviour.\n\nValentine and Curumilla had made themselves the tutors of the\nhacendero's children, who, for their part, felt a deep friendship for\nthe hunters. Don Pablo had frequently made long hunting parties in the\ndesert with them; and it was to them he owed the certainty of his aim,\nhis skill in handling weapons, and his knack in taming horses.\n\nNo secrets existed between Don Miguel and the hunters: they read in his\nmind as in an ever open book. They were the disinterested confidants of\nhis plans; for these rude wood rangers esteemed him, and only required\nfor themselves one thing--the liberty of the desert. Still, despite the\nsympathy and friendship which so closely connected these different\npersons, and the confidence which formed the basis of that friendship,\nDon Miguel and his children had never been able to obtain from the\nhunters information as to the events that had passed prior to their\narrival in this country.\n\nFrequently Don Miguel, impelled, not by curiosity, but merely by the\ninterest he felt in them, had tried, by words cleverly thrown into the\nconversation, to give them an opening for confidence; but Valentine had\nalways repelled those hints, though cleverly enough for Don Miguel not\nto feel offended by this want of confidence. With Curumilla they had\nbeen even more simple. Wrapped in his Indian stoicism, intrenched in his\nhabitual sullenness, he was wont to answer all questions by a shake of\nthe head, but nothing further.\n\nAt length, weary of the attempt, the hacendero and his family had given\nup trying to read those secrets which their friends seemed obstinately\ndetermined to keep from them. Still the friendship subsisting between\nthem had not grown cold in consequence, and it was always with equal\npleasure that Don Miguel met the hunters again after a lengthened ramble\nin the prairies, which kept them away from his house for whole months at\na time.\n\nThe hunter and the Mexican were seated by the fire, while Curumilla,\narmed with his scalping knife, was busy flaying the two jaguars so\nskillfully killed by Don Miguel, and which were magnificent brutes.\n\n\"Eh, _compadre!_\" Don Miguel said with a laugh; \"I was beginning to lose\npatience, and fancy you had forgotten the meeting you had yourself given\nme.\"\n\n\"I never forgot anything, as you know,\" Valentine answered seriously;\n\"and if I did not arrive sooner, it was because the road is long from my\njacal to this clearing.\"\n\n\"Heaven forbid that I should reproach you, my friend! Still I confess to\nyou that the prospect of passing the night alone in this forest only\nslightly pleased me, and I should have been off had you not arrived\nbefore sunset.\"\n\n\"You would have done wrong, Don Miguel: what I have to tell you is of\nthe utmost importance to you. Who knows what the result might have been\nhad I not been able to warn you?\"\n\n\"You alarm me, my friend.\"\n\n\"I will explain. In the first place let me tell you that you committed,\na few days back, a grave imprudence, whose consequences threaten to be\nmost serious for you.\"\n\n\"What is it?\"\n\n\"I said one, but ought to have said two.\"\n\n\"I am waiting till you think proper to express yourself more clearly,\"\nDon Miguel said with a slight tinge of impatience, \"before I answer.\"\n\n\"You have quarrelled with a North American bandit.\"\n\n\"Red Cedar.\"\n\n\"Yes; and when you had him in your power you let him escape, instead of\nkilling him out and out.\"\n\n\"That is true, and I was wrong. What would you? The villain has as tough\na life as an alligator. But be at ease. If ever he fall into my hands\nagain, I swear that I will not miss him.\"\n\n\"In the meanwhile you did do so--that is the evil.\"\n\n\"Why so?\"\n\n\"You will understand me. This man is one of those villains, the scum of\nthe United States, too many of whom have lived on the frontier during\nthe last few years. I do not know how he contrived to deceive your New\nYork agent; but he gained his confidence so cleverly that the latter\ntold him all the secrets he knew about your enterprise.\"\n\n\"He told me so himself.\"\n\n\"Very good. It was then, I suppose, that you stabbed him?\"\n\n\"Yes, and at the same time I plucked out his claws; that is to say, I\nseized the letters he held, and which might compromise me.\"\n\n\"A mistake. This man is too thorough-paced a scoundrel not to foresee\nall the chances of his treason. He had a last letter, the most important\nof all; and that you did not take from him.\"\n\n\"I took three.\"\n\n\"Yes, but there were four. As the last, however, in itself was worth as\nmuch as the other three, he always wore it about him in a leathern bag\nhung round his neck by a steel chain; you did not dream of looking for\nthat.\"\n\n\"But what importance can this letter, I do not even remember writing,\npossess, that you should attach such weight to it?\"\n\n\"It is merely the agreement drawn up between yourself, General Ibanez,\nand Mr. Wood, and bearing your three signatures.\"\n\n\"_Con mil demonios!_\" the hacendero exclaimed in terror. \"In that case\nI am lost; for if this man really possesses such a document, he will not\nfail to employ it in order to be revenged on me.\"\n\n\"Nothing is lost so long as a man's heart beats in his breast, Don\nMiguel. The position is critical, I allow, but I have saved myself in\nsituations far more desperate than the one you are now in.\"\n\n\"What is to be done?\"\n\n\"Red Cedar has been about again for two days. His first care, so soon as\nhe could sit a horse, was to go to Santa Fe, the capital of New Mexico,\nand denounce you to the Governor. That has nothing to surprise you from\nsuch a man.\"\n\n\"Then I can only fly as speedily as I can?\"\n\n\"Wait. Every man has in his heart at least one of the seven deadly sins\nas a bait for the demon.\"\n\n\"What are you driving at?\"\n\n\"You will see. Fortunately for us, Red Cedar has them all seven, I\nbelieve, in the finest stage of development. Avarice, before all, has\nreached its acme with him.\"\n\n\"Well?\"\n\n\"This happened. Our man denounced you to the governor as a conspirator,\netc., but was careful not to give up the proofs he possessed in support\nof the denunciation at the outset. When General Isturitz, the governor,\nasked him for these proofs, he answered that he was ready to supply them\nin exchange for the sum of one hundred thousand piastres in gold.\"\n\n\"Ah!\" the hacendero said, with a breath of relief, \"and what did\nIsturitz say?\"\n\n\"The general is one of your most inveterate enemies, I grant, and he\nwould give a good deal for the pleasure of having you shot.\"\n\n\"That is true.\"\n\n\"Yes, but still the sum appeared to him, as it really is, exorbitant,\nthe more so as he would have to pay it all himself, as the government\ndoes not recognise transactions of that nature.\"\n\n\"Well, what did Red Cedar do then?\"\n\n\"He did not allow himself beaten; on the contrary, he told the general\nhe would give him a week to reflect, and quietly left the Cabildo.\"\n\n\"Hum! And on what day was this visit paid?\"\n\n\"Yesterday morning; so that you have six days still left for action.\"\n\n\"Six days--that is very little.\"\n\n\"Eh?\" the Frenchman said, with a shrug of his shoulders impossible to\ndescribe. \"In my country--\"\n\n\"Yes, but you are Frenchmen.\"\n\n\"That is true: hence I allow you twice the time we should require. Come,\nlet us put joking aside. You are a man of more than common energy; you\nreally wish the welfare of your country, so do not let yourself be\ncrushed by the first reverse. Who knows but that it may all be for the\nbest?\"\n\n\"Ah, my friend, I am alone! General Ibanez, who alone could help me in\nthis critical affair, is fifty leagues off. What can I do? Nothing.\"\n\n\"All. I foresaw your objection. Eagle-wing, the Chief of the Coras, has\ngone from me to warn the general. You know with what speed Indians\ntravel; so he will bring us the general in a few hours, I feel\nconvinced.\"\n\nDon Miguel regarded the hunter with mingled admiration and respect.\n\n\"You have done that, my friend?\" he said to him as he warmly pressed his\nhand.\n\n\"By Jove!\" Valentine said, gaily, \"I have done something else too. When\nthe time arrives I will tell you what it is. But let us not lose an\nhour. What do you intend to do for the present?\"\n\n\"Act.\"\n\n\"Good: that is the way I like to hear you talk.\"\n\n\"Yes, but I must first come to an understanding with the general.\"\n\n\"That is true; but it is the least thing,\" Valentine answered, as he\nlooked skyward, and attentively consulted the position of the stars. \"It\nis now eight o'clock. Eagle-wing and the man he brings must be at\nmidnight at the entrance of the _Canyon del Buitre_. We have four hours\nbefore us, and that is more than we require, as we have only ten leagues\nto go.\"\n\n\"Let us go, let us go!\" Don Miguel exclaimed eagerly.\n\n\"Wait a moment; there is no such hurry. Don't be alarmed; we shall\narrive in time.\"\n\nHe then turned to Curumilla, and said to him in Araucano a few words\nwhich the hacendero did not understand. The Indian rose without\nreplying, and disappeared in the density of the forest.\n\n\"You know,\" Valentine continued, \"that I prefer, through habit,\ntravelling on foot; still, as under present circumstances minutes are\nprecious, and we must not lose them, I have provided two horses.\"\n\n\"You think of everything, my friend.\"\n\n\"Yes, when I have to act for those I love,\" Valentine answered with a\nretrospective sigh.\n\nThere was a moment's silence between the two men, and at the end of\nscarce a quarter of an hour there was a noise in the shrubs, the\nbranches parted, and Curumilla re-entered the clearing, holding two\nhorses by the bridle. These noble animals, which were nearly untamed\n_mustangs_, bore a striking resemblance to the steeds of the Apaches, on\nwhose territory our friends now were. They were literally covered with\neagle plumes, beads, and ribbons, while long red and white spots\ncompleted their disguise, and rendered it almost impossible to recognise\nthem.\n\n\"Mount!\" Don Miguel exclaimed so soon as he saw them. \"Time is slipping\naway.\"\n\n\"One word yet,\" Valentine remarked.\n\n\"Speak.\"\n\n\"You still have as chaplain a certain monk by the name of 'Fray\nAmbrosio.'\"\n\n\"Yes.\"\n\n\"Take care of that man--he betrays you.\"\n\n\"You believe it?\"\n\n\"I am sure of it.\"\n\n\"Good! I will remember.\"\n\n\"All right. Now we will be off,\" Valentine said, as he buried his spurs\nin his horse's flanks.\n\nAnd the three horsemen rushed into the darkness with headlong speed.\n\n\n[1] \"Tiger-Slayer,\" etc. Same publishers.\n\n\n\n\nCHAPTER XII.\n\nEL MESON.\n\n\nThe day on which our story commences the village of the Paso del Norte\npresented an extraordinary appearance. The bells were ringing out full\npeals, for the three hundredth anniversary of its foundation was\ncelebrated. The population of Paso, greatly diminished since the\nproclamation of Mexican independence, was hurrying to the churches,\nwhich flashed with silver and gold. The houses were decorated with rich\ntapestry, and the streets strewn with flowers.\n\nToward nightfall the inhabitants, whom the intolerable heat of the\ntropical sun had kept prisoners in the interior of the houses, flocked\nout to inhale the sharp perfumes of the desert breeze, and bring back a\nlittle fresh air into their parched lungs. The town, which had for\nseveral hours appeared deserted, suddenly woke up: shouts and laughter\nwere heard afresh. The walks were invaded by the mob, and in a few\nminutes the _mesons_ were thronged with idlers, who began drinking\npulque and mezcal, while smoking their cigarettes, and strumming the\njarabe and vihuela.\n\nIn a house of poor appearance, built like all its neighbours, of earth\nbricks, and situated at the angle formed by the Plaza Mayor and the\nCalle de la Merced, some twenty-five fellows, whom it was easy to\nrecognise as adventurers by the feather in their hats, their upturned\nmoustaches, and specially by the long bronzed-hilted sword they wore on\nthe thigh, were drinking torrents of aguardiente and pulque at the\ngambling tables, while yelling like deaf men, swearing like pagans, and\nthreatening at every moment to unsheathe their weapons.\n\nIn a corner of the room occupied by these troublesome guests two men,\nseated opposite each other at a table, seemed plunged in deep thought,\nand looked round them absently, not thinking about drinking the contents\nof their glasses, which had not been emptied for more than half an hour.\nThese two men presented the most striking contrast. They were still\nyoung. The first, aged twenty-five at the most, had one of those frank,\nhonest, and energetic faces which call for sympathy, and attract\nrespect. His pallid brow, his face of a delicate hue, surrounded by his\nlong black curls, his straight and flexible nose, his mouth filled with\na double row of teeth of dazzling whiteness, and surmounted by a slight\nbrown moustache, gave him a stamp of distinction, which was the more\nstriking owing to the strict, and perhaps common, style of his attire.\n\nHe wore the costume of the wood rangers; that is to say, the Canadian\n_mitasse_, fastened round the hips, and descending to the ankle; _botas\nvaqueras_ of deer skin, fastened at the knee; and a striped zarape of\nbrilliant colours. A panama straw hat was thrown on the table, within\nreach of his hand, by the side of an American rifle and two\ndouble-barrelled pistols. A machete hung on his left side, and the hilt\nof a long knife peeped out of his left boot.\n\nHis companion was short and thick-set; but his well-knit limbs and his\noutstanding muscles indicated no ordinary strength. His face, the\nfeatures of which were commonplace enough, had a cunning look, which\nsuddenly disappeared to make room for a certain nobility whenever under\nthe influence Of any sudden emotion; his eyebrows contracted; and his\nglance, ordinarily veiled, flashed forth. He wore nearly the same garb\nas his comrade; but his hat stained with rain, and the colours of his\nzarape faded by the sun, evidenced lengthened wear. Like the first one\nwe described, he was well armed.\n\nIt was easy to see at the first glance that these two men did not belong\nto the Hispano-American race, indeed, their conversation would have\nremoved any doubts on that head, for they spoke in the French dialect\nemployed in Canada.\n\n\"Hum!\" the first said, taking up his glass, which he carelessly raised\nto his lips. \"After due consideration, Harry, I believe we shall do\nbetter by mounting our horses again, and starting, instead of remaining\nin this horrible den, amid these _gachupinos_, who croak like frogs\nbefore a storm.\"\n\n\"Deuce take your impatience!\" the other replied ill-temperedly. \"Can't\nyou remain a moment at rest?\"\n\n\"You call it a moment, Harry. Why, we have been here an hour.\"\n\n\"By Jove! Dick, you're a wonderful fellow,\" the other continued with a\nlaugh. \"Do you think that business can be settled all in a moment?\"\n\n\"After all, what is our game? For may the old one twist my neck, or a\ngrizzly give me a hug, if I know the least in the world! For five years\nwe have hunted and slept side by side. We have come from Canada together\nto this place. I have grown into a habit--I cannot say why--of referring\nto you everything that concerns our mutual interests. Still I should not\nbe sorry to know, if only for the rarity of the fact, why on earth we\nleft the prairies, where we were so well off, to come here, where we are\nso badly off.\"\n\n\"Have you ever repented, up to today, the confidence you placed in me?\"\n\n\"I do not say so, Harry. Heaven forbid! Still I think--\"\n\n\"You think wrong,\" the young man sharply interrupted. \"Let me alone, and\nbefore three months you shall have three times your hat full of massive\ngold, or call me a fool.\"\n\nAt this dazzling promise the eyes of Dick, the smaller of the hunters,\nglistened like two stars. He regarded his comrade with a species of\nadmiration.\n\n\"Oh, oh!\" he said in a low voice, \"It is a placer, is it?\"\n\n\"Hang it!\" the other said, with a shrug of his shoulders, \"were it not,\nshould I be here? But silence, our man has arrived.\"\n\nIn fact, a man entered at this moment. On his appearance a sudden\nsilence fell on the meson; the adventurers gambling and cursing at all\nthe tables, rose as if moved by a spring, respectfully took off their\nplumed hats, and ranged themselves with downcast eyes to let him pass.\nThe man remained for an instant on the threshold of the venta, took a\nprofound glance at the company, and then walked toward the two hunters.\n\nThis man wore the gown of a monk; he had the ascetic face, with the\nharsh features and sharply-marked lines, that forms, as it were, the\ntype of the Spanish monks of which Titian has so admirably caught the\nexpression on his canvas. He passed through the adventurers, holding out\nright and left his wide sleeves, which they reverentially kissed. On\napproaching the two hunters he turned round.\n\n\"Continue your sports, my sons,\" he said to the company; \"my presence\nneed not disturb your frolics, for I only wish to speak for a few\nmoments with those two gentlemen.\"\n\nThe adventurers did not let the invitation be repeated, but took their\nplaces again tumultuously, and soon cries and oaths recommenced with\nequal intensity. The monk smiled, took a butaca, and seated himself\nbetween the two hunters, while bending a searching glance on them. The\nlatter had followed with a mocking eye all the interludes of this little\nscene, and without making a movement, they let the monk seat himself by\ntheir side. So soon as he had done so, Harry poured him out a large\nglass of pulque, and placed within his reach the squares of maize leaf\nand tobacco.\n\n\"Drink and smoke, senor padre,\" he said to him.\n\nThe monk, without any observation, rolled a cigarette, emptied the glass\nof pulque at a draught, and then leaning his elbows on the table and\nbending forward, said,--\n\n\"You are punctual.\"\n\n\"We have been waiting an hour,\" Dick observed in a rough voice.\n\n\"What is an hour in the presence of eternity?\" the monk said with a\nsmile.\n\n\"Let us not lose any more time,\" Harry continued. \"What have you to\npropose to us?\"\n\nThe monk looked around him suspiciously, and lowered his voice.\n\n\"I can, if you like, make you rich in a few days.\"\n\n\"What is the business?\" Dick asked.\n\n\"Of course,\" the monk continued, \"this fortune I offer you is a matter\nof indifference to me. If I have an ardent desire to obtain it, it is,\nin the first place, because it belongs to nobody, and will permit me to\nrelieve the wretchedness of the thousands of beings confided to my\ncharge.\"\n\n\"Of course, senor padre,\" Harry answered seriously. \"Let us not weigh\nlonger on these details. According to what you told me a few days back,\nyou have discovered a rich placer.\"\n\n\"Not I,\" the monk sharply objected.\n\n\"No consequence, provided that it exists,\" Dick answered.\n\n\"Pardon me, but it is of great consequence to me. I do not wish to take\non myself the responsibility of such a discovery. If, as I believe,\npeople will go in search of it, it may entail the death of several\npersons, and the church abhors bloodshed.\"\n\n\"Very good: you only desire to profit by it.\"\n\n\"Not for myself.\"\n\n\"For your parishioners. Very good; but let us try to come to an\nunderstanding, if possible, for our time is too precious for us to waste\nit in empty talk.\"\n\n\"_Valgame Dios_!\" the monk said, crossing himself, \"How you have\nretained the impetuosity of your French origin! Have a little patience,\nand I will explain myself.\"\n\n\"That is all we desire.\"\n\n\"But you will promise me--\"\n\n\"Nothing,\" Dick interrupted. \"We are honest hunters, and not accustomed\nto pledge ourselves so lightly before knowing positively what is asked\nof us.\"\n\nHarry supported his friend's words by a nod. The monk drank a glass of\npulque, and took two or three heavy puffs at his cigarette.\n\n\"Your will be done,\" he then said. \"You are terrible men. This is the\naffair.\"\n\n\"Go on.\"\n\n\"A poor scamp of a gambusino, lost, I know not how, in the great desert,\ndiscovered at a considerable distance off, between the Rio Gila and the\nColorado, the richest placer the wildest imagination can conceive.\nAccording to his statement the gold is scattered over the surface, for\nan extent of two or three miles, in nuggets, each of which would make a\nman's fortune. This gambusino, dazzled by such treasures, but unable to\nappropriate them alone, displayed the greatest energy, and braved the\nutmost perils, in order to regain civilised regions. It was only through\nboldness and temerity that he succeeded in escaping the countless\nenemies who spied, and tracked him on all sides; but Heaven at length\nallowed him to reach Paso safe and sound.\"\n\n\"Very good,\" Dick observed. \"All this may very possibly, be true; but\nwhy did you not bring this gambusino, instead of talking to us about the\nplacer, of which you know as little as we do? He would have supplied us\nwith information which is indispensable for us, in the event of our\nconsenting to help you in looking for this treasure.\"\n\n\"Alas!\" the monk replied, hypocritically casting his eyes down, \"the\nunhappy man was not destined to profit by this discovery, made at the\nprice of so many perils. Scarce two days after his arrival at Paso, he\nquarrelled with another gambusino, and received a stab which sent him a\nfew hours later to the tomb.\"\n\n\"In that case,\" Harry observed, \"how did you learn all these details,\nsenor padre?\"\n\n\"In a very simple way, my son. It was I who reconciled the poor wretch\nin his last moments with Heaven; and,\" he added, with an air of\ncompunction splendidly assumed, \"when he understood that his end was at\nhand, and that nothing could save him, he confided to me, in gratitude\nfor the consolation I bestowed on him, what I have just told you,\nrevealed to me the situation of the placer, and for greater certainty\ngave me a clumsy chart he had drawn out on the spot. You see that we can\nproceed almost with certainty.\"\n\n\"Yes,\" Harry said, thoughtfully; \"but why, instead of first applying to\nthe Mexicans, your countrymen, did you propose to us to help you in your\nenterprise?\"\n\n\"Because the Mexicans are men who cannot be trusted, and before reaching\nthe placer we should have to fight the Apaches and Comanches, on whose\nterritory it is situated.\"\n\nAfter these words, there was a lengthened silence between the three\nspeakers: each was reflecting deeply on what he had just heard. The monk\ntried to read with cunning eye the impression produced on the hunters by\nhis confidence; but his hopes were deceived. Their faces remained\nunmoved. At length Dick spoke in a rough voice, after exchanging a\nmeaning look with his comrade.\n\n\"All that is very fine,\" he said; \"but it is absurd to suppose that two\nmen, however brave they may be, can attempt such an enterprise in\nunknown regions peopled by ferocious tribes. It would require at least\nfifty resolute and devoted men, otherwise nothing could be possible.\"\n\n\"You are right, and hence I did not calculate on you alone. You will\nhave determined men under your orders, chosen carefully by myself, and I\nshall also accompany you.\"\n\n\"Unluckily, if you have counted on us, you are mistaken, senor padre,\"\nHarry said, peremptorily. \"We are honest hunters; but the trade of a\ngambusino does not at all suit us. Even if we had a chance of gaining an\nincalculable fortune, we would not consent to take part in an expedition\nof gold seekers.\"\n\n\"Not even if Red Cedar were at the head of the expedition, and consented\nto take the direction?\" the monk said in a honeyed voice, and with a\nside glance.\n\nThe hunter started, a feverish blush suffused his face, and it was in a\nvoice choked by emotion that he exclaimed,--\n\n\"Have you spoken with him about it?\"\n\n\"Here he is; you can ask him,\" the monk answered.\n\nIn fact, a man was entering the meson at this moment. Harry looked down\nin confusion, while Dick tapped the table with his dagger and whistled.\nA smile of undefinable meaning wandered over the monk's pallid lips.\n\n\n\n\nCHAPTER XIII.\n\nRED CEDAR.\n\n\nRed Cedar was more than six feet in height; his enormous head was\nfastened to his square shoulders by a short and muscular neck, like a\nbull's; his bony members were covered with muscles hard as ropes. In\nshort, his whole person was a specimen of brute strength at its\nculminating point.\n\nA fox-skin cap, pressed down on his head, allowed escape to a few tufts\nof coarse greyish hair, and fell on his little grey eyes, which were\nclose to a nose that was hooked like the beak of a bird of prey; his\nwide mouth was filled with white, large teeth; his cheekbones were\nprominent and purpled; and the lower part of his face disappeared in a\nthick black beard, mingled with grey hairs. He wore a hunting shirt of\nstriped calico, fastened round the waist by a strap of brown leather,\nthrough which were passed two pistols, an axe, and a long knife; a pair\nof leggings of tawny leather, sewed at equal distances with hair, fell\ndown to his knees; while his legs were protected by Indian moccasins,\nornamented with a profusion of beads and bells. A game bag of fawn skin,\nwhich seemed full, fell over his right hip; and he held in his hand an\nAmerican rifle, studded with copper nails.\n\nNo one knew who Red Cedar was, or whence he came. About two years prior\nto the period of our story opening he had suddenly made his appearance\nin the country, accompanied by a wife of a certain age--a species of\nMegaera, of masculine form and repellant aspect; a girl of seventeen;\nand three vigorous lads, who resembled him too closely not to be his\nown, and whose age varied from nineteen to twenty-four.\n\nRed Cedar himself appeared to be fifty-five at the most. The name by\nwhich he was known had been given to him by the Indians, of whom he had\ndeclared himself the implacable enemy, and boasted that he had killed\ntwo hundred. The old woman was called Betsy; the girl, Ellen; the eldest\nson, Nathan; the second, Sutter; and the last, Shaw.\n\nThis family had built a shanty in the forest, a few miles from Paso, and\nlived alone in the desert, without having entered into any relations\nwith the inhabitants of the village; or the trappers and wood rangers,\nits neighbours. The mysterious conduct of these strangers had given rise\nto numerous comments; but all had remained without reply or solution,\nand after two years they remained as perfect strangers as on the day of\ntheir arrival.\n\nStill, mournful and sad stories were in circulation on their account:\nthey inspired an instinctive hatred and involuntary terror in the\nMexicans. Some said in a whisper that old Red Cedar and his three sons\nwere nothing less than \"scalp hunters;\" that is to say, in the public\nesteem, people placed beneath the pirates of the prairies, that unclean\nbreed of birds of prey which everybody fears and despises.\n\nThe entry of Red Cedar was significant; the otherwise unscrupulous men\nwho filled the venta hurriedly retired on his approach, and made room\nfor him with a zeal mingled with disgust. The old partisan crossed the\nroom with head erect; a smile of haughty disdain played round his thin\nlips at the sight of the effect his presence produced, and he went up to\nthe monk and his two companions. On reaching them he roughly placed the\nbutt of his rifle on the ground, leaned his two crossed hands upon the\nbarrel, and after bending a cunning glance on the persons before him,\nsaid to the monk in a hoarse voice,--\n\n\"The deuce take you, senor padre! Here I am: what do you want with me?\"\n\nFar from being vexed at this brutal address, the latter smiled on the\ncolossus, and held out his hand to him, as he graciously made answer,--\n\n\"You are welcome, Red Cedar; we were expecting you impatiently. Sit down\nby my side on this butaca, and we will talk while drinking a glass of\npulque.\"\n\n\"The deuce twist your neck, and may your accursed pulque choke you! Do\nyou take me for a wretched abortion of your sort?\" the other answered as\nhe fell into the seat offered him. \"Order me some brandy, and that of\nthe strongest. I am not a babe, I suppose.\"\n\nWithout making the slightest observation, the monk rose, went to speak\nwith the host, and presently returned with a bottle, from which he\npoured a bumper for the old hunter. The latter emptied the glass at a\ndraught, put it back on the table with a sonorous \"hum!\" and turned to\nthe monk with a grimacing smile.\n\n\"Come, the devil is not always so black as he looks, senor padre,\" he\nsaid, as he passed his hand over his mouth to wipe his moustache. \"I see\nthat we can come to an understanding.\"\n\n\"It will only depend on you, Red Cedar. Here are two worthy Canadian\nhunters who will do nothing without your support.\"\n\nThe Hercules took a side glance at the young men.\n\n\"Eh!\" he said, \"what do you want with these children? Did I not promise\nyou to reach the placer with my sons only?\"\n\n\"He, he! You are powerfully built, both you and your lads, I allow; but\nI doubt whether four men, were they twice as strong as you are, could\ncarry out this affair successfully. You will have numerous enemies to\ncombat on your road.\"\n\n\"All the better! The more there are, the more we shall kill,\" he\nanswered with a sinister laugh.\n\n\"Senor padre,\" Dick interrupted, \"as far as I am concerned, I care\nlittle about it.\"\n\nBut he was suddenly checked by a meaning glance from his mate.\n\n\"What do you care little about, my pretty lad?\" the giant asked in a\nmocking voice.\n\n\"Nothing,\" the young man answered drily. \"Suppose I had not spoken.\"\n\n\"Good,\" Red Cedar remarked; \"it shall be as you wish. Here's your\nhealth.\"\n\nAnd he poured the rest of the bottle into his glass.\n\n\"Come,\" said Harry, \"Let us have but few words. Explain yourself once\nfor all, without beating about the bush, senor padre.\"\n\n\"Yes,\" Red Cedar observed, \"men ought not to waste their time thus in\nchattering.\"\n\n\"Very good. This, then, is what I propose. Red Cedar will collect within\nthree days from this time thirty resolute men, of whom he will take the\ncommand, and we will start immediately in search of the placer. Does it\nsuit you in that way?\"\n\n\"Hum!\" Red Cedar said. \"In order to go in search of the placer we must\nknow a little in what direction it is, or deuce take me if I undertake\nthe business!\"\n\n\"Do not trouble yourself about that, Red Cedar; I will accompany you.\nHave I not got a plan of the country?\"\n\nThe colossus shot at the monk a glance which sparkled under his dark\neyelash, but he hastened to moderate its brilliancy by letting his eyes\nfall.\n\n\"That is true,\" he said with feigned indifference; \"I forgot that you\nwere coming with us. Then you will leave your parishioners during your\nabsence?\"\n\n\"Heaven will watch over them.\"\n\n\"Eh! It will have its work cut out. However, that does not concern me at\nall. But why did you oblige me to come to this meson?\"\n\n\"In order to introduce you to these two hunters, who will accompany us.\"\n\n\"I beg your pardon,\" Dick observed, \"but I do not exactly see of what\nuse I can be to you in all this: my aid, and that of my mate, do not\nappear to me to be indispensable.\"\n\n\"On the contrary,\" the monk answered quickly, \"I reckon entirely on\nyou.\"\n\nThe giant had risen.\n\n\"What!\" he said, as he roughly laid his enormous hand on Dick's\nshoulder, \"You do not understand that this honourable personage, who did\nnot hesitate to kill a man in order to rob him of the secret of the\nplacer, has a terrible fear of finding himself alone with me on the\nprairie? He fears that I shall kill him in my turn to rob him of the\nsecret of which he became master by a crime. Ha, ha, ha!\"\n\nAnd he turned his back unceremoniously.\n\n\"How can you suppose such things, Red Cedar?\" the monk exclaimed.\n\n\"Do you fancy that I did not read you?\" the latter answered. \"But it is\nall the same to you. Do as you please: I leave you at liberty to act as\nyou like.\"\n\n\"What! You are off already?\"\n\n\"Hang it! What have I to do any longer here? All is settled between us.\nIn three days thirty of the best frontiersmen will be assembled by my\ncare at Grizzly Bear Creek, where we shall expect you.\"\n\nAfter shrugging his shoulders once again he went off without any salute,\nor even turning his head.\n\n\"It must be confessed,\" Dick observed, \"that the man has a most\nvillainous face. What a hideous fellow!\"\n\n\"Oh!\" the monk answered with a sigh, \"The exterior is nothing. You\nshould know the inner man.\"\n\n\"Why, in that case, do you have any dealings with him?\"\n\nThe monk blushed slightly.\n\n\"Because it must be so,\" he muttered.\n\n\"All right for you,\" Dick continued; \"but as nothing obliges my friend\nand myself to have any more intimate relations with that man, you must\nnot mind, senor Padre, if--\"\n\n\"Silence, Dick!\" Harry shouted, angrily. \"You do not know what you are\ntalking about. We will accompany you, senor padre. You can reckon on us\nto defend you if necessary, for I suppose that Red Cedar is right.\"\n\n\"In what way?\"\n\n\"You do not wish to trust your life defencelessly in his hands, and you\nreckoned on us to protect you. Is it not so?\"\n\n\"Why should I feign any longer? Yes, that man terrifies me, and I do not\nwish to trust myself to his mercy.\"\n\n\"Do not be alarmed; we shall be there, and on our word as hunters, not a\nhair of your head shall fall.\"\n\nA lively satisfaction appeared on the monk's pale face at this generous\npromise.\n\n\"Thanks,\" he said warmly.\n\nHarry's conduct appeared so extraordinary to Dick, who knew the lofty\nsentiments and innate honor of his comrade, that, without striving to\nfathom the motives which made him act thus, he contented himself by\nbacking up his words by an affirmative nod of the head.\n\n\"Be assured, caballeros, that when we have reached the placer, I will\ngive you a large share, and you will have no cause to regret\naccompanying me.\"\n\n\"The money question has but slight interest with us,\" Harry answered.\n\"My friend and I are free hunters, caring very little for riches, which\nwould be to us rather a source of embarrassment than of pleasure and\nenjoyment. Curiosity alone, and the desire of exploring strange\ncountries, are sufficient to make us undertake this journey.\"\n\n\"Whatever the reason that makes you accept my proposals, I am not the\nless obliged to you.\"\n\n\"Now you will permit us to take leave of you, and we shall hold\nourselves at your orders.\"\n\n\"Go, gentlemen; I will not keep you longer. I know where to find you\nwhen I want you.\"\n\nThe young men took up their hats, slung their rifles on their shoulders,\nand left the meson. The monk looked after them.\n\n\"Oh!\" he muttered, \"I believe I can trust to those men: they have still\nin their veins a few drops of that honest French blood which despises\ntreachery. No matter,\" he added, as if on reflection; \"I will take my\nprecautions.\"\n\nAfter this aside, he rose and looked around him. The room was full of\nadventurers, who drank or played at _monte_, and whose energetic faces\nstood out in the semi-obscurity of the room, which was scarce lighted by\na smoky lamp. After a moment's reflection the monk boldly struck the\ntable with his clenched fist, and shouted in a loud voice:\n\n\"Senores caballeros, I invite you to listen to me. I have, I fancy, an\nadvantageous proposal to make to you.\"\n\nThe company turned their heads; those who were gambling for a moment\nabandoned their cards and dice; the drinkers alone kept in their hands\nthe glasses they held; but all approached the monk, round whom they\ngrouped themselves curiously.\n\n\"Caballeros,\" he continued, \"if I am not mistaken, all present are\ngentlemen whom fortune has more or less ill-treated.\"\n\nThe adventurers, by an automatic movement of extraordinary regularity,\nbowed their heads in affirmation.\n\n\"If you wish it,\" he continued with an imperceptible smile, \"I will\nundertake to repair the wrong by it done you.\"\n\nThe adventurers pricked up their ears.\n\n\"Speak, speak, senor padre!\" they shouted with delight.\n\n\"What is the affair?\" a man with a hang-dog face said, who stood in the\nfront ranks.\n\n\"A war party which I intend to lead shortly into Apacheria,\" the monk\nsaid, \"and for which purpose I need you.\"\n\nAt this proposition the first ardor of the adventurers visibly cooled\ndown. The Apaches and Comanches inspire an invincible terror in the\ninhabitants of the Mexican frontiers. The monk guessed the effect he had\nproduced; but he continued, as if not observing anything:--\n\n\"I take you all into my service for a month, at the rate of four\npiastres a day.\"\n\nAt this magnificent offer the eyes of the adventurers sparkled with\ngreed, fear gave way to avarice, and they all exclaimed,--\n\n\"We accept, reverend father!\n\n\"But,\" the man continued who had already spoken, \"we shall be happy,\nsenor padre, if, before starting, you would give us your holy\nbenediction, and absolve us from the few sins we may have committed.\"\n\n\"Yes,\" the company yelled, \"we shall be happy if you consent to that,\nreverend father.\"\n\nThe monk appeared to reflect: the adventurers, anxiously waited.\n\n\"Well, be it so,\" he answered after a moment. \"As the work in which I am\nabout to employ you is so meritorious, I will give you my blessing, and\ngrant you absolution of your sins.\"\n\nFor a few minutes there was a shout and exclamations of joy in the room.\nThe monk demanded silence, and when it was restored he said,--\n\n\"Now, caballeros, give me each your name, that I may find you when I\nneed you.\"\n\nHe sat down and began enrolling the adventurers, who, with the men Red\nCedar supplied, would form the band with which he hoped to reach the\nplacer. We will leave the worthy monk for a few moments, and follow the\ntwo Canadian hunters.\n\n\n\n\nCHAPTER XIV.\n\nTHE TWO HUNTERS.\n\n\nHarry and Dick, whom we saw seated at a table in the meson with Red\nCedar and Fray Ambrosio, were however, very far from resembling those\ntwo men morally. They were free and bold hunters, who had spent the\ngreater part of their life in the desert, and who, in the vast solitude\nof the prairie, had accustomed themselves to a life free and exempt from\nthose vices which accompany a town residence.\n\nFor them gold was only the means to procure the necessary objects for\ntheir trade as hunters and trappers; and they never imagined that the\npossession of a large quantity of that yellow metal they despised would\nplace them in a position to enjoy other pleasures than those they found\nin their long hunts of wild beasts--hunts so full of strange incidents\nand striking joys.\n\nThus Dick had been to the highest degree surprised when he saw his\nfriend eagerly accept the monk's offer, and agree to go in search of the\nplacer; but what even more surprised him was Harry's insisting that Red\nCedar must take the lead of the expedition. Though no one could\npositively accuse the squatter, owing to the precautions he took, of\nleading a life of rapine and murder, still the mysterious conduct he\naffected, and the solitude in which he lived with his family, had cast\non him a shadow of reprobation.\n\nEvery one regarded him as a scalp hunter, and yet no one would have\nventured to affirm the odious deeds of which he was accused. The result\nof the general reprobation that fell on the squatter, and which we know\nto be fully merited, was that he and his family were placed under a ban\nby the frontier hunters and trappers, and every one fled not only their\nsociety, but any contact with them. Dick was thoroughly acquainted with\nhis friend's upright character and nobility of heart. Hence his conduct\nunder the present circumstances seemed to him perfectly\nincomprehensible, and he resolved to have an explanation with him.\n\nThey had scarce quitted the meson ere Dick bent down to his companion,\nand said, while looking at him curiously,--\n\n\"We have been hunting together for five years, Harry, and up to the\npresent I have ever let myself be guided by you, leaving you free to act\nas you pleased for our mutual welfare. Still this evening your conduct\nhas appeared to me so extraordinary that I am obliged, in the name of\nour friendship, which has never suffered a break up to this day, to ask\nyou for an explanation of what has occurred in my presence.\"\n\n\"For what good, my boy? Do you not know me well enough to be certain\nthat I would not consent to do any dishonourable deed?\"\n\n\"Up to this evening I would have sworn it, Harry: yes, on my honor I\nwould have sworn it--\"\n\n\"And now?\" the young man asked, stopping and looking his friend in the\nface.\n\n\"Now,\" Dick answered, with a certain degree of hesitation, \"hang it all!\nI will be frank with you, Harry, as an honest hunter should ever be. Now\nI do not know if I should do so: no, indeed I should not.\"\n\n\"What you say there causes me great pain, Dick. You oblige me, in order\nto dissipate your unjust suspicions, to confide to you a secret which is\nnot my own, and which I would not have revealed for anything in the\nworld.\"\n\n\"Pardon me, Harry, but in my place I am convinced you would act as I am\ndoing. We are very far from our country, which we shall never see again,\nperhaps. We are responsible for each other, and our actions must be free\nfrom all double interpretation.\"\n\n\"I will do what you ask, Dick, whatever it may cost me. I recognise the\njustice of your observations. I understand how much my conduct this\nnight must have hurt you and appeared ambiguous. I do not wish our\nfriendship to receive the least wound, or the slightest cloud to arise\nbetween us. You shall be satisfied.\"\n\n\"I thank you, Harry. What you tell me relieves my bosom of a heavy load.\nI confess that I should have been in despair to think badly of you; but\nthe words of that intriguing monk, and the manners of that worthy\nacolyte, Red Cedar, put me in a passion. Had you not warned me so\nquickly to silence, I believe--Heaven pardon me!--that I should have\nended by telling them a piece of my mind.\"\n\n\"You displayed considerable prudence in keeping silence, and be assured\nyou will completely approve me.\"\n\n\"I do not doubt it, Harry; and now I feel certain I deceived myself. I\nfeel all jolly again.\"\n\nWhile speaking thus the two hunters, who were walking with that rapid\nstep peculiar to men habituated to traverse great distances on foot, had\ncrossed the village, and found themselves already far in the plain. The\nnight was magnificent--the sky of a deep blue. An infinite number of\nglistening stars seemed floating in ether. The moon spread its silvery\nrays profusely over the landscape. The sharp odour of the flowers\nperfumed the atmosphere. The two hunters still walked on.\n\n\"Where are we going now, Harry?\" Dick asked. \"I fancy we should do\nbetter by taking a few hours' rest, instead of fatiguing ourselves\nwithout any definite object.\"\n\n\"I never do anything without a reason, friend, as you know,\" Harry\nanswered; \"so let me guide you, and we shall soon arrive.\"\n\n\"Do as you think proper, my boy; I shall say nothing.\"\n\n\"In the first place you must know that the French hunter, Koutonepi, has\nbegged me, for reasons he did not tell me, to watch Fray Ambrosio. That\nis one of the motives which made me be present at this night's\ninterview, although I care as little for a placer as for a musk-rat's\nskin.\"\n\n\"Koutonepi is the first hunter on the frontier; he has often done us a\nservice in the desert. You acted rightly, Harry, in doing what he\nasked.\"\n\n\"As for the second reason that dictated my conduct, Dick, you shall soon\nknow it.\"\n\nHalf talking, half dreaming, the young men reached Buffalo Valley, and\nsoon entered the forest which served as a lair for the squatter and his\nfamily.\n\n\"Where the deuce are we going?\" Dick could not refrain from saying.\n\n\"Silence!\" said the other: \"We are approaching.\"\n\nThe darkness was profound in the forest: the density of the leafy dome\nunder which they walked completely intercepted the light of the\nmoonbeams. Still the Canadians, long accustomed to a night march,\nadvanced as easily through the chaos of creepers and trees tangled in\neach other as if they had been in open day. On reaching a certain spot\nwhere the trees, growing less closely together, formed a species of\nclearing, and allowed an uncertain and tremorous light to pass, Harry\nstopped, and made his comrade a sign to do the same.\n\n\"This is the place,\" he said. \"Still, I as the person I have come to see\nexpects me to be alone, and your unexpected presence might cause alarm,\nhide yourself behind that larch tree: above all, be careful not to stir\ntill I call I you.\"\n\n\"Oh, oh!\" the hunter said, with a laugh, \"have you perchance led me to a\nlove meeting, Harry?\"\n\n\"You shall judge,\" Harry replied laconically. \"Hide yourself.\"\n\nDick, greatly troubled, did not need the invitation to be repeated: he\nconcealed himself behind the tree his friend had indicated, and which\nwould have sheltered a dozen men behind its enormous stem. So soon as\nHarry was alone, he raised his fingers to his lips, and at three\ndifferent intervals imitated the cry of an owl with such perfection that\nDick himself was deceived, and mechanically looked up to seek the bird\nin the tall branches of the tree by which he stood. Almost immediately,\na slight noise was audible in the shrubs, and a graceful and white form\nappeared in the glade. It was Ellen, who rapidly walked toward the young\nman.\n\n\"Oh, it is you, Harry!\" she said with joy. \"Heaven be blessed, I was\nafraid you would not come, as it is late.\"\n\n\"It is true, Ellen: pardon me. I made all possible speed, however; and\nit is not my fault that I did not arrive sooner.\"\n\n\"How good you are, Harry, to take so much trouble for my sake! How can I\never recognise the continual services you do me?\"\n\n\"Oh! Do not speak about them. It is a happiness for me to do anything\nagreeable to you.\"\n\n\"Alas!\" the maiden murmured, \"Heaven is my witness that I feel a deep\nfriendship for you, Harry.\"\n\nThe young man sighed gently.\n\n\"I have done what you asked of me,\" he said suddenly.\n\n\"Then it is true my father is thinking about leaving this country to go\nfurther still?\"\n\n\"Yes, Ellen, and into frightful countries, among the ferocious Indians.\"\n\nThe girl gave a start of terror.\n\n\"Do you know the reason of his going?\" she continued.\n\n\"Yes; he is about to look for a gold placer.\"\n\n\"Alas! Who will protect me, who will defend me in future, if we go\naway?\"\n\n\"I, Ellen!\" the hunter exclaimed impetuously. \"Have I not sworn to\nfollow you everywhere?\"\n\n\"It is true,\" she said sadly; \"but why should you risk your life on the\ndistant journey we are about to undertake? No, Harry, remain here; I\ncannot consent to your departure. From what I have heard say, the band\nmy father commands will be numerous--it will have scarce anything to\nfear from the Indians; while, on the other hand, you, compelled to hide\nyourself, will be exposed alone to terrible danger. No, Harry, I will\nnot permit it.\"\n\n\"Undeceive yourself, Ellen. I shall not be forced to conceal myself; I\nshall not be alone, for I am a member of your father's band.\"\n\n\"Is it possible, Harry?\" she exclaimed, with an expression of joy that\nmade the young man quiver.\n\n\"I enrolled myself this very evening.\"\n\n\"Oh!\" she said, \"Then in that case we can often meet?\"\n\n\"Whenever you please, Ellen, as I shall be there.\"\n\n\"Oh! Now I am anxious to be away from here, and wish we had already\nstarted.\"\n\n\"It will not be long first, set your mind at rest. I am convinced that\nwe shall start within the week.\"\n\n\"Thanks for the good news you bring me, Harry.\"\n\n\"Are your father and mother still unkind to you, Ellen?\"\n\n\"It is nearly always the same thing; and yet their conduct toward me is\nstrange. It often seems to me incomprehensible, as it is so marked with\npeculiarities. There are moments in which they seem to love me dearly.\nMy father especially caresses and embraces me, and then all at once, I\nknow not why, repulses me rudely, and looks at me in a way that causes\nme to shudder.\"\n\n\"That is indeed strange, Ellen.\"\n\n\"Is it not? There is one thing above all I cannot explain.\"\n\n\"Tell it me, Ellen; perhaps I can do so.\"\n\n\"You know that all my family are Protestants?\"\n\n\"Yes.\"\n\n\"Well, I am a Catholic.\"\n\n\"That is certainly curious.\"\n\n\"I wear around my neck a small golden crucifix. Every time accident\nmakes this trinket glisten before my father and mother they grow\nfurious, threaten to beat me, and order me to hide it at once. Do you\nunderstand the meaning of this, Harry?\"\n\n\"No, I do not, Ellen; but, believe me, leave everything to time; perhaps\nit will enable us to find the clue to the mystery which we seek in vain\nat this moment.\"\n\n\"Well, your presence has rendered me happy for a long time, Harry, so\nnow I will retire.\"\n\n\"Already?\"\n\n\"I must, my friend. Believe me that I am as sad as yourself at this\nseparation; but my father has not yet returned, and may arrive at any\nmoment. If he noticed that I was not asleep, who knows what might\nhappen?\"\n\nWhile saying the last words the girl held out her delicate hand to the\nhunter, who raised it to his lips passionately. Ellen withdrew it\nsuddenly, and bounding like a startled fawn, darted into the forest,\nwhere she soon disappeared, giving the young man a parting word, which\ncaused him to quiver with joy:--\n\n\"We shall meet soon.\"\n\nHarry stood for a long time with his eyes fixed on the spot where the\nseductive vision had disappeared. At length he uttered a sigh, threw his\nrifle over his shoulder, and turned as if to depart. Dick was before\nhim. Harry gave a start of surprise, for he had forgotten his friend's\npresence; but the latter smiled good-humouredly.\n\n\"I now comprehend your conduct, Harry,\" he said to him; \"you were right\nto act as you did. Pardon my unjust suspicions, and count on me\neverywhere and always.\"\n\nHarry silently pressed the hand his friend offered him, and they walked\nback rapidly in the direction of the village. As they emerged from the\nforest they passed, a man who did not see them. It was Red Cedar. So\nsoon as he had gone a short distance Harry stopped his companion, and\npointing to the squatter, whose long black shadow glided through the\ntrees, said, as he laid his hand on his shoulder,--\n\n\"That man hides in his heart a horrible secret, which I am ignorant of,\nbut have sworn to discover.\"\n\n\n\n\nCHAPTER XV.\n\nFRAY AMBROSIO.\n\n\nThe monk remained for a long time in the room of the meson, taking down\nthe names of the adventurers he wished to enrol in his band. It was late\nwhen he left it to return to the Hacienda de la Noria; but he was\nsatisfied with his night's work, and internally rejoiced at the rich\ncollection of bandits of the purest water he had recruited.\n\nThe monks form a privileged caste in Mexico: they can go at all hours of\nthe night wherever they please without fearing the numerous \"gentlemen\nof the road,\" scattered about the highways. Their gown inspires a\nrespect which guarantees them from any insult, and preserves them better\nthan anything from unpleasant rencontres. Besides, Fray Ambrosio, as the\nreader has doubtless already perceived, was not the man to neglect\nindispensable precautions in a country where, out of ten persons you\nmeet on your road, you may boldly assert that nine are rogues, the tenth\nalone offering any doubts. The worthy chaplain carried under his gown a\npair of double-barrelled pistols, and in his right sleeve he concealed a\nlong _navaja_, sharp as a razor, and pointed as a needle.\n\nNot troubling himself about the solitude that reigned around him, the\nmonk mounted his mule and proceeded quietly to the hacienda. It was\nabout eleven o'clock.\n\nA few words about Fray Ambrosio, while he is peacefully ambling along\nthe narrow path which will lead him in two hours to his destination,\nwill show all the perversity of the man who is destined to play an\nunfortunately too important part in the course of our narrative.\n\nOne day a gambusino, or gold seeker, who had disappeared for two years,\nno one knowing what had become of him, and who was supposed to be dead\nlong ago, assassinated in the desert by the Indians, suddenly reappeared\nat the Paso del Norte. This man, Joaquin by name, was brother to Andres\nGarote, an adventurer of the worst stamp, who had at least a dozen\n_cuchilladas_ (knife stabs) on his conscience, whom everybody feared,\nbut who, through the terror he inspired, enjoyed at the Paso, in spite\nof his well-avouched crimes, a reputation and species of impunity which\nhe abused whenever the opportunity offered.\n\nThe two brothers began frequenting together the mesones and ventas of\nthe village, drinking from morn till night, and paying either in gold\ndust enclosed in stout quills, or in lumps of native gold. The rumour\nsoon spread at Paso that Joaquin had discovered a rich placer, and that\nhis expenses were paid with the specimens he had brought back. The\ngambusino replied neither yes nor no to the several insinuations which\nhis friends, or rather his boon companions, attempted on him. He\ntwinkled his eyes, smiled mysteriously, and if it were observed that, at\nthe rate he was living at, he would soon be ruined, he shrugged his\nshoulders, saying:--\n\n\"When I have none left I know where to find others.\"\n\nAnd he continued to enjoy his fill of all the pleasures which a wretched\nhole like Paso can furnish.\n\nFray Ambrosio had heard speak, like everyone else, of the gambusino's\nasserted discovery; and his plan was at once formed to become master of\nthis man's secret, and rob him of his discovery, were that possible.\n\nThe same evening Joaquin and his brother Andres were drinking, according\nto their wont, in a meson, surrounded by a crowd of scamps like\nthemselves. Fray Ambrosio, seated at a table with his hands hidden in\nthe sleeves of his gown, and hanging head, appeared plunged in serious\nreflections, although he followed with a cunning eye the various\nmovements of the drinkers, and not one of their gestures escaped him.\n\nSuddenly a man entered, with his hand on his hip, and throwing in the\nface of the first person he passed the cigarette he was smoking. He\nplanted himself in front of Joaquin, to whom he said nothing, but began\nlooking at him impudently, shrugging his shoulders, and laughing\nironically at all the gambusino said. Joaquin was not patient, he saw at\nthe first glance that this person wished to pick a quarrel with him; and\nas he was brave, and feared nobody, man or devil, he walked boldly up to\nhim, and looking at him fixedly in his turn, he said to him, as he\nthrust his face in his:\n\n\"Do you seek a quarrel, Tomaso?\"\n\n\"Why not?\" the latter said impudently, as he set his glass on the table.\n\n\"I am your man. We will fight how you please.\"\n\n\"Bah!\" Tomaso said carelessly, \"let us do things properly, and fight\nwith the whole blade.\"\n\n\"Be it so.\"\n\nThe combats which take place between the adventurers are truly like\nthose of wild beasts. These coarse men, with their cruel instincts, like\nfighting beyond all else, for the smell of blood intoxicates them. The\nannouncement of this duel caused a thrill of pleasure to run through the\nranks of the leperos and bandits who pressed round the two men. The fun\nwas perfect: one of the adversaries would doubtless fall--perhaps\nboth--and blood flow in streams. Cries and yells of delight were raised\nby the spectators.\n\nThe duel with knives is the only one that exists in Mexico, and is\nsolely left to the leperos and people of the lowest classes. This duel\nhas its rules, which cannot be broken under any pretext. The knives\nusually employed, have blades from fourteen to sixteen inches in length,\nand the duelists fight according to the gravity of the insult, with one,\ntwo, three, six inches, or the entire blade. The inches are carefully\nmeasured and the hand clutches the knife at the marked spot.\n\nThis time it was a duel with the whole blade, the most terrible of all.\nWith extraordinary politeness and coolness the landlord had a large ring\nformed in the middle of the room, where the two adversaries stationed\nthemselves, about six paces from each other at the most.\n\nA deep silence hung over the room, a moment previously so full of life\nand disturbance; every one anxiously awaited the _denouement_ of the\nterrible drama that was preparing. Fray Ambrosio alone had not quitted\nhis seat or made a sign.\n\nThe two men rolled their zarapes round their left arm, planted\nthemselves firmly on their outstretched legs, bent their bodies slightly\nforward and gently placing the point of the knife blade on the arm\nrounded in front of the chest, they waited, fixed on each other flashing\nglances. A few seconds elapsed, during which the adversaries remained\nperfectly motionless: all hearts were contracted, all bosoms heaving.\n\nWorthy of Callot's pencil was the scene offered by these men, with their\nweather-stained faces and harsh features, and their clothes in rags,\nforming a circle round two combatants ready to kill each other in this\nmean room, slightly illumined by a smoky lamp, which flashed upon the\nblue blades of the knives, and in the shadow, almost disappearing in his\nblack gown, the monk, with his implacable glance and mocking smile, who,\nlike a tiger thirsting for blood, awaited the hour to pounce on his\nprey.\n\nSuddenly, by a spontaneous movement rapid as lightning, the adversaries\nrushed on each other, uttering a yell of fury. The blades flashed, there\nwas a clashing of steel, and both fell back again. Joaquin and Tomaso\nhad both dealt the same stroke, called, in the slang of the country, the\n\"blow of the brave man.\" Each had his face slashed from top to bottom\nwith a gaping wound.\n\nThe spectators frenziedly applauded this magnificent opening scene: the\njaguars had scented blood, and were mad.\n\n\"What a glorious fight!\" they exclaimed with admiration.\n\nIn the meanwhile the two combatants, rendered hideous by the blood that\nstreamed from their wounds and stained their faces, were again watching\nfor the moment to leap on one another. Suddenly they broke ground; but\nthis time it was no skirmish, but the real fight, atrocious and\nmerciless. The two men seized each other round the waist, and entwined\nlike serpents, they twisted about, trying to stab each other, and\nexciting themselves to the struggle by cries of rage and triumph. The\nenthusiasm of the spectators was at its height: they laughed, clapped\nhands, and uttered inarticulate howls as they urged the fighters not to\nloose their hold.\n\nAt length the enemies rolled on the ground still enclasped. For some\nseconds the combat continued on the ground, and it was impossible to\ndistinguish who was the conqueror. All at once one of them, who no\nlonger had a human form, and whose body was as red as an Indian's,\nbounded to his feet brandishing his knife. It was Joaquin.\n\nHis brother rushed toward him to congratulate him on his victory, but\nall at once the gambusino tottered and fainted. Tomaso did not rise\nagain: he remained motionless, stretched out on the uneven floor of the\nmeson. He was stark dead.\n\nThis scene had been so rapid, its conclusion so unforeseen, that, in\nspite of themselves, the spectators had remained dumb, and as if struck\nwith stupor. Suddenly the priest, whom all had forgotten, rose and\nwalked into the centre of the room, looking round with a glance that\ncaused all to let their eyes fall.\n\n\"Retire, all of you,\" he said in a gloomy voice, \"now that you have\nallowed this deed worthy of savages to be accomplished. The priest must\noffer his ministry, and get back from Satan, if there be still time, the\nsoul of this Christian who is about to die. Begone!\"\n\nThe adventurers hung their heads, and in a few moments the priest was\nleft alone with the two men, one of whom was dead, the other at the last\ngasp. No one could say what occurred in that room; but when the priest\nleft it, a quarter of an hour later, his eyes flashed wildly. Joaquin\nhad given his parting sigh. On opening the door to go out Fray Ambrosio\njostled against a man, who drew back sharply to make room for him. It\nwas Andres Garote. What was he doing with his eye at the keyhole while\nthe monk was shriving his brother?\n\nThe adventurer told no one what he had seen during this last quarter of\nan hour, nor did the monk notice in the shade the man he had almost\nthrown down.\n\nSuch was the way in which Fray Ambrosio became master of the gambusino's\nsecret, and how he alone knew at present the spot where the placer was.\n\n\n\n\nCHAPTER XVI.\n\nTWO VARIETIES OF VILLAINS.\n\n\nNow that the reader is well informed touching Fray Ambrosio, we will\nfollow him on his road home from the meson. The night was calm, silent\nand serene. Not a sound troubled the silence, save the trot of the mule\nover the pebbles on the road, or at times, in the distance, the snapping\nbark of the coyotes chasing in a pack, according to their wont, some\nstraggling hind.\n\nFray Ambrosio ambled gently on, while reflecting on the events of the\nevening, and calculating mentally the probable profits of the expedition\nhe meditated. He had left far behind him the last houses of the village,\nand was advancing cautiously along a narrow path that wound through an\nimmense sugar cane field. Already the shadow of the tall hacienda walls\nstood out blackly in the horizon. He expected to reach it within twenty\nminutes, when suddenly his mule, which had hitherto gone so quietly,\npricked up its ears, raised its head, and stopped short.\n\nRoughly aroused from his meditations by this unexpected halt, the monk\nlooked about for some obstacle that might impede his progress. About ten\npaces from him a man was standing right in the middle of the path. Fray\nAmbrosio was a man not easily to be frightened: besides, he was well\narmed. He drew out one of the pistols hidden under his gown, cocked it,\nand prepared to cross-question the person who so resolutely barred his\nway. But the latter, at the sharp sound of the setting hammer, thought\nit prudent to make himself known, and not await the consequences of an\naddress nearly always stormy under similar circumstances.\n\n\"Halloh!\" he shouted in a loud voice, \"Return your pistol to your belt,\nFray Ambrosio; I only want to talk with you.\"\n\n\"_Diavolo_!\" the monk said, \"the hour and moment are singularly chosen\nfor a friendly conversation, my good fellow.\"\n\n\"Time belongs to nobody,\" the stranger answered sententiously. \"I am\nobliged to choose that which I have at my disposal.\"\n\n\"That is true,\" the monk said as he quietly uncocked his pistol, though\nnot returning it to his belt. \"Who the deuce are you, and why are you so\nanxious to speak with me? Do you want to confess?\n\n\"Have you not recognised me yet, Fray Ambrosio? Must I tell you my name\nthat you may know with whom you have to deal?\"\n\n\"Needless, my good sir, needless; but how the deuce is it, Red Cedar,\nthat I meet you here! What can you have so pressing to communicate to\nme?\"\n\n\"You shall know if you will stop for a few moments and dismount.\"\n\n\"The deuce take you with your whims! Cannot you tell me that as well\ntomorrow! Night is getting on, my home is still some distance off and I\nam literally worn out.\"\n\n\"Bah! you will sleep capitally by the side of a ditch, where you could\nnot be more comfortable. Besides, what I have to say to you does not\nadmit of delay.\"\n\n\"You wish to make a proposal to me, then?\"\n\n\"Yes.\"\n\n\"What about, if you please?\"\n\n\"About the affair we discussed this evening at the Paso.\"\n\n\"Why, I fancied we had settled all that, and you accepted my offer.\"\n\n\"Not yet, not yet, my master. That will depend on the conversation we\nare about to have, so you had better dismount and sit down quietly by my\nside; for if you don't do it, it will come to nothing.\"\n\n\"The deuce take people that change their minds every minute, and on whom\none cannot reckon more than on an old surplice!\" the monk growled with\nan air of annoyance, while, for all that, getting off his mule, which he\nfastened to a shrub.\n\nThe squatter did not seem to remark the chaplain's ill temper, and let\nhim sit down by his side without uttering a syllable.\n\n\"Here I am,\" the monk went on, so soon as he was seated. \"I really do\nnot know, Red Cedar, why I yield so easily to all your whims.\"\n\n\"Because you suspect that your interest depends on it: were it not for\nthat, you would not do so.\"\n\n\"Why talk thus in the open country, instead of going to your house,\nwhere we should be much more comfortable?\"\n\nRed Cedar shook his head in denial.\n\n\"No,\" he said; \"the open is better for what we have to talk about. Here\nwe need not fear listeners at out doors.\"\n\n\"That is true. Well, go on; I am listening.\"\n\n\"Hum! You insist upon my commanding the expedition you project?\"\n\n\"Of course. I have known you a long time. I am aware that you are a sure\nman, perfectly versed in Indian signs; for, if I am not mistaken, the\ngreater part of your life has been spent among them.\"\n\n\"Do not speak about what I have done? The question now concerns you, and\nnot me.\"\n\n\"How so?\"\n\n\"Good, good! Let me speak. You need me, so it is to my interest to make\nyou pay as dearly as I can for me.\"\n\n\"Eh?\" the monk muttered, as he made a grimace. \"I am not rich, gossip,\nas you are aware.\"\n\n\"Yes, yes; I know that, so soon as you have a few piastres or ounces,\nthe monte table strips you of them immediately.\"\n\n\"Hang it! I have always been unlucky at play.\"\n\n\"For that reason I do not intend asking you for money.\"\n\n\"Very good. If you have no designs on my purse we can easily come to an\nunderstanding. You may speak boldly.\"\n\n\"I hope that we shall easily understand one another, the mere so as the\nservice I expect from you is almost a mere nothing.\"\n\n\"Come to the point, Red Cedar: with your deuced way of twining your\nphrases together in the Indian way, you never make an end of it.\"\n\n\"You know that I have a deadly hatred against Don Miguel Zarate?\"\n\n\"I have heard some say about it. Did he not lodge his knife somewhere in\nyour chest?\"\n\n\"Yes, and the blow was so rude that I all but died of it; but, thanks to\nthe devil, I am on my legs again, after remaining three weeks on my back\nlike a cast sheep. I want my revenge.\"\n\n\"I can't help saying you are right: in your place, may Satan twist my\nneck if I would not do the same!\"\n\n\"For that I count on your help.\"\n\n\"Hum! that is a delicate affair. I have no cause of complaint against\nDon Miguel--on the contrary: besides, I do not see how I can serve you.\"\n\n\"Oh! very easily.\"\n\n\"You believe so?\"\n\n\"You shall see.\"\n\n\"Go on, then; I am listening.\"\n\n\"Don Miguel has a daughter?\"\n\n\"Dona Clara.\"\n\n\"I mean to carry her off.\"\n\n\"Deuce take the mad ideas that pass through your brain-pan, gossip! How\nwould you have me help you in carrying off the daughter of Don Miguel,\nto whom I owe so many obligations? No, I cannot do that, indeed.\"\n\n\"You must, though.\"\n\n\"I will not, I tell you.\"\n\n\"Measure your words well, Fray Ambrosio, for this conversation is\nserious. Before refusing so peremptorily to give me the help I ask,\nreflect well.\"\n\n\"I have reflected well, Red Cedar, and never will I consent to help you\nin carrying off the daughter of my benefactor. Say what you like,\nnothing will ever change my resolution on that head, for it is\ninflexible.\"\n\n\"Perhaps.\"\n\n\"Oh! Whatever may happen, I swear that nothing will make me alter.\"\n\n\"Swear not, Fray Ambrosio, for you will be a perjurer.\"\n\n\"Ta, ta, ta! You are mad, my good fellow. Don't let us waste our time.\nIf you have nothing else to say to me, I will leave you, though I take\nsuch pleasure in your society.\"\n\n\"You have become scrupulous all of a sudden, my master.\"\n\n\"There is a beginning to everything, compadre; so let us say no more,\nbut good-bye.\"\n\nAnd the monk rose.\n\n\"You are really going?\"\n\n\"_Caray_! Do you fancy I mean to sleep here?\"\n\n\"Very good. You understand that you need not count on me for your\nexpedition?\"\n\n\"I am sorry for it; but I will try to find someone to take your place.\"\n\n\"Thank you.\"\n\nThe two men were standing, and the monk had put his foot in the stirrup.\nRed Cedar also appeared ready to make a start. At the moment of\nseparation a sudden idea seemed to occur to the squatter.\n\n\"By the way,\" he said carelessly, \"be kind enough to give me some\ninformation I require.\"\n\n\"What is it now?\" the monk asked.\n\n\"Oh! a mere trifle,\" the squatter remarked indifferently. \"It concerns a\ncertain Don Pedro de Tudela, whom I think you formerly knew.\"\n\n\"Eh!?\" the monk exclaimed, as he turned, with his leg still in the air.\n\n\"Come, come, Fray Ambrosio,\" Red Cedar continued in a jeering voice,\n\"let us have a little more talk together. I will tell you, if you like,\na very remarkable story about this Don Pedro, with whom you were\nacquainted.\"\n\nThe monk was livid; a nervous tremor agitated all his limbs; he let\nloose his mule's bridle, and followed the squatter mechanically, who\nseated himself tranquilly on the ground, making him a sign to follow his\nexample. The monk fell, suppressing a sigh, and wiping away the drops of\ncold perspiration that beaded on his forehead.\n\n\"Eh, eh!\" the squatter continued at the end of a moment, \"we must allow\nthat Don Pedro was a charming gentleman--a little wild, perhaps; but\nwhat would you have? He was young. I remember meeting him at Albany a\nlong time ago--some sixteen or seventeen years ago--how old one\ngets!--at the house of one--wait awhile, the name has slipped my\nmemory--could you not help me to it, Fray Ambrosio?\"\n\n\"I do not know what you mean,\" the monk said in a hollow voice.\n\nThe man was in a state that would have produced pity; the veins in his\nforehead were swollen ready to burst; he was choking; his right hand\nclutched the hilt of his dagger; and he bent on the squatter a glance\nfull of deadly hatred. The latter seemed to see nothing of all this.\n\n\"I have it!\" he continued. \"The man's name was Walter Brunnel, a very\nworthy gentleman.\"\n\n\"Demon!\" the monk howled in a gasping voice, \"I know not who made you\nmaster of that horrible secret, but you shall die.\"\n\nAnd he rushed upon him, dagger in hand.\n\nRed Cedar had known Fray Ambrosio a long time, and was on his guard. By\na rapid movement he checked his arm, twisted it, and seized the dagger,\nwhich he threw a long distance off.\n\n\"Enough,\" he said in a harsh voice. \"We understand one another, my\nmaster. Do not play that game with me, for you will be sick of it, I\nwarn you.\"\n\nThe monk fell back on his seat, without the strength to make a sign or\nutter a syllable. The squatter regarded him for a moment with mingled\npity and contempt and shrugged his shoulders.\n\n\"For sixteen years I have held that secret,\" he said, \"and it has never\npassed my lips. I will continue to keep silence on one condition.\"\n\n\"What is it?\"\n\n\"I want you to help me in carrying off the hacendero's daughter.\"\n\n\"I will do it.\"\n\n\"Mind, I expect honest assistance; so do not attempt any treachery.\"\n\n\"I will help you, I tell you.\"\n\n\"Good! I count on your word. Besides you may be easy, master; I will\nwatch you.\"\n\n\"Enough of threats. What is to be done?\"\n\n\"When do we start for Apacheria?\"\n\n\"You are coming, then?\"\n\n\"Of course.\"\n\nA sinister smile played round the monk's pale lips.\n\n\"We shall start in a week,\" he said.\n\n\"Good! On the day of the start you will hand over the girl to me, one\nhour before our departure.\"\n\n\"What shall I do to compel her to follow me?\"\n\n\"That is not my business.\"\n\n\"Still--\"\n\n\"I insist.\"\n\n\"Be it so,\" the monk said with an effort. \"I will do it; but remember,\ndemon, if I ever hold you in my hands, as I am this day in yours, I\nshall be pitiless and make you pay for all I suffer at this moment.\"\n\n\"You will be right to do so--it is your due; still I doubt whether you\nwill ever be able to reach me.\"\n\n\"Perhaps.\"\n\n\"Live and learn. In the meanwhile I am your master, and I reckon on your\nobedience.\"\n\n\"I will obey.\"\n\n\"That is settled. Now, one thing more; how many men have you enlisted\nthis evening?\"\n\n\"About twenty.\"\n\n\"That's not many; but, with the sixty I shall supply, we shall have a\nvery decent band to hold the Indians in check.\"\n\n\"May Heaven grant it!\"\n\n\"Don't be alarmed, my master,\" the squatter said, re-assuming the\nfriendly tone which he employed at the outset of the conversation; \"I\npledge myself, to lead you straight to your placer. I have not lived ten\nyears with the Indians not to be up to all their tricks.\"\n\n\"Of course,\" the monk answered as he rose, \"You know, Red Cedar, what\nwas agreed upon; the placer will be shared between us. It is, therefore,\nto your interest to enable us to reach it without obstacle.\"\n\n\"We shall reach it. Now that we have nothing more to say to each other\nand have agreed on all points--for we have done so, I think?\" he said\nsignificantly.\n\n\"Yes, all.\"\n\n\"We can part, and go each home. No matter, my master! I told you that I\nshould succeed in making you alter your mind. Look you, Fray Ambrosio,\"\nhe added in impudent tone, which made the monk turn pale with rage;\n\"people need only to understand one another to do anything.\"\n\nHe rose, threw his rifle over his shoulder, and turning away sharply,\nwent off with lengthened strides. The monk remained for a moment as if\nstunned by what had happened. Suddenly he thrust his hand under his\ngown, seized a pistol, and aimed at the squatter. But ere he had time to\npull the trigger his enemy disappeared round a turning, uttering a\nformidable burst of laughter, which the mocking echo bore to his ear,\nand revealed to him all the immensity of his impotence.\n\n\"Oh!\" he muttered as he got in the saddle, \"How did this fiend discover\nthe secret which I believed no one knew?\"\n\nAnd he went off gloomy and thoughtful. Half an hour later he reached the\nHacienda de la Noria, when the gate was opened for him by a trusty peon,\nfor everybody was asleep. It was past midnight.\n\n\n\n\nCHAPTER XVII.\n\nEL CANYON DEL BUITRE.\n\n\nWe will now return to the hacendero, who, accompanied by his two\nfriends, is galloping at full speed in the direction of Valentine's\njacal. The road the three men followed led them further and further from\nthe Paso del Norte. Around them nature grew more abrupt, the scenery\nsterner. They had left the forest, and were galloping over a wide and\narid plain. On each side of the way the trees, becoming rarer, defiled\nlike a legion of phantoms. They crossed several tributary streams of the\nDel Norte, in which their horses were immersed up to the chest.\n\nAt length they entered a ravine deeply imbedded between two wooded\nhills, the soil of which, composed of large flat stones and rounded\npebbles, proved that this spot was one of those _desaguaderos_ which\nserve to carry off the waters in the rainy season. They had reached the\nCanyon del Buitre, so named on account of the numerous vultures\nconstantly perched on the tops of the surrounding hills.\n\nThe defile was deserted, and Valentine had his cabin not far from this\nspot. So soon as the three men had dismounted, Curumilla took the horses\nand led them to the jacal.\n\n\"Follow me,\" Valentine said to Don Miguel.\n\nThe latter obeyed, and the two men began then climbing the escarped\nflanks of the right hand hill. The climb was rude, for no road was\ntraced; but the two hunters, long accustomed to force a passage through\nthe most impracticable places, seemed hardly to perceive the difficulty\nof the ascent, which would have been impossible for men less used to a\ndesert life.\n\n\"This spot is really delicious,\" Valentine said with the complacent\nsimplicity of a landowner who boasts of his estate. \"If it were day, Don\nMiguel, you would enjoy from this spot a magnificent view. A few hundred\nyards from the place where we are, down there on that hill to the right,\nare the ruins of an ancient Aztec camp in a very fine state of\npreservation. Just imagine that this hill, carved by human hands, though\nyou cannot see it in the darkness, is of the shape of a pyramidal cone:\nits base is triangular, the sides are covered with masonry, and it is\ndivided into several terraces. The platform is about ninety yards long\nby seventy-five in width, and is surrounded on three sides by a\nplatform, and flanked by a bastion on the north. You see that it is a\nperfect fortress, constructed according to all the rules of military\nart. On the platform are the remains of a species of small teocali,\nabout twenty feet high, composed of large stones covered with\nhieroglyphics sculptured in relief, representing weapons, monsters,\nrabbits, crocodiles, and all sorts of things; for instance, men seated\nin the oriental fashion, and wearing spectacles. Is not that really\ncurious? This little monument, which has no staircase, doubtless served\nas the last refuge to the besieged when they were too closely\nbeleaguered by the enemy.\"\n\n\"It is astonishing,\" Don Miguel answered, \"that I never heard of these\nruins.\"\n\n\"Who knows them? Nobody. However, they bear a considerable likeness to\nthose found at Jochicalco.\"\n\n\"Where are you leading me, my friend? Are you aware that the road is not\none of the pleasantest, and I am beginning to feel tired?\"\n\n\"A little patience: in ten minutes we shall arrive. I am leading you to\na natural grotto which I discovered a short time back. It is admirable.\nIt is probable that the Spaniards were unacquainted with it, although\nthe Indians, to my knowledge, have visited it from time immemorial. The\nApaches imagine it serves as a palace to the genius of the mountain. At\nany rate, I was so struck by its beauty that I abandoned my jacal, and\nconverted it into my residence. Its extent is immense. I am certain,\nthough I never tried to convince myself, that it goes for more than ten\nleagues under ground. I will not allude to the stalactites that hang\nfrom the roof, and form the quaintest and most curious designs; but the\nthing that struck me is this: this grotto is divided into an infinite\nnumber of chambers, some of them containing pools in which swim immense\nnumbers of blind fish.\"\n\n\"Blind fish! You are jesting, my friend,\" Don Miguel exclaimed, and\nstopped.\n\n\"I am wrong: blind is not the word I should have employed, for these\nfish have no eyes.\"\n\n\"What! No eyes?\"\n\n\"None at all; but that does not prevent them being very dainty food.\"\n\n\"That is strange.\"\n\n\"Is it not? But stay--we have arrived.\"\n\nIn fact, they found themselves in front of a gloomy, gaping orifice,\nabout ten feet high by eight wide.\n\n\"Let me do the honours of my mansion,\" Valentine said.\n\n\"Do so, my friend.\"\n\nThe two men entered the grotto: the hunter struck a match, and lit a\ntorch of candlewood. The fairy picture which suddenly rose before Don\nMiguel drew from him a cry of admiration. There was an indescribable\nconfusion: here a gothic chapel, with its graceful soaring pillars;\nfurther on, obelisks, cones, trunks of trees covered with moss and\nacanthus leaves, hollow stalactites of a cylindrical form, drawn\ntogether and ranged side by side like the pipes of an organ, and\nyielding to the slightest touch varied metallic sounds which completed\nthe illusion. Then, in the immeasurable depths of these cavernous halls,\nat times formidable sounds arose, which, returned by the echoes, rolled\nalong the sides of the grotto like peals of thunder.\n\n\"Oh, it is grand, it is grand!\" Don Miguel exclaimed, struck with fear\nand respect at the sight.\n\n\"Does not man,\" Valentine answered, \"feel very small and miserable\nbefore these sublime creations of nature, which God has scattered here\nas if in sport? Oh, my friend! It is only in the desert that we\nunderstand the grandeur and infinite omnipotence of the Supreme Being;\nfor at every step man finds himself face to face with Him who placed him\non this earth, and traces the mark of His mighty finger engraved in an\nindelible manner on everything that presents itself to his sight.\"\n\n\"Yes,\" Don Miguel said, who had suddenly become thoughtful, \"it is only\nin the desert that a man learns to know, love, and fear God, for He is\neverywhere.\"\n\n\"Come,\" said Valentine.\n\nHe led his friend to a hall of not more than twenty square feet, the\nvault of which, however, was more than a hundred yards above them. In\nthis hall a fire was lighted. The two men sat down on the ground and\nwaited, while thinking deeply. After a few moments the sound of\nfootsteps was audible, and the Mexican quickly raised his head.\nValentine did not stir, for he had recognised his friend's tread. In\nfact, within a moment the Indian chief appeared.\n\n\"Well?\" Valentine asked him.\n\n\"Nothing yet,\" Curumilla laconically answered.\n\n\"They are late, I fancy,\" Don Miguel observed.\n\n\"No,\" the chief continued, \"it is hardly half past eleven: we are before\nour time.\"\n\n\"But will they find us here?\"\n\n\"They know we shall await them in this hall.\"\n\nAfter these few words each fell back into his thoughts. The silence was\nonly troubled by the mysterious sounds of the grotto, which re-echoed\nnearly at equal intervals with an horrific din. A long period elapsed.\nAll at once, ere any sensible noise had warned Don Miguel, Valentine\nraised his head with a hurried movement.\n\n\"Here they are,\" he said.\n\n\"You are mistaken, my friend,\" Don Miguel observed; \"I heard nothing.\"\n\nThe hunter smiled.\n\n\"If you had spent,\" he said, \"like we have, ten years in the desert,\ninterrogating the mysterious voices of the night, your ear would be\nhabituated to the vague rumours and sighs of nature which have no meaning\nto you at this moment, but which have all a significance for me, and, so\nto speak, a voice every note of which I understand, and you would not\nsay I was mistaken. Ask the chief: you will hear his answer.\"\n\n\"Two men are climbing the hill at this moment,\" Curumilla answered\nsententiously. \"They are an Indian and a white man.\"\n\n\"How can you recognise the distinction?\"\n\n\"Very easily,\" Valentine responded with a smile. \"The Indian wears\nmoccasins, which touch the ground without producing any other sound than\na species of friction: the step is sure and unhesitating, as taken by a\nman accustomed to walk in the desert, and only put down his foot firmly:\nthe white man wears high-heeled boots, which at each step produce a\ndistinct and loud sound; the spurs fastened to his boots give out a\ncontinuous metallic clink; the step is awkward and timid; at each moment\na stone or crumble of earth rolls away under the foot, which is only put\ndown hesitatingly. It is easy to see that the man thus walking is\naccustomed to a horse, and does not know the use of his feet. Stay! They\nare now entering the grotto: you will soon hear the signal.\"\n\nAt this moment the bark of the coyote was raised thrice at equal\nintervals. Valentine answered by a similar cry.\n\n\"Well, was I mistaken?\" he said.\n\n\"I know not what to think, my friend. What astonishes me most is that\nyou heard them so long before they arrived.\"\n\n\"The ground of this cave is an excellent conductor of sound,\" the hunter\nanswered simply: \"that is all the mystery.\"\n\n\"The devil!\" Don Miguel could not refrain from saying; \"You neglect\nnothing, I fancy.\"\n\n\"If a man wants to live in the desert he must neglect nothing: the\nsmallest things have their importance, and an observation carefully made\nmay often save a man's life.\"\n\nWhile these few words were being exchanged between the two friends the\nnoise of footsteps was heard drawing nearer and nearer. Two men\nappeared: one was Eagle-wing, the Chief of the Coras; the second,\nGeneral Ibanez.\n\nThe general was a man of about thirty-five, tall and well-built, with a\ndelicate and intelligent face. His manners were graceful and noble. He\nbowed cordially to the hacendero and Valentine, squeezed Curumilla's\nhand, and fell down in a sitting posture by the fire.\n\n\"Ouf!\" he said, \"I am done, gentlemen. I have just ridden an awful\ndistance. My poor horse is foundered, and to recover myself I made an\nascent, during which I thought twenty times I must break down; and that\nwould have infallibly happened, had not friend Eagle-wing charitably\ncome to my aid. I must confess that these Indians climb like real cats:\nwe _gente de razon_[1] are worth nothing for that trade.\"\n\n\"At length you have arrived, my friend,\" Don Miguel answered. \"Heaven be\npraised! I was anxious to see you.\"\n\n\"For my part I confess that my impatience was equally lively, especially\nsince I learned the treachery of that scoundrelly Red Cedar. That humbug\nof a Wood sent him to me with so warm a recommendation that, in spite of\nall my prudence, I let myself be taken in, and nearly told him all our\nsecrets. Unfortunately, the little I did let him know is sufficient to\nhave us shot a hundred times like vulgar conspirators of no\nconsequence.\"\n\n\"Do not feel alarmed, my friend. After what. Valentine told me today, we\nhave, perchance, a way of foiling the tricks of the infamous spy who has\ndenounced us.\"\n\n\"May Heaven grant it! But nothing will remove my impression that Wood\nhas something to do with what has happened to us. I always doubted that\nAmerican, who is cold as an iceberg, sour as a glass of lemonade, and\nmethodical as a Quaker. What good is to be expected from these men, who\ncovet the possession of our territory, and who, unable to take it from\nus at one lump, tear it away in parcels?\"\n\n\"Who knows, my friend? Perhaps you are right. Unfortunately, what is\ndone cannot be helped, and our retrospective recriminations will do us\nno good.\"\n\n\"That is true; but, as you know, man is the same everywhere. When he has\ncommitted a folly he is happy to find a scapegoat on which he can lay\nthe iniquities with which he reproaches himself. That is slightly my\ncase at this moment.\"\n\n\"Do not take more blame on yourself, my friend, than you deserve; I\nguarantee your integrity and the loyalty of your sentiments. Whatever\nmay happen, be persuaded that I will always do you justice, and, if\nneeded, defend you against all.\"\n\n\"Thanks, Don Miguel. What you say causes me pleasure and reconciles me\nwith myself. I needed the assurance you give me in order to regain some\nslight courage, and not let myself be completely crushed by the\nunforeseen blow which threatens to overthrow our hopes at the very\nmoment when we expected to find them realised.\"\n\n\"Come, come, gentlemen,\" Valentine said, \"the time is slipping away, and\nwe have none to waste. Let us seek to find the means by which to repair\nthe check we have suffered. If you permit me I will submit to your\napproval a plan which, I believe, combines all the desirable chances of\nsuccess, and will turn in our favour the very treachery to which we have\nfallen victims.\"\n\n\"Speak, speak, my friend!\" the two men exclaimed, as they prepared to\nlisten.\n\nValentine took the word.\n\n\n[1] Literally, \"men of reason\"--a graceful expression the whites employ\nto distinguish themselves from the Indians, whom they affect to consider\nbrute beasts, and to whom they do not even grant a soul.\n\n\n\n\nCHAPTER XVIII.\n\nFATHER SERAPHIN.\n\n\n\"Gentlemen,\" said Valentine, \"this is what I propose. The treachery of\nRed Cedar, in surrendering to the Government the secret of your\nconspiracy, places you in a critical position, from which you cannot\nescape save by violent measures. You are between life and death. You\nhave no alternative save victory or defeat. The powder is fired, the\nground is mined under your feet, and an explosion is imminent. Well,\nthen, pick up the glove treachery throws to you--accept frankly the\nposition offered you. Do not wait till you are attacked, but commence\nthe contest. Remember the vulgar adage, which is perfectly true in\npolitics, and specially in revolution--that 'the first blow is half the\nbattle.' Your enemies will be terrified by your boldness--dashed by this\nuprising which they are far from expecting, especially now, when they\nimagine they hold in their hands all the threads of the conspiracy--an\nerror which makes them put faith in the revelations of a common spy, and\nwill ruin them if you act with skill--above all, with promptitude. All\ndepends on the first blow. It must be terrible, and terrify them: if\nnot, you are lost.\"\n\n\"All that is true; but we lack time,\" General Ibanez observed.\n\n\"Time is never lacking when a man knows how to employ it properly,\"\nValentine answered peremptorily. \"I repeat, you must be beforehand with\nyour adversaries.\"\n\nAt this moment the sound of footsteps was heard under the vault of the\ncave. The most extreme silence at once reigned in the chamber where the\nfive conspirators were assembled. Mechanically each sought his weapons.\nThe steps rapidly approached, and a man appeared in the entrance of the\nhall. On seeing him all present uttered a cry of joy and rose\nrespectfully, repeating, \"Father Seraphin!\"\n\nThe man advanced smiling, bowed gracefully, and answered in a gentle and\nmelodious voice, which went straight to the soul,--\n\n\"Take your places again, gentlemen, I beg of you. I should be truly\nvexed if I caused you any disturbance. Permit me only to sit down for a\nfew moments by your side.\"\n\nThey hastened to make room for him. Let us say in a few words who this\nperson was, whose unexpected arrival caused so much pleasure to the\npeople assembled in the grotto.\n\nFather Seraphin was a man of twenty-four at the most, although the\nfatigues he supported, the harsh labours he had imposed on himself, and\nwhich he fulfilled with more than apostolic abnegation, had left\nnumerous traces on his face, with its delicate features, its gentle and\nfirm expression, imprinted with a sublime melancholy, rendered even more\ntouching by the beam of ineffable goodness which escaped from his large,\nblue and thoughtful eyes. His whole person, however, exhaled a perfume\nof youth and health which disguised his age, as to which a superficial\nobserver might have been easily deceived.\n\nFather Seraphin was a Frenchman, and belonged to the order of the\nLazarists. For five years he had been traversing as an indefatigable\nmissionary, with no other weapon than his staff, the unexplored\nsolitudes of Texas and New Mexico, preaching the gospel to the Indians,\nwhile caring nothing for the terrible privations and nameless sufferings\nhe incessantly endured, and the death constantly suspended over his\nhead.\n\nFather Seraphin was one of those numerous soldiers, ignored martyrs of\nthe army of faith, who, making a shield of the Gospel, spread at the\nperil of their lives the word of God in those barbarous countries, and\ndie heroically, falling bravely on their battlefield, worn out by the\npainful exigencies of their sublime mission, aged at thirty, but having\ngained over a few souls to the truth, and shed light among the ignorant\nmasses.\n\nThe abnegation and devotion of these modest men, yet so great in heart,\nare too much despised in France, where however, the greater number of\nthese martyrs are recruited. Their sacrifices pass unnoticed; for, owing\nto the false knowledge possessed of beyond-sea countries, people are far\nfrom suspecting the continual struggles they have to sustain against a\ndeadly climate. And who would credit it? The most obstinate adversaries\nthey meet with in the accomplishment of their mission are not among the\nIndians, who always nearly welcome them with respect, if not joy, but\namong the men whom their labours benefit, and who ought to aid and\nprotect them with all their might. There is no vexation or humiliation\nwhich they do not endure from the agents of Mexico and the American\nUnion, to try and disgust and compel them to abandon the arena in which\nthey combat so nobly.\n\nFather Seraphin had gained the friendship and respect of all those with\nwhom accident had brought him into contact. Charmed with meeting a\nfellow countryman in the midst of those vast solitudes so distant from\nthat France he never hoped to see again, he had attached himself closely\nto Valentine, to whom he vowed a deep and sincere affection. For the\nsame motives, the hunter, who admired the greatness of character of this\npriest so full of true religion, felt himself drawn to him by an\nirresistible liking. They had frequently taken long journeys together,\nthe hunter guiding his friend to the Indian tribes across the desolate\nregions of Apacheria.\n\nSo soon as Father Seraphin had taken his place near the fire, Eagle-wing\nand Curumilla hastened to offer him all those slight services which they\nfancied might be agreeable to him, and offered him a few lumps of roast\nvenison with maize tortillas. The missionary gladly gratified the two\nchiefs, and accepted their offerings.\n\n\"It is a long time since we saw you, father,\" the hacendero said. \"You\nneglect us. My daughter asked me about you only two days ago, for she is\nanxious to see you.\"\n\n\"Dona Clara is an angel who does not require me,\" the missionary replied\ngently. \"I have spent nearly two months with the Comanche tribe of the\nTortoise. Those poor Indians claim all my care. They are thirsting for\nthe Divine Word.\"\n\n\"Are you satisfied with your journey?\"\n\n\"Sufficiently so, for these men are not such as they are represented to\nus. Their instincts are noble, and, as their primitive nature is not\nadulterated by contact with the vicious civilization that surrounds\nthem, they easily understood what is explained to them.\"\n\n\"Do you reckon on staying long among us?\"\n\n\"Yes; this last journey has fatigued me extremely. My health is in a\ndeplorable state, and I absolutely need a few days' rest in order to\nregain the requisite strength to continue my ministry.\"\n\n\"Well, father, come with me to the hacienda; you will remain with us,\nand make us all truly happy.\"\n\n\"I am going to make that request to you, Don Miguel. I am delighted that\nyou have thus met my wishes. If I accept your obliging offer, it is\nbecause I know I shall not incommode you.\"\n\n\"On the contrary, we shall be delighted to have you among us.\"\n\n\"Ah! I know the goodness of your heart.\"\n\n\"Do not make me better than I am, father: there is a spice of egotism in\nwhat I am doing.\"\n\n\"How so?\"\n\n\"Hang it! By labouring at the education of the Indians you render an\nimmense service to the race I have the honor of belonging to; for I,\ntoo, am an Indian.\"\n\n\"That is true,\" the priest answered with a laugh. \"Come, I absolve you\nfrom the sin of egotism, in favour of the intention which makes you\ncommit it.\"\n\n\"Father,\" Valentine then said, \"is the game plentiful in the desert just\nat present?\"\n\n\"Yes, there is a great deal: the buffaloes have come down from the\nmountains in herds--the elks, the deer, and the antelopes swarm.\"\n\nValentine rubbed his hands.\n\n\"It will be a good season,\" he said.\n\n\"Yes, for you. As for myself, I have no cause of complaint, for the\nIndians have been most attentive to me.\"\n\n\"All the better. I ever tremble when I know you are among those red\ndevils. I do not say that of the Comanches, who are warriors I esteem,\nand have always displayed the sincerest affection for you; but I have a\nterrible fear lest those villains of Apaches may play you a wicked trick\nsome fine day.\"\n\n\"Why entertain such ideas, my friend?\"\n\n\"They are correct. You cannot imagine what treacherous and cruel cowards\nthose Apache thieves are. I know them, and carry their marks; but do not\nfrighten yourself. If ever they ventured on any extremities against you,\nI know the road to their villages: there is not a nook in the desert\nwhich I have not thoroughly explored. It is not for nothing I have\nreceived the name of the 'Trail-hunter.' I swear to you I will not leave\nthem a scalp.\"\n\n\"Valentine, you know I do not like to hear you speak so. The Indians are\npoor ignorant men, who know not what they do, and must be pardoned for\nthe evil they commit.\"\n\n\"All right--all right!\" the hunter growled. \"You have your ideas on that\nscore, and I mine.\"\n\n\"Yes,\" the missionary replied with a smile, \"but I believe mine be\nbetter.\"\n\n\"It is possible. You know I do not discuss that subject with you; for I\ndo not know how you do it, but you always succeed in proving to me that\nI am wrong.\"\n\nEverybody laughed at this sally.\n\n\"And what are the Indians doing at this moment?\" Valentine continued.\n\"Are they still fighting?\"\n\n\"No; I succeeded in bringing Unicorn, the principal chief of the\nComanches, and Stanapat (the Handful of Blood), the Apache sachem, to an\ninterview, at which peace was sworn.\"\n\n\"Hum!\" Valentine said incredulously, \"that peace will not last long, for\nUnicorn has too many reasons to owe the Apaches a grudge.\"\n\n\"Nothing leads to the supposition, at present, that your forebodings\nwill be speedily realised.\"\n\n\"Why so?\"\n\n\"Because, when I left Unicorn, he was preparing for a grand buffalo\nhunt, in which five hundred picked warriors were to take part.\"\n\n\"Ah, ah! and where do you think the hunt will take place, father?\"\n\n\"I know for a certainty, because, when I left Unicorn, he begged me to\ninvite you to it, as he knew I should see you shortly.\"\n\n\"I willingly accept, for a buffalo hunt always had great attractions for\nme.\"\n\n\"You will not have far to go to find Unicorn, for he is scarce ten\nleagues from this place.\"\n\n\"The hunt will take place, then, in the neighbourhood?\"\n\n\"The meeting-place is Yellowstone Plain.\"\n\n\"I shall not fail to be there, father. Ah! I am delighted, more than you\ncan suppose, at the happy news you have brought me.\"\n\n\"All the better, my friend. Now, gentlemen, I will ask you to excuse me;\nfor I feel so broken with fatigue that, with your permission, I will go\nand take a few hours' rest.\"\n\n\"I was a fool not to think of it before,\" Valentine exclaimed with\nvexation as he struck his forehead. \"Pardon me, father.\"\n\n\"I thought for my brother,\" said Curumilla. \"If my father will follow me\nall is ready.\"\n\nThe missionary thanked him with a smile and rose, bowed to all present,\nand supported by Eagle-wing, he followed Curumilla into another chamber\nof the grotto. Father Seraphin found a bed of dry leaves covered with\nbear skins, and a fire so arranged as to burn all night. The two Indians\nretired after bowing respectfully to the father, and assuring themselves\nthat he needed nothing more.\n\nAfter kneeling on the ground of the grotto Father Seraphin laid himself\non his bed of leaves, crossed his arms on his chest, and fell into that\nchildlike sleep which only the just enjoy. After his departure Valentine\nbent over to his two friends.\n\n\"All is saved,\" he said in a low voice.\n\n\"How? Explain yourself,\" they eagerly answered.\n\n\"Listen to me. You will spend the night here; at daybreak you will start\nfor the Hacienda de la Noria, accompanied by Father Seraphin.\"\n\n\"Good! What next?\"\n\n\"General Ibanez will proceed, as from you, to the governor, and invite\nhim to a grand hunt of wild horses, to take place in three days.\"\n\n\"I do not understand what you are driving at.\"\n\n\"That is not necessary at this moment. Let me guide you; but, above all,\narrange it so that all the authorities of the town accept your\ninvitation and are present at the hunt.\"\n\n\"That I take on myself.\"\n\n\"Very good. You, general, will collect all the men you can, so that they\ncan support you on a given signal, but hide themselves so that no one\ncan suspect their presence.\"\n\n\"Very good,\" Don Miguel answered; \"all shall be done as you recommend.\nBut where will you be all this while?\"'\n\n\"You know very well,\" he answered with a smile of undefinable meaning.\n\"I shall be hunting the buffalo with my friend Unicorn, the great chief\nof the Comanches.\"\n\nHastily breaking off the interview, the hunter wrapped himself in his\nbuffalo robe, stretched himself before the fire, closed his eyes, and\nslept, or feigned to sleep. After a few minutes' hesitation his friend\nimitated his example, and the grotto became calm and silent as on the\nday of the creation.\n\n\n\n\nCHAPTER XIX.\n\nUNICORN.\n\n\nBefore retiring to rest Father Seraphin, on the previous evening, had\nwhispered a couple of words in the Indians' ears. The sun had scarce\nbegun to rise a little above the extreme blue line of the horizon ere\nthe missionary opened his eyes, and after a short prayer hurried to the\nhall in which his companions had remained. The four men were still\nasleep, wrapped in their furs and buffalo skins.\n\n\"Wake up, brothers,\" Father Seraphin said, \"for day is appearing.\"\n\nThe four men started up in an instant.\n\n\"My brothers,\" the young missionary said in a gentle and penetrating\nvoice, \"I thought that we ought, before separating, to thank God in\ncommon: for the blessings He does not cease to vouchsafe to us--to\ncelebrate our happy meeting of last night. I have, therefore, resolved\nto hold a mass, at which I shall be happy to see you with that purity of\nheart which such a duty demands.\"\n\nAt this proposition the four men exclaimed gladly their assent.\n\n\"I will help you to prepare the altar, father,\" Valentine said; \"the\nidea is excellent.\"\n\n\"The altar is all ready, my friends. Have the kindness to follow me.\"\n\nFather Seraphin then led them out of the grotto.\n\nIn the centre of a small esplanade in front of the cave an altar had\nbeen built by Eagle-wing and Curumilla on a grassy mound. It was very\nsimple. A copper crucifix planted in the centre of the mound, covered by\na cloth of dazzling whiteness; on either side of it two block-tin\ncandlesticks, in which burned candles of yellow tallow, a Bible on the\nright, the pyx in the centre--that was all.\n\nThe hunter and the two Mexicans knelt piously, and Father Seraphin\ncommenced offering the holy sacrifice, served devotedly by the two\nIndian chiefs.\n\nIt was a magnificent morning; thousands of birds, hidden beneath the\nfoliage, saluted the birth of day with their harmonious songs; a fickle\nbreeze poured through the branches, and refreshed the air; in the\ndistance, far as eye could extend, undulated the prairie, with its\noceans of tall grass incessantly agitated by the hurried foot falls of\nthe wild beasts returning to their dens; and on the naked side of this\nhill, at the entrance of this grotto--one of the marvels of the New\nWorld--a priest, simple as an apostle, was celebrating mass on a grass\naltar under the eye of Heaven, served by two poor savages, and having as\nsole congregation three half-civilised men.\n\nThis spectacle, so simple primitive, had something about it imposing and\nsublime, which inspired respect and summoned up dreams of ancient days,\nwhen the persecuted church took refuse in the desert, to find itself\nface to face with God. Hence the emotion experienced by the witnesses of\nthis religious act was sincere. A beam of happiness descended into their\nsouls, and it was with real effusion that they thanked the priest for\nthe pleasant surprise he had reserved for them. Father Seraphin was\ndelighted at the result he had attained. Seeing the truly profound faith\nof his friends, he felt his courage heightened to continue the rude and\nnoble task he had imposed on himself.\n\nThe mass lasted about three quarters of an hour. When it was finished\nthe missionary placed the poor holy vessels in the bag he constantly\ncarried with him, and they returned to the grotto for breakfast. An hour\nlater, Don Miguel, General Ibanez, and the missionary took leave of\nValentine, and mounted on their horses, which Curumilla had led to the\nentrance of the ravine. They started at a gallop in the direction of the\nPaso del Norte, whence they were about twenty leagues distant. Valentine\nand the two Indian chiefs remained behind.\n\n\"I am about to leave my brother,\" Eagle-wing said.\n\n\"Why not remain with us, chief?\"\n\n\"My pale brother no longer requires Eagle-wing. The chief hears the\ncries of the men and women of his tribe who were cowardly assassinated,\nand demand vengeance.\"\n\n\"Where goes my brother?\" the hunter asked, who was too thoroughly\nacquainted with the character of the Indians to try and change the\nwarrior's determination, though he was vexed at his departure.\n\n\"The Coras dwell in villages on the banks of the Colorado. Eagle-wing is\nreturning to his friends. He will ask for warriors to avenge his\nbrothers who are dead.\"\n\nValentine bowed.\n\n\"May the Great Spirit protect my father!\" he said. \"The road is long to\nthe villages of his tribe. The chief is leaving friends who love him.\"\n\n\"Eagle-wing knows it: he will remember,\" the chief said with a deep\nintonation.\n\nAnd, after pressing the hands the two hunters held out to him, he\nbounded on his horse, and soon disappeared in the windings of the\ncanyon.\n\nValentine watched his departure with a sad and melancholy look.\n\n\"Shall I ever see him again!\" he murmured. \"He is an Indian: he is\nfollowing his vengeance. It is his nature: he obeys it, and God will\njudge him. Every man must obey his destiny.\"\n\nAfter this aside the hunter threw his rifle on his shoulder and started\nin his turn, followed by Curumilla. Valentine and his comrade were on\nfoot: they preferred that mode of travelling, which seemed to them sure,\nand quite as quick as on horseback. The two men, after the Indian\ncustom, walked one behind the other, not uttering a syllable; but toward\nmidday the heat became so insupportable that they were obliged to stop\nto take a few moments' repose. At length the sunbeams lost their\nstrength, the evening breeze rose, and the hunters could resume their\njourney. They soon reached the banks of the Rio Puerco (Dirty River),\nwhich they began ascending, keeping as close as they could to the banks,\nwhile following the tracks made since time immemorial by wild animals\ncoming down to drink.\n\nThe man unacquainted with the splendid American scenery will have a\ndifficulty in imagining the imposing and savage majesty of the prairie\nthe hunters were traversing. The river, studded with islets covered with\ncottonwood trees, flowed silent and rapid between banks of slight\nelevation, and overgrown with grass so tall that it obeyed the impulse\nof the wind from a long distance. Over the vast plain were scattered\ninnumerable hills, whose summits, nearly all of the same height, present\na flat surface; and for a greater distance northward the ground was\nbroadcast with large lumps of pebbles resembling gravestones.\n\nAt a few hundred yards from the river rose a conical mound, bearing on\nits summit a granite obelisk one hundred and twenty feet in height. The\nIndians, who, like all primitive nations, are caught by anything\nstrange, frequently assembled at this spot; and here the hecatombs are\noffered to the Kitchi Manitou.\n\nA great number of buffalo skulls, piled up at the foot of the column,\nand arranged in circles, ellipses, and other geometrical figures, attest\ntheir piety for this god of the hunt, whose protecting spirit, they say,\nlooks down from the top of the monolith. Here and there grew patches of\nthe Indian potato, wild onion, prairie tomato, and those millions of\nstrange flowers and trees composing the American flora. The rest of the\ncountry was covered with tall grass, continually undulating beneath the\nlight footfall of the graceful antelopes or big horns, which bounded\nfrom one rock to the other, startled by the approach of the travellers.\n\nFar, far away on the horizon, mingling with the azure of the sky,\nappeared the denuded peaks of the lofty mountains that serve as\nunassailable fortresses to the Indians: their summits, covered with\neternal snow, formed the frame of this immense and imposing picture,\nwhich was stamped with a gloomy and mysterious grandeur.\n\nAt the hour when the _maukawis_ uttered its last song to salute the\nsetting of the sun, which, half plunged in the purple of evening, still\njaspered the sky with long red bands, the travellers perceived the tents\nof the Comanches picturesquely grouped on the sides of a verdurous hill.\nThe Indians had, in a few hours, improvised a real village with their\nbuffalo skin tents, aligned to form streets and squares.\n\nOn arriving at about five hundred yards from the village the hunters\nsuddenly perceived an Indian horseman. Evincing not the slightest\nsurprise, they stopped and unfolded their buffalo robes, which floated\nin the breeze, as a signal of peace. The horseman uttered a loud cry. At\nthis signal--for it was evidently one--a troop of Comanche warriors\ndebouched at a gallop from the village, and poured like a torrent down\nthe sides of the hill, coming up close to the motionless travellers,\nbrandishing their weapons, and uttering their war yell.\n\nThe hunters waited, carelessly leaning on their guns. Assuredly, to a\nman not acquainted with the singular manners of the prairie, this mode\nof reception would have seemed overt hostilities. But it was not so;\nfor, on coming within range of the hunters, the Comanches began making\ntheir horses leap and curvet with that grace and skill characteristic of\nthe Indians, and deploying to the right and left, they formed a vast\ncircle, inclosing the two unmoved hunters.\n\nThen a horseman quitted the group, dismounted, and rapidly approached\nthe newcomers: the latter hastened to meet him. All three had their arm\nextended with the palm forward in sign of peace. The Indian who thus\nadvanced to meet the hunters was Unicorn, the great chief of the\nComanches.\n\nAs a distinctive sign of his race, his skin was of a red tinge, brighter\nthan the palest new copper. He was a man of thirty at the most, with\nmasculine and expressive features; his face possessed a remarkable\nintelligence, and was stamped with that natural majesty found among the\nsavage children of the prairie; he was tall and well built; and his\nmuscular limbs evidenced a vigour and suppleness against which few men\nwould have contended with advantage.\n\nHe was completely painted and armed for war; his black hair was drawn up\non his head in the form of a casque, and fell down his back like a mane;\na profusion of wampum collars, claws of grizzly bear, and buffalo teeth\nadorned his breast, on which was painted with rare dexterity a blue\ntortoise, the distinctive sign of the tribe to which he belonged, and of\nthe size of a hand.\n\nThe rest of his costume was composed of the _mitasses_, fastened round\nthe hips by a leathern belt, and descending to the ankles; a deerskin\nshirt, with long hanging sleeves, the seams of which, like those of the\nmitasse, were fringed with leather strips and feathers; a wide cloak, of\nthe hide of a female buffalo, was fastened across his shoulders with a\nbuckle of pure gold, and fell down to the ground; on his feet he had\nelegant moccasins of different colours, embroidered with beads and\nporcupine quills, from the heels of which trailed several wolf tails; a\nlight round shield, covered with buffalo hide, and decorated with human\nscalps, hung on his left side by his panther skin quiver full of arrows.\nHis weapons were those of the Comanche Indians; that is to say, the\nscalping knife, the tomahawk, a bow, and an American rifle; but a long\nwhip, the handle of which painted red, was adorned with scalps,\nindicated his rank as chief.\n\nWhen the three men were close together they saluted by raising their\nhands to their foreheads; then Valentine and Unicorn crossed their arms\nby passing the right hand over the left shoulder, and bowing their heads\nat the same time, kissed each other's mouth after the prairie fashion.\nUnicorn then saluted Curumilla in the same way; and this preliminary\nceremony terminated, the Comanche chief took the word.\n\n\"My brothers are welcome at the village of my tribe,\" he said. \"I was\nexpecting them impatiently. I had begged the Chief of Prayer of the\npalefaces to invite them in my name.\"\n\n\"He performed his promise last night. I thank my brother for having\nthought of me.\"\n\n\"The two stranger great hunters are friends of Unicorn. His heart was\nsad not to see them near him for the buffalo hunt his young people are\npreparing.\"\n\n\"Here we are! We set out this morning at sunrise.\"\n\n\"My brothers will follow me, and rest at the council fire.\"\n\nThe hunters bowed assent. Each received a horse, and at a signal from\nUnicorn, who had placed himself between them, the troop started at a\ngallop, and returned to the village, which it entered to the deafening\nsound of drums, chikikouis, shouts of joy from the women and children\nwho saluted their return, and the furious barking of the dogs. When the\nchiefs were seated round the council fire the pipe was lit, and\nceremoniously presented to the two strangers, who smoked in silence for\nsome minutes. When the pipe had gone the round several times Unicorn\naddressed Valentine.\n\n\"Koutonepi is a great hunter,\" he said to him; \"he has often followed\nthe buffalo on the plains of the Dirty River. The chief will tell him\nthe preparations he has made, that the hunter may give his opinion.\"\n\n\"It is needless, chiefs,\" Valentine replied. \"The buffalo is the friend\nof the redskins: the Comanches know all its stratagems. I should like to\nask a question of my brother.\"\n\n\"The hunter can speak; my ears are open.\"\n\n\"How long will the chief remain on the hunting grounds with his young\nmen?\"\n\n\"About a week. The buffaloes are suspicious: my young men are\nsurrounding them, but they drive them in our direction before four or\nfive days.\"\n\nValentine gave a start of joy.\n\n\"Good,\" he said. \"Is my brother sure of it?\"\n\n\"Very sure.\"\n\n\"How many warriors have remained with the chief?\"\n\n\"About four hundred: the rest are scattered over the plain to announce\nthe approach of the buffaloes.\"\n\n\"Good! If my brother likes I will procure him a fine hunt within three\ndays.\"\n\n\"Ah!\" the chief exclaimed, \"then my brother has started some game?\"\n\n\"Oh!\" Valentine answered with a laugh, \"Let my brother trust to me, and\nI promise him rich spoils.\"\n\n\"Good! Of what game does my brother speak?\"\n\n\"Of _gachupinos_[1]. In two days they will meet in large numbers not far\nfrom here.\"\n\n\"Wah!\" said the Comanche, whose eyes sparkled at this news, \"My young\nmen will hunt them. My brother must explain.\"\n\nValentine shook his head.\n\n\"My words are for the ears of a chief,\" he said.\n\nWithout replying, Unicorn made a signal: the Indians rose silently, and\nleft the tent. Curumilla and Unicorn alone remained near the fire.\nValentine then explained to the Comanche, in its fullest details, the\nplan he had conceived, in the execution of which the aid of the Indians\nwas indispensable for him. Unicorn listened attentively without\ninterrupting. When Valentine had ended,--\n\n\"What does my brother think?\" the latter asked, fixing a scrutinising\nglance on the impassive countenance of the chief.\n\n\"Wah!\" the other replied, \"the paleface is very crafty. Unicorn will do\nwhat he desires.\"\n\nThis assurance filled Valentine's heart with joy.\n\n\n[1] Wearers of shoes--a name given by the Indians to the Spaniards at\nthe conquest.\n\n\n\n\nCHAPTER XX.\n\nTHE HUNT OF WILD HORSES.\n\n\nDon Miguel Zarate and his two friends did not reach the hacienda till\nlate. They were received in the porch by Don Pablo and Dona Clara, who\nmanifested great joy at the sight of the French missionary, for whom\nthey felt a sincere esteem and great friendship. Spite of all his care,\nFray Ambrosio had always seen his advances repelled by the young people,\nin whom he instinctively inspired that fear mingled with disgust that is\nexperienced at the sight of a reptile.\n\nDona Clara, who was very pious, carried this repulsion to such a pitch\nthat she only confessed her faults and approached the holy table when\nFather Seraphin came to spend a few days at the hacienda.\n\nFray Ambrosio was too adroit to appear to notice the effect his presence\nproduced on the hacendero's children: he feigned to attribute to\ntimidity and indifference on religious matters what was in reality a\nstrongly expressed loathing for himself personally. But in his heart a\ndull hatred fermented against the two young folk, and especially against\nthe missionary, whom he had several times already attempted to destroy\nby well-laid snares.\n\nFather Seraphin had always escaped them by a providential chance; but in\nspite of the chaplain's obsequious advances, and the offers of service\nhe did not fail to overwhelm him with each time they met, the missionary\nhad thoroughly read the Mexican monk. He had guessed what fearful\ncorruption was hidden beneath his apparent simplicity and feigned piety:\nand while keeping to himself the certainty he had acquired, he remained\non his guard, and carefully watched this man, whom he suspected of\nincessantly planning some dark treachery against him. Don Miguel left\nhis children with the missionary, who immediately took possession of him\nand dragged him away, lavishing on him every possible attention. The\nhacendero retired to his study with General Ibanez, when the two men\ndrew up a list of the persons they intended to invite; that is to say,\nthe persons Valentine proposed to get out of the way, though they were\ninnocent of his scheme. The general then mounted his horse, and rode off\nto deliver the invitations personally. For his part Don Miguel sent off\na dozen peons and vaqueros in search of the wild horses, and to drive\nthem gradually toward the spot chosen for the hunt.\n\nGen. Ibanez succeeded perfectly: the invitations were gladly accepted,\nand the next evening the guests began arriving at the hacienda, Don\nMiguel receiving them with marks of the most profound respect and lavish\nhospitality.\n\nThe governor, General Isturitz, Don Luciano Perez, and seven or eight\npersons of inferior rank were soon assembled at the hacienda. At sunrise\na numerous party, composed of forty persons, left the hacienda, and\nproceeded, accompanied by a crowd of well-mounted peons, towards the\nmeet. This was a vast plain on the banks of the Rio del Norte, where the\nwild horses were accustomed to graze at this season. The caravan\nproduced the most singular and picturesque effect with the brilliant\ncostumes of the persons who composed it, and their horses glittering\nwith gold and silver. Starting at about four a.m. from the hacienda,\nthey reached four hours later a clump of trees, beneath whose shade\ntents had been raised and tables laid by Don Miguel's orders, so that\nthey might breakfast before the hunt.\n\nThe riders, who had been journeying for four hours, already exposed to\nthe rays of the sun and the dust, uttered a shout of joy at the sight of\nthe tents. Each dismounted: the ladies were invited to do the same,\namong them being the wives of the governor and General Isturitz, and\nDona Clara, and they gaily sat down round the tables.\n\nToward the end of the breakfast Don Pablo arrived, who had started the\nevening previously to join the vaqueros. He announced that the horses\nhad been started, that a large manada was now crossing the Plain of the\nCoyotes, watched by the vaqueros, and that they must make haste if they\nwished to have good sport. This news augmented the ardor of the hunters.\nThe ladies were left in camp under the guard of a dozen well-armed\npeons, and the whole party rushed at a gallop in the direction indicated\nby Don Pablo.\n\nThe Plain of the Coyotes extended for an enormous distance along the\nbanks of the river. Here and there rose wooded hills, which varied the\nlandscape that was rendered monotonous by the tall grass, in which the\nriders disappeared up to their waists. When the hunting party reached\nthe skirt of the plain Don Miguel ordered a halt, that they might hold a\ncouncil, and hear the report of the leader of the vaqueros.\n\nThe races of wild horses that nowadays people the deserts of North\nAmerica, and especially of Mexico, is descended from Cortez' cavalry.\nHence it is a pure breed, for at the period of the Spanish conquest only\nArab horses were employed. These horses have multiplied in really an\nextraordinary manner. It is not rare to meet with manadas of twenty and\neven thirty thousand head. They are small, but gifted with an energy and\nvigour of which it is impossible to form a fair idea without having seen\nthem. They accomplish without fatigue journeys of prodigious length.\nTheir coat is the same as that of other horses, save that during winter\nit grows very long, and frizzy like the wool of sheep. In spring this\nspecies of fur falls off. The American horses may be easily trained.\nGenerally, so soon as they find themselves caught they easily submit to\nthe saddle.\n\nThe Mexicans treat their steeds very harshly, make them journey the\nwhole day without food or drink, and only give them their ration of\nmaize and water on reaching the bivouac, where they let them wander\nabout the whole night under guard of the _nena_, a mare whose bell the\nhorses follow, and will never leave. It is not from any cruel motive,\nhowever, that the Mexicans treat their horses thus, for the riders are\nvery fond of their animals, which at a given moment may save their\nlives. But it seems that this mode of treatment, which would be\nimpracticable in Europe, is perfectly successful in Mexico, where the\nhorses are much better off than if treated in a more gentle way.\n\nThe leader of the vaqueros made his report. A manada of about ten\nthousand head was two leagues off on the plain, quietly grazing in the\ncompany of a few elks and buffaloes. The hunters scaled a hill, from the\ntop of which they easily saw on the horizon a countless mob of animals,\ngrouped in a most picturesque way, and apparently not at all suspecting\nthe danger that threatened them.\n\nTo hunt the wild horses men must be like the Mexicans, perfect centaurs.\nI have seen the _jinetes_ of that country accomplish feats of\nhorsemanship before which our Europeans would turn pale.\n\nAfter the vaquero's report Don Miguel and his friends held a council,\nand this is the resolution they came to. They formed what is called in\nMexico the grand circle of the wild horses; that is to say, the most\nskilful riders were echeloned in every direction at a certain distance\nfrom each other, so as to form an immense circle. The wild horses are\nextremely suspicious: their instinct is so great, their scent is so\nsubtle, that the slightest breath of wind is sufficient to carry to them\nthe smell of their enemies, and make them set off at headlong speed.\nHence it is necessary to act with the greatest prudence, and use many\nprecautions, if a surprise is desired.\n\nWhen all the preparations were made the hunters dismounted, and dragging\ntheir horses after them, glided through the tall grass so as to contract\nthe circle. This manoeuvre had gone on for some time, and they had\nsensibly drawn nearer, when the manada began to display some signs of\nrestlessness. The horses, which had hitherto grazed calmly, raised their\nheads, pricked their ears, and neighed as they inhaled the air. Suddenly\nthey collected, formed a compact band, and started at a trot in the\ndirection of some cottonwood trees which stood on the banks of the\nriver. The hunt was about to commence.\n\nAt a signal from Don Miguel six well-mounted vaqueros rushed at full\nspeed ahead of the manada, making their lassoes whistle round their\nheads. The horses, startled by the apparition of the riders, turned back\nhastily, uttering snorts of terror, and fled in another direction. But\neach time they tried to force the circle, horsemen rode into the midst\nof them, and compelled them to turn back.\n\nIt is necessary to have been present at such a chase, to have seen this\nhunt on the prairies, to form an idea of the magnificent sight offered\nby all these noble brutes, their eyes afire, their mouths foaming, their\nheads haughtily thrown up, and their manes fluttering in the wind, as\nthey bounded and galloped in the fatal circle the hunters had formed\nround them. There is in such a sight something intoxicating, which\ncarries away the most phlegmatic, and renders them mad with enthusiasm\nand pleasure.\n\nWhen this manoeuvre had lasted long enough, and the horses began to grow\nblinded with terror, at a signal given by Don Miguel the circle was\nbroken at a certain spot. The horses rushed, with a sound like thunder,\ntoward this issue which opened before them, overturning with their\nchests everything that barred their progress. But it was this the\nhunters expected. The horses, in their mad race, galloped on without\ndreaming that the road they followed grew gradually narrower in front of\nthem, and terminated in inevitable captivity.\n\nLet us explain the termination of the hunt. The manada had been cleverly\nguided by the hunters toward the entrance of a canyon, or ravine, which\nran between two rather lofty hills. At the end of this ravine the\nvaqueros had formed, with stakes fifteen feet long, planted in the\nground, and firmly fastened together with cords of twisted bark, an\nimmense corral or inclosure, into which the horses rushed without seeing\nit. In less than no time the corral was full; then the hunters went to\nmeet the manada, which they cut off at the risk of their lives, while\nthe others closed the entrance of the corral. More than fifteen hundred\nmagnificent wild horses were thus captured at one stroke.\n\nThe noble animals rushed with snorts of fury at the walls of the\ninclosure, trying to tear up the stakes with their teeth, and dashing\nmadly against them. At length they recognised the futility of their\nefforts, lay down, and remained motionless. In the meanwhile a\ntremendous struggle was going on in the ravine between the hunters and\nthe rest of the manada. The horses confined in this narrow space made\nextraordinary efforts to open a passage and fly anew. They neighed,\nstamped, and flew at everything that came within their reach. At length\nthey succeeded in regaining their first direction, and rushed into the\nplain with the velocity of an avalanche. Several vaqueros had been\ndismounted and trampled on by the horses, and two of them had received\nsuch injuries that they were carried off the ground in a state of\ninsensibility.\n\nWith all the impetuosity of youth Don Pablo had rushed into the very\nheart of the manada. Suddenly his horse received a kick which broke its\noff foreleg, and it fell to the ground, dragging its rider with it. The\nhunters uttered a cry of terror and agony. In the midst of this band of\nmaddened horses the young man was lost, for he must be trampled to death\nunder their hoofs. But he rose with the rapidity of lightning, and quick\nas thought seizing the mane of the nearest horse, he leaped on its back,\nand held on by his knees. The horses were so pressed against one another\nthat any other position was impossible. Then a strange thing\noccurred--an extraordinary struggle between the horse and its rider. The\nnoble beast, furious at feeling its back dishonoured by the weight it\nbore, bounded, reared, rushed forward; but all was useless, for Don\nPablo adhered firmly.\n\nSo long as it was in the ravine, the horse, impeded by its comrades,\ncould not do all it might have wished to get rid of the burden it bore;\nbut so soon as it found itself on the plain it threw up its head, made\nseveral leaps on one side, and then started forward at a speed which\ntook away the young man's breath.\n\nDon Pablo held on firmly by digging his knees into the panting sides of\nhis steed; he unfastened his cravat, and prepared to play the last scene\nin this drama, which threatened to terminate in a tragic way for him.\nThe horse had changed its tactics; it was racing in a straight line to\nthe river, resolved to drown itself with its rider sooner than submit.\nThe hunters followed with an interest mingled with terror the moving\ninterludes of this mad race, when suddenly the horse changed its plans\nagain, reared, and tried to fall back with its rider. The hunters\nuttered a shout of agony. Don Pablo clung convulsively to his animal's\nneck, and, at the moment it was falling back, he threw his cravat over\nits eyes with extraordinary skill.\n\nThe horse, suddenly blinded, fell back again on its feet, and stood\ntrembling with terror. Then the young man dismounted, put his face to\nthe horse's head, and breathed into its nostrils, while gently\nscratching its forehead. This operation lasted ten minutes at the most,\nthe horse panting and snorting, but not daring to leave the spot. The\nMexican again leaped on the horse's back, and removed the bandage; it\nremained stunned--Don Pablo had tamed it[1]. Everybody rushed toward the\nyoung man, who smiled proudly, in order to compliment him on his\nsplendid victory. Don Pablo dismounted, gave his horse to a vaquero, who\nimmediately passed a bridle round its neck, and then walked toward his\nfather, who embraced him tenderly. For more than an hour Don Miguel had\ndespaired of his son's life.\n\n\n[1] This mode of taming horses is well known to the Indians, and we\nsubmit the fact to our readers without comment.\n\n\n\n\nCHAPTER XXI.\n\nTHE SURPRISE.\n\n\nSo soon as the emotion caused by Don Pablo's prowess was calmed they\nbegan thinking about returning. The sun was rapidly descending in the\nhorizon: the whole day had been spent with the exciting incidents of the\nchase. The Hacienda de la Noria was nearly ten leagues distant: it was,\ntherefore, urgent to start as speedily as possible, unless the party\nwished to run the risk of bivouacking in the open air.\n\nThe men would easily have put up with this slight annoyance, which, in a\nclimate like that of New Mexico, and at this season of the year, has\nnothing painful about it; but they had ladies with them. Left one or two\nleagues in the rear, they must feel alarmed by the absence of the\nhunters--an absence which, as so frequently happens when out hunting,\nhad been protracted far beyond all expectation.\n\nDon Miguel gave the vaqueros orders to brand the captured horses with\nhis cipher; and the whole party then returned, laughing and singing, in\nthe direction of the tents where the ladies had been left. The vaqueros\nwho had served as beaters during the day remained behind to guard the\nhorses.\n\nIn these countries, where there is scarce any twilight, night succeeds\nthe day almost without transition. As soon as the sun had set the\nhunters found themselves in complete darkness; for, as the sun descended\non the horizon, the shade invaded the sky in equal proportions, and, at\nthe moment when the day planet disappeared, the night was complete. The\ndesert, hitherto silent, seemed to wake up all at once: the birds,\nstupefied by the heat, commenced a formidable concert, in which joined\nat intervals, from the inaccessible depths of the forest, the snapping\nof the _carcajous_ and the barking of the coyotes mingled with the\nhoarse howling of the wild beasts that had left their dens to come down\nand drink in the river.\n\nThen gradually the cries, the songs, and the howling ceased, and nothing\nwas audible save the hurried footfalls of the hunters' horses on the\npebbles of the road. A solemn silence seemed to brood over this abrupt\nand primitive scenery. At intervals the green tufts of the trees and the\ntall grass bowed slowly with a prolonged rustling of leaves and\nbranches, as if a mysterious breath passed over them, and compelled them\nto bend their heads. There was something at once striking and terrible\nin the imposing appearance offered by the prairie at this hour of the\nnight, beneath this sky studded with brilliant stars, which sparkled\nlike emeralds, in the presence of this sublime immensity, which only\nsuffered one voice to be heard--that of Deity.\n\nThe young and enthusiastic man to whom it is given to be present at such\na spectacle feels a thrill run over all his body: he experiences an\nundefinable feeling of happiness and extraordinary pleasure on looking\nround him at the desert, whose unexplored depths conceal from him so\nmany secrets, and display to him Divine Majesty in all its grandeur and\nomnipotence. Many a time during our adventurous journeys on the American\ncontinent, when marching at hazard during these lovely nights so full of\ncharms, which nothing can make those comprehend who have not experienced\nthem, we have yielded to the soft emotions that overcame us. Isolating\nand absorbing ourselves within ourselves, we, have fallen into a state\nof beatitude, from which nothing had the power of drawing us.\n\nThe hunters so gay and talkative at the start, had yielded to this\nomnipotent influence of the desert, and advanced rapidly and silently,\nonly exchanging a few syllables at lengthened intervals. The profoundest\ncalm still continued to reign over the desert; and while, owing to the\nastonishing transparency of the atmosphere, the eye could embrace a\nhorizon, nothing suspicious was visible.\n\nThe fireflies buzzed carelessly round the top of the grass, and the\nflickering fires burning before the tents to which the hunters were\nbound could be already seen about half a league ahead. At a signal from\nDon Miguel the party, which had, up to the present, only trotted, set\nout at a long canter; for each felt anxious to leave a scene which, in\nthe darkness, had assumed a sinister aspect.\n\nThey thus arrived within a hundred yards of the fires, whose ruddy glow\nwas reflected on the distant trees, when suddenly a fearful yell crossed\nthe air, and from behind every bush out started an Indian horseman\nbrandishing his weapons, and making his horse curvet round the white\nmen, while uttering his war cry. The Mexicans, taken unawares, were\nsurrounded ere they sufficiently recovered from their stupor to think\nabout employing their weapons. At a glance Don Miguel judged the\nposition: it was a critical one. The hunters were at the most but\ntwenty: the number of Comanche warriors surrounding them was at least\nthree hundred.\n\nThe Comanches and Apaches are the most implacable foes of the white\nrace. In their periodical invasions of the frontiers they hardly ever\nmake any prisoners: they mercilessly kill all who fall into their hands.\nStill the Mexicans rallied. Certain of the fate that awaited them, they\nwere resolved to sell their lives dearly. There was a moment of supreme\nexpectation before the commencement of the deadly combat, when suddenly\nan Indian galloped out of the ranks of the warriors, and rode within\nthree paces of the little band of Mexicans. On arriving there he\nstopped, and waved his buffalo robe in sign of peace. The governor of\nthe provinces prepared to speak.\n\n\"Let me carry on the negotiations,\" Don Miguel said. \"I know the Indians\nbetter than you do, and perhaps I shall succeed in getting out of this\nawkward position.\"\n\n\"Do so,\" the governor answered.\n\nGeneral Ibanez was the only one who had remained calm and impassive\nsince the surprise: he did not make a move to seize his weapons; on the\ncontrary, he crossed his arms carelessly on his chest, and took a\nmocking glance at his comrades as he hummed a seguidilla between his\nteeth. Don Pablo had placed himself by his father's side, ready to\ndefend him at the peril of his life. The Indian chief took the word.\n\n\"Let the palefaces listen,\" he said; \"an Indian sachem is about to\nspeak.\"\n\n\"We have no time to spare in listening to the insidious words which you\nare preparing to say to us,\" Don Miguel replied in a haughty voice.\n\"Withdraw, and do not obstinately bar our passage, or there will be\nblood spilt.\"\n\n\"The palefaces will have brought it on themselves,\" the Comanche\nanswered in a gentle voice. \"The Indians mean no harm to the pale\nwarriors.\"\n\n\"Why, then, this sudden attack? The chief is mad. We do not let\nourselves be so easily deceived as he seems to suppose: we know very\nwell that he wants our scalps.\"\n\n\"No; Unicorn wishes to make a bargain with the palefaces.\"\n\n\"Come, chief, explain yourself; perhaps your intentions are as you\ndescribe them. I do not wish to reproach myself with having refused to\nlisten to you.\"\n\nThe Indian smiled.\n\n\"Good!\" he said. \"The great white chief is becoming reasonable. Let him\nlisten, then, to the words Unicorn will pronounce.\"\n\n\"Go on, chief; my comrades and myself are listening.\"\n\n\"The palefaces are thieving dogs,\" the chief said in a rough voice;\n\"they carry on a continual war with the redskins, and buy their scalps\nas if they were peltry; but the Comanches are magnanimous warriors, who\ndisdain to avenge themselves. The squaws of the white men are in their\npower: they will restore them.\"\n\nAt these words a shudder of terror ran along the ranks of the hunters;\ntheir courage failed them; they had only one desire left--that of saving\nthose who had so wretchedly fallen into the hands of these bloodthirsty\nmen.\n\n\"On what conditions will the Comanches restore their prisoners?\" Don\nMiguel asked, whose heart was contracted at the thought of his daughter,\nwho was also a prisoner. He secretly cursed Valentine, whose fatal\nadvice was the sole cause of the frightful evil that assailed him at\nthis moment.\n\n\"The palefaces,\" the chief continued, \"will dismount and arrange\nthemselves in a line. Unicorn will choose from among his enemies those\nwhom he thinks proper to carry off as prisoners; the rest will be free,\nand all the women restored.\"\n\n\"Those conditions are harsh, chief. Can you not modify them?\"\n\n\"A chief has only one word. Do the palefaces consent?\"\n\n\"Let us consult together for a few moments at any rate.\"\n\n\"Good! Let the white men consult. Unicorn grants them ten minutes,\" the\nchief made answer.\n\nAnd turning his horse, he went back to his men. Don Miguel then\naddressed his friends.\n\n\"Well; what do you think of what has occurred?\"\n\nThe Mexicans were terrified: still they were compelled to allow that the\nconduct of the Indians was extraordinary, and that they had never before\nevinced such lenity. Now that reflection had followed on the first\nfeeling of excitement, they understood that a struggle against enemies\nso numerous was insensate, and could only result in rendering their\nposition worse than it was before, and that the chiefs conditions, harsh\nas they were, offered at least some chance of safety for a portion of\nthem, and the ladies would be saved.\n\nThis last and all powerful consideration decided them. Don Miguel had no\noccasion to convince them of the necessity of submission. Whatever\nstruggle it cost them, they dismounted and arranged themselves in a\nline, as the chief had demanded, Don Miguel and his son placing\nthemselves at the head.\n\nUnicorn, with that cool courage characteristic of the Indians, then\nadvanced alone toward the Mexicans, who still held their weapons, and\nwho, impelled by their despair, and at the risk of being all massacred,\nwould have sacrificed him to their vengeance. The chief had also\ndismounted. With his hands crossed on his back, and frowning brow, he\nnow commenced his inspection.\n\nMany a heart contracted at his approach, for a question of life and\ndeath was being decided for these hapless men: only the perspective of\nthe atrocious tortures which menaced the ladies could have made them\nconsent to this humiliating and degrading condition. The Unicorn,\nhowever, was generous: he only selected eight of the Mexicans, and the\nrest received permission to mount their horses, and leave the fatal\ncircle that begirt them. Still, by a strange accident, or a\npremeditation of which the reason escaped them, these, eight\nprisoners--among whom were the governor, General Isturitz, and the\ncriminal judge, Don Luciano Perez--were the most important personages in\nthe party, and the members of the Provincial Government.\n\nIt was not without surprise that Don Miguel observed this; the\nComanches, however, faithfully fulfilled their compact, and the ladies\nwere at once set at liberty. They had been treated with the greatest\nrespect by the Indians, who had surprised their camp, and seized them\nalmost in the same way as they had done the hunters--that is to say, the\ncamp was invaded simultaneously on all sides. It was a matter worthy of\nremark in an Indian ambuscade that not a drop of blood had been spilt.\n\nAfter the moments given up to the happiness of seeing his daughter again\nsafe and sound, Don Miguel resolved to make a last attempt with Unicorn\nin favour of the unhappy men who remained in his hands. The chief\nlistened with deference, and let him speak without interruption; then he\nreplied with a smile whose expression the hacendero tried in vain to\nexplain,--\n\n\"My father has Indian blood in his veins; the redskins love him: never\nwill they do him the slightest injury. Unicorn would like to restore him\nimmediately the prisoners, for whom he cares very little; but that is\nimpossible. My father himself would speedily regret Unicorn's obedience\nto his Wish; but, in order to prove to my father how much the chief\ndesires to do a thing that will be agreeable to him, the prisoners will\nnot be ill-treated, and will be let off with a few days' annoyance.\nUnicorn consents to accept a ransom for them, instead of making them\nslaves. My father can himself tell them this good news.\"\n\n\"Thanks, chief,\" Don Miguel answered. \"The nobility of your character\ntouches my heart: I shall not forget it. Be persuaded that, under all\ncircumstances, I shall be happy to prove to you how grateful I am.\"\n\nThe chief bowed gracefully and withdrew, in order to give the hacendero\nliberty to communicate with his companions. The latter were seated sadly\non the ground, gloomy and downcast. Don Miguel repeated to them the\nconversation he had held with Unicorn, and the promise he had made with\nrespect to them. This restored them all their courage; and, with the\nmost affectionate words and marks of the liveliest joy, they thanked the\nhacendero for the attempt he had made in their favour.\n\nIn fact, thanks to the promise of liberating them for a ransom at the\nend of a week, and treating them well during the period of their\ncaptivity, there was nothing so very terrifying about the prospect; and\nit was one of those thousand annoyances to which men are exposed by\naccident, but whose proportions had been so reduced in their eyes, that,\nwith the carelessness which forms the staple of the national character,\nthey were the first to laugh at their mishap.\n\nDon Miguel, however, was anxious to retire; so he took leave of his\ncompanions, and rejoined the chief. The latter repeated his assurances\nthat the prisoners should be free within a week, if they consented each\nto pay a ransom of one thousand piastres, which was a trifle. He assured\nthe hacendero that he was at liberty to withdraw whenever he pleased,\nand he should not oppose his departure.\n\nDon Miguel did not allow the invitation to be repeated. His friends and\nhimself immediately mounted their horses, together with the ladies, who\nwere placed in the centre of the detachment; and after taking leave of\nUnicorn, the Mexicans dug their spurs into their horses, and started at\na gallop, glad to have got off so cheaply. The campfires were soon left\nfar behind them, and General Ibanez then approached his friend, and\nbending down to his ear, whispered,--\n\n\"Don Miguel, can the Comanches be our allies? I fancy that they have\nthis night given a bold push to the success of our enterprise.\"\n\nThis thought, like a ray of light, had already crossed the hacendero's\nbrain several times.\n\n\"I do not know,\" he said with a clever smile; \"but at any rate, my dear\ngeneral, they are very adroit foes.\"\n\nThe little band continued to advance rapidly toward the hacienda, which\nwas now no great distance, and which they hoped to reach before sunrise.\nThe events we have described had occurred in less than an hour.\n\n\n\n\nCHAPTER XXII.\n\nTHE MEETING.\n\n\n\"By Jove!\" General Ibanez said, \"it must be confessed that these red\ndevils have done us an immense service without suspecting it. It might\nbe said, deuce take me, that they acted under a knowledge of facts. This\nUnicorn, as the chief is called, is a precious man in certain\ncircumstances. I am anxious to cultivate his acquaintance, for no one\nknows what may happen. It is often good to have so intelligent a friend\nas him at hand.\"\n\n\"You are always jesting, general. When will you be serious for once?\"\nDon Miguel said with a smile.\n\n\"What would you have, my friend? We are at this moment staking our heads\nin a desperate game, so let us at any rate keep our gaiety. If we are\nconquered, it will be time enough then to be sad, and make bitter\nreflections about the instability of human affairs.\"\n\n\"Yes, your philosophy is not without a certain dose of fatalism, which\nrenders it more valuable to me. I am happy to see you in this good\ntemper, especially at a moment when we are preparing to play our last\ncard.\"\n\n\"All is not desperate yet, and I have a secret foreboding, on the\ncontrary, that all is for the best. Our friend the Trail-hunter, I feel\nconvinced, has something to do, if not all, with what has happened to\nus.\"\n\n\"Do you believe it?\" Don Miguel asked quickly.\n\n\"I am certain of it. You know as well as I do these Indios Bravos, and\nthe implacable hatred they have vowed against us. The war they wage with\nus is atrocious; and for them to be suddenly changed from wolves into\nlambs requires some powerful motive to make them act thus. People do not\nlay aside in a moment a hatred which has endured for ages. The\nComanches, by the choice they made, know the importance of the prisoners\nthey have seized. How is it that they consent so easily to give them up\nfor a trifling ransom? There is some inexplicable mystery in all this.\"\n\n\"Which is very easy to explain though,\" a laughing voice interrupted\nfrom behind the shrubs.\n\nThe two Mexicans started, and checked their horses. A man leaped from a\nthicket, and suddenly appeared in the centre of the track the little\nband of hunters was following. The latter, believing in a fresh attack\nand treachery on the part of the Comanches, seized their weapons.\n\n\"Stop!\" Don Miguel said sharply, \"the man is alone. Let me speak with\nhim.\"\n\nEach waited with his hand on his weapon.\n\n\"Hold!\" Don Miguel continued, addressing the stranger, who stood\nmotionless, carelessly resting on his gun. \"Who are you, my master?\"\n\n\"Do you not recognise me, Don Miguel? and must I really tell you my\nname?\" the stranger answered with a laugh.\n\n\"The Trail-hunter!\" Don Miguel exclaimed.\n\n\"Himself,\" Valentine continued. \"Hang it all! You take a long time to\nrecognise your friends.\"\n\n\"You will forgive us when you know all that has happened to us, and how\nmuch we must keep on our guard.\"\n\n\"Confound it!\" Valentine said laughingly, as he regulated his pace by\nthe trot of the horses, \"do you fancy you are going to tell me any news?\nDid you not really suspect from what quarter the blow came?\"\n\n\"What!\" Don Miguel exclaimed in surprise, \"did you--\"\n\n\"Who else but I? Do you think the Spaniards are such friends of the\nIndians that the latter would treat them so kindly when meeting them\nface to face in the desert?\"\n\n\"I was sure of it,\" General Ibanez affirmed. \"I guessed it at the first\nmoment.\"\n\n\"Good heavens! Nothing was more simple. Your position, through Red\nCedar's treachery, was most critical. I wished to give you the time to\nturn round by removing, for a few days, the obstacles that prevented the\nsuccess of your plans. I have succeeded, I fancy.\"\n\n\"You could not have managed better,\" exclaimed the general.\n\n\"Oh!\" Don Miguel said with a reproachful accent, \"why did you hide it\nfrom me?\"\n\n\"For a very simple reason, my friend. I wished that in these\ncircumstances your will and conscience should be free.\"\n\n\"But--\"\n\n\"Let me finish. Had I told you of my plan, it is certain that you would\nhave opposed it. You are a man of honor, Don Miguel: your heart is most\nloyal.\"\n\n\"My friend--\"\n\n\"Answer me. Had I explained to you the plan I formed, what would you\nhave done?\"\n\n\"Well--\"\n\n\"Answer frankly.\"\n\n\"I should have refused.\"\n\n\"I was sure of it. Why would you have done so? Because you would never\nhave consented to violate the laws of hospitality, and betray enemies\nyou sheltered beneath your roof, though you knew all the while that\nthese men, on leaving you, would have considered it their duty to seize\nyou, and that they watched your every movement while sitting by your\nside, and eating at your table. Is it not so?\"\n\n\"It is true; my honor as a gentleman would have revolted. I could not\nhave suffered such horrible treachery to be carried out under my very\neyes.\"\n\n\"There! You see that I acted wisely in saying nothing to you. In that\nway your honor is protected, your conscience easy, and I have in the\nmost simple fashion freed you for some days from your enemies.\"\n\n\"That is true; still--\"\n\n\"What? Have the prisoners to complain of the way in which they have been\ntreated?\"\n\n\"Not at all; on the contrary, the Comanches, and Unicorn in particular,\ntreated them most kindly.\"\n\n\"All is for the best, then. You must congratulate yourself on the\nunexpected success you have achieved, and must now profit by it without\ndelay.\"\n\n\n\"I intend to do so.\"\n\n\"You must act at once.\"\n\n\"I ask nothing better. All is ready. Our men are warned, and they will\nrise at the first signal.\"\n\n\"It must be given immediately.\"\n\n\"I only ask the time to leave my daughter at the hacienda; then\naccompanied by my friends, I will march on Paso, while General Ibanez,\nat the head of a second band, seizes Santa Fe.\"\n\n\"The plan is well conceived. Can you count on the persons who follow\nyou?\"\n\n\"Yes; they are all my relatives or friends.\"\n\n\"All for the best. Let us not go further. We are here at the place where\nthe roads part; let your horses breathe awhile, and I will tell you a\nplan I have formed, and which, I think, will please you.\"\n\nThe small party halted. The horsemen dismounted, and lay down on the\ngrass. As all knew of the conspiracy formed by Don Miguel, and were his\naccomplices in different degrees, this halt did not surprise them, for\nthey suspected that the moment for action was not far off, and that\ntheir chief doubtless wished to take his final measures before throwing\noff the mask, and proclaiming the independence of New Mexico. On\ninviting them to hunt the wild horses, Don Miguel had not concealed from\nthem Red Cedar's treachery, and the necessity in which he found himself\nof dealing a great blow, if he did not wish all to be hopelessly lost.\n\nValentine led the hacendero and the general a short distance apart.\nWhen they were out of ear-shot the hunter carefully examined the\nneighbourhood; then within a few minutes rejoined his friends, whom his\nway of acting considerably perplexed.\n\n\"Caballeros,\" he said to them, \"what do you intend doing? In our\nposition minutes are ages. Are you ready to make your pronunciamento?\"\n\n\"Yes,\" they answered.\n\n\"This is what I propose. You, Don Miguel, will proceed direct on Paso.\nAt about half a league from that town you will find Curumilla, with\ntwenty of the best rifles on the frontier. These men, in whom you can\ntrust, are Canadian and Indian hunters devoted to me. They will form the\nnucleus of a band sufficient for you to seize on Paso without striking a\nblow, as it is only defended by a garrison of forty soldiers. Does that\nplan suit you?\"\n\n\"Yes; I will set about it at once. But my daughter?\"\n\n\"I will take charge of her. You will also leave me your son, and I will\nconvey them both to the hacienda. As for the other ladies, on reaching\nthe town, they will merely go to their homes, which I fancy, presents no\ndifficulty.\"\n\n\"None.\"\n\n\"Good! Then that is settled?\"\n\n\"Perfectly.\"\n\n\"As for you, general, your men have been echelonned by my care in\nparties of ten and twenty along the Santa Fe road, up to two leagues of\nthe city, so that you will only have to pick them up. In this way you\nwill find yourself, within three hours, at the head of five hundred\nresolute and well-armed men.\"\n\n\"Why, Valentine, my friend,\" the general said laughingly, \"do you know\nthere is the stuff in you to make a partisan chief, and that I am almost\njealous of you.\"\n\n\"Oh! that would be wrong, general: I assure you I am most disinterested\nin the affair.\"\n\n\"Well, my friend, I know it: you are a free desert hunter, caring very\nlittle for our paltry schemes.\"\n\n\"That is true; but I have vowed to Don Miguel and his family a\nfriendship which will terminate with my life. I tremble for him and his\nchildren when I think of the numberless dangers that surround him, and I\ntry to aid him as far as my experience and activity permit me. That is\nthe secret of my conduct.\"\n\n\"This profession of faith was at least useless, my friend. I have known\nyou too intimately and too long to doubt your intentions. Hence, you\nsee, I place such confidence in you, that I accept your ideas without\ndiscussion, so convinced am I of the purity of your intentions.\"\n\n\"Thanks, Don Miguel; you have judged me correctly. Come, gentlemen, to\nhorse, and start. We must separate here--you, Don Miguel, to proceed by\nthe right-hand track to Paso; you, general, by the left hand one to\nSanta Fe; while I, with Don Pablo and his sister proceed straight on\ntill we reach the Hacienda de la Noria.\"\n\n\"To horse, then!\" the hacendero shouted resolutely; \"And may God defend\nthe right!\"\n\n\"Yes,\" the general added; \"for from this moment the revolution is\ncommenced.\"\n\nThe three men returned to their friends. Don Miguel said a few words to\nhis children, and in an instant the whole party were in the saddle.\n\n\"The die is cast!\" Valentine exclaimed. \"May Heaven keep you,\ngentlemen!\"\n\n\"Forward!\" Don Miguel commanded.\n\n\"Forward!\" General Ibanez shouted, as he rushed in the opposite\ndirection.\n\nValentine looked after his departing friends. Their black outlines were\nsoon blended with the darkness, and then the footfalls of their horses\ndied out in the night. Valentine gave a sigh and raised his head.\n\n\"God will protect them,\" he murmured; then turning to the two young\npeople, \"Come on, children,\" he said.\n\nThey started, and for some minutes kept silence. Valentine was too busy\nin thought to address his companions; and yet Dona Clara and Don Pablo,\nwhose curiosity was excited to the highest pitch, were burning to\nquestion him. At length the girl, by whose side the hunter marched with\nthat quick step which easily keeps up with a horse, bent down to him.\n\n\"My friend,\" she said to him in her soft voice, \"what is taking place?\nWhy has my father left us, instead of coming to his house?\"\n\n\"Yes,\" Don Pablo added, \"he seemed agitated when he parted from us. His\nvoice was stern, his words sharp. What is happening, my friend? Why did\nnot my father consent to my accompanying him?\"\n\nValentine hesitated to answer.\n\n\"I implore you, my friend,\" Dona Clara continued, \"do not leave us in\nthis mortal anxiety. The announcement of a misfortune would certainly\ncause us less pain than the perplexity in which we are.\"\n\n\"Why force me to speak, my children?\" the hunter answered in a saddened\nvoice. \"The secret you ask of me is not mine. If your father did not\nimpart his plans to you, it was doubtless because weighty reasons oppose\nit. Do not force me to render you more sorrowful by telling you things\nyou ought not to know.\"\n\n\"But I am not a child,\" Don Pablo exclaimed. \"It seems tome that my\nfather ought not to have thus held his confidence from me.\"\n\n\"Do not accuse your father, my friend,\" Valentine answered gravely:\n\"probably he could not have acted otherwise.\"\n\n\"Valentine, Valentine! I will not accept those poor reasons,\" the young\nman urged. \"In the name of our friendship I insist on your explaining\nyourself.\"\n\n\"Silence!\" the hunter suddenly interrupted him. \"I hear suspicious\nsounds around us.\"\n\nThe three travellers stopped and listened, but all was quiet. The\nhacienda was about five hundred yards at the most from the spot where\nthey halted. Don Pablo and Dona Clara heard nothing, but Valentine made\nthem a sign to remain quiet; then he dismounted and placed his ear to\nthe ground.\n\n\"Follow me,\" he said. \"Something is happening here which I cannot make\nout; but it alarms me.\"\n\nThe young people obeyed without hesitation; but they had only gone a few\npaces when Valentine stopped again.\n\n\"Are your weapons loaded?\" he sharply asked Don Pablo.\n\n\"Yes.\"\n\n\"Good! Perhaps you will have to make use of them.\"\n\nAll at once the gallop of a horse urged to its utmost speed was audible.\n\n\"Attention!\" Valentine muttered.\n\nStill the horseman, whoever he might be, rapidly advanced in the\ndirection of the travellers, and soon came up to them. Suddenly\nValentine bounded like a panther, seized the horse by the bridle and\nstopped it dead.\n\n\"Who are you, and where are you going?\" he shouted, as he put a pistol\nbarrel against the stranger's chest.\n\n\"Heaven be praised!\" the latter said, not replying to the question.\n\"Perhaps I shall be able to save you. Fly, fly, in all haste!\"\n\n\"Father Seraphin!\" Valentine said with stupor, as he lowered his pistol.\n\"What has happened?\"\n\n\"Fly, fly!\" the missionary repeated, who seemed a prey to the most\nprofound terror.\n\n\n\n\nCHAPTER XXIII.\n\nTHE ABDUCTION.\n\n\nRed Cedar and Fray Ambrosio had not remained inactive since their last\ninterview up to the day when Don Miguel set out to hunt the wild horses.\nThese two fellows, so suited to understand each other, had manoeuvred\nwith extreme skill. Fray Ambrosio, all whose avaricious instincts had\nbeen aroused since he had so artfully stolen from poor Joaquin the\nsecret of his placer, had assembled a formidable collection of the\nbandits who always swarm on the Indian frontiers. In a few days he found\nhimself at the head of one hundred and twenty adventurers, all men who\nhad cheated the gallows, and of whom he felt the more sure as the secret\nof the expedition was concealed from them, and they fancied they formed\na war party engaged to go scalp hunting.\n\nThese men, who all knew Red Cedar by reputation, burnt to set out, so\nconvinced were they of carrying out a successful expedition under such a\nleader. Only two men formed an exception to this band of scoundrels, the\nsmallest culprit of whom had at least three or four murders on his\nconscience. They were Harry, and Dick, who, for reasons the reader has\ndoubtless guessed, found themselves, to their great regret, mixed up\nwith these bandits. Still we must say, in justice to Fray Ambrosio's\nsoldiers, that they were all bold hunters, accustomed for many a year to\ndesert life, who knew all its perils, and feared none of its dangers.\n\nFray Ambrosio; apprehending the effects of mezcal and pulque on his men,\nhad made them bivouac at the entrance of the desert, at a sufficiently\ngreat distance from the Paso del Norte to prevent them easily going\nthere. The adventurers spent their time joyously in playing, not for\nmoney, as they had none, but for the scalps they intended presently to\nlift from the Indians, each of which represented a very decent sum.\nStill Fray Ambrosio, so soon as his expedition was completely organised,\nhad only one desire--to start as speedily as possible; but for two days\nRed Cedar was not to be found. At length Fray Ambrosio succeeded in\ncatching him just as he was entering his jacal.\n\n\"What has become of you?\" he asked him.\n\n\"What does that concern you?\" the squatter answered brutally. \"Have I to\nanswer for my conduct to you?\"\n\n\"I do not say so: still, connected as we are at this moment, it would be\nas well for me to know where to find you when I want you.\"\n\n\"I have been attending to my business, as you have to yours.\"\n\n\"Well, are you satisfied?\"\n\n\"Very much so,\" he answered with a sinister smile. \"You will soon learn\nthe result of my journey.\"\n\n\"All the better. If you are satisfied, I am so too.\"\n\n\"Ah, ah!\"\n\n\"Yes, all is ready for departure.\"\n\n\"Let us be off--tomorrow if you like.\"\n\n\"On this very night.\"\n\n\"Very good. You are like me, and don't care to travel by day on account\nof the heat of the sun.\"\n\nThe two accomplices smiled at this delicate jest.\n\n\"But before starting,\" the squatter continued, becoming serious again,\n\"we have something left to do here.\"\n\n\"What is it?\" Fray Ambrosio asked with candor.\n\n\"It is wonderful what a short memory you have. Take care: that failing\nmay play an awkward trick some day.\"\n\n\"Thanks! I will try to correct it.\"\n\n\"Yes, and the sooner the better: in the meanwhile I will refresh your\nmemory.\"\n\n\"I shall feel obliged to you.\"\n\n\"And Dona Clara, do you fancy we are going to leave her behind?\"\n\n\"Hum! Then you still think of that?\"\n\n\"By Jove! More than ever.\"\n\n\"The fact is it will not be easy to carry her off at this moment.\"\n\n\"Why not?\"\n\n\"In the first place, she is not at the hacienda.\"\n\n\"That is certainly a reason.\"\n\n\"Is it not?\"\n\n\"Yes; but she must be somewhere, I suppose?\" the squatter said with a\ncoarse laugh.\n\n\"She has gone with her father to a hunt of wild horses.\"\n\n\"The hunt is over and they are on their return.\"\n\n\"You are well informed.\"\n\n\"It is my trade. Come, do you still mean serving me?\"\n\n\"I must.\"\n\n\"That is how I like you. There cannot be many people at the hacienda?\"\n\n\"A dozen at the most.\"\n\n\"Better still. Listen to me: it is now four in the afternoon. I have a\nride to take. Return to the hacienda, and I will come there this evening\nat nine, with twenty resolute men. You will open the little gate of the\ncorral, and leave me to act. I'll answer for all.\"\n\n\"If you wish it it must be so,\" Fray Ambrosio said with a sigh.\n\n\"Are you going to begin again?\" the squatter asked in a meaning voice as\nhe rose.\n\n\"No, no, it is unnecessary,\" the monk exclaimed. \"I shall expect you.\"\n\n\"Good: till this evening.\"\n\n\"Very well.\"\n\nOn which the two accomplices separated. All happened as had been\narranged between them. At nine o'clock Red Cedar reached the little\ngate, which was opened for him by Fray Ambrosio, and the squatter\nentered the hacienda at the head of his three sons and a party of\nbandits. The peons, surprised in their sleep, were bound before they\neven knew what was taking place.\n\n\"Now,\" Red Cedar said, \"we are masters of the place, the girl can come\nas soon as she likes.\"\n\n\"Eh?\" the monk went on. \"All is not finished yet. Don Miguel is a\nresolute man, and is well accompanied: he will not let his daughter be\ncarried off under his eyes without defending her.\"\n\n\"Don Miguel will not come,\" the squatter said with a sardonic grin.\n\n\"How do you know?\"\n\n\"That is not your business.\"\n\n\"We shall see.\"\n\nBut the bandits had forgotten Father Seraphin. The missionary, aroused\nby the unusual noise he heard in the hacienda, had hastily risen. He had\nheard the few words exchanged between the accomplices, and they were\nsufficient to make him guess the fearful treachery they meditated. Only\nlistening to his heart, the missionary glided out into the corral,\nsaddled a horse, and opening a door, of which he had a key, so that he\ncould enter or leave the hacienda as his duties required, he started at\nfull speed in the direction which he supposed the hunters must follow in\nreturning to the hacienda. Unfortunately, Father Seraphin had been\nunable to effect his flight unheard by the squatter's practised ear.\n\n\"Malediction!\" Red Cedar shouted, as he rushed, rifle in hand, toward a\nwindow, which he dashed out with his fist, \"We are betrayed.\"\n\nThe bandits rushed in disorder into the corral where their horses were\ntied up, and leaped into their saddles. At this moment a shadow flitted\nacross the plain in front of the squatter, who rapidly shouldered his\nrifle and fired. Then he went out: a stifled cry reached his ear, but\nthe person the bandit had fired at still went on.\n\n\"No matter,\" the squatter muttered; \"that fine bird has lead in its\nwing. Sharp, sharp, my men, on the trail!\"\n\nAnd all the bandits rushed off in pursuit of the fugitive.\n\nFather Seraphin had fallen in a fainting condition at Valentine's feet.\n\n\"Good heavens!\" the hunter exclaimed in despair, \"what can have\nhappened?\"\n\nAnd he gently carried the missionary into a ditch that ran by the side\nof the road. Father Seraphin had his shoulder fractured, and the blood\npoured in a stream from the wound. The hunter looked around him; but at\nthis moment a confused sound could be heard like the rolling of distant\nthunder.\n\n\"We must fall like brave men, Don Pablo, that is all,\" he said sharply.\n\n\"Be at your ease,\" the young man answered coldly.\n\nDona Clara was pale and trembling.\n\n\"Come,\" Valentine said.\n\nAnd, with a movement rapid as thought, he bounded on to the missionary's\nhorse. The three fugitives started at full speed. The flight lasted a\nquarter of an hour, and then Valentine stopped. He dismounted, gave the\nyoung people a signal to wait, lay down on the ground, and began\ncrawling on his hands and knees, gliding like a serpent through the long\ngrass that concealed him, and stopping at intervals to look around him,\nand listen attentively to the sounds of the desert. Suddenly he rushed\ntowards his companions, seized the horses by the bridle, and dragged\nthem behind a mound, where they remained concealed, breathless and\nunable to speak.\n\nA formidable noise of horses was audible. Some twenty black shadows\npassed like a tornado within ten paces of their hiding place, not seeing\nthem in consequence of the darkness.\n\nValentine drew a deep breath.\n\n\"All hope is not lost,\" he muttered.\n\nHe waited anxiously for five minutes: their pursuers were going further\naway. Presently the sound of their horses' hoofs ceased to disturb the\nsilence of the night.\n\n\"To horse!\" Valentine said.\n\nThey leaped into their saddles and started again, not in the direction\nof the hacienda, but in that of the Paso.\n\n\"Loosen your bridles,\" the hunter said: \"more still--we are not moving.\"\n\nSuddenly a loud neigh was borne on the breeze to the ears of the\nfugitives.\n\n\"We are lost!\" Valentine muttered. \"They have found our trail.\"\n\nRed Cedar was too old a hand on the prairie to be long thrown out: he\nsoon perceived that he was mistaken, and was now turning back, quite\ncertain this time of holding the trail. Then began one of those fabulous\nraces which only the dwellers on the prairie can witness--races which\nintoxicate and cause a giddiness, and which no obstacle is powerful\nenough to stop or check, for the object is success or death. The\nbandits' half wild horses, apparently identifying themselves with the\nferocious passions of their riders, glided through the night with the\nrapidity of the phantom steed in the German ballad, bounded over\nprecipices, and rushed with prodigious speed.\n\nAt times a horseman rolled with his steed from the top of a rock, and\nfell into an abyss, uttering a yell of distress; but his comrades passed\nover his body, borne along like a whirlwind, and responding to this cry\nof agony, the final appeal of a brother, by a formidable howl of rage.\nThis pursuit had already lasted two hours, and the fugitives had not\nlost an inch of ground: their horses, white with foam, uttered hoarse\ncries of fatigue and exhaustion as a dense smoke came out of their\nnostrils. Dona Clara, with her hair untied and floating in the breeze,\nwith sparkling eye and closely pressed lips, constantly urged her horse\non with voice and hand.\n\n\"All is over!\" the hunter suddenly said. \"Save yourselves! I will let\nmyself be killed here, so that you may go on for ten minutes longer, and\nbe saved. I will hold out for that time, so go on.\"\n\n\"No,\" Don Pablo answered nobly; \"we will be all saved or perish\ntogether.\"\n\n\"Yes,\" the maiden remarked.\n\nValentine shrugged his shoulders.\n\n\"You are mad,\" he said.\n\nAll at once he started, for their pursuers were rapidly approaching.\n\n\"Listen,\" he said. \"Do you two let yourselves be captured; they will not\nfollow me, as they owe me no grudge. I swear to you that if I remain at\nliberty I will deliver you, even if they hide you in the bowels of the\nearth.\"\n\nWithout replying Don Pablo dismounted, and Valentine leaped on to his\nhorse.\n\n\"Hope for the best!\" he shouted hoarsely, and disappeared.\n\nDon Pablo, so soon as he was alone with his sister, made her dismount,\nseated her at the foot of a tree, and stood before her with a pistol in\neither hand. He had not to wait long, for almost immediately he was\nsurrounded by the bandits.\n\n\"Surrender!\" Red Cedar shouted in a panting voice.\n\nDon Pablo smiled disdainfully.\n\n\"Here is my answer,\" he said.\n\nAnd with two pistol shots he laid two bandits low; then he threw away\nhis useless weapons, and crossing his arms on his breast said,--\n\n\"Do what you please now; I am avenged.\"\n\nRed Cedar bounded with fury.\n\n\"Kill that dog!\" he shouted.\n\nShaw rushed toward the young man, threw his nervous arms around him, and\nwhispered in his ear,--\n\n\"Do not resist, but fall as if dead.\"\n\nDon Pablo mechanically followed his advice.\n\n\"It is all over,\" said Shaw. \"Poor devil! He did not cling to life.\"\n\nHe returned his knife to his belt, threw the supposed corpse on his\nshoulders, and dragged it into a ditch. At the sight of her brother's\nbody, whom she supposed to be dead, Dona Clara uttered a shriek of\ndespair and fainted. Red Cedar laid the maiden across his saddle-bow,\nand the whole band, starting at a gallop was soon lost in the darkness.\nDon Pablo then rose slowly, and took a sorrowful glance around.\n\n\"My poor sister!\" he murmured.\n\nThen he perceived her horse near him.\n\n\"Valentine alone can save her,\" he said.\n\nHe mounted the horse, and proceeded toward the Paso, asking himself this\nquestion, which he found it impossible to answer:--\n\n\"But why did not that man kill me?\"\n\nA few paces from the village he perceived two men halting on the road,\nand conversing with the greatest animation. They hurriedly advanced\ntoward him, and the young man uttered a cry of surprise on recognising\nthem. They were Valentine and Curumilla.\n\n\n\n\nCHAPTER XXIV.\n\nTHE REVOLT.\n\n\nDon Miguel Zarate had marched rapidly on the Paso, and an hour after\nleaving Valentine he saw flashing in the distance the lights that shone\nin the village windows. The greatest calmness prevailed in the vicinity;\nonly at times could be heard the barking of the dogs baying at the moon,\nor the savage miawling of the wild cats hidden in the shrubs. At about\none hundred yards from the village a man suddenly rose before the small\nparty.\n\n\"Who goes there?\" he shouted.\n\n\"_Mejico e independencia!_\" the hacendero answered.\n\n\"_?Que gente?_\" the stranger continued.\n\n\"Don Miguel Zarate.\"\n\nAt these words twenty men hidden in the brushwood rose suddenly, and\nthrowing their rifles on their shoulders, advanced to meet the horsemen.\nThey were the hunters commanded by Curumilla, who, by Valentine's\norders, were awaiting the hacendero's arrival to join him.\n\n\"Well,\" Don Miguel asked the chief, \"is there anything new?\"\n\nCurumilla shook his head.\n\n\"Then we can advance?\"\n\n\"Yes.\"\n\n\"What is the matter, chief? Have you seen anything alarming?\"\n\n\"No; and yet I have a feeling of treachery.\"\n\n\"How so?\"\n\n\"I cannot tell you. Apparently everything is as usual: still there is\nsomething which is not so. Look you, it is scarce ten o'clock: generally\nat that hour all the mesones are full, the ventas are crammed with\ngamblers and drinkers, the streets flocked with promenaders. This night\nthere is nothing of the sort: all is closed--the town seems abandoned.\nThis tranquillity is factitious. I am alarmed, for _I hear the silence_.\nTake care.\"\n\nDon Miguel was involuntarily struck by the chief's remarks. He had known\nCurumilla for a long time. He had often seen him display in the most\ndangerous circumstances a coolness and contempt for death beyond all\npraise: hence some importance must be attached to the apprehensions and\nanxiety of such a man. The hacendero ordered his party to halt,\nassembled his friends, and held a council. All were of opinion that,\nbefore venturing to advance further, they should send as scout a clever\nman to traverse the town, and see for himself if the fears of the Indian\nchief were well founded.\n\nOne of the hunters offered himself. The conspirators concealed\nthemselves on either side the road, and awaited, lying in the shrubs,\nthe return of their messenger. He was a half-breed, Simon Munez by\nname, to whom the Indians had given the soubriquet of \"Dog-face,\" owing\nto his extraordinary likeness to that animal. This name had stuck to the\nhunter, who, _nolens volens_, had been compelled to accept it. He was\nshort and clumsy, but endowed with marvellous strength; and we may say\nat once that he was an emissary of Red Cedar, and had only joined the\nhunters in order to betray them.\n\nWhen he left the conspirators he proceeded toward the village whistling.\nHe had scarce taken a dozen steps into the first street ere a door\nopened, and a man appeared. This man stepped forward and addressed the\nhunter.\n\n\"You whistle very late, my friend.\"\n\n\"A whistle to wake those who are asleep,\" the half breed made answer.\n\n\"Come in,\" the man continued.\n\nDog-face went in, and the door closed upon him. He remained in the house\nhalf an hour, then went out, and hurried back along the road he had\ntraversed.\n\nRed Cedar, who wished before all to avenge himself on Don Miguel Zarate,\nhad discovered, through Fray Ambrosio, the conspirators' new plan.\nWithout loss of time he had taken his measures in consequence, and had\nmanaged so well that, although the general, the governor, and the\ncriminal judge were prisoners, Don Miguel must succumb in the contest he\nwas preparing to provoke. Fray Ambrosio, to his other qualities, joined\nthat of being a listener at doors. In spite of the distrust which his\npatron was beginning to display toward him on Valentine's\nrecommendation, he had surprised a conversation between Don Miguel and\nGeneral Ibanez. This conversation, immediately reported to Red Cedar,\nwho, according to his usual custom, had appeared to attach no importance\nto it, had been sufficient, however, to make the squatter prepare his\nbatteries and countermine the conspiracy.\n\nDog-face rejoined his companions after an hour's absence.\n\n\"Well?\" Don Miguel asked him.\n\n\"All is quiet,\" the half-breed answered; \"the inhabitants have retired\nto their houses, and everybody is asleep.\"\n\n\"You noticed nothing of a suspicious nature?\"\n\n\"I went through the town from one end to the other, and saw nothing.\"\n\n\"We can advance, then?\"\n\n\"In all security: it will only be a promenade.\"\n\nOn this assurance the conspirators regained their courage, Curumilla was\ntreated as a visionary, and the order was given to advance. Still\nDog-face's report, far from dissipating the Indian chief's doubts, had\nproduced the contrary effect, and considerably augmented them. Saying\nnothing, he placed himself by the hunter's side, with the secret\nintention of watching him closely.\n\nThe plan of the conspirators was very simple. They would march directly\non the Cabildo (Town hall), seize it, and proclaim a Provisional\nGovernment. Under present circumstances nothing appeared to be easier.\nDon Miguel and his band entered the Paso, and nothing occurred to arouse\ntheir suspicions. It resembled that town in the \"Arabian Nights,\" in\nwhich all the inhabitants, struck by the wand of the wicked enchanter,\nsleep an eternal sleep. The conspirators advanced into the town with\ntheir rifle barrels thrust forward, with eye and ear on the watch, and\nready to fire at the slightest alarm; but nothing stirred. As Curumilla\nhad observed, the town was too quiet. This tranquillity hid something\nextraordinary, and must conceal the tempest. In spite of himself Don\nMiguel felt a secret apprehension which he could not master.\n\nTo our European eyes Don Miguel will perhaps appear a poor conspirator,\nwithout foresight or any great connection in his ideas. From our point\nof view that is possible; but in a country like Mexico, which counts its\nrevolutions by hundreds, and where _pronunciamentos_ take place, in most\ncases, without sense or reason because a colonel wishes to become a\ngeneral, or a lieutenant a captain, things are not regarded so closely;\nand the hacendero, on the contrary, had evidenced tact, prudence, and\ntalent in carrying out a conspiracy which, during the several years it\nhad been preparing, had only come across one traitor. And now it was too\nlate to turn back: the alarm had been given, and the Government was on\nits guard. They must go onwards, even if they succumbed in the struggle.\n\nAll these considerations had been fully weighed by Don Miguel; and he\nhad not given the signal till he was driven into his last intrenchments,\nand convinced that there was no way of escape left him. Was it not a\nthousand fold better to die bravely with arms in their hands, in support\nof a just cause, than wait to be arrested without having made an attempt\nto succeed? Don Miguel had sacrificed his life, and no more could be\nexpected of him.\n\nIn the meanwhile the conspirators advanced. They had nearly reached the\nheart of the town; they were at this moment in a little, dirty, and\nnarrow street, called the Calle de San Isidro, which opens out on the\nPlaza Mayor, when suddenly a dazzling light illumined the darkness;\ntorches flashed from all the windows; and Don Miguel saw that the two\nends of the street in which he was were guarded by strong detachments of\ncavalry.\n\n\"Treachery!\" the conspirators shouted in terror.\n\nCurumilla bounded on Dog-face, and buried his knife between his\nshoulders. The half-breed fell in a lump, quite dead, and not uttering a\ncry. Don Miguel judged the position at the first glance: he saw that he\nand his party were lost.\n\n\"Let us die!\" he said.\n\n\"We will!\" the conspirators resolutely responded.\n\nCurumilla with the butt of his rifle beat in the door of the nearest\nhouse, and rushed in, the conspirators following him. They were soon\nintrenched on the roof. In Mexico all the houses have flat roofs, formed\nlike terraces. Thanks to the Indian chief's idea, the rebels found\nthemselves in possession of an improvised fortress, where they could\ndefend themselves for a long time, and sell their lives dearly.\n\nThe troops advanced from each end of the street, while the roofs of all\nthe houses were occupied by soldiers. The battle was about to begin\nbetween earth and heaven, and promised to be terrible. At this moment\nGeneral Guerrero, who commanded the troops, bade them halt, and advanced\nalone to the house on the top of which the conspirators were intrenched.\nDon Miguel beat up the guns of his comrades, who aimed at the officer.\n\n\"Wait,\" he said to them; and, addressing the general, \"What do you\nwant?\" he shouted.\n\n\"To offer you propositions.\"\n\n\"Speak.\"\n\nThe general came a few paces nearer, so that those he addressed could\nnot miss one of his words.\n\n\"I offer you life and liberty if you consent to surrender your leader,\"\nhe said.\n\n\"Never!\" the conspirators shouted in one voice.\n\n\"It is my place to answer,\" Don Miguel said; and then turning to the\ngeneral, \"What assurance do you give me that these conditions will be\nhonourably carried out?\"\n\n\"My word of honor as a soldier,\" the general answered.\n\n\"Very good,\" Don Miguel went on; \"I accept. All the men who accompany me\nwill leave the town one after the other.\"\n\n\"No, we will not!\" the conspirators shouted as they brandished their\nweapons; \"we would sooner die.\"\n\n\"Silence!\" the hacendero said in a loud voice. \"I alone have the right\nto speak here, for I am your chief. The life of brave men like you must\nnot be needlessly sacrificed. Go, I say; I order you--I implore it of\nyou,\" he added with tears in his voice. \"Perhaps you will soon take your\nrevenge.\"\n\nThe conspirators hung their heads mournfully.\n\n\"Well?\" the general asked.\n\n\"My friends, accept. I will remain alone here. If you break your word I\nwill kill myself.\"\n\n\"I repeat that you hold my word,\" the general answered.\n\nThe conspirators came one after the other to embrace Don Miguel, and\nthen went down into the street without being in any way interfered with.\nThings happen thus in this country, where conspiracies and revolutions\nare on the order of the day, as it were. The defeated are spared as far\nas possible, from the simple reason that the victors may find themselves\ntomorrow fighting side by side with them for the same cause. Curumilla\nwas the last to depart.\n\n\"All is not ended yet,\" he said to Don Miguel. \"Koutonepi will save you,\nfather.\"\n\nThe hacendero shook his head sadly.\n\n\"Chief,\" he said in a deeply moved voice, \"I leave my daughter to\nValentine, Father Seraphin, and yourself. Watch over her: the poor child\nwill soon have no father.\"\n\nCurumilla embraced Don Miguel silently and retired; he had soon\ndisappeared in the crowd, the general having honourably kept his word.\n\nDon Miguel threw down his weapons and descended.\n\n\"I am your prisoner,\" he said.\n\nGeneral Guerrero bowed, and made him a sign to mount the horse a soldier\nhad brought up.\n\n\"Where are we going?\" the hacendero said.\n\n\"To Santa Fe,\" the general answered, \"where you will be tried with\nGeneral Ibanez, who will doubtless soon be a prisoner like yourself.\"\n\n\"Oh!\" Don Miguel muttered thoughtfully, \"who betrayed us this time?\"\n\n\"It was still Red Cedar,\" the general answered.\n\nThe hacendero let his head sink on his chest, and remained silent. A\nquarter of an hour later the prisoner left the Paso del Norte, escorted\nby a regiment of dragoons. When the last trooper had disappeared in the\nwindings of the road three men left the shrubs that concealed them, and\nstood like three phantoms in the midst of the desolate plain.\n\n\"O heavens!\" Don Pablo cried in a heart-rending voice, \"my father, my\nsister--who will restore them to me?\"\n\n\"I!\" Valentine said in a grave voice, as he laid his hand on his\nshoulder. \"Am I not the TRAIL-HUNTER?\"\n\n\n\n\nPART II.--EL PRESIDIO DE SANTA FE.\n\n\n\n\nCHAPTER I.\n\nEL RANCHO DEL COYOTE.\n\n\nAbout a month after the events we have described in the first part of\nthis veracious history, two horsemen, well mounted, and carefully\nenwrapped in their cloaks, entered at a smart trot the town of Santa Fe\nbetween three and four o'clock in the afternoon.\n\nSanta Fe, the capital of New Mexico, is a pretty town, built in the\nmidst of a laughing and fertile plain. One of its sides occupies the\nangle formed by a small stream: it is surrounded by the _adobe_ walls of\nthe houses by which it is bordered. The entrance of each street is\nclosed by stakes in the form of palisades; and like the majority of\ntowns in Spanish America, the houses, built only one story high in\nconsequence of the earthquakes, are covered with terraces of well-beaten\nearth, called _azoteas,_ which are a sufficient protection in this\nglorious climate, where the sky is constantly pure.\n\nIn the time of the Castilian rule Santa Fe enjoyed a certain importance,\nowing to its strategic position, which allowed an easy defence against\nthe incursions of the Indians; but since the emancipation of Mexico this\ncity, like all the other centres of population in his unhappy country,\nhas seen its splendour vanish forever, and despite the fertility of its\nsoil and the magnificence of its climate, it has entered into such a\nstate of decadence that the day is at hand when it will be only an\nuninhabited ruin. In a word, this city, which fifty years back contained\nmore than ten thousand inhabitants, has now scarcely three thousand,\neaten up by fevers and the utmost wretchedness.\n\nStill during the last few weeks Santa Fe had appeared to emerge, as if\nby magic from the lethargy into which it is ordinarily plunged; a\ncertain degree of animation prevailed in its usually deserted streets;\nin short, a new life circulated in the veins of this population, to\nwhom, however, all must appear a matter of indifference. The fact was\nthat an event of immense importance had recently taken place in this\ntown. The two leaders of the conspiracy lately attempted had been\ntransferred to safe keeping at Santa Fe.\n\nThe Mexicans, ordinarily so slow when justice has to be dealt, are the\nmost expeditious people in the world when a conspiracy has to be\npunished. Don Miguel and General Ibanez had not pined long in prison. A\ncourt martial, hurriedly convened, had assembled under the presidency of\nthe governor, and the two conspirators were unanimously condemned to be\nshot.\n\nThe hacendero, through his name and his position, and especially on\naccount of his fortune, had numerous partisans in the province: hence\nthe announcement of the verdict had caused a profound stupor, which\nalmost immediately changed into anger, among the rich land owners and\nthe Indians of New Mexico. A dull agitation prevailed throughout the\ncountry; and the governor, who felt too weak to hold head against the\nstorm that threatened him, and regretted that he had carried matters so\nfar, was temporising, and trying to evade the peril of his position\nuntil a regiment of dragoons he had asked of the Government arrived, and\ngave strength to the law. The condemned men, whom the governor had not\nyet dared to place in _capilla_, were still provisionally detained in\nthe prison.\n\nThe two men of whom we have spoken, rode without stopping through the\nstreets of the town, deserted at this hour, when everybody is at home\nenjoying his siesta, and proceeded toward an unpretending rancho, built\non the banks of the stream, at the opposite end of the town from that by\nwhich they entered.\n\n\"Well,\" one of the horsemen said, addressing his comrade, \"was I not\nright? You see everyone is asleep: there is nobody to watch us. We have\narrived at a capital moment.\"\n\n\"Bah!\" the other answered in a rough voice, \"Do you believe that? In\ntowns there is always somebody watching to see what does not concern\nhim, and report it after his fashion.\"\n\n\"That is possible,\" the first said, shrugging his shoulders\ndisdainfully. \"I care about it as little as I do for a stringhalt\nhorse.\"\n\n\"And I, too,\" the other said sharply. \"Do you imagine that I care more\nthan you do for the gossips? But stay; I fancy we have reached the\nrancho of Andres Garote. This must be the filthy tenement, unless I am\nmistaken.\"\n\n\"It is the house. I only hope the scamp has not forgotten, the meeting I\ngave him. Wait a minute, senor padre; I will give the agreed-on signal.\"\n\n\"It is not worth while, Red Cedar. You know that I am always at your\nexcellency's orders when you may please to give them,\" a mocking voice\nsaid from inside the rancho, the door of which immediately opened to\ngive admission to the newcomers, and allowed a glimpse of the tall\nfigure and intelligent face of Andres Garote himself.\n\n\"_Ave Maria purisima!_\" the travellers said, as they dismounted and\nentered the rancho.\n\n\"_Sin pecado concebida!_\" Andres replied, as he took the bridles of the\nhorses and led them to the corral, where he unsaddled them and gave each\na truss of alfalfa.\n\nThe travellers, fatigued by a long journey, sat down on butacas arranged\nagainst the wall, and awaited the host's return, while wiping their dank\nforeheads and twisting a maize cigarette between their fingers. The room\nin which they were had nothing extremely attractive about it. It was a\nlarge chamber with two windows, protected by iron bars, the greasy panes\nallowing but a doubtful light to pass. The naked and smoky walls were\ncovered with clumsily-painted pictures, representing various holy\nobjects. The furniture only consisted of three or four halting tables,\nthe same number of benches, and a few butacas, the torn and harsh\nleather of which evinced lengthened use. As for the floor, it was merely\nof beaten earth, but rendered uneven by the mud incessantly brought in\nupon the feet of visitors. A door carefully closed led to an inner room,\nin which the ranchero slept. Another door was opposite to it, and\nthrough this Andres speedily entered after giving the horses their\nprovender.\n\n\"I did not expect you yet,\" he said as he entered; \"but you are welcome.\nIs there anything new?\"\n\n\"My faith, I know nothing but the affair that brings us. It is rather\nserious, I fancy, and prevents us attending to anything else,\" Red Cedar\nremarked.\n\n\"_Caspita_! what vivacity, compadre!\" Andres exclaimed. \"But, before\ntalking, I hope you will take some refreshment at any rate. There is\nnothing like a cup of mezcal or pulque to clear the brain.\"\n\n\"Not to forget,\" Fray Ambrosio said, \"that it is infernally hot, and my\ntongue is glued to my palate, as I have swallowed so much dust.\"\n\n\"_Cuerpo de Dios_!\" Andres said as he went to look for a bottle among\nseveral others arranged on a sort of bar, and placed it before the\ntravellers. \"Pay attention to that, senor padre; for it is serious, and\nyou run a risk of death, _caray!_\"\n\n\"Give me the remedy, then, chatterer,\" the monk replied as he held out\nhis glass.\n\nThe mezcal, liberally poured out, was swallowed at a draught by the\nthree men, who put back their glasses on the table with a \"hum\" of\nsatisfaction, and that clinking of the tongue peculiar to topers when\nthey are swallowing anything that tickles the throat.\n\n\"And now suppose we talk seriously,\" Red Cedar said.\n\n\"At your orders, senores caballeros,\" Andres replied. \"Still, if you\nprefer a hand at monte, you know that I have cards at your service.\"\n\n\"Presently, senor Andres, presently. Everything will have its turn. Let\nus first settle our little business,\" Fray Ambrosio judiciously\nobserved.\n\nAndres Garote bowed his head in resignation, while thrusting back into\nhis pocket the pack of cards he had already half drawn out. The three\nmen made themselves as comfortable as they could, and Red Cedar, after\ncasting a suspicious glance around him, at length took the word.\n\n\"You know, caballeros,\" he said, \"how, when we thought we had nothing to\ndo but proceed straight to Apacheria, the sudden desertion of nearly all\nour gambusinos checked us. The position was most critical for us, and\nthe abduction of Dona Clara compelled us to take the utmost\nprecautions.\"\n\n\"That is true,\" Andres Garote observed with an air of conviction.\n\n\"Although certain influential persons protect us under the rose,\" Red\nCedar continued, \"we are compelled to keep in the shade as far as we\ncan. I therefore sought to remedy the gravest points in the business. In\nthe first place, the girl was hidden in an inaccessible retreat, and\nthen I began looking for comrades to take the place of those who\nabandoned us so suddenly.\"\n\n\"Well?\" the two men interrupted him sharply.\n\n\"At this moment,\" Red Cedar calmly continued, \"when the placers of\nCalifornia call away all the men belonging to the profession, it was\ncertainly no easy task to collect one hundred men of the sort we want,\nthe more so as we shall have to fight the Indios Bravos in our\nexpedition. I did not care to enlist novices, who at the sight of the\nfirst Apache or Comanche savages, would bolt in terror, and leave us in\nthe lurch on the prairies. What I wanted were resolute men, whom no\nfatigue would disgust, and who, once attached to our enterprise, would\nfollow it out to the end. I have, therefore, during the past month, been\nrunning about to all the frontier presidios; and the devil has come to\nmy help tolerably well, for the evil is now repaired, and the band\ncomplete.\"\n\n\"I hope, Red Cedar,\" Fray Ambrosio asked, \"that you have not spoken\nabout the placer to your men?\"\n\n\"Do you take me for a fool! No, padre,\" the squatter answered sharply,\n\"no, no. A hundred thousand reasons urge us to be prudent, and keep the\nexpedition secret. In the first place, I do not wish to make the fortune\nof the Government while making our own. An indiscretion would ruin us\nnow, when the whole world only dreams of mines and placers, and Europe\nsends us a mob of lean and starving vagabonds, greedy to grow fat at our\nexpense.\"\n\n\"Famously reasoned,\" said Andres.\n\n\"No, no, trust to me. I have assembled the finest collection of picaros\never brought together for an expedition, all food for the gallows,\nruined by monte, who do not care for hard blows, and on whom I can fully\ncount, while being very careful not to drop a word that can enlighten\nthem as to the spot whither we propose leading them; for, in that case,\nI know as well as you do that they would abandon us without the\nslightest scruples, or, as is even more probable, assassinate us to gain\npossession of the immense treasures we covet.\"\n\n\"Nothing can be more just,\" Fray Ambrosio answered. \"I am quite of your\nopinion, Red Cedar. Now what have you resolved on?\"\n\n\"We have not an instant to lose,\" the squatter continued. \"This very\nevening, or tomorrow at the latest, we must set out. Who knows whether\nwe have not already delayed our start too long? Perhaps one of those\nEuropean vagabonds may have discovered our placer, for those scoundrels\nhave a peculiar scent for gold.\"\n\nFray Ambrosio cast a suspicious glance at his partner.\n\n\"Hum!\" he muttered, \"that would be very unlucky, for hitherto the\nbusiness has been well managed.\"\n\n\"For that reason,\" Red Cedar hastened to add, \"I only suggest a doubt\n--nothing more.\"\n\n\"Come, Red Cedar,\" the monk said, \"you have yourself narrated all the\nembarrassments of our position, and the countless difficulties we shall\nhave to surmount before reaching our object. Why, then, complicate the\ngravity of our situation still more, and create fresh enemies\nneedlessly?\"\n\n\"I do not understand you, senor padre. Be good enough to explain\nyourself more clearly.\"\n\n\"I allude to the young girl you carried off.\"\n\n\"Ah, ah!\" Red Cedar said with a grin, \"Is that where the shoe pinches\nyou, comrade? I am vexed at it; but I will not answer your question. If\nI carried off that woman, it was because I had pressing reasons to do\nso. These reasons still exist; that is all I can tell you. All the\nbetter if these explanations are sufficient for you; if not, you must\nput up with them, for you will get no others.\"\n\n\"Still it appears to me that, regarding the terms on which we stand to\neach other--\"\n\n\"What can there be in common between the abduction of Dona Clara and the\ndiscovery of a placer in the heart of Apacheria? Come, you are mad, Fray\nAmbrosio; the mezcal is getting to your head.\"\n\n\"Still--\" the monk insisted.\n\n\"Enough of that!\" Red Cedar shouted as he roughly smote the table with\nhis clenched fist. \"I will not hear another word on the subject.\"\n\nAt this moment two smart blows were heard on the carefully-bolted door.\n\nThe three men started, and Red Cedar broke off.\n\n\"Shall I open?\" Andres asked.\n\n\"Yes,\" Fray Ambrosio answered: \"hesitation or refusal might give an\nalarm. We must foresee everything.\"\n\nRed Cedar consented with a toss of his head, and the ranchero went with\nan ill grace toward the door, which was being struck as if about to be\nbeaten in.\n\n\n\n\nCHAPTER II.\n\nTHE CUCHILLADA.\n\n\nSo soon as the door was opened two men appeared on the threshold. The\nfirst was Curumilla; the other, wrapped up in a large cloak, and with\nhis broad-brimmed hat drawn over his eyes, entered the room, making the\nIndian chief a sign to follow him. The latter was evidently a Mexican.\n\n\"_Santas tardes!_\" he said as he raised his hand to his hat, but not\nremoving it.\n\n\"_Dios las de a usted buenas!_\" the ranchero answered. \"What shall I\nserve to your excellencies?\"\n\n\"A bottle of mezcal,\" the stranger said.\n\nThe newcomers seated themselves at the end of the room, at a spot which\nthe light reached in such a weakened state that it was almost dark. When\nthey were served each poured out a glass of liquor, which he drank; and\nleaning his head on his hands, the Mexican appeared plunged in deep\nthought, not occupying himself the least in the world about the persons\nnear him. Curumilla crossed his arms on his chest, half closed his eyes,\nand remained motionless.\n\nStill the arrival of these two men, especially the presence of the\nstranger, had suddenly frozen the eloquence of our three friends. Gloomy\nand silent, they instinctively felt that the newcomers were enemies, and\nanxiously waited for what was about to occur. At length Red Cedar,\ndoubtless more impatient than his comrades, and wishful to know at once\nwhat he had to expect, rose, filled his glass, and turned toward the\nstrangers.\n\n\"Senores caballeros,\" he said, imitating that exquisite politeness which\nthe Mexicans possess in the highest degree. \"I have the honor of\ndrinking to your health.\"\n\nAt this invitation Curumilla remained insensible as a granite statue:\nhis companion slowly raised his head, fixed his eye for a moment on the\nspeaker, and answered in a loud and firm voice,--\n\n\"It is needless, senor, for I shall not drink to yours. What I say to\nyou,\" he added, laying a stress on the words, \"your friends can also\ntake for themselves if they think proper.\"\n\nFray Ambrosio rose violently.\n\n\"What do you say?\" he exclaimed in a threatening voice. \"Do you mean to\ninsult me?\"\n\n\"There are people whom a man cannot mean to insult,\" the stranger\ncontinued in a cutting voice. \"Remember this, senor padre--I do not wish\nto have any dealings with you.\"\n\n\"Why so?\"\n\n\"Because I do not please--that is all. Now, gentlemen, do not trouble\nyourselves about me, I beg, but continue your conversation: it was most\ninteresting when I arrived. You were speaking, I believe, about an\nexpedition you are preparing: there was a question too, I fancy, when I\nentered, about a girl your worthy friend, or partner--I do not know\nwhich he is--carried off with your assistance. Do not let me disturb\nyou. I should, on the contrary, be delighted to learn what you intend\ndoing with that unhappy creature.\"\n\nNo words could render the feeling of stupor and terror which seized on\nthe three partners at this, crushing revelation of their plans. When\nthey fancied they had completely concealed them by their cunning and\nskill, to see them thus suddenly unveiled in all their extent by a man\nwhom they did not know, but who knew them, and in consequence could only\nbe an enemy--this terrified them to such a degree that for a moment they\nfancied they had to do with the spirit of evil. The two Mexicans crossed\nthemselves simultaneously, while the American uttered a hoarse\nexclamation of rage.\n\nBut Red Cedar and Fray Ambrosio were men too hardened in iniquity for\nany event, however grave in its nature, to crush them for long. The\nfirst moment past, they recovered themselves, and amazement gave way to\nfury. The monk drew from his vaquera boot a knife, and posted himself\nbefore the door to prevent egress; while Red Cedar, with frowning brow\nand a machete in his hand, advanced resolutely toward the table, behind\nwhich their bold adversary, standing with folded arms, seemed to defy\nthem by his ironical smile.\n\n\"Whoever you may be,\" Red Cedar said, stopping two paces from his\nopponent, \"chance has made you master of a secret that kills, and you\nshall die.\"\n\n\"Do you really believe that I owe a knowledge of your secrets to\nchance?\" the other said with a mocking accent.\n\n\"Defend yourself,\" Red Cedar howled furiously, \"If you do not wish me to\nassassinate you; for, _con mil diablos!_ I shall not hesitate, I warn\nyou.\"\n\n\"I know it,\" the stranger replied quietly. \"I shall not be the first\nperson to whom that has happened: the Sierra Madre and El Bolson de\nMapimi have often heard the agonising cries of your victims, when\nIndians were wanting to fill up your number of scalps.\"\n\nAt this allusion to his frightful trade the squatter felt a livid pallor\ncover his face, a tremor agitated all his limbs, and he yelled in a\nchoking voice,--\n\n\"You lie! I am a hunter.\"\n\n\"Of scalps,\" the stranger immediately retorted, \"unless you have given\nup that lucrative and honourable profession since your last expedition to\nthe village of the Coras.\"\n\n\"Oh!\" the squatter shouted with an indescribable burst of fury, \"He is a\ncoward who hides his face while uttering such words.\"\n\nThe stranger shrugged his shoulders contemptuously, and let the folds of\nhis mantle fall sharply.\n\n\"Do you recognise me, Red Cedar, since your conscience has not yet\nwhispered my name to you?\"\n\n\"Oh!\" the three men exclaimed in horror, and instinctively recoiling\n\"Don Pablo de Zarate!\"\n\n\"Yes,\" the young man continued, \"Don Pablo, who has come, Red Cedar, to\nask of you an account of his sister, whom you carried off.\"\n\nRed Cedar was in a state of extraordinary agitation: with eyes dilated\nby terror, and contracted features, he felt the cold perspiration\nbeading on his temples at this unexpected apparition.\n\n\"Ah!\" he said in a hollow voice, \"Do the dead, then, leave the tomb?\"\n\n\"Yes,\" the young man shouted loudly, \"they leave their tomb to tear your\nvictims from you. Red Cedar, restore me my sister!\"\n\nThe squatter leaped like a hyena on the young man, brandishing his\nmachete.\n\n\"Dog!\" he yelled, \"I will kill you a second time.\"\n\nBut his wrist was suddenly seized by a hand of iron, and the bandit\ntottered back to the wall of the rancho, against which he was forced to\nlean, lest he should roll on the ground. Curumilla, who had hitherto\nremained an impassive witness of the scene that took place before him,\nhad thought the moment for interference, had arrived, and had sharply\nhurled him back. The squatter, with eyes injected with blood, and lips\nclenched by rage, looked around him with glaring worthy of a wild beast.\nFray Ambrosio and the ranchero, held in check by the Indian chief, did\nnot dare to interfere. Don Pablo walked with slow and measured step\ntoward the bandit. When he was ten paces from him he stopped, and looked\nfixedly at him.\n\n\"Red Cedar,\" he repeated in a calm voice, \"give me back my sister.\"\n\n\"Never!\" the squatter answered in a voice choked by rage.\n\nIn the meanwhile the monk and the ranchero had treacherously approached\nthe young man, watching for the propitious moment to fall on him. The\nfive men assembled in this room offered a strange and sinister scene by\nthe uncertain light that filtered through the windows, as each stood\nwith his hand on his weapon, ready to kill or be killed, and only\nawaiting the opportunity to rush on his enemy. There was a moment of\nsupreme silence. Assuredly these men were brave. In many circumstances\nthey had seen death under every aspect; and yet their hearts beat as if\nto burst their breasts, for they knew that the combat about to commence\nbetween them was without truce or mercy. At length Don Pablo spoke\nagain.\n\n\"Take care, Red Cedar,\" he said. \"I have come to meet you alone and\nhonourably. I have asked you for my sister several times, and you have\nnot answered; so take care.\"\n\n\"I will sell your sister to the Apaches,\" the squatter howled. \"As for\nyou, accursed one, you shall not leave this room alive. May I be\neternally condemned if your heart does not serve as a sheath to my\nknife!\"\n\n\"The scoundrel is mad!\" the young man said contemptuously.\n\nHe fell back a pace, and then stopped.\n\n\"Listen,\" he continued. \"I will now retire, but we shall meet again; and\nwoe to you then, for I shall be as pitiless to you as you have been to\nme. Farewell!\"\n\n\"Oh! you shall not go in that way, my master,\" replied the squatter, who\nhad regained all his boldness and impudence. \"Did I not tell you I would\nkill you?\"\n\nThe young man fixed upon him a glance of undefinable expression, and\ncrossed his arms boldly on his chest.\n\n\"Try it,\" he said in a voice rendered harsh by the fury boiling in his\nheart.\n\nRed Cedar uttered a yell of rage, and bounded on Don Pablo. The latter\ncalmly awaited the attack; but, so soon as the squatter was within reach\nhe suddenly took off his mantle, and threw it over his enemy's head,\nwho, blinded by the folds of the thick garment, rolled about on the\nground, unable to free himself from the accursed cloth that held him\nlike a net. With one bound the young man was over the table, and\ntroubling himself no further about Red Cedar, proceeded toward the door.\n\nAt this moment Fray Ambrosio rushed upon him, trying to bury his knife\nin his chest. Feeling not the slightest alarm, Don Pablo seized his\nassailant's wrist, and with a strength he was far from anticipating,\ntwisted his arm so violently that his fingers opened, and he let the\nknife fall with a yell of pain. Don Pablo picked it up, and seized the\nmonk by the throat.\n\n\"Listen, villain!\" he said to him. \"I am master of your life. You\nbetrayed my father, who took pity on you, and received you into his\nhouse. You dishonour the gown you wear by your connection with\ncriminals, whose ill deeds you share in. I could kill you, and perhaps\nought to do so; but it would be robbing the executioner to whom you\nbelong, and cheating the garrote which awaits you. This gown, of which\nyou are unworthy, saves your life; but I will mark you so that you shall\nnever forget me.\"\n\nAnd placing the point of the knife on the monk's livid face, he made two\ngashes in the shape of a cross along the whole length and breadth of his\nface.\n\n\"We shall meet again!\" he added in a thundering voice, as he threw the\nknife away in disgust.\n\nAndres Garote had not dared to make a move: terror nailed him motionless\nto the ground beneath the implacable eye of the Indian warrior. Don\nPablo and Curumilla then rushed from the room and disappeared, and ere\nlong the hoofs of two horses departing at full speed from the town could\nbe heard clattering over the pavement.\n\nBy the aid of the ranchero, Red Cedar presently succeeded in freeing\nhimself from the fold of the cloak that embarrassed him. When the three\naccomplices found themselves alone again an expression of impotent rage\nand deadly hatred distorted their faces.\n\n\"Oh!\" the squatter muttered, grinding his teeth, and raising his fist to\nheaven, \"I will be revenged.\"\n\n\"And I too,\" said Fray Ambrosio in a hollow voice, as he wiped away the\nblood that stained his face.\n\n\"Hum! I do not care,\" Andres Garote said to himself aside. \"That family\nof the Zarates is a fine one; but, _caray_! it must be confessed that\nDon Pablo is a rough fellow.\"\n\nThe worthy ranchero was the only one chance had favoured in this meeting\nby letting him escape safe and sound.\n\n\n\n\nCHAPTER III.\n\nTHE HUNTERS.\n\n\nAt about two leagues from Santa Fe, in a clearing situated on the banks\nof the stream which borders that town, and on the evening of the same\nday, a man was seated before a large fire, which he carefully kept up,\nwhile actively engaged in making preparations for supper. A frugal meal,\nat any rate, this supper! It was composed of a buffalo hump, a few\npotatoes, and maize tortillas baked on the ashes, the whole washed down\nwith pulque.\n\nThe night was gloomy. Heavy black clouds coursed athwart the sky, at\ntimes intercepting the sickly rays of the moon, which only shed an\nuncertain light over the landscape, which was itself buried in one of\nthose dense mists that, in equatorial countries, exhale from the ground\nafter a hot day. The wind blew violently through the trees, whose\nbranches came in contact, with plaintive moans: and in the depths of the\nwoods the miawling of the wild cats was mingled with the snarl of the\ncoyotes and the howls of the pumas and jaguars. All at once the sound of\ngalloping horses could be heard in the forest, and two riders burst into\nthe clearing. On seeing them the hunter uttered an exclamation of joy,\nand hurried to meet them. They were Don Pablo and Curumilla.\n\n\"Heaven be praised!\" the hunter said. \"Here you are at last. I was\nbeginning to grow alarmed at your long absence.\"\n\n\"You see that nothing has happened to me,\" the young man answered,\naffectionately pressing the hunter's hands.\n\nDon Pablo had dismounted, and hobbled his own horse and Curumilla's near\nValentine, while the Indian chief busied himself in preparing the\nsupper.\n\n\"Come, come,\" the hunter said gaily, \"to table. You must be hungry, and\nI am dying of inanition. You can tell me all that has occurred while we\nare eating.\"\n\nThe three men went to the table; that is, they seated themselves on the\ngrass in front of the fire, and vigorously assailed their meagre repast.\nDesert life has this peculiarity--that in whatever position you may find\nyourself, as the struggles you go through are generally physical rather\nthan moral, nature never resigns her claims: you feel the need of\nkeeping up your strength, so as to be ready for all eventualities. There\nis no alarm great enough to prevent you from eating and drinking.\n\n\"Now,\" Valentine asked presently, \"what have you done? I fancy you\nremained much longer than was necessary in that accursed town.\"\n\n\"We did, my friend. Certain reasons forced me to remain longer than I\nhad at first intended.\"\n\n\"Proceed in regular order, if you have no objection. I fancy that is the\nonly way of understanding each other.\"\n\n\"Act as you please, my friend.\"\n\n\"Very good: the chief and I will light our Indian pipes while you make\nyour cigarette. We will sit with our backs to the fire, so as to watch\nthe neighbourhood, and in that way can converse without apprehension.\nWhat do you say, Pablo?\"\n\n\"You are always right, my friend. Your inexhaustible gaiety, your honest\ncarelessness, restore me all my courage, and make me quite a different\nman.\"\n\n\"Hum!\" Valentine said, \"I am glad to hear you speak so. The position is\nserious, it is true; but it is far from being desperate. The chief and I\nhave many times been in situations were our lives only depended on a\nthread: and yet we always emerged from them honourably--did we not,\nchief?\"\n\n\"Yes,\" the Indian answered laconically, drawing in a mouthful of smoke,\nwhich he sent forth again from his mouth and nostrils.\n\n\"But that is not the question of the moment. I have sworn to save your\nfather and sister, Pablo, and will do so, or my carcass shall be food\nfor the wild beasts of the prairie; so leave me to act. Have you seen\nFather Seraphin?\"\n\n\"Yes, I have. Our poor friend is still very weak and pale, and his wound\nis scarce cicatrised. Still, paying no heed to his sufferings, and\nderiving strength from his unbounded devotion to humanity, he has done\nall we agreed on. For the last week he has only left my father to hasten\nto his judges. He has seen the general, the governor, the\nbishop--everybody, in short--and has neglected nothing. Unfortunately\nall his exertions have hitherto been fruitless.\"\n\n\"Patience!\" the hunter said with a smile of singular meaning.\n\n\"Father Seraphin believes for certain that my father will be placed in\nthe capilla within two days. The governor wishes to have done with\nit--that is the expression he employed; and Father Seraphin told me that\nwe have not a moment to lose.\"\n\n\"Two days are a long time, my friend; before they have elapsed many\nthings may have occurred.\"\n\n\"That is true; but my father's life is at stake, and I feel timid.\"\n\n\"Good, Don Pablo; I like to hear you speak so. But reassure yourself;\nall is going on well, I repeat.\"\n\n\"Still, my friend, I believe it would be wise to take certain\nprecautions. Remember it is a question of life or death, and we must\nmake haste. How many times, under similar circumstances, have the best\narranged plans failed! Do you think that your measures are well taken?\nDo you not fear lest an unhappy accident may derange all your plans at\nthe decisive moment?\"\n\n\"We are playing at this moment the devil's own game, my friend,\"\nValentine answered coldly. \"We have chance on our side; that is to say,\nthe greatest power that exists, and which governs the world.\"\n\nThe young man lowered his head, as if but slightly convinced. The hunter\nregarded him for a moment with a mixture of interest and tender pity,\nand then continued in a soothing voice,--\n\n\"Listen, Don Pablo de Zarate,\" he said. \"I have said that I will save\nyour father, and mean to do so. Still I wish him to leave the prison in\nwhich he now is, like a man of his character ought to leave it, in open\nday, greeted by the applause of the crowd, and not by escaping furtively\nduring the night, like a vile criminal. Hang it all! Do you think it\nwould have been difficult for me to enter the town, and effect your\nfather's escape by filing the bars or bribing the jailer? I would not do\nit. Don Miguel would not have accepted that cowardly and shameful\nflight. Your father shall leave his prison, but begged to do so by the\ngovernor himself, and all the authorities of Santa Fe. So regain your\ncourage, and no longer doubt a man whose friendship and experience\nshould, on the contrary, restore your confidence.\"\n\nThe young man had listened to these words with even increasing interest.\nWhen Valentine ceased speaking he seized his hand.\n\n\"Pardon me, my friend,\" he answered him. \"I know how devoted you are to\nmy family; but I suffer, and grief renders me unjust. Forgive me.\"\n\n\"Child, let us forget it all. Was the town quiet today?\"\n\n\"I cannot tell you, for I was so absorbed in thought that I saw nothing\ngoing on around me. Still I fancy there was a certain agitation, which\nwas not natural, on the Plaza Mayor, near the governor's palace.\"\n\nValentine indulged once again in that strange smile that had already\nplayed round the corners of his delicate lips.\n\n\"Good!\" he said. \"And did you, as I advised, try to gain any information\nabout Red Cedar?\"\n\n\"Yes,\" he answered with a start of joy, \"I did; and I have positive\nnews.\"\n\n\"Ah, ah! How so?\"\n\n\"I will tell you.\"\n\nAnd Don Pablo described the scene that had taken place in the rancho.\nThe hunter listened to it with the utmost attention, and when it was\nfinished he tossed his head several times with an air of\ndissatisfaction.\n\n\"All young people are so,\" he muttered; \"they always allow their passion\nto carry them beyond the bounds of reason. You were wrong, extremely\nwrong, Don Pablo,\" he then added. \"Red Cedar believed you dead, and that\nmight have been of great use to us presently. You do not know the\nimmense power that demon has at his disposal: all the bandits on the\nfrontier are devoted to him. Your outbreak will be most injurious to\nyour sister's safety.\"\n\n\"Still, my friend--\"\n\n\"You acted like a madman in arousing the slumbering fury of the tiger.\nRed Cedar will persist in destroying you. I have known the wretch for a\nlong time. But that is not the worst you have done.\"\n\n\"What is it, then?\"\n\n\"Why, madman as you are, instead of keeping dark, watching your enemies\nwithout saying a word--in short, seeing through their game--by an\nunpardonable act of bravado you have unmasked all your batteries.\"\n\n\"I do not understand you, my friend.\"\n\n\"Fray Ambrosio is a villain of a different stamp from Red Cedar, it is\ntrue; but I consider him even a greater scoundrel than the scalp hunter.\nAt any rate, the latter is purely a rogue, and you know what to expect\nfrom him: all about him bears the stamp of his hideous soul. Had you\nstabbed that wild beast, who perspires blood by every pore, and dreams\nof naught but murder, I might possibly have pardoned you; but you have\ncompletely failed, not only in prudence, but in good sense, by acting as\nyou have done with Fray Ambrosio. That man is a hypocrite. He owes all\nto your family, and is furious at seeing this treachery discovered. Take\ncare, Don Pablo. You have made at one blow two implacable enemies, the\nmore terrible now because they have nothing to guard against.\"\n\n\"It is true,\" the young man said; \"I acted like a fool. But what would\nyou? At the sight of those two men, when I heard from their very lips\nthe crimes they had committed, and those they still meditate against us,\nI was no longer master of myself. I entered the rancho, and you know the\nrest.\"\n\n\"Yes, yes, the cuchillada was a fine one. Certainly the bandit deserved\nit; but I fear lest the cross you so smartly drew on his face will cost\nyou dearly some day.\"\n\n\"Well, let us leave it in the hand of Heaven. You know the proverb, 'It\nis better to forget what cannot be remedied.' Provided my father escape\nthe fate that menaces him, I shall be happy. I shall take my precautions\nto defend myself.\"\n\n\"Did you learn nothing further?\"\n\n\"Yes; Red Cedar's gambusinos are encamped a short distance from us. I\nknow that their chief intends starting tomorrow at the latest.\"\n\n\"Oh, oh! Already? We must make haste and prepare our ambuscade, if we\nwish to discover the road they mean to follow.\"\n\n\"When shall we start?\"\n\n\"At once.\"\n\nThe three men made their preparations; the horses were saddled, the\nsmall skins the horseman always carries at his saddle-bow in these dry\ncountries were filled with water, and five minutes later the hunters\nmounted. At the moment they were leaving the clearing a rustling of\nleaves was heard, the branches parted, and an Indian appeared. It was\nUnicorn, the great sachem of the Comanches. On seeing him the three men\ndismounted and waited. Valentine advanced alone to meet the Indian.\n\n\"My brother is welcome,\" he said. \"What does he want of me?\"\n\n\"To see the face of a friend,\" the chief answered in a gentle voice.\n\nThe two men then bowed after the fashion of the prairie. After this\nceremony Valentine went on:\n\n\"My father must approach the fire, and smoke from the calumet of his\nwhite friends.\"\n\n\"I will do so,\" Unicorn answered.\n\nAnd drawing near the fire, he crouched down in Indian fashion, took his\npipe from his belt, and smoked in silence. The hunters, seeing the turn\nthis unexpected interview was taking, had fastened up their horses, and\nseated themselves again round the fire. A few minutes passed thus, no\none speaking, each waiting till the Indian chief should explain the\nmotive of his coming. At length Unicorn shook the ashes from his\ncalumet, returned it to his belt, and addressed Valentine.\n\n\"Is my brother setting out to hunt buffaloes again?\" he said. \"There are\nmany this year on the prairies of the Rio Gila.\"\n\n\"Yes,\" the Frenchman replied, \"we are going hunting. Does my brother\nintend to accompany us?\"\n\n\"No; my heart is sad.\n\n\"What means the chief? Has any misfortune happened to him?\"\n\n\"Does not my brother understand me, or am I really mistaken? It is that\nmy brother only really loves the buffaloes, whose meat he eats, and\nwhose hides he sells at the _tolderia_?\"\n\n\"Let my brother explain himself more clearly; then I will try to answer\nhim.\"\n\nThere was a moment of silence. The Indian seemed to be reflecting\ndeeply: his nostrils were dilated, and at times his black eye flashed\nfire. The hunters calmly awaited the issue of this conversation, whose\nobject they had not yet caught. At length Unicorn raised his head,\nrestored all the serenity to his glance, and said in a soft and\nmelodious voice,--\n\n\"Why pretend not to understand me, Koutonepi? A warrior must not have a\nforked tongue. What a man cannot do alone, two can attempt and carry\nout. Let my brother speak: the ears of a friend are open.\"\n\n\"My brother is right. I will not deceive his expectations. The hunt I\nwish to make is serious. I am anxious to save a woman of my colour; but\nwhat can the will of one man effect?\"\n\n\"Koutonepi is not alone: I see at his side the best two rifles of the\nfrontier. What does the white hunter tell me? Is he no longer the great\nwarrior I knew? Does he doubt the friendship of his brother Haboutzelze,\nthe great sachem of the Comanches?\"\n\n\"I never doubted the friendship of my brother. I am an adopted son of\nhis nation. At this very moment is he not seeking to do me a service?\"\n\n\"That service is only half what I wish to do. Let my brother speak the\nword, and two hundred Comanche warriors shall join him to deliver the\nvirgin of the palefaces, and take the scalps of her ravishers.\"\n\nValentine started with joy at this noble offer.\n\n\"Thanks, chief,\" he said eagerly. \"I accept; and I know that your word\nis sacred.\"\n\n\"Michabou protects us,\" the Indian said. \"My brother can count on me. A\nchief does not forget a service. I owe obligations to the pale hunter,\nand will deliver to him the gachupino robbers.\"\n\n\"Here is my hand, chief: my heart has long been yours.\"\n\n\"My brother speaks well. I have done what he requested of me.\"\n\nAnd, bowing courteously, the Comanche chief withdrew without adding a\nword.\n\n\"Don Pablo,\" Valentine exclaimed joyously, \"I can now guarantee your\nfather's safety: this night--perhaps tomorrow--he will be free.\"\n\nThe young man fell into the hunter's arms, and hid his head on his\nhonest chest, not having the strength to utter a word. A few minutes\nlater, the hunters left the clearing to go in search of the gambusinos,\nand prepare their ambuscade.\n\n\n\n\nCHAPTER IV.\n\nSUNBEAM.\n\n\nWe will now go a little way back, in order to clear up certain portions\nof the conversation between Valentine and Unicorn, whose meaning the\nreader can not have caught.\n\nOnly a few months after their arrival in Apacheria the Frenchman and\nCurumilla were hunting the buffalo on the banks of the Rio Gila. It was\na splendid day in the month of July. The two hunters, fatigued by a long\nmarch under the beams of the parching sun, that fell vertically on their\nheads, had sheltered themselves under a clump of cedar wood trees, and,\ncarelessly stretched out on the ground, were smoking while waiting till\nthe great heat had passed, and the evening breeze rose to enable them to\ncontinue their hunt. A quarter of elk was roasting for their dinner.\n\n\"Eh, _penni_,\" Valentine said, addressing his comrade, and rising on his\nelbow, \"the dinner seems to be ready; so suppose we feed? The sun is\nrapidly sinking behind the virgin forest, and we shall soon have to\nstart again.\"\n\n\"Eat,\" Curumilla answered, sharply.\n\nThe meat was laid on a leaf between the two hunters, who began eating\nwith good appetite, and indulging in cakes of _hautle_. These cakes,\nwhich are very good, are certainly curious. They are made of the pounded\neggs of a species of water bug, collected by a sort of harvest in the\nMexican lakes. They are found on the leaves of the _toule_ (bulrush),\nand the farina is prepared in various ways. It is an Aztec preparation\n_par excellence_, for so long back as 1625 they were sold on the\nmarketplace of the Mexican capital. They form the chief food of the\nIndians, who consider them as great a dainty as the Chinese do their\nswallow nests, with which this article of food has a certain resemblance\nin taste. Valentine had taken a third bite at his hautle cake when he\nstopped, with his arm raised and his head bent forward, as if an unusual\nsound had suddenly smitten his ear. Curumilla imitated his friend, and\nboth listened with that deep attention that only results from a\nlengthened desert life; for on the prairie every sound is\nsuspicious--every meeting is feared, especially with man.\n\nSome time elapsed ere the noise which startled the hunters was repeated.\nFor a moment they fancied themselves deceived, and Valentine took\nanother bite, when he was again checked. This time he had distinctly\nheard a sound resembling a stifled sigh, but so weak and hollow that it\nneeded the Trail-hunter's practised ear to catch it. Curumilla himself\nhad perceived nothing. He looked at his friend in amazement, not knowing\nto what he should attribute his state of agitation. Valentine rose\nhurriedly, seized his rifle, and rushed in the direction of the river,\nhis friend following him in all haste.\n\nIt was from the river, in fact, that the sigh heard by Valentine had\ncome, and fortunately it was but a few paces distant. So soon as the\nhunters had leaped over the intervening bushes they found themselves on\nthe bank, and a fearful sight presented itself to their startled eyes. A\nlong plank was descending the river, turning on its axis, and borne by\nthe current, which ran rather strongly at this point. On this plank was\nfastened a woman, who held a child in her clasped arms. Each time the\nplank revolved the unhappy woman plunged with her child into the stream,\nand at ten yards at the most from it an enormous cayman was swimming\nvigorously to snap at its two victims.\n\nValentine raised his rifle. Curumilla at the same moment glided into the\nwater, holding his knife blade between his teeth, and swam toward the\nplank. Valentine remained for a few seconds motionless, as if changed\ninto a block of marble. All at once he pulled the trigger, and the\ndischarge was re-echoed by the distant mountains. The cayman leaped out\nof the water, and plunged down again; but it reappeared a moment later,\nbelly upwards. It was dead. Valentine's bullet had passed through its\neye.\n\nIn the meanwhile Curumilla, had reached the plank with a few strokes,\nwithout loss of time he turned it in the opposite direction from what it\nwas following; and while holding it so that it could not revolve, he\npushed it onto the sand. In two strokes he cut the bonds that held the\nhapless woman, seized her in his arms, and ran off with her to the\nbivouac fire.\n\nThe poor woman gave no signs of life, and the two hunters eagerly sought\nto restore her. She was an Indian, apparently not more than eighteen,\nand very beautiful. Valentine found great difficulty in loosening her\narms and removing the baby; for the frail creature about a year old, by\nan incomprehensible miracle, had been preserved--thanks, doubtless to\nits mother's devotion. It smiled pleasantly at the hunter when he laid\nit on a bed of dry leaves.\n\nCurumilla opened the woman's mouth slightly with his knife blade, placed\nin it the mouth of his gourd, and made her swallow a few drops of\nmezcal. A long time elapsed ere she gave the slightest move that\nindicated an approaching return to life. The hunters, however, would not\nbe foiled by the ill-success of their attentions, but redoubled their\nefforts. At length a deep sigh burst painfully from the sufferer's\noppressed chest, and she opened her eyes, murmuring in a voice weak as a\nbreath!\n\n\"_Xocoyotl_ (My child)!\"\n\nThe cry of the soul--this first and supreme appeal of a mother on the\nverge of the tomb--affected the two men with their hearts of bronze.\nValentine cautiously lifted the child, which had gone to sleep\npeacefully on the leaves, and presented it to the mother, saying in a\nsoft voice:\n\n\"_Nantli joltinemi_ (Mother, he lives)!\"\n\nAt these words, which restored her hope, the woman leaped up as if moved\nby a spring, seized the child, and covered it with kisses, as she burst\ninto tears. The hunters respected this outpouring of maternal love: they\nwithdrew, leaving food and water by the woman's side. At sunset the two\nmen returned. The woman was squatting by the fire, nursing her child,\nand lulling it to sleep by singing an Indian song. The night passed\ntranquilly, the two hunters watching in turn over the slumbers of the\nwoman they had saved, and who reposed in peace.\n\nAt sunrise she awoke; and, with the skill and handiness peculiar to the\nwomen of her race, she rekindled the fire and prepared breakfast. The\ntwo men looked at her with a smile, then threw their rifles over their\nshoulders, and set out in search of game. When they returned to the\nbivouac the meal was ready. After eating, Valentine lit his Indian pipe,\nseated himself at the foot of a tree, and addressed the young woman.\n\n\"What is my sister's name?\" he asked.\n\n\"Tonameyotl (the Sunbeam),\" she replied, with a joyous smile that\nrevealed the double row of pearls that adorned her mouth.\n\n\"My sister has a pretty name,\" Valentine answered. \"She doubtless\nbelongs to the great nation of the Apaches.\"\n\n\"The Apaches are dogs,\" she said in a hollow voice, and with a flash of\nhatred in her glance. \"The Comanche women will weave them petticoats.\nThe Apaches are cowardly as the coyotes: they only fight a hundred\nagainst one. The Comanche warriors are like the tempest.\"\n\n\"Is my sister the wife of a cacique?\"\n\n\"Where is the warrior who does not know Unicorn?\" she said proudly.\n\nValentine bowed. He had already heard the name of this terrible chief\npronounced several times. Mexicans and Indians, trappers, hunters, and\nwarriors, all felt for him a respect mingled with terror.\n\n\"Sunbeam is Unicorn's wife,\" the Indian girl continued.\n\n\"Good!\" Valentine answered. \"My sister will tell me where to find the\nvillage of her tribe, and I will lead her back to the chief.\"\n\nThe young woman smiled.\n\n\"I have in my heart a small bird that sings at every instant of the\nday,\" she said in her gentle and melodious voice. \"The swallow cannot\nlive without its mate, and the chief is on the trail of Sunbeam.\"\n\n\"We will wait the chief here, then,\" Valentine said.\n\nThe hunter felt great pleasure in conversing with this simple child.\n\n\"How was my sister thus fastened to the trunk of tree, and thrown into\nthe current of the Gila, to perish there with her child? It is an\natrocious vengeance.\"\n\n\"Yes, it is the vengeance of an Apache dog,\" she answered. \"Aztatl (the\nHeron), daughter of Stanapat, the great chief of the Apaches, loved\nUnicorn--her heart bounded at the mere name of the great Comanche\nwarrior; but the chief of my nation has only one heart, and it belongs\nto Sunbeam. Two days ago the warriors of my tribe set out for a great\nbuffalo hunt, and the squaws alone remained in the village. While I\nslept in my hut four Apache thieves, taking advantage of my slumber,\nseized me and my child, and delivered us into the hands of Stanapat's\ndaughter. 'You love your husband,' she said with a grin: 'you doubtless\nsuffer at being separated from him. Be happy: I will send you to him by\nthe shortest road. He is hunting on the prairies down the river, and in\ntwo hours you will be in his arms, unless,' she added with a laugh, 'the\ncaymans stop you on the road.'--'The Comanche women despise death,' I\nanswered her. 'For a hair you pluck from me, Unicorn will take the\nscalps of your whole tribe; so act as you think proper;' and I turned my\nhead away, resolved to answer her no more. She herself fastened me to\nthe log, with my face turned to the sky, in order, as she said, that I\nmight see my road; and then she hurled me into the river, yelling:\n'Unicorn is a cowardly rabbit, whom the Apache women despise. This is\nhow I revenge myself.' I have told my brother, the pale hunter,\neverything as it happened.\"\n\n\"My sister is a brave woman,\" Valentine replied: \"she is worthy to be\nthe wife of a renowned chief.\"\n\nThe young mother smiled as she embraced her child, which she presented,\nwith a movement full of grace, to the hunter, who kissed it on the\nforehead. At this moment the song of the maukawis was heard at a short\ndistance off. The two hunters raised their heads in surprise, and looked\naround them.\n\n\"The quail sings very late, I fancy,\" Valentine muttered suspiciously.\n\nThe Indian girl smiled as she looked down, but gave no answer. Suddenly\na slight cracking of dry branches disturbed the silence. Valentine and\nCurumilla made a move, as if to spring up and seize their rifles that\nlay by their side.\n\n\"My brothers must not stir,\" the squaw said quickly: \"it is a friend.\"\n\nThe hunters remained motionless, and the girl then imitated with rare\nperfection the cry of the blue jay. The bushes parted, and an Indian\nwarrior, perfectly painted and armed for war, bounded like a jackal over\nthe grass and herbs that obstructed his passage, and stopped in face of\nthe hunters. This warrior was Unicorn. He saluted the two men with that\ngrace innate in the Indian race; then he crossed his arms on his breast\nand waited, without taking a glance at his squaw, or even appearing to\nhave seen her. On her side the Indian woman did not stir.\n\nDuring several moments a painful silence fell on the four persons whom\nchance had assembled in so strange a way. At length Valentine, seeing\nthe warrior insisted on being silent, decided he would be the first to\nspeak.\n\n\"Unicorn is welcome to our camp,\" he said. \"Let him take a seat by the\nfire of his brothers, and share with them the provisions they possess.\"\n\n\"I will take a seat by the fire of my paleface brother,\" he replied;\n\"but he must first answer me a question I wish to ask of him.\"\n\n\"My brother can speak: my ears are open.\"\n\n\"Good!\" the chief answered. \"How is it the hunters have with them\nUnicorn's wife?\"\n\n\"Sunbeam can answer that question best,\" Valentine said gravely.\n\nThe chief turned to his squaw.\n\n\"I am waiting,\" he remarked.\n\nThe Indian woman repeated, word for word, to her husband the story she\nhad told a few minutes before. Unicorn listened without evincing either\nsurprise or wrath: his face remained impassive, but his brows were\nimperceptibly contracted. When the woman had finished speaking, the\nComanche chief bowed his head on his chest, and remained for a moment\nplunged in serious thought. Presently he raised his head.\n\n\"Who saved Sunbeam from the river when she was about to perish?\" he\nasked her.\n\nThe young woman's face lit up with a charming smile.\n\n\"These hunters,\" she replied.\n\n\"Good!\" the chief said, laconically, as he bent on the two men glances\nfull of the most unspeakable gratitude.\n\n\"Could we leave her to perish?\" Valentine said.\n\n\"My brothers did well. Unicorn is one of the first sachems of his\nnation. His tongue is not forked: he gives his heart once, and takes it\nback no more. Unicorn's heart belongs to the hunters.\"\n\nThese simple words were uttered with the majesty and grandeur the\nIndians know so well how to assume when they think proper. The two men\nvowed their gratitude, and the chief continued:--\n\n\"Unicorn is returning to his village with his wife: his young men are\nawaiting him twenty paces from here. He would be happy if the hunters\nwould consent to accompany him there.\"\n\n\"Chief,\" Valentine answered, \"we came into the prairie to hunt the\nbuffalo.\"\n\n\"Well, what matter? My brothers will hunt with me and my young men; but\nif they wish to prove to me that they accept my friendship, they will\nfollow me to my village.\"\n\n\"The chief is mounted, while we are on foot.\"\n\n\"I have horses.\"\n\nAny further resistance would have been a breach of politeness, and the\nhunters accepted the invitation. Valentine, whom accident had brought on\nto the prairies of the Rio Gila and Del Norte, was in his heart not\nsorry to make friends there, and have allies on whose support he could\nreckon in case of need. The squaw had by this time risen: she timidly\napproached her husband, and held up the child, saying in a soft and\nfrightened voice,--\n\n\"Kiss this warrior.\"\n\nThe chief took the frail creature in his muscular arms, and kissed it\nrepeatedly with a display of extraordinary tenderness, and then returned\nit to the mother. The latter wrapped the babe in a small blanket, then\nplaced it on a plank shaped like a cradle, and covered with dry moss,\nfastened a hoop over the place where its head rested, to guard it from\nthe burning beams of the sun, and hung the whole on her back by means of\na woolen strap passing over her forehead.\n\n\"I am ready,\" she said.\n\n\"Let us go,\" the chief replied.\n\nThe hunters followed him, and they were soon on the prairie.\n\n\n\n\nCHAPTER XXIX.\n\nTHE ADOPTION.\n\n\nSome sixty Comanche warriors were lying in the grass awaiting their\nsachem, while the tethered horses were nibbling the tall prairie grasses\nand the tree shoots. It could be seen at the first glance that these men\nwere picked warriors, selected for a dangerous expedition. From the\nheels of all dangled five or six wolf tails--marks of honor which only\nrenowned warriors have the right to wear.\n\nOn seeing their chief, they hurriedly rose and leaped into their\nsaddles. All were aware that their sachem's wife had been carried off,\nand that the object of their expedition was to deliver her. Still, on\nnoticing her, they evidenced no surprise, but saluted her as if she had\nleft them only a few moments previously. The war party had with it\nseveral horses, which the chief ordered to be given to his squaw and his\nnew friends; then, at a signal from him, the whole party started at full\nspeed, for the Indians know no other pace than the gallop.\n\nAfter about two hours' ride they reached the vicinity of the village,\nwhich could be smelt some time before reaching, owing to the habit the\nComanches have of placing their dead on scaffoldings outside the\nvillages, where they moulder away: these scaffoldings, composed of four\nstakes planted in the ground, terminated in a fork, while from poles\nstuck up near them hung skins and other offerings made by the Indians to\nthe genius of good.\n\nAt the entrance of the village a number of horsemen were assembled,\nawaiting the return of the sachem. So soon as they perceived him they\nburst into a formidable yell, and rushed forward like a whirlwind,\nshouting, firing guns, and brandishing their weapons. Unicorn's band\nfollowed this example, and there was soon a most extraordinary\nconfusion.\n\nThe sachem made his entry into the village in the midst of shouts,\nbarking of dogs, and shots; in short, he was accompanied to the square\nby an indescribable row. On reaching it the warriors stopped. Unicorn\nbegged the hunters to dismount, and guided them to his cabin, which he\nmade them enter before him.\n\n\"Now,\" he said to them, \"brothers, you are at home: rest in peace, eat\nand drink. This evening I will come and talk with you, and make you a\nproposal which I sincerely hope you will not reject.\"\n\nThe two hunters, wearied by the long ride they had made, fell back with\nextreme satisfaction on the beds of dried leaves which awaited them.\n\n\"Well,\" Valentine asked Curumilla, \"penni, what do you say about what is\nhappening to us?\"\n\n\"It may be good.\"\n\n\"Can it not?\"\n\n\"Yes.\"\n\nOn which Curumilla fell asleep, and Valentine soon followed his example.\nAs he had promised, toward evening Unicorn entered the cabin.\n\n\"Have my brothers rested?\" he asked.\n\n\"Yes,\" Valentine answered.\n\n\"Are they disposed to listen to me?\"\n\n\"Speak, chief; we are listening.\"\n\nThe Comanche sachem then squatted near the fire, and remained for\nseveral minutes, with his head bent forward and his eyes fixed on the\nground, in the position of a man who is reflecting. At length he raised\nhis head, stretched forth his arm as if to give greater authority to the\nwords he was about to utter, and began thus:--\n\n\"Brother, you and your friend are two brave warriors. The prairies\nrejoice at your arrival among us; the deer and the buffaloes fly at your\napproach; for your arm is strong, and your eye unerring. Unicorn is only\na poor Indian; but he is a great warrior among the Comanches, and a much\nfeared chief of his tribe. You have saved his wife, Sunbeam, whom the\nApache dogs threw into the Gila, and whom the hideous alligators were\npreparing to devour. Since his wife, the joy of his hearth, and his son,\nthe hope of his old days, have been restored to him, Unicorn has sought\nin his heart the means to prove to you his gratitude. He asked the Chief\nof Life what he could do to attach you to him. Unicorn is terrible in\ncombat; he has the heart of the grizzly bear for his enemies--he has the\nheart of the gazelle for those he loves.\"\n\n\"Chief,\" Valentine answered, \"the words you utter at this moment amply\nrepay us for what we have done. We are happy to have saved the wife and\nson of a celebrated warrior: our reward is in our hearts, and we wish\nfor no other.\"\n\nThe chief shook his head.\n\n\"No,\" he said; \"the two hunters are no longer strangers for the\nComanches; they are the brothers of our tribe. During their sleep\nUnicorn assembled round the council fire the chiefs of his nation, and\ntold them what has passed. The chiefs have ranged themselves on\nUnicorn's side, and have ordered him to make known to the hunters the\nresolution they have formed.\"\n\n\"Speak, then, chief,\" Valentine said, \"and believe that the wishes of\nthe council will be commands to us.\"\n\nA smile of joy played round the chief's lips.\n\n\"Good!\" he said. \"This is what was agreed on among the great chiefs. My\nbrothers the hunters will be adopted by the tribe, and be henceforth\nsons of the great Comanche nation. What say my brothers?\"\n\nA lively feeling of pleasure made Valentine quiver at this unexpected\nproposition. To be adopted by the Comanche tribe, was obtaining the\nright of hunting over the whole extent of the immense prairies which\nthat powerful nation holds through its indomitable courage and the\nnumber of its warriors. The hunter exchanged a glance with his silent\ncomrade and rose.\n\n\"I accept for myself and friend,\" he said as he held out his hand to the\nchief, \"the honor the Comanches do us in admitting us into the number of\nthe sons of their warlike nation. We shall prove ourselves worthy of\nthis marked favour.\"\n\nUnicorn smiled.\n\n\"Tomorrow,\" he said as he rose, \"my brothers will be adopted by the\nnation.\"\n\nAfter bowing gracefully to the hunters he took leave of them and\nwithdrew. The next daybreak the chiefs entered the cabin. Valentine and\nCurumilla were ready, and had long been acquainted with the trials they\nwould have to undergo. The neophytes were conducted into the great\nmedicine hut, where a copious meal was prepared. It consisted of dog\nmeat boiled in bear fat, tortillas, maize, and hautle cakes. The chiefs\nsquatted in a circle, while the squaws waited on them.\n\nWhen the meal was ended all rose. Unicorn placed himself between the\nhunters, laid his hands on their heads, and struck up the great war\nsong. This song was repeated in chorus by the company to the sound of\nthe war whistles, the drums and the _chikikouis._ The following is the\ntranslation of the song:--\n\n   \"Master of Life, regard us with a favourable eye.\n   We are receiving two brothers in arms who appear to have sense.\n   They display vigour in their arms.\n   They fear not to expose their bodies to the blows of their enemies.\"\n\nIt is impossible for anyone who has not been present at the ceremony to\nform even a distant idea of the frightful noise produced by their hoarse\nvoices mingled with the shrill and discordant instruments: it was enough\nto produce a deafness. When the song was ended each took his seat by the\ncouncil fire.\n\nThe hunters were seated on beaver skins, and the great war calumet was\npresented to them, from which each took several puffs, and it went the\nround. Unicorn then rose, and fastened round the neck of each a wampum\ncollar, and another made of the claws of the grizzly bear. The Indians,\nduring this time, had built near the medicine lodge a cabin for the\nsweating, and when it was finished the hunters took off their clothes\nand entered it. The chiefs then brought two large stones which had been\npreviously made red hot, and after closing the hut carefully, left the\nneophytes in it.\n\nThe latter threw water on the stones, and the steam which arose almost\nimmediately produced a profuse perspiration. When this was at its height\nthe hunters ran out of the hut, passed through the double row of\nwarriors, and leaped into the river, according to the usual fashion.\nThey were immediately drawn from the water, wrapped in blankets, and led\nto Unicorn's hut, in order to undergo the final trial, which is also the\nmost painful. The hunters were laid on their backs, then Unicorn traced\non their chests with a sharp stick dipped in water in which gunpowder\nhad been dissolved, the figure of the animal serving as _totem_\n(protector) to the tribe. Then with two spikes fastened to a small piece\nof wood, and dipped in vermillion, he proceeded to prick the design.\n\nWhenever Unicorn came to a place that was too hard he made an incision\nin the flesh with a gun-flint. The places that were not marked with\nvermillion were rubbed in with powder, so that the result was a red and\nblue tattooing. During the course of this operation the war songs and\nchikikouis were constantly heard, in order to drown the cries which the\natrocious pain might draw from the patients; but the latter endured it\nall without even a contraction of the eyebrows evidencing the pain they\nmust have felt.\n\nWhen the tattooing was over the wounds were cauterised with rotten wood\nto prevent suppuration; they were washed with cold water, in which had\nbeen infused a herb resembling box, a great deal of which the Indians\nmix with their tobacco to reduce the strength. The trial we have\ndescribed is so painful to endure, that nearly always it is only\naccomplished at intervals, and often lasts a week. This time the hunters\nendured it bravely during the six hours it lasted, not uttering a cry,\nor giving a sign of weakness. Hence the Indians, from this moment,\nregarded them with a species of respect; for with them courage is the\nfirst of qualities.\n\n\"My brothers are children of the tribe,\" the chief said, offering each a\nhorse. \"The prairie belongs to them. These coursers will bear them to\nthe most remote limits of the desert, chasing the wild beasts, or\npursuing the Apache dogs.\"\n\n\"Good!\" Valentine answered.\n\nAt one bound the two hunters were in their saddles, and made their\nhorses perform the most elegant and graceful curvets. This last and\nheroic deed, after all they had suffered during the course of the day,\nraised to their full height the joy and enthusiasm of the Comanches, who\napplauded with frenzied shouts and yells all they saw their new brothers\nexecute. After remaining nearly an hour on horseback they dismounted,\nand followed the chiefs into the medicine lodge; and when each had taken\nhis seat round the council fire, and the calumet had again been smoked,\nUnicorn rose.\n\n\"The Master of Life loves His Comanche sons, since He gives them for\nbrothers such warriors as Koutonepi and Curumilla. Who can equal their\ncourage! Who would dare to contend with them! On their approach the\ngrizzly bear hides at the extremity of its den; the jaguar bounds far\naway on seeing them; the eagle itself, which looks the sun in the face,\nflies from their unerring bullet. Brothers, we congratulate ourselves on\ncounting you among our warriors. Henceforth we shall be invincible.\nBrothers, give up the names you have up to this day borne, and assume\nthose we now give you. You, Koutonepi, are henceforth Quauhtli, and bear\nthe name of that eagle, whose courage and strength you possess. You,\nCurumilla, will be called Vexolotl, and the cock will be proud to see\nthat you have taken possession of its name.\"\n\nThe two hunters warmly thanked their new brothers, and were led back by\nthe chiefs to their cabin, who wished them a pleasant night after so\nrude a day. Such was the way in which Valentine and Curumilla, to whom\nwe shall continue to give their old names, formed the acquaintance of\nUnicorn, and the result of it.\n\n\n\n\nCHAPTER VI.\n\nTHE MISSIONARY.\n\n\nWith time the relations existing between the hunters and the Indians\nwere drawn closer, and became more friendly. In the desert physical\nstrength is the quality most highly esteemed. Man, compelled to struggle\nincessantly against the dangers of every description that rise each\nmoment before him, is bound to look only to himself for the means to\nsurmount them. Hence the Indians profess a profound contempt, for sickly\npeople, and weak and timid nerves.\n\nValentine easily induced Unicorn to seize, during the hunt of the wild\nhorses, the Mexican magistrates, in order to make hostages of them if\nthe conspiracy were unsuccessful. What the hunter foresaw happened. Red\nCedar had opposed stratagem to stratagem; and, as we have seen, Don\nMiguel was arrested in the midst of his triumph, at the very moment when\nhe fancied himself master of the Paso del Norte.\n\nAfter Valentine, Curumilla, and Don Pablo had seen, from their hiding\nplace in the bushes, the mournful escort pass that was taking Don Miguel\nas a prisoner to Santa Fe, they held a council. Moments were precious;\nfor in Mexico conspirators have the sad privilege over every other\nprisoner of being tried quickly, and not left to pine. The prisoner must\nbe saved. Valentine, with that promptitude of decision which formed the\nsalient point of his character, soon arranged in his head one of those\nbold schemes which only he could discover.\n\n\"Courage!\" he said to Don Pablo. \"As long as the heart beats in the\nbreast there is hope, thank Heaven! The first hand is lost, I allow; but\nnow for the second game.\"\n\nDon Pablo had entire faith in Valentine: he had often been in the\nposition to try his friend. If these words did not completely reassure\nhim, they at least almost restored his hope, and gave him back that\ncourage so necessary to him at this supreme moment, and which had\nabandoned him.\n\n\"Speak, my friend,\" he said. \"What is to be done?\"\n\n\"Let us attend to the most important thing first, and save Father\nSeraphin, who devoted himself for us.\"\n\nThe three men started. The night was a gloomy one. The moon only\nappeared at intervals: incessantly veiled by thick clouds which passed\nover its disc, it seemed to shed its sickly rays regretfully on the\nearth. The wind whistled through the branches of the trees, which\nuttered mysterious murmurs as they came into collision. The coyotes\nhowled in the plain, and at times their sinister form shot athwart the\nskyline. After a march of about an hour the three men approached the\nspot where the missionary had fallen from the effect of Red Cedar's\nbullet; but he had disappeared. An alarm mingled with a frightful agony\ncontracted the hunter's hearts. Valentine took a despairing glance\naround; but the darkness was too dense for him possibly to distinguish\nanything.\n\n\"What is to be done?\" Don Pablo asked sadly.\n\n\"Seek,\" Valentine replied sharply: \"he cannot be far.\"\n\nCurumilla had already taken up the trail, and had disappeared in the\ngloom. The Araucano had never been a great speaker naturally: with age\nhe had grown almost dumb, and never uttered a word save when absolutely\nnecessary. But if the Indian did not talk, he acted; and in critical\nsituations his determination was often worth long harangues. Don Pablo,\nobedient to Valentine's orders, threw his rifle over his shoulder, and\nprepared to execute them.\n\n\"Where are you going?\" the hunter asked him, as he seized his arm.\n\n\"To look for Father Seraphin.\"\n\n\"Wait.\"\n\nThe two men stood motionless, listening to the mysterious sounds of the\ndesert, that nameless melody which plunges the soul into a soft reverie.\nNearly an hour passed thus, nothing revealing to the hunters that\nCurumilla's search had proved successful. Valentine, growing impatient\nat this long delay, was also preparing to go on, at once the weak,\nsnapping cry of the walkon rose in the air.\n\n\"What's that?\" Don Pablo asked in surprise.\n\n\"Silence!\" Valentine muttered.\n\nA second time the walkon sang, but this time stronger, and much nearer.\nValentine raised his fingers to his lips, and imitated the sharp, shrill\nyell of the ocelot twice, with such perfection that Don Pablo started\ninvoluntarily, and looked round for the wild beast, whose eyes he\nfancied he could see flashing behind a thicket. Almost immediately the\nnote of the walkon was heard a third time. Valentine rested the butt of\nhis rifle on the ground.\n\n\"Good!\" he said. \"Do not be alarmed, Don Pablo. Curumilla has found\nFather Seraphin.\"\n\nThe young man looked at him in amazement. The hunter smiled.\n\n\"They will both arrive directly,\" he said.\n\n\"How do you know?\"\n\n\"Child!\" Valentine interrupted him, \"In the desert the human voice is\nmore injurious than useful. The song of birds, the cry of wild beasts,\nserve us as a language.\"\n\n\"Yes,\" the young man answered simply, \"that is true. I have often heard\nit stated; but I was not aware you could understand one another so\neasily.\"\n\n\"That is nothing,\" the hunter answered good-humouredly: \"you will see\nmuch more if you only pass a month in our company.\"\n\nIn a few moments the sound of footsteps became audible, at first faint,\nthen gradually coming nearer, and two shadows were dimly drawn on the\nnight.\n\n\"Halloa!\" Valentine shouted as he Raised and cocked his rifle, \"friend\nor foe?\"\n\n\"_Pennis_ (brothers),\" a voice answered.\n\n\"It is Curumilla,\" said Valentine. \"Let us go to meet him.\"\n\nDon Pablo followed him, and they soon reached the Indian, who walked\nslowly, obliged as he was to support, almost carry, the missionary.\n\nWhen Father Seraphin fell off his horse he almost immediately lost his\nsenses. He remained for a long time lying in the ditch, but by degrees\nthe night cold had brought him round again. At the first moment the poor\npriest, whose ideas were still confused, had cast anxious glances around\nhim, while asking himself how he came there. He tried to rise; but then\na poignant pain he felt in his shoulder reminded him of what had\noccurred. Still he did not despair. Alone, by night in the desert,\nexposed to a thousand unknown dangers, of which the least was being\ndevoured by wild beasts, without weapons to defend himself, too weak,\nindeed, to attempt it, even if he had them, he resolved not to remain in\nthis terrible position, but make the greatest efforts to rise, and drag\nhimself as well as he could to the Paso, which was three leagues distant\nat the most, where he was sure of finding that care his condition\ndemanded.\n\nFather Seraphin, like the majority of the missionaries who generously\ndevote themselves to the welfare of humanity, was a man who, under a\nWeak and almost feminine appearance, concealed an indomitable energy,\nand a resolution that would withstand all trials. So soon as he had\nformed his plan he began carrying it out. With extreme difficulty and\natrocious pain he succeeded in fastening his handkerchief round his\nshoulder, so as to check the hemorrhage. It took more than an hour\nbefore he could stand on his legs: often he felt himself fainting, a\ncold perspiration beaded at the root of his hair, he had a buzzing in\nhis ears, and everything seemed to be turning round him; but he wrestled\nwith the pain, clasped his hands with an effort, raised his tear laden\neyes to heaven, and murmured from the bottom of his heart,--\n\n\"O God! Deign to support thy servant, for he has set on thee all his\nhopes and confidence.\"\n\nPrayer, when made with faith, produces in a man an effect whose\nconsequences are immediate; it consoles him, gives him courage, and\nalmost restores him the strength that has deserted him. This was what\nhappened to Father Seraphin. After uttering these few words he set out\nboldly, supporting his tottering footsteps with a stick, which a\nprovidential chance had placed in his way. He walked thus for nearly\nhalf a league stopping at every instant to draw breath; but human\nendurance has limits beyond which it cannot go. In spite of the efforts\nhe made, the missionary at length felt his legs give way under, him; he\nunderstood that he could not go further; and he sank at the foot of a\ntree, certain that he had attempted impossibilities, and henceforth\nresigning to Providence the care of saving him.\n\nIt was at this moment Curumilla arrived near him. The Indian aided him\nto rise, and then warned his comrades of the success of his search.\nFather Seraphin, though the chief offered to carry him, refused, and\nwished to walk to join his friends; but his strength deserted him a\nsecond time, he lost his senses, and fell into the arms of the Indian,\nwho watched him attentively; for he noticed his increasing weakness, and\nforesaw his fall. Valentine and Curumilla hastily constructed a litter\nof tree branches, on which they laid the poor wounded man, and raising\nhim on their shoulders, went off rapidly. The night passed away, and the\nsun was already high on the horizon, and yet the hunters--were marching.\nAt length, at about eleven o'clock, they reached the cavern which served\nValentine as a shelter, and to which he had resolved to carry his\npatient, that he might himself nurse him.\n\nFather Seraphin was in a raging fever; his face was red, his eyes\nflashing. As nearly always happens with gunshot wounds, a suppurating\nfever had declared itself. The missionary was laid on a bed of furs, and\nValentine immediately prepared to probe the wound. By a singular chance\nthe ball had lodged in the shoulder without fracturing the blade bone.\nValentine drew it; and then helped by Curumilla, who had quietly pounded\noregano leaves, he formed a cataplasm, which he laid on the wound, after\nfirst carefully washing it. Scarcely had this been done ere the\nmissionary fell into a deep sleep, which lasted till nightfall.\n\nValentine's treatment had effected wonders. The fever had disappeared,\nthe priest's features were calmed, the flush that purpled his cheeks had\ngiven place to a pallor caused by the loss of blood; in short, he was as\nwell as could be expected. On opening his eyes he perceived the three\nhunters watching him anxiously. He smiled, and said in a weak voice,--\n\n\"Thanks, my brothers, thanks for the help you have afforded me. Heaven\nwill reward you. I feel much better.\"\n\n\"The Lord be praised!\" Valentine answered. \"You will escape, my father,\nmore cheaply than I had dared to hope.\"\n\n\"Can it be possible?\"\n\n\"Yes, your wound, though serious, is not dangerous, and in a few days\nyou can, if you think necessary, resume your avocations.\"\n\n\"I thank you for this new good, my dear Valentine. I no longer count the\ntimes I have owed my life to you. Heaven, in its infinite goodness, has\nplaced you near me to support me in my tribulations, and succour me in\ndays of danger.\"\n\nThe hunter blushed.\n\n\"Do not speak so, my father,\" he said; \"I have only performed a sacred\nduty. Do you feel strong enough to talk for a few minutes with me?\"\n\n\"Yes. Speak, my friend.\"\n\n\"I wished to ask your advice.\"\n\n\"My talents are very slight: still you know how I love you, Valentine.\nTell me what vexes you, and perhaps I may be able to be useful to you.\"\n\n\"I believe it, my father.\"\n\n\"Speak, then, in Heaven's name, my friend; for, if you have recourse to\nme, the affair must be very serious.\"\n\n\"It cannot be more so.\"\n\n\"Go on: I am listening.\"\n\nAnd the missionary settled himself on his bed to hear as comfortably as\nhe could the confession the hunter wished to make to him.\n\n\n\n\nCHAPTER VII.\n\nTHE INTERVIEW.\n\n\nAt daybreak the next morning Curumilla started for Unicorn's village. At\nsunset he returned to the cavern, accompanied by the Comanche chief. The\nsachem entertained the most profound respect for Father Seraphin, whose\nnoble character he could appreciate, and felt pained at the state in\nwhich he found him.\n\n\"Father,\" he said to him as he kissed his hand. \"Who are the villains\nwho thus wounded you, to whom the Master of Life has imparted the secret\nto make us happy? Whoever they may be, these men shall die.\"\n\n\"My son,\" the priest answered gently, \"I will not pronounce before you\nthe name of the unhappy man who, in a moment of madness, raised his hand\nagainst me. My God is a God of peace; He is merciful, and recommends His\ncreatures to forget injuries, and requite good for evil.\"\n\nThe Indian looked at him in amazement. He did not understand the soft\nand touching sublimity of these precepts of love. Educated in the\nsanguinary principles of his race--persuaded, like all redskins, that a\nwarrior's first duty is revenge--he only admitted that atrocious law of\nthe prairies which commands, \"Eye for eye, tooth for tooth\"--a terrible\nlaw, which we do not venture, however, utterly to condemn in these\ncountries, where ambushes are permanent, and implacable death stands at\nevery corner of the road.\n\n\"My son,\" Father Seraphin continued, \"you are a great warrior. Many a\ntime you have braved the atrocious tortures of the stake of blood, a\nthousand fold more terrible than death itself. Often have you, with a\npleasure I excuse (for it is in your nature), thrown down your enemy,\nand planted your knee on his chest. Have you never pardoned anybody in\nfight?\"\n\n\"Never!\" the Indian answered, his eye sparkling with satisfied pride.\n\"Unicorn has sent many Apache dogs to the happy hunting grounds: their\nscalps are drying at the door of his cabin.\"\n\n\"Well,\" the missionary said gently, \"try clemency once, only once, and\nyou will know one of the greatest pleasures God has granted to man on\nearth--that of pardoning.\"\n\nThe chief shook his head.\n\n\"No,\" he said; \"a dead enemy is no longer to be feared. Better to kill\nthan leave him means to avenge himself at a later date.\"\n\n\"My son, you love me, I believe?\"\n\n\"Yes. My father is good; he has behaved well to the Comanches, and they\nare grateful. Let my father command, and his son will obey.\"\n\n\"I have no right to give you an order, my son. I can only ask a favour of\nyou.\"\n\n\"Good! My father can explain himself. Unicorn will do what he desires.\"\n\n\"Well, then,\" said the missionary with a lively feeling of joy, \"promise\nme to pardon the first unhappy man, whoever he may be, who falls into;\nyour hands, and you will render me happy.\"\n\nThe chief frowned, and an expression of dissatisfaction appeared on his\nfeatures. Father Seraphin anxiously followed on the Comanche's\nintelligent countenance the different shadows reflected on it as in a\nmirror. At length the Indian regained his stoicism, and his face grew\nserene again.\n\n\"Does my father demand it?\" he asked in a gentle voice.\n\n\"I desire it.\"\n\n\"Be it so: my father shall be satisfied. I promise him to pardon the\nfirst enemy whom the Manitou causes to fall beneath the point of my\nlance.\"\n\n\"Thanks, chief,\" the missionary exclaimed joyfully, \"thanks! Heaven will\nreward you for this good idea.\"\n\nThe Indian bowed silently and turned to Valentine, who had been\nlistening to the conversation.\n\n\"My brother called me, and I came. What does he want of Unicorn?\"\n\n\"My brother will take his seat at the council fire, and smoke the\ncalumet with his friend. Chiefs do not speak without reflecting on the\nwords they are about to utter.\"\n\n\"My brother speaks well, and I will take my seat at his fire.\"\n\nCurumilla had lighted a large fire in the first grotto of the cavern.\nThe four men left Father Seraphin to take a few moments' rest, and\nseated themselves round the fire, when the calumet passed from hand to\nhand. The Indians never undertake anything important, or commence a\ndiscussion, without first smoking the calumet in council, whatever may\nbe the circumstances in which they are placed. When the calumet had gone\nthe round Valentine rose.\n\n\"Every day,\" he said, bowing to the chief, \"I appreciate more and more\nthe honor the Comanches did me in adopting me as a son. My brother's\nnation is powerful; its hunting grounds cover the whole surface of the\nearth. The Apaches fly before the Comanche warriors like cowardly\ncoyotes before courageous men. My brother has already several times done\nme a service with that greatness of soul which distinguishes him, and\ncan only belong to a warrior so celebrated as he is. Today I have again\na service to ask of my brother, and will he do it me? I presume so; for\nI know his heart, and that the Great Spirit of the Master of Life dwells\nin him.\"\n\n\"Let my brother explain,\" Unicorn answered. \"He is speaking to a chief;\nhe must remove the skin from his heart and let his blood flow red and\nbright before a friend. The great white hunter is a portion of myself. I\nshould have to be prevented by an arrant impossibility if I refused any\nrequest emanating from him.\"\n\n\"Thanks, brother,\" Valentine said with emotion. \"Your words have passed\nfrom your lips into my breast, which they have rejoiced. I am not\nmistaken. I see that I can ever count on your well-tried friendship and\nhonest aid. Acumapicthzin de Zarate, the descendant of the Mexican\nkings, the friend of the redskins, whom he has ever protected, is a\nprisoner to the gachupinos. They have carried him to Santa Fe in order\nto put him to death, and deprive the Indians of the last friend left\nthem.\"\n\n\"And what does my brother want?\"\n\n\"I wish to save my friend.\"\n\n\"Good!\" the chief answered. \"My brother claims my help to succeed in\nthat project, I suppose?\"\n\n\"Yes.\"\n\n\"Good! The descendant of the Tlatoanis shall be saved. My brother can\nfeel reassured.\"\n\n\"I can count, then, on my brother's aid?\" Valentine asked quickly.\n\nThe chief smiled.\n\n\"Unicorn holds in his hands Spaniards who will answer for the life of\nthe prisoner.\"\n\n\"That is true!\" Valentine exclaimed as he struck his forehead. \"Your\nidea is a good one, chief.\"\n\n\"My brother will leave me to act. I answer for success on my head.\"\n\n\"_Caramba!_ Act as you please, chief. Still, were it only form's sake, I\nshould not be sorry to know what you intend doing.\"\n\n\"My brother has a white skin, but his heart is Indian. Let him trust to\nthe prudence of a chief; Unicorn knows how to treat with the\ngachupinos.\"\n\n\"Doubtless.\"\n\n\"Unicorn will go to Santa Fe to speak with the chief of the white men.\"\n\nValentine looked at him in amazement. The chief smiled.\n\n\"Have I not hostages?\" he said.\n\n\"That is true,\" Valentine remarked.\n\nThe chief went on:--\n\n\"The Spaniards are like chattering old women, prodigal of seductive\nwords, but Unicorn knows them. How many times already has he trodden the\nwarpath on their territory at the head of his warriors! They will not\ndare to deceive him. Ere the sun has twice accomplished its revolution\nround the tortoise whose immense shell supports the world, the chief of\nthe Comanches will carry the bloody arrows to the whites, and propose to\nthem peace or war. Is my brother satisfied?\"\n\n\"I am. My heart is full of gratitude toward my red brother.\"\n\n\"Good! What is that to Unicorn? Less than nothing. Has my brother\nanything else to ask of me?\"\n\n\"One thing more.\"\n\n\"Let my brother explain himself as quickly as possible, that no cloud\nmay remain between him and his red brother.\"\n\n\"I will do so. Men without fear of the Great Spirit, urged by some mad\ndesire, have carried off Dona Clara, the daughter of the white chief\nwhom my brother pledged to save.\"\n\n\"Who are these? Does my brother know them?\"\n\n\"Yes, I know them only too well. They are bandits, at the head of whom\nis a monster with a human face, called Red Cedar.\"\n\nAt this name the Indian started slightly, his eye flashed fire, and a\ndeep wrinkle hollowed his forehead.\n\n\"Red Cedar is a ferocious jaguar,\" he said with concentrated passion.\n\"He has made himself the scourge of the Indians, whose scalps he\ndesires. This man has no pity either for women or children, but he\npossesses no courage: he only attacks his enemies in the dark, twenty\nagainst one, and when he is sure of meeting with no resistance.\"\n\n\"My brother knows this man, I see.\"\n\n\"And this man has carried off the white gazelle?'\n\n\"Yes.\"\n\n\"Good! My brother wishes to know what Red Cedar has done with his\nprisoner?\"\n\n\"I do wish it.\"\n\nThe Indian rose.\n\n\"Time is slipping away,\" he said. \"Unicorn will return to his friends.\nMy brother the hunter need not feel alarmed: a chief is watching.\"\n\nAfter uttering these few words the chief went down into the cavern,\nmounted his horse, and disappeared in direction of the desert. Valentine\nhad every reason to be satisfied with his interview with the Comanche\nchief; but Father Seraphin was less pleased than the hunter. The worthy\npriest, both through his nature and his vocation, was not disposed to\nemploy violent measures, which were repugnant to him: he would have\nliked, were it possible, to settle everything by gentleness, and without\nrunning the risk of bloodshed.\n\nThree weeks elapsed, however, ere Unicorn appeared to be effectually\ncarrying out the plan he had explained to Valentine, who only learnt\nindirectly that a strong party of Comanche warriors had invaded the\nMexican frontiers. Father Seraphin, though not yet completely cured, had\ninsisted on proceeding to Santa Fe to take some steps to save Don\nMiguel, whose trial had gone on rapidly, who was on the point of being\nexecuted. For his part Don Pablo, half mad with uneasiness, also\ninsisted, in spite of Valentine's entreaties and remarks, on entering\nSanta Fe furtively, and trying to see his father.\n\nThe night on which we found Valentine in the clearing Unicorn visited\nhim for the first time in a month: he came to inform him of the success\nof the measures he had taken. Valentine, used to Indian habits,\nunderstood half a word: hence he had not hesitated to announce to Don\nPablo as a positive fact that his father would soon be free.\n\n\n\n\nCHAPTER VIII.\n\nTHE PRISON.\n\n\nDon Miguel had been transferred to the prison of Santa Fe. Europeans,\naccustomed to philanthropic manners, and regarding human life as of some\nvalue, cannot imagine what atrocities the word \"prison\" contains in\nMexico. In countries beyond sea the penitentiary system is not even in\nits infancy; for it is completely ignored, and has not even been\nsuggested yet. With the exception of the United States, prisons are in\nAmerica what they were at the period of the Spanish dominion; that is to\nsay, filthy dens, where the wretched prisoners suffer a thousand\ntortures.\n\nAmong ourselves, so long as a man is not proved guilty, he is assumed to\nbe innocent; but over there, so soon as a man is arrested, he is\nconsidered guilty, and consequently every consideration and all pity\nvanish, to make room for brutal and barbarous treatment. Thrown on a\nlittle straw in fetid holes, often inhabited by serpents and other\nunclean animals, the prisoners have more than once been found dead at\nthe expiration of twenty-four hours, and half devoured. We have\nwitnessed scores of times atrocious tortures inflicted by coarse and\ncruel soldiers on poor fellows whose crimes, in our country, would have\nmerited a slight chastisement at the most. Still, in the great centres\nof populations, the prisons are better managed than in the towns and\nvillages; and in this land, where money is the most powerful lever, a\nrich man easily succeeds in obtaining all he wishes, and rendering his\nposition at any rate tolerable.\n\nDon Miguel and General Ibanez had managed to be confined together by the\nexpenditure of many entreaties and a heavy sum of gold. They inhabited\ntwo wretched rooms, the entire furniture of which consisted in a halting\ntable, a few leather covered butacas, and two benches which served them\nas beds. These two men, so powerful by nature, had endured without\ncomplaint all the humiliation and insults inflicted on them during their\ntrial, resolved to die as they had lived, with head erect and firm\nheart, without giving the judges who had condemned them the satisfaction\nof seeing them turn weak at the last moment.\n\nIt was toward evening of the same day on which we saw Valentine in the\nclearing. Darkness fell rapidly, and the only window, a species of\nnarrow slit that served to light the prison, allowed but a weak and\ndubious light to penetrate. Don Miguel was walking with long strides up\nand down his prison, while the general, carelessly reclining on one of\nthe benches, quietly smoking his cigarette, watching with childish\npleasure the light clouds of bluish smoke which rose in a spiral to the\nceiling, and which he constantly blew asunder.\n\n\"Well,\" Don Miguel said all at once, \"it seems it is not for today\neither.\"\n\n\"Yes,\" the general said, \"unless (though I do not believe it) they wish\nto do us the honor of a torchlight execution.\"\n\n\"Can you at all account for this delay?\"\n\n\"On my honor, no. I have ransacked my brains in vain to guess the reason\nthat prevents them shooting us, and I have given it up as a bad job.\"\n\n\"Same with me. At first I fancied they were trying to frighten us by the\ncontinued apprehension of death constantly suspended over our heads like\nanother sword of Damocles; but this idea seemed to me too absurd.\"\n\n\"I am entirely of your opinion: still something extraordinary must be\noccurring.\"\n\n\"What makes you suppose that?\"\n\n\"Why, for the last two days our worthy jailer, Tio Quesada, has become,\nnot polite to us--for that is impossible--but less brutal. I noticed\nthat he has drawn in his claws, and attempted a grin. It is true that\nhis face is so little accustomed to assume that expression, that the\nonly result he obtains is to make a wretched grimace.\"\n\n\"And you conclude from that?\"\n\n\"Nothing positive,\" the general said. \"Still I ask myself whence comes\nthis incomprehensible change. It would be as absurd to attribute it to\nthe pity he feels for our position as to suppose the governor will come\nto ask our pardon for having tried and condemned us.\"\n\n\"Eh?\" Don Miguel said with a toss of his head. \"All is not over--we are\nnot dead yet.\"\n\n\"That is true; but keep your mind at rest--we shall be so soon.\"\n\n\"Our life is in God's hands. He will dispose of it at His pleasure.\"\n\n\"Amen!\" the general said with a laugh, as he rolled a fresh cigarette.\n\n\"Do you not consider it extraordinary that, during the whole month we\nhave been here, our friends have not given a sign of life?\"\n\nThe general shrugged his shoulders carelessly.\n\n\"Hum!\" he said, \"a prisoner is very sick, and our friends doubtless\nfeared to make us worse by the sight of their grief: that is why they\nhave deprived themselves of the pleasure of visiting us.\"\n\n\"Do not jest, general. You accuse them wrongfully, I feel convinced.\"\n\n\"May Heaven grant it! For my part, I heartily forgive them their\nindifference, and the oblivion in which; they have left us.\"\n\n\"I cannot believe that Don Valentine, that true-hearted and noble-minded\nman, for whom I ever felt so deep a friendship, has not tried to see\nme.\"\n\n\"Bah! How, Don Miguel, can you, so near death as you are, still believe\nin honourable feelings in any man?\"\n\nAt this moment there was a great clash of iron outside, and the door of\nthe room was opened sufficiently to afford passage to the jailer, who\npreceded another person. The almost complete obscurity that prevailed in\nthe prison prevented the condemned men from recognising the visitor, who\nwore a long black gown.\n\n\"Eh, eh!\" the general muttered in his comrade's ear, \"I believe that\nGeneral Ventura, our amiable governor, has at length made up his mind.\"\n\n\"Why so?\" Don Miguel asked in a low voice.\n\n\"_Canarios!_ he has sent us a priest, which means that we shall be\nexecuted tomorrow.\"\n\n\"On my word, all the better,\" Don Miguel could not refrain from saying.\n\nIn the meanwhile the jailer, a short, thick-set man, with a ferret face\nand cunning eye, had turned to the priest, whom he invited to enter,\nsaying in a hoarse voice,--\n\n\"Here it is, senor padre: these are the condemned persons.\"\n\n\"Will you leave us alone, my friend?\" the stranger said.\n\n\"Will you have my lantern? It is getting dark, and when people are\ntalking they like to see one another.\"\n\n\"Thanks; you can do so. You will open when I call you by tapping at the\ndoor.\"\n\n\"All right--I will do so;\" and he turned to the condemned, to whom he\nsaid savagely, \"Well, senores, here is a priest. Take advantage of his\nservices now you have got him. In your position there is no knowing what\nmay happen from one moment to the other.\"\n\nThe prisoners shrugged their shoulder's contemptuously, but made no\nreply. The jailer went out. When the sound of his footsteps had died\naway in the distance, the priest, who had till this moment stood with\nhis body bent forward and his ear on the watch, drew himself up, and\nwalked straight to Don Miguel. This manoeuvre on the part of the\nstranger surprised the two gentlemen, who anxiously awaited what was\nabout to happen. The lantern left by the jailer only spread a faint and\nflickering light, scarcely sufficient to distinguish objects.\n\n\"My father,\" the hacendero said in a firm voice, \"I thank the person\nwho sent you to prepare me for death, for I anxiously wished to fulfil\nmy duties as a Christian before being executed. If you will proceed with\nme into the adjoining room I will confess my sins to you: they are those\nwhich an honest man ordinarily commits; for my heart is pure, and I have\nnothing to reproach myself with.\"\n\nThe priest took off his hat, seized the lantern, and placed it near his\npale face, whose noble and gentle features were suddenly displayed in\nthe light.\n\n\"Father Seraphin!\" the prisoners exclaimed with a surprise mingled with\njoy.\n\n\"Silence!\" the priest ordered quickly. \"Do not pronounce my name so\nloudly, brothers: everyone is ignorant of my being here except the\njailer, who is my confidant.\"\n\n\"He!\" Don Miguel said with a stupor; \"the man who has been insulting and\nhumiliating us during a month!\"\n\n\"That man is henceforth ours. Lose no time, come. I have secure means to\nget you out of prison, and to leave the town ere your evasion can be\neven suspected: the horses are prepared--an escort is awaiting you.\nCome, gentlemen, for the moments are precious.\"\n\nThe two prisoners interchanged a glance of sublime eloquence; then\nGeneral Ibanez quietly seated himself on a butaca, while Don Miguel\nreplied,--\n\n\"Thanks, my father. You have undertaken the noble task of soothing all\nsorrow, and you do not wish to fail in your duty. Thanks for the offer\nyou make us, which we cannot, however, accept. Men like us must not give\nour enemies right by flying like criminals. We fought for a sacred\nprinciple, and succumbed. We owe it to our countrymen and to ourselves\nto endure death bravely. When we conspired we were perfectly well aware\nof what awaited us if we were conquered. Once again, thanks; but we will\nonly quit this prison as free men, or to walk to punishment.\"\n\n\"I have not the courage, gentlemen, to blame your heroic resolution: in\na similar case I should act as you are doing. You have a very slight\nhope still left, so wait. Perchance, within a few hours, unforeseen\nevents will occur to change the face of matters.\"\n\n\"We hope for nothing more, my father.\"\n\n\"That word is a blasphemy in your mouth, Don Miguel. God can do all He\nwills. Hope, I tell you.\"\n\n\"I am wrong, father: forgive me.\"\n\n\"Now I am ready to hear your confession.\"\n\nThe prisoners bowed. Father Seraphin shrived them in turn, and gave them\nabsolution.\n\n\"Hola!\" the jailer shouted through the door. \"Make haste; it is getting\nlate. It will soon be impossible to leave the city.\"\n\n\"Open the door,\" the missionary said in a firm voice.\n\nThe jailer appeared.\n\n\"Well?\" he asked.\n\n\"Light me and lead me out of the prison. These caballeros refuse to\nprofit by the chance of safety I came to offer them.\"\n\nThe jailer shook his head and shrugged his shoulders.\n\n\"They are mad,\" he said.\n\nAnd he went out, followed by the priest, who turned on the threshold and\npointed to heaven. The prisoners remained alone.\n\n\n\n\nCHAPTER IX.\n\nTHE EMBASSY.\n\n\nOn the selfsame day that Father Seraphin went to the prison to propose\nan escape to the condemned, a very strange circumstance had aroused the\nentire population of Santa Fe. At about midday, at the moment when the\ninhabitants were enjoying their siesta, and the streets, calcined by the\nbeams of a tropical sun, were completely deserted, a formidable whoop,\nthe terrible war yell of the Comanche Indians, burst forth at the\nentrance of the town.\n\nThere was a general alarm, and everybody barricaded himself in his\nhouse, believing in a sudden assault of the savages. Presently an\nimmense clamour, and cries of distress and despair uttered by a\nterrified population, could be heard throughout the town. Several times\nalready the Comanches, in their periodical incursions, had come near\nSanta Fe, but never so closely as this time; and the remembrance of the\ncruelties they had practised on the hapless Spaniards who fell into\ntheir hands was still present to every mind.\n\nIn the meanwhile a few inhabitants, bolder than the rest, or having\nnothing to lose, proceeded with the greatest precautions toward the spot\nwhence the shouts were heard; and a singular spectacle presented itself.\nA detachment of dismounted Comanche warriors, about two hundred strong,\nwas marching in close column, flanked on either wing by two troops, each\nof fifty horses. About twenty paces in front caracoled Unicorn.\n\nAll these men had a martial aspect which was really remarkable: all were\nstrangely painted, well adorned, and in their full war costume. The\nhorsemen were loaded with all sorts of arms and ornaments: they had a\nbow and quiver on their backs, their guns slung and decorated with their\nmedicine bags, and their lances in their hands. They were crowned with\nmagnificent black and white eagle feathers, with a falling tuft. The\nupper part of the body, otherwise naked, was covered by a coyote skin\nrolled up and worn across the shoulder; their bucklers were ornamented\nwith feathers, cloth of different colours, and human scalps. They were\nseated on handsome saddlecloth of panthers' skins, lined with red, which\nalmost covered the horses' backs. According to the prairie fashion, they\nhad no stirrups.\n\nUnicorn brandished in his right hand the long medicine lance, the\ndistinctive mark of the powerful \"dance of the prairie dogs.\" It was a\nstaff in the shape of a crook, covered with an otter skin, and decorated\nthrough its entire length with owl feathers. This talisman, which he had\ninherited, possessed the power of bringing under his orders all the\nwarriors of his nation scattered over the prairies: hence on all grand\noccasions he never failed to carry it. He wore a shirt made of the skin\nof the bighorn, embroidered on the sleeves with blue flowers, and\nadorned on the right arm with long stripes of rolled ermine and red\nfeathers, and on the left arm with long tresses of black hair cut from\nthe scalps he had raised. Over his shoulders he had thrown a cloak of\ngazelle skin, having at each end an enormous tuft of ermine. On his\nforehead the chief had fastened two buffalo horns, which with the blue,\nred, and green paint plastered on his face, gave him a terrible aspect.\nHis magnificent horse, a mustang full of fire, which he managed with\ninimitable grace and skill, was painted red in different fashions: on\nits legs were stripes like a zebra, and on either side the backbone were\ndesigned arrowhead, lances, beavers, tortoises, &c. The same was the\ncase with the face and the haunches.\n\nThere was something at once imposing and striking in the appearance\npresented by this band of ferocious warriors as they advanced though the\ndeserted streets of the city, brandishing their tremendous weapons, and\nuttering at intervals their sinister war cry, which they accompanied by\nthe shrill sound of long whistles made of human thigh bones, which they\nwore suspended by strips of wild beast hide.\n\nBy this time the Comanches had penetrated to the heart of the city,\ndriving before them, though without violence, the few inhabitants who\nhad ventured to get in their way. They marched in good order, not\nturning to the right or left to plunder, and doing no reprehensive\naction.\n\nThe Spaniards, more and more surprised at the haughty and bold attitude\nof the Indians, and their exemplary conduct, asked themselves with\nterror what these men wanted, and what reason had led them to invade\ntheir frontiers in so sudden and secret a way, that the scouts the\nMexican Government pays to watch them had no knowledge of their march.\nAs usually happens in such cases, terror gradually gave way to\ncuriosity. In the first place the leperos and adventurers dared to\napproach the Indians; then the inhabitants, if not completely\ntranquilised, still reassured by their peaceful attitude, mingled with\nthe groups; so that when the Comanche war party arrived on the Plaza\nMayor; it was followed by a crowd of Spaniards, who regarded them with\nthe restless and stupid curiosity only to be found among the masses.\n\nThe Comanches did not appear to notice the excitement they created. As\nsoon as they were on the Plaza Mayor they halted, and remained\nmotionless, as if their feet had suddenly grown to the ground. Unicorn\nmade a sign with his talisman; a warrior quitted the ranks, and rode up\nto the sentry standing in front of the governor's palace, who regarded\nthe singular scene with a dazed air.\n\n\"Wah!\" the Indian said sarcastically, as he lightly touched the soldier\nwith the end of his lance. \"Is my brother asleep, that he does not hear\na warrior addressing him?\"\n\n\"I am not asleep,\" the soldier answered, as he fell back a pace. \"What\ndo you want?\"\n\n\"The great sachem of the Comanches, the cacique whom the red children\ncall Haboutzelze, has come to speak to his great white father, the chief\nof the frontier palefaces.\"\n\n\"What does he want with him?\" the soldier asked, not knowing what he\nsaid, so much had the unexpected sight of the redskin disturbed him.\n\n\"Is my brother a chief?\" the Indian asked cunningly.\n\n\"No,\" the soldier answered, greatly confused by this lesson.\n\n\"Well, then, let him close his ears as regards those the Great Spirit\nhas set above him, and deliver the message I give him in the sachem's\nname.\"\n\nWhile the Comanche was exchanging these few words with the sentry,\nseveral persons, drawn out of the palace by the unusual disturbance they\nheard, mingled with the crowd. Among them were several officers, one of\nwhom advanced to the Indian horseman.\n\n\"What does my brother want?\" he asked him.\n\nThe warrior saw at the first glance that this time he had to do with a\nchief. He bowed courteously, and answered.\n\n\"A deputation of the great Comanche nation desires to be introduced to\nmy great white father.\"\n\n\"Good! But all the warriors cannot enter the palace,\" the officer said.\n\n\"My brother is right. Their chiefs alone will go in: their young men\nwill await them here.\"\n\n\"Let my brother be patient. I will go and deliver his message in all\nhaste.\"\n\n\"Good! My brother is a chief. The Spider will await him.\"\n\nThe officer disappeared in the interior, while the Spider planted the\nend of his long lance in the ground, and remained with his eye fixed on\nthe gate of the palace, not evincing the slightest impatience.\n\nThe new governor of Santa Fe was a general of the name of Don Benito\nVentura. He was ignorant as a fish, stupid and haughty as a heathcock.\nLike the majority of his colleagues in this eccentric country, he had\ngained his general's epaulettes by repeated pronunciamentos, managing\nto gain a step by every revolution, while never having seen more fire\nthan that of the thin husk _pajillo_ he constantly had in his mouth. To\nsum him up, he was very rich, a wonderful coward, and more afraid of\nblows than aught else in the world. Such he was morally: physically he\nwas a plump little man, round as a barrel, with a rubicund face, lighted\nup by two small grey eyes.\n\nThis worthy officer perspired water and blood when the duties of his\nstation obliged him to put on the uniform, every seam of which was\noverlaid with gold lace: his chest literally disappeared under the\ninfinity of crosses of every description with which each president had\nhonoured him on attaining power. In a word, General Ventura was a worthy\nman, as fit to be a soldier as he was to be a cardinal; and he had only\none object, that of being President of the Republic in his turn; but\nthis object he ever pursued without Once swerving from his path.\n\nIf he accepted the governorship of New Mexico, it was for the simple\nreason that, as Santa Fe was a long distance from Mexico, he had\ncalculated that it would be easy for him to make a _pronunciamento_ in\nhis own favour, and become, _ipso facto_, president. He was not aware, on\ncoming to Santa Fe, that the province he was about to govern was\nincessantly menaced by Indian forays. Had he known it, however\nadvantageous the post of governor might, be for his schemes, he would\nhave refused point blank so perilous an honour.\n\nHe had learned with the utmost terror the entrance of the Comanches into\nthe town, and when the officer intrusted with the Spider's message\npresented himself before him he had literally lost his head. It took all\npossible trouble to make him comprehend that the Indians came as\nfriends, that they merely wished to have a palaver with him, and that\nsince their coming their conduct had been most honourable and exemplary.\nFortunately for the Spanish honour, other officers entered the apartment\nin which was the governor, attracted to the palace by the news, which\nhad spread with the speed of a train of powder through Santa Fe, of the\nappearance of an Indian detachment.\n\nWhen the general saw himself surrounded and supported by the officers of\nhis staff his terror was slightly toned down, he regained his presence\nof mind and it was with a calm and almost dignified demeanor that he\ndiscussed the question whether it was proper to receive the Indian\ndeputation, and in what manner it should be done. The other officers,\nwho, in the course of their professional career, had had many a skirmish\nwith the redskins, felt no inclination to anger them. They produced in\nsupport of their opinions such peremptory reasons, that General Ventura,\nconvinced by their arguments gave the officer who brought the message\norders to bring the three principal Indian chiefs into the palace.\n\n\n\n\nCHAPTER X.\n\nTHE PRESENTATION.\n\n\nIt needed the thorough knowledge the Comanches possessed of the terror\nthey inspired the Mexicans with to have dared to enter in so small a\nbody a town like Santa Fe, where they might expect to find a\nconsiderable garrison.\n\nThe general officer sent by General Ventura had performed his duty.\nUnicorn and two other chiefs dismounted, and followed him into the\npalace; while the Indian warriors, in spite of the heat of the sunbeams\nthat played on their heads, remained motionless on the spot where their\ncaciques bade them wait.\n\nThe general desired, by a certain display of strength, to impose on the\nredskin deputies; but unfortunately, as is always the case in Mexico,\nthe garrison, which on paper represented eight hundred men, was in\nreality only composed of sixty at the most--a very small number for a\nfrontier town, especially under the present circumstances. But if\nsoldiers were lacking, to make up for it there was no paucity of\nofficers; for about thirty were assembled at the palace, which allowed\none officer to every two privates. This detail, which might appear\nexaggerated, is, however, strictly correct, and shows in what a state of\nanarchy this hapless country is plunged. The thirty officers, attired in\ntheir splendid uniforms, that glistened with gold and decorations, were\narranged round the general, while three posts of ten men each held the\ndoors of the halls of reception.\n\nWhen the preparations were completed the ambassadors were introduced.\nThe Indian chiefs, accustomed for a long period to Spanish luxury,\nentered without testifying the slightest surprise. They bowed with\ndignity to the assembly, and, crossing their arms on their chests,\nwaited till they were addressed. The general regarded them with an\nastonishment pardonable enough, for this was the first time he had found\nhimself in the presence of these untamable redskins, whose terrible\nrenown had so often made him shudder.\n\n\"What reason can have been so powerful as to oblige my sons to come and\nsee me?\" he asked in a gracious and conciliating tone. \"Let them make\ntheir request, and, if I can do so, I shall be most ready to satisfy\nit.\"\n\nThis opening, which the governor fancied to be very politic, was, on the\ncontrary, most awkward, as it offended the pride of those he addressed,\nand whom he had the greatest interest in humouring. Unicorn took a step\nforward. A sarcastic smile played on his lips, and he replied in a voice\nslightly tinged with irony,--\n\n\"I have heard a parrot speak. Are the words addressed to me?\"\n\nThe general blushed up to the eyes at this insult, which he did not dare\nretaliate.\n\n\"The chief has not understood my words,\" he said. \"My intentions are\ngood, and I only wish to be agreeable to him.\"\n\n\"The Comanches do not come here to ask a favour,\" Unicorn answered,\nhaughtily. \"They know how to avenge themselves when insulted.\"\n\n\"What do my sons want then?\"\n\n\"To treat with my father for the ransom of the white chiefs who are in\ntheir power. Five palefaces inhabit the cabin of the Comanches. The\nyoung men of the tribe demand their punishment, for the blood of the\npalefaces is agreeable to the Master of Life. Tomorrow the prisoners\nwill have ceased to live if my father does not buy them off today.\"\n\nAfter these words, uttered in a firm and peremptory tone, there was a\nmoment of supreme silence. The Mexican officers reflected sadly on the\nfearful fate that threatened their friends. Unicorn continued:--\n\n\"What does my father say? Shall we fasten our prisoners to the stake of\nblood, or restore them to liberty?\"\n\n\"What ransom do you ask?\" the general said.\n\n\"Listen, all you chiefs of the palefaces here present, and judge of the\nclemency and generosity of the Comanches. We only, wish, for the life of\nthese five men, the life of two men.\"\n\n\"That is little, I allow,\" the general remarked; \"and who are the two\nmen whose lives you ask?\"\n\n\"The palefaces call them, the first, Don Miguel Zarate; the second,\nGeneral Ibanez.\"\n\nThe general started.\n\n\"These two men cannot be delivered to you,\" he answered; \"they are\ncondemned to death, and will die tomorrow.\"\n\n\"Good! My prisoners will be tortured this night,\" the chief replied\nstoically.\n\n\"Confound it!\" the general sharply exclaimed, \"Is there no other\narrangement possible? Let my brothers ask me a thing I can grant them,\nand--\"\n\n\"I want those two men,\" the chief quickly interrupted. \"If not, my\nwarriors will themselves deliver them; and in that case the Comanche\nchiefs cannot prevent the injury their warriors may commit in the town.\"\n\nOne of the officers present at this interview was aroused by the tone\nUnicorn had affected since the beginning of the audience. He was a brave\nold soldier, and the cowardice of his comrades shamed him. He rose at\nthis point.\n\n\"Chief,\" he said in a firm voice, \"your words are very haughty and\nfoolish for the mouth of an ambassador. You are here, at the head of\nscarce two hundred warriors, in the heart of a town peopled by brave\nmen. Despite all my desire to be agreeable to you, if you do not pay\ngreater respect to your audience, prompt and severe justice shall be\ninflicted on your insolence.\"\n\nThe Indian chief turned toward the new speaker, whose remarks had\naroused a sympathetic murmur.\n\n\"My words are those of a man who fears nothing, and holds in his hands\nthe life of five men.\"\n\n\"Well,\" the officer retorted sharply, \"what do we care for them? If they\nwere such fools as to let you capture them, they must suffer the\nconsequences of their madness; we cannot pay for them. Besides, as you\nhave already been told, those you claim must die.\"\n\n\"Good! We will retire,\" Unicorn said haughtily. \"Longer discourse is\nneedless; our deeds shall speak for us.\"\n\n\"A moment!\" the general exclaimed. \"All may yet be arranged. An affair\nlike the present cannot be settled all in a hurry; we must reflect on\nthe propositions made to us. My son is a chief, and will grant us\nreasonable time to offer him a reply.\"\n\nUnicorn bent a suspicious glance on the governor.\n\n\"My father has spoken wisely,\" he presently made answer. \"Tomorrow at\nthe twelfth hour, I will come for the final answer of the palefaces. But\nmy father will promise me not to order the punishment of the prisoners\ntill he has told me the decision he has come to.\"\n\n\"Be it so,\" the general answered. \"But what will the Comanches do till,\nthen?\"\n\n\"They will leave the town as they entered it, and bivouac on the plain.\"\n\n\"Agreed on.\"\n\n\"The Master of Life has heard my father's promise. If he break his word\nand possess a forked tongue, the blood shed will fall on his head.\"\n\nThe Comanche uttered these words in a significant tone that made the\ngeneral tremble inwardly; then he bowed to the assembly, and left the\nhall with his companions. On reaching the square the chiefs remounted\ntheir horses and placed themselves at the head of their warriors. An\nhour later the Comanches had left the town, and camped within two\ngunshots of the walls, on the banks of the river. It was after this\ninterview that Unicorn had the conversation with Valentine which we\nrecently described.\n\nStill, when the Mexican officers were alone with the general, their\ncourage returned all at once, and they reproached him for the little\ndignity he had displayed before the Indians, and specially for the\npromise he had made them. The general listened to them calmly, with a\nsmile on his lips, and contented himself with answering them, in a tone,\nof indescribable meaning,--\n\n\"The promise you allude to pledges me to nothing. Between this and\ntomorrow certain things will happen to free us from the Comanches, and\nlet us dispense with surrendering the prisoners they demand so\ninsolently.\"\n\n\n\n\nCHAPTER XI.\n\nPSYCHOLOGICAL.\n\n\nAbout half a league to the west of Santa Fe three men and a woman were\nseated behind a dense clump of trees, which sheltered while rendering\nthem unseen, over a _bois-de-vache_ fire, supping with good appetite,\nand chatting together. The three men were Red Cedar's sons; the female\nwas Ellen. The maiden was pale and sad: her dreamy eye wandered around\nwith a distraught expression. She listened hardly to what her brothers\nsaid, and would certainly have been greatly embarrassed to describe the\nconversation, for her mind was elsewhere.\n\n\"Hum!\" Sutter said, \"what the deuce can keep the old one so long? He\ntold us he should be back by four o'clock at the latest; but the sun is\njust disappearing on the horizon, and he has not come yet.\"\n\n\"Pshaw!\" Nathan said with a shrug of the shoulders. \"Are you afraid that\nsomething has happened to him? The old chap has beak and nails to defend\nhimself; and since his last turn up with Don Miguel, the fellow who is\nto be shot tomorrow at Santa Fe, he has kept on his guard.\"\n\n\"I care very little,\" Sutter replied brusquely, \"whether father is here\nor not; but I believe we should do well not to wait longer, but return\nto the camp, where our presence is doubtless necessary.\"\n\n\"Nonsense! Our comrades can do without us,\" Shaw observed. \"We are all\nright here, so suppose we stop the night. Tomorrow it will be day. Well,\nif father has not returned by sunrise, we will go back to camp. Harry\nand Dick can keep good order till our return.\"\n\n\"In truth, Shaw is right,\" Nathan said. \"Father is at times so strange,\nthat he might be angry with us for not having waited for him; for he\nnever does anything lightly. If he told us to stay here, he probably had\nhis reasons.\"\n\n\"Let us stay, then,\" Sutter remarked carelessly. \"I ask for nothing\nbetter. We shall only have to keep the fire up, and so one of us will\nwatch while the others sleep.\"\n\n\"Agreed on,\" Nathan replied. \"In that way, if the old man comes during\nour sleep, he will see that we waited for him.\"\n\nThe three brothers rose. Sutter and Nathan collected a pile of dry wood\nto maintain the fire, while Shaw intertwined a few branches to make his\nsister a sufficient shelter for the night. The two elder brothers thrust\ntheir feet toward the fire, wrapped themselves in their blankets, and\nwent to sleep, after advising Shaw to keep a bright lookout, not only\nagainst wild beasts, but to announce the old squatter's approach. Shaw,\nafter stirring up the fire, threw himself at the foot of a larch tree,\nand letting his head sink on his chest, plunged into deep and painful\nmeditation.\n\nThis poor boy, hardly twenty years of age, was a strange composite of\ngood and evil qualities. Reared in the desert, he had grown up like one\nof its native trees, thrusting out here and there branches full of\npowerful sap. Nothing had ever thwarted his instincts, no matter what\ntheir nature might be. Possessing no cognizance of justice and\ninjustice, he had never been able to appreciate the squatter's conduct,\nor see the injury he did society by the life he led. Habituated to\nregard as belonging to himself all that he wished for, allowing himself\nto be guided by his impressions and caprices, never having felt any\nother fetter than his father's despotic will, this young man had at once\na nature expansive and reserved, generous and avaricious, gentle and\ncruel: in a word, he possessed all the qualities of his vices; but he\nwas, before all, a man of sensations. Endowed with a vast intellect,\nextreme audacity, and lively comprehensions, he would have been\nindubitably a remarkable man, had he been born in a different position.\n\nHis sister Ellen was the only member of his family for whom he\nexperienced sympathy; and yet it was only with extreme reserve that he\nintrusted his boyish secrets to her--secrets which, during the last few\ndays, had acquired an importance he did not himself suspect, but which\nhis sister, with the innate intelligence of woman, had already divined.\n\nShaw, as we have said, was thinking. The young savage's indomitable\nnature revolted against an unknown force which had suddenly sprung up in\nhis heart--mastered and subdued him in spite of all his efforts. He was\nin love! He loved, ignorant even of the meaning of the word love, which\ncomprises in this nether world all earthly joy and suffering. Vainly he\nsought to explain his feelings; but no light flashed across his mind, or\nillumined the darkness of his heart. He loved without desire and without\nhope, involuntarily obeying that divine law which compels even the\nroughest man to seek a mate. He was dreaming of Dona Clara. He loved\nher, as he was capable of loving, with that passionate impetuosity, that\nviolence of feeling, to which his uncultivated mind adapted him. The\nsight of the maiden caused him a strange trouble, which he did not\nattempt to account for. He did not try to analyse his feelings, for that\nwould have been impossible; and yet at times he was a prey to cold and\nterrible fury, when thinking that the haughty maiden, who was even\nunconscious of his existence, would probably only spurn and despise him\nif she knew it. He was yielding to these crushing thoughts, when he\nsuddenly felt a hand laid on his shoulder. On turning, Ellen stood\nbefore him, upright and motionless, like the white apparitions of the\nGerman legends. He raised his head, and bent an inquiring glance on his\nsister.\n\n\"You are not asleep, Ellen?\"\n\n\"No,\" she answered in a voice soft as a bird's song. \"Brother, my heart\nis sad.\"\n\n\"What is the matter, Ellen? Why not enjoy a few hours of that repose so\nnecessary for you?\"\n\n\"My heart is sad, I tell you, brother,\" she went on. \"In vain do I seek\nsleep--it flies far from me.\"\n\n\"Sister, tell me the cause of your sufferings, and perhaps I can appease\nthe grief that devours you.\"\n\n\"Can you not guess it?\"\n\n\"I do not understand you.\"\n\nShe looked at him so sternly that he could not let his eyes fall.\n\n\"On the contrary, you understand me too well, Shaw,\" she said with a\nsigh. \"Your heart rejoices at this moment at the misfortune of the woman\nyou should defend.\"\n\nThe young man blushed.\n\n\"What can I do?\" he murmured faintly.\n\n\"Everything, if you have the firm will,\" she exclaimed energetically.\n\n\"No,\" Shaw went on, shaking his head with discouragement; \"the person of\nwhom you speak is the old man's prisoner. I cannot contend against my\nfather.\"\n\nEllen smiled contemptuously.\n\n\"You seek in vain to hide your thoughts from me,\" she said harshly. \"I\nread your heart as an open book: your sorrow is feigned, and you really\nrejoice at the thought that in future you will constantly be by Dona\nClara's side.\"\n\n\"I!\" he exclaimed with an angry start.\n\n\"Yes, you only see in her captivity a means to approach her. Your\nselfish heart is secretly gladdened by that hope.\"\n\n\"You are harsh to me, sister. Heaven is my witness that, were it\npossible, I would at once restore her the liberty torn from her.\"\n\n\"You can if you like.\"\n\n\"No, it is impossible. My father watches too closely over his prisoner.\"\n\n\"He will not distrust you, but allow you to approach her freely.\"\n\n\"What you ask of me is impossible.\"\n\n\"Because you will not, Shaw. Remember that women only love men in\nproportion to the sacrifices they make for them: they despise cowards.\"\n\n\"But how to save her?\"\n\n\"That is your affair, Shaw.\"\n\n\"At least give me some advice which will help me to escape from the\ndifficult position in which I find myself.\"\n\n\"In such serious circumstances your heart must guide you, and you must\nonly ask counsel of it.\"\n\n\"But the old one?\" Shaw said hesitatingly.\n\n\"Our father will not know your movements. I take on myself to prevent\nhim noticing them.\"\n\n\"Good!\" the young man remarked, half convinced; \"but I do not know where\nthe maiden is hidden.\"\n\n\"I will tell you, if you swear to do all in your power to save her.\"\n\nThere was a moment of silence.\n\n\"I swear to obey you, Ellen. If I do not succeed in carrying the girl\noff, I will at any rate employ all my intellect to obtain that result.\nSpeak, then, without fear.\"\n\n\"Dona Clara is confined at the Rancho del Coyote: she was intrusted to\nAndres Garote.\"\n\n\"Ah, ah!\" the young man said, as if speaking to himself, \"I did not\nfancy her so near us.\"\n\n\"You will save her?\"\n\n\"At all events I will try to free her from the hands of the man who\nguards her.\"\n\n\"Good!\" the maiden remarked; \"I now recognise you. Lose no time: my\nfather's absence alarms me. Perhaps at this moment he is preparing a\nsafer hiding place for his prisoner.\"\n\n\"Your idea is excellent, sister. Who knows whether it is not too late\nnow to tear from the old man the prey he covets?\"\n\n\"When do you intend to start?\"\n\n\"At once: I have not a moment to lose. If the old man returned I should\nbe compelled to remain here. But who will keep watch while my brothers\nsleep?\"\n\n\"I will,\" the maiden answered resolutely.\n\n\"Whence arises the interest you feel in this woman, sister, as you do\nnot know her?\" the young man asked in surprise.\n\n\"She is a woman, and unhappy. Are not those reasons sufficient?\"\n\n\"Perhaps so,\" Shaw remarked doubtfully.\n\n\"Child!\" Ellen muttered, \"Can you not read in your own heart, the motive\nof my conduct toward this stranger?\"\n\nThe young savage started at this remark.\n\n\"It is true!\" He exclaimed passionately. \"Pardon me, sister! I am mad;\nbut I love you, and you know me better than I do myself.\"\n\nAnd rising hurriedly, he kissed his sister, threw his rifle over his\nshoulder, and ran off in the direction of Santa Fe.\n\nWhen he had disappeared in the gloom, and the sound of his footsteps had\ndied out in the distance, the girl fell on the ground, muttering in a\nlow, sad voice:\n\n\"Will he succeed?\"\n\n\n\n\nCHAPTER XII.\n\nDIAMOND CUT DIAMOND.\n\n\nRed Cedar did not remain long under the effect of the startling insult\nhe had received. Pride, wrath, and, before all, the desire to avenge\nhimself restored his strength, and a few minutes after Don Pablo\nZarate's departure the squatter had regained all his coolness and\naudacity.\n\n\"You see, senor padre,\" he said, addressing the monk, \"that our little\nplans are known to our enemies; we must, therefore, make haste if we do\nnot wish to see persons break in here, from whom it is of the utmost\nimportance to conceal ourselves. Tomorrow night at the latest, perhaps\nbefore, we shall start. Do not stir from here till my return. Your face\nis too well known at Santa Fe for you to venture to show it in the\nstreets without imprudence.\"\n\n\"Hum!\" the monk muttered, \"That demon, whom I fancied dead, is a rude\nadversary. Fortunately we shall soon have nothing more to fear from his\nfather, for I hardly know how we should get out of it.\"\n\n\"If the son has escaped us,\" Red Cedar said with an ugly smile, \"that is\nfortunately not the case with the father. Don't be alarmed; Don Miguel\nwill cause us no further embarrassment.\"\n\n\"I wish it most earnestly, _canarios!_ for he is a determined man; but I\nconfess to you that I shall not be entirely at my ease till I have seen\nhim fall beneath the bullets of the soldiers.\"\n\n\"You will not have long to wait. General Ventura has ordered me to go\nand meet the regiment of dragoons he expects, in order to hurry them on,\nand bring them into the town this very night, if possible. So soon as\nthe governor has an imposing force at his disposal he will no longer\nfear a revolt on the part of the troops, and give the order for\nexecution without delay.\"\n\n\"May Heaven grant it! But,\" he added with a sigh of regret, \"what a pity\nthat most of our scamps deserted us! We should have almost arrived at\nthe placer by this time, and been safe from the vengeance of our\nenemies.\"\n\n\"Patience, senor padre; all is for the best, perhaps, trust to me.\nAndres, my horse.\"\n\n\"You will start at once, then?\"\n\n\"Yes. I recommend you to watch carefully over our prisoner.\"\n\nThe monk shrugged his shoulders.\n\n\"Our affairs are tolerably well embarrassed already; then why burden\nourselves with a woman?\"\n\nThe squatter frowned.\n\n\"That is my business,\" he exclaimed in a peremptory tone. \"Keep all\nstupid observations to yourself. A thousand devils! I know what I am\nabout. That woman will possibly prove our safeguard at a later date.\"\n\nAnd mounting his horse, Red Cedar galloped out of Santa Fe.\n\n\"Hum!\" Andres Garote said as he watched him depart, \"what a diabolical\neye! Though I have known him several years, I never saw him like that\nbefore. How will all this end?\"\n\nWithout further remarks he arranged matters in the rancho, repairing as\nwell as he could the disorder caused by the previous struggle; then he\ntook a look round him. The monk, with his elbows on the table and a\ncigarette in his mouth, was drinking the fluid left in the bottle,\ndoubtless to console himself for the _navajada_ with which Don Pablo had\nfavoured him.\n\n\"Why, senor padre,\" the ranchero said in an insinuating voice, \"do you\nknow that it is hardly five o'clock?\"\n\n\"Do you think so?\" the other answered for the sake of saying something.\n\n\"Does not the time seem to you to go very slowly?\"\n\n\"Extraordinarily so.\"\n\n\"If you liked we could easily shorten it.\"\n\n\"In what way?\"\n\n\"Oh, for instance, with these.\"\n\nAnd Andres drew from his boot a pack of greasy cards, which he\ncomplacently spread out on the table.\n\n\"Ah! That is a good idea,\" the monk exclaimed with sparkling eyes. \"Let\nus have a game of monte.\"\n\n\"At your orders.\"\n\n\"Don Andres, you are a most worthy gentlemen. What shall we play for?\"\n\n\"Ah, hang it! That is true; we must play for something,\" the ranchero\nsaid, scratching his head.\n\n\"The merest trifle, simply to render the game interesting.\"\n\n\"Yes, but to do that man must possess the trifle.\"\n\n\"Do not let that trouble you. If you permit me I will make you a\nproposal.\"\n\n\"Do so, senor. You are a remarkable clever man, and can have none but\nbright ideas.\"\n\nThe monk bowed to his flattering insinuation.\n\n\"This is it: we will play, if you like for the share of the gold we\nshall receive when we reach the placer.\"\n\n\"Done!\" the ranchero shouted enthusiastically.\n\n\"Well,\" the monk said, drawing from his pocket a pack of cards no less\ndirty than the others, \"we can at any rate kill time.\"\n\n\"What! You have cards too?\" the ranchero remarked.\n\n\"Yes, and quite new, as you see.\" Andres bowed with an air of\nconviction.\n\nThe game began at once, and soon the two men were completely absorbed in\nthe combinations of the _seis de copas,_ the _as de bastos_, the _dos de\noro_, and the _cuatro d'espadas_. The monk, who had no necessity to\nfeign at this moment, as he was in the company of a man thoroughly\nacquainted with him, yielded frenziedly to his ruling passion. In\nMexico, and throughout Spanish America, the _angelus_ rings at sunset.\nIn those countries, where there is no twilight, night arrives without\ntransition, so that ere the bell has done tinkling the gloom is dense.\nAt the last stroke of the angelus the game ceased, as if by common\nagreement between the two men, and they threw their cards on the table.\n\nAlthough Garote was a passed master in trickery, and had displayed all\nhis science, he found in the monk so skilful an adversary that, after\nmore than three hours of an obstinate struggle, they both found\nthemselves as little advanced as at the outset. The monk, however, on\ncoming to the rancho, had an object which Red Cedar was far from\nsuspecting.\n\nFray Ambrosio rested his arms on the table, bent his body slightly\nforward, and while carelessly playing with the cards, which he amused\nhimself by sorting, he said to the ranchero, as he fixed a scrutinising\nglance upon him,--\n\n\"Shall we talk a little, Don Andres?\"\n\n\"Willingly,\" the latter replied, who had partly risen, but now fell back\non his chair.\n\nBy a secret foreboding Andres Garote had guessed that the monk wished to\nmake some important proposal to him. Hence, thanks to that instinctive\nintuition which rogues possess for certain things, the two men read each\nother's thoughts. Fray Ambrosio bit his lips, for the gambusino's\nintelligence startled him. Still the latter bent upon him a glance so\nfull of stupid meaning, that he continued to make a confidant of him, as\nit were involuntarily.\n\n\"Senor Don Andres,\" he said in a soft and insinuating voice, \"what a\nhappiness that your poor brother, on dying, revealed to me the secret of\nthe rich placer, which he concealed even from yourself!\"\n\n\"It is true,\" Andres answered, turning slightly pale; \"it was very\nfortunate, senor padre. For my part, I congratulate myself on it daily.\"\n\n\"Is it not so? For without it the immense fortune would have been lost\nto you and all else.\"\n\n\"It is terrible to think of.\"\n\n\"Well, at this moment I have a horrible fear.\"\n\n\"What is it, senor padre?\"\n\n\"That we have deferred our departure too long, and that some of those\nEuropean vagabonds we were speaking of just now may have discovered our\nplacer. Those scoundrels have a peculiar scent for finding gold.\"\n\n\"_Caray_, senor padre!\" Andres said, striking the table with a feigned\ngrief (for he knew very well what the monk was saying was only a clever\nway of attaining his real point), \"that would drive me mad--an affair so\nwell managed hitherto.\"\n\n\"That is true,\" Fray Ambrosio said in corroboration. \"I could never\nconsole myself.\"\n\n\"_Demonios_! I have as great an interest in it as yourself, senor\npadre,\" the gambusino replied with superb coolness. \"You know that an\nuninterrupted succession of unfortunate speculations robbed me of my\nfortune, and I hoped thus to regain it at a stroke.\"\n\nAt these words Fray Ambrosio had incredible difficulty in repressing a\nsmile; for it was a matter of public notoriety that senor Don Andres\nGarote was a lepero, who, as regarded fortune, had never possessed a\nfarthing of patrimony; that throughout his life he had never been aught\nbut an adventurer; and that the unlucky speculations of which he\ncomplained were simply an ill luck at monte, which had recently stripped\nhim of 20,000 piastres, acquired Heaven alone knew how. But senor Don\nAndres Garote was a man of unequalled bravery, gifted with a fertile and\nready mind, whom the accidents of life had compelled to live for a\nlengthened period on the _llanos_ (prairies), whose paths he knew as\nthoroughly as he did the tricks of those who dwelt on them. Hence, and\nfor many other reasons, Andres Garote was an invaluable comrade for Fray\nAmbrosio, who had also a bitter revenge to take on the monte table,\nbecause he pretended to place the most sincere faith in what it pleased\nhis honourable mate to say touching his lost fortune.\n\n\"However,\" he said, after an instant's reflection, \"supposing that the\nplacer is intact, and that no one has discovered it, we shall have a\nlong journey to reach it.\"\n\n\"Yes,\" the gambusino remarked, significantly; \"the road is difficult and\nbroadcast with perils innumerable.\"\n\n\"We must march with our chins on our shoulders, and finger on the rifle\ntrigger--\"\n\n\"Fight nearly constantly with wild beasts or Indians--\"\n\n\"In a word, do you not believe that the woman Red Cedar has carried off\nwill prove a horrid bore?\"\n\n\"Dreadfully so,\" Andres made answer, with an intelligent glance.\n\n\"What is to be done?\"\n\n\"Hang it! That is difficult to say.\"\n\n\"Still we cannot run the risk, on account of a wretched woman, of having\nour hair raised by the Indians.\"\n\n\"That's true enough.\"\n\n\"Is she here?\"\n\n\"Yes,\" the gambusino said, pointing to a door; \"in that room.\"\n\n\"Hum!\"\n\n\"You remarked--\"\n\n\"Nothing.\"\n\n\"Could we not--\"\n\n\"What?\"\n\n\"It is perhaps difficult,\" Andres continued, with feigned hesitation.\n\n\"Explain yourself.\"\n\nThe gambusino seemed to make up his mind.\n\n\"Suppose we restore her to her family?\" he said.\n\n\"I have thought of that already.\"\n\n\"That is strange.\"\n\n\"It must be all managed very cleverly.\"\n\n\"And the relations pay a proper ransom.\"\n\n\"That is what I meant to say.\".\n\nThere was a silence.\n\nDecidedly these two honourable persons were made to understand one\nanother.\n\n\"But who is to undertake this delicate mission?\" asked the monk.\n\n\"I, _con mil demonios!_\" the gambusino exclaimed, his eyes sparkling\nwith greed at the thought of the rich ransom he would demand.\n\n\"But if Red Cedar were to find out,\" the monk remarked, \"that we\nsurrendered his prisoner?\"\n\n\"Who will tell him?\"\n\n\"I am sure I shan't.\"\n\n\"Nor I.\"\n\n\"It is very easy; the girl will have escaped.\"\n\n\"Quite true.\"\n\n\"Do not let us lose time, then. You have a horse?\"\n\n\"I have two.\"\n\n\"Bravo! You will place Dona Clara on one, and mount the other yourself.\"\n\n\"And go straight to the Hacienda de la Noria.\"\n\n\"That is it. Don Pablo will be delighted to recover his sister, whom he\nexpected never to see again, and will not haggle over the price he pays\nfor her deliverance.\"\n\n\"Famous! In that way we run no risk of not reaching the placer, as our\nparty will only consist of men.\"\n\n\"Excellently reasoned!\"\n\nAndres Garote rose with a smile which would have caused the monk to\nreflect, had he seen it; but at the same moment the latter was rubbing\nhis hands, saying in a low voice, and with a most satisfied air,--\n\n\"Now, my scamp, I've got you.\"\n\nWhat secret thought possessed these two men, who were carrying on a\nmutual deceit, none save themselves could have said. The gambusino\napproached the door of the room where Dona Clara was confined, and put\nthe key in the lock. At this moment two vigorous blows were dealt on the\ndoor of the rancho, which had been carefully bolted after Red Cedar's\ndeparture. The two accomplices started.\n\n\"Must I open?\" Andres asked.\n\n\"Yes,\" the monk answered; \"hesitation or refusal might create alarm. In\nour position we must foresee everything.\"\n\nThe ranchero went to open the door, which the newcomer threatened to\nbreak in. A man walked in, who took a careful glance around, then doffed\nhis hat and bowed. The confederates exchanged a glance of vexation on\nrecognising him, for he was no other than Shaw, Red Cedar's youngest\nson.\n\n\"I am afraid I disturb you, gentlemen,\" the young man said, with an\nironical smile.\n\n\"Not at all,\" Andres made answer; \"on the contrary, we are delighted to\nsee you.\"\n\n\"Thanks!\"\n\nAnd the young man fell back into a butaca.\n\n\"You are very late at Santa Fe,\" the monk remarked.\n\n\"It is true,\" the American said, with some embarrassment; \"I am looking\nfor my father, and fancied I should find him here.\"\n\n\"He was so a few hours back, but he was obliged to leave us.\"\n\n\"Ah!\"\n\nThis exclamation was rather drawn from the young man by the necessity he\nfelt of replying, than through any interest he took in the information\nafforded him. He was evidently preoccupied; but Fray Ambrosio did not\nappear to notice it, as he continued,--\n\n\"Yes: it appears that his Excellency the Governor ordered your father to\ngo and meet a regiment of dragoons intended to reinforce the garrison,\nand hasten its march.\"\n\n\"That is true; I forgot it.\"\n\nThe monk and the miner did not at all understand the American's conduct,\nand lost themselves in conjectures as to the reasons that brought him to\nthe rancho. They guessed instinctively that what he said about his\nfather was only a pretext or means of introduction; and that a powerful\nmotive, he would not or dared not avow, had brought him. For his part,\nthe young man, in coming to the Rancho del Coyote, where he knew that\nDona Clara was imprisoned, expected to find Andres alone, with whom he\nhoped to come to an understanding in some way or another. The presence\nof the monk disturbed all his plans. Still, time was slipping away he\nmust make up his mind, and, before all, profit by Red Cedar's\nprovidential absence, which offered him an opportunity he could hardly\ndare to hope again.\n\n\n\n\nCHAPTER XIII.\n\nA STORMY DISCUSSION.\n\n\nShaw was not timid, as we have said--he ought rather be accused of the\nopposite excess; he was not the man, once his resolution was formed, to\nlet anything soever turn him from it. His hesitation was not long; he\nsuddenly rose, and violently stamping his rifle butt on the ground,\nlooked at the two men, while saying in a firm voice,--\n\n\"Be frank, my presence here at this hour astonishes you, and you ask\nyourselves what cause can have brought me.\"\n\n\"Sir,\" the monk said, with a certain degree of hesitation rendered\nhighly natural by the young man's tone.\n\n\"Pardon me,\" Shaw exclaimed, interrupting him, \"the cause you will seek\nin vain. I will tell you: I have come to deliver Dona Clara.\"\n\n\"Can it be possible?\" the two men exclaimed with stupefaction.\n\n\"It is so; whether you like it or not, I care little. I am the man to\nhold my own against both of you, and no one can prevent me restoring the\nmaiden to her father, as I have resolved on doing.\"\n\n\"What do I hear?\" said Fray Ambrosio.\n\n\"Hum!\" the young man continued quickly, \"Believe me, do not attempt any\nuseless resistance, for I have resolved, if needs must, to pass over\nyour bodies to success.\"\n\n\"But we have not the slightest wish--\"\n\n\"Take care,\" he interrupted him in a voice full of menace and frowning,\n\"I will only leave this house accompanied by her I wish to save.\"\n\n\"Sir,\" the monk remarked, in an authoritative voice which momentarily\nquelled the young savage, \"two words of explanation.\"\n\n\"Make haste!\" he answered, \"For I warn you that my patience is\nexhausted.\"\n\n\"I do not insist on your listening any length of time. You have come\nhere, you say, with the intention of delivering Dona Clara?\"\n\n\"Yes,\" he answered impatiently, \"and if you attempt to oppose it--\"\n\n\"Pardon me,\" the monk interrupted, \"such a determination on your part\nnaturally surprises us.\"\n\n\"Why so?\" the young man said, raising his head haughtily.\n\n\"Because,\" Fray Ambrosio answered tranquilly, \"You are the son of Red\nCedar, and it is at least I strange that--\"\n\n\"Enough talking,\" Shaw exclaimed violently; \"will you or not give me up\nher I have come to seek?\"\n\n\"I must know, in the first place, what you intend doing with her.\n\n\"How does that concern you?\"\n\n\"More than you imagine. Since that girl has been a prisoner I\nconstituted myself--if not her guardian, for the dress I wear forbids\nthat--her defender; in that quality I have the right of knowing for what\nreason you, the son of the man who tore her from her family, have come\nso audaciously to demand her surrender to you, and what your object is\nin acting thus?\"\n\nThe young man had listened to those remarks with an impatience that\nbecame momentarily more visible; it could be seen that he made\nsuperhuman efforts to restrain himself. When the monk stopped, he looked\nat him for a moment with a strange expression, then walked up so close\nas almost to touch him, drew a pair of pistols from his girdle and\npointed them at the monk.\n\n\"Surrender Dona Clara to me,\" he said, in a low and menacing voice.\n\nFray Ambrosio had attentively followed all the American's movements, and\nwhen the latter put the pistol muzzles to his chest, the monk, with an\naction rapid as lightning, also drew two pistols from his girdle, and\nplaced them, on his adversary's chest. There was a moment of supreme\nexpectation, of indescribable agony; the two men were motionless, face\nto face panting, each with his fingers on a trigger, pale, and their\nbrows dank with cold perspiration. Andres Garote, his lips curled by an\nironical smile, and his arms crossed, carelessly leaned against a table,\nwatching this scene which had for him all the attractions of a play.\n\nAll at once the door of the rancho, which had not been fastened again\nafter the squatter's entry was violently thrown back and a man appeared.\nIt was Father Seraphin. At a glance he judged the position and boldly\nthrew himself between the foemen, hurling them back, but not uttering a\nword. The two men recoiled, and lowered their weapons, but continued to\nmenace each other with their glances.\n\n\"What!\" the missionary said in a deep voice, \"Have I arrived just in\ntime to prevent a double murder, gentlemen? In Heaven's name, hide those\nhomicidal weapons; do not stand opposite each other like wild beasts\npreparing for a leap.\"\n\n\"Withdraw, father; you have nothing to do here. Let me treat this man as\nhe deserves,\" the squatter answered, casting at the missionary a\nferocious glance--\"his life belongs to me.\"\n\n\"Young man,\" the priest replied, \"the life of a fellow being belongs\nonly to God, who has the right to deprive, him of it; lower your\nweapons\"--and turning to Fray Ambrosio, he said to him in a cutting\nvoice, \"and you who dishonour the frock you wear, throw away those\npistols which sully your hands--a minister of the altar should not\nemploy other weapons than the Gospel.\"\n\nThe monk bowed, and caused his pistols to disappear, saying in a soft\nand cautious voice, \"My father, I was compelled to defend my life which\nthat maniac assailed. Heaven is my witness that I reprove these violent\nmeasures, too frequently employed in this unhappy country; but this man\ncame into the house with threats on his lips; he insisted on our\ndelivering a wretched girl whom this caballero,\" he said, pointing to\nthe gambusino, \"and myself did not think proper to surrender.\"\n\nAndres corroborated the monk's words by a nod of the head.\n\n\"I wish to save that young girl from your hands,\" Shaw said, \"and\nrestore her to her father.\"\n\n\"Of whom are you speaking, my friend?\" the missionary asked with a\nsecret beating of his heart.\n\n\"Of whom should I speak, save Dona Clara de Zarate, whom these villains\nretain here by force?\"\n\n\"Can it be possible?\" Father Seraphin exclaimed in amazement. \"Dona\nClara here?\"\n\n\"Ask those men,\" Shaw answered, roughly, as he angrily struck the butt\nof his rifle against the ground.\n\n\"Is it true?\" the priest inquired.\n\n\"It is,\" the gambusino answered.\n\nFather Seraphin frowned, and his pale forehead was covered with febrile\nruddiness.\n\n\"Sir,\" he said, in a voice choking with indignation. \"I summon you, in\nthe name of that God whom you serve, and whose minister you lay claim to\nbeing, to restore at once to liberty the hapless girl whom you have so\nunworthily imprisoned, in defiance of all laws, human and divine. I\nengage to deliver her into the hands of those who bewail her loss.\"\n\nFray Ambrosio bowed; he let his eyes fall, and said in a hypocritical\nvoice--\n\n\"Father, you are mistaken as regards myself. I had nothing to do with\nthe carrying off of that poor child, which on the contrary, I opposed to\nthe utmost of my power; and that is so true, father,\" he added, \"that at\nthe moment when this young madman arrived, the worthy gambusino and\nmyself had resolved, at all risks, on restoring Dona Clara to her\nfamily.\"\n\n\"I should wish to believe you, sir; if I am mistaken, as you say, you\nwill forgive me, for appearances were against you; it only depends on\nyourself to produce a perfect justification by carrying out my wishes.\"\n\n\"You shall be satisfied, father,\" the monk replied. At a signal from him\nGarote left the room. During the few words interchanged between the two\nmen, Shaw remained motionless, hesitating, not knowing what he ought to\ndo; but he suddenly made up his mind, threw his rifle over his shoulder,\nand turned to the missionary.\n\n\"Father,\" he said respectfully, \"my presence is now needless here.\nFarewell; my departure will prove to you the purity of my intentions.\"\n\nAnd turning suddenly on his heel, he hurried out of the rancho. A few\nmoments after his departure the gambusino returned, Dona Clara following\nhim.\n\nDona Clara no longer wore the dress of the whites, for Red Cedar, in\norder to render her unrecognizable, had compelled her to don the Indian\ngarb, which the maiden wore with an innate grace which heightened its\nstrange elegance. Like all Indian squaws, she was attired in two white\nchemises of striped calico--the one fastened around the neck, fell to\nthe hips; while the other, drawn in at the waist, descended to her\nankles. Her neck was adorned with collars of fine pearls, mingled with\nthose small shells called wampum, and employed by the Indians as money.\nHer arms and ankles were surrounded by wide circles of gold, and a small\ndiadem of the same metal relieved the pale tint of her forehead.\nMoccasins of deer hide, embroidered with wool and beads of every colour\nimprisoned her small and high-arched feet.\n\nAs she entered the room, a shadow of melancholy and sadness spread over\nher face, adding, were that possible, a further charm to her person. On\nseeing the missionary, Dona Clara uttered a cry of joy, and rushed\ntoward him, fell into his arms, and murmured in a heart-rending voice:--\n\n\"Father! save me! save me!\"\n\n\"Be calm, my daughter!\" the priest said to her, gently. \"You have\nnothing more to fear now that I am near you.\"\n\n\"Come!\" she exclaimed, wildly, \"Let us fly from this accursed house, in\nwhich I have suffered so greatly.\"\n\n\"Yes, my daughter, we will go; set your mind at rest.\"\n\n\"You see, father,\" Fray Ambrosio said, hypocritically, \"that I did not\ndeceive you.\"\n\nThe missionary cast at the monk a glance of undefinable meaning.\n\n\"I trust that you spoke truly,\" he replied; \"the God who gauges hearts\nwill judge you according to works. I will rescue this maiden at once.\"\n\n\"Do so, father; I am happy to know her under your protection.\"\n\nAnd picking up the cloak which Don Pablo left after blinding Red Cedar,\nhe placed it delicately on the shuddering shoulders of Dona Clara, in\norder to conceal her Indian garb. Father Seraphin drew her arm through\nhis own, and led her from the rancho. Ere long they disappeared in the\ndarkness. Fray Ambrosio looked after them as long as he could see them,\nand then re-entered the room, carefully bolting the door after him.\n\n\"Well,\" Andres Garote asked him, \"what do you think, senor Padre, of all\nthat has happened?\"\n\n\"Perhaps things are better as they are.\"\n\n\"And Red Cedar?\"\n\n\"I undertake to render ourselves as white in his sight as the snows of\nthe Caffre de Perote.\"\n\n\"Hum! it will be difficult.\"\n\n\"Perhaps so.\"\n\n\n\n\nCHAPTER XIV.\n\nTHE MYSTERY.\n\n\nOn leaving the Rancho del Coyote, Red Cedar dug his spurs into his\nhorse's flanks, and galloped in a south-western direction. So soon as he\nwas out of the town he turned to the left, took a narrow path that ran\nround the walls, pulled up his horse, and advanced with the utmost\ncaution. Throwing suspicious glances on either side, he went on thus for\nabout three-quarters of an hour, when he reached a house, in one of the\nwindows of which burned three wax tapers.\n\nThe lights thus arranged were evidently a signal for the squatter, for\nso soon as he came to the house he stopped and dismounted, attached his\nhorse to a larch-tree, and prudently concealing himself behind a\nthicket, imitated thrice at equal intervals the hu-hu of an owl. The\nlights burning in the window were extinguished, as if by enchantment.\n\nThe night was gloomy, only a few stars studded the vault of heaven; a\nleaden silence brooded over the plain, which appeared quite solitary. At\nthis moment a voice could be heard from the house which Red Cedar was\nwatching so carefully. The squatter listened; the speaker leaned for a\nsecond out of the window looked cautiously round, and disappeared\nmuttering loud enough for the American to overhear--\n\n\"All is quiet in the neighbourhood.\"\n\n\"Still,\" the squatter said, without showing himself, \"the coyotes prowl\nabout the plain.\"\n\n\"Are you coming or going?\" the man at the window continued.\n\n\"Both,\" the squatter answered, still hidden behind his bush.\n\n\"You can come on, for you are expected.\"\n\n\"I know it; hence here I am.\"\n\nWhile making this answer, the squatter left his hiding place, and placed\nhimself before the door with folded arms, like a man who has nothing to\nfear.\n\nThe door was cautiously opened; a man emerged, carefully wrapped up in,\na wide cloak, which only allowed eyes to be seen, that flashed in the\ngloom like a jackal's. This person walked straight up to Red Cedar.\n\n\"Well,\" he asked, in a low voice, \"have you reflected?\"\n\n\"Yes.\"\n\n\"And what is the result of your reflections?\"\n\n\"I refuse.\"\n\n\"Still?\"\n\n\"More than ever.\"\n\n\"Take care.\"\n\n\"I do not care, Don Melchior, for I am not afraid of you.\"\n\n\"No names!\" the stranger exclaimed, impatiently.\n\n\"We are alone.\"\n\n\"No one is ever alone in the desert.\"\n\n\"That is true,\" Red Cedar muttered. \"Let us return to our business.\"\n\n\"It is simple--give and give.\"\n\n\"Hum! You get to work very fast; unfortunately it cannot be so.\"\n\n\"Why not?\"\n\n\"Why, because I am growing tired of constantly taking in my nets game by\nwhich others profit, and which I ought to keep as a safeguard.\"\n\n\"You call that girl a guarantee?\"\n\n\"By Heaven! what else do you mean to make of her?\"\n\n\"Do not compare me with you, scoundrel!\"\n\n\"Where is the difference between us? I am a scoundrel, I grant; but, by\nheaven, you are another, my master, however powerful you may be.\"\n\n\"Listen, caballero!\" the stranger answered, in a cutting voice. \"I will\nlose no more of my time in discoursing with you. I want that girl, and\nwill have her, whatever you may do to prevent me.\"\n\n\"Good; in that case you declare war against me?\" the squatter said, with\na certain tinge of alarm, which he tried in vain to conceal.\n\nThe stranger shrugged his shoulders.\n\n\"We have known one another long enough to be perfectly well acquainted;\nwe can only be friends or foes. Is not that your opinion?\"\n\n\"Yes.\"\n\n\"Well, then, hand Dona Clara over to me, and I will give you the papers\nwhich--\"\n\n\"Enough!\" the squatter said, sharply. \"Have you those papers about you?\"\n\nThe stranger burst into a laugh.\n\n\"Do you take me for such a fool?\" he said.\n\n\"I do not understand you.\"\n\n\"I will not insult you by believing you. No, I have not those papers\nabout me. I am not such an ass as to risk assassination at your hands.\"\n\n\"What would your death profit me?\"\n\n\"Hang it all! If it were only my scalp you would be sure to receive at\nleast fifty dollars for it.\"\n\nAt this mournful jest the squatter began laughing.\n\n\"I did not think of that,\" he said,\n\n\"Listen to me, Red Cedar, and print the words on your memory.\"\n\n\"Speak.\"\n\n\"In a month from today, hour for hour, day for day, wherever you may be,\nI shall present myself to you.\"\n\n\"For what purpose?' the squatter asked impudently.\n\n\"To repeat my demand with reference to the prisoner.\"\n\n\"Then, as now, I shall reply No, my master.\"\n\n\"Perhaps so. Live and learn. Now good-bye, and may the devil, your\npatron saint, preserve you in good health until our next meeting. You\nknow that I have you tight; so consider yourself warned.\"\n\n\"Good, good! Threats do not frighten me. _Demonios_, since I have been\ntraversing the desert, I have found myself opposed to enemies quite as\ndangerous as you, and yet I managed to get quit of them.\"\n\n\"That is possible, Red Cedar; but believe me, meditate carefully on my\nwords.\"\n\n\"I repeat that your threats do not frighten me.\"\n\n\"I do not threaten, I warn you.\"\n\n\"Hum! Well, then, listen in your turn. In the desert, every man armed\nwith a good rifle has nothing to fear from whomsoever.\"\n\n\"What next?\" the stranger interrupted him, in a sarcastic voice.\n\n\"Well, my rifle is excellent, I have a sure aim, and I say no more.\"\n\n\"Nonsense, you are mad! I defy you to kill me!\"\n\n\"Hang it, though, what can be your motive for wishing to have this girl\nin your power?\"\n\n\"That is no affair of yours. I have no explanations due to you. Enough\nfor you to know that I want her.\"\n\n\"You shall not have her.\"\n\n\"We shall see. Good-bye, Red Cedar.\"\n\n\"Good-bye, Don Melchior, or whatever be the name you please to bear.\"\n\nThe stranger made no reply, but turned his head with a gesture of\ncontempt, and whistled. A man emerged from the house, holding a horse by\nthe bridle; at one bound the stranger reached the saddle, and ordered\nthe servant to withdraw.\n\n\"Farewell, _Compadre_, remember our appointment.\"\n\nAnd loosing his reins, the stranger started at a gallop, not\ncondescending even to turn his head. Red Cedar looked after him with an\nindescribable expression of rage.\n\n\"Oh,\" he muttered in a low voice, \"demon! Shall I never free myself from\nyour clutches?\"\n\nAnd with a motion rapid as thought he shouldered his rifle, and aimed at\nthe departing man. All at once the latter turned his horse, and stood\nright opposite Red Cedar.\n\n\"Mind not to miss me!\" he cried, with a burst of laughter that caused a\ncold perspiration to bead on the bandit's forehead.\n\nThe latter let his rifle fall, saying in a hollow voice: \"He is right,\nand I am mad! If I only had the papers!\"\n\nThe stranger waited for a moment calm and motionless; then he started\nagain and soon disappeared in the darkness. Red Cedar stood with his\nbody bowed forward, and his ears on the watch, so long as the horse's\nhoofs could be heard; then he returned to his own steed, and bounded\ninto the saddle.\n\n\"Now to go and warn the dragoons,\" he said, and pushed on.\n\nThe squatter had scarce departed ere several men appeared from either\nside; they were Valentine, Curumilla, and Don Pablo on the right;\nUnicorn and Eagle-wing on the left. Valentine and his friends were\nastonished at meeting the Comanche chief, whom they believed gone back\nto his camp; but the sachem explained to them, in a few words, how, at\nthe moment he was crossing the spot where they now were, he had heard\nRed Cedar's voice, and concealed himself in the shrubs in order to\noverhear the squatter's colloquy with his strange friend. Valentine had\ndone the same; but, unfortunately, the party had been greatly\ndisappointed, for the squatter's conversation remained to them an\nenigma, of which they sought the key in vain.\n\n\"'Tis strange,\" Valentine remarked, as he passed his hand several times\nacross his forehead. \"I do not know where I have seen the man just now\ntalking here with Red Cedar, but I have a vague reminiscence of having\nmet him before, where and under what circumstance I try, though in vain,\nto recall.\"\n\n\"What shall we do?\" Don Pablo asked.\n\n\"Hang it, what we agreed on;\" and turning to the chief, he said, \"Good\nluck, brother, I believe we shall save our friend.\"\n\n\"I am sure of it,\" the Indian replied, laconically.\n\n\"May heaven hear you, brother,\" Valentine continued. \"Act! While, on\nyour side, you watch the town for fear of treason. We then will ambush\nourselves on the road the gambusinos must take, in order to know\npositively the direction in which they are proceeding. Till tomorrow,\nchief!\"\n\n\"Stop!\" a panting voice exclaimed, and a man suddenly appeared in the\nmidst of them.\n\n\"Father Seraphin!\" Valentine said in a surprise. \"What chance brings you\nthis way?\"\n\n\"I was looking for you.\"\n\n\"What do you want with me?\"\n\n\"To give you some good news.\"\n\n\"Speak! Speak quickly, father! Has Don Miguel left his prison?\"\n\n\"Alas! Not yet; but his daughter is free!\"\n\n\"Dona Clara free!\" Valentine shouted joyously. \"Heaven be blessed! Where\nis she?\"\n\n\"She is temporarily in safety, be assured of that; but let me give you a\nwarning, which may perhaps prove useful to you.\"\n\n\"Speak! Speak!\"\n\n\"By order of the governor, Red Cedar has gone to meet the regiment of\ndragoons, coming up to reinforce the Santa Fe garrison.\"\n\n\"_Caramba_,\" Valentine said, \"are you sure of your statement, father?\"\n\n\"I am: in my presence, the men who carried off Dona Clara spoke about\nit.\"\n\n\"All is lost if these soldiers arrive.\"\n\n\"Yes,\" the missionary said; \"but, how to prevent it?\"\n\nCurumilla lightly touched the leader's arm.\n\n\"What do you want, chief!\"\n\n\"The Comanches are warriors,\" Curumilla answered, curtly.\n\n\"Ah!\" Valentine exclaimed, and tapping his forehead with delight, \"that\nis true, chief; you save us.\"\n\nCurumilla smiled with pleasure.\n\n\"While you go in pursuit of the soldiers,\" said Don Pablo, \"as I can be\nof no service to you, I will accompany Father Seraphin to my poor\nsister, whom I have not seen so long, and am eager to embrace.\"\n\n\"Do so,\" Valentine answered. \"At daybreak you will bring Dona Clara to\nthe camp, that I may myself deliver her to her father.\"\n\n\"That is agreed.\"\n\nValentine, Curumilla, and Unicorn rushed out in the plain, while Father\nSeraphin and Don Pablo returned to the town. The two gentlemen, anxious\nto join the girl, did not perceive that they were closely watched by an\nindividual, who followed their every movement, while careful not to be\nseen by them. It was Nathan, Red Cedar's eldest son.\n\nHow was that man there?\n\n\n\n\nCHAPTER XXXIX.\n\nTHE AMBUSCADE.\n\n\nThe nigh breeze had swept the clouds away; the sky, of a deep azure, was\nstudded with an infinity of stars; the night was limpid, the atmosphere\nso transparent as to allow the slightest varieties of the landscape to\nbe distinguished. About four leagues from Santa Fe, a numerous band of\nhorsemen was following a path scarce traced in the tall grass, which\napproached the town with countless turns and windings. These horsemen,\nwho marched in rather decent order, were nearly 600 in number, and\nformed the regiment of dragoons so anxiously expected by General\nVentura.\n\nAbout ten paces ahead rode four or five officers gaily chatting\ntogether, among whom was the colonel. The regiment continued its march\nslowly, advancing cautiously, through fear of losing its way in a\nperfectly strange country. The colonel and his officers who had always\nfought in the States bordering the Atlantic, found themselves now for\nthe first time in these savage countries.\n\n\"Caballeros,\" the colonel suddenly remarked, \"I confess to you that I am\ncompletely ignorant as to our whereabouts. Can any one of you throw a\nlight on the subject? This road is fearful, it seems to lead nowhere,\nand I am afraid we have lost our way.\"\n\n\"We are all as ignorant as yourself on that head, colonel,\" an officer\nanswered, \"not one of us could say where we are.\"\n\n\"On my word!\" the colonel went on, taking a glance of satisfaction\naround, \"We are not in a hurry to reach Santa Fe. I suppose it makes\nlittle difference whether we get there today or tomorrow. I believe that\nthe best thing for us to do is to bivouac here for the rest of the\nnight; at sunrise we will start again.\"\n\n\"You are right, colonel,\" the officer said, whom he seemed to address\nmost particularly, \"a few hours' delay is of no consequence, and we run\nthe risk of going out of our course.\"\n\n\"Give the order to halt.\"\n\nThe officer immediately obeyed; the soldiers, wearied with a long\nnight's march, greeted with shouts of joy the order to stop. They\ndismounted. The horses were unsaddled and picketed, campfires were\nlighted, in less than an hour the bivouac was arranged.\n\nThe colonel, in desiring to camp for the night, had a more serious fear\nthan that of losing his way; it was that of falling in with a party of\n_Indios bravos._\n\nThe colonel was brave, and had proved it on many occasions; grown gray\nin harness, he was an old soldier who feared nothing in the world\nparticularly; but accustomed to warfare in the interior of the Republic,\nhad never seen opposed to him any but civilised foes, he professed for\nthe Indians that instinctive fear which all the Mexicans entertain, and\nhe would not risk a fight with an Apache or Comanche war party in the\nmiddle of the night, in a country whose resources he did not know, and\nrun the risk of having his regiment cut to pieces by such Protean\nenemies. On the other hand, he was unaware that the governor of Santa Fe\nhad such pressing need of his presence, and this authorised him in\nacting with the utmost precaution. Still, as soon as the bivouac was\nestablished, and the sentries posted, the colonel sent off a dozen\nresolute men under an Alferez, to trot up the country and try to procure\na guide.\n\nWe will observe, in passing, that in Spanish America, so soon as you\nleave the capitals, such as Lima, or Mexico, roads, such as we\nunderstand them in Europe, no longer exist; you only find paths traced,\nin nine cases out of ten, by the footprints of wild beasts, and which\nare so entangled one with the other, that, unless you have been long\naccustomed to them, it is almost impossible to find your way. The\nSpaniards, we grant, laid out wide and firm roads, but since the War of\nIndependence, they had been cut up, deteriorated and so abandoned by the\nneglect of the ephemeral governments that have followed each other in\nMexico, that with the exception of the great highways of communication\nin the interior of the country, the rest had disappeared under the\nherbage.\n\nThe little squad of troopers sent out to beat up the country had started\nat a gallop, but it soon reduced its pace, and the soldiers and sergeant\nbegan laughing and talking, caring little for the important mission with\nwhich they were intrusted. The moon rose on the horizon, shedding her\nfantastic rays over the ground. As we have said, it was one of those\nlovely nights of the American desert full of strange odours. A majestic\nsilence hovered over the plain, only disturbed at intervals by those\nsounds, without any known cause, which are heard on the savannahs, and\nwhich seem to be the respiration of the sleeping world. Suddenly the\nmockingbird sung twice, and its plaintive and soft song resounded\nmelodiously through the air.\n\n\"Hallo,\" one of the dragoons said, addressing his comrade, \"that's a\nbird that sings very late.\"\n\n\"An evil omen,\" the other said with a shake of his head.\n\n\"_Canarios_! What omen are you talking about, comrade?\"\n\n\"I have always heard say,\" the second, speaker remarked sententiously,\n\"that when you hear a bird sing on your left at night it predicts\nmisfortune.\"\n\n\"The deuce confound you and your prognostics.\"\n\nAt this moment the song, which appeared previously some distance off,\ncould be heard much more close, and seemed to come from some trees on\nthe side of the path the dragoons were following. The Alferez raised his\nhead and stopped, as if mechanically trying to explain the sound that\nsmote his ears; but all became silent again, so he shook his head and\ncontinued his conversation. The detachment had been out more than an\nhour. During this long stroll, the soldiers had discovered nothing\nsuspicious; as for the guide they sought, it is needless to say that\nthey had not found him, for they had not met a living soul. The Alferez\nwas about to give orders to return to camp, when one of the troopers\npointed out to him some heavy, black forms, apparently prowling about\nunsuspiciously.\n\n\"What on earth can that be?\" the officer asked, after carefully\nexamining what was pointed out to him.\n\n\"_Caspita_,\" one of the dragoons exclaimed, \"that is easy to see; they\nare browsing deer!\"\n\n\"Deer!\" said the Alferez, in whom the hunter's instinct was suddenly\naroused, \"there are at least thirty; suppose we try to catch some.\"\n\n\"It is difficult.\"\n\n\"Pshaw!\" another soldier shouted, \"It is light enough for each of us to\nsend them a bullet.\"\n\n\"You must by no means use your carbines,\" the Alferez interposed\nsharply; \"if our shots, re-echoed through the mountains, caught the ears\nof the Indians, who are probably ambushed in the thickets, we should be\nruined.\"\n\n\"What is to be done, then?\"\n\n\"Lasso them, _caspita_, as you wish to try and catch them.\"\n\n\"That is true; I did not think of that.\"\n\nThe dragoons, delighted at the opportunity of indulging in their\nfavourite sport, dismounted, fastened their horses to the roadside trees\nand seized their lassos. They then advanced cautiously toward the deer,\nwhich continued grazing tranquilly, without appearing to suspect that\nenemies were so near them. On arriving at a short distance from the\ngame, the dragoons separated in order to have room for whirling their\nlassos, and making a covering of each tree, they managed to approach\nwithin fifteen paces of the animals. Then they stopped, exchanged\nglances, carefully calculated the distance, and, at a signal from their\nleader, sent their lassos whizzing through the air.\n\nA strange thing happened at this moment, however. All the deer hides\nfell simultaneously to the ground, displaying Valentine, Curumilla, and\na dozen Comanche warriors, who, profiting by the stupor of the troopers\nat their extraordinary metamorphosis, hunted the hunters by throwing\nlassos over their shoulders and hurled them to the ground. The ten\ndragoons and their leader were prisoners.\n\n\"Well, my friends,\" Valentine said with a grin, \"how do you like that\nsort of fun?\"\n\nThe startled dragoons made no reply, but allowed themselves to be bound;\none alone muttered between his teeth:--\n\n\"I was quite sure that villain of a mockingbird would bring us ill luck;\nit sang on our left. That never deceives, _Canarios!_\"\n\nValentine smiled at this sally. He then placed two fingers in his mouth\nand imitated the cry of the mockingbird with such perfection, that the\nsoldier looked up at the trees. He had scarce ended, when a rustling was\nheard among the bushes, and a man leaped between the hunters and their\nprisoners. It was Eagle-wing, the sachem of the Coras.\n\n\n\n\nCHAPTER XVI.\n\nA FRIENDLY DISCUSSION.\n\n\nAfter leaving his enemy (for the mysterious man with whom he had so\nstormy a discussion could be nothing else), Red Cedar set out to join\nthe regiment, and hasten its arrival according to the orders he had\nreceived. In spite of himself, the squatter was suffering from\nextraordinary nervousness, and involuntarily he went over the various\npoints of the conversation with the person who took such precautions in\ncommunicating with him. The threats he had proffered recurred to his\nmind. It appeared as if the bandit, who feared nothing in the world, had\ngood reason, however, for trembling in the presence of the man who, for\nmore than an hour, had crushed him with his irony. What reason could be\nso powerful as to produce so startling a change in this indomitable\nbeing? No one could have said; for the squatter was master of his\nsecret, and would have mercilessly killed anybody he suspected of having\nread even a portion of it.\n\nThe reason was, at any rate, very powerful; for after a few minutes of\ndeep thought, his hand let go the reins and his head fell on his breast:\nthe horse, no longer feeling the curb, stopped and began nibbling the\nyoung tree shoots. The squatter did not notice this halt; he was\nthinking, and hoarse exclamations now and then came from his chest, like\nthe growling of a wild beast. At length he raised his head.\n\n\"No,\" he shouted, as he directed a savage glance at the starlit sky,\n\"any struggle with that demon is impossible. I must fly, so soon as\npossible, to the prairies of the far west. I will leave this implacable\nfoe; I will fly from him, as the lion does, carrying off my prey in my\nclaws. I have not a moment to lose. What do I care for the Spaniards and\ntheir paltry disputes? General Ventura will seek another emissary, for\nmore important matters claim my attention. I must go to the Rancho del\nCoyote, for there alone I shall find my revenge. Fray Ambrosio and his\nprisoner can supply me with the weapons I need for the terrible contest\nI am compelled to wage against that demon who comes straight from hell,\nand whom I will send back there.\"\n\nAfter having uttered these words in a low voice, in the fashion of men\nwont to live in solitude, Red Cedar appeared to regain all his boldness\nand energy. He looked savagely around, and, burying his spurs in his\nhorse's flanks, he started with the speed of an arrow in the direction\nof the rancho, which he had left but a few hours previously, and where\nhis two accomplices still remained.\n\nThe monk and the gambusino, delighted at the unforeseen termination of\nthe scene we recently narrated, delighted above all at having got rid of\nDona Clara without being immediately mixed up in her escape, tranquilly\nresumed their game of _monte_, and played with that mental satisfaction\nproduced by the certainty of having nothing to reproach themselves with,\ndisputing with the utmost obstinacy for the few reals they still\nhappened to have in their pockets. In the midst of a most interesting\ngame, they heard the furious gallop of a horse up the paved street.\nInstinctively they stopped and listened; a secret foreboding seemed to\nwarn them that this horse was coming to the rancho, and that its rider\nwanted them.\n\nIn truth, neither Fray Ambrosio nor Andres Garote had a quiet\nconscience, even supposing, which was very doubtful, that either had a\nconscience at all, for they felt they were responsible to Red Cedar for\nDona Clara. Now that the maiden had escaped like, a bird flying from its\ncage, their position with their terrible ally appeared to them in all\nits desperate gravity. They did not conceal from themselves that the\nsquatter would demand a severe account of their conduct, and despite\ntheir cunning and roguishness, they knew not how they should get out of\nit. The sharp gallop of the approaching horse heightened their\nperplexity. They dared not communicate their fears to each other, but\nthey sat with heads bent forward, foreseeing that they would soon have\nto sustain a very firm attack.\n\nThe horse stopped short before the rancho; a man dismounted, and the\ndoor shook beneath the tremendous blows of his fists.\n\n\"Hum!\" the gambusino whispered, as he blew out the solitary candle that\nillumined the room. \"Who the deuce can come at this advanced hour of the\nnight! I have a great mind not to open.\"\n\nStrange to say, Fray Ambrosio had apparently regained all his serenity.\nWith a smiling face, crossed arms, and back leaned against the wall, he\nseemed to be a perfect stranger to what perplexed his mate so furiously.\nAt Garote's remark an ironical smile played round his pale lips for a\nsecond, and he replied with the most perfect indifference--\n\n\"You are at liberty to act as you please, gossip; still I think it my\nduty to warn you of one thing?\"\n\n\"What is it?\"\n\n\"That, if you do not open your door, the man, whoever he may be, now\nbattering it, is very capable of breaking it in, which would be a\ndecided nuisance for you.\"\n\n\"You speak very much at your ease, senor Padre,\" the gambusino answered,\nill-temperedly. \"Suppose it be Red Cedar?\"\n\n\"The greater reason to open the door. If you hesitate, he will begin to\nsuspect you; and then take care, for he is a man capable of killing you\nlike a dog.\"\n\n\"That is possible; but do you think that, in such a case, you will\nescape with clean hands?\"\n\nFray Ambrosio looked at him, shrugged his shoulders, but made no further\nanswer.\n\n\"Will you open, _demonios_?\" a rough voice shouted.\n\n\"Red Cedar!\" both men whispered.\n\n\"I am coming,\" Andres replied, in a voice which terror caused to\ntremble.\n\nHe rose unwillingly, and walked slowly towards the door, which the\nsquatter threatened to tear from its hinges.\n\n\"A little patience, caballero,\" the gambusino said, in that honeyed\nvoice peculiar to Mexicans when they meditate some roguery. \"Coming,\ncoming.\"\n\nAnd he began unbarring the door.\n\n\"Make haste!\" the squatter howled, \"For I am in a hurry.\"\n\n\"Hum! It is surely he!\" the gambusino thought. \"Who are you?\" he asked.\n\n\"What! Who am I?\" Red Cedar exclaimed, bounding with wrath. \"Did you not\nrecognise me, or are you having a game with me?\"\n\n\"I never have a game with anyone,\" Andres replied, imperturbably: \"but I\nwarn you that, although I fancy I recognise your voice, I shall not open\ntill you mention your name. The night is too far advanced for me to risk\nreceiving a suspicious person into my house.\"\n\n\"I will break the door down.\"\n\n\"Try it,\" the gambusino shouted boldly, \"and by our Lady of Pilar I will\nsend a bullet through your head.\"\n\nAt this threat the squatter rushed against the door in incredible fury,\nwith the evident intention of breaking it in; but, contrary to his\nexpectations, though it creaked and groaned on its hinges, it did not\ngive way. Andres Garote had indulged in a line of reasoning which was\nfar from being illogical, and revealed a profound knowledge of the human\nheart. He had said to himself, that, as he must face Red Cedar's anger,\nit would be better to let it reach its paroxysm at once so as to have\nonly the decreasing period to endure. He smiled at the American's\nsterile attempts, then, and repeated his request.\n\n\"Well, then,\" the other said, furiously, \"I am Red Cedar. Do you\nrecognise me now, you devil's own Gachupino?\"\n\n\"Of course; I see that I can open without danger to your Excellency.\"\n\nAnd the gambusino hurriedly drew back the bolts.\n\nRed Cedar rushed into the room with a yell of fury, but Andres had put\nout the light. The squatter stopped, surprised by the gloom which\nprevented him distinguishing any object.\n\n\"Hallo!\" he said. \"What is the meaning of this darkness? I can see\nnothing.\"\n\n\"_Caspita_!\" Andres replied, impudently, \"Do you think I amuse myself o'\nnights by watching the moon? I was asleep, compadre, when you came to\narouse me with your infernal hammerings.\"\n\n\"That is possible,\" the squatter remarked; \"but that was no reason for\nkeeping me so long at your door.\"\n\n\"Prudence is the mother of security. We must not let every comer enter\nthe rancho.\"\n\n\"Certainly not; I approve of that. Still, you must have recognised my\nvoice.\"\n\n\"True. Still I might be mistaken; it is difficult to know anyone through\nthe thickness of a door; that is why I wished you to give your name.\"\n\n\"Very good, then,\" Red Cedar said, as if tired of combating arguments\nwhich did not convince him. \"And where is Fray Ambrosio?\"\n\n\"Here, I suppose.\"\n\n\"He has not left the rancho?\"\n\n\"No; unless he took advantage of your arrival to do so.\"\n\n\"Why should he do that?\"\n\n\"I don't know; you question, and I answer; that's all.\"\n\n\"Why does he not speak, if he is here?\"\n\n\"He is possibly asleep.\"\n\n\"After the row I made, that is highly improbable.\"\n\n\"Hang it, he may be a hard sleeper.\"\n\n\"Hum!\" the squatter snorted, suspiciously; \"Light the candle.\"\n\nAndres struck a match, and Red Cedar looked eagerly round the room Fray\nAmbrosio had disappeared.\n\n\"Where is the monk?\" the American asked.\n\n\"I do not know: probably gone.\"\n\nThe squatter shook his head.\n\n\"All this is not clear,\" he muttered; \"there is treachery behind it.\"\n\n\"That is possible,\" the gambusino answered, calmly.\n\nRed Cedar bent on Andres eyes that flashed with fury, and roughly seized\nhim by the throat.\n\n\"Answer, scoundrel?\" he shouted. \"What has become of Dona Clara?\"\n\nThe gambusino struggled, though in vain, to escape from the clutch of\nthe squatter, whose fingers entered his flesh, and pressed him as in a\nvice.\n\n\"Let me loose,\" he panted, \"you are choking me!\"\n\n\"Where is Dona Clara?\"\n\n\"I do not know.\"\n\nThe squatter squeezed more tightly.\n\n\"You do not know!\" he yelled.\n\n\"Aie!\" Andres whined, \"I tell you I do not know.\"\n\n\"Malediction!\" Red Cedar went on. \"I will kill you, _picaro_, if you are\nobstinate.\"\n\n\"Let that man go, and I will tell you all you wish to know,\" was said in\na firm voice by a hunter, who at this moment appeared on the threshold.\n\nThe two men turned in amazement.\n\n\"Nathan!\" Red Cedar shouted on recognising his son. \"What are you doing\nhere?\"\n\n\"I will tell you, father,\" the young man said, as he entered the room.\n\n\n\n\nCHAPTER XLI.\n\nNATHAN.\n\n\nNathan was not asleep, as Ellen supposed, when she urged on Shaw to\ndevote himself to liberate Dona Clara, and he had listened attentively\nto the conversation. Nathan was a man of about thirty years of age, who,\nboth physically and morally, bore a marked resemblance to his father.\nHence the old squatter had concentrated in him all the affection which\nhis uncultivated savage nature was capable of feeling. Since the fatal\nnight, when the chief of the Coras had avenged himself for the burning\nof his village and the murder of its inhabitants, Nathan's character had\ngrown still more gloomy; a dull and deep hatred boiled in his heart\nagainst the whole human race; he only dreamed of assassination: he had\nsworn in his heart to revenge on all those who fell into his hands the\ninjury one man had inflicted on him; in a word, Nathan loved none and\nhated everything.\n\nWhen Shaw had disappeared among the bushes, and Ellen, after taking a\nfinal glance around to convince herself that all was in order,\nre-entered the hut that served her as a shelter, Nathan rose cautiously,\nthrew his rifle over his shoulder, and rushed after his brother. Another\nreason urged him to foil Shaw and Ellen's plans; he had a double grudge\nagainst Don Miguel--the first for the stab the Mexican gentlemen had\ngiven his father; the second because Don Miguel had compelled him to\nleave the forest in which his family had so daringly installed itself.\n\nConvinced of the importance of the affair, and knowing the value the\nsquatter attached to carrying off the maiden, who was a most precious\nhostage for him, Nathan did not lose a moment, but reached Santa Fe by\nthe most direct route, bounding with the agility of a tiger cat over the\nobstacles that beset his path. Presently he reached an isolated house,\nnot far from which several men were conversing together in a low voice.\nNathan stopped and listened; but he was too far off, and could\ndistinguish nothing. The squatter's son, reared in the desert, was\nthoroughly versed in all its stratagems; with the piercing eye of a man\naccustomed to night journeys in the prairie, he recognised well-known\npersons, and his mind was at once made up.\n\nHe laid himself on the ground, and following the shadow cast by the\nmoon, lest he might be perceived by the speakers, he advanced, inch by\ninch, crawling like a serpent, stopping at intervals lest the waving of\nthe grass might reveal his presence, in short, employing all the\nprecautions usual under such circumstances. At length he reached a clump\nof Peru trees only a few yards distant from the spot where the men he\nwished to overhear were standing. He then got up, leaned against the\nlargest tree, and prepared to listen. His expectations were not\ndeceived; though a few words escaped him here and there, he was near\nenough perfectly to catch the sense of the conference. This conversation\nwas, in truth, most interesting to him; a sinister smile lit up his\nface, and he eagerly clenched the barrel of his rifle.\n\nPresently the party broke into two. Valentine, Curumilla, and Unicorn,\ntook the road leading to the open country, while Don Pablo and Father\nSeraphin returned toward the town. Valentine and his two friends almost\ntouched the young man as they passed, and he instinctively carried his\nhands to his pistols; they even stopped for a moment and cast suspicious\nglances at the clump that concealed their foe. While conversing in\nwhispers, Unicorn drew a few branches aside and peered in; for some\nseconds Nathan felt an indescribable agony; a cold perspiration stood at\nthe root of his hair and the blood coursed to his heart; in a word, he\nwas afraid. He knew that if these men, his mortal enemies, discovered\nhim, they would be pitiless to him and kill him like a dog. But this\napprehension did not last longer than a lightning flash. Unicorn\ncarelessly let the leafy curtain fall again, saying only one word to his\ncomrades:--\n\n\"Nothing.\"\n\nThe latter resumed their march.\n\n\"I do not know why,\" said Valentine, \"but I fancy there is someone\nhidden there.\"\n\n\"No,\" the chief answered, \"there is nobody.\"\n\n\"Well, be it so,\" the hunter muttered, with a toss of his head.\n\nSo soon, as he was alone, Nathan drew two or three deep breaths, and\nstarted in pursuit of Don Pablo and the missionary, whom he soon caught\nup. As they did not suppose they were followed, they were conversing\nfreely together.\n\nIn Spanish America, where the days are so warm and the nights so fresh,\nthe inhabitants, shut up at home so long as the sun calcines the ground,\ngo out at nightfall to breathe a little pure air; the streets, deserted\nin consequence of the heat, are gradually peopled; benches are placed\nbefore the doors, on which persons recline to smoke and gossip, drink\norangeade, strum the guitar, and sing. Frequently the entire night is\npassed in these innocent amusements, and folks do not return home till\ndawn, in order to indulge in the sleep so grateful after this long\nwatch. Hence the Hispano-American towns must be especially visited by\nnight, if you wish to judge truthfully the nature of this people--a\nstrange composite of the most discordant contrasts, who only live for\nenjoyment, and only accept from existence the most intoxicating\npleasures. Still, on the night to which we refer, the town of Santa Fe,\nusually so laughing and chattering, was plunged into a gloomy sadness,\nthe streets were deserted, the doors closed; no light filtered through\nthe hermetically closed windows; all slept or at least feigned to sleep.\nThe fact was, that Santa Fe was at this moment in a state of mortal\nagitation, caused by the condemnation of Don Miguel Zarate, the richest\nland owner in the province--a man who was loved and revered by the whole\npopulation. The agitation took its origin in the unexpected apparition\nof the Comanche war detachment--those ferocious enemies whose cruelties\nhave become proverbial on the Mexican frontier, and whose presence\npresaged nothing good.\n\nDon Pablo and his companion walked quickly, like persons anxious to\nreach a place where they knew they are expected, exchanging but a few\nwords at intervals, whose meaning, however, caught up by the man who\nfollowed them, urged them still more not to let them out of sight. They\nthus traversed the greater part of the town, and on reaching the Calle\nde la Merced, they stopped at their destination--a house of handsome\naspect.\n\nA weak light burned at the window of a ground floor room. By an\ninstinctive movement, the two gentlemen turned round at the moment of\nentering the house but Nathan had slipped into a doorway, and they did\nnot perceive him. Father Seraphin tapped gently; the door was at once\nopened, and they went in. Nathan stationed himself in the middle of the\nstreet, with his eye ardently fixed on the only window of the house lit\nup. Ere long, shadows crossed the curtains.\n\n\"Good!\" the young man muttered; \"But how to warn the old one that the\ndove is in her nest?\"\n\nAll at once, a heavy hand was laid on his shoulder, and Nathan turned,\nfiercely clutching a bowie knife. A man was before him, gloomy, silent\nand wrapped in the thick folds of his cloak. The American started.\n\n\"Go your way,\" he said in a menacing voice.\n\n\"What are you doing here?\" the stranger asked.\n\n\"How does that concern you? The street is free to all.\"\n\n\"No.\"\n\nThis word was pronounced with a sharp accent. Nathan tried in vain to\nscan the features of the man with whom he had to deal.\n\n\"Give way,\" he said, \"or blood will surely be shed between us.\"\n\nAs sole reply, the stranger took a pistol in his right hand, a knife in\nhis left.\n\n\"Ah!\" Nathan said, mockingly, \"You mean fighting.\"\n\n\"For the last time, withdraw.\"\n\n\"Nonsense, you are mad, senor Caballero; the road belongs to all, I tell\nyou. This place suits me, and I shall remain.\"\n\n\"I wish to be alone here.\"\n\n\"You mean to kill me, then?\"\n\n\"If I must, yes, without hesitation.\"\n\nThe two speakers had exchanged these words in a low and hurried voice,\nin less time than we have employed to write them. They stood but a few\npaces apart with flashing eyes, ready to rush on each other. Nathan\nreturned his pistol to his belt.\n\n\"No noise,\" he said; \"the knife will do; besides, we are in a country\nwhere that is the only weapon in use.\"\n\n\"Be it so,\" the stranger replied; \"then, you will not give way to me?\"\n\n\"You would laugh at me if I did,\" the American said with a grin.\n\n\"Then your blood will be on your own head.\"\n\n\"Or on yours.\"\n\nThe two foemen each fell back a pace, and stood on guard, with their\ncloaks rolled round their left arms. The moon, veiled by clouds, shed no\nlight; the darkness was perfect; midnight struck from the cathedral; the\nvoice of the _serenos_ chanting the hour could be heard in the distance,\nannouncing that all was quiet. There was a moment's hesitation, which\nthe enemies employed in scrutinising each other. Suddenly Nathan uttered\na hoarse yell rushed on his enemy, and threw his cloak in his face, to\nput him on his guard. The stranger parried the stroke dealt him, and\nreplied by another, guarded off with equal dexterity. The two men then\nseized each other round the waist, and wrestled for some minutes,\nwithout uttering a word; at length the stranger rolled on the ground\nwith a heavy sigh; Nathan's knife was buried in his chest. The American\nrose with a yell of triumph--his enemy was motionless.\n\n\"Can I have killed him?\" Nathan muttered.\n\nHe returned his knife to his vaquera boot, and bent over the wounded\nman. All at once he started back, for he had recognised his brother\nShaw.\n\n\"What is to be done now?\" he said; but then added carelessly, \"Pshaw!\nall the worse for him. Why did he come across my path?\"\n\nAnd, leaving there the body of the young man, who gave no sign of life--\n\n\"Well, Heaven knows, I ought not, and could not have hesitated,\" he\nsaid.\n\nShaw lay to all appearance dead, with pale and drawn cheeks, in the\ncentre of the street.\n\n\n\n\nCHAPTER XLII.\n\nTHE WOUNDED MAN.\n\n\nNathan proceeded straight to the Rancho del Coyote, where his unexpected\narrival was a blessing for Andres Garote, whom the old squatter was\ntreating very roughly. On hearing his son's words, Red Cedar let go of\nthe gambusino, who tottered back against the wall.\n\n\"Well,\" he asked, \"where is Dona Clara?\"\n\n\"Come with me, father,\" the young man answered; \"I will lead you to\nher.\"\n\n\"You know her hiding place, then?\"\n\n\"Yes.\"\n\n\"And so do I,\" Fray Ambrosio shouted, as he rushed into the room with\ndiscomfited features; \"I felt sure I should discover her.\"\n\nRed Cedar looked at him in amazement, but the monk did not wince.\n\n\"What has happened to her?\" the squatter said presently, as he looked\nsuspiciously from the monk to the gambusino.\n\n\"A very simple matter,\" Fray Ambrosio answered, with an inimitably\ntruthful accent; \"about two hours back your son Shaw came here.\"\n\n\"Shaw!\" the squatter exclaimed.\n\n\"Yes, the youngest of your sons; he is called so, I think?\"\n\n\"Yes; go on.\"\n\n\"Very good. He presented himself to us as coming from you to remove our\nprisoner.\"\n\n\"And what did you do?\" the squatter asked, impatiently.\n\n\"What could we do?\"\n\n\"Why, oppose the girl's departure.\"\n\n\"_Caspita_! Do you fancy we let her go so?\" the monk asked,\nimperturbably.\n\nThe squatter looked at him in surprise--he no longer understood\nanything. Like all men of action, discussion was to him almost a matter\nof impossibility; especially with an adversary so crafty as the one he\nhad before him. Deceived by the monk's coolness and the apparent\nfrankness of his answers, he wished to make an end of it.\n\n\"Come,\" he said, \"how did all this finish?\"\n\n\"Thanks to an ally who came to your son's help, and to whom we were\nobliged to bow--\"\n\n\"An ally! What man can be so bold as to dare--\"\n\n\"Eh!\" the monk sharply interrupted Red Cedar, \"that man is a priest, to\nwhom you have already bowed many a time.\"\n\n\"You are jesting, senor Padre,\" the squatter exclaimed, savagely.\n\n\"Not the least in the world. Had it been anyone else, I should have\nresisted; but I, too, belong to the Church; and, as Father Seraphin is\nmy superior, I was forced to obey him.\"\n\n\"What!\" the squatter said, with a groan, \"Is he not dead?\"\n\n\"It appears,\" the monk remarked, ironically, \"as if those you kill are\nall in good state of health, Red Cedar.\"\n\nAt this allusion to Don Pablo's death, the squatter stifled a cry of\nanger, and clenched his fists.\n\n\"Good!\" he said; \"If I do not always kill, I know how to take my\nrevenge. Where is Dona Clara, at this moment?\"\n\n\"In a house no great distance from here,\" Nathan answered.\n\n\"Have you seen her?\" the squatter asked.\n\n\"No; but I followed Don Pablo and the missionary to that house, which\nthey entered, and as they were ignorant that I was close to them, their\nconversation left me no doubt as to the whereabouts of the girl.\"\n\nAn ill-omened smile momentarily lit up the old bandit's features.\n\n\"Good!\" he said; \"as the dove is in her nest, we shall be able to find\nher. What o'clock is it?\"\n\n\"Three in the morning,\" Andres interjected. \"Day will soon break.\"\n\n\"We must make haste, then. Follow me, all of you.\" Then he added, \"But\nwhat has become of Shaw? Does anyone of you know?\"\n\n\"You will probably find him at the door of Dona Clara's house,\" Nathan\nsaid, in a hollow voice.\n\n\"How so? Has my son entered into a compact with my enemies?\"\n\n\"Yes; as he arranged with them to carry off your prisoner.\"\n\n\"Oh! I will kill him if he prove a traitor!\" the squatter shouted with\nan accent that made the blood run cold in the veins of his hearers.\n\nNathan fell back two steps, drew his knife from his boot, and showed it\nto his father.\n\n\"That is done,\" he said, harshly. \"Shaw tried to stab me, so I killed\nhim.\"\n\nAfter these mournful words, there was a moment of silence in the rancho.\nAll these men, though their hearts were steeled by crime, shuddered\ninvoluntarily. Without, the night was gloomy; the wind whistled sadly;\nthe flickering light of the candle threw a weird light over the scene,\nwhich contained a certain degree of terrible poetry. The squatter passed\nhis hard hand over his dank brow. A sigh, like a howl, painfully forced\nits way from his oppressed chest.\n\n\"He was my last born,\" he said, in a voice broken by an emotion he could\nnot control. \"He deserved death, but he ought not to have received it at\nhis brother's hands.\"\n\n\"Father!\" Nathan muttered.\n\n\"Silence!\" Red Cedar shouted, in a hollow voice, as he stamped his foot\npassionately on the ground; \"What is done cannot be undone; but woe to\nmy enemies' family! Oh! I feel now that I can take such vengeance on\nthem as will make all shudder who hear it spoken of!\"\n\nAfter uttering these words, which were listened to in silence, the\nsquatter walked a few steps up the rancho. He approached a table, seized\na bottle half full of mezcal that stood on it, and emptied it at a\ndraught. When he had finished drinking, he threw down the bottle, which\nbroke with a crash, and said to his mates in a hollow voice--\n\n\"Let us be off! We have wasted too much time here already!\"\n\nAnd he rushed out of the rancho, the others following close at his\nheels.\n\nIn the meanwhile, Don Pablo and Father Seraphin were in the house. The\npriest had taken the maiden to the house of an honest family which owed\nhim great obligations, and was too happy to receive the poor sufferer.\nThe missionary did not intend, however, to let her be long a burthen to\nthese worthy people. At daybreak he intended to deliver her to certain\nrelations of her father, who inhabited a hacienda a few leagues from\nSanta Fe.\n\nDona Clara had been placed in a comfortable room by her hosts. Their\nfirst care had been to make her doff the Indian robes for others more\nsuitable to her birth and position. The maiden worn out by poignant\nemotions of the scene she had witnessed, was on the point of retiring to\nbed, when Father Seraphin and Don Pablo tapped at the door of her room.\nShe hastily opened it, and the sight of her brother, whom she had not\nhoped to see so speedily, overwhelmed her with joy.\n\nAn hour soon slipped away in pleasant chat. Don Pablo was careful not to\ntell his sister of the misfortune that had befallen her father; for he\ndid not wish to dull by that confession the joy the poor girl promised\nherself for the morrow. Then, as the night was advancing, the two men\nwithdrew, so as to allow her to enjoy that rest so needed to strengthen\nher for the long journey to the hacienda, promising to come and fetch\nher in a few hours. Father Seraphin generously offered Don Pablo to pass\nthe night with him by sharing the small lodging he had not far from the\nPlaza de la Merced, and the young man eagerly accepted. It was too late\nto seek a lodging at a locanda, and in this way he would be all the\nsooner with his sister next morning. After a lengthened leave-taking,\nthey, therefore, left the house, and, so soon as they were gone, Dona\nClara threw herself, ready dressed, into a hammock hanging at one end of\nthe room, when she speedily fell asleep.\n\nOn reaching the street, Don Pablo saw a body lying motionless in front\nof the house.\n\n\"What's this?\" he asked, in surprise.\n\n\"A poor wretch whom the ladrones killed in order to plunder him,\" the\nmissionary answered.\n\n\"That is possible.\"\n\n\"Perhaps he is not quite dead,\" the missionary went on; \"it is our duty\nto succour him.\"\n\n\"For what good?\" Don Pablo said, with an air of indifference; \"if a\nsereno were to pass he might accuse us of having killed the man.\"\n\n\"Nay, sir,\" the missionary observed, \"the ways of the Lord are\nimpenetrable. If He allowed us to come across this unhappy man, it was\nbecause He judged in His wisdom that we might prove of use to him.\"\n\n\"Be it so,\" the young man said; \"let us look at him, as you wish it. But\nyou know that in this country good actions of such a nature generally\nentail annoyance.\"\n\n\"That is true, my son. Well, we will run the risk,\" said the missionary,\nwho had already bent over the wounded man.\n\n\"As you please,\" Don Pablo said, as he followed him.\n\nShaw, for it was he, gave no signs of life. The missionary examined him,\nthen rose hastily, seized Don Pablo's arm, and drew him to him, as he\nwhispered--\n\n\"Look!\"\n\n\"Shaw!\" the Mexican exclaimed, in surprise; \"What could that man be\ndoing here?\"\n\n\"Help me, and we shall learn. The poor fellow has only fainted; and the\nloss of blood has produced this semblance to death.\"\n\nDon Pablo, greatly perplexed by this singular meeting, obeyed the\nmissionary without further remark. The two men raised the wounded lad,\nand carried him gently to Father Seraphin's lodging, where they proposed\nto give him all the help his condition required.\n\nThey had scarce turned the corner of the street, when several men\nappeared at the other extremity. They were Red Cedar and his\nconfederates. On arriving in front of the house they stopped: all the\nwindows were in the deepest obscurity.\n\n\"Which is the girl's room?\" the squatter asked in a whisper.\n\n\"This one,\" Nathan said, as he pointed to it.\n\nRed Cedar crawled up to the house, drove his dagger into the wall,\nraised himself to the window, and placed his face against a pane.\n\n\"All is well! She sleeps!\" he said, when he came down. \"You, Fray\nAmbrosio, to one corner of the street; you, Garote, to the other, and do\nnot let me be surprised.\"\n\nThe monk and the gambusino went to their allotted posts. When Red Cedar\nwas alone with his son he bent and whispered in his ear--\n\n\"What did you do with your brother after stabbing him?\"\n\n\"I left him on the spot where he fell.\"\n\n\"Where was that?\"\n\n\"Just where we now stand.\"\n\nThe squatter stooped down to the ground, and walked a few steps,\ncarefully examining the bloody traces left on the pebbles.\n\n\"He has been carried off,\" he said, when he rose again. \"Perhaps he is\nnot dead.\"\n\n\"Perhaps so,\" the young man observed, with a shake of his head.\n\nHis father gave him a most significant look.\n\n\"To work,\" he said coldly.\n\nAnd they prepared to escalade the window.\n\n\n\n\nCHAPTER XLIII.\n\nINDIAN DIPLOMACY.\n\n\nWe will return, for the present, to Valentine and his comrades.\n\nThe sudden apparition of the sachem of the Coras had produced a certain\ndegree of emotion among the hunters and the Comanches. Valentine, the\nfirst to recover from his surprise, addressed Eagle-wing.\n\n\"My brother is welcome,\" he said, as he held out his hand, which the\nIndian warmly pressed, \"What news does the chief bring us?\"\n\n\"Good,\" the Coras answered laconically.\n\n\"All the better,\" the hunter said gaily; \"for some time past all we have\nreceived has been so bad that my brother's will create a diversion.\"\n\nThe Indian smiled at this sally, but made no remark.\n\n\"My brother can speak,\" Valentine continued; \"he is surrounded by none\nbut friends.\"\n\n\"I know it,\" the chief answered, as he bowed gracefully to the company.\n\"Since I left my brother two months have passed away: I have worn out\nmany moccasins amid the thorns and brambles of the desert; I have been\nbeyond the Great Lakes to the villages of my nation.\"\n\n\"Good; my brother is a chief; he was doubtless well received by the\nsachems of the Coras of the Great Lakes.\"\n\n\"Mookapec is a renowned warrior among his people,\" the Indian answered\nproudly; \"his place by the council fire of the nation is pointed out.\nThe chiefs saw him with joy: on his road he had taken the scalps of\nseven gachupinos: they are now drying before the great medicine lodge.\"\n\n\"It was your right to do so, chief, and I cannot blame you. The\nSpaniards have done you harm enough for you to requite them.\"\n\n\"My brother speaks well; his skin is white, but his heart is red.\"\n\n\"Hum,\" observed Valentine; \"I am a friend to justice; vengeance is\npermissible against treachery. Go on, chief.\"\n\nThe hunter's comrades had drawn nearer, and now formed a circle round\nthe two speakers. Curumilla was occupied silently, as was his wont, in\ncompletely stripping each Spanish prisoner, whom he then bound in such a\nway that the slightest movement was impossible.\n\nValentine, although time pressed, knew too well the Redskin character to\ntry and hurry Eagle-wing on. He felt certain that the chief had\nimportant news to communicate to him; but it would have been no use\ntrying to draw it from him; hence he allowed him to act as he pleased.\nUnicorn, leaning on his rifle, listened attentively, without evincing\nthe slightest impatience.\n\n\"Did my brother remain long with his tribe,\" Valentine continued.\n\n\"Two suns. Eagle-wing had left behind him friends to whom his heart drew\nhim.\"\n\n\"Thanks, chief, for the pleasant recollections of us.\"\n\n\"The chiefs assembled in council to hear the words of Eagle-wing,\" the\nCoras continued. \"They shuddered with fury on hearing of the massacre of\ntheir children; but Mookapec had formed his plan, and two hundred\nwarriors are assembled beneath his _totem_.\"\n\n\"Good!\" said Valentine, \"the chief will avenge himself.\"\n\nThe Indian smiled.\n\n\"Yes,\" he said, \"my young men have their orders, they know what I mean\nto do.\"\n\n\"Very good; in that case they are near here?\"\n\n\"No,\" the chief replied, with a shake of his head. \"Eagle-wing does not\nmarch with them; he has hidden himself under the skin of an Apache dog.\"\n\n\"What does my brother say?\" Valentine asked with amazement.\n\n\"My white brother is quick,\" Unicorn said, sententiously; \"he will let\nMookapec speak. He is a great sachem, and wisdom dwells in him.\"\n\nValentine shook his head, however, and said--\n\n\"Hum! Answering one act of treachery by another, that is not the way in\nwhich the warriors of my nation behave.\"\n\n\"The nation of my brother is great, and strong as the grizzly bear,\"\nUnicorn said; \"it does not need to march along hidden paths. The poor\nIndians are weak as the beaver, but like him they are very cunning.\"\n\n\"That is true,\" Valentine replied, \"cunning must be allowed you in\ndealing with the implacable enemies who surround you. I was wrong; so go\non, chief; tell us what deviltry you have invented, and if it is\ningenious. Well, I will be the first to applaud it.\"\n\n\"Wah, my brother shall judge. Red Cedar is about to enter the desert, as\nmy brother doubtless knows?\"\n\n\"Yes.\"\n\n\"Does my brother know the _Gringo_ has asked the Apaches for a guide?\"\n\n\"No, I did not.\"\n\n\"Good. Stanapat, the great chief of the Apaches, sent a Navajo warrior\nto act as guide to Red Cedar.\"\n\n\"Well?\"\n\n\"The Navajo was scalped by Eagle-wing.\"\n\n\"Ah, ah! Then Red Cedar cannot set out?\"\n\n\"Yes, he can do so when he likes.\"\n\n\"How so?\"\n\n\"Because Eagle-wing takes the place of the guide.\"\n\nUnicorn smiled.\n\n\"My brother has a deal of wisdom,\" he said.\n\n\"Hum!\" Valentine remarked, with some show of ill-humour. \"It is\npossible, but you play for a heavy stake, chief. That old villain is as\ncrafty as ten monkeys and ten opossums united. I warn you that he will\nrecognise you.\"\n\n\"No.\"\n\n\"I wish it; for if he does, you are a lost man.\"\n\n\"Good, my brother can be easy. Eagle-wing is a warrior; he will see the\nwhite hunter again in the desert.\"\n\n\"I wish so, chief; but I doubt. However, act as you please. When will\nyou join Red Cedar?\"\n\n\"This night.\"\n\n\"You are going to leave us?\"\n\n\"At once. Eagle-wing has nothing more to confide to his brother.\"\n\nAnd, after bowing courteously to the company, the Coras chief glided\ninto the thicket, in which he disappeared almost instantaneously.\nValentine looked after him for some time.\n\n\"Yes,\" he said at last, with a thoughtful air, \"his project is a daring\none, such as might be expected from so great a warrior. May heaven\nprotect him, and allow him to succeed! Well, we shall see; perhaps all\nis for the best so.\"\n\nAnd he turned to Curumilla.\n\n\"The clothes?\" he said.\n\n\"Here they are,\" the Aucas answered, laconically, as he pointed to an\nenormous heap of clothing.\n\n\"What does my brother mean to do with them?\" Unicorn asked.\n\n\"My brother will see,\" Valentine said, with a smile, \"each of us is\ngoing to put on one of those uniforms.\"\n\nThe Comanche drew himself up hastily.\n\n\"No,\" he said, \"Unicorn does not put off the dress of his people. What\nneed have we of this disguise?\"\n\n\"In order to enter the camp of the Spaniards without being discovered.\"\n\n\"Wah! For what good? Unicorn will summon his young men to cut a passage\nthrough the corpses of the gachupinos.\"\n\nBut Valentine shook his head mournfully.\n\n\"It is true,\" he remarked, \"we could do so. But why shed blood\nneedlessly? No; let my brother put confidence in me.\"\n\n\"The hunter will act rightly. Unicorn knows it, and he leaves him free;\nbut Unicorn is a chief, he cannot put on the clothes of the palefaces.\"\n\nValentine no longer insisted, as it would have been unavailing; so he\nagreed to modify his plan. He made each of his comrades put on a dragoon\nuniform, and himself donned the clothes stripped from the Alferez. When\nall this metamorphosis was as complete as possible, he turned to\nUnicorn.\n\n\"The chief will remain here,\" he said, \"to guard the prisoners.\"\n\n\"Good,\" the Comanche answered. \"Is Unicorn, then, a chattering old\nwoman, that warriors place him on one side?\"\n\n\"My brother does not understand me. I do not wish to insult him, but he\ncannot enter the camp with us.\"\n\nThe chief shrugged his shoulders disdainfully.\n\n\"The Comanche warriors can crawl as well as serpents. Unicorn will\nenter.\"\n\n\"Let my brother come, then, since he wishes it.\"\n\n\"Good; my brother is vexed; a cloud has passed over his face. He is\nwrong; his friend loves him.\"\n\n\"I know it, chief, I know it. I am not vexed, but my heart is sad to see\na warrior thus run the risk of being killed without any necessity.\"\n\n\"Unicorn is a sachem; he must give an example to his young men on the\nwarpath.\"\n\nValentine gave a nod of assent.\n\n\"Here are the horses of the palefaces,\" Curumilla said; \"my brother will\nneed them.\"\n\n\"That is true,\" the hunter answered, with a smile; \"my brother is a\ngreat chief--he thinks of everything.\"\n\nEveryone mounted, Unicorn alone remaining a-foot. Valentine placed the\nAlferez by his side.\n\n\"Caballero,\" he said to him, \"you will act as our guide to the camp. We\ndo not wish to take the lives of your countrymen; our intention is\nsimply to prevent them following us at present. Pay attention to my\nwords: if you attempt to deceive us, I blow out your brains. You are\nwarned.\"\n\nThe Spaniard bowed, but made no reply. As for the prisoners, they had\nbeen so conscientiously tied by Curumilla that there was no chance of\ntheir escaping. The little band then set out, Unicorn disappearing among\nthe trees. When they came a short distance from the bivouac, a sentry\nchallenged, \"Who goes there?\"\n\n\"Answer,\" Valentine whispered the Alferez.\n\nHe did so. They passed, and the sentry, suddenly seized by Curumilla,\nwas bound and gagged in the twinkling of an eye, all the other sentinels\nsharing the same fate. The Mexicans keep up a very bad watch in the\nfield, even in the presence of an enemy; the greater reason, then, for\nthem to neglect all precaution when they fancy themselves in safety.\nEverybody was asleep, and Valentine and his friends were masters of the\ncamp. The regiment of dragoons had been surprised without striking a\nblow.\n\nValentine's comrades dismounted; they knew exactly how to act, and did\nnot deviate from the instructions given by their leader. They proceeded\nfrom picket to picket, removing the horses, which were led out of camp.\nWithin twenty minutes all had been carried off. Valentine had anxiously\nfollowed the movements of his men. When they had finished, he raised the\ncurtain of the colonel's tent, and found himself face to face with\nUnicorn, from whose waist-belt hung a reeking scalp. Valentine could not\nrepress a movement of horror.\n\n\"What have you done, chief?\" he asked, reproachfully.\n\n\"Unicorn has killed his enemy,\" the Comanche replied, peremptorily.\n\"When the leader of the antelopes is killed, his flock disperses; the\ngachupinos will do the same.\"\n\nValentine drew near the colonel. The unhappy man, fearfully mutilated,\nwith his brain laid bare, and his heart pierced by the knife of the\nimplacable Indian, lay stark dead, in a pool of blood, in the middle of\nthe tent. The hunter vented a sigh at this sorry sight.\n\n\"Poor devil!\" he said, with an air of compassion.\n\nAfter this short funeral oration, he took away his sabre and epaulettes,\nleft the tent, followed by the Indian chief, and rejoined his comrades.\nThe horses were led to the Comanche camp, after which Valentine and his\nparty wrapped themselves in their blankets, and slept calmly till\ndaybreak. The dragoons were no longer to be feared.\n\n\n\n\nCHAPTER XLIV.\n\nTHE STRANGER.\n\n\nFather Seraphin and Don Pablo we left bearing the wounded man to the\nmissionary's lodging. Although the house to which they were proceeding\nwas but a short distance off, yet the two gentlemen, compelled to take\nevery precaution, employed considerable time on the journey. Nearly\nevery step they were compelled to halt, so as not to fatigue too greatly\nthe wounded man, whose inert limbs swayed in every direction.\n\n\"The man is dead,\" Don Pablo remarked, during a halt they made on the\nPlaza de la Merced.\n\n\"I fear so,\" the missionary answered, sadly; \"still, as we are not\ncertain of it, our conscience bids us to bestow our care on him, until\nwe acquire the painful conviction that it avails him nought.\"\n\n\"Father, the love of one's neighbour often carries you too far; better\nwere it, perhaps, if this wretch did not come back to life.\"\n\n\"You are severe, my friend. This man is still young--almost a boy.\nTrained amid a family of bandits, never having aught but evil examples\nbefore him, he has hitherto only done evil, in a spirit of imitation.\nWho knows whether this fearful wound may not offer him the means to\nenter the society of honest people, which he has till now been ignorant\nof? I repeat to you, my friend, the ways of the Lord are inscrutable.\"\n\n\"I will do what you wish, father. You have entire power over me. Still,\nI fear that all our care will be thrown away.\"\n\n\"God, whose humble instruments we are, will prove you wrong, I hope.\nCome, a little courage, a few paces further, and we shall have arrived.\"\n\n\"Come on then,\" Don Pablo said with resignation.\n\nFather Seraphin lodged at a house of modest appearance, built of adobes\nand reeds, in a small room he hired from a poor widow, for the small sum\nof nine reals a month. This room, very small, and which only received\nair from a window opening on an inner yard, was a perfect conventual\ncell, as far as furniture was concerned, for the latter consisted of a\nwooden frame, over which a bull hide was stretched, and served as the\nmissionary's bed; a butaca and a prie-dieu, above which a copper\ncrucifix was fastened to the whitewashed wall. But, like all cells, this\nroom was marvellously clean. From a few nails hung the well-worn clothes\nof the poor priest, and a shelf supported vials and flasks which\ndoubtless contained medicaments; for, like all the missionaries, Father\nSeraphin had a rudimentary knowledge of medicine, and took in charge\nboth the souls and bodies of his neophytes.\n\nThe father lit a candle of yellow tallow standing in an iron\ncandlestick, and, aided by Don Pablo, laid the wounded man on his own\nbed; after which the young man fell back into the butaca to regain his\nbreath. Father Seraphin, on whom, spite of his fragile appearance, the\nfatigue had produced no apparent effect, then went downstairs to lock\nthe street door, which he had left open. As he pushed it to, he felt an\nopposition outside, and a man soon entered the yard.\n\n\"Pardon, my reverend father,\" the stranger said; \"but be kind enough not\nto leave me outside.\"\n\n\"Do you live in this house?\"\n\n\"No,\" the stranger coolly replied, \"I do not live in Santa Fe, where I\nam quite unknown.\"\n\n\"Do you ask hospitality of me, then?\" Father Seraphin continued, much\nsurprised at this answer.\n\n\"Not at all, reverend father.\"\n\n\"Then what do you want?\" the missionary said, still more surprised.\n\n\"I wish to follow you to the room where you have laid the wounded man,\nto whose aid you came so generously a short time back.\"\n\n\"This request, sir--\" the priest said, hesitating.\n\n\"Has nothing that need surprise you. I have the greatest interest in\nseeing with my own eyes in what state that man is, for certain reasons\nwhich in no way concern you.\"\n\n\"Do you know who he is?\"\n\n\"I do.\"\n\n\"Are you a relation or friend of his?\"'\n\n\"Neither one nor the other. Still, I repeat to you, very weighty reasons\ncompel me to see him and speak with him, if that be possible.\"\n\nFather Seraphin took a searching glance at the speaker.\n\nHe was a man of great height, apparently in the fullest vigour of life.\nHis features, so far as it was possible to distinguish them by the pale\nand tremulous moonbeams, were handsome, though an expression of\nunbending will was the marked thing about them. He wore the dress of\nrich Mexican hacenderos, and had in his right hand a magnificently\ninlaid American rifle. Still the missionary hesitated.\n\n\"Well,\" the stranger continued, \"have you made up your mind, father?\"\n\n\"Sir,\" Father Seraphin answered with firmness, \"do not take in ill part\nwhat I am going to say to you.\"\n\nThe stranger bowed.\n\n\"I do not know who you are; you present yourself to me in the depths of\nthe night, under singular circumstances. You insist, with strange\ntenacity, on seeing the poor man whom Christian charity compelled me to\npick up. Prudence demands that I should refuse to let you see him.\"\n\nA certain annoyance was depicted on the stranger's features.\n\n\"You are right, father,\" he answered; \"appearances are against me.\nUnfortunately, the explanation you demand from me justly would make us\nlose too much precious time, hence I cannot give them to you at this\nmoment. All I can do is to swear, in the face of Heaven, on that\ncrucifix you wear round your neck, and which is the symbol of our\nredemption, that I only wish well to the man you have housed, and that I\nam this moment seeking to punish a great criminal.\"\n\nThe stranger uttered these words with such frankness, and such an air of\nconviction, his face glistened with so much honesty, that the missionary\nfelt convinced: he took up the crucifix and offered it to this\nextraordinary man.\n\n\"Swear,\" he said.\n\n\"I swear it,\" he replied in a firm voice.\n\n\"Good,\" the priest went on, \"now you can enter, sir; you are one of\nourselves; I will not even insult you by asking your name.\"\n\n\"My name would teach you nothing, father,\" the stranger said sadly.\n\n\"Follow me, sir.\"\n\nThe missionary locked the gate and led the stranger to his room, on\nentering which the newcomer took off his hat reverently, took up a post\nin a corner of the room, and did not stir.\n\n\"Do not trouble yourself about me, father,\" he said in a whisper, \"and\nput implicit faith in the oath I took.\"\n\nThe missionary only replied by a nod, and as the wounded man gave no\nsign of life, but still lay much in the position he was first placed in,\nFather Seraphin walked up to him. For a long time, however, the\nattention he lavished on him proved sterile, and seemed to produce no\neffect on the squatter's son. Still, the father did not despair,\nalthough Don Pablo shook his head. An hour thus passed, and no\nostensible change had taken place in the young man's condition; the\nmissionary had exhausted all his stock of knowledge, and began to fear\nthe worst. At this moment the stranger walked up to him.\n\n\"My father,\" he said, touching him gently on the arm, \"you have done all\nthat was humanly possible, but have not succeeded.\"\n\n\"Alas! No!\" the missionary said sadly.\n\n\"Will you permit me to try in my turn?\"\n\n\"Do you fancy you will prove, more successful than I?\" the priest asked\nin surprise.\n\n\"I hope so,\" the stranger said softly.\n\n\"Still, you see I have tried everything that the medical art prescribes\nin such a case.\"\n\n\"That is true, father; but the Indians possess certain secrets known\nonly to themselves, and which are of great efficacy.\"\n\n\"I have heard so. But do you know those secrets?\"\n\n\"Some of them have been revealed to me; if you will permit me, I will\ntry their effects on this young man, who, as far as I can judge, is in a\ndesperate condition.\"\n\n\"I fear he is, poor fellow.\"\n\n\"We shall, therefore, run no risk in trying the efficacy of my superior\nremedy upon him.\"\n\n\"Certainly not.\"\n\nThe stranger bent over the young man, and regarded him for a moment with\nfixed attention; then he drew from his pocket a flask of carved crystal,\nfilled with a fluid as green as emerald. With the point of his dagger he\nslightly opened the wounded man's closed teeth, and poured into his\nmouth four or five drops of the fluid contained in the flask. A strange\nthing then occurred; the young man gave vent to a deep sigh, opened his\neyes several times, and suddenly, as if moved by supernatural force, he\nsat up and looked around him with amazement. Don Pablo and the\nmissionary were almost inclined to believe in a miracle so extraordinary\ndid the fact appear to them. The stranger returned to his dark corner.\nSuddenly the young man passed his hand over his dank forehead, and\nmuttered in a hollow voice:--\n\n\"Ellen, my sister, it is too late. I cannot save her. See, see, they are\ncarrying her off; she is lost!\"\n\nAnd he fell back on the bed, as the three men rushed towards him.\n\n\"He sleeps!\" the missionary said in amazement.\n\n\"He is saved?\" the stranger answered.\n\n\"What did he want to say, though?\" Don Pablo inquired anxiously.\n\n\"Did you not understand it?\" the stranger asked of him.\n\n\"No.\"\n\n\"Well, then, I will tell you.\"\n\n\"You!\"\n\n\"Yes, I; listen! That lad wished to deliver your sister!\"\n\n\"How do you know?\"\n\n\"Is it true?\"\n\n\"It is; go on.\"\n\n\"He was stabbed at the door of the house when she sought shelter.\"\n\n\"What next?\"\n\n\"Those who stabbed him wished to get him out of the way, in order to\ncarry her off a second time.\"\n\n\"Oh, that is impossible!\"\n\n\"It is the fact.\"\n\n\"How do you know it?\"\n\n\"I do not know it, but I can read it plainly.\"\n\n\"Ah!\" Don Pablo exclaimed in despair, \"my father--let us fly to my\nsister's aid!\"\n\nThe two gentlemen rushed from the house with a presentiment of\nmisfortune. When the stranger found himself alone with the wounded man,\nhe walked up to him, wrapped him in his cloak, threw him over his\nshoulders as easy as if he were only a child, and went out in his turn.\nOn reaching the street, he carefully closed the door, and went off at a\ngreat rate, soon disappearing in the darkness. At the same instant the\nmelancholy voice of the sereno could be heard chanting--\n\n_\"Ave Maria purisima! Los cuatro han dado! Viva Mejico! Todo es\nquieto!_\"[1]\n\nWhat irony on the part of accident was this cry after the terrible\nevents of the night!\n\n\n[1] Hail, most pure Mary! It has struck four. Long live Mexico! All is\nquiet.\n\n\n\n\nCHAPTER XLV.\n\nGENERAL VENTURA.\n\n\nIt was about six in the morning. A dazzling sun poured down its\ntransparent rays on the streets of the Presidio of Santa Fe, which were\nalready full of noise and movement at that early hour of the morning.\nGeneral Ventura was still plunged in a deep sleep, probably lulled by\nagreeable dreams, judging from the air of beatitude spread over his\nfeatures. The general, reassured by the speedy arrival of the dragoons\npromised him, fancied he had nothing more to fear from mutineers who had\nhitherto inspired him with lively apprehensions. He thought, too, that\nby the aid of the reinforcements, he could easily get rid of the\nComanche, who, on the previous day, had so audaciously bearded him in\nthe very heart of his palace.\n\nHe slept, then, that pleasant morning sleep, in which the body, entirely\nrested from its fatigue, leaves the mind the entire liberty of its\nfaculties. Suddenly the door of the sleeping room in which the worthy\ngovernor reposed, was torn violently open, and an officer entered.\nGeneral Ventura, aroused with a start, sat up in his bed, fixing on the\nimportunate visitor a glance, at first stern, but which at once became\nuneasy on seeing the alarm depicted on the officer's features.\n\n\"What is the matter, senor Captain Don Lopez?\" he asked, trying in vain\nto give firmness to his voice, which trembled involuntarily from a\nforeboding of evil.\n\nCaptain Lopez was a soldier of fortune, who had grown grey in harness,\nand contracted a species of rough frankness, that prevented him toning\nthe truth down under any circumstances, which fact made him appear, in\nthe General's eyes, a bird of very evil omen. The captain's arrival,\ntherefore, doubly disquieted the governor. In the first place, through\nhis alarmed face; and secondly, the reputation he enjoyed. To the\ngeneral's query the captain only replied the following three storm laden\nwords--\n\n\"Nothing that's good.\"\n\n\"What do you mean? Have the people rebelled??\"\n\n\"On my word, no! I do not fancy they even dream of such a thing.\"\n\n\"Very well, then,\" the general went on, quite cheered by the good news,\n\"what the deuce have you to tell me, captain?\"\n\n\"I have not come to tell you anything,\" the other said, roughly. \"There\nis a soldier outside who has just come from I don't know where, and who\ninsists on speaking with you. Shall I bring him, or send him about his\nbusiness.\"\n\n\"One moment,\" exclaimed the general, whose features had suddenly become\ngloomy; \"who is the soldier?\"\n\n\"A dragoon, I fancy.\"\n\n\"A dragoon! Let him come in at once. May heaven bless you, with all your\ncircumlocution! The man, doubtless, brings me news of the arrival of the\nregiment I am expecting, and which should have been here before.\"\n\nThe captain shrugged his shoulders with an air of doubt.\n\n\"What is it now?\" the general said, whom this expressive pantomime\neminently alarmed; \"What are you going to say?\"\n\n\"Nothing, except that the soldier looks very sad to be the bearer of\nsuch good news.\"\n\n\"We shall soon know what we have to depend on. Let him come in.\"\n\n\"That is true,\" said the captain, as he went off.\n\nDuring this conversation the general had leaped from his bed, and\ndressed himself with the promptness peculiar to soldiers. He now\nanxiously awaited the appearance of the trooper whom Don Lopez had\nannounced to him. In vain he tried to persuade himself that the captain\nwas mistaken, and that the soldier had been sent to tell him of the\narrival of the regiment. In spite of himself, he felt in his heart a\nspecies of alarm which he could not account for, and yet nothing could\ndissipate.\n\nA few minutes were thus passed in febrile restlessness. All at once a\ngreat noise was heard in the Plaza Major. The general went to a window,\npulled aside a curtain, and looked out. A tumultuous and dense crowd was\nthronging every street leading to the square and uttering sharp cries.\nThis crowd, momentarily increasing, seemed urged on by something\nterrible, which the general could not perceive.\n\n\"What is this?\" the general exclaimed; \"And what can be the meaning of\nthis disturbance?\"\n\nAt this moment the shouts grew louder, and the detachment of Comanche\nwarriors appeared debouching by the Calle de la Merced, and marching in\ngood order, and at quick step, upon the palace. On seeing them the\ngeneral could not restrain a start of surprise.\n\n\"The Indians again!\" he said; \"How can they dare to present themselves\nhere? They must be ignorant of the arrival of the dragoons. Such\nboldness is incomprehensible.\"\n\nHe let the curtain fall, and turned away. The soldier whom the captain\nhad announced to him stood before him, waiting the general's pleasure to\nquestion him. The general started on perceiving him. He was pale; his\nuniform was torn and stained with mud, as if he had made a long journey\non foot through brambles. The general wished to clear up his doubts;\nbut, just as he was opening his mouth to ask the man a question, the\ndoor flew back, and several officers, among whom was Captain Don Lopez,\nentered the room.\n\n\"General,\" the captain said, \"make haste! You are expected in the\ncouncil hall. The Indians have come for the answer you promised to give\nthem this morning.\"\n\n\"Well! Why this startled look, gentlemen?\" the general said, severely.\n\"I fancy the town has not yet been set on fire. I am not at the orders\nof those savages, so tell them that I have no time to grant them an\naudience.\"\n\nThe officers gazed at the general with a surprise they did not attempt\nto conceal, on hearing these strange and incomprehensible words.\n\n\"Good, good,\" Captain Lopez said, roughly, \"the town is not yet fired,\n'tis true; but it might be so, erelong, if you went on in this way.\"\n\n\"What do you mean?\" the general asked, as he turned pale. \"Are matters\nso serious?\"\n\n\"They are most serious. We have not a moment to lose, if we wish to\navoid heavy disasters.\"\n\nThe general started.\n\n\"Gentlemen,\" he then said, in an ill-assured voice, \"it is our duty to\nwatch over the safety of the population. I follow you.\"\n\nAnd taking no further heed of the soldier he had ordered to be sent in,\nhe proceeded towards the council hall.\n\nThe disorder that prevailed without had at length gained the interior of\nthe palace. Nothing was to be heard but shrieks or exclamations of anger\nand terror. The Mexican officers assembled in the hall were tumultuously\ndiscussing the measures to be adopted in order to save a contest and the\ntown. The entrance of the governor produced a healthy effect upon them,\nin so far that the discussion, which was degenerating into personalities\nand reproaches, dictated by individual fear, suddenly ceased, and\ncalmness was restored.\n\nGeneral Ventura regretted in his heart having counted on imaginary help,\nand not having listened to the sensible advice of some of his officers,\nwho urged him the previous day to satisfy the Indians by giving them\nwhat they asked. In spite of the terror he felt, however, his pride\nrevolted at being compelled to treat on equal terms with barbarians, and\naccept harsh conditions which they would doubtless impose on him, in the\nconsciousness of having the upper hand.\n\nThe governor, in entering the hall, looked around the assembly\nanxiously. All had taken their places, and, externally at least, had\nassumed that grace and stern appearance belonging to men who are\npenetrated with the grandeur of the duties they have to perform, and are\nresolved to carry them out at all hazards. But this appearance was very\ndeceptive. If the faces were impassive the hearts were timorous. All\nthese men, habituated to a slothful and effeminate life, did not feel\ncapable of waging a contest with the rude enemies who menaced them so\naudaciously, even at the doors of the governor's palace.\n\nUnder present circumstances, however, resistance was impossible. The\nIndians, by the fact of their presence on the square, were masters of\nthe town. There were no troops to oppose to them; hence, the only hope\nwas to make the easiest terms possible with the Comanches. Still, as all\nthese men wished to save appearances at any rate, the discussion began\nanew. When everyone had given his opinion, the governor rose, and said\nin a trembling voice--\n\n\"Caballeros, all of us here present: are men of courage, and have\ndisplayed that quality in many difficult circumstances. Certainly, if\nthe only thing, was to sacrifice our lives to save the hapless townsmen,\nwe would not hesitate to do so, for we are too well imbued with the\nsoundness of our duties tot hesitate; but, unhappily, that sacrifice\nwould not avail to save those whom we wish before all to protect. Let us\ntreat, then, with the barbarians, as we cannot conquer them. Perhaps in\nthis way we shall succeed in protecting our wives; and children from the\ndanger that menaces them. In acting thus, under the grave circumstances\nin which we find ourselves, we shall at least have the consolation of\nhaving done our duty, even if we do not obtain all we desire.\"\n\nHearty applause greeted this harangue, and the governor, turning to the\nporter, who stood motionless at the door, gave orders to introduce the\nprincipal Indian chiefs.\n\n\n\n\nCHAPTER XLVI.\n\nTHE COMANCHES.\n\n\nValentine and his friends awoke at daybreak. The Comanches were already\nprepared to start; and Unicorn, dressed in his great war costume,\npresented himself to the hunter.\n\n\"Is my brother going?\" Valentine asked him.\n\n\"Yes,\" the sachem answered. \"I am returning to the Presidio to receive\nthe answer of the chief of the palefaces.\"\n\n\"What is my brother's intention, should his demand be rejected?\"\n\nUnicorn smiled.\n\n\"The Comanches have long lances,\" he said; \"the palefaces will not\nrefuse.\"\n\n\"My anxiety will be extreme till you return, chief; the Spaniards are\nperfidious; take care they have not planned some treachery.\"\n\n\"They would not dare,\" Unicorn said, haughtily. \"If the chief, whom my\nbrother loves, is not delivered to me safe and sound, the Spanish\nprisoners shall be tortured on the plaza of Santa Fe, the town burned\nand sacked. I have spoken; my brother's mind may be at rest.\"\n\n\"Good! Unicorn is a wise chief; he will do what is necessary.\"\n\nIn the meantime the Comanche warriors had formed their ranks, and only\nawaited the signal of the sachem to start. The Spanish prisoners taken\nduring the night were placed in the centre bound and half naked.\nSuddenly a disturbance was heard in the camp, and two men rushed panting\ntoward the spot where stood Valentine, the sachem and Curumilla. They\nwere Don Pablo and Father Seraphin, their clothes in disorder, their\nfeatures haggard, and their faces glistening with perspiration. On\nreaching their friends, they fell, almost in a fainting state, on the\nground. The proper attentions were at once paid them, and the missionary\nwas the first to recover. Don Pablo seemed stupefied; the tears poured\nincessantly down his cheeks, and he could not utter a word. Valentine\nfelt strangely alarmed.\n\n\"Good heavens!\" he exclaimed, \"What has happened? Don Miguel--?\"\n\nThe missionary shook his head.\n\n\"No,\" he said, \"nothing has happened to him, as far as I know.\"\n\n\"Heaven be praised! But what is the matter, father? What misfortune have\nyou to announce to me?\"\n\n\"A frightful one, indeed, my son,\" the missionary replied, as he buried\nhis face in his hands.\n\n\"Speak, in Heaven's name! Your delay is killing me.\"\n\n\"Dona Clara--\"\n\n\"Well!\" he hunter said, sharply.\n\n\"Was captured again last night by Red Cedar, and torn from the refuge\nwhere I placed her.\"\n\n\"Oh!\" Valentine exclaimed, with concentrated fury, as he stamped his\nfoot, \"Always that demon--that accursed Red Cedar. My curses on him! But\ntake courage, father; let us first save Don Miguel, and then I swear to\nyou that I will restore his daughter to him.\"\n\nUnicorn advanced.\n\n\"Master of prayer,\" he said to Father Seraphin, in a soft and impressive\nvoice, \"your heart is good. The Comanches love you. Unicorn will help\nyou. Pray to your God. He will protect us in our researches, since He\nis, as you say, so powerful.\"\n\nThen the chief turned to Don Pablo, and laid his hand firmly on his\nshoulder.\n\n\"Women weep,\" he said; \"men avenge themselves. Has not my brother his\nrifle?\"\n\nOn feeling the Comanche's hand laid on him--on hearing these words--the\nyoung man quivered as if he had received an electric shock. He drew\nhimself up, and fixed on the chief his eyes burning with the fever of\nsorrow.\n\n\"Yes,\" he said, in a broken voice, \"you are right, chief, and,\" passing\nhis hand over his eyes, with a gesture of rage, \"let us leave tears to\nwomen, who have no other weapons to protect their weakness. I am a man,\nand will avenge myself.\"\n\n\"Good. My brother speaks well: he is a warrior; Unicorn esteems him; he\nwill become great on the war path.\"\n\nDon Pablo, crushed for a moment, had regained all his energy; he was no\nlonger the same man; he looked around him.\n\n\"Where are you going?\" he asked.\n\n\"To Santa Fe, to deliver your father.\"\n\n\"I will go with you.\"\n\n\"Come,\" said Unicorn.\n\n\"No,\" Valentine interposed, authoritatively. \"Your place is not there,\nDon Pablo; leave the Comanche warriors to act as they please; they do\nnot need your help to carry out their plans properly. Remain with me.\"\n\n\"Command me, my friend,\" the young man said with resignation; \"I have\nperfect confidence in your experience.\"\n\n\"Good. You are reasonable. Brother,\" he added, turning to the chief,\n\"you can start. The sun is already high in the horizon; may Heaven grant\nthat you may succeed!\"\n\nUnicorn gave the signal for departure. The Comanches uttered their war\nyell, while brandishing their arms, and started at a quick amble, the\nonly pace they know. Curumilla then rose, and wrapped himself in his\nbuffalo robe; Valentine watching him, inquiringly.\n\n\"Does my brother leave us?\" he said.\n\n\"Yes,\" the Araucano answered, laconically.\n\n\"For long?\"\n\n\"For a few hours?\"\n\n\"Where is my brother going?\"\n\n\"To look for the camp of Red Cedar's gambusinos,\" the Indian replied\nwith a cunning smile.\n\n\"Good,\" Valentine said, gleefully. \"My brother is a wise chief; he\nforgets nothing.\"\n\n\"Curumilla loves his brother; he thinks for him,\" the chief answered,\nsimply.\n\nAfter uttering these words, the Unicorn bowed gracefully, and proceeded\nin the direction of the Paso del Norte, soon disappearing in the\nwindings of the road. Valentine looked after him for a long while. When\nhe no longer saw him, he let his head fall pensively on his chest,\nmurmuring in a low voice--\n\n\"Good, intelligent fellow! Heart of gold! The only friend left me! The\nonly one remaining of my old and faithful comrades! Louis, my poor\nLouis, where are you now?\" A deep sigh burst from his bosom, and he\nremained absorbed in a gloomy reverie.\n\nAt length Valentine raised his head, passed his hand over his brow, as\nif to dispel these sad thoughts, and turned to his friends.\n\n\"Pardon me,\" he said, \"but I, at times, give way to my thoughts in that\nfashion. Alas! I, too, have suffered; but let us leave that,\" he added,\ngaily. \"Bygones must be bygones. Let us attend to your affairs.\"\n\nHe made them a sign to sit down by his side on the grass, rummaged his\nalforjas and produced some slight food, which he laid before them.\n\n\"Eat,\" he said to them; \"we do not know what awaits us within the next\nfew hours, and we must recruit our strength. When you have satisfied\nyour appetite, you will tell me all about Dona Clara being carried off\nagain, for I must have the fullest details.\"\n\nWe will leave the three now conversing, and join the Comanches and\nUnicorn again.\n\nWhen the Comanches reached the Plaza Mayor, opposite the Cabildo, they\nhalted. At an order from Unicorn, the prisoners were completely stripped\nof their clothing and placed some distance in front of the first rank of\nIndians, each of them having at his side a fully armed Indian ready to\nmassacre him mercilessly at the slightest sign from Unicorn. When the\npreparations were completed, and the Comanches had stationed sentinels\nat each corner of the streets, opening in the square, in order not to be\ntaken in reverse, and surrounded by the Spaniards, if they felt any\ninclination for fighting, the Spider, the chief who had already\nperformed the duty of flag of truce, pranced up to the gate of the\npalace, and demanded speech with the governor.\n\nThe officer of the guard, who was no other than Don Lopez, politely\nrequested the Indian warrior to wait a few moments, and then proceeded\nin all haste to General Ventura. We have seen what took place, and,\nafter a delay of nearly half an hour, Captain Don Lopez returned. It was\ntime, for the Comanches were beginning to grow tired of waiting, and\nwere preparing to force the passage which was not voluntarily granted\nthem. After some preliminary explanations, Captain Lopez informed the\nSpider that the general, surrounded by his staff, was awaiting, in the\nhall of audience, the sachem of the nation and his three principal\nwarriors.\n\nThe Spider communicated this answer to Unicorn, who gave a nod of\nassent, dismounted, and entered the Cabildo.\n\n\n\n\nCHAPTER XXIII.\n\nNEGOTIATIONS.\n\n\nWhen Unicorn entered the council chamber, preceded by Captain Lopez, and\nfollowed by the three Indian chiefs, the deepest silence prevailed among\nthe Spanish officers assembled to meet him. The governor, seated in a\nchair placed in the centre of the hall, was looking nervously round him,\nwhile tapping on the arm of the chair with the fingers of his right\nhand. Still, his countenance was tolerably composed; nothing externally\nrevealed the terror that devoured him. He answered by a nod the\nceremonious bow of the Comanches, and drew himself up as if intending to\naddress them; but if such were his desire, Unicorn did not grant him\ntime to do so. The sachem draped himself in his buffalo robe with that\nmajestic grace possessed by all those untamed sons of the desert, drew\nhis head up proudly, and walked toward General Ventura, who watched him\napproach with an anxious eye. On coming within four paces of the\ngovernor, Unicorn stopped, crossed his arms on his chest, and took the\nword.\n\n\"I salute my father!\" he said, in a loud and fierce voice. \"I have come,\nas was agreed on yesterday, to fetch the answer he owes me.\"\n\nThe general hesitated for an instant.\n\n\"I am waiting!\" the Indian went on, with a frown that augured ill.\n\nThe general, forced almost into his last entrenchments, saw that the\nhour for surrender had at length arrived, and that no way of escape was\nleft him.\n\n\"Chief,\" he answered, in anything but a firm voice, \"your behavior\nnaturally surprises me. To my knowledge the Spaniards are not at war\nwith your nation; the whites have not done anything of which you have a\nright to complain. For what reason do you come, then, against the sworn\nfaith, and when nothing authorises you, to invade a defenceless town,\nand interfere in matters that only concern ourselves?\"\n\nThe sachem understood that the Spaniard was trying to shift the question\non to other ground; he saw the snare offered him, and was not to be\ncaught.\n\n\"My father does not answer my request,\" he said. \"Still, in order to\nhave finished at once with the recriminations he brings up, I will\nanswer his questions peremptorily, separating them one from the other.\nIn the first place, my father knows very well that the palefaces and\nredskins have been in a constant state of warfare since the arrival of\nwhite men in America. This war may have slightly relaxed at intervals,\nbut has never really ceased. Our two races are hostile; the struggle\nwill not end between them until one of the two families, whether white\nor red, has given place to the other by its general extinction.\nSecondly, my father said that nothing has been done of which we had a\nright to complain. My father is mistaken, we have a cause, the\nimprisonment of Don Miguel Zarate, who, himself an Indian, has never\nbelied his origin. Hence my father must no longer ask by what right I am\nhere, for that is perfectly established; it is that which every honest\nman possesses of defending an innocent person who is oppressed. Now that\nfact is cleared up, let us pass to another. When I came here yesterday,\nmy father gave me to understand that my propositions would be accepted,\nand the exchange of prisoners carried out.\"\n\n\"It is possible, chief,\" the general replied; \"but things are so in this\nworld, no one knows today what he will do tomorrow. With night\nreflection has come, and, in short, your propositions have appeared to\nme unacceptable.\"\n\n\"Wah!\" the Indian said, though not testifying his surprise otherwise.\n\n\"Yes,\" the general continued, growing animated, \"I should be ashamed to\ngrant them, for I should have the appearance of only yielding to\nthreats. No, it cannot be. The two gentlemen you claim are guilty, and\nshall die; and if you venture to oppose the execution of the just\nsentence of the court, we will defend ourselves, and God will protect\nthe good cause.\"\n\nThe Mexican officers warmly applauded this haughty response, which they\nwere far from expecting. They felt their courage rekindled, and did not\ndespair of obtaining better conditions. A smile of disdain played round\nthe chiefs haughty lips.\n\n\"Good,\" he said; \"my father speaks very loudly. The coyotes are bold\nwhen they hunt the buffalo in packs. My father has carefully reflected,\nand is determined to accept the consequences of his answer. He wishes\nfor war, then?\"\n\n\"No,\" the general quickly interposed, \"heaven forbid! I should be glad\nto settle this matter amicably with you, chief, but honor forbids me\nsubscribing those disgraceful proposals which you did not fear to lay\nbefore me.\"\n\n\"Is it really honour that has dictated my father's answer?\" the Indian\nasked, ironically. \"He will permit me to doubt it. In short, whatever be\nthe reason that guides him, I can but withdraw; but, before doing so, I\nwill give him news of a friend whom he doubtless impatiently expects.\"\n\n\"What means that word, doubtless?\"\n\n\"This,\" the Indian said, sharply. \"The warriors whom my father expected\nto arrive to his aid this day have been dispersed by my young men, as\nthe autumn breeze sweeps away the leaves. They will not come.\"\n\nA murmur of surprise, almost of terror, ran through the assembly. The\nsachem let the long folds of his buffalo robe fall back, tore from his\ngirdle the bleeding scalp that hung there, and threw it at the general's\nfeet.\n\n\"That,\" he said, gloomily, \"is the scalp of the man who commanded my\nfather's warriors! Does the chief of the palefaces recognise it? This\nscalp was raised by me from the head of the man who was to arrive, and\nwho, at this hour, has set out for the happy hunting grounds of his\nnation.\"\n\nA shudder of terror ran round the room at the sight of the scalp; the\ngeneral felt the small dose of courage that had still animated him\noozing out.\n\n\"Chief,\" he exclaimed, in a trembling voice \"is it possible you have\ndone that?\"\n\n\"I have done it,\" the sachem answered, coldly. \"Now, farewell. I am\nabout to join my young men, who are impatient at my long absence.\"\n\nWith these words the Comanche haughtily turned his back on the governor,\nand walked toward the door.\n\n\"A few moments longer, chief,\" the general said; \"perhaps we are nearer\nan understanding than you suppose.\"\n\nThe Comanche gave the speaker a glance which made him quiver.\n\n\"Here is my last word,\" he said. \"I insist on the two prisoners being\nhanded over to me.\"\n\n\"They shall be.\"\n\n\"Good; but no perfidity, no treachery.\"\n\n\"We will act honourably,\" the general replied, not dreaming, of resenting\nthe insult conveyed in the Indian's words.\n\n\"We shall see. My warriors and myself will remain on the square till my\nfather has performed his promise. If, within an hour, the palefaces are\nnot free, the prisoners I hold will be pitilessly massacred, and the\n_altepetl_ plundered. I have spoken.\"\n\nA gloomy silence greeted these terrible threats. The pride of the\nMexicans was quelled, and they at length recognised that nothing could\nsave them from the vengeance of the Comanche chief. The general bowed in\nassent, not having strength to answer otherwise. The sight of the scalp\nhad paralyzed in him all desire to contend longer. Unicorn left the\nhall, mounted his horse again, and calmly awaited the fulfilment of the\npromise made to him.\n\nWhen the Indians had left the council chamber, the Mexicans rose\ntumultuously, for each feared the execution of the chief's threats.\nGeneral Ventura was pressed on all sides to make haste, and run no risk\nof breaking his word. When the governor saw that his officers were as\nterrified as himself, he re-assumed his coolness, and cleverly profited\nby this state of mind, in order to throw the responsibility off himself,\nand appear only to act under the impulse of others.\n\n\"Caballeros,\" he said, \"you have heard this man. You understood as well\nas I did the menaces he dared to offer us. Shall such an insult be left\nunpunished? Will you allow yourselves to be thus braved in the heart of\nthe town by a handful of scoundrels, and not attempt to inflict on them\nthe chastisement they deserve? To arms, caballeros, and let us die\nbravely, if it must be so, sooner than suffer this stain on the old\nSpanish honor our fathers transmitted to us!\"\n\nThis warm address produced the effect the general anticipated from it;\nthat is to say, it redoubled, were that possible, the terror of the\nhearers, who had long been acquainted with their chiefs cowardice, and\nknew how little he could be depended on. This sudden warlike order\nseemed to them so unusual, and before all so inopportune, that they\npressed him to accept without delay the proposals dictated by the\nsachem.\n\nThis was all the governor wanted. He had the minutes of the council at\nonce drawn up, when it was signed by all present, he put it in his\npocket.\n\n\"As you insist,\" he said, \"and nothing can induce you to offer an\nhonourable resistance, I will myself proceed to the prison, in order to\navoid any misunderstanding, and have the doors opened for Don Miguel\nZarate and General Ibanez.\"\n\n\"Make haste, pray?\" the officers answered.\n\nThe general, glad in his heart at having got out of the scrape so well,\nleft the Cabildo, and walked across the square to the prison, which\nstood on the opposite side. The Comanches were motionless as statues of\nFlorentine bronze, leaning on their weapons, with their eyes fixed on\nthe chief, ready to carry out his orders.\n\n\n\n\nCHAPTER XXIV.\n\nFREE.\n\n\nDon Miguel and General Ibanez were completely ignorant of what was going\non outside, and the rumours of the town did not reach their ears. Had\nthey deigned to question their jailer, the latter, who was beginning to\nfear for himself the effects of the ill-treatment he had made the two\ngentlemen undergo, would doubtless not have hesitated to give them all\npossible information, for the sake of regaining their favour; but each\ntime this man presented himself before them, and opened his mouth to\nspeak, they turned their backs contemptuously, giving him a sign to\nwithdraw at once, and be silent.\n\nOn this day, according to their wont, the two prisoners had risen at\nsunrise, and then, with incredible coolness, began conversing on\nindifferent topics. Suddenly a great noise was heard in the prison; a\nclang of arms reached the prisoners' ears, and hurried footsteps\napproached the rooms in which they were confined. They listened.\n\n\"Oh, oh!\" said Ibanez, \"I fancy it is for today at last.\"\n\n\"Heaven be praised!\" Don Miguel answered; \"I am glad they have made up\ntheir minds to bring matters to a conclusion.\"\n\n\"On my honour, and so am I,\" the general said, gaily; \"time was\nbeginning to hang heavy in this prison, where a man has not the\nslightest relaxation. We are going to see again that splendid sun which\nseems afraid of showing itself in this den. _Viva Cristo_! I feel\ndelighted at the mere thought, and gladly pardon my judges.\"\n\nStill the noise drew nearer and nearer, and confused voices were mingled\nwith the echoing steps in the passage, and the rattling of sabres.\n\n\"Here they are,\" said Don Miguel; \"we shall see them in a minute.\"\n\n\"They are welcome if they bring us death, that supreme solace of the\nafflicted.\"\n\nAt this moment a key creaked in the lock, and the door opened. The two\nprisoners fell back in surprise on seeing the general, who rushed into\nthe cell followed by two or three officers. Assuredly, if the prisoners\nexpected to see anybody, it was not the worthy General Ventura. Ibanez'\nsurprise was so great at this unexpected apparition, that he could not\nrefrain from exclaiming, with that accent of caustic gaiety which formed\nthe basis of his character--\n\n\"What the deuce do you want here, Senor Governor? Have you, too,\nsuddenly become a frightful conspirator, such as we are accused of\nbeing?\"\n\nBefore answering, the general fell back into a chair, wiping away the\nperspiration that trickled down his forehead, such speed had he\ndisplayed in coming to the prison. Three or four officers stood\nmotionless on the threshold of the widely open door. The condemned men\ncould not at all understand the affair.\n\n\"Have you by any chance, my dear governor,\" General Ibanez said, gaily,\nthough not believing a word of it, \"come to restore us to liberty? That\nwould be a most gallant action, and I should feel deeply indebted to you\nfor it.\"\n\nGeneral Ventura raised his head, fixed on the prisoners eyes sparkling\nwith joy, and said, in a panting voice--\n\n\"Yes, my friends, yes; I _would_ come myself to tell you that you are\nfree; I would not yield to anyone else the pleasure of announcing the\ngood news.\"\n\nThe prisoners fell back in amazement.\n\n\"What!\" General Ibanez exclaimed, \"You are speaking seriously?\"\n\nDon Miguel attentively looked at the governor, trying to read in his\nface the reasons of his conduct.\n\n\"Come, come,\" General Ventura cried, \"this hole is frightful; do not\nremain any longer in it.\"\n\n\"Ah!\" Don Miguel remarked, bitterly, \"You find it frightful; you have\nbeen a long time in discovering the fact; for we have lived in it nearly\na month, and the thought never once occurred to you of disturbing our\nrepose.\"\n\n\"Do not be angry with me, Don Miguel,\" the governor answered eagerly,\n\"it was greatly against my will you were detained so long; had it only\ndepended on me you would have been free; but, thanks to Heaven, all is\nsettled now, and I have succeeded in having justice done you. Come away;\ndo not remain a moment longer in this pestilential den.\"\n\n\"Pardon me, Caballero,\" Don Miguel said coldly, \"but, with your\npermission, we will remain a few moments longer in it.\"\n\n\"Why so?\" General Ventura asked, opening his eyes to their fullest\nextent.\n\n\"I will tell you.\"\n\nDon Miguel pointed to a chair, and sat down himself. Ibanez following\nhis example. There was a moment of deep silence between these three men\nas they strove to read each other's real secret thoughts.\n\n\"I am waiting your pleasure to explain yourself,\" the governor at last\nsaid, as he was anxious to get away, and time pressed.\n\n\"I am about doing so,\" Don Miguel answered; \"you have come to tell us we\nare free, sir; but you do not say on what conditions.\"\n\n\"What do you mean by conditions?\" the general asked, not understanding\nhim.\n\n\"Of course,\" Ibanez went on, supporting his friend; \"and these\nconditions, too, must suit us; you must see, my dear sir, we cannot\nleave this delightful place without knowing the why or wherefore. _Viva\nCristo_! We are not vagabonds to be got rid of in that way; we must know\nif we are justified in accepting the proposals you have just made.\"\n\n\"The general is right, sir,\" the hacendero said in his turn; \"the care\nof our honor does not permit us to accept a liberation which might stain\nit; hence, we shall not leave this prison until you have given us an\nexplanation.\"\n\nThe governor hardly knew whether he was on his head or his heels; he had\nnever before had to deal with such obstinate prisoners. He racked his\nbrains in vain to discover why it was that men condemned to death could\nso peremptorily decline their liberty. His ideas were too narrow, his\nheart was too cowardly for him to comprehend the grandeur and nobility\nin this determination on the part of two men, who preferred an honourable\ndeath to a branded life which they only owed to the pity of their\njudges. Still, he must induce them to quit the prison, for time was fast\nslipping away, and their obstinacy might ruin everything. Hence, General\nVentura made up his mind like a man.\n\n\"Gentlemen,\" he said, with feigned admiration, \"I understand what\nnobleness there is in your scruples, and am happy to see that I was not\nmistaken in the greatness of your character. You can leave this prison\nin full security, and take once more the station that belongs to you in\nthe world. I will lay no conditions on you; you are free, purely and\nsimply. Here are the documents connected with your trial, the proofs\nproduced against you; take them and destroy them, and accept my sincere,\napologies for all that has passed.\"\n\nWhile saying this, the governor drew from his breast an enormous bundle\nof papers, which he offered Don Miguel. The latter declined them with an\nair of disgust; but General Ibanez, less scrupulous or wiser in his\ngeneration, eagerly clutched them, looked through them to see that the\ngovernor was not deceiving him, and then threw them into the _brasero_,\nstanding in the middle of the room. In less than four minutes, all this\nundigested mass was consumed. General Ibanez watched them burning with a\ncertain degree of pleasure, for he began to feel himself really free.\n\n\"I am waiting for you, gentlemen,\" said the governor.\n\n\"One word more, by your leave,\" the hacendero remarked.\n\n\"Speak, sir.\"\n\n\"On leaving this prison, where are we to go?\"\n\n\"Wherever you please, gentlemen. I repeat to you that you are perfectly\nfree, and can act as you think proper. I do not even ask your word of\nhonor to enter into no further conspiracy.\"\n\n\"Good sir,\" Don Miguel said, holding out his hand to General Ventura,\n\"your conduct affects me--thanks.\"\n\nThe governor blushed.\n\n\"Come, come,\" he said, to hide his embarrassment on receiving this so\nill-deserved praise.\n\nThe prisoners no longer hesitated to follow him.\n\nIn the meanwhile, the news of Don Miguel's deliverance had spread\nthrough the town with the rapidity of a train of gunpowder. The\ninhabitants, reassured by the continence of the Comanches, and knowing\nthat they had only come to save a man, in whose fate the entire\npopulation felt interested, had ventured to leave their houses, and at\nlength thronged the streets and squares; the windows and roofs were\nfilled with men, women, and children, whose eyes, fixed on the prison,\nawaited the moment of Don Miguel's appearance. When he did so,\ntremendous shouts greeted him.\n\nUnicorn walked up to the governor.\n\n\"My father has kept his promise,\" he said, gravely, \"I will keep mine;\nthe white prisoners are free; I now depart.\"\n\nThe governor listened to these words with a blush; the sachem returned\nto the head of his war party, which rapidly retired, followed by the\nshouts of a mob intoxicated with joy. Don Miguel, perplexed by the scene\nwhich had taken place in his presence, and who began to suspect a\nmystery in the governor's conduct, turned to him to ask an explanation\nof the Indian chief's words--an explanation the governor luckily\nescaped, owing to the eagerness of the people who flocked up to\ncongratulate the prisoners on their release.\n\nOn reaching the gate of the Cabildo, General Ventura bowed courteously\nto the two gentlemen, and hurried into his palace, happy at having\nescaped so cheaply, and not tearing with his own hands the cloak of\ngenerosity which he had paraded in the sight of his prisoners.\n\n\"What do you think of all that?\" the hacendero asked his friend.\n\n\"Hum!\" General Ibanez muttered, \"The governor's conduct seems to me\nrather queer; but, no matter, we are free. I confess to you, my friend,\nthat I should have no objection to go a little distance from this place,\nthe air of which, despite General Ventura's protestations, appears to me\nremarkably unhealthy for us.\"\n\nAt this moment, and ere Don Miguel could answer, the general felt a\nslight touch on his shoulders; he turned and saw Curumilla before him,\nwith a smiling face. Don Miguel and the general suppressed a cry of joy\nat the sight of the grave and excellent Indian.\n\n\"Come!\" he said to them, laconically.\n\nThey followed him, with some difficulty, through the crowd that\naccompanied them with shouts, and whom they were obliged to stop and\nthank. On reaching a small street near the square, and which was nearly\ndeserted, Curumilla led them to a house before which he stopped.\n\n\"It is here,\" he said, as he tapped twice.\n\nThe door opened, and they entered a courtyard, in which were three ready\nsaddled horses, held by a groom, which they at once mounted.\n\n\"Thanks, brother,\" the hacendero said, warmly, as he pressed the chiefs\nhand; \"but how did you learn our deliverance?\"\n\nThe Araucano smiled pleasantly. \"Let us go,\" he said, making no other\nanswer.\n\n\"Where to?\" Don Miguel asked.\n\n\"To join Koutonepi.\"\n\nThe three men started at full speed. Ten minutes later they were out of\nthe town, and galloping across the plain.\n\n\"Oh!\" General Ibanez said, gaily, \"How pleasant the fresh air is! How\ngood it is to inhale it after remaining for two months stifled between\nthe walls of a prison!\"\n\n\"Shall we soon arrive? Don Miguel asked.\n\n\"In an hour,\" the chief answered.\n\nAnd they went on with renewed speed.\n\n\n\n\nCHAPTER XLIX.\n\nTHE MEETING.\n\n\nOn reaching a spot where the trail they were following formed a species\nof fork, Curumilla stopped, and the two gentlemen imitated him.\n\n\"That is your road,\" the Araucano chief said. \"At the end of that path\nyou will see Koutonepi's bivouac fire. I must leave you here.\"\n\nAfter uttering these words, Curumilla turned his horse and started,\nafter giving them a parting wave of the hand. The Unicorn was not much\nof a talker naturally; generally, he did more than he said. His friends,\nconvinced that urgent necessity could alone have forced him thus to\nbreak through his habits, made no observation, but let him go. When they\nwere alone, they gently relaxed the pace of their horses, and proceeded\nat a canter.\n\nGeneral Ibanez was radiant. He inhaled the fresh air Of the desert,\nwhich dilated his wide chest, revelling in his liberty. He thought of\nnothing but enjoying the present, regardless of the past, which, with\nhis careless character, he had already forgotten, only to dream of the\nfuture, which he gazed on through a prism of brilliant hues. Don Miguel,\non the contrary, felt, during the last few moments, a sad melancholy\ninvade his mind. Not able to account for the emotion he experienced, he\nhad a species of secret presentiment that a misfortune was suspended\nover his head. In vain did he try to dispel these ideas, but they\nconstantly returned more obstinately than ever and it was with a sort of\ndread that he advanced in the direction where he was to meet Valentine,\nalthough he was his best friend, so much did he fear that he would greet\nhis arrival with evil tidings.\n\nThe two gentlemen went on thus for nearly half an hour without\nexchanging a syllable; but, just as they turned a corner in the path,\nthey saw a horseman about thirty paces in front of them, barring the\nroad, and apparently waiting for them. The Mexicans examined him\nattentively. He was a tall man, well armed, and wearing the garb of the\nrich hacenderos; but, singularly enough, a black velvet mask prevented\nthem distinguishing his features. By an instinctive movement Don Miguel\nand his friend moved a hand to their holsters, but they were empty.\n\n\"What is to be done?\" the hacendero asked the general.\n\n\"Go on, of course. We have just escaped too great a peril for us to fear\nthis. Even in the event of the mysterious being planted there before us,\nlike an equestrian statue, trying to play us a trick, which is not\nimpossible.\"\n\n\"Let us trust to Heaven,\" Don Miguel muttered, and pushed on.\n\nThe distance separating them from the stranger was soon cleared. On\ncoming within five yards of him, they stopped.\n\n\"_Santas tardes_, caballeros,\" said the stranger, in a friendly voice.\n\n\"_Santas tardes_!\" the gentlemen answered, in accord.\n\n\"I salute you, Don Miguel Zarate, and you, General Ibanez,\" the stranger\nthen said. \"I am happy to see you at length safe and sound out of the\nclaws of that worthy General Ventura, who, if he could, would certainly\nhave played you a trick.\"\n\n\"Caballero,\" Don Miguel made answer, \"I thank you for the kind words you\naddress to me, and which can only come from a friend's lips. I should be\npleased if you would take off the mask that conceals your features, so\nthat I may recognise you.\"\n\n\"Gentlemen, if I removed my mask you would be disappointed, for my\nfeatures are unfamiliar to you. Do not be angry with me for keeping it\non; but, be assured that you are not mistaken with regard to me, and I\nam really your friend.\"\n\nThe two Mexicans bowed courteously to each other, and the stranger went\non.\n\n\"I knew that so soon as you were free you would hasten to join that\nworthy hunter Valentine, whom the trappers and gambusinos along the\nfrontier have christened the 'Trail-hunter.' I placed myself here, where\nyou must infallibly pass, in order to make you a communication of the\nutmost importance, which interests you extremely.\"\n\n\"I am listening, sir,\" Don Miguel responded with secret alarm; \"and I\nbeg you to accept, beforehand, my sincere thanks for the step you have\ntaken on my behalf.\"\n\n\"You will thank me when the proper time comes, Don Miguel. Today I only\nwarn you: at a later date I hope to aid you, and my help will not prove\nuseless.\"\n\n\"Speak, sir! You excite my curiosity to the highest pitch, and I am\nanxious to learn the news of which you have condescended to be the\nbearer.\"\n\nThe stranger shook his head sadly, and there was a moment's silence.\nThis meeting of three horsemen, one of whom was masked, in this deserted\nplace, where no sound troubled the imposing silence of solitude, had\nsomething strange about it. At length the mask spoke again.\n\n\"Two months have elapsed, Don Miguel, since, through the treachery of\nRed Cedar, you were arrested and made prisoner at the Paso del Norte.\nMany events of which you are ignorant have occurred since then; but\nthere is one I must inform you of at once. On the very night of your\narrest, at the moment you laid down your arms, your daughter was carried\noff by Red Cedar.\"\n\n\"My daughter!\" the hacendero exclaimed; \"And Valentine to whom I\nconfided her, and who was responsible for her safety?\"\n\n\"Valentine attempted impossibilities to save her; but what can one man\neffect against twenty?\"\n\nDon Miguel shook his head mournfully.\n\n\"After researches, long, sterile, and extraordinary efforts, a man\nprovidentially aided by Father Seraphin, at length succeeded last night\nin taking Dona Clara from her ravishers; but Red Cedar, advised by some\nextraordinary chance, entered the house where the maiden had sought\nshelter, and carried her off again.\"\n\n\"Oh! I will avenge myself on that man!\" the hacendero shouted,\npassionately.\n\nThe stranger's eyes flashed with a lurid light though the holes in his\nmask.\n\n\"You will find your son and Father Seraphin with Valentine. Red Cedar\nintends to start this evening at the head of a band of gambusinos, to go\ninto the deserts of the Rio Gila in search of a placer, which his\naccomplice, Fray Ambrosio, had indicated to him.\"\n\n\"Fray Ambrosio!\" the hacendero repeated, in stupor.\n\n\"Yes. Your former chaplain, who served as spy to the squatter, revealed\nyour plans to him, and provided him the means to enter the hacienda and\ncarry off your daughter.\"\n\n\"Good,\" Don Miguel said, in a hollow voice. \"I will remember.\"\n\n\"Red Cedar, I know not with what design, is taking your daughter with\nhim into the desert.\"\n\n\"I will follow him, were it for a thousand leagues,\" Don Miguel said,\nresolutely. \"Thanks to you for having instructed me so fully. But whence\ncomes the interest you take in me so gratuitously, since, as you say, I\ndo not know you?\"\n\n\"You shall learn at a later date, Don Miguel. Now, before I leave you,\none last word--an earnest warning.\"\n\n\"I listen attentively, caballero.\"\n\n\"Do not tell anyone--not even the French hunter, not even your son--of\nour meeting. Let this secret be buried in your breast. When you reach\nthe far west, if you see before you, at one of your bivouacs, a piece of\nmahogany bearing the impress of a horse's shoe, rise at midnight, and\nleave the camp, not letting anyone see you. When you have gone one\nhundred paces in the tall grass, whistle thrice; a similar whistle will\nanswer you, and then you will learn many things important for you to\nknow, but which I cannot tell you today.\"\n\n\"Good. Thanks. I will do what you tell me.\"\n\n\"You promised it?\"\n\n\"I swear it on my word as a gentleman,\" Don Miguel said, as he took off\nhis hat.\n\n\"I accept your oath. Farewell.\"\n\n\"Farewell.\"\n\nThe stranger dug his spurs into his horse's sides and the animal started\noff as if impelled by a tornado.\n\nThe two gentlemen looked after him for a long time, admiring the grace\nand ease of his movements; at length, when horse and rider had\ndisappeared in the distance, Don Miguel went on again pensively, while\nsaying to the general--\n\n\"Who can that man be?\"\n\n\"I know no more than you do. _Viva Cristo_!\" his friend answered, \"but I\nassure you I will know, even if to do so I have to search all the\nthickets and caverns in the desert.\"\n\n\"What,\" Don Miguel exclaimed, \"do you intend to come with me?\"\n\n\"Did you ever doubt it, Don Miguel? If so, you insulted me. You will\nneed all your friends to go in search of your daughter, and inflict on\nthat demon of a gringo squatter the chastisement he deserves. No, no; I\nwill not leave you under such circumstances, for that would be\ncommitting a bad action; besides, I shall not be sorry,\" he added with a\nsmile, \"to get out of the sight of the government for a time.\"\n\n\"My friend, I thank you,\" the hacendero said, as he took his hand. \"I\nhave long known that you were entirely devoted to me; I am pleased to\nreceive this new proof of your friendship.\"\n\n\"And you accept it?\" the general asked gaily.\n\n\"Most heartily; the help of an iron arm like yours must be most useful\nto me under the painful circumstances in which I am placed.\"\n\n\"That is settled, then; we will start together, _Mil rayas!_ and I swear\nwe will deliver Dona Clara.\"\n\n\"May Heaven grant it,\" the hacendero said, sadly.\n\nThe conversation then dropped, and the two friends proceeded in silence.\nA quarter of an hour later they reached the Trail-hunter's bivouac.\n\n\n\n\nCHAPTER XXVI.\n\nDONA CLARA.\n\n\nValentine had been warned, nearly an hour previously, by Unicorn of the\nresult of the negotiations with the governor of Santa Fe, and the\nimmediate liberation of the prisoners; he was, therefore, expecting\nthem. Though they were ignorant where to find him, Valentine presumed\nthat the chief would leave some Indian to direct them, and, therefore,\ndid not feel at all surprised at seeing them. So soon as he noticed\ntheir approach he walked to meet them, followed by Don Pablo and the\nmissionary, while the hacendero and his comrade on their side pricked\non to join them sooner.\n\nA few hours were spent, after the first greetings were over, in a\nconference, of which the poor child so audaciously carried off was the\nsole subject. Valentine drew up with his friends the plan of the\ncampaign against Red Cedar, which was so daring that it would have made\na European nervous; but the free adventurers who were about to carry it\nout in no way feared the mysterious dangers of the desert which they\nwere going to confront. We say, free, because Father Seraphin had taken\nleave of his friends and found Unicorn, with whom he wished to go to the\nComanche villages, in the hope of spreading the light of the Gospel\nthere. Still, he did not despair about, meeting his friends in the\nprairies, whither he was himself proceeding. Toward evening, Curumilla\narrived. The Araucano was covered with dust, and his face damp with\nperspiration; Not uttering a word, he sat down by the fire, took his\ncalumet from his girdle, and began smoking. Valentine let him do so\nwithout asking a question, but so soon as he saw him absorbed in his\npipe, he laid his hand on his shoulder.\n\n\"Well?\" he said to him.\n\n\"Curumilla has seen them.\"\n\n\"Good; are they numerous?\"\n\n\"Ten times the number of fingers on my two hands, and one more.\"\n\n\"_Caramba!_\" Valentine exclaimed, \"Are they so many as that? We shall\nhave a tough job in that case.\"\n\n\"They are bold hunters,\" the chief added.\n\n\"Hum! Do you know when they will start?\"\n\n\"This evening, when the new moon rises.\"\n\n\"Ah, ah! I read their plan,\" the hunter said. \"They intend crossing the\nford of the Toro before day.\"\n\nCurumilla bowed his head in affirmation.\n\n\"That is true,\" Valentine remarked; \"once the ford is passed they will\nbe in the desert, and have comparatively nothing to fear, or at least\nthey suppose so. I must confess,\" he added, addressing his friends,\n\"that Red Cedar is a remarkably clever scoundrel; nothing, escapes him,\nbut this time he has a' tough adversary. I have my revenge to take on\nhim, and, with the help of Heaven, it shall be exemplary.\"\n\n\"What shall we do?\" Don Miguel asked.\n\n\"Sleep,\" Valentine answered, \"we have still several hours before us, so\nlet us profit by them; in the new life we are beginning, we must neglect\nnothing, the body and mind must repose, so that we may act vigorously.\"\n\nCurumilla had slipped away but now returned, bringing with him two\nrifles, pistols, and knives.\n\n\"My brothers had no weapons,\" he said, as he laid his load before the\nMexicans.\n\nThe latter thanked him heartily; for, owing to the foresight of\nCurumilla, who thought of everything, they could now enter the desert\nboldly. Two minutes later the five men were fast asleep, and we will\ntake advantage of their slumber to return to Red Cedar, whom we left on\nthe point of climbing through Dona Clara's window, while Fray Ambrosio\nand Andres Garote were watching at either end of the street.\n\nAt one bound the bandit was in the room, after breaking open the window\nwith a blow of his fist. Dona Clara, suddenly aroused, leaped from the\nbed, uttering fearful cries at the sight of the terrible apparition\nbefore her.\n\n\"Silence,\" Red Cedar said to her, in a threatening voice, as he placed\nthe point of his knife on her chest, \"one cry more, and I kill you like\na dog.\"\n\nThe maiden, trembling with fright, looked pitifully at the bandit; but\nRed Cedar's face wore such an expression of cruelty, that she understood\nhow little she had to hope from this man. She addressed a silent prayer\nto Heaven, and resigned herself to her fate. The bandit gagged the poor\nchild with the rebozo that lay on the bed, threw her over his shoulder,\nand clambered out of the window again. So soon as he put foot on the\nground, he whistled lightly for his comrades to rejoin him, which they\ndid immediately, and, still carrying his burthen, he proceeded with them\nin the direction of the Rancho del Coyote.\n\nDuring the walk, which was not a long one, the bandits did not meet a\nsoul. Andres opened the door and lit a candle; the ruffians entered, and\nthe door was carefully bolted again. Thus, after only a few hours of\nliberty, the wretched girl had fallen once more into the hands of her\nravishers, and placed again by them in the wretched room where she had\nspent so many days in prayer and weeping. Red Cedar carried Dona Clara,\nwho was in a half-fainting state, to her room, removed the rebozo, and\nthen returned to the bar.\n\n\"There;\" he said, with satisfaction, \"that is all right; the sheep has\nreturned to the fold. What do you say, reverend father? This time let us\nhope she will not escape us.\"\n\nThe monk smiled.\n\n\"We shall do well in not remaining here long,\" he said.\n\n\"Why so?\"\n\n\"Because this hiding place is known and will soon be visited.\"\n\nThe squatter shrugged his shoulders.\n\n\"Listen! Fray Ambrosio,\" he said, with a sinister grimace, which he\nintended for a smile. \"I predict that, rogue as you are, you run a great\nchance of dying in a fool's skin, if you are not flayed beforehand,\nwhich may easily be the case.\"\n\nThe monk shuddered. Red Cedar's gaiety had the peculiarity of being even\nmore fearful than his anger. The squatter sat down on a bench, and\nturned to the gambusino.\n\n\"Drink!\" he said roughly.\n\nGarote fetched a jar of mezcal, which he placed before his terrible\naccomplice. The latter, not taking the trouble to pour the liquor into a\nglass, raised the jar to his lips, and drank till breath failed him.\n\n\"Hum!\" he said, with a click of his tongue, \"That's pleasant tipple when\nyou're thirsty. Listen to my orders, my dear children, and try to carry\nthem out to the letter; or, if not, your roguish hides will bear the\nblame.\"\n\nThe three men bowed silently.\n\n\"You, Nathan,\" he went on, \"will come with me, for you are not wanted\nhere, but your presence is necessary at. Cerro Prieto, where our\ncomrades are encamped.\"\n\n\"I will follow you,\" the young man replied, laconically.\n\n\"Good! Now, you others, bear this carefully in mind:--Our enemies will\nnever suppose that I have made such a mistake as to bring my prisoner\nback here; for that is so absurd, that the idea will never enter their\nheads; so you can be at ease, and no one will trouble your peace of\nmind. Tomorrow, so soon as the moon rises, you will make the girl put on\nan Indian dress, mount her, and come to me at Cerro Prieto. Immediately\nafter your arrival we shall start.\"\n\n\"Good!\" Fray Ambrosio answered. \"We will take care.\"\n\n\"I expect so; for, if you do not, I wouldn't give a _cuartillo_ for your\naccursed hide, my reverend friend.\"\n\nAfter uttering these friendly words, the squatter seized the jar of\nmezcal, emptied it at a draught, and sent it flying across the room,\nwhere it broke to pieces.\n\n\"Good bye till tomorrow,\" he then said, \"come, Nathan.\"\n\n\"Till tomorrow,\" they answered.\n\nThe squatter and his son left the rancho, and walked on silently side by\nside, plunged in gloomy reflections produced by the events of the night.\nThey soon left the town. The night was gloomy, but darkness did not\nexist for squatters accustomed to find their way anywhere, and never\ndreaming of going astray. They walked thus for a long time, with slung\nrifle, not exchanging a word, but listening to the slightest noise and\nsounding, the darkness with their tiger-cat eyes. All at once they heard\nthe firm footfall of a man coming towards them. They cocked their\nrifles, ready for any emergency. A voice was then heard, though the\nperson to whom it belonged was invisible.\n\n\"My brothers must not fire; they would kill a friend.\"\n\nThe words were Apache--a language well known to the squatters.\n\n\"Tis an Indian,\" said Nathan.\n\n\"Do you think I did not recognise him?\" Red Cedar replied, brutally;\n\"then,\" he added, in the same dialect, \"there are no friends in the\nshadow of the desert. My brother must get out of my path, or I will kill\nhim like a coyote.\"\n\n\"Is it thus,\" the Indian continued, \"that the 'maneater' receives the\nguide whom Stanapat, the Great Chief of the Apaches, sends him? In that\ncase, good-bye. I will retire.\"\n\n\"One moment,\" the squatter said, sharply, as he lowered his rifle, and\nmade his son a sign to follow his example. \"I could not guess who you\nwere. Advance without fear and be welcome, brother, for I was anxiously\nexpecting you.\"\n\nThe Indian stepped forward. He wore the costume and characteristic paint\nof the Apache warriors; in a word, he was so well disguised, that\nValentine himself could not, have recognised in him his friend,\nEagle-wing the Chief of the Coras, though it was he.\n\nRed Cedar, delighted at the arrival of his guide, received him in the\nmost affable manner. He had long been acquainted with Stanapat, the most\nferocious warrior of all the Indian nations that traverse the immense\nregions of the Rio Gila, and whom we shall presently visit. After\nseveral questions, which Eagle-wing answered without hesitation or once\ntripping, Red Cedar, convinced that he was really the man the Apache\nchief had promised to send him, dismissed all doubt, and conversed with\nhim in the most friendly spirit, inquiring after certain warriors he had\nformerly known.\n\n\"What is my brother's name?\" he asked, in conclusion.\n\n\"The Heart of Stone!\" Eagle-wing replied.\n\n\"Good!\" the squatter said, \"My brother has a grand name. He must be a\nrenowned warrior in his tribe.\"\n\nA short time after, the three men reached the camp of the gambusinos,\nestablished in a formidable position on the top of a rock called the\nCerro Prieto (Black Mountain). The miners greeted Red Cedar's arrival\nwith the most lively joy, for his presence announced a speedy departure;\nand all these semi-savages, the greater part of whose life had been\nspent in the prairies, were anxious to quit civilization to re-assume\ntheir adventurous career, which was so full of charms and strange\nincidents.\n\n\n\n\nCHAPTER XXVII.\n\nEL VADO DEL TORO.\n\n\nRed Cedar reasoned correctly when he told Fray Ambrosio and Garote that\nDona Clara was in safety at the rancho, and no one would dream of\nseeking her there. In truth, Valentine knew the squatter's cunning too\nwell to suppose that he would commit the impudence of bringing his\nprisoner back to the very spot where she was discovered.\n\nThe squatter's two accomplices passed the day quietly in playing, on\ncredit, at monte; each cheating with a dexterity which did honor to\ntheir knowledge of that noble game. No one came to disturb them, or cast\nan indiscreet glance into this famous den, which, in the bright\nsunshine, had an air of respectability pleasant to look on, and amply\nsufficient to dispel all suspicions. About nine in the evening, the\nmoon, though new, rose magnificently on a deep blue sky, studded with\nbrilliant stars.\n\n\"I fancy it is time to get ready, gossip,\" Fray Ambrosio said, \"the moon\nis peering through the trees in your neighbour's garden.\"\n\n\"You are right, senor Padre, we will be off; but let me, I implore you,\nfirst finish this deal; it is one of the most magnificent I ever\nwitnessed. _Caspita!_ I will bet a nugget as big as my thumb on the\nseven of clubs.\"\n\n\"I'll back the two of spades. Something tells me it will turn up first,\nespecially if you pull up the sleeves of your jacket, which must be\nhorribly in the way when dealing.\"\n\n\"Oh dear, no, I assure you; but stay, what did I tell you? There is the\nseven of clubs.\"\n\n\"That is really extraordinary,\" Fray Ambrosio replied, with feigned\nsurprise, for he was not duped by the gambusino's trickery; \"but I fancy\nwe had better make haste.\"\n\n\"Decidedly,\" said Andres, as he hid his greasy cards in his vaquera\nboots, and proceeded to the room in which Dona Clara was confined. She\nfollowed him out, weeping bitterly.\n\n\"Come, come,\" the gambusino said to her, \"dry your tears, senorita; we\ndo not mean you any harm. Hang it all! Who knows but this may end\nperhaps better than you expect; ask that holy monk what he thinks.\"\n\nFray Ambrosio bowed an assent, but the maiden made no response to the\ngambusino's consolation; she allowed herself to be disguised\nunresistingly, but still continued to weep.\n\n\"In truth, it is absurd,\" the worthy Andres muttered, in an aside to\nhimself, while attiring his prisoner and looking covetously at the\npearls with which she was adorned, \"to waste gold and pearls in this\nfashion; would it not be much better to use them in buying something\nserviceable? What she has on her is worth at least three thousand\npiastres--what a splendid game of monte a fellow could have with that\nsum--and if that demon of a Red Cedar had only been willing--well, we\nshall see presently.\"\n\nWhile making these judicious reflections, the gambusino had completed\nthe maiden's Indian toilet. He perfected the disguise by throwing a\nzarape over her shoulders; then giving a parting glance round his\ndomicile, he put in his pocket a pack of cards accidentally left on the\ntable, drank a large glass of spirits, and left the room, followed by\nDona Clara and the monk, who, in spite of the varying incidents of the\nlast few days had regained all his good humour, doubtless owing to the\ngood company in which he was, and the game of monte--that inveterate\npassion in every Mexican.\n\nDona Clara was placed on a horse; Andres and the monk also mounted, and\nleaving the house to the problemical care of Providence, the gambusino\ngave the signal for departure. He made a wide circuit, to avoid passing\nthrough the Presidio, and then started at a gallop in the direction of\nthe Cerro Prieto.\n\nRed Cedar had lost no time, and all was ready for departure. The\nnewcomers did not even dismount, but so soon as they were sighted, the\ncaravan, composed, as we have stated of some hundred and twenty resolute\nmen, after forming in Indian file, started in the direction of the\nprairies, having first prudently detached two scouts to watch the\nneighbourhood.\n\nNothing is so mournful as a night march in an unknown country, covered\nwith snares of every description, when you fear least the ever-watchful\nenemy may pounce on you from every bush. Thus, the gambusinos, restless,\nand starting at the slightest rustling of the leaves, advanced silently\nand gloomily, with their eyes fixed on the clumps that grew along the\nwayside, rifle in hand, ready to fire at the slightest suspicious\nmovement. They marched, however, for upwards of three hours, and nothing\nhappened to justify their fears; a solemn calmness continued to prevail\naround them. Gradually these apprehensions were dissipated; they began\ntalking in a suppressed voice, and laughing at their past terrors, when\nthey reached, on the banks of the Del Norte, the _vado_, or Ford del\nToro.\n\nIn the interior of Southern America, and specially in New Mexico, a\ncountry still almost entirely unknown, the means of communication are\n_nil_, and consequently bridges may be looked for in vain. There are\nonly two methods of crossing even the widest rivers--looking for a ford,\nor, if you are in a great hurry, forcing your horse into the oft-times\nrapid current, and trying to reach the other bank by swimming.\n\nThe squatter had selected the first method, and in a few minutes the\nwhole party was in the water. Although the ground of the ford was\nuneven, and at times the horses were up to their chests, and compelled\nto swim, the gambusinos managed to get across safely. The only persons\nleft on the bank were Red Cedar, Eagle-wing, the guide, Dona Clara, and\nAndres Garote.\n\n\"It is our turn now, Heart of Stone,\" the squatter said, addressing\nEagle-wing; \"you see that our men are in safety, and only await us to\nset out again.\"\n\n\"The squaw first,\" the Indian replied, laconically.\n\n\"That is true, chief,\" the squatter said, and, turning to the prisoner,\n\"Go across,\" he said to her, coarsely.\n\nThe maiden, not deigning to answer, boldly made her horse enter the\nriver, and the three men followed. The night was dark, the sky covered\nwith clouds, and the moon, constantly veiled, only shone forth at\nlengthened intervals, which rendered the passage difficult and even\ndangerous, as it did not allow objects to be distinguished, even at a\ndistance. Still, after a few seconds, Red Cedar fancied he saw that Dona\nClara's horse was not following the line traced by the ford, but was\nturning to the left, as if carried away by the current. He pushed his\nhorse forward, to assure himself of the reality of the fact; but\nsuddenly a vigorous hand seized his right leg, and before he could even\nthink of resisting, he was hurled back into the water, and his throat\nseized by an Indian. Andres Garote hurried to his assistance.\n\nDuring this time, Dona Clara's horse, probably obeying a hidden impulse,\nwas proceeding still further from the spot where the gambusinos had\nlanded. Some of them, at the head of whom were Dick, Harry, and the\nsquatter's three sons, perceiving what was going on, returned to the\nwater, to proceed to their chiefs help, while the others, guided by Fray\nAmbrosio, galloped down the river bank, in order to cut off retreat,\nwhen Dona Clara's horse landed.\n\nAndres Garote, after several fruitless efforts, succeeded in catching\nRed Cedar's horse, which he brought to him at the moment when the latter\nhad scalped his enemy. The American got into his saddle again, reached\nthe bank, and tried to restore some order among his band, while actually\nwatching the incidents of the silent drama being played in the river\nbetween Eagle-wing and the young Spanish girl.\n\nThe Coras sachem had urged his steed in pursuit of Dona Clara's, and\nboth were following almost the same line down the stream, the former\nstriving to catch up the latter, who, for her part, was doing her utmost\nto widen the distance between them. Suddenly the Coras horse gave a\nleap, while uttering a snort of pain, and began madly beating the water\nwith its forelegs, while the river was tinged with blood around it. The\nchief, perceiving that his horse was mortally wounded, leaped from the\nsaddle, and leant over the side, ready to leap off. At this moment, a\nhideous face appeared flush with water, and a hand was stretched out to\ngrasp him. With that imperturbable coolness that never deserts the\nIndians, even under the most critical circumstances, the Coras seized\nhis tomahawk, split his enemy's skull open, and glided into the river.\n\nA formidable war yell was, at this moment, heard from the forest, and\nsome fifty shots were fired from both banks at once, illumining the\nscene with their fugitive flashes. A multitude of redskins rushed on the\ngambusinos, and a terrible fight commenced. The Mexicans, taken\nunawares, defended themselves at first poorly, giving ground and seeking\nshelter behind trees; but, obeying the thundering voice of the squatter,\nwho performed prodigies of valor while exciting his comrades to sell\ntheir lives dearly, they regained courage, formed in close column, and\ncharged the Indians furiously, beating them down with the butts of their\nmuskets, or slashing them with their machetes.\n\nThe combat was short; the redskins, who were only a party of marauding\nPawnees, seeing the ill-result of their surprise, grew discouraged, and\ndisappeared as rapidly as they had come. Two minutes later calmness and\nsilence were so perfectly re-established, that had it not been for a few\nwounded gambusinos, and several Indians stretched dead on the\nbattlefield, the strange scene would have appeared as a dream.\n\nSo soon as the Indians were routed, Red Cedar bent an eager glance up\nthe river; on that side the struggle was also over, and Eagle-wing,\nmounted behind the young lady, was guiding her horse to the bank, which\nit soon reached.\n\n\"Well?\" the squatter asked.\n\n\"The Pawnees are cowardly coyotes,\" the Coras answered, pointing to two\nhuman scalps that hung all bloody from his girdle; \"they fly like old\nwomen, so soon as they see the war plume of a warrior of my nation.\"\n\n\"Good!\" the squatter said, gleefully, \"My brother is a great warrior; he\nhas a friend.\"\n\nThe Coras bowed with a smile of indescribable meaning. His object was\ngained; he had acquired the confidence of the man he meant to destroy.\nDona Clara, Ellen, and the squatter's wife were placed in the centre of\nthe caravan, and the band started again.\n\nAn hour later, a second party of horsemen also crossed the Vado del\nToro. It was much less numerous than the first, as it consisted of only\nfive men, but they were Valentine, Curumilla, Don Miguel, his son, and\nGeneral Ibanez. The real struggle was about to commence: behind them\nthey left the civilised world, to find themselves face to face on the\ndesert with their enemies.\n\n(Those of our readers who take an interest in the Trail-hunter, we must\nask to follow his adventures through a second volume, to be called--THE\nPIRATES OF THE PRAIRIES.)\n\n\nTHE END\n\n\n\n\n\nEnd of the Project Gutenberg EBook of The Trail-Hunter, by Gustave Aimard\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n# Copyright\n\nCopyright \u00a9 2018 by Sunil Amrith\n\nCover design by Rebecca Lown\n\nCover image \u00a9 View Stock\/Getty Images\n\nCover \u00a9 2018 Hachette Book Group, Inc.\n\nHachette Book Group supports the right to free expression and the value of copyright. The purpose of copyright is to encourage writers and artists to produce the creative works that enrich our culture.\n\nThe scanning, uploading, and distribution of this book without permission is a theft of the author's intellectual property. If you would like permission to use material from the book (other than for review purposes), please contact permissions@hbgusa.com. Thank you for your support of the author's rights.\n\nBasic Books\n\nHachette Book Group\n\n1290 Avenue of the Americas, New York, NY 10104\n\nwww.basicbooks.com\n\nFirst Edition: December 2018\n\nPublished by Basic Books, an imprint of Perseus Books, LLC, a subsidiary of Hachette Book Group, Inc. The Basic Books name and logo is a trademark of the Hachette Book Group.\n\nThe Hachette Speakers Bureau provides a wide range of authors for speaking events. To find out more, go to www.hachettespeakersbureau.com or call (866) 376-6591.\n\nThe publisher is not responsible for websites (or their content) that are not owned by the publisher.\n\nThe Library of Congress has cataloged the hardcover edition as follows:\n\nNames: Amrith, Sunil S., 1979\u2013 author.\n\nTitle: Unruly waters : how rains, rivers, coasts and seas have shaped Asia's history \/ Sunil Amrith.\n\nDescription: First edition. | New York : Basic Books, 2018. | Includes bibliographical references and index.\n\nIdentifiers: LCCN 2018020666 (print) | LCCN 2018036442 (ebook) | ISBN 9780465097739 (ebook) | ISBN 9780465097722 (hardcover)\n\nSubjects: LCSH: Water\u2014Social aspects\u2014Asia\u2014History. | Bodies of water\u2014Social aspects\u2014Asia\u2014History. | Water and civilization. | Asia\u2014Civilization\u2014Environmental aspects.\n\nClassification: LCC DS12 (ebook) | LCC DS12 .A54 2018 (print) | DDC 950\u2014dc23\\\n\nLC record available at https:\/\/lccn.loc.gov\/2018020666\n\nISBNs: 978-0-465-09772-2 (hardcover), 978-0-465-09773-9 (ebook)\n\nE3-20181029-JV-NF\n\n# CONTENTS | Cover\n\n---|---  \n|\n\nTitle Page\n\n|\n\nCopyright\n\n|\n\nDedication\n\n|\n\nList of Maps\n\n|\n\nMaps\n\n|\n\nA Note on Names and Terminology\n\nONE | The Shape of Modern Asia\n\nTWO | Water and Empire\n\nTHREE | This Parched Land\n\nFOUR | The Aqueous Atmosphere\n\nFIVE | The Struggle for Water\n\nSIX | Water and Freedom\n\nSEVEN | Rivers Divided, Rivers Dammed\n\nEIGHT | The Ocean and the Underground\n\nNINE | Stormy Horizons\n\nEPILOGUE | History and Memory at the Water's Edge\n\n|\n\nAcknowledgments\n\n|\n\nAbout the Author\n\n|\n\nAlso by Sunil Amrith\n\n|\n\nArchives and Special Collections\n\n|\n\nNotes\n\n|\n\nIndex\n_For Theodore and Lydia_\n\n# LIST OF MAPS\n\nHimalayan Rivers\n\nSouth Asia's Major Rivers\n\nNortheast Monsoon\n\nSouthwest Monsoon\n\nBritish India, 1900\n\nThe Partition of India, 1947\n\nThe Brahmaputra\/Yarlung Tsangpo\n\nIndia's Dams\n\nThe Coastal Mega Cities of Asia\n\nMap of Mumbai\n\nMap showing the winds during the northeast monsoon, which blows from December to March.\n\nMap showing the winds during the southwest monsoon, from June to September.\n\n# A NOTE ON NAMES AND TERMINOLOGY\n\nMANY OF THE PLACES I WRITE ABOUT IN THIS BOOK HAVE BEEN known by different names at different points in time. As a rule, I have used the names that correspond to the period I am writing about\u2014to cite a few examples, I use Bombay, Madras, Calcutta, Poona, and Rangoon when I am discussing the colonial period and the early decades after independence; I switch to Mumbai, Chennai, Kolkata, Pune, and Yangon, respectively, when I am talking about more recent history, as those names were formally changed in the 1990s. I adopt a similar strategy when it comes to country names: for example, I use Ceylon and Malaya when discussing the colonial period, and Sri Lanka and Malaysia when writing about the post-independence era.\n\nFor clarity I have transliterated words from South Asian languages in a way that reflects common practice in the region rather than employing the formal diacritical marks favored by scholars of South Asian languages.\n\nA NASA satellite image from October 27, 2002, showing a Himalayan mountain range and the rivers that descend from the Tibetan Plateau into North India. CREDIT: Jacques Descloitres, MODIS Rapid Response Team, NASA\/GSFC\n\n# ONE\n\n# THE SHAPE OF MODERN ASIA\n\nLOOKING DOWN FROM ORBIT, THE LENS OF A NASA SATELLITE LANDS upon this patch of Earth. In the upper half of the picture lies the curve of a Himalayan mountain range, fringed by the iridescent lakes of the Tibetan plateau.\n\nThe satellite picture is a snapshot of a single moment on October 27, 2002. But there are layers of history embedded within it. It shows us the outcome of a process that unfolded in deep time. Approximately 50 million years ago, the Himalayas were created by the collision of what would become the Indian peninsula, which had detached from Madagascar, with the Eurasian landmass. The island buckled under the edge of Eurasia, pushed up the Tibetan Plateau, and eradicated a body of water later named the Tethys Sea. \"Geology, looking further than religion,\" E. M. Forster wrote in _A Passage to India,_ \"knows of a time when neither the river [Ganges] nor the Himalayas that nourish it existed, and an ocean flowed over the holy places of Hindustan.\" Volcanic activity under the Indian Ocean kept the pressure up, forcing layers of rock to crumple under the Indian margin to create the largest mountain chain on Earth.\n\nSo massive are the mountains, so heavy is their concentration of snow, ice, heat, and melting water that they shape Earth's climate. Asia's great rivers are a product of this geological history. They flow south and southeast, and they have shaped the landscape that is visible here: the force of the rivers descending from the mountains eroded rock, creating the gorges and valleys. Over centuries the rivers have carried silt and sediment from the mountains; they have deposited them along Asia's valleys and floodplains to sustain large human populations. Writing in the 1950s, guided by maps and not yet by satellite photographs, geographer Norton Ginsburg described Asia's \"mountain core\" as the \"hub of a colossal wheel, the spokes of which are formed by some of the greatest rivers in the world.\"\n\nAnd then your eye comes to rest on what was invisible to the satellite but is now superimposed\u2014evidence of a more recent history lies in the borders that dissect the rivers, their shapes governed by bureaucratic, not environmental, logic. Within the frame of this image alone, the mountains run through southwestern China, Nepal, Bhutan, and northeastern India. The rivers are more unruly; they spill beyond the frame of the photograph. From mountain peaks flow ten great rivers that serve a fifth of humanity\u2014the Tarim, the Amu Darya, the Indus, the Irrawaddy, the Salween, the Mekong, the Yangzi, the Yellow River, and, at the heart of this photograph, the Ganges and the Brahmaputra. The Himalayan rivers run through sixteen countries, nourished by countless tributaries. They traverse the regions we carve up as South, Southeast, East, and Central Asia; they empty out into the Bay of Bengal, the Arabian Sea, the South and East China Seas, and the Aral Sea.\n\nLook at the left of the picture and you can see a more compressed history. The haze of pollution that hangs over North India is a composite \"brown cloud\" of human-produced sulfates, nitrates, black carbon, and organic carbon. Aerosol concentrations over the Indian subcontinent are the highest in the world, especially in the winter months when there is little rain to wash the skies clean. Individual particles remain in the atmosphere only for a matter of weeks, but cumulatively the cloud lasts for months\u2014what we see here is a fleeting archive of every domestic stove, every truck and auto-rickshaw exhaust pipe, every factory smokestack and crop fire that burned across the Gangetic plain after the end of the monsoon rains that year. But the location of the cloud, and its contributing sources, testify to a longer twentieth-century history of population growth, urban expansion, and uneven economic development through that belt of northwestern India. Over time, a constant succession of transient \"brown clouds\" may have attenuated rainfall over South Asia over the past half century, transforming the water cycle that binds the clouds, the mountains, and the rivers.\n\nFinally, look at the snow on the mountain peaks visible from outer space. The time horizon this gestures toward is the future. The descent of water is vulnerable, now, to the ascent of carbon. As Earth's surface warms, the Himalayan glaciers are melting; they will melt more rapidly in the decades ahead, with immediate consequences for the flow of Asia's major rivers\u2014and for the planet's climate.\n\nASIA IS HOME TO MORE THAN HALF THE WORLD'S POPULATION, but it contains less freshwater than any continent except Antarctica. A fifth of humanity lives in China, a sixth in India; but China has only 7 percent, and India 4 percent, of the world's freshwater\u2014and within both countries that water is distributed unevenly. The quality as well as the quantity of water is under strain from a multiplicity of new demands and uses. Asia's rivers are choked by pollutants and impounded by large dams. An estimated 80 percent of China's wells contain water unsafe for human consumption; in India, groundwater is poisoned by fluoride and arsenic, or made undrinkable and unhealthy by salinity.\n\nThe effects of climate change are already manifest. They compound the water-related risks that Asia's peoples already face. Most predictions hold that the Himalayan rivers will swell as the planet warms and the ice thaws; and then, around the middle of this century, they will start to dry out for part of the year. Existing inequalities will deepen: wet regions will get wetter, and dry regions will get drier. Within that broad pattern, there will be an increase in variability and a rise in extreme weather. The effects of planetary warming have already begun to interact with regional drivers of climate change\u2014changes in land use, aerosol emissions, and \"brown clouds\"\u2014to multiply uncertainty. Coastal regions in particular face a cascade of threats: heat stress, flooding, rising sea level, and more intense cyclonic storms. Most at risk is the coastal crescent at the southern and eastern edge of the Eurasian landmass, home to the greatest concentration of the world's population. All twenty cities in the world with the largest populations vulnerable to rising sea levels are in Asia. Most threatened, because numbers are compounded by high levels of poverty and inequality, are Mumbai and Kolkata in India, Dhaka in Bangladesh, Jakarta in Indonesia, and Manila in the Philippines.\n\nAll the while, statesmen and engineers plot water's final subjugation by technology. Over the next decade, more than four hundred large dams will be built on the Himalayan rivers\u2014by India, China, Nepal, Bhutan, and Pakistan\u2014to feed the region's hunger for electricity and its need for irrigation. New ports and thermal power plants line the coastal arc that runs from India, through Southeast Asia, to China. India and China have embarked on schemes to divert rivers to bring water to their driest lands: costing tens or hundreds of billions of dollars, they are the largest and most expensive construction projects the world has ever seen. At stake in how these plans unfold is the welfare of a significant portion of humanity. At stake is the future shape of Asia, the relations among its nations. Each of these risks, each of these responses, is rooted in ideas, institutions, and choices that earlier generations have made\u2014that is to say, they are shaped by Asia's modern history.\n\n# I\n\nTo understand why Asia is the part of the world most vulnerable to climate change, why South Asia in particular stands at the front line, we need to turn to the history of water. Across the heartland of Asia\u2014from Pakistan in the west, through India and Southeast Asia, to China in the east\u2014the control of water has underpinned an increase in human population and an expansion in longevity that would have been unimaginable even in the middle of the twentieth century. In a warming world, Asia is distinctive for its sheer scale, and distinctive for the scale of inequality among its peoples. Both are rooted in the quest for water, which is a vital feature of modern Asian history, and one that we have ignored.\n\nThe struggle for water in modern history is a global story. We can tell a version of it set in the western United States, or in Germany, or in the Soviet Union, which was an Asian as well as a European power. But nowhere has the search for water shaped or sustained as much human life as in India and China. Their demographic weight is not a fact of nature. It is an outcome of history, a history in which the control of water was pivotal. Today that control is more rigorous than ever, thanks to intensive hydraulic engineering, but the foundations of that control are fragile. Nowhere is the multiplier effect of any destabilization in the material conditions of life greater than it is in Asia. This, too, demands a historical explanation. As rains grow erratic and storms more intense, as rivers change course and wells dry up, the hard-won gains of half a century are vulnerable to reversal. The force of planetary warming combines with the material legacy of earlier quests to control water. Warming seas meet coastal zones that sag under the weight of growing cities, many of them founded as colonial ports in the eighteenth and nineteenth centuries. River deltas are sinking, starved of sediment by large dams upstream that were built in the 1950s and 1960s. We live with the unintended consequences of earlier generations' dreams and fears of water.\n\nThe origins of these dreams and fears, the longevity of the policies and infrastructures to which they gave rise, are the subject of this book. _Unruly Waters_ tells the story of how the schemes of empire builders, the visions of freedom fighters, the designs of engineers\u2014and the cumulative, dispersed actions of hundreds of millions of people across generations\u2014have transformed Asia's waters over the past two hundred years.\n\nThis is not the way we usually understand Asia's modern history. Since the 1990s, identity and freedom have been the dominant themes in historical writing: these have oriented the study of Asia as much as anywhere else. The late 1980s and the 1990s witnessed an upsurge in struggles for democracy in China, Indonesia, the Philippines, and Burma. In trying to explain the weakness or the persistence of authoritarian states, historians looked to political and intellectual history to capture alternative understandings of freedom, especially as earlier clusters of ideas were reinvigorated after the end of the Cold War. In the study of South Asia, the theme of identity has loomed largest. In India in the 1990s, political mobilization along caste lines\u2014and growing recognition of the deep wounds that caste still inflicts on Indian society\u2014clashed with the spectacular rise of a violent and exclusionary Hindu nationalism to focus historians' attention on the cleavages of culture and community that continue to divide South Asia.\n\nThese histories shed light on struggles for recognition and justice that are unfinished; they pinpoint inequalities that endure. But there is much that we have missed. Novelist Amitav Ghosh points out the irony that twentieth-century literary fiction proved oblivious to the growing crisis of climate change at the very moment of its escalation\u2014a solipsistic turn at a moment when the material world was in the process of irrevocable transformation. With only a few exceptions, the same charge can be leveled at those of us who write history. My premise here is that the transformation of Asia's environment, and in particular its ecology of water, may be as consequential in modern history as the political and cultural transitions that have compelled our attention\u2014and it is consequential, not least, for its impact on both culture and politics.\n\nOutside the specialized field of environmental history, the disappearance of nature from most broad accounts of historical change has been marked. It is also recent. In the 1970s and 1980s, agrarian history was a vibrant field. In those decades, discussions of water and agriculture in Asia fought to shake off the ghost of the German Marxist sociologist Karl Wittfogel. Wittfogel had argued in the 1950s that the need for centralized control over irrigation lay at the heart of \"hydraulic societies\" like China, ancient Egypt, and India, predisposing them all to absolutist government, or what Marx had called \"oriental despotism.\" Wittfogel's generalizations crumbled under closer examination. The agrarian histories written in the 1970s and 1980s emphasized the variability of arrangements through which different Asian societies harnessed the power of water. They all insisted on the importance of irrigation, but traced no simple relationship between that hydraulic fact and political forms. Browse through any study of South Asian or East Asian agriculture written in those decades: water is omnipresent. Historians of China were inclined to take a very long view, showing how the control of water shaped Chinese society and civilization over millennia; historians of South Asia were more likely to emphasize discontinuity\u2014and especially the rupture that came with British colonialism, which forced the Indian countryside more fully into the global capitalist economy. Whether on the scale of millennia or of decades, this work exudes a rich sense of landscape. It is alive with a sense of the seasons changing, of the shifting flow of rivers, of the threat that floods or drought posed to human survival. This tradition of historical writing disappeared most conspicuously from the study of South Asia, where the turn to cultural history swept all before it. But in other fields, too, historians decamped to the cities, leaving rural history behind as they turned to urban culture and politics, to intellectual history, to histories of cosmopolitanism and travel and migration. They did so just when a mounting water crisis began to pose an existential threat.\n\nThere are two main ways in which my view departs from the perspective of earlier work on the Asian countryside. The first is to see water as more than just a resource. In the pages that follow, the effects of new economic pressures and new technologies on water itself\u2014on the water cycle, on the toxicity of water, on ideas about the value of water\u2014are as important as the effect of water resources on agricultural output, which is what economic historians were primarily concerned with. As Asia's waters were transformed, water was understood in new ways by meteorologists, hydrologists, and oceanographers. Recent scientific research, made possible by advances in imaging technology and statistical capacity, has transformed the possibility of understanding water and climate historically, bringing us to archives we had scarcely thought to look at. The great French historian Marc Bloch believed that human history lived \"behind the features of landscape\" as much as it lived in \"tools and machinery\" and in institutions. It lives, too, behind the chemical content of river water samples; behind satellite images of the water that lies underground; behind the composition of the smog that hovers above South Asia every winter, altering its rainfall. It lives in the changing ocean currents and winds.\n\nIn Fernand Braudel's three-fold conception of historical time, the first, slowest-moving layer was the time of nature and the seasons: a \"history of constant repetition, ever-recurring cycles.\" His perspective influenced histories of the Indian Ocean, for example, in which the regularly reversing monsoon winds provide a basic material backdrop, enabling long-distance trade and shaping the agricultural cycle. But over the last two hundred years nature has been altered by human intervention to such a profound extent that that stability and \"constant repetition\" cannot be assumed. By the end of the twentieth century it became possible to ask\u2014as this book will ask\u2014not only how climate has shaped us, but how we have affected the climate.\n\nMy second departure stems from a more elastic sense of geography. Like most history-writing until the end of the twentieth century, agrarian history took the nation-state for granted, though often the most meaningful unit of study was the region-within-the-nation: South China or Java, the Bengal or Mekong deltas. To put water at the heart of the narrative is to demand that we adopt a more flexible conception of space. Rivers pay no heed to human frontiers; but political boundaries have had a material effect on their flow. The quest to understand climate has led meteorologists and engineers and geographers to think beyond borders; but they have faced countervailing pressure to fix their plans and dreams in place. Water draws our attention not only to the two-dimensional space between points on a map\u2014as when we trace the crooked line of a river\u2014but also to depth and altitude, which turn out to matter more than historians have realized.\n\nWhat we end up with is not an alternative to the well-known narrative of modern Asia shaped by empire and capitalism, forged by anticolonial revolution, remade in the second half of the twentieth century by ambitious new states. Rather, water adds another dimension to that familiar story. Asia's waters have long been a gauge for rulers' ambition, a yardstick of technological prowess\u2014and a dump for the waste products of civilization. Water is, in a sense, a \"sampling device\" for other sorts of change, even as changes in water ecology have had a direct effect on millions of people's lives. We can trace many of Asia's political transitions through the effects they had on water: from the global reach of the British empire in the nineteenth century, to the projects of national reconstruction that the Indian and Chinese states carried out in the twentieth. But the history of water is more than a mirror to human intentions. The history of water shows that nature has never truly been conquered. Water has served as a material constraint on every Promethean plan of growth and plenty. The sheer ferocity of a wet climate\u2014a climate of monsoons and cyclones\u2014remains a source of fear, and no fear is as great as the fear of water's absence, in drought. The cultural history of water is one of reverence as much as hubris. And water has its own chronology\u2014the chronology of the seasons; the episodic chronology of sudden, intense disasters; the imperceptible chronology of cumulative damage, as manifested in the effects of human activity on the oceans.\n\n# II\n\nEnvironmental history derives its richness from a close attention to particular landscapes\u2014the most profound works have often been local and regional in scope, ranging from the study of a single village to a city, a forest, or a river. Only at that limited scale can we truly tease out the relationships between nature and human society. But the scale of environmental change has ballooned; its pace has accelerated. Connections between environmental crises have multiplied: the causes of harm and risk in any given locality may lie far away. We need a larger view. In a 2009 article, \"The Great Himalayan Watershed,\" historian of China Kenneth Pomeranz took up the challenge: \"For almost half the world's population,\" he wrote, \"water-related dreams and fears intersect in the Himalayas and on the Tibetan plateau.\" The Himalayan rivers bind the futures of the significant portion of the world's population that depend on them; conflicts over their course, and their use, threaten to ratchet up tensions between bordering states, especially India, Pakistan, and China.\n\nThe scale and interconnection of Asia's water crises provide a starting point for _Unruly Waters_. But this is not only a view from the Himalayan peaks; still less is it the omniscient view from a satellite image, for one characteristic of the satellite view is that there are no people in it, even if signals of the human imprint are everywhere apparent. This is a history of Asia's waters with India at its heart\u2014and there are three compelling reasons why India is an illuminating vantage point from which to tell a story that crosses regional and national boundaries.\n\nThe first is India's centrality to the history of the British Empire; and empire's centrality, in turn, to the history of climate change. The conquest of most of the world by European powers in the nineteenth century forced a fundamental transformation in the human relationship with the rest of nature. Asian and African lands were drawn more closely into a global capitalist economy. Their absorption was underpinned by imperial gunboats and colonial taxes, but it was driven, too, by new opportunities for enrichment and advancement. India was at the sharp edge of change\u2014exploited more intensively and on a larger scale than almost anywhere else, and pivotal to the further thrust of imperial power into Asia. From European trading companies' earliest expansion into the islands of the Atlantic and the Caribbean in the early modern era, they thrived on the exploitation of \"cheap nature\" as well as coerced labor. The pace of change stepped up in the nineteenth century. The period from the 1840s to the 1880s witnessed the global triumph of industrial capitalism; in Eric Hobsbawm's words, \"an entirely new economic world was added to the old and integrated into it.\" India's fields and its waters were pushed harder to sustain the colonial state\u2014which depended on agricultural taxes\u2014and to produce the raw materials that fed Europe's industrial machine and its working classes: cotton, jute, indigo, sugar, tea, and coffee. Each of these thirsty crops generated new demands for water.\n\nFrom India, imperial power and investment spread east and west across the Indian Ocean. British ships, filled with Indian troops, set sail in 1839 to bombard China, to force the Chinese government to allow the sale of Indian opium to Chinese consumers\u2014a traffic that was vital to the East India Company's financial health. A reordering of the entire region between India and China followed. By the last quarter of the nineteenth century, from Burma to Vietnam, Asia's demography changed as migration opened new frontiers of settlement; its ecology altered to accommodate the spread of cash crops for export. Many of Asia's largest coastal cities\u2014Mumbai, Calcutta, Chennai, Dhaka, Hong Kong, Jakarta\u2014began life as colonial ports, built to sustain the global trading networks on which European empires thrived.\n\nImperial India reached further than the present boundaries of the Indian nation-state, and further, too, than the region we now define as South Asia\u2014present-day India, Pakistan, Bangladesh, Nepal, Bhutan, and Sri Lanka. But British India was also more internally variegated than independent India. Areas under direct British control existed amid a patchwork of other forms of polity, known collectively as \"princely states,\" all of which retained a degree of sovereignty while submitting to overall British domination. Both within the Indian subcontinent and beyond its shores, water constituted the connective tissue of imperial power. In the British imagination, India extended across the vastness of the Indian Ocean, connected to China and Southeast Asia (the \"East Indies\") through the flow of its rivers and the span of its climate. The ability to imagine India on that scale was, itself, a product of the nineteenth century and its new ways of seeing\u2014maps, censuses, surveys, and photographs. It depended on the compression of space by the railway and the steamship. The contraction of those larger geographies in the twentieth century is a recurrent theme in this book.\n\nIn another sense, too, India's experience of imperialism cast a long shadow over the history of Asia's waters. British colonialism was a source of enduring trauma for many Indians, including for the educated elite that led India's nationalist movement in the first half of the twentieth century. Beyond the outright violence that the British government of India deployed, this trauma resided in a sense of profound social and economic destabilization. The effects of British policies combined with drought at various moments in the nineteenth century to create famines that killed millions. At the core of anticolonial thought in and beyond India was a clear imperative: \"never again.\" In China, too, the experience of a \"century of humiliation\" at the hands of European powers, beginning with the catastrophic Opium Wars, left political leaders with a deep and urgent drive for self-sufficiency and self-reliance. The control of water was central to almost every scheme that arose from this quest for development. Memories of the nineteenth century lie beneath the fervor with which India built 3,500 large dams, and China 22,000, in the decades after independence. The memory of subordination by European empires continues to shape Indian and Chinese foreign policy: it orients their approach to agriculture; it even underpins their responses to climate change.\n\nIf India's role in empire is one reason to put it at the heart of this story, the second is India's political history after independence. Alone among Asia's newly independent or postrevolutionary states, India has been a democracy continuously since 1947, for all but three years. Clamorous and vibrant and flawed, Indian democracy has coexisted with glaring social and economic inequalities; the Indian state has often behaved in an authoritarian manner, not averse to exercising the powers it inherited wholesale from the British Raj. In their pursuit of water at any cost, there has been little to distinguish Asian states with different political systems and with varied ideological complexions. Still, the depth and diversity of India's public sphere has been unique. Debates about water in India were never limited to disagreements between experts behind closed doors (as they were in China until the 1980s). They threaded through newspaper columns; they animated social movements; they filled the publications of environmental organizations. Many of these ideas echoed beyond India, and in turn Indian observers marshaled examples and gathered data from around the region. One vehicle for the movement of ideas about water and technology was the cinema. In the second half of the twentieth century India developed a popular film industry that exceeded Hollywood in size and matched it in influence in the postcolonial world: Indian films drew large audiences across Asia, the Middle East, and Africa. To an extent that has no parallel, Indian cinema captured the hopes and fears that fired visions of \"development\" across the Third World. Water was a recurrent theme.\n\nThe third, and perhaps the most fundamental, reason to tell this story looking out from India is a climatic one. The Indian subcontinent is the crucible of the monsoon. And the monsoon is the thread that runs through _Unruly Waters._ In its simplest definition the monsoon is \"a seasonal prevailing wind.\" There are other monsoons, in northern Australia and in North America; none is as pronounced, as marked in its reversal between wet and dry seasons, as the South Asian monsoon. More than 70 percent of total rainfall in South Asia occurs during just three months each year, between June and September. Even within that period, rainfall is not consistent: it is compressed into a total of just one hundred hours of torrential rain across the summer months. Despite a vast expansion in irrigation since 1947, 60 percent of Indian agriculture remains rain-fed, and agriculture employs 60 percent of India's population. Unlike China, unlike most large countries in the world, India's population will continue to be predominantly rural by the mid-twenty-first century. No comparably large number of human beings anywhere in the world is so dependent on such intensely seasonal rainfall. In the first decade of the twentieth century, the finance minister in the imperial government declared that \"every budget is a gamble on the rains\"; more than a century later, leading environmental activist Sunita Narain reversed the terms but retained the substance of the observation: \"India's finance minister _is_ the monsoon,\" she declared.\n\nStorm clouds, characteristic of the weeks leading up to the burst of the monsoon. CREDIT: NurPhoto\/Getty Images\n\nClimate is woven into the fabric of Indian social, economic, and political thought in a way that it is not (or is no longer) elsewhere. In the late twentieth century that claim would have raised hackles among scholars of South Asia; it might still do so today. A fundamental assumption of modernity was that we had mastered nature. The notion of India in thrall to the monsoon would seem to perpetuate a colonial idea of India's irredeemable backwardness. To emphasize the power of the monsoon would be to portray Indian lives as so many marionettes moved by a climatic puppetmaster. That is how this story would have been understood a generation ago. But to our eyes now, alarmed by the planetary crisis of climate change, a reminder of nature's power has different implications. This is not a story of geography as destiny. It is a story of how the _idea_ of geography as destiny provoked, from the mid-nineteenth century on, a whole series of social, political, and technological responses within and beyond India. The monsoon is significant precisely because it has been a unique source of human concern, fear, and adaptive ingenuity. The desire to liberate India from its climate powered hydraulic engineering on an ever-increasing scale, with consequences far beyond India's borders. The struggle to understand the monsoon's dynamics motivated scientific research that remains at the core of our understanding of global climate. Living with the monsoon, India never had the luxury of the climate-blindness that has seeped into many other societies' worldviews. The history of how Indians have understood and coped with the monsoon may have wider lessons at a moment when climate can no longer be ignored, anywhere in the world\u2014in this sense, at least, India is not behind the world but ahead of it. The lessons are not always heartening. As we will see, awareness of the monsoon's enduring power has coexisted with inertia, with negligence, with decisions to put more people in harm's way, and with maneuvers by the wealthy and powerful to insulate themselves from risk.\n\nThe South Asian monsoon has effects far beyond South Asia. We know this, at least in part, because of climate research undertaken in India in the twentieth century. Sir Gilbert Walker, a pioneer of global climate science, wrote in 1927 that \"the climate of India is of special interest, not merely as that of the greatest tropical region in the British Empire, but also because it seems to have been designed by nature with the object of demonstrating physical processes on a huge scale.\" That sense of scientific opportunity, combined with the pressing material need to understand the monsoon, inspired a century of study in India. Charles Normand, Walker's successor as head of the Indian weather service, insisted that the monsoon is \"an active, not a passive feature in world weather.\" Subsequent research has confirmed his view\u2014the Asian monsoon is entwined with many aspects of the global climate. It has an important influence on global atmospheric circulation. The future behavior of the South Asian monsoon has implications for the whole world. Arguably no other part of the global climate system affects more people, more directly.\n\nAND SO, STARTING OUT FROM AND RETURNING TO INDIA, WE FOLLOW the monsoon, the mountain rivers, and ocean currents\u2014straying into Chinese waters, traveling down the Mekong, skirting the coastal arc of Asia, and coming back to the heart of South Asia.\n\nThis is a story with many possible beginnings. The recently excavated Liangzhu Ancient City, along the lower Yangzi River delta, reveals the vast scale of hydraulic engineering undertaken along China's coast five thousand years ago. The elaborate step-wells of Rajasthan and Gujarat, and the anicuts (dams) along the rivers of South India, are testament to a long struggle to cope with the monsoon. But our starting point lies in the nineteenth century, when the scale and interconnectedness of Asia's waters first became visible, in tandem with unprecedented pressure to put water to work. The concatenation of political, economic, and environmental transformations this set in train continue to shape modern Asia.\n\n# TWO\n\n# WATER AND EMPIRE\n\nALONG THE BANKS OF THE GODAVARI RIVER, CLOSE TO THE MIDSIZED city of Rajahmundry on the eastern coast of India, stands a museum dedicated to Sir Arthur Thomas Cotton. Pointing the way to the museum, a bronze statue of the man watches over a busy bridge across the river; he is mounted on a horse, head cocked, eyes on the horizon.\n\nArthur Cotton was born in Surrey, one of eleven children. He joined the East India Company's forces in 1819, as second lieutenant in the Madras Engineers. Two years later, he was seconded to the chief engineer of Madras, and developed his lifelong fascination with water. He was an evangelical Christian, a stern and devout man. His career began with a marine survey of the Pamban channel, off the coast of Madras. In the 1840s, Cotton renovated and restored the ancient dam at Kallanai, along the Kaveri River that flows east from the Western Ghats mountain range to the Bay of Bengal; the Kaveri's fertile delta was, and still remains, the agrarian heartland of Tamil-speaking South India. Cotton's attention moved north of the Kaveri, to the Krishna and Godavari rivers that meet the Indian Ocean in the region of Andhra, farther up India's eastern seaboard. In 1852, Cotton completed a barrage, or dam, over the Godavari River at Dowleswaram, which regulated the flow of the river using large gates. Henry Morris, the district's chronicler, described the barrage as the \"noblest feat of engineering skill which has yet been accomplished in British India.\" It was a \"gigantic barrier thrown across the river from island to island, in order to arrest the unprofitable progress of its waters to the sea.\" The Godavari delta gets scarcely a mention in most general histories of British India. It witnessed no major battles or massacres; it was home to few members of India's nationalist intelligentsia; its urban centers were relatively small. But the region epitomizes the transformation of India's waters in the nineteenth century.\n\nStatue of Sir Arthur Cotton at the Arthur Cotton Museum, near Rajahmundry, India. CREDIT: Sunil Amrith\n\nThe Cotton museum, close to the barrage, conveys both unaffected enthusiasm and palpable neglect. The photographs are faded. Monuments of water technology from the 1850s\u2014pulleys and simple pumps\u2014are dotted around the complex. There is something almost accidental in their placement, as if they had been forgotten there. But the museum is well attended by groups of schoolchildren and by young couples. Most of the explanatory text in the museum is in Telugu. The message is unambiguous: Arthur Cotton saved the Godavari delta. His bold engineering skill turned it from a poor region into an expanse of irrigated fertility. Frescoes on the wall tell the story: before Cotton, this land was stalked by famine and leached by drought; thanks to his munificence it became the \"rice bowl\" of India, secure from the fluctuations of climatic fortune. On a local bus the next day, my neighbor turned to me and\u2014prompted by nothing more than the lush landscape around us, and a sense that I was a visitor\u2014told me the story again. \"Everything here,\" he said, his arm sweeping across the horizon, \"is here because of Cotton _dora_ ('Boss Cotton'); he was a very great person.\" A Telugu language biography of Cotton was published a few years ago. Farmers' associations are named after him. Every year on his birthday, cultivators gather to garland his statue. In 2009 a small delegation from Andhra, including a former cabinet minister, traveled to England to locate his grave, which they found in a quiet corner of Dorking, Surrey. Such veneration of a colonial Englishman is unusual in contemporary India: curiously at odds with the movement to rename cities and streets and buildings to erase the stain of imperialism. It reflects a sense that water has a value beyond ideology, beyond politics\u2014beyond history.\n\nFresco inside the Arthur Cotton Museum showing Andhra at the mercy of the elements before Cotton's engineering feats. CREDIT: Sunil Amrith\n\nBoat on the Godavari River. CREDIT: Sunil Amrith\n\nThe geography of empire in India was sculpted by wind and water. Until the nineteenth century, the only India that Europeans knew, the only India they were interested in, was the India that was wet. They sailed to India's coasts, swept there by the direction of the monsoon winds; in the eighteenth century, they moved upriver into the Ganges valley, heartland of the successive Indian empires of the Mauryas, the Guptas, the Afghans, and finally the Mughals. By 1800, the English East India Company had defeated its remaining Indian challengers: the Marathas in the west, and Tipu Sultan's kingdom of Mysore in the south. Following the Napoleonic Wars, British power commanded the Indian Ocean. But the British faced the same hydraulic dilemmas of every South Asian empire before them. The sea routes between India and the world were governed by the reversal of the winds. Communication between the coasts and the interior was slow; India's mighty rivers could only be traveled up at certain times of the year; roads were poor. The East India Company's revenues were tied to the cycle of planting and harvesting. Only gradually did the Company incorporate arid zones into its domain\u2014the Deccan and the southeastern edge of the Peninsula by 1800; and then, by the middle of the nineteenth century, India's northwest frontier.\n\nOver the next half century British engineers and administrators and investors sought to master nature, as a step toward connecting India's interior more closely to its coastal ports and from there to the rest of the world. The quest to understand water in India fused the efforts of adventurers and engineers, mariners and scientists. They were driven by curiosity and by necessity. Some sought profit and renown. Others followed their private enthusiasms. Not all of them served the colonial state. Their work would not have been possible without the ingenuity of Indian assistants, observers, draftsmen, recorders, porters, and soldiers, whose achievements have been effaced, for the most part, from the historical record. Women were scarce in this scientific world, but the few who were involved made contributions of lasting significance. The science of water in nineteenth-century India traced the descent of the rivers, the tracks of the storms, and the path of the rains. Each of these crossed the borders of British India. Knowledge of each brought awareness of interdependencies and inequalities on a regional scale. Each provoked new kinds of political intervention.\n\n# I\n\nThe word \"monsoon\" appeared in English first in the late sixteenth century, derived from the Portuguese _mon\u00e7\u00e3o_. It comes from the Arabic _mawsim_ (for \"season\"), which also provides the word for \"season,\" _mausam,_ in Urdu and Hindi. In its simplest definition, it is a weather system of regularly reversing winds, characterized by pronounced wet and dry seasons. There are many monsoon systems around the world, but the Asian monsoon is by far the greatest in scale and consequence, and the Indian subcontinent is its zone of most intensive activity.\n\nSouth Asia lies at the heart of the monsoon system because of the geological history that has left the Indian peninsula protruding from Eurasia into the vastness of the Indian Ocean. India lies at the edge of the continental landmass that dominates the northern hemisphere, facing a southern hemisphere that is mostly water. The monsoons have evolved over tens of millions of years. They have left an archive of their natural history on the seabed and on land. Tiny algae, diatoms, and single-celled marine plants called radiolara show that the monsoons first appeared in the Miocene era, soon after the Himalayas irrupted from the collision of the island Indian peninsula and the Eurasian landmass. Traces embedded in tree rings tell us that the Asian summer monsoon has strengthened during warm interglacial periods, as during the Medieval Warm Period up to the fourteenth century, and weakened during periods of planetary cooling, as during the Little Ice Age that lasted from the middle of the sixteenth to the early eighteenth centuries.\n\nAs early as 1686, English astronomer Edmund Halley identified the basic driving force of the monsoon as the differential heating of the sea and the land\u2014he saw it as a gigantic sea breeze. In the summer months, land temperatures rise more rapidly than the sea warms. Winds are driven from areas of high pressure over the sea to areas of low pressure over land. \"The Air which is less rarified or expanded by heat and consequently more ponderous,\" Halley wrote, \"must have a Motion towards those parts thereof, which are more rarified, and less ponderous, to bring it to an Equilibrium.\" Halley's understanding missed one crucial dimension, which was understood by George Hadley in the eighteenth century\u2014Earth's rotation affects the winds, causing them to veer right in the northern hemisphere and left south of the equator.\n\nSo as the Asian landmass begins to heat up in the spring, the warming air above it rises, and cooler, moist ocean air moves in to take its place. The monsoon winds blow from the southwest, curving and doubling back to grip India, pincerlike, from both the Arabian Sea and the Bay of Bengal. The air sweeping in from the ocean contains vast stores of solar energy in the form of evaporated water, which is released as the vapor condenses as rain: the release of this stored energy sustains the power of the monsoon. The monsoon makes landfall in Kerala and Sri Lanka in late May or early June, reaching the Bengal delta by the end of the month and moving steadily inward. The arrival, or \"burst,\" of the monsoon is presaged by a period of unsettled weather and frequent thunderstorms. When it comes it can be spectacular. It brings welcome relief after months of building heat; it sustains the land's capacity to feed India.\n\nTorrential rainfall cools the earth's sodden surface as the peaks of temperature and rainfall move steadily inland, finally petering out as they reach the far northwest of India and Pakistan. The Himalayas are a crucial part of the monsoon system. The elevation of the Tibetan Plateau leads it to warm rapidly and so drives the differentials of pressure and temperature that power the monsoon system; but the mountains themselves act as a colossal barrier to the winds, essentially sealing India off from the rest of Asia, and concentrating the monsoon rains to the south of the mountains, along the Gangetic plain.\n\nAs the temperature contrast between land and sea begins to even out, the system returns to equilibrium, and another period of transition begins. As winter advances the Asian landmass cools more rapidly than the ocean. The winds now reverse to blow from the northeast, creating dry conditions over much of Asia between November and March. But neither in summer nor in winter is the monsoon uninterrupted. The wet season is characterized by frequent suspensions in rainfall, known as \"breaks\"; the \"dry\" winter monsoon brings the bulk of the year's rainfall for a few regions, including the Tamil Nadu coast in southeastern India. The periods of transition, as the winds reverse, are prime time for the devastating cyclones that visit the Bay of Bengal regularly. As we will see throughout _Unruly Waters,_ the quest to understand the monsoon, which began in earnest in the second half of the nineteenth century, has been riddled with obstacles. The study of the monsoon remains filled with uncertainty.\n\nIN THE LONG SWEEP OF INDIAN HISTORY THE MONSOON IS BOTH an internal and an external frontier. The monsoon has shaped the limits of cultivation and the distribution of crops. It has facilitated communication between some places and barred it between others. Its ecological niches have created economic unevenness\u2014the stuff of which political power is made. The reach of the monsoon also marks the junction, the ecological nexus, between two very different ideas of India. One is as a settled agrarian empire; the other, as the outward-looking heart of the Indian Ocean world. The pattern of the monsoon draws a rough vertical line down the middle of the Indian subcontinent, dividing the drier west\u2014part of a Eurasian \"arid zone\" stretching across Central Asia and as far as the Sahara\u2014from the wet, marshy east, which stretches beyond, to Southeast Asia, to form a region that twentieth-century geographers called \"monsoon Asia.\" The bulk of India's population has always lived to the east of that line.\n\nThe line dividing wet and dry zigzags across the subcontinent. The arid zone reaches down from Rajasthan in the northwest to the Deccan plateau at the heart of central India. The Deccan lies in the \"rain shadow\" created by the hulking Western Ghats mountain range\u2014the rain clouds that sweep in from the Arabian Sea collide with the high mountains and disgorge their contents, leaving little for the plains beyond. From there arid bands snake down to the very far southeast of India, interspersed with more fertile coastal or riverine belts. The frontier corresponds to the ancient division\u2014still visible today, though now modified by technology\u2014between the major staple crops that have fed Indians for centuries: rice in the monsoon zone, and wheat or millet in the drier region.\n\nThe great rivers of South Asia modify the patchwork of wet and dry. They interact with the monsoon in a hydraulic cycle of colossal proportions. Because of their fertility, their ability to sustain life and to produce a surplus, the Gangetic plains have been the heart of every Indian empire. Their rich alluvial soils have supported a large population for centuries. The river system watered crops and provided an artery of transportation and trade, if never on the same scale or with the same reach as the Yangzi River system in China and its ancient complex of canals.\n\nGiven their immense power both to sustain and to destroy life, India's rivers have been among the most revered on Earth. The river Ganges\u2014often styled as \"mother,\" or \"Ma Ganga\"\u2014is the archetypical sacred river, spiritual source of all of India's rivers, writes scholar of Hinduism Diana Eck. In some sense, all of India's other rivers are microcosms of the Ganges. For millennia, the Ganges has been a site of pilgrimage, most especially at the point of its confluence with the Yamuna River at Prayag. For many Hindus, _moksha_ \u2014liberation from the cycle of rebirth\u2014has been believed to come from bathing in the waters of the Ganges, or being cremated on it banks. The purity of the waters of the Ganges ( _gangajal_ ) has long been accepted and valued by people across India. Hindu scriptures contain many versions of the origin myth of the Ganges, known as the _avatarana,_ or descent to Earth. In the version of the story in the _Ramayana_ and the _Mahabharata,_ the unruly Ganges tumbles from heaven, tamed as it flows through the serpentine locks of Shiva's hair before it spills onto the plains of India. In all of these stories, the Ganges epitomizes liquid _shakthi,_ the energy that sustains the universe. The Ganges is not alone, it stands at the apex of a land of sacred waters. In many regions of India, rivers have been personified; their flow helps people to imagine how distant places are connected to one another. In many spiritual traditions in South Asia, the rivers have been thought to channel the power of all the water in the world, from the clouds to the oceans.\n\nTHE LINE BETWEEN PLENITUDE AND SCARCITY MIRRORED THE trajectory of the rain-bearing winds and followed the paths of the rivers. Over centuries India's rulers built irrigation canals, storage tanks, channels, and dams. These shifting arrangements bore little resemblance to Wittfogel's ideal type of a \"hydraulic society.\" Regional leaders and imperial administrations spearheaded construction projects, but so too did local lineages, temple complexes, and landowners. In the pre-modern period the most widespread infrastructures of irrigation were found in Sri Lanka and in central and southern India. The biggest of them were elaborate hydraulic systems, individual waterworks linked in a larger web. And some of the dams, like the sixteenth-century Daroji reservoir in arid northern Karnataka, were large even by modern standards. Spurts of hydraulic ambition alternated with stasis, construction with disrepair. The power that arose from the control of water spread unevenly, liable to seizure or decay.\n\nWater was never far from the minds of the mounted conquerors from the highlands of Inner Asia who stormed their way to the Gangetic plain in the second millennium to forge a new political power in India. The heart of their power lay at the frontier between the monsoon and the arid lands. They harnessed the benefits of both. Established in 1206, the Delhi Sultanate was the first Persian-Islamic state in South Asia. Though it collapsed in the second half of the 1300s, riven by internal division and threatened by fresh invasions from the northwest, the reach of the sultanate's power into the heartland of the Indian subcontinent was a prelude to the Mughals' even greater empire.\n\nThe Mughals were a Turko-Mongol dynasty with roots in present-day Uzbekistan. They unified much of the Indian subcontinent during the two centuries when they were at the height of their powers. Zahir-ud-din Muhammad Babar (1483\u20131530), known as Babur, was the first Mughal emperor. He claimed descent from Timur (Tamerlane), the Turkic conqueror, and on his mother's side from Chingghis Khan. Driven from Samarkand, Babur established a new kingdom in Kabul, Afghanistan. From there he launched an assault on the Indian subcontinent, where he established the Mughal Empire in 1526. From the age of twelve, he kept a diary from which he later composed the _Babur Nama,_ one of the earliest autobiographies in the Islamic tradition. He was a meticulous observer. He was driven by naked ambition. He was not averse to brutality. He was a lover of nature. The _Babur Nama_ is filled with references to water. Babur's primary interest was in water as both ornament and practical necessity in constructing the gardens that he loved. In the Mughal tradition of landscape architecture, gardens played both a symbolic and an aesthetic role: they were places of beauty and sensual pleasure. Their proportions embodied the principles of order and harmony.\n\nBabur's interest in water went beyond the requirements of his exquisite gardens. He commented on the entire system of irrigation at work as he advanced into North India. The cultivation of gardens and the sustenance of agriculture were related endeavors. \"The greater part of the Hindustani country is flat,\" he observed of the Yamuna valley. \"Many though its towns and cultivated lands are, it nowhere has running waters\"\u2014by \"running waters,\" Babur meant the canals well known in the Central Asian lands of his birth. Rather, \"rivers and, in some places, stagnant waters\" in wells or tanks, irrigated the Indian plains. He saw that \"autumn crops grow by the downpour of rain themselves,\" but that \"some vegetables\" had to be \"watered constantly.\" Babur observed cultivators at work. He was struck particularly by the method that the British would later dub the Persian Wheel. \"In Lahore, Dibalpur and those parts people water by means of a wheel,\" he wrote:\n\nThey make two circles of ropes long enough to suit the depth of the well, fix strips of wood between them, and on these fasten pitchers. The ropes with the wood and attached pitchers are put over the well-wheel. At one end of the wheel axle a second wheel is fixed, and close to it another on an upright axle. This last wheel the bullock turns. Its teeth catch in the teeth of the second, and thus the wheel with the pitchers is turned. A trough is set where the water empties from the pitchers and from this the water is conveyed everywhere.\n\nFurther down the Yamuna valley, toward Agra, he noticed that \"people water with a bucket\"\u2014leather buckets lifted by yoked oxen\u2014which he described as \"a laborious and filthy way.\"\n\nThe Mughal realm expanded ceaselessly between 1560 and 1605, and again between 1630 and 1690. Its territories stretched from Gujarat in the west to Bengal in the east, and far into South India. The Mughals mobilized long-distance trading networks that followed the caravan routes to the western edge of central Asia and beyond. They stored wealth in precious metals. Once the Mughals conquered the Gangetic plains, they filled the state's coffers from productive, densely settled agrarian lands and large populations. The Mughals inaugurated a rigorous system of land taxation. The administration relied on an interlocking system of larger landowners ( _zamindars_ ), who served as tax collectors, and, below them, subordinate holders of rights to the land. The state used land grants to reward its loyal officials and to co-opt local elites into the system.\n\nMughal military and financial prowess came from the mounted martial traditions of dry lands. As many as one in five men served in local military forces, often as seasonal labor in a climate where agriculture was uncertain and horses were plentiful. But India's unruly waters had a bearing on Mughal military strategy, as they would later constrain the options of the British. As they approached the Bengal delta, Mughal armies struggled beyond Rajmahal\u2014their horses were ineffective in the humid conditions; they had to use boats. The _Akbar Nama,_ Abul Fazl's account of the Emperor Akbar's reign (1556\u20131603), describes the challenge of climate. In 1574, as Akbar's forces captured the city of Patna\u2014near the site of the ancient Ganges River port of Pataliputra\u2014they \"chose the river route, in this season full of turbulence, and with constant rain and tempest.\"\n\nAside from military campaigns, the instability of India's rivers were a source of woe to local inhabitants. Historian Irfan Habib's _An Atlas of the Mughal Empire_ \u2014a painstaking work of recovery\u2014pieced together, from the minutiae of Persian sources and European travel accounts, how often, how abruptly, rivers changed course. This turmoil came from the huge loads of silt the rivers carried down from the Himalayas\u2014when the rivers were in full force they threw up sandbanks and islands as obstacles around which the waters found a new path; silt deposits raised riverbeds and pushed the rivers into new channels. At other times, sudden changes in course were driven by violent earthquakes. The only response was flight\u2014abandoned settlements lined the banks of vanished rivers, as when the Ganges \"deserted\" the once-flourishing town of Kanauj in the early sixteenth century. People had no choice but to move with the migrating waters. Beginning in the early seventeenth century, the Ganges began to shift eastward. A major earthquake in 1762 and another in 1769\u20131770 jolted the river away from its channel, forcing it into contact with new tributaries: the Tista and the Jamuna, the Jelangi and the Mathabhanga, the Kirtinasa and the Naya Bhangini. Even the names that rivers bore could be a testament to their instability: \"Naya\" means new, suggesting memories of an old Bhangini. The coastline shifted with the rivers: Habib reconstructs the coast of Gujarat through \"decaying ports\": places that had once been on the coast, that now lay silted up.\n\nAs their realm expanded to reach India's coasts, the Mughals incorporated within their realm port towns that faced the eastern and western Indian Ocean. Long before the arrival of Europeans, merchants in India had trading links that spanned the Indian Ocean rim. Indian textiles filled marketplaces across Southeast Asia and China, the Mediterranean and West Africa. Many regions of India\u2014Gujarat, Bengal, and the Coromandel coast\u2014thrived on long-distance trade. Textiles brought forth medicinal products, spices, local crafts, and large quantities of precious metals. On one estimate the Indian economy absorbed 20 percent of the world's supply of silver between 1600 and 1800. Throughout Southeast Asia's era of commercial expansion in the sixteenth century, Indian traders from the coasts of Gujarat, Madras, and Bengal shipped cloth to Pegu and Tennasserim in Burma, to the thriving port of Melaka on the Malay peninsula, and to the Indonesian islands of Sumatra and Java.\n\nThe Portuguese apothecary Tom\u00e9 Pires observed in Melaka in 1512 that ships from Bengal brought \"five white cloths, seven kinds of _sinabafos,_ three kinds of _chautares, beatilhas, beirames_ and other rich materials. They will bring as many as twenty kinds.\" In their holds came \"very rich bed-canopies, with cut-cloth work in all colors and very beautiful,\" and \"wall hangings like tapestry.\" Pires concluded that \"Bengali cloth fetches a high price in Melaka, because it is a merchandise all over the east\"\u2014from Melaka these textiles would make their way to markets across the Indonesian archipelago. In return, Indian traders exported from Melaka \"camphor and pepper\u2014an abundance of these two\u2014cloves, mace, nutmeg, sandalwood, silk, seed-pearls a large quantity, copper, tin, lead, quicksilver, large green porcelain ware from the Liukiu [Japan's Ryukyu islands], opium from Aden... white and green damasks, _enrolados_ from China, caps of scarlet-in grain and carpets; krises and swords from Java are also appreciated.\" The variety of Indian cloth gave rise to a lexicon that seeped into languages of trade everywhere: longcloth and salemporis, moris and gingham, dungarees and guinea cloth and kaingulong. Indian weavers targeted diverse markets. Their weaves, patterns, colors, and designs were all adapted to local tastes.\n\nThe Indian Ocean's trading world reached deep into the interior. It intersected with circuits of commercial exchange that went overland to Central Asia. In India, as elsewhere, the sixteenth and seventeenth centuries saw the deepening commercialization of the countryside. From 1500, South India saw the rise of a merchant class whose diversified business included overseas trade, collecting and keeping a share of local tax revenues, and financing local rulers' military ambitions. Global demand for Indian cotton set up a chain of transactions that linked the cotton-growing countryside to the port towns. Cultivators became more dependent on credit from urban merchants to finance the next year's crop; gold and silver currency came into more widespread use. Agrarian commercialization fed the Mughal state's treasury. The state's demand for cash taxes, historian Victor Lieberman observes, \"acted like a giant pump, sucking foodstuffs from the countryside into towns and cites.\"\n\nThis was the world that European trading companies entered; initially, they were players among many others. The monsoon winds brought the first Portuguese ships to India at the end of the fifteenth century. Arriving at Malindi on the eastern coast of Africa, Vasco da Gama sought advice from Indian traders there. They counseled him on the pattern of the winds that would take him across the Indian Ocean to Calicut. A local pilot guided his voyage. Chartered companies from Europe's western edge took to the sea because the landed power of the three great Islamic empires\u2014the Ottoman, the Safavid, and the Mughal\u2014blocked their path across central Eurasia. So they found an alternative route to the profitable cotton textiles of India, the spices of the Indonesian archipelago, and the ceramic manufactured goods of China. Their ships rounded the Cape of Good Hope and crossed the Indian Ocean. They arrived on the coast of western India to find a region already open to the world, tied by commerce to the littoral of the Indian Ocean right up to the Mediterranean. Coastal India already, in the sixteenth century, faced both ways: connected to the far reaches of the Indian Ocean, but also to mountainous Central Asia.\n\nSuccessive invaders of South Asia had come on horse from the northwest. Now, parvenu claimants to power approached by sea. Conquest was neither easy nor initially attractive. Monopoly was what the Europeans sought. In staking a claim to exclusivity they militarized the Indian Ocean's sea lanes in a new way. Roiled by the Dutch revolt against the Spanish crown at home, the Portuguese soon met competition from the Dutch and English East India Companies in the Indian Ocean. European power was concentrated in a scattering of \"factories\" along the Indian coast. Each served as a dormitory, a trading post, and a warehouse all in one. European power rested on relationships delicately negotiated with local rulers. Competition among Europeans fueled the companies' expansion\u2014armed conflict at home and rivalry on the high seas heightened commercial contests in India.\n\nBy the eighteenth century European chartered companies had mobilized relationships and resources that spanned the globe. They paid for their purchases in Asia with silver from the mines of Potos\u00ed (in present-day Bolivia). They transported Indian cottons to the coast of West Africa, and exchanged them for enslaved human beings. Europeans possessed many advantages as they inserted themselves into the fissiparous politics of regional kingdoms that had risen to fill the political void of a declining Mughal Empire. As well as precious metals, they could offer their local allies the security promised by the sophisticated weaponry at their disposal. The companies raised a lot of capital, quickly, aided by their structure as joint-stock firms. They intervened in local succession disputes. They did deals with bankers and _zamindars_. British, French, and Dutch companies became enmeshed at the frontier between the coast and the interior.\n\nThe transformative moment came in the second half of the eighteenth century. British maritime power pushed inland. Not long after the Battle of Plassey in 1757\u2014which stabilized the British \"bridgehead\" in Bengal\u2014the East India Company received the _diwani,_ or the right to the land revenues of Bengal, the most fertile province of the Mughal Empire. Bengal's agrarian wealth funded a violent cycle of English expansion. The Company's army rose to become one of the largest military forces in the world. The English were the first truly to harness both of the monsoon frontiers, welding maritime and landed wealth. Between 1757 and 1857, British control expanded up the Ganges valley from Bengal; the most rapid period of expansion came in the 1790s and 1800s, as British confidence and ambition were fueled by the worldwide war against France. The Company subdued the proudly independent southern Indian kingdom of Mysore in 1799, the Gangetic kingdom of Awadh and the southern domain of Arcot in 1801, the Dutch-dominated island of Sri Lanka by 1815, the Maratha lands in western India by 1818, and the coast of Burma by 1826. The late Christopher Bayly, whose early work on the merchants and markets of North India is unsurpassed, observed that at all times the Ganges valley remained \"the main axis of Britain's Asian Empire.\" The valley's commerce \"pointed northward to the high regions of Central Asia,\" while \"huge quantities of cotton, opium, and indigo bound for China and Europe\" flowed downriver to Calcutta, along with hides, oilseeds, and the saltpeter used to make gunpowder. Shipments of rice, opium, and tobacco went upstream toward India's northwest frontier.\n\nAs British power forced its way into the interior, into southern and western India, the distribution of water created different possibilities for agricultural, fiscal, and therefore political expansion. For scientists and travelers, mastering the map of India's water became a matter of curiosity; for revenue administrators it was a matter of urgency.\n\n# II\n\nOne of the earliest maps of the British domains in India was the work of James Rennell (1742\u20131830), surveyor-general of Bengal. Given the centrality of the Ganges to British power in India, Rennell set out to map the Ganges, from its source to the Bengal delta. He described the descent of the river from the \"vast mountains of Thibet\" to the Indian plains, where it \"serves the capacity of a _military way_ through the country,\" before reaching the ocean in \"a labyrinth of rivers and creeks.\" Rennell was impressed by the power of India's rivers: \"next to earthquakes,\" he wrote, \"perhaps the floods of the tropical rivers produce the quickest alterations in the face of our globe.\" He described the \"extensive islands\" of silt (known locally as _chars_ ) formed \"during an interval far short of that of a man's life.\" Rennell saw that \"it is no new thing for the rivers in India to change their course.\" He described how the confluence of the Kosi and the Ganges had migrated forty-five miles in a short span of time. The mighty Brahmaputra, India's second-largest river, had \"varied its course still more.\" Rennell described the Brahmaputra as unknown, forbidding, and turbulent. Where the Brahmaputra met the Ganges, in the Bengal delta, he described \"a body of running fresh water, hardly to be equalled in the old hemisphere.\" Its governing influence was the monsoon. The characteristic pre-monsoonal storms, known as northwesters, were \"the most formidable enemies that are met with in this inland navigation.\" Of enemies there were plenty. Traveling upriver, he wrote, a regular \"budgerow\" ( _bajra_ )\u2014large boats with cabins that covered their length, most common along the Ganges\u2014\"hardly exceeds 8 miles a day, at ordinary times.\"\n\nMap of the Ganges and Brahmaputra as they meet in the Bengal delta, from James Rennell, _A Bengal Atlas: Containing Maps of the Theatre of War and Commerce on That Side of Hindostan_ (London, 1781). CREDIT: Courtesy of the Map Collection, Harvard College Libraries\nRennell was not alone in his fascination with, even fear of, India's climate. As they established gardens as a site of botanical and commercial experiment, Company officials in India took a greater interest in the weather\u2014none more so than William Roxburgh (1751\u20131815). Roxburgh studied anatomy and surgery at Edinburgh University at a time of intellectual ferment; he left Edinburgh in 1772 to join the East India Company's ship _Houghton_ as surgeon's mate on a voyage to India. The following year, he signed on for another voyage, which took him via Saint Helena and the Cape to Madras. Upon his arrival at Fort Saint George in 1776, Roxburgh began a meteorological diary. He equipped himself with a portable barometer \"made by RAMSDEN\" and an indoor thermometer supplied by Nairne and Blunt\u2014scientific instruments, like texts and theories, moved along imperial shipping lines. His outdoor thermometer he placed \"under a small, shady tree.\" His observations, three each day, devised a scale for describing the winds: \"gentle, brisk, stormy, and what we call a tufoon in India.\" The rain gauge he had initially proved worthless; he assured his correspondents that he had installed a better model on the roof of his house on the hospital grounds. Based initially in Madras, Roxburgh moved south down the coast to the small port of Nagore, long connected to Southeast Asia by Tamil Muslim merchants; from there, he settled in Samulcottah (Samalkot) on the Godavari delta, midway along India's eastern coast, until finally in 1793 he became director of Calcutta's botanical garden. Roxburgh followed many pursuits. He made a small fortune as a private trader. He kept an \"experimental botanical plantation\" where he grew indigo and pepper, breadfruit and sugarcane; he collaborated with botanist Johann Gerhard K\u00f6nig, who was stationed at the Danish settlement of Tranquebar on the Madras coast.\n\nRoxburgh was a keen observer of life around him. He took an interest in how the monsoon's rhythms shaped farming on the land. \"The rains generally set in, in June,\" he wrote in his description of the growing season of the Godavari delta, \"towards the end of that month, the coarse or early Paddy, is sown, and in July the better sorts, or great crop.\" He described how \"our rains continue from the time they set in, June, 'till about the middle of November; July and August, are generally our wettest months: in October and November the weather is more stormy, being the period we call the Monsoon.\" In his usage, \"monsoon\" was the period of change, as the winds switched from southwest to northeast. \"The cultivator has to depend on the rains,\" he said again; \"the more favorable they are, the better is the crop.\" Higher, more arid lands, \"as in every other part of India,\" were given over to \"dry grain.\" He undertook a detailed study of hardy crops that thrived in dry conditions, locally and around the world. He ordered samples from across the empire; he planted them in his experimental garden.\n\nRoxburgh became an astute observer of South India's climate, both in its regularity and its extremes. He could not himself escape its risks. In 1787, a severe cyclone struck the Godavari delta; it destroyed Roxburgh's home, his herbarium, his library, and most of his personal wealth. His family escaped narrowly with their lives. He observed at close quarters the prolonged drought that brought famine to the region in the late 1780s and early 1790s. In 1791 Roxburgh wrote to his friend, celebrated English naturalist Joseph Banks, that \"the famine of these provinces begins to rage with double violence, owing to a failure of our usual rains.\" Two years later, Roxburgh's friend Andrew Ross declared that \"the dreadful effects of the famine here have... far exceeded any description from us.\" He saw that \"in many places where populous villages formerly stood, there is at present neither vestige of man or beast.\" Roxburgh looked for patterns in the data he had collected\u2014he sought to understand the cycles of the seasons, and variations from year to year. During his years on the Coromandel coast, Roxburgh had collected an extent of meteorological data on the Madras coast that one historian describes as \"unrivalled elsewhere until the 1820s except among indigenous Chinese observers.\" The early initiatives of Roxburgh and his colleagues formed a foundation on which India's modern meteorology was built.\n\nHe began to wonder, like so many others of his generation and education, whether India's nature might benefit from \"improvement.\" He wanted to harness the water that \"passes annually unemployed into the sea.\" He was \"astonished\" to find not the \"least trace of any work, ancient or modern, for retaining, or conveying the water to fertilize their Paddy Lands\" in the region; the result was that \"the cultivators here depend entirely on rains, when they fail, a famine is, and must ever be, the consequence.\" Roxburgh observed, and sketched, the Godavari delta; he imagined it transformed. \"In consequence of the favourable level and descent of lands,\" he wrote, \"we clearly see the infinite benefit that must arise from the waters of large rivers when a method of making them subject to the will of man is affected.\" The solution he saw was to use natural basins to store large quantities of water as the Godavari descended from the hills.\n\n# III\n\nHowever we understand India's economic transformation in the nineteenth century, water is at the heart of it. The flow of water\u2014the flow of India's rivers, their seasonality, their propensity to change course\u2014constrained how India's producers could respond to new market opportunities and new compulsions. Britain's industrialization had benefited from an elaborate network of canals; India's economic development, by contrast, was limited by the difficulty and expense of water transportation. China had a far more extensive network of canals than India at the time, but unlike in Britain, energy sources were far from the waterways. The availability of water enabled a changing landscape of cash crop production, for some; water's absence tested others' capacity for bare subsistence. Water was instrumental to making the Indian soil produce more of the commodities the world demanded.\n\nIn the 1830s and 1840s, India's British rulers still faced constraints that would have been familiar to their predecessors. Transport was slow\u2014and dangerous. \"No part of the inland navigation of India is so dreaded or dangerous,\" wrote the botanist Joseph Hooker in 1848, \"as the Ganges at its junction with the Cosi\"; in the rainy season the Cosi \"pours so vast a quantity of detritus into the bed of the Ganges that long islets are heaped up and swept away\"; boats \"are caught in whirlpools formed without a moment's warning.\" The monsoon governed not only the harvest\u2014and threatened the possibility of harvest failure\u2014but also threatened the health of Europeans. Cholera, malaria, and other ailments led many British officials in India to an early grave. Water was still a source of both awe and foreboding for British residents in India. A medical topography of Calcutta published in 1837 stated that \"without taking into view the expanse of the Bay [of Bengal], the coup d'oeil of a good map of Bengal will at once show how bountiful nature has been to that country, by means of her majestic rivers with innumerable tributaries.\" But these waters were at the same time the source of \"aqueous exhalations\"\u2014a product of the \"commerce of land and water\" in a monsoon climate\u2014that menaced life. The author James Martin saw a clear \"connexion of the rainy season with disease,\" and suggested that \"among Europeans, the diseases of the rainy season assume a character of diminished vital action.\" Throughout the nineteenth century, fears persisted of whether Europeans could survive tropical climates.\n\nTo make India productive, to integrate it more fully with the global capitalist economy that was in formation, to exploit more effectively its natural resources to feed Britain's industrialization, British engineers and investors and administrators looked to master the unevenness of water, its extreme seasonality in India; and they sought to conquer space. Both of these quests unfolded between the 1830s and 1870.\n\nHALF A CENTURY AFTER ROXBURGH'S TIME, THE GODAVARI DELTA was still \"entirely without any general system of irrigation, draining, embankments or communications.\" This was the verdict of Arthur Thomas Cotton (1803\u20131899), the museum to whose memory opened this chapter. Like Roxburgh's before him, Cotton's problem was the distribution of rainfall across the landscape. His task: \"counteracting the irregularity of natural supplies of water.\" \"One year a portion of the whole crop... is destroyed by the overflowing of the rivers,\" Cotton observed, \"in another, the crop is destroyed by a failure of the rains over three-fourths of the district.\" He was convinced that \"not an acre... need be dependent at all\" upon the rains if a comprehensive system of irrigation were introduced. He insisted that the Godavari delta needed not a piecemeal restoration of existing irrigation works, but rather \"works of a general nature.\" Perennial irrigation; an improvement in the \"roads and bridges\" of the region; a restoration of the port of Kalinga (\"Coringa\") so that it could fulfill its potential as \"incomparably the best port\" between Hooghly and Trincomalee\u2014such investments in infrastructure would free the district from its uneven and capricious rainfall.\n\nCotton made a fervent case for government intervention. India was unlike Britain, he argued; the rules governing public expenditure could not be considered akin to the principles of household economy. The problem was that \"there is almost literally no capital to enable landowners to make improvement.\" An outlay of three hundred or four hundred thousand rupees each year by the state \"would put life and activity into the whole district\"\u2014in time, revenue would flow into the treasury far exceeding what the state might spend. Possessed by evangelical self-confidence\u2014nothing less than a sense of destiny\u2014Cotton went further. He condemned what he saw as his countrymen's \"proneness... to lower ourselves to the level of natives\" instead of \"diligently applying the means which God has placed in our hands to benefit the countries He has given us charge of.\" Cotton found the support and the money for his grand scheme. In 1852, he completed his barrage at Dowleswaram. But his dreams were bigger. Cotton imagined a network of canals that would, one day, bring the Himalayan rivers to the southern tip of the peninsula. He also saw that the rivers had unrealized potential for navigation. In 1867, Cotton dreamed of a link between the Brahmaputra River\u2014its upper reaches were still at that time unknown to British explorers\u2014and the Yangzi. \"The throwing open of all India to all China, the access of a country containing 200 millions of people to the produce of a country occupied by 400 millions,\" he wrote, would be \"a work of such magnitude as that nothing approaching it has ever been seen in the world.\"\n\nIn the British imagination as well as in administration, Peninsular India was quite distinct from the \"heartland\" of Gangetic India. Separated by half a century, William Roxburgh and Arthur Cotton in turn sought to mold a riverine landscape that attached the dry interior of the Deccan plain to the coast of the Bay of Bengal. They sought both to harness and to overcome the political inheritance that distinguished South India from the north. In the former, political power was contested within a system of small states that arose to fill the void of the troubled Mughal Empire; the hydraulic landscape was dispersed in thousands of tanks, wells, dams, and weirs, many of them now lay in a state of disrepair after decades of warfare\u2014not least the warfare that accompanied English expansion. But Cotton's counterparts along the Ganges were no less anxious to see what could be done to \"improve\" nature: to repair or replace the hydraulic remnants that scattered the valley. They faced different challenges, they chose different solutions, but they shared many assumptions with their counterparts in the south. Just two years after Cotton's barrage was complete, a project still more monumental opened its floodgates: the Ganges Canal, at the time (and still today) the largest in the world.\n\nThe Ganges Canal was the creation of Proby Cautley\u2014Arthur Cotton's contemporary, classmate, and eventually his bitter rival. Cautley arrived in India in 1819 as an artilleryman. A few years after his arrival, the first Anglo-Burma war in 1824 drew many of the East India Company's engineers across the Bay of Bengal; their absence created new openings in India for those without formal training. Like so many Company officers, Cautley was an autodidact. He learned his craft through practice and observation. Working in different ecological settings, Cautley and Cotton embraced different hydraulic approaches. By the 1860s, they fought their battles in a bitter and public war of pamphlets. Cotton accused Cautley of making fundamental mistakes in the design of the Ganges Canal; at stake was not only prestige, but also a debate over the ownership and financial management of India's hydraulic works. Along the Ganges, as everywhere else in India, the infrastructure of water control long preceded British rule. But in the nineteenth century British engineers turned the Ganges valley into one of the most \"thoroughly engineered\" landscapes in the world.\n\nThe Gangetic plain's hydraulic transformation began with the Company's effort to restore the old Yamuna Canal's supply of water to Delhi. The waterworks dated back to pre-Mughal times: Delhi's water infrastructure owes much to the rule of Sultan Iltutmish in the thirteenth century. He ordered the construction of an elaborate web of tanks and step wells. The Mughals brought them to a new level of sophistication. They built a complex of ornate gardens along the banks of the Yamuna River, laid out around the tombs of Mughal leaders. They watered their new capital at Shahjahanabad from a canal and an interlocking system of smaller canals and drains. Emperor Akbar ordered the renovation of the West Yamuna Canal\u2014first built by the ruler Firoz Shah\u2014for irrigation, and extended it to Delhi. Akbar's Canal Act of 1568 declared the canal's aims to be \"to supply the wants of the poor,\" to \"leave permanent marks of the greatness of my Empire by digging canals,\" and to ensure that \"the revenues of the Empire will be increased.\" The British found the canal gone to ruin, yet traces of its sophisticated engineering remained. In 1820, British engineers restored the water supply to Delhi through the West Yamuna Canal. They followed quite consciously in the footsteps of Mughal architects.\n\nWith this success in hand, local administrators turned to the restoration of the eastern branch of the Yamuna Canal. Second in command of this project was young Proby Cautley, who had no prior experience of hydraulic engineering. Cautley was open, perhaps unusually open, to learning from local practices: he suggested adapting local well-building techniques to provide a stronger foundation for bridges than usual European methods could sustain in the soils of the Gangetic plain. As he took charge of the canal project, Cautley ordered the construction of rest houses every ten or twenty miles along the path\u2014in keeping with the old Mughal tradition of _caravansarais_ along the Grand Trunk Road. Besides water, Cautley's interests encompassed archaeology, paleontology, and botany. In 1831, while supervising the digging of a well as part of the canal project, he discovered evidence of an ancient settlement at Belka. With even more enthusiasm, Cautley and his colleague Hugh Falconer began collecting fossils of mammals and birds and fish, eventually shipping to the British Museum in London a collection that took up 214 crates. The history of science in nineteenth-century India often saw the blurring of lines between disciplines.\n\nBy the middle of the 1830s, though, Cautley was first and foremost a water engineer. In 1835, he became the Company's superintendent of canals. His predecessor in that role, John Colvin, had left him with an idea: to build a canal to bring the waters of the Ganges to the arid Doab (the name means \"between two rivers\") that lay between the Ganges and the Yamuna. Early investigations concluded that the canal would be too expensive\u2014and probably an engineering challenge too far. The calculus of costs and benefits, so central to the Company's mode of administrative thought, changed in 1837 when a major famine devastated the drought-prone Doab. By 1840 plans were in place to build what would become the Ganges Canal.\n\nThe centerpiece of the canal complex was a headworks at Haridwar, where the Ganges meets the plains. Its most complex feat of design was the Solani aqueduct, which ran sixteen miles below Haridwar\u2014civil engineer G. W. MacGeorge, author of an 1894 treatise on the infrastructure of British India, called it the \"most interesting and remarkable modern structure in India.\" The technical challenges were formidable. The project created a hybrid landscape as an artificial \"river\" crisscrossed Himalayan streams that in the summer became torrents. \"To carry the great canal\u2014itself a small river\u2014across such a country,\" one British engineer observed, \"to see it pass silently on, uninterrupted and uninjured by these torrents\" was \"a triumph of art and engineering ability.\" Above all, it was a feat of labor. The works were labor-intensive; machines played little role in the initial stages. The canal was the work of thousands who molded and fired bricks, their kilns fed by timber from local forests. Earthworkers ( _bildars_ ) dug the canal. Hundreds of men were deployed in transporting materials. Much of the work was organized by local contractors, who recruited workers from across the region. We know almost none of their names. Historian Jan Lucassen, in studying a strike by the brickworkers in 1848\u20131849, has uncovered a few of their stories. When their employers tried to cut wages, brickworkers first deserted the site, and later set fire to a number of encampments.\n\nAt the time of its opening in 1854, the Ganges Canal was more than seven hundred miles long. A pamphlet, \"A Short Account of the Ganges Canal,\" was distributed in English, Hindi, and Urdu at the opening. It declared that \"the great motive by which the British government was led to sanction the Ganges Canal\" was \"to secure to its people, in the country between the rivers Ganges and Jumna, an immunity from the pains and losses that famine brings with it.\" The famine of 1837 and 1838 was still fresh in the spectators' memories. For Company administrators, those memories involved the loss during the famine of land revenues, and relief expenditures amounting to over 5 million pounds\u2014financial loss was the spur to action, however sincere the humanitarian considerations might have been.\n\nA year after the canal opened, the _North American Review,_ a Boston literary journal, published an account of the Ganges Canal and its opening ceremonies. It evoked the \"double sanctity\" that the \"mysterious river of the farthest East\" now possessed\u2014the Ganges had long been revered and worshipped as a divine river, a place of pilgrimage for people from the distant corners of India; now it was newly (or doubly) blessed by the bounty of technology. The canal was hailed as \"the largest of its kind in the world, adapted for navigation as well as for irrigation\"; it was \"designed not less for the benefit of a remote future than of the present age.\" The inauguration of the canal drew large crowds. \"From the most distant parts of India pilgrims came up this year,\" the American correspondent wrote, \"when the revered Ganges was about to leave her ancient and hallowed channel for one formed for her by the hands of strangers.\" Quoting from a \"private account\" that had come into the journalist's possession, he described how the aqueduct's embankments were \"lined by our own work-people, to the number of more than thirty-five thousand men,\" in \"long lines of stout forms.\" The military presence was strong, for here as in every development of infrastructure in India, military imperatives were paramount. \"The infantry were on the tops of the aqueduct parapets\" while \"the artillery were stationed on a high piece of ground.\" The crowd gathered to celebrate the new canal was estimated at not fewer than five hundred thousand people.\n\nThe canal was a monument to imperial power, a symbol of English conquest over India's land and water. In his opening speech, Lieutenant Governor John Colvin\u2014Cautley's predecessor, and the originator of the idea for the Ganges Canal\u2014declared that \"we have an answer... to the old reproach, that the British have left no permanent mark upon the soil of India to attest the power, the wealth, and the munificence of their nation.\" But the canal also marked a symbolic step up in the justification of British rule on humanitarian grounds. In the eyes of the American journalist, it was \"difficult to conceive of a more impressive service\" than the opening prayers consecrating the complex. Seen through the observer's evangelical imagination, the entire Ganges Canal complex was the work of a \"few hundred Christians in the heart of a foreign country, surrounded by many thousand heathens\"\u2014a \"work of civilization... for the benefit of these unenlightened multitudes.\" He declared a new \"era of intelligent and liberal government\" that \"regards and cherishes the interests of the governed.\" He conceded that, for all its benefits, the advance of British rule in India had been attended by \"the bitter consequences of evil\" and \"past misgovernment.\" But the tide had turned: \"The night in which false religion, tyranny, and war have enveloped India,\" he wrote, \"is giving place to the day of Christianity, good government, and peace.\"\n\nNot long after the completion of the Ganges Canal, the Indian Rebellion of 1857 brought an end to the East India Company's rule. A mutiny within the army spiraled into widespread social protest that spread across North India; the old Mughal emperor, Bahadur Shah, was the rebels' symbolic leader. The rebellion was suppressed with spectacular violence. The British government took control of India from the East India Company. The colonial state intervened more extensively in the countryside. It used the law to reconfigure property rights and to reshape relations between landlords and tenants, men and women, Hindus and Muslims, dominant and subordinate castes. It used force to settle mobile people, and the force of punitive contracts to mobilize labor for the plantations of Southeast Asia. In 1869, the government's approach to the land found clear expression in Lord Mayo's dictum that \"every measure for the improvement of the land enhances the value of the property of the State\"; especially so, he added, because \"the duties which in England are performed by a good landlord fall in India, in a great measure, upon the government.\"\n\nIN THE ODES OF THE EMPIRE'S PRAISE SINGERS, THE ACHIEVEMENTS of British engineers in nineteenth-century India stood without parallel. But what truly was new about the hydraulic fever of the second half of the nineteenth century? For Cotton, it was the capacity to design the world anew through \"works of a general nature.\" But the ancient tank irrigation of southern India was just as ambitious, just as systemic. Landscapes of water have always been shaped by human intervention. Water historian Terje Tvedt warns us against the conceit that the \"conquest of nature\" is a modern phenomenon. But there can be no question that the scale of the works designed and built in the nineteenth century were without precedent. Steam power broke the physical limits of earlier modes of construction\u2014even though, as we have seen, old methods were used extensively as better adapted to local ecology.\n\nThe British justification of large-scale public works, too, had new dimensions. In precolonial India, though not as markedly as in China, the control of water lent legitimacy to local rulers. Irrigation works bolstered local resilience to drought, and ensured states' coffers remained full. Maximizing revenue remained the be-all and end-all of British rule in India, from start to finish; behind every investment in infrastructure lay the aim of extraction. Like many local rulers before them the British government of India used irrigation works to signal their benevolence, to demonstrate their power, to satisfy their own vanity. But some British architects of water went further than this. Driven by an aggressive evangelical Christianity, engineers like Arthur Cotton saw their mission as going far beyond the sustenance of revenue for the state. Irrigation, alongside other technologies, would usher in the social and moral transformation of rural India: midwife to an ever-expanding universe of commerce and trade. The moral argument for infrastructure in India laid deep roots in the nineteenth century\u2014it would not be long before it was turned against British rule.\n\nIn earlier times the benefits of hydraulic infrastructure resided entirely at the local, or at most the regional, level. In this sense, perhaps the most far-reaching change that British imperial engineering brought was its spatial expanse\u2014in the imagination of British engineers and administrators and investors, the irrigation of a particular district of the Godavari delta or the Gangetic plain would have repercussions across the globe, as more and more of the products of India's soil found their buyers in the markets of London and Liverpool, Hamburg and New York.\n\n# IV\n\nThe irregular availability of water was one challenge; the conquest of space was another. To take the augmented products of India's irrigated lands to market, India's great rivers had to be made navigable. In the first three decades of the nineteenth century, little had changed since cartographer James Rennell's description in the 1790s. The inland waterways of Bengal sustained \"a system of regional trade that served a population of some 60 millions.\" An array of specialist vessels plied the waterways of the Bengal delta: salt boats, the boats used by woodchoppers in the Sundarbans, the small craft for the traffic in betel leaf, and the distinctive port lighters that serviced European merchantmen, loading and unloading their cargoes. Higher up the hierarchy in this catalog of rivercraft were the _bajra_ preferred by European employees of the Company, with a large sail on a single mast. Most luxurious was the pinnace, reserved for higher officials and the wealthiest Indian merchants. The river teemed with life, animated by the labor of boatmen with a panoply of specialized skills. A distinctive Anglo-Indian lexicon emerged to describe work on the river\u2014the product of British translations and mistranslations, transcriptions and mispronunciations of local words. _Serangs_ and _tindals_ were boatswains; _manjhees_ and _seaconnies_ were steersmen; _dandees,_ expert oarsmen, _lascars,_ sailors. Each group knew the river intimately.\n\nBut fluctuations in the Ganges River's flow between the wet and the dry season, the heavy loads of silt that it carries, forging and undoing sandbanks and shoals along its course\u2014these all made it treacherous for larger vessels. Sandbanks deceived the most seasoned boatmen. Insurance firms charged the same premium on freight going from Calcutta up the Ganges to Allahabad as they did to London. If anything, Company rule had slowed traffic along the river, because of its punitive taxation. This was the view of Charles Trevelyan, who traveled down the Ganges and the Yamuna in 1830 to report on the oppressions inflicted by countless customs posts, acting in the Company's name, along the riverbanks. \"These streams, intersecting, as they do, the whole of the Bengal Provinces from one end to the other and terminating in the Sea port of Calcutta,\" Trevelyan wrote, \"must be the great channels and high roads of trade of the country.\" But they fell short of that potential. Trevelyan found that the number of customs inspections was enough \"not only to embarrass the navigation of the Jumna, but to close it entirely for nearly half its course from the hills.\" He noted how few of the \"great staples\" of salt, cotton, ghee, and asafoetida traveled by river\u2014instead, merchants resorted to \"tedious and expensive land carriage.\" Only one cargo of salt arrived in Agra from Delhi in the year 1830 even though vast quantities were traded yearly from Delhi \"for the consumption of our Eastern Provinces.\" In his sloping copperplate hand, Trevelyan was damning: \"It may appear extraordinary,\" he wrote, \"that the Officers who are charged with the collection of the customs should possess so imperfect an idea\" of the inspections and exactions conducted in their name. Those who suffered most were the \"poorer class of merchants and traders who can ill afford to pay.\" Trevelyan noted that \"speed is the Life of Trade.\" Like so many other Company officers, he was a trader himself: he spoke from experience. Trevelyan's report played a key role in the East India Company's abolition of internal customs duties.\n\nBy the time Trevelyan made his voyage downriver, the Ganges was the site of some of the earliest experiments with steam technology in India. British administrators and businessmen looked eagerly to the expansion of trade and navigation upriver. The first steam engine to arrive in Calcutta, in 1817 or 1818, was there to clean the river Hooghly. The eight-horsepower engine from Birmingham powered revolving buckets to clear the river of the silt it carried from the hills. A few years later, the steamboat _Diana,_ launched in July 1823 by the Calcutta shipbuilding firm Messrs Kyd & Co, drew large crowds to witness its maiden voyage. As a commercial proposition it failed. War spurred technical improvement. In 1825 the company launched a military expedition against the Burmese kingdom after tension along India's expanding northeastern frontier. The old dredger was converted into a warship; unprofitable _Diana_ was pressed into service to carry medical supplies and the wounded between India and Arakan, on the eastern littoral of the Bay of Bengal. It was deployed up the Irrawaddy River, where local people labeled the ship the \"fire devil.\"\n\nBy the 1830s, steamboat agents had set up shop at every point along the river\u2014most of them as a sideline to their primary occupations. Many profited from the arrival of steam. J. P. Leslie was by day a pleader at the Allahabad High Court; he made himself the government's agent at the port and charged commission for overseeing the loading and unloading of cargo. Carr, Tagore, & Co., managing agents, secured the contract to supply coal to the government's Steam Department from their mines in Burdwan, in eastern Bengal. The Ganges was a microcosm of India's economic transformation. Steamboats began to carry to Calcutta the revenues on which the East India Company's administration depended\u2014the proceeds of the annual harvest. Steamboats were filled with \"boxes loaded with five thousand rupee coins,\" each one \"roped, ticketed, and sealed with lead and wax\" and guarded by one or more soldiers. Upriver, private money traveled from Calcutta merchants to Patna, Benaras, and Allahabad\u2014advances on the crops eagerly awaited by traders in London and Liverpool and New York. The Ganges was a conduit for India's economic integration with the world. The bulk of the cargo traveling up from Calcutta consisted of the material paraphernalia of British imperialism in India\u2014arms, medical supplies, printing presses, seals for opium agents, compasses and theodolites for the staff of the Indian Survey, the mammoth project to map and survey every inch of British territory in India. Consignments of three commodities dominated steam traffic along the Ganges: cotton, indigo, and opium. Each was of global importance. Long transported to Calcutta on country boats, indigo was small enough to be stashed away\u2014it was a favored way for company officers to smuggle their ill-gotten gains out of the country. From 1836, consignments began to travel downriver by steamboats.\n\nBy the end of the 1830s, ironclad steamers traversed the 780 miles between Calcutta and Allahabad in three weeks, but beyond Allahabad they found their passage blocked. The Ganges continued to challenge the power of steam. Even under experienced pilots, steamboats ran aground. They foundered on the river's treacherous shoals and sandbanks. Silt blocked the path of large vessels, confining them to the most easily navigable stretches of the river. More nimble vessels lacked the power to push upriver. And steamboats were expensive. The most valuable commodities formed the bulk of the cargo on steamboats along the Ganges. But steam freight was too costly for most merchants. Bulk goods\u2014rice, sugar, saltpeter, linseed, hemp, and hides\u2014continued to travel on country boats, or overland. Far from supplanting earlier uses of the river, the steamboat took its place in a varied economy of energy and transportation: the oldest and the newest technologies coexisted and competed with one other. Often the river itself\u2014its currents, its seasonality, its contours\u2014set the boundaries of what was possible or financially viable.\n\nIn the end it was not road but rail that emerged as the biggest competitor to steam vessels on the river. Because of British investors' embrace of the railroad, steam transportation along the Ganges never really flourished.\n\nTHE WORLD OVER, THE COLLAPSE OF SPACE AND TIME BY THE RAILROAD, the steamship, and the telegraph underpinned the transition to industrial capitalism. Writing of the same period in the nineteenth century, environmental historian William Cronon describes how Chicago's tentacles reshaped the entire landscape of the American Midwest. Cronon writes of the \"railroads' liberation from geography\"\u2014their ability to operate \"quite independently of the climatic factors that had bedeviled earlier forms of transportation.\" But how much did this apply in India? Could rail operate \"independently\" of the monsoon, where river transport could not?\n\nIndia's railway dreams were born in the 1830s; by the 1840s, these visions had become a sort of \"mania.\" The construction of India's railway network began in the 1850s and reached its zenith in the last quarter of the nineteenth century, financed by private investors at public risk, their returns guaranteed by the state. By this time railway lines snaked across Europe and North and South America. Now speculators eyed India with anticipation. After false starts and burst bubbles, construction began in the 1850s. In 1853 the Marquess of Dalhousie, governor-general of India, inaugurated the construction of India's railways: investors were guaranteed a rate of return of 5 percent. As ever, military needs loomed largest in the government's calculations. As Dalhousie told Parliament in a statement that inaugurated India's railway age, a countrywide rail network \"would enable the Government to bring the main bulk of its military strength to bear upon any given point in as many days as it would now require months.\" He envisaged the \"commercial and social advantages\" of the railway as \"beyond all present calculation.\" Beyond calculation, too, were the \"extent and value of the interchange which may be established with people beyond our present frontier.\"\n\nWithin three decades, civil engineer George W. MacGeorge estimated that the railway had reduced India to \"one-twentieth of its former dimensions.\" Trains could cover up to six hundred kilometers a day. Bullock carts could manage, at best, twenty to thirty kilometers a day; river boats could cover sixty-five kilometers a day going downstream, but even fewer than a bullock cart when battling upriver. The railway grid consolidated the state's control over Indian territory. India's cotton and indigo, jute and opium moved faster to the ports of Bombay and Calcutta for export. The railway implanted the colonial state upon the recently conquered lands of the northwest.\n\nAmong those who watched with interest was Karl Marx, who saw the railway as a necessary tool to break down feudalism and social division. Rail collapsed distance. Rail integrated markets. Marx quoted from a British observation just a few years earlier, in 1848, that \"when grain was selling from 6\/- to 8\/- a quarter at Khandesh, it was sold at 64\/- to 70\/- at Poona, where people were dying in the streets\"; the sole reason was that \"the clay-roads were impracticable.\" In Marx's view, one of the railway's potential benefits lay in what it could do to the hydraulics of the land: it could \"easily be made to subserve agricultural purposes,\" he thought, \"by the formation of tanks\" along the railway embankments, \"and by the conveyance of water along different lines.\" A few years later, in 1860, railway engineer Edwin Merrall published a riposte to Sir Arthur Cotton's condemnation of expensive railway construction in India; it was a defense of the value of railways, faced with Cotton's alternative\u2014investing in India's waterways. For Merrall, too, water was central. Merrall insisted that the railway could overcome climatic variation, operating \"at all seasons of the year,\" whereas rivers swelled during the monsoon and dwindled in the dry season, leaving few months of the year when they could be traversed safely. India regularly suffered from famine, \"from the failure of the periodical rains\"; but \"such scarcity,\" he argued, \"is not general, but partial and local,\" and could \"very easily be met by an increased supply of food from other and more fortunate districts.\" The railway's greatest promise was to connect the driest parts of India with those \"which never want water.\"\n\nIt is difficult to write about India's railways without a flurry of astonishingly large numbers. Over the second half of the nineteenth century, the Indian railways expanded to encompass twenty-four thousand miles of track, and India possessed the fourth-largest railway network in the world: by far the largest in Asia. India's railways demanded a \"prodigious consumption of mineral or vegetable food, in the shape of coal, coke, or wood,\" vast quantities of iron and steel, and the complex manipulation of water to supply the engines. Because so many of the materials were imported, Indian railway expansion was a fillip to British industry.\n\nBut what did this mean for villagers in a district facing drought or flood? Some observers worried from the outset that, contrary to the dominant view, transportation was no panacea for social and economic inequality. Writing in 1851, C. H. Lushington, railway commissioner, saw that the lands of the Ganges valley were \"sublet in very small portions\"; they were worked by the \"poor and needy\" who were \"without capital who live from hand to mouth.\" They lacked the stocks of grain that would \"make it worth their while\" to seek distant markets. He worried that railway lines would cut through small holdings, dividing them further; he feared the \"serious and tangible injuries\" that would come from the way the railway lines interfered with drainage by building over natural floodplains. A quarter of a century later, many of his fears would prove prescient.\n\nThe railways reached deep into the interior. They carried the force of the state and the pull of global markets to even the smallest villages. An Indian economist writing in the mid-twentieth century described the railways as sparking a \"revolution in the economic pattern of the country\": the \"age-old walls of localized economy,\" he wrote, \"were collapsing.\" Recent analysts reach a similar conclusion based on the modeling of district-level data. Dave Donaldson estimates that the arrival of the railroad in any given district raised real income by 16 percent, thanks to \"previously unexploited gains from trade due to comparative advantage.\" He echoes a sense widely shared in the late nineteenth century: \"Districts that had been largely closed economies opened up as they were penetrated by railroads.\" His data show that the dependence of local grain prices and even local mortality rates on local rainfall vanished at last with the arrival of the railways. But these numbers tell us little about how resources were shared within each village, district, or household. Markets grew more integrated, but many without land or capital were not well positioned to benefit from this expansion. And what effect did these transformations have on those\u2014the lower castes, women and children, those ill or disabled\u2014who had to look outside the market, to benefits in kind or to always-tenuous customary entitlements, for their security and survival?\n\nRailway bridge over the Godavari River. CREDIT: Sunil Amrith\n\nHowever dazzling its scale, the rail network met the needs of a colonial export economy, driving produce to the river mouth ports. It was designed to take India's jute and cotton and tea and coal to where merchants needed them; it was designed to transport labor to the plantations and mines and factories. The massive expansion in mobility both within and beyond India in this period coexisted with deepening pockets of social and geographical immobility. Swaths of India remained off the railway map. The inequality between regions that benefited from the change and those left behind grew starker. Where there were no rail connections, roads also tended to be poor, and waterways poorly maintained. The National Sample Survey of India showed that, even a century after 1870, no fewer than 72 percent of journeys in many parts of rural India were undertaken on foot. Always, the newest and the oldest technologies depended on one another: as historian David Arnold observed, \"The railways relied on country carts to bring raw cotton and other cash crops to the railheads or distribute grain to needy villages in times of famine.\"\n\nBut in this age of evangelical faith in the \"civilizing\" effects of capitalism, of which British rule was the \"Providential\" vehicle in India, the railway seemed to augur an \"extraordinary awakening\": a \"wonderfully rapid subversion of previous habits of life and thought\" in India. The engineer George MacGeorge's encomium concluded that \"one of the most rigid and exclusive caste systems in the world\" had been \"penetrated on every side by the power of steam.\" Many Indian observers shared his enthusiasm. Madhav Rao, chief minister of the princely states of Travancore, Indore, and then Baroda, wrote of the \"glorious change the railway has made,\" as \"populations which had been isolated for unmeasured ages, now easily mingle in civilized confusion\"\u2014it offered no less than the prospect of India becoming \"a homogeneous nation.\" Not only did the railway change people's behavior, it transformed the landscape: trains ran through the rainy season and the dry, they crossed bridges across rivers and mountains, connected humid and arid regions. As they cut a track through the Western Ghats, railway engineers had turned \"the stupendous natural inequalities of the precipitous hills into a series of uniform inclined surfaces,\" and \"the whole rugged and inhospitable region has been smoothed down.\"\n\nBehind the self-regarding, if genuine, heroism of the engineers' accounts lies the forgotten heroism of those Indian workers who built the steel lines, the canals, the bridges. Countless among them paid with their lives. However fervently the railway engineers believed they could import their methods from England\u2014just as they imported steam engines, coal, locomotives, tracks, sleepers, and even prefabricated bridges\u2014what emerged in practice was a hybrid approach to building infrastructure. India's ecology could not easily be \"smoothed down.\" Whether through railway lines or irrigation canals, reshaping the Indian landscape was a colossal feat of work.\n\nIt was India's hydrology that challenged every scheme. Building the rail line from Howrah to Burdwan in Bengal, engineers faced an \"inland sea\" of water channels that required, in response, \"viaducts, bridges, culverts, and flood openings\" on a scale that no engineering project in the nineteenth century had ever attempted. Building bridges and aqueducts over the Himalayan rivers demanded ingenuity and a great deal of improvisation. Designs had to take into account the \"immense volumes of water periodically brought down\" by the great rivers through \"seasons of flood,\" and the \"erratic and unstable character of their channels.\" The rivers \"scoured\" the piers and abutments of bridges: torrents of water scraped away their foundations. So the engineers harnessed the knowledge of the people who knew the land most intimately. Even with the arrival of steam dredges and sand pumps, the terrible labor of divers and hand-diggers was vital; the piers of India's railway bridges often reached one hundred feet below the water's surface. Other skills came into play. Work was overseen by men who had experience as seafarers, as _lascars,_ in the British merchant marine. Their mastery of the winds, the tides, the currents; their mastery of a language of command\u2014these were put to new use to reshape India's inland seas, far from the ocean air.\n\nFew believed as fervently as Rudyard Kipling in Britain's imperial mission. But in his short story \"The Bridge Builders\"\u2014based on his experience watching the Kaisar-i-Hind bridge being built across the Sutlej River, a tributary of the Indus in the northwest\u2014what emerges most strongly is a sense of fragility before nature, and the deep dependence of British engineers on local expertise. The driving force in the story is the character of Peroo, a _lascar_ from Kachch, \"familiar with every port between Rockhampton and London.\" His mastery of the sailing ship had found a new outlet:\n\nThere was no one like Peroo, serang, to lash, and guy, and hold, to control the donkey-engines, to hoist a fallen locomotive craftily out of the borrow-pit into which it had tumbled; to strip, and dive, if need be, to see how the concrete blocks round the piers stood the scouring of Mother Gunga, or to adventure upstream on a monsoon night and report on the state of embankment-facings.\n\nAs a storm threatens the bridge he has designed, chief engineer Findlayson falls into an opium-induced hallucination, haunted by the question of whether his bridge would survive the onslaught of the water: he asks himself, \"What man knew Mother Gunga's arithmetic?\"\n\nIn the story the bridge survives. But the fears were well founded. India's ecology of water threatened not only the stability of bridges but the lives of the thousands of men who built them. Railway workers faced punishing conditions; infectious diseases were a constant threat. New infrastructure diverted watercourses, altered drainage channels, modified the water cycle; new risks were not far behind, posed above all by malaria. However far the power of steam had advanced, the monsoon rivers retained the capacity to surprise. In 1868, Sibganj, a \"great grain market\" along the Ganges, was destroyed: \"A northward movement of the river in 1868 swept away the bank on which the market stood.\" Traders moved on; they set up at Karik, six miles to the northeast.\n\n# V\n\nIf India's empire builders needed a reminder of their fragility, it came from the ferocity of the monsoon climate.\n\nIn October 1864, a \"cyclone of unparalleled fury\" struck Calcutta and the coastal districts of Bengal. The \"rivers raged and tossed like a sea\" and left the city \"in ruins.\" \"Far as the eye can see,\" a British correspondent wrote, \"there is unbroken waste and gloom.\" The cyclone originated in the Bay of Bengal, to the west of the northern Andaman Islands, on October 2. That morning, from the deck of the _Conflict,_ sailors observed that \"the stars had a sickly appearance.\" The sailors saw that the sun \"rose blood red.\" The cyclone had built up in the southwest, its effects felt in Ceylon and Port Blair a few days earlier; it gathered force as it approached the Andamans. From the Andamans the cyclone swept up the Bay of Bengal, traveling at ten miles an hour toward the mouth of the Hooghly River. As the cyclone approached the coast of Bengal, the steamer _Martaban_ was at anchor in the Saugor Roads. By the morning of October 5, the wind-whipped vessel drifted with its \"jibboom gone and likewise the fore-royal and top-gallant masts.\" By the afternoon the gusts had eased, the captain wrote, \"leaving us a total wreck.\" The crew realized they had been \"dragged 17 miles across the banks at low tide.\"\n\nAnother vessel, the _Ally,_ foundered. It had departed from Calcutta on October 4, carrying 335 migrants bound for Mauritius. They were indentured workers, hundreds among the hundreds of thousands of indentured laborers from India bound to labor on the sugar plantations that met the British Empire's taste for sweetness. The ship was overturned by a gale. Only twenty-two of the emigrants and seven of the ship's crew survived.\n\nHowever visibly Calcutta was affected, the storm was worse in rural Bengal. It swept through coastal districts and moved inland to the northeast, finally fizzling out over Assam on October 7. Few people survived to bear witness. The storm surge generated \"great sea waves... which, on reaching shallow waters, were piled up to a height greatly exceeding that of the highest spring tides, when they broke over the low lying lands at the mouths of the Hooghly and Godavery.\" A lighthouse keeper wrote to Calcutta in despair: \"I cannot accurately state what the loss of life has been by the Cyclone and inundation, but I am afraid the fatal malady has carried off more.\" Disease killed many more than the initial flood. \"Every tank, pond and well,\" he wrote, \"is stagnant with decaying matter.\" More than fifty thousand people died. Flooding drove millions from their homes.\n\nThis account of the storm, penned in 1867, comes from Henry Francis Blanford. He was born in London in 1834. His father, William Blanford, owned a workshop manufacturing gilt moldings\u2014one of innumerable small manufacturers propelling Britain's industrialization. In 1851, Henry joined London's Royal School of Mines, and from there traveled to the Bergakademie in Freiberg to continue his study of mining. Along with his brother William Thomas Blanford, Henry joined the Geological Survey of India in 1855. Their first assignment was to explore the Talchir coalfield in Orissa, in eastern India\u2014vital in this new era when India's hunger for coal was spurred by the railways. The Blanfords' inquiries in Orissa established some of the key groundwork for the later discovery of Gondwana, the supercontinent that fused the southern hemisphere's landmasses together with the Indian subcontinent and the Arabian Peninsula. Gondwana broke up into eastern and western segments\u2014separating Africa, South America, and Australia\u2014while the Indian subcontinent's northward drift and collision with the Eurasian continent, around 50 million years ago, created the Himalayas.\n\nThe following year, 1856, Henry took over as curator of the new Museum of Geology in Calcutta, and supervised the official Geological Survey of India, charged with exploring India's geology with a view to exploiting its mineral resources. He spent the rest of the 1850s in southern India, investigating the stratigraphy and paleontology of rock formations between Tiruchirapalli and Pondicherry. After a sojourn back in Europe to recover from ill health brought on \"by the exposure incidental to geological surveying in India,\" Blanford returned in 1862 to teach physics and chemistry at Presidency College, Calcutta.\n\nAround that time, Blanford became involved with the Asiatic Society. Founded by famed Orientalist and linguist Sir William Jones in 1784, for \"enquiring into the History, Civil and Natural, the Antiquities, Arts, Sciences and Literature of Asia,\" the Asiatic Society of Calcutta emerged as the most influential among a network of learned societies in the colonial world. Its journal was a storehouse of cultural, linguistic, and scientific research. The inclusion of meteorology among its subjects was due in large part to the work of a retired ship's captain and president of the Marine Courts of Calcutta, Henry Piddington (1797\u20131858). Inspired by the work of Colonel Henry Reid\u2014pioneer of American meteorology and author of _An Attempt to Develop the Law of Storms_ \u2014Piddington's interest in the characteristic storms of the Indian Ocean was deeply practical. His aim is clear from the title of his 1848 treatise, _The Sailor's Horn-Book for the Law of Storms,_ which he dedicated to \"mariners of all classes in all parts of the world.\" In his catalog of different types of storms, Piddington proposed a new word, \"cyclone,\" to describe those driven by \"circular or highly curved winds.\" He derived his term \"from the Greek _kukloma_ (which signifies amongst other things the coil of a snake).\" The new science of cyclones demanded attention from sailors, he wrote, \"for it is... a question of life and death, of safety or ruin.\" He described a \"storm wave,\" of the kind that struck Bengal in 1864, as a \"mass of water... driven bodily along with the storm or before it\"; crashing upon bays and estuaries, they caused \"dreadful inundations.\" Piddington published in the Asiatic Society's journal a series of ships' logs, from which he derived his work on the forces driving the Bay of Bengal's cyclones. Blanford began lecturing at Presidency College, Calcutta, a few years after Piddington's death; he studied Piddington's writings and developed an interest in the science of storms. Given his prominence in Calcutta's world of science, given his expressed interest in the weather, Blanford was an obvious candidate to lead the society's inquiry into the great cyclone of 1864. James Gastrell, Blanford's collaborator in compiling the report on the great cyclone, was the Asiatic Society's treasurer, and he also worked as the deputy surveyor-general for the government of India. What began as a report for the Asiatic Society became an official inquiry.\n\nTo reconstruct the path of the storm, Gastrell and Blanford pored over the logbooks of ten ships dispersed across the Bay of Bengal through the storm. The storm's chroniclers tallied the ships' barometric pressure readings with sailors' descriptions of sky and sea; they plotted these against the ships' likely positions to track the storm's path. The record of the _Moneka_ gives a sense of the terse precision of the ship's log, heavy with foreboding:\n\nFrom midnight to noon, light and variable winds from north-west to west, with cloudy weather; sea more composed, but south south-west swell as lively as ever. No rain this day. Barometer 29.74. Thermometer 82\u02da. From noon until midnight, light and unsteady winds from west by north, with cloudy weather; sky looking very black and lowering to the north and north north-east, with a high rolling sea from the same quarter. Sea rose very quickly; observed lightning in the north north-west. Barometer inclined to fall. Midnight, gently increasing wind from west, with gloomy appearances to north north-east. Sea still very heavy from that quarter. Ship pitching, bows under.\n\nGastrell and Blanford traced the cyclical motion of the winds by comparing the logs of the _Conflict_ and the _Golden Horn_. The two vessels began the afternoon of the storm one hundred miles apart; by midnight, they were at most twenty or thirty miles apart, having been blown toward each other by the rotating cyclone. Sailors' most common response to the power of the storm was awe. The power of steam was no match for the cyclone. The _Alexandra_ was a steam tug, at the mouth of the Hooghly River when the storm hit; traveling into a headwind, its engines \"were set going with seven revolutions, at full power,\" making no progress. The \"frightful roar of the hurricane\" drowned out even the din of the steam engine. The winds overcame the ship: a \"moaning sound,\" a \"sudden blast from the northwest,\" and suddenly the ship was lying on its side.\n\nVital though ships' readings were, their barometers were not always standardized, and their readings were difficult to compare with one another. The ships' records had to be read against measurements from land observatories. Gastrell and Blanford had access to the records of sixteen land stations, from Agra and Benares on the Gangetic plain to Kandy in the mountains of Ceylon and Port Blair on the Andaman Islands; farthest south was the station in Singapore. They were few in number and widely spaced. Their records had to be supplemented by the private journals kept by individuals such as one Mr. Barnes of Kandy. He noted on October 1, 1864: \"raw and very damp, low scud _nimbi_ covering the greater part of the sky, (dense _cirro-cumuli_ beyond), and moving generally from west south-west, the wind veering from west to south and back.\"\n\nTo follow the storm after it made landfall, the investigators relied on eyewitnesses: lighthouse keepers, railway stationmasters, European missionaries and district officials, captains of riverboats, government engineers. Gastrell and Blanford were tireless in their work of archiving the storm. Accounts of the devastation were heartbreaking. The storm's detectives searched these reports for telling details; they were intrigued by accounts that described the changing color of the sky, the shifting quality of the light, as the cyclone advanced. It was the best record they had of the formation and movement of clouds. Observers saw clouds of \"dark lead\" and indigo; they saw blackened afternoon skies lit up with \"balls of fire,\" and nocturnal landscapes glowing with eerie light.\n\nStorm investigation was a form of narrative, and nature was the protagonist. Gastrell and Blanford depicted a battle of forces. A few days before the storm's arrival on the coast of Bengal, \"the northerly current retreated before its stronger opponent, now forcing its way up the east of the Bay.\" That current was in turn \"opposed\" by the Yamadoung Mountains of Arakan on Burma's western coast\u2014an obstacle the winds circumvented as they \"curved round\" them. The story of the storm took form akin to a travel account: it had a point of origin, an itinerary, and a destination\u2014and it left a trail of destruction in its wake. The investigators depicted the storm's \"tracks\" using wood-block print, in two dimensions on a map. But their picture of the storm\u2014bringing together pressure readings, wind speeds, latitude, and longitude\u2014was fully three-dimensional. They were as interested in its vertical as its horizontal dimensions: the dance of rotating winds, the churning ocean currents, the contours of landscape.\n\nGastrell and Blanford were limited to retrospective reconstruction. What they hoped for was instantaneous information. Their goal was to track future storms as they unfolded, through a network of monitoring stations linked by telegraph. They emphasized the \"great importance\" of meteorological telegraph stations \"along both coasts\" of India; ideally, they wanted an archipelago of observatories encompassing Ceylon and the Burma coast, \"if possible as far down as Port Blair on the east side of the Bay.\" From the very detail of their inquiry into the storm of 1864, they sought an expanded sense of correlation and consequence across space and time, using \"indications from distant stations.\" What were the telltale signs, in the Andamans or in Ceylon, that presaged trouble in Bengal a few days later? How could some form of warning be delivered in time? Their view of the world was one that linked land, sea, and atmosphere. The Indian Ocean, on this view, was a weather factory: the source of India's climate.\n\nWITH THE ADVANCEMENT OF STORM SCIENCE CAME A DIFFERENT way of thinking about space\u2014and about India's place in the world. A new understanding of the monsoon emerged from the fusion of maritime and terrestrial observation. In the same decades, British explorers and scientists began to study the Himalayas. From there, botanist Joseph Hooker observed the monsoon from the other side\u2014from the mountain peaks. In Sikkim, Hooker witnessed a watery realm reaching from the ocean to the atmosphere; the sky was a mirror to the sea. \"The ocean-like appearance of this southern view,\" he wrote, \"is even more conspicuous in the heavens than on land, the clouds arranging themselves after a singularly sea-scape fashion.\" \"Upon what a gigantic scale does nature here operate,\" Hooker exclaimed. He described a climatic system where \"vapours raised from an ocean whose nearest route is 600 kilometres distant are safely transported without the loss of one drop of water to support the rank luxuriance of this far distant region.\" And then, \"the waste waters are returned by the rivers to the oceans, and again exhaled, exported, recollected and returned.\"\n\nThe enduring power of the monsoon to cause distress and upset political calculation, within and far beyond India, would become amply clear in the 1870s.\n\n# THREE\n\n# THIS PARCHED LAND\n\nBETWEEN 1876 AND 1879 THE DECCAN PLATEAU IN THE SOUTH and parts of northwestern India suffered famine as intense as any ever recorded. Twenty years later, in 1896 and 1897, drought ravaged millions of lives again, this time across a large expanse of central India. Before they had recovered, another serious famine struck those same regions in 1899 and 1900. Crops withered. Cattle perished. Tanks ran dry. Employment vanished. Those with the least power in society\u2014the landless, the aged and infirm, women and children\u2014were the first to find that they could earn no money with which to purchase the food that made it to market, its price swollen by scarcity and rumor. People moved to the cities, where some of them survived on private charity; hundreds of thousands moved to British famine camps, where they received meager rations and a cash wage for strenuous labor building roads, digging ditches, breaking stones. Rarely in the voluminous reportage on the famines do we read the actual name of a person who died. They succumbed to starvation; weakened by hunger, they fell to cholera, to plague, to the catchall \"fevers\" that medical officers inscribed as their \"cause of death.\"\n\n\"The rains failed.\" The phrase recurs in almost every account of these catastrophes, always intransitive. What this meant is that the rains failed to behave as they were expected to\u2014they failed to fall when, as much, or where they usually did. The rains failed to obey the patterns upon which human societies had organized their material lives. The suffering of those years reached as far as China, Java, Egypt, and Brazil's northeast. In China five northern provinces\u2014Shandong, Zhili, Shanxi, Henan, and Shaanxi\u2014bore the brunt of the suffering. Between 9.5 and 13 million people died in China, most of them from diseases that spread hand-in-glove with starvation. We know now that failures of rainfall in the late nineteenth century were caused by El Ni\u00f1o Southern Oscillation (ENSO) events of exceptional intensity. El Ni\u00f1o is a quasiperiodic rise in sea surface temperatures in the equatorial Pacific, with effects on global atmospheric circulation. Local fishers off the coast of Peru had identified the phenomenon as early as the seventeenth century, and had called it El Ni\u00f1o (Christ child), since it tended to appear near Christmas time. Those who tried to make sense of it in the nineteenth century had only a growing sense that the droughts were global in reach\u2014somehow connected.\n\nMore searching questions followed. The rains failed\u2014did individuals, societies, and governments fail, too? In an era when technology promised to collapse both time and space\u2014as so many enthusiastic observers of rail and steam foresaw\u2014need drought always turn to starvation? In an era when the British Empire proclaimed its superiority and its benevolence, did the colonial authorities act with foresight and justice? Could the famines have been prevented? These questions animated supporters and critics of colonial policy, economists, and meteorologists; they haunted administrators who carried the guilt for starvation in their districts. Drought and famine sparked many discussions about the future of water, for it was water's absence that had spelled disaster. Water unleashed new claims upon states by journalists and humanitarians and engineers; water unleashed new claims by states upon their subjects. Here, in the realm of ideas\u2014in competing visions of the future, in the articulation of fears about nature and hopes of betterment\u2014lay the lasting legacies of the nineteenth century's nightmares. We live with them still.\n\n# I\n\nThe first portent of trouble in southwestern India came with the drought that set in over the princely state of Mysore in 1875. The following year, rainfall during the southwest monsoon was much lower than usual over the whole Deccan plateau. By October, as local supplies were exhausted, the first murmurings of famine began to be heard in districts across Madras and Bombay. In the western Indian countryside of Bombay Presidency, the alarm was raised early on by workers of the Poona Sarvajanik Sabha. Founded in 1870, the association was, in the words of its constitution, a \"mediating body\" between the state and \"the people.\" The Sabha was a bold experiment in representative politics\u2014each member had to produce a _mukhtiarnama_ (a power of attorney) signed by at least fifty people, authorizing him to speak on their behalf. Dominated by the landed and the wealthy, exclusively male in membership, the Sabha flourished under the leadership of Mahadev Govind Ranade, a judge and social reformer who carried a reputation as an orator when he moved from Bombay to Poona in 1871. The Sabha pioneered a new tradition of social investigation in India. The British kept close watch, sensing that the Sabha, in the words of one official, \"threatens to grow into an _imperium in imperio,_ \" noting that \"popular representation is a sharp weapon, and a very perilous one to play with.\" The Sabha's growth coincided with the worst famine to hit the region in living memory\u2014from the start, the Sabha played an important role in drawing attention to the suffering.\n\nIn a series of letters to the government of Bombay, written in the last few months of 1876, the Sabha gave an account of the famine's spread: one of the letters insisted that it contained \"details accessible only to those who, like the agents and correspondents deputed by the Sabha, live among, and form part of, the people overtaken by this calamity.\" The Sabha's workers mirrored the \"tours\" undertaken by British officials through their districts, but they presented a view closer to the ground. The dispatches are brief; village by village, they chart a looming catastrophe. One of the entries reads:\n\nPangaum, Oct 11\u2014No rain except the first showers.... Drinking water scarce, as in the hot season. The tank will last for 3 months. No new water in the wells.... Neighbouring villages in a worse condition. The only relief work is to be commenced at Mohol. Great distress is expected of the respectable and poor people. Grains should be imported and sold gratis.\n\nThe northeast monsoon brought no respite that winter, extending the drought to southeastern India. One contemporary observer noted that prices \"sprang at a bound\" to levels previously unknown. In a brutal reversal of what had been thought would be the impact of the railways, newly laid rail lines raised prices as grain was \"hurriedly withdrawn by rail and sea from the more remote districts,\" channeled to urban markets where speculators fed on fears of shortages to reap higher prices. In the summer of 1877, the monsoon was slow to begin and then patchy, followed by a deluge late in the season which destroyed many of the limited crops that had managed to take root. Water's absence, followed by a short burst of excess, brought cultivators to ruin across Madras Presidency, their reserves depleted by the previous year's crop failures. That summer the drought spread north; central India and parts of the northwest saw their lowest rainfall ever on record. They had few reserves to fall back on because so much grain had been exported from India, drawn by high prices on the London market. What began as a localized drought in Mysore became a catastrophic famine. The winter rains of 1877 brought some relief, but only in the summer of 1878 did \"normal\" rainfall produce a good harvest. The drought coincided with the most severe El Ni\u00f1o event in 150 years; its effect was global.\n\nEverywhere drought sparked a rapid rise in food prices accompanied by an abrupt loss of agricultural employment. Landless laborers in the worst-affected districts were the first to feel the effects. What followed was a collapse in the livelihoods and incomes of the most vulnerable sections of the population. Many people undertook long journeys in search of relief. By the end of 1876, starvation began to kill those who were most vulnerable. Illness thrived where immunity had been weakened by widespread starvation. Cholera and dysentery accompanied the movement of people, and social disruption contributed to their spread. The return of the rains in 1877, and then more fully in 1878, saw another spike in death: most likely as a result of malaria, which thrived in the sudden change in the ecology of water after a long dry spell.\n\nIN THE MIDST OF INDIA'S DISASTER, IN MAY 1877, RICHARD STRACHEY presented a lecture in London on the \"Physical Causes of Indian Famines.\" Strachey was from an aristocratic family deeply connected with empire. From the 1840s, as a member of the Bengal Engineers, he took an interest in irrigation. Meteorology was among his most enduring interests, and from 1867 to 1871 he served as director-general of irrigation. He would go on to lead the government of India's Famine Enquiry Commission of 1880. He began his lecture by depicting a constant struggle between the forces of life and death: \"Among the most active of forces are the conditions of local climate, and notably those of atmospheric heat and moisture.\" Many contemporary observers saw climate as an active force in the world. In its late-nineteenth-century English usage, \"agency\"\u2014from the medieval Latin _agentia_ \u2014was used as often to refer to natural as to human actions. India was overcome by the \"agency of drought,\" preyed on by \"agencies of destruction.\" These \"devastating forces of tempest, drought, flood and disease\" rendered human life fragile in their wake.\n\nAs the drought gripped the Deccan region, administrators and journalists and missionaries followed its path. Drought appears in their accounts as an unwelcome visitor, leaving telltale marks upon the land: \"The whole country is bare and brown,\" one letter described; \"tanks, which at this season of the year ought to be wide sheets of water, are now nothing but vast expanses of dry mud.\" Drought left its fingerprint on market prices. The most widely traveled famine observer was Sir Richard Temple, sent to tour the affected districts by the imperial government in Calcutta. He had one overriding goal\u2014to spend as little money on famine relief as possible. Critics charged that he barely dismounted from his carriage: he swept through the land and saw only what confirmed his prejudices. But his own pen left us a vivid description of monsoon failure. Temple recounts the progress of drought as if it were on a journey. \"On the right or southern bank of the Toongabhadra river,\" he reported, \"the drought developed all its most destructive agencies, and showed its greatest force all along the frontier.\" Drought \"visited\" the city of Madras, and then \"rested for some time on the districts of South Arcot, Tanjore, and Trichinopoly, and threatened them with evil.\" It had the force of a marauding army as it \"extended itself with havoc throughout the southern peninsula, laying waste the districts of Madura and Tinnevelly, right down to the sea-shore near Cape Comorin.\" Worse was to come. By the middle of 1877, \"all hope of the south-west monsoon was given up,\" the Madras government wrote to the British Indian capital in Calcutta. The government of southern India pleaded for resources. Endowed with a cruel propensity to tease, the clouds tantalized only to disappoint: \"Very heavy showers would fall with a dash from blackened skies over a small area, whilst all around the skies continued as iron.\"\n\nDrought was an \"agency\" of famine; its prime characteristics were violence and caprice. In many eyes, a Christian God brought or withheld rain: \"I now see but little chance\" of rain, wrote Mr. Price, collector of Cuddapah district, \"except by a special dispensation of Providence.\" Some invoked Hindu cosmology: the rains rested in the power of \"Indra and Vayu, the Watery Atmosphere and the Wind,\" who were \"still the prime dispensers of weal or woe to the Indian races.\" Others turned to the language of science to describe the physical drivers of climate: \"The true cause of all movements of the atmosphere which we describe as wind is wholly mechanical, being difference of pressure at neighboring places,\" Strachey declared. Mechanical or divine, to see climate as an active force was also to attribute a certain inevitability to drought. Famine appeared to be a natural characteristic of India, no less than its landscape. At best, governments and communities could adapt to the certainty of periodic monsoon failure. In the shadows of this resignation lay a British reluctance to acknowledge that now, to an extent unimaginable fifty years earlier, the means were at hand to mitigate the impact of famine quite drastically. That, after all, was the boosters' claim for the railways. But faced with the scale of the crisis, faced with demands to spend more money, it was easier to insist that famine was, and ever would be, India's climatic fate.\n\nIF THE DROUGHT WAS A FORCE OF NATURE, IT ALSO PRESENTED AN all-too-human crisis. Prolonged and exceptional in its severity, it threw into sharp relief the fractures of society. It exposed the fragile infrastructures of economic life. It pressed upon the limitations of the physical infrastructure\u2014the word only came into widespread use in English in the early twentieth century, from the French\u2014that had impressed so many observers of India's landscape. In the late nineteenth century, human dependence on water\u2014rains and rivers, wells and streams\u2014began to be posed as a moral and political challenge.\n\nIn India, as in many parts of the world, the weather reflected moral concerns. Throughout Christian Europe, extreme weather and geological events\u2014floods, droughts, eruptions, earthquakes\u2014were seen as manifestations of divine judgment. China had a deep tradition of \"moral meteorology,\" as historian Mark Elvin describes: \"Rainfall and sunshine were thought to be seasonal or unseasonal, appropriate or excessive, according to whether human behavior was moral or immoral\"; the conduct of the Emperor mattered most of all. \"The people of the empire bring floods, droughts, and famines on themselves,\" the Yongzheng emperor declared in a decree of 1731. Because extremes of climate were unevenly distributed, a moral geography of rain and drought could be discerned\u2014areas that suffered most, on this view, were those where standards of public behavior or administration had slipped. North America, too, had its version of \"moral meteorology.\" In the 1870s and 1880s, ideas about rainfall and virtue underpinned conflicting views of how land should be allocated in the Great Plains of the United States. The idea arose that \"rain follows the plow\": industrious white settlers would transform the land, and their labor would in turn bring rain. Settlers, historian Richard White writes, saw themselves as \"the agents of climate change.\" Aridity was a form of cultural or spiritual malaise.\n\nA form of \"moral meteorology\" pervaded the writings of missionaries in India who observed the great drought\u2014but it acquired a radical edge, as they suggested it was a judgment not on the morality of Indian society but rather on that of the British government. Florence Nightingale wrote from India in 1878 that \"the land of India is not especially subject to famine\"; she insisted that \"the cultivators of the soil are industrious; the native races compare favorably with other races in capacity to take care of themselves.\" The true nature of the problem was simple: \"We,\" the British, \"do not care for the people of India.\" \"Famine in India is no invincible foe,\" another observer wrote\u2014\"in a climate whose great danger is drought, and where Nature therefore teaches the necessity of precaution, we reap only a legitimate punishment when we suffer the penalties of imprudent neglect.\" Here was a reversal of the language of Malthus: the \"precaution\" was lacking, the \"imprudence\" manifest on the part of the British rulers of India and not its people. An American writer in the _New York Times_ went so far as to describe a \"state of society in India whose only parallel in recent times was to be found in American slavery.\" In a bold Tamil work depicting the great famine in verse, Villiyappa Pillai, court poet of the small kingdom of Sivagangai, turned the conventional view on its head. His bitingly satirical poem, published at the end of the nineteenth century, depicts the lord Sundarweswara (Siva) confessing to the starving people of the area that he was helpless in the face of their suffering\u2014he directs them instead just to write a letter to the local _zamindar_. The \"agencies of destruction\" were not climatic\u2014they were human.\n\nWho personified, or what embodied, the transformation sweeping India and leaving millions vulnerable to a failure of the rains? A clear culprit emerged from critical accounts of the famines of the 1870s: \"the new class of capitalists.\" They were described by W. G. Pedder, an official who had worked in Gujarat and Bombay, as \"men possessed by no ennobling ideas of public duty, cowards by caste and confession, citizens in no sense beyond that of benefiting society by selfish accumulation.\" Pedder insisted: \"The dearth was one of money and labour rather than of food; the cultivators were without the resources their own fields should have furnished, the labourers could not obtain work and wages.\" The core of Indian cultivators' vulnerability, he observed, was their \"constant relations with a mercantile class\" who combined \"the functions of general shopkeepers, dealers in agricultural produce, bankers, and money-lenders,\" known as _banias,_ or _saukars_. Already indebted to the local moneylender, cultivators found themselves trapped without reserves if the harvest should fail. Pedder juxtaposed the wily \"unscrupulousness\" of the lenders with the \"ignorance and timidity of the peasants.\"\n\nA lack of capital was the root cause of cultivators' vulnerability. The railway commissioner Lushington had seen this as early as the 1850s, when he cautioned against undue optimism about the railways. Andrew Wedderburn, the collector of Coimbatore district in Madras\u2014a humane and direct voice railing against official inaction during the famine\u2014saw that \"some villages have sold their brass vessels, their ornaments (even including their wives' 'talis'), their field implements, the thatch of the roofs, the frames of their doors and windows.\" In Bombay, G. L. Hynes, the master of the mint, saw something similar: \"Silver ornaments and melted country silver discs are pouring in,\" he wrote, to the tune of 9 lakhs (900,000) of rupees each month.\n\nThe leaders of the Poona Sarvajanik Sabha also had the malign influence of the moneylender firmly in their sights. In the first issue of their journal, in 1878, they published their analysis of the famine. They argued that India's cultivators \"are found to be too poor, too hopeless of retaining their independence, too inextricably involved in debt to be able to undertake agricultural improvements.\" The famine had served to \"throw the Ryots [cultivators] more and more into the hands of the Sowcars [moneylenders], and leave them little ground to hope a change for the better.\" Families fortunate enough to have sufficient resources to survive three seasons of drought now found \"the savings of years utterly exhausted.\" They, too, had no choice but to turn to the moneylenders\u2014and, at once, \"from free men they have been degraded into slaves.\"\n\nWHILE HUMANITARIANS, SOCIAL REFORMERS, AND EVEN SOME COLONIAL officials attributed India's vulnerability to famine to the grip of social and economic inequality, other observers drew a direct connection between nature's \"agencies of destruction\" and human actions. In the eighteenth century, a group of European naturalists known as \"desiccationists,\" many of whom had traveled and worked in tropical lands, argued that cutting down trees caused drought. Deserts, on their view, were but ruined forests. This view gained prominence through the work of Alexander von Humboldt, who wrote in 1819 that \"by the felling of trees that cover the tops and the sides of mountains, men in every climate prepare at once two calamities for future generations; the want of fuel and a scarcity of water.\" Desiccationist views were common in British India by the middle of the nineteenth century; very often they were used to condemn local pastoralism and to restrict the use of forests by India's tribal peoples, known today as _adivasis_. Justified by arguments for conservation, the colonial state encroached upon India's forests, claiming more and more forest land, as well as uncultivated \"wastelands\" for itself; this limited the use rights of local people in a punitive way.\n\nThe strongest iteration of the desiccationist view was penned by an anonymous correspondent to _Macmillan's Magazine_ in 1877. He called himself \"Philindus\"\u2014an echo, probably deliberate, of the liberal publication _Friend of India_. His dispatch reached a wide audience, discussed not only in England but as far away as Japan. Philindus begins his piece by recalling an episode at \"the most sacred sanctuary of South Indian vapidity: I mean the bar of the Madras Club,\" where he heard two men, \"Jones and Brown of the Civil Service,\" denigrating Arthur Cotton's achievements as driven by vanity and wasteful expenditure. After a strong defense of Cotton's genius, echoing Cotton's belief that water was the key to India's security and prosperity, Philindus set out his main claim: the \"disastrous action of drought in Southern India\" owed much to the \"enormous extent\" to which \"the jungles of the Carnatic, and of the Peninsula generally, have been cut down during the past century.\" He invoked the authority of the American geographer George Perkins Marsh, whose influential 1865 book contained the first global account of environmental transformation by human hands. Marsh had seen that \"as forests are cut down, the springs which flow from them, and consequently the water-courses which are fed by these, diminish in number, continuity and volume.\" Philindus approved: he was quite certain that \"the most crying of the evils\" afflicting South India was \"the increasing desiccation of the country from the reckless destruction of its trees and forests.\" In the first of their \"Famine Narratives,\" published for India's growing reading public, the Sabha's reformers invoked a similar sense of drought being caused by human intervention: \"Owing to denudation of forests and the absorption of waste lands under the mischievous system of a wrongly conceived revenue settlement,\" they argued, \"the occasional rain that falls is never retained by the soil.\"\n\nAt the peak of the great famine, the Government of India's Forest Act of 1878 brought India's forests under public ownership. Was this the avant-garde of global conservation, or a land grab that dispossessed India's most marginal people? It was both. The attempt to limit the pace at which India's forests were being cut down began with a material concern about rapid depletion of a profitable resource, vital not least to India's railways. By the 1870s, the ecological argument pushed in the same direction, spurred by concerns about the long-term climatic harm that deforestation brought. By the same token, the Forest Act compounded the creeping attack on the rights of India's _adivasis_. Wrote Valentine Ball, an Irish geologist and anthropologist who had spent many years in central India: \"The reservation of forest tracts which prohibits the inhabitants from taking a blade of grass from within the boundaries\" had the effect of leaving people \"cut off from... food sources throughout wide areas.\" Forests were a refuge particularly in lean times: a source of tubers, fruits, and other foodstuffs that staved off starvation, even if they were no defense against hunger. The encroachment of the state upon India's forests threatened the lifestyles as well as the livelihoods of _adivasi_ communities.\n\n# II\n\nThe famine provoked a searching look at Indian society; most of all, it turned the spotlight on the state. The British government's inaction, its seeming indifference to Indian lives, catalyzed criticism from within and from outside.\n\nThe most immediate charge was that the government of India had neglected the most basic precautions in advance of the drought, driven by a damaging parsimony. It had neglected southern India's web of tanks; by the 1870s, many of them lay in ruins. One British observer lamented that \"the former rulers of India, if not so great or so powerful, yet had more of that simple craft and homely benevolence which show themselves in storing the rain and diverting the torrent to the first necessities of man.\" In these humble tasks, the British colonial state failed. Their failure was evident when the rains at last came, late in the summer of 1877\u2014so much rain that \"the famine year of 1877 will appear in the Madras meteorological records as the year of the heaviest rainfall in a long period.\" The maintenance of tanks had fallen victim to the state's drive to cut spending, but many saw that \"when economy is required, [tanks] are the very last thing that should be tampered with.\" The neglect of India's infrastructure of water allowed \"a flow towards the sea of a precious fluid which represents in passing away unused a sacrifice of human lives.\" Florence Nightingale observed that the Madras rains of 1877 \"were lost, because the tanks were left unfinished in the autumn of 1876; the order having been issued for the stoppage of all public works\" as a measure to cut government spending. The result: \"Millions of tons of precious water so ran to waste,\" and millions of people starved for want of water. From this point on, the \"waste\" of India's water came to be a battle cry for humanitarians, engineers, and critics of colonial policy.\n\nA greater indictment of colonial rule arose from what the famine revealed about the conditions of life in India\u2014it was as if the catastrophe lifted a veil covering the everyday workings of Indian society and economy. The reason that so many people in India were so vulnerable to the failure of the rains, many argued, was that British domination had impoverished them, eroded their defenses. \"A people must be poverty-stricken beforehand to be thus absolutely cut down by want of food,\" an American observer of the famine wrote. The idea that British misrule had undermined Indian economic independence was not novel; Adam Smith had been damning in his assessment in _The Wealth of Nations_. By the 1860s, the notion of a \"drain of wealth\" from India was most closely associated with the writings of Dadabhai Naoroji, a Parsi merchant and scholar who would go on to be elected the first Indian member of the British Parliament. Naoroji penned a devastating verdict on the economic effects of British rule in India. The famines of the 1870s sharpened his scalpel. The vast bulk of India's people were \"living from hand to mouth,\" he wrote, so much so that \"the very touch of famine carries away hundreds of thousands.\" Despite this, he argued, the Indian peasantry bore a \"crushing\" burden of land tax\u2014India essentially paid for its own colonization through the \"home charges\" remitted each year to Britain. \"Every single ounce of rice... taken from the 'scanty subsistence'\" of India's people, Naoroji charged, \"is to them so much starvation.\" Heedless of India's suffering, British rule, in contravention of its own stated principles, \"moves in a wrong, unnatural, and suicidal groove.\"\n\nNaoroji's analysis was all the more powerful for being couched in the language of loyalty to empire: his fury was measured, every sentence backed by the colonial government's own statistics. Like so many political economists at the time, Naoroji was fond of fluid metaphors\u2014he wrote of a \"drain\" of wealth from India, of \"flows\" in the wrong direction. India's closeness to the brink of disaster and the fragility of its people's subsistence arose from its acute dependence on water. In the view of so many critics of British policy in the 1870s, to mitigate that dependence\u2014to secure life against the \"very touch of famine\"\u2014ought to be the overriding concern of government.\n\nAS THE DISASTER UNFOLDED, THE MOST PRESSING CHARGE AGAINST the British government was that it had failed to provide adequate relief to starving people: it was too slow, too callous, too divided within itself, too concerned with economy, or simply too incompetent. The deliberate withholding of relief by the state, underpinned by an unrelenting faith in free markets and notwithstanding plentiful stocks of food in other regions of India\u2014this remains at the core of historians' later attempts to provide a moral reckoning of India's nineteenth-century famines.\n\nWilliam Digby (1849\u20131904) wrote the most detailed account of the Madras famine, in two exhaustive volumes that appeared in 1878. Digby was a journalist and campaigner. From a humble start at his local paper, the _Isle of Ely and Wisbech Advertiser,_ Digby edited the _Ceylon Observer_ and, from 1877, the _Madras Times_. Digby's account is a tragedy in slow motion: a story of warnings ignored, precautions abandoned, decency abrogated. A recurrent theme in his account of the famine is the conflict between local officials, many of them humane and observant, and an imperial government hidebound by ideology. As the first reports of starvation deaths began to filter through to the government in the last months of 1876, every village magistrate in North Arcot district received a warning from the collector that they would be \"held responsible for the safety of individuals whose deaths may have been occasioned by starvation.\" But responsibility was precisely what the imperial government wished to avoid. Late in 1876, as it seemed more likely that large-scale famine relief would be necessary, the Madras government tried to bolster its granary by purchasing food in secret, acting through Messrs Arbuthnot & Co. The Madras government was at pains to do so anonymously so as not to interfere with the market, but its actions drew the ire of the imperial government, which ordered an immediate halt to the practice: \"The supreme authorities objected to interference with trade,\" Digby observed laconically.\n\nFamines had recurred periodically in India through the nineteenth century. British responses were ad hoc: they depended on the proclivities of the officials in charge at the time; they drew on local precedents and memories of earlier dearth; they were shaped by the security or fragility of colonial control over the affected areas. The most recent crisis before the disasters of 1876\u20131878, the Bihar famine of 1873\u20131874, was anomalous: on that occasion British intervention in the face of food shortages was unusually energetic and effective. Rather than imposing controls on the grain trade, the government imported directly 480,000 tons of rice from Burma, which by that time was emerging as India's new rice frontier. With regular and plentiful rainfall in the Irrawaddy delta, Burma seemed immune from the fluctuations of climatic fortune that bedeviled Indian agriculture. More locally, the state purchased grain incognito, acting through agency houses to stockpile food without creating panic in the market. Local people were employed on relief works for a cash wage. By contrast with common British practice, eligibility tests for relief were neither exacting nor punitive: the administration trusted the knowledge of village leaders and local-level officials. Even by the standards of, say, the late twentieth century, British intervention in the Bihar famine was a success. The official in charge of relief was Richard Temple.\n\nFar from celebrating the policy's success in Bihar, public reaction in England was harsh. _The Economist_ condemned the large expenditure on famine relief, concerned that it had led Indians to believe that it was \"the duty of the Government to keep them alive.\" Worse was to follow. A tract titled _The Black Pamphlet of Calcutta,_ circulated in 1876, turned its fire on Temple in particular, describing his policies as \"an economic catastrophe, a culmination of unthrift and unreason.\" It was published anonymously but its author was soon revealed as Charles O'Donnell, an Irishman who served in the colonial government of India. In these critics' eyes, the famine was a fiction, because for all the money spent, hardly a single death from starvation was recorded; they refused to see that this may rather have confirmed the policy's wisdom.\n\nTemple was stung by humiliation. An ambitious man, he was determined to learn from the experience, and more determined to see that it did not hold back his advancement. When Temple was appointed the viceroy's envoy to the famine districts in 1877, there were murmurings that he might be too generous. He set out to prove them wrong. As Digby pointed out, \"Sir Richard was commissioned to the distressed districts to economise, and it was known... that he would exercise economy.\" Temple fought to trim the scale of relief that the Madras government was providing to its starving subjects\u2014in what became notorious as the \"Temple Wage,\" he cut wages on relief works to (or even below) the level of basic subsistence. A newspaper in Ceylon pointed out with dark humor that the scale of rations provided in the famine relief camps of Madras was significantly lower than those enjoyed by a prisoner, Juan Appu, \"recently convicted of knocking out the brains of a near relative.\" In a missive \"from the affected districts,\" a missionary correspondent took aim at Temple's \"extra-economical theory of managing a famine\"\u2014perhaps there is a double meaning here, suggesting both that Temple's policy was excessively stingy, and that it was \"extra-economical\" in the sense of being motivated by ideology more than by economy. The letter concluded: \"The duty of a great Government is not only to prevent its subjects from dying of starvation but to save life.\"\n\nOthers around the world drew different lessons from India's famine. In Shanghai, as news of the horror of North China's famine reached the port city's elite, a feature in _Shenbao_ \u2014one of China's first modern newspapers\u2014praised the British response to India's famine to underline the paper's charge that the Chinese state had failed to relieve its suffering subjects. The reference to India was mostly rhetorical, a way to call attention to the Chinese state's weakness. It was not based on a deep understanding of the Indian famine; rather, _Shenbao_ 's account of the Indian famine drew largely on the accounts of Shanghai's British press, which reflected the ideological orthodoxy in support of Temple's parsimony.\n\nBY THE TIME THE SUMMER MONSOON ARRIVED IN 1878, SWATHS OF India lay in ruins. At a most cautious estimate, 5 million people had died from starvation or from disease. When the British government appointed a commission of inquiry to investigate the catastrophe, it was all set to be a whitewash. The viceroy, Lord Lytton, sought vindication in the face of widespread criticism at home and abroad; the finance minister, John Strachey, pushed for the appointment of his brother, Richard, as chair of the committee. Despite its origins, despite a heavy dose of self-justification, the Famine Enquiry Commission's 1880 report was an \"intellectual and administrative masterpiece,\" in the words of one of the most astute observers of hunger and famine in late-twentieth-century India. Under Strachey's leadership, the commission included (as \"English\" member) the Irishman James Caird, Madras official H. E. Sullivan, and C. Rangacharlu and Mahadeo Wasadeo Barve, officials of the princely states of Mysore and Kohlapur. They toured the country. They accumulated hundreds of hours of testimony. One theme dominated their report\u2014water.\n\nThe famine commissioners' diagnosis of the root cause of India's famines was unequivocal: \"All Indian famines,\" they wrote, were \"caused by drought.\" The commissioners saw that India's task was to find means for the \"protection of the people of India from the effects of the uncertainty of the seasons.\" The seasonality and unpredictability of the monsoon was at the heart of the matter. The report's opening sketch of India's geography takes the form of a narrative map of water, an account, by now familiar, of the frontiers of wet and dry lands. Strachey's long-standing interest in meteorology was one reason why the committee's report went beyond justifying imperial policy: Strachey saw, here, an occasion for the elevation of meteorology to a position of greater prominence in India's future. To Strachey's dismay, there was vigorous debate within the committee. The final report was accompanied by a dissenting note by Caird and Sullivan. They disagreed with the majority view that the government of India was right not to intervene in the grain trade. Caird and Sullivan condemned the reliance on harsh \"tests\" to determine eligibility for relief; above all, they advocated for the establishment of public granaries in remote districts, where private trade was unlikely to reach. India's suffering, they saw, came from a \"want of timely preparation to meet a calamity, which though irregular in its interval, is periodical and inevitable.\"\n\nIf there was disagreement on how far the state should intervene in markets, on the \"inevitability\" of drought all agreed. Turning to the question of whether droughts may better be predicted, the famine commission dismissed an idea that was fashionable at the time\u2014the theory that sunspots, dark and cool patches on the sun's surface caused by magnetic flux, followed eleven-year cycles correlated with droughts across large parts of Earth. The theory was championed by figures including the economist W. Stanley Jevons, famed logician and proponent of the \"marginalist revolution\" in economics, and William Wilson Hunter, editor of the _Gazetteer of India_. The commissioners, however, concluded that sunspots \"cannot be said to be in any sufficient degree established, still less to be generally accepted\"\u2014they had been \"contested on various grounds, such as that the evidence is directedly opposed to them.\" Instead, the commissioners lauded the patient work of meteorological observation. \"As at present no power exists of foreseeing the atmospheric changes effective in producing the rain-fall, or of determining beforehand its probable amount in any season,\" the commissioners concluded, \"the necessity becomes greater for watching with close attention the daily progress of each season as it passes, for ascertaining with accuracy and promptitude the actual quantity of rain in all parts of the country.\" They observed that \"within the last few years a very satisfactory system of meteorological observations has been established all over British India\"\u2014there were more than one hundred rainfall observation stations across the country by 1880, and these tracked the progress and development of the monsoon across the country. The famine commissioners insisted that \"it is of primary importance\" that this infrastructure \"shall be maintained in complete efficiency.\"\n\nThe most important institutional innovation of the commission lay in the Famine Codes, which sought to break the link between drought and famine. They consisted of prescriptions for local government officials faced with scarcity. Their ultimate tool was the provision of large-scale public works in famine-affected areas to generate employment income, and to attract food supplies to the area by boosting purchasing power. The codes were, by definition, an emergency measure, but they also stimulated thinking about how to reduce India's dependence on the vagaries of the monsoon. \"Among the means that may be adopted for giving India direct protection from famine arising from drought,\" the famine commission concluded, \"the first place must unquestionably be assigned to works of irrigation.\" The famine commission's overriding conclusion was that India's water resources needed to be managed better.\n\n# III\n\nTwenty years after the suffering of the 1870s, India experienced two more major famines that marked what historian Ira Klein has aptly called a \"grim crescendo of death.\"\n\nThe drought of 1896 was felt first in the \"black soil\" region of Bundelkhand in central India, in the nineteenth century a region at the frontier of cotton production for export. By the end of the year, the summer rains had fallen short; the suffering spread across central India, reaching up toward Punjab, down to Madras, and east to upper Burma. The return of the rains in 1897 then unleashed a lethal epidemic of malaria. The famine coincided with an epidemic of bubonic plague that arrived in Bombay in 1896. The epidemic would persist for a decade, thriving on the large-scale migration sparked by famine, feeding on a population weakened by hunger, spreading along the railway lines to rural areas. In the eastern region of Chota Nagpur, creeping deforestation and colonial forestry laws had imperiled local _adivasi_ communities, which had no local source of subsistence left, even as they fell through the cracks of the minimal safety net provided by colonial relief works. Mass starvation ensued. Only three years later, the same regions of central India faced another failure of the summer monsoon. Rainfall in 1899 was the lowest ever recorded in India. The drought covered an expanse of a million square kilometers of territory\u2014central India was again worst hit\u2014affecting tens of millions of people. In Bombay, the famine of 1899 and 1900 was the worst of the nineteenth century.\n\nBy the 1890s, photographic technology had become cheaper and more portable than it had been two decades earlier. Haunting images of starving people in India circulated through missionary and humanitarian networks around the world. Fund-raising drives garnered millions of pounds in donations. Missionaries, writers, and photographers traveled to India. Among them was the American George Lambert, representing the Home and Foreign Relief Commission that drew its members from Ohio, Pennsylvania, Indiana, Michigan, Illinois, Nebraska, and Kansas. The famines of the 1890s helped to bring about a new global humanitarian sensibility on the part of middle-class publics in Britain, the United States, and Europe\u2014a sense of identification with the suffering of distant strangers. But the same imagery that brought forth donations often reinforced the idea of a helpless India at the mercy of the elements, diverting attention from the political and economic subordination to British interests that left such a large number of Indians vulnerable to the monsoon. The question that had raised itself in the 1870s remained: was famine a \"natural\" disaster, or a political one?\n\nThe backdrop to so many of the photographs, and to so many famine travelogues by European and American missionaries and journalists, was the sheer dryness of the land. In 1899, the _Times of India_ 's correspondent in the princely state of Kathiawar, in Gujarat, suggested that even a photograph might not be enough to convey the absence of water:\n\nWere I an artist of the impressionist school and did I wish to represent the scene, I should dash in yellowish grey, a long diminishing streak, which would be the road throwing up the heat that made the distance shimmering and indistinct; a great splash of reddy-brown on either side would indicate the land where the crops should be; and above all a liberal dash of blue from the horizon to the top of my canvas would be the sky. I do not think I ever hated blue before; but I do now.\n\nVaughan Nash (1861\u20131932), a British journalist and correspondent for the _Manchester Guardian,_ described \"tracts of dismal sun-cracked desert,\" and \"brown wilderness spreading to right and left\"\u2014there was \"no water in the wells, no water in the rivers,\" and the people he met at famine camps had their \"lips and throats too parched for speech,\" so much that \"the silence is unbroken.\" The famine camps offered a bare minimum. At a famine camp outside Poona, he wondered \"whether the people can subsist on this penal allowance without ripening for cholera and other famine diseases.\" \"There must be something wrong with India,\" he concluded, \"when one finds a collapse like this.\"\n\nThe \"collapse\" was not as severe as it had been in the 1870s. The Famine Codes had taken effect in 1883\u2014each province had its own, modeled on the template proposed by the 1880 famine commission. Once a district officially declared that it had crossed from \"scarcity\" into \"famine,\" the codes' machinery started up: public works were initiated to provide employment to boost local incomes, combined with relief for those not able to work. However, local governments worked under enduring, sometimes intolerable, pressure to economize; there was a clear incentive for district officials not to declare famines, and many waited until it was too late in 1897 and again in 1899. Nash pointed out that India's Famine Codes were \"excellent on paper,\" but in reality local governments were \"short of administrators, short of doctors, short of medical assistants, short of material.\" The infrastructure of relief was creaky, but it was extensive. At the peak of the famine of 1896\u20131897, 6.5 million people were receiving public relief. The monsoon failures of the 1890s were more severe than those of the 1870s\u20141899 saw the greatest shortfall\u2014but mortality was lower. Nevertheless, at the very least a million people died.\n\nThe more concerted response to famine in the 1890s cannot be ascribed to imperial benevolence, though there is no doubting the good intentions of many local officials. Rather the colonial state was newly aware of pressure from Indian civil society\u2014from journalists, lawyers, industrialists, and activists who came together in a growing number of associations: professional associations, caste associations, religious associations, reformist associations. They met in study clubs and book rooms, in university halls and public parks; they expressed their views in an expanding universe of print, in multiple Indian languages as well as in English. If the British were quick to see the malign influence of \"agitators\" behind every criticism of their rule, they could not avoid growing public scrutiny of their actions\u2014or their failure to act. The Poona Sarvajanik Sabha, which had played such an active role in documenting the 1876 famine, adopted a more confrontational approach under the leadership of Bal Gangadhar Tilak, who took over in 1890. When before they petitioned government, now the Sabha convened large public meetings, speaking directly to _kisans_ (farmers) about their rights to relief. The Sabha's local informants produced reports that contradicted official statistics and fueled criticism. Unable to trust government to intervene, Indian civic leaders took famine relief into their own hands, often working with charities overseas. The private charity of wealthy families had always played a vital role in providing food to the hungry. Arguably it mattered more, for most of Indian history, than government policy; very often it was religiously motivated. But indigenous charity organized itself on a larger scale in the 1890s, mirroring as well as challenging state infrastructures. The two most prominent Hindu reformist movements of the age\u2014the Arya Samaj and the Ramakrishna Mission\u2014undertook large-scale charity work for the first time during the famines of the 1890s. The famines of the late nineteenth century spurred the development of pan-Indian political anger and activism.\n\n# IV\n\nMore than a century after the great famines, the question of responsibility still haunts us. For Mike Davis, author of a path-breaking global history of the famines, the answer is present even in his title, _Late Victorian Holocausts_. \"Imperial policies towards starving 'subjects' were often the exact moral equivalent of bombs dropped from 18,000 feet,\" Davis writes; the millions who died during the late Victorian famines were \"murdered... by the theological application of the sacred principles of Smith, Bentham and Mill.\" Yes, the rains failed on a colossal scale; but what turned drought into disaster was imperial policy: in the long term, by undermining the resilience of rural communities in the process of dragging India into modern capitalism; in the short term, by denying relief to starving people because of an unflinching refusal to interfere with \"free\" markets. Far from alleviating famine, the railways encouraged speculation, sucked food out of regions where it was needed most, and hastened the spread of epidemics.\n\nWithout absolving colonial high officials of callousness, other writers paint a more ambivalent picture. They point to how life and livelihood in rural India had long been acutely dependent on rainfall; as long as India's infrastructure remained patchy, this would continue to be the case. \"Famines were frequent and devastating\" in colonial India, writes geographer Sanjoy Chakravorty, \"but were they more frequent or more devastating than famines in pre-colonial regimes? Very doubtful.\" Economists continue to believe that \"railroads dramatically mitigated the scope for famine in India\" and made Indian lives \"less risky,\" but they point out that it took until the early twentieth century for these effects to be widely felt.\n\nIn his global history of the nineteenth century, J\u00fcrgen Osterhammel draws a contrast between the Indian and Chinese famines of the 1870s. Osterhammel calls the Indian famine a \"crisis of modernization,\" which is to say, it was a crisis brought on by the uneven impact of global markets on the Indian countryside. The Chinese famine, by contrast, he called more a \"crisis of production than a crisis of distribution.\" The affected parts of North China were already under strain; they inhabited an \"ecologically precarious niche, where for centuries state intervention had been able to ward off the worst consequences of disastrous weather conditions.\" Still reeling from the massive rebellions that rocked China in the 1850s and 1860s\u2014the Taiping Rebellion the largest of them\u2014and under pressure from European encroachment, the Qing state was much less capable than it was in the past of responding to the crisis. Osterhammel is cautious in his assessment of the historical consequences of the great famines of the 1870s. They brought no real change, he argues. In China there was \"no really significant increase in political or social protest\"; in India, British rule \"held firm.\"\n\nBut beyond the immense suffering they caused, there is a sense in which the famines were profoundly consequential for the future. The catastrophes of the late nineteenth century left many people\u2014Indian economists and British administrators, water engineers and humanitarian reformers\u2014with an acute anxiety about climate and water. To borrow a phrase from an earlier work of Davis's\u2014a book about California and not India\u2014climate was at the heart of a new \"ecology of fear.\"\n\n# FOUR\n\n# THE AQUEOUS ATMOSPHERE\n\nIN OCTOBER 1876, JUST AS ALARM SPREAD ACROSS MADRAS OVER THE failure of the rains, India's eastern seaboard was struck by the worst cyclones ever recorded. Two storms followed in rapid succession: the first hit Vishakapatnam on the Orissa coast; the second inundated the Meghna delta in eastern Bengal. The loss of life was incalculable\u2014the greatest toll was taken by the storm surge that accompanied the cyclone as it hit eastern Bengal. The commissioner of Dacca division, surveying the devastation, wrote of one locality that \"not a single hut and hardly a post was left standing\"; it was, he said, \"too soon to attempt to compute with anything like accuracy the loss of life which has occurred.\" In district after district, local people estimated that 40 or 50 percent of the local inhabitants had died. In another village on his journey the commissioner listed the victims not by their names but by their positions: \"Moonsif, rural sub-registrar, native doctor, post-master, court sub-inspector, abkaree darogah, two abkaree burkundauzes, seven constables, a mohurir of the moonsif's court, and a post-office peon.\"\n\nJohn Eliot, meteorologist of Bengal, set out to archive the storm. He relied on the usual combination of ships' logs and eyewitness accounts, supplemented now by records from the many land-based observatories that had been established over the preceding decade. Eliot conveyed the ferocity of the storm as it built over the Bay of Bengal: \"This piled-up mass of water advanced under the pressure of the acting forces towards the head of the Bay\" at twenty miles an hour. He estimated that the storm contained latent energy from the evaporation of water over the Bay of Bengal \"equal to the continuous working power of 800,000 steam-engines of 1,000 horse-power.\" Eliot proceeded to narrate an epic battle of forces between the storm wave rushing in from the ocean, and the Himalayan rivers\u2014the combined power of the Ganges and the Brahmaputra\u2014seeking an outlet to the sea. \"These two vast and accumulating masses of water opposed each other over the shallows of the estuary,\" he wrote; their \"struggle and contention for mastery\" brought death and destruction to millions of people. Eventually, the \"larger and more powerful mass of water forming the storm-wave\" overcame the river waters. It deluged the islands at the mouth of the Meghna, one of three rivers that constitutes the Ganges delta; the islands were themselves \"formed chiefly from the _detritus_ of the Himalayas deposited over the area in which the tidal and river waters wage incessant warfare.\"\n\nThis was a vision of India's climate shaped by water in every dimension: the descent of water from the Himalayan rivers and the ascent of water vapor from the Bay of Bengal and the winds stirring the ocean surface and transporting clouds to shore. On what scale should the climate of India be understood? The problem, Eliot noted, was that \"so little is known of the action and independent motion of the aqueous vapour in the atmosphere, and of its relations to the atmosphere of dry air.\" Tracking alongside the story of the disastrous famines, this chapter charts the quest to understand the monsoon that unfolded in the last quarter of the nineteenth century. Famine spurred the development of Indian meteorology. As knowledge of the monsoons grew, so, too, did awareness that India's climate was shaped by distant influences. As India's boundaries hardened, it mattered more to understand the rivers that crossed them. New knowledge of water raised new questions about India's place in Asia\u2014and uncomfortable questions about how far science could conquer nature.\n\n# I\n\nMeteorology was an international science by the 1870s: the telegraph had allowed the world's weather to be tracked with unprecedented immediacy. In 1873, the United States and a number of European countries agreed to form the International Meteorological Organisation (IMO). Like many international associations at the time, the IMO was a voluntary initiative, founded on an aspiration for greater cooperation between national weather services in the sharing of information. Like many international associations at the time, its concerns were dominated by those of industrialized, imperial powers. Britain claimed predominance in international meteorology because it represented a vast empire of climatic variation.\n\nAmong the priorities of the new international meteorology was to devise a common and standardized language in which to describe the weather anywhere in the world. Especially daunting was the challenge of finding words to describe clouds in all of their variety, in their mutability and evanescence, in all of their profoundly local manifestations. The tools of Linnaean classification struggled to capture the texture of the skies. The basic cloud types that we still use\u2014the puffy white _cumulus;_ the gray blanket of _stratus;_ the wispy _cirrus;_ the dark rain cloud, _nimbus_ \u2014date from the early nineteenth century, in the parallel but independent work of Luke Howard in England and of the French statistician Jean-Baptiste Lamarck. The midcentury advent of photography was a fillip to cloud watchers, giving them the tool to capture clouds in a fleeting instant. Under the leadership of Swedish meteorologist Hugo Hildebrandsson, director of the Uppsala observatory, the IMO published its first international cloud atlas in 1892. The atlas aimed to standardize the cloud observations that professional and amateur meteorologists and cloud-watchers were compiling the world over. Hildebrandsson and colleagues illustrated their atlas primarily with photographs, each illustrating a typical instance of a particular cloud form, even if each individual cloud that had been photographed would have changed shape moments later. As historian of science Lorraine Daston observes, long after the effort to standardize observations, knowledge of clouds and weather remained a profoundly local affair.\n\nMeteorologists in the nineteenth century studied the genesis of storm systems in the Indian Ocean. CREDIT: Illustration by Matilde Grimaldi\n\nFormal classification could not always capture the nuance of clouds in different climates. In agrarian societies (and so across much of Asia) the mutability of clouds had an immediate bearing on people's fortunes and the sky was a series of signs to be read, or warnings to be heeded. Every Indian language contains a rich lexicon to describe clouds, capturing their relation to the seasons and to the landscape. In Tamil, _mazhaichaaral_ invokes the drizzle from clouds that gather atop hills; _aadi karu_ are the dark clouds that gather in the month of _aadi,_ promising a good harvest to come. Notwithstanding the promise of meteorological advances, British officials in India often turned to local, or what they called \"folk\" knowledge, when they wanted to understand how the weather shaped the harvest. Historian Shahid Amin found in the local archive of Gorakhpur district, in the northern Indian state of Uttar Pradesh, a handwritten account from 1870, penned by a British district officer, compiling local aphorisms about the seasons. The sayings take the form of instructions to farmers issued by each season, personified by its name in the Sanskrit calendar. The hot summer season of \"Jeth\" (in May and June) says: \"Be undaunted by the heat of the season. Make ready your threshing flood; work hard and gather the produce before the rains set in.\" The winter season of \"Magh\" warns farmers to \"leave your cane mill and drive the water full into your fields. If God be pleased to give you rain you will be truly blessed.\" Other regional traditions across India had their own stores of wisdom about the clouds and the rains. In the words of a Tamil proverb, \"If clouds withhold their gifts and grant no rain, the treasures fail across the ocean's wide domain\"\u2014local wisdom conveyed an awareness of the connectedness of the weather across large areas. Cultivators searched the skies for signs of foreboding. \"Oh farmer!\" another proverb pleads, \"get out of the field with the young seedlings in your hand, should you see the first crescent moon in [the month of] _Arpisi_.\" Right up to the 1930s, alongside the development of meteorology, local governments in British India collected and published proverbs about climate and weather and cultivation.\n\nBut the monsoon failures of the 1870s were so total, so devastating, that new answers were sought from the new science of climate.\n\n# II\n\nFaced with the total failure of the rains in 1876 and 1877, India's meteorologists sought an explanation. Leading the quest was Henry Blanford, the geologist-turned-meteorologist who had risen to prominence with his study of the Calcutta cyclone of 1864, and who was now director of the Indian Meteorological Office. In his regular report on India's climate and rainfall for the year 1876\u2014written with factual detachment in the midst of disaster\u2014Blanford ascribed the drought to the \"remarkable and unseasonable persistence of dry northwest winds\"\u2014winds he had studied a few years earlier. Blanford observed two abnormal forces at work: the first was exceptionally high pressure across northern and western India; the second was a sharper than normal temperature contrast between northwestern and eastern India. He concluded that \"some cooling influence more potent than usual was at work, probably in the Punjab and on the northern mountain zone.\" The following year, again, Blanford reported that \"the land winds have been so persistent in the upper provinces and on the plateau south of the Ganges, as to cause an almost complete failure of the summer rains in that region.\" As he sought to understand what had happened, what Blanford needed above all was data.\n\nBy the time of the famine commission report in 1880, India had more than one hundred meteorological observation centers. In the decade that followed, Madras, for instance, maintained eighteen observatories under the directorship of Elizabeth Isis Pogson. She was the daughter of Norman Pogson, an astronomer who was director of the Madras Observatory for decades. Isis was taken on in 1873 in the role of \"computer,\" earning the salary of a \"cook or a coachman.\" The family grew up in poverty, and Isis was forced to work, as well as looking after her siblings upon her mother's death. By the 1880s, as meteorology developed as a branch of science separate from astronomy, she was placed in charge of Madras's network of monitoring stations.\n\nPogson was zealous, inspecting regularly as many of the rain monitoring stations as she could. Her reports exposed the shaky edifice upon which India's weather data were based. Weather observatories tended to be built on hospital grounds, under the responsibility of the local medical officer; some were more enthusiastic than others about this addition to their duties. At Cochin, Pogson found that the local station needed better fencing, \"to prevent stray cattle straying into the shed\"; she personally arranged for supplies from Oakes and Company of Madras. She battled vandals as much as cattle. In Cuddapah, \"the grass minimum thermometer had only been in use for six days when it was... found broken outside the hospital compound, evidently done out of sheer mischief.\" From Kurnool, she had to report that the data were \"perfectly useless\" because of the positioning of the apparatus. There, the local postmaster had to double as the meteorological assistant, and he struggled with the job. \"He was very willing and anxious to learn,\" Pogson wrote, \"but... could not possibly undertake to record\" the data \"as his combined duties as Postal and Telegraph Master were too much.\"\n\nThe traces she has left in the archive are filtered through the technical language of meteorology and contained within columns and tables of official forms. As far as I know she left no personal papers\u2014we can only speculate about how Isis Pogson experienced being a rare woman within the scientific apparatus of British India. As a young science struggling for legitimacy meteorology was likely more open, a little freer from prevailing hierarchies, than more established fields. Meteorology was among the \"field sciences\" that, as historian Kapil Raj shows, were more open to local knowledge than the laboratory sciences. But the obstacles Pogson faced getting her due recognition as a pioneer of global meteorology are telling. In 1886, she was nominated for membership of the Royal Astronomical Society; she was turned down after the council decided that the use of the masculine pronoun throughout the Society's charter meant that women could not be admitted as fellows. She finally became a member only in 1920, after she had returned to England, leaving India and meteorology behind.\n\nThe Indian staff of the meteorological department, too, found a degree of openness they would not have encountered elsewhere in the bureaucracy of the Raj, though this was always weighed down by the knowledge that they could never rise beyond a subordinate position. Much was left in their hands by an institution that was young, understaffed, and underfunded. The most senior Indian meteorologist under Blanford, Lala Ruchi Ram Sahni, wrote a memoir in the 1930s, by which time he had risen to prominence as a patriot and a social reformer in Punjab; his recollections give us a rare insight into everyday life in the cockpit of monsoon science. Ruchi Ram recalled the global reach of Indian meteorology even at that early stage; Blanford would invite him home regularly to sit down and discuss the latest research findings \"made in Russia, America, or somewhere else.\" Blanford's emphasis was always \"on the interdependence of the weather in different parts of the world.\" Ruchi Ram recalled that this \"made a deep impression on me in its widest implications.\"\n\nBut Ruchi Ram concluded his account of the meteorological department on a more personal note. He suggested that \"if all Englishmen were like Mr Blanford, the social and political relations between the two races [British and Indian]... would have been quite different from what, unfortunately, we find today.\" He absolved Blanford of any sense of racial arrogance. Ruchi Ram's most powerful memory was \"trifling,\" but revealing. Most British officials in those days, he recalled, would bark orders at their subordinates, keeping them waiting, and standing\u2014but not Blanford. Blanford, Ruchi Ram wrote, \"never once shouted to me from his chair, or even sent for me through the _chaprasi_.\" Instead, \"the old man would get up from his seat, and opening the door that separated our rooms, would say gently, 'Lala Ruchi Ram.'\"\n\nLooking forward, it is surprising to note how many Indian intellectuals spent time working in the meteorological department in the early twentieth century. They included Chintamani Ghosh, founder of the influential nationalist periodical _The Modern Review,_ and Prasanta Chandra Mahalanobis, a master statistician who would play a leading role in making economic policy in independent India. In part, this may have been down to the way meteorology posed a daunting challenge of statistical analysis, which attracted many of India's brightest minds; in part, it may have come from a sense that understanding the monsoon was of vital importance to India's future. And perhaps the Meteorological Department also left them freer from restrictions and pressures. On that count, Ruchi Ram learned one important lesson from Blanford: \"He would ask me not to do this or that work myself, but got it done by one of the clerks so as to find [me] more time for self-study.\" But that is to get ahead of the story.\n\nAS RECORDS OF THE MONSOON ACCUMULATED, METEOROLOGISTS looked to capture its \"normal\" characteristics. In the late 1870s, Henry Blanford described the monsoon as a self-contained system; a climatological force that shaped and demarcated the Indian subcontinent: \"India, together with the circumadjacent seas, is, in the main, a secluded and independent area of atmospheric action.\" In Blanford's vision, the monsoons were an active force\u2014he saw that \"the goal of the monsoon, the place of low barometer to which its course is directed, is constantly changing.\" Blanford viewed the monsoon as driven by the \"primary contrast of land and water.\" He confirmed what was by then well known, that the driving force of the monsoon was the difference in solar heat received by the land surface of India at different points in the year, and its contrast with the relative heating or cooling of the Indian Ocean. But Blanford was able, now, to introduce new complexities to the science of the monsoon. He showed that the monsoon was not, as many had believed, \"one current flowing alternately to and from Central Asia,\" but rather that it was formed from the intersection of \"several currents, each having its own land centre.\" He described an alternate opening and closing of the Indian subcontinent to wider atmospheric forces. In the periods of transition, as the winds reversed\u2014between March and May, and again in October and November\u2014Blanford observed that \"the interchange of air currents between land and sea is, in a great measure, restricted to India and its two seas.\" But once the southwest and northeast monsoons had set in, they connected the Indian \"wind system,\" as Blanford called it, with \"those of the Sunda Islands and Australia, and, at one season, the trade winds of the South Indian Ocean.\"\n\nDriven by clear laws, Blanford believed that the monsoons were strongly predictable. \"Order and regularity are as prominent characteristics of our atmospheric phenomena,\" he wrote, \"as are apparent caprice and uncertainty those of their European counterparts.\" Beneath this broad predictability, however, the monsoon was characterized by its unevenness. Within any given monsoon season, Blanford observed, rains were not \"persistent and unvarying\"; rainfall was subject to \"prolonged periods of suspension\" as well as \"regular interruptions known as 'breaks.'\" Even more striking was the monsoon's spatial unevenness: meteorologists found \"a great diversity of rainfall\" in different parts of India. \"No country in the world,\" Blanford insisted, \"furnishes such contrasts.\" Even as the broad contours of the monsoon seemed amenable to prediction, uncertainty was a defining climatic feature in many parts of the country, and \"those provinces which have the lowest rainfall are also those in which it is most precarious.\"\n\nMeteorological anxiety about the unevenness of India's rainfall shaped perceptions of the land itself. In the most lyrical passage in his guide to India's weather, Blanford contrasted tropical with temperate landscapes:\n\nInstead of feeding perennial springs, and nourishing an absorbent cushion of green herbage, the greater part flows off the surface and fills the dry beds of drains and watercourses with temporary torrents. In uncultivated tracts, where jungle fires have destroyed the withered grass and bushy undergrowth, and have laid bare the soil and hardened its surface, this action is greatly enhanced;... not only is water lost for any useful purpose, but by producing floods, becomes an agent of destruction. Under any circumstances, the character of the rainfall is hardly compatible with economical storage and expenditure in any high degree; and much more, therefore, than in temperate regions is it incumbent on us to safeguard such provident arrangements as nature has furnished for the purpose.\n\nThe rhythms governing the distribution of rainfall proved an enduring mystery. Indian meteorologists pored over the correlations of monsoon failure across different parts of the country. Blanford observed the \"curious relations in the way in which certain provinces are prone to vary alike,\" suffering drought or excessive rainfall simultaneously, \"while others vary in the opposite direction.\" The 1880 famine commission observed that on five occasions over a century, severe droughts on the Indian Peninsula were followed, a year later, by drought on the plains of North India. The causal mechanisms at work eluded meteorology until well into the twentieth century. Much uncertainty still remains.\n\nAs clues mounted, Blanford looked back at his annual reports for 1876 and 1877, and he grasped the significance of two tentative observations he had made at the time. The first was that the years of the great drought had also seen unusually heavy snowfall over the Himalayas, later than usual in the winter. Blanford investigated this puzzle over the years that followed. By 1884, he was convinced that the \"extent and thickness of the Himalayan snows exercise a great and prolonged influence on the climatic conditions and weather of the plains of North-Western India.\" He suggested that keeping a close watch on Himalayan snowfall might hold the key to predicting the strength of the summer's monsoon to follow. But he was also quick to acknowledge that the forces at work might be far larger. Between 1876 and 1878, he wrote, \"excessive pressure was shown to affect so extensive a region, that it would be unreasonable to attribute it to the condition of any tract so limited as a portion of the Himalayan chain,\" vast though the mountains were. Blanford's calculations showed that high pressure had prevailed across \"extra-tropical Asia... and in Australia.\"\n\nWeather scientists across the British empire sought to pool their expertise and their information. Isis Pogson's detailed account of her library's holdings in Madras gives us a glimpse of these global connections. It included the proceedings of the First International Meteorological Congress in Vienna and reports from observatories in Batavia and Singapore and Manila. The development of monsoon science probably owed more to imperial and interimperial networks within Asia and Oceania than to wider international ones. As Blanford pursued his intuition about the great drought, he relied heavily on \"private correspondence\" with district officials in the Himalayas, and with meteorologists across the British Empire. He wrote to his counterparts at other stations across the Indian and Pacific oceans asking them to furnish him with data on atmospheric pressure from 1876 to 1878. Charles Todd, chief meteorologist of South Australia, was quick to respond with records from South Australia and the Northern Territories. Todd and Blanford both saw that their data correlated. By 1888, Todd concluded that \"there can be little or no doubt that severe droughts occur as a rule simultaneously\" over India and Australia. Information filtered in to Blanford from island observatories, too, which had long been central to British and French ecological investigations: Mauritius, Reunion, the Seychelles, and Ceylon. It was clear, by the 1880s, that the scale of influences on India's climate reached far beyond India's shores.\n\nThe famine commission of 1880 had expressed hope that the development of meteorology may provide some advance warning of monsoon failure. In 1881, Blanford was asked to come up with concrete proposals to implement the famine commission's recommendations for the development of India's meteorological infrastructure. His priority was the establishment of more monitoring stations. But he also looked to the more systematic collection of data from ships: a strengthening of the earliest maritime roots of meteorology. Information from ships was \"urgently required\" not only to track storms, but also \"to throw light on the causes of the variations of the south-west monsoon rainfall.\" From 1881, data was collected systematically from every ship entering the port of Calcutta.\n\nIn 1882, Blanford began to produce his first, tentative monsoon forecasts. A long-range monsoon forecast was a fundamentally different enterprise from the storm warnings that had dominated the concerns of Indian meteorologists. Especially with the aid of the telegraph, the approach of storms was now immediately visible. Cyclones were dramatic; their impact was urgent. Forecasting a year's monsoon, by contrast, required a more fundamental understanding of climate and climatic variation, founded on the slow analysis of a wide range of parameters on longer timescales. Despite his own awareness that India's climate was subject to oceanic or even planetary influences\u2014a phenomenon that we now know as \"teleconnection\"\u2014Blanford chose to base his forecasts on one primary indicator: snowfall in the Himalayas. From 1885, the Indian Meteorological Office's annual monsoon forecasts were published in the _Gazette of India_ \u2014and for the first few years, they proved accurate, at least as a broad-brush indication of whether monsoon rainfall was likely to be normal, excessive, or deficient.\n\n# III\n\nThe investigation of the oceanic and planetary influences on India's climate became an enduring concern for John Eliot (1839\u20131908), Henry Blanford's successor as director of the Indian Meteorological Office. The son of a schoolmaster and a graduate of Cambridge, Eliot began his career in India lecturing at the engineering college in Roorkee, which Proby Cautley had established at the head of the Ganges Canal; he moved on to Muir College in Allahabad, where he also served as director of the local meteorological observatory. In 1874, he took up a position as professor of physical science at Presidency College, Calcutta, where Blanford had also taught\u2014and Eliot took over Blanford's role as meteorological reporter to the government of Bengal. In 1886, Eliot again succeeded Blanford, now as the meteorological reporter to the government of India\u2014effectively the head of India's meteorological service\u2014and held that position until 1903. Tall and heavy-set and prone to bouts of illness, Eliot was also an \"accomplished musician\" on the piano and organ. In contrast with Blanford, Eliot was known, by his Bengali staff, as \"the native hater.\" Blanford's most senior Indian officer, Lala Ruchi Ram Sahni, decided he would rather quit his job and move to the Punjab Education Department than work for the irascible and prejudiced Eliot.\n\nLike Blanford, Eliot first served in Bengal. Like Blanford, his early work was on the cyclones that threatened the Indian coast. Even as prolonged drought stalked the land, sudden tropical storms continued to pose a recurrent threat, as Eliot had seen during the cyclones of 1876. Eliot pursued simultaneously the two strands of Indian meteorology that Blanford handed on to him: the study of extreme weather events, and the quest to forecast each year's monsoon. The first proved easier than the second to achieve.\n\nEliot's greatest influence on the field came from his understanding of cyclones. A few years after taking over as chief meteorologist of British India, Eliot published his _Handbook of Cyclonic Storms in the Bay of Bengal_. The nautical roots of monsoon science remained evident: Eliot's book was, above all, a practical guide for seafarers. It gained readers across Asia. Among those who learned from Eliot's book\u2014calling it both \"masterful\" and instructive\"\u2014was Father Jos\u00e9 Algu\u00e9 (1856\u20131930), a Spanish Jesuit meteorologist who led the Manila Observatory and stayed on after the 1898 American conquest of the Philippines to head the weather bureau. From a network of observatories across the western Pacific, local weather watchers grappled with the power and the unpredictability of tropical storms known as \"typhoons\" in the Chinese-speaking world: storms identical in nature to the cyclones of the Indian Ocean and the hurricanes of the Atlantic. The Japanese government invested in a centralized system of weather observation in keeping with its modernizing thrust after the Meiji Restoration of 1868. Elsewhere private bodies took the initiative\u2014particularly the Jesuits, who founded a series of meteorological observatories across and beyond the Spanish empire: at the Real Colegio de Bel\u00e9m in Havana (in 1857), at Ateneo Municipal de Manila (in 1865), and at Zikawei (Xujiahui) on the outskirts of Shanghai (in 1872). From 1869, the British-run Chinese Maritime Customs Service developed a network of meteorological observation. As in India, the practical value of storm forecasts for mariners provided the spark. Robert Hart, director of the Chinese customs, wrote of his hope of \"throwing light on natural laws and... bringing within the reach of scientific men facts and figures from a quarter of the globe which, rich in phenomena, has heretofore yielded so few data for systematic generalisation.\" These observatories exchanged information and developed a network of observation across the Pacific coast of Asia; but this remained walled off from the Indian Ocean, even as data began to reveal the connectedness of Asia's climate across its whole expanse. Algu\u00e9 was a towering figure, and he found much in Eliot's work on the Bay of Bengal to echo his own studies on the Philippine archipelago.\n\nIn his 1904 treatise on _The Cyclones of the Far East,_ a revised English version of a Spanish text written in 1897 (penned under the \"roar of the cannon and the rumors of war\" that \"rob the mind of the calmness which is so necessary in a work of this kind\"), Algu\u00e9 insisted that \"there is no tropical storm which is developed or felt in the sea or on the coasts of China which has not exercised some influence upon this Archipelago.\" The Manila Observatory built weather monitoring stations across the Philippines, staffed by Filipino volunteers and a growing cadre of trained local technicians. Algu\u00e9 aimed to educate the public about the tropical storms that posed a recurrent threat to their lives and livelihoods. But just like Blanford and Eliot in India, Algu\u00e9 imagined a broader climatic region. The telegraph allowed for the transmission of instantaneous weather information. The Manila Observatory could now warn the China coast of approaching storms; meteorologists could \"watch\" storms developing in the Pacific. A French journalist wrote admiringly of the \"completeness with which the Asiatic continent, from Cape St. James to the mouth of the Amur river, is safeguarded against surprises thanks to the meteorological services of Japan and the Philippines.\" \"Owing to the opening up of the Far East in recent years,\" Algu\u00e9 wrote, he had revised his work on the Philippines to give it \"a greater compass.\" That \"compass\" reached beyond the South China Sea and toward the Bay of Bengal.\n\nPart of Algu\u00e9's book was devoted to an account of \"two very remarkable storms\" described by Eliot. In a feat of meteorological detection, Algu\u00e9 matched up Eliot's accounts of the Port Blair cyclone that hit the Andaman Islands in November 1891, and the Chittagong cyclone of October 1897, with his own records in Manila. Of the storm of 1891, Eliot had simply written that there was an \"absence of information\" on the cyclone's origins in the South China Sea. Algu\u00e9 found a small item in the _Bulletin_ of the Manila Observatory for October 1891 that might provide the missing context: \"Very probably the typhoon which was felt on the 30th and 31st in Singora and other cities,\" Manila's meteorologists wrote, \"then traversed the Peninsula of Malaca after running through the Gulf of Siam, to obtain new strength in the Bay of Bengal.\" Ships' logs allowed observers in the Philippines to track the path of the storm down the South China Sea, until they lost sight of it in the Gulf of Siam\u2014which is precisely where Eliot began his account. Eliot picked up the storm in Siam, where on November 1, 1891, a storm wave flooded Chaiya, and \"387 religious buildings and 4,238 other buildings were more or less completely destroyed.\" It moved out over the Andaman Sea causing devastation in Port Blair, and fizzled out over the east coast of India.\n\nAlgu\u00e9 reconstructed the Chittagong cyclone of 1897 with equal precision. There, too, Eliot began his account of the storm on the Malay Peninsula, but Algu\u00e9 traced it back to the seas around the Philippine archipelago. He combed through accounts from Jesuit observers in the Philippines; he tracked the storm's path through the logs of the German steamer _Sachsen,_ heading from Singapore to Hong Kong, and the British ship _Faichiow,_ traveling from Bangkok to Hong Kong.\n\nAlgu\u00e9's account reveals how little communication there was at the time between weather observatories in the South China Sea and those in the Indian Ocean. Each body of water seemed a closed system, each with its own characteristic storms\u2014the typhoon seas and the cyclone seas. The expansion of telegraphic communication allowed for a new sense of scale to emerge, a new way of envisaging weather in time and space. Algu\u00e9's map of the two \"remarkable storms\" presents a different Asia: an Asia of storm tracks that traversed sea and land, crossing imperial borders; a coastal rim from the Philippines in the east to India in the west that shared risks to an extent previously unimagined.\n\n\"Two Remarkable Cyclones\": a map from Algu\u00e9's study of typhoons and cyclones. CREDIT: Rev. Jos\u00e9 Algu\u00e9, _The Cyclones of the Far East_ (Manila: Bureau of Public Printing, 1904)\n\nBut where Eliot succeeded in illuminating the nature and the threat of cyclones, his long-range forecasts of the South Asian monsoon fared less well. The number of weather monitoring stations in India grew from 135 in 1887 to 230 in 1901. By the turn of the twentieth century, the meteorological office issued five daily weather reports\u2014one for India as a whole, and one for each major region (including one for the Bay of Bengal). Eliot's most significant innovation was to introduce what he called an \"extra-Indian\" dimension to his forecasts. He incorporated into his forecasts data from the southern Indian Ocean; he was particularly convinced of a correlation between pressure in Mauritius and monsoon rainfall in India. Eliot's forecasts grew increasingly elaborate through the 1890s as they \"extended to thirty printed foolscap pages.\"\n\nHowever, Eliot's forecasts failed to warn of the climatic disasters that arrived in India in 1896\u20131897, and again in 1899\u20131900. Both, later research would show, were strong El Ni\u00f1o years; both droughts reached beyond India to affect China, Southeast Asia, and Australia. In 1899, Eliot's forecast predicted that monsoon rainfall would be \"on the average of the whole area... slightly above the normal.\" As it proved that year the shortfall from \"the normal\" was worse than ever before. But even had Eliot predicted the drought accurately, it is unlikely that the British colonial government would have had the willingness or the drive to intervene on the scale that would have been necessary to avert starvation.\n\n# IV\n\nWhat did it imply to think of Asia as an integrated climatic system? According to the evolving understanding of storms and monsoons, Asia appeared as an expanse of depth and altitude put in motion by the circulation of air. It was a land- and seascape defined by nature rather than by empires, its boundaries dictated by the winds and the mountains. But as soon as this picture of climate was translated into two-dimensional maps, the weight of political boundaries became evident. The first _Climatological Atlas of India,_ compiled by John Eliot after his retirement from India, began with a map of winds and pressure across an interlinked oceanic and continental system. It showed how the climate of India was shaped by the transfer of heat and energy between the Eurasian continent and the vastness of the Indian Ocean. This was in keeping with Eliot's own, evolving understanding of India's monsoon. But the flurry of maps in the atlas\u2014monthly maps of temperature and humidity and rainfall, cloud cover and wind direction and wind speed\u2014confined themselves to the territorial expanse of British India. In map after map, the territory of British India was shaded a different color from the surrounding mass in order to stand out; even the arrows showing wind speed are limited to the subcontinent, as though the winds were self-contained. Only the map of storm tracks stretches out toward the Bay of Bengal, as if the ocean were but an external source of weather as it affects the land.\n\nClimate science was forced into contact with geopolitics. Ideas about India's climate echoed, and informed, broader debates about India's place in the world. The networks of storm warnings along the coastal crescent from India to China mirrored Asia's maritime geography. The names of the stations that broadcast telegraph reports were the names of the great ports; the tracks of the tropical cyclones they monitored were the tracks of busy shipping lanes. Research on the longer-term regularities of India's _climate,_ as opposed to episodic weather, pointed in a different direction. India's climatology emphasized its distinctiveness, even its isolation. As Blanford put it, the monsoon system rendered India \"a secluded and independent area of atmospheric action.\" Ideas about climate coincided with new understandings of both geology and geopolitics. Blanford had begun life as a geologist, and in the late nineteenth century others in that field delved deeper into India's natural history. They argued that India was a breakaway fragment of the lost supercontinent of Gondwana that had collided with Eurasia in what was, on a geological timescale, the recent past. In the realm of geopolitics, at the same time, British strategists were increasingly worried by threats to their dominance that came not from sea, but from land, through the mountains of Central Asia\u2014the threat from Russia above all. These arguments about India\u2014each of them depicted visually in the form of maps\u2014came together to produce a \"subcontinental\" as opposed to what had been essentially a maritime view of India. The use of the term \"the Indian subcontinent\" dates from the early twentieth century. The Himalayas were crucial to this vision of India. They came more clearly into view in the last two decades of the nineteenth century: their role in India's climate, their place as the source of India's rivers, their strategic importance to India's security.\n\nThe official compendium of India's history and geography, the _Imperial Gazetteer_ \u2014edited by William Wilson Hunter, a keen meteorologist as well as an ethnographer, and published in eight volumes in 1881\u2014insisted that \"by India we now imply not merely the wide continent which stretches southward from the Himalayas to Cape Comorin, but also the vast entourage of mountainous plateaux and lofty ranges.\" India, Hunter insisted, \"can no longer be considered apart from that wide hinterland of uplands.\" This was a political as much as a geographical imperative. \"India,\" he wrote, \"must be held to include those outlying territories over which the Indian administration extends its control, even to the eastern and southern limits of Persia, Russia, Tibet, and China.\" It was overland, across the mountain passes from the northwest, that every invading force\u2014save the British\u2014had arrived in India. India's imperial rulers continued to fear a resurgent threat from their rivals among landed Eurasian empires. But Hunter's concern was with the future as much as with a repetition of the past. In \"a future of railways developments,\" with a \"rush of motor traffic,\" it was possible that \"the land approaches to India\" would \"rival those of the sea\"\u2014\"then will some of these again become the highways of the eastern world.\" By that time, \"we shall take out tickets in London for Herat, and change at Kandahar for Kabul or Karachi.\" Among these frontiers, the least explored but one \"potentially destined to play an important part in Indian history\" was \"the great highway of the Brahmaputra valley from the plateau of Tibet to the plains of Assam.\"\n\nHunter drew two conclusions. The first was that \"the material wealth of India largely depends\" on its \"capacity for the storage of that water supply which carried fertility to its broad plains.\" The other was that British India's security depended on \"guarding the gateways and portals of the hills,\" preventing \"those landward irruptions\" that had reshaped Indian history on many occasions in the past. These propositions would endure; they would outlast the British Empire that Hunter and his contemporaries wanted above all to preserve. And here is the contradiction at their heart: envisioning India through its rivers encouraged a far-reaching consideration of its connections with distant places, of \"world highways\" (or potential \"highways\") that situated India in relation to the flow of goods, people, money, and water to and from China, Tibet, and the expanse of Central Asia. By contrast, to see the Himalayas as a natural barrier, one always under threat of breach, was to advocate for a vision of India as a bounded place, an \"amphitheater\" sealed off from the rest of Asia. Natural frontiers became synonymous with the security of the realm. Hunter's concern with the economic value of water to Indian agriculture reinforced this bounded view. Water was a resource to be stored, possessed, harnessed, and put to work: the essence of India's \"material wealth.\"\n\nHunter's was a view moving up to the mountains, and away from the ocean. The last part of the nineteenth century saw a final push of Himalayan exploration, which had begun a century earlier. When Trelawney Saunders drew his map of India's mountains and river basins in 1870, he noted that Tibet was still _terra incognita;_ Arthur Cotton argued the same when he dreamed of a canal link between India and China. Locating the source of Asia's great rivers was the final frontier in the spate of expeditions undertaken by European explorers in the nineteenth century. It was not until his expedition of 1905\u20131908 that the Swede Sven Hedin finally discovered the source of both the Indus and the Brahmaputra on the Tibetan Plateau. Describing his first sight of the Brahmaputra, Hedin fell into rapture. \"Above the dark-grey ridge rises a world of mountains which seems to belong to the heavens rather than the earth,\" he wrote of the northern Himalayas, \"between them and the dark grey crest, comparatively near to us, yawns an abyss, a huge fissure on the earth's crust, the valley of the Brahmaputra or the Tsangpo.\" He described the water: \"Bluish-green and almost perfectly transparent, it flows slowly and noiselessly in a single bed to the east, while here and there fishes are seen rising.\"\n\nA decade after Frederick Jackson Turner's famous address to the American Historical Association, on the \"closing\" of the American frontier, the British imperial geographer and strategist Halford Mackinder made the point on a much larger scale. In 1904, just the year before Hedin's expedition, Mackinder argued that \"geographical exploration is nearly over.\" There were no \"blank spaces\" left on the map of the world. In Asia, he witnessed \"the last moves of the game first played by the horsemen of Yermak the Cossack and the Shipmen of Vasco da Gama.\" Mackinder foresaw that the heartland of Eurasia would, again, become the pivot of global power. \"A generation ago steam and the Suez Canal appeared to have increased the mobility of sea-power relatively to land-power,\" he declared, but now transcontinental railways were \"transmuting the conditions of land-power.\" In this light the mountainous frontier of the Himalayas still appeared remote and forbidding. But it was now clear that they might contain vast water resources\u2014resources to be captured for the development and security of the plains.\n\nAnd here we have the paradox that deepened in these years: water and climate were boundless. Their boundlessness became clearer with every advance in the technologies of measurement. Yet, as the next chapter will show, they came under ever tighter but more fragmented territorial control. We turn now to the fevered quest for water that gripped India, and most of Asia, in the early decades of the twentieth century.\n\n# FIVE\n\n# THE STRUGGLE FOR WATER\n\nIN _I NVISIBLE CITIES_, HIS TALE OF AN IMAGINED ENCOUNTER BETWEEN the Italian explorer Marco Polo and the Mongol emperor Kublai Khan, Italian novelist Italo Calvino describes the fictional city of Isaura like this:\n\nIsaura, city of the thousand wells, is said to rise over a deep, subterranean lake. On all sides, wherever the inhabitants dig vertical holes in the ground, they succeed in drawing up water, as far as the city extends, and no farther. Its green border repeats the dark outline of the buried lake; an invisible landscape conditions the visible one; everything that moves in the sunlight is driven by the lapping wave enclosed beneath the rock's calcerous sky.\n\nCalvino's exquisite depiction of \"Isaura\" distills the themes of this chapter, which is concerned with the search for new sources of water in India. At the turn of the twentieth century, India still reeled from the famines of the 1870s and the 1890s. Advances in meteorological science had pointed to the awesome power of the monsoon climate, and sketched its continental span. Indian economists and British administrators, water engineers and industrialists joined in a search for water to secure India from climatic vulnerability. They looked for water underground. They built dams to harness the power of rivers. They explored new ways to store rainwater. Across India, water formed \"invisible landscapes\" that shaped \"the visible one.\" In trying to exploit water more intensively, administrators and engineers came to see it as a bounded resource. Control was their aim, but it proved an elusive goal. For in those same decades, climate science pointed in the opposite direction: it threw up the enormity of the scale on which nature worked. The study of the monsoon became ever more spectacularly global. Understanding that the climate of any given part of Asia was subject to remote oceanic and atmospheric influences still too complex fully to comprehend, meteorology called into question the confidence with which many believed nature could be conquered.\n\n# I\n\nIn almost any account of modern Indian history, the last two decades of the nineteenth century appear as a moment of political awakening. The formation of the Indian National Congress in 1885 represents, in most textbooks, the beginning of organized nationalism, albeit a tentative nationalism still dominated by an urban and professional elite, still outwardly loyal to the empire. The famines of the last quarter of the nineteenth century catalyzed criticism of colonial rule. They made brutally evident the insecurity of life endured by most of India's people; they called into question the effectiveness of the colonial government; above all, they laid bare whose interests the British had in mind when they governed India. \"Britain has appropriated thousands of millions of India's wealth for building up and maintaining her vast British Indian Empire... [and] has thereby reduced the bulk of the Indian population to extreme poverty, destitution, and degradation,\" said Dadabhai Naoroji, an early proponent of economic nationalism in India, and the first Indian to be elected to the British Parliament, in a speech at the Plumstead Radical Club in London in 1900. Naoroji argued that it was Britain's \"bounden duty in common justice and humanity to pay from her own exchequer the costs of all famines and diseases caused by such impoverishment.\" The experience of drought and famine gave rise to new ways of thinking about state and economy, nature and climate. The material consequences of those ways of seeing continue to shape our world.\n\nBy the 1890s, many Indian economists saw the monsoon as a limiting condition on India's future development. Mahadev Govind Ranade\u2014social reformer and High Court judge and early leader of the reformist Poona Sarvajanik Sabha\u2014declared in 1890 that in many parts of the country \"the last margin has been reached, and millions die or starve when a single monsoon fails.\" Ranade believed that only a complete transformation of land and water could protect India from future disasters. He lamented the \"ruralizing\" of India\u2014the destruction of local industries as a result of Lancashire textiles flooding the market. Ranade felt that only an industrial future, with some measure of protection for local industries from foreign competition, would free India from its vulnerability, which arose, he believed, from the country's acute dependence on agriculture. He wrote admiringly of the Cultivation System in the Dutch East Indies, a coercive Dutch colonial policy that forced cultivators to set aside a portion of their lands for the production of export crops. Whether or not Ranade was aware of the system's harshness, which provoked protest and resistance in Java, he saw it as a welcome alternative to the British worship of free trade and even as a boost to Indonesian industry.\n\nRomesh Chander Dutt\u2014Bengali economist and poet, civil servant, and translator of Hindu epics\u2014went further than Ranade in pinpointing responsibility for India's famines. In 1901 Dutt began his open letter to the viceroy, Lord Curzon, with the \"melancholy\" observation that 15 million people had died in famines in India within Dutt's own lifetime. He dismissed British observers' fixation on the predatory moneylender as the chief cause of distress to Indian cultivators in times of drought. \"The money-lender is the result, not the cause, of the poverty of the cultivators,\" Dutt wrote, suggesting that Indian farmers fell back on credit only because their earnings were insufficient to cover the heavy burden of taxation. Turning to Malthusian fears about India's population growth, Dutt pointed out that India's population grew more slowly in the nineteenth century than England's. The \"immediate cause of famines in almost every instance is the failure of the rains,\" he said, and the threat of famine would persist \"until we have a more extensive system of irrigation.\" But he distinguished the \"immediate\" cause of famine from the root cause\u2014the reason why a failure of the rains should bring so many to the brink of starvation. That root cause was the \"chronic poverty of the cultivators, caused by the over-assessment of the soil.\" Developing the argument that Naoroji had made earlier, Dutt argued that the weight of the land tax\u2014used to finance the \"most expensive foreign government on earth,\" bloated by its imperial adventures beyond India's shores\u2014was the main cause of India's poverty. In 1903 Dutt published his masterpiece, an economic history of India in two volumes. Written with flair and backed by reams of statistics, Dutt's book argued that Britain had \"drained\" India of its wealth and contributed to the devastation of its industries. In his catalog of misguided colonial policies, however, Dutt expressed his great admiration for Arthur Cotton, whom he thought to possess \"a reputation higher than that of any other engineer who has ever worked in India.\" Dutt agreed with Cotton that India's salvation lay in irrigation. The problem, Dutt discerned, was that colonial administrators had ignored Cotton; they \"returned again and again to the narrower view, based on the immediate financial return of works constructed.\" The British government of India was so concerned with avoiding unnecessary expense that it was trapped in a short-term view of India's economic development, unwilling to invest in expensive infrastructure that might, in time, have paid off.\n\nAT THE BEGINNING OF THE TWENTIETH CENTURY, THE INDIAN Irrigation Commission undertook a grand tour of India's water. Famine was a raw and recent memory. Colonial officials and Indian critics alike believed that irrigation was essential to protect India from a recurrence of the horror. The viceroy, Lord Curzon, appointed Scotsman Colin Scott-Moncrieff to lead the investigation. Scott-Moncrieff was famous for directing the hydraulic transformation of the Nile in the 1880s through a restoration of the Nile Barrage, but he had started his career with the Bengal Engineers and had served as Burma's chief engineer in the 1870s. Twenty years later, he returned to India. His brief was as simple as it was daunting\u2014to determine how far irrigation could protect India from the vagaries of its climate.\n\nThe British colonial government of India has been described as \"ethnographic\": it collected obsessively information about every aspect of life within its domain; it classified India's people by caste and faith; it published reports of \"moral and material progress.\" In the first three decades of the twentieth century, four commissions of inquiry\u2014on irrigation, agriculture, banking, and labor\u2014cemented the state's knowledge of its territory and its population. This vast enterprise of information-gathering generated authoritative tomes, divided by province, backed with appendices. To read them now is to leap into the minutiae of economic life in India, village by village. The commissions published extensive volumes of testimony, both oral and written, which are in turn prosaic and lyrical, combative and dull. It is easy to forget how small these commissions were, how dependent they were on knowledge provided by others. Scott-Moncrieff was joined by Thomas Higham, India's chief inspector of irrigation; by Denzil Ibbetson, chief commissioner of the Central Provinces; by John Muir-Mackenzie, chief secretary to the government of Bombay; and by a sole Indian, Rajaratna Mudaliar, a member of the Madras Legislative Council. The five men met in Lahore, and over two periods of six months each, in 1901 and 1902, they traveled together across India. They covered more than five thousand miles; they interviewed 425 witnesses in 91 sittings. \"This is pretty hard work,\" Scott-Moncrieff wrote to a relative in England, \"listening to witnesses and asking questions for six hours is, I can assure you, rather fatiguing\"; but he conceded that \"we travel very luxuriously, generally in special trains\" and, at every stop, \"we are most hospitably entertained in the generous old Indian fashion.\"\n\nFinally they settled in Lucknow, by the banks of the Ganges, and wrote their report surrounded by meteorological charts, statements, petitions, and two years' worth of impressions.\n\nOnce again, they began with the rains: \"Not only a main factor in determining the value of irrigation\" but also \"the primary source of all means of supplying it.\" The irrigation commissioners went back to Henry Blanford's writings, their main source on India's struggle with the monsoon. They drew attention to water's \"unequal distribution throughout the seasons, its still more irregular distribution over the surface of the country, and its liability to failure or serious deficiency.\" They turned, then, to India's geology, which had been Blanford's original concern. Each of India's main geological regions, the expanse of alluvial and crystalline soils, offered possibilities, and challenges, for irrigation. The commission's data showed that 20 percent of the cultivable land of British India was under irrigation by the turn of the twentieth century, from the large canals of the Gangetic plain to the 626,000 wells watering Madras Presidency.\n\nThey concluded that India offered \"a wide but not unlimited field\" for ambitious engineers. They listed the many obstacles, from the topography of the landscape to the cost of irrigation works. The British government retained the fixation on parsimony that had driven famine policy in the 1870s. In hundreds of pages of testimony to the irrigation commission, witnesses were asked, again and again, about whether investment in irrigation would ever pay for itself. The commissioners dreamed of a time when farmers would pay for water \"what it is really worth\"; they searched for ways to justify irrigation expenditure to the state, ever anxious about the charge of extravagance. They also saw as a problem water's very expanse, in relation to the lines of jurisdiction that ran through India. \"The manner in which the various states and territories are intermingled,\" they wrote, were an \"obstacle\" to their visions. The commissioners expressed fulsome thanks to the princely rulers that had hosted them on their tour, but the patchwork of sovereignty that interspersed regions of direct British control with princely territories caused anxiety. The problem was enduring: the \"only suitable site for a storage work,\" they observed, \"may lie in a territory whose people would not only derive no benefit, but might even be put to considerable loss.\" They had diagnosed a source of hydraulic inequality that would sharpen over the century ahead.\n\nAmong the resources the irrigation commissioners looked to was India's ocean of water under the ground. By the late nineteenth century the quest for subterranean water had attained new fervor. A breakthrough in technology brought with it a sense of untapped possibility: engine-driven pumps promised to reach much deeper underground than the manual ways that were little changed from Babur's time. After the famines of the 1870s, agricultural officials in the most arid districts encouraged the construction of wells by landowners. \"The main difficulty... in well construction is not the discovery of water,\" wrote W. C. Bennett, director of agriculture in the Northwest Provinces, \"the water level is known locally all over the Provinces.\" The problem was to determine where the soil could support deep wells, given the \"extreme capriciousness and uncertainty\" of the clay stratum beneath the surface.\n\nFrom the start, and in contrast with government-built canals, wells were in private hands; \"policy,\" such as it was, involved encouraging landowners to dig wells. The government's role, if it had any, was limited to providing credit and information. Already by the 1880s, Bennett estimated that almost 58 percent of the irrigated area in the province he administered was under well irrigation; canals, which received infinitely more money and attention, accounted for only 24 percent. To his mind the importance of wells could not be overstated. They usually held enough water \"to carry the people through one season of failure of the rains\"\u2014though he cautioned that if the rains should fail two years running, as they had in 1876 and 1877, the wells would run dry. Water prospectors began to imagine the map of India with an added dimension\u2014the underground. As well as roads and railways, wrote an engineer from the Bombay Deccan, it would help if maps could show the \" _water_ levels as well as showing depth of spring levels below the surface.\"\n\nBy the early twentieth century, Madras was at the forefront of a water-mining boom. A keen observer of this economic revolution was Alfred Chatterton, a British engineer who had made his career in the Madras government. He served as superintendent of industrial education and insisted on the development of a department of industry in the province to encourage local manufacturing. He was quietly sympathetic to _swadeshi_ initiatives to support local producers and boycott imports. To that end he opened a government pencil factory in Kurukkupet, which promised to boost local industry.\n\nIrrigation was among Chatterton's main interests as an engineer. He was the author of a book on lift irrigation (the extraction of water from wells), and the possibilities he saw before him, as he surveyed the Madras countryside, seemed limitless. On a tour of Trichinopoly he reported that there had been a \"very large number of applications\" for government loans for the purchase of engines and pumps. He appended a list of all of the firms supplying oil engines in Madras, a total of 125 in the province; the largest dealer was Massey & Co., which in 1904 supplied ninety-one engines of a total of 869 horsepower. Chatterton was impressed: \"It is obvious from this that the oil-engine has come to stay in the Madras Presidency and that it is suited to the conditions of the country,\" he wrote. \"I am sanguine enough to think that this is merely the beginning of things and that in the next year or two the use of oil-engines will increase very rapidly for irrigation work.\"\n\nThat year, the northeast monsoon had brought scanty rainfall; the new technology was put immediately to the test. \"The engine was started 568 times,\" reported an agricultural station at Melrosepatnam, \"and ran for 2074 hours.\" Most of the engines ran on liquid fuel\u2014that is to say, petroleum\u2014which was half or even a quarter the price of kerosene oil. The main source of supply was from the Borneo oil fields, imported by Best and Company and held in a four-thousand-ton storage tank in Madras. In Borneo, as in the fields of the Burmah Oil Company, a new era had begun.\n\nIndia tumbled into the embrace of fossil fuels. Very early in the twentieth century, oil unleashed a vision of plenitude based on the extraction of water\u2014a vision that continues to shape contemporary India. \"The oil engine and pump do work,\" Chatterton observed, \"[utilizing] to an extent absolutely unknown previously the quantity of water available for irrigation.\" The abundance of water beneath the soil lay in \"a succession of spring channels, one below the other.\" Their exploitation had once relied on a vast corps of voluntary or _corv\u00e9e_ labor, but now \"much larger quantities of water could be abstracted from the river-beds by means of engines and pumps.\" Already there were three-quarters of a million wells scattered across Madras. Most of them were in the hands of small landowners. The potential for further development seemed vast. Chatterton evoked a terrain with \"many deep old furrows filled with sand which mark the ancient course of river-beds\" that were \"full of water.\" Water engineers imagined a whole country underground\u2014a network of vanished rivers still running with water, waiting to be forced to the surface. On this view, the \"problem of subterranean water\" was part of a broader problem of energy: its extraction depended on \"distributing power cheaply... by wire ropes, compressed air or electricity.\"\n\n# II\n\nNowhere in India was the quest for water more transformative than in Punjab. In a remarkably short time, Punjab rivaled the Gangetic plains and the river valleys of the southern peninsula as India's agrarian heartland. The impetus for the change came from British attempts to bring water to what they claimed as \"wastelands\" in western Punjab\u2014a land at the limits of the monsoon, described by James Douie, an experienced district official in the province, as \"in reality part of the great desert extending from the western Sahara to Manchuria.\" Here, \"irrigation was not designed to assist agriculture and diminish the losses from seasonal vagaries, but to create [agriculture] where it did not exist.\"\n\nBetween 1885 and 1940, the colonial government built nine Canal Colonies\u2014townships of settlement each built around an irrigation canal\u2014through a stretch of western Punjab between the Beas and Sutlej rivers and the Jhelum, covering an area of 13 million acres. The largest of them was the Chenab settlement, started in 1892; the most challenging for the engineers was the massive Triple Canal project, completed in 1915 through difficult terrain. More than a million people moved voluntarily to these new colonies from the more densely populated lands of eastern Punjab; they inaugurated a revolution in the production of wheat, cotton, and sugar. The Canal Colonies attempted to create a new landscape\u2014and a new society. They were among the first of their kind, establishing a pattern that would become common throughout the world in the twentieth century: a state-financed and state-directed resettlement of people in the service of agricultural \"development.\" The British officials who envisaged the project in the 1880s were ambitious. They sought to create a new sort of Indian village, populated by cultivators carefully selected for their vigor and ability, providing a new dynamo for agrarian capitalism.\n\nSpeaking to an audience at London's Royal Society of the Arts in 1914, James Douie\u2014a career civil servant who had spent thirty-five years in Punjab, his roles including chief secretary to the government, and settlement commissioner\u2014lauded the achievements of Punjab's engineers. In his mind, the \"colonization\" of \"waste\" land in the Punjab was part of a global process. \"When one speaks of colonization the mind turns at once to new countries,\" he conceded: \"the prairies of the United States, or the pampas of Argentina.\" Many of those large movements of people, too, had involved the search for new sources of water to make arid lands productive. And in Punjab a similar movement was underway. Douie's account made clear how fully the Canal Colonies were the product of an engineer's vision backed by the power of the state\u2014right down to \"allotting land in complete squares.\" People \"who love the irregular fields of the homeland, with their hedges of black and white thorn or wilding rose, may think these unenclosed rectangular fields dull,\" he admitted\u2014but they were designed for utility. In a society where the rules of inheritance allotted an equal share of land to every son, \"twenty-five equal rectangular fields can be divided, accurately, easily, and cheaply.\"\n\nThe state reserved the power to decree not only the shape of each parcel of land but also the composition of its population. The language of science lent authority to ideas about physical capacity and heredity\u2014many British officials were obsessed with categorizing India's people according to the size of their skulls. The impetus to conflate physical appearance with innate character was especially strong in Punjab, which had become the prime recruiting ground for the British army in the aftermath of the 1857 rebellion. To make this strategic shift appear natural, army recruiters, high officials, and a number of their local allies proffered an enduring myth that Sikhs\u2014alongside the Gurkhas of Nepal and in contrast with \"effeminate\" Bengalis\u2014were a \"martial race.\" Relating how he had made his choice of future colonists from among the large number of applicants in any given village, Douie was straightforward: \"I looked at their chests,\" he said, as he proceeded to show his audience in London a series of slides of unclothed Punjabi bodies. One of his fellow officers had focused on the applicants' hands.\n\nThe British gamble paid off. By 1915, the financial commissioner of Punjab wrote of Chenab district that \"the land revenue of this tract... exceeds any other district in India.\" Punjab became the engine of agrarian growth in India. By 1931, 46 percent of all canal-irrigated land in British India was in Punjab; Madras came a distant second. Prosperity from cultivation spurred local industrialization with the rise of cotton ginning factories. With land grants to families who had served in the army, the British shored up loyal support in the region. Many local men saw themselves as protagonists in what one of them called \"man's conquest over nature.\" But social transformation was never free from tension. In 1907 protest erupted in the Canal Colonies against legislation giving the government sweeping new powers to regulate settlement and land use. Colonists complained about the inadequate supply of irrigation water to their fields; they chafed against a lower-level bureaucracy that accumulated arbitrary power; they objected to the infringement of customary rights in the name of scientific management.\n\nBRITISH OFFICIALS LIKE DOUIE SAW PUNJAB'S CANAL COLONIES IN light of a worldwide process of frontier colonization. He referred repeatedly to Saskatchewan and Manitoba, to Australia and the American prairie. He could just as well have looked across to China. In China's far northeast, a comparable process of peasant colonization was underway. It dwarfed the movement in Punjab. Between 28 and 33 million Chinese migrants moved to Manchuria after 1850; the movement accelerated in 1890, simultaneous with the development of Punjab's Canal Colonies. Many went to work in mines and on the railways, but most Chinese migrants to Manchuria went as cultivators. By the 1920s, the soybean made up 80 percent of the region's exports. A relatively small proportion of the migrants owned land on a freehold basis; many more leased their land or worked as sharecroppers. Expanses of Manchuria were owned by Chinese official organizations, or by private and semiprivate companies. Large landowners accumulated holdings. And if the Chinese state did not play as active a role in encouraging migration as did the British colonial government in northwestern India, it was far from absent.\n\nAs in Punjab, family was the \"engine of migration\" to Manchuria. Families in Shandong and Hebei sent young men to Manchuria as part of a diversified strategy for family survival. But there the expectation of return was almost universal, unlike in Punjab, where families moved permanently to new settlements and over shorter distances than their Chinese counterparts destined for Manchuria. As in Punjab, most Chinese migrants to Manchuria moved in small groups of kinsmen or fellow villagers. They went where uncles, cousins, or others from their villages had blazed a trail. The railway and the steamship made their journeys more affordable, and took them to places previously inaccessible. When this happened on a large enough scale, whole \"villages across the sea\" emerged, each resembling a northern Chinese village transplanted to Manchuria. In historical perspective, these movements represent the final closing of a global frontier\u2014the crescendo of a process of migration, settlement, and colonization that had begun a few centuries earlier. From Punjab to the American West, colonization depended on water; in Manchuria, too, irrigation was vital to the expansion of settlement and production. Everywhere, colonization displaced communities of people\u2014often pastoralists\u2014who were already there. Their livelihoods and their cultural habits were disregarded as these \"empty\" lands were settled in the name of agriculture and civilization.\n\nThe very designation of western Punjab as \"waste\" stripped local pastoralists of their claim to the land: this was no neutral description of a landscape, but rather a justification for what administrators and engineers wished to create. Pastoralists in western Punjab soon found their livelihoods under threat. Where once they knew the landscape intimately, now British engineers and new settlers had captured the water. In the words of a local official, many local people were \"driven to migrate by the gradual impoverishment of their villages.\" Many other \"calamities for future generations,\" as Alexander von Humboldt had called them, lay in store. One was malaria\u2014changes in the hydrology of the land created conditions ripe for its spread. A local petitioner complained that the Canal Colonies had \"injuriously modified the climate\" so that \"malaria always prevails there.\" Another enduring problem that emerged from Punjab's great experiment was waterlogging so severe that it made some lands impossible to cultivate.\n\n# III\n\nThe transformation of India's waters went far beyond Punjab. Bombay, Madras, and the United Provinces were all sites of intervention and experiment spurred by the recent memory of famine. Hydraulic engineers built on the nineteenth-century schemes like the dams designed by Arthur Cotton along the Krishna, Godavari, and Kaveri rivers of southern India. In Punjab, the land allocated to the Canal Colonies was already in government hands. Elsewhere, a prolonged process of land acquisition was underway, facilitated by 1894 legislation making it easier for the government to take over private land for \"public\" purposes. While the 1894 Land Acquisition Act aimed to ease railway construction, it came into force just as water engineering projects proliferated across India. By the early twentieth century a long and painful process was underway, displacing families and whole communities to make way for water infrastructure projects\u2014it continues to unfold to this day.\n\nSitting in the Maharashtra Archives in Mumbai, housed in the decaying Gothic splendor of Elphinstone College, I saw thick files full of petitions and disputes over land, water, and compensation. \"Attempts were made to acquire the land amicably but the owners refuse reasonable terms,\" an official notice of 1889 read, and so it would be seized: \"Permission may be granted to take possession of the land on the expiration of 15 days from the publication of this notice.\" That was just the beginning; just how much land was taken over this way, piece by piece? Some years later, in 1903, Sakharam Balaji, a farmer who stood to lose his lands, wrote in anguish to the local government office. He had lent support to the construction of a local dam by supplying it with materials, \"straining my every nerve\" in the process. He was then stunned to find out that the dam would drown his two hundred acres of land. \"Money given to me by the Government in light of compensation will not suffice,\" he declared, for where else could he go? \"Land if properly cultivated and repaired will last long,\" he said, and he had invested handsomely in his own. The archival file is complete with crossings-out and handwritten insertions as Sakharam (or perhaps his scribe) sought the right words. \"My occupation is nothing but husbandry,\" he argued, \"and my maintenance is solely dependent on this husbandry.\" His concern was well founded. Displacement meant more than the loss of land\u2014it was a loss of livelihood, an uprooting of life. Sakharam was fortunate. The local government ruled in his favor and resituated the dam. But many others would suffer irreparable loss.\n\nGroups of landowners came together to protest their dispossession. The residents of Belgaum district in Bombay Presidency\u2014\"the most loyal and dutiful, but at the same time the poorest, subjects of His Majesty\"\u2014feared that a dam, linked to a series of canals, would \"inundate almost all the lands of our valleys causing the residents to shift themselves for their lives to some other distant tract not yet known.\" They made a powerful claim, couched in the language of justice; they challenged the government on its own terms: \"The inhabitants of the above-named villages have equal rights to secure the advantages of benign rule of British Government, as the villages which are to be profited by this canal.\" The petition ends with more than a hundred signatures in Marathi.\n\nTHE PEOPLE OF BELGAUM WERE DISPLACED, IN PART, BECAUSE OF A cascade of new demands for water from India's growing cities. Whereas the decades from the 1840s to the 1880s had seen India reduced to a \"colonial\" economy\u2014focused on the export of raw materials, and the import of manufactured goods\u2014the tide had started to shift by the 1890s. While factories in India were concentrated in a handful of cities\u2014Bombay, Ahmedabad, Kanpur, Coimbatore\u2014Indian industrial capitalism hit its stride in the early decades of the twentieth century, boosted by the suspension of imports and the voracious military consumption of manufactured goods during the First World War. The French geographer Jules Sion observed after the war: \"In Bombay, almost all the factories have Indian owners, directors, engineers. The country's capitalists compete with foreigners for mining concessions. Many vast plantations of tea and coffee are managed entirely by locals. This economic nationalism faces an obstacle in the entrenched habits and ignorance of the masses,\" he observed, \"But it has already proved its vitality.\"\n\nWith urbanization came new demands on water. Bombay was the first city in British India to have a municipal water supply, which started operation in 1860. But the city's growth quickly outstripped the scheme's capacity. In 1885, the Tansa project aimed to augment urban water supply; when it opened in 1892, it supplied the city with 77 million liters of water a day, though this still reached only a small proportion of its residents. The project came with unwanted consequences. Water supply in the absence of adequate drainage led to waterlogging in many neighborhoods of the city, contributing to the conditions that allowed the bubonic plague epidemic of 1896 to spread. As the city of Bombay grew\u2014its population reached 1.2 million by 1920\u2014its tentacles reached ever further into the countryside of Maharashtra in search of water for its residents.\n\nAs industrialists grew more prominent, both within the colonial government and within the nationalist movement, their voices grew louder. Among their most pressing needs: water and power. The use of water for irrigation was \"well understood,\" the Indian Industrial Commission reported in 1918, but now water had a \"double object\"\u2014irrigation and electricity generation. The era of \"white gold,\" as hydroelectric power had come to be known in Europe and America, came to India.\n\nIndia's first hydroelectric plant was built outside the Raj, in the princely state of Mysore, in 1903. The Sivasamudram Dam along the Kaveri River was built to supply electricity to the Kolar gold mines nearby, which were India's largest. India's biggest industrial firm, Tata and Sons, followed closely behind. Having made a fortune exporting opium to China in the first half of the nineteenth century, the Tatas were among the earliest to move into the cotton industry. Perennially in need of electricity to power their factories in Bombay, the Tatas looked to harness the heavy monsoon rains that fell on the Western Ghats mountain range. Theirs was already a self-consciously global enterprise. The Tatas' close ties with the United States meant that American models and expertise were foremost on their minds when they decided to build a hydroelectric plant of their own at Khopoli, in the Western Ghats, completed in 1915. Within three years, the plant was supplying Bombay's cotton mills with forty-two thousand horsepower of electricity, for twelve hours a day.\n\nOne of the enduring heroes of India's attempt to transform its waters was an engineer named Mokshagundam Visvesvaraya. Visvesvaraya was born to a poor Brahmin family in the princely state of Mysore in 1860. He studied engineering in Pune's College of Science, and served the government of Bombay for twenty-five years as an engineer in the Department of Public Works. Visvesvaraya was ascetic in his habits, a firm believer in hard work and self-help. His first big achievement was the construction of a new water pipeline in the town of Nasik. In the course of redesigning the water supply of the much larger city of Pune, Visvesvaraya patented a new system of automatic sluice gates, which remained in use for decades afterward; in keeping with his dedication to public service, he gave up any claim to royalties. From the outset, Visvesvaraya was obsessed with the \"waste\" of water. He grappled with \"how to bring under control the irregular distribution of water to crops... and its wasteful use by cultivators.\" He found that \"the cultivators were unaccustomed to control\" by engineers and wise administrators. To assert that \"control\" over water would be a lifelong battle for Visvesvaraya.\n\nVisvesvaraya's horizons expanded in the first decade of the twentieth century, when he began to travel the world. His first trip abroad was to the country that impressed him more than any other: Japan. Visvesvaraya spent three months in Japan in 1898. He wrote a book based on his impressions, but \"did not... think the time was opportune to publish it\"; he was a civil servant, and not one for overt political confrontation. Yet seeing Japan had led him to conclude just how little the British were doing, in comparison, for the development of India. \"Since all industrial progress in Japan has been achieved in comparatively recent years,\" he wrote, \"she offers to India the most direct and valuable lessons obtainable in material advancement and reconstruction.\" The greatest lesson, he thought, was that the Japanese state had taken a direct and interventionist role in fostering economic development.\n\nVisvesvaraya's career took him to West as well as to East Asia. In 1906, the government of Bombay deputed him to the British protectorate of Aden, to investigate the city's water supply. Poring over the city's sanitary records, he found an alarmingly high death rate; he was quick to conclude that \"a system of pipe sewers is the only satisfactory method.\" The authorities accepted his recommendation that a series of wells be dug to supply the city from the River Lahex. Upon retirement from government service\u2014and perhaps frustrated by the racial glass ceiling of the Raj, which meant that he could never become chief engineer of Bombay\u2014Visvesvaraya toured the world. He visited western Italy, Russia, and North America to study dams and irrigation techniques. Arriving in New York, he found \"an association of Indian traders and businessmen there, men of energy, vitality and ambition.\"\n\nAfter a year of roving consultancy, Visvesvaraya was persuaded in 1909 to take up the position of chief engineer of Mysore. Among the princely states, Mysore was self-consciously progressive; a succession of dewans (prime ministers) had, with the Maharaja's encouragement, instituted educational and infrastructural reforms. A few years after his appointment as chief engineer, Visvesvaraya assumed the role of Dewan: he made primary education compulsory; he invested in infrastructure and sanitation. Given free rein and a generous budget, Visvesvaraya scaled up his ambitions. He dreamt his masterpiece: the Krishnarajasagar Dam along the Kaveri River. It was Visvesvaraya's plan for a \"multi-purpose\" project\u2014the dam would irrigate one hundred acres of land, it would provide power to the Kolar gold fields, and it would electrify the city of Bangalore. The problem, as we will see, is that the British had their own plans for the same river.\n\nCHANNELING RIVERS AND RAINS, PUMPING WATER FROM UNDERGROUND\u2014there was a third frontier in view at the turn of the twentieth century: the ocean. Frederick Nicholson joined the Madras civil service in 1869, aged twenty-three and fresh from Oxford. Over the next three decades he would serve as collector of Tinevelly (Tirunelveli), Madras, and Coimbatore; he witnessed the great famines of the 1870s, which shaped his view of the world; he fell in love with the Nilgiri hills, where eventually he retired and lived until his death in 1936. At the turn of the twentieth century, Nicholson turned his attention to the fisheries of Madras. He wrote in 1899 that \"when we despair of food independent of climate for a rapidly-increasing population, of industries for non-agriculturists, of manure for deteriorating soils, we may thank God that we have yet got the fisheries to develop.\" \"Food independent of climate\" was precisely what India's quest for water sought to bring about\u2014Nicholson saw this more starkly than most. \"The sea yields its harvests in enormous quantities wholly irrespective of droughts and seasonal catastrophes,\" Nicholson wrote, \"and the food, being highly nitrogenous and concentrated, is of extreme value.\" As insurance against a fickle monsoon, the fisheries were hard to beat. He told the Lahore Industrial Conference in 1909 that, when he had first arrived in India, \"the time was not ripe for devising for the distant and vague harvest of the sea what was barely coming into contemplation for the harvest of the soil under foot\"; but now, the time was ripe for \"adding the harvest of the sea to the harvest of the soil.\" Nicholson made a crucial distinction: the government's interest in fisheries was purely as a source of food, and not as a source of revenue\u2014which is how the British had treated India's land from the earliest days of the East India Company's rule.\n\nNicholson traveled to Japan and to Denmark, seeking inspiration for the development of the fisheries of Madras. Like Visvesvaraya, Nicholson thought that Japan's example was most apposite for Madras. There, an \"ancient\" fishing industry had been transformed by \"scientific foresight,\" backed by an energetic government and generous investment; Nicholson suggested that India's fishing industry in 1907 was where Japan's had been before the Meiji Restoration of 1867. Nicholson created the Madras Department of Fisheries in 1907 and served as its honorary director. His genuine sympathy for poor fishing communities melded with a widely shared view that held \"poor,\" \"ignorant,\" and caste-bound hereditary fishers culturally inferior to landed cultivators. The result was a policy of gradualism. Nicholson doubted whether the rapid advance of fishing technology was possible in India; he preferred to build incrementally on existing practices using existing social structures; he was an enthusiast for cooperative societies as a way to foster a stronger sense of collective action among fishers.\n\nNicholson's successor, James Hornell (1865\u20131949), devoted years of his life to understanding the fisheries of India's eastern coast. At the turn of the twentieth century, after a decade working on the isle of Jersey, Hornell traveled to Ceylon to survey the marine fisheries there. From 1908 to 1924, he played a leading role in running the Madras Fisheries Department; he undertook detailed studies of coastal fisheries, on the economy of fishing and the changing composition of the catch; he developed a particular fascination with indigenous fishing vessels along the coasts of the Indian and Pacific Oceans, on which he published over a hundred articles in his lifetime. More than Nicholson, Hornell's caution about the wholesale technological transformation of India's fisheries was based on respect for fishers' traditions and their ways of life. In 1917, Hornell described the daily scene on the shore at Tuticorin, long a center of India's pearl fishing industry. \"There is no wholesale fish market except the beach, there are no companies or large owners controlling each a number of boats, and while there are certainly some fish salesmen and traders, these men seldom or never keep any accounts,\" he reported. \"The catch is usually thrown in a heap on the beach and the 'lot' as it lies is sold by auction\u2014the buyers must appraise its value by eye, and make their bids accordingly.\" But change was on the horizon; at sea, as on land, new technology and new commercial ambitions strained against the limits that British policies, and British interests, had placed upon them.\n\n# IV\n\nThe quest for water altered India's economic balance. For centuries, it had been a matter of common sense that India's wealth lay in its river valleys\u2014they were the most densely settled, the most intensively cultivated, the most prosperous regions. They had always been the core regions of political power. Wealth and poverty, it so often seemed, were a function of geography. The meteorological divisions of India mapped onto the standard of living. \"The most densely populated, and therefore the most fertile regions are those of silt deposit, the population becoming most dense towards the river deltas,\" wrote agricultural hydrologist Edward Buck in 1907. It appeared an immutable fact of nature\u2014but even by the time Buck was writing, the link between population density and agricultural productivity was no longer so clear. By the 1910s, the largest of Punjab's Canal Colonies brought more revenue to the government than any other district in India.\n\nArtificial irrigation sparked an agricultural boom in the drier regions of India's west and south: a boom focused on the production of high value crops for export. Public investment in irrigation poured into regions and crops most likely to bring the state revenue and to bring farmers profit. Punjab and Sind in the northwest, parts of Gujarat and Bombay Presidency in the west, and some parts of Madras\u2014the western region, around Coimbatore, and coastal Andhra in particular\u2014flourished. Those regions still remain India's most prosperous. The gap in productivity between these favored regions and the rest of the country has grown more marked in the second half of the twentieth century, but the roots of a fundamental reversal in India's economic geography lie in the water boom of the late nineteenth and early twentieth centuries.\n\nThe \"ancient\" zones benefited far less than the irrigated areas from new markets and new technologies. Riverine Bengal, the fabled wealth of which had drawn in the East India Company in the seventeenth century, now saw a protracted decline in its relative economic position within India. Agricultural productivity in the valleys of Tamil Nadu declined precipitously. A \"tide of indebtedness\" consumed smallholders in both regions, and also along the Gangetic valley. In both the boom regions and those left behind, the control of water as well as the control of credit concentrated land in fewer hands. Dry regions that had not benefited from irrigation did worst of all. By the 1920s, most agricultural land in India was still not irrigated, it was rain-fed. Already by the early twentieth century, irrigated lands produced four times as much as those that depended on rainfall. A recent survey by economic historians emphasizes the point: for most of India, most of the time, rainfall was the most important factor shaping agricultural output, and yields were probably among the lowest in the world.\n\nTwo large commissions of inquiry in the 1920s\u2014the Royal Commission on Agriculture in India, and the Banking Enquiry Commission, both on the model of the 1901 irrigation commission but on an even larger scale\u2014revealed Indian cultivators' continuing vulnerability to climatic fluctuations. Irrigation had advanced rapidly, but most Indian farmers did not benefit from it. \"Except in the north-west,\" the report on agriculture maintained, \"the whole country is dependent on the monsoon and all major agricultural operations are fixed and timed by this phenomenon.\"\n\nThe main response of Indian cultivators to this vulnerability was to borrow money. The scale of rural indebtedness in India emerged from the inquiry into India's banking system, a colossal exercise in information-gathering that went province by province, reporting its results in 1930 in dozens of thick volumes. \"From the sowing of the seed to the sale of the products,\" one of hundreds of witnesses in the United Provinces of North India testified, \"it is the indigenous moneylenders and bankers who enable the produce of the villages to be brought to the market.\" The monsoon shaped the rhythms of the money market. Cultivators' need for credit converged on certain crucial times in the season: credit for seed and manure in October and November, after the rains; credit to pay agricultural labor at harvest time. In many parts of rural India there was \"an exaggerated alternation of over-work and unemployment.\" Account after account from rural India made the same point. \"The annual rainfall is scanty and uncertain and irrigation is nominal,\" wrote a local official from Meerut district in the United Provinces; and \"a peasant proprietor once entrapped by the _mahajan_ [moneylender] can never extricate himself.\" Many borrowers faced 24 percent compound interest. \"Ninety-five per cent of the agricultural classes are in debt,\" wrote an administrator in Mathura district, \"due to successive failures of crops in this district.\" Only a few respondents disagreed. The collector of Kaira district in Bombay Presidency told the agricultural inquiry that monsoon failure was not one of the main causes of indebtedness: \"The frequency of very bad seasons in which the cultivator would be left completely insolvent is not very great,\" he noted.\n\nThe most creative response to the fact that most Indians still depended on the rains for their livelihoods came from J. S. Chakravarti, who worked for years in Mysore. He was a colleague of the great engineer Visvesvaraya; Chakravarti, too, flourished in the southern Indian princely state that was bolder than any part of British-ruled India in its approach to the problems of water. Chakravarti worked for Mysore's State Insurance Committee for many years, and in the 1910s rose to the position of controller and financial secretary to the government of Mysore. He advocated a system of drought insurance in preference over generalized crop insurance: while the failure of a particular crop could be down to bad practice or neglect on the part of individual farmers, a failure in the rains affected everybody. Where rainfall dipped below a certain proportion, say 35 percent, of average, cultivators would receive a payout. Chakravarti saw rainfall insurance as \"intimately connected with three sciences, _viz._ economics, meteorology, and agriculture.\" His starting point was that \"Indian agriculture is dependent almost entirely on rainfall\" and that the \"quantity of rain during the year and its distribution as regards time are almost the only essential factors\" in determining the incomes of farmers; absence (or excess) of rain at certain critical periods in the growing season could be devastating. And what Chakravarti called the \"rainfall factor\" was \"uncontrollable by human exertions.\" Only the state could carry out an insurance scheme on the scale he envisaged, though he hoped that private providers might eventually enter the market. The case for insurance was clear. \"Agricultural insurance will also be famine insurance,\" Chakravarti declared, for \"under the present circumstances, a famine in India does not generally mean grain-famine, but money-famine, due to enforced unemployment of the agriculturist.\" Chakravarti's scheme gathered dust; one Indian commentator observed toward the end of the twentieth century that Chakravarti's scheme was far in advance of what the World Bank had come up with by the early 1990s. The primary reason for this neglect is that a very different approach to mitigating climatic risk emerged in mid-twentieth century India\u2014an approach that emphasized technological solutions to the problem of water.\n\nReflecting on the state of India's development, the members of the Indian Industrial Commission wrote confidently in 1918 that \"the terrible calamities which from time to time depopulated wide stretches of country need no longer be feared.\" In a monsoon climate, \"failure of the rains must always mean privation and hardship,\" but it no longer need lead to \"wholesale starvation and loss of life.\" This conveyed a strong sense that something fundamental had changed in India over the first two decades of the twentieth century. The risk posed by climate had been mitigated by both policy (the early-warning system of the Famine Codes) and by technology (railways and irrigation). As long as India remained predominantly agrarian, some level of risk would remain, but the commissioners envisaged a future in which industrialization would provide new employment and greater security, as India's population moved from the countryside to the cities. In the 1870s, the idea that famine was inevitable in India prevailed among British administrators. By the 1920s, most observers believed that India had conquered famine. But anxieties about water did not go away.\n\n# V\n\nIndia's engineers fought to assert their sovereignty over the monsoon; climate science demonstrated, in those same years, that the monsoon moved to planetary rhythms\u2014rhythms far beyond human control. The pioneer of Indian monsoon meteorology in the early twentieth century was a brilliant mathematician and a modest man: \"kindly, liberal minded, wide of interest, and a very perfect gentleman.\"\n\nGilbert Thomas Walker was born in Rochdale, Lancashire, in 1868, the fourth child in a family of eight. He grew up in Croydon, just outside London, where his father was the borough's chief engineer. Gilbert received a scholarship to the elite St. Paul's School in 1881, and went on to a distinguished undergraduate career in mathematics at Trinity College, Cambridge. His talents were idiosyncratic. At Trinity he left behind him \"the legend of his prowess in throwing boomerangs on the Cambridge Backs\"\u2014his study of their aerodynamics marked the beginning of his fascination with the physics of the atmosphere; he acquired the nickname \"Boomerang Walker.\" In 1890, Walker suffered a breakdown in his health; he spent three summers recovering in Switzerland, where he developed a passion for skating. In time, both hobbies would nourish his insights into the world's weather. Returning to Trinity as a fellow in mathematics, Walker's work focused on electromagnetism. Formalizing his fascination with boomerangs, he also wrote an essay on the aerodynamics of sports and games. There was little in his background or experience to suggest that within a few years, he would be director of the Indian Meteorological Department.\n\nJohn Eliot was approaching retirement. His models for monsoon forecasts had grown more complex during his years in charge of the Indian weather service, but they were erratic in accuracy and had failed to predict the droughts of 1899 and 1900. Convinced, still, that he was on to something, Eliot searched for an able statistician as his successor, someone who could make sense of the profusion of pressure, temperature, and wind readings dispatched from observatories across the Indian and Pacific oceans. Walker's reputation reached Eliot's notice, perhaps through shared Cambridge networks; the chance to develop a new field of inquiry, in an unfamiliar land, was attractive to the younger man. Before sailing for India, Walker toured meteorological observatories in Europe and the United States: this was his crash course in the science of weather. Visiting field stations in the Midwest, Walker was especially impressed by the sophisticated techniques deployed by the US Weather Bureau.\n\nWalker arrived in India just three years after the last major famine. Eliot's failures had dented public and official confidence in meteorology. In 1904, Walker took over as chief. He kept a low profile for the next four years: marshaling resources, recruiting staff, bolstering a skeletal meteorological department. In those early years in Simla\u2014the Himalayan summer capital of the Raj, where British officials rushed to escape the pre-monsoon heat\u2014Walker found time for his two beloved hobbies. He was often spotted flinging boomerangs on Annandale, the only stretch of flat ground in Simla. He designed a low canvas screen to keep the ice cold on Simla's skating rink\u2014it was so effective that, by January, \"the ice was too hard to be skated on with pleasure,\" and the rink's owner asked him to remove it. Walker's mind leapt to make new connections. Many years later he revealed that his experience with the ice rink had led him to understand the \"extreme transparency of the air to heat radiated from the ground during the very dry winter periods\" over North India.\n\nFrom these beginnings came a vital breakthrough in climate science.\n\nWALKER'S APPROACH TO UNDERSTANDING AND PREDICTING THE monsoon was resolutely empirical. He relied on a vast \"human computer\"\u2014that is, on the labor of his Indian staff led by Hem Raj\u2014to process the numbers. \"The relations between weather over earth are so complex,\" Walker felt, that \"it seems useless to try to derive them from theoretical considerations\"\u2014the monsoon was too complex. Rather, Walker sought to amass and analyze as much weather data as he could, from all over the world. This had been Blanford's challenge: to determine how far the monsoon's causes as well as its effects were confined to \"India and its seas.\" Blanford's first forecasts relied on Himalayan snowfall; while he was increasingly aware of remote influences on India's climate, he assumed a closed system. Eliot moved to incorporate influences from across the Indian Ocean into his model, but he misunderstood the relationships at work. Walker, with more statistical tools (and more staff) at his disposal, broadened his parameters. His team processed a quantity of data that would have been inconceivable a generation earlier. The numbers told a clear and a new story: they suggested that \"the monsoon system extended to a pan-oceanic,\" even planetary, scale.\n\nWalker took aim, first, at the desiccationists. He showed that there was little evidence to support their claim that human activity, and particularly deforestation, had modified India's climate in the nineteenth century. However much the denudation of forests may affect the climate and the soil moisture of a particular locality, the scale of the monsoon system far outstripped such local influences. As his thoughts turned to monsoon prediction, Walker looked west of India. The Nile had long been on the minds of India's hydraulic engineers, as comparison, inspiration, or competition; Walker turned his attention to the annual Nile flood for different reasons. \"Inasmuch as the Nile flood is determined by the monsoon rainfall of Abyssinia,\" he wrote, \"and as the moist winds which provide this rainfall travel in the earlier portion of their movement side by side with those which ultimately reach the north of the Arabian Sea\" so there was a \"tolerably close correspondence\" between the extent of the Nile flood and the strength of the Indian monsoon. This was \"seasonal foreshadowing\" at work\u2014a term Walker preferred to the more confident \"forecasting.\"\n\nWalker's statistical prowess paved the way for his most startling discovery. Mining data from across the world, Walker noticed that \"there is a swaying of pressure on a big scale backwards and forwards between the Pacific Ocean and the Indian Ocean, there are swayings, on a much smaller scale, between the Azores and Iceland, and between the areas of high and low pressure in the North Pacific.\" The \"Southern Oscillation,\" as Walker named it, had a \"much greater\" influence on \"world weather\" than the other two. He called these pivotal areas of high and low pressure \"centers of action.\" Walker had identified an inverse relationship of atmospheric pressure at sea level across the Pacific Ocean, measured by readings from stations at Darwin and Tahiti. The usual pattern was for high pressure in Tahiti and low pressure in Darwin, driving the winds from east to west. The pressure contrast across the Pacific drove the storied westerly \"trade winds\"; but they were prone to periodic reversals that could last for one or two seasons. This was the mystery at the heart of Walker's findings. Changes in the location and intensity of the \"centers of action\" shaped the world's climate\u2014but what prompted these changes?\n\nWalker's immediate challenge, as head of the Indian weather service, was to determine how these \"centers of action\" affected India. The broad contours of the picture were clear by the early 1920s. \"Abundant Indian rains,\" he wrote,\n\n... tend to be associated with low pressure in India, Java, Australia and S. Africa; with high pressure in the central Pacific Ocean (Samoa and Honolulu) and South America (Chile and the Argentine); with previously scanty rainfall in Java, Zanzibar, Seychelles and South Rhodesia; and with low temperature in the Aleutian islands.\n\nHe sought correlations in time as well as in space; there were \"foreshadowings\"\u2014in Zanzibar or in the Aleutian Islands\u2014of dearth or plenty in India; Walker probed the \"lags\" of a season, or two, in the relationships he discovered. But the forces at work proved elusive: \"I cannot help believing that we shall gradually find out the physical mechanism by which these [oscillations] are maintained,\" Walker said in 1918. A few years later he told the audience at his presidential address to the Royal Meteorological Society that \"variations in activity of the general oceanic circulation\" would likely be \"far reaching and important.\" It would take another forty years for it to become clear just how \"far reaching and important\" the oceanic dimension really is.\n\nThe role of official meteorologist had left Walker little time for basic research; he took what opportunities he could. The First World War brought new challenges. The meteorological office had always scrabbled for resources; during the war, Walker's deputies were transferred out. G. C. Simpson and Charles Normand were sent to Mesopotamia in 1916, where Normand took charge of military meteorology. Their expertise formed part of a wider infusion of British Indian personnel into Iraq to accompany an even larger number of troops: hydraulic engineers and entomologists, together with a great many vessels built in Indian shipyards. In their absence, it fell to Hem Raj to keep Indian meteorology running. A veteran Indian officer in the department with a \"photographic memory\" for weather charts, Hem Raj oversaw the department's day-to-day operations. Walker later paid warm tribute to \"R.B. Hem Raj, who sacrificed his life in the cause of the allies by concealing a serious illness in order that he might continue his important assistance in an under-staffed office.\" Even as research ground to a halt in Walker's office, the war brought significant advances in global meteorological understanding, thanks primarily to the work of Vilhelm Bjerknes in Bergen\u2014together with his son Jacob and Halvor Solberg\u2014on the development of midlatitude cyclones. Bjerknes and his team brought a metaphor from the battlefields to their understanding of weather \"fronts,\" as they illuminated the dynamic interaction of warm and humid air currents with polar currents. It would be Jacob, five decades later, who finally identified the El Ni\u00f1o phenomenon, expanding on Gilbert Walker's insights.\n\nHowever wide he cast his statistical net, Walker remained a keen observer of his surroundings. He developed an interest in the flight of vultures and kites, which he watched through a telescope in Simla. He noticed that the birds knew where to look for updrafts, allowing them to ascend to up to two thousand feet without flapping their wings. This insight inspired Walker to take an interest in the physics of cloud formation; he even wanted to take up gliding upon his return to England, and was sad to find that \"at 65 his reactions were too slow to allow him to be a successful glider pilot.\" Walker left India in 1924, twenty years after taking over the Meteorological Department. He became professor of meteorology at Imperial College, London. Freed from the practical responsibilities of the Indian weather service, Walker turned his attention in the 1920s to developing his understanding of what he called \"world-weather\"\u2014to which he saw the Southern Oscillation as crucial. Walker remained eclectic in his interests and his methods. In 1927 he warned of the danger of over-specialization: \"There is, to-day, always a risk that specialists in two subjects, using languages full of words that are unintelligible without study,\" he wrote, \"will grow up not only, without knowledge of each other's work, but also will ignore the problems which require mutual assistance.\" There is a hint, here, of the quest to defend meteorology as a science, against those who would see it as mere observation. However global his perspective became, Walker never left the monsoon behind. One of his successors as director of Indian meteorology, Charles Normand, explained many years later why this was. \"The Indian Monsoon,\" Normand wrote, describing Walker's work, \"stands out as an active and not a passive feature in world weather.\" India, on this view, was a driving force in the world's climate. The irony was that \"Walker's worldwide survey ended by offering a promise for the prediction of events in regions other than India,\" since India's experience seemed \"more efficient as a broadcasting tool than an event to be forecast.\"\n\nSoon after the war, the advent of long-distance flights between Europe and Asia in the 1920s shed new light on atmospheric dynamics while demanding more comprehensive forecasts for aviators. From an observatory in Agra, the Indian weather service could now send balloons up to twenty thousand feet, which returned measurements of upper air conditions. Flight through clouds was giving scientists \"insight into cloud formation\"; aerial photography provided a new perspective on the vertical dimension of the weather\u2014\"we see how different a cloudscape seen from above is from one viewed from the ground.\"\n\nEVEN BEFORE THE WAR'S END, THE PROSPECT OF POLITICAL CHANGE was in the air in India. Under pressure from the eruption of mass political protest in India, to which we will return in the next chapter, the reforms enacted under the Government of India Act of 1919 devolved many responsibilities to the provincial level\u2014a system known as \"dyarchy.\" It expanded representative government, though electorates remained small. In parallel to this, a change in personnel began to be felt across the colonial administration\u2014a process of \"Indianization,\" as it was called, in the bureaucracy. Lower-level judges and immigration officials, health inspectors and government scientists were more likely, after 1920, to be Indian rather than British men\u2014but men they mostly remained. And there were limits to how high Indians could rise, which in turn fueled middle-class support for the nationalist movement.\n\nMeteorology moved in the same direction. In the early 1920s, new Indian officers were hired to senior positions in the Meteorological Department: G. Chatterjee, of Presidency College, Calcutta, took over the Upper Air Observatory in Agra; S. K. Banerjee, a noted mathematician, joined the department in Simla\u2014he would go on to become the first director of the Indian Meteorological Department after independence. In Bengal, a young statistician called Prasanta Chandra Mahalanobis, fresh from Cambridge, was hired to work at the observatory in Calcutta; his impact on the path of India's economic development would be immense. In 1921, Gilbert Walker wrote to Delhi, insisting that \"it is natural for the government to insist that a serious attempt be made to find and train Indians who shall be capable of carrying out the work of the Department,\" noting that there was little to distinguish the abilities of British from Indian staff. \"On political grounds,\" he noted, \"it is obvious that this policy of Indianization must be loyally accepted.\" J. H. Field, Walker's successor, went further. In 1925, Field pleaded for more resources\u2014he wanted six new posts, and a more energetic program of research. He justified his demands by insisting on the talent of Indian meteorologists, and by pointing to the new demands that would arise from air links between Europe and Asia. \"India has now for the first time an opportunity to show what an Indianised department can do,\" he declared. \"The opportunity is magnificent and unique: if the controlling department of the government will only rise to the height of this occasion and give me what I ask, it is to be expected that my Indian staff will justify demands as an outstanding example of efficiency in running their own concern.\" In an age of austerity Field did not secure the resources he had asked for, but the \"Indianization\" of the meteorology department was underway. The Indian officials who joined the department in the 1920s were \"nationalists to the core,\" writes D. R. Sikka, director of the Indian Meteorological Department in the late twentieth century. This did not prevent them from being \"loyal to the department,\" believing it to stand above politics. But in the 1920s and 1930s India's rainfall became ever more deeply political.\n\nIN HIS ACCOUNT OF ISAURA, ITALO CALVINO WROTE THAT THE city's water gods inhabited the whole vertical expanse of water. They lived, he wrote,\n\nin the buckets that rise, suspended from a cable, as they appear over the edge of the wells, in revolving pulleys, in the windlasses of the norias, in the pump handles, in the blades of the windmills that draw the water up from the drillings in the trestles that support the twisting probes, in the reservoirs perched on stilts over the roofs, in the slender arches of the aqueducts, all the columns of water, the vertical pipes, the plungers, the drains, all the way up to the weathercocks that surmount the airy scaffoldings of Isaura, a city that moves entirely upward.\n\nThe fictional Isaura is timeless, its waters unchanging. In early twentieth-century India, technology transformed water in every dimension. Electric pumps extracted water from the depths and balloons measured the moisture of the upper atmosphere; big dams harnessed the descent of rivers, for irrigation and flood control and to generate electricity. A vision of India that \"moves entirely upward\" sat alongside the older (and flatter) maritime conception of India at the heart of an imperial web of sea-lanes. The next chapter will turn to the story of how struggles for water were intensified by Asian nationalisms in the 1930s.\n\n# SIX\n\n# WATER AND FREEDOM\n\nAN ANCIENT DICTUM\u2014THAT THE CONTROL OF WATER CONFERRED political power\u2014acquired new meaning, and new urgency, in an age of nationalism. From India to China, water was at the heart of programs for political renewal and national development in the 1920s and 1930s. The rising generation of leaders in Asia included engineers, architects, and physicists, alongside lawyers and schoolteachers. Many of them felt that the conquest of nature in the early twentieth century had not gone far or fast enough. For inspiration they looked to the world's rising powers. They studied the New Deal in the United States. American technological modernity was epitomized by the Tennessee Valley Authority, which gathered together previously disparate approaches to flood control, river navigation, electricity generation, soil conservation, irrigation, and public health. Asian nationalists drew lessons from the breakneck industrialization and the colossal engineering schemes of the Soviet Union, not least because the Soviet Union was also a major Asian country that had attempted to reengineer landscapes that resembled those of China's and India's northwestern reaches, and has done so at a pace unprecedented in global history.\n\nIn India, in China, across Southeast Asia, nationalist movements were unstable social coalitions. Their leaders struggled to create a sense of unity and purpose, while acknowledging the fractures of social and economic inequality and addressing regional disparities. Many divisions emerged over the control and sharing of resources\u2014among which water was often the most vital. The 1920s and 1930s were characterized by deepening contacts and solidarities among anticolonial and revolutionary movements across and beyond Asia. But when it came to tangible material questions, like sharing water, they began to draw firm lines around their respective domains. In these decades between the world wars, the seeds were sown for water conflicts that would intensify in the second half of the twentieth century after Asian nations won their freedom from colonial rule.\n\n# I\n\nAs we saw in the last chapter, India's landscape was reshaped by the quest for water in the first two decades of the twentieth century. In the accounts of engineers the construction of canals and dams, and the pumping of underground water were purely a technical process, outside politics. Few of them, British or Indian, departed from the assumption that the British colonial government would lead the charge. But the rise of nationalism raised new questions about who would benefit from these changes in India's land and water.\n\nIndian nationalism emerged as a powerful mobilizing force in the first decade of the twentieth century. The Swadeshi movement arose in protest against a 1905 British plan to partition the province of Bengal, ostensibly for economic reasons, but also to divide what the colonial government perceived to be a threatening locus of political organization. \"Swadeshi,\" meaning \"home-made,\" began with the boycott of British goods in favor of locally produced products, but it burgeoned into a diverse movement that included those\u2014branded \"terrorists\" by the British\u2014who advocated the violent overthrow of the colonial government. The protesters achieved their immediate goal: the British revoked their decision to break up the province. But the Swadeshi mobilization was transient and fragile; it splintered into mutually hostile factions. It was largely confined to the province of Bengal, and even there, it was dominated by elite Hindus to the exclusion of Muslims. The Swadeshi movement mirrored similar uprisings elsewhere. Across Asia, the early twentieth century saw a wave of boycotts and demonstrations and strikes. The same year the Swadeshi movement began, Shanghai witnessed a widespread boycott of American goods in response to the wave of violence and discriminatory legislation directed against Chinese immigrants in the United States in the early twentieth century. By the 1910s, these stirrings of unrest had turned into mass movements.\n\nIn India, the most effective and visible political leader was a lawyer named Mohandas Karamchand Gandhi. Born to a merchant family in Porbandar, a port town in Gujarat on India's western coast, Gandhi spent decades outside India. He studied law in London between 1888 and 1891. There he came under the influence of the spiritualist Theosophy movement; he discovered vegetarianism; he experienced a political and spiritual awakening that led him to the study of Indian philosophy and religion. In 1893, Gandhi took up an offer of a job as a lawyer in South Africa. He quickly came to lead protests against the race-based exclusions and restrictions faced by the Indian community in South Africa\u2014a diverse group that included Gujarati merchants and traders, concentrated in Durban and Johannesburg, as well as indentured workers from Tamil Nadu and Bihar, who worked on the sugar plantations of Natal. Gandhi, like most of South Africa's Indian community, supported the British in the South African War, a brutal contest between English and Afrikaner settlers. Hopes that the British would reward Indian support after the war proved short-lived. The rapprochement between the English and the Afrikaners in the postwar settlement led to a tightening of restrictions on the Indian community, including a requirement for them to carry identity cards (\"passes\")\u2014though, always, the colony's African majority faced discrimination that was far worse. Immersed in reading Tolstoy and Thoreau, Gandhi experimented with communal living at a settlement named Phoenix. He started a printing press. He honed his political tactics\u2014a form of nonviolent civil disobedience that he dubbed _satyagraha_ (\"struggle for truth.\")\n\nDuring his South African years, Gandhi developed a critical account of British rule in India. He published _Hind Swaraj_ in 1909, a treatise that took the form of a dialogue with an imagined reader. Gandhi took aim not only at the violence and tyranny that underpinned British rule in India but also, more radically, at its material effects. \"India's salvation consists of unlearning what she has learnt during the past fifty years or so,\" he wrote. \"The railways, telegraphs, hospitals, lawyers, doctors and such like have all to go, and the so-called upper class have to learn to live consciously and religiously and deliberately the simple life of a peasant.\" Gandhi concluded that \"machinery is the main symbol of modern civilization; it represents a great sin.\" In rejecting \"telegraphs, hospitals, lawyers [and] doctors,\" Gandhi was being provocative; he aimed to shock his readers into asking questions about the ultimate ends of India's embrace of industrial modernity.\n\nGandhi's analysis stood at odds with the rush to secure India against vulnerability to nature\u2014a process which, we have seen, involved many Indians alongside British water engineers and administrators. Over the years, Gandhi developed further the idea that India's freedom lay in living with the rhythms of nature. He was repulsed by India's cities, though his vision of the country as an agglomeration of \"village republics\" was largely a myth drawn from the writings of British orientalists like Henry Maine. As a symbolic figure, as a tactician, Gandhi was unrivaled within the Indian nationalist movement. His economic ideas remained marginal. They stood as a quiet counterpoint to the powerful tune of more technology, more control, more progress. Most of Gandhi's associates and many of his followers had a different view of what India needed\u2014the \"simple life of the peasant\" was precisely what they aimed to relegate to the past.\n\nWhen Gandhi returned to India in 1915, he hurled himself into political activity. He had already acquired the honorific \"Mahatma\" (\"great soul\"); his reputation as an effective organizer and powerful speaker had traveled with him from South Africa. He began on a small scale, interceding on behalf of indigo workers in Champaran, Bihar, who were protesting their exploitative working conditions. By 1917, Gandhi was India's preeminent politician. He jolted the Indian National Congress to life, expanded its membership, forged a coalition of rural and urban supporters. Gandhi rejected the class-based mobilization of the left in favor of an emphasis on conciliation; among his supporters were India's largest industrialists, including the Birla family. Gandhi launched his mass Non-Cooperation Movement in 1919, in protest against the slow pace of political reform in British India and directed in particular against the prolongation of the state's wartime emergency powers. A campaign of protests and boycotts, fasts and vigils, lasted until 1922, when Gandhi called it off following an act of violence in the small town of Chauri Chaura, where Congress supporters had attacked a police station. A cycle of repression and concession would unfold over the subsequent two decades. The British government of India locked Gandhi and his lieutenants up in prison on many occasions, interspersed with periods of negotiation.\n\nThe nationalist upsurge spanned Asia. The Non-Cooperation Movement in India raged at the same time that China saw an outpouring of social and political protest against the territorial concessions that Japan, victorious on the Allied side, had gained in China after the First World War. A wide coalition of youth and students and activists came together in a loose grouping known as the May Fourth Movement. In Vietnam and in Indonesia, too, the 1920s saw the rise of new political and social movements directed, respectively, against French and Dutch colonial rule. In all three countries, unlike in India, communism emerged among the most powerful and most compelling of political movements.\n\nAsia's nationalist movements spoke the language of freedom and sovereignty, and it is on the richness and multiplicity of these concepts that historians have focused their attentions. But there was always a strong material underpinning to the ambitions of Asia's nationalists. It is here that the history of nationalism intersects with the battle to bring unruly waters under control. Nationalist leaders needed water, mineral resources, and fossil fuels to realize their plans for industrialization, to make good on their promises of an end to hunger and poverty. A new sense of confidence crept into visions of Asia's future. Consider this contrast: In 1909, the imperial finance minister of India had characterized each of his budgets as a \"gamble on the rains,\" conveying a sense of fatalism about the power of nature over economy and society. Twenty years later, Jawaharlal Nehru\u2014Cambridge-educated lawyer and scion of an elite Allahabad family, son of pioneering nationalist Motilal Nehru, and by the 1920s Gandhi's most trusted younger colleague, and one of India's most influential and charismatic politicians\u2014declared that \"modern science claims to have curbed to a large extent the tyranny and the vagaries of nature.\" Nehru was clear about the material urgency behind every vision of freedom. \"Our desire for freedom is a thing more of the mind than the body,\" Nehru said, but most Indians suffered \"hunger and deepest poverty, and empty stomach and a bare back.\" For the masses, \"freedom is a vital bodily necessity.\"\n\nIn China, as in India, water was a vital ingredient of freedom. The control of water was essential to China's emergence from a century of humiliation at the hands of imperial powers, which had culminated in 1911 with the collapse of the Qing dynasty. Sun Yat-sen, architect of China's republican revolution, applied himself to the problem of China's development, even as rival regional polities tore China apart. In _The International Development of China,_ Sun set forth an expansive vision of China as what he called an \"economic ocean\" for the world. His book was replete with maps of rivers diverted, maps of rail lines laid, maps of ports dredged and electricity generated. Water was at the heart of his vision. Sun told a meeting in Guangdong in 1924: \"If we could utilize the water power in the Yangtze and Yellow rivers to generate one hundred million horsepower of electrical energy, we would be putting twenty-four hundred million men to work!\" Sun predicted that \"when that time comes, we shall have enough power to supply railways, motor cars, fertilizer factories and all kinds of manufacturing establishments.\"\n\nIn contrast with India, where the focus had long been on irrigation, Chinese river engineering in the early twentieth century focused on flood control. Though China, too, had suffered from the great droughts of the 1870s and 1890s, it had also experienced disastrous river flooding on a scale unknown in India. By the 1920s, the Yellow River, famously silt-laden, posed a particular challenge\u2014a challenge embraced by an international corps of engineers. Two Americans, John Freeman and O. J. Todd, played a central role; their Chinese prot\u00e9g\u00e9s included Li Yizhi (1882\u20131938), whose stature in China was akin to Visvesvaraya's in India. Li studied in Berlin and then visited hydraulic projects across Europe; he was aided in his work by a new cadre of Chinese graduates from MIT and other leading American institutions. \"To manufacture cotton into yarn, to grind grain into flour, to light cities and otherwise modernize this part of Shansi,\" Todd wrote, \"will be part of the benefit that these Yellow River Falls may confer on the nearby country.\" His ambition found many echoes across China, in India, and in other parts of the colonized world. But beneath that ambition was an enduring sense of fragility.\n\nTO HARVEST WATER WAS TO REDRESS THE INEQUALITIES OF nature\u2014to even out the uneven reach of the monsoon, to ensure against the particular unpredictability of the rains in the places that needed it most. But water was also an engine of inequality between people, between classes and castes, between city and country, between regions. The command of water underpinned the accumulation of land. The control of water was a source of power; its absence, a source of enduring exclusion. In the first three decades of the twentieth century, water was at the material heart of many struggles for freedom\u2014but freedom for whom?\n\nThe question arose forcefully in the western Indian town of Mahad, near Poona, in March 1927. The local Dalit community\u2014those excluded from the Hindu caste system, once known as Untouchables, whose daily lives were marred by residential and occupational segregation as well as by violence and material deprivation\u2014were denied access by upper-caste Hindus to a local tank containing drinking water. Although a court had ruled that this exclusion was illegal, it continued\u2014as it did in countless towns and villages across India, as it still does today. Dalit leader Bhimrao Ambedkar\u2014a brilliant lawyer from a poor family in western India, who had received scholarships to study at the London School of Economics and Columbia University\u2014led a march to the tank. He drank a symbolic cup of water from the reservoir. The retaliation from local caste Hindus, who felt their social dominance under threat, was brutal and immediate. Dalits were attacked; many lost their jobs. \"We now want to go to the Tank only to prove that, like others, we are also human beings,\" Ambedkar declared, as he launched a _satyagraha_ with four thousand volunteers. At the last minute, he called off the movement, trusting in the courts to deliver justice for his community. It took a decade for Ambedkar's trust to be vindicated, when a further ruling insisted the tank be opened to all\u2014contradicting the caste Hindus' claim that it was private property, and therefore that they were free to exclude whomever they chose from the tank's waters.\n\nIn the broadest terms there remained a tension at the heart of the Indian nationalist movement. As one political theorist has described it, it was a tension between, on the one hand, \"social freedom from caste domination,\" and, on the other hand, an overriding emphasis on the immediacy of \"political freedom from colonial rule,\" deferring or subsuming those other struggles. Ambedkar and Gandhi would find themselves on opposite sides of that debate, coming into conflict in the 1930s over whether Dalits should have separate representation in the legislative councils of British India, which India's Muslims already had. It is no coincidence that both leaders invoked the symbolic as well as material power of water. For his part, one of Gandhi's most effective and iconic campaigns was the \"salt march\" to the sea at Dandi, in 1930. Choosing the British salt tax as the symbolic focus of his _satyagraha,_ Gandhi observed that \"next to air and water, salt is perhaps the greatest necessity of life.\" The vital properties of salt linked the coastal ecosystem with the lives of millions inland. Gandhi's was an argument about climate and society\u2014the poorest, who labored outdoors in the heat, were most in need of salt. Where Ambedkar's march on the tank had drawn attention to water as an indicator of profound social inequality, Gandhi used it as a symbol of unity. Within and beyond India, competing claims on water and resources escalated in the 1930s.\n\n# II\n\nHow far could Asia's environmental inheritance be molded? What was the potential of technology to transform Asia, to make use of water and to make water available to all? Contending answers to these questions played out in the decades between the two world wars. Iron confidence in the conquest of nature, expressed by engineers and scientists and nationalists, alternated with a sense of vulnerability before nature's power and its unpredictability. As new knowledge of the monsoon became more widely known, climate itself provided a new way to think about Asia, its boundaries\u2014and its future.\n\n\"I wish to treat the monsoon as a way of life,\" wrote Japanese philosopher Watsuji Tetsuro (1889\u20131960) in the late 1920s; this was \"something that a hygrometer cannot do.\" And so the fullest expression of the idea that the monsoon constituted the essence of India came not from a European but a Japanese observer. Watsuji was a scholar of Japanese ethics and aesthetics. He translated the works of S\u00f8ren Kierkegaard into Japanese. He traveled to Germany in 1927 to study with Martin Heidegger. His journey took him by way of Southeast Asia, India, and the Middle East. During and after his journey he wrote _F\u016bdo,_ loosely translated as \"climate\"; it was Watsuji's response to Heidegger's _Sein und Zeit_ (Being and Time). _F\u016bdo_ was not translated into English until 1961. It is unlikely that the book was widely known in India. But India was central to the book's argument that climate shaped culture, society, and history. _F\u016bdo_ is unusual in contrasting India's climate not primarily with Western Europe's, but with Japan's and China's. Watsuji's work was part of a larger intellectual and political movement in Japan to think about Asian societies\u2014their similarities and contrasts\u2014in light of European domination of the world, and in light of Japanese ambitions for regional supremacy.\n\nThe humidity of a monsoon climate, Watsuji believed, \"does not arouse within man any sense of a struggle against nature,\" unlike in desert lands. The \"distinctive character... of human nature in the monsoon zone,\" he insisted, \"can be understood as submissive and resignatory.\" This was in part because of the monsoon climate's doubleness: it \"typifies the violence of nature\" with its huge storms, \"the power is so vast that man is obliged to abandon all hope of resistance\"; but this is \"a threat filled with power\u2014a power capable of giving life.\" In Watsuji's eyes India represented the most extreme manifestation of a monsoon climate. \"It is the rainy season, brought by the monsoon, that has done most to create the resignation of the Indian,\" he noted. He observed that \"over two-thirds of India's 320 millions (a fifth of the world's population) are farmers and grow their crops thanks to the monsoon\" and so \"whether it is late, whether it lasts its due time\" are \"matters of great moment.\" India's masses, Watsuji argued, had \"no means of resistance against nature.\" There was \"no escape for India's people from such insecurity of life.\" This insecurity brought about \"a lack of historical awareness, a fullness of feeling and a relaxation of will power.\"\n\nThis was a familiar pattern of argument: a familiar set of stereotypes about Indians as lazy and emotional. Nineteenth-century British liberals claimed that Indians lacked the rationality for self-government. They were too close to nature. Watsuji drew on this intellectual tradition; but in his writing we also see a distinct sense of Japan's historic mission to \"save\" Asia from European domination and from its own backwardness. \"The people of the South Seas have never made any appreciable cultural progress,\" he declared, but \"there would be startling advances if some way were found to break this mold and set this teeming power in motion.\" The resignation of Indians, he wrote, \"prompts in us and draws out from us all our own aggressive and masterful characteristics.\" It was \"on such grounds that the visitor to India is made to wish impulsively that the Indian would take up his struggle for independence.\" A struggle, by implication of this circular reasoning, that could only be guided by peoples whose climates had endowed them with different traits. Watsuji implied that the Japanese were better placed to lead this charge than Europeans. Westerners could never truly understand the monsoon, whereas Japan had its own experience of tropical climates on the southern fringes of the archipelago, and on its model colony of Taiwan. Watsuji was not alone. Between the wars, many Asian students, scientists, and political leaders contemplated the relationship between nature and power, between nature and empire, between nature and nation. Watsuji Tetsuro concluded that for India's future, \"change depends upon the conquest of climate.\" Stripped of its moral, even spiritual, connotations, that conquest was, ultimately, a question of technology.\n\nA less abstract perspective, but one that shared Watsuji's concern with how climate and ecology shaped culture, came from the Bengali sociologist and economist Radhakamal Mukerjee, a professor at Lucknow University who devoted much time and many pages in the 1920s and 1930s to the problems of rural India. Mukerjee was deeply concerned with water. In recent years the eccentric and eclectic Mukerjee has been recovered by historians as a prophet of an ecologically sensitive and localist approach to development\u2014but he cuts an ambiguous figure. He was a committed eugenicist; he absorbed the racial and environmental determinism of his time and then inverted it, calling, for instance, for _lebensraum_ for the \"teeming millions\" of India and China. Nevertheless, his was a rare voice of concern about India's environmental balance at a time of rapid development\u2014and his concerns were more tangible and specific than, say, Gandhi's. \"Man, tree, and water cannot be regarded as separate and independent,\" Mukerjee wrote; he decried \"crimes\" against nature that would in turn \"[let] loose destructive forces.\" Wise development, Mukerjee argued, would pay heed to the \"natural balance of man with the organic and inorganic world around him.\" Only in that balance could human society find \"security, well-being, and progress.\"\n\nMukerjee's prescription for India's future came from his close study of the riverine landscape of his native Bengal. Drawing on the work of the Russian anarchist geographer L\u00e9on Metchnikoff, who in 1889 published a wide-ranging history of riverine civilizations (including the Ganges valley), Mukerjee thought of river basins as living entities. Each river was \"a synthesis or epitome of all the possible environmental variations and influences\"; each river's \"properties, colorations and varied taste,\" as well as its \"plastic or destructive power,\" was a product of climate and geology. Mukerjee's diagnosis was that the vital force of Bengal's rivers had been eviscerated by more than a century of British rule. He observed the deterioration of soil quality from overintensive cultivation. Others had observed this worrying trend, and ascribed it to the pressure of population. But the root problem, as Mukerjee saw it, was that \"agriculture comes to be influenced rather [more] by the state of the market than by an arranged succession of crops which may replenish the soil.\" Pressing on the ecology of land and water, the demands of the colonial state and capitalists for the products of the soil had left the Bengal delta \"moribund.\" But where did the roots of its revival lie? For Mukerjee, as also for the British hydraulic engineer William Willcocks\u2014famous as the architect of the first Aswan Dam on the Nile River\u2014part of the answer lay in recovering and reviving local traditions of irrigation and water management. For others, as we will see, only the wholesale transformation of nature by technology would match the scale of the challenge.\n\nAnother part of Mukerjee's concern echoed the debate of the early twentieth century about India's place in the world\u2014a debate that ranged across many fields of science and politics\u2014over whether India was better seen as a bounded territory or as part of an oceanic realm. Of all of the ways that human beings had \"gained a gradual mastery of the waters,\" he argued, \"by far the most significant development is trade by sea.\" India's maritime connections, Mukerjee observed, \"usher[ed] in an oceanic civilization superseding the fluvial.\" The resources of the river valleys were \"narrow and limited\" in comparison with oceanic commerce, which \"extends as wide as the world.\" The more that traffic on the sea-lanes sucked up the produce of the river valleys, the sharper their decline became. Demand from distant markets upset what Mukerjee called \"ecological balance.\" But he was optimistic. He felt that the excesses of \"oceanic civilization\" were now apparent; he looked forward to the moment when \"man becomes more agriculturally inclined than ever before and atones for his past neglect.\" He was to prove prescient\u2014though the motive force of a return to agriculture, and a revival of the river valleys, was not atonement so much as necessity.\n\nIF THE INTEGRATION OF INDIA WITH REGIONAL AND GLOBAL MARKETS had placed new demands upon soil and water, the collapse of those markets in the 1930s created new dilemmas. For the first two decades of the twentieth century, many rural communities in India had relied on resources from overseas to survive\u2014exports from the rice fields of Burma, and the remittances that came back to India from the wages of migrant workers in Burma, Malaya, and Ceylon. This was the key to a puzzle that historian Christopher Baker confronted in a brilliant and neglected 1981 essay on the economic integration and subsequent disintegration of Asia. In India, as in China and Java, the 1920s marked the \"critical point\" when \"land ran out,\" Baker wrote. What demographers have struggled to explain is that, despite dire warnings in the 1920s, no Malthusian crisis ensued. To the contrary, population growth gathered pace even as agricultural yields declined year after year. The answer, Baker saw, lay in the interconnected regional economy that provided a lifeline for the densely settled agrarian heartlands of southeastern India or southern China, providing new opportunities for long-distance migration for their young men and a smaller but still significant number of young women. The expansion of rice cultivation along the Irrawaddy, Mekong, and Chao Phraya river basins after 1870 added around 14 million acres of new rice-growing lands in fifty years. The opening of this final frontier of cultivation was accompanied by vast migration from India and China to Southeast Asia. More than 20 million passenger journeys traversed the Bay of Bengal, and a similar number the South China Sea, in the half century after 1870. Migrants went to work on the rubber plantations and tin mines of Malaya, on the tobacco fields of Sumatra, on the docks and in the mills and factories and on the streets of the growing port towns of Singapore and Rangoon, Penang and Surabaya. Many of these journeys were temporary, their pattern circular. Violence was never far from the experience of migrant workers; they traveled under a variety of arrangements and agreements, founded on debt. But Southeast Asia provided a horizon of opportunity, however fragile; year after year, the number of new arrivals in Southeast Asia outstripped the number of people heading back home.\n\nThe global economic depression of the 1930s changed everything as it disconnected the regional economies of South and Southeast Asia. The depression made the inequalities of colonial capitalism starkly visible. Frustration about rising unemployment and intolerable debt found an outlet in anti-immigrant sentiment; mass political movements began to speak of redistribution. The collapse of global commodity markets led to a reversal of the flows of migration that had become entrenched over sixty years. The number of Indians departing Burma and Malaya exceeded the number of arrivals between 1930 and 1933; the same was true of the Chinese throughout Southeast Asia, despite the fact that China in those years suffered both from civil strife and from escalating Japanese military intervention. More than six hundred thousand people left Malaya between 1930 and 1933. They had to fend for themselves when they returned home. The Indian government's agent in Malaya noted that repatriation to India in times of distress \"is proving less and less effective as a remedy against unemployment.\" Tamil workers in Malaya received no relief, and \"their suffering is merely transferred from Malaya to South India.\" John Furnivall, Burma-based British scholar, administrator, and Fabian socialist, wrote with prescience in 1939 that \"we can already see that 1930 marks the... close of a period of sixty years, beginning with the opening of the Suez Canal, and, although less definitely, the close of a period of four hundred years from the first landing of Vasco da Gama in Calicut.\"\n\n# III\n\nWhen the colonial government of Madras opened the Mettur Dam along the Kaveri River in 1934, it was for a brief moment the largest dam in the world. It had been in the works for almost two decades. The dam could \"boast of controlling works that leave those at Assouan [Aswan] well behind,\" one newspaper report declared, revealing how far water engineering had become a global endeavor. Mettur was three times the Aswan dam's length, standing 5,300 feet long, 176 feet high, 171 wide, and boasting a sixteen-foot roadway on top. The idea invoked, again and again, was control. \"Rivers in India are not all tidy instruments,\" the columnist observed. India's rivers had a will of their own: \"not many of them are content just with carrying water from mountain to sea,\" he wrote, \"they love to spill it on the way... to damage while they enrich the lands they flow through.\" He drew a clear lesson: \"They do not restrain themselves, and must be restrained.\" Not all observers were so sanguine. In a handwritten note on a file in the Tamil Nadu archives in Chennai, a civil servant who signed off as \"SA\" took a dim view of what seemed to be the hubris behind the Mettur Dam:\n\nThe Superintending Engineer's report is too self satisfied, or takes too much for granted, the infallibility of the officers of the department and the rank ignorance and prejudice of the ryots [farmers]. I do not think the position in Tanjore district is quite so very simple as the... report makes it out to be. While I am behind no one else in my admiration of the skill of our engineers who have built the great dam at Mettur and have succeeded in opening out the possibility of providing efficient irrigation facilities for large tracts of Country, the problem in Tanjore district has yet to be studied with sympathy and local knowledge.\n\nBut SA's was the voice of a minority. The \"restraint\" of water, more than any other single solution, promised to address so many of the problems that came together to create a sense of agrarian crisis. Declining soil fertility, falling crop yields, the closing of overseas frontiers for migration and the depression's shock to trade\u2014all of these combined with a broader sense of the enduring unpredictability of a monsoon climate. But every scheme to control water had the potential to create conflict between users upstream and downstream, between beneficiaries and losers. Since those unequal benefits often fell on either side of a political boundary, attempts to control water sharpened awareness of borders. Everywhere, as claims multiplied on river water for irrigation and power, so too did efforts to claim water as territory. The Mettur Dam had run into just this problem\u2014that is why it was so long in the making. The Kaveri River flowed through both Madras Presidency and the princely state of Mysore. Mysore was quicker than Madras to attempt to harness the river, thanks in large part to the work of the engineer M. Visvesvaraya. But if Visvesvaraya's plans for his Krishnarajasagar Dam were realized, the British claimed, Mettur would not have enough water. The tangle that ensued became the first, and certainly not the last, territorial dispute over water in modern India. The first treaty between the governments of India and Mysore over water dated back to 1892; at that time of agricultural intensification, it was already clear that conflict might lie ahead. Unable to reconcile their dispute over Krishnarajasagar, both sides went to arbitration by the imperial government of India and agreed upon a technical solution: the tribunal decided on the exact quantities of water that Mysore and Madras were entitled to. Neither side was entirely satisfied, but they signed an agreement in 1924. The distribution of Kaveri water has continued to haunt the Indian states of Tamil Nadu and Karnataka since 1947\u2014a point of recurrent conflict, both within and outside of the courts.\n\nMany of Asia's leaders believed that centralized planning would balance the needs of different constituencies. Planning would address the conflicts that arose between regions and communities; it would distribute resources in the most equitable and efficient way. Water resource planning took its place alongside economic planning in China as well as in India. In 1933, and in the aftermath of disastrous floods two years earlier, Chiang Kai-shek, who was by that time in command of a large part of China, assembled the Yellow River Conservancy Commission. Just as Indian engineers started to imagine the uses of water beyond irrigation, so China's planners too moved toward multipurpose water projects. Foreign engineers were drawn to the challenge that the Yellow River's control posed. Eminent German hydrologist Hubert Engels set up a Yellow River research center in Dresden; the League of Nations, too, lent its support and expertise to the Yellow River commission. China was an independent republic in the 1930s but faced a growing threat from the territorial expansion of the Japanese empire. In India, the Congress party\u2014which won large majorities in the elections of 1937, held under an expanded franchise\u2014began to think about India after British rule, even if the arrival of freedom seemed to lie in the distant future. The Congress party's National Planning Committee, convened in 1938 by Jawaharlal Nehru, brought together a coalition of left-leaning nationalists, Gandhian thinkers, industrialists, and scientists including Radhakamal Mukerjee. It saw itself as a state-in-waiting. The group formed several subcommittees, of which one dealt with \"River Training and Irrigation,\" chaired by Nawab Ali Nawaz Jung, chief engineer of the princely state of Hyderabad. The committee reported that \"it is important that our rivers should be developed to the greatest possible extent and effectively utilised.\" It was a task that could not wait: \"Conservation of water by storage,\" they concluded, \"has become a matter vital to the future\" of India.\n\nTHROUGHOUT THE 1930S, INDIAN AND CHINESE PLANS TO CONTROL water proceeded with each oblivious to the other\u2014and Chinese plans were soon consumed by the crisis of war with Japan. It would be a long time before their river engineering projects put them on a collision course. But there were portents of trouble to come. In the early 1930s, there was a flare-up of tension on the fringes of British and Chinese control\u2014on the border between Burma, still ruled as part of British India, and Yunnan. A secret British intelligence file went into great detail on \"Chinese Claims to the Irrawaddy Triangle\"; it was filled with correspondence and translated pamphlets and newspaper articles, all deployed as evidence that the Chinese state was making a \"fantastic claim... to the whole of Burma north of latitude 25\u02da35N, right up to the Assam border.\" On the Chinese side, William Credner, a geographer sympathetic to Chinese nationalism and based at Sun Yat-sen University, undertook an expedition to the Irrawaddy triangle along with three Chinese officials in 1930. They sought to address the \"long-outstanding question of the undemarcated northern and southern sections of the Yunnan-Burma boundary,\" left undefined in a treaty of 1894. They protested successive British military expeditions in the area, which the British justified on the grounds of suppressing a local slave trade. The Chinese party \"advanced far into the uncivilized and remote districts,\" he wrote, and undertook the \"plentiful collection of information\"\u2014not only on the \"boundary question,\" but also on the \"topography of the region.\" For now, the border was important as a symbolic marker of Chinese sovereignty: \"It is hoped that Yunnanese of all classes will unite in striving to prevent the territory from being treated as a British colony again,\" an intercepted Chinese memorandum declared. But there was also a hint, in the close attention to landscape and the flow of rivers, that frontier regions would become vital for other reasons, too: for their water and their mineral riches.\n\nThe question of borders arose, in a different sense, in India's fisheries. By the 1930s, V. Sundara Raj had succeeded James Hornell as the first Indian director of the Madras Fisheries Department. Unlike his predecessors, he looked forward to the wholesale transformation of India's fisheries by technology. Writing at the height of the Depression, as the regional economy had contracted and patterns of inter-regional migration had reversed, Sundara Raj worried that the Ceylon government had begun \"deep-sea fishing experiments\" with a trawler, in what he saw as water belonging to Madras. He pointed to Malaya, too, and the \"great awakening in these sister states\"; his concern was that \"other Governments will exploit the Madras fishing grounds.\" He repeated his request, denied the first time around, for a trawler and a cutter, to commence his own deep-sea exploration. Sundara Raj saw \"intensive ocean research and exploration of ocean grounds\" as a global trend\u2014he cited examples from Japan, Canada, and the United States.\n\n# IV\n\nFrom the late 1930s Asia was embroiled in war. China's experience of war was most prolonged, and most traumatic. Beginning with the annexation of the northeastern Chinese region of Manchuria in 1931, the Japanese empire advanced, propelled by the actions of local military commanders. Japan's rulers eyed China's mineral resources, its strategic position, and its territorial expanse. General Chiang Kai-shek's very success in gaining control over China presented a threat to Japanese ambitions, which had been well served by China's internecine strife in the 1920s. In 1937, simmering conflict erupted into full-scale war. Under pressure of conflict, the best laid plans for the development of water resources went awry, with catastrophic effects. In June 1938, retreating Chinese troops breached the Yellow River dikes in Huangyuankou, in Henan Province, to stop the Japanese advance on the Nationalist stronghold of Wuhan. It was, in the words of one historian, \"the single most environmentally damaging act of warfare in world history.\" Its dikes breached, the Yellow River rushed southeast, spilling into the Huai River system and drowning tracts of flat land on its way. More than eight hundred thousand people were killed, and 4 million displaced, by this desperate act of hydraulic sabotage.\n\nThe war in Asia spread in December 1941, when Japanese forces simultaneously attacked Pearl Harbor and swept through Southeast Asia. Within a year, the Japanese empire had absorbed a region that had, since the nineteenth century, been divided among imperial powers. They conquered British-ruled Malaya and Burma, the Dutch East Indies, French Indochina, and the American-ruled Philippines. The fall of Burma brought the threat of a Japanese invasion to India's borders.\n\nThough Indian territory saw little fighting, it became a vast supply base and center of operations for the Allied war effort in Asia\u2014and Indian troops constituted a sizeable contingent of Allied forces in every theater of war. The war also transformed Indian politics. Incensed that the British government had declared war on India's behalf without consulting Indian politicians, the Congress party resigned from the provincial governments it had controlled since the elections of 1937. After the failure of negotiations with a British delegation led by Labour politician Sir Stafford Cripps in August 1942, Gandhi launched another mass campaign of civil disobedience\u2014the Quit India Movement. Parts of North India became ungovernable. The British responded by dropping bombs on civilians to quell the revolt. In their search for support as Congress party leaders languished in prison, the British turned elsewhere. The war boosted the power and the standing of the Muslim League and its leader, Muhammad Ali Jinnah, who had in 1940 passed the \"Pakistan resolution,\" calling for the establishment of a homeland for India's Muslims\u2014though how, where, and when were questions left deliberately unclear. The British were forced to concede that India would gain freedom, in some shape or form, after the war.\n\nAS HISTORIAN SRINATH RAGHAVAN HAS SHOWN, THE WAR LED TO A vastly expanded role for the state in the economy, laying the groundwork for the apparatus of planning in independent India.\n\nAmong other fields, the war gave a boost to meteorology, as India became a hub of military aviation. The Indian Meteorological Department grew fast: its budget trebled between 1939 and 1944, and it established a new base of operations on a thirty-acre campus along Delhi's Lodhi Road. It proved difficult to find and train enough staff to keep pace with the expansion of facilities. Some of India's leading meteorologists suffered loss and hardship during the war. Most of the staff of the Burma Meteorological Department were Indian; and when Japanese bombing raids began on Rangoon, they joined the exodus of up to a half-million Indian refugees\u2014most of whom walked back to India, through jungle and mountains, into Assam. The director of Burma's weather service, S. C. Roy, walked from Rangoon to Imphal. One of his deputies, S. N. Ghosh, survived the long trek only to be killed in a Japanese bombing raid on the Indian border. The war saw the recruitment of a new cadre of meteorologists in India\u2014the generation that would staff India's weather service after independence. The meteorology department had three times as many staff by the end of the war as it did at the start. In 1944, Charles Normand retired as its director after thirty-one years working for the department; his successor, S. K. Banerji, was the first Indian to head the meteorological service. The war saw the beginning of aircraft weather reconnaissance over the Bay of Bengal, through a series of flights between Madras and the Andaman Islands. It also witnessed a breakthrough in communications technology. The India Meteorological Broadcast Center was established at the Royal Air Force base in Nagpur, in central India. The Royal Air Force and the United States Air Force installed the first teleprinters in India for the transmission of weather data.\n\nThe development of meteorology was oriented by military needs. In weather forecasting as in medicine, civilian applications for the new technologies were a low priority. However much the new technologies promised, India's experience of the war shattered the complacent assumption, pervasive by the 1930s, that nature had been conquered.\n\nWHEN THE JAPANESE INVADED BURMA IN 1942, BRITISH INDIA lost 15 percent of its total rice supply. In some areas that took large imports of Burmese rice, like Madras, the shortfall was overcome by local production. But in Bengal a long-term decline in the rural economy came together with natural disaster, compounded by wartime political bungling, to cause a catastrophic famine\u2014the first in India since the early twentieth century. The return of starvation to Bengal came as a traumatic shock. From the time of the Indian Industrial Commission in 1918, most observers took for granted that famine had been confined to India's past. In the 1920s and 1930s, nutritional scientists and health officials began to think about food as a way to enhance life rather than simply to sustain it\u2014their concern moved from absolute starvation to malnutrition. \"The days when we could cast the blame on the gods for all our ills are past,\" Nehru had written in 1929.\n\nDuring the winter monsoon of 1942, a fearsome cyclone struck eastern Bengal, flooding fields and destroying crops. \"In violence and devastation it surpassed any other natural calamity that befell this country,\" one contemporary account observed, \"it forced from the Bay a high tidal wave\" that reached 140 miles per hour. The cyclone \"swept the standing crops, blew off the roofs, uprooted most of the trees, demolished the huts\"; the floods that followed \"washed away nearly [three quarters] of the livestock, and some 40,000 human beings.\" Unnerved by the prospect of a Japanese invasion from Arakan, local officials imposed a scorched-earth policy, denying local cultivators the boats they used to transport rice. Internal divisions paralyzed the Bengal government. Driven by Winston Churchill's animus toward India, the British cabinet ignored every warning. They continued to export Indian rice to feed troops in other theaters of war. They refused to deploy Allied ships to send relief to Bengal. As shortages intensified, the most vulnerable people\u2014landless laborers, fishers, women, and children\u2014starved. Calcutta's relative wealth sucked in from rural Bengal rice that could have fed those in dire need.\n\nThe vulnerability of Bengal's poor, like their debts, had compounded over decades. During the Depression, smallholders unable to repay loans had lost much of their land. The productivity of Bengal's lands had declined in the twentieth century as railway embankments stemmed the flow of rivers and invasive water hyacinth choked streams. By 1942, the crisis was acute. As scarcity closed in, patrons deserted their sharecroppers, choosing to pay them in cash rather than in kind just when inflation made rice unaffordable. Families abandoned their weaker members. Hit by the successive blows of a loss of imports, \"boat denial\" by the state, a devastating cyclone, and a lack of relief, the economy and society of Bengal collapsed.\n\nEven the conservative _Statesman_ newspaper of Calcutta published photographs of starving children and abandoned corpses\u2014scenes reminiscent of the 1870s and the 1890s when the great El Ni\u00f1o droughts had combined with the churning effects of capitalism to deliver disaster. These images met with stony-faced inaction by the British government. This time, Indian observers held the British government directly responsible for starvation. \"It was a man-made famine which could have been foreseen and avoided,\" Jawaharlal Nehru wrote from Ahmednagar jail. He was sure that \"in any democratic or semi-democratic country, such a calamity would have swept away all the governments concerned with it.\" But just as disturbing was the callousness of wealthy Indians. Nehru expressed disgust at the \"dancing and feasting and a flaunting of luxury\" in Calcutta while millions starved. S. G. Sardesai, a Communist activist, decried the \"unbridled profiteering\" of hoarders and speculators, and argued that \"total mobilization means vigorous and just procurement of the genuine surplus from rural areas, vigorous price controls, and total rationing in cities.\" When they finally secured London's commitment to relief in the autumn of 1943, Indian officials had to raise the alarm that Bengal's continued starvation could endanger the war effort.\n\nWhen the Stanford University economists V. D. Wickizer and M. K. Bennett examined Asia's rice economies in 1941, they surveyed the wreck of what had once been an integrated system. In their analysis, they used the term \"Monsoon Asia\" as \"a convenient designation for a specific group of countries in which monsoonal climatic conditions profoundly influence both agriculture and economic life.\" \"Monsoon Asia\" was bound together by climate, by the direction of the winds, and by the trade in rice, but divided by empires. Wickizer and Bennett witnessed \"Monsoon Asia\" splintering further as a result of depression and now war. They wrote of their hope for \"a reversal of the recent trend towards economic nationalism.\" Their recipe for regional sustainability was for a return to the free commerce in rice, augmented by capital investment. But their projection of \"unfavorable\" conditions proved much closer to the eventual outcome. \"If peace should come with important territorial changes in Monsoon Asia,\" they argued, \"changes in the political composition of Monsoon Asia following the termination of present wars might readily result in a rather sudden shift and re-orientation, completely reversing the tendencies of the past decade or more.\"\n\nThe most enduring political consequence of the Bengal famine was the decisive rejection of any postwar return to the old ways of unregulated markets and inter-regional trade in rice. Indian planners and politicians, technocrats and populists, all emphasized the need for self-sufficiency in the future. Water was vital to their plans. Starvation's return to India scarred Nehru's generation of leaders. Having asserted that national sovereignty would alleviate the problem of starvation, Nehru and his contemporaries were haunted by the prospect of failure. \"We live continually on the verge of disaster in India, and indeed disaster sometimes overwhelms us,\" Nehru wrote. The same year, Patna University economist and demographer Gyan Chand declared that \"ours is a death-ridden country. We might very well adopt the human skull as our national emblem.\"\n\nAS THE WAR APPROACHED ITS END, THE EXPERIENCE OF FAMINE IN India\u2014and also in China and Vietnam\u2014came together with the force of rising expectations. In the eyes of many Asian observers, only the wholehearted embrace of state planning, wedded to powerful technology and under the control of nationalist rather than colonial forces, would address the colossal vulnerability of Asia's people to privation and starvation, both of which the war had laid bare.\n\nEven British planners began to contemplate large schemes to transform water. The Bhakra Dam, in Punjab, was first proposed in 1944. It was a monument to British plans for India's postwar reconstruction at a time when few believed the Raj would collapse so quickly after the war's end. The project had its skeptics. \"For advertisement reasons some authorities in India have published optimistic forecasts of the time in which they propose to construct high dams,\" one official scrawled in an archival note. \"There is sound opinion, unbiased by connection with these projects, which considers these forecasts fantastic.\" But speed and scale were what Indian nationalists wanted.\n\nOne of the voices in favor of a planned conquest of India's rivers was the scientist Meghnad Saha. Saha was born in 1893 in an East Bengal village, to a lower-caste family without education or resources, and with several children to feed. His scientific aptitude was evident from childhood; he won a series of scholarships that led him to Calcutta University in the 1910s. He studied in England and then in Germany, returning to a position at Allahabad University, one of India's most distinguished institutions. Saha's pioneering contributions to astrophysics gained him widespread recognition, notably for his paper on \"Ionisation in the Solar Chromosphere.\" By the 1930s he was no longer content to confine his work to the laboratory. He founded the journal _Science and Culture_ to reach a wider public; he became a missionary for scientific development, and a strident critic of Mahatma Gandhi's suspicion of modern technology. \"We do not for a moment believe that better and happier conditions of life,\" he wrote, can be secured by \"reverting back to the spinning wheel, the loin cloth, and the bullock cart.\" One of Saha's central concerns was with water, and his dreams of water were grand.\n\nSaha's essay on \"Flood,\" published in _Science and Culture_ in 1943, at the height of the war, envisaged wholesale environmental transformation. He described the decline of the Damodar River: diverted by railway embankments, its course had moved toward Calcutta, and now it threatened to inundate the city. Drawing on a global range of examples and references\u2014the 1913 floods in the Miami valley, and, above all, the Tennessee Valley Authority\u2014Saha argued that the key lay in a \"radical solution\" to make the Damodar \"a perennial\" rather than a seasonal river: that is, to \"liberate\" it from the monsoon. He argued for the adaptation of the American approach: \"to regard the whole river basin as a unitary area, and coordinate plans of flood control with those of irrigation, development of backward agricultural areas, development of hydroelectric power, and improvement of navigation.\" Elaborating on his plans in another article the following year, written together with his colleague Kamalesh Ray, Saha expressed optimism: \"Nature, vested interests and thoughtless managements made a once prosperous valley a wilderness, but Nature, Man and Science can again make it a smiling garden,\" they wrote. Saha was scathing in response to those, like Radhakamal Mukerjee, who had argued that restoring forests and local efforts at soil conservation would strip the Damodar of its destructive power. Saha called the claim that deforestation affected rainfall \"absurd\"\u2014a claim for which there was \"not a single iota of positive proof.\" If changes in forest cover and land use had any effect at all on local climate, Saha argued, they \"must be extremely small compared to the huge monsoon currents which are responsible for the precipitation on the Damodar Valley.\" Working at the cutting edge of planetary science, Saha was well aware of new work on the monsoons, and their integration with other parts of Earth's climate. The scale of India's climate was so vast as to render any local modifications in the water cycle trivial in importance. Saha pointed out that rainfall in the Damodar valley was determined by \"atmospheric conditions in the Bay of Bengal... which are generally thousands of feet in depth\", \"local conditions\" could have little effect upon them.\n\nRainfall was beyond human intervention. But human intervention to transform the landscape could neutralize the threat posed by uncertain rainfall, securing rivers from the alternating lack and excess of water. And here, Saha was confident about the future. \"We are fortunate to live at a time when the large scale experience of thousands of dams constructed in the USA since 1915 are at our disposal,\" he said; he believed that the global circulation of ideas and technology, a process of learning, would come to India's aid. In valorizing the American and Soviet models, Saha indicated that his dreams for India extended beyond anything the sluggish British colonial state could carry out. He envisaged the construction of dams in eastern India that would last for \"hundreds of years.\"\n\nIN THE 1920S AND 1930S, WATER BOTH CONNECTED AND DIVIDED Asia. A new awareness of the dynamics of climate made clear the extent to which Asia's coastal arc shared vulnerability to powerful cyclones that crisscrossed its seas. In the fields of geography and climatology, the idea of \"monsoon Asia\" arose to highlight the common rhythms of rural life governed by extreme seasonality. To think of Asia as bound together by water in every dimension\u2014the rains, rivers, and seas\u2014was to suggest that material conditions transcended the borders between empires. But those borders hardened in the decades between the wars. The depression of the 1930s broke many links in the chain that held monsoon Asia together: barriers to movement proliferated and migration patterns were overturned; commodity markets collapsed and the trade in rice declined. These reversals made the question of who controlled water all the more important.\n\nThe Second World War provided new tools and revived old fears. The trauma of famine and social breakdown met a newfound confidence in the power of state planning and big technology to reshape economy, society, and the environment. The next chapter turns to the struggles waged by India and by other independent Asian nation-states to understand and conquer water.\n\n# SEVEN\n\n# RIVERS DIVIDED, RIVERS DAMMED\n\nBETWEEN 1945 AND 1950 THE MAP OF ASIA WAS REDRAWN. THE war overturned imperial rule in Asia. The prestige of European powers in South and Southeast Asia never recovered from their rapid collapse before the Japanese advance in 1942. Economically ruined by the war, European empires could not afford to hold on to their colonial territories by force without American backing\u2014which was forthcoming only when it furthered US interests in the deepening Cold War. Most importantly, emboldened and militarized Asian leaders refused to contemplate any return to the old order.\n\nIn the war's aftermath, new states were forged from the ruins of empires. In 1947, British India was divided into the independent states of India and Pakistan: a bloody partition along religious lines that cost millions of lives and ruined millions more. In 1948, Burma and Ceylon gained their independence from the British in mostly peaceful transfers of power\u2014but Burma immediately faced multiple internal revolts, by Communist guerrillas and by the Karen and Kachin ethnic minorities. The Dutch tried to hold Indonesia by force. They were driven out by Sukarno's nationalist forces in 1949 after a protracted anticolonial war that also saw failed uprisings by Communist, Islamic, and regional secessionist forces. In Vietnam, the Communist leader Ho Chi Minh declared independence in 1945, moving into the vacuum left by the sudden surrender of the Japanese at the end of the war. But the French were determined to return to Vietnam, and with growing American support they waged war against the Viet Minh until Ho's victory at the battle of Dien Bien Phu in 1954. In East Asia, too, the war's end brought revolutionary change. The civil war between Chiang Kai-shek's Nationalist forces and Mao Zedong's Communist army culminated in Mao's victory and the foundation of the People's Republic of China in 1949, together with a de facto Nationalist state on the island of Taiwan.\n\nThe partition of India in 1947, showing the division of the Indus and Ganges basins.\n\nThe borderland between India and China, showing the Brahmaputra\/Yarlung Tsangpo river.\n\nAsia's political transformation was so rapid, so dramatic, so violent, that few at the time gave any thought to its environmental consequences. With the benefit of hindsight, we can see that those consequences were profound. The impact of these midcentury partitions and borders on Asia's waters are a vital, and neglected, part of any history of the second half of the twentieth century. We have hardly begun to reckon with their effects, both positive and negative, on the lives of a significant proportion of humanity.\n\nDam building, more than any other project, epitomized Asia's new leaders' confidence in their ability to tame nature. India had fewer than three hundred large dams at independence; by 1980, it had more than four thousand. Dams were the single largest form of public investment in modern India, swallowing considerably more government expenditure than health care or education. Dams promised to liberate India from the capricious monsoon; they promised finally to free it from the specter of famine that had struck so often, and so harshly, in the colonial era. India was not alone: the enthusiasm for dam building was global. Under Mao, China built large dams on a scale that eclipsed India's efforts: an estimated twenty-two thousand after 1949, almost half of all the large dams in the world. Along the Mekong River, dams formed part of the American strategy to shore up the anti-Communist state of South Vietnam following the French defeat at Dien Bien Phu.\n\nThese projects proceeded in parallel; by the 1960s they came into contention. Dams tried to make rivers conform to political borders by impounding their waters and diverting them to serve the needs of national development. But as multiple projects and competing ambitions arose, upstream and downstream, dams made tangible the material interdependence that transcended borders. At the time of independence, few in India had thought much about the fact that many of India's rivers originated in Chinese territory. Only when both sides' ambitions for river development swelled did the cross-border flow of rivers appear as a threat.\n\nIn the postcolonial age, large dams carried enormous symbolic weight. They epitomized dreams of development. More than any other technology, they promised the mastery of nature. In the global history of dam building, India played a pivotal role. India's experience exemplified the scale of the challenge facing the Third World, but also the scale of ambition that new states upheld. Because of the unevenness of the monsoon, India's rulers were obsessed with water. So, too, were the legions of foreign experts who arrived to help India's quest. Unlike China, India benefited from aid from both sides in the Cold War: India's developmental plans were a terrain for competition between the United States and the Soviet Union. India's water engineers received advice from around the world, and in turn they shared their expertise through the United Nations and other international bodies. The broader influence of India's addiction to large dams was cultural as much as it was political, as Indian cinema captured the imagination of viewers across Asia and Africa. Some of the most iconic Hindi films of the age were set against a backdrop of India's struggles for water; their stories resonated far beyond India.\n\nThe conquest of water by concrete behemoths came at enormous cost. Some of those costs were evident from the outset\u2014the displacement of people from their homes, the flooding of villages and forests by new reservoirs. Others became clearer with time. Few in the 1950s or 1960s could see just how fundamentally dams would transform Asia's ecology of water. It was in that era that Asia's states and peoples started on a collision course toward the water-related crises they face today.\n\n# I\n\nPartition was a particular kind of British decolonization, which came about as an attempt to engineer, in an extremely compressed period, nation-states with clear and decisive ethnic majorities out of previously heterogeneous colonial territories. It was implemented first in Ireland, and then twice in the 1940s: on the Indian subcontinent, and in Palestine. While political tensions between the Indian National Congress and the Muslim League ran high in the 1930s, it was only after the end of the Second World War that it became likely that India's future would be a divided one. Until then, the League's claim to represent all of India's Muslims\u2014divided by language and region, by class and politics\u2014rang hollow. In the aftermath of war, a spiral of violence supercharged political negotiations, and accelerated the timetable for independence. The British, fearing entanglement in an Indian civil war, and reeling from economic crisis, sought to leave as quickly as possible, no matter what the cost. Last-ditch negotiations failed when the Congress leadership were unwilling to concede to Muslim League leader Muhammad Ali Jinnah's demand for a weak federal government with power resting in provincial hands. On June 3, 1947, British prime minister Clement Attlee announced the plan to partition the subcontinent into India and Pakistan. Lord Louis Mountbatten was appointed the last viceroy of India, charged with overseeing the division. The job of drawing the border fell to Cyril Radcliffe, a lawyer with no prior experience of India. Closeted with a small boundary commission, supplied with maps and census returns, his task was to draw a line to carve off the Muslim-majority areas of Punjab and Bengal from the rest of British India, thereby creating the western and eastern wings of Pakistan\u2014divided by more than one thousand miles of Indian territory. The location of the border was not announced until the day after independence, on August 15, 1947.\n\nNobody had predicted the colossal scale of upheaval that followed. In just over a month, between September and October 1947, more than 849,000 refugees entered India on foot. A further 2.3 million crossed the Punjab border by train. Trains were attacked by armed mobs on both sides of the border\u2014their packed carriages became chambers of death. The Indian and Pakistani armies, by mutual consent, crossed the frontier into each other's territory to lead convoys of refugees back to safety. Arriving refugees were settled in what had been deemed \"evacuee property\"\u2014many who had sought temporary refuge from the violence returned to find that their homes had been seized, their departure construed as an intention to emigrate. South Asia's cities swelled with new arrivals, Delhi and Karachi and Calcutta above all. Around 20 million refugees crossed Radcliffe's border, more than half of them in Punjab.\n\nTHE SIFTING OF INDIA'S RELIGIOUS MAJORITIES AND MINORITIES IN 1947 was also, as one historian describes it, a \"division of nature.\" Radcliffe himself was aware of the problem in Punjab: his border, he recorded, was \"complicated by the existence of canal systems, so vital to the life of Punjab but developed only under the conception of a single administration.\" His solution pleased nobody. Canals were severed from their headworks. In Punjab, Partition broke the carefully planned canal network laid down over a half century. Bengal had no need of the intricate irrigation systems of arid Punjab, watered as it was by the monsoon and by the Himalayan rivers. But there, the border tried to contain a naturally volatile waterscape. As the geologists and bridge builders of the nineteenth century had found, Bengal's rivers changed course suddenly; _chars,_ or sandbanks, emerged with the deposit of silt and vanished with the coming of floods. The _chars_ were so fertile as to be desirable land for cultivation\u2014if they arose along the riverine borders, were they now part of India or Pakistan? For those who inhabited this braided landscape of land and water, the answer had vital consequences. The Bengal border ran through the sacred Ganges and the turbulent Brahmaputra. In 1947 there was little infrastructure to stem the flow of this water, but there were many plans in place. What would happen in the future, when engineers on both sides eyed new ways of harnessing the waters?\n\nFor some, the unity of nature was set against the human divisiveness of Partition. The socialist Rammanohar Lohia wrote of his astonishment that Nehru was willing to divide India's great river basins out of political expediency. Saadat Hasan Manto, the most incisive and enduring chronicler of Partition in fiction, turned to the problem of water in his 1951 short story \"Yazid.\" The story's opening image is almost shocking: \"The riots of 1947 came and went. In much the same way as spells of bad weather come and go every season.\" In his first two sentences, Manto evokes the indifference of nature to human suffering; he signals the insignificance of human folly faced with the cycle of the seasons; he also draws the suggestion that Partition's violence may have been as \"natural\" as the rains\u2014a message colored with irony, since it runs counter to so much of Manto's fiction, which depicts Partition as the monumental consequence of petty and all-too-human decisions. The most memorable exchange in the story takes place between the sage village midwife, Bakhto, and Jeena, wife of the protagonist, Karimdad. One day Bakhto arrives with news that \"the Indians were going to 'close' the river.\" Jeena is nonplussed: \"What do you mean by closing the river?\" When Bakhto replies, plainly, \"they will close the river that waters our crops,\" Jeena laughs in disbelief: \"You talk like a mad woman... who can close a river; it's a river, not a drain.\"\n\nPARTITION AFFECTED EVERY PART OF GOVERNMENT, EVERY INSTITUTION. The Indian Meteorological Department, too, was divided in 1947. One of the pressing tasks for the partition of Indian meteorology was the exchange of observational data\u2014all original records relating to the weather of Pakistan, wherever in (undivided) India they were held, were transferred to the new Pakistani meteorological service. Both sides held that climatological data were \"records of common interest,\" as if to acknowledge that the monsoons respected no human frontiers. They supplied each other with duplicate copies. And then there was the question of the instruments upon which weather science rested. These, too, were divided: the Indian Meteorological Department reported that \"out of the stock of instruments and stores held at [headquarters in] Poona and Delhi, stocks between 20 to 25% of each item were to be given to Pakistan.\" A simple list conveys a deep rupture:\n\nAs a result of the partition, 2 type A Forecast Centres, 1 type C Forecast Centre, 5 Auxiliary Centres, 8 Aerodrome reporting stations, 3 Radio-sonde stations, 14 Pilot Balloon Observatories, 82 surface observatories, and 1 seismological station were transferred to the Pakistan Meteorological Service.\n\nIncidentally, the list also conveys how dense the infrastructure of meteorology in British India had become by the end of the war. There is a poignant sense of how hard meteorologists fought to keep doing their work, regardless of the chaos and violence around them. \"Interim arrangements were made,\" they noted, \"for the issue of storm warnings, etc. for certain regions falling in Pakistan.\"\n\nLike meteorologists, engineers and economists looked at the material knots tying the two new states together and many of them believed that a future of cross-border cooperation was inevitable. Just a year after the event, C. N. Vakil, a professor of economics at Bombay University, wrote a pamphlet on _The Economic Consequences of Partition_. It was prosaic in the face of colossal upheaval. The facts, he thought, made it \"easy to appreciate the need for an agreed economic policy between the two Dominions now and in the future\"\u2014that both states formally remained Dominions within the British Empire until 1950 provided a measure of political cover for negotiations to take place. Against \"the atmosphere of communal bitterness as well as increasing mistrust,\" Vakil believed that \"fundamental economic forces in the two Dominions are likely to work in the direction of mutual inter-dependence.\" But he acknowledged the real possibility that \"political forces\" would win out; he saw that India and Pakistan could end up in a state of \"economic warfare.\" The darker edge to his pamphlet came in his wish to inform \"the layman\" of the economic \"weapons\" at India's disposal if \"warfare\" it was to be.\n\nIMMEDIATELY AFTER PARTITION ENGINEERS ON BOTH SIDES MUDDLED through. In the midst of crisis they kept the water running. In December 1947, the chief engineers of East and West Punjab signed a Standstill Agreement to maintain supplies to the Bari Doab, one of the Indus River canals ruptured by the border: the headworks fell on the Indian side of the border and most of the canal in Pakistan. When Punjab's Canal Colonies were built they had been conceived as a unitary system, its hydraulic parts each useless in isolation; now the engineers had to improvise. And then the water stopped. On April 1, 1948, the day the makeshift agreement expired, India shut off the water supply to the canal. The fears that Manto depicted in his fiction mirrored historical events\u2014\"it's a river, not a drain.\" But the rivers, too, were national now.\n\nThe sudden stoppage raised alarm on the Pakistani side. In the midst of the spring sowing season, the waters to the Upper Bari Doab and Dibalpur canals stopped, disrupting cultivation and threatening the harvest. Residents of Lahore saw the canal that bisected their city empty of water: before their eyes was a visceral sign of Pakistan's vulnerability. East Punjab's engineers, on the Indian side, shut off the canal water to Pakistan without the approval of the central government in Delhi; Nehru himself worried that \"this act will injure us greatly in the world's eyes.\"\n\nThe conflict over Indus waters joined the territorial conflict between India and Pakistan over the princely state of Jammu and Kashmir, whose Hindu ruler had chosen to join India rather than Pakistan under considerable pressure from the Indian side, and against the wishes of the territory's predominantly Muslim population. The tussle over Kashmir erupted into military conflict within months of Partition, following the invasion of the territory by Pathan militia from the northwest with covert support from the Pakistani state. Upon Kashmir both India and Pakistan projected their anxious sense of truncation, in the sense that both sides ended up with less territory than they thought they ought to have, as a result of a hasty partition that satisfied nobody. Both sides came to see control over Kashmir as a vindication of their founding ideologies: for India, the extension of a secular and democratic polity; for Pakistan, the achievement of a Muslim homeland in South Asia. The views of Kashmiris, then as now, were ignored. But the Kashmir dispute also had a hydraulic dimension: of the five tributaries of the Indus River, one, the Jhelum, originates in the Kashmir valley; another, the Chenab, flows through Jammu. Gnawing at India and Pakistan, through their inability to find a solution in Kashmir, was a quest to control the state's water.\n\nPakistan and India each made their case before a global audience. The dispute over the Indus attracted international attention because it seemed like just the sort of water conflict that many others could face as the map of the world was redrawn. Pakistan's delegation to the United Nations declared in 1950 that \"the withholding of water essential to an arid region to the survival of millions of its inhabitants\" was \"an international wrong and a peculiarly compelling use of force contrary to the obligations of membership in the United Nations.\" The Indian argument, by contrast, was that India \"has the right under the Partition, as also in equity, over the waters of rivers flowing through her territory.\" India's lawyers also advised that the provisions of international law were ill-suited to \"the case of two countries, which have come into existence from the partition of a previously existing national unit\"\u2014which rested on the idea that British India was a \"national unit,\" a strange claim for Indian nationalists to make just a year after independence.\n\nBoth sides used Partition to bolster their arguments. India insisted that Radcliffe's line had given Punjab's richest agricultural lands to Pakistan, including the Canal Colonies\u2014lands farmed mostly by Sikh and Hindu cultivators who now found themselves uprooted as refugees in India. Partition had \"disrupted [a] unitary system of canal irrigation and therefore the entire economy of the area,\" executed with \"complete disregard of physical or economic factors.\" In this light, the Indians argued, it was their prerogative to make best use of the water resources that remained. India's advocates portrayed eastern Punjab as the victim of a partition that had been imposed upon it \"to satisfy the ideology of Mr. Jinnah and his Muslim League.\" With little warning, East Punjab \"found itself an economic unit, and a very much underdeveloped area.\" Its survival depended on wresting control over \"the life-giving waters from the Himalayas\" that had, through British canals, \"been unfairly diverted to increase the prosperity of distant tracts\" that now lay across the border. The Pakistanis retorted that India \"wishes to make a desperate attempt to escape the economic consequences of partition\"\u2014which Pakistan, as a new state, had no choice but to face. The Pakistani submission to the tribunal of arbitration gave Partition a material as well as an ideological dimension: \"Apart from religious and cultural considerations, one of the main objects of partition is to enable the residents of the two Dominions to use and develop their economic resources for their own benefit.\" They closed with a goad: \"East Punjab should have the courage to face the economic consequences of a political standing by itself.\"\n\nEach Punjab \"found itself an economic unit\"\u2014the phrase suggests that this happened as one might \"find oneself\" in an unfamiliar destination after getting on the wrong train. Divided provinces, like the divided nations of which they were part, had to stand alone where once they were part of a larger whole. The vogue for planning demanded a simplified model of the economy upon which plans could be made. This cemented the vision of an Indian economy set apart from the whole web of connections that tied India to Southeast Asia and beyond. Partition stymied many plans: it struck at the mutual dependence of the jute growers of eastern Bengal and the export houses of Calcutta, at the ties between coal producers of eastern India and Pakistan's factories, at the carefully calibrated use of water by farmers along the length of the canals of Punjab. These new \"economic units\" unleashed a desperate competition for water: the precondition for every vision of prosperity.\n\nIn both India and Pakistan, Partition generated a sense of loss and a feeling of vulnerability. Following India's water stoppage, water engineering became an urgent priority in Pakistan. Pakistan's engineers designed a new canal project known as the BRBD (Bambanwala-Ravi-Bedian-Dibalpur); it would run parallel to the partition border, a \"canal designed to sever Pakistan's [water] supply from India.\" A volunteer corps of laborers rallied to the cause of the new canal as an act of national defense\u2014it came to be known as the Martyrs' Canal. For India, the loss of the productive agrarian lands of western Punjab hastened the push to develop its eastern reaches. Plans for a large dam at Bhakra, on the drawing board since the early twentieth century, now became a priority. Old fears of famine had never gone away; they were reactivated by Partition. Eastern Punjab needed new sources of water to keep its most vulnerable districts secure from a failure in the rains.\n\nThe Indian water stoppage lasted a matter of weeks. Negotiations between the two sides resumed at the end of April 1948. In exchange for payment, India agreed to continue supplying canal water to Pakistan for an unspecified period, during which \"alternative sources\" would be developed. Both sides clashed repeatedly, their claims often directed at international observers. Pakistan proposed international arbitration; India insisted it was a domestic matter. In 1951 the American David Lilienthal\u2014a senior official in the Tennessee Valley Authority, and now a globetrotting development consultant\u2014toured India and Pakistan; he took a particular interest in the Indus water dispute. Lilienthal contrasted \"politics and emotion\" with \"engineering or professional principles.\" He described how Partition, driven by emotion and not by reason, \"fell like an ax\" upon the Indus basin. But, he added, \"the river pays no attention to Partition\u2014the Indus, she 'just keeps running along' through Kashmir and India and Pakistan.\" He wrote a long piece for _Collier's_ magazine warning the American foreign policy establishment that Kashmir was \"another Korea in the making.\" Lilienthal had proposed that water management be removed from the political realm. He had faith in the shared training and professional camaraderie of India's and Pakistan's water engineers; he believed in their ability to work together for a \"cooperative,\" technical, solution. In Partition's aftermath, such apolitical solutions were as attractive as they were unrealistic.\n\nTHE PARTITION OF INDIA MARKED THE BEGINNING, NOT THE END, of the division of Asia's waters. A year after Mao Zedong's army overpowered Chiang Kai-shek's Nationalists in the Chinese Civil War, inaugurating the People's Republic of China in October 1949, Chinese forces invaded Tibet. The annexation of Tibet was a thorn in India's relations with China. Many Indian politicians, including members of Nehru's cabinet, urged him to take a hard line, but Nehru opted for a path of conciliation, recognizing that India was not in a position to take any action. What went almost unremarked at the time was that the annexation of Tibet in 1950 also gave China control over much of Asia's freshwater. The Indus was divided between India and Pakistan\u2014but its source is on the Tibetan Plateau, which was now ruled as part of China. From the Tibetan Plateau flow the Brahmaputra (known in Tibet as the Yarlung Tsangpo), the Salween, the Mekong, and also the Yangzi. The source of the great rivers still seemed, in 1950, remote, wild, untouched by the modern world. It is no surprise that water was mostly invisible through the process of dividing Asia into modern nation-states. In the second half of the twentieth century, water resources would become increasingly important to the process of marking and laying claim to the earth, increasingly pivotal to conflicts between Asian states. In 1950 water was not, or not yet, a cause of conflict except between India and Pakistan. But their effects on shared water resources would be among the most far-reaching consequences of Asia's midcentury territorial disputes.\n\nStill, the power of nature, paying no heed to new borders, was on full display in August 1950, when\u2014on Indian Independence Day, the fifteenth\u2014a powerful earthquake tore through the borderlands straddling India, East Pakistan, Tibet, and Burma. The earthquake was one of the ten most powerful ever recorded, caused by the collision of two continental plates. Its epicenter was in Rima, Tibet\u2014and came just three months before the Chinese invasion\u2014but the bulk of the damage fell on the northeastern Indian state of Assam. Even as politicians were busy redrawing the map of Asia, the earthquake altered the landscape and devastated human lives. As if to underscore the remoteness of the earthquake's epicenter from centers of political power, relief was slow to arrive. The earthquake blocked the course of many tributaries of the Brahmaputra, changing the river's course. The British botanist and explorer Francis Kingdon Ward was traveling in Tibet at the time, and penned one of the few eyewitness accounts from the earthquake's epicenter. He wrote that \"the immediate result of the earthquake was to pour millions of tons of rock and sand into all the main rivers... displacing millions of cubic feet of water.\"\n\nEvery scheme to engineer water had to contend with the instability of Asia's mountain rivers; with the growing confidence of postcolonial engineers, caution began to be set aside.\n\n# II\n\nIn 1951, India carried out its first census after independence. It was at that time the largest census ever undertaken in the world. The average life expectancy in India stood at just 31.6 years for men, and 30.25 years for women. In the United States at the same time, that figure was 65.6 years for men, and 71.4 for women. For every 1,000 live births in India at the time, more than 140 infants died. This was an indictment of two centuries of British rule, since the \"abstract number which is the average human life span,\" as philosopher Georges Canguilhem noted, revealed much about \"the value attached to life in a given society.\" In China, after more than a decade of war, life expectancy was no higher. Nothing illustrates so plainly the magnitude of the challenge before the governments of Asia's new states. For India and China, as for Pakistan and Burma and countries all along the great crescent, harnessing water was a priority in their quest to transform the conditions and expectations of life.\n\nIn his introduction to the census, commissioner R. A. Gopalaswami pinpointed what he saw as a turning point in India's population history, around 1921. Until then India's population had grown slowly. The terrible famines of the late nineteenth century, the prevalence of infectious diseases like plague and malaria, the devastating and rapid toll of the influenza epidemic of 1918, which killed between 12 and 13 million people in India\u2014taken together, they produced a grim toll of premature death and debilitating illness. After his account of the influenza, Gopalaswami's narrative reaches its pivot: \"We now reach the turning point,\" he wrote, where after 1921, \"we hear no longer about abnormal deaths.\" From that point on, he argued, famine and mass epidemics ceased to be the killers that they were in India. Some of the credit he gave to the mobilizing power of Indian nationalism, some to administrative improvements that came from lessons the British had learned from earlier disasters. Gopalaswami presented a picture of India's climate that was no longer the threat that it was: \"Though the usual cycles of vicissitudes of the seasons continued and the brown and yellow belts of the country continued to suffer from droughts... there was no extraordinary calamity\" in the years between the wars. Good policy was matched with good fortune, since \"nature also seems to have been kindlier\" in the two decades after 1921, with fewer major droughts. The Bengal famine of 1943 was a devastating reminder that famine could return to India. But it did not invalidate the longer-term pattern, and prevailing wartime conditions made it exceptional; rather, Bengal \"gave a sharp jolt\" to India's leaders, and reminded them of the need for vigilance.\n\nGopalaswami went on to consider the implications of India's rapid population growth for the country's future. The most alarming statistic, in his mind, was that the area of cultivated land per capita in India had fallen by 25 percent in thirty years: land had run out, and yields were declining. He ended his discussion on a note of foreboding, placing India's experience in global context. It could well be the case, he wrote, that \"we are passing through the last stage of that exceptional phase in the growth of mankind in numbers which was introduced mainly by the opening up of the New World and partly by the creation of a world market.\" Gopalaswami steered clear of Malthusian alarm. Like many of his generation he believed in the power of the state, in the marriage of wise planning with technology, to solve social and economic problems.\n\nThere were three possible responses to the challenge of feeding India. The independent Indian state tried them all. To the extent that the productivity of the land suffered from the very small plots of land held by the majority of cultivators, land redistribution seemed a promising solution. Soon after independence, Nehru's government proposed to abolish the _zamindari_ system, the practice by which large landowners acted as intermediaries between the state and cultivators, wielding the power to collect taxes, a key feature of both Mughal and British administration. Although it had to confront some vested interests, _zamindari_ abolition, which had to be approved state by state, was relatively straightforward\u2014in the political culture of free India, _zamindars_ epitomized the old feudal order that independence was meant to sweep away. But at most 6 percent of land in India changed hands under these reforms. The chief beneficiaries were most often farmers who were already relatively well off. Any energy behind land redistribution in India fizzled by the mid-1950s: by that time rural landowners had cemented themselves as an important constituency in Indian electoral politics; they closed ranks to defend their interests.\n\nThe second approach, implemented with considerable success, was for the state to intervene more actively in the food economy. A commitment not to intervene in markets had been a shibboleth of British administration in India after the end of the East India Company. As we have seen, that iron confidence in markets shaped the British approach to the famines of the 1870s and 1890s, in which so many millions of Indians died. But the Second World War reversed that faith abruptly. India saw the rise of an elaborate apparatus of food control that lasted well beyond independence. The American T. W. Schultz, one of the pioneers of development economics, remarked in 1946 that \"no country in the world, with perhaps the exception of Russia, has gone so far [as India] in controlling basic food distribution.\" By 1946, close to eight hundred cities and towns were covered by the rationing scheme. In 1947 the wartime \"Grow More Food\" campaign was resurrected, and in 1949 the government of India set the campaign's goal as the attainment of national self-sufficiency in food grains by 1952. In the late 1940s and early 1950s, the Indian state purchased 4.3 million tons of food from its own farmers, and just short of 3.5 million tons abroad. An elaborate network of transportation and storage was established, providing the skeleton of India's public distribution system that, to this day, remains vital to the food security of hundreds of millions. As we will see, when serious drought threatened northern India in the mid-1960s, and Maharashtra in the early 1970s, the state's food distribution proved its worth and averted famine.\n\nBut the approach that received by far the most attention, and the most funding, was the quest to intensify agricultural production\u2014to grow more food on the same amount of land. Gopalaswami observed that this could be done, in Indian conditions, by increasing the spread of double cropping\u2014in which two crops were grown each year, one in the winter and one in the summer\u2014through the expanded use of fertilizer; and, above all, by expanding year-round irrigation to free agriculture from dependence on the monsoon. In 1951, India launched the first of its five-year plans for economic development\u2014influenced by Soviet central planning, but maintaining a mixed economy. Fifteen percent of total expenditure under the first plan went to irrigation, and a majority of that to what were called \"major and medium irrigation projects.\" At their apex stood the large multipurpose dams that would transform India's rivers in the years to come. The mastermind behind the Indian planning commission was the brilliant Bengali statistician Prasanta Chandra Mahalanobis, who had spent some years in the 1920s working for the meteorological department at Alipore observatory in Calcutta, and who had published a statistical essay on floods in Orissa.\n\nTHROUGH THE LATE-NINETEENTH-CENTURY CLIMAX OF IMPERIAL globalization, Asia had been reconfigured laterally, as steamships and railways connected distant places. Now, however, vertical space came to matter more. Engineers began to think about the gradient of each river's fall so they could be harnessed for hydropower; geologists aimed to measure the depth of water resources in underground reservoirs. The reorientation of Asia along a vertical axis\u2014as if the map were now drawn in three dimensions\u2014had everything to do with the conquest of water. And the conquest of water, ultimately, promised the enhancement of life and the diminution of premature death.\n\n# III\n\nInaugurating the Hirakud Dam on the Mahanadi River in eastern India, just a year after he became prime minister, Nehru described the scene as a \"fascinating vision of the future which fills one with enthusiasm.\" He wrote that \"a sense of adventure seized me and I forgot for a while the many troubles that beset us\"\u2014the \"troubles,\" that is, of mass refugee movements after Partition, of multiple insurgencies, of hostilities with Pakistan, of governing a new and heterogeneous nation. The sight of Hirakud convinced Nehru that \"these troubles will pass\" but \"the great dam and all that follow from it will endure for ages to come.\"\n\nThe Indian state wasted no time pursuing its ambitious agenda. At independence, India had only thirty dams higher than thirty meters. Most colonial works of hydraulic engineering had been on a smaller scale: fifteen- to twenty-meter-high tanks and bunds, linked to a network of canals. During the Second World War, the colonial state began to think big\u2014and the plans of the 1940s paved the way for India's largest hydraulic schemes in the 1950s. The largest of them were large water storage works at Bhakra, in Punjab, Hirakud in Orissa, Tungabhadra and Nagarjunasagar in the Deccan, and the Damodar valley project in Bengal. They epitomized the imperative of multipurpose development. Each scheme promised a cluster of benefits: year-round irrigation; water storage to even out the concentration of the monsoon rains; embankments to prevent flooding during torrential rain; river navigation; and hydroelectric power. Each scheme had its own priority among these uses\u2014the Damodar project was designed with flood control primarily in mind; Bhakra's particular symbolic importance came from its role in compensating for India's loss of Punjab's Canal Colonies at Partition, by creating a new hydraulic infrastructure around the dam complex. The projects were technically demanding as well as expensive. The Damodar valley project owed most to outside funding and expertise: it received a US$18.5 million loan from the World Bank in 1950, and its technical advisors included David Lilienthal. For the most part, India's large dams were funded by the state's tax revenues, their costs recouped later through an \"improvement cess,\" a levy on landowners who benefited most from the dams. But large dams very often ran over budget and behind schedule.\n\nThe large schemes were the most visible manifestation of India's attempt to remake nature. Many smaller projects sprang to life after independence: irrigation dams in the Vindhya hills, the Pykara hydroelectric scheme in Tamil Nadu, the Sarda Canal in Uttar Pradesh, the Sengulam project in Kerala. Beyond these lay innumerable canals restored, power lines laid, tanks and irrigation channels built or resurrected. Many of these schemes built on colonial precedents. What was new was their scale, but also the language in which they were justified. As a perceptive visitor to India observed at the time, the colossal projects \"stand out for their quality of newness,\" even if their impact on water availability and power generation was less than the cumulative impact of many smaller initiatives. The mega-projects marked a true departure: \"dependable in the worst monsoon, dynamic in the most backward region.\" Above all, \"they stand for something India could not build, and did not will, before she became a nation.\" The moral fervor behind the quest to harness India's waters came from this sense of historic opportunity. So, too, did the planners' willingness to force through their schemes, whatever the cost.\n\nThe Bhakra Nangal project, in Punjab, was India's most prominent engineering scheme. It stood 680 feet high, the second-tallest dam in the world at the time; it consumed 500 million cubic feet of concrete. Bhakra's location in the partitioned province of Punjab added resolve and poignancy to its promise of a more secure future. The dam had first been proposed in 1944, and construction began soon after independence. Facing the future, India's water engineers began with a familiar problem: \"One of the chief characteristics of rainfall is its unequal distribution over the country,\" they wrote, and \"another important characteristic is the unequal distribution of precipitation over the year.\" They set out to free India from the seasons.\n\nA. N. Khosla (1892\u20131984) stood behind many of India's plans to harness water after independence. He was the first chair of the Central Water and Power Commission of India, a graduate of the Roorkee College of Engineering and a stalwart of the Punjab irrigation department. He imagined the future in, by his own admission, \"fantastic\" terms. In an address delivered on All-India Radio in 1951, Khosla declared that \"it will be no idle dream to contemplate the linking up of the Narmada with the Ganga through the Sone, or with the Mahanadi over the Amarkantak plateau, and thus connect the Arabian Sea with the Ganga and the Bay of Bengal right through the heart of India.\" Old dreams of reshaping India's geography\u2014Arthur Cotton's dreams\u2014gained new life after independence.\n\nDreams of hydraulic engineering were inseparable from dreams of freedom. Kanwar Sain, Khosla's successor as head of India's water authority, wrote that \"the river valley projects constitute the biggest single effort since independence to meet the material wants of the people, for from irrigation springs ultimately the sinews of man, from power the sinews of industry.\" He voiced the hopes of many of India's planners and architects when he declared that the dams \"are indeed the symbols of the aspirations of new India, and the blessings that stream forth from them are the enduring gifts of this generation to posterity.\" His words were followed, in a public information pamphlet, by page after page of statistics: kilowatts generated and projected, hectares irrigated, gallons of water stored, tons of concrete expended. Large numbers were a form of rapture. For Khosla and for Sain, as for so many of their peers, Bhakra was the showpiece.\n\nJawaharlal Nehru addressing a large crowd at the dedication of the Bhakra Dam, October 1963. CREDIT: Bettmann\/Getty Images\n\nJawaharlal Nehru and Zhou Enlai celebrating the New Year on a special train taking them back to Delhi after a visit to the Bhakra Dam, December 31, 1956. CREDIT: Bettmann\/Getty Images\n\nOpening the Nangal Canal in 1954, Nehru's reverence was palpable. \"What place can be greater than Bhakra Nangal,\" he wondered, \"where thousands of men have worked or shed their blood and sweat and laid down their lives as well? Where can be holier than this?\" Nehru spoke at length in Hindi and more briefly in English, before a crowd of thousands. \"We talk about Mother India,\" he said, and now \"Mother India is in labour, producing and creating things.\" At the time, India was ablaze with demands for the creation of linguistic states out of the composite provinces inherited from British India. \"We talk so much about changing the provinces, expanding them, shortening them, disintegrating them,\" Nehru said at Nangal, his irritation unconcealed. \"I do not mind our people getting terribly excited about it,\" he said, \"and forgetting the major things.\" But Nehru made a clear contrast between the \"major things\"\u2014\"Making a new India... putting an end to the poverty of India\"\u2014and what he called \"petty disputes.\" Nehru turned to the theme of revolution. \"A revolution does not mean the breaking of heads,\" he insisted; of India's own gradual, nonviolent revolution, he declared that, with independence, \"we finished it in a way in the political sphere... we have to continue it in the social and the economic sphere.\" Bhakra became a symbol of India's ambitions. It was an obligatory stop on the itinerary of every official visitor. On the last day of 1956, Nehru took Chinese premier Zhou Enlai to Bhakra\u2014the two had built a rapport over the previous two years, though it would turn out to be short-lived. \"These are the new temples of India where I worship,\" Nehru told Zhou. \"I am deeply impressed,\" Zhou replied.\n\nThe excitement of the Bhakra project was captured in a 1957 documentary made by the government of India's Films Division. It was produced by Ezra Mir, born Edwyn Meyers: an Indian Jewish filmmaker who began by making propaganda films for the British during the Second World War, and who went on to make seven hundred documentaries in the 1950s and 1960s. The Films Division was charged with bringing \"new India\" to life on screen. Its productions included films about India's freedom struggle and profiles of political leaders and musicians. Above all, the films dramatized the quest for \"development\"\u2014for health and water, food and education. Public information films played in cinemas across the country before the feature films that people had bought tickets for\u2014they reached an audience of millions.\n\nThe Bhakra film was an epic. Its visual language came from a tradition that had circulated globally during the war: Soviet and Path\u00e9 newsreels suggested the form, the genre, the structure in which India's propagandists worked\u2014and which they made their own. The narrator's voice is clipped and serious. The soundtrack begins with nineteenth-century European music: brassy and bright, like a march. \"For centuries,\" the narrator begins, \"gazing upon the parched lands of Punjab and Rajasthan, we have dreamed of reclaiming the desert.\" The struggle at the heart of the film is clear from the outset, as suddenly the background music turns to an Indian folk theme and the film cuts to a scene of women lining up at a well, whose \"search for water, \"we are told, \"was never-ending.\" The solution, the film's narrator declared, lay in the \"unused, wasted\" waters of the Sutlej River, flowing down from the Himalayas. And then \"at last,\" as if inevitable, \"the decision was taken\u2014the Sutlej must be tamed.\" The film cuts to the image of a young, studious-looking engineer, slide rule in hand: he is the protagonist of this drama. The narrator spins out superlatives, the music turns to a fanfare of trumpets, the camera shows us Bhakra's sheer size: it is \"massive,\" \"stupendous,\" \"mighty,\" \"a miracle.\" It held out for India \"the promise of an exciting, dramatic future.\"\n\nThe film was finished before the dam. It shows us a worksite of ceaseless, noisy activity: the rumble of drills; the explosion of blasted rock; the clatter of a conveyor belt ferrying bucket after bucket of material up to the dam; the clink of hammers and the heft of spades as imported machines\u2014some of them reassembled, piece by piece, on location\u2014dovetail with the oldest kinds of human work. India's dams were a lucrative source of contracts for foreign engineering firms like Hazra & Co. of Chicago, who provided material, equipment, and many consultants. But dam building was also a spur to local industry; the largest cement factory in Asia came up to feed the \"colossus\" that was Bhakra. Toward the end of the film, we see the changing of shifts at the end of a working day. Darkness descends on the site and \"the lights of the great dam are switched on, glowing like stars.\" At moments in the film, the dam seems to transcend technology, evoking a deeper and more ancient sense of wonder\u2014it was a \"miracle.\"\n\nUntil the last few minutes of the film, the only voice we hear, apart from the narrator's, is a brief clip of Harvey Slocum\u2014the straight-talking, autodidact American dam builder who supervised Bhakra's \"army\" of Indian engineers, and who died suddenly on site in 1961. But then, finally, our focus shifts to the makers of Bhakra. At the end of a shift, \"men emerge from every corner.\" We see a worker arriving home as his three children run to greet him. As he enters his modest quarters, his wife rises to offer him water, a smile on her face. It is the first intimate or domestic scene in the film\u2014the first sight of a woman, or of children; in the background the orchestra is replaced by a simple folk melody played by a flute. The parting message is one of national unity: Bhakra represents the nation. \"Never in the long history of India,\" the narrator declares, \"have there been so many men from different parts of the country working together for a common purpose.\" The effort of creating Bhakra united Indians where the politics of region and linguistic identity divided them. The film closes with the voices of the workers, each speaking his own language. B. Srinivasan faces the camera; he speaks in Tamil without captions or voiceover; his speech is stiffly awkward before the camera. \"My name is Srinivasan. I have worked here for a year and a half. My home is in Madurai district.\" We hear from workers in Punjabi, Marathi, and Bengali, some of their words drowned out by the noise of construction.\n\nHarvey Slocum, the American dam builder who supervised construction at Bhakra. CREDIT: James Burke\/Getty Images\n\nThe Bhakra Dam under construction. CREDIT: James Burke\/Getty Images\n\nIndia's new dams attracted thousands of visitors from near and far. They became landmarks on the landscapes they had altered beyond recognition. Their clean lines and monumental size reminded some observers of Buddhist stupas. Aesthetically, as well as symbolically, they were the \"temples of new India.\" But most Indians in the 1950s had never visited a big dam. Most Indians did not read the _Indian Journal of Power and River Valley Development,_ the pages of which told a heroic story of India's hydraulic adventures. The place where most Indians encountered the grandeur of India's water projects was on screen. Public information films made an impression on many minds, but the feature films of Hindi cinema really captured people's hearts\u2014they, more than any pamphlet filled with statistics, gave India's hydraulic revolution emotional content. And in the process, they reached beyond India's shores to make India's dams a symbol for hope and progress across postcolonial Asia and Africa.\n\nConstruction workers on the Bhakra Dam site came from all over India and had many specialized skills. CREDIT: James Burke\/Getty Images\n\nWhere many a foreign consultant ended up \"a White Man's Burden character,\" as David Lilienthal described himself in his diary, one American observer of India's water projects was admiring, even at times uncritical: Henry Hart, a University of Wisconsin political scientist. Having worked as a \"minor administrative officer\" in the Tennessee Valley Authority as a young man in the 1930s, having seen a \"newly-harnessed river brought to life\" and the rural southern landscape of his childhood transformed, Hart lived in independent India between 1952 and 1954. He sought the answer to a question that weighed on many American minds as the Cold War intensified: \"Can a revolution... be _built_?\" Hart traveled the length of the country, funded by a Fulbright fellowship and a grant from the Ford Foundation. His book, _New India's Rivers,_ remains the most detailed and sympathetic account we have of India's great hydraulic experiment.\n\nHart's eye was drawn to the workers who were reduced to \"an army\" in so many depictions of India's dam fever; he dedicated his book to \"all who died without seeing the new India they built.\" And where many observers saw only the romance of technology, Hart's romanticism sought artisanal skill. At the Tungabhadra Dam site, for which Hart saves his richest description, he encountered Vellu Pillai, a fifty-three-year-old stonecutter who chiseled granite from the quarry to exacting specifications. Hart discovered that Vellu Pillai came from a family of stonecutters in Thanjavur. His had been \"a life of dressing stones for well-linings and walls\"\u2014for \"ten prosperous years,\" he had carved temple deities. Evoking the ruined capital of the Vijayanagar empire, just a few miles away, Hart saw the dam resurrecting ancient skills. \"The design itself was novel,\" he wrote,\" but the teamwork of brain and hand was a renaissance.\" Hart speaks to no women workers, but their presence and their labor are visible in his account and especially in the photographs that illustrate his book; their stories remain untold.\n\nThe centripetal force of India's water projects drew workers, skills, and materials from across India\u2014and beyond. Hart observed that some of the earliest workers on the Tungabhadra worksite were Telugu porters who had arrived from Burma during the war; they were among the half-million Indian refugees who had walked through the hilly jungles back to India when the Japanese bombs fell. Many died on the arduous journey on foot across the mountains into Assam. In the 1920s and 1930s, Burma was a frontier for Telugu-speaking migrant workers from coastal Andhra: they provided much of the labor on Rangoon's docks, they pulled rickshaws, laid roads, worked in the rice mills. As long-established patterns of migration shut down during and after the war, new ones opened up along India's river valleys. From the moment the war ended, the large projects' capacity to generate employment was a strong argument in their favor. Water projects were themselves a way to prevent the resumption of Indian labor migration overseas, which so many Indian nationalists had come to oppose as exploitative. \"At present a great deal of Indian labour is being sent to Burma and other places,\" noted a water engineer surveying the Ramaprasadasagara scheme along the Godavari. \"An irrigation scheme of this magnitude would prevent the exodus of labor.\"\n\nSome workers traveled to the dam sites in groups. Teams of porters, masons, and stonecutters arrived, some of them following an old practice of migration from place to place in search of work. Others ended up building the dams of India after a chance encounter. In a wartime hangar along the Tungabhadra River, India's first factory for the manufacture of sluice gates was overseen by Mr. Eswariah; his \"sheer mechanical intelligence\" had been discovered by the dam's chief engineer, Srirangachari, in a \"Madras highway repair shop.\"\n\nOn many accounts of India's dam projects, as in the public information film on Bhakra, a sense of mission drove the workers. There can be no doubting that idealism animated the efforts of many who toiled on the dam sites. But so, too, did the need to earn a living. For only so long could appeals to sacrifice on behalf of the nation mask poor pay and harsh conditions. The workers on the Hirakud Dam learned how quickly the postcolonial state was willing to deploy force to keep construction going. In 1954, Hirakud's workers established a union of their own to challenge the officially recognized association of workers. They grappled over pay rates and the rhythms of work. After a breakdown in negotiations, the district magistrate ordered armed police to disperse a group of workers who were headed toward the chief engineer's residence, reportedly with the intention of harming him. Following a _lathi_ (baton) charge, fifty workers were hospitalized; two died the following day. In Hart's account, which reflects the official view, the strike was the work of \"agitators\" affiliated with the Communist Party. As he tells the story, India's climate itself played a role. \"On any great outdoor work built in a monsoon climate,\" he declared, \"the hot, pre-monsoon months are the tense season.\" And so, in the end, the strike appears in Hart's account as a mere interruption: \"On the Monday after the bloody Friday, men began going back to work. By Wednesday, the dam began to rise again, full speed.\" It is as though India's dams were a juggernaut, with a life and force of their own. That is certainly how they must have seemed to those who stood in the way of the engineers' plans.\n\nNot every commentator was unequivocally in favor of the large dams. The Bengali journalist Kapil Prasad Bhattacharjee was among the earliest to call into question the approach of the Damodar Valley Corporation. As a student in Paris in the 1930s, he was influenced by the work of French hydrologists. Schooled in the economic nationalism of Dadabhai Naoroji and Romesh Dutt, Bhattacharjee argued that the Damodar valley project would perpetuate a colonial effort to keep India poor by keeping it an agrarian economy. He worried for the future of Calcutta as a port; for Bhattacharjee, the worst effect of the Damodar projects would be to silt up the Hooghly River. He felt that more could be achieved by repair and restoration\u2014the \"proper maintenance of old canals, tanks, lakes\"\u2014than through expensive projects dependent on foreign engineering expertise. Voices like Bhattacharjee's were in a very small minority in the 1950s, not only in India, but all over the world. He worried about the economic and the ecological effects of dam building; even Bhattacharjee had little to say about their human consequences.\n\nAS INDIA SOUGHT TO TAME ITS RIVERS, THE DISPUTE WITH PAKISTAN over who controlled the Indus intensified. It did not take long for David Lilienthal's idealistic vision of water beyond politics\u2014his proposal for the shared development of the Indus basin\u2014to fail. In early 1954, the World Bank proposed an alternative solution, which gave shape to the treaty that eventually came about. In place of Lilienthal's idea, the bank proposed to finish the job of Partition by dividing the water of the Indus and its tributaries completely, if not neatly, between India and Pakistan. The eastern rivers\u2014the Beas, the Ravi, and the Sutlej\u2014went to India; the western rivers, the Chenab and the Jhelum, went to Pakistan. Since the latter, too, originated in Indian territory, India would have the right to use their water for irrigation, transportation, and power generation, up to a limited volume. The negotiations dragged on for six years: India's position of strength as the upper riparian continued to trouble Pakistan. But much remains obscure about the process. Despite a commitment to greater transparency, the bank declassifies documents on a case-by-case basis; many of my requests were denied, as they contained \"classified material provided by member states.\" Dam building remains a sensitive subject\u2014its history raises uncomfortable questions about the present and the future.\n\nAs ambitions for river development intensified, older water disputes reasserted themselves within India. British India had always existed amid a patchwork of other forms of sovereignty\u2014the absorption of the former princely states into independent India raised its own problems where water was concerned. The Kaveri River dispute between the princely state of Mysore and the British-ruled Madras Presidency dates back to 1891, when the Maharaja of Mysore first proposed to make use of the river's water in his domains. The failure of the two sides to agree on how to share the Kaveri's water stalled M. Visvesvaraya's great Krishnarajasagar Dam, which began to rise only after a 1924 agreement that apportioned fixed amounts of the river's capacity to each state. In 1956, the internal map of India was redrawn: some of the old provinces of British India were broken up into new states, their boundaries corresponding roughly to the boundaries of linguistic regions. The old Madras Presidency was divided into Tamil Nadu, Karnataka, Kerala, and Andra Pradesh, each with its own state legislature\u2014and each with a new set of claims upon the central government for resources. In this context, the Kaveri dispute resurfaced. The government of Karnataka reopened the argument, arguing that changing political and economic realities demanded a revision of the earlier agreement. In particular, the ambitious plans each state had to expand irrigation\u2014under the auspices of India's overarching five-year plans\u2014spawned a new set of claims upon the Kaveri's waters. The Kaveri dispute remains India's longest-running water conflict; it is far from resolved. Competing claims to ownership over water, in the postcolonial era, were not only national: they were just as likely to be regional.\n\n# IV\n\nMehboob Khan's 1957 melodrama, _Mother India,_ remains one of the world's best-known films. It is, among other things, Indian cinema's great water epic. The film opens with the shot of an aged Radha, the film's protagonist and the eponymous \"Mother India,\" touching a clump of earth to her mouth; she raises it above her head, hands trembling. Behind her: tractors, power lines, roads\u2014the churn of progress. A procession of construction equipment roars in the foreground, drowning out the music. The camera pans to a shot of a large dam rising. A jeep arrives in the village, full of khaki-clad, white-capped men\u2014functionaries of the ruling Congress party. They tell Radha that the new dam will bring water to her village; they want her, as the community's most respected elder, to inaugurate it. She refuses in all humility, until, head lowered, she allows the politicians to lead her to the dam. They place a ceremonial garland around her neck. At the very moment she is about to pull the lever to open the dam's gates, the film slips into flashback. Radha's life comes to symbolize the struggle of the Indian nation for freedom.\n\nEarly in the film Radha's husband is maimed in an accident; scenting her vulnerability, the predatory moneylender makes advances to Radha, which she rebuffs. The capricious power of nature over Radha's life and livelihood is a recurrent theme. In one of the film's central song sequences, we see Radha working in the fields with her children, pausing to feed them a meager meal of porridge, eating none herself. But she is proud, unbowed. As the song comes to an end, storm clouds build in the sky. The sky darkens, the screen crackles with lightning, the wind rises and the rain pours. The family's makeshift shelter collapses. Floods destroy the village. The crops lay ruined. Radha's youngest child dies. Even in extremis, Radha makes her surviving children reject the moneylender's offer of food. Redemption comes when the villagers fight back the flood; they come together to harvest a good crop the following season. In their dance, the camera zooms out to show us the massed villagers form the shape of a map of undivided India. Radha's song implores the villagers not to abandon their land.\n\nIn opening the film with the large dam, and a vision of prosperity, _Mother India_ suggests that India's victory was, in part, a triumph over the monsoon. These were the freedoms that India fought for, and won: freedom from want, freedom from exploitation\u2014and freedom from the vagaries of nature. The publicity pamphlet for the film used a quotation from the German Orientalist Max Mueller to encapsulate its central message: \"If I were to look over the whole world to find out the country most richly endowed with all the wealth, power, and beauty that nature can bestow\u2014I should point to India.\" In _Mother India,_ vulnerability to the weather is confined to the unhappy past; it represents an old, unchanging India, juxtaposed against an India where technology and political freedom would triumph over nature.\n\nMehboob Khan (1907\u20131964) was born in Baroda state and moved to Bombay as a young man to work for a noted horse supplier to the film industry; his first job was repairing horseshoes. He rose as a producer in the era of silent film. Khan was an active participant in the Progressive Writers Movement, a group of left-leaning writers, dramatists, and film producers who forged an Indian art that reflected the social conditions of the country. In common with many others of his generation, Khan's strong commitment to Indian nationalism melded with an outward-looking internationalism: he was voracious in his absorption of artistic inspiration from diverse places; he believed in a sense of shared struggle, across what would come to be known as the Third World, against imperial exploitation. The logo of his production company incorporated a hammer and sickle, a symbol that was discreetly removed before the film was submitted for the Oscars.\n\nFrom the start, Khan envisaged an international audience for _Mother India_. His initial working title was _This Land Is Mine,_ and he wanted to work with Sabu Dastogir, an Indian actor successful in Hollywood; the plan fell through, but the ambition to reach the wider world remained. _Mother India_ was not the first Hindi film to strike a chord with audiences across Asia, Africa, and Eastern Europe. Raj Kapoor's _Awaara_ (1951) was a global blockbuster, the story of an unfeeling and autocratic judge and his estranged son, who becomes a vagrant. The film's strong message of social justice, its memorable theme song (\"Awaara Hoon\"), and its visual splendor all compelled audiences across Asia and Africa; reportedly, Mao Zedong was a fan. Mehboob Khan followed Raj Kapoor's success with _Mother India,_ which was popular across Francophone and Anglophone West Africa, in Ethiopia, across Asia and the Middle East, and in the Soviet Union and Greece. Egyptian audiences were enthusiastic about _Mother India,_ which had immediate resonance in another country where dams symbolized modernity.\n\nThe film's influence endures. Anthropologist Brian Larkin describes a scene in northern Nigeria in the 1990s, where Lebanese distributors had been importing Hindi films for four decades. \"It is Friday night in Kano, and _Mother India_ is playing at the Marhaba Cinema,\" Larkin writes. \"Outside, scalpers are hurriedly selling the last of their tickets to the two thousand people lucky enough to buy seats in the open-air cinema of this city on the edge of Africa's Sahel desert.\" Throughout the screening, \"people sing along to the songs in Hindi, they translate the dialogue into Hausa and speak the actors' lines for them.\" Forty years after its release, _Mother India_ 's appeal transcends generations. \"I have been showing this film for decades,\" a local distributor told Larkin, \"it can still sell out any cinema in the north.\"\n\nHow did a Hindi film about one woman's lifelong struggle against nature and exploitation prove so resonant across so much of the world? We are used to thinking of \"development\" as something imposed from on high, by all-powerful states upon unsuspecting populations. Since the 1980s, a lot of writing about \"development\" in the postcolonial world has been heavy with irony: we know, now, how so many of those schemes turned out. Their costs are too evident; their consequences, intended and unintended, have mounted. But beneath the grand plans were also simple dreams of a better life. _Mother India_ touched millions of people because it told a humane and humanizing story about dreams of water and plenty, dreams of security\u2014dreams of a future that was better than the past.\n\n# V\n\nA darker reality lurked behind the glossy dreams of dams and plenty. On a rock of colonial-era legislation, the Land Acquisition Act of 1894, millions of lives in independent India foundered. Passed to foster railway development, the law gave the state the right to compulsory purchase of land for \"public benefit\"\u2014the law of \"eminent domain,\" a version of which most modern states have retained. The postcolonial state pressed the act immediately into service. Bhakra, Hirakud, the Damodar valley: each of these projects began in 1948; each of them needed more than one hundred thousand acres of land. So, too, did the new steel plants of Bhilai, Rourkela, and Durgapur.\n\nEven before the end of the war, intensive discussions were underway about the numbers of people who would be displaced by the Bhakra Nangal Dam. One Punjab official wrote in February 1945: \"We cannot obviously allow the whole scheme to be wrecked by a few obstinate people who may refuse to move.\" If the \"majority\" were \"not strongly averse to the proposal,\" then those against it would \"if necessary [be] ejected by force.\" Pieces of paper came to bear immense value; a Punjab government official dismissed cultivators' fears of dispossession by pointing out that \"they appear to have no right in the lands they cultivate at present.\" He could \"sympathize wholeheartedly\" with those \"ousted from their ancestral lands,\" but the \"cold truth\" was that \"their interests cannot be allowed to impede\" a project \"which, with its irrigation and hydroelectric potentialities, is likely to carry so many benefits to wide areas of the country.\" The architects of Bhakra were under no illusions about the scale of displacement the project would cause. However attractive the alternative land offered villagers, one official wrote, \"it is extremely doubtful whether it will be a satisfactory form of gilt-edged security.\" Where could people go? How would they recover their livelihoods? And what of their bonds of community\u2014their bonds with the piece of the earth they knew best? Too often, irreversible projects ploughed ahead with those questions unasked.\n\nDisplacement never went unchallenged. In December 1950, ten thousand villagers from an area soon to be submerged by the Tungabhadra Dam came together in a public meeting. They issued a list of demands, topped by their demand for adequate compensation for their loss of land and income. Among those who needed to be resettled were not only registered landowners, but also \"houseless coolies, weavers, and ryots [cultivators]\" who faced the \"difficult present circumstances.\" Two years later, the government offered them a better deal: 30 percent over and above the amounts awarded by the Land Acquisition officers, but only to those \"who withdraw their cases from the civil courts.\" In this rare willingness to compromise lies a clue to just how many people fought their dispossession in the courts of independent India. A few months after the meeting of Tungabhadra villagers, a touring official wrote to Hyderabad's minister of revenue: \"I told the people gathered in a pretty large number at Manur that they should not stand in the way of the construction of the project but should consider it a great sacrifice on their part, since by the sufferings, if at all, of a small number the country is going to prosper.\"\n\nThis was the crux of the utilitarian argument\u2014the greatest good for the greatest number\u2014that authorized India's great water projects at any cost: in families displaced, in villages drowned and futures ruined. Through most of the 1950s, India's courts agreed. When the Maharaja of Darbhanga, Kameshwar Singh, tried to resist the acquisition of part of his Bihar estate, the High Court ruled that \"the Legislature is the best judge of what is good for the community... and it is not possible for this Court to say that there was no public purpose behind the acquisition contemplated by the impugned state.\" Convinced of its mission, the Indian state felt little need even to document the displaced.\n\nThe true scale of displacement would only become clear later on. Through the painstaking work of activists like Walter Fernandes and scholars like Sanjoy Chakravorty, working through fragmentary archives and court cases, we now have a sense of the numbers of people that India's dams chased from their homes. From independence to the present day, that number is likely to exceed 40 million. It bears repeating: 40 million people in India have been displaced by dams alone. More than 50 million people have been displaced by the state's development projects writ large. India's _adivasis_ have borne the brunt of this displacement, and they are also least likely to have received any form of compensation. One reason that almost every official figure vastly underestimates the numbers displaced is that they count only those who own land that has been acquired by the state through compulsory purchase. They do not include the large number of landless people in India who \"depend on the acquired land for income\"\u2014tenants, wage laborers, service providers. The latter group have seldom if ever received compensation. The other way in which the true losses sustained by displaced communities exceed official calculations is that dams often submerged common property\u2014forests, grazing grounds, and other grounds considered \"wasteland\" by the state. The commons were already under strain by the end of the nineteenth century, more likely than ever to be enclosed and possessed as private property\u2014but the poorest groups in society, _adivasis_ above all, still depended on them in the 1950s. Through a painstaking assembly of data, Chakravorty has estimated that water projects are by far the greatest cause of population displacement in independent India. The \"core problem,\" he writes, is that \"the population that benefitted from the development projects is fundamentally different from the population that was displaced or disrupted.\" The benefits of large dams have gone downstream; power from hydroelectric plants has gone to cities and factories and to farmers who \"still have land [on which] to run their pumps.\"\n\nThe environmental consequences of large dams have been considerable: forests have been drowned, soils salinated, rivers blocked in midflow, deltas starved of silt, natural drainage hindered\u2014leading, ironically, to more severe flooding. But the sheer scale of the impact of large dams on Asia's ecology of water would only become clear toward the end of the twentieth century.\n\nNehru himself began to have a change of heart at the end of the 1950s. \"For some time past,\" he said, speaking about India's dam fever, \"I have been beginning to think that we are suffering from what we may call 'disease of gigantism.'\" He proceeded to tell his audience of engineers that \"the small irrigation projects, the small industries and the small plants for electric power will change the face of the country, far more than a dozen big projects in half a dozen places.\" But a massive failure in the monsoons precipitated a shift in political priorities.\n\n# VI\n\nThe United Nations' Economic Commission for Asia and the Far East (ECAFE) started life in Shanghai in 1947. The commission set about collecting information on economic conditions across Asia. Its first survey came out in 1948\u2014an \"incomparably more thorough and informative account of the economic life of the region\" than any ever attempted\u2014and painted a picture of a continent in ruins after the war. ECAFE's original regional members were Pakistan, India, Burma, Thailand, the Philippines, and China. At a time when much of Asia remained under imperial control, \"nonregional\" members\u2014the imperial powers and the Soviet Union\u2014had a major presence, as did \"associate,\" nonvoting, members that were still under colonial rule or postwar occupation\u2014Ceylon Malaya, Vietnam, Cambodia, Laos, Hong Kong, the Republic of Korea, and Japan. The intersection of Asia's postwar revolutions with the early Cold War created immediate political tensions. After the Communist victory in the Chinese civil war in 1949, ECAFE decamped from Shanghai to Bangkok; like the rest of the UN system, it recognized the Republican government in Taiwan as the rightful representative of China.\n\nAsia in the 1940s was torn three ways: between colonial powers still clinging to power, newly independent states hungry for the technology and financing that would allow them to realize their grand plans, and the new superpowers in competition for their allegiance. ECAFE's deputy director, the American C. Hart Schaaf, made the bold claim that, amid these tensions, ECAFE was primarily the creature of new Asian states. He insisted that \"in the most extensive and populous region on earth,\" the \"most conspicuous political fact\" was \"a new dynamic nationalism\"; he suggested that ECAFE had not only \"witnessed\" but also \"facilitated\" this movement. One indication of this stance was ECAFE's appointment of an Indian director, P. Lokanathan, a Madras University economist who had been editor of the influential periodical _Eastern Economist._ Invoking the judgment of an imagined \"future historian of ECAFE,\" Hart Schaaf argued that ECAFE's role in the region would in time be seen as vital.\n\nAsia faced a \"revolution of rising expectations\" that cut across revolutionary and nonrevolutionary, capitalist and Communist states. ECAFE embraced water as a major regional priority, and one that could bring people together: it was an area where tangible results could be achieved. In 1950 the organization convened a meeting in Bangkok on flood control in Asia. Hydraulic engineers, city planners, and hydrologists from across the region came together to compare notes; through the mediation of international agencies, \"often... a particular national project becomes regional.\" In his search for a metaphor that would explain the benefits of regional cooperation, Hart Schaaf turned to \"the concept of an act or event which sets in motion a cumulative process of great momentum,\" adding that it was \"around [this idea] that Lord Keynes and others have constructed their thinking about the 'multiplier.'\" ECAFE's achievements were piecemeal, perhaps relatively small, but they were not insignificant. The agency epitomized the ideal known as \"functionalism,\" which Lilienthal had put forth in response to the Indus dispute: that technical matters, from public health to water management, could be removed from the political arena and solved cooperatively by experts whose professional camaraderie transcended geopolitical fault lines. But the politics of national sovereignty, like the geopolitics of superpower rivalry, could not be avoided. The very absence of the People's Republic of China from the ECAFE table made the organization's claims to a comprehensive regional perspective ring hollow. In 1954, for the first time, ECAFE's _Economic Survey_ included \"Mainland China,\" compiling statistics from official sources in the PRC: it was among ECAFE's most keenly read publications, as Asian planners, economists, and statesmen had the occasion to compare their countries' rates of \"progress\" with that of the revolutionary behemoth.\n\nThe curiosity of Asia's leaders and engineers about China\u2014fascination combined with suspicion\u2014went beyond the capacity of an ECAFE report to satisfy. Nowhere more so than in China's largest neighbor, India. The government of India had been among the first to recognize the People's Republic of China in 1949. Nehru's first ambassador to China, the historian K. M. Panikkar, was keenly impressed by China's revolution, and by Mao. The Chinese government's invasion and annexation of Tibet in October 1950 alarmed and surprised India, which regarded a semi-independent and friendly Tibet as a buffer along its border with China. Nehru came under pressure from others within the Congress to take a harder line; but a realistic sense of India's inability to intervene prevailed. Relations between the two countries warmed in the 1950s in line with Nehru's foreign policy of nonalignment with either bloc in the Cold War. Among the Indians most keen to learn about developments in China were the country's leading hydraulic engineers. In May 1954, Kanwar Sain, chairman of India's Central Water Commission, and K. L. Rao embarked on an official mission to China to inspect and to report back on China's water projects\u2014flood control in particular. Sain and Rao were among the first outsiders to see, firsthand, China's hydrological experiment; and because they were favorably disposed toward the Chinese government, they had extensive access to information.\n\nSain and Rao arrived in China on May 4, 1954, and stayed two months. They spent much of their time on the water, making many parts of their journey along the Yangzi by boat. Sain and Rao were chief protagonists of India's own colossal efforts to control water; the scale of work in China dazzled them. They undertook their tour of China half a century after the Indian Irrigation Commission had traveled through India in search of water. Theirs was part of the same quest: the famines of the late nineteenth century had unleashed in India a desperate and continuing search for sources of water to mitigate dependence on the monsoon. Sain and Rao were trained in the colonial tradition of Indian water engineering. But now they represented an independent nation, and for inspiration they looked not to Europe but to revolutionary China. Consider the contrast between the two tours: where the irrigation commission traveled with a retinue of servants in a specially chartered train, Sain and Rao were given strictly limited foreign exchange. They were accompanied by just two interpreters and two officials. They were impressed by the modesty, the lack of ostentation they saw\u2014and by the absence of the tight social hierarchies of India, to say nothing of caste. Another thing puzzled Sain and Rao. For generations, Indian intellectuals had believed that the most fundamental bond between India and China was Buddhism. But \"to our repeated questions about the Buddha\" came a standard response: \"Chairman Mao is our Buddha.\" In a new era, India and China needed a new language, a new basis for their interaction\u2014it came naturally to two hydraulic engineers to see that basis in the shared problem of water.\n\nDistinguishing his perspective from the negativity about China that he had imbibed reading the accounts of \"foreign diplomats,\" Sain commented on the high standards of sanitation he saw everywhere. An austere Indian engineer in the tradition of Visvesvaraya, Sain noted with satisfaction \"an absence of headlines in the newspapers highlighting murder, scandals, or disgraceful lives.\" Most of all, he was impressed by the \"clear-cut vision\" state officials had of a \"new China of their dreams,\" which in turn instilled in the people \"unbound faith and confidence in the wisdom, goodness, and creative policy of the government.\" Sain contrasted the Shanghai he had seen in 1939, on his way home from the United States\u2014a cosmopolitan and decadent city, in his mind\u2014with the city he saw in 1954. \"The bright lights had gone out,\" he wrote, but he meant this as a compliment. Shanghai now looked \"more typically Chinese,\" shorn of Western concessions and colonial settlements. Like so many other port cities along the littorals of the Indian Ocean and the South China Sea, it was now \"integrated [more] closely with the economy of the interior rather than dependent on foreign luxury trade.\"\n\nSain's and Rao's report contains an extended list of every Chinese official they met, from ministers to field engineers to water scientists at the College of Hydraulic Engineering for Eastern China, in Nanjing. They were struck by the quality of China's hydraulic engineers. They delighted in the firm emphasis given to technical education in China. They praised the Chinese capacity for improvisation, building huge dams from local materials when imports were in short supply. Traveling through China, Sain's and Rao's thoughts turned naturally to comparisons with India. There were clear differences in the challenges each country faced. One sharp contrast between India and China was climatic\u2014once again, what made India distinctive was the monsoon. \"Unlike India, hemmed in by the Himalayas,\" they wrote, \"China is open to Central Asia\"; this meant that, in the summer, \"China unlike India is not the single objective of the air circulation of a whole ocean.\" China received \"less heavy and less concentrated rainfall\" than India, and its rain was \"much more equally spread across the interior.\" By contrast, China's rivers were more menacing than India's, more prone to burst their banks. India's great need was irrigation; China's was flood control. Both countries eyed an industrial future, and the promise of hydroelectric power attracted them both.\n\nIndia and China shared a sense of urgency. They shared a conviction that water held the key to security and prosperity\u2014these translated into an addiction to mammoth projects. In both countries, bigger was better. Just as the pamphlets of India's Central Water Commission and the documentaries of its Films Division extolled the pace and the scale of dam building in India, so too did the Chinese state and its engineers take pride in their compression of time. In the five years since the liberation, they boasted, \"250 major and thousands of minor irrigation projects\" had begun in China, adding 9.2 million acres of irrigated land. Most striking to the Indian visitors were the \"remarkable speeds of construction\" China had achieved through the mobilization of labor on a scale \"unknown in recent times.\"\n\nSoon after the establishment of the People's Republic in 1949, Mao's government made the Yellow River a priority. Known for centuries as \"China's sorrow,\" the Yellow River was notoriously prone to flooding. Within a year after the liberation, both the Huai and Yellow rivers experienced catastrophic floods. In the interests of national reconstruction, they had to be tamed and conquered. Where India drew expertise and aid from both the Americans and the Soviets, and where Nationalist water engineers in China in the 1920s and 1930s had maintained close links with the United States and Germany, after 1949 Chinese hydraulic engineers combined Russian technical assistance with local ingenuity. In parallel with India's race to build Bhakra Nangal, the most ambitious Chinese dam was the Sanmenxia on the Yellow River.\n\nLike Bhakra, the Sanmenxia had its origins in the 1930s; like Bhakra, a sense of urgency that followed the revolutionary upheaval of the 1940s brought it to the top of the agenda. The dam was located near the border between Shanxi and Henan, designed by Soviet engineers. The Soviets proposed a concrete gravity dam across the Yellow River, with a reservoir 360 meters above sea level. The initial plan would have displaced more than 800,000 people from their homes, and flooded 3,500 square kilometers of land. As plans for the dam went into circulation, between 1955 and 1957, Chinese experts debated it at length. During a fleeting moment of political openness under Mao's \"Hundred Flowers\" campaign, hydraulic engineer Huang Wanli\u2014trained as a meteorologist at Cornell and Iowa in the 1930s, and then an aide at the Tennessee Valley Authority\u2014raised the alarm. He argued in favor of a lower dam, with a smaller reservoir. He hinted that the Soviet plans had not undertaken a detailed analysis of costs and benefits. He felt that a smaller scheme, which displaced fewer people, would be less risky. He feared that the dam's design was no match for the heavy loads of silt that the Yellow River carried: the danger, as he saw it, was that the dam's reservoir would silt up, making the dam useless\u2014or, worse, dangerous.\n\nThe Hundred Flowers campaign was short-lived; as people spoke more freely, Mao disliked what they said. His retribution was swift. Huang was condemned and humiliated. He was deemed a \"rightist\" and sent for \"re-education.\" His predictions proved uncomfortably accurate. Within a few years of the Sanmenxia Dam's completion in 1960, it was clear that its reservoir was clogged with sediment. As Sino-Soviet relations soured after 1960, it became easier to blame a faulty Soviet design for the dam's problems. Acknowledging the scale of the problem, Zhou Enlai ordered a reconstruction and renovation of the scheme, at huge cost. At the same time, the human cost of the dam was immense. Just as Bhakra and Hirakud began the decades-long displacement of Indians by large dams, Sanmenxia led to the forcible relocation of an estimated 280,000 people.\n\nOther aspects of China's experience had no parallel in India, as Sain and Rao were quick to notice. The mobilization of labor in China was on a scale unknown in India, and this also set China on a path to water engineering quite different from the one established by their Soviet allies. Mao had prevailed in the Chinese civil war because of his stunning success in mobilizing popular support and enthusiasm\u2014first in the vanguard of the anti-Japanese resistance, and then deployed against Chiang Kai-shek's Nationalist army. This commitment to mass action never left Mao. It shaped deeply his government's approach to water management. The first achievement of the people's energies, mobilized at the county level, village by village, was the two-hundred-kilometer People's Victory Canal, linking the Yellow River and the Wei. The _People's Daily_ ran features on the exertions of model workers who had broken records and distinguished themselves in devotion to the cause. By the mid-1950s, as China's collectivization drive gathered pace, Mao's government laid ever-greater emphasis on irrigation from the ground up. This fervor reached a peak during the Great Leap Forward, when every county was set to work building its own dams and irrigation ditches. Zhou Enlai made a rare official acknowledgment of the disastrous consequences of this approach in 1966, when he said, \"I fear that we have made a mistake in harnessing and accumulating water and cutting down so much forest cover... Some mistakes can be remedied in a day or a year, but mistakes in the field of water conservancy and forestry cannot be reversed for years.\"\n\nOf course, much of this took place after Sain's and Rao's visit. They saw no signs of danger in China's quest for water. In table after table they compared China with India\u2014how much concrete their dams consumed, how much water their reservoirs could hold. The speed of canal excavation was where China's achievements were most dramatic in comparison with India's. The Indian engineers' conclusion was wistful: \"In India, where similar human force is available, it should be possible to attain similar speeds... by proper organization and creation of enthusiasm among the people\"; they chose not to mention that \"proper organization\" would have demanded a level of coercion that the Indian government was unwilling (or unable) to muster.\n\nThe most revealing part of Sain's and Rao's report is a verbatim record of a speech by China's director of water conservancy, given at the end of their stay, in which he sought to address the questions that had arisen during their visit. Director Hao positioned the People's Republic firmly within an ancient tradition of water management in China. \"The record of exploitation of water by the Chinese people,\" he wrote, \"dates back to ancient times.\" But under the \"corrupt\" and \"feudalistic\" rule of a decaying empire, compounded by the failures of the Nationalist government, \"the hydraulic constructions [of China] were seriously ruined owing to long years of negligence.\" The Communist state claimed the mantle of imperial power over water\u2014it was a revival as well as a revolution. China's archives of water control were on display everywhere the visitors went: at Tsinghua University, they were shown an eight-hundred-year-old text \"containing excellent plans\" of the Yellow River. Hao spent as much time telling his Indian visitors about small projects as he did extolling the gargantuan ones. His was a story of repair and renovation as much as creation. He spoke of the myriad ponds and dikes that conveyed water to the fields of southern China. He spoke of the spread of simple technology\u2014a \"Liberation-type waterwheel\" that outdid the age-old technologies still in use.\n\nSain and Rao returned to India filled with enthusiasm. For all that they grasped the complexity of China's approach to water, their message back home was a simple one: it was a message of scale, speed, and control. Their main \"lesson\" from the Yellow River for the management of the notoriously flood-prone Kosi River of Bihar was the lesson of centralized command; China's intensive emphasis on small projects fell by the wayside in their accounts. Soon after his return Sain was summoned by Nehru, and \"closeted with him for about an hour.\" Nehru, soon to depart for China himself, asked for Sain's impressions. Listening closely, he pressed harder: how would Sain describe China in one sentence? Sain recalls that he gave Nehru an unscripted answer: \"At present China is behind India in every field, but I feel that at the rate they are progressing, China may be ahead of India in 10-15 years.\" Nehru \"made no comment,\" Sain remembered, \"but I could see from his face that he did not relish this reply.\"\n\nIndia's most eminent water engineers returned from China with a sense that the two countries shared fundamental problems, and that there were lessons they could learn from China. But there were ominous portents, too. In his speech, Director Hao had described how China's water projects had been \"extended to the border regions of our fraternal minorities and helped to promote national unity.\" There was never an attempt on the Chinese side to disguise the fact that water was intrinsic to political power. The conquest of water meant the conquest of space. With the control over water came the projection of state power over peoples with a different vision of water's uses: the people of Xinjiang, the people of Tibet. Unspoken was the thought that some day the \"border regions\" in question may include China's borders with India.\n\nSain and Rao faced a problem when they returned with the first-ever maps of China's water projects to be seen outside China. The Indian Ministry of External Affairs was rankled by what Sain and Rao had missed: the maps they had been given by Chinese officials claimed as Chinese territory a large swath of the borderland that the Indian state saw as integral to India. The maps were destroyed, redrawn to accord with India's understanding of its territorial boundaries. Sain later wrote in his memoirs that he was grateful this had been caught before the volume was published\u2014if it had not, it would have been a source of \"great embarrassment\" a few years later, when India and China went to war over just those borders.\n\nOne of Sain's enduring impressions of his trip had been \"how the Chinese people loved and admired the Russians.\" \"The bookstalls are generally full of Russian books and journals,\" he wrote; Russian expertise was offered without condescension and without strings\u2014or so it seemed. He had the opportunity to see for himself a year later, when he led an Indian delegation to the Soviet Union, sponsored by the United Nations' Technical Assistance Administration. Again, the visit of Indian technical experts was followed by an official visit by Nehru. Sain wrote a sweeping account of just how rapid the Soviet Union's economic progress had been since the revolution. Inspecting its hydraulic projects, he concluded that \"the interests of the power engineers have been accorded pride of place,\" a dominance he traced back to Lenin's emphasis on electrification as the key to socialism\u2014flood control and irrigation, India's and China's other great needs, were less valued. But Sain's conclusion was clear: China had much more directly to teach India than the Soviets.\n\nAS CHINA'S EXPERIENCES INSPIRED INDIA'S ENGINEERS, SO INDIA'S experiences became a model for the rest of Asia. A year after his visit to the Soviet Union, ECAFE's director, Lokanathan, commissioned Sain to join a UN mission to survey the Mekong. Like the Brahmaputra, the Salween, and the Yangzi, the Mekong originates on the Tibetan Plateau. In the twentieth century, it has been Asia's quintessential \"transboundary river,\" running through China, Burma, Thailand, Laos, Cambodia, and Vietnam before spilling into the South China Sea. The Mekong, the ECAFE commission noted with understatement, was \"a perennial river of great importance.\" That importance was clear to the US government, which maintained an escalating financial and military presence in South Vietnam after the end of the French-Vietnamese war in 1954, which had resulted in the division of the country. The US Bureau of Reclamation, the domestic agency responsible for hydraulic engineering, had a global presence by the 1950s. Its engineers surveyed Thailand, the Philippines, and Indonesia, looking for hydroelectric potential. They had surveyed the Mekong, too, but their initial verdict was lukewarm.\n\nConvinced that a substantively international effort was needed, the UN went back. Sain was joined on the commission by Y. Kubota, the president of Japan's Nippon Koei corporation; G. Duval, a former colonial official; M. Sakaita, an engineering geologist from Japan; and the Dutchman W. J. van der Oord, a navigation specialist. They toured the Mekong in April and May 1956. The commission placed its faith in two large hydroelectric projects, one on the Tonle Sap in Cambodia, the other at Nam Lik in Laos. \"The prick has gone too deep to be halted\"\u2014this is how Sain described his sense that large-scale hydraulic engineering was inevitable, now, in the Mekong as elsewhere in Asia, given the bold claims that had been made on behalf of big dams, and given the hunger for progress and development that he saw wherever he traveled. The following year, Sain joined another commission coordinated by ECAFE and led by Raymond Wheeler, former chief engineer of the US Army Corps of Engineers. Wheeler's account of the mission harked back to the language of colonial exploration. \"There were no maps of the country,\" he wrote, \"we had to make them... Nobody had any data on river flow, or even any idea how to keep data.\" Wheeler described the Mekong as \"truly a virgin river.\" Historian David Biggs notes that the commission proposed \"a cascade of hydroelectric dams and irrigation schemes in the valley from the Chinese border southward to the Mekong Delta.\" The Mekong commission signified an opening for private interests who stood to profit from the dam-building rush; Japanese, Korean, and Taiwanese companies, in particular, stepped in with materials and personnel.\n\nThe Mekong commission was quickly overshadowed by the escalation of American involvement in Indochina. The US Bureau of Reclamation placed its faith in what it called \"impact type projects,\" the grandest of them being the Pa Mong Dam, upstream of the Laotian capital, Vientiane. Vast, ambitious, planned to the last detail\u2014the dam never materialized, as the United States became mired in military conflict in Vietnam that engulfed Vietnam's neighbors as well. There was a close bond between American support for dam building in Asia and American strategic imperatives in the Cold War. But Kanwar Sain\u2014a patriotic Indian engineer at the pinnacle of his profession, enamored of China but with close personal and professional links to the US Bureau of Reclamation\u2014chose to spend a decade of his career with the Mekong commission, trying to coordinate the development of Asia's most international river. In his memoirs, he hints that the material reward of working for the United Nations was one clear incentive. But his motivations went deeper than the money. Sain believed, like so many of his generation, that taming the waters was a goal beyond national sovereignty\u2014and beyond ideology. Working for ECAFE alongside many former colonial civil servants and engineers now turned development consultants, Sain held to a vision of Asian nations working together to claim their rightful place in the community of nations. In a memoir that is detached, even clinical, in tone, a rare moment of emotion comes when Sain describes his \"pilgrimage\" to the site of Angkor Wat, in Siem Reap, Cambodia, while on his first Mekong mission: \"I was very much moved by the ancient glory and culture of India reflected in Angkor Wat,\" he wrote. Just as many of India's water engineers presented their \"new temples\" as standing within an ancient historical tradition of water engineering, so Sain appealed to a deep history of cultural exchange across borders to provide ballast for his vision of an Asia united by water\u2014or by water engineers.\n\nIn a sense, Sain's faith was eventually vindicated. The Mekong commission outlasted the American war. It received a new lease of life in the 1990s and now stands as one of the most important, if not always the most effective, river-regulating bodies in the world.\n\n# VII\n\nThe \"multiplier\" that ECAFE invoked to justify its work on cross-border river valley development could have the opposite effect: as projects and ambitions escalated, so did the potential for conflict. After a high point of warmth in their relationship in the mid-1950s\u2014the era of \"Hindi-Chini _bhai bhai_ \" (India and China are brothers)\u2014the territorial conflict between India and China intensified. The unmarked and mountainous frontier between India and China became contentious as both states intensified their presence in the borderlands. New infrastructure brought border regions within easier reach of Beijing and Delhi; military forces were stationed there; migration from the plains brought new settlers, often ethnically distinct from the people who inhabited the uplands. The spark for conflict was the construction of a Chinese road linking Xinjiang and Tibet\u2014a road that passed within what India considered to be its territory. Indian intelligence did not find out about the road until 1957, by which time its construction was well advanced. India insisted on the sanctity of agreements made under the British; the Chinese charged that India now stood as the beneficiary of British imperial aggression. In a pained and lengthy letter to Zhou Enlai, which was later published by the Indian government, Nehru countered that \"the boundaries of India were... settled for centuries by history, geography, custom and tradition.\" He turned, then, to water: \"The water-parting formed by the crest of the Himalayas is the natural frontier\" between India and China, \"accepted for centuries as the boundary.\"\n\nWater was not, by 1960, perceived as a source of conflict, but recently declassified Indian sources show that there were fears about the future. Rumors were rife. In exile in India after the failed Tibetan uprising of 1959, the Dalai Lama raised the alarm in a public meeting. He charged that the Chinese state was \"planning to build high dams across the Brahmaputra and Indus group of rivers in the Tibet region,\" and that the Chinese \"had these schemes in view ever since they came to Tibet.\" He asked, pointedly, \"how far such projects undertaken unilaterally would be in the interests of India.\" The Indian foreign ministry responded cautiously to a report on the Dalai Lama's speech. \"We have... no information so far about any proposal of the Chinese government to construct dams across the Indus or Brahmaputra before the rivers leave Tibet,\" one official wrote, but he saw the \"necessity of being alert in this manner.\" Indian officials were well aware that \"there is a great fall in the Brahmaputra just before it enters India\" with \"potential for power and irrigation.\" But they were reassured by the thought that it would take \"huge resources to make anything of it\"; any plans the Chinese had \"will certainly take a long time.\" The Indian trade mission in Gyantse, Tibet\u2014which clearly doubled as a source of intelligence\u2014concurred. They reported to Apa Pant, the chief political officer in the Indian protectorate of Sikkim, that \"construction of dams and reservoirs on the river is likely to involve huge resources including manpower, which the Chinese authorities will be able to utilize only after they have brought in large numbers of Chinese for settlement.\"\n\nThe Indian Ministry of External Affairs was concerned enough to involve colleagues in the irrigation department. In a letter marked \"top secret,\" K. K. Framji, chief engineer of India's irrigation department, reassured the foreign ministry that \"substantial or imminent diversions by China for irrigation purposes in the Tibet region do not appear to be practicable.\" The construction of storage dams for power generation might even benefit India as they \"would be helpful in mitigating floods in Assam or East Pakistan.\" But he then raised a darker prospect: \"If the Chinese hydro-electric schemes are so projected as to divert substantial quantities of Brahmaputra flows away from the present course into adjoining valleys,\" this would be \"a significant loss of valuable water resources to India, and even more so, to Pakistan.\" He concluded on a hopeful note: \"No doubt we will be given timely information regarding any observed or reported activities towards any such diversion.\"\n\nEvents soon overtook these concerns about the future of water. In 1959, India infuriated the Chinese government by granting asylum to the Dalai Lama; from that point, tensions on the border between India and China ran high. Both sides built up their military forces along the border; India pursued a \"Forward Policy,\" stationing troops north of the McMahon line, the 1914 frontier that had marked the boundary between Tibet and British India. Taking the Indians by surprise, Chinese military forces launched attacks on both the eastern and western flanks of the border region on October 20, 1962. As the world was transfixed by the Cuban missile crisis, Indian and Chinese forces fought in the high Himalayas. But it was no contest: the Indian military was no match for Chinese forces, who won decisive victories. A month after the offensive, the Chinese declared a unilateral cease-fire and withdrew their forces to the \"line of actual control\"\u2014or the de facto border.\n\nThe war with China marked a humiliating defeat for India. The Indian army was ill-equipped, ill-prepared; China's invasion seemed to mock the effort Nehru had put into fostering good relations between the two countries. Nehru's political legitimacy at home was battered. With hindsight, 1962 appears as the beginning of the end of the Nehru era in Indian politics. In the opening pages of his first novel, _Such a Long Journey,_ set in Bombay of the 1960s and 1970s, Rohinton Mistry evokes a widespread sense that \"the war with China froze Jawaharlal Nehru's heart, then broke it. He never recovered from what he perceived to be Chou En-lai's betrayal.\" Nehru's frank, even desperate, plea for American military assistance in the war dented his commitment to nonalignment in the Cold War. India's defeat on the international stage coincided with a rising chorus of criticism at home, raising questions about the economic and political strategy Nehru had pursued in the fifteen years since independence. Was there a better way?\n\nJAWAHARLAL NEHRU DIED IN 1964. HIS WILL AND TESTAMENT EXPLAINED why he wanted his ashes to be scattered in the Ganges upon his death. \"I have been attached to the Ganga and the Jumna rivers in Allahabad ever since my childhood,\" he wrote, \"and, as I have grown older, this attachment has also grown.\" The Ganges, he wrote, \"is the river of India, beloved of her people\": bearer of \"her hopes and fears, her songs of triumph, her victories and her defeats.\" Evoking the identity of the river with the very geography of India, Nehru wrote that each glimpse of the Ganges \"reminds me of the snow-covered peaks and the deep valleys of the Himalayas, which I have loved so much, and of the rich and vast plains below, where my life and work have been cast.\" Nehru was adamant that \"my desire to have a handful of my ashes thrown into the Ganga at Allahabad has no religious significance.\" Water still had imaginative power to evoke the sacred, to shape nations' perceptions of their limits. The Ganges remained the essence of India, the Himalayas India's natural boundary, even in an age when India's \"new temples\" were large dams.\n\n# EIGHT\n\n# THE OCEAN AND THE UNDERGROUND\n\nIN _T HE LIVES OF OTHERS_, SET IN BENGAL IN THE LATE 1960S AND early 1970s, Neel Mukherjee brings a novelist's imaginative sympathy to evoke what it feels like to depend on the rains. Supratik, the novel's protagonist, has become involved in the Naxalite movement, a violent revolutionary movement of Maoist inspiration, committed to rural revolution in India. As part of his work of political outreach, Supratik, a privileged city boy, spends time in the countryside, growing accustomed to its rhythms. He inhabits a rural Bengal that is impoverished, indebted, under the heel of landlords\u2014and governed by the monsoon. As they prepare the seedbeds to transplant paddy saplings, Supratik's host, Kanu, gazes at the sky. His questions are perennial ones: \"Would it arrive this year? his eyes seemed to be asking; would it be late? would it be enough? There was both anxiety and resignation in his face.\" And then, this time, the rains came. Supratik continues: \"It was exactly as I remembered from childhood\u2014sheets of water coming down for hours and hitting the ground with such force that you thought the road would dissolve\u2014except here the ground, which is earth, does dissolve.\"\n\nOn the surface there is little to connect this imaginative account of the lived experience of climate with the meteorological data being accumulated by international scientists at the time. Human lives and voices could not be fed into climate models, where readings of pressure and wind and moisture could. But they told different facets of the same story.\n\nThis chapter tells two intersecting stories\u2014the connections between them were not fully evident at the time. The first is the story of monsoon science in the 1960s; the second is the political and economic history of India's mid-1960s droughts. The International Indian Ocean Expedition ran between 1959 and 1965. It used new technology to uncover the forces underpinning the monsoon; it resituated South Asia in relation to the vastness of the Indian Ocean; it established new links between the countries along the ocean's rim, including a web of weather-monitoring stations. The scientific expedition raised the alarm that human activity was starting to make itself felt in the oceans\u2014perhaps even that it was altering Earth's climate. At just the moment when Asian states were sloughing off the web of connections that linked them across borders and seas, satellites and aerial photographs and deep sea probes projected a view of Asia shaped by a vast, connected climate system\u2014a system with very tangible consequences for the large development schemes that states around the Indian Ocean rim had embarked on.\n\nIn the same years the monsoon came urgently back into view in India, which was in the grip of drought for three pivotal years in the mid-1960s. India's enthusiasm for the Indian Ocean Expedition, in common with many other countries in the region, was driven primarily by short-term concerns with a looming food crisis, which threatened political unrest. The Indian Ocean Expedition promised a survey of the sea as a set of \"material resources to be exploited.\" Most of all, it promised more accurate weather forecasts. India's response to the crisis of the mid-1960s was to intensify its quest for water. The dam building of the 1950s had not gone far or fast enough. Old fears of the monsoon climate resurfaced, regardless of the advances in climatic understanding that the new science promised. The government adopted a package of agricultural reforms that included high-yielding crops, vast quantities of chemical fertilizer, and the more intensive exploitation of groundwater using electric pumps.\n\nIN HIS 1981 PRESIDENTIAL ADDRESS TO THE AMERICAN HISTORICAL Association, on \"The Challenge of Modern Historiography,\" historian of the Atlantic world Bernard Bailyn spoke of the relationship between what he called latent and manifest processes in history. The former he described as \"events that contemporaries were not fully or clearly aware of... however much they might have been forced unwittingly to grapple with their consequences.\" He described the relationship between latent and manifest events like this:\n\nThe events I am referring to were known, if at all, only vaguely by contemporaries or by previous historians to have _been_ events; they are being discovered as particular happenings now for the first time. Taken together, they form a new landscape, a landscape like that of the ocean floor, assumed to have existed in some vague way by people struggling at the surface of the waves but never seen before as actual rocks, ravines, and cliffs. And like the newly discovered ocean floor\u2014so rich, complex, and busy a world in itself\u2014the world of latent events can be seen to be part of, directly involved with, the manifest history of the surface world itself.\n\nBailyn's oceanographic metaphor is especially apt in this chapter, where it takes on literal as well as symbolic meaning. We can now, for the first time, integrate the discovery of the Indian Ocean's effects on the monsoon, the early signs of climate change, with the manifest political and economic transformations of India and other parts of Asia in the 1960s and 1970s.\n\nThe lessons of the new climate science\u2014that Asia was intensely vulnerable, increasingly interconnected, bound by growing risks from the destabilization of its climate\u2014went unheeded before a renewed quest to conquer nature. The 1960s and 1970s were the decades that pushed India and other parts of Asia more fully toward a crisis of water.\n\n# I\n\nBy the 1960s the Indian Ocean was largely invisible to states in South Asia who looked no further than the waters immediately off their coasts. Migrants had once traversed the sea with few restrictions, in a pattern of circular migration. Now they faced an obstacle-strewn space governed by passports and visa restrictions. As India prepared for the International Conference on the Law of the Sea, to be held in Geneva under the auspices of the United Nations in early 1958, it was clear to Indian negotiators that the meeting would have what they called \"far-reaching consequences.\" At stake was the renegotiation of the customary three-mile limit on the extent of each state's \"territorial waters,\" a legal conception that came into widespread use around this time. Lawyers at India's Ministry of External Affairs pinpointed the core conflict: \"The countries which support the three mile limit,\" they noted, \"own over 80% of the world tonnage and are therefore interested in maintaining freedom of the seas.\" India, on the other hand, along with many developing countries, claimed a greater expanse of water of its coasts \"on the ground of security or for economic reasons such as the preservation of exclusive fishing rights for their nationals.\" Among the pressing reasons for a change were technological developments that allowed for the exploitation of resources further offshore, not least fishing by large trawlers. India was equally concerned with the \"conditions under which the waters of a bay can be regarded as internal waters,\" given the \"close linking of the waters to the land domain\" and the \"utility of the bay to the economic needs of the country.\" Although negotiations over the UN Convention on the Law of the Sea would continue through the early 1980s, the final agreement recognized the claims of countries pressing for an extended definition of territorial waters, to which was added a wider exclusive economic zone. The sea came, more and more, to resemble the land\u2014as a form of territory.\n\nTo scientists, the Indian Ocean was \"the largest unknown area on earth.\" Paul Tchernia, who worked in the physical oceanography laboratory of the National Museum of Natural History in Paris, described it as the \"forlorn ocean.\" Returning from a voyage through the Indian Ocean en route to and from the Antarctic, he suggested that an international investigation of the Indian Ocean should be incorporated into the activities of the UN's International Geophysical Year in 1957\u20131958: a massive exercise in coordinated data gathering that transformed knowledge of Earth's physical processes. Tchernia's suggestion came too late to include the Indian Ocean in that giant program, but there was a convergence of interest in investigating the least well-studied among the world's oceans. The catalyst came from a meeting of the Special Committee on Oceanic Research, set up by the International Council of Scientific Unions. It met for the first time at the Woods Hole Oceanographic Institution in coastal Massachusetts in August 1958. Among its champions was the Scripps Institution of Oceanography's Roger Revelle (1909\u20131991), a pioneer in the study of global warming and the effects of carbon emissions on the oceans.\n\nMidcentury oceanographers were drawn to the Indian Ocean for the same reason that medieval traders could cross it\u2014the seasonal reversal of the monsoon winds. This pattern of reversing winds made the Indian Ocean unique; this made it a \"model of the world ocean,\" upon which scientists could test their \"wind-driven models.\" Many scientists who lived on the ocean's rim, especially those in government service, had more immediate interests. The ocean's fisheries held the potential to address concerns about food shortages in Asia and Africa; its mineral wealth had barely begun to be exploited. Unlocking the mechanism of the ocean's influence on climate could provide the key to food security and economic development.\n\nFrom the start there was tension between short-term and long-term aims of the project; between the search for quick results and the patient accumulation of data on which to build scientific understanding. Oceanographer Henry Stommel approached the wild enthusiasm for the new Indian Ocean Expedition with skepticism. He published a few editions of an anonymous newsletter he called _Indian Ocean Bubble_ \u2014named to invoke the eighteenth-century speculative craze known as the South Sea Bubble\u2014with the implication that his colleagues' craze for the Indian Ocean, too, was built on speculation. Its circulation was limited to a short list of oceanographers. Its final editorial was honest to a fault. \"I think there is only a very remote chance that the Expedition will help improve fisheries and alleviate the poverty of the people in many Indian Ocean countries,\" Stommel wrote. He found it \"disheartening\" to see \"oceanography join the long line of pressure groups acting\u2014under the guise of humanitarianism\u2014to advance their own interests.\" Those \"interests\" were \"in themselves legitimate, but essentially unrelated to the moral and 'socio-economic' issues which they pretend to serve.\"\n\nIn the end the International Indian Ocean Expedition involved forty ships from thirteen countries. Its capacious agenda encompassed what the mission's official chronicle called \"moral and 'socio-economic issues\" as well as the \"interests\" of oceanography in basic research. The list of countries involved does not map easily onto the geography of the Cold War. Many large states bordering the Indian Ocean were enthusiastic participants in the program, including hostile neighbors India and Pakistan, as well as Indonesia and Australia. The United States played a leading role, involving scientists from the Scripps and Woods Hole institutes of oceanography, as well as the US Weather Bureau and the navy. The British, too, were heavily involved, given that they still had a substantial colonial and strategic presence in the region in the early 1960s. The Soviet Union contributed the 6,500-ton _Vityaz,_ the largest ship in the program. The Indian Ocean Expedition also marked the rebirth of German oceanography after the war, and it showed the resurgent scientific and technical prowess of Japan, which contributed two vessels, the _Kagoshima Maru_ and the _Umitaka Maru_.\n\nThe Indian ships on the expedition reflected\u2014in their origins, their shape, their materials\u2014different epochs of seafaring history. The _Kistna,_ its \"sleek lines betraying... naval origins,\" as one observer put it, was built as a naval frigate in 1943, a product of the Second World War's fillip to Indian industry. Now armed with an Edo echo sounder with a range of six thousand fathoms, the ship was fitted for oceanographic research, but it came with a warning: \"Austere living conditions; not fit for women scientists. No salt water bath fitted.\" The smallest vessel in the expedition was the _R.V. Conch,_ which belonged to the University of Kerala. It represented a much older tradition of shipbuilding: it was a small ship built of hardwood, in the long tradition of coastal craft that had threaded together India's western coast for centuries. By contrast the trawler _R.V. Varuna_ was brand-new, purpose-built in Norway in 1961 in connection with the Indo-Norwegian fisheries project. Despite its novelty, it came with the same \"men only\" warning as the naval frigate: \"Women scientists cannot be housed.\" From the earliest days of the spice traders, the Indian Ocean was crossed predominantly by men\u2014some things were very slow to change, and the loss to Indian Ocean science has been considerable.\n\nThe expedition's research aims encompassed the study of ocean currents and littoral drift; an investigation of ocean chemistry, salinity, and temperature; the exploration of marine life, and especially fisheries; the study of wind and atmospheric conditions and rainfall. Much of the excitement came from the new technologies that allowed scientists to see the sea anew. Sonar technology allowed them to hear enough to map the Indian Ocean's sea floor with heightened accuracy\u2014their images evoked an underwater continent as varied in its topography as the land above. Advances in satellite technology provided synoptic pictures of cloud cover and precipitation. Computers allowed scientists to process quantities of data beyond all precedent: Klaus Wyrtki of the University of Hawaii oversaw the production of an oceanographic atlas, which processed data from twelve thousand hydrographic stations stored on two hundred thousand computer cards.\n\nAmong all of the Indian Ocean Expedition's endeavors, one observer wrote, \"none shows more contrast between past and present than meteorology.\" The Indian Ocean Expedition marked the most intensive investigation of the South Asian monsoon since the days of Gilbert Walker, now with a raft of new tools. Fascinating though it was to glimpse the ocean's floor, for many scientists the most urgent priority for the Indian Ocean Expedition was to provide a better picture of Asia's climate. Almost a century after the establishment of the Indian Meteorological Department, scientists wrote in 1962 that \"inadequate knowledge of the large-scale influences on weather have always hampered weather forecasting.\" The need to understand the monsoon \"has become even greater and more urgent,\" they argued, \"in view of the large scale development plans of many of the countries in the field of agriculture, exploitation of water resources, flood control programmes, and programmes for ameliorating the consequences of weather extremes.\" Economic planning, they wrote, demanded \"accurate advance information on the onset of the rains, its variations from day-to-day\" and \"the occurrence of spells of heavy rain and breaks.\"\n\nIn his 1927 presidential address to the Royal Meteorological Society, Gilbert Walker had speculated that \"variations in activity of the general oceanic circulation\" would likely be \"far reaching and important\" in understanding the world's climate. It was not until the 1960s, bolstered by data collected during the International Geophysical Year and the Indian Ocean Expedition, that his insight would be developed. Walker's pioneering work on the lateral connection between the climates of Indian Ocean and the Pacific\u2014his Southern Oscillation\u2014now acquired a vertical dimension. The Indian Ocean Expedition focused on understanding the exchange of energy between the ocean and the atmosphere, driven by the monsoon winds. Piece by piece, scientists sought to understand the large-scale monsoon circulation of the Indian Ocean. The reversal in the direction of the monsoon winds had been well known for centuries, but it was more difficult, one scientist wrote, to \"determine the vertical limits\u2014than the horizontal\u2014of the monsoon influence.\" Changes on Earth's surface were linked with changes in the deep sea, and in the upper atmosphere.\n\nA crucial component of the Indian Ocean Expedition was the International Meteorological Centre that was set up in Bombay in 1963, at the Colaba observatory, which was first built as an astronomical observatory by the East India Company in 1826. Along with India, the project received support from Ceylon, Indonesia, Japan, the Malagasy Republic, Malaya, Mauritius, Pakistan, Thailand, the states of East Africa, the United States, and the United Kingdom. Its prized possession was an IBM 1620 \"computor\" (as the word was then spelled), financed by the United Nations Special Fund. The center's director was tropical meteorologist Colin Ramage. In 1958 Ramage moved from a position as deputy director of the Royal Observatory in Hong Kong\u2014where he had studied the South China Sea's typhoons\u2014to a professorship at the University of Hawaii at Manoa, where he directed a US Air Force\u2013funded research station on meteorology. \"Just as every viewer has his personal rainbow,\" wrote Colin Ramage, \"so every meteorologist seems to possess a personal and singular understanding of what is meant by 'monsoon.'\" The one point of agreement, Ramage noted, was that the Indian monsoon is the largest and most dramatic. For all the international involvement, the core of the International Meteorological Centre's personnel came directly from the Indian Meteorological Department, which contributed one hundred staff.\n\nAlready during the UN's International Geophysical Year in 1957\u20131958, Indian meteorologists had made a signal contribution. Among them was Anna Mani. Born in 1918 in the princely state of Travancore, Anna Mani studied physics at Presidency College Madras and then worked at Nobel laureate C. V. Raman's laboratory at the Indian Institute of Science, Bangalore. In 1945, she received a scholarship to Imperial College\u2014I heard many stories of Anna Mani from the Indian meteorologists I met, including a story, perhaps apocryphal, of how she endured her voyage to Southampton as one of the few women on a ship full of demobilized troops. Mani joined the meteorological department in 1948, and during the International Geophysical Year, she took charge of a network of stations to measure solar radiation across India. Just the year before the International Meteorological Centre was established, India had augmented its research capacity with the establishment of the Indian Institute of Tropical Meteorology in Poona, which remains one of the country's preeminent institutions. Under the Indian Ocean Expedition, those most intimately familiar with the monsoon now contributed to research on a global scale.\n\nIn a pamphlet published by the World Meteorological Organization, Ramage described the Colaba center at work:\n\nThroughout the night, staff in the small, air-conditioned communications room have been receiving broadcast coded weather reports from the Indian Ocean region in morse code and on teleprinters. Pictures of charts analysed a few minutes before in the meteorological centres at Nairobi, Moscow, Sangley Point and Canberra unroll from facsimile printers. Across the compound of Colaba Observatory in the Signal Office of the western Regional Meteorological Centre, other teleprinters disgorge figure-crammed sheets of paper containing detailed information on Indian weather, and on the weather over the whole eastern hemisphere north of the equator.\n\nIn this scene the Indian Ocean came alive, where as a zone of trade and as a political idea it was dead. The sea was connected in a new way by weather maps and the flow of data through fascimile printers. In the late nineteenth century, the expanded collection of data transmitted through the telegraph allowed the first synoptic weather maps of large regions to be drawn. The Colaba center's work reflected a new geography as well as advances in technology. Overlaid on the old British imperial networks of weather reporting were new centers of knowledge and power, including Moscow and even Vladivostock. This nocturnal hive of activity, in a small corner of south Bombay, gave substance to the idea of the Indian Ocean as a vast weather system stretching beyond national boundaries. Not all of the exchange was unfettered. Thousands of reports came in, but \"problems in radio transmission meant that the centre got less than half the observations made.\" Nevertheless, Ramage was reassured, \"copies of all observations were sent by mail,\" furnishing a paper archive of minute observations of the Indian Ocean's climate, even if, by the time they arrived, \"they were not much good for forecasting.\" More disappointing was the fate of a floating automatic weather station provided by the US government and anchored in the Bay of Bengal by the Indian navy: \"After a few months, its radio quit and it was neither seen nor heard from again.\"\n\nThe new quest to understand the monsoon relied on two breakthroughs in technology: aerial video and satellite photography. Five research aircraft were based in Bombay to support the International Meteorological Centre's work: one belonged to Woods Hole, the other four to the US Weather Bureau, including two large DC-6 airplanes. As in the nineteenth century, cyclones held a particular fascination for meteorologists\u2014and now improved storm forecasting would benefit the growing numbers of people who lived in South Asia's coastal cities. The US Weather Bureau's two DC-6 aircraft flew the first aerial reconnaissance mission into a cyclone in the Indian Ocean (though there had been many such missions over the Atlantic): Ramage was in the scientific observer's seat on one of them. Ramage's plane flew through the storm at 20,000 feet; the other was way down at 1,500 feet above the ocean's surface. \"I thought the aircraft was falling to pieces,\" Ramage wrote, \"we dropped 300 feet in a single second\"\u2014a common complaint of nervous flyers in turbulence, but in Ramage's case it was likely accurate. Most awesome, for this lifelong student of tropical storms, was the experience of flying into the eye of a monsoon depression. He described flying \"into an amphitheatre of multi-layered nimbo-stratus cloud. In the centre only thin milky cloud above us and almost none below, and five minutes later we were once again in rain clouds.\"\n\nWhen Ramage and Indian meteorologist C. R. Raman (brother of Nobel laureate in physics C. V. Raman) produced their atlas of Indian Ocean meteorology, they were able to draw 144 charts based on 194,000 shipboard observations and 750,000 balloon ascents into the upper atmosphere. The ability to visualize the weather in new ways was especially compelling. Scientists installed cameras on commercial and military aircraft flying through the region to take time-lapse films of the clouds they encountered in flight over the Indian Ocean. Ramage described how time-lapse cameras had previously been used \"spectacularly in science films to compress into a few seconds the blooming cycles of flowers\"; now they were placed on aircraft to photograph clouds at the rate of one frame every three seconds. On a six-hour flight, they would record every cloud on thirty meters of 16 mm film. Watching the videos later, Ramage wrote, \"The viewer gets the rather exciting impression of flying at about 50 times the speed of the aircraft.\"\n\nAn even more promising development was underway by the end of the expedition\u2014daily satellite photographs of the Indian Ocean. \"We now have for the first time,\" Ramage enthused, \"the opportunity to attempt a complete description of the whole atmospheric distribution over the Indian Ocean.\" That \"complete description\" is what Ramage and Raman attempted, chart by chart, in their meticulous work of climatic reconstruction. The most exciting prospect lay just over the horizon of possibility. The force of the monsoon came from the exchange of energy between air and sea\u2014it was now possible to study this complex process. The promise of satellite photography was that it might \"elucidate the role of the monsoons in the total atmospheric circulation.\"\n\nDespite his optimism, Ramage delivered a modest assessment of progress. The goal of Henry Blanford and Gilbert Walker\u2014a long-range forecast of the monsoon\u2014remained elusive. Even incremental improvements in forecasting, Ramage thought, would help in \"aiding flood prevention and control and in enabling irrigation engineers to make the best possible use of stored water,\" as well as helping the fishers of the Indian Ocean rim to take advantage of lulls in the monsoon, up to a week long, to take to sea. But the prospect of an accurate long-range forecast, Ramage lamented, was \"as remote as ever.\" The vast accumulation of data had not altered a truth well known\u2014that \"the atmosphere is turbulent and chaotic.\" There was no substitute for patient observation. The best meteorologists could do was to keep doing what was embedded in their practice: to use \"long climatological records and detailed statistics to come up with a sort of odds on what the next season's rainfall will be.\" His conclusion was sober:\n\nThe apparently rhythmic nature of rain-and-break, rain-and-break, during the summer monsoon encourages us to delve more deeply into the underlying causes of the rhythm and in particular the causes for interruptions or changes in the rhythm. Finding the rhythm of a total season, however, seems almost certainly beyond our immediate grasp.\n\nMORE OMINOUS SIGNS EMERGED FROM THE INDIAN OCEAN EXPEDITION. Two years before that expedition began, Revelle had written, with his colleague the geochemist Hans Seuss, that human beings were conducting, unwittingly, a \"large scale geophysical experiment\" with the world's climate. \"Within a few centuries,\" Revelle and Seuss wrote, \"we are returning to the atmosphere and oceans the concentrated organic carbon stored in sedimentary rocks over hundreds of millions of years.\" One of Revelle's students, Charles Keeling, was the first to start systematic measurements of atmospheric carbon the following year, in 1958. Revelle and colleagues had long-range goals for their study of the Indian Ocean: they wanted to see how far the Indian Ocean was a \"dump for the waste products of industrial civilization.\" And they sought to determine \"the role of the ocean in climatic change, especially in absorbing the carbon dioxide spewed into the atmosphere when fossil fuels are burned.\" We have forgotten how important the Indian Ocean was to documenting anthropogenic climate change, prompting early stirrings of alarm. The data from the Indian Ocean voyages suggested that the sea and the atmosphere were being affected by human activity on land. But these \"long range\" problems were then distant from the level of human experience. The time horizons of oceanic research were incommensurable with those of planning for food security. Because the long-range monsoon forecast remained elusive, because understandings of climate grew more complex, it was easier to focus on what could be contained and controlled\u2014one river valley at a time.\n\nThe Indian Ocean Expedition generated a picture of the South Asian monsoon that was an integral part of the global climate system. South Asia's climate was part of a large-scale interchange of energy and moisture between the ocean and the atmosphere. This expansion in the scale and complexity of understanding sat uneasily with the confidence, so prevalent in Asia among dam builders and planners, that nature could be anticipated, or its effects engineered away. A 1968 pamphlet on the monsoon\u2014written by P. K. Das, director of the Indian Meteorological Department, and published as part of a National Book Trust series, \"India: The Land and People\"\u2014contrasted the tendency of Indian farmers and poets to see the monsoon in vast, even cosmic, terms with \"more rational techniques, such as the scientific control of river valleys.\" In a sense, with its emphasis on turbulence and chaos and complexity, the new monsoon science resonated more easily with what Das saw as the superstitious view than with a view of the monsoon as simply a variable to be controlled.\n\n# II\n\nThe final year of the International Indian Ocean Expedition coincided with the worst failure of the South Asian monsoon in decades. For two successive years, in 1965 and 1966, large parts of India suffered from drought. The drought coincided with India's first major political transition after independence, following the death of Jawaharlal Nehru at the end of May 1964, and it pushed forward a change in economic strategy that was already underway. The two years after Nehru's death made clear the limits of India's progress toward self-sufficiency in food.\n\nNehru was succeeded by the diminutive and mild-mannered Lal Bahadur Shastri, a party stalwart from the Hindi heartland of North India. When Shastri took office, alarm about India's food situation was widespread. There had been many flare-ups of protest in the cities, dubbed \"bread riots\" in the media. The Congress party's distinctive style of accommodative politics came under strain after the disaster of the China war. Groups that had long formed part of the Congress coalition\u2014middle- and upper-caste landowners, urban workers, industrialists\u2014were no longer content to defer to the urban elite, no longer willing to keep a lid on their conflicts with each other. Some of them found a voice outside the Congress umbrella. Under attack from left and right, Shastri, who was easy to underestimate, undertook a series of quiet but decisive reforms. Political scientist Francine Frankel, who was doing research in India at the time, describes the result: \"A series of undramatic initiatives in economic policy that went virtually unnoticed at the time cumulatively altered the entire approach to India's development strategy.\"\n\nThe core of Nehru's approach had been a push for import-substituting industrialization. India in the 1950s had developed as a mixed economy, but one in which the public sector played a leading role, especially at the \"commanding heights\" of the economy. This was the strategy championed in the 1950s by the planning commission, to which Nehru gave considerable autonomy under the leadership of master statistician (and sometime meteorologist) P. C. Mahalanobis, dubbed \"the Professor.\" The Indian countryside was given two roles to play in this \"drama,\" as Mahalanobis insisted that the second five-year plan must be: The first was to ensure food security to a country still scarred by the memory of colonial famines, with the ultimate aim being self-sufficiency in food grains. The second was to generate foreign exchange through the export of nonfood crops to pay for the imported machinery that India would need until its own factories could make them. Jute and cotton were two of India's most valuable exports.\n\nBut at odds with the emphasis on self-reliance came an increasing reliance, through the 1950s, on food aid from the United States. From the time of its institution in 1954, Public Law 480, or PL-480\u2014known widely as the \"food for peace\" scheme\u2014disposed of the large agricultural surpluses of the American Midwest in the postcolonial world, on preferential terms. India was by far the largest recipient of this aid. Indian imports of American wheat grew from two hundred thousand tons in 1954 to more than 4 million tons by 1960. Given the ups and downs of the Indo-American relationship in the 1950s, this struck many Indian politicians as an uncomfortable level of dependence on an unreliable patron. The importance of American food aid was only one indication that the Indian government's agricultural strategy had faltered. Looking back, in the late 1960s, an official report acknowledged the problem. \"All the efforts at achieving self-sufficiency in foodgrain production during the three Plan periods did not fully succeed,\" the Indian ministry of agriculture acknowledged; instead, a sharp drop in agricultural production in the early 1960s came as a \"great shock to everyone concerned with agriculture.\" After 1961, per capita income in India did not increase, and by the mid-1960s the availability of food per capita was lower than it was in 1956.\n\nThe man charged with addressing this \"shock\" was C. Subramaniam. He was born in 1910 to a farming family in Coimbatore district in Madras, a prosperous region of irrigated export agriculture at the edge of the Western Ghat mountains and a center of India's textile industry. Subramaniam was a prot\u00e9g\u00e9 of the veteran Madras leader of the Congress party, C. Rajagopalachari. He joined the Indian freedom movement as a young man; he was one of an army of Congress party workers imprisoned by the British during the Quit India movement in 1942. He spent the 1950s in the state government of Madras, until Nehru appointed him to the coveted industries portfolio in his cabinet. For Shastri to move him to agriculture, something of a Cinderella ministry, seemed a step down. But Subramaniam embraced the challenge. He came to represent a strand of economic thinking in India that had always run alongside, sometimes in tension with, the planning commission's \"industry first\" approach. Subramaniam believed, contrary to the planning commission, that the Indian countryside was the key to security and progress. Subramaniam drew on ideas that had been in circulation from the late nineteenth century. They were prominent in the writings of India's early \"economic nationalists,\" and found new expression in the 1920s and 1930s in detailed studies of agricultural economics.\n\nIn the circumstances of the early 1960s, a rediscovery of rural India's importance converged with a line of thought that American development experts pressed upon India. From the late 1950s the World Bank and many American government observers began to urge that India should pay more attention to agriculture, even at the expense of scaling back its grand industrial vision. Specifically, they advocated for markets to play a greater role in Indian agricultural policy, which would in turn spur investment in new technologies. They were skeptical of the Nehru government's emphasis on agricultural cooperatives; they argued, instead, that India could boost its food production most rapidly by providing incentives to farmers with capital, those who already benefited from larger landholdings and irrigation facilities, even if this came at the price of higher levels of inequality in the countryside. Wielding the powers of persuasion and veiled threat that they gained from India's dependence on American food aid, these outside experts gained a sympathetic hearing from within the Indian government, which had its own \"America lobby\" as well as pro-Soviet faction.\n\nFrom the outset Subramaniam believed that new technologies were the key to the transformation of India. One of his first initiatives was to strengthen the moribund Indian Council for Agricultural Research, and to bolster agricultural education in India. Early in his tenure as agriculture minister, Subramaniam was impressed by reports of the stunning results shown by Rockefeller-sponsored experiments in Mexico with high-yielding varieties of maize and wheat, and by experiments with new hybrid strains of rice in Taiwan and the Philippines. Could they work in India? A team of scientists led by the Canadian plant pathologist R. Glenn Anderson had already initiated a series of experimental stations in India with pilot projects in Delhi, Ludhiana, Pusa, and Kanpur; when one hundred kilograms of seed, flown in from Mexico, arrived in India in 1964, they were ready to test them in Indian conditions. Here lay the roots of what would come to be known as the \"Green Revolution,\" which would transform Indian and global agriculture in the final third of the twentieth century.\n\nBut first Subramaniam had to prevail over his cabinet colleagues who remained committed to the Congress's stated goal of moving toward a \"socialist pattern of society\"\u2014or, at least, to rapid industrialization first and foremost. T. T. Krishnamachari was the planning commission's most vocal proponent of focusing on heavy industry. His concern was primarily with keeping food prices down for urban workers. To achieve this, he argued for a national food distribution system based on a system of price controls and monopoly food procurement by the state. Subramaniam pointed out that this infrastructure of food control, which had its roots in the wartime economy, was \"uneconomical\"\u2014government prices for compulsory procurement were so low that they gave farmers no incentive to invest in new technology. The two sides crossed swords over inequality. Subramaniam recalled in his memoirs that his opponent argued against high-yielding varieties because \"this strategy would lead to greater social tension within the rural areas, because benefits would be unequally distributed.\" Subramaniam's response was to ask his critics \"what other option we had.\" To those who argued that he was caving to American pressure, Subramaniam countered that only a new approach to agriculture would save India from subjection, for he feared that \"once we became dependent on these imported foodgrains other political strings would be attached to them.\" The turn to high-yielding varieties made Indian agriculture dependent, instead, on large imports of chemical fertilizer\u2014and more dependent than ever on new sources of water.\n\nThe crux of Subramaniam's strategy was to \"concentrate modern inputs in irrigated areas.\" This is a prosaic way to describe a fundamental change. From the nineteenth century India's geography of water had shaped plans for the country's future. Now the difference between irrigated and rain-fed lands would be accepted as a necessary inequality\u2014even a matter of strategy.\n\nSUBRAMANIAM'S APPROACH TO AGRICULTURE GAINED A FILLIP from the overwhelming sense of economic and political crisis surrounding India in 1965. By September 1965 it was clear that the year's summer rainfall was far short of normal: aggregate agricultural production was 17 percent down on the previous year. In a speech to persuade the chief ministers of India's states of his policy, Subramaniam spoke of a \"race against time.\" Facing down criticism from the Communist H. N. Mukherjee in Parliament, who accused Subramaniam of forcing through his strategy under American pressure, the agriculture minister accused his questioner of exploiting \"the psychological hour when the monsoon has failed,\" and preying on the \"fears\" of the country. India's vulnerability to the monsoons, it seemed, was as deep as ever. \"How helplessly we are at the mercy of the elements,\" a newspaper editorial lamented in 1965, arguing that all India had to show for the previous decade of development efforts were some \"shallow and tentative improvements in irrigation.\"\n\nEconomic crisis merged with political turmoil. A series of military skirmishes on the border escalated into war with Pakistan, from August 5 to September 22, 1965. The catalyst was Pakistani military infiltration into Kashmir, undertaken in the hope of sparking a rebellion against Indian rule. Both sides claimed victory after a UN-brokered cease-fire; in contrast with the China war, however, the Shastri government's military campaign was greeted as a success at home. Shastri urged struggle on both fronts with his resonant phrase: \"Jai Jawan! Jai Kisan!\"\u2014victory to the soldier, and victory to the farmer. An immediate consequence of the war was the abrupt cessation of American aid both to India and to Pakistan.\n\nUS president Lyndon Baines Johnson had already assumed direct charge of PL-480 shipments; with characteristic bluntness he called it the \"short tether.\" He would ship only enough grain to India to meet immediate needs: a direct and unabashed form of political leverage, which angered the Indian government. Subramaniam played an essential role in negotiations over restarting the shipments after the war. He had developed a reputation in Washington as an Indian leader favorable to the United States, and whose policy views tallied with American advisors' ideas. Subramaniam forged a rapport with Orville Freeman, Johnson's secretary of agriculture. The two men met at a UN Food and Agriculture Organization meeting in Rome in November 1965 and signed a secret agreement to accelerate India's agrarian reforms in exchange for expanded American food aid. Taking over as prime minister after Shastri's sudden death in Tashkent, where he had gone to sign the peace treaty with Pakistan, Nehru's daughter Indira Gandhi visited the United States in 1966 to cement this deal. She received a warm reception from LBJ, but she felt deep humiliation at having to go to the United States as a supplicant. \"I don't ever want us to beg for food again,\" Indira Gandhi told an aide in December 1966.\n\nWhen the summer monsoon failed for a second successive year, in 1966, India's food situation worsened. The state of Bihar, one of India's poorest, was most directly affected. Facing Republican congressmen who were hostile to aid and hostile to India, the Johnson administration emphasized the scale of the food emergency in India\u2014evoking the specter of famine for the first time since independence. In order to push through a bill boosting aid to India, Johnson told Freeman that he wanted the American public to know \"that people were being hauled away dead in trucks, and that they needed food.\" By contrast Indira Gandhi's government chose not to declare a famine in Bihar for fear of the domestic political fallout. Instead, the central government dismissed early reports of starvation from Bihar as hyperbole\u2014just as the British imperial government had been wont to do. Famine was too much a symbol of a dark past, its conquest too vital to the political legitimacy of the Indian state, to concede this, the first famine since India's independence. But the scale of suffering in Bihar threatened to explode and on April 20, 1967, the government of India declared the existence of famine in Palamau and Hazaribag districts; five more districts were added in time, along with others that suffered \"scarcity.\" The nineteenth-century Famine Codes, revised incrementally over the years, came into effect. The full force of the state swung into action to counter food shortages in Bihar. PL-480 shipments from the United States were vital, distributed in twenty thousand fair-price shops. In keeping with traditional practice, the government initiated public works on a large scale to provide employment to augment local incomes and to encourage food imports from other regions of India. Under the leadership of the socialist Jayaprakash Narain, the Bihar Relief Committee mobilized an army of volunteers, as well as donations. A year before the Biafra crisis of 1968, often held up as a watershed in the development of a global humanitarian consciousness, the Bihar crisis reached the wallets, and the television screens, of a public far away.\n\nIndira Gandhi in January 1967, soon after she became India's prime minister. CREDIT: Express Newspapers\/Getty Images\n\nBy most measures, the Indian and American response to dearth in Bihar was a success. Despite major food shortages, far fewer people died than during the nineteenth-century famines; the official death toll was in the region of 2,300 people. Even if this was an underestimate, the contrast with India's experience of famine under colonial rule was stark.\n\nBut the story has a strange and unexpected coda. Housed in the Johnson Presidential Library in Austin, Texas, is an innocuously named box of archival files: \"India Memos and Misc., 1 of 2, volume 8.\" I learned about its contents in a fascinating article by two historians of Cold War science, Ronald Doel and Kristine Harper. Doel and Harper argue that the environmental sciences were vital to the Johnson administration's foreign policy, and to its projection of American power overseas. From the 1950s, the deepening American involvement in Vietnam had generated a deepening interest in the hydrology of the Mekong River. As the US military intervened more intensively in Vietnam after 1962, mastering nature became strategically vital. American medics experimented with new drugs for the control of malaria, a prerequisite for jungle warfare: they devised mefloquine. Some went further: they had visions of intervening to alter patterns of rainfall to disrupt the agricultural base of North Vietnam. The United States had seen a long series of attempts at weather control in the twentieth century, a history characterized more by outlandish schemes than by any measurable success. Now it became a plank of military and diplomatic strategy. The Bihar drought occurred just when secret American plans for weather control in Vietnam were in testing. President Johnson connected the two, as he became the most unlikely of experts on the South Asian monsoon. Johnson would write in his memoirs that, as he looked at weather charts before approving each PL-480 shipment to India, he came to know \"exactly where the rain fell and where it failed to fall in India.\"\n\nIn January 1967, Pierre St. Amand arrived in Delhi with others from the Naval Ordnance Test Station on a highly classified mission; indeed, only a decade ago did the work of Doel and Harper bring it to light. Nicknamed Project Gromet, the scheme\u2014with the secret approval of Indira Gandhi's government\u2014aimed to inject silver iodide into \"large, high-altitude cold clouds\" to force precipitation. Official acknowledgment of the program came in the form of a sly and prosaic statement: \"Scientists from the United States and India are cooperating in a joint agro-meteorological research project, localized in Eastern U.P. [Uttar Pradesh] and Bihar to study the cloud physics and rain producing mechanisms over these areas of India which have incurred several droughts during the last few years.\" The problem was that, in January, the skies over Bihar were virtually cloudless. They expanded their quest toward Punjab. US ambassador Chester Bowles, an Indophile, wrote in secret that \"both we and the Indians want to demonstrate that if we can [force precipitation] India's food and agriculture need not be entirely at the mercy of weather vagaries.\" Soon after that, the archival trail runs cold; \"GROMET quietly died,\" its historians conclude.\n\nDoel and Harper were interested in Project Gromet primarily for what it tells us about US foreign policy under Johnson. Seen in the light of India's long history of struggle with the monsoon, it acquires other layers of meaning. In a sense, Project Gromet was the antithesis of the Indian Ocean Expedition. Where the new science emphasized the complexity of the monsoon climate, rooting it in teleconnected land-ocean-atmosphere interactions on a planetary scale, the attempt to make rain in Bihar epitomized the logic of control and containment. Even then, the architects of the plan feared the consequences of engineered rain in India having an unwanted impact across the border in Pakistan.\n\nTHE YEAR 1967 WAS A TURNING POINT IN INDIA'S POLITICAL HISTORY. Having ruled India with a comfortable parliamentary majority since independence, the Congress party took a drubbing at the polls in India's third general election. While they remained in power in New Delhi, their majority was reduced\u2014more significantly, they lost control of state governments in states including Tamil Nadu (to the regional party born of the anti-caste movement, the Dravida Munnetra Kazhagam), Kerala (to the Communists), and West Bengal (to a coalition including the Communist Party). A generation after independence, the mantle of Mahatma Gandhi and Jawaharlal Nehru was no longer enough to muster support for the party that had led the nationalist movement. The coalition of social forces that underpinned Congress dominance was fragile; in many places it unraveled. A confidential assessment by the British High Commission in Delhi reported to London that the election results reflected \"impatience with the chronic failure to deal with rising prices, low wages [and] food shortages, amounting to famine in certain areas.\"\n\nOccasionally, conventional accounts of this political moment include climate among their explanations for the change. India's political and economic transformation in the 1960s, writes political scientist Ashutosh Varshney, owes much to the \"serendipity of the monsoon.\" In recent years, it has become common once again for historians to see climate as a force underpinning political events. But to think of India's political transformation as \"caused\" by the failure of the monsoon would be unduly simplistic. The outcome of the 1967 elections reflected the hopes of India's voters; it was the outcome of new languages of political mobilization deployed by India's parties; it distilled new struggles for power and justice and recognition unleashed by mass democracy, which could no longer be contained within a dominant party system. A richer picture emerges if we see climate not as an external force determining human outcomes, but rather as a source of all-too-human fears and anxieties. Only in the context of century-long fears about India's monsoon climate\u2014the deep historical association of monsoon failure with famine\u2014can we understand why so many experienced the drought of the mid-1960s as evidence of political failure.\n\n# III\n\nIndira Gandhi was one of few heads of state to attend the UN's first conference on the human environment, held in Stockholm in June 1972. In her rousing speech to the plenary session, she set out ecological problems that were already a matter of public discussion in India:\n\nWe share your concern at the rapid deterioration of flora and fauna. Some of our own wildlife has been wiped out, miles of forests with beautiful old trees, mute witnesses of history, have been destroyed. Even though our industrial development is in its infancy, and at its most difficult stage, we are taking various steps to deal with incipient environmental imbalances. The more so because of our concern for the human being\u2014a species which is also imperiled. In poverty he is threatened by malnutrition and disease, in weakness by war, in richness by the pollution brought about by his own prosperity.\n\nBut her diagnosis of the root cause differed from that of many of the conference's promoters, whose vision was consumed by dark Malthusian nightmares in the Third World, epitomized by biologist Paul Ehrlich's _Population Bomb,_ published in 1968. The opening lines of Ehrlich's book described a \"stinking hot night\" in Delhi. \"As we crawled through the city, we entered a crowded slum area... the streets seemed alive with people,\" he wrote: \"People eating, people washing, people sleeping. People visiting, arguing, and screaming... People, people, people, people.\"\n\nIn response, Mrs. Gandhi set out a position that saw environmental degradation as primarily a problem of poverty: a problem of distribution, not of numbers. She reminded her audience that \"we inhabit a divided world.\" She attributed historical responsibility for the despoliation of Earth where it rightly belonged: with the wealthy countries of the world. \"Many of the advanced countries of today have reached their present affluence by their domination over other races and countries,\" she said, and through \"the exploitation of their own natural resources.\" The rich world \"got a head start through sheer ruthlessness, undisturbed by feelings of compassion or by abstract theories of freedom, equality or justice.\" But now the poor countries were being told that they could not do the same. \"The riches and the labour of the colonized countries played no small part in the industrialization and prosperity of the West,\" she reminded her audience, but in the 1970s, \"as we struggle to create a better life for our people, it is in vastly different circumstances, for obviously in today's eagle-eyed watchfulness we cannot indulge in such practices even for a worthwhile purpose.\" The development of a middle class in India, or even just the provision of minimally decent standards of living to its poorest citizens, took place with a growing awareness of finite resources. \"We do not wish to impoverish the environment any further,\" she insisted, \"and yet we cannot for a moment forget the grim poverty of large numbers of people.\" Her most resonant phrase, and the one for which her speech is remembered, was in the form of a question: \"Are not poverty and need the greatest polluters?\"\n\nShe raised the stakes, asking: \"How can we speak to those who live in villages and in slums about keeping the oceans, the rivers and the air clean when their own lives are contaminated at the source?\" She resisted the binary choice of development _or_ environmental protection. \"The environment cannot be improved in conditions of poverty,\" she declared, and \"nor can poverty be eradicated without the use of science and technology.\" In \"science and technology\" lay India's great hope.\n\nGandhi concluded by describing to the audience how she saw India's quest since independence. \"For the last quarter of a century,\" she said, \"we have been engaged in an enterprise unparalleled in human history\u2014the provision of basic needs to one-sixth of mankind within the span of one or two generations.\" In this evocation of speed, urgency, and scale lay Gandhi's recognition of the demographic and material transformation that was sweeping the world. The edge of Malthusian panic remained, despite Indira Gandhi's eloquent rebuttal at Stockholm. In time just such a sense, that population growth was an exorable and destabilizing force, fed her own fears of conspiracy. In the context of labor unrest and judicial challenges to the legitimacy of her election victory, they underpinned the siege mentality that led her to declare a state of emergency in 1975, suspending India's democratic constitution for the first and (so far) the only time, using a colonial-era provision. It would lead Indira Gandhi's government to enact a brutal population control program involving gross abuses, including forced sterilizations.\n\nIn Indira Gandhi's vision, many disparate concerns came together to constitute an overarching problem of \"the environment\"\u2014population growth and the finitude of natural resources; concerns about the impact of rapid development on human health; concerns about species extinction and disappearing habitats. The international conference coincided with the earliest attempts to confront these challenges at home in India. Mrs. Gandhi's government sponsored the Water Act of 1974, which was among the first attempts to deal with an environmental issue on a national basis in India. The act created pollution control boards at both the state and the national levels; the boards were given the authority to determine permissible levels of pollution, setting limits on the composition and quantity of effluent that factories, for instance, could discharge into water bodies. The legislation took an expansive view of water, covering \"streams, inland waters, subterranean waters, and sea or tidal waters.\" But the law proved, and it continues to prove, difficult to implement. Pollution control bodies did not have the will or the power to prosecute powerful local industrialists. The Water Act faced many challenges in the courts, often on the grounds that it violated the constitutional right to carry on a trade or business. For instance, in 1981 lawyers for a firm called Aggarwal Textile Industries, challenging a ruling from Rajasthan state's pollution control board, argued that \"the problem of prevention of water pollution is a problem of vast magnitude and... it would be beyond the means of an individual to prevent or control the pollution resulting from an industry set up by him.\" In a significant number of cases, offenders were allowed to continue their polluting activities.\n\nDESPITE DAWNING AWARENESS OF THE QUESTION OF SUSTAINABILITY, the crux of India's response to the crisis of food and population lay in a massive increase in the use of water. The early twentieth century's faith in the possibility of expansion without limit was reinvigorated. The precondition for the growth of the Green Revolution in India was an expansion in irrigation. And the water available from surface irrigation works, however ambitious in scale, was insufficient. Cultivators in arid parts of India had known for centuries, and British administrators recognized in the nineteenth century, that South Asia's groundwater resources provided the most immediate insurance against drought. Many regions of South Asia had ancient and elaborate systems of well irrigation, though this infrastructure had fallen into disrepair in many places by the nineteenth century. The advantages of groundwater resources are manifest in South Asia: groundwater is locally available on demand and requires far less infrastructure than surface irrigation works; groundwater is spared the large-scale water loss from evaporation that reservoirs experience. Groundwater resources are more resilient to episodic monsoon failure. Until the 1960s, groundwater could not be mobilized on a large enough scale to meet India's food requirements. The arrival of the tubewell changed that decisively.\n\nTubewells are a humble technology, unlikely harbingers of a hydrological revolution. A tubewell is a well driven by an electric pump, consisting of a long stainless steel tube that is bored into an underground aquifer. Historian and architect Anthony Acciavatti argues that they represent an \"inversion\" of the monumental water technologies: dams and canals. In contrast with dams, the tubewell has a \"minimal footprint and maximum draft of water, it creates undreamed of independence and three-dimensional chaos.\"\n\nAn electric tubewell, of the kind that proliferated in India in the 1960s. CREDIT: Illustration by Matilde Grimaldi\n\nThe Indian government encouraged the adoption of high-yielding varieties of wheat and rice by subsidizing the capital costs of infrastructure for the intensive exploitation of groundwater. State electricity boards reduced the cost of electricity. By the 1970s, unable to bear the cost of monitoring energy use by millions of farmers across widely dispersed areas, state electricity boards opted for flat tariffs. This gave large farmers an incentive to use as much energy as possible to extract water from underground. As a result, agriculture's share of total energy use in India grew from 10 percent in 1970\u20131971 to 30 percent by 1995, even as state electricity boards accumulated huge losses. By 2009, groundwater accounted for 60 percent of India's irrigated area, and surface irrigation for only 30 percent. Just as they shared the waters of the Indus, India and Pakistan came to share a new dependence on groundwater. Embracing the same package of hybrid seeds and intensive fertilizer use, Pakistan's food security became even more reliant on irrigation than India's. All the while, and despite the declining importance of surface irrigation, the profusion of large dams continued. Dam construction was unstoppable, even as underground water now supplied the greater share of water for irrigation. Large dams had acquired enormous symbolic power, to the point where they epitomized the conquest of nature by technology. Three decades after India's independence there were also many vested interests in the engineering and construction industries committed to the continued proliferation of dams. Their social and ecological costs multiplied through the 1970s; as we shall see in the next chapter, their costs provoked widespread resistance in the 1980s.\n\nUtopian technological schemes for the capture of India's waters flourished in the 1970s, alongside\u2014and as though unaware of\u2014growing understanding of the scale, power, and unpredictability of climate. Although Roger Revelle was a pioneer of oceanography and an architect of the 1960s' Indian Ocean Expedition, he turned in the 1970s to more practical matters. Revelle moved from Scripps to Harvard to found the Harvard Center for Population and Development Studies. In 1975 he wrote an essay with his Indian colleague V. Lakshminarayana on what they called the \"Ganges Water Machine.\" They expressed concern that \"deeply embedded cultural, social, and economic problems inhibit modernization of agriculture and fuller utilization of water resources\" in India. They envisaged that \"the introduction of technological changes on the required scale might break the chains of tradition and injustice that now bind the people in misery and poverty.\" They had in mind a technological mirroring of the vast, interconnected hydraulic system that linked the monsoon rains, the Himalayan rivers, and the waters underground\u2014an expanded network of bunds, dams, and, above all, the massive expansion of groundwater pumps. The same year, K. L. Rao, a veteran of Indian irrigation, published an even grander plan. He returned to the dream of Sir Arthur Cotton, irrigation pioneer of the nineteenth century, in proposing a large scheme to transfer water\u2014through a network of canals\u2014from the wettest to the driest parts of India.\n\nIndia's experience of water-driven growth in the 1970s found echoes across Asia. The 1970s also saw rapid growth in Chinese agriculture, as China developed its own path toward a green revolution. The unprecedented expansion in food production in China in the 1970s built upon the extension of agricultural research stations right down to the level of local communes. Like the Indian government, the Chinese state viewed the rapid growth in population and the pressure on arable land with alarm, reversing the pronatalism that characterized the first two decades after the revolution. As in India, Chinese farmers' adoption of high-yielding seed varieties depended on large quantities of chemical fertilizer. In the 1970s, the Chinese fertilizer industry expanded through a dispersed network of small-scale factories. High-yield dwarf rice varieties spread especially rapidly in China in the 1970s, boosting harvests. And in China, as in India, the agricultural growth of the 1970s depended on mining underground water. Electric pumps played almost as significant a role in expanding irrigation in China as they did in India. In 1965, there were approximately half a million mechanized irrigation and drainage devices in China; by 1978, there were more than 5 million. But in other ways, the Chinese approach to growing more food diverged from India's. In keeping with the Maoist emphasis on mass political mobilization, China's agricultural strategy was more broadly based than India's. Historian Sigrid Schmalzer describes it as a \"patchwork of methodologies,\" in which mechanization coexisted with labor-intensive terracing, chemical fertilizer with traditional practices of night-soil collection and the application of pig manure. The water- and fertilizer-fueled growth of the 1970s laid the groundwork for China's further agricultural expansion in the 1980s; but with the end of agrarian collectivization and the arrival of market reforms, rural inequalities, too, grew wider.\n\n# IV\n\nThe water inequalities that India has always faced deepened in the 1960s and 1970s; they were accentuated by the uneven spread of tubewells. From the late 1960s, as the Green Revolution took off, the drier regions of India's northwest and southeast emerged as the centers of agricultural growth\u2014a result of groundwater exploitation, fueled by electrification to allow the use of high-yielding seeds. The water-rich areas of India's northeast, by contrast, continued to rely on rainfall, utilizing relatively inefficient diesel pumps for shallow groundwater irrigation; they remained at risk of regular waterlogging, but lacked the infrastructure to use the surplus water for storage or groundwater recharge. Where large dams promised to create energy through hydroelectric power, pumps used it in large quantities to mine water.\n\nThese inequalities were manifest between 1970 and 1973, when parts of western and central India experienced three successive years of drought. The western state of Maharashtra was worst affected. In the World Bank's archives in Washington is a fifty-page typescript account of the Maharashtra drought; scrawled at the top is a handwritten instruction: \"Circulate.\" The author was agricultural economist Wolf Ladejinsky (1899\u20131975). He was born to a Ukrainian Jewish family who fled the Russian civil war to the United States in 1922. Ladejinsky studied at Columbia University and joined the US Department of Agriculture's foreign service, coming to specialize in Asia's agrarian problems. He served in the American occupation of Japan, where he played a key role in overseeing land reform, as he then did in Taiwan. Ladejinsky came under suspicion in the McCarthy era, but President Eisenhower defended him and appointed him to direct land reform in South Vietnam in the late 1950s, just as American involvement there was deepening. Through the 1960s, Ladejinsky continued to focus on the problems of rural Asia, working with the World Bank. He was anti-communist but saw the importance of land reform in societies where landholdings were highly concentrated in a few hands.\n\nLadejinsky had worked in India on numerous occasions, and the World Bank sent him back in 1972 to investigate a drought that threatened hunger in Maharashtra. \"This is an occasion when the writer intends to keep his emotions in leash,\" he promised at the outset of his note, fearing that his \"credibility may suffer from dramatizing the incontrovertible cruelty of nature\u2014no rain and no crops.\" He observed that \"historically the struggle of the Maharashtra farmer has been one of quest for water.\" In an echo of British commentary in the late nineteenth century, he personified the monsoon as a force, as when he referred to \"the monsoon playing truant.\" He described his journey out to the countryside from Poona; it was not long before he found himself traveling through a landscape that \"leaves one shaken about the perversity of nature.\" Lack of drinking water was a problem everywhere. More than shortages of food, it was a lack of water that, Ladejinsky saw, led farmers to uproot themselves and migrate in search of work. He described the sight of water tankers surrounded by people at famine relief camps\u2014tankers that had been donated by oil companies as an act of charity. Words mattered, Ladejinsky argued: neither the state nor the central government wished to invoke the term \"famine\"\u2014seen as a relic of a dark colonial past\u2014but their use of the mild term \"scarcity\" masked the severity of the crisis.\n\nThe drought in Maharashtra showed how little the hydrologic revolution of the 1950s and 1960s had touched many parts of rural India. In his detailed study of the drought, economist Jean Dr\u00e8ze noted that only 8 percent of land in the region was irrigated. For those on rain-fed lands, Dr\u00e8ze wrote, \"the meagre harvest of coarse grains remain a gamble on the monsoon and the land offers a spectacle of desolation and dust during the slack season.\" The drought led to a 14 percent drop in food availability; the threat of starvation was very real. It was averted by concerted government response, arguably one of the most effective in the history of independent India. The Food Corporation of India organized the transportation of wheat from other parts of the country. It was sold at subsidized prices through thirty thousand fair-price shops distributed across the state. At the same time, a large program of public works generated employment; up to 2 million people a day attended these works, building roads and bridges and digging wells. This boosted local incomes and in turn pushed up food prices in Maharashtra, attracting supplies from beyond the state\u2014often illegally, since the government had barred the interstate trade in grain during the crisis. Those illicit supplies, even at inflated prices, helped to compensate for the shortfall.\n\nIn the end it was not big technology but rather the unheralded public distribution system of India that averted catastrophe in Maharashtra. The drought showed how patchy and uneven the reach of water engineering was. It showed the importance of public policy and prompt intervention. But these were not the lessons learned. The drought did nothing to dent confidence in the idea that all India needed was irrigation, now from deeper and deeper underground.\n\n# V\n\nScientists' understanding of the monsoon advanced in the 1960s and 1970s, spurred by the data collected by the Indian Ocean Expedition and by the International Geophysical Year that had preceded it, in 1957\u20131958. At the heart of monsoon science now were two phenomena. The first was \"moist processes\"\u2014most simply, the release of latent heat and the effect of clouds on radiation. The second was the coupling of ocean and atmosphere. There was a new awareness that the monsoon system formed, as one meteorologist put it, a \"complex of seemingly disparate parts: two fluids, the mobile air and the changing ocean below.\" Increasingly sophisticated computer models could turn each of these processes on and off in an attempt to isolate and investigate different variables.\n\nThe most important breakthrough came with the work of Jacob Bjerknes, a Norwegian meteorologist at UCLA\u2014son of Vilhelm Bjerknes. Father and son were both part of the team that had, in the 1910s, discovered and named the phenomenon of polar fronts from their observatory in Bergen. Now, using data generated by the expeditions of the International Geophysical Year, Jacob Bjerknes determined the mechanism driving a phenomenon that Gilbert Walker had first observed in the 1920s, during and just after his time as director of the Indian meteorological service. Walker had called it the Southern Oscillation, an oscillating contrast in atmospheric pressure across the Pacific Ocean, as measured in Darwin and Tahiti. But Walker had been unable to determine the cause of this swing in pressure; Bjerknes discovered that the answer lay in the waters of the Pacific Ocean.\n\nIn the 1960s, Jacob Bjerknes discovered the El Ni\u00f1o Southern Oscillation (ENSO)\u2014the illustrations show its \"warm\" (El Ni\u00f1o) and \"cool\" (La Ni\u00f1a) phases. CREDIT: Illustration by Matilde Grimaldi\nBjerknes found that the key to the southern oscillation lay in the periodic warming of sea surface temperatures in the eastern Pacific Ocean, and he dubbed it El Ni\u00f1o in keeping with the term that fishers had given the phenomenon. This warming reverberates throughout the world's climate. Most of the time, the waters of the western Pacific, off Indonesia, are warmer than in the eastern Pacific\u2014this drives the easterly \"trade winds,\" but Bjerknes saw that these were a surface manifestation of an overturning circulation in the upper atmosphere at higher latitudes. He named this the Walker Circulation in honor of Sir Gilbert. During an El Ni\u00f1o episode the contrast narrows as the waters of the eastern Pacific warm up; in response, the Walker Circulation weakens, since it is driven by that difference in temperature and pressure. Less intensive surface winds reduce the ocean's churn, leading to less of the colder water from the depths welling up to the surface; this sustains the abnormal warmth in the eastern Pacific, and so flattens the usual temperature contrast across the ocean\u2014attenuating further the Walker Circulation.\n\nThis disruption in circulation has consequences for rainfall in the western Pacific and the Indian Ocean\u2014and even the North Atlantic. El Ni\u00f1o years tend to be associated with weak monsoons in Asia, and with excessive rainfall in South America. Bjerknes dubbed the overall oceanic-atmospheric system the El Ni\u00f1o Southern Oscillation (known as ENSO), of which El Ni\u00f1o was the phase of ocean surface warming and La Ni\u00f1a (girl child) of cooling; years that exhibit neither extreme are known as \"neutral.\" La Ni\u00f1a has the opposite effect of El Ni\u00f1o, strengthening the temperature contrast between the eastern and western Pacific, strengthening the Walker Circulation, and bringing more rain than usual to Asian shores.\n\nThe discovery of ENSO marked a breakthrough for understanding the Asian monsoon. Once it had been identified, historical climatologists showed that many of the worst droughts in Asian history\u2014including the droughts that brought famine in the 1870s and the 1890s, and also the Maharashtra drought of 1972\u20131973, discussed earlier in this chapter\u2014coincided with El Ni\u00f1o events. But the causal relationship between ENSO and the monsoon is complex. There is some evidence to suggest that an especially strong or weak monsoon might foreshadow rather than follow the corresponding phase in the ENSO cycle. Tropical meteorologist Peter Webster argues that there may be truth in Charles Normand's suggestion\u2014made in the 1950s after his retirement as head of India's meteorological service and before Bjerknes had discovered El Ni\u00f1o\u2014that India's weather was more use in predicting what was in store for other parts of the world than it was itself amenable to prediction.\n\nKnowledge of ENSO raised new questions about the periodicity of drought in Asia\u2014a question that, as we have seen, had provoked much discussion in the 1870s. It reinforced the sense, drawn from the Indian Ocean Expedition, that Asia's climate was fiendishly complex, associated with many other parts of the planet's climate. ENSO is quasiperiodic; it recurs but the intervals between events vary and are not easy to predict. The early 1970s also brought new knowledge of internal climatic variability on shorter timescales. In 1971, Roland Madden and Paul Julian, based at the US National Center for Atmospheric Research, discovered what came to be known as the Madden-Julian Oscillation (MJO): an oscillation in surface pressure and wind direction over large areas, with consequences on a planetary scale. The MJO has a clear periodicity; it is, as meteorologist Adam Sobel has put it, \"a signal that emerges above the meteorological noise.\" Migrating from west to east, from the Indian Ocean to the Pacific, the MJO lasts between thirty and sixty days, and its intensity varies from year to year. In its \"active\" phase, the MJO brings heavy rain, and a heightened chance of tropical cyclones; in its \"suppressed\" phase, it interrupts the monsoon flow, even reversing the wind direction, bringing clear skies. The MJO is associated particularly with the northern winter; but scientists also discovered another intraseasonal oscillation in the northern summer months, which propagates northward rather than eastward. Known as the Boreal Summer Intraseasonal Oscillation, its connection with or independence from the MJO has been the subject of debate, but it, too, is thought to play a vital role in the fluctuations of rainfall over Asia each summer. In the 1980s further research was done to uncover the mechanisms at work behind these intraseasonal oscillations, though some uncertainty remains. These intraseasonal oscillations might well explain the alternation between active and break periods in any monsoon season, which has such vital and direct effects on agriculture.\n\nAdvances in technology and understanding did little to revise Colin Ramage's verdict, at the end of the Indian Ocean Expedition, that little headway had been made in forecasting the monsoon in a practical sense. Progress had been made in understanding the system on a large scale. But what mattered most to Asian farmers were the rhythms of rainfall within a given monsoon season\u2014the relationship between what meteorologists call \"active\" and \"break\" periods of the monsoon. So finely attuned is Asian agriculture to the monsoon that the most devastating effects on cultivation often come from unexpected breaks in the midst of the summer rains, even when rainfall overall is plentiful\u2014the skies brighten suddenly, and crops do not receive the water they need to thrive at a critical phase in their life cycle.\n\nA further push to crack the monsoon's code came in 1979, as part of a worldwide effort called the Global Weather Experiment\u2014the Indian Ocean component of that came to be known as the Monsoon Experiment, or MONEX. The scale of the operation was vast, even larger than the Indian Ocean Expedition of the 1960s, and more fully equipped with satellite technology. It encompassed 3,400 land stations, 800 upper air observatories, 9 weather ships, 7,000 merchant ships, and 1,000 commercial aircraft drawn in to record observations, 100 dedicated research aircraft, 50 research ships, 5 weather satellites, and 300 balloons. Despite this scale, despite the dazzling advances in equipment, Webster noted that much older ways of knowing the monsoon\u2014the instinctive knowledge of mariners\u2014were still in evidence in 1979; he saw that _dhows,_ the traditional sailing vessels of the northwestern Indian Ocean that had for centuries harnessed the monsoon currents, were still widely used.\n\nCOLIN RAMAGE, DIRECTOR OF THE METEOROLOGICAL COMPONENT of the Indian Ocean Expedition, returned to India in the 1970s. In his spare time while there on assignment, he turned amateur historian and wrote a short and provocative essay on how his predecessor John Eliot\u2014the second director of Indian meteorology, after Henry Blanford\u2014had failed spectacularly to forecast the crushing drought of 1899\u20131900. Ramage delved into the archives of the _Times of India_ and wrote a powerful account of the famine that followed. \"The government refused with religious fervor to modify the holy writ of laissez-faire,\" he declared. Like many critics of imperial policy at the time, he saw that the railways had done as much to worsen as to alleviate the famine, by making it easier for speculators to ship grain out to areas where purchasing power was higher. He praised the Indian government's efforts to protect its citizens against the threat of famine, but noted that, despite best efforts, India remained dependent on food imports, as it had since the 1920s. His conclusion was ominous. \"Can we be sure that such a devastating famine will not recur?\" he asked; not since Indian independence had there been a drought as severe as the drought of 1899. He ended his essay on that note, leaving implicit the underlying question: what would happen if another drought of that magnitude were to materialize?\n\nBut another threat was now on the horizon. In 1979, the same year as MONEX, the World Meteorological Organization held its first World Climate Conference. The conference declaration recognized the need to \"foresee and prevent potential man-made changes in climate that might be adverse to the well-being of humanity.\" From the 1980s, the combination of climate change and other environmental threats compounded the water-related risks faced by billions of people in Asia.\n\n# NINE\n\n# STORMY HORIZONS\n\nIN THE 1980S THE FULL PROPORTIONS OF ASIA'S WATER CRISIS BECAME manifest\u2014including its vertical dimension. In that decade, satellite images and remote sensing data revealed to scientists how fully human activities had transformed the physical environment, reaching from the underground waters to the upper atmosphere. Much of the impact was visible without the aid of sophisticated technology. It became viscerally clear in the quality of the air that people breathed and the water that they drank, in the strangulation of the rivers they lived by and lived from. In 1984, the chemical pollutants that coursed through the holy Ganges reached such concentrations that a stretch of water caught fire: it became a river of flames.\n\nAsia's waters were subject to unprecedented demands that came from the convergence of two large processes. The first, which began in the 1950s, was population growth. India's population grew from just under 370 million people in 1950 to 684.8 million in 1980, an increase of 185 percent; through the 1970s alone, India added 131 million new citizens. China's population grew more slowly, but from a larger base: from 562 million people in 1950 to just under 988 million by 1980, and with an absolute increase of close to 166 million people in the 1970s. Although the rate of population growth in both India and China had slowed considerably by the 1980s, not least because of coercive population \"control\" measures in both countries, the cumulative impact of the previous decades' growth has been manifest. Belying the fears of the Malthusian prophets, the effects of this expansion on the biosphere were initially limited by very low levels of income per capita\u2014and limited, too, by deliberate efforts by both the Indian and Chinese governments to hold back consumption to generate savings for future investment in industrial development. But then arrived the second major transformation, which began in the 1980s: the rapid economic growth of Asia's two largest countries, China and India, both of them following a path forged by other countries in East and Southeast Asia a decade or more earlier, but on an altogether different scale.\n\nThe bonfire of socialist austerity began first in China, where the reforms of Deng Xiaoping enshrined the notion that \"to get rich is glorious.\" From 1978 to 2012, the Chinese economy grew at an average annual rate of 9.4 percent, \"the fastest sustained expansion by a major economy in history.\" The Indian economy was slower to accelerate, but by the 1980s average annual growth was around 5 percent. Following an emergency loan from the International Monetary Fund to meet a critical shortage of foreign exchange, the Indian economy underwent a process of liberalization after 1991, orchestrated by the economist Manmohan Singh. This involved a dismantling of elaborate regulations governing private investment and trade, dubbed the \"License-Permit Raj.\" High growth followed, picking up in the late 1990s; but it was accompanied by galloping inequality. India has remained home to more poor people, in absolute terms, than any country on Earth. In India, more than in China, the ecological threats generated by new prosperity intensified the more familiar, water- and weather-related risks of extreme poverty. Unlike China, India's population has remained predominantly rural, and will continue to be so by the middle of the twenty-first century. The destabilization of Asia's water ecology, which accelerated in the 1980s, put more people at risk in India and in neighboring Bangladesh than anywhere else.\n\nThis chapter shows how, starting in the 1980s, Asia's waters submitted to a concatenation of demands from industry, from agriculture, and from the needs of booming cities. The mining of groundwater exceeded the capacity of the hydraulic cycle to replenish aquifers. A hunger for energy led to a renewed interest in hydroelectric power. States and private investors eyed the upper reaches of the great rivers. From the 1980s hydraulic projects converged upon the Himalayas. The most promising lowland dam sites were exhausted by the 1970s; the steep drops of the mountain rivers made them ideal for power generation. As a cluster of competing projects lined up along the rivers' descent from the 1980s, the potential for conflict grew. States acquired the capacity to deny water to others downstream; not so much the technical capacity, since dam technology had changed relatively little from the 1950s, but rather the financial and infrastructural capacity\u2014and above all, the will. New demands on resources, and new demands for water, came from the revival of trade between South, Southeast, and East Asia, which had ebbed in the 1950s and 1960s.\n\nThe final ingredient in this cocktail of ecological destabilization came with the accelerating effects of climate change. Already in 1982, environmental activists in India invoked what they called a \"rather futuristic problem\"\u2014the \"possibility of global climatic change taking place by the end of the century because of increasing carbon dioxide in the atmosphere.\" They raised an ominous prospect: \"It is quite possible... that agriculture as practiced for centuries in India may have to change and crop outputs may become a matter of even greater uncertainty than today.\" Since then, the scientific consensus on the reality of anthropogenic climate change has been overwhelming. Climate change is no longer a \"futuristic\" problem\u2014its effects are here, now. And its effects menace the coastal rim that stretches from India to China.\n\nClimate change affects water in every form: it affects the rain clouds and the Himalayan glaciers, the flow of rivers and the shape of coastlines, the level of the ocean and the intensity of cyclones. Climate change is irreducibly historical. As historian and Marxist theorist Andreas Malm observes, \"The storm of climate change draws its force from countless acts of combustion over, to be exact, the past two centuries.\" But the current crisis is a product of history in another sense too. The acute impact of climate change on Asia, and on South Asia in particular, will play out across a landscape shaped by the past\u2014shaped by the cumulative effects of social inequality, shaped by the borders of the mid-twentieth century, shaped by infrastructures of water control. And it will be shaped by the legacy of ideas from the past, including ideas about climate and the economy.\n\n# I\n\nWater was a core ingredient in Asia's experience of what economist Angus Deaton calls the \"great escape\" from scarcity. The intensification of agriculture driven by the Green Revolution\u2014a package of high-yielding seed varieties and extensive fertilizer use, sustained by vast quantities of water\u2014augmented food production to an extent that would have been unthinkable even one generation earlier. Between 1970 and 2014, India's production of cereals grew by 238 percent, compared with a 182 percent expansion in population over the same period. This took place with only a marginal increase in the quantity of land given over to food crops. In China the expansion was more dramatic still: a 420 percent increase in cereal output with no increase in land area under cultivation. Just a decade after the desperate recourse to American food aid during the monsoon failures of the 1960s, India became a food surplus country.\n\nIntangible though it was, an unshakable sense took hold among the Indian elite that the threat of an uncertain monsoon had receded. It was a sense expressed by writer and newspaper editor Khushwant Singh in a 1987 essay on the monsoon in Indian literature. Singh ranged widely across Indian epics and poetry to show how deeply the monsoon had shaped Indian cultural sensibilities over hundreds of years. But he concluded that, in recent decades, \"India has taken enormous strides toward freeing herself from dependence on the vagaries of the monsoons.\" Technology led the charge: India had \"raised enormous dams, laid thousands of miles of irrigation canals, and dug innumerable electrically operated tube-wells to supply water to her farms.\" A sense of security brought disenchantment. \"There is no longer the same agony waiting through long summer months of searing heat to catch a glimpse of the first clouds,\" he argued. The monsoon had vanished from Indian literature; it \"no longer stirs the imagination of the poet or the novelist with the same intensity it used to.\"\n\nThose closer to rural India had a different view. The same year as Singh, the modernist artist Jyoti Bhatt, trained in the influential Baroda school in Gujarat and immersed in local artistic traditions, wrote that for all of the improvements in weather forecasting, the ability to predict the character of a whole monsoon season remained elusive. In folk culture, if not in high poetry, the monsoon's mysteries lived on. Bhatt described an annual festival in the arid lands of Kutch and Saurashtra, in Gujarat, celebrating Bhadali\u2014the daughter of a shepherd and a gifted diviner of rain. The festival was bound up with anxious expectation. Villagers in Gujarat, he wrote, \"keep observing and interpreting various omens, signs, and factual symptoms around them.\" They relied on the \"collected experience of many generations\" to decide when to plant their crops each year. Bhatt was agnostic about how far these rituals helped farmers, but at the very least he saw that they provided more excitement and drama than \"watching a Door Darshan [the state broadcaster] weather forecast based on data received from Insat, on a small TV screen.\"\n\nAlso in 1987, but on a larger scale and in the language of economics rather than poetry, Harry Oshima revisited the old region of \"monsoon Asia.\" Hawaii-born Oshima (1918\u20131998) wrote his dissertation on the national income statistics of Asia's new states; he worked for the United Nations in the 1950s, and served as the Rockefeller Foundation's representative in the Philippines in the 1970s. Oshima found that monsoon Asia's coherence had been shattered by a transformation in the relationship between water and productivity. Oshima began with a timeless vision: across the coastal and deltaic sweep from South Asia to East Asia, the intense seasonality of rainfall created common patterns of agriculture\u2014labor-intensive paddy cultivation, high population density, a preponderance of small farms. He wrote, too, of the \"philosophy\" of the \"monsoon economy\"\u2014an ethos of \"harmony... compromise, moderation, diligence, and cooperation.\" In writing this, Oshima echoed the language of an earlier era, which drew a straight line from climate to culture. But the period since 1970 had broken deep historical patterns. There had been an unexpected differentiation in income levels across the region, which was now, in Oshima's view, \"crystallizing with a few modifications into the three basic regions of... East, Southeast, and South Asia.\" Oshima was least sanguine about South Asia's prospects; pessimism about India was widespread among economists at the time. South Asia was effectively now the residue of \"monsoon Asia\"; everywhere else, industrial growth and intensive irrigation had powered an escape from the monsoon. But even in India, it was clear by the 1980s that something fundamental had changed.\n\nIN INDIA, THE REVOLUTION IN FOOD PRODUCTION DEPENDED, above all else, on groundwater. As we have seen, the first experiments with using motorized pumps to extract groundwater in India date from the late nineteenth century, but until the 1960s, their use was negligible. The greatest growth came in the use of private tubewells: there were half a million in use across India in 1968; that number had grown to 5 million by 1994. As the exploitation of groundwater increased, so too did the depth that tubewells had to reach. Investment in tubewells has been almost entirely private, in contrast with dams and other surface irrigation works that have been publicly funded. But under Indira Gandhi's government in the 1970s, landowners were encouraged to utilize groundwater and install tubewells through the provision of subsidized or even free electricity; state electricity boards were left to set prices, and many of them incurred heavy losses. The use of groundwater proceeded with no regulation. Large farmers, with the capital to invest in technology and with the large landholdings to benefit from irrigation, dug deeper than their neighbors, capturing groundwater for their private use and even selling it on to others. Cheap electricity provided an incentive for farmers to extract as much groundwater as they could, with little thought for replenishing the aquifers. Tushaar Shah, a leading expert on groundwater policy in India, has described it as \"an atomistically managed water-scavenging irrigation regime involving tens of millions of pump owners who divert surface and groundwater at will.\" In all, nearly three-quarters of the expansion in India's irrigated cropland since independence has come from groundwater, and much of the expansion came in the 1970s and 1980s.\n\nThe effects of this boom in water mining were clearest in Punjab. Already by the 1910s, Punjab was India's most prosperous agricultural region; the elaborate system of canals built by the British made its arid lands productive. By the early twenty-first century, Punjab produced 20 percent of India's wheat and 42 percent of its rice, on just 1.5 percent of the country's land area. Punjab possessed only around eleven thousand tubewells in the late 1960s, on the eve of the Green Revolution\u2014that number would grow more than 100-fold to 1.3 million over the next forty years. Groundwater provides two-thirds of Punjab's water supply. But the water table has declined perilously since the late 1970s. The intensive use of pesticides in farming has contaminated water sources, and this is widely acknowledged to be responsible for a substantial increase in the incidence in cancers in the area. In the western Indian state of Gujarat, another region where agriculture is dependent on groundwater, the water table has dropped by 1.4 meters each year from the late 1970s through the end of the 1980s, and at an even faster rate since then.\n\nIf the monsoon no longer inspired India's poets, as Khushwant Singh observed in the 1980s, the infrastructure of groundwater extraction has become an unavoidable feature of the landscape in ways that have left their mark on South Asian literature. In a powerful short story published early this century, Pakistani-American writer Daniyal Mueenuddin evokes the landscape of that part of Punjab that formed part of Pakistan after Partition\u2014the agrarian heart of a country even more dependent on irrigation than India. The protagonist is Nawabdin, the village electrician; his special talent was \"a technique for cheating the electric company by slowing down the revolutions of electric meters.\" This mattered deeply, because electricity was the lifeblood of agriculture\u2014\"In this Pakistani desert, behind Multan, where the tube wells ran day and night, Nawab's discovery eclipsed the philosopher's stone.\" In that simple detail, as Mueenuddin sets the scene, we glimpse a vast agrarian transformation.\n\nINDIA AND CHINA HAVE MUCH IN COMMON IN THEIR RELIANCE ON groundwater to secure an increase in food production, in their vulnerability to the depletion of water sources, in the economic geography of their water use, and in the sheer scale of change they have experienced since the 1980s. But China has grown much faster than India, and India has been even more vulnerable than China to water- and climate-related risks, as a result of its greater dependence on agriculture, its higher levels of poverty, and, to return to a theme that has recurred throughout _Unruly Waters,_ because of the particular characteristics of the monsoon.\n\nIf China's use of groundwater since the 1970s has not been quite as prodigious as India's, it is not far behind. Underground aquifers provide water to 40 percent of China's farmland, and drinking water to 70 percent of the population of China's arid north and northwest. Across the North China Plain, groundwater levels have dropped by approximately 1 meter a year since 1974, a rate of depletion comparable with that of Punjab. Like India, China's groundwater is contaminated. A study undertaken by the Chinese government in the 2000s showed that 90 percent of China's groundwater was polluted, and 60 percent severely polluted with heavy metals and fertilizer and chemical waste.\n\nIn the broad sweep of history, China and India have undergone comparable shifts in their economic geography\u2014in both cases, groundwater and other sources of irrigation were the driving force of change. Historian David Pietz points out that China, in the second half of the twentieth century, underwent a \"reversal of food production patterns that pertained for most of the imperial period.\" The dry North China Plain now produces 60 percent of China's wheat and 40 percent of its corn on 22 percent of its land, and just 4 percent of its water resources. This has led to the transfer of what hydrologists call \"virtual water\"; that is to say the water that is embedded in crops, from water-scarce to water-abundant areas. This is a story that parallels, in nature and in timing, the emergence of arid Punjab as India's agricultural powerhouse. As the economist Harry Oshima noted in the 1980s, even before the scale of China's and India's transformation was evident, the old geography of monsoon Asia had been shattered. It had been shattered, above all, by new sources of water. At any point until the late nineteenth century, it would have been self-evident that agrarian wealth in Asia lay in areas of abundant rain\u2014the essence of monsoon Asia was the intensity of cultivation, especially rice cultivation, that the monsoon climate allowed. In a remarkably short space of time\u2014the forty or fifty years after 1960\u2014this pattern had been reversed by technology, and by fossil fuels.\n\nThe terrible paradox is that this stunning expansion in food production was achieved in a way that cannot be sustained. Groundwater resources are under acute strain in the regions of Asia that most depend on them. A study using data from the NASA Gravity Recovery and Climate Experiment satellites showed that, between 2002 and 2008, groundwater depletion in northwestern India\u2014the heartland of the Green Revolution\u2014amounted to 109 cubic kilometers of water, an amount that exceeds the storage capacity of India's largest reservoir. Freshwater availability per capita in India is projected to fall to 1,335 cubic meters by 2025, in comparison with a global average of 6,000 cubic meters. Groundwater has been the cornerstone of India's and China's food security since the 1970s\u2014but for how long?\n\nTHE SUSTAINABILITY OF INDIA'S GROUNDWATER BOOM IS ONLY ONE aspect of a deeper crisis of water. It was clear from the earliest years of the Green Revolution that one consequence of the new approach to agriculture was deepening rural inequality. In his commentary on the Maharashtra drought of 1970 to 1973, economist Wolf Ladejinsky had seen how sharp the contrast was between irrigated and nonirrigated lands. Long-standing fault lines between wet and dry, rain-fed and groundwater-supplied lands grew deeper. Access to water was both a cause and a symptom of inequality.\n\nIn the 1980s, recognition of the extent of water inequality energized an intellectual and political movement that called into question the fundamental pillars of India's development strategy. Disagreements over economic policy were common enough in the 1950s and 1960s. India's policymakers included committed planners as well as those in favor of free markets. But they disagreed about the means, and not the ends of development. By the end of the 1960s, India faced a radical alternative, in the shape of a Maoist insurgency that began in West Bengal and soon spread to other parts of the country. The insurgents, led by an urban elite committed to the romance of revolution, believed that only the violent dispossession of India's landowning class could bring about substantive change. Paradoxically, they drew inspiration from China at just the moment when Chinese agriculture changed course, embracing its own version of the Green Revolution. Others looked to India's past, to the history of water, for inspiration as they considered alternative economic models.\n\nMahatma Gandhi was a clear source of inspiration for many of those who, in the 1980s, challenged the assumptions of the Indian state. Though their influence on economic policy was muted, Gandhians continued after independence to urge upon India a different model of development\u2014more rooted in rural communities, less wedded to monumental technology. They called for a holistic approach to development that emphasized both social and ecological equilibrium. The essence of their philosophy was encapsulated in Gandhi's 1946 pronouncement that \"the blood of the villages is the cement with which the edifice of the cities is built. I want the blood that is today inflating the arteries of the cities to run once again in the blood vessels of the villagers.\" The 1970s saw the rise of the Chipko movement that brought together concerns with environmental degradation in Himalayan forests with the assertion of forest peoples' rights to the resources on which their livelihoods depended. The movement was explicitly Gandhian in inspiration, and women played a leading role within it.\n\nThat spirit infused a new approach to India's water problems in the 1980s, an approach that looked back to a golden age of local, sustainable water management, embedded in the ancient practices of rural India. Just as Gandhi evoked a largely mythic notion of India as a collection of village republics\u2014an idea that he drew primarily from Western writers\u2014environmental activists in the 1980s harked back to an ecologically responsible, traditional India. It mattered little that this vision bore little resemblance to the picture that was emerging from historical and archaeological research. Archaeologist Kathleen Morrison puts it this way: \"nostalgic\" environmentalists evoked \"a mode of life that I have simply been unable to reconstruct even as my work has expanded to incorporate three thousand years of agrarian history.\"\n\nIn those same decades, research on the history and diversity of India's water practices painted a more complex picture. Water management was often tied to the exercise of royal power. Irrigation works depended on coerced labor. Access to common property was governed by the exclusions of caste\u2014and those commons were under pressure even before the nineteenth century. The valorization of communitarian solutions could often serve to legitimize inequality and oppression. Many of the architects of independent India, including Jawaharlal Nehru, had seen this clearly. B. R. Ambedkar, leader of India's Dalits and architect of the Indian constitution, was no rural romantic: \"What is the village but a sink of localism, a den of ignorance, narrow-mindedness, and communalism,\" he had asked in the Constituent Assembly in 1948. The divergence between these different visions of India's past reminds us that water has a public as well as a scholarly history\u2014throughout the 1980s, idealized narratives about water management in the past had rhetorical and strategic value for the debates of the present, and they informed contending visions of the future.\n\nEven if it was little more than a useful fiction, the idea of a return to a more ecologically harmonious past motivated many strands of the Indian environmental movement, which emerged in earnest in the early 1980s. The movement's foundational text was the _First Citizen's Report on the State of India's Environment,_ written by Anil Agarwal and his colleagues at Delhi's Centre for Science and Environment, which Agarwal had founded in 1980. Agarwal was by then one of India's most influential environmentalists. Born in the northern industrial city of Kanpur in 1947, to a landowning family, Agarwal studied mechanical engineering at the Indian Institute of Technology there. His work as the science correspondent of the _Hindustan Times_ in the 1970s brought his writing to international attention. The citizen's report paid close attention to the water crisis facing rural India. A few years later, in 1985, a report by the center urged the importance of recovering and repairing India's ancient practices of harvesting the waters of the monsoon. The report's authors, Agarwal and his prot\u00e9g\u00e9 Sunita Narain, went further; in their view, nothing less than a revitalization of rural India would reverse India's slide toward ecological degradation and social crisis. They described India's traditional villages as \"integrated agro-sylvo-pastoral entities,\" dependent on common property resources: the rivers and the lakes and the forests. They claimed that the Indian state's approach to development\u2014top-down, reliant on big technology\u2014had \"torn asunder this integrated character of the villages.\" In this was more than a trace of the holistic \"rural sociology\" of Radhakamal Mukerjee in the 1920s. Agarwal and Narain argued that \"the process of state control over natural resources that started with colonialism must be rolled back.\" Their prescribed solution was the final decolonization of rural India, a reversal of the process that began in the mid-nineteenth century with the British search for India's water wealth.\n\nA similar commitment to elevating traditional practices and indigenous knowledge underpins the most wide-ranging and influential condemnation of the Green Revolution, written by environmental activist Vandana Shiva. Shiva emerged as a distinctive voice in Indian debates in the 1980s, and in the 1990s she would go on to become internationally influential within the antiglobalization movement. Trained as both a physicist and a philosopher, Shiva started the Research Foundation in Science, Technology and Ecology in the Himalayan town of Dehra Dun in 1982. In the opening pages of her book, _The Violence of the Green Revolution,_ published in 1991, Shiva looked back on the 1980s, and described that as the decade when Asian societies came under the grip of what she described as \"an ecological crisis and the threat to life support systems posed by the destruction of natural resources.\" Taking aim at the idea that the Green Revolution had brought about an agricultural \"miracle,\" Shiva highlighted its costs. Many of these were well known, but Shiva's forceful prose gave them new prominence. She called Punjab a \"tragedy,\" a cautionary tale of the folly of \"breaking out of nature's limits and variabilities.\" She argued that the use of high-yielding seeds, pesticides, and ever-more water had left Punjab with \"diseased soils, pest-infested crops, water-logged deserts, and indebted and discontented farmers.\" She challenged the Indian state's obsession with technological solutions to social and ecological problems; echoing Agarwal and Narain, she implied that India had not rid itself of its colonial legacy. Juxtaposing the quest for water with her emphasis on conservation, she posed it as a struggle between \"diversity, decentralization and democracy,\" on one side, against \"uniformity, centralization, and militarization\" on the other.\n\nShiva's book epitomized a new sort of environmental thinking in India. But it also reflected a new set of intellectual and political connections that crossed Asia's borders in the 1980s. Her book was published by the Third World Network, which was based in Penang, Malaysia. Formed in 1984, the Third World Network was an offshoot of the Consumers' Association of Penang\u2014which, founded in 1970, was one of Asia's earliest pressure groups devoted to a broad range of causes ranging from fair prices and housing to food safety. It was the group's report on the _State of Malaysia's Environment_ that had inspired Anil Agarwal to undertake a similar exercise in India after attending a conference in Penang. The Third World Network marked the incorporation of environmental concerns fully under the umbrella of issues on which Asian activists made common cause. The network\u2014which reached beyond Asia to encompass groups in Africa and Latin America, with many allies among activists and nongovernmental organizations in Europe and North America\u2014brought together a commitment to social and economic justice with a new concern about sustainability. The Third World Network helped to bring Shiva's work to a wide audience among activists in Asia and beyond. For her part, Shiva applied her analysis to Asian societies writ large, and not just to Punjab. Even if the 1970s' movement for a New International Economic Order at the United Nations had proved short-lived, marginalized by the Anglo-American turn toward privatization in the 1980s under Reagan and Thatcher, ethical claims on behalf of what we now call the Global South lived on. The sense of a unified Third World fighting against the legacies of colonialism as well as new forms of imperial power began to crumble as parts of Asia began to experience very rapid economic growth, but it continued to influence movements for environmental justice that focused on the ways poverty heightened environmental vulnerability and inequality worsened environmental harm.\n\nThe networks of activism that linked environmentalists across Asia's borders turned, in the early 1990s, to the problem of climate change. In a 1991 text that remains influential to this day, one of the earliest and most eloquent expressions of the argument for global environmental justice, Anil Agarwal and Sunita Narain wrote about the problem of _Global Warming in an Unequal World_. Their opening sentence is powerful and stark: \"The idea that developing countries like India and China must share the blame for heating up the earth and destabilizing its climate... is an excellent example of _environmental colonialism_.\" They pointed out that historical responsibility for the accumulation of carbon in the atmosphere lay entirely with the advanced industrial countries of the world; they highlighted the hypocrisy of those countries now telling India and China to cut their emissions, when in per capita terms, India's or China's emissions were miniscule. Their conclusion was that \"the Third World today needs far-sighted political leadership\" to resist the calls by Western political leaders and environmentalists to \"manage the world as one entity,\" which could only be a mask for exploitation as long as the world remained so unequal and so divided.\n\nThe pamphlet was published just on the eve of India's economic liberalization: a series of market reforms that followed an emergency IMF loan, secured when India faced a crisis of foreign exchange. In its language, it belongs firmly in the era that was closing, though few saw it at the time. The idea of the Third World, invoked repeatedly, had already started to come unstuck; with the collapse of the Soviet Union, it dissolved. What Agarwal and Narain could not have anticipated was just how rapidly the Indian environment, with the Indian economy, would be transformed by a new openness to global capitalism. China's own economic transformation was well underway in 1991, but its colossal scale and its colossal implications for the world were only slowly becoming evident. With good reason, the 1991 pamphlet called for a united front against the powerful nations in international negotiations over climate and emissions. But it was blind to the proliferation of cross-border challenges confronting Asia\u2014in the realm of water above all.\n\nTHE EXCHANGE OF IDEAS ABOUT SHARED ENVIRONMENTAL CONCERNS took place alongside a focus on deeply local problems. As the crisis of rural India became more visible, it appeared to be rooted in climatic and social characteristics that were distinctive to the Indian subcontinent, familiar to observers going back to the nineteenth century\u2014the deep and particular unevenness of water's distribution, and the pervasive social and caste inequalities that limited people's access to water.\n\nJournalist Palagummi Sainath (b. 1957) spent the 1980s working on the Bombay tabloid _Blitz_. In 1993, he received a _Times of India_ fellowship that he chose to spend traveling through India's poorest districts. He traveled one hundred thousand kilometers over a few years, more than five thousand of them on foot. The _Times_ published his dispatches in installments, at a time when ever-less reporting from impoverished rural India reached a metropolitan audience that was now in the grip of economic expansion. Sainath wanted to move beyond what he saw as the media's focus on \"the spectacular\" and to highlight \"the long-term trends that spell chaos [but that] don't make good copy.\" Sainath's articles were collected and published as a book in 1996 with the deeply ironic title _Everybody Loves a Good Drought_. In his many articles on water, Sainath drew attention to the opportunities for profit that water scarcity brought to a new cabal of \"water lords.\" Sainath observed something familiar from earlier times\u2014absolute scarcity of water was not always the problem, its distribution was. \"Simply put,\" he wrote, \"we have several districts in India that have an abundance of rainfall\u2014but where one section, the poor, can suffer acute drought.\" With the insight that came from his immersion in rural India, Sainath distinguished between \"agricultural drought\" and \"meteorological drought,\" arguing that the latter was not necessary for the former to bite\u2014there were droughts that were \"real,\" and droughts that were \"rigged.\"\n\nThe Indian countryside reeled from a double burden. Many farmers, those inhabiting the 60 percent of India's farmland without irrigation, suffered under the age-old burden of their dependence on an uncertain monsoon. But high-intensity farming brought its own burdens. When the history of late-twentieth-century India is written from a perspective of greater distance, alongside vertiginous economic transformation there will be a less visible, shameful, story: the story of an epidemic of farmer suicides on a scale that may be without parallel in the world. Sainath was among the first to bring this silent crisis to public attention. Starting in the late 1990s, an estimated seventeen thousand farmers each year have taken their lives\u2014at least two hundred thousand deaths from suicide between 1997 and 2010. At the root of the intolerable pressure that many of India's farmers labor under are their growing debts\u2014debts for purchases of seed and fertilizer and pesticide and fuel for groundwater pumps.\n\nMeanwhile access to water continues to be an indicator of the most fundamental social inequalities. In a comprehensive survey of the practice of untouchability in rural India, undertaken at the start of the twenty-first century, Delhi sociologist Gyansham Shah and colleagues found that Dalits in rural India regularly face exclusion from access to basic public services\u2014and of these, the authors found, the most important by far was the denial of access to water. No fewer than 48 percent of villages surveyed reported such denial. Pervasive upper-caste beliefs about the polluting effects of Dalits having contact with water sources leads to systematic discrimination, enforced by violence. The practices that Shah and colleagues documented ranged from absolute exclusion from tubewells and tanks to Dalits being forced to wait until everyone else had taken their water before being allowed limited access. Ninety years after Ambedkar's march on the tank at Mahad, unequal access to water remains pervasive in India. And caste discrimination explains why such inequalities are sharper in India than anywhere else in Asia.\n\nSainath's dispatches from rural India in the 1990s date from a time when climate change was not foremost among India's concerns. To revisit his urgent reportage two decades later, when the signs of climate change are everywhere, reminds us that, in India and elsewhere in Asia, climate change comes on top of a mountain of intersecting ecological and economic crises that has been building since the 1980s.\n\n# II\n\nIf the ocean underground began to recede as a result of the unsustainable use of groundwater in the 1980s, Asia's running waters\u2014its rivers\u2014were more visibly scarred. The rivers were the conduit between the countryside and the insistent demands of growing cities. Since the late nineteenth century, the engineering of rivers\u2014damming, diverting, impounding them\u2014has governed efforts to redistribute water. The twentieth-century quest for the \"white gold\" of hydroelectric power intensified that quest. In India and China alike, the abuse of rivers provoked a new environmental consciousness in the 1980s. The dreams of the 1950s and 1960s gave way to an unfolding nightmare. The circumstances under which the Indian and Chinese environmental movements emerged were very different from those of their counterparts in North America, Europe, and Japan earlier in the century. In Asia, rapid growth followed, rather than preceded, awareness of scarcity and natural limits. And a sense of fragility before the power of nature, a sense that hard-won gains were under threat, led authorities in India and China to the defiant, even violent defense of large technological solutions to the problem of water.\n\nIn India, the crisis of river pollution was clearly visible by the 1980s. In their first report on India's environment, Anil Agarwal and his colleagues at the Centre for Science and Environment wrote that \"river pollution in India has reached a crisis point. A list of India's polluted rivers reads like a roll of the dead.\" They described the holy Ganges as a \"network of cesspools,\" and came up with a grim list of industries responsible for the damage: \"DDT factories, tanneries, paper and pulp mills, petrochemical and fertilizer complexes, rubber factories...\" A few years later, Darryl D'Monte, a pioneer of Indian environmental journalism and a contributor to the Centre for Science and Environment's report, declared that the \"destruction of life support systems along the Himalayas\" constituted \"the world's single biggest ecological crisis.\"\n\nIn 1985, a campaigning lawyer, M. C. Mehta, took up the river's cause. Mehta, born in a small village in Jammu and Kashmir state, worked as a public interest litigation lawyer in India's supreme court. A 1984 visit to the Taj Mahal awakened him to the harm being done to the monument by polluting factories nearby\u2014by the early 1980s, the Taj Mahal's lustrous marble had been stained a dirty yellow. Mehta filed a public interest case against the offending industries. The following year, he turned his attention to the pollution of the Ganges. In a series of landmark cases, heard weekly over several years, Mehta succeeded in having three hundred factories closed and five thousand forced to install cleaner technology; the court ordered 250 municipalities to install sewage treatment plants. Mehta's were the most significant cases brought under India's Water Act of 1974; their proceedings revealed the extent of river pollution in India by the 1980s. In its 1988 judgment on a case Mehta brought against the owners of tanneries in the industrial town of Kanpur, the Supreme Court of India noted that \"any further pollution of the river is likely to lead to a catastrophe.\" They noted the relentless discharge of sewage and chemical effluent into the river. In another case the same year, Mehta took on Kanpur Municipality. He brought as evidence a report from the Industrial Toxicology Research Centre, which showed that the water of the Ganges was completely unfit for human consumption. Mehta won his cases; the polluting industries were ordered to amend their ways. But in comparison with the scale of the problem of river pollution in India, these were small victories in an enormous battle.\n\nBeginning in the 1980s, river pollution in India reached crisis proportions. CREDIT: Dominique Faget\/Getty Images\n\nPROPELLED BY ECONOMIC TRANSFORMATION FAR MORE RAPID than India's, China's rivers endured a comparable assault. In China, too, under tighter political constraints, the 1980s saw the emergence of an environmental movement\u2014and there, too, water was a prime concern. One of China's first private environmental organizations, Friends of Nature, was formed by Liang Congjie in 1993. Liang Congjie's grandfather was Liang Qichao, the prominent late-nineteenth-century reformer; his father was an architectural conservationist who suffered brutal persecution during the Cultural Revolution. Liang Congjie took a keen interest in environmental issues from the 1980s. His approach initially was cautious; his activism began with seemingly innocuous targets, like his campaign to save the chiru, or Tibetan antelope. But Friends of Nature, like its counterparts in India, harnessed the power of information to illuminate a slow crisis. The organization began to publish the _China Environment Yearbook,_ which was akin to the _State of the Indian Environment_ reports, if less overtly critical of the government. Anxieties about water pollution and water shortages multiplied in China in the 1980s and 1990s, prompted by the breakneck pace of urbanization.\n\nIn the 1990s, around the same time that P. Sainath undertook his investigative tour of rural India, journalist Ma Jun published a series of articles for Hong Kong's _South China Morning Post_ on the state of China's waters, culminating in his influential 1999 book _China's Water Crisis_. He wrote of his realization that government officials and engineers were \"trying to rob nature of the last drop of water to serve economic expansion.\" He noted that \"while most people regarded the floods, dry spells, and sandstorms as some sort of evil force that demanded even larger engineering projects, I began to view them as nature's way of retaliating for man's reckless attempt to conquer and harness nature.\" He described how the flow of the Yellow River\u2014the \"mother river,\" cradle of Chinese civilization\u2014began to decrease from 1972. In 1997, for a period of 330 days, the Yellow River failed altogether to reach the ocean. Ma Jun called the abuse of China's rivers a \"heinous crime,\" as he made an emotional appeal to the power of the rivers and the reverence with which they had been treated for centuries. \"To rescue the dying rivers with our devotion and work would be our most glorious effort,\" he urged.\n\nThe most shocking passages of his book described the pollution of the lower Yangzi River\u2014we can see similarities in both tone and content with the writings of Indian environmentalists. Ma Jun described the river as \"a vast open sea of garbage and sewage.\" It was clogged by rubbish from the dense traffic on the river: \"Styrofoam lunch boxes and vegetable scraps, toilet waste, cooking oil, machine oil and industrial muck.\" Worst of all was the pollution from the cities. Ma Jun wrote that \"day by day, week by week, month by month, in a monotonously inexorable fashion, they throw or pour the detritus of 400 million people into the waterway.\" In China, as in India, minor triumphs, small cleanups, have failed to stem a hurricane of waste.\n\nFACED WITH MULTIPLE WATER CRISES, THE INDIAN AND CHINESE states fell back on the strategy that they had favored since the 1940s\u2014to turn, again and again, to large-scale hydraulic engineering. Ecological and social harm reinforced each other in the case of large dams, which continued to be constructed on an ever-expanding scale even after groundwater became a far more important source of water for irrigation. While they drowned forests and flooded fields, they also displaced millions of people. Concerns with the social suffering caused by large engineering schemes joined fears about water pollution to form a second major strand of environmental activism. Here, too, India was at the vanguard. The 1980s saw an escalation in the scale and reach of protest against large dams in India.\n\nIn 1978, India sought World Bank assistance for the mammoth Narmada project, which called for the construction of 30 large, 135 medium-sized, and 3,000 small dams along the Narmada River, which flowed west through the states of Madhya Pradesh, Maharashtra, and Gujarat to the Arabian Sea. In 1985, the World Bank committed US$450 million to the project, around 10 percent of its total cost. Little had changed since the 1950s in the Indian government's iron certainty that the benefits of the project would outweigh its costs. On their initial estimate, seven thousand families would be displaced by the project. Plans were made for rehabilitation, but in keeping with common practice, only those with formal title to their lands were included. As the true scale of displacement and environmental destruction emerged, resistance grew. In the late 1980s, a cluster of nongovernmental organizations\u2014a broad coalition that included human rights groups, environmentalists, students, and local people's associations\u2014came together to form the Narmada Bachao Andolan, or NBA (\"Save Narmada Movement\"), led by the social activist Medha Patkar. Patkar, born in Mumbai in 1954 to parents who were active in the nationalist and labor movements, studied social work at the prestigious Tata Institute of Social Sciences, but abandoned the doctoral dissertation she had started as she became more involved with the struggles of marginalized communities along the Narmada valley. Under her leadership, the NBA harnessed the power of Gandhian nonviolent protest and drew on a rich vein of ideas that insisted on people's sovereignty over their lands and landscapes. Among their most resonant techniques was the \"monsoon Satyagraha\"\u2014silent demonstrations held as the river's waters rose during the monsoon, slowly submerging the protesters until they stood waist-deep in water. It appealed symbolically to the power of climate and seasonality, which the dams sought to engineer away.\n\nThe NBA succeeded in harnessing international support. In the United States, Lori Udall of the Environmental Defense Fund took up the fight. Patkar met with the World Bank in 1987, and pressure on the bank from international supporters of the Narmada movement led it, in 1991, to initiate an inquiry into the project. The bank's decision to withdraw funding for the project in 1993 was a victory for the Narmada movement, and marked a shift in the bank's previously uncritical support for large dams. But the Indian government's response was defiant. Alongside nonviolent resistance, the Narmada movement took to the courts: they had some early success, and then, from the late 1990s, faced a series of defeats as the Supreme Court ruled in favor of the project's continuation. The World Bank's withdrawal served to harden the government's resolve to find private finance for the Narmada project. When the Sardar Sarovar Dam was finally declared open in autumn 2017 by Indian prime minister Narendra Modi\u2014who had strongly supported the Narmada project when he was chief minister of Gujarat, condemning environmentalists as \"anti-development\" and purveyors of a \"campaign of misinformation\"\u2014he took pains to point out that \"with or without the World Bank, we completed this massive project on our own.\"\n\nEnvironmental activist Medha Patkar, leader of the Narmada Bachao Andolan, joins a protest against the construction of a court complex at Pipliyahana Reservoir near Indore. CREDIT: _Hindustan Times_ \/Getty Images\n\nResistance to the Narmada Dam drew attention to the harm, both environmental and social, that arose from India's post-independence addiction to large dams. Research undertaken by scientists and activists in the 1980s and 1990s showed that these problems combined to devastating effect. In the first two decades after India's independence, an estimated half-million hectares of forest was submerged by dams; this loss of land accelerated in the 1970s and 1980s with ever-larger projects like the Narmada scheme and the equally controversial Polavaram Dam in Andhra, along the Godavari River. The dams themselves suffered from the failure of their designs to take into account the quantity of silt the rivers carried. Their architects had underestimated the extent of the problem, as silt-heavy rivers clogged up reservoirs. The lifespans of the great Bhakra and Hirakud Dams\u2014two of the first to be built after independence\u2014was significantly reduced by higher rates of siltation than planners anticipated. Large dams also caused a major problem with waterlogging\u2014inundating agricultural land beyond its capacity to absorb moisture and so rendering it infertile. An estimated thirty-three thousand hectares of productive land were lost as a result of the Tungabhadra Dam. Here we see yet another contrast\u2014as India's arid regions drew down their water tables by pumping groundwater, well-watered areas near the headworks of large dams suffered from excess. As they interfered with the ecology of water, dams also created the conditions for water- and insect-borne diseases to thrive. Many studies around large dams in India and elsewhere showed a significant rise in the incidence of malaria, as large reservoirs and canals provided conditions for the anopheles mosquito to flourish.\n\nAll the while, the social disruption caused by large dams continued unabated. As we saw, the most comprehensive estimate for the number of people that have been displaced by dams in India since independence reaches 40 million people. The projects with the highest cost in lives disrupted were those of the 1980s: two of the largest dams in the Narmada project, the Sardar Sarovar and the Narmada Sagar, displaced two hundred thousand people each. A high proportion of those displaced were marginalized _adivasis_ (tribal peoples), who had little power to negotiate adequate compensation from the state. The fate of these internal refugees\u2014refugees from water development projects\u2014too often goes unrecorded. The work of journalists like P. Sainath and environmental campaigning organizations like the NBA has brought some of their stories to light. Novelist Arundhati Roy found a wide international audience with a visceral essay on the profound costs of India's addiction to large dams, though her polemical style also drew criticism. Others have turned to fiction to depict their suffering. In 2001, Vairamuthu\u2014a prolific lyricist who has written the words to more than seven thousand Tamil film songs\u2014wrote a novel, _Kallikaatu Ithihaasam_ (\"Saga of the Drylands\"), which won him the Sahitya Akademi award in 2003, India's highest literary honor. Vairamuthu chose a historical setting from his childhood to explore the suffering of those displaced in the name of progress. As a child, in the 1950s, Vairamuthu lived in one of fourteen villages flooded by the Vaigai Dam in Madurai, in southern Tamil Nadu. The wide attention his work received had a striking contemporary resonance in the 2000s, when debates about dams and displacement raged in India.\n\nAFTER ITS NARMADA DEBACLE, THE WORLD BANK SUPPORTED THE creation of a World Commission on Dams in 1997, charged with assessing the benefits and costs of dam building worldwide over the previous half century. The commission's membership included strong supporters as well as opponents of dams, among them Medha Patkar\u2014but its report, when it appeared in 2000, was more critical of large dams than most critics expected it to be. The commissioners estimated that on average dams were 56 percent overbudget, and that they delivered less irrigation water and hydropower than they promised. The commission's assessment of their environmental impact was equally bleak. Challenging the view that hydropower was an ecologically preferable alternative to the use of fossil fuels, the commission's studies pointed to the large greenhouse gas emissions from rotting vegetation in the reservoirs of large dams. It also pointed to the ecological consequences that Indian scholars and environmentalists had long highlighted\u2014dams altered river flow to the detriment of aquatic habitats; they interfered with the paths of migratory fish. By impounding silt, they robbed lands downstream of fertility.\n\nOne of the consultants to the World Commission on Dams was Ramaswamy Iyer, a career civil servant who served as India's secretary of water resources in the mid-1980s. Iyer's intellectual rigor and honesty set him apart, reflected in his willingness to change his mind. As water secretary, Iyer had taken for granted the value of large dams. He played an important role in pushing through government approval for the Narmada project. But by the end of the 1980s, he began to be influenced by what he called \"newly emerging concerns about environmental impacts and the displacement of large numbers of people.\" Environmental thinking began to influence government decisions in the late 1980s, he recalled, but the growing force of popular opposition to dams led to what he describes as a \"retreat from enlightenment\" in the 1990s. Indian administrators and policymakers came to view Medha Patkar and all that she represented with antagonism, particularly after the World Bank's withdrawal of funding for the Narmada project. The Indian government's response to the World Commission on Dams was brusque dismissal. The cavalcade of arguments about the harmful effects of large dams fell on deaf ears. Searching and thoughtful in his analysis, Iyer turned to history for illumination. The fundamental problem, he discerned, was the persistence of a deep legacy of water engineering, going back to Arthur Cotton; this had bequeathed to India a Western tradition of water engineering, to which Iyer had no objection, \"but also the underlying Promethean attitude to nature,\" which he had started to see as more problematic. To that tradition was added a distinctively postcolonial addiction to what Iyer called the \"magic spell of gigantism.\"\n\nFar from retreating from dam construction, the Indian state redoubled its efforts in the 2000s. It embarked on a scheme to link India's rivers, through one of the largest and most expensive construction projects in human history. It plans to spend at least US$80 billion on a project to link thirty-seven of its rivers through 14,000 kilometers of canals, transferring 170 billion cubic meters of water across India. Among the promised benefits of the scheme are an additional thirty to thirty-five gigawatts of electricity and better water supply for irrigation. The roots of the river linkage scheme lie in the nineteenth century, in the dreams of Arthur Cotton. More proximately, it was the brainchild of irrigation engineer K. L. Rao, who had worked alongside Kanwar Sain and A. N. Kholsa to launch India's dam-building revolution after independence. The idea gained traction in the 2000s, under the coalition government dominated by the Hindu nationalist Bharatiya Janata Party. In 2012, the Indian Supreme Court decreed that it was a matter of \"national interest\" and that the project should be completed as quickly as possible. Environmentalists have raised concerns about the project's consequences and its disruption to already fragile hydrological systems.\n\nIn the years leading up to his death in 2015, Ramaswamy Iyer remained an eloquent critic of the scheme. \"The project is in essence an attempt to redesign the entire geography of the country,\" he wrote; \"underlying it is the old hubristic idea of 'conquest of nature.'\" He argued that the water diversion project was based on a simplistic and dangerous view of India's hydrology; even to divide India simply into \"water surplus\" and \"water deficit\" areas, in ignorance of local ecology, was absurd. The problem went deeper, Iyer thought: \"Rivers are not human artefacts; they are natural phenomena, integral components of ecological systems, and inextricable parts of the cultural, social, economic and spiritual lives of the communities concerned.\" Iyer gave voice to a view of water ecology that was at odds with the conceptions of the Indian state, but it was a view of which we have seen echoes throughout _Unruly Waters_. It was a view that sustained numerous local initiatives that pushed against the juggernaut of large dams, including careful local efforts to restore ancient irrigation systems, through a system of small and simple check dams, in arid regions of Rajasthan.\n\nBut still, \"gigantism\" prevails. At the time of writing, the river-linking project has been given renewed emphasis, though it is years behind schedule, and it is far from clear that it will ever be realized. The counterpart project in China is much further advanced. China's own river diversion project seeks to redress the country's inequalities in the distribution of water by redistributing it on a massive scale. The South to North line, from Danjiangkou reservoir in central China to Beijing, opened in 2014; it is the most expensive infrastructure project the world has ever seen. Two-thirds of Beijing's tap water now comes from Danjiangkou, almost nine hundred miles away. Another arm of the diversion project, the \"eastern route\" that follows the old Grand Canal, opened in 2013. Already, the diversion scheme has brought similar problems to those predicted in India\u2014heightened water conflicts, wastage, social disruption, and substantial ecological harm to riverine ecosystems. The most ambitious part of the project\u2014the western line, linking the headwaters of the Yellow and Yangzi rivers across the Tibetan Plateau\u2014lies in the future, and it is the most likely to cause problems for China's neighbors.\n\nThe ecological and social effects of dams have been well documented; over the past decade, new scientific research suggests that, cumulatively, the world's dams exercise a fundamental geological impact on Earth. The sheer scale of water engineering in the second half of the twentieth century is changing the shape of the world's most densely populated river deltas, which are now denied up to a third of the sediment from flowing rivers that have, over thousands of years, built up the deltas and replenished them. On one estimate, reservoirs have increased by 600 or 700 percent the amount of water held in the world's major rivers, but much less of it now reaches the sea. The once mighty Indus, like the Yellow River, is now a trickle by the time it reaches the Arabian Sea\u2014dammed and diverted into a web of canals, many of them first built by the British in the late nineteenth century. The effect of hydraulic engineering has been to put coastal settlements\u2014and mega-cities, above all\u2014at greater risk of flooding, even before we take into account the effects of climate change and sea level rise.\n\nAS THE RISKS OF CLIMATE CHANGE BECOME INCREASINGLY EVIDENT, water becomes ever-more central to political and strategic conflicts at the heart of Asia. In the face of ecological uncertainty and strategic competition, the Himalayas are home to the greatest concentration of dam construction projects in the world. In historical perspective this marks the final frontier in a conquest of water that began in the nineteenth century. From the 1880s, as European and Indian explorers reached the source of Asia's great rivers in the Himalayas, it was known that the interaction of the Himalayan rivers and the monsoons held the key to Asia's water supply. As territorial borders sharpened up in that era\u2014the border between British India and Tibet, for example, was marked by the McMahon line in 1914\u2014there came the faint knowledge that struggles over water may lie in wait. Even when imperialism was overthrown in Asia after the end of the Second World War, the problem of transboundary rivers arose directly only in relation to the partition of India. As late as 1960, as we have seen, Indian intelligence agents dismissed reports that the Chinese were planning to dam the Brahmaputra, arguing that they had neither the labor nor the infrastructure to do so.\n\nThat Indian assessment was not mistaken. But things changed rapidly in the 1980s. It was in that decade that the Chinese state's dam-building ambitions fixated upon the Tibetan Plateau, source of Asia's rivers. By the 1980s, the large-scale settlement of Han Chinese in Tibet had changed the composition of the region's population; the construction of roads and railways made it less remote from the lowlands and river valleys. Above all, China's frenetic economic growth produced a demand for energy\u2014and an uncomfortable dependence on imported oil\u2014that the hydroelectric potential of the mountain rivers promised to meet. As long as the Himalayan source of Asia's great rivers remained remote and forbidding, it mattered little who formally controlled them; but if the unruly waters were to be tamed, it mattered profoundly who brought them under control.\n\nAs I write this, more than four hundred large dams are planned in the Himalayan regions of India, Nepal, Bhutan, and Pakistan. Construction is already underway on many of these projects. A further one hundred dams are planned on the Chinese side, where so many of the rivers originate.\n\nIf these projects come to fruition, there will be a dam every thirty-two kilometers along the Himalayan rivers, making it the most heavily dammed region in the world. A secretive complex of public and private interests converge and compete to harness the waters of the high mountains. The hunger for energy is widely shared across the region, though demand is driven primarily by the voracious needs of China and India. Geopolitical rivalries play out in the negotiations over who will build the dams, and on whose terms. As multiple dams line up along the same river valleys, the risk to downstream users is grave. In the Indian states of Arunachal Pradesh and Assam, there are fears about Chinese plans for the Brahmaputra upstream in Tibet, where it is called the Yarlung Tsangpo. Downstream, Bangladesh is most vulnerable of all. Already in the 1980s, Bangladesh protested the effects of India's Farakka barrage, built in 1975 to divert water from the Ganges to the Bhagirathi-Hooghly, in part to revive the port of Calcutta that had, since the mid-twentieth century, suffered from severe silting. By reducing the river's flow to Bangladesh, the dam had an impact on soil fertility, irrigation, and health. With an increasing number of dam projects upstream, the risk to Bangladesh has multiplied.\n\nIn one respect, the latest wave of dam construction departs from the precedents of the 1950s and 1960s\u2014it is financed in a different way. Until the 1990s, large dams in India and China were financed primarily by the governments, with India receiving additional funding from international financial institutions like the World Bank, and China, until the split with the Soviet Union in 1961, benefiting from Soviet aid. The new rush to build dams depends more heavily on private capital. In India, public sector organizations like the National Hydroelectric Power Company and the North Eastern Electric Power Company play a major role in dam construction; but so too do private companies like Tata Power (architect of one of India's earliest hydroelectric dams, in the 1910s), and Reliance Energy. State governments have raised capital from domestic markets as international organizations have backed away from funding large dams. But the biggest shift is the role of China. China's dam-building industry, in the late 1990s and early 2000s emerged as a major force in the world. Given the scale of China's own dam building, the depth of engineering expertise in China rivals anywhere in the Western world; and that expertise has been matched by money. By 2008, ten Chinese companies were involved in thirteen dam projects in Nepal and nine in Pakistan, many of them financed by Chinese state-owned and private banks. When India's leading hydraulic engineers had visited China in 1954, they had found their Chinese counterparts dependent on Russian expertise, having to make do and improvise. By the end of the century, the Chinese dam industry led the world.\n\nSuch is the rush for growth that warnings about the impact and the potential risks of these new Himalayan dams have been brushed aside. Environmental assessments on many of the projects have been cursory at best. Given that the dams are entwined with geopolitical and security considerations, given that governments around the region fear popular protest against the dams\u2014which has been widespread not only in India, but increasingly in neighboring countries, too\u2014considerable secrecy shrouds the plans. Even data about river flow across borders is guarded as a state secret. The Himalayan region is less densely populated than the river valleys, but the same problems that accompanied the large dams of the twentieth century are likely to follow here\u2014drowned lands and displaced people. Large reservoirs are less common at these heights than in the lowlands, but diversions to the course of rivers affect life on the river. Mountain species are under threat from the loss of their habitats\u2014already, the brown bear, the snow leopard, the musk deer, the golden mahseer, and the snow trout are imperiled. Much of the power generated by the large dams will be sold to large cities far away, while many local livelihoods are imperiled. The Lower Subansari Hydroelectric Project in the northeastern Indian state of Assam, one of India's most controversial\u2014it has been stalled repeatedly by local protests\u2014threatens the passage of country boats that carry a lot of local trade. The submergence of forest lands will deny local people their main source of firewood. Historian Rohan D'Souza describes the Brahmaputra as a \"moving inland ocean\" bound together by the rhythms of subsistence fishing and floodplain agriculture\u2014a system that is under threat from the dam. The now familiar problem of siltation menaces many of the dams. But this is also one of Earth's most active seismic zones, with earthquakes of 8.0 or more on the Richter scale not uncommon. Fan Xiao, a geologist from Sichuan and a brave opponent of recent mountain dam projects in China, fears that dams will become \"a source of permanent grief and regret for future generations yet unborn.\"\n\nTHE GRAVEST RISK OF ALL\u2014TO THE DAMS, TO THE HIMALAYAS, TO billions of people downstream\u2014comes from climate change. Climate change affects the Himalayan glaciers two ways: by changing patterns of snowfall and by hastening the process of melting. Research findings are complex\u2014not all glaciers are in retreat, and, more seriously, there are very few monitoring stations and few long-term studies. While research has been ongoing in the Chinese Himalayas since the 1990s, the Indian side has been virtually untouched by scientists. The inclusion (and later retraction) of a careless claim by the Intergovernmental Panel on Climate Change (IPCC) about the speed at which the Himalayan glaciers are melting was wielded by climate change skeptics to try to discredit the organization's work. But the consensus is overwhelming that the warming of the planet has led to a recession of the Himalayan glaciers since the mid-nineteenth century and at an accelerating pace in recent decades, if not uniformly everywhere across the mountain range. Most models predict that river flow will be augmented in the short term by the melting of the glaciers\u2014bringing more frequent and severe floods, and even the risk of catastrophic dam collapses. Few observers believe that the designs for the large Himalayan dams have taken into account the uncertainties of climate change. The dangers are greater still given the heightened possibility of extreme rainfall\u2014which, as we shall see, is likely. Around the middle of the twenty-first century, by 2050 or 2060, scientists predict that the dry season flow of the major Himalayan rivers will see significant declines. Not only will this diminution make many of the planned dams ineffective, it will put many lives and livelihoods at risk. More than 1.3 billion people rely directly on the Himalayan rivers for water; 3 billion people rely on the food, water, and energy the Himalayan rivers provide. Changes in the flow and behavior of the rivers as a direct result of the warming of the glaciers threatens a significant proportion of humanity.\n\n# III\n\nThe monsoon has been a continuous thread through _Unruly Waters_ \u2014and it is with the monsoon that we conclude.\n\nThe breakthroughs in tropical meteorology of the late twentieth century shed new light on the scale and complexity of internal variability in the monsoon on multiple timescales\u2014from the quasiperiodic impact of the ENSO system to the intraseasonal variations attributed to the Madden-Julian Oscillation. In recent years, the focus of scientific research has been on how the effects of anthropogenic climate change interact with the monsoon's natural variability in dangerous and unpredictable ways.\n\nThe most fundamental forces driving the monsoon, as we have seen, are the thermal contrast between the land and the ocean, and the availability of moisture. Climate change affects both of these drivers of wind and rain. The warming of the ocean's surface is likely to augment the amount of moisture the monsoon winds pick up on their journey toward the Indian subcontinent. But if the ocean surface warms more rapidly than the land, which appears to be happening in equatorial waters, this would narrow the temperature gradient that drives the winds, and so weaken circulation. Put simply, many climate models predict that the first of these processes will predominate: \"wet gets wetter\" as a result of greenhouse gas emissions. They predict, that is to say, that the moist monsoon lands will see an increase in rainfall. But the monsoon is an intricate phenomenon, as meteorologists have long known. It is increasingly clear that monsoon rainfall is affected not only by planetary warming but also by transformations on a regional scale, including the emission of aerosols\u2014from vehicles, crop burning, and domestic fires\u2014and changes in land use. The urgent challenge for climate science is to disentangle and to understand these global and regional influences on the behavior of the monsoon. And so far, the monsoon has proved much harder to capture in models than, say, global temperatures.\n\nThe availability of detailed records of climate and rainfall in India\u2014which themselves are a product of the history of Indian meteorology going back to the efforts of Henry Blanford and his colleagues in the late nineteenth century\u2014have allowed scientists to reconstruct in detail the monsoon's behavior over the last sixty years. The picture these data present is complex, and in some ways surprising. Average summer rainfall over India has declined by around 7 percent since 1950. But what lies behind this trend? The cause of the decline in rainfall lies in the pattern of India's development since independence. Its explanation, that is to say, lies in the province of economic history.\n\nIn the late 1990s, research vessels observed exceptionally high concentrations of aerosols in the northern Indian Ocean. Satellite images showed a stain that spread across the Gangetic plain and over the Indian Ocean\u2014researchers called it the \"brown cloud,\" an accurate if not a poetic description of the haze. Between January and March 1999, a large team of investigators set out to understand this brown cloud, taking readings from their base at the Kaashidhoo observatory on one of the most remote islands of the Maldives. The project was led by Veerabhadran Ramanathan, an Indian oceanographer based at the Scripps Institute in La Jolla, California. One of the scientists involved was Dutch atmospheric chemist Paul Crutzen, who around the same time also coined the phrase \"the Anthropocene,\" referring to a new geological epoch in which human activity is the most important influence on Earth's physical processes.\n\nThe project found that the haze was a noxious composite of sulfate, nitrate, black carbon, dust, and fly ash as well as naturally occurring aerosols including sea salt and mineral dust. Three-quarters of the composition of the brown cloud could be attributed directly to human activity especially concentrated along the densely populated Gangetic plain and northwestern India. In this region, where up to 80 percent of the population remains rural, and where many rural families continue to be deprived of electricity, much of the black carbon is produced by domestic burning of biomass\u2014wood, crop residue, dung, and coal\u2014used primarily for cooking. Open crop burning accounts for the rest. The stoves used in households are inefficient and combustion is incomplete, producing large amounts of soot. Apart from their likely effects on regional climate, these emissions also poison human bodies. On one estimate, more than four hundred thousand premature deaths each year in India can be attributed to indoor pollution. Black carbon combines, in the brown cloud, with sulfates and other aerosols\u2014and the Gangetic plain bears an additional burden in this respect, as a result of pockets of intensive industrial and extractive activity. Since the late nineteenth century, the Indo-Gangetic plain has been the core region of India's extractive industries, built around the rich coal and mineral deposits in the Chota Nagpur region. Further along Yamuna River, the Delhi region is one of India's fastest-growing metropolitan areas, and its largest in absolute terms. Emissions have increased exponentially since the 1970s as India's population has grown, as its economy has expanded, as inequalities within and among regions have widened. The Gangetic plain suffers from a double pathology: the sulfur, carbon, and nitrogen dioxide emissions that accompany energy-intensive growth are combined with the black carbon that comes from the use of cheaper, dirtier fuels by millions without access to electricity. If India leads in black carbon, China, too, has a brown cloud problem, with sulfates from factory emissions dominating the mix there.\n\nAll of this is shifting the monsoon's patterns. Aerosols absorb solar radiation, allowing less of it to reach Earth's surface. This cools the land, diminishes the temperature contrast between the land and sea, and weakens the atmospheric circulation that sustains the summer monsoon. Changes in circulation over the Indian subcontinent in turn affects the tightly integrated air-sea interaction that binds the Asian continent with the Indian Ocean, a system that already contains plenty of internal variability. Because of the way the Asian monsoon is linked to other parts of the planet's climate, it is possible that aerosols over South Asia and China have global consequences. When all of these effects are coupled with the impact of global warming on the ocean and the atmosphere, the instabilities multiply. Far from counteracting the effect of greenhouse gases in any simple sense, the impact of aerosols complicates them.\n\nA further driver of regional climate change is rapid changes in land use. Over the last 150 years, forest cover over most parts of Asia has declined dramatically. The intensification of agricultural production in India, and the use of more water for irrigation, have affected the moisture of the soil, its capacity to absorb or reflect heat. Crops reflect more solar radiation than forests, which tend to absorb it; the greater reflexivity of land planted with crops makes it cooler, once again weakening the temperature differentials that drive circulation and rainfall. Tropical meteorologist Deepti Singh points out that climate models have often failed to predict the monsoon's behavior in part because they are too abstract to take into account the \"complex topography, temperature and moisture gradients in the region that can influence the monsoon circulation.\" The models omit, that is, precisely the details of landscape and microclimate that the meteorologists of a century earlier were so deeply interested in, which they depicted in their detailed local and regional maps of India's climate.\n\nWe are left with the most bitter of ironies. Many of the measures taken to secure India against the vagaries of the monsoon in the second half of the twentieth century\u2014intensive irrigation, the planting of new crops\u2014have, through a cascade of unintended consequences, destabilized the monsoon itself. When the geographers of the early twentieth century wrote of \"monsoon Asia,\" they saw the monsoon as sovereign\u2014it shaped the lives of hundreds of millions of people, who waited on its every move. Monsoon Asia means something quite different now, when the monsoon's behavior, increasingly erratic, responds to human intervention.\n\nAT ONE LEVEL, THE STORY OF HOW THE MONSOON HAS CHANGED since the 1950s is a story of India's resilience. India has experienced _more_ droughts since the 1940s than in the half century before that, a half century that saw so many devastating famines. Even on a shorter timescale, there are signs of progress. In 2014 and 2015, India experienced two successive years of drought that were as severe as the monsoon failures of 1965 and 1966, which\u2014as we have seen\u2014India could only ride out with massive external aid. In 2014\u20132015, there was no noticeable drop in agricultural production, which observers attribute to better planning\u2014but also to much better forecasts, enabled by the advances in meteorological understanding, and technology, that took root in the 1970s and 1980s. Intraseasonal oscillations\u2014the MJO and the Boreal Summer Intraseasonal Oscillation\u2014have become more amenable to prediction, improving forecasts on a timescale of two to four weeks. Alongside a general drying trend, the monsoon has grown more prone to extremes over the past several decades. If India has received less rain overall, more of it has come in torrents. Between 1981 and 2000, wet spells have been more intense, while droughts have been more frequent but less intense. From the nineteenth century, understanding and predicting the fearsome cyclones that visit the Bay of Bengal with regularity prompted the development of meteorology in India\u2014just as the menace of typhoons spurred research in the Philippines and along the China coast. Predictions of the impact climate change will have on the development of cyclones are as uncertain as those that seek to model the overall behavior of the monsoon. The same countervailing forces are at work: warming seas are, in theory, likely to produce more cyclones\u2014but not if the seas are warming faster than the land. A more definite finding is that the Bay of Bengal's cyclones have grown in intensity in recent decades, as have hurricanes in the Atlantic and tropical storms in other parts of the world. In the Bay of Bengal, scientists predict that climate change will, in the coming century, lead to fewer but more powerful cyclones\u2014though it is possible that the Arabian Sea, not known for cyclones, could see an increase.\n\nNowhere in the world have tropical storms affected more lives than in Bangladesh. In the 1860s and 1870s, severe cyclones in that region of eastern Bengal, then part of British India, spurred the development of meteorological science. In the second half of the twentieth century, cyclones have been more frequent and just as devastating. Approximately 40 percent of global storm surges in the last fifty years have hit Bangladesh, including the two with the highest death tolls, in 1970 and in 1991. Five of the ten worst storms to affect any part of Asia in the twentieth century have struck Bangladesh. But the past twenty years have witnessed a dramatic reduction in cyclone mortality in Bangladesh. Cyclone Sidr, which struck Bangladesh in 2007, was as severe\u2014in terms of wind speed and rainfall\u2014as cyclone Bhola of 1970, but the death toll was one hundred times smaller. An estimated five hundred thousand people died in the cyclone of 1970; in 2007, that number was below five thousand. In part this is a tribute to improvements in forecasting. The Bangladesh Meteorological Department's ability to track cyclones as they develop in the Bay of Bengal improved significantly, with assistance from a Japanese satellite as well as data from the US National Oceanic and Atmospheric Administration. After the fearsome cyclone of 1991, the Bangladesh government embarked on the construction of thousands of cyclone shelters, which have saved millions of lives. Cyclone warnings have become more effective, helped greatly by the spread of mobile phones to even the poorest villages. Changes in the landscape have also played a role. Coastal embankments have kept floodwaters out, although their impact on local ecology has been more controversial. An extensive program of mangrove reforestation has helped to restore one of the most effective natural flood defenses to parts of Bangladesh's low-lying coast. But in Bangladesh, as in India and elsewhere in Southeast Asia, real gains in protecting people from tropical storms contend with a series of new risks\u2014risks that the weather will become more extreme and less predictable, and manufactured risks in the form of unregulated coastal construction, rising population density, and galloping social inequality.\n\nResearch is underway into what governs the increasingly erratic, increasingly extreme behavior of the monsoon. It likely stems from the interaction, on multiple levels and over different timescales, of planetary warming, regional climate change, and natural variability. Recent advances in the oceanographic study of the Bay of Bengal make clear how much the sea's chemistry itself affects climate. The bay is less salty than most bodies of water because of the vast discharge of freshwater from the Himalayan rivers, and because it receives more rainfall than any other sea. This has implications for ocean circulation, temperature differentials, and the interaction of ocean and atmosphere, but the forces at work are still the subject of intensive research. Some of that research looks to the deepest past for clues about the future. Satellites allowed for a new appreciation of climatic forces in the late twentieth century; now the seabed is the next frontier. In 2015, the 470-foot vessel _JOIDES Resolution,_ equipped with a 200-foot drilling tower, collected sediment cores from the sea floor under the Bay of Bengal. Scientists seek a record of the monsoon's behavior going back 15 million years, embedded in the sunken fossils of microorganisms known as plankton foraminifera that once inhabited the surface water and now lie buried. The project aims to use that deep historical data to predict the monsoon's future behavior under conditions of global warming, by examining how the monsoon has responded to historic changes in temperature, salinity, sea level, and atmospheric carbon. It is ironic, perhaps fitting, that the _Resolution_ was once an oil-drilling vessel, now converted to the more benign purpose of oceanographic investigation.\n\nON JULY 26, 2005, 37.2 INCHES OF RAIN BATTERED THE CITY OF Mumbai, most of it between 2:30 p.m. and 7:30 p.m. A third of the city was flooded. Cellular phone networks crashed. The airport was shut when its runways flooded. Close to 150,000 people were stranded in stations on Mumbai's massive commuter rail network, which came to a standstill. Almost one thousand people died, tens of thousands were made homeless. The government was completely unprepared when faced with this extreme amount of rain, even though such downpours were not completely without precedent in Mumbai's history. To many observers, this was a freak of nature\u2014or an act of God.\n\nMap of Mumbai, showing it flanked by the Arabian Sea to the west and Thane Creek to the east\u2014much of contemporary Mumbai sits on reclaimed land. CREDIT: Illustration by Matilde Grimaldi\n\nBut a citizens' commission assembled by the city's nongovernmental organizations to report on the floods reached a different conclusion. The commission argued that Mumbai had put itself in harm's way. After decades of relentless growth and expansion, Mumbai had few natural drainage channels left. They had been concreted over. There was nowhere for the water to go. Storm drains were clogged with waste, tidal flats had been built upon. What environmental regulations there were on paper could not rein in a boom in unauthorized construction\u2014in a city where prime real estate was worth more than in New York or Hong Kong. The destruction of the mangroves of Mahim Creek\u2014which stretched to seven hundred acres as late as 1930\u2014for highway construction and urban development robbed the city of a natural buffer between land and sea.\n\nA taxi under water during the Mumbai floods of July 26, 2005. CREDIT: _Hindustan Times_ \/Getty Images\n\nBeyond the suffering that the storm caused, it also provided a stark warning. If, as scientists predict, the \"once-in-a-hundred-year\" storm is likely, in the future, to materialize every ten or twenty years, or perhaps more regularly than that, Mumbai is acutely vulnerable, along with so many cities at the water's edge. The threat to the coasts, once again, comes from the sea\u2014fueled no longer just by natural patterns but also by human activity that is once regional and planetary in its sources and its effects.\n\nThe prospect of the next big storm hitting Mumbai is the alarming picture that Amitav Ghosh sketches, powerfully, in his nonfiction work on climate change, _The Great Derangement_. Ghosh forces us to imagine Mumbai in a superstorm:\n\nAt this point waves would be pouring into South Mumbai from both its sea-facing shorelines; it is not inconceivable that the two fronts of the storm surge would meet and merge. In that case the hills and promontories of South Mumbai would once again become islands, rising out of a wildly agitated expanse of water.\n\nBut in the face of catastrophe that is \"inconceivably large,\" Ghosh argues that most states, like most human beings, are guided by \"the inertia of habitual motion.\"\n\nIn recent years, the view that we should live with and adapt to the natural hydraulic risks of littoral zones\u2014to say nothing of how these risks are worsening with climate change\u2014has infiltrated the worlds of architecture and design. Mumbai architects and urban theorists Anuradha Mathur and Dilip da Cunha insist that the colonial and postcolonial practice of drawing a firm boundary between land and water in Bombay stems from a fundamental misreading of the fluid coastal landscape. Mumbai during the monsoon demands to be seen \"in cross-sectional depth,\" they argue, not in the two dimensions of maps and plans. When the monsoon comes, \"there is too little time and too much water to make an orderly exit through courses delineated on maps.\" In their vision, the city in monsoon becomes a fluid, mutable organism at the boundary between land and water, shaped by the interplay of \"the monsoon clouds above through the labyrinthine world of creeks, to the web of aquifers beneath.\" Only if we understand this, design for it, adapt to it, they argue, can we live with, rather than trying to engineer away, risk.\n\nWorking within a tradition that goes back a century\u2014to Blanford and Isis Pogson and Ruchi Ram Sahni\u2014Indian meteorologists have a distinctive understanding of the climate risks facing India today. However much patterns of rainfall may be changing, they suggest, the monsoon has _always_ presented a risk to South Asia: the fundamental source of escalating risk today lies in foolhardy policies.\n\nThis is what emerged in my conversations with S. Raghavan, formerly a senior officer in the Indian Meteorological Department, now retired in Chennai, where we met at his home. His father was a large farmer in an arid tract of rural Tamil Nadu. Raghavan grew up with an intimate knowledge of water and crops; long before he became a meteorologist, he recognized the rhythms of the monsoon. After taking a degree in physics at Madras University, Raghavan received three job offers: one from All India Radio, one from the auditor general's office, and one from the meteorological department. With little knowledge of meteorology, he took that option, excited in part by the chance to work with the latest technology, including the radiosondes he had seen on display at a stall in his college's engineering fair. At the height of the Cold War, Raghavan was sent to the United States on a government scholarship to study radar technology. When Delhi's Safdarjung Airport received its first radar in 1957, Raghavan was put in charge of its operation. In 1972, he returned to Madras to take charge of radar meteorology there, equipped with a cyclone warning radar purchased from Japan, which arrived only after a long tussle with the customs department at a time when India had stringent restrictions on imports and foreign exchange. That year, a serious cyclone struck the coastal town of Cuddalore. It was the first cyclone in India to be tracked by radar as it approached; with accurate information and early warnings, casualties were minimal. It was then, Raghavan said, that \"I realized that I was doing some service to society.\"\n\nBut just as forecasting capacities improved, in the 1980s, the risk posed by extreme weather in India multiplied. The cause was manufactured, not climatic, he said. The reason so many millions who live in coastal India are in danger, he told me, was because governments, planners, developers, and citizens had completely neglected the ordinary climatic risks that coastal South Asia faces. \"Time and time again, we put ourselves in harm's way,\" he said. Raghavan does not believe that the risks of cyclonic storms, for example, can be engineered away; at best they can be prepared for. He believes in early warning, and for that there is no substitute for the patient observation of weather fronts developing in the Bay of Bengal. His mission in retirement is the production of a Tamil lexicon of climatological terms, in the same way that the colonial meteorologists of the late nineteenth century were not averse to collecting local proverbs. He regularly gives talks to schools and residential societies about climate and weather. He described the destruction of cities' natural drainage and storm defenses. As late as the 1940s, he remembered seeing Chennai's Cooum River busy with traffic, including boats carrying salt from Andhra; it had become a \"cesspit,\" he told me. Drains were blocked by a \"plastics explosion\"; the destruction of mangroves had taken away natural protection against storm surges. \"Our own actions are responsible\" for the crisis, he said.\n\nThe soft-spoken Mr. Raghavan was careful and precise in his judgments; as we talked, he often turned to his shelves to find a book, or to consult a folder of press clippings that he had maintained over many years. The day after we met he sent me a PowerPoint presentation he had made for a recent lecture. But there was no mistaking the emotion in what he said\u2014he was both sad and angry at the way the risks of a monsoon climate had been disregarded. His was a view of the weather, and the climate, that was rarely about control\u2014it was about adapting to known and felt risks. The long quest of India's meteorologists to understand the monsoon continues to shape their responses to a changing climate.\n\nTHESE ARE NOT, FOR THE MOST PART, THE LESSONS THAT ARE BEING learned. The map of cities at risk resembles a series of beads on a necklace threaded along the coastline of Asia. One study predicts that by 2070, nine out of the ten cities with the most people at risk from extreme weather will be in Asia\u2014Miami is the only non-Asian inclusion. The list includes Kolkata and Mumbai in India, Dhaka in Bangladesh, Guangzhou and Shanghai in China, Ho Chi Minh City and Hai Phong in Vietnam, Bangkok in Thailand, and Yangon in Myanmar. Each one of these cities will confront any change in the interaction of land and water, winds and rain over Asia's oceans. Just two years after the Mumbai floods, it was the turn of Jakarta, the Indonesian capital and the fastest-sinking city in the world, pulled down by the weight of construction, by the extraction of groundwater, and by the rising sea. Jakarta is sinking by between three and six inches every year. The storm in 2007 washed over the sea walls built to protect the city. Half the city was underwater, displacing 340,000 people from their homes. In Jakarta and in each of Asia's coastal megacities, climate change compounds a cavalcade of risks that are severe in and of themselves\u2014hasty development driven by property speculation and new forms of middle-class consumption, crumbling health and sanitary infrastructures, and a lack of preparedness and precaution, are all symptoms of profound social and economic inequalities both among and within nations. Of all the countries in the world, few are more directly under threat than low-lying Bangladesh.\n\n# IV\n\nThe struggle for water transcends Asia's borders. The Himalayan rivers, dammed and diverted and vulnerable to changes in glacier cover, flow through many nation-states on their descent to the sea. Planetary warming is a result of the historical emissions of fossil fuels\u2014initially and cumulatively by the wealthy and industrialized countries of the world, but also, and increasingly since the 1980s, by China and India. Global warming interacts with and compounds the effects of regional climate change. Aerosol emissions from the Gangetic plain or from fast-industrializing areas of China have effects far beyond India's or China's borders, creating a series of brown clouds that blanket the Indian Ocean and affect rainfall far away. Climate change creates problems of distance\u2014between the source of pollution and its consequences\u2014but it also creates new forms of proximity in the form of shared risks and interdependence. The image of the Himalayas as \"Asia's water tower\" conveys both the scale of the hydraulic system that binds much of Asia, and the scale of the threat that they face from the destabilization of that source of so much water. By the 1960s, the sea itself was a form of territory. The Bay of Bengal was the crucible of the earliest monsoon science in the nineteenth century; it remains the crucible of monsoon science today. But it is a very different sort of space. It is crowded. It is contested. It is walled off by borders in the sea as much as on land. Even international cooperation in oceanographic research on the monsoon has to confront the reality of borders at sea. A major project between 2013 and 2015 set out to investigate the Bay of Bengal and its role in monsoon circulation; it brought together American, Indian, and Sri Lankan scientists. Their research vessels roamed the Bay for two years, taking an enormous number of measurements of ocean salinity, temperature, currents, and chemistry. Yet the map of their voyages, a dense set of tracks that the ships followed over those years, is divided up by a thin line marking the extent of territorial waters and exclusive economic zones; some, like the border of Myanmar, the ships could not cross for political reasons.\n\nIf borders at sea are forbidding, those on land are even more so. Throughout Asia one of the ways in which communities have coped with extreme weather has been to move\u2014often temporarily, and not necessarily over long distances. For regions that are threatened by climate change and water-related risks, borders create barriers to mobility. \"Climate refugees\" are much discussed in current legal and political debates. But the Red Cross rightly stresses that the \"populist term 'climate refugees' is profoundly misleading\": environmental drivers of migration act \"in conjunction with economic, social and political factors, and [are] linked to existing vulnerabilities,\" and it is \"conceptually difficult to establish a precise category of environmental or climate migrant.\" It would be a mistake to separate a discussion of \"climate migration\" from a broader consideration of regional patterns of mobility.\n\nThere is an odd historical resonance to some of the pronouncements about climate and migration. In the nineteenth century, too, many observers saw the movement of people across the Indian Ocean and the South China Sea as driven by climate\u2014not by climate change but by climate's natural volatility. The use of liquid metaphors to describe migration remains pervasive: a language of \"floods\" and \"tides\" and \"waves\" and \"flows.\" Many of the region's migrants today come from places and from communities that have been mobile in the past. This is hardly a surprise. Some of the places most threatened by environmental catastrophe are also places\u2014the coasts and the great river deltas\u2014that have the longest histories of migration. But other affected regions lack the accumulated family connections, knowledge, experience, and access to credit to allow them to move. Forced immobility can be as dangerous, as traumatic, as forced migration. Controls on mobility have intensified since the middle of the twentieth century, and they are likely to harden: hysteria in India about \"illegal migration\" from Bangladesh, for instance, has led to the securitization and fortification of the border, though many people risk their lives to cross it out of desperation. The slow effects of climate change are as likely to leave people stranded, unable to move, as they are to spark a rush of \"climate refugees.\"\n\nA recent study by the World Bank makes clear that although cross-border migration receives more attention, vastly more people migrate within their own countries than migrate internationally. The overwhelming majority of people who are displaced by climate change over the next three decades will move internally\u2014an estimated 40 million people in South Asia alone, and 143 million people globally. In the depersonalized language common to climate policy documents, the World Bank concludes that \"several hotspots of climate in- and out-migration are in transboundary areas\" of South Asia, and that these \"must be explored for their opportunities and managed for their challenges.\" But what does this really mean? It means the options facing people whose lives are threatened by drought or deluge will be constrained by borders as well as by poverty, gender, caste, or a lack of opportunity. It means that the closest refuge, if it should lie across a border, may not be a refuge at all. It means that many routes that make social, cultural, or ecological sense to people\u2014routes embedded in family histories, routes across regions that have not always been divided by borders\u2014will be blocked. It means that those who are compelled to cross closed frontiers in search of security will face unprecedented risks.\n\nGIVEN THE WEIGHT OF BORDERS, ARE THERE PROSPECTS OF CLOSER regional cooperation to confront the problems of water and the threat of climate change? If so, these prospects are modest in scope and ambition. Existing regional institutions\u2014the Association of Southeast Asian Nations (ASEAN), the newer and smaller Bay of Bengal Initiative for Multisectoral Technical Cooperation (BIMSTEC)\u2014are focused overwhelmingly on the development of infrastructure and the promotion of trade. Though environmental protection is not absent from their concerns, it is not a high priority. When policy documents refer to climate, it is often as a metaphor, as in the often expressed hope of creating a \"climate friendly to investment.\" And when new infrastructure projects threaten ecological and social harm\u2014as do so many of the port projects that proliferate along the Bay of Bengal's coasts\u2014they have almost always proceeded regardless, except where they have met with significant public protest. Nevertheless the second half of the twentieth century did create agreements and institutions to manage water across borders, and these need to be strengthened wherever possible. Though flawed, the Indus Treaty signed in 1960 between India and Pakistan, with the World Bank's mediation, has largely worked. The two hostile neighbors have for the most part worked cooperatively to manage that shared river, though there have been periodic surges of tension between them. The Mekong River Commission, created in the 1950s by the UN, has outlasted the Cold War. Though the commission has often failed to prevent reckless development along the river, the growing involvement of China in its discussions suggests that states are taking more seriously the shared threats they face.\n\nBut the most promising initiatives to address shared risks may lie in the realms of science and civil society. From the start, climate science in Asia has been a cosmopolitan enterprise. In the late nineteenth century, observatories and scientists across imperial borders exchanged data and theory and reports. This is not to say that climate science stands apart from politics. It never has. In the nineteenth century, the development of meteorology was deeply entwined with imperial interests. But climate science provided a way of visualizing Asia beyond borders, as a vast and connected climatic space, bound together in every dimension\u2014the oceans, the air, and the land. Meteorologists saw that the same storms menaced the Philippines as India. Growing knowledge of climatic connections inspired attempts to share warnings if not coordinate responses. In an era of nation-states, that level of cross-border cooperation among scientists has continued\u2014and it is more vital than ever. Even as an organization like BIMSTEC is hampered by political tensions among its member states, it has made small but tangible gains in coordinating the sharing of early warnings to bolster disaster preparedness. As meteorologists' ability to forecast storms has advanced with improvements in satellite technology as well as better models, that information is now more readily accessible to a wide public\u2014mobile phones are ubiquitous across South Asia; even the smallest fishing vessels are now equipped with GPS technology.\n\nSome of the most promising recent efforts to increase cooperation across borders to tackle Asia's water problems have focused on the sharing of information. As we have seen, data concerning the hydraulics of the Himalayan rivers are a closely guarded secret. The Third Pole, a nongovernmental organization based in London and New Delhi, dedicated to understanding and communicating the cross-border water issues faced by Asian states\u2014with a focus on the Himalayan rivers\u2014has compiled as much information as is available on river flow and on climatic trends. Using open source data, it has created a new mapping platform that allows for the sharing of data on river flow and hydropower, glaciers, and groundwater. This is now readily available to journalists, activists, and scholars. These maps of the Himalayan region transcend borders, emphasizing shared ecological challenges. The ability to visualize the risks holds the promise of stimulating a more coordinated response; it might even inspire new solidarities that come from a sense of shared vulnerability. These efforts to pool information have begun to mobilize public participation by so-called citizen scientists. Season Watch, an Indian organization, encourages its members, including schoolchildren, to submit detailed daily observations of climate, thereby linking very local experiences of changing seasonal cycles with changes on a regional and global scale. This effort extends to the preservation of local archives. The World Meteorological Organization has urged the importance of \"data rescue\"\u2014the recovery of records and logs of rainfall and temperature, often handwritten, preserved in local repositories and threatened by physical deterioration. These are potentially invaluable to climatologists looking for long-term patterns of change. Those very archives, as we have seen throughout this book, are full of evidence of the ways in which, in earlier times, not only storms and currents but scientific information crossed borders.\n\nFrom the 1980s onward, there has been close cooperation among environmental activists across Asia. They have pooled information, campaigned together, and recognized that environmental degradation\u2014including but not limited to climate change\u2014is a menace they all face. Historian Prasenjit Duara sees reasons for hope in the organizations of what he calls \"network Asia\"\u2014the web of NGOs, some of them religiously motivated and others resolutely secular, coming together to confront problems of water and climate. The environmental movement has at times been genuinely transregional, yet in both India and China, it has become clear in recent years that environmental organizations are vulnerable to crackdowns by the state. There is also an imaginative barrier to cross. As we have seen, the power of environmental activism very often comes from the ability to evoke a sense of emotional attachment to particular landscapes. Narratives about the past have been fundamental to the rise of environmentalism in India and elsewhere in Asia, but the pasts they have appealed to are profoundly local ones; their narratives juxtapose an earlier age of ecological innocence with the depredations of colonialism and modernity. An appeal to nationalism has been, and remains, one powerful way that environmentalists can mobilize public support. But this can make it more difficult to work across borders.\n\nAt a time of environmental crisis, local histories and national responses are insufficient on their own. It is now easy to see on a map, or in an alarming graphic, the scale of water-related risks that Asia faces. It is clear that those risks pay no heed to borders. The promise of a new sort of environmental history, a more connected and expansive history of Asia's unruly waters, is to fill that space with cultural and political meaning\u2014to show that the landscape of Asia's mountain rivers and its monsoons have also constituted a space of migration, a zone of trade, a path of pilgrimage.\n\nThroughout history, water has both connected and divided Asia. The rivers and oceans have been thoroughfares of trade as well as zones of imperial domination. In the nineteenth century, when European empires dominated the world, Asia's hydrology underpinned many of the commodities that fueled global industrial capitalism. The storms that have always menaced coastal regions always crossed frontiers, but states have responded to them in different ways. As connections across Asia frayed in the mid-twentieth-century decades of nationalism and war, water, too, came under ever-tighter territorial control. One reason why almost all of Asia's new nation-states tried so boldly to harness water was to gain self-sufficiency in a postcolonial era in which their autonomy was nevertheless called into question by the machinations of the superpowers in the Cold War. They were spurred to do so by memories of water's lack\u2014bitter memories of famine and suffering within living memory. They were spurred, especially in India, by a fear of the monsoon climate and the power it had over human life. \"For us in India scarcity is only a missed monsoon away,\" Prime Minister Indira Gandhi said\u2014and this sense of a battle against enormous natural forces inspired in her, as in so many others, a tug between despair and optimism that science and technology held the key to liberation. Over time that insistence on self-sufficiency combined with a sense of perpetual crisis led to a narrowing of vision and a willful blindness to the consequences of repeated attempts to conquer nature. Today, the inability of states to think beyond their borders imperils lives and denudes the political imagination.\n\nIf there is one consistent lesson in _Unruly Waters,_ it is that water management never has been, and can never be, a purely technical or a scientific question; neither can it be addressed on a purely national scale. Ideas about the distribution and management of water are deeply inflected with cultural values, with notions of justice, with ideas and fears about nature and climate\u2014including very old fears about the monsoon, which grows more capricious. The battle continues to understand the monsoons and mountain rivers that shape Asia.\n\n# EPILOGUE\n\n# HISTORY AND MEMORY AT THE WATER'S EDGE\n\n\"THERE IS THE SCIENTIFIC AND IDEOLOGICAL LANGUAGE FOR WHAT IS happening to the weather,\" writes novelist Zadie Smith, \"but there are hardly any intimate words,\" no words that capture the sense of loss that climate change brings with it. \"The weather has changed, is changing,\" Smith writes, \"and with it so many seemingly small things... are being lost.\" Faced with the forbidding scale of climate change, many responses are profoundly local. Indian farmers, deeply attuned to the tenor of the skies, are changing when they plant their seeds. But changes in the weather also bring a sense of disorientation\u2014a loss of one's bearings. Everywhere I traveled over the eight years I have been working on this book, I heard stories about the weather\u2014stories of how it is not what it used to be. In many cases, these stories were prompted by a particular landscape that was familiar once, and is now unrecognizable. \"Look there,\" I was told by a longtime resident of Thanjavur on a trip through Tamil Nadu in 2012, \"when I was young, the river ran full, now it is completely dry.\"\n\nThere are many other kinds of loss that climate change threatens us with. A changing monsoon affects every form of life that depends on it. From the Gurukula botanical sanctuary in Wayanad, northern Kerala, Suprabha Seshan and her colleagues cultivate endangered plants native to the ecosystem of the Western Ghats, the western Indian mountain range that receives some of the most intensive rainfall during the summer monsoon. \"We refer to these plants as refugees,\" she writes; many have been rescued from areas where forests have already been cut down. \"The weather features regularly in our speech,\" she writes. The gardeners' work depends on an intuitive knowledge of the weather. But these patterns are changing. Seshan observes that \"ever since I have been here, about 24 years now, I have heard people talking about how the monsoon has gone awry, that it is no longer what it used to be. We also know this from scientific data, but crucially for us, we know this from the behaviour of the plants and animals in our sanctuary.\" Here, meteorological research and local perceptions match. Everyone is sure that the southwest monsoon has weakened\u2014and has become more unpredictable. Local species are \"confused,\" Seshan writes, by the weather's signals. Temperatures are too high for some mountain species to thrive, and rising temperatures bring new diseases. \"I worry,\" Seshan concludes, \"that the monsoon, with its moods and savage powers, might altogether cease.\"\n\nALONG THE COAST OF SOUTHEASTERN INDIA, TOO, ARE MANY SIGNS of irreversible change. In a small village near Pondicherry, earlier this decade, I met Mr. Rathnam, a fisherman in his fifties whose family have been fishers in the area for generations. On both sides of the narrow strip of beach on which we sat were granite sea walls. \"If not for these walls,\" he said, \"the sea would have taken this settlement long ago.\" The beach has been eaten away over the past twenty years, most noticeably by the construction of a large new port in Pondicherry, a few minutes down the coast. The Pondicherry port marked the beginning of an explosion in port construction in India, with dozens of ports currently planned for India's eastern and western seaboards. They eye the newly flourishing commercial opportunities of the Indian Ocean's littoral, which is vibrant again after falling into decline for the second half of the twentieth century. The ports cause enormous upheaval to the coastline. \"Where these boats are now,\" Mr. Rathnam said, pointing to the beach, \"those were all houses. Look, you can see the remains of the floors of houses.\" I saw little fragments jutting out from the soil, a small archive of coastal environmental history.\n\nThe remains of a house: an archive on the landscape of coastal erosion. CREDIT: Sunil Amrith\n\nHe is convinced that the sea is changing in ways beyond what is visible to the eye, beyond the visibly changing shape and extent of the beach. The weather is \"unpredictable,\" he said; \"the seasons seem to mean nothing now.\" He was convinced the monsoons are shifting, and in his narrative the Indian Ocean tsunami of 2004 was the moment when \"everything changed.\" The tsunami was a geological phenomenon, caused by an undersea earthquake, but to Mr. Rathnam it seemed a portent of fundamental change. He paused his story. \"I don't understand the sea anymore,\" he said, suddenly: the sea that he has known, intimately and instinctively, for a lifetime. I asked him what does the future holds. \"Nothing,\" he said; \"there will be no fish left to catch.\"\n\nA narrowing stretch of beach near Pondicherry, eroded by the construction of a new port. CREDIT: Sunil Amrith\n\nClimate change is not the most obvious or proximate cause of his distress. Here, as elsewhere in Asia, the effects of climate change compound a crisis already far advanced\u2014a product of reckless development and galloping inequality. Mr. Rathnam's livelihood has been threatened by the concentration of power in the hands of a small number of highly capitalized owners of large trawlers. There are fewer fish to catch because of what a recent report calls \"an uncontrolled addition of fishing boats between 1965 and 1998.\" The size of the catch has collapsed, and its composition has changed: fewer large predators, fewer fish that command high prices on the market. The dramatic fall in their incomes has pushed many small fishers ever deeper into debt. Development along the large highway down the coast from Chennai has led to a spike in property speculation, fueling a construction boom that flouts coastal zone regulations. A tidal wave of plastic, and effluent from factories and power plants, floats out to sea. Compounding each of those challenges, climate change is also now making itself felt. Rising sea surface temperatures in the Bay of Bengal have exceeded the boundaries that can sustain many forms of aquatic life.\n\nOne of the questions I asked Mr. Rathnam that day on the beach, was, \"What happened to the family that lived in this house, and others like them?\" One part of the answer, I expected\u2014there had been a large movement of younger people to the growing cities, and to Chennai in particular.\n\nBut there was another part of the answer that I did not expect. In light of the work I had spent the previous decade doing on migration across the Indian Ocean, I had the feeling of a very familiar map of migration being drawn before my eyes. All but the very poorest households in the village, Mr. Rathnam told me, has at least one family member overseas. A similar story emerged in the neighboring hamlet. Older routes of migration have been reinvigorated\u2014plenty of sons and nephews in the village were in Singapore and Malaysia, working in construction. Others had taken more recent paths. Many work on fishing fleets in the Persian Gulf. Colonial connections, too, continue to shape people's trajectories in Pondicherry, which was a French-ruled enclave within British India\u2014one older fisher turned to his memories of the \"French time,\" and then enumerated his family members now living in Paris. Old geographies still matter. In this part of South India, people experience and imagine climate change at home in relation to a constellation of distant places; family histories of mobility are reactivated as a means of support or insurance. But borders are harder than ever to cross. Every day, South Indian fishermen, struggling to make a living, stray into Sri Lankan territorial waters in search of fish; many have been arrested and detained by the Sri Lankan coast guard.\n\nPeople experience climate change in space as well as in time. They mark change in terms of their memories of the seasons as they used to be, or of epochal storms that now seem portents of the future. But they also mark it through traces on the landscape, through memories of old houses and old neighbors. Traces of those earlier times lie embedded as debris at the water's edge.\n\n# ACKNOWLEDGMENTS\n\nTHE RESEARCH I DID FOR THIS BOOK BETWEEN 2012 AND 2015 received funding under the European Union's Seventh Framework Programme (FP\/2007\u20132013\/ERC Grant Agreement 284053) from the European Research Council, held at Birkbeck College, London. Since 2015, my work has been supported by Harvard University's Faculty of Arts and Sciences; I am grateful to deans Michael Smith, Nina Zipser, and Laura Fisher for providing me with the resources that make my work possible. More recently I have benefited from the support of the Infosys Science Foundation and the MacArthur Foundation, and their extraordinary generosity has helped me bring this project to a conclusion. I would also like to thank Carol Richards and the late David Richards for their generous support of the Center for History and Economics.\n\nThe research for this book would not have been possible without the help of many fine archivists and librarians. Within the wonderful Harvard library system, I would especially like to thank Fred Burchstead, Ramona Islam and Richard Lesage, Laura Linard at the Baker Library, and everyone at the Map Collection. The staff at the Asian and African Studies reading room at the British Library have helped me over many years and many projects. In India I would like to thank the staff of the National Archives of India and the Nehru Memorial Museum in Delhi; Kiran Pandey at the Centre for Science and Environment's library; the staff of the Tamil Nadu State Archives in Chennai; and the Maharashtra State Archives Department in Mumbai. Elsewhere, I would like to thank the staff of the National Maritime Museum in Greenwich, the World Bank Archives in Washington, DC, the National Archives of Myanmar in Yangon, and the National Archives of Sri Lanka, Colombo.\n\nOver the years I have been working on this book, there has been a revolution in digitization. Sources that I spent weeks or months locating are now, as I write this, freely available online. A particular treasure for historians of South Asia has been the digitization of the library of the Gokhale Institute of Politics and Economics. That remarkable institution's collection of reports, occasional papers, and theses span a century and are now available to historians everywhere; this is truly a public service. The National Archives of India, too, has embarked on an ambitious program of digitization; access came too late for me to make use of these digitized materials in this book, but it will make new kinds of work possible in the future.\n\nI have benefited from excellent research assistance over the course of writing this book. I would like to thank Sneha, who was indefatigable in finding sources at the National Archives in Delhi, and to two Harvard undergraduates, Aaisha Shah and Ellie Lasater-Guttman, who tracked down many textual and visual materials for me.\n\nI am grateful to the many people who have shared their stories and memories with me over the course of writing this book. From among the extraordinary community of Indian meteorologists who have dedicated their careers to understanding the monsoon, I am especially grateful to Ranjan Kelkar, S. Raghavan, and S. R. Ramanan, who were generous with their time and spent many hours talking to me. They also shared unpublished material with me, and facilitated access to the wealth of material in the libraries of the Regional Meteorology Centre, Chennai, and the India Meteorological Department in Pune. There are many more people to thank who preferred to speak informally or to remain anonymous: my understanding of water and climate would have been poorer without the opportunity to learn from many fishers and cultivators and government officials in Tamil Nadu and Pondicherry. I would particularly like to thank R. Rajamanickam for an illuminating introduction to coastal communities around Pondicherry, and for setting up some initial interviews.\n\nWriting about the history of meteorology has taken me far from my areas of expertise, and I am deeply grateful for the advice of colleagues in the fields of tropical meteorology and climate science. Professor Peter Webster of Emory University and Marena Lin of Harvard's Department of Earth and Planetary Sciences were generous in answering questions and sharing unpublished work with me; I am particularly grateful to Professor Adam Sobel of Columbia University who read drafts of two chapters, providing invaluable feedback, clarifying my understanding, and saving me from errors.\n\nMy wonderful literary agent, Don Fehr, has been supportive and encouraging throughout the process. Don made it possible for me to work with an editorial \"dream team\" of Brian Distelberg at Basic Books and Simon Winder at Penguin. Brian's thorough and deeply insightful comments on a ramshackle draft pushed me to sharpen the argument and tighten the structure. Simon's edits were brilliant, honing in on where I had been evasive and suggesting many fruitful comparisons and connections. The final version has benefited enormously from the meticulous and thoughtful line editing of Roger Labrie, and from Bill Warhop's sensitive copyediting. Melissa Veronesi oversaw the production process with grace and efficiency.\n\nLOOKING BACK AT MY CAREER OVER THE PAST FIFTEEN YEARS, nothing makes me happier than that Emma Rothschild, the scholar from whom I have learned more than from anyone else, is now my closest colleague. Nobody has done more to support and inspire my work. I depend on her advice even more now than when I first arrived on her doorstep as a graduate student.\n\nI began this book while I was at Birkbeck College, University of London, where I taught for nine years. I would like to thank my former colleagues at Birkbeck, especially Chandak Sengoopta and Hilary Sapire. Since arriving at Harvard in 2015, I have accumulated many debts. I have been lucky to have the support of three exceptional deans: Diana Sorensen followed by Robin Kelsey in the Division of Arts and Humanities, and Claudine Gay in the Division of Social Science. Thomas Skerry eased my move to Harvard in many ways. At the Department of South Asian Studies, Cheryl Henderson's kindness and efficiency make everything possible; I would also like to thank Parimal Patil for his support during his time as the department's chair. At the Department of History, I have been fortunate to work with Rob Chung, Kimberly Richards O'Hagan, and their team. I am grateful to David Armitage and Dan Smail for their kindness during their respective terms as chair. At the Center for History and Economics, it is truly a pleasure to work with Emily Gauthier and Jennifer Nickerson\u2014their dedication sustains the amazing intellectual community of the center. I would also like to express my thanks to the staff of the Harvard Asia Center, the Weatherhead Center for International Affairs, the Harvard Academy for International and Area Studies, and the Harvard University Center for the Environment.\n\nFor their warmth and camaraderie and for many kinds of help, I would like to thank friends and colleagues at Harvard: Sugata Bose, Allan Brandt, Richard Delacy, Arunabh Ghosh, David Jones, Gabriela Soto Laveaga, Kenneth Mack, Durba Mitra, Jonathan Ripley, Charles Rosenberg, Amartya Sen, Ajantha Subramanian, Michael Szonyi, and Karen Thornber.\n\nThe greatest joy of moving to Harvard has been the opportunity to work with a remarkable group of students. My undergraduate students' commitment to positive change, their courage, and their talent give me optimism for the future in these dark times. I have also been privileged to work with many outstanding graduate students: Mou Banerjee, Aniket De, Yuting Dong, Shireen Hamza, Neelam Khoja, Kiran Kumbhar, Lei Lin, Amulya Mandava, Tsitsi Mangosho, Mircea Raianu, Priyasha Saksena, Hannah Shepherd, and, beyond Harvard, Jack Loveridge and Lucas Mueller. I am especially grateful to the four students I've worked with most closely over the last three years: Divya Chandramouli, Hardeep Dhillon, Sarah Kennedy Bates, and Iris Yellum\u2014I have learned far more from them than they have from me.\n\nDispersed though they are across the world, I am always grateful for the friendship of Isabel Hofmeyr, Maya Jasanoff, Diana Kim, Sumit Mandal, Mahesh Rangarajan, Taylor Sherman, Naoko Shimazu, Benjamin Siegel, Kavita Sivaramakrishnan, Glenda Sluga, Eric Tagliacozzo, and A. R. Venkatachalapathy\u2014each of them is a model of scholarly integrity, generosity, and kindness. This time again, I owe a particular debt to Tim Harper.\n\nFor helpful conversations, ideas, or invitations to present my work, I would like to thank: Seema Alavi, Michiel Baas, Abhijit Banerjee, Ritu Birla, Anne Blackburn, Dipesh Chakrabarty, Joya Chatterji, Rohit De, Prasenjit Duara, David Engerman, Amitav Ghosh, Ramachandra Guha, Anne Hansen, Namrata Kala, Akash Kapur, Adil Hasan Khan, Sunil Khilnani, T. M. Krishna, Michael Laffan, Melissa Lane, David Ludden, Amala Mahadevan, Rochona Majumdar, Farina Mir, Kazuya Nakamizo, Michael Ondaatje, Prasannan Parthasarathi, Jahnavi Phalkey, Gyan Prakash, Srinath Raghavan, Bhavani Raman, Jonathan Rigg, Harriet Ritvo, Tansen Sen, Tomoko Shiroyama, Mrinalini Sinha, Vineeta Sinha, Helen Siu, K. Sivaramakrishnan, Smriti Srinivas, Julia Stephens, Kohei Wakimura, Roland Wenzlhumer, and Nira Wickramasinghe. I join many people around the world in mourning the loss of Christopher Bayly, who remains a guiding light.\n\nTHIS IS THE FIRST BOOK I HAVE WRITTEN SINCE BECOMING A PARENT, and I could not do my work without the work of many others. I am full of admiration for the creative and nurturing teachers at Radcliffe Child Care Center in Cambridge, Massachusetts, and I appreciate deeply the care of Marlene Boyette, Pearl Kerber, and Uyen-Nguyen Tran. I am grateful to the many people who have made Cambridge, Massachusetts, feel like home: Ian Miller and Crate Herbert, who have welcomed us from the start; Priyanka Shankar; the kind neighbors on our street; the community of Radcliffe parents, and especially Laura Muir and Danny Pallin; all at Cambridge Friends Meeting.\n\nMy family has sustained me through this time of many transitions. Barbara Phillips has been a caring presence and she has often put her own plans on hold and traveled a long way to help look after the children. Megha Amrith has been many things\u2014an intellectual inspiration, a listening ear, a trusted source of advice, a wonderful travel companion, and a loving and enthusiastic aunt to my children. Over these years we are lucky to have added Andreas Werner and his parents to our extended family. Jairam and Shantha Amrith have given me everything\u2014and they continue to support me in everything that I do.\n\nNothing I have ever achieved would have been possible without the love and generosity of Ruth Coffey. The origins of this book lie in conversations I had with Ruth a decade ago, when she was studying for a master's degree in environmental management in London; those conversations have developed over many travels together in South and Southeast Asia. Over the past few years my absence on research trips has placed an additional burden of childcare on her, and she has taken it on with grace while embarking on a judicial career in England, teaching law at Harvard, and pursuing many other projects. She is my anchor and my guiding light.\n\nTheodore was born when this book was in its early stages, and he has traveled many miles with me in search of water stories. I am blessed by the exuberant joy he brings to every day. He started to read just as I was writing the last sentences of this book\u2014he asked me the other day if it was finished, and suggested that my next book should be written for children. Lydia arrived as this project neared its conclusion, and she has made my life richer and more full of wonder. I dedicate this book to them both, with love and gratitude.\n\n**S UNIL AMRITH** is the Mehra Family Professor of South Asian Studies and Professor of History at Harvard University and a 2017 MacArthur Fellow. The prize-winning author of _Crossing the Bay of Bengal_ (2013), as well as several other books and articles, he lives in Cambridge, Massachusetts.\n\n# ALSO BY SUNIL AMRITH\n\n_Crossing the Bay of Bengal: The Furies of Nature and the Fortunes of Migrants_\n\n_Migration and Diaspora in Modern Asia_\n\n_Decolonizing International Health: India and Southeast Asia, 1930\u201365_\n\n# ARCHIVES AND SPECIAL COLLECTIONS\n\nBRITISH LIBRARY, LONDON\n\n_India Office Records_\n\nEast India Company Board of Control Records\n\nEast India Company Factory Records\n\nEconomic Department Records\n\nMarine Records\n\nPublic & Judicial Department Records\n\nPolitical & Secret Records\n\nOfficial Publications Series\n\nNATIONAL MARITIME MUSEUM, GREENWICH\n\nBritish India Steam Navigation Company papers\n\nIrrawaddy Flotilla Company papers [uncataloged]\n\nNATIONAL ARCHIVES OF INDIA, NEW DELHI\n\nDepartment of Revenue and Agriculture\n\nDepartment of Revenue, Agriculture and Commerce\n\nDepartment of Education, Health and Lands\n\nDepartment of Industries and Labour: Meteorology\n\nHome Department, Judicial Branch & Public Branch\n\nMeteorological Department\n\nMinistry of External Affairs\n\nMinistry of Irrigation\n\nMinistry of States\n\nPolitical Department\n\nTAMIL NADU STATE ARCHIVES, CHENNAI\n\nFisheries Department\n\nPublic Works Department\n\nMAHARASHTRA STATE ARCHIVES DEPARTMENT, MUMBAI\n\nPublic Works Department: Irrigation, 1868\u20131909\n\nINDIA METEOROLOGICAL DEPARTMENT, PUNE\n\nMiscellaneous reports, charts, and memoirs\n\nWORLD BANK GROUP ARCHIVES, WASHINGTON, DC\n\nDamodar Multi-Purpose Project, India: Administration, Correspondence, and Negotiations, 1953\u20131957\n\nDrought Prone Areas Project, India: Correspondence, 1973\u20131985\n\nIndus Basin Dispute, General Negotiations and Correspondence, 1949\u20131960\n\nUttar Pradesh Tube Wells Projects, India: Correspondence, 1961\u20131992\n\n# NOTES\n\nCHAPTER ONE: THE SHAPE OF MODERN ASIA\n\n1. E. M. Forster, _A Passage to India_ (London: Edward Arnold, 1924), 116; Pranay Lal, _Indica: A Deep Natural History of the Indian Subcontinent_ (New Delhi: Allen Lane, 2017), 258.\n\n2. Norton Ginsburg, ed., _The Pattern of Asia_ (New York: Prentice-Hall, 1958), 5\u20136.\n\n3. V. Ramanathan et al., \"Atmospheric Brown Clouds: Impact on South Asian Climate and Hydrological Cycle,\" _Proceedings of the National Academy of Sciences_ 102 (2005): 5326\u20135333.\n\n4. Asia Society, _Asia's Next Challenge: Securing the Region's Water Future, a Report by the Leadership Group on Water Security in Asia_ (New York: Asia Society, 2009), 9; C. J. V\u00f6r\u00f6smarty et al., \"Global Threats to Human Water Security and River Biodiversity,\" _Nature_ 467 (September 30, 2010): 555\u2013561; Chris Buckley and Vanessa Piao, \"Rural Water, Not City Smog, May be China's Pollution Nightmare,\" _New York Times,_ April 11, 2016; Malavika Vyawahare, \"Not Just Scarcity, Groundwater Contamination Is India's Hidden Crisis,\" _Hindustan Times,_ March 22, 2017.\n\n5. Intergovernmental Panel on Climate Change, _Climate Change 2014: Impacts, Adaptation, and Vulnerability_ (Geneva: IPCC, 2014); World Bank, _Turn Down the Heat: Climate Extremes, Regional Impacts, and the Case for Resilience_ (Washington, DC: World Bank, 2013); Deepti Singh et al., \"Observed Changes in Extreme Wet and Dry Spells During the South Asian Summer Monsoon,\" _Nature Climate Change_ 4 (2014): 456\u2013461.\n\n6. Benjamin Strauss, \"Coastal Nations, Megacities, Face 20 Feet of Sea Rise,\" Climate Central, July 9, 2015, accessed January 12, 2018, www.climatecentral.org\/news\/nations-megacities-face-20-feet-of-sea-level-rise-19217.\n\n7. Mike Davis, _Ecology of Fear: Los Angeles and the Imagination of Disaster_ (New York: Metropolitan Books, 1998); David Blackbourn, _The Conquest of Nature: Water, Landscape, and the Making of Modern Germany_ (New York: W.W. Norton, 2006).\n\n8. Dipesh Chakrabarty, \"The Climate of History, Four Theses,\" _Critical Inquiry_ 35 (2009): 197\u2013222.\n\n9. Amitav Ghosh, _The Great Derangement: Climate Change and the Unthinkable_ (Chicago: University of Chicago Press, 2016).\n\n10. Karl Wittfogel, _Oriental Despotism: A Comparative Study of Total Power_ (New Haven, CT: Yale University Press, 1957).\n\n11. Much of the scholarship on Chinese environmental history over the long term is surveyed in Mark Elvin, _The Retreat of the Elephants: An Environmental History of China_ (New Haven, CT: Yale University Press, 2004); see also Peter Perdue, _Exhausting the Earth: State and Peasant in Hunan, 1550\u20131850_ (Cambridge, MA: Harvard University Press, 1986), and Kenneth Pomeranz, _The Making of a Hinterland: State, Society, and Economy in Inland North China, 1853\u20131937_ (Berkeley: University of California Press, 1993). On India, key works include Dharma Kumar, _Land and Caste in South India: Agricultural Labour in the Madras Presidency During the Nineteenth Century_ (Cambridge: Cambridge University Press, 1965); C. J. Baker, _An Indian Rural Economy: The Tamilnad Countryside, 1880\u20131955_ (Oxford: Clarendon Press, 1984), and Sugata Bose, _Agrarian Bengal: Economy, Social Structure and Politics, 1919\u20131947_ (Cambridge: Cambridge University Press, 1986).\n\n12. Marc Bloch, _The Historian's Craft,_ trans. Peter Putnam (New York: Knopf, 1953), 26.\n\n13. Fernand Braudel, \"Histoire et sciences sociales: la longue dur\u00e9e,\" _Annales, economies, soci\u00e9t\u00e9s, civilisations_ 13 (1958), 725\u2013753; K. N. Chaudhuri, _Trade and Civilisation in the Indian Ocean: An Economic History from the Rise of Islam to 1750_ (Cambridge: Cambridge University Press, 1985), quotation from Braudel on p. 23.\n\n14. \"Sampling device,\" from Charles E. Rosenberg, _Explaining Epidemics and Other Studies in the History of Medicine_ (Cambridge: Cambridge University Press, 1992), 279.\n\n15. Kenneth Pomeranz, \"The Great Himalayan Watershed: Water Shortages, Mega-Projects and Environmental Politics in China, India, and Southeast Asia,\" _Asia Pacific Journal: Japan Focus_ 7 (2009): 1\u201329.\n\n16. Raj Patel and Jason W. Moore, _A History of the World in Seven Cheap Things: A Guide to Capitalism, Nature, and the Future of the Planet_ (Berkeley: University of California Press, 2017), 44\u201363.\n\n17. Eric Hobsbawm, _The Age of Capital, 1848\u20131875_ (London: Weidenfeld and Nicolson, 1975), 48.\n\n18. Ramachandra Guha, _India After Gandhi: The History of the World's Largest Democracy_ (London: HarperCollins, 2007).\n\n19. Peter D. Clift and R. Alan Plumb, _The Asian Monsoon: Causes, History and Effects_ (Cambridge: Cambridge University Press, 2008), vii.\n\n20. Sunita Narain, Science and Democracy Lecture, Harvard University, December 4, 2017.\n\n21. Gilbert T. Walker, \"The Meteorology of India,\" _Journal of the Royal Society of Arts_ 73 (1925): 838\u2013855, quotation on p. 839; Charles Normand, \"Monsoon Seasonal Forecasting,\" _Quarterly Journal of the Royal Meteorological Society_ 79 (1953): 463\u2013473, 469; A. Turner and H. Annamalai, \"Climate Change and the South Asian Monsoon,\" _Nature Climate Change_ 2 (2012): 587\u2013595.\n\n22. Bob Yirka, \"Earliest Example of Large Hydraulic Enterprise Excavated in China,\" Phys.org, December 5, 2017, accessed December 15, 2017, phys.org\/news\/2017-12-earliest-large-hydraulic-enterprise-excavated.amp.\n\nCHAPTER TWO: WATER AND EMPIRE\n\n1. \"Madras Government request the Court of Directors' sanction for the expenditure of 5000 rupees on deepening the Pamban Channel between India and Ceylon,\" October 1833\u2013March 1835, Board's Collections: British Library [hereafter BL] India Office Records [hereafter IOR], F\/4\/1523\/60207.\n\n2. H. Morris, \"A Descriptive and Historical Account of the Godavery District in the Presidency of Madras,\" (1878): BL, IOR, V\/27\/66\/18.\n\n3. E. Halley, \"An Historical Account of the Trade Winds and the Monsoons, Observable in the Seas Between and Near the Tropicks, with an attempt to assign the physical cause of the sail winds,\" _Philosophical Transactions of the Royal Society of London_ 16 (1686): 153\u2013168.\n\n4. The clearest explanations of the monsoon can be found in Peter J. Webster, \"Monsoons,\" _Scientific American_ 245 (1981): 108\u2013119; and Peter D. Clift and R. Alan Plumb, _The Asian Monsoon: Causes, History and Effects_ (Cambridge: Cambridge University Press, 2008).\n\n5. Jos Gommans, _Mughal Warfare: Indian Frontiers and Highroads to Empire, 1500\u20131700_ (London: Routledge, 2003), chapter 1; Jos Gommans, \"The Silent Frontier of South Asia, c. AD 1100\u20131800,\" _Journal of World History_ 9, no. 1 (1998): 1\u201323.\n\n6. Victor Lieberman, _Strange Parallels: Southeast Asia in Global Context, c. 800\u20131830,_ vol. 2, _Mainland Mirrors: Europe, Japan, China, South Asia, and the Islands_ (Cambridge: Cambridge University Press, 2009), 632\u2013636.\n\n7. Diana Eck, \"Ganga: The Goddess in Hindu Sacred Geography,\" in _The Divine Consort: Radha and the Goddesses of India,_ ed. John Hawley and Donna Wulff (Boston: Beacon Press, 1982), 166\u2013183; Diana Eck, _India: A Sacred Geography_ (New York: Harmony, 2012); Anne Feldhaus, _Connected Places: Religion, Pilgrimage and the Geographical Imagination in India_ (New York: Palgrave Macmillan, 2003).\n\n8. Karl Wittfogel, _Oriental Despotism: A Comparative Study of Total Power_ (New Haven, CT: Yale University Press, 1957); Kathleen D. Morrison, \"Dharmic Projects, Imperial Reservoirs, and New Temples of India: An Historical Perspective on Dams in India,\" _Conservation and Society_ 8 (2010): 182\u2013195; Kathleen D. Morrison, _Daroji Valley: Landscape, Place, and the Making of a Dryland Reservoir System_ (New Delhi: Manohar Press, 2009).\n\n9. Peter Jackson, _The Delhi Sultanate_ (Cambridge: Cambridge University Press, 1999); Sunil Kumar, _The Emergence of the Delhi Sultanate, 1192\u20131286_ (New Delhi: Oxford University Press, 2007).\n\n10. James L. Wescoat Jr., \"Early Water Systems in Mughal India,\" _Environmental Design: Journal of the Islamic Environmental Design Research Centre_ 2, ed. Attilo Petruccioli (Rome: Carucci Editions, 1985), 51\u201357.\n\n11. _Babur Nama,_ trans. Annette Susannah Beveridge (New Delhi: Penguin, 2006), 93, 264\u2013265. For later discussion of Mughal-era irrigation, see Irfan Habib, _The Agrarian System of Mughal India (1556\u20131707)_ (London: Asia Publishing House, 1963), 24\u201336.\n\n12. Muzaffar Alam and Sanjay Subrahmanyam, eds., _The Mughal State, 1526\u20131750_ (New Delhi: Oxford University Press, 1998); Lieberman, _Strange Parallels,_ 636\u2013637.\n\n13. John F. Richards, _The Mughal Empire_ (Cambridge: Cambridge University Press, 1995); Lieberman, _Strange Parallels_.\n\n14. Gommans, _Mughal Warfare_.\n\n15. Gommans, _Mughal Warfare_ ; _The Akbarnama of Abu'l Fazl_ , vol. 3, trans. H. Beveridge (Calcutta: Asiatic Society, 1910), 135\u2013136.\n\n16. Irfan Habib, _An Atlas of the Mughal Empire_ (New Delhi: Oxford University Press, 1982), reference to Kanauj on plate 8B.\n\n17. Prasannan Parthasarathi and Giorgio Riello, \"The Indian Ocean in the Long Eighteenth Century,\" _Eighteenth-Century Studies_ 48 (Fall 2014): 1\u201319.\n\n18. Anthony Reid, \"Southeast Asian Consumption of Indian and British Cotton Cloth, 1600\u20131850,\" in _How India Clothed the World: The World of South Asian Textiles, 1500\u20131850,_ ed. Giorgio Riello and Tirthankar Roy (Leiden: Brill, 2009), 31\u201352.\n\n19. Armando Coresao, trans., _The Suma Oriental of Tom\u00e9 Pires_ (London: Hakluyt Society, 1944), 3:92\u201393.\n\n20. Sinappah Arasaratnam, _Merchants, Companies and Commerce on the Coromandel Coast 1650\u20131740_ (New Delhi: Oxford University Press, 1986), 98\u201399.\n\n21. Sanjay Subrahmanyam and C. A. Bayly, \"Portfolio Capitalists and the Political Economy of Early Modern India,\" _Indian Economic and Social History Review_ 25 (1988): 401\u2013424; Richards, _The Mughal Empire_ ; Alam and Subrahmanyam, eds., _The Mughal State_ ; \"giant pump\" from Lieberman, _Strange Parallels,_ 694\u2013696.\n\n22. David Ludden, \"History Outside Civilisation and the Mobility of South Asia,\" _South Asia_ 17 (1994): 1\u201323.\n\n23. H. V. Bowen, John McAleer, and Robert J. Blyth, _Monsoon Traders: The Maritime World of the East India Company_ (London: Scala, 2011).\n\n24. C. A. Bayly, _Indian Society and the Making of the British Empire_ (Cambridge: Cambridge University Press, 1988).\n\n25. C. A. Bayly, _Rulers, Townsmen and Bazaars: North Indian Society in the Age of British Expansion_ (Cambridge: Cambridge University Press, 1983), 1; Murari Kumar Jha, \"The Rhythms of the Economy and Navigation along the Ganga River,\" in _From the Bay of Bengal to the South China Sea,_ ed. Satish Chandra and Himanshu Prabha Ray (New Delhi: Manohar, 2013), 221\u2013247.\n\n26. James Rennell, _Memoir of a Map of Hindoostan; or, The Mogul Empire_ (London: M. Brown, 1788), 280.\n\n27. T. F. Robinson, \"William Roxburgh, 1751\u20131815: The Founding Father of Indian Botany\" (PhD dissertation, University of Edinburgh, 2003).\n\n28. \"A Meteorological Diary, & c. Kept at Fort St. George in the East Indies. By Mr William Roxburgh, Assistant-Surgeon to the Hospital at Said Fort. Communicated by Sir John Pringle, Bart. P.R.S.,\" _Philosophical Transactions of the Royal Society of London_ 68 (1778): 180\u2013193.\n\n29. Robinson, \"William Roxburgh.\"\n\n30. Alexander Dalrymple, ed., _Oriental Repertory_ (London: G. Biggs, 1793\u20131797), 2:58\u201359.\n\n31. Record of proceedings at Fort Saint George, February 8, 1793, Madras Public Consultations, January 28\u2013March 8, 1793, BL IOR, P\/241\/37.\n\n32. Letter from William Roxburgh to Joseph Banks, August 30, 1791: BL, IOR, European Manuscripts, EUR\/K148, ff. 243\u201347; Andrew Ross cited in Robinson, \"William Roxburgh,\" 224n4.\n\n33. William Roxburgh, \"Remarks on the Land Winds and their Causes,\" _Transactions of the Medical Society of London_ (1810), 189\u2013211.\n\n34. Richard H. Grove, _Green Imperialism: Colonial Expansion, Tropical Island Edens and the Origins of Environmentalism, 1600\u20131860_ (Cambridge: Cambridge University Press, 1995), 399\u2013400.\n\n35. Richard Drayton, _Nature's Government: Science, Imperial Britain, and the \"Improvement\" of the World_ (New Haven, CT: Yale University Press, 2000).\n\n36. Letter from William Roxburgh to Andrew Ross, February 14, 1793, in Dalrymple, _Oriental Repertory,_ 73.\n\n37. Dalrymple, _Oriental Repertory,_ 56; Robinson, \"William Roxburgh,\" quotations on pp. 237\u2013238, 241.\n\n38. Jurgen Osterhammel, _The Transformation of the World: A Global History of the Nineteenth Century_ , trans. Patrick Camiller (Princeton, NJ: Princeton University Press, 2014), 656; E. A. Wrigley, _Energy and the English Industrial Revolution_ (Cambridge: Cambridge University Press, 2010), 91\u2013112; Terje Tvedt, _Water and Society: Changing Perceptions of Societal and Historical Development_ (London: I.B. Tauris 2016), 19\u201344.\n\n39. Joseph Dalton Hooker, _Himalayan Journals: Notes of a Naturalist in Bengal, the Sikkim and Nepal Himalayas, the Khasia Mountains & c._ (London: J. Murray, 1854), 1:87.\n\n40. James Ranald Martin, _Notes on the Medical Topography of Calcutta_ (Calcutta: G.H. Huttmann, 1837), 90\u201393; on climate and racial thinking, see David Arnold, _The Tropics and the Traveling Gaze: India, Landscape and Science, 1800\u20131856_ (Seattle: University of Washington Press, 2005).\n\n41. Arthur Thomas Cotton, \"Report on the Irrigation, & c., of Rajahmundry District\" [1844], House of Commons Sessional Papers, XLI (1850), quotation on pp. 4\u20135 of Cotton's report.\n\n42. Cotton, \"Report on the Irrigation, & c., of Rajahmundry District,\" 13.\n\n43. Arthur Cotton, \"On a Communication between India and China by the line of the Burhampooter and Yang-tsze,\" _Journal of the Royal Geographical Society_ 37 (1867): 231\u2013239, quotation on p. 232.\n\n44. For a detailed account of their rivalry, see Alan Robertson, _Epic Engineering: Great Canals and Barrages of Victorian India,_ ed. Jeremy Berkoff (Melrose, UK: Beechwood Melrose Publishing, 2013).\n\n45. Anthony Acciavatti, _Ganges Water Machine: Designing New India's Ancient River_ (New York: Applied Research and Design Publishing, 2015), 120.\n\n46. James L. Wescoat Jr., \"The Water and Landscape Heritage of Mughal Delhi,\" accessed June 22, 2016, www.delhiheritagecity.org\/pdfhtml\/mughal\/JW-the-water-and-landscape-heritage-of-mughal-delhi-Oct8.pdf.\n\n47. Henry Yule, \"A Canal Act of the Emperor Akbar, with some notes and remarks on the History of the Western Jumna Canals,\" _Journal of the Asiatic Society of Bengal_ 15 (1846).\n\n48. Proby Cautley, \"On the Use of Wells, etc. in Foundations as Practiced by the Natives of the Northern Doab,\" _Journal of the Asiatic Society of Bengal_ 8 (1839): 327\u2013340.\n\n49. Proby Cautley, _Report on the Central Doab Canal,_ BL, IOR, V\/27\/733\/3\/1.\n\n50. G. W. MacGeorge, _Ways and Works in India: Being an Account of Public Works in that Country from the Earliest Times up to the Present Day_ (London: Archibald Constable & Company, 1894), 153.\n\n51. B. H. Tremenheere, \"On Public Works in the Bengal Presidency,\" _Minutes of Proceedings of the Institute of Civil Engineers_ 17 (1858): 483\u2013513.\n\n52. Proby T. Cautley, _Report on the Ganges Canal Works: from their Commencement until the Opening of the Canal in 1854,_ 3 vols. (London: Smith, Elder & Co, 1860), 3:2.\n\n53. Jan Lucassen, \"The Brickmakers' Strike on the Ganges Canal in 1848\u20131849,\" _International Review of Social History_ 51 (2006) supplement: 47\u201383.\n\n54. Ganges Canal Committee, _A Short Account of the Ganges Canal_ (Calcutta: Ganges Canal Committee, 1854), 3; the Hindi version was published as _Ganga Ki Nahar Ka Sankshepa Varnana_ (Agra: Ganges Canal Committee, 1854).\n\n55. \"Short Account of the Ganges Canal,\" _North American Review,_ October 1855, 81.\n\n56. David Washbrook, \"Law, State and Agrarian Society in Colonial India,\" _Modern Asian Studies_ 15, no. 3 (1981): 648\u2013721; Mayo quoted in David Ludden, _India and South Asia: A Short History_ (London: Oneworld, 2014), 150.\n\n57. David Mosse (with assistance from M. Sivan), _The Rule of Water: Statecraft, Ecology, and Collective Action in South India_ (New Delhi: Oxford University Press, 2003), 29; Terje Tvedt, \"'Water Systems': Environmental History and the Deconstruction of Nature,\" _Environment and History_ 16 (2010): 143\u2013166, quotation on p. 160.\n\n58. Amitav Ghosh, \"Of Fanas and Forecastles: The Indian Ocean and Some Lost Languages of the Age of Sail,\" _Economic and Political Weekly_ , June 21, 2008, 56\u201362.\n\n59. Henry T. Bernstein, _Steamboats on the Ganges: An Exploration in the History of India's Modernization through Science and Technology_ (Bombay: Orient Longmans, 1960), 7\u20138, 13\u201316.\n\n60. \"Impediments to the Traffic on the Ganges and Jumna, arising from the number of customs chokeys,\" (February 5, 1833), BL, IOR, F\/4\/1506.\n\n61. Bernstein, _Steamboats,_ 28\u201331.\n\n62. Bernstein, _Steamboats,_ 84\u201399.\n\n63. David Arnold, _Science, Technology and Medicine in Colonial India_ (Cambridge: Cambridge University Press, 2000).\n\n64. Bernstein, _Steamboats,_ 99.\n\n65. William Cronon, _Nature's Metropolis: Chicago and the Great West_ (New York: W.W. Norton, 1991), 74; Richard White, _Railroaded: The Transcontinentals and the Making of Modern America_ (New York: W.W. Norton, 2011).\n\n66. Ian Kerr, _Engines of Change: The Railroads That Made India_ (Westport, CT: Praeger, 2007).\n\n67. Minute by Lord Dalhousie to the Court of Directors, April 20, 1853, in _Railway Construction in India: Select Documents,_ ed. S. Settar (New Delhi: Indian Council of Historical Research, 1999), 2:23\u201357.\n\n68. MacGeorge, _Ways and Works,_ 221.\n\n69. Karl Marx, \"The Future Results of the British Rule in India,\" _New York Daily Tribune,_ August 8, 1853; Edwin Merrall, _A Letter to Col. Arthur Cotton, upon the Introduction of Railways in India upon the English Plan_ (London: E. Wilson, 1860), quotations on pp. 8 and 47.\n\n70. MacGeorge, _Ways and Works,_ 422\u2013426, quotation on p. 426; Ian J. Kerr, _Engines of Change: The Railroads That Made India_ (Westport, CT: Praeger, 2007).\n\n71. C. H. Lushington, quoted in Tarasankar Banerjee, _Internal Market of India, 1834\u20131900_ (Calcutta: Academic Publishers, 1966), 90\u201391.\n\n72. Quotation from Banerjee, _Internal Market,_ 323; Dave Donaldson, \"Railroads of the Raj: Estimating the Impact of Transportation Infrastructure,\" (working paper, MIT\/NBER, 2010); Robin Burgess and Dave Donaldson, \"Railroads the Demise of Famine in Colonial India,\" (working paper, LSE\/MIT\/NBER, 2012).\n\n73. On immobility see Joya Chatterji, \"On Being Stuck in Bengal: Immobility in the 'Age of Mobility,'\" _Modern Asian Studies_ 51 (2017): 511\u2013541; quotation from Arnold, _Science, Technology and Medicine,_ 110.\n\n74. MacGeorge, _Ways and Works,_ 220\u2013221, 358; Madhav Rao, cited in Kerr, _Engines of Change,_ 4.\n\n75. Arnold, _Science, Technology, and Medicine_.\n\n76. MacGeorge, _Ways and Works,_ 328\u2013331; Kerr, _Engines of Change,_ 47\u201351.\n\n77. Rudyard Kipling, \"The Bridge Builders,\" in _The Day's Work_ (New York: Doubleday & McClure, 1899), 3\u201350.\n\n78. W. W. Hunter, _Statistical Account of Bengal_ (London: Trubner & Co., 1877), 14:31. On railways and the ecology of malaria: Iftekhar Iqbal, _The Bengal Delta: Ecology, State, and Social Change, 1840\u20131943_ (Basingstoke: Palgrave\/Macmillan, 2010), 117\u2013139.\n\n79. _New York Observer and Chronicle,_ December 8, 1864.\n\n80. J. E. Gastrell and Henry F. Blanford, _Report on the Calcutta Cyclone of the 5th of October 1864_ (Calcutta: O.T. Cutter, 1866), 11, 31\u201332.\n\n81. Gastrell and Blanford, _Calcutta Cyclone,_ 139.\n\n82. Gastrell and Blanford, _Calcutta Cyclone,_ 109, 127.\n\n83. Henry Piddington, _The Sailor's Horn-Book for the Law of Storms_ (London: John Wiley, 1848); description of \"storm wave\" in Henry Piddington, _The Horn-Book of Storms for the Indian and China Seas_ (Calcutta: Bishop's College Press, 1844), 20; Piddington's inspiration was William Reid, _An Attempt to Develop the Law of Storms By Means of Facts_ (London: John Weale, 1838).\n\n84. Gastrell and Blanford, _Calcutta Cyclone,_ 11\u201313.\n\n85. Gastrell and Blanford, _Calcutta Cyclone,_ 4.\n\n86. Gastrell and Blanford, _Calcutta Cyclone,_ 70.\n\n87. Gastrell and Blanford, _Calcutta Cyclone,_ 14\u201315.\n\n88. Gastrell and Blanford, _Calcutta Cyclone,_ 108.\n\n89. Hooker, _Himalayan Journals,_ 1:97.\n\nCHAPTER THREE: THIS PARCHED LAND\n\n1. Mike Davis, _Late Victorian Holocausts: El Ni\u00f1o Famines and the Making of the Third World_ (London: Verso, 2001); on the China famine, see Kathryn Edgerton-Tarpley, _Tears from Iron: Cultural Responses to Famine in Nineteenth-Century China_ (Berkeley: University of California Press, 2008).\n\n2. _The Constitution of the Poona Sarvajanik Sabha and its Rules_ (Poona, 1870); quote from S. R. Mehrotra, \"The Poona Sarvajanik Sabha: The Early Phase (1870\u20131880),\" _Indian Economic and Social History Review_ 9 (1969): 293\u2013321; C. A. Bayly, _Recovering Liberties: Indian Thought in the Age of Liberalism and Empire_ (Cambridge: Cambridge University Press, 2012).\n\n3. Poona Sarvajanik Sabha, \"Famine Narrative, No. 1,\" October 21, 1876.\n\n4. _Medical and Sanitary Report of the Native Army of Madras, for the Year 1875_ (Madras: Government Press, 1876), 49; _Report of the Indian Famine Commission_ , 2 vols. (London: HM Stationery Office, 1880); William Digby, _The Famine Campaign in Southern India_ (London: Longmans, Green & Co., 1878), 1:6; W. W. Hunter, _The Indian Empire: Its People, History and Products_ (London: Trubner & Co., 1886), 542. For a perspective from current climate science: Edward R. Cook et al., \"Asian Monsoon Failure and Megadrought over the Last Millennium,\" _Science_ 328 (2010): 486\u2013489.\n\n5. Arup Maharatna, \"Regional Variation in Demographic Consequences of Famines in Late 19th Century and Early 20th Century India,\" _Economic and Political Weekly_ 29 (June 4, 1994): 1399\u20131410; Arup Maharatna, _The Demography of Famines: An Indian Historical Perspective_ (New Delhi: Oxford University Press, 1996); Tim Dyson, \"On the Demography of South Asian Famines, I,\" _Population Studies_ 45 (1991): 5\u201325.\n\n6. Richard Strachey, \"Physical Causes of Indian Famines,\" May 18, 1877, _Notices of the Proceedings of the Meetings of the Members of the Royal Institution of Great Britain_ 8 (1879): 407\u2013426.\n\n7. \"The Famine, Letter from the Affected Districts,\" _The Examiner,_ March 24, 1877, 363.\n\n8. Digby, _Famine Campaign,_ 1:67\u201368, 1:155\u2013156.\n\n9. Digby, _Famine Campaign,_ 1:174\u2013175.\n\n10. J. Norman Lockyer and W. Hunter, \"Sun-Spots and Famines,\" _The Nineteenth Century,_ (November 1877), 601.\n\n11. Strachey, \"Physical Causes,\" 411.\n\n12. Mark Elvin, \"Who Was Responsible for the Weather? Moral Meteorology in Late Imperial China,\" _Osiris_ 13 (1998): 213\u2013237; Richard White, _The Republic for Which It Stands: The United States During Reconstruction and the Gilded Age, 1865\u20131896_ (New York: Oxford University Press, 2017), 425\u2013427.\n\n13. \"The Causes of Famine in India,\" _New York Times (NYT),_ August 25, 1878, 6.\n\n14. \"The Famine: Letter from the Affected Districts,\" _The Examiner,_ March 24, 1877, 363.\n\n15. \"Causes of Famine,\" _NYT,_ August 25, 1878.\n\n16. Villiyappa Pillai, _Panchalakshana Thirumukavilasam_ [1899] (Madurai: Sri Ramachandra Press, 1932).\n\n17. W. G. Pedder, \"Famine and Debt in India,\" _The Nineteenth Century_ (September 1877).\n\n18. Digby, _Famine Campaign,_ 1:172\u2013174.\n\n19. Letter from Sarvajanik Sabha Rooms to S. C. Bayley, Additional Secretary to the Government of India, April 1, 1878, in Poona Sarvajanik Sabha, _Journal_ 1 (1878).\n\n20. \"Letter from the Affected Districts\" (1877), 363.\n\n21. Richard H. Grove, _Green Imperialism: Colonial Expansion, Tropical Island Edens and the Origins of Environmentalism, 1600\u20131860_ (Cambridge: Cambridge University Press, 1995); Diana K. Davis, _The Arid Lands: History, Power, Knowledge_ (Cambridge, MA: MIT Press [2016]).\n\n22. Cited in Davis, _Arid Lands,_ 83.\n\n23. \"Causes of Famine,\" _NYT,_ August 25, 1878.\n\n24. Philindus, \"Famines and Floods in India,\" _Macmillan's Magazine,_ November 1, 1877, 236\u2013256; George Perkins Marsh, _Man and Nature, or, Physical Geography As Modified by Human Action_ (New York: Scribner, 1865).\n\n25. \"Famine Narrative no. 1,\" October 21, 1876, in Poona Sarvajanik Sabha, _Journal_ 1.\n\n26. Ramachandra Guha, \"An Early Environmental Debate: The Making of the 1878 Forest Act,\" _Indian Economic and Social History Review_ 27 (1990): 65\u201384.\n\n27. Valentine Ball, \"On Jungle Products Used as Articles of Food in Chota Nagpur,\" in _Jungle Life in India: Or, the Journeys and Journals of an Indian Geologist_ (London: Thos. De La Rue & Co., 1880), 695\u2013699.\n\n28. George Chesney, \"Indian Famines,\" _Nineteenth Century_ 2 (November 1877): 603\u2013620.\n\n29. Digby, _Famine Campaign,_ 1:148\u2013150.\n\n30. Florence Nightingale, \"A Missionary Health Officer in India,\" _Good Words,_ January 20, 1879, 492\u2013496.\n\n31. \"Causes of Famine,\" _NYT,_ August 25, 1878.\n\n32. Dadabhai Naoroji, _Poverty of India_ (London: Vincent Brooks, Day and Son, 1878), 42\u201343, 66.\n\n33. See especially Davis, _Late Victorian Holocausts_.\n\n34. Chandrika Kaul, \"Digby, William (1849\u20131904),\" _Oxford Dictionary of National Biography_.\n\n35. _Report of the Indian Famine Commission, Part 1, Famine Relief_ (London: Stationery Office, 1880): 9\u201310; Jean Dr\u00e8ze, \"Famine Prevention in India\" (working paper 45, WIDER: United Nations University, Helsinki, 1988), 45.\n\n36. \"Wasting Public Money,\" _The Economist,_ July 4, 1874.\n\n37. _The Black Pamphlet of Calcutta. The Famine of 1874. By a Bengal Civilian_ (Calcutta: William Ridgeway, 1876).\n\n38. Digby, _Famine Campaign,_ 1:48.\n\n39. \"Letter from the Affected Districts,\" (1877), 363.\n\n40. Edgerton-Tarpley, _Tears from Iron,_ 152\u2013153.\n\n41. Lance Brennan, \"The Development of the Indian Famine Code,\" in _Famine as a Geographical Phenomenon,_ ed. Bruce Currey and Graeme Hugo (Dordrecht: D. Reidel, 1984), 91\u2013112.\n\n42. Dr\u00e8ze, \"Famine Prevention in India.\"\n\n43. On Caird, see Peter J. Gray, \"Famine and Land in Ireland and India, 1845\u20131880: James Caird and the Political Economy of Hunger,\" _Historical Journal_ 49 (2006): 193\u2013215.\n\n44. W. Stanley Jevons, \"Sun-Spots and Commercial Crises,\" _Nature_ 19 (1879): 588\u2013590; Lockyer and Hunter, \"Sun-Spots and Famines.\"\n\n45. _Report of the Indian Famine Commission, Part 1,_ 7.\n\n46. _Report of the Indian Famine Commission, Part 2, Measures of Protection and Prevention_ (London: Stationery Office, 1880), 9.\n\n47. _Report of the Indian Famine Commission, Part 2,_ 150\u2013151.\n\n48. Ira Klein, \"When the Rains Failed: Famine, Relief, and Mortality in British India,\" _Indian Economic and Social History Review_ 21 (1984): 185\u2013214, quotation on p. 185.\n\n49. _Report of the Indian Famine Commission, 1898_ (London: Stationery Office, 1898); _Report of the Indian Famine Commission, 1901_ (London: Stationery Office, 1901).\n\n50. George Lambert, _India, the Horror-Stricken Empire: Containing a Full Account of the Famine, Plague, and Earthquake of 1896\u20137; including a Complete Narrative of the Relief Work through the Home and Foreign Commission_ (Elkhard, IN: Mennonite Publishing Co., 1898).\n\n51. Cited in C. S. Ramage, _The Great Indian Drought of 1899_ (Boulder, CO: Aspen Institute for Humanistic Studies, 1977), 4.\n\n52. Vaughan Nash, _The Great Famine and Its Causes_ (London: Longmans, Green and Co., 1900), 11\u201314, 18\u201319, 27, 47.\n\n53. Jon Wilson, _The Chaos of Empire: The British Raj and the Conquest of India_ (New York: Public Affairs, 2016), 341\u2013347; Georgina Brewis, \"'Fill Full the Mouth of Famine': Voluntary Action in Famine Relief in India, 1896\u20131901,\" _Modern Asian Studies_ 44 (2010): 887\u2013918.\n\n54. Davis, _Late Victorian Holocausts,_ 22, 9.\n\n55. Sanjoy Chakravorty, _The Price of Land: Acquisition, Conflict, Consequence_ (New Delhi: Oxford University Press, 2013), 88.\n\n56. Robin Burgess and Dave Donaldson, \"Railroads and the Demise of Famine in Colonial India\" (working paper, 2012), available at http:\/\/dave-donaldson.com\/wp-content\/uploads\/2015\/12\/Burgess_Donaldson_Volatility_Paper.pdf.\n\n57. Jurgen Osterhammel, _The Transformation of the World: A Global History of the Nineteenth Century,_ trans. Patrick Camiller (Princeton: Princeton University Press, 2014), 208\u2013209.\n\n58. Mike Davis, _Ecology of Fear: Los Angeles and the Imagination of Disaster_ (New York: Metropolitan Books, 1998).\n\nCHAPTER FOUR: THE AQUEOUS ATMOSPHERE\n\n1. J. Elliott, _Vizagapatam and Backergunge Cyclones_ (Calcutta: Bengal Secretariat Press, 1877), 165\u2013167. The most common spelling of his name is Eliot, which is the variant I use in the text, but in this publication it appears as Elliott.\n\n2. Elliott, _Vizagapatam,_ 158, 182.\n\n3. Elliott, _Vizagapatam,_ 159.\n\n4. Elliott, _Vizagapatam,_ 183.\n\n5. Paul N. Edwards, \"Meteorology as Infrastructural Globalism,\" _Osiris_ 21 (2006): 229\u2013250; on Britain's role, see Katharine Anderson, _Predicting the Weather: Victorians and the Science of Weather Prediction_ (Chicago: University of Chicago Press, 2005).\n\n6. Luke Howard, _Essay On the Modification of Clouds_ [1803], 3rd ed. (London: John Churchill & Sons, 1865); Jean-Baptiste Lamarck, \"Nouvelle d\u00e9finition des termes que j'emploie pour exprimer certaines formes des nuages qu'il importe de distinguer dans l'annotation de l'\u00e9tat du ciel,\" _Annuaire M\u00e9t\u00e9orologique pour l'an XIII de la R\u00e9publique Fran\u00e7aise_ 3 (1805): 112\u2013133; H. Hildebrandsson, A. Riggenbach, and L. Teisserenc de Bort, eds., _Atlas International des Nuages_ (Paris: IMO, 1896); for further discussion, see Lorraine Daston, \"Cloud Physiognomy: Describing the Indescribable,\" _Representations_ 135 (2016): 45\u201371, and Richard Hamblyn, _Clouds: Nature and Culture_ (London: Reaktion, 2017).\n\n7. University of Madras, _Tamil Lexicon_ (Madras: University of Madras, 1924\u20131936), 219, 1680; William Crooke, _A Glossary of North Indian Peasant Life_ , ed. Shahid Amin (New Delhi: Oxford University Press, 1989), Appendix D, \"A Calendar of Agricultural Sayings\"; C. A. Benson, \"Tamil Sayings and Proverbs on Agriculture,\" _Bulletin_ , Department of Agriculture, Madras No. 29, New Series (1933), paragraphs 144, 163, 168, 213, 311. I have modified some of the translations from the Tamil.\n\n8. Henry F. Blanford, \"Winds of Northern India, in Relation to the Temperature and Vapour-Constituent of the Atmosphere,\" _Philosophical Transactions of the Royal Society_ 164 (1874): 563.\n\n9. India, Meteorological Department, _Report on the Meteorology of India in 1876_ (Calcutta: Office of the Superintendent of Government, 1877).\n\n10. India, Meteorological Department, _Report on the Meteorology of India in 1877_ (Calcutta: Office of the Superintendent of the Government, 1878).\n\n11. \"Administrative Report of the Meteorological Reporter to the Government of India, 1884\u201385,\" BL, IOR, V\/24\/3022, quotations on pp. 5\u201314.\n\n12. Kapil Raj, _Relocating Modern Science: Circulation and the Construction of Knowledge in South Asia and Europe, 1650\u20131900_ (Basingstoke: Palgrave\/Macmillan, 2007); Mandy Bailey, \"Women and the RAS: 100 Years of Fellowship,\" _Astronomy & Geophysics_ 57 (February 2016): 19\u201321.\n\n13. _Memoirs of Ruchi Ram Sahni: Pioneer of Science Popularisation in Punjab,_ ed. Narender K. Sehgal and Subodh Mahanti (New Delhi: Vigyan Prasar, 1997), 15\u201317.\n\n14. _Memoirs of Ruchi Ram Sahni,_ 16.\n\n15. Henry F. Blanford, _Meteorology of India: Being the Second Part of the Indian Meteorologist's Vade-Mecum_ (Calcutta: Government Printer, 1877), 48.\n\n16. Henry F. Blanford, _A Practical Guide to the Climates and Weather of India, Ceylon and Burmah and the Storms of the Indian Seas_ (London: MacMillan and Co., 1889), 42.\n\n17. Blanford, _Meteorology of India,_ 144\u2013145.\n\n18. Blanford, _Meteorology of India,_ 48.\n\n19. Blanford, _Practical Guide,_ 64.\n\n20. Henry F. Blanford, _The Rainfall of India,_ India Meteorological Memoirs vol. 3 (Calcutta: Government Printer, 1886\u20131888), 76.\n\n21. Blanford, _Rainfall of India,_ 79\u201381.\n\n22. Henry F. Blanford, \"On the Connexion of the Himalaya Snowfall with Dry Winds and Seasons of Drought in India,\" _Proceedings of the Royal Society of London_ 37 (1884): 3\u201322.\n\n23. List of library holdings in \"Administration Report of the Meteorological Department in Western India for the year 1880\u201381,\" BL, IOR, V\/24\/3023.\n\n24. Blanford, \"On the Connexion of the Himalaya Snowfall.\"\n\n25. Richard Grove, \"The East India Company, the Raj and El Ni\u00f1o: The Critical Role Played by Colonial Scientists in Establishing the Mechanisms of Global Climate Teleconnections, 1770\u20131930,\" in _Nature and the Orient: The Environmental History of South and Southeast Asia,_ ed. Richard Grove, Vineeta Damodaran, and Satpal Sangwan (New Delhi: Oxford University Press, 1998), 301\u2013323.\n\n26. John Eliot, _Climatological Atlas of India_ (Edinburgh: J. Bartholomew & Co., 1906), xi\u2013xii.\n\n27. _Dictionary of National Biography,_ 1912 supplement (London: Smith, Elder & Co., 1912).\n\n28. _Memoirs of Ruchi Ram Sahni,_ 23.\n\n29. John Eliot, _Handbook of Cyclonic Storms in the Bay of Bengal_ (Calcutta: Bengal Secretariat Press, 1890).\n\n30. Rev. Jose Algu\u00e9, _The Cyclones of the Far East,_ 2nd ed. (Manila: Philippines Weather Bureau, 1904), 219.\n\n31. Robert Hart, \"Documents Relating to 1. The Establishment of Meteorological Stations in China; and 2. Proposals for Co-operation in the Publication of Meteorological Observations and Exchange of Weather News by Telegraph along the Pacific Coast of Asia\" [1874], published in _Chinese Maritime Customs Project Occasional Papers,_ no. 3, ed. Robert Bickers and Catherine Ladds (Bristol: University of Bristol, 2008); for further discussion, see Robert Bickers, \"'Throwing Light on Natural Laws': Meteorology on the China Coast, 1869\u20131912,\" in _Treaty Ports in Modern China: Law, Land, and Power,_ ed. Robert Bickers and Isabella Jackson (London: Routledge, 2016), 179\u2013200.\n\n32. Agust\u00edn Udi\u00e1s, \"Meteorology of the Observatories of the Society of Jesus,\" _Archivum Historicum Societatis Iesu_ 65 (1996): 157\u2013170; James Francis Warren, \"Scientific Superman: Father Jos\u00e9 Algu\u00e9, Jesuit Meteorology, and the Philippines under American Rule, 1897\u20131924,\" in _Colonial Crucible: Empire in the Making of the Modern American State,_ ed. Alfred W. McCoy and Francisco A. Scarano (Madison: University of Wisconsin Press, 2009), 508\u2013522.\n\n33. _Cosmos,_ no. 1091 (1906), 717\u2013719: cited in Warren, \"Scientific Superman,\" 515.\n\n34. Algu\u00e9, _Cyclones,_ 3.\n\n35. Algu\u00e9, _Cyclones,_ 219\u2013229; Eliot, _Handbook of Cyclonic Storms_.\n\n36. Charles Normand, \"Seasonal Monsoon Forecasting,\" _Quarterly Journal of the Royal Meteorological Society_ 79 (1953): 463\u2013473; Eliot, _Climatological Atlas_ , xiii.\n\n37. Eliot, _Climatological Atlas_.\n\n38. _The Imperial Gazetteer of India,_ rev. ed. (Oxford: Clarendon Press, 1901), 5\u20136, 19\u201322.\n\n39. Sven Hedin, _Trans-Himalaya: Discoveries and Adventures in Tibet_ (New York: Macmillan, 1909), 1:279, 1:284.\n\n40. Halford Mackinder, \"The Geographical Pivot of History,\" _Geographical Journal_ 4 (1904): 421\u2013444.\n\nCHAPTER FIVE: THE STRUGGLE FOR WATER\n\n1. Italo Calvino, _Invisible Cities,_ trans. William Weaver [1974] (London: Vintage, 1997), 17.\n\n2. Dadabhai Naoroji, _Poverty and Un-British Rule in India_ (London: Swan, Sonnenschein & Co., 1901), 648\u2013653.\n\n3. M. G. Ranade, _Essays in Indian Economics: A Collection of Essays and Speeches_ (Bombay: Thacker & Co., 1899), quotation on p. 66.\n\n4. R. C. Dutt, _Open Letters to Lord Curzon on Famines and Land Assessments in India_ (London: K. Paul, Trench & Tr\u00fcbner, 1900), quotations on pp. 1, 17; R. C. Dutt, _The Economic History of India in the Victorian Age_ (London: K. Paul, Trench & Tr\u00fcbner, 1904), 172.\n\n5. Mary Albright Hollings, _The Life of Colin Scott-Moncrieff_ (London: J. Murray, 1917), 298.\n\n6. Bernard S. Cohn, _Colonialism and Its Forms of Knowledge: The British in India_ (Princeton, NJ: Princeton University Press, 1996); Nicholas B. Dirks, _Castes of Mind: Colonialism and the Making of Modern India_ (Princeton, NJ: Princeton University Press, 2001).\n\n7. _Report of the Indian Irrigation Commission, 1901\u20131903_ (London: HM Stationery Office, 1903), 1:2\u20134; Hollings, _Colin Scott-Moncrieff,_ 299.\n\n8. _Report of the Indian Irrigation Commission_ , 1:5\u201314.\n\n9. _Report of the Indian Irrigation Commission_ , 1:16, 1:124\u2013125.\n\n10. Letter from W. C. Bennett, Director of the Department of Agriculture and Commerce, North-West Provinces and Oudh, to the Secretary, Board of Revenue, NW Provinces, May 27, 1883, Maharashtra State Archives Department, Mumbai [hereafter MSA], Public Works Department: Irrigation Branch [hereafter PWD: Irrigation], v. 406 (1868\u20131890), M167\u2013169.\n\n11. Bennett to Board of Revenue, NW Provinces, May 27, 1883, MSA, PWD: Irrigation, v. 406, M167\u2013169.\n\n12. Letter from W. W. Goodfellow, Superintending Engineer, Belgaum to the Secretary to the Government, Public Works Department, Bombay, October 17, 1883, MSA, PWD: Irrigation, v. 406, M199.\n\n13. V. Sriram, \"Made in Madras,\" _The Hindu,_ November 16, 2014.\n\n14. Letter from A. Chatterton to the Secretary to the Commissioner of Revenue Settlement, Department of Land Records and Agriculture, May 23, 1905, MSA, PWD: Irrigation, v. 272 (1904\u20131909), M164\u2013165.\n\n15. Letter from A. Chatterton to the Director of Agriculture, Poona, July 15, 1906, MSA, PWD: Irrigation, M199\u2013215.\n\n16. \"in reality part of the great desert\": in James Douie, \"The Punjab Canal Colonies,\" lecture delivered at the Royal Society of Arts on May 7, 1914, _Journal of the Royal Society of Arts_ 62 (1914): 611\u2013623, quotation on p. 612; \"irrigation was not designed\": in E. H. Calvert, _The Wealth and Welfare of the Punjab_ (Lahore, 1922), 123.\n\n17. Douie, \"Punjab Canal Colonies,\" 614.\n\n18. Mrinalini Sinha, _Colonial Masculinity: The 'Manly Englishman' and the 'Effeminate Bengali' in the Late Nineteenth Century_ (Manchester: Manchester University Press, 1995).\n\n19. Douie, \"Punjab Canal Colonies,\" 615\u2013616.\n\n20. M.W. Fenton, Financial Commissioner, in 1915: quoted in Indu Agnihotri, \"Ecology, Land Use, and Colonisation: The Canal Colonies of Punjab,\" _Indian Economic and Social History Review_ 33 (1996): 37\u201358.\n\n21. Quoted in David Gilmartin, _Blood and Water: The Indus River Basin in Modern History_ (Berkeley: University of California Press, 2015), 168.\n\n22. Gilmartin, _Blood and Water,_ 175\u2013176.\n\n23. Thomas Gottschang and Diana Lary, _Swallows and Settlers: The Great Migration from North China to Manchuria_ (Ann Arbor: University of Michigan Press, 2000).\n\n24. John F. Richards, _Unending Frontier: An Environmental History of the Early Modern World_ (Berkeley: University of California Press, 2003).\n\n25. Hung Chung Chang, \"Crop Production in China, with Special Reference to Production in Manchuria\" (Master of Science thesis, University of Michigan Agricultural College, 1922).\n\n26. Indu Agnihotri, \"Ecology, Land Use, and Colonisation: The Canal Colonies of Punjab,\" _Indian Economic and Social History Review_ 33 (1996): 37\u201358.\n\n27. Petition from Sakharam Balaji, undated (ca. 1903), MSA, PWD: Irrigation, v. 124, \"Petitions\" (1899\u20131903).\n\n28. \"The humble memorial of the inhabitants of the within mentioned villages in the Belgaum Taluka, of the Belgaum District,\" [undated, 1903], MSA, PWD: Irrigation, v. 124, \"Petitions\" (1899\u20131903).\n\n29. J. Sion, _Asie Des Moussons,_ book 9 of the _G\u00e9ographie Universelle,_ ed. P. Vidal De La Blanche (Paris: Librairie Armand Colin, 1928), 2:363.\n\n30. Matthew Gandy, _The Fabric of Space: Water, Modernity and the Urban Imagination_ (Cambridge, MA: MIT Press, 2014), 114\u2013119; Ira Klein, \"Urban Development and Death: Bombay City, 1870\u20131914,\" _Modern Asian Studies_ 20 (1986): 725\u2013754; Hector Tulloch, _The Water Supply of Bombay_ (Roorkee: Thomason College Press, 1873).\n\n31. Indian Industrial Commission, _Report_ (Calcutta: Government Printing, 1918).\n\n32. Indian Industrial Commission, _Report,_ 57\u201362.\n\n33. M. Visvesvaraya, _Memoirs of My Working Life_ [1951] (New Delhi: Government of India Publications Division, 1960), 9.\n\n34. M. Visvesvaraya, _Reconstructing India_ (London: P.S. King & Son, 1920), 127.\n\n35. Visvesvaraya, _Memoirs,_ 115\u2013124.\n\n36. S. Muthiah, \"Madras Miscellany,\" _The Hindu,_ November 24, 2014.\n\n37. Extract from an Official Note of 1899 on the Desirability of Developing the Agricultural Department, _Madras Fisheries Bureau,_ Bulletin No. 1; F. A. Nicholson, \"The Marine Fisheries of the Madras Presidency,\" paper read at Lahore Industrial Conference, 1909: contained in BL, IOR, V\/25\/550\/3.\n\n38. F. A. Nicholson, _Note on Fisheries in Japan_ (Madras: Government Press, 1907).\n\n39. James Hornell, _A Statistical Analysis of the Fishing Industry of Tuticorin,_ Madras Fisheries Bulletin vol. 11, report no. 3 (Madras: Government Press, 1917). For another perspective, see the insightful discussion of Nicholson and Hornell in Ajantha Subramanian, _Shorelines: Space and Rights in South Asia_ (Stanford: Stanford University Press, 2009), 107\u2013124.\n\n40. Edward Buck, \"Report on the Control and Utilization of Rivers and Draignage for the Fertilization of the Land and Mitigation of Malaria\" (1907), MSA, PWD: Irrigation, v. 267 (1904\u20131909).\n\n41. Christopher J. Baker, _An Indian Rural Economy: The Tamilnad Countryside, 1880\u20131955_ (Oxford: Clarendon Press, 1984); Sugata Bose, _Peasant Labour and Colonial Capital: Rural Bengal Since 1770_ (Cambridge: Cambridge University Press, 1993); the \"tide of indebtedness\" was a phrase used by a colonial official in Dhaka, quoted by Bose; Haruka Yanagisawa, _A Century of Change: Caste and Irrigated Lands in Tamilnadu, 1860s\u20131970s_ (New Delhi: Manohar, 1996). On low yields in rain-fed agriculture, see Latika Chaudhary, Bishnupriya Gupta, Tirthankar Roy, and Anand V. Swamy, eds., _A New Economic History of Colonial India_ (New York: Routledge, 2016), 100\u2013116.\n\n42. Royal Commission on Agriculture in India, _Abridged Report_ (Bombay: Government Central Press, 1928), 5.\n\n43. _Report of the United Provinces Provincial Banking Enquiry Committee, 1929\u201330_ (Allahabad: Government Press, 1930), 2:119, 234.\n\n44. _Report of the United Provinces Provincial Banking Enquiry Committee, 1929\u201330_ (Allahabad: Government Press, 1930), 3:137.\n\n45. Royal Commission on Agriculture in India, _Evidence Taken in the Bombay Presidency,_ vol. 2, part 1 (London: Stationery Office, 1927), 342.\n\n46. J. S. Chakravarti, \"Agricultural Insurance,\" _Agricultural Journal of India_ 12 (1917): 436\u2013441, quotations on pp. 436\u2013437; J. S. Chakravarti, _Agricultural Insurance: A Practical Scheme Suited to Indian Conditions_ (Bangalore: Government Press of Mysore, 1920). The comment on Chakravarti's prescience is from P. K. Mishra, \"Is Rainfall Insurance a New Idea? Pioneering Work Revisited,\" _Economic and Political Weekly_ 30 (1995): A84\u2013A88.\n\n47. Indian Industrial Commission, _Report,_ 4.\n\n48. P. A. Sheppard, revised by Isabel Falconer, \"Walker, Gilbert Thomas,\" _Oxford Dictionary of National Biography_ , https:\/\/doi.org\/10.1093\/ref:odnb\/36692.\n\n49. G. I. Taylor, \"Gilbert Thomas Walker, 1868\u20131958,\" _Biographical Memoirs of Fellows of the Royal Society_ 8 (1962): 166\u2013174.\n\n50. D. R. Sikka, \"The Role of the India Meteorological Department, 1875\u20131947,\" in _Science and Modern India: An Institutional History, c.1784\u20131947,_ ed. Uma Das Gupta (New Delhi: Pearson, 2010), chapter 14.\n\n51. Gilbert T. Walker, \"The Meteorology of India,\" _Journal of the Royal Society of Arts_ 73 (July 1925): 838\u2013855, quotation on p. 839.\n\n52. Gilbert T. Walker, \"Correlation in Seasonal Variations of Weather, VIII. A Preliminary Study of World-Weather,\" _Memoirs of the Indian Meteorological Department_ 24, part 4 (1923): 75\u2013131, quotation on p. 75.\n\n53. Michael Bardecki, \"Walker Circulation,\" in _Encyclopedia of Global Warming and Climate Change_ , ed. S. George Philander, 2nd ed. (New York: Sage, 2005), 1:1073.\n\n54. Gilbert T. Walker, \"On the Meteorological Evidence for Supposed Changes of Climate in India,\" _Indian Meteorological Memoirs_ 21, part 1 (1910): 1\u201321.\n\n55. Gilbert T. Walker, \"Correlation in Seasonal Variations of Weather, II,\" _Indian Meteorological Memoirs_ 21, part 2 (1910): 21\u201345, quoted in J. M. Walker, \"Pen Portraits of Past Presidents\u2014Sir Gilbert Walker, CSI, ScD, MA, FRS,\" _Weather_ 52 (1997): 217\u2013220, quotation on p. 219.\n\n56. Richard W. Katz, \"Sir Gilbert Walker and a Connection Between El Ni\u00f1o and Statistics,\" _Statistical Science_ 17 (2002): 97\u2013112.\n\n57. Walker, \"Correlation, VIII\" (1923), 109.\n\n58. Walker, \"Meteorology of India,\" 843.\n\n59. Gilbert T. Walker, \"Correlation in the Seasonal Variations of Weather,\" _Quarterly Journal of the Royal Meteorological Society_ 44 (1918): 223\u2013234; Gilbert T. Walker, \"The Atlantic Ocean,\" _Quarterly Journal of the Royal Meteorological Society_ 53 (1927): 71\u2013113, quotations on p. 113.\n\n60. Priya Satia, \"Developing Iraq: Britain, India and the Redemption of Technology in the First World War,\" _Past and Present_ 197 (2007): 211\u2013255.\n\n61. Walker, \"Meteorology of India,\" 849.\n\n62. Taylor, \"Gilbert Thomas Walker,\" 171.\n\n63. Gilbert T. Walker, Review of _Climate Through the Ages: A Study of Climatic Factors and Climatic Variations_ by C.W.P. Brooks, _Quarterly Journal of the Royal Meteorological Society_ 53 (1927): 321\u2013323.\n\n64. Gilbert T. Walker, \"On Monsoon Forecasting in India,\" _Bulletin of the American Meteorological Society_ 19 (1938): 297\u2013299.\n\n65. Charles Normand, \"Monsoon Seasonal Forecasting,\" _Quarterly Journal of the Royal Meteorological Society_ 79 (1953): 463\u201373, quotations on p. 469.\n\n66. Walker, \"Meteorology of India,\" 838\u2013855, quotation on p. 848.\n\n67. Sikka, \"India Meteorological Department\"; the quotations from Walker and Field draw on their papers deposited in the office of the director-general of meteorology in India, which were not available for consultation by researchers.\n\n68. Calvino, _Invisible Cities_ , 17.\n\nCHAPTER SIX: WATER AND FREEDOM\n\n1. M. K. Gandhi, _Hind Swaraj and Other Writings,_ ed. Anthony J. Parel (Cambridge: Cambridge University Press, 1997), 131.\n\n2. Jawaharlal Nehru to B. J. K. Hallowes (Deputy Commissioner, Allahabad, and President of the Famine Relief Fund of Gonda), June 26, 1929, in _The Essential Writings of Jawaharlal Nehru,_ ed. S. Gopal and Uma Iyengar (Delhi: Oxford University Press, 2003), 12.\n\n3. Jawaharlal Nehru, \"The Basis of Society,\" Presidential Address to Bombay Youth Congress, Poona, December 12, 1928, in _Essential Writings of Jawaharlal Nehru,_ 1:8\u201310.\n\n4. Sun Yat-sen, _The International Development of China_ (New York: Knickerbocker Press, 1922).\n\n5. Sun Yat-sen, \"Third Principle of the People: People's Livelihood,\" cited in Deirdre Chetham, _Before the Deluge: The Vanishing World of the Yangtze's Three Gorges_ (New York: Palgrave Macmillan, 2002), 117.\n\n6. David A. Pietz, _The Yellow River: The Problem of Water in Modern China_ (Cambridge, MA: Harvard University Press, 2015), chapter 3; quotations on pp. 93\u201394.\n\n7. This account of the Mahad protest draws on Christophe Jaffrelot, _Dr Ambedkar and Untouchability: Analysing and Fighting Caste_ (London: Hurst and Company, 2005), 47\u201348.\n\n8. Sudipta Kaviraj, \"Ideas of Freedom in Modern India,\" in _The Idea of Freedom in Asia and Africa,_ ed. Robert H. Taylor (Stanford, CA: Stanford University Press, 2002), 120\u2013121.\n\n9. M. K. Gandhi, \"Salt Tax,\" _Young India,_ February 27, 1930, in _Collected Works of Mahatma Gandhi_ (New Delhi: Government of India Publications Division, 1970), 48:499\u2013500.\n\n10. Robert Carter and Erin McCarthy, \"Watsuji Tetsur\u00f4,\" in _The Stanford Encyclopedia of Philosophy,_ ed. Edward N. Zalta (Winter 2014 Edition), http:\/\/plato.stanford.edu\/archives\/win2014\/entries\/watsuji-tetsuro\/.\n\n11. Watsuji Tetsuro, _A Climate: A Philosophical Study,_ trans. Geoffrey Bownas (Tokyo: Ministry of Education, 1961), 18\u201320.\n\n12. Watsuji, _Climate,_ 25\u201326.\n\n13. Watsuji, _Climate,_ 22\u201323, 38.\n\n14. Watsuji, _Climate,_ 39.\n\n15. Mukerjee makes an appearance in, among others: C. A. Bayly, _Recovering Liberties: Indian Thought in the Age of Liberalism and Empire_ (Cambridge: Cambridge University Press, 2012), and Alison Bashford, _Global Population: History, Geopolitics, and Life on Earth_ (New York: Columbia University Press, 2014).\n\n16. Radhakamal Mukerjee, \"Social Ecology of a River Valley,\" _Sociology and Social Research_ 12 (1927): 341\u2013347, quotations on p. 342; Radhakamal Mukerjee, _Regional Sociology_ (New York and London: Century and Co., 1926); Radhakamal Mukerjee, _The Changing Face of Bengal: A Study in Riverine Economy_ (Calcutta: University of Calcutta Press, 1938).\n\n17. L\u00e9on Metchnikoff, _La Civilisation et Les Grands Fleuves Historiques_ (Paris: Hachette, 1889); Mukerjee, \"Social Ecology,\" quotation on 342; William Willcocks, _Ancient System of Irrigation in Bengal and its Application to Modern Problems_ (Calcutta: University of Calcutta Press, 1930).\n\n18. Mukerjee, \"Social Ecology,\" 345\u2013347.\n\n19. C. J. Baker, \"Economic Reorganization and the Slump in South and Southeast Asia,\" _Comparative Studies in Society and History_ 23 (1981): 325\u2013349.\n\n20. I explore these migrations in detail in Sunil S. Amrith, _Crossing the Bay of Bengal: The Furies of Nature and the Fortunes of Migrants_ (Cambridge, MA: Harvard University Press, 2013), especially chapters 4 and 5.\n\n21. Confidential letter from the Agent of the Government of India in British Malaya to the Government of India, April 3, 1933: NAI, Department of Education, Health and Lands: Overseas, file no. 206-2\/32\u2014L&O.\n\n22. J. S. Furnivall, _Netherlands India_ (Cambridge: Cambridge University Press, 1939), 428.\n\n23. \"World's Largest Dam Opened,\" _The Statesman,_ August 22, 1934.\n\n24. Handwritten memo by \"SA,\" April 30, 1938, appended to the file of correspondence following the Chief Engineer's \"Note on the Beneficial Effects of the Stanley Reservoir to Cauvery Delta Irrigation,\" Tamil Nadu State Archives, Chennai, Government Order 547-I, 27\/2\/1936.\n\n25. Handwritten note in Government of Madras Public Works Department, Government Order number 375, February 24, 1938. Tamil Nadu State Archives, Chennai [TNSA].\n\n26. Pietz, _Yellow River,_ chapter 3.\n\n27. _National Planning Committee No. 2: Being an Abstract of the Proceedings and other Particulars Relating to the National Planning Committee_ (Bombay: K.T. Shah, 1940), 43.\n\n28. \"Burma-China Frontier: Chinese Claim to the Irrawaddy Triangle\" (1933), BL, IOR, L\/P&S\/12\/2231, enclosing William Credner's article in _Eastern Miscellany_ (Shanghai), January 10, 1931; P. M. R. Leonard and V. G. Robert, _Report on the Fourth Expedition to the \"Triangle\" for the Liberation of Slaves_ (Rangoon: Government Printing, 1930), quotation in text from notes in archival file.\n\n29. _Madras Fisheries Bulletin, 1918\u20131937_ (Madras: Government Printing, 1938), 2; see Ajantha Subramanian, _Shorelines: Space and Rights in South Asia_ (Stanford: Stanford University Press, 2009), 120\u2013124.\n\n30. Micah Muscolino, \"Yellow River Flood, 1938\u201347,\" DisasterHistory.org, accessed March 3, 2018, www.disasterhistory.org\/yellow-river-flood-1938-47; Micah S. Muscolino, _The Ecology of War in China: Henan Province, the Yellow River, and Beyond, 1938\u20131947_ (Cambridge: Cambridge University Press, 2015).\n\n31. Christopher Bayly and Tim Harper, _Forgotten Armies: The Fall of British Asia, 1941\u201345_ (London: Allen Lane, 2004).\n\n32. Srinath Raghavan, _India's War: The Making of Modern South Asia, 1939\u201345_ (London: Allen Lane, 2016).\n\n33. India Meteorological Department, _Hundred Years of Weather Service (1875\u20131975)_ , bound typescript in the library of the Regional Meteorological Centre, Chennai, consulted in February 2015.\n\n34. Sunil S. Amrith, \"Food and Welfare in India, c. 1900\u20131950,\" _Comparative Studies in Society and History_ 50 (2008): 1010\u20131035; the quotation from Nehru is in a letter to B. J. K. Hallowes, June 26, 1929, see note 2 above.\n\n35. _The Ramakrishna Mission: Bengal and Orissa Cyclone Relief, 1942\u201344_ (Howrah: Ramakrishna Mission, 1944), 1\u20132.\n\n36. Bayly and Harper, _Forgotten Armies,_ 282\u2013291.\n\n37. Note to Famine Commission (1944): Papers of L. G. Pinnell, British Library, Asian and African Studies Collection, European Manuscripts: MSS Eur D 911\/7.\n\n38. On ecological decline, see Iftekhar Iqbal, _The Bengal Delta: Ecology, State, and Social Change, 1840\u20131943_ (Basingstoke: Palgrave\/MacMillan, 2010), chapter 8. On famine, see Sugata Bose, \"Starvation Amidst Plenty: The Making of Famine in Bengal, Honan and Tonkin, 1942\u201345,\" _Modern Asian Studies_ 24, no. 4 (1990): 699\u2013727; Amartya Sen, _Poverty and Famines: An Essay on Entitlement and Deprivation_ (Oxford: Oxford University Press, 1981); Paul Greenough, _Prosperity and Misery in Modern Bengal: The Famine of 1943\u20134_ (New York: Oxford University Press, 1982); and Bayly and Harper, _Forgotten Armies,_ 282\u2013291.\n\n39. Jawaharlal Nehru, _The Discovery of India_ [1946] (New Delhi: Oxford University Press, 2003), 496\u2013498; S. G. Sardesai, _Food in the United Provinces_ (Bombay: People's Publishing House, 1944), 19, 36\u201337.\n\n40. V. D. Wickizer and M. K. Bennett, _The Rice Economy of Monsoon Asia_ (Stanford, CA: Stanford University Press, 1941), 1, 189.\n\n41. Nehru, _Discovery of India,_ 535; Gyan Chand, _Problem of Population_ (London: Oxford University Press, 1944), 10.\n\n42. File note on Bhakra Dam Project, February 23, 1945, NAI, Political Department, I A Branch: file no. 21(22)\u2014IA\/45.\n\n43. File note by T.A.W. Foy, October 31, 1946, NAI, 21(22)\u2014IA\/45.\n\n44. Meghnad Saha, editorial, _Science and Culture_ 1 (1935): 3\u20134.\n\n45. Meghnad Saha, \"Flood,\" _Science and Culture_ 9 (September 1943): 95\u201397.\n\n46. Meghnad Saha and Kamalesh Ray, \"Planning for the Damodar Valley\" (originally published in _Science and Culture,_ 10 [1944]), in _Collected Works of Meghnad Saha,_ ed. Santimay Chatterjee (Bombay: Orient Longman, 1987), 2:115\u2013144, quotations on pp. 116, 132, 135.\n\n47. Saha and Ray, \"Planning for the Damodar Valley,\" 132, 135.\n\nCHAPTER SEVEN: RIVERS DIVIDED, RIVERS DAMMED\n\n1. The best narrative account of the period is in Christopher Bayly and Tim Harper, _Forgotten Wars: The End of Britain's Asian Empire_ (London: Allen Lane, 2007); on the early Cold War in Asia, see Odd Arne Westad, _The Cold War: A World History_ (New York: Basic Books, 2017), chapter 5.\n\n2. For a concise overview, see Yasmin Khan, _The Great Partition: The Making of India and Pakistan_ (New Haven, CT: Yale University Press, 2007).\n\n3. Vazira Fazila-Yacoobali Zamindar, _The Long Partition and the Making of Modern South Asia: Refugees, Boundaries, Histories_ (New York: Columbia University Press, 2007); Joya Chatterji, _The Spoils of Partition: Bengal and India, 1947\u201367_ (Cambridge: Cambridge University Press, 2007); for a moving set of testimonies, see Urvashi Butalia, _The Other Side of Silence: Voices from the Partition of India_ (London: Hurst, 2000).\n\n4. David Gilmartin, _Blood and Water: The Indus River Basin in Modern History_ (Berkeley: University of California Press, 2015), 206.\n\n5. Government of India, Press Information Bureau, \"Facts about Canal Dispute\"\u2014enclosure in a letter from S. V. Sampath to all Indian Missions abroad, September 27, 1949: National Archives of India [hereafter NAI], Ministry of External Affairs [hereafter MEA], File 6\/1\/7-XP (P)\/49.\n\n6. Joya Chatterji, \"The Fashioning of a Frontier: The Radcliffe Line and Bengal's Border Landscape, 1947\u201352,\" _Modern Asian Studies_ 33, no. 1 (1999): 185\u2013242.\n\n7. Rammanohar Lohia, _The Guilty Men of India's Partition_ (Allahabad: Kitabistan, 1960).\n\n8. Ayesha Jalal, _The Pity of Partition: Manto's Life, Times, and Work Across the India-Pakistan Divide_ (Princeton: Princeton University Press, 2013).\n\n9. Saadat Hasan Manto, \"Yazid,\" in _Naked Voices: Stories and Sketches_ , trans. Rakhshanda Jalil (New Delhi: Roli Books, 2008), 106.\n\n10. India Meteorological Department, _Hundred Years of Weather Service (1875\u20131975)_ , bound typescript in the library of the Regional Meteorological Centre, Chennai, consulted in February 2015, p. 55.\n\n11. C. N. Vakil, _Economic Consequences of the Partition,_ 2nd ed. (Bombay: National Information and Publications, 1949), 3\u20134.\n\n12. Gilmartin, _Blood and Water,_ 206.\n\n13. Daniel Haines, _Rivers Divided: Indus Basin Waters in the Making of India and Pakistan_ (Oxford: Oxford University Press, 2016), chapter 3.\n\n14. Cited in Haines, _Rivers Divided,_ 51.\n\n15. India, Ministry of External Affairs, Directive on Canal Water Dispute Between India and Pakistan, NAI, MEA, File 6\/1\/7-XP (P)\/49.\n\n16. Directive on Canal Water Dispute, NAI, MEA, File 6\/1\/7-XP (P)\/49.\n\n17. Manu Goswami, _Producing India: From Colonial Economy to National Space_ (Chicago: University of Chicago Press, 2004).\n\n18. Gilmartin, _Blood and Water,_ 212; Bashir A. Malik, _Indus Waters Treaty in Retrospect_ (Lahore: Brite Books, 2005), 104.\n\n19. David E. Lilienthal, \"Kashmir: Another 'Korea' in the Making?,\" _Collier's_ 128, no. 5 (1951): 58.\n\n20. \"Today in Earthquake History: Assam, 1950,\" Seismo Blog, Berkeley Seismology Lab, accessed March 1, 2018, http:\/\/seismo.berkeley.edu\/blog\/2017\/08\/15\/today-in-earthquake-history-assam-1950.html; M. C. Podder, \"Preliminary Report on the Assam Earthquake of 15th August 1950,\" _Bulletin of the Geological Survey of India,_ Series B 2 (1950): 1\u201340; Francis Kingdon Ward, \"Aftermath of the Assam Earthquake of 1950,\" _The Geographical Journal_ 121 (1955): 290\u2013303.\n\n21. _Census of India, 1951,_ vol. 1, Part 1A (New Delhi: Government Press, 1953).\n\n22. Georges Canguilhem, _The Normal and the Pathological,_ trans. Carolyn R. Fawcett (New York: Zone Books, 1989), 161.\n\n23. _Census of India 1951,_ vol. 1, Part 1A: 126\u2013131.\n\n24. _Census of India 1951,_ vol. 1, Part 1A: 150.\n\n25. Sanjoy Chakravorty, _The Price of Land: Acquisition, Conflict, Consequence_ (New Delhi: Oxford University Press, 2013), see especially chapter 7.\n\n26. Report of American Famine Mission to India, led by T. W. Schultz, cited in Henry Knight, _Food Administration in India, 1939\u201347_ (Stanford, CA: Stanford University Press, 1954), 253; Government of India, Foodgrains Policy Committee, _Interim Report_ (New Delhi, 1948); Government of India, Foodgrains Policy Committee, _Final Report_ (New Delhi, 1948); _Report of the Foodgrains Enquiry Committee, 1957_ (New Delhi: Ministry of Food & Agriculture, 1957), 26\u201327.\n\n27. Jawaharlal Nehru's letter to India's chief ministers, April 15, 1948, in _Letters for a Nation: From Jawaharlal Nehru to His Chief Ministers, 1947\u20131963,_ ed. Madhav Khosla (New Delhi: Allen Lane, 2014), 147\u2013148. On the Hirakud Dam, see Rohan D'Souza, \"Damming the Mahanadi River: The Emergence of Multi-Purpose River Valley Development in India (1943\u201346),\" _Indian Economic and Social History Review_ 40 (2003): 81\u2013105.\n\n28. Henry C. Hart, _New India's Rivers_ (Bombay: Orient Longmans, 1956), 250.\n\n29. India, Central Water-Power, Irrigation & Navigation Commission, _Quinquennial Report, April 1945\u2013March 1950_ , p. 2: BL, IOR, V\/24\/4496.\n\n30. There is an illuminating account of Khosla's career in Daniel Klingensmith, _One Valley and a Thousand: Dams, Nationalism, and Development_ (New Delhi: Oxford University Press, 2007).\n\n31. A. N. Khosla, \"Our Plans,\" _Indian Journal of Power and River Valley Development_ [ _IJPRVD_ ], June 1951: 1\u20134.\n\n32. India, Central Water & Power Commission, _Major Water & Power Projects of India_ (Bhagirath Pamphlet 1, June 1957).\n\n33. Jawharlal Nehru, speech at the opening of the Nangal Canal, July 8, 1954, in _Jawaharlal Nehru: An Anthology,_ ed. Sarvepalli Gopal (New Delhi: Oxford University Press, 1980), 213\u2013215.\n\n34. \"Nehru Shows Chou India's Dam Project,\" _New York Times,_ January 1, 1957, 4.\n\n35. On the history of the Films Division, see Peter Sutoris, _Visions of Development: Films Division of India and the Imagination of Progress, 1948\u201375_ (London: Hurst, 2016), which includes a discussion of Ezra Mir's career; see also, Judith Pernin et al., \"The Documentary Film in India, 1948\u20131975,\" undated, Hong Kong Baptist University, last accessed May 13, 2018, http:\/\/digital.lib.hkbu.edu.hk\/documentary-film\/india.php#footnote.\n\n36. _Bhakra Nangal,_ dir. N.S. Thapa, Government of India Films Division (1958).\n\n37. On Lilienthal, see Klingensmith, _One Valley and a Thousand._\n\n38. Hart, _New India's Rivers,_ 97.\n\n39. Hugh Tinker, \"A Forgotten Long March: The Indian Exodus from Burma, 1942,\" _Journal of Southeast Asian Studies_ 6 (1975): 1\u201315; Sunil S. Amrith, _Crossing the Bay of Bengal: The Furies of Nature and the Fortunes of Migrants_ (Cambridge, MA: Harvard University Press, 2013), chapter 6.\n\n40. Letter from the Secretary to the Government of Punjab, Public Works Department, to the Secretary to the Governor General, June 18, 1945, NAI, file no. 21(22)\u2014IA\/45; Proceedings of a meeting held on January 31, 1946, between representatives of Madras and Hyderabad: Government of India, Political Branch: Hyderabad Residency; NAI, file no. 92(2), 1946.\n\n41. Hart, _New India's Rivers,_ 115.\n\n42. \"Lathi Charge on Strikers,\" _Times of India,_ January 31, 1954, 9.\n\n43. Hart, _New India's Rivers,_ 178\u2013184.\n\n44. Ashis Nandy, \"Dams and Dissent: India's First Modern Environmental Activist and His Critique of the DVC Project,\" _Futures_ 33 (2001): 709\u2013731.\n\n45. World Bank Group Archives, Washington DC [hereafter WBA], File no. 1787276, Indus Basin Dispute, General Negotiations, 1949\u201352, Correspondence; File no, 1787280, Notes of Mission, September 1\u201316, 1954; File no. 1787263, Chronology of Indus Waters Dispute. Files 1787269 and 1787270 (Indus Basin Dispute, Working Party, Correspondence vol. 3 & 4), for example, both had several items that had been removed as unsuitable for declassification.\n\n46. M. V. V. Ramana, _Inter-State River Water Disputes in India_ (Hyderabad: Orient Longman, 1992), chapter 4.\n\n47. Sumathi Ramaswamy, _The Goddess and the Nation: Mapping Mother India_ (Durham, NC: Duke University Press, 2009), 244.\n\n48. _Mother India,_ dir. Mehboob (1957).\n\n49. Quoted in Ramaswamy, _Goddess and the Nation,_ 243.\n\n50. Sangita Gopal and Sujata Moorti, _Global Bollywood: Travels of Hindi Song and Dance_ (Minneapolis: University of Minnesota Press, 2008), 60.\n\n51. Brian Larkin, \"Bollywood Comes to Nigeria,\" accessed November 14, 2017, www.samarmagazine.org\/archive\/articles\/21.\n\n52. James C. Scott, _Seeing Like a State: Why Certain Schemes to Improve the Human Condition Have Failed_ (New Haven, CT: Yale University Press, 1998); Arturo Escobar, _Encountering Development: The Making and Unmaking of the Third World_ (Princeton, NJ: Princeton University Press, 1995).\n\n53. Chakravorty, _Price of Land,_ 113\u2013114; Rohan D'Souza, \"Framing India's Hydraulic Crises: The Politics of the Modern Large Dam,\" _Monthly Review_ 60 (2008): 112\u2013124.\n\n54. File note, February 23, 1945, Bhakra Dam Project, Government of India, Political Department, IA Branch, NAI, file no. 21(22)\u2014IA\/45.\n\n55. File note, Anon., March 12, 1945, Bhakra Dam Project, Government of India, Political Department, IA Branch: NAI, file no. 21(22)\u2014IA\/45.\n\n56. File note, Anon., March 12, 1945, NAI, file no. 21(22)\u2014IA\/45.\n\n57. Proceedings of a meeting held on January 31, 1946, between representatives of Madras and Hyderabad: Government of India, Political Branch: Hyderabad Residency; NAI, file no. 92(2), 1946, quotations from this file.\n\n58. P. Chaturvedi and A. Dalal, _Law of Special Economic Zone: National and International Perspective_ (Kolkata: Eastern Law House, 2009), 342, cited in Chakravorty, _Price of Land,_ 115.\n\n59. Walter Fernandes and Enakshi Ganguly Thukral, eds., _Development, Displacement and Rehabilitation_ (New Delhi: Indian Social Institute, 1989); Esther Duflo and Rohini Pande, \"Dams,\" _Quarterly Journal of Economics_ 122 (2007): 601\u2013646; Satyajit Singh, _Taming the Waters: The Political Economy of Large Dams in India_ (New Delhi: Oxford University Press, 1997), 182\u2013203; Chakravorty, _Price of Land,_ quotations on pp. 123\u2013130.\n\n60. Singh, _Taming the Waters,_ 133\u2013158.\n\n61. Jawaharlal Nehru, \"Social Aspects of Small and Big Projects,\" Inaugural address at the 29th annual meeting of the Central Board of Irrigation and Power, New Delhi, November 17, 1958, in Baldev Singh, _Jawaharlal Nehru on Science and Society: A Collection of His Writings and Speeches_ (New Delhi: Nehru Memorial Museum and Library, 1990), 172\u2013175.\n\n62. United Nations, Economic Commission for Asia and the Far East (ECAFE), _Economic Survey of Asia and the Far East, 1948_ (Bangkok: ECAFE, 1949); C. Hart Schaaf, \"The United Nations Economic Commission for Asia and the Far East,\" _International Organization_ 7 (1953): 463\u2013481, quotation on p. 468.\n\n63. Hart Schaaf, \"Economic Commission for Asia\" (1953).\n\n64. Hart Schaaf, \"Economic Commission for Asia\" (1953), 481.\n\n65. UN, ECAFE, _Economic Survey of Asia and the Far East, 1954_ (Bangkok: ECAFE, 1955); chapter 10 covers the People's Republic of China.\n\n66. Kanwar Sain and K. L. Rao, _Report on the Recent River Valley Projects in China_ (New Delhi: Government of India Central Water and Power Commission, 1955), their full itinerary appears in Appendix F; \"Mao is our Buddha\" reported in Kanwar Sain, _Reminiscences of an Engineer_ (New Delhi: Young Asia Publications, 1978), 208\u2013209.\n\n67. Sain, _Reminiscences of an Engineer,_ 208\u2013209.\n\n68. Sain and Rao, _River Valley Projects in China,_ 206\u2013207; list of Chinese officials in Appendix G.\n\n69. Sain and Rao, _River Valley Projects in China,_ 154.\n\n70. Judith Shapiro, _Mao's War Against Nature: Politics and Environment in Revolutionary China_ (Cambridge: Cambridge University Press, 2001), chapter 1.\n\n71. Cited in Philip Ball, _Water Kingdom: A Secret History of China_ (Oxford: The Bodley Head, 2016), 225.\n\n72. Sain and Rao, _River Valley Projects in China,_ 162.\n\n73. For an elaboration of the continuity argument, see Ball, _Water Kingdom,_ chapter 8.\n\n74. Sain and Rao, _River Valley Projects in China._ Hao's speech is reproduced in Appendix A.\n\n75. Sain, _Reminiscences of an Engineer,_ 210.\n\n76. Christopher Sneddon, _Concrete Revolution: Large Dams, Cold War Geopolitics, and the US Bureau of Reclamation_ (Chicago: University of Chicago Press, 2015); David Biggs, _Quagmire: Nation-Building and Nature in the Mekong Delta_ (Seattle: University of Washington Press, 2010).\n\n77. Biggs, _Quagmire,_ 172.\n\n78. Sneddon, _Concrete Revolution_.\n\n79. Sain, _Reminiscences of an Engineer,_ 388\u2013392.\n\n80. Nehru's letter to Zhou Enlai, September 26, 1959, published in _India-China Conflict_ (New Delhi: Indian Ministry of External Affairs, 1964).\n\n81. Letter from B. C. Mishra, Ministry of External Affairs to Apa B. Pant, Political Officer of the Government of India, Gangtok, Sikkim, October 7, 1960, in India, Ministry of External Affairs, \"Construction of dam on Brahmaputra and Indus group of rivers by the Chinese\": NAI, MEA, file F no. 4(75)\u2014T 60.\n\n82. Letter from R. S. Kapoor, Indian trade agent, Gyantse, Tibet to Apa Pant, Political Officer of the Government of India, Gangtok, Sikkim, December 15, 1960: NAI, MEA, file F no. 4(75)\u2014T 60.\n\n83. Letter marked \"top secret,\" from K. K. Framji, Chief Engineer & Joint Secretary, Ministry of Irrigation and Power to B. C. Mishra, DS (China), Ministry of External Affairs, January 5, 1961: NAI, MEA, file F no. 4(75)\u2014T 60.\n\n84. Rohinton Mistry, _Such a Long Journey_ (London: Faber & Faber, 1991), 10.\n\nCHAPTER EIGHT: THE OCEAN AND THE UNDERGROUND\n\n1. Neel Mukherjee, _The Lives of Others_ (London: Vintage, 2014), 195\u2013197.\n\n2. Indian National Committee on Oceanic Research [hereafter INCOR], _International Indian Ocean Expedition: Indian Scientific Programmes, 1962\u20131965_ (New Delhi: Council of Scientific and Industrial Research, 1962), 15.\n\n3. Bernard Bailyn, \"The Challenge of Modern Historiography,\" _American Historical Review_ 87 (1982): 1\u201324, quotations on pp. 10\u201311.\n\n4. Sunil S. Amrith, _Crossing the Bay of Bengal: The Furies of Nature and the Fortunes of Migrants_ (Cambridge, MA: Harvard University Press, 2013), chapter 7.\n\n5. India, Ministry of External Affairs, Memorandum on the International Conference of Plenipotentiaries on the Law of the Sea [undated, probably late 1957]. NAI, MEA: UN II Section, file no. 9(6) UN II\/57.\n\n6. Daniel Behrman, _Assault on the Largest Unknown: The International Indian Ocean Expedition_ (Paris: UNESCO Press, 1981), 10\u201311; G. E. R. Deacon, \"The Indian Ocean Expedition,\" _Nature_ 187 (August 13, 1960): 561\u2013562.\n\n7. Warren S. Wooster, \"Indian Ocean Expedition,\" _Science,_ n.s., 150 (October 15, 1965): 290\u2013292.\n\n8. INCOR, _International Indian Ocean Expedition,_ 15.\n\n9. _The Indian Ocean Bubble_ , issue 5, March 1, 1960, Woods Hole Oceanographic Institution Open Access Server, last accessed March 10, 2018, https:\/\/darchive.mblwhoilibrary.org\/handle\/1912\/218.\n\n10. Behrman, _Assault,_ 27.\n\n11. Behrman, _Assault,_ 52.\n\n12. INCOR, _International Indian Ocean Expedition,_ 1\u20135.\n\n13. Klaus Wyrtki, _Oceanographic Atlas of the International Indian Ocean Expedition_ (Washington, DC: National Science Foundation, 1971).\n\n14. Behrman, _Assault,_ 64.\n\n15. INCOR, _International Indian Ocean Expedition,_ 44.\n\n16. Gilbert T. Walker, \"The Atlantic Ocean,\" _Quarterly Journal of the Royal Meteorological Society_ 53 (1927): 113.\n\n17. Deacon, \"Indian Ocean Expedition\"; INCOR, _International Indian Ocean Expedition,_ 12; Wyrtki, _Oceanographic Atlas,_ 7.\n\n18. C. S. Ramage, _Monsoon Meteorology_ (London: Academic Press, 1971), 1; Thomas A. Schroeder, \"A Personal View of the History of the Department of Meteorology, University of Hawaii at Manoa\" (2006), accessed June 1, 2016, www.soest.hawaii.edu\/met\/history.pdf.\n\n19. Sanchari Pal, \"Anna Mani Is One of India's Greatest Woman Scientists,\" _The Better India,_ January 21, 2017, accessed May 1, 2018, https:\/\/www.thebetterindia.com\/83063\/anna-mani-scientist-meteorology-ozone-wind-energy\/.\n\n20. C. S. Ramage, _Meteorology in the Indian Ocean_ (Geneva: World Meteorological Association, 1965).\n\n21. Ramage, _Meteorology in the Indian Ocean_.\n\n22. Behrman, _Assault,_ 67.\n\n23. Behrman, _Assault,_ 65; C. S. Ramage and C. R. Raman, _Meteorological Atlas of the International Indian Ocean Expedition_ (Washington, DC: US Government Printer, 1972).\n\n24. Behrman, _Assault,_ 66.\n\n25. Ramage, _Meteorology in the Indian Ocean_.\n\n26. Roger Revelle and H. E. Suess, \"Carbon Dioxide Exchange Between Atmosphere and Ocean and the Question of an Increase of Atmospheric CO2 During the Past Decades, \" _Tellus_ 9 (1957): 18\u201327, quotation on pp. 19\u201320. On oceanography and the discovery of climate change, see Naomi Oreskes, \"Changing the Mission: From the Cold War to Climate Change,\" in _Science and Technology in the Global Cold War,_ ed. Naomi Oreskes and John Krige (Cambridge, MA: MIT Press, 2014), 141\u2013187.\n\n27. Behrman, _Assault,_ 11\u201312.\n\n28. P. K. Das, _The Monsoons_ (New Delhi: National Book Trust, 1968), 6.\n\n29. Francine Frankel, _India's Political Economy, 1947\u20131977: The Gradual Revolution_ (Princeton, NJ: Princeton University Press, 1980), 247\u2013248.\n\n30. Statistics on Indian wheat imports from Nick Cullather, _The Hungry World: America's Cold War Battle Against Poverty in Asia_ (Cambridge, MA: Harvard University Press, 2010), 144; _Economic Survey of Indian Agriculture for 1966\u201367_ (New Delhi: Government of India, 1969); Frankel, _India's Political Economy,_ 293.\n\n31. David Ludden, _An Agrarian History of South Asia_ (Cambridge: Cambridge University Press, 1999), especially chapter 4.\n\n32. David C. Engerman, _The Price of Aid: The Economic Cold War in India_ (Cambridge, MA: Harvard University Press, 2018); Cullather, _Hungry World_.\n\n33. Cullather, _Hungry World,_ 207.\n\n34. C. Subramaniam, _Hand of Destiny,_ vol. 2, _The Green Revolution_ (Bombay: Bharatiya Vidya Bhavan, 1993), 137\u2013138.\n\n35. Frankel, _India's Political Economy,_ 270.\n\n36. Subramaniam, _Hand of Destiny,_ vol. 2, 154, 165\u2013167.\n\n37. \"Years Before a Revolution,\" _Times of India,_ August 22, 1965.\n\n38. Quoted in Mahesh Rangarajan, \"Striving for a Balance: Nature, Power, Science and Indira Gandhi's India, 1917\u20131984,\" _Conservation and Society_ 7 (2009): 299\u2013312.\n\n39. Cullather, _Hungry World,_ 223.\n\n40. Paul R. Brass, \"The Political Uses of Crisis: The Bihar Famine of 1966\u20131967,\" _Journal of Asian Studies_ 45 (1986): 245\u2013267, 249.\n\n41. Ronald E. Doel and Kristine C. Harper, \"Prometheus Unleashed: Science as a Diplomatic Weapon in the Lyndon B. Johnson Administration,\" _Osiris_ 21 (2006): 66\u201385.\n\n42. James R. Fleming, _Fixing the Sky: The Checkered History of Weather and Climate Control_ (New York: Columbia University Press, 2010); Kristine C. Harper, _Make It Rain: State Control of the Atmosphere in Twentieth Century America_ (Chicago: University of Chicago Press, 2017).\n\n43. Lyndon B. Johnson, _The Vantage Point: Perspectives of the Presidency, 1963\u20131969_ (New York: Holt, Rinehart and Winston, 1971), 226.\n\n44. Doel and Harper, \"Prometheus Unleashed,\" 80, 83.\n\n45. Rajni Kothari, \"The Congress 'System' in India,\" _Asian Survey_ 4 (1964): 1161\u20131173; Confidential Despatch from British High Commission, Delhi to London, 3 March 1967, in United Kingdom National Archives (UKNA), \"India\u2014Political Affairs\u2014Internal\" FO 37\/35.\n\n46. Ashutosh Varshney, _Democracy, Development, and the Countryside: Urban-Rural Struggles in India_ (Cambridge: Cambridge University Press, 1998), 57.\n\n47. Geoffrey Parker, _Global Crisis: War, Climate Change & Catastrophe in the Seventeenth Century_ (New Haven, CT: Yale University Press, 2013); Sam White, _The Climate of Rebellion in the Early Modern Ottoman Empire_ (Cambridge: Cambridge University Press, 2011).\n\n48. Indira Gandhi, \"Man and Environment,\" speech at the Plenary Session of United Nations Conference on Human Environment, Stockholm, June 14, 1972: full text available at http:\/\/lasulawsenvironmental.blogspot.com\/2012\/07\/indira-gandhis-speech-at-stockholm.html, last accessed May 14, 2018.\n\n49. Paul Ehrlich, _The Population Bomb_ (New York: Ballantine Books, 1968), 15\u201316.\n\n50. Indira Gandhi, \"Man and Environment,\" speech at the Plenary Session of United Nations Conference on Human Environment, Stockholm, June 14, 1972; Jairam Ramesh, \"Poverty Is the Greatest Polluter: Remembering Indira Gandhi's Stirring Speech in Stockholm,\" _The Wire,_ June 7, 2017, accessed November 30, 2017, https:\/\/thewire.in\/144555\/indira-gandhi-nature-pollution\/.\n\n51. On coercive population control in India, see Matthew Connelly, _Fatal Misconception: The Struggle to Control World Population_ (Cambridge, MA: Harvard University Press, 2008); Emma Tarlo, _Unsettling Memories: Narratives of the Emergency in Delhi_ (London: Hurst, 2003).\n\n52. Shyam Divan and Armin Rosencranz, eds., _Environmental Law and Policy in India: Cases, Materials and Statutes_ (New Delhi: Oxford University Press, 2001), 167\u2013241; the case cited is Aggarwal Textile Industries v. State of Rajasthan, S.B.C. Writ Petition No. 1375\/80, March 2, 1981, presented in Divan and Rosencranz, _Environmental Law,_ 187.\n\n53. Anthony Acciavatti, \"Re-imagining the Indian Underground: A Biography of the Tubewell,\" in _Places of Nature in Ecologies of Urbanism,_ ed. Anne Rademacher and K. Sivaramakrishnan (Hong Kong: Hong Kong University Press, 2017), 206\u2013237, quotation on p. 207.\n\n54. Tushaar Shah, _Taming the Anarchy: Groundwater Governance in South Asia_ (New York: Routledge, 2008); Tushaar Shah, \"Climate Change and Groundwater: India's Opportunities for Mitigation and Adaptation,\" _Environmental Research Letters_ 4 (2009): 1\u201313.\n\n55. Roger Revelle and V. Lakshminarayana, \"Ganges Water Machine,\" _Science,_ n.s., 188 (1975): 611\u2013616, quotation on p. 611; K. L. Rao, _India's Water Wealth: Its Assessment, Uses, and Projections_ (New Delhi: Orient Longman, 1975).\n\n56. Joshua Eisenman, \"Building China's 1970s Green Revolution: Responding to Population Growth, Decreasing Arable Land, and Capital Depreciation,\" in _China, Hong Kong, and the Long 1970s: Global Perspectives,_ ed. Priscilla Roberts and Odd Arne Westad (New York: Palgrave Macmillan, 2017), 55\u201386.\n\n57. Sigrid Schmalzer, _Red Revolution, Green Revolution: Scientific Farming in Socialist China_ (Chicago: University of Chicago Press, 2016), quotation on p. 13.\n\n58. Francine Frankel, _India's Green Revolution: Economic Gains and Political Costs_ (Princeton, NJ: Princeton University Press, 1971); N. K. Dubash, _Tubewell Capitalism: Groundwater Development and Agrarian Change in Gujarat_ (New Delhi: Oxford University Press, 2002); Shah, \"Climate Change and Groundwater.\"\n\n59. L. J. Walinsky, ed., _Agrarian Reform As Unfinished Business: The Selected Papers of Wolf Ladejinsky_ (Washington, DC: World Bank, 1977); G. Rosen, \"Obituary: Wolf Ladejinsky (1899\u20131975),\" _Journal of Asian Studies_ 36 (1976): 327\u2013328.\n\n60. Wolf Ladejinsky, \"Drought in Maharashtra (Not in a Hundred Years),\" typescript contained in World Bank Archives (WBA), file number 1167800, Drought Prone Areas Project\u2014India\u2014Correspondence vol. 1.\n\n61. Jean Dr\u00e8ze, \"Famine Prevention in India\" (working paper 45, WIDER: United Nations University, Helsinki, 1988), 69\u201375.\n\n62. John A. Young, \"Physics of the Monsoon: The Current View,\" in _Monsoons,_ ed. Jay S. Fein and Pamela L. Stephens (New York: John Wiley & Sons, 1987), 211\u2013243, quotation on p. 211; see the discussion of \"moist processes\" in Peter J. Webster, \"Monsoons,\" _Scientific American_ 245 (1981): 108\u2013119; on modeling, see Kirsten Hastrup and Martin Skrydstrup, eds., _The Social Life of Climate Change Models: Anticipating Nature_ (London: Routledge, 2012).\n\n63. Jacob Bjerknes, \"A Possible Response of the Atmospheric Hadley Circulation to Equatorial Anomalies of Ocean Temperature,\" _Tellus_ 18 (1966): 820\u2013829; Jacob Bjerknes, \"Atmospheric Teleconnections from the Equatorial Pacific,\" _Journal of Physical Oceanography_ 97 (1969): 163\u2013172.\n\n64. P. J. Webster, H. R. Chang, and V. E. Toma, _Tropical Meteorology and Climate_ (Oxford: Wiley Blackwell, in press), chapter 14.\n\n65. R. A. Madden and P. R. Julian, \"Detection of a 40\u201350 Day Oscillation in the Zonal Wind in the Tropical Pacific,\" _Journal of the Atmospheric Sciences_ 28 (1971): 702\u2013770; R. A. Madden and P. R. Julian, \"Description of Global-Scale Circulation Cells in the Tropics with a 40\u201350 Day Period,\" _Journal of the Atmospheric Sciences_ 29 (1972): 1109\u20131123.\n\n66. David M. Lawrence and Peter J. Webster, \"The Boreal Summer Intraseasonal Oscillation: Relationship between Northward and Eastward Movement of Convection,\" _Journal of the Atmopheric Sciences_ 59 (2002): 1593\u20131606.\n\n67. Adam Sobel, _Storm Surge: Hurricane Sandy, Our Changing Climate, and Extreme Weather of the Past and Future_ (New York: Harper Wave, 2014), 9\u201320.\n\n68. On MONEX, see Behrman, _Assault,_ 64; Webster, Chang, and Toma, _Tropical Meteorology,_ chapter 14.\n\n69. C. S. Ramage, _The Great Indian Drought of 1899,_ Occasional Paper, Aspen Instiute for Humanistic Studies, Program on Science, Technology, and Humanism (1977), quotations on pp. 4, 6.\n\n70. _Declaration of the Climate Conference_ (Geneva: World Meteorological Organization, 1979), 1.\n\nCHAPTER NINE: STORMY HORIZONS\n\n1. World Bank Group, \"China Overview,\" accessed February 12, 2018, www.worldbank.org\/en\/country\/china\/overview.\n\n2. Sumit Ganguly and Rahul Mukherjee, _India Since 1980_ (Cambridge: Cambridge University Press, 2012), chapter 3.\n\n3. Anil Agarwal, Kalpana Sharma, and Ravi Chopra, _The State of India's Environment, 1982: A Citizens' Report_ (New Delhi: Centre for Science and Environment, 1982), 20.\n\n4. Naomi Oreskes, \"The Scientific Consensus on Climate Change,\" _Science_ 306 (December 2004): 1686.\n\n5. Intergovernmental Panel on Climate Change, _Climate Change 2014: Impacts, Adaptation, and Vulnerability_ (Geneva: IPCC, 2014); World Bank, _Turn Down the Heat: Climate Extremes, Regional Impacts, and the Case for Resilience_ (Washington, DC: World Bank, 2013).\n\n6. Andreas Malm, _The Progress of This Storm: Nature and Society in a Warming World_ (London: Verso, 2018), 5.\n\n7. Angus Deaton, _The Great Escape: Health, Wealth, and the Origins of Inequality_ (Princeton, NJ: Princeton University Press, 2013).\n\n8. Hannah Ritchie, \"Yields vs. Land Use: How the Green Revolution Enabled Us to Feed a Growing Population,\" _Our World in Data,_ August 22, 2017, accessed February 10, 2018, https:\/\/ourworldindata.org\/yields-vs-land-use-how-has-the-world-produced-enough-food-for-a-growing-population.\n\n9. Khushwant Singh, \"The Indian Monsoon in Literature,\" in _Monsoons,_ ed. Jay S. Fein and Pamela L. Stephens (New York: John Wiley & Sons, 1987), 35\u201350, quotations on p. 48.\n\n10. Jyoti Bhatt, \"Divination of Rainy Days: An Annual Festival in Gujarat\" [1987], in Asia Art Archive, Hong Kong: Jyoti Bhatt Archive, last accessed April 24, 2018, https:\/\/aaa.org.hk\/en\/collection\/search\/archive\/jyoti-bhatt-archive-english\/object\/divination-of-rainy-days-an-annual-festival-in-gujarat.\n\n11. University of Hawaii at Manoa Economics Department, \"Harry T. Oshima (1918\u20131998),\" accessed February 16, 2018, www.economics.hawaii.edu\/history\/faculty\/oshima.html.\n\n12. Harry T. Oshima, _Economic Growth in Monsoon Asia: A Comparative Survey_ (Tokyo: University of Tokyo Press, 1987); Harry T. Oshima, \"Seasonality and Underemployment in Monsoon Asia,\" _Philippine Economic Journal_ 19 (1971): 63\u201397.\n\n13. Statistics from A. Vaidyanathan, _Water Resources of India_ (New Delhi: Oxford University Press, 2013); T. Shah, \"Climate Change and Groundwater: India's Opportunities for Mitigation and Adaptation,\" _Environmental Research Letters_ 4 (2009): 1\u201313, quotation on p. 3.\n\n14. Meera Subramanian, _A River Runs Again: India's Natural World in Crisis, from the Barren Cliffs of Rajasthan to the Farmlands of Karnataka_ (New York: PublicAffairs, 2015), 9\u201366, on Punjab; on Gujarat, see David Hardiman, \"The Politics of Water Scarcity in Gujarat,\" in Amita Baviskar ed. _Waterscapes: The Cultural Politics of a Natural Resource_ (New Delhi: Permanent Black, 2006), 39\u201362.\n\n15. Daniyal Mueenuddin, \"Nawabdin Electrician,\" in _In Other Rooms, Other Wonders_ (New York: W.W. Norton, 2009), 13\u201328, quotation on p. 13.\n\n16. Jane Qiu, \"China Faces Up to Groundwater Crisis,\" _Nature_ 466 (2010): 308.\n\n17. David A. Pietz, _The Yellow River: The Problem of Water in Modern China_ (Cambridge, MA: Harvard University Press, 2015), 264\u2013265; M. Webber et al., \"The Yellow River in Transition,\" _Environmental Science and Policy_ 11 (2008): 422\u2013429.\n\n18. M. Rodell, I. Velicogna, and J. S. Famiglietti, \"Satellite-Based Estimates of Groundwater Depletion in India,\" _Nature_ 460 (2009): 999\u20131002.\n\n19. M. K. Gandhi, \"Some Mussooree Reminiscences,\" _Harijan,_ June 23, 1946, 198; Ramachandra Guha, _The Unquiet Woods: Ecological Change and Political Protest in the Himalaya_ (New Delhi: Oxford University Press, 1989).\n\n20. Kathleen D. Morrison, \"Dharmic Projects, Imperial Reservoirs, and New Temples of India: An Historical Perspective on Dams in India,\" _Conservation and Society_ 8 (2010): 184.\n\n21. Ambedkar's statement was delivered in India's Constituent Assembly on November 4, 1948; for research on the complexity of water management in pre-modern India, see David Mosse, _The Rule of Water: Statecraft, Ecology, and Collective Action in South Asia_ (New Delhi: Oxford University Press, 2003); Haruka Yanagisawa, _A Century of Change: Caste and Irrigated Lands in Tamil Nadu, 1860s\u20131970s_ (New Delhi: Manohar, 1996); for an overview that is skeptical of the idea that colonial rule was an absolute ecological watershed, see Mahesh Rangarajan, \"Environmental Histories of India: Of States, Landscapes, and Ecologies,\" in _The Environment and World History,_ ed. Kenneth Pomeranz and Edmund Burke III (Berkeley: University of California Press, 2009), 229\u2013254.\n\n22. Agarwal, Chopra, and Sharma, _The State of India's Environment, 1982_ ; A. Agarwal and Sunita Narain, eds., _The State of India's Environment, 1984\u201385: A Second Citizens' Report_ (New Delhi: Centre for Science and Environment, 1985), quotation from \"Statement of Shared Concern\"; The Centre for Science and Environment also produced a documentary film on water harvesting: _Harvest of Rain,_ dir. Sanjay Kak (1995), Centre for Science and Environment, 1995; Tim Forsyth, \"Anil Agarwal,\" in _Fifty Key Thinkers on Development,_ ed. D. Simon (London: Routledge, 2005), 9\u201314.\n\n23. Vandana Shiva, _The Violence of the Green Revolution: Third World Agriculture, Ecology and Politics_ (London: Zed Books, 1991), 11.\n\n24. The authors of the first Indian report cite the inspiration of Penang in their preface: Agarwal, Chopra, and Sharma, _State of India's Environment, 1982_ ; on the Third World Network, see its website, accessed February 1, 2018, www.twn.my\/twnintro.htm; on the Consumers' Association of Penang, see Matthew Hilton, _Prosperity for All: Consumer Activism in an Era of Globalization_ (Ithaca, NY: Cornell University Press, 2009); on the rise and fall of the New International Economic Order, see Nils Gilman, \"The New International Economic Order: A Reintroduction,\" _Humanity_ (Spring 2015): 1\u201316.\n\n25. Anil Agarwal and Sunita Narain, _Global Warming in an Unequal World: A Case of Environmental Colonialism_ (Delhi: Centre for Science and Environment, 1991).\n\n26. P. Sainath, _Everybody Loves a Good Drought: Stories from India's Poorest Districts_ (New Delhi: Penguin, 1996), quotations on pp. 319\u2013320.\n\n27. Aseem Shrivastava and Ashish Kothari, _Churning the Earth: The Making of Global India_ (New Delhi: Viking, 2012), 176\u2013183; P. Sainath, \"Farm Suicides: A 12-Year Saga,\" _The Hindu,_ January 25, 2010; P. Sainath, \"The Largest Wave of Suicides in History,\" _The Hindu,_ February 16, 2009; Akta Kaushal, \"Confronting Farmer Suicides in India,\" _Alternatives_ 40 (2016): 46\u201362.\n\n28. Gyansham Shah, Harsh Mander, Sukhadeo Thorat, Satish Deshpande, and Amita Baviskar, _Untouchability in Rural India_ (New Delhi: Sage, 2006), 75.\n\n29. Agarwal, Chopra, and Sharma, _State of India's Environment, 1982,_ 20\u201323.\n\n30. Darryl D'Monte, _Temples of Tombs? Industry versus Environment, Three Controversies_ (New Delhi: Centre for Science and Environment, 1985), 15.\n\n31. M. C. Mehta v. Union of India (Kanpur Tanneries), _All India Reporter_ (1988), SC 1037; M. C. Mehta v. Union of India (Municipalities), _All India Reporter_ (1988), SC 1115: cases cited in _Environmental Law and Policy in India: Cases, Materials and Statutes,_ ed. Shyam Divan and Armin Rosencranz (New Delhi: Oxford University Press, 2001), 210\u2013225. Mehta's biography from the M. C. Mehta Foundation, http:\/\/mcmef.org\/m-c-mehta\/; details of his career are available in his citation for the Goldman Prize, which he won in 1996, www.goldmanprize.org\/recipient\/mc-mehta\/.\n\n32. Judith Shapiro, _China's Environmental Challenges_ (London: Polity, 2012), 112\u2013118.\n\n33. Ma Jun, _China's Water Crisis_ , trans. Nancy Yang Liu and Lawrence R. Sullivan (Norwalk, CT: EastBridge\/International Rivers, 2004), quotations on pp. vii\u2013xi, 79\u201380; first published in Chinese as _Zhongguo shui weiji_ (Beijing: China Environmental Sciences Publishing House, 1999).\n\n34. \"India\u2014Mr McNamara's Meeting with the Indian Finance Minister,\" Memorandum of September 27, 1978: WBA, Contacts with Member Countries: India\u2014Correspondence 09, folder 1771081.\n\n35. See the entry on Medha Patkar on the website of the Goldman Environmental Prize, which she won in 1992. Accessed March 1, 2018, www.goldmanprize.org\/recipient\/medha-patkar\/.\n\n36. \"Bankwide Lessons Learned from the Experience with the India Sardar Sarovar (Narmada) Project,\" World Bank report, May 19, 1993, accessed March 19, 2018, http:\/\/documents.worldbank.org\/curated\/en\/221941467991015938\/Lessons-learned-from-Narmada.\n\n37. Smita Narula, \"The Story of Narmada Bachao Andolan: Human Rights in the Global Economy and the Struggle against the World Bank,\" _New York University Public Law and Legal Theory Working Papers_ 106 (2008); Balakrishnan Rajagopal, \"The Role of Law in Counter-hegemonic Globalization and Global Legal Pluralism: Lessons from the Narmada Valley Struggle in India,\" _Leiden Journal of International Law_ 18 (2005): 345\u2013355; Alf Gunvald Nilsen, _Dispossession and Resistance in India: The River and the Rage_ (London: Routledge, 2010). Modi's comments quoted in \"54 Years On, Modi Opens Sardar Sarovar Dam,\" _FirstPost,_ September 18, 2017, accessed March 20, 2018, www.firstpost.com\/politics\/sardar-sarovar-dam-inaugurated-narendra-modi-alleges-conspiracy-to-stop-project-congress-calls-it-election-gimmick-4054251.html.\n\n38. A synthesis of the research of the 1980s and 1990s appears in Satyajit Singh, _Taming the Waters: The Political Economy of Large Dams in India_ (New Delhi: Oxford University Press, 1997), especially 133\u2013163 on ecological consequences.\n\n39. On displacement, see Sanjoy Chakravorty, _The Price of Land: Acquisition, Conflict, Consequence_ (New Delhi: Oxford University Press, 2013), especially Appendix A9.2; and Singh, _Taming the Rivers,_ 182\u2013203. For a global perspective on dam displacement, see International Committee of the Red Cross, _World Disasters Report 2012: Focus on Forced Migration and Displacement_ (Geneva: Red Cross, 2012). On the disproportionate impact of dams on marginalized communities, see Esther Duflo and Rohini Pande, \"Dams,\" _Quarterly Journal of Economics_ 122 (2007): 601\u2013646.\n\n40. Arundhati Roy, \"The Greater Common Good,\" _Outlook,_ May 24, 1999; for a critique, see Ramachandra Guha, \"The Arun Shourie of the Left,\" _The Hindu,_ November 26, 2000.\n\n41. Vairamuthu, _Kallikaatu Ithihaasam_ (Chennai: Thirumagal, 2001).\n\n42. _Dams and Development: A New Framework for Decision Making. The Report of the World Commission on Dams_ (London: Earthscan, 2000).\n\n43. Ramaswamy R. Iyer, \"The Story of a Troubled Relationship,\" _Water Alternatives_ 6 (2013): 168\u2013176, quotations on pp. 169, 175; on Iyer's career, see Amita Baviskar's obituary: \"He Watered the Arid Fields of Administration with Intellectual Rigour and Honesty,\" _The Wire,_ September 11, 2015, last accessed May 2, 2018, https:\/\/thewire.in\/environment\/watering-the-arid-fields-of-administration-with-intellectual-rigour-and-honesty.\n\n44. Ravi S. Jha, \"India's River Linking Project Mired in Cost Squabbles and Politics,\" _The Guardian,_ February 5, 2013, accessed May 4, 2018, www.theguardian.com\/environment\/2013\/feb\/05\/india-river-link-plan-progress-slow; Supreme Court of India. Writ Petition (Civil) No. 512 of 2002 in Re. Networking of Rivers, judgment, accessed May 14, 2018, http:\/\/courtnic.nic.in\/supremecourt\/temp\/512200232722012p.txt; Y. A. Alagh, G. Pangare, and B. Gujja, _Interlinking of Rivers in India: Overview and Ken-Betwa Link_ (New Delhi: Academic Foundation, 2006).\n\n45. Ramaswamy R. Iyer, \"River Linking Project: A Disquieting Judgment,\" _Economic and Political Weekly,_ April 7, 2012, 33\u201340, quotations on p. 37.\n\n46. Meera Subramanian, _A River Runs Again: India's Natural World in Crisis, from the Barren Cliffs of Rajasthan to the Farmlands of Karnataka_ (New York: PublicAffairs, 2015).\n\n47. \"China Has Built the World's Largest Water Diversion Project,\" _The Economist,_ April 5, 2018; on the scheme in historical perspective, see Kenneth Pomeranz, \"The Great Himalayan Watershed: Water Shortages, Mega-Projects and Environmental Politics in China, India, and Southeast Asia,\" _Asia Pacific Journal: Japan Focus_ 7 (2009), accessed February 1, 2018, https:\/\/apjjf.org\/-Kenneth-Pomeranz\/3195\/article.html.\n\n48. C. J. V\u00f6r\u00f6smarty et al, \"Battling to Save the World's River Deltas,\" _Bulletin of the Atomic Scientists_ , 65, 2 (2009): 31\u201343; James Syvitski, \"Sinking Deltas Due to Human Activities,\" _Nature Geoscience_ 2 (2009): 681\u2013686; Roger L. Hooke, \"On the History of Humans as Geomorphic Agents,\" _Geology_ 28 (2000): 843\u2013846.\n\n49. Pomeranz, \"The Great Himalayan Watershed.\"\n\n50. Shripad Dharmadhikary, _Mountains of Concrete: Dam Building in the Himalayas_ (Berkeley: International Rivers, 2008); Douglas Hill, \"Trans-boundary Water Resources and Uneven Development: Crisis Within and Beyond Contemporary India,\" _South Asia: Journal of South Asian Studies_ 36 (2013): 243\u2013257; John Vidal, \"China and India 'Water Grab' Dams Put Ecology of the Himalayas in Danger,\" _The Observer_ , August 10, 2013.\n\n51. Dharmadhikary, _Mountains of Concrete_.\n\n52. Dharmadhikary, _Mountains of Concrete_ ; R. Grumbine and M. Pandit, \"Threats from India's Himalaya Dams,\" _Science_ 339 (2013): 36\u201337; Rohan D'Souza, \"Pulses Against Volumes: Transboundary Rivers and Pan-Asian Connectivity,\" in _Heading East: Security, Trade, and Environment Between India and Southeast Asia,_ ed. Karen Stoll Farrell and Sumit Ganguly (Oxford: Oxford University Press, 2016); Jane Qiu, \"Flood of Protest Hits Indian Dams,\" _Nature_ 492 (2012): 15\u201316; Fan Xiao quoted in Charlton Lewis, \"China's Great Dam Boom: A Major Assault on Its Rivers,\" _Yale Environment 360_ , November 4, 2013, accessed March 1, 2018, https:\/\/e360.yale.edu\/features\/chinas_great_dam_boom_an_assault_on_its_river_systems.\n\n53. T. Bolch et al., \"The State and Fate of the Himalayan Glaciers,\" _Science_ 336 (2012): 310\u2013314; World Bank, _Turn Down the Heat_ ; Dexter Filkins, \"The End of Ice: Exploring a Himalayan Glacier,\" _New Yorker,_ April 4, 2016.\n\n54. S. P. Xie et al., \"Towards Predictive Understanding of Regional Climate Change,\" _Nature Climate Change_ 5 (2015): 921\u2013930.\n\n55. A. Turner and H. Annamalai, \"Climate Change and the South Asian Monsoon,\" _Nature Climate Change_ 2 (2012): 587\u2013595; M. Bollasina, Y. Ming, and V. Ramaswamy, \"Anthropogenic Aerosols and the Weakening of the South Asian Summer Monsoon,\" _Science_ 334 (2011): 502\u2013505; Deepti Singh, \"South Asian Monsoon: Tug of War on Rainfall Changes,\" _Nature Climate Change_ 6 (2016): 20\u201322; R. Krishnan et al., \"Deciphering the Desiccation Trend of the South Asian Monsoon Hydroclimate in a Warming World,\" _Climate Dynamics_ 47 (2016): 1007\u20131027.\n\n56. J. Lelieveld, P. J. Crutzen, V. Ramanathan et al., \"The Indian Ocean Experiment: Widespread Air Pollution from South and Southeast Asia,\" _Science_ 291 (2001): 1031\u20131036; P. J. Crutzen and E. F. Stoermer, \"The Anthropocene,\" _Global Change Newsletter_ 41 (2000): 17\u201318.\n\n57. V. Ramanathan, \"Atmospheric Brown Clouds: Impact on South Asian Climate and Hydrological Cycle,\" _Proceedings of the National Academy of Science_ 102 (2005): 5326\u20135333; H. V. Henriksson et al., \"Spatial Distributions and Seasonal Cycles of Aerosols in India and China seen in Global Climate-Aerosol Model,\" _Atmospheric Chemistry and Physics_ 11 (2011): 7975\u20137990.\n\n58. Bollasina, Ming, and Ramaswamy, \"Anthropogenic Aerosols\"; Theodore G. Shepherd, \"Atmospheric Circulation as a Source of Uncertainty in Climate Change Projections,\" _Nature Geoscience_ 7 (2014): 703\u2013708.\n\n59. Singh, \"South Asian Monsoon,\" 21; Krishnan et al., \"Deciphering the Desiccation Trend\"; D. Niyogi, C. Kishtawal, S. Tripathi, and R. Govindaraju, \"Observational Evidence that Agricultural Intensification and Land Use Change May Be Reducing the Indian Summer Monsoon Rainfall,\" _Water Resources Research_ 46 (2010), https:\/\/doi.org\/10.1029\/2008WR007082.\n\n60. D. Singh, M. Tsiang, B. Rajaratnam, and N. Diffenbaugh, \"Observed Changes in Extreme Wet and Dry Spells During the South Asian Summer Monsoon,\" _Nature Climate Change_ 4 (2014): 456\u2013461; Krishnan et al., \"Deciphering the Desiccation Trend\"; B. N. Goswami, S. A. Rao, D. Sengupta, and S. Chakravorty, \"Monsoons to Mixing in the Bay of Bengal: Multiscale Air-Sea Interactions and Monsoon Predictability,\" _Oceanography_ 29 (2016): 28\u201337.\n\n61. Adam Sobel, _Storm Surge: Hurricane Sandy, Our Changing Climate, and Extreme Weather of the Past and Future_ (New York: Harper Wave, 2014), 203\u2013232; Amitav Ghosh, _The Great Derangement: Climate Change and the Unthinkable_ (Chicago: University of Chicago Press, 2016), 41\u201343.\n\n62. Data from the EM-DAT International Disaster Database, last accessed April 22, 2018, www.emdat.be.\n\n63. Ubydul Haque et al., \"Reduced Deaths from Cyclones in Bangladesh: What More Needs to Be Done?,\" _Bulletin of the World Health Organization_ 90 (2012): 150\u2013156, doi: 10.2471\/BLT.11.088302.\n\n64. A. Mahadevan et al., \"Freshwater in the Bay of Bengal: Its Fate and Role in Air-Sea Heat Exchange,\" _Oceanography_ 29 (2016): 72\u201381.\n\n65. \"Seafloor Holds 15 Million Years of Monsoon History,\" accessed April 10, 2018, https:\/\/news.brown.edu\/articles\/2015\/02\/monsoons.\n\n66. Concerned Citizens' Commission, _Mumbai Marooned: An Inquiry into the Mumbai Floods, 2005_ (Mumbai: Conservation Action Trust, 2006).\n\n67. Ghosh, _Great Derangement,_ 50\u201351.\n\n68. Anuradha Mathur and Dilip da Cunha, \"The Sea and Monsoon Within: A Mumbai Manifesto,\" in _Ecological Urbanism,_ ed. Mohsen Mostafavi with Gareth Doherty (Cambridge, MA: Harvard University Graduate School of Design\/Lars M\u00fcller, 2010), 194\u2013207.\n\n69. Susan Hanson et al., \"A Global Ranking of Port Cities with High Exposure to Climate Extremes,\" _Climatic Change_ 104 (2011): 89\u2013111; Orrin H. Pilkey, Linda Pilkey-Jarvis, and Keith C. Pilkey, _Retreat from a Rising Sea: Hard Choices in an Age of Climate Change_ (New York: Columbia University Press, 2016), 65\u201374.\n\n70. International Federation of Red Cross and Red Crescent Societies, \"Indonesia: Jakarta Floods,\" Information Bulletin 4.2007, September 26, 2007; Pilkey, Pilkey-Jarvis, and Pilkey, _Retreat from a Rising Sea,_ 70\u201371.\n\n71. On the contemporary geopolitics of the Bay, see Sunil S. Amrith _Crossing the Bay of Bengal: The Furies of Nature and the Fortunes of Migrants_ (Cambridge, MA: Harvard University Press, 2013), chapter 8.\n\n72. A. Mahadevan et al., \"Bay of Bengal: From Monsoons to Mixing,\" _Oceanography_ 29 (2016): 14\u201317, map on p. 16.\n\n73. International Federation of Red Cross Societies, _World Disasters Report 2012: Focus on Forced Migration and Displacement_ (Geneva: IFRC, 2013), 231.\n\n74. Amrith, _Crossing the Bay of Bengal,_ chapter 8.\n\n75. Joya Chatterji, \"Dispositions and Destinations: Refugee Agency and 'Mobility Capital' in the Bengal Diaspora, 1947\u20132007,\" _Comparative Studies in Society and History_ 55 (2013): 273\u2013304; IFRC, _World Disasters Report 2012,_ 38.\n\n76. _Groundswell: Preparing for Internal Climate Migration_ (Washington, DC: World Bank, 2018).\n\n77. World Bank, \"Policy Note #2: Internal Climate Migration in South Asia\" (2018), accessed May 14, 2018, https:\/\/openknowledge.worldbank.org\/bitstream\/handle\/10986\/29461\/GroundswellPN2.pdf?sequence=7&isAllowed=y.\n\n78. ASEAN, _Master Plan on ASEAN Connectivity 2025_ (Jakarta: Asean Secretariat, 2016); Constantino Xavier, _Bridging the Bay of Bengal: Towards a Stronger BIMSTEC_ (New Delhi: Carnegie India, February 2018).\n\n79. Aparna Roy, \"Bay of Bengal Diplomacy,\" _The Hindu,_ October 10, 2017.\n\n80. Season Watch, accessed May 1, 2018, www.seasonwatch.in\/.\n\n81. Prasenjit Duara, _The Crisis of Global Modernity: Asian Traditions and a Sustainable Future_ (Cambridge: Cambridge University Press, 2014).\n\n82. Gandhi quoted in Singh, \"Indian Monsoon in Literature,\" 50.\n\nEPILOGUE: HISTORY AND MEMORY AT THE WATER'S EDGE\n\n1. Zadie Smith, \"Elegy for a Country's Seasons,\" _New York Review of Books,_ April 3, 2014.\n\n2. Namrata Kala, \"Learning, Adaptation, and Climate Uncertainty: Evidence from Indian Agriculture,\" working paper, August 2017, accessed March 3, 2018, https:\/\/namratakala.files.wordpress.com\/2017\/08\/kala_learning_aug2017_final.pdf, last.\n\n3. Suprabha Seshan, \"Once, the Monsoon,\" June 22, 2017, accessed March 10, 2018, https:\/\/countercurrents.org\/2017\/06\/22\/once-the-monsoon\/.\n\n4. M. Rajshekhar, \"Why Tamil Nadu's Fisherfolk Can No Longer Find Fish,\" _Scroll,_ July 8, 2016, accessed April 15, 2018, https:\/\/scroll.in\/article\/808960\/why-tamil-nadus-fisherfolk-can-no-longer-find-fish, the quotation in the text is from Rajshekhar's report; E. Vivekanandan, \"Impact of Climate Change in the Indian Marine Fisheries and Potential Adaptation Options,\" in _Coastal Fishery Resources of India: Conservation and Sustainable Utilisation_ (Cochin, India: Society of Fisheries Technologists, 2010), 169\u2013184; Amitav Ghosh and Aaron Savio Lobo, \"Bay of Bengal: Depleted Fish Stocks and Huge Dead Zone Signal Tipping Point,\" _The Guardian,_ January 31, 2017.\n\n# INDEX\n\naerial video, 239\u2013240\n\naerosols (brown cloud), 2\u20133, 305\u2013307\n\nAgarwal, Anil, 282\u2013283, ,\n\nagriculture\n\nCanal Colonies, 122\u2013124,\n\nand climate change,\n\nclouds,\n\ncontrol of water,\n\ncyclone of 1942 and WWII, 168\u2013169\n\nand drought, 68\u201369,\n\ndrought insurance,\n\nand economy, , 133\u2013136\n\nand electricity, 257\u2013258, ,\n\nand famine, 73\u201374,\n\nfood aid from US,\n\nfood prices, 68\u201369\n\nfood production, , 279\u2013280\n\nfoodgrain self-sufficiency, 243\u2013244\n\nfrontier colonization, 122\u2013125\n\ngovernment policy in 1950s and 60s, 243\u2013247\n\nand Green Revolution, 245\u2013247, , , , 283\u2013284\n\ngroundwater and wells, 256\u2013258, , 276\u2013280\n\nand inequalities,\n\nintensification in India, 192\u2013193, ,\n\nand irrigation, , , 133\u2013135, , , 258\u2013259\n\nland redistribution and _zamindari_ abolition,\n\nand meteorology, 36\u201338\n\nand migration within Asia, 160\u2013161\n\nand money lending, 73\u201374, ,\n\nand monsoons, , , , ,\n\nPartition and canals, ,\n\nrain and rainfall, , 134\u2013136, ,\n\nrice economies in WWII,\n\nsuicide problem,\n\nwater as resource,\n\nAkbar, Emperor, ,\n\n_Akbar Nama,_\n\nAlgu\u00e9, Jos\u00e9, 104\u2013107\n\nAmbedkar, Bhimrao\/B.R., 154\u2013155,\n\nArthur Cotton museum, ,\n\nAsia\n\nclimate change, , , 303\u2013304,\n\ncontrol of water, 5\u20138,\n\ndam building post-WWII, 177\u2013179\n\nearthquake of 1950,\n\neconomic conditions post-WWII, 213\u2013214\n\nend of imperial rule, 175\u2013177\n\ngroundwater resources,\n\nhistory and history-writing, 5\u20138, 9\u201310\n\nidentity and freedom,\n\nas integrated climatic system, 108\u2013109, 173\u2013174\n\nmegacities by the ocean, 270\u2013271 (map)\n\nmigration and agriculture, 160\u2013161\n\nnationalism, 147\u2013148, 151\u2013153\n\nrivers, , , , 289\u2013292, 290 (fig.) ( _see also_ specific areas' rivers)\n\nrole of monsoons, 155\u2013157,\n\nshared water resources,\n\nwater crisis of 1980s, , 272\u2013274, 280\u2013281\n\nwater schemes and regional cooperation, 214\u2013215\n\nweather control schemes by US, 250\u2013251\n\nAsiatic Society of Bengal, 60\u201361\n\natmospheric pressure, 140\u2013141\n\nAttlee, Clement,\n\nAustralia, drought science,\n\n_Awaara_ (movie),\n\nBabur (Zahir-ud-din Muhammad Babar), 27\u201328,\n\nBailyn, Bernard,\n\nBaker, Christopher, 159\u2013160\n\nBalaji, Sakharam, 126\u2013127\n\nBall, Valentine,\n\nBanerji, S. K.,\n\nBangladesh, , 309\u2013310\n\nBari Doab canal,\n\n_Baur Nama,_\n\nBay of Bengal\n\ncyclones, , 60\u201361, , 308\u2013309\n\nand monsoon science, ,\n\nBayly, Christopher,\n\nBelgaum district (Bombay),\n\nBengal\n\nborders and nature, 181\u2013182\n\nBritish control and trade, , 47\u201348\n\ncyclones and storms, 58\u201359, 60\u201363, 91\u201392, 168\u2013169,\n\nfamine and crisis in WWII, 168\u2013170\n\nand Partition, 148\u2013149, 179\u2013180, 181\u2013182\n\nBennett, M. K.,\n\nBennett, W. C., 119\u2013120\n\nBhakra Dam and Bhakra Nangal project, 197 (fig.), 202 (fig.)\n\nconstruction and workers, 199\u2013201, 201 (fig.), 202 (fig.)\n\ndedication, 196 (fig.)\n\ndescription and background, , 196\u2013198\n\ndisplacement of people, 210\u2013211\n\nfilm, 198\u2013201\n\nBhatt, Jyoti, 275\u2013276\n\nBhattacharjee, Kapil Prasad,\n\nBihar\n\nfamines, 79\u201380, 248\u2013250\n\nrain control,\n\nBjerknes, Jacob, , 262\u2013264\n\nBjerknes, Vilhelm, ,\n\nBlanford, Henry Francis\n\nclimate and rainfall reports,\n\nmonsoon and storm science, 60\u201363, 99\u2013103, ,\n\nscience and geology, 59\u201360,\n\nBlanford, William Thomas,\n\nBloch, Marc,\n\nBombay (now Mumbai), 67\u201368, 126\u2013127, ,\n\nborders\n\nAsia as integrated climatic system, 108\u2013109, 173\u2013174\n\nand climate change, 317\u2013319, 329\u2013330\n\ncompetition for water resources, 162\u2013163\n\ncontrol of water, 164\u2013165\n\ncooperation for water schemes and climate, 319\u2013323\n\nand dams,\n\nand fisheries,\n\nand Himalayan rivers, ,\n\nIndia\u2013China border, 164\u2013165, 177 (map), 225\u2013226, 227\u2013228\n\nand Partition, 180\u2013182\n\nBoreal Summer Intraseasonal Oscillation,\n\nBowles, Chester,\n\nBrahmaputra, , 35 (map), , 226\u2013227\n\nBraudel, Fernand,\n\nBRBD (Bambanwala-Ravi-Bedian-Dibalpur) project,\n\n\"bread riots\" (1960s),\n\n\"The Bridge Builders\" (Kipling),\n\nbridges for railway, 56\u201357\n\nBritain, , 175\u2013176\n\nBritish India\n\nin 1900, 18 (map)\n\narmy,\n\nborder with China, 164\u2013165\n\nBritish self-interest in India, 77\u201378, 114\u2013116\n\nBritish expansion, 33\u201334\n\ncaste relations, 154\u2013155\n\ncities expansion, 127\u2013128\n\ncivil society and humanitarian aid, 84\u201385, 86\u201387\n\nclimatology atlas, 108\u2013109\n\nclouds and rain,\n\ncontrol of water, , , 38\u201345, 46\u201347, , 164\u2013165\n\ncustoms duties, 48\u201349\n\ndata collection on India, 96\u201397, 117\u2013118, 134\u2013135\n\ndessicationism, 74\u201375,\n\ndrain of wealth by British, 77\u201378\n\ndrought science, , , 72\u201373\n\neconomy, 11\u201312, 38\u201339, 133\u2013134, 160\u2013161\n\nend of imperial rule, 175\u2013176, 180\u2013181\n\nfamine ( _see_ famines)\n\nfishing industry, 131\u2013133\n\nGanges, , 34\u201335\n\nGanges canal, 41\u201342, 43\u201345\n\ngeology, 59\u201360\n\ngeopolitics of water, 110\u2013111\n\nhydroelectricity, 128\u2013129\n\nimpact on water in India and Asia, 11\u201313\n\nIndian Rebellion of 1857, 45\u201346\n\nIndianization of officials, 143\u2013144\n\nindustrial development, 128\u2013129,\n\nirrigation schemes, 19\u201320, , , , 122\u2013124,\n\nland acquisition and displacement of people, 126\u2013127,\n\nmeteorology of cyclones and monsoon, 36\u201338, 60\u201364, 92\u201393, 98\u2013104, , , 139\u2013143\n\nmigration for agriculture in 1930s, 160\u2013161\n\nand monsoon, 21\u201322, 36\u201337, , 58\u201359\n\nmunicipal water supply,\n\nnationalism, 114\u2013115, 148\u2013153\n\nas oceanic realm, 131\u2013132,\n\npolitical reforms, ,\n\nprincely states breakup, 206\u2013207\n\nracial relations,\n\nrailway, 51\u201358\n\nsources of water, 113\u2013114\n\nstate-directed settlement, 122\u2013125\n\nsteamboats, 49\u201351\n\nSwadeshi movement, 148\u2013149\n\ntaxation, ,\n\ntechnological development, 56\u201357\n\nunderground water, 119\u2013121\n\nvessels for trade, 47\u201348, 49\u201350\n\nwater transportation, 38\u201339\n\nWWII period, 166\u2013167\n\nYamuna Canal restoration, 42\u201343\n\nbrown cloud, 2\u20133, 305\u2013307\n\nBurma, dam workers in India, 203\u2013205\n\nBurma Meteorological Department,\n\nBurma (under British rule), 164\u2013165, , 175\u2013176\n\nCaird, James,\n\nCalcutta (now Kolkata), 49\u201351, 58\u201359\n\nCalvino, Italo, ,\n\nCanal Act (1568, India),\n\nCanal Colonies, 122\u2013124, 125\u2013126, , ,\n\ncanals\n\nin India, 41\u201345, 197\u2013198\n\nand Partition, 184\u2013185, , 187\u2013188\n\n_See also_ specific canals\n\ncapitalism\n\nand famines, 73\u201374, 87\u201388\n\nand fisheries, 328\u2013329\n\nin India, 11\u201312, ,\n\ncarbon and CO2 in ocean, 241\u2013242\n\ncaste, 154\u2013155, 287\u2013288\n\nCautley, Proby, 41\u201343\n\nCentral Water and Power Commission of India, ,\n\nCentre for Science and Environment, ,\n\nCeylon (later Sri Lanka), fisheries and borders,\n\nChakravarti, J. S., 135\u2013136\n\nChakravorty, Sanjoy, ,\n\nChatterton, Alfred, ,\n\nChennai. _See_ Madras\n\nChiang Kai-shek, ,\n\nChina\n\nagricultural growth and irrigation in 1970s\u201380s,\n\nborder with India, 164\u2013165, 177 (map), 225\u2013226, 227\u2013228\n\ncivil war and creation of PRC, 176\u2013177\n\ncontrol of water,\n\ncyclone and storm forecasts, 104\u2013105\n\ndam building, , , 300\u2013302\n\ndam financing, 301\u2013302\n\ndroughts and famines, , ,\n\neconomic growth of 1980s,\n\nenvironmental movement, , 290\u2013291\n\nflood control, , 217\u2013218\n\nfood production, ,\n\nfood security,\n\ngroundwater and wells,\n\nlife expectancy,\n\nmigration for agriculture in 1930s, 160\u2013161\n\n\"moral\" meteorology,\n\nnationalism, , 152\u2013153\n\npopulation growth,\n\nrelationship with India, 215\u2013218, 220\u2013223,\n\nresource planning for water,\n\nriver pollution, 291\u2013292\n\nstate-directed settlement, 124\u2013125\n\nTibet interests, 226\u2013227,\n\nwar with Japan, 165\u2013166\n\nwater crisis, 278\u2013279, 291\u2013292\n\nwater diversion project, 298\u2013299\n\nwater schemes and management, , 218\u2013222, , 298\u2013299\n\nworkers mobilization, 219\u2013220\n\n_China Environment Yearbook,_\n\n_China's Water Crisis_ (Ma Jun),\n\ncities, 127\u2013128, 270\u2013271 (map)\n\n_See also_ coastal regions and cities\n\ncitizen scientists,\n\ncivil society\n\nenvironmental cooperation, 320\u2013322\n\nfamines of 1890s, 84\u201385, 86\u201387\n\nclimate (weather)\n\nagency of climate, 70\u201371\n\nAsia as climatic system, 108\u2013109, 173\u2013174\n\nand clouds,\n\ncomplexity, 265\u2013266\n\nand culture, 155\u2013158\n\ncycles changed by human actions, , 74\u201376\n\ndessicationism, 74\u201375,\n\nand Indian Ocean Expedition, 230\u2013231,\n\ninternational cooperation, 320\u2013322\n\nmigration and refugees, 317\u2013319\n\nand monsoons, 15\u201316\n\nas moral concern, 71\u201373\n\nand politics, 252\u2013253\n\nand population of India, 190\u2013191\n\nrisks, 313\u2013315\n\nand understanding of Asia, 155\u2013157\n\n_See also_ meteorology\n\nclimate change\n\nand activism, 285\u2013286\n\nand agriculture,\n\nand borders, 317\u2013319, 329\u2013330\n\ncarbon and CO2 in ocean, 241\u2013242\n\nand cyclones' intensity, 308\u2013309\n\nimpact in Asia, , , 303\u2013304,\n\ninternational cooperation, 319\u2013320\n\nland cover and forests,\n\nand life cycles,\n\nand loss, 325\u2013330\n\nand monsoons, 304\u2013305, , 308\u2013309\n\nas risk, , 273\u2013274, 303\u2013304\n\nand water crisis, , 316\u2013317\n\n_Climatological Atlas of India_ (Eliot), 108\u2013109\n\nclouds, 93\u201395, ,\n\ncoastal regions and cities\n\nchanges and losses, 326\u2013330\n\nclimate change impact,\n\nstorm risks and preparation, 312\u2013313, 315\u2013316\n\ncolonialism, 11\u201312\n\n_See also_ British India\n\nColvin, John, ,\n\nCongress party (Indian National Congress), , , 166\u2013167, , ,\n\nresource planning for water, 163\u2013164\n\ncontrol of water\n\nin agrarian history,\n\nin Asia, 5\u20138,\n\nand borders, 164\u2013165\n\nBritish India, , , 38\u201345, 46\u201347, , 164\u2013165\n\nand colonialism,\n\nGodavari River, , 39\u201340\n\nin India's history, 26\u201327,\n\nas quest, 5\u20136\n\n_See also_ irrigation\n\nCotton, Arthur Thomas, 19 (fig.)\n\ncontrol of water and irrigation, 39\u201341, , , ,\n\nGanges canal, 41\u201342\n\nGodavari River, 17\u201320, 39\u201340\n\nlegacy of, , ,\n\nand railway,\n\nCredner, William,\n\nCrutzen, Paul,\n\ncultivators. _See_ agriculture\n\nculture, and climate, 155\u2013158\n\nCurzon, Lord,\n\ncyclones and storms\n\n1864 storm, 58\u201359, 60\u201363\n\n1876 storm, 91\u201392\n\n1891 and 1897 storms, 106\u2013107\n\n1942 storm, 168\u2013169\n\nBangladesh, 309\u2013310\n\nand climate change, 308\u2013309\n\n\"cyclone\" as new word,\n\nradar tracking,\n\nscience and forecasting, 60\u201363, 104\u2013106, , , 314\u2013315\n\n_See also_ monsoons\n\n_The Cyclones of the Far East_ (Algu\u00e9), 105\u2013106, 107 (fig.)\n\nDalai Lama, ,\n\nDalhousie, Marquess, 51\u201352\n\nDalits, , , 287\u2013288\n\nDamodar River and Valley, and project, 172\u2013173, ,\n\ndams\n\nand borders,\n\nbuilding in Asia, 177\u2013179\n\nin China, , , 300\u2013302\n\ncosts, 179\u2013180\n\ndevelopment in India, 177\u2013178, , 193\u2013196, 201\u2013203, 207\u2013208, , , 297\u2013298,\n\ndisplacement of people, 210\u2013211, 212\u2013213, , 295\u2013296\n\nenvironmental impact, , 296\u2013297, , 302\u2013303\n\nfinancing, 301\u2013302\n\nin India, 178 (map)\n\nproblems from, , 294\u2013297, , 302\u2013303\n\nprotests and resistance, 292\u2013294,\n\nsymbolism,\n\nworkers in India, 199\u2013201, 202 (fig.), 203\u2013205\n\n_See also_ hydroelectricity; specific dams and projects\n\nDas, P. K.,\n\nDavis, Mike,\n\ndeforestation, 74\u201376, , 172\u2013173,\n\ndessicationists, 74\u201375,\n\n_Diana_ (steamboat),\n\nDigby, William, 78\u201379,\n\ndisplacement of people\n\ndams, 210\u2013211, 212\u2013213, , 295\u2013296\n\nand land acquisition, 126\u2013127, 210\u2013211,\n\ndiversion of rivers, , 297\u2013299\n\nD'Monte, Darryl,\n\nDoel, Ronald, ,\n\nDonaldson, Dave,\n\nDouie, James, 122\u2013123,\n\nDowleswaram dam, ,\n\nDr\u00e8ze, Jean,\n\ndrought of 1876\u20131879 in India\n\ndescription, 65\u201367\n\ndrought science, , , 72\u201373, ,\n\nand famine, 65\u201374\n\nand human intervention, 73\u201376\n\nrain and rainfall, , 67\u201369\n\nsunspots theory, 82\u201383\n\ndroughts\n\n1896, 84\u201386\n\n1899\u20131900,\n\n1950s to today,\n\nmid-1960s, 230\u2013231, 242\u2013243, ,\n\n1970 to 1973, 260\u2013262\n\nagricultural _vs._ meteorological droughts,\n\nand agriculture, 68\u201369, ,\n\nand El Ni\u00f1o events,\n\nforecasts,\n\ninsurance in agriculture,\n\nand monsoons, 68\u201369, 230\u2013231\n\nand politics, 252\u2013253\n\nscience of, , , 72\u201373, , 95\u201396, , , 230\u2013231\n\nand water distribution, 286\u2013287\n\nDutt, Romesh Chander, 115\u2013116\n\nearthquake of 1950,\n\nEast India Company, , 45\u201346, 48\u201349\n\nEconomic Commission for Asia and the Far East (ECAFE), 213\u2013215, 223\u2013224\n\neconomy\n\nagriculture and cultivators, , 133\u2013136\n\nin British India, 11\u201312, 38\u201339, 133\u2013134, 160\u2013161\n\nconditions in post-WWII Asia, 213\u2013214\n\nfood economy and control, , ,\n\ngrowth of 1980s in India, 272\u2013273\n\nand irrigation, 133\u2013135\n\nand land, 160\u2013161\n\nmarket collapse of 1930s, 159\u2013161\n\nand Partition, 183\u2013184, 186\u2013187\n\npolicy in 1950s and 60s, 243\u2013244,\n\nand railway, , 53\u201355\n\nrice economies in WWII,\n\nEhrlich, Paul,\n\nEl Ni\u00f1o, , , 263 (fig.), 264\u2013265\n\nEl Ni\u00f1o Southern Oscillation (ENSO), 264\u2013265\n\nelectricity. _See_ hydroelectricity\n\nEliot, John\n\natlas of climatology of India, 108\u2013109\n\ncyclone and storm science, , , ,\n\nmeteorology, 103\u2013104\n\nmonsoons forecasts, 107\u2013108, ,\n\nenvironment and nature\n\nactivism and environmental movement, 281\u2013286, , 290\u2013291, 321\u2013322\n\nclimate and culture, 157\u2013159\n\nconference of 1972, 253\u2013255\n\ncooperation cross-border, 320\u2013322\n\ndams' impact, , 296\u2013297, , 302\u2013303\n\nimpact of post-WWII changes,\n\nimpact on history of Asia, 6\u20137\n\nmigration and migrant workers, 329\u2013330\n\nand Partition, 181\u2013182\n\nproblems in 1970s India, 253\u2013254,\n\nprotests against dams, 292\u2013294\n\ntraditional water management, 281\u2013283\n\nvulnerability of India, , , 134\u2013135, , 207\u2013208,\n\nwater legislation in India, 255\u2013256\n\nenvironmental crises, impact,\n\nepidemics of 1890s in India,\n\nerosion, 327\u2013328\n\nEurope, trade to India and Asia, 31\u201333\n\nFamine Codes, , ,\n\nFamine Enquiry Commission (1880), 81\u201382,\n\nfamine of 1876\u20131878 in India\n\ndescription, 65\u201366, 68\u201369\n\nand drought, 65\u201374\n\nenquiry commission, 81\u201383\n\nand human intervention, 72\u201376\n\nand inequality, 73\u201374\n\nin Madras, 78\u201379\n\nand meteorology, 69\u201370, 82\u201383, 102\u2013103\n\nand monsoon, ,\n\nrelief and response, 78\u201381, 82\u201383\n\nand state failure, 76\u201379, 81\u201382\n\nfamines\n\nagriculture and cultivators, 73\u201374,\n\nand capitalism, 73\u201374, 87\u201388\n\nconsequences,\n\nand food imports,\n\nand irrigation, ,\n\nand meteorology, 69\u201370, 82\u201383, 102\u2013103\n\nresponsibility for, , , , 87\u201388, 115\u2013116, 169\u2013170,\n\nsolutions to, 136\u2013137\n\nand water schemes,\n\n_See also_ specific famines\n\nfamines of 1890s in India, 84\u201387\n\nfarmers and farming. _See_ agriculture\n\nField, J. H.,\n\nFilms Division of India,\n\n_First Citizen's Report on the State of India's Environment_ (Agarwal and Narain), 282\u2013283\n\nfisheries, 131\u2013133, , 328\u2013329\n\nfood\n\nfeeding challenge in India, 191\u2013193\n\nfood aid, , , 249\u2013250\n\nimports in famines,\n\npolicy and strategy in India, 243\u2013247\n\nprices, 68\u201369,\n\nproduction, 274\u2013275, 279\u2013280\n\nriots of 1964, 274\u2013275\n\nsecurity and control, , ,\n\nself-sufficiency, , 243\u2013244, 322\u2013323\n\nFood Corporation of India, 261\u2013262\n\nForest Act (1878, India),\n\nforests and deforestation, 74\u201376, , 172\u2013173,\n\nFreeman, Orville,\n\nFriends of Nature, ,\n\nfrontiers. _See_ borders\n\nfunctionalism,\n\nFurnivall, John,\n\nGandhi, Indira, 248\u2013249, 249 (fig.), 253\u2013255,\n\nGandhi, Mohandas Karamchand \"Mahatma,\" 149\u2013151, , ,\n\nGanges, 35 (map)\n\nand British power, , 34\u201335\n\ncanal, 41\u201342, 43\u201345\n\ncentrality and reverence in India, 25\u201326, 44\u201345,\n\nchanges in course,\n\ncustom duties and taxation, 48\u201349\n\ncyclone of 1876,\n\nmonsoons' power, 57\u201358\n\nnavigation and trade, 47\u201348,\n\npollution, 289\u2013290\n\nand spirituality,\n\nsteamboats, 49\u201350\n\nGangetic plain, and brown cloud, 305\u2013306\n\nGastrell, James, 61\u201363\n\nGeological Survey of India,\n\ngeology, 1\u20132, 59\u201360,\n\nGhosh, Amitav,\n\nGinsburg, Norton,\n\nglaciers, and climate change,\n\n_Global Warming in an Unequal World_ (Agarwal and Narain), ,\n\nGlobal Weather Experiment,\n\nGodavari River, irrigation and control of water, 18\u201320, , 39\u201341\n\nGondwana,\n\nGopalaswami, R. A., 188\u2013189,\n\nGovernment of India Act (1919),\n\ngroundwater, for irrigation and food production, 256\u2013258, , 276\u2013280\n\nGujarat, groundwater and water table,\n\nHalley, Edmund,\n\nHaridwar headworks,\n\nHarper, Kristine, ,\n\nHart, Henry, 202\u2013203,\n\nHart, Robert,\n\nHart Schaaf, C., 214\u2013215\n\nhaze (brown cloud), 2\u20133, 305\u2013307\n\nHedin, Sven,\n\nHem Raj, R. B.,\n\nHildebrandsson, Hugo, and colleagues, 93\u201394\n\nHimalayan rivers, x\u2013xi (map), xviii (fig.)\n\nborders and countries, ,\n\nclimate change impact, , 303\u2013304\n\ndams and dam building, , 299\u2013303\n\ndata and information sharing, ,\n\ndescription and history,\n\nearthquake of 1950,\n\nenvironmental crises,\n\nexploration and mapping,\n\nin geopolitical view, 110\u2013112\n\nand pollution, 2\u20133,\n\nas shared water resources, , 303\u2013304\n\nHimalayas, , 63\u201364, 109\u2013112, 303\u2013304\n\nHirakud Dam, 193\u2013194, 204\u2013205\n\nhistory and history-writing\n\nlatent and manifest events,\n\nviews on water and its impact, 5\u201310\n\nHooker, Joseph, 63\u201364\n\nHornell, James, 132\u2013133\n\nHuang Wanli,\n\nHumboldt, Alexander von,\n\n\"Hundred Flowers\" campaign,\n\nHunter, William Wilson, 109\u2013111\n\nhydroelectricity, 128\u2013129, 257\u2013258, , ,\n\nIltutmish, Sultan,\n\nImperial India. _See_ British India\n\nimperialism, impact on Asia, 11\u201312\n\nIndia \/ Indian subcontinent\n\nambition post-independence, 193\u2013195, , 197\u2013198, 207\u2013208\n\nclimatology and politics, 108\u2013109\n\nelections of 1967,\n\ninfrastructure neglect,\n\nas inspiration for water schemes, 223\u2013224\n\nlife expectancy, 189\u2013190\n\npolitical awakening, 114\u2013115, 148\u2013149\n\npopulation history and growth, 190\u2013191, ,\n\nrepresentative politics, 67\u201368\n\nstate of emergency of 1975,\n\nas \"subcontinent,\" 109\u2013110\n\ntraditional water management, 281\u2013283\n\nas vantage point for water history, 10\u201316\n\nvulnerability towards nature, , , 134\u2013135, , 207\u2013208,\n\nwater crisis, 278\u2013283,\n\nwater debates post-independence,\n\nwater diversion project, 297\u2013298\n\nwater legislation, 255\u2013256\n\nwater schemes post-WWII, 171\u2013173, 194\u2013195, , 221\u2013222, 292\u2013293\n\nwater table,\n\nwet and dry\/arid line,\n\n_See also_ British India; specific topics, events, and waters\n\nIndian Industrial Commission, ,\n\nIndian Irrigation Commission, 117\u2013120,\n\nIndian Meteorological Office\/Department\n\nchanges at, 141\u2013142, 143\u2013144,\n\ndirectors, , , ,\n\nIndian staff and Indianization, 97\u201399,\n\nin International Meteorological Centre,\n\nfor military in WWII,\n\nmonsoon forecasts,\n\nand Partition, 182\u2013183\n\nIndian National Congress. _See_ Congress party\n\nIndian Ocean\n\ncoastal megacities, 270\u2013271 (map),\n\nknowledge and study ( _see_ Indian Ocean Expedition)\n\ntsunami of 2004, 327\u2013328\n\nIndian Ocean Expedition\n\ncarbon and CO2, 241\u2013242\n\nand climate science, 230\u2013231,\n\nforecasts of monsoons, 240\u2013241\n\nIndian ships in study,\n\ninternational cooperation, 234\u2013235\n\nmeteorology study, 236\u2013237\n\nmonsoons and winds, 233\u2013234, 236\u2013237, 239\u2013241,\n\ntechnology in study, 239\u2013241\n\nIndian Rebellion of 1857, 45\u201346\n\nIndian Supreme Court,\n\nIndianization of officials, 143\u2013144\n\nIndonesia, end of imperial rule,\n\nIndus Treaty, , 319\u2013320\n\nIndus waters\n\ndivision, , 319\u2013320\n\nand Partition, 185\u2013186, 187\u2013188, 205\u2013206,\n\nsource in China,\n\nindustrial capitalism. _See_ capitalism\n\nindustrial development in British India, 128\u2013129,\n\ninequality\n\nand agriculture,\n\nand famine, 73\u201374\n\nand railway, , 55\u201356\n\nrural India, 280\u2013281\n\nand water, , , 287\u2013288\n\nInternational Conference on the Law of the Sea (1958),\n\nInternational Meteorological Centre (at Colaba), , 238\u2013239\n\nInternational Meteorological Organisation (IMO), 93\u201394\n\nIrrawaddy triangle, borders and control of water, 164\u2013165\n\nirrigation\n\nand agriculture, , , 133\u2013135, , , 258\u2013259\n\nand economy, 133\u2013135\n\nand famine, ,\n\nGodavari River, 18\u201320, 39\u201341\n\ngroundwater and wells, 119\u2013121, 256\u2013258, , 276\u2013280\n\ninvestment and intervention, , 118\u2013119\n\noverview as problem,\n\nand Partition,\n\nPunjab and Canal Colonies, 122\u2013124, 125\u2013126, , , 184\u2013185,\n\nrain and rainfall,\n\nirrigation schemes\n\nearly India, 26\u201327, ,\n\nBritish India, 19\u201320, , , , 122\u2013124,\n\nindependent India, 194\u2013196, 256\u2013257, 258\u2013259\n\nIyer, Ramaswamy, 296\u2013297,\n\nJakarta,\n\nJammu and Kashmir, and Partition,\n\nJapan, , , 165\u2013166,\n\nJinnah, Muhammad Ali, ,\n\nJohnson, Lyndon Baines \"LBJ,\" , 250\u2013251\n\n_JOIDES Resolution_ (ship),\n\nJulian, Paul,\n\nKashmir, , 247\u2013248\n\nKaveri River, dispute, 162\u2013163, 206\u2013207\n\nKhan, Mehboob, , 208\u2013209\n\nKhosla, A. N., 195\u2013196\n\nKipling, Rudyard,\n\nKolkata. _See_ Calcutta\n\nKrishnamachari, T. T.,\n\nKrishnarajasagar dam, , ,\n\nLa Ni\u00f1a, 264\u2013265\n\nLadejinsky, Wolf, 260\u2013261,\n\nLakshminarayana, V.,\n\nland\n\nacquisition and displacement of people, 126\u2013127, 210\u2013211,\n\nand economy, 160\u2013161\n\nredistribution and _zamindari_ abolition,\n\nLand Acquisition Act (1894, India) and officers, , ,\n\nLarkin, David, ,\n\nlaw of the sea, and territorial waters, 232\u2013233\n\nlegislation in India, 255\u2013256\n\nLi Yizhi,\n\nLiang Congjie, 290\u2013291\n\nLilienthal, David, 187\u2013188, , ,\n\nLokanathan, P., ,\n\nMa Jun, 291\u2013292\n\nMacGeorge, George W., , ,\n\nMackinder, Halford, 111\u2013112\n\nMadden, Roland,\n\nMadden-Julian Oscillation (MJO), 265\u2013266\n\nMadras (now Chennai)\n\nfamine, 78\u201379, 80\u201381\n\nfisheries, , ,\n\nwater mining, 120\u2013121\n\nMadras Presidency, 162\u2013163, , 206\u2013207\n\nMahad (India), struggle for freedom,\n\nMahalanobis, Prasanta Chandra, ,\n\nMaharashtra, drought, 260\u2013262,\n\nmalaria,\n\nMalaya (now Malaysia), migrant workers,\n\nManchuria, 124\u2013125\n\nMani, Anna, 237\u2013238\n\nManila Observatory, , 105\u2013106\n\nManto, Saadat Hasan,\n\nMao Zedong, , , , 219\u2013220\n\nMaoist insurgency, 280\u2013281\n\nMarsh, George Perkins,\n\nMarx, Karl,\n\nmegacities in Asia, 270\u2013271 (map),\n\nmegaprojects in India, 193\u2013194\n\nMehta, M. C., 289\u2013290\n\nMekong River, 178\u2013179, 223\u2013225,\n\nMelaka, textiles, 30\u201331\n\nMerrall, Edwin, 52\u201353\n\nmeteorology\n\nand agriculture, 36\u201338\n\nAsia as integrated climatic system, 108\u2013109\n\nclouds study, 93\u201395\n\ndata collection and sharing, 96\u201397, 101\u2013102, 104\u2013105, , ,\n\ndrought science, , , 72\u201373\n\nand famines, 69\u201370, 82\u201383, 102\u2013103\n\nmonsoon science, 36\u201338, 60\u201364, 92\u201393, 98\u2013104, , , 139\u2013143, 236\u2013237, 304\u2013306, , ,\n\n\"moral\" meteorology, 72\u201373\n\nand Partition,\n\npolitics and climate, 108\u2013109\n\nas science,\n\nships in research, , , 61\u201362, , , 104\u2013105, ,\n\nin WWII, 167\u2013168\n\n_See also_ climate; Indian Meteorological Office\/Department\n\nMettur Dam, 161\u2013162\n\nmigration and migrant workers, 160\u2013161, 317\u2013319\n\nmilitary,\n\n_See also_ Second World War (WWII)\n\nModi, Narendra, 293\u2013294\n\n_Moneka_ (ship),\n\n\"Monsoon Asia,\" ,\n\nMonsoon Experiment (MONEX), 266\u2013267\n\nmonsoons, xiv\u2013xv (maps)\n\nand agriculture, , , , ,\n\nand Asia, 155\u2013157,\n\nin British India, 21\u201322, 36\u201337, , 58\u201359\n\nchanges and shifts, , 327\u2013328\n\nand climate, 15\u201316\n\nclimate change, 304\u2013305, , 308\u2013309\n\nand culture, 155\u2013158\n\ndata collection, 61\u201362, , 101\u2013102, 104\u2013105, , ,\n\ndefinition and terminology, ,\n\ndescription, 13\u201314, 23\u201324\n\nand disease,\n\nand drought, 68\u201369, 230\u2013231\n\nand ENSO, 264\u2013265\n\nfailure of 1965 and 1966, 242\u2013243, ,\n\nfamine of 1876\u20131878 in India, ,\n\nforecasts, 102\u2013103, 107\u2013108, , 139\u2013140, 142\u2013143, 240\u2013241, 266\u2013267, , ,\n\nhistory and evolution, 22\u201323\n\nimpact and human responses to,\n\nas lived experience, 229\u2013230\n\nand meteorological science, 36\u201338, 60\u201364, 92\u201393, 98\u2013104, , , 139\u2013143, 236\u2013237, 304\u2013306, , ,\n\nplanning and preparation for,\n\npower and dangers, 57\u201358\n\npredictability and patterns, , , 306\u2013307, ,\n\nand railway in India, 52\u201353\n\nand rain science, 95\u201396, 100\u2013101, 140\u2013141, 304\u2013305\n\nrice economies in WWII,\n\nand rivers in India, 25\u201326\n\nrole and influence in India, 13\u201316, 24\u201325, 156\u2013157, 275\u2013276\n\nscience of, 63\u201364, 92\u201393, 262\u2013266,\n\nand seabed,\n\nstatistical analysis, 139\u2013140\n\nas system and pattern, 23\u201325\n\nand transportation, , 52\u201353\n\nwet and dry line in India,\n\nand wind, 23\u201324, 99\u2013100, , 236\u2013237\n\n_See also_ cyclones and storms\n\n_Mother India_ (movie), 207\u2013208, 209\u2013210\n\nMueenuddin, Daniyal,\n\nMughal Empire, 27\u201330,\n\nMukerjee, Radhakamal, 157\u2013159\n\nMukherjee, H. N.,\n\nMukherjee, Neel, 229\u2013230\n\nMumbai (was Bombay), 310\u2013312, 311 (map), 312 (fig.),\n\nmunicipal water supply,\n\nMuslim League, ,\n\nMysore, , , 162\u2013163,\n\nNangal Canal, 197\u2013198\n\n_See also_ Bhakra Dam\n\nNaoroji, Dadabhai, 77\u201378,\n\nNarain, Sunita, ,\n\nNarmada Bachao Andolan (NBA or \"Save Narmada Movement\"), 293\u2013294\n\nNarmada project, 292\u2013294,\n\nNash, Vaughan, ,\n\nnationalism\n\nawakening in India, 114\u2013115, 148\u2013149\n\nand castes, 154\u2013155\n\nin China, , 152\u2013153\n\ndevelopment in Asia, 147\u2013148, 151\u2013153\n\nfractures between states, ,\n\nGandhi's rise and work, 149\u2013151\n\nand power and freedom, , 154\u2013155\n\nand self-sufficiency,\n\nand water, , 152\u2013154\n\nwater schemes post-WWII, 171\u2013173\n\nnature. _See_ environment and nature\n\nNehru, Jawaharlal\n\nannexation of Tibet, ,\n\nBhakra Dam and Nangal Canal, 196\u2013197 (figs.), 197\u2013198\n\neconomic strategy,\n\nfamine in Bengal in WWII, 169\u2013170\n\non freedom,\n\nHirakud Dam, 193\u2013194\n\nland redistribution and _zamindari_ abolition,\n\nand Partition, ,\n\npolitical waning and death,\n\nrelationship with China, , , ,\n\nresource planning,\n\non size of projects,\n\nand starvation problem, ,\n\nwater schemes in China,\n\nNicholson, Frederick, 131\u2013132\n\nNightingale, Florence, ,\n\nNile River,\n\nNormand, Charles, , , 142\u2013143, ,\n\n_North American Review,_ and Ganges Canal, 44\u201345\n\nnortheast monsoon, xiv (map),\n\noceans\n\nchanges and erosion, 327\u2013328\n\nseabed and monsoons,\n\nterritorial waters and law of the sea, 232\u2013233\n\n_See also_ Indian Ocean\n\noil engines for water extraction, 120\u2013121\n\nOshima, Harry, ,\n\nOsterhammel, J\u00fcrgen,\n\nPacific Ocean, and monsoons,\n\nPakistan\n\ncanals, 184\u2013185, , 187\u2013188\n\neconomy, 186\u2013187\n\nIndus waters, 185\u2013186\n\nirrigation and agriculture,\n\nKashmir clashes, 247\u2013248\n\nmeteorology,\n\nPartition, 180\u2013181, 183\u2013188\n\nPartition, 176 (map)\n\nand borders, 180\u2013182\n\ncanals and water schemes, , 184\u2013185, , 187\u2013188\n\ndescription, , 180\u2013181\n\nimpact on government and institutions of India, 182\u2013184\n\nand Indus waters, 185\u2013186, 187\u2013188, 205\u2013206,\n\nPatkar, Medha, , 294 (fig.),\n\nPedder, W. G.,\n\nPersian Wheel method,\n\n\"Philindus,\"\n\nPhilippines, cyclone science, , 105\u2013106\n\nphotography of droughts, 84\u201385\n\nPiddington, Henry,\n\nPillai, Vellu,\n\nPillai, Villiyappa,\n\nPires, Tom\u00e9, 30\u201331\n\nPL-480 (Public Law 480), , ,\n\nplants, impact of climate change,\n\nPogson, Elizabeth Isis, 96\u201397,\n\npollution\n\nbrown cloud, 2\u20133, 305\u2013306\n\nimpact on water cycle, 2\u20133\n\nrivers, 2\u20133, 289\u2013292, 290 (fig.)\n\nwater legislation in India, 255\u2013256\n\nPomeranz, Kenneth,\n\nPondicherry, changes and erosion, , 328 (fig.)\n\nPoona Sarvajanik Sabha (\"the Sabha\"), 67\u201368, , 75\u201376, 86\u201387\n\npopulation growth, 190\u2013191, , , ,\n\nPortugal, trade in India, 31\u201332\n\npoverty, and ecological problems,\n\nProject Gromet,\n\nPublic Law 480 (PL-480), , ,\n\nPunjab\n\nagricultural problems, 277\u2013278, 283\u2013284\n\ndisplacement of people, 210\u2013211\n\ngroundwater and wells, 277\u2013278\n\nirrigation and canals, 121\u2013124, 125\u2013126, , , 184\u2013185,\n\nPartition, 180\u2013181, 184\u2013185, 186\u2013187\n\nQuit India Movement,\n\nRadcliffe, Cyril, ,\n\nRadcliffe's line, 176 (map),\n\nRaghavan, S., 314\u2013315\n\nrailway in India\n\ndevelopment and expansion, 51\u201352, , 54\u201356\n\nand hydrology, 56\u201357\n\ninequality and change, , 55\u201356\n\nmonsoons and climate, 52\u201353\n\nand technology, 56\u201357\n\nin trade and economy, , 53\u201355\n\nrain and rainfall\n\nand agriculture, , 134\u2013136, ,\n\ncloud injections,\n\ndata collection in British India, 96\u201397, , 134\u2013135\n\ndecline in India,\n\ndrought of 1876\u201379, , 67\u201369\n\nand El Ni\u00f1o, 264\u2013265\n\nfailure in India and China, 66\u201367, , , 242\u2013243, , ,\n\nand irrigation,\n\n\"moral\" meteorology,\n\nMumbai storm of 2005, 310\u2013312\n\npattern and description, 13\u201314, 23\u201324, ,\n\nplanning and preparation for,\n\nrainfall insurance,\n\nscience and monsoons, 95\u201396, 100\u2013101, 140\u2013141, 304\u2013305\n\nand water infrastructure,\n\nRamage, Colin, , 239\u2013241, ,\n\nRaman, C. R.,\n\nRamanathan, Veerabhadran,\n\nRanade, Mahadev Govind,\n\nRao, K. L.\n\nirrigation plans, ,\n\nwater schemes in China, 216\u2013218, , 220\u2013222\n\nRennell, James, 34\u201336\n\nresource planning, for water, 163\u2013164\n\nRevelle, Roger, , ,\n\nrice economies in WWII,\n\nrivers\n\nchanges and diversions, , 29\u201330, 297\u2013299\n\nand monsoon, 25\u201326\n\npollution, 2\u20133, 289\u2013292, 290 (fig.)\n\nvessels for trade, 47\u201348, 49\u201350\n\n_See also_ specific rivers and areas\n\nRoxburgh, William, 36\u201338\n\nRoy, Arundhati,\n\nRuchi Ram, Lala, 98\u201399,\n\nrural India\n\ndrought, 286\u2013287\n\ninequalities, 280\u2013281\n\nmonsoon rains and water problems, , 281\u2013283, 286\u2013287\n\nRussia, water schemes, 218\u2013219, 222\u2013223\n\nSaha, Meghnad, 171\u2013173\n\nSain, Kanwar\n\nBhakra Dam,\n\nMekong River commission, 223\u2013225\n\nwater schemes in China, 216\u2013218, , 220\u2013223\n\nSainath, Palagummi, 286\u2013287,\n\n\"salt march,\"\n\nSanmenxia Dam, 218\u2013219\n\nsatellite photography, for science, , 240\u2013241\n\nScott-Moncrieff, Colin, ,\n\nseabed and monsoons,\n\nSeason Watch,\n\nSecond World War (WWII), 166\u2013169, ,\n\nSeuss, Hans,\n\nShah, Firoz,\n\nShah, Gyansham, and colleagues, 287\u2013288\n\nShah, Tushaar,\n\nShastri, Lal Bahadur, ,\n\nships, information on monsoons and meteorology, , , 61\u201362, , , 104\u2013105, ,\n\nShiva, Vandana, 283\u2013284\n\nSimpson, G. C.,\n\nSingh, Deepti,\n\nSingh, Khushwant, ,\n\nSingh, Manmohan,\n\nSion, Jules,\n\nSlocum, Harvey, , 200 (fig.)\n\nSmith, Zadie,\n\nSolani aqueduct,\n\nSouth Africa, 149\u2013150\n\nSouth Asia's rivers, xii\u2013xiii (map)\n\nSouth to North line project, 298\u2013299\n\nSouthern Oscillation, , ,\n\nsouthwest monsoon, xv (map), 13\u201314, 23\u201324\n\nSoviet Union. _See_ Russia\n\nSrinivasan, B., 199\u2013200\n\nsteamboats, 49\u201351\n\nStommel, Henry,\n\nstorms. _See_ cyclones and storms\n\nStrachey, Richard, , , 81\u201382\n\nSubramaniam, C., 244\u2013247,\n\nsuicides of cultivators,\n\nSullivan, H. E.,\n\nSun Yat-sen, 152\u2013153\n\nSundara Raj, V.,\n\nsunspots theory of drought, 82\u201383\n\nSupratik (in _The Lives of Others_ ), 229\u2013230\n\nSutlej River,\n\nSwadeshi movement, 148\u2013149\n\nTansa project,\n\nTata and Sons,\n\ntaxation, 48\u201349,\n\nTchernia, Paul,\n\ntechnology, impact on water, ,\n\nTemple, Richard, , 80\u201381\n\nterritorial waters, and law of the sea, 232\u2013233\n\ntextiles trade, 30\u201331\n\nThird Pole,\n\nThird World, 284\u2013285\n\nThird World Network, 284\u2013285\n\nTibet, , , 226\u2013227,\n\nTibetan Plateau, xviii (fig.), 1\u20132, 188\u2013189\n\nTodd, Charles,\n\ntrade\n\nboats for, 47\u201348, 49\u201350\n\nearly trade, 30\u201333\n\nand railway, , 53\u201355\n\ntransportation, and monsoons, , 52\u201353\n\nTrevelyan, Charles, 48\u201349\n\ntsunami of 2004, 327\u2013328\n\ntubewells, , 256\u2013257, 257 (fig.),\n\nTungabhadra Dam and River, , ,\n\ntyphoons. _See_ cyclones and storms\n\nUN conference on environment (1972), 253\u2013255\n\nUN Convention on the Law of the Sea,\n\nunderground aquifers. _See_ groundwater\n\nUnited Nations, Indus dispute,\n\nUnited States\n\nagricultural policy in India,\n\nBihar famine of 1966\u20131967, , 249\u2013250\n\nfood aid to India, ,\n\nMekong River and interests in Asia, , ,\n\nin monsoon study, ,\n\nweather control schemes, 250\u2013251\n\nVairamuthu, 295\u2013296\n\nVakil, C. N., 183\u2013184\n\nVasco da Gama, 31\u201332\n\nVietnam, , 178\u2013179,\n\n_The Violence of the Green Revolution_ (Shiva), 283\u2013284\n\nVisvesvaraya, Mokshagundam, 129\u2013131,\n\nWalker, Gilbert\n\nbackground and interests, 137\u2013138,\n\nbreakthroughs in weather science, , 140\u2013141, ,\n\nclimate in India, 15\u201316\n\nIndianization of staff,\n\nmonsoon science and forecasts, 139\u2013140, 142\u2013143, ,\n\nrainfall in India, 140\u2013141\n\nWater Act (1974, India), 255\u2013256,\n\nwater control. _See_ control of water\n\nwater crisis, , 272\u2013274, 278\u2013283, 291\u2013292, 316\u2013317\n\nwater mining. _See_ groundwater\n\nwater table in India,\n\nWatsuji Tetsuro, 155\u2013157\n\nweather. _See_ climate\n\nweather control schemes, 250\u2013251\n\nWebster, Peter, ,\n\nwells, for irrigation, 119\u2013121, 256\u2013258,\n\nWheeler, Raymond,\n\nWickizer, V. D.,\n\nwind\n\nand monsoons, 23\u201324, 99\u2013100, , 236\u2013237\n\nand trade to India, 31\u201332\n\nWittfogel, Karl,\n\nWorld Bank\n\nagricultural policy in India,\n\nclimate migration,\n\nand dams, , , , ,\n\nMaharashtra drought, 260\u2013261\n\nWorld Commission on Dams, ,\n\nWorld War II. _See_ Second World War\n\nYamuna River and Canal, , 42\u201343\n\nYangzi River, 40\u201341, 291\u2013292\n\nYarlung Tsangpo. _See_ Brahmaputra\n\nYellow River, , , , 218\u2013219,\n\nYunnan, control of water, 164\u2013165\n\nZhou Enlai, 197 (fig.), , , , \n\n# CONTENTS\n\n  1. Cover\n  2. Title Page\n  3. Copyright\n  4. Table of Contents\n  5. Dedication\n  6. List of Maps\n  7. Maps\n  8. A Note on Names and Terminology\n  9. ONE The Shape of Modern Asia\n  10. TWO Water and Empire\n  11. THREE This Parched Land\n  12. FOUR The Aqueous Atmosphere\n  13. FIVE The Struggle for Water\n  14. SIX Water and Freedom\n  15. SEVEN Rivers Divided, Rivers Dammed\n  16. EIGHT The Ocean and the Underground\n  17. NINE Stormy Horizons\n  18. EPILOGUE History and Memory at the Water's Edge\n  19. Acknowledgments\n  20. About the Author\n  21. Also by Sunil Amrith\n  22. Archives and Special Collections\n  23. Notes\n  24. Index\n\n# Navigation\n\n  1. Begin Reading\n  2. Table of Contents\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n#\n\n_By the same author_\n\nA Short History of Tractors in Ukrainian   \nTwo Caravans\n\n# We Are All Made   \nof Glue\n\n## MARINA LEWYCKA\n\n### FIG TREE   \n_an imprint of_   \nPENGUIN BOOKS\n\n# Contents\n\n1 Adhesives in the Modern World\n\n1 The gluey smell\n\n2 Pheromones\n\n3 Shelf life\n\n4 Bonding dissimilar materials\n\n5 Fish\n\n6 Sticky brown stuff\n\n2 Adventures with Polymers\n\n7 Pick and mix\n\n8 Biopolymer\n\n9 Rubber\n\n10 Polymerisation\n\n11 Black treacle\n\n12 Marine biological glues\n\n13 No job too small\n\n14 Reindeer meat and dried fish\n\n15 The Bad Eel\n\n16 The handyman\n\n17 The care package\n\n18 Sherry\n\n3 Bonding\n\n19 Christmas with all the trimmings\n\n20 The festive season\n\n21 The Adhesives party\n\n22 Changing the locks\n\n23 Stress fractures\n\n24 Experimenting with Velcro\n\n25 The attraction between adhesives and adherends\n\n26 A gummy smile\n\n4 Adhesives Around the Home\n\n27 The breeze-block fortress\n\n28 Ancient and inexplicable\n\n29 The Abomination\n\n30 The broken gutter\n\n31 The epoxy hardener\n\n32 UPVC\n\n33 Avocados and strawberries\n\n34 The glue exhibition\n\n35 Uses of superglue\n\n36 The adhesion consultant\n\n37 A trip to B&Q\n\n38 Without walls\n\n39 Home improvements\n\n5 If Only It Came in Tubes\n\n40 Heavy as watermelons\n\n41 Cyanoacrylate AXP-36C\n\n42 The right glue for the materials\n\n43 Unpromising adherends\n\n44 Water creases\n\n45 The dance of the polymers\n\n46 Smoke circles\n\n47 The penthouse party\n\n48 A lot of bargains\n\nAcknowledgements\nFIG TREE\n\nPublished by the Penguin Group\n\nPenguin Books Ltd, 80 Strand, London WC2R 0RL, England\n\nPenguin Group (USA) Inc., 375 Hudson Street, New York, New York 10014, USA\n\nPenguin Group (Canada), 90 Eglinton Avenue East, Suite 700, Toronto, Ontario, Canada M4P 2Y3  \n(a division of Pearson Penguin Canada Inc.)\n\nPenguin Ireland, 25 St Stephen's Green, Dublin 2, Ireland (a division of Penguin Books Ltd)\n\nPenguin Group (Australia), 250 Camberwell Road, Camberwell, Victoria 3124, Australia   \n(a division of Pearson Australia Group Pty Ltd)\n\nPenguin Books India Pvt Ltd, 11 Community Centre, Panchsheel Park, New Delhi \u2013 110 017, India\n\nPenguin Group (NZ), 67 Apollo Drive, Rosedale, North Shore 0632, New Zealand   \n(a division of Pearson New Zealand Ltd)\n\nPenguin Books (South Africa) (Pty) Ltd, 24 Sturdee Avenue,   \nRosebank, Johannesburg 2196, South Africa\n\nPenguin Books Ltd, Registered Offices: 80 Strand, London WC2R 0RL, England\n\nwww.penguin.com\n\nFirst published 2009\n\nCopyright \u00a9 Marina Lewycka, 2009\n\nThe moral right of the author has been asserted\n\n'I Will Survive' words and music by Dino Fekaris and Freddie Perren \u00a9 copyright 1978   \nPerren-Vibes Music Company\/PolyGram International Publishing Incorporated, USA.   \nUniversal Music Publishing Limited. Used by permission of Music Sales Limited.   \nAll rights reserved. International copyright secured.\n\nAll rights reserved\n\nWithout limiting the rights under copyright reserved above, no part of this   \npublication may be reproduced, stored in or introduced into a retrieval system,   \nor transmitted, in any form or by any means (electronic, mechanical, photocopying,   \nrecording or otherwise), without the prior written permission of both the   \ncopyright owner and the above publisher of this book\n\nISBN: 978-0-14-192134-1\nTo my father, Petro Lewyckyj  \nPoet, engineer, eccentric  \nOctober 1912 \u2013 November 2008\n\n# 1   \n_Adhesives in the  \nModern World_\n\n# 1\n\n# The gluey smell\n\nThe first time I met Wonder Boy, he pissed on me. I suppose he was trying to warn me off, which was quite prescient when you consider how things turned out.\n\nOne afternoon in late October, somewhere between Stoke Newington and Highbury, I'd ventured into an unfamiliar street, and come upon the entrance of a cobbled lane that led in between two high garden walls. After about fifty metres the lane opened out into a grassy circle and I found myself standing in front of a big double-fronted house, half derelict and smothered in ivy, so completely tucked away behind the gardens of the neighbouring houses that you'd never have guessed it was there, crouched behind a straggly privet hedge and a thicket of self-seeded ash and maple saplings. I assumed it was uninhabited \u2013 who could live in a place like this? Something was carved on the gatepost. I pulled the ivy aside and read: Canaan House. Canaan \u2013 even the name exuded a musty whiff of holiness.\n\nA cloud shifted and a low shaft of sunshine made the windows light up momentarily like a magic show. Then the sun slipped away and the flat dusky light exposed the crumbling stucco, the bare wood where the paint had peeled away, rag-patched windows, sagging gutters, and a spiny monkey puzzle tree that had been planted far too close to the house. Behind me, the gate closed with a clack.\n\nSuddenly a long wailing sob, like the sound of a child crying, uncoiled in the silence. It seemed to be coming from the thicket. I shivered and drew back towards the gate, half expecting Christopher Lee to appear with blood on his fangs. But it was only a cat, a great white bruiser of a tomcat, with three black socks and an ugly face, who emerged from the bushes, tail held high, and came towards me with a purposeful glint in his eye.\n\n'Hello, cat. Do you live here?'\n\nHe sidled up, as though to rub himself against my legs, but just as I reached down to stroke him, his tail went up, his whole body quivered, and a strong squirt of eau-de-tomcat suffused the air. I aimed a kick, but he'd already melted into the shadows. As I picked my way back through the brambles I could smell it on my jeans \u2013 it had a pungent, faintly gluey smell.\n\nOur second encounter was about a week later, and this time I met his owner, too. One evening at about eleven o'clock, I heard a noise in the street, a scraping and scuffling followed by a smash of glass. I looked out of the window. Someone was pulling stuff out of the skip in front of my house.\n\nAt first I thought it was just a boy, a slight sparrowy figure wearing a cap pulled down low over his face; then he moved into the light and I saw it was an old woman, scrawny as an alley cat, tugging at some burgundy velour curtains to get at the box of my husband's old vinyls half buried under the other junk. I waved from the window. She waved back gaily and carried on tugging. Suddenly the box came free and she fell backwards on to the ground, scattering the records all over the road, smashing a few of them. I opened the door and rushed out to help her.\n\n'Are you okay?'\n\nScrambling to her feet, she shook herself like a cat. Her face was half hidden under the peak of the cap \u2013 it was one of those big jaunty baker boy caps that Twiggy used to wear, with a diamant\u00e9 brooch pinned on one side.\n\n'I don't know what type of persons is throwing away such music. Great Russian composers.' A rich brown voice, crumbly like fruitcake. I couldn't place the accent. 'Must be some barbarian types living around here, isn't it?'\n\nShe stood chin out, feet apart, as if sizing me up for a fight.\n\n'Look! Tchaikovsky. Shostakovich. Prokofiev. And they throw all in a bin!'\n\n'Please take the records,' I said apologetically. 'I don't have a record player.'\n\nI didn't want her to think I was a barbarian type.\n\n'Thenk you. I adore especially the Prokofiev piano sonatas.'\n\nNow I saw that behind the skip was an old-fashioned pram with big curly springs into which she'd already loaded some of my husband's books.\n\n'You can have the books, too.'\n\n'You heff read them all?' she asked, as though quizzing me for barbarian tendencies.\n\n'All of them.'\n\n'Good. Thenk you.'\n\n'My name's Georgie. Georgie Sinclair.'\n\nShe tipped her head in a stiff nod but said nothing.\n\n'I've not lived here long. We moved down from Leeds a year ago.'\n\nShe extended a gloved hand \u2013 the gloves were splitting apart on the thumbs \u2013 like a slightly dotty monarch acknowledging a subject.\n\n'Mrs Naomi Shapiro.'\n\nI helped her gather the scattered records and stow them on top of the books. Poor old thing, I was thinking, one of life's casualties, carting her worldly possessions around in a pram. She pushed it off down the road, swaying a little on her high heels as she went. Even in the cold outside air I could smell her, pungent and tangy like ripe cheese. After she'd gone a few yards I spotted the white tomcat, the same shaggy bruiser with three black socks, leeching out of the undergrowth of next door's garden and trailing her down the pavement, ducking for cover from time to time. Then I saw there was a whole cohort of shadowy cats slipping off walls and out of bushes, slinking along behind her. I stood and watched her go until she turned a corner and disappeared from sight, the Queen of the Cats. And I forgot about her instantly. I had other things to worry about.\n\nFrom the pavement I could see the light still on in Ben's bedroom window and the computer monitor winking away as he surfed the worldwide waves. Ben, my baby boy, now sixteen, a paid-up citizen of the web-wide world. 'I'm a cyber-child, Mum. I grew up with hypertext,' he'd once told me, when I complained about the time he was spending online. The square of light blinked from blue to red to green. What seas was he travelling tonight? What sights did he see? Up so late. On his own. My heart pinched \u2013 my gentle, slightly-too-serious Ben. How is it that children of the same parents turn out so differently? His sister Stella, at twenty, had already grabbed life by the horns, wrestled it to the ground, and was training it to eat out of the palm of her hand (along with a changing m\u00e9nage of hopeful young men) in a shared rented house near Durham University which, whenever I phoned, seemed always to have a party going on or a rock band practising in the background.\n\nIn the upstairs window the coloured square winked and disappeared. Bedtime. I went in and wrote my husband a curt note asking him to come and remove his junk, and I put it in an envelope with a second-class stamp. First thing next day, I telephoned the skip hire company.\n\nSo let me explain why I was putting my husband's stuff on a skip \u2013 then you can decide for yourself whose fault it was. We're in the kitchen one morning \u2013 the usual rush of Rip getting off to work and Ben getting off to school. Rip's fiddling with his BlackBerry. I'm making coffee and frothing milk and burning toast. The air is full of smoke and steam and early-morning bustle. The news is on the radio. Ben is thumping around upstairs.\n\nMe: I've bought a new toothbrush holder for the bathroom. Do you think you might find a moment to fix it on the wall?\n\nHim: (Silence.)\n\nMe: It's really nice. White porcelain. Sort of Scandinavian style.\n\nHim: What?\n\nMe: The toothbrush holder.\n\nHim: What the fuck are you talking about, Georgie?\n\nMe: The toothbrush holder. It needs fixing on to the wall. In the bathroom. (A helpless little simper in my voice.) I think it's a rawplug job.\n\nHim: (Deep manly sigh.) Some of us are trying to do something really worthwhile in the world, Georgie. You know, something that will contribute to human progress and shape the destiny of future generations. And you witter on about a toothbrush.\n\nI can't explain what came over me next. My arm jerked and suddenly there were flecks of milky froth everywhere \u2013 on the walls, on him, all over his BlackBerry. A gob of froth had caught in the blond hairs of his left eyebrow and hung there, quivering goopily with his rage.\n\nHim: (Furious.) What's got into you, Georgie?\n\nMe: (A shriek.) You don't care, do you? All you care about is your bloody world-changing destiny-shaping bloody work!\n\nHim: (Shaking his head in disbelief.) As it happens, I do care very much. I care about what happens in the world. Though I can't say I care deeply about a toothbrush.\n\nMe: (Watching, fascinated, as the gob of froth works itself loose and starts to slide.) A toothbrush _holder_.\n\nHim: What the fuck's a toothbrush _holder_?\n\nMe: It's... ah! (There she goes... Splat!)\n\nHim: (Self-righteously rubbing his eye.) I don't see why I should put up with this.\n\nMe: (Flushed with achievement.) No one's asking you to put up with it. Why don't you just go? And take your bloody BlackBerry with you. (Not that there was the slightest chance that he'd have left _that_ behind.)\n\nHim: (Hoity-toity.) Your outbursts of hysteria are not very attractive, Georgie.\n\nMe: (Lippy.) No, and you're not attractive either, you big self-inflated fart.\n\nBut he was attractive. That's the trouble \u2013 he was. And now I've well and truly blown it, I thought, as I pictured Mrs Shapiro pootling away up the street with his precious collection of great Russian composers tucked away in her pram. \n\n# 2\n\n# Pheromones\n\nI was sitting at my desk, staring at the rain and trying to finish off the November edition of _Adhesives in the Modern World_ , when the skip lorry arrived. Adhesives can sometimes, I admit, be quite boring, so it was nice to be distracted. I watched it reversing and clanking into position, lowering the chain loops to winch up the overflowing skip, dangling it in the air with the damp spare mattress, the dishevelled papers, limply flapping magazines, the bin bags of clothes and the boxes that contained all the soggy detritus of his Really Important Work, and crashing it down on the back of the truck with a satisfying thud. When it was ready I went out and paid the skip man, and I must confess I did feel a pang of extreme apprehension as it trundled out of view. I knew Rip would be furious.\n\nWhen he'd got back from work that day \u2013 the day of the toothbrush holder \u2013 I'd calmed down but he was still in a rage. He started piling up his stuff in his car.\n\nMe: (Nervous.) What are you doing?\n\nHim: (Stony-faced.) I'm leaving. I'm going to stay with Pete.\n\nMe: (Clinging. Pathetic. Despicable. Self-hating.) Don't go, Rip. I'm sorry. It's only a toothbrush holder. I'll put it up myself. Tell you what (little giggle), I'll learn to do rawplugs.\n\nHim: (Clenched jaw.) But it's not just that, is it?\n\nMe: What d'you mean? (A terrible truth dawns on me.) Are you...?\n\nHim: (Sigh of boredom.) There's no one else, if that's what you're thinking. Just...\n\nMe: (Relief.) Just... me?\n\nHim: (Looking at his watch.) I'd better get going. I told Pete I'd be there at seven.\n\nMe: (Feeling like a despicable worm too low even to crawl out of its miserable hole, but putting on a show of nonchalance.) Fine. If that's what you feel. Fine by me. Give my regards to Pete.\n\nPete was Australian, Rip's squash partner, and a senior colleague on the Progress Project. We called him Pectoral Pete, because he always wore tight white T-shirts and big white trainers and made loud jokes about lesbians. In spite of that, I quite liked him. He and his wife Ottoline lived in a tall-windowed house overlooking a square in Islington, with a top-floor flat that they sometimes rented out. I went and stood outside one evening, looking up into the lighted windows. They couldn't see me standing down there in the dark with tears pouring down my face.\n\nIt lasted for a few weeks, the crying phase. Then rage took over.\n\n'I'll come back for the rest of my things,' Rip had said as he left.\n\nBut he didn't. The shoes in the hall \u2013 I gave them a kick each time I went past \u2013 the old clothes in the wardrobe \u2013 they still carried a faint whiff of him \u2013 the back copies of _The Economist_ and the _New Statesman_ stacked up against the wall, the filing cabinets bulging with progress. Even his used underpants he'd left in the laundry basket. What was I supposed to do \u2013 take them out and wash them?\n\nI didn't want him cluttering up my new independent life with his old discarded stuff. I'll be fine, I told myself. I'll get over it. I'll meet someone else. And just to convince myself that I really meant it, I hired a skip. Perhaps I should have taken it all to Oxfam, but I didn't have a car and it just seemed too complicated. And besides, if I had done, this story might never have been written, because it was the skip that brought Mrs Shapiro into my life.\n\nAbout an hour after the skip had gone, the doorbell rang. So soon! I stood frozen, paralysed by the enormity of what I'd done. I listened as the bell rang again, a long, persistent, I-know-you're-in-there ring. No, best not to answer it. But what if he looked in through the window and saw me standing there? Maybe I should take my shoes off and silently sneak up to the bedroom. But what if he looked in through the letter box and saw me creeping up the stairs? What if he saw my silhouette in the window? I tiptoed into the corridor, lay down on the floor out of the sight line of any of the windows, and held my breath.\n\nThe doorbell rang again and again and again. Obviously he wasn't fooled. Then the letter box clattered. Then silence. As I lay stretched out on the floor watching the light fade on the ceiling, I could feel my heartbeat slowing down and my breathing getting calmer. After a while, a song drifted into my head.\n\n' _You thought I'd lay down and die_. _Oh no, not I! I will survive!'_ Gloria Gaynor. It was one of Mum's favourites. How did it go? ' _At first I was afraid, I was petrified.'_ I started to sing. ' _I didn't know if I could something something without you by my side_... _something change the locks_... _I will survive!_ ' I'd forgotten most of the other words, but I still knew the chorus, ' _I will survive! I will survive!_ ' I belted it out over and over again.\n\nThat's how Ben found me when he got back from school, lying flat on my back in the corridor, singing at the top of my voice. He must have let himself in so quietly that I didn't hear the door; then I looked up and saw his face looking down at me.\n\n'Are you all right, Mum?' His eyes squinted with concern.\n\n'Course I am, love. Just... enjoying a musical interlude.'\n\nI clambered up from the floor and looked out of the window. The street was empty. It was raining again. There were no signs the skip had ever been there apart from a few shards of black vinyl on the road. Then I noticed a leaflet on the doormat. Ben picked it up curiously. _The Watchtower. Watch and pray for ye know not when the time is._\n\n'What's this about?'\n\n'It's the Jehovah's Witness magazine. It's about the end of the world, when Jesus returns, and all the true believers get whisked up to heaven.'\n\n'Hm.' He flicked through it, and to my surprise he stuck it in his pocket and clomped upstairs to his room.\n\nWhat a shame. I could have done with a comforting heart-to-heart with some nice Jehovah's Witnesses.\n\nThe doorbell rang again as Ben and I were about to sit down for tea. Ben answered it.\n\n'Hi, Dad.'\n\n'Hi, Ben. Is your mother in?'\n\nNowhere to hide this time. I had to face him across the table. Pectoral Pete was with him. They were both wearing their jogging gear. They must have run all the way over from Islington. I could smell the sweat on them. The whole kitchen reeked of pheromones, and I felt a mortifying stab of lust \u2013 my traitor hormones letting me down just when I thought I was beginning to get things under control.\n\nHim: (Lounging in his chair and stretching his legs out as if he owned the place.) Hi, Georgie. I got your message. I've come to rescue my stuff.\n\nMe: (Oh, help! What have I done?) It's too late. They took the skip away this morning.\n\nHim: (Eyes round and blinky. Mouth open in a little round O that makes him look like a hooked trout.) You're kidding. (Yes, definitely more trouty than destiny-shaping. Ha ha!)\n\nMe: Why would I be kidding? (His hair seems to have receded a bit, too. Good. He's not as gorgeous as he thinks he is.)\n\nHim: (Disbelieving.) They took the records? My great Russian composers?\n\nMe: (A sly smirk.) Mmhm.\n\nHim: (Even more disbelieving.) My first-fifteen rugby boots?\n\nMe: All the junk. (How can a man who discards his loyal and devoted wife without a frisson of sentiment get all dewy-eyed about a pair of mouldy old football boots?)\n\nHim: (World-weary sigh.) Why are you being so childish, Georgie?\n\nChildish? Me? I picked up a plate of pasta. I could feel that twitching in my arm again. Pete was grinning with embarrassment, trying to bury his face in the _Guardian_. Then I caught the frightened look in Ben's eyes \u2013 poor Ben, he didn't need to see his parents behaving like this. I put the pasta down, bolted out of the room and ran up the stairs; I threw myself on to my bed, blinking the tears out of my eyes. I will survive. I will grow strong. I will change the locks. Look at Gloria Gaynor \u2013 she turned her heartbreak into a song that sold millions. As I sat there listening to the voices down below, and wishing I'd kept my cool, an appealing thought floated into my head. I can't sing, but I can write.\n\nIn fact I was already halfway there. I had a working title and a terrific nom de plume. My mind lingered on a seductive image of myself as a published author, trendy in crumpled linen with a stylish leather bag full of proofs slung casually over my shoulder, jetting around the world with an entourage of poet toyboys. Rip would be revealed to the world as a self-obsessed workaholic, pitifully underendowed, with an insatiable Viagra habit and dandruff. His wife would be beautiful and long-suffering, with a fabulous bum.\n\n' _Forget! Survive!_ ' Gloria Gaynor's voice seemed to chide in my head. ' _You'll waste too many nights thinking how he did you wrong. Change the locks! Grow strong!'_\n\nAnd to be fair, she had a point. My previous attempts at fiction, twelve and a half full exercise books, were stowed away in a drawer, along with a file of hoity-toity rejection slips.\n\nDear Ms Firestorm,\n\nThank you for sending _The Splattered Heart_ for our consideration. Your book has some colourful characters and displays an impressive array of adjectives, but I regret to say I was unable to summon sufficient enthusiasm...\n\nThat sort of thing is bad for morale, and my morale was already low. But it was no use \u2013 a seed of optimism had lodged itself in my heart, and the opening lines were already sprouting in my head. There was one empty exercise book left.\n\n# The Splattered Heart   \nChapter I\n\n_It was past midnight when Rick rolled exhaustedly on to his broad, muscular slightly podgy back and casually ran his powerful fingers with their chewed-down fingernails through his thick curly, naturally blond discreetly highlighted hair._\n\nOkay, I know it's not your Jane Austen. Maybe Ms Insufficient Enthusiasm had a point about the adjectives. I sat staring at the page. Had I developed writer's block already? Downstairs I heard voices in the hall and the click of the latch. Then my bedroom door opened a crack.\n\n'Are you all right, Mum? Aren't you having any dinner?'\n\n# 3\n\n# Shelf life\n\nAfter Rip moved into Pectoral Pete's top-floor flat we agreed that Ben should spend half a week with each of us. One day I noticed him with his watch and his pencil ticking off the days on the calendar. Sunday, Monday, Tuesday: Dad. Wednesday, Thursday, Friday: Mum. Saturday \u2013 that's the tricky one \u2013 one week with Dad, one week with Mum. We broke him in half and divided him between us. I could see the frown of concentration on his face as he tried to work out which week we were in. He was determined to be fair to both of us.\n\nAs the rage against Rip congealed in my heart, I was sometimes taken over by a numbness so intense it felt like pain. On the days when Ben wasn't here I found it almost unbearable to be in the house alone. The silence had an intrusive jangling quality, like a persistent tinnitus. When I walked from room to room I could hear my footsteps on the laminate floor. When I ate, I could hear the scraping of my knife and fork on the plate in the echoing kitchen. At first I tried having the radio on or playing music, but that made it worse: I knew the silence was there even though I couldn't hear it.\n\nWhen the silence got too much I'd take a walk, just to get out of the house. Wearing my comfy trodden-down trainers and an ancient brown duffel coat with a wide flapping hood and sleeves like bat's wings, I flitted about at dusk, peeping through lighted windows into other people's lives, catching them eating an evening meal, or sitting on a sofa watching TV, and tried to remember what it felt like to be still stuck together. Maybe I should have been prettying myself up and keeping my eye open for another man, but the wing-sleeves of the coat enfolded me like arms, and at that time, it was the only comfort I had. The look was not so much Bat Woman as batty-woman, but it didn't matter because I never met anyone I knew, and anyway, the coat made me invisible.\n\nOne afternoon I went as far as Islington, thinking I'd get a few things I needed from Sainsbury's and catch the bus back. It was about four o'clock, and the sticker lady was doing her end-of-day reductions. A crowd was milling around her like a piranha tank at feeding time. Mum had always been a great advocate of sell-by-date shopping, and I remembered with a twist of nostalgia how, when I was little, she used to send me scampering along the aisles looking out for the bright red REDUCED stickers that pouted like scarlet kisses on the cling film. She didn't think much of Listernia and Saminella, and even an unpleasant experience with some mature crabstix didn't dampen her enthusiasm. She would pat her elasticated middle. 'Waste not, want not.' Mum always looked after her pennies as if they came from heaven. Funny how long after you leave home you still carry a bit of your parents around inside you. Now, without the certainty of Rip's salary landing in our joint bank account with a generous kerchung! each month, I understood that sharp edge of insecurity that Mum must have felt all her life. Or maybe I was just so dejected at the time that I felt a queasy kinship with the curled-at-the-corners pastries, the sad rejected chicken wings. Anyway, I pushed forward to join the crush.\n\nThe sticker lady was working incredibly slowly, spewing out labels that kept jamming the machine. No sooner had she stuck a new label on something than a hand reached out of the throng and grabbed it from her. The reduced items weren't even reaching the shelf. I noticed it always seemed to be the same hand \u2013 a bony, gnarled, jewel-encrusted hand, darting out and snatching. Turning to follow it with my eyes, I spotted an old woman diving in low beneath the shoulders of two fat ladies. Her hair was tucked up into a jaunty Scotch plaid cap with a heart-and-arrow diamant\u00e9 brooch pinned on one side, and a straggle of black curls escaping under the brim. She was reaching and grabbing like a virago. It was Mrs Shapiro.\n\n'Hello!' I called.\n\nShe looked up and stared at me for a moment. Then she recognised me.\n\n'Georgine!' she cried. She pronounced it with hard 'G's and an 'eh' sound at the end. Gheorghineh! 'Good afternoon, my darlink!'\n\n'Good to see you, Mrs Shapiro.'\n\nI leaned and gave her a peck on each cheek. In the enclosed space of the groceries aisle, her smell was ripe and farty like old cheese, with a faint hint of Chanel No. 5. I could see the looks on the faces of the other shoppers as they backed away and let her through. They thought she was just a bag lady, a batty-woman. They didn't know she collected books and listened to the great Russian composers.\n\n'Plenty good bargains today, darlink!' Her voice was breathless with excitement. 'One minute full price, next minute half price \u2013 same thing, no difference. Always tastes better when you pay less, isn't it?'\n\n'You should meet my mum. She always likes a bargain. She says it's because of the war.'\n\nI guessed she was a bit older than Mum \u2013 in her late seventies, maybe. More wrinkled, but more energetic. She was of the age when she should have been wearing those extra-wide-fitting bootees held on with Velcro, but in fact she was tottering about daintily on peep-toe high-heeled shoes like a lady of style, the grubby toes of her grey-white cotton ankle socks poking out in front.\n\n'Not only the war, darlink. I heff learnt in my life to make the ends meet. A hard life is a good teacher, isn't it?'\n\nHer cheeks were flushed, her eyes focused and alert, her brow slightly furrowed with the effort of mental arithmetic as the new labels were stuck on top of the old.\n\n'Come on, Georgine \u2013 you must grebbit!'\n\nI squeezed in beside one of the big ladies and grabbed at a passing chicken korma, reduced from \u00a32.99 to \u00a31.49. Mum would have been proud of me.\n\n'You heff to be quick! You like sossedge? Here!'\n\nMrs Shapiro snatched a pack of sausages reduced to 59p out of the hand of a bewildered pensioner, and tossed it into my basket.\n\n'Oh... thanks.'\n\nThey looked unappetisingly pink. Seizing me by the wrist, she pulled me towards her and whispered in my ear, 'Is okay. Jewish. No sossedge.' The pensioner was staring at the sausages in my basket.\n\n'You Jewish also, Georgine?' She must have noticed me eyeing the sausages with distaste.\n\n'No. Not Jewish. Yorkshire.'\n\n'Ach, so. Never mind. Can't help it.'\n\n'Have you been playing the records, Mrs Shapiro? Are they all right? Not too scratched?'\n\n'Great records. Glinka. Rimsky-Korsakov. Mussorgsky. Such a music. Take you straight up into heaven.' Her bony hands spread theatrically in the air, the rings glittering, the varnished fingernails bright like little bunches of cherries. Close up I saw that the red highlights in her cheeks, which I'd mistaken for a flush of excitement, were actually two circles of rouge, one with a clear thumbprint in the middle.\n\n'Shostakovich. Prokofiev. Myaskovsky. My Arti has played with them all.'\n\n'Who's Arti?' I asked, but she was distracted by a 79p quiche Lorraine.\n\nI didn't like to admit that classical music wasn't my thing \u2013 I always thought of it as Rip's look-at-me-I'm-doing-the-hoovering music. I'm a Bruce Springsteen and Joan Armatrading fan myself.\n\n'I don't think I have much of an ear for music.'\n\nRip used to tease me that I was tone deaf and even my singing-in-the-bath efforts were painful to the cultivated ear.\n\n'Not all great art is for the messes, darlink. But you would like to learn, would you?' She batted her azure eyelids. 'I will play for you. You like the fish?'\n\nAs she said the word, I noticed a fishy undernote welling up through the cheese-and-Chanel. It was coming from her trolley. I saw that among her bargain produce were several packs of fish, all REDUCED. I hesitated. This fish definitely smelled off. Even Mum would have given it a miss.\n\n'You come in my house, I will cook them for you.'\n\nPoor old thing, she must be lonely, I thought.\n\n'I'd love to, but...' But what?\n\nI was trying to muster an excuse when she let out a blood-curdling shriek.\n\n'No no! You teef!'\n\nThere was a sudden angry scuffle in the aisle and a clang of shopping trolleys being barged. The pensioner from whose hand she'd grabbed the sausages had sneakily tried to pinch them back out of my basket. Mrs Shapiro snatched them from him and brandished them in the air.\n\n'Teef! You pay for you own sossedge full price if you want it!'\n\nDefeated and humiliated the pensioner slunk away. She turned towards me, flushed with triumph.\n\n'I am liffing not far from you. Big house. Big garden. Too many trees. Totley Place. Kennen House. You come on Saturday seven o'clock.'\n\n'Have you got a Nectar Card?' asked the girl at the checkout, swiping my bargains over the bar-code reader (where did that vile-looking cheese sauce come from?).\n\nI shook my head, and muttered something Rip-like about the surveillance society. Behind me, Mrs Shapiro had got into an argument with someone else in the queue and I was planning a quick getaway.\n\n'Bravo, darlink! These surveyors are getting everywhere,' she cried, barging her way towards the exit, bashing the legs of the man in the next queue with her trolley. He was a big man with a stubble of close-cropped blond hair, built like a rugby player. He turned round and gave her an unsmiling stare.\n\n'Sorry, sorry, darlink.' The crimson lipstick flashed. The blue eyelids fluttered. The man shook his head, as though saddened by the presence of lunatics.\n\nHe made his way through the checkout and out into the car park. I watched him load his purchases into a massive black tinted-windowed four-by-four parked in a disabled bay in front of Mrs Shapiro's pram. Immediately behind him, a blue Robin Reliant had pulled in tight, sideways on. It had a disabled badge in the window. He put the four-by-four \u2013 it looked like one of those American military Humvee things \u2013 into reverse and started to back up, but the Robin Reliant was blocking him in. On the other side, Mrs Shapiro was loading her purchases into her pram. He edged forward and stuck his head out of the window.\n\n'Can you just shift your pram, lady, so I can pull round?'\n\n'One moment, please!' Mrs Shapiro cried. 'I need a new reduction!' She'd found a bruise on a not-reduced apple, and was heading off back into the shop to negotiate a discount.\n\nWhile I was waiting, the owner of the Robin Reliant returned. He was a little shrivelled man, propping himself up with a stick. He got into the Reliant, took a meat pie out of a paper bag, and started to eat. The man in the Humvee beeped his horn loud and long, but the meat-pie man carried on eating. The Humvee started to reverse, slowly slowly, until its rear bumper touched the door panel of the Reliant. Tunk! There was a distinct jolt as it made contact. By now a few people had gathered on the pavement to watch. I spotted the two fat ladies from the sticker scrum, eating biscuits out of a bag. The _Big Issue_ seller had come round from the front of the store, and a girl who'd been handing out leaflets when I arrived. They were all yelling at him to stop. The meat-pie man was taking his time, savouring every bite.\n\nSuddenly the Humvee driver slammed into forward, swung the wheel round as far as it would go, and started inching his chrome bumper towards where I was standing by Mrs Shapiro's pram. There was something about the fixed set of his jaw and the way his eyes stared straight ahead, refusing to look at me, that made me livid. I positioned myself defiantly in front of the pram, gripping the handle tight, with my own shopping bags wedged between my feet. I hadn't picked this fight, but I was prepared for martyrdom. The driver beeped his horn and carried on inching. He was going to barge the pram right out of the way with his bully bull bars!\n\nThen Mrs Shapiro emerged beaming from the super-market, brandishing the apple, which now had a REDUCED sticker on it.\n\n'They give me five pence off!'\n\nShe pulled a packet of cigarettes and a box of matches out from under the hood of the pram, offered me one \u2013 which I declined \u2013 and lit up.\n\n'Thenk you, Georgine, for waiting.' She nodded in the direction of the _Big Issue_ salesman and the leaflet girl, and whispered loud enough for them to hear, 'Looks like gypsies, isn't it? They want to steal my shoppings?'\n\n'No, they're...'\n\n'Just shift your bloody pram, you old bat!' snarled the Humvee driver out of his window.\n\n'Don't you dare talk to her like that, you big bully!' I hissed back.\n\n'What he is saying, Georgine?'\n\n'I think he wants you to move the pram, Mrs Shapiro, so that he can get his car out. But just take your time.'\n\nShe fluttered her azure eyelids at him.\n\n'Sorry, sorry, darlink.'\n\nSwaying a little on her heels, she manoeuvred the pram out of the way and tottered off down the road towards Chapel Market, still puffing away. \n\n# 4\n\n# Bonding dissimilar materials\n\nWhen I got home, I put the kettle on for a cup of tea and phoned Mum to tell her about my pram adventure. I knew she'd be as intrigued by Mrs Shapiro as I was. (Dad, on the other hand, would approve of my befriending a vulnerable old lady.) Mum had turned seventy-three in October and time was weighing down on her. Her eyesight was beginning to deteriorate ('immaculate degeneration') and the doctor had told her she shouldn't drive any more. Dad had been struck down with the 'waterworks mither'. Her son, my brother Keir, five years divorced, with two sons he hardly ever saw, was posted in Iraq. And now I was splitting up with my husband. Just at the time when she should have been sailing into a rosy sunset, everything on her horizon seemed stormy and unsettled.\n\nTo cheer her up, I launched into a description of my bargains.\n\n'Chicken korma, Mum. Reduced from \u00a32.99 to \u00a31.49.'\n\n'Oh, lovely. What's a chicken corner?'\n\nMum isn't stupid, but she's partially deaf \u2013 my nana had measles during her pregnancy. Dad and I tease her because she refuses to wear a hearing aid. ('People'll say I'm an alien if I start going around wi' bits of wire coming out of my head.' Actually, where I come from, in Kippax, they might.)\n\n'Chicken korma. It's Indian. Sort of spicy and creamy.'\n\n'Oah, I don't know if your dad'd fancy that.' Her voice sounded flat and defeated.\n\nI tried another tack.\n\n'Have you read any good books recently, Mum?'\n\nIn the right mood, this is her favourite topic, a guilty pleasure we share. When I was sixteen, Dad had given me a copy of _The Ragged Trousered Philanthropists_ , which I'd pretended to enjoy but had secretly found depressing and tedious, and Mum had introduced me to Georgette Heyer and Catherine Cookson, whom I pretended to despise, but secretly devoured.\n\n'Always look out for the underdog,' Dad had said.\n\n'There's nowt to beat a happy ending,' said Mum.\n\n'I just finished _Turquoise Temptation_ ,' she sighed down the phone. 'But it were rubbish. Too much heavy breathing and ripped-up underwear.' A pause. 'Have you seen owt of Euridopeas?'\n\nI knew she secretly hoped we would get together again. I didn't tell her he'd been round to pick up his stuff.\n\nWhen Rip and I first fell in love, I sometimes used to imagine us as romantic characters in a great tempestuous love story set against the turbulent background of the miners' strike, transgressing boundaries of wealth and class to be together. I was his door into an exotic world where noble savages discussed socialism while soaping each other's backs in t' pit baths. He was my door into Pemberley Hall and Mansfield Park. We were so full of illusions about each other, maybe it was bound to end in a splattering.\n\nAfter Mum had rung off, I made myself a cup of tea and picked up my pen.\n\n#\n\n# The Splattered Heart   \nChapter 2\n\n_It was a sunny October day, and Rip ck's mind was on carnal things as his mini Porsche nosed its way roared up over the Roaches hills still brilliant with dazzling autumn colour. After a few miles after Leek..._ (Should I change the location as well as the names? I tried to cast my mind back to my journalism course with frisky Mrs Featherstone, but I couldn't for the life of me remember what she'd said about libel.) _... the road turned sharply to the right, and Gina saw the entrance to a driveway, with a cattle grid and two stone gateposts, and there at the bottom of the valley, a good mile away, was Holtham House Holty Towers, sailing like a stone galleon in a shimmering red-green-and-gold sea._ (Pause for admiration; that was good \u2013 the galleon bit.) _Despite herself, Gina was impressed by found herself inexorably drawn towards the house stately pile and she could not help noticing that these people obviously had a bob or two the stunning period features. So this is how the other half live, she thought. Actually, she found it quite appealing. How disgusting._\n\nIn fact Rip was always much less troubled by the differences between our two families than I was.\n\nMe: (Whisper.) You never told me they were so posh.\n\nHim: (Murmur.) But when you have money, you realise how little it really matters.\n\nMe: (Loud whisper.) Yes, but it matters if you haven't got enough.\n\nHim: (Quietly confident.) Inequality only matters if it makes people _feel_ unequal.\n\nMe: Yes, but... (But that's a load of crap.)\n\nHim: You don't feel unequal, do you, Georgie?\n\nMe: No, but... (Of course I bloody do. I don't know what to do with all the knives and forks. I feel as though they look down their hoity-toity noses at me. But I can't admit it, can I, without seeming like a complete loser? So I'd better keep my mouth shut.)\n\nHim: Mmm. (Kisses me tenderly on the lips, then we end up in bed. Which is always nice.)\n\n# 5\n\n# Fish\n\nIt was already dark on Saturday evening when I made my way up the lane to Canaan House for my dinner date. As I moved out of range of the spooky sodium glow from the street lamp on Totley Place, the shadows closed in on me, and I must admit I felt a tremor of apprehension. What was I letting myself in for?\n\nThe night was cold and starry. The moonlight etched silvery outlines of trees and the gables of Canaan House on to the darkness. But even in that ashy light there was something cheerfully eccentric about its hodgepodge of styles: Victorian bay windows, a Romanesque entrance porch with twirly columns supporting chubby rounded arches, exuberant Tudor chimneys, and a mad Dracula turret with pointed Gothic windows stuck on one side. I wouldn't go so far as to say I was _inexorably drawn_ , but I did quicken my step. The garden path was almost overgrown with brambles but a narrow trail led towards the porch. I pulled my duffel coat around me and looked for signs of light up ahead. Had she forgotten I was coming?\n\nThe house itself was dark but I had a sense of eyes watching me. I stopped and listened. I could hear nothing but a faint rustling of leaves that could have been the wind. There was a smell of earth, mouldering vegetation and a musky foxy stink. I took a couple of steps closer to the house, and as I approached the porch a cat burst out of the undergrowth on to the path in front of me. And another. And another. I couldn't count how many cats there were in that soft seething throng, rubbing up against me, purring and mewing, their eyes glinting gold and green, as if I'd stepped into a teeming shoal of furry fishes.\n\nThe front door had a frosted glass panel through which I could now see a faint sliver of light far away inside. There was a bell to one side. I pressed it and heard it ringing somewhere in the depths of the house. The sliver of light widened into a crack and then into a rectangle as a door opened. I heard shuffling footsteps, a safety chain being unlocked, then Mrs Shapiro opened the door.\n\n'Georgine! Darlink! Come in!'\n\nIt's hard to describe the stench that hit me as I stepped over the threshold. I almost gagged and I had to struggle to keep the look of disgust off my face. It was a smell of damp and cat pee and shit and rot and food mould and house filth and sink gunge and, cutting through all that, a rank, nauseating, fishy stink. This last smell, I realised with a sinking feeling, was dinner.\n\nThe cats had slunk in beside me \u2013 there were only four of them after all \u2013 and dashed up ahead into the back of the house. Mrs Shapiro clapped her hands as though to chase them away and smiled indulgently.\n\n'Little pisskes!'\n\nShe was wearing a long-sleeved dress in carmine velvet, shaped at the waist and daringly cut away at the front and back to reveal her wrinkled shoulders and the loose skin of her chest. A double string of pearls gleamed around her throat. Her dramatic black curls were piled on top of her head with a collection of tortoiseshell combs, and she'd painted on a dash of matching carmine lipstick \u2013 not all of it on her lips. I was wearing jeans and a baggy pullover under my brown duffel coat. She stepped back on her high heels and eyed me critically.\n\n'Why you wearing this old shmata, Georgine? Is not flettering for a young woman. You will never get a man this way.'\n\n'I... er... I don't need...' I stopped. Maybe a man is what I needed after all.\n\n'Come. I will find you something better.'\n\nShe led me into the wide tiled entrance hall, from the centre of which a polished mahogany staircase curved away to the next floor. Underneath the staircase were piles of black bin bags, bursting with \u2013 I don't know, really, what they contained, but I could see clothes and books and electrical items and crockery and bedding spilling out where the bags had split. To one side was parked the old high-sprung pram, now apparently full of bundled rags, on which a couple of stripy felines were dozing. She shooed them away and started to root among the bundles. After a few moments she began to tug at a piece of dark green stuff which, when she pulled it out, turned out to be a dress in some heavy silky fabric with long scalloped sleeves.\n\n'Here,' she held it up to my chin, 'this I think is more flettering for you.' I looked at the label \u2013 it was a size 12 \u2013 my size \u2013 and a Karen Millen. In fact it was a gorgeous dress. Where on earth had she got this from?\n\n'It's lovely, but...' Actually, when I thought about it, I could guess where she'd got it from \u2013 she must have pulled it out of a skip. '... but I can't possibly take it.'\n\nWho would put a dress like this in a skip? Then I thought of Rip's clothes, which I'd put in the skip, and in a flash I understood \u2013 another heart had been splattered somewhere.\n\n'Is too big for me,' she said. 'Will look better on you. Take it, please.'\n\n'Thank you, Mrs Shapiro, but...' I brushed away the cat hairs that were clinging to the silky fabric. As I shook it out, I could smell the faint sweat and expensive perfume of its previous owner, and I wondered what had driven her lover to get rid of the dress.\n\n'Try it! Try it! No need to be emberressed, darlink.'\n\nDid she expect me to put it on straightaway? Obviously she did. She stood over me as I stripped down to my knickers in the cold foul-smelling hall and slipped the dress, still slightly warm from the sleeping cats, over my head. It slid down over my shoulders and hips as though it was made for me. Why was I doing this? I asked myself. Why didn't I just put on my own clothes and firmly but politely say goodnight? I thought of escaping, I really did. Then I thought of the trouble she must have gone to, to prepare the meal, and how let-down she would feel. And I remembered my empty house and the bright pink sausages in the fridge and _Casualty_ on TV. And by then it was too late.\n\n'Wait, I will zip it!' I could feel her hands, bony like claws, on my skin as she wrenched the zip up behind me. 'Beautiful, darlink. You already looking much better. You are a nice-looking woman, Georgine. Nice skin. Nice eyes. Good figure. But look at your hair. Looks like a sheep's popo. When you last been at the hairdresser?'\n\n'I can't remember. I...' I remembered the way Rip used to look at me, the way he would run his fingers through my hair when he kissed me.\n\n'You want I will put some lipstick on you?'\n\n'No, really, Mrs Shapiro.'\n\nShe hesitated, looking me up and down.\n\n'Okay. For tonight is okay. Come, please.'\n\nI followed her through a door into a long gloomy room where an oval-shaped mahogany table had been spread with a white cloth and two places set at one end with cutlery, napkins and glasses. In the centre of the tablecloth, the large white tomcat was curled up asleep.\n\n'Raus, Wonder Boy! Raus!' (She pronounced it Vunder Boy.) She clapped her hands.\n\nThe cat stretched one muscular black-socked leg behind its ear and began licking its private parts. Then it scratched about, sending bits of fluff flying everywhere. Then it rose to its feet, stretched itself a couple of times, jumped down from the table, and sauntered around the room.\n\n'This is Wonder Boy. Looks like he has made a little wish in the corner.' There was a wet patch on the wall by the door, more or less at the height of Wonder Boy's tail, that reminded me of our first meeting. She reached out and scratched him behind the ears, and he let out a purr like a motorbike starting up. 'He is my darlink. Soon you will meet Violetta and the Stinker. The pram babies you have already met. Mussorgsky is somewhere hiding. He is a little bit jealous of the Wonder Boy. Borodin you will not see. He comes only to take the food. Seven altogether. My little femily.'\n\nI handed her the bottle of wine I'd brought. White Rioja. Nice with fish. We both struggled with the corkscrew, but she got it open and poured us each a glass.\n\n'To bargains!' she said. We clinked.\n\n'Can I help you with anything?' I was nervous about what could be happening in the kitchen, but she gestured me severely to a chair.\n\n'You are my invited guest. Please, Georgine, tek a seat.'\n\nClose up I could see that the tablecloth was not white at all but a sort of mottled greyish yellow, bristling with cat hairs of many colours. The napkins weren't white either, they had pink and red blotches that could have been wine or beetroot or tomato soup. While Mrs Shapiro busied herself in the kitchen, I discreetly tried to clean away the gunge that was encrusted between the prongs of my fork, and to study the room I was in. The only light came from a single long-life bulb screwed into a brass chandelier whose other five bulbs were defunct. On the wall opposite the door was a marble fireplace, and above it a large gilt-framed mirror so spotted and clouded that, when I stood up to take a peep at myself in the green dress, I seemed faded and grey, sadder and older than my mental image of myself, my eyes hollow and too dark, my hair wind-snaggled and too curly, the dress so different to anything I'd worn for ages that I hardly recognised myself. I turned away quickly as if I'd seen a ghost. On the facing wall were two tall windows that seemed to be boarded up behind the curtains, and between them hung a photograph in black and white, an old-fashioned studio portrait of a young man in evening dress with sharp clean features, fair curly hair swept back from a high forehead, and in his left hand, held up against his cheek, the neck of a violin. He had startlingly pale eyes staring out of the picture that caught and held my gaze as though he was present in the room. Strangely, although the photograph was in black and white, it seemed more vivid and alive than my own image in the mirror.\n\nAs I studied the photograph I became aware of a smell \u2013 a faintly fishy aroma that seemed to have wafted into the room. I looked round and saw that Mrs Shapiro was standing in the doorway carrying a large silver tray on which were two steaming bowls.\n\n' _Soupe de poisson_. _Cuisine fran\u00e7aise_ ,' she beamed, placing one bowl in front of me and seating herself opposite me with the other. I looked into the bowl. It was a thin scummy-looking liquid in which some greyish gobs of matter were floating part-submerged.\n\n'Please start. Don't wait.'\n\nI dipped my spoon in. Probably it won't kill me, I told myself. I've eaten worse than this in Kippax. Across the table, Mrs Shapiro was slurping away with gay abandon, pausing only to dab her lips with her napkin. Ah \u2013 that's what those red blotches were. I found that if I held my breath as I swallowed I could manage the liquid. The grey gobs I tried to mash up in the bottom of the bowl so it wasn't obvious how much I'd left.\n\n'Lovely,' I said, trying to find a clean corner of napkin to pat my mouth.\n\nThe second course was in some ways better and in some ways worse. It was better because there were boiled potatoes and leeks in a white sauce which, although lumpy, looked reasonably edible; it was worse because the fish, a whole curled-at-the-corners fillet of something hard, brown and yellow, smelled so sickening that I knew I would never be able to bring myself to swallow it. Even Mum never served anything as bad as this.\n\nAs I was poking away at the potatoes and leeks I felt a sudden warm pressure in my groin. I looked across the table at Mrs Shapiro. She smiled. The pressure turned into pounding, rhythmic and insistent. What the hell was going on?\n\n'Mrs Shapiro...'\n\nShe smiled again. I could feel a vibration accompanied by a strange rasping sound like a car engine trying to start on a cold day. Now through the silky stuff of the dress I felt a sharp prick of claws on my thighs. I slipped my hand down under the tablecloth and touched warm fur. Then I had an idea.\n\n'Mrs Shapiro, that photo,' I pointed to the wall behind her, 'who's it of?'\n\nAs she turned her back for a moment, I slid the fillet off my plate on to the floor, and gave the cat a shove.\n\n'That is my husband,' she turned towards me and clasped her hands together, 'Artem Shapiro. My beloved Arti.'\n\nBeneath the table the purring had intensified, and turned into a satisfied chomping.\n\n'Was he a musician?'\n\n'One of the greatest, darlink. Before the war. Before the Nazis got him into the camp.'\n\n'He was in a concentration camp?'\n\n'Besides the Baltic Sea. Many Jews from all over Europe ended there. Even some we knew from Hamburg.'\n\n'Your family was from Hamburg?'\n\n'Left in 1938.'\n\n'But Artem \u2013 he got away, too?'\n\n'This story is too long, Georgine. Too long and too long ago.'\n\nThe young man in the picture held me with his pale intense eyes. I noticed how elegantly his fingers were clasped around the neck of the violin. In _The Splattered Heart_ the heroine's lover would have hands like this, I thought. Ms Firestorm was already on the prowl, looking out for a great romance set against the turbulent background of World War Two.\n\n'Please tell me, Mrs Shapiro. I love stories.'\n\n'Yes, this is a loff story,' she sighed. 'But I do not know if it will heff a happy ending.'\n\nThe story she started to tell me that night did turn out to be a love story of sorts, and though she related it in her funny hobbled English, my imagination filled in the spaces between the words so vividly that afterwards I couldn't remember what she'd said and what I had imagined.\n\nArtem Shapiro, her husband, she told me, was born in 1904 in the small town of Orsha in a country that sometimes belonged to Poland, sometimes to Russia, sometimes to Lithuania, and most of the time was just a place where people \u2013 Jewish people, anyway \u2013 got on quietly with their business, keeping their heads down during the years of wars and pogroms and the political machinations of the great powers.\n\n'It is our way. We believed if we kept quiet we would survive.'\n\nHis father was a violin maker, quite a successful one, and he thought the boy would learn the trade, too, but one day Artem picked up the instrument and began to play, and that was how it started. Every day for an hour or two after working with his father, he would sit outside in the backyard and play the popular tunes he heard on the street. Then he tried to improvise his own tunes. The neighbours would drop whatever they were doing, and hang over the fence to listen. It was not long before he began to show real promise as a violinist.\n\n'Darlink, everybody who was listening was astonished. They could not believe that such a young boy would be playing like this.'\n\nWhen Artem was in his teens, the family moved to Minsk, the capital of Byelorussia. His parents paid for him to have lessons with a violin teacher, and it was the teacher who suggested that the young man should go to St Petersburg, or Leningrad as it was by then, several hundred kilometres to the east, to study at the conservatoire.\n\n'He was tooken to it like a duck into the water!' she said, gobbling up the vile brown-yellow fish with apparent enthusiasm as she talked.\n\nAfter the revolution, Leningrad was a hub of political and cultural life; musicians, writers, artists, film-makers, philosophers were caught up in the ferment of political ideas. Many had revolutionary sympathies and were eager to put their art to the service of the people. One of these was Sergei Prokofiev, who met the talented young violinist from Orsha when he conducted the orchestra in which Artem was playing.\n\n'Arti, too, wanted to bring the great music in front of the masses.'\n\nHe had learned his socialist sympathies from his father, who was a Jewish Bundist, she explained. Before I could ask what a Bundist was, she rattled on, 'So long as you were not saying something bad about the Bolsheviks, in that time you could play what music you liked.'\n\nBy the late 1930s Artem was playing lead violin with the People's Orchestra and had just started to perform as a soloist. But as Stalin's grip tightened musicians, too, were booted into line. Mrs Shapiro frowned and wolfed down her fish.\n\n'Like poor Prokofiev. He had to repent, isn't it? When I listen to the seventh symphony I think always of how they made him change the ending.'\n\nThe false sense of security afforded by the Molotov\u2013 Ribbentrop pact meant that Russia did not anticipate the German invasion in the summer of 1941. So when Artem heard that his father was ill, he felt safe enough to set out to visit his family in Minsk in June that year. Byelorussia was at that time in the eastern part of the former Polish territories, which had recently been annexed by Russia, and rumours were flying about of what had happened to Jews in the German-occupied west. Artem hitched a ride on a goods train heading west at exactly the time when every Jew in Europe who could flee was heading east, just as the pact collapsed and the German armies swept eastwards through Poland into the Soviet Union.\n\n'But he was reunited with his family?'\n\n'Yes. His parents and two sisters still were there. But the Nazis were building a wall of barbed wires around the streets in Minsk where the Jews lived so no one would run away.'\n\n'A ghetto?'\n\n'Ghetto. Prison. Same thing. But ghetto is worse. Too many peoples crammed inside. No food. Potato peels and rats they were eating. And every day soldiers were shooting people in the streets. Other ones died from diseases. Some suicided themselves out of despair.'\n\nMrs Shapiro's voice had grown so quiet that I could hear a tap dripping in the kitchen, and the scuffle of a feline scratching itself under the table.\n\n'But what happened to Artem's family?'\n\nBy the time Artem arrived in Minsk, the population was already swollen by the thousands of Jews who had fled eastwards, as well as by German Jews for whom there was no longer room in the German and Polish ghettoes or concentration camps. Despite starvation and the periodic typhus and cholera epidemics that raged through the ghetto, and daily summary executions, sometimes of hundreds of people at a time, they just weren't dying fast enough. Shooting them all would use up too much ammunition. Then a local Nazi commandant came up with a clever idea to kill Jews efficiently without wasting precious bullets.\n\nOne morning some forty Jews were picked up at random off the streets, herded to a woody spot on the outskirts of town, and forced to dig a pit. Then they were roughly roped together and pushed into the pit they had dug. Russian prisoners of war were ordered to bury them alive.\n\n'But the Bolshie Russians refused to do it, so the Jews had to be shot in the end and the Russians also. So even more bullets was used up, isn't it?'\n\nArtem's father was among the forty.\n\nTo save bullets and time, mobile gassing vans were set up to tour the district. But why waste all that human labour potential, when the munitions factories were struggling to find workers? It was decided that able-bodied Jews like Artem should be made to contribute to the war effort.\n\n'So they put him in the camp.'\n\n*\n\nThe camp to which Artem was dispatched was a labour camp, not an extermination camp, though it was no holiday camp either, buffeted by cold winds off the Baltic, squatting inside its cage of barbed wire under a perpetually pewter sky. In this miserable spot, a number of German companies, including some that are now household names, contracted to use the cheap labour facilities. Those inmates who could work could eat, the others died.\n\nBut the Lithuanian guards were lax and lazy, and couldn't always be bothered with the security procedures demanded by their new bosses. One early morning on his way to work Artem came upon a guard, still tipsy from the night before, pissing against a wall \u2013 he had chosen a private spot where the wall turned a corner behind the barracks. Artem knew immediately that this was his chance; live or die, he had to take it. Although he was weak from months of semi-starvation, surprise was on his side. He picked up a stone and bludgeoned the Lithuanian over the head; then he stole his uniform and papers.\n\n'And he ran away on all his fastest legs into the forest, to join the partisans.'\n\nShe paused and reached for a cigarette. Beneath the table a fight had broken out over the remnants of my fish. There was a screeching and thrashing of tails.\n\n'Raus, Wonder Boy! Raus, Stinker! Raus, Violetta!' She tried to kick them under the table but her feet got tangled in the tablecloth and she sat back with a sigh of resignation.\n\n'What happened next?' I prompted.\n\nStraightening herself out, she lit her cigarette.\n\n'Ach, Georgine, I cannot tell this story while we are eating good food and thinking of these poor hungry peoples. Another time I will finish it. Better now I should play you some music. Great Russian composers. You would like?'\n\nI nodded. Under the table, hostilities had been suspended while the cats waited for their next course. Wonder Boy was licking his bottom again. Violetta was rubbing herself against my legs. Mrs Shapiro gathered the plates and tottered off into the kitchen, leaving her cigarette smouldering in a saucer. I was beginning to feel a bit strange. Between us, we'd almost polished off the bottle of wine. The dim long-life bulb cast fuzzy downward shadows on the table and walls that made everything seem faded and unreal \u2013 or maybe it was the images from the terrible story working on my imagination.\n\nAfter a while I became aware of a sound in the next room, a low mournful sound, like a voice calling from the netherworld. For a moment I thought it was a cat, then I realised it was music \u2013 such soft, sad music \u2013 that had crept in quietly through the open door. At first it was a single violin, then it was joined by others, and then a tune emerged, throbbing with melancholy and repeating itself over and over, growing louder and higher. For some reason I found myself thinking of Rip \u2013 of Rip and me together, of Rip and me making love, our hands and bodies fumbling for each other in the dark, always finding each other, always coming together, each time the same yet each time different, repetitions and variations.\n\nNow the tempo of the music changed; it became louder, more violent, with cymbals clashing and drums throbbing away like a headache, and the violins racing up and down, faster and faster, arguing with each other, contradicting each other, in a turmoil of sound. I thought of Rip again, and I remembered the terrible fury and churning of our last argument. No, I realised, it wasn't just the music. A gut-churning feeling was building up in my stomach right now. Then Mrs Shapiro reappeared in the doorway with another tray.\n\n'Now we heff a dessert.'\n\n'Er...'\n\nShe placed the tray on the table. It looked like some sort of shop-bought pie, still in its foil dish. I could handle that \u2013 I'd grown up on this sort of food. There was a tub of REDUCED cream on which the sell-by date was clearly visible. I did a quick calculation. Only two days overdue. I'd eaten worse. '... just a little.'\n\nI tasted the pie cautiously. It seemed perfectly all right. I had only a tiny drop of cream, which also seemed all right.\n\n'You like it?' asked Mrs Shapiro.\n\n'Yes, lovely. Delicious. What is it?'\n\n'Prokofiev. _Symphonic Song_. Wait. It will get better.'\n\nAs I listened the tempo of the music changed again. It became graciously flowing and jubilant; the original melody had returned, but with more depths and heights of emotion, as though it was leaping over itself, over the contradictions and arguments, over the terrible drumbeat and stomach-churning turmoil, into a new world, a new happy world where everything was going to be all right again for ever and ever. Tears welled up in my eyes; heavy and warm, they rolled down my cheeks.\n\nThe music stopped and silence seeped into the room. Across the table from me, I saw that Mrs Shapiro was dabbing at her eyes with her napkin. Then she fumbled in her bag for her cigarettes and matches, lit up again, inhaled and then gave a long sigh.\n\n'We heff been living here in this house playing music together. I was playing piano, he was playing violin. Such great music we made together. Now I am living here alone. Life goes on, isn't it?'\n\nI could feel my tears welling up again. How much better it would be, I thought, to love and be loved like this until parted by death, and even after death, than to feel love shrivel and die while life goes on around you, dreary and loveless. Oh heck, there goes my splattered heart again.\n\n'Why are you crying, Georgine? You have lost someone, too?'\n\n'Yes. No. It's not the same. My husband... he walked out on me, that's all.'\n\n'You are still young, you will find somebody else.'\n\nI wiped my tears and smiled. 'If only it was so easy.'\n\n'Darlink, I will help you.'\n\nThe next thing I can remember is throwing up on my own doorstep. I was still wearing the green dress, with my jeans underneath and my pullover and duffel coat on top. I felt terrible. My head was throbbing, and I was alternating violently, frighteningly, between burning hot and clammy cold. Above my head, stars were spinning in the blackness. I knelt on the stone steps and puked again. Then I felt a warm furry presence beside me. It was Violetta. She must have followed me home. 'Hello, cat.' I stretched out my hand to stroke her, and she arched her back and purred, rubbing herself against me. Then she started lapping up the sick off the doorstep. \n\n# 6\n\n# Sticky brown stuff\n\nOn the Sunday morning after my dinner with Mrs Shapiro, I woke up at about ten o'clock. There was a horrible taste in my mouth and a bowl with some slimy stuff by the bed. I must have thrown up in the night, but I couldn't remember anything. My head was throbbing. A hard beam of sunlight was hammering between the curtains where they didn't quite meet, like a chisel splitting my brain. I got up and attempted to fiddle with the curtains, but as soon as I stood up dizziness brought me down again, and I flopped back on the bed. The ceiling seemed to be moving backwards and forwards above me, like in an earthquake. I pulled the covers over my head, but that brought on a panic of suffocation. What had I been dreaming about? I had an image of people roped together, pushed into a pit to be buried alive. Was that a nightmare? No, it was worse than a nightmare \u2013 it was something that had really happened.\n\nI staggered into the bathroom and gulped cold water from the tap, then splashed some on to my face and returned to the bedroom. The light was too harsh. I rooted through my drawer for something to cover my eyes and found a pair of black knickers; I slipped them over my head like a hood. The waist elastic just reached the tip of my nose. I lay back down on the bed and let the darkness enfold me. That was better. If Rip had been there he'd have laughed at me. If Rip had been there he'd have made me a cup of tea and comforted me. I remembered that music, the bounding, soaring, happily-ever-after melody that had carried me along in its arms last night. Was that a dream? Yes, it was.\n\nAt our wedding, the organist had played the _Arrival of the Queen of Sheba_ and Dad overcame his scruples about religion enough to walk me up the aisle on his arm. It was the first time Rip's parents and mine had met, and it was all excruciatingly polite. Rip had discreetly removed the engraving of the Staffordshire coal mine which some ancestor had owned in 1882, and I'd persuaded Dad not to wear his National Union of Mineworkers tie. Mr Sinclair engaged Dad in conversation about rugby, drawing on his own school experiences but skirting around the fact that the sport had been named after his own school, and Dad did his best to keep the conversation up, skirting around the differences between rugby union and rugby league. Mrs Sinclair complimented Mum on her hat, and Mum asked her for the recipe for the chocolate peripherals; Mrs Sinclair skirted around the question without revealing that everything, including the profiteroles, had come from a catering firm in Leek. Mum didn't say anything about the olives on the canap\u00e9s, but I could see her eyeing them with suspicion. It was 1985, remember, and olives hadn't yet reached Kippax. To be on the safe side, she slipped them under a cushion. Later, I saw Mrs Sinclair shaking hands with the vicar, with three olives adhering to her behind.\n\nI poured myself a large glass of water, and went back to bed. I must have dozed off, for when I woke up later, in the afternoon, I felt much better. I went downstairs to rummage in the fridge for something to eat and ended up pouring myself a glass of wine instead. My stomach was still feeling delicate from its Saturday night trauma, and probably it would have been sensible to stick to tea and toast, but I needed something to cheer me up. An after-mood from the nightmare still clung to the edges of my mind. And I was missing Ben. Three more days until he'd be back with me. Carrying my glass of wine I went back upstairs, and I noticed that the door to his room was slightly open, so for no particular reason I went inside.\n\nIt smelled of Ben, or to be precise it smelled of Ben's socks; and here they were, in a waiting-to-be-washed heap near the door. Also in heaps on the floor were his school clothes, his not-school clothes, the books he was halfway through reading, the books he would never read, exercise books, notebooks, and loose papers that might have once been in books, a collapsed stack of DVDs, a mound of CDs and various bits of mysterious electronic gear. A triangle of dried-out pizza with two symmetrical bites, one on each side, was propped against a half-empty bottle of lime-green fluid on the mouse mat. On the walls were posters of the Arctic Monkeys and Amy Winehouse, and a _Lord of the Rings_ poster featuring a close-up of Orc dentistry. My eyes roved around the busy cluttered space and I smiled to myself \u2013 dear Ben.\n\nThe desk was a tip of crumpled papers, broken pens, chewed-down pencils, bottle tops, gum, wrappers, flyers, tissues, all splattered with some kind of sticky brown stuff \u2013 it may have been congealed hot chocolate \u2013 that was also daubed on the keyboard of his computer, and even on the monitor, where a Windows logo was whirling mindlessly around. A small photograph was stuck to the bottom of the monitor with Blu-tack. I leaned forward to look, and my heart squeezed in my chest. It was Ben and Stella. They were sitting on a park bench surrounded by greenery, grinning their heads off.\n\nI bent down to get a closer look at the photo \u2013 Ben's innocent open-mouthed grin; Stella's pretty smile, more posed and self-conscious \u2013 and my sleeve caught the glass of wine, which splashed everywhere, mingling with the brown stuff. I took a tissue from my pocket and started to mop, taking care not to disturb anything, because part of me was thinking that I didn't want Ben to know I'd been poking around in his room. As I wiped the mouse, the computer suddenly whirred into life and the screen came up \u2013 a black background with a single word flashing in red, animated with dancing flames: _Armageddon_. It looked like some stupid computer game.\n\nAfter that fish dinner I avoided Mrs Shapiro for a couple of weeks, then I forgot about her. Life carried on with its limping rhythm: Ben, not-Ben, Ben, not-Ben. I was learning to walk with the limp and I slept better with the black knickers. Sometimes, to cheer myself up, I fantasised about revenge. In _The Splattered Heart,_ feisty Gina, having discovered Rick's infidelities, was also planning something dramatically unpleasant involving extra-spicy Madras vegetable curry and\/or a subtler approach based on fish soup diluted with pee.\n\nI was sitting at my laptop one dull afternoon in November, trying to write about adhesives but sneaking back every few minutes to the exercise book which was open on my desk, when the phone rang.\n\n'Mrs Georgina Sinclair?' An unfamiliar woman's voice, squeaky like a rusty gate.\n\n'Yes. Sort of. Who's speaking?'\n\n'I'm Margaret Goodknee from the Whittington Hospital.'\n\nMy hands went cold and my heart started to thump. 'What's happened?'\n\n'We have a Mrs Naomi Shapiro in A&E.'\n\n'Oh, dear.'\n\nI must confess, all I felt was relief. Not Ben. Not Stella.\n\n'On her admission form she's named you as her next of kin.'\n\n# 2   \n_Adventures with Polymers_\n\n# 7\n\n# Pick and mix\n\n'Why me?' I wondered, half curious and half irritated, as I made my way down a long busy ward looking for Mrs Shapiro. 'Doesn't she have anyone closer?'\n\nI found her at last, shrunk down into the hospital bed, with only her little face peeping out above the sheet, and her black curls straggling over the pillow. The silver line along the parting was several centimetres wide, but apart from that, without her weird make-up, she actually looked better than before.\n\n'Mrs Shapiro? Naomi?'\n\nHer face lit up with a smile of recognition, and she reached out her hand from under the covers to hold mine.\n\n'Georgine? Thenk Gott you come. You heff to get me out of here.'\n\n'I'll do my best, Mrs Shapiro. When you're better. What happened?'\n\n'Slipped on the ice. Wrist brokken.'\n\nShe waved her left hand at me, which was plastered and strapped, the fingers protruding from the dressing like bent grey twigs with splashes of chipped nail varnish at the tips.\n\n'You heff to get me out. Food is terrible. They mekking me eat sossedge.'\n\n'Shall I tell them you want a kosher diet?'\n\n'Kosher pick and mix. No hem, no sossedge. But bekkon I like.' She winked a mischievous eye. 'A little bit of something does you good, isn't it?'\n\nThe sister in charge was a small brisk unsmiling woman with scraped-back hair who sniffed at the idea of pick and mix, so I asked her to put Mrs Shapiro down for kosher. She scribbled it in the file, then she added, 'She doesn't seem to be registered with a GP. We need her NHS card or some form of ID to verify her entitlement.' She must have seen my jaw tighten. 'It's the rule now. Just a box I have to tick.'\n\nWhen I came back to her bedside, Mrs Shapiro was sitting up looking chirpy and trying to get into conversation with the woman in the next bed, who was lying on her back, breathing through an oxygen mask.\n\n'Mrs Shapiro,' I asked, 'are you registered with a doctor?'\n\n'What for I need the doctor?' She was in a fighting mood. 'These young boys, what do they know? Only to ask schmutzig questions. When you last been on the toilet? Please stick out the tongue. What kind of a doctor says this? In Germany we had Doctor Schinkelman \u2013 this was a real doctor.' A faraway look had come into her eyes. 'Plenty medicine. Always red. Tasted of cherries. And plenty tablets for _Mutti_.'\n\n'But do you have a medical card? Any form of ID?'\n\nShe sighed dramatically and passed her good hand across her brow.\n\n'Seventy year I been living in this country, nobody ask me for no card.'\n\n'I know,' I soothed. 'It's like Sainsbury's \u2013 the surveillance society. But you need something to show how long you've been living here. What about the bills on the house? Council tax? Gas?'\n\n'All papers are in the bureau. Maybe they will find something.' She sat up and blinked rapidly. 'They are looking into my house?'\n\n'I'm sure it's just a formality. I'll go and get them, if you prefer.'\n\nShe turned around, gesturing with her strapped-up hand.\n\n'Key to my house is in the coat.'\n\nIn the bedside locker was a dark brown astrakhan coat with a turned collar and cuffs, elegantly fitted at the waist, and conspicuously moth-eaten, with bare patches down to the leather all along the back. She saw me examining it.\n\n'You like this coat? You can heff it, Georgine.'\n\n'It's very nice but...'\n\nIt smelled of old cheese.\n\n'Please. Tek it. I heff another. What's the matter \u2013 you don't like it?'\n\n'... I think it's a bit small for me.'\n\n'Try. Try it.'\n\nI made a show of taking off my duffel coat, and trying to squeeze myself into it. It had a heavy satin lining torn under the armpits, with a sheen of grease around the buttons and cuffs, but still it had a residual touch of luxury. Once, about fifty years ago, it had been a fabulous coat.\n\n'It suits you good, darlink. Tek it. Is better than your coat.'\n\nTrue, my brown Bat Woman duffel coat, even in its 1985 heyday, had been in a lower league.\n\n'It's lovely. Thank you. But look, it doesn't fit.' I pretended to struggle with the buttons.\n\n'You must be more elegant, Georgine. And look at your shoes. Why you don't wear mit heels?'\n\n'I'm sure you're right, Mrs Shapiro. But I like to be comfortable.' I slipped my hands into the generously deep satin-lined pockets. 'Where's the key?'\n\n'Always in the pocket. You must be more elegant if you will catch a man, Georgine.'\n\nI rifled through the pockets. There was a disgusting snot-caked handkerchief with traces of dried blood, a box of matches, a cigarette butt, a sticky boiled sweet with bits of fluff stuck to it, half a crumbled biscuit that had covered everything in greyish crumbs, and a pound coin. No key.\n\n'Should be in there. Maybe is fallen in the leaning.'\n\nThe key had slipped through a hole in the pocket, and was shaking around in the hem of the lining, along with a stub of black eyebrow pencil, two more cigarette butts, an apple core, and some loose change. I fished them all out through the hole and put them in the other pocket.\n\n'Here it is. I'll have a look in your bureau and see if I can find something official to keep them happy.'\n\n'You must look only in the bureau. Not everywhere poking, Georgine.' She was smoothing the bedclothes with a nervous movement. 'Darlink, I am worrying about the Wonder Boy. If you go to my house, you will please put some food for him? Other cats can catch, but this poor boy he is always hungry. And next time you come, Georgine, you bring some cigarettes mit you, okay?'\n\n'I don't think smoking is allowed in hospital, Mrs Shapiro.'\n\n'Nothing is allowed.' She breathed a dramatic sigh. 'Only sleeping and eating sossedge.'\n\nIn the next bed, the woman with the oxygen mask had started to make a horrible gurgling noise. A couple of nurses rushed up, and drew the curtain around the bed. The gurgling continued. There was a clatter of instruments, and low voices talking urgently.\n\n'You heff to get me out, Georgine.' Mrs Shapiro gripped my wrist again. 'Place is full of krankies. Everybody dying.'\n\nI stroked her hand until her grip relaxed. 'You'll be home soon. Would you like me to bring you anything?'\n\nShe gave me an appealing look.\n\n'If you could bring the Wonder Boy...'\n\n'I don't think they allow pets in here.' Especially not Wonder Boy, I was thinking, with his disgusting habits. 'What about your photo of Artem? Would you like to have it with you? I'm sure they'd allow that.'\n\nShe shook her head. 'Too many teefs in here. But Wonder Boy nobody will steal.'\n\nWell, she was right about that. Rather than getting drawn into a plan to smuggle Wonder Boy into the hospital, I changed the subject, thinking maybe reminiscence would settle her, for old people often feel more at home in the past than in the present. And I was curious to know the end of the story she'd started to tell me that night over the fish dinner, twisted up in her convoluted English.\n\n'You never finished telling me the story about Artem. How he got to England. How you met.'\n\nLetting go of my wrist she sank back on to her pillow.\n\n'It is a long _megillah_ , Georgine.'\n\n'You said he ran away to join the partisans in the forest.'\n\n'Yes, in Naliboki. Almost six months he was living mit the Pobeda partisans.'\n\nShlomo Zorin and his Pobeda band of partisans had set up a family camp along the same lines as the Bielski camp in a clearing in the vast Naliboki forest in Byelorussia. Here they sheltered any Jews who made their way there, and even sent scouts back into the ghettoes to organise escapes. Artem Shapiro undertook several of these missions, using stolen papers; his bright blond hair, inherited from his grandfather, allowed him to pass himself off as a Christian.\n\n'Such a beautiful blondi, he was. He could pass easy.' Mrs Shapiro's voice wavered. 'So one day he made his journey back to Minsk.'\n\nEarly in the autumn, while there was still plenty to eat in the woods and before the snows started, Artem set out to find his mother and sisters, thinking to lead them back to the forest with him. But the Minsk ghetto, when he arrived, seemed like a ghost town of living skeletons shuffling around the once-familiar streets with death in their eyes. He learned from a former neighbour that his mother was dead \u2013 she had died of starvation, or maybe of a broken heart, shortly after he had been taken away. One of his sisters had died of typhus. No one knew what had happened to the other sister. Someone told him that she'd been taken to Auschwitz; someone else told him that she'd used her mother's gold teeth to bribe a local brigand, and had got away, 'To Sweden. Or mebbe to England.'\n\nAfter that visit to Minsk, something broke apart inside Artem's heart. All the music died. A terrible chorus of wailing filled his head night and day, so he could neither sleep nor work nor think. In a situation where morale is crucial, he felt himself becoming a drag on the Pobeda camp, undermining everyone's spirits with his own misery. One morning, after a night of wailing dreams, he smashed his violin against a tree. Then he said goodbye to Zorin and headed eastwards through the silent snow-laden forests towards his birthplace at Orsha. Maybe he was hoping to make contact with surviving members of his family, but when he arrived in the spring of 1942 the Orsha ghetto had already been liquidated. Thousands of Jews had been shot and the remainder had been herded on to freight trains.\n\n'They put them on the trains but they been transported to nowhere. They been poisoned where they waited, in the wagons. The Russian prisoners dug up a mass grave and buried them.' She paused. Her breath came slow and rattly. 'Truly they wanted to kill us all.'\n\nArtem did not return to Zorin. Such a fury possessed him that simply surviving in the forest was no longer enough. The chorus of wailing resolved itself into a single long howl \u2013 the howl of a wounded animal ready to kill. He headed north to join up with a group of Russian partisans who were harrying the German army, which had by now encircled Leningrad. The first time he ambushed a German jeep with a tree felled across the road, he taunted them with savage delight: ' _Ich bin der ewige Jude!_ '\n\n'Shut up that nonsense!' bawled Velikov, the commander of the unit. 'Just shoot!'\n\nThe partisans were trying to open up a supply route into the beleaguered city. It was a dangerous mission, for the German grip on Leningrad and the Finnish corridor was almost total, but by early 1943 Meretskov had brought the front forward from the east, and a few supplies started to get through. Artem was with a group of partisans who were driving a sleigh loaded with potatoes and beet across the frozen Lake Lagoda when they came under fire from a German patrol. The other three perished instantly, along with their stubby-legged Mongolian pony, but Artem was only wounded in the shoulder. He knew that running away over the ice in winter would be certain death; instead he crawled into the sleigh, hid under the wolfskins which covered the beets, and waited for his destiny to catch up with him. Either the Germans would take him, or the Russians would rescue him, or he would freeze to death. Everyone knows that hypothermia is a pleasant drowsy death. At least I won't die of hunger, he thought. He waited and listened, trying to staunch his wound with a cloth wrapped around a lump of ice. He could hear shooting and voices calling, but they seemed to be getting further away, not closer.\n\n'Then started the snow to fall.'\n\nHe must have fainted or drifted off to sleep, for he lost track of how long he'd lain there, when he was jerked into sudden consciousness by a sharp jolt of the sleigh. He peeped from under the snow-heavy wolfskins, and saw that it had been harnessed to what he thought was another pony that was trotting over the ice into the whirling blizzard. Seated above and behind him, he could hear two men talking. He caught the sound of laughter and a whiff of cigarette smoke. Were they talking German or Russian? He couldn't tell.\n\n'And all this time the pony was walking in the snow and the ice, and the snow was falling all the time, and the pony was walking on in the freezing snow and on and on over the ice and on and on...'\n\nShe stopped. I waited for her to continue. I thought she must be remembering and maybe she found the memories too painful to talk about. But after a while I heard a gentle snoring and I realised she'd fallen asleep.\n\n'When do you think Mrs Shapiro might be able to come home?' I asked the sister at the desk on my way out.\n\n'It's too early to say. We'll see how she gets on,' she replied without looking up.\n\n'But it's only a broken wrist, isn't it?'\n\n'I know, but we'll have to assess her home situation. We don't want her to go back and have another fall. At her age, she might be better off in residential care.'\n\n'Why, how old is she?'\n\n'She told us she was ninety-six.' She looked up. Our eyes met, and mine must have betrayed my astonishment. 'Isn't she your gran?'\n\n'No, she's just a neighbour. I live a couple of streets away. I don't really know her that well.'\n\nCould Mrs Shapiro really be ninety-six? But why would she lie about her age?\n\n'Another reason it'd be useful to have some ID.'\n\n# 8\n\n# Biopolymer\n\nI spotted Wonder Boy lurking in the porch of Canaan House as I walked up the path. He was ripping the guts out of a bird he'd caught \u2013 it looked like a starling. It was still alive, struggling between his paws. Feathers were everywhere. He bolted off into the bushes as he saw me coming, the bird still flapping in his jaws. This cat can well take care of himself, I thought. Usually I'm fond of cats but there was something horrible about Wonder Boy. I tried to imagine catching him, stuffing him in a bag, and taking him on the bus to the hospital. No way.\n\nThe key Mrs Shapiro had given me was only a Yale; in fact any enterprising burglar could have just smashed the frosted glass and put his hand through to turn the lock. I pushed the door open against a heap of mail that had piled up on the inside. As soon as I stepped into the hall the stink hit me, a bitter must of cat pee, damp and rot. I put my handkerchief up to my nose. Out of nowhere, Violetta materialised around my ankles, mewing pitifully. Poor thing \u2013 she must have been locked in the house for at least three days. I picked up the mail and flicked through it in case there was anything that needed attention, but it all seemed to be junk. There was even an offer for a Sainsbury's Nectar Card.\n\nI followed Violetta through to the kitchen. A chaos of dirty plates, dead cups with remains of disgusting brown fluids, empty tins and greasy ready-meals packaging was spread across every grimy surface. In a cracked pot sink under the window, a stack of unwashed dishes and congealed food remnants was soaking in scummy water on to which a cold tap was drip-drip-dripping. The gas cooker was crusted with dark brown gunge, and so old that it had levers instead of knobs. There was an Aga, but it was unlit and seemed to be used for storing old newspapers. A dank mouldy chill pervaded everything. I shivered. Even in my warm duffel coat I was cold.\n\nI hunted around and found a dozen cat-food tins in a cupboard. I spooned some out into a bowl for Violetta and she wolfed it down, almost choking in her desperation. Then I unlocked the back door \u2013 the key was on the inside \u2013 refilled the bowl, and put it out on the step. Wonder Boy appeared, hissed at Violetta, batted her out of the way, and polished it off. A few other scrawny moggies were hanging around too. I fed them all \u2013 there must have been a good half-dozen of them, miaowing and rubbing themselves against me. A couple of them sneaked indoors between my legs. I locked the kitchen door and returned to the house.\n\nThe bureau Mrs Shapiro had been talking about was in a downstairs room which could have been a study. The window had been boarded up behind drawn curtains, so the only light was from a lone surviving candle-bulb in the heavy gilt candelabra that cast a feeble glow over the old-fashioned floral wallpaper, floor-to-ceiling bookcases, Persian rugs, and a tiled fireplace above which an ornate ormolu mirror would have reflected the blocked-out view over the garden. Even in the gloomy light I could see it was a lovely room. The smell was different, too, musky and dusty, with only a faint trace of cat pee. There was a spoon-back armchair and two desks \u2013 a mahogany kneehole desk by the window, and a tall oak bureau-bookcase in an alcove beside the chimney breast.\n\nI decided to start here. I have to confess that, even then, Ms Firestorm was looking over my shoulder and whispering in my ear, there must be a story here \u2013 maybe a better story than _The Splattered Heart._\n\nThe bureau was full of papers, mostly bills in the name of Naomi Shapiro, and some, the older ones, in the name of Artem Shapiro, and bank statements from a joint-name account. The most recent of these, to my astonishment, showed a balance of just over \u00a33,000. The oldest I could find dated back to 1948. There was, it seemed, a small monthly income from an annuity, as well as Mrs Shapiro's widow's pension going into the bank. I took a selection of statements at random; would these give the hospital the information they needed? In the same drawer, held together with a rubber band, was a bundle of receipts including one for \u00a325, dated 26th October from Felicity NU2U Dress Agency, and one dated 16th October for \u00a323 from P. Cochrane, Antique and Secondhand Emporium, New North Road. That explained the pram.\n\nThere must be something else, I thought, something personal to show a date or place of birth, of baptism or marriage, education or employment. You can't live a whole life that's only recorded through bills and receipts. The kneehole desk was crammed full of stationery, crumpled notepaper, dried-up pens and stubs of pencils, receipts, old tickets, train timetables years out of date, a library card, also out of date, and assorted out-of-date leaflets about pensions and benefits: the useless bits of officialdom we cart with us through life. One drawer housed a correspondence with the Council about the monkey puzzle tree, which Mrs Shapiro had wanted to cut down, although apparently it had a tree preservation order on it.\n\nIn the last drawer there was a thick brown envelope stuffed with official-looking papers. This was what I'd been searching for. An odd-looking passport, light blue with a black stripe on one corner. Artem Shapiro; date of birth 13th March 1904; place of birth Orsha; date of issue 4th March 1950, London. Ration book: Artem Shapiro 1947. Driving licence: Artem Shapiro 1948. Abbey National Life Insurance plan: Artem Shapiro 1958. Death certificate: Artem Shapiro 1960; cause of death: cancer of the lung. Knowing his story, I felt a special tug of intimacy as I turned the flimsy typewritten paper over in my hands. So that was how his journey ended: the ghetto, the barbed-wire camp, the silent forests, the ice-bound lake. I folded the death certificate and put it back, hoping he'd died in his sleep, cosseted with morphine.\n\nBut what about her? A Co-op savings book was the only document that had her name on it: Mrs N. Shapiro 13th July 1972. There has to be something else, I thought; and I remembered what she'd said \u2013 _you must look only in the bureau_. So if anything had been deliberately hidden, I wouldn't find it here.\n\nIn a frenzy of curiosity, I poked through the other rooms. The sideboard in the dining room where I'd eaten the death-defying fish dinner yielded nothing but plates and cutlery. The sitting room was dark, the windows boarded up, and the light switch didn't work. I'd need a torch to search in here. Under the staircase, behind the pram, a narrow door opened on to a stone stair leading down to the basement. A wave of trapped musty air rose up towards me. I felt with my hand along the wall for the light switch, and a fluorescent strip light juddered into life, flickering madly on and off, plunging the low-ceilinged room alternately into light and pitch darkness.\n\nIt seemed to be some kind of workshop. A glass-fronted cabinet was fixed to the wall with rows of tools neatly arranged, the blades now tarnished with rust. Below it was a workbench with a variety of clamps. Bits of strangely carved wood were hanging from hooks. I realised after a few moments that they were panels and necks of unfinished violins. There was a pot of dried-up glue, a small dried-up brush sticking out of it. The glue was clear and amber-coloured, still exuding a faint sickly whiff. Animal glue. Bio-polymer. Used for woodworking, veneers and inlays, until better modern synthetic glues came along.\n\nMy boss Nathan once told me that the Nazis had made glue from human bones. Lampshades from human skin; mattresses stuffed with human hair. Nothing wasted. I was beginning to feel dizzy. Maybe it was the strobe effect of the faulty fluorescent tube, or the memories trapped in the ghost-breathed air.\n\nI made my way back up the stone stairs. As my fingers felt for the light switch I turned back towards the workshop, and that's when I saw a flash of colour on top of the tool cabinet \u2013 a couple of millimetres of blue just visible above the architrave. Curious, I went back down and pulled up a chair to have a look. It was an oblong tin, a bit rusty, with a picture of Harlech Castle surrounded by an improbably blue Welsh sky. I lifted it down and eased it open. It was the sort of tin that would once have held toffees or shortbread biscuits, but all it had in it now was a few photographs. I slipped it under my arm and went back up into the light.\n\nFrom the hall, a wide staircase with a curved mahogany banister led up to the first floor. As I mounted the treads, still clutching the tin, a threadbare Axminster carpet secured by brass rods released clouds of dust under my feet. The same mahogany handrail galleried the first-floor landing, off which nine doors opened. One of them was slightly ajar. I pushed it further open. A scurry of movement. Two lean stray cats bolted out between my legs. The room was large and light, with a double window overlooking the front garden, and dominated by a massive art-deco walnut double bed on which a tattered-eared tomcat \u2013 he had the same moth-eaten look as Mrs Shapiro's astrakhan coat \u2013 was curled up asleep. Raising his shaggy head he followed me with his eyes as I came in. The stench in here was terrible. Phew! I opened a window. 'Shoo! Shoo! Piss off!' I tried to chase him out but he just looked at me with contempt. Eventually he uncurled himself, jumped down from the bed flicking his tail grumpily from side to side, and sauntered towards the door.\n\nThis, I guessed, was Mrs Shapiro's bedroom, for her clothes were scattered everywhere \u2013 the Scotch plaid baker boy cap, the peep-toe high heels, and on the floor by the bed a pair of peach camiknickers trimmed with cream lace, a faint stain yellowing the silk. The walnut wardrobe, carved with artdeco sunbursts, was full of clothes on satin-padded hangers, reeking of moth-balls, stylish and expensive like costumes in a Humphrey Bogart movie. A matching sunburst dressing table stood in one corner, with a triple-hinged mirror facing the window through which I had a view of the garden. I rifled through layers of ancient decomposing make-up and musty, slightly stinky underwear. There was nothing of interest, so I sat down on the edge of the bed, opened the Harlech Castle tin, and spread out the six photographs.\n\nMost were in black and white, but the top one was in sepia, creased and tattered at the edges. It was a family portrait from the turn of the century: the mother in a lace-collared dress cradling a baby, the father with a beard and a tall hat, and two children, a little girl wearing a flouncy dress and a strikingly blond toddler in white pantaloons and an embroidered shirt. There was writing on the back that didn't seem to make sense. Until I realised it was in Cyrillic script. All I could make out was the date: _1905_. He must have carried it with him, hidden in a pocket or a lining, all that way.\n\nNext, a wedding photograph caught my eye: a tall man, fair and handsome, grasping the hand of a pretty woman with ardent eyes and thick curly black hair pinned up beneath a crown of white blossom. They were gazing out of the photograph, wide-eyed, half smiling, as though taken by surprise at their own happiness. The man I recognised as Artem Shapiro. But who was the woman? An attractive heart-shaped face with wide-set dark eyes and a full, generous mouth. I studied it carefully, for people's faces do change as they age, but, really, there could be no doubt. The woman in the photo was not Naomi Shapiro.\n\nI was still staring at the photo when suddenly I heard a sound outside in the garden \u2013 voices, and the clack of the gate. My heart thumped. Quickly I slipped the photos into my bag, closed the tin and shoved it on top of the wardrobe out of sight. In one of the panes of the triple mirror I could see a reflection of the window and, through it, the garden. A man and a woman were standing on the path; they were standing and gazing at the house. The woman was a stout redhead, wearing a vivid green jacket; the man was stocky and redcheeked, wearing a blue parka, smoking a cigarette. The man stubbed his cigarette out on the path and spoke to the woman. I couldn't catch his words, but I saw her toothy laugh. By the time I came down to the door they'd gone. \n\n# 9\n\n# Rubber\n\nThere was a different nurse on duty when I went back up to the hospital next time. She examined the papers I showed her without comment, ticked a box on Mrs Shapiro's notes, and passed them back to me.\n\n'How's she doing?' I asked.\n\n'Fine. She'll be ready to go as soon as we can get her home assessment done.' She flicked through the notes. 'I understand you have the key to her house. I'll get Mrs Goodknee to ring you for an appointment.'\n\nMrs Goodknee again. I imagined someone in a miniskirt with chubby dimpled knees.\n\nMrs Shapiro was sitting up in bed, her hair combed back tidily, the hospital nightgown, antiseptic green, buttoned up to her throat. She seemed well; the stay in hospital had fattened her up. Her cheeks were rosy and her eyes looked bluer \u2013 yes, her eyes were definitely blue.\n\n'Hello. You look good, Mrs Shapiro. Are they feeding you well? Are they still making you eat sausages?'\n\n'Not sossedge. Now is better. Now is chickens and fry pottetto. Did you bring the Wonder Boy?'\n\n'I tried, but he ran away,' I lied.\n\nI wanted to ask her about the photographs, but I held back because I didn't want to admit that I'd been rifling through her house and had discovered the hidden tin. I would have to find another way of worming the story out of her.\n\nWe sipped the thick, bitter tea that came around on the trolley and munched our way through the box of chocolates I'd brought in my role as next of kin.\n\n'Mrs Shapiro, I'm worried that your house is... well... don't you think it's a bit big for you to manage? Wouldn't you be happier in a nice cosy flat? Or in a home where you'd have someone to look after you?'\n\nShe looked at me with wide-eyed horror, as though I'd put a curse on her.\n\n'Why for you say this to me, Georgine?'\n\nI couldn't find polite words to explain my concern about the smell and the gunge and the crumbling fabric of the house, so I just said, 'Mrs Shapiro, the nurse thinks you might be too old to live on your own.' I studied her face. 'She told me you're ninety-six.'\n\nHer mouth twitched. She blinked. 'I am not going nowhere.'\n\n'Mrs Shapiro, how old are you really?'\n\nShe ignored my question.\n\n'What would heppen to my dear cats?' A stubborn look had come over her. 'How is the Wonder Boy? Next time you must bring him.'\n\nI told her about Wonder Boy's starling \u2013 'That notty boy!' \u2013 and Violetta's plaintive mewing \u2013 'Ach! Always she is singing _La Traviata_!' \u2013 and the cat that sneaked upstairs to sleep on her bed. 'That is Mussorgsky. Maybe it is my fault, I allow it. Darlink, sometimes I am so lonely in the night.'\n\nShe glanced at me, and my face must have given something away, because she said, 'You also are lonely, Georgine, are you? I can see in your eyes.'\n\nI nodded reluctantly. I was the one who was supposed to be asking the questions. But she squeezed my hand. 'So, tell me about your husband \u2013 the one who was running away.'\n\n'Oh, it's a long story.'\n\n'But not so long as mine, isn't it?' An impish smile. 'It was a story of loff at first sight?'\n\n'Actually, it was, Mrs Shapiro. Our eyes met across a crowded room.'\n\nIn fact it was a courtroom in Leeds, where two miners from Castleford were on trial for a picket-line scuffle. Rip was defending; he was still doing his articles and volunteering at the Chapeltown Law Centre. I was a junior reporter on the _Evening Post_. After the verdict was announced \u2013 they were cleared \u2013 we went for a celebratory drink. Later Rip drove me home to my parents' bungalow in Kippax, and we made love in front of the fire. I remembered how I'd teased him about his name.\n\nMe: (Twisting my fingers into his curls.) Knock, knock.\n\nHim: (Fumbling with my bra, his mouth wet on my ear.) Who's there?\n\nMe: (Pulling him down on top of me.) Euripides.\n\nHim: (Hand up my skirt.) Euripides who?\n\nMe: (Giggling between kisses.) You rippe dese knickers off...\n\nSo he did. It was strange, because we hardly knew each other, yet it was as if we'd known each other for ever.\n\n'And your parents, what did they say? They were a little surprised, isn't it?'\n\n'Fortunately we were dressed by the time they got back. Mum fell for him at once. He could really put on the charm. Dad thought he was a class enemy. You see Rip was from a moneyed family, and I thought he might patronise my parents. But he was nice... respectful.'\n\nShe flicked her head impatiently. 'So tell me more about loff.'\n\n'Well...' the memories tightened in my throat, 'you could say it's a tempestuous story of forbidden love between an almost-aristocrat and a humble girl from a mining village.'\n\nShe nodded. 'This is a good beginning.'\n\nThey'd been out to the Miners' Welfare in Castleford \u2013 a retirement do for a fellow pit-deputy. There'd been a singalong and speeches, and then more beer was drunk. Dad was glassy-eyed and unusually talkative. Mum, who'd drawn the driving straw, was also not stone-cold sober.\n\nDad: (Mutters to Mum.) What the heck's our Georgie brought home?\n\nMum: (Whispers to me.) You've landed a good fish here, Georgie.\n\nMe: (Embarrassed, to Rip.) Meet my parents, Jean and Dennis Shutworth.\n\nRip: (All charm and golden curls.) Rip Sinclair. Delighted to meet you.\n\nDad was wearing his best three-piece suit, the waistcoat all buttoned up. The only concession he ever made to slackness was a slightly loosened tie. Mum, on the other hand, had long since surrendered to the lure of the elasticated waistband, but she'd made a special effort for the occasion, with a Cupid's bow of cerise lipstick and a dab of Je Reviens behind her ears.\n\nMum: (Taking extra care with her vowels.) Rip. That's an unusual name.\n\nRip: (A dimply self-deprecating grin.) It's short for Euripides. My parents had great hopes for me. (His smile makes my heart jump about all over the place. I'm in love.)\n\nDad: (Whispers to me.) Not your type, Georgie.\n\nMe: (Whispers to Dad.) You've got it wrong. He's not like that. He's on our side.\n\nDad: (Jaw tight. Silence.)\n\nMum: (Getting in quick.) Would you care to join us for some tea?\n\n'And so he drank the tea?' Mrs Shapiro stifled a yawn. 'Mit your parents? This is quite normal in Germany.'\n\n'No. In Yorkshire, tea also means dinner.'\n\nMum had pulled the giant-sized pack of oven chips out of the freezer, shook the contents into a Pyrex dish, and slipped a dozen pre-cooked BBQ-flavour Chicken Drumstix under the grill, heated up a tin of Jackson's own-brand mushroom soup in the microwave and poured it over the drumsticks. My heart shrank into its boots. 'Chicken chez-sewer,' she said, sprinkling them liberally with salt, in case Mr Jackson had been stingy in that regard. Rip made a great show of pleasure, chomping noisily and wiping his mouth on a torn-off square of paper towel. Mum was completely charmed.\n\nWe'd all squeezed round the bench and table in the kitchen. Rip was wedged between Dad and the corner. I was sitting on the other side with Mum.\n\nDad: (Still suspicious.) So what do you do for a living?\n\nRip: (A strange look of panic has come over him.) I'm training to be a... (he catches my eye)... Johnny...\n\nMe: (What's going on? Why has he turned suddenly weird?)\n\nMum: (From the cooker. Awed.) That sounds interesting.\n\nRip:... a solicitor. (Dad is chomping a drumstick. Rip is gesturing to me behind the table \u2013 a hand movement that looks a bit like wanking.)\n\nMe: (Proud.) He was defending the miners in court today, Dad.\n\nDad: (Determined not to be impressed.) You mean Jack Fairboys and Robbie Middon?\n\nRip: (Giving me a kick under the table.) Yes, Jack and Rob. Rub. Rub... ber.\n\nDad: (Gives him a funny look.) They got off, din't they?\n\nRip: (Shifty.) Absolutely. Scot free.\n\nDad: (Concentrating on the ketchup bottle.) Lads being lads. Should never have gone to court.\n\nRip: (More covert under-the-table wanking.) No case to answer. On the floor. By the fire. Justice was done.\n\nMe: (A light dawns.) Excuse me, Mum...\n\nI squeezed out past her and into the sitting room. There it was on the floor by the fire, glistening and slippery. I picked it up and tossed it into the embers. There was a brief sizzle and a smell of burning rubber. In the kitchen I could hear Mum saying, 'I like a man with a good appetite. Euridopeas! Well I never!'\n\nI looked across to see whether Mrs Shapiro had got the joke, but her eyes were closed, and I realised she'd drifted off to sleep long ago.\n\nWhen I got home, at about three o'clock, there was a message on the answering machine from Mrs Goodknee. Would I be so kind as to ring her \u2013 a tinny, middle-aged voice. I rang, and got another answering machine. I left a message. Then I made myself a cup of tea, and took it up to my room. I'd taken the six photographs out of my bag and spread them across the floor in front of the window like playing cards. Now, crouching down beside them, I frowned as I tried to puzzle out the story I was sure was there.\n\nFirst, the Shapiro family in which Artem was the toddler, taken in 1905. Then the wedding photograph \u2013 a different woman. Artem Shapiro must have been married to someone else before Naomi. The same couple, Artem and the mystery woman, were pictured in another photograph standing in front of a fountain. There was snow on the ground. He was wearing a cap pulled down low over his eyes and smoking a cigarette, grinning broadly at the camera. She was wearing a tight-waisted coat and a beret cocked rakishly to one side, looking up at him. There was something scrawled on the back: _Stockholm Drott_... I couldn't make out the rest of the word.\n\nThere was a group photo, a man and four women wearing formal clothes seated around a piano. _Wechsler family, London 1940_ it said on the back. I looked closely, but the faces were too small to be distinct. In another of the photographs I recognised Canaan House with the monkey puzzle tree, quite a bit smaller than now, in the background. Two women were standing in front of the porch. The taller of the two looked like the brown-eyed woman in the wedding photograph. The other, curly-haired and elfin small, I didn't recognise. I turned the picture over. On the back was written _Highbury 1948_. I looked more closely \u2013 although the facial features were indistinct, there was something familiar about the defiant feetapart pose of the smaller woman. I remembered the slight boyish figure in the light of the street lamp, pulling things out of the skip. Naomi. So they'd been together in Canaan House, they'd known each other.\n\nI recognised the taller woman in another photo; this time she was alone, standing in an arched stone doorway, wearing a flowery dress, her dark eyes squinting into the sun, smiling. On the back was written _Lydda 1950_. That's a pretty name, I thought, for a pretty woman. But who was she?\n\nDownstairs, the front door slammed; the house shook. Ben coming home from school at half past four. I heard the thud of his school bag in the hall, the slap of his parka on the floor, and the thump-thump of his footsteps on the stairs. A few minutes later, I heard the Windows welcome chimes. He hadn't even said hello. I felt something in my chest fall away and flap against my heart. Sweeping the photos together into a pile, I went down to the kitchen, made two cups of tea and carried them upstairs. I knocked on the door of his room but he didn't answer, so I pushed it open with my foot. Ben was sitting at his desk staring at the computer monitor. I caught a quick glimpse of the screen \u2013 a flash of red writing on a black background. A single word, picked out in flickering white flames, leapt out at me: _Armageddon_. Then with a click of the mouse the screen changed to a Microsoft sky.\n\n'Ben...'\n\n'What?'\n\n'What's the matter, love?'\n\n'Nothing.'\n\nI reached out and ruffled his hair. He flinched under my touch and I withdrew my hand quickly.\n\n'It's okay to feel upset, Ben. It's a hard time for all of us.'\n\n'I don't feel upset.'\n\nHe was still staring silently at the screen, his hands clenched into fists, resting on the front of the keyboard as if waiting for me to go away. The blue light of the monitor caught the curve of his cheek and his upper lip, lightly shadowed with dark, soft down.\n\n'Is it school? How's the new class?'\n\n'Okay. Fine. Cool.'\n\nThe move from Leeds to London had been hard on Ben. He'd resented being plucked out of his group of friends, some of whom he'd known since pre-school playgroup, and having to fight his way into the unwelcoming circles of his North London comprehensive. He never brought any friends home, but a few times he'd come back from school later than usual, and muttered something about having been with someone called Spike. Spike \u2013 what kind of a name was that? Although I burned with curiosity, I knew better than to press for details.\n\n'What would you like for tea, love?'\n\n'Anything. Spaghetti.'\n\n'Okay. About half an hour?'\n\n'I'll come down, Mum. All right?' he said without looking up, in a voice that meant 'leave me alone'.\n\nI went into the kitchen and poured myself a glass of Rioja, feeling my failure sink inside me like a stone. Failed wife. Failed mother. Friendless \u2013 for my old Leeds friends were Rip's friends, too. I tried ringing Stella in Durham, but she was out, and the resident rock band was in session. Mum had enough troubles of her own \u2013 I'd ring her when I was feeling better. I downed the Rioja in a couple of gulps, and poured another. Maybe I should get a cat \u2013 or seven or eight.\n\nNo, I'd just have to pull myself together and make new friends here in London. In fact I'd made one already. (The Rioja slipped down, warmly reassuring.) Sure, her food hygiene left something to be desired, but we were mates. And I had online work colleagues, too, whom I'd known for years but never met. I'd drop in at the _Adhesives_ office in Southwark one day and say hello. I was particularly curious to meet Nathan, the boss. He had a soft, confiding voice when we spoke on the phone, as though he was sharing a secret, not just passing on technical information. I'd no idea what he looked like, but I imagined someone hunkily intelligent, with horn-rimmed glasses and a sexy white lab coat. Penny told me he was single, so I was in with a chance, and he lived with his elderly father, which seemed gentle and caring.\n\nPenny was the admin manager; I'd never met her either, but she liked to gossip on the phone in her booming voice, filling me in about all the other people I'd never met: Sheila the office junior; Paul and Vic, who took care of the technical side and, alternately, of Sheila; Mardy Mari, the cleaner from hell; Lucy from design, who was a Jehovah's Witness and got up Mari's nose. Then there were the other freelancers like me, whose intimate lives she dispensed without inhibition.\n\nRip's new Progress Project colleagues were frighteningly high-powered. I'd met some of them at a Christmas party last year. He'd introduced me to a couple called Tarquin and Jacquetta (Mum would think they were a kind of food bug) and Pectoral Pete and his wife Ottoline. He was bulging pectorally out of a loud check jacket. She was like a china doll \u2013 dainty and expressionless, with a perfect bow-shaped scarlet mouth and a voice that tinkled like cut glass. Rip had spent most of the evening out in the corridor tapping at his BlackBerry.\n\nI poured myself another Rioja. My cheeks began to glow pleasantly. I went and fetched my exercise book.\n\n#\n\n# The Splattered Heart   \nChapter 3\n\n_'Darling, I have to attend to some really important work on my BlackBerry,' said remarked Rick one evening._\n\n_'Of course, beloved,' Gina said murmured softly._ (Vary your vocabulary, Mrs Featherstone used to say.) _'Your work is really important and must always take precedence over everything else.'_\n\n_'How lucky I am to have such an understanding wife,' said he uttered, and kissed her on her cheek before disappearing._ (I know this is a bit unbelievable, but it is fiction.)\n\n_An hour later Gina was surprised to hear a ringing sound suspiciously like Rick's BlackBerry coming from the study. But Rick was nowhere to be seen._\n\nSuddenly I felt the pressure of a warm hand on my shoulder.\n\n'When's dinner ready?'\n\nQuickly, I closed my exercise book and pushed the almost-empty wine bottle aside.\n\n'Sorry, Ben. Just catching up on a bit of work.'\n\nHe frowned. 'You should go easy on that stuff, Mum.'\n\n'What, this?' I giggled. 'It's only a little Rioja.' Was he worried that I would turn into an unfit mother? I caught the anxious look in his eyes, and pulled myself together. Maybe he had a point.\n\nWe cooked dinner together. Pasta with anchovies, broccoli and Parmesan \u2013 a recipe Mrs Sinclair had taught me. Dad once boasted that he had never eaten broccoli in his life, and never intended to. Mum said that anchovies \u2013 anch _oo_ vies she called them \u2013 made her breath smell. Parmesan they did eat \u2013 they sprinkled it out of a cardboard container straight on to tinned spaghetti hoops. Mum said it gave them a touch of distinction.\n\nBen slurped his spaghetti noisily, pulling a goofy face to make me laugh, like when he was a little boy pretending to eat worms. From the next room, we could hear the television booming, the chimes of the evening news. I wasn't really paying attention; I was still thinking about Rip \u2013 his BlackBerry obsession, my toothbrush-holder obsession. How had we let our happiness be ruined by such trivial things?\n\n'Why do they do that?' Ben asked suddenly. His face clouded over and he seemed to hunch himself lower over his plate.\n\n'What?'\n\n'Suicide bombers \u2013 why do they blow themselves up?'\n\nHe was listening to an item on the news.\n\n'It's because... when people are desperate... it's the way they draw attention...'\n\nThe warm glow from the Rioja had worn off and a gnawing headache was burrowing into my skull. 'It's when you want to hurt someone so much you don't care if you hurt yourself, too.' Desperate. I remembered the frothed milk splattered all over the kitchen.\n\n'But why do _that_? It's gross.' Ben was still staring into his plate, twirling the remaining strands of spaghetti around his fork. Then he said, without looking up, 'It's like... There was this kid at school who cut his arms with a razor.'\n\n'Oh, Ben. Why...?' I felt a rush of anxiety \u2013 I knew the cruelty kids could unleash on each other.\n\n'Dunno. Like you said. Drawing attention.'\n\nMy heart lurched. A buried image from my own schooldays had pushed its way up into my mind. Kippax. It must have been about 1974. A girl cut her arms in the toilets.\n\n'Ben, if you're feeling...'\n\n'It's all right, Mum. I'm all right. Don't stress.'\n\nHe smiled fleetingly, loaded his plate into the dishwasher, and slouched off upstairs.\n\n# 10\n\n# Polymerisation\n\nNext morning I found myself nursing a headache from last night's Rioja, worrying about Ben, and wrestling with a chain of polymers. Polymerisation is the key to the chemistry of adhesion \u2013 it's when a single molecule suddenly grabs on to two other similar molecules on each side, to make a long chain. A bit like line dancing. Not what you feel like first thing in the morning. Then the phone rang. It was Mrs Goodknee, trying in her squeaky voice to get me to hand over the key so she could do her home assessment. I insisted we meet at Canaan House and look around it together. We arranged to meet at noon.\n\nI wanted to give Mrs Shapiro the best chance I could, so I went over there about an hour earlier to prepare for her visit. I'd filled a plastic bucket with cleaning stuff, air-freshener spray, and a pair of rubber gloves, and set off at a brisk pace. Instead of the batty-woman outfit I was wearing a smart grey jacket which I hoped would make an appropriately serious impression. The hard wintry air made my centrally-heated lungs gasp with the shock of the cold at every breath, and the brightness of the light stung my hungover eyes, but I forced myself to look up at the sky. The clouds had cleared and a shaft of low sunlight gilded the upstairs windows of the terrace of houses along my road. My heart lifted. Winter sunshine \u2013 it seemed like a gift, a promise of warmer days to come. I started to hum, ' _Here comes the sun_... _na na nah na_...'\n\nThere was another clump of bird feathers on the path at Canaan House \u2013 a pigeon, this time. I kicked them out of the way. The cats must have been waiting for me because the moment I approached the house they all appeared, clamouring around me, miaowing with their pink hungry mouths. I fed them outside, taking care not to let them sneak indoors.\n\nThen I got to work on the kitchen. I took off my smart jacket, pulled on the rubber gloves and cleared the festering detritus out of the sink. I filled up a couple of bin bags with packaging (mostly labelled REDUCED), and the oozing contents of the disgusting fridge. To think that I'd eaten food stored in this fridge, prepared on this table, cooked in these pans \u2013 I was lucky to be alive. Maybe it wasn't the fish that had nearly killed me that night but some lethal bug endemic to this kitchen, to which Mrs Shapiro had long since become immune. In the bottom of the fridge I found three black, wizened human fingers. It took me a moment to realise they were carrots.\n\nI poured bleach into the sink and swept the floor in the kitchen and hall, removing a pile of cat poo mouldering beside the telephone table. Still fifteen minutes to go before midday. I went upstairs to Mrs Shapiro's room, opened the windows, sprayed the air-freshener around and picked up the clothes on the floor and shook the bed covers out of the window. As an afterthought, I pushed the Harlech Castle tin further back on top of the wardrobe where it would be completely out of sight. I'd worked up a bit of a sweat with all the exertion, and my cheeks glowed self-righteously.\n\nI was admiring my handiwork when I heard a woman's voice in the garden. I froze and listened. She must have been standing directly below the open window. It was an ugly, metallic voice, like a rusty gate, and she was talking loudly, the way people do into their mobile phones.\n\n'I'm just going in to have a look around.' (A pause, while she listened to the voice at the other end.) 'I'll let you know.' (Pause.) 'It's an old biddy who lives here. She'll be going into a home.' (Pause.) 'I don't know yet. I'll get a good valuation.' (Pause.) 'Hendrix.' (Pause.) 'Cash. Five grand.' (Pause.) 'Damian.' (Pause.) 'I'll find out. And I'll ask about the tree. I'd better go now.' (Pause.) 'Bye-ee!'\n\nA few moments later, I saw her walking back up the path smoking a cigarette. I recognised her at once as the redhead who'd been in the garden the other day \u2013 that toxic-green jacket. Its quilted texture reminded me of lizard skin. She stopped by the gate \u2013 she was waiting for me, thinking I'd be coming up the road. I didn't want her to see me emerging from the house so I grabbed my jacket, let myself out of the kitchen door, locked it behind me, and looked for another way out. A mossy path led down through the long back garden to a derelict mews block at the back. Beside it was a gate. It was bolted, but I managed to force it open and found myself on another cobbled alley that must once have been an access to the mews, now overgrown with brambles that led back on to Totley Place. As I turned into the lane I could see Mrs Goodknee waiting for me at the gate, flicking through a file.\n\n'Hi. I'm Georgie Sinclair. Sorry I'm late.'\n\nShe must have been in her mid-forties, about the same age as me, perhaps even a bit younger, but she had a stiff over-groomed style that made her seem middle-aged. I couldn't see her knees, but I doubted they were dimpled and chubby. She handed me a business card. Ah.\n\n'Margaret Goodney. I'm a senior social worker at the hospital. Thank you for coming. Have you got the key?' Her Essex vowels squeezed themselves into a bland corporate dialect.\n\nShe followed me up the path. Fortunately the cats had gone off to do their own catty things. Only pretty, friendly Violetta appeared, rubbing herself against our legs.\n\n'Hello, kitty kitty,' Mrs Goodney squeaked. 'Who's a pretty kitty, then?'\n\nShe took a spiral-bound notebook out of her shoulder bag and turned to a new page. _Canaan House, Totley Place_ , she wrote at the top. Then she underlined it twice.\n\n'A bit of a jungle, isn't it? That tree needs to be cut down.'\n\n'It's got a preservation order on it.'\n\nShe made a note.\n\nSeeing Mrs Shapiro's house through Mrs Goodney's social worker eyes made me realise how pathetic my clear-up efforts had been. Her nose wrinkled the moment we walked in through the door.\n\n'Poo! It's like the black hole of Calcutta in here.'\n\nThe air-freshener had worn off already. Her heels click-clicked on the loose tiles in the hall. Her eyes darted around. She made a note on every room we walked into. Her note on the dining room read: _Good proportions. Original fireplace._ Her note on the kitchen read: _Total refurbishment._ She saw me craning to see what she'd written, and flicked the page.\n\n'A house this size is a liability,' she said, not unkindly. 'She'd be much happier in a nice care home.' She made another note. 'Mm. No food in fridge. That's a sure sign of self-neglect.'\n\n'I cleared the fridge.'\n\n'What did you do that for?'\n\n'It was going mouldy.'\n\n'That's what I mean. We have to do what's best for her, don't we, Mrs...?'\n\n'Sinclair. Call me Georgie. Doesn't she have any say in the matter?'\n\n'Oh, yes, of course we have to get her consent. That's where you could be very helpful, Mrs Sinclair.'\n\nI felt a flush spreading up my cheeks. Was she going to offer me five grand? But she just smiled her toothy smile.\n\nAs we stepped into the bedroom, she quivered and put her hand to her nose. Mussorgsky had managed to get in there ahead of us and had taken up his position on the bed. He raised his head and yowled as we came in. Violetta had sneaked in with us and was lurking in the doorway, giving Mussorgsky the eye.\n\n'These cats \u2013 they'll have to go.'\n\n'They're her friends. She gets lonely.'\n\n'Yes, companionship \u2013 that's another of the advantages of residential care.'\n\nShe made a note in her book.\n\nOn the floor by the bed were Mrs Shapiro's peach silk camiknickers, pretty but whiffy, which I'd overlooked in my whirlwind clean-up. She bent down and picked them up, held them for a moment between finger and thumb, then let them fall.\n\n'She fancies herself, doesn't she?'\n\nI saw her wipe her fingers discreetly on a tissue. I can't explain why, but it was that contemptuous finger-wiping gesture that really made me hate her.\n\nThe bathroom came as a shock to both of us. The smell was definitely human, not cat pee. The toilet bowl, originally white porcelain patterned with blue irises, was now brown-stained, cracked and encrusted. The stain had seeped in a damp acrid circle into the rotting floorboards, which had partly collapsed under the toilet bowl, making it lean at an alarming angle. Hanging loose from the wall was a basin in the same iris design, with green-yellow drip-trails beneath the taps. A large enamel claw-foot bath stood under the window, with an old-fashioned shower head above it. The grime circles inside the bath grew in layers, like trunk rings in an ancient tree.\n\n'It'll all have to come out,' she murmured, jotting in her notebook. 'What a shame.'\n\nDownstairs in the hall, she stretched out her hand to shake mine.\n\n'Thank you very much, Mrs... Georgie. I'll go and write my report.'\n\n'You're going to put her into a home, aren't you?' I blurted.\n\n'Of course my recommendation is entirely confidential.' She pursed her lips. 'But I think residential care could be an appropriate option. We have to do what's best for her, not what suits us, don't we, Georgie?'\n\n'What do you mean \u2013 what suits us?'\n\n'It can be hard for a carer to let go, when the time comes. They like to think they're doing it all for the other person, when really they're just being selfish, trying to hang on to their caring role even when they're no longer needed, because they want to feel valued.'\n\nShe smiled a bland professional smile. I felt like strangling her with her repulsive reptilian outfit and stuffing her nasty cube-heel shoes into her squeaky creaky gob.\n\n'So you think I'm just a selfish bitch with a cat-poo fetish?'\n\nShe glanced at me sharply, decided I must be joking, and her lips twitched in a thin smile.\n\n'We wouldn't want to be held responsible if she had another accident, would we?'\n\nShe turned and click-clicked down the path.\n\nAs soon as I got home, I took out the card Mrs Goodney had given me, phoned the number, and asked for Damian Hendrix. There was a long pause.\n\n'This is the hospital social work department,' a woman's voice told me. 'Are you sure you don't want the local authority Social Services?'\n\nI looked up the local authority number in the phone book and tried again.\n\n'Could I speak to Mr Damian Hendrix?'\n\n'I'm sorry, we don't have anyone by that name here. What was it about?'\n\n'It's about an old lady going into a home.'\n\n'Hold on, I'll put you through to elderly.'\n\nThe line crackled.\n\n'Elder-lee!' a cheerful voice chimed in my ear.\n\n'I'm looking for Mr Damian Hendrix.'\n\n'Mm-mm. No Hendrixes here. Are you sure you've got the right name?'\n\n'I'm sure about the Damian. Have you got any Damians?'\n\n'Mm-mm...' I heard the voice call to someone else in the room, 'Eileen, 'ave we got any Damians?'\n\n'Only 'im in't store,' said Eileen.\n\n'Only one who works in the resource centre,' said the cheery voice.\n\n'No, it must be someone else. Thanks.'\n\nI put the phone down.\n\nEileen \u2013 that voice \u2013 she must be from Yorkshire. I felt a little stab of homesickness, remembering how I'd felt when we moved down from Leeds to London, after Rip was offered the job on the Progress Project. We'd hovered for weeks like lost souls in the limbo of estate agent offices, looking for a place that might one day feel like home. We'd been dismayed at London prices, and at how poky the houses were \u2013 at least, the ones we could afford. The squat Edwardian semi we finally bought had seemed brighter than most. It had been all done out for a quick sale by the builder, painted in neutral shades to compliment (sic) the stunning period features, with laminate floors (authentic-oak-style), a granite worktop (Uba Tuba) and fitted well-known-brand kitchen appliances. It smelled of newness and fresh plaster. It had no character at all. I'd liked it at the time \u2013 it had seemed like a fresh canvas on to which we'd paint our new life. But that's not how it worked out. Maybe things had been going wrong for ages and, like damp seeping into a basement, I just hadn't noticed the warning signs.\n\nLater that afternoon, as I was walking down the local parade of a dozen or so shops, I remembered the other reason we'd chosen our house. This little neighbourhood had seemed an intimate island of friendliness in the vast anonymous bustle of London. There was the Turkish bakery, strangely famous for its Danish pastries; the Song Bee, our favourite takeaway, recently opened by two young women specialising in Chinese and Malaysian cuisine; Peppe's Italian delicatessen; Acne Al, as Ben called the newsagent by the bus stop; and two estate agents, a local branch of Wolfe & Diabello on the corner where I was standing, and across the road Hendricks & Wilson.\n\nThen it dawned on me. Hendricks! Should I barge in and make a scene? Instead, on impulse, I pushed open the door of Wolfe & Diabello. If Mrs Goodney was going to get her little Damian to do a valuation on Canaan House, at least I could get one, too, for comparison.\n\nA small bosomy girl with sleek blonde hair and careful eyes was sitting at a desk by the window. Her name badge said Suzi Brentwood.\n\n'My aunt's thinking of selling her house. Could you give us a preliminary valuation?'\n\n'Of course.' She flashed her little pearly teeth at me. 'I'll make an appointment with one of the partners. Is next Friday all right? We're a bit busy. What's the address?'\n\nHer eyebrows rose a fraction when I told her.\n\n# 11\n\n# Black treacle\n\nBy the time Friday came round it was raining again, a miserable December drizzle that stained the streets and rooftops melancholia grey. I was starting to regret my appointment with the estate agent, and I thought of ringing up to cancel, but something about Canaan House had pricked my imagination. As Ms Firestorm would say, I was _inexorably drawn_.\n\nI left home in a rush without an umbrella, and the hood of my duffel coat kept slipping back as I ran, so I arrived breathless and thoroughly bedraggled. As I turned the corner into Totley Place I saw a black sports car, a low-slung meanlooking machine, skulking with predatory menace on the road outside Canaan House. Skulking like a Wolfe, though I soon saw it was in fact a Jaguar. When I came close the driver's door opened, and a long lean form uncurled itself on to the pavement. Tall, dark, handsome. I stopped and caught my breath. There was something oddly familiar about him.\n\n'Mrs Sinclair?'\n\nI nodded. He raised a quizzical eyebrow and proffered his hand, which was warm and firm. My heart flipped like a hooked fish. I became aware of a pleasant sensation in my pelvic area.\n\n'You must be Mr Wolfe,' I said, trying to shake the rain out of my wet-sheep hair.\n\n'No, I'm Mark Diabello.' His smile made rugged creases in his craggily handsome cheeks. The cleft in his square, manly chin dimpled seductively. His dark and smouldering eyes seemed to gaze right into my soul \u2013 or perhaps right into my underwear. I noticed the pleasant pelvic glow once more. 'It means beautiful day, I've been told.'\n\nHis voice was like black treacle \u2013 sweet, with a hard mineral edge.\n\n'Not like today then.' I batted my wet eyelashes. What was happening to me? This man was an estate agent, and definitely not my type. 'Er... unusual name. Italian?'\n\nI was regretting that I'd worn my batty-woman clothes.\n\n'Spanish. My father was an itinerant mandolin player.'\n\n'Really?' He was still smiling, and I couldn't tell from his face whether he was joking, but the idea was, mmm, appealing. 'I've got the key,' I mumbled. 'Do you want to look around?'\n\nThe cheeks crinkled into a smile. The eyes smouldered. I gazed into them. My poor fish-heart tugged feebly at the line, but it was caught.\n\nWonder Boy, Violetta and their mates had congregated at the front door. I let them in and fed them in the kitchen because it was too wet outside. It was bitterly cold indoors, a dank pungent chill which hit you with the stink of stale cat food mingled with other odours which were quite a lot worse. Then I became aware of another, more pleasant smell, faint and spicy like expensive soap. That was _him_. Inexorably drawn, I followed along as he wandered around the house, murmuring to himself under his breath. He had a little instrument like a torch with a laser beam that bounced enticingly against the walls of the rooms, to measure the size. I watched, transfixed. Click. Flash. If I asked nicely, would he let me have a go with it? He wrote the details on the back of what looked like a crumpled till receipt.\n\nHe seemed completely unfazed by the smell. Even when he stepped in a pile of fresh cat poo in the hall (how did that get there?) he just bent down and cleaned it off with an immaculate white cotton handkerchief from his breast pocket. I watched, awestruck, as he deposited it in the kitchen bin.\n\n'I could live in a place like this,' he murmured huskily in his deep manly mineral-edged voice that spoke directly to my hormones, bypassing my brain completely. I realised now where we'd met before \u2013 in the pages of _The Splattered Heart_. He was just as I'd imagined the hero. Except that in my book the hero was a poet, not an estate agent.\n\n'Character. That's what you so rarely get in the housing market nowadays.'\n\nWe were standing together in the entrance porch at the end of his tour. The rain had stopped and the weak wintry sun was putting in a brief appearance, so it was warmer out here than indoors, and much less smelly.\n\n'Ornate plasterwork; period arches; decorative corbels. I mean, don't get me wrong, Mrs Sinclair, there's a lot needs doing. You'd have to do it sensitively, of course. Keep all those stunning period features. Get a couple of designers in to give you ideas. You could open up the attic, for instance. Make a fabulous penthouse suite.' A flame flickered in the depths of his eyes.\n\n'Everybody seems to fall for this house.'\n\n'It's the potential. You can see the potential. You'd have to cut that tree down, for starters.'\n\n'It's got a preservation order on it.'\n\n'Doesn't matter. You just pay the fine. The tree gets cut; the Council gets its cut. Everybody's smiling.'\n\nI hadn't much liked the tree myself, but now it suddenly felt like an old friend.\n\n'You can't do that!'\n\n'So when's your aunty planning to put it on the market?'\n\n'She just wanted an idea of its value, in case she decides to sell. What do you think?'\n\nHe looked at the notes he'd scribbled on the receipt, crinkling his eyes and furrowing his handsome brow in a way that was faintly reminiscent of Aristotle. Well okay, only very faintly.\n\n'Half a million, maybe?'\n\nI don't know exactly what I'd been expecting, but our own semi with its three poky bedrooms and narrow strip of garden had cost almost that. He saw the look on my face.\n\n'The area brings it down. And we're looking at a cash buyer, of course, not a mortgage. I'll put it in writing for you.'\n\nI gave him my address. We shook hands. He climbed into his hungry-looking car, and was gone in two puffs of hot air from the chunky twin exhausts.\n\nI walked back slowly, still feeling slightly giddy from my encounter. As I came up the street, I could see that Ben was already home, the blue square of his monitor winking away through the window as he tried to navigate the lonely cyber-seas, teeming with who knows what pirates and sharks. My mother-heart tightened with a little squeeze of sadness: it wasn't good for him to spend his evenings up there on his own.\n\n'Hey, Ben, shall we go to the pictures tonight? We could go and see Daniel Craig as James Bond.'\n\nSean Connery, Roger Moore, Pierce Brosnan. While my mother-heart ached for Ben, my woman-hormones were still tingling for Mark Diabello.\n\n'Sounds like crap.'\n\n'It probably is crap, but it might be entertaining.'\n\n'I don't find crap entertaining, Mum? But we can go if you want to?' I noticed there was something different about his voice \u2013 a new rising inflection at the end of his sentences \u2013 questioning, or apologetic. I wondered whether he was like this when he stayed with Rip. Somehow I imagined that life in Islington would be an endless round of stimulating activities and highbrow conversations, and it was only with me that he spent his hours closeted with his computer. I would have rung Rip to ask him if we'd been on better terms, but we weren't, and I didn't.\n\nInstead of going out, we ordered dinner from the Song Bee and ate it by the gas fire in front of the TV. It was a cop drama, I can't remember what. I'd just been thinking that the male lead looked a bit like Mr Diabello when suddenly Ben turned to me.\n\n'Mum, do you believe in Jesus?'\n\nHis question hit me out of the blue. I drew a slow breath.\n\n'I don't know. I'm not sure what I believe, Ben.' What was this all about? I wondered. 'I believe Jesus was a real person, if that's what you mean.'\n\n'No, what I mean is, Mum, do you believe Jesus'll save you at the end of the world?'\n\n'Ben, love, the world isn't going to end.'\n\nI had a sudden memory of how I'd been at his age \u2013 I believed that nuclear war would wipe out the human race before I'd even had a chance to lose my virginity. We'd sat around on Saturdays in the Kardomah caf\u00e9 in Leeds, my mates and I, fantasising about how we would spend our last four minutes after the final warning.\n\n'It's like... I really love you, Mum. You and Dad. I don't want...' He was mumbling, as though his mouth was full of sand. 'All you have to do is accept Jesus into your life?' His eyes, when he looked up at me, were wide, the pupils dilated, as if fixed on some private nightmare.\n\n'The signs are there, Mum? All the signs are in place?' That strange questioning inflection \u2013 it was as though someone else, an alien, had got inside him, and was speaking through his mouth, staring at me through his eyes.\n\n'The world's been around for a long time, Ben. Don't worry.'\n\nI pulled him into my arms and hugged him tight. He stiffened against me at first, but I held him close until I felt him relax, his head resting on my shoulder. Whatever it is, I thought, he'll grow out of it.\n\nNext day I overcame my pride and phoned Rip.\n\n'I'm worried about Ben. Can we talk?'\n\n'I'm just in the middle of something. Can I ring you back in half an hour?'\n\nBut he didn't.\n\nBen didn't go round to Rip's until late on Saturday. He spent the day upstairs on his computer, and I spent the day working on _The Splattered Heart_. Outside, the rain lashed at the garden, and the wind made a spooky whistling noise through the ill-fitting secondary glazing, but inside we had the central heating on and Snow Patrol keeping up a soft background rumble. Each time I walked into Ben's room, he minimised the screen he was looking at. We took turns to bring one another cups of tea, and treated ourselves to Danish pastries from the Turkish bakery and dim sum for lunch from the Song Bee. I needed the extra sustenance; now we were getting down to the nitty-gritty: _The Splattered Heart, Chapter 4_.\n\nIt was hard to decide whether Rick should be a lust-crazed sex fiend or a minutely endowed, impotent Viagra case. I crossed out a whole page, and started to think about Rip. No, it wasn't sex that had gone wrong between us, but whatever had gone wrong had taken the shine out of the sex, too. Keep the romance in your marriage, Mum's magazines used to say, and they advised strategies like wearing sexy underwear, and greeting your hubby in your negligee when he got back from work. Actually, I tried that once, but he didn't notice.\n\nHe called round at six to pick Ben up in his Saab convertible and they went straight off to the cinema \u2013 they were going to see Daniel Craig as James Bond. After they'd gone, that horrible silence settled on the house, like a coffin lid closing. \n\n# 12\n\n# Marine biological glues\n\nIt wasn't until Monday morning that I remembered I hadn't fed Mrs Shapiro's cats. I heard a familiar yowling sound in the garden, and when I looked out of the upstairs window, there was Wonder Boy lurking under the laurel bush. He was looking up at the window with a reproachful look on his face. All around him was a mass of grey and brown feathers, sodden in the rain. Seeing him here in my garden made me furious \u2013 I didn't want him killing my birds; in fact I didn't want him at all. I pulled on my brown duffel coat and my wellies and strode off round to Totley Place. He followed me, slinking along at a distance, ducking into a gateway or garden if I stopped and looked back. Then I noticed that the Stinker was following me, too; and another scrawny tabby. I was turning into the Queen of the Cats. The other cats were waiting for me in the porch when I arrived, a purry, enthusiastic reception committee. None of them looked particularly wet.\n\nThere were three weird things I noticed on that visit. The first was a pile of fresh cat poo, almost in the same place where Mr Diabello had stepped in it the other day. It had a distinctive curled macaroon shape, unlike all the other brown shrivelled-sausage deposits that I found around the house from time to time. I was sure that I'd got all the cats out of the house when we left. Who was the culprit \u2013 and how had he got in? I cleared it up and counted them as they milled around my legs \u2013 one, two, three, four, five, six, seven. When I left, I'd make sure to count them out.\n\nAs I straightened up, my eye fell on a picture on the wall directly above where the cat poo had been. It was a photograph, rather grainy and washed-out, of an arched stone doorway with a cross on top, Corinthian columns on each side, and above the door a carving of a man on horseback with a spear. Something about it was familiar. I must have looked at it dozens of times without really seeing it. What I saw now was that it was the same arched doorway as in one of the Harlech Castle tin photographs, the one with the dark-eyed woman: Lydda. The columns and the harshness of the light made me think of Greece.\n\nThe third thing I noticed, when I went through to the kitchen to feed the cats, was that the key to the back door, which should have been inside in the lock, was missing. Someone had taken it. I realised in a flash that it could only have been wicked, wolfy Mr Diabello.\n\nI fed the cats quickly and rushed home in a rage, but just as I picked up the phone to vent my fury at Wolfe & Diabello, it rang in my hand. It was Penny, the admin manager from _Adhesives_ , wanting to know whether I'd received the press release about the new research into marine biological glues. The truth was, it had come two days ago, and I hadn't even looked at it. I mumbled something vague and apologetic, but she saw right through me.\n\n'What's going on, Georgie?' she boomed. 'Something's not right, darling, I can tell. Is it that husband of yours again?'\n\n'No. It's another devious man.'\n\nI explained about the missing key and the dodgy estate agent.\n\n'Hm.' I could hear Penny breathing on the other end of the phone. Nothing about her was quiet. 'Don't rush into anything, darling. You could be wrong about the key, then you've blown your chances with that sexy man.'\n\nHow did she know he was sexy? Was I that obvious?\n\n'You should get a second opinion, darling. Two second opinions. One about the price of the house, and one about the social worker.'\n\nThe same thing had happened to her Aunty Floss, she said, who'd been put in a home by the Council and died six months later of unspecified complications.\n\n'God bless. I'm sure she's up there in heaven, tippling sherry and looking down and cursing those bloody scumbags who got her house.'\n\n'Do they allow tippling and cursing in heaven?' I giggled.\n\n'Well, if they don't, darling, I'm not going there.'\n\nThe thought of all that tippling and cursing cheered me up, and I promised Penny I'd get on with the marine biological glues \u2013 yes, straightaway \u2013 but first I'd take her advice and try to get another social worker assessment.\n\nMrs Goodney, I knew, worked at the hospital, not the Council, so next day I telephoned the Council's social services department again. I explained to the cheery 'Elder-lee!' voice that an elderly neighbour had gone into hospital and needed an assessment before she could go home.\n\n'Mm-mm. Can you hold the line a minute? (Eileen, what's 'er what does 'ome visits?)'\n\nEileen's voice, muffled by distance, said something that sounded like 'Bad Eel'.\n\n'She's out on 'er coffee break,' I heard her say.\n\n'You need to speak to Muz Bad Eel. I'm afraid she's in a meeting. Can I take your number and ask her to call you back?'\n\nBad Eel. I pictured someone slim and slippery, with scarlet lipstick and a small silver gun tucked inside a frilly garter.\n\nI spent the whole morning at my desk, looking out of the window at the wind chasing stray leaves around the dank patch of lawn, and waiting for the Bad Eel to phone me back. I was supposed to be working on the press release Penny had sent about marine biological glues. Some company was developing a synthetic version of the glue that bivalves such as mussels and oysters use when they cling to the rocks. One of the strongest bonds in nature, apparently. They use fine thread-like tentacles called byssus, which are rich in phenolic hydroxyls. Phenolic hydroxyls: something about those words just turned my brain to glue.\n\nI started thinking about bivalves living down there in the dappled light, how they filter the algae from the water, how they close themselves up against the sea. It must be wonderful to be a bivalve, to be able to shut yourself away in your own mother-of-pearl-lined world, hanging on to the rock while the waves and tides churn outside. Ms Firestorm showed up to help me out. _Cloistered in their shimmering watery depths, the loyal bivalves cling passionately together_... Yes, we could learn a lot from bivalves. I realised I wasn't very interested in commercial applications, and when the other elusive marine creature still hadn't called by lunchtime, I wrapped up warm against the wind and set out for the hospital.\n\nMrs Shapiro was sitting in the day room when I arrived, wearing a pinafore-style hospital dressing gown tied at the back and a pair of woolly socks on her feet. I felt a pang of guilt. Probably it was my responsibility, as her next of kin, to bring in some suitable hospital gear for her. I'd have to remember for next time.\n\nA tattered magazine was open on her knee, but she wasn't reading; she seemed to be engaged in a fractious and incoherent argument with another old lady sitting beside her.\n\n'But 'er were on this ward when she shouldn't of been,' the old lady was saying vehemently, 'and new sister said it weren't 'er business anyway.'\n\n'Well, if they was no longer there someone must heff tooken them.'\n\n'No, because she weren't supposed to be. That's what I'm sayin' to yer.'\n\nShe looked up and saw me in the doorway.\n\n'That's 'er there. Ask 'er.'\n\nMrs Shapiro turned, and stretched out her hands to me.\n\n'Georgine, you got to get me out of here. All this people is mad.'\n\n'She's talkin' tripe,' said the old lady, and heaving herself up out of the chair she minced off along the ward, muttering aloud.\n\n'What's going on?' I asked.\n\n'She is a bonker,' said Mrs Shapiro. 'Brain been amputated.'\n\nThe old lady stopped, turned, flicked two fingers at us, and carried on.\n\n'How are you doing, Mrs Shapiro?' I pulled up a chair beside her. 'I thought you'd be going home by now.'\n\n'I am not going nowhere,' said Mrs Shapiro. 'They say I must go into the oldie-house. I tell them I am not going nowhere.' She folded her arms determinedly across the front of the green dressing gown. The argument with the old lady was obviously just a warm-up for a much bigger argument to come.\n\nThere was a new sister on duty, a young girl who looked hardly older than Ben.\n\n'What's happened with the home assessment?' I asked.\n\n'The report's just come through. They're recommending residential care. I'm afraid she's not very happy about it.'\n\n'I really don't see why she needs residential care. She was managing fine.'\n\n'Yes, but you know, once they start falling, they can very easily lose their confidence. Especially at her age.'\n\nShe brushed a stray hair off her face and looked over her shoulder towards the nurses' station. I could see there were a dozen things she needed to be doing more urgently than talking to me.\n\n'What if she refuses to go?'\n\n'We can't discharge her into an unsafe situation.'\n\n'So she just stays here?'\n\n'She can't stay here. She's blocking an emergency bed that someone else could use.'\n\n'So what are the options?'\n\n'Look, I think you'd better talk to Mrs Goodney. The social work office is over by physio.'\n\nI went back to sit with Mrs Shapiro. 'It's okay,' I said. 'I'll get them to do another assessment.'\n\n'Thenk you, darlink,' she said, gripping my hands. 'Thenk you very much. And my dear cats, how are they?'\n\n'They're fine. But Wonder Boy seems to be killing a lot of birds.'\n\n'Ach, poor darlink, he is upset. You must bring him here. Next time. You promise, Georgine?'\n\nI mumbled something evasive, but just then the tea lady appeared with the trolley.\n\n'You heff no kr\u00e4utertee?' said Mrs Shapiro grumpily. 'Okay, I tek this horse's piss. No milk. Three sugar.'\n\nCradling her cup in her hands, she settled back on the pillows.\n\n'Now, Georgine, your running-away husband. You heffn't finished telling me.'\n\n'I did tell you. It was so boring you fell asleep.'\n\nShe caught my eye and gave a little laugh.\n\n'You told me about your parents. That was quite boring, isn't it? But what about the husband? He was a good man? You were happy in loff?'\n\n'We were happy at first. But then... I don't know... He got absorbed in his work. I had babies. Two \u2013 a girl and a boy.'\n\nAnd a miscarriage in between. Then I started writing a book.\n\nAfter the miscarriage I'd given up my job and started freelancing. Rip had taken his articles but found the solicitor's work tedious and applied for a job in the northern office of a national charity. He was keen and committed, out and about all over the place, so one of us had to be home-based. The freelancing didn't fit easily around children so, inspired by my earlier introduction to Mum's preferred reading matter, I decided to try my hand at romantic fiction. I got a couple of short stories published in a women's magazine and after that encouraging start I plugged away at a romantic novel \u2013 it was about a plucky young heroine who is inexorably drawn to a grand but gloomy house inhabited by a handsome, moody, extremely rich poet (I know, but it _is_ fiction) who falls in love with her but alas dies of a mysterious ailment on the eve of their wedding, which is terribly tragic, but then she falls in love with the local schoolteacher who lives in a cute rose-covered cottage and is penniless but has a good sense of humour and is great in bed.\n\nI thought I'd got the genre spot-on, and it grieved me that no one wanted to publish it. I tried changing the font, changing the ink colour, I changed my nom de plume, but the rejection slips just kept coming.\n\n' _Splettered Heart_. This is a good title for a book, Georgine. Povverful.'\n\n'Thank you. Rip thought it was too melodramatic.'\n\n'Ach! He is a man. What does he know?'\n\n'He thought I should call it _The Shattered Heart_ or _The Broken Heart_ , but I thought that was a bit clich\u00e9d.'\n\n'Exactly so. And it has been published?'\n\n'No. Not yet.'\n\n'But you must not give it up.'\n\n'I'm completely rewriting it. A new version. But it's hard to find the time. I've got another job now, writing for online trade magazines.'\n\n'Lane tred? What is this?'\n\n'It's a group \u2013 _Adhesives in the Modern World_ , _Ceramics in the Modern World_ , _Prefabrication in the Modern World_ , things like that. I work on all of them, but mainly _Adhesives_. I've been doing it for about nine years.'\n\n'But this is fascinating!'\n\n'Well, it's just for the building trade. It's not exactly world shattering.'\n\n'Too much shattering is going on nowadays, Georgine. Building is much better.'\n\nNathan had conducted a cursory interview over the telephone, during the course of which he'd asked me, among other things, what my favourite pudding was (Bakewell), whether I'd ever been to Prague (no), and which team I supported (Kippax Killers, of course), and told me after five minutes that I was just the person he was looking for.\n\n'Glue,' he'd said. 'Don't worry, it'll grow on you.'\n\nRomantic it wasn't, but it paid the bills, and it meant I could be at home for the kids. Strangely enough, it did grow on me.\n\n'So that's my story so far. Not very exciting, really.'\n\n'Well, we will heff to see if we can make you a happy ending.' She raised her teacup. 'To happy endings!'\n\nOn my way home from the hospital, I dropped in at Canaan House to feed the cats and do a quick tidy-up in case the Bad Eel should deign to visit. The wind was still blustering, swirling up dead leaves and litter on the pavement. Wrapping my coat tight around me, I turned into Totley Place. At once I saw there was something unusual there \u2013 something brightly coloured at the entrance to the cobbled lane that led up to Canaan House. As I drew closer my heart began to beat with rage and trepidation. Yes, it really was what I'd suspected, half hidden there among the creepers \u2013 a large green-and-orange For Sale sign, with the name written in bold black letters: Wolfe & Diabello.\n\nIt was stuck into the ground beside the wall. I grabbed the post and heaved. It held firm, so I pushed and pulled it backwards and forwards, to loosen it up. Then I got round behind it, scrambling through a climbing dog rose that clung to the wall. Surely Mr Diabello hadn't done this, the thorns picking at his Italian-styled suit? It must have been some strong-arm minion in a white van, hammering the post into the ground with a mallet. I'd worked myself up into a frenzy, but still it wouldn't budge. If anyone had seen me, they'd have thought I'd gone mad. I grabbed the post in both hands, arched my back and bent my knees for one last heave. It slid out of the ground as smoothly as a knife out of butter. I slid with it, staggered, lost my balance, and fell backwards into the rose bush. A thorn jagged my cheek. Wonder Boy appeared yowling out of the undergrowth. It started to rain.\n\n*\n\nI'd been all fired up to storm into the Wolfe & Diabello office and demand an explanation, but I called in at home to pick up my raincoat, and the phone was ringing as I opened the door. It was Rip.\n\n'Hi, Georgie, I just wanted to have a quick word about Christmas.'\n\nI steeled myself. 'Fire ahead.'\n\n'I wondered if you'd made any plans?'\n\n'Not really. Why? Have you?' I felt a quiver of dread \u2013 Christmas: the time when families are supposed to be together. Would I be able to survive a Christmas on my own?\n\n'I was wondering about going up to Holtham with Ben and Stella...'\n\n'Fine.' Actually I felt like drowning myself in a tub of lukewarm piss, but I managed to put on a brave show of nonchalance. 'Do that. Fine by me.'\n\n'What about you?'\n\n'I haven't really thought about it.'\n\nAfter he'd put the phone down, I went up to my bedroom, flung myself down on the bed, and let the tears flood into my eyes. I sobbed and sobbed until my chest ached and my shoulders heaved and my nose ran with snot \u2013 I sobbed for my broken marriage and my broken family, all the hurts and humiliations I'd ever endured in life, my ailing parents, my absent brother, my too-far-away daughter, the general sorrows of humankind, starving babies in Africa, kids who self-mutilate, suicide bombers and their victims, they all came washing saltily in on the same vast relentless indivisible tide of human misery. I thought about the bivalves, the curved pearly walls inside their shells, the greenish light filtered through seawater; whatever the extraordinary glue was that enabled them to hold so tight while the storms swirled around them, that's what I needed now. \n\n# 13\n\n# No job too small\n\nBy next day, the fight had gone out of me a bit, but I decided to walk across to Wolfe & Diabello anyway. I needed to clear my head, and I still had a couple of bones to pick with them. It was another raw, blustery December day, the sky teeming with grey scurrying clouds. I pulled my hood up, and put my head down into the wind, and maybe that's why I didn't see it until I almost stumbled over it \u2013 a post lying across the pavement. Attached to the post was a For Sale sign. Not Wolfe & Diabello, but Hendricks & Wilson. That was odd \u2013 it had been windy in the night, but not _that_ windy. Even odder \u2013 as I turned the corner, there was another one, stuck into a hedge, a few hundred metres up the road. Then further along, I spotted another lying in a skip.\n\nThere was no one in the Wolfe & Diabello office when I went in. I opened and closed the door again, making it 'ping', but still nothing happened. The third time I did it, Suzi Brentwood emerged from a door at the back; I thought I spotted a shifty look flit across her face before her professional smile composed itself.\n\n'Hello, Mrs... How may I help you?'\n\n'My aunty is thinking of selling her house before Christmas,' I said very loudly.\n\nAs if by magic, the door at the back of the office opened, and Mr Diabello appeared.\n\nHe was wearing the same dark stylish suit, a clean freshly folded handkerchief peeping out of the breast pocket.\n\n'Hello, Mrs Sinclair. What can we do for you?'\n\n'The For Sale sign in the garden at Canaan House \u2013 you put it there?'\n\nHe smiled, that irresistible cheek-creasing smile.\n\n'We have to keep one step ahead of the competition.'\n\n'What do you mean?'\n\n'We heard on the grapevine that Hendricks had sent a valuer in.'\n\nIt must have been Damian, I thought. But how did he get in?\n\n'No harm in that, Mrs Sinclair. It's a free market. Shop around. See who can offer you the best deal. But, you know, I felt after our chat the other day that you deserve a \u2013 how can I put it? \u2013 a more focused view of the service we offer here at Wolfe & Diabello.' His eyes smouldered with dark fire. His quizzical eyebrows quizzed.\n\nMs Firestorm popped up briefly to take a look, and she was well impressed. 'Deserve. Focused. Service.' She repeated the words slowly in her head. They sounded deeply sexy. But they still didn't make sense.\n\n'You mean you just marched up and stuck a For Sale sign in someone's front garden without their permission?'\n\n'It's a bit cut-throat around here,' he murmured apologetically. 'Hendricks & Wilson \u2013 I don't like to say this about another estate agent, but they aren't the most reputable in the business. Underhand tactics. Stealing our customers. You'll never believe this, but sometimes they even go round and pull our sale boards out. What valuation did he give you, by the way?'\n\nI looked him in the eye.\n\n'He said she should be able to get a million for her house. At least a million. Maybe more.'\n\nHe didn't bat an eyelid.\n\n'I'm sure we could match that for you, Mrs Sinclair. And we could agree a special rate on the commission.' His handsome nostrils flared tantalisingly. A hint of a smile played at the corners of his sensual mouth. 'If your aunty decides to sell before Christmas.'\n\nI could have swooned into his rugged manly arms at that point, but then I remembered my second issue.\n\n'The key. You stole the key.'\n\n'Pardon me?'\n\n'The back-door key. To the kitchen. It was in the door.'\n\nHis eyes seemed to widen a fraction.\n\n'I think you've made a mistake.'\n\n'No, I haven't. You took it. It must have been you.'\n\nHis brooding brow furrowed.\n\n'Mrs Sinclair, it wasn't me, I assure you. Have you considered the other possibility?'\n\n'What other possibility?'\n\nHis mouth tightened. His head twitched.\n\n'Them.' His head twitched again, a sort of reflex jerk to the left. 'Hendricks.'\n\n'It couldn't have been them.'\n\nThen I thought back. I was in the kitchen feeding the cats. Mr Diabello was wandering around scribbling on the back of his receipt. I was feeding the cats in the kitchen because it was raining. I didn't open the back door. Was it locked? Was the key in the lock? I couldn't remember. When was the last time I was sure I'd seen the key? Was it when I was showing Mrs Goodney around? I realised I was totally confused.\n\n'I'll look into it.' Maybe I'd misjudged him after all. 'If I've made a mistake I apologise,' I said stiffly.\n\nAnyway, all I need to do, I thought to myself, is change the lock. Where do you get a new lock? My mind went blank. Then I remembered a commercial I'd seen on TV. B&Q. For some reason, the thought was pleasantly appealing. The nearest branch to me was in Tottenham.\n\nIt wasn't till next day, as I made my way in through the sliding glass doors past the displays of Christmas baubles and endof-line kitchen units, that I realised what it was that drew me to B&Q: it was the men. Yes, although Rip was both handsome and brainy, he was definitely deficient in the DIY department. There's something deeply attractive about a man with a screwdriver in his hand, I was thinking. If you wanted to be Freudian about it, you could say it was a father-fixation, for Dad was always fixing things about the house, while Mum brought him cups of tea and Keir and I got under his feet. These B&Q types reminded me of the men in Kippax \u2013 not destiny-shaping men; not even craggily handsome splatter-your-heart-type men; but nice ordinary blokes wearing jeans and pullovers with comfortable shoes, their pockets bulging with tape measures and hand-drawn diagrams on bits of paper; sometimes a bit paunchy; even a tattoo here and there. Who cares? So long as they weren't always dashing off somewhere to change the world. Maybe if I hung around, one of them would come along to measure me up, would compliment my tasteful decor, be stunned by my period features.\n\nI should come here more often, I resolved, as I made my way through the mysterious aisles. There, on my left, was a whole section of rawplugs. I glanced at them quickly \u2013 they seemed alien, frightening things, with their poky plastic shells, their complicated colours and numbers \u2013 the sheer _rawness_ of them. But the worst thing is, that you have to make the hole in the wall with an electric drill, then you have to hammer the right-sized rawplug into the right-sized hole, and you can't just use any old screw \u2013 you have to know the right size and type. I held my breath and hurried past.\n\nAt last I found my way to the section that displayed locks \u2013 there were dozens of them. I picked up one or two at random, trying to remember what the one on Mrs Shapiro's door had looked like. It was definitely not a Yale type of lock; it was the other type \u2013 the type with a big key. Yes, a mortise. The trouble was, there were so many different models and sizes.\n\nA man was browsing among the hinges and doorknobs at the end of the aisle \u2013 a small tubby Asian man. I caught his eye and smiled a sweet damsel-in-distress smile. He came over at once.\n\n'You need help?'\n\nHis eyes sparkled darkly. With his neat moustache and beard, he looked like a well-groomed hamster.\n\n'I'm looking for a lock. Mortise. With a big key. Only I've lost the key.'\n\n'You know what type? Union? Chupp?'\n\nI shook my head.\n\n'You don't know this? You must know. Otherwise impossible to replace it.'\n\n'It's for a back door.'\n\n'What it looks like? Can you describe?'\n\n'I can't remember exactly. I think it's a bit like this one. Or that one.'\n\nI pointed randomly.\n\n'In my country we have a saying, knowledge is the key. But you have no knowledge and no key.' He sighed, fished in his trouser pocket, and handed me a small dog-eared business card \u2013 the sort you can get printed at the railway station.\n\n_HANDIMAN  \n **Mr Al Ali**  \nTelefon 07711 733106  \nNo job too small_\n\n# 14\n\n# Reindeer meat and dried fish\n\nThe phone was ringing when I got home. I could hear it through the door as I fumbled with my key, but by the time I picked it up, they'd rung off. There was a message on the answering machine.\n\n'Hello Mrs Sinclair. This is Cindy Bad Eel from Social Services, returning your call.'\n\nI rang back immediately, but I just got another answering machine. I left a message asking her to ring as soon as she got back.\n\nNext day, she still hadn't rung, so I tried Social Services again.\n\n'Elder-lee!'\n\n'Could I speak to Mrs Bad Eel please.'\n\n'It's Muz. Not Missis.'\n\n'Well, can I speak to her anyway?'\n\n'Hold on a minute.'\n\n('Eileen, where's Muz Bad Eel?' 'She's just 'ere. 'Old on. Who is it?')\n\n'May I ask who's speaking please?'\n\n'It's Georgie Sinclair. I rang about the old lady going into a home.'\n\n('It's that woman about t' old woman.' 'She says she'll ring back in a minute.')\n\n'She's just in a meeting. She'll ring you as soon as she gets out.'\n\n'No \u2013 please tell her it's urgent. I need to speak to her now.'\n\nThere was a lot of muttering and crackling in the background, then a new voice came on the line \u2013 a low, smooth, sultry voice with a slight drawl in the vowels.\n\n'Hello-o. This is Cindy Bad Eel.'\n\n'Oh, hello Mrs Bad Eel. Ms. I really need your help \u2013 I mean, a friend of mine needs your help.' I was gabbling, fearful that she would hang up. 'Mrs Naomi Shapiro. She's in hospital. She broke her wrist. Now they won't let her go home. They want to put her in a home.'\n\n'Slo-ow down, please. Who am I speaking to?'\n\n'My name is Georgie Sinclair. I left a message for you.'\n\n'So you did, Ms Sinclair. Slow down. Take a deep breath. Now, count one, two, three, four. Hold. Breathe out. One, two, three, four. Rela-ax! That's better. Now \u2013 would you describe yourself as her carer \u2013 an informal carer?'\n\n'Yes \u2013 yes, a carer. Informal. That's definitely what I am.'\n\nWaves of calm engulfed me. I suddenly felt very caring.\n\n'How old is the lady?'\n\nI hesitated. 'I don't know exactly. She's quite elderly, but she was getting along fine.'\n\n'But you say she had an accident?'\n\n'The accident was in the street, not in her house. She slipped on the ice. It could have happened to anyone.'\n\n'And you say she had a home circumstances assessment visit?'\n\n'It was someone from the hospital. Mrs Goodney. The house was a bit untidy, but it wasn't _that_ bad.'\n\nThere was a long silence. I started to anticipate her response, her stock of excuses for doing nothing. Her phone-answering track record had not been impressive. Then she spoke again, slowly.\n\n'It isn't for us to judge another person's lifestyle choices. I will visit the house, but I need her permission. Which hospital is she in?'\n\nAs soon as I'd put the phone down, I ran into my bedroom and stuffed a few things into a carrier bag \u2013 Stella's old dressing gown, a spare pair of slippers, a hairbrush, a nightie \u2013 and set off for the hospital. I wanted to forewarn Mrs Shapiro, and make sure she said the right things. I didn't want her to blow this chance on another bout of cussedness.\n\nThe rain had stopped, but there were still puddles in the road as I raced to the bus stop, and big damp clouds were hanging just above the rooftops like billowing grey washing. I was the only person on the top deck of the Number 4 bus as it lurched and swayed along the now-familiar roads, brushing against the dripping trees, so close to the houses I could see right into people's bedrooms. I recalled my lonely afternoons wandering the streets peering enviously into other people's lives. What had all that been about? It seemed an age ago. Now Mrs Shapiro and Canaan House were keeping me so fully occupied I hardly had time to think of anything else.\n\nIn the bus shelter outside the entrance to the hospital there was the usual little knot of people huddled over their cigarettes. I'd passed them before without really noticing, but this time, a voice called out to me.\n\n'Hey! Georgine!'\n\nI had to look twice before I recognised Mrs Shapiro. She was enveloped in a pink candlewick dressing gown several sizes too big for her, and so long that it trailed on the ground. Below it, just peeping out in front, was a pair of outsize slippers \u2013 the sort that children wear, with animal faces on the front. I think they were _Lion King_ s. Ben once had a similar pair. Her companion was the bonker lady with whom she'd been arguing last time. Now they seemed to be getting on like a house on fire. They were sharing a cigarette, passing it between them, taking deep drags.\n\n'Mrs Shapiro \u2013 I didn't recognise you. That's a nice dressing gown.'\n\n'Belongs to old woman next to me. Dead, isn't it?' She grabbed the cigarette from the bonker lady, who'd had more than her fair share of puffs. 'Cigarettes was in the pocket.'\n\n'Nice slippers, too.'\n\n'Nurse give them to me.'\n\n'She give me these,' said the bonker lady, lifting up the hem of her dressing gown to show off a pair of fluffy powder-blue wedgie-heeled mules. Her toes were protruding out of the ends, with the most horrible thick crusty yellow toenails I'd ever seen.\n\n'Them should heff been for me,' said Mrs Shapiro sulkily.\n\nWe left the bonker lady to finish the cigarette and made our way back to the ward, where I handed over my carrier bag of things; she took only the hairbrush, and gave the rest back to me.\n\n'I have better night cloth-es in my house. Real silk. Not like this shmata. You will bring one for me, next time, Georgine? And Wonder Boy. Why you didn't bring the Wonder Boy?'\n\n'I don't think they'd let him in. He's not very...'\n\n'They heff too many idiotic prejudices. But you are not prejudiced, are you, my Georgine?' she wheedled. 'You are so clever mit everything. I am sure you will find a way.'\n\n'Well, of course, I'll try my best,' I lied.\n\nThe ward was busy with visitors, so I pulled two chairs up by the window in the day room. It was a square featureless room near the entrance to the ward, with green upholstered chairs dotted randomly around, a TV fixed too high on the wall, and a window that looked out on to a yard. It smelled of disinfectant and unhappiness.\n\n'Mrs Shapiro, I've asked for another assessment from Social Services. Someone's going to come and visit you. She's called Ms Bad Eel.'\n\n'This is good. Bed Eel is a good Jewish name.'\n\nThis surprised me, but what did I know? We didn't have any Jewish people in Kippax.\n\n'Tell her I've got the key and I'll meet her there to show her around. She has my phone number but I'll write it down for you again.' I wrote my number on a scrap of paper, and she stuffed it into the pocket of the candlewick dressing gown. 'If anyone says anything to you about going into a residential home, just tell them you're having another assessment. That should keep them quiet.'\n\nShe leaned across and clasped my hand.\n\n'Georgine, my darlink. How can I thenk you?'\n\n'There is one problem. She's certain to ask how old you are.'\n\nShe looked at me \u2013 a clear, canny look. She knew I knew she wasn't ninety-six.\n\n'What I should say?'\n\n'Mrs Shapiro, I'll help you if I can. But you have to tell me the truth.'\n\nShe hesitated, then leaned up and whispered close to my ear, 'I am only eighty-one.'\n\nI didn't say anything. I waited. After a moment she added, 'I told them I am more older.'\n\n'Why did you tell them that?'\n\n'Why? I don't know why.' She shook her head with a stubborn little flick. 'I heff never met anybody asking so much questions, Georgine.'\n\n'I'm sorry \u2013 it's because I come from Yorkshire. Everybody's nosy up there.'\n\nI tried to recall the picture of the two women in front of the house. _Highbury 1948_. I did a quick calculation. She would have been about twenty-three when it was taken.\n\n'So do you know your date of birth?' I probed. 'She's bound to ask you that.'\n\n'Eight October nineteen hundert twenty-five.' A quick, precise answer. But was it the truth?\n\nI wanted to question her more, but I didn't want to confess that I'd already searched beyond the bureau in the study and that I'd found the photos in the Harlech Castle tin hidden in the workshop. I had questions to ask about Lydda. Who was she? When did Artem marry her? What had happened to her? And I was aching to know who'd hidden the tin, and from whom.\n\nWe were the only ones in the day room, but the TV was blaring away in the corner. I looked for a remote control to turn the volume down, but I couldn't find it, so I switched it off and settled myself into an armchair in listening mode.\n\n'You didn't finish telling me about Artem.'\n\n'You heffn't told me about your running-away husband. Why he was running away?'\n\n'It's your turn, Mrs Shapiro. I'll tell you my story next time.'\n\n'Ach, so.' She laughed. 'Where heff I gotten to?'\n\n'The pony...'\n\n'Yes, the pony that was trotting on the ice. But you see it was not a pony, it was a reindeer. The reindeer people took him away mit them.'\n\nThe S\u00e1mi men who had hitched up Artem's sleigh were from Lapland. Part traders and part bandits, they made forays down across the ice to exchange smoked fish, reindeer meat and furs for wheat or tobacco or vodka or whatever they could find. When they discovered him under the wolfskins, they debated whether to kill him; but as he opened his eyes, he smiled to find himself still alive, and started to sing a Russian peasant song.\n\n' _Ochi chornye, ochi strastnye_...' Mrs Shapiro's voice quavered. 'It is a beautiful song about the loff for a woman mit black and passionate eyes. He used to sing it often.'\n\nThe song saved his life. The faint croaky voice of the wounded soldier made the men laugh, so they took him with them to their settlement in a vast snowy wilderness beyond the Arctic Circle, where the white horizon merged into the long pale sky. He was treated first as a prisoner, then as a curiosity, and finally as a great source of entertainment.\n\nHe stayed with them for several months living on a bed of skins in the corner of a fishy, smoky, snow-covered hut, eating reindeer meat and drinking some horrible herbal concoction which they also poured on to his wound. When he had drunk a few cupfuls, he would start to sing \u2013 Jewish songs from his childhood in Orsha, partisan songs from the time in the woods, Russian folk songs, even a few arias. The men slapped their thighs and threw their heads back with laughter. The women giggled and retreated into their furs, watching him curiously with their strange cat-like eyes. At night he studied the mysterious coloured lights playing across the sky and tried to work out his position from the stars. When he was fully recovered, and smudgy light broke into the sky on the southern horizon for a few hours each day, the S\u00e1mi people offered to take him back to Russia. He explained with gestures that he wanted to go the other way, towards Sweden. So they took him to a place where he could see the next S\u00e1mi settlement over the Swedish border, gave him a small sleigh and a bag of dried fish, and sent him on his way.\n\n'He was looking for his sister. But she was already gone. Maybe she never was there. In that time Sweden was full of Jews who were running away from the Nazis. Everybody was looking for somebody or passing on the news of somebody.'\n\n'So when did you meet him? Did you go to Sweden, Mrs Shapiro?'\n\nShe started to say something, then stopped. A sad-looking lady attached to a drip tube had just walked into the day room, trailing her bag of fluid behind her. We watched her for a few moments in silence, then Mrs Shapiro whispered, 'That is enough for today. Now is your turn, Georgine. This your husband \u2013 why he was running away? There was another woman?'\n\nThe drip lady was searching for the TV remote control. I hesitated. I didn't want to go into details about the rawplugs and the toothbrush holder, but I found myself saying, 'I don't think so. He said there was no one else. He was too obsessed with his work.'\n\nMrs Shapiro was looking at me quizzically. She obviously preferred the 'other woman' hypothesis.\n\n'Why you think this?'\n\n'He was always full of big ideas. He wanted to change the world. I think he was just bored with domesticity.'\n\nThere, I'd said it. Even putting it into words made me feel better. Mrs Shapiro wrinkled her nose.\n\n'Ach, so. This is a typical story. He wants to change the world but he doesn't want to change the neppies, isn't it?'\n\n'Sort of. The children were already out of nappies.' I wanted to explain that it was the same roving, inquisitive spirit that had brought him to me in the first place. 'When we met, I was different to the other people he knew. He used to call me his rambling Yorkshire rose.'\n\n'Don't worry, my Georgine.' She grinned merrily. 'When I am mended we will go rembling again.'\n\nThe drip lady had slumped into an armchair and was gazing mournfully at the fluid in her drip bag, which looked like watered-down tea. Mrs Shapiro threw her a contemptuous look.\n\n'Too many krankies in here.' She sniffed. 'So this husband \u2013 when he is finished mit the rembling, you think he is coming back?'\n\n'I don't think so. I threw all his stuff in the skip.'\n\n'Bravo!' She clapped her hands. 'So what he said then?'\n\n'He said...' (I put on a hoity-toity voice.) '... why are you being so childish, Georgie?'\n\nShe rocked back in her chair and shrieked with laughter.\n\n'This running-away husband is quite a schmuck, isn't it?'\n\nIt was such a jolly, raucous laugh that I found myself laughing, too. Our laughter must have carried right down the ward, for a few minutes later the bonker lady came waltzing in to see what was going on, dancing around and lifting up the hem of her dressing gown to flaunt her new slippers. She winked at me, pulled a cigarette out of the pocket, and waved it under Mrs Shapiro's nose.\n\n'Look what one of the porters give me. Mind, I 'ad to drop my knickers down for 'im in the lift. I says if yer give me the packet you can 'ave yer wicked way wiv me. 'E says no thanks, missis, I've seen better on the mortuary trolley.'\n\nMrs Shapiro let out another shriek, and that set the bonker lady off, cackling and waltzing around and flashing her appalling toenails, and that made me laugh some more, and even the sad drip lady managed a dribbly chuckle. We were all clutching our sides, screeching and hooting like a flock of mad geese, when the ward sister came along and ticked us off. On the bus on the way home I felt a strangely pleasant aching sensation in my chest. I realised I hadn't laughed as much as this since... since Rip had left. \n\n# 15\n\n# The Bad Eel\n\nThe Bad Eel phoned me back a couple of days later. We made an appointment to meet at the house. As before, I went an hour earlier, with some cleaning things. The Phantom Pooer had been at work again; there were two fresh macaroon-shaped deposits in the hallway. I cleared them away and did a quick round with a duster and a brush, paying special attention to the bedroom and bathroom, though the latter was really a lost cause. I did what I could and sprayed the air-freshener around liberally. Although the weather was dry, I couldn't feed the cats by the back door because I didn't have the key, so I fed them in the kitchen, and counted them again. There were only five. Wonder Boy was in there, right at the front, batting the Stinker out of the way. Borodin crept in, his belly low to the ground, snatched his food and disappeared. One of the pram babies, I noticed, had a weepy eye. Mussorgsky and Violetta were missing. Violetta appeared at the front door a few moments later, her pretty tail swaying as she walked, and behind her was a person who could only have been the Bad Eel.\n\nThe first disappointment was that she didn't look at all like an eel. In fact she was uninhibitedly exuberantly plump, with curves that bulged in soft roly-poly layers beneath a tight stretchy blancmange-pink outfit which revealed each elastic-line of her startlingly skimpy underwear. She held her hand out to me. Each finger was like a meaty little chipolata sausage.\n\n'Hello, Mrs Sinclair. I'm Cindy Baddiel.'\n\nShe stressed the second syllable. That was the next disappointment. She wasn't a _bad_ eel at all. Her honey-gold hair fell in loose curls from two large butterfly clips above her ears. Her eyes were the colour of angelica; her skin was like peaches; she smelled of vanilla. Despite my disappointment, there was something very edible about her.\n\nI must have been staring rudely. Violetta broke the silence between us with a chatty miaow. We both bent to stroke her at the same time, our heads touched together, and we laughed, and after that, everything was easy. She strolled around the house. ('Lo-ovely. Pe-erfect.') She greeted the Stinker like an old flame. ('Well, hello-o, boy.') She did flinch for a moment in the bathroom, but her only comment was, 'There's no accounting for cultural diversity.'\n\n'One thing surprises me,' she remarked, as we were walking back down the stairs. 'She doesn't seem to be getting support from the Jewish community. Usually they're good at looking after their elderly.'\n\nThe same thought had once occurred to me, but I understood now that Mrs Shapiro was, like myself, someone who'd come unstuck.\n\n'I suppose it's her personal choice.' She'd taken a little notebook out of her bag \u2013 it had a picture of a floppy-eared Labrador puppy sitting on a cushion \u2013 and a biro with a very chewed end, and was writing something down.\n\nAt the end, when we were standing in the hall, I asked her the question that had been pressing at the back of my mind since my meeting with Mrs Goodney.\n\n'What would happen to her house, if she had to go into a home?'\n\n'Oh, I don't think it'll come to that.'\n\n'But if it did, would the Council take it from her?'\n\n'Oh no, we don't do that! Where did you get that idea?' She shook her golden curls. 'If someone goes into a care home, we assess their financial situation. If they have assets of more than twenty-one thousand pounds, then they have to pay the full cost of their care.' She was still scribbling in her notebook as she talked. Her voice was so soothing that I found it hard to concentrate on what she was saying. 'Below that, the Council picks up the bill. It can be quite expensive \u2013 four or five hundred pounds a week \u2013 so we try to maintain people's independence in their own home. It's usually what they prefer, too \u2013 familiar surroundings \u2013 chosen lifestyle.' She gave me a peachy smile.\n\n'Twenty-one thousand pounds? That's not much, is it? So would their house \u2013 this house for example \u2013 would that be classed as an asset?'\n\n'If no one else is living there, and the person is in a home, it could be sold to cover the home fees.' She was still making notes, pausing ruminatively, looking around her and chewing on the end of her biro.\n\n'But what if the person didn't want to sell?'\n\n'Don't worry.' She took my hand and squeezed it between her little chipolatas. 'I can see no reason for her to go into residential care at this stage. I'm going to recommend a means-tested care package that'll support her continuing to live at home.'\n\nI held back my impulse to say I was sure she didn't need a care package. There was something about her that made me want to take a big juicy bite, but I hugged her instead. It was irresistible, really, that soft pink bolster of flesh. Probably she was used to it, because she just stood there and smiled.\n\n'You're very demonstrative, Mrs Sinclair,' was all she said.\n\nMrs Shapiro, on the other hand, was disappointed in Ms Baddiel.\n\n'Not Jewish. Too fet.'\n\nShe shook her head with a grumpy face.\n\nI'd rushed around to the hospital immediately to tell her the good news, and we were sitting in the day room again, in front of the window. The bonker lady kept wandering in and out, making smoking gestures at me, trying to catch my eye, but I ignored her.\n\n'She said you can have a care package in your own home.'\n\n'Vat is this peckedge? Vat is in it?'\n\nShe wrinkled up her nose, as though she could smell it already.\n\n'Well, maybe a home help, to help you keep the place clean. Someone to help with your shopping and cooking.'\n\n'I don't want it. These people are all teefs.'\n\nI tried to persuade her, worried she'd lose her chance to get back home through her own stubbornness, but she looked at me with a little smile.\n\n'You are a clever-kn\u00f6del, Georgine. But I heff another news for you. I heff hed a visitor.'\n\nShe produced a card from the pocket of the candlewick dressing gown, a garish orange-and-green card, with a bold black inscription across the top in mock-Gothic letters: _Wolfe & Diabello_. Beneath, in smaller letters, a name: _Mr Nick Wolfe_.\n\n'Quite a charming man, by the way. He has made me an offer to buy up my house.'\n\nI gasped. My breath was really taken away. These people, they don't miss a trick.\n\n'Mr Wolfe! How much did he offer you?'\n\nShe turned the card over. On the back, written in blue biro, was the figure: _\u00a32 million_.\n\n'Very nice-looking man, by the way. Would be a good husband for you, Georgine.'\n\nI felt suddenly out of my depth. The social workers, the nurses, I could handle them; but men who flashed around those amounts of money scared the pants off me.\n\n'It's a lot of money. What did you say?'\n\n'I said I will think about it.'\n\nShe caught my eye and smiled impishly.\n\n'What for I need two millions? I am too old. I already heff all what I need.'\n\nThe nurse \u2013 it was the brisk young woman I'd met on my first visit \u2013 was happy with the care package, and a date was set for Mrs Shapiro to return home. I promised I'd be there to meet her, and would drop in regularly until she was settled. There was one more thing I needed to sort out before she came home. I didn't want Damian or Mr Diabello \u2013 whichever one of them had the key \u2013 barging into the house while she was there on her own. I must get that Asian handyman to change the back-door lock. I rang the number on the card and made an appointment with him for the next day. \n\n# 16\n\n# The handyman\n\nMr Ali arrived on a bicycle. I'd been expecting a man in a van, so I didn't notice him at first, wobbling quietly up the lane. He was smaller and tubbier than I recalled, and he was wearing a pink-and-mauve striped woolly hat pulled right down over his ears, which was sensible, because the morning was cold. It was hard to tell how old he was; his face looked young, but his beard and moustache were heavily flecked with grey. He didn't look at all like a handyman \u2013 for one thing he didn't seem to have any tools.\n\nHe jumped off his bike, removed the cycle clips from his ankles, straightened out his trouser bottoms \u2013 they were grey flannel, with neat creases down the fronts \u2013 and greeted me with a polite nod of the head. I noticed now that there was a small leather bag \u2013 it could have been a woman's handbag \u2013 on a long strap slung across his chest, with the head of a hammer poking out at one side.\n\n'I have come to fixitup lock,' he announced.\n\nHe pushed his bicycle up the path, and propped it in the porch at the front door.\n\n'Jews live here?'\n\nThere was something sharp in his voice that took me aback.\n\n'Yes. How did you know?'\n\n'Mezuzah.' He pointed out something that looked like a small tin roll pinned on to the door frame. It had been painted over, and I hadn't noticed it before.\n\n'Strange thing for me,' he muttered. 'Never mind. Here in London is no broblem.'\n\nHe took his pink-and-mauve cap off \u2013 I saw now that his black hair was also threaded through with grey \u2013 and stuck it in his pocket, along with the cycle clips.\n\n'You Jewish?'\n\nI shook my head. 'Yorkshire. It's almost a religion.'\n\nHe gave me a funny look \u2013 I don't think he realised it was a joke. His dark eyes darted around, taking in the details.\n\n'Every house speaks its history to one who knows how to listen.'\n\nSo, not your typical handyman, I thought.\n\n'Where you have this broblem lock?'\n\nThere was something cutely hamster-like, I thought, about the way he sometimes confused his 'p's and 'b's, though I have no evidence that hamsters actually do this.\n\nI led him through to the kitchen. The back door was heavy pine, painted to look like walnut, with two panels of blue engraved glass.\n\n'For this one you have lost the key?'\n\n'That's right.'\n\n'Hm.' He stroked his beard. 'Locked up.'\n\n'Yes \u2013 that's why I called you.'\n\n'Hm. Only way to open must be with force. You want me to do this?'\n\n'I... I don't know. I thought maybe you could unscrew something.'\n\n'This type of lock sits inside door frame, not screwed on outside.'\n\n'Oh, I see.' Actually, now he pointed it out, it was totally obvious.\n\n'But usually,' he said, stroking his beard again, 'usually there exists more than one key for every door.' He turned the door handle up and down. 'You do not have another key? You have lost it, too?' He sounded reproachful, as though I'd been unreasonably careless.\n\n'It's not my house. I'm just feeding the cats while the owner's in hospital.'\n\n'The key that proves the ownership of the house.'\n\nI was beginning to feel annoyed. I wanted a handyman, not a philosopher.\n\n'I think it's been stolen. Really, if you can't help, Mr Ali, I don't want to waste any more of your time.'\n\n'Certainly I can help. But better not to break down the door if we can open it by some other way. You have looked for another key?'\n\n'Where should I look?'\n\nI was thinking that a lad in a van might have been easier to deal with. He looked at me as though I was completely stupid.\n\n'How can I know this? I am a handyman, not a detecteef.'\n\nHe scanned the room with his hamster eyes, then he started opening cupboard doors and pulling out drawers, rifling through the mouldy tea towels and crusty cutlery.\n\nIn the built-in pine cupboard at the side of the chimney breast was a jumble of crockery, pots, tins, jars, bowls, vases, candlesticks, and other stuff which could loosely be described as bric-a-brac. Mr Ali stood up on a chair and went through it all methodically, working from the top down, taking each item from the shelf, shaking it, and replacing it. Inside an ornate silver coffee pot on the middle shelf, he found a bundle of obsolete ten-shilling notes and a bunch of keys.\n\n'Try.' He passed them to me. One of the keys fitted the back door.\n\n'So now your broblem is fixit,' he beamed.\n\n'Yes, thank you very much, Mr Ali.' I resisted a sudden urge to stroke his little hamster head. 'But I'd still like you to change the lock, if you can, so the person who took the other key can't use it.'\n\nHe rubbed his chin. 'I understand. In that case I must buy new lock.'\n\nHe replaced his cycle clips and wobbled off down the road.\n\nAs soon as he was out of sight, I took the opportunity to continue my investigation of the house. I wasn't sure what I was looking for, but I was driven by the conviction that there must be a stash of documents or letters somewhere that would provide the key to Mrs Shapiro's story, and the identity of the mystery woman with beautiful eyes. However, apart from Mrs Shapiro's bedroom, all the upstairs rooms were sparsely furnished, with nowhere much to hide anything, and I began to feel disheartened.\n\nFrom an upstairs window, I watched the cold shadows sidle into the garden. A few cats were still prowling around; I caught sight of Wonder Boy in the bushes beside the mews block and Violetta sitting on the roof of a ruined outhouse. The bedroom I was in had a bleak institutional feel, compared with the stinky decadence of Mrs Shapiro's room. Mrs Sinclair's old burgundy-coloured curtains, which I'd put in the skip, had been spread as bedcovers on the two single beds. I checked the drawers, but they were empty, and there was nothing under the mattresses. I drew a blank. When I looked out of the window again, some minutes later, Wonder Boy was up on the outhouse roof, too; he appeared to be raping Violetta. I banged on the window and he slunk away.\n\nMr Ali was gone for ages, maybe an hour, and I was getting fed up of hanging around in the dank empty smelly house. Next time I need a handyman, I was thinking, I'll get someone out of Yellow Pages. I returned to Mrs Shapiro's bedroom and sat on the edge of the bed, staring at the garden path through the mirror, wishing he would hurry up. And that's when my eye fell on another drawer in the dressing table, a low, curved, concealed drawer without a handle, beneath the mirror. I hadn't noticed it before, and I realised it had been designed not to be noticed. I eased it open. It was full of jumbled jewellery \u2013 necklaces, earrings, brooches. A lot of it seemed rather grotty and broken, but there were one or two pieces that looked as though they might be valuable. Was it wise for her to keep it here in the house? As I lifted out a blue bead necklace I saw there was a photograph underneath the jewellery at the bottom of the drawer. I pulled it out to add to my collection, but it was only a landscape in black and white of a not very appealing hillside, barren and rocky, planted with terraces of shrubby trees. In the valley below was a scattering of flat rooftops. It looked like Greece. I turned it over. On the back was written _Kefar Daniyyel_ and two lines of verse.\n\n_I send my love across the sea_\n\n_And pray that you will come to me_\n\nNaomi\n\nAnother name: Daniyyel. How did he come into the story? Had Naomi had a secret lover? There was a long person-shaped shadow in the foreground \u2013 it must be the photographer standing with his back to the sun. So who had taken the photo?\n\nThen I heard the tink-tink of a bicycle bell outside, and a moment later Mr Ali reappeared.\n\n'Very sorry for delay. I was looking everywhere for right size of the lock. Old-style lock not easy to find.'\n\nIt took him less than ten minutes to lever out the old lock and fit the new one. I took one of the new keys and put it back on the key ring in the coffee pot; the other I put in my pocket with a smile. In my imagination, I pictured Mrs Goodney and Damian tiptoeing round to the back of the house at dusk, fiddling and fiddling with the old key, trying to get it to fit. In the end they gave up and stomped away, tripping on brambles and ending up covered in cat poo. Serve them right.\n\nI settled up with Mr Ali \u2013 he asked for ten pounds, plus the cost of the lock, but I persuaded him to take twenty \u2013 and thanked him profusely.\n\n'Always better,' he said, packing his tools back into his shoulder bag, 'first to try the non-violent solution.'\n\n# 17\n\n# The care package\n\nIt was quite late next evening \u2013 it must have been after ten o'clock \u2013 when the telephone rang.\n\n'Is that Mrs Sinclair?'\n\nA grating voice, familiar, but I couldn't place it.\n\n'Speaking.'\n\n'This is Margaret Goodney from the social work department at the hospital.'\n\nSurely she wasn't still at work.\n\n'Oh, hello, Mrs Goodney. Is everything all right?'\n\nI smirked to myself. Maybe she and Damian had already tried their key and failed to get in.\n\n'I think you know what I'm ringing about.'\n\n'No. I don't know. Please tell me.'\n\nIn my mind's eye I could picture her at the other end of the telephone, smoking a cigarette, wearing her lizard-green quilted jacket, covered in cat poo.\n\n'I know what you're up to.'\n\n'Excuse me?'\n\n'That ridiculous care package you and Mrs Whatsit've cobbled together. You should keep your nose out of this. Leave it to the professionals.'\n\n'Ms Baddiel is a professional.'\n\n'She's not a professional.' An ugly nasal sneer. 'She's a box-ticker. Those local authority social workers don't know what real social work is.'\n\nBefore I could muster a reply, she struck again.\n\n'You won't get away with it, you know. If I have to, I'll call the police in.'\n\nI was completely thrown.\n\n'I'm sorry, I have no idea what you're talking about.'\n\n'You persuaded her to name you as next of kin, didn't you? We've seen it all before, you know \u2013 someone befriends a vulnerable old person, then the next thing we know, they've altered the will and the new friend gets the lot.'\n\nMy adrenaline was up. I could feel my heart starting to race.\n\n'Nobody has altered any will.'\n\n'But that's what you're after, isn't it \u2013 the house?' she hissed.\n\nI suppose I should have put the phone down, but I was too shocked.\n\n'I'm not after anything.'\n\n'Being all friendly, going round and cleaning up, feeding the cats.'\n\n'It's called being a good neighbour. Looking out for the vulnerable in our society. Wouldn't you do the same?'\n\n'Nobody does all that without expecting something in return.' Her malevolent rusty-gate voice made me wince. 'You're not family. In fact you hardly seem to know her. And all of a sudden you go barging into her life, taking over her affairs.'\n\n'You're accusing me of...'\n\n'I'm not accusing you of anything, Mrs Sinclair. I'm just saying that if you were to be found to be applying undue pressure or benefiting in an improper way from this relationship, then it would be a matter for the police.'\n\nIt took a moment for the sheer audacity of it to sink in.\n\n' _I'm_ the one who should be reporting _you_. You and Damian. I know your little plan. Then you have the nerve to ring me up in the middle of the night and accuse me...'\n\n'I'm not accusing you, Mrs Sinclair. Get this straight, will you? I'm just advising you of the consequences that could follow from certain actions.'\n\nShe put the phone down. In the silence that followed, I could hear the clock ticking, and the faint ker-chunga-chunga coming from Ben's room. I realised my hands were shaking.\n\nDespite the veiled threat in Mrs Goodney's phone call, Mrs Shapiro was discharged before the end of the week and returned home by taxi to an ecstatic welcome from Violetta, a languid welcome from Mussorgsky, and a dead pigeon from Wonder Boy. The other four were all there too, rubbing against her legs, rolling on their backs and purring like trail bikes.\n\nI'd cleaned up the mess in the hall, put a fan heater on to take the chill off the place, brought some shopping in, and put a vase of flowers on the hall table. I'd also replaced the key in the lock, so she'd be able to use the back door. She looked in good shape, and excited to be back. She took off her astrakhan coat and emptied out a carrier bag which contained the pink candlewick dressing gown and one high-heeled shoe. The other was lost. She was still wearing the _Lion King_ slippers on her feet.\n\nI made a pot of coffee and some sardines on toast \u2013 probably not a good idea I soon realised \u2013 and we sat at the table in the kitchen. The cats circled around, attracted by the smell of the sardines, and I wiped some bread in the oil and put it down for them. They gobbled it up in a flash, and carried on circling. Wonder Boy leapt up on to Mrs Shapiro's lap, and started kneading her thighs vigorously with his big bruiser paws; from time to time, he reached one out and snatched a piece of sardine-on-toast from her plate. Violetta sat on my knee, purring sweetly when I stroked her.\n\n'You heff been a very good friend for me, Georgine. Without you I'm sure they would heff put me away into the oldie-house.'\n\nWe clinked our cups together.\n\n'To friendship.'\n\nBut something still niggled at the back of my mind. Each time I looked at her I found myself wondering about the other woman in the photo, Lydda.\n\n'Don't you have a family, Mrs Shapiro? Any sisters? Or brothers? Anyone who could look after you?'\n\n'Why for I need someone to look after me? All was okay before this accident.'\n\n'Any grown-up children? Or even cousins?' I persisted.\n\n'I am not need nobody. I am okay.' She bit fiercely into a piece of toast.\n\n'But even if you're all right now, you're not getting any younger and...'\n\n'I think I will sue the Council.'\n\n'... of course I'm happy to help, but...'\n\n'They should be tekking care better of the pavements. They think we elect them only for giving our money away to immigrants? I am paying rets on this house sixty year. I think they must pay me a compensation.'\n\n'Well, before we get on to that...'\n\n'Yes, I will sue for the compensation. I will go to Citizen Advice this afternoon.'\n\n'I don't think you should go out anywhere just yet, Mrs Shapiro. Wait till you're a bit better. And the lady is coming this afternoon, from the Council. Remember? Your care package?'\n\n'Peckedge schmeckedge.'\n\n'But I think you should...'\n\n'I don't want no peckedge. Definitely no peckedge.'\n\nMrs Shapiro's care package was a thin dour Estonian woman called Elvina with blackheads and an economics degree. She did make some impact on the chaos in the kitchen, and the house looked generally cleaner but, as if in response, the Phantom Pooer redoubled his efforts, and now as often as not there were two little macaroon-shaped deposits each day, one in the hall, and one in the kitchen just behind the door. Elvina shouted at the cats in Estonian, and went for them with the broom. Mrs Shapiro called her a Nazi collaborator and sent her packing a week before Christmas, claiming she had stolen a silver coffee pot and some cat biscuits. \n\n# 18\n\n# Sherry\n\nA couple of days before Christmas, I set off for Canaan House to deliver my Christmas present \u2013 a little basket of scented soap and body lotion that I thought Mrs Shapiro would like. A nippy wind flicked my hair against my cheeks and made my batty-woman coat flap against my legs. There were no leaves left on the trees, but tattered shreds of plastic bags fluttered from the branches like pennants, and bits of wind-driven litter skittered along the street in front of me.\n\nAs I turned the corner, I saw a massive four-by-four, black with darkened windows, tractor-sized tyres, and doubtless a global-warming-sized engine, parked at the bottom of the lane. It looked vaguely familiar, but I couldn't place it. I quickened my step. Violetta was waiting for me in the porch, her fur fluffed out against the cold. I rang the bell.\n\nThere was a long silence, then footsteps, then Mrs Shapiro appeared at the door. She was wearing full make-up and a rather stylish striped jersey, with brown slacks, and a different pair of high-heeled shoes \u2013 these were snakeskin, with peeptoes and slingbacks, a couple of sizes too large. Her left wrist was still strapped, and in the other hand she was holding a cigarette.\n\n'Georgine! My darlink!' She grabbed me in her arms, the cigarette waving dangerously close to my hair. 'Come in! Come in! I heff a visitor!'\n\nI followed her through the chilly hall \u2013 yes, there was a little deposit in the usual place \u2013 to the kitchen, where the fan heater was on at full blast and a kettle was steaming away on the gas stove. There was the usual smell of cat piss and decay, and, above it, a new smell, musky and potent, of perfumed aftershave. A man was sitting at the kitchen table. He was turned away from me, but even from his back I could see he was a big man, broad, with close-cut blond hair, and muscles that pushed against the seams of his clothes. He rose to his feet and turned to greet me as I came in. He rose and rose \u2013 he must have been well over six feet tall, and heavily built, like a slightly-gone-to-seed rugby player \u2013 and then our eyes met. A flash of mutual recognition passed between us and in that moment we made an unspoken pact to forget that we had ever met before.\n\n'Nicky,' said Mrs Shapiro, fluttering her derelict eyelashes at him, 'this is my dear friend Georgine.' She turned to me. 'This is my new friend Mr Nicky Wolfe.' She obviously didn't recognise him at all.\n\n'Pleased to make your acquaintance.'\n\nHe gripped my hand \u2013 his palm was moist and meaty \u2013 and pumped it up and down.\n\n'Hello, Mr Wolfe.'\n\nI don't automatically think of sex when I meet a man, but I did with him; I thought it would be quick, painful and humiliating. I would be Violetta to his Wonder Boy. There was that look in his eyes.\n\n'Call me Nick, please.'\n\n'Hello, Nick. You must be from the estate agents.'\n\n'Got it in one. How did you guess?'\n\n'Mrs...' I usually addressed her formally, but I had to impress on him that we were close. '... Naomi showed me your card. She said you'd made an offer for her house.'\n\n'An offer I hope she won't be able to refuse.' He leered at her.\n\n'Georgine, darlink. Will you heff a drink?'\n\nMrs Shapiro's cheeks were flushed beneath the two little circles of rouge.\n\n'A cup of tea would be nice.'\n\nThe kettle was still hissing away, filling the kitchen with steam. Then I saw that on the table, amidst all the clutter, were a sherry bottle and two glasses, his full, hers empty.\n\n'I heff only kr\u00e4utertee. From herbs.'\n\n'That's fine.'\n\n'Why not heff a little aperitif?'\n\n'It's a bit early for me, Naomi.' I loaded my voice with reproach. 'It's not yet ten o'clock.'\n\n'Is it so early?' She looked around with wide scandalised eyes, and giggled. 'You are a very notty man, Mister Nick.'\n\nHe chuckled, a rapist's chuckle. 'Never too early for a bit of fun.'\n\nI turned the gas off and poured the boiling water from the kettle over a tattered tea bag in a cracked and stained porcelain cup. It tasted like not-very-clean pond water. Actually, I could have murdered a glass of sherry.\n\n'Happy Christmas \u2013 I mean, festive season \u2013 Naomi.' I passed her my little package.\n\n'Thenk you, darlink.' She held it up to her nose and breathed in, closing her eyes with pleasure. 'But I must find something for you!'\n\nHer eyes wandered around the kitchen, resting for a moment on a REDUCED packet of biscuits on the counter, a squashed box of Maltesers, a half-eaten packaged cake.\n\n'Oh, no. Please. You're too kind; I don't need anything. What will you be doing for... for the festive season, Naomi? Will you be all right on your own?'\n\n'Darlink, I will not be on my own. First Christmas we will celebrate, then Hanukkah. Turkey breast and latkes. Pick and mix non-stop festivity, isn't it, Wonder Boy?'\n\nBut Wonder Boy was nowhere to be seen.\n\n'I'd better be going. I'll leave you two ladies to your fun.' Nick Wolfe towered over both of us. 'I've still got three valuations before I can knock off. An estate agent's work is never done.'\n\n'Please, Nicky, you heffn't finish your drink.' Mrs Shapiro had gone all fluttery again.\n\nHe picked up the full glass and downed it in one go. I could see that with his body mass, it wouldn't make the slightest impact.\n\n'But you must take your bottle.' She pushed it towards him.\n\n'I wouldn't dream of it, Naomi. Please keep it as a small token of my regard.'\n\nHe sidestepped her with a rugby player's deftness and moved through the door out into the hall. Would he step in the cat poo? No, he didn't. Pity.\n\nShe saw him out to the front door. As I waited in the kitchen I became aware of a strange unsettling animal noise coming from the study. I went to investigate. There, in front of the fireplace, was Wonder Boy, his back arched, his muscular haunches pumping up and down, rasping and grunting on top of a small brown fluffy cat that lay motionless on the fender. Was it squashed dead, poor thing? I looked more closely... no, it wasn't a cat, it was one of the _Lion King_ slippers.\n\n'He is quite an adorrable man, isn't it?' Mrs Shapiro minced back into the kitchen with a radiant look on her face. 'Next time I will invite him, you also must come. You must put on a bit of mekkup, darlink. And better clothes. I heff a nice coat I will give you. Why you always wearing this old brown shmata?'\n\n'It's kind of you to think of me, Mrs Shapiro, but...'\n\n'No need to be shy, Georgine. When you see a good man, you must grebbit.'\n\n'... are you sure you wouldn't like a nice cup of herbal tea?'\n\n'No, thenk you, darlink. I heff enjoyed my aperitif.'\n\nA sound like a satisfied grunt came from the direction of the study.\n\nNext morning \u2013 it was Christmas Eve \u2013 I was woken up at seven o'clock by the phone ringing. I guessed straightaway who it was.\n\n'Georgine? Is this you? Come quick. Something is heppening to my votter.'\n\n'Is it leaking? Is it a burst pipe?' I muttered groggily, wishing she could have waited an hour before ringing me.\n\n'No, nothing. I turn on the tap and nothing happens.'\n\n'Look,' I said, 'I'm not an expert on plumbing. But I know a handyman. Would you like me to give him a ring?'\n\nThere was a pause.\n\n'How much he is charging?'\n\n'I don't know. It depends what the problem is. He's very nice. His name's Mr Ali.'\n\nThere was another silence.\n\n'Is he a Peki?'\n\n'Yes. No. I don't know. Look, do you want me to ring him or not?'\n\n'Is okay. I will ring my good friend Mister Nick.'\n\nShe put the phone down. I rang Wolfe & Diabello but only got an answering machine. A few minutes later Mrs Shapiro phoned back.\n\n'Is not there. Only answering machine in the office. These people are too lazy, isn't it? Sleeping all morning instead of working. What is the number of your Peki?'\n\nWhen I went round to Totley Place at about ten o'clock I saw that Mr Ali's bicycle was already propped up in the porch and he was sitting in the kitchen drinking a cup of that vile pond water. He stood up when I came into the room and greeted me warmly.\n\n'Fixit broblem, Mrs George.'\n\n'What was it?'\n\n'Something peculiar,' said Mrs Shapiro. 'Someone has turned off the votter tap outside. Mr Ali has found it underneath the back door. Such a clever-kn\u00f6del!'\n\n'Water was all off,' nodded Mr Ali, beaming. 'Now back on.'\n\n'But why?'\n\n'How can I know this?' He shrugged mildly. 'I am a handyman, not a pseecholog.'\n\n'That _is_ strange,' I said. My mind started to race. Who would do a thing like that?\n\nMr Ali finished his tea and stood up to go.\n\n'Any more broblem, you telephone to me, Mrs Naomi.' (He pronounced it Nah-oh-me.)\n\n'But wait, I must pay you. How much it is?' Mrs Shapiro fumbled in a brown leatherette shopping bag that was under the table.\n\n'Is okay. You no pay this time. I did nothing. Only turn tap.'\n\n'But I must give you something for coming to my house.'\n\n'You have given me cuppa tea.'\n\nHe slung his bag of tools over his shoulder and I rose to show him out to the door.\n\n'Thank you for your help,' I said, following him out into the hall.\n\nSuddenly he stopped by the quaint little phone table with its barley-twist legs. I thought at first he'd stepped in the cat poo, then I saw that his eyes were fixed on the framed photograph of the stone arch. He leaned forward for a closer look.\n\n'It looks quite old, doesn't it?' I said chattily, though I had no idea of its age.\n\n'Church of Saint George,' he said. 'In Lydda.'\n\n'Lydda.' A place, not a person. 'You've been there?'\n\n'One time, I went back. Looking for my family.' He said it so quietly it was almost a whisper. 'I was born nearby to that place.'\n\n'In Greece?' I was surprised. He didn't look Greek.\n\nHe shook his head. 'Palestine.'\n\nBefore I could think what to say, he'd disappeared through the door. I heard the tink-tink of his bicycle bell as he pushed it down the path. Mrs Shapiro was beaming when I came back into the kitchen.\n\n'Very good Peki,' she said.\n\nI didn't tell her he was a Palestinian.\n\nMy mind was still whirling. This story was turning out to be much more complicated than I'd thought. Nothing was what it appeared to be. Lydda was a place not a person; Mr Ali was from Palestine not from Pakistan; and someone had turned Mrs Shapiro's water off. Why? A practical joke? Or harassment? The more I thought about it, the more I was sure it must have been Mr Wolfe. He must have noticed the stopcock when he was snooping around. He knew how vulnerable she was. He had sat there, on that chair in the kitchen, plying her with sherry and flattering her. That was the carrot. And at night-time, when she was on her own, he had applied a bit of stick.\n\nThe phone rang at that moment. Mrs Shapiro shuffled out to the hall. I could see her through the open door, gesticulating as she talked.\n\n'Nicky! You got my messedge at your office... thenk you for ringing... Is okay. Votter problem is solved, but you can come anyway. Georgine is here... Ach, so. Never mind. Any time you want to drink coffee mit me, you are very welcome. Yes, and happy Christmas to you, Nicky.' She fluttered her eyelids as she talked, as though he was there in the room. When she put the phone down, she turned to me.\n\n'Very nice man. Would be a perfect husband for you, Georgine. Rich. Hendsome. What you say?'\n\nI laughed. 'Not quite my type.'\n\n'Ach, you young girls! Nowadays, you heff too much choices. In my days, if you seen a good man, you had to grebbit.'\n\n'Is that what you did, Mrs Shapiro? Like the sausages at Sainsbury's?' I teased.\n\nHer face clouded over. She started fishing around in the ashtray for a cigarette butt, frowning as she tried to work out which was the longest one.\n\n'You know, in the wartime so many men were getting killed. If you seen one you liked, you must grebbit quick.'\n\n# 3   \n_Bonding_\n\n# 19\n\n# Christmas with all the trimmings\n\nI went back to Kippax for Christmas, though I wasn't in a particularly celebratory mood. It was my first Christmas away from Ben and Stella, and there was a sore cavity in my heart as if from a couple of extracted teeth. I was worried about Mum and Dad, too, and the worst thing was, I could sense they were also worried about me. Dad was still quite poorly; the latest hernia operation had knocked him back, but he was determined not to let it show, and he prowled about the bungalow in his new Santa slippers pinning up the Christmas decorations. I caught him wincing once or twice when he thought no one was looking. The heating and the TV were both on full blast. Mum was wandering around the kitchen in a daze, wearing a pair of jokey reindeer antlers, wondering what she'd done with the bread sauce, and still insisting steadfastly that she had to do Christmas right, with all the trimmings.\n\nWhen I lived at home, Mum and I always used to sneak off to the midnight service at St Mary's on Christmas Eve. Mum liked to join in the carols. Her voice, piercing and slightly off key, used to make me cringe with embarrassment. But as I got older, I cultivated a blank what-d'you-think-you're-looking-at stare for the people in the pews in front when they craned round to see who was making the racket. Dad stayed obdurately at home and played his old Woody Guthrie records. Keir, my younger brother, went out with his mates to the pub. But this year, even the lure of loud singing couldn't persuade Mum to go out in the cold, and we all settled down on the sofa in front of the TV.\n\nOn Christmas Day, instead of the traditional turkey, we had a traditional-style turkey breast roast, which came with a sachet of bread sauce \u2013 which Mum had lost. Dad made the gravy out of granules mixed with warm water. He put a pinny on specially for the occasion.\n\n'I bet you never thought I'd turn into a new man, Jean,' he said to Mum.\n\n'No, I din't,' said Mum. 'Which are the new bits?'\n\nShe was busy defrosting the chipolatas in the microwave. They were from Netto's bargain range. They reminded me of the Bad Eel's fingers, pink and plump. When I bit into one, pink juice oozed out.\n\nMum had set a place for Keir at the dinner table \u2013 he got the Loch Lomond placemat, which had always been my favourite. I was stuck with Edinburgh Castle again.\n\n'To absent friends!' She raised a glass of tepid Country Manor \u2013 her third. 'And death to Iraqis.' The reindeer antlers had slipped down her forehead and were pointing forward, as though she was getting ready to rut.\n\n'Mum,' I whispered, 'Keir's supposed to be liberating them, not killing them.'\n\nBut it was too late. Dad leaned back in his chair and thumped his hands on the table.\n\n'Got no business to be there at all.' His voice was loud enough to be heard next door. 'If they 'adn't shut all t' pits, they wouldn't be so mad for oil now, would they?'\n\nThe war had thrown into sharp relief the differences between them: Mum passionately loyal to her family, Dad stubbornly loyal to his principles.\n\n'Don't start now, Dennis. It's Christmas.' Mum reached out and laid her hand on his arm. She was wearing all her rings: gold, sapphire and diamond.\n\n'Aye, but it in't Christmas for them, is it?' said Dad, always the internationalist.\n\nThe Christmas lights on the tree winked on and off, competing with the turned-down TV in the corner, where King's College choirboys were soundlessly singing their heads off.\n\n'What d'you think of this turkey breast?' asked Mum, changing the subject. 'It were on special.'\n\nBut Dad wasn't to be deflected.\n\n'I'd sooner 'ave 'ad Tony Blair trussed up and roasted. Wi' all 'is gizzards in.'\n\nMum leaned over to me and whispered loudly, 'I don't know what it is, but Christmas always gets 'im gooin'.'\n\nAs she said it, I had a sudden vivid image from another Christmas \u2013 it was long before the strike, I must have been about ten at the time, and Keir five. A group of carol singers had come to the door. They were kids from the local school. They rang on the doorbell, and when Dad answered it, they started to sing in their little squeaky voices:\n\n_We three kings of Orient are,_\n\n_Bearing gifts we travel afar._\n\nDad stood patiently and waited for them to finish. When they got to the end of the second verse, they fell silent. Probably that was as much as they knew. Dad reached in his pocket and gave them some change. Then just as they were mumbling their thanks, he burst into song:\n\n_The people's flag is deepest red,_\n\n_It shrouded oft our martyred dead..._\n\nHis voice was deep and loud. Keir and I crept away and hid behind the sofa. The children stood there gawping. When he got on to the bit about the limbs growing stiff and cold, they suddenly turned and made a dash for it, and didn't look back until they'd got to the end of the street.\n\n'What d'you do that for, Dennis?' Mum scolded.\n\n'They should teach 'em in school,' said Dad mildly. 'Proper history, not fairy tales.'\n\nWhen we went back to school after the Christmas holidays, the kids were there waiting for me.\n\n'Your dad's potty,' they said.\n\n'No 'e in't.' I stood and faced them out. 'It's because your singing were crap.'\n\nI saw Dad wince now, as he shifted in his chair, and a stab of his pain got to me, too. Dear Dad \u2013 he'd never been afraid to jump in deep, and he'd always done what he thought was right, regardless of the consequences. I thought with a pang of sadness of Rip, Stella and Ben, spending their Christmas at Holtham without me. The food would be better, the gifts more extravagant, the decor subdued and tasteful. There would be no Santa slippers or reindeer antlers, no political arguments, no Highland-scene placemats or plastic tree with winking coloured lights. Stella would wallow in the jacuzzi and flirt shamelessly with her grandpa. Ben would come back with some hi-tech gizmo for his computer, which he would discreetly hide in his bedroom so as not to upset me.\n\n'Never mind, duck,' said Mum, reading my face. 'There's nowt like being wi' yer own family at Christmas.'\n\nWe clinked our glasses together, Mum's filled with the last of the Country Manor, Dad's and mine with Old Peculier. The mystery of the bread sauce was solved when Dad poured it over the Christmas pudding.\n\n*\n\nThat night I lay awake in my old bed, listening to the voices in the next room. Mum had left a thick Danielle Steel out for me, but I couldn't get into it, my mind was wandering into the past, retracing the journey I'd made away from my family.\n\nIt was books that had changed my life \u2013 had catapulted me out of the coal-smoke semis of Kippax into university and a wider world beyond. When the careers teacher at Garforth Comp had asked me what I wanted to do, I said I wanted to be a writer. 'Writing's a great hobby,' he'd sighed, like one who knew. 'But you'll need a day job, too.'\n\nI took an English degree at Exeter, then a postgraduate course in journalism at the London College of Printing. I was the first in my family to go to university \u2013 I know it's a clich\u00e9 now, but it wasn't a clich\u00e9 for us. After a traineeship on the _Dulwich Post_ , I came back up to Yorkshire for a junior reporter's job on the Bradford _Telegraph and Argus_ , in order to be nearer to home. Then I got a lucky break on the _Evening Post_ in Leeds. Somewhere along the line, it happened so gradually that I didn't notice, I stopped talking Yorkshire and thinking Kippax. Nowadays you might just notice the flatness in my vowels when I say 'bath'; and marrying Gavin Connolly is no longer the pinnacle of my ambition. They didn't resent it. Mum kept a scrapbook of news clippings with my byline, which she brought out on any excuse. They were proud of me, my mum and dad.\n\nThey'd made their own journeys, too. Dad had gone into the coal industry after the war, and when he'd had a pint or two he would wax lyrical over the post-war settlement, when the family, the community, the pit, the union, the government, the nation, the United Nations, had all flowed seamlessly one into the other. It had given him all he had, and he'd done his bit in return, studying at night school to get his deputy's ticket, reading his textbooks at the coal face by the light of his lamp, because it was his belief that he should use his abilities on behalf of those less able than himself. When Ledston Luck closed in 1986, along with 160 other pits that closed in the wake of the strike, men like my dad and my brother were thrown out of that embracing society into a different kind of world. 'Maggie's Britain', Dad called it, and he never said it without a sneer. It wasn't his country any more.\n\nKeir \u2013 he was just twenty-one at the time \u2013 survived by finding himself a new family: the Royal Engineers filled the gap left by the pit and the NUM. There were postcards of Keir building bridges in exotic places surrounded by smiling dark-skinned children; Keir wearing civvies and drinking a beer against the backdrop of an awesome snow-capped mountain; Keir and his mates grinning under their helmets, posing beside a jeep in a desert somewhere. 'Look where 'e is now, our lad,' Mum would murmur, running her fingers over the glossy prints.\n\nDad was shifted to the Selby coalfield, and when that closed, too, he was still young enough to get the redundo, old enough to get a decent pension and free coal for life, and bolshie enough to take on the chairmanship of the local Labour Party. He dedicated his life to the overthrow of Maggie, and pursued it with the same dogged diligence as he'd once sniffed out the firedamp. He never quite forgave Keir for joining the army, nor me for marrying Rip, but he never gave up on us either, just as he never gave up on the Labour Party, even when Tony Blair turned out to be, as he called him, the mini-Maggie.\n\nMum, on the other hand, had blossomed during the eighties. She loved the shoulder pads and the jewellery. She loved gadding about on coaches during the strike, and shouting at the top of her voice. Afterwards, rather than accept defeat, she put her new-found organising skills to good use and enrolled in a night class in Castleford to learn bookkeeping. Just as Dad was squaring up to retirement, Mum was opening the door on a new career, doing the books for Pete's Plaice, Annie's Antiques, Sparky Steve, All-night Abdul, Curl Up and Dye, and various other small businesses that came and went in the former pit villages. For the first time in her life she was financially independent. Then around the time Rip and I were splitting up, she started to lose the central vision in her left eye. It would be slow, the doctor said, but progressive.\n\nSomewhere in the darkened house, a door clicked and a toilet flushed. It must be Dad, battling with his prostate as stubbornly as he'd once battled with global capitalism. Then there was a second flush and a sound of voices in the kitchen. The whistle of the kettle, the clink of cups. In the daytime, they'd put on a brave face for me, but at night all their worries crept back and beset them. They couldn't get to sleep, so they were having a cup of tea together. My mum and dad.\n\nI lay there in the dark, listening to the murmur of their voices and thinking about Christmas. Really, when you consider, it's not a very nice story. Okay, a baby was born, and there were angels, and a star in the sky \u2013 that bit's not so bad \u2013 but what about poor Mary having to travel all that way on a donkey \u2013 in her condition? The three kings with their sinister gifts? Then the slaughter of the innocents? And that was just the beginning; after that, there were crucifixions, a resurrection \u2013 with Armageddon and the Second Coming still to look forward to.\n\nMy mind flashed to the conversation I'd had with Ben about Jesus and the end of the world, the look of fear in his eyes as he tried to rationalise the irrational. Yes, Christmas is a dangerous time, I thought. Sometimes it can be better just to stay at home until it all blows over. \n\n# 20\n\n# The festive season\n\nWhen I got back to London on the day after Boxing Day, Wonder Boy was waiting for me at the front door with a dead bird. I supposed it was his idea of a gift, so I let him into the kitchen and gave him a saucer of milk, even though I'd previously resolved not to encourage him. Well, it was Christmas. He thanked me by lifting his tail and spraying against the dishwasher. Thanks, Wonder Boy.\n\nBen wasn't going to be back for a few days. Even _Adhesives_ was having a break \u2013 the next issue wasn't due out until early March. Nathan phoned me to wish me a happy New Year and share a joke with me.\n\n'What beats glue when it comes to bonding?' he murmured in his conspiratorial voice.\n\n'I don't know. Tell me.'\n\n'Hybrid bond. Glue and a screw. Geddit?' I imagined him with his white coat casually unbuttoned, chuckling glueily at the end of the phone. After he'd hung up, the silence of the house closed in around me.\n\nThat first night at home, I tossed and turned in my half-empty double bed and wished I was back in my old room in Kippax with the TV on too loud and Mum and Dad making cups of tea in the middle of the night. Of course, I knew that if I'd been there I'd only be wishing I'd stayed here \u2013 it wasn't here, and it wasn't Kippax \u2013 the bug was inside me, gnawing away.\n\nIt's at moments like this that you seek consolation in literature. I made myself a cup of tea and reached for my exercise book.\n\n# The Splattered Heart   \nChapter 5\n\n_Christmas at Holty Towers was an orgy of gluttony and conspicuous consumption which Gina found dangerously tempting absolutely disgusting. Mrs Sinclair Sinster gave Mr Sinclair Sinster a yacht a private jet a Rolex watch a silver hip flask set of golf clubs, although he already had four sets, because he already also had everything else._\n\nActually, let's face it, I have no idea what those sort of people would give each other. Although the Sinclairs weren't the super-rich Sinsters of _The Splattered Heart_ , Ben and Stella were their only grandchildren, and they did tend to go overboard on the gifts at Christmas. Stella accepted everything with effusive thanks and, when she was old enough, wheedled the receipts out of the donors and took the items back to exchange for the things she really wanted. Ben accepted everything guiltily and donated the unwanted gifts to the Animal Sanctuary, where he'd developed a special relationship with a rescued donkey called Dusty. Ben and Stella; so dear, so different. I closed up my exercise book and lay quietly in the dark, calling up their faces into my mind, missing them.\n\nThe day before New Year's Eve, the phone rang at about five minutes to midnight. It hauled me up abruptly out of a deep groggy sleep. I fumbled for the receiver and the bedside light, and managed to knock my glass of water on to the floor.\n\n'Hello?'\n\n'It's me.' The voice sounded muffled and squeaky.\n\n'Who's that?'\n\n'It's me. Ben.'\n\n'Ben! Whatever's the matter? D'you know what time it is?'\n\n'Mum, will you be in tomorrow? I'm coming home. I forgot my key.'\n\nHis voice sounded unfamiliar \u2013 slightly croaky, with a touch of London that I hadn't noticed before.\n\n'Of course. But I thought you were staying until after New Year.'\n\n'I was. But now I'm coming back tomorrow. The train gets in at ten past three.'\n\nThere was just the hint of a tremor as he spoke. If I hadn't been his mother, I wouldn't have noticed it.\n\n'Do you want me to meet you at Paddington?'\n\n'No, it's okay. I'll get the bus.'\n\n'Is everything all right?'\n\n'Yeah. Fine.'\n\n'But why...?'\n\n'I'll tell you when I see you.'\n\nClick.\n\nAfter that, it was at least an hour before I could get back to sleep. Something must have happened, I thought. There must have been a row.\n\nIn fact it was about half past four by the time Ben got back next day. Either the train was late, or there'd been no bus. I found myself glancing at the clock, waiting with the same anxious eagerness as I'd once waited for Rip to come home after a business trip. Then the doorbell rang, and there he was, my boy, standing on the doorstep in the wintry dusk, with his bulging backpack and a carrier bag in each hand. My heart bounced with joy, even though it'd been only just over a week since we'd said goodbye.\n\n'Hi, Mum.'\n\n'Hi, Ben.'\n\nHe dumped his bags down in the hall and stood there, grinning stiffly with his arms by his sides while I hugged him, tolerating this embarrassing ritual, but not actively taking part. He looked both thinner and taller, as if he'd sprouted up an inch or two in the last week. There was a shadow of a moustache on his upper lip. His hair had grown, too, and he had it tied up in a little red kerchief knotted behind his ears, pirate-style. This was new.\n\nHe'd only taken the backpack when he went away, so the extra stuff in the carrier bags must have been presents. There was even a present for me from the Sinclairs \u2013 an enormous box of Belgian chocolates, a bit similar to the one that I'd sent up for them, but bigger and more expensive.\n\n'How was your Christmas?' I asked.\n\n'Fine.'\n\nThere was something scarily grown-up about the way that Ben had handled the separation between me and Rip; it filled me with admiration and awe. He never played us off against one another \u2013 he was fiercely loyal to both of us. But I was burning with spiteful not-grown-up curiosity to find out what had happened at Holtham at Christmas.\n\n'So what made you come back early?' I said it very casually.\n\n'Oh, I just got fed up.'\n\nI might have believed him and just left it at that, but I remembered the phone call, his trembling voice at two minutes to midnight. That was more than just fed up.\n\n'And Stella? Was she there?'\n\n'Yeah. But then she left. I think she went to stay with her boyfriend.'\n\nI'd sent a present for her, a hand-made silk shawl in different shades of rose \u2013 she would look lovely in it \u2013 it was her colour. I was hoping she'd ring, but all I'd got was a text message. _Thanx mum great prezzy happy xmas c u soon xxx._\n\nAlthough I'd left him a message before Christmas, it wasn't until the morning of New Year's Eve that Mark Diabello had phoned me back. I remembered I'd been trying to get to the bottom of Mrs Shapiro's turned-off water, and I was sure that either he or Nick Wolfe was responsible.\n\n'Mrs Sinclair. What can I help you with? Did you see your aunty over Christmas?'\n\nSo, okay, I hadn't been quite truthful either.\n\n'Look, Mr Diabello, I just want to know what's going on. You offer Mrs Shapiro half a million for her house. Then you up it to a million, just like that. Then your partner offers her two million.'\n\nThere was only a second of hesitation.\n\n'With a unique property like this, Mrs Sinclair, it's difficult to arrive at an accurate evaluation, because there's nothing out there on the market to compare it with. At the end of the day, the market value is \u2013 how can I put it? \u2013 whatever the highest bidder will pay. That's why I suggest we float it on the market and see what offers come in. Does that make sense?'\n\nActually, it sounded pretty plausible.\n\n'Then he goes round in the middle of the night and turns the stopcock off.'\n\n'Nick did that?'\n\n'I'm sure it was him. He'd been round there the same morning, plying Mrs Shapiro with sherry.'\n\nA pause.\n\n'I don't think you should jump to any conclusions, Mrs Sinclair. Do you mind if I call you Georgina?'\n\nDid I mind? Didn't I mind? I couldn't hear myself think above the chatter of my hormones.\n\n'I'll have a word with him if you like. Sometimes he... he does get a bit carried away. He falls for a property, and he forgets that it belongs to somebody else.' He hesitated. His voice changed. 'You know, this may surprise you, Georgina, but being an estate agent is a labour of love. You go into this game because you're passionate about property. The elegant terraces, the cosy cottages, the grand mansions and the stylish apartments \u2013 each property is a life to be lived \u2013 a dream come true for someone. Our job is to match the dream to the property.'\n\n'So now you deal in dreams?' I was trying to sound hard-headed, but as he spoke, I was thinking, there's something exciting about black treacle \u2013 it's subtler, more complex than bland sugary golden syrup.\n\n'We try to make dreams come true, Mrs Sinclair.' There was a breath like a sigh on the other end of the phone. 'But you spend most of your time flogging ex-council maisonettes to people who dreamed of something better, and converted buy-to-lets to amateur landlords who want to make a quick buck. Your passion goes cold; you just keep doing it for the money. Then once in a while something really special comes along, something you can lose your heart to. And your brains. Like Canaan House.'\n\nAs I said, I'm not a woman who automatically thinks of sex when she talks to a man, but Mr Wolfe seemed to have started a trend, and I found myself wondering what it would be like with Mr Diabello. And, mmm, I have to say, it was quite a lot nicer. But \u2013 I shushed my revving hormones \u2013 he _was_ still an estate agent, and probably a crook.\n\n'It's not a property \u2013 it's a home. It's not for sale,' I snapped.\n\nIt wasn't until he'd put the phone down that I realised the disjuncture in what the two of them had been saying. Mark Diabello had been talking about selling the house at its market value, whatever that was. But Nick Wolfe had wanted to buy it.\n\n'What are you doing for New Year's Eve, Mum?'\n\nBen came and sat down on the arm of my chair, interrupting my thoughts.\n\n'I don't know \u2013 I hadn't thought about it. It's tonight, isn't it?'\n\nIf Christmas is a time when families get together, New Year's Eve is a time for celebrating friendship \u2013 and most of my friends were up in Leeds.\n\n'I haven't made any plans, Ben. We could cook something special, crack open a bottle of wine, watch the celebrations on TV. What would you like to do?'\n\nHe shuffled about on the arm of the chair.\n\n'I was wondering about going out with some mates from school...'\n\n'Yes, do that. I'll...' My heart leapt. I thought fast. '... I'll go and see Mrs Shapiro.'\n\n'... but I'll stop in if you want. If you're going to be on your own.'\n\n'No, no. Go for it. That's great.'\n\nI didn't want him to guess that my heart was crowing. He had friends \u2013 he was part of a crowd \u2013 my poor broken-in-half boy \u2013 he'd spend New Year's Eve getting drunk and throwing up in the gutter, not sitting at home in front of the TV with his mum.\n\n'Mrs Shapiro and me \u2013 we'll down a bottle of sherry and sing raucous songs. It'll be a ball.'\n\nActually, I was thinking, I'd be happy to have a break from Mrs Shapiro and her smelly entourage, and spend the evening in on my own.\n\nThen at about six o'clock the phone rang. My heart sank. I was sure it would be Mrs Shapiro. But it was Penny from _Adhesives_.\n\n'Hiya, Georgie \u2013 have you got any plans for tonight?' she boomed. 'I'm having a bit of a bash round at my place. Some of the work gang'll be there. Just bring a bottle, and your dancing shoes.'\n\nShe told me the address, just off Seven Sisters Road. I hadn't realised she lived quite close by. I wondered briefly what to wear, then I remembered the green silk dress. I had intended to get it dry-cleaned, but what the hell. \n\n# 21\n\n# The Adhesives party\n\nI could hear the music as I turned the corner into the street. Penny greeted me at the door with a hug, helped me out of my coat and took the bottle of Rioja out of my hand. She was petite and curvaceous, in her mid-forties I would guess, wearing a short black skirt covered with swirls of sequins and a low-cut red top that plunged right down to her bra. Her short curly hair was dramatically bleached and fluffed up on top of her head, making her look like a buxom elf.\n\n'Thanks for inviting me, Penny. It's great to meet you at last.'\n\nI kissed her on each round warm cheek and followed her through into a room where the lights were turned off and a PA in the corner was pumping out such a volume that I had to put my hands up to my ears. The room was packed with people all swaying and shuffling and the air was thick with several types of smoke.\n\n'They're all in here.' Penny was swaying her hips as she talked. 'Nathan's brought his dad.'\n\nShe gave me a little shove. I lurched forwards. I hadn't really been feeling in a party mood, but suddenly the atmosphere caught me, and shuffling in time to the beat I worked my way through the press of bodies further into the room.\n\n'This is Sheila.' Penny introduced me to a girl of about Stella's age, wearing a little strip of red satin \u2013 the minimum amount of material that you could call a dress \u2013 and smooching with a young black guy, about six feet tall, slim and gorgeous. He was holding a wine glass in one hand and a cigarette in the other. There was a lot of hip-thrusting going on. Penny pushed past them and led me deeper into the room.\n\n'Over there, that tall guy. That's Emery, one of the freelancers on _Prefabrication_. I told you about his little operation?' she whispered.\n\n'No, er, what...?'\n\nI wondered what she'd told them about me.\n\n'Here, meet Paul. Paul, this is Georgie. You know, from _Adhesives_.'\n\nPaul was slightly built with a shy stoop and a yin-yang tattoo on his forearm.\n\nHe nodded in my direction and carried on dancing, mesmerised by the tiny dark girl spinning her torso in front of him. When I turned round again Sheila had disappeared, and the slim gorgeous guy was thrusting towards me. I felt my knees droop and my pelvis liquefy but somehow the rhythm got hold of my feet, and I found my hips doing unfamiliar gyrations. He moved in closer.\n\n'Hi, beauty. I'm Penny's cousin,' he shouted above the boom of the music. 'Darryl Samson. I'm a doctor.'\n\nHaving a doctor like that would be enough to keep anybody in bed, I thought. A bit different to seedy Dr Polkinson at the Kippax surgery.\n\n'I'm surprised any of your patients bother to get better.'\n\nHis laugh was deep and juicy.\n\n'I'm Georgie. I'm a... writer.'\n\n'No kiddin'!'\n\nI could feel his hips \u2013 and not just his hips \u2013 pressing up close against me. Then Penny appeared at my side, grabbed me by the hand and pulled me away.\n\n'Come on \u2013 you need a drink.' She threw Darryl a warning look, and he spread his palms with an apologetic smile.\n\n'Take care with that one. He's my sister's brother-in-law. Don't believe anything he tells you.'\n\n'Is he a doctor?'\n\n'Ha!' She threw her head back. 'I've had a few complaints. He told Lucy he was a gynaecologist. And she believed him.'\n\nWhen I looked back, he was moving across the floor with the same languid insolence as Wonder Boy, thrusting himself in between Paul and the girl with the spinning torso, and in no time they were grooving together, pelvis to pelvis. I stood in the drinks room clutching my glass of red wine and feeling mildly annoyed with Penny, when suddenly she dived into the crowd and pulled someone else towards me. 'Georgie, here's someone you gotta meet.'\n\nI stared. This was incredible. Horn-rimmed glasses. Deep blue eyes. Dark hair swept back from brainy forehead. Yes, definitely hunkily intelligent \u2013 all he needed was a white coat. And maybe a few inches. Okay, he was a bit short \u2013 but did that matter? Was I so shallow that I couldn't fancy a man half an inch shorter than me? I was pondering on this when the small intelligent hunk stretched out his hand.\n\n'Hi. I'm Nathan.'\n\n'I'm Georgie.' I felt myself blush. 'Good to meet you at last.'\n\n'The Chattahoochee Rose.'\n\n'What?'\n\n'Georgia. You know, on the Chattahoochee River.'\n\n'Oh. Geography's not my strong point,' I mumbled. Already I'd revealed myself as an ignoramus. I noticed he was wearing a midnight-blue silk shirt that matched his eyes, and that the dark designer stubble that shadowed his chin and jaw was attractively flecked with silver.\n\n'Awesome dress.'\n\n'Thank you. It came from...' There was a small vomit stain on one sleeve, but probably he hadn't noticed.\n\n'I've been looking forward to meeting you, Georgia.' That low, confiding voice, with maybe just a touch of the mid-Atlantic about the vowels. I realised that our only topic of conversation over the years had been glue. Should I mention my thoughts about polymerisation?\n\n'Me, too. I was thinking about what you said...' I remembered his New Year's joke. Glue and a screw. No, that wasn't the right way to begin. 'I mean, after all these years. You know, talking about adhesives over the phone. I thought you must be...' No, that wasn't right, either. I blushed.\n\n'Mr Bond?'\n\n'Something like that.'\n\nThen an elderly man I hadn't noticed before, thin and wiry, with a bushy white beard and a glass of red wine in his hand, moved in beside me.\n\n'Aren't you going to introduce me to your young lady, Nathan?'\n\nI thought I saw a quick glimmer of annoyance flash through Nathan's eyes, but he just said, 'Tati, this is my colleague Georgia. Georgia, meet my father.'\n\n'Georgia! Aha! State or republic?'\n\n'Er...' Was this another geography quiz? I hadn't done geography since I was fourteen. At Garforth Comp in those days you had to choose between history and geography. I felt myself turning pink under Nathan's curious gaze.\n\nI was saved by the chimes of Big Ben. The lights came on. Corks popped and everyone held their glasses out. Nathan grabbed a bottle and topped us both up. I took a great gulp that went straight to my head. Putting his glass aside, the old man crossed his arms, took my left hand in his right one with a surprisingly firm grasp, and reached the other hand out to Nathan. Then he took a deep breath and started to sing. _Should auld acquaintance be forgot_... The room went still... _and never brought to mind_... His voice reverberated unexpectedly deep and mellow. What happened next was a bit like polymerisation \u2013 suddenly the individual people-molecules milling about in the room grabbed hands and formed a long covalent chain. Soon we were all holding crossed hands and swaying, everybody kissing everybody. I even got a quick snog with Darryl. That was nice. Then Sheila pulled him away and the old man pushed in and covered my face with his bristles. He started kissing me vigorously, a whiskery, spicy kiss \u2013 vindaloo. I struggled, but his grip was tight. Nathan came to my rescue.\n\n'Happy New Year, Georgia,' he murmured, as though it was our special secret. For a moment, he held me in his arms. Our lips met. The room started to spin. But the old man squeezed in between us, coming in for another round, so I pulled myself away, grabbed my coat from the pile in the other room, and was out in the street in a flash.\n\nIt was incredibly cold. I started to run. The streets were full of revellers, and the sky was full of stars.\n\nThe house, when I got home, was quiet, dark, and warm. I didn't put the lights on. I flung off my coat and shoes, lay down on the bed, and almost immediately fell asleep. I woke up two hours later feeling cold, with a disgusting taste in my mouth. It was a mixture of rough wine and vindaloo. But it set me thinking how long it was since I'd been kissed. Actually, it had done me good. I should get out more often.\n\nI had a wash, cleaned my teeth, put my nightie on, and went back to bed. I tried to call Ben, but his mobile was switched off. I suppose he didn't want his mum ringing to embarrass him. I drifted off to sleep wondering where he was, and thinking of New Year's Eve in Kippax in 1980, when I'd snogged Karl Curry, and wondering where _he_ was now.\n\nI woke up again just after dawn and wandered across the landing to see whether Ben was back. The curtains were drawn and the light was out. The air had a musty smell of sleep and old socks. But he wasn't in his bed. A red light was flashing on his computer \u2013 it was the screen saver whizzing about \u2013 a garish vertigo-inducing pattern of white-and-red swirls. I went to shut it down, and as I touched the mouse, the screen he'd been looking at came up.\n\nI remembered it was the same red-on-black text as before. This time, the single word flashing in red on black in a circle of dancing flames was _Antichrist_. What was this rubbish he was looking at? Out of curiosity, I hit the 'back' button, and found myself in some sort of chat forum. There were only two names: Benbo and Spikey.\n\nSpikey: hey benbo happy newyear this is the year of antichrists rein watchout\n\nBenbo: who do you think is the antichrist putin or bush?\n\nSpikey: putin is the king of the north who will join forces with the king of the south at the battle of armagedon daniel 11:40\n\nBenbo: where is armagedon?\n\nSpikey: its in the north of isreal\n\nBenbo: phew quite a long way from highbury who is the king of the south?\n\nSpikey: gadafi or sadam hassain or osama binladin take your pick\n\nBenbo: do you think obl is still alive?\n\nSpikey: check out _<http:\/\/www.dramusic.com\/endtime\/prophesies\/obllives.html>_ he has gout in his toes but is ok apart from that\n\nBenbo: i think saddam is still alive did you notice something wird about those hanging photos the angle of the head is wrong and the eyes when someone is hanged their eyes bulge out from the pressure but saddams eyes look normal i think the head has been copied and pasted from a different photo\n\nSpikey: your right if the pictures are fake mybe the execusion was fake too have you seen _<http:\/\/www.saddamhusseinlives.com>_?\n\nBenbo: i read somewhere that prince charles is the anitchrist because of the duchy of cornwall bar codes\n\nSpikey: 666 is the mark of the beast check this link Antichrist\n\nBenbo I supposed was Ben. How did he know so much about hanging? But who was Spikey? Whoever he was, I didn't think much of his spelling.\n\nI clicked on the last link, which took me to the webpage of someone who called himself Isiah. He was a middle-aged man with a crew cut, drooping eyelids and a chunky wooden cross on a chain around his neck. Beneath the picture was a banner heading:\n\n# WHO IS THE ANTICHRIST?\n\nMany Christians used to believe that Communism was the Antichrist, and _Armageddon_ would be nuclear war between Russia and America. However, it seems that now the forces of Islam and Christianity are lining up for a definative battle before the third Temple is rebuilt in Jerusalem and Christ comes back to rule the earth in all His power and glory.\n\nInfact all the signs are that the Antichrist, Satan the great Deceiver, is already stalking the earth. 'Take heed that no one deceives you. For many will come in My name, saying, \"I am the Christ,\" and will deceive many.' (Matthew 24: 4\u20135)\n\nIn the Book of Revelation the _Mark of the Beast_ is revealed as 666.\n\nI rubbed my eyes. It was too early in the morning for this sort of stuff. But I was curious about how Ben spent his hours cloistered up here. There was a list of names, each underlined with a link and marked with a little flaming crest.\n\n_Osama Bin Laden_\n\n_Saddam Hussein_\n\n_Pope Benedict XVI aka Joseph Ratzinger_\n\n_Vladimir Putin_\n\n_Prince Charles of Wales_\n\nI opened the last link.\n\nThis English aristocrat is a surprise candidate \u2013 but look at the evidance. His full official name both in English and Hebrew adds up to _666_ as described in the ancient Hebrew Gematria, and his heraldic symbols are based on the _beasts_ of Daniel and Revelation. Also, he really is a prince, as predicted in Daniel 9. Rome is obviously the new Babylon, and the evil _European Union_ is the new Holy Roman Empire. It's constitution is under discussion, and Prince Charles could one day be it's ruler. Infact the fact that he seems unlikely is the strongest argument in his favor, because as the Bible tells us in Revelation 12: 9 'The Devil and Satan deceives the whole world.' Check out _www.greaterthings.com\/News\/PrinceCharles\/index.html_.\n\nUp to this point I'd been reading with a kind of fascinated horror, but the bit about Prince Charles made me laugh out loud. Poor lad, I thought. And the grammar and spelling. How could anyone take seriously anything spelled infact, definative, evidance? I must definately (ha ha) pull Ben's leg about this. Out of curiosity, I clicked on the 666 link.\n\nThe Mark of the Beast may already be in your home. Take a look at the bar code which is on every product you purchase. You may have bought goods marked with the Beast's sign 666 including products sold from Prince Charles's own sinister Duchy of Cornwall brand. Check out _www.av1611.org\/666\/barcode.html._\n\nSmiling to myself, I clicked on Start, Shutdown, then I went downstairs and put the kettle on. When I took my coffee through to the sitting room, I found Ben there, asleep on the sofa, clutching a large traffic cone to his chest, dead to the world. He stirred and opened his eyes.\n\n'Happy New Year, Mum.'\n\n'Happy New Year, Ben. What's with the traffic cone?'\n\nHe looked down at his chest, and shook his head in surprise.\n\n'I've no idea, Mum.' He grinned sleepily. 'Absolutely no idea.'\n\nBefore I could ask him about the webpages, he'd drifted off to sleep again, his feet sticking over the end of the sofa, the traffic cone still cradled in his arms.\n\nThe light was flashing on my answering machine.\n\n'Georgia. It's Nathan. Tati says sorry about last night. He gets a bit carried away when he's had a drink. Hope you got home all right. Happy New Year.'\n\nI was going to ring him back, but I would probably end up making a fool of myself. Quit while you're ahead, I thought. Instead, I phoned Penny and left a message on her answering machine.\n\n'Great party. Thanks.'\n\nThat was it, then: Christmas and New Year, the festive season over. I'd survived.\n\n# 22\n\n# Changing the locks\n\nOne of the hardest things I found, after Rip left, was sleeping by myself in that great empty bed. In the day I could keep myself busy, but at night the hours seemed to swell and expand, losing their definition. It wasn't just sex I missed, it was having someone warm to cuddle up to, a solid presence beside me on the cruel nightmare ride from dusk to daybreak. Sometimes I would wake to find myself snuggled up to the spare pillow, my arms and legs locked around it.\n\nAbout three weeks into the New Year I came downstairs very early in the morning to make myself a cup of tea after a restless sleep. I'd woken up before dawn to find my pillow wet with tears. I could remember nothing about my dream except a faceless malevolent shadow dragging towards me. Somewhere in the still-dark streets a siren was wailing, a persistent, unsettling call like a sinister bird of the night. It was cold, the central heating hadn't come on yet. I shivered as I poured the tea, and was about to go back to bed when the phone rang. It was Mrs Shapiro.\n\n'Georgine \u2013 please come quick. There is a burglary. Door is brokken.'\n\nFeeling mildly irritated, I got dressed, put on my coat and went around straightaway. It had started to snow \u2013 not proper snowflakes, but miserable powdery stuff flaking down out of the sky like frozen dandruff. Mrs Shapiro answered the door wearing her pink dressing gown and _Lion King_ slippers, her hair dishevelled, lipstick smeared on hastily. Violetta was hanging around, miaowing at her feet. She led me through to the kitchen. It was bitterly cold. One of the pretty Victorian blue glass panels on the back door had been smashed and an icy draught was whistling through. The key on the inside had been stolen. Nothing else seemed to be missing.\n\n'Maybe it was your Peki. Maybe he is a teef.'\n\n'Why would it be him?' I couldn't keep the irritation out of my voice. 'He didn't even charge you for coming out last time. He didn't steal anything, did he? You should be grateful, Mrs Shapiro, but all you do is moan.'\n\nOkay, I know it wasn't a very nice thing to say, but I wasn't feeling very nice.\n\n'Hm. But if not the Peki who can it be?' She gave poor Violetta a petulant little kick and shuffled across to put the kettle on.\n\n'It could be anybody. A burglar or anybody.' I saw the look of terror flit across her face, and wished I'd held my tongue. I hadn't told her that Mr Ali had already changed the lock once \u2013 I hadn't wanted to alarm her. But now I was alarmed myself.\n\n'But why they want to frighten me? Why they don't come into the house? Why they just tek the key?' She looked as if she was working herself up into a state.\n\n'It might be someone who's planning to come back.' It was hard to imagine the sheer malevolence of someone who would terrorise a defenceless old lady in her own home. 'Listen, you'd better get the glass mended and the lock changed today. You should call Mr Ali. Unless you know of anybody better.'\n\nShe started poking around in her disgusting cupboards looking for the pond-water tea. Her back was turned towards me.\n\n'He is a clever-kn\u00f6del, this Peki,' she muttered.\n\nIrritation and concern vied for dominance in my mind, and irritation was getting the upper hand. She poured the hot water into a jug and dangled a limp greyish tea bag in it by its string. After a moment, she looked up at me and said, 'But I think I will ask Mr Wolfe. My Nicky.'\n\nThen she gave me a sly little grin as if to say, I may be eighty-one but I can still wind you up. And she succeeded.\n\n'That's fine. Just the job. You and your Mr Wolfe can sort it out. I don't know why you bothered to call me at all.'\n\nAll of a sudden, my annoyance overwhelmed me. I stood up abruptly and made for the door. I'd had enough of her constant demands and her petty prejudices and her silly mysteries. I couldn't stand the stink in her house a moment longer, and I certainly didn't want to sit there in the cold, drinking her weak pondy tea when I'd left my own cup of tea to go cold in the kitchen. Let her sort herself out, I thought. I wanted to get back to my bed.\n\nOnce home, I heated the tea up in the microwave and climbed into my bed fully clothed. Outside the window, a feeble dawn was just breaking through a bruised purple sky, with long red streaks like bleeding cuts smearing the surface of the clouds. I pulled down the black knickers over my eyes to keep the light out, and tried to will myself back to sleep, but I was too wound up to drift off, and too tired to get up. That dream or nightmare that had woken me was still pushing at the edges of my consciousness \u2013 the malevolent figure with a blank eyeless face. I shuddered. For some reason I remembered the website Ben had been looking at \u2013 the Antichrist, the Deceiver, stalking the earth unrecognised, spreading evil and fear. It didn't seem quite so funny now.\n\nThen the phone rang.\n\n'Don't be engry mit me, Georgine. I am only jokking. I am an old woman. Please, telephone to Mr Ali. I heff lost the number.'\n\n'Okay, okay.'\n\nShe phoned me back a few hours later to tell me that Mr Ali had been and boarded up the back door and changed the lock. He had put a new mortise lock on the front, in addition to the Yale, and had fitted bolts to both doors.\n\n'You will be as safe as prison,' he'd said.\n\n'How much did he charge you?' I asked.\n\n'I give him ten pound. Plus he mek me pay full price for locks and bolts.'\n\nShe said it with a grumble in her voice, as though she felt she'd been overcharged.\n\n'You should be grateful,' I said, though she clearly wasn't.\n\n'You are still engry mit me, Georgine, isn't it? Don't be engry. You are the only friend I heff.'\n\n'No. I'm not angry, Mrs Shapiro.'\n\nAnd it's true, I wasn't angry with her any more. I had other things on my mind.\n\nRip had returned from a business trip and had phoned around lunchtime to say he was coming to pick Ben up after work tomorrow. Even after all this time, his phone calls still agitated me. I needed time to get myself into the right frame of mind to face him on the doorstep. Upstairs I could hear the thud-thud of footsteps followed by the thud-thud of music \u2013 Ben's morning getting-up rituals, though it was well past midday. That boy could sleep for England. Something else \u2013 I still hadn't found out what had happened at Holtham at Christmas.\n\nThe doorbell rang a bit earlier than I'd expected on Monday afternoon. I went to answer it with my ready-for-anything smile fixed on my face. But it wasn't Rip on the doorstep, it was Mark Diabello. His black Jaguar was parked by the gate, and he was smiling a ready-for-anything smile, too.\n\n'Hello, Mrs Sinclair. Georgina.' The deep creases in his rugged cheeks crinkled craggily. 'I hope you don't mind my dropping round like this. I've been following up on some of the concerns you raised in our last chat, and I wanted to bring you up to date.'\n\nMaybe if I hadn't been expecting Rip to appear at any moment I wouldn't have asked him to come in. But it seemed too good an opportunity to miss.\n\n'That's kind of you, Mr Diabello. Can I offer you a coffee?'\n\n'Call me Mark, please.'\n\nHe followed me inside, looking around him as I led him through to the sitting room.\n\n'I showed a client round this place when it was first on the market. You've done wonders to it, if I may say so. Added all your little feminine touches.'\n\n'Thank you.'\n\nAs far as I was aware, I'd added no touches to it whatsoever, apart from unloading my furniture and hanging some curtains up.\n\nI positioned him on the sofa by the bay window, where he could be seen from the road. Then I put the kettle on and spooned some coffee into the cafeti\u00e8re.\n\n'Milk? Sugar?'\n\n'Black with four sugars.'\n\nI laughed. 'It'll taste like black treacle.'\n\n'Mm. That's how I like it.'\n\nHe must have noticed that I kept glancing towards the window because he said, 'I hope I'm not making you nervous, Georgina.' Black treacle with a hard mineral edge.\n\n'No, not at all,' I blustered, feeling intensely nervous.\n\nThen a car horn beeped outside \u2013 I recognised the distinctive note of Rip's Saab.\n\n'Please excuse me.' I went to the bottom of the stairs and shouted, 'Ben! Rip's here!'\n\n'Coming!'\n\nA moment later Ben appeared, with his shoelaces still undone, his shirt hanging out, and his big backpack over his shoulder. God knows what he carted around in it because he always seemed to wear the same clothes. I went out to the car with him, my ready-for-anything smile fixed in position. But Rip just pulled the inside lever to open the boot and sat in his Saab, waiting for Ben to put his backpack in. He didn't even wind the window down. I couldn't even tell whether he'd noticed the black Jag or the man sitting in the window. I wanted to hammer on the window with my fists, I wanted to kick in the glossy, dark green door panels. But Ben was waving goodbye, so I blew him a kiss and went back inside, slamming the door.\n\nMy face must have been livid when I returned to the sitting room, for Mr Diabello gave me a sharp look and said, 'All going to plan?'\n\n'Not exactly.'\n\nHis left eyebrow lifted a fraction, and his cheeks tightened, and I realised from that look that he had understood everything about my situation. I blushed as if he'd walked in on me naked in my bedroom. He was a man, I remembered with a shiver, who could read people's dreams.\n\n'Want to talk about it?' His voice oozed sympathy. 'I can recommend a good solicitor.'\n\n'No. No, it's not at that stage yet.' As I said the words I realised that probably it was at that stage, and probably I did need legal advice. But the thought of a friend of Mark Diabello's crawling all over the intimacies of my life made me cringe. 'Just tell me what you came to tell me.'\n\n'Yes \u2013 you were concerned that my partner, Nick Wolfe, might be behaving... how can I put it?... improperly.'\n\n'Harassing an old lady in order to force her out and get possession of her house.'\n\nMy coffee had gone cold, but I sipped it anyway, to avoid looking at him. His gaze was making me feel uncomfortable and sweaty, like sitting under a spotlight. I could feel my cheeks going pink.\n\n'I've had words with Nick. He admits he's fallen for the house, and has maybe been a bit too... er... enthusiastic in approaching Mrs Shapiro. But he denies absolutely having done anything improper.'\n\n'But he admits to plying her with sherry. Hoping she'd sign a bit of paper that he just happened to have in his briefcase?'\n\nHowever annoyed I got with Mrs Shapiro, I wasn't going to stand by and let these two shysters take her to the cleaners.\n\n'I think the sherry was meant as a goodwill gesture. A gift. He didn't mean her to open it up and start drinking it straightaway. That was her idea. By all accounts, she was giving him the eye.'\n\n'Oh, come off it! She's eighty-one. Anyway, why would he bring her a gift?'\n\n'A token of appreciation for a valued client.'\n\n'But she's not a client. He just turned up at her hospital bedside.'\n\n'From what Nick says, she was a willing party. More than willing. Positively eager. He also told me, by the way, that she's not in fact your aunt.'\n\nHe looked up at me from lowered eyes, a small smile playing around his... how would you describe his lips? Not full and sensuous. No. But definitely... kissable.\n\n'Okay, so I made that up. But it doesn't change anything.'\n\n'It does raise the issue of what _your_ interest is in the property.'\n\n'I haven't any interest. I just don't want to see an old lady be ripped off. Someone must have told him about the house.' Then I realised. ' _You_ must have told him.'\n\nOur eyes met. I noticed for the first time that his were not brown, as I'd previously thought, but dark sea-green, with sparks of gold and obsidian in their depths.\n\n'I did mention our conversation to him. I didn't expect him to get quite so excited about it. He's a very passionate man, you know. It's our motto at Wolfe & Diabello. Passionate about property.'\n\n'Passionate' \u2013 there was something about the way he lingered over the word.\n\n'He thought Mrs Shapiro deserved a more focused view of his services?'\n\n'Exactly so.'\n\n'Like me?'\n\n'That's up to you, Georgina.'\n\n'Thank you. It's been nice talking to you.' I stood up abruptly, knocking my empty coffee cup over. He stood up, too, brushing past me as he made his way towards the door. I felt a shiver \u2013 or was it a shudder?\n\n'The pleasure's all mine,' he said.\n\nWhen I looked out of the window, I saw it was snowing again outside.\n\nAfter he'd gone, I sat down on the sofa and breathed deeply. _In \u2013 two \u2013 three \u2013 four. Out \u2013 two \u2013 three \u2013 four._ For some reason, my heart started to thump. Yes, I knew, in my sensible core, that the last thing I needed was a man like Mark Diabello in my life \u2013 a treacle-voiced estate agent with black and gold in his eyes. But I was unhappy and furious and needy. And it was so long since someone had looked at me with desire. And a little voice in the back of my head was whispering \u2013 why not? \n\n# 23\n\n# Stress fractures\n\nIt was still snowing that same powdery snow next day when I walked past the Islington branch of Wolfe & Diabello on my way to the bus stop. I'd been down to pick up a new laser cartridge, some more exercise books and a box of Choco-Puffs. (I think they're disgusting, but Ben likes them, and I'm in competition with whatever he gets in Islington.) I glanced in through the window and saw Nick Wolfe, bending over the desk of a young blonde woman who could have been a clone of Suzi Brentwood. On impulse, I pushed the door and went in. They both looked up as the door pinged.\n\n'Mr Wolfe. I'm glad you're in. Have you a minute?'\n\nThe blond stubble on his scalp gleamed as he straightened up. He led me into an office at the back and pulled out two chairs.\n\n'What can I do you for, Georgette?' He smiled wolfily.\n\nI explained my concern about the mains water tap and the back-door keys, keeping my voice carefully neutral and avoiding any hint of accusation.\n\n'You spoke to my colleague Mark Diabello about this, didn't you?'\n\nHis voice was slightly plummier than Mark's. I guessed he'd been to public school, whereas Mark had pulled himself up the hard way. Like me. He glanced pointedly at his watch. I ignored the hint.\n\n'What I can't understand is what you and Mr Diabello are up to.' I was smiling sweetly, looking him straight in the eye. 'He wants to sell it for half a million. Then he puts it up to a million. Then you go barging into the hospital with an offer of two million.' I spoke fast, conscious that his eyes were fixed on me in a not-very-friendly way. 'You must admit, it's a bit... worrying.'\n\n'Look, Mrs... Georgette. To be frank, I don't really know what it's got to do with you. It's up to Mrs Shapiro what she does with her house, isn't it? I understand she's not even related to you.' He glanced at his watch again. 'I made Mrs Shapiro what I consider to be a very fair offer. More than fair. Generous. I don't know what Mark told you, but let's get one thing straight.' There was a bullying note in his voice that made me flinch. 'Just because it's floated on the market doesn't mean it reaches its market price. Nor that the person who makes the initial purchase is the ultimate buyer, if you see what I mean.'\n\nWhat _did_ he mean? In the close space of the office, I could smell his musky aftershave, and beneath it, a strong almost feral odour that reminded me of Wonder Boy.\n\n'You mean Mark Diabello buys it for half a million, and sells it on to somebody else for two million, trousering the difference?'\n\n'I did not say that, Georgette.' He emphasised every syllable forcefully. 'That is not what I said.' He looked at his watch again, and then stood up. 'If you'll excuse me, I have business to attend to.'\n\nI stood outside on the pavement reeling. It had turned dark in the last half-hour, and a few stray snowflakes were spinning like scattered thoughts in the orange-tinged light. One or two of the shops had already closed up, but I noticed that Hendricks & Wilson was still open. Well, what did I have to lose?\n\nAlthough the two shopfronts looked similar from the outside, the interiors were startlingly different. Whereas Wolfe & Diabello had been all glass and chrome with laminate flooring and halogen lights, in the style of a city bistro, Hendricks & Wilson had red carpet and leather armchairs with brass wall-lights in the style of a gentlemen's club. I suppose it was meant to feel traditional and reassuring, but it just seemed ridiculously pompous in such a small space. A thin youth with spiky gelled hair was sitting at a computer, staring intently at the monitor. He looked up and smiled as I came in.\n\n'I'm looking for Damian,' I said.\n\n'That's me,' he beamed. His teeth were slightly crooked, and he looked reassuringly gormless. 'How can I help?'\n\nI hadn't really prepared what I was going to say, so I tried the familiar line about my aunty selling a house in Totley Place. I watched his face carefully, but there was no sign of recognition. It seemed that whatever Mrs Goodney had been planning, she hadn't put it into action. Maybe I'd frightened her off.\n\n'I think you need to speak to one of the partners about something like that. Would you like me to make an appointment?' He reached for a large red-bound desk diary.\n\nI hesitated. Did I really need any more estate agents in my life?\n\n'Couldn't you give me just a rough idea?'\n\n'Hm.' He chewed a fingernail. 'Tell you what \u2013 I'll drive past on my way home tonight and take a look.'\n\n'Thanks. I'll give you a ring tomorrow. Thanks, Damian.'\n\n'How did you know...?'\n\nI quickly made for the door.\n\nWhen I phoned Damian the next morning, I was even more convinced that he wasn't involved in the dirty tricks. He wouldn't give me a valuation, but he said, 'A big site like that in the heart of Highbury \u2013 it has development potential. You're talking millions. You'll have to speak to Mr Wilson.'\n\n'I don't think my aunty would want it to be developed. But thank you for your help.'\n\nI hung up quickly before he could ask me any questions.\n\nIf Damian wasn't involved, that meant it must be Wolfe & Diabello. Rage was burning in my head. I tried to calm myself down with Ms Baddiel's breathing exercises. _In \u2013 two \u2013 three \u2013 four. Out \u2013 two \u2013 three \u2013 four._ Wolfe & Diabello. What a pair of gobshites. I phoned their office \u2013 my hands were shaking so much I dialled a couple of wrong numbers before I finally got through. Neither of the partners was there. I left a message with Suzi Brentwood.\n\n'Please can you get one of them to ring me back. No, I can't say what it's about. Just tell them I know what's going on. Tell them they're a couple of sleazy double-crossing crooks.'\n\nIt was Mark Diabello who phoned me back, within ten minutes.\n\n'I got your message, Georgina. Strong language. What did we do to upset you?'\n\n'It's not what you did, it's what I did. I got another valuation.'\n\n'So you should, Georgina. And?'\n\n'And he said it was a development site with potential. He said it could be worth several millions.'\n\n'Who said that?'\n\n'Somebody. Somebody from Hendricks's.'\n\n'The office junior? They always make wild guesses.'\n\n'No. Someone highly qualified. And reputable. Not a conman like you.'\n\n'You're a very emotional woman, Georgina. I like that. But you've forgotten what I said.'\n\n'What did you say?'\n\n'I said I'd match any genuine valuation.'\n\nHad he said that? It's true, I'd forgotten.\n\n'But the other one \u2013 your sidekick \u2013 he offered her two million.'\n\n'I can't speak for my partner. But I said I'd match their valuation. I think you owe me an apology, Georgina.'\n\n' _I_ owe _you_ an apology?'\n\nI put the phone down. I was shaking. Then I thought back over our previous conversations. Yes, maybe I'd been a bit hasty. Even a bit rude. I remembered now, he _had_ said something about matching Hendricks & Wilson's valuation, but that had been in a different context. And it's true, Damian did seem to be the office junior. But what he'd said rang true. Actually, what all of them said rang true. That was the trouble. How was I to know who to believe?\n\n_'Stress fractures can occur in adhesive bonds when the materials have different coefficients of thermal expansion.'_\n\nI'd been staring at the sentence on my monitor for at least half an hour, as a cup of tea went cold on my desk, thinking maybe that's what had gone wrong between Rip and me. He's slow to get angry, but when he does, he stays hot much longer. I flare up quickly but quickly cool down again. My mind tripped back to that morning's conversation with Mark Diabello \u2013 yes, maybe I had flared up too quickly then. Maybe I should have given him the benefit of the doubt. What exactly _had_ he said? I couldn't remember. The glue had got to my brain.\n\nIt was time to break for lunch. I wandered over to investigate the fridge. There were two eggs, a slice of bread and the remains of a supermarket bag of rocket salad. In the door was an opened bottle of Rioja. Should I? Shouldn't I?\n\nI was whipping up some scrambled eggs, when the doorbell rang. Mark Diabello was standing on the doorstep with a bottle of champagne in his hand. It wasn't just any old supermarket champagne, either, it was Bollinger. Maybe it was just a trick of the light, but I could swear his eyes were smouldering. Deep sea-green, with flickers of obsidian and gold. Something in my heart did a funny little skip.\n\n'A token of appreciation for a valued client,' he murmured.\n\n'I'm not your client.'\n\n'But you could be.'\n\n'I don't think so. But come in.'\n\nI went and fetched two wine glasses from the kitchen. I didn't have proper champagne flutes. We clinked.\n\n'I like you, Georgina. You're different.'\n\nThe smile-creases in his cheeks deepened. My heart did that wayward skip again.\n\n'Have you come to offer me a more focused view of your services?'\n\n'Would you like that?'\n\nI didn't say yes. But I didn't say no.\n\nWe ended up in my bedroom. He led the way. Of course, he was an estate agent, he knew where to go. It all happened astonishingly fast, with the well-oiled precision of a top-of-the-range Jaguar. He gave me just the right amount of champagne, kissed me in just the right way, holding me firmly but carefully under the chin. Then at exactly the right moment one hand moved down from my chin to my left breast. The other worked its way up between my legs. There was something reassuringly impersonal about it all. His hands found their way unerringly to the right places. His fingers were strong and supple. There was no fumbling with clothes \u2013 they just fell away. His body was hard and hairy. He produced a condom from his pocket. If I'd had time to think, I might have thought \u2013 what the hell am I doing? But I didn't think about anything. My brain was full of bubbles. My skin tingled like electricity. My body purred in his hands. I'd like to say I recoiled in disgust at the sheer efficiency of it. But actually, it was fantastic.\n\nI can't remember what happened next \u2013 well, okay, I can, but I'm too embarrassed to write it down. Look, he was the only man apart from Rip I'd slept with in twenty years. It was as though I'd slipped out of my familiar skin and become a different person, someone whose body waved and fluttered like a piece of silk in a storm.\n\nAfterwards we lay together watching the shadows lengthen in the garden and he pulled me into his arms and stroked my hair, murmuring sweet meaningless words. Then he reached into the breast pocket of his jacket, which was hanging on the back of the chair, and passed me a clean white handkerchief.\n\nWe didn't talk much. It wasn't about us as people. He left before Ben got back from school. I thought I might feel dirty, or used, or disgusted with myself, but I guess I realised deep down that having sex with someone else was part of a sticky repair process I had to go through. What was it Nathan had said? You get better bonding with glue and a screw. Maybe there's something in that. After he'd gone, all I felt was a great mass of melancholy like a rain cloud swelling over my heart. I didn't want him to see me cry, but as I heard his Jag pull away I let the tears come. I couldn't even have said why I was crying, or what it was that had stirred up such a storm in me. Maybe the sex had just loosened me up so there was no rigidity to hold back the tears.\n\n*\n\nAbout half an hour later, I heard the door-click of Ben letting himself in. I dried my eyes, pulled my clothes on, and went down to greet him.\n\n'You okay, Mum?' He looked at me intently. 'You seem a bit sort of... weird.'\n\nThe scrambled eggs were still on the hob, yellow and congealed.\n\n'Weird in what way?'\n\n'Sort of hyper. Hyper-manic.'\n\n'It must be all the coffee I've been drinking. I'm having a sticky patch with _Adhesives_. Ha ha. How about you? How's life in...' (I censored a number of sarcastic epithets.) '... Islington?'\n\n'It's okay. Dad's a bit hyper, too.'\n\nHe poured the milk over the Choco-Puffs and sat down with his spoon.\n\n'Oh, is he?'\n\nI craved these snippets of information, but loyal Ben handed them out very sparingly.\n\n'He says he's starting a new project?'\n\nThere was that rising inflection in his speech again. I found it troubling. It didn't sound like my Ben.\n\n'Not the Progress Project?'\n\n'He says it's progressing to a higher level?'\n\n'Yes, he's always had high aspirations.'\n\nA sarcastic note must have crept into my voice. Ben's look warned me that I was in danger of transgressing the subtle boundaries he'd drawn up between his two worlds.\n\nThat night, after Ben had gone to bed, I poured myself a glass of wine and reached for my exercise book. They were banqueting again at Holty Towers.\n\n# The Splattered Heart   \nChapter 6\n\n_Spurned by her errant husband, heartbroken Gina at last found love fulfilment consolation in the arms of an itinerant mandolin player with obsidian cerulean sapphire amethyst jade lapis lazuli eyes._ (Thanks, Mr Roget.) _He brought her beautiful gifts \u2013 hand-embroidered Spanish underwear garters hankies mantillas._\n\nAs I closed up my exercise book an hour later, I realised that the wine bottle was empty, and I'd opened another one. This was no good. Maybe Ben was right \u2013 I should go easy on the Rioja. The house was full of silence. I listened. Faintly, I could hear a car passing on the road and the tick-tick-tick of the water in the radiators. That was all. Holty Towers, the ecstasy and drama, the sumptuous meals and mandolin music, was a world away. \n\n# 24\n\n# Experimenting with Velcro\n\nMark Diabello came back again the following Wednesday, this time without the champagne, but with a bunch of flowers \u2013 red roses \u2013 and a small gift box, which I took to be chocolates. I was waiting for him, wearing a rather revealing top, which I'd bought the day before, and some lacy panties under a sleek clinging skirt, which I'd also bought the day before. I caught sight of myself in the hall mirror, my flushed cheeks and brilliant eyes, and didn't recognise myself. I could feel myself starting to melt as he kissed me. It took us about five minutes to get from the doorstep to the bedroom.\n\nHe was already undressing me as we fell on to the bed, his hands working with the same target-driven efficiency. When his shirt came off I could smell his body, soapy, sweaty, musky, and another smell, faintly chemical and disconcerting. What was it? I pressed my face to his skin. Sulphur? Chlorine. And in a flash I was sixteen years old, back in the locker room at the International Pool in Leeds, locked in a cubicle with Gavin Connolly, locked in his arms, lost in love.\n\n'Have you been swimming?'\n\n'How can you tell?'\n\n'You smell of chlorine.'\n\n'Don't you like it?'\n\n'I do like it. A lot.'\n\n'I'm a diver. High board.'\n\n'That must be so terrifying!'\n\n'It is. You just have to shut your eyes and plunge.'\n\nI imagined him, hard and lean and straight as a pike, hurtling into the water. I shut my eyes and plunged.\n\n'Aren't you going to open your present?' he murmured.\n\nI reached for the little box that had slipped down at the side of the bed, and pulled the ribbon off. Something red and silky slipped out. I held it up. It was a tiny pair of panties, shiny red satin, trimmed with black lace. I stared. Blimey! Were they for me? I'd never owned anything like this before. I wasn't even sure I liked them.\n\n'Aren't you going to put them on?'\n\nI wriggled into them and felt them flutter like moth's wings against my thighs. There was something odd about them \u2013 the gusset \u2013 it was open. Surely that defeated the whole purpose...? What's the point of panties without a gusset?\n\nShe soon found out. Not me, not Georgie Sinclair, no, it was a different woman \u2013 someone sexy and shameless who frolicked around in red satin panties trimmed with black lace and an opening in the gusset, who smelled of sex, whose body melted like warm sugar in the arms of a dark handsome stranger who appeared on her doorstep and made love to her one afternoon.\n\nThe dark handsome stranger lay with the sexy woman, carrying his weight on one elbow. His other hand was exploring the opening in the gusset. She could smell the chlorine on his skin.\n\n'Look, there's something else in the box,' he said.\n\nThe sexy woman fumbled shamelessly in the box and pulled out \u2013 what the hell were they? Two loops of red padded satin trimmed with black lace. Garters? No, there was a Velcro fastening.\n\n'You naughty little slag,' he whispered. 'Let me...'\n\nHe leaned over her and fastened her wrists to the bedhead, pressing down on her, pressing all the breath out of her till she had to cry out. She came almost at once, before he'd even entered her.\n\nIt was warm and steamy in the locker room, and we were wet and slippery, and then we towelled each other dry and got into our damp chlorine-sticky clothes. What had happened to Gavin Connolly? What had happened to Georgie Shutworth? I couldn't help myself \u2013 I started to cry. Mark Diabello dabbed my eyes with his hankie and kissed me lingeringly on my throat and neck.\n\n'You're a very beautiful woman, Georgina. Has anyone ever told you that?'\n\nI wanted to believe him. I almost believed him; but a cool whisper in my head reminded me that he probably slept with dozens of women, and said that to all of them. Then something from another age stirred in my mind, Rip's voice, husky against my cheek: 'If ever any beauty I did see, which I desired and got, 'twas but a dream of thee.' How long ago was that?\n\n'You'd better go now. It's nearly four o'clock.'\n\n'What happens at four o'clock? D'you turn into a pumpkin?'\n\n'No, I turn into a mother.'\n\nJust after four, the key turned in the latch, and I did turn into a mother.\n\n'Hi, Mum.'\n\nBen flung his bag down and let me hug him, turning his head to one side. He looked tense and pale.\n\n'Everything okay?'\n\n'Fine. Cool.'\n\nHe wasn't looking at me. His eyes were fixed on the window.\n\n'Do you want a sandwich? Some Choco-Puffs?'\n\n'Nah. I'll just have water.'\n\nHe drank resting both elbows on the table, his brown curls falling across his eyes.\n\n'I've been feeling a bit sort of... weird.'\n\nI kicked Mr Diabello right out of my mind and sat down opposite him.\n\n'How do you mean, weird?'\n\n'I've been having these weird feelings.'\n\nI could feel my pulse starting to pound, but I kept my voice soft and easy.\n\n'What feelings, Ben?'\n\n'Sort of... liminal.'\n\n'Liminal?'\n\nI had no idea what he was talking about. I waited, listening.\n\n'Like we're living in liminal times. You can see it in the light, Mum \u2013 look \u2013 it's like it's seeping in from the edges of another world.'\n\nHe pointed at the window. I looked round. Between the houses, a low shaft of pinky sunlight was lighting up a bank of purple cumulus from below. The brick buildings and leafless trees were all backlit, cast in shadow, despite the vivid light. I could see what he meant \u2013 it did look unearthly.\n\n'It's winter, Ben. The sun's always low in the sky at this time of year. Further north, in Scandinavia, they don't have any daytime at all.'\n\nHe looked up with a flicker of a smile.\n\n'You're so literal-minded, Mum.'\n\nThe clouds rearranged themselves and the shaft of light disappeared, but still there was a fiery glow on the underbelly of the sky.\n\n'I keep having these feelings, like the world's going to end soon.' He paused, gulping a mouthful of water. 'Like we're coming to the end of time?'\n\n'Ben, you should have said...'\n\n'So I Googled _End of Time._ And that's when I realised it wasn't just me?'\n\nThat's what they do, his generation, I thought. They don't talk to their parents or friends like we did \u2013 they look on the internet.\n\n'There's like all these signs \u2013 predictions in the Bible about the end of time? Wars, earthquakes, floods, plagues and that \u2013 it's all starting to come true?' His voice was strained and crackly.\n\n'But you don't believe all that stuff about prophecies, do you, Ben?'\n\n'No, but... well, yes... I just think, like, if that many people believe it, there could be something in it?'\n\n'But those things \u2013 wars, earthquakes, floods, plagues \u2013 they've been happening since the beginning of recorded history.'\n\n'Yeah, I know, but it's all speeded up now. Floods and earthquakes \u2013 like, there's one every year. And AIDS, SARS, avian flu \u2013 all these new diseases. It's all started coming true. Like in the Bible, it's predicted the Jews'll return to Israel, and they did. You know, in 1948. After the Holocaust and that? That was the beginning of all the wars in the Middle East. The invasion of Lebanon. You can read it for yourself, Mum \u2013 it's all there in the Bible. And it's not just Jews and Christians? A lot of Muslims think their great prophet is coming? Like they call him the Last Imam?' The rising inflection in his voice seemed to challenge me to disagree.\n\nHow could I explain without sounding pompous that just because millions of people believe something doesn't make it true.\n\n'Why didn't you tell me, Ben, that you were having these feelings? Or Rip?'\n\n'I thought you'd think it was mad? You wouldn't listen? You and Dad \u2013 you never listen to anybody.' He dropped his voice to a mumble. 'Like, you're so sure you know everything already?'\n\nHe didn't say it as an accusation, but it stung like one. We were so preoccupied with our own lives and problems that we'd failed to hear our own son's cry for help.\n\n'I'm sorry, Ben. You're right \u2013 we don't always listen. D'you want to talk about it now?'\n\n'Nah, it's all right, Mum.' He grinned sheepishly, swallowing the rest of his water. 'I feel all right now. I think I'll have some Choco-Puffs.'\n\nAfter he'd gone upstairs, I sat in the kitchen with a glass of wine wondering where we'd gone wrong. We'd brought him up to respect difference \u2013 diversity. To disrespect someone's faith would cause offence. At his primary school in Leeds, Rip and I, like good middle-class parents, had cheered enthusiastically as the children celebrated Christmas and Eid and Diwali. All belief was equally valid. Christianity, Islam, Hinduism, Judaism, astrology, astronomy, relativity, evolution, creationism, socialism, monetarism, global warming, damage to the ozone layer, crystal healing, Darwin, Hawking, Dawkins, Nostradamus, Mystic Meg, they were all out there vying with each other in the marketplace of ideas. How was anyone to know which was true and which wasn't? \n\n# 25\n\n# The attraction between   \nadhesives and adherends\n\nSometime in the night it started to snow. When I pulled back my curtains in the morning everything was white, and I felt a sudden burst of happiness, like I'd felt as a child waking up on a snowy day. No school; snowball fights with my brother; tobogganing on a tea tray down the slag heaps. In those days, before the advent of four-by-fours and working online, snow meant holiday, anarchy, delight.\n\nIn the garden, even the horrible yellow-spotted laurel bush was touched with magic, the leaves and branches bowing gracefully under their overcoat of snow. I noticed a movement \u2013 it seemed like three small black creatures hopping along, then I realised they were black feet attached to a white body. Wonder Boy prowled along the edge of the garden, tiptoed across the lawn, and took up his position under the laurel bush, staring up at the house. He was reminding me I should drop in on Mrs Shapiro today.\n\n'Look, Ben,' I said, when he came down for breakfast. 'It's snowing. You could take the day off school.'\n\n'It's okay, Mum. I feel better today. I've got to work on my technology project. The bus'll be running.'\n\nHow did he get to be so sensible? I hugged him.\n\n'Take care.'\n\nAfter he'd left, I sat down and tried to concentrate on an article for _Adhesives_. 'The attraction between surfaces in adhesive bonding.'\n\n_'Powerful attractive forces develop between the adhesive and the adherend which may be adsorptive, electrostatic or diffusive.'_\n\nThere was something quite romantic, I was thinking, about those gluey time-enduring forces, bonds so strong that they could outlive the materials themselves.\n\nMmm. My mind started to drift. It was no good. _Adhesives_ would have to wait \u2013 I wanted to get outside before the snow disappeared.\n\nI phoned Mrs Shapiro, to see whether she needed anything from the shops. There was no reply, so I pulled on my wellies and my coat and went out anyway. The sun was low but brilliant, dusting every white surface with a sparkle of gold, but the snow had already started to melt and there were mini-avalanches all around as it slipped off roofs and branches. Wonder Boy followed me down the road. I lobbed a snowball at him, but he ducked out of the way.\n\nWhen I got to Canaan House, I saw that the snow had pulled an end of the gutter down, and melting snow was dripping down the porch. Maybe I would have to get Mr Ali in again. There were footprints in the snow leading away from the house. I knocked on the door just in case but I wasn't surprised that there was no reply. She must have gone out already. Wonder Boy trotted up the path, sat down in the porch and started to yowl.\n\n'What's the matter?'\n\nI reached down to stroke him, but he hissed and went for me with his claws. I gave him a kick with my welly and went off to do my shopping.\n\nIn the afternoon I phoned Mrs Shapiro again. Still no reply. This was odd. I began to get worried. Why had she gone out so early in the snow? Then Ben came back from school and I got on with cooking supper. I'll ring later, I thought.\n\nAt about seven o'clock in the evening, the phone rang. It was an old woman's voice, hoarse and throaty.\n\n'She's in 'ere.'\n\n'Sorry?'\n\n'Yer pal. She's in 'ere. But she ent got 'er dressin' gahn wivver.'\n\n'I'm sorry, I think you've got the wrong number.'\n\n'Nah, I ent. She give it me. You're 'er what comes to 'ospital, in't yer? Wivve posh voice? She give me yer number. That lady wivve pink dressin' gahn. She says she wants 'er dressin' gahn agin. And 'er slippers.'\n\nI realised in a flash that it must be the bonker lady.\n\n'Oh, thank you for contacting me. I'll...'\n\n'An' she says can yer bring some ciggies wiv yer when yer come.'\n\nThe telephone beeped a few times then went dead. She must have been calling from the hospital payphone.\n\nI glanced at the clock. There was maybe half an hour of visiting time left. I'd given back the key to her house, so I bundled together my own slippers, a nightdress and Stella's dressing gown.\n\n'I'll see you in a bit, Ben,' I called upstairs as I set off for the bus.\n\nThe snow had already melted and the air was surprisingly mild. I walked quickly, avoiding the slushy patches on the pavements. The newsagent by the bus stop was still open. Should I get some cigarettes? Or would it make me into a peddler of disease and death? Probably. But anyway, I did.\n\nThe bonker lady was hanging around in the foyer when I arrived. I saw her approach a departing visitor and cadge a cigarette off him. She was still wearing her fluffy blue mules, now more grey than blue, and her toes poking out into the cold air looked bluish-grey too, the yellow toenails crustier than ever. Feeling like a smuggler delivering contraband, I handed her the cigarettes and she pocketed them swiftly. 'Fanks, sweet'eart. She's in Eyesores.'\n\nIt took me a while to track Mrs Shapiro down to Isis ward. I could see at once that she was in a bad way. Her cheek was bruised, one eye almost closed up, and she had a dramatic bandage around her head. She reached out and gripped my arm.\n\n'Georgine. Thenk Gott you come.' Her voice was weak and croaky.\n\n'What happened?'\n\n'Fell down in the snow. Everything brokken.'\n\n'I've brought the things you asked for.' I took the things out of the bag and put them in her bedside cabinet. 'Your friend phoned me.'\n\n'She is not my friend. She is a bonker. All she wants is cigarettes.'\n\n'But what happened? I telephoned earlier to see whether you needed anything.'\n\n'Somebody telephoned to me in morning. Said my cat was in a tree stuck up in the park.'\n\n'Who telephoned you? Was it somebody you know?'\n\n'I don't know who. I thought it was Wonder Boy stuck. Poor Wonder Boy is not good up the trees.'\n\n'Was he stuck?'\n\n'Don't know. Never seen him. Somebody bumped me, I slipped and fallen. They put me back in the krankie house.'\n\nVisiting time was over, and people were already making their way towards the door.\n\n'Look out for the Wonder Boy. You will feed him again, will you, Georgine? Keys are in pocket, same like before. Thenk you, Georgine. You are my angel.'\n\nI must say, I felt rather grumpy for an angel. Neighbourliness is all very well, but there are limits. Still, I took the keys out of her astrakhan coat pocket again and joined the tide of visitors flowing towards the exit. Had it really been an accident? I wondered on the way home. Or had someone lured her out into the snow and pushed her over? What was it Mrs Goodney had said? 'Wouldn't want to be held responsible if she had another accident...?'\n\nBen was still up when I got back.\n\n'Somebody phoned for you,' he said.\n\n'Did they leave a message?'\n\n'He said can you call him. Mr Diabello.'\n\n'Oh, yes, the estate agent.' I kept my voice absolutely expressionless. 'I'm trying to get him to value Mrs Shapiro's house.'\n\n'Funny name.'\n\n'Yes, that's what I thought. It's a bit late. I'll call tomorrow.'\n\nShould I or shouldn't I? I remembered with a shiver the brazen conduct of the Shameless Woman frolicking in scarlet panties, wriggling wantonly in the grip of Velcro \u2013 was that really me? \n\n# 26\n\n# A gummy smile\n\nOn Saturday, after Ben had left with Rip, I changed into my old jeans, put a torch and a screwdriver into my bag just in case, and walked over to Canaan House. This was my chance to have another good poke around. I was determined to find out Mrs Shapiro's real age and to discover the identity of the mystery woman in the photo. There were two places I hadn't investigated yet \u2013 the sitting room with the boarded-up bay window and broken light, and the attic. I fed the cats and cleaned up the poo in the hall. Then I started to search systematically.\n\nThe catch on the sitting-room door was faulty and it hung slightly ajar. I pushed it open. The smell \u2013 feral, feline, fetid \u2013 was so overpowering I almost backed away, but I held my handkerchief over my nose and stepped inside, shining the torch around with my free hand. The beam fell on a high ornate ceiling with its defunct chandelier, a huge marble fireplace with a fall of soot spilling on to the hearth and, on the mantelpiece, an ornate gilt clock whose hands had stopped at just before twelve. There were two sofas and four armchairs, all draped in white sheets, a carved mahogany sideboard with glasses and decanters \u2013 one decanter still with a few centimetres of brown viscous liquid that smelled like paint-stripper \u2013 and over by the window a grand piano, also under a sheet. I played the torchlight around the walls; they were hung with paintings \u2013 gloomy Victorian oils of Highland scenes, storms at sea, and dying animals \u2013 quite different to the intimate clutter of personal pictures and photos that covered the walls in the other rooms.\n\nThe bay window was covered by heavy fringed brocade curtains; an ugly box-shaped pelmet, covered in the same brocade, was sagging away from the wall, and when I looked up I could see why. A huge crack ran from the ceiling above the lintel right down to the floor, with a cold draught whistling through it. At the base, where it disappeared into the ground, it must have been several centimetres wide. It must be the roots of the monkey puzzle tree that had caused the damage, I thought. No wonder she wanted to cut it down.\n\nSeating myself at the grand piano, I raised the dust sheet, uncovered the keyboard \u2013 it was a Bechstein \u2013 and struck a few keys. The melancholy out-of-tune twang reverberated in the silence of the room. There were books of music in the piano stool \u2013 Beethoven, Chopin, Delius, Grieg. Not the sort of stuff they listen to in Kippax. In the front of the Grieg piano concerto a name was handwritten in copperplate script, with old-fashioned curlicues on the capitals: Hannah Wechsler. In the front of the Delius lieder was another name: Ella Wechsler. I remembered the photograph of the Wechsler family seated around the piano. Who were they? As I leafed through the music a piece of paper fluttered to the floor. I picked it up and held the torchlight to it. It was a letter.\n\nKefar Daniyyel near Lydda 18th June 1950\n\nMy Dearest Artem,\n\nWhy you do not reply to my letters? Each day I am thinking of you, each night I am dreaming of you. All the time I am wondering if I was doing the right thing coming here and leaving you in London. But I cannot undo my decision. For this will be our place of safety, my love, the place where our people gathered out of every country where we have been exiles can be living finally at peace. Here in our Promised Land our scattered nation who have been swirling round the globe for many centuries like clouds of human dust are finally come to rest. If only you would be here, with us, Artem.\n\nYou cannot imagine, my love, the joyful spirit of working not for wages or profits but for building a community of shared belief. We will get old and die here, but we will build a future for our children. They will grow up fearless and free in this land we are making for them \u2013 a land without barbed wires, out of which no person ever again will drive us away.\n\nAt last we are moving from our temporary house in Lydda into our new moshav here at Kefar Daniyyel on a west-facing hillside overlooking the town. A few hectares of barren wasteland and a trickle of water, an empty abandoned place, but it will be our garden. In the east the sun is rising over the mountains of Judaea and in the west is setting above the coastal plain with its fields of wheat and citrus groves. At night we see the lights sparkling in the valley like Havdalah candles.\n\nIn the morning before the sun is too hot we are working outside clearing stones away from the hillside and preparing terraces for autumn planting. Yitzak has obtained some seedstones for a new type of fruit-tree called avo-kado which he believes we can establish here once the irrigation problem is solved. The men are laying a water pipe that will bring life in this empty land that previously supported only a few dozen of people and their miserable livestocks.\n\nMy love I have some big news I hope will persuade you to come now even if you did not want to before, for our love will have a fruit. Arti you will be a father. I am with a child. Many evenings when the air is cool I am going up to the hilltop at Tel Hadid and watching the sun setting over the sea, and thinking of you living there beyond the sea and your baby growing here inside of me. My dear love please come and be with us if you can, or if you cannot come at this time please write me here at Kefar Daniyyel and I will understand.\n\nWith warm kisses,\n\nNaomi\n\nThere was a smudge at the bottom of the page, which might have been a kiss, or a tear.\n\nThe letter was written in small neat handwriting on both sides of two sheets of thin paper folded together. Had it been hidden, or lost? I read it through again. Her English had been better then, I thought. I folded it back along the creases, and put it in my bag. I could picture the poor sad girl sitting out on the hillside carrying her baby inside her, watching the sun setting over the sea, and dreaming of her lover. But the story still refused to fall into place. Did he go to her in the end? Or was it Naomi who came back? Was he married to someone else? And what happened to the baby?\n\nCuriously I rifled through the music books to see whether any more letters would fall out. The Delius songbook fell open at a page that had an English translation beneath the German original.\n\n_I have just seen two eyes so brown_\n\n_In them my joy, my world I found_\n\nThere was a time when Rip used to call me his brown-eyed girl, and I remembered how he would sing along to the Van Morrison tape when we were driving in the car to France, with Ben and Stella strapped in the back, and our bulky frame tent and Camping Gaz strapped to the roof rack. And I would squeeze his hand, and the kids would roll their eyes and snort in derision at this display of adult soppiness. What happens to love? Where does it go, when it's not there any more? Rip's love had all dribbled away into the Progress Project. Probably I was also to blame for letting it happen \u2013 for letting my eyes become less brown.\n\nThe light of the torch was beginning to flicker; the battery must be running low. I switched it off and climbed the stairs to the first-floor landing. The nine doors were all closed. I gathered together Mrs Shapiro's grey-white satin nightdress, pink candlewick dressing gown and _Lion King_ slippers from her bedroom. Standing on tiptoe, I could just see that the Harlech Castle tin was still on top of the wardrobe where I'd left it. Then I closed the door behind me and opened the door that led up to the attic.\n\nI'd not been up here before, and I'd sneered inwardly when Mark Diabello had talked about the penthouse suite, but as I climbed the steep dog-leg stair, light poured in from two high rounded gable-end windows and a vast light-well in the roof, revealing wide beamed eaves branching off into rooms with sloping ceilings and magnificent views over the treetops towards Highbury Fields.\n\nThe rooms, however, were full of junk \u2013 heaps and bundles and cardboard boxes, all piled dustily on top of each other. My heart sank. It would take ages to search through this lot. I opened one of the boxes at random \u2013 it was full of books. I pulled one out and flicked through the pages. _Saint Teresa of Avila: A Life of Devotion._ Not my sort of thing. Other boxes contained crockery, cutlery and some rather ghastly china ornaments. A cupboard that looked promising had nothing in it but rubber bands and jam-jar lids \u2013 dozens of them \u2013 and some pre-war recipe books and magazines. There were no documents or photos, or letters or a diary that would fill in the gaps in Mrs Shapiro's story.\n\nOn my left a narrow doorway opened on to a spiral staircase that led up into a small round room. This, I realised, was the fanciful little turret perched on the west side of the house. It was barely large enough to fit an armchair, and that's almost all that was in there, a wide armchair upholstered in faded blue velvet with claw feet and a scrolled back, and by it a little carved table in front of the window. As I sat down on the chair a cloud of dust rose up around me making me sneeze. I looked out over the jungly rain-washed garden, imagining how pleasant it would be to sit here on a quiet afternoon with a cup of tea, a Danish pastry and a good book; and then out of the blue I felt an intense sensation of presence \u2013 of someone else who had sat here looking out of the window just as I was. Whose chair had this been? Who else had sat here looking down into the garden? My restless hands had been stroking the velvet and now I felt something unexpectedly hard against my fingertips \u2013 a coin. It was one of those big old-fashioned pennies with a picture of Queen Victoria, pushed down the side of the chair. I carried on feeling with my fingers and pulled out a paper clip, a cigarette butt, and a small crumpled photograph. I smoothed it out. It was a picture of a baby, a beautiful brown-eyed baby. I couldn't tell whether it was a girl or a boy. Somebody's hands were holding it up under the arms as it grinned gummily into the camera.\n\n'Yoo-hoo! Anybody there?'\n\nI jumped. I'd left the front door open, I remembered. Guiltily, I shoved the coin and the photo back down the sides of the chair and made my way down the stairs. Mrs Goodney was standing in the hall with a smug smile on her face.\n\n'I thought I'd find you here. Having a good snoop around, are we?'\n\nShe was wearing the same pointy cube-heel shoes and an ugly raincoat with a slightly scaly texture in almost the same shade of lizard green. I suppose someone had once told her the colour suited her.\n\n'Mrs Shapiro asked me to feed her cats. She gave me the key.'\n\n'Feeding them in the bedrooms? I don't think so.'\n\nI blushed, more with fury than embarrassment, but I kept my mouth shut.\n\n'Anyway, you can hand the keys over now, because we've established that you're not in fact the next of kin. She's got a son.'\n\nI caught my breath \u2013 the baby! But there was something about the way Mrs Goodney looked at me that made me think she was bluffing. Or fishing for information. Well, that was a game two could play.\n\n'But I don't think he'll be coming over from Israel to feed the cats.'\n\nShe blinked, a quick reptilian blink.\n\n'We have our links with international agencies, you know. We'll be inviting him to help sort out his mother's business when the house goes up for sale.'\n\n'You can't put it up for sale without her consent.' Or could she?\n\n'Of course, he'll have an interest in the property, too \u2013 the son.' She watched me closely with her lizard eyes. 'In the meantime, she's in the care of Social Services. She said, by the way, that she doesn't want you to visit her any more.'\n\nHer words sent a tremor through me. Had Mrs Shapiro really said that? It was possible \u2013 she was cussed enough \u2013 but somehow I just didn't believe it.\n\n'So,' Mrs Goodney held out her hand for the keys, 'I'll be taking over the care of the cats.'\n\nVioletta appeared, as if on cue, purring and rubbing herself against Mrs Goodney's legs, and I noticed Mussorgsky creeping towards the bottom of the stairs, waiting for the right moment to sneak up to the bedroom. I realised Mrs Shapiro's bed was their love nest. I realised also from the way Mrs Goodney looked at them that her idea of care would be to call in the Council's pest control department.\n\n'I'm not going to hand over the keys without her written permission.' I tried to make my voice sound snooty, but this just annoyed her more.\n\n'I can always get a court order, you know,' she snapped back.\n\n'Fine. Do that.'\n\nCould she?\n\nAfter she'd gone, I locked up the house carefully, putting the new key from the back door on to my key ring, grabbed the carrier bag I'd filled with Mrs Shapiro's stuff (of course that would have been the perfect reason for me to be looking around upstairs, but you never think of those things at the time, do you?) and headed straight for the hospital. I raced around the antiseptic maze of corridors looking for Isis ward. But when I got there, she'd gone. Someone else was in her bed. I looked up and down all the bays, but she was nowhere to be seen.\n\nThe nurse on duty was another teen-child, thin and harassed.\n\n'Where's Mrs Shapiro? She was in that bed.' I pointed.\n\nThe teen-child looked vague.\n\n'She's gone into a nursing home, I think.'\n\n'Can you tell me where she's gone? I brought some things for her.'\n\n'You'll have to ask in the social work department. It's in the same block as physio.'\n\nShe pointed vaguely in the wrong direction. Just the thought of the smug smile on Mrs Goodney's face if I went there to ask made my blood boil.\n\nMaybe the bonker lady would know. I hadn't seen her in the lobby when I arrived, and when I went back to the ward where I'd first met her with Mrs Shapiro, I couldn't find her anywhere. I thought she might be down in the foyer cadging cigarettes, but she wasn't there either. I walked over to the porters' desk, but I realised I didn't even know her name. Then on my way out I spotted her hanging around outside the main entrance doors, by the No Smoking sign. She seemed to be involved in an argument with a couple of youths wearing baseball caps, one of whom had his leg in plaster. She grabbed me as I came out.\n\n'They've tooken me cigs off of me.'\n\n'Who? The nurse?' About time, I was thinking.\n\n'No, them.' She pointed to the youths, who were both smoking hard, their heads bent over their hands, as though their lives depended on it.\n\nI went up to them. 'Did you...?'\n\n'She's bonkers,' said the one who had his leg in plaster. They ignored me and carried on smoking.\n\n'You're better off without them,' I tried to console her. 'They're not good for you.'\n\nShe stared at me silently, a look that combined desolation and contempt.\n\n'Okay, I'll get you some more. Do you know where my friend is? Mrs Shapiro? The lady in the pink dressing gown?'\n\n'They've tooken 'er away. This mornin'. She give 'em some proper lip, too. You should've 'eard 'er carryin' on. Swearin' an' all. An' I thought she were a lady!'\n\nShe tutted disapproval.\n\n'Do you know where she is?'\n\n'Never 'eard naffink like it. Filfy tongue, she's got on 'er. They gonna put 'er in ar 'ome. Best place for 'er.'\n\n'Do you know the name of the home? Where it is?'\n\n'Nightmare 'ouse.'\n\n'Nightmare?' That didn't sound good.\n\n'It's where they all go. Bin there meself. Up Lea Bridge way. Not many comes aht alive.' She shook her head ominously and started to cough.\n\n'Thanks. Thank you very much.'\n\nI made to go, but she held on to my coat.\n\n'Yer won't forget me ciggies, will yer?'\n\nThere was no Nightmare House in the telephone directory or on the internet. (Well, there was, but it turned out to be a video game.) I telephoned Ms Baddiel and left a message, but she didn't ring back. Eileen said mysteriously that she was 'on a case'. I was furious and frustrated. Should I go to the police and tell them my friend had been kidnapped by Social Services? I could just imagine their faces. Should I write to my MP? See a lawyer? It came to me that the only person who might be able to help us was Mark Diabello. He'd know what happened in situations like this; and he had a strong interest in making sure Mrs Shapiro's house wasn't sold from under her.\n\nEver since our last encounter, I'd been avoiding him and not returning his calls. It's not just that I'd decided he wasn't my type \u2013 I'd come to the conclusion that I wasn't his, either, and that I was only one of dozens of women he slept with in the line of business. I guessed he was probably much more interested in Canaan House than in me. Still, I swallowed my misgivings and dialled his number. It rang just once.\n\n'Hello, Georgina.' (My number must be on his mobile.) 'Nice to hear from you.'\n\nThere was something in his voice that reminded me of... Velcro. I felt a flush rise in my cheeks. If I got him to help me, would we end up in bed again? And was that what I really wanted? I pushed the awkward questions to the back of my mind.\n\n'Mrs Shapiro's disappeared,' I blurted. 'She's in a nursing home, but I don't know where.'\n\nHe didn't seem surprised. 'Leave it with me, Georgina. When are we going to...?'\n\n'Thanks, Mark. Got to dash. Someone's at the door...'\n\nBefore he could get back to me, though, Mrs Shapiro found a way of contacting me herself. One day when I went around to Canaan House, I found a letter on the mat inside the door; I almost didn't pick it out among the junk mail. It was a used envelope, originally addressed to a Mrs Lillian Brown at Northmere House, Lea Gardens Close. The address had been crossed out, and Mrs Shapiro's address written in. There was nothing inside the envelope except a scrap of paper torn off from the corner of a newspaper, with two words scrawled in what looked like black eyebrow pencil \u2013 _HELP ME_. \n\n# 4   \n_Adhesives Around the Home_\n\n# 27\n\n# The breeze-block fortress\n\nNorthmere House was not really a house at all, but a squat square two-storey institution, purpose-built out of plastered breeze block punctured at regular intervals by square windows that opened wide enough for ventilation but not for escape. The only access to the interior was via a sliding glass door operated by a button behind the reception desk, which was guarded by a fierce middle-aged woman in a corporate uniform.\n\n'Can I help you?' she barked.\n\n'I've come to see Mrs Shapiro.'\n\nShe tapped a few strokes on her keyboard and said, without raising her eyes from the screen, 'She's not allowed visitors.'\n\n'What do you mean she's not allowed visitors? This isn't a prison, is it?'\n\nMy voice was a bit too shrill. Calm down, I told myself. _In \u2013 two \u2013 three \u2013 four_...\n\n'That's what it says on her notes. No visitors.'\n\n'Why not? Who made that decision?'\n\n'It'll be up to matron.'\n\n'Can I speak to her?'\n\nShe looked up at me finally, a cold, indifferent look.\n\n'She's in a meeting.' She indicated a row of pink upholstered chairs along the wall. 'You can wait if you like.'\n\n'And if I just go for a wander around while I'm waiting?'\n\nI tried to sound cool, but my heart was pumping away, making my voice wobble.\n\n'I'll have to call security.'\n\nFrom a window in the lobby I could see through to a central courtyard with a square of trim corporate grass, surrounded by a concrete path which led nowhere, and four benches, one at each side. The access was by way of another pair of sliding glass doors on the far side of the courtyard, presumably also button operated. Through the glass, I had a glimpse of a corridor, with doors opening off it. In one of these breeze-block cubicles Mrs Shapiro would be sitting in her bed waiting for me to free her. Somehow I had to get a message through to reassure her that I was trying. She would still be bandaged up, I supposed, and hopefully receiving some kind of medical care in here.\n\nI sat on a pink chair and waited for a while, wondering what to do. The place was eerily quiet, the sounds all muffled by the thick pink carpet and closed double doors, the air dead, with a synthetic smell that was sweetish and chemical. From time to time, a lift discharged someone into the lobby and the guard-dog lady pressed her button to let them exit the building. Some wore nurses' uniforms, some the same corporate skirt and jacket as the guard-dog lady, and there was a woman with a stethoscope who looked as though she might be a doctor. They all seemed busy and preoccupied. It dawned on me that the impassioned human-rights-violation speech I was composing in my head was going to cut absolutely no ice here.\n\nOn a low table next to the pink chairs was a bowl of polished waxy fruit, no doubt intended to reassure families that their incarcerated relatives would be getting a wholesome diet. I picked up a bright green apple \u2013 it was the same colour as Mrs Goodney's jacket \u2013 and bit into it, hard. The sound of my crunching filled the lobby. The guard-dog lady glared at me. When I'd finished it, I placed the apple core on the reception desk and left.\n\nWalking to the bus stop, I racked my brains for ways of springing Mrs Shapiro. I imagined a video-game scenario with the two of us haring along the corridors dodging security guards and ampoule-armed matrons, violins playing wildly on the soundtrack as we burst out through the sliding glass doors down to the Lea Bridge Road and on to a passing bus.\n\nThere's something magical about sitting in the front seat up on top of a double-decker bus, wending among the treetops. I could feel the tension seeping out of my shoulders and neck as we swayed along high above the road, like riding an elephant. As we crossed the bridge I caught a glimpse of the slim glassy curves of the River Lea as it slipped into London. All around me the sky was full of scudding clouds that fleetingly turned to pink when they caught the sunlight \u2013 not the dead chemical pink of Northmere House, but a bright transient gleam of colour like an unexpected smile. I thought of the young woman pregnant with her baby, sitting on the stony hillside, watching the sunset redden over the western sea, waiting for her lover. Now she was locked up in that breeze-block fortress waiting for me to release her.\n\nThe bus jolted and turned as we came out from among the treetops at Millfield Park, and for a moment the whole skyscape opened up in front of me, turbulent, vivid, with apocalyptic shafts of light breaking through the clouds. Somewhere it was raining. A coloured arc glimmered briefly and disappeared. For some reason, tears came into my eyes. I remembered my strange conversation with Ben. Liminal. A time of transition. The threshold of a new world. Poor Ben \u2013 why did he take everything to heart so?\n\nMondays were my worst days for missing Ben \u2013 two days still to go. They never warn you how much your children are going to hurt you; they never warn you about that needle-keen love-pain that gets in under your ribs and twists around just when you're trying to get on with your life. It was already four o'clock \u2013 home time. Would Ben be back at Rip's by now, eating Choco-Puffs and talking about his day at school? At the next bus stop, a load of schoolkids clambered on and joined me on the top deck, gabbling and laughing and throwing stuff at one another. Did they worry about Armageddon and liminal times? Actually, with kids, you can never tell.\n\nAs soon as I got home, I put the kettle on and while it was coming to the boil I listened to the messages on my answering machine. There was one from Mark Diabello asking me to ring when I had time, one from Nathan at _Adhesives in the Modern World_ , reminding me of the new deadline, one from Pectoral Pete \u2013 no idea what that was about \u2013 and a bald peremptory three-word message from Rip, 'Ring me straightaway.' Like hell I would. I tried to delete the one from Rip and accidentally deleted them all. Now I'd have to remember to phone everyone back. Another time. I put a tea bag into the cup and looked in the fridge for milk. Drat. I'd run out. I was still fuming at Rip's message \u2013 at the tone of his voice. Once, not so long ago, he'd have left a message with love. What had happened to all that tenderness?\n\nI hunted around for some powdered milk, and ended up pouring myself a glass of wine instead. Then another. The silence of the kitchen closed in on me. Two days still to go. Then the phone rang. It was Mark Diabello.\n\n'Georgina, you're at home. I've been... er... making a few enquiries. Shall I come round?'\n\nI should have made an excuse and put the phone down, but the wine had made me weepy, and the treacly sweetness in his voice filled me with unexpected longing. No, not for sex \u2013 I just wanted someone to be nice to me.\n\n'Sorry I didn't ring you back. I've been feeling...'\n\nI didn't get the end of the sentence out. A big sob rose up in my throat and washed the words away. He was around within ten minutes.\n\nI suppose I'd been hoping for a little tenderness, but I could see from the way Mark Diabello looked at me on the doorstep that sex was what was on offer. He led me straight into the bedroom, where he noted with a murmur of approval that the satin and Velcro handcuffs were still in position from last time. Then his shirt was off, and my top was off and his trousers were off and my skirt was up and... what happened next was far too disgusting to describe. He went through all the stages like someone working through a car service manual, and I surrendered with all the abandon of a Ford Fiesta having its eighty-thousand-mile service.\n\nAs the bedclothes cooled against my skin and my eyes adjusted to the dimness in the room I noticed that his clothes were folded up on the chair, while mine were all tangled in the duvet. Circling me in his arms, he stroked the hair back from my forehead.\n\n'Georgina, you're a very sensitive woman. I like that.'\n\n'I like you, too.'\n\nI forced myself to say it, but the words felt wooden and clunky in my mouth. I rested my cheek on his damp chest that smelled of sweat and musky soap and chlorine.\n\nHe ran a finger down my cheek. 'You're special. I mean... different. I'd like to see more of you, Georgina.'\n\n'Mmm,' I murmured non-committally.\n\nThe touchy-feely talk was probably fake, I'd concluded, and all he wanted from me was sex.\n\nWe hadn't spoken about Mrs Shapiro and Canaan House last time, as if by tacit agreement, as if our relationship floated above the world and its sordid concerns in its own enclosed bubble. But there was something so _purposeful_ about those neatly folded clothes.\n\n'You know, Mark, I still wonder about that house...'\n\n'What do you wonder, sweetheart?'\n\n'... what you and your partner are up to.'\n\n'I could ask you the same thing, you know, Georgina. Why did you come to me in the first place to have it valued? She's not your aunty. It's obvious she doesn't want to sell \u2013 so why the sudden interest on your part?' He propped himself up on one elbow, studying my face. 'I keep asking myself \u2013 what's in it for you? Why did you start this whole thing?'\n\nI gasped. He thought... he thought _I_ was like _him_. Mrs Goodney, I remembered, had made the same accusation.\n\n'I didn't start it.' I had a sudden vivid recollection of the rusty-gate voice talking into the mobile phone. I remembered the phrase she'd used to describe Mrs Shapiro \u2013 an old biddy. 'It was the social worker who started it. She wanted to put Mrs Shapiro in a home and make her sell the house. She was going to have it valued by Damian at Hendricks & Wilson. I heard her say it.'\n\nHe sat up, his limbs suddenly taut.\n\n'You should have told me that before. It's a well-known scam. All the estate agents have their contacts in Social Services. That's how we get to hear of properties with potential before they go on the market \u2013 old people going into homes, deceased estates, mortgage foreclosures. There might be a client in the wings, an investor or a developer, who'll pay a good price for the tip-off.'\n\nMy brain was struggling to keep up. The shameless red panties were crumpled under the bedclothes. Then I remembered something else.\n\n'Actually, that social worker had a man with her the first time. He could have been a builder \u2013 I think she was showing him the house. She must have been talking to him on the phone. But surely... what if Mrs Shapiro has a family?'\n\n'They do a deal with the family, Georgina \u2013 cash sale, no questions asked \u2013 the family get their hands on the money, and they get the house off their hands. There's always someone in every family that'll take that line. People \u2013 how can I put it? \u2013 in my line of business you tend to see them at their worst.'\n\n'But I still don't understand why the family goes along with it.'\n\n'If their old dad or aunty goes into a nursing home, the money from selling their house is supposed to pay the home fees, right? At five hundred quid a week or more, that can soon gobble up an entire estate that the family hoped they would inherit. But when the money runs out, the Council takes over the payment for the nursing home. So they get the valuer to put in a false low valuation. He gets his cut. They sell it cheap to an associate, based on the low valuation. The relatives pay the nursing-home fees until the money from the phoney sale's all gone, and the Council takes over the payments. After a few months, they can put the property back on the market at its true value, and they pocket the difference.'\n\nI tried to follow what he said, but all I could see was a gyre of money and bricks swirling around in my head. I was wishing I'd kept my mouth shut.\n\n'But that's just a rip-off.'\n\n'You're very innocent, sweetheart. I like that.'\n\nHe kissed me on the forehead in a way that made me feel suddenly queasy.\n\n'You'd better go now. Ben'll be back soon. Anyway, I don't think she has any family.'\n\nHe threw me a sharp look, as if he knew I was lying about Ben, and reached across for his underpants \u2013 sleek dark Lycra that perfectly defined his manly parts, as the shameless woman might have observed \u2013 but she'd gone off somewhere, and Georgie Sinclair was back home.\n\n'So the social worker could just be flying solo,' he said.\n\n'You mean, robbing solo?'\n\n'That's one way of seeing it. But look at it from the social worker's point of view \u2013 they don't get paid much, do they?' He slipped his arms into his shirtsleeves. 'Not many perks. And it's a pretty thankless job. Then once in a lifetime an opportunity like this comes along. Who's she robbing? There's no family. The old lady doesn't need millions, she just needs a nice, safe, clean home. Why not help her and help yourself at the same time?'\n\nI was shocked. 'Aren't social workers supposed to care for the elderly?'\n\nHe laughed, a cold laugh. 'Nobody cares for anybody in this world, Georgina.'\n\nHe was buttoning up his shirt now. The bleakness in his voice was like the mineral aftertaste of black treacle. I felt an unexpected pang of pity. Poor Mr Diabello with his sleek beautiful body and his sleek shiny Jaguar \u2013 condemned to live in a universe where nobody cares. I kissed his wrist where the black hairs curled out from under the starched white cuff of his shirt.\n\n'I thought you cared for me.'\n\n'That's different. _You're_ different, Georgina.'\n\nHe bent down and kissed me so gently that I was just beginning to think he might mean it after all, and my undisciplined hormones started up their chatter. Then he raised his head and I saw the glint of his eyes darken from gold to obsidian. 'So just out of interest \u2013 what did Hendricks & Wilson value it at?'\n\n'Seven million,' I hazarded.\n\n'You're lying to me.'\n\n' _You_ might be lying to _me_.'\n\nHe laughed, tilting back his head to knot up his tie, so I could see the attractive growth of five o'clock shadow dappling the handsome cleft in his chin. The Velcro was chafing against my wrists.\n\n'Mark, you've forgotten...'\n\n'Oh, yes.' He reached out and undid the fastenings. They dangled limply from the headboard as he made his way out into the dusk, and I retrieved my tangled clothes. \n\n# 28\n\n# Ancient and inexplicable\n\nIt poured with rain next day, and I sat at my laptop trying to think about adhesives. Bonding. For some reason, my mind kept drifting to Velcro \u2013 fascinating stuff. All those sexy little hooks. After a while, I gave up trying to work, put my wellies on, and went round to feed Mrs Shapiro's cats. They were waiting for me as I approached Canaan House, circling disconsolately out in the rain. The porch where they usually waited was one huge puddle. I looked up and saw that water was now pouring down from the broken gutter I'd first noticed nearly a fortnight ago, and splashing straight into the porch. I fed the cats in the kitchen, and shooed them out through the back door. I noticed Violetta sneaking round the back towards the derelict outhouses, and a few minutes later Mussorgsky slunk off in the same direction. I watched to see whether Wonder Boy would follow, but he was still hanging around for the last scrapings out of the tin. I dished it out slowly, to give the lovers the best chance I could. Then I went home via the Turkish bakery and treated myself to a Danish pastry.\n\nAs soon as I got in I phoned Mr Ali. He was hesitant at first when I described the problem.\n\n'I am a handyman not a builder. Big ladders needed for this job.'\n\nBut he agreed to take a look. Next, I rang Northmere House. I was annoyed but not surprised to discover that Mrs Shapiro was barred from receiving phone calls as well as visitors. No doubt her mail would be censored, too.\n\nFortified by my cup of tea and Danish pastry, I returned to my desk. Adhesives. Bonding. Bondage. Mark Diabello. The trouble was, I caught myself thinking, as I stared at the screen of my laptop, that we didn't have anything at all in common. Once the initial excitement of sex wore off, I found him \u2013 I hadn't been able to admit this to myself before \u2013 a bit, well, boring. Maybe that was the trouble with _The Splattered Heart_. Those romantic hero types can be limited in their appeal. What I needed was someone I could talk to: someone intellectual; preferably someone hunkily intellectual.\n\nI'd deleted Nathan's message without writing down the new deadline. Should I ring him to check? I hesitated. He already thought I was pretty stupid. I pictured him sweeping back his black hair in exasperation from his craggily intelligent brow \u2013 he was sitting down at his desk so you couldn't tell he was rather short. Anyway, size doesn't matter, does it? I dialled his number.\n\n'Nathan, I'm sorry, I deleted your message by mistake. What's the new deadline?'\n\nHe sighed and tutted in a way that suggested he wasn't really cross.\n\n'March twenty-fifth. D'you think you'll be able to have it ready in time, Georgie girl?'\n\n'I think so. Actually, Nathan,' I lowered my voice, 'I keep on getting distracted.'\n\n'Oh? Anything interesting?' he breathed. I wavered. No, better not mention Velcro.\n\n'Nathan, have you ever heard of a place called Lydda?'\n\n'You mean Lydda near Tel Aviv? Where the airport is? They call it Lod nowadays.'\n\n'See? I remembered you were good on place names.'\n\n'Are you thinking of going off on holiday? You'd better finish the April _Adhesives_ before you go anywhere,' he added with mock sternness. 'The coast's nice down there. I've got some cousins who live at Jaffa.'\n\nSomehow, it hadn't registered with me before that Nathan was probably Jewish, too. The thing is, in Kippax, everybody comes from Kippax. Mr Mazzarella who ran the chippie and his wife who ran the ice-cream van were the only exotic people in town.\n\n'No, I've been visiting an elderly Jewish lady who lives near me. She's got an old photo in her hall of Lydda.' There was silence on the other end of the phone. 'I thought it was a person. I didn't realise it was a place,' I mumbled. 'That's all.'\n\n'You thought it was a person? Like Georgia?'\n\n'I told you geography's not my strong point.'\n\nSomehow I'd managed to make a complete prat of myself again. But even as I said it, I was thinking \u2013 why does Mrs Shapiro have a photo of Lydda?\n\n'There was a terrorist attack there in 1972. A bunch of Japanese terrorists gunned down a load of people at the airport. You might have read about it,' said Nathan.\n\nI searched back through my memory. I would have been twelve years old at the time. Just finding my feet at Garforth Comp. It must have been one of those tragedies in a faraway place that flits across the television screen and vanishes in a day, wringing less grief than the death of Lionheart the school rabbit.\n\n'What did they do that for?'\n\n'They were avenging two Palestinian hijackers who'd been gunned down by the Israelis.'\n\nMy mind blanked over. Palestinians and Israelis killing each other \u2013 an enmity as ancient and inexplicable as Wonder Boy and Violetta. Somebody else's problem, not mine. \n\n# 29\n\n# The Abomination\n\nNext day I waited for a break in the rain to dash across to Canaan House on my feline mercy mission. They were all there, waiting for me, circling and purring. There's something very nice about getting such a warm furry welcome, even when you know it's really just the food they want \u2013 it isn't love at all. Maybe the emotions don't matter \u2013 maybe if Rip was just a bit more warm and furry when he talked to me, I thought, I'd be able to cope with the lack of feeling.\n\nI fed them in the kitchen, then just as I was about to lock up and go home, the rain started again, big heavy drops, presaging a downpour. I could have made a run for it, but getting back to _Adhesives_ just didn't seem that appealing. I excused myself by thinking I should check the roof for leaks, and made my way up into the attic. Despite the disrepair in the rest of the house, the roof was surprisingly sound. There was a place at the front, more or less above the bay window, where a couple of slates were missing and water was dripping in. I hunted among the junk for a container to catch the drips and found a pretty Victorian chamber pot with a blue-iris design, similar to the pattern in the bathroom.\n\nIn the turret room, the ceiling showed no damp patches. I settled myself into the blue armchair to wait for the rain to pass, and felt round the edges for the baby photo. It was still there. I took it out and studied it. Mrs Sinclair had once said, shortly after Stella was born, that in her opinion all babies were alike. I'd been outraged at the time; but now, looking at this crumpled photo of a bald gummy baby, I thought she had a point. Only the lovely dark baby-wide eyes stood out. I gazed back at them, and something from long-ago 'O' level biology popped into my head: the brown-eye gene is dominant; the blue-eye gene is recessive. So this baby must have had at least one brown-eyed parent. Mrs Shapiro's eyes were blue. And so were Artem Shapiro's.\n\nNow my curiosity was truly aroused. With my fingers, I explored the crevice around the edges of the armchair. There was a lot of fluff, cat hair and miscellaneous debris that stuck in my nails. At last, near the left armrest, I came across what felt like paper. It couldn't have fallen in by accident \u2013 it must have been pushed down deliberately \u2013 it must have been hidden. With one hand I held back the blue upholstery, and with the other I dug two fingers in deep enough to catch hold of one end and pull it out. It was a letter, concertinaed up, on the same flimsy notepaper as the one I'd found in the piano stool.\n\nKefar Daniyyel near Lydda 26th November 1950\n\nMy Dearest Artem,\n\nI am writing with some wonderful news for you. Our baby was born on 12th November, a little boy. Every day I watch him grow a little more beautiful like his father. Truly he has your face, Arti, but he has my brown eyes. I am often talking to him about his daddy in London, and he smiles and lifts up his little hands in the air, as if he is understanding everything. I have called him Chaim after our great president Chaim Weizmann. One day your daddy will come here to us I promise to him. Why do you not come, Arti? Why do you not write? Have you forgotten about us?\n\nWe are so eagerly waiting for you, to wrap you up with our love. My dear one the air here is so good and clean after the horrible smogs of London I am sure that your health will be improved straightaway. My friend Rachel is expectant also. You cannot imagine how it is good after half a century of death to be surrounded with new life. You will be feeling like at home among these olim who have made aliyah from every corner of the world. Many here at Daniyyel are from Manchester and everyone speaks English, though the big thing now is to learn again our own ancient tongue.\n\nThere is so much of our people's history in this red earth and in these white stones that are lying across the landscape like the bones of our forebears, sometimes I imagine their spirits sitting beside us on the hillside in the evening to watch the sun going down and the first stars to rise in the east. Finally after so much suffering they are in peace. When the wind whispers over the hilltop it is like the voices of our dead singing their kaddish prayers. Six million souls who have come home. Dear one, I am still remembering our house in Highbury and our happy evenings by the piano, and then my eyes are full of tears. Why do you not write?\n\nWith all my love,\n\nNaomi\n\nI read and reread the letter as I sat waiting for the rain to ease off. Then I folded it up and pushed it back down the side of the chair with the photograph. Who _was_ Naomi? She must have been the pretty brown-eyed woman \u2013 the mother of the baby. But then who was the old lady in Northmere House? How did the two Naomis fit together?\n\nThe rain showed no sign of easing off: the water streamed down the windows as though someone was playing a hose on them. There was something almost apocalyptic about this never-ending downpour. Was it one of the prophetic signs of the End Times? Ben would probably know. I glanced at my watch. It was three o'clock, almost time for him to be back. In the end, I just resigned myself to getting soaked, and made a dash for home.\n\nWhen I got in, I rubbed my hair dry with a towel, put on dry clothes, and guiltily sat down at my laptop. Okay. Concentrate. Glue. ' _Adhesive curing is the change from a liquid to a solid state_.' Sometimes the science of stickiness can be boringly obvious. Maybe it was time to start another novel \u2013 a novel about an old lady who lives in a huge crumbling house with seven cats, and a secret. I pushed the dissident thought out of my mind and forced myself to focus. _Adhesives in the Modern World_ was what paid the bills. Something else was niggling at the back of my brain. Ben seemed to be home later than usual.\n\nWhen at last I heard his key in the latch, I folded up my laptop and went downstairs to greet him. As I came into the hall, I stopped and caught my breath. I saw a stranger standing there \u2013 a bald weirdo who'd broken into my house.\n\n'Hi, Mum.' He grinned embarrassedly and hung up his wet coat. 'Don't stare like that.'\n\n'What...?'\n\nAll his hair, his lovely brown curls, had gone. His skull, knobbly and pale, looked obscenely naked.\n\n'It looks very...'\n\nHe met my eyes. 'Don't say it, Mum.'\n\nI put my hand over my mouth. We both laughed.\n\n'D'you want some Choco-Puffs?'\n\nHe shook his head.\n\n'I don't know why you always get those for me. Dad gets them, too. I hate them.'\n\n'I thought you liked them.'\n\n'I used to. I've gone off them now. They taste funny. Sort of metallic?'\n\n'So what would you like?'\n\n'S'all right. I'll get it.'\n\nHe made himself some toast and spread it with peanut butter a centimetre thick, a layer of strawberry jam on top of that, then a sprinkling of cocoa powder. I'd expected him to take it up to his room, but he pulled out a chair and sat down at the kitchen table. Outside, the rain splashed and gurgled, overflowing the gutter. Surely such heavy rains in February were something new? I must remember to ask him. I poured myself a cup of tea. Ben, since our liminal conversation, had been drinking only water.\n\n'So isn't it a bit... cold?'\n\nHe gave me a look of mild reproach. 'Yeah. But when you think our Lord was crucified, it sort of puts it in perspective?'\n\nThe rising inflection made him sound defensive. I felt a flutter of panic.\n\n'Is it something you think about a lot, Ben?'\n\nHe opened his school bag, unzipped an inner pocket, and pulled out a book. With a shock of recognition I saw it was Rip's old school Bible \u2013 black, gilt-edged, with the crest of his public school inside the front cover. He leafed through to a page that was bookmarked with an old bus ticket.\n\n'When ye therefore shall see the... Abomination of Desolation...' he stumbled on the clunky words, 'spoken of by Daniel the prophet, stand in the holy place, then let those who are in Judaea flee to the mountains. Let him which is on the housetop not come down to take anything out of his house. Neither let him which is in the field return back to take his clothes.' He read carefully, looking up from time to time to check I was still listening. 'And then shall they see the Son of Man coming in the clouds of heaven, with power and great glory.'\n\nHe paused to take a bite of toast. I had a sudden image of the skyscape I'd seen from the top of the bus. Those gleaming galloping clouds \u2013 they _were_ like chariots of glory.\n\n'And he shall send his angels with a great sound of a trumpet, and they shall gather together his elect from the four winds, from the uttermost part of the earth to the uttermost part of heaven.' When I didn't say anything, he added, 'Mark chapter thirteen? Verses fourteen to twen'y-seven?'\n\n'Ben...'\n\nIn the silence between us, a sweet curly-haired child hovered on the edge of extinction. I wanted to hug him in my arms. I wanted him to be my little boy again, to tell him stories about rabbits and badgers, but he was somebody different.\n\n'I'm not saying it's all rubbish, Ben. That language \u2013 it's very powerful. But don't you think it refers to things that happened a long time ago?'\n\n'The Abomination that bringeth Desolation isn't a long time ago, Mum \u2013 it's in the future \u2013 soon. Some nutter'll drop a nuclear bomb on the Temple Mount at Jerusalem. The holy place. That stuff about fleeing into the hills, not going back for anything, not even picking up your coat. The mushroom cloud. It's all there.' He reached over for the cocoa powder and gave his toast another dusting, then he licked his finger and circled it round in the surplus cocoa on the edges of his plate.\n\n'But...' How can you take this stuff seriously? I wanted to say. And yet I realised with a pang of apprehension that Ben was far from alone, and it was my own cosy secular world view that was in retreat before a sweeping global tide of belief.\n\n'Daniel predicted it first. In the Old Testament? Then Matthew and Mark picked up on it? They didn't even know about nuclear weapons, but the way they describe it... it's kind of uncannily accurate?' His voice, crackly and insistent, seemed alien.\n\n'But isn't it just symbolic? You're not meant to take it literally, Ben.'\n\nHis eyes brightened with zeal. He licked his fingers again.\n\n'Yeah, that's what it is. Symbolic. You've got to interpret the signs? They're happening all over the world, the signs of the End Times? If you know what to look out for?'\n\nWithout his crown of brown curls, the dark downy hair on his lip and chin seemed to stand out more against the pallor of his skin. He looked like a stranger \u2013 a stranger trying to impersonate someone I knew intimately.\n\n'But they're nutcases, Ben, the people who run those websites.'\n\nI shouldn't have let my exasperation show. His voice became whiny and defensive.\n\n'Yeah, a'right, some may be a bit nutty. But the big guys that run the world \u2013 they all know it's going to happen? George Bush 'n' Tony Blair? Why d'you think they're always praying together? Why d'you think they're so, like, totally obsessed with the Middle East? Why are they getting so stressed about Iran going nuclear? _They_ know it's the prophecy of the Second Coming that's working out in our time? Like, we're the last generation?'\n\nHe slapped two bits of toast together into a sandwich, and licked at the peanut butter that squeezed out around the crusts.\n\n'Want to know why America supports Israel? Because in the Bible it says when the chosen people go back to their promised land, like they did in 1948, that's the start of the End Times.' He bit into the sandwich with a crunch. 'It's sad cases like you and Dad that'll get left behind.'\n\n'Left behind what?'\n\n'The rapture? The Second Coming? When the elect get taken up to heaven, and all the sad gits with their _Guardian_ s and their anti-war placards'll be left behind to stew in the tribulations.' A spot of jam had oozed on to the edge of his plate. He licked it off. 'George Bush's pal Tim LaHaye wrote a book called _Left Behind_. It's all in there.'\n\n'Just because George Bush believes it doesn't make it true.'\n\n'Yeah, but maybe they know something you don't? Like, they've got their sources of information? The website's got five million subscribers?' He gave me a look that was both angry and pitying. 'Don't be so blind, Mum.'\n\nThen he took a swig of water, got up abruptly taking his bag and his Bible, and stomped off to his room, his pale skull bobbing up and down as he climbed the stairs.\n\nMy stomach clenched into a knot. I finished drinking my tea and went upstairs to my bedroom. I sat on my bed with a pillow behind my back and opened up my laptop. In the rain-washed light from the window, the blue sky of my desktop image seemed absurdly optimistic. I typed _End of Time_ into Google, just as Ben said he'd done. There were literally millions of entries. I started opening a few at random, following links, and all of a sudden I found I'd stepped over a threshold into an eerie parallel world I'd never even guessed existed. Ben was right \u2013 there were millions of people out there scouring their Bibles and actively trying to calculate the timetable for the end of the world from clues in the text.\n\nAt first I felt piqued. Why hadn't I read about any of this in the _Guardian_? Or heard about it on Radio Four? Why hadn't Rip told me? Then I started to feel scared. Some sites had bizarre names like _teotwawki_ (The End of the World As We Know It), _escapeallthesethings_ , _raptureready_. Millions of people clearly _were_ getting ready. The Old Testament prophecies of Daniel and Ezekiel, the four Gospels of the New Testament and the Book of Revelation were quoted again and again in rambling blogs by individuals offering their own personal interpretations of the prophecies, and in huge complex sites with links to dozens of organisations. There were even sites that marketed 'End Times Products'. One link led to a quotation from a speech by George Bush: 'We are living in a time set apart.' It was highlighted in red and animated with horrid little flames, sharp like razor teeth. Another linked mysteriously to a page called 'How to Fix Self-Tanning Mistakes'.\n\nAlone in my dusky room, with only the fan of my laptop purring away intermittently, and these creepy fundamentalists as my guides, I could feel the boundaries of reason start to dissolve and notions from the irrational hinterland encroach into my consciousness. Was this what Ben had felt? I remembered my dream, the formless malevolent spirit, and despite myself I shuddered. Everything in this other world seemed illusive, like a nightmare in which everyday things like bar codes, seen through a prism of unreason, take on a sinister skew, while war, disease, terrorism, global warming \u2013 the scourges of our age \u2013 are seized on with glee as signals of the Second Coming. A man who called himself Jeremiah \u2013 his website showed him with a neat little goatee beard and a Scotch plaid cap similar to Mrs Shapiro's \u2013 explained that the parable of the fig tree \u2013 ' _When his branch is yet tender, and putteth forth leaves, ye know that summer is nigh' \u2013_ referred to seasonal changes from global warming, which were a sign of imminent rapture. Power up the central heating and the air conditioning! Roll on, four-by-fours! Fly by jet! Consume! As the earth warms up and the fig trees blossom, those lucky ones, the elect, will be seized and whisked off into heaven! His smug little smile said it all.\n\nHow come I didn't know about any of this? I thought back to my religious knowledge lessons at Kippax Primary School, Mrs Rowbottom wearing her mauve bobble-knit jumper and a porcelain rose brooch; the smell of closely packed children, and Lionheart the school rabbit snoozing in his cage; the little bottles of pre-Thatcher milk waiting in the crates by the door. We'd learned about forgiveness and mercy. We'd learned about the wheat and the tares, and the Prodigal Son. I'd even got a gold star for my drawing of the Good Samaritan. Mum had proudly put it on the fridge in the kitchen, even though Dad was a subscriber to the opium-of-the-masses theory of religion.\n\nWhen we were a bit older, we discussed motes and beams, and learned to recite the Beatitudes and Saint Paul's _faith, hope and charity_ epistle off by heart. It had all seemed very uplifting and benign. I had no idea about all this other stuff. Had Mrs Rowbottom known? If so, she'd seemed unperturbed.\n\nJeremiah's website had a Promised Land link which took me to a whole page of links to both Christian and Jewish sites discussing God's promise to the Jews. When was that promise to be fulfilled? In God's time, in the prophesied future? Or now, in the present-day Middle East? Was the rebuilding of the third Temple in Jerusalem a metaphor for spiritual rebirth? Or was it about bricks and mortar? The cyber-arguments raged. Something else Ben had said came into my mind. When the chosen people go back into Israel, in 1948, that's the start of the End Times. My mind flashed to the letter in the piano stool. _Our Promised Land._ The date on the letter was 1950. When I'd read it first, it had seemed like a quaint voice from another age. Now past, present and future were in terrifying collision.\n\nAnd it wasn't only Christians and Jews who were preoccupied with the Second Coming. Ben had said something about a Last Imam. Google came up with more than a million links to websites anticipating the imminent return of the Imam al-Mahdi. It all seemed a long way from the Prince of Wales's bar codes.\n\nAs I surfed from one link to another, the light from my laptop screen threw an eerie coloured glow on the walls and ceiling. I was beginning to understand why Ben was so rattled. Compared with the vast inevitability of this Rapture machine, the world of our own little secular family seemed puny and insubstantial. Dusk dimmed into darkness outside the window, where flurries of rain still pattered on the glass. Yes, the rain. I forgot to ask him about the rain. \n\n# 30\n\n# The broken gutter\n\nBy Saturday the rain had stopped but the pavements were still wet, and soft heavy drops dripped from the overhanging trees as I walked along to Canaan House, where I'd arranged to meet Mr Ali to take a look at the gutter. I'd set out a little early, hoping to catch him on his own. I wanted to ask him about Lydda; I wanted to find out about Islam and the Last Imam. But as I turned into Totley Place, I spotted a small battered red van parked in the lane that led to Canaan House, and then I heard men's voices in the garden, shouting. I quickened my step. The shouting got louder \u2013 I couldn't tell what they were shouting \u2013 it wasn't in English anyway. Violetta dashed out to greet me; she was running around in circles, mewing.\n\nAs I approached the gate, I glimpsed between the trees a terrifying sight \u2013 Mr Ali was dangling in the air, like a rather tubby Tarzan wearing a pink-and-mauve knitted hat. He was hanging on for grim death to a length of cast-iron gutter that had come away from the wall. I watched transfixed as he tried to reach with his toes for the window ledge, bawling something in a foreign language. All that was holding him up was a rusty iron bracket at one end, and a twine of ivy that had clambered over the roof and luckily got a grip on the chimneys. On the ground, floundering among the wet brambles, two young men in flowing white robes and Arabic headgear were grappling with an extendable aluminium ladder that had come apart.\n\nThey heaved it this way and that, their robes snagging in the brambles. The ladder definitely seemed to be winning. At last they slotted the three sections together and wielded it in Mr Ali's direction, trying to catch him as he hung with just one toe now resting precariously on the window ledge, and the other kicking at the air. But they swung the ladder too wide, then overcorrected and swung it too far the other way. Mr Ali let out a tirade of furious words. I could see the bracket straining under his weight and the ivy coming away from the bricks. If they didn't get their act together fast, he was going to plummet some thirty feet on to the stone terrace in front of the house. I held my breath, and a thought clicked in my head \u2013 these young men, they really are unbelievably useless.\n\nIn the end they managed to get the ladder under Mr Ali, but it was too short to reach the ground. So one of them held it up to Mr Ali's flailing foot, while the other one tried to extend it from below, jerking the catch-hooks down over the rungs, Mr Ali shrieking in terror at each jerk. Then he jerked too hard and the ladder fell apart again. Wonder Boy was sitting in the porch watching, flicking his tail with excitement, a beastly look on his face.\n\nI stood on the path, petrified, thinking I should definitely keep out of this. I didn't want to distract the men's attention for a single second, for I could see that a momentary lapse might be fatal. But just as they'd almost reassembled the ladder, the one at the front lost concentration at exactly the moment that Wonder Boy decided to make a dash for it. Swerving to avoid the cat, his foot caught in a loop of bramble and he staggered forward, holding the ladder high and crashing the top section right through the bedroom window inches away from Mr Ali, completely smashing away the bottom frame. A shower of glass tinkled on to the flagstones.\n\nMr Ali was still balancing with one leg on the window sill and one thrashing the air, yelling his head off. Then he spotted me by the gate. Our eyes met. It was too late for me to back away. He called down to the two men in the garden, and they looked round, shouting and beckoning. So I ran over to help. I grabbed one end of the ladder, determined to show that although I was a woman, I was not utterly useless like them. But it was much heavier than I thought. As I swayed under its weight, the other end swung round, clonking one of them on the head. He staggered back into the bushes and lay there, motionless. I rushed to his aid. Oh, heck! Had I killed him? Mr Ali and the other young man had gone silent, too. Wonder Boy, who had come over to investigate, gazed up at me and I thought I glimpsed in his slitty yellow eyes a look of... was it respect?\n\nAfter a few moments the young man pulled himself out of the bushes, no harm done, and between the three of us we managed to extend the ladder to its full length and get it up securely against the wall below Mr Ali's feet. He climbed down, bawling at the other two \u2013 he was literally spitting with rage. Then just as his feet touched the ground, all the fight seemed to go out of him, and he slumped down, his head resting on his knees, breathing deeply.\n\n'This job is for a younger, fitter man. Not double excel gentleman my age.'\n\n'But you _did_ excel, Mr Ali. Keeping so calm,' I said, though calmness, to be honest, was not the first word that sprang to mind.\n\n'No, size XXL, Mrs George.' He clasped his arms around his hamster tummy. 'My wife feeds me too much. No good for climbing up ladders.'\n\nI laughed. 'Next time, you should get one of the other two to go up the ladder.'\n\nHe shook his head with a melancholy sigh, but said nothing.\n\nThe other two were perched uncomfortably on the triple edge of the ladder. They had got out a packet of cigarettes, and were lighting up. I wondered why they were wearing those bizarre outfits \u2013 they looked more like extras out of _Lawrence of Arabia_ than any Palestinians I'd seen on TV. They were younger than Mr Ali, taller, and incredibly handsome in a dark-flashing-eyes white-flashing-teeth kind of way. (Tut. Isn't this an utterly incorrect stereotype? Get a grip, Georgie. They're young enough to be your sons.)\n\n'Hello,' I smiled. 'I'm Georgie.'\n\nThey nodded their heads and flashed their teeth at me. It was clear they didn't speak a word of English. Mr Ali struggled to his feet.\n\n'Allow me to make introductions. Mrs George, this is Ishmail, my nephew. He is completely useless. This is his friend Nabeel. He is also completely useless.'\n\nThe useless young men nodded and flashed their teeth. 'What a misfortune at my age to have two complete Uselesses for my assistants.'\n\nThen he spoke in Arabic to them, and something about the way he looked at me suggested that he was saying I was pretty useless, too. They nodded politely at me and smiled some more.\n\nWhen they'd finished their cigarettes and stubbed them out on the ground, they put the ladder up against the wall and Nabeel held the bottom of the ladder while Ishmail started to mount it, his feet tangling in his robe.\n\n'No, no, no!' yelled Mr Ali, jumping up, then he yelled something in Arabic. It was obvious even to me that the ladder was too short and the angle too steep to be safe. 'We must get a bigger ladder. I told you this one is no good.'\n\nThe Uselesses heaved the no-good ladder on to the roof-rack of the van with a lot of puffing and shouting, then sat on the step of the porch and lit up again. They were grinning like a pair of naughty kids and batting each other with a folded newspaper printed in Arabic. Mr Ali reached across and confiscated it.\n\n'This house \u2013 it needs too much work,' he sighed. There was a big damp patch on his trousers from sitting on the wet ground. 'I do not know if I can do it with these Uselesses.'\n\n'I'm sure you can,' I said, making my voice especially calm, which I felt was called for in the situation. 'There's no hurry. I think Mrs Shapiro will be away for a while.'\n\n'You think? Hm.'\n\nHe paused and gave me an oblique look. The Uselesses were still sitting in the porch, but now they'd started arguing in loud voices and shoving each other off the step. Then Mussorgsky appeared at the broken bedroom window (how had he got in there?) and began yowling with gusto, and Wonder Boy yowled back from the garden, a smug selfsatisfied yowl.\n\n'You know, Mrs George, I am thinking is a pity so big house must stand empty.'\n\nMr Ali stroked his neat beard and looked at me thoughtfully again. 'This, my nephew Ishmail \u2013 he has no place to stay. Sleeping on floor in my apartment. Drive my wife mad. This other useless one, too, sometimes sleeping in there.'\n\nI could see what he meant \u2013 they would drive me mad, too.\n\n'Well... I don't know what Mrs Shapiro would think...' I started. Then it occurred to me that these two might be useless at house repairs, but they could do a great job of keeping the likes of Mrs Goodney and Nick Wolfe at bay. And they could feed the cats. 'It would have to be on the strict understanding that they move out when Mrs Shapiro comes back.'\n\n'No problem. Even if they stay for short time it will make big difference for my wife. Give her chance to clean up.'\n\nI wondered what Mrs Shapiro would say if I told her they were Palestinian.\n\n'I am sorry they have no money for paying rent. But they will repair the house. Everything will be fixitup like new.' He saw the look on my face. 'I supervise, of course.'\n\nI suppose I should have said no there and then, but there was something irresistibly cuddly about Mr Ali. And besides, I was on the scent of another story.\n\n'Where did you learn all your building skills, Mr Ali? In Lydda?'\n\nHe shook his head.\n\n'No. We were sent away from Lydda. Do you not know what happened there?'\n\n'You mean the terrorist attack? I know about that,' I said, pleased with myself.\n\n'Ha! All of the world knows this.' He seemed annoyed. 'Terrorists shooting on innocent Israelis. But you know why? You know what happened before?'\n\nI shook my head. 'Tell me.'\n\nIn a clearing in the brambles Wonder Boy and Mussorgsky were now hissing and going for one another with their claws. Violetta was hovering close by, waving her tail and making little yelping noises of encouragement, though I couldn't work out which one she was encouraging. Mr Ali flicked the newspaper at them to chase them away.\n\n'In 1948 all Palestinians were sent out from Lydda. Not only Lydda \u2013 many many towns and villages in our country were destroyed. To make way for Jews. People still are living in the refugee camps.'\n\nHe went suddenly quiet.\n\n'But... but you learned to be a builder, yes?' I encouraged, wanting to reassure myself that something positive had come out of all that displacement, all that history clogged up with memories of unrighted wrongs.\n\n'In Ramallah I trained for engineer.' (He pronounced it inzhineer.) 'Here in England I must make new examinations. But I am old and time has tipped his bucket on me. This useless one,' he pointed at his nephew, 'he will study for engineering, too. Aeronautical.'\n\n'Aeronautical?'\n\nThat sounded quite brainy. I tried to imagine going up in an aeroplane engineered by Ishmail, and felt an uncomfortable tightening in my chest.\n\n'He has a scholarship.' He had lowered his voice to a proud whisper. 'Other one, I don't know. Now they are both learning English. First-class English language course nearby to here \u2013 Metropolitan University, next door from Arsenal Stadium.'\n\nThe Uselesses, realising that they were being talked about, chipped in.\n\n'Arsenal. Yes, please.'\n\nYes, it would need to be a first-class English course, I thought.\n\n'So why did _you_ come to England, Mr Ali? I mean, wasn't your family over there?'\n\n'You are asking difficult questions, Mrs George.'\n\nI could see he didn't really want to talk, but I was still filling in the gaps in the story.\n\n'I'm sorry. It's a Yorkshire habit. Where I come from, everybody knows everybody's business.'\n\nHe hesitated, then continued. 'You know, after my youngest son died, I saw no hope. No possibility of end to this conflict. I wanted only to come away from this place. I have a good friend, Englishman, he was a teacher in Friends School in Ramallah. He helped me to come here.'\n\n'Your son died...?'\n\nSuddenly, my nosiness had led me into a darker avenue than I'd intended.\n\n'He had a burst appendix.' He stared at the ground as though his son's face was pictured there. 'We were in Rantis, visiting wife's family. We wanted to take him to hospital in Tel Aviv but we were delayed at the checkpoint. My wife was weeping and pleading with the soldiers \u2013 one soldier \u2013 he was a boy of eighteen but he had a power of life or death over us. He was playing with his power. He said we must go back to Ramallah. When we got there it was too late.' His eyes glinted with a harder brightness. 'How can I forgive? My son was fourteen years old.'\n\nOn a corner of the newspaper, he started to draw a map.\n\n'This was five years ago. Now with wall is worse. You can see. Green Line. Wall line.' He drew another snaking line. I stared at the map \u2013 the crazy curling line \u2013 and felt a flutter of panic. Maps. Not my thing. But why did it snake around so much. In fact, why was there a line at all?\n\n'So you wanted to leave...?'\n\n'Now my daughter is married with this Englishman. I have three grandchildren.' He smiled briefly. 'Drive my wife mad.'\n\nI thought I'd like to meet his wife one day.\n\nThe Uselesses had finished their cigarettes, and gone off to sit in the van. They must have had a CD player in there, because I could hear strains of Arabic music, sweet and melancholy, drifting incongruously over the damp lawn and dripping brambles.\n\nBut maybe all places have their histories of sadness and displacement, I was thinking. People move in, others move on; new lives and new communities spring up among the stones of the old. In school, we'd learned about the history of Kippax, how in the 1840s miners from Scotland and Wales had been recruited as scabs to break up the union in County Durham \u2013 desperate hungry men sucking the marrow out of the bones of other desperate hungry men. When the seam at Ledston Luck was opened up, their grandchildren and great-grandchildren were brought down from County Durham to Yorkshire and settled in Kippax. There are men who shape destiny, who draw lines on maps and shift populations about; and there are men like Dad and Mr Ali who live their lives in the interstices of the grand plans of others, labouring to provide food and shelter for their families.\n\n'So, what you say, Mrs George?' Mr Ali interrupted my thoughts. 'They stay here and fixitup the house?'\n\n'I don't know,' I said weakly. My heart ached for sad exiled Mr Ali and his charming useless assistants, but I owed a duty of care to Mrs Shapiro, and the scenario with the ladder had filled me with apprehension. 'Maybe if you fix the gutter first, it'll give me time to have a word with Mrs Shapiro.'\n\n'Tomorrow,' he said, 'we come with the new gutter and big ladder. You will see.'\n\n'And, er, the window. That needs fixing, too, now.'\n\n# 31\n\n# The epoxy hardener\n\nSometimes when I try to understand what's going on in the world, I find myself thinking about glue. Every adhesive interacts with surfaces and with the environment in its own particular ways; some are cured by light, some by heat, some by the exchange of subatomic particles, some simply by the passage of time. The skill in achieving a good bond is to match the appropriate adhesive to the adherends to be bonded.\n\nAcrylics, for example, are known to be fast curing, and they don't require as much surface preparation as epoxies, which have high cohesive strength but a slower cure rate. Epoxy adhesives have two components: the adhesive itself, and a hardening agent, which accelerates the process. On Friday, I was sitting at my laptop, pondering this profound philosophical duality, when a cunning thought slipped into my head. What I needed to re-bond with Mrs Shapiro was a hardening agent. And who could be harder than Mr Wolfe?\n\nFlushed with inspiration, I rummaged in the desk drawer for a card and wrote a get-well-soon note to Mrs Shapiro, adding that I was doing my best to visit her and advising her under no circumstances to sign anything until we'd talked. I mentioned that I'd found some builders who might be staying at the house while they did some work there \u2013 I freely admit, I didn't go into much detail. I told her the cats were doing extremely well and that Wonder Boy was missing her (well, probably he was, in his own brutal and selfish way). I enclosed a stamped addressed envelope and a blank sheet of paper, put it all in an envelope with the card, and sealed it. Then I walked down to the office of Wolfe & Diabello. A quick reconnoitre in the car park round the back told me that Mark Diabello was out and Nick Wolfe was in.\n\nIn the small office, his physical presence was overwhelming; he seemed to fill the whole room, pushing me back against the wall. He greeted me with a bruising hand-grip and asked me what he could do me for. (Either he thought that old clich\u00e9 was still amusing, or his unconscious was speaking.) I told him in my specially friendly voice that Mrs Shapiro had been asking after him. On a yellow Post-it note, I scribbled the address of Northmere House and, handing him my envelope, said that if he found the time to call round, would he drop off the card from me, too.\n\n'Fine,' he said.\n\nThen I went home and got on with _Adhesives in the Modern World_. The article I was editing was about the importance of good joint design in bonding. You see, however good the glue, a poorly designed joint can snooker you. End to end joints should be overlapped if possible, or tongued and grooved, or mortised and tenoned. Or you could go for a hybrid joint \u2013 I remembered Nathan's joke, glue and a screw. You should always prepare the surfaces to maximise the bonding area. _'Surface atraction is increased, by rouhgening or scraching the surfaces to be bonded.'_\n\nThe article had been written by a young man who knew his glues but seemed to have a total contempt for spelling and punctuation. What do they teach them in school these days? I tutted to myself. Ben was just as bad.\n\nI found myself worrying about how he'd got on in school today. He'd struggled to settle into his new class when we'd moved down from Leeds; in fact the New Year's email chat with the strange semi-literate Spikey was the nearest I'd got to meeting any of his friends. I was anxious that his shaved head and religious leanings could make him a target for bullies, and while we were having tea that evening I tried to raise it with him.\n\n'What did they say at school, then, when you turned up with your new hair-do \u2013 your no-hair-do?'\n\n'Oh, nothing.'\n\nWithout his brown curls his face looked different. The brown hair was my genetic legacy, but those arched eyebrows, with their slightly haughty lift, and the intense blueness of the eyes \u2013 I could see more of Rip in him now.\n\n'Didn't the kids take the mick?'\n\nHe shrugged. 'Yeah, a bit, but I don't care. Jesus suffered taunts an' that, din't he?'\n\nYes, and look what happened to him \u2013 I held back the thought, and loaded my voice with maternal concern. 'But wasn't it a bit... horrible? I mean, kids can be very cruel.'\n\n'Nah,' he said. 'It's all earthly stuff. Don't bother me. Brings me closer to Our Lord.'\n\nWhen he'd finished his meal, he laid down his knife and fork, put his hands together briefly and closed his eyes. Then he picked up his bag and disappeared upstairs. Maybe I should have been pleased that he wasn't stealing cars or taking drugs, but there was a scary intensity about him that was almost like an aura of martyrdom. I felt a stab of guilt. Was it our failure as parents that had led him to seek out a different kind of certainty? Sometimes I felt I wasn't grown-up enough myself to be a parent \u2013 I always seemed to be just a step ahead, making it up as I went along.\n\nRip didn't have any such uncertainties \u2013 he always knew what was right, and committed himself to making it happen. It was one of the things I'd loved about him \u2013 his commitment. Yes, perhaps I had been wrong not to take more of an interest in his work. But what exactly was it that he did? Something about global systems for iterating progress. Or iterating systematic progressive globalisation. Or globalising iterative progressive systems. I understood each word on its own, but together, they had the same effect on my brain as phenolic hydroxyls. I'd made some notes once, ages ago, on a bit of paper, while he was explaining it to me, thinking I'd get my head round it in my own time, that one day we would converse about progress, globalisation, systems and suchlike. It was in the desk drawer somewhere, jumbled up with the old rubber bands and the out-of-ink biros.\n\nOn impulse, I picked the phone up and dialled his number. A young woman answered \u2013 I nearly didn't recognise her voice.\n\n'Stella?'\n\n'Mum?'\n\nThe pain of missing her caught me off guard like a thump in the chest.\n\n'Aren't you supposed to be at uni?' (Why was she visiting Rip and not me?)\n\n'I... It's reading week. I just came down to see...' I guessed from her hesitation that it might be something to do with her complicated love life. 'Do you want to speak to Dad?'\n\nHer voice \u2013 so sweet \u2013 still reedy like a child's, but with an adult's self-assurance. She'd always been a daddy's girl. Sometimes their closeness made me envious.\n\n'Yes \u2013 no. Stella, can _we_ talk? We always seem to communicate by messages and texts.'\n\n'So?' A prickly tone. She didn't want me making her feel guilty.\n\n'Listen, I'm worried about Ben. Have you noticed anything different about him?'\n\nI realised she wouldn't have seen his haircut yet, but she and Ben were close \u2013 they'd fought and loved one another all through their childhoods, just as Keir and I had done.\n\n'He's always been a bit mental, my little bro.'\n\nShe was always so confident in her judgements.\n\n'But does he seem unhappy to you?'\n\n'He's cool, Mum. He's got religion in a big way, that's all \u2013 like I had Leonardo DiCaprio when I was his age.'\n\n'That's what I mean \u2013 religion \u2013 it doesn't seem quite normal for sixteen.'\n\n'I don't know what's wrong with you, Mum. He could be shooting up or nicking cars, and you're stressing about him reading the Bible.'\n\nMaybe she's right, maybe that's all it is, I thought, a schoolboy phase. But there was something terrifying about his intensity, the strained look on his face, the dilated eyes.\n\n'He talks about the end of the world as though it's going to happen any minute now.'\n\n'Yeah, Dad keeps on at him about it. They had a big row over Christmas. Then Grandpa got stuck in.'\n\n'I wondered what that was about.'\n\n'Ben started banging on about religion.'\n\n'What did he say?'\n\n'Something about miring the sanctity of Christmas with alcohol and consumerism. They all laughed. Ben got really upset and tried to shut them up.'\n\n'Poor Ben.' I kept my voice even, but I could feel my rage boiling up in me.\n\n'It was gross. Grandpa called him a pansy.'\n\n'What did Ben say?'\n\n'He said, I forgive you, Grandpa.' She giggled. I giggled, too. I tried to imagine my father-in-law's face.\n\n'Good for him.'\n\nBen hadn't told me because he'd wanted to spare my feelings.\n\n'Stella, it's lovely to talk to you. Have you finished your teaching practice?'\n\n'Yeah. It was nearly enough to turn me into a mass child-murderer. I don't know if teaching's really me.' There was a slight whininess in her voice that I recognised, too. 'But anyway, I'll stick with it till the end of the course, then decide. Don't worry about Ben, Mum. He'll be fine.'\n\nWhen I put the phone down I was filled with a wonderful sense of ease, as if a sack of rocks had just rolled from my shoulders; I wanted to run out into the street and hug everybody. Instead, I burst into Ben's room and hugged him.\n\n'You all right, Mum?' He lifted his head from the computer.\n\n'I've just been talking to Stella.'\n\n'What did she say?'\n\n'Oh... she said she wasn't sure about teaching \u2013 whether it was right for her.'\n\nHe gave me a long intense look.\n\n'You need to calm down, Mum. You're getting hyper again.'\n\n# 32\n\n# UPVC\n\nOn Saturday morning, after Ben had left for Rip's, I got a phone call from Mr Ali.\n\n'You can come and see, Mrs George. House is all fixitup.'\n\nThey were waiting for me when I arrived \u2013 all three of them, plus the cats. The Uselesses were wearing jeans and baseball caps. I don't know what had happened to their Arabic gear. Mr Ali was grinning with pride.\n\n'See?'\n\nUpstairs, where the old Victorian window had been smashed, a brand new double-glazed white uPVC top-opening window unit had been fitted \u2013 it was a bit short for the opening, which had been bricked up with breeze blocks to make it fit. There was a new gutter running the length of the house, also in white uPVC. The brambles had been hacked back to make room for a white uPVC table and chair set, and a white uPVC birdbath sat in the centre of the lawn. Wonder Boy was sitting beside it, surrounded by feathers, licking his chops and looking very pleased with himself.\n\n'It's... er... lovely...' I put on a smile.\n\nThe useless ones beamed.\n\n'You let them stay, they will fixitup everything for you,' said Mr Ali.\n\n'Maybe... maybe not too many repairs. Just essential things. Maybe the woodwork just needs rubbing down and a lick of paint.'\n\n'Baint, yes,' he nodded enthusiastically, and said something in Arabic. The useless ones nodded enthusiastically, too.\n\n'I'll give you a ring. I need to get a spare set of keys cut,' I said, playing for time, thinking maybe Mrs Shapiro would be back soon.\n\nBut on Wednesday morning there was a letter for me on the door mat. I recognised my own handwriting on the envelope. The letter inside was written with thick blue marker-pen \u2013 the sort Mum used for marking her Bingo cards.\n\nDearest Georgine\n\nThank you for your Card and for you sending my Nicky to comfort me in Prison. He is quite adorable! He was coming with Champagne and white Roses. A real Gentleman! We were talking for Hours about Poetry Music Philosophy the Time was passing too quick like flowing Water under a Bridge and I am always asking myself what matters it if there gives a Gulf in our Ages so long as there gives a Harmony inbetween our Souls. It was like so with Artem he was twenty years my older but we have found Joy together. I wonder if I would ever find such a Joy again with another Man to feel the arms of a Mans around me and the warmth of a good Body close beside mine better than Cats. He has said he will come again now every Hour is dragging too long I wait for him to come and you also my dear Georgine. How have I escaped Transportation and Inprisonment in all my Life only to face it now alone in my Older Age. They are wanting me to sign a Confession before I can return to my Home. They are saying I must give the Power of Returning but my Nicky also is saying I must not sign nothing so I am putting a brave Resistance. I must stop the Nurse comes soon with my Injection. Please help me.\n\nYour dear Friend\n\nNaomi Shapiro\n\nI read it through a couple of times. Then I tried to read between the lines. Then I phoned Mr Wolfe.\n\n'Thanks for taking my card round. How was she? She looked awful in hospital. I was surprised they let her out so quickly.'\n\n'Bit of bruising. Gash to the head. Nothing too serious. We had a good laugh.'\n\n'She seems to be very fond of you.' I was wheedling for information.\n\n'Yes. And you know, in a funny way, I've grown quite fond of her, too.'\n\nThere was a glibness in his voice, as though it was something he'd been practising.\n\n'Do you know anything about this confession she's been asked to sign?'\n\n'Sorry?'\n\n'Something about power of returning.'\n\n'Ah. Yes. They're wanting her to sign a Power of Attorney.'\n\n'What does that mean?' It sounded ominous.\n\n'It means they \u2013 whoever she signed it over to \u2013 would have the power to sign legal documents on her behalf.'\n\n'Like the sale of a house, for instance?'\n\n'Got it in one.'\n\nI felt my heart starting to race. Things seemed to be spiralling out of control again, but I kept my voice steady.\n\n'What can we do to stop that?'\n\n'I've been wondering that myself.'\n\nWhatever he had in mind, he obviously wasn't going to share it. I needed to find out what he knew, without giving away too much myself. Then I thought of something that would put him on the back foot.\n\n'Did she tell you about her son? Apparently he's coming over from Israel. That'll be a great help, won't it?'\n\nI thought I heard a sharp intake of breath on the other end of the phone.\n\n'Indeed.'\n\nThere was something else I needed to know.\n\n'By the way, did you have any trouble getting in? They seem to have quite strict security.'\n\n'Oh, yes, they told me she wasn't allowed any visitors.'\n\n'So...?'\n\n'I just told them not to be so bloody ridiculous.'\n\nSo that's how it's done, I thought.\n\nAn hour or so later the phone rang. It was Mark Diabello.\n\n'Hi, Georgina. Glad I've caught you at home. Listen, I think I've got the answer to your dilemma.'\n\n'What dilemma?' I tried to remember our last conversation. It was something unpleasant and incomprehensible about bricks and money.\n\n'How to avoid Mrs Shapiro having to sell up if she goes into a home. Apparently the Council can just put a charge on her house. It's like a mortgage \u2013 the house is sold after the person dies, and that's when the Council calls in the debt. The residue, if any, goes to the estate.'\n\n'You mean the debt to cover the nursing-home fees? Nobody told me about that.'\n\n'Well, they wouldn't, would they?'\n\n'But the thing is, Mark, she doesn't need to be in a nursing home at all. She's fine at home. She likes her independence.'\n\n'You'd better get her back home as soon as you can, then. Or get someone else to live in the house till she gets back. These things've got a way of picking up their own momentum.'\n\n'Tell me about it.'\n\nThe whole house saga had picked up far too much momentum, as far as I was concerned, and he'd been among those pushing it along.\n\n'How about over dinner tonight, sweetheart?'\n\nThere was an earnest note in his voice that made me feel guilty; but I steeled myself.\n\n'I can't. I'm meeting... somebody. And I've got a lot of work on at the moment \u2013 something I'm trying to write,' I added quickly.\n\n'You're a very active woman. I like that.' A sigh or a crackle on the line. 'As it happens, I do a bit of writing myself. Poetry.'\n\n'Really?' Despite myself, I was intrigued. The hero of the original _Splattered Heart_ had been a poet, too. 'Will you show me?'\n\n'I'd love to. When...?'\n\n'I'll ring you.' I put the phone down.\n\nI'd arranged to meet Mr Ali and the Uselesses in the afternoon, and I still hadn't got a set of keys cut, so I walked down to the cobbler's on the Balls Pond Road then back up to Totley Place. It had turned cold again, a spiteful, stabbing cold, with a mean wind shaking the naked branches of the trees against a washed-out sky and flinging swirls of litter and dead leaves against my legs. At least the rain had held off.\n\nIt was just after two when I arrived at Canaan House. The red van was already parked outside, and the three of them were hunched up in the front, the Uselesses puffing at cigarettes, Mr Ali reading a newspaper. The house looked startlingly different, the white plastic window with its breeze-block base seemed to wink at me like a diseased eye. As soon as they saw me they jumped down, talking excitedly in Arabic, and followed me up the path, carrying their stuff with them in dozens of carrier bags. They looked as though they were planning to move in for a while. There were sleeping bags, books, clothes, a CD player, and even an old PC. In one carrier bag I spotted what looked like the Arabic outfits \u2013 obviously they hadn't given up on them yet. I showed them upstairs.\n\nWhile they were unpacking and sorting their stuff out, I walked around the house with Mr Ali, pointing out the things I thought needed fixing: the missing slates on the porch, the broken latch on the door to the sitting room and the faulty light, the peeling wallpaper in the dining room, the dripping taps in the bathroom and kitchen, the cracked toilet bowl, and the huge gaps around the edges of doors and window frames where the wind whistled in. Those were just the obvious things.\n\n'Hm. Hm,' he said, writing it all down in a notebook. 'All will be fixitup good, Mrs George.'\n\nHe'd never seen inside the whole house before. As his bright hamster eyes explored the details of the rooms he made little murmurs of amazement. 'Hm. Hm.' When we went up into the attic, he gasped. 'In here we could make beautiful benthouse suite.'\n\n'Let's just concentrate on the essentials to start with,' I said.\n\nDown in the hall, he stopped once more in front of the picture of the church at Lydda, his arms folded in front of him. I tried to read the emotion in his face, but he was turned in profile, so all I could see was the shadow of a furrow in his brow.\n\n'You know, exactly next door to this church was a mosque. Cross and crescent standing side by side in peace.'\n\n'Tell me more about Lydda,' I said. 'Is your family still there?'\n\n'Do you not know about Nakba?'\n\n'Nakba? What's that?'\n\n'Hm. You are completely ignorant.' He said it with a sigh, in the same way that he'd introduced the Uselesses. 'In my country we say that ignorance is the warm bath in which it is comfortable to sit but dangerous to lie down.'\n\n'I'm sorry. I'll make some tea if you tell me.'\n\nI put the kettle on, rinsed two of the less grotty cups as thoroughly as I could under the tap, and put a kr\u00e4utertee tea bag into each of them. We sat on the wooden chairs at the kitchen table. Fortunately I'd cleared away the remnants of Mrs Shapiro's last unfinished meal. He drank his pond water with three heaped spoonfuls of sugar, so I put the same into mine \u2013 obviously this was the secret. We stirred and sipped.\n\n'So you were going to tell me about your family,' I prompted.\n\n'I will tell you how they left Lydda. But you know the history \u2013 about British Palestine Mandate?'\n\n'Well, just a bit. Actually, not a lot.'\n\nHe sighed again.\n\n'But you know about Jewish Holocaust?'\n\n'Yes, I know about that.'\n\n'Of course, everybody knows about sufferings of the Jews.' He sniffed irritably. 'Only suffering of Palestinian people nobody knows.'\n\n'But I want to know, Mr Ali. If you'll tell me.'\n\nThis story \u2013 I could see by now that it was going to be much more complicated than a Ms Firestorm-type romance. But it had somehow got under my skin.\n\nMr Ali blew on his tea and took a sip, sucking the sweet liquid off the ends of his moustache.\n\n'You know in the end of the war, after what they have done to the Jews, the whole world was looking for a Jewish homeland? And the cunning British say \u2013 look, we will give them this land in Palestine. Land without people, people without land. Typical British, they give away something which does not belong to them.' He looked up to make sure I was still paying attention. I nodded encouragingly. 'This land is not empty, Mrs George. Palestinian people have been living there, farming our land, for generations. Now they say we must give half of it up to the Jews. Did you not learn about it in school?'\n\n'No.' I was embarrassed by my ignorance. Geography, okay I had an excuse. But I'd done history at 'O' level. 'In history lessons we learned about Kings and Queens of England. Henry the Eighth and his six wives.'\n\n'Six wives? All at one time?'\n\n'No. He killed two and he divorced two, and one died.'\n\n'Typical British behaviour. Same with us. Some killed. Some sent away into exile. Some died.' Mr Ali shook his head crossly and took a gulp of tea, scalding his mouth and sucking in air to cool himself down.\n\n'But that was a long time ago.'\n\n'No. Nineteen forty-eight. Same like the Romans did to Jews, Jews did to Palestinians. Chased them out. We call it Nakba. It means disaster in your language.'\n\n'No, I mean Henry the Eighth was a long time ago.'\n\n'Before Romans?'\n\n'No, after the Romans, but before... Never mind.' I saw the bemused look on his face.\n\n'It's all just history, isn't it?'\n\nThis seemed to make him even more annoyed.\n\n'You have learned nothing in school. Apart from a man with six wives. History has no borders, Mrs George. Past rolls up into present rolls up into future.' He made agitated roly-poly movements with his hands. 'Young Israelis also are ignorant. In school, their teachers tell them Jews came into an empty land, but not how this land was made empty.'\n\nI thought about the letter in the piano stool. Yes, that's what _she_ wrote \u2013 a barren and empty land.\n\n'So was it like... the Nazis and the Jews?'\n\n'No, not like Nazis,' he tutted angrily. 'You must not exaggerate. Israelis do not plan to exterminate all Arab people, only to drive them out of the land.'\n\n'But the Jewish people need a homeland, too. Don't they?'\n\nHe sighed. His mouth curled down.\n\n'But why in Palestine? Palestinian people never made any harm to Jews. Pogrom, ghetto, concentration camp \u2013 Europeans made all this. So why they make their revenge on us?'\n\n'It _was_ their land, wasn't it? Before the Romans sent them away?'\n\n'This land belongs to many peoples. All nomadic peoples wandering here and there, following their sheeps. Palestine, Lebanon, Syria, Jordan, Egypt, Arabia, Mesopotamia. Who knows where everybody was coming from?'\n\nMy mind blanked out. All those places \u2013 how on earth did they fit together? I would have to look it up on the internet.\n\n'They will tell you Palestinians abandoned their farms and houses and ran away because their leaders told them. No, they ran away because of terror. Israeli state was made by terrorists. You think only crazy Arabs are terrorists?' Mr Ali was becoming intensely un-hamster-like.\n\n'I'm sorry to be so ignorant. At school we just learned British history.'\n\n'So you must know about Balfour Declaration?'\n\n'A bit.' I couldn't admit how little that bit was. 'Wasn't it about partitioning the Middle East at the end of the First World War?'\n\nI'd seen _Lawrence of Arabia_ once, with Peter O'Toole. He was great. Those eyes. But I'd never understood who betrayed whom over what. I remembered the bit where he fell off the motorbike. That was sad.\n\n'Balfour said to meet Jewish aspirations without prejudicing rights of Palestinians.'\n\nThere was something about those words that reminded me vaguely of the Progress Project. He took a gulp of pond water, and continued.\n\n'But Palestinian people still are sitting in refugee camps. They have lost their lands, fields, orchards. They have no work, no hope. So they sit in refugee camps and dream of revenge.' His eyes were glittering with unusual ferocity. 'They have no weapons, so they make their children into weapons.'\n\nI put the kettle on again, wondering about Ben. How had he blundered into this thorny Biblical world?\n\n'Isn't there a prophecy, Mr Ali? Don't the Jews have to rebuild the Temple in Jerusalem, where the Messiah will come back? The third Temple?'\n\n'Their book says they must rebuild the Temple. But it is not possible at this time, because on this site now stands our mosque \u2013 al-Aqsa Mosque. Next to the Dome of the Rock. One of our most holy places.'\n\n'But is it true that Muslims, too, are waiting for the Last Imam? The Imam al-Mahdi. Do you believe that, Mr Ali?'\n\nHe hadn't struck me as a man of extreme beliefs \u2013 beyond an extreme misplaced belief in white uPVC.\n\n'I will answer your question. Mostly Shia believe in the return of al-Mahdi. I am Sunni.' He gave me a curious look. 'You learned about this in school?'\n\n'No. On the internet.'\n\nI saw now that the hard glitter in his eyes was a trick of the light, and when he turned towards me his face was gentle and sad. I took a deep breath.\n\n'Actually it was my son who told me. He found all this stuff on the internet. Weird sites about the end of time. The Antichrist. Armageddon. Great armies and battles. The Abomination, whatever that is. He's so preoccupied with it... I was worried, that's all. I wanted to understand what it was about.'\n\nThe kettle whistled, and I made us another round of kr\u00e4utertee. Mr Ali spooned three more sugars into his cup and stirred, looking at me gravely.\n\n'Mrs George, the young are ready to believe anything that will lead them into heaven. Even to die for it. And there are always some whisperers who will say to them that death is the gateway to life.'\n\n'You mean...?'\n\nI shivered as though a cold draught had touched my neck. I had a sudden image of Ben \u2013 my lovely curly-haired Ben \u2013 his eyes radiant with conversion, his boyish body strapped up to that deadly payload, attempting a little smile or a joke as he said goodbye. The thought made me feel sick.\n\nUpstairs I could hear the young men \u2013 they'd managed to set up the CD player and bursts of wild jangling music were swirling downstairs. They were thudding about as though they were dancing, but probably they were just walking around. Ben, who's quite slim, always thuds like an elephant when he walks.\n\n'Do not worry about your son, Mrs George. He will grow up before too long. Ishmail and Nabeel used to talk also about these things when they lived under occupation. Now they talk about football.'\n\nThe thudding upstairs turned into thundering on the stairs, and a few moments later the Uselesses appeared in the hall. They said something in Arabic to Mr Ali, and he translated for me.\n\n'They want to say thank you. This is a very good place.'\n\nHis eyes were twinkling again.\n\n'There's something else they have to do,' I said. 'They must feed the cats.'\n\nI showed them the cupboard in the kitchen where the cat food was kept. They nodded enthusiastically.\n\n'And they have to clear up the mess.'\n\nI led them back into the hall and pointed out a small deposit the Phantom Pooer had left in the usual place. I'd spotted it earlier but not got around to cleaning it up. The taller one \u2013 I think he was Mr Ali's nephew, Ishmail \u2013 shuddered and put his hand over his nose and mouth. I shrugged and offered a sympathetic smile, but I was thinking, that's nothing \u2013 you wait till you find one of the big fresh ones. The other one, Nabeel, said something loud and urgent in Arabic. Mr Ali said something loud and urgent back. They argued like that, back and forth, for a few minutes. Then Ishmail went and got a piece of kitchen roll and started to wipe it up, but somehow just managed to spread it around even more. Mr Ali shook his head.\n\n'Completely useless.'\n\nAnyway, in the end, the cat poo was wiped up, and it was time for me to go, and I took the keys I'd had cut out of my pocket.\n\n'If anyone comes to the house, anyone you don't know, you mustn't let them in.'\n\nMr Ali translated it into Arabic, and they nodded emphatically.\n\n'No in. No in.'\n\nThey made waving 'keep out' gestures with their hands. I gave them the keys. And I must admit I felt a pang of extreme apprehension. The least bad thing that could happen would be that the repairs would be done less or more uselessly, and the house would be bedecked with white uPVC. The worst case didn't bear thinking about. Who were these young men? I didn't know anything about them. They could be illegal immigrants. They could be terrorists. Mr Ali could be the leader of a terrorist cell. A terrorist disguised as a hamster. He smiled.\n\n'Don't worry, Mrs George. All will be fixitup good for you. I will subervise.'\n\n# 33\n\n# Avocados and strawberries\n\nThe following Saturday afternoon I made my way down to Sainsbury's in Islington for my big weekly shop. Although there's a closer Sainsbury's in Dalston, this one is on a direct bus route. At the top of the end aisle, I spotted a crowd milling around \u2013 it was the sticker lady doing her reductions \u2013 and out of habit I made my way to join them. Without Mrs Shapiro there, it was all much more refined, just a bit of genteel basket-barging when something exciting turned up. One woman was helping the sticker lady by gathering up the sell-by-dates from the counters and passing them to her for re-stickering, standing over her to make sure she got first pick. What a cheek. Even Mum didn't do that. Still, I managed to get some good bargains on cheeses, and a plastic box with three avocados reduced to 79p, perfect apart from a dent in the lid. I remembered the letter I'd found in the piano stool at Canaan House \u2013 avo-kado she'd called them. They must have been newly discovered at the time. Mum called them advocados. Given her aversion to anything exotic, I'd been surprised to find she'd quite taken to them. She served them with defrosted prawns doused in salad cream. Even Dad ate them.\n\nThere were some bargains in the fresh-produce aisles, too. Bananas, slightly spotted \u2013 tastier that way \u2013 reduced to 29p; nets of oranges on buy-one-get-one-free; plastic-box strawberries flown in from somewhere or other, pretty but flavourless. I remembered the strawberries Dad used to grow on the allotment at Kippax \u2013 the fresh, intense flavour, the kiss of summer on your tongue, the occasional slug to keep you on your toes. Keir and I would go down after school and fill a bowl up for tea, then fight over them all the way home.\n\nNo, even at half price, these strawberries weren't worth it. Where can you get strawberries so early in March, I was wondering, as I made my way out of the store. A young woman was handing out leaflets near the entrance \u2013 I must have missed her on the way in. I took one from her hand absent-mindedly and was about to stick it in with my shopping when the words jumped off the page at me: BOYCOTT ISRAELI GOODS.\n\nSeeing my interest, she pushed a sheet of paper towards me on a clipboard.\n\n'Will you sign our petition?'\n\n'What's it about?'\n\n'We want the government to make a commitment to stop serving Israeli-sourced products in the Houses of Parliament. Until Israel accepts UN Resolution 242.'\n\n'Isn't that a bit...?' I stopped myself. The word that had come into my mind was 'pointless'. She looked so solemn, her pale eyes fixed on me as she talked.\n\n'It's all grown on stolen land. Watered with stolen water,' she said.\n\n'I know, but...' But what? But I didn't want to think about it \u2013 I wanted to get home with my shopping. 'But, I mean, it all happened so long ago. It was terrible, I know. The Nabka.' (Or was it Nakba?) 'But isn't it just \u2013 what they had to do?'\n\n'That's crap!' Then she checked herself. 'Sorry, I shouldn't get so worked up.' I realised she was very young \u2013 hardly older than Ben. Her hair was cut short and teased up into little spikes on top of her head. 'But it's not just something that happened long ago. It's still happening. Every day. They're stealing Palestinian land. Bulldozing Palestinian houses. Bringing in Jewish settlers. From Moscow and New York and Manchester.' She spoke very fast, gabbling as though frightened of losing my attention.\n\n'That can't be true.' Surely if it was true, I thought, somebody would put a stop to it.\n\n'It _is_ true. The International Court of Justice says it's illegal. But America supports them. And Britain.'\n\n'Why would anyone want to leave New York to go and live in the middle of a desert?'\n\n'They believe God gave them the land. To make an Israeli state. The people who were there before, the Palestinians, they've cleared them off. Those that are left, they've walled them in. Given them a few poxy reservations. Like the American Indians. The Australian Aborigines. They think if they make life hard enough, they'll just vanish away. Inconvenient people. Who just happen to be in the way. Of somebody else's dream.'\n\n'But you can't wind back the clock, can you?'\n\n'Why not? You'd only need to go back to 1967. Before the Six Day War. You know, the Green Line. Gaza and the West Bank.'\n\nThis was all getting a bit too geographical for me. What Green Line? But there was something very disarming about her earnestness. I ran my eyes down the leaflet. On one side was a crude map, showing a thin straight line between Israel and Palestine, and another line, drawn in green, some way to the right, showing the Palestinian land that had been occupied after the Six Day War. There was a gap between the two lines. And there was a third line, hatched in grey, a contorted snaking line on the right-hand side of the green line. Right is east: left is west, I reminded myself. The key said: Line of separation wall. I forced myself to study it, remembering the map Mr Ali had drawn and wondering why maps had suddenly taken on such importance. The more I stared, the less sense it seemed to make.\n\nI turned the leaflet over. On the other side were pictures of Israeli produce. Avocados. Lemons. Oranges. Strawberries. Well, at least I hadn't bought the strawberries.\n\n'But surely if they're on the sell-by date? If they're reduced...?'\n\nShe fixed me with a solemn look. 'Have you any idea how much water it takes to grow strawberries in the desert? Where do you think it all comes from?'\n\nSuddenly her head swivelled around, and following her gaze I saw a police car draw up and two officers get out \u2013 a man and a woman. They made their way towards us. They looked very young, too.\n\n'Would you mind moving on now?' said the man. 'You're causing an obstruction.'\n\n'No, we're not,' I said, though I could see he was really addressing the girl. She was shuffling her leaflets and her clipboard into a bag.\n\n'We've had a complaint,' said the woman officer, almost apologetically.\n\n'We're just chatting,' I said. 'About avocados. Surely we're allowed to stand on the pavement and chat?'\n\nThe policewoman smiled and said nothing. I looked round to the girl, but she'd disappeared.\n\nI was still wondering about the contents of my carrier bags as I made my way back towards the bus stop at Islington Green. After all, it was just the supermarket clearing excess stock. It would be wasteful to throw it all away. Wouldn't it? What would Mum have done? I remembered an incident during the last miners' strike. It was the winter of 1984, bitterly cold. Firewood was in short supply. I'd brought home a bag of coal that I'd bought at a petrol station. Dad had refused to have it in the house.\n\n'We're not burning no scab coal,' he'd said. 'I'd sooner freeze.'\n\nHe'd taken it outside, and tipped it into the dustbin. Next morning, though, when I went to put the rubbish out, it was gone. Mum didn't say anything, but I always wondered whether it was she who'd scooped it out of the bin in the night. Waste not want not.\n\nThere was quite a queue at the bus stop. The sun had gone, a cold wind had sprung up, and I was beginning to feel hungry. I hunted around in my shameful shopping bags and broke off a ripe banana \u2013 at least they were okay to eat \u2013 weren't they? I noticed a couple standing with their backs to me looking into a shop window. The man was tall, fair, solidly built; there was something oddly familiar about him. His head was slightly out of proportion to his body. I realised with a shock of recognition that it was Rip. I hadn't noticed before how big his head was. Gorgeous, but too big. Like Michelangelo's David. The woman was small, even in her high heels, with a sleek dark bob and scarlet lipstick. I stared. It was Ottoline Walker. What was going on? Where was Pectoral Pete? She was wearing a tightly buttoned coat that showed off her curves. I could see her reflection in the shop window. They were holding hands. She was laughing at something, looking up at him. The little bitch! He bent down and kissed her.\n\nSomething inside me snapped. A sound rose in my chest, swelled up and forced its way out \u2013 aaah! yaaah! \u2013 a high-pitched wail, rasping at my throat. They turned. Everybody turned. I lurched across the pavement. _Wait! In \u2013 two \u2013 three_... Oh, sod that! The banana pitched forward and mushed into a soft slippery paste in her face. She struggled, but the banana in my hand \u2013 it just kept going round and round. It forced its way up into her nostrils. It smeared the slut-scarlet lipstick all around her mouth. It made soft feathery streaks in her eyebrows. Rip's mouth opened wide \u2013 that round trouty look \u2013 O! Then he grabbed my arm.\n\n'Georgie! Stop! Have you gone mad?'\n\nWhat a stupid question.\n\n'Aaah! Yaaah!'\n\nNext she turns on me, sputtering.\n\n'What have I done to deserve this?'\n\nThat voice \u2013 her parents must have spent a fortune teaching her to talk like that. Spoiled brat. You can tell from her voice she's used to getting everything she wants.\n\n'You just thought you could have him, didn't you? You didn't stop to think of me. Me and Ben and Stella. He belongs to us, not you.'\n\n'What d'you mean?'\n\nThere's a bit of banana hanging down from her nose like a big creamy bogey. It makes me laugh.\n\n'We were just inconvenient people, getting in the way of your lovely dream.'\n\nI'm laughing like mad now, splitting my sides at the sheer symmetry of everything.\n\nThen \u2013 this is good \u2013 the Scarlet-mouthed Slut scrapes the mush off her face with her hands and starts to smear it over Rip, over his clothes and his hair. And he says, 'Ottie! Stop! What's the matter with you?'\n\nAnd she says, 'What's the matter with _you_? You told me it was okay. You told me she didn't mind. You lied to me.' She's wailing, too. 'You told me she'd gone off with another man! In a Jaguar!'\n\n'She did. She is.' He backs away. 'You're both bloody mad. Both of you!' He backs away and breaks into a run. She runs after him, stumbling on her bitch-stilettos. And I run, too. I'm wearing my batty-woman trainers, so I can almost keep up. I run after him up the street, dodging through the startled pedestrians.\n\n'Aaah! Yaaah!'\n\nBut he's fast, Rip, fast and fit, ducking and weaving through the Saturday crowd. He shakes us both off.\n\nIn the end, I have to give up. I've lost sight of him. I'm panting for breath, my chest heaving, my throat raw from screaming. My head is spinning. Everything's spinning. I stop and catch my breath, leaning forwards on to my knees. Then I straighten up and turn around. I've lost sight of her, too. She's disappeared somewhere, into her bitch-lair. Still panting, I make my way back down Upper Street towards The Green. About halfway down, on the pavement, I stumble across a discarded black suede stiletto shoe. I kick it into the road, and a Number 19 squashes it flat.\n\nThe crowd at the bus stop has thinned out. I look for my shopping bags where I left them on the pavement. But they've disappeared. Someone has picked them up and taken them. The settler avocados. The blood-soaked oranges. All gone.\n\nActually, it was worth it, I thought to myself, as I sat in the kitchen and poured a glass of wine. Okay, I'd made a fool of myself and I'd lost my week's shopping. But it was worth it just to see that creamy banana bogey hanging from her nostril. It was worth it to see his trout-mouth \u2013 O! To see him run.\n\nI couldn't face going back into Islington, so I just went out and got a bit of shopping at Highbury Barn. When I got back I saw that the answering machine was blinking. There was a message from Ms Baddiel. She was sorry she hadn't been in touch before. She'd been on a course (not a case!). It seemed odd that she'd phoned on a Saturday, but maybe she'd left the message before and I just hadn't noticed. I rang her back straightaway but she wasn't there. The second message was from Nathan. He wanted to know if I'd like to go to the Adhesives Trade Fair in Peterborough tomorrow with him and his father. I pressed Delete. I know I'm sad, but I'm not _that_ sad. I poured myself another glass of wine and settled down in front of the television. _Casualty_ would be on soon.\n\nAs my euphoria wore off, I realised that there was only one more glass left in the bottle, and that if I finished it off then there would be nothing to stop me drinking a whole bottle again tomorrow night. And the night after. And then I'd be well on the road to becoming an Unfit Mother. _Casualty_ was not satisfying \u2013 too much shouting and argy-bargy. What had happened to the heroic drama of life and death? What had happened to that dishy Kwame Kwei-Armah? I recalled my spree of shouting and bad behaviour earlier that afternoon with a prick of shame. Really, people don't want to watch that sort of thing. It's not gentile, as Mum would say.\n\nThen the reality of three Ben-less days loomed, and I started to think that maybe a trade fair in Peterborough was what I needed after all. Maybe Nathan's father would be okay when sober. And the more I thought about it, the more I realised that short men can be incredibly sexy. I dialled Nathan's number. As he picked up the phone at the other end ('Nathan Stein speaking') I heard in the background the familiar theme as the trailer credits rolled away \u2013 he'd been watching _Casualty_ , too. \n\n# 34\n\n# The glue exhibition\n\nNathan picked me up next day at ten o'clock. I'd been trying to imagine what kind of car he would turn up in, but the last thing I'd expected was an open-top sports car, a Morgan, pale blue. He greeted me with a hug. I dropped my knees a bit so our cheeks were just at the same height.\n\n'Sorry, my father couldn't make it.'\n\n'So it's just you and me?' My heart skipped.\n\n''Fraid so. Can you put up with me for a whole day?' (Could I just!) 'You'll need a warmer coat than that.' (I'd already put on my smart grey jacket over my revealing top.) 'And a scarf or something. Otherwise your hair'll blow away.'\n\nI changed into my brown duffel coat, fastened it up to my chin, and tied a scarf down over my ears.\n\n'Sit tight!' he said.\n\nWe whizzed up the Holloway Road and out on to the A1, the wind slapping my head, my eyes stinging, my ears ringing. Shops. Houses. Trees. Flats. Houses. Trees. Whoosh! We couldn't talk; I tried to open a conversation but my words just got blown away. All I could do was watch Nathan's hands on the wheel and gearstick \u2013 he was wearing fingerless leather driving gloves \u2013 and his hunky profile as he concentrated on the road. His silver-flecked designer-stubbly jaw was clenched in a daredevil look of defiance. My stomach was clenched in a knot. I was trying to decide whether it would be better to die instantly or to live out my life in a wheelchair.\n\nPeterborough emerged suddenly out of a fenland mist, the elegant towers and arches of its cathedral swanning above the rooftops. I'd never been here before. The exhibition centre was on the outskirts, a low featureless hangar of a building. The car park was almost empty. Nathan pulled up near the entrance, switched the engine off, and turned to me with a dimply smile.\n\n'Did you enjoy that, Georgia?'\n\nI smiled weakly. I couldn't bring myself to say yes, even to him.\n\nThe exhibition itself was nowhere near as exciting as the journey. It was basically a display of tubes and phials with long technical explanations mounted on card, and samples of things glued together, mainly materials \u2013 laminates stuck to concrete, glass stuck to wood, steel stuck to steel. We seemed to be the only punters, apart from a man in a black-and-white shell suit who was walking round taking notes. Our footsteps click-clacked in the echoing space. Well, what did I expect? The most interesting thing was a car, an old Jaguar, glued to a metal plate on its roof which was bolted to a chain suspended from the ceiling, so it dangled there in mid-air, spinning slowly if you touched it, held up by the power of adhesion.\n\n'Wow! That's amazing!'\n\n'Yes, I'll have to remember that next time I want to hang my car up,' said Nathan.\n\nI had a sudden thought.\n\n'Nathan, do you think you could use glue to stick something like, say, a toothbrush holder on to bathroom tiles?'\n\n'Absolutely. There are a number of purpose-made adhesives. Look for brands with \"nails\" in the name. No-nails. Goodbye-nails.'\n\n'But you wouldn't use nails in a bathroom. It'd have to be rawplugs, wouldn't it?'\n\nHe gave me a sideways grin. 'You mean instead of cooked plugs?'\n\n'What d'you mean?'\n\n'They're called Rawlplugs, Georgie.'\n\n'Rawlplugs?'\n\n'But you're right about one thing \u2013 they're on their way to obsolescence. Adhesives can do many of the same things nowadays.'\n\nMy heart bounced up. Rawplugs were history!\n\nNathan was wandering around with a notebook, an intelligent frown furrowing his brow. I kept very close, hoping he would take my hand or slip an arm around my shoulder. Should I ask after his father? Should I mention _Casualty_? I cleared my throat.\n\n'Did you enjoy...?'\n\n'Hey, look at this, Georgia.'\n\nHe'd stopped to examine a photograph on display near the cyanoacrylates. It was a very distressing full-colour closeup of a bottom stuck to a blue plastic toilet seat. From the angle it had been taken, you couldn't tell whether it was a man's bottom or a woman's. It had obviously been shot in a hospital: there was somebody in the background wearing surgical gloves and a mask. Just imagine if that was you \u2013 it would be bad enough getting stuck on the toilet and having to call for help, and then having blokes with tools break down the door, unbolt the toilet seat and rush you to hospital, and people phoning up \u2013 they would phone an expert like Nathan in this situation \u2013 for advice about solvents. And all the time you'd be wondering who put the glue there; in fact you'd probably be able to guess. You'd be fuming. Fuming but helpless. Then you'd have to be photographed for medical records. Everyone would be solemn and respectful, but behind your back they'd be laughing their heads off.\n\nThe explanation card at the side of the picture simply read:\n\n# CYANOACRYLATE AXP-36C   \nA PRACTICAL JOKE\n\n'Deary me,' said Nathan.\n\nActually, that's not a bad idea, I thought.\n\nThe next stand was a display about the history of glue. There were pictures of trees with gum or resin oozing out and dark-skinned men catching it in little cups. There was a picture which showed Aztec builders mixing blood into their mortar. The explanation card said the Aztec structures were so strong they would withstand an earthquake. It seems that blood is sticky stuff, too \u2013 stickier than water.\n\nI tried another tack.\n\n'You seem very close to your father...' I ventured.\n\n'Ah, yes. Tati.' He paused. I waited for him to continue, but he just wandered on, looking at the exhibits.\n\n'Have you always lived with him?'\n\n'Not always.'\n\nI followed him round the stand, casually brushing against him when he stopped at the corner of the display, but he didn't seem to notice.\n\n'My parents live in Yorkshire,' I said. 'I miss them. But I couldn't live with them.'\n\n'I don't know that I can live with Tati much longer.'\n\nI brushed against him again, this time more determinedly. Surely my intentions must be totally obvious. He opened his notebook and scribbled something down.\n\n'It might make a nice article for _Adhesives in the Modern World_, Georgia,' he suggested. 'Something about the history of adhesion. Glue past and present. What d'you think?'\n\nMaybe he just didn't fancy me. Maybe I wasn't intelligent enough for him. Maybe he was involved with someone else. The thought filled me with gloom.\n\n'Mmm. Good idea.'\n\n'Or even glue past, present and future.'\n\nThe designer stubble on his chin gleamed with dashes of silver as he spoke.\n\n'I don't think I could do the future bit.'\n\nI was thinking of Mrs Shapiro. When you see a good man you have to grebbit quick. Should I just grab him?\n\n'You could just speculate. Glue made from recycled carrier bags. Glue made from liposuction by-products. Glue made from stray cats and dogs. Glue made from boiled-up illegal immigrants. Melted-down social undesirables.' He gave me a sideways grin. 'No?'\n\n'Like you told me once the Nazis made glue out of Jews?'\n\n'Very good glue it was, too. Now Jews are trying to make glue out of Palestinians. But with less success.' He dropped his voice to a whisper. 'They say God told them to.'\n\nI stared at him. How could he joke about that? He saw the look in my eye.\n\n'Sorry, it's only metaphorical glue. A sticky mess. And I mean the Israeli state, not the Jews. We have to distinguish.'\n\n'Really?' What the hell was he talking about? 'I'm not sure I understand...'\n\n'I'm what they call a self-hating Jew. A gay self-hating Jew.'\n\nAh! Gay! That explained everything! I smiled inwardly, grateful that he'd told me before I'd made an utter fool of myself. But why the self-hatred? Could it be because he was gay?\n\n'Do you really hate yourself, Nathan?'\n\n'As much as custard.'\n\n'Custard's one of my favourite things,' I hurried to reassure him.\n\n'Mine, too. Especially made with eggs and vanilla with a sprinkling of nutmeg.'\n\n'So why...?' Maybe it was his height. 'You know...'\n\n'Sorry, Georgia, I didn't mean to inflict my obsessions on you. Self-hating is just a label the neo-Zionists use for people who disagree with them; you're either an anti-Semite or a self-hating Jew.'\n\nHe gave me a hunkily intelligent grin, pushing back his horn-rimmed glasses that had slipped down his gorgeous nose. Gay. What a shame!\n\n'We just got it out of a tin. Bird's.' I heard my voice prattling on, filling the silence. 'But they weren't anti-Semites, my parents. My dad's a socialist. He once thumped someone for calling the man in the fish and chip shop a wop. Mum's more... more of an anarchist, I suppose. She'd thump anybody for anything.'\n\nEven as I said it, I was thinking about the banter of the men in the Miners' Welfare at Kippax. Poofs. Gays. Queers. Pansies. They were the casual everyday slights that were the currency of contempt down our way. Dad might not be an anti-Semite, but I'd never heard him threatening to thump someone for using those words. Mum on the other hand had once ticked Keir off for calling one of his teachers a poofter. 'He's very nice, your Mr Armstrong, even if he is hormosexual.'\n\n'What about _your_ father?' I asked.\n\n'Yes, well, Tati moved in with me after Mother died, and Raoul moved out. It's sort of put paid to my love life.'\n\n'Is he rude to your friends?'\n\n'Oh, no. He just sings.'\n\nI laughed. 'That sounds nice.'\n\n'It is. But there are only so many lieder a person can take.' He murmured conspiratorially, 'I keep hoping a nice widow will take him off my hands.'\n\nWe'd stopped in front of another photo \u2013 it was a little girl whose hands were stuck together. She was crying, her mouth open, her eyes screwed up in pain.\n\n'Oh, dear. As it says in the manual, one of the disadvantages of adhesive bonding is that disassembly is usually not possible without destruction of the component parts,' Nathan remarked drily.\n\nIt was one of the things about adhesives that had always secretly troubled me. I stared. There was something so hopeless about the mess the girl was in that my heart went out to her.\n\n'I know what you mean by self-hating, Nathan. I hate myself sometimes.'\n\n'Do you, Georgia?'\n\n'Yes. I mean, I often feel stupid. Or hopeless. Or despicable. Or I just wish I was somebody else.' My voice was wobbling pathetically. 'I feel as though I've made a mess of my life.'\n\nWhat would it have been like, I wondered, to grow up on custard made with eggs and vanilla and nutmeg, instead of on Bird's powder and oven chips? Would I have been a different kind of person, more articulate and witty? Would I have had a high-powered career, or a string of bestselling novels? Would my husband not have left me? The trouble is, I was bonded to Rip; cyanoacrylate: a permanent bond. He was the only man I'd really loved, and however much I raged against him I knew I would never love anyone in that way again. I felt tears brimming into my eyes. Nathan slipped his arm around me and gave me a friendly squeeze.\n\n'Glue can be messy stuff.'\n\nI rested my head on his shoulder, which was at just the right height if I bent my knees a bit, and let the tears roll down the sides of my nose, big and warm. Nathan didn't say anything. He just stood there and let me cry. After a while I pulled a crumpled ball of tissue out of my pocket and dabbed my eyes.\n\n'Nathan, there's something I'd like to ask you.'\n\n'Fire ahead.'\n\n'Would you mind, on the way home, driving more slowly?'\n\n# 35\n\n# Uses of superglue\n\nI woke up next day feeling full of life. It was late \u2013 almost nine o'clock \u2013 and intermittent bursts of sunshine were pushing in beneath the elastic of the black knickers. The crying yesterday had refreshed me, like the rain in the night, and so much exposure to the possibilities of glue had fired me up with new enthusiasm for my work. Sitting up in bed I switched my laptop on. The article I was working on was about medical uses of adhesives. Cyanoacrylate (superglue) had been used effectively in emergency battlefield situations in Vietnam to hold wounds together until they could be sutured properly. Now a number of companies were trying to develop specialist adhesives to be used in place of suture. Human bonding.\n\nThere were two technical problems, it seemed, to be overcome. One, how to get the sides to hold together for long enough for bonding to take place. Two, how to achieve separation without tearing the flesh.\n\nThen I remembered. Cyanoacrylate AXP-36C. I fumbled in the bedside drawer for a scrap of paper to write it down on before I forgot. I tried to picture Rip's face when he realised he was stuck. I tried to picture his bottom, the agony of tearing flesh as he tried to free himself. Who would rescue him? Who would call the ambulance? Ottoline Walker? Or would it be me? Would I laugh? Would I minister gently to his adhered behind? So many possibilities!\n\nI put aside medical uses of adhesives, just for a moment, and opened my exercise book.\n\n# The Splattered Heart   \nChapter 7\n\n_One evening, as the Sinster family_ were was _were sitting down to their sumptuous_ tea dinner _evening meal in the vast candle-lit dining hall surrounded by deer's antlers and other dead things,_ they heard _the_ plangent pungent poignant _melodious_ twanging tinkling twinkling _(oh, sod this) sound of a mandolin assaultailled their eager ears and a moment later a tall dark handsome figure clad_ only _(clad only \u2013 what was I thinking of!)_ _in a swirling velvet cloak strode into the hall. After he had finished his performance Mrs Sinster threw him a few coins from her silk purse and said, 'Oh, Mr Mandolin Player, please come again. I am fascinated by your_ large mandolin _charming folk culture.'_\n\nPoor Mrs Sinclair \u2013 was I being a bit unfair? When I'd first met the Sinclairs, their world had seemed so alien and intimidating \u2013 governed by unspoken rules and veiled assumptions \u2013 but she had really tried to make me feel at home, had inducted me kindly into the arcane mysteries of napkin rings and the _Daily Telegraph_ crossword, and I suppose I must have seemed a sullen and ungracious daughter-in-law. At the time, it had irked me that they appeared to have no idea how privileged their lives were. It had irked me the way Mr Sinclair asked, in a hushed voice, whether I'd really met Arthur Scargill; I'm no great fan of the comb-over king, but the way the Sinclairs went on, you'd think he was the Antichrist himself.\n\nIt had taken me a long time to realise that the Sinclairs were probably as scared of me as I was of them. Okay, it can't have helped that on my third visit to Holtham I'd worn a large yellow badge with 'The enemy within' in bold letters. They must have seen me as an outrider of a sinister army bent on destroying order, decency, _Horse and Hound_ , and everything else they held dear. It wasn't long after the end of the miners' strike, and I thought they needed shaking up a bit \u2013 well, that's my excuse. Rip had tried to persuade me to take it off, but when I insisted, had stuck up for me valiantly and tried to explain to his bewildered parents what it was about.\n\n'But if it's supposed to be a _secret_ enemy, I can't understand why she's wearing a badge,' I overheard Mrs Sinclair whispering to Rip.\n\nYes, perhaps I was being a bit hard on Rip, too. But all's fair in love and fiction. I pressed on.\n\nSurprised in a compromising position with the mandolin player, Gina is expelled from Holty Towers. She protests that it was only a response to Rick's philandering, and determines to seek revenge by glueing his bottom to a toilet seat. The secret is to match the right adhesive to the adherends. Hurray! That would mean another visit to B&Q (strictly for research, of course). The trouble is, I couldn't help feeling a touch of sympathy for Rick. After all, he was just a weak and deluded male \u2013 easily led by the cunning spotty Spanish maid \u2013 he couldn't really help it. And Gina should have known better than to get involved with that wayward mandolin player. Something else was bothering me. I tried to focus on the image of Rip's bottom in the toilet seat, but the other photo from the glue exhibition kept intruding: the little girl, her screwed-up eyes as she tried to pull her hands apart; her scream.\n\nHauling myself out of bed, I stood at the window and looked down over the garden, stretching my arms above my head and waggling my shoulders, which were still stiff from the cold and tension of yesterday's car journey. The ground was wet, and the leaves on the laurel bush were dazzling with captive raindrops, but the sun kept coming in and out behind the rain clouds, casting fleeting rainbows across the sky. At the far end of the garden, a haze of mauve crocuses had spread, almost overnight. Birds were hard at work, hopping about in pairs with bunches of grass in their beaks.\n\nThen I spotted Wonder Boy slinking along the edge of the fence, making his stealthy way towards the blackbird couples. I banged on the window and they flew away. Wonder Boy looked up and gave me a long reproachful stare. I felt a pang of guilt. Okay, a visit to Mrs Shapiro was long overdue, I wanted to say to him, but it wasn't exactly easy, was it? The _HELP ME_ note Mrs Shapiro had sent was on my bedside table \u2013 I'd just scribbled the glue code on the envelope. As I looked at it with its scrawled-out name and address, I had a brainwave. \n\n# 36\n\n# The adhesion consultant\n\nAfter lunch, I dressed myself up in a red jacket that had belonged to Stella \u2013 I had to leave the buttons undone \u2013 and a glittery Oxfam scarf, and pulled a woolly hat down low over my hair. I put on bright red lipstick and an old pair of sunglasses by way of disguise \u2013 and made my way to the bus stop on the Balls Pond Road. Though in fact when I arrived at Northmere House I saw that my disguise was redundant, for there was a different guard-dog lady at the reception desk.\n\n'Can I help you?' she barked.\n\n'I've come to see Mrs Lillian Brown.'\n\nShe consulted her list. 'Are you family?'\n\n'A cousin. Once removed.' Well, I could have been.\n\n'Would you sign in please? Room twenty-three.'\n\nShe pressed the button that opened the sliding door. And in I went \u2013 into the muted realm of the pink carpet, the sickly chemical air, the rows of closed doors from behind which, from time to time, a television blared eerily. On the other side of the corridor was the long plate-glass sliding door which gave on to the courtyard with its square of grass and four benches, now all damp with rain. A demented bleeper sounded constantly in the background, reminding the absent staff that behind one of these closed doors, someone desperately needed help.\n\nI knocked on the door of number twenty-three. There was no reply so I pushed it open. The room was small and overheated, with a terrible deathly smell. A massive television set, volume on at full blast, dominated the room, so it took me a moment to notice the tiny figure lying motionless on the bed.\n\n'Mrs Brown?'\n\nThere was no reply. I shouted louder, 'Mrs Brown? Lillian?'\n\nI tiptoed over to the bed. She was lying there with her eyes closed. Her hand, I saw, was clutched around the bleeper on its cable. I couldn't tell whether she was breathing.\n\nI backed out and let the door close behind me. My chest was thumping. A fat woman in a pink corporate uniform was coming down the corridor.\n\n'In here,' I said.\n\n'Are you Mrs Brown's niece?' She seemed to be unaware of the bleeping alarm.\n\n'Actually, I'm...'\n\n'I hope you're not smuggling cigarettes.' She scrutinised me fiercely.\n\n'Oh, no. Nothing like that.'\n\n'Because last home I worked at, someone give an old lady a fag and some matches, and it all went up in flames.'\n\n'Oh, dear. Was anyone hurt?'\n\n'We was saved by a dog.'\n\n'Really?'\n\n'A mongrel,' she snorted. 'And then they tried to smuggle in a gearbox.'\n\n'A gearbox? What for?'\n\n'Beats me. Anyhow, matron got rid of it. Said it weren't hygienic.' Her face softened for a moment. 'It were a shame really, poor old man. Still, 'e got 'is revenge.' She chortled. 'Anyhow, we don't allow that sort of thing in 'ere. We got rules.'\n\n'Er... I think this lady needs some help...'\n\nBut she'd already vanished up the corridor. As I watched the door close behind her, I noticed there was now someone sitting out on one of the benches in the courtyard in the rain, a solitary hunched figure wearing a powder-blue dressing gown and matching peep-toe slippers, puffing away at a cigarette. It was the bonker lady.\n\nI banged on the window and waved. She looked up and waved back. But when I slid open the door and went out to join her in the courtyard, she put on a sulky face.\n\n'You never brought me cigs.'\n\n'I did,' I lied. 'You weren't there.'\n\nShe sniffed as though she knew it wasn't true.\n\n'Are yer lookin' for 'er? Yer pal?'\n\n'Mrs Shapiro. Yes.'\n\n'She's in solitary. She in't allaared visitors.'\n\n'Why not?'\n\n'Bin a naughty gel, ent she?'\n\n'Why? What's she done?'\n\nShe stubbed out her cigarette on the path and threw the butt into the middle of the lawn, where there was already quite a scattering.\n\n'It's what she ent done. She won't sign the Powah. Keeps refusin'. Bonkers, if yer ask me. They won't let 'er aht till she signs it.'\n\n'Do you know which room she's in?'\n\nThe rain had almost stopped. She pulled a cigarette packet out of her dressing-gown pocket and looked inside. There were only two left.\n\n'Yer won't forget me cigs next time, will yer?'\n\n'No. I promise.'\n\nShe placed one of the cigarettes between her lips and let it rest there for a few moments, savouring the anticipation, before she took the box of matches from the other pocket.\n\n'Twen'y-seven.'\n\n'Thanks.'\n\n'If she in't there, she'll be watchin' telly in twen'y-three. That's my room. They all watch telly in there.'\n\n'Isn't there a day room?'\n\n'Yeh, there is. But the telly's crap.'\n\nMrs Shapiro's room was just as small as the other one, and just as hot, but the smell was more sickly than deathly, and there was no television. She looked dreadful. She was lying on her bed, fully clothed, staring at the ceiling. Her hair was wild and matted, the silver line now a highway, her skin loose and baggy, folding in deep yellowy wrinkles around her mouth and chin.\n\n'Mrs Shapiro?'\n\n'Georgine?'\n\nShe struggled to her feet groggily and stared at me.\n\n'How are you?' I hugged her. She seemed so frail, like a bird. All bones.\n\n'Thenk Gott you come.'\n\n'I'm sorry I didn't come before. I tried, but they wouldn't let me in.'\n\n'Did you bring me cigarettes?'\n\n'Sorry. I forgot.'\n\n'Never mind. Good you heff come, Georgine. I do not want to die in here!'\n\nShe sat down on the edge of the bed and immediately started to cry, her skinny shoulders shaking. How small and bent she seemed. I sat beside her, stroking her back until the sobs turned to sniffles. Then I passed her a tissue.\n\n'We've got to get you home. But I don't know how.'\n\n'Too much guards in this place. Like in prison.' She blew her nose, then opened up the tissue to examine the snot. It had a horrible greenish colour. 'How are my dear cats?'\n\n'They're fine. Waiting for you. I've got some young men staying there, looking after them. Fixing the house up.' I saw the look of alarm on her face. 'Don't worry. As soon as you're ready to come home, they'll leave.'\n\nThe smell in the room was making me feel faint. I stood up and opened a window. The soupy overheated air stirred, and we could hear the traffic on the Lea Bridge Road, and voices of children playing somewhere nearby. Mrs Shapiro took a deep breath, and her eyes seemed to brighten a bit.\n\n'Thenk you, darlink.' She squeezed my hand, studying me with her wrinkled eyes. 'You looking better, Georgine. Nice lipstick. Nice scarf. You got a new husband yet?'\n\n'Not yet.'\n\n'Maybe soon I will heff a new husband.' She smiled archly to see the look of surprise cross my face. 'Nicky is saying he wants to marry mit me.'\n\n'Mr Wolfe?'\n\nI gasped. The scheming devil! I remembered how fluttery she'd been when he'd sat in her kitchen plying her with sherry.\n\n'I was thinking he would be for you the perfect husband, Georgine. But you heff showed little interest. So maybe this is an opportunity for me.' Her smile now was coyly flirtatious. She had cheered up considerably. 'What you think? Should I marry my Nicky?'\n\n'Does he know how old you are?'\n\n'I tolt him I was sixty-one.' She caught my eye and giggled. 'I am too notty for you, Georgine, isn't it?'\n\n'You are a bit naughty, Mrs Shapiro.'\n\n'Why get ready for the grave? It will catch you soon enough anyway, isn't it? Why not to enjoy the moment as it flies.' She flapped her hands like birds' wings. 'You know Goethe?'\n\nI shook my head. Then I thought of something.\n\n'Maybe it's because...' I remembered his intake of breath on the phone. 'I told him you had a son.'\n\nA son who would inherit her estate. Unless, of course, she remarried.\n\nShe looked at me sharply. 'How you know about this son?'\n\n'The social worker told me. Mrs Goodney.'\n\nShe stopped. I pretended to be looking out of the window. Go on, go on! I was silently willing her, but she went quiet.\n\nAfter a few moments, she said, 'Ach, this woman. All she thinks about is how to shvindel me. I told her I heff a son because she was wanting me to sign the Power of Returning. I said my son will be returning. He will heff the house.'\n\n'But he's not your son, is he?' I said gently.\n\nThere was a pause. 'Not mine. No.'\n\n'So who was his mother?'\n\nShe sighed. 'This whole _megillah_ is too long for you. You will be falling asleep before I tell it.'\n\n'But tell me anyway.'\n\n'It was the other one. Naomi Shapiro.'\n\nLittle by little, I drew it out of her. Her real name was Ella Wechsler, she said, pronouncing it carefully, as though not quite sure it belonged to her. She was born in 1925 in Hamburg. I calculated that would make her eighty-one. Her family was Jewish, but of the pick and mix variety. Speck but no sausages. Sabbath and Sunday. Christmas as well as Hanukkah \u2013 not that all this made any difference to the Nazis, when the time came. Her father, Otto Wechsler, ran a successful printing business; her mother, Hannah, was a pianist; her two older sisters, Martina and Lisabet, were students. Their house, a solid four-storey villa in the Grindel Quarter, was a hanging-out place for musicians, artists, heartbroken lovers, dreamers, travellers arriving or departing, four cats, and a German maid called Dotty. There was always coffee _mit schlagsahne_ , always music and conversation going on. She chuckled.\n\n'We were better at being German than the Germans. I thought this life was normal. I did not know such happiness was not permitted to Jews, Georgine. I did not know what it means to be a Jew until Herr Hitler told me.'\n\nBut by 1938, Hitler's message was loud and clear \u2013 clear enough for the family to realise they had to get out of Germany before things got worse.\n\n'You see in that time Hitler was thinking only how to clear out the Jewish people from Germany. The plan for exterminations came after.'\n\nThe Wechslers \u2013 Ella, Martina, Lisabet, and their parents \u2013 fled to London. Ella was nearly thirteen years old, Martina was seventeen, Lisabet twenty. In 1938 the Wechslers had been able to bribe their way out of Germany, but England did not hold out her arms in welcome. The 1905 Aliens Act meant that they could only come to Britain if they already had a job to come to.\n\n'Even the English they did not want us. Too many Jews were running away from pogroms in Poland, Russia, Ukrainia. Everybody thought it was a big sport to chase the Jews, isn't it?'\n\nThrough a cousin on his mother's side, Otto Wechsler had managed to secure a job in a print shop on Whitechapel Road \u2013 it was a huge ancient Heidelberg press which he coaxed back into life. The owner, Mr Gribb, was a widower from Elizavetgrad who had changed his name from Gribovitch when his family fled the pogroms in 1881. Hannah Wechsler became his housekeeper. Lisabet worked in a bakery. Martina trained as a nurse. Ella went to the Jewish school in Stepney. They lived in a poky two-roomed flat above the print shop ('Everything we touched was bleck from the ink') in the heart of the East End Jewish community, and they counted themselves blessed.\n\nThey received coded letters via Switzerland from their family describing the impact of the Nuremberg Laws, the enforced wearing of yellow stars, the terror of Kristallnacht, the expropriation of businesses, the expulsion from the professions (Cousin Berndt turned out of his surgery and made to sweep leaves in the park), the public humiliations, the increasingly ugly assaults in the streets (Uncle Frank's front teeth broken by a cheering, jeering gang of schoolboys). Actions that an individual would find morally repugnant became amusing when there was a crowd cheering you on. Then the mass transportations started, and the letters stopped.\n\nI felt the tremor in Mrs Shapiro's shoulders, the long catch of her breath. We were still sitting side by side on the bed. The light had faded in the window, and the roar of traffic outside intensified with the onset of rush hour. But we were in a different world.\n\n'Tell me about Artem. When did you meet him?'\n\n'In 1944 he arrived in London. In the spring. Eyes crazy like a madman. Still asking if they had seen his sister.'\n\nSkeletal, louse-ridden, hollow-eyed, he'd fetched up at the Newcastle docks on a British merchant ship that had snuck out from Gothenburg with a cargo of butter and ball bearings. The Seamen's Mission had taken him in and he was passed on, via Jewish relief organisations, to the flat in Whitechapel Road. He stayed with them for a year, helping to run the printing press and sleeping on a camp bed at the back of the workshop. He was clever with his hands. He didn't say much \u2013 he spoke Russian and only a few phrases of German and English \u2013 but his silence, brooding and mysterious, seemed to the girls to speak volumes. In his spare time, he started to make a violin. Lisabet, Martina and Ella watched him working with the fretsaw and the glue, his head bowed over the workbench, a thin self-rolled cigarette hanging from his lip, humming to himself. By then, Ella was eighteen, Martina was twenty-three and Lisabet twenty-six. All of them were a bit in love with him and a bit in awe of him.\n\n'Did he finish the violin?'\n\n'Yes. Gott knows where he got the strings. But in Petticoat Lane at that time you could buy all what you needed. When he was playing, it was like the angels in heaven. Sometimes I or _Mutti_ accompanied mit the piano.'\n\nI remembered the music in the piano stool. Delius. 'Two Brown Eyes'. Ella Wechsler. Her name was written in the front of the songbook, but the brown eyes had belonged to somebody else.\n\n'Do you still play the piano, Mrs Shapiro? Ella?' Somehow, the new name didn't seem to fit the old lady I'd grown fond of.\n\n'Look at my hends, darlink.'\n\nShe held them out in front of her, bony, with swollen joints and shrivelled brown-stained skin. I took them and warmed them in mine. They were so cold.\n\n'And Naomi? Who was she?'\n\nI had such a strong image from the photographs of the sweet heart-shaped face, the tumble of brown curls, the playful eyes. Mrs Shapiro didn't reply. She was gazing into a place beyond the dusky window. When at last she spoke, all she said was, 'Naomi Lowentahl. She was rather tall.'\n\nThen she went quiet again. I didn't interrupt. I knew she'd tell me in her own time.\n\n'Yes, nice looking. Always mit red lipstick, nice shmata. Who would heff thought she would be the type to go away digging in the ground in Israel?' Her mouth twitched. Another silence. She withdrew her hands from mine and started to fiddle with her rings. 'Some people said she was beautiful. Eyes always blazing like a fire. Yes, she was like a person on fire. She was in loff with Arti, of course.'\n\n'And he...?'\n\nShe sniffed. 'Yes. And he.'\n\nArtem Shapiro and Naomi Lowentahl were married in the synagogue at Whitechapel in October 1945, after the end of the war. Ella, Hannah and Otto Wechsler went to the wedding. Lisabet was away in Dorset on her own honeymoon with a Polish Jewish airman. Martina had been killed by a V2 rocket raid in July 1944, on her way home from the Chest Hospital in Bethnal Green \u2013 one of the last air raids of the war. But Mr Gribb put on a good spread for the couple. People came from all over Stepney just to get a bit of chicken.\n\nA sharp rap on the door made us both jump. Then without waiting for an answer, the woman in the pink uniform, the same one I'd met earlier, barged into the room.\n\n'Tea time, Mrs Shapiro.'\n\nShe caught sight of me.\n\n'You'll have to leave,' she said. 'Mrs Shapiro in't allowed visitors.'\n\n'I'm not a visitor. I'm a...' I thought fast. 'I'm an adhesion consultant.'\n\n'Oh.' That stopped her in her tracks. She looked me up and down, trying to assess my status. 'I thought you was Mrs Brown's niece. You'll 'ave to make an appointment through matron.'\n\n'Of course.' I stood up and put on a Mrs Sinclair-ish voice. 'If you could just leave us now. We've almost finished our consultation.'\n\n'I'll have to report it to matron.' She shook her head. 'We can't just 'ave people wandering in off the streets.'\n\nWhen we were alone again, Mrs Shapiro gripped my hands.\n\n'You will keep my secret, Georgine?'\n\n'Of course I will.'\n\n'What should I do?'\n\n'Don't sign anything. Don't marry Nicky.'\n\n'But if I am married, they will heff to let me go home, isn't it?'\n\n'I'll try to get you out.'\n\n'If I will say no to him, he will stop coming. Is better if I say maybe yes and maybe no.' She winked.\n\n'You're naughty, Mrs Shapiro,' I laughed. 'How does he manage to get in? Doesn't the matron stop him?'\n\n'He told them he is my solicitor.'\n\n'Ah. Clever. But...'\n\nActually, I thought, what she needs is a proper solicitor.\n\nThere was a sudden rush of footsteps and voices in the corridor. I kissed Mrs Shapiro on the cheeks and quickly said goodbye, just as they reached the door. The pink-overalled lady was in front, followed by a big green-cardied woman and a security guard. Their faces were flushed with purpose. But before they could say anything they were distracted by a ghastly scream from down the corridor outside number twenty-three. I ran out \u2013 we all ran out \u2013 to see the bonker lady waving her hands in the air and yelling, ''Elp! 'Elp! There's a dead body in 'ere!'\n\nThey forgot all about me in the ensuing chaos. I slipped out through the sliding door while someone else was rushing in, and kept my head down as I walked to the bus stop on the Lea Bridge Road. All the way home on the top deck of the bus, I was working out a plan to get Mrs Shapiro out. \n\n# 37\n\n# A trip to B&Q\n\nNext morning, I phoned Ms Baddiel. Amazingly, she answered on the first ring.\n\n'Oh, thank goodness I've got hold of you. Something terrible's happened. Mrs Shapiro's been kidnapped,' I gabbled. I didn't want to complicate things by mentioning the body.\n\n'Sssh. Ca-alm down, Mrs Sinclair. Now, take a deep breath for me. Hold. Two \u2013 three \u2013 four. Breathe out with a sigh. Two \u2013 three \u2013 four, and rela-ax.'\n\nI did as she instructed. My stomach-knot eased and my fists turned back into hands.\n\n'That's perfect. Now, you were saying...?'\n\nI tried to explain that Mrs Shapiro had been kidnapped and held against her will until she agreed to sign away her house. I avoided directly accusing Mrs Goodney of theft, but Ms Baddiel was more concerned that Mrs Shapiro's lifestyle choices were being violated.\n\n'There are a number of options open to her. If she is to live at home, the house needs to be made suitable. It's easy to move a bed downstairs and convert a living room into a bedroom. The problem is usually to create a downstairs bathroom. Alternatively, of course, she could install a lift. Even a stairlift.'\n\n'Mm. Yes. Good idea. I've got some men in there at the moment, fixing it up. I could ask...'\n\n'Perfect.'\n\nI tried to picture Mr Ali and the Uselesses installing a stairlift. Mmm. No.\n\n'There used to be grants available for that kind of work, but unfortunately now it usually has to be self-financed. Has she got any funds, do you know?'\n\nI thought of the receipts from the secondhand traders I'd found in her bureau drawer.\n\n'I'm not sure. I'll ask her.' Though I knew as sure as hell she wouldn't tell me. My heart sank. Then I imagined trying to persuade her to have a stairlift installed.\n\n'And we could increase her care package. I take it that worked out all right?'\n\n'Yes. Fine. Fantastic.'\n\nWe arranged to meet at the house on Friday. I wanted time to be sure that the Uselesses had made some progress, and to check that the place was at least habitable. Ms Baddiel undertook to visit Northmere House in the meantime, and to challenge the terms of Mrs Shapiro's incarceration.\n\n'It's a violation of human rights,' she said confidently in her peachy voice.\n\nOn Wednesday afternoon I set out to visit Mrs Shapiro at Northmere House again. I walked down to the Balls Pond Road to get the Number 56, and I must have dozed off on the bus (or _sunk into a reverie_ , as Ms Firestorm would put it) for when I looked out of the window we were already on the Lea Bridge Road, and I realised I'd missed my stop. I rang the bell hastily and raced down the stairs, and when the bus finally came to a halt I found myself standing near a familiar jolly orange-and-grey building. Another branch of B&Q! It must be destiny, I thought.\n\nThe B&Q store was tattier than the one at Tottenham and almost empty, silent with a hush of reverence \u2013 like a temple, I thought, dedicated to some peculiar male cult. The high ceilings and echoing aisles, the air of solemn devotion, the acolytes walking with bowed heads, the obscure objects of veneration, the mysteries. Apart from me, there was only one other woman in the place, a stunningly pretty Asian girl with a sparkling nose stud, on one of the checkouts. With the air of a slightly bored priestess, she pointed me in the direction of the adhesives on aisle twenty-nine.\n\nCyanoacrylate AXP-36C. I had pulled the crumpled Mrs Brown envelope out of my pocket and started to look at the labels on the packaging. It was easy enough to distinguish between the PVAs, the epoxies and the acrylates, but there didn't seem to be one with that precise formulation. A number of them carried warnings about misuse. I browsed the packets, looking for the ones with the direst warnings.\n\nAfter a while, a nice blokey type appeared and asked me if I needed any help. I showed him my paper. He studied it for a few moments with a puzzled frown, then asked, 'What's it for, sweetheart?'\n\nI noticed that he had a Kent NUM tattoo on his forearm. How strange, I thought; if I had met him instead of Mr Ali when I was first looking for the lock, there would have been a different point of connection, and quite a different story.\n\n'It's for... er... just, you know, general use.' I smiled mysteriously, picked up a few superglues, and put them in my shopping basket with a nonchalant air.\n\nAnother discovery I made, by the way, at the end of the adhesives aisle, is that duck tape has nothing at all to do with ducks. No quacking or waddling involved. In fact it's duct tape. What a disappointment.\n\nOut of interest, I passed by to look at the toilet seats. Although they had exotic names \u2013 Chamonix, Valencia, Rossini \u2013 they weren't in fact very exciting. There were no musical ones or ones which lit up, as I'd seen advertised in the Sunday papers \u2013 seats designed to attract a curious bottom. I'd have to look on the internet. Ideally, I should get one that played a ridiculous but catchy tune like 'Jingle Bells' or 'The Birdie Song', that would keep going until the person got up \u2013 _if_ they _ever_ got up!\n\nBy the time I remembered I'd meant to call at Northmere House on the way back, I'd already overshot the stop again. That would have to wait for another day. I was filled with a pleasant feeling of satisfaction on my way home, sitting in my favourite seat, upstairs at the front, with my purchases in a bag on my knee, and enjoying the changing patterns of clouds and light as the bus lumbered down the Lea Bridge Road.\n\nAt Clapton a group of schoolboys got on, jostling and giggling. I didn't notice at first that they were wearing small skull caps. They crowded on to the top deck, and made a rush for the other front seat, all four of them, barging with their backpacks and trying to shove each other out of the way. Mrs Shapiro's story was still fresh in my mind, and I wanted to talk to them, to ask about their parents and grandparents \u2013 about the countries they'd left and the journeys they'd made. But why should they have to worry about any of that old painful stuff? These lads \u2013 they didn't have the air of exiles. They were gossiping about one of their teachers who, apparently, had been spotted at a Westlife concert wearing a dress that revealed too much. Let them be, I thought. Let them be happy. As we thundered along among the treetops, I closed my eyes and felt through my eyelids the brilliant spring light flicker over my face: dark-light-dark-light-dark-light. When I got off at Balls Pond Road, a few stops later, I could still hear their peals of laughter as the bus pulled away. Let them be happy while they can.\n\nAs I turned the corner into my road, I saw there was a car parked outside my house. A black car. A Jaguar. I stopped. How long had he been waiting for me? Since the debacle with Nathan, I'd been feeling a sort of blank emptiness inside me. Now I felt my heart quicken, a beat between panic and pleasure. Or maybe I was just inexorably drawn. I carried on walking, wondering what I should say. As I got closer, the driver's-side door opened, and he stepped out on to the pavement, all lean and hungry six-feet-something of him, with a bunch of flowers in his hand \u2013 blue irises. My heart did a skip.\n\n'Doing a spot of DIY, are you, Georgina?' He was looking at my B&Q carrier bag with interest. 'Have you got time for a quick word? About Canaan House? There are some... er... developments you should know about.'\n\n'Developments?'\n\nI glanced at my watch. It was just turned three o'clock.\n\n'It'll have to be quick. Ben'll be back soon.'\n\nI noticed he had a fresh white handkerchief in his jacket pocket, and despite my resolution, a tremor like a Pavlovian response ran through me.\n\n'I thought you should know... my colleague, Nick Wolfe. You were right. His intentions are not honourable. _Very_ not honourable.'\n\n'You'd better come in.'\n\nHe followed me into the house. I shoved the B&Q bag in the bottom of a cupboard in the mezzanine study on my way down into the kitchen, and put the kettle on. While it boiled, I arranged the irises in a vase. They reminded me of Mrs Shapiro's toilet bowl. He stood very close beside me, watching. I could sense the heat of his body through the centimetre of air between us, and that pleasant pelvic glow \u2013 it was the shameless woman, putting in a surprise guest (gusset) appearance.\n\n'Tell me,' I said.\n\n'Yes. Nick. He's \u2013 how can I put it? \u2013 he's got obsessed with Canaan House. He's commissioned an architect; had plans drawn up to turn it into a gated community. Luxury flats. Done out to the highest spec. Penthouse suite. Basement gym. Enclosed Japanese-style garden with pebble and stone water-feature. The full monty. Plus six mews studios.'\n\nI took a deep breath. I could smell the expensive soap, and beneath it the chlorine.\n\n'Okay. And so what's he planning to do with Mrs Shapiro?'\n\n'He's planning to marry her.'\n\nHe delivered his punchline with a slight lift of the eyebrows. I pretended to be shocked, but inside I was smiling.\n\n'Apparently they struck up quite a friendship, and one day he asked her age. She had him on that she was sixty-one. Well, that roused his suspicions, so he sneaked a look at her medical records in the nursing home. They gave her age as ninety-six.'\n\n'No! Really?' I feigned surprise.\n\n'He thought \u2013 well, at that age her life expectancy \u2013 how can I put it? \u2013 it left a lot to be desired. A couple of years, at most. He reckoned he was on to a good thing.'\n\n'Did he tell you she has a son?'\n\n'He mentioned something along those lines. That's why he's in a hurry to tie the knot. If she's married to him, he gets the lot when she pops her clogs. Unless she's made a will, of course.'\n\n'The son's supposed to be coming all the way from Israel. He obviously thinks he's on to a good thing, too. But I don't know if he's really her son. Her husband was married before, you know.'\n\nBefore what? That's what I couldn't work out. If Ella Wechsler had married Artem Shapiro, her name would have become Ella Shapiro. But why had she changed her first name from Ella to Naomi? Why would someone change their whole name?\n\n'If she wasn't married to him,' I was thinking aloud, 'if she was just living with him...'\n\n'Mm. Good point. Would she still have a claim on the house?' I could see his mind working in the gold flickering of his eyes.\n\n'Does it make any difference, who was married to who? Surely, if she's lived there all these years, the house is hers?'\n\n'It depends on how the deeds were drawn up.' He was stirring the sugar into his coffee, tinkling the spoon against the china and looking at me with those vari-coloured eyes. I could feel myself melting inside. 'It'd be interesting to sneak a look, Georgina. Do you know where they're kept by any chance?'\n\nThey were probably among the sheaves of paper up in the attic. 'I haven't a clue,' I said, squeezing my tea bag sexily and fishing it out with a provocative little flick of the spoon.\n\n'It might be possible to find out from the Land Registry,' he murmured.\n\nHe finished his coffee and stood up, leaning in the doorway, smiling darkly. 'Shall we...?'\n\nHe led, I followed.\n\n'You said you were going to show me your poems,' I said, teasing, but to my surprise he produced a slim cream envelope from the pocket of his jacket \u2013 not the handkerchief one, the one inside the lining.\n\n'I've written one specially for you, Georgina.'\n\nThe envelope was slightly warm and curved to the contours of his body. I opened it curiously as he undressed me. There was a poem, written out by hand, the letters squat and confident on the creamy paper.\n\n_I wandered through the city streets_\n\n_My heart was burdened down with care,_\n\n_And then I saw thee standing there_\n\n_With raindrops sparkling on thy hair._\n\n_Sweet Saint Georgina, thou art_\n\n_The dragon slayer of my heart._\n\n_Tell me thou love me, for I know_\n\n_We'll never be apart._\n\nI couldn't stop myself; I cringed; then I covered it up with an embarrassed kiss.\n\n'Mmm. That's lovely,' I said.\n\n'Glad you like it, sweetheart. Have you got the...?'\n\n'The...?'\n\nI fumbled in my bedside drawer for the shameless accessories, and slipped them on. He checked the gusset. He tightened the satin handcuffs. Thank heavens for IKEA slatted headboards. Where would we be without them? thought the Shameless Woman as she sighed and lay back on the pillows. But the poem \u2013 the ugly doggerel \u2013 jangled in my head. I tried to abandon myself to shamelessness but it was no use. 'Sweet Saint Georgina...Tell me thou love me...' And to think I had once dreamed of having an entourage of poet toy-boys! In the end I just had to fake it. Afterwards, when I was lying tense and sweaty in his arms, and he was stroking my hair and doing his hankie thing, I had a sudden memory of the first night Rip and I had spent together in his attic flat in Chapeltown. We'd lain together in the crumpled sheets looking at the candlelight flickering on the sloping ceiling, and he'd reached down a well-thumbed book from his shelf and read me John Donne's 'The Sunne Rising'. ' _She is all states, and all princes I. Nothing else is._ '\n\nWhat had happened to _that_ Rip \u2013 not the always-in-a-hurry destiny-shaping Progress Project Rip, but the other Rip who was as bouncy as a puppy, curious, funny, eager, idealistic, who read Donne and Marvell when we made love, and brought me Marmite on toast in the morning? What had happened to _him_? A pain like the shock of bereavement hit me right in the heart, making me flinch. What was I doing here? Why was I in bed with this man?\n\n'Why did you use those words, thee and thou?' I asked.\n\n'Don't you like them?'\n\n'I do, but... they're a bit old-fashioned.'\n\n'You strike me as being \u2013 how can I put it? \u2013 quite an old-fashioned girl, Georgina.' He ran a finger down my cheek. 'I can change it if you don't like it, sweetheart.'\n\nThe trouble is, I realised, I only wanted him wicked and wolfy. I didn't want this touchy-feely gooey stuff. And I definitely didn't want the poetry.\n\n'No, leave it. It's fine as it is. But... it should be thou lovest, not thou love.'\n\nAs soon as I said it, I wished I hadn't. I didn't mean it as a put-down \u2013 it was just my Eng Lit degree popping out in the wrong place.\n\n'Lovest?' He sounded utterly crushed.\n\n'But it's fine as it is. Romantic. Please! Don't change anything!'\n\nBut he was already sitting up and putting on his neatly folded clothes.\n\n'Mark, you've forgotten...'\n\n'Lovest!'\n\nThe door closed with a quiet click and he was gone.\n\nI lay there for a while thinking about the poem. It wasn't just the archaisms that bothered me, it was the flaky metaphor of Saint George and the dragon, and that ugly fore-shortened last line, like a broken tooth. You'd have thought he could have found a couple of spare syllables to patch it up with. A sudden vivid memory caught me off guard: it was the first time Rip and I went down to Holtham at Christmas. Rip slipped his hand between my thighs as I drove and read me Donne's poem 'Nocturnal upon St Lucy's Day' as we crossed the wintry Pennines, rough with browned-off heather beneath which new shoots were already pushing into the black oozing peat. ' _I am every dead thing, In whom love wrought new alchemy._ ' I was so overcome with passion that we had to stop in a lay-by. It's not easy to make love in the back of a Mini, but I remembered how our bodies closed together like two shells of a bivalve.\n\nRiding in on the memory came an intense pang of longing for Rip \u2013 for his warm solid body, his alert clever mind. In spite of the Sinclair confidence that bordered on arrogance, in spite of the Progress Project and the destiny-shaping work, in spite of the dereliction of DIY duties and the irritating BlackBerry habits, in spite even of the Scarlet-mouthed Slut, he was still Ben and Stella's dad; yes, and he was still the man I loved. Maybe it was time to stop messing around with other men and start glueing together my marriage.\n\nJust then, the front door slammed. It must be Ben letting himself in. I sat up and... no, I tried to sit up, but my wrists were still firmly strapped to the headboard. I tugged. Nothing happened. Irritated, I pulled harder, but the Velcro held fast.\n\n'Mum?' Ben called from the kitchen.\n\n'Hi, Ben. I'm just finishing something off. I'll be with you in a minute.'\n\nFor God's sake. It was only Velcro. But because of the way it was fastened on my wrists, when I tugged I was just pulling tighter on to the join. I tried to squeeze up my hands and slip them through the loops, but there was no slack. I could hear the crickle-crickle of the Velcro hooks under strain. Then the crickling stopped. My thumb joints were still in the way. My wrists were getting sore. My arms were aching. My heart was racing. _Don't panic. In \u2013 two \u2013 three \u2013 four. Out \u2013 two \u2013 three \u2013 four._\n\n'D'you want some tea, Mum?'\n\n'Lovely, thanks. NO! No, it's all right. Just put the kettle on. I'll come down.'\n\nNext I tried using my teeth. I found that if I strained and wriggled, I could get my mouth within an inch of my left wrist. Half an inch. But no more. I tried the other side. That was worse. My arms weren't long enough. Or maybe they were too long. I went back to the left side. I strained and strained. If I stuck my tongue out I could even touch the Velcro with the tip of it \u2013 I just couldn't get it with my teeth. When my shoulder felt as though it was going to break, I gave up. Exhausted, I lay back on the pillows and considered my options. Then I realised I had no options. Well, the only option was to call out to Ben for help. That wasn't really an option. I'd rather die. Then I became aware of another unpleasant sensation. I needed a pee.\n\n'Kettle's boiled!'\n\n'Right! Thanks!'\n\nI could tell Ben it was an accident. Oh, yeah. I could pretend I'd been trying out an experiment. Playing a game. Practising for a pantomime. Like you do. Trouble was, the duvet was down around my knees, and I was still wearing the red panties. And nothing else. There was nothing for it but to go back to the crickling. Each little crickle-crickle was a hook opening, I told myself. Just take it slowly. Forget about the bladder. Concentrate on the wrists. Concentrate on one wrist at a time. I seemed to have more power in my right wrist. I found that by moving the thumb joint and flexing my fingers up and down I could increase the crickling. Cricklecrickle-crickle. Crickle-crickle-crickle. The more gently I did it, the more it crickled. I could move my right thumb quite a bit now. I could fold it into my palm and ease it... ease it... yes, there it goes. My right hand was free. I reached across and freed my left hand. Then I grabbed my dressing gown and dashed to the toilet.\n\n'Is everything all right, Mum?'\n\n'Yes. Just get that kettle on.'\n\nTwo minutes later I strolled into the kitchen wearing my jeans and jumper and an insouciant smile on my face. I poured the hot water over the tea bag.\n\n'Thanks, Ben. Just something I had to get finished by today.'\n\nHe studied me curiously. I slipped my hands behind my back so he wouldn't see the raw marks on my wrists.\n\n'Are you all right, Mum? You look a bit... red.'\n\n'Red?' I blushed.\n\n'Have you been in a fight?'\n\n'No. Not exactly. Why?'\n\n'You seem \u2013 sort of \u2013 irregular.'\n\nIt wasn't until I had another pee at bedtime that I spotted the red lace-trimmed panties still crumpled up on the floor in the toilet. Had Ben noticed them when he went upstairs? Should I say anything? Should I pretend they were Stella's? (Shame on you, Georgie!) Or should I just keep quiet? That's what I did. \n\n# 38\n\n# Without walls\n\nBen and I had taken to sometimes having our tea in front of the gas fire in the sitting room with the television on in the background \u2013 a comfy Kippax habit, which we'd adopted now there were just the two of us. So there we were on Thursday balancing our plates on our knees and watching the seven o'clock news \u2013 the usual gloom, doom and trivia. I was about to flick the remote when an item came up about the nuclear missile defence shield that the Americans were supposed to be stationing in Poland to stop missiles from Iran. I know my geography's a bit shaky, but wasn't that, like, the wrong continent? Then I noticed Ben had gone very still.\n\n'I wouldn't worry,' I said. 'I'm sure it won't work, anyway.'\n\nBen was staring at the screen.\n\n'It's the prophecy. Gog and Magog.' His voice was almost a whisper. 'They're getting ready for the missiles.'\n\n'What missiles?'\n\nBen put his plate aside, slid off the sofa, and knelt in front of me.\n\n'Mum, I'm begging you. Take Jesus into your heart.'\n\nHe stretched out his hands to me as if he was pleading or praying \u2013 my poor broken-in-half boy. I took his hands \u2013 they were shaking. I knew that nothing I said would be the right thing, so I kept quiet and just held his hands tight in mine. Then he closed his eyes, and started to speak \u2013 it was more like a chant \u2013 in that grating up-talk inflection.\n\n'Ezekiel thirty-eight? Thus saith the Lord God? Behold, I am against thee, O Gog, Prince of Meshech and Tubal. I will turn thee back, and put hooks into thy jaws, all thine army, horses and horsemen? Persia, Ethiopia, and Libya with them? Gomer, and all his bands; the house of Togar-mah of the north quarters, and all his bands? All of them clothed in all manner of armour _,_ shields and swords?'\n\nIt reminded me of the _Lord of the Rings_ poster on his wall: the Orcs with their sub-prime dentistry, the vast exotic computer-enhanced armies marching into the field. I would have dismissed it as lads' fantasy, but for what came next.\n\n'In the last days I will bring thee against my land? Thou shalt come into this land, that is brought back to peace from the sword, gathered out of many people to dwell amongst the mountains of Israel, which had been a wasteland before, but is brought forth out of the nations. And they shall dwell safely? All of them? Dwelling without walls, and having neither bars nor gates?'\n\nHis voice was wobbling.\n\n'Oh, Ben...' I squeezed his hands. Phrases from Naomi's letter from Israel \u2013 the letter I'd found in the piano stool, which I'd reread so many times, flashed into my mind. _Our place of safety_... _of barren wasteland_... _our people gathered out of every country where we have been exiles_... _a land without barbed wires._ But Mr Ali had told me there were walls now, and checkpoints, and barbed wire.\n\n'And I will rain upon him an overflowing rain, and great hailstones, fire, and brimstone.' Ben's eyes were still closed. Then he looked up at me. 'Take Jesus, Mum. Please? Before it's too late?'\n\n'Okay, Ben. Okay.'\n\nHe was pale and trembling.\n\n'But you don't believe it, do you?' He shook his shaved head \u2013 covered now with fine dark stubble \u2013 in a gesture that could have been frustration or despair.\n\n'Well...'\n\n'You're just saying it to keep me happy, aren't you?' His eyes were liquid, backed up with tears. 'What's the point? What's the point in being saved, if everybody... like everybody you really love is going to be damned?'\n\nThe television was still burbling in the background, and I flicked the remote to turn it off \u2013 to cut off that terrifying stream of madness that kept leaching into our own little fireside world.\n\n'Come here, you.' I pulled him up on to the sofa beside me, and put my arms round his shoulders, squeezing him tight. 'It's just talk and posturing. It'll all blow over.'\n\nI said it with a confidence I didn't feel, putting on a brave face for Ben, for a part of me was scared, too. However much my rational mind dismissed the gibberish of prophets, there was a dark cave hidden away beneath my brain where the monsters slept, fears and nightmares chained up since childhood, but still with a residual power to instil dread. We sat together, holding hands and listening to the silence settling back into the room. It was raining again outside, a soft pit-pat, not a downpour. I could hear Ben's breathing getting slower. His hands were very cold.\n\nSuddenly, outside in the street, we heard the sound of a car pulling up, tyres splashing in the rain, a diesel engine idling, footsteps on the path, a knock on the door. Ben and I stared at each other. There was another louder knock, then a man's voice \u2013 an unfamiliar voice \u2013 'Anybody there?'\n\nI got up and opened the door. I didn't recognise the man standing there, a bulky, dark-skinned man. But after a moment, I realised he was the driver of the taxi that had pulled up outside on the road. Then the door of the taxi opened, and out clambered Mrs Shapiro.\n\n'Georgine!' she exclaimed. 'Please \u2013 help me! Do you heff some money for the taxi?'\n\n'Of course,' I said. 'How much?'\n\n'Fifty-four pound,' said the taxi driver. He wasn't smiling.\n\n'Isn't that a bit...?'\n\n'It should be more than that. We been going round in circles for hours.'\n\nI went to look in my purse. I had forty pounds and some change.\n\n'Ben, can you help?'\n\nHe was standing behind me, trying to work out what was going on.\n\n'I'll have a look.'\n\nHe went upstairs. I remembered some loose coins in my duffel coat pocket. And there were some pounds I'd put in the Barnardo's envelope by the door. Ben came down with a fiver. Mrs Shapiro fished a pound coin out of the lining of her astrakhan coat. Between us we rustled together \u00a352.73. The taxi driver took it crossly, mumbled something, and disappeared.\n\n'Come in, come in,' I said to Mrs Shapiro.\n\n'Thenk you,' she said. 'Some persons are living in my house. Will not let me in.'\n\nAs she crossed the threshold, Wonder Boy appeared out of the darkness and slunk in beside her.\n\nShe sat by the fire cradling a mug of tea in her hands, which Ben had brought on a tray, with some chocolate digestives.\n\n'Thenk you, young man. Charming. I am Mrs Naomi Shapiro. Please \u2013 tek a biscuit.'\n\nWonder Boy stretched himself out in front of the fire and started rubbing himself up against the _Lion King_ slippers, making a gruff rasping noise which was as close as he could get to purring. Through sips of tea and mouthfuls of biscuit crumbs, she told us the story of her escape.\n\nAfter the discovery of the dead body, the bonker lady had become totally bonkers.\n\n'Crezzy. Brain completely rotted away.'\n\nNot content with hanging around in the corridor cadging cigarettes from visitors, she would embellish her patter with an invitation.\n\n'I show you the dead body if you give me a ciggie.'\n\nThat upset the staff \u2013 they thought it was giving a bad impression of the home. From time to time, just to wind them up, she would rush down the corridor yelling ''Elp! 'Elp! There's a dead body in 'ere!' It came to a head when a party of relatives accompanying their aged mother on an inspection tour were accosted by the bonker lady, who somehow led them to believe (they were all smokers) that finding corpses was almost a daily occurrence. The staff member who was showing them round lost her rag and tried to push the bonker lady back into her room.\n\n'But she was fighting them like a tiger. Clawving and skretching mit the hends!'\n\nIn the end the security guard had to be called. Then matron arrived in her green cardie with an ampoule of sedative and a needle, but the bonker lady kept struggling and yelling, ''Elp! 'Elp! They gonner kill me!'\n\nThe relatives, rattled by so much violence, tried to call the police on a mobile phone. By now all the residents \u2013 those of them who were upright \u2013 had crowded into the corridor and were cheering the bonker lady on. In all the kerfuffle Mrs Shapiro managed to slip unnoticed through the door into the lobby and out on to the Lea Bridge Road, where a passing taxi whisked her to safety.\n\n'And here I am, darlinks!' she exclaimed, flushed with the excitement of her adventure. 'Only problem is some persons are living in my house. We must evict them now!'\n\nShe put her empty cup down and rose to her feet. I tried to persuade her to stay for a bite to eat, and even offered her a bed for the night, but she was desperate to get home. Wonder Boy had stopped purring and was thrashing his tail against the floor.\n\nWe set off down the road, Mrs Shapiro leading the way \u2013 it was surprising how fast she could move in those _Lion King_ slippers \u2013 Ben and I lagging behind, and Wonder Boy bringing up the rear. It was quite dark and cold, the air still damp from the recent rain. As we turned into Totley Place a couple of the other cats appeared out of the bushes and tagged along, too. Violetta was waiting for us in the porch, ecstatic with pleasure at Mrs Shapiro's return. Wonder Boy hissed, batted her with his paws, and sent her packing.\n\nThere were lights in some of the windows, and this was surprising in itself, because I'd never before seen Canaan House lit up so brightly from the inside. I noticed that the front door had been painted yellow and the broken floor tiles in the porch replaced with what looked like modern bathroom tiles. While Mrs Shapiro was fumbling for her key, I rang on the doorbell.\n\nIt was Mr Ali's nephew, Ishmail, who answered the door. He recognised me at once, and beaming broadly gestured to us to come inside.\n\n'Welcome! Welcome!'\n\nHe'd learned another word. The inside of the house had been painted, too, in white and yellow. It looked lighter and fresher, and smelled much better. I saw Mrs Shapiro looking around, and tried to judge the expression on her face. She seemed to be quite pleased.\n\n'You've been busy,' I said to Ishmail. 'This is Mrs Shapiro. She's the owner of the house. She's come home now, so I'm afraid you'll have to leave. It's what we agreed. Remember?'\n\nHe smiled and nodded blankly. He obviously had no idea what I was on about. I tried again, talking more loudly, with accompanying gestures.\n\n'This lady \u2013 live here \u2013 come back \u2013 you must go \u2013 go now.' I pointed at Mrs Shapiro and made shooing hand movements.\n\n'Yes. Yes.' He smiled and nodded.\n\nThen Nabeel appeared on the scene, and joined in the smiling and nodding, offering his three words of English.\n\n'Hello. Please. Welcome. Hello. Please. Welcome.'\n\n'Hello. Yes, please. Welcome,' said Ishmail.\n\nI went through my pointing and shooing routine. They smiled and nodded.\n\n'Hello. Yes. Please.'\n\nWe were getting nowhere.\n\nThen Ishmail \u2013 you have to credit him with some intelligence \u2013 got his mobile phone out, keyed a number, and started talking in Arabic to the person at the other end. After a few moments he passed the phone to me. It was Mr Ali.\n\n'You'll have to tell them to leave,' I said. 'Now that Mrs Shapiro's home. They can't stay. You promised \u2013 remember? I'm really sorry. I thought we'd have some warning, but...' I was getting a bit hysterical.\n\nI passed the phone to Ishmail. He listened for a few moments, then uttered a stream of Arabic, then listened again, then passed the phone to me.\n\n'Tonight too late. I have no van.' Mr Ali's voice sounded faint and crackly. 'Please let them to stay for tonight. Tomorrow I come with van.'\n\n'Okay,' I said. 'Just tonight. I'll talk to Mrs Shapiro. Mr Ali, thank you for the work you've done \u2013 the painting \u2013 it looks wonderful.'\n\n'You like this yellow colour?'\n\n'Very much.'\n\n'I knew you would like it.' He sounded pleased.\n\nMrs Shapiro had lost patience with our three-way conversation, and had disappeared somewhere. Ben and Nabeel had wandered off into the study, where a television had been rigged up with an internal aerial. They were watching football, sitting side by side grinning and cheering when a goal was scored. Nabeel pointed to himself and said, 'Hello! Please! Arsenal!' Ben pointed to himself and said, 'Hello, Leeds United!'\n\nI found Mrs Shapiro in her bedroom. She was curled up in bed with Wonder Boy, Violetta, Mussorgsky and one of the pram babies. Wonder Boy had actually got under the covers with her. They were all purring, and Mrs Shapiro was snoring. \n\n# 39\n\n# Home improvements\n\nNext morning, I woke up with the feeling that I had something important to do, but I couldn't remember what it was. I'd left Mrs Shapiro sleeping at the house last night, and I thought maybe I should go back this morning and check up on her. Then the phone rang. It was Ms Baddiel, reminding me of our meeting. After I'd put the phone down, I had a bright idea. I picked it up again, and dialled Nathan's number.\n\n'I wonder whether you could give us some advice. About the use of modern adhesives in home improvements. This morning. Eleven o'clock.' I gave him the address.\n\n'Great. I'll bring the DIY demonstration kit.'\n\n'Bring your father, too.'\n\nI smiled as I put the phone down. Matchmaking is a game that two can play.\n\nI went up there a bit earlier to make sure everything was shipshape for Ms Baddiel, and to supervise the departure of the Uselesses \u2013 I hoped they'd be all packed up and ready to go. When I rang on the bell at about half past ten, it was Ishmail who opened the door again and invited me in. The house was pleasantly warm, and smelled of woodsmoke, freshly brewed coffee and cigarettes. I followed him through to the study at the back of the house where a fire had been lit in the hearth. They were burning sheaves of papers and bits of old wood \u2013 including some of the boards that had been taken down from the windows. The television was on, and a sofa, still draped in a white dust sheet, had been dragged through from the sitting room. On the sofa sat Mrs Shapiro and Nabeel. They were smoking and drinking coffee from the silver pot and watching _The Hound of the Baskervilles_ in black and white on the television. Mrs Shapiro was wearing her candlewick dressing gown and her _Lion King_ slippers. Violetta was curled up on her lap, Mussorgsky was on Nabeel's lap and Wonder Boy was stretched out on the rug in front of the fire. It was a scene of cosy decadence.\n\n'Georgine! Darlink!' She swivelled round and patted the empty space at the end of the sofa. 'Come and drink a coffee mit us.'\n\n'Maybe later,' I said. 'We have to get ready. The social worker's coming.'\n\n'What for I need the social work?' Mrs Shapiro sniffed. 'I heff my young men.'\n\n'But they're going home now, Mrs Shapiro. They have to go.'\n\nOn the screen, the hound started roaring terrifyingly. Wonder Boy pricked up his ears and started swinging his tail. Mrs Shapiro gripped my hand.\n\n'This dog is a monster. Same like the matron in the Nightmare House. Grrah! I will not go back to this place. Never.'\n\n'No, definitely not. But _this_ social worker is nice. She'll help you to stay at home. It's Ms Baddiel. You met her before. Remember?'\n\n'I remember. Not Jewish. Too fet.'\n\nShe'd lost interest in our conversation, and was watching the fearsome hound racing over the darkening moors.\n\nIshmail thrust a cup of coffee into my hands. It was thick, black and bitter. He handed me the sugar bowl and though I don't usually take sugar I helped myself to a couple of heaped spoonfuls. I declined the cigarette he offered me, but Mrs Shapiro took it and lit it from the end of the one that was still smouldering in the ashtray at her feet.\n\n'What is this brown boots?' she asked, coughing a little.\n\nAs I was trying to explain the significance of the black and brown boots in the plot, the doorbell rang.\n\nThe other three were completely gripped by the drama so I got up to answer it. Ms Baddiel was standing there. She was wearing a floaty silk aquamarine coat, and her honey-gold hair was twisted up in a loose braid. Behind her on the porch stood Nathan, with a large attach\u00e9 case under his arm, and Nathan's Tati, looking very spruce in a collar and tie. They had obviously introduced themselves already.\n\n'Nathan's come along to advise us about adhesives,' I said. 'In case there are any urgent repairs that need doing.'\n\n'Perr-fect.' She followed me through to the study, sniffing the air and looking around her, taking in all the improvements. 'Lovely.'\n\nMrs Shapiro hardly looked up as we came into the room, her eyes were fixed on dashing Basil Rathbone on the screen, but Ishmail, with impeccable politeness, jumped up and offered Ms Baddiel his corner of the sofa.\n\n'Hello, Mrs Shapiro.' She leaned forward towards the old lady. 'How are you doing? I understand you've had some adventures.'\n\n'Ssh!' Mrs Shapiro held her finger to her lips. 'The hund is killing.'\n\nHalf an hour or so later, as the final credits rolled, she turned to us and said in a croaky voice, 'I heff seen this film once before. Mit Arti. When we were still in loff. Before the sickness snetched him away. So long ago. What has heppened to all the years?'\n\nThere were tears in the corners of her eyes. Ms Baddiel leaned forward and hugged her in her plump arms. Then she reached in her bag for a vanilla-scented tissue.\n\n'It's all right now. You can let it all out. Take a deep breath. Hold. Breathe out with a sigh. There. Perfect.'\n\nVioletta stretched her paws and rubbed her head against Mrs Shapiro's thigh. Tati put a piece of wood \u2013 it looked worryingly like an antique chair leg \u2013 on the fire and reached down to stroke Wonder Boy, who rolled on his back, legs thrown apart abandonedly, and started to purr. Nathan and I exchanged smiles. Nabeel went and made another pot of coffee. Ishmail offered round a packet of Camel cigarettes.\n\n'Are you her carer?' Ms Baddiel asked.\n\n'Hello. Yes. Please.' He flashed his lovely teeth at her.\n\nShe took out her Labrador-puppy notebook and wrote something down. Then Nabeel came back from the kitchen with a steaming coffee pot and fresh cups.\n\n'And you? You're a carer, too?'\n\n'Hello. Yes. Welcome!'\n\n'Well, you may be entitled to claim the Carer's Allowance,' she said. 'One of you. The Carer's Allowance is payable if you spend at least thirty-five hours a week looking after someone who is in receipt of Attendance Allowance. Are you claiming Attendance Allowance, Mrs Shapiro?'\n\n'What for I need attendents?' said Mrs Shapiro. She was still sniffling a bit.\n\n'Well, you know,' Ms Baddiel offered her another tissue, 'after what you've been through, Mrs Shapiro, I think you deserve a bit of help. Of course it's up to you, entirely.'\n\nA skinny tabby cat jumped up into her lap. She ran her little chipolatas over its fur making it purr so much it started to dribble and she had to get another tissue out. Nathan's Tati was sitting watching all this with such a solemn look on his face I thought she'd have to hand him a tissue, too.\n\nThen the doorbell rang again. Ishmail was already on his feet so he went to answer it. I heard him talking animatedly, and another quieter voice replying. A moment later, Mr Ali joined us in the study. He and Ishmail were still arguing in Arabic, and now Nabeel joined in. Mr Ali turned to Mrs Shapiro.\n\n'They are saying they want to stay here. They are saying they can baint all house and fixitup and help you make it clean. I will supervise of course. You pay only for materials.'\n\nI saw a quick flicker pass through Mrs Shapiro's eyes. She said nothing.\n\n'You know in our culture we have great respect for old people,' Mr Ali pressed on. 'But I think mebbe you do not like to have young men into your house, Mrs Naomi?'\n\nEveryone's gaze was now focused on Mrs Shapiro. She looked around cannily. Her eyes were still moist but her cheeks were flushed with excitement, or maybe with too much strong coffee, and I could see her mouth twitch as she weighed up her options.\n\n'I donnow. I donnow.' She put one hand dramatically to her brow, and ran the other through Wonder Boy's shaggy belly-fur. 'Wonder Boy, what you think?' Wonder Boy purred ecstatically. 'Okay. We try it.'\n\nThere was a general exhalation of breath.\n\nMr Ali led us on a guided tour around the house to show us the improvements he'd made. The dingy hall looked much brighter under its coat of white paint, and the loose floor tiles had been fixed or replaced with shiny white bathroom tiles. I noticed with dismay as we climbed the stairs that the grand old mahogany banisters and handrail had been painted with yellow gloss to match the front door, but Mrs Shapiro didn't seem to mind.\n\nHowever, the most spectacular change was in the bathroom. The original chipped and cracked white tiles had been retained, but beneath them an entire new bathroom suite had been installed. Well, it wasn't exactly new \u2013 it looked as if it dated back to the sixties and had been taken out of a house undergoing renovation \u2013 two houses, in fact. There was a wide rose-pink washbasin and matching lavatory complete with pink plastic seat cover, and under the window an avocado-green bath with curved chrome handrails. The rotten floorboards under the lavatory had been patched up, and a piece of lino in blue-and-white mosaic covered the whole floor. If you were colour-blind, it would have been lovely.\n\nAs my eyes scanned the room, they fell on a white porcelain toothbrush holder fixed on to the wall above the basin. I bent closer to take a surreptitious look while everyone was oohing and aahing over the bath. Yes, it was definitely the same one. There was a small chip on one side \u2013 must be from where I'd tossed it into the skip. It was quite stylish \u2013 clean lines, Scandinavian-style. But really, at the end of the day, it was just a toothbrush holder. To imagine I'd once got so worked up over it!\n\nThen Mr Ali turned the taps on and off to demonstrate that they all worked. As he flushed the lavatory, steam rose. He stared into the toilet pan with a puzzled frown.\n\n'Some small mistake. Maybe wrong beep. Soon fixitup.'\n\n'But the hot water is much better!' cried Mrs Shapiro. 'You are a very clever-kn\u00f6del, Mr Ali.'\n\nHe beamed at her. 'Colours you like?'\n\n'The pink is nice colour,' she said. 'Better than the green.'\n\n'Lovely,' I said.\n\n'Lovely,' agreed Ms Baddiel, who had seen \u2013 and smelled \u2013 the original.\n\n'They've developed a new kind of flexible non-crack tile adhesive based on a thixotropic gel,' said Nathan, producing a tub of something from his demonstration pack. 'Should you be thinking of replacing the tiles.'\n\nNathan's Tati cleared his throat and sang a verse of the 'Toreador Song' from _Carmen_ that resonated in the small space.\n\n'Good acoustics!' he said. Everyone applauded, apart from Nathan.\n\nThe bedroom the Uselesses were sharing was the one with the white uPVC window. It had been replastered over the breeze block and actually, from inside, it didn't look so bad. The walls were freshly painted white, and the beds neatly made with the burgundy velour curtains for bedcovers. Their shoes, folded clothes and carrier bags were lined up tidily against one wall. I caught Nathan's eye.\n\n'Admirably spic and span,' was all he said.\n\nMrs Shapiro's bedroom was untouched, the wallpaper a faded-out colourless fawn with small nondescript flowers picked out in muddy taupe.\n\n'We will baint it up next. What colour you like it?' asked Mr Ali.\n\nShe pressed her fingers against her brow as she tried to envision a new room.\n\n'What about the penthouse?' I whispered to Mr Ali. 'Have you started up there?'\n\n'Not yet. Still clearing rubbish. Boys burning it. But slow.'\n\n'They're burning all the papers?' I had an image of priceless historical records going up in smoke. 'Mrs Shapiro? Aren't some of your belongings up there?'\n\n'Is all the rubbish belonging previous inhebitents,' she said dismissively. 'Was some type of religious persons living here before. Orsodox or Kessolik I don't know. They heff left behind all their rubbish and run away.'\n\n'They ran away?'\n\n'In the bombing. They ran and left it all behind. Yes, eau de nil.'\n\n'But who...?'\n\n'Eau de nil is the most charming colour for the bedroom, isn't it?'\n\n'An admirable choice,' murmured Nathan's Tati sonorously into Mrs Shapiro's ear, brushing her cheek with the tips of his whiskers.\n\nAs we came back down the stairs, he held out his arm for Mrs Shapiro, and she rested her weight on it lightly. She seemed to be blushing more than usual under all the rouge. My plan was working!\n\nThe last room we went into was the large sitting room downstairs at the front \u2013 the one with the grand piano. The stench in there made us recoil as we stepped inside, and now it was obvious why the room had been out of use for so long. Mr Ali had removed the boards from the window, and in the daylight we could see the sagging ceiling and a great crack in the bay, so wide that you could see daylight on the other side and the green of the monkey puzzle tree. A trail of muddy paw prints led from the base of the crack across the carpet towards the door with its broken latch. So this explained the mystery of how the Phantom Pooer got in and out \u2013 even though I still didn't know which one was the culprit. In fact that was the least of our problems.\n\nNathan, Nathan's Tati and Mr Ali went over to examine the crack, rubbing their chins solemnly and pacing up and down with lowered eyes, the way men do in B&Q.\n\n'There are new types of heavy-duty fast-setting foam fillers, called structural methacrylates, suitable for construction work...' Nathan began hesitantly.\n\n'But this does not fixitup the problem,' Mr Ali scratched his head. 'First we must find out what causes. Maybe this tree...'\n\nThey were looking into the break in the floorboards below the ruptured skirting board. 'We could cut the tree down, dig the roots out, then pump the gap full of methacrylate foam,' suggested Nathan.\n\n'Concrete may be better,' said Mr Ali. 'But pity to cut up such a fine tree.'\n\n'Mind the gap,' Nathan's Tati murmured to Mrs Shapiro, who had come over to have a look, placing his hand on her shoulder and letting it linger there.\n\n'What do you think, Mrs Shapiro? Should we cut the tree down?' asked Ms Baddiel.\n\nMrs Shapiro looked shifty. 'No. Yes. Maybe.'\n\nI remembered her correspondence with the Council's tree department.\n\n'It may have a preservation order on it,' I said. 'Shall I contact the Council and find out?'\n\nEveryone seemed pleased with this suggestion. As we stood staring into the crack, a skinny feline head poked up between the floorboards and the Stinker eased himself into the drawing room. Crouching low, he looked around the semicircle of human legs, found a suitable gap, and made a dash for the door.\n\n'Raus! Little pisske! Raus!' cried Mrs Shapiro, waving him on, but you could see she didn't mean it. A cheerful, almost skittish mood had come over her; she was revelling in the presence of so many visitors \u2013 or maybe of one visitor in particular. She moved over to the piano, lifted the cover and tinkled a few notes. Even those out-of-tune keys seemed to come alive under her touch. To my amazement, without any music to read, she started to play the 'Toreador Song', embellishing it with broken chords and little trills, and Nathan's Tati, standing behind her, gave us a full baritone rendition \u2013 he was more in tune than the piano. Nathan joined in the choruses. At the end, Mrs Shapiro sat back, placing her gnarled ring-encrusted hands together with a sigh.\n\n'Hends no good, isn't it?'\n\n'Nonsense, Naomi,' said Tati, taking her hands and holding them in his.\n\nThen we all made our way back towards the entrance hall to say our goodbyes. Nabeel had to intervene to halt a hissing and scratching match between Mussorgsky and Wonder Boy \u2013 despite his initial aversion to cat poo, he had turned out to be quite a cat lover. Mr Ali talked to his nephew softly in Arabic and embraced him in a hamstery hug. Mrs Shapiro sidled up to me and, nodding her head towards Nathan, whispered, 'He is your new boyfriend, Georgine?'\n\n'Not my boyfriend. Just a friend.'\n\n'Good thing,' she whispered. 'He is too petit for you. But quite intelligent. The father also is charming. Pity he is too old for me.'\n\nAfter they'd all gone, Mrs Shapiro and her attendants went back to sit by the fire, leaving me alone in the hall for a moment, and that's when I noticed that the framed photograph of Lydda which used to hang above the hall table had disappeared. There was nothing but a nail sticking out of the wall to show it had been there. Who had moved it? I was still puzzling over it when suddenly I heard the distinct clack of the front gate. I thought it must be one of the others coming back for something they'd forgotten so I opened the door. Coming down the path towards the house was Mrs Goodney in her lizard-green quilted jacket and her pointy shoes, with an important-looking black briefcase under her arm. Behind her came a dark thickset man I'd never seen before, middle-aged, wearing a crumpled brown suit. Neither of them was smiling. There was something odd about the way the man was looking at me: his eyes seemed asymmetrical.\n\nMrs Goodney stopped in her tracks when she saw me standing in the doorway. She eyed me up for a few moments. Then she continued her advance. Now a third person, a tall spindly youth, appeared on the garden path and made his way towards us. It was Damian, the young man from Hendricks & Wilson, his hair sticking up with gel, his suit a bit too short in the legs. Blue socks. He was looking up and around, studying the house, avoiding my eyes.\n\n'Feeding the cats again, are we?' said Mrs Goodney to me. I was so taken aback by her rudeness that I forgot to ask her what she was doing here. She turned to Damian and smiled toothily.\n\n'Glad you could make it, Mr Lee. The gentleman just needs an initial estimate of value at this stage.'\n\nThe thickset man nodded. He was looking at the house in frank amazement, his misaligned eyes sliding around this way and that. Then I realised one of them was made of glass.\n\n'Must be vort a bit, eh?' he said. 'Big house like this. Good part of London town. I am somewhat impressed.' His English was better than Mrs Shapiro's, if a bit pedantic, with just a slight guttural accent.\n\nDamian took a dog-eared notebook out of his pocket and started to make notes with a stub of pencil. He was still avoiding my eyes.\n\n'Unfortunately it's not worth as much as you think. It's in poor condition, as you can see.' Mrs Goodney was simpering at the glass-eye man. 'I've had a reputable builder to view it and he reports that it needs a substantial amount of money spending on it to bring it up to present-day standards. I'll show you his report if you like.' The glass-eye man sniffed discontentedly, but Mrs Goodney placed one plump red-nailed gold-ringed hand on his arm and the other on Damian's. 'Don't worry, Mr Lee'll quote you a good price. Won't you, Mr Lee?'\n\nDamian nodded and chewed the end of his pencil.\n\n'This is what you do for your five grand, is it, Damian?' I hissed. He ignored me and carried on chewing.\n\n'Seems like she's already had some builders in. Cowboys, by the look of it.' Her eye had fallen on the uPVC window on the first floor.\n\n'He's not a cowboy,' I blurted out. They all stared at me. 'He's...'\n\nThen I noticed that their gaze had shifted away from me to a point somewhere beyond my left shoulder. I turned round. Mrs Shapiro was standing there, and behind her, Nabeel and Ishmail.\n\n'Hello, Mrs Shapiro,' Mrs Goodney's rusty-gate voice squeaked with fake cheeriness. 'What are you doing here, sweetie? You're supposed to be...'\n\n'I am come home. Finish mit Nightmare.'\n\n'But you can't stay here on your own. This house isn't safe for you, poppet.'\n\n'Poppet schmoppet.' She pulled herself up into her five-feet-tall, chin-out-fighting pose and looked the social worker in the eye. Her cheeks were still flushed from the excitement of the morning. 'I heff my Attendents. I will claim the Attendents Allowance.'\n\nThe young men standing behind her flashed their teeth and their eyes at everybody. Violetta, who seemed to have snuck up with Mussorgsky, was hovering around our feet rubbing herself against Mrs Shapiro's legs and purring. Unexpectedly, she arched her back and hissed at Mrs Goodney, who almost \u2013 you could see it in her face \u2013 hissed back.\n\nSuddenly the man with the glass eye stepped forward and fixed Mrs Shapiro with his disconcerting gaze.\n\n'Ella? You are Ella Wechsler?'\n\nMrs Shapiro drew back. I couldn't see her face, but I could hear her throaty intake of breath. 'You are mistooken. I am Naomi Shapiro.'\n\n'You are not Naomi Shapiro.' His voice was gravelly. 'She was my mother.'\n\n'I don't know what you talking about.' Mrs Shapiro elbowed past me, reached out, and slammed the door.\n\nThey didn't go away for about half an hour. Standing inside the freshly painted hall, the four of us listened to them ringing on the doorbell and rattling the letter box. Then we heard their voices as they walked round the outside of the house and started rapping on the kitchen door. Somewhere in the depths of the house, Wonder Boy started to yowl. Eventually they gave up.\n\nI didn't leave until I was sure the coast was clear. I walked home slowly, trying to make sense of what had happened. He must be the real Naomi Shapiro's son \u2013 the child she wrote about in her letters, the gummy brown-eyed baby in the photo \u2013 this thickset, ugly middle-aged man who had embodied all the idealism and hopes of his beautiful mother. But who was she? And how had Mrs Goodney contacted him? Maybe this was why I'd found no documents or papers in the house \u2013 Mrs Goodney had got there first. Had got them and used them to summon up this genie from the past.\n\nAs soon as I got home I went up to my bedroom, and spread the photos out on the floor. Baby Artem; the wedding photo; the couple by the fountain; the woman in the archway; the two women at the Highbury house; the Wechsler family; the moshav near Lydda. At half past four, Ben wandered in to see what I was doing, and pointed out something so obvious I should have noticed it before.\n\n'I wonder why he's carrying a gun.'\n\n'Who?'\n\n'The man who took the photo. Look.'\n\nHe pointed to a dark patch on the stony foreground in the landscape photograph. It was the shadow of the photographer \u2013 the sun was behind him, and you could see the outline of the head and shoulders, the arms raised to hold the camera to the eye, and something long and straight hanging down from one shoulder. Yes, it could be a gun.\n\nHe picked up the photograph of the woman standing in the stone archway and turned it over.\n\n'Who's she?'\n\n'I think she must be Naomi Shapiro.'\n\n'The old lady down the road?'\n\n'No, someone else.'\n\n'It says Lydda.'\n\n'That's a place. In Israel.'\n\n'I know, Mum. It's in one of the prophecies. It's supposed to be where the Antichrist returns.' His voice had gone husky.\n\n'Don't be daft, Ben,' I said. Then I saw the look in his eyes. 'Sorry \u2013 I didn't mean _you're_ daft, I meant _it's_ daft. All that Antichrist stuff. Putin and the Pope. The Prince of Wales and his evil bar codes.' I was trying to sound jokey, but Ben didn't smile.\n\n'The Muslims call him Dajjal? He's got one eye? He gets killed by Jesus in this massive battle at the gates of Lydda?' There were beads of sweat on his forehead.\n\n'Ben, it's all...' The word on my lips was 'rubbish', but I held back.\n\n'I know you don't believe in it, Mum. I'm not gonna argue about it, all right? I'm not even sure I believe all of it myself. But I know there's something in it. I just know. Like, I can feel it coming?' \n\n# 5   \n_If Only It Came in Tubes_\n\n# 40\n\n# Heavy as watermelons\n\nI walked round to Canaan House the next day, hoping to have a chance to speak to Mr Ali. I wanted to ask him about Lydda. After my unsettling talk with Ben last night, I'd logged on to the internet to look up information about the prophecies relating to Lydda. This story \u2013 I wasn't sure where it was leading me, but now, because of Ben, it had become my story, too, and I knew I had to follow it through.\n\nThe sun was shining for once, a hard clear brightness, with even a touch of warmth, and I could smell the trees and shrubs catching their silky breaths as if taken by surprise: this is it at last \u2013 a real spring day. Around the margins of the lawn, daffodils were poking their yellow heads up between the cut-back loops of bramble that had already started to regrow. Mr Ali was there, standing up on a ladder painting the outside of Mrs Shapiro's bedroom window, singing wordlessly to himself. Wonder Boy was supervising him, sitting on one of the white uPVC chairs in the garden with his tail wrapped round his legs.\n\n'Hello, Mr Ali!' I called. 'Is everything okay?'\n\nHe came back down the ladder and wiped his hands on a piece of cloth from the pocket of his blue nylon overall.\n\n'Hello, Mrs George. Nice day!'\n\nActually, I realised that Wonder Boy wasn't supervising Mr Ali at all; he was supervising a couple of thrushes which were hard at work building their nest among a thicket of ivy in one of the ash trees. I watched them come and go with their bits of moss and dry grass. Wonder Boy was watching too, flicking the tip of his tail.\n\n'Tomorrow I borrow the van, we take Mrs Shapiro to choose a colour of paint for inside.'\n\n'That's good.'\n\n'How is your son?'\n\n'He's okay, but...' I hesitated. An image of Ben slipped into my mind, his waxy face, the fear in his eyes. He'd gone off to bed last night without eating anything. I'd knocked on the door of his room, but it was locked from the inside. I was beginning to doubt whether this was normal teenage behaviour, something he would grow out of.\n\n'Mr Ali, that picture in the hall \u2013 of Lydda. Was it you who took it down?'\n\n'Lydda.' He stuck his paintbrush in a pot of turpentine and swirled it around. 'In the old times this town was famous for its beautiful mosques. But do you know, Mrs George, that this town is a special place to you also? Is home town of your Christian Saint George. You are named from him, I think?'\n\nI didn't want to admit that in fact I'd been named after George Lansbury. It was Dad's idea, and Mum hadn't been able to think of a suitably inspirational female socialist icon to suggest instead.\n\n'Really? Saint George the dragon slayer came from Lydda?'\n\n'You can see his picture carved above the door of the church.'\n\nSweet Saint Georgina. I recalled Mark Diabello's poem with a shudder. But Ben had also talked about a one-eyed devil.\n\n'The picture of Lydda that was in the hallway, why did you take it down, Mr Ali?'\n\n'Why you are always asking questions, Mrs George?' He wasn't exactly being rude, but the easy friendliness of our previous conversation had gone. 'Everything is okay. Sun is shining. I am working. Everybody is happy. Now you start asking questions, and if I tell you the truth you will not be happy.'\n\n'You were going to tell me about your family, remember? What happened in Lydda?'\n\nHe didn't say anything. He was concentrating on cleaning his brushes. Then he pulled up one of the white plastic chairs and sat down at the table. Wonder Boy had slunk off; I saw him sitting directly beneath the thrushes' tree, staring up into the branches. I shooed him away and sat myself down opposite Mr Ali. He put aside the brushes, poured some of the turpentine on to his hands, rubbed them together and wiped them on his piece of cloth.\n\n'You want to know? Okay. I will tell you, Mrs George.' He put the cloth back in his pocket and folded his arms across his XXL tummy. 'I come from Lydda. I had one brother, born the same time.'\n\n'A twin?'\n\n'If you will please stop interruption, I will tell you.'\n\nMustafa al-Ali, the man I knew as Mr Ali, was born in Lydda in 1948 \u2013 this much he knew. He didn't know his mother's name, nor that of his twin brother, nor even his exact date of birth, but he reckons he was a few months old on 11th July 1948.\n\n'Why, what happened then?'\n\n'Have patience. I will tell you.'\n\nLydda was at that time a busy town of some 20,000 inhabitants that had grown up over centuries in the fertile coastal plain between the mountains of Judaea and the Mediterranean sea. But that summer, the summer of Nakba, the town was filled up with refugees from Jaffa and smaller towns and villages all up the coast. 'You can imagine how everybody was jittering, talking about expulsions and massacres.'\n\nOne late morning in July, when everything was hot and still, and even cats and sparrows had gone off to look for shade, there was a sudden roar of engines overhead. People who looked up saw a flight of planes swoop low out of the glimmering sky. Then the explosions started. One after the other after the other, as the planes began unloading their bombs on the sleepy little town. Houses, shops, mosques, market stalls. One after the other after the other. There was nowhere to flee to. No bomb shelters. No anti-aircraft guns. People just scurried around like frenzied ants. Some caught a blast and fell in the street. Some died when rubble collapsed on them. Some sat tight in a corner and covered their heads and prayed.\n\n'But their purpose mainly was not to kill,' Mr Ali continued, fixing me with his eyes. 'They wanted to drive us out, with terror.'\n\nNext day, as people were emerging from the rubble to inspect the damage and bury their dead, a battalion with mounted machine guns suddenly rolled into town at high speed. At first they thought it was the Jordanian army, come to defend them, but all at once the machine guns let rip, barrels blazing, bullets flying in all directions. Men, women and children were gunned down \u2013 some 200 fell and died in the streets. Others fled in fright.\n\n'You can read it on your internet, Mrs George. How it was reported in American newspapers. Blitzkrieg. Ruthlessly brilliant. Corpses riddled with bullets by roadside. All this was done to create terror. This is how they emptied Lydda of its population.'\n\nSome tried to take safety in the great Dahmash Mosque. But later that night, the people who lived nearby heard round after round of gunfire coming from the building. Next day 176 bodies were found inside.\n\nAs dawn broke, soldiers ran from house to house, banging on the doors with their rifle butts and ordering those inside to leave at once.\n\n'\"Go! Go to King Abdullah!\" the soldiers shouted. What they mean is \u2013 get out of this country, and leave it for us! Go to Jordan! Flee to any Arab country that will take you! You never heard about this?'\n\nI shook my head. 'Go on.'\n\nThe terrified population, expelled from their homes, grabbed what they could and fled. The al-Ali family \u2013 the women and children, for their father had disappeared \u2013 were dragged out of their house on to the street, given only a few minutes to grab their valuables. Soldiers were herding everyone out into the streets, shoving them with the barrels of their guns if they were too slow, shooting them if they resisted.\n\n'Where are we going?' the mother had asked, grabbing her children to her in all the chaos.\n\nSomeone had said, 'They're taking us to Jordan,' and someone else had said, 'We're going to Ramallah.'\n\nThey were marched to the outskirts of the town, the soldiers firing shots in the air to make them run.\n\n'Go! Run to Abdullah in Jordan!'\n\nAs they passed through a cordon, soldiers searched them, stripping them of their possessions. Ahead of them, one of their neighbours, recently married, who quibbled about surrendering his savings, was shot dead before the horrified eyes of his new bride. After that, no one protested. The al-Ali family were robbed of their money, their gold jewellery, their watches, even their silver coffee cups. All they were allowed to keep was a bundle of clothes, some bread and olives and a bag of oranges.\n\n'Run! Run!' The soldiers fired volleys of shots above their heads. But the asphalt road was barred, and they were forced to make their way eastwards across the stubbly, newly harvested fields.\n\nBy now it was midday and the heat was intense, the sky so blue and hard it seemed to glimmer like lazurite. In the coastal plain, the temperatures in July can easily reach forty degrees. There was no shade at all \u2013 only a few prickly thorn bushes growing among the rocks. Beyond the plain stretched a long hill, and they could see a miserable procession of their fellow townspeople already stumbling towards the stony horizon.\n\nMr Ali paused. He sat back in his chair and stared at the sky, his eyes wrinkled up as though to keep out too much brightness.\n\n'Each time I remember this story, my heart turns into stone.'\n\n'Go on,' I said.\n\nThe al-Ali family joined the procession walking across the fields, stepping out briskly at first, buoyed up by their anger, and confident that this was just a temporary situation, that soon the Arab armies would drive out the intruders and they would be able to return to their home. After a couple of hours, as they mounted what they thought was the crest of a hill, only to find another steeper one stretching out ahead of them, their hearts sank. Sitting with their backs to the sun, the women with their scarves pulled over their heads for shade, they ate some of their bread and olives, and quenched their thirst on the oranges. They had brought so little water \u2013 who would think of carrying water instead of silver and gold? All around them, other families were sitting, too exhausted and dehydrated to move, while others abandoned the possessions they could no longer carry, and plodded on up the hill in the searing sun.\n\nAs the day waned, they came into the small village of Kirbatha. There was a well there \u2013 but no bucket. The women took off their scarves, tied them together, and lowered them down till they tipped the little black circle of water, then pulled them up and sucked the water from the damp cloth.\n\nThe third day of the march was the worst. The women's sandals were already falling apart, their feet were bleeding and swollen. Netish thorns and blue field-thistles snagged at their skirts and legs.\n\n'Go,' said his mother to her older son, Tariq. 'Go on ahead and find us some water to drink. Maybe there is a village up there with a well.'\n\nBut there was no water. All along the way people were fainting from thirst and exhaustion. On a rocky scree the boy came across a woman staggering under the weight of a huge bundle. Two watermelons, it looked like; and he thought, if she drops them, I'll pick them up and take them back to my mother. But as he drew closer the woman sank to the ground and he saw that she was carrying two babies.\n\n'Help me, brother,' she pleaded. 'My boys are too heavy for me. I cannot carry them.'\n\nThe boy hesitated. He was only fourteen years old, and he already had his mother and sisters to look after; but it was clear this woman was not going to make it.\n\n'Take just one of them,' she said in a voice that was barely more than a whisper.\n\nTariq looked at the two babies. They looked terribly red and wrinkled, their eyes screwed shut against the light. How could he choose? Then one of them stirred and opened its dark bright eyes, which seemed to stare straight into his. The woman, seeing him waver, wrapped the baby in her shawl and thrust it into his arms.\n\n'Go on ahead. Don't wait for me. Go. I'll meet you in Ramallah.'\n\nMr Ali went silent. I gazed at the green sunlit garden, the busy thrushes, the bursting daffodils, but I could feel a desert wind on my cheek, and all I could see was dry rocks and thorn bushes.\n\n'That was you? The baby in the bundle?'\n\nHe nodded.\n\nA door opened and from the interior of the house I heard the sweet jangle of Arabic music and the noisy patter of daytime television. Then Mrs Shapiro appeared on the doorstep wearing her dressing gown and her _Lion King_ slippers.\n\n'Will you take a coffee mit us?'\n\nMr Ali didn't reply. His eyes were fixed somewhere else.\n\n'My name is Mustafa,' he said quietly. 'It means one who is chosen. My brother Tariq told me this story.'\n\nI wanted to touch him, to take his hand or put my arm round his shoulder, but there was a reserve about him, a self-containedness, that made me hold back.\n\n'Did he tell you what happened to the other baby?' I asked.\n\nMr Ali shook his head. 'He told me only that the soldier who shot the bridegroom had on his arm a tattoo \u2013 a number.'\n\nMr Ali's story had cast a shadow over me, and I found I couldn't join in with the cheerful gossip over coffee. I caught his eye once or twice, and I kept wanting to ask him what had happened to the al-Alis; whether they had all made it to Ramallah, and whether he, Mustafa, had ever found his mother and brother. But in my heart I knew the answer.\n\nI was troubled, too, by the story of the soldier with the number tattooed on his arm \u2013 what was in his mind when he shot the young bridegroom? How could a Jew who was himself a survivor of the death trails of Europe act with such casual cruelty against the hapless civilians of his promised land? What had happened in his heart? Then I started to wonder about Naomi herself \u2013 when she had let herself be photographed in the archway at Lydda, did she really not know what had taken place there two years before? Or did she know, and consider it a necessary price?\n\n'What are you thinking about, Georgine?' Mrs Shapiro reached across and patted my hand. 'Is it your running-away husband, darlink? Don't worry, I heff a plan.'\n\n'No. I'm thinking about... how hard it is to live in peace together.'\n\nShe threw me an oblique look. 'Ach, this is too serious.' She lit a cigarette for herself and one for Nabeel. 'Better to enjoy the happiness of today.'\n\nAfter we'd finished our coffee I left to go home. The sun was still shining and Wonder Boy was still sitting patiently under the tree, gazing at the thrushes' nest. Mr Ali was back up his ladder. Inside the house, Nabeel was clattering pans and playing music, and Ishmail was vacuuming. A westerly breeze stirred the tips of the saplings and made the daffodils dance. But I kept thinking about the twin babies in the bundle, heavy as watermelons \u2013 the one who was chosen, and the one who was not.\n\nIf only I had Mrs Shapiro's gift for living in the present, I thought, as I walked home past the front gardens greening with new growth; trees, shrubs, weeds, grass \u2013 everything was coming to life. Near the corner of my street, a willow tree was sticking out its silvery buds through a railing. I thoughtlessly snapped off a pussy-paws twig, and my mind flashed back to the bunches of pussy willows and catkins we used to bring in to decorate our classroom at Junior School in Kippax. Soon it would be Easter. I remembered Mrs Rowbottom's plonkety-plonk on the piano and our thin wobbly voices as we sang ' _There is a green hill far away_ '. How that hymn had scared me as a child. It had seemed a harsh intrusion into the happy world of Easter bunnies and foil-wrapped eggs. I knew now, as I hadn't known at the time, that those hills were not green at all \u2013 they were rocky and barren. I'd been puzzled, then, by the absence of a city wall; now I realised that so many walls had been built and knocked down and rebuilt again over the centuries, that time itself had lost track of what belonged to whom.\n\n' _He hung and suffered there._ ' Yes, the history of that place was steeped in cruelty. Mrs Rowbottom had glossed over the details of what happened during the crucifixion and tried to convince us that ' _without a city wall_ ' meant 'outside'. But when I asked Dad he said, 'War and religion \u2013 they both 'ave an unquenchable thirst for human blood. They feed off each other like nuggins.'\n\nMum rolled her eyes to the ceiling.\n\n''E's off again.'\n\n'What's...?'\n\n'Dennis, she's only nine.'\n\nI never did find out what a nuggin was.\n\nMum always waited until closing time on Easter Saturday to buy chocolate eggs for us, when those that were left were reduced to half price.\n\n'What d'you want them fancy eggs for, Jean?' Dad said. 'We're remembering an execution, not celebrating a birthday.'\n\nBut he ate them anyway. He had a real fondness for chocolate.\n\n# 41\n\n# Cyanoacrylate AXP-36C\n\nOn Sunday I'd planned to make the most of the fine weather and do some gardening, to get some good dirt under my fingernails, have a go at the nasty spotted laurel bush and see off the fat brown slugs. But it felt like I spent the whole day on the phone, and each phone call left me feeling more upset.\n\nThe first call was at nine o'clock (on Sunday morning \u2013 would you believe it!) from Ottoline Walker, the Scarlet-mouthed Slut.\n\n'Hello? Georgie Sinclair? Is that you?'\n\n'Who's speaking?' I kind of recognised the voice already.\n\n'It's me. Ottoline. We met. You remember?'\n\nLike hell I remembered. Big banana bogey. Ha ha.\n\n'Yes, I remember. Why are you ringing me?'\n\n'It's about Rip...' (Well, it would be, wouldn't it?) '... I just wanted to tell you I had no idea you were still... sort of... involved.'\n\n'Sort of married, actually.'\n\n'He told me it was over between you two ages ago. He told me you didn't mind...'\n\n'He told _me_ he was advancing human progress.'\n\n'Oh. I see.' There was a pause on the other end of the phone as she fumbled for a response. 'Look, I'm really sorry. It sort of changes things... I mean, when you're in love, you don't always do the right thing... you don't think about the consequences for other people.' She paused. I said nothing. 'I believe in commitment, you know.'\n\n'Like you were committed to Pete. And now you're committed to Rip.'\n\n'That's not what I mean. You make it sound terrible.'\n\n'Actually...' I stopped myself. I didn't want her to have the satisfaction of knowing how much she'd hurt me.\n\n'Ben doesn't know, if that's what you're wondering.'\n\n'What about Pete? Does he know?' I almost called him Pectoral Pete.\n\n'He found out. Poor Pete. It was awful. He was going to kill himself. Then he was going to kill Rip.'\n\nShe sounded as though she was sniffling on the phone, or maybe I imagined it. Anyway, for a moment I felt sorry for her.\n\n'You'll not get much commitment from Rip. He's committed to the Progress Project.'\n\nThere was a silence. In the background, I could hear music on the radio \u2013 a woman singing blues.\n\n'That's the other thing I wanted to ask you. This Progress Project. What is it, exactly?'\n\n'Didn't Pete tell you?'\n\n'He did, he talked about it endlessly. But he wasn't very good at explaining. I somehow couldn't get it.'\n\n'It _is_ quite complicated.'\n\n'But then with Rip it was just the same. All those big words. I realised it must be me who's a bit thick.'\n\nShe gave a little self-deprecating giggle that was quite endearing.\n\n'Er... hold on a minute. I've got written it down.' Where was that bit of paper? I rummaged in the desk drawer. 'Here it is.' I read aloud. 'The human race faces unprecedented challenges as we enter the millennium of globalisation. We need to iterate new synergies if we are to make progress in meeting the aspirations of the developing world, while clearly understanding at the same time that nothing shall prejudice the economic achievements of the developed world.'\n\nThere was another pause. The woman blues singer let out a long throbbing moan.\n\n'That's it?'\n\n'Isn't it enough?'\n\n'Well, sort of, I suppose. What does it mean exactly?'\n\n'Why don't you ask _him_?'\n\nShe made that sound again on the end of the phone. It could have been a sniffle, or a giggle. I put the phone down.\n\nGrabbing my secateurs, I pulled on my gardening gloves and stomped out into the garden. The sun was shining, but my head was full of dark clouds. Still fired up with thoughts of Rip and the Scarlet-mouthed Slut, I hacked away pitilessly at the ugly laurel bush \u2013 Wonder Boy's favourite haunt \u2013 grinding the fallen leaves into the mud. What gave her the right to ring me on a Sunday morning to cadge sympathy? Snip. Still sort of involved! Snip. I believe in commitment! Snip. Snip. I should have just put the phone down as soon as I heard her voice, instead of letting myself get drawn into conversation. Now I felt so wound up and angry that all thoughts of peace in the world had evaporated like water in the desert. And yet I had felt a frisson of fellow-feeling, and I was secretly glad to discover that despite her big scarlet mouth and her slut stilettos, it was the Progress Project that was his real mistress.\n\nAfter an hour or so the phone rang once more. I carried on snipping and let it ring until the answering machine clicked on. Then a minute later it started ringing again. And again. This was some persistent bastard. I put the secateurs down and went to answer it.\n\n'Hello, Georgina, I've been trying to get hold of you.'\n\nThat voice. I shivered as though a cool hand had touched my bare skin. It was the first time we'd spoken since the episode with the poem and the Velcro handcuffs. 'Have you got a minute? I just wanted to let you know that I've heard back from the Land Registry about Canaan House.'\n\nI took a deep breath. Despite my resolution, I could feel that warm red-panties glow coming over me again. I mustn't let my hormones take over.\n\n'And...?'\n\nHe explained that the house was unregistered, and that if Mrs Shapiro wanted to sell it she would need to register it, for which she would need the deeds. I had to force myself to concentrate on what he was saying.\n\n'What about that son you mentioned, Georgina? The son in Israel? Maybe he knows where they are.' He was still angling for information.\n\n'I met him the other day.'\n\nI told him an edited version of our doorstep encounter. I didn't mention Mr Ali and the Attendents, but I told him about Damian.\n\n'Damian Lee from Hendricks & Wilson. There he was, chewing on his pencil and pretending he was making a valuation.'\n\n'Ah!' Mark Diabello caught his breath. 'That explains the BMW I saw parked round the back of their offices.'\n\n'So Damian's job is...?'\n\n'To persuade the son to let the social worker's friendly builder have the house for, say, a quarter of a million, then disappear back off to Israel with the cash in his pocket.'\n\n'Just like you tried to persuade me?'\n\n'That was different. I wasn't working for the buyer. Tsk. Naughty Damian.' His voice oozed disapproval. 'I told you they were crooks. And it's only a 1 Series two-door hatch.'\n\n'You mean, just a starter model, really.'\n\nI tried to picture Damian with his gelled-up hair sitting at the wheel of a secondhand BMW. The little shit!\n\nAt about five o'clock, just as I was trying to decide what to have for tea, Rip rang. I listened to his facing-unprecedented-challenges voice leaving a message on my answering machine, telling me to ring him immediately. Well, let him wait. He still thought he could boss me about. Typical. Probably he was ringing to tell me he wanted to take the kids up to Holtham at Easter with the Scarlet-mouthed Slut \u2013 'He told me it was over between you two ages ago. He told me you didn't mind.' There was something about the tone of Rip's voice on the answering machine that reminded me of... glue. Cyanoacrylate AXP-36C. I thought of the B&Q package stowed in the mezzanine study and smiled to myself. Peace in the world was all very well, but no way was it going to extend to Rip and me. No way. When someone hurts you like that, what you want is revenge, not peace.\n\nI didn't ring back. I went upstairs to my room and got out my exercise book.\n\n# The Splattered Heart   \nChapter 8  \nGINA'S REVENGE\n\n_Early next morning, heartbroken Gina made her tearful way to the Castleford branch of B &Q. The sight of the jolly orange-clad building made her broken heart leap with smile. Inside it was vast and creepily echoing like a church, and full of weird men prowling around the aisles, eyeing lovely curvaceous Gina lustfully, and wiggling their saucy screwdrivers suggestively. She made her way to the extensive adhesives section. At last her eyes lit on a tube of glue that said in large letters:_ DANGER! AVOID CONTACT WITH SKIN.\n\nI stopped. The picture of the little girl at the glue exhibition stuck in my mind. Human bonding. Messy stuff.\n\nThe last phone call came just as I was getting ready for bed. I knew it was Mum \u2013 she usually rings about this time \u2013 but I was taken aback by the flatness in her voice.\n\n'Your dad's been took poorly,' she said. 'He's got to have that operation on 'is prostrate. Doctor says it could make 'im imputent.'\n\nI could just imagine poor Dad with that long-suffering look on his face and seedy Dr Polkinson telling him what did he expect at his age and we've all got to die of something. The operation date wasn't fixed, but it would be sometime soon after Easter. My mind went into overdrive at once, trying to work out the logistics of going up to Kippax, leaving Ben with Rip, and meeting my deadline for Nathan.\n\n'D'you want me to come up to Kippax, Mum?'\n\n'It's all right, duck. I know you're busy.'\n\n'Mum...'\n\nI was racking my brains for some cheerful or uplifting comment, when Mum chipped in.\n\n'Did you hear about that friend of yours, Carole Benthorpe?'\n\n'She wasn't my friend, Mum.' I shuddered as I remembered her watery reproachful eyes. 'Her dad was a scab.'\n\nCarole Benthorpe _had_ been my friend once, before the miners' strike \u2013 the short Heath strike of 1974, not the year-long Thatcher strike of 1984\u20135. 'Scabs take the gain without the pain,' Dad had said. 'You never get a scab giving up his wage rise that were won on't backs o't strikers.'\n\nThere were only four scabs in Kippax, and Carole's father was one of them. After that she didn't have any friends.\n\n'Dad always told me not to talk to scabs.'\n\n'Aye, and he's right,' said Mum. 'But she weren't a scab, were she? She were only a little kid.' She sighed. All this had suddenly become too heavy. 'Anyroad, all I was going to say is, she won the Jackson's Saleswoman of the Year award. It were in t' _Express_. She got a weekend in Paris.'\n\n'Oh, that's so brilliant! Good for her!'\n\nI felt an unexpected burst of joy for Carole Benthorpe, not about Jackson's or the award, but because she'd survived what we did to her.\n\nThat cold winter in 1974 \u2013 the men hanging around in knots on the streets instead of disappearing underground as they were supposed to, the women pawning their rings and chuntering about how they would make do without a wage coming in. One day after school some of the kids waylaid Carole Benthorpe on her way home. They jostled and taunted her as she walked along, then things got a bit rough and a couple of lads pushed her into the icy tadpole ponds by the back lane. Everybody cheered and laughed as they watched her flounder. Me, too \u2013 I'd stood and laughed with the others. I recalled with a shock of remorse how great it had felt to be part of that cheering, jeering gang. Carole Benthorpe crawled out covered in slime and ran home, all wet and bawling. Next day in the school toilets she carved the word SCAB into her forearm with a Stanley knife.\n\n'If you see her, Mum, give her my love.'\n\n'Oh, I never see 'er. She lives up Pontefract way now.'\n\n# 42\n\n# The right glue for the materials\n\nI was half-heartedly trundling the vacuum cleaner around the house on Monday afternoon, worrying about Mum and Dad, when the phone rang. I thought it might be Mum with some more news of Dad's operation, but it was Mrs Shapiro.\n\n'Come quick, please, Georgine. Chaim is mekking trouble.'\n\nI realised I'd been half expecting it. Apparently Mrs Shapiro and Ishmail had gone off with Mr Ali in the red van to choose some paint at B&Q in Tottenham. Nabeel had stayed behind to start sanding down the woodwork and the kitchen door had been left unlocked. They got back at about four o'clock with their five litres of matt emulsion \u2013 'Eau de nil \u2013 very charming colour \u2013 you will see it' \u2013 to find Nabeel and Chaim Shapiro wrestling on the carpet in the dining room.\n\n'Fighting like the tigers. You must come, Georgine, and talk to them.'\n\n'But what's that got to do with me?'\n\n'Why you are always arguing, Georgine? Please come quick.'\n\nBy the time I got there, the wrestling, if it had ever really happened, was over and there was an uneasy truce around the dining-room table. Mr Ali was sitting on one side of the table, flanked by the Uselesses, and opposite them sat Chaim Shapiro, leaning back heavily, his arms and legs splayed out as though the chair was too small for him, cracking his knuckles from time to time. Mrs Shapiro sat next to him, chain-smoking and fidgeting with her rings. Wonder Boy was sitting on a chair at the head of the table, looking very magisterial. I could hear their voices arguing as I came in through the front door, which had been left on the latch for me, but as I entered the dining room they went quiet. I sat down at the other end of the table, opposite Wonder Boy.\n\n'Hello, everybody!' I said, looking round with a cheery smile. No one smiled back. The atmosphere was like curdled milk. Maybe we should start with Ms Baddiel's breathing exercises, I thought, just to calm us all down.\n\nMrs Shapiro poured me a glass of water from a jug and introduced the newcomer to me as Chaim Shapiro, adding, 'This is Georgine, my good neighbour.'\n\nHe pounced on me at once, demanding to know why I had invited these strangers into his house \u2013 I winced at the emphasis \u2013 ' _my_ house' \u2013 but before I could get a word out, Mrs Shapiro pounced back.\n\n'Is not your house, Chaim. I been living here sixty year paying rets.'\n\n'Shut up your mouth, Ella. You have no feet to stand on, letting Arabs come into your home.'\n\n'You shut up the mouth,' Mrs Shapiro snapped. He ignored her.\n\n'So, Miss Georgiana. Please, we are awaiting your explanation,' he rasped in a breathy voice not unlike Wonder Boy's purr. 'Speak up now or for ever hold your pieces.'\n\nI started to explain that the house needed repair and renovation and that's why Mr Ali and his assistants had been called in. He gave a dubious sniff and rocked back in his chair. Then there was the issue of security, I told him, describing the stolen key and the turned-off water main and hinting at Mrs Goodney's involvement. That made him sit up. The eyebrow above his glass eye started to twitch.\n\n'That Goody with her young stick-up-the-hair-nik, they think I am made of short planks. They think I will sell them my house cheap so they can make some quick bucks out of me. But I have a different plan.'\n\n'Is not your house, Chaim.'\n\n'It is my father's house. Father likes son.'\n\n'My house,' hissed Mrs Shapiro. 'When your father died, he give it me.'\n\n'So what's your plan, Mr Shapiro?' I interrupted, to move the conversation on.\n\n'My plan is to undertake some major renovations here in _my_ house.' There were sharp intakes of breath all around. Wonder Boy's tail started to flick. 'In fact I am something of a do-it-myself enthusiast. I have already purchased a tool kit.' He looked around the table, but nobody met his eye. I glanced across at Mr Ali, but his face was impassive.\n\n'Chaim, darlink, your mother would be eating her own kishkes to hear you speaking like this. She was giving up everything to build the new Israel. Beautiful homeland for the Jews. Why you are not staying there? Why you are coming back now and putting me on to the street?' There was a wheedling note in her voice.\n\n'Nobody is putting you on to any street, Ella. You are putting yourself on to the street living with these Arabs.'\n\n'These are my Attendents.'\n\n'Ella, you have lost your screws. All Arabs are the same \u2013 they are only waiting for the opportunity to push Jews into the sea.'\n\nAcross the table, Mr Ali was whispering something to Ishmail. The Attendents' faces were sullen.\n\n'Nobody is pushing me into the sea. The sea is a long way from here, Chaim. Sea is at Dover. I heff been there mit Arti.' Her chin was sticking out defiantly.\n\n'I know this Dover Beach. Where ignorant armies splash at night,' Chaim Shapiro tutted, taking little sips from his glass of water as if to cool himself down.\n\nMrs Shapiro stared at him. Then she leaned across and whispered to me, 'What is he talking about, Georgine?'\n\n'It's a poem.'\n\n'A poem? Is he med?'\n\n'I am talking about terrorism, Ella. Look at my blinded eye. What I was doing? Nothing. Sitting minding my own businesses.' He was cracking his knuckles furiously as he talked, from nervousness or anger.\n\n'We are in London now, Chaim. Not in Tel Aviv.'\n\n'And you see they have commenced bombing here in London.'\n\nMr Ali translated for Ishmail, who leaned over and whispered to Nabeel. All three of them were scowling.\n\n'We are already in the darkened plain.'\n\n'Darlink Chaim, this is a house, not an aeroplane. Please, be a little calm. And these are my Attendents, not suicideniks. See, they are even animal lovers.'\n\nNabeel had reached across and was stroking Wonder Boy behind the ears, whose rhythmic purring was a soothing background to the fractious discussion. If only someone would stroke Chaim Shapiro behind the ears, I thought.\n\nNow Mr Ali spoke, his voice splintering with anger. 'Arabs, Christians, Jews been living side by side for many generations. Making businesses together. No broblem. No bogrom. No concentration camp. Even we selled you some of our land. But this is not enough. You want whole bloody lot.'\n\nChaim Shapiro ignored him and turning towards me explained in a teachery tone, 'All Palestinians have the same story. They come along with some old key, saying this is the key to my house. You must move out immediately! But when my mother came to Israel nobody was living there. It was empty as a desert. Abandoned. All the inhabitants had scarpered.'\n\n'Driven out with gunpoint!' Mr Ali tried to shout, but his voice was shaking and it ended in a little squeak. The last time I'd seen him so mad was when he was sitting on the wet grass at the bottom of the ladder.\n\n'If you want to live alongside us in our land, all you must do is to stop attacking us. Is that not fair enough?' Chaim smirked and spread his hands theatrically.\n\n_In \u2013 two \u2013 three \u2013 four. Out \u2013 two \u2013 three \u2013 four._\n\n'Look, we're not going to solve all the world's problems today,' I said cheerily. 'But it's quite a big house. Especially if we convert the penthouse suite. Maybe everyone can live here together.'\n\nThey all turned towards me, and I could feel myself turning crimson under their collective gaze. In fact everybody had gone a bit beetrooty, even Mr Ali. Wonder Boy was snarling like a dog, swinging his fat tail from side to side.\n\n'I do not want to share my house with three Arabs,' Chaim Shapiro grouched.\n\n'Chaim,' said Mrs Shapiro appeasingly, 'the Peki is not living here. He is only a visitor.'\n\n'You do not understand the Arab mentality, Ella. They will not let us in peace. Do you think Israel would exist today if half its population was Arab, and trying to destroy it from within?'\n\nI felt a stab of anger, remembering the twin babies, heavy as watermelons, and the soldier with the number tattooed on his arm.\n\n'But you can't expect people to give up their homes and land and not fight back!'\n\nMr Ali translated for the benefit of the Attendents, who nodded fervently in my direction. Chaim Shapiro's face was sweating, his good eye blinking rapidly.\n\n'Ha! Then we have the right of self-defence! Every time you strike Israel we will strike back harder. You give us homemade rocket-launchers, we give you US-made helicopter gunships. Bam bam bam!' He aimed his hands like a gun across the table. Then, turning to me, he added, 'As your immortal bard William Shakespeare said, to do great right, we have to do a small wrong! It isn't pretty, but it is necessary, Miss Georgiana.'\n\nWhen I said nothing, he lurched forward and slapped the table suddenly like a volley of gunfire. 'Bam bam bam! Bam bam bam!'\n\nWonder Boy, who was still sitting on the chair at the head of the table, flattened his ears at the noise and hissed, showing his horrible fangs. Then he leapt up on to the table in fighting pose, his back arched, his tail puffed out, and with a yowl he flew at Chaim Shapiro, going at his face with his claws. Chaim Shapiro fought back, trying to pull the big cat off, but Wonder Boy clung tight, his tail thrashing, his claws lashing. Mrs Shapiro shrieked frenziedly at both of them.\n\n'Halt! Chaim! Stop this smecking! Wonder Boy! Raus!'\n\nThe cat hissed and fled, knocking over the jug of water that trickled down on to our legs. Chaim Shapiro pulled out a handkerchief and dabbed at his bleeding cheek. When he looked up, we saw that his glass eye had swivelled round grotesquely in its socket. Only the white was showing, staring out blankly like a hard-boiled egg.\n\nEveryone went quiet, as if shocked at how quickly the confrontation had flared up, and a sudden thought lit up like a light bulb in my head: these people \u2013 they're all completely mad. In another part of the house, we could hear a menacing yowl \u2013 Wonder Boy sizing up his next victim (feline, I suppose, for Mrs Shapiro's slippers were on her feet). It was Mrs Shapiro who spoke first. I noticed an appraising look in her eyes as she leaned over to Chaim and patted his arm.\n\n'Darlink Chaim, there is no need to fight. If you heff no home you can live here mit us. You can take any room what you like \u2013 except of mine, of course. You can make all your beautiful renovations, mit your tool kit. Build in kitchen units. Dishwashers. Meekrowaves. My Nicky has told me everything what is needed for the modern kitchen.' She took his hand and gave it a squeeze. 'We will make dinner parties mit cultured conversations. Concerts in the evenings. Even we will heff poetry recitals if that is what you like.' I could see his face softening as he pictured these delectable scenes. 'You are my Arti's son, Chaim. This is always your home whenever you want. But my Attendents also must stay here mit me.'\n\nHer voice was so seductive that I might almost have applied for residency myself, even though I knew, as Chaim did not, about the Phantom Pooer. Chaim, I could tell, was already seduced.\n\n'Ella, I can see you are quite a little home-pigeon, and I will gladly accept your invitation to take up my residence with you. And if the Arabs must stay, maybe we can divide the house between us. They keep to the top part of the house, and we stay in our part.' He beamed magnanimously across the table.\n\n'Hm! Next you will build a wall,' said Mr Ali drily. 'Checkpoint on the stairs. Then you will steal some more rooms for settlements.'\n\nIshmail and Nabeel smiled confusedly.\n\n'Have you got a sticking plaster, Mrs Shapiro?' I asked, to defuse the tension.\n\nChaim's cheek was bleeding badly \u2013 Wonder Boy had taken quite a swipe. She scuttled off to find one. Mr Ali and the Attendents had convened a separate meeting in the kitchen. I could hear the clink of the coffee pot, and soon after the smell of freshly brewed coffee drifted into the room. So for a few minutes, Chaim Shapiro and I were alone together. He took off his jacket, hung it on the back of his chair, and undid the top button of his white shirt. He was sweating profusely under the arms. Without the jacket he seemed to shrink in size. His bulk, I realised, was mainly shoulder pads.\n\nThe eye that looked at me \u2013 his good eye \u2013 was dark and sad, but it reminded me of the blazing brown eyes of the young woman in the photographs, and his round pudgy face ended in a little pointed chin like a crude copy of hers. I was still thinking that someone should stroke him behind the ears, but instead, I leaned forward and said, 'You remind me of your mother.'\n\nHe turned towards me and his look changed entirely, lit up with a smile so sweet and childlike it seemed to have strayed on to the wrong face.\n\n'You knew my mother?'\n\n'I didn't know her,' I said. 'I've seen her photo. You look like her.'\n\n'I wish you could have met her. Everybody who met her loved her.' He was smiling that same baby smile, his heavy cheeks dimpling with pleasure at the memory.\n\n'And your father...'\n\n'Yes, Artem Shapiro. The musician. She was always talking about him, like the legs of a donkey.'\n\n'... why didn't he join her in Israel?' I found myself holding my breath.\n\n'He was too sick. Lungs kaput. Ella was looking after him. Here in this house.'\n\nThe death certificate had said lung cancer.\n\n'And your mother never went back to him?'\n\n'She wanted to build a garden in the desert. Can you imagine \u2013 with her naked hands? She would never leave until it was finished.' A shadow settled over him and he seemed to shrivel up even more inside the white polyester shirt. 'Then she got sick. Blood sickness. She died when I was ten years old. A few months after my father.'\n\nI remembered the date on the letter from Lydda. Chaim was born in 1950, so she must have died in 1960.\n\n'I'm sorry. To lose your whole family... And then your injury...'\n\nI wanted to ask how it had happened. I guessed he didn't know without looking in a mirror that the glass eye was turned the wrong way in its socket.\n\n'But my family was the moshav \u2013 father, mother, sister, brother. After she died I stayed there with them. Everybody was family in our new nation.'\n\nIt must have been the same moshav she wrote about in the letter, the stony hillside where she'd cradled her newborn baby in her arms, looking out to the west and waiting for her husband. I still had her photo at home in my bedroom. I'd bring it for him next time.\n\n'Was she from Byelorussia too?'\n\n'No, she came from Denmark. But they met in Sweden. They were married in London. And I was born in Israel.' He smiled that chubby dimply smile. 'Naomi Shapiro. She was a person who knew how to dream.'\n\n'She dreamed of a promised land?'\n\n'Our homeland. Zion.' His cheeks dimpled again. 'Home sweet home.'\n\nBut something was niggling me. Why does everyone go on about homeland? Surely what really matters is the people we're attached to? Ben and Stella were my homeland \u2013 yes, and Rip. I tried to imagine what it would be like to love a country more than them. I thought about the woman in the photographs \u2013 those dark eyes blazing with conviction. She'd left her love behind to find the homeland of her dreams and someone else \u2013 another Naomi Shapiro \u2013 had stepped into her place.\n\n'But isn't it your homeland, too, Chaim? More so, because you were born there? Haven't you got a family? Friends? Colleagues? I can't understand why you want to make your life here.'\n\nAt your age, I meant to add, but didn't want to seem rude.\n\n'I was a teacher for thirty years. English language and literature.' He shuffled in his chair. 'Now I am retired. Not married. What woman wants to marry a one-eyed man?'\n\n'Oh, I don't know...' Mrs Shapiro will soon sort you out, I was thinking.\n\nShe had reappeared with a rather grubby curling-at-the-corners sticking plaster, which she applied to his cheek with a little pat.\n\n'Now your home is mit us, isn't it?'\n\nI noticed that there were a couple of cat hairs adhering underneath.\n\n'Thank you, Ella. My mother told me you were very solicitous to my father in his illness. And encouraging him to go to Israel upon his recovery. She showed me the letter you wrote.'\n\nI glanced across at Mrs Shapiro.\n\n'It was very long ago,' she said. Some inscrutable emotion flitted across her face and she gave a little shrug. 'Sometimes is better to let the past alone.'\n\n'Yes, long ago.' He sat back heavily in his chair. 'You know, Ella, this country, this Israel, it is not the same country she dreamed of. It should have been a beautiful country \u2013 prosperous, modern, democratic. Founded on justice and the rule of law. But _they_ have spoiled it with their fanaticism.'\n\nHe gestured with his head towards the kitchen where Mr Ali and the Attendents were still chatting in Arabic. There was a clink of coffee being poured.\n\n'You know, Miss Georgiana, no teacher wants to have blood of children on his hands. Not even of little stone-throwing Arab ratscallions.'\n\nBut I wasn't really listening, my mind had drifted back to what he was saying before Wonder Boy had lashed out with his claws. To do a great right, do a little wrong. That was Bassanio, in _The Merchant of Venice_. I'd done it for 'A' level. But what was it that Portia had said? Something about the quality of mercy. When mercy seasons justice. That was it.\n\n'So what do you think is the solution?' I asked.\n\n'There is no solution. I can see no possibility of peace in my lifetime.' He sank lower in his chair, resting his chin on his hands. 'So long as they continue with their attacks, we will continue our defences. We are trapped in tits for tats. It is impossible for someone so sensitive like myself to live life this way.'\n\n'But... it's never too late, is it? For peace? I mean, if only the will is there...'\n\nI was thinking even as I spoke that the words sounded good, but they were probably tosh. The will for peace \u2013 Rip and I had still not managed to work it out, had we?\n\n'Too late for me, Miss Georgiana.' He sighed. 'For at my back I always hear time's horse-drawn chariot galloping near.'\n\n'Wing\u00e8d.' I couldn't stop myself, but he was lost in his thoughts and didn't hear. Maybe I should introduce him to Mark Diabello. They would have the same taste in poetry.\n\n*\n\nI noticed, as I walked home in the early evening, that the silver buds of the pussy willow had opened out, and flaunted golden flecks of pollen in their fur. The air was soft and moist. A fine spring rain dampened my face and settled like mist on my hair; it glistened on the leaves and fell in slow heavy drops from the overhanging branches. Everything was cool and green. It was a different world to that of Chaim Shapiro and Mustafa Ali \u2013 but it was the same world. We all had to learn to live here somehow.\n\nI'd felt so full of pity when Mr Ali had told me his story, if I'd had a gun, I would have gone out myself to seek revenge for his lost home and violated family. Now I was beginning to feel sorry for this sad crumpled man, this one-eyed orphan of his mother's broken dreams. My parents had taught me always to look out for the underdog, but even underdogs can snap and snarl. How could I know who'd started it? Whose fault it really was? Maybe that was the wrong question to ask in the first place. If you could just get the human bonding right, maybe the other details \u2013 laws, boundaries, constitution \u2013 would all fall into place. It was just a case of finding the right adhesive for the adherends. Mercy. Forgiveness. If only it came in tubes.\n\nIt wasn't until I was almost at home that I remembered I'd never asked Chaim how he had lost his eye. Had he been caught up in the revenge attack at Lydda airport? I recalled my conversation with Ben a few days ago \u2013 the ancient prophecy of the battle between Jesus and the Antichrist at the gates of Lydda which was supposed to precede the end of the world. An airport _is_ a kind of a gate to a city \u2013 isn't it? But surely the terrorists wouldn't have known the words of the prophets. I felt a small quake of dread in my guts. How could the present reach back into the past? What mysterious tendrils of causation could have brought about this connection? No wonder Ben was so rattled. And Dajjal, the devil with one eye? But Chaim Shapiro was no devil; he was a casualty, too \u2013 a stray soul who had lost his mother too young. Without his shoulder pads, he was just a sweaty middle-aged man in a polyester shirt. Still, I felt a shudder as if an ancient hand had tapped my shoulder and a voice from another world had whispered, ' _Armageddon_.' \n\n# 43\n\n# Unpromising adherends\n\nAs I approached my house, the daylight was already fading and I could see through Ben's window that his computer monitor was on, the screen saver flickering white, red and black. That was strange. Ben was supposed to be with Rip. Maybe he'd forgotten to turn it off before he left. Or maybe he'd come back early.\n\n'Hi, Ben!' I shouted up the stairs as I came in through the door. There was no reply. I put the kettle on, then I went up and tapped on the door of his room. No answer. So I pushed it open.\n\nThere was that musty smell of socks and trainers, and there was the screen saver whizzing around in the dusk, hurling its dizzying pattern against the walls. White! Red! Black! White! Red! Black! Whoosh! Whoosh! Whoosh! The walls lit up, burst into flames, blackened to char. My ears were filled with a terrifying sound that I thought at first was the computer until I recognised the roar of blood beating in my own head. From across the room, a lumbering monster with hideous teeth lurched towards me \u2013 Ben's Orc poster, fleetingly illuminated by the flare of the screen. Then I saw Ben. He was lying on the floor between the bed and the desk, crumpled like a bundle of rags among his scattered clothes.\n\n'Ben!' I screamed. But as if in a nightmare the word came out of my mouth as a voiceless croak.\n\nThen I realised it wasn't just the jerky light; Ben was moving, twitching. Head thrown back, eyes open and rolled back in their sockets like Chaim Shapiro's glass eye, flecks of foam or vomit dribbling from the corners of his mouth. I stumbled towards him, and as I did so, I knocked the chair which was snagged around the cable of the mouse, and the screen he'd been viewing came up \u2013 the same fiery red on black screen with dancing flames and one flashing word: _Armageddon._\n\nI screwed up my eyes and reached across to pull the plug out of the socket. The room went dark. I switched the light on. Ben moaned and flailed with his arms and legs. A sickly sour smell was coming from him. A trickle of moisture darkened his trousers and puddled on the floor. I lay down beside him and folded him in my arms, stroking his cheeks and his forehead, whispering his name. I wasn't sure whether it was the right thing to do, but I held him tight until he lay still and his breathing slowed down. Then I phoned for an ambulance.\n\nThe next stage all happened very fast, in a whirl of panic and brisk paramedics and blue flashing lights. I tried ringing Rip from the ambulance but there was no reply so I sent him a text. After a few minutes Ben came round. He lifted his head from the stretcher, looking around him with a dazed expression.\n\n'Where am I?'\n\n'You're on your way to hospital.'\n\n'Oh.' He seemed disappointed.\n\n'I'm your mum.'\n\n'I know that.'\n\nI held his hand, whispering little mother-words as we ripped through the evening streets, siren howling.\n\n*\n\nThe ward they admitted him to was the same one Mrs Shapiro had been in that first time. The sister \u2013 I didn't recognise her \u2013 came and drew the curtain around us. It was frightening to be on the inside of that drawn curtain. I remembered the gurglings that had come from the next bed as the lady of the pink dressing gown passed away. The doctor who came round to see us seemed hardly older than Ben \u2013 in fact he had the same gelled-up hairstyle as Damian.\n\n'It seems like he's had a fit,' he said. His voice had a nasal Liverpool twang.\n\n'What \u2013 epilepsy?'\n\n'Could be. Could just be a one-off.'\n\n'But why?'\n\n'Too early to tell. We'll have a better idea when we've done the MRI scan.'\n\n'When will that be?'\n\n'Tomorrow. He'll see the neurologist. Let him sleep it off tonight. We'll keep an eye on him, don't worry. It's not that uncommon, you know, in young people his age.'\n\nHe smiled awkwardly, fiddling with the stethoscope that hung round his neck. He was trying to be kind, but he was too young to be convincing.\n\nThen the curtain parted and Rip and Stella came in. Rip ignored me, and I think I might have done a runner if Stella hadn't come straight up and hugged me.\n\n'What's up with Ben, Mum?'\n\nHow pretty she was, but so thin \u2013 too thin. She smelled of apple shampoo and neroli. I held her and stroked her hair which sheafed down her back like dark silk. I wanted to burst into tears, but I forced a cheerful grin on to my face.\n\n'Something happened \u2013 he had a fit, or something. I think he's going to be okay.'\n\nStella squeezed her brother's hand. 'Yer daft little beggar.' She was putting on a thick Leeds voice, the voice of their shared childhood banter.\n\nHe opened his eyes and looked around with a beatific smile on his face.\n\n'Hey, everybody!' Then he drifted off again.\n\nRip stood framed by the curtain, trying to hector the doctor into conversation, demanding explanations and clarifications which the young man was clearly unable to give, and all the time carefully avoiding meeting my eyes. When the doctor left Rip came and sat on the other side of the bed, still ignoring me, and took Ben's other hand, leaning over and talking in a sickly cooey-cooey voice. I got up and walked out.\n\nI went as far as the swing doors, then I stopped. I knew I was being ridiculous. I turned round and went and sat in the day room to calm down, clenching and unclenching my hands \u2013 _in \u2013 two \u2013 three \u2013 four_ ; _out \u2013 two \u2013 three \u2013 four \u2013_ breathing in the heavy medical air that was thick with all the anxiety and grief that had been exuded in this room. I remembered the drip lady, and the wild cackling. It seemed an age ago.\n\nA minute later, the door swung open and Stella came in. Her face was red and blotchy. At first I thought she was upset; then I realised she was furious.\n\n'Mum, you're mental \u2013 you and Dad \u2013 you've got to stop acting like kids. We're sick of it, me and Ben. We want you to... I dunno... like, grow up.'\n\nShe was chewing at a strand of hair that had straggled across her face, just like she'd done as a child. I stared at her. She was twenty years old, as skinny as a twig, and she was wearing a skirt that showed her knickers when she bent over; and I'd carried her in my belly and fed her at my breast, and here she was telling _me_ to grow up.\n\n'Yes, but what about _him_?' I whined.\n\n'Him, too. I've told him, too. Both of you. You've got to stop it.'\n\nShe sounded just like Mrs Rowbottom reprimanding Gavin Connolly for flicking pellets.\n\n'But he started it.'\n\n'Doesn't matter who started it. We're fed up of it. _And_ it's not doing Ben any good.'\n\nShe brushed back the hair from her face and tried to look stern.\n\n'Okay. Well, I will if he will. But I'm not...'\n\n'So just go back in there, and smile at him, and... I dunno... just be _normal_ , Mum.'\n\nSo I did. I smiled at Rip, and he smiled at me, a bit awkwardly, and he explained that he'd had to move out of Pete's place, and he'd tried to ring me to tell me that Ben was coming home earlier than expected, but I'd not rung him back. When an accusing note slid into his voice, Stella threw him a warning look.\n\n'Dad!'\n\nShe would make a great teacher, this girl.\n\nWhen I think of the turning point, the point from which it all started to get better again, I think of that Monday in March, that scene in the curtained cubicle at the hospital, Ben sitting up and trying to remember what had happened, Stella perched on the edge of the bed tickling Ben's toes through the bedclothes and making him laugh. It reminded me of the glue exhibition, with me and Rip sitting awkwardly on each side of the bed like lumpy unpromising adherends, and Ben and Stella in the middle holding us together like two blobs of glue.\n\nWe sat together like that in the neurologist's office next day, Rip, Ben and I, with Ben in the middle. The neurologist took us through a series of questions, and asked us about the circumstances of Ben's fit. When I described the whirling screen saver and the flashing flames of the Armageddon website, he told us about a cluster of 685 cases of epilepsy in 1997 in Japan that had apparently been triggered by a single Pok\u00e9mon episode on television.\n\n'It's possible for photosensitivity to trigger an epileptic seizure,' he said, peering at us through his small rimless glasses. 'What we can't tell at this stage is whether it will happen again.' He turned to Ben. He had a surprisingly mischievous smile for a neurologist. 'Try and be more selective about which sites you visit, young man. It's wild out there in cyberspace.'\n\n'Right,' Ben nodded. He was embarrassed by all the attention.\n\nBut there must be more to it than that, I thought. I remembered our liminal conversation, the haunted look in his eyes.\n\n'I can understand the computer flashing could set something off,' I said. 'But what about...?' I cast my mind back. 'Sometimes you said you were feeling strange when you got back from school, before you'd even turned the computer on. Don't you remember, Ben?'\n\nHe blinked and frowned.\n\n'Yeah. It was when I was on the bus. We passed these trees. I could see the sun through the branches.' He described a long road where low winter sunlight flickered through branches of an avenue of trees as he sat on the upper deck of the bus. 'That's when I started having, like, _feelings_.'\n\n'But when you've stayed with me in Islington you've been perfectly all right.' There was an edge of accusation in Rip's voice, as though I'd caused the problem.\n\n'I got a different bus.'\n\nThe neurologist nodded. 'If you find yourself in that situation another time, young man, just try closing one eye.'\n\nSo that's all there was to it \u2013 the generations of prophets, the reign of the Antichrist, the tribulations, the Abomination of Desolation, Armageddon, the fearsome battle of all the armies of the world, the rebuilding of the Temple at Jerusalem, the end of time with trumpet clarions and fiery chariots, the return of the Messiah, the rapture of the elect \u2013 it was all down to a frequency of flashing lights, a temporary short circuit in the wiring of the brain. All you had to do was close one eye.\n\nI felt both relief and disappointment. For there was a part of me that yearned to believe \u2013 to surrender to the irrational, to be swept away by the rapture.\n\n'So all that religious stuff is just nonsense?' Rip's voice was irritatingly smug. I wanted to kick him, to shut him up, but I saw Ben wasn't listening. He was studying a chart of the brain pinned on the wall at the side of the neurologist's desk.\n\n'It's now believed that some prophets and mystics were in fact epileptics,' the neurologist said. 'There's thought to be a physiological explanation behind much religious experience.'\n\nRip misread the look on my face, and leaned across to squeeze my hand.\n\n'Why didn't you tell me Ben was having these problems? You should have told me, Georgie.'\n\n'I...' _In \u2013 two \u2013 three \u2013 four. Out \u2013 two \u2013 three \u2013 four_. 'You're right \u2013 I should have.'\n\nI squeezed his hand back.\n\nAs we left the hospital, Rip asked me rather sheepishly whether it would be okay if he moved back in temporarily, and I replied rather grumpily that it made no difference to me but I was sure Ben would appreciate it. Yes, I was pleased, on the whole; things were going in the right direction. But I was surprised to find that my feelings were ambivalent. I'd got a life of my own now, and I wasn't ready to give it up. When Rip was around, he had a way of taking over. While he'd been away, I'd been remembering all the things I missed about him, but being with him again reminded me of all the things that irritated me. It occurred to me that maybe he felt the same way about me. So there was still a lot that needed to be resolved between us. He brought his stuff over from Islington in his Saab later that afternoon, and set himself up on a camp bed in the little mezzanine study. We tiptoed around each other, being excessively polite and considerate.\n\nHim: Would you like a cup of tea, darling?\n\nMe: That would be lovely, darling.\n\nThat kind of tosh.\n\nI had to clear out the spare room to make space for Stella, who'd be coming home soon for Easter. Buried at the bottom of one of the drawers I found an envelope of photos. Rip and me on our wedding day: Rip was wearing a top hat and tails. His hair curled down on his collar and he had curly sideburns. I was wearing a cartwheel hat and a fitted dress with big shoulders and slut-style high heels. My pregnant bulge was clearly visible. We looked ridiculous \u2013 and ridiculously happy. Then a picture of Rip and me and baby Stella in a buggy walking round Roundhay Lake. Then Rip and me and five-year-old Stella and baby Ben on the beach at Les Sables d'Olonne. Rip and me and Ben and Stella and Mum, taken one Christmas at Kippax. Rip and I were wearing Santa hats; Mum was wearing reindeer antlers; Ben was wearing his new _Lion King_ slippers and a gawky smile \u2013 what a funny little kid he'd been; Stella \u2013 she must have been thirteen \u2013 was pouting red lipstick at the camera, wearing a figure-hugging red top with a tinsel wreath draped around her shoulders. Dad wasn't in the picture \u2013 he must have been behind the camera. The Christmas tree wearing millennium-themed baubles was clearly visible in the background. I pored over the photos, then slid the envelope under my mattress. It seemed like a good omen.\n\nAt the end of term Stella came home, and from being empty, the house suddenly became full. It was Stella who told me, over a quiet cup of tea, that Ottoline had thrown Rip out. He'd spent the night before coming to the hospital in a hotel. That's why Ben had come home unexpectedly that Monday.\n\n'Ben says he overheard them having a row. Apparently she told him he had a poor attitude to commitment,' she murmured in a grave voice, lowering her head, so if I hadn't been looking I wouldn't have seen the flicker of a grin at the corners of her mouth.\n\nStella made the most of her holiday, sleeping in late and taking long showers, sometimes twice a day, clogging up the plughole with her long hair and filling the house with the smell of apple shampoo. Ben filled the house with techno music and thumped around cheerfully, no longer glued to the computer. Rip went off to work every morning, just as he had before, and in the evenings he sat at his desk and filled the house with brainwaves. We took turns to cook \u2013 we had two teams: Rip and Stella, who cooked mainly Thai curries, and Ben and I who cooked mainly Italian. Then Ben announced one day that he'd become a vegetarian, and we spent ages adapting and devising recipes for him. I once caught him sitting at the table and poring over a book with that same intense concentration that he had once read the Bible, but it turned out to be a cookery book: _One Hundred Recipes to Save the Planet_. The knobbly skull had disappeared under a growth of new brown curls, which he wore tied back with a red bandana.\n\nThe neurologist had suggested Ben change the screen saver, and warned him off websites with animation. He advised him to get a flat-screen monitor, which apparently runs at a different frequency, and not to sit too close to the television. We watched anxiously, to see whether he could handle his condition without medication or whether he would need to take anti-epileptic drugs.\n\nRip and I fell into a pattern of sharing the same space while keeping out of each other's way. We didn't actually divide the house but we learned each other's habits and avoided unnecessary contact. It wasn't positively amicable, but it wasn't hostile, either. Sometimes, on Stella's insistence, we all watched TV together.\n\n'Just try to be _normal_ , okay?' she coached us.\n\nRip and I sat on armchairs on opposite sides of the fireplace, with resolutely normal expressions on our faces, while Ben and Stella sprawled on the sofa, their arms and legs casually intertwined. From time to time one of them would try to shove the other off.\n\nAt Easter, we didn't go to Kippax or to Holtham. We stayed at home, and Rip and I made a tentative stab at collaboration, hiding a trail of miniature Easter eggs around the house for Ben and Stella. They whooped around, pretending to be surprised. The radio was on in the background, and at one point I heard a church congregation with miserable whiny voices singing that hymn. _There is a green hill far away_... _He hung and suffered there._ I switched it off quickly. Why let that morbid long-ago stuff spoil a nice family holiday? \n\n# 44\n\n# Water creases\n\nOn the Tuesday after Easter I nipped into the local Turkish supermarket and bought a large Easter egg, reduced to half price. It was a hideous-looking thing covered in mauve foil with Space Invaders figures wielding ray guns all over the packaging. Someone somewhere must have thought this was an appropriate Easter gift for a little boy \u2013 in fact maybe it _was_ surreally appropriate to the new reality of the Holy Land \u2013 but at least it had been left on the shelf by discerning parents, for it was the only egg they had. I carefully peeled off the REDUCED sticker, wrapped the egg in tissue, and set off for Canaan House.\n\nIt was a fresh, cold day, with splashes of sunlight spilling through raggy clouds. Small bright buds were bursting on the ash-tree saplings in the garden at Canaan House \u2013 it seemed as though they'd appeared overnight \u2013 and the white plastic garden furniture gleamed invitingly.\n\nNobody answered the doorbell when I rang. I crouched down and peered in through the letter box. There were no signs of human life, though a couple of felines were dozing in the pram which was parked under the stairs. I thought I caught a glimpse of movement at the end of the corridor, and then I noticed something very alarming \u2013 water seemed to be dripping down from a crack in the ceiling and collecting in a pool on the hall floor. A moment later, Chaim Shapiro appeared in his shirtsleeves. I rang the doorbell again to get his attention, but he just looked up at the leaking ceiling, shouted something into the back of the house, then vanished up the stairs. The drip of water had intensified into a trickle. Suddenly Nabeel and Mr Ali materialised, legs first, running down the stairs and shouting at each other. I rang the bell again, and Mr Ali came and opened the door. I thought he'd opened it for me, but he raced right past me, out through the door and round to the back of the house. I followed him, and watched as he started frantically pulling away at the grass and weeds near the kitchen door to reveal a small metal hatch cover, which he removed. Still shouting at Nabeel, who was behind us, he rolled up his sleeve and reached into the hole in the ground.\n\n'What's happening?' I asked Nabeel.\n\nNabeel flashed his beautiful eyes, pointed a finger upwards, and shouted back to Mr Ali. Then he raced back to the front of the house. I followed behind. The two tabbies in the pram in the hall were awake by now. They roused themselves, stretched mardily, and slunk out into the garden, their ears flat with irritation at being disturbed. Then Mrs Shapiro turned up, tottering on her high heels, waving a cigarette in her hand.\n\n'Ah! Georgine! Thenk Gott you come!' She flung her arms around me.\n\n'What's going on?'\n\n'Votter creases! I was telephoning to you!'\n\n'Water creases?'\n\n'They are trying to mek votter pipe diversion into the penthouse suite. Chaim! Chaim!' she yelled up the stairs. 'What you doing? Heffn't we got enough votter pissing down already?'\n\nThe trickle of water had become a steady stream; I noticed that the water was pleasantly warm. The hall was filling up with steam like a bathroom. Above us, the plaster ceiling was beginning to bow, while Mrs Shapiro was mopping determinedly but hopelessly at the puddle with a silk blouse she'd pulled out of the pram, kneeling down on all fours and holding her cigarette between her lips. Now Mr Ali appeared in the doorway. He shook his head with a philosophical air and sighed as he gazed at the stream of water, which was fast becoming a torrent.\n\n'It comes out of the tank. Not men's water,' he explained to Mrs Shapiro. Then he shouted something at Nabeel, who bowed his head and slouched off upstairs. Mr Ali shrugged apologetically. 'Completely useless.'\n\nI was still puzzling over the gender status of the hot water when Ishmail and Chaim Shapiro came running down the stairs, almost colliding with Nabeel on the way up. Chaim pointed at the water coming through the ceiling, and shouted, rather unnecessarily, 'Water water everywhere!'\n\n'Men's now off, but water still coming out,' Mr Ali shouted back.\n\nIshmail shouted at Nabeel. Mrs Shapiro shouted at Chaim, who shouted back at her. I shouted at him to shut up. Soon everybody was shouting at everybody else. Somewhere in the house, Wonder Boy started to yowl. Mrs Shapiro had given up trying to mop the floor with the silk blouse, and started flicking it at her stepson.\n\n'Is all your fault. You wanted to make votter separation. Jewish votter, Arab votter. So! Now we have pissing votter.'\n\n'Not my fault, Ella. Useless Arabs cut the wrong pipe.'\n\nThen the doorbell rang.\n\nWe all fell silent and looked at the door. Through the frosted glass I could see a tall dark figure looming. Nobody moved. The bell rang again. I opened the door. It was Mark Diabello.\n\n'Hello...' He stared at the scene in the hall, taking in the flushed faces peering through the clouds of steam, the wet floor and the pouring water. 'Georgina, I just wanted to...'\n\n'Come in. We're having a bit of a water crisis...'\n\n'Who is this?' asked Mrs Shapiro, pulling herself up straight and smiling at the handsome stranger. 'Are you another Attendent?'\n\n'Let me introduce Mr Wolfe's partner,' I said. 'Mark Diabello.'\n\n'My Nicky's partner? How charming!' She fluttered her eyelids.\n\nHe stepped forward, proffering his hand, his chin-dimple winking, his smile-creases crinkling, his green-gold-black eyes flickering non-stop.\n\n'Delighted, Mrs Shapiro. If I could just trouble you for a second \u2013 the house deeds...'\n\nAt that moment, there was a horrible wrenching sound above our heads. Everyone looked up. One of the ornate Doric-style plaster corbels supporting the Romanesque arch where the water had come through had started to crack away. Even as we watched, the crack widened. The corbel slipped sideways and slid. Mr Diabello seemed to stagger as he took a step back. His knees sagged. His mouth opened, but no sound came out. Then he fell to the ground with a thud. He had been stunned by a stunning period feature.\n\nPoor Mr Diabello. By the time the ambulance arrived he was sitting up on the wet floor, propped against the wall beneath the grey mark where the picture of Lydda had hung \u2013 it was the cat-poo spot, though any lingering cat poo would have long since been washed away \u2013 pressing a clean white handkerchief to a gash in his head.\n\nYet after his accident, a strange exhausted peace fell on the house. The water finally stopped running when the hot-water tank that held the immersion heater had emptied out. Ishmail got a broom and started sweeping the water out of the hall through the front door \u2013 there must have been several gallons. The cats danced around the eddies of water, excited by all the action but not wanting to get their paws wet. Mrs Shapiro danced around, too, making encouraging noises. Nabeel went into the kitchen to make a pot of coffee. As the door swung open, I overheard a snippet of conversation.\n\nMr Ali: Where you get your tool kit, Chaim?\n\nChaim Shapiro: B&Q. You want to see it?\n\nI sat with Mr Diabello until the ambulance arrived.\n\n'I thought you might be here. I came to see you, Georgina,' he murmured. 'I didn't realise your hubby was back.'\n\n'Yes. I should have told you. I'm sorry. You and me \u2013 it's over between us, Mark.' I squeezed his hand as they led him away to the ambulance. 'But it was fun.'\n\n'Mrs Shapiro,' I said, keeping my voice casual, 'do you happen to know where the deeds for this house are kept?'\n\nMr Ali and Chaim Shapiro had gone off to B&Q together in manly silence and we were having a companionable cup of coffee by the fire in the study, with Prokofiev's piano sonatas tinkling on the record player and the soggy _Lion King_ slippers steaming away on the fender.\n\n'What for I need deeds?' She looked at me through narrowed eyes.\n\n'Apparently the house isn't registered with the Land Registry.'\n\n'On this house I been paying rets sixty years no problem.'\n\n'Mr Diabello said it would be better to register in case you want to sell up at any time.'\n\n'I am not selling nothing.'\n\n'Of course there's no reason why you should sell.' There was no point in arguing with her. 'But it would be better for you if the house was registered in your name, Mrs Shapiro. Then no one could take it away from you.'\n\nShe reached in her bag for a cigarette and stuck it between her lips.\n\n'You think Chaim wants to take it away from me?'\n\n'Everybody wants it. Chaim. Mrs Goodney. Even Mr Wolfe and Mr Diabello. It's a desirable property.'\n\n'And what about you, Georgine?'\n\nShe said it casually, fumbling in her bag for the matches, not looking at me. I wondered whether it was an accusation.\n\n'It's a really lovely house,' I said, 'but I already have a house of my own.'\n\n'When I am dead, Georgine, darlink, you can heff it.'\n\nI laughed. 'It's kind of you, but it's too big for me. Too many problems.'\n\n'You can heff it, so long as you will liff in it and pay the rets.'\n\nShe gripped my hand and pulled me towards her. Suddenly she was intensely serious.\n\n'This house \u2013 it belongs to no one. Artem found it empty. Abandoned. Inhebitents ran away.'\n\n'But why...?'\n\n'You know, Artem was just new married. He was needing somewhere to live.'\n\n'With Naomi?'\n\nShe avoided my eyes. 'It was the wartime. German bombings. People running everywhere.'\n\nAbove our heads there was a clang of copper pipes and a tirade of words. Chaim and Mr Ali must have got back from B&Q. There was suddenly a lot of running up and down stairs and clattering and shouting going on in the background.\n\n'So they moved in?'\n\n'Such a beautiful house, isn't it? Even a piano. Bechstein. Sometimes _Mutti_ and I came to play on it. He played on the violin, we accompanied mit the piano.'\n\n'Two brown eyes.'\n\n'You know, Georgine, I was only a young girl. I didn't know anything \u2013 I knew only that I was in loff.' She pursed her lips and puffed a couple of smoke rings; they drifted towards the fire on the warm draught, and vanished in the flames. 'When you are in loff, when you heff an idea in your head, you are not always thinking about the consequence.'\n\nMy mind tripped back to my conversation with the Scarlet-mouthed Slut, her tentative apology that I'd accepted with such bad grace.\n\n'You thought being in love made it okay?'\n\n'I thought only that I could not live without him. And she was no good for him, that one. Always she was nagging him to go to Israel. A poor man like this with ruined lungs. What use he will be in Israel?'\n\n'So she went on her own?'\n\n'She was blazing like a person on fire. She could not sit still. Always talking of Zion \u2013 of making a homeland for all the Jews of the world. But he wanted only to die in peace.' A bit of wood shifted on the fire and clouds of ash drifted on to the fender. 'Already he was dust.'\n\n'Didn't you feel...?'\n\nShe shrugged and tossed her head in a vague gesture. 'I was looking after him. He could not be on his own. He was saying he will go there when he is better.'\n\nWhat I really wanted to ask was \u2013 did she feel guilty? For stealing Naomi's husband, and Chaim's father.\n\n'She wrote to him from Israel, didn't she?'\n\nShe nodded. 'Yes. Those letters. I burned them all.'\n\nHer face was turned away towards the fire, so I could not read her expression.\n\n'Not all of them.'\n\n# 45\n\n# The dance of the polymers\n\nIt wasn't until I got home that evening that I realised I still had the Space Invaders Easter egg in my bag. I unwrapped it and put it at the back of the cupboard. It was so vile that I couldn't bring myself to give it to Stella or Ben.\n\n'Who was that man?' Stella asked as we were clearing up together after a Thai curry dinner. We were alone together in the basement kitchen. Rip and Ben were watching football upstairs.\n\n'What man?'\n\n'That smooth creepy guy in the Jag who came round this afternoon while you were out?' Her lip curled with disapproval.\n\n'Oh, he must have been the estate agent. He wants to buy a house from an old lady I know who lives at Totley Place. Why?'\n\n'Daddy answered the door. They both seemed a bit surprised to see each other.' She gave me a hard look. 'He had a bunch of flowers. White roses.'\n\n'Really? They were probably for someone else.'\n\n'No, he left them. They're in my room. I told Daddy they were for me.'\n\n'Thanks, Stella. You can keep them. I don't want them.'\n\nShe grinned, a quick glimmer of a grin.\n\nNext day Rip gave me a peck on the cheek before he left for work, and maybe that's what made it difficult to write about Gina's revenge. Although I had some glue stuff to catch up on, I was determined to finish Chapter 8, so I set my laptop aside and opened up my exercise book.\n\n#\n\n# The Splattered Heart   \nChapter 8   \nGINA'S REVENGE (continued)\n\n_Disguised as an_ itinerant window-cleaner rag-and-bone woman Avon lady she made her way to Holty Towers and in the dead of night, she tiptoed through to the luxurious ensuit onsite ensued (bloody Microsoft! \u2013 I was using the spellchecker on my laptop because my dictionary was still propping up a shelf in the mezzanine study) _bathroom and got the deadly tube vial phial out of her Avon box and squeezed a thin layer of extra-strong adhesive on to the seat of the toilet lavatory. Then she turned the cold tap on in the basin so that it ran in a steady stream. Tinkle tinkle tinkle. A smile suffused her rosy lips._\n\nBut something wasn't right. I was starting to feel a bit sorry for Rick. Okay, so he had his flaky moments, but there was something endearing about him, wasn't there? Those blond tumbled curls. The vulnerability of the sleeping man. And Gina \u2013 wasn't she a bit off the rails too, falling for that dodgy mandolin player? What a pair of idiots Rick and Gina were. Why couldn't they just sort their differences out and stick together? I realised that something inside me had shifted \u2013 I was no longer very interested in revenge. I was ready to move on.\n\nI closed my exercise book and clicked open the _Adhesives_ document I was meant to be working on. 'The Chemistry of Adhesive Bonding.' On New Year's Eve, when we'd joined hands, like molecules grabbing hold of each other, and sung 'Auld Lang Syne', I'd had a flash of insight into polymerisation. Now I had discovered something even better \u2013 polymerisation depends on sharing. An atom which is short of an electron looks out for another atom that's got the right sort of electron (it's called covalency, for the chemically inclined), then the atom grabs the electron it needs. But no theft or nastiness is involved. The two atoms end up sharing the electron, and that's what holds all the atoms together in one beautiful long endlessly repeating dance \u2013 the beauty of glue!\n\nCanaan House was still on my mind, and I started thinking about the two Naomis, each trying to grab Artem. Had there been sharing and dancing? Or was it a case of theft and nastiness? Would Artem have made a different choice if he'd read Naomi's letters? Would Ella's heart have been broken instead? Burning the letters seemed such a monstrously wicked thing to do; yet I couldn't think of her as a wicked woman. It's as though love gives you a special licence to do anything you like. In the end death, the ultimate fracture line, split Ella and Artem apart. And Canaan House itself had been part of the dance, too, shared by one couple, then another. But whom did it really belong to? There were still some parts of the story that weren't clear. There must be a way of finding out.\n\nAfter lunch \u2013 four radishes and half a bagel with a bit of crusty cheese was all I could muster from the fridge \u2013 I nipped up to use the loo, and that's when I realised the other thing that was wrong with 'Gina's Revenge'. Men and women \u2013 we're different. Men stand up to pee.\n\nIn the afternoon the rain stopped long enough for me to pull on my Bat Woman coat and wing off down to the library on Fielding Street, just off Holloway Road. The reference library was up on the top floor, a hushed high-ceilinged room susurrating with the nasal snifflings of damp people and the dry rustle of pages being turned. The wet weather had brought in all the homeless folk, whose moist unwashed smell mingled with the musty odour of books and the municipal aroma of wax and disinfectant. Silent hunched figures eyed each other furtively above the pages. Ms Firestorm would have a field day in here.\n\n'I'm trying to find out the history of a house near where I live. It's called Canaan House. In Totley Place.'\n\nThe woman at the counter raised her eyes from her computer.\n\n'That's an interesting name. There was quite a fashion in Victorian times, you know, to give places Biblical names. There's no end of Bethels and Zions. And there's a Jordan Close in Richmond. Different Jordan, of course,' she giggled mousily.\n\n'Are there some old maps or anything like that?'\n\n'They've moved the local history archive to the Finsbury Library. We've just got a small local history section over there on the right.'\n\nOf the twenty or so volumes, the only specifically local book was one called _Walter Sickert's Highbury_. I flicked through the chapter headings and illustrations. On page 79 was a lithograph of a large house with a tree in front of it \u2013 the more I stared, the more sure I was that it was the same house with the same monkey puzzle tree, but much smaller. The caption read: 'The Monkey Puzzle House, home of Miss Lydia Hughes, whose portrait Sickert painted in 1929 when he was living in nearby Highbury Place.' Perhaps the name of the house had been changed. I looked in the index and browsed through the chapters, but there was no more information.\n\nThen my eye fell on a slim booklet in a yellow card cover: _A History of Christian Witness in Highbury_. It was obviously self-published. I took it through to the reading room and sat down at one of the desks. The booklet was mainly a rather dull list of Anglican and Catholic churches with scratchy line drawings, but the last chapter was devoted to what the author called the 'sects': Methodists, Baptists, Congregationalists, Quakers, Unitarians, Presbyterians, Seventh Day Adventists, Jehovah's Witnesses, Pentecostals, Sandemanians, Christadelphians, Swedenborgians, Latter Day Saints, Plymouth Brethren. So many different faiths all waiting, as Ben was, for the Day of Judgement that would bring about a new Heaven and a new Earth \u2013 not just over there in that dry, thorny, tortured land, but here in damp, leafy Highbury. Still waiting. Well, let them wait, I thought.\n\nTowards the end of the chapter was a short entry that read: 'A Teresian community was established in the late 1930s in a house in Totley Place. It was evacuated following an air raid in 1941 and the community dispersed.' I felt a rush of excitement \u2013 this could be it! But there was no more. The author was a Miss Sylvia Harvey. The book was published in 1977, thirty years ago. I scribbled the details on a piece of paper. The room was so quiet that you could hear the squeak of my pen as I wrote. There was no other sound apart from the snuffling and rustling and an occasional intermittent gurgle of the water cooler, like a dyspeptic gut. It reminded me that Dad's operation had been due today. I wondered how he'd got on.\n\nOver in the far corner by the magazines and newspapers, a tall heavily built man was wrestling with the _Financial Times_. He was sitting with his back towards me. He had curly grey hair \u2013 no, it was blond, streaked with grey. I stared. At first I thought my eyes were deceiving me, but there was no mistaking him. It was Rip. Beside him on the floor were his briefcase and our large blue Thermos flask. I wanted to rush up to him and put my hands over his eyes to surprise him, but something held me back \u2013 something about the way he was sitting \u2013 that sagging posture, staring straight ahead, his big shoulders hunched. He looked defeated. He wasn't even reading the newspaper, I realised, he was just passing the time. He was passing the time in the library because he didn't want us to know he wasn't at work.\n\nI took the booklet across to the desk.\n\n'How can I trace this author?' I asked in a low voice.\n\nThe woman smiled vaguely. 'You could try the telephone directory. Or the internet. Would you like me to have a look?'\n\n'No, it's all right. Thanks for your help.'\n\nI gathered my things together as quietly as I could and tiptoed out through the door.\n\n# 46\n\n# Smoke circles\n\nI'd already started cooking dinner when Rip came in just before six o'clock. It was something elaborate involving tofu and lemon grass. Stella was out and Ben was stretched out on the sofa with a book. Since his seizure, he'd been avoiding the computer, and only watched television occasionally.\n\n'D'you want a hand, Mum?' he'd shouted down to me. His voice sounded deeper, less croaky, than a couple of weeks ago. How quickly he'd changed.\n\n'It's okay,' I'd shouted back.\n\nI liked to see him with his nose stuck in a book, as I'd been at his age, though when he came down to eat later on, I saw that the book was _Revenge of the Busty Biker Chicks_.\n\n'Hi, Ben! Hi, Georgie!' Rip called as he came in, then he went straight up into the mezzanine study. I could hear him pottering around in there, playing music. Half an hour later, I stuck my head round the door.\n\n'Dinner's ready.'\n\n'What's all this, Georgie?'\n\nHe was standing in the middle of the room holding a B&Q carrier bag in his hand.\n\n'Where did you find that?' Then I remembered. I'd shoved it in the cupboard when Mark Diabello came round.\n\n'Are you planning a bit of DIY?' He was looking at me intently, curiously. I could feel myself turning red.\n\n'No, not DIY. Collage.'\n\n'Collage?'\n\nI smiled inwardly at the incredulity in his voice.\n\n'You know \u2013 sticking things. It's a form of art.'\n\nOur eyes met. He grinned. I grinned. We stood grinning at each other across a bridge of lies. I would never tell him that I'd seen him in the library, that I'd glimpsed his vulnerability. I reached out my arms and took a tentative step forward. There was a faint crackle and a smell of scorching, and Ben called from the kitchen, 'Come on, you two! The rice's burning!'\n\nDad always used to say, 'I like a bit of burned,' which was just as well, because Mum often obliged. Sometimes she went too far, like the first Sunday lunch Rip had with us at Kippax, when she placed a charred and shrivelled chicken in front of Dad for him to carve.\n\n'Poor little bugger looks like 'e's been cremated,' said Dad.\n\n'Nowt wrong wi' cremation,' said Mum. 'Keeps you regular.'\n\nI hadn't told Mum yet that Rip had moved back in \u2013 I didn't want to tempt fate \u2013 but I rang her after dinner to find out how Dad's operation had gone. She was in an ebullient mood.\n\n'They did a biopic. Doctor says it in't cancer.'\n\n'Oh, that's good. How's he feeling?'\n\n'Full of chips. Food were lovely in 'ospital. Got into a blazing argument with the bloke in the next bed about Iraq. Keir's coming home, by the way. Did I tell you?'\n\n'No, you didn't. That's good news, too.'\n\nIt would be good to see Keir again. Since he'd joined the army, our worlds had drifted apart; nowadays all we had in common was our shared childhood, but Mum resolutely held us together like the family glue.\n\n'She sent us some lovely flowers, by the way, your Mrs Sinclair. And a card. Best wishes for your recovery.'\n\n'I didn't know she knew about Dad.'\n\n'Oh, we keep in touch. She rings up from time to time. Or I ring her.'\n\n'Really?'\n\nThis was complete news to me. I tried to imagine what Mum and Mrs Sinclair would talk about. Then I realised they probably talked about us.\n\nI poured another glass of wine and put my feet up on the sofa while Rip and Ben put the rice pan to soak and cleared up in the kitchen. Then the phone rang.\n\n'Georgine, come quick! We heff an invitation!'\n\nMrs Shapiro's breathy voice shrilled down the telephone, but I was going nowhere.\n\n'What've we been invited to?'\n\n'Wait! Let me see \u2013 aha, here it is! We are invited to a funeral!'\n\nMy heart lurched. The last thing I needed was bad news.\n\n'Oh, dear. Who is it?'\n\n'Wait! It is here! What is this? I cannot read this name. Looks like Mrs Lily and Brown, ninety-one years old, passes peacefully in the sleep at the Nightmare House.'\n\nSo she never did break free, poor thing.\n\n'Who is this Brown Lily?'\n\n'She's the old lady you made friends with in the hospital. And at Northmere House. You know \u2013 who was always asking for cigarettes?'\n\n'This one who got the dead woman slippers? She is not my friend \u2013 she is a bonker.'\n\n'But it's nice that you've been invited to her funeral. Her family must have remembered you.'\n\n'What is so nice about a funeral?'\n\n'Don't you want to go?'\n\n'Certainly we must go!'\n\nThe crematorium was in Golders Green, miles away beyond Hampstead Heath. I mentioned this to Nathan, and suggested he might like to come along with his Tati.\n\n'He'll enjoy it,' I said. 'There's sure to be plenty of singing.'\n\nSomehow, the four of us fitted into Nathan's Morgan, even though it was really a glorified two-seater. Nathan and Mrs Shapiro sat in front. She was wearing a long black coat that smelled pleasantly of mothballs and Chanel No. 5 \u2013 better than the stinky astrakhan \u2013 and a chic little black beret with a veil and a feather. Nathan's Tati squeezed into the back with me. He was wearing a raincoat and a Bogart-style trilby. I was wearing my smart grey jacket and a black scarf. The car struggled under the weight of us all as we crawled up the Finchley Road. It was a Saturday morning in April, the air warm and sparkling in the slanting sunlight. In the residential streets the front gardens were already frothy with cherry blossom.\n\nNathan's Tati took Mrs Shapiro's hand to guide her up the step to the crematorium, and she acknowledged the gesture with a gracious nod. There were only two other people in the chapel when we arrived: a grey wispy-looking woman who introduced herself as Mrs Brown's niece, Lucille Watkins, and her father, Mrs Brown's brother. He was tall and lean, with rosy cheeks and a twinkle in his eye \u2013 one of those wiry sprightly ninety-somethings who go on for ever.\n\n'Charlie Watkins,' he introduced himself, lingering over Mrs Shapiro's chipped-varnish fingers which she extended graciously to him. 'I think we met at the 'ospital once. Did you know our Lily well?'\n\nOut of the corner of my eye, I saw Nathan's Tati watch him, bristling with annoyance.\n\n'Not well,' Mrs Shapiro replied, fluttering her eyelids. 'Only from smoking. And from slippers. She got the dead-woman slippers.'\n\n'Smokin' like a kipper!' he chuckled, nodding towards the flower-covered coffin in front of us. 'That sounds like our Lily.'\n\nI wasn't particularly surprised when Ms Baddiel turned up, too, just as the service was about to start.\n\n'It's always so-o sad when a client passes away,' she murmured, searching in her oversize bag for a packet of tissues.\n\nThere was music playing in the chapel, spooky-sounding organ music that made you feel as though you were already halfway into the next world. The coffin with its large wreath of lilies rested on an ornate catafalque to the left of the altar. A plaque on the wall solemnly reminded us _Mors janua vitae_. Death is the gateway to life. Where had I heard that before? Tall leaded windows filtered and chilled the sunshine leaking in from outside, turning it into a cool greenish fluid. It reminded me of the bivalves, clinging on under the sea.\n\nWe spread out around the pews, trying to make ourselves look like more than seven. Mrs Shapiro sat in the front row, and Nathan's Tati took up his position beside her. Nathan and Ms Baddiel sat in front on the other side. The niece and her father spread out in the middle, and I sat at the back. How sad, I was thinking, to have just seven people at your funeral, two of whom had never even met you. A thin man in a black suit droned through a short liturgy and disappeared. We all looked around, wondering whether this was all. Then suddenly there was a rustling behind us; the organ music stopped mid-note and gave way to a jolly lilting big-band number. Ba-doop-a-doop-a! Ba-doop-a-doop-a!\n\nYou could hear everybody gasp. Charlie Watkins rose to his feet and did a little hip-swing in the pew, then he squeezed out past his daughter and bopped up to the lectern. As the music faded away, he cleared his throat and began.\n\n'Ladies and gen'lemen, we're 'ere to celebrate the life of a great lady, and a great dancer, Lily Brown, my sister, who was born Lillian Ellen Watkins in 1916 in Bow. She was the youngest of three sisters and two brothers (he was reading from a sheet of paper he'd fished out of his jacket pocket, modulating his voice like an actor). Now I'm the only one what's left, and all that past life, the 'appiness and sorrow, the triumphs and disappoin'ments, is all washed away on the tides of time.' He fumbled in his pocket for a handkerchief. There was a general shuffling in the pews. This wasn't at all what we'd expected. He blew his nose and continued. 'Even when she was a young gel, our Lily danced like an angel.'\n\nThe Watkinses were a Music Hall family. Charlie described how Lily enrolled for dance classes at the City Lit, got pregnant, ran off to Southend, then came back to London a year later, without the baby and without the boyfriend. Her breakthrough came when she got a place in the chorus line at Daly's. He paused, snuffling into his hankie \u2013 it wasn't for effect, the emotion was genuine \u2013 then he leaned forward, departing from his script.\n\n'I seen 'er up there on the stage, kickin' like she could kick the bollocks off a giraffe.'\n\nIn the front pew, I could see Ms Baddiel quiver like a soft jelly, dabbing at her eyes with a tissue while Nathan slipped a solicitous arm around her shoulder. There was something about that gesture that sent a pang of longing through me \u2013 not longing for Nathan \u2013 that was in the past \u2013 but for the warmth of human comfort.\n\nLily settled in Golders Green, married and lost a soldier.\n\n'That's when she took to smokin',' said Charlie, 'puffin' away like she wanted to be up in 'eaven.'\n\nHe blew his nose again and raised his eyes. 'Ladies and gen'lemen, I ask you to pray for the soul of Lillian Brown. May she dance with the angels.'\n\nThe band music started up again, Ba-doop-a-doop-a! Ba-doop-a-doop-a! Then with a clatter of rollers, the coffin disappeared through the wooden doors. I thought of the old woman I'd known as the bonker lady, trying to keep her image in my mind's eye as the coffin rattled away, and despite the cheerful music, tears welled up in my eyes. What cruel tricks time plays on us! Before the cigarettes and the crusty toenails, before the deep-grooved wrinkles and the crumpled mind, there'd been another Mrs Brown \u2013 a young woman who danced in one of the most beautiful chorus lines in London, who lived life to the full, who could kick the bollocks off a giraffe.\n\nBa-doop-a-doop-a! Ba-doop-a-doop-a! Ms Baddiel and Nathan and Nathan's Tati were swaying in time to the music and fluttering the tissues which Ms Baddiel had handed out. The niece and Charlie Watkins were sobbing and bopping, and I found my feet, too, were pulled by the irresistible rhythm. Only Mrs Shapiro was standing stock-still \u2013 her back was to me, so I couldn't see the look on her face. Suddenly a current of air caught the folding door behind which the coffin had disappeared, making it swing forward, and, I swear I'm not making this up, as it gusted towards us a puff of grey smoke eddied out into the chapel, circling and wreathing around us before it drifted away.\n\nThe sunlight stung our eyes as we shuffled outside into the Garden of Rest and walked in a sad tight knot between the flower beds. Mrs Shapiro lit a cigarette and sat on a bench puffing away, as if in honour of her fractious former smoking companion. I wandered along looking at the names on the memorial plaques on the walls \u2013 there were so many. Some names I recognised \u2013 Enid Blyton, Peter Sellers, Anna Pavlova, Bernard Bresslaw (Mum's favourite actor), H. G. Wells (one of Dad's gurus), Marc Bolan (died so young!) and alongside them all the hundreds of anonymous dead, jostling together for a bit of memory space. Soon enough we'll all be anonymous except to the few people who knew us, I was thinking, until they in their turn become anonymous, too.\n\nThat's the thing about funerals \u2013 even if you hardly know the person who died, the closeness of death itself makes you melancholy. I recalled the people whose mysterious lives had brushed against mine \u2013 beautiful Lily Brown, before she became the bonker lady; Mustafa al-Ali, the chosen one, and his anonymous twin who died on the hillside; Artem Shapiro who had trekked across the Arctic; Naomi Shapiro of the blazing eyes; and the old lady I thought of as Naomi Shapiro, but who was really someone else. Were they exceptional people, or was it the time they lived through that made them seem exceptional? Had our safe post-war world stripped all the glamour and heroism out of life (sob) leaving us with the husks (sob) \u2013 consumer goods wrapped up in stylish packaging (sob, sob)? By now the tissue Ms Baddiel had given me was completely soggy. Blinded with tears, I stumbled on a step, stubbed my toe on a stone plinth and almost fell into the pond.\n\nCharlie Watkins was clutching his daughter's arm, his tall thin frame shaking with each breath. I wanted to ask him what had happened to the baby \u2013 had she aborted it or given it up for adoption? I wanted to know about Mr Brown \u2013 was he the one who brought her here to Golders Green? Had he loved her? Did he stay with her to the end? But Charlie'd crumpled the bit of paper back into his pocket, and his eyes were full of tears. He pointed up at the chimney, where a faint wisp of smoke curled into a perfect circle, wavered in the wind, and was gone.\n\n'There she flies! Our angel!'\n\nWheeee! A high-pitched whining sound carried on the air, like the distant whirr of angel wings. We all stopped and looked around. It was an eerie sound, as if her spirit was amongst us, trying to speak from another world.\n\nCharlie's daughter leaned over and whispered in his ear, 'Dad, you're whistling!'\n\n'Sorry. Sorry.'\n\nHe reached up and adjusted his hearing aid.\n\nThat gesture broke the tension. Everybody laughed, brushed their tears away, and started to move purposefully towards the car park. It's all very well thinking about the passing of time and the presence of death, but there's work to be done, dinners to be cooked, life to be lived. I put the soggy tissue away, and that's when my fingers touched something hard and long at the bottom of the jacket pocket. It was a key. I fished it out. Where had that come from? When was the last time I'd worn this jacket? Then I remembered. It was when I first met Mrs Goodney over at Canaan House.\n\nIt wasn't until we got to the car park that we realised Mrs Shapiro was missing. With a mutter of irritation, we split up to scour the gardens. Everybody was ready for home by now. A cool wind had sprung up, and all the emotion had made us hungry and tired. It was Nathan's Tati who found her. She'd strayed right out of the crematorium and across Hoop Lane into the Jewish cemetery. He'd come across her wandering among the graves and led her back solicitously, supporting her on his arm.\n\n'She keeps going on about some artist,' he whispered to me. 'Poor old thing.'\n\n# 47\n\n# The penthouse party\n\nIt was Mrs Shapiro's idea to hold a house-warming party for the penthouse suite. We drew up the guest list together one morning over a cup of coffee in the kitchen. The sun had come out, and a mild blossom-scented breeze wafted in through the open back door. Mrs Shapiro was in an effervescent mood. Her hair was pinned up and she was wearing a crumpled not-very-white cotton blouse with her smart brown slacks and the _Lion King_ slippers. She saw me looking at them, and gave a little shrug.\n\n'They are quite ugly, isn't it? But Wonder Boy adores them.'\n\n'Mm,' I said.\n\n'We can invite the charming old man from the crematorium. He is good at singing. Pity he is so old. And his petit son.'\n\n'Good idea. Who else?'\n\nIt seemed incredible that Mr Ali and the Uselesses had managed to install a functioning shower and toilet and three Velux windows in the attic rooms, without further mishap \u2013 but it was true. They'd moved their stuff up there, and all the junk \u2013 what was left of it \u2013 was piled up in a side room whose ceiling was too low to make a useful living space.\n\n'It will be a musical soir\u00e9e. Or maybe it will be a garden party. What you think?'\n\n'I think we should be flexible. You can never tell what the weather's going to do.'\n\n'You are very wise, Georgine,' she nodded, as though I'd offered some great insight into the human condition.\n\nUpstairs we could hear thudding and hammering as Chaim and the Uselesses put the finishing touches to the floorboards. They'd hired a sander for the day without realising the amount of preparation that was needed. Mr Ali had gone off on some mysterious errand to B&Q. I noted how tidy everything was in the kitchen, a stack of washing-up still covered in soapsuds draining at the side of the sink.\n\n'Maybe when they've finished the penthouse suite we can discuss some kitchen improvements with Chaim and Mr Ali.'\n\n'What for I need improvement?'\n\n'Remember what you said \u2013 dishwasher, microwave?'\n\nShe looked at me in astonishment. Obviously her previous plans had vanished from her head, and something else was preoccupying her.\n\n'Now, Georgine, this party will be a good opportunity for you to find a new husband.'\n\n'Oh, really?' You had to give her credit for persistence.\n\n'We will invite my Nicky and the other one also, the hendsome one. Maybe more hendsome even than Nicky, isn't it?'\n\n'Yes, very handsome, but...'\n\nI hadn't yet told her that I was not looking for a new husband, I just wanted to recondition the one I'd already got.\n\n'You must make more effort, Georgine, if you want to catch a man. You are a nice-looking woman, but you heff let yourself go. You must wear something nice. I heff a nice dress, red spotted mit white collar. Will look nice on you. And lipstick. You must wear a nice lipstick in metching colour. I heff one you can borrow. Will go good mit this dress.'\n\nI smiled non-committally, remembering the grotty decomposing make-up in her bedroom drawer.\n\nAfter a while the banging from upstairs stopped and Chaim put his head round the door. He was wearing jeans and a T-shirt, and there were bits of sawdust in his hair and eyebrows.\n\n'What shall we do with all the junk, Ella? The belongings from the previous inhabitants?'\n\n'The ones that ran away?' I teased.\n\n'This is no laughable matter. All over Europe Jews are coming and demanding their property back.'\n\n'Like the Palestinians with their keys?' I smiled smugly. He looked cross.\n\n'You \u2013 you are not a Jew, Miss Georgiana. You cannot understand what it means.'\n\n'It's a Yorkshire thing \u2013 calling a spade a spade.'\n\n'A spade is like a spade?'\n\n'But they were not Jews living here, Chaim,' Mrs Shapiro intervened soothingly. 'Why you are always mekking problems? Leave the junk where it is. Sit down and drink a coffee mit us.'\n\nChaim pulled out a chair rather nervously. Wonder Boy had sidled in with his ears pricked back and was lurking under the table, his tail quivering.\n\n'Raus, Wonder Boy! Go and make your little wish elsewhere!' Mrs Shapiro shooed him away.\n\nSuddenly a horrible juddering whine shook the whole house. It was the sander springing into life. Wonder Boy set off a competing yowl of protest. Chaim Shapiro jumped to his feet.\n\n'You know, I better give those boys my hand. They are completely useless.' He grinned at me. 'A spade is like a spade.'\n\nWhen we were alone again, Mrs Shapiro leaned across and whispered to me, 'He is crezzy, isn't it? He was hit in the eye mit a piece of glass, you know. Some boys were throwing stones at the bus. I think also a piece went in his brain.'\n\nAfter we'd drawn up the party list, we divided the duties. Mrs Shapiro said she would ring Wolfe & Diabello, and reluctantly agreed to invite Ms Baddiel, too. I was delegated to call Nathan and his Tati. I picked up the phone as soon as I got home.\n\n'Your father's made a conquest, Nathan.'\n\n'Wonderful! I knew from the moment you met that you were made for each other. You'll be a fine stepmother, Georgia.'\n\nThe thought tickled me. 'Yeah, I'll lock you up and feed you on poisoned apples. Don't you want to know who it really is?'\n\n'I think I can guess. It's your old lady, Mrs Shapiro?'\n\n'Has he said anything?'\n\n'He says it's a pity she's so old.'\n\n'That's what _she_ says about _him_. Anyway, you're both invited to a party.'\n\nI told him the day \u2013 it was a Saturday, about four o'clock.\n\n'Put it in your diary. It might be a musical soir\u00e9e or a garden party.'\n\n'So hard to know what to wear,' he murmured cheesily.\n\n'If it's any help, I shall be wearing a red dress with white polka dots and a white collar.'\n\nI would have rung off then, but my conversation with Chaim was still on my mind and I suddenly remembered the glue exhibition.\n\n'Nathan, you know what you said about being called a self-hating Jew?'\n\n'Did I say that?'\n\n'You did. I thought it was because you were gay. Or sm...' I stopped myself. '... or something.'\n\n'Look, Georgia, some people get excited about what sets them apart. I get excited about what bonds people together. That's all.'\n\n'But... isn't it something about not believing in a Jewish homeland?'\n\n'That place you come from \u2013 Kippers \u2013 is that your homeland?'\n\nThere was an edge of irritation in his voice.\n\n'Kippax, not kippers. It should really be called Oven Chippax.' Even as I said it, I felt a pang of shame at my disloyalty. 'It's just that people go on about their homeland as if it was the biggest thing in their lives. It seems strange to me...' I could hear Nathan's prickly silence on the other end of the phone. But when he spoke, his voice was sad, not prickly.\n\n'That was Tati's generation. Zion was their big dream. It was a good dream, too. But they found you can't build dreams with guns. Just nightmares. Does that answer your question?'\n\nI paused. It did and it didn't.\n\n'I expected the Jews would be... you know, after all that suffering... more compassionate.'\n\n'Why would suffering make anyone compassionate? It doesn't work like that, Georgia. Abused children often grow up to become abusers themselves. It's what they learn.'\n\n'Mmm. But...'\n\n'And if you've convinced yourself that you're really the victim, or even just potentially the victim \u2013 well, it gives you a free rein, doesn't it? You can kill as many people as you like.'\n\nBut we didn't bully Carole Benthorpe because we'd been abused, I wanted to say. We did it because we truly believed that something \u2013 something higher than us \u2013 gave us the right.\n\n'It's like glueing a joint, Georgia. That article you edited. Surface attraction is increased by roughing up the surfaces to be bonded. Like an abusive relationship. It's the mutual damage that holds the two sides together.'\n\nI'd never heard such passion in his voice. Gay. What a shame!\n\nOur conversation stayed in my head after I'd put the phone down. But we didn't have a wall in Kippax, I was thinking. When the strike ended the community was split, and the bitterness of betrayal and defeat was on everyone's mind. People badmouthed their neighbours. Taunts and bricks were thrown, cars were scratched, drunks and kids picked fights. But still life went on. You had to go to the same schools, shop in the same shops, dig in the same allotments, sit eyeball to eyeball in the doctor's surgery \u2013 and after a while the habit of living together slowly turned into peace. Eventually a generation comes along that doesn't remember what the conflict was ever about. Maybe forgiveness isn't such a big deal, after all. Maybe it's just a matter of habit.\n\nLater that day, as I settled down with a cup of tea and a Danish pastry, I remembered the other thing I'd meant to ask Nathan: Danish. _She_ was Danish. I knew nothing at all about Denmark except the pastries. And Hamlet, of course. Why had she left Denmark? What had happened there during the war? I'd have to ask him next time.\n\nIn the end, the party was neither a musical soir\u00e9e nor a garden party \u2013 it was a barbecue. That was Ishmail and Nabeel's idea, and they got so excited about it that no one had the heart to argue with them, though personally I thought the combination of scorched half-raw meat, bugs from Mrs Shapiro's kitchen and barbecue lighter fuel was potentially lethal. Anyway, they built an improvised barbecue in front of the house out of spare bricks and some metal racks they got out of an old oven which Mrs Shapiro had spotted on a skip. They got a job lot of cheap Halal lamb chops and chicken wings from a butcher on Dalston Lane and Mrs Shapiro produced some discoloured burgers of unknown provenance from the depths of her fridge. I made a mental note to avoid those.\n\nI'd suggested that Mr Ali invite his wife, but apparently when she heard that Ishmail and Nabeel were involved she declined.\n\n'Give her headache,' Mr Ali explained.\n\nHowever, she sent along a huge bowl of hummus laced with olive oil and sprinkled with fresh coriander leaves.\n\n'What is this thing?' Mrs Shapiro poked her finger in and licked it, wrinkling her nose, then I saw a smile of pleasure spread across her face.\n\nHave you ever noticed the similarity between BBQ and B&Q? My theory is that that's why men feel the urge to take over the cooking on these occasions. It's what Rip would call synergy. At one point all four of them \u2013 Chaim, Mr Ali, Ishmail and Nabeel \u2013 were crowded round the smoking barbecue, puffing and flapping to try and get it lit. Ishmail and Nabeel took turns splashing squirts of lighter fuel on to the smouldering charcoal, then jumping back howling with laughter as the flames flared up. In fact they managed to splash a fair bit of lighter fuel on themselves, too. Probably Mrs Ali was wise to stay away. I watched them from the window of Mrs Shapiro's bedroom, where I was trying on the red-and-white spotted dress, while Mrs Shapiro fussed around for the right shade of lipstick.\n\nWe were blessed with the weather. The sun had come out after lunch, and stayed out all afternoon. The thrush was up in his tree, his chest puffed out, singing his war song, and all seven of Mrs Shapiro's cats, plus a few feline guests from the neighbourhood, were circling, attracted by the smell of the cooking meat. Mrs Shapiro and I chopped up salads and split pitta breads and set out plates and glasses on the white uPVC table. A spare table from the study and some dining chairs had been carried out on to the grass, too.\n\nNathan and his Tati were the first to arrive. Nathan had brought two bottles of Blind River Pinot Noir, and his Tati had brought a bunch of blue irises for Mrs Shapiro.\n\n'Thenk you so much!' Her bright blue eyelids fluttered ecstatically. That was a good start. 'Will you heff a drink?'\n\nShe was wearing the same brown slacks and striped jersey in which she'd first entertained Mr Wolfe, with her high-heeled slingbacks that kept sticking into the grass as she tottered about. Her hair was freshly dyed and elaborately pinned up with three tortoiseshell combs. In fact she looked quite elegant. I was wearing the little red-and-white number. Nathan looked me up and down.\n\n'Nice dress.'\n\n'Thanks. I like your trousers. We match.'\n\nHe was wearing red trousers with what looked like a white waiter's jacket.\n\nMs Baddiel, when she arrived, was wearing a flowing muslin garment which might have been a coat or a dress or a skirt and top \u2013 it was impossible to say how it all fitted together \u2013 tie-dyed in swirling shades of amber, bronze and gold. It fluttered lightly in the breeze, making her look delicate and ethereal, despite her size. I saw Mark Diabello eyeing her with interest as he came up the path, and felt a small stab of annoyance. Okay, so I'd given him the push, but he was supposed to be eyeing _me_ , not her. He was wearing the same dark suit as always, the white handkerchief winking invitingly in his jacket pocket. The Shameless Woman poked her head up briefly, and thought an utterly shameless thought: I bet they don't do the red open-gusset panties in _her_ size.\n\n'Nice dress, Georgina. Suits you.' He pecked me on the cheek and handed me a packet of Marks & Spencer's sausages and a bottle of champagne.\n\n'Oh, lovely. Mrs Shapiro'll like those.'\n\n'Is your hubby coming?'\n\n'Yes, later,' I lied. Actually I hadn't invited him. It wasn't because of Mark. It was because he'd have come out of a sense of duty and then complained that he was missing the football. Besides \u2013 I don't know \u2013 I just wanted to keep Canaan House and its eccentric inhabitants to myself.\n\n'Nick's coming later, too. He had some... er... work to catch up on.'\n\n'Mark, there's something I think you should know. Something you and Nick should know. Only... I don't know whether I should tell you.'\n\nHe raised a quizzical eyebrow.\n\n'You're being very mysterious, Georgina.'\n\nIf I hadn't already had a couple of glasses of wine I might have kept quiet, but I blurted out, 'The deeds to the house... there aren't any. Her husband just moved in. It was abandoned. After a bombing raid. Actually, I don't think he was even her husband.'\n\nA strange look came over him. His eyes flickered through many changes of colour, and the smile-creases in his cheeks twitched furiously. He looked as though he was about to explode. Then I realised he was trying to stop himself from laughing.\n\n'No title! Wait till I tell Nick!'\n\n'But can't she... I don't know... what about squatter's rights?'\n\nHe burst into a chuckle. 'No title! Ha ha! No, maybe on second thoughts I won't tell him! Where's the old lady?'\n\nMrs Shapiro and Tati had disappeared into the house. They'd opened the window in the study and moved the old gramophone up to it, so we could hear the music in the garden. Now they were poring through Mrs Shapiro's collection, trying to decide what to play. You could see them through the window, talking and laughing together. They chose an orchestral piece that sounded vaguely familiar. It might have been one of Rip's old vinyls. What will she say, I wondered with a pang of conscience, when I ask to have them back?\n\n'Penny sends her apologies.' Nathan sidled up to me. 'Her cousin Darryl's getting married.'\n\n'That's nice.' I felt a small prick of regret.\n\n'Who's the guy in the brown suit?'\n\nOver by the barbecue, Mr Ali and Chaim Shapiro were cooking and arguing. Seeing them together like that, I was struck by how alike they were. Chaim was stabbing at the chicken wings to see whether they were done. Mr Ali shoved a mouthful of lamb chop into his mouth and beamed as he caught my eye, patting his tummy.\n\n'XXL.'\n\n'Trouble with you Arabs,' Chaim was saying, 'is you always pick bad leaders.'\n\n'You Jews put all the good ones in prison.'\n\nMr Ali speared another lamb chop on a skewer and brandished it in the air. The chicken wings were beginning to smoke. Chaim flipped them over.\n\n'We put only terrorists in prison.'\n\n'You not heard of Nelson Mandela? You want peace you free Marwan Barghouti,' said Mr Ali, emphasising his point with the skewered lamb chop.\n\n'This Barghouti \u2013 is he Hamas or Fatah?' Chaim picked up a smoking chicken wing with his fingers \u2013 ouch! hot! \u2013 and bit into it with a crunch, sucking cool air into his mouth.\n\n'Hamas, Fatah \u2013 all listen to Barghouti!' The lamb chop flew off Mr Ali's skewer and whizzed over our heads. It landed on the ground and he speared it up, covered with bits of grass, and started flashing it about again. 'He only can bring peace.'\n\n'Mr Ali, Chaim, this is my colleague Nathan Stein,' I butted in. They stopped in mid-sentence, and turned towards us.\n\n'Come! Eat something!' Chaim waved a chicken wing at him.\n\n'We are discussing politics,' said Mr Ali. When he looked round at us, I could see he had a grin on his face and bits of barbecue sauce in his beard. They both looked as though they were enjoying themselves. On the barbecue, things were sizzling away.\n\n'Discussion is the better part of valour!' added Chaim.\n\nIn the middle of the grass, Mussorgsky and the Stinker were fighting over a chicken bone. The neighbourhood guest cats, the ones with proper homes to go to, were looking on askance at this display of bad behaviour.\n\n'Very tasty.' Nathan took a bite of the chicken wing.\n\n'Better than a poke in the eye with a sharp piece of glass, eh?' said Chaim, and bellowed with laughter at his own joke.\n\nDenmark, I reminded myself. Mustn't forget to ask him about Denmark.\n\nIshmail had rigged up an ingenious rotating spit, but the chops and wings were too bony to spear, and the sausages just split. Brainy but useless. It's often the way. Now he and Nabeel were racing about with plates of charred meat as it came off the barbecue, flashing their smiles as they offered it around. They were in a skittish mood, and kept on barging into each other and dropping bits on the floor. Wonder Boy was in the bushes attempting to rape one of the visiting guest cats (little did he know that it was to be his last fling!), Mrs Shapiro was sitting on one of the white chairs with her feet up on another, smoking a cigarette and discreetly feeding the uncooked M&S sausages to the cats, who were snatching and snarling. Tati was sitting at the table beside her, slugging back red wine and eating a burger \u2013 it must be one of the ones from the back of her fridge \u2013 I hoped he had a strong constitution. Mark Diabello was topping up the glasses. Ms Baddiel was keeping everyone supplied with tissues.\n\n'I work mainly with old people,' I overheard her explaining peachily to Mark Diabello. 'Sorting out their housing needs to enable them to live independently.'\n\n'Fascinating,' he murmured. 'I'm in housing myself.'\n\nThe music poured out into the garden, wheeling and soaring above it all.\n\nEverything that happened next happened very quickly, so I may have got the order of events slightly wrong, but it was something like this. The thrush started it. From his perch in the ash tree he'd spotted a piece of pitta bread that had fallen on to the ground. Wonder Boy, having satisfied his lust, was lurking in the bushes watching the bird. As the thrush swooped down Wonder Boy flattened his nose to the ground and wriggled into pouncing position, his tail twitching. The bird went for the bread. The cat went for the bird. I grabbed the first thing that came to hand \u2013 it was a lamb chop \u2013 and lobbed it at Wonder Boy. It arced through the air spinning like a boomerang. Normally I'm hopeless at throwing, but this time I scored a direct hit. Wonder Boy let out a yowl and leapt sideways right under the feet of Nabeel who was carrying a plate of chicken wings up the garden. Nabeel barged into Ishmail, who lurched and stumbled against the barbecue, which collapsed scattering hot coals everywhere, setting fire to the barbecue lighter fuel that hadn't been screwed up properly and had spilled on the ground right under the open study window, where a curtain was flapping in the breeze. The cats fell on the scattered chicken wings in a frenzy. Wonder Boy grabbed the biggest one and raced off down the path. The wind gusted; the curtain caught and blazed. Mark Diabello sprayed champagne over the flames, but it was too little too late. Outside on the lane there was a screech of brakes and a thud. The flames leapt through the window. Nick Wolfe appeared at the gate, holding up Wonder Boy's limp lifeless body gingerly by one leg. Mrs Shapiro jumped up, screamed and fainted. Tati tried to give Mrs Shapiro mouth-to-mouth resuscitation. The fire spread from the curtains to some loose papers on the bookshelf under the window. The music slurred and stopped. Mr Ali phoned the fire brigade on his mobile phone but failed to make himself understood. The fire roared through the study and into the hall. Nathan phoned the fire brigade on his mobile phone, and did manage to get through. I just stood there watching, clenching my hands into fists, wishing I could recall the flying lamb chop, and feeling terribly terribly terribly guilty.\n\nHours later, after the fire brigade had been and gone, and Mrs Shapiro had been carted off into temporary accommodation accompanied by Ms Baddiel, and Ishmail and Nabeel had gone home to Mrs Ali, and Chaim had gone home with Nathan and his Tati, and Wolfe and Diabello had finished off the booze and slunk off back to their lair, I walked home through the balmy dusk. _In \u2013 two \u2013 three \u2013 four. Out \u2013 two \u2013 three \u2013 four._ I breathed deeply, noticing that the air, despite its taint of London traffic, carried the sweetness of rising sap and fresh growth. I noticed peonies in the front gardens, and the greenness of leaves newly uncurled. I noticed that my hands were clenched into fists, and that my palms carried deep imprints of my fingernails. I uncurled them and let them relax. They hung like new leaves. When I got to my front door I noticed that Violetta was there beside me.\n\nBen and Rip were home already. They'd been out to the football, and now they were drinking a beer and watching the television \u2013 a round-up of the week's news.\n\n'Good party?' asked Rip, without looking up.\n\n'Great.' I came and slumped on the sofa. Violetta jumped up on to my lap, purring.\n\n'Look at this,' said Rip, pointing at the screen. 'Who would have believed it?' Two men were being interviewed, grinning in front of a bank of cameras and microphones. One of them looked a bit like the Reverend Ian Paisley. I had no idea who the other one was. 'Those two old bastards!'\n\n'Who are they?'\n\n'Ian Paisley and Martin McGuinness,' said Ben, who'd been watching the item from the beginning. 'They've done a deal.'\n\n'Really? You mean, in Northern Ireland?'\n\nI tried to think back to a time when that conflict hadn't been in the news. How had this peace thing happened? How come I hadn't noticed? I remembered something about a woman whose hair fell out. She'd died while Rip and I were still living in Leeds, hadn't she?\n\n'Who'd have thought it was possible? Peace has broken out!' Rip turned to face me. He was smiling, then the smile broadened into a lop-sided grin. 'What the fuck are you wearing, Georgie?'\n\n'Oh, I thought I'd dress up for the party.'\n\nMy jeans and jumper and Bat Woman coat had been swallowed up by the blaze \u2013 or even if they were still there, the firemen had barred access to them.\n\n'You... You've changed, Georgie. You're different.'\n\nHe was still staring at me, as though he hadn't seen me before.\n\n'Less...?'\n\n'More...'\n\n'I've been experimenting...' I hesitated. How could I explain that in the last six months I'd been Georgine, Georgina, Georgette, Mrs George and Miss Georgiana? Not to mention Ms Firestorm and the Shameless Woman. '... with different ways of being myself.'\n\n'It suits you, Mum,' said Ben. 'Sort of retro.'\n\nLater that night, after the football highlights on TV had ended, and the thud-thud of Ben's music was quiet, and Violetta had wolfed down a tin of tuna and curled up on the sofa, I lay in bed, reflecting on what had happened at Canaan House that day, and tuning in to the silence around me. And that's when I heard a faint crackling sound \u2013 so faint that if I consciously tried to listen, it disappeared \u2013 it was the crackle of brainwaves coming from the mezzanine study. I put on my slippers and dressing gown and went to investigate. There was a sliver of light under the door. I tapped softly.\n\n'Come in.'\n\nRip was sitting at the computer in his boxer shorts, a cold cup of coffee at his elbow, staring at the screen.\n\n'You're working late.'\n\n'Got a report to finish,' he said, without looking round.\n\n'Progress Project?'\n\n'No. I'm done with the Progress Project.'\n\nI glanced over his shoulder and I could see quite clearly on the monitor that he was working not on a report but on his CV. He didn't even try to close or minimise the window.\n\n'Is it... are you okay, Rip?'\n\n'What do you think?'\n\nI slipped an arm around his shoulder \u2013 it was a habit of physical affection that bypassed the picky brain and the unreliable emotions. How warm his skin was, how big his shoulders; yet there was something about the way he was leaning forward in his chair, sagging almost, that struck a sudden chime of pity in me. I stroked his hair.\n\n'You're tired. You should go to bed.'\n\n'I need to get this done. It has to be in tomorrow.'\n\n'What's it for?'\n\n'Something called the Synergy Foundation.'\n\nI can't explain why, but my heart sank. Synergy Foundation. What the hell was that? It sounded like something you put on your face. _In \u2013 two \u2013 three \u2013 four_...\n\n'That sounds interesting. Shall I make you another coffee?'\n\n'That'd be nice.'\n\nI went down to the kitchen and made two cups of coffee. Then I remembered the Space Invaders Easter egg lurking at the back of the cupboard.\n\n'D'you fancy a bit of chocolate?'\n\nI smashed up the egg inside its foil wrapper, and we polished off the sickly chocolate between us. An hour later, when he crawled into the low canvas camp bed, I crept in beside him. \n\n# 48\n\n# A lot of bargains\n\nCanaan House is now a building site. The destruction from the fire wasn't extensive, but after the fire brigade had gone the surveyors checking the damage found an unexploded bomb left over from the war, buried deep in the roots of the monkey puzzle tree. The whole street had to be evacuated while the bomb squad carried out a controlled explosion. We all stood behind red-and-white barrier tape and watched. It was a bright windy day and dust blew everywhere \u2013 that's all that was left in the end, dust. Mrs Shapiro was weeping quietly, and when I put my arm around her to comfort her I suddenly started sobbing, too. In fact I think I cried more than she did.\n\n'You know, dear Georgine, you were right,' she said, patting her eyes with the disgusting hankie from the pocket of the astrakhan coat. 'This house was too big for me. Too many problems. Too many memories. Like caught in a trap. Now is the time for moving on.'\n\nFortunately, Mark Diabello had managed to get the title registered in Mrs Shapiro's name, using the evidence of her sixty years as a ratepayer to justify her claim, so she was able to sell the site to a developer for a substantial sum. Only Mark knows how substantial, and he is sworn to secrecy.\n\nShe has bought herself a lovely apartment in a sheltered housing development in Golders Green \u2013 sadly no pets allowed \u2013 and she has set up Chaim and Mussorgsky in a flat in Islington. Violetta has stayed with me. We keep each other company, and in a quiet moment when everybody is out we sit on the sofa together and share our smelly memories. I sometimes ask myself whether she misses Wonder Boy, but somehow I don't think she does. The remainder of the money from Canaan House went, with the remaining feline residents, to the Cats Protection League. Mrs Shapiro won't tell anyone how much, but I'm sure it was more than enough to keep any number of lean and hungry moggies in pet food for the rest of their stinky little lives.\n\nI never got a chance at the party to ask Chaim about Denmark, but we meet up one Saturday in September at a caf\u00e9 on Islington Green, near where he has his flat. It's raining again \u2013 it seems to have been raining most of the year \u2013 but he's found a cosy corner near the window and is flicking through a travel brochure. I almost don't recognise him at first, he looks so completely different to the man in the brown suit. He's wearing black jeans and a blue open-necked shirt, and stylish rimless glasses. Unless you knew, you wouldn't even notice that he has a glass eye. I shake out my umbrella and we hug, my wet cheek against his bristly one, and order our coffees. He tells me about his new job at a travel agent's specialising in Holy Land tours, but I'm not in the mood for small talk. I want to put the last piece of my Canaan House puzzle into place.\n\nI pull the photo out of my bag \u2013 the one of the young woman standing in the stone archway \u2013 and push it across the table to him.\n\n'This is for you, Chaim. Tell me about her.'\n\nHe picks up the photo and studies it, and that sweet dimply baby smile creeps over his face.\n\n'Yes, this is her. Smiling like a Monalissa.'\n\n'You said she came from Denmark.'\n\n'Do you know the story of the Danish Jews? It was like no other Jews in Europe.'\n\n'Tell me.'\n\nHe has taken off his glasses and is sitting back in his chair. The photo is still in his hand, but it's as though he's gazing right through it into another time and place.\n\n'Naomi Lowentahl was her name. She was born in 1911 in Copenhagen. Wonderful wonderful Copenhagen! You been there?'\n\n'No.' I shake my head.\n\n'Nor me. Maybe I will make a trip next year. They lived in the Jewish quarter. My grandparents are buried there, in the Jewish cemetery. She was the youngest of three children. Their mother died when she was ten. Like me.' A cloud drifts over his face. 'But she still had her father and two older brothers. She was spoiled like a rotter, I think, having all those men running around after her.'\n\nNaomi's father, Chaim's grandfather, was a mathematician at the university, and Naomi herself taught maths in a high school. Her brothers were active in the Zionist movement in the 1930s, which had taken root in the Jewish communities of Europe, seeded by anti-Semitism and persecution.\n\n'And Naomi?'\n\n'Naomi was sometimes with her brothers, sometimes with her father. My grandfather was one of those who believed that Jews could be assimilated as equal citizens into the countries where they lived. He believed the storm clouds gathering over Europe would simply be blown away by winds of progress and enlightenment.' He pauses, fiddling with his glasses. 'But alas, eventualities were to prove him tragically wrong.'\n\nWhen the Germans invaded Denmark in 1940, those arguments took on a new urgency. The Danish Government had come to an agreement with the occupiers \u2013 Danish butter and bacon in exchange for self-government. Nor would they hand over their Jews. 'Jews and Christians, we are all Danes,' they said.\n\nDespite the agreement, there was little active support for the Nazis, and by 1943 the agreement with Germany started to break down. The Nazis began to make secret preparations to round up and exterminate all 7,000 of the Jews in Denmark. Chaim smiled. 'They thought their \"final solution\" would be not so final when these insolent Danish Jews are strutting around Scots free.'\n\nIn fact it was a German attach\u00e9 who foiled the Nazi deportation plan by leaking the details to a Danish politician. What was to have been a swift and secret operation was thwarted when the Danish people simply said no. Not here in Denmark. Not to _our_ Jews.\n\nSpontaneously, haphazardly, as word got around, friends, neighbours and colleagues offered help, money, transport, and places to hide. They would have no truck with the horrors that defiled the rest of Europe. It was Naomi's head of department, a Lutheran, who called at her apartment late on the night of 29th September and warned her that there were two passenger boats moored at the docks with orders to take 5,000 Danish Jews away \u2013 they were due to sail on 1st October. He advised her to go to the Bispebjerg Hospital, where a shelter had been set up.\n\nShe and her elderly father stuffed what they could into a suitcase and made their way to the hospital. They found a quiet corner in the psychiatric unit and watched with apprehension as more and more of the city's Jewish population arrived, alone or in groups, bewildered, anxious, carrying their most precious possessions in cardboard suitcases. In the end some 2,000 people were crammed into the psychiatric wards, the nurses' accommodation, and anywhere else they could be fitted in. Secrecy was impossible, and there was no need for it \u2013 everyone in the hospital from the director to the porters was involved. The staff looked after them and fed them from the hospital kitchens, and as word spread, gifts of food and money poured in from local people. Day and night, ambulances drove them to secret hiding places on the coast.\n\nOther Jews, including her brothers, were hidden in churches, schools, libraries, and many in private houses by their neighbours. In holiday villages all along the northern coast, support groups sprang up to shelter the fleeing Jews while they waited for a boat and the right weather conditions to cross to neutral Sweden. Even the coastguards were in on it.\n\nSquashed up in the stinking hull of a fishing boat with twelve others, Naomi and her father made the short crossing to Sweden on 3rd October 1943. They were stopped by a German patrol, but the fisherman sucked obtusely on his pipe and offered them a pair of herrings, while under the hatch beneath his feet the passengers held their breath. The fisherman thought it was a great adventure and posed with his beaming human cargo in the Swedish port before setting off home to pick up another load.\n\n'I have the picture,' says Chaim. 'I will show you.'\n\nSweden was teeming with refugees, and seething with talk of resistance, of freedom, of an international union of Jews, of safety, of Zion. In the refugee centre in Gothenburg she was reunited with her brothers. Although they were Zionists, they had struck up a friendship with a young socialist Bundist from Byelorussia. His name was Artem Shapiro.\n\n'Did they fall in love on sight?'\n\nChaim grins. He has a little frothy moustache on his upper lip from the cappuccino.\n\n'I haven't a foggy. Remember, I was not born yet.'\n\nOf more than 7,000 Jews in Denmark, the Nazis got fewer than 500, and even most of those survived in Theresienstadt, for the Danish authorities sent medicines and food over for them. Those Jews who returned to Denmark at the end of the war found their homes intact and looked after, their gardens watered, even their dogs and cats sleek and well fed.\n\nI don't know why it's the thought of those plump Copenhagen moggies that finally makes me choke up and reach for a tissue as Chaim gets to the end of his story. I've seen the pictures of the stick-like walking dead of Belsen, the heaps of corpses, the terrible piles of children's shoes. I know all that happened, but I want to believe that something else is possible.\n\n'Thank you, Chaim. You told me what I wanted to know.'\n\nIt's still drizzling as I make my way across Islington Green towards Sainsbury's. I've arranged for Rip to pick me up in the car park at one o'clock, so I have a couple of hours to do my shopping. He's gone off to a meeting with the team at the Finsbury Park Law Centre, where he's starting his new job next week. The Synergy Foundation turned him down.\n\nIt seems that if you hang around for long enough in Sainsbury's on a Saturday morning, the whole world passes by \u2013 or maybe my imagination has filled in the gaps. I see the same _Big Issue_ seller, lurking under the canopy by the entrance, and there goes the Robin Reliant man crossing the road with his stick. The Boycott Israeli Goods girl is there, brandishing her clipboard, though her hair has grown a bit and she's now collecting signatures on a petition to save the whales. Ben is there with her \u2013 he often comes down here on a Saturday morning \u2013 and his hair is longer, too, twisted into incipient dreadlocks and tied behind his neck with the red pirate-style scarf.\n\n'Hey, Mum!'\n\nI stop to sign their petition, though I've already signed it several times. The girl looks a bit sheepish, maybe thinking I despise her defection, but I just smile, because I understand now that everything \u2013 whales and dolphins, Palestinians and Jews, stray cats, rainforests, mansions and mining villages \u2013 they're all interconnected, held together by some mysterious force \u2013 call it glue, if you like.\n\nWhile I'm picking up some beer for Rip, I spot Mark Diabello and Cindy Baddiel lingering hand in hand in the wine department. He's wearing a check shirt and beige trousers, and I notice that a small bulge is developing above the waistband, and his hair has grey streaks at the temples, but as he turns towards me I feel that pleasant pelvic glow \u2013 yes, he's still the hero of _The Splattered Heart_.\n\n'Hello, Georgina!' He greets me with a kiss on each cheek, and Ms Baddiel hugs me in her roly-poly arms. She looks exactly the same. I check discreetly for signs of Velcro burns on her wrists \u2013 shame on you, Georgie! \u2013 but they are plump and wholesome.\n\n'Thanks for everything you did with getting the house registered,' I say to Mark. 'How's it going?'\n\nA couple of months ago, Wolfe & Diabello mysteriously disappeared from the high street to be replaced by Wolfe & Lee. Mark tells me he is now running a housing association for ex-offenders.\n\n'It's \u2013 how can I put it? \u2013 more satisfying.'\n\nThe mineral edge in his voice makes me shiver.\n\n'I'm glad it all worked out.'\n\n'Take care,' they say.\n\nHere's someone I don't want to see. It's Mrs Goodney pushing her trolley towards me. I'd duck out of her way and avoid her if I could, but the aisle is narrow, and there's nowhere else to go, so I just stand still and smile.\n\n'Hello,' she says. 'I didn't expect to see you here.'\n\n'No. Nor me.' I'm still trying to decide whether to be friendly. 'How are things up at the hospital?'\n\n'Oh, I gave all that up. Too much hassle. No one thanks you for anything.' She sighs. 'It was for her own good, you know. Do-gooders like you, you have this romantic idea that old people want to stay in their crumbly grotty houses until they die. But they don't. They want somewhere small that's easy to keep warm and clean, with all mod cons. Making the move is always a wrench. They may need a bit of help. But once they've done it, they never want to go back. Anyway, I'm running a little nail bar now, up Stoke Newington Church Street.' She glances down at my hands. 'Drop in one day.'\n\nAt the deli counter I bump into Nathan and Raoul, gravely discussing the comparative merits of olive and avocado oils. Nathan has his arm round Raoul's shoulder in that casual gesture with which he once comforted me, though Raoul is several inches taller than he is, and only half as handsome. They greet me with warm hugs, and bring me news of Mr Ali, who has just installed a new jacuzzi at their flat in Hoxton. Ishmail is still living with the al-Alis out Tottenham way, and is due to start his engineering course later in the month, but Nabeel has gone back to Palestine. His older brother was killed during an Israeli air strike on Gaza only a week after our barbecue \u2013 a bystander casualty \u2013 and now Nabeel is the head of the family. Gentle animal-loving coffee-making Arsenal-supporting Attendent Nabeel \u2013 my heart aches \u2013 it's hard to imagine him as head of anything.\n\n'Come and have dinner with us one day,' says Nathan.\n\n'I'd love to. Will you make French-style egg custard with vanilla?'\n\n'We need to get some vanilla,' says Raoul seriously. 'We used it all up on that bavarois, remember?'\n\n'Look out for Tati and Ella,' says Nathan. 'They're around here somewhere.'\n\nSure enough, there they are, pushing the high-sprung pram down one of the aisles, leaning together like a pair of newlyweds. I watch her lift her face up as he bends to give her a whiskery kiss and whisper something in her ear. She laughs, and rests her head against him. The way they're gazing into the pram, you'd think there was a baby in there, but when I peep inside, all I see is a lot of bargains. \n\n# Acknowledgements\n\nMany people have contributed to the making of this book. Thanks to Ewen Kellar for first getting me interested in adhesives, to Cathy Dean for DIY discussions, to Mikey Rosato for giving me the low-down on estate agents, and thanks to all those who have shared tales of bad lovers, bad pets, bad plumbing and other bad things \u2013 you know who you are, and I won't embarrass you by naming and shaming. Thanks to those who have helped me to appreciate the many-layered complexity of the Middle East \u2013 Raja Shehadeh, Donald Sassoon, Saleh Abdel Jawad, Naomi Ogus, Eitan Bronstein, Graham Birkin \u2013 I have learned such a lot from talking with you. Thanks to Merilyn Feickert for Biblical references, and to Val Binney and Steve Blomfield for helping me to understand epilepsy.\n\nWriting can be a lonely business, and I'm very grateful to the many people who gave me hospitality and nurture while I was writing, especially the Widgers, the Pierces, Anne MacLeod and family, Mina Hosseinipour, Janine Edge, Theo and Viv at Bushy Park and the Sisters of Compassion at Whanganui River Road. Thanks to the Tyldesleys for looking after me and to Dave and Sonia for putting up with me.\n\nFinally, a big thank you to my agent Bill Hamilton and my editor and publisher Juliet (Ms Whiplash) Annan, for being so horribly tough and for not letting me get away with anything. And thanks to the wonderful team at Penguin for everything else. \n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nKokoro yoga\n\nMARK DIVINE\n\nCDR, U.S. Navy SEAL\n\nand CATHERINE DIVINE\n\nForeword by Gary Kraftsow\n\n  ST. MARTIN'S GRIFFIN   NEW YORK\nBegin Reading\n\nTable of Contents\n\nAbout the Authors\n\nCopyright Page\n\n**Thank you for buying this**\n\n**St. Martin's Press ebook.**\n\nTo receive special offers, bonus content,\n\nand info on new releases and other great reads,\n\nsign up for our newsletters.\n\nOr visit us online at\n\nus.macmillan.com\/newslettersignup\n\nFor email updates on Mark Divine, click here.\n\nFor email updates on Catherine Divine, click here.\nThe author and publisher have provided this e-book to you for your personal use only. You may not make this e-book publicly available in any way. **Copyright infringement is against the law. If you believe the copy of this e-book you are reading infringes on the author's copyright, please notify the publisher at:<http:\/\/us.macmillanusa.com\/piracy>**.\nMAXIMIZE YOUR HUMAN POTENTIAL   \nAND DEVELOP   \nTHE SPIRIT OF   \nA WARRIOR\n\n# FOREWORD\n\nYoga is growing in popularity worldwide at an unprecedented rate. The great majority of enthusiasts and practitioners, however, primarily see yoga as a system of exercise oriented toward development of the physical body through the performance of postures known as asana.\n\nTo a lesser extent, but increasing steadily, is the emergence of the field of \"yoga therapy\" as a system of self-care. Perhaps one of the most significant roles of yoga therapy in the context of modern health care is helping with the paradigm shift from illness-based and health practitioner\u2013based care to wellness-based and self-based care.\n\nThe scope of yoga therapy extends from:\n\n **A form of adapted movement therapy to manage structural conditions to**\n\n **A method of sympathetic\/parasympathetic regulation via specialized breathing practices to help manage common symptoms of chronic illness such as stress, sleeplessness, fatigue, and pain management to**\n\n **A system of mental health care via an integrated use of breathing practices, self-inquiry, and meditation to help balance emotions, clarify thoughts, and support behavioral change.**\n\nModern scientific and medical research is demonstrating the incredible health benefits of these practices for anatomy, physiology, and the brain. Beyond these more physical benefits, the deeper work of yoga helps us surface our unconscious patterns, gain control over our desires, feelings, thoughts, and behavior. Through these practices we can deepen our self-understanding and gain mastery over our bodies and minds. With that as a foundation, we can access the higher states of awareness that lead to deep wisdom and compassion, and enable us to tap into and actualize our highest potential.\n\nMark Divine speaks from this deeper and more integrated understanding of yoga in Kokoro Yoga. While yoga therapy can function as a kind of life raft, helping those lost in the ocean of suffering, Kokoro Yoga is a kind of a launchpad for those who want to blast off into the unexplored regions of their own potential.\n\nSharing his own personal journey from would-be Wall Street professional, to martial artist, to Navy SEAL, to creator of SEALFIT, Mark clearly illustrates the power of the integrated approach to self-development passed on by the ancients.\n\nAs a starting point, Mark shares with us his self-reflections that \"his career path was incongruent with his ideals.\" This insight initiated his journey of self-discovery and self-development. From martial arts training to Navy SEAL training, he continues to listen to his inner voice. \"Though trained to kill,\" he shares, his heart led him to \"discover the path of the peaceful warrior.\"\n\nMark lays out the foundation of this path throughout his book, linking his own insight and understanding to ancient yogic teachings drawn from key texts such as the _Bhagavad Gita,_ the Yoga Sutras of _Patanjali,_ and the _Taittiriya Upanishad_. He explains the importance and necessity of building an ethical foundation as the root of self-development. He systematically walks us through the component parts of integrated self-development, including training the body, the breath, and the mind. He emphasizes key characteristics of the successful warrior, including the ability to stay \"calm, energized, in control of one's emotions, focused, ready for the mission, and able to manage the stress\" that arises in any situation.\n\nMark reflects on his own experience as he gets older, and the importance of continually rebalancing his training to avoid injuries and burnout. Through his ongoing reflective self-awareness as he trains, he realizes what the ancients said: Our practice must change and evolve to reflect our own stage in life.\n\nEarly in the book, Mark speaks of a saying among the SEALs, \"Take care of your gear, and it will take care of you.\" Similarly, the ancient yogis used to say, \"Take care of dharma, and dharma will take care of you.\" As I read through Mark's book, it became clear that he has truly realized this ideal. Mark's personal journey on the path of Kokoro Yoga, working multidimensionally to optimize his potential at every level\u2014physical, mental, emotional, intuitive, and spiritual\u2014led to his discovery of his own _svadharma_ and self-definition as a \"world-centric warrior and servant of humanity.\" A true yogi, Mark has realized through his own efforts that the purpose of human development leading to self-mastery is altruistic; that we then are able to serve others better!\n\nThis book explains clearly what self-development means at each level. It offers clear training instructions and tactics that will guide those committed to an ongoing path of self-development and personal growth. More than anything else, this book is a manual for self-empowerment, sharing in contemporary language an ancient path that enables each individual to actualize his or her potential and live life with meaning and purpose.\n\n\u2014Gary Kraftsow\n\nAuthor of _Yoga for Wellness_ and _Yoga for Transformation_\n\nDecember 2015\n\nOakland, California\n\n# CHAPTER 1\n\n# AN M4 AND A YOGA MAT\n\n# My path to a complete warrior art\n\n_In the beginner's mind there are many possibilities, but in the expert's there are very few._  \n\u2013SHUNRYU SUZUKI\n\nINTO THE BREACH\n\nIN 1991 I WAS A NAVY OFFICER, RECENTLY GRADUATED FROM SEAL TRAINING (BUD\/S) AS THE HONOR MAN IN MY CLASS, EARNING THE COVETED NAVY SEAL TRIDENT. SOON I WAS ASSIGNED TO SEAL TEAM 3, TASKED TO GO TO IRAQ TO FIGHT IN OPERATION DESERT STORM. FORTUNATELY FOR MANY, THAT WAR ENDED BEFORE WE DEPLOYED, and at SEAL Team 3 I would complete 6 more years of active duty in a relatively peaceful period of our history. Although I would visit the Middle East a number of times from 1991 to 1997, I wouldn't get the call to go to another turbulent Iraq until 2004, when I was serving as a reserve officer.\n\nLike most in the reserves during that time, it wasn't a surprise for me to get mobilized for duty during what was being called the war on terror. I knew it was coming but was not sure when.\n\nAt 41, my days as a gun-slinging operator were behind me. It didn't make sense for me to go back to a shooting SEAL task unit. So it was cool that my mission would be to lead a fairly complicated study for the U.S. Navy, involving the integration of the U.S. Marine Corps (USMC) into the special ops community (also known as SOCOM). I was to shadow a detachment of 100 handpicked U.S. Marines, intelligence and recon guys called SOCOM Detachment 1 who were to conduct a proof-of-concept deployment under the watchful eyes of SEAL Team 1.\n\nIt was a big deal. Twenty years earlier, the Marines had declined to be a part of the joint program to form the Special Operations Command, which included the Navy's SEAL teams, the Army's special forces and a Ranger battalion, and the Air Force's special ops teams such as Pararescue. But after 9\/11, as the Marine Corps watched particularly hot missions\u2014and the money to support them\u2014flow to SOCOM, they started to rethink their position. The secretary of defense, Donald Rumsfeld, put the brakes on an effort to fast-track the process. He didn't want to mess around with the 20 years of intricate and complicated coordination work that had already happened between the initial units of SOCOM. A thorough study and evaluation was deemed important to make sure that they didn't screw up what had taken years to get to work well. Though the USMC was ready to throw the 100-man team into combat, validation was a good idea before sending a newly formed concept team from a conventional-minded military service into the murky SOF (Special Operations Forces) world. And as you might imagine, there were a lot of charged convictions and emotions when it came to who was taking orders from whom between the SEALs and Marines. The call I got for the job was from Commander Mike Lumpkin, who was then the Naval Special Warfare Group 1 operations officer and had just rolled out of the position as deputy commander of Special Operations Forces, overseeing the 2,000 special operators in Baghdad. (In 2013, Mike Lumpkin became the assistant secretary of defense for SOLIC.)\n\nPrior to deployment to the combat zone with SEAL Team 1, I would lead the organization of the predeployment training certification for SEAL Team 1 with the 100 U.S. Marines from SOCOM Det 1 in tow. This was a good project for me. For one thing, I was intrigued with the underlying matrix of leadership that would need to be worked out between the SEALs and the Marines. Since my years on SEAL Team 3, I had expanded my views and beliefs regarding the definition of a \"warrior,\" leaving behind most parochial and tribal viewpoints on who is the best branch of the military, or who is the best special operator. Even though I was a SEAL, through and through, I would be able to offer an impartial viewpoint in conducting the exercise and ensuing study. My job was made easy by the fact that the Marines were great guys and solid operators\n\nThe certification exercise was a big success, and the time finally came for me to deploy to Baghdad to continue part two of my job, the study of the Marine team in SOF combat. Things happened fast. I paid a visit to the supply depot in Coronado, California, to get my weapon and gear, said good-bye to my family, and before I knew it I was on a flight to Bahrain\u2014with a bunch of polished new gear and an M4 rifle that I hadn't had time to take to the range. As an active-duty SEAL, shooting \"my\" weapon seemed to be a constant. I got real intimate with my primary and secondary weapons. But in the reserves we did not get issued our own weapons, so I literally had to check one out of the armory before I left. Any military member will understand how important it is to sight in your weapon and get comfortable with its idiosyncrasies. In addition, the life of the active-duty SEAL involves around-the-clock training and sharpening skills as an individual and as part of a team. It is a day-in, day-out, year-round affair. As an officer in the SEAL reserves, however, we didn't get to shoot nearly as much, nor did we get issued our own weapon to sleep with.\n\nThat was a big concern of mine, along with the web gear I was to use. I had brand-new web gear that wasn't broken in and customized to fit my frame. I needed to \"run and gun\" with the gear to ensure I would know where the ammo pouches would be in a pinch, and to make sure they wouldn't fly off in a firefight. On active duty, I got really comfortable with my equipment and knew I could rely on it. We had a saying: \"Take care of your gear, and it will take care of you!\" But here I was, about to deploy into a war zone, and I was looking at a bunch of plastic bags encasing brand-new, untouched equipment and a weapon I hadn't even shot yet. My pucker factor\u2014military jargon for adrenalin\u2014was rising.\n\nRatcheting up the stress was the news coming out of Iraq. On March 31, 2004, a friend of mine, Stephen \"Scott\" Helvenston, was one of four Blackwater military contractors that were in a convoy ambushed by insurgents in Fallujah. Scott and the others were killed in a horrifying manner, made worse for me by the fact that I saw him the day before he deployed weeks earlier. This was to be his last deployment with Blackwater. The graphic imagery startled me, knowing that I would soon be stepping into that same area where I could easily be the next target.\n\nTo make matters even worse, in mid-May, days before my deployment, a militant group posted a video of the decapitation of Nick Berg, an American radio-tower repairman from Pennsylvania. The video, which I immediately regretted watching, made me sick to my stomach. The stark reminder that we were fighting an enemy who seemed nuts, believing they were in the right to perform such deranged and hideous acts, steeled me as I stepped onto the C-130.\n\nOn my way to Baghdad, I stopped in Bahrain for a couple of days while awaiting final transport to the war zone. There I met up with a civilian analyst, from the Center for Naval Analysis, assigned to write the USMC side of the same report I was working on. He was to go to the Green Zone (the so-called secure area in Baghdad that the American military worked) with me. We discussed the project and our approaches as we waited for our ride.\n\nThe C-130 was scheduled to depart at 0500 hours. As I waited for the analyst to share a ride to the airfield, he approached me and said, \"Mark, I won't be going. I have a bad feeling about this.\" Great, I thought,... Wonder if he knows something I don't!\n\nWell, I was going anyhow. I couldn't lose face with my teammates and I was a tough SEAL officer, right? Climbing aboard the turboprop transport workhorse, the C-130, which the U.S. military uses to transport troops and equipment, I was never more nervous in my life. Keenly aware that anything could happen I felt on high alert. As the windowless C-130 roared into the air, I considered how things were stacking up. My civilian counterpart may have been spooked by another story in the press of how an Australian soldier had caught a bullet through his ass while on an aircraft leaving Baghdad. It was just someone shooting from the ground. A bullet had ripped through the fuselage and killed him. The ominous signs were getting the best of me.\n\nSitting across from me was a one-star Marine general working feverishly on a presentation with an aide. It was a 2-hour flight. After we lifted off, I couldn't bear sitting so I looked around the plane and spotted an open space by some cargo netting in the ramp area. My thoughts were set to full speed and I needed to do something to calm down. Remembering how calm I felt after my yoga sessions back home, I went to the open space near a stack of pallets and started doing a deep-breathing exercise and a few forward folds and backbends. This led to a full-blown yoga session in the middle of the bumpy ride in the C-130. (Later in my reserve career, I made it a point to practice yoga on military transports whenever I could. Often I had other members of my SEAL team or other military passengers join me, but I am pretty sure this was a first in military history!) The one-star Marine general must have been thinking: That SEAL officer is obviously green to combat and scared shitless. I didn't care. The yoga began to calm my mind and helped me regain control of my emotions. I felt much better as we turned our nose toward the Iraqi desert.\n\nBy the time we landed in Baghdad, I wasn't in a perfect Zen state by any means\u2014we were in a combat zone after all\u2014but I was far more calm, present, and centered, and ready for what came next.\n\nThat was a good thing. I hadn't been on the ground more than 15 minutes when I heard someone shout, \"Incoming!,\" followed by the unmistakable whistle of a mortar flying toward us. I had only heard mortars while in training, not combat, and in training they are whistling away from you. Trust me when I say it sounds very different when it is coming full bore at you! It exploded about a quarter mile away. Okay, I said to myself. Welcome to combat.\n\nLater, a couple of SEAL team guys drove up to retrieve me\u2014loaded for bear for the 45-minute ride through bad-guy land\u2014they gave me a sign to lock and load my M4 (I didn't have the guts to tell them I hadn't even sighted it in yet) and off we went to the SEAL compound at one of Saddam Hussein's former palace grounds.\n\nThat yoga session on the C-130 was my first official session of what I called Warrior Yoga (I later changed the name to Kokoro Yoga to avoid a trademark infringement). I realized in that moment that yoga presented a powerful toolkit for my own warrior development.\n\nSEIDO: THE BEGINNING OF MY JOURNEY\n\nSo there I was, breathing slowly and deeply into a Sun Salutation in the cargo area of a C-130 on my way to a combat zone. I was too focused on the moment to ask the obvious question: How did I get here?\n\nAs random and seemingly out of place my initial session of Kokoro Yoga might sound, it was a significant point of arrival in a long and steady search I had been conducting both during my active-duty time with the SEALs and after.\n\nThe search was in some ways a circular one, trying to reconnect with the kind of integrated warrior training that had initially infused me with the awareness and courage to let go of a big-money CPA career I had taking shape on Wall Street for the rigorous challenge of becoming and being a Navy SEAL. It started with what had been a growing sense of inner doubt about what I was setting out to do with my life, a voice I largely ignored as I began to climb the corporate ladder. I was in it for the money, in other words. The prospects for my success were bright. One of the chief rewards came from my family who appreciated that I was conforming to an ideal they had for me. Although I wasn't acknowledging it at the time, my career was incongruent with my ideal. There was a growing weight on my shoulders as my future in high finance stretched out before me. I was walking home one night from work when my train of thought was disrupted by a series of shouts coming from a second-floor window of a seven-story building on West 23rd Street. Intrigued, I walked up a flight of stairs and into what would become a truly disruptive force in my life: the Seido Karate dojo run by Grandmaster Tadashi Nakamura.\n\nNakamura had formerly been deployed to the United States from Japan to lead a style of karate known for tournament fighting, Kyokushinkai. Nakamura had become disenchanted with the lack of dimension in the training and left despite intense pressure from Japan. Nakamura went about creating Seido Karate with the intent of focusing on human development rather than sheer fighting prowess. The word _seido_ is Japanese for \"sincere way.\" This was my first exposure to the martial arts, and I soon found that Seido was a practice that truly integrated body, mind, and spirit training. Unlike other martial arts I would become acquainted with over the years, Seido was unique in that it didn't just talk about the mental and spiritual aspects\u2014it was actually part of the training. Meditation and spiritual talks and discussions on mental development were part of the routine, along with the fighting practice. It was through this work that I was able to connect with the sincere voice within my being and understand that I was meant for something different than taking a place in the family business, as was expected.\n\nSeido not only unlatched access to an inner wisdom that led me to join the Navy and become a SEAL, it also prepared me in a foundational way for what is considered the most arduous and demanding military training program in the world.\n\nThe five guiding principles of Seido Karate training are as follows:\n\n1. ETHICAL FOUNDATION. The ethical foundation of Seido is based upon what's called \"bushido,\" also known as the Way of the Warrior, a series of moral standards embraced by samurai warriors, like honor, frugality, and loyalty. As you read the next chapter in this book, you'll note that the first two levels of yoga, or limbs, are also staked in an ethical foundation.\n\n2 INTEGRATED DEVELOPMENT. As mentioned, Seido didn't just pay lip service to the concept of integrating mental and spiritual training into the daily practice, Nakamura emphasized this fusion: \"My purpose in founding Seido Karate was to show what I feel is the true essence, the kernel of true karate: the training of body, mind, and spirit together in order to realize the fullness of human potential.\"\n\n3 SPIRITUAL AWARENESS. Zen meditation is core to the Seido practice. As esoteric as this may sound, the meditation and spiritual lectures helped me develop the awareness and humility to thrive through the ego-busting stress of BUD\/S and Hell Week.\n\n4 CRUCIBLE TRAINING. Frequent tests and challenges are part of the Seido program and work to help push students to new levels of performance and to comprehend the magnitude of the potential lying within. For example, an annual crucible session at the dojo might include thousands of kicks and punches. bAnother common crucible session was conducted over the course of days at a monastery, where we would fill our days endlessly cycling back and forth from meditation to karate work.\n\n5 FORGING MENTAL TOUGHNESS. In Seido, we were worked hard and steadily toward developing resiliency and a mentally tough attitude where we never backed down. As Nakamura explained: \"Seido seeks to develop in each student a 'non-quitting' spirit. No matter what the obstacle or difficulty\u2014emotional, physical, financial\u2014we want students to feel that, though there may be setbacks, they will never be overcome by any of these problems.\"\n\nThe unified training of Seido proved to be invaluable the day I stepped onto the path to become a Navy SEAL. Because of the relentless difficulty of BUD\/S, of having to go 100 mph for the better part of a year, from Hell Week to drownproofing to SEAL Qualification Training, I survived the staggeringly high failure rate. Actually it was because of the integrated warrior training, which I took so seriously, that I did more than just survive. I was able to thrive, finishing as honor man, # 1 graduate, of my class.\n\nSEARCH FOR A COMPLETE WARRIOR ART\n\nIn joining the Navy SEALs, I was leaving Wall Street behind\u2014a good thing for me. A sacrifice, however, was leaving behind my training at the Seido World Headquarters on West 23rd Street in Manhattan. In departing for the SEALs, I took with me a desire to find another practice that had a similar comprehensive approach to human development that Seido did. Being on a SEAL team was all that I had imagined, of course. But as an operator you get very focused, for obvious reasons, on shooting, fighting, and mission success. You won't find time dedicated to a spiritual practice on the schedule. So my search for something similar to Seido put me on a quest.\n\nSCARS (Special Combat Aggressive Reactionary System) was my first stop. SCARS was developed by a Vietnam vet named Jerry Peterson from a lethal hand-to-hand combat system called Kung Fu San Soo. Peterson had stripped away all of the cultural elements into what you might call a clear science of how to offensively fight to win. The training was brutal\u2014a 30-day, 10-hour-day program to become certified to teach SCARS. I loved the techniques and the training was fun, but whereas Seido was about developing moral character and spirit, SCARS was about fighting and surviving. In fact, SCARS training came with a warning: Do not use unless someone must die. In the end, I had more than 1,000 hours of training in SCARS when I left the active-duty SEAL teams. In a story that illuminates why I was motivated to continue my search for another Seido-like program, my wife, Sandy, had become a therapist for the Navy. One assignment sent her to a U.S. Navy vessel in Australia that would soon be returning home. Her job was to help the sailors prepare for the jarring realities of returning to civilian life after months at sea. At a dinner in the officers' cabin, the commander of the ship was asking about Sandy's background, and it came up that she was married to someone also in the Navy.\n\n\"Who are you married to?\" she was asked.\n\n\"Lieutenant Commander Mark Divine,\" Sandy answered.\n\nOne of the junior officers went off: \"Mark Divine! Mark Divine. I know him... he's a SEAL and SCARS master. He could kill us all with his pinky finger!\"\n\nAfter I finished laughing when Sandy told me this story, I began to think about it. As much as I didn't want to be known as some sort of CPA to be feared in a corporate audit, I also didn't find it appealing to be known as a master of the science of killing. I knew that humble warriors are the last to pick up a weapon. I was becoming a more peaceful warrior, even as a SEAL officer.\n\nIn transitioning from active duty to the SEAL reserve force, I started training in a Goju-Ryu karate dojo, which had similar roots to Seido. I earned my black belt quickly, in part because I already knew most of the physical moves. But there was no meditation or spiritual training, and when I was recalled in 1999 back to active duty for a stint in Egypt and the Middle East, I never returned to Goju-Ryu.\n\nAfter the 1-year tour of duty, Sandy and I adopted our son, Devon, and moved to North County, San Diego, about a 40-minute drive up the coast from the SEAL base in Coronado. My search continued. I began to study with Sensei Shane Phelps, a ninjutsu master. Sensei Phelps was trying to get his ninjutsu studio, Temple of the Full Autumn Moon, off the ground when I started training with him. I helped him write a business plan, in fact. He had one of the most sensational backgrounds you're ever going to find. He fought in the Vietnam War and then went on to serve 7 years as a Navy SEAL. He worked for the United Nations as a peacekeeper in places like Syria and Lebanon, and also as an antiterrorism agent of the CIA. He got his BA at Stanford went on to earn a masters degree in comparative religion at Harvard and a Master of Divinity at Yale. Before his Western schooling in the Ivy League, however, he spent 2 years studying Tai Chi and meditating at a Buddhist monastery in China. Shane has long been an awesome example of what I call the 20x factor.\n\nTo this day I love the art of ninjutsu. It is an incredible combination of some 40 different types of martial arts, with a variety of weapons and both internally (oriented toward the psychological and spiritual) and externally oriented arts (the more physical-leaning of the martial arts). In the negative column, I found the training frustratingly slow-going and fragmented. I was working toward my black belt when Shane's school suddenly ran out of money and closed its doors. He began working with only private clients, and so my search continued for the complete warrior workout.\n\nIt was during this phase of my journey when I discovered yoga.\n\nTHE WAY OF THE PEACEFUL WARRIOR\n\nYoga in the West is viewed through a variety of lenses... for most it is a form of exercise. Pilates, Power Yoga, Core Yoga, and Hot Yoga are good examples of this movement. Others may consider it a mystical practice bound to Hinduism, or as a place to train Cirque du Soleil athletes. Since the late 1990s there has been a boom of yoga studios around the country giving rise to millions walking to and from group classes with mats jutting out of their backpacks in a quest to stretch, bend, sweat, and look great naked. I was soon to learn that yoga offered much more.\n\nMy introduction to yoga came through reading a classic titled _Autobiography of a Yogi_ , by Paramahansa Yogananda. I figured anyone with the word \"Yoga\" in his name must know what he was talking about. Funny thing, the book had nothing to do with stretching and twisting your body into a pretzel. What Yogananda brought to life was a powerful philosophy of living and developing oneself spiritually. I was intrigued, as I had just left ninjutsu and couldn't find another program near my home that inspired me. So the thought that perhaps yoga could fill that void popped into my head after reading the book. The spiritual component was something that I was seeking, even though I was not drawn toward the Hindu mythology glued to the yoga programs I had seen to date. I consider myself a Christian and wondered if there would be a conflict. However, I recalled training at the Zen Mountain Monastery with my karate team back in 1989. The head monk, Daido, said that Buddhism as a philosophy was in complete alignment with Christianity. From what I had read, yoga was similar in that it was not a religion, but a philosophy of living as well and a science of personal development. I thought it could be in complete alignment with any religious conviction. Armed with that theory, my journey into yoga began.\n\nFive years before my deployment to Baghdad I mustered the courage to walk into a Hot Yoga studio in Encinitas, California. In Hot Yoga they crank the temperature up to 105 degrees as you twist and boil your way through 26 poses. The first thing I noticed walking out of the yoga class, dripping wet, was how good I felt. The 90 minutes of standing and seated poses in the sauna-like studio yielded some incredible detoxification and deep-stretching benefits.\n\nNot being one to shy from a gut check, I immediately signed up for their challenge of a Hot Yoga class every day for 60 days. The challenge for me was not so much the discipline, but rather that the classes were chock-full of very attractive women bending and twisting in spandex. Not only was it hard to concentrate, but also my preconditioned notion of what men do and what women do for fitness, was put to the test. I had to trust my intuition that this was a worthy pursuit and shift my attention inward to keep focused on the training effect. I found that this inward focus developed greater awareness and deepened my intuition. It was an experience quite different from my years studying martial arts. In fighting and the martial arts, the focus is mostly outward, except when meditating before and after class. In yoga, it is meant to be inward. Rather than scanning the room or my opponent for opportunities and threats, I was attending to my breathing and the nuances of moving into and staying in the pose. I began to notice that if I went into a session with a scattered mind, the practice settled my mind and connected me to a deeper part of my character.\n\nBut Hot Yoga was just a launching pad into this amazing new world. Though a fine introduction to yoga, the precise repeating of the same 26 poses each session, in the same sequence, with the instructors uttering the exact same words each class\u2014became mind-numbing to me. I felt a need for variety and silence in my practice, and I could not get it there. I soon began to wonder if I needed a studio at all. The movements were familiar enough to me after 15 years of martial arts that I thought I would be able to train on my own.\n\nI found two yoga DVDs to use at home. One was by Baron Baptiste, called _Power Yoga_ emphasizing core strength and balance, and another by Shiva Rea, emphasizing a fluid, dancelike sequence and breathing. I really enjoyed both as they expanded my repertoire and deepened my knowledge. I would rotate them and add an occasional visit to the Hot Yoga studio to get my sweat on. This went on for 2 years before I stumbled into Ashtanga Yoga, which blew my mind open. Discovering Ashtanga Yoga was a turning point in my yoga studies.\n\nI am lucky to live and work in a town that virtually screams health and fitness. Encinitas, California, brims with world-class endurance athletes, and well-known surfers and skateboarders. Guess what else may be found in Encinitas? Some of the most-qualified yoga teachers in the world. In a conversation with a friend I was asked if I trained with Ashtanga Yoga legend Tim Miller. She said the name with such reverence, she might as well have called him \"Master Tim Miller.\" Tim is the first American to be certified in Ashtanga Yoga by Sri K. Pattabhi Jois. Tim had to train for many years and make several long trips to India, virtually begging Jois for the honor. It was clearly not given out lightly, especially to an American. I found Tim's studio literally across the street from my office. He was the \"real deal,\" and he became my next sensei.\n\nAshtanga Yoga was derived from the teachings of the famed yoga master Krishnamacharya. He taught Sri K. Pattabhi Jois a progressive system of increasingly challenging series of poses, six series in total, designed for young athletes and military groups. It had a rigid structure that the young men were not to deviate from. After all, good order and discipline are required in the training of new warriors. Jois named it Ashtanga, borrowing the term from Patanjali's Yoga Sutras (more on that in chapter 2). I was drawn to the Ashtanga system because it seemed to share a developmental ladder similar to a martial art belt-ranking system. Though you don't test and get promoted in Ashtanga, you do work progressively through the series of poses over the years. I first approached it with my Western goal-oriented mind, thinking I had found my new martial art and that I was going to \"get my black belt in Ashtanga.\"\n\nMy first session of Ashtanga kicked my ass and rekindled the warrior flame within me. It took me\u2014an elite athlete, martial artist, and yoga practitioner\u20141 hour and 45 minutes to get through, and the session was so demanding I almost lost all bodily functions. I've found the true yoga, I thought with elation, as I crawled off the mat. Later, I would attend two 100-hour teacher trainings in the first and second series with Tim. But as the Iraq War heated up in 2004, duty came knocking again and I replaced my yoga attire with the uniform of the Navy SEALs for the third time.\n\nWelcome to Yoga Saddam\n\nIn Baghdad, my yoga session aboard the C-130 stayed with me as I settled into life in the combat zone. My work routine mirrored the \"battle rhythm\" of the Navy SEAL task group where I set up shop. I would awake around 9:00 a.m. and work till 2:00 or 3:00 a.m. Sleep was a luxury few enjoy in combat.\n\nExercise was another challenge. SEALs will always improvise to find a way to train, even when operating on combat missions that go late into the night. In my situation it was largely impractical to go to the gym, which was located at Camp Victory and required a combat drive in an armored humvee to get to. It was not worth the risk or time. So I began running around the compound, a 3-mile loop, and doing body weight PT (physical training). Soon I felt the itch for yoga, but there were certainly no yoga classes (that I was aware of) being held anywhere in Baghdad, or Iraq. Another nonstarter. So I again decided to follow my intuition and just go it alone based upon what I had learned from Hot Yoga, Power Yoga, and Ashtanga Yoga.\n\nFinding a small patch near one of Saddam Hussein's former palaces, next to a lake, I set up shop. It wasn't as picturesque as this might sound\u2014for starters, the pool and house were blanketed with pockmarks from a firefight\u2014but it had some trees to provide shade in the desert heat and was removed enough that I wouldn't get awkward stares from the other warriors on base. I skipped breakfast every morning and found refuge at my new training spot. Equipped with a mat, my M4 (now sighted in), and a kettlebell, I started playing with different combinations of yoga poses, functional interval workouts, self-defense moves, and breathing and visualization exercises. The visualization was always of me at home with my family after leaving Baghdad (a version of the \"future me\" visualization I teach in this book). I would listen closely to my body and train from 45 to 90 minutes depending on what my intuition told me I needed. When I was finished with the practice I felt amazingly clear and calm.\n\nAs the weeks progressed this practice became my center post in the storm of combat. One day, CDR Wilson, the commanding officer of SEAL Team 1 stopped by to observe my training. Though he was intrigued, I couldn't get him to join me... the demands on his time were simply too much for him to make that leap. Or perhaps he thought what I was doing was a little bit odd, and he didn't want to risk his men thinking I had converted him! I could only explain what it felt like, but a new \"initiate\" must experience the practice for him or herself to truly understand the vast benefits of yoga. Now, years later, I realize how valuable this practice would be for warriors in the field to manage stress, win in their minds, and avoid the devastating effects of PTSD.\n\nWhile at the height of the Iraq War, I started each workday feeling calm, energized, in control of my emotions, present, and ready for the mission. My mental facilities were sharp, as were my skills in dealing with the stressful environment. These benefits were, in my opinion, a direct result of the daily yoga practice.\n\nFast-Forward\n\nMy experience in Baghdad was profound and propelled me into taking my life in an entirely new direction. When I returned home I amped up my Ashtanga Yoga practice, and launched US CrossFit and the SEALFIT integrated training program. By 2013 I had a worldwide reputation for success in training SEAL and special ops candidates and other elite athletes through SEALFIT and a mental toughness program called Unbeatable Mind. Based on the training program I wrote three books, two of which became bestsellers. A 20,000-square-foot training center in Encinitas, California, became my laboratory. I could be heard saying, \"I eat my own dog food,\" because I endeavored to train for 2 to 3 hours a day doing a combination of SEALFIT and Kokoro Yoga (and still do to this day).\n\nIn 2014, I turned 51 and my body was telling me that I needed to rebalance my training. The combination of the hard-hitting SEALFIT program, with hard-core Ashtanga Yoga, worked well\u2014until I turned 50! Now, it was leading to small injuries and burnout. I needed to find balance in my personal practice, not just for my own comfort, but also so that I could teach athletes and warriors of all ages, not just the younger set. Though I love the Ashtanga practice and community, the rigidity of the routines and difficulty of the poses made me concerned that I would get seriously injured and sidelined as I got older. The warrior's way is to train every day that you are alive, and I planned to be training until 150\u2014then drop dead on the training floor in Savasana (corpse pose!). Thus as I evolved, I wanted my yoga to work for all stages of life, for differing intentions, and for different types of people. A new approach was in order.\n\nI found my next mentor in Gary Kraftsow, founder of American Viniyoga\u2014also adapted from Krishnamacharya's teaching. (Krishnamacharya taught a third application of yoga to B. K. S. Iyengar, which is popular in the West.) These three systems (Ashtanga, Viniyoga, and Iyengar) all seem very different to the observer, but to Krishnamacharya they were just \"yoga\" taught for different applications and different phases of life. This made sense to me: In the SEALs we used what worked, discarded what didn't, and strove to adapt our training to our situation, environment, and age. I adopted some Viniyoga training methods so that Kokoro Yoga could be more flexible and balanced.\n\nWhen asked by my athletes to put my method yoga into a fixed form that could be trained at home or in the field, I was hesitant at first. I always molded it to the audience. And who was I to write a book about yoga in the shadow of such incredible teachers and mentors? But one of my students, a former Marine, asked how his Marines and other military members could train in Kokoro Yoga. He implied that they would be open to try yoga if it came from a warrior like myself, who they trusted to give them practical training to improve their survivability and ability to manage combat-related stress. I received a similar message from my CrossFit \"fire-breathing\" friend Greg Amundson. He felt that the athletic community needed a yoga that could complement their athleticism through durability, spinal health, and breathing. Finally, I got the blessing from Gary, who felt that this community of warriors desperately needed yoga. I agreed, and this book is my humble attempt to serve.\n\nWhether you are a Navy SEAL running toward the sounds of gunfire or an athlete seeking maximum performance in your sport, a dedicated daily practice of Kokoro Yoga will help you to perform at your peak. If you are suffering from combat (or any shock) related stress, it will allow you to recover your peace of body and mind.\n\nAs I found during my time in Baghdad, and have continued to discover to this day, there is incredible value to be absorbed from integrated, full-spectrum training. Do you desire to be more flexible, gain core strength, and be more durable? Yoga will absolutely bring it. Do you want to gain composure under pressure and a calm mind? Yoga will bring it. But that's just the beginning. For the athlete, the military operator, the corporate executive, the artist, the auto mechanic, the firefighter, the student, the homemaker, the parent, I believe Kokoro Yoga is a Trojan horse ready to unleash a host of unforeseen benefits, ultimately leading to the highest levels of consciousness. I know, it sounds too good to be true, but if you stay with me and begin a daily routine that meets your practical needs, body type, and goals, then you will be planting the seeds for a powerful future.\n\nUltimately, this book is about mastering yourself at all levels so that you can become all you were meant to be. Someone who is willing to say yes to the right mission, and say no to the status quo. Someone who can transcend the various strains of neuroses, which today's media would love to have you feed on, and be \"sheepdog strong,\" so you can serve and protect others. Someone who accelerates their development to the highest, integrated stages of consciousness\u2014and become a world-centric warrior and servant to all of humanity.\n\n# CHAPTER 2\n\n# THE PURSUIT OF MAXIMUM HUMAN POTENTIAL\n\n_When you are inspired by some great purpose, some extraordinary project, all your thoughts break their bonds. Your mind transcends limitations, your consciousness expands in every direction, and you find yourself in a new, great, and wonderful world. Dormant forces, faculties, and talents become alive, and you discover yourself to be a greater person by far than you ever dreamed yourself to be._  \n\u2013PATANJALI\n\nOLDEST PERSONAL DEVELOPMENT SYSTEM\n\nJust what is yoga? If you think yoga is a group stretching program, or a fitness program for the ladies, then I suggest a visit to northern India to watch how yoga is practiced by the yogi warriors. There you would see how yoga shares its beginnings with the martial traditions, and if you open your mind to fresh understanding of what yoga is, you'll begin to appreciate the depth and breadth of what is the world's oldest and most complete self-development program.\n\nThe word _yoga_ means \"to yoke, to unify, or to integrate.\" At the mental level it means to integrate your ego mind with your witnessing mind. (Maybe we can call that your soul?) It also means to integrate your body, mind, and spirit into a whole, as well as to live life in a more integrated, balanced manner with simplicity, spirituality, and nonattachment to material distractions. Ultimately I like to focus on yoga as a means to develop mastery of the self, so we can serve others better. We can do this if we integrate fully, connect to our spiritual selves and advance consciousness to the highest level available to us in our lifetime. The word \"Kokoro\" means warrior spirit, or to merge heart and mind into your actions. So Kokoro Yoga is a warrior development application of this ancient self-realization system.\n\nIt's speculated that yoga was first practiced in the fifth or sixth centuries BCE, but it is more plausible that it is thousands of years old. While yoga in the West is most often associated with fitness, or our reverence for Mahatma Gandhi for his yogic philosophy of nonviolent leadership, or Paramahansa Yogananda, founder of the Self-Realization Fellowship, it has a deep, complicated place in India's warrior history in a way that demonstrates the depth and breadth of yoga as a living philosophy of spirituality and science of the mind. As William Pinch detailed in _Warrior Ascetics and Indian Empires_ , his account of yoga history in India, he wrote: \"Crucial to the transition to wide scale military entrepreneurship in the eighteenth century was the ability of the yogi to be many things at once\u2014to be Muslim and Hindu, emperor and mendicant, ascetic and archer, soldier and spy.\" Pinch also details the age of India's military labor market and the role yogi warriors or \"armed ascetics\" had in India's history from 1500 to the present.\n\nConsider the following description of the broad, open-source code nature of yoga by Suketu Mehta, a professor at New York University, who wanted to emphasize how yoga's dimensionality goes beyond fitness or preparing for a fight:\n\nThe yoga that most Americans are aware of is Hatha Yoga, only one of the various types of yoga. Krishna in the Bhagavad-Gita defines the others: Raja, Karma, Bhakti, and Jnana yoga. Yoga is diverse and profound\u2014volunteering at a soup kitchen is yoga; raising your voice in praise in a gospel choir is yoga; trying to understand how the galaxies shift and why the poor lack shoes is also yoga.\n\nThe above quote clearly points to yoga being far more than a series of physical movements designed to get you fit and looking good. Jnana Yoga is the yoga of the intellect. Through deep study of scripture, gaining knowledge and wisdom and understanding the working of one's own mind, enlightenment is attained. This is the path many intellectuals in modern religious traditions take. Bhakti Yoga is the path of utter devotion and love for God in a way meaningful to the seeker. This is the path that suits anyone who prefers an \"I-Thou\" relationship with God. Karma Yoga is the path of action. Through one's dedication to duty and service through action, karma is purified and spiritual evolution occurs. Hatha Yoga is the physical training system for personal mastery that, as Mehta points out, most modern offerings are based upon. In its purest form, Raja Yoga includes Hatha and the study of all eight levels of training described in Patanjali's Yoga Sutras (more on that soon). Where does our yoga fit in this mix?\n\nKokoro Yoga is designed for anyone who wishes to tap into his or her warrior archetype: those who are driven toward action, passionate about service, and committed to continuous personal growth. It is Raja Yoga that combines Karma Yoga (the yoga of action) with Hatha Yoga (the yoga of personal mastery). We seek to train all of the eight stages of development for self-mastery. As mentioned, these Eight Limbs were defined by Patanjali in the BCE era. Early in my training it was that work that opened my mind to yoga being more than a fitness regimen. I experienced its power as an integrated development system, similar in some ways to what I had experienced with Seido Karate.\n\nTHE EIGHT LIMBS\n\n_It is said that if you know your enemies and know yourself, you will not be imperiled in a hundred battles; if you do not know your enemies but do know yourself, you will win one and lose one; if you do not know your enemies nor yourself, you will be imperiled in every single battle._ \u2014SUN TZU\n\nThe Yoga Sutras are a collection of 196 aphorisms (in Sanskrit known as sutras\u2014a thread of specific principles described in terse, cogent pieces of writing). The bulk of the sutras deal with the science of mental development. Yet Patanjali also describes eight levels of the yoga path. They are like rungs on a ladder. The eight levels are to be progressed in a \"transcend and include\" manner toward a complete integration. This means that each level transcends the level preceding it, but doesn't leave the training and benefits behind. They are included in the upward spiral of development. In sum we are working toward the highest level of integration, which we will call self-mastery. Self-mastery means developing each and every level into a unified whole, so we can experience life at its fullest. Here is a brief description of the Eight Limbs:\n\nLEVEL 1: YAMA. ETHICAL DISCIPLINES. Yama is a code of morality and character to study, align with, and ultimately live by. It can be viewed as a set of ethical disciplines to guide our interactions with other humans. We will discuss some of these disciplines later in the book, and they include a code of restraint, or balance; a code of nonviolence; a code of truthfulness and honesty; a code of nonstealing; a code of noncoveting\u2014meaning the elimination of greed and developing contentment with few possessions and nonindulgence.\n\nLEVEL 2: NIYAMA. PERSONAL DISCIPLINES. These are individual disciplines taken on to set the foundation for mastery of your body and mind, leading to purity and contentment. They include: detoxifying the body and the mind; a deep introspection and healthy use of mental capacities; also, a dedication to a spiritual practice (this can be attending your church, for example). For me, my spiritual practice is yoga and meditation.\n\nLEVEL 3. ASANA. FUNCTIONAL MOVEMENT. This level is what most Westerners think of when they think of yoga. The true purpose of this rung is to prepare the student's body so that he or she can sit in concentration and meditation for long periods of time. For our purposes, this is the movement branch of Kokoro Yoga. It includes the customary poses seen in many yoga studios, and also some functional movements from my SEALFIT program, or CrossFit and even the martial arts.\n\nLEVEL 4. PRANAYAMA. BREATH CONTROL. The fourth level is dedicated to the regulation of breath and harnessing the natural energy all around us. This level is where we begin to transcend from knowing yoga as a physical and ethical practice to experiencing a profound spiritual evolution. Working with the breath is free medicine, bringing optimal health and even great power. Yoga works with prana, or \"life force,\" through the breath. It is identical in this way to practices like Tai Chi (chi meaning \"life energy\") and Qigong (\"life energy cultivation\"). Through breath control practices like our box breathing drills, the energy centers of the body are tapped into, linked, and energized through the conscious movement and control of the breath.\n\nLEVEL 5. PRATYAHARA. MASTERY OF THE SENSES. From the extrasensory capacities developed by the deaf and blind, we know that by shutting off one sense, other senses will expand in extraordinary ways to compensate. Thus through level five we seek to train and regulate our senses. By mastering the senses, we can better control those random urges\u2014think about what propels greed and overeating and stuffing our garage with crap we don't really need. By shutting off the senses we can sharpen our capacity to listen to the sixth sense, intuition.\n\nLEVEL 6. DHARANA. CONCENTRATION. This level is about developing deep powers of concentration. We sharpen our mind and develop single-point focus. In practice, this is about brushing away the noise of the unruly mind by strengthening our capacity to lock on to one thing and block out the rest. Martial artists work binding their thoughts, with laserlike focus, to movements. In Kokoro Yoga we will concentrate on the poses, the breath, and an object.\n\nLEVEL 7. DHYANA. MEDITATION. Dhyana is about developing presence. Whereas concentration is a coherent, singular focus, a state of meditation is the absence of thought\u2014settling your witnessing, perceiving mind on the object or subject of your choosing until there is a transfer of information. It is letting go of active thought and being in complete presence. It is at this level of yoga where deep levels of consciousness and awareness are obtained and further trained. In sports, this meditative state is known as flow, or the zone of peak performance, where time stands still and a flow state ensues.\n\nLEVEL 8. SAMADHI. UNION, INTEGRATION. This is considered to be the level of spiritual enlightenment, of the union of true self with the ego self. My friend Ken Wilber, creator of the Integral Theory and author of _The Theory of Everything_ , explains that when integrated (\"enlightened\" in Eastern terminology) you will tether with your \"soul self\" which then becomes the center of your consciousness versus your limited ego-thinking self. When this happens we end separation and are able to express ourselves most authentically to the world, able to take perspectives on our own perspectives as well as those of others, and can experience life as a blissful connection with all sentient beings.\n\nHISTORICAL INFLUENCES\n\nNow that you have a sense of the depth and breadth of yoga, let's get a grasp of the historical roots. However, considering the hundreds or thousands of years that yoga has existed, a definitive history of yoga is beyond the capacity of this book. What I do wish to impart is that yoga has had many expressions and can be considered an open-source project taken on by warriors, athletes, and spiritual aspirants over many eons. The project continues today with millions of Western practitioners adopting its methods and philosophy.\n\nArjuna: A Warrior at a Crossroads\n\n_The mind is restless, turbulent, strong and unyielding... as difficult to subdue as the wind. For the uncontrolled there is no wisdom. Nor for the uncontrolled is there power of concentration. And for him without concentration there is no peace. And for the un-peaceful how can there be happiness_ \u2013ADAPTED FROM THE _BHAGAVAD GITA_ (KRISHNA SPEAKING   \nTO ARJUNA)\n\nA text that routinely appears in comparative religion classes throughout the Western world is the _Bhagavad Gita_. The Hindu epic, composed circa the seventh century BCE, presented the many faces of yoga and how they appeal to certain characters such as the warrior Arjuna. A central theme is the importance of performing one's simple, daily work and duties with a critical and spiritual binding with renunciation\u2014of not being attached to material rewards. The _Gita_ is a warrior's manual that encourages the reader to actively confront and take on evil in the world rather than turning the other cheek.\n\nAt the heart of the _Gita_ is a dialogue between Arjuna, a master archer warrior, and Krishna, an earthly manifestation of the Indian deity Vishnu, representing the voice of God. A major battle is about to erupt between Arjuna's tribe and his cousin's tribe. Arjuna is confronted with the fact that his mission requires fighting against members of his family\u2014the sort of inner conflict that Americans recall from their history of the Civil War. Arjuna's skill with the bow and arrow is so exceptional that he can pierce the eye of a bird in flight, but in the _Gita_ he struggles with questions of morality, ethics, and his duty as a warrior, seeking answers from Krishna. Arjuna ultimately chooses not to fight, sacrificing his life rather than betray his thoughts and feelings for his family. The dialogue with Krishna finally offers a path to spiritual freedom that is not total renunciation of the fruits of one's labor nor abstinence from performing one's work\u2014rather, spiritual perfection is pursued by performing one's duty in life without looking for rewards and with detachment to the results. Arjuna's sacrifice informs the inner struggle of warriors of all races and generations, including the modern warrior who strives to do the right thing in spite of the consequences.\n\nBodhidharma and Kalaripayattu\n\nIn surveying the expansion of yoga from the ancient East to the modern West, we should also look at the shared roots yoga has with the martial arts. The legend of Bodhidharma, a son of an Indian King in the sixth century, sheds light on both of these topics. Bodhidharma traveled by foot and boat from his homeland, where he eventually arrived at the Shaolin Monastery in the Henan Province of China, the famous Buddhist temple known for the development of Kung Fu. Martial arts history holds that after Bodhidharma arrived at the temple, he spent 9 years in seated meditation, inspiring the monks with both his fierce discipline and deep spiritual powers. Bodhidharma began to teach the monks the practice of what would be known as Zen meditation, but the monks were academics and didn't have the physical capacities to perform extended meditation.\n\nBodhidharma created what you might call a health and fitness program to complement the meditation practice, a series of 18 flowing yoga postures very similar to Hatha Yoga. The movements also shared properties of the Indian martial art known as Kalaripayattu\u2014which is considered by some to be the oldest fighting art in human history. Bodhidharma has been credited by some for the creation of Shaolin Boxing and his contribution to Zen Buddhism. There is ample evidence that the Chinese had developed martial systems far before Bodhidharma arrived, but the key point I'd like to make is that through this form of educational dissemination, the ancient practice of yoga was transmitted across borders and eventually throughout the world.\n\nYoga and Gymnastics\n\nIn sorting out the modern offerings of yoga in the West, it can be helpful to reverse engineer the influences and reinterpretations involved. One of the more interesting discussions, sparked by author Mark Singleton, suggests that there was a blending of West into East as modern yoga evolved. Singleton's book, _Yoga Body,_ pins some of his research on a fitness trend that occurred in Europe in the nineteenth century. It was then that the Scandinavian system of gymnastics became popular throughout Europe in the 1800s. Called \"primitive gymnastics\"\u2014and showing up in the YMCA network of gymnasiums\u2014it was a bodyweight exercise regimen, attractive to civilians for the health benefits and attractive to the fighting forces for the fitness edge it could provide. The poses used within the Danish system are vividly similar to the poses you'd find in an introduction to yoga class.\n\nFollowing is an interesting assessment by writer Matthew Lee Anderson on the _Yoga Body_ :\n\nMark Singleton analyzed Niels Bukh's _Primary Gymnastics_ (1925) and found that \"at least 28 of the exercises in the first edition of Bukh's manual are strikingly similar (often identical) to yoga postures occurring in Pattabhi Jois' Ashtanga sequence or in Iyengar's _Light on Yoga_. Both Jois and Iyengar were students of T. Krishnamacharya, who taught yoga in the Indian royal palace and whose classes were categorized as \"physical culture\" or \"exercise\" in the official palace records. By that point, the Danish gymnastic system had reached such a level of popularity that it had been incorporated into the British Army and into the Indian YMCA.\n\nYoga Comes to American Shores\n\nThe European explosion of yoga and gymnastics for exercise was a precursor to several key yoga voices crossing the pond to introduce Americans to the mysteries of yoga. One of the most influential was Paramahansa Yogananda, from Uttar Pradesh, India, who came to the United States in 1920. Through Yogananda's establishment of the Self-Realization Fellowship and his book, _Autobiography of a Yogi_ (incidentally this is the only book Steve Jobs kept on his personal iPad), he introduced the philosophy of yoga, and meditation to millions. Other teachers that have had a big impact on the popularity of yoga in the United States include Mr. Universe winner Walt Baptiste, followed by his son Baron. Baron has more recently popularized Power Yoga, a variant of Ashtanga Vinyasa (flowing) Yoga. With Power Yoga he weaves mind-and-body empowerment practice including meditation and \"active self-enquiry.\" Baron became a performance coach for the Philadelphia Eagles, an early pioneer in applying yoga methods for sports performance.\n\nYoga continues to evolve to this day. As Kaitlin Quistgaard, former editor-in-chief of _Yoga Journal_ , put it: \"With around 15 million Americans practicing yoga, there's room for diversity and for the practice to evolve in more than one direction at a time.\"\n\nPERFORM LIKE A MASTER\n\nOne of the chief purposes of this book is to make a cogent case for why an integrated somatic practice like Kokoro Yoga can serve up profound performance improvements for first responders, military operators, CrossFitters, endurance athletes, and, frankly, anyone committed to self-mastery in the service of others. The path of Kokoro Yoga is to blend physical, mental, emotional, intuitive, and spiritual training to positively impact performance on the playing field, peace of mind in the battle zone, presence in a boardroom, and fitness while on the road.\n\nSo what does peak performance look like in a flow state?\n\nThe quarterback sizzling first-down strikes in a championship game\u2014his eyes pierced in concentration, as if in another universe altogether\u2014relaxed and oblivious to the crowd noise and the millions watching on TV; the SEAL team silently executing the Bin Laden raid in cool, fluid precision despite the stress and chaos of the combat zone; the CrossFitter in a fury during the workout of the day (WOD) as she continues to execute pull-ups and thrusters with power and grace, despite the onslaught of pain and fatigue cooking within the muscles and through her lungs. These are just a few examples of the connection between states of mind and performance, where it's abundantly clear that mental preparation, mental toughness, and accessing flow figure into physical performance capacity.\n\nThis door swings both ways: As 2013 World Chess Champion Viswanathan Anand said in an interview with _Complete Well Being,_ \"In chess, since we have to prepare for about 7\u20138 hours [of competition] a day, physical fitness is as essential as mental fitness.\" He added that emotional control also figures into the nature of the peak performance state. \"For me, as it is for everyone, feelings determine performance,\" he said. \"Only when you are happy or feel good about yourself can you play well. I try to keep my life simple, as chess itself is very complicated.\" So, according to the chess master, physical, mental, and emotional aspects all play into peak performance.\n\nIn other words, to generate and maintain top performance, we must supercharge the physical with mental and the emotional aspects of training. This is true whether you are training to be a chess master, a Navy SEAL, or an elite corporate warrior. Another compelling example from the world stage is Japan's Ichiro Suzuki, Major League Baseball's ten-time All-Star. Here are a few details about Ichiro's integrated approach to his sport:\n\n_Ichiro Suzuki painstakingly cleans every piece of his equipment, including shoes and glove, before each game. He has done this for 22 seasons, during an amazing career that has stretched across two continents. He keeps his bats in a humidor-like carrying case._\n\n_His physical preparation is just as thorough. He begins his gameday routine with weight lifting and a series of stretching exercises and yoga-like poses... After games, he records his thoughts on the day in a personal journal._ \u2013RICHARD JUSTICE, \"MODEL OF EXCELLENCE: ICHIRO REACHES 4,000 HITS HIS WAY,\" MLB.COM\n\nAnd this from another source:\n\n_When he gazed in the general direction of the crowd he was staring a hole right through us. He was totally locked in. He was in a place far away, some sort of deep font of performance excellence._ \u2013JUSTIN PANSON, \"ICHIRO, WHAT IS THE MEANING OF LIFE?\" CONFLUENCE STUDIO\n\nWhat Ichiro has is what I call an unfettered mind, which comes from a special form of training. I call that training integrated warrior development. This program is a structured version of such a regimen, in that it integrates the mental, emotional, physical, and spiritual intelligences to prepare for \"battle,\" similar to how Ichiro did in his training.\n\nTRAIN FOR LIFE\n\nMaybe it is abundantly clear by now that Kokoro Yoga is more than a workout. It is training that can meet you where you are at, and stay with you for life. You will change the methods to suit your needs over time. Which poses, functional exercises, sequences, breathing, concentration, and visualization drills you use will depend upon your age, body type, season of the year, and intentions. My hope is also that you appreciate that the training is presented from a Western warrior's point of view. The language is familiar and avoids cultural nuances found in most other yoga training\u2014such as the Sanskrit pose names, Hindu mythology, and chanting. The following core principles define the full experience of our system, each of which will be explored in greater detailed later in this book:\n\nBOTH A PRACTICE AND A LIFESTYLE: Kokoro Yoga can be both a practice, providing powerful tools to aid in your physical and mental training, as well as a set of principles for living an enriched, unbeatable life.\n\nFLEXIBLE AND VARIABLE: Variety is the spice of life and Kokoro Yoga is flexible, providing for maximum variability to meet your specific needs. Doing the same rigidly fixed form year in and year out will lead to a rut, burnout, and injury. Variety is a good thing and can be found by changing up pose sequences for a particular effect, changing the duration based on your time and intention, and also flexing your practice into the times of the day that you train. While taking a modular approach to the practice allows for maximum variety and learning, it can also lead to avoiding things that are challenging and could help you break through to a new level. You will explore this concept as your practice evolves.\n\nINTENTIONAL: Rather than just hitting a yoga class for a workout, we are clear about the intention for our training, and the desired result for each session. For instance, using the Hip Mobility Drill after a CrossFit WOD is intended to open the hips, develop flexibility and durability, calm the nervous system after ramping it up, and immediately trigger the recovery process. Doing a session in the morning when you wake up is intended to facilitate spinal health, stimulate your nervous system, clear your mind, and charge you positively to provide a foundation of excellence for your day. Kokoro Yoga can be performed to enhance athleticism, to develop strong leadership traits, and for spiritual enlightenment. In the case of a warrior, all of these goals are relevant and will be developed with the methods employed in these pages.\n\nBALANCED: Train hard, train soft. Train long, train short. We use yoga to find balance in our bodies, minds, and lives. We seek a balance between effort and surrender, between work and recovery. If you have a bone-crushing CrossFit workout, then you would do a moderate restorative yoga session to balance the effort and energy of the workout. However, if you are on vacation and yoga is your primary training, due to space and time constraints (which is often my case), then balance may mean including a bodyweight WOD module into an hour-long challenging session (see the Fit Warrior sequence in chapter 5)... similar to how I trained in Baghdad.\n\nADAPTABLE: We adapt the pose to our bodies rather than try to adapt our body to the pose. I learned this by contrasting how Ashtanga Yoga is taught versus Viniyoga. In Ashtanga the pose is the pose, and you exert extreme effort to contort your body to the pose structure. This approach, if performed unwisely or when not prepared, can lead to injury. In Viniyoga, the approach is to move into the pose with repetitions to refine the functional movement pattern, and then to stay in the full expression of the pose that is appropriate for you in that moment. We have adopted this sensible model for Kokoro Yoga, where you approach a pose with your own level of flexibility, mobility, functional fitness, and prior injuries that you need to adapt for. Even the time of day and season of the year will influence how our bodies and minds react to a pose, and how we should wisely approach it.\n\nINTEGRATED: Already discussed at length, our premise is that accelerated growth occurs when we embody training in an integrated manner. You will be developing yourself physically, mentally, morally, emotionally, intuitionally, and spiritually. These intelligences are deeply connected and when you actively integrate the training of them they will conspire to unlock your full potential as a human.\n\nFIXED AND FLOWING: We use an approach that embraces both static form and flowing movement. What I mean by this is that we will do some poses with no flow, and others we will flow with the breath between poses, but then stay in some poses for a fixed amount of time or breath cycles to allow our bodies to sink deeper into the pose. This allows for a deepening of concentration and awareness. In addition, there are some flowing poses that are mostly about the breathing technique that has a particular effect on your body or mind. These poses look more like Tai Chi than traditional yoga. Keep in mind that yoga, in its original form thousands of years ago, was a warrior and spiritual development training method that has deep connections with Tai Chi, Qigong, and other martial arts as it spread west from India into Tibet and regions now known as China, Korea, and Japan.\n\nTHE RESULTS\n\nIndeed, Kokoro Yoga is designed to do far more for you than help you work up a sweat. It is about developing a warrior spirit, merging your heart and mind into your actions so you can achieve your maximum human potential. In the next chapter, we'll take a thorough look at the strategies that allow for the integration of the training to work its magic.\n\n# CHAPTER 3\n\n# THE STRATEGIES\n\n_Happiness is when what you think, what you say, and what you do are in harmony._  \n\u2013MAHATMA GANDHI\n\nNAVY SEALS RELY ON BOTH STRATEGIES AND TACTICS TO ACCOMPLISH THEIR MISSION. STRATEGIES ARE THE OVERARCHING METHODS EMPLOYED TO DIRECT THE ENERGIES, FOCUS, AND RESOURCES OF THE TEAM. TACTICS ARE THE SPECIFIC METHODS THE TEAMS WILL EMPLOY TO GET THE ACTUAL JOB DONE. IN THIS CHAPTER I WANT TO SPEND TIME NOW ON THE CORE STRATEGIES WITHIN THE KOKORO YOGA PROGRAM\u2014WHY WE DO THINGS A CERTAIN WAY. YOU'LL NOTICE THAT the strategies are linked to the wisdom of the original Eight Limbs of yoga, informed through my personal experience training thousands of modern-day warriors since 2007. With an understanding of the strategic engines firmly established, we'll then take a look at the ground-level tactics in chapter 4: the nuts, bolts, and basic methods of how we execute these strategies.\n\nSTRATEGY 1: DEVELOP A PERSONAL ETHOS\n\nExcellence outcomes in life are built upon a personal philosophy, or code, of excellence. I call this a personal ethos. Defining this ethos requires deep introspection and skillful methods leading to a continuous pursuit of self-knowledge and growth. Training without a personal ethos can leave you directionless, not able to answer the question, _Why?,_ when faced with life's many challenges.\n\n\"The mass of men lead lives of quiet desperation,\" wrote Henry David Thoreau, exposing a truth that holds true today. Finding stillness to eliminate the noise and distractions that obscure the voice of our soul from being heard is not easy, but crucial for us to find and fulfill our purpose. Rather than reacting like a pinball to the myriad of forces and stressors rushing at us every day, we must take time to question deeply what we are meant for, what we are passionate about, and how we can serve meaningfully through our unique gifts.\n\nWe must question what societal and family beliefs make the best fit, and begin to orient ourselves to universal laws and positive (love) energy so that we can allow our uniqueness and beauty to flow out of us like a river. We start this process by asking deep questions, working to separate from ego identification with our roles, challenges, issues, and obsessions. As we gain clarity on the Yamas and Niyamas introduced in the last chapter, we connect deeper with our witnessing self, your \"soul self.\" In deep connection with your witness you are said to act from your \"True Self\" as opposed to your \"False Self,\" which is guided by your ego.\n\nA good place to enact your code of conduct is to examine and align with the five precepts called Yamas, that lead the way to excellence and are a contemplation as well as an ethical practice. Aligning with these precepts will produce a profound grounding effect on your life. These are codes of restraint and self-regulations that have us acknowledge our interconnected relationship with all sentient beings and the environment, and to align with them in a spirit of harmony. They are:\n\n1. Nonviolence to oneself or others. This code of conduct poses a challenge for warriors. Keep in mind that duty plays a role, as Arjuna learned.\n\n2. Speaking, thinking, and acting truth.\n\n3. Not hoarding unnecessarily. Being honest with what belongs to you, what belongs to others, and what belongs to the common good.\n\n4. Balancing your energy in work, play, and relationships.\n\n5. Not grasping for or getting attached or clinging to ideas, material objects, or relationships.\n\nThese five precepts, when asked as questions, provide a powerful guide for our actions and thoughts. Establishing a code that is grounded with these five precepts will lead to a balanced and positive relationship with all stakeholders in the great web of life. The code in action is your integrity. They are to be practiced in thoughts, words, and deeds. Becoming aware of how you live the precepts is important. How do you act truthful? How do you speak truthfully? How do you think of truth? Truth is different in the domains of action, speech, and thought. To have integrity means they are aligned.\n\nWe next want to examine and refine our habits or Niyamas, which ground the journey toward personal mastery. These are:\n\n  Developing purity and control of the body and mind. We accomplish this with proper fueling and a daily practice of functional fitness and asana, or some other skillfully executed method of somatic movement training.\n\n  Contentment of one's circumstances in life. We are where we are for a reason, so be okay with it while working methodically to improve your own condition and the condition of the world... one relationship at a time.\n\n  Taking control over your desires, the constant grasping at pleasure and striving to avoid pain. Neither pleasure nor pain is good or bad on its own, but it is the craving or avoidance that makes it so. Consider the avoidance of the pain associated with intense physical training. Choosing to avoid this kind of temporary discomfort means that we won't develop our warrior body and mind. Or being constantly drawn to the pleasure of sweets. Consistently folding to this weakness means that we sacrifice control and we slowly kill ourselves with sugar.\n\n  Self study and study of sacred texts. This is one aspect of the deeply spiritual component of yoga. It's about what we choose to focus our mind upon, whether our internal states of mind and emotion, or a parable of Jesus, we come to understand deeply. Our awareness of reality and knowledge of the nature of things increases.\n\n  Surrender to a higher power. This is your concept of God, whether you have a religious orientation or not. If not, Gaia, the Tao, or \"Spirit that runs through all things\" will do just fine. Let go of the mental grasping and contracting of ego self, and surrender to a \"now\" presence where you can connect to this power. This practice will lead to more energy and keep you in alignment with your True Self.\n\nThe discipline habits are meant to tame the body, mind, and senses so that we can advance our training through concentrated study and meditation. Similar to the first five precepts, they characteristically arise in thought, speech, and action as if there were three elements of practice needing to be trained together. For instance, do you think you are content? Do you talk about contentment and do you act content? Upon deep reflection you may find that there are subtle differences in each of these, and this awareness leads you to work toward their alignment.\n\nWrite down any insights that come to you as you consider your relationship to the above precepts and disciplines. Review them weekly and note what comes up in your silence practices. You will be developing a powerful personal ethos through this process. Ultimately you want to always be able to answer the following questions: Why am I doing this? Why is this happening to me? Am I aligned with my ethos?\n\nThis level of self-study naturally draws us even deeper inward, to places we previously didn't know existed. This is where we must travel to and spend time exploring the root answer of the questions about your reality. When you can answer the question, Why?, you can better answer the questions: What am I going to do about this? How and with whom? Of course at the heart of these questions lies the one question that is the very essence of what it means to be human:\n\nWho am I?\n\nTaking time to settle into these ponderous issues is the next step in the awakening process of the first strategy. These are questions stoked with power; leave them unasked, or unanswered, and you're sure to drift along as I did in the first 24 years of my life. Asking them, and reflecting on what comes up in silence, will slowly but definitely lead to the construction of a deep personal ethos. As daunting as this may seem, I want you to know how rewarding and dynamic this process is, and how it opens you up like a flower awakening to the dawn. The answers you have to these questions today will evolve over time as you continue to work the program. They will inform the choices you make, the projects you take on, and the values you establish. You will cultivate honor and integrity as you align your actions with your growing sense of ethos and purpose in life. As you become more comfortable with this insight work you will gain the traction and energy that comes from clarifying your identity and knowing what you stand for.\n\nThe development of your personal ethos counts on silence to be your guide. We plant the questions in our minds and trust the daily rituals to draw answers up from the depths, as if through a magnetic force. New insights about the questions and deeper awareness will bubble to the surface and reveal themselves as insights, new ideas, feelings, or images.\n\nThe following are the core questions to ask before your practice sessions, particularly the morning and evening rituals (discussed in chapter 8):\n\nWho am I?\n\nWhy am I here\u2014what is my purpose?\n\nWhat unique passion can I offer the world?\n\nWhat and whom do I love?\n\nAm I positive and helpful?\n\nHow can I boldly serve my family, team, country, world?\n\nWhat do I value and want to bring more of into my life?\n\nWhat do I want less of in my life?\n\nA great example of a morning ritual would be to ask these questions and follow with box breathing, a short yoga sequence, and then to meditate on aligning your personal ethos with Yamas and Niyamas. Over time you will be able to articulate the answers to the above questions clearly. You will know beyond a shadow of a doubt just who you are right now, where you're at in your life, and where you need to go. It's from this rich material that you will build your powerful and driving vision for your life. This first strategy of developing a personal ethos is critical to living the warrior's way, so don't shortcut it or think you already have all the answers. Humility is to admit that you do not know everything... and in fact, the more you learn, the less you seem to know about the mysteries of the world! Now on to the next strategy: to develop body awareness, control, and optimal physical health.\n\nSTRATEGY 2: FUNCTIONAL CONDITIONING\n\nThe second strategy is to optimize your functional movement capacity. We do this with classic \"asana\" poses intelligently sequenced, as well as with mindfully performed functional exercises. One of the paradigm shifts we've seen in the last decade has been an embrace of functional fitness. In Kokoro Yoga, we rely on traditional poses of yoga\u2014those you see in any yoga class. But we also rely on functional movements of CrossFit and SEALFIT to close gaps, or weaknesses, in physical conditioning found in other yoga systems. This does not mean you have to become a CrossFit or SEALFIT athlete to do this program. The movements are simple and scalable for all practitioners. It simply means that this yoga could be your complete integrated training system for both \"working out\" and \"working in.\" Functional movements are comprised of exercises that involve compound movements and universal motor recruitment patterns\u2014like push-ups, burpees, and squats\u2014as opposed to traditional gym equipment. The term \"universal\" refers to the natural use of full-body patterns that mimic physical movements in our daily lives, such as digging a ditch, shoveling snow, hiking in rugged terrain, and chopping wood.\n\nThe conditioning modules we present get quick results by performing functional movements with high-intensity, building cardiovascular endurance, strength, stamina, and power. This work also leads to deep awareness of the body and greater confidence in the performance of work and play outdoors.\n\nThe functional movements we depend upon for the Fit Warrior and Protective Warrior sequences are classic body-weight gymnastics and \"cardio-kickboxing\" movements. Having said that, on occasion kettlebells, dumbbells, or sand bags can be utilized to augment the impact of the training. The work is performed in the context of the yoga sequence as an embedded module. Doing the yoga then going to the gym is old school. If you do not have a functional fitness regimen at the moment, then this program will get you on your way by performing three rigorous sessions a week, using Fit Warrior and Protective Warrior sequences. This program will deliver excellent functional fitness and health, all geared toward supporting you on that journey to excellence.\n\nFIVE LAYERS OF BEING\n\nBefore moving on to the next strategy, this is a good place to insert some comments about the five dimensions of the self. These were called the \"pancamayas\" (five pervasive layers) in the ancient Vedic text Taittiriya Upanishad wherein the Rishis introduced ancient yoga philosophy to the world. These dimensions are aspects of our being to access, align, and develop in a holographic manner through the science of yoga. Modern science is making progress understanding the first three, but lacks credible means to penetrate the latter two, which are the realm of deep interior development.\n\nTHE PHYSICAL DIMENSION\n\nThe outermost of the layers is literally called the food layer by the ancient yogis. This layer is built by food and can become food if we find ourselves in the jungle alone. It is the aspect of our beings that is trained with the second strategy of Kokoro Yoga, functional movement. But it also requires effective fueling and sleeping habits for the care of and nurturing of the body to allow us to fulfill our purpose and live in happiness. When we take care of the physical being we are able to journey inward without our bodies being distracted with pain, injuries, or disease, such as when we sit in meditation. The physical body has been the focus of Western fitness, but has lacked much acknowledgment of the other layers. In Kokoro Yoga we strive to train and develop all layers simultaneously for optimal effect.\n\nENERGETIC DIMENSION\n\nThe next dimension is the vital, energetic layer. This is the vital force that turns the lights on in our human house. The energy is produced internally through metabolism, and is also brought in from the outside through the breath, thoughts, and our environment. Kokoro Yoga seeks to refine and train this level of our being for optimal energy flow through pathways in the body called Nadis. Through Pranayama, or breath control training (i.e., box breathing), we get intimate with the energy body, explore it, and roam further inward to additional layers.\n\nMENTAL DIMENSION\n\nThe next dimension is the mental. This level is for cognizing, learning, and learning how to learn. In this dimension we take in information from our senses, process it as thoughts and emotions, and then seek meaning in the world. The mental aspect of our being is the judger, feeler, and supervisor of our lives. In modern terms this would be the rational mind, centered in the neocortex. The mental being is tuned to sniff for danger and opportunities (in fact all inbound information is initially processed through the amygdala, the main purpose of which is to discriminate between things to fear and not fear).\n\nNegativity can prevail in the mental layer with obsessive, negative associations and weak thinking\u2014a disease of our modern world. Through the practice of Kokoro Yoga we strive to gain control over this crucial dimension, train it to be positive and uncovering truth, then to connect with the next dimension, the wisdom of personality. As this happens the mind is cleared of doubt and illusion and we can connect with our true self\u2014that aspect of your mind that receives clear reflection of reality from deep wisdom. A fully developed mental dimension allows us then to express our uniqueness fully, uncluttered by fear or deception.\n\nWISDOM DIMENSION\n\nThe next is the wisdom dimension, that aspect of our being that can discern an innate, felt knowingness of purpose and drives your deeper, \"true self,\" or authentic, personality. Excessive thinking in the mental dimension can get in the way of accessing this wisdom layer. The deep wisdom that lies in all of us and is found by cultivating sacred silence. This dimension is felt in our heart and belly regions and discriminates between good and bad, useful and not useful. Deep conditioning, patterns, and habits are grooved into this layer over this lifetime, and possibly even past lives.\n\nWhen accessed, the wisdom dimension brings us into deep connection to others, all of humanity and the earth. The sensations from this layer are beyond thoughts\u2014and are always positive. The wisdom dimension is \"the courage wolf\" that lies within us, that generator of love in all of its forms. Through our integrated training we can connect with our wisdom, dimensions, eradicate negative patterns, and uncover the love that lies within. Then we are empowered to go even deeper within, in search of the eternal center of bliss consciousness.\n\nBLISS DIMENSION\n\nThe bliss dimension is the most interior of the layers and it connects to the seat of consciousness itself. When connected to, this layer is experienced as radical bliss, peace, and joy beyond the mind, independent of any stimulus that may trigger a happy or blissful mental reaction. In short, it is your capacity for happiness without any extrinsic reason for being happy. The ancient yogis said that the bliss experienced as this dimension is accessed is 1,000 times more powerful than the highest form of mentally experienced joy. Wow.\n\nIn the silence of deep meditation, in service to others, and in moments of total presence in a purposeful, passionate pursuit, this bliss is released. We are experiencing a connection to the center of our soul. Hard to put into words, but it is called different names by various traditions (i.e., Atman, Christ Consciousness). It is our eternal center that was never born and never dies. Accessing the bliss dimension is the direct experience of a long-term integration practice and it will unlock our full potential as it allows our unique beauty to express itself fully.\n\nSTRATEGY 3: MENTAL AND EMOTIONAL CONTROL\n\nThe second Sutra of Patanjali is \"yogah citta vrtti nirodah.\" This means that yoga (integration) comes from the ability to concentrate the mind until all thoughts are under one's control. Then we can direct the mind exclusively toward an object or subject and sustain that without any distractions for as long as desired. This speaks to the power of yoga for forging mental and emotional control. If we consider that emotions are primal and stored thoughts experienced not as words in our heads, but as feelings in our bodies, then we can surmise that Patanjali included emotional control in this definition.\n\nOne of the defining qualities of being human is our gift to experience feelings that lead to meaning. This doesn't happen by just thinking something over: rather, it's the connection between real experiences, the thoughts that come with those experiences, and the emotions provoked by those thoughts. This is the space where we have an opportunity to cultivate deep learning and connection. It is also the most overlooked and undertrained area of development for many of us.\n\nThoughts and emotions are part of the mental dimension. Through concentration, meditation, and visualization practice we seek to manage our mental and emotional selves so we can direct thoughts and emotions positively and intentionally toward fulfilling our personal ethos. An example is how we deal with fear. When we experience fear such as with a life-threatening situation, through our training we learn to face the fear and transmute the mental and emotional energy of it into a more positive and focused energy (such as determination), which will help to survive and end the threat. Navy SEALs are modern yoga practitioners in that they are masters at this form of mental and emotional management.\n\nIn this strategy we seek a level of control over our lives that is uncommon in our busy, modern world. Your concentration, focus, and overall awareness will improve as a result of the daily practice of the integrated training. With this deeper connection you will have at your command greater control to detect and direct your thoughts and emotions in a purpose-filled direction.\n\nSTRATEGY 4: BREATH MASTERY\n\nI learned during my SEAL training just how powerful mastery of the breath could be. I am confident that this one strategy led to my big success there, and later as a Special Ops and business leader. It is now the first thing I teach to aspiring Navy SEAL candidates at SEALFIT. Breathing is a unique system in the human body; unlike, for example, digestion, breathing has both an involuntary control mechanism as well as voluntary, similar to a 747 in that it can shift between being piloted and being left on autopilot. If you are a military special operator, or have a background with martial arts or yoga, it's likely that you inherently understand the strategy of mastering the breath. If not, you will find the most immediate practical benefits in your life from this strategy. When you begin the box breathing practice you will note an immediate impact in the form of lower stress, a heightened, alert focus, and a calmer mind. Strategy 4 strives to accomplish the following three aims:\n\n1. First, air is our primary source of fuel so learning how to get the most out of each \"bite\" can lead to many health and longevity benefits (see chapter 7 for more on the benefits). Most people are unaware of this simple truth, as well as of their own breathing patterns. The average person is accustomed to panting some 25,000 shallow breaths throughout the day, or 16 to 20 breaths per minute. If this is you, then you should know that in doing so you are using just a fraction of the natural lung capacity you are born with, and you are keeping your body in a constant state of agitation in the process.\n\n2. Controlling the breath controls our stress, or fight, flight, or freeze response. Shallow, choppy, and irregular chest breathing stimulates the fight-or-flight, sympathetic nervous system, ratcheting up urgency and stress. A calm, deep, and measured belly breath, on the other hand, stimulates the rest and digest, parasympathetic system, dialing down stress, slowing the heart rate, and calming the entire body down.\n\n3. Breath control is a bridge to mental control. When I learned as a martial artist to control my breathing during intense fighting bouts it proved invaluable later during the chaos of combat. As noted above I found that it both calmed my body and mind\u2014bringing them immediately under control, allowing me to focus on my targets and make good decisions.\n\nTo help achieve these aims, you'll be performing breath control exercises in the training sessions. These exercises will link breath to the movements. You will note in the sequencing that we specify when to breathe in and out, so that you ride the breath like a surfer riding a long wave. Over time you will cultivate greater energy, the same energy that martial artists refer to as \"chi\" and yogis refer to as \"prana.\" We will get into the practical steps for mastering the breath in the next chapter on tactics.\n\nSTRATEGY 5: SENSE MASTERY\n\nWhen I trained in the martial art of ninjutsu, I spent a lot of time developing what we called sensual awareness. No\u2014it wasn't so we could be better in bed (though it didn't hurt). Rather this was about developing the sense \"doors\" by opening them to receive more information. Through this training I became more nuanced in interpreting the received information and making more sense out of it. Was that tingling sensation a signal of danger? Was the ripple in energy someone entering the room, disrupting my energy matrix? Many warriors who have seen combat have experienced enhanced sensory awareness as an adaptation to the risk.\n\nHowever, when the warriors get back to \"normal life\" they note that those sense doors seem to close again. I believe that we can train to keep these doors open so that our intuition is always clear and accessible\u2014allowing us to make better decisions and be more effective warriors and leaders.\n\nIn performing the poses and flowing between poses, you are meant to focus your mind and senses inward. For example, you'll want to tune down your sense of hearing and narrow your sight to a single-point focus. Or you will soften your gaze or close your eyes altogether and focus on the sound of your breathing. Or perhaps you will let your inner awareness rest on a point in the body, such as a joint, muscle, or energy center (cakra). Intuition, as intelligence, is not experienced verbally, but sensationally and via imagery. Therefore, within this work of mastering the senses, we will be developing the language of imagery and sensations, rather than words. This is difficult to convey in a book, of course. It is your job to constantly scan your body for sensations, insights, and images from your intuition. Turning the senses inward during practice helps us to tune and expand the senses more fully for our overall awareness of the nature of self, others, and the world.\n\nSTRATEGY 6: INTEGRATION\n\nThe above five strategies work together toward supporting the final strategy that I call integration. Kokoro Yoga drives toward achieving an integration and union of the five dimensions as well as our True Self with our clarified egoic self. But what does this integration feel like? At the external \"service\" level it is experienced as effortless success, meaning that the entire world seems to conspire to support your mission and purpose in life. At the internal \"self-mastery\" level it is experienced as a world-centric consciousness, peaceful and mastery over your mind and emotions. The Japanese concept of _shibumi,_ \"effortless perfection,\" is a nice way to express this experience. You will effortlessly perfect your trade or art, as you are meant to with your unique and innate skills and talents, and in doing so you will find great joy in your work.\n\nIntegration allows you to unlock a flow state at will. When I train this with long-term coaching clients the process is greatly accelerated due to the intensity and enmeshment of all the senses in the training. Students learn how to turn on and off flow through the big four skills of breath control, positivity, visualization, and micro-goals. Flow is the psychological term coined by Mihaly Csikszentmihalyi, a psychology professor at the University of Chicago and author of _Flow: The Psychology of Optimal Experience._ In his book, Csikszentmihalyi defines this flow as a state of absorption with the moment and with what one is doing, and he makes an acute comparison to yoga: \"The similarities between [integration in] yoga and flow are extremely strong; in fact it makes sense to think of yoga as a very thoroughly planned flow activity. Both try to achieve a joyous, self-forgetful involvement through concentration, which in turn is made possible by a discipline of the body.\"1\n\nAs he later said to _Wired_ magazine, \"The ego falls away. Time flies. Every action, movement, and thought follows inevitably from the previous one, like playing jazz. Your whole being is involved, and you're using your skills to the utmost.\"\n\nCsikszentmihalyi notes that the relationship between the difficulty of a task we take on, and our training or experience in performing said task, impacts our ability to remain present and access flow. The ideal is to have the challenge slightly greater than our training level\u2014so pushing the envelope with training, then pushing the envelope performing, has a positive upward spiral effect on performance, at both a personal and team level, because it creates the conditions for flow to \"present\" itself.\n\nSteven Kotler writes in his book, _The Rise of Superman,_ that there are also several environmental, psychological, social, and creative triggers for flow to occur. Deep embodiment is one of the environmental triggers. That means being able to pay attention to multiple sensory inputs simultaneously. Time becomes relative\u2014there is no concept of past or future that might otherwise be dogging us in the form of regret or worry.\n\nNow, with these six overarching strategies in mind, I'd like to run through the tactics in the next chapter\u2014the essential tools and techniques you'll be employing to effectively train with this program.\n\n\u2014\u2014\u2014\u2014\u2014\u2014\n\n1Mihaly Csikszentmihalyi, _Flow: The Psychology of Optimal Experience_ (New York: Harper & Row, 1990), 105.\n\n# CHAPTER 4\n\n# THE TACTICS\n\n_A gem cannot be polished without friction, nor a man perfected without trials._  \n\u2013CHINESE PROVERB\n\nWHEREAS A STRATEGY IS A HIGH-LEVEL ORIENTATION OR PLAN FOR HOW TO ACCOMPLISH A MISSION, A TACTIC IS A GROUND-LEVEL METHOD, TECHNIQUE, OR PROCEDURE FOR MOVING THE DIAL IN THE CHOSEN STRATEGIC DIRECTION. FOR INSTANCE, IN SPECIAL OPERATIONS THE STRATEGY WOULD BE TO USE SPEED TO SURPRISE THE ENEMY. A TACTIC THE SEALS USED WAS TIME-SENSITIVE PLANNING WHERE WE PLANNED LITERALLY ON THE FLY AND COULD BE ON TARGET swifter than any other SOF unity. For our purposes here, with the strategy of mastering the breath, one tactic we will employ is box breathing, and another will be connecting breath with movement. Strategies without solid tactics lack punch. Tactics without an effective strategy don't get us through to mission accomplishment. We have ten core tactics that are employed to ensure our strategies put us on target.\n\nTACTIC 1: SEQUENCING AND ADAPTATION\n\nThe sequencing in Kokoro Yoga refers to the order of poses in a given training session. By adeptly choosing and sequencing a series of poses for a purposeful session, along with variables such as how long you hold each individual pose and the breathing patterns used in the session, you can achieve different physical and mental results supporting your purposes. For example, a guiding purpose of the morning ritual is to stimulate your nervous system and enliven the spine so you are ready to move into your day alert, energized, and feeling powerful. Therefore, this practice includes poses, a sequence, and breathing patterns different from what you would perform in the evening ritual, where the intention is to move toward calmness and rest.\n\nPart of my personal education in teaching yoga has included many hours in learning effective sequencing from a master, Gary Kraftsow. It's an art that's beyond the scope of this book, so we've created programming for a number of common-use scenarios, including the morning and evening practice, pre- and post-workout practice, stress management (Peaceful Warrior), functional fitness (Fit Warrior and Protective Warrior), warriorlike focus (Zen Warrior), and warrior energy development (Jedi Warrior).\n\nLet's talk more about the principle of adaptation. I don't subscribe to the belief that the body must be contorted into some ideal pose. There's no demand being made that you have to funnel your body into the dream lotus position, or be able to execute a headstand that rewards you with a perfect 10 score. I have seen such a mind-set lead to frustration, injury, or both. Let us reverse that thinking. Rather than adapting your body to a pose, adapt the pose to what your body is ready for and capable of. I'd like you take into account who you are today and what you are dealing with as you approach your practice. Do you have old injuries that need to be considered? Most of us do. What about mobility restrictions? I'd be surprised if you don't have any. How does your age and experience impact your pose? All of these factors, and more, count. Please do not use this as an excuse to avoid riding the edge a bit, but know that it is fine not to be \"perfect\" and to leverage support tools, if necessary, like a folded blanket or a yoga block, to assist you into getting into a version of a pose that's right for you, at that moment.\n\nTACTIC 2: CULTIVATING STILLNESS\n\nThere are several ways I want you to think about the tactic of cultivating stillness. At the physical level, let's picture you in the midst of performing a sequence of poses in your morning ritual. You will transition from pose to pose with a flowing style of movement. But in the full expression of the pose, we will take a few moments, measured with the breath, to practice stillness where we focus on being mindfully present. An example is when we hold the Downward-Facing Dog (aka Downward Dog or Down Dog) position for 8 or 10 breaths.\n\nAnother way to employ this tactic is the more advanced skill of finding stillness while moving. This is a place where we refrain from any mental and emotional distractions and sink our awareness into the movement itself. We aren't wondering about our ability to do the pose, or checking out the other students. Our mind is fully merged with the action; subject and object become one, experienced as embodied presence and stillness while moving. This deep internal awareness helps to unlock that flow state described earlier. A third way that we practice this tactic is in seated box breathing, visualization, and meditation drills. Even though our bodies are seemingly in a still position, there's still a lot of motion (i.e., our breathing patterns, fidgeting, and thinking of things you don't want or need to think about). During these sessions we want to pay attention to the motion in body, mind, and emotions, and strive to move toward stillness.\n\nTACTIC 3: BREATH CONTROL\n\nBreath control training will change your body and mind in remarkably positive ways. It will tune your nervous system and allow you to activate the parasympathetic or sympathetic nervous systems at will, helping you to perform in a stressful environment or to excel in competition. It will also place in your toolbox an immediate action drill to calm and focus your mind when necessary. Breath control training creates another \"aha\" moment for all of my students. It is so powerful and impacts you so quickly that you will wonder why such an important skill was kept from us our entire lives. I have selected some simple, yet profound, breathing practices to jump-start your training.\n\n3-Part Breath\n\nThe first is to retrain the way we breathe from the ground up, or should I say the belly up. We will segment the breath into three parts to develop a new pattern of deep diaphragmatic breathing, one that utilizes our entire lung capacity. I call this the \"3-Part Breath.\" To begin, place your hand on your belly, and take a deep inhalation through the nose. Picture your belly as if it were a balloon, and as you breathe in, inflate the belly balloon. Stop when your belly is fully extended, before you activate your diaphragm to deepen the breath even further.\n\nNow when you exhale, blow all air out from the balloon through your nose while drawing the belly button back toward your spine; this will help you expunge all the remaining air. Repeat this first step 3 times. Okay, after 3 cycles, on the following inhalation, fill the belly up with air as in step one, but when the belly feels full, use your diaphragm to draw in more air, expanding the rib cage. Now when you expunge the breath, start with the air in the rib cage by allowing the ribs to compress back together, and then expel the remaining air from the belly. Repeat this second step a total of 3 times. The final stage of the 3-Part Breath is to once again fill the belly and activating the diaphragm to bring air into the rib cage, then you're going to top things off by expanding your rib cage and filling your chest all the way up to the collarbone with more air. Your upper chest will rise with expansion; the lungs will be completely full. You will note an energizing effect from the pressure. The sequence on the exhale will go from the top down: start by expelling the air from the upper chest, then the rib cage, then the belly. Notice the flow of relaxation that floods into your body on the exhale. Use this drill a couple times a day until you get the hang of breathing this way normally.\n\n1:2 Breath\n\nThis breath control pattern is super simple and has the effect of calming you down and slowing your heart rate. This breathing technique uses a long, slow exhalation that triggers the parasympathetic, rest, and digest system into action. It counteracts your fight-or-flight system, which will calm down after just a few cycles. It is also a nice drill to begin to learn to control the duration of your inhale and exhale. It helps to practice this before you start to add holds such as with the box breathing drill.\n\nThe process is simply to exhale twice as long as you inhale. For instance, as you inhale through the nose using the full belly breath to a count of 3, you exhale out to a count of 6. Over time, as your skill increases, you can take it to a 4:8 or 5:10 and so on. To yield some dramatic improvements with the 1:2 Breath, try a 30-day challenge of practicing it for 10 minutes each day. This is a great breathing pattern to use when in a stressful situation to gain control over your body and mind.\n\nBox Breathing\n\nThe third breathing tactic will add a retention of the breath after inhale and exhale. This is my favorite breathing practice due to the powerful effect it has after only a short time of practicing. I gave it the name box breathing when I started doing it back in 2002 because of the four-sided pattern of the practice. It is something you can do anywhere and anytime you are not performing a highly complex task. I practice it in my morning ritual, before a workout, while standing in line, stuck in traffic, and whenever else I can. Along with training more powerful breathing musculature, it slows down the rate that you breathe and deepens your concentration skills. When you perform box breathing, even for 5 minutes, you are left with a deeply calm body and an alert, focused state of mind.\n\nTo begin the practice, expel all of the air from your chest. With empty lungs, retain this state for a 4-count hold. Press all the air out and then perform your inhalation, through the nose, to a count of 4. With the lungs full, hold for a count of 4. When you hold the breath, do not clamp down and create back pressure. Rather, maintain an expansive, open feeling even though you are not inhaling. When ready, release the hold and allow the exhale to flow out smoothly through your nose to a count of 4. This is one circuit of the box breathing practice. I recommend you do it for a minimum of 5 minutes, and no more than 20 minutes. I have found that the best approach is to do a single, dedicated practice of 10 to 20 minutes a day, then do a few 1- or 2-minute \"spot drills\" as opportunities present themselves during the day. Box breathing with this 4-4-4-4 ratio has a neutral energetic effect: It's not going to charge you up or put you into a sleepy relaxed state. But it will, as mentioned, make you very alert and grounded, ready for action. As your breathing threshold improves you can increase the duration of the ratio, such as 5-5-5-5 and so on.\n\nThreshold Training\n\nWould you like to increase the threshold of your lung capacity? Your breathing threshold is the duration of a complete breathing cycle, including any holds. It is determined by a few factors, such as the power of your breathing musculature, your lung capacity (volume of air they can hold), and the efficiency of gas transfer dictated by the strength of your heart's cardiac output (heart rate and stroke volume). If you have a threshold of 4 seconds then you are breathing 15 breaths per minute. Up to 20 cycles per minute is common, but don't read that as healthy. After years of practice, I now average 4 to 6 breaths per minute when not paying attention, and have a threshold of 55 seconds for a single breath, meaning I don't get winded or agitated with a 55 second breath count over a minimum of 12 cycles. The yoga masters believed that the human life span was determined by the number of breaths taken. According to them, slowing your breathing down would lengthen your life span. I don't know about you, but I am in\u2014at 4 breaths per minute versus 16 I should live 4 times as long! At any rate, my experience is that slowing your breathing rate down is very healthy and I think you will find this to be true for yourself as well.\n\nLet's begin threshold training with a ratio of 1:2:2:1. So a 3-second inhale, 6-second hold, 6-second exhale, and a 3-second hold is a 18-count breath threshold, assuming you can repeat this cycle comfortably a minimum of 12 times in a row. This would be just shy of three breaths per minute, which is a good target to work toward. For those who need to really extend your threshold, such as SEAL trainees, you will want to work toward a 60-second threshold. Note that you will be able to hold your breath longer in one-off situations when you prepare with sustained deep breathing and meditation. **Warning: never practice breath-hold training in the water alone.** I know it sounds obvious, but some special ops trainees have foolishly tried this and are not alive to read this book as a result.\n\nTACTIC 4: FUNCTIONAL CONDITIONING\n\nThis tactic is one of the unique aspects of Kokoro Yoga\u2014where we combine simple functional movements from SEALFIT training into an asana (pose) practice. This combination truly does turn the yoga session into an integrated workout. We should consider traditional yoga poses to be functional movements, (when done properly) but adding a module of interval exercises to your training session, such as with the Fit Warrior sequence, will build work capacity, strength, stamina, and durability. You may recall from chapter 1 that this is exactly what I did in Baghdad\u2014adding 20-minute circuits of burpees, push-ups, jumping jacks, and Mountain Climbers in the middle of my yoga session. Another quick example: add a sequence of 20 sets of 5 push-ups, then 10 sets of 10 squats, and 5 sets of 20 sit-ups to break up your overall routine. In Appendix A, I share more sample functional workouts that you can use to employ this tactic. If this training inspires you and you have not been exposed to SEALFIT, I encourage you to check out my companion book, _8 Weeks to SEALFIT,_ to learn a more challenging functional training regimen.\n\nTACTIC 5: COMBAT CONDITIONING\n\nThis tactic brings yoga back to it's martial roots. I believe that a warrior must have a basic understanding of how to defend oneself. The foundation for self-defense is your mind, but it also requires the ability to make a weapon out of your body. Then you strike a desired target on another human being with those weapons if the need calls. Though Kokoro Yoga is not meant to be a martial art in the traditional sense, this tactic provides a fun and effective means to develop your punching and kicking weapons. You will get your heart rate up for a sustained period and break a good sweat, and you don't need a black belt to add this to your routine. If you can throw a punch and launch a kick, I hereby declare you ready to include the combat conditioning tactic to your practice.\n\nA sample session would look like this: After a sequence of standing poses, you will insert 10 snap kicks with your right leg, then 10 left, followed by 10 punches with your right fist, and then 10 punches left. Next, you move on to some more complex combinations, or go back to your asana, then do it again a few more times during the training session. The possibilities are endless while being very empowering and fun. In Appendix B you will find more details on the movements and potential combinations.\n\nTACTIC 6: AWARENESS AND RELAXATION\n\nAwareness and relaxation work comes at the end of the session. With this tactic, your whole focus will turn inward with a full-body scan and breathing exercise that will prepare you for a meditation and visualization to mark the end of a session. The relaxation will help to integrate all of the benefits of the training session. As we do this, our internal sense of awareness deepens, leading to more sensitivity, composure, and depth. Recall the first strategy of developing a personal ethos? Well, deepening awareness is one way to gain insight into our deepest yearnings, leading to important growth...when we are ready.\n\nThis particular tactic is usually performed while in the Resting Pose (aka Dead Man Pose), though it can also be done in a seated meditation pose. With your eyes closed, you'll scan your body from toes to head, searching out pockets of tension, discomfort, and lingering emotions. The idea is not to judge these items but rather tune into each with neutral awareness. I should mention that this tactic is also a great place to work on your emotional development. As you scan you may notice emotions, and can take a few moments to explore the emotion in a detached way, examining the roots of the emotion. Now shift to a seated meditation position and move your awareness up your body, take time to pause at each of the six energy \"cakra\" centers:\n\nROOT: located at the base of your spine, associated with security and groundedness.\n\nSACRAL: located about 3 inches above the root in your lower abdomen, below the navel, near the sacrum, associated with creativity.\n\nBELLY: located just behind the belly button, associated with feelings of personal power.\n\nHEART: located near the solar plexus and heart, associated with empathy and connecting to others.\n\nTHROAT: located in the throat region behind the Adam's apple, associated with sincere communication.\n\nBROW: the third eye, so to speak, in the center head above cervical spine, associated with insight and wisdom.\n\nCROWN: located at the top of the head, associated with universal connection and intelligence.\n\nAfter this, take time to be aware of the surroundings and connect with the energy all around you.\n\nTACTIC 7: CONCENTRATION AND MEDITATION\n\nConcentration and meditation form another crucial tactic, and though they may sound one and the same, from the outside, there is a distinct difference in what is going on inside. Concentration is the skill of being able to maintain deep focus on one thing for long periods of time. It is one of the skills you are working on during your pose sequences and breathing exercises, where you are absorbed in a single, specific thing, rather than letting the mind run. As mentioned, daily asana and box breathing drills are tremendous at improving your concentration. When you are concentrating, you are deeply engaged in observing the object of attention, so that you can maintain that focus for long periods of time without distraction. Concentrating with this depth is particularly useful for creative work, reading, problem-solving, or observing a target. In concentration your eyes will be focused intently, and your center of consciousness is in your critical thinking mind. Even if just practicing box breathing, you are focused intently on the breathing pattern, the quality of the breath, and the experience itself.\n\nMeditation differs in that you will observe an object, a mantra, or some concept with complete absorption and presence, not using your critical thinking mind, rather allowing for direct perception to occur. Ultimately you will sense a merging with the object of your attention. This is a good place to mention that your mind works in five distinct ways: critical thinking (judging, analyzing, deciding), direct perception (just observing and knowing things without judging), imagining things, accessing memory, and dreaming. Interestingly, the way you use your eyes and the region of your brain that is active differs with each mode. When you meditate your eyes will be closed, or if open they will be soft, wide, and not focused. When you concentrate your eyes are piercing, focused.\n\nThe practice of meditation is difficult but, ironically, practicing a form of concentration first makes it easier. That is because training your mind in concentration allows us to collapse the myriad of thoughts to just one or two and helps us to still the active mind. When we begin to meditate we are less distracted. Within a few months of consistent concentration practice, you will note that you are more present. Your mind has developed resistance to being swept away with worries and distractions. You will begin to identify with the witnessing part of you, rather than with the thoughts themselves.\n\nNot all meditation is focused on an object, mantra, or concept. In mindful meditation, an adaptation of a Buddhist meditative practice, we seek a state of non-judging awareness where sensory input, such as a car honking its horn or a thought of an important task, is noticed without judgment, attachment, and upending our equanimity. We just notice the disturbance and instantly let it go. As we get into higher and higher stages of meditation, we are able to shut down the noise of the mind entirely and rest in a deeply absorbed state. At that level of progress you will begin to experience shibumi, which was described in the last chapter. You will want to experiment with objects of concentration and forms of meditation to determine what is best for your personality and stage of development. Seeking guidance from an expert is very helpful with this tactic.\n\nTACTIC 8: VISUALIZATION\n\nI have relied on visualization practice since I used it to both get into and through the rigorous SEAL training back in 1990. With Kokoro Yoga we strive to train the brain's ability to create and hold imagery so that we can imagine, then see with clarity, something we desire to achieve in the future, or to learn from in the past. This tactic specifically works on your mental and wisdom dimensions. With visualization we can create imagery for an intended purpose, and repeatedly revisit said imagery to develop a skill, heal, gain insights on something, or to develop confidence and momentum toward completing a goal. A familiar and powerful use of visualization training is to prepare for a competition by visualizing your best performance, whether real or imagined. This practice will cut a mental, emotional, and nervous system groove upon which to follow as you perform. Research has proven that most skills can be improved through visualization. For example, imagining perfect technique in successful free throw basketball shots, we can markedly enhance the development of that skill.\n\nI'd like to emphasize some of the more important uses for visualization work. As alluded to, I first employed this tactic when I was preparing for Basic Underwater Demolition SEAL training. For 9 months I spent time daily visualizing myself as a Navy SEAL, as having successfully completed the program, and wearing the SEAL Trident. The visualization slowly rendered within me an overwhelming level of belief that I was destined to be a SEAL. By the time I arrived in Coronado to begin the training, I had already won in my mind. The training was familiar and I had a deep sense of confidence that I would succeed. Nothing could get in my way, and I credit the visualization training for reinforcing my mental toughness and resiliency. Visualization practice can sow seeds deep within to help you achieve goals, develop healthy self-esteem, and untangle psychological blocks. I have also seen its effectiveness in treating PTSD and emotional abuse. There are several visualization exercises described in chapter 6 to get you started. This tactic is typically employed at the close of a training session, but can also be done as a stand-alone drill.\n\nTACTIC 9: SPOT DRILLS\n\nKokoro Yoga is designed for busy individuals who may at first wonder how they're going to jam yet another activity into their already overwrought schedules. This ninth tactic will allow you to integrate several drills into a single ritual or workout, and to do your training \"on the spot\" in the many free spots that open up during the day. This tactic also comes from my original experience in Baghdad. Time was extremely limited\u2014I had to skip breakfast for training in the first place. But during the long days and nights, I would still get physically fatigued, mentally cluttered, or emotionally charged. So I relied on spot drills to get back into balance and maintain a link with my integrated training. Bottom line, if you can breathe, then you can train! Most of the drills take just 1 or 2 minutes, and even on the most hectic days you can be innovative and locate a few minutes here or there that might otherwise go wasted. Waiting in line, browsing the net, driving your car, visiting the restroom, on board an airplane\u2014if you aren't engaged in a conversation, you can be doing a 1-minute (or longer) movement, breathing, concentration, memory, mental agility, or visualization practice. When flying, there's no harm in going to the back of the cabin and figuring out a pose or two that you can do. At the office, set your alarm so that on the hour, you take a minute to do a Sun Salutation or box breathing. Develop the habit of choosing 2 in the morning and 2 in the afternoon. Here are a few ideas to get you started:\n\n  Short breathing drill\n\n  Sun Salutation A\n\n  1 minute of mindful awareness meditation\n\n  1-minute body scan\n\n  What wolf are you feeding? check-in. Ask if you are in a positive frame of mind or stuck in the negative. If positive, reinforce it. If negative, interdict and redirect to positive.\n\n  Recite a powerful mantra or affirmation. My favorite: \"Feeling good, looking good, oughta be in Hollywood.\"\n\n  Listen to an empowering podcast.\n\n  Read a poem, bible verse, or a paragraph in an inspiring book.\n\n  Visualize the alphabet in large, neon letters, drawing each letter mentally.\n\n  Use a brain-training app for mental speed, agility, and improving memory.\n\n  Ask what you are grateful for.\n\n  Chant.\n\n  Express gratitude to a coworker, spouse, child, or boss.\n\n  Smile for a full minute.\n\n  Laugh out loud for a while\u2014laughing yoga is a powerful positive boost and a blast!\n\nSpot drilling is a habit that you can weave into your life. After all, life is just a big training ground. Over time, these spot drills will accelerate and compound your training, making a profound difference in your life.\n\nTACTIC 10: AUSTERE AND TRAVEL TRAINING\n\nI've included this tactic because I want you to realize, first of all, you don't need spandex pants, a yoga studio, or $300 of equipment to practice Kokoro Yoga. And second\u2014as indicated in the previous tactic\u2014no matter how busy you are or how much you travel, the original nature of yoga was to be versatile so that you can make any location suitable for a great training session. If you have 5 minutes to an hour, and access to any open space, you are good to go. As far as yoga studios, getting outside has myriad perks. A nature trail is a wonderful place to do your practice. I personally love to step away from the gym and go down to the beach to train among the waves and seagulls. When traveling, the airport concourse and the patch of free space in front of my hotel bed are my studios.\n\nAs far as equipment, you really don't need much\u2014even a yoga mat is a luxury sometimes. Ninety-nine percent of what you need is carried on your body and between your ears. Straps, blocks, or towels are easy to come by and a meditation bench is easy to make. For functional fitness equipment I encourage you to scrounge through your garage or basement to find things to pick up and move that will serve you well. If you want to buy the gear go ahead, but it can also be fun to employ the austerity measure of this tactic and improvise to overcome. If austere training is new, it may surprise you to learn just how fun and challenging training with a bag of sand can be!\n\nBottom line: This training allows for no excuses. Now let's learn the classic poses and get moving.\n\n# CHAPTER 5\n\n# CORE SEQUENCES\n\n_Watching [Norman]do his practice was like watching an Olympic gymnast work out._  \n\u2013BERYL BENDER BIRCH, DESCRIBING HIS FIRST IMPRESSIONS OF ASHTANGA YOGA IN 1955\n\nKOKORO YOGA HAS STEADILY EVOLVED OVER THE YEARS SINCE ITS INCEPTION IN THE CARGO HOLD OF MY C-130 FLIGHT TO A WAR ZONE. THROUGH THIS DEVELOPMENT PERIOD I HAVE CONTINUED TO EDUCATE MYSELF IN THE PHILOSOPHY AND THE PRACTICE OF THE ANCIENT ARTS AND DEVELOPED PROGRAMS TO ASSIST OTHERS IN LEARNING. MY DRIVE HAS BEEN TO INSPIRE OTHERS TO LIVE BY A PERSONAL ETHOS AND DEVELOP A DISCIPLINED TRAINING IN THE FIVE MOUNTAINS\u2014Both of which are anchored in the ancient yoga traditions as well as elite modern warrior training. Personal development of this nature closes the openings of our weaknesses and leverages our strengths, ultimately balancing physical, mental, emotional, intuitional, and spiritual aspects of our lives. This chapter will delve into the actual training routines called sequences. Here you will learn the core sequences to employ in your daily lives as you develop your own training plan and personal practice.\n\nTRAINING AND PRACTICE\n\nTo experience the full benefits of a practice, one must select a training regimen with appropriate tools, and practice daily, weekly, and monthly. Often the training is just referred to as \"practice\" \u2014Ashtanga Yoga founder Sri K. Pattabhi Jois reminds us that: \"Yoga is 99% practice, 1% theory\" and \"practice, and all else is coming!\" The effective employment of yoga requires that we understand and properly employ the training methods, as well as develop a personal practice suitable to our lives. Training methods will be discussed in this chapter, and developing a personal practice will be the focus of chapter 8.\n\nTraining and practice are definitely related and are often used synonymously, but they are different concepts. As mentioned, training refers to a regimen, a method, and also an event, such as a training session or seminar. Training covers strategies, tactics, and tools available for practitioners of the system of development. It is highly unlikely that a practitioner will use all of the strategies, tactics, and tools available. Rather, he or she will select the appropriate ones for them and construct a training regimen to meet their specific objectives. This is where training begins to overlap with the concept of a personal practice. Even though two people may be using the same tools and similar training regimen, they will each approach it in an individualized manner of personal practice. Rather than just pulling a sequence or drill out of a box and doing it for a training session, you will select those most appropriate for your life situation and weave them into a daily, weekly, and monthly practice that supports and improves your lifestyle. That is why it is called a personal practice\u2014it is deeply personal to you and no two personal practices are the same.\n\nAt any rate, in deciding how to construct your training plan you will begin with your intentional needs (which will change over time), your fitness level (which will also evolve), and your work\/life situation. So rather than just showing up randomly at a yoga studio and getting your sweat on, you ask yourself: What is my intention with the practice overall and for this particular period of my life. This includes what warrior archetype you seek to develop further. For example, my stepdaughter Catherine, who leads our teacher training program and has coauthored this book, answers this question with: \"I am a warrior of peace, and it is my intention to spread love into the world through my teaching and personal practice.\" Catherine's deep experience teaching and practicing yoga and meditation brings a gentle side to her warrior archetype. She organizes her training to support that archetype and development. It includes a selection of gentle and restorative sequences, meditation, and Ashtanga and Viniyoga skills.\n\nMy training plan differs because as a retired SEAL and teacher of warriors, I strive to remain grounded in the combat warrior way as well as develop my kokoro spirit. I need to be a protector and ready for battle at all times. I strive to ride my razor's edge so I can always lead by example. I do this to stay true to my stand and align with my purpose of mastering myself so I can serve and inspire others. My principles include earning my trident of respect every day, so I endeavor for daily improvement. All of this is in search of continuous, conscious evolution that will help me authentically serve others in their own journey. To support my training I do the morning and evening rituals every day, as well as teach or perform the more rigorous Protective Warrior and Fit Warrior sequences once or twice a week. I also train with box breathing, meditation, my mantra, and visualization once or more times a day. We will look at how Catherine and I organize these training methods into our personal practice in chapter 8.\n\nSEQUENCING\n\nSequences are one of the primary tools of an effective yoga system. Sequences combine poses, breathing methods, functional fitness, combat conditioning moves, visualization and meditation into focused and intentional training regimens. Recall that sequencing the movements into an intelligent flow, that fulfills a distinct purpose, is the first strategy of Kokoro Yoga. Learning to sequence effectively takes many years of experience, so we have designed sequences for you that meet the most common applications of a personal practice. One sequence will challenge the elite warrior, another will appeal to the peaceful warrior and yet another will support the warrior in need of psychological and spiritual recovery. Applying the right sequence at the right moments in your training plan, and in life, is potent medicine.\n\nWhat follows are select sequences to get you started on your journey. Please explore them with the intuitive sense of your needs and let experience be your guide. It is a good idea to review the pose descriptions and pictures in chapter 7 before trying out these sequences.\n\n# The Core Sequences\n\nThe core sequences are practice sessions of varying length that meet specific training objectives. Your objectives will vary, but we wanted to cover a few that every healthy practitioner can benefit from.\n\nI am particularly excited to offer a sequence for warriors recovering from intense stress or career-related psychological wounds. I have no doubt about the value of this sequence in assisting wounded warriors to recover and find balance and peace again.\n\nThe sequences are included in written form at the end of this chapter, but it is awkward to describe the sequences themselves with words. Therefore we have included photos (with accompanying captions) to help visually demonstrate the pose\/sequence.\n\nIn each of the core sequences the poses are linked with the breath. Additionally some poses are held for a set number of breaths (indicated in the written version). One of the distinctions of Kokoro Yoga is the adaptable nature of the poses, so that even a beginner can do any of the sequences by modifying: Some pose modifications are shown.\n\nPeaceful Warrior\n\nThe Peaceful Warrior is approximately 45 minutes long and will take you deep into an inner state of peace and leave you feeling very grounded. It will provide a moderate level of endurance training and has a broad focus on spinal and joint health as well as energetic alignment. Bring your Peaceful Warrior attitude to this practice.\n\nSEATED MEDITATION\u2014SET INTENTION\n\nPW1\n\nALTERNATE NOSTRIL BREATHING\n\nPW2\n\nPW3\n\nPW4\n\nCAT COW ROTATION\n\nPW5\n\nPW6\n\nDOWNWARD DOG\n\nTRANSITION BACK TO DOWN DOG\n\nPW7\n\nKNEELING WARRIOR\u2014EACH SIDE\n\nPW8\n\nWIDE ANGLE FORWARD FOLD\n\nPW9\n\nROLL UP TO STANDING\n\nPW10\n\nBRINGING DOWN THE HEAVENS 3X\n\nPW11A\n\nSUN SALUTATION A\u20142X\n\nPW11b\n\nWARRIOR 1\n\nPW12\n\nWARRIOR 2\n\nPW13\n\nSIDE ANGLE POSE\n\nPW14\n\nEXALTED WARRIOR\n\nPW15\n\nPW16\n\nPW17\n\nPUSH-UP PLANK\n\nUPWARD-FACING DOG\n\nPW18\n\nDOWNWARD DOG\n\nREPEAT LEFT SIDE\n\nPW19\n\nSHARP WARRIOR\n\nPW20\n\nSUN SALUTATION A\n\nPW21\n\nMODIFICATION\n\nTREE POSE\n\nPW22\n\nDANCER POSE\n\nPW23\n\nSUN SALUTATION A TO DOWNWARD DOG\u2014THEN COME TO A SEAT\n\nPW24\n\nSEATED FORWARD FOLD\n\nPW25\n\nTABLETOP\n\nPW26\n\nCLEANSING WARRIOR (EACH SIDE)\n\nPW27\n\nPW28\n\nPW29\n\nBRIDGE OR WHEEL\u20142X\n\nRECLINED PIGEON (EACH SIDE)\n\nPW30\n\nHAPPY BABY POSE\n\nPW31\n\nHEALTHY WARRIOR\n\nPW32\n\nBUTTERFLY 1 AND 2\n\nPW33\n\nRECLINED WARRIOR POSE\n\nPW34\n\nRESTING POSE\n\nPW35\n\nSEATED MEDITATION\u2014REVISIT INTENTION AND BREATH\n\nCOMPASSION AND PEACE MEDITATION\n\nPW36\n\nZen Warrior\n\nThe Zen Warrior is a moderately challenging 30-minute long sequence. It will cultivate strength, flexibility, and enhance your ability to access a moving meditative state. It covers the standing warrior poses and breathing exercises, as well as some nice core development, backbends, and spinal twists. Most importantly, bring your serene and victorious warrior attitude into this practice. Use the time to sharpen your mental sword and find victory in your mind, like the samurai on the Zen bench prior to battle. Breath awareness is a key to maintain throughout the practice, as it will help awaken the flow state.\n\nMOUNTAIN POSE\n\n(5X 1\u20132 BREATHS)\n\nZW1\n\nSUN SALUTATION A\u20143X\n\nZW2\n\nSUN SALUTATION B\u20143X\n\nZW3\n\nBRINGING DOWN THE HEAVENS\n\nZW4\n\nKICKING WARRIOR\n\nZW5\n\nMOUNTAIN POSE\n\nZW6\n\nARCHER WARRIOR\n\nZW7\n\nSCOOPING THE MOON\n\nZW8\n\nWARRIOR MUSASHI\n\nZW9\n\nSUN SALUTATION A\n\nZ10\n\nDOWNWARD DOG\u2014COME TO KNEES\n\nMODIFIED\n\nFULL POSE\n\nCAMEL\n\nZ11\n\nRECLINED WARRIOR POSE\n\nZW12\n\nWISE WARRIOR TWIST\n\nZW13\n\nROCK BACK AND FORTH TO STANDING\n\nZW14\n\nZW15\n\nSUN SALUTATION A TO PUSH-UP\u2014LOWER ONTO BELLY\n\nZW15\n\n1-ARM LOCUST\u2014EACH SIDE\n\nZW16\n\nLOCUST\n\nZW17\n\nBOW\n\nZW18\n\nREVERSE PUSH-UP\n\nZW19\n\nPUSH-UP\u20145X\n\nZW20\n\nUP DOG OR COBRA\n\nZW21\n\nGRATEFUL WARRIOR\n\nZW22\n\nCOW\n\nZW23\n\nCAT\n\nZW24\n\nZW25\n\nRESTING POSE\n\nFOLLOW WITH STRENGTH AND GROUNDING MEDITATION\n\nZW26\n\nFit Warrior\n\nThe length of this sequence depends upon how long you do the functional fitness module. It is typically about a 45-minute dynamic flow with an intense interval-training component. It is definitely on the challenging end of the spectrum but can be scaled according to your needs. The session will take you deep inside your mental training space, while providing a kick-ass, fun workout to supplement your functional fitness training. This is a great sequence to use if you are a road warrior with limited access to weights and other tools; it is excellent for women who don't want to \"throw weights around\"; and for beginners or older trainees who are uncomfortable with weighted training. This sequence can be performed in two ways. This first is where the fitness segments are inserted into the middle as a distinct module. The second is to add the fitness movements throughout the sequence between the yoga poses, such as doing 10 push-ups after every other Up Dog, or 20 squats between the transition from one side to the other on warrior poses. Appendix B has a number of exercise combinations to choose from, or you can let your intuition guide you!\n\nFW1\n\nFW2\n\nCOW\n\nFW3\n\nCAT\n\nFW3\n\nREPEAT OTHER SIDE\n\nDOG DOWN\n\nFW4\n\nFORWARD FOLD\n\nFW5\n\nMOUNTAIN\n\nFW5\n\nSUN SALUTATION AX2\n\nFW6\n\nSUN SALUTATION AX2\n\nFW7\n\nTRIANGLE\n\nFW8\n\nTWISTED TRIANGLE\n\nFW9\n\nSHARP WARRIOR\n\nFW10\n\nFW11\n\nFW11\n\nWIDE ANGLE FORWARD FOLD\n\nFW12\n\nFW12\n\nTWISTED WIDE ANGLE FORWARD FOLD\n\nMOUNTAIN\u2014REPEAT SEQUENCE ON LEFT SIDE\n\nFW12\n\nFIT MODULE X3\n\nFW13\n\nFit Module\n\nFM1\n\nFM2\n\nFM3\n\n10 BURPEES\n\nFM4\n\nFM5\n\nFM6\n\nFM7\n\n20 JUMPING SQUATS\n\nFM8\n\nFM9\n\nFM10\n\n20 MOUNTAIN CLIMBERS\n\nFM11\n\nFM12\n\nFLUTTER KICKS\n\nFM13\n\nFM14\n\n20 SIT-UPS\n\nFM15\n\nFM16\n\n20 PUSH-UPS\n\nSCOOPING FOR THE MOON\n\nFW14\n\nFIGHTING WARRIOR\n\nFW15\n\nBRINGING DOWN THE HEAVENS\n\nFW16\n\nDOWN DOG\n\nFW17\n\nPIGEON\n\nFW18\n\nCAMEL\n\nFW19\n\nCOW\n\nFW20\n\nCAT\n\nFW20\n\nCLEANSING WARRIOR\n\nFW21\n\nHEALTHY WARRIOR\n\nFW22\n\nBACK ALIGNMENT\n\nFW22\n\nBRIDGE\n\nFW23\n\nRESTING POSE\n\nFW24\n\nSEATED MEDITATION\n\nFW25\n\nBOX BREATHING 3-6-6-3 5X\n\nFW26\n\nFighting Warrior\n\nThis sequence is about 45 minutes to an hour long and is as challenging as the Fit Warrior but differs in that it uses combat conditioning movements instead of functional exercises for the exercise part. This allows us to get a great \"cardio\" workout while practicing our strikes and kicks. (See the appendix for a description of the fight moves.) This sequence is excellent for anyone with fighting skills, or who enjoys cardio kickboxing.\n\nYou will bring a fighting-warrior attitude, as if you were in a sensitive situation and you will need to \"bring it\" to solve it. Visualizing an opponent during the sequences develops our visual acuity. As with Fit Warrior, there are two ways to perform this sequence type. The first is to sprinkle in the fight moves between the yoga poses, and the second is to do the standing poses to warm up, then do the fight sequences altogether, and finally finish with the seated poses. Both are equally effective, but the first keeps your heart rate down in an endurance training zone, while the latter ramps it up to an intensity zone. Choose whichever suits your needs and get busy! You don't need to be an experienced martial artist to have a blast with this sequence.\n\nBOX BREATHING 5X5\n\nFGW1\n\nSUN SALUTATION A\n\nFGW2\n\nWARRIOR 1 WITH BREATH\n\nFGW3\n\nCHOPPING WOOD\n\nFGW4\n\nWARRIOR BREATH\u2014100X\n\nFGW5\n\nPUNCHING WARRIOR\u201420X EACH SIDE\n\nFGW6\n\nKICKING WARRIOR\u201420X EACH SIDE\n\nFGW7\n\nCOMBINATION MODULE OF KICK AND PUNCH\u201420X EACH SIDE\n\nFGW8\n\nMOUNTAIN POSE\u20145 BREATHS\n\nFGW9\n\nSUN SALUTATION A\n\nFGW10\n\nBREATH OF FIRE\n\nFGW11\n\nVICTORIOUS WARRIOR VISUALIZATION\u2014EITHER STANDING OR SITTING\n\nFGW12\n\nJedi Warrior\n\nThis sequence is 20 minutes long utilizing breathing and movements from the Chinese \"warrior-yoga\" art of Qigong, with select poses to support joint mobility and structure, while also enhancing our energetic body's health. Like many martial arts, we will experience this work at the physical level before experiencing it at the subtle energy level. This sequence is great for internal organ health as well as muscular strength and conditioning. Over time this sequence will train deeper awareness, and the attitude is that of one seeking mastery, much like the Jedi Knight training of Luke Skywalker in the movie _Star Wars_.\n\nBOX BREATHING 5X5\u2014SET INTENTION\n\nJW1\n\nROM DRILLS\n\nJW2\n\nSUN SALUTATION A\u20142X\n\nJW3\n\nSUN SALUTATION B\u20142X\n\nJW4\n\nPLANK\u2014HOLD FOR 30 SECONDS\n\nJW5\n\nWINDMILL\n\nJW6\n\nQIGONG OR \"WATER MILL\" FORWARD FOLD\u20143 MINUTES\n\nJW7\n\nKI-UB\n\nJW8\n\nJW8\n\nWIDE ANGLE FORWARD FOLD\n\nJW9\n\nSUN SALUTATION A\n\nJW10\n\nPLANK\u2014HOLD FOR 1 TO 3 MINUTES\n\nJW11\n\nDOWNWARD DOG\n\nJW12\n\nEXALTED WARRIOR\u2014EACH SIDE\n\nJW13\n\nJW14\n\nPUSH-UP\u2014HOLD FOR 5 BREATHS\n\nJW15\n\nDOWNWARD DOG\n\nJW16\n\nFORWARD FOLD\u20145 BREATHS\u2014ROLL UP TO STANDING\n\nJW17\n\nHORSE STANCE\u2014HOLD FOR 3 TO 5 MINUTES\n\nJW18\n\n9 BREATHS\u20143 SETS\n\n1. RELAX  \n2. SELF-HEALING VISUALIZATION  \n3. GRATITUDE\n\nJW19\n\nMOUNTAIN POSE\u2014REVISIT INTENTION\n\nJW20\n\nSEATED OR STANDING MEDITATION\u2014JEDI WARRIOR VISUALIZATION\n\nJW21\n\nAgeless Warrior\n\nThis sequence is a comprehensive and balanced series of poses designed to optimize spinal and energetic health for longevity. _The focus is vitality, balance, and inspiration._ The universality of being ageless is to embody devotion to a greater cause than oneself, focus to execute tasks at any phase of life, and gratitude for the challenges and blessings that life brings to one as they walk the path of a warrior. Do this sequence once or twice a week and use the future me visualization to help you move toward your ageless self.\n\nBOX BREATHING \u20141:2:2:1 (5X)\n\nAW1\n\nOPENING THE BOOK OF WISDOM (WHILE IN MOUNTAIN POSE)\n\nAW2\n\nWARRIOR 1 WITH BREATH 5X\n\nAW3\n\nWARRIOR 2 WITH BREATH 5X (ARMS STRAIGHT BY EARS)\n\nAW4\n\nWARRIOR 2 (WITH BACK PALM UP)\n\nAW5\n\nEXALTED WARRIOR\n\nAW6\n\nSIDE ANGLE POSE\n\nAW7\n\nWIDE ANGLE POSE (TRANSITION TO OTHER SIDE AW1\u2013AW8)\n\nAW8\n\nTWISTED WIDE ANGLE\n\nAW9\n\nROLL UP TO STANDING\n\nAW10\n\nHORSE STANCE\u2014HOLD FOR 1 MINUTE\n\nAW11\n\nSUN SALUTATION A\n\nAW12\n\nSTANDING MEDITATION WITH FUTURE SELF IN PERFECT HEALTH VISUALIZATION\n\nAW13\n\nRecovering Warrior (PTSD)\n\nThis sequence is a 15-minute restorative practice with a heavy focus on breathing, imagery, and holding select poses for a long time. The practice is designed to restore balance to the nervous system and calm an agitated mind. The mental attitude and imagery is one of total health and healing of the body, mind, and spirit. Performing one of the meditation or visualization practices immediately after this sequence is important.\n\nRESTING POSE\u20141:2 OR 4:8 BREATH (10X)\n\nRW1\n\nCLEANSING WARRIOR\n\nRW2\n\nROCK BACK AND FORTH 3X TO ENERGIZE YOUR SPINE\n\nRW3A\n\nRECLINED PIGEON\u2014EACH SIDE\n\nRW3B\n\nBRIDGE\n\nRW4\n\nTABLETOP\n\nRW5\n\nRW6A\n\nRW6B\n\nSEATED FORWARD FOLD\n\nBUTTERFLY 1\n\nRW7\n\nBUTTERFLY 2\n\nRW8\n\nWISE WARRIOR\u2014EACH SIDE\n\nRW9\n\nRW10A\n\nRW10B\n\nSEATED FORWARD FOLD\n\nSEATED MEDITATION WITH ALTERNATE NOSTRIL BREATHING\n\nRW11\n\nSTILL WATER VISUALIZATION,   \nFOLLOWED BY \"FUTURE ME\" IN PERFECT HEALTH\n\nRW12\n\nHip Mobility Drill\n\nThis sequence focuses on opening the hips, twisting the spine, and engaging the core to provide stability in the pelvic region. It is particularly useful if you have been hiking or rucking for long distances, or done any weighted squatting.\n\n**Each pose will be held for 3 to 5 breaths.**\n\nSUN SALUTATION A UNTIL DOWNWARD DOG\n\nRIGHT SIDE FIRST\n\nHMD1\n\nWARRIOR 1\n\nHMD2\n\nWARRIOR 2\n\nHMD3\n\nSIDE ANGLE POSE\n\nHMD4\n\nKNEELING WARRIOR, AKA ARCHER POSE\n\nHMD5\n\nTWISTING WARRIOR\n\nHMD6\n\nPIGEON\n\nHMD7\n\nPLANK\n\nHMD8\n\nPUSH-UP 5X\n\nHMD9\n\nUP DOG\n\nHMD10\n\nDOWNWARD DOG\n\nREPEAT ON LEFT SIDE\n\nHMD11\n\nMeditation and Visualization Practice\n\nMeditation is a key tactic in Kokoro Yoga. The overall progression of the training is meant to take you from the outer, externally focused physical level to the energetic, more internally focused breath\/prana level, and finally into the deep interior of your heart-mind through meditation and visualization. These two practices are closely related working hand in glove to develop your mind and access your heart for optimal performance and growth.\n\nLet's discuss meditation in more detail. Many in the West think meditation is just sitting quietly on a mat with eyes closed, or chanting a mantra. Another popular view is to follow a guided visualization using a CD or an app. These are valuable practices, for sure, but they represent the very beginning of your work. Ultimately meditation is a broad term for how we can use our inner mental skills to develop our heart-mind to better see truth, access wisdom, and experience love. There is a definite process to this though.\n\nAs mentioned earlier, the first step in the process is to deepen your concentration skills. Concentration is developed through the sequence practice, through breath control training, and through a practice whereby you deeply concentrate on one thing, such as an object or symbol. The object of focus can be external, such as a candle, a symbol like a cross, or an image like a yantra or mandala or a sunset. The object of focus can also be internally generated, such as focusing on the movement of breath, a count, sound, mantra, prayer, koan (thought question), scriptural verse, poem, or image. The best idea is to just choose something that is easy for you to focus on and deeply meaningful. The point of concentration practice is to train your thinking mind to be still. Over time you will be able to focus for long periods. This pays big dividends when it comes to achieving your goals in life. Concentration practice requires a deliberate effort and is typically done after a movement practice or as a stand-alone event.\n\nAs your concentration deepens, so does your awareness of what is going on around you and inside of you. You are now connecting to your perceiving mind, which knows without being taught something. Truth about nature is experienced through this part of your mind. The practice of mindfulness meditation is a nice tool to expand this growing awareness even more and to train your mind to click into a place of action instead of reaction. This technique can be used walking on the beach, washing dishes, or while shopping. It is a great way to extend your training throughout the day. The excuse of not having time to train falls apart when you consider that box breathing and mindfulness meditation can be performed almost anywhere and anytime.\n\nVisualization is meditation with imagery that you create and direct. Visualizing Jesus Christ while you meditate is an example. With Kokoro Yoga we want to develop our visual acuity and create powerful visions of our desired future states and accomplishments. We also use visualization to practice skills or to heal a wound. The power of visualization is awesome when the mind has been trained to concentrate and be still.\n\nPatanjali told us that the purpose of yoga is to develop control over your mind, so you can get control over your life. So after we can concentrate deeply and maintain presence, then what? Well, the mind will continue to grasp after material objects, concepts, beliefs, and desires. The grasper (us) gets confused with the thing being grasped and leads us continually to false identification with things we accomplish or things we think we control outside of ourselves, or those cherished beliefs we cling to. Identification with a career, spouse, role, fancy car, etc., are examples, and when they change, which is inevitable, we fall apart because we have identified with them as \"us.\" But none of the external things are us, and none of the beliefs or internal patterns are us, either. This always leads to suffering of some form or another, because none of these things last forever, and none of them has anything to do with who we really are.\n\nThe mind truly is a wily trickster, though. This instrument of perception of ours tricks us into thinking it is the boss, when in fact it works for your deeper self, your witnessing soul-centered self (recall the bliss layer from chapter 3). Insight, or self-reflection meditation is our method to disentangle the grasper from that which is grasped, and to clarify the mind so that it reflects reality clearly while serving you without complaint. This method is to have you meditate on your patterns and behavior and to work relentlessly to align with your spirit. You will reflect daily in meditation on alignment with your ethical stand and disciplines (recall the Yamas and Niyamas). Slowly you will be firmly seated in the driver's seat, steering your mind relentlessly toward your eternal witness's vision for your life's purpose. Your ego will be in its rightful place, retaining its healthy, positive qualities, dropping the negative and needy grasping for control. This final stage of meditation is how we develop into higher stages of healthy, integrated consciousness.\n\nThrough meditation practice we move from distraction to attention and then attention to understanding completely that which we concentrate on. You become a scientist studying the inner workings of your own human nature, and as your mind penetrates the inner workings of your own nature you begin to understand all of human nature. Information is exchanged between you and your object of focus, whether that is an image of Christ or your own disruptive patterns and you crack open your innate, intuitive intelligence. This stage of your development will bring great clarity and peace.\n\nBeginning Meditation Technique\n\nTo begin a meditation or visualization practice you can start with a simple breath awareness exercise. This can be done anywhere, at your desk or in bed when you wake up for instance. Simply close your eyes and pay attention to your breath for a few moments.\n\nIt is advisable to create a \"sacred space\" in your home where you can do your yoga training, especially the meditation part. Having this space and using it every day will help you to focus as you implement your personal practice. It will also help keep others in your family or tribe from distracting you (more on how to develop a personal practice in the last chapter).\n\nWhen you \"sit\" for meditation it is important that your spine is erect and not compressed in any way. The easiest way to accomplish this is to sit on the edge of a chair. Many think that there is some magic to sitting cross-legged on the floor. Not so\u2014the reason this was done was because there weren't many chairs around back in the day! If you do sit on the floor, then you can use a small zazen bench (which is what I use) or stack some pillows to sit on. To create the sacred feeling, it helps to create an \"altar\" of sorts with a meaningful picture and candle. Plenty of information is available on how to begin a meditation practice, so I won't belabor the point. Let's now take a glance at a few of the insight meditation practices.\n\nFull-Body Scan Stress Reduction\n\nSit comfortably, with your spine erect in such a way that it's as if a balloon is attached to the dome of your skull with a string, gently lengthening your spine upward. Your eyes can either be open or softly closed. Take a series of slow, deep breaths, and perform a light scan from the top of the skull downward, imagining a tingling flow of energy that relaxes you. Starting with the forehead, relax each section of your head and body with light attention, breathing into the awareness and exhaling tension. In this way, relax the muscles of the face, the neck, and the shoulders, observing the tension melt and drip away. Move through your chest, your abdomen, your lower back, and pelvic girdle, through to your upper legs, knees, and then lower legs. Finish with a scan of your feet, both the tops and the arches, and finally, the toes. Pay attention to the state of peace, relaxation, and energy that is now moving through you and in tune with your slow, deep breathing. Your body is completely relaxed; your mind is completely relaxed. Note that this relaxation sequence can be used for each of the visualization and meditation practices.\n\nNow picture yourself on a footpath that you've taken before, on a pristine, beautiful day, where you can feel the warmth of the sun on your face and the rustle of a breeze moving through your hair. Walk on the path and soak in the stunning beauty of the moment. Pause to pick up a rock from the path, bring it up to your nose, and smell the grounded earthiness of your path. Place it back down and continue walking around a bend where you see and hear a freshwater stream. Breathe in as you let your mind sink into the beauty and power of the pure blue water. Allow the stream to calm your mind, your spirit, your body. This will take you to a place (over a long consistent practice) of neutral, nonjudgmental observation. If a thought stirs, you greet it with, \"Not here, not now,\" and allow it to float away. It's in this place that you feel the witness expand with bliss and gratitude, an immense connection to everything and everyone. You allow a moment to soak in this deep, all-powerful silence.\n\nFuture Me\n\nWith the future me exercise, you are going to project a vision of your ideal self, at a point in the future, to build upon the connection with the internal witness and also carve a track for the subconscious to follow. The subconscious domain of your being cannot tell the difference between a real or an actual experience, so the more acute detail and emotional intensity you can pack this visualization with, the more effective it will be. The task here it to cultivate a powerful image of who you intend to become\u2014the warrior on the path of mastery, a year or several years down the road. The time really depends on where you are in your life. As an aspiring SEAL my future me visualization was 1 year out, when I earned the coveted SEAL Trident. Today my future me is 10 to 20 years out when I have fulfilled my current vision.\n\nWhen your mind is focused, after a short box breathing or body scan practice, conjure up a vision of a future version of yourself in the process of fulfilling your life purpose, or a momentous goal. You know that you have made it, having achieved a dream that required massive and dedicated action. You see it in full color with as much detail as possible, with real emotion. In this inner space you feel a humble acceptance, knowing you have aligned with your spirit and served a greater good. You're living in a highly tuned state of health and energy, and the goodness of the moment circulates with your blood. You feel fantastic and alive. This is the new you, now. Remain here for at least 5 minutes living the vision of your future self.\n\nReturning your attention to the breath, experience the grace of the moment with humble gratitude. You return to the present moment, opening your eyes, highly energized, confident, at peace, with all your senses firing, and with the imbedded sense of having accomplished something worthwhile and important.\n\nGrounding\n\nStarting out in a comfortable seat with your eyes closed, gaze softly within. The spine is in alignment and the core is engaged. There is a softness and relaxation in the jaw and shoulders, as you sit upright and engaged. Begin to tune your awareness to the breath. Notice how it feels to breathe in and out through the nose. Notice if there is more ease when you inhale or exhale. You are just noticing, not judging or trying to change anything. Now begin to intentionally smooth out the edges of the breath, match the length of inhale and exhale to a count of 5 on each side so that 1 round of breath takes 10 seconds. Focus on this until it feels natural and you are in a state of relaxation and expanded awareness. Now connect to the area in the pelvic floor and the base of the spine. Take a few conscious breaths into your pelvic bowl and begin to visualize a bright red ball of energy that is glowing and radiating healing. Visually create roots growing from this ball the width of your torso down into the earth. Allow the roots with each breath to grow deeper and deeper into the earth until you have reached the core center of this planet and hook into it. Breathe gently into your roots. Then like you would drink through a straw to sip liquid, start to sip in the earth energy through the roots you have created. Allow the earth's energy to be drawn up into your body filling you with strength and vitality. See the origin of your roots as this glowing red orb get brighter and brighter as you recharge your body.\n\nThis is a meditation exercise you can also do standing barefoot in the park or on a beach or a mountain. It may also be used for curing issues like fatigue, depression, lethargy, and for times when you feel out of sorts or not yourself.\n\nWillpower\n\nFind a quiet place where you can settle into a comfortable seat. Your eyes can be open or closed. If the eyes are open, then keep a very soft gaze; maybe light a candle and gaze at the flame. If the eyes are closed, gaze inward and down toward the tip of the nose to assist in concentration. Place both hands over your belly and breathe into the belly expanding it out like a balloon and then exhale out through the mouth releasing tension. Inhale into the belly expanding it out and then exhale through the mouth expunging negative attitudes and thoughts. Inhale one more time into the belly expanding it out and then exhale out through the mouth with an ahhh sound, releasing unconscious thoughts that are not serving you well. You do not need to know what the tension or the thoughts are; the intention of letting go is enough. Relax your hands and place them palms down on your knees. Begin to breathe in and out through the nose at a natural pace. Now focus on your own internal energetic fire that is constantly burning off toxins as well as driving your will to be present and put forth effort in the world. Ask yourself: What is my purpose? As the answer arises, connect with your willpower energy in the belly and begin to fuel your purpose from this place of focus and passion. Anchor your purpose into the fire within and remain seated for a few more minutes in silence. Breathe with an awareness on your belly and the power that resides within.\n\nCompassion\n\nLying flat on your back, place your right hand on your belly and your left hand on your heart center. Close your eyes softly and begin to tune into the frequency of your energetic heart center. This center traditionally is the place of universal love and compassion. Feel the physical heart beat in the belly and the heart regions. Turn your focus to the energy of your heart, in the center of your chest, and connect the energy of your breath with the vibration of your heart and feel as if your physical body is beginning to melt like snow in a warming sun. As the physical body softens and begins to let go of attachments relax the arms beside the body. Now ask the heart center: _What am I holding onto that no longer serves me?_ As the question is answered let go of the story around the event, the person, or yourself and how any pain came to be. Just sit in the knowledge that holding on to negative emotions only restricts us from thriving at optimal levels.\n\nWith this attitude say to yourself 3 times in silence: _I now release_ ________ _with love and acceptance. I acknowledge that_ __________ _served its purpose for my growth and evolution._ Once you are finished with this affirmation begin to envision a ball of light glowing in your heart center. Breathe with the new awareness of compassion and love that you and everything is perfect right now in this moment. Feel your heart energy expand as you breathe compassion into your heart. Envision this glowing ball of light energy begin to spread through your entire body as you breathe naturally and with ease. If any resistance arises during the process and you notice any negative thoughts creeping in, interdict them with the statement: _This is a thought, it is not who I am._ Then move back into the visualization and continue to breathe for 5 more minutes with an open heart and a calm body and mind. When you are finished, sit up gently and bring your hands to your heart thanking it for its wisdom.\n\nSequence Appendix for Reference\n\n1. PEACEFUL WARRIOR\u2014MOST POSTURES HELD OR SEQUENCED THROUGH 5\u201310 BREATHS\n\n**PW1:** Seated Meditation\u2014Nostril Breathing\n\n**PW2:** Alternate Nostril Breathing\u2014Start inhale on left side, equal part count on inhale and exhale, retain breath at equal part count after inhale. Example: inhale through left nostril, close both nostrils retain, exhale right, inhale right nostril, close both nostrils retain, exhale left. Repeat for 7\u201314 rounds ending on exhale through the left nostril.\n\n**PW3:** Cow Pose\u2014Inhale\n\n**PW4:** Cat Pose\u2014Exhale\n\n**PW5:** Child's Pose\u2014Move into on exhale\n\n**PW6:** Transition through to Cow\u2014Move through on inhale\n\n**PW7:** Down Dog\u2014Move into on exhale\n\n**PW8:** Kneeling Warrior Each Side\u2014Move into on inhale, transition back to Down Dog on exhale\n\n**PW9:** Wide Angle Forward Fold\u2014Move into on exhale\n\n**PW10:** Roll up to standing while exhaling\n\n**PW11A:** 3 Part Movement Bringing Down the Heavens: 1. Inhale arms out and up, 2. continue to inhale as you lift heels and interlace fingers, 3. Exhale lower arms back alongside.\n\n**PW11B:** Sun Salute A 2X\n\n**PW12:** Warrior 1: Move into on Inhale\n\n**PW13:** Warrior 2: Move into on Inhale\n\n**PW14:** Side Angle Pose: Move into on Exhale\n\n**PW15:** Exalted Warrior: Move into on Exhale\n\n**PW16:** Plank: Move into on Inhale\n\n**PW17:** Push-up: Move into on Exhale\n\n**PW 18:** Upward Facing Dog: Move into on Inhale\n\n**PW19:** Downward Facing Dog: Move into on Exhale; Repeat on Left Side\n\n**PW 20:** Sharp Warrior: Move into on Exhale\n\n**PW 21:** Sun Salute A\n\n**PW 22:** Tree Pose: Move into on Inhale\n\n**PW 23:** Dancer: Move into on Inhale\n\n**PW 24:** Sun Salute A as a transition to seated position\n\n**PW 25:** Seated Forward Fold: Move into on Exhale\n\n**PW 26:** Table Top: Move into on Inhale\n\n**PW 27:** Cleansing Warrior: Bring knees into chest on exhale inhale straighten arms and bring knees away from chest\n\n**PW 28:** Bridge: Move into on Inhale\n\n**PW 29:** Bridge or Wheel: Move into on Inhale\n\n**PW 30:** Reclined Pigeon: Move into on Exhale\n\n**PW 31:** Happy Baby Pose: Move into on Inhale\n\n**PW 32:** Healthy Warrior: Move into on Exhale\n\n**PW 33:** Butterfly 1&2: Move into on Exhale\n\n**PW 34:** Reclined Warrior Pose: Move into on Exhale\n\n**PW 35:** Resting Pose: Breathe at natural pace, let go of victorious breath or any counted breathing pattern.\n\n**PW 36:** Seated Meditation\u2014Focus on revisit intention\/breath followed by compassion and peace meditation: Breathe at natural pace, let go of victorious breath or any counted breathing pattern.\n\n2. ZEN WARRIOR\u2014MOST POSTURES HELD OR SEQUENCED THROUGH 5\u201310 BREATHS\n\n**ZW1:** Mountain Pose: Breathe with Focus\n\n**ZW 2:** Sun Salute A 3x\n\n**ZW 3:** Sun Salute B 3x\n\n**ZW 4:** Brining Down the Heavens: 1. Inhale arms out and up, 2. continue to inhale as you lift heels and interlace fingers, 3. Exhale lower arms back alongside.\n\n**ZW 5:** Kicking Warrior: Inhale in Warrior 1 then Move into kick on Exhale\n\n**ZW 6:** Mountain Pose: Breathe with Focus\n\n**ZW 7:** Archer Warrior: 1. Inhale in Warrior 2 bring back hand so palms face each other 2. Through a pinhole shape of the mouth exhale in spurts and as you pull back the metaphoric bow\n\n**ZW 8:** Scooping the Moon: Move into on Inhale, Exhale as back of the hand returns to in front of forehead\n\n**ZW 9:** Warrior Musashi: Move into on Inhale and Exhale Strike\n\n**ZW 10:** Sun Salute A\u2014Finish in Down Dog and come to knees\n\n**ZW 11:** Camel: Move into on Inhale\n\n**ZW 12:** Reclined Warrior Pose: Move into on Exhale\n\n**ZW 13:** Wise Warrior Twist: Move into on Exhale and find length of spine on the Inhale\n\n**ZW 14:** Rock back and forth to standing in mountain: Rock back on inhale and forward on exhale\n\n**ZW 15:** Sun Salute A to push up lower on the belly\n\n**ZW 16:** 1 Arm Locust: Move into on Inhale\n\n**ZW 17:** Locust: Move into on Inhale\n\n**ZW 18:** Bow: Move into on Inhale\n\n**ZW 19:** Reverse Push-up: Move into on Inhale\n\n**ZW 20:** 5x Push-up: Exhale as you lower\n\n**ZW 21:** Up Dog or Cobra: Move into on Inhale\n\n**ZW 22:** Grateful Warrior-Child's pose with palms up: Move into on Exhale\n\n**ZW 23:** Cow: Move into on Inhale\n\n**ZW 24:** Cat: Move into on Exhale\n\n**ZW 25:** Seated Meditation position for Box breathing: Nostril breathing\n\n**ZW 26:** Stay in Seated or come to Resting Pose for Grounding and strength meditation: Breathe at natural pace, let go of victorious breath or any counted breathing pattern.\n\n3. FIT WARRIOR\u2014MOST POSTURES HELD OR SEQUENCED THROUGH 5\u201310 BREATHS, WHEN IN FIT MODULE THEN EACH MOVEMENT IS DONE ON AN INDIVIDUAL BREATH\n\n**FW 1:** Box Breathing: Nostril breathing\n\n**FW 2:** Seated Meditation: Breathe at natural pace, let go of victorious breath or any counted breathing pattern.\n\n**FW 3:** Cat\/Cow: Cow on Inhale and Move into on Exhale\n\n**FW 4:** Downward Facing Dog: Move into on Exhale after 5 breaths step feet forward\n\n**FW 5:** Forward Fold come to standing\n\n**FW 6:** Sun A 2x\n\n**FW 7:** Sun B 2x\n\n**FW 8:** Triangle: Move into on Exhale\n\n**FW 9:** Twisted Triangle: Move into on Exhale\n\n**FW 10:** Sharp Warrior: Move into on Exhale\n\n**FW 11:** Wide Angle Forward Fold: Move into on Exhale\n\n**FW 12:** Wide Angle Twist: Move into on Exhale, when complete on both sides come to mountain\n\n**FW13:** Fit Module x3\n\n**FM1:** Burpees, bring hands to floor\n\n**FM2:** Jump back to plank position\n\n**FM3 AND 4:** Push up\n\n**FM5:** Jump Forward\n\n**FM6:** Jump up and clap hands overhead\n\n**FM7:** Jumping squats, go into squat position, knees track over toes and the weight in heels\n\n**FM8:** Jump up and land back in squat to repeat\n\n**FM9:** Mountain Climbers, plank position and jump on foot forward to outside of hand\n\n**FM10:** Engaging core jump leading foot back as the back foot comes forward, count of 4 for 1 count of mountain climbers\n\n**FM11:** Flutter kicks, laying on back with palms on ground while hands are supporting back, chest is lifted, legs hovering a few inches above the ground, lift one leg higher to start\n\n**FM12:** as the leading leg lifts the other leg lowers slightly, count of 4 for 1 count of flutter kicks\n\n**FM13:** Sit-Ups, start on your back with feet together hands overhead\n\n**FM14:** With engaged abdominal muscles come up and touch your toes\n\n**FM15:** Push-Ups, plank position inhale\n\n**FM16:** Lower chest to ground on exhale, return to plank\n\n**FW14:** Scooping the Moon: Move into on Inhale, Exhale as back of the hand returns to in front of forehead\n\n**FW15:** Fighting Warrior-Punching: Inhale in Warrior and Exhale to punch\n\n**FW16:** Bringing Down the Heavens: 1. Inhale arms out and up, 2. continue to inhale as you lift heels and interlace fingers, 3. Exhale lower arms back alongside.\n\n**FW17:** Downward Facing Dog: Move into on Exhale\n\n**FW18:** Pigeon: Move into on Exhale\n\n**FW19:** Camel: Move into on Inhale\n\n**FW20:** Cat\/Cow: Cow on Inhale and Move into on Exhale\n\n**FW21:** Cleansing Warrior: Move into on Exhale and start with right side\n\n**FW22:** Healthy Warrior with movement: Move into on Exhale\n\n**FW23:** Bridge: Move into on Inhale\n\n**FW24:** Resting Pose: Breathe at natural pace, let go of victorious breath or any counted breathing pattern.\n\n**FW25:** Seated Meditation: Breathe at natural pace, let go of victorious breath or any counted breathing pattern.\n\n**FW26:** Box Breathing: Nostril Breathing\n\n4. FIGHTING WARRIOR\u2014MOST POSTURES HELD OR SEQUENCED THROUGH 5\u201310 BREATHS\n\n**FGW1:** Box Breathing: Nostril Breathing\n\n**FGW2:** Sun Salute A\n\n**FGW3:** Warrior 1 with Breath: Start with arms alongside the body and Move into Warrior 1 with arms out to side and elbows bent on Inhale\n\n**FGW4:** Chopping Wood: Move into on Inhale and Chop on Exhale\n\n**FGW5:** Warrior Breath 100x: Inhale arms up, exhale Pull arms down quickly, Inhale arms out in front, exhale arms in quickly. This is designed to build focus and heat.\n\n**FGW6:** Punching Warrior 20x each side: Move into on Inhale, exhale punch\n\n**FGW7:** Kicking Warrior 20x each side: Prep on inhale, Exhale kick\n\n**FGW8:** Combination Module of kick on punch 20x each side: Prep on inhale, Exhale kick and punch\n\n**FGW9:** Mountain Pose 5 breaths: Nostril Breathing\n\n**FGW10:** Sun Salute A\n\n**FGW11:** Seated Position with Breath of Fire: Inhale, Quick rapid Exhale as you pulse belly in and out\n\n**FGW12:** Victorious Warrior Visualization seated or standing meditation\n\n5. JEDI WARRIOR\u2014MOST POSTURES HELD OR SEQUENCED THROUGH 5\u201310 BREATHS\n\n**JW1:** Box Breathing and set intention: Nostril breathing\n\n**JW2:** ROM Drills: Nostril breathing\n\n**JW3:** Sun Salute A 2x\n\n**JW4:** Sun Salute B 2x\n\n**JW5:** Plank hold 30 seconds: Nostril breathing\n\n**JW6:** Windmill: Start on Inhale the twist is on an Exhale\n\n**JW7:** Qigong Forward Fold\u2014Hold 3 minutes: Nostril Breathing\n\n**JW8:** Ki Ub: Inhale as you Breathe into belly, Exhale as you bring hands to low back and then Inhale as you look up and Exhale as you bring hands down back of legs to the feet, Inhale as you bring hands up the calves and thighs and Exhale as the hands return back to the belly.\n\n**JW9:** Wide Angle Forward Fold: Move into on Exhale\n\n**JW10:** Sun Salute A to plank position\n\n**JW11:** Plank hold for 1\u20133 minutes: Nostril breathing\n\n**JW12:** Down Dog: Move into on Exhale\n\n**JW13:** Exalted Warrior each side: Move into on Inhale\n\n**JW14:** Plank: Move into on Inhale\n\n**JW15:** Push-up: Move into on Exhale\n\n**JW16:** Down Dog-walk feet forward: Move into on Exhale\n\n**JW17:** Forward Fold-roll up to standing: Exhale as you roll up\n\n**JW18:** Horse Stance: Move into on Exhale\n\n**JW19:** 9 Breath\u20143 sets: Inhale as you lean back, Exhale as you fold torso forward\n\n**JW20:** Mountain-Revisit intention: Nostril Breathing\n\n**JW21:** Seated or Standing Meditation\u2014Jedi Warrior Visualization: Breathe at natural pace, let go of victorious breath or any counted breathing pattern.\n\n6. AGELESS WARRIOR\u2014ALL POSTURES HELD OR SEQUENCED THROUGH 5\u201310 BREATHS\n\n**AW1:** Box Breathing: Nostril breathing\n\n**AW2:** Mountain Pose; Repeat 5x\n\n**AW3:** Warrior 1 w\/Breath Repeat 5x\n\n**AW4:** Warrior 2: Arms straight by the ears: Move into on Inhale, Exhale round upper back and wrap arms around the body\n\n**AW5:** Warrior 2: Back Palm Up: Move into on Exhale\n\n**AW6:** Exalted Warrior: Move into on Exhale\n\n**AW7:** Side Angle Pose: Move into on Exhale\n\n**AW8:** Wide Angle Forward Fold: Move into on Exhale\n\n**AW9:** Twisted Wide Angle Pose: Twist on Exhale\n\n**AW10:** Roll up to Standing: Roll up on Exhale\n\n**AW11:** Horse Stance: Move into on Exhale\n\n**AW12:** Sun Salute A\n\n**AW13:** Standing Meditation of Future Self in Perfect Health (option to do seated or in lying meditation form): Breathe at natural pace, let go of victorious breath or any counted breathing pattern.\n\n7. RECOVERING WARRIOR (PTSD SUPPORT AND HEALING)\u2014MOST POSTURES HELD OR SEQUENCED THROUGH 5\u201310 BREATHS\n\n**RW1:** Resting Pose: Breathe at natural pace, let go of victorious breath or any counted breathing pattern.\n\n**RW2:** Cleansing Warrior: Both knees come in on Exhale, Inhale straighten arms, Repeat 5x\n\n**RW3A:** Rock Back and Forth 3x finish lying on back\n\n**RW3B:** Reclined Pigeon Each Side: Move into on Exhale\n\n**RW4:** Bridge: Move into on Inhale\n\n**RW5:** Table Top: Move into on Inhale\n\n**RW6:** Seated Forward Fold A and B: Move into on Exhale\n\n**RW7:** Butterfly 1: Move into on Exhale\n\n**RW8:** Butterfly 2: Move into on Exhale\n\n**RW9:** Wise Warrior: Move into on Exhale\n\n**RW10:** Seated Forward Fold A and B: Move into on Exhale\n\n**RW11:** Seated Meditation with Alternate Nostril Breathing: Alternate Nostril Breathing\u2014Start inhale on left side, equal part count on inhale and exhale, retain breath at equal part count after inhale. Example: inhale through left nostril, close both nostrils retain, exhale right, inhale right nostril, close both nostrils retain, exhale left. Repeat for 7\u201314 rounds ending on exhale through the left nostril.\n\n**RW12:** Still Water Visualization\n\n8. HIP MOBILITY DRILL\u2014MOST POSTURES HELD OR SEQUENCED THROUGH 5\u201310 BREATHS\n\n**HMD1:** Sun A until Down Dog\n\n**HMD2:** Warrior 1: Move into on Inhale\n\n**HMD3:** Warrior 2: Move into on Inhale\n\n**HMD4:** Side Angle Pose: Move into on Exhale\n\n**HMD5:** Kneeling Warrior: Move into on Inhale\n\n**HMD6:** Twisting Warrior: Move into on Exhale\n\n**HMD7:** Pigeon: Move into on Exhale\n\n**HMD8:** from Down Dog Plank: Move into on inhale\n\n**HMD9:** Push-up: Move into on Exhale\n\n**HMD10:** Up Dog: Move into on Inhale\n\n**HMD11:** Down Dog\u2014Repeat on Left side and once finished with Left Side complete Sun A and return to standing\n\n*For all sequences except the PTSD we encourage you to add inversions at the end before meditation if you have learned them from an instructor and feel safe at home to do them on your own. Poses encouraged if suitable are headstand, handstand, shoulder stand, forearm balance, and plow pose.\n\n# CHAPTER 6\n\n# POSES AND MOVEMENTS\n\n_Yoga, an ancient but perfect science, deals with the evolution of humanity. This evolution includes all aspects of one's being, from bodily health to self-realization. Yoga means union\u2014the union of body with consciousness and consciousness with the soul. Yoga cultivates the ways of maintaining a balanced attitude in day-to-day life and endows skill in the performance of one's actions._  \n\u2013B. K. S. IYENGAR, _ASTADALA YOGAMALA_\n\nCONSIDER THIS CHAPTER YOUR MASTER REFERENCE SOURCE FOR DETAILED INSTRUCTIONS AND PHOTOS FOR ALL KOKORO YOGA POSES AND MOVEMENTS PRESENTED IN THIS BOOK.\n\n1. MOUNTAIN POSE\u2014TADASANA\n\nStanding with feet together or hip-width apart, the spine is in alignment and abdominals engaged. Shoulders are back and the chest is uplifted, the body is strong, and you are balanced on all four corners of your feet. This is the foundation for all standing poses.\n\n2. STANDING FORWARD FOLD\u2014UTTANASANA\n\nStand with feet together or hip-width apart. Draw the belly button into the spine and engage quadriceps and abdominal muscles while folding forward. If you have lower-back issues, keep the knees slightly bent; otherwise maintain strong, straight legs with weight evenly distributed throughout the feet.\n\n3. TRIANGLE POSE\u2014TRIKONASANA\n\nFeet are a minimum of 3 feet apart, with heels in alignment. Leading foot is open at a 90-degree angle, and the back foot at a 45-degree angle. Hips are open and all muscles of the body are engaged. Leading hand is on the floor, shin, or calf. The opposite arm is long and straight. Gaze is set at the fingertips.\n\n4. TWISTED TRIANGLE POSE\u2014PARIVRTTA TRIKONASANA\n\nFeet are at a minimum 3 feet apart with heels aligned. The leading foot is at 90 degrees and the back foot is at 65 degrees. Hips are aligned and the twist happens from the base of the spine through the top of the head. Leading arm is extended and straight in the air with eyes focused on fingertips. The opposite hand is on the outside of the leading foot or on the shin or thigh.\n\n5. WIDE-ANGLE FORWARD FOLD\u2014PRASARITA PADOTTANASANA\n\nFeet are at 3 to 4 feet apart, heels slightly out, hands are on the floor between the feet, press into the outer edges of the feet, bring top of the head to the ground or a block, lift the sit bones toward the ceiling as you lift the kneecaps engaging the thighs.\n\n6. TWISTED FORWARD FOLD\u2014PARIVRTTA PADOTTANASANA\n\nStart with the Wide-Angle Forward Fold. Begin with left hand in the middle of the feet, twist torso bringing the right hand up toward the ceiling or on the sacrum, switch to other side.\n\n7. SHARP WARRIOR\u2014PARSVAKONASANA\n\nFeet are at a minimum 2.5 feet apart, leading foot is at a 90-degree angle, back foot is at a 65-degree angle, torso folds forward with an active or rounded spine, hips are aligned, hands are in prayer position between the shoulder blades with palms pressing together (or palms placed on the hips or floor for more balance). Bring the forehead to the knee or chin to shin, and gaze is at the tip of the nose.\n\n8. WARRIOR 1\u2014VIRABHADRASANA A\n\nFeet are at a minimum 3 feet apart with the leading foot at a 90-degree angle and the back foot at a 65-degree angle, hips are aligned and the leading leg is bent so that the thigh is parallel to the ground, the back leg is straight, torso is in a slight upper backbend, arms are lifted alongside the head, actively extended by the ears with the hands together or apart; the gaze is forward or up to the fingertips.\n\n9. WARRIOR 1 WITH BREATH MOVEMENT\u2014VIRABHADRASANA A WITH MOVEMENT\n\nSee here for visual cues from the Fighting Warrior sequence.\n\nStart in Warrior 1, bring arms to relax alongside the body, lift arms up over the head, exhale, bend at the elbows drawing the shoulder blades down the back, inhale moving into a gentle upper backbend, exhale, return to starting position, and repeat 5 to 10 times.\n\n10. WARRIOR 2\u2014VIRABHADRASANA B\n\nFeet are at a minimum 3 feet apart with the leading foot at 90 degrees and the back foot at a 45-degree angle, hips are open and the leading leg is bent so that the thigh is parallel to the ground and the ankle and knee are in alignment, back leg is straight, torso is upright with an active spine, arms are extended out strong parallel to the ground and in alignment with the shoulders, the gaze is over the leading shoulder to the fingertips.\n\n11. EXALTED WARRIOR\u2014VIRABHADRASANA WITH BACKBEND\n\nStart in Warrior 1. Leading hand reaches up and back as you move deep into the lunge and take an upper backbend. Opposite hand wraps around the waist and the hand is placed on the lower back or on the opposite hip.\n\n12. DILIGENT WARRIOR\u2014PARIVRTTA VIRABHADRASANA\n\nSee here for visual cues from the Hip Mobility Drill supporting sequence.\n\nFeet are at a minimum 3 feet apart with the leading foot at 90 degrees and the back foot at a 65-degree angle. Hips are aligned and the leading leg is bent so that the thigh is parallel to the ground, knee and ankle are in alignment, and the back leg is straight, hands are in prayer position at the heart center and the opposite elbow to the leading leg is assisting in a deep twist that starts at the base of the spine spiraling up to the top of the head, the gaze is at the tip of the nose, and the abdominals are engaged.\n\n13. KNEELING WARRIOR 1, 2, 3\u2014ANJANEYASANA\n\nSee here for visual cues from the Hip Mobility Drill supporting sequence.\n\nStart in high lunge, bring back leg, knee and top of foot onto the ground; arms straight by the ears leading leg knee to track past the ankle and over the toes; interlace fingers behind the back and squeeze shoulder blades, elbows, and palms toward one another, continue to lift chest and take a deeper backbend; release the hands coming out of the backbend as you bend the back leg, take the same hand as back leg and grab the outside of the ankle or foot bringing the heel to the hip (this stretches the quadriceps), opposite hand rests lightly on leading thigh.\n\n14. RESTING WARRIOR\u2014SUPTA BADDHA KONASANA\n\nStart in Butterfly 1. With a bolster or rolled-up blankets parallel to the spine, gently lean back so that you are supported by the props and can completely relax, palms face up, eyes closed.\n\n15. STANDING SIDE ANGLE POSE\u2014UTTHITA PARSVAKONASANA\n\nStarting in Warrior 2 reach out with your leading hand and either rest your elbow on the bent thigh or place your hand on the outside of your foot, extend your other arm at an angle in alignment with your back leg with your palm facing the floor and the fingers together. Gaze up toward your palm or elbow.\n\n16. SUN SALUTATION A\u2014SURYA NAMASKARA A\n\nStart in Mountain Pose. Inhale, arms over the head, and look up. Exhale, Forward Fold, inhale, extend the back and look forward. Exhale, plant your hands on the floor, and step or jump back into plank and continue into a push-up. Inhale, Up Dog, exhale, Downward Dog (take 5 breaths in Downward Dog). After your last exhale, look forward and walk or jump to the top of your mat, inhale, extend back, and look forward. Exhale, Forward Fold, inhale, strong legs arms out and around as you come back to standing bringing arms over the head. Exhale, back to Mountain Pose.\n\nREPEAT ON LEFT SIDE AFTER VINYASA\n\n17. SUN SALUTATION B\u2014SURYA NAMASKARA B\n\nStanding in Mountain Pose, inhale, arms over the head, and look up as you sit into Chair Pose. Exhale, Forward Fold, inhale, extend the back, and look forward. Exhale, plant your hands, and step or jump back into plank and continue into a push-up. Inhale, Up Dog, exhale, Downward Dog, inhale, step your right foot forward, and land in Warrior 1 and hold the pose for 3 breaths. On exhale bring hands to the ground, step back, and go through a push-up. Inhale to Up Dog, exhale, Downward Dog, inhale, step the left foot forward into Warrior 1 and hold the pose 3 breaths. On exhale bring hands to the ground, step back, and go through a push-up. Inhale to Up Dog, exhale, Downward Dog, and hold the pose for 5 breaths. After your last exhale, look forward and walk or jump to the top of your mat, inhale, extend back, and look forward. Exhale, Forward Fold, inhale, strong legs arms out and around as you come back to Chair. Exhale, back to Mountain Pose.\n\n18. UPWARD-FACING DOG\u2014URDHVA MUKHA SVANASANA\n\nHands are under the shoulders or slightly back and the hips and thighs are lifted as you press the tops of your feet into the ground. Your gaze is upward with a long neck as you take an upper backbend.\n\n19. COBRA POSE\u2014BHUJANGASANA\n\nSee here for visual cues from the Zen Warrior sequence.\n\nHands are alongside the upper rib cage or under the shoulders and the thighs are on the ground. You are still pressing into the tops of the feet as you lift only the chest off the ground in a gentle backbend.\n\n20. CHAIR POSE\u2014UTKATASANA\n\nSee here for visual cues from the Sun Salutation B sequence.\n\nFeet are either together or hip-width apart. Hips, knees, and ankles are all in alignment, active spine, and the weight is slightly in the heels of the feet so that you can see all 10 toes, the arms are straight by the ears, and the gaze is upward.\n\n21. GRATEFUL WARRIOR\u2014BALASANA\n\nBring your knees wide apart, rest your forehead on the floor or a block, arms are stretched out in front of you or alongside the body. This pose is a posture of relaxation and surrender.\n\n22. BRIDGE POSE\u2014SETU BANDHA SARVANGASANA\n\nLying flat on the back with knees bent, feet hip-width apart, arms alongside the body, palms are facing the ground, feet firmly planted, and the knees in alignment with the heels, press into the ground with palms, shoulders, and feet. Lift hips to the sky while engaging the inner-thigh muscles. Arms and hands can remain in their initial position or if you need more support, bring hands to the lower back and elbows onto the floor.\n\n23. WHEEL POSE\u2014CHAKRASANA OR URDHVA DHANURASANA\n\nLying flat on the back with knees bent and hands under the shoulders with fingertips facing the heels, elbows are pointed to the ceiling, at the same time press palms and feet into the ground to lift the body into a backbend as the shoulder blades draw toward one another and away from the ears, and the inner thighs and hamstrings engage.\n\n24. CAMEL POSE\u2014USTRASANA\n\nStanding on your knees with the toes tucked under, hips, knees, and ankles are all aligned, and hamstrings are engaged. Hands are as if you were putting your fingertips in your back pockets, rotating the inner thighs toward one another. Imagine your body creating the shape of a rainbow and begin and maintain the arch from the upper back as you move into the backbend.\n\n25. BOW\u2014DHANURASANA\n\nLying on your belly, bend your knees, and hold on to the outside of your ankles with the thumbs pointing down. Bring feet together or at hip-width apart, keep knees at hip-width apart, press belly and hips into the ground, and lift feet toward the ceiling. Inner thighs and hamstrings are engaged as well as the back muscles, reach the chest forward, and gaze softly up.\n\n26. LOCUST\u2014SALABHASANA\n\nThis backbend starts lying on your belly, legs straight, and big toes together. Back of the hands are on the ground alongside the hips, press into the ground with belly, hands, and hips, as you use the posterior chain to lift the body.\n\n27. BACK ALIGNMENT\u2014APANASANA (WITH OR WITHOUT MOVEMENT)\n\nLying on the back with the knees drawn into the chest, hands are on the knees and the arms are straight, inhale, exhale, use abdominals to press your sacrum into the ground as you bend the elbows and hug the knees close to the chest, inhale, return to start position, repeat 5 to 10 times, without movement; bend elbows and hug knees into chest.\n\n28. CLEANSING WARRIOR\u2014EKA PADA APANASANA\n\nSee here for visual cues from the Peaceful Warrior sequence.\n\nLying on the back with the knees drawn into the chest, straighten the left leg to the ground so that the heel is on the ground and both feet are still flexed, take both hands below the right knee on the shin and bend elbows engaging the biceps and triceps, switch to the other side.\n\n29. HEALTHY WARRIOR\u2014SUPTA MATSYENDRASANA\n\nLying on the back with both knees drawn into the chest, take the hands to the knees, and draw the knees into the body. Engage abdominals as you twist extending arms out like the letter T.\n\n30. BUTTERFLY 1\u2014BADDHA KONASANA 1\n\nSitting with feet together and knees bent, take your hands to the outside of the feet and open them up so that the pinky toes are touching and the big toes are moving outward toward the ground, knees are actively moving toward the floor, abdominals engaged, chin to chest.\n\n31. SHOULDER OPENER\u2014BADDHA KONASANA WITH GOMUKHASANA ARMS\n\nSit in Butterfly 1, inhale, bring arms out to the side, stretch up 1 arm alongside the head, lower the other arm as it externally rotates, and you bring that hand between the shoulder blades, bend the lifted arm and meet the hands interlacing the fingers. If this pose is unachievable, grab a strap as a prop for assistance.\n\n32. HEADSTAND\u2014SIRSASANA\n\nBegin with knees on the ground and create a base by bringing the hands, interlaced wrists, and elbows on the ground in a triangle shape. Palms create a cup for the back of your head as you place the top of your head on the ground, shift legs into a Downward Dog position, walk the legs in toward the body, lift legs into the air using core strength. No weight on the head, action is in the arms and core.\n\n33. SHOULDER STAND\u2014SARVANGASANA\n\nStart lying flat on your back with arms over your head, inhale, exhale, bring your legs behind you as you bring the arms along the side of the body, toes meet the floor behind you, the legs are very active, the neck still has its natural curve, and the weight is in the shoulders. Place palms on lower back, inhale, bring the legs over the head in alignment with the hips, feet are pointed.\n\n34. RECLINED PIGEON\u2014VARIATION ON BACK OF SUPTA RAJAKAPOTASANA\n\nLying on the back with knees bent and feet on the ground at hip width, take the right ankle to the opposite knee, feet are flexed, thread right arm through the thighs, and interlace the fingers at the back of the left thigh, draw the left knee into chest (leg can remain bent or straightened), continue to draw the right knee away from the body and left knee toward the body.\n\n35. HAPPY BABY POSE\u2014BALASANA\n\nLying on your back, bring knees into the chest and then straighten the legs up so that the feet are facing ceiling, separate legs a bit wider than hip-width apart, bend the knees taking hands to the back of the thighs, calves, ankles, or feet. Bring knees to the outside of the rib cage as you actively press feet toward the sky while pulling down with your hands.\n\n36. SEATED MEDITATION\u2014SUKHASANA\n\nSitting on a chair, bench, pillow, block, blanket, or the ground, make sure you are comfortable, and your abdominals and spine are active. Top of the head is like an antenna open and receptive, abdominals are engaged, and the palms rest on the thighs or knees facing up or down.\n\n37. RESTING \"DEAD MAN\" POSE\u2014SAVASANA\n\nThis pose is where you let go completely and allow yourself to be present and still. You are lying flat on your back with your arms relaxed by your sides, palms facing up, and feet relaxed. This pose is a practice of nonattachment and release. Practice breathing without effort; surrender yourself into your breathing.\n\n38. ALTERNATE NOSTRIL BREATHING\u2014NADI SHODHANA\n\nSit in a comfortable position with the spine active and abdominals engaged. Left palm is resting on the left knee; right hand has the middle and pointer finger tucked in so that the ring, pinky, and thumb are all extended. To start, exhale the air out completely, close the right nostril lightly with the thumb, and inhale through the left. Close both nostrils using the thumb and the ring finger and hold the breath. Keeping the left nostril closed, exhale through the right. You can do this sequence with or without retentions.\n\n39. TWISTING WARRIOR\u2014PARIVRTTA PARSVAKONASANA VARIATION\n\nStart in Warrior 1. Shift back foot so that you are on the ball of the foot and pressing out through the heel in a lunge. Inhale, exhale, move hands to prayer position at the heart center, engage the abdominals as you hook opposite elbow to leading leg, twist from the base of the spine through the top of the head, eyes gazing gently over the shoulder, engage the legs, stay strong in your lines of energy.\n\n40. HUMBLE WARRIOR\n\nFeet are about 2.5 to 3 feet apart with the leading foot at a 90 degree angle and the back foot at a 65 degree angle. Hips are squared and the leading leg is bent so that the thigh is parallel to the ground and the back leg is straight. Arms are behind with the fingers interlocked while squeezing the shoulders together. The warrior is bowing forward so that the leading shoulder is on the inside of the leading leg and the chin is sealed to the chest. The gaze is at the nose.\n\n41. SEATED FORWARD FOLD\u2014PASCHIMOTTANASANA\n\nSitting strong with legs out in front, feet are active, inhale, reach arms above the head, exhale, Forward Fold placing hands or a strap around the feet, keep drawing the belly button toward the spine as you press out through the balls and heels of your feet.\n\n42. WISE WARRIOR 1\u2014MARICHYASANA\n\nSitting upright, start with folding left leg so that the heel is alongside the right hip. Bring the right foot to the outside of the left thigh, right knee facing the ceiling, place your right hand behind you. Inhale, left arm up, exhale. Engaged in the abdominal twist, bring your left elbow or hand to the outside of your right knee, top of the head parallel to the ceiling, and both sit bones on the ground. Wise Warrior 1 variation\u2014with the bottom leg straight.\n\n43. TREE POSE\u2014VRKSASANA\n\nStart in Mountain Pose. Shift weight onto left leg and lift the right leg up. Take the lifted leg's foot to the inside of the standing leg's calf or thigh. Do not put the foot on the knee joint, press the leg into the foot and the foot into the leg. Hands are in prayer position at the front of the chest.\n\n44. VICTORIOUS WARRIOR\u2014VIRASANA\n\nStart on your knees, bring the knees together or at hip-width apart, and sit on the inside of your feet so that the heels touch your hips. Press the tops of your feet into the ground and rotate your inner thighs toward one another, knees are touching the ground (if they are not, sit on a block), hands are on the knees, and the chin is on the chest. Gaze is at the tip of the nose.\n\n45. TABLETOP\u2014PURVOTTANASANA VARIATION\n\nStart lying on your back, bring feet to the ground, knees and ankles in alignment, bring hands under the shoulders with palms on the floor, press into the hands and the feet as you lift your body up, engage the posterior chain, gaze at the tip of the nose.\n\n46. QIGONG, OR WATER MILL WITH FORWARD FOLD\n\nUse blocks if necessary. Looks like Forward Fold, but it so much more.\n\nMain components:\n\nWeight on balls of feet\n\nLegs pressing together, ankles, calves, knees, thighs\n\nFingers clasped, hands pressing into blocks\n\nBreathing deep to lower Dantian\n\nActive through pose\n\nInhale, remember to press legs, exhale, press down with hands\n\nExit roll spine up\n\nPress down with hands, keep pressing legs\n\n47. BRINGING DOWN THE HEAVENS\n\nSee here for visual cues from the Peaceful Warrior sequence.\n\nBend the knees. Inhale and with very soft muscles arc the hands out to each side, out and then up to the face with palms slightly turned toward the body, exhale and lower hands down the midline. The intention is to be completely relaxed and balance the energies of Sky\/Heaven and Earth\/Yin Yang. On the inhale, imagine collecting earth energy and when the hands reach the level of the shoulders, imagine taking in the energy of the sky. On the exhale intend to balance those energies in the your body. Pose is also done to help wipe the slate clean exercise.\n\nSTANDING QIGONG MEDITATION\n\nKnees are bent slightly, weight on the front 1\/3 of your feet. Hips are tucked gently. Lengthen spine up from the base to the top of your head. Soften all the muscles and pay attention to the space that your body occupies. Breath is slow and deep to the lower abdomen.\n\n48. NARROW HORSE STANCE\n\nFeet are shoulders width and toes are very slightly pointing in. Knees bent to 90 degrees or as close as you can get. Arms are actively pressing out in front, right hand over left. Shoulders are actively pressing down. Eyes are focused straight ahead looking at the back of your hands. Knees should be active pressing in like you are trying to hold a medicine ball between them. Keep your spine as straight as possible. This is a survival exercise, so do your best to maintain form for 3 minutes: It will be challenging. To finish, turn palms into fists, rise up on your toes while bringing your fists into you ben elbows; pulling your hands in and drawing the energy. Then step your feet together, turn palms down, and press down with both hands holding them with fingertips facing forward, wrists extended. Press the legs together as in the Water Mill\u2014extended forward bend. Take 3 deep breaths while continuing to press the legs together.\n\n49. DANCING WARRIOR\u2014NATARAJASANA\n\nStanding with feet together, shift weight to the left foot and lift right foot toward your buttocks. Take your right hand and grab the inside or outside of your right foot. Bring the left arm straight by your head and begin to kick your right foot into your right hand as you reach your chest forward and create an arch with the back: 50 percent back, 50 percent forward.\n\n50. DOWNWARD-FACING DOG\u2014ADHO MUKHA SVANASANA\n\nWith your hands and feet on the ground your body is creating an inverted-V position. Quadriceps are engaged and your heels are on the ground or moving in that direction. All knuckles of your hands are pressing into the ground as you engage your abdominals and lift your sit bones toward the ceiling.\n\n51. PLANK\n\nStrong, stafflike position with the whole body. Hands are pressing firmly into the ground. It is like the beginning of a push-up. Press the heels toward the wall behind you as you balance on your toes. Whole body is engaged.\n\n52. WARRIOR BREATH\n\nWarrior Breath is a strong inhale and exhale through the nose to the upper chest. The focus is on the inhale. Slightly rock your upper body back as you inhale. Slightly rock forward with the exhale. The most important component to remember is that it is a warrior's breath, so it is strong. Take 9 (or more breaths for each set). On the last breath hold for 3 seconds, swallow down, and round out your abdomen. Keep abdomen gently pressing out as you exhale slowly through your lips. The exhale is called the Eternal Exhale. Some people take 2 to 3 minutes to exhale completely. During this exhale relax and visualize.\n\n# CHAPTER 7\n\n# PHYSIOLOGICAL AND PSYCHOLOGICAL BENEFITS OF YOGA\n\nYoga is not a religion. It is science, science of well-being, science of youthfulness, science of integrating body, mind and soul.  \n\u2013AMIT RAY\n\nIN THIS CHAPTER WE WILL TAKE A LOOK AT CURRENT RESEARCH ON THE MANY BENEFITS OF YOGA, AND KOKORO YOGA IN PARTICULAR. BUT BEFORE WE DIG THROUGH SOME OF THE MODERN RESEARCH THAT HAS EMERGED IN RECENT YEARS THAT HAS HELPED SCIENTISTS UNDERSTAND WHAT A YOGA PRACTICE CAN DELIVER, I FIRST WANT TO ENCOURAGE YOU TO BE YOUR OWN SCIENTIST: TO HONOR ANY SKEPTICISM YOU MAY HAVE AND FIND A BALANCED APPROACH to being analytical in your appraisal of yoga, but to also be rigorous and dedicated to testing it out for yourself.\n\nI'm a big believer in trusting through verification. By giving this training a concentrated, objective, personal trial, you'll be able to verify the benefits I'm discussing. What I don't want to see you do is to slide into the trap of prejudging yoga as an excuse to not give it a valid try.\n\nLack of time also should not be a barrier. Kokoro Yoga is designed for a busy individual going 100 mph, and I wouldn't have it any other way. So even if it's 5 or 10 minutes a day for 3 months, I encourage you to conduct your own science by testing out the program and noting how it works for you.\n\nYoga has thousands of years of subjective science within its foundation, but for the Western audience seeking validation from a peer-reviewed research study, it's only in recent years that we have plowed deeper into the subject. Scientists and medical professionals\u2014in pursuit of effective, low-cost health-care solutions, are using high-tech tools to uncover the most detailed and objective data in history. For example, they are now able to chart changes in brain growth and function (neoplasticity), hormonal and electrical patterns associated with thoughts and emotions, and changes in gene expressions from exercise that lead to physiological improvements (epigenetics). There's much about yoga that I believe is still out of the current range of modern science to test, but I'm confident that if you integrate Kokoro Yoga rituals and disciplines into your life, you won't care about what the scientists say, you will just love the benefits and growth you are experiencing.\n\nBENEFITS ACROSS THE SPECTRUM\n\nThe following is a holistic review of the benefits, both physiological and psychological, of what you can expect to get from a disciplined, daily practice:\n\nBody Control\n\nYoga develops body control through an acute awareness of body position and movement. This control extends to awareness of how you move safely and effectively throughout each day: from standing to walking, running, lifting, or even the inherently unhealthy act of sitting. It will allow you to maintain connection with and move from your center (the Hara in Japanese and Dantian in Chinese), and extend to control over your physiological adaptation to stress.\n\nCore Strength\n\nAs mentioned earlier, yoga shares a common history with the martial arts, and engaging the core of your body while moving is a focus of the movement practice. If you have some training in martial arts, then you know what I mean by this. Others will have to experience it. You will learn to sink into a stable position of deep balance and engage all the muscles of your core as your primary source of physical strength and stability. Too often we focus on the extremities (arms and legs) in training, ignoring the critical source of power of the core, which is more than one's abs. It is the entire body minus the limbs and head. Building a strong core starts with deep connection to the muscles protecting the spine, engaging the root near your tailbone, and extending to all of the supporting muscles of your glutes, lower back, abs, chest, and obliques.\n\nConcentration\n\nIn yoga practice, you sharpen your ability to concentrate on one thing, and for long periods of time, by focusing your mind on a single point, such as your breathing or the structure of a pose. Research clearly supports how this brand of exercise elevates the ability to concentrate. In one study, 60 subjects were involved in a 3-month retreat where they practiced focusing on the breath. The group was split into those who participated in the practice and those who were on a wait-list group that weren't. The test they were given involved watching lines flash on a computer screen. Each time they saw a line that they believed was shorter than the others, their job was to click the mouse, an exceptionally boring task where the mind was apt to wander. Those performing concentration practice consistently demonstrated significantly greater capacity to focus than those who weren't. As reported when the study was published in 2010, the researchers at University of California, Davis, wrote, \"Training produced improvements in visual discrimination that were linked to increases in perceptual sensitivity and improved vigilance during sustained visual attention.\"\n\nFlexibility and Durability\n\nFor many, this is the sum total of what yoga is expected to deliver. Through the work of the sequences, you develop flexibility of your muscles and durability of your spine, joints, and connective tissues.\n\nEnergy Integrity\n\nControlling and enhancing energy, which can be conceptualized as life force, through breath control and visualization drills. Learning to draw energy into the body, to move it, and enliven the body at a cellular level, comes from a long-term practice of yoga.\n\nDetoxification\n\nOne of the first benefits I registered in my initial exposure to Hot Yoga was the detoxification of the muscles, the blood, and the organs. We accomplish this in Kokoro Yoga without the need for a superheated room by using breathing techniques and, of course, vigorous movement. These detox benefits can be keen for desk warriors in the world. Prolonged sitting not only degrades the tissues of the hip capsule, but also weakens and shortens all of the machinery of the trunk that protects the spine. As you can feel after unhinging yourself from the window seat after a long flight, prolonged periods of sitting shut off the lymphatic system and retard the removal of waste products from the muscles, connective tissues, and nervous system.\n\nImproved Immune Function\n\nHard-charging athletes typically have some experience in what is often referred to as \"overtraining syndrome.\" Overtraining means, obviously, the athlete has been overdoing it: relentless training without adequate recovery. Symptoms include chronic fatigue, iron deficiency, states of depression, upper-respiratory infections, and a sharp decline in performance. This syndrome is not limited to the sports arena: special operators, first responders, and those in high-stress business pursuits can also get caught in this psychophysical quicksand. When decline in performance begins to show up in the training journal, the highly motivated individual can react by training even harder, making things worse.\n\nIn specific regard to overtraining and the prevention of overtraining, first consider the vast complexity of the immune system, with multilayers of protection throughout the body that works tirelessly to screen out or kill viruses, microbes, pathogens, and cancers.\n\nAs you'll note from the chart, the quality of your immune function is affected not just by training but by just about every part of your life that you can imagine. Stress is stress, and if on top of your training stress, you are absorbing vast amounts of stress, incurred from either bad diet, lack of sleep, poor relationships, and the like, then your immune system will ultimately fall into what is known as immunosuppression.\n\nAlong with the basic habits of good health, like a smart diet, drinking plenty of water, and getting adequate sleep, consistent yoga practice will help boost your immune function. A 2013 Bloomberg article reported that, \"Scientists are getting close to proving what yogis have held to be true for centuries\u2014yoga and meditation can ward off stress and disease.\" The story centered on a 5-year study being conducted at the Harvard Medical School that was examining the effects of yoga and meditation on brain activity and gene function.\n\nJohn Denninger, a psychiatrist at Harvard Medical School, is leading the study on how the practices affect genes and brain activity in the chronically stressed. His latest work follows a study he and others published earlier this year showing how so-called mind-body techniques can switch on and off some genes linked to stress and immune function. Using neuroimaging and genomics technologies, Denninger and his colleagues are finding that yoga can switch specific genes, related to immune function, on and off, and how yoga can play a powerful role in reducing hypertension and preventing depression. \"There is a true biological effect,\" he told Bloomberg. \"The kinds of things that happen when you meditate do have effects throughout the body, not just in the brain.\"\n\nIn another key study conducted at Ohio State, surgical nurses, who experienced high levels of stress through their work and proximity to death, were shown to have a 40 percent decrease in salivary alpha amylase, a stress marker, after practicing yoga on a consistent basis.\n\nLongevity\n\nNeuroscientists have come to believe that the toll of aging on the human body and brain is an outcome of disuse rather than use. In fact, in pioneering research conducted in the 1970s at the University of California (Berkeley and San Francisco), it was discovered that the brain responds favorably, at the microscopic level, to movement and stimulating experience. It was when usage came to a stop that the part of the brain associated with the usage weakened.\n\nIn this study, the scientists determined that the brain was \"neuroplastic,\" a term meant to describe how connections between brain cells are circuitlike\u2014strengthening, changing, or weakening in response to how and how much the circuits are being used.\n\nThe implications of the neuroplasticity model are huge. In a long-term study performed at Cardiff University in the United Kingdom, researchers took a detailed look at the effects of exercise and lifestyle choices on 2,235 men, ages 45 to 49, over the course of 30 years. The results were powerful: consistent exercise was the most cogent factor in reducing the risk of dementia by 60 percent. In reporting on the magnitude of this study, _The Wall Street Journal_ said, \"Imagine if there were a drug that could reduce the risk of dementia by 60%. It would be the most talked-about drug in history.\"\n\nBrain exercise has been shown to improve cognitive computing power against the grain of decline normally associated with aging. After completing 10 1-hour brain exercise sessions, 2 per week over the course of a month and a half, the subjects demonstrated remarkable effects on their ability to think, reason, and function even 10 years after the brain exercise workouts.\n\nThat's just the beginning when it comes to what a mental training exercise, like meditation, can do. Studies at the University of California, Davis, have looked at the effect that meditation has on telomerase activity in the brain\u2014telomerase is known as the \"immortality enzyme.\"\n\nThe key action can be found in the infinitesimal corners of genes in what are called \"telomeres.\" Telomeres are found at the ends of chromosomes in sequences of DNA. Typically, they shorten every time the cell divides, and when they get too short, the cell dies. Telomerase actively rebuilds and lengthens the telomeres, consequently promoting a longer cell life, and studies have suggested that by higher amounts of telomerase equates to improved states of mental and physical health, and that it can have a direct role in preventing stress-related aging rates.\n\nResearchers at UCLA found that a mere 12 minutes of yoga per day, increased telomerase by 43 percent.\n\nIncreased Stamina and Endurance\n\nIn a study published in 2013, researchers in Boston took a high-tech look at what they termed the \"relaxation response\" that occurs from yoga and meditation. Using blood samples taken before and after a meditation session, the blood was analyzed to extract gene transcription profiles. The results were powerful and all-encompassing: \"Practice enhanced expression of genes associated with energy metabolism, mitochondrial function, insulin secretion and telomere maintenance, and reduced expression of genes linked to inflammatory response and stress-related pathways.\" The researchers found that the greatest benefits were harvested after a long-term practice, with improved energy metabolism and mitochondrial function some of the especially appealing benefits that an endurance athlete might be interested in.\n\nCognitive Performance\n\nA New York Academy of Sciences review showcased the value of yoga in regards to cognition and mental performance, with researchers emphasizing that yoga may be an effective intervention for the elderly dealing with a decline in memory and other aspects of brain performance. The research team concluded: \"Studies involved a wide variety of meditation techniques and reported preliminary positive effects on attention, memory, executive function, processing speed, and general cognition.\"\n\nDecrease in Cellular Inflammation\n\nDecreasing cellular inflammation is important for overall health, but also for peak performance. There have been some breakthrough studies being performed using blood testing that are answering the question of why yoga has a positive effect on cellular inflammation. Since cellular inflammation is one of the root causes of type 2 diabetes and chronic diseases like cancer (as well as Alzheimer's, which is predicted to join the diabetes epidemic), this finding is particularly forceful. In the athletic training world, a lot of attention is paid to the effect diet has on cellular inflammation. Basically, a crappy, high-carb diet (rich in processed foods and sugars) sends the body into a state of hyperglycemia, or chronic state of high-blood sugar, because of the fatigue associated with overtapping the insulin response that occurs when we eat a high-carb meal. Hyperglycemia is the precursor to diabetes and a host of related chronic diseases that you want to prevent at all costs. In addition to actively getting a handle on your diet, research shows that yoga and meditation can be a powerful anti-inflammatory. In a study published in _The Clinical Journal of Oncology,_ Ohio State scientists looked at three different cytokine levels in the blood in breast cancer survivors. The cytokines analyzed were distinct proteins commonly used as markers for cellular inflammation levels. After 12 weeks of yoga practice, the subjects of the study showed a 10 to 15 percent lowering in all three cytokine markers.\n\nDepression and PTSD\n\nA 2008 RAND Corporation study indicated that one out of five combat troops that had returned from the wars in Iraq or Afghanistan met the criteria for PTSD. Incorporating an assortment of studies supporting that yoga is an effective management tool in regards to chronic depression and PTSD, psychiatric researchers at the Boston University School of Medicine took an in-depth look at the mechanisms involved with PTSD and has proposed that there are \"far-reaching implications for the integration of yoga-based practices in the treatment of a broad array of disorders exacerbated by stress.\"\n\nLower Back Pain\n\nIn 2008, the Medical Expenditure Panel Survey collected a data set that suggested that 100 million American adults were affected by chronic pain, back pain, and arthritis. The study estimated that the health-care costs due to these maladies, when you add up the productivity costs and health-care costs, is in the range of $560 to $635 billion\u2014 _annually._ In 2011, the largest study ever conducted on the subject, published by the _Archives of Internal Medicine,_ found that 12 weeks of yoga diminished symptoms and improved overall back function. As far as research goes, yoga has been highlighted as an answer to lower back pain, a problem that costs the United States billions in terms of lost productivity from the workforce.\n\nPresence\n\nPower is found by focusing on the right thing, right now. A quality of the peak performance flow state is to be fully in the moment. Not distracted by emotionally charged memories or the fear of what may happen in the future, but being comprehensively engaged with the present moment. Kokoro Yoga training will guide you on how to use your mind powerfully in the future and past, so that you don't need to dwell there, but can release those time-based mental constructs and stay in the here and now\u2014and perform.\n\nFitness and Wellness\n\nAs discussed in this chapter, research has found that yoga is helpful in treating high-blood pressure, diabetes, back pain, sleep problems, depression, anxiety, stress, and more. But the benefits can be enhanced with more rigorous interval training. Before I began my practice of yoga, I tried to use yoga programs for my fitness needs. I soon learned that they were a poor substitute for real functional fitness, the type required of warriors and athletes. Though it has some cardio benefit, traditional yoga focuses mostly on balance, flexibility, and core strength. Because it is lacking in strength, stamina, work capacity, and durability, we introduced more intense functional fitness routines to Kokoro Yoga.\n\nA POWERFUL FUTURE\n\nMy intent throughout this chapter was to explore the benefits and take a short dive into the science behind the principles in this book. I believe Kokoro Yoga is a Trojan horse ready to unleash a host of benefits, ultimately leading to the highest levels of performance and even consciousness itself. I know, it may sounds too good to be true, but if you stay with me and begin a daily routine that meets your practical needs, body type, and goals, then you will be planting the seeds for a powerful future.\n\nREFERENCES\n\nA quick note: I included this chapter as review of the more recent research that is currently being conducted in regards to yoga, and as such have included references so you can validate what's being learned.\n\nBhasin, M. K., J. A. Dusek, B-H Chang, M. G. Joseph, J. W. Denninger, et al. \"Relaxation Response Induces Temporal Transcriptome Changes in Energy Metabolism, Insulin Secretion and Inflammatory Pathways.\" _PLoS ONE_ vol. 8, no. 5 (2013): e62817. doi:10.1371\/journal.pone.0062817.\n\n\"Does Yoga Really Do the Body Good?\" American Council on Exercise, September\/October 2005. www.acefitness.org\/getfit\/studies\/YogaStudy2005.pdf.\n\nElwood, P., J. Galante, J. Pickering, S. Palmer, A. Bayer, et al. \"Healthy Lifestyles Reduce the Incidence of Chronic Diseases and Dementia: Evidence from the Caerphilly Cohort Study.\" _PLoS ONE_ vol. 8, no. 12 (2013): e81877. doi:10.1371\/journal.pone.0081877.\n\nGaskin, D. J., Richard, P. \"The Economic Costs of Pain in the United States.\" _Journal of Pain_ vol. 13, no. 8 (2012): 715\u201324.\n\n\"Harvard Yoga Scientists Find Proof of Meditation Benefit.\" Bloomberg News, November 21, 2013. www.bloomberg.com\/news\/articles\/2013-11-22\/harvard-yoga-scientists-find-proof-of-meditation-benefit.\n\n\"Invisible Wounds of War. Psychological and Cognitive Injuries, Their Consequences, and Services to Assist Recovery.\" RAND Corporation, January 15, 2008. www.rand.org\/content\/dam\/rand\/pubs\/monographs\/2008\/RAND_MG720.pdf.\n\nJacobs, T. L., Epel, E. S., Lin, J., Blackburn, E. H., Wolkowtiz, O. M., et al. \"Intensive meditation training, immune cell telomerase activity, and psychological mediators.\" _Psychoneuroendocrinology_ vol. 36, issue 5 (2011): 664\u201381.\n\nGlaser, J., Bennett, J. M., Andridge, R., Peng, J., et al. \"Yoga's Impact on Inflammation, Mood, and Fatigue in Breast Cancer Survivors: A Randomized Controlled Trial,\" _Journal of Clinical Oncology_ vol. 32, (January 2014): 1040\u20131049.\n\nLavretsky, H., P. Siddarth, N. Nazarian, N. St. Cyr, D. S. Khalsa, et al. \"A pilot study of yogic meditation for family dementia caregivers with depressive symptoms: Effects on mental health, cognition, and telomerase activity.\" _International Journal of Geriatric Psychiatry_ vol. 28, no. 1 (2013): 57\u201365.\n\nMacLean, K. A., Ferrer, E., Aichele, S. R. Bridwell, D. A., Zanesco, A. P., et al. \"Intensive Meditation Training Improves Perceptual Discrimination and Sustained Attention.\" _Psychological Science_ vol. 21, no. 6 (2010): 829\u201339.\n\nGothe, Neva P. \"The effects of an 8-week Hatha yoga intervention on executive function in older adults.\" _Journals of Gerontology_ 69 (2014): 1109\u201316.\n\nRosenzweig, M. R. \"Aspects of the Search for Neural Mechanisms of Memory.\" _Annual Review of Psychology_ 47 (1996): 1\u201332.\n\n# CHAPTER 8\n\n# DEVELOPING A PERSONAL PRACTICE\n\n_Today I will do what others won't, so tomorrow I can accomplish what others can't._  \n\u2013NFL GREAT JERRY RICE\n\n\"TODAY I WILL DO WHAT OTHERS WON'T, SO TOMORROW I CAN ACCOMPLISH WHAT OTHERS CAN'T.\" THIS QUOTE IS WORTH REPEATING. IT'S WORTH TAPING TO YOUR MIRROR DURING YOUR FIRST MONTHS OF TRAINING. WHY? ONCE YOU'VE APPLIED THE DRUMBEAT OF CONSISTENCY WITH A SET OF KOKORO YOGA SEQUENCES AND RITUALS, AND HAVE MAINTAINED THE PROGRAM LONG ENOUGH FOR THE BENEFITS TO CATCH HOLD\u2014AS IT BECOMES MESHED WITH YOUR daily routine in a way that you can't imagine missing out on\u2014there will be no want or need to burn time trying to assess the next shiny thing or obsess about the latest double-blind, peer-reviewed research study on the subject on whether this works or not. The thing is: You'll see it, feel it, and know it.\n\nUnfortunately, what typically happens when someone gets all revved up to start a personal development program is this: After drifting through a period with too much work, stress, and poor eating habits, self-loathing sets in. You feel like you hit a threshold where you just can't take it any longer so you begin to search. A provocative Facebook ad promises immediate transformation so you jump into a new program. The opening stage of any such undertaking is fraught with peril, as a move toward deep-rooted change in habits or our being is often met with a tidal wave of internal resistance. After 1 or 2 months, your initial high motivation meets with the ultimate truth that this takes work\u2014the daily, routine type of work. You tell yourself that maybe this wasn't \"it\" and that you just don't have time for this stuff... you lose faith and once again begin to be push-pulled in a myriad of directions. The slow, steady erosion of physical, mental, and spiritual well-being continues anew. Sad story, but it doesn't have to be that way. It is time to truly upgrade yourself not just through faith, but with massive commitment.\n\n_\"Do. Or do not. There is no try,\"_ says Jedi Master Yoda of _Star Wars_ fame, imploring Luke Skywalker, his disciple, to build personal power through daily disciplined practice. Trying is not committing. Commitment leads to courage, which gets us moving toward developing the skills and competencies for the task of fulfilling your mission and purpose. Fear is met head-on then tossed aside as confidence and momentum build.\n\nTrying, on the other hand, immediately gives you an out\u2014a back door to run through when things don't go as planned (which is always). Many languages have no word for \"try,\" and now is a good time to erase it from yours. Language can be a prison for the rational mind, limiting your range of thought and action. Better to avoid words like \"try,\" \"could,\" \"should,\" and \"would,\" and adopt the attitude that once a decision is made, a redline is crossed\u2014a line that can't be retraced. We simply must proceed with a \"fire-in-the-belly\" commitment and an \"all-in\" mind-set. _Do. Or do not._\n\nThe SEALs say that you must \"earn your Trident every day.\" The Trident is their symbol of commitment to excellence and they proudly wear it on their chest. Everyone in the sphere that they work recognizes and honors that symbol for the individual behind it and the ethos that they represent. \"Self-mastery in service to humanity\" sums it up. That is the same type of ethos you will be developing with this program, and you will need to sharpen your sword daily on the path.\n\nThe simplest way to look at sharpening the sword is through daily disciplined physical training and practice, a foundational principle of Kokoro Yoga. In opening up your mobility and spinal health, by increasing energy through powerful breathing techniques and recharging your physical capacity through functional movements, you are renewed with energy and growing confidence that becomes a self-perpetuating force. This leads to a clear mind, an open heart, and a connection to your true self, allowing a vision of the passion, purpose, and principles of your unique personal ethos to arise. Further integration accelerates you toward your goals, with positive energy radiating outward that attracts a team of warriors ready to support you in a mutually symbiotic, interconnected web destined to co-create a better world.\n\nYoga is the multilayered warrior development discipline I had been searching for ever since the day I left my NYC dojo to become a Navy SEAL. But as it turns out, I had to create it to find it. For thousands of years, yoga has been an open source project to develop a complete personal development program, but it has been largely hidden from view in modern times. Practicing Kokoro Yoga on a daily basis for weeks, months, and years offers you a single approach in which you sharpen the array of inner resources we're all born with across the spectrum of mind-body-spirit. When we sharpen our sword daily, everyone and every moment becomes our teacher. We open ourselves to the reality that we know little about the true nature of the universe and meaning of life, so we lower our heads, soften our gaze, and become lifelong learners.\n\nThe Japanese concept of \"kokoro,\" denotes the spiritual aspect of our training. Kokoro represents our warrior's spirit and translates as \"merge heart and mind in action.\" Kokoro training begins as a solitary journey without a true summit, and at every peak you successfully climb you will experience an unforeseen plateau, followed by a fresh climb. But on the mountain path you will taste previously untouchable realms of power and connect with fellow travelers. It is a journey of the soul, and one that you must make alone.\n\nA CUSTOM PRACTICE\n\nYour practice will be personal and change over time. Every individual coming to yoga has different needs, inspiration, mental and emotional development, moral bearing, body type, injury matrix, and level of fitness and spiritual beliefs. A personal practice is optimized when it can be customized. In fact, that is one of the unique and pleasant ways that Kokoro Yoga is different from most other forms or systems of yoga.\n\nA huge challenge for us is that our lives are frenetic; racing to the yoga studio to catch a class 5 times a week can exacerbate the problem into a rolling boil. So why not train in your own bedroom, or on the beach, or in the woods? Let's customize our training so we can build \"stick-to-it habits\" that discipline ourselves to train daily. Let me share how Catherine and I customize our personal practices so you can get a sense of where to go with yours.\n\nRecall that Catherine is training the Peaceful Warrior within her\u2014so she can spread love through her teaching and personal practice. Catherine organizes her personal practice to support that archetype, and it includes a weekly dose of 2 to 3 Ashtanga group classes in the primary and secondary series and teaching 3 restorative Peaceful Warrior classes. It is supplemented with daily meditation and spot drills. She supports it with a functional fitness routine of 3 CrossFit classes and 1 to 2 moderate walks or runs. All of this is woven into her weekly schedule as it ebbs and flows. She allows for flexibility to add or subtract based on her work and travel schedule.\n\nMy personal practice is based on my warrior ethos. I practice a short morning and evening ritual and do spot drills daily (box breathing and visualization drills) to ensure that I cover all Five Mountains in my practice. I teach Kokoro Yoga at SEALFIT HQ once a week, focusing on the Fighting or Fit Warrior sequences. The meat and potatoes of my training are 2-3 moderate to long sessions and 3\u20134 SEALFIT WODS and 1 self-defense class where I integrate mental, emotional, and awareness skills into the training. I train for roughly 3 hours a day all told, and allow for flexibility to be spontaneous, jump into training with my team, or take a rest day. I also strive to challenge myself with something new and different every month that will test my limits.\n\nOf course you will design your personal practice based on your intuitive sense of where you need to develop skills and close weakness gaps. Your personal practice must sync both with your warrior spirit and your lifestyle. That is the beauty of this practice\u2014it is uniquely designed to mold to your personality and intuitively felt needs.\n\nRITUALS, SPOT DRILLS, AND RETREATS\n\nAs I have made abundantly clear, committing to, and sticking to, a personal practice requires the development of new habits. Habits must become ritualistic for them to sink in. It may seem overwhelming to add one more thing to your schedule now... but don't despair! Kokoro Yoga has three key training methods that ensure your personal practice will become a discipline and then a habit. They are: 1) rituals, 2) spot drills, and 3) retreats.\n\nThe ritual is a practice session that you do every day at the same time to provide a foundation for excellence during the day. We have the morning ritual and the evening ritual. Both are integrated training sessions, seeping with power, providing important bookends to your day. If you do no other training besides these two rituals, you are good to go. I will describe the morning ritual first, followed by the evening ritual.\n\nThe Morning Ritual\n\nIt is common to wake up, grab your smartphone to check e-mails while preparing your coffee, and allow your mind and emotions to spin up. The more uncommon, Kokoro Yoga approach, is to commit to win in your mind before you step foot onto the day's battlefield. The morning ritual is an integrated session that includes refining your ethos while training positivity, breath control, visualization, and functional movement. Here is the process:\n\n  When you awake, begin to connect with your breathing and body by performing a quick body scan while breathing deeply. Recite silently to yourself words to this effect: I like myself, I like myself, I've got this, another great day, I am grateful for this day, etc. This immediately grounds you in presence and positivity.\n\n  Your next act when you get out of bed is to get a glass of fresh water, and with gratitude for the water and earth, drink it.\n\n  Now sit in your training space and check in with your ethos: Your ethos (refer back to chapter 3) should be written in a place you can refer to daily\u2014mine is on my iPhone. Simply read the purpose statement and scan your principles. Reflect upon your vision and mission. Consider how you will align with your ethos on this day so that you reinforce it and move the dial toward fulfilling it. This is a good opportunity to deepen your commitment to the important things in your life, or reorient your mind away from things that are distracting you, moving you away from your purpose.\n\n  Perform box breathing for 5 to 20 minutes. Select a ratio that will balance you (1:1:1:1) or charge you up (1:2:2:1). If you are working on a specific breathing pattern, you can do that here instead.\n\n  After the box breathing, it is time to perform your functional movement. This will be one of the pose sequences or spot drill. You can select any other movement practice you would prefer here\u2014like body weight interval or combat conditioning sequences, a jog, or a walk. The key with this part of the ritual is to move the blood and energy in your body. Also it will stimulate and strengthen your spine and nervous system. You will start the day with an integrated body, mind, and spirit.\n\n  After the movement component, sit back down and perform a short visualization session. My preferred method is to visualize my ideal future having accomplished my major life purpose and vision, and then to \"dirt dive\" my day. In the dirt dive I see myself accomplishing every project, task, and interaction with effortless perfection. You should endeavor to see yourself accomplishing all tasks and projects beyond expectations. All interactions leave you and others elevated. You are positive and in a flow state throughout the day. You have won in your mind.\n\n  You are now ready for your coffee or tea and healthy breakfast. The ritual will help you maintain positive energy and focus as you start your workday.\n\nThe Evening Ritual\n\nWhereas the morning ritual is designed to charge you up and ground you for optimal performance in the day, the evening ritual is meant to integrate the positive results of the day, distill the lessons learned, and wind your mind and nervous system down for a great night of sleep. Here is the process:\n\n  Prepare your time and space for the ritual\u2014that means take care of the day's business, put the kids to bed (if applicable), and ensure an uninterrupted time to practice. Have your journal handy.\n\n  Get comfortable and perform 5 to 10 minutes of box breathing. Choose a calming ratio such as 1:0:2:0 or 1:1:2:1.\n\n  Perform a calming sequence such as Peaceful Warrior, Recovering Warrior or Hip Mobility Drill, or if you want just go into Happy Baby or Pidgeon Pose for 5 minutes with a deep awareness of your breath.\n\n  In your mind perform a \"recapitulation visualization\" where you will visualize yourself back at the start of your day, just after the morning ritual. Moving forward, review all events of the day and mentally note positive interactions and accomplishments. Jot at least three victories down in your journal. Then note the things that didn't go so well. Ponder the lesson, or silver lining, of each event. Write down what you learned, then mentally \"let it go\" by forgiving yourself and committing to doing better next time. Bottom line: You never want to go to bed with unfinished business or regrets!\n\n  Perform the evening meditation described below, as time allows.\n\n  Drink a nice, cool glass of water with gratitude for your health, for the day, for life, and for anyone or anything else you can think of.\n\n  Go to bed and sleep like a baby!\n\nEvening Meditation: Begin with 5 rounds of alternate nostril breathing. This breathing drill will balance and align the left and right hemispheres of the brain. When finished with that, come back to deep breathing without controlling the breath and notice the shifts in your mind and body. Now reflect on impermanence and the cycle of change that a day represents. One day, one lifetime is what Grandmaster Tadashi Nakamura would tell me, nodding to the cycle of life and how each day is a precious opportunity to be enlightened and aligned in purpose.\n\nBegin to visualize a bright red ball of energy glowing and radiating health near your pelvic floor. Visually create roots that grow from this ball down into the earth. Allow the roots to grow deeper and deeper into the earth, until you have reached the center of the planet and connect to the core. Allow the earth's energy to be drawn up into your body filling you with strength and vitality. Close your practice with gratitude.\n\nThe morning and evening rituals will bookend your day in a way that is so powerful that not only will you notice the difference almost immediately, so will everyone else in your life, who will also benefit. I cannot emphasize enough how these rituals are the crux of building your personal practice. Going to a yoga studio a few times a week is great, but doing the morning and evening ritual will have a far more profound and positive impact on your development, and your life!\n\nSPOT DRILLS\n\nBREATHING DRILLS\n\nBox Breathing\n\nWarrior Breath\n\nEnergy Breath\n\nAlternate Nostril Breathing\n\nMENTAL CHECKUP DRILLS\n\nFeed the Courage Wolf\n\nEnergizing Visualization\n\nVictorious Warrior Visualization\n\nFuture Me Visualization\n\nMOVEMENT DRILLS\n\nEnergizing Drill\n\nHip Mobility Drill\n\nCalming Drill\n\nBody Blast (choose 1 round of a functional drill, or do 50 to 100 of 1 of the following: squats, push-ups, pull-ups, or burpees)\n\nBring Down the Heavens\n\nBouncy Breath\n\nChopping Wood or Musashi Strikes\n\nSpot Drills\n\nWhereas the morning and evening ritual bookend your day and ensure excellence in your personal practice, the spot drills are another way to supercharge your training on a daily basis. The spot drills are designed to do training \"on the spot\" based upon the time and space you have, and what you feel you need.\n\nFor instance, if you are feeling fatigued in midafternoon, you can step away from your desk and perform the spot drill of 1-2-2-1 Breathing or Sun Salutation. If you need to calm yourself and prepare for an important speech, then 5 minutes of box breathing is a nice spot drill. The spot drill is like a Swiss Army knife, a multitool always handy to be certain your integrated training is covering all Five Mountains as you evolve your personal practice. You could schedule these drills, or set reminders to do them throughout the day, or just do them whenever you intuitively feel the need. Above are some of my favorites.\n\nRetreats\n\nBack in my active-duty Navy SEAL days, we did daily ritualistic training unique to preparing for the specialized missions of a SEAL team. But we all looked forward to the next \"away time\" where we would immerse ourselves 100 percent into learning away from the daily clutter of our lives. On these immersive training adventures we would retrace the basics of whatever skill we were learning, and then take it to a new level, or learn a new skill altogether. We always came home refreshed and energized, even though the training was often grueling for weeks on end.\n\nWe achieve this same \"away time\" immersion in Kokoro Yoga through what we call a retreat. It is important to retreat from busy daily life to immerse yourself in refining a skill or learning something new. The retreat doesn't have to be for yoga; it can come from any of the 5 mountains. For example, some of retreats that I have enjoyed include martial arts, silent meditation, self-defense, emotional therapy, mindfulness, survival, tracking, intuition, memory, and breathing retreats. You can also retreat to charge ahead after a challenge such as a SEALFIT 20X challenge, or a Spartan Race. In any case, the possibilities with the retreat method, as a tool in the integrated training model, are endless. This method of training is incredibly rejuvenating and rewarding...I encourage you to commit to your next one immediately.\n\nSTARTING A PERSONAL PRACTICE\n\nIf you are new to yoga, as I know many of you are, it can be daunting to figure out how and where to start. Let's just say that you start at the beginning\u2014the beginning that is right for you. For me it was to attend a 60-day Hot Yoga challenge to jump-start my training. Catherine started by leaping right into a 500-hour teacher training! For you it will be different. The key is to fully commit, make it what I call a \"burn-your-boats\" behind you commitment, the sort where you commit to your significant other, your kids, your inner-sanctum teammates, that you won't let them down.\n\nCommitment gives you the courage to dive in and explore. I would recommend you start by clarifying your personal ethos and begin the morning ritual. In my book _Unbeatable_ _Mind_ and the online training program by that name, I have a series of questions and exercises that help you uncover your ethos. I recommend using that process\u2014because it works. At any rate, you need a place to dig your heels into a stand that strengthens your commitment to mastery and resolve to maintain a personal practice day in and day out.\n\nCRAWL, WALK, RUN\n\nIn SEAL tactical training we learned to crawl first with a new skill. We were careful not to try to go from zero to hero and risk injury or mission failure. Honing the basics proved to be the most valuable when we actually entered combat. Being able to perform the basics without thinking was crucial to survival and mission success. This is metaphorically true for your yoga practice. Be patient enough to train the basics for far longer than others.\n\nWhen the basics are mastered, then you will move on to the crawl phase to layer in new skills and knowledge. This is where you will work in greater sync with your teammates. You can eventually run fast with the skill, employing them with precision. But then we would always come back to train the basics again! The brilliance in this method is that it evolves you step-by-step from a place of unconscious incompetence, to conscious incompetence, to conscious competence, and, finally, to unconscious competence where the skills flow with effortlessness. Of course this type of training requires patience and perseverance, two qualities you will need to develop in your practice. It is important to find deep intrinsic motivation, which is why it needs to be grounded in your personal ethos. Relating your training to your ethos is a great intrinsic motivator (i.e., get 1 percent better every day, master yourself so you can serve others, etc.). When you can do this, your personal practice will take on a new level of meaning. Missing it will interfere with your growth as a human, risk your mission and let your team down. We don't do that!\n\nWHERE TO TRAIN\n\nKokoro Yoga can occur in many places. The most common will be in your home or at work in a spot you have designated or created for your personal practice. I think the home is best because you get to control access and noise (Unless your spouse or kids don't respect your space very well, which is a different issue!). Creating a special place to train and using it every day is powerful. You can include items for you eyes to focus on, such as a candle or an iconic picture. You will have your mat, chair, or bench ready for action. Over time this place will become special to you and have an energy that helps you focus. Try to avoid changing spots or randomly practicing in a different place every time.\n\nTraining at work can be a good option if you are lucky enough to work at a conscious company that has dedicated personal development space. Regardless, if you can get access to an empty room, or a park nearby, you can practice. Of course finding a local yoga studio near you is also an excellent option, especially if you resonate with their community. Ideally you will find a combination of these three to provide you ample opportunity to get your practice in every day.\n\nTHE THREE MOMENTS\n\nEvery practice session has three distinct moments: the beginning, the middle, and the end. The beginning is the preparation and entering into the practice. This will be different for each location. At home the beginning moment will be designed to protect your time and to set up the space. Make sure the phone is on silent and your family or roommate is aware that it is \"practice time.\" It is not a good idea to eat within 30 minutes before a practice. Ensure the room is warm so you won't be distracted by the cold, but not so warm it puts you to sleep. Make sure you have the right tools so you don't have to stop to retrieve them. You get the picture.\n\nThe middle moment is the actual practice itself. This is the moment you have been waiting for\u2014so make the most of it\u2014and really practice. A session with a distracted mind, interruptions, chatting with someone, or being crawled on by the kids is not ideal. However don't beat yourself up if you are taken off track by your monkey mind or monkey kids, but use those opportunities to practice a different aspect of your ethos, such as forgiveness, patience, and playfulness.\n\nThe end moment is when you wind down and integrate the practice, leaving it with an expression of gratitude and grace. This is where many shortchange themselves. They perform an amazing yoga practice, then as soon as the last pose is done they get up, grab their mat, and check their e-mail. Please don't do that! Take time in the Resting Pose to integrate all that occurred in your practice. The ancient yogis felt that this moment was the most important part of the practice because it allows the benefits to literally sink in at all levels\u2014physical, energetic, mental, wisdom, and bliss\u2014the five dimensional bodies come together as one.\n\nThe end moment is also the perfect time to perform visualization work. Generally speaking, we want to start practice with the \"hard\" work of movement, and then move toward the softer, inner work. Movement precedes breath control, followed by visualization and meditation, in that order. But you may note that I put box breathing before the movement in the sequences. The reason is that I have found it useful to settle and sync the body, mind, and emotions before the movement because it deepens our practice. We are so agitated from the myriad of stressors coming at us day in and out that we can't get enough box breathing in to soothe our nerves and calm our body and our mind. So we kick off our practice with box breathing to ensure that the movement, and entire practice, is stronger.\n\nWhen you are completely finished, you are still not done. The transition back to the \"real\" world is also part of your practice. Avoid getting spun up into a conversation right away or checking your e-mail or engaging in some mental drama after your practice. Use the transition time to refine your awareness and grace. Keep quiet, smile, and walk slowly, with awareness. Don't be weird about it, just enjoy the heightened sense of connection and integration, and notice how good it feels. If you do get spun up into another story, notice the difference! Keep in mind that the objective is to develop mental, emotional, and spiritual depth for your personal mastery... so that you may fulfill your purpose, live with passion, remain in alignment with your principles, and be of service to others. It makes sense to continue the deep awareness that the practice brought you for as long as possible, doesn't it? Why not make your entire day a practice session? Why not make every action (even every breath) a practice and leave the world better off for it? Why not indeed!\n\nSTAYING THE COURSE\n\nThough getting started is a challenge for many, staying the course is often the bigger challenge. We expect immediate results without much effort these days. The good news is that you will see immediate results if by \"immediate\" you mean 1 to 3 months. Trust me, Kokoro Yoga requires effort, and you will also see results quickly. But after the excitement of the initial positive changes\u2014such as a sense of calm, more control, and more energy\u2014goes away, don't be surprised to find that deeper change seems to evade you. The reason for this is that deeper levels of growth lay dormant until the necessary integration moment occurs\u2014that magical crossroad when the right lessons are learned, and the training penetrates to the right depth, to provide the key that unlocks the new insights or next stage of consciousness.\n\nPositive growth is then suddenly experienced as an \"aha\" moment, or a sense of knowingness you did not have prior. It just shows up at the right time, often when we are challenged and dig deep. Sometimes we experience it through others by the new way they treat us\u2014with a level of respect we previously didn't experience. They say we have a \"look in the eye\" that wasn't there before, but of course you can't see it yourself, but you feel it. These are moments to savor and humbly acknowledge that the training is performing its magic. If you are deeply religious, give credit where it is really due, to your Creator.\n\nBottom line: Transformation is a holistic and nonlinear process that occurs when things line up, when the timing is right.\n\nAnother important point I would like to bring up is that we must \"translate\" our past negative energy into neutral or positive energy in order to transcend to higher stages of consciousness. This means we must overcome shadow self-regrets, shame, and guilt to gain understanding and acceptance of our negative, shadow selves. We do this so that we can end the negative contraction and \"graduate\" to transcend to higher stages of development. If we try hard to transcend without doing the translation work, we will get pulled back down into negative territory sooner or later. Progress is thwarted by our shadow self, leaving us feeling frustrated and stuck. Translation (shadow work), the gift of Western psychotherapy, and transformation (enlightenment work), the gift of warrior and spiritual practices, conspire together to facilitate our growth to maximum potential as a human. If you just focus on the transcendent practices of meditation, without diving into your shadow self to translate, then transmute mental and emotional negativity, then you will set yourself up for disappointment.\n\nDISCIPLINE, COMPETENCE, CONFIDENCE\n\nCommitment to a daily practice unlocks courage. Courage expresses itself in yoga not as a roar of \"I CAN!\" but the quiet discipline of just showing up and putting out with a heartfelt effort, every day. Over time that act of discipline leads to a competence in the skills, knowledge, and nuances of the training. Ultimately we gain a concrete confidence in ourselves, a confidence that favorably impacts all that we do. We become more successful, and that success reinforces the confidence. We have unlocked the upward spiral of growth, which has no upper limit.\n\nWHO IS ON YOUR TEAM?\n\nIn spite of what you may believe from the movies, a Navy SEAL, or other elite military operator, seeks to master the skills of the warrior not so that they can be a lone operator, but so they can be an effective teammate. Great teams are great because they are filled with individuals working toward greatness. They are willing to align personal needs and desires to the team's mission.\n\nSimilarly, in your training, it will help you to find a swim buddy or team to train with. The team-training effect is a critical aspect of the lifestyle I am espousing here. A great team will hold you to a higher standard, call you out on your bull, and offer opportunities to face your fears and develop emotional power. On the contrary, you may find yourself unsupported in your efforts by your supposed team. If that is the case, then I propose you find new teammates. No question, this is an awkward situation if those individuals live under the same roof. Let your newfound confidence and awareness guide the development of a supportive team.\n\nAVOIDING RUTS AND INJURIES\n\nKokoro Yoga is designed to maximize longevity and to avoid injuries or burnout. The customizable nature of the training and adaptation of the poses and integrated nature will help you maintain a balance throughout your training ups and downs. Having said that, it is possible, if not likely, that eventually injuries may occur. These come from accidents or from dysfunctional movement patterns that grease the groove of an injury. These dysfunctional movement patterns are subtle and hidden from view from us, but can be \"outed\" by a skilled teacher.\n\nThis points to the need to have a teacher to check our work. Attending a yoga studio once a week (or even once a month) to have alignment adjustments and a watchful eye is a good idea. As mentioned above, periodically attending a yoga seminar or retreat is another good idea. Moving into a pose with several repetitions to align your structure, before holding your position, is a tip that Gary Kraftsow taught me. This method will help ferret out dysfunctional movement patterns.\n\nIn spite of all the above, what do you do if an injury occurs? First, it helps to assess the magnitude. Maybe it is the Navy SEAL in me, but I have generally noted that many people overestimate the extent of an injury and take unnecessary precautions, sidelining themselves for weeks or months. It is hard to reengage with your personal practice after 3 months on the bench. My Ashtanga teacher, Tim Miller, says that you need to learn the difference between \"integrating pain\" and \"disintegrating pain.\" Having said that, this is not a recommendation to do anything stupid. Please get injuries examined and follow the advice of your doctor, after checking your intuition and agreeing with it. Remember, the medical docs will always err way on the side of caution to ensure liability coverage. Most of the time you are able to continue training with modifications. An additional professional to consult might be the most experienced yoga professional, chiropractor, or naturopathic doctor in your hood. Don't let injury take you away from your practice\u2014it is part of staying in the game over the long haul.\n\nWhat if you experience burnout? Burnout is dangerous because it can lead to injury and quitting altogether. Ideally, you will organize your practice with enough variety and regenerative sessions so that you don't experience burnout. But if you do, then what?\n\nRetreat, retreat! Yes, take time to retreat to recharge your batteries. When you begin to feel burnout, instead of taking time off altogether, consider changing things up in your training schedule, and plan an extended period of integrative practice. This can be a day alone, a weekend seminar, or a multiday training. There are an enormous number of excellent seminars and retreats to choose from. Recently I did a 6-day silent Yin Yoga retreat in Hawaii. It completely fired me up to deepen my practice and got me out of a minor rut from the holidays. Finally, to hammer this point home, it helps to check in with your ethos every day to remind yourself why you practice. Maintaining that connection with your \"Why\" is deeply motivating, providing you the drive and determination to stay the course.\n\nTRUTH, WISDOM, LOVE\n\nSo where are we going with all this? What is the destination? There is none. The way of yoga is a journey, not some elusive destination. When we unlock growth, we set ourselves on a path toward an ever-expanding sense of self. The uncanny nature of the growth is that the higher we go up the Five Mountains, the more we can see. The more we see, the more we understand just how little we truly know about the world. Great humility settles in, and we find more grace and joy in the small things in life. Details we didn't even notice as youths, or scoffed at as important business executives, suddenly seem important. And they are.\n\nMy directive to you is to you stay focused on the path and not the destination. Put your attention and intention into the daily sharpening of the sword. This is where the good stuff is. You will get glimpses of new plateaus and gain enlightening insights to guide future steps. Day by day, step-by-step, that divine heart of your inner warrior, your spirit, and the witness to your life, will begin to merge with your physical heart. You will be consumed with the courage wolf and positivity will be your constant companion as you experience ever-expanding waves of truth, wisdom, and love. Your thoughts and actions will be filled with passion, directed toward purpose, guided by worthy principles, and backed by your stand. You will move as fast as the wind, be as quiet as the forest, as daring as fire, and as immovable as a mountain itself.\n\nLife will become about finding perfection in the moment, about discovering truth, unlocking the capacity to love, and to live with wisdom in deep connection to all of life. This is our journey, our way. It is time that we all step into our uniqueness, gain mastery over our body and minds, and allow our spirits to speak to the world. It is time to fulfill our purpose with passion, and live with honor, courage, and commitment.\n\nWe are deeply humbled that you have chosen to join this journey.\n\n# APPENDIX A\n\n# FUNCTIONAL CONDITIONING\n\nThe following functional conditioning movements are for the Fit Warrior sequences. You will choose a handful of these to add to your practice. Appendix B has combat conditioning movements, and Appendix C has some combinations of both fitness and combat conditioning movements to get you started. Change it up and have fun with these!\n\nFor more guidance on proper technique and methodology for functional fitness, see www.sealfit.com\/videos and my book _8 Weeks to SEALFIT._\n\nRemember the golden rule of virtuosity when it comes to maximizing your conditioning modules: good form is the first priority and intensity comes second. In other words, don't make the mistake of compromising good technique to bang out a few more reps.\n\nPUSH-UPS\n\nStandard Push-up\n\n  Elbows close to body, core engaged.\n\n  Chest touches ground, elbows locked out at top.\n\nScaled Push-up\n\n  Hand position elevated on a box or wall.\n\n  Chest touches box or wall.\n\nPlyometric Push-up\n\n  Explosive UP, hands and feet leave the ground.\n\n  In place or moving across the floor forward or backward.\n\nDive Bomber Push-ups\n\n  Top of push-up, send hips up, moving in a V.\n\n  Scoop the chest down and forward low, with hips following. Lock arms, then reverse to the start.\n\nClapping Push-ups\n\n  Explosive UP, hands leave the ground, and clap in front of or behind the body.\n\nAlligator Push-ups\n\n  Offset grip (fore and aft), do a push-up, crawl forward, and repeat on the other side.\n\nHand-Placement Variations\n\n  Diamond (formed with thumbs and index fingers touching).\n\n  Wide.\n\n  On fists.\n\n  Offset (fore and aft).\n\nWave-offs (Arm Haulers)\n\n  On belly, arms, and legs lifted. Hands touch in front, sweep back, and touch behind back.\n\nStatic Plank Hold\n\n  Either top of push-up position or on elbows with straight body or in pike as lean and rest.\n\nSingle Arm\n\n  Center-weighted arm under body, other behind back.\n\n  Can scale on a box or well-secured chair.\n\nHandstand Push-up\n\n  Upside down against wall, arms locked out.\n\n  Lower with control to touch head to ground or elevated pad. Lift back up to straight arms.\n\n  Can scale with feet on a box and the body in pike position.\n\n  Handstand hold is also an option.\n\nSQUATS AND JUMPS\n\nStandard Squat\n\n  Feet shoulder width, initiate by driving hips back. Weight on heels.\n\n  Proud chest, eyes forward, knees out, tracking over toes.\n\n  Bottom is reached when the hip crease is below the top of the knee.\n\nJump Squat\n\n  Accelerate through the top, hips open, feet leave the ground.\n\n  Soft landing, core engaged.\n\nSquat Hold\n\n  Hold at the bottom to work mobility or slightly above parallel to work strength.\n\nSingle Leg Squat (Pistol)\n\n  Drive hips back, unweighted leg in front.\n\n  Send weight back.\n\nTuck Jumps\n\n  Jump straight up, pulling knees up high.\n\n  Quiet, springy landing.\n\nStar Jump\n\n  Squat down, touch hands to floor.\n\n  Spring up with the feet leaving the floor and arms out, body forms an X.\n\nJumping Jacks\n\n  Stand up straight, with arms at the side.\n\n  Jump feet and arms wide, with elbows bending at the top.\n\nFroggies\n\n  Squat and touch fingertips to the ground.\n\n  Jump into the top of a wide-stance jumping jack.\n\nLUNGES\n\nStandard Lunge\n\n  Big step forward, knee tracks over foot.\n\n  Back leg lightly touches the ground.\n\n  Eyes forward, upright torso.\n\n  Return to standing.\n\nJump Lunges\n\n  Jump from one lunge to another with core engagement and control.\n\nWalking Lunges\n\n  Connect lunges to move forward or backward.\n\nCORE\n\nSit-ups\n\n  On back, soles of feet together.\n\n  Engage core and lift to sitting.\n\nFlutter Kicks\n\n  On back, fists under glutes, legs straight.\n\n  Lift and lower legs, keeping them straight (scissor).\n\nPlank Toe Taps\n\n  On elbows, body straight.\n\n  Lift one leg and tap toes to heel\n\n  Lower and repeat with the other.\n\nSide Hip Raises\n\n  On one elbow or locked out arm, body straight, side torso facing down.\n\n  Lower hip to tap the ground and raise to return to a straight body.\n\nBicycle Crunches\n\n  Sit in a V, touch right elbow to left knee while extending left leg.\n\n  Repeat.\n\nLeg Levers and Holds\n\n  On back, fists under glutes, legs straight.\n\n  Lift and lower straight legs or hold at various heights.\n\nSupermans\n\n  On belly, arms outreached, palms face each other or down.\n\n  Lift arms and legs together, engaging core and reaching legs back.\n\nJump Throughs\n\n  Pike Position\/Downward-Facing Dog.\n\n  Bend knees, lift heels, look forward.\n\n  Jump feet through to sitting, crossing the ankles when the feet are passing between the arms.\n\nMULTIJOINT MOVEMENTS\n\nDown-Ups\n\n  Drop to your back or belly (spontaneous choice).\n\n  Jump back up to a choice combat stance.\n\n  Repeat ad infinitum.\n\nBurpees\n\n1. Hands to ground.\n\n2. Feet jump out to top of push-up position.\n\n3. Chest to ground.\n\n4. Lift chest, jump feet to the bottom of a squat.\n\n5. Jump up, open hips, claps hands over head.\n\n8-Count Body Builders\n\n1. Hands to ground.\n\n2. Feet jump out to top of push-up position.\n\n3. Chest to ground.\n\n4. Top of push-up.\n\n5. Jump feet wide.\n\n6. Jump feet back to normal push-up position.\n\n7. Jump feet to the bottom of a squat.\n\n8. Jump up, open hips, claps hands over head.\n\nMountain Climbers\n\n  Top of push-up position.\n\n  Right foot jumps to outside of right hand.\n\n  Left foot jumps to outside of left hand as right foot jumps back to starting position.\n\n  Repeat.\n\n# APPENDIX B\n\n# COMBAT CONDITIONING\n\nWe talked about \"combat conditioning\" in chapter 4: It's a fantastic way to insert a fun and highly charged aerobic element using offensive and protective movements into your Kokoro Yoga routine. I'll say this again because it's so important: You don't need to have a martial arts background to throw a few punches and kicks. Throwing some punches and kicks (the essence of combat conditioning) can be done as a stand-alone workout or inserted as a conditioning module in your routine.\n\nThe combat conditioning module is not a requirement, of course. Rather, it's offered as a method to include a specific brand of heart-pounding cardio into the 1 hour (or 30 minutes) of time that you have to train on a jam-packed day. Combat conditioning moves can also be intermixed with functional fitness moves. As long as you're consistent in changing things up, you'll avoid the rut of stagnation that can occur from doing the exact same routine every time you train. Constantly varying your functional conditioning is a way to shock the body into delivering the highest return on your investment when it comes to physiological response to your training.\n\nKicks\n\n  Front: lift knee, kick quickly straight ahead through the ball of the foot (vary the height), quick recoil.\n\n  Side: lift knee, kick to side through the heel (quick out-in).\n\nPunches\n\n  Straight punch: fists, arms up to protect head, offset stance, drive rear side arm forward, rotate hips.\n\n  Jab: fists, arms up to protect head, offset stance, drive front side arm forward.\n\n  Hook: bent elbow punch from the side, use hips and upper-back muscles.\n\nDucks and Feints\n\n  Side-to-side moves, squat, head moves.\n\nSprawl\n\n  Jump the legs back.\n\n  Arch the back, hands to ground, control the descent with the chest striking first and contact rolling down the front of the body (like the rockers on a rocking horse).\n\n# APPENDIX C\n\n# CONDITIONING MODULES\n\nPicking and choosing from the movements listed in Appendix A and Appendix B, you can craft your own conditioning modules that you can insert into the middle of a Kokoro Yoga session. The following workout patterns can be used as templates.\n\n10! UP or DOWN\n\n  Pick 2 or more movements.\n\n  10 reps each, then 9 each... continuing down to 1 each.\n\n  Can start at 1 and work up the ladder as well.\n\n  Can do 10 of 1 movement and 1 of another and work the ladders in opposite directions.\n\nx Rounds of y Reps\n\n  5 rounds of 20 reps each.\n\n  3 rounds of 15 reps each.\n\nAMRAP\n\n  As many rounds as possible (in X time).\n\n  For example, 12 minutes AMRAP of 5 burpees + 5 Mountain Climbers.\n\nEMOM\n\n  Every minute on the minute.\n\nRoll the Dice\n\n  Pick 2 to 6 movements and assign dice numbers.\n\n  Roll the dice and choose the corresponding movement.\n\n  Roll it again for rep count, double or triple it.\n\nA Few Sample Modules:\n\n10! UP: standard push-ups, L+R straight punches, sit-ups\n\n5 rounds of 20 reps each of: jumping jacks, squats, Supermans, lunges\n\nAMRAP 8 minutes: 4 burpees, 6 front kicks, 8 wave-offs, 10 Mountain Climbers\n\nEMOM 10 minutes: 3 star jumps, 4 dive bomber push-ups, 5 2-count flutter kicks (scale up with 4-count flutter kicks)\n\nALSO BY MARK DIVINE\n\n_8 Weeks to SEALFIT:  \nA Navy SEAL's Guide to Unconventional Training for Physical and   \nMental Toughness_\n\n_Unbeatable Mind:  \nForge Resiliency and Mental Toughness to Succeed at an Elite Level_\n\n_The Way of the SEAL:  \nThink Like an Elite Warrior to Lead and Succeed_ (with Allyson E. Machate)\n\n# ABOUT THE AUTHORS\n\n**MARK DIVINE**\n\nA _New York Times, Wall Street Journal,_ and Amazon.com bestselling author, Mark Divine is the founder of SEALFIT and Unbeatable Mind. Mark served for twenty years as a Navy SEAL, retiring as a Commander in 2011. He is an accomplished martial artist with black belts in Seido and Goju Ryu Karate, as well as military hand-to-hand combat certification in SCARS\/San Soo Kng Foo and a senior ranking in Saito Ninjutsu.\n\nMark came to yoga from the martial arts in 1999, studying with Ashtanga Yoga's Tim Miller, where he received teacher training in the first and second series. He later received a 500-hour teaching certification from Gary Kraftkow's American ViniYoga Academy. In 2007 Mark created the innovative Kokoro Yoga program to train Navy SEAL and other SOF candidates the science of mental development through SEALFIT. Since then, he has trained thousands from all walks of life to embrace the full spectrum of physical, mental, emotional, and spiritual training that yoga offers. You can sign up for author updates here.\n\n**CATHERINE DIVINE**\n\nAfter years of the stress of work as a professional cook, Catherine found in yoga a path toward healing. Yoga soon became her medicine as well as the focus of a new path in life. She is now an experienced teacher and spiritual activist, flowing with enthusiasm for the healing properties yoga brings to each individual. In collaboration with stepfather Mark Divine, the founder of Kokoro Yoga, Catherine is the head facilitator for programming, teacher training, and yoga retreats for the program. Catherine has multiple teaching certifications and has trained in various styles under some of the best teachers in the world, including Tim Miller and Gary Kraftsow. Her key focus as a teacher is to bring a community together under the common thread of the Peaceful Warrior, seeking compassion, love, and environmental awareness. You can sign up for author updates here.\n**Thank you for buying this**\n\n**St. Martin's Press ebook.**\n\nTo receive special offers, bonus content,\n\nand info on new releases and other great reads,\n\nsign up for our newsletters.\n\nOr visit us online at\n\nus.macmillan.com\/newslettersignup\n\nFor email updates on Mark Divine, click here.\n\nFor email updates on Catherine Divine, click here.\n\n# CONTENTS\n\nTitle Page\n\nCopyright Notice\n\nForeword\n\n1. An M4 and a Yoga Mat\n\n2. The Pursuit of Maximum Human Potential\n\n3. The Strategies\n\n4. The Tactics\n\n5. Core and Supporting Sequences\n\n6. Poses and Movements\n\n7. Physiological and Psychological Benefits of Yoga\n\n8. Developing a Personal Practice\n\nAppendix A: Functional Conditioning\n\nAppendix B: Combat Conditioning\n\nAppendix C: Conditioning Modules\n\nAlso by Mark Divine\n\nAbout the Authors\n\nCopyright\nThe information in this book is not intended to replace the advice of the reader's own physician or other medical professional. You should consult a medical professional in matters relating to health, especially if you have existing medical conditions, and before starting any new fitness regimen. The author and the publisher do not accept responsibility for any adverse effects individuals may claim to experience, whether directly or indirectly, from the information contained in this book.\n\nKOKORO YOGA. Copyright \u00a9 2016 by Mark Divine and Catherine Divine. Foreword \u00a9 2016 by Gary Kraftsow. All rights reserved. Printed in the United States of America. For information, address St. Martin's Press, 175 Fifth Avenue, New York, N.Y. 10010.\n\nLibrary of Congress Cataloging-in-Publication Data\n\nNames: Divine, Mark. author. | Divine, Catherine, author.\n\nTitle: Kokoro yoga : maximize your human potential and develop the spirit of a warrior \/ by CDR Mark Divine, US Navy SEAL, and Catherine Divine; foreword by Gary Kraftsow.\n\nDescription: New York, NY : St. Martin's Griffin, 2016.\n\nIdentifiers: LCCN 2015045659 | eISBN 978-1-4668-7522-7 (e-book)\n\nSubjects: LCSH: Yoga. | Mental health. | Mind and body. | BISAC: HEALTH & FITNESS \/ Yoga. | SELF-HELP \/ Motivational & Inspirational.\n\nClassification: LCC B132.Y6 D575 2016 | DDC 613.7\/046\u2014dc23\n\nLC record available at <http:\/\/lccn.loc.gov\/2015045659>\n\nOur eBooks may be purchased in bulk for promotional, educational, or business use. Please contact the Macmillan Corporate and Premium Sales Department at 1-800-221-7945, ext. 5442, or by e-mail at MacmillanSpecialMarkets@macmillan.com.\n\nFirst Edition: April 2016\n\n# Contents\n\n  1. Cover\n  2. Title Page\n  3. Copyright Notice\n  4. Foreword\n  5. 1. An M4 and a Yoga Mat\n  6. 2. The Pursuit of Maximum Human Potential\n  7. 3. The Strategies\n  8. 4. The Tactics\n  9. 5. Core and Supporting Sequences\n  10. 6. Poses and Movements\n  11. 7. Physiological and Psychological Benefits of Yoga\n  12. 8. Developing a Personal Practice\n  13. Appendix A: Functional Conditioning\n  14. Appendix B: Combat Conditioning\n  15. Appendix C: Conditioning Modules\n  16. Also by Mark Divine\n  17. About the Authors\n  18. Newsletter Sign-up\n  19. Contents\n  20. Copyright\n\n## Guide\n\n  1. Cover\n  2. Copyright\n  3. Table of Contents\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n_Jess and Betty Jo Hay Series_\n_The_ MAN WHO WROTE _the_ PERFECT NOVEL\n\nJohn Williams, _Stoner_ , and the Writing Life\n\nCHARLES J. SHIELDS\n\nUniversity of Texas Press\n\nAustin\nCopyright \u00a9 2018 by Charles J. Shields\n\nPublished by University of Texas Press under license from Lebowski\n\nPublishers\/Overamstel Uitgevers B.V.\n\nAll rights reserved\n\nFirst English-language edition, 2018\n\nRequests for permission to reproduce material from this work should be sent to:\n\nPermissions\n\nUniversity of Texas Press\n\nP.O. Box 7819\n\nAustin, TX 78713\u20137819\n\nutpress.utexas.edu\/rp-form\n\nLIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA\n\nNames: Shields, Charles J., 1951\u2013, author.\n\nTitle: The man who wrote the perfect novel : John Williams, Stoner, and the writing life \/ Charles J. Shields.\n\nDescription: First English-language edition. | Austin : University of Texas Press, 2018. | Includes bibliographical references and index.\n\nIdentifiers: LCCN 2018012653\n\nISBN 978-1-4773-1736-5 (cloth : alk. paper)\n\nISBN 978-1-4773-1737-2 (library e-book)\n\nISBN 978-1-4773-1738-9 (nonlibrary e-book)\n\nSubjects: LCSH: Williams, John, 1922-1994. | Authors, American\u201420th century\u2014Biography. | Williams, John, 1922\u20131994. Stoner.\n\nClassification: LCC PS3545.I5286 Z86 2018 | DDC 813\/.54 [B]\u2014dc23\n\nLC record available at <https:\/\/lccn.loc.gov\/2018012653>\n\ndoi:10.7560\/317365\n_To my wife_\nContents\n\nIntroduction\n\nPART I. _Nothing But the Night_\n\nChapter One. He Comes from Texas\n\nChapter Two. \"Ho, Ho! Wasn't I the Character Then?\"\n\nChapter Three. Rough Draft\n\nChapter Four. Key West\n\nChapter Five. Alan Swallow\n\nChapter Six. Love\n\nPART II. _Butcher's Crossing_\n\nChapter Seven. The Winters Circle\n\nChapter Eight. \"Natural Liars Are the Best Writers\"\n\nChapter Nine. _Butcher's Crossing_\n\nChapter Ten. Fiasco\n\nPART III. _Stoner_\n\nChapter Eleven. \"It Was That Kind of World\"\n\nChapter Twelve. \"The Williams Affair\"\n\nChapter Thirteen. _Stoner_\n\nPART IV. _Augustus_\n\nChapter Fourteen. Bread Loaf and \"Up on the Hill\"\n\nChapter Fifteen. The Good Guys\n\nChapter Sixteen. \"Long Life to the Emperor!\"\n\nPART V. _The Sleep of Reason_\n\nPoem. \"An Old Actor to His Audience\"\n\nChapter Seventeen. \"How Can Such a Son of a Bitch Have Such Talent?\"\n\nChapter Eighteen. In Extremis\n\nEpilogue. John Williams Redux\n\nAcknowledgments\n\nNotes\n\nWorks Consulted\n\nA John Williams Bibliography\n\nIndex\nIntroduction\n\nDriving slowly down North Woolsey Avenue in Fayetteville, Arkansas, hunting for an address, Anne Marie Candido had no idea what kind of house she was looking for. John Williams, she had been given to understand, was a novelist and former professor at the University of Denver. He'd moved to Fayetteville for his health\u2014the lower altitude was easier on his breathing. A friendship between him and John Harrison, dean of the libraries at the University of Arkansas, had led to Williams agreeing to donate his papers to the university's special collections.\n\nBut that had been a year ago, in 1987, and all of Williams' materials were still awaiting disposition in cardboard boxes at the library. When Candido, a PhD in literature, heard there was grant money available to hire someone to organize the collection, she applied for the position\u2014anything to do with books and manuscripts sounded interesting.\n\nShe hadn't read any of Williams' novels, though. She wondered whether she should have before coming over to his house. Maybe it was presumptuous or discourteous to arrive without having done her homework. Her husband, a professor of English, was a fan of Williams', and he had been glad to praise him: Williams' novels were outstanding\u2014especially _Stoner_ , about a Missouri farm boy who becomes a professor. Williams was from Texas during the Depression, so it could be autobiographical. He won the National Book Award for _Augustus_ , a fictional life of the Roman emperor. But the committee split the fiction award between him and John Barth that year\u2014the first time it had ever happened in that category. Some said Williams got half as a kind of consolation prize for the committee having ignored _Stoner_ ten years earlier. And there was one other novel, too, which was about buffalo hunting\u2014 _Butcher's Crossing_ , published early in his career. Too bad his books were out of print, but they had just never caught on for some reason.\n\nCandido slowed down to see the house numbers on North Woolsey, a humble street like an afterthought without curbs or streetlights. She expected some kind of rambling old place, fit for a professor in retirement, an author whose career had peaked some time ago. And then there it was\u2014number 1450. Set back in the bare January trees, a small house with a large, steeply pitched roof like a building on a farm that had been remodeled. It looked rural, somehow.\n\nFor the next three months, she arrived at John Williams' home every other week with her special file folder of mysteries. Items in his papers, such as letters without dates, handwritten drafts of correspondence, photos of unidentified people\u2014oddments from his life that might be important. She wanted to be thorough. So she sat at the dining room table, observing him as he read from the folder.\n\nHe was a small man, petite, in his late sixties with thick hair that was still resolutely dark. Could he be part Native American? He was cordial, almost courtly. He took care to be presentable for their sessions. House slippers, green slacks, and tangerine shirt with socks to match on one occasion. His voice was deep, a baritone with a trained, musical quality. He chuckled at his own droll comments. Most striking, though, were his eyes: very large, light blue, and owlish, behind a pair of heavy, black-framed glasses that she never saw him without.\n\nNow and then while they were working together, Nancy, his wife, would pause during her comings and goings to check on them. She was tall and much younger than he was. As a couple, they seemed comfortable around each other, Nancy making the point that the mystery file was \"John's thing.\"\n\nIt was curious how uninterested he was about dates and the details of his life. Often he would just smile and say, \"I don't remember\u2014I can't recall,\" as if some kind of story had already been told, and these typewritten carbons and penciled notes were like wood shavings from a completed cabinet, or lint from the office he'd occupied for thirty years in the English Department at Denver. Sometimes, as they talked, he took a mist inhaler from his pants pocket and pumped a puff of vapor into his mouth. He was a chain-smoker, never quit. If he wanted to fetch something from his second-floor study, she waited. She heard him struggling to breathe between his slow footfalls on the stairs.\n\nShe grew to like him. His fields of study had been Elizabethan poetry and creative writing. By coincidence, her father had attended the University of Denver and majored in English, and he was an English Renaissance scholar too. She invited John and Nancy to her house for dinner the next time he was visiting.\n\nThe two men did have much in common, and she and her husband enjoyed hearing a pair of older heads reminisce about their careers in academe. Both World War II veterans, they'd been in graduate school in the 1950s. And they'd both known many of the same people\u2014names that had kept appearing in Williams' papers\u2014the literary critic Yvor Winters and his wife, the novelist Janet Lewis; the poets Wallace Stegner, John Ciardi, and J. V. Cunningham; and Alan Swallow, the owner of a small press in Denver. So many names from long ago that the room, illuminated by the candlelight, seemed inhabited by listening ghosts, eager to hear themselves mentioned.\n\nShe found it hard to reconcile the English Department talk with a photograph of John she'd come across in his papers. It was a black-and-white studio portrait from the 1940s, the kind you'd have to make an appointment for. He was in his early twenties then. In the picture, he was wearing a lightweight sports jacket with a small-checkered pattern, over a button-down white shirt with a dark, regimental-stripe tie. His face was turned slightly to the side, his smooth cheek reflecting some of the light that fell favorably on his narrow lip mustache, parted in two artistic brushstrokes, left and right\u2014the image of a young man who wants to give the impression of sophistication. She hadn't seen it on the book jacket on any of his novels, so maybe it was taken before he'd been published.\n\nNow he was across the table, half a century later. She caught herself thinking how lined his face had become since that photograph. Proof once again of her belief that sometimes you can tell just by looking at people how interesting their lives must have been.\nPART I\n\n_Nothing But the Night_\n**CHAPTER ONE**\n\nHe Comes from Texas\n\n_Don't read so much, you'll make your brain tired_.\n\n\u2014JOHN WILLIAMS' GRANDFATHER, 1930S\n\nTo the jobless men killing time in Wichita Falls, Texas, in 1939, John Ed Williams\u2014the son of the janitor at the post office\u2014invited a second look as he passed by after school to his job assisting customers at the Lovelace Bookstore.\n\nStudents like the Williams boy were becoming a fairly common sight downtown, ever since Hardin Junior College had opened just a few years earlier. In the early 1920s, college classes of about seventy-five students had met on the third floor of the high school at the intersection of Avenue H and Coyote Boulevard. But after oil tycoon John G. Hardin plowed close to a million dollars into a whole new campus on forty acres of former pastureland in the city's south end, going to college had become a possibility for almost two thousand students. Wichita Falls, formerly just a railhead in northern Texas for cattle drives, now had a two-year college for the grandchildren of pioneers\u2014a point of pride among the residents\u2014and it had been built, it needed to be said, right in the middle of the worst economic depression in the nation's history. The banks had gone bust; the Great Plains were ripped up from overfarming; and during the long, ruinous droughts, mountainous dust storms the color of dirty snow blew across northern Texas.\n\nBut here was a flamboyant young man who apparently expected better times. John Ed wore a blazer and pleated trousers. Instead of a tie, he wore a silk scarf, as if he were imitating some kind of English country squire or poet. He acted as if he were from another time and place. And indeed, in his mind at least, he was.\n\nJohn Edward Williams' grandparents, Elbert G. Walker, from Virginia, and his wife, n\u00e9e Laura Belle Lee, from Tennessee, were southerners. Elbert was twenty-five, and Laura Belle only sixteen, when they married in 1886. Within a few years they headed west, drawn by the Oklahoma Land Rush of 1889, which opened for settlement two million acres of the best free public land in the United States. The Walkers' two daughters, Emma and her younger sister, Amelia\u2014John Ed's mother\u2014were born in 1896 and 1898, respectively, in Oklahoma's Choctaw Indian Territory.\n\nElbert Walker's line of work was market gardening\u2014raising vegetables for local sale. But he seemed convinced that there was always a better situation to be had elsewhere. For the next twenty years, the Walkers circled the lower Great Plains, picking up and starting over each time. By 1910, they had left Oklahoma and settled on a farm in Texas outside of Young, a prosperous hamlet of cotton farmers and cattle ranchers. Then, ten years on, they could be found dirt farming in Logan, Arkansas, in the Arkansas River Valley, where many of their neighbors had come from Kentucky and Tennessee. Amelia and Emma\u2014both young ladies now in their early twenties\u2014lived at home, although Emma taught at the school. But Elbert was still not satisfied with his lot, and he decided, in his early fifties, that things would work out better if they went down to Clarksville in northeast Texas, where his mother had settled until her death in 1908, and they moved there.\n\nHe had reason to believe he might be right, because Clarksville was doing well in the 1920s. Its 3,500 residents enjoyed the services of two banks, two newspapers, two flourmills, and an electric power plant that illuminated homes and businesses. The biggest employer in town was a factory that crushed cottonseed for its light golden oil, filling the air with an aroma that was sweet and nutty. On Saturdays, farmers sold watermelons, livestock, and vegetables in the town square from trucks and wagons. In the center, above them, a statue of a Confederate soldier stood high on a twenty-foot pedestal facing north\u2014its face and uniform eroded smooth like a pillar of salt from the weather\u2014perpetually ready to repulse another Yankee invasion. Sunday mornings, the ministers of seven churches, five white and two black, delivered sermons to their congregations.\n\nIt was in Clarksville that Amelia Walker, twenty-five, met J. E. Jewell, a gentleman a few years older than she. They married on January 21, 1921. Nineteen months later, she gave birth to a son, John Edward, named after his father.\n\nLike the Walkers, J. E. Jewell was comfortable with impermanence. He shared his father-in-law's yearning to find El Dorado along the way to somewhere else. Together, the two men led their families to their next destination, Wichita Falls\u2014250 miles north\u2014the scene of the first memories of the little boy who would become John Ed Williams.\n\nIn the early 1920s, Wichita Falls was infected with oil fever. Ever since the Lucas No. 1 Well near the town of Beaumont on the Gulf Coast had struck \"black gold\" in 1901, oil fever had been boiling over in Texas. Fresh water that smelled like rotten eggs meant there was a pocket of oil below somewhere\u2014maybe a whole sea of it\u2014trapped inside a salt dome. Giant drill bits, centered by A-frame derricks, twisted down deeper and deeper into the earth like iron hypodermic needles. At Sour Lake Springs, where the water stank like burning sulfur, a black sticky plume of petroleum flared a hundred feet high. The oil reservoir was so huge that before long, derricks sprouted everywhere, like rows of wooden Eiffel Towers, hundreds of them, serviced by thousands of roustabouts who rinsed the dust and grit from their throats in fifty-two saloons.\n\nIn 1918, \"Texas tea\" erupted near the edge of the Wichita Falls city limits. Fortune-seekers stampeded the boardinghouses, willing to pay ridiculous rates. If no rooms were available, newcomers slept in shacks, in tents, and in cars and trucks, deluging the local schools with bewildered children. Bank deposits increased by 400 percent. Street corners became outdoor exchanges where buyers and sellers wrangled over real estate, water permits, and deeds to mines. Companies that existed on paper only advertised themselves with \"come-and-get-it\" names like Over the Top, Sam's Clover Leaf, Bit Hit, and O Boy!\n\nThe Walkers and the Jewells followed the flood. J. E. found an apartment for his wife and infant son, and a job selling animal feed and agricultural supplies at Miracle Coal & Feed Store. The Walkers, now without their elder daughter, Emma, who had stayed behind to teach in Clarksville, rented a farm a few miles outside the city on Walnut Road. But at least everyone was in the proximity of riches.\n\nThe work at the feed store was steady, but J. E. wanted a better life. The _Wichita Falls Times_ carried classified advertisements hinting that a clever man could climb higher\u2014like the one promising \"An opportunity to make some real money. Quick action, too. If you have a few hundred dollars in cash, see Mr. McKinney at the Morgan Bldg.\" The nature of the opportunity for \"real money\" didn't need explaining. You'd have to be living on the moon not to know it had to do with oil.\n\nWith opportunities for \"quick action\" also came frauds, double-talk, and the old switcheroo, however. The get-rich-quick game was not for the credulous. One of the biggest scams in memory had turned the Wichita Fall city fathers into a laughingstock in 1919. A property title researcher named J. D. McMahon offered to answer the demand for office space by building the first skyscraper in Texas at the corner of Seventh and La Salle Streets. Local investors envisioned a business beehive, a tower of ringing phones and ticker-tape machines opposite the prestigious St. James Hotel. After eagerly inspecting the blueprints, they raised the equivalent of $2.7 million to build it. But the completed Newby-McMahon Building, though handsomely appointed, was only 40 feet high, 10 feet wide, and 18 feet deep. The staircase took up a quarter of the interior. Outraged, the investors sued McMahon, only to have the judge rule against them when it was pointed out that the blueprints clearly stated that the wonder of Wichita Falls would be 480 _inches_ high, not feet.\n\nAccording to a story told in the family, J. E. made his move as an investor one day at an open-air exchange downtown where transactions consisted of handshakes, wads of cash, and contracts signed on the spot. He purchased some land located outside the city, intending to turn it around for profit by selling it to another speculator. He found a prospective buyer, showed him the property, and got paid in cash. On the way home, he stopped at a gas station to fill up. It's said that he mentioned his good luck. If he did, it marked him as a tinhorn who talked too much, like the hapless Swede in Stephen Crane's \"The Blue Hotel,\" who literally bragged himself to death. A hitchhiker approached and asked for a lift into town; Jewell, maybe feeling magnanimous, told him to get in. At some point during the ride, the stranger stuck a gun in Jewell's ribs, forced him to pull over, and killed him. A roll of bills that had already changed hands once that day ended up in still another man's pocket. The murderer continued off into the wilderness on foot.\n\nHowever, the real casualty might have been the truth. An examination of newspapers both in Clarksville, where the Jewells were married, and in Wichita Falls, where they lived, turns up nothing about the murder of a John Edward Jewell. Court records make no mention of it in either location. Furthermore, the robbery scenario is problematic. Why would the killer abandon a car with a fresh tankful of gas and a dead driver?\n\nPerhaps the last glimpse of J. E. Jewell, alive and heading for the red horizon, appears at the bottom of the front page of the August 12, 1924, edition of the _Vernon Record_ , published fifty miles from Wichita Falls: \"Cupid is taking a vacation according to officials of the County Clerk's office. Only two marriage licenses have been issued\u2014one for a Miss Rose Lee Owen and her fianc\u00e9, J. W. Todd; the second for a Mrs. L. L. Moreland and J. E. Jewell.\"\n\nVery little has been recorded about John Williams' father, and much less was remembered. Many years later, when Williams was filling out his own marriage license, on the line asking for his father's birthplace he wrote \"Unknown.\" \"Unknown\" will have to stand for Mr. Jewell's final disposition in the world, too.\n\nWithout a husband, Amelia listed herself in the Wichita Falls phone directory with \"(wid)\" after her name, indicating she was a widow. The suffix \"(wid)\" served to safeguard the reputation of an unmarried woman with a child. On the other hand, since the abbreviation was voluntary, it could also be interpreted as a kind of plea from a young woman possessing just a year of high school, and raising a small child alone. A studio photograph taken of her about that time shows Mrs. Jewell, quite lovely and dark-haired, gently balancing John Ed, a blond toddler, who is demonstrating, with a slightly surprised look, that he can stand on his mother's lap.\n\nNot long after that, Amelia received a gentleman caller\u2014George Clinton Williams.\n\nTen years older than Amelia, George Williams hailed from Tyler, Texas. During World War I he had served in the National Guard, claiming an exemption from the draft as the sole supporter of a wife and a daughter in Dallas. His people were in banking, but he had shunned the family's expectations, preferring instead to get paid on Fridays for shift work. He had driven a city bus, toted bricks in construction, and picked crops. He was squat, with a broad back and large hands.\n\nGeorge proposed and Amelia accepted. For a woman not quite thirty, marriage meant a husband, a wage earner, and a father for her boy. It was another fresh beginning, the sort her migrant parents had counted on so many times in their lives. Romance may have played a part too, but the reality is that Amelia's options were few.\n\nBut George, a heavy drinker, couldn't make a go of it, not even with an oil boom all around. Several times, he and Amelia jumped the rent in the middle of the night before the landlord arrived with the sheriff in the morning. Six months after they had moved into Wichita Falls proper, George and Amelia, with John Ed in tow, arrived with their belongings back at Grandfather Walker's farm just as he was putting in the spring vegetables. By then, another hard-luck relative had joined the Walker household as well: an elderly aunt who rocked all day in the parlor except to announce, now and then, \"Well, I believe I'll go to the bathroom!\" In May 1925, Amelia gave birth to John Ed's sister, who was given a boy's name, George Rae, because her father would have preferred a son.\n\nDefeat seemed to follow Amelia. When John was four, a terrible and touching image of his mother impressed itself on his memory. He heard a sound coming from the closet in the hallway. Opening the door, he saw his mother kneeling behind a curtain of swaying coats and jackets, hiding like a child, her face in her hands, sobbing.\n\nBecause they were poor, and the work they did was manual labor, silence was like a physical presence for the Walkers and Jewells. Williams later wrote, of the farm family in _Stoner_ , \"In the evenings, the three of them sat in the small kitchen lighted by a single kerosene lamp, staring into the yellow flame; often during the hour or so between supper and bed, the only sound that could be heard was weary movement of a body in a straight chair and the soft creak of a timber giving a little beneath the age of the house.\" Fortunately, there was a radio, and John Ed was allowed to listen to it with the adults, \"watching it,\" he remembered, as if voices became faces in the yellow glow of the dial.\n\nAlso, there were a few magazines to read. For the price of ten cents, his mother could escape into the pages of some of her favorites. She doted on _Ranch Romances_ , _Clues_ , _Love Romances_ , and _Sweetheart Stories_. Rather than allow her son to become too interested in swooning and kisses, however, Amelia introduced him to _Flying Stories_ , a monthly magazine featuring the Skywaymen\u2014leather-helmeted, square-jawed pilots\u2014who crossed the Arctic, flew low over thrashing waves in hurricanes, and rescued fellow aviators from their tattered biplanes in midair. In _The Blue Book of Fiction and Adventure_ , there were better stories, by contributors such as Sax Rohmer, Agatha Christie, and Edgar Rice Burroughs. From the school library, he borrowed every western by Zane Grey he could get his hands on, including the most famous, _Riders of the Purple Sage_. Grey, a midwestern dentist who had suffered under his father's authority (and beatings), found an outlet for his passions writing novels about an imagined American West with a wildness almost untouched by formal law and order, and governed instead by an unwritten code of loyalty, generosity, fair play, and integrity. Grey's idealized frontier restored men and women to natural grace, as if they had returned to Eden:\n\nVenters turned out of the gorge, and suddenly paused stock-still, astounded at the scene before him. The curve of the great stone bridge had caught the sunrise, and through the magnificent arch burst a glorious stream of gold that shone with a long slant down into the center of Surprise Valley. Only through the arch did any sunlight pass, so that all the rest of the valley lay still asleep, dark green, mysterious, shadowy, merging its level into walls as misty and soft as morning clouds.\n\nWilliams would one day challenge the paradise myth of the Old West in _Butcher's Crossing_ , especially Emerson's belief that living close to nature imparts virtue to human souls. But during his boyhood, John and his mother shared a wholesale love of stories\u2014the more melodramatic and more vivid the better\u2014for their power to spirit them away from the humdrumness of rural life.\n\nPerhaps that's why Amelia Walker waited until John Ed was nine years old to tell him the most important story of his life. Her decision to give away the secret of his birth might have been an act of love, because there are indications that John was unhappy around George Williams, who was often described as \"a difficult man.\" Revealing the truth might have been a mother's attempt at reassurance.\n\nGiven her love of romance, it's likely she presented it as a tale that her reader-son would appreciate. The setting and plot weren't uncommon in the West\u2014the Walkers were humble people, farmers, who had worked hard all their lives. But a mysterious stranger and a tragedy had changed everything. John's last name wasn't Williams\u2014not legally, because George hadn't adopted him\u2014it was Jewell. George was his stepfather; and his little sister, George Rae, who had just turned five, was his half-sister. His father, J. E. Jewell, had been murdered when John was still an infant, ambushed and killed while he was on his way home to his family, years ago, and the killer had never been caught.\n\nWilliams later claimed this revelation about his identity\u2014a thunderclap from his mother's lips\u2014didn't affect him, because George felt \"unfamiliar\" to him. But it's hard to believe that a nine-year-old boy could have been so philosophical. Personal identity is fundamental. Immediately after the creation of the world, God gives Adam the responsibility of assigning names to things to distinguish himself from other creatures. And the psychological work of constructing self-identity is repeated every time one child asks another, \"What's your name?\" In fairy tales, the unveiling of a secret, _hidden_ identity can be momentous\u2014the ugly duckling who becomes a swan, the despised stepdaughter who fits the glass slipper, or, more recently, the boy who learns he's a \"wizard, and a thumpin' good one\"\u2014because it cracks open self-perception, usually revealing fantastic, unrealized potential.\n\nBut in John Williams' case, news that he was not the person he thought he was called into question the importance of telling the truth. Until the moment of his mother's confession, the adults in his family had been deliberately lying to him. They had all been complicit in perpetuating a fiction about him. Whether they did so out of love or shame, the gist was the same: they had known all along, but he didn't. The lesson was that if everyone believes something is true then it might as well be. Lies, fictions, whoppers, and fibs could create meaning, and the best ones were the most convincing. Hadn't teachers called out \"John Williams?\" for the roll every morning, and hadn't he answered, \"Here!\"? For John Ed, the future novelist, the experience was profound, if terribly painful.\n\nAbout this time, Grandfather Walker decided his grandson needed some toughening-up. Although the boy spent time outdoors doing his chores before and after school, he spent a lot of time in the house reading. When he talked, he stammered, as if he was fearful. \"Don't read so much,\" Walker cautioned him. \"You'll make your brain tired.\"\n\nSo when John was ten, come the first cold day of winter, Walker got down his .22 rifle and escorted his grandson across the yard over to the pig pen. \"If you're going to eat pork, you should know where it comes from.\" They discussed which one was the healthiest-looking pig, then the old man climbed into the pen, took a couple of short boards, and used them to swat the animal on the rump and sides, steering it toward the fence. It ambled up to John, sticking out its muddy snout to snuffle at him. Walker leashed the pig to the fence post so it would stay there.\n\n\"Shoot it,\" he said. The old man tapped at a spot right between his eyes.\n\nJohn Ed raised the barrel of the rifle, aimed through the fence boards, and pulled the trigger. The rifle jumped and the report sent the other pigs scampering away. The old man knelt, reached with a butcher knife through the fence, and sliced the pig's jowl, releasing a splash of blood on its front feet. It collapsed butt first on the dirt then lolled halfway to one side, bleeding out through the gash. They got the truck, tied the rope around the pig's hind trotters, and dragged it over to the barn.\n\nThey filled an oil barrel standing on bricks with buckets of water from the pump, set fire to kindling underneath, and waited until it turned into a scalding bath. When Walker judged the water was hot enough, he looped the free end of a rope through a pulley overhead on a barn joist. Then they hauled the pig up into the air by its hind legs, swung it into place, and lowered it like it was diving into the boiling water. After a few minutes, they pulled it out again, steaming and clean. John Ed helped scrape the bristles off the hide until the flesh was white. With a smaller knife, Walker cored the rectum, pulled it out, went around to the front, split the crotch, and tied off the pizzle so there wouldn't be pee everywhere.\n\nIn _Butcher's Crossing_ , Williams' description of an experienced buffalo hunter skinning a carcass carries the same precision and matter-of-factness:\n\nThe hide parted neatly with a faint ripping sound. With a stubbier knife, he cut around the bag that held the testicles, cut through the cords that held them and the limp penis to the flesh; he separated the testicles, which were the size of small crab-apples, from the other parts of the bag, and tossed them to one side; then he slit the few remaining inches of hide to the anal opening.\n\n\"I always save the balls,\" he said. \"They make mighty good eating, and they put starch in your pecker.\"\n\nThen Walker took a hacksaw, held the pig's head steady by one ear, so the carcass wouldn't swing, and sawed through the neck. It separated and fell heavily to the floor of the barn. He shoved it away with his boot. He inserted the butcher knife into the neck hole and flayed upward through the chest, stopping to pull the rib cage apart, and stepped back out of the way. The gut sack leaned out and tumbled through, landing wetly at his feet.\n\nIt was hog butchering, a routine task on a farm, but Williams, observing it with a writer's eye, noted a procedure that had a beginning, middle, and end. How a beast falls from a gunshot and surrenders to the knife; how its flesh comes off like rind on a fruit; how slaughtering an animal was done rapidly and clinically.\n\nAs John Ed was leaving childhood and beginning adolescence in the early 1930s, his family was swept into a new life. In 1932, the third year of the Great Depression, Franklin Delano Roosevelt's confidence about the future\u2014his campaign song was \"Happy Days Are Here Again\"\u2014carried him into the White House. During his first one hundred days in office, billions of dollars rolled out from Washington across the nation through the New Deal federal recovery and reform programs.\n\nThe Williamses were among the beneficiaries. In Wichita Falls, at the corner of Tenth and Lamar Streets, built where a decrepit Methodist Church had been, an enormous new post office advertised for custodians. George Williams applied and landed a civil service job at a pay rate equal to an entry-level accountant. He and Amelia rented a house on Lee Street. In 1938, John Ed's grandparents, Elbert and Laura Walker, left their hardscrabble farm and joined them. For over fifty years, Grandmother Walker had followed her husband from one place to another: from Tennessee in the 1880s, to Oklahoma Indian Territory, to Texas, to Arkansas, and then back to Texas again. But now, at sixty-eight, she was dying from cancer, and she passed away soon after moving from the farm into town.\n\nNot long after her death, George purchased a good-sized house on Broad Street for Amelia, the children, and their grandfather. It was their first residence not owned by someone else.\n**CHAPTER TWO**\n\n\"Ho, Ho! Wasn't I the Character Then?\"\n\n_We linger for an instant in a morass of stupidity and conceit and then we graduate from high school_.\n\n\u2014JOHN WILLIAMS, AGE SEVENTEEN, 1939\n\nThe subject of a biography should be the person's search for identity. Answering the question \"Who am I?\" is the great work of life, and all of a person's efforts are, in some way, responses to that unvoiced question, which begins to be heard in childhood as soon as children perceive themselves as different and apart from everyone else. The amount of freedom to spend creating the opus of the self isn't the same for everyone, but at least a basic sense of \"Who am I?\" belongs to any child who knows where he lives, who his parents are, who loves him, who doesn't, and so on.\n\nJohn Williams' discovery at age nine that he was someone else\u2014not his father's son, not a Williams, and not related to his little sister, George Rae, in the way he had been led to believe\u2014upended his world. The narrative of who he was, the one he had been telling himself all along, was actually a story about another boy who was like him, but not him. He had the skin of John Williams on him, and the adults in his family\u2014his mother, stepfather, and grandparents\u2014had agreed to present him that way to everyone, including himself. It would be hard to imagine a more convincing early lesson for a future writer about the power of words to suspend disbelief.\n\nIn the meantime, the question \"Who am I?\" was reopened just when he was entering adolescence, a period when young people try out various faces on the world to see how they're received. Because of what he'd found out, he was more susceptible than most to choosing a radically new appearance, one that would compensate for how he had been fooled, one that would give others the impression of competence and self-possession. Ideally, too, it would be in sync with his love of books and stories, and his fascination with the heroism of airmen and frontiersmen.\n\nOne afternoon when he was thirteen, in the spring of 1936, he saw such a character\u2014a persona, really\u2014at the movies in Wichita Falls. The experience enchanted him.\n\nJohn was invited by an aunt to bring a friend to see the film _A Tale of Two Cities_ , starring Ronald Colman as Sidney Carton. Before buying their tickets, they stopped for lunch, and the boys ordered glasses of cold buttermilk, waiting to drink it until their lady chaperone excused herself to use the restroom. Once the coast was clear, John produced a cough syrup bottle filled with whiskey\u2014probably from his stepfather's stash\u2014and sociably poured half of the contents into his friend's glass and the rest into his. By the time his aunt returned from freshening up, the boys were wiping off buttermilk mustaches and grinning.\n\nAs a prank, it was a good one for a pair of adolescent boys. And the spiked buttermilk gave Williams his first real glow from liquor in public, besides adding to the excitement of watching a splendid Hollywood version of Dickens' nineteenth-century novel. Westerns were the usual matinee fare, and audiences were delighted by cowboy idols on the screen and their cold \"violence without rage,\" as Williams later called it. But Ronald Colman's performance thrilled him in ways no gunfighter ever had.\n\nIn _A Tale of Two Cities_ , Sidney Carton is an eighteenth-century English barrister wasting his talent with drinking and self-pity. But then he takes the case of French aristocrat Charles Darnay, arrested on charges of spying for the French. During the proceedings, Carton meets Darnay's love interest, Lucie Manette. Carton falls in love with her and wishes he were a better man, someone worthy of a woman like her. Ironically, Carton and Darnay resemble each other enough to be brothers, an allusion to the type of person Carton might have been. For Lucie's sake, Carton makes the case for Darnay's innocence brilliantly and brings in a verdict of not guilty, after which Darnay and Lucie marry and return to Paris.\n\nBut sometime later, Darnay is arrested again, this time caught in the net of the Reign of Terror, and he's sentenced to die. Carton sees a chance to atone for his failed life and ensure that he will live honorably in Lucie's memory. Because they look so much alike, he can impersonate Darnay and take his place on the guillotine.\n\nAs he waits to be called at the foot of the guillotine\u2014an innocent man exchanging his life for another's\u2014Carton, the former cynic, is at peace. \"It is a far, far better thing that I do, than I have ever done; it is a far, far better rest I go to than I have ever known.\" As he ascends the steps, the heavens above the rooftops of Paris come into view, underlining his redemption.\n\nWilliams was smitten\u2014he thought he'd seen a parable unfold with a message just for him. On the one hand, _A Tale of Two Cities_ is a story about injustice, oppression, and one man's Christ-like sacrifice for another, but it appealed to him because it is also a fable about two identities, like the prince and the pauper story.\n\nCarton and Darnay are two halves, dark and light. One embodies lost chances and bad luck; the other, his double, represents goodness, rectitude, and success. Ronald Colman delivered the character of Carton with a combination of grace, sex appeal, and vulnerability. His insouciant response to heartbreak is a jaunty tip of the hat. Inwardly, he suffers, but outwardly he appears cavalier, a little mad, the poet with a cloven hoof. But by bearing the worst of things with dignity, right up until the moment of his death, he turns misfortune into a moral triumph.\n\nHere was a message, a view of life that Williams\u2014a smallish boy who felt different from others\u2014could take to heart. Maybe it was possible to will yourself to be the kind of person you wanted to be. Practicing that person, performing him, despite other people's skepticism, might bring off a new invention, which would be you, the hero of your own life.\n\nA week later, his eighth-grade English teacher at Reagan Junior High, a Miss Annie Laurie Smith, assigned an essay on a topic of the students' choosing. John Ed pounced on Colman's performance as the subject of his. Miss Smith, head of the English Department, was his favorite teacher, and he tried hard to impress her.\n\nA few days later, he listened abashed and proud as she read his paper aloud to the class. She finished by saying that his analysis of Colman as Sidney Carton was \"the work of a college student.\" Not only did she praise it, she honored it by placing it in her special display case in the hallway reserved for outstanding work.\n\nHe never forgot that moment. \"It was one of the first compliments I had received about anything I'd done,\" he later said. In the stroke of a day, he had been elevated from ordinary to, as one of his classmates put it, \"someone special among us.\"\n\nMoving into town as a result of his stepfather getting a job at the post office put John Ed within the boundaries of Wichita Falls Senior High School. With Miss Smith's praise still ringing in his ears, he began his freshman year in 1936.\n\nFrom then on, he counted himself as writing with \"serious purpose and intent.\" He joined the school newspaper and submitted poems to the literary magazine. Writing, he said, was \"something I could do, and I felt some confidence in it.\" From the school and public library he borrowed armloads of titles by Willa Cather, John Steinbeck, and William Faulkner, fastening on Thomas Wolfe as his favorite author. Reading _Look Homeward Angel_ introduced him to the journey of the young artist-hero Eugene Gant, who rises above circumstances in many ways similar to Williams' own. It felt like a \"religious experience,\" he said, realizing that his upbringing didn't disqualify him from becoming a writer, which he thought would be a romantic life.\n\nHe fell upon the school library with such ferociousness, in fact, that the _Wichita Times Record_ ran a feature about the student \"poet-critic\" who was reading \"four to five novels, poetry collections, or biographies a week.\" The reporter posed the young dynamo beside a table stacked high with books, contrasting Williams' size with his appetite for reading. \"You, reader, are the victim of an optical illusion. Those books aren't over John Ed Williams' head. He has read them all and a great many more.\" The high school librarian spoke proudly of him, confirming that John Ed was \"the heaviest and most constant reader of the entire student body. No other student even approaches him in the number of books read.\"\n\nThe joke comparing his size to the height of the books behind him was an on-the-spot inspiration of the photographer, probably, but it called attention to Williams' height. In vain he waited for the growth spurt that teenagers usually get. But by the time he could drive it was plain that he would not be a \"tall lank boy\" like Wolfe's Eugene Gant. He was leveling off in the mid-five-foot range and possessed of a stubbornly small frame that resisted the effects of exercise that might add to his shoulders and chest. Aware that tall men get a kind of built-in respect that is not available to others, he worked at ratcheting up his presence and masculinity instead.\n\nHe was careful about his appearance and laid out his clothes as ensembles, a habit that never left him. His cinematic hero Ronald Colman appeared in movie adaptations of James Hilton's novel _Lost Horizon_ and Anthony Hope's _Prisoner of Zenda_ , modeling the kind of sophistication that Williams wished to have. In the breast pocket of his sports jacket, he tucked a folded handkerchief. He let his hair grow long like Eugene Gant's, combing it carefully before the mirror every morning. To conquer his stammer, he enrolled in drama courses, learning elocution and how to produce a baritone voice that came from his midsection. He discovered from school productions that he was a good actor, and wrote a one-man \"imaginary sketch of Abraham Lincoln's musings some troubled afternoon in the White House,\" which he performed at a school assembly. It's a sign of his willpower that he would write and perform an original piece before the student body, believing that he had every right to seek their attention because he was talented.\n\nIn the high school yearbook, the _Coyote_ , he listed the two things he liked: first, \"Me,\" and, as would befit a Byronic young man like himself, the color \"Black.\" (One afternoon cleaning his .22 rifle at home, he shot himself in the shoulder and was taken by ambulance to the hospital. After that episode, he strolled around the downtown jauntily smoking a cigarette, his other arm in a sling. Who could doubt that he was mad, bad, and dangerous to know?)\n\nSex raised its tumid head, of course. At the Lovelace Bookstore on Ninth Street where he worked after school, an older boy, T. G. Willis, manager of the stamp-collecting department, loaned him a censored edition of D. H. Lawrence's _Lady Chatterley's Lover_ , which had been rinsed for American readers of \"fuck,\" \"orgasm,\" and \"penis.\" But it was still instructive: \"And he took the leaves from her hair, kissing her damp hair, and the flowers from her breasts, and kissed her breasts, and kissed her navel, and kissed her maidenhair, where he left the flowers threaded.\"\n\nFor rougher stuff, there were drugstore magazines, such as _Spicy Adventure_ , _Spicy Mystery_ , or _Spicy Detective_ , copies of which he sometimes picked up on his way home. _Spicy Detective_ carried a comic strip starring \"Sally Sleuth,\" a maverick crime fighter. The point of the comic was to have her clothes come off just a few panels into the story. In one episode, crooks tied her by her wrists to a chair back; she was wearing nothing except panties, and they whipped her.\n\nBy his senior year, he had acquired a reputation as a kind of finer soul among his classmates. He was editor of the school paper, and the previous year, he had been awarded first place in the state poetry contest. \"Hail and farewell!\" he wrote in an essay for the 1939 yearbook. \"We linger for an instant in a morass of stupidity and conceit and then we graduate from high school.\" He had finished in three years instead of four. (\"Ho, ho!\" he laughed when that was read aloud to him many years later. \"Wasn't I the character then?\")\n\nIn August, he enrolled at Hardin Junior College. A young man now, and a college student, he adapted the attire of an artist, wearing a knotted scarf above his open shirt\u2014an outfit that gave the broken-down farmers and unemployed businessmen on the street corners of Wichita Falls reason to believe that the younger generation was going to hell.\n\nDespite the hard times of the 1930s, Hardin Junior College offered students a wide range of academic choices. But Williams fed his imagination and career ambitions mainly from the buffet of extracurricular clubs and activities. By the end of his first semester, he was so involved in things that he was flunking freshman English. His excuse was, \"I had already read most of what we were studying. And I had read a lot of it in high school.\"\n\nHe preferred putting his energies into things that showcased his talents. He added his name to the roster of every organization dedicated to self-expression: the literary club, the student senate, the student newspaper, and the Blue Curtain drama group. He made friends with two students who were serious about becoming actors\u2014Marjorie Coleman and Jack Newsom. On Thanksgiving, they performed for broadcast on KWFT, a new radio station in Wichita Falls, an original drama by Williams called _Dr. Cooper Speaking_ , about an English headmaster. It went so well that he wrote and directed one every week for KWFT after that, for which he would \"kind of produce it, play a part myself and get local actors around to play the others.\" George Rae, his half-sister and a teenager now, basked in the afterglow of her brother's prominence in the local arts scene. She began telling her high schools friends that she, too, planned to act, commencing an affectionate rivalry with her brother.\n\nBut come September 1939, he didn't reenroll at Hardin. The reason was money, he said, which is surprising, because he had a part-time job at a bookstore, and his stepfather, George, was a civil service employee who, according to the 1940 census, was one of the better-paid workers on Broad Street.\n\nWilliams later said too many people in Wichita Falls cared about money. It was \"absolutely pervasive. . . . They were conscious of the lack of money and the possession of money . . . a kind of consciousness about the _existence_ of money that seemed involved with identity.\" It could be that despite his stylish clothes\u2014paid for with his own money\u2014and his accomplishments in school, some of his peers kept in mind that he was the son of the post office janitor. It wasn't lack of money that kept him from reenrolling in college, perhaps, but the social side of student life.\n\nIn any case, he spent what would have been the fall semester of his sophomore year preparing to stage a piece of theater for the public\u2014Thornton Wilder's _Our Town_ , the winner of the 1938 Pulitzer Prize for drama. With his friend Jack Newsom as codirector, the two young men held open auditions. To play the townspeople in crowd scenes, John prevailed on twenty-four members of his family's church, First Presbyterian, to take the parts. For the role of Emily Webb, the female lead, they cast their friend Marjorie Coleman, who had been acting in summer stock in upstate New York.\n\n_Our Town_ was so innovative for its time that the _Wichita Daily Times_ gently primed readers about what to expect: \"Instead of a rising curtain to open the show, there will be the stage manager\u2014Jack Newsom in this production. The stage manager starts it off by making the audience acquainted with Grovers Corners, New Hampshire, right there on the stage behind him. Then the natives begin to stir about in the morning air, and the play is on.\" Because the \"entire play is of such disarming simplicity,\" there would be no sets, and actors would mime the props. All that was necessary was the audience's imagination, the newspaper guaranteed.\n\nThree weeks after staging _Our Town_ , Williams presented a costumed biblical pageant for Christmas at First Presbyterian and narrated the Nativity story, with his deep, mellifluous voice filling the sanctuary.\n\nFor a young man of nineteen, he was off to a quick start in the arts. He thought he saw his way ahead, although optimism among Americans was scarce in the spring of 1941. Conservatives in Congress had buried Roosevelt's New Deal under counter-legislation; newspaper headlines predicted war in Europe; and rainfall in dusty northern Texas was just half the yearly average. A crash in farm prices would trigger another financial quake and further depress the job market. But to young Williams\u2014budding actor, writer, poet, he wasn't sure which yet\u2014the stars were aligning for him, at least. He had a sense of knowing what he was about.\n\nHis reading kept pace with his ambitions, too. He upgraded from boys' magazines to _Liberty_ magazine and the _Saturday Evening Post_. He also discovered the \"little magazines,\" as they are still called, such as the _Partisan Review_ , where writers talked about the craft of writing and the book reviews suggested to him what to read next. Wolfe, his favorite author, was replaced by Conrad Aiken. Aiken introduced him to psychological fiction, which led on to Proust. He grazed in the public library and purchased books at a discount at the Lovelace Bookstore, reading whatever he liked. \"It wasn't anything I had to study. It seemed like kind of a fraud, it was so much fun.\" He read for enjoyment\u2014\"And I think that finally, when I began to read so-called more 'serious' things . . . they became a great deal more authentic than if I had gone to them thinking they were supposed to be terribly good.\"\n\nLikely his sister, George Rae, would be the only one interested in talking about books at the dinner table on Broad Street in the spring of 1941. Their mother, Amelia, listening to them, had only completed the eighth grade; Grandfather Walker, who had fallen back on the kindness of his daughter and son-in-law, was a simple man. And George Williams, of course, would be tired from finishing a ten-hour day cleaning the post office.\n\nIt's interesting to speculate what George Williams might have thought as he listened to his stepson hold forth about another extravaganza he was involved in. George had turned his back on his banking family and everything that tribe had stood for. How did he regard John Ed's slightly raffish clothes and his talk about poetry, novels, and his actor friends? His stepson, just like the stage-struck teenager in Willa Cather's short story \"Paul's Case,\" had \"high, cramped shoulders and a narrow chest,\" and large blue eyes that he used \"in a conscious, theatrical sort of way.\" George, a stocky man with a broad back from years of labor, may well have regarded Amelia's boy as a kid who needed to concern himself less with putting on make-up for plays or striking arty poses, and more with finding a steady job to support himself. Williams never mentioned his stepfather, an indication, perhaps, that the relationship was unpleasant.\n\nIn the spring of 1941, John Ed struck out on his own. Beckoning him was radio\u2014and more experiences like performing _Dr. Cooper Speaking_. National broadcasting had become a kingdom of the air, featuring soap operas, symphony concerts, news commentary, variety shows, and eyewitness reporting. Such was the popularity of radio that the problem of overlapping programs created wild cacophonies of voices and screeching music on the dial. By order of the federal government, over seven hundred broadcast stations shifted their frequencies to untangle the airwaves.\n\nWilliams' voice and timing were perfect for the medium, and he had already demonstrated that he could write dramatic scripts. In April he appeared on stage a final time in the Wichita Players' production of Henrik Ibsen's _Pillars of Society_ , and then he was gone, to add his voice to those competing to be heard, and not returning to Wichita Falls, as it would turn out, until after World War II.\n\nThe summer of 1941, listeners to station KRRV, \"The Voice of the Red River Valley,\" broadcasting from Sherman, Texas, were introduced to a new morning announcer\u2014\"Jon Williams,\" the on-air handle for John Ed Williams.\n\nThe format of KRRV was folksy but proper. The small staff of women running the station, some of them volunteers, worked to make radio an uplifting instrument of the home. On Sundays, \"Jon Williams\" set up the equipment for live broadcasts from area churches. During the week, he interviewed local politicians and civic leaders who wanted to say a few words to KRRV's listeners about a wholesome topic, anything from religious freedom to the firemen's annual bake sale. September found him broadcasting live from the county fair. From a booth beside the pickle display, he queried farmers about their crops, and asked homemakers for tips about how to make a dollar stretch in the kitchen. Listeners could hear the cheers and whistles of ranchers and farmers in the background who were watching Negro football teams play on a field of mown hay.\n\nIn KRRV's family-oriented broadcast day, there was no room for Williams' original sketches or news commentary. But in Denton, Texas, just north of Dallas, Harwell V. \"Shep\" Shepard, the owner of KDNT, could use an extra hand.\n\nThese were the wildcatting days of radio, and entrepreneurs hurried to sink broadcast towers into the ground ahead of the competition. \"Shep\" Shepard, born and raised in Denton, had tried his hand at other things before the radio business, including running a funeral home. But broadcasting appealed to him as a better business idea, because Dallas, a big market for advertising, would be within range of a low-power station in Denton. With a borrowed investment of $6,000, Shepard hand-wired the cottage adjacent to his home at 400 Ross Street. Above it, local plumbers constructed a 160-foot broadcast tower. It fell over, then rose again. On June 1, 1938, KDNT-AM went live. To pay the station's bills, Shepard called on advertisers in a 1932 Plymouth sedan with the station's call letters and thunderbolts painted in gold on the doors. His wife, \"Red,\" handled the bookkeeping.\n\nWhen Williams arrived to interview for a job as a newsman, Shepard showed him the new United Press International Teletype machine clattering away in a corner of the studio, generating enough bulletins from around the world for news broadcasts throughout the day. Williams was impressed and wanted the job. But Shepard also needed someone who could write advertising copy for merchants purchasing airtime. If Williams was interested in that\u2014on commission\u2014then they had a deal. They shook on it.\n\nA week later, \"Jon Williams\" was the station's main announcer and engineer. He reported the news, wrote advertising copy, played records, and ran the instrument board for his own broadcasts. Some Saturday nights, he stayed late at the station to set up the studio like a barn dance, adding hay bales for atmosphere. Those nights, traveling western swing bands took their turns at the standing microphone, singing and telling corny jokes until midnight for the listeners of KDNT.\n\nThe job had an unexpected bonus, though. The station also carried the _Saturday Night Variety Show_ from the campus of North Texas State College in Granbury, southwest of Dallas. Williams' official reason for making the two-hour drive down to Granbury was to set up the equipment in the auditorium before the show. But for socializing and meeting smart girls, there wasn't anything like North Texas State College for miles around. Over 90 percent of the student body was female. During one of his trips to campus, he met Alyeene Rosida Bryan, a junior majoring in music and drama.\n\nAlyeene's father was a lumberman in Granbury, her mother a housewife. There was something a little exotic in her Mexican-Irish features\u2014dark eyes and tight ringlets of hair that fell to her shoulders. She acted, sang, and performed as a soloist at recitals and weddings. Like John, she had been editor of her high school newspaper, and she wrote poetry, too, which she showed him, knowing he had won prizes for his. She was said to be a little melodramatic, and abided by a rarefied southern code of behavior common to the better class of young ladies. If John hadn't given her proper warning that he was coming to campus, she let it be known, through her roommate answering the door, that she \"was not accepting callers\" that evening\u2014could he call again?\n\nThey married six months after their first date, on Easter Sunday, April 5, 1942, in the Granbury Methodist Church. Both families went to lengths to see that it was a traditional ceremony, despite the demands of the war. The church altar was decorated with irises and lilacs in tall baskets. Jack Newsom, Williams' actor friend, was best man. The bride wore an afternoon dress of taffeta instead of a bridal gown, and carried a white Bible. Her bridesmaids wore pins: a pink enamel heart pierced by a wooden match with the words \"It's a Match\u2014Alyeene and Jon, April 5.\"\n\nIn addition to changing his name, Williams had also burnished his educational achievements. The _Hood County Tablet_ printed on its \"Weddings\" page that the bridegroom had earned a bachelor's degree from the University of New Mexico and done \"graduate work in Chicago.\" These credentials were later amended to \"attended the University of New Mexico\" in the wedding announcement appearing in the _North Texas State_ campus newspaper, which was read by many of Alyeene's friends who knew Jon.\n**CHAPTER THREE**\n\nRough Draft\n\n_\"Take any man,\" I said, reaching across the table and lighting your cigarette, \"study him carefully. . . . Take him, season with a little imagination and sympathy\u2014and you'll have a novel.\"_\n\n\u2014JOHN WILLIAMS, DRAFT FOR A STORY, MARCH 1944\n\nScanning the newspapers in September 1942 for items he might use for broadcast, Williams noticed repeated calls for men to enlist in the Army Air Corps. Since the bombing of Pearl Harbor ten months earlier, recruiting officers were advertising for volunteers to join up before they were drafted, especially ones with certain skills. The drumbeat on page two of the _Lubbock Morning Avalanche_ was typical:\n\nIf you're between the ages of 18 and 50 and physically able to pursue any kind of mechanical trade in civilian life, and if you're interested in enlisting directly into the United States Army Air Corps, you're urged to drop in at the recruiting station and airplane engine exhibit at 1012 Main Street today. . . . Men with any mechanical or radio technical experience may be qualified.\n\nIn this instance, specialists were needed for crew duty in bombers and transport planes. During World War I, air combat had been fought between the pilots of single-engine biplanes made of canvas and plywood. They fired at each other, raked the mud trenches below with machine-gun fire, and hand-dropped bombs the size of footballs. But now, heavy aluminum and steel bombers required crews of six to ten, including a radio man, for high-altitude, long-distance missions over Europe, Asia, and the South Pacific. An officer quoted in the article still had one foot in the previous war when he said that radiomen contributed as much as pilots to \"helping the Doughboys.\"\n\nFor John, military service would defer his career just as opportunity seemed to be opening up. His boss, Harwell Shepard, had a request in at the federal level to boost KDNT's broadcast output from 100 to 250 watts. A stronger signal would carry Williams' newscasts down past Dallas all the way west to San Antonio and east to Beaumont, doubling the listening area. If a national network like CBS heard him, it might syndicate his public affairs reporting. Shepard was nevertheless resigned to going the duration of the war with no increase of signal, given the emergency state of things and the bureaucracy in Washington.\n\nWilliams decided there was no point in delaying his military service. If he waited until he was drafted, he might be assigned to the infantry. But if he enlisted in the Army Air Corps, he could take the exam for radio technician, which he knew something about.\n\nMarried only five months, Williams enlisted \"somewhat reluctantly\" in the Army Air Corps at the recruiting station in Dallas on October 6, 1942, quite possibly the same place where his stepfather had registered for World War I in 1917. No patriotic surge of feeling swept Williams into the armed services, however. \"We had to get into the goddamn war\" was his attitude. And the majority of Americans felt the same way. As one historian wrote of lack of choice in the matter, \"If there was no practical alternative, there was certainly no moral one either. Britain and the Commonwealth were carrying the battle for all civilization, and the overwhelming majority of Americans, led in the late election by their president, wished to help them.\"\n\nJohn's young wife, Alyeene, would have to finish her teaching degree at North Texas State without him. A week after he enlisted, she saw him off at the bus depot as he departed for basic training. She had hoped he would be able to attend her solo performance coming up at Granbury Methodist Church, but recruits were never granted leave during the two months of basic training. Three hours later, he arrived at Sheppard Field, still under construction on three hundred acres of former pastureland outside Wichita Falls.\n\nHe was a twenty-year-old bantamweight, and boot camp was fierce. His pack and rifle tugged against his 104-pound frame during ten-mile marches, but he passed the radio technician examination. Alyeene came by bus from Denton in mid-November to say goodbye before he transferred to the Air Corps Technical Training Command in Greensboro, North Carolina. Then, on the morning of Christmas Eve, she set out from Dallas by train to visit him, arriving exhausted in Greensboro on Friday, Christmas night, after thirty hours of traveling, with only two days remaining on John's three-day pass.\n\nSaturday was warm for December but rainy, and they spent most of it alone. The movie theaters, restaurants, taverns, and stores were open on Sunday, which the strict religious \"blue laws\" normally would have prohibited. But it was wartime and restrictions had been lifted. Come very early Monday morning, they parted at the Southern Railway train station while it was still dark. By the time he reported for roll call, she had already begun the two-day trip back to Texas.\n\nHis orders, when they came through in mid-January 1943, assigned him to the 10th Air Force, 443rd Troop Carrier Group, 1st Troop Carrier Squadron. From San Francisco he would leave aboard a troopship bound on a fifty-day voyage to Bombay, the port city on the Arabian Sea. A train would take him to Calcutta, followed by truck transport to Sookerating Airfield on India's border with Tibet in the Assam Valley.\n\nSookerating was the Allies' closest airbase for supplying General Chiang Kai-shek, who was directing the Chinese National Army fighting the Japanese. Allied transport planes flew the five-hundred-mile resupply route every day, sometimes twice, soaring over the highest mountain range in the world, the thirty-thousand-foot wall of the Himalayas, nicknamed \"the Hump\" by pilots. The only overland route, the Burma Road, was in the hands of the enemy. Journalist Eric Sevareid visited the airfield at Chabua in the Assam Valley and found it \"a dread and dismal place. . . . They were trying to do too much with far too little. Pilots were overworked, and when they had made the perilous flight to China and back the same day, having fought storm and fog and ice, they simply fell into their cots as they were, unshaved and unwashed, to catch a few hours of unrefreshing sleep before repeating the venture the next day.\"\n\nIn March, Williams debarked in the port of Bombay to begin a train journey of eight hundred miles across mountains, plains, and desert toward Eastern India and the thickly jungled province of Bengal. For five days and nights, he passed through \"small, indescribably filthy villages,\" so described by an army engineer who also traveled the same route. Using his duffel bag for a pillow, radioman Williams tried to sleep on his wooden bench in the drowsy heat, but was awakened whenever the one-note locomotive whistle shrieked at people or animals on the tracks. Each time the train stopped, villagers rushed forward selling trinkets, or begging for food or a little _baksheesh_ , money.\n\nIn Sookerating, Williams got to work supervising \"tea parties\" of local laborers who loaded C-47 Skytrains\u2014\"Gooney Birds,\" the crews called them\u2014using blocks and tackles, and sometimes elephants sleepy with opium to keep them calm. Jeeps, trucks, grenades, coffee, gasoline, bombs, spare parts, and ambulances\u2014anything useful to the Chinese army, including generals\u2014went over the Hump. Once the aircraft had been closed up and made ready, Williams set aside his clipboard and changed into his role as radioman. First, he tested the beacon box, which beamed a secret signal identifying the airplane as friendly, and installed it in the tail section. Then he took his seat behind the cockpit and checked the communications equipment to his left and in front of him. Over the next three and a half hours, his job would be to communicate with ground operators using the daily code, take time checks, make entries in his log, and provide position reports to the navigator plotting the course.\n\nThe airbase was so near the foot of the Himalayas that, to climb fast enough for takeoff, pilots locked the brakes, waiting for the twin Pratt & Whitney 1,200-horsepower engines to roar to a thunderous, whining pitch\u2014until everything on board was shaking. The instant the brakes were released, the plane leaped forward and rushed down the runway, gathering speed and lifting its two-ton cargo into the sky. Even so, C-47s had a ceiling altitude of only twenty-three thousand feet, and if the pilot or navigator lost his way, some of the Himalayan peaks were still half a mile higher.\n\nAs the plane rose out of Assam Valley, and above the steam bath of the jungle, the men pulled on fleece-lined leather jackets. Temperatures inside the plane slowly dropped to below freezing. There were no heaters, because they weighed too much. From the Upper Assam Valley to the Yunnanese plateau, they first crossed the Mishmis Hills, which are between two and three miles high. At this point, Williams flipped on his oxygen and began using a mask. He was a chain-smoker, so he pushed the mask off-center, allowing enough oxygen to spill out the side to keep a cigarette lit in the corner of his mouth. Through his headset he sometimes heard the voice of a Japanese disc jockey the men nicknamed \"Hanoi Hannah.\" In between playing American songs and big band music, she delivered personal messages. \"To the boys on N99131\u2014turn back, fellas. Lt. Jim Kelley, turn that plane around. Because if you don't, you're all going to die on this flight.\" Williams and the rest of the crew were more concerned about the weather. From June to October, the monsoon season brought anvil-headed thunderclouds with violent updrafts that could flip a plane on its back. During night thunderstorms, the propellers sometimes became wrapped in the blue orbs of St. Elmo's fire. Spidery filaments of electricity crackled around the windows of the cockpit like miniature lightning. Occasionally, when a pent-up charge arced with the airplane's steel frame, an explosive blue fireball knocked out Williams' radio equipment and stopped communication with the ground.\n\nWinter storms brought another kind of terror. The C-47 was a workhorse, but when the wings were glazed with ice, it waddled drunkenly. At those times, Williams came forward and grabbed the retraction lever between the pilots, running the landing gear up and down to help reduce the plane's tipsiness. The wheels partly down acted like a keel, but they also created an additional drag that might cause the plane to stall. A sudden drop in altitude sent loose objects slamming against the cabin roof. If the plane crashed in the jungle, tribes of headhunters\u2014the tattooed Naga people\u2014were down there to meet them.\n\nIn a novel left uncompleted at his death, _The Sleep of Reason_ , Williams included a scene of an Army Air Corps desk officer, Capt. Parker, hitching a ride on a C-47 with an enlisted man named Matthews:\n\nThey sat apart from each other, wedged tightly between lashed crates that nearly filled the interior of the plane's long body, and when the C-47 began its long, shuddering takeoff down the runway, [Matthews] was glad that Parker could not see his hands clutching the slats of the crates on either side of him.\n\nBut when they were airborne and Matthews became more used to the odd, unsteady floating sensation, which he found not altogether unpleasant, he looked across the aisle and was astonished by Capt. Parker's face. It was a pale, yellow contortion with staring eyes and open mouth. Beads of sweat glistened on his forehead, and he breathed in heavy gasps that could be heard above the roar of the airplane.\n\nFor a moment, Matthews thought that he was ill and he started to speak to him, but he realized in the instant before he spoke that what he saw was a terror as pure as he had ever imagined.\n\nAfter the Mishmis Hills, Williams' plane traversed the Hukawng Valley and the muddy meadows of the Upper Irrawaddy River. Next came the Kaolikung range, which hid the cavernous gorge of the Salween River below, walled on the other side by the four-mile-high Hump itself. When at last the silvery Tali Lake came into view, the crew was ten minutes from a plate of eggs and a cup of coffee on the Yunnanese plateau. A sign over the mess hall sounded slightly incredulous: \"You Made It Again.\"\n\nHe said he was shot down once. A shell from a spring-loaded Japanese knee mortar struck the plane on its final approach. He found himself on the ground with his back up against a tree, a pain in his side from broken ribs. Pieces of the plane were hanging in the jungle canopy. The pilot and copilot survived, but the five men in the rear were killed. Williams and the pilots found their way through the underbrush until they hit a safe stretch of the Burma Road. Once back, he volunteered to join a party to retrieve the dead men's dog tags. On the way, they ran out of rations, so they skinned a monkey and roasted it. He remembered it \"looked like a baby on a spit.\" A Japanese patrol passed by so close that their white puttees could be seen flashing through the jungle foliage.\n\nBut the Japanese didn't wear white puttees in the jungle: they wore khaki leggings, wrapped to their knees. And among the names of men flying out of Sookerating who crashed\u2014over a hundred were killed\u2014\"John Edward Williams\" doesn't appear. Also, if he suffered broken ribs, he would have received a Purple Heart. But his name is not among the list of recipients. What, then, to make of Williams' account of being shot down by the enemy?\n\nHis bravery is unquestionable, of course. The flight over the highest mountain range in the world was the graveyard of the air. He \"almost got used to people being killed.\" He came down with malaria, the effects of which plagued him for the rest of his life, and a touch of scrub typhus near his eye, a marshland disease that produces skin lesions and high fever. He endured a lot, and his superiors thought well enough of him to promote him to sergeant.\n\nBut to come home without a few good stories would have been unthinkable to Williams. Crafting a tale\u2014embellishing it, adding surprises\u2014becomes an opportunity to get a little better at the art. And the best stories, the memorable ones, hardly ever fall into the teller's lap fully formed. \"Remember Charlie,\" Ernest Hemingway wrote to his friend Charles Poore in 1953, \"in the first war all I did mostly was hear guys talk; especially in hospital and convalescing. Their experiences get to be more vivid than your own. You invent from your own and from all of theirs.\" Call it lying, but writers call it creating fiction.\n\nThe round scar on his shoulder, for instance, was left from shooting himself by accident at age seventeen. When one of his daughters asked him years later, pointing to the mark, \"What's that?\" he said, \"I got that when I was a co-pilot in the jungle and the Japs shot us down.\"\n\n\"Dear John . . .\"\n\nSometime in the early spring of 1944, Alyeene wrote him to say she wanted a divorce. There was somebody else\u2014that's what he claimed she said, although she wouldn't remarry for ten years. Or maybe she chose a reason that was irrevocable but short, one that would fit on a couple pages of overseas V-mail. In any case, they would have to wait until he returned to make the divorce official, because that was the law.\n\nWhat could he do about it, and how should he respond from the faraway edge of eastern India? For the rest of his life, he took a sporting attitude about his first marriage. \"Oh, I don't even count that one,\" or, \"That one was my fault.\" He said he grieved about it for about two weeks.\n\nBut the failure with Alyeene was one in a series of attempts at respectability. He had dropped out of junior college and tried his hand at radio plays and stage direction. Next, he had started a career in broadcasting, which turned out to be a disappointing job of covering county fairs and selling advertising. To compensate, he bestowed on himself a nonexistent degree from the University of New Mexico.\n\nAbout the time of her letter, Williams had a studio portrait taken in Calcutta. Like a movie star, he is looking up and off to his left, the light falling on his face, accentuating his large eyes and close-cropped mustache. Most men and women in uniform preferred a straight-ahead shot like a yearbook photo, but Williams went for effect. He's dressed in a sharply pressed airman's tunic, his right shoulder thrusting forward with the China-Burma-India campaign patch, a striped shield on whose blue field rests the Kuomintang Sun of China and the Star of India. There's a bid for transcendence about the way he looks up into the light. It isn't accidental. It's part performance, part personal narrative\u2014he means to say that he's different, and he's artistic. A romantic. He's still Ronald Colman in spirit, the gentleman swashbuckler. But the facts of Williams' young life so far did not shine favorably on him becoming successful.\n\nWhile he was in Calcutta, Williams purchased half a dozen composition booklets, like the kind he'd used for school: red covers, blue-lined pages with a vertical red line for the margins. Writers tend to be particular about their materials, and by reaching back for the tools he first used for essays and poems, he was indicating a wish to reconnect.\n\nHe spent some his off-hours in the tent he shared with two other airmen, laying on his cot writing. If it was getting dark, he had to reach up and light the kerosene lantern hanging from the ridgepole. The gray mongoose they let hang around to kill snakes appeared at sunset begging for scraps. In his attempts at fiction, at first he emulated his favorite authors. He wanted to invoke Conrad Aiken's psychological language that had impressed him so much, and the interiority of Proust. He hoped to sound both literary and sophisticated to the reader, and show his characters' thoughts. From a page in one of his composition notebooks, dated March 4, 1944:\n\n\"Take any man,\" I said, reaching across the table and lighting your cigarette, \"study him carefully, know him so well that you premeditate his basic reactions. Understand his likes and dislikes, his prejudices and convictions; know a little about his background, his environment. Take him, season with a little imagination and sympathy\u2014and you'll have a novel.\"\n\nYou leaned back in your chair and sipped slowly at your drink. You didn't say anything. You put your drink back on the table and drug [ _sic_ ] heavily on your cigarette and let the smoke dribble out of your lazy, skeptical mouth.\n\n\"I'll go even further,\" I insisted. \"Take any man\u2014fat or lean, wise or foolish, good or bad\u2014take one day, or one night of that man's life. Know what he does during that appointed time, understand why he does it\u2014and you'll have a novel.\"\n\nHe kept revising, adding, starting again. \"I didn't know what I was doing,\" he later said.\n\nThe best passages he copied out and mailed home to his family. What they thought about such missives from the edge of the jungle is anyone's guess.\n\nHe would be returning to the United States in six months or so\u2014at the end of the year, or in early 1945, at least. His marriage was over and he would be single again. The GI Bill would give him the financial wherewithal to attend college, but he didn't know where someone with ambitions should enroll. Through a friend, he wrote to a college instructor, Satyavati C. Jordan at Thoburn College in Lucknow, India, hoping for some advice. Mrs. Jordan was a woman in her forties who had been educated at Baker University in Kansas and Northwestern University near Chicago. Her father was the first Indian bishop elected to the Methodist Episcopal Church.\n\nShe replied that she would be \"very glad indeed\" to meet Sergeant Williams, and extended an invitation to the Sunday evening service at the Victorian-style Lal Bagh English Methodist Church, where she was the organist. \"I think you would like our service. . . . I can take you around the Isabella Thoburn College and we can have a visit.\"\n\nWilliams was nearing the end of his military service and on the cusp of deciding about the direction of his life. Perhaps during his walk with Mrs. Jordan through the gardens on campus, he mentioned that his mother was ill with tuberculosis. His family had moved while he was away from Wichita Falls to Pasadena, California, to be near his sister, who was performing at the Pasadena Playhouse. When he returned, everything would be different.\n\nMrs. Jordan responded with another invitation to dinner to meet her family, and followed up by letter with the names of two friends\u2014a journalist in Tennessee, and a poet in Kansas\u2014who might help him think through his choices. \"Remember, when you come this way again,\" she told him, \"you have friends here\u2014we'll be looking for you.\"\n\nBack in his tent at the airfield, he had kept a letter from his actor friend Jack Newsom. He liked to reread it because Newsom was back home and preparing big plans for the future. Jack was in advertising, selling radio airtime in Florida, and he'd gotten a tip that the worldwide Mutual Broadcasting System was going to tie in with a brand new station, WKWF, in tropical Key West:\n\nWilliams, at present I have things lined up so that the minute you step foot back into this country we can at once pack up and head for the most wonderful [radio] station in Florida you ever laid eyes on. . . . I really got going and met the right people and as a result have things lined up like we used to dream about. Now here's the deal. You have been fucking around over there long enough. . . . [Y]ou are a big boy now and must settle down. If you will just get a move on, we shall take up where we left off.\n**CHAPTER FOUR**\n\nKey West\n\n_I knew I could make a living as a writer for radio, but it meant you just hacked it out. I grew up a little bit, and realized this was a crappy way to live your life\u2014to be a radio announcer and hawk products or whatever_.\n\n\u2014JOHN WILLIAMS, 1981 INTERVIEW\n\nArriving stateside again in the spring of 1945, he was in a hurry. Life was precious, and yet paradoxically, not very important in the scheme of things. Sometimes, when the big C-47 cargo plane put down in China, its rubber wheels smoking on impact, he would feel an extra bump as they taxied, followed by a disgusted shout from the cockpit\u2014\"Aw _shit_! We hit another one!\" A Chinese civilian had rushed in front of the plane because he believed that the blurred, chopping blades would destroy his evil spirits. Instead, he disappeared into the slipstream of nothingness and left gore on the runway.\n\nThe former \"Jon\" Williams, the smooth-faced boy of twenty, had disappeared over the Himalayas with the sound of Hanoi Hannah crooning in his headset. In his place a veteran had returned home, sporting a mustache, sometimes plagued by bad dreams or periods of cold sweats from the effects of malaria. And who relied on drinking\u2014alone, if no one was handy\u2014to take his mind off things.\n\nHis resources for civilian life were slim\u2014one year of junior college; a year of professional broadcasting; sixty days of Army Air Corps radio school; sergeant's stripes, and a dozen composition notebooks filled with drafts of a novel. He wasn't in a position to do whatever he wanted; he would have to take what looked promising.\n\nAs soon as his ship dropped anchor in New York Harbor, he began wrapping up what still remained of his prewar life. From New York, he headed southwest to Wichita Falls to finalize his divorce from Alyeene. In the three years since he'd seen her, she had changed from a teenager into a young woman supporting herself as a bank teller. Then he continued on to Pasadena, California, northeast of Los Angeles near Hollywood, to visit his family in their new home on North Summit Avenue. He also wanted to collect the sum of his poker winnings, and the bet he'd won at the Royal Calcutta Turf Club. His parents were supposed to put the money aside for him, but it turned out they hadn't. It had gone instead into the down payment on their new house. He didn't begrudge what they had done\u2014the climate was better for his mother, and George Rae was performing at the Pasadena Playhouse, which benefited her acting career. But he left again quickly, explaining that he didn't care for California, and there might be a job in radio waiting for him in Florida.\n\nBy June, he was seated at the control board of \"the most wonderful station in Florida you ever laid eyes on,\" as Jack Newsom had described it: WKWF, \"the nation's southernmost radio station,\" in Key West, Florida. Actually, it was still a small operation in a bungalow, and owned by a local businessman with political aspirations, but John Williams was more knowledgeable than anyone in the studio.\n\nKey West, an island four miles long and a mile wide, suited him well\u2014it hit the median somehow. The sleepy, sea-level island, hanging from a string of sandbars at the tip of Florida, provided a good transition to civilian life. Key West is almost precisely the same latitude as Calcutta, and he was accustomed to the languor and sultriness. There was an air of exoticism about the town, too. Sailors, pilots, and technicians from the Naval Air Station and the Fleet Sonar School consorted with the population of bohemians, sidewalk strollers, and sun-worshippers. Few other places in the United States had Key West's admixture of ordinary and strange, but it was palpable here at the endpoint of an archipelago jutting into the Caribbean where tropical sunlight shone down on the old romance of an eastern bazaar.\n\nAnd some of Key West's residents were writers, including the poets Elizabeth Bishop, Wallace Stevens, and Robert Frost, the playwright Tennessee Williams, and the novelist Ernest Hemingway. John Williams collected local anecdotes about \"Papa\" Hemingway, which he repeated over the years. If he had to make a living as a broadcaster in order to write, then Key West was a good loose fit, until he got his bearings.\n\nSometime that fall, he finished the novel he'd begun fitfully in Sookerating, titling it _Nothing But the Night_. It had gone through six rewrites on the way to completion, and he hated revising. To avoid it as much as possible, he rehearsed passages in his mind before committing words to paper, a method he used for the rest of his life. Unlike his author-hero Thomas Wolfe, who could write thousands of words at a single sitting, a good day for Williams was a paragraph he liked.\n\n_Nothing But the Night_ is an ambitious novel, steeped in psychological realism\u2014illuminating characters' interior lives by reaching into their subconscious. Having read and admired Proust and Conrad Aiken, Williams hoped to reach even further by adding daubs of psychoanalytic theory: pretty formidable stuff for a new writer. _Nothing But the Night_ , which plays out in one Joycean day and night, explores a disturbing, perhaps abnormal relationship between a young man and his mother, opening the doors to related pathologies such as masochism, sadism, and a desire to subjugate women.\n\nIt begins with a dream unfolding. The unnamed dreamer is observing a party going on. Among the guests is a pale, flaccid young man. He makes the others uncomfortable, and they try to ignore him: \"To these people he was a noise without meaning, an explosion with disruption.\" Gradually, to his shock, the dreamer realizes that he's the detestable young man everyone dislikes. They surround him, menacingly. Then, as a kind of biblical stoning, they beat him into senselessness.\n\nThe dream over, Arthur Maxley, a young man of leisure, begins waking up. He's expecting the maid to arrive soon to clean his disordered room, but before she arrives he decides he should go out for a walk. He distracts himself by repeating the beginning of the Lord's Prayer, like a chant, until he pares it down to \"Father, Father, Father . . .\" \"What an ugly word,\" he says to himself.\n\nStopping into a diner for breakfast, Maxley comes across as neurotically aesthetic. Smells and images affront him. \"He wiped a fleck of breadcrumb from the soiled cover of a chair and sat down at a table near the front. . . . The menu was one of those typewritten affairs, and this must have been a fourth or fifth carbon, well used and smudged by previous customers. He sniffed delicately and dropped it on the table.\" He primly but silently scolds his waitress, telling her to please \"try to hide the remnants of last night's debauch.\" On all sides, the world looks like an ugly place to him, full of gross behavior.\n\nReturning to his apartment, he plays a tantalizing game of keep-away with the maid. \"He raised his free hand and let the back brush negligently across her breasts.\" Behind her back, she's hiding a letter for him. It's from Arthur's wealthy father, Hollis Maxley, who's in town and wants to meet him for dinner. In a flashback, Arthur remembers a scene of seeing his mother and father in a room, and \"that terrible thing which he could never purge from the darkest reaches of his mind,\" and how he had screamed, \" _Mother Mother Mother_ until he was in a hoarse stupor.\" Their bond seems faintly Oedipal:\n\nSometimes she would fling her arms about him, lie beside him, tousle his hair, and whisper to him. At other times she would seem distracted, absent, not really there beside him at all. Then she would hold him to her briefly, talk to him in short stops and starts. But the rarest moments of all\u2014and to him, the startlingly beautiful\u2014were the times when she floated into his room like a white angel, sat beside him, held him softly, saying little, gazing at his eager moon-bathed face with great tenderness and calm . . .\n\nBut there was always the goodnight kiss.\n\nThey would linger over that. And when her lips had left his face, he would remain so, his eyes closed, an unconscious smile hovering over his mouth.\n\nAfter receiving his father's invitation for dinner, about which he isn't happy, he meets a gay friend for drinks. This is an edgy scene for the 1940s. (The bartender throws them both out. \"What kind of place do you think this is?\") Williams seems intent on salting the manuscript with passing references to characters' psychological states. Arthur continues on his way after drinks to meet his father, Hollis, at a posh hotel.\n\nThe elder Maxley, it turns out, wants to make amends, to heal a rift with his son. Something has happened which neither of them wants to revisit, so they only allude to it. Just as Arthur begins to feel sorry for his father, a beautiful woman stops by the table. Awkwardly, it becomes apparent that she and Arthur's father are having an affair. She apologizes for interrupting and leaves. Arthur feels nauseated because the woman reminds him of his mother. He refuses to accept his father's explanation that \"they all look like _her_. I try to make myself believe that . . .\" Angered by what he perceives as his father's betrayal\u2014or, his realization that his father has a sexual connection to his mother that he can never overcome\u2014Arthur leaves and goes out into the night.\n\nHe wanders into a nightclub, where a young woman, abandoned by her date, plops down drunkenly beside him. Arthur's unhappiness is triggered again unexpectedly when on the nightclub's stage, a female dancer performs a highly erotic number. \"The smooth muscles of her belly turned and twisted; her body throbbed uncontrollably in effortless spasms.\" Her face turns into the face of Arthur's mother and he rises to his feet, crying out in pain. He recalls in a dreamlike sequence what he witnessed as a child: his mother shooting his cringing father, a repressed wish to kill his father acted out. But then \"she inserted the still-smoking barrel into her mouth. He heard the muffled report and saw the head jerk backward.\"\n\nArthur whisks the girl away from the nightclub in a taxi. Upstairs in her room, some kind of urge for revenge, or maybe just disgust, compels Arthur to lash out and strike her. She falls to the floor, naked, blood coming from the corner of her mouth. The bedroom door bursts open and in comes a large, rough man who pushes Arthur out of the room and downstairs into the darkness.\n\nFor the first time we find out that Arthur is \"little,\" like Williams. \"You know what's going to happen, don't you?\" says the man, who calls him \"sonny.\" Mutely, without protest\u2014like a guilty son at the hands of his violent father\u2014Arthur submits to a beating. His passivity suggests he's aware of having broken a taboo, because \"his hands were leaden weights that hung at his side.\" When it's over, he crawls along the sidewalk and finds his glasses, then gets to his feet unsteadily. Without a word, he walks away, \"toward where the darkness converged and there was no light, where the night pressed in upon him, where nothing waited for him, where he was, at last, alone.\" The novel ends with a sense that Arthur got what he came for\u2014or what he deserved\u2014and that the violence associated with his punishment was ordained.\n\nNow that _Nothing But the Night_ was finished, Williams needed a reader whose opinion he trusted\u2014someone who knew about fiction and literature and not just an obliging friend. By chance, just such a person appeared at the radio station: a vacationing professor from the University of Alabama who chatted on-air once a week about books, writers, and summer reading. Williams, listening behind the glass in the control room, heard a language he was hungry for.\n\nGeorge K. Smart\u2014\"Ken\" to his friends, held a doctorate in American literature from the University of Alabama. An outgoing gentleman, Smart resembled a balding southern salesman with a preference for bow ties and light-colored suits. Beneath his friendliness, however, was a serious scholar whose specialty was William Faulkner. Ten years older than Williams, Smart had a tendency to attract or seek out young people in need of advice. He and Williams arranged to meet at the La Conga Bar on Duval Street in the oldest part of Key West.\n\nWhat Smart gleaned from listening to Williams was that here was a bright, thoughtful, ex-GI who was sincere about his desire to write, but who did not have much of an idea about how to proceed. Smart had been an assistant professor at Harvard, but his instincts told him that his younger friend probably wouldn't enjoy it\u2014it was too much of a leap. He recommended the University of Alabama, and for several reasons. First, he was in the English Department there. Also, he and his wife, Virginia, had an extra room that Williams was welcome to use. Smart could put a word in with the admissions office, too. But most important for Williams' benefit, Alabama offered one of the few creative writing seminars in the country. It was taught by Hudson Strode, who had been successful bringing several of his students' novels to market. Williams could get the kind of guidance he needed. \"Alabama is a friendly, easy-come-easy-go place,\" Smart assured him. \"It's big, rambly [ _sic_ ], and extroverted.\"\n\nBut for the present, at least, Williams wasn't sure he wanted to commit to more schooling. A little magazine of contemporary art and literature in New York, _The Tiger's Eye_ , had accepted one of his poems. He had also completed a one-hour radio play; and there was the manuscript Smart had just read, _Nothing But the Night_. If there was any way he could write for a living, maybe he owed it to himself to try. Smart agreed that the novel was ready to send out, and gave Williams the name of an acquaintance, a senior editor at Harper and Brothers, Edward Aswell\u2014who had also been Thomas Wolfe's editor, by an auspicious coincidence.\n\nThe decision over whether to reenroll in college or continue writing independently was temporarily delayed, however, when Williams received word in December that his mother was seriously ill with tuberculosis. She was forty-seven and had been in and out of sanitariums since he was in high school, the first time for a minor operation in Paris, Texas. The owner of WKWF replied to Williams' resignation with a letter praising his talents, which, it was understood, Williams could use as a recommendation to find another position in radio.\n\nBut he never used it, or looked for another opening in broadcasting. Despite his friend Jack Newsom's cheery letter about making good on their adolescent plans, he realized now that he didn't want to pick up where he'd left off in 1942, before he'd enlisted. \"I knew I could make a living as a writer for radio, but it meant you just hacked it out,\" he told an interviewer. \"I grew up a little bit, and realized this was a crappy way to live your life\u2014to be a radio announcer and hawk products or whatever.\"\n\nHe headed to California to see about his mother.\n**CHAPTER FIVE**\n\nAlan Swallow\n\n_I think Denver is a good idea. This guy Swallow sounds like a very good Joe_.\n\n\u2014GEORGE K. SMART TO JOHN WILLIAMS, 1947\n\nIn the spring of 1946, residents of Pasadena became accustomed to seeing a slight, dark-haired young man in a biscuit-colored uniform and black bow tie wearing a peaked cap with a badge. He would be walking quickly between houses to peer at the meters and take a reading for the Southern California Gas Company. Williams kept up a good pace, covering about four hundred properties a day. He followed a different route each week to complete his territory, averaging about four miles of walking daily before clambering into the truck to head back to the central office.\n\nHe was living with his stepfather; his sister, George Rae; and his grandfather, Elbert Walker, who was almost ninety, at 363 North Summit. George Williams had retired from the Wichita Falls Post Office on a civil service pension and was taking care of Amelia, whose tuberculosis kept her resting in bed most days. The first antibiotic for the disease, streptomycin, was introduced that year. But it was too late for Amelia; in July, she died.\n\nWith her death, John was the last member of the evanescent Jewell family of Clarksville, Texas. His nature was to be reserved about personal matters, and later he preferred not to reminisce about his mother. When asked about her, his embarrassed silence gave the impression that \"he loved his mother too much to talk about her.\" Ten years after her death, he wrote \"Memories: Texas, 1932,\" which begins with the poet recalling how he had needed his mother when he was nine years old, the age, probably not coincidentally, at which she told him the secret of his birth:\n\n_They are gone now, but I remember_\n\n_The narrow room and the nine-year-old boy_\n\n_Who lay at night on a bed so wide_\n\n_That hands and feet could find no falling_\n\n_From softness into the dark around him_.\n\n_Deep in the drowse of childhood, he heard_\n\n_His mother's voice in the next room_ ,\n\n_Unmeaning, soft as his first comfort_ ;\n\n_And thought of what he could not know\u2014_\n\n_And his room_\n\n_shifted in its darkness, a presence_\n\n_Gathered where he could not see_ ,\n\n_His arms flailed at the blackness; and he called_\n\n_From his terror, \"Mother!\"\u2014and called_\n\n_Again._\n\nThe summer his mother died, a rejection letter arrived from Edward Aswell, the editor at Harper and Brothers whom Smart had recommended. Aswell believed _Nothing But the Night_ was irretrievable. \"Your central character is presented in such a way that the reader is unsympathetic toward him. Thus it is difficult to care very deeply about what happens to him.\" Smart tried to humor his young friend out of his disappointment. \"I know you have good sense enough not to pay any attention to his twaddle about 'sympathetic.' You might write him back that he, Aswell, is an unsympathetic character.\" The poems Williams circulated didn't meet with any better luck. Editors pointed to insufficiencies, such as a lack of \"development,\" or language that was too \"anonymous.\" In August, yet another rejection of a different sort occurred. Taking Smart's advice, Williams had applied to the University of Alabama. A letter from the admissions office informed him that he had not been accepted for the fall semester. Thus, there was nothing to look forward to, no good news. For the time being, then, his suddenly diminished world would consist of being a gas meter reader, divorced, sleeping on the couch in his stepfather's house and writing at the kitchen table.\n\nMonths passed until a bright spot appeared. Williams had heard that Swallow Press in Denver, Colorado, was on the lookout for manuscripts that had been overlooked by eastern publishers. He thought he'd give it a try and sent _Nothing But the Night_ to publisher Alan Swallow. Judging from the letterhead on his reply in November, apparently Swallow was a member of the English Department at the University of Denver. And apparently, he didn't have a secretary. Every capital letter jumped higher than the rest of the sentence. But there was sincerity in the warm tone. \"I was glad to see your novel come in yesterday. I look forward eagerly to reading it, and you may be sure I will do all I can for it. Manuscripts have piled up on me more than I anticipated, so that I am running about three weeks for consideration. However, I shall try to handle yours in two weeks or less.\"\n\nThe promise to \"do all I can for it\" was unexpected\u2014there was no suggestion that Swallow Press might consider other work, if he cared to submit it. Williams put the letter aside and waited.\n\nIn the meantime, George Rae was trying to get her brother to stay in California. The University of Alabama didn't have to be his only choice. There was no lack of colleges in the area, for one thing. Also, he'd done some acting and directing, and the big film studios in Culver City, Burbank, and Hollywood were minutes away. She was proud of John and supplied him with reasons to stay, including fixing him up with one of her friends from the Pasadena Playhouse, a nineteen-year-old brunette in junior college with freckles and a beaming smile, Yvonne Elyce Stone.\n\nYvonne was on the lookout for adventure\u2014some kind of change. She had been born in Los Angeles and never lived very far from it, spending her childhood in Beverly Hills, then her teenage years in Salinas, after which she returned to Los Angeles. Yvonne's mother, who was divorced from her father, was a bit of a crackpot. After being introduced to Yvonne's new boyfriend, Mr. Williams, and listening to his conversation, she decided he was a communist.\n\nIt was Williams' way to always have a woman in his life\u2014sometimes more than one, never going for long without a female in the background, supporting his efforts.\n\n\"Don't work too hard, baby, and be sure to write me,\" Yvonne wrote on a postcard after they had been dating for several months. It was postmarked Pacific Groves, where she was enjoying the beaches near San Francisco\u2014four hundred miles up the California coast from Los Angeles. She dropped him another postcard the next day from Carmel, telling him coquettishly, \"I finally got my suitcase and am wearing clothes again.\"\n\nSwallow followed up his initial response to _Nothing But the Night_ with two typewritten, single-spaced pages of criticism for Williams, going beyond what an editor-publisher would normally do. \"It starts out with a scene that I call surrealist, continues with a long stretch of minute emotional writing\u2014all torturous\u2014and then, approximately the last third, comes a solid psychological writing that one grasps, understands, and finds exciting.\" What concerned him was that \"the novel lacks a certain consistency or observable direction of development. . . . [I]t seems to prey a good bit on rather deliberate obscurantism in the early parts, and then, toward the end, becomes rather simple in comparison, a good ending, but not tied together thematically and stylistically as well.\"\n\nHe suggested that Williams \"may well be a writer who needs to throw away about two or three novels before the thing starts clicking.\" In other words, perhaps he hadn't served his apprenticeship yet, and Swallow added that although his press \"sponsor[ed] early work, unusual and experimental work,\" he had to be careful with his money. His advice\u2014\"if you care to hear it\"\u2014was \"that you try the novel many other places\" and benefit from the experience of having a number of editors read it.\n\nSwallow's letter was at once generous, instructive, and respectful of a new writer's hopes, but ultimately he confessed that he couldn't do it justice at present and wouldn't stand in Williams' way. It was a remarkable response, but then, Alan Swallow was an unusual person.\n\nA glimpse of him as a boy in Wyoming captures his personality and, as his friend Yvor Winters once said, Swallow's \"curious inner violence that contributed to his success and to his death.\" In 1927, he was twelve years old and speeding down a road on an Indian motorcycle that he had repaired himself. As he leaned into a curve, he thrust out his arms, threw back his head to the sun, and let the wind take its chances with him.\n\nHe had been born in Powell, Wyoming, in 1915, within sight of the eastern slope of the Rocky Mountains. His parents raised beets and sheep three miles outside of town on formerly Shoshone wetlands that had been drained by the federal government. The farm belonged to his grandmother; she had been working the land since her husband, a railroad detective, had died from an accidental gunshot wound to the stomach. Swallow's sister Vera recalled \"the gasoline lamps, the kerosene lights . . . the house so cold in the mornings, the wood burning stoves, the wonderful scent of fresh bread and cinnamon rolls, the baby chicks behind the stove in the spring.\" Alan was a stout boy with reddish-blond hair and a habit of standing with his hands in his pockets. He was quiet, a loner, and liked working on junked cars and tractors, thinking that he might want to go to college to become a mechanical engineer. And then at sixteen, he had an experience that\u2014in an overused, but necessary phrase\u2014changed his life.\n\nHe was working at a gas station in Gardiner, Montana\u2014the north entrance to Yellowstone National Park\u2014and passing the time by reading titles in the Little Blue Books series. The publisher was the Jewish American socialist, atheist, and social reformer Emanuel Haldeman-Julius, whose low-priced paperbacks were intended for working men and women. Printed with slate blue covers and priced at ten cents apiece, the Little Blue Books offered thought-provoking, self-improving subject matter intended to elevate readers, and perhaps lift them out of poverty, such as Thomas Paine's _The Age of Reason_ , Boccaccio's _Tales from the Decameron_ , _Hints on Writing Poetry_ by Clement Wood, Margaret Sanger's _What Every Girl Should Know_ , _A Guide to Emerson_ by H. M Tichenor, _Do We Need Religion?_ by Joseph McCabe, and _How to Be an Orator_ by the socialist governor of Illinois, John Peter Altgeld. Ultimately, there were over a thousand titles in the series.\n\nSwallow purchased a boxful of used copies and read them all. As interesting to him as the subject matter was the concept of selling high-quality literature for the price of pocket change. If he had a press, he could publish uncopyrighted classics for his list, then advertise for submissions, review them, and grow larger. His stock could be printed on a hand press in small runs to save money. The post office would take care of the rest.\n\nIn April 1932 he received a scholarship to the University of Wyoming, but by then engineering as a major had fallen by the wayside in favor of English literature. By his sophomore year, he was mimeographing and distributing his own literary magazine, _Sage_. He married his sister's friend Mae Elder, a Powell girl, in 1936, the year he graduated from Wyoming. For graduate school, and ultimately a doctorate, Alan enrolled at Louisiana State University, where Robert Penn Warren, Allen Tate, and Cleanth Brooks were leaders in the New Criticism movement\u2014which stressed close reading to see how a work, particularly poetry, functioned as a self-contained piece of art. Mae worked in the office of the New Critics' literary quarterly, the _Southern Review_ , proofreading manuscripts, keeping the accounts, collating the printed pages, and preparing magazines for mailing\u2014experiences which brought nearer the reality of Alan becoming a publisher one day.\n\nIn 1939, he borrowed $100 from his father and purchased a used electric Kelsey Excelsior Press, which ran like a miniature locomotive. A coupling rod drove the main wheel, making all the parts move and clank, including the platen that pressed the paper against the bed of type. In fact, Alan was as enamored of the craft-mechanical side of publishing as he was of books. \"I don't object to setting the type,\" he wrote to a friend. \"In fact I rather like it,\" he later said. He enjoyed printing, collating, and stapling in a \"workmanlike fashion some of the things I would be wanting to publish.\" In 1940, in a garage in Baton Rouge, he published _Signets: An Anthology of Beginnings_ , featuring poems, stories, and artwork by student contributors. It was the first to bear the imprint \"Alan Swallow, Publisher.\"\n\nAfter accepting a position teaching at the University of New Mexico, he expanded his publishing fivefold, still relying heavily on Mae, though the marriage lopsidedly favored what was important to Alan. \"Although she has no particular interest in literature,\" he admitted, \"she helped me with the publishing work, and she has suffered by being left to her own devices, while I worked night after night at the press or the typewriter.\" From his noisy Kelsey Excelsior came chapbooks and quarterlies of poetry under various short-lived imprints\u2014Big Mountain Press, Sage Books, Modern Verse, Swallow Pamphlets\u2014typical of his tendency to leap from one project to the next. As someone said of him, just having \"an idea was always a call to action.\"\n\nHe ran Swallow Press according to his principles: first, he used his own judgment to choose what to publish, instead of being driven by what was popular. He was proud of being a maverick. Second, he would keep a lookout for writers from the West. The eastern literary establishment tended to have a double standard: writers who lived in the West were treated as regionalists. Nebraska-native Mari Sandoz's novel _Old Jules_ , set in the Sand Hills of the Plains, had been rejected by every major New York publisher before it won a fiction contest and became a best seller in 1935. It seemed that an author was taken seriously only if he had the fairy dust of eastern cosmopolitanism on him\u2014such as Owen Wister, author of _TheVirginian_, a groundbreaking western. Swallow was out to prove that literature originating in the West deserved to be honored.\n\nIn addition to being a small-press publisher, Swallow was indeed an English professor at the University of Denver, as Williams had surmised when he received Swallow's reply about _Nothing But the Night_ , typed on English Department stationery. The overlap was a result of the university administration urging him to create a university press. Excited, but unwilling to abandon his own imprints, he had decided he could do both. By 1946, he was simultaneously running the University of Denver Press and negotiating with William Morrow and Company in New York to distribute Swallow Press books. At his brick, split-level suburban home at 2679 South York Street, which he shared with Mae and their infant daughter, Karen, he happily anticipated how he would need a bigger garage to accommodate all the business and inventory that would be coming his way.\n\nIn the meantime, Williams had revised _Nothing But the Night_ several times and shopped it around to other publishers, as Swallow had recommended, but the responses all harped on one thing: it just wasn't strong enough. \"We think you have writing talent,\" said an editor at Harcourt, Brace and Company, \"and we think it would be a mistake to start off with this book as your first long work. It is an excellent exercise.\" Ken Smart, at Alabama, urged him not to give up, writing, \"There are still a lot of people rooting hard for you to break down the barriers, and for Christ's sake don't get too discouraged.\" And in fact, a \"second-look\" letter had arrived from Swallow in April 1947, justifying Smart's faith in Williams.\n\n\"My lateness may have destroyed any opportunity I had to see what you have done with the re-worked novel,\" Swallow wrote rather breathlessly. \"But I am eager and interested.\" He said he was creating a category of titles in his catalog called The Short Fiction Club. \"Selections for the club will come in part from other publishers, but primarily I will publish for it under my private imprint. I don't care a damn about making money on it.\" He thought Williams' novel would fit nicely among his titles. And he had something even better to offer: if Williams would consider enrolling at the University of Denver to complete his undergraduate degree, he promised to help him in any way he could. \"I'd be damned glad if you could come to Denver next fall. . . . The writing program would interest you, particularly, I think; and especially if you plan to stay for the MA, too, for which we have a good program.\" The English Department was actually quite small, with fewer than a dozen full- and part-time instructors. But Swallow had a hunch that a personal invitation to an undecided young man with an unpublished novel might just do the trick.\n\nWilliams replied that he and Yvonne were marrying in August and things were up in the air until then, but he would have his credits transferred from Hardin Junior College, just in case. A snag occurred when the University of Denver admissions office refused to accept his unofficial transcript, but Swallow rushed in. \"The transcript matter has probably been cleared up by now. And I imagine you have received admission,\" he wrote. \"That leaves the most troublesome problem, of course\u2014housing.\" But \"we are certain to be able to get [you] a pretty pleasant sleeping and study room, very likely with [the] privilege of getting your own breakfast, so that the two of you would have to eat out two meals a day at our cafeteria or elsewhere. So if you are willing, both you and your wife, to put up with the inconvenience . . .\"\n\nYvonne, twenty, and John, twenty-four, were married on August 28 in Pasadena by a Methodist minister. Less than a week later, they left on a combination honeymoon and road trip, heading to Denver and a new life, like millions of other postwar couples just starting out.\n\n\"I think Denver is a good idea,\" Smart wrote when he heard the news. \"This guy Swallow sounds like a very good Joe.\"\n\nDenver sits twelve miles east of the foothills of the Rocky Mountains, where Cherry Creek runs into the South Platte River. Its story has the romance of the Old West. Prospectors arrived on the banks of the South Platte panning for gold when it was still Kansas Territory. The Pike's Peak Gold Rush in 1858\u2014a decade after the California Gold Rush\u2014brought saloon operators, blacksmiths, carpenters, developers, and speculators swarming like bees to the mining camp on the Platte named for James Denver, the Kansas territorial governor. Within a few years, Denver was a town with streets laid out on a grid aligned with the four points of the compass. In 1870 the transcontinental railroad came through, ensuring that the location would become one of the commercial hubs of the Pike's Peak region. It became the state capital in 1876.\n\nThe region had such an abundance of agricultural and mineral resources\u2014the Colorado Silver Boom occurred in 1879\u2014that by the early part of the twentieth century, four thousand boxcars of goods and raw materials left Denver annually. Yet the city remained the \"reluctant capital\" of the Rocky Mountains West. It was \"Colorado's cow town,\" worried about losing its frontier independence, \"contentedly disinterested in its own continuing growth, abhorrent of risk-taking, chary of progress,\" according to historians writing in 1949. After World War II, the old-timers looked on unhappily as thousands of ex-servicemen, like Williams, enrolled at \"GI Tech,\" as the university was called. When a reporter asked Denver's seventy-two-year-old mayor, Benjamin F. Stapleton, how he planned to address the housing shortage, he replied, \"Oh, well, if all these people would only go back where they came from, we wouldn't have a housing shortage.\"\n\nBut they had come to stay. Enrollment at the university shot up to ten thousand, its highest ever. When the poet Alan Stephens, who later became one of Williams' friends, enrolled in 1949, he found his classes dominated by \"some exceptionally able and resolute people. Being amongst them was like being in an electrical storm\u2014the air seemed charged with dangerous intelligence. Exciting days.\" The dormitories overflowed and students resorted to living in Quonset huts, old hotels, basements, and converted garages. Williams applied for a $90 per month subsistence grant available to veterans, enough to rent the one-room apartment that Swallow located for him and Yvonne at 1000 South Clarkson Street.\n\nWilliams took a stroll not long after arriving to take the measure of his new life. Everywhere the university was scrambling to expand. Nineteen new buildings were under construction for the liberal arts and the sciences, physical science, engineering, and physical education. In _Stoner_ , Williams describes young William Stoner, until recently a farmer, exploring the campus of the University of Missouri for the first time:\n\nFor several minutes after the man had driven off, Stoner stood unmoving, staring at the complex of buildings. He had never before seen anything so imposing. The red brick buildings stretched upward from a broad field of green that was broken by stone walks and small patches of garden. Beneath his awe, he had a sudden sense of security and serenity he had never felt before. Though it was late, he walked for many minutes about the edges of the campus, only looking, as if he had no right to enter.\n\nThe same sense of strangeness and exhilaration also fell upon Williams at the University of Denver. None of his people had ever attended a university. He had grown up poor, and had received mediocre grades in junior college because he had been too busy with his own plans to care. He thought of himself as \"the most unlikely person possible to enter an academic setting.\"\n**CHAPTER SIX**\n\nLove\n\n_All of my belongings had been moved into a locked closet and there were some items that belonged to another female all over the apartment_.\n\n\u2014YVONNE WILLIAMS, UNPUBLISHED MEMOIR\n\n\"We are settled in a little four room apartment near the center of town,\" Yvonne wrote to friends at the end of their first academic year in the spring of 1948. \"The first two months we were here we lived in a stinking room. But with the right connections and a small amount of capital we were able to get this apartment and also buy the furniture that is in it. John has a study to himself and we have a front room, bedroom, kitchen and bathroom. It's on the fourth floor and not too bad.\"\n\nTheir new apartment, at 2058 California Street, was convenient for Yvonne, because she worked in the offices of an oil company that was within walking distance. She hated the job. John was \"doing pretty good in his studies,\" she wrote. \"I think it was the right choice of schools. The main reason being that Alan Swallow is here. He and John get along magnificently; you couldn't ask for a better set up. Throughout the past months they have become very good friends. Alan has bestowed upon John many of the duties of a publisher and has just recently asked him to be Associate Editor of the Swallow Press. Which shows that Alan must admire John's work and ideas.\"\n\nThe two men were indeed getting along well. On afternoons when Williams didn't have class, he walked the four blocks from campus to South York Street and let himself into Alan's combination garage and publishing house. Business was good, and fortunately for Alan, the English Department had reduced his teaching load in return for his work managing the fledgling University of Denver Press.\n\nWith the radio playing, the two men\u2014and whomever else Alan had enlisted, usually graduate students\u2014spent the hours arranging metal type in trays, running the hand press, collating pages, distributing the type back into its proper cases, and boxing finished books for shipping.\n\nAmong the students pitching in was twenty-two-year-old Avalon \"Lonnie\" Smith, from Silverton, Colorado, who had recently graduated from Stephens College, an all-women's college in Columbia, Missouri. She knew John in another capacity outside of Swallow Press, too, as the poetry editor of _Foothills_ , the campus literary magazine, to which she contributed verses. She took note of the up-and-coming writer. His essay on the poetry of J. V. Cunningham was slated for the summer 1950 issue of _Arizona Quarterly_. Not only that, but Swallow's smartly dressed associate editor also had a novel in the book catalog they were assembling, _Nothing But the Night_ , dedicated to Ken Smart. They were mailing dozens of review copies; Alan rarely took out advertisements, convinced that in the great scriptorium where good books were cherished, good literature eventually would meet its reward.\n\n\"Holy Jesus Christ, man!\" Smart wrote, when he received his copy. \"Multiple congratulations to you. I hope your wife realizes she has a talented husband whose chief accomplishment is knowing how to dedicate books. I am touched and overcome. . . . You have my permission to dedicate anything you like to me, including three or four thousand-dollar bills.\"\n\nSales of the book were \"not too good and not too bad,\" Yvonne reported. \"It feels swell to be old married folks but it doesn't seem like six months it seems more like six years. But it's wonderful. Wouldn't trade it for anything.\" The downside was that John wasn't home very often, and she worked downtown during the week, nine-to-five. He went to school and stayed out most evenings. When he was home weekends, she couldn't get him to go out and have fun.\n\nBack in Los Angeles, George Rae, who was twenty-two now and working as a keypunch clerk in the US Office of Price Administration, was having fun, too. She was in love.\n\nHer boyfriend, Willard \"Butch\" Marsh, was only a month older than John\u2014about the same size but stockier, with black hair and black-framed glasses. They looked enough alike to be brothers. Also, both men had been Army Air Corps sergeants during the war; both had been married once; and both wrote fiction, although Butch also needed \"plenty of time to blow jazz, man.\" Before he was drafted, he had put himself through three years of school at Chico State College north of Sacramento, California, performing on trumpet and trombone with Will Marsh and the Four Collegians. The GI Bill had paid for a senior year at San Francisco State, but he had dropped out because he wanted to write full-time\u2014an ambition still subsidized by playing gigs in nightclubs.\n\nGeorge Rae and Butch were living together in his three-bedroom, white stucco house in West Hollywood on North Gardner Street. \"We can't get married until August, because his divorce isn't final until then. . . . Daddy doesn't know anything about it\u2014not even where I live,\" she wrote to John. Should John object to this subterfuge, his sister put him on notice: \"I've never been so happy in my whole life as I have been in the two weeks we have lived here, and, of course, Butch is the most wonderful person I have ever known. I never really thought I could care about anyone as much as I do him. He's a really fine person. I know you will like him, you better!\"\n\nJohn sent her paramour a copy of _Nothing But the Night_ at George Rae's insistence, she thinking, probably, that it would give the two men something to talk about. Marsh replied to Williams' gift with a seven-page, single-spaced typewritten critique, establishing his credentials as a writer who was every bit as serious (and talented, presumably) as his girlfriend's brother was.\n\nHe started off by listing the small magazines for poetry and stories he had submitted to, such as the _Southwest Review_ , _Matrix_ , _Sibylline_ , and _Narrative_. Helpfully, he provided summaries of the editorial needs of other magazines, in case Williams might want to try them. Finally, he got into the meat of his letter on page three: \"I was only a little way into [ _Nothing But the Night_ ] when I wondered why it had been turned down by a couple of major publishers. . . . [A]t the same time I simultaneously respected, envied and was amused by your complete violation of the only real taboo in novel writing. . . . One simply does not have a yen to fuck Mother, unless one is Poppa.\" He gave his opinion on Williams' tone, his word choices, his sentences, and whether the ending was satisfactory.\n\nThen he warned Williams about publishing with Swallow Press: \"I don't think it is going to do you any good as a writer, and may hurt you a hell of a lot.\" Marsh had never heard of Alan Swallow and couldn't find any of his titles in bookstores. But then, perhaps that was because Swallow Press mainly only published writers who were eggheads\u2014the kind that the average reader wouldn't appreciate\u2014the phonies and aesthetes. Butch thought he knew the type: \"Lucinda Longhaire has been writing some of the most provocative and stimulating prose of the past decade, and is author of _Duet for Trombone and Tombstone_ and _Years Like Turds_. These challenging novels may be obtained by sending $1.75 to Watercloset Publications, or from Miss Longhaire, 69 Finger Blvd., Greenwich Village, NY. Miss Longhaire's previous appearance in _Thunderjug_ was with an essay on Gertrude Stein's use of the split infinitive.\"\n\nSpeaking for himself, Butch said he would be a dumb \"son-of-a-bitch\" if, like John, he busted his \"ass writing a novel and then, after one or two rejections by major publishers, gave it away to a well-meaning amateur with utterly no means of promoting it.\" He had twenty-eight stories and eight poems finished; and when he had fifty stories, he would blitz the publishing world, warning editors that he was not about to sit around for three months waiting for form letters. If they liked what they saw, they had better say so, and then he would send them his novel (as yet unwritten). He concluded, \"Jealously Yours, Butch.\"\n\nWilliams wasn't upset by Marsh's gratuitous critique of his novel, or his career. He could have pointed out that he'd just had a poem, \"Drouth\"\u2014about the regenerative power of tilling the earth\u2014accepted by _The American Scholar_ , which trumped any of the little off-set magazines Marsh was courting. But he replied in a friendly way, thanking him for taking the time to read his work so carefully, though disagreeing with him on a few points, too. For his sister's sake, he would withhold judgment on her bellicose boyfriend, who seemed so cocksure of himself.\n\nCooped up in the apartment in the evenings by herself, Yvonne was getting fed up with how much time John spent working. If he was at home, he was either studying or writing; or if he was away during the day, she left the door unlocked after she'd already gone to bed. She lay awake nights, thinking about this state of affairs. \"John and I never did anything together and that wasn't enough for this twenty-one-year-old. I had belonged to a little theater group in Pasadena, California as well as attending a radio school. Alone in a strange new place wasn't enough for me.\" The reluctance to socialize wasn't coming from her: She enjoyed meeting her husband's friends\u2014artists and writers mainly. Fortunately, invitations to dinner parties at the Swallows' home came regularly. In April 1948, as the school year was drawing to a close, Alan invited them over to meet a young friend of his from New England who was on his way to the Southwest, Douglas Woolf.\n\nWoolf was guaranteed to be interesting. He was a cousin of Leonard Woolf, Virginia Woolf's husband, and shared with him the same long, thin face, wide forehead, and strong nose. Born in New York City in 1922, he had been raised in Larchmont, a wealthy enclave outside the city, and in Connecticut. He came from a rather eccentric family of some means, achievement, and social standing. His father was a world-renowned expert on Persian rugs; his stepmother was Thomas Edison's granddaughter. During World War II, Woolf had served as an American Field Service ambulance driver in North Africa as well as in the Army Air Corps, experiences he later used in his 1962 novel, _Wall to Wall_. He was also an early Beat poet, rejecting literary tradition and writing Zen-like verse:\n\n_Memory_\n\n_isn't just of the past_\n\n_I went there_\n\n_I saw that_\n\n_Memory is of the future too_\n\n_I'll go there_\n\n_I'll do that_\n\n_Perhaps I'll see you_ ,\n\n_too_\n\nThe poet Robert Creeley was struck by how Woolf seemed to let the world come to him, without expectations. \"Douglas Woolf was an uncannily reflective person,\" Creeley wrote, \"as though he chose to take his color and shape from the surrounding world rather than to force upon it his own determinations and judgment. However, what he did exercise, unremittingly, was an acute perception, _his_ witness, _his_ recognition, _his_ fact in being there, wherever there was or might be.\"\n\nAt the Swallows' dinner with Woolf in attendance, conversation-starved Yvonne enjoyed listening to him. \"Where does your accent come from?\" she asked.\n\n\"Harvard.\" (\"Har- _var_ -dians,\" Williams sarcastically called them.)\n\n\"Oh.\" She thought she should have known that, for some reason.\n\nAs everyone was saying their goodbyes later, Woolf mentioned that he didn't have a car; neither did the Williamses. But he was staying in a hotel near their apartment, so the three of them caught a streetcar. Yvonne slid over on the seat to make room for Douglas to sit beside her, \"and we talked about the various entertainments in Denver. He told me of an upcoming concert at the park and we decided to go the following Sunday. John was going to be busy that day so Doug and I made arrangements to go.\"\n\nThey stayed most of the afternoon at the concert, walking around the park afterward. Yvonne was pleased that Douglas was \"interested in what I said and did.\" She arranged a double date with a friend of hers from work, but John paid more attention to her friend than Doug did. A few days later, Woolf phoned her at the oil company office and suggested they grab a bite during her lunch hour. John's reaction when Yvonne suggested they invite Woolf over for dinner was a shrug of the shoulders. \"When Doug arrived John wasn't home yet,\" she said. \"We waited for quite a while, even went for a walk to the Platt River.\" By the time they came back, the meat loaf was burned and there was still no sign of John. She didn't know what to make of it. She always assumed he was working at Swallow Press, or perhaps he had \"his own social life which didn't include me. I didn't know for sure. I just knew that for a relatively new bride, very na\u00efve, I needed more attention than I was getting from John and I enjoyed Doug's company.\"\n\nThe lunch dates continued, sometimes over martinis; other times, after Yvonne had finished work, they met at a drugstore counter for chocolate mint Cokes, their favorite treat. At some point, the friendship turned serious; when Doug told her he was heading to Tucson, Arizona, in June, she told him she would meet him there on her way back from a trip to California to see her parents. In Douglas Woolf's novel _Wall to Wall_ , Yvonne is the barely fictionalized Vivien; the narrator, Claude, runs away with her to the deserts of the Southwest. Vivien pleads, \"'Twenty-one is too old to go anywhere alone, you know that. I want to go with someone. I don't mean as a bride, I'm not so gauche as that, but as a mistress or paramour or concubine or companion or friend or pal or anything else. I just don't want to be left alone! I want to get out of here!' She said it again for all the wide-faced flowers to hear: 'I want to get out of here!'\"\n\nThat summer, too, the Swallows were away from Denver for three months. Alan had accepted an eight-week lectureship at the University of California, Berkley. But he had perfect confidence in John, and temporarily handed him the reins of the press as if it were his own. From Berkeley, Swallow redirected a submission to him. \"John Williams, my associate editor, has taken a great interest in these poems of yours. I am returning the manuscript to Williams in Denver, and you will hear from him next.\"\n\nAs associate editor, into Williams' hands came submissions of poetry and fiction from both established writers and those who were on their way to making a reputation. In the former category were Louise Brogan, Caroline Gordon, Thomas McGrath, Archibald MacLeish, Henry Miller, Wallace Stevens, Allen Tate, and Mark Van Doren. In the category of new and notable writers were Herbert Gold, Weldon Kees, Janet Lewis, and Ana\u00efs Nin. When it was literally hot off the press, Williams read a collection of critical essays by Yvor Winters published by Swallow Press\u2014 _In Defense of Reason_ , a reply to the Romantic poet Shelley's _In Defense of Poetry_. Winters argued for a moral versus a romantic theory of literature, and for practicing rationality in art over feeling. He attacked Walt Whitman's \"loose and sprawling poetry to 'express' the loose and sprawling American continent,\" adding, \"In fact, all feeling, if one gives oneself (that is, one's form) up to it, is a way of disintegration; poetic form is by definition a means to arrest the disintegration and order the feeling.\" Winters' perspective so impressed Williams that he left behind his romantic, scarf-wearing persona in Wichita Falls forever. He counted himself as a Winterian from then on, and took a special interest, as Winters did, in English Renaissance poetry and rationalism.\n\nThus, Williams, the \"most unlikely person possible to enter an academic setting,\" was finding himself quite at home in university life. He enjoyed his coursework. And by temporarily taking over Swallow's chair as editor and publisher of the press, he was being schooled in the freshest contemporary literature.\n\nBut his fiction, still a bit overwrought with psychology and exquisite language, hadn't changed. _Mademoiselle_ magazine rejected his first-person short story \"The Summer\" with an unusual amount of scorn, saying, \"It offered more poetically than fiction-wise. The theme, for example, is developed so cynically that the piece takes on the nature of a philosophical treatise. Similarly, the characters are bloodless, paper-people. . . . Our whole feeling is that the story could, and should have, generated power within itself, but even its natural pace has been slowed by [a] too stylish treatment.\"\n\n\"The Summer\" is largely an interior monologue by an unnamed narrator who recalls spending a vacation, when he was sixteen years old, with friends of his family at their cottage on a \"foam-tipped cobalt lake.\" It's a coming-of-age story. \"For I was doubly in love that summer, first with Tom and then with Doris; or first with Doris, then with Tom,\" says the narrator. \"It was impossible to determine which was more important.\" He is infatuated with Romantic poetry, reporting: \"I imagined myself sinking into the blue cold lake behind our cabin, a poem on my lips.\" The style is breathless and word-drunk, which smothers the quieter story about the boy's first love. But Williams' early penchant for big words and abstractions reveals something important. In his later fiction, too, there tends to be a lag between a character's thought and emotion\u2014overthinking gets in the way of feeling. Later, in Williams' first major novel, _Butcher's Crossing_ , the young buffalo hunter Will Andrews runs from the room rather than let a young woman seduce him:\n\n\"What is it?\" Francine said sleepily. \"Come back.\" \"No!\" he said hoarsely, and flung himself across the room, stumbling on the edge of the rug. \"My God! . . . No. I'm sorry.\" He looked up. Francine stood dumbly in the center of the room; her arms were held out as if to describe a shape to him; there was a look of bewilderment in her eyes. \"I can't,\" he said to her, as if he were explaining something. \"I can't.\"\n\nIn _Stoner_ , Williams' second major novel, middle-aged Professor Stoner struggles to express himself to Katherine, his graduate student, with whom he's falling in love. Stoner's presentation of himself as a suitor is sadly funny as he uses his classroom approach to discuss their \"problem\":\n\n\"I was perhaps selfish. I felt that nothing could come of this except awkwardness for you and unhappiness for me. You know my\u2014circumstances. It seemed to me impossible that you could\u2014that you could feel for me anything but\u2014\"\n\n\"Shut up,\" she said softly, fiercely. \"Oh, my dear, shut up and come over here.\"\n\nIn the final scene of \"The Summer,\" the story kicked to the curb by _Mademoiselle_ , it's nighttime, and the narrator is sitting in the front seat of the car beside his friend Tom. The vacation is ending, and Tom's sister never did return his love, or sexual desire, because he didn't let her know how he felt:\n\n\"Well, it was a good summer?\" [Tom's] face was placid, calm, unconcerned, the composed eyes staring straight into the darkness. I did not speak. He looked at me. \"Wasn't it a good summer?\"\n\n\"Yes,\" I said bitterly, too agonized to think. \"Oh, yes.\" Then I turned away [. . .] to look back up into the impenetrable past darkness, searching always, vainly, fruitlessly, without hope, another face somewhere lost, somewhere shrouded, somewhere dead and half-forgotten in the dim irretrievable past.\n\nWilliams could build his fiction around thought warring with feeling, which creates tension, and to suggest that emotions are ineffable, beyond characters' reach. But he was also writing poetry, and the intellectual distance, the control, he wielded in fiction did not serve him well as a poet, which will be addressed later.\n\nIn July, George Rae received a phone call from Yvonne, who was back in Pasadena, visiting her parents, and wanted to meet for lunch. Had her courage not failed her, she intended to tell her sister-in-law that she was falling out of love with John, and there was another man in her life. What George Rae heard instead was that Yvonne was more mature now. \"I hope you and Yvonne get your difficulties straightened out,\" she wrote to her brother the day after seeing Yvonne. \"She seems to have developed into quite a nice kid\u2014no longer the bobby-soxer type. . . . And she was missing you like the devil already.\" By then, Yvonne had called John to say she needed time \"to think about our strange marriage.\"\n\nFrom Pasadena, Yvonne went by train to Tucson, and Doug met her at the station. For a few weeks, they lived the kind of life Yvonne had been expecting to have with John. They enrolled in courses for the second half of the summer term at the University of Arizona and did their homework together. Earlier, Doug had renounced his family's wealth in favor of art, and Yvonne was willing to join him in that. Lugging a suitcase full of sample wares in the summer heat, they sold plastic household items door-to-door: tablecloths, shower curtains, plates, knives and forks. She loved the adventures they were having. \"We would work the same neighborhood and meet in a little mom and pop grocery store to cool off with a cold soft drink. When we weren't going to school or walking the streets selling, we would often go for nighttime swims at El Conquistador Hotel. We took buses and at times even hitchhiked to town. Doug and I had lots of fun together.\"\n\nIn August, John phoned. He was having second thoughts about their break-up, and he wanted to give the marriage another try. George Rae and Butch were getting married in Los Angeles, and he wanted Yvonne to attend the ceremony with him. Although he didn't say so, perhaps the reason he wanted her to go with him was that he didn't want to face a lot of questions about why she was not there. Gallantly, Doug paid for Yvonne's bus ticket home to Denver and saw her off.\n\nJohn wasn't in the apartment when she arrived, so she let herself in. But as she entered, she had a sense of being in the wrong place. \"All of my belongings had been moved into a locked closet and there were some items that belonged to another female all over the apartment.\" Putting her suitcase on the bed, she noticed a crumpled piece of paper in the wastebasket. It was a note in John's handwriting: \"Better get your stuff out of the apartment today. My wife is coming back from the insane asylum.\"\n\nShe called Doug and he wired her money for airfare to Tucson. She filed for a quick divorce in Florida\u2014which was available only to state residents, but she met the requirement by signing a lease for a rented room in Miami. She was not someone who was starry-eyed about the institution of marriage and the courts; she suspected \"all they ever wanted was money, anyway.\" The divorce became final in February 1949.\n\nGeorge Rae loyally took her brother's side. \"It's a tough stinking deal about Yvonne, but it is slightly understandable with the bitch-mother she had. I guess early training sooner or later shows.\" Butch added a postscript, figuratively throwing his arm around John's shoulder as a comrade-in-arms, saying, \"I don't suppose it would be quite safe for me to express a wistful envy for your status as a bachelor; not unless I intend to start getting my own breakfasts\u2014a prospect which I am too cowardly to face.\" He recommended Williams go to a free legal aid clinic and explain that he was an ex-GI enrolled in college, broke, and needing help with a divorce.\n\nWithout Yvonne, though, John didn't want to attend their wedding, news his sister was not pleased to hear. \"You've got to\u2014we have everything planned and you must! Besides, I want to see you, and I want you to meet my boy. So, you see, you can't get out of coming.\" But he demurred, saying he had too much work. \"Damn! Damn! Damn!\" his sister replied, disappointed and hurt. \"You don't have enough time between semesters to make the trip, do you? This is very upsetting\u2014we had planned so on you coming.\" When she asked for a postcard update\u2014\"Write to us, you stinker, and let us know what goes\"\u2014he sent her one with a paragraph of typed gibberish that ended, \"Nyet, nyet, klumph.\"\n\nButch, bothered that George Rae was upset, picked a fight with John by bad-mouthing Swallow Press again. \"I will have to risk stirring your hackles by pointing out what seems to me to be an embarrassing inconsistency in your stand: your own book has been the only publication of Swallow's that I've read that I would trouble to wipe my ass on. . . . All of the selections of The Short Fiction Group I've seen . . . have been so consistently pitiful.\"\n\nGeorge Rae feared a permanent break and tried making peace before it was too late: \"I hope by this time you have stopped bristling with anger at our last letter. Butch really isn't a violently commercial boy. . . . It is true that he hasn't liked the recent Short Fiction selections. But you can't completely condemn him for that. He feels that Swallow is giving you a bad deal, etc. Well, enough of this\u2014but I wish you two could meet\u2014then you would stop fighting\u2014I know you would, because you would find that you were pretty much in agreement.\"\n\nWilliams, as a peace offering, sent a signed copy of his poetry collection, just published by Swallow Press, _The Broken Landscape_ , which included the poem \"The Lovers\":\n\n_This cluttered room knows form tonight. Before_\n\n_My window, casual and tense you stand_ ,\n\n_While to the north an erosive moon explores_\n\n_The freezing sky, and crumbles on your hand_.\n\n_Now we, though silent in our rafted room_ ,\n\n_May meet, although we do not speak, or brush_\n\n_As unique petals on earth's blackened rim_ :\n\n_Here distance is made finite in times hush_ ,\n\n_And here the shrinking dark conspires with space_.\n\n_But you must turn and shatter me, invite_\n\n_Dispersion through your body's aching space_.\n\n_\u2014Thus all is huge again, and depthless night._\n\nAcknowledging its receipt, Butch replied, \"We were both surprised to learn, from the dust jacket, that you are 'Currently attending the graduate school of the University of Denver.' Why didn't you tell us you had graduated?\" Butch didn't remark on, or perhaps he didn't notice, that the volume was dedicated to \"Lonnie\"\u2014Avalon Smith, who worked at Swallow Press.\n\nIn March 1949, a month after Yvonne's divorce was final, John and Lonnie married in Provo, Utah, where her parents lived, in a private ceremony, again at a Methodist minister's home.\nPART II\n\n_Butcher's Crossing_\n**CHAPTER SEVEN**\n\nThe Winters Circle\n\n_Yvor Winters cut a giant's figure. . . . [N]o one whom Winters touched as a poet, critic, colleague or teacher could remain indifferent to his passionate convictions about poetry_.\n\n\u2014KENNETH FIELDS, WINTERS' COLLEAGUE, _STANFORD MAGAZINE_ , 2000\n\nAs John Williams' marital problems waxed and waned, quarterly statements began indicating that Swallow Press was in real trouble. It wasn't because of anything Williams did or failed to do; it was because Swallow had his fingers in half a dozen pies\u2014teaching classes on creative writing, directing the university's creative writing program, and managing both his own and the university press. He was overwhelmed, and his business showed it.\n\nOn the positive side, he had a keen eye for spotting worthy submissions, which time alone would prove. Swallow Press carried a few strong titles, such as Janet Lewis' _The Wife of Martin Guerre_ , which Williams admired tremendously; and Allen Tate's _On the Limits of Poetry_. But Swallow's business acumen was almost nonexistent. And in his haste to publish what he liked, Swallow Press books were riddled with typographical errors. Mark Harris, one of Williams' fellow students at Denver, and later the author of _Bang the Drum Slowly_ , wondered how Swallow could carry on. \"His publishing struck me as crazy,\" he later said. \"He published poetry and criticism by people I had never heard of, assumed financial losses, and was constantly in search of warehouse space for thousands of books he never sold.\"\n\nNevertheless, Swallow tended to operate from a happy frame of mind. All would be well as long as he was moving in literary circles and his Swallow Press books were going out the door. He wrote to Williams, in fine spirits, \"Glad everything is going so well there.\" He was in San Francisco now, after his summer teaching stint at Berkeley. There was \"a poet on every block, and about every tenth poet has written some rather decent poems,\" he said. \"And the influences and crosscurrents are remarkable and almost impossible to straighten out, I'm afraid. . . . The wonderful thing about it is that they all, on the surface, get along well and help each other create the atmosphere of much activity going on; privately, they scratch each other like cats.\"\n\nNearly at the top of the intellectual heap of cats who were into poetry was Williams' hero, the poet Yvor Winters and husband of the novelist Janet Lewis. Winters was the \"sage of Palo Alto,\" Swallow called him, and Winters would not have minded the appellation. From his high place at Stanford University forty-five miles south of San Francisco, \"Yvor Winters cut a giant's figure,\" said Kenneth Fields, a student of his who later became a colleague. \"He stood out from the crowd and was often viewed with awe\u2014or, occasionally, near-loathing. Whatever the reaction, seemingly no one whom Winters touched as a poet, critic, colleague or teacher could remain indifferent to his passionate convictions about poetry.\" His reputation was at its peak in the late 1940s, and there was nothing he liked better than defending it with a good intellectual spat. Critic Stanley Edgar Hyman said Winters' \"heart and mind seem firmly back in the London of 1700,\" where \"violent oracular tradition\" ruled. Winters warned his adversaries, \"I have spent my entire life in the remote west, where men are civilized but never get within gunshot of each other.\"\n\nSome of it was bravado. He had not always been such an Augustan; or even a Californian, for that matter, much as he loved the West. His roots were in Chicago, where his father had been a well-to-do stock and grain broker on the Chicago Stock Exchange. As an undergraduate at the University of Chicago during World War I, where he met his future wife, Janet Lewis, he had cofounded the Chicago Poetry Club to redress the lack of modern poetry in the university's English courses. He was writing experimental verse then and would have counted himself an imagist. Had he stayed connected to the university somehow, he might have become a Chicago poet, instructor, and literary critic, a contemporary of Carl Sandburg's.\n\nBut during his junior year at Chicago, Winters was diagnosed with tuberculosis and sent by his family to St. Vincent's Sanatorium in Santa Fe, New Mexico, for a dry-climate treatment. Writers of his generation, including Lewis, went to Paris if they could, but due to Winters' health\u2014and perhaps a break with his family\u2014he supported himself for three years by teaching poor children in Raton, New Mexico, an experience that turned him into an advocate for social causes. He published the first of two collections of mainly free verse during those years, _The Immobile Wind_ (1921) and _The Magpie's Shadow_ (1922), both of which were well received. He was completely at home in modernism at that time, considering poetry as escapist and druggy, \"a permanent gateway to walking oblivion.\" His early poems were moody, concrete as opposed to abstract, and not overly concerned with ideas.\n\nFrom Paris, Lewis wrote him to say that she, too, had been diagnosed with tuberculosis: What did he think she should do? He found her a position as a tutor at the Sunmount Sanatorium in Santa Fe. A passionate correspondence between them followed, and they married in 1926. From New Mexico, they went to the University of Colorado, Boulder, where he earned degrees in Romance languages and Latin. He disowned his youthful work, which had followed in the tradition of the Romantic poets. To Harriet Munroe, founder of _Poetry_ magazine, he expressed his views humorously, but succinctly:\n\n_May hell wipe out divinities_\n\n_And a-cerebral infinities_\n\n_And other asininities_ ,\n\n_Praise hell, praise hell, praise hell._\n\nNow he was writing poetry with the aims of the English Renaissance poets, to go \"from oblivion to definition\"\u2014without vagaries, or Romantic flights. Winters now had a \"rage for order.\" Poetry should reflect certainty and clear moral choice, he argued, arrived at by thought, not by an \"impulse from a vernal wood,\" as Wordsworth would have it. The venerated English poet Thomas Gray had waxed sentimental on nature and death in \"Elegy Written in a Country Churchyard\" (1751). But Winters, a New Critic, wanted nothing hovering outside the text itself\u2014no ghosts, no Thanatos, no nostalgia, no fearful impulses. In one of his best poems, \"On a View of Pasadena from the Hills\" (1931), he gazes with satisfaction on his tidy garden, the product of thoughtful planning, carefully maintained and hemmed in by \"seeping concrete walls \/ Such are the bastions of our pastorals!\"\n\nA poem, he insisted, should reveal meanings in human experience that were expressed in precise terms. The problem distorting modern poetry was extravagant and obscure language. He called for replacing it with a return to the classical or plain style\u2014sometimes called the Native style in English Renaissance poetry\u2014which uses concrete words rather than abstract ones, and native words rather than borrowed ones. Plain style called for modulating emotion versus abandoning oneself to mystery, obscurity, or the supernatural. As was said of the Greek historian Herodotus, his writing was, \"clear, rapid, euphonious, marvelously varied according to variations of his subject matter; he can write in a plain and simple manner, with short sentences loosely strung together, but he can also build up elaborate periodic structures making effective use of many poetical words.\"\n\nFor teaching to students his approach to poetry, Winters assembled a new canon of poems from the 1500s through the 1950s. Dashing expectations, he largely excluded the poets of the English Petrarchan School\u2014their language was too rich, he claimed, and most of their substance was clever rhetoric\u2014though he did include two of its most famous representatives, Sir Philip Sidney and Edmund Spenser. Instead he elevated plain-style poets of the period, including George Gascoigne, who \"deserves to be ranked,\" he said, \"among the six or seven greatest lyric poets of the century, and perhaps higher.\" Other members of his pantheon included Sir Walter Raleigh, Ben Jonson, Thomas Nashe, Thomas Wyatt, and John Donne. And then, reaching down into the realms of obscurity, Winters pulled up Barnabe Googe, George Turberville, and Fulke Greville and set them among the best, too, where he believed they belonged.\n\nIt was a bold project, examining four hundred years of poetry, and his methods were hard to understand. \"He gives only his conclusions,\" complained Hyman, \"almost never with any evidence approaching adequacy, and in a form in which it is not possible to argue with him or even understand what he is trying to say.\" The poet Kenneth Rexroth was more blunt: Winters was \"responsible for some of the most wrong-headed and eccentric criticism ever written.\"\n\nBut Williams accepted Winters' _In Defense of Reason_ as a manifesto. It was fresh, yet old school and reactionary, and it was anarchic, but held in place by Winters' uncompromising doctrine:\n\nThe Romantic theory assumes that literature is mainly or even purely an emotional experience, that man is naturally good, that man's impulses are trustworthy, that the rational faculty is unreliable to the point of being dangerous or possibly evil. The Romantic theory of human nature teaches that if man will rely upon his impulses, he will achieve the good life. When this notion is combined, as it frequently is, with a pantheistic philosophy or religion, it commonly teaches that through surrender to impulse man will not only achieve the good life but will achieve also a kind of mystical union with the Divinity: this, for example, is the doctrine of Emerson. Literature thus becomes a form of what is known popularly as self-expression.\n\nWilliams was not being dogmatic by taking sides during exchanges of robust, provocative criticism like this. Some of the sharpest minds on American university campuses were attracted to English and American literature after World War II. Criticism by I. A. Richards, William Empson, T. S. Eliot, D. H. Lawrence, Lionel Trilling, Allen Tate, and F. R. Leavis swept away the older nineteenth-century approaches of philology and literary history. \"You waited for their essays,\" said the poet Robert Lowell, \"and when a good critical essay came out it had the excitement of a new imaginative work.\" By allying himself with the Winters Circle, Williams sought to join a school of poets that included Edgar Bowers, Thom Gunn, Helen Pinkerton, Philip Levine, Margaret Peterson, Donald Hall, J. V. Cunningham, Catherine Davis, Donald Justice, Alan Stephens, and Robert Pinsky.\n\nWilliams also heard in Winters' summons to return to a restrained, classical style of writing a call to stoicism\u2014thinking one's way to clarity and understanding; practicing rational acceptance of what is. Life promises nothing and will never yield, especially not to those who attempt to please or mollify it by losing their integrity. Ecstasy was self-indulgent; giving over to feeling meant renouncing self-possession; and every line of poetry or sentence should reflect, Williams came to believe, the writer's will, rather than awe.\n\n\"How is it really all working out?\" Ken Smart wrote from the University of Miami in early 1949. \"How do you feel about the future\u2014I mean would you ever seriously like to do any teaching, do you plan to stay writing as a major undertaking, or are you like so many people, content to make plans for the next ten hours only?\"\n\nWilliams, newly married, was juggling his time between writing fiction and poetry; working as an editor at a small press; and, if he continued in graduate school, possibly laying the groundwork for a career in academia. Of the three, teaching at the university level would provide a stable base, one that would also allow him time to write. He didn't anticipate that writing and teaching would interfere with each other. \"My God, since when is working for a living selling out?\" he asked. \"Writing isn't something you do in the mind of God, or in some abstraction; you write in the world, in a room, on a table, with a typewriter or a pencil. You've got to eat in order to live; and if it distracts you less from your writing to eat well than to eat badly, then you ought to eat well.\" Besides, teaching would strengthen him as a man of letters who understood texts, historical periods of authorship, and so on, unlike Butch, who believed that talent, inspiration, and time were enough for a writer. \"You boys in Denver are missing a great opportunity in not providing me with a crate of muscatel and a tape-recorder,\" Butch, the jazzman\/writer, exalted. \"I could turn out pages of fantasy in the joyous mode: you could sell it, not in thin pamphlets, but by the laundry-basket full, and make a fortune.\"\n\nWilliams ignored the taunt; he was convinced that a writer needed to be better read than any scholar. Earning a PhD and teaching would provide what he needed and was what he sought to accomplish, working \"in a room, on a table, with a typewriter . . .\" He'd like to be an author-academic. Alan Swallow had introduced him at a party to someone in that category, who happened to be a member of the Winters Circle, besides\u2014poet and essayist J. V. Cunningham.\n\nCocktail parties for the literary-minded were one of the perks of being friends with Alan Swallow. He invited students and colleagues over to meet Swallow Press authors. Sometimes the gathering was an autograph party; other times, just a Friday afternoon shindig after work. Williams attended regularly; like his fellow graduate students, he appreciated the chance to feel a \"sense of connection with the world of achievement.\"\n\nIt was generous of Swallow and Mae to act as hosts. Social situations were not Swallow's forte. Novelist Mark Harris remembered his almost painful attempts at making conversation: \"Among fellow professors and talkative students, he was slow at repartee and banter, giving the impression of a man either humorless or a little deaf.\" Mae Swallow felt self-conscious too, because book-talk was not something she kept up with. So she occupied herself with providing a nice event. Her graciousness put everyone at ease and offset Alan's awkwardness, as he stood in the middle of the living room, a whiskey in one hand and flipping peanuts in his mouth with the other, listening and nodding.\n\nIn attendance now and then was J. V. (James Vincent) Cunningham. Raised in Denver in a working-class family, Cunningham was Jesuit educated. Speaking of himself in the third person, he said, \"the tradition that surrounded him and formed much of the texture of his early years was the tradition of Irish Catholics along the railroads of the West. . . . He was a Catholic by tradition, training, and deep feeling. . . . Hence, his own identity he fenced off, and though it formed part of the terrain it had its property lines.\" Cunningham was an intense but detached listener, witty at times, not grave. It was true that some part of him was \"fenced off\"; there was a hint of the patrician in how he deliberately held himself apart\u2014an unspoken assumption that intellectual integrity and self-reliance called for being reserved. Irving Howe, the literary and social critic, sensed tension beneath his polished surface: \"Cunningham lived with, believing in and suffering from, an inordinate pride. Pride was the defense a serious man put up against the world\u2014pride and a fifth of bourbon. Pride was a sin, but an enabling sin: it helped one get through one's time.\"\n\nCunningham's father had been a steam-shovel operator. The family had arrived in Denver from Montana when Jim was four. At fifteen, he graduated with top honors in Latin and Greek from Regis Jesuit High School. But then his father was killed in an accident at work, and his plans for college were set aside. To help support his mother and siblings, he worked as a copyboy for the _Denver Post_ writing headlines, and then as a runner at the Denver Stock Exchange. The Crash of 1929 upended his hopes a second time. Jobless, in September 1930 he and his brother took to the road in an old car. Small-town newspapers and trade magazines paid him by the article for items of interest to merchants, farmers, and businesspeople.\n\nCunningham wrote to Yvor Winters at Stanford asking for help, because Winters had befriended him when Jim was still in high school and they had corresponded about poetry. Winters urged his young, penniless friend to come from his freezing cabin in northern Arizona to sunny Palo Alto; he could stay in the garden greenhouse temporarily, rent-free, and Winters would see to it that he was admitted to Stanford. He told him which train to take to San Francisco, and he was there waiting on the platform when Cunningham arrived.\n\nCunningham was ten years older than Williams, but Williams could imagine becoming like him one day\u2014a professor at the University of Chicago, a teacher, a scholar, and a poet. True to the Winters Circle, Cunningham's verses had a kind of \"cold grace\" about them. He mistrusted emotion. The emphasis was on meter, rhyme, and precise use of language, giving his stanzas the burnish of seventeenth-century poetry, traits that Winters praised. His verse was classically epigrammatic, and his satire had the sting of Juvenal or Martial. In \"For My Contemporaries,\" he salutes (but smiles behind his hand) the groaning efforts of poets to bare their souls:\n\n_How time reverses_\n\n_The proud in heart!_\n\n_I now make verses_\n\n_Who aimed at art_.\n\n_But I sleep well_.\n\n_Ambitious boys_\n\n_Whose big lines swell_\n\n_With spiritual noise_ ,\n\n_Despise me not!_\n\n_And be not queasy_\n\n_To praise somewhat_ :\n\n_Verse is not easy_.\n\n_But rage who will_.\n\n_Time that procured me_\n\n_Good sense and skill_\n\n_Of madness cured me._\n\nThe madness he has been cured of, the \"spiritual noise,\" was his Catholicism. Without his faith, he was a \"renegade\" now, he said, although there is a trace of loss in his poetry, a sense that what had been good and consoling could never be recovered. His first wife, the poet Barbara Gibbs, with whom he had a daughter, bitterly compared him in \"Accusatory Poem\" to \"shifting chips of brilliant colored glass,\" like a shattered stained-glass window.\n\nWilliams submitted his first critical essay, \"J. V. Cunningham: The Major and the Minor,\" to _Poetry_ magazine in the fall of 1949. The poet Hayden Carruth, the senior editor, might have returned it automatically because Cunningham was a contributing editor, and it would look like an inside job. \"I cannot print any commentary\u2014particularly such an extended one\u2014on members of my staff,\" he wrote. Instead, he responded with a two-page letter about the career politics of publishing criticism in journals. It included some pointed advice:\n\nYou have a chance to do Jim [Cunningham] some good, but you must do it in the right way. Winters has done Jim much more harm than good\u2014though out of the most honest and generous motives\u2014by pushing him too hard, too belligerently, when others were ignoring him. The result is that Jim has come to be known as Winter's boy, and of course, that is neither true nor just. . . . What I mean to say is that your essay should be published out of the Winters's orbit, yet not in the other camp. It should be published in the East. If I were you, I'd try _Hudson Review_ first.\n\nAs a sidebar, Carruth added that he couldn't accept Williams' claim that there has been no major poem since Milton: \"Even limiting ourselves to English, which I suppose you intend, there have been other major poems.\" Without a stronger premise to defend, the rest of Williams' argument regarding major and minor poets didn't follow: \"You are overstepping your evidence by about ten thousand miles.\"\n\nCarruth's gentle cuffing of him in the ring of heavyweight criticism did Williams a favor. Six months later, in the spring of 1950, the editors at the _Arizona Quarterly_ accepted his new draft of the Cunningham essay, saying they were \"unanimous in their praise.\" They, too, however, admitted to being a little confused about Williams' major\/minor dichotomy. What did he mean, \"'major' poetry is intrinsically preferable to 'minor' poetry, since there is between them no real difference\"? But it didn't bother the editors enough to ask for a revision.\n\nAppearing in the _Arizona Quarterly_ seemed a good omen, among others, that he should continue for a doctorate. Swallow recommended the University of Missouri, Columbia, where there was a new program combining creative writing with literary scholarship. On Alan Swallow's advice, he applied to Missouri. Columbia beckoned for another reason, as well: Lonnie could complete an advanced teaching degree there.\n\nWilliams was following the path of Winters, Cunningham, and Swallow. He felt a kinship with them\u2014intellectually and personally. In particular, the Winters and Cunningham philosophy that combined life and art\u2014independence of thought as a matter of integrity, the power of reason over emotion, and the plain style as an extension of stoicism expressed in prose and poetry\u2014strongly appealed to him.\n\nAnd yet, although he wanted to belong to the Winters Circle, he seemed not to have realized how different his goals were from those of most academics in the humanities, who cared nothing about writing novels, and looked skeptically at professors who did. Ironing out problems of character development, pacing, motive, and theme generally didn't interest them. Nor would they, moreover, when they were in their mid-twenties, like Williams, have dreamed of showing fiction to an agent.\n\nWilliams had only to look at the three men he admired for glimpses of the scholar's life. J. V. Cunningham would distinguish himself with the monograph _Woe or Wonder: The Emotional Effectof Shakespearean Tragedy_, published by Swallow Press. Winters, the older he got, would crossly fight off all claimants to the throne he thought was rightly his as the preeminent American critic of poetry. And Swallow, the professor\/independent publisher, might have served as an example of trying to straddle too much, of having too many masters. There was hardly time enough in Swallow's life to do anything well. Regardless of their different circumstances, none of these academics had a professional inclination to spend years on a novel, which John Williams fully intended to do, and hold down a professorship in the bargain.\n\nAs he and Lonnie prepared to leave for Columbia in the fall of 1950, he packed the manuscripts and story ideas he was working on.\n**CHAPTER EIGHT**\n\n\"Natural Liars Are the Best Writers\"\n\n_I write of human experience so that I may understand it and thereby force myself into some kind of honesty_.\n\n\u2014JOHN WILLIAMS, 1964 INTERVIEW\n\nAshland Gravel Road in Columbia, Missouri, where newlyweds John and Lonnie Williams found an apartment in the fall of 1950, was only a few blocks from campus. Years before, it had been out in the country\u2014explaining why, in Williams' novel, when young Bill Stoner enrolls at Missouri in a time set decades earlier, he boards at his relatives' farm but can walk to class. What struck Stoner so powerfully in the middle of the university's quadrangle is still there:\n\nSometimes, in the evenings, he wandered in the long open quadrangle, among couples who strolled together and murmured softly; though he did not know any of them, and though he did not speak to them, he felt a kinship with them. Sometimes he stood in the center of the quad, looking at the five huge columns in front of Jesse Hall that thrust upward into the night out of the cool grass; he had learned that these columns were the remains of the original main building of the University, destroyed many years ago by fire. Grayish silver in the moonlight, bare and pure, they seemed to him to represent the way of life he had embraced, as a temple represents a god.\n\nThe implication is that the University of Missouri, the first university west of the Mississippi, is a kind of Athens on the edge of the wilderness. It's fitting that Stoner, the son of a farmer, should come to the brink of his vegetable world and view the columns as a kind of outpost of the ancient tradition of thought and inquiry. Enrolled to study agriculture, Stoner grows lonely in this strange world of metaphysics, discussion, and analysis.\n\nWilliams, on the other hand, was accustomed to the sociability offered by college life. His first semester, he joined a club, the Tabard Inn, consisting of about twenty-five students and professors, all men, who met twice a month in a banquet room at the Moon Valley Villa, a dine-and-dance restaurant on the eastern edge of Columbia. A \"Chaucerian cell,\" they called themselves in \"spiritual affiliation\" with \"Chaucer's pilgrims who set out from a grog house yclept 'Ye Tabard Inn.'\" The club's purpose was to offer camaraderie among men who liked conversation with a literary bent. Within a year of joining, Williams was part of the inner circle. And it was probably here, while the members lingered past midnight to prolong their revels, that he heard the story of a decades-old feud between two professors of English at Missouri, which became one of the inspirations for _Stoner_.\n\nYears earlier, about the time of World War I, Robert L. Ramsay and A. (Arthur) H. R. Fairchild had been colleagues and friends at the University of Missouri. Both were \"learned,\" in the quaint phrase of the day, with doctorates from Johns Hopkins and Yale, respectively. Later, their animosity toward each other became legendary.\n\nRamsay, a southerner born in Sumter, South Carolina, taught courses in the history of the English language. He corresponded with just about every American and English scholar of language of his day. His hobby was investigating the origin of place names in Missouri\u2014whether they were Native American, French, Spanish, or English, and so on. His love of etymology carried over into his linguistics classes, which were demanding. He wrote energetically on the board, connecting various forms of words and their origins, with chalk lines of red, yellow, or purple, getting the dust on the sleeves of his jacket and sometimes on his snow-white beard.\n\nIn his short-story writing course, students were surprised to hear him say that if they were natural liars, then they might succeed as writers. \"Many men start life with a natural gift for exaggeration,\" he pointed out, \"but in a short time develop a morbid and unhealthy faculty of truth-telling. If, when you look over your past life and recall those occasions when you ventured to use a lie, you find that you met humiliating failure, then you will not make a writer.\"\n\nThere were other prerequisites for being a writer, he said, such as a yen for \"poking [your] nose into other people's business.\" A writer should also be \"an ardent lover of gossip and a confirmed listener to all sorts of scandal.\" A penchant for cruelty helped, as well: \"A genuine artist will never hesitate to deprive his characters of any happiness, to hurt them, mutilate, torture, or even kill them.\" If these traits made some students uncomfortable, he reassured them: don't worry, because \"you may at least be a good citizen.\"\n\nHis gently ironic sense of humor extended to oral examinations of candidates seeking a master's degree. \"Well, do you think you've written a masterpiece?\" Professor Ramsay would inquire, looking skeptically at the student's thesis.\n\n\"Oh, no, sir,\" was the usual, humble reply.\n\n\"No? Why not? What is a masterpiece?\"\n\nThe student, trying to recall everything he or she could about guilds and journeymen, would explain that a masterpiece was work done to qualify a craftsman for the rank of master.\n\n\"So then,\" Ramsay would say, smiling, \"you have written a _master_ piece, haven't you?\"\n\nRamsay's colleague in the English Department, Fairchild, an Elizabethan specialist, was a Canadian from Toronto who limped from an unsuccessful operation in middle age. His lectures tended to be formal and arid\u2014\"austere,\" as someone called them. His logical mind expressed the link between education and success as a syllogism. \"In the last analysis,\" he concluded, \"the success of a man in any line of work must depend upon his knowledge of human nature. A man who limits his capacity for understanding human nature limits his ability to deal with his fellows. A widened point of view, a knowledge of our fellowmen, can only be drawn from good books.\" In 1915, Fairchild married a Miss Workman from Grand Rapids, Michigan, who became a renowned hostess and chairperson of many activities. Invitations to their home for social occasions were prized.\n\nFor a time, Ramsay and Fairchild gave public lectures together about developing taste in books, poetry, and Shakespeare\u2014topics of broad interest to residents of a college town who enjoyed a little cultural spillover from the campus. Seated on the dais, side by side, glancing at their notes, they were the epitome of university men\u2014two scholars, both alike in dignity. But then a split occurred, resulting in an antagonism that became, as one department member said, \"intense, protracted, and largely inexplicable.\"\n\nDuring Fairchild's chairmanship, he slighted Ramsay. Ramsay usually taught the works of Milton to undergraduates. But Fairchild, who believed he was better qualified to teach Milton because he was the department's Renaissance expert, took the course away. Ramsay's fellow instructors rallied around him and created a course at the graduate level called, \"Miltonic Criticism.\" For three decades, Professor Ramsay happily taught the course under the department chair's nose, but Fairchild never forgot how Ramsay had played mutinous Fletcher Christian to his Captain Bligh.\n\nA second quarrel, over splitting the department, came to a head in 1937. Fairchild agreed with a lengthy report from a senior speech professor pointing out that treating speech and drama as separate from the study of literature was becoming a national trend in higher education. But Ramsay didn't see it that way; he blamed Fairchild's imperiousness for creating a rift, writing to him, \"Your overbearing attitude and unwillingness to share supreme control [of departmental matters] was, I believe, the primary cause that drove our fourteen brother teachers of speech, oratory, and dramatics to press for the setting up of a separate department\u2014a step I cannot help thinking disastrous for our students and our common interests.\"\n\nThe fight between Ramsay and Fairchild had become notorious by now, and the bad blood trickled into every corner of the department, impossible to ignore. In fact, it poisoned the oral examination of one of Ramsay's doctoral candidates.\n\nRamsay's student arrived at the examination prepared to defend his thesis about the importance of local color for creating regional literature, such as the dialect found in William Faulkner, John Steinbeck, or Willa Cather. Ramsay and Fairchild, and the rest of the thesis committee, took their seats. Everything seemed to be going well, until it was Fairchild's turn to begin asking questions. Feigning the bewilderment of Socrates, he pressed the candidate to define his terms: the words \"regional\" and \"local\" and even \"literature\"\u2014what did they mean? For example, _Uncle Tom's Cabin_ , would that be considered \"great regional literature\"?\n\nNervously, the student tried to reply as best he could; but Fairchild continued to badger him, probing, until at last he was ready to spring his trap. \"Since Aristotle has established the principle that great literature must be universal,\" he said, \"local color writing, which is your subject, must by its nature be something less than great\u2014wouldn't you agree?\"\n\nRamsay and other examiners were aghast. They understood what Fairchild was doing\u2014his trick question was intended to demean Ramsay and his hobby of collecting the \"local color\" of place names, which Fairchild thought was a childish pursuit. The candidate was being sacrificed for malicious reasons in front of his adviser's horrified eyes. But before Fairchild could thrust home, Ramsay interceded, and other instructors came to the rescue as well, until the thesis and the student seemed to be on safe ground again. Fairchild retired not long after that, in 1946, having accused Ramsay of stealing postage stamps from the office.\n\nWhen Williams arrived at the University of Missouri in 1950, there was still the odor of brimstone hanging about the English Department. With few changes, the feud would become a set piece of _Stoner_ , appearing in the novel as the decades-long antagonism between Bill Stoner and his nemesis, Hollis Lomax, a professor of Romantic poetry, who not only limps like Fairchild, but has a humped back, too.\n\nEverything was \"going great guns!\" back in Denver, Swallow reported in May 1951. \"The writing program and the progress of the university press are the bright spots. Yet the rest promises to be a shambles if we can't hold it together.\" A distance of seven hundred miles wasn't enough to keep Swallow's problems from becoming Williams', as well. Would he mind, Swallow queried, taking delivery on a five-hundred-pound shipment of printing paper already on its way to his apartment? \"I hope you will find it not too inconvenient.\" An agreement followed, indicating that the cardboard boxes of paper belonged to him and not to Swallow Press. \"I'd appreciate it much if you could sign the enclosed,\" Swallow wrote to Williams, \"\u2014and perhaps your landlord would go along with signing to reinforce the situation.\"\n\nSwallow's difficulties, usually the result of lurching from one project to another, tried Williams' patience at times, but Swallow was meticulous about returning favors and using his influence whenever he could. To thank Lonnie, who was compiling his _Index to Little Magazines_ practically for free, Swallow wrote a letter of recommendation to support her application to graduate school at Missouri. And when Williams completed a novel he'd begun in Denver, called _Splendid in Ashes_ , he sent it to Swallow, confident that his friend would reply with a candid and detailed opinion.\n\nThe title _Splendid in Ashes_ comes from Thomas Browne's 1658 _Hydriotaphia, Urn Burial_ : \"Man is a noble animal, splendid in ashes, and pompous in the grave.\" The book \"smells on every page of the sepulcher,\" said American essayist Ralph Waldo Emerson, whose poem \"Threnody,\" about the death of his son, is one of the greatest memorializing poems of the nineteenth century.\n\nWilliams was intrigued with the possibilities of starting a novel eulogistically by describing a life that has ended. _Splendid in Ashes_ begins that way, with the main character, Douglas Morely, already dead, which will be the identical method used on the first page of _Stoner_.\n\nMorely, in _Splendid in Ashes_ , is a solitary person, observing life from the other side of an emotional wall. Standing apart from everyone else, he rebuffs attempts to break through to him. He's a Nietzschean hero, straining against the boundaries of a conventional life. All that defines him is creative freedom.\n\nSwallow praised it, and thought it so good that it deserved a larger readership than he could reach with Swallow Press. That's what he told Williams, although he may have been trying to avoid having to turn down a friend's submission because it wouldn't sell. _Splendid in Ashes_ was another piece of experimental fiction, similar to _Nothing But the Night_ , copies of which were in boxes unsold in Swallow's garage. For a second opinion of Williams' newest novel, Swallow put him in touch with Barthold Fles in New York, a Dutch American literary agent and former book reviewer for _The New Republic_. Fles' clients included Bertolt Brecht, Heinrich Mann, Jessica Mitford, and Bruno Walter.\n\nFles' response to _Splendid in Ashes_ was rapturous. \"Both my wife and I, as well as my assistant,\" he wrote Williams, \"have now read your novel. It is a pleasure to be able to tell you that we are unanimous in considering it one of the most distinguished pieces of fiction that we have read for a long time. It was an experience to read it.\" It brought to mind a few parallels. \"We mentioned _Wuthering Heights_ , _Under the Volcano_ and some French novels, when we discussed the book. We feel confident of finding a first rate publisher for this.\" In short order, Fles distributed _Splendid in Ashes_ to every major publisher in New York and Boston.\n\nIt would have been better if he hadn't, and had offered Williams some editorial advice instead. But Fles was taking the approach of throwing spaghetti against a wall, because something had to stick. Within weeks of receiving a copy of the novel, many prestigious American publishing houses\u2014Random House, Knopf, Harpers, Simon and Schuster, and a dozen more\u2014returned it. And generally their criticisms were the same. Remarks by Harry Brague at Charles Scribner's and Sons were typical:\n\nThere is a great deal of superlative writing in it, and the characterizations for the most part are skillfully developed. But somehow when you have gotten through reading the manuscript you are left with a puzzled feeling as to what the author is attempting to say, and the reader is puzzled as to just what Douglas Morley's problem was and not a little impatient over the fact that so many people should spend so much time worrying about a basically uninteresting and irritating character. . . . The book begins by informing the reader that Douglas is dead, and as it progresses the reader is given no reason for either sorrow or elation over the fact.\n\nTo us it seems a shame that a writer of Mr. William's [ _sic_ ] obvious capabilities and potentialities should have spent so much time delineating a character who is basically not worth it.\n\nPerhaps Barthold Fles hadn't read this letter when he replied ebulliently to Williams from Beverly Hills:\n\nYour letter updated caught up with me here, which is most unfortunate, because I would certainly have come down from Iowa to see you had it come in time. As it was, we were both anxious to get to the West Coast, because of a pending movie deal about which we are in constant telephonic touch with Hollywood. . . . We consider your novel, _Splendid in Ashes_ , one of the best we have seen in years and a real discovery.\n\n_Splendid in Ashes_ continued to stagger through Manhattan after that, being turned away from every door. From an editor at E. P. Dutton: \"Unfortunately, we think that in the present market this manuscript is just too long and too pretentious. This does not mean that there is not a great deal which is powerful and thoughtful, but the material seems to us somehow too elaborate and somewhat too posing.\"\n\nFinally, Fles threw up his hands. \"John, this has had endless submissions\u2014twenty in all\u2014and it's just impossible to do in today's fiction market.\" Bowing out with a tip of his hat, Fles expressed his regrets. \"Sorry, but we tried hard\u2014I tried hard.\" Williams never heard from him again. As Ana\u00efs Nin said when she quit Fles, dissatisfied, \"Bonjour, friend, and good-bye, literary agent.\"\n\n\"Dear Slaves,\" Butch Marsh began a letter to John and Lonnie in October 1952. He and George Rae had relocated to Mexico and were living the life of literary bums. With \"both typewriters clacking, and the jug of tequila diminishing as we go\"\u2014they were writing to congratulate Lonnie and John on the birth of their first child, Katherine. (John said, \"Except for the mustache, she looks like me, God help her.\") The Marshes were living in a rented house in the town of Ajijic, south of Guadalajara, a \"quietish, friendly, clean little town on the banks of the country's largest lake.\" They had two floors and seven rooms to themselves above, at street level, an ice factory. They were on the main floor, and upstairs on the third floor was a glassed-in studio overlooking a roof garden with a view of the town, where Butch worked all day. \"We find that we can seem to live quite well, in our cozily disordered way, for about seventy-five bucks a month, including everything.\"\n\nHe was getting a lot of work done, Butch informed John, in the new digs, having an \"almost pathological obsession with getting everything I turn out into print, somewhere or other\u2014even to the point of using goofy pen names if I am ashamed of both the story and the magazine.\" His output since moving to Mexico included a fifty-eight-thousand-word novel written in forty days (rejected); a poem published in the _Arizona Quarterly_ , and a short story carried in the _Antioch Review_. Random House returned his second novel \"for commercial reasons. [The editor] didn't want I should have a novel that would only get mild praise, when I had such manifest gifts, etc. I tell you, there wasn't a dry eye in the house.\" An editor at G. P. Putnam suggested he revise and then resubmit the manuscript, but by then Butch had lost interest. Now he was a quarter of the way into his third novel, salvaging some of the material from the first. Added to this, he was generating stories and poems constantly, banking on the odds that he could make a few hundred dollars a month at it.\n\nHe confessed, tongue-in-cheek, to feeling contrite about the difference in their situations. \"I hope you understand that we are not gloating about living in a country where we, of all people, are considered millionaires.\"\n\nThe rough teasing was accepted. Williams didn't begrudge his brother-in-law the freedom to write all day, because he was after bigger game than selling stories to monthly magazines. Many writers were paying the bills that way (before Upton Sinclair became a novelist, he wrote eight thousand words a day for pulp magazines, seven days a week). But the respect of his peers was important to Williams, and he would never submit to _Colliers_ or _Argosy_ any more than he would to _Spicy Detective_. Besides, added to the demands on him as a new father as well as a student, novelist, and poet, he had a doctoral thesis to write.\n\nHis subject was English literature, and his work at Swallow Press, together with Swallow's influence, had led him to the English Renaissance. The press had published, and Williams had read, a revision of Swallow's master's thesis at Louisiana State, \"Principles of Wyatt's Composition,\" retitled as _Some Poems of Sir Thomas Wyatt_. Williams was so taken with Wyatt's \"They Flee from Me\" that he memorized it, savoring its cadence by reciting the poem aloud. His introduction to Wyatt had coincided with reading Winters' _In Defense of Reason_ , also published by the press, which included the essay \"The Anatomy of Nonsense,\" which criticized Romantic poets for using \"words denoting emotions . . . loosely and violently, as if the very carelessness expressed emotion.\"\n\n_In Defense of Reason_ introduced Williams to Winters' canon of the greatest English-language poems, where he found, among the reexamined Elizabethans, Fulke Greville (1st Baron Brooke), a member of the sixteenth-century Essex-Pembroke circle of playwrights, versifiers, and essayists. Winters claimed Greville was one of the most important lyric poets of the age, because his work resembled Drayton's, \"although it is far better.\" \"Furthermore,\" he wrote, \"in [Greville's] later work he became a greater poet in every way than any of the associates of his youth. It is my opinion that he should be ranked with Jonson as one of the two great masters of the short poem in the Renaissance.\"\n\nGreville is not unknown to Elizabethan scholars. As an aristocrat, courtier, soldier, spymaster, patron, dramatist, historian, and poet, he certainly led a colorful life. Some claim he was one of Shakespeare's collaborators. But after his death in 1628, Greville was deemed a lesser light of Elizabethan England's golden age of poetry. He was \"uncertainly admired\" by Coleridge, Lamb, and Hazlitt; and C. S. Lewis labeled him \"imperfectly golden.\" More attention has always been paid to Greville's closest friend, Philip Sidney, in whose memory Greville wrote _The Life of the Renowned Philip Sidney_. Because of that book, Greville is better remembered as a biographer than as a poet.\n\nBy choosing Greville for his dissertation, Williams would be moving toward acceptance in the Winters Circle. By siding with Winters over Greville's importance, he would be linking arms with the famous critic's assessment of an overlooked poet. If his dissertation was then published as a book, he would have a title on the shelf of scholarship declaring that he was a Winterian.\n\nIntense reading of the poetry of that era had a salutary effect on his writing style. The benefit of studying the plain style, practiced by many of the greater English poets of the Golden Age, was seeing its genuineness. Editors had faulted him for being tedious and overwrought. But the more he came to appreciate the merits of using simple sentence structures and concrete words and actions, the better he was able to recognize the problems with _Nothing But the Night_ and _Splendid in Ashes_. A sentence like, \"I shook the rain from my hat and walked into the room,\" from Mickey Spillane's _I, the Jury_ , is purposeful and vivid. Nothing might seem further removed from the traits of classical prose as it was practiced by some Elizabethans than pulp fiction\u2014Seneca from Sam Spade\u2014but both styles demonstrate the virtues of thinking clearly and writing honestly. When handled well, the result can be poetical.\n\nWilliams' challenge was to say what he meant, and to resist his desire to impress. His inclination toward flair, embellishment, and exaggeration\u2014\"lying\" a bit while telling a good anecdote or polishing up his experiences\u2014did not belong at his desk while he was writing. (The good-humored Professor Ramsay at Missouri oversimplified when he compared fiction-writing to lying.) Winters' beliefs about restraint and rational stoicism commanded Williams to write honorably. \"I have found that the knowledgeable act of writing a poem or a novel or an essay,\" Williams said, \"is a means not of self-expression (which is of little importance in a literary sense) but of contemplation. I write of human experience so that I may understand it and thereby force myself into some kind of honesty.\" The demands of the craft altered his character during the hours he spent trying to establish a real world for his characters to exist in, because truth is paramount in creating readers' belief.\n\nIt was a nice coincidence, and a boost to his career, when the editor of the literary quarterly the _Western Review_ , Ray B. West, contacted him in early 1953, requesting a review of Winters' _Collected Poems_ from Swallow Press. Williams entitled his piece\u2014fitting for Winters' emphasis on intellect over feeling\u2014\"The Goddess of Mind.\" But West shelved the Winters' review for the time being, pleading a backlog of submissions. \"It caught me at the end-of-semester rush,\" West apologized. \"Yes, it did come too late for the summer issue. I will use it in the fall.\"\n\nDisappointed, Williams asked a fellow student, Alan Stephens, a friend from his Denver days, to give the review to Janet Lewis, Winters' wife, who was coming to campus for the Missouri Writers' Workshop. As the workshop's director, Williams thought it might be inappropriate to ask a favor involving her husband when he had invited her as a guest speaker. But Stephens didn't mind helping his friend.\n\nWinters had a reputation of \"provoking out of people their ineptitudes,\" as Swallow put it. Replying to Stephens, Winters gave Williams' review a high grade. \"The review by John Williams I liked a great deal, partly, I suppose, because it praised me, but partly also because it gave a very good account of what I was trying to do.\" But Stephens' poems annoyed him, because they were too similar to his own. Some poets would be flattered to see that younger ones were emulating them, but Winters considered imitation lazy. It was stealing his work, which made him roar.\n\n\"There are certain poems in this lot, which are so obviously plagiarized that they could hardly be published; and I cannot see the virtue of that kind of thing as exercise,\" Winters wrote to Stephens. \"If you wish to write well, you must learn to adjust language to subject; you can do this only if the subject is your own and you have come to understand it.\" He upbraided Stephens for appropriating from him: \"When you swipe a subject and a bundle of imagery from someone else, it is not your own, you have not mastered it, and the original poem, in which the subject is well handled is perpetually in your way, so that you can depart from the original only by moving out into inferior treatment. . . . There is too much miscellaneous diction and imagery mopped up from my own verse, even where the subjects are original.\" Stephens, for his efforts, felt abashed, but Williams had the pleasure of finding out even before his review was published that Winters appreciated his critique.\n\nAnd then the unthinkable happened. Whether West, at the _Western Review_ , discovered that Williams was an editor for Swallow Press, and was worried about a conflict of interest, or he simply didn't get around to reading Williams' review for months, he returned it abruptly that fall. \"We all liked it, but agreed finally, that it took too long to make its point.\" He was \"going to have someone else do the review.\"\n\nWilliams was mortified. He was in Winters' good graces now; the \"sage of Palo Alto\" would be expecting to see the review in print. Quickly, Williams tried to interest other magazines and journals in accepting it, including _Poetry_ and the _Sewanee Review_ , explaining plaintively that he had been led to believe he had been \"commissioned\" to write the review. A note of desperation followed: \"I feel very strongly that this review should have a hearing, and I assure you that I am not alone in that feeling. I believe that I have something to say about Winters' poetry that has not been said before, and I believe that what I have to say is of some importance.\"\n\nBut it was no use\u2014it was too late. Reviews of _Collected Poems_ had been assigned already. Autumn passed, and Williams never heard whether Winters noticed that the favorable review of _Collected Poems_ he had read wasn't published. But one thing was certain: he was aware of a young professor named John Williams who admired his work.\n\nWilliams' dissertation, \"The World and God: The Poems and Dramas of Fulke Greville,\" was accepted in June 1954, and he passed his oral examinations. Alan Swallow added a further measure of happiness by informing him that there were two positions available in the English Department at the University of Denver, and Alan was recommending him and Alan Stephens.\n\nTo celebrate, the Williamses chose a long vacation to Mexico. Butch and George Rae, in Ajijic, had been asking for a visit. \"In case you should ever have a sabbatical or a little time off why don't you and Lonnie and Kathy come down here for it?\" And so, with another couple from the university, Florence and William Hamlin, the Williamses drove to San Miguel de Allende for a three-month stay, with the Marshes dropping in on weekends for parties.\n\nThe days were spent relaxing, reading, driving a dusty seven miles to a private pool for a swim, and drinking. To ensure that everyone had enough tomato juice for their morning Bloody Marys, Butch and John went around to every _mercado_ in town and bought up their stock, with the result that hotel owners had to come to them, pleading for a share. Florence Hamlin was delighted by John's sense of humor. \"We rode the train to Mexico City to see the bullfights,\" she later said in an interview. \"We laughed the entire way. It was just ridiculous. We laughed all the way going to the bullfights. There were six of them in one afternoon.\" They all read Ernest Hemingway's _Death in the Afternoon_ , which is both a treatise on bullfighting and a manual for fiction. \"Find what gave you emotion,\" Hemingway exhorts, \"what the action was that gave you excitement. Then write it down making it clear so that the reader can see it too. Prose is architecture, not interior decoration, and the Baroque is over.\"\n\nFlorence Hamlin, observing Williams, noticed he was happier outside the university. Without the pressure of coursework for the first time in seven years, living loosely like Butch\u2014up early to write; reading in the shade during the long afternoons\u2014he was a pleasure to be around. Hamlin \"liked John so much,\" but found she couldn't get close to Lonnie. \"I didn't dislike her,\" she said, \"but I don't think I was fond of her. She was rather cold. John was the one that everyone liked. He was wonderful, just a delightful man.\"\n**CHAPTER NINE**\n\n_Butcher's Crossing_\n\n_The essential American soul is hard, isolate, stoic, and a killer. It has never melted_.\n\n\u2014D. H. LAWRENCE, _STUDIES IN CLASSIC AMERICAN LITERATURE_ , 1923\n\nAt the University of Denver, Alan Swallow was no longer teaching or directing the creative writing program when Williams returned in August 1954 from vacationing in Mexico. An affair with the wife of a university trustee had cost him everything, and he had resigned acrimoniously. Behind him, the administration had closed the University of Denver Press.\n\nBut he was content. \"As you all know,\" he wrote to Williams, \"one of the prides I had in my publishing was its independence, the fact that it was frankly only the taste of myself and some friends to aid me. But the damned thing has mushroomed.\"\n\nOnce he was released from teaching, his days were devoted entirely to running Swallow Press. Bookstore owners in the Denver area became accustomed to seeing him deliver books in his MG roadster with the top down, smoking a cigar. For bigger hauls, he used his Rambler station wagon, loaded with books. The basement of his home became his sanctuary. There, he typed long, rambling letters to friends and clients, or stayed up late drinking whiskey and reading car magazines. \"I'll not apply for a job elsewhere,\" he assured Williams, \"but try to stick it out here in Denver for a year or two, living from publishing, writing, and probably manuscript criticism and consultation.\" Before his awkward departure from the university, Swallow had recommended Williams as the new director of the creative writing program. After all, besides being an instructor and published author, Williams also brought experience running a public conference about writing, an innovation in its day.\n\nAt the University of Missouri, Williams had run a ten-day conference. All the members of the English Department, and their wives, received personal invitations to evening social events, which added, as one of their colleagues gallantly put it, \"dignity and charm to the occasion.\" Participants received individual critiques from English Department instructors and from guest speakers, many of whom were prominent writers of the day, such as Katherine Anne Porter, Nelson Algren, Walter van Tilburg Clark, and James T. Farrell. The guest lecture was free to the public. By the end of the conference, scholars of literature, teachers, and administrators were \"flattered to be included for talk with the celebrated, and came away with only the most positive feelings about the worth of the Writing Program.\"\n\nIt was a success at Missouri, and in Denver, Williams would hold an even grander event. He would expand the conference to four weeks, adding summer writing courses to the offerings. Later, his experience went into serving as president of the Association of Writers & Writing Programs (AWP), the largest professional organization of its kind in the United States. The AWP played a foundational role in establishing creative writing as an academic discipline.\n\nSwallow believed in Williams and tended to be more relaxed around him than he was with others. He often confided in him. Swallow could be impatient, sometimes \"harsh\" and unable to \"tolerate 'stupid' people who weren't interested in the literary world.\" One of his former students, Gerald Chapman, who later joined the Denver English Department and rose to chairman, said the friendship between Swallow and Williams operated on several levels: \"Alan was John's mentor, in a way. And they had a very great deal in common, including a drinking problem, but I think Alan probably felt John identified with Alan.\"\n\nThe West was Swallow's passion, and they discussed misperceptions of it: how screenwriters had bowdlerized the region's history so much that it became a Christian parable about retaking Eden from heathens; or a paean to \"justice at the end of a gun,\" as movie-makers would have it\u2014buckaroos and sheriffs in silk shirts, blazing away with pearl-handled revolvers. Pulp western magazines and paperbacks\u2014comic strips, too\u2014perpetuated these fantasies. Williams found himself thinking about a show-biz-produced American history \"in which 'The West' does not exist, did not ever, exist. It's a dream of the East\u2014almost as if The East made up The West.\"\n\nThe real Old West had lasted only a few decades. It was already part of legend in 1883 when William Frederick \"Buffalo Bill\" Cody\u2014a former army scout during the Indian Wars\u2014premiered _Buffalo Bill's Wild West_ , a three-hour spectacle that toured the country annually for thirty years. Part circus and part historical romance, \"Buffalo Bill's Wild West\" was a nationalist extravaganza about taking and securing the West.\n\nAs a brass band struck up \"The Star-Spangled Banner\" and audiences got to their feet, cowboys on horseback, whistling and war whooping, stampeded a procession of elk, cattle, and buffalo past the grandstand. Behind them, former Pony Express riders came riding full-tilt, snatching mailbags from startled rodeo clowns who fell over in amazement. Annie Oakley, skipping into the middle of the arena to thunderous cheers, shooed the clowns away, bowed, and shot fifty glass balls being thrown high in the air by her husband, the handsome Frank Butler. Next, at the far end of the ring, a red, white, and blue curtain rose slowly, revealing a sentimental scene: a settlers' cabin and a little family dancing to fiddle music. But then a band of \"wild Indians\" attacked. A trumpet blast announced the timely arrival of the US cavalry, but the savage enemy was trying to make off with a white child. So Buffalo Bill himself came barreling into the melee driving a stagecoach, the reins clenched in his teeth as he fired at the retreating Indians with a pistol in either hand. And at last came the ear-splitting climax: a reenactment of Custer's Last Stand at the Little Big Horn River, featuring a troop of die-hard US bluecoats commanded by a blond-wigged Lieutenant Colonel George Armstrong Custer, fighting the Lakota Sioux surrounding them to a dramatic, staged \"death.\" Some of the Native Americans in the arena had been at the actual battle.\n\nTo millions of Americans\u2014and Europeans who saw the show in Belgium, Germany, Italy, the Netherlands, Great Britain, and France\u2014 _Buffalo Bill's Wild West_ was an epic story of heroes and villains contending gloriously in a showdown of good versus evil. The knock-off industry it created of B films, cowboy ballads, and third-rate novels about gunslingers helped to explain why eastern literary critics tended to dismiss western fiction as frivolous stuff.\n\nSwallow was willing to wager that he could give the West its due in American literature. He had seen the rise of the Southern Agrarian movement and its standard-bearers\u2014William Faulkner, Katherine Anne Porter, Tennessee Williams, and Zora Neale Hurston, among others. Now he \"hoped that the time in the South was achieved,\" said Gerald Chapman, \"and it was the time for the West.\" According to Chapman, \"[Swallow] was interested in cultivating the Western literature, and John picked up a lot of that.\"\n\nWilliams' interest as a novelist, however, was how to redress in fiction the misperceptions about the West. Cultural critic Lewis Mumford, in his influential book _The Golden Day_ (1949) suggested that the \"vast gap between the hope of the Romantic Movement and the reality of the pioneer period is one of the most sardonic jests of history.\" The Westward Movement, he said, was the \"epic march of the covered wagon, leaving behind it deserted villages, bleak cities, depleted soils, and the sick and exhausted souls. . . . The truth is that the life of the pioneer was bare and insufficient: he did not really face Nature, he merely evaded society.\"\n\nWhat if a young man, Williams wondered, steeped in Romantic notions about nature, expecting to meet the divine and invisible, arrives on the Great Plains during the height of the Westward Movement and heads into the wilderness, like Ishmael going to sea in a whaling ship. On the Great Plains, in a wilderness as large as Mexico, what would a New England intellectual flower do? How would a \"Har- _var_ -dian,\" accustomed to holiday caroling in Boston, cope with a Rocky Mountain winter?\n\nHistorical details would be essential, and Williams had never extensively researched a novel before. From his Texas boyhood he could recall a rusted gate squeaking in the wind, or the smell of a creek that was flat, warm, and ankle-deep. But to re-create the West in the years immediately after the Civil War, to depict a balanced conflict, a fair fight between poetical abstractions and reality, fidelity to time and the environment would be key.\n\nWilliams collected maps of the Pike's Peak region in 1860 showing the locations of wagon trails, cattle drives, and towns. The territories in those days were surprisingly unmarked by human habitation. If a rider left Fort Larned in Kansas, for example, heading north, and stopped for supplies at Fort Kearney in Nebraska before continuing on to Fort Randall in the Dakota Territory, he could go for days without seeing another human being. Williams chose a working title that reflected the vastness: _The Naked World_.\n\nBut there were great interior seas of buffalo. \"Tremendous numbers,\" Williams wrote in his notes, \"whole landscape sometimes seemed a mass of buffalo.\" An estimated 20 million buffalo lived on the western plains in 1850, the most numerous single species of large wild mammals on Earth. An adult buffalo weighed about two thousand pounds, stood six feet high at the hump, and measured ten feet in length. By the 1840s, the market for buffalo meat and robes among whites was flourishing, and in the Kansas Territory during the 1850s buffalo were hunted for sport. In the winter of 1872\u20131873, more than 1.5 million buffalo hides were loaded onto trains bound for St. Louis, Chicago, Cleveland, and New York. The despoiling continued until by 1889 only 541 buffalo could be found alive in the United States, sheltering at the foot of the Rocky Mountains.\n\nA parallel exploitation had taken place in fiction, Williams believed, regarding how the story of the Western Movement had been told. He blamed the despoiling on \"literary racketeers\" and \"hired hacks,\" writers who were \"contemptuous of the stories they have to tell, of the people who animate them and of the settings upon which they are played.\" There were better westerns\u2014a recognized genre by the 1940s. Walter van Tilburg Clark ( _The Ox-Bow Incident_ ), A. B. Guthrie Jr. ( _The Big Sky_ ), and Jack Shaefer ( _Shane_ ) had developed moral complexity. But Williams was convinced that \"each of these novelists is, in his own way, guilty of mistaking the real nature of his subject,\" which should be the interior lives of those who went west. Many had \"no precise ideological motive for [their] exploitation\" of the land except to change their lives somehow. That experience, more challenging to show than adventure, could be better presented by a mythic quest, \"one that is essentially inner.\"\n\nIn literature, myth dares to enter areas that are ambiguous to the soul. The external world becomes a backdrop, offering no law or guidance to the wanderer. And in the American West, a cowpoke could become Ulysses on the illimitable sea\u2014a character driven by the rage to survive and to conquer the dictates of conscience. The outcome of myth, as the _Odyssey_ shows, \"is always mixed,\" Williams said, because \"its quest is for an order of the self that is gained at the expense of knowing at last the essential chaos of the universe.\"\n\nWilliams anticipated western revisionist novels such as Oakley Halls' _Warlock_ , Thomas Berger's _Little Big Man_ , Larry McMurtry's _Horseman, Pass By_ , and Cormac McCarthy's Border Trilogy. But after a year of research on his novel-in-progress, he moved with the English Department into a three-story, red-brick building that had been a student dormitory, and he lost most of his notes for _The Naked World_.\n\n\"You seem to be nicely settled. I guess we are too, as much as we ever are,\" the Marshes wrote, congratulating John and Lonnie on the birth of their daughter Pamela during John's first semester teaching. Butch was pleased to report that his luck in Ajijic, Mexico, was holding out. \"Blessed be the name of Ellery Queen,\" he said, praising the mystery magazine that had paid well for one of his stories. He was also selling science fiction, a new avenue for him, and the money was enough to keep the Marshes solvent for a few more weeks and to party with the expat writers and artists in town. George Rae mentioned that a friend of theirs, science-fiction writer Theodore Cogswell, whom John knew from the writer's workshop at the University of Missouri, would be applying to graduate school in English at the University of Denver. \"Tracy [his wife] and the kids are going to her family or some damn place,\" she said, indicating that she knew him well enough to know of his marital problems.\n\nIt was hard to forget Ted Cogswell once you met him. Handsome, unserious, and offbeat, except when he was depressed, he had joined the Young Communist League (YCL) when he was seventeen and shipped out on the _Aquitania_ for Spain in June 1937, to serve in the Spanish Republican Army. He returned after two years (\"Most of the time I was hauling cabbages,\" he reported), completed college, and enlisted in the Army Air Force during World War II, assigned to airfields in India, Burma, and China. His early science-fiction stories, published during the 1950s, proved to be his most popular, beginning with \"The Spectre General\" and \"The Wall Around the World.\" When Butch and George Rae met him in Mexico, he was an instructor at the University of Minnesota, preparing to motorcycle to the 1953 World Science Fiction Convention in Philadelphia, which would be attended by \"all the Young Turks,\" including Kurt Vonnegut, Robert Silverberg, Harlan Ellison, Isaac Asimov, and Robert Sheckley.\n\nThat the Marshes would know a maverick like Cogswell, or be going to bullfights with Norman Mailer, was part of living in Ajijic and why they loved it. The pretty lakeside town had been popular with writers since the 1920s. D. H. Lawrence had written _The Plumed Serpent_ there, and Somerset Maugham, _The Razor's Edge_. In the 1940s, Tennessee Williams played poker many nights at the Old Posada hotel, an experience that became part of _A Streetcar Named Desire_. George Rae hoped their house might hold a bit of literary kismet for them, too. \"Another bit of interesting anecdote,\" George Rae wrote:\n\nIt was given out by our landlady a few days ago that [Robert Penn] \"Red\" Warren wrote _All the King's Men_ in the place where we are living now. She said he was a \"nice person with red hair who drank a lot\u2014and gave wonderful parties.\" She had no idea of what she did to me\u2014I'm still going around sniffing the same air and touching the same walls as Robert Penn Warren and Butch is hoping that some of his wonderful writing will carry over to his novel.\n\nBeing in a small community of writers, Butch and George Rae were just as likely to end up as characters in somebody else's novel, which they did. Ellen Bassing's 1963 novel set in Ajijic, _Where's Annie?_ , features Butch as \"Willie Chester\" and George Rae as \"Sam.\" One afternoon, a passerby in the novel peeps in on them while the _chicharras_ (locusts) are clicking in the heat:\n\nShe paused and looked around. She was almost at Willie Chester's house. He was enshrined there on his patio only half hidden among the tel\u00e9fono vines, typing away. He wrote. Merciful God, how he wrote. A story every day he said, good, bad, indifferent, sensational, like a nondiscriminating machine, learning, he said, with each one he wrote, but writing them so fast, so terribly, frighteningly fast. And he sold some of them, not many. That he sold any was alarming. He had no reverence, no respect, no fear of his own possible or impossible talent. He wrote; it was the answer to everything for him. And there, in the patio behind him, was his wife . . . Sam. Of course Sam wasn't really her name\u2014that is, her parents didn't name her Sam when she was born\u2014but it was the name she had chosen for herself and which she requested everyone to use. It wasn't used very often, because nobody ever had anything to ask of her and if anybody did, he just called her \"you.\" Sam was behind Willie, circling about in a stained and tattered leotard, steadily but badly practicing her ballet. Did she woo and win him with her twittering, soiled dancing? Oh, turn my eyes from the vision of their lives . . .\n\nButch would retaliate later, in _A Week with No Friday_ , by turning Ellen Bassing into Martha Blissing, \"a lady novelist with a lousy memory.\" Her husband, Beau, a \"fairly entertaining slob,\" has poodles he can't afford to feed and is possessed of one gift: he can sing \"Blue Skies, Nothing But Blue Skies,\" backward.\n\nLosing his notes for _The Naked World_ turned into \"a most fortunate accident,\" Williams realized when he began writing the first chapters. \"I might have tried to force what I had learned into the novel; as it was, I simply wrote the novel, and I found that when I needed a fact, it would come to me,\" he later said. \"In other words, the accident forced a method on me, and the information was, I hope, wholly subservient to the more important concerns of the novel.\"\n\nAs a counterpoint to Emerson and the transcendentalists\u2014the inspirations for his young hero going west\u2014he had been reading American authors who treated nature as an indifferent presence\u2014neither opposed to, nor supportive of, human hopes. He admired Stephen Crane, Jack London's _Call of the Wild_ , and especially Willa Cather's _O, Pioneers!_ and _My Antonia_. Cather's desire to suggest \"the inexplicable presence of the thing not named, of the overtone divined by the ear but not heard by it,\" evoked the ancient Greek stage, where forces play out, but are not seen. Naturalists such as Cather didn't seek cause behind events: they found truth from depicting life faithfully. She said of _My Antonia_ \u2014the story of several immigrant families becoming Nebraska settlers, \"In it there is no love affair, no courtship, no marriage, no broken heart, no struggle for success. I knew I'd ruin my material if I put it in the usual fictional pattern. I just used it the way I thought absolutely true.\"\n\nThe naturalists' influence on Williams was apparent when he sent a completed first draft of _The Naked World_ to Janet Lewis at the end of the summer of 1958. \"It struck me that the method you use is that of Crane in the 'Open Boat,'\" she replied. \"You have characters who exist only in the moments of the story, of the immediate account of events. They have no pasts and no futures. This makes the events very intense, and that is exactly right for both experiences. In fact it would seem to be the main point in both experiences. . . . Congratulations. It is the most extraordinary reversal of procedure from your first novel.\"\n\nWilliams' protagonist, Will Andrews, drops out of Harvard in the 1870s and, armed with a small bequest, travels from Massachusetts to Kansas and the desolate town of Butcher's Crossing, a tiny, hardscrabble hamlet. Will is brimming with Romantic ideas about nature, and he's eager for an adventure that will unlock \"the Wildness\" in his \"unalterable self.\" He wants to become \"a part and parcel of God, free and uncontained.\" Shortly after his arrival, looking for adventure, he impulsively agrees to bankroll a man named Miller, a seasoned hunter, who promises to lead one of the last great buffalo hunts. With them will go Charley Hoge, a wagoner, to haul supplies, and Fred Schneider, a skilled buffalo skinner. Hoge's right arm ends at the wrist, where he lost a hand to frostbite, an omen of what awaits. They saddle up and Miller leads the expedition over the prairie. Yet the spiritual experience Andrews was expecting is nowhere to be found. He feels numbed by \"the routine detail of bedding down at night, arising in the morning, drinking black coffee from hot tin cups, packing bedrolls upon gradually wearying horses,\" the rituals that mark the passing of one monotonous day, and then another. At last, Miller finds the secret pass he was looking for\u2014a sort of chink in the ordinary world\u2014that leads into a Colorado valley that seems to be known only to God:\n\nA quietness seemed to rise from the valley; it was the quietness, the stillness, the absolute calm of a land where no human foot had touched. Andrews found that despite his exhaustion he was holding his breath; he expelled the air from his lungs as gently as he could, so as not to disturb the silence.\n\nMiller tensed, and touched Andrews's arm. \"Look!\" He pointed to the southwest.\n\nThe buffalo are innumerable, a dark, undulating shadow as if cast by a passing cloud on the sunlit grassland. Miller becomes obsessed with killing every last one of the animals for their hides, and after weeks of shooting and skinning, the valley becomes an abattoir of carcasses, while \"the remaining herd strayed placidly among the ruins of their fellows,\" until nearly all the creatures are dead and Miller is sated. Then, from the clouds, a single snowflake portentously falls\u2014the only hint of supernatural agency that Williams will allow\u2014heralding an apocalyptic winter that arrives within hours. The men are trapped for months under the snow. Hoge, already prone to religious fever, shows signs of going mad. The men survive, but Schneider, the skinner, is killed crossing a river on the way back; the wagon, piled obscenely high with hides, falls over into the water's torrent and gets washed away. The uselessness and cupidity of the whole venture is underlined when they reach Butcher's Crossing and learn that the buffalo hides are worthless\u2014the bottom has fallen out of the market from oversupply. Miller makes a pyre of hundreds of rotting hides, stored in a corral, and sets them afire, the burning stink and sparks rising into the night like an Old Testament sacrifice to appease the anger of Jehovah. Looking on, an easterner appalled by the irrationality and chaos, cries out at the destruction. \"Go back,\" Miller tells him evenly. \"Get out of this country. It doesn't want you.\"\n\nWill Andrews, purged now of sentimental notions about nature learned from books, doesn't want to accept that he's been lied to:\n\n\"'No,' Andrews said. A vague terror crept from the darkness that surrounded them, and tightened his voice. 'That's not the way it is.'\"\n\nWilliams started looking for an agent. _The Naked World_ had taken four years to write, his most sustained effort. For some reason, he turned for advice not to his university colleagues, or to Alan Swallow, but to someone whose situation he identified with: Morton M. Hunt, a former Army Air Corps pilot he had been friends with in Burma, who was now freelancing in New York. \"About an agent for you,\" Hunt replied. \"There are a number of large agencies with glittering names, to which I might recommend you, but I know about them only their general reputation, and have no personal intimate contacts with most of them. I prefer to suggest to you a smaller agency, where I am personally well acquainted with the principal partner, and know her to be a woman of considerable sensitivity, charm, and creative ability.\"\n\nHer name was Marie Rodell. \"She is utterly ethical, well thought of in the business, and perfectly capable of handling contract details, Hunt told him.\" Hunt had already spoken to her by phone, in fact, before responding to Williams, and she had said she would be delighted to read the manuscript.\n\n\"Wild man, wild,\" Butch wrote when he heard the news.\n**CHAPTER TEN**\n\nFiasco\n\n_I'm almost physically sick . . . to be classified as a \"Western\" novelist could be damn near ruinous to me as a teacher and a scholar. And don't think the gleeful word won't get around_.\n\n\u2014JOHN WILLIAMS, 1960\n\nMarie Rodell had been an agent for ten years when she received Williams' manuscript for _The Naked World_ in 1958. She was \"world-wise, well-traveled, a sophisticated New Yorker fluent in four languages who moved comfortably in many of New York's most elite literary and publishing circles,\" said a biographer of Rachel Carson, author of the ecology classic _Silent Spring_ and one of Rodell's clients. Rodell had been an editor at several large houses until after the war, when layoffs of women in publishing, to make room for veterans, forced her to strike out on her own. Briefly, she'd been married to a playwright. Because the statuette of Edgar Allan Poe awarded to her by the Mystery Writers of America reminded her of her ex-husband, she kept it in a closet. Attractive, of average height, with auburn hair that she wore upswept, she was a poker player, chain smoker, and gourmet cook. She was comfortable around men\u2014who liked the contrast between her femininity and her bawdy sense of humor\u2014but her views about men and women were conservative, making her a good match for the traditionally minded Williams. Rodell had resigned as Betty Friedan's agent after reading _The Feminine Mystique_ in manuscript. \"I think she was threatened by my book, as many women were,\" Friedan later wrote. \"She was a great devotee of Freud and she wanted me to throw out the whole chapter on Freud.\"\n\nShe was not an aggressive agent. She used a light touch with clients' work\u2014reviewing their manuscripts, and offering her reactions and suggestions here and there for improvement, but leaving the close editing of style and syntax to publishing house editors. Her job was selling the book, and she liked coming up with catchy titles. \"She does not fight or argue quite as fiercely as some for the last inch of advantage,\" Morton Hunt had written to Williams, \"but I don't conceive that to be important to you. When you sell 150,000 copies of the new book and are ready to do another, that's when you have to do the hard fighting, not now.\"\n\n\"Dear Mr. Williams,\" Rodell replied in October after having his manuscript for a month, \"Both Miss Daves [her assistant] and I have read _A Naked World_ and like it very much. You have made a meaningful as well as a highly readable novel of the story, and we'd be delighted to represent you and it.\" Williams replied that he was pleased because he was convinced _A Naked World_ was \"something above the run-of-the-mill novel.\" She asked what else he might be working on. He was reluctant to say because it was hard to summarize. The next one would be about a college professor:\n\nTo all outward appearances, he is a failure; he is not a popular teacher; he is one of the less distinguished members of his department; his personal life is a shambles; his death by cancer at the end of an undistinguished career is meaningless. But the point of the novel will be that he is a kind of saint; or, stated otherwise, it is a novel about a man who finds no meaning in the world or in himself, but who does find meaning and a kind of victory in the honest and dogged pursuit of his profession.\n\nIf she had any reservations about a novel by a college professor whose hero was a failed college professor dying of cancer, she didn't say. Instead, Rodell responded warmly to the news that he was already working on his next book.\n\nSeeing John go from strength to strength in his career, Butch had decided perhaps he should give teaching writing a try, too. Having hit a dry spell at the typewriter in Ajijic, he wrote, \"Poverty can be amusing, if you like your humor on the ironic side.\" Not that his \"weirdly precarious existence\" as an independent writer for ten years had been fruitless. He had a dependable market selling to pulp and men's magazines such as _Playboy_ , _Nugget_ , _Dude_ , _Swank_ , and _Rogue_. His smart-alecky, dame-wise characters\u2014projections of himself, really\u2014were suited to male readers. His better work\u2014mysteries, science fiction, and domestic dramas\u2014had appeared in two best-of-the-year short story anthologies, including Random House's _Saturday Evening Post Stories 1954_.\n\nBut for George Rae's sake, he believed it was incumbent on him to finish his bachelor's degree as a precaution. She had been able to \"enjoy leisure with poverty\" long enough. With the help of an adviser at the University of Iowa\u2014Paul Engle, soon to become famous as the director of the Iowa Writers' Workshop\u2014Marsh's three years' worth of college credits put him within a semester of receiving his undergraduate degree. But Engle, seeing an opportunity to add an experienced, published writer to his faculty, also offered him a paid assistantship teaching creative writing, provided he stayed to complete his masters. The only other \"real professional\" on the faculty, Marsh discovered, was Vance Bourjaily, author of _The End of My Life_ \u2014a respected postwar novel.\n\nSo September 1958 found Willard Marsh\u2014former jazz musician, bohemian, and expat writer\u2014trudging across the Iowa campus as a college instructor. To students, he appeared to be a slightly pudgy, shorthaired, mid-thirties gentleman with a stern expression, wearing thick glasses and letting a cigarette hang from the corner of his mouth. For his brother-in-law's delectation, he described a normal day:\n\nI rise at a civilized hour, unhungover, don a clean shirt and even tie, walk a few clean streets of East Overshoe (unlittered by burro shit and drunken Mexicans) to a room on the campus where twelve to twenty eager youths want to learn how to sell their first short story to the _Saturday Evening Post_ and\/or the _Yale Review_ , and thereby instantly and perpetually become that dashing figure, The Professional Writer.\n\nI figure I'm just the cat to show them how. I don't know who was teaching those things at East Overshoe before I came along, but compared to me, he doesn't know his ass from a hot rock. Could _he_ tell them how to con an editor into thinking he's seeing a rewrite of a story he rejected last year? Could he even know which editor should see that story in the first place? Get in and get that big slick dollar, children, like I used to before I became as dated as a celluloid collar; screw that arty-farty literary twaddle. Get in there where the tall cane grows.\n\nUnlike Williams, Marsh didn't think he could ever become \"a very nice, dedicated prodigy-Ph.D. like yourself.\" His ambition was to teach well, get the master's degree, and then return to Ajijic as soon as possible, where he could sit at his desk every day and listen to how his typewriter blended with the _chicharras_ buzzing outside, while in the kitchen George Rae prepared another pitcher of margaritas with lemons and limes, rattling with ice cubes.\n\nWilliams' hopes for _A Naked World_ received an unexpected boost when he received a call from the senior editor at Macmillan, Cecil Scott, saying he happened to be in Denver\u2014could he have a look at the novel? Morton Hunt, who had recommended Rodell as his agent, must have tipped him off, because Rodell hadn't sent out the manuscript yet. Williams said he could, but he instantly wrote to Rodell as a precaution. \"I made it quite clear to him that you were handling the book, and that if he became seriously interested that he would have to work through you.\"\n\nScott, an erudite Londoner, read about forty pages and seemed excited, Williams thought. A few days later, a note from him arrived, written on the stationery of the posh Hotel Adolphus in Dallas. He had forgotten to mention that Macmillan was offering a new fiction award worth $7,500. _A Naked World_ would make an excellent candidate.\n\nIt was a bit overwhelming: unexpected compliments from a major publishing house, plus the chance at a prize worth two years of his salary as a college professor. Rodell assured her anxious client there was nothing to worry about. Scott had phoned her, wanting to know whether any other houses had seen it. She told him cagily that \"the manuscript was under consideration elsewhere,\" but she would note that Macmillan was interested. \"They [Macmillan] are very far from my favorite house,\" she informed Williams. \"The president retains the attitude most publishers had fifty years ago (but have since been persuaded out of it) that if his house wants to bestow the grace of its imprint on an author's work, that should satisfy him. . . . Since there are many other good houses who don't share this attitude, I stay away when possible.\" Williams, never before having reached this level in his writing career, put his faith in his new agent; apparently she was a cool customer when it came to doing business: \"I do not wish to urge a judgment of mine in an area where you are more knowledgeable,\" he told her.\n\nTwo months passed, but other publishers didn't share Cecil Scott's enthusiasm. \"He may be a writer,\" replied an editor at W. W. Norton. \"The main problem, it seems to me, is that the hero's motivation is not made fully clear. The author simply attributes to him a vague yearning to find the meaning of life, not much better defined than that.\" _Nothing But the Night_ had been criticized for the same kind of shortcomings. In fact, at Scribner's, Harry Brague, who had also disliked _Nothing But the Night_ , rejected _The Naked World_ because it, too, seemed \"watery.\" A rejection from an editor at Viking took a devastatingly condescending tone, as if he were holding the manuscript away from him with a pair of tongs. \" _A Naked World_ is not a novel. . . . In fact, the book is almost completely static. The hero, William Andrews, leaves Harvard and comes west, and we don't know why. He goes out buffalo hunting and all he learns is how to skin one, and then he leaves we know not whither. I hope Mr. Williams will one day decide to tell just a story.\"\n\nMystified, Williams wondered what he should do. He had spent almost four years on the novel\u2014time that might have been spent submitting to journals and advancing his academic career. Thinking that perhaps he should retreat to firmer ground, he asked Rodell if she'd like to see his doctoral dissertation, \"The World and God: The Poems and Dramas of Fulke Greville.\" Greville, the Elizabethan poet and courtier, hadn't received \"a book-length study\" since 1903\u2014\"and that one was quite inadequate,\" Williams urged.\n\nRodell was not enchanted by the idea of trying to sell a book about a minor poet in tights, a doublet, and feathered hat. She asked instead if he had any short stories they could offer to magazines such as _Esquire_. He mailed half a dozen, adding, apologetically, \"I don't feel that I write very good short stories. I have never taken the form with the seriousness I'm sure it deserves. I'm afraid I reserve my best energies for my novels.\" In the meantime, his novel about the college professor had come to a halt, he said, because he was under \"academic pressures.\" Likely, he was getting the jitters and couldn't write without some good news.\n\nRodell advised giving editors another month to reply. Then they should take advantage of Cecil Scott's interest and enter Macmillan's fiction contest. As a courtesy, they would defer any offers until the winner was announced. It would be a gamble. Macmillan might not offer a contract if they lost, but those were the stakes.\n\nWilliams began having doubts about the novel's title. Perhaps _A Naked World_ was a bit existential-sounding\u2014too pretentious. Maybe just _Butcher's Crossing_ , the name of a town, would be better. He asked Cecil Scott's opinion, who replied, \"Frankly, I do not like _Butcher's Crossing_ as well as _A Naked World_. It seems to me a good title. Obviously the jacket would give some clue to the subject matter and I would prefer _A Naked World_ because it would remove any suspicion in the prospective reader's mind that this was another conventional Western. So for the present at least I would prefer to leave the original title.\" With the results of the contest only a week away now, Williams decided to leave well enough alone.\n\nOut of two thousand manuscripts submitted to the Macmillan contest, _A Naked World_ came in second. To Williams, Rodell dashed off, \"I'm so sorry!\" And during the judging, the publisher Little, Brown and Company had made an offer higher than Macmillan's. Cecil Scott counteroffered with even better terms, because \"we have the most substantial faith not only in this book but in the author's future.\" Rodell urged Williams to accept the Macmillan offer. Or perhaps she didn't\u2014it was hard to tell. She tossed the ball back to Williams. \"I do not want my own prejudices against Macmillan to influence you unduly.\" Comparing the two publishers, she said, \"Both are honorable imprints. Enthusiasm at Macmillan seems to be more unanimous than at Little Brown.\" Williams, a novice, \"rather dazedly\" agreed, his only concern being that Macmillan make a sincere effort to promote the book, even though it had come in second, \"to give it a chance to gain some attention. . . . I presume they will not neglect that.\"\n\nFor a second opinion, he turned to Janet Lewis, who complimented him by pointing out that her advance from Doubleday for _The Ghost of Monsieur Scarron_ was much less than what Macmillan was offering him, and her novel just missed becoming a Book-of-the-Month Club selection. In the end, she said, publishing is \"something of a gamble always.\" She wished him well. That settled, he accepted Macmillan's offer, his first time doing business with a major trade publisher.\n\nWith _A Naked World_ slated for publication in the spring of 1960, still eight months off, he returned to his next novel, about a mid-western college professor. The time needed to plan it would be about a year, he estimated, and a prestigious Guggenheim Fellowship, awarded to those \"who have demonstrated exceptional capacity for productive scholarship or exceptional creative ability in the arts,\" would relieve him of teaching. He decided to ask J. V. Cunningham, now at Brandeis University outside of Boston, a previous recipient of a Guggenheim, if he could use his name as a reference on his application.\n\nThe deferential tone of the letter is uncharacteristic for Williams, who was thirty-eight at the time and an associate professor after six years at the University of Denver. But having Cunningham's respect was important to him. \"I realize that you don't know a great deal about my abilities as a novelist\u2014I assume that you have not read my first novel, published by Alan, and I fervently hope that my assumption is correct; it isn't a very good novel, and it certainly does not adequately represent, or even foreshadow, what I am trying to do now.\" Not immodestly, he mentioned that Janet Lewis compared him to Stephen Crane. Cunningham assured him of his support.\n\nIt was the start of some good luck. The University of New Mexico invited him to apply for a position in the English Department. He supplied a two-page autobiographical sketch listing his work in _The American Scholar_ , _Arizona Quarterly_ , and _A Journal of Modern Culture_ , among others. Having a novel that was scheduled for publication looked impressive, too, by any measure. Then came a letter from a former colleague and friend from the University of Missouri, who said he would be happy to consider Williams' dissertation on Fulke Greville as a scholarly book for the university press.\n\nHe seemed to be reaching a milestone, a turning point. To celebrate, he went camping in the Rockies, ten thousand feet up and off a dirt road, about a mile from Clear Lake in the Arapahoe National Forest, near a ghost town. While he was camping, he would \"descend briefly a couple of times a week\" from the mountains to check on the family, and then he was off again. He didn't clear his plans with Lonnie because seeking permission from his wife was not in his nature: where he went, and how long he would be away, was his prerogative. He stayed the latter half of June and all of July, fishing in the Cherry Creek reservoir and cooking his catches over an open fire.\n\nCome September, he went to New York to meet Marie Rodell in person for the first time, and to have lunch with Cecil Scott. They were still dickering about the best title for the novel. Rodell didn't care for _A Naked World_. \"I'm inclined to think the word 'Hunt' should be in it and perhaps 'Valley'? No. 'Lost Valley?' Better. If it sets up in book buyers a subconscious association with _Lost Horizon_ it can't hurt sales!\" Scott was pulling for _A Naked World_. Williams offered _The Crossing_ , _The Hunt_ , _Hunt in the Valley_ , and _The Western Path_. \"I don't like _The Western Path_ ,\" Rodell cautioned, \"because I think we have to be careful never to let careless reviewers or bookstore personnel classify this as a Western.\" Scott, wanting to bring the discussion to an end so he could begin thinking about marketing, told Williams, \"Personally I like the best of any of your suggested titles, and better than any title I have come up with myself, _Butcher's Crossing_ . . . _Butcher's Crossing_ on the right kind of jacket will give the reader a certain clue to the contents. It is a phrase which is easy to remember and it is also mellifluous.\" Scott, at last, won the day, and he accompanied Williams over to the Macmillan offices on Fifth Avenue to meet the staffers who were already working to meet the _Butcher's Crossing_ publication date set for late March.\n\nA few weeks before its release, Williams took a sentimental trip. He drove 875 miles south from Denver to Clarksville, Texas, intersecting with the fictional paths of his four buffalo hunters in _Butcher's Crossing_ across Kansas and into the Colorado mountains. In Clarksville, he located the house where he had been born, and walked the brick streets that his mother and father had known. In Paris, Texas, the first town west of Clarksville, the newspaper editor heard of his return and ran an article about the exciting career of \"Dr. Williams.\" An elderly gentleman in Clarksville, who had known the Williams family well, felt moved to write to him in a crabbed hand, congratulating him on his \"wonderful achievements in the educational field,\" adding, \"Your Aunt Emmie [Walker] was a wonderful teacher, also.\" On the eve of publication, advance orders of _Butcher's Crossing_ to bookstores exceeded everyone's expectations.\n\nMarie Rodell broke the news. \"Brace yourself for a bad shock next Sunday. The _New York Times_ has reviewed _Butcher's Crossing_ as a western, and the idiot who writes the column has of course no small inkling of what the book is about.\" She was furious. \"It's too, too revolting. I told Cecil that if the reviewers had been informed this was the runner-up in the contest, and if the jacket hadn't looked like a western, this wouldn't have happened.\" On the cover, a man and woman in Hollywood western dress stand in the foreground; behind them, far below on the plain, a wagon train of pioneers passes by. The cover announced, \"Through blinding heat across an unyielding land they trekked. Men in search of a burning vision.\" There was no trek in the novel; it was a buffalo hunt. There was no vision, just the desire to get rich from buffalo hides. There was no Conestoga wagon train with settlers walking beside it, only four men with a string of packhorses behind them. And the only woman in the story is a prostitute who appears at the beginning and at the end.\n\nThe review, by Nelson C. Nye, founder of the Western Writers of America (his pen names were Clem Colt and Drake C. Denver), damned the book. It \"is practically plotless, an account of four men who go out to hunt buffalo, find them, slaughter them and are caught by cold weather. The work abounds in graphic descriptions, even to the variegated colors of blades of grass. The story, however, contains little excitement and moves as though hauled by a snail through a pond of molasses. You can leave it anytime, a lot of people will.\"\n\nWilliams was horrified. The damage was irreparable and he knew it. A review carried by one of the most respected newspapers in the United States\u2014a source that booksellers and buyers relied on\u2014had branded _Butcher's Crossing_ an unreadable bore. \"I'm almost physically sick,\" he wrote to Rodell. \"I have just received your letter, have thought about it for an hour, and I get sicker by the moment. God knows, I might be angry at a bad, imperceptive review\u2014but it would be anger and nothing more. But this is really inexcusable\u2014of the _Times_ , of course, but mainly of Macmillan. It would have been much better had the book been entirely ignored\u2014at least it would have been better from my point of view.\" The situation was a \"great deal more serious than it might appear.\" As the director of Denver's creative writing program, he had ventured into mainstream publishing, confident that it was within his power to compete with the best contemporary novelists. He had intended to revive a genre, to contribute to literature. Instead, \"to be classified as a 'Western' novelist could be damn near ruinous to me as a teacher and a scholar. And don't think the gleeful word won't get around after next Sunday. One might say that anyone who read the book would realize the falsity of the classification\u2014but who the hell's going to read it after such a tantalizing invitation not to?\" He was heartbroken, and if had the money, he would purchase every copy and destroy them.\n\nAt Macmillan, Scott took the unusual step of objecting strongly to the editor of the book section of the _New York Times_ , arguing that the novel should not have been reviewed in a column about westerns. \" _Butcher's Crossing_ is not a 'Western' although its background is the early west, any more than [A. B. Guthrie Jr.'s] _The Big Sky_ is a 'Western.'\" Paul Engle of the Iowa Writers' Workshop, Scott pointed out, had called it \"a superb tale of the Old West\" on the front page of the book section of the _Chicago Tribune_ , and the _Denver Post_ had hailed it as \"one of the finest novels of the West ever to come out of the West.\" Scott commiserated with Williams by inveighing against idiot reviewers, and the tendency of people to suspect anything new and different, and tried to persuade him that no serious readers would really pay attention to Nye's column anyway. \"The thing you must simply not allow yourself to be is discouraged.\"\n\nBut the novel's chances were ruined. The summer passed and _Butcher's Crossing_ sold only a few thousand copies. Come fall, the Guggenheim Foundation turned down his application for a fellowship, leaving him to wonder whether there was a connection between being scorned in the _New York Times_ and not receiving the award.\n\n_John Edward Jewell, later to become John Williams, with his mother Amelia, about 1924_.\n\n_Williams' birthplace, Clarksville, Texas, when he was a child. The statue in the square is of a Confederate soldier facing north in perpetual defense_.\n\n_\"The Corner,\" in Wichita Falls, Texas, where land speculation and oil deals were made on a handshake. Williams' father, John Jewell, would have known this place well_.\n\n_Williams at Hardin Junior College in Wichita Falls, wearing a silk scarf out of admiration for Ronald Colman, who played Sydney Carton in the 1935 film version of_ A Tale of Two Cities.\n\n_For a short time, and during his first marriage, Williams pursued a career in radio news broadcasting under the name \"Jon Williams\" in Denton, Texas_.\n\n_Williams served as a radio operator aboard US Army Air Corps cargo planes in northwestern India, flying \"the Hump\" over the Himalayas into China_.\n\n_Yvonne (Stone) Woolf, shortly after her divorce from Williams in 1949. She married Beat writer Douglas Woolf that year_.\n\n_George Rae, Williams' sister, and her husband, Willard \"Butch\" Marsh, in a photograph taken for identification purposes during one of their frequent trips to Mexico_.\n\n_Florence Roberts, whose husband was an English professor at the University of Missouri, the setting for_ Stoner _; John Williams; and Williams' third wife, Avalon Smith, during a vacation to San Miguel de Allende, Mexico, in 1954_.\n\n_Williams, when he was writing_ Butcher's Crossing _, shortly after joining the faculty at the University of Denver_.\n\n_Buffalo hunters in Texas about 1870_. Butcher's Crossing _, published in 1960, was dismissed as a Western, instead of what Williams intended: a debunking of Emersonian ideas about nature and the frontier_.\n\n_Poet and classicist J. V. Cunningham, a friend of Williams' and the model for professor Bill Stoner in_ Stoner.\n\n_The 1966 faculty of the Bread Loaf Writers' Conference_. Rear, left to right _: William Sloane, John Frederick Nims, William Hazlett Upson, David Wagoner, John Aldridge, X. J. Kennedy_. Center, left to right _: Brock Brower, Dan Wakefield, Eunice Blake, John Ciardi, Seymour Epstein, William Lederer_. Front, left to right _: John Williams, Robert Pack, Edward Alexander Martin. The conference was the first time Williams was in the company of other fiction writers and poets, and it helped assuage his loneliness. Robert Pack led a small countercultural revolution that overthrew most of the old guard, including Williams, by 1972_.\n\n_Williams in his office, a converted dormitory room at the University of Denver, in 1967, two years after publishing_ Stoner.\n\n_Williams and his fourth wife, Nancy, in the early 1970s_.\n\n_Williams near the end of his life. Novelist Joanne Greenberg (_ I Never Promised You a Rose Garden _) said, \"He had a face like a five-day rain.\"_\nPART III\n\n_Stoner_\n**CHAPTER ELEVEN**\n\n\"It Was That Kind of World\"\n\n_They wanted to know who was I, apart from papers and credentials\u2014\"Can you do it? Can you do the work? Could you teach and inspire students?\"_\n\n\u2014ROBERT D. RICHARDSON\n\nJohn's foray into mainstream publishing, where the landscape was familiar to Butch Marsh, left him bewildered. Likewise, Butch was finding that academe was another country. \"Being a faculty wife is a rather strange experience for me,\" George Rae confided to John, \"and here in the South it is a full time job. This place is crawling with women's clubs, literary societies (made of illiterates), and social groups. The sole occupation of the members of these groups is giving Teas, 'Coffee-drop-ins,' and 'little luncheons.' However, I'm holding up pretty well, and the people are really extremely nice and kind.\"\n\nShe and Butch had left the University of Iowa after he had completed the coursework for his masters, a few months before the publication _Butcher's Crossing_ , and relocated to Rock Hill, South Carolina, where he was now an assistant professor of creative writing at Winthrop College. Their experiment with college teaching was continuing as they weighed the benefits of security versus living paycheck-to-paycheck from Butch's fiction. He had stockpiled about fifty stories, but was too busy teaching to submit them. He fantasized about a New York editor just dropping by their little apartment at 830 College Avenue and choosing the best for a short story collection. \"It isn't as if I'm never going to have a decent novel of my own someday, so I feel I'm worth the visit.\"\n\nThey had purchased a car, their first\u2014a two-door 1952 Nash Rambler, a favorite of suburbanites\u2014sleek, modern, and \"frighteningly cute,\" George Rae said. The wife of the English Department chairman had befriended them and was playing fairy godmother, loaning \"Mrs. Willard Marsh,\" as she was referred to, enough furniture, sofa pillows, and dishware to set up housekeeping. She took George Rae under her wing, \"helping me learn the ropes,\" as George Rae put it. \"She also tells me that I may begin to say 'no' to invitations. Thank heavens.\" On weekends, there was \"some very pleasant socializing with the younger, drinking segment of the faculty.\" The chairman's wife had made it quite clear that the Marshes could stay as long as they liked\u2014indefinitely. Ajijic receded into the distance.\n\nJohn could have told them that if Winthrop College was cliquish, that was to be expected; it was no different from the English Department at the University of Denver. Everyone belonged to a subset. In the early 1960s, for example, Elizabeth Richardson accompanied her then husband, the biographer Robert D. Richardson, to the University of Denver to begin his instructorship in the English Department. After they arrived, \"one or two other women made a real effort to include me,\" she said. \"But I had zero in common with them. We had young kids, so several times we went out to Cherry Creek Reservoir and sat in the sand while the kids played and most of the conversation was complaining about husbands. After a while that got\u2014well, I was a biology major and all those women were English majors, so it seemed I couldn't be part of a conversation that transcended those topics.\"\n\nMen in the University of Denver English Department\u2014the faculty was male throughout most of the 1960s\u2014had a different experience. Newcomers were invited to join a kind of band of brothers who were outdoorsy, drank hard, and were at the top of their game. Robert Richardson, a New Englander and Harvard-educated, was excited by the West as \"open country.\" For the job interview, he said, \"John did to me what he did to everyone\u2014he took me to a cabin with some faculty members and a few wives and we got drunk. It was a weekend retreat with a cookout, mostly men. They wanted to know who was I, apart from papers and credentials\u2014'Can you do it? Can you do the work? Could you teach and inspire students?'\"\n\nIt was hard not to warm to John if he liked you. \"Always very nattily dressed. Spiffy dresser,\" said Richardson. \"He was low-key, and loved to laugh. 'Hi- _lar_ -ious!' he would say, and 'And in this country, it was _bound_ to happen.' His small stature didn't matter to him. He felt no need to work against that. He was extremely kind.\" If John didn't like someone, he tended to be prickly, according to Fred Inglis, a cultural historian; more often he just ignored them. But when he felt sympathy for a person, he tended to extend his help and friendship as a kind of respect. A deciding factor was whether the person seemed genuine. Inglis came to Denver on a fellowship from Oxford University because John had worked out the details with his tutor, a mutual friend. As a Briton arriving at the foot of the Rockies, Inglis felt a little disoriented at first, but \"John and I got on instantly,\" he later recalled. \"He was good at that. The friendship was plainly stated, and you knew it was there\u2014you could just reach out a hand and touch it for reassurance without ever making a great to-do about it. He was somebody who lived at the front of a friendship.\"\n\nIt was not only the bonhomie of the department that induced English instructors to come to Denver, either. Williams offered them an opportunity to help build the \"Harvard of the West.\" He was developing a PhD program in creative writing that had the potential to become one of the best in the nation, requiring proficiency in two languages, passing an oral examination, and submitting a thesis\u2014in this case, a novel or a short story or poetry collection. As a result, said Richardson, there were \"bright, hard-working students\" in the program who earned the respect of instructors who taught English literature and literary criticism. \"The department ran with a kind of smooth unity.\"\n\n\"It was that kind of world,\" said Gerald Chapman, \"and John helped us keep it alive. He didn't care for the ponderousness at all, or any pretension would really turn him off quickly. But he loved people who were willing to pitch in and be there.\"\n\nThe ethos of the English Department, being all male, was in agreement about something else, too\u2014sex. Male chauvinism was taken for granted, and affairs, which included sleeping with students, weren't worth mentioning except as gossip. \"This was the decade when the east-to-west academic exodus was in full flood,\" wrote Harvard professor Daniel Aaron in his memoir, _The Americanist_ :\n\nPalo Alto and Berkeley had become gilded Botany Bays for middle-aged East Coast professors in flight from sagging marriages undermined by their irregular romances with students: The West Coast represented a paradigm shift, new love in a new clime. Here was the stuff for a spate of tragic-comic college novels with an archetypal plot: a professor falls in love with a young woman prettier, smarter, and more exciting than his shopworn wife, who has drudged for him, raised their brood in mean surroundings, and grown obsolescent in the process; the professor feels he has earned his eminence and can no longer deny himself what Providence has decreed, so he dumps the wife with varying degrees of anguish and remorse.\n\nDuring the summers, Williams left for weeks at a time to stay in the mountains where he wrote and fished, accompanied now and then by his third child, Jonathan, who was thrilled to wake up one morning, during hunting season, \"with about three inches of snow on us\u2014that was pretty much of an adventure!\" Williams purchased a cabin near the town of Pine, Colorado, in remote territory favored by hunters. By going up to the cabin on Thursday, he could get three days of writing done on the screened-in porch.\n\nLonnie was not in the picture for much of John's life outside of home. To the wife of a department member, the third Mrs. Williams came across as \"a Colorado girl, kind of small-town, bright, but not a very forward personality.\" Gerald Chapman \"didn't know Lonnie well. I had dinner at their home and I saw her on various occasions. She and John had some pretty bad quarrels.\" More than once, Lonnie had wanted to leave a faculty party, but John, drunk and annoyed with her, had wanted to stay. It was known that John was having an affair with Shirley White, the English Department secretary. The romance had been going on for about a year when one day, in the spring of 1960, one of Williams' students\u2014a tall, rosy-cheeked young woman in her mid-twenties\u2014asked him to sign her copy of _Butcher's Crossing_.\n\nHer name was Nancy Ann (Gardner) Leavenworth. She was living in Denver with her parents and her four children, having left her husband, James Leavenworth, who had suffered a nervous breakdown. Her father was a Denver elementary school principal, her mother a housewife, and Nancy and her sister had been raised in a conservative home and educated at local Catholic schools. During junior high, 1946\u20131947, she had attended extracurricular classes for young ladies that taught domestic arts, including how to speak politely and practice social graces that would suit her as a wife and mother. In 1950, she had enrolled as an English major at Grinnell College\u2014a small liberal arts college in Grinnell, Iowa\u2014partly because her parents were pleased that it was only an overnight train ride away. When she met James Leavenworth, he was a junior, not long out of the army, and working toward a degree in comparative literature. At the end of her freshmen year, they married after she became pregnant.\n\nJames showed promise as a writer, winning awards at Grinnell for original plays and musicals. He might have pursued a career in college teaching, or gone into the arts, but married with two children when he graduated in 1953, he accepted a job writing for General Electric's trade magazine in Western Springs, Illinois. After that, the Leavenworths relocated to upstate Schenectady, New York, to General Electric's huge, campus-like research facility, and then back to Illinois again\u2014typical of a young corporate couple on their way up after the war.\n\nIt was around that time that James' behavior and personality changed. His voice lost its highs and lows and he began speaking in a monotone. He shambled, heavy-footed, despite being tall and long-limbed, as if he were a stranger to his body. After losing his job, he became obsessed with a solution to everything. The Leavenworth family needed to escape to the mountains, where they could start over, he insisted. He would be a \"mountain man\"\u2014answering to no one, and he would provide by building a house, and they could live off the land, the way nature intended. To get ready for hardship, he began sprinkling dirt on his food, because that was what mountain men ate.\n\nNancy fled with the children on a train to Denver, having witnessed the \"cruelest, saddest thing\u2014he didn't know what was wrong.\" James followed them, wandering about the city helplessly, protesting, wanting the children back, until Nancy's aunt, a pediatrician, prescribed medication to quiet his mind, enough to allow him to return to his family home in southern Michigan. Their divorce was uncontested.\n\nBy the time Nancy enrolled in Professor Williams' poetry class in the spring of 1960, thinking she might finish her degree in English, Williams had entered a new, and rather splendid, sartorial phrase, perhaps to counter the damage from\u2014or in defiance of\u2014the review in the _New York Times_ that had labeled him the author of a terrible western novel. His new plumage stated unequivocally that he was an artist. He replaced his usual Windsor-knotted tie with a dark blue, polka-dotted cravat. For social occasions, around his small waist he drew a cummerbund, purchased in the men's clothing section at the Denver, the best department store in town, which he wore under a blazer (two-piece suits made him look petite). For the final touch, he added a dab of Brylcreem to his hair and combed it before the mirror up and away from his forehead into a pompadour. Before going out the door, he inserted a fresh cigarette into a short holder made of pear wood and brass. By fashioning himself as a cultured, sophisticated loner, like the Hollywood leading man of his youth, Ronald Colman, he restored his self-confidence.\n\nHe was a presence as he entered the classroom. \"John was a fairly small man,\" remembered a student, \"with dark hair, glasses, a creased face, and a salt-and-pepper goatee. Being a poet, I thought he looked like a bohemian Black Mountain Poet, perhaps Robert Creeley without the eye patch. . . . He had a certain flair, and an elegance of manner, though with a slight rough edge by way of Texas.\"\n\nNancy felt her instructor's big, booming voice conveyed important things. He read aloud from _Butcher's Crossing_ now and then, to illustrate his points about rhythm and sense. \"I loved to hear him talk about it,\" she said. \"I liked his authority in that it was all earned. He knew what he was talking about. And his wit, of course. It was never dull with John, ever.\"\n\nNancy signed up for as many classes taught by Professor Williams as the quarter-system schedule would allow, becoming dependable company in his classrooms, as if she were proof of her conviction about him and his talent. Reading Melville's _The Confidence-Man_ , she found a quote that fit young Will Andrews' experience in _Butcher's Crossing_. She copied it out: \"Aye, and poets send out the sick spirit to green pastures, like lame horses turned out unshod to the turf to renew their hoofs. A sort of yarb-doctors in their way, poets have it that for sore hearts, as for sore lungs, nature is the grand cure. But who froze to death my teamster on the prairie? And who made an idiot of Peter the Will Boy?\"\n\nShe took it to his office, along with her copy of _Butcher's Crossing_ to sign. He read the quote, smiled, and took the book from her hand. On the title page, he wrote, \"To Nancy, in memory of that mad, mad night on the Cap D'Antibes, when . . .\" and no more. He handed it back to her.\n\nShe understood that he was declaring his intentions.\n\nHe began courting her\u2014asking her out for coffee, coming over to her house to cook his specialty from his India days, chicken curry and rice. Afterward he would serve her tea and they would talk. He didn't mention Lonnie, and she didn't ask about her. Shirley White, the English Department secretary, had left to take a position at the University of New Mexico, but Nancy wanted to make sure that affair was over. He went to Albuquerque to break it off. On dates, John said hello to people occasionally, but he didn't introduce her, wanting to keep their relationship private. As in _Stoner_ : \"He had no talent for dissimulation, nor did it occur to him to dissemble his affair with Katherine Driscoll; neither did it occur to him to display it for anyone to see. It did not seem possible to him that anyone on the outside might be aware of their affair, or even be interested in it.\"\n\nOne afternoon while they were sitting in her parents' kitchen, Nancy confessed something that she had been trying to keep from him. She wanted to have his baby, but that wasn't possible, not with his three children and her four. \"I wasn't depressed, I was sad and passionate,\" she later said. He propped his elbows on the table and began telling her a story. \"It was about a perfect child\u2014ours. It was never born, it just materialized somehow and it was ours.\" After that, whenever he came over, he continued the story where he had left off. She didn't have to ask him; in the evenings when he came by to ask how her day had been, he talked about the baby. The child had a name and did funny, endearing things. After a while, she began to laugh when she knew exactly what the baby would do and how they would respond. \"He courted me for a year before we made love,\" Nancy recalled. \"I wasn't ready. But by the time the year was up, I was just insane about him. I adored the guy.\"\n\nProgress on the novel about a college professor named William Stoner was going \"slowly,\" Williams wrote to Marie Rodell in September 1960, \"but there's nothing to worry about, I think. I hope to have a draft by Spring.\" Professor Stoner was having a love affair with a graduate student named Katherine, moving him into an emotional sphere he had never before experienced. Williams' working title for the novel was _A Matter of Light_. Meanwhile, he had high hopes of publishing a collection of his poems, titled _The Shape of the Air_. He needed a victory after the debacle of _Butcher's Crossing_ ; and, as he told Rodell, \"I don't want to sound snobbish about this, but I have spent a good twenty years of my life in a fairly concentrated study of the problem of poetry and poetic method, and I know what I am talking about.\"\n\nWilliams refused to accept that the ideas and methods advanced by the up-and-comers such as John Berryman, Elizabeth Bishop, Gwendolyn Brooks, or Theodore Roethke were legitimate. Like other practitioners of the New Criticism, in \"poor simple Walt Whitman,\" as Alfred Kazin said of that school, Williams could find \"no Donne-like tension, paradox, or ambiguity.\" He greatly admired the 1930s modernist American poet William Carlos Williams, and endorsed his doctrine about the importance of concrete objects\u2014\"no ideas but in things\"\u2014because it dovetailed with Winters' emphasis on rationality. But further on the timeline of poetry he would not go. Consequently, as the 1960s bloomed, Williams' conservatism as a poet made him difficult to publish. Moreover, his convictions prevented him from becoming a better poet. To write verse well\u2014the kind that is sublime and flashes an image of a thing or experience on the reader's inner eye\u2014he would have needed to be more in touch with those Wordsworthian impulses that he resisted. If he felt them, he forced them and racked them into a frame with so many beats per line. Mastery of technique is not art, or there would be statues of forgers, counterfeiters, rock 'n' roll tribute bands, and plagiarists. Williams' insights about life and human nature, the empathy\u2014the _interiority_ of experience\u2014that made him a superior novelist, aren't evident in his verse. He maintained a distance from the material. He kept his greatest strengths out of his poetry.\n\nIt showed. \"I am very much afraid that our final verdict has been the same as with the earlier manuscript of poetry which you sent us,\" wrote Cecil Scott from Macmillan. \"There simply has not been the sort of enthusiasm on the part of our readers that would warrant our embarking on publication.\" The rejection annoyed Williams, who was still smarting about the fate of _Butcher's Crossing_. He wondered whether Macmillan's continued negativity about his poetry meant they should offer the new novel, _A Matter of Light_ , to another publisher. He expressed these thoughts to Marie Rodell, writing, \"I think they did rather a bad job in packaging [ _Butcher's Crossing_ ], that is to say, in a sense they misrepresented it, by the jacket, the jacket copy, and the advertising.\" But she demurred, saying it wasn't worth breaking with Macmillan \"just in order to get the poems published.\" Eventually, after rejections of his poetry collection had arrived from Harcourt Brace, Scribner's, Viking Press, and New Directions, Williams conceded that perhaps, as a last resort, they might try a small press.\n\nScott, realizing he was in a position to do a favor for an author who must be unhappy with him, assisted with a related project instead. Williams hadn't given up trying to publish his dissertation about Fulke Greville. But editors hadn't shared his conviction that a book focusing on a lesser-known Elizabethan poet, even edited for the trade market, was needed. Not even his friend and former colleague at the University of Missouri William Peden, editor of the university press, expressed much enthusiasm after he read the manuscript. \"Nothing but bad news,\" Peden wrote to him. \"Very regretfully, I have been unable to include your study of Greville in our publications schedule for the next couple of years.\"\n\nBut then Williams saw a practical way of folding Greville into a larger concept\u2014one that would both serve his career as a scholar and honor, at the same time, the portion of the Winters canon that dealt with the 1500s\u2014an anthology, _English Renaissance Poetry: A Collection of Shorter Poems from Skelton to Jonson_. He explained to Scott how, by being specific, it would fill a niche: \"The anthologies [currently] available are either critical collections based on what is essentially 19th century taste; or 'representative' collections which include snippets of virtually everything, important or unimportant; or 'period' anthologies designed primarily for lower division survey courses, and inadequate for anything beyond that.\" Professors who taught the Elizabethans would welcome it in their classrooms, he knew from experience. The era was a staple of upper-level and graduate courses\u2014and anthologies that became the cornerstones of instructors' syllabi could become classics in their own right, enjoying long and profitable lives in college bookstores.\n\nScott proposed the idea to Macmillan's college textbook department, but the editors, thinking that the collection would be just another paperback of best-loved poems, or something like it, decided the cost couldn't be justified. So Scott, unwilling to give up, went out of his way for Williams. \"Would you like me to drop a line to Pyke Johnson at Anchor Books?\" he asked. Anchor was an imprint of one of Macmillan's competitors, Doubleday.\n\nNo sooner had Williams given his permission\u2014literally within a few days of Scott contacting his fellow editor\u2014than an offer arrived. \"Your idea for an anthology of Renaissance poetry strikes us as a very interesting one,\" Pyke Johnson wrote, \"and one that we would be interested in publishing as an Anchor Book.\" He offered a generous advance\u2014better than many midlist novelists were receiving\u2014along with a first printing of ten thousand books, with an expectation of reaching thirty thousand. He only needed to know when the manuscript would be delivered to finalize a contract. Williams replied that he could finish it in a year\u2014midsummer of 1963, because he already had all the material. He was steeped in Winters, and relied, for classroom instruction, on his three important essays about sixteenth-century lyric poets, which had appeared in issues of _Poetry_ magazine in 1939.\n\nIt's easy to imagine him rubbing his hands with satisfaction. His admiration for Winters was constant, and his friendship with Winters' wife, Janet Lewis, indicated that Williams might reasonably count himself as part of the Winters Circle. If he published an anthology essentially canonizing the Elizabethan portion of the canon, then he would have done yeomen's work in the service of the \"sage of Palo Alto.\"\n\nWilliams exceeded his self-imposed deadline by six months, and the ease with which the anthology came together was the beginning of a tide of small but good things rising around him. To Marie Rodell he reported, in February 1963, that \"the novel is coming along so well that I hesitate even to speak of it. When I spoke to you in New York, I predicted optimistically a June 1 completion date for the first draft. Well, that is no longer optimistic; I will almost certainly have it done by then, maybe sooner.\" He walked in on his typist, a junior in history, while she was finishing \"typing chapter 15, and discovered great huge tears coursing down her cheeks. I shall love her forever.\"\n\nIn the English Department, the feeling of fellowship was running high. Gerald Chapman, in his new capacity as department chairman, wrote a letter of praise in support of Williams' application to attend a summer session on Elizabethan poetry at Oxford:\n\nProfessor Williams is, without question, one of the more brilliant artist-scholars in the Rocky Mountain region, and, some may think, in the United States. Only a very few in university circles can be a novelist, a poet, a research scholar, an editor, and a teacher, all with high excellence. Professor Williams is responsible for our having one of the two organized doctoral programs in Creative Writing throughout the United States. . . . He is currently finishing a third novel and planning a fourth. He has published poems, essays and reviews in almost every important journal in the country; his anthology _English Renaissance Poetry_ is being published by Anchor Books this spring.\n\nHis choice of words is telling. He characterizes Williams as an \"artist-scholar\" and lists his achievements beginning with his literary ones. Privately, Chapman didn't regard his friend as \"that much of a scholar.\" Later, he said, \"His anthology on poetry is an excellent one. But when we talked, he thought he knew more about the Renaissance than he really did. John didn't have a scholarly mind.\" In any case, Williams was accepted into the six-week summer program at Oxford. Chapman had sweetened the whole experience by giving Williams the fall quarter off to pursue his writing. After Oxford, Williams planned go to Rome for several weeks to think about another novel he was considering.\n\nThe trip was important for another reason, too. Lonnie would be accompanying him. Although the marriage was damaged, he felt obligated to share this reward with the person who had supported his career, regardless of the state of their relationship now. He booked passage from New York for the two of them, because the \"prospect of a few days on board ship, with nothing to do, is damned attractive right now,\" he wrote to Rodell. Besides, he said, he had developed a dread of flying.\n\n_English Renaissance Poetry_ was slated for publication in May 1963, a month before their departure for England. Williams' introduction to the anthology, written with students in mind, is an approachable, easygoing discussion, as if he were a professor giving an overview of what the class will cover. He explains that in Renaissance poetry \"there was a fairly conscious progression from one set of principles to another, from one method to another, throughout the century,\" and then sets out to examine those principles and methods. The introduction itself exemplifies the plain, or Native, style that Williams favors, because, in Thomas Wyatt's phrase, it's \"graven with diamonds, in letters plain.\" Williams dismisses the Petrarchan style because \"the rhetoric is very nearly the whole poem.\" Each of the twenty-three poets represented in the anthology receives a brief biography that is very interesting and colorful, preparing the ground for the verse to follow, and there are madrigals, too. It isn't just an anthology of verse: it's about writing poetry and the joy of reading it.\n\nFrom Anchor Books, Pyke Johnson flew out to Denver to meet Williams for the first time and convey his congratulations on the upcoming publication of the anthology. The two walked around campus, and Williams told Johnson the story of how his interest in Renaissance poetry had begun, principally due to his friendship with Alan Swallow and his reading of Yvor Winters' _In Defense of Reason_.\n\nIn mid-June, Williams took the train two thousand miles to New York and met Lonnie, who preferred to fly, at Idlewild Airport (now John F. Kennedy International). Ahead of him, he had mailed a completed draft of _A Matter of Light_ to Rodell at her office in Manhattan on East Forty-Eighth Street, just off Fifth Avenue. She was not at all sanguine about the appeal of _A Matter of Light_ , a college novel whose protagonist, the narrator admits on the first page, was eminently forgettable:\n\nStoner's colleagues, who held him in no particular esteem when he was alive, speak of him rarely now; to the older ones, his name is a reminder of the end that awaits them all, and to the younger ones it is merely a sound which evokes no sense of the past and no identity with which they can associate themselves or their careers.\n\nRodell's pessimism was not unjustified\u2014it was based on what people were reading. Novels at the top of the best-seller lists were richly adorned with description and dialogue\u2014lively stories that favored European settings: Daphne du Maurier's _The Glass Blowers_ ; Morris L. West's _The Shoes of the Fisherman_ ; and _The Moon-Spinners_ by Mary Stewart. Barring exceptions that year, such as J. D. Salinger's _Raise High the Roof Beam, Carpenters, and Seymour: An Introduction_ , and G\u00fcnter Grass's _The Tin Drum_ , Americans generally preferred fiction that prized plot over character, and trusted genres they recognized at a glance: romance, historical fiction, biography, history, and so on.\n\n\"I may be totally wrong,\" Rodell said, speaking as someone who had agented one of the most talked-about books of the previous year, Rachel Carson's _Silent Spring_ , \"but I don't see this as a novel with high potential sale. Its technique of almost unrelieved narrative is out of fashion, and its theme to the average reader could well be depressing.\" She knew the market and readers' tastes. _A Matter of Light_ was going against the grain of contemporary literature.\n\nWilliams, however, seemed not to be listening, replying:\n\nOh, I have no illusions that it will be a \"best seller\" or anything like that; but if it is handled right (there's always that [excuse])\u2014that is, if it is not treated as just another \"academic novel\" by the publisher as _Butcher's Crossing_ was treated as \"western,\" it might have a respectable sale. The only thing I'm sure of is that it's a good novel; in time it may even be thought of as a substantially good one. A great deal more is going on in the novel than appears on the surface, and its technique is a great deal more \"revolutionary\" than it appears to be.\n\nAnd so, with that, he prepared to sail.\n\nThe MS _Berlin_ had weighed anchor shortly before noon in New York Harbor on a clear, hot day in late June to begin its weeklong journey across the Atlantic to Southampton. As a diversion, Williams kept a journal during the crossing. He intended to relax by jotting down his impressions of what he saw and heard\u2014good practice for a writer. But strangely, strains of unpleasant feelings related to the World War II seem to have intruded on his thoughts.\n\nHe may have accidentally invoked them by choosing for the voyage exactly the same kind of composition booklets he'd used in Burma, and this was his first oceanic trip since shipping out for India in 1943. Warming up his pen by journaling, he noticed that shadows of the war seemed everywhere. \"I had known it was a German ship, but I had not really anticipated that most of the passengers would be German. But\u2014my God!\u2014they are,\" he wrote. A sense of history, and the rise and fall of power, had stolen aboard somehow. He caught a little boy goose-stepping up and down the corridor outside his stateroom door. The resentful expressions he saw on the faces of the ships' German laborers, \"the deck hands, lowest of the crew members,\" who picked up the cigarette butts and trash, made him think of the concentration camps: \"These are the men\u2014the sergeants, the corporals\u2014who did the physical shoving of the victims into the ovens\u2014laughing perhaps, making jokes\u2014while our ship's officers stood by and shook their heads regretfully, after they had signed the orders.\"\n\nA topic of conversation in the ship's baronial and high-ceilinged passenger lounge\u2014\"English style in oiled walnut and green leather, a peculiar kind of wilderness\"\u2014was John F. Kennedy's arrival in West Berlin that week. His refrain of \"Ich bin ein Berliner!\" within sight of the Berlin Wall, on the knife-edge of the Soviet hegemony, would be met with roars of acclamation from tens of thousands of West Berliners, some of whom, just twenty years earlier, had shouted themselves hoarse swearing their allegiance to the Thousand-Year Reich.\n\nIt made Williams think about the mystery of identity. He had heard a story, shortly after the publication of _Butcher's Crossing_ , about a scandal during the reign of Caesar Augustus. Despite being emperor of the Western world, he couldn't force his daughter Julia to obey him. Though he trusted her with private secrets, she would not be reined in. Her behavior gave his enemies reason to hope that he was vulnerable. In 2 BC, Augustus exiled his only child to a tiny island called Pandateria, removing her from society. Here in mid-ocean, Williams thought about the human dimension of the conflict: on the throne Augustus was emperor; in his own home, he was a father with a difficult daughter, and the ramifications shook the known world. In Augustus' story, in postwar Germany, and in the novel he was working on, _Stoner_ , the question of identity, and which one was truest, intrigued him. It spoke to the person who had been John Jewell, then \"Jon\" Williams, and the effort to become what he wanted to be.\n**CHAPTER TWELVE**\n\n\"The Williams Affair\"\n\n_I don't wish any communication with Williams unless it is to plead for mercy. . . . I will wreck this book unless you provide proper acknowledgment_.\n\n\u2014YVOR WINTERS, 1963\n\nThe Williamses unpacked their travel belongings in their rooms at 26 St. Michael's Street in Oxford, a rather dingy three-story rooming house run by a Mrs. McArdle in a neighborhood that wore, as someone said about the face of postwar England, \"an insulted look.\" Their disappointment with the living arrangements contributed to strains between them, until, after just three weeks, Lonnie announced that she was going home early. \"I believe she misses the kids,\" Williams explained to Marie Rodell, minimizing what her sudden departure suggested about the state of their marriage. Williams stayed on, enjoying the tourist-crowded streets of Oxford and discovering, to his surprise, that \"the teachers here in the University are no better than you will find in a decent graduate department in the States\u2014perhaps not quite so good, really, though they do speak better and are a great deal wittier and handsomer.\"\n\nIn the meantime, Rodell had given _A Matter of Light_ to Cecil Scott at Macmillan. As she predicted, the response was a flat-out no. In fact, Scott and his associate editor, Al Hart, were willing to let Williams go. \"While both of us agree that the writing is excellent, we simply cannot foresee a satisfactory sale, the sort of sale that John would expect and feel entitled to. We understand, too, that this will mean our losing John as an author, which disturbs us because quite obviously he has a bright future in front of him.\"\n\nWilliams was unfazed, and, as Rodell prepared to send the manuscript to other publishers, he expressed relief they were finally free of Macmillan, writing, \"I can't really believe that it's going to be impossible for us to find a publisher for the novel, and I suspect that almost anyone we get will be an improvement.\" He asked her to wait until he could tighten the manuscript a bit more, but she had already sent it on to Little, Brown and Company, along with a letter that sounded like regret: \"Here is John Williams' new novel, _A Matter of Light_. This is not the final draft; John wants to do more with the wife's motivations, for one thing.\"\n\nWhile he was away in Oxford, Williams' editor at Doubleday Anchor, Pyke Johnson, also took a vacation for most of July. _English Renaissance Poetry_ had been in college bookstores since May, ready for the fall semester, and sales of the first printing of ten thousand had been brisk. When Johnson returned to the office, on his desk was an airmail letter from Williams\u2014a nice coincidence for bringing him up to date about the anthology's success. But there was also a small sheaf of correspondence in chronological order, on top of which was a three-page letter from Yvor Winters.\n\n\"Dear Sirs,\" Winters began frostily, \"There has recently come to my attention an anthology published by yourselves entitled _English Renaissance Poetry_ , edited by John Williams. I regret to inform you that you have been taken. The book is in a large and serious measure pirated from my own publications.\" To substantiate his claims, Winters had asked a colleague to compare the titles of poems from the Winters Canon against _English Renaissance Poetry_ 's table of contents. The overlap was about 80 percent. \"You will find not only a surprising number of the titles in Williams' anthology,\" Winters continued, \"but a theory of the history of the sixteenth century lyric which Williams takes over bodily and which no one except myself (or so far as I know has ever) propounded. . . . Williams discusses many of the poets exactly in my terms. This outline of 16th century poetry is original and it is correct, but it is mine, not Williams'. In so far as he departs from it, he departs into fog.\" Winters said he would have protested to the English Department chairman at the University of Denver, too, if it weren't for the fact that \"the poor bastard has a wife and two or three kids, and they need the job.\"\n\nUnderneath this opening salvo was a reply to Winters from Johnson's assistant. \"John Williams is presently abroad, so I am afraid we will not be able to contact him about this matter until the fall. Mr. Johnson, the editor of the volume is also on vacation for a few weeks, but I am sure he will give your letter his prompt attention when he returns.\" Winters, beside himself with anger, replied accordingly. Into a file he labeled \"The Williams Affair,\" he placed a carbon of his response, which carried a threat of retaliation:\n\nI have your courteous brush-off letter of yesterday. There is not the slightest reason why this matter should wait until Mr. Williams gets back from his tour of Europe or Mr. Johnson from his vacation. In a matter as scandalous as this Doubleday should act at once. If Doubleday does not act, there are more ways I can act than by suing. You are someone's secretary, and doubtless have little understanding of my access to the literary press. I have easy access, however; and I am familiar with the laws of libel. If you people insist on slow action, I will make a quick scandal out of it. I am one of the best-known poets, critics, and scholars now living.\n\nJohnson's assistant, rattled by talk of libel, asked another editor to intercede, who assured Winters that the matter was being taken very seriously; but he refused to be mollified. \"I will not wait for Williams to return from his junket in Europe. . . . Besides, there is nothing he can say, and he knows it.\" He demanded that Anchor Books redress his grievance, and if they didn't, he would \"activate the grapevine\" against Williams and drag his name through the mud.\n\nPyke Johnson, looking over this furious exchange that had taken place during his absence, considered what to do. He returned Winters' original three-page letter to his assistant and asked her to retype it, minus the personal attacks on Williams. Then, to accompany the redacted copy, he included a business-like memo to Williams, reducing Winters' fury to a problem that needed looking into. \"I arrived back from vacation this morning and found your nice note of July 22nd,\" he wrote Williams. \"Unfortunately, there was also on my desk some material that I bring to your attention most reluctantly, but also most urgently. You will find it enclosed in the form of excerpts from a long letter from Yvor Winters. . . . I am now writing him to say we cannot consider any action until we hear from you.\"\n\nWinters could accept that someone would plagiarize intellectual content from him\u2014his ego prepared him for that. But what galled him was the lack of attribution, which he believed was deliberate and typical of Williams. \"There is no question about Williams knowing my work,\" he wrote Johnson. \"Some years ago, my publisher taught at the University of Denver, and Williams was one of his students. Swallow introduced Williams to my work, and Williams has followed it closely ever since. I understand he has been teaching from it.\" Johnson chose not to share that allegation with Williams.\n\nIndeed, however, Williams' students did know of their professor's proclivity for borrowing. One of them, the poet Heather McHugh, who later became a MacArthur Fellow, wondered \"why a working novelist might occasionally lecture from verbatim hand-transcripts out of a standard text he never acknowledges.\" She recognized substantial quotations, complete with turns of phrase and rhetoric, from _Literary Criticism: A Short History_ by William Kurtz Wimsatt and Cleanth Brooks. \"At the time I felt a holy righteousness. . . . [I]f that's teaching, I thought indignantly, why even I could do it!\" But to Williams, it was a matter of priorities: he was a writer first and college instructor second. He had taken shortcuts since the beginning of his teaching career: from uncritically embracing Winters' theories about poetry, to piggybacking on Alan Swallow's Wyatt dissertation for his own on Fulke Greville, to compiling a poetry anthology incorporating Winters' scholarship.\n\nNor was it a matter of being forced to serve one master or the other. It was taken for granted that he would publish fiction\u2014he was the director of the creative writing program at Denver, after all\u2014and it would redound to the university's credit if he became well known. But the reality was that finishing a novel took him four or five years, and he would not take any more time away from his writing than necessary to teach. Unlike Professor St. Peter in one of his favorite Willa Cather novels, _The Professor's House_ , he would not say that he had \"done full justice to his university lectures, and at the same time carried on an engrossing piece of creative work.\" Still, his conscience was clear. Choosing between \"conflicting integrities,\" Williams said, \"that is, I think, essential drama; it is also essential life\u2014at least, essential civilized life. . . . Life forces us into compromise, and that has nothing to do with integrity. . . . Integrity is a private affair, essentially selfish.\"\n\nHis reply to Winters was of a gentleman caught in an error. \"I cannot tell you how miserable I am about this whole unhappy affair,\" he wrote, but he could not \"accept the implications of piracy and dishonesty.\" There was no way to avoid duplication among the greats of a particular era, although \"a large number of my selections do not correspond to your choices.\" And then, in a particularly telling passage of the long letter, he flew his true banner: he is a writer, first and foremost, making a living as a teacher:\n\nMy only motive for putting together this anthology was to make available some of the poems that you and I and others have admired. I do not think of myself as a critic or a scholar in the professional sense of those words, and I have no ambition to be thought of as such. I am primarily a novelist and secondarily a poet, though perhaps not a very good poet; but I am a teacher, and I put together the anthology as a service to my own students and to others who might not otherwise become acquainted with these poems.\n\nWithout admitting to guilt, other than to being influenced by Winters' seminal work on the English Renaissance, he offered to include \"in subsequent printings of the volume an explicit statement of my indebtedness to you and to others, in which I will also attempt to suggest that you are not to be held responsible for the distortions I may make to your or anyone else's ideas.\" He completed a draft of an acknowledgment page for the anthology's second printing, slated for September, paying respect to Winters, who had inspired the \"marked increase of critical interest\" in the period.\n\nWinters ignored the draft Williams sent him, and to Johnson, dismissed the offer of \"some kind of dust-throwing acknowledgement.\" In the meantime, Anchor Books got busy inserting slips into unshipped books explaining that the volume was based largely on Winters' work. As an extra gesture of goodwill, Johnson mailed Winters a gift: a set of recently published selections from Anchor Books' Seventeenth Century Series, plus an edited edition of _The Psalms of Sir Philip Sidney and the Countess of Pembroke_. \"I think that the editing of these volumes will come somewhat closer to meeting your standards,\" Johnson wrote to him. Several weeks passed and Johnson saw no more \"vituperative\" letters from Winters on his desk, a man with a \"monstrous ego,\" in his opinion, so he trusted that the matter was closed.\n\nTo Winters, however, it would never be closed; he wrote to friends that he would \"take care\" of Williams in due time. In _Forms of Discovery_ , published four years after Williams' anthology, Winters recounted the whole tale as a \"most interesting case,\" adding, archly, \"I mention Professor Williams, because I do not wish to give the impression that I am borrowing a large part of my book from him.\" Williams' anthology would be useful in classrooms, he admitted, but \"unfortunately, he often uses inferior versions of poems written by Wyatt and Ralegh.\" Disputation was not so much an art for Winters as it was a brawl with endless rounds until death sounded the final bell.\n\nNancy was surprised that John took it so hard. \"Yvor was a bit nuts, but that situation upset the living hell out of John,\" she said later. To have the person he most wanted to impress take his efforts as an insult, and worse, accuse him of plagiarism, was terrible\u2014in fact, beyond the exact reverse of his hopes, because his integrity became the issue. If he had written some ham-fisted criticism, or made a poor decision in his life as a teacher or administrator, it could be written off as a mistake. But Winters' attack implied that he had spent a long time in an endeavor that was intellectually dishonest, and that he had intended to get away with it.\n\nTwice now, Williams had been dismissed as an upstart, a pretender in the royal court of literature. His first major novel, _Butcher's Crossing_ , had been whipped out of town in the pages of the _New York Times_ as a pretentious western; and his _English Renaissance Poetry_ had been damned as arrogant by a famous critic. These humiliations could not be taken lightly. They were happening without the consolation that things would turn out in the long run, or that he would one day be vindicated. Readers of histories and biographies have the advantage of knowing the end of the story, but to the person living it, the darkness is all around. As Williams approached middle age, under the excuse that perhaps it was time he owned up to the limit of his abilities, he might have struck his colors and given up, believing that it's nobler to kill your dreams than to let someone else do it.\n\nIt's worth asking, then, why he was considering a third novel, this time set in the classical world. Was it because he was still casting around for a genre he could succeed in? Hadn't he found his niche yet, after all that time? When he was asked about this, his answer was simple: he liked the challenge; he enjoyed the adventure of testing himself each time in a new, imaginary world. \"I try never to repeat myself. . . . Why do it again, if you've done it once?\"\n\nWilliams completed his studies at the University of Oxford in mid-August 1963. Taking the train to Dover, he booked an overnight berth for the Channel crossing and continued on to Rome, eager to begin putting ideas on paper for a prospective novel about the emperor Augustus. For six weeks, he worked uninterrupted in a room at the Bellavista Milton on the Via di Porta Pinciana overlooking the Villa Borghese gardens. The decor was a bit garish and the wine overpriced, as if the establishment were trying to be worthy of its location two blocks from the fashionable Via Veneto, but the solitude delighted him.\n\n\"It's amazing what you can get done when you're alone,\" he wrote Rodell, \"and when you don't have desperate students breathing down your neck. I'm seeing the things I need to see for this Augustus novel, and I'm ahead of my schedule on the revision of the William Stoner novel.\" Regarding the latter, he felt certain, more than ever, that it was a solid piece of work, although the \"hardest thing I have ever done.\"\n\nWilliams believed in the Stoner book because it expressed his conviction that novels should \"imitate in form the natural world\"; or, as he put it another way, \"This happens, and then this happens, and then this happens.\" Time, the iron clock of the universe, goes in one direction, and the business of the novelist is to create \"the realized and presented history of a person, or persons, moving through sensibly experienced space and time, between the recognized intervals of birth and death.\" The closer one adhered, when writing a novel, to life as it's experienced by individuals day by day, he believed, the truer it would be. Joyce's _Ulysses_ and Woolf's _Mrs. Dalloway_ he admired for their technical virtuosity, but the sensibility of those works was intended to be poetic, and poems are about the \"chaos of experience,\" or sudden realized moments or impressions. This was inauthentic, solipsistic\u2014and, stylistically, \"a one-note samba,\" in one of his favorite phrases used in class, meaning that the effort is clever but ultimately uninteresting. Moreover, it expressed the author's personal truth, which may not be everyone's, or even a few persons'. The novel's strength was that it gave the novelist room to explore, to think in large ways about human nature and moral problems. It graced the author with opportunities to pause along the path of telling the story to observe the way people act, the choices they make, and thereby to illuminate\u2014not in a heavy-handed way, but with craft\u2014certain verities about life.\n\nCome the end of September, he boarded the SS _Constitution_ at Southampton for New York to resume his former life in Denver. The last he'd seen of Lonnie, she was on her way out the door in Oxford, after they'd been together for only three weeks. No doubt the tension in his marriage was rising into a wave that would break over him when he arrived home. \"I imagine by this time I'm an almost total stranger to the kids,\" he mused. His daughter Katherine would be starting high school soon; Pamela was in middle school, and Jonathan was still in elementary school.\n\n\"He came in late at night,\" Jonathan remembered, \"and we saw him the next morning, and he had been gone for like three months.\" At the breakfast table, the children asked about what he'd seen and done, but he didn't regale them with stories. Jonathan didn't expect that he would. \"You could go to him and tell him things and he'd be respectful and interact. But he would never come up to you and say: What did you do in school today? How was your day? What did you do today? It just never happened.\" Via the intuition that children develop about their parents, it was understood they were unhappy, which cast a pall over most things. Within days after his return, John and Lonnie began quarreling again. \"They always fought,\" Jonathan said. \"There were times when I remember huddling in the basement and hearing them fight\u2014about what, I have no clue what the fights were about.\"\n\nWhen at last Williams had had enough, he stormed out of the house, off to campus, or a bar, or in search of Nancy. At least he was in love with Nancy, who adored him. There was that to sustain him as he began the routines of the fall semester again\u2014teaching, attending department meetings, and reviewing applications to the creative writing program\u2014while in the background his third marriage was becoming increasingly hard for him to tolerate.\n\nAlso on his arrival back in Denver, there was a letter waiting for him from his brother-in-law, Butch, who was ensconced once again in Ajijic, footloose and writing full time. He had resigned from Winthrop College, and had taught creative writing briefly at the University of Southern California\u2014a \"pretty congenial place\"\u2014where they tried to get him to stay on. But it had never been his ambition to teach: it was the writing life for him, and so he and George Rae were back to being literary bums. \"How was it on the Oxford\/Rome axis?\" Butch asked. \"We had a wildly relaxing, wildly productive summer in Ajijic, during which time I got so much accomplished on the novel that I can have the mother in the mails before the year's end.\" The novel was _Week with No Friday_ , a marijuana- and booze-infused portrait of the expatriate life in Ajijic among writers and artists. \"The advance on the novel, if it's publishable, might get us another year. Therefore, \"operating under the belief that thinking beyond six or seven years in the thermonuclear age is pointless, we have seized the day.\n\n\"Bring us up to date on your own itinerary down life's highway.\"\n**CHAPTER THIRTEEN**\n\n_Stoner_\n\n_And though I may seem to take something away from Stoner in the end at his death, I don't really; I give him more than he has had before, and more than any of us ever gain\u2014his own identity_.\n\n\u2014JOHN WILLIAMS, 1966\n\nNews from his agent about _A Matter of Light_ \u2014renamed now _A Matter of Love_ in an effort to give it a fresh start\u2014was not encouraging. After a desultory period at the end of 1963 when a series of editors returned the manuscript to Marie Rodell, a response from Simon and Schuster in March confirmed what she had been warning Williams about from the beginning: the story of Professor Stoner was depressing. \"Several others here have now read the manuscript,\" she wrote him, \"and we're all in agreement that it is a book to be respected highly but that it has such a pale grey character that it would be most unlikely to earn its keep in hard covers and almost impossible to sell to a paperback house. . . . What a simpatico writer\u2014yet what a problem he presents.\"\n\nFalling down the ladder of large publishers, rung by rung, _A Matter of Love_ passed through the hands of Pyke Johnson as well, editor of the contentious poetry anthology, who said, \"I found it a very moving story, one that might have been told of several professors I met along the academic path, as well as people completely outside the Groves.\" But ultimately, Johnson concluded, it wasn't right for him.\n\nUntil now, Williams had been campaigning for his work with zeal, trying to put the sunniest light even on rejections. To Rodell, he speculated that perhaps the book was destined for a more \"literary house.\" It might be slow to find a home, but \"I still cannot convince myself that we are going to have an impossible time getting it placed. Whatever its 'commercial' possibilities, I believe finally that its quality will, if nothing else, shame someone into wanting to do it. I may be na\u00efve; but I cannot help believing that somewhere, someone will feel compelled to publish a good novel.\"\n\nHe kept up the good front, but within days of _A Matter of Love_ being characterized as having \"a pale grey character\" and being \"impossible to sell,\" he also received a dispiriting letter from the director of the University of Missouri Press, quoting an outside reader's opinion of his poetry collection, _The Shape of the Air_. \"The poems reflect his academic experience,\" the reader had decided. \"Most of the poems . . . lack the vitality which distinguishes the exciting poet from the hand of the craftsman, the passion of the poet is missing. . . . The imagery in the epigraphs is banal and the philosophical content, on which their merit might conceivably rest, is scarcely worthy of serious consideration.\"\n\nRemarks of that sort seemed to be telling him that he was an academic trying to become something he was not. Signs pointed to the advisability of dropping anchor in the harbor of academia and forgetting about the high seas of literature altogether. A memo from the dean, received the same day as the rejection about _The Shape of the Air_ , congratulated him on being promoted to full professor for the fall semester. This came on the heels of the university receiving a grant from the Rockefeller Foundation to create a national literary quarterly, and Williams had been offered the position of editor. He would have the freedom to set the direction of the publication, solicit contributions, review submissions, and put the quarterly into the hands of readers around the country\u2014a far more influential role than having his verse published in little magazines.\n\nHis academic career, as opposed to his efforts as a novelist, had a wholeness and symmetry that reflected well on him. Here he was, director of the creative writing program, and soon to be editor of a small press quarterly\u2014very similar to Alan Swallow's achievements at that age. It had taken over fifteen years, but he had equaled or exceeded his mentor in most respects. And yet, it was practically guaranteed that very little would change in his professional life during the coming years: there would be classes to prepare for, students to advise, meetings to attend, dissertations to read, and then the lassitude of summers before it started all over again. By the time he was an old man, his adventures in fiction and poetry would seem, in retrospect, like things he had dabbled in, fond ideas and so on.\n\nPerhaps it was time to take stock. He was forty-two: the age at which Professor Stoner, Williams wrote, \"could see nothing before him that he wished to enjoy and little behind him that he cared to remember.\" After twenty years of trying to make a name for himself as a novelist, he was nowhere. \"In those days you could put literary Denver in a phone booth,\" said Joanne Greenberg, whose 1964 novel, _I Never Promised You a Rose Garden_ , would become an international best seller. \"And I met John at some literary event. I met him because he was this small, well-made, dapper guy\u2014drink in one hand, cigarette in the other\u2014holding forth. He was very theatrical. 'Who is that?' I asked. 'He writes Westerns,' somebody said.\"\n\nStudents who sought him out for advice found him occupying an untidy office down the hall from the English Department, which was still housed in a former dormitory. No framed awards hung from his walls indicating that anyone would be fortunate to have the benefit of his experience as novelist and poet. When he stayed late at night, he drank from a bottle of whiskey he kept in a desk drawer. A note from one of his graduate students expressed thanks for his help, but also sympathy for someone who was clearly discontented: \"Your class lectures were consistently and intensely stimulating. Your opinions, presented in your capacity as my adviser, in some cases saved me much time and energy in many extremely profitable pursuits in the academic field and the field of creative writing.\" The young man was readying a collection of poems for publication, starting out on the path that Williams had begun many years ago. \"In closing let me say that I am aware that, for various reasons, we never became friends. I am aware, too, that there is much anger, and bitterness, and even vehement contempt for what you see in the world. Nevertheless, I do hope that you also feel, for me, at least a modicum of the warm regard I feel for you.\"\n\nOne afternoon, Williams sat quietly at Nancy's house, preoccupied about something. He crushed his cigarette in a tray and exhaled a brooding cloud of smoke. \"I don't need to write novels,\" he said.\n\nShe was stunned by how dismissive he sounded. \"He was tough. If he said he might quit, he might. He was that type, he could just say, 'Okay, that's it,'\" she later said. \"But I didn't want him to stop\u2014writing meant everything to him and I was worried that if he quit, he would never be happy.\"\n\nAt 625 Madison Avenue, in the manic heart of \"Mad Men\" advertising agencies, Viking Press occupied a corner of the book world that authors dreamed about joining. On its list were writers who received splashy reviews in magazines that were bellwethers of new and exciting literature\u2014 _Harper's_ , _The Atlantic_ , the _Saturday Review_ , or _Time_ , for instance\u2014writers who were in the vanguard of a cultural shift in America, such as Saul Bellow, William S. Burroughs, Robert Coover, Ken Kesey, and Thomas Pynchon. The Viking catalog also included respected works of intellectual history and criticism by Hannah Arendt, Barbara Tuchman, and Lionel Trilling, as well as poetry by Phyllis McGinley, Marianne Moore, and Siegfried Sassoon.\n\nIf you were asked to meet with a Viking editor, you took the elevator up to the sixteenth floor, where the office doors were to the left, down the hall. As you entered, you had the impression of a private club. Recessed lights in the ceiling illuminated dark walnut bookcases, with book jackets displayed on the shelves like prints in an art gallery. The receptionist was seated behind a desk that resembled a three-sided pulpit made of the same dark wood; above her hung the company's nickel-plated logo: a Viking ship under full sail, designed by illustrator Rockwell Kent and chosen by the company's founder Harold Ginzburg as a symbol of enterprise, adventure, and exploration in publishing.\n\n\"Oh, Mr. Williams,\" the receptionist might say, \"you're expected,\" and you were invited to wait in one of the leather club chairs, opposite which were rows of black-and-white photographs mounted on the wall in silver frames: James Joyce, Graham Greene, John Steinbeck, Stefan Zweig, Sherwood Anderson\u2014authors from the early days. When you were told you could go in now, you followed a paneled corridor past the office of the founder's son, Thomas Ginzburg, a trim man with Mediterranean features whose suits and shirts were hand-tailored. The chairs in his office were covered in chintz; and, just for whimsy, he had a model train that ran. Although he had the final say on any manuscript an editor wanted to purchase, he relied on one of his best editors\u2014considered brilliant in fiction publishing: Corlies M. Smith.\n\n\"Cork,\" as his friends called him, was another patrician of the book world. He had attended the Episcopal Academy outside Philadelphia, a boys' school that was nearly as old as the United States itself. Tall and good-looking, he'd worked at Lippincott in Philadelphia after graduating from Yale in 1951. The editorial side of publishing at that time was the preserve of Ivy League graduates in tweed suits and button-down shirts\u2014a style Smith personally preferred because, as he said to a colleague, \"this is not about fashion.\" When he left Lippincott in 1963\u2014at one point he had threatened to resign if any of fourteen uses of \"screw\" and \"fuck\" were removed from Barbara Probst Solomon's _The Beat of Life_ \u2014he brought Thomas Pynchon with him to Viking.\n\nIn meetings, he expressed his opinions about authors and submissions in a collegial way to avoid giving the impression that he believed he had unerring instinct. But he was extraordinarily perceptive\u2014a good judge of what mattered, whether the title in question was destined to be hardback literary fiction or a trade paperback for the mass market. Once he got down to working with an author, his method was hands-on, or getting \"close-in.\" His edits were returned with reports referring to specific pages; and, if necessary, he would recommend getting together to discuss changes.\n\nIn June 1964, Smith received a manuscript from Marie Rodell, whom he had known since Lippincott days. It was _A Matter of Love_ by John Williams. The title was bland, he thought, but he settled in to give it a read.\n\nFrom the first pages, it was clear that Williams had written a novel that was, as he had intended, \"somewhat against the 'fashionable' novel.\" American fiction in the 1960s tended to declare heatedly, as Irving Howe complained, that \"something about the experience of our age is unique, a catastrophe without precedent.\" The story of Professor Stoner ran against that grain. It was strangely profound, despite its subject\u2014the life of an unremarkable man. It proceeded with a deliberate, unhurried step, almost at the stately pace of Greek drama, because, as Williams himself said, he wanted to \"hold the attention of the reader without resort to gimmicks and inventions,\" with a narrative \"moving primarily upon that level on which we are moved in life.\"\n\nFor his main character, William Stoner, Williams had looked to his colleague J. V. Cunningham. Cunningham was from a working-class background and had entered university life because an older man, Yvor Winters, had pointed out the way to him. Cunningham's first marriage, to the poet Barbara Gibbs\u2014called \"disastrous\" by one who knew him well\u2014had produced a daughter, Margie, and he had assumed custody of her.\n\nOn the first page of the novel, Williams begins at a point not often dared by writers: the story is already over, and the narrator is merely reporting the events that happened. He had tried it once before in _Splendid in Ashes_. But now, the unemotional tone, the understatement, dispenses with all attempts at literariness. \"William Stoner entered the University of Missouri as a freshman in the year 1910, at the age of nineteen. Eight years later, during the height of World War I, he received his Doctor of Philosophy degree and accepted an instructorship at the same University, where he taught until his death in 1956.\"\n\nThe hero's surname, \"Stoner,\" suggests the obdurate life he had led growing up with his parents as dirt farmers in Missouri, an existence of unbroken labor and silence. The longest speech he ever heard his father make occurs one night at the kitchen table; the subject is college. The older man repeats the county agent's remark that \"they got new ideas, ways of doing things they teach you at the University.\" Perhaps young Stoner should enroll in college and study agriculture. Left unsaid is the expectation that, as the couple's only child, he will return to run the farm.\n\nOnce enrolled at the University of Missouri, however, \"the required survey of English literature troubled and disquieted him in a way nothing had ever done before.\" He's startled awake by poetic language. A sonnet by Shakespeare, read aloud by his instructor, Professor Sloane, speaks to him across the span of three hundred years, and Stoner experiences a sense of existing, of being aware of himself:\n\nHe looked away from Sloane about the room. Light slanted from the windows and settled upon the faces of his fellow students, so that the illumination seemed to come from within them and go out against a dimness; a student blinked, and a thin shadow fell upon a cheek whose down had caught the sunlight. Stoner became aware that his fingers were unclenching their hard grip on his desk-top. He turned his hands about under his gaze, marveling at their brownness, at the intricate way the nails fit into his blunt finger-ends; he thought he could feel the blood flowing invisibly through the tiny veins and arteries, throbbing delicately and precariously from his fingertips through his body.\n\nUntil that moment, Stoner has lived enveloped by the insensate world of nature. But language makes it possible for him to reach new consciousness. Words permit reasoning, which can be used to concretize elusive qualities of life. To make real _futility_ , Williams describes the home where Stoner grew up: \"The floors [of the house] were of unpainted plank, unevenly spaced and cracking with age, up through which dust steadily seeped and was swept back each day by Stoner's mother.\" The image is mythic in its power to express, like the task of Sisyphus, the terror of endlessness. The psychological depth Williams had tried to achieve in _Nothing But the Night_ , and the forcefulness that eludes him in writing verse, he invokes to tremendous, moving effect in _Stoner_. It's as if he has taken a carpenter's plane to the solipsistic, useless overwriting of _Nothing But the Night_. He \"strips the narrative of its rhetoric,\" in Hilton Als' phrase, and shaves down each sentence to Flaubertian exactness.\n\nStoner switches his major to English literature without telling his parents, and informs them when they arrive on campus to see him graduate that he will not be returning with them to the farm. They receive his news with the same stoical acceptance as they would the fact that inadequate rainfall that season has killed the fields. They are not people of words. He \"watched his father's face, which received those words as a stone receives the repeated blows of a fist. . . . [His mother] was breathing heavily, her face twisted as if in pain, and her closed fists were pressed against her cheeks.\" His decision not to return has put them under a sentence condemning them to work without the benefit of his help and his \"new ideas,\" contrary to what the county agent promised, for the rest of their lives.\n\nStoner receives his PhD in English literature in 1918, the year he meets Edith Bostwick, the daughter of a local banker. From the outset, it's clear that their romance is flawed. Stoner's idealized expectations about love are taken from books. In his courses, \"Tristan, Iseult the fair, walked before him; Paolo and Francesca whirled in the glowing dark; Helen and bright Paris, their faces bitter with consequence, rose from the gloom.\" These are only counterfeits of love that a mature couple might discard in time. But what's fatal to Bill Stoner and Edith Bostwick's relationship is that they lack a shared, personal language, the intimate \"little language\" of lovers, as Jonathan Swift called it. Their inarticulateness is painful:\n\n\"It was a very nice reception,\" Edith said faintly. \"I thought everyone was very nice.\"\n\n\"Oh, yes, of course,\" Stoner said. \"I meant . . .\" He did not go on. Edith was silent.\n\nHe said, \"I understand you and your aunt will be going to Europe in a little while.\"\n\n\"Yes,\" she said.\n\n\"Europe . . .\" He shook his head. \"You must be very excited.\"\n\nShe nodded reluctantly.\n\n\"Where will you go? I mean\u2014what places?\" . . .\n\nStoner was silent for a moment.\n\nHe attempts to corner her into talking; and then suddenly, all in a rush, she begins \"to tell him about herself, as he had asked her to do. He wanted to tell her to stop, to comfort her, to touch her. He did not move or speak.\" She continues on and on, mechanically relating a litany of facts about herself, deposing everything about her life because, as we will come to understand, she feels contempt for sincerity, for love, for everything that's weak or vulnerable. Her talk is obliging noise. She's a disappointed romantic, and carries on the courtship with an imperceptible sneer.\n\nWilliams' portrait of Edith has been criticized as misogynistic. But he implies that she has been a victim, and the blame belongs to an abusive man, her father. \"'They were very close,' Edith's mother said mysteriously. 'Much closer than they seemed.'\" On his wedding night, Stoner \"put his hand upon [Edith] and felt beneath the thin cloth of her nightgown the flesh he had longed for. He moved his hand upon her; she did not stir; her frown deepened.\" After intercourse, she hurries into the bathroom and vomits. Much later, on the day of her father's funeral, Edith returns to her parents' home after the service, to the bedroom she had as a girl, and divides her personal items into two piles: one is for gifts and notes from school friends and relatives, and the other is for what \"her father had given her and of things with which he had been directly or indirectly connected.\" Everything from him\u2014dolls, clothing, pictures\u2014she smashes, pounds, or burns in the fireplace, \"methodically, expressionlessly.\" This exorcism only partly succeeds. After the ritual, she goes to the opposite pole, beginning her life again, this time as an artsy, free-spirited, unconventional woman. The effect is pathetic: \"she seemed happy, though perhaps a bit desperately so.\"\n\nThe marriage will be unhappy, Stoner realizes, because they are fundamentally antagonistic. Edith Bostwick is winter; he, a former farmer, is summer. Their union is impossible, unnatural even. \"In her white dress she was like a cold light coming into the room.\" Her face \"was like a mask, expressionless and white.\" When the couple moves to a house near campus because Edith demands it, she \"wanted it white, and he had to put three coats on so that the dark green would not show through,\" as though burying their relationship under snow. For two months, she is wildly sexual, waiting \"crouched in the semidarkness,\" like a predator. But there is no tenderness; and her \"eyes were wide and staring,\" as if sex were an assault. Her strange behavior stops immediately when she becomes pregnant. After months of sickness, she delivers a baby girl, Grace, with whom Stoner \"falls instantly in love\"; but for a year, Edith acts troubled and upset when she tries to hold her infant daughter in her arms.\n\nAbout that time, a new instructor joins the English Department named Hollis N. Lomax, a specialist in nineteenth-century literature. Holding a PhD from Harvard, he delays his appearance until the first department meeting. Then he enters, a figure from Gothic Romance:\n\nSomeone whispered, \"It's Lomax,\" and the sound was sharp and audible through the room.\n\nHe had come through the door, closed it, and had advanced a few steps beyond the threshold, where he now stood. He was a man barely over five feet in height, and his body was grotesquely misshapen. A small hump raised his left shoulder to his neck, and his left arm hung laxly at his side. His upper body was heavy and curved, so that he appeared to be always struggling for balance; his legs were thin, and he walked with a hitch in his stiff right leg. For several moments he stood with his blond head bent downward, as if he were inspecting his highly polished black shoes and the sharp crease of his black trousers. Then he lifted his head and shot his right arm out, exposing a stiff white length of cuff with gold links; there was a cigarette in his long pale fingers. He took a deep drag, inhaled, and expelled the smoke in a thin stream. And then they could see his face.\n\nIt was the face of a matinee idol.\n\nLong ago, John Ed Williams, the teenaged poet of Wichita Falls, had been dazzled by matinee idol Ronald Colman, and had taken him for the epitome of the Romantic hero. But in the character of Lomax, Williams parodies the sentimentality of Romanticism for teaching that feelings can be relied on as a guide to truth. \"I am Lomax,\" the visitor intones, after a theatrical pause, in a \"deep and rich\" voice\u2014a customary thing to say, but meant to seem profound because it was delivered melodramatically. There is an air of charlatanism about Lomax. For Professor Stoner, whose field is logical thought and the poetry of the classically influenced English Renaissance, his antithesis has arrived.\n\nLomax avoids his colleagues, although he is \"ironically pleasant\" to them. His classes are popular because he performs: he trades on the cult of personality. And then, on an unusually cold night in September\u2014Edith-weather, in other words\u2014Lomax surprises everyone by attending a house-warming party at the Stoners' home. Getting quite drunk, he stays till almost dawn, talking about his loneliness, his inadequacies, and how literature had been a refuge. Before he leaves, he gives Edith a chaste kiss on the lips, and whispers something to her, as if they understand one another. The next day, Stoner tries to follow up with overtures of friendship; but Lomax\u2014threatened by how his drunkenness made him drop his Byronic mask of alienation and self-sufficiency\u2014snubs him \"with an irony that was like cold anger.\" Lomax's resentment never lessens; he becomes half of the ancient Fairchild-Ramsay feud at the University of Missouri. Probably because Fairchild limped, Williams unconsciously wrote \"Fairchild\" sometimes in his manuscript instead of \"Lomax.\"\n\nStoner carries on, a conscientious but not inspiring teacher. Despite his love of language, he pulls back uncertainly from conveying his passion for it, as if that would be admitting too much. He publishes a book in his field, awed by his temerity at daring to make a bid for immortality. When he holds it in his hands, it seems \"delicate and alive, like a child.\" His affectionate relationship with his daughter, Grace, six years old now, fulfills him, too. The simplicity of his love for her, his instincts as a father, lead him to believe \"that it might be possible for him to become a good teacher.\"\n\nBut Edith cannot live comfortably without conflict. Like her complement, Lomax, she is histrionic, a born thespian, because drama gives her a role. She resents the feeling of calm between Stoner and Grace. After her absence following her father's death, when she returns home, Edith says to her daughter, \"'Gracie, honey,' . . . in a voice that seemed to [her husband] to be strained and brittle, 'did you miss your mommy? Did you think she was never coming back?'\" Surprised that such a frightening thought had never occurred to Grace, Edith determines to give her the education about family relationships that Stoner is failing to give. One evening, when Grace and her father are in his study, \"laughing together, senselessly, as if they both were children,\" Edith enters and tells Grace not to disturb her father. Confused, Grace leaves the room. The \"enormity\" of his wife's \"surprise attack\" suddenly becomes clear to Stoner. From now on, Edith will hold Grace hostage in a never-ending game of making her complicit in hurting her father. Wearily, he surrenders to his wife's stratagems, and concedes his happiness at home. \"There are wars and defeats and victories of the human race that are not military and that are not recorded in the annals of history,\" Williams writes. But in another contest, this one in the English Department, Stoner realizes he cannot stand by.\n\nOne of Lomax's doctoral students, Charles Walker, comes to Stoner to ask a favor. His area is Romantic poetry, and he's a younger version of his adviser: his left hand hangs \"stiffly at his side, and his left foot dragged as he walked.\" He wants Stoner's permission to enroll in an overloaded, and challenging, seminar on the influence of the Latin tradition on English literature. Professor Lomax has assured him he can handle the work. Stoner listens and decides to give this impulsive young man what he wants.\n\nOn the first day of class, Walker is late. Then he interrupts with questions for the sake of drawing attention to himself. His remarks don't stem from curiosity; he wants to demonstrate that Romantic poetry, \" _real_ poetry,\" has no need of a classical perspective. After class, Stoner seeks reassurance from Lomax that Walker can indeed do the work. Lomax extolls his prot\u00e9g\u00e9's brilliance, adding, with \"cheerful malevolence, 'As you may have noticed, he is a cripple.'\" The insinuation is that Walker should be given extra consideration; and if Stoner doesn't agree, Lomax will take it personally. Thus Lomax has brushed aside the question of Walker's intellectual ability and made _feelings_ the issue.\n\nWalker's dependably off-the-point remarks continue in class, and when it becomes clear that his classmates regard him as a nuisance, he expresses \"outrage and resentment.\" The last straw, however, is an audacious stunt he tries to pull off that makes a mockery of academic study. He pretends to be reading his final paper aloud to the class, when he's really only extemporizing. In a sad imitation of Lomax's fervent style, he makes a rebuttal to a well-researched and thoughtful paper presented earlier by a student named Katherine Driscoll, whose genuine scholarship has earned Stoner's gratitude and respect. Walker's unserious response shows contempt for both him and Driscoll. Stoner gives him an F, to which Walker warns, \"You have not heard the last of this.\"\n\nThe confrontation comes, as it did between professors Fairchild and Ramsay, during Walker's preliminary oral examination\u2014with Stoner, Lomax (now the interim department chair), and two other academics at the table. It's clear that Lomax has coached his candidate, whose ignorance is appalling, but Stoner will not let the travesty get by. As a scholar, it's his responsibility to defend the profession, whose primary aim is to elucidate truth through knowledge and reasoning. Predictably, when Stoner refuses to agree to giving Walker a \"pass\" on the exam, Lomax accuses him of \"holding incipiently prejudiced feelings\" against Walker. But Stoner stands his ground\u2014at the cost of his career, as it turns out. A corrupt compromise, handed down from higher in the administration, allows Walker to take his oral preliminaries again, this time with a committee selected by the new English Department chairman, Hollis Lomax. Edith laughs at her husband's principles: \"Honestly, things are so important to you. What _difference_ could it make?\" When Stoner receives his teaching schedule for the new term, Lomax has assigned him courses appropriate for a beginning instructor.\n\nFollowing this defeat, Williams releases Stoner into a love affair with graduate student Katherine Driscoll; it's a release in the sense that love frees him to learn about what he is capable of. Williams emphasized the need for characters to have a moral identity that is not fixed, but constantly redefining itself, as a way of illustrating the complexity of people's relationships. This is not an argument that art should be moral: it was a practical consideration Williams had learned as an author for developing the identity of characters striving against an indifferent world.\n\nSome of Williams' favorite examples came from Henry James. (He once surprised one of his graduate students by bringing her a stack of James novels. \"Never mind what they tell you in class\u2014just read and study these. You have to learn it on your own.\") Williams subscribed to James' moral realism, where the \"imagination of loving\"\u2014the phrase used in _The Portrait of a Lady_ \u2014creates circumstances of passion involving moral choices the character can explore.\n\nAlthough the affair between Stoner and Katherine Driscoll becomes known, \"what surprised them both was that it did not seem to matter. No one refused to speak to them; no one gave them black looks; they were not made to suffer by the world they had feared. They began to believe that they could live in the place they had thought to be inimical to their love, and live there with some dignity and ease.\" However, running against society's strictures has consequences, or, as Williams liked to say, \"You pay your money and you take your choice, and you find at last that you can't have it both ways.\" Chairman Hollis Lomax is prepared to use the affair as an excuse to ruin their academic careers. Stoner is tenured and middle-aged, but Katherine is twenty years younger. For her sake, Stoner breaks off the relationship, and she quietly leaves town. He adjusts himself to accepting loneliness as a condition of his life. \"Stoner does endure too much, he accepts too much, and he doesn't fight enough,\" Williams wrote a friend. \"But if these are vices, they are vices that are merely the obverse of certain virtues\u2014virtues that allow him to endure in one world [his profession] at the expense of his happiness in another [his marriage].\"\n\nThe years pass, and after learning he has cancer, Stoner retires from teaching. Edith regards his illness as a tiresome inconvenience. Grace, who comes to say goodbye, has become a passive, colorless person; she drinks because she feels rejected. In the final moments of his life, Stoner gazes out the window of his sickbed on a summer afternoon, and then \"a sense of his own identity came upon him with a sudden force, and he felt the power of it. He was himself, and he knew what he had been.\" Williams, who never knew his father, who exaggerated his adventures and created a rather fey persona of the _artiste_ to cover his insecurities, gives Stoner the gift of self-knowledge, without the interfering fog of moral expectations or a Christian redemption. As his hero gently tells Katherine, when their love affair must end, \"We have come out of this, at least, with ourselves. We know that we are\u2014what we are.\"\n\nCork Smith at Viking offered a contract for _A Matter of Love_ in mid-July 1964. He wasn't crazy about the title (\"Unfortunately, I have no brilliant suggestions\"), so in the ensuing months, after a three-way correspondence between Smith, Marie Rodell, and Williams, they settled on _Stoner_ instead. \"I couldn't be more pleased with [Viking],\" Williams wrote to Rodell. \"Their first list is excellent and never so large as to be unmanageable. On the basis of all I know about the books they publish and seem to like I probably could not have chosen a house that would have pleased me better.\"\n\nIt was the beginning of a series of good things happening to him. Alan Swallow accepted a collection of his poems and brought them out under the title _The Necessary Lie_ as part of the press's Verb Poetry Series; Williams' anthology, _English Renaissance Poetry_ (with the acknowledgment to Winters added), was a hit among college instructors, as he had predicted, and went into a second edition.\n\nIn April, _Stoner_ appeared, lightly reviewed. Williams went to New York for an autographing party hosted by Viking and did a few interviews. Two strange things happened, and had Williams been thin-skinned, or superstitious, he might have taken offense. A limousine picked him up at his hotel for a talk show on radio. As he got in, \"Chuckles the Clown\" was sitting in the rear seat, on his way to the same radio program. He was in full circus regalia: a small man with a red rubber nose, black fright wig, huge white ruff, big shoes, and a baggy red-and-white polka-dotted suit made of silk. Williams regarded his companion's costume and thought to himself, \"But this is radio.\" He was chauffeured next to a circus-themed television show, hosted by \"a fat person named Stubby Kaye,\" which turned out to be a Saturday kiddie show called _Shenanigans_. \"I can't seem to get away from clowns,\" he mentioned dryly to Smith.\n\nBy June, sales of _Stoner_ stood at 1,700, less than half of Butch Marsh's _Week with No Friday_ about artists living a bacchanalian life in Mexico, which had been published the month before _Stoner_. \"The book is moving, but slowly,\" Cork Smith admitted. \"This in no way diminishes our feelings about the novel. You were right and we were right.\"\n\nThen, nearly a year after _Stoner_ 's release, in February 1966, a review by Irving Howe in the _New Republic_ hailed it as an overlooked discovery:\n\nGiven the quantity of fiction published in this country each year, it seems unavoidable that most novels should be ignored and that among these a few should nonetheless be works of distinction. _Stoner_ , a book that received very little notice upon its appearance several months ago, is, I think, such a work: serious, beautiful and affecting. . . . Mr. Williams writes with discipline and strength: he is devoted to the sentence as a form, and free from the allure of imagery. . . . I think there should be a few thousand people in this country who will find pleasure in the book.\n\nThe day after the review appeared, Williams arrived early at the English Department office dressed in his best. Howe's praise had come late, but it was reason to celebrate. He knew that a few of his colleagues subscribed to the _New Republic_ , so he made himself available in the outer office beside the faculty mailboxes. All day he sat in the reception area opposite the secretary at her desk, drinking coffee and smoking, as his fellow instructors walked past, to and from class. A few congratulated him, but most said nothing. A friend in the department, Robert Pawlowski, wasn't surprised. \"Most knew that great jealousy lived in a number of his colleagues and he stoically suffered them,\" Pawlowski later said. At five o'clock, when the halls had emptied out, Williams went into his office and closed the door behind him.\n\nThe review in the _New Republic_ didn't make much of a difference in sales. Cork Smith mailed out thirty thousand flyers to high school English teachers, hoping they would make _Stoner_ assigned reading. But that didn't improve the numbers, either.\nPART IV\n\n_Augustus_\n**CHAPTER FOURTEEN**\n\nBread Loaf and \"Up on the Hill\"\n\n_I would dearly love to find a kind of colleague with less than the rather staggering total of twenty-five years of service to the Bread Loaf Writers Conference. . . . There are things in_ Stoner _that make me think you might be such a person_.\n\n\u2014WILLIAM M. SLOANE TO JOHN WILLIAMS, 1965\n\nNancy didn't know many \"Lonnie people,\" nor did she want to know them. That part of John's life was his business and she trusted him to take care of it. She had acquiesced to his desire not to flaunt their relationship out of respect for Lonnie and the children. Among his friends, Williams was straightforward about having two households\u2014to Nancy's youngest child, he was her mother's friend, \"John Worms,\" but other than that, the professor and his mistress shared an unspoken bargain about the outside world and keeping it out. That included Alan and Mae Swallow.\n\nFrom what Nancy gathered, Alan had been largely responsible for John's career. John and Lonnie had met while working for Swallow Press. And after they married, the Swallows visited them a few times in Columbia, Missouri. Since then, for going on fifteen years, the two couples had been regulars out for dinner in Denver.\n\nAlan had slipped from the Denver scene somewhat, because of a traffic accident. Cruising noisily down the streets of Denver on his Indian motorcycle, he'd swerved to miss a turning car and caught his right leg on the handle of the passenger door. There had been a chance he would lose it up to the knee; instead, the surgeon reconnected the shinbone with a steel pin. Still, his foot flailed as he walked. To cope with the pain, he drank and \"worked like two men,\" as a friend said, to continue mailing out an average of fifty thousand books a year, in addition to publishing seventy-five new titles. Six months after the motorcycle crash, he collapsed in the living room on Christmas morning from a heart attack. From his bed he wrote letters to author friends, apologizing for having a body that couldn't keep up with him.\n\nAdded to the physical strain, though, was Swallow's spiritual isolation. The academic community in Denver had gradually forgotten him after he resigned in 1954 over the affair with a trustee's wife. At first, he had enjoyed the days spent humming along to the radio in the garage, chewing on a cigar, and corresponding with authors and booksellers, regaling them in multiple-page letters about his prospects. And it was good to see the boxes of finished books, smelling of fresh ink, delivered by a truck backing into his driveway. A delivery meant at least a week of late nights spent wrapping orders, individually or in parcels, and making sure to slip a handwritten invoice inside each one. Stacks of paperwork to be sorted sat on top of the file cabinets.\n\nSwallow Press truly absorbed him, but he missed the old days of noisy autographing parties at his home\u2014he didn't know whom to invite any longer, apart from old friends. Being stricken from the faculty mailing list meant he was out of the loop when it came to special lectures, social events, and English Department matters. He was just a \"townie\" now. While he'd been director of the creative writing program, publisher of the University of Denver Press, and owner-publisher of Swallow Press, there had been a wonderful synergy that put him at the center of everything he cared about.\n\nHis frustration over his estrangement surfaced in a reply to John Williams in March 1963 about some poetry Williams had submitted. Addressing his erstwhile student in the tone of a teacher, Swallow scolded Williams for being too easily distracted and not following through on his work. \"There are some good poems here, nicely done on real subject matter. But on the whole, I was disappointed. Too many poems seem to me to lack enough subject or theme to become interesting and test the accomplishment of versification.\" He chided him for continuing to devote energy to fiction. \"It seems possible that . . . in the last ten years or so, you have so concentrated your thought on fiction that the poetic concern has not blossomed as much as it would have otherwise.\"\n\nIn the meantime, Saturdays were sale days at the garage on York Street. In April 1966, Alan once again wrestled the folding tables into position for displays of discounted books\u2014ones that hadn't sold or were spoiled for some reason, with the prices written in pencil on the title page. The walk-in trade wasn't brisk\u2014mainly graduate students and people who stopped by to browse\u2014but he could do paperwork while the shop was open. He assumed his place at a card table near the front with an adding machine at his elbow for checkout. He saw a woman coming up the drive. It was Nancy.\n\nThere was no one else around, only the two of them, and after saying hello, Nancy walked slowly around the tables, her fingertips finding covers of books lying on their backs. She chose a few, holding them in the crook of her arm and pressing them against her sweater, until she was carrying five or six poetry and essay books. Alan rose and came to where she was at the far end of the garage. He stood by expectantly. She glanced up at him and offered the books for checkout. Instead, he tossed them on the table and grabbed her by the shoulders.\n\n\"Suddenly he slammed me against the wall and pressed himself against me,\" Nancy later said in an interview years later. \"I've never been so scared. He was a barrel-chested guy\u2014not tall, but solid. And there was nothing I could do. I kept saying, 'No, Alan, don't!' and he was groping. I'd not been flirting. I'm not a flirt. I said, 'No, Alan,' enough times that he just as suddenly backed off.\"\n\nShe hurried back to her car, her hands shaking as she tried to unlock door. In the interview, she said, \"Alan was probably thinking, 'This is John's whore,' because he was a friend of Lonnie's. I don't know. He could have killed me.\"\n\nShe drove away wondering whether she should tell John about what had happened. At first she thought she wouldn't, but she was too upset. She went to the apartment he kept near campus and told him everything. He left her there to wait and was gone an hour.\n\n\"And that was the end. He never spoke to Alan again,\" she said. \"After all that nurturing, and everything that Alan meant to John. I kept asking myself, 'Did I do anything? No, I just wanted to look at books.'\"\n\nCome June, John looked forward to being on his own for a few weeks once Lonnie and the children departed for Mexico to visit Uncle Butch and Aunt George Rae. He hadn't seen his sister and brother-in-law for twelve years, and he'd wrangled an invitation as part of a summer itinerary that would give him maximum time to do what he wanted. First, he needed to put the third issue of the _Denver Quarterly_ to bed; then, at the end of July, he would fly down to Mexico for a couple of weeks with his family in Ajijic. Finally, he'd be off to the Bread Loaf Writers' Conference in Vermont for most of August.\n\nButch was in fine fettle when John arrived. _Week with No Fridays_ was selling well\u2014unlike _Stoner_ , which had fallen off the edge of the earth after the first few weeks. In fact, Butch's publisher had taken out a big advertisement in the _New York Times_. Butch had half a dozen photographs taken for publicity\u2014an expat American writer living the sweet life\u2014his sports coat thrown nonchalantly over one shoulder as he strolled around Ajijic. While John was visiting, they talked about Butch's next novel, which would be set in San Francisco, and the published short stories from little magazines he was going to pitch as a collection. John was all for it, and he reciprocated by talking about his next project: a novel about power, consisting of correspondence, he was thinking, between Roman senators, generals, and certain family members who benefited or suffered from the rise of Augustus.\n\nIt was to his credit that he never begrudged his brother-in-law's incremental successes\u2014partly out of loyalty to George Rae, who adored Butch\u2014but also because John was wholly focused on his own work. There were too many authors, living and dead, to envy. He was striving for respect as a novelist, and it didn't take anything from him to wish his brother-in-law well. Then, after a couple of weeks spent being dad and husband in Ajijic\u2014the kids' complaints about being bored got on his nerves\u2014Williams flew back to Denver and into Nancy's arms before continuing on to the Bread Loaf Conference in Vermont.\n\nHe'd been invited to Bread Loaf the year before, unfortunately too late for an instructorship for that session. But William M. Sloane, an editor and fantasy writer who helped run the event, made it clear that Williams was on the \"must invite\" list. \"I would dearly love to find a kind of colleague with less than the rather staggering total of twenty-five years of service to the Bread Loaf Writers Conference,\" he joked, referring to his own tenure there since World War II, \"so that some of what I believe in about this conference can be handed along to someone else who might possibly care very much about it after a year or two of experience. There are things in _Stoner_ that make me think you might be such a person.\" Williams accepted a position teaching poetry and fiction; the fee was an honorarium\u2014a few hundred dollars\u2014but the prestige of teaching at a conference reckoned to be the best in the country meant a lot to a writer in mid-career.\n\nHe wished Nancy could accompany him, but she had children and a large vegetable garden that was just coming into season. The strain of trying to have everything he wanted, to meet all of his commitments, personal and professional, was getting to him. \"I don't know whether or not I told you about my summer; it's a bitch,\" he wrote to an acquaintance. \"I had hoped to get some work done on my new novel, but the prospects look highly unlikely. . . . We don't need money\u2014we need time.\" In August 1966, he flew to New York and hitched a ride with the wife of a Bread Loafer instructor and her friend. Together they drove up through Concord, New Hampshire, and then northwest into central Vermont, to an upland valley shaped like a diamond in the Green Mountains.\n\nIt was midafternoon when they arrived, the day before the start of the conference, and the grounds were nearly deserted. A few attendees were registering at the rambling, turn-of-the-century Bread Loaf Inn, a gift from a local philanthropist to Middlebury College along with thirty thousand acres of timber forest. Smaller buildings\u2014cottages, a barn, a library\u2014were within walking distance, giving the place the appearance of a streetless village seated on an expanse of grass, dappled with old spreading trees. He was surprised to find the setting so bucolic. It was located \"some ten or fifteen miles from Middlebury, the nearest town of any consequence,\" he later wrote. \"It is a land-locked island of a wilderness so soft that it seems almost contrived. A dozen or so buildings\u2014from tiny cottages to spacious three-story houses\u2014are scattered randomly upon many acres of gently rolling fields, and in the near distance Bread Loaf Mountain lies squat upon the horizon.\"\n\nMiddlebury College had convened the first summer school session here in 1920. It was then called the Bread Loaf School of English and offered graduate-level study in English and American literature. Many of the attendees were teachers. Robert Frost, who lived in nearby South Shaftesbury, took an interest because he dreamed of owning a farm that would attract writers annually to a literary summer camp. He joined the first faculty of invited guests during the summer in 1922. Eventually, the two weeks remaining after the official end of summer school became the Bread Loaf Writers' Conference, devoted exclusively to creative writing. And for almost thirty years since then, Frost had been its guiding spirit: Frost, the poet of the life and landscape of New England whose flinty personality made younger writers nervous. Then, in 1963, the first \"Frostless summer\" after a brief period of interim directors, he passed the baton to his heir apparent, the poet John Ciardi.\n\nCiardi had attended Bread Loaf before World War II and almost every year thereafter beginning in 1947, making him \"the conference's anchor in teaching poetry as surely as his deep baritone found a place in the evening sing-alongs,\" wrote David Haward Bain in his history of Bread Loaf, _Whose Woods These Are_. Ciardi did have one weakness, however\u2014his \"utter (sometimes dismaying) lack of modesty\" about his financial success, despite coming from a poor Italian neighborhood in Boston's North End neighborhood. Now the poetry editor of the _Saturday Review_ and host of _Accent_ , a weekly television magazine program, said he was the first poet in America to own a pink Cadillac DeVille. The car, his proudest possession, was parked outside the fine old Bread Loaf Inn the afternoon Williams checked in.\n\nCiardi handpicked his faculty. As an editor of the _Saturday Review_ , he had definite ideas about who belonged at Bread Loaf: primarily middle-aged white males who were part of the literary establishment and given to looking like Oxford dons, with sports jackets and briar pipes. Seymour Epstein, who didn't have a college degree, thought he had misheard when Ciardi said, cheerily, \"We'll have to have you up on the hill next summer.\" Epstein smiled and nodded, thinking he meant, \"We'll have you up to dinner sometime.\" But Epstein was typical of the kind of Bread Loaf instructor Ciardi was looking for: bright and broadly accepted. Epstein had just published _Leah_ , a vivid description of midcentury Manhattan through the eyes of a woman searching for love. Having been tapped by Ciardi himself, he accepted the invitation to come up to \"the hill\" as a new instructor that summer of 1966, the same as Williams. There was only one woman on staff that year: a children's book editor.\n\nJohn unpacked his suitcase, hung up his clothes in a closet that was \"as big as many a bedroom,\" and then went downstairs for a walk \"into the amber softness of the late Vermont afternoon.\" Drinks were available at Treman cottage, reserved for faculty, and he headed in that direction.\n\nDan Wakefield, a freelance journalist and author of _Revolt in the South_ , a 1961 collection of reports on the civil rights movement, had never heard of John Williams as he came through the door of the cottage. He later described him as a \"short, wiry, intense man with black hair, a sharp beard, and glasses,\" recalling that \"a fellow staff member identified him as the author of _Stoner_ , 'a novel that was supposed to be terrific.'\" Williams opened a beer and went over to a group making small talk to join the conversation. Within minutes, he and Wakefield got into a nasty argument over the merits of a minor political figure. The atmosphere in the room spoiled, Williams returned to his room and went to bed early, wondering \"not for the first time nor the last time in my life why I was where I was.\"\n\nAt breakfast the next morning, Ciardi, wearing a tweed jacket that seemed to billow on his huge, pear-shaped frame, welcomed the staff and students. His mission, he said, was to abide by Frost's \"first principle,\" as he called it, \"the center of the Bread Loaf idea: let them come together; let the writers, and the teachers, and the would-be writers, the hopeful and the hopeless come together; let good writers lead their discussion and set the terms of it; and let the talk be of writing from inside the writing process.\" Wakefield, glancing at _The Crumb_ , the daily conference newsletter placed each morning beside everyone's plate, saw that Williams was giving a reading later, and he decided to attend\u2014\"only from a sense of noblesse oblige, tinged with curiosity.\"\n\nHe took a seat at the appointed time, waiting for the presenter to begin. A few more Bread Loafers ambled in. Then, for twenty minutes, Williams read from _Stoner_ in his deep, rumbling voice, trained for radio broadcasts that went out into the gigantic Texas night, and \"the passages read were so eloquent, so moving in their understated passion,\" Wakefield said, \"that I rushed out after the lecture, bought the book, and spent the rest of the day reading it.\" That evening, seeing his adversary, the \"short, wiry, intense man,\" nursing a drink alone at Treman cottage, Wakefield went over and extended his hand. \"Look,\" he said, \"I don't give a _shit_ about politics.\" The two men remained friends for the rest of Williams' life.\n\nThe Bread Loaf Conference was a respite and a kind of safe harbor for Williams. In the English Department at Denver, he was a bit of an oddball, a raconteur at parties, an academic who wrote novels. But transported to the foot of Bread Loaf Mountain, he was admired. \"I was in awe of John,\" said one of the students years later. \"Deep voice, chain smoker, very intense. He would give a thoughtful answer about the weather. Rock-like, stone-like face, and then suddenly he would break into a big smile. He was gracious, but he said what he thought. He was surprised that people treated him as a true novelist and not an academic.\"\n\nThat fall, too, Alan Swallow attended a symposium on historical fiction in Pullman, Washington. Anyone encountering him there wouldn't have guessed that he was anything other than in good health. Novelist Frederick Manfred shook hands with him warmly. \"His leg had at last healed and he was walking around again like a young man. His face had once more the glow of health. He was lively and happy and full of gentle witticisms,\" he remembered. \"We all had a great time together. We were like a bunch of rancher brothers who, after looking for breaks in the fence all day, meet at dusk and sit around the campfire telling stories.\"\n\nReturning home, Swallow plunged back into the pile of Swallow Press work that was waiting. On Thanksgiving morning 1966, Mae woke up early to start breakfast. Hearing the motor of the electric typewriter whirring in the basement, she assumed that Alan had stayed up all night, and she took him down a cup of coffee. He was seated at his desk, bent forward with his head resting on the typewriter like it was a steel pillow, dead at fifty-one from a heart attack.\n**CHAPTER FIFTEEN**\n\nThe Good Guys\n\n_Well, I cleaned out my office, and guess what? I found a dead student_.\n\n\u2014JOHN WILLIAMS\n\n\"It never rains but it pours,\" Williams wrote merrily to his Bread Loaf friend Seymour Epstein in February 1967. \"And since you mention it, I _will_ tell you about my triumphs, my successes, my hopes, my dreams, my fears.\" He would be leaving in grand style on an ocean liner in May to research _Augustus_ , courtesy of a grant. He would return at summer's end for Bread Loaf, then retreat to his mountain cabin in Pine on sabbatical. Come the first of the year, he would begin as writer-in-residence at Smith College in Northampton, Massachusetts. Hence, he would be gone from the University of Denver for an entire year. Sounding a little bemused by this rush of opportunities, he wrote to Epstein, \"It is a strange new world.\"\n\nIt was hard to believe that just two years earlier, _Stoner_ had come and gone with only Howe's belated review in the _New Republic_ praising it. The following year, 1966, he wrote to another friend, seemed to Williams \"the worst year of my life, in some ways, as far as time is concerned,\" because teaching and editing the _Denver Quarterly_ had kept him from working on his new novel about Rome. Now, suddenly, he was feeling a measure of success. Unexpectedly, \"for me at least,\" he wrote Epstein, \" _Stoner_ has paid off handsomely,\" meaning that the novel had played a role in getting him the grant. \"Now, I should have finished the _Augustus_ novel by the summer of 1968.\" But he was going to need a replacement to take his classes, and asked Epstein, who accepted his offer of a temporary position in Denver's creative writing program.\n\nThe Augustus novel, as he imagined it, would depict Rome and the life of the emperor through imagined letters and journal entries. Major figures of the era would be put in conversation and conflict with one another: Julius Caesar, Marcus Agrippa, Maecenas, Cicero, Brutus, Mark Antony, Cleopatra, Strabo, Nicolaus of Damascus, Horace, Ovid, Virgil, and Augustus' daughter Julia. The structure would be \"webbed,\" he said. That is, it would \"not be straightforwardly chronological, but will cut back and forth in time, getting at Augustus's character from many different angles and points of view,\" using the novelist's art to illustrate the evolution of Octavian from a mild-mannered young man to Caesar Augustus, ruler of the Western world. \"He will become what he will become,\" says his tutor, the Greek Stoic Athenodorus, \"out of the force of his person and the accident of his fate.\"\n\nBut Williams would not try to describe the pageantry of banner-waving Roman legions or ships burning on the Ionian Sea. Too much of that would be a distraction from his main theme, which is how \"force of person\" meets the grinding forces of circumstance to forge an identity. His method of showing Augustus' development was his favorite: the _bildungsroman_ , where the journey is away from security toward self-formation. In _Butcher's Crossing_ , Will Anderson goes West to test his notions of himself against the world; in _Stoner_ , the farm boy Bill Stoner has an epiphany over a sonnet that sets the direction of his life. Octavian is still a teenager when the catastrophic news reaches him that his uncle, Julius Caesar, has been murdered. His young friends study him for a sign of what he will do, how he will react. \"For a long time we watched him, a slight boyish figure walking on the deserted field, moving slowly, this way and that, as if trying to discover a way to go.\"\n\nAnd for the second time, as he had in _Stoner_ , Williams would turn to the pain caused by a father-daughter relationship that has failed. He detected in Augustus' behavior toward his only daughter and biological child, Julia, \"some mystical identification between [her] and Rome; to the one he gave ethical training, education, and love; to the other he gave laws, a sense of order, and the dedication of his immense talent for ruling.\" When she is eleven, Augustus takes Julia's education and development into his own hands, and as she grows, he confides matters to her that he would trust to few others.\n\nDespite his carefulness, however, Julia defies him with public licentiousness that is politically dangerous. Macrobius records Augustus' remark: \"There are two wayward daughters that I have to put up with, the Roman commonwealth and Julia.\" For the sake of keeping order, he exiles her and her mother to an island. Five years later, she is allowed to return to the mainland, but her father never forgives her betrayal, or how she had placed pleasure above duty. \"As we know now, and as her father sensed then,\" Williams said, \"Rome went the way of Julia, not the way of Augustus.\"\n\nThat Williams would be concerned about the pain of being estranged from a child\u2014deeply enough to make it a theme in two novels\u2014would have surprised most of those who knew him. He preferred not to talk about personal matters. Colleagues and friends didn't know much about his war experiences, for instance, outside of a few anecdotes he repeated until they were threadbare. And although he and his fellow instructor Gerald Chapman were both raised in North Texas during the Depression, the coincidence never inspired Williams to talk about growing up in Wichita Falls. \"I think for John, that was the furthest topic of interest,\" Chapman said.\n\nBut about his eldest child, Katherine, he was uncharacteristically forthcoming. He exercised full bragging rights as a father, crowing about her successes at school, what she said about a book she was reading, and so on. How much he loved her was apparent to everyone in his circle. To him, a father's break with his child was one of the most traumatic things he could conceive of\u2014perhaps he drew on feelings about his own disappeared father, John Jewell. Katherine was a projection of him, part of his legacy. As a child, she realized her father's approval was somehow tied to writing. At seven, she begged not to have to write a thank-you note to Marie Rodell for a gift because she was afraid it wouldn't be good enough. By the time she was an adolescent, she regaled him at the dinner table with what was going on in her English class. John and Lonnie's middle child, Pamela, resented how her sister steered conversations to what their father preferred to discuss\u2014books, authors, ideas. \"My father was a snob,\" she said bitterly. The youngest, Jonathan, who was interested in mechanical things, also felt left out. \"I remember sitting around the dinner table and Kathy and my father would be talking about academic stuff. I got bored with it. It really never interested me at all. I think he resented that a little, because I wasn't following his path.\"\n\nIn May, Williams prepared to sail from New York on the brand new SS _France_ , the biggest ocean liner in the world, for a two-month stay in Europe. On a Saturday evening, friends arrived at his room on the promenade deck to see him off, among them Cork Smith, who related to Marie Rodell that the drinking and merry-making was \"notable.\" A week later, the ship arrived in Le Havre, and Williams caught a train to Stuttgart, where he purchased, factory-fresh, a maroon Mercedes-Benz 230 four-door sedan: front-wheel drive, six cylinders, four-speed manual transmission, radial tires, and a retracting sunroof. Tooling through the Alps on his way to Milan and thence to Rome, he cut the figure of a man in his prime. He registered again at the Bellavista Milton. The location had proved lucky for working on the final draft of _Stoner_ four years before; now he could get down to the business of planning _Augustus_.\n\nIn the early stages of _Butcher's Crossing_ and _Stoner_ , it had surprised him how much casting a spell of verisimilitude in fiction depends on fact. A novel about buffalo hunting is historical beyond dispute, and he had deeply researched the western frontier. But he had to do almost as much research about the first half of the twentieth century in Missouri for _Stoner_. What would the salary have been of a professor in the 1920s? Would private cars have been common in a small town? The category \"historical novel\" was so all-inclusive that he could see \"no difference between the novel of history and the novel of the near present.\" When he lectured about his work, he sometimes emphasized this point by quoting the Belgian novelist Marguerite Yourcenar, the author of a novel about the late Roman Empire, the _Memoirs of Hadrian_ (1951): \"Those who put the historical novel in a category apart are forgetting that what every novelist does is only to interpret, by means of the techniques which his period affords, a certain number of past events; his memories, whether personal or impersonal, are all woven of the same stuff of history itself. The work of Proust is a reconstruction of a lost past quite as much as is _War and Peace_.\"\n\nWilliams undergirded his new novel, although set two thousand years in the past, with facts. To feed a Roman army of sixty thousand men for three months, he calculated, it would take three hundred barrels of olive oil, three hundred tons of grain, one hundred tons of dried fish, fifty tons of cheese, a thousand casks of wine, and sundry other items. To cover them with warm cloaks in the chilly forests of Gaul would take a quarter of a million yards of heavy wool. But he was careful to avoid \"antiquarianism\"\u2014clogging the text with picayune details\u2014which carries the risk of lulling the author into thinking \"that history has created his characters, and that he need do no more than to accede to history, and to repeat and embellish the record, with whatever accuracy he chooses.\"\n\nHe wanted his characters to push forward, out from the factual background, to have the quality of living in their own present, grappling with conflicts between public and private life, between barbarism and civilization, between duty and love. \"The difficulty of writing a historical novel is in the history,\" he said. \"You and your readers know what has happened. So I had a technical challenge in creating a sense of discovery in what happened to the characters.\" The consequences of decisions characters make, they can only guess at, not knowing the future. Williams has Julius Caesar, only weeks before his assassination, write to his adopted son, Augustus, saying that his mind is troubled about his place in history, despite all that he has achieved. \"I have conquered the world, and none of it is secure. I have shown liberty to the people, and they flee it as if it were a disease; I despise those whom I can trust, and love those best who would most quickly betray me. And I do not know where we are going, though I lead a nation to its destiny.\" This is showing \"a way of thinking, the life of the mind,\" in Yvor Winters' phrase, and Williams learned it from studying the methods of the English Renaissance poets.\n\nUnlike those poets, however, Williams does not give his Romans comfort from religious belief. About the soul, gods, and spirituality in general, he expressed little interest, personally or in fiction, as if the question were settled for him. \"I knew he was an atheist and he knew I was a practicing Jew,\" said Joanne Greenberg. \"He once said, 'You're an idiot, but you have good instincts.' John was not a believer in any way, shape or form.\" One of his graduate students in the early 1970s, who later became a Catholic nun, had \"the sense that John thought salvation was not available to him.\" As she got to know him better as her adviser for her dissertation, she discovered \"he knew a lot about religious life. He treated me and talked with me as if he understood. But he couldn't believe. Perhaps his experiences in the war had something to do with that.\"\n\nWilliams wasn't against religion or conflicted about it; for him, the black universe was eyeless, uninhabited. As Augustus says in the final pages of the novel, \"I have come to believe that in the life of every man, late or soon, there is a moment when he knows beyond whatever else he might understand, and whether he can articulate the knowledge or not, the terrifying fact that he is alone, and separate, and that he can be no other than the poor thing that is himself.\"\n\nHoled up at the Bellavista Milton in Rome, Williams alternated working and sightseeing: one day at his desk, fortified with a pot of coffee, and capped by a bottle of wine that evening on the hotel terrace; the next day, driving to nearby locations that would play a part in the new novel. Longer excursions took him around the edges of the Mediterranean. \"I wanted to experience the quality of the air,\" he later said. \"I went to a large number of places where the Roman Empire extended. I indulged myself in a leisurely, long, meditative look at the places about which I would be writing\u2014Turkey, Yugoslavia\u2014which was Macedonia in the Roman Empire\u2014and the Greek Isles.\"\n\nBy the end of July, he had finished outlining most of the novel and written chapter summaries. He made arrangements to ship his car to New York so it would be waiting when he arrived. Once home in Denver, he began the process of looking into a larger house for the family, and then in August he rushed off to Bread Loaf to teach at the 1967 session. Back again in September, he went up into the foothills of the Rockies to his cabin near the town of Pine in a hard-to-find spot named Sphinx Park. As autumn turned to winter, he could imagine from the windows of his study that the deep, snow-covered conifer forests cleaving to the slopes were the Teutoburg Forest, where the Gauls lay in ambush for the unsuspecting commander Publius Quinctilius Varus and his legions.\n\nButch Marsh tapped out a three-page, stream-of-consciousness letter to John\u2014with a pitcher of daiquiris at his elbow\u2014the gist of which was that he was in mid-career but not where he expected to be at this point. _Week with No Friday_ had gone into paperback and was now spinning around on vertical metal book racks in bus stations and drugstores, but his publisher, Harper and Row, had rejected his second novel, the one set in San Francisco, and he was broke again.\n\nFor more than twenty years, since getting out of the Army Air Corps in 1945, Butch had been writing at white heat\u2014stories, poems, novels\u2014figuring that making it as a full-time author was like playing horses at the racetrack: you spread your bets. Maybe something that took a long time to write wouldn't sell, but a story dashed out in a day might hit the jackpot. Now he was burning out: he was never more than a few hundred bucks ahead of being broke. Much as he hated to say it, he was considering teaching in the United States again.\n\nEver since John and Butch had begun corresponding, the brothers-in-law had been arguing a point: not about who was the better writer, but over the issue of teaching and writing. Butch taught as a stopgap, although he was good at it. Everywhere he'd been on a faculty\u2014Winthrop College, the University of Iowa, the University of Southern California\u2014he had received high marks and compliments. But he couldn't get any writing done in the meantime.\n\nJohn didn't see the two occupations as mutually exclusive. \"There's no conflict between the roles,\" he tried convincing Butch, \"really, except the inevitable one of time, time to do your best in both roles, each of which is rather demanding. In many ways, each role supports the other: as a teacher, I am paid to think about and discuss with others those concerns that are most important to me as a novelist; as a novelist, I can bring some authority to bear on what I have to say about literature.\"\n\nMaybe Butch's dilemma was really just a matter of too few choices, he suggested. \"The freedom one thinks one finds in Mexico can be a kind of prison; that is to say, though one is free to make choices there, the number of choices possible is fairly limited.\" He encouraged Butch to mail his r\u00e9sum\u00e9 to fifty universities, at least. If he was invited to interview, it was almost certain the job was his, \"unless you pee on the floor or wear too much rouge.\" There was no disgrace in teaching; it \"will help you as a writer, and it is a decent life.\"\n\nButch replied that he would think about it. He had a title anyway for the story collection he was putting together\u2014 _Beachhead in Bohemia_.\n\nIn January 1968, Williams got ready to depart for Smith College to take up a position as writer-in-residence. The family had just finished moving into a housing development in the Denver suburb of Aurora. He had been a distant father, but even with his marriage to Lonnie uncertain, he did his best to provide for her and the children. He had purchased a lot on a cul-de-sac in Aurora for a split-level, four-bedroom home that would have two fireplaces and three baths. Jonathan, fascinated as he pored over the blueprints showing the house's wiring and ventilation systems, considered the privilege of being involved \"a very big deal.\" As John was packing his belongings in the Mercedes for the two-thousand-mile trip, a letter arrived from John Ciardi inviting him to return to Bread Loaf again for a third summer. Williams dashed off a reply saying he would, of course, adding, \"The New Year hasn't been that goddamn happy\"\u2014referring indirectly to friction with Lonnie. But August was months away, and in the meantime, Williams was due at Smith.\n\nWilliams took a leisurely few days to make the drive. The college had offered him an apartment in Northampton that was on the first floor of a grand, turn-of-the-century home on Bedford Terrace. He could walk through the backyard to reach the campus on the street behind. \"I've only been here a few days,\" he reported to Wakefield, \"but it appears that I'm actually going to be able to get some work done. Which will be a novelty, given the last several months.\" He liked the ambience of Northampton, a small New England town, founded by seventeenth-century English colonists, \"but the girls seem to be so goddam _young_ ; I have been remarkably chaste and aloof all winter, and nary a girl has pressed a grape in my mouth.\" By the end of February he had completed the first chapter of _Augustus_ , and by mid-March, he sensed he had passed \"the magic point beyond which it will be all right.\"\n\nHe wrote home fairly regularly, too, but when Jonathan asked his mother what Dad had said, her reply at one point was, \"No, you can't read this one.\" He knew his parents were unhappy, and overhearing a conversation between his mother and a close friend of hers\u2014the wife of an English professor\u2014about divorce worried him. He called his father for an update.\n\n\"Hello?\" a woman answered. He hung up, thinking he had the wrong number. He dialed again.\n\n\"Yes, hello?\" It was Nancy, whom he'd never met, but whose voice he recognized later. Until then, he hadn't known there was a woman other than his mother in his father's life.\n\nWhen John returned in May from Smith College, he moved in with Nancy; several nights a week he spent staying over in Aurora with the children. Lonnie wrote on envelopes addressed to him, \"Doesn't live here anymore.\"\n\n\"I don't think we were in that house for more than a year before we moved into a condominium instead and Dad saw us there,\" Jonathan said. \"No, I'm sure we weren't.\"\n\nThe August 1968 session of Bread Loaf crackled with dissent. The earlier part of the year had seen the assassinations of Martin Luther King Jr. and Robert Kennedy; civil rights demonstrations overlapped with anti\u2013Vietnam War protests. The little summer camp for writing was not immune to feelings of restlessness and impatience with authority. Williams was a member of a subset of staffers who called themselves the Good Guys: Dan Wakefield; the poet Maxine Kumin; Miller Williams and Robert Pack, also poets; the magazine journalist Brock Brower, and the novelist Harry Crews. They were a hard-drinking clique, and not everyone looked kindly on their revels, although the Beatnik-bohemian atmosphere at the conference had been growing stronger since the early 1960s. But nothing was going to cramp the Good Guys' style, and as far as their nightly inebriation was concerned, they had the support of Ciardi, who was of the opinion that \"it is simply impossible to gather writers, editors, agents, etc. without alcohol.\" The older heads were not going to accommodate the complaints of counterculture-types attending Frost's \"literary summer camp.\"\n\nThree weeks before the start of the session, Williams had received a memo addressed to the staff from William Sloane in which he tried to prepare instructors for a change of attitude among their students. Beginning the conference with \"the old [Bernard] De Voto approach,\" of delivering a lecture about theory, \"no longer impresses the young who approach writing with the belief that the reader owes them a hearing and that the reader is supposed to absorb the story or the material in such a manner and way as the artistic sensibilities of the writer determine.\" In other words, the reader should have to work harder to understand the author's meaning, instead of the author trying to court or please the reader. \"Fiction is in trouble because everybody thinks it is no longer necessary to regard readers as a volunteer audience but a captive one and this is contrary to nature and fact.\"\n\nWilliams believed he knew exactly what the problem was. \"Anyone who could afford a ballpoint pen or a typewriter was allowed to think of himself as a poet or a novelist; talent and craft were suspect, more often than not described as 'elitist,' a curse-word of the period; and literacy was thought by many to be a species of corruption, a loss of innocence or a kind of damnation.\" He was schooled in New Criticism, which emphasizes the formal properties of literature, and the idea that self-expression automatically deserved respect, regardless of its merits, was absolutely anathema to him.\n\nHowever, he was right that the term \"elitist\" was used to object to many things perceived as undemocratic, including how the staff and attendees were treated as separate but equal. Why was dinner delayed until the instructors had finished their cocktails at Treman cottage? Why were the instructors seated at their own high table in the dining hall?\n\nCiardi, in his sonorous, baritone voice\u2014like a captain in command speaking to a wayward crew\u2014made it plain that he didn't want to hear any more grumbling about differences in rank. When one of the instructors wanted to know why Ciardi had refused his request to invite a student for a drink at Treman, the director shot him a warning look. \"Don't get too near a psychic buzz saw,\" he said darkly. He was blunt with the administrators of Middlebury College, who had heard rumors of discontent. Ciardi dismissed it as bellyaching. Bread Loaf, he said, was \"intense, rapid, informal, and more ego-driven, in most cases, than scholarly. Throw a faculty member to one of the dining room tables with conference members and he would be devoured. Insist on such a policy and half the staff would refuse to come back.\"\n\nPrivately, some wondered if that might not be such a bad thing.\n\nThe patriarchy of Bread Loaf was typical of academe, and the arts, for most of the twentieth century. The composition of the faculty at Bread Loaf closely resembled the full-time faculty of the Department of English at the University of Denver: a dozen white, middle-aged men\u2014about a third of them World War II veterans\u2014and one or two women, depending on the year. At Denver in 1968, a female graduate student, pursuing a master's degree in English literature, smiled when her favorite instructor said, \"Well, Sally, you're only a girl, but you're the best we've got.\" Sally Boland, who later cofounded the women's studies program and chaired the English Department at a small private university, later thought about his remark. \"The amazing thing, now I look back on it, is not what he said to me, but rather what I said to myself, which was: 'Maybe if I can learn to think more like a man, I'll do better.'\"\n\nOver the years, John Williams had contributed to building Denver's English Department and its creative writing program, both of which were uncontestably male and white. Seymour Epstein, after a year of filling in for him while he was on sabbatical, was welcomed to the club, so to speak, with a permanent position in 1969. For several years after that, they drove out to Bread Loaf together. \"Nice not to have the wives along,\" Williams said to him, sweeping along the interstate in his sports car. \"Now we don't have to talk dirty!\" Around three or four in the afternoon, it was \"as though a light came on,\" Epstein said, and they pulled over in search of a bar because Williams was in need of a drink.\n\nThere was camaraderie, a bond between buddies in the department who, despite some cultural differences, were from the same American phylum. Former faculty members of the Denver English Department in the 1960s recall it as a period of harmony, a decade when a kind of _esprit de corps_ guided the direction of things\u2014a \"smooth unity,\" as the instructor and biographer Robert Richardson characterized the times. The connection between men like them\u2014raised during the Depression, tempered by the war years, influenced by mass culture in the 1950s\u2014was very close when it came to interests, ambitions, and what they could expect in life.\n\nWilliams felt easiest around men and colleagues who enjoyed a good laugh and bumped along well together. His candid description in a letter accompanying a recommendation for an instructor hints at the kind of person who matched the disposition of the department: \"I think I should tell you that he was an exceedingly good undergraduate teacher; how well he might do in graduate courses, I'm just not sure. At least he might do an adequate job, and he might very well do better. . . . He's not a 'swinger' like we were in the old days, but he is in no sense prudish, stuffy, or whatever. He's easy to get along with, does not have the kind of neurotic ego that we often find among our peers.\"\n\nA newcomer to the department in September 1969, Peggy McIntosh\u2014one of only two women on the English faculty\u2014saw Williams _in situ_ in the environment he helped create as he walked the halls between classes, decked out in his ascot and sports jacket, and entering debates about Vietnam by pointing out, \"Now your Chinese\u2014as I learned when I was over there in '44 . . .\" McIntosh was young, had been raised in New Jersey, and was a graduate of Radcliffe and Harvard. Her days as a leading feminist\u2014and author of a famous essay that introduced the terms \"white privilege\" and \"male privilege\" into cultural criticism\u2014were still some years off. She recalled how her arrival in Denver was regarded as an imposition:\n\nI was this Anglo blonde\u2014short, semi-pretty girl, looking like a girl with my hair in a bun, and I imagined that I just provoked the hell out of John Williams. I imagine he just couldn't stand it that somebody who looked like me and who was interested in Emily Dickinson had come into the department. He never gave me the time of day. He was not interested in conversation with me. Never said anything supportive to me. I was afraid of him. He was so dour, and resentful of women coming into what had been men's territory, that I avoided him. He was one of those who felt, as many men now feel, that the world is closing in on them, with all these \"women and minorities\"\u2014an illogical phrase\u2014[taking] the place that they were raised to think is rightfully theirs. And I think he felt threatened by the federal government putting pressure on college administrations to hire women.\n\nIt was true that the largely male citadel of academe in the 1960s felt blitzed by feminist forces. Robert Richardson's wife at the time remembered her then husband's disgust that a woman had been chosen over him for a position at Harvard. \"They _had_ to hire a woman, that's all,\" he had said.\n\nThe bombardment from the Left was also aimed at the canon of English and American literature, which Williams was determined to answer with salvos of his own. On that score he wouldn't yield. One of his younger colleagues found his counterattack bracing. \"To put it mildly,\" he later wrote, \"Williams had Olympian standards in judging fiction. One could mention virtually any well-known novelist and Williams' reaction would be the same. 'An awfully nice guy,' he'd say, suggesting that he knew everyone worth knowing.\"\n\nIf Williams went out of his way to cold-shoulder McIntosh, it was because he resented federal rules about equality in hiring, which he perceived as wrongheaded and politically motivated\u2014an incursion to be resisted. They were a band of brothers in the department, and of generally one mind about what to teach and how to teach it. For his part, according to McIntosh, he thought she \"mothered\" her students. McIntosh thought, \"He didn't give a damn about his.\" Perhaps he knew she was listening the day he wandered into the English Department office and announced, \"Well, I cleaned out my office, and guess what? I found a dead student.\"\n\nWilliams held the same convictions about the importance of keeping high and low culture separate. From his office window at the University of Denver, he looked out and saw evidence that \"the arrows of the barbarians,\" in his words, were raining down. He was not apolitical; but teach-ins about social justice on the lawn outside struck him as not part of the mission of the university. He worried that English departments were choosing inferior works for their syllabi, too, and teaching too much literary theory, \"whereas the whole point of the novel is the experience of reading the novel. Standards have been declared extraliterary.\"\n\nEverywhere the intellectual life was imperiled, as he saw it. Beyond the University of Denver campus, certain neighborhoods, such as \"Desolation Row,\" consisting of two blocks of head shops and hippie-related stores on East Colfax Avenue, had become hippie enclaves. Williams disliked the counterculture. It placed emotion higher than reason by venerating Thoreau, Whitman, Emerson, transcendentalism, Brook Farm, arts and crafts, and communes. The issues and the times were different, but the slogan among the young, \"If it feels good, do it,\" caught the essential spirit. At Stanford University, Winters was refusing to participate in creative writing courses any longer because he deplored student attitudes. Wallace Stegner had resigned in disgust as director of the Stanford program because Tom Wolfe's book _The Electric Kool-Aid Acid Test_ had made Ken Kesey ( _One Flew over the Cuckoo's Nest_ ) and his Merry Pranksters synonymous with the university.\n\nWilliams was not against modern literature\u2014the purpose of the _Denver Quarterly_ , which he founded and edited, was to explore the idea of the modern in original fiction and verse. But like his intellectual heroes, Winters and J. V. Cunningham, he was alarmed by how quickly new fiction, especially experimental fiction, had been embraced by college English departments. Higher education should be teaching \"approved tradition of the elders,\" as Cunningham put it, for the sake of grounding students in essential texts. Modernism was too heady, too sophisticated, and too likely to be imitated by students. As for everything published after World War II\u2014Susan Sontag, Kurt Vonnegut, Norman Mailer, G\u00fcnter Grass, Richard Brautigan\u2014that was the kind of new writing one discussed at a party over a beer, or read in _The New Yorker_ , in Cunningham's opinion, and Williams echoed it. \"The university should be an insulation against the specious kinds of values that are always in the world,\" Williams maintained. \"A kind of protection against them. . . . Once a university becomes what universities often say they are\u2014a reflection of the will of the community, or something like that\u2014well, it's dead. The university more often than not ought to be _opposed_ to the aims of the community, the aims of the world around it.\"\n\nThis was not only the principle he stood behind as an academic: it was also his creed as a novelist. At a party in Denver one night, Williams listened to a hippie artisan explain how he ran his jewelry business. The key to success, the young man said, was to \"psych out the market\"\u2014figure out what kind of jewelry and art people were into\u2014and then make what the public wanted. Williams pointed his finger at the young man and enunciated slowly to make sure he was understood: \"No, you are wrong. That is not what you do, as an artist. You figure out what the thing is you most want and need to do, and you do it. You don't try to 'please the public.' You do your best.\"\n\nAt least Williams could turn away from his office window and think about his work on _Augustus_. The ancient voices he was invoking through fiction had endured for two thousand years, which testified to the greatness of what they had to say; by comparison, student takeovers, riots, and peace demonstrations would become a parenthesis in history.\n\nEven so, the golden days of the Good Guys at Bread Loaf were numbered\u2014two more summers at most\u2014and the authority of the \"war generation\" would come to an end. In a photograph taken during the 1970 session, director Ciardi\u2014large, graying, bespectacled\u2014is seen arguing with a long-haired young man who looks resentful. The issue could have been the war in Vietnam, but actually there was no one on the faculty who was defending it. Instead, the \"rabble-rousing,\" as Ciardi called it, stemmed from differences of opinion about the relationship between politics and art, and the responsibility of an artist.\n\nBut Ciardi wanted none of that invading Bread Loaf on his watch. Transient problems of the world must not come to Middlebury. \"I feel,\" he said, \"I am trustee for a number of ghosts.\"\n**CHAPTER SIXTEEN**\n\n\"Long Life to the Emperor!\"\n\n_The world is not a poem_.\n\n\u2014JOHN WILLIAMS, _AUGUSTUS_\n\nAfter twenty years of marriage, John and Lonnie Williams divorced in January 1969. They had shared the new house for less than a year; now John was living permanently at his apartment, and the house was too large for Lonnie and their three children. She found a condominium in Denver six miles north of the University of Denver campus, near the Metropolitan State College, where she had been hired as an instructor. John came over for dinner two nights a week to check on the children. He paid child support. Tight-lipped, as usual, about his personal affairs, however, he didn't tell Nancy how much it was, and she didn't ask. \"He had a checking account I never saw,\" she later said. \"I made a point of not knowing.\"\n\nNancy had been living in a small house in the Park Hill neighborhood that she had purchased after moving out of her parents' home a few years earlier, near Denver's East High School, where she taught English. She put it up for sale, and she and John started looking for a home of their own. She had toyed on and off with the idea of leaving Denver and starting afresh, but John refused to consider moving away from his children, perhaps thinking of his own childhood. \"And I realized,\" Nancy said, \"Would I stay with someone who would do that?\"\n\nIn March 1970, they chose a home located on a quarter-acre of land at the end of South Madison Street, ten blocks from campus. John liked the \"great amount of room in the house itself,\" and he liked how the property was landscaped\u2014with \"fantastic plantings,\" he told Marie Rodell. He described \"a couple of spruces and pines that go up thirty or forty feet,\" adding, \"I'll send you a photo, when I have time to make one.\" The backyard was big enough for a garden, and he paced off a plot that was twenty by sixty feet. That summer, he planted tomatoes and other vegetables, along with a pumpkin vine for a neighbor child who said he always wanted \"the biggest one,\" but without pesticides, because his grandfather, a farmer, hadn't used them.\n\nIn December, after a few months in the house, he and Nancy married, and John's transition from one family to another was complete. He now had a role in the lives of Nancy's three children. For her part, Nancy felt that Pamela and Kathy Williams, his daughters, resented her. \"To the girls, I was the wicked stepmother,\" she said. \"I took him away from their mother.\"\n\nThe large garden and house spoke to a part of Williams that liked appearing consequential, someone with roots who knew what he was all about. Parked in the driveway was the imported Mercedes, a similar declaration. He would drive it for the rest of his life, another twenty years, until it became a fixture in Denver, synonymous with John Williams, novelist and professor of literature. And he amended the history of his life to fit the image. When questions of decorum came up, he spoke seriously of an inbred southern code he abided by because he was from Texas\u2014a better biography than growing up poor on a dirt farm in the shadow of oil rigs.\n\nIt was a carefully constructed identity, a presentation. Nancy understood its importance when he told her, \"In high school, I was Ronald Colman\"\u2014not \"I _thought_ I was.\" The ascots he tucked under his shirt collars, the fine wool jackets, and the pleated trousers were suitable attire for courting respect. Nancy noticed he was especially careful about his appearance when meeting strangers for the first time, and suspected this \"outer John\" was a cover-up for shyness. Other women, too, noticed how careful he was about his demeanor. He walked gracefully, reminding one woman who knew him at the time \"of the way men from India move: skinny Indian men, loping. It was different from other men. He wasn't jerky.\" Joanne Greenberg remembered him at a party crossing his legs at the ankles, and lowering himself slowly to a cushion on the floor, \"almost like he was making a bow. Almost feminine.\" Friends said he seemed happiest when he was playing chef in his own kitchen, whipping up his best dishes for dinner guests\u2014soups, stews, chili, biscuits\u2014and beaming at the compliments. Once, when he was seated at someone else's table, at the mention of dessert he announced dramatically, \"I make a brilliant mousse!\" unaware of the pun and provoking an outburst of laughter.\n\nOn campus, his preoccupation with himself could be overbearing at times, but he was proud of belonging to the community of scholars, to the academy, and being a novelist besides. He would never consider throwing it over for the obscure life of the artist in the garret. He must have the certainty of making a living and the automatic status that came with being a tenured professor. \"I sometimes hear people, writers, say, 'I will not sell out,' meaning for money,\" he told an interviewer. \"And I always ask them if they've had any offers lately.\" The fact that he had a beautiful new home and a lovely bride was proof to him that he'd lost nothing by combining the academic life and writing. He felt like he was at the top of his game:\n\nIf I thought that the academic life in itself, was bad for my writing, then I'd quit. . . . [I]t has become one of the many platitudinous ideas that certain kinds of writers throw about nowadays: the academy is the death of creativity, the enemy of poetry, the corrupter of poets, and so forth. . . . And I don't like whiners, either; some of the poets and novelists that keep muttering about the academy remind me of the boy who would run away from home, if only dad would loan him the station wagon.\n\nMeanwhile, work went well on _Augustus_. On days when he didn't have to teach, Williams liked to get up early and be in his study by eight o'clock. He preferred Nancy to be home when he was writing. Occasionally he would get up from his typewriter to see what she was doing, or call out, \"Are you there?\" She kept disruptions at bay, answering the phone and saying that her husband wasn't available. \"I would try to keep his work time inviolate,\" she said. Sometimes she would see him outside, standing in the garden, not looking at anything in particular, and she would think, \"He's stuck on something.\" He would rather work out a passage silently in his head than make a run at it and have to revise\u2014although he would revise, if necessary, to sharpen the effects. \"Since I plan carefully,\" he wrote, \"I am not faced with the problem of major reorganization after a first draft is completed. . . . Though it seems to be so, this is not a mysterious process; it is simply the result of a long period of concentration and work.\"\n\nHe broke for lunch, by which time he usually had finished a page. Two or three pages in a single morning were a triumph. The rest of the afternoon he spent planning for the next day's work. Then, promptly at five, he stopped and went into the kitchen and poured himself a drink and lit a cigarette before ambling over to the sofa with his glass to watch television. He often told Nancy he might just as well have been a carpenter or a plumber. Writing was a job.\n\nThings weren't as rosy for Butch. Running out of money, he had reluctantly taken John's advice and gone job-hunting. North Texas State University had invited him for an interview. As a candidate for a position in the English Department, he didn't have a PhD. But with stories in more than seventy periodicals, including _Antioch Review_ , _Prairie Schooner_ , _Northwest Review_ , _Yale Review_ , _Esquire_ , _Playboy_ , and _Saturday Evening Post_ , he clearly he was the genuine article, a real writer coming in out of the rain. North Texas State hired him. He and George Rae left Mexico and moved into a prim, whitewashed apartment building called Coral Isles on a side street in Denton. For four semesters and two sessions of summer school in 1968\u20131970, he met his classes, graded papers, and attended faculty meetings. Since he never did anything by halves, he didn't have time to write anything publishable while he was teaching.\n\nAll that looked to change when he and George Rae returned to Mexico, but he was not in the best of shape by then. Butch's heart had been giving him trouble. He was a bit thickset at forty-eight, but not overweight, with large, freckled hands he cupped around a match while lighting a cigarette against the night breezes off Lake Chapalupa. In the collection of stories he was preparing, _Beachhead in Bohemia_ , published posthumously, there was one about a man in his forties who feels twinges in his chest: \"If pain hit when he was dozing, he would awaken instantly because it was the left side. And even though the pain was in the wrong location, all that his lulled mind could initially report was terror on his left. It was humiliating to be braced in a constant cringe.\" Butch was glad to be back in the sweet life of writing full-time again, but the pressure was on. He'd gone stale creatively and his inventory of salable stories was low. He was drinking a lot in Ajijic\u2014George was, too\u2014but it seemed to be the only way he could calm down after another day at the typewriter. As he filled his glass, his hand trembled from too much nicotine.\n\nGeorge Rae insisted they make an appointment with a specialist in Guadalajara, the nearest large city. A Dr. Rivera checked him over and admitted him for observation to Hospital Angeles Del Carmen, a brand new and modern facility, just off the office plaza in the center of town. As he was resting in his room for further tests, Butch died from a heart attack on May 27, 1970. The death certificate issued by the American consulate a few days later identified Willard N. Marsh as a vacationing retiree, not as a writer, whose last known address was Denton, Texas.\n\nGeorge couldn't remain in Ajijic without him. \"I'm moving back to San Miguel de Allende,\" she wrote John. \"I've gotten a little apartment at the Quinta Loreto where Willard and I lived before in 1967 and '68. Ajijic has just become too dismal for me. It's too full of old retired people who read nothing, see nothing, and do nothing but drink. I didn't mind it when I had Willard, but without him, I just can't take it. If I stay around such people much longer, I'm afraid I'll become just like them\u2014or go mad.\"\n\nShe saw to it that he was interred in the Ajijic municipal cemetery. Then, two years later, a real estate developer cut through the grounds with a street, naming it De Las Flores and desecrating many graves. One of those that disappeared was the last resting place of Willard \"Butch\" Marsh, novelist, short story writer, and former jazzman.\n\nAbout the time of his brother-in-law's death, Williams was reminded again of how ephemeral a writer's work can be, when he discovered that _Stoner_ had gone out of print after only a year. His publisher hadn't notified him, either\u2014a surprising oversight by his usually careful editor, Cork Smith. The novel's sales had never been robust, but the discourtesy bothered him. It was dismissive, similar to how _Butcher's Crossing_ had been lumped in with westerns. He mentioned his frustration to Dan Wakefield, his fellow instructor at Bread Loaf.\n\nAs a favor, Wakefield mounted a campaign to get _Stoner_ reissued by asking some notable authors and college instructors to write statements endorsing the idea, which he would then forward to Marie Rodell as a friendly petition. Excerpts from their remarks could also serve as blurbs on the cover of a reissue. The one submitted by the poet Miller Williams spoke for many: \" _Stoner_? Funny you should ask. Sad that you should have to. . . . I have one copy, which is the only one I've been able to find. It has been read not only by my wife and me, but nine other people whose names I can't recall at the moment. . . . There is no doubt in my mind but that _Stoner_ would have the wide reputation it deserves as an American classic if it were possible for readers to buy the book.\"\n\nMerrily, Wakefield collected the statements of support\u2014\"I got a beautiful response,\" he told John\u2014and added his own cover letter to Rodell, whom he'd never met. It began: \"A number of us who are admirers of John Williams' novel _Stoner_ have been disturbed to find it out of print.\" He sent photocopies off to her, and the originals to Williams, feeling the pleasure that comes from doing a good deed.\n\nRodell was offended at being told her business. \"Dear Mr. Wakefield,\" she replied, \"Thank you for your letter of October first and its enclosures.\" She only wished, she added archly, that the book's advocates had concentrated more on Stoner's literary values, and less on how they couldn't get copies for their college classrooms. Although she didn't say so, she had inferred from receiving a handful of unsolicited requests to reprint _Stoner_ from \"distinguished writers and teachers\" that Williams had been complaining about her. The damage seemed to be spreading when she also received a note from the publishing entrepreneur Seymour Lawrence, who said he'd heard from Wakefield that John Williams' new novel about Augustus Caesar might be available. Could he read the manuscript? It wasn't available, Rodell informed him curtly\u2014it was already under contract to Viking: \"I'm sorry you were misinformed.\"\n\nWilliams stepped in and apologized on behalf of his friends. He had been reading aloud from the manuscript of _Augustus_ during workshop sessions at Bread Loaf, he explained, and there \"are many people who have seemed excited about the novel\u2014staff members, visitors, etc.\u2014and it has occasioned some talk.\" Thus, he accepted some of the blame; he was just doing his job as a good client.\n\nAs it turned out, Wakefield's crusade did help. In January 1971, another publisher, a small paperback house, reissued _Stoner_. Rodell thanked Wakefield for the letters he had so \"diligently collected\" and assured him that the editor who bought the book \"has become as confirmed a 'Stonerite' as you or I.\"\n\nAt the end of the term in May 1972, John and Nancy drove to Mexico to check on George Rae. She was living in Ajijic again, part-owner of a discotheque, of all things, and dating Ted Cogswell, the science fiction and fantasy writer who had struck up a friendship with the Marshes years earlier. Nancy knew him not at all, and she was not impressed. In fact, she found him rather frightening. \"When George and Ted were together they were drunk a good part of the time,\" she said. \"She'd be completely stone drunk. . . . She stayed at our house. Ted would come over and see her.\"\n\nTed liked to ride around Ajijic on his motorcycle wearing a Mickey Mouse costume, complete with the bulbous head, big ears, and imbecilic cartoon expression. Driving around drunk one day, he stopped by the Williams' apartment looking for George Rae. Nancy was alone. For some reason, he had a knife and showed it to her. Then he forgot about it and roared away on the motorcycle, leaving the knife on top of the refrigerator. When John returned, Nancy insisted that Ted not be allowed in the house again.\n\nApart from George's taste in boyfriends, Nancy was curious about the bond between her and John. Although they were only half-siblings, she noticed they were strikingly alike\u2014they had same petite size and pale coloring, and a similar style of speaking\u2014as alike as \"two peas in a pod.\" And if John seemed\u2014as someone had said about him, \"gnomish,\" with his goatee, deep voice, and penetrating blue eyes\u2014George Rae seemed like a pixie. They had the same theatrical air. In fact, when George Rae talked about how she wanted to get back into theater eventually, John's rough teasing and dismissiveness carried a tone of envy. After all, it had been his ambition before it was hers.\n\nNancy and John returned to Denver in midsummer, and a few months later, George Rae and Ted married in San Miguel de Allende. The Williamses didn't attend; but a young American writer witnessed the event and the wild send-off that followed:\n\nLike everyone else in the wedding party at the Episcopalian church, including the priest, George Rae was an atheist. The ceremony was followed by the wettest reception I've ever attended, and that was followed by Ted and George getting into her Jeep and heading for their honeymoon suite in Puerto Vallarta. An hour later the reception was winding down when the Jeep came roaring back, stopping between the plaza and the police station for Ted to shove his screaming, cursing bride out on the cobblestones, where bride and groom were immediately arrested for being drunk and very disorderly. Placed in separate cells, Ted made a pillow out of his boots and went to sleep on the thin mattress of the cell's wooden cot. George propped her mattress against her cell's door, set it on fire, and screamed bloody murder until the cops turned them loose at dawn to continue on their honeymoon.\n\nSomeone snapped a picture of the pair on their wedding day, standing on a second-floor balcony, wearing matching leather safari jackets like adventurers. Ted is making a face, with one arm around George Rae and flipping the camera the finger with the other.\n\nThat summer, the axe fell on John Ciardi as director of the Bread Loaf Writers' Conference. Complaints from attendees and a few younger instructors continued to reach the Middlebury College administration. The antagonism was partly generational and ego-driven; Ciardi bristled at talk of deposing the older heads and their lordly \"maestro system.\" Defiantly, he typed a letter of resignation and submitted it, undated, to the president of Middlebury, adding, \"When you want it to be time, just fill in the date and I'll be gone.\" It was time. The president called his bluff and accepted his resignation. Ciardi would be given the 1972 session as his last to run.\n\nThe new director would be Robert Pack, also one of the Good Guys, but Pack was determined to recruit fresh blood. The criteria for choosing the staff would not depend on personal friendship or amiability, or how long they had been \"up on the hill.\" They would have to be good teachers and demonstrate respect for beginning writers, both published and unpublished. John Williams had the requisite qualifications. \"I was very fond of John,\" Pack said. \"He was a good teacher, a good raconteur, and charming, but he had a drinking problem.\" And that was the deciding factor\u2014he was not among those invited back after 1972 for what would have been his seventh year.\n\nIt might appear that too much emphasis is being placed on alcohol abuse, as it affected John Williams and his circle of friends. He himself expressed contempt for \"lushes.\" \"John had a sixth sense about who was one,\" according to Dan Wakefield; as though compared to _that_ guy, he didn't have a problem. \"He didn't slur his words, or wobble when he walked. He didn't show that he was drunk,\" said Wakefield.\n\nNo one can say when the culture of hard drinking became associated with writing, but liquor and literature seemed to go together: five American authors who won the Nobel Prize during the postwar era were alcoholics\u2014Sinclair Lewis, Eugene O'Neill, William Faulkner, Ernest Hemingway, and John Steinbeck. A character in O'Neill's _Long Day's Journey into Night_ rhapsodizes by quoting Baudelaire: \"Be always drunken. Nothing else matters: that is the only question. If you would not feel the horrible burden of Time weighing on your shoulders and crushing you to the earth, be drunken continually.\" A conviction that drinking was somehow tied to the seer's gift\u2014a sign of being in touch with the gods, a spur to prophecy\u2014drew many willing young artists into the stumbling ranks of drunks during those years. Wakefield said it was an expectation: \"We were told by other, older writers that if you're going to be a serious writer, you have to be a serious drinker.\" Williams said as much to a young graduate student, a woman who came to his office to discuss her stories with him. \"He opened a desk drawer and poured a shot of Wild Turkey into a glass,\" she reported. \"'If you don't have a drink, you aren't a serious writer.' It seemed like a test and I took it.\"\n\nPeople in publishing took it for granted, too, that booze and the making of books went hand in hand, with potentially unfortunate results for Williams when he submitted the finished manuscript of _Augustus_ to Cork Smith in the spring of 1972.\n\n_Augustus_ had taken nearly seven years to complete, and he was confident that it was \"really a very good novel,\" predicting to Cork Smith, tongue-in-cheek, \"that it might have a good chance to do well on the market, despite that fact.\" Having read the early sections as they came in, Smith agreed with Marie Rodell that it was \"one hell of a book and should be exactly as John Williams wants it.\" And Smith made a brilliant suggestion about formatting. He recommended omitting numbered chapters\u2014they seemed too modern. Instead, there should be Book 1, Book 2, Book 3, and an Epilogue. Williams concurred immediately, because it also fit with how he had broken the story into three sections, almost like three acts. He was glad the manuscript was in the hands of an editor who understood the spirit of the novel.\n\nAnd then, suddenly, two months after Williams submitted the final draft, Smith stepped down from Viking. He was being furloughed home from the office indefinitely. His alcoholism was out of control, and the publisher, Thomas Ginzburg, had given him the choice of either drying out or losing his job. It would be up to Ginzburg to decide when, or if, he was ready to return; in the meantime, Smith told Williams, he was going to be \"standing back and taking a longish look at myself.\"\n\nThe news upset Williams. Smith had championed _Stoner_ from the beginning, and he'd midwifed _Augustus_ right up to the point of publication. Now, who would step forward, and who could be trusted to do right by the new book all the way to market? As it turned out, the manuscript went to Alan D. Williams, a skilled and versatile editor whose gifts included the ability to establish a good rapport with authors across various genres. His \"open-sensibility,\" as a fellow editor called it, made him approach a work submitted to him with respect; it was doubly lucky for John that _Augustus_ went to him, because he was well versed in classical literature. Under his guidance, Robert Fagles' translations of the _Iliad_ and the _Odyssey_ later became standards.\n\nWhile John was waiting for his new editor to read _Augustus_ for the first time (\"I hope you have read the novel; I hope you liked it,\" he told Alan Williams), Rodell let him know that Curtis Publishing, owner of a line of inexpensive paperbacks, wanted to reprint _Butcher's Crossing_. The novel had been out of print for ten years; reprinting it would at least guarantee a second chance. Plus, Williams stood to make a little money from the advance. But \"the real stumbler,\" she cautioned, was that the sales staff insisted on marketing it as a western, because \"it has to be described as something!\" they said.\n\nWilliams absolutely refused. He'd rather the book not come out at all. Then Rodell received a similar offer from Popular Books\u2014same advance, same approach: _Butcher's Crossing_ would be advertised as a paperback western. And again, he refused, even though his career as a novelist currently was at a standstill. It had been almost twenty-five years since his first novel had appeared, _Nothing But the Night_ , which he disavowed as beginner's work. Next, _Stoner_ had barely made a ripple when it dropped into the pond of forgotten books. But allowing _Butcher's Crossing_ to be reprinted and advertised as a western might permanently label him a second-rate writer. The enjoyment of being able to say he had a novel in print wasn't compensation enough for the blow to his self-respect. He could only hope that _Augustus_ , when it was published, would be his vindication.\n\nCork Smith's leave of absence didn't affect the march of _Augustus_ toward publication, after all. \"LONG LIFE TO THE EMPEROR!\" Rodell telegrammed Williams when it appeared in late 1972. The reviews were more widespread, more flattering, and in much greater number than for either of his two previous books. Orville Prescott, the principal reviewer for the _New York Times_ , sent a personal note: \"I think it ranks with Thornton Wilder's _The Ides of March_ as a work of literature and I know it is truer to the facts of Roman history.\" J. V. Cunningham, ever the classicist, in addition to telling Williams it was an \"astonishing book,\" complimented him on \"the observance of decorum in style.\"\n\nAs was John's custom each time he had a new novel out, he took up a prominent spot in the English Department office, where he sat all day smoking and drinking coffee, expecting that some of his colleagues would congratulate him. He waited, making small talk with people coming in to pick up their mail or take a break between classes. The hours passed pleasantly enough, but nothing was said to him about the book. And finally, as the weak December light coming in through the windows began to fade, he went down the nearly empty hallway to his office and closed the door.\nPART V\n\n_The Sleep of Reason_\n**AN OLD ACTOR TO HIS AUDIENCE**\n\nJOHN WILLIAMS\n\n_Ford Maddox Ford: 1873\u20131939_\n\n_Sirs, I address you out of age, my voice_\n\n_Gone slack and hoarse, who stood before you once_\n\n_With some grace and carriage. Ah, time . ._.\n\n_The face that once was marble now_\n\n_Is flesh. Motion is impure, and we_\n\n_Must move, although we break. The voice that was_\n\n_Your master is your servant now, reminding you_\n\n_Of its ancient art that once cast up_\n\n_A substance that could move you out of time_ ,\n\n_Our mortal blemish. And you\u2014the wise and foolish_\n\n_Who listen to an old man's wheezing voice\u2014_\n\n_Suffered your destruction like a pleasure_\n\n_Scarcely to be borne, desiring to be deceived_\n\n_Out of the falsehood of your time and place_.\n\n_But now I am old, am old, and suppliant_\n\n_To your most gracious whim. We are the relics_\n\n_Of our ruined past\u2014although I see you now_\n\n_As if you were not changed, as if you were_\n\n_As I created you once long ago_\n\n_Out of the pride and arrogance_\n\n_Of my spent youth. To whom do I speak, if not_\n\n_Myself? If not my own, whose faces stare_\n\n_At me? Had you given me laurel once_ ,\n\n_I would have worn it most carelessly_\n\n_And spoken my echoing lines in its despite_.\n\n_But now this pate is bald; bald pates have need_\n\n_Of bay, for warmth and show. I ask_\n\n_Your kindness now, and ask forbearance of_\n\n_These loosening years; they make men foolish_ ,\n\n_Who were never wise. I stand before you_ ,\n\n_Stripped of years, a beggar_.\n\n_And yet a supplicant_ ,\n\n_I would remind you, who has given service_\n\n_To you all. Out of these creaking boards_\n\n_I once created worlds that you could not conceive_\n\n_And peopled them with what you might have been_ ,\n\n_Showing a fairer image of yourself_\n\n_Than you would dare to dream, and given you_\n\n_Some instant plucked from time that was your own_.\n\n_From your deep heart's most lonely need, I have_\n\n_Dissembled shadows that became your selves_\n\n_And let them stroll as if they were alive_\n\n_In the Roman ruins of your northern fields_.\n**CHAPTER SEVENTEEN**\n\n\"How Can Such a Son of a Bitch Have Such Talent?\"\n\n_Ah life, that amateur performance_.\n\n\u2014JOHN WILLIAMS\n\nOn April 11, 1973, the _New York Times_ carried a story about controversy in the book world. \"In an unprecedented display of public disagreement, the 1973 National Book Award judges announced yesterday that they had split the fiction prize between John Barth's _Chimera_ and John Williams' _Augustus_.\" This had never happened before in the organization's twenty-four-year existence. But lately, nothing seemed immune from dissent. The week before the announcement, the _Saturday Review_ had predicted that literary politics would decide the fiction prize because the judges fell into two camps: postmodernists (literary critic and historian Leslie A. Fielder, along with essayist and novelist William Gass), and traditionalists (Evan S. Connell, philosophical novelist Walker Percy, and book critic Jonathan Yardley).\n\nThe magazine was right about the likelihood of disagreement: the meeting was \"noisy and argumentative,\" according to the _Times_. The previous year, the historian and journalist Garry Wills had walked out of his committee's meeting when he refused to endorse his fellow judges' choice of the hippie bible, _The Whole EarthCatalog_, as the contemporary affairs winner. And now, as the judges in different categories adjourned, not only was the award for fiction split, but also the one for the best history. This had never happened before, either.\n\nHowever, as the book reviewer Jonathan Yardley, a courtly young man from North Carolina, stepped up on the dais in the Biltmore Grand Ballroom in New York to announce the winners, he tried to convey that nothing could have been more natural than a tie. The novels, _Chimera_ and _Augustus_ , he said, were both books of \"uncommon quality . . . similar in subject matter[,] but . . . represent dissimilar approaches to the writing of fiction.\" _Chimera_ was about transforming myth into reality; _Augustus_ brought the violent times of imperial Rome to life. Consequently, Barth and Williams would each get half the award money: $500 apiece (which wasn't much more than each of the judges had been paid to read the books). No explanation was given as to why there were two history prizes.\n\nThe double deadlock wrecked the organization\u2014not immediately, the big awards ceremony would still go forward\u2014but in the coming weeks. With the publicity value of an author winning cut in half, publishers protested by withdrawing their financial support. No more free books for the judges to read, or luncheons, hotels, transportation, and all the rest. The National Book Committee was forced to disband, and it was not until two years later, when a caretaker administrator for the organization \"begged\" prospective judges not to split awards, that the contest resumed.\n\nIn the meantime, news that John Williams had won a major literary award arrived in Denver \"on little cat feet,\" as Joanne Greenberg put it, thinking of Carl Sandburg's poem \"Fog.\" Despite the fact that Williams was the first and only Coloradan ever to receive the National Book Award, the _Denver Post_ , the largest newspaper in the state, failed to send a reporter to get his reaction. Likewise, in the English Department, there was no, as one instructor put it, \"ecumenical coming together in celebration of John.\" Instead, there was a lot of headshaking behind closed doors. \"Oh my God,\" one English professor later said, \"if he was difficult to live with before!\" Some of it was envy, but Williams' colleagues knew that receiving the laurels for fiction would mean that he would now be delivering his growly pronouncements about literature with even greater authority; the little genie wrapped in cigarette smoke would never go back into the brass lamp.\n\nThe editor of the university alumni magazine wasn't glad to hear the news, either. Normally, a faculty member receiving a national honor was tailor-made for encouraging alums to donate more to the endowment; over the years, however, the editor had tried to avoid Professor Williams. He was impossible. She disapproved of his romantic affairs, and of the way he missed class because he was hung over. She passed the assignment to a newcomer on the staff, a graduate student in history, Carol DeBoer-Rolloff, who later became a biographer and a professor at Brown (as Carol DeBoer-Langworthy).\n\nCarol didn't know Williams, but she had heard of him. \"A lot of people talked about John on campus,\" she said. \"People liked to gossip about him. He was considered outrageous.\" When she knocked on his office door for their appointment, she opened it gingerly, not sure what kind of person she would find inside. The room was long and narrow\u2014a rabbit hole that smelled of smoke and coffee, with sloppily arranged books on shelves and cardboard boxes on the floor. Seated at a desk was a dark-haired man with a head that seemed too big for his body. He looked up at her with enormous blue eyes that swam behind a pair of thick, black-framed glasses. His face was heavily lined. \"This guy is a philanderer?\" was her first thought. He invited her to take the only chair available. In the middle of his desk, jutting like a rock from a tide pool of papers, was a large, dark gray typewriter.\n\nThe interview went smoothly, though he enjoyed talking about books more than he did describing his past. In between remarks, he coughed loudly as he pulled on his cigarette, or had to stop to clear his throat before continuing. Most experimental novels, he said, seemed dreadfully stale and forced, and they were always better the first time around. It was so much easier dealing with theories of fiction, political issues, and so forth than with relationships\u2014that was the problem with the current state of fiction.\n\n\"What do you plan to talk about in your speech at the National Book Awards?\" she asked.\n\nHis relaxed manner changed suddenly as he leaned forward. \"A defense of the goddamn novel,\" he said.\n\nThe Williamses' visit to New York City for the ceremony was triumphant. They checked into the forty-seven-story Waldorf Astoria Hotel, known for its striking Art Deco design, lavish dinner parties, galas, and international conferences. The next morning a photojournalist arrived at their suite to take pictures of John for _Time_ magazine. Then they went to lunch with Marie Rodell, where John resisted the temptation to start drinking too early. He could wait. Cork Smith, back at Viking part-time, had provided him with an itinerary that included a \"Boozerama\" at the Tavern on the Green in Central Park after the awards, followed by late-evening drinks and a buffet at publisher Thomas Ginzburg's apartment on Madison Avenue.\n\nAt the Lincoln Center for the Performing Arts that evening, hundreds of people from the publishing industry attended a reception, where, among other things, Williams was introduced to his co-winner, John Barth, for the first time. Called \"Jack\"\u2014a bald, serious-looking man with sideburns down to his jaw\u2014Barth was a professor of English at the State University of New York at Buffalo. Only a few years younger than Williams, he had been nominated twice before for the National Book Award. His newest novel, _Chimera_ , was a fabulist, highly theoretical work, not Williams' cup of tea at all. But asked by a reporter about the controversy over splitting the award, they were in the mood to be good sports. If anything, they said, the decision demonstrated that fiction was alive and well, and that literature was roomy enough to accommodate the novel in many forms. Later, on the stage for the program, they sat together. When Williams' turn came to speak, he chose the high road; instead of launching into \"a defense of the goddamn novel,\" he expressed his deep pleasure at being selected and predicted a ringing future for fiction. \"My friend Brock Brower, whose novel _The Late Great Creature_ was one of the nominees for this award,\" he remarked, \"is said to have said, 'Listen, there's only one stable institution in this country. It's not Princeton . . . it's not marriage, it's the novel.'\"\n\nWhen he arrived home in Denver, there was a warm letter from Barth, who was glad that what might have been a \"sticky situation turned out to be really a delightful one.\"\n\nUntil now, John Williams had been a kind of extra or chorus member on the literary scene, a spear-carrier in the opera of American fiction, who played his part and then the dusty curtain fell down. So it had been for _Nothing But the Night_ , _Butcher's Crossing_ , and _Stoner_ , all three of which had come and gone with barely a tip of the hat from the public. Also, his success with poetry had been modest. A small magazine would accept a few stanzas occasionally\u2014but even his publisher, Viking, had returned a sheaf of his poems, only a month after the National Book Awards, with an apologetic note saying they preferred younger poets whose work was more experimental.\n\nNothing, however, could take away the significance of receiving a major literary award for _Augustus_ \u2014it proclaimed his rightful place in the annals of contemporary novelists. It was a vindication, and it seemed to him that he was now entitled to extra consideration from the university. After all, over the course of nearly twenty years, he had taught, edited the _Denver Quarterly_ , directed the creative writing program, and published literary criticism. These duties weren't out of the ordinary for an academic, of course; but his stature as a novelist added luster to the institution as a whole, and especially the English Department. He deserved a raise, or a bonus\u2014at the very least, more time to write. And, not coincidentally, two weeks after receiving the National Book Award, Brandeis University in Boston had invited him as a visiting professor for two semesters, beginning in September of that year, 1973. He would have the privilege of selecting his students and limiting the enrollment to \"numbers agreeable\" to him. The time to act on his prerogatives was clearly now\u2014he was in demand\u2014and he went off to see his department chairman, Gerald Chapman, to make his case.\n\nChapman was not surprised by Williams' visit; he had been expecting it. Williams asked to be relieved of some of his duties; he wanted a lighter schedule, such as having the spring quarter off, so he could begin his summer early. And Chapman appreciated how important it was to John that he be treated with extra consideration.\n\nBut as Chapman listened to Williams' requests for privileges that were unusual, he knew he'd have to draw the line. The English Department was small, and special treatment was impossible to give. The opportunity to teach for Brandeis could be arranged; but as for needing time to write and so on, Chapman said, \"We all have our problems.\" Williams left in a rage, complaining bitterly about high-handed \"Har-VARD-ians\" treating him like a peon. And it wasn't just the English Department. He had also approached the campus library about depositing his papers there, but they weren't interested; Nancy blamed \"some prude who didn't approve of John's reputation.\" His publisher reported \"sales resistance\" from Denver bookstores about carrying _Augustus_. \"What have you been doing and what have you not been doing that you should have been doing?\" Rodell asked him.\n\nThe offer from Brandeis had come at a good time, and it had the effect of sending John away for a cooling-off period. Arriving in Boston in the late summer of 1973, he and Nancy had the good luck to find an apartment around the corner from Dan Wakefield. Cunningham was still teaching at Brandeis, and that friendship was renewed, too. Nothing would please Williams more than if guest-teaching at Brandeis turned into a permanent position. And to increase the likelihood of that happening, he planned to present himself a little differently.\n\nIn the English Department at Denver, he had been adamant that experimental and modernist poets were not yet part of the canon, and had argued against creating new courses that would start with T. S. Eliot\u2014the disease-bringer in the corruption of poetry, in his opinion. By comparison to the older poets, most of the modernists, he insisted, fell short. But at Brandeis, he offered two courses in fiction that were, for him, quite a departure from his traditionalist stance about teaching the greats. During the fall semester, he taught \"Modern Fiction: Form and Theory,\" covering major European and American novelists during the first half of the twentieth century; and then in the spring, he tried out \"Contemporary Criticism and the Contemporary Novel,\" with readings from Lionel Trilling, Leslie Fiedler, Richard Poirier, and the Marxist literary historian Gy\u00f6rgy Luk\u00e1cs. The novels assigned were by Thomas Pynchon, Alain Robbe-Grillet, John Barth, Jorge Luis Borges, and several others. He was well thought of there, but his audition as an instructor didn't bring about an offer to join the faculty.\n\nTexas, on the other hand, embraced him as a native son. After winding up his work at Brandeis, he and Nancy went to Houston, where he received the Texas Institute of Letters award for fiction, given to authors who resided in the state or who had spent their formative years in Texas. Following the ceremony, they returned by way of Wichita Falls to see the farm where John had spent his childhood. It was still there, and had hardly changed in forty years. There stood the barn where he had slaughtered his first pig; behind the house were the acres where his grandfather had planted vegetables. Looking around, he talked about spending every moment he could outdoors, just to get away. \"I remember feeling sorry for my parents,\" he said to Nancy, \"because they had no privacy\" living with his grandparents. From this little farm, he had found his way to becoming a professor and the winner of the National Book Award. But instead of being praised for his literary achievement, he felt resented at Denver. They turned toward home.\n\nWilliams' students became accustomed to him appearing at the last minute for his nine o'clock class, his hair, combed straight back, still wet from showering to shake off the previous night's drinking. \"That damn metabolism of yours that need forego nothing,\" Ciardi had once said admiringly. One of Williams' doctoral dissertation students wasn't so sure. \"It was a terrible assault on his body,\" he said years later. And according to one of his seminar students, his health was \"very tenuous\": \"Sometimes in class it would almost bring him to his knees. He was very frail. He smoked heavily and would be coughing.\" The high altitude, the raw winters in Denver from October to April, and the effort of dealing with deep snows on their property\u2014the last house on a dead-end street\u2014were becoming too much for him.\n\nAt parties, Williams became boorish while drunk and could be tiresome. \"He wanted to come over and sit next to you and blah, blah, blah, blah, blah at you,\" said one of the English professors. Another colleague tried to be philosophical. \"How can such a son of a bitch be such a great writer? Well, he'll piss on all our graves, that's for sure.\" At times it seemed inconceivable that John Williams, the man getting insensible with bourbon in the corner, could also be the author of _Stoner_ or _Augustus_ , novels of almost magisterial restraint and control. Joanne Greenberg attributed his contradictions to a secret that he beat down by drinking\u2014that it was hard for him to act tough, to pretend that he couldn't care less what people thought. \"John's admiration of the Romans and Augustus was what he wanted to be. The reticence, the capacity to endure pain, the somewhat cynical approach to life. But he was too sensitive for that.\"\n\nHis students learned to accept that Professor Williams was moody. Normally, he was quiet and straightforward. If a student in his poetry-writing seminar presented a piece that wasn't very good, reading it aloud for criticism, he would listen, \"gravely,\" one former student recalled, \"and then, without remotely condemning it, open up the conversation to what the rest of us thought. It was a most civilized occasion.\" But he could also be testy, uncompromising, and miserly with praise. \"Anything other than the obvious to add, Mr. Weaver?\" he inquired of another student, who was constantly raising his hand. Sometimes, he could be entertaining, reading verse aloud or reciting it in his deep, sepulchral voice. It was also known that he wouldn't object to a student bringing a jug of wine to pass around at his late-afternoon seminars, and then a Johnsonian atmosphere of rambling conversation replaced the assigned readings. Warmed by the wine, when the breezes of storytelling were running high in him, Williams would invite the class to join him at the tavern down the street. \"You repaired to the Stadium Inn and it was wonderful to be with this unassailable figure!,\" said a former student. Williams liked a good bull session and would retell favorite anecdotes about books and authors until it grew dark and long past dinner. His office desk disappeared under a mound of papers: applications to the graduate program, requests from students asking for updates on their work, and correspondence. His replies often began with the same regret: \"I apologize for the delay in responding\" (six weeks); \"I am sorry to be so long answering your letter\" (three months). He delegated the editorial side of the _Denver Quarterly_ , but when his overworked assistant demanded to be paid, and the college administration refused his request, Williams' volunteer quit. A student from Stanford University, Baine Kerr, who came to see him about the creative writing program, was taken aback by the appearance of his office. \"It was a disaster,\" he said. \"Books piled everywhere. Eight or ten stained coffee cups and papers scattered all around.\" Kerr was thinking about transferring to Denver, and his initial impression, based on the work environment of the program's director, at least, wasn't favorable. \"But when he started talking about writing,\" Kerr said, \"I wanted to be a part of what he was offering.\"\n\nWilliams' dilatoriness about attending to the creative writing program eventually caught up with him. An advisee of his, a candidate for a doctorate who resided in Canada, had been mailing chapters of his novel to him for review. As the weeks passed, the envelopes landed on the white drift of unopened mail atop Williams' desk. Hearing nothing, the student submitted his finished manuscript in fulfillment of part of the requirement for the degree. On his dissertation committee was Seymour Epstein, who realized\u2014after he located the novel in Williams' slush pile and read it\u2014that its inexperienced author hadn't received any guidance, and now his manuscript was completely unacceptable. Furious, Epstein made phone calls up and down the line in the College of Arts and Sciences to figure out what to do. A compromise was reached whereby the candidate would substitute his short stories as a collection, with a preface to fulfill the requirement. In the nick of time, a hastily agreed upon solution saved the day; otherwise, the reputation of the whole creative writing program, and the value of a PhD from it, could have been jeopardized. As of 1975, Williams was no longer director of the program.\n\nThat year, too, his agent Marie Rodell died. He had become very close to her over the years, stopping in New York City when he could to take her out to lunch. They signed their letters \"Love,\" to one another, and John, despite the disaster of _Butcher's Crossing_ , had trusted her implicitly. For him, she was synonymous with his career as a published author\u2014from his cautious beginnings to winning the National Book Award. Her assistant Frances Collin took over the agency, but it wasn't the same, of course. The two events coming close together\u2014Marie's death and being relieved of the creative writing program\u2014left him feeling disconnected, restless.\n\nOne afternoon, while he was serving on an oral examination committee, a small but telling incident occurred. Williams was one of four instructors present to listen to a student in English literature who would be defending a dissertation on fabulism in short stories, using _Don Quixote_ as a bridge between fantasy and realism. Eager to impress, the candidate took the full forty-five minutes to make his presentation, answering at length every question he was asked. Slowly, the sky began to darken outside, and it looked as if that Colorado was about to have one of its late spring rains mixed with snow. The wind rose and the budding trees swayed. Then the blue-gray clouds thrashed the window with a downpour that made the glass rattle.\n\nWithout a word, Williams got up from his chair and went to look the storm, while the voice of the student droned on behind him. To no one but himself, but overheard by everyone in the room, Williams said quietly, \"Oh, my tomatoes . . .\"\n**CHAPTER EIGHTEEN**\n\nIn Extremis\n\n_What I saw was the work\u2014the hours and hours. So much of his life was dedicated to the work_.\n\n\u2014NANCY WILLIAMS\n\nHaving a famous author on the faculty who was unhappy, and vocal about it, presented the University of Denver with a problem, which it solved with a grand gesture. Effective at the beginning of the school year in 1976, Williams occupied an endowed chair. He would only have to teach two quarters each academic year, for three years, and have every third quarter off. John and Nancy began using those ten weeks to visit, first, the Gulf Coast of Texas, then Portugal; several times they circled back to Key West, as if they were experimenting with flyways they would use later as migrating birds to a safe place where John could work on his fourth novel. Of their peregrinations to various locations for the sake of John's health, and his peace of mind, it seemed at last that Key West would suit them the best.\n\nNevertheless, there was gossip in the English Department office about Professor Williams taking off for Florida whenever he pleased\u2014further evidence, said some knowingly, that the National Book Award had conferred a specialness on him and he was taking advantage of it. His old friend Seymour Epstein said that Williams had become \"insufferable,\" and he was not alone in thinking so. His advisees either had to accept that he would be away for long periods, or, as some did, switch advisers.\n\nIgnoring the complaints, Williams went forward with his plans to relocate. With their children now grown or in college, John and Nancy began moving sideways into semi-retirement. They sold the big house on Madison Street in Denver and moved into a condominium on Pearl Street, where John could stay when he taught during the spring and fall. In Key West, they purchased a small white frame house on the corner of Florida and Duncan Streets, where they \"spent about six months remodeling, refurbishing, furnishing, cursing, and so forth,\" as John put it in a letter to his friend Fred Inglis. He put colored spotlights at the foot of the palm trees shining up into the leaves to catch the Caribbean feel; Nancy noted to a friend, \"it is _de rigueur_ here to display a number of exotics in tubs arranged cordially on the patio, if you have a patio,\" which they did, made of mossy brick. When John thought of what it would be like having to return to campus and put up with the pettiness\u2014to say nothing of the effect of pollution and cold on his lungs\u2014he began to dread the idea. \"There are no more than two or three other people that I give a damn about seeing back there,\" he told Inglis.\n\nThe verdant little town perched on the flat, fish-shaped island of white sand held strong and agreeable associations for John. It was here that he had arrived shortly after the war, because it was both familiar and exotic. Key West had represented a new beginning in those days as he was finishing his first novel, a time when several roads had been open to him. He might have continued as a news announcer, using his near-perfect voice and delivery to build a career in radio; or he might have taken George Smart's advice and enrolled at the University of Alabama, and then applied for a spot in Hudson Strode's fiction workshop. He also might never have left the island, and become a denizen of the beach-and-town scene, spending his days on a stool at the bar in a local taproom, drinking and telling anyone would who listen how he had an idea for a book.\n\nKey West felt like a homecoming for John for another reason, too; it was a \"winter repository of some very good writers, who are also very good friends,\" he wrote to Inglis. The Williamses stayed for a month at John Ciardi's new three-bedroom bungalow on Windsor Lane, partly because the corpulent poet with the air-conditioned Cadillac was always eager to show off his wealth. \"It's you I love,\" Nancy would say to him, \"not your money!\" The two men reminisced, ho-ho-hoing about Bread Loaf days, as she languished in the heat, but \"John sweats and is happy,\" she said.\n\nDan Wakefield came down from Boston hoping to interview John for _Ploughshares_ , but Williams wouldn't hear of it until they had drunk deep in the waters of the laid-back life in Key West. Wakefield eventually got his interview, which he published in _Ploughshares_ :\n\nFor three days he managed to avoid [the interview] altogether, as he took me around to some of the bars, beaches, and restaurants of the island, dropping in on old friends like the poets Richard Wilbur and John Ciardi, meeting his new friend and Key West neighbor Peter Taylor, the short story writer, going to a party at poet James Merrill's house, drinking wine and talking and eating the Conch Chowder that is the local specialty and John has now added to his culinary repertoire, a favorite right up there with his Texas Jailhouse Chili.\n\nThe working title of Williams' new novel was _The Sleep of Reason_ , taken from an epigraph by Francisco Goya: \"The sleep of reason brings forth monsters.\" The setting is Washington, DC, during the Nixon years. The central figure is Paul Mathews, in his middle fifties, a senior curator of the paintings in a small but distinguished Washington museum. As a nineteen-year-old corporal during World War II, he had been captured by the Japanese in Burma and held prisoner for several months. It was a period he tries not to think about.\n\n_The Sleep of Reason_ opens on a day when Mathews is trying to establish the authenticity of a painting that has just arrived at the museum, _Peter at the Tomb of the Resurrected Christ_ by the great Paduan painter of the Quattrocento, Andrea Mantega. To Mathews, something about it is indefinably wrong, but he can't put his finger on it. His mind is further made uneasy by the unexpected appearance of a man he knew at the time of his capture: former Office of Strategic Services captain Dave Parker, now a rather seedy and defeated-looking operative for the intelligence community. Parker hints that he wants something from his former subordinate. Memory and the long-ago past in art and history have suddenly overlapped, and the question of authenticity\u2014what is true and what isn't\u2014becomes the theme of the novel.\n\nWhen Baine Kerr, the young man transferring from Stanford who had been enchanted by Williams' passion for writing, went out deep-sea fishing with his instructor one night, Williams talked about the novel\u2014\"its aesthetics, and his hope that this would be his masterpiece.\" Usually, he was reticent about personal things, but on the fishing boat with Kerr, he talked about his childhood years and his parents' poverty, and the hurt he'd felt over the Winters affair, especially because he had tried to promote the work of Winters' wife, Janet Lewis. Back at the house, he read aloud a chapter from his new work, his rhythmic, rumbling voice accentuating the cadence of the sentences. It was memorable, Kerr thought\u2014it was an honor to be treated as an equal. \"It was a friendship,\" said Kerr's wife at the time, Cindy, \"with literature at its center.\"\n\nDisturbing, though, was how Williams' drinking was affecting his behavior. A few years earlier, Kerr had thought nothing of his instructor pouring a glass for himself from the wine jug at seminars, or accompanying his students to the Stadium Inn for a bull session, with pitchers of beer circling the table. But he was not prepared for the \"vehemence of his views and his verbal violence\" when he was dead drunk. Saddest of all, some of his rage was directed at Nancy. He would become repetitive, and when she tried to correct him on some point, he would lash out, calling her profane names and telling her to shut up. Kerr vacillated between admonishing his old instructor in his own house or holding his tongue. \"It was very excruciating at times\u2014to be at the dinner table and hear that,\" he said. \"You wanted to protect and defend Nancy from abuse, but I didn't know how to do that. Why did she take it? Nancy devoutly believed in his genius. She had an understanding of the dark places where that came from\u2014his war experiences, and his childhood.\" Looking on, Cindy concluded that Nancy's devotion to her husband\u2014normally a \"sweet man, never given to ridiculing others\"\u2014stemmed from her belief that he would leave a legacy, and it was up to her to stick by him.\n\nA few weeks after the Kerrs returned to Denver, Williams fired off a letter to the English Department excoriating them for daring to amend the curriculum to include form and theory classes with a modern emphasis while he was out of touch in Florida. But having thrown his lightning bolt to no effect, he once again withdrew into silence. He had more important things to think about. The literature committee at Yaddo, the four-hundred-acre artist colony in upstate New York, had invited him to stay as their guest from the next May through September, or for as long as he liked during that time. The contrast seemed symbolic\u2014the annoyance of academic politics versus spending months in the company of fellow writers and artists.\n\nBut first he would have to attend to a health problem. During a check-up for his emphysema, an X-ray showed the presence of a dark spot on his right lung.\n\nGiven Williams' history of smoking, his physician said the abnormality was almost certainly cancerous. A flexible probe sent painfully into the bottom third of his lung was unable to retrieve tissue for a biopsy, so he was scheduled for an operation to remove \"a wedge,\" his doctor called it. Williams preferred to have the procedure done in Denver.\n\nNancy had been through a similar health issue earlier, and their support of one another at times like this was unsparing. Following an operation to remove a melanoma, her incision had become infected and she had experienced toxic shock syndrome. During the three and a half weeks she remained in the hospital, John canceled classes and asked friends to help with the children, who were still at home then. Waking up from time to time in her hospital bed, she would look over at the chair by the window and he would be there. \"John never left my side,\" she said.\n\nAdmitted to Denver's Presbyterian\u2013St. Luke's Medical Center, Williams sat propped up in bed, wearing a bright blue bathrobe and looking oddly out of uniform without a sports jacket and trousers. His large eyes took in everything from behind his black-framed glasses with an expression of mild boredom and resignation. A few hours later, after the procedure, he was wheeled back into his room. The small nodule in his lung wasn't cancerous, it turned out; but now, he told Fred Inglis, \"I have a nice scar like a saber wound that violates my fair white body.\" He was given the usual advice to stop smoking (which he ignored), and was told to spend as little time in Denver as possible\u2014Key West was a better climate (advice he gladly took). And, the doctor said, it would take a few months to recover his energy (it took a year).\n\nHe spent the first half of 1980 resting, on medical leave from the university. During that time, he and Nancy paid a visit to Boston and Dan Wakefield. Wakefield had assumed that after Williams' lung operation, he would have given up chain smoking, but he hadn't. \"John would take a puff, then have a paroxysm of coughing, then squirt some medication down his throat, then take another puff, and on and on through the afternoon and night. The next morning the house smelled like gas. I went out to the Public Garden clenching my pipe in my teeth but not smoking and threw away my pipe in a big garbage can and never smoked again. It was like shock treatment.\"\n\nIn July, Williams arrived in upstate New York for a two-month stay at Yaddo to work on _The Sleep of Reason_. \"There is no rule against looking for inspiration,\" said the retreat's welcoming booklet, \"but Yaddo cannot guarantee that anyone will find it.\" The writing went slowly\u2014a handful of sentences before noon. He killed time after lunch by sitting next to the pool in an open shirt and silk scarf, smoking and thinking. A swimmer asked him one afternoon if he was John Williams, who wrote _Stoner_. \"So you read it!\" he said affably. As 4:00 p.m. approached, he ambled over to the Great Hall, invariably the first to arrive for cocktails before dinner. Above the fireplace where he liked to stand was a glass mosaic depicting a phoenix. He reached for the bottle of Scotch on the hospitality table and waited for others to arrive. Once, the _New Yorker_ artist Roxie Munro encountered him on his way to cocktail hour and asked him how his day went. \"Great!,\" he said. \"I wrote eight perfect sentences.\" Weekends, he went to the famous racetrack nearby to watch the horses run. He sat on the porch of the clubhouse, where drinks would stay cool in the shade all day.\n\nThe John Williams who entered the classroom in the early 1980s pulled behind him a small silver oxygen tank on wheels. He used the clear mask attachment, which he held in one hand, alternating between inhaling through it and then taking a drag from a burning cigarette. Like the radio operator he had been during World War II on supply planes in Burma, he puffed away instinctively. Seated at his desk in the classroom, he spoke, smoked, and took a breath of oxygen in a rhythm, the way he had at thirty thousand feet flying over the Himalayas.\n\nWilliams was on the glide path now, finding enjoyment in serving as an example of a thrice-published novelist in the role of master to his pupils. \"He tended to teach anecdotally; he told wonderful stories that evoked, vividly, the writers, their craft, and their circle,\" said one former student. With _The Sleep of Reason_ stalled at one hundred pages, nostalgia, for Williams, was overtaking inspiration. There was no hurry to finish the manuscript, in any case; Corlies Smith, his longtime editor at Viking, had left to become editorial director at Ticknor and Fields, and his much younger replacement, Amanda Vaill, had a cordial relationship with him, but that was all.\n\nFinally, circumstances seemed to indicate that 1985 would be a good year for him to retire. He was sixty-three; he wasn't the head or editor of anything any longer, and he didn't want the responsibility. Although he hadn't published a novel for thirteen years, he was honored by the Academy of Arts and Letters, which inducted him as a member and awarded him $5,000. (Ciardi hinted to the secretary of the academy, \"He is delighted, of course, and would like to announce the award in Denver papers as a sort of crown on his career there.\") He also received a prestigious Guggenheim Fellowship, worth $25,000, to continue writing _The Sleep of Reason_. To help him clear out his office, he hired a local book dealer to take what he wanted from the shelves.\n\nThen, just as he was preparing to return to Key West, he received a phone call from out of the past. His second wife, Yvonne, was in town, and she wanted to see him.\n\nYvonne was now in her late fifties, and looking much younger than the gentleman pulling the oxygen canister behind him who entered the Cruise Room of Denver's Oxford Hotel, casting a fishy eye around the establishment in search of her. The hotel had been a flophouse when John and Yvonne were living together in Denver; now it was the swankiest accommodation of its kind in the city and listed on the National Register of Historic Places. John made his way over to Yvonne's table and eased into the seat opposite her. There was a young woman beside her he didn't recognize. Yvonne introduced her to John as her daughter Gale.\n\nAfter their divorce in February 1949, Yvonne had married Douglas Woolf, in March. She had now been his wife and untiring supporter for over twenty years. The pattern of their life together had been set the autumn before that, in 1948, when she had fled Denver to meet him in Tucson, Arizona, where, to put food on the table, they had sold plastic housewares door-to-door in the withering heat. Douglas Woolf was an itinerant, and a loner who believed in a semi-mystical way that his surname was part of his character. Yvonne was one of those persons born to be the sidekick, intercessor, friend, and lover of someone who might be a genius. She was tough to an amazing degree.\n\nThey had traveled around the West like gypsies. Douglas would take temporary jobs that required no commitment other than to show up: migrant farm worker, ice cream seller, beer and hot dog vendor at baseball games, egg man delivering off a truck; Yvonne worked as a bookkeeper, clerk, or product demonstrator in department stores. When they got enough cash, they would throw it all over and take to the woods or the desert with their two girls and camp out\u2014blankets spread over tree branches for shade, or bedding down in ghost towns. Using the car as his workplace, he wrote and published two novels about escaping the bourgeois life from the perspective of the footloose hero, _Wall to Wall_ and _Fade Out_ , and a short story collection, _Signs of a Migrant Worrier_.\n\nBut Douglas was an alcoholic, and Yvonne had left him in the late 1960s. She was in Denver to attend a conference as part of her job with the federal bankruptcy courts and thought she'd say hello to John.\n\n\"I have another daughter besides Gale,\" she said. \"Gale was born in 1949.\"\n\nJohn blushed. \"What month, 1949?\"\n\nThis moment was Yvonne's true reason for the reunion. \"Let's have a little fun with him,\" she had said to her daughter, as they watched her former husband make his way to the table.\n\nBefore she answered John's question, she paused, as if counting from September 1948 when she'd left him. \"Hmm, October.\"\n\nThey could see him doing the arithmetic in his head. And then, \"Oh, certainly\u2014I see.\"\n\nWith Denver behind him, Williams returned to the island, \"out of the snow and smog and into the warmth and air.\" The university had offered him an honorary degree, but his response had been, \"How much?\" In other words, could the award translate into money, to help him get on with his writing? In any case, no award of any kind was forthcoming, and Williams interpreted it as another example of being snubbed.\n\nKey West was becoming an even better place for the community of writers now. The annual Key West Literary Seminar\u2014four days of readings, conversations, lectures, panel discussions, and parties\u2014held its first conference in 1983, and Ciardi persuaded Williams to serve on the faculty for the following year. He posed on the beach for a group photo, beaming among the Pulitzer Prize and National Book Award winners\u2014quite an auspicious start for a new event\u2014and exactly the kind of thing, he joked, that he meant to get away from in retirement.\n\nBut he had come to the party too late, in a way. You needed to be young, or at least healthy, to really enjoy the place. And Williams was neither. He needed hospital services periodically because of his bad lungs; moreover, the sea-level mugginess was oppressive, especially when he was hung over. Key West couldn't offer him what he needed professionally, either. There was no large library, no big bookstore.\n\nNancy began to have similar complaints. In Denver, she had been the director of the Rocky Mountain Women's Institute, and Key West, which was becoming more commercialized and touristy, left her bored. \"I was just hanging out for a year,\" she said. \"It was awful. The small island gets smaller.\" To move again, though, seemed unthinkable after purchasing a home and fixing it up. They had only been settled for less than two winters. Even if they decided to relocate, they couldn't decide where they would be better off.\n\nThe monotony was broken by an invitation from the poet Miller Williams, asking John to deliver a series of lectures at the University of Arkansas. There would also be a few old friends from Bread Loaf on the program\u2014a partial convening of the Good Guys. And if that weren't reason enough to visit Fayetteville, Miller wanted to discuss something else. As director of the University of Arkansas Press, he was interested in bringing _Butcher's Crossing_ and _Stoner_ back into print.\n\nWith alacrity, John accepted.\n\nFayetteville is located in the Ozarks in a forested region of highlands, plateaus, rivers, and lakes covering much of the southern half of Missouri and an extensive portion of northwestern and north central Arkansas. The winter weather is mild and generally dry. When the rain stops in midsummer, the high heat begins. Eight hundred miles northwest, within an easy two-day drive, is Denver.\n\nNancy found the university town of forty thousand \"old and [with] a beauty and character of its own, not grand but rather pretty and nice.\" As an experiment, she and her husband began looking for a house to rent, and they found one available from John Clellon Holmes, founder of the creative writing program there. Holmes' semiautobiographical novel from 1952, _Go_ , had been the first to depict the restlessness, disillusionment, and drug-charged lives of the Beats.\n\nAlthough Holmes had only just turned sixty, he was on his last legs\u2014an alcoholic who was returning to New England where he had been raised to spend what would be the last two years of his life. Allen Ginsburg's long, declamatory poem _Howl_ , which alludes to _Go_ , begins with the line \"I saw the best minds of my generation destroyed by madness,\" and Holmes' journals, which he intended to publish, were filled with tales of self-destruction.\n\nBut now he had become a portly, grandfatherly-looking man with glasses. He and his wife, Shirley, showed their visitors the property. It was near the center of town and the university campus, an older home across from a small park with four bedrooms, two baths, and a fireplace. Holmes took them down into the basement and gave the furnace a derogatory kick, saying it didn't work very well. The house didn't have insulation, and the heating bills during the winter would be higher than the monthly rent. But the Williamses, having spent a good part of their lives in Denver, thought they could use the fireplace to take the chill off.\n\nThey signed a lease for their \"little gray home in the Ozarks,\" as Nancy called it, in 1985 and began the arduous business of moving from Key West fifteen hundred miles away. Fayetteville had a \"full complement of medical specialists,\" a research-grade library on campus, and the built-in advantage of a few old friends in town. It seemed perfect. And the University of Arkansas had indeed decided to reprint _Stoner_ and _Butcher's Crossing_ , long out of print, which would be instrumental, though Williams wouldn't live to see it, in creating renewed interest in his work. \"So I seem to be returning to where I began, with a smallish press,\" John noted with satisfaction. \"I'm just as pleased; it has a nice symmetry.\" By early spring, they were settled in the new house, and spending most mornings having their first cups of coffee in bed, until the ancient furnace bestirred itself to heat the rooms.\n\nNear the end of Williams' finest novel, _Stoner_ , the University of Missouri English Department gives a farewell dinner, a grave affair, which tired and ill Professor William Stoner attends. Some of Williams' former colleagues at the University of Denver thought he should have received more recognition at the end of his career. He was, after all, the most famous person in the English Department\u2014the first Colorado author to have won the National Book Award. He had taught for almost thirty years and had been a major figure in adding creative writing to the academy. A junior instructor, remembering how grateful he had been for Williams' encouragement when he had first arrived in Denver, set out to coordinate \"A Celebration of John Williams\" for a Saturday in March 1986. The event would pay tribute to his body of work with a symposium; Nancy later liked to call it \"John Day.\"\n\nTo anchor the event, John Ciardi was invited. He planned to deliver the keynote as part of a six-city speaking tour, beginning with a group of librarians in Topeka, Kansas, who were paying him a large fee. He needed a wheelchair to make his connecting flights at the airport. But it became too much effort, and he couldn't make it back to Denver. \"What good is [the money] to a dead man?\" he wrote to his biographer, Edward M. Cefelli. \"I just don't have Sgt. Ciardi's resilience these days\u2014nor the bastard's legs.\" He canceled his appearance at \"John Day,\" and three days later he died of a heart attack at home.\n\nRichard Yates, however, did arrive. He would be a panelist for the symposium. Yates, the author of postwar novels such as _Revolutionary Road_ and _The Easter Parade_ , about the struggles of his generation to adjust to domestic life and peace, was America's \"least famous great writer,\" according to _Esquire_. For ten years he had been living in cockroach-infested apartments, checking himself into the Boston veterans' hospital when he thought he was going mad. Because his reputation preceded him, he was put under the supervision of Williams' former graduate student James Clark, who later said he'd been told \"'to keep him company' and 'out of trouble,' which I took to mean 'relatively sober.'\"\n\nThat evening, Williams arrived at the Denver Public Library to begin a reading of a chapter from his unfinished novel, _The Sleep of Reason_. Before an audience of a hundred or so, he came up to the front of the room, wheeling his oxygen tank behind him although his breathing had been better than usual recently, and took a seat at the high table so he could be seen. Yates, who had gotten drunk at dinner, started to cause a disturbance; Clark had to quiet him before Williams could begin.\n\n\"I'm going to read from a novel that's in progress,\" he said\u2014Chapter 2, because it was \"self-contained,\" he told the audience. He'd been working on the story for ten years, in fact, possibly more. \"My energies are not what they once were.\" It had been difficult for him to make progress. When he went into his study, he might work, or he might try lubricating the muse with a drink. He had completed one hundred pages or so\u2014not much more than what he had in hand when his former student Baine Kerr had listened to him read an excerpt in Key West several years before.\n\nFor the next forty minutes, he read his selection slowly, in his warm, dark voice, never varying or stumbling. He'd been over and over it, hundreds of times. But there's something missing in _The Sleep of Reason_ , the story of art museum curator Paul Mathews. There's a tone of weariness. Chapter 2, the one Williams read aloud, proceeds at an even pace as Mathews recalls a mysterious mission he participated in as a nineteen-year-old corporal\u2014he was to deliver a briefcase of secret information to a jungle location in Burma. But there is no sense of inevitability, no urgency about events. The details are vivid, drawn as they are from the author's experiences. And, strange as it may sound, perhaps that's the problem. Williams had an aversion to writing about himself. He told Nancy, \"Fiction and autobiography don't go together in any sensible way. I bore myself when it's about me.\" Careful of the persona he maintained as a man of letters, he seems reluctant to drag himself, as the narrator, back into the jungle muck. He'd been alluding to his stretch in the Army Air Corps at Sookerating for years. He'd talked and talked about it, and embellished his experiences a bit. But, as he said in an interview, \"it's difficult to lie when you write a novel.\" His integrity wouldn't allow it, drinking wouldn't inspire it, and so _The Sleep of Reason_ refused to be written.\n\nGeorge Rae's second husband, Ted Cogswell, died in 1987. They were living in Pennsylvania when he passed away, and \"the idea of another cold, cloudy, snowy winter alone here is more than I can bear,\" she wrote to the Williamses. \"So I'm thinking of heading down toward south Texas when I unload the house.\" Or maybe Mexico, but she was afraid to be alone there. John said she should come to Fayetteville and be near them. When she arrived, it was clear that heavy drinking had affected her. She told Nancy she could understand what the birds were saying. She wanted to try acting again.\n\nUnfinished business bothered John, too. \"I need to ask a favor of you,\" he wrote to his agent Frances Collin in 1989 when he was sixty-seven, \"but the asking needs a little background.\" His textbook anthology, _English Renaissance Poetry_ , published in 1963, had become a classic in classrooms during the ten years it had been in print. He pointed out that \"a book can sell only a few hundred copies and still be an 'important' literary text.\" It had been allowed to go out of print, and he was trying to get an answer from the publisher about how to have the rights revert to him. His letters had gone unanswered. The University of Arkansas Press wanted to reprint it. \"There is some urgency involved,\" he wrote. The problem of rights was taken care of easily; but by involving himself in the matter, he felt he was working, urging on the writing career that, by this time, was over. During a return visit to Key West, he added \"OLD?!! Humbug! Nonsense! Young whippersnappers!!\" to the bottom of a postcard to his Denver friends. He and Nancy were vacationing with his daughter Kathy, a PhD in literature now and a college professor.\n\nThat was the last time John would be there.\n\nMedication was giving him strange dreams. He dreamed that he and Nancy were staying in a casino. They needed money. A friend had told him that playing \"the one-armed bandit\"\u2014the slot machines\u2014would be a good investment. Their investment would pay 10 percent regularly. But he couldn't leave his room to go down to the game room, so he asked Nancy to go instead. She tried, but she lost all the money and came back to him for more. He woke up worried.\n\nHe fell in 1992 and required a long hospital stay. To be near the university hospital, John and Nancy moved into the smallest house they'd ever owned. As John had grown weaker and less able to negotiate the features of larger homes\u2014stairs, high cabinets, and so on\u2014they had moved several times within Fayetteville, trying to find a floor plan that would accommodate him. His world had become more and more circumscribed, and they were now in their third house in less than ten years.\n\nIn January 1994, George Rae stopped by to check on John and found him lying on the floor like he was asleep. Nancy agreed it was time for him to receive hospice care in his bedroom. A visiting nurse assigned to him insisted that he get all the morphine he wanted. If he was having a good day, Nancy called some of his friends, including Miller Williams from his Bread Loaf days, and told them to bring over beer and sandwiches. They sat by his bed talking about sports.\n\nJohn said to Nancy, \"I never expected to live this long.\" She sat beside him.\n\n\"We said goodbye a hundred times,\" Nancy said. \"He'd be lying there, and it was so hard to breathe. He'd say it was almost not worth it. But I would hear that word 'almost.'\"\n\nHe died of respiratory failure on March 3, 1994, at the age of seventy-two.\n**EPILOGUE**\n\nJohn Williams Redux\n\nObituaries about John Williams seemed more interested in how the National Book Award was split for the first time in its history in the year he received it for _Augustus_ , 1973, and the \"unusual display of public disagreement among the judges,\" than in what his works had contributed to American literature. Perhaps it was because his novels were out of joint with the times and mid-twentieth-century literature in general.\n\nNone of his three major works\u2014 _Butcher's Crossing_ , _Stoner_ , or _Augustus_ \u2014held a mirror up to present-day society the way the struggles of Saul Bellow's _Herzog_ did, or James Baldwin's short story collection about race, _Going to Meet the Man_ , both of which were being talked about in 1965, the year _Stoner_ appeared. That year, too, while readers pored over Alex Haley's _The Autobiography of Malcolm X_ , Alabama state troopers had clubbed civil rights marchers to their knees in Selma; riots had broken out in the Los Angeles neighborhood of Watts, and the first American combat troops arrived in Vietnam. Williams' stories about a buffalo hunt, an undistinguished professor, and a Roman emperor seemed almost belligerently indifferent to what was going on. Thus his uniqueness, which might have distinguished him in other, less restive times, became the millstone that sank him.\n\nEvery decade or so, the name \"John Williams\" and _Stoner_ would reemerge, the way a summer drought sometimes reveals a forgotten edifice standing on the bottom of an ancient lake. People had heard a rumor of it, and there it was again\u2014intriguing, puzzling, a curiosity from the past. In 1973, C. P. Snow had asked about _Stoner_ in the _Financial Times_ , \"Why isn't this book famous?\" In 1981, Dan Wakefield had combined an overview of Williams' career as an author with an interview of him in the literary quarterly _Ploughshares_. Morris Dickstein, a literary and cultural historian, devoted a 2007 _New York Times_ article, \"The Inner Lives of Men,\" to _Stoner_ , acclaiming it as \"the perfect novel.\" Then somehow, by a process that was \"mysterious, even alchemical,\" said a commentator on National Public Radio, _Stoner_ rose from the depths to become a best seller in Europe by 2013.\n\nBut there was more to it than that. The process of resurrecting _Stoner_ , and thereby John Williams, was by no means a matter of magic; it started as a result of conversations between people who love books.\n\nCrawford Doyle Booksellers, on a stretch of Madison Avenue on the Upper East Side of New York\u2014a neighborhood near the Metropolitan Museum of Art that used to be replete with independent bookshops\u2014has survived the slow incursion of boutiques, art galleries, and caf\u00e9s on the block since 1995. The street-level shop has a mix of rare and contemporary books in the window and a bargain bin outside. Past the door, it has the book-walled coziness of a floor-to-ceiling private library with an up and downstairs. The husband and wife owners, Judith Crawford and John Doyle, are handsellers\u2014bibliophiles who act as guides for customers hunting for something. To a woman who inquired about a nonfiction study of the Paris sewer system, Judith replied, \"Which one are you looking for? There were two published in English, and I could get you the one by Harvard's Professor Reid within the week.\"\n\nOne day at the shop in the early 2000s, Doyle happened to mention to Edwin Frank, editor of the _New York Review of Books_ Classics series, that he couldn't carry enough copies of _Stoner_ , a title he liked to recommend. Perhaps the publishing side of the _Review_ should consider adding it to their selected series of overlooked titles. Doyle and his wife had done well with the _Review_ 's curated collection of titles by Georges Simenon, Jessica Mitford, Nikolai Gogol, Stefan Zweig, and many others.\n\nFrank reached Nancy Williams through one of John's former students. From her, he learned that the University of Arkansas Press had reprinted _Stoner_ in 1993. He bought up their surplus stock and reissued _Stoner_ in 2006 under the _New York Review of Books_ Classics imprint.\n\nA book reissued with a spanking new cover is a little like wearing a new suit to the office: it gets attention. Frank's tight, handsome-looking edition of _Stoner_ , graced by American realist painter Thomas Eakins' _The Thinker, Portrait of Louis H. Kenton_ , caught the asceticism of the story; Irish novelist John McGahern's introduction redoubled the effect by informing readers that the novel ahead was \"about work, the hard unyielding work of the farms; the work of living within a destructive marriage and bringing up a daughter with patient mutability in a poisoned household; the work of teaching literature to mostly unresponsive students. How Williams manages to dramatize this almost impossible material is itself a small miracle.\"\n\nFrank was a bit disappointed when sales equaled about what they were when Viking published it in 1965\u2014a few thousand copies. Admittedly, he said, \"It's not an easy book to pitch\u2014a midcentury, midwestern novel about a man who is a medievalist and whose life is a failure.\" It was Morris Dickstein's praise in the _New York Times_ the following year gave the _New York Review of Books_ edition \"a jump and got it going,\" Frank said.\n\nMeanwhile, French novelist Anna Gavalda had read Colum McCann's list in the _Guardian_ of his favorite top ten novels, with _Stoner_ in first place. \"I have bought at least fifty copies of it in the past few years, using it as a gift for friends,\" McCann wrote. \"It is universally adored by writers and readers alike.\"\n\nGavalda purchased a copy in English, and she wished at the end that she had written it herself. Stoner's \"rectitude, his intelligence, his finesse, his tenderness. I didn't warm up to him, I fell in love with him. I like men who don't talk a lot, but who are attentive to the slightest detail.\" She persuaded her publisher, Le Dilettante, to license the French rights in 2007. But attempts to find a satisfactory translator brought her around to \"what I already knew, that William Stoner\u2014it was me, and it was up to me to stick to it.\" The task of rendering the novel into French\u2014\"I took liberties so that it would be as beautiful in my language as it is in his\"\u2014would take several years while she continued with her own writing. When it was released in 2011, the French edition of _Stoner_ became a best seller.\n\nEven before the French edition, however, Gavalda's attention to the work sparked interest elsewhere in Europe. In Spain, Tito Exp\u00f3sito, at Ediciones Baile del Sol, read an interview with Gavalda and decided, \"if she liked this novel, and I liked Gavalda, then surely I would also like _Stoner_.\" In 2009, the first translation of _Stoner_ in Europe appeared in Spanish; and then in February 2012, Elido Fazi of Fazi Editore published the first Italian edition. The Fazi edition drew critical acclaim in the Italian newspaper _Corriere della Sera_ by Paolo Giordano, a winner of Italy's most prestigious literary award, the Strega Prize, along with praise by Irene Bignardi in the Italian newspaper _La Repubblica_ ; Mario Fortunato in a weekly magazine, _L'Espresso_ ; Roberto Bertinetti in the business newspaper _Il Sole 24 Ore_ ; and Niccol\u00f2 Ammaniti, winner of the 2007 Strega Prize. A new word entered the world of books and publishing: \"Stonermania.\" Although the phenomenon began as a word-of-mouth recommendation from readers, it wasn't long before the character of William Stoner also began appearing in articles and discussions not strictly about the novel, but on the theme of the importance in a person's life of rectitude and incorruptibility.\n\nBut strangely, readers in the United States still seemed resistant to _Stoner_. During one of his regular visits to New York, Oscar van Gelderen of Lebowski Publishers in Amsterdam heard that some of the younger editors at HarperCollins were reading it for their own pleasure. Van Gelderen purchased a copy at a bookshop, went to his hotel, and read the book in one sitting. Despite the story being \"spectacularly unspectacular,\" he was surprised by how good it was. \"Stoner is a teacher. And then he dies. Well, let's hope the author is very good-looking,\" he thought, \"and in his or her mid-thirties to help sell that kind of story,\" which wasn't the case, of course. There would be no talk shows, no promotional brainstorming with the author because, as Van Gelderen was later informed, John Williams had been dead for twenty years.\n\nBut the novel was being read in Europe, and becoming popular in Italy, France, and Israel, even if Williams wasn't being honored in his own land. Consequently, after acquiring the rights, Van Gelderen became, as he put it, a \"Jehovah's Witness for _Stoner_.\" Lebowski Publishers placed it as the lead title in their September 2012 catalog, gave the book a strong, iconic cover, and sent galleys to booksellers that summer accompanied by a printed \"love letter\" about the novel from the publisher. It was the start of a six-month-long campaign, Van Gelderen said, \"from door to door, from one bookseller to the next, from one journalist to the next,\" to reintroduce a forty-seven-year-old book as though it were a new, contemporary piece of hot fiction. Salespeople were instructed to ask booksellers for blurbs, to get them involved, and to offer customers a money-back guarantee. Customers did come back\u2014not because they were dissatisfied, but to buy another copy. \"I wanted booksellers to feel proud that they were up on the latest\u2014give them a reason to say to customers, 'Listen, this is something special.'\" Van Gelderen posted the eye-catching cover of a gray-bearded older man against a jet-black background close to four hundred times on Facebook and Twitter.\n\nAfter six months, by March 2013, _Stoner_ was the best-selling book in the Netherlands, and it remained at the top of the list for five weeks in a row\u2014an unprecedented record for a \"lost classic.\" Van Gelderen continued to promote the book at the April London Book Fair. Sales of the book were so striking that journalists in the United States, at _The Millions_ , and _Publishers Weekly_ , for example, wrote about the success in Holland, which kick-started more reviews and _Stoner_ articles in the United Kingdom and the United States.\n\nClara Nelson, who was then with Penguin Random House in the United Kingdom, seeing that sales for the Dutch edition were taking off\u2014two hundred thousand copies\u2014decided to adapt Lebowski's approach of intense exposure, but keep the spotlight trained on _Stoner_ even longer. \"We aimed to do a piece of publicity every week for a year in the United Kingdom national press,\" part of which included giving Williams \"a voice again through champions in the literary community.\" During the campaign, reviews by Julian Barnes and Bret Easton Ellis, along the lines of \"upon first looking into Williams' _Stoner_ ,\" had a domino effect of persuading journalists that they ought to find out what all the fuss was about. Bryan Appleyard, a nonfiction author and reviewer, said in the _Sunday Times_ , \"This is the story of the greatest novel you have never read. I can be confident you have never read it because so few people have. In recent weeks, I have come across academics specializing in American literature who have never even heard of it. Yet it is, without question, one of the great novels in English of the twentieth century. It's certainly the most surprising.\" Ian McEwan echoed the same opinion on BBC Radio in June, telling his listeners that if they were readers who kept up on the latest books everyone was talking about, here was one that may have gotten past them.\n\nWaterstones named it Book of the Year in 2013, by which time rights had been sold in twenty-one countries\u2014including China\u2014and by riding the best-seller lists in Germany, France, Israel, Holland, and the United Kingdom, _Stoner_ 's success encouraged some publishers to bring out _Butcher's Crossing_ and _Augustus_ as well.\n\nEdwin Frank at New York Review Books has a theory about why _Stoner_ , in particular, is embraced in Europe, more so than in the United States. \"I think it's of an era that occurred _before_ its publication,\" he said. \"There's an existentialist edge to it, and I would point that out to European publishers, because I was confident that _Stoner_ would find a European audience for that reason. It's an American book like an Edward Hopper painting. It has that long-shadowed, lonely feeling. Loneliness is a big part of twentieth-century fiction. You might put _Stoner_ in the company of _The Plague_ , _The Stranger_ , and other enduring, existentialist books of that era.\" Cristina Marino, who obtained the book for Fazi Editore, believes that Italians don't share the optimism of Americans. \"We are more accepting of human failings, of people being fragile.\"\n\nFrank believes that _Stoner_ 's slow rise in popularity in the United States has been largely a word-of-mouth phenomenon\u2014proof of the fundamental importance of readers recommending and discussing books, especially at a time when social media promotes flash fiction, listicles, videos and pictures, and nonliterary discourse. \"It's a book about a person who loves books, and published at a time when people feel passionately that they need to defend the precincts of book culture,\" he said.\n\nFrom the last page of the novel, when Professor Stoner is dying:\n\nHe opened the book; and as he did so it became not his own. He let his fingers riffle through the pages and felt a tingling, as if those pages were alive. The tingling came through his fingers and coursed through his flesh and bone; he was minutely aware of it, and he waited until it contained him, until the old excitement that was like terror fixed him where he lay. The sunlight, passing his window, shone upon the page, and he could not see what was written there. The fingers loosened, and the book they had held moved slowly and then swiftly across the still body and fell into the silence of the room.\nAcknowledgments\n\nWilliams' papers at the University of Arkansas, Fayetteville, are deep and detailed. They contain extensive personal and official correspondence, notes and manuscripts of all the novels, official documents, photocopies of newspaper clippings, printed materials, essays on Williams' work, interviews, photographs, and a scrapbook. Papers belonging to some of his literary friends are available elsewhere\u2014including Yvor Winters, Alan Swallow, and J. V. Cunningham, at Stanford University, the University of Syracuse, and the University of Chicago, respectively. Papers belonging to a few of Williams' colleagues can be found at the University of Denver.\n\nSpecial thanks go to Jim Clark, the Red River Historical Society; Lindsay M. Morecraft, Special Collections, University of Iowa Archives; the University of Arkansas Special Collections Library; Geoffrey Stark, Reading Room supervisor, Special Collections, University of Arkansas; Halley Grogan, Texas State Library and Archives Commission; Katherine Crowe, curator, Special Collections and Archives, University of Denver; Mark A. Greene, director, American Heritage Center, University of Wyoming; Giana Ricci, Tamiment Library and Robert F. Wagner Labor Archives; Lita Watson, Wichita County Historical Commission; Chloe Morse-Harding, Robert D. Farber University Archives and Special Collections, Brandeis University; David K. Frasier, the Lilly Library, Indiana University; Becky Morrison, Wichita Falls Public Library; and Laura Russo, Howard Gotlieb Archival Research Center, Boston University. Tony Burton, literary historian of American writers in Mexico, was most helpful.\n\nWilliams' friends, colleagues, fellow authors, former students, and family members who graciously made themselves available for interviews include Michelle Latiolais, Cindy Carlisle, David Milofsky, Joe Nigg, Robert Richardson, Robert Pack, Dan Wakefield, Sherman Leavenworth, Sandra Cordon, Martin and Joyce Shoemaker, Jim Clark, Angela Ball, Brock Bower, Anne Marie Candido, Victor Castellani, David Haward Bain, Steve Wiegenstein, Doug Devaux, Gerald Chapman, R. H. Epstein, Thomas E. Kennedy, Miriam Epstein, Nancy Esterlin, Gale Woolf, Ben Kilpea, Eric Gould, Joanne Greenberg, Steve Heller, Fred Inglis, Jean James, Baine P. Kerr, Carol DeBoer-Langworthy, Nancy L. Easterlin, Heather McHugh, Jonathan Williams, Pamela Williams, Katherine Williams, Peggy McIntosh, David Milofsky, Roxie Munro, Edward M. Cifelli, Sherry Christie, Mike Dabrishus, David Myers, David Nemec, Jay Neugeboren, Joe Nigg, Robert Pawlowski, Alan Prendergast, Burton Raffel, Bin Ramke, Elizabeth Richardson, Florence Roberts, Joan Saalfeld, Amanda Vaill, Gordon Weaver, Michael White, Geary Hobson, Timothy Steele, Marc Yacht, Sandra Braman, William Giraldi, and William Zaranka.\n\nMatthew Carter, William Zaranka, Robert T. Tally, and Gerald Chapman generously made time to read the manuscript or answer questions related to it. Deb Stone uncovered documents and public records that had eluded discovery; Linda Justice transcribed interviews with lightning speed; graduate students Isobel Strobing at Boston University, and J. Edward Shockley at Midwestern State University (formerly Hardin Junior College), assisted in research.\n\nAt Folio Literary Management, Molly Jaffa and Jeff Kleinman, my agent, were the keys to connecting with Oscar van Gelderen at Lebowski Publishers and my perceptive, encouraging editor Stijn de Vries.\n\nFrances Collin, friend and partner to John Williams' agent Marie Rodell, and Collin's associate, Sarah Yake, spurred the creation of this book immensely by sharing materials and correspondence from their files. Their belief in this biography never wavered and led directly to its publication. Elido Fazi and Valentina Bortolamedi of Fazi Editore, Cristina Marino of Rizzoli, Edwin Frank of New York Review Books, Patricia Reimann of DTV Literature, Netta Gurevitch of Yedioth Books, Tito Exp\u00f3sito of Ediciones Baile del Sol, Frances MacMillan of Random House UK, Clara Nelson of Michael O'Mara Books, Claude Tarr\u00e8ne of Le Dilettante, novelist and translator Anna Gavalda, and Oscar van Gelderen graciously retraced their steps in explaining how they rediscovered _Stoner_.\n\nNancy Williams spent several days answering questions about her late husband and sharing her memories. Without her, much of her late husband's past would have been lost forever.\nNotes\n\n**PART I. _NOTHING BUT THE NIGHT_**\n\nChapter One. He Comes from Texas\n\n. Nancy Williams, interview, October 10, 2014.\n\n. Williams' marriage, his second, to Yvonne Elyse Stone, State of California, Certificate of Registry of Marriage, #26401, September 2, 1947.\n\n. Nancy Williams, interview, October 10, 2014.\n\n. Ibid.\n\n. Dedication to George Rae Williams by her husband, Willard Marsh, in _Week with No Friday_ (New York: Harper and Row, 1965).\n\n. Nancy Williams, interview, October 10, 2014.\n\n. John Williams, _Stoner_ (New York: New York Review Books, 2006).\n\n. Dan Wakefield, \"John Williams, Plain Writer,\" _Ploughshares_ 7, nos. 3\u20134 (Fall\/Winter 1981): 15.\n\n. Ibid.\n\n. Frank Gruber, _Zane Grey: A Biography_ (Mattituck, NY: Amereon, 1969), 213.\n\n. Nancy Williams, interview, October 10, 2014. Nancy said, \"Things began to make sense to John after that about his family.\"\n\n. Wakefield, \"John Williams, Plain Writer,\" 15.\n\n. Nancy Williams, interview, October 10, 2014.\n\n. Ibid. The description of slaughtering the pig is based on common practice.\n\n. John Williams, _Butcher's Crossing_ (New York: New York Review Books, 2007).\n\nChapter Two. \"Ho, Ho! Wasn't I the Character Then?\"\n\n. Nancy Williams, interview, October 9, 2014.\n\n. John Williams, \"The 'Western': Definition of the Myth,\" _The Nation_ 43, no. 17 (November 18, 1961).\n\n. Nancy Williams, interview, October 9, 2014.\n\n. The scene fades to black and the words appear: \"I am the resurrection and the life\"\u2014an example of Hollywood hitting the audience over the head and pitching the film to churchgoers who might not approve of movies. The novel ends differently.\n\n. Dan Wakefield, \"John Williams, Plain Writer,\" _Ploughshares_ 7, nos. 3\u20134 (Fall\/Winter 1981).\n\n. Note from an interview conducted by Jody McCall with Williams, apparently for a local newspaper. Williams Papers, University of Arkansas, Fayetteville (Williams Papers hereafter), Series 10 (Interviews), Box 27, Folders 9\u201310.\n\n. Brian Wooley, \"An Interview with John Williams,\" _Denver Quarterly_ (Winter 1986): 13.\n\n. Wakefield, \"John Williams, Plain Writer,\" 15.\n\n. \"These Books Not Really Over His Head,\" _Wichita Daily Times_ , March 6, 1938.\n\n. Bin Ramke, interview, October 7, 2014. Ramke was a colleague of Williams' at the University of Denver.\n\n. \"Young Wichitan Accidentally Shot,\" _Wichita Daily Times_ , June 30, 1940.\n\n. McCall notes, Williams Papers.\n\n. Wooley, \"Interview with John Williams,\" 30.\n\n. Wakefield, \"John Williams, Plain Writer,\" 15.\n\n. Ibid., 18.\n\n. Newsom and Coleman had landed parts for the summer at the county playhouse in Suffern, NY. Vesta Kelling, \"Actors Take a Busman's Holiday\u2014And Act!,\" _Evening Independent_ , July 12, 1941, 6, 9. Newsom became a Hollywood writer; Coleman's interest in theater trailed off during the war.\n\n. \"Playgoers Await Unique 'Our Town,'\" _Wichita Daily Times_ , December 8, 1940.\n\n. Wooley, \"Interview with John Williams,\" 15.\n\n. Michael Shannon, \"The History of KDNT Radio in Denton, Texas, Part I: 1938\u201346,\" Dallas\u2013Fort Worth Radio & Television History, n.d., dfwretroplex.com.\n\n. Martin (Alyeene's nephew) and Joyce Shoemaker, interview, June 4, 2015.\n\n. \"Miss Bryan, Mr. Williams Married Methodist Church,\" _Hood County Tablet_ , April 9, 1942.\n\n. Ibid.\n\nChapter Three. Rough Draft\n\n. \"Mechanics Still Are Wanted by Air Corps,\" _Lubbock Morning Avalanche_ , September 26, 1942.\n\n. Dan Wakefield, \"John Williams, Plain Writer,\" _Ploughshares_ 7, nos. 3\u20134 (Fall\/Winter 1981): 21.\n\n. Conrad Black, _Franklin Delano Roosevelt, Champion of Freedom_ (New York: PublicAffairs, 2003), 603\u2013605.\n\n. _Hood County Tablet_ , November 19, 1942, 8.\n\n. _Hood County Tablet_ , December 24, 1942, 1.\n\n. Donovan Webster, _The Burma Road: The Epic Story of the China-Burma-India Theater in World War II_ (New York: Harper Perennial, 2004), 129.\n\n. Thomas Ray Foltz, \"My Life as a GI Joe in World War II,\" China-Burma-India: Remembering the Forgotten Theater of World War II, 2007, cbi-theater.com\/gijoe\/gijoe.html.\n\n. Ibid.\n\n. Theodore White, \"The Hump: The Historic Airway to China Was Created by US Heroes,\" _Life_ , September 11, 1944.\n\n. Memoirs of World War II radiomen report that this was common practice.\n\n. Douglas F. Devaux, \"China, Burma and India from the Back Seat: Memories from the China-Burma-India Theater,\" China-Burma-India: Remembering the Forgotten Theater of World War II, 2001 (adapted for internet 2016), cbi-theater.com\/backseat\/backseat.html.\n\n. Still, the efforts of the Hump-fliers were weakened by corruption on the other end. The supplies that landed in China didn't always end up in the proper hands\u2014the bane of international charity\u2014and the black market was booming in and around the drop-off point of Kunming. Unscrupulous hoarders built fortunes from selling supplies shipped from Indian bases. As Donovan Webster writes in _The Burma Road: The Epic Story of the China-Burma-India Theater in World War II_ (New York: Farrar, Straus and Giroux, 2003), \"troops were living on gruel, Spam, and rice, while those close to Hump deliveries in China grew fat on American-bought pork, beef, and chicken\" (128).\n\n. Williams, reading aloud from his manuscript during an honorary symposium about him, March 29, 1984, CD-ROM, included in the Williams Papers collection.\n\n. Alan Prendergast, \"Sixteen Years After His Death, Not-So-Famous Novelist John Williams Is Finding His Audience,\" _Westword_ , November 3, 2010.\n\n. Nancy Williams, interview, October 10, 2014.\n\n. USAAF\/USAF Aircraft Accidents 1942\u20131955 for India: Military Aviation Incident Reports, aviationarcheology.com; National Archives, College Park, Maryland, Lists of Allied Air Crashes, compiled 09\/1939\u201303\/1945, ARC Identifier 7373711\/MLR Number A1 2109-C Series, Record Group 92: Records of the Office of the Quartermaster General, 1774\u20131985.\n\n. Wakefield, \"John Williams, Plain Writer,\" 21.\n\n. Barbara Tuchman, _Stilwell and the American Experience in China_ (New York: Grove Press, 1970); Nancy Williams, interview, October 10, 2014. Nancy said, \"His nightmares continued for years and gradually lessened\u2014the guilt.\"\n\n. James McWilliams, \"The Examined Lie: A Meditation on Memory,\" _American Scholar_ , Summer 2015.\n\n. Katherine Williams, email, May 2, 2015.\n\n. Dan Wakefield, interview, November 11, 2013.\n\n. Nancy Williams, interview, October 10, 2014.\n\n. From a booklet dated March 1944, Williams Papers. That month the 1st and 2nd Troop Carrier Squadrons flew seventeen low-altitude missions, parachuting supplies to Merrill's Marauders, an Allied guerrilla force in the Burmese jungle\u2014more than twenty tons every other day. Instead of keeping a soldier's diary of events, Williams may have preferred writing fiction as a gesture of faith that he would survive to be published one day.\n\n. Brian Wooley, \"An Interview with John Williams,\" _Denver Quarterly_ (Winter 1986): 5.\n\n. Satyavati C. Jordan to Williams, March 25, 1945, Williams Papers.\n\n. Satyavati C. Jordan to Williams, March 29, 1945, Williams Papers.\n\n. Jack Newsom to Williams, October 22, 1944, Williams Papers.\n\nChapter Four. Key West\n\n. Nancy Williams, interview, October 9, 2014. When Williams was in India, he and the men drank something the locals brewed\u2014they called it \"spot-bottle\"\u2014drops of which stained their uniforms white.\n\n. Nancy Williams, interview, October 9, 2014.\n\n. Brian Wooley, \"Interview with John Williams,\" _Denver Quarterly_ (Winter 1986): 25. If Williams wrote half a dozen sentences he liked in a day's work, he was satisfied.\n\n. Edward Stone, _A Certain Morbidness: A View of American Literature_ (Carbondale: Southern Illinois University Press, 1969), 18.\n\n. All quotations here and in the passage that follows are from John Williams, _Nothing But the Night_ (Fayetteville: University of Arkansas Press, 1990).\n\n. Before the war, Smart had published a study of private libraries in colonial Virginia that other scholars would still be citing sixty years later. About this role as an adviser to students, later, at the University of Miami, a student named Marc Yacht sought out Smart, his creative writing teacher. Yacht was in pre-med and wanted to be a doctor, but he was rethinking that decision. Smart had been a member of the pre-med club at the University of Alabama, so he understood. Yacht decided to drop out for a while, but later he enrolled in medical school and enjoyed a long career as a physician. Marc Yacht, MD, interview, June 13, 2015.\n\n. Smart was a heavy drinker and intimated that it was a problem in his later correspondence.\n\n. George K. Smart to Williams, July 7, 1946, Williams Papers.\n\n. Aswell succeeded Maxwell Perkins as administrator of the Thomas Wolfe estate in 1947.\n\n. _Paris News_ , September 10, 1937, 9.\n\n. John M. Spottswood to Williams, December 6, 1945, Williams Papers.\n\n. Dan Wakefield, \"John Williams, Plain Writer,\" _Ploughshares_ 7, nos. 3\u20134 (Fall\/Winter 1981): 16.\n\nChapter Five. Alan Swallow\n\n. Ten years later, the house fell beneath the bulldozers digging for the Ventura Freeway.\n\n. \"Texas Memories: 1932,\" _Denver Quarterly_ (Winter 1986): 129. Nancy Williams said, \"John loved his mother too much to talk about her.\" Nancy Williams, interview, October 9, 2014.\n\n. Edward Aswell to Williams, June 19, 1946, and George K. Smart to Williams, July 7, 1946, Williams Papers.\n\n. Alan Swallow to Williams, November 2, 1946, Williams Papers.\n\n. Gale Woolf (Yvonne's daughter), interview, June 8, 2015.\n\n. Yvonne (Stone) Woolf to Williams, May 21 and 22, 1947, Williams Papers.\n\n. Alan Swallow to Williams, December 14, 1946, Williams Papers.\n\n. Dale W. Nelson, _The Imprint of Alan Swallow: Quality Publishing in the West_ (Syracuse, NY: Syracuse University Press, 2010), 159.\n\n. Ibid., 11.\n\n. The Blue Books were favorites of William S. Burroughs, Jack Kerouac, Richard Byrd (the polar explorer), Louis L'Amour (who said, \"The Little Blue Books were a godsend to wandering men and no doubt to many others\"), Harlan Ellison (who called them \"moveable schoolrooms at ten cents a shot\"), Saul Bellow, Gore Vidal, and Margaret Mead (who took a suitcase full of them to the Samoan Islands). Studs Terkel, who was raised in a Chicago boardinghouse, loved overhearing discussions between boarders about titles in the series. FBI director J. Edgar Hoover regarded the political, religious, and economic tracts as threats to the nation and arranged to have Haldeman-Julius arrested for tax evasion. The popularity of the Little Blue Books declined, however, during the Red Scare of the 1950s.\n\n. In 1938, Warren Brooks published a textbook, _Understanding Poetry_ , that codified many of the New Critical ideas into a coherent approach to literary study. The book, and its companion volume, _Understanding Fiction_ (1943), revolutionized the teaching of literature in universities well into the 1960s.\n\n. Quoted in Richard Ellman, \"Publisher for Poets,\" _Saturday Review_ , July 22, 1961, 33\u201334; Nelson, _Imprint of Alan Swallow_ , 57, 78.\n\n. Nelson, _Imprint of Alan Swallow_ , 75.\n\n. Donna Ippolito and Shirley Kopatz, \"Alan Swallow: Platten Press Publisher,\" _Journal of the West_ 8 (1969): 476.\n\n. Ibid., 476.\n\n. Robert Giroux to Williams, January 13, 1947, Williams Papers.\n\n. George K. Smart to Williams, May 5, 1947, Williams Papers.\n\n. Alan Swallow to Williams, April 29, 1947, Williams Papers.\n\n. Ibid.\n\n. Alan Swallow to Williams, August 5, 1947, Williams Papers.\n\n. George K. Smart to Williams, September 9, 1947, Williams Papers.\n\n. Historians Charles Graham and Robert Perkin, as well as Mayor Stapleton, are quoted in R. Laurie Simmons and Thomas H. Simmons, \"Historic Resources of Downtown Denver,\" National Register of Historic Places, Multiple Property Documentation Form, US Department of the Interior, National Park Service, <http:\/\/legacy.historycolorado.org\/sites\/default\/files\/files\/OAHP\/crforms_edumat\/pdfs\/646.pdf>.\n\n. \"Our Literary Alumni,\" _University of Denver Magazine_ , September 1964.\n\n. John Williams, _Stoner_ (New York: New York Review Books, 2006), 7.\n\n. Michelle Latiolais, interview, December 14, 2013. Latiolais was one of Williams' graduate students at the University of Denver.\n\nChapter Six. Love\n\n. Yvonne Williams to Elbert and Adair (?), April 25, 1948, Williams Papers. Elbert and Adair had a little boy, Harry, according to Yvonne's letter, so this is not John's grandfather Elbert Walker, who was in his nineties.\n\n. She edited Swallow's annual _Index to Little Magazines_ from 1948 to 1951.\n\n. George K. Smart to Williams, September 7, 1948, Williams Papers.\n\n. Yvonne Williams, April 25, 1948, Williams Papers.\n\n. Yvonne Woolf, \"DW [Douglas Woolf]: A Memoir\" (unpublished, undated), provided by Gale Woolf, her daughter.\n\n. Willard Marsh to Williams, May 4, circa 1958, Williams Papers.\n\n. George Rae Williams to Williams, March 16, 1948, Williams Papers.\n\n. Willard Marsh to Williams, March 23, 1948, Williams Papers.\n\n. Woolf, \"DW\"; Yvonne Williams, April 25, 1948, Williams Papers.\n\n. In _Wall to Wall_ , Woolf wrote:\n\nThere was nothing unusual about the truck parked by itself a little beyond the camp, it was an ordinary open-bed truck, Dodge, similar to the old-fashioned garbage truck. . . . No garbage truck I'd ever seen had smelled like that. I knew at once what was causing it, but I walked over for a look anyway, or perhaps because. The driver must have been waiting in the chowline with everyone else . . . so I was all alone. . . . He had a full load of soldiers in his truck, thrown in, their arms and legs and heads in various positions and attitudes that I'd never seen before. They were mostly French, a few Arabs, and despite their uniforms they didn't look very important any more. Later I learned that if you watched men die, especially if you've known them at all, they still look important afterward no matter what you have to do with them, but I was inexperienced then.\" (Douglas Woolf, _Wall to Wall_ [New York: Grove Press, 1962], 80\u201381)\n\n. Douglas Woolf, _Hypocritic Days and Other Tales_ , edited by Sandra Braman (Santa Barbara, CA: Black Sparrow Press, 1993), 396.\n\n. Robert Creeley, \"Reading Douglas Woolf's Ya! & John-Juan,\" Dalkey Archive Press website, www.dalkeyarchive.com\/reading-douglas-woolfs-ya-john-juan.\n\n. Gale Woolf, interview, June 8, 2015.\n\n. Woolf, \"DW\"; Gale Woolf, interview, June 8, 2015.\n\n. Woolf, _Wall to Wall_ ; Gale Woolf, interview, June 8, 2015.\n\n. Alan Swallow to John Pauker, August 17, 1948, Williams Papers. Pauker, a commentator for the Voice of America during World War II, later became a poet, playwright, editor, and translator.\n\n. Yvor Winters, _Primitivism and Decadence: A Study of American Experimental Poetry_ (New York: Arrow Editions, 1937).\n\n. Billie Watson to Williams, July 9, 1948, Williams Papers.\n\n. John Williams, _Butcher's Crossing_ (New York: New York Review Books, 2007), 63.\n\n. John Williams, _Stoner_ (New York: New York Review Books, 2006), 193.\n\n. Undated draft of \"The Summer,\" Williams Papers.\n\n. George Rae Williams to Williams, July 11, 1948, Williams Papers; Woolf, \"DW.\"\n\n. Woolf, \"DW.\"\n\n. Gale Woolf, interview, June 8, 2015.\n\n. Woolf, \"DW.\"\n\n. Willard Marsh to Williams, October 14, 1948, Williams Papers.\n\n. Letters from George Rae and Willard Marsh to Williams, October 11 and 14, 1948, Williams Papers.\n\n. George Rae Marsh to Williams, December 4, 1948, Williams Papers.\n\n. Willard Marsh to Williams, December 4, 1948, Williams Papers.\n\n. George Rae Marsh to Williams, December 10, 1948, Williams Papers.\n\n. John Williams, _The Broken Landscape_ (Denver: Swallow Press, 1949), 23.\n\n. Willard Marsh to Williams, January 1, 1949, Williams Papers.\n\n**PART II. _BUTCHER'S CROSSING_**\n\nChapter Seven. The Winters Circle\n\n. Dale W. Nelson, _The Imprint of Alan Swallow: Quality Publishing in the West_ (Syracuse, NY: Syracuse University Press, 2010), 86. Not all of Swallow's books were for intellectuals. Virgil Scott's _The Hickory Stick_ (1947) was crime fiction. On August 20, 1947, _Kirkus Reviews_ said, \"Self-portrait of a heel, this, in its cold criminal conduct, its lust without love, its tough, tense narrative stands up well against (but does not necessarily derivate from) James Cain.\"\n\n. Quoted in Nelson, _Imprint of Alan Swallow_ , 98.\n\n. Alan Swallow to Williams, August 26, 1948, Williams Papers.\n\n. Kenneth Fields, \"True to His Word,\" _Stanford Alumni Magazine_ , November-December 2000, <https:\/\/alumni.stanford.edu\/get\/page\/magazine\/article\/?article_id=39395>. The memory of hearing Winters reading verses moved Robert Lowell to say, \"His voice and measures still ring in my ears. They pass [poet A. E.] Housman's test for true poetry: if I remembered them while shaving, I would cut myself.\" Kenneth Fields, \"Winters's Wild West,\" _Los Angeles Review of Books_ , September 10, 2013, <https:\/\/lareviewofbooks.org\/article\/winterss-wild-west\/#!>.\n\n. Fields, \"Winters's Wild West.\" See also Stanley Edgar Hyman, \"Yvor Winters and Evaluation in Criticism,\" in _The Armed Vision: A Study in the Methods of Modern Literary Criticism_ (New York: Alfred A. Knopf, 1955), 23\u201353. It was true: Winters was born three centuries too late, and he was a thoroughly Johnsonian character. His marriage to Janet Lewis, however, was tender and they were devoted to each other.\n\n. David Yezzi, \"The Seriousness of Yvor Winters,\" _New Criterion_ , June 1997, 26.\n\n. R. L. Barth, ed., _The Selected Letters of Yvor Winters_ (Athens: Swallow Press \/ Ohio University Press, 2000).\n\n. A. Alvarez, \"Yvor Winters,\" in _Beyond All This Fiddle: Essays, 1955\u20131967_ (New York: Random House, 1968), 255\u2013259 (originally published in _The New Statesman_ , 1960).\n\n. Hugh Lloyd-Jones, Regius Professor of Greek at Oxford, quoted in Joseph Epstein, \"Father of History: Herodotus and the Human Dimension in the Past,\" _Weekly Standard_ , October 20, 2014.\n\n. Yvor Winters, \"Individual Poets and Modes of Poetry: The 16th Century Lyric in England. A Critical and Historical Reinterpretation,\" in Paul J. Alpers, ed., _Elizabethan Poetry: Modern Essays in Criticism_ (New York: Oxford University Press, 1967), 98; originally published in _Poetry Magazine_ 53 (1939).\n\n. Quoted in Yezzi, \"Seriousness of Yvor Winters\"; Kenneth Rexroth, _American Poetry in the Twentieth Century_ (New York: Seabury Press, 1971), 92\u201393.\n\n. Yvor Winters, _In Defense of Reason_ (Denver: Swallow Press, 1947).\n\n. Frederick Seidel, \"Robert Lowell, The Art of Poetry No. 3,\" _Paris Review_ (Winter-Spring 1961).\n\n. Winters was also a moral figure to his students, which meant he had as profound an influence on them as any academic knowledge they picked up in his classes. He and Janet Lewis vigorously protested the internment of Japanese Americans during World War II. They were founding members of the California branch of the NAACP (National Association for the Advancement of Colored People); they organized the retrial of a man unjustly convicted of murder; and they were both concerned with the plight and history of Native Americans.\n\n. George K. Smart to Williams, circa early 1949, Williams Papers.\n\n. Martha Hume, \"Artist of Diversity: John Williams,\" _Dust_ (Winter 1966): 18.\n\n. Willard Marsh to Williams, January 20, 1949, Williams Papers. Marsh sometimes poured himself a drink, or several, when he wrote letters, so his tone has to be taken with a grain of salt.\n\n. Nelson, _Imprint of Alan Swallow_ , 85.\n\n. Ibid., 98.\n\n. Jean James, interview, June 20, 2015. Her husband, Stuart James, was later chair of the English Department at the University of Denver. She said,\n\nAlan, he was impatient, with people who weren't interested in ideas, who weren't interested in the literary world. I don't think he tolerated stupid people very well. He was just kind of harsh in many ways. He never was to Stewart or to me, but I think he was to others. I saw that at many gatherings. He liked working. And many writers, or people of that ilk, that's the way they are, they are always thinking about writing. They are always thinking about ideas. And so to be in a world where they would consider it frivolous, they just don't want to bother.\n\n. J. V. Cunningham, _The Collected Essays of J. V. Cunningham_ (Chicago: Swallow Press, 1976), 421.\n\n. Irving Howe, _A Margin of Hope_ (New York: Harcourt Brace Jovanovich, 1982), 190.\n\n. Timothy Steele, \"An Interview with J. V. Cunningham,\" _Iowa Review_ (Fall 1985).\n\n. Francis Fike, \"Cold Grace: Christian Faith and Stoicism in the Poetry of J. V. Cunningham,\" _Renascence: Essays on Values in Literature_ (Milwaukee, WI: Marquette University Press, 2007), 141\u2013158.\n\n. J. V. Cunningham, \"For My Contemporaries,\" from _The Exclusions of a Rhyme: Poems and Epigrams_ (Athens: Ohio University Press\/Swallow Press, 1960).\n\n. Hayden Carruth to Williams, October 31, 1949, Williams Papers.\n\n. Desmond Powell to Williams, April 6, 1950, Williams Papers.\n\nChapter Eight. \"Natural Liars Are the Best Writers\"\n\n. John Williams, _Stoner_ (New York: New York Review Books, 2006), 15.\n\n. Quoted in \"Natural Liars Are the Best Writers,\" _Moberly (MO) Monitor-Index_ , October 8, 1936.\n\n. Leon T. Dickinson, _An Historical Sketch of the Department of English, University of Missouri\u2013Columbia_ (Columbia: Department of English, University of Missouri, 1986).\n\n. \"A.H.R. Fairchild Talks on Value of Knowledge of Human Nature as Learned from Books,\" _Columbia (MO) Evening Missourian_ , October 30, 1922.\n\n. Dickinson, _Sketch_.\n\n. Ibid.\n\n. Ibid.\n\n. Alan Swallow to Williams, February 9, 1951, Williams Papers.\n\n. Barthold Fles to Williams, September 10, 1951, Williams Papers.\n\n. Alan Swallow to Williams, September 19, 1951, Williams Papers.\n\n. Harry Brague to Williams, November 11, 1951, Williams Papers. The legendary Maxwell Perkins hired Brague in 1946. He was Kurt Vonnegut's editor for _Player Piano_ and became Hemingway's editor in the late 1950s. A colorful description of New York publishing as a decrepit old boys' club after the war appears in an essay by Charles Scribner Jr., \"I, Who Knew Nothing, Was in Charge,\" _New York Times_ , December 9, 1990.\n\n. Barthold Fles to Williams, July 19, 1952, Williams Papers.\n\n. Harry Shaw to Barthold Fles, June 21, 1954, Williams Papers.\n\n. Barthold Fles to Williams, June 21, 1954, Williams Papers.\n\n. Ana\u00efs Nin, _Fire: From \"A Journal of Love, 1934\u20131937_ (Reprint, New York: Houghton Mifflin Harcourt, 1995), 280.\n\n. Willard Marsh to Williams, October 22, 1952, Williams Papers.\n\n. Willard Marsh to Williams, undated, Williams Papers.\n\n. Willard Marsh to Williams, October 22, 1952, Williams Papers.\n\n. A number of Williams' students over the years remembered his delight in reciting Wyatt during class.\n\n. Yvor Winters, _In Defense of Reason_ (Denver: Swallow Press, 1947). One of the editions is available online at archive.org.\n\n. Yvor Winters, _Forms of Discovery_ (Denver: Swallow Press, 1967), 44. Twenty years after publishing _In Defense of Reason_ , Winters still thought so much of Greville that he devoted ten pages to him in _Forms of Discovery_.\n\n. Williams, \"Fulke Greville: The World and God,\" _Denver Quarterly_ (Summer 1975). \"Imperfectly golden\" seems like a generous assessment of a poet who addressed his mistress as \"Faire dog\" in \"Sonnet II.\"\n\n. \"Our Literary Alumni,\" _University of Denver Magazine_ , September 1964.\n\n. Ray B. West to Williams, June 8, 1953, Williams Papers.\n\n. Alan Swallow to Williams, February 22, 1953, Williams Papers.\n\n. Yvor Winters to Alan Stephens, August 21, 1953, Williams Papers.\n\n. Ray B. West to Williams, circa late 1953, Williams Papers.\n\n. Williams to The Editors, _Poetry_ , January 20, 1954.\n\n. Willard Marsh to Williams, March 18, 1953, Williams Papers.\n\n. Florence Roberts, interview, June 6, 2015.\n\n. Ibid.\n\n. Ibid.\n\nChapter Nine. _Butcher's Crossing_\n\n. Alan Swallow to Williams, February 8, 1953, Williams Papers.\n\n. Martha H. Hume, \"Alan Swallow: In Memoriam,\" _Small Press Review_ (Spring 1967); Alan Swallow to Williams, January 1, 1954, Williams Papers.\n\n. Though commonplace today, in the 1950s there were no professional associations of writers in the United States, and only a handful of master of fine arts (MFA) programs. Some faculty resisted convening panels and workshops about writing for market, maintaining it was outside the mission of academe. Others looked askance at writing courses whose purpose was to produce writers; it sounded ironic\u2014maybe even a little Kafkaesque. Allen Tate, a leader of the New Criticism who later taught creative writing at Princeton, voiced a complaint in 1964 that has become perennial: \"The academically certified Creative Writer goes out to teach Creative Writing, and produces other Creative Writers who are not writers, but who produce still other Creative Writers who are not writers.\"\n\n. Leon T. Dickinson, _An Historical Sketch of the Department of English, University of Missouri-Columbia_ (Columbia: Department of English, University of Missouri\u2013Columbia, 1986).\n\n. Associated Writing Programs Newsletter, \"John Williams to Head AWP,\" 1974. The association supports writing programs in over 500 colleges and universities as well as 130 writers' conferences and centers.\n\n. Jean James, interview, June 20, 2015. Ms. James came to Denver as a student in 1959 and married Stuart James, an English instructor who became department chairman.\n\n. Gerald Chapman, interview, May 10, 2015.\n\n. Dan Wakefield, \"John Williams, Plain Writer,\" _Ploughshares_ 7, nos. 3\u20134 (Fall\/Winter 1981): 19.\n\n. Program for \"Buffalo Bill's Wild West and Congress of Rough Riders of the World,\" 1893, <https:\/\/archive.org\/details\/buffalobillswild00buff>. For several seasons of the show, Sitting Bull, the Lakota holy man who had a vision predicting victory over the soldiers, walked somberly into the middle of the ring as the finale to the mock battle. Unexpectedly, he became a figure of respect among whites. Indian agency police killed him in 1890, fearing that he would use his influence to further inspire the mystical Ghost Dance movement.\n\n. Gerald Chapman, interview, May 10, 2015.\n\n. Lewis Mumford, _The Golden Day: A Study in American Experience and Culture_ (New York: Liverwright, 1926), 79.\n\n. Williams, \"The 'Western': Definition of the Myth,\" _The Nation_ , November 18, 1961.\n\n. Ibid.\n\n. Ibid.\n\n. Revisionist western films include _3:10 to Yuma_ (1957, 2007); _Jeremiah Johnson_ (1972); _Dances with Wolves_ (1990); and _Unforgiven_ (1992).\n\n. Willard and George Rae Marsh to Williams, November 11, 1954, Williams Papers.\n\n. Theodore R. Cogswell to Art Landis, March 14, 1978, Arthur Landis Papers, Tamiment Library and Robert F. Wagner Labor Archives, New York.\n\n. This is the convention, renamed the Milford Science Fiction Convention in Kurt Vonnegut's _God Bless You, Mr. Rosewater_ , at which Rosewater blurts out drunkenly to the assemblage, \"I love you sons of bitches. You're all I read anymore.\"\n\n. George Rae Marsh to Williams, January 25, 1955, Williams Papers.\n\n. Eileen Bassing, _Where's Annie?_ (New York: Random House, 1956), 54. This sketch of George Rae as a ditzy muse does injustice to her career as a writer and playwright. As George Rae Williams, she wrote five published plays: _Mind over Mumps: A One-Act Farce_ (Franklin, OH: Eldridge Publishing Company, 1951); _Augie Evans: Private Eye: A One-Act Farce_ (Franklin, OH: Eldridge Publishing Company, 1951); _Leave It to Laurie: A Comedy in One Act_ (Minneapolis: Northwestern Press, 1952); _Keeping It in the Family: A Comedy in One Act_ (Minneapolis: Northwestern Press, 1953); and _A Will and a Way: A Three Act Comedy_ (Franklin, OH: Eldridge Publishing Company, 1962). As George Rae Cogswell, she wrote (with her second husband, Ted) the short story \"Contact Point\" (1975), and they contributed a coauthored story to _Six Science Fiction Plays_ (New York: Pocket Books, 1975). In 1979, as Georgia Cogswell, she published _Golden Obsession_ (New York: Kensington, 1979). Tony Burton, \"George Rae Marsh (Williams), aka Georgia Cogswell (1925\u20131997),\" Sombrero Books, September 14, 2015, sombrerobooks.com\/?p=2689.\n\n. Martha Hume, \"John Williams: Artist of Diversity,\" _Dust_ (Winter 1966): 24.\n\n. Willa Cather, _Not Under Forty_ (New York: Alfred A. Knopf, 1936), 50.\n\n. Flora Merrill, \"A Short Story Course Can Only Delay, It Cannot Kill an Artist, Says Willa Cather,\" reprinted in _Nebraska State Journal_ (April 25 1925): 11.\n\n. Janet Lewis to Williams, September 3, 1958, Williams Papers.\n\n. John Williams, _Butcher's Crossing_ (New York: New York Review Books, 2007), 117.\n\n. Ibid., 250.\n\n. Morton M. Hunt to Williams, August 28, 1958. Hunt authored popular nonfiction books about healthy approaches to relationships and living.\n\nChapter Ten. Fiasco\n\n. Linda Lear, _Rachel Carson: Witness for Nature_ (New York: Houghton-Mifflin, 1997).\n\n. Betty Friedan, _Life So Far_ (New York: Simon and Schuster, 2006), 138. Rodell was also Martin Luther King Jr.'s literary agent during the publication of his first book, _Stride Toward Freedom_.\n\n. Morton M. Hunt to Williams, August 28, 1958, Williams Papers.\n\n. Marie Rodell to Williams, October 20, 1958, Frances Collin Papers, private collection (Collin Papers hereafter). Collin became a partner in the agency after Rodell retired.\n\n. Williams to Rodell, September 1958, Williams Papers.\n\n. Willard Marsh to Williams, undated, circa 1958, Williams Papers.\n\n. Ibid.\n\n. Ibid.\n\n. Ibid.\n\n. Willard Marsh to Williams, August 8 1958, Williams Papers.\n\n. Williams to Rodell, October 31, 1958, Collin Papers.\n\n. During his career, until his retirement in 1965, Scott edited James Michener's _Tales of the South Pacific_ ; Elizabeth Stevenson's _The Crooked Corridor_ ; and Barbara Tuchman's _The Guns of August_ , all winners of the Pulitzer Prize.\n\n. Rodell to Williams, November 31, 1958, Collin Papers.\n\n. Williams to Rodell, November 7, 1958, Williams Papers.\n\n. Editor at W. W. Norton to Joan Daves, December 9, 1958, Collin Papers; Harry Brague to Rodell, January 30, 1959, Collin Papers.\n\n. Pascal Covici to Rodell, March 2, 1959, Collin Papers.\n\n. Williams to Rodell, February 18, 1959, Collin Papers. Williams was a good judge of his own ability. The short story form wasn't his fort\u00e9. His novel-length voice sounded unhurried, removed even, and a sense of immediacy is critical in short stories.\n\n. Cecil Scott to Williams, May 7, 1959, Williams Papers.\n\n. Cecil Scott to Rodell, May 12, 1959, Collin Papers; Rodell to Williams, May 13, 1959, Collin Papers; Rodell to Williams, May 13, 1959, Williams Papers. By reversing herself about Macmillan, she confused Williams. He would have been content with Little, Brown. She wrote, \"It's a very good house, I think; at least, I have no serious quarrel with the quality of their list.\"\n\n. Williams to J. V. Cunningham, June 11, 1959, Williams Papers.\n\n. Ibid.\n\n. Williams to Rodell, June 18, 1959, Collin Papers. Nancy Williams said there was no point in asking John where he was going. \"Out,\" he responded.\n\n. Rodell to Williams, July 20 and 22, 1959, Williams Papers; Williams to Rodell, August 2, 1959, Williams Papers; Rodell to Williams, August 6, 1959, Williams Papers; Williams to Rodell, August 11, 1959, Williams Papers. All of this correspondence was about choosing a title.\n\n. W. L. Rice to Williams, April 19, 1960, Williams Papers. Rice waited a few weeks after the article appeared to write to Williams.\n\n. Rodell to Williams, March 29, 1960, Williams Papers.\n\n. Nelson Nye, \"Roundup,\" _New York Times_ , April 4, 1960.\n\n. Williams to Rodell, March 31, 1960, Collin Papers.\n\n. Cecil Scott to Francis Bacon, _New York Times_ , March 31, 1960, Collin Papers.\n\n. Cecil Scott to Williams, April 12, 1960, Williams Papers.\n\n**PART III. _STONER_**\n\nChapter Eleven. \"It Was That Kind of World\"\n\n. George Rae Marsh to Williams, January 18, 1960, Williams Papers.\n\n. Willard Marsh to Williams, circa early 1960, Williams Papers.\n\n. Nash Ramblers resemble cars drawn by Robert Crumb in his 1960s underground comics. The model Butch and George Rae owned was the same as the one driven by Lois Lane in the _Superman_ television series.\n\n. Robert B. Richardson, interview, December 12, 2013.\n\n. Ibid.\n\n. Fred Inglis, interview, September 5, 2014.\n\n. Robert B. Richardson, interview, December 12, 2013.\n\n. Gerald Chapman, interview, May 10, 2015.\n\n. Daniel Aaron, _The Americanist_ (Ann Arbor: University of Michigan Press, 2007), 144\u2013145. Aaron was the author of _Writers on the Left_ (New York: Columbia University Press, 1992), his classic study of the radical American activists and novelists of the 1920s and '30s.\n\n. Jean James, interview, June 20, 2015; Gerald Chapman, interview, May 10, 2015.\n\n. Nancy Williams, interview, October 9, 2014.\n\n. Jonathan Williams, interview, October 9, 2014.\n\n. Jim Clark, letter, February 27, 2015.\n\n. Nancy Williams, interview, October 9, 2014.\n\n. \"Yarb\" is a vernacular mispronunciation of \"herb\"; \"yarb-doctors\" were folk medicine healers.\n\n. Nancy Williams, interview, October 9, 2014.\n\n. Nancy Williams, interview, October 10, 2014.\n\n. Williams to Marie Rodell, December 14, 1960, Collin Papers.\n\n. Alfred Kazin, _Writing Was Everything_ (Cambridge, MA: Harvard University Press, 1995), 10.\n\n. Bin Ramke, interview, October 7, 2014. Ramke was a friend and former colleague of Williams at Denver. His collection of poems, _The Difference Between Night and Day_ (1978), won the Yale Younger Poets Prize; _The Massacre of the Innocents_ (1994) and _Wake_ (1998) were awarded the Iowa Poetry Prize.\n\n. Cecil Scott to Williams, September 14, 1960, Williams Papers.\n\n. Williams to Rodell, September 26, 1960, Williams Papers.\n\n. William Peden to Williams, July 21, 1960, Williams Papers.\n\n. John Williams, \"Prospectus: An Anthology of the Shorter Poems of the English Renaissance,\" undated, Williams Papers.\n\n. Pyke Johnson to Williams, May 7, 1962, Williams Papers.\n\n. Williams to Rodell, February 2, 1963, Collin Papers.\n\n. Williams to Rodell, June 13, 1963, Williams Papers.\n\n. Gerald Chapman to R. C. Hadley, March 8, 1963, Williams Papers; Chapman, interview, May 10, 2015.\n\n. Williams to Rodell, April 2, 1963, Collin Papers.\n\n. John Edward Williams, _English Renaissance Poetry: A Collection of Shorter Poems from Skelton to Jonson_ (New York: New York Review of Books, 2016), xxiv.\n\n. Pyke Johnson to Williams, July 29, 1963. \"I can't help remembering how you and I discussed Mr. Winters during our rambles around the Denver campus.\"\n\n. Rodell to Williams, June 11, 1963, Collin Papers.\n\n. Williams to Rodell, June 13, 1963, Williams Papers.\n\n. John Williams, \"'63 Journal\" (in his handwriting), beginning June 21, 1963, Williams Papers.\n\nChapter Twelve. \"The Williams Affair\"\n\n. Williams to Marie Rodell, July 19, 1963, Collin Papers.\n\n. Cecil Scott to Rodell, July 12, 1963, Collin Papers. Al Hart was later Ian Fleming's editor at Macmillan.\n\n. Williams to Rodell, July 19, 1963, Collin Papers.\n\n. Rodell to Alan Williams, July 17, 1963, Collin Papers.\n\n. Yvor Winters to Doubleday Anchor Books, July 19, 1963, Yvor Winters and Janet Lewis Papers, Department of Special Collections and University Archives, Stanford University Libraries, Stanford, California (Winters\/Lewis Papers hereafter).\n\n. Johnson's assistant was Helen D'Alessandro, who later became Harlan Ellison's editor.\n\n. Yvor Winters to Helen D'Alessandro, July 23, 1963, Winters\/Lewis Papers.\n\n. Yvor Winters to Anne Freedgood, July 25, 1963, Winters\/Lewis Papers.\n\n. Winters to \"Dear Sirs,\" July 19, 1963, Winters\/Lewis Papers.\n\n. Recounting the situation two years later to a professor friend, Winters was still angry:\n\nAnd his students know [he cribs from my lectures]. One of my students brought me a copy of the Williams anthology shortly after it appeared, almost two years ago. . . . I wrote to Anchor and told them they had been taken. They checked. They finally made Williams write a page of acknowledgement, which I accepted, although it was pretty sleazy. . . . Williams teaches at the U. of Denver; Swallow tells me that he has been lecturing verbatim from my essays for a long time, without acknowledgment, and that his students know it. He . . . publish[es] poems here and there which are ghastly imitations and piracies, mostly of me. (Yvor Winters to Gerald Graff, December 21, 1965, email from David Myers, November 15, 2013 [Myers collection hereafter])\n\n. Heather McHugh, email, May 12, 2015.\n\n. Martha Hume, \"John Williams: Artist of Diversity,\" _Dust_ (Winter 1966): 16.\n\n. Williams to Yvor Winters, August 8, 1963, Williams Papers. While the draft acknowledgment was making its way to California, Winters told Johnson on the phone, \"I don't wish any communication with Williams unless it is to plead for mercy. . . . I will wreck this book unless you provide proper acknowledgment.\" Pyke Johnson to Williams, August 9, 1963, Williams Papers.\n\n. Pyke Johnson to Williams, August 14, 1963, Williams Papers. \"I don't see what else we can do now,\" Pyke wrote, \"and I am beginning to doubt whether we will ever be able to satisfy him. However, let's try.\"\n\n. Pyke Johnson to Yvor Winters, September 25, 1963, Winters\/Lewis Papers.\n\n. Pyke Johnson to Williams, August 26, 1963, Williams Papers.\n\n. Yvor Winters to Gerald Graff, December 21, 1965, Myers collection.\n\n. Nancy Williams, interview, October 10, 2014.\n\n. Hume, \"Artist of Diversity: John Williams,\" 17.\n\n. Williams to Rodell, November 21, 1963, Collin Papers.\n\n. John Williams, \"The Future of the Novel,\" in _Contemporary Literary Experience_ , David Madden, ed. (Pasadena, CA: Salem Press, 1974).\n\n. \"Our Literary Alumni,\" _University of Denver Magazine_ (September 1964). Williams couldn't abide the solipsism of the Beat writers and poets.\n\n. Williams to Rodell, September 16, 1963, Williams Papers.\n\n. Jonathan Williams, interview, October 9, 2014.\n\n. Several years later, Dan Wakefield recalled visiting Williams at his home. He said, \"Lonnie was really icy and you could feel the tension. Very unpleasant. Finally John said, 'Let's get out of here.' We drove over to the home of a woman who was lovely. And I then realized, 'This is his girlfriend!'\"\n\n. Willard Marsh to Williams, October 15, 1963, Williams Papers.\n\nChapter Thirteen. _Stoner_\n\n. Marie Rodell to Williams, March 11, 1964, Williams Papers.\n\n. Pyke Johnson to Williams, March 26, 1964, Williams Papers.\n\n. Williams to Rodell, November 21, 1963, Collin Papers.\n\n. Robert L. Morris to Williams, March 16, 1964, Williams Papers.\n\n. The University of Denver received $50,000 to establish the quarterly, which seemed like an enormous amount at the time\u2014the equivalent of four or five instructors' salaries. _The Denver Quarterly_ , of which Williams was the first editor, celebrated its fiftieth year in 2015 and has long been considered a leading literary magazine. Williams discovered he enjoyed creating it from the ground up\u2014contacting writers for submissions, planning the look of the publication, and editing the inaugural issue slated for early 1965. It suited him because he made the quarterly reflect his opinions. Concerned about the \"insidious ways that middle class culture\" was unduly influencing popular taste, he urged contributors to submit \"good essays done on such things as the 'new avant-garde' novel, new-wave cinema, the New York Review of Books, the literary magazines (in general or in particular), modern art, virtually all of the performing arts, and so forth.\"\n\n. John Williams, _Stoner_ (New York: New York Review Books, 2006), 181.\n\n. Joanne Greenberg, interview, October 8, 2014.\n\n. Bob Johnson to Williams, June 5, 1963, Williams Papers.\n\n. Nancy Williams, interview, October 10, 2014.\n\n. Amanda Vail, interview, September 9, 2014. Vail, later an acclaimed biographer, was the editor at Viking who later replaced Smith.\n\n. Gerald Howard, \"Jacket Required: Thomas Pynchon's V,\" Graywolf Press December 12, 2013, <https:\/\/www.graywolfpress.org\/blogs\/jacket-required-thomas-pynchons-v>. Howard, who worked with Smith, later became an executive editor at Doubleday.\n\n. Amanda Vail, interview, September 9, 2014.\n\n. Undated response to questionnaire submitted by a student to Williams, Williams Papers; Williams to Jeremy Larner, April 22, 1966, Williams Papers; Irving Howe, \"The Culture of Modernism,\" _Commentary_ , November 1, 1967.\n\n. Responses to questionnaire, Williams Papers.\n\n. David Myers, interview, December 16, 2013. Myers, a student of Cunningham's, said, \"I know for a fact that Cunningham was the model for Stoner. Stoner's marriage to Edith is Cunningham's wife, Barbara Gibbs\" (the poet). Williams spoke often of _My Mortal Enemy_ (1926), Willa Cather's portrait of a marriage that disintegrated over many years. Regarding Cunningham's relationship with his daughter, Timothy Steele said, \"He raised her, and, as far as I could judge, they had a good relationship.\" Email, July 3, 2015. Steele is the editor of _The Poems of J. V. Cunningham_ (Athens, OH: Swallow Press, 1997).\n\n. Williams, _Stoner_ , 3.\n\n. Ibid., 6.\n\n. Ibid., 13.\n\n. Ibid., 5.\n\n. The novelist Baine P. Kerr was a student of Williams': \"He would explain brilliantly Flaubert's methods\u2014light glinting off leaves, smells.\" Kerr, interview, October 8, 2014. Hilton Als, in _Joan Didion: The Center Will Not Hold_ , Griffin Dunne, director (2017).\n\n. Williams, _Stoner_ , 23.\n\n. Responses to questionnaire, Williams Papers.\n\n. Williams, _Stoner_ , 16.\n\n. Ibid., 51.\n\n. Ibid., 53.\n\n. Ibid., 117, 119.\n\n. Ibid., 65\u201366, 85\u201386, 95.\n\n. Ibid., 91.\n\n. Ibid., 92.\n\n. Ibid., 99.\n\n. Williams Papers, Box 11.\n\n. Williams, _Stoner_ , 111\u2013112.\n\n. Ibid., 37.\n\n. Ibid., 131.\n\n. Ibid., 137.\n\n. Ibid., 148.\n\n. Ibid., 174.\n\n. Moral identity is by no means just a literary construct. How it defines who we are is clear from an incident in medical history. In 1848, an explosion launched a thirteen-pound tamping iron through the skull of a twenty-five-year-old railroad worker named Phineas Gage, taking a chunk of his brain with it. Formerly mild-mannered and responsible, Gage emerged from the accident impulsive and foul-tempered. His character changed so markedly that those who knew him said he was \"no longer Gage.\" Recent experiments conducted in human behavior demonstrate that \"moral traits\u2014more than any other mental faculty\u2014are considered the most essential part of identity, the self, and the soul.\" And so it is with creating strong characters in fiction who operate from a moral identity that is established through choice, behavior, and response to conflict. Nina Strohminger and Shaun Nichols, \"The Essential Moral Self,\" _Cognition_ 131 (2014): 159\u2013171.\n\n. Sr. Joan Saalfeld, email, February 26, 2016.\n\n. Michelle Latiolais, interview, December 14, 2013.\n\n. Christof Wegelin, \" _The Imagination of Loving: Henry James's Legacy to theNovel_ by Naomi Lebowitz\" (review), _Southwest Review_ (Spring 1966): 197\u2013199. Winters used a similar method for putting teeth into his poetry. \"Every poem of Winters',\" wrote the critic Alvin B. Kernan, \"is a dramatization of the conflict between the moral perceiver and a spiritually empty world which is nevertheless so solid and real that the confrontation cannot be avoided\" (John Fraser, \"Leavis, Winters and 'Tradition,'\" in _Southern Review_ , Vol. 7, No. 4, Autumn 1971, 963\u2013985.\n\n. Williams, _Stoner_ , 203.\n\n. Williams to Jeremy Larner, April 22, 1966, Williams Papers.\n\n. Williams to Fred Inglis, January 10, 1966, Williams Papers.\n\n. Williams wrote, \"Grace is one of those mysterious people, as I suggest in the novel, whose moral nature is of such frailty and delicacy that it must be nurtured with infinite care. The fact is, the world does not nurture with such care, and such people are usually destroyed. The question of whether or not this should be so is irrelevant; it simply _is_ so [his italics]. If all the world could have loved Grace as Stoner did, she might have been saved; but the world did not.\" Williams to Fred Inglis, January 10, 1966, Williams Papers.\n\n. Williams, _Stoner_ , 215.\n\n. Marie Rodell, quoting Williams in a letter to Cork Smith, August 3, 1964, Collin Papers.\n\n. Dan Wakefield, interview, November 13, 2013. To Cork Smith, Williams wrote, \"I can't seem to get away from clowns.\" Williams to Smith, May 21, 1965, Williams Papers. Stubby Kaye was not as incidental a performer as Williams thought. He was a Broadway musical actor whose role in _Guys and Dolls_ was reprised in the 1955 film version with Marlon Brando and Frank Sinatra.\n\n. Cork Smith to Williams, October 19, 1965, Williams Papers.\n\n. Irving Howe, \"The Virtues of Failure,\" _The New Republic_ , February 22, 1966.\n\n. Robert Pawlowski, email, April 19, 2015. Jean James about the scene in the English Department\u2014Williams seated alone in a chair all day\u2014from her husband, Stuart, the department head, in interview on June 18, 2016.\n\n**PART IV. _AUGUSTUS_**\n\nChapter Fourteen. Bread Loaf and \"Up on the Hill\"\n\n. Nancy Williams, interview, October 10, 2014; Robert B. Richardson, interview, December 12, 2013.\n\n. Dale W. Nelson, _The Imprint of Alan Swallow: Quality Publishing in the West_ (Syracuse, NY: Syracuse University Press, 2010), 108; Frederick Manfred, \"Alan Swallow, Poet and Publisher,\" _Chicago Tribune_ , December 18, 1966.\n\n. Alan Swallow to Williams, March 13, 1963, Williams Papers.\n\n. Nancy Williams, interview, October 10, 2014.\n\n. Willard Marsh to Williams, June 2, 1965, Williams Papers.\n\n. William Sloane to Williams, July 25, 1965, Williams Papers.\n\n. Williams to Wayne Carver, May 13, 1966, Williams Papers. Williams taught a writing seminar at Carleton College in Northfield, Minnesota, at Carver's invitation.\n\n. John Williams, \"Looking for John Ciardi at Bread Loaf,\" in Vincente Clemente, ed., _John Ciardi: Measure of the Man_ (Fayetteville: University of Arkansas Press, 1987).\n\n. David Haward Bain and Mary Smyth Duffy, eds., _Whose Woods These Are: A History of the Bread Loaf Writers' Conference, 1926\u20131992_ (Hopewell, NJ: Ecco Press, 1993), 61.\n\n. Ibid., 95.\n\n. Williams, \"Looking for John Ciardi at Bread Loaf.\"\n\n. Dan Wakefield, \"John Williams, Plain Writer,\" _Ploughshares_ 7, nos. 3\u20134 (Fall\/Winter 1981): 9.\n\n. Williams, \"Looking for John Ciardi at Bread Loaf.\"\n\n. Dan Wakefield, interview, November 13, 2013.\n\n. Jay Neugeboren, interview, September 3, 2014.\n\n. Manfred, \"Alan Swallow, Poet and Publisher.\"\n\nChapter Fifteen. The Good Guys\n\n. Williams to Seymour Epstein, February 17, 1967, Williams Papers. He also expressed his satisfaction with Marie Rodell in this letter, despite the contretemps he'd suffered in connection with _Butcher's Crossing_ , saying she was \"the most honest and straightforward agent I have ever known, a real professional, and totally respected among the editors and publishers. . . . _If_ she talks to you about your writing, she will talk to you about it as if she were your friend, not your critic, and she will have respect for you, what you do, and your own decisions.\"\n\n. Williams to Harry H. Taylor, Ball State University English Department, June 7, 1966, Williams Papers.\n\n. Williams to Marie Rodell, June 6, 1966, Collin Papers.\n\n. John Williams, \"Statement of Project\" to the Rockefeller Foundation, April 12, 1966, Williams Papers.\n\n. John Williams, _Augustus_ (New York: New York Review Books, 2014), 22.\n\n. John Williams, \"Statement of Project\" to the Rockefeller Foundation, April 12, 1966, Williams Papers.\n\n. Gerald Chapman interview, July 12, 2015.\n\n. Fred Inglis, interview, June 22, 2014. Inglis was a visiting professor from England during the late 1960s and became one of Williams' closest friends.\n\n. Williams to Rodell, September 14, 1959, Collin Papers.\n\n. Jonathan Williams, interview, October 9, 2014. Katherine's career is similar to her father's. She received her PhD from the CUNY Graduate School, which specializes in seventeenth-century, romantic, and Victorian British literature. She received a bachelor's degree in French from the University of Denver. As of this writing, she is an associate professor and chair of the English Department at New York Institute of Technology.\n\n. Rodell to Williams, June 6, 1967, Williams Papers.\n\n. John Williams, \"Fact in Fiction: Problems for the Historical Novelist,\" _Denver Quarterly_ 7, no. 4 (Winter 1973): 1\u201312. Williams was influenced in his thinking by a paper given by Janet Lewis at the 1966 University of Denver summer writing workshop. It was titled \"The Problems of the Historical Novelist.\" Winters\/Lewis Papers.\n\n. Williams, \"Fact in Fiction,\" 5.\n\n. Ibid.\n\n. Richard Johnson, \"Interview: John Williams,\" _Empire_ , _Denver Post_ , June 9, 1985. As Harold Rosenberg wrote, \"A generation is fashion: but there is more to history than costume and jargon. The people of an era must either carry the burden of change assigned to their time or die under its weight in the wilderness.\" \"Death in the Wilderness,\" in _The Tradition of the New_ (New York: Da Capo Press, 1994), 255.\n\n. Williams, _Augustus_ , 18\u201319. Williams' view of life seems almost Russian at times, especially Tolstoyan, as in _The Death of Ivan Ilyich_ , where the hero seeks some kind of transcendence that will reveal his purpose before he dies.\n\n. Joanne Greenberg, interview, October 8, 2014; Sr. Joan Saalfeld, SNJM (Sisters of the Holy Names of Jesus and Mary), interview, October 7, 2014.\n\n. Williams, _Augustus_ , 293.\n\n. Johnson, \"Interview: John Williams.\"\n\n. Williams to Willard Marsh, October 11, 1967, Willard Marsh Papers, Special Collections, University of Iowa Libraries, Iowa City, Iowa.\n\n. Williams to John Ciardi, January 22, 1968, Williams Papers.\n\n. Ibid.\n\n. Williams to Dan Wakefield, March 8, 1968, Wakefield Papers, Lilly Library, Indiana University.\n\n. Jonathan Williams, interview, October 9, 2014.\n\n. Blake Bailey recounts the legend of Richard Yates, who, in a fit of mania abetted by alcohol, climbed up on the roof of Treman cottage and assumed the spread-arm position of Christ crucified. _A Tragic Honesty: The Life and Work of Richard Yates_ (Reprint, New York: Picador, 2004). He had to be talked down.\n\nAccording to David Haward Bain and Mary Smyth Duffy, eds., _Whose Woods These Are: A History of the Bread Loaf Writers' Conference, 1926\u20131992_ (Hopewell, NJ: Ecco Press, 1993), the \"torrid romance of 1963\" was between the forty-one-year-old novelist John Hawkes and Joan Didion, who was twenty-eight years old at the time and had come to Bread Loaf on a fellowship. \"'They were like tragic teenagers,' snickered one Bread Loafer. _'Two weeks left to live!'_ \u2014before they had to go back to their respective lives\" (p. 89, italics in original). A staff member suffered a nervous breakdown during Ciardi's tenure as director. \"God did not intend for all those writers to gather in one place at one time,\" he warned from his hospital bed (p. 101).\n\n\"We are all madmen, which we already knew,\" Wakefield wrote to Williams in 1969.\n\n. Bain and Duffy, eds., _Whose Woods These Are_ , 98.\n\n. William Sloane, July 18, 1968, Williams Papers.\n\n. Bain and Duffy, eds., _Whose Woods These Are_ , 102.\n\n. Ibid., 98.\n\n. Sally Boland, \"Acceptance Speech for the 1997 Kalikow Award,\" June 29, 1997, Plymouth State University, Plymouth, New Hampshire.\n\n. Alan Prendergast, \"Sixteen Years After His Death, Not-So-Famous Novelist John Williams Is Finding His Audience,\" _Westword_ , November 3, 2010.\n\n. Williams to William Hamlin, Chairman, Division of Humanities, University of Missouri at St. Louis, December 7, 1965. Hamlin and his wife, Florence, accompanied John and Lonnie on the vacation to Mexico in 1954, which should be taken into account regarding the breezy tone of Williams' letter.\n\n. Gerald Chapman, interview, May 20, 2015.\n\n. Peggy McIntosh, \"White Privilege and Male Privilege: A Personal Account of Coming to See Correspondences through Work in Women's Studies,\" Working Paper (Wellesley College, Center for Research on Women), 1988, no. 189.\n\n. Peggy McIntosh, interview, March 17, 2015.\n\n. Elizabeth Richardson, interview, May 18, 2015. The woman chosen over Richardson was American literary critic Helen Vendler.\n\n. David Milofsky, \"John Williams Deserves to Be Read Today,\" _Denver Post_ , July 1, 2007.\n\n. Peggy McIntosh, interview, March 17, 2015.\n\n. Brian Wooley, \"An Interview with John Williams,\" _Denver Quarterly_ (Winter 1986): 19.\n\n. Johnson, \"Interview: John Williams.\"\n\n. Timothy Steele, email, March 1, 2015. Steele is a poet and author. He edited _The Poems of J. V. Cunningham_ (Athens, OH: Swallow Press, 1997).\n\n. Wooley, \"Interview with John Williams.\"\n\n. Dan Wakefield, \"A Matter of Style,\" _Denver Quarterly_ (Winter 1986): 139.\n\n. _Stoner_ fans often wonder whether Bill Stoner and John Williams are similar. In their convictions about the academy, they are. Julian Barnes wrote, discussing Professor Stoner, \"He becomes a teacher, 'which was simply a man to whom his book is true, to whom is given a dignity of art that has little to do with his foolishness or weakness or inadequacy as a man'. Towards the end of his life, when he has endured many disappointments, he thinks of academe as 'the only life that had not betrayed him'. And he understands also that there is a continual battle between the academy and the world: the academy must keep the world, and its values, out for as long as possible.\" Julian Barnes, \"Stoner: The Must-Read Novel of 2013,\" _Guardian_ , December 13, 2013.\n\n. John W. Aldridge, _After the Lost Generation: A Critical Study of the Writers of Two World Wars_ (New York: McGraw-Hill, 1951), ix. Aldridge, who taught alongside Williams at Bread Loaf, was himself a World War II combat veteran who had been awarded the Bronze Star. Ciardi had been a machine-gunner on a bomber over Japanese islands.\n\n. Bain and Duffy, eds., _Whose Woods These Are_ , 103.\n\n. Ibid., 106.\n\nChapter Sixteen. \"Long Life to the Emperor!\"\n\n. Nancy Williams, interview, October 4, 2014.\n\n. Ibid.\n\n. Williams to Marie Rodell, April 3, 1970, Collin Papers.\n\n. Nancy Williams said, \"We never insisted that the kids like or see each other. We would have his kids over for dinner a couple times a month, maybe. John didn't discipline his very much\u2014they weren't misbehaving children. But he expected that his and mine would do what he said. Never raised his voice or hand to them.\" Apparently, John was a sterner and angrier parent when he was married to Lonnie. His son, Jonathan, recalled being punished with a belt a few times. During the holidays, both sets of grandparents\u2014Lonnie's and Nancy's parents\u2014came to the house and socialized.\n\n. Few of his friends knew the story of how \"John Jewell\" had become \"John Williams.\"\n\n. Carol DeBoer-Langworthy, interview, October 3, 2014; Joanne Greenberg, interview, October 8, 2014. None of the many people interviewed, however, believed that Williams' small stature had anything to do with his demeanor. In other words, he didn't try to compensate for it in any obvious way.\n\n. Robert Pawlowski, email, February 28, 2015.\n\n. Martha Hume, \"John Williams: Artist of Diversity,\" _Dust_ (Winter 1966): 18.\n\n. Ibid.\n\n. Williams' typed responses to a questionnaire, undated. Williams Papers.\n\n. Willard Marsh, \"Forwarding Service,\" in _Beachhead in Bohemia_ (Baton Rouge: Louisiana State University Press, 1969), 143.\n\n. George Rae Williams to Williams, October 25, 1970, Williams Papers.\n\n. Tony Burton, \"Willard 'Butch' Marsh (1922\u20131970) and His Novel About 1950s Ajijic,\" Sombrero Books, October 26, 2015, sombrerobooks.com\/?p=2682.\n\n. Miller Williams, September 1970, Williams Papers.\n\n. Dan Wakefield to Williams, October 1, 1970, Williams Papers.\n\n. Rodell to Dan Wakefield, October 9, 1970, Collin Papers.\n\n. Seymour Lawrence to Rodell, September 8, 1970, Collin Papers.\n\n. Rodell to Lawrence, September 11, 1970, Collin Papers.\n\n. Rodell to Dan Wakefield, January 27, 1971, Williams Papers.\n\n. Nancy Williams, interview, October 11, 2015.\n\n. Ibid.\n\n. Jerry Murray, \"Memories of Mack and Jeannette\" (April 2009), www.efanzines.com\/EK\/eI43\/index.htm#murray.\n\n. David Haward Bain and Mary Smyth Duffy, eds., _Whose Woods These Are: A History of the Bread Loaf Writers' Conference, 1926\u20131992_ (Hopewell, NJ: Ecco Press, 1993), 105.\n\n. Robert Pack, interview, December 13, 2013.\n\n. Dan Wakefield, interview, August 3, 2014.\n\n. Susan Cheever, _Drinking in America: Our Secret History_ (New York: Twelve Books, 2015). Cheever, the daughter of the writer John Cheever, wrote, \"By the time my father's generation of writers started publishing in the years after World War II, being a writer almost always meant being a drunk\" (p. 165).\n\n. Dan Wakefield, interview, August 3, 2014. The Irish poet Dylan Thomas died of alcoholic poisoning in 1953, a few days after drinking eighteen whiskeys at the White Horse Tavern in Greenwich Village, a favorite waterhole for writers.\n\n. Michelle Latiolais, interview, December 14, 2013.\n\n. Williams to Corlies Smith, February 17, 1972, Collin Papers.\n\n. Smith to Williams, March 1, 1972, Williams Papers.\n\n. Smith to Williams, April 4, 1972, Collin Papers.\n\n. Williams to Rodell, June 14, 1972, Collin Papers.\n\n. Williams to Alan Williams, July 8, 1972, Williams Papers; Rodell to Williams, June 8, 1972, Collin Papers.\n\n. Dan Wakefield, \"John Williams, Plain Writer,\" _Ploughshares_ 7, nos. 3\u20134 (Fall\/Winter 1981): 12.\n\n. Orville Prescott to Alan D. Williams, July 15, 1972, Williams Papers; J. V. Cunningham to Williams, October 8, 1972, Williams Papers.\n\n**PART V. _THE SLEEP OF REASON_**\n\nChapter Seventeen. \"How Can Such a Son of a Bitch Have Such Talent?\"\n\n. Eric Pace, \"2 Book Awards Split for the First Time,\" _New York Times_ , April 11, 1973. The National Book Awards are given to one book (and author) annually in each of four major categories: fiction, nonfiction, poetry, and young people's literature. There are other categories, but they generally fall under the main four. Each panel considers hundreds of books.\n\n. John B. Breslin, \"Year of the Big Book: NBA 1973,\" _Saturday Review_ , May 5, 1973, 408\u2013409.\n\n. Pace, \"2 Book Awards Split.\"\n\n. Ibid.\n\n. William Cole, \"The Last of the National Book Awards?\" The Guest Word, _New York Times_ , May 4, 1975, 288.\n\n. Joanne Greenberg, interview, October 8, 2014.\n\n. William Zaranka, interview, October 7, 2014. Over the years, Zaranka served as an English professor, program chair of the creative writing program, dean of liberal arts and sciences, and provost of the university. Philip Doe, who also taught in the department, said, \"John winning the award wasn't universally greeted with joy\" (Philip Doe, interview, October 8, 2014).\n\n. Carol DeBoer-Rolloff, later DeBoer-Langworthy, became a lecturer in literary biography at Brown. The editor, Martha Rupp, explained to Rolloff why she didn't want to write the article.\n\n. Carol DeBoer-Langworthy, interview, October 3, 2014.\n\n. Ibid.\n\n. Ibid. Williams' remarks about disliking experimental fiction are paraphrased from the interview he gave to Martha Hume for _Dust_ in 1966, but he repeated the same sentiments consistently over the years.\n\n. John F. Baker, \"Academe Dominates a Low-Key NBA,\" _Publishers Weekly_ , April 30, 1973, 33.\n\n. John Barth to Williams, April 18, 1973, Williams Papers.\n\n. Williams to Durrett Wagner at Swallow Press, May 2, 1973. Williams summarizes Viking's reasons for rejecting his poems in his letter to Wagner. Four partners purchased Swallow Press and moved the business to Chicago following Alan's death in 1966. Suzanne Alaura Klinger, \"The Swallow Press Archives: A Finding Guide with Background on the Swallow Press and Its Significance\" (master's thesis, University of Chicago, 1984).\n\n. Robert O. Preyer, Brandeis University English Department chairman, to Williams, April 25, 1973, Williams Papers. The poet Adrienne Rich held the one-year appointment at the time.\n\n. Philip Doe, interview, October 8, 2014.\n\n. Robert Pawlowski, email, February 28, 2015.\n\n. Nancy Williams, interview, October 10, 2014.\n\n. Marie Rodell to Williams, May 25, 1973, Collin Papers.\n\n. Baine Kerr, interview, October 8, 2014.\n\n. Nancy Williams, interview, October 10, 2014.\n\n. Eric Gould, interview, March 19, 2015; Jonathan Williams, interview, October 9, 2014; David Milofsky, interview, December 13, 2013. Gould taught with Williams for many years. Milofsky was a professor in the department who knew John late in his career.\n\n. John Ciardi to Williams, June 18, 1971, Williams Papers; Joan Saalfeld, interview, October 7, 2014.\n\n. Michael White, interview, October 3, 2014.\n\n. Burton Raffel, interview, June 15, 2015; Miriam Epstein, interview, June 14, 2015.\n\n. William Zaranka, interview, March 2, 2015.\n\n. Joanne Greenberg, interview, October 8, 2014.\n\n. Fred Inglis, interview, June 22, 2014; Steve Heller, interview, March 2, 2015. Heller said the \"Anything other than the obvious . . .\" remark was directed at the novelist and short story writer Gordon Weaver, who used to tell the story as a joke on himself.\n\n. Interviews with Williams' former students: Joan Saalfeld, October 7, 2014; Angela Ball, June 14, 2015; Nancy Esterlin, June 27, 2015; Heather McHugh, by email, May 12, 2015; Philip Doe, October 8, 2014; Michelle Latiolais, December 14, 2013; Michael White, October 3, 2014; and William Zaranka, October 7, 2014.\n\n. Baine Kerr, interview, October 8, 2014.\n\n. William Zaranka, interview, October 7, 2014.\n\n. Joseph Nigg, interview, December 12, 2013. Nigg later became the author of popular books for young people about mythical creatures and lost places.\n\nChapter Eighteen. In Extremis\n\n. John, almost sixty, had mellowed toward children. The grandchildren were allowed free rein. Jonathan and his wife came to pick up their son, and the little boy \"had trashed the apartment,\" Jonathan later said. \"Oh, we're so sorry, we'll pick it up!,\" he told his father. But Nancy and John replied, \"No. Don't bother. We had a wonderful time watching him.\"\n\n. Williams to Fred Inglis, Fall 1980, Williams Papers.\n\n. Nancy Williams to Peggy and Burton Feldman, June 5, 1979, Burton Feldman Papers, University of Denver.\n\n. Williams to Fred Inglis, Fall 1980, Williams Papers.\n\n. Ibid.\n\n. William Cifelli, _John Ciardi: A Biography_ (Fayetteville: University of Arkansas Press, 1997), 416. Nancy Williams, interview, October 9, 2014. Windsor Lane, near the top of Solares Hill, the highest point in Key West, is now called the Writer's Compound. It's a private neighborhood of restored shanties, cottages, and bungalows connected by brick walkways and lush gardens. Former residents include Richard Wilbur, John Hersey, Ralph Ellison, and John Ciardi.\n\n. Dan Wakefield, \"John Williams: Plain Writer,\" _Ploughshares_ 7, nos. 3\u20134 (Fall\/Winter 1981).\n\n. Baine Kerr, interview, December 16, 2013; Cindy Carlisle, interview, December 18, 2013.\n\n. Baine Kerr, interview, October 8, 2014; Cindy Carlisle, interview, December 18, 2013. Said Inglis, \"I've seen him get very aggressively tight once or twice with Nancy and she just wept copiously when it happened. And that was very difficult and embarrassing, about what to do and where to look\" (Fred Inglis, interview, September 5, 2014).\n\n. Williams to Burton Feldman, a colleague at the University of Denver, October 24, 1979, Burton Feldman Papers, University of Denver (Feldman Papers hereafter); William Zaranka, interview, October 7, 2014.\n\n. His son, Jonathan, believed Williams had an addictive personality: he went at writing, drinking, and smoking with the same ferocious intensity.\n\n. Nancy Williams, interview, October 9, 2014.\n\n. Williams to Fred Inglis, Fall 1980, Williams Papers.\n\n. Dan Wakefield, email, April 11, 2016.\n\n. Roxie Munro, interview, October 2, 2014.\n\n. Steve Heller, interview, March 2, 2015. Heller, a first-timer and still in college\u2014technically, too young be there\u2014scheduled himself to read one of his stories to an audience of Pulitzer and National Book Award winners. Touchingly, \"it was well-received.\" David Nemec, interview, February 23, 2015.\n\n. Jim Clark, letter, February 27, 2015.\n\n. Bin Ramke, interview, October 7, 2014.\n\n. John Ciardi to Margaret Mills, National Institute of Arts and Letters, February 20, 1985, Williams Papers.\n\n. Douglas Woolf, _Hypocritic Days and Other Tales_ , edited by Sandra Braman (Santa Barbara, CA: Black Sparrow Press, 1993), 393\u2013399.\n\n. Gale Woolf, interview, June 7, 2015. Yvonne's partner beginning in 1980 was an Irish American sportsman in Washington State. The relationship that lasted twenty-five years, until his death in 2005.\n\n. Alan Prendergast, \"Sixteen Years After His Death, Not-So-Famous Novelist John Williams Is Finding His Audience,\" _Westword_ , November 3, 2010; Nancy Williams, interview, October 9, 2014.\n\n. Williams to Fred Inglis, Fall 1980, Williams Papers; Arlo Haskell, \"A Day at the Beach, 1984,\" Key West Literary Seminar, April 29, 2008 (updated September 26, 2016), www.kwls.org\/key-wests-life-of-letters\/a_day_at_the_beach_in_key_west.\n\n. Nancy Williams, interview, October 10, 2014.\n\n. Nancy Williams to Margaret and Burton Feldman, April 29, 1986, Feldman Papers.\n\n. Williams to Edward Loomis (handwritten draft, circa 1988), Williams Papers. Loomis was a professor of English at the University of California at Santa Barbara.\n\n. Williams to the Feldmans, April 29, 1986, Feldman Papers; Nancy Williams, interview, October 10, 2014.\n\n. It must be said that regardless of the friction that existed between Williams and some of the others, the symposium was made possible through grants from the Colorado Endowment for the Humanities; the support of the University of Denver English Department of the Faculty of Arts and Humanities; and the _Denver Quarterly_ , which devoted its winter 1986 issue entirely to him and his work, including the first John Williams bibliography. It was an effort that took almost a year to plan.\n\n. Cifelli, _John Ciardi_ , 476.\n\n. Blake Bailey, _A Tragic Honesty: The Life and Work of Richard Yates_ (Reprint, New York: Picador, 2004), 558.\n\n. Jim Clark, email, February 27, 2015.\n\n. Clark wrote, in the email of February 27, 2015:\n\nSadly, Mr. Yates fell completely off the wagon in Denver. He made a bit of a spectacle of himself at the main program at the Denver Public Library, and when I stopped by the hotel to pick him up to take him to the airport, he was nowhere to be found. I asked the desk clerk where I could find Mr. Yates, the writer, as I was supposed to take him to the airport. \"You need to speak to the Manager,\" she replied. I learned that Mr. Yates had collapsed in the lobby a little earlier, suffering delusions and seizures. He ended up staying in the hospital in Denver for about two weeks, drying out.\n\n. Prendergast, \"Sixteen Years After His Death.\"\n\n. Audio recording made at the Denver Public Library, March 29, 1986, included in Williams Papers.\n\n. Nancy Williams, interview, October 10, 2014.\n\n. _Ozarks at Large: Arkansas Voices_ , interview with John Williams broadcast on National Public Radio station KUAF, 1993.\n\n. George Rae to Williams, March 3, 1987, Williams Papers.\n\n. The collage of anecdotes comes from a letter from George Rae Marsh to the Williamses, March 21, 1987, Williams Papers; Williams to Frances Collin, May 10, 1989, Collin Papers; Nancy, John, and Kathy Williams to Margaret and Burton Feldman, January 29, 1990, Feldman Papers; Nancy Williams to the Feldmans (the dream), November 14, 1990, Feldman Papers; interviews with Nancy Williams, October 8\u201310, 2014; and Prendergast, \"Sixteen Years After His Death.\"\n\nEpilogue. John Williams Redux\n\n. Edwin Frank, interview, October 1, 2015.\n\n. Colum McCann, \"Colum McCann's Top 10 Novels on Poets,\" _Guardian_ , October 2, 2006.\n\n. Astrid Gagneur, Interview with Anna Gavalda, Myboox, September 19, 2011.\n\n. Ibid.\n\n. Oscar Van Gelderen, interview, October 1, 2015.\n\n. Ibid.\n\n. Edwin Frank, interview, October 1, 2015; Cristina Marino, interview, October 2, 2015.\n\n. Edwin Frank, interview, October 1, 2015.\n\n. John Williams, _Stoner_ (New York: New York Review Books, 2006), 277\u2013278.\nWorks Consulted\n\nAaron, Daniel. _The Americanist_. Ann Arbor: University of Michigan Press, 2007.\n\nAldridge, John W. _After the Lost Generation: A Critical Study of the Writers of Two World Wars_. New York: McGraw-Hill, 1951.\n\nAlvarez, A. \"Yvor Winters.\" In _Beyond All This Fiddle: Essays, 1955\u20131967_. New York: Random House, 1968. Originally published in _The New Statesman_ , 1960.\n\nBailey, Blake. _A Tragic Honesty: The Life and Work of Richard Yates_. Reprint, New York: Picador, 2004.\n\nBain, David Haward, and Mary Smyth Duffy, eds. _Whose Woods These Are: A History of the Bread Loaf Writers' Conference, 1926\u20131992_. Hopewell, NJ: Ecco Press, 1993.\n\nBassing, Eileen. _Where's Annie?_ New York: Random House, 1956.\n\nCheever, Susan. _Drinking in America: Our Secret History_. New York: Twelve Books, 2015.\n\nCifelli, William. _John Ciardi: A Biography_. Fayetteville: University of Arkansas Press, 1997.\n\nClemente, Vincente, ed. _John Ciardi: Measure of the Man_. Fayetteville: University of Arkansas Press, 1987.\n\nCreeley, Robert. \"Reading Douglas Woolf's Ya! & John-Juan.\" Dalkey Archive Press website, www.dalkeyarchive.com\/reading-douglas-woolfs-ya-john-juan.\n\nCunningham, J. V. _The Collected Essays of J. V. Cunningham_. Chicago: Swallow Press, 1976.\n\n. \"For My Contemporaries.\" In _The Exclusions of a Rhyme: Poems and Epigrams_. Athens: Ohio University Press \/ Swallow Press, 1960.\n\nDickinson, Leon T. _An Historical Sketch of the Department of English, University of Missouri\u2013Columbia_. Columbia: Department of English, University of Missouri, 1986.\n\nEllman, Richard. \"Publisher for Poets.\" _Saturday Review_ , July 22, 1961.\n\nEpstein, Joseph. \"Father of History: Herodotus and the Human Dimension in the Past.\" _Weekly Standard_ , October 20, 2014.\n\nFike, Francis. \"Cold Grace: Christian Faith and Stoicism in the Poetry of J. V. Cunningham.\" In _Renascence: Essays on Values in Literature_. Milwaukee, WI: Marquette University Press, 2007.\n\nFraser, John. \"Leavis, Winters and 'Tradition.'\" _Southern Review_ (Autumn 1971).\n\nGruber, Frank. _Zane Grey: A Biography_. New York: Amereon, 1969.\n\nHoward, Gerald. \"Jackets Required: Thomas Pynchon's V.\" Graywolf Press, December 12, 2013, <https:\/\/www.graywolfpress.org\/blogs\/jacket-required-thomas-pynchons-v>.\n\n. \"Pynchon from A to V: Gerald Howard on _Gravity's Rainbow_.\" _Bookforum_ (Summer 2005), www.bookforum.com\/archive\/sum_05\/pynchon.html.\n\nHume, Martha. \"Artist of Diversity: John Williams.\" _Dust_ (Winter 1966).\n\nHyman, Stanley Edgar. \"Yvor Winters and Evaluation in Criticism.\" In _The Armed Vision: A Study in the Methods of Modern Literary Criticism_. New York: Alfred A. Knopf, 1955.\n\nJohnson, Richard. \"Interview: John Williams.\" _Empire_ , _Denver Post_ , June 9, 1985.\n\nKlinger, Suzanne Alaura. \"The Swallow Press Archives: A Finding Guide with Background on the Swallow Press and Its Significance.\" Master's thesis, University of Chicago, 1984.\n\nLear, Linda. _Rachel Carson: Witness for Nature_. New York: Houghton-Mifflin, 1997.\n\nMerrill, Flora. \"A Short Story Course Can Only Delay, It Cannot Kill an Artist, Says Willa Cather.\" Reprinted in _Nebraska State Journal_ , April 25, 1925.\n\nMilofsky, David. \"John Williams Deserves to Be Read Today.\" _Denver Post_ , July 1, 2007.\n\nMumford, Lewis. _The Golden Day: A Study in American Experience and Culture_. New York: Liverwright, 1926. Online.\n\nNelson, Dale W. _The Imprint of Alan Swallow: Quality Publishing in the West_. Syracuse, NY: Syracuse University Press, 2010.\n\nNye, Nelson. \"Roundup.\" _New York Times_ , April 4, 1960.\n\n\"Our Literary Alumni.\" _University of Denver Magazine_ (September 1964).\n\n_Poetry_ , LII excerpted in Paul. J. Alpers, ed. _Elizabethan Poetry: Modern Essays in Criticism_ (Oxford: Oxford University Press, 1967).\n\nRexroth, Kenneth. _American Poetry in the Twentieth Century_. New York: Seabury Press, 1971.\n\nSeidel, Frederick. \"Robert Lowell, The Art of Poetry No. 3.\" _Paris Review_ (Winter-Spring 1961).\n\nSimmons, R. Laurie, and Thomas H. Simmons. \"Historic Resources of Downtown Denver.\" National Register of Historic Places, Multiple Property Documentation Form. US Department of the Interior, National Park Service.\n\n_Source Book of Information on Tabard Inn_. Columbia, MO: Press of the Crippled Turtle, 1951.\n\nStanford, Donald E. \"Yvor Winters, 1900\u20131968,\" _Southern Review_ (Summer 1968).\n\nStone, Edward. _A Certain Morbidness: A View of American Literature_. Carbondale: Southern Illinois University Press, 1969.\n\nStrohminger, Nina, and Shaun Nichols, \"The Essential Moral Self.\" _Cognition_ 131 (2014).\n\nSwallow, Alan. _An Editor's Essays of Two Decades_. Seattle: Experiment Press, 1962.\n\nWakefield, Dan. \"John Williams, Plain Writer.\" _Ploughshares_ 7, nos. 3\u20134 (Fall\/Winter 1981).\n\nWhite, Theodore. \"The Hump: The Historic Airway to China Was Created by US Heroes.\" _Life_ , September 11, 1944.\n\nWilliams, John. \"Fact in Fiction: Problems for the Historical Novelist.\" _Denver Quarterly_ 7, no. 4 (Winter 1973).\n\n. \"In the American Grain: The Importance of William Carlos Williams.\" _Colorado Review_ (Fall 1997).\n\n. \"The 'Western': Definition of the Myth.\" _The Nation_ , November 18, 1961.\n\nYvor Winters. \"Individual Poets and Modes of Poetry: The 16th Century Lyric in England. A Critical and Historical Reinterpretation.\" In Paul J. Alpers, ed., _Elizabethan Poetry: Modern Essays in Criticism_. New York: Oxford University Press, 1967. Originally published in _Poetry Magazine_ 53 (1939).\n\nWooley, Brian. \"An Interview with John Williams.\" _Denver Quarterly_ (Winter 1986).\n\nWoolf, Douglas. _Hypocritic Days and Other Tales_. Edited by Sandra Braman. Santa Barbara, CA: Black Sparrow Press, 1993.\n\nYezzi, David. \"The Seriousness of Yvor Winters.\" _New Criterion_ (June 1997).\nA John Williams Bibliography\n\n**NOVELS**\n\n_Nothing But the Night_. Denver: Swallow Press, 1948; Fayetteville: University of Arkansas Press, 1990; London: Vintage, 2018. Published in translation in Dutch (Lebowski), German (DTV), and Italian (Fazi), and forthcoming in Catalan from Ediciones 62.\n\n_Butcher's Crossing_. New York: Macmillan, 1960; London: Victor Gollancz, 1960; London: Odham's Press, 1961; New York: Dolphin Books, 1962; London: Panther Books, 1963; Boston: Gregg Press, 1978; Fayetteville: University of Arkansas Press, 1987; New York: New York Review Books, 2007; London: Vintage, December 2013. Published in translation in Bulgarian (Labyrinth), Chinese (Horizon Media), Danish (Lindhardt & Ringhof), Dutch (Lebowski), Finnish (Bazar), French (Piranha), German (DTV), Hebrew (Yedioth), Italian (Fazi), Korean (Opus), Norwegian (CappelenDamm), Portuguese (Brazil) (Absurdaventura), Serbian (Plato), Spanish (Lumen\/Random House Mondadori), Swedish (Natur & Kultur), and Turkish (Koton Kitap).\n\n_Stoner_. New York: Viking, 1965; New York: Lancer Books, 1966; New York: Pocket Books, 1972; London: Allen Lane, 1973; Fayetteville: University of Arkansas Press, 1987; New York: New York Review Books, 2006; New York: New York Review Books, 2015 (50th Anniversary Hardcover edition); London: Vintage Classics, 2012 (American Greats series). Published in translation in Albanian (IDK), Arabic (Dar Athar), Bulgarian (Labyrinth), Catalan (Ediciones 62), Chinese (CM Publishing, Horizon Media), Croatian (Fraktura), Czech (Kniha Zlin), Danish (Lindhardt and Ringhof), Dutch (Lebowski, best seller), Finnish (Bazar), German (DTV, best seller on the _Der Spiegel_ list for over a year), Greek (Gutenberg), Hebrew (Yedioth), Icelandic (Draumsyn), Italian (Fazi), Korean (Random House), Latvian (Zvaigzne ABC), Lithuanian (Baltos Lankos), Norwegian (Cappelen Damm), Polish (Sonia Draga), Portuguese (Brazil) (Editora Absurdaventura), Portuguese (Portugal) (Dom Quixote), Romanian (Polirom), Russian (Astrel), Slovak (Artforum), Slovene (Mladinska), Spanish (Argentina) (Fiordo Editorial via Ediciones Baile del Sol), Spanish (Spain) (Ediciones Baile del Sol), Swedish (Natur & Kultur), Turkish (Koton Kitap), and Ukrainian (Ranok).\n\n_Augustus_. New York: Viking, 1972; London: Allen Lane, 1973; New York: Dell Press, 1973; New York: Penguin Books, 1979; Fayetteville: University of Arkansas Press, 1993; New York: New York Review Books, 2014; London: Vintage, 2013. Published in translation in Bulgarian (Labyrinth), Catalan (Ediciones 62), Chinese (Horizon and Chi Ming), Croatian (Fraktura), Danish (Lindhardt and Ringhof), Finnish (Bazar), French (Piranha), German (DTV), Greek (Dardanos), Hebrew (Yedioth), Italian (Fazi, Frasinelli, Sperling and Kupfer), Japanese (Sakuhin Sha), Korean (GU-Fic), Latin (Lebowski), Norwegian (CappelenDamm), Portuguese (Brazil) (Editora Radio Londres), Spanish (Ediciones Pamies), Swedish (Natur & Kultur), and Ukrainian (Ranok).\n\n**BOOKS OF POETRY**\n\n_The Broken Landscape_. Denver: Swallow Press, 1949.\n\n_The Necessary Lie_. Denver: Verb Publications, 1965.\n\n**FICTION APPEARING IN PERIODICALS**\n\n\"Short Story 2.\" _Story_ 33, no. 130 (Spring 1960).\n\n\"The Sleep of Reason.\" _Ploughshares_ 7, nos. 3\u20134 (October 1981): 23\u201360.\n\n**POETRY APPEARING IN PERIODICALS**\n\n\"Love Poem.\" _New Mexico Quarterly_ 16, no. 3 (Autumn 1946): 359.\n\n\"Time, Place and Circumstance.\" _Interim_ 2, nos. 3\u20134 (1946): 34\u201337.\n\n\"Variation on a Theme by Donne.\" _Yale Poetry Review_ 5 (Autumn 1946): 12.\n\n\"The Waiting Man.\" _Matrix_ 9, nos. 3\u20134 (Winter 1946\u20131947), 37.\n\n\"The Harbinger of Sunstroke.\" _Experiment_ 3, no. 1 (Spring 1947): 191\u2013192.\n\n\"Statement.\" _University of Kansas City Review_ 13, no. 4 (Summer 1947): 327.\n\n\"Weapon.\" _Experiment_ 3, no. 1 (Spring 1947): 192.\n\n\"The Braggart Mind.\" _Experiment_ 4, no. 2 (1948): 148.\n\n\"The Fishes.\" _Berkeley: A Journal of Modern Culture_ 5 (1948): 5.\n\n\"The Forest Is All Blown.\" _The Tiger's Eye: On Arts and Letters_ 5 (October 1948): 16.\n\n\"The Land.\" _Experiment_ 4, no. 1 (1948): 239.\n\n\"Ode.\" _Arizona Quarterly_ 4, no. 2 (1948): 148.\n\n\"The Pocketing Season.\" _Experiment_ 4, no. 1 (1948): 240.\n\n\"Casualty.\" _Poetry Book Magazine_ 13 (1949): 5.\n\n\"The Wall.\" _Poetry_ 74, no. 6 (Summer 1949): 319.\n\n\"The Dancer.\" _Poetry Book Magazine_ 4, no. 4 (Summer 1952): 10.\n\n\"Song of the Arctic Explorers.\" _Talisman_ 2 (1952): 44.\n\n\"Act Five, Scene Two.\" _Prairie Schooner_ 1 (1953): 17.\n\n\"The Affliction.\" _Prairie Schooner_ 1 (1953): 20.\n\n\"The Demons.\" _Talisman_ 4 (Winter 1953): 22\u201327.\n\n\"The Lovers.\" _Western Review_ 17, no. 2 (Winter 1953): 101\u2013102.\n\n\"The Steps of Love.\" _Prairie Schooner_ 1 (1953): 18.\n\n\"Voyage.\" _Prairie Schooner_ 1 (1953): 19.\n\n\"On the Key to My Bookcase.\" _Western Review_ 18, no. 3 (Spring 1954): 198.\n\n\"A Summer Day.\" _Between Worlds_ 1, no. 1 (Summer 1960): 46\u201347.\n\n\"The Name of Death.\" _Colorado Quarterly_ 9 (Winter 1961): 224.\n\n\"Passage.\" _Colorado Quarterly_ 9 (Winter 1961): 225.\n\n\"The Sparrow.\" _Western Humanities Review_ 15 (Winter 1961): 32.\n\n\"A Return.\" _Voices: A Journal of Poetry_ 178 (May\u2013August 1962): 26.\n\n\"Cold Coffee.\" _South Dakota Review_ 1, no. 1 (December 1963): 35.\n\n\"The Progress of the Soul.\" _South Dakota Review_ 1, no. 1 (December 1963): 33\u201334.\n\n\"Seascape with Figure.\" _Verb: A Broadsheet in the Flat Style_ 2, nos. 3\u20134 (November\u2013December 1964): 6\u20137.\n\n\"A Winter Garden.\" _South Dakota Review_ 1, no. 1 (December 1963): 36.\n\n\"Letters from Rome.\" _Poetry Northwest_ 7, no. 3 (Autumn 1966): 27\u201328.\n\n\"At Bernini's Fountains.\" _Southern Review_ 3, no. 2 (Spring 1967): 406.\n\n\"Letter to a Friend.\" _Denver Quarterly_ 3, no. 2 (Summer 1968): 74\u201378.\n\n**POEMS APPEARING IN ANTHOLOGIES**\n\n\"Six Poems: The Dead; The Meaning of Violence; For My Students, Returning to College; A History; The Skaters; The Leaf.\" In _New Poems by American Poets #2_ , ed. Rolfe Humphries. New York: Ballantine Books, 1957, 169\u2013172.\n\n\"An Airman, Falling, Addresses His Sweetheart.\" In _The Sound of Wings_ , ed. Joseph B. Roberts and Paul L. Brand. New York: Henry Holt, 1957.\n\n\"Nightmare.\" In _One of the Family_ , ed. Fred Inglis. London: Ginn, 1971.\n\n\"Four Poems: Letter from the Atlantic; At the Theater; The Skaters; Cold Coffee.\" In _Contemporary Poetry in America_ , ed. Miller Williams. New York: Random House, 1972, 64\u201365.\n\n\"The Skaters.\" In _Hosannah the Home Run!_ , ed. Alice Fleming. Boston: Little, Brown, 1972.\n\n\"Letter from Oxford.\" In _The Scene_ , ed. Fred Inglis. Cambridge: Cambridge University Press, 1972.\n\n\"The Skaters.\" In _Direction_ , ed. Theodore Clymer, Leo P. Ruth, Peter Evanechko, and Julia Higgs. New York: Random House, 1974.\n\n\"On Reading Aloud My Early Poems.\" In _Western Wind: An Introduction to Poetry_ , ed. John Frederick Nims. New York: Random House, 1974, 359.\n\n\"Two Poems.\" In _Poets of Western America_ , ed. Clinton F. Larson and William Stafford. Provo: Brigham Young University Press, 1975.\n\n**EDITORIAL**\n\n_Twentieth Century Literature: A Scholarly and Critical Journal_ , 1954\u20131956.\n\n_University of Denver Quarterly: A Journal of Modern Culture_ , 1966\u20131970.\n\n_English Renaissance Poetry_. Garden City: Doubleday, 1963. New York: Anchor Books, 1973. New York: W. W. Norton, 1974.\n\n**ESSAYS APPEARING IN PERIODICALS**\n\n\"J. V. Cunningham: The Major and the Minor.\" _Arizona Quarterly_ 6, no. 2 (Summer 1950): 132\u2013146.\n\n\"The 'Western': Definition of Myth.\" _The Nation_ 43, no. 17 (November 18, 1961): 401\u2013406.\n\n\"Concerning Little Magazines: Something Like a Symposium\" (editorial statement). _Carleton Miscellany_ 7, no. 2 (Spring 1966): 1974\u20131976.\n\n\"Editorial Statement.\" _Denver Quarterly_ 1, no. 1 (Spring 1966).\n\n\"Henry Miller: The Success of Failure.\" V _irginia Quarterly Review_ 44, no. 2 (Spring 1968): 225\u2013245. (Also anthologized twice; see below.)\n\n\"Fact in Fiction: Problems for the Historical Novelist.\" _Denver Quarterly_ 7, no. 4 (Winter 1973): 13\u201336.\n\n\"Fulke Greville: The World and God.\" _Denver Quarterly_ (Summer 1975).\n\n**ESSAYS APPEARING IN ANTHOLOGIES**\n\n\"The Western: Definition of the Myth.\" In _The Popular Arts: A Critical Reader_ , ed. Irving Deer and Harriet A. Deer. New York: Charles Scribner's Sons, 1967.\n\n\"Henry Miller: The Success of Failure.\" In _American Literary Anthology #3_ , ed. George Plimpton and Peter Ardery. New York: Viking Press, 1970. Reprinted in _Contemporary Literary Criticism_ , ed. Carolyn Riley. Detroit: Gale Research Co., 1973.\n\n\"The Future of the Novel.\" In _Contemporary Literary Experience_ , ed. David Madden. Pasadena, CA: Salem Press, 1974.\n\n**WORKS ABOUT JOHN WILLIAMS**\n\nSelected Reviews\n\n_Nothing But the Night_\n\n_Sewanee Review_ 57, no. 1 (Winter 1949): 127\u2013135 (F. Cudworth Flint).\n\n_Sewanee Review_ 56, no. 4 (Autumn 1964): 671\u2013684 (Robert B. Heilman).\n\n_Butcher's Crossing_\n\nBignardi, Irene. \"Che cosa si impara leggendo un western.\" _La Repubblica_ , March 20, 2013.\n\nBlackburn, Virginia. \"How the West Was Slaughtered: _Butcher's Crossing_ Review.\" _The Express_ , January 5, 2014.\n\nBrenner, Jack. \"Butcher's Crossing: The Husks and Shells of Exploitation.\" _Western American Literature_ 8, no. 4 (Winter 1973).\n\nFoulds, Alan. \"Butcher's Crossing Is Not at All Like Stoner\u2014But Just as Superbly Written.\" _The Spectator_ (January 18, 2014).\n\nHeilman, Robert B. \"The Western Theme: Exploiters and Explorers.\" _Partisan Review_ (March 1961).\n\nKemp, Peter. \"Shock and Gore: _Butcher's Crossing_.\" _Sunday Times_ , January 5, 2014.\n\nLezard, Nicholas. \"Is John Williams' Wild West Novel Better Than the Hit _Stoner_?\" _Guardian_ , January 7, 2014.\n\nMalvaldi, Marco. \"Verso la grande frontiera: Un viaggio di formazione che anticipa Cormac McCarthy.\" _La Stampa_ , March 23, 2013.\n\n_Partisan Review_ 28, no. 2 (March-April 1961): 286\u2013297 (Robert B. Heilman).\n\nVan Essen, Rob. \"Moby Dick in the Wild Western.\" _Boeken_ (November 22, 2013).\n\n_Stoner_\n\nAkbar, Arifa. \"The Quiet Professor Who Finally Became a Bestseller.\" _The Independent_ , June 4, 2013.\n\nAlmond, Steve. \"Lost & Found: Stoner.\" _Tin House_ (June 2009).\n\nBarnes, Julian. \"Stoner: The Must-Read Novel of 2013.\" _Guardian_ , December 13, 2013.\n\n_Best Sellers_ 32 (May 1, 1972): 71.\n\nBigsby, Christopher. \"Stoner: A Classic Tale of a 'Small' Academic Life.\" _Times Higher Education UK_ (September 12, 2013).\n\nBland, Archie. \"John Williams' Stoner Has Achieved Late Popular Acclaim But Is the Dead Author's Second Novel Another Forgotten Work of Brilliance?\" _The Independent_ , December 6, 2013.\n\n_Booklist_ 61 (June 1, 1965): 953.\n\n_Books & Bookmen_ 18 (July 1973): 134.\n\nChevilley, Philippe. \"Romain Am\u00e9rican: _Stoner_ de John Williams.\" _Les Echos_ , September 5, 2011.\n\nCripps, Ed. \" _Stoner_ : The Literary Rediscovery of the Year.\" _Huffington Post_ , July 28, 2013.\n\n_Daily Mail_ (London). \"Stone Me, It's a Truly Great Read.\" August 17, 2013.\n\nDeats, Sara Munson, and Lagretta Tallent Lenker, eds. _Aging and Identity: A Humanities Perspective_. Westport, CT: Praeger, 1999.\n\nDickstein, Morris. \"The Inner Lives of Men.\" _New York Times Book Review_ , June 17, 2007.\n\n\"Gavalda, son coup de _Stoner_.\" _Elle_ (September 9, 2011).\n\n_Guardian Weekly_ 108 (May 5, 1973): 25.\n\nGuerrera, Antonello. \"Stoner Mania.\" _La Repubblica_ , August 8, 2013.\n\nHabash, Gabe. \"A 'Perfect' American Novel Strikes Gold Overseas.\" _Publishers Weekly_ (April 27, 2013).\n\nHeathcock, Alan. \"Three Books to Take to a Fistfight.\" NPR, April 28, 2011.\n\n_Kirkus Reviews_ 33 (March 1, 1965): 264.\n\nKreider, Tim. \"The Greatest American Novel You've Never Heard Of.\" _The New Yorker_ , \"Page-Turner\" (blog), October 21, 2013.\n\n_Library Journal_ 90 (April 1, 1965): 1749.\n\n_Listener_ 89 (June 21, 1973): 840.\n\nMcCann, Colum. \"Book of a Lifetime: _Stoner_ by John Williams.\" _The Independent_ , June 15, 2013.\n\n. \"Stoner by John Williams.\" _The Independent_ , June 15, 2013.\n\nMontague, Sarah. \"Ian McEwan: Stoner di Williams tocca la verita: Come la grande letteratura.\" _La Repubblica_ , August 10, 2013.\n\nNeuhoff, Eric. \"Un homme ordinaire qui ne ressemble a personne.\" _Le Figaro_ (September 1, 2011).\n\n_New Yorker_ 41 (June 12, 1965): 155.\n\nNivet, Jean-Fran\u00e7ois. \"R\u00e9v\u00e9lation Am\u00e9ricaine: _Stoner_ de John Williams, traduit de l'Anglais par Anna Gavalda.\" _L'Humanit\u00e9 Supplement_ (January 5, 2012).\n\nNucci, Matteo. \"Da _Stoner_ a _Augusto_ , la vita anonima di un genio letterario perduto e ritrovato.\" _Il Venerdi_ (November 1, 2013).\n\n_Observer_ , April 22, 1973, 33.\n\nSnow, C. P. \"Good Man and Foes.\" _Financial Times_ , May 24, 1973.\n\n_Southern Review_ (Winter 1967): 186\u2013196.\n\nStark, John. \"The Novels of John Williams.\" _Hollins Critic_ 17, no. 4 (1980): 12\u201313.\n\n_Sunday Mirror_ (London). \"Books,\" August 4, 2013.\n\nSutherland, John. \"Literature Needs More Lazarus Miracles Like _Stoner_.\" _The Telegraph_ , July 13, 2103.\n\n_Times Literary Supplement_ , May 18, 1973, 545.\n\n_Virginia Quarterly Review_ 41 (Autumn 1965): cxx.\n\n_Augustus_\n\n_America_ 128 (May 5, 1973): 408.\n\n_Book World_ 6 (October 29, 1972): 15.\n\n_Economist Survey_ 249 (November 10, 1973): 16.\n\n_Kirkus Reviews_ 40 (August 15, 1972): 975.\n\n_Library Journal_ 97 (September 1, 1972): 2756.\n\n_Listener_ 90 (October 4, 1973): 459.\n\n_National Observer_ 11 (December 9, 1972): 25.\n\n_New Yorker_ 48 (November 25, 1972): 199.\n\n_New York Times_ 122 (October 28, 172): 29.\n\n_New York Times Book Review_ (April 8, 1973): 30 (John Leo).\n\n_New York Times Book Review_ (August 26, 1979): 31.\n\n_Publishers Weekly_ 202 (September 18, 1972): 71.\n\n_Saturday Review_ 1 (January 13, 1973): 66.\n\n_Time_ 100 (December 18, 1972): 107.\n\n_Times Literary Supplement_ (January 11, 1974): 25.\n\n_Wall Street Journal_ 181 (January 24, 1973): 18.\n\n_The Broken Landscape_\n\n_Arizona Quarterly_ 6, no. 3 (1950): 269.\n\n_Partisan Review_ 17, no. 2 (February 1950): [189]\u2013193 (Randall Jarrell).\n\n_Necessary Lie_\n\n_Southern Review_ (Winter 1967): 197\u2013228 (G. Lensing).\n\nCritical Studies\n\nBrenner, Jack. \"Butcher's Crossing: The Husks and Shells of Exploitation.\" _Western American Literature_ (February 1973): 243\u2013259.\n\nHartt, Julian. \"Two Historical Novels.\" _Virginia Quarterly Review_ (Summer 1973): 450\u2013458.\n\nHeilman, Robert B. \"Butcher's Crossing.\" _Partisan Review_ 28, no. 2 (March\u2013April 1961): 286\u2013297.\n\n. \"The Western Theme: Exploiters and Explorer.\" _Partisan Review_ (March\u2013April 1962): 286\u2013297.\n\nHobson, Geary. \"Reassessing John Williams' _Stoner_ : Twelve Years After.\" _New America_ 3, no. 2 (1977).\n\nHowe, Irving. \"The Virtues of Failure.\" _New Republic_ (February 12, 1966): 19\u201320.\n\nLeo, John. \"A Portrait Shaded toward the Stoic.\" (Review\/discussion of _Augustus_.) _New York Times Book Review_ (April 8, 1973): 30.\n\nNelson, Robert J. \"Accounts of Mutual Acquaintances to a Group of Friends: The Fiction of John Williams.\" _Denver Quarterly_ (Winter 1973): 13\u201336.\n\nSnow, C. P. \"Good Man and Foes.\" _Financial Times_ (London), May 24, 1973.\n\nStamper, Major Rex. \"John Williams: An Introduction to the Major Novels.\" _Mississippi Review_ 3, no. 1 (1974): 89\u201398.\n\nStark, John. \"The Novels of John Williams.\" _Hollins Critic_ 17, no. 4 (October 1980). (List of books by Williams, 12\u201313, is inaccurate.)\n\nWalsh, George. \"John Williams.\" In _Twentieth Century Western Writers_ , ed. James Vinson. Detroit: Gale Research, 1982, 815\u2013816.\n\nWerner, Craig. \"John Williams.\" _Dictionary of Literary Biography_. Detroit: Gale Research Co., 1980), 371\u2013377.\n\nOf Additional Interest\n\n_Book World_ (January 27, 1974): 4.\n\n_Book World_ 9 (September 16, 1973): 13.\n\nBreslin, John B. \"Year of the Big Book\u2014NBA 1973.\" _America_ 128 (May 5, 1973): 408\u2013409.\n\nPace, Eric. \"The National Book Award in Fiction: A Curious Case.\" _New York Times Book Review_ (May 6, 1972): 16\u201317.\n\n. \"Two Book Awards Split for the First Time.\" _New York Times_ , April 11, 1973, 38.\n\nInterviews\n\n\"Artist of Diversity.\" _Dust_ (Winter 1966).\n\n\"John Williams.\" _Denver Quarterly_ 20, no. 3 (Winter 1986).\n\nWakefield, Dan. \"John Williams, Plain Writer.\" _Ploughshares_ 7, nos. 3\u20134 (Fall\/Winter 1981): 9\u201322.\nIndex\n\n_Page numbers in italics refer to photographs_.\n\nAaron, Daniel: _The Americanist_ ,\n\nAiken, Conrad, , ,\n\nAjijic, Mexico, , , 105\u2013, 114\u2013, , 161\u2013, , 212\u2013\n\nalcoholism, 216\u2013, , , 281n27\n\nAldridge, John, __\n\nAlgren, Nelson,\n\nAls, Hilton,\n\n_American Scholar_ , ,\n\nAmmaniti, Niccol\u00f2,\n\nAnchor Books, 147\u2013, 154\u2013,\n\nAppleyard, Bryan,\n\n_Antioch Review_ , ,\n\nArendt, Hannah,\n\nAristotle,\n\n_Arizona Quarterly_ ,\n\nAsimov, Isaac,\n\nAssociation of Writers & Writing Programs (AWP),\n\nAswell, Edward, ,  262n9\n\n_Augustus_ (Williams): inspiration for, 151\u2013; National Book Award for, viii, 223\u2013, 226\u2013, , 232\u2013, , , 281n1; submission and publication of, 217\u2013, ; synopsis of, 193\u2013; writing of, 159\u2013, 192\u2013, , ,\n\nBaldwin, James: _Going to Meet the Man_ ,\n\nBarnes, Julian, , 279n44\n\nBarth, John, ; _Chimera_ , 223\u2013, ; Williams' first meeting with, 226\u2013\n\nBassing, Ellen, 107\u2013; _Where's Annie?_ , , 269n20\n\nBellow, Saul, ; _Herzog_ ,\n\nBerger, Thomas: _Little Big Man_ ,\n\nBerryman, John,\n\nBertinetti, Roberto,\n\nBignardi, Irene,\n\nBishop, Elizabeth, ,\n\nBlake, Eunice, __\n\nBorges, Jorge Luis,\n\nBourjaily, Vance,\n\nBowers, Edgar,\n\nBrague, Harry, , , 267n11\n\nBrandeis University, , 227\u2013\n\nBread Loaf Writers Conference, __ , 187\u2013, , , 200\u2013, , 213\u2013, , , ,\n\nBrecht, Bertolt,\n\nBrogan, Louise,\n\nBrooks, Cleanth, ,\n\nBrooks, Gwendolyn,\n\nBrower, Brock, __ , ; _The Late Great Creature_ ,\n\nBrowne, Thomas: _Hydriotaphia, Urn Burial_ ,\n\n\"Buffalo Bill\" Cody, 102\u2013\n\nBuffalo Bill's Wild West, 102\u2013, 269n9\n\nbuffalo hunting, viii, , , , 109\u2013, , 120\u2013, __ , ,\n\nBurroughs, Edgar Rice,\n\nBurroughs, William S., , 263n10\n\n_Butcher's Crossing_ (Williams), viii, , , , __ , __ , , , , ; final title decision about, 117\u2013; _New York Times_ review of, as a western, 120\u2013, , , , , ; reprintings of, 218\u2013, ; submission and publication of, 112\u2013; synopsis of, 109\u2013; writing of (working title _The Naked World)_ , 104\u2013, , , 113\u2013\n\nButler, Frank,\n\nCandido, Anne Marie, vii\u2013viii\n\nCarruth, Hayden, 82\u2013\n\nCarson, Rachel: _Silent Spring_ , ,\n\nCather, Willa, , ; _My Antonia_ , ; _O, Pioneers!_ , ; \"Paul's Case,\" ; _The Professor's House_ ,\n\nCefelli, Edward M.,\n\nChapman, Gerald, 101\u2013, 139\u2013, , ,\n\nCharles Scribner's and Sons, , ,\n\nChiang Kai-shek,\n\nCiardi, John, ix, __ , 188\u2013, , 201\u2013, , , , , , ,\n\nClark, James,\n\nClark, Walter van Tilburg, ,\n\nClarksville, Texas, 4\u2013, , , __\n\nCody, William Frederick \"Buffalo Bill,\" 102\u2013\n\nCogswell, Theodore, , 214\u2013, , 270n20\n\nColeman, Marjorie, ,\n\nCollin, Frances, , , 270n4\n\nColman, Ronald: in _Lost Horizon_ , ; in _Prisoner of Zenda_ , ; in _A Tale of Two Cities_ , 16\u2013; Williams' emulation of, , , __ , , ,\n\nConnell, Evan S.,\n\nCoover, Robert,\n\nCrane, Stephen: \"The Blue Hotel,\" ; Williams' admiration for, ,\n\nCrawford, Judith, 250\u2013\n\nCrawford Doyle Booksellers (New York City), 250\u2013\n\nCreeley, Robert, ,\n\nCrews, Harry,\n\nCunningham, James Vincent (J. V.), , , __ , , , ; \"For My Contemporaries,\" ; marriage of, ; Williams' essay on, 82\u2013; Winters Circle and, 79\u2013; _Woe or Wonder: The Emotional Effect of Shakespearean Tragedy_ , 83\u2013\n\nCurtis Publishing,\n\nCuster, George Armstrong,\n\nDavis, Catherine,\n\nDeBoer-Rolloff, Carol, , 281n8\n\n_Denver Quarterly_ , , , , , , 274n5, 284n28\n\nDickstein, Morris, ,\n\nDoe, Philip, 281n7\n\nDonne, John, ,\n\nDoubleday, , , 154\u2013\n\nDoyle, John, 250\u2013\n\nDrayton, Michael,\n\ndu Maurier, Daphne: _The Glass Blowers_ ,\n\nEakins, Thomas,\n\nEliot, T. S., ,\n\nElizabethans, , 94\u2013, , 146\u2013\n\nEllis, Bret Easton,\n\nEllison, Harlan, , 263n10, 273n6\n\nEmerson, Ralph Waldo, , , ; _Butcher's Crossing_ and, , __ ; on _Hydriotaphia, Urn Burial_ (Browne), ; \"Threnody,\"\n\nEmpson, William,\n\nEngle, Paul, ,\n\nEpstein, Seymour, __ , , 192\u2013, , 231\u2013; _Leah_ ,\n\n_Esquire_ , , ,\n\nExp\u00f3sito, Tito,\n\nFagles, Robert,\n\nFairchild, A. H. R., , 87\u2013, ,\n\nFarrell, James T.,\n\nFaulkner, William, , , , ,\n\nFazi, Elido,\n\nFazi Editore, ,\n\nFielder, Leslie A.,\n\nFields, Kenneth, ,\n\nFirst World War. _See_ World War I\n\nFles, Barthold, 91\u2013\n\nFortunato, Mario,\n\nFrank, Edwin, , 254\u2013\n\nFriedan, Betty: _The Feminine Mystique_ ,\n\nFrost, Robert, , , ,\n\nGage, Phineas, 275n38\n\nGascoigne, George,\n\nGass, William,\n\nGavalda, Anna, 251\u2013\n\nGibbs, Barbara: \"Accusatory Poem,\" , , 274n15\n\nGinzburg, Harold,\n\nGinzburg, Thomas, , ,\n\nGold, Herbert,\n\nGooge, Barnabe,\n\nGordon, Caroline,\n\nGoya, Francisco,\n\nG. P. Putnam,\n\nGrass, G\u00fcnter, ; _The Tin Drum_ ,\n\nGray, Thomas,\n\nGreat Depression, 13\u2013, 22\u2013, ,\n\nGreenberg, Joanne, __ , , , , ; _I Never Promised You a Rose Garden_ ,\n\nGreville, Fulke (1st Baron Brooke), ; _The Life of the Renowned Philip Sidney_ , ; Williams' dissertation on, 94\u2013, , , , ,\n\nGrey, Zane: _Riders of the Purple Sage_ , ; Williams' childhood reading of,\n\nGunn, Thom,\n\nGuthrie, A. B., Jr.: _The Big Sky_ , ,\n\nHaldeman-Julius, Emanuel, , 263n10\n\nHaley, Alex: _The Autobiography of Malcolm X_ ,\n\nHall, Donald,\n\nHalls, Oakley: _Warlock_ ,\n\nHamlin, Florence, ,\n\nHamlin, William,\n\nHarcourt Brace, ,\n\nHardin, John G.,\n\nHardin Junior College, , 20\u2013, , __\n\nHarris, Mark, 73\u2013, ; _Bang the Drum Slowly_ ,\n\nHarrison, John, vii\n\nHart, Al, 153\u2013\n\nHeller, Steve, 282n28, 283n16\n\nHemingway, Ernest, , , , 267n11; _Death in the Afternoon_ , 98\u2013\n\nHerodotus,\n\nHilton, James: _Lost Horizon_ , ,\n\nHolmes, John Clellon, ; _Go_ ,\n\nHomer: _Iliad_ ,\n\nHomer: _Odyssey_ , ,\n\nHope, Anthony: _Prisoner of Zenda_ ,\n\nHowe, Irving, , , ,\n\n_Hudson Review_ ,\n\nHunt, Morton M., 110\u2013, , , 270n27\n\nHurston, Zora Neale,\n\nHyman, Stanley Edgar, ,\n\nIbsen, Henrik: _Pillars of Society_ ,\n\nInglis, Fred, , 234\u2013,\n\nIowa Writers' Workshop, ,\n\nJames, Henry,\n\nJewell, Amelia (n\u00e9e Walker; Williams' mother), 4\u2013, 8\u2013, , , , __\n\nJewell, J. E. (Williams' father), 5\u2013, , __\n\nJohnson, Pyke, , , , , , , 273n13\n\nJonson, Ben, ,\n\nJordan, Satyavati C.,\n\nJoyce, James, ; _Ulysses_ , ,\n\nJustice, Donald,\n\nKazin, Alfred,\n\nKees, Weldon,\n\nKennedy, X. J., __\n\nKent, Rockwell,\n\nKernan, Alvin B., 276n41\n\nKerr, Baine, , 236\u2013, , 275n20\n\nKesey, Ken, ; _One Flew over the Cuckoo's Nest_ ,\n\nKey West, Florida, 40\u2013, 233\u2013, , 241\u2013, , ; Writer's Compound (Windsor Lane), , 283n6\n\nKumin, Maxine,\n\nLawrence, D. H., , ; _Lady Chatterley's Lover_ , ; _The Plumed Serpent_ ,\n\nLawrence, Seymour,\n\nLeavenworth, James, 141\u2013\n\nLeavis, F. R.,\n\nLebowski Publishers, 253\u2013\n\nLederer, William, __\n\nLe Dilettante,\n\nLevine, Philip,\n\nLewis, C. S.,\n\nLewis, Janet, ix, , 73\u2013, , , 118\u2013, , ; _The Ghost of Monsieur Scarron_ , ; marriage of, to Winters, ix, 74\u2013, 266n5; _The Wife of Martin Guerre_ ,\n\nLewis, Sinclair,\n\n_Liberty_ ,\n\nLittle, Brown and Company, ,\n\nLittle Blue Books, , 263n10\n\nLomax, Hollis N., , 172\u2013\n\nLondon, Jack: _Call of the Wild_ ,\n\nLowell, Robert, , 265\u2013266n4\n\nLuk\u00e1cs, Gy\u00f6rgy,\n\nMacLeish, Archibald,\n\nMacmillan, 115\u2013, 120\u2013, 145\u2013, , 271n19\n\n_Mademoiselle_ , 65\u2013,\n\nMailer, Norman, ,\n\nmale chauvinism,\n\nManfred, Frederick,\n\nMann, Heinrich,\n\nMarsh, George Rae (n\u00e9e Williams; Williams' half-sister), , , __ , 214\u2013, ; acting career of, ; birth and childhood of, , , , , ; and death of brother, ; marriage of, to \"Butch\" Marsh, 68\u2013, , , 106\u2013, , 137\u2013, 161\u2013, , 211\u2013; marriage of, to Ted Cogswell, 215\u2013; writing career of, 269\u2013270n20\n\nMarsh, Willard \"Butch,\" __ ; _Beachhead in Bohemia_ , , ; education of, , ; grave of, ; illness and death of, 211\u2013; marriage of, to George Rae, 68\u2013, , , 98\u2013, 105\u2013, , 137\u2013, 161\u2013, , 211\u2013; teaching career of, , 198\u2013, 211\u2013; _A Week with No Friday_ , , , , ,\n\nMartin, Edward Alexander, __\n\nMaugham, Somerset: _The Razor's Edge_ ,\n\nMcCann, Colum, 251\u2013\n\nMcCarthy, Cormac,\n\nMcEwan, Ian,\n\nMcGinley, Phyllis,\n\nMcGrath, Thomas,\n\nMcHugh, Heather,\n\nMcIntosh, Peggy, 204\u2013\n\nMcMahon, J. D., 6\u2013\n\nMcMurty, Larry: _Horseman, Pass By_ ,\n\nMelville, Herman: _The Confidence Man_ ,\n\nMiddlebury College, , , ,\n\nMiller, Henry,\n\nMilton, John, ,\n\nMissouri Writers' Workshop, ,\n\nMitford, Jessica, ,\n\nMoore, Marianne,\n\nmoral identity, , 275n38\n\nMumford, Lewis: _The Golden Day_ ,\n\nMunro, Roxie,\n\nMunroe, Harriet, 75\u2013\n\n_Naked World, The_ (Williams). See _Butcher's Crossing_ (Williams)\n\nNashe, Thomas,\n\nNational Book Award, viii, 223\u2013, 226\u2013, , 232\u2013, , , , 281n1\n\nNational Book Committee,\n\nNelson, Clara,\n\nNew Criticism, , , , , 263n11, 268n3\n\nNew Directions,\n\n_New Republic_ , , 178\u2013,\n\nNewsom, Jack, , , , , , , 260n16\n\n_New Yorker_ ,\n\n_New York Review of Books_ , , 254\u2013\n\nNims, John Frederick, __\n\nNin, Ana\u00efs, ,\n\n_Nothing But the Night_ (Williams), , , , , , ; Butch's critique of, 60\u2013; submission and publication of, , 48\u2013, , 59\u2013; synopsis of, 41\u2013; writing of,\n\nNye, Nelson C., 121\u2013\n\nOakley, Annie,\n\nO'Neill, Eugene, ; _Long Day's Journey into Night_ ,\n\nPack, Robert, __ , ,\n\nPawlowski, Robert,\n\nPeden, William,\n\nPercy, Walker,\n\nPeterson, Margaret,\n\nPetrarchan School of poetry, 76\u2013,\n\nPinkerton, Helen,\n\nPinsky, Robert,\n\n_Ploughshares_ , ,\n\n_Poetry_ , 75\u2013, , ,\n\nPoirier, Richard,\n\nPopular Books,\n\nPorter, Katherine Anne, ,\n\nProust, Marcel, , , ,\n\nPynchon, Thomas, 166\u2013,\n\nRaleigh, Walter,\n\nRamsay, Robert L., 86\u2013, , ,\n\nRexroth, Kenneth,\n\nRichards, I. A.,\n\nRichardson, Elizabeth,\n\nRichardson, Robert D., 138\u2013, 203\u2013\n\nRobbe-Grillet, Alain,\n\nRodell, Marie (Williams' agent), 111\u2013, , , 213\u2013, 217\u2013, , ; death of, ; _Stoner_ 's publication and, 144\u2013, 147\u2013, 153\u2013, , 163\u2013, ,\n\nRoethke, Theodore,\n\nRomanticism, 77\u2013, , ,\n\nRomantic poetry, 65\u2013, , , , 174\u2013\n\nRoosevelt, Franklin Delano, 13\u2013; and the New Deal, ,\n\nSalinger, J. D.: _Raise High the Roof Beam, Carpenters and Seymour: An Introduction_ ,\n\nSandoz, Marie: _Old Jules_ ,\n\nSassoon, Siegfried,\n\n_Saturday Evening Post_ , , 114\u2013,\n\n_Saturday Evening Post Stories 1954_ ,\n\nScott, Cecil, 115\u2013, , 145\u2013, 153\u2013\n\nScribner's. _See_ Charles Scribner's and Sons\n\nSecond World War. _See_ World War II\n\nSevareid, Eric,\n\n_Sewanee Review_ ,\n\nSheckley, Robert,\n\nShepard, Harwell V. \"Shep,\" ,\n\nSidney, Philip, ,\n\nSilverberg, Robert,\n\nSinclair, Upton,\n\nSitting Bull, 269n9\n\nSloane, William M., __ , , ,\n\nSmart, George K. \"Ken,\" 44\u2013, , 48\u2013, 54\u2013, , , , 262nn6\u20137\n\nSmith, Annie Laurie (Williams' eighth-grade English teacher),\n\nSmith, Avalon \"Lonnie\" (Williams' third wife), , , __\n\nSmith, Corlies M. \"Cork,\" 167\u2013, 177\u2013, , , 217\u2013, ,\n\nSmith College, , 199\u2013\n\nSnow, C. P.,\n\nSolomon, Barbara Probst: _The Beat of Life_ ,\n\nSontag, Susan,\n\n_Southern Review_ ,\n\nSpenser, Edmund,\n\n_Spicy Adventure_ ,\n\n_Spicy Detective_ , ,\n\n_Spicy Mystery_ ,\n\nSpillane, Mickey: _I, the Jury_ ,\n\nStanford University, , 80\u2013, , , ; Merry Pranksters and,\n\nStapleton, Benjamin F.,\n\nStegner, Wallace, ix,\n\nSteinbeck, John, , , ,\n\nStephens, Alan, , , 96\u2013\n\nStevens, Wallace, ,\n\nStewart, Mary: _The Moon-Spinners_ ,\n\nstoicism, , ,\n\n_Stoner_ (Williams), xiii, , , , 85\u2013, 89\u2013, __ , __ ; autographing party for, 177\u2013; Dutch edition of, 253\u2013; existentialism of, ; French edition of, ; as _New York Review of Books_ Classics imprint, ; as \"the perfect novel,\" , ; reprintings of, , , ; review of, by Irving Howe, 178\u2013, ; similarities between Williams and title character of, 279n44; submission and publication of, 163\u2013, 177\u2013; synopsis of, 168\u2013; translations and international editions of, 252\u2013; as Waterstones Book of the Year, ; working title, _A Matter of Light_ , for, , 149\u2013, 153\u2013, 163\u2013; working title, _A Matter of Love_ , for, , , ; writing of, , 149\u2013, 153\u2013,\n\nStrode, Hudson, ,\n\nSwallow, Alan, , 50\u2013, , 100\u2013, , , , 183\u2013; death of, ; early years of, 51\u2013; education of, ; first published book of, ; influence of Little Blue Books on, ; marriage of, to Mae, ; publishing principles of, 53\u2013\n\nSwallow, Mae (n\u00e9e Elder), 52\u2013, , ,\n\nSwallow Press, , 53\u2013, 58\u2013, 64\u2013, 69\u2013, 73\u2013, , , 90\u2013, , 96\u2013, 183\u2013, , 265n1, 282n14\n\n_Tale of Two Cities, A_ (film), 16\u2013, __\n\nTate, Allen, , , , 268n3; _On the Limits of Poetry_ ,\n\n_Tiger's Eye, The_ ,\n\nTrilling, Lionel, , ,\n\nTuchman, Barbara,\n\nTurberville, George,\n\nUniversity of Denver: _Denver Quarterly_ , , , , , , 274n5, 284n28; Williams as professor at, 98\u2013, , , , 138\u2013, 183\u2013, , 202\u2013, 228\u2013; Williams as student at, ix, 55\u2013\n\nUniversity of Missouri: Missouri Writers' Workshop, , ; and Ramsay-Fairchild feud, 86\u2013, , ,\n\nUpson, William Hazlett, __\n\nvan Doren, Mark,\n\nvan Gelderen, Oscar,\n\nViking Press, , , 166\u2013, , , , , , , , 274n10, 282n14\n\nVonnegut, Kurt, , , 267n11; _God Bless You, Mr. Rosewater_ , 269n18\n\nWakefield, Dan, __ , 189\u2013, 200\u2013, 213\u2013, 216\u2013, , , , ; _Revolt in the South_ ,\n\nWalker, Amelia (Williams' mother). _See_ Jewell, Amelia (n\u00e9e Walker; Williams' mother)\n\nWalker, Elbert G. (Williams' grandfather),\n\nWalker, Laura Belle (n\u00e9e Lee; Williams' grandmother),\n\nWalter, Bruno,\n\nWarren, Robert Penn, ; _All the King's Men_ ,\n\nWebster, Donovan, 261n12\n\nWest, Morris L.: _The Shoes of the Fisherman_ ,\n\nWest, Ray B., ,\n\n_Western Review_ , ,\n\nWestward Movement, 103\u2013\n\n_Whole Earth Catalog_ , 223\u2013\n\nWichita Falls, 5\u2013, , , ; \"The Corner,\" __ ; Hardin Junior College, , 20\u2013, , __ ; Newby-McMahon Building, 6\u2013; Texas oil boom, 5\u2013; Wichita Players,\n\nWilder, Thornton: _The Ides of March_ , ; _Our Town_ ,\n\nWilliams, Alyeene Rosida (n\u00e9e Bryan; Williams' first wife): divorce of, from Williams, , ; marriage of, to Williams, 26\u2013\n\nWilliams, Avalon \"Lonnie\" (n\u00e9e Smith; Williams' third wife): _Broken Landscape_ dedicated to, ; divorce of, from Williams, 199\u2013, ; education of, , ; first meeting of, with Williams, ; marriage of, to Williams, , , , 98\u2013, , , 140\u2013, , 148\u2013, , , 185\u2013, , 274n25, 280n4\n\nWilliams, George Clinton,\n\nWilliams, George Rae. _See_ Marsh, George Rae (n\u00e9e Williams; Williams' half-sister)\n\nWilliams, John: birth of, ; at Brandeis (visiting professor), 227\u2013; at Bread Loaf, __ , , 200\u2013, , 213\u2013, , , , ; and \"Celebration of John Williams,\" 244\u2013; composition notebooks of, 36\u2013, ; courses taught at Brandeis by, ; death of, ; dissertation of, on Greville, 94\u2013, , , , , ; as editor at _Denver Quarterly_ , , , , , , 274n5; as editor at Swallow Press, 64\u2013; emulation of Ronald Colman by, , , , , , ; endowed chair of, ; enlistment of, in Army Air Corps, 28\u2013; European trip of, 195\u2013; final illness of, 237\u2013, , ; Guggenheim Foundation application of, 118\u2013, ; Guggenheim Foundation Fellowship of, 239\u2013; at Hardin Junior College, 20\u2013, , __ ; in high school, 18\u2013; and hog butchering with grandfather, 12\u2013; in India, 30\u2013; influence of _A Tale of Two Cities_ (film) on, 16\u2013; in junior high school, 15\u2013; in Key West, 40\u2013, 233\u2013; as lecturer at Berkeley, ; Lovelace Bookstore after-school job of, , , ; marriage of, to Alyeene, 26\u2013, 29\u2013, , ; marriage of, to Lonnie, , , , 98\u2013, , , 140\u2013, , 148\u2013, , , 185\u2013, , 274n25, 280n4; marriage of, to Nancy, , , 214\u2013, 228\u2013, 233\u2013, 242\u2013, ; marriage of, to Yvonne, 55\u2013, 58\u2013, 62\u2013, 67\u2013; military service of, 28\u2013; with mother, __ ; and mother's death and illness, 45\u2013; and Oxford University summer session, 148\u2013; as parent and grandparent, , 280n4, 283n1; personal identity of, 10\u2013, 15\u2013; portraits of, 35\u2013, __ , __ ; as radio announcer at KRRV and KDNT (as \"Jon Williams\"), 24\u2013, , ; as radio announcer at WKWF, , 40\u2013; relationship of, with George Rae, ; as Smith College writer-in-residence, , 199\u2013; and staging of _Our Town_ on KWFT, ; and Texas Institute of Letters award for fiction, ; at University of Arkansas (guest lecturer), ; at University of Denver (professor), 98\u2013, , __ , __ , 138\u2013, 183\u2013, , 202\u2013, 228\u2013; at University of Denver (student), ix, 55\u2013; and Wichita Players performance, ; _Wichita Times Record_ student feature piece on, 18\u2013; \"Winters affair\" and, , 154\u2013, , 273n10, 273n13\n\n\u2014works of: _The Broken Landscape_ (poetry collection), ; _Dr. Cooper Speaking_ (radio drama), , ; \"Drouth\" (poem), ; _English Renaissance Poetry_ (edited anthology), 146\u2013, , , , ; \"J. V. Cunningham: The Major and the Minor\" (critical essay), 82\u2013; \"The Lovers\" (poem), ; \"Memories: Texas, 1932\" (poem), ; _The Necessary Life_ (poetry collection), ; \"An Old Actor to His Audience\" (poem), 221\u2013; _The Shape of the Air_ (poetry collection), , ; _The Sleep of Reason_ (unfinished novel), , 235\u2013, 239\u2013, 245\u2013; _Splendid in Ashes_ (unpublished novel), 90\u2013, , ; \"The Summer\" (short story), 65\u2013. See also _Augustus_ ; _Butcher's Crossing_ ; _Nothing But the Night_ ; _Stoner_\n\nWilliams, Jonathan (Williams' son), , , , 199\u2013, 280n4, 283n1\n\nWilliams, Katherine (Williams' daughter), , , , 194\u2013, , , 277n10\n\nWilliams, Miller, , , ,\n\nWilliams, Nancy Ann Leavenworth (n\u00e9e Gardner; Williams' fourth wife), ix, __ , 158\u2013, , , , , , , ; education of, 141\u2013; first meeting of, with Williams, 141\u2013; marriage of, to Williams, , , 214\u2013, 228\u2013, 233\u2013, 242\u2013,\n\nWilliams, Pamela (Williams' daughter), , , ,\n\nWilliams, Tennessee, , ; _A Streetcar Named Desire_ ,\n\nWilliams, William Carlos,\n\nWilliams, Yvonne Elyce. _See_ Woolf, Yvonne Stone (n\u00e9e Williams)\n\nWillis, T. G.,\n\nWills, Garry,\n\nWimsatt, William Kurtz,\n\nWinters, Yvor, ix, , , , 275\u2013276n41; \"The Anatomy of Nonsense,\" ; _Collected Poems_ , 96\u2013; _In Defense of Reason_ , , , 94\u2013, ; _Forms of Discovery_ , ; _The Immobile Wind_ , ; influence of, on Williams, 94\u2013, 145\u2013; _The Magpie's Shadow_ , ; marriage of, to Janet Lewis, ix, 74\u2013, 266n5; as moral figure for students, 266n14; \"On a View of Pasadena from the Hills,\" ; Williams and (\"Winters affair\"), , 154\u2013, , 273n10, 273n13\n\nWinters Circle, 74\u2013, ,\n\nWister, Owen: _The Virginian_ , 53\u2013\n\nWolfe, Thomas, ; influence of, on Williams, 18\u2013, , ; _Look Homeward, Angel_ , ,\n\nWolfe, Tom: _The Electric Kool-Aid Acid Test_ ,\n\nWoolf, Douglas, 62\u2013; marriage of, to Yvonne, __ , ; _Wall to Wall_ , , , , 264n10\n\nWoolf, Leonard,\n\nWoolf, Virginia, ; _Mrs. Dalloway_ ,\n\nWoolf, Yvonne Stone (n\u00e9e Williams), , __ , 240\u2013; divorce from Williams, 69\u2013; marriage to Williams, 55\u2013, 58\u2013, 62\u2013\n\nWordsworth, William, ,\n\nWorld War I, 28\u2013, ; George Williams and, , ; _Stoner_ and,\n\nWorld War II, , 262n23; and attack on Pearl Harbor, ; and Bread Loaf, ; and Hump-fliers, 31\u2013, __ , 261n12; and John Williams, 29\u2013, 151\u2013, ; and Leonard Woolf, 62\u2013; and _The Sleep of Reason_ , ; and Ted Cogswell,\n\nW. W. Norton,\n\nWyatt, Thomas, , , , , ; \"They Flee from Me,\"\n\nYacht, Marc, 262n6\n\n_Yale Review_ , ,\n\nYardley, Jonathan, 223\u2013\n\nYates, Richard, 244\u2013, 278n25, 284n32; _The Easter Parade_ , ; _Revolutionary Road_ ,\n\nYourcenar, Marguerite: _Memoirs of Hadrian_ ,\n\nZaranka, William, 281n7\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":"\n\n\n\nProduced by Chris Curnow, Josephine Paolucci and the Online\nDistributed Proofreading Team at http:\/\/www.pgdp.net. (This\nfile was produced from images generously made available\nby The Internet Archive.)\n\n\n\n\n\n\n\n\n\"PHIZ\"\n\n(H. K. BROWNE)\n\nA Memoir.\n\n\n_From PUNCH, July 22nd, 1882._\n\n\"Phiz.\"\n\nHABLOT K. BROWNE, ARTIST. BORN, 1815. DIED, JULY, 1882.\n\n    The Lamp is out that lighted up the text\n      Of DICKENS, LEVER--heroes of the pen.\n    _Pickwick_ and _Lorrequer_ we love, but next\n      We place the man who made us see such men.\n    What should we know of _Martin Chuzzlewit_,\n      Stern _Mr. Dombey_, or _Uriah Heep_?\n    _Tom Burke of Ours?_--Around our hearths they sit,\n      Outliving their creators--all asleep!\n\n          No sweeter gift ere fell to man than his\n          Who gave us troops of friends--delightful PHIZ!\n\n    He is not dead! There in the picture-book\n      He lives with men and women that he drew;\n    We take him with us to the cozy nook\n      Where old companions we can love anew.\n    Dear boyhood's friend! We rode with him to hounds;\n      Lived with dear _Peggotty_ in after years;\n    Missed in old Ireland where fun knew no bounds;\n      At _Dora's_ death we felt poor _David's_ tears!\n\n          There is no death for such a man--he is\n          The spirit of an unclosed book! immortal PHIZ!\n\n[Illustration]\n\n\n\n\n\"PHIZ\"\n\n(HABLOT KNIGHT BROWNE)\n\nA Memoir.\n\nINCLUDING\n\n_A Selection from his Correspondence and Notes on his Principal Works._\n\nBY\n\nFRED. G. KITTON.\n\nWITH A PORTRAIT, AND NUMEROUS ILLUSTRATIONS.\n\nLONDON:\nW. SATCHELL & CO.,\n19, TAVISTOCK STREET COVENT GARDEN.\n\nMDCCCLXXXII.\n\n\nLONDON:\nG. NORMAN AND SON, PRINTERS, HART STREET,\nCOVENT GARDEN.\n\n\n\n\nPREFACE.\n\n\nTaking into consideration the ability of the Artist whose name has\nbecome identified with the works of DICKENS, of LEVER, and of AINSWORTH;\nand who has contributed in the course of the present century more\nlargely (perhaps with the single exception of CRUIKSHANK) to the\nembellishment of popular books than any other known illustrator; it\nwould seem an inexcusable omission, almost amounting to neglect, if the\nlife and labours of the late HABLOT KNIGHT BROWNE met with no more\nworthy recognition than the fleeting comments of the daily press.\n\nSuch, at least, is my opinion; and as a humble tribute to the memory of\nan able and industrious draughtsman, and fertile designer, I place on\nrecord the more generally interesting particulars of an honourable and\nexemplary career.\n\nTo Mr. W. G. BROWNE and Dr. EDGAR BROWNE, sons of the deceased artist,\nmy best thanks are due for a kindly interest in my work, manifested more\nespecially by the loan of many interesting letters dashed off on various\noccasions by \"Phiz\" in the wildest spirit of fun; and a willing consent\nto their appearance in print.\n\nI have also to acknowledge the courtesy of Messrs. H. SOTHERAN & CO.,\nfor permission to copy for publication a few letters written by \"Phiz\"\nto CHARLES DICKENS, which are now published for the first time. For the\nPortrait (copied from a photograph, perhaps the best of the very few now\nin existence) I am indebted to the Proprietors of _The Graphic_.\n\nAnd lastly, the Author desires to associate with this brochure the name\nof his friend, Mr. GEORGE REDWAY, who has rendered much valuable\nassistance in bringing it before the public.\n\nFRED. G. KITTON.\n\n25, PAULTONS SQUARE,\nCHELSEA, S.W.\n\n_August, 1882_.\n\n\n\n\nLIST OF PLATES.\n\n\nPortrait of \"Phiz\" (H. K. Browne)               FRONTISPIECE\n\nThe Departure                                To face page  8\n\nArtist's \"Fancies for Mr. Dombey\"                    \"    11\n\nSam Weller and his Father                            \"    14\n\nTail-piece to _Barnaby Rudge_                        \"    16\n\nDick Swiveller and the Lodger                        \"    20\n\nDeath of Quilp                                       \"    26\n\nThe Rioters                                          \"    30\n\nNOTE.--With the exception of the Portrait, and the \"Dombey fancies,\" the\nabove engravings are printed from electro-types of the original blocks,\nwhich were first published in _Master Humphrey's Clock_ (1840-1).\n\n\n\n\n\"PHIZ\" (H. K. BROWNE) A MEMOIR.\n\n\n\"Fizz, Whizz, or something of that sort,\" humorous TOM HOOD would say,\nwhen trying to recall the pseudonym that has since become so familiar by\nmeans of the innumerable works of art to which it was appended. At the\ntime HABLOT[A] KNIGHT BROWNE first used this quaint _soubriquet_, it was\ncustomary to look upon book-illustrators as second, or even third-rate\nartists--mere hacks in fact; and for this reason they usually suppressed\ntheir real names, in order to give themselves the opportunity of earning\nthe title of _artist_, when producing more ambitious results as\npainters. Occasionally, whether by accident or design, the subject of\nthis memoir would affix his real name to his illustrations; and the\npublic were consequently under the impression that the two signatures\nwere those of different artists, and were even wont to remark that\n\"_Browne's work was better than that of 'Phiz!_'\"\n\nIt is not, perhaps, generally known that the artist's first _nom de\ncrayon_ was \"NEMO,\" which to some extent bears out the above statement\nthat a book-illustrator was considered a \"nobody.\" Mr. BROWNE himself,\nin referring to the _Pickwick Papers_, gave the following\nexplanation:--\"I think I signed myself as 'NEMO' to my first etchings\n(those of No. 4) before adopting 'Phiz' as my _soubriquet_, to\nharmonize--I suppose--better with Dickens' 'Boz.'\" It is only on the\nearliest printed plates in some copies of the _Pickwick Papers_ that the\nsignature of \"NEMO\" can be faintly traced.\n\nHABLOT KNIGHT BROWNE, son of William Loder Browne, a descendant from a\nNorfolk family, was born on the 12th of July, 1815, at Kennington,\nLondon. He was educated at a private school in Norfolk, and from an\nearly age evinced a taste for drawing, which, being recognized by his\nrelatives, induced them to apprentice him to FINDEN, the well-known\nline-engraver. An anecdote is told of him during his apprenticeship\nwhich will bear repetition. Finding BROWNE very painstaking and\nconscientious, his master usually sent him with engraved plates to the\nprinter, in order that he might superintend the operation of\nproof-taking. As printers usually take their own time over such matters,\nthe youth found that this waiting the pressman's pleasure tried his\npatience too much. It therefore occurred to him that to spend the\ninterval in the British Museum, hard by, would be much more suited to\nhis tastes. On his returning with the proofs, FINDEN would praise the\nboy's diligence, little thinking what trick had been practised on him.\n\nLine-engraving, however, did not find much favour with the future\n\"Phiz,\" the process being too tedious; for FINDEN would probably occupy\nsome weeks to produce a small plate, which by the quicker process of\netching, could have been executed in as many hours. He accordingly\nsuspended operations in that quarter, and, in conjunction with a young\nkindred spirit, hired a small attic, and employed his time in the more\nfascinating pursuit of water-colour drawing, which he continued to\nfollow with remarkable assiduity until a few days before his death.\n\nThese juvenile disciples of the brush then worked hard at drawing in\ncolour. BROWNE paid his share of the rent in drawings, which he produced\nrapidly; indeed, there was a solemn compact between the co-workers to\n\"do three a day\"--they subsisting, meanwhile, on the simplest fare. At\nthis time he attended the evening class at the \"Life\" School in St.\nMartin's Lane, and was a fellow-pupil with ETTY, the famous painter of\nthe \"nude.\" It was BROWNE'S great delight to watch this talented student\nat work, and he considerably neglected his own studies in consequence.\n\nAt the age of seventeen, or thereabouts, he succeeded in gaining a medal\noffered for competition by the Society of Arts for the best\nrepresentation of an historical subject; and was again fortunate in\nobtaining a prize, from the same Society, for a large etching of \"John\nGilpin.\" Mr. GEORGE AUGUSTUS SALA, himself an artist of no small\nability, remembers to have seen, in a shop-window in Wardour Street, a\ncertain print by a young man named HABLOT BROWNE, representing the\ninvoluntary flight of John Gilpin, scattering the pigs and poultry in\nhis never-to-be-forgotten ride.\n\n[Illustration]\n\nBy the time he had attained his twentieth year he had acquired\nconsiderable facility with the pencil. CHARLES DICKENS, but three years\nhis senior, and with whom the name of \"Phiz\" is inseparably connected,\nhad just then made a wonderful reputation by his \"Sketches,\" which first\nappeared, at intervals, during 1834-5, and were afterwards published in\nbook form, illustrated by the renowned GEORGE CRUIKSHANK.\n\nIn 1836, there appeared in print a pamphlet of some forty or fifty\npages, entitled _Sunday under Three Heads--As it is; as Sabbath Bills\nwould make it; as it might be made_; \"By Timothy Sparks; illustrated by\nH. K. B.;\" and dedicated to the Bishop of London. The author was CHARLES\nDICKENS, whose satire was levelled at Sir Andrew Agnew and the extreme\nSabbatarian party, and had immediate reference to a bill \"for the better\nobservance of the Sabbath,\" which the House of Commons had recently\nthrown out by a small majority. The illustrations in this little work\nwere drawn by HABLOT BROWNE, and are very choice examples of\nwood-engraving of the school that existed half a century ago. Its\noriginal price was one shilling, but having become very scarce, it is\nnow worth more than its weight in gold.\n\nThese early productions of BROWNE'S pencil at once introduced him to\npublic notice, and DICKENS showed his appreciation of their excellence\nby selecting him as the illustrator of the _Pickwick Papers_, which\nappeared during the early part of that year. It is well known to the\nreaders of Forster's _Life of Dickens_, that the idea of \"Pickwick\" was\nsuggested to the author by ROBERT SEYMOUR, whose tastes induced him to\netch a few plates of sporting subjects to which DICKENS was to supply\nthe text. Thus commenced that immortal work known as _The Posthumous\nPapers of the Pickwick Club_. SEYMOUR produced seven illustrations, when\nhe committed suicide, which obliged the publishers to make arrangements\nwith another artist. R. W. BUSS[B] succeeded SEYMOUR, and etched two\nplates, which DICKENS, who had by this time assumed the control of the\nwork, thought so unsatisfactory (as indeed they were), that he declined\nhis further services. Here a fresh opening was created, and WILLIAM\nMAKEPEACE THACKERAY competed with HABLOT KNIGHT BROWNE for the post;\nboth submitting to DICKENS' inspection some specimens of their work.\n\nThe choice fell upon \"Phiz,\" the artist whose ability has so admirably\nproved the wisdom of the selection; and THACKERAY thereupon determined\nto adopt another profession, with what happy results let _Esmond_\ntestify. Who could say whether _Vanity Fair_ would ever have been\nwritten had this mighty penman been chosen to succeed BUSS? It is\ncurious to note THACKERAY'S great anxiety to become an artist; he even\nwent abroad to study, but SALA tells us that \"Mr. THACKERAY drew,\nperhaps, rather worse than he had done before beginning his continental\nstudies, although at that time he actually supplied a series of etchings\nto illustrate DOUGLAS JERROLD'S _Men of Character_, which were prodigies\nof badness.\"\n\nWhen \"Phiz\" had been selected as the illustrator of the _Pickwick\nPapers_, his generous rival was the first to tell him the good news, and\noffer his congratulations.\n\n\"Phiz\" may now be said to have fairly commenced his career as a\nbook-illustrator. His sense of humour corresponded so exactly with that\nof DICKENS, that a mere suggestion enabled him to vividly represent the\nscenes described by the author. It has been remarked (and truly) that in\nmany cases the plates do not correspond with the text; but this can be\naccounted for. DICKENS, then an enthusiastic young author, and somewhat\nimpetuous in his demands for drawings, would arrive unexpectedly at\nBROWNE'S studio, hurriedly read a few pages of manuscript, and\nexclaiming, \"Now, I want you to illustrate that,\" would take an abrupt\ndeparture, carrying the manuscript off with him. As soon as the artist\ncould collect his faculties, he would try to recall the scene so hastily\ndescribed, and endeavour to put it on paper. DICKENS himself, in his\npreface to the _Pickwick Papers_, gives a similar explanation, viz.--\"It\nis due to the gentleman, whose designs accompany the letterpress, to\nstate that the interval has been so short between the production of each\nnumber in manuscript and its appearance in print, that the greater\nportion of the illustrations have been executed by the artist from the\nauthor's verbal description of what he intended to write.\" It is\ntherefore not surprising that a few errors, in such details as the\nnumber of boys in a procession,[C] or the dress of an individual, should\noccur.\n\n[Illustration]\n\nOf DICKENS' Novels, _Martin Chuzzlewit_ contains, perhaps, our etcher's\nmost vigorous productions, but the small woodcut illustrations in\n_Master Humphrey's Clock_ are very praiseworthy, and without doubt\nconduced greatly to the popularity of the book.\n\nThe illustrations in the _Pickwick Papers_ are on the whole inferior to\nmany which \"Phiz\" subsequently executed. But an exception must be made\nin favour of the artist's realization of the character of Sam Weller,\nthan which, even SEYMOUR'S happy invention of Mr. Pickwick did not more\neffectually ensure the popularity of DICKENS' comic epic and give it a\n\"deathless date.\"\n\nThe extraordinary demand for copies of the _Pickwick Papers_\nnecessitated a re-etching of the copper-plates, which, owing to friction\ncaused by the printer's hand, had become very much worn. This\nreproduction will account for any slight difference in the details of\nthe illustrations; for the repetition of subjects once etched, was a\ntask by no means congenial to the artist; and this no doubt induced him\nto say, some years afterwards, in a letter to one of his sons, \"O! I'm\na' weary, I'm a' weary of this illustrating business.\"\n\nArtists frequently experience great difficulty in realizing, to the\nauthor's satisfaction, the description of scenes and characters. An\nillustration is here given showing BROWNE'S various \"fancies for Mr.\nDombey,\" all of which failed to please DICKENS, who also expressed his\ndisapprobation of this artist's treatment of another subject in _Dombey\nand Son_. \"I am really distressed,\" writes he, \"by the illustration of\nMrs. Pipchin and Paul. It is so frightfully and wildly wide of the mark.\nGood Heaven! in the commonest and most literal construction of the text,\nit is all wrong. She is described as an old lady, and Paul's 'miniature\narm-chair' is mentioned more than once. He ought to be sitting in a\nlittle arm-chair down in the corner of the fire-place, staring up at\nher. I can't say what pain and vexation it is to be so utterly\nmisrepresented. I would cheerfully have given a hundred pounds to have\nkept this illustration out of the book. He never could have got that\nidea of Mrs. Pipchin if he had attended to the text. Indeed, I think he\ndoes better without the text; for then the notion is made easy to him in\nshort description, and he can't help taking it in.\"\n\nAs the tale proceeded, the artist more than compensated for his\nunsuccessful rendering of this incident; and with \"Micawber,\" in _David\nCopperfield_, he obtained the author's entire approbation, who says,\n\"Browne has sketched an uncommonly characteristic and capital Mr.\nMicawber for the next number.\" Again, with reference to an illustration\nin _Bleak House_, \"Browne has done Skimpole, and helped to make him\nsingularly unlike the great original.\"[D]\n\nOf the private life of \"Phiz\" little is known. His extreme nervousness\nand dislike to publicity was often misconstrued as pride; and DICKENS\neven had considerable difficulty in occasionally persuading him to meet\na few friends and spend a pleasant evening. When he did accept such\ninvitations, he invariably tried to seclude himself in a corner of the\nroom, or behind a curtain. His desire for a quiet, unobtrusive life,\ninduced him to pass most of his time in country retirement, all business\nmatters in town being transacted by an intimate friend.[E] Authors or\npublishers wishing to have a personal interview with \"Phiz\" were\ncompelled to visit him at his residence, a few miles from town, and many\nwere the _contretemps_ on dark nights as they crossed a bleak moor to\nreach their destination. His sons looked forward to the time when\nvisitors were expected, in order to hear the stories of wild adventure\nwhich generally befell them, and to laugh at their discomfiture.\n\n\"Phiz\" had been from his boyhood accustomed to horses, and frequently\nhunted with the Surrey hounds. To this circumstance is due the extreme\nfacility with which he delineated the horse in action in the hunting\nfield and elsewhere. At one time he contributed sketches to _The\nSporting Gazette_. This industrious artist was never known to take a\nlengthened holiday, but occasionally spent a few days at the seaside,\nwhere, no doubt, his pencil was fully employed. A letter, written while\nstaying at Margate, to his son Mr. Walter G. Browne (whom, for some\nunknown reason he styled \"Doctor\"), shows his innate sense of humour.\n\n     _Tuesday, June 19_, 6A, CRESCENT PLACE, MARGATE.\n\n     MY DEAR DR.,\n\n     \"I haave [Transcriber's note: haave has two macrons over the\n     a's to denote a very long a is the correct pronunciation]\n     my W. C. White:[F]--but I have no white _collars_--and\n     as I am swelling it about without a necktie--mine having\n     mysteriously disappeared, left behind in a bath\n     probably--perhaps it would be coming it too strong\n     to appear without collars also, and it is hardly warm enough\n     for it either. Your P.O. is from the Miscellany--to H. K.\n     Browne--from Mr. Barrett--Xtian name unknown--and no matter.\n     Any blocks that come, forward on. Send me a * * * * * *\n     before I return. I did some very good shades myself--of\n     myself--unconsciously--yesterday evening. The baths run\n     along one side of the High Street, flush with the\n     pavement--and I found when I had nearly finished my toilet\n     that the gas-burner was so ingeniously placed, that it was\n     impossible for any bather to avoid casting gigantic studies\n     of the nude upon the window blind.--This sort of thing.--\"\n\n[Illustration]\n\n       *       *       *       *       *\n\n[Here follow several other sketches of the bather in various attitudes].\n\nHis appreciation of fun is thus referred to by DICKENS in a letter to\nMrs. Dickens, dating from the Lion Hotel, Shrewsbury. \"Thursday, Nov.\n1st, 1838.--We were at the play last night. It was a bespeak--'The Love\nChase,' a ballet (with a phenomenon!), divers songs, and 'A Roland for\nan Oliver.' It is a good theatre, but the actors are very funny. Browne\nlaughed with such indecent heartiness at one point of the entertainment,\nthat an old gentleman in the next box suffered the most violent\nindignation.\"\n\nIn 1837, \"Phiz\" accompanied DICKENS to Flanders, for a ten days' summer\nholiday; and in 1838 they went to Yorkshire, a journey which resulted in\nthe production of _Nicholas Nickleby_.\n\nThe following year he made one of a party of four, and visited, with\nDICKENS, MACREADY and FORSTER, nearly all the London prisons. These\njoint tours of Author and Artist could not fail to assist the\nrealization of the scenes they intended to depict.\n\nIt is an interesting fact in connection with the career of \"Phiz,\" that\nhe would never agree to draw from the living model,--all his\nrepresentations of moving crowds, and the various types of humanity,\nwhich his etchings exhibit, being drawn from recollection. He would\nsometimes make a few jottings in pencil--mere memoranda--when anything\nstruck him as being worthy of reproduction, but beyond that he depended\non his excellent memory. For example, he would go to Epsom on the Derby\nDay without taking a pencil even, and, on returning home, would draw to\nthe life exact portraits of any conspicuous or eccentric character he\nhad seen on the course.\n\nAs previously stated, BROWNE was extremely fond of water-colour drawing,\nand executed some thousands during his life; not unfrequently a day's\nwork would be represented by three or four of these productions. They\nwere not caricatures, as one might suppose, but rural scenes _a la\nWatteau_, and allegorical subjects. This fact controverts the statement\nmade in a daily paper, that \"unfortunately, without a text to\nillustrate, 'Phiz' never had half-a-dozen ideas in his head\" (!). For\nmany years he was a constant contributor of pictures--figure subjects of\na humorous and dramatic character--to the Exhibitions of the British\nInstitution, and of the Society of British Artists. Among his more\nambitious efforts was a cartoon of considerable dimensions, representing\n\"A Foraging Party of Caesar's Forces surprised by the Britons,\" which\nappeared as No. 65 at the Westminster Hall Exhibition of 1843. This,\nnotwithstanding the \"scratchy\" manner of its execution, displayed\nremarkable skill and abundant energy of design. At the same gathering\nanother cartoon was attributed to him, of which the energy bordered on\ncaricature; it was named, \"Henry II defied by a Welsh Mountaineer.\"\n\n[Illustration]\n\nAt one time \"Phiz\" received an extraordinary commission to reproduce in\nwater-colour all his illustrations to the Novels of DICKENS. The Artist\nreminded his patron of the magnitude of the undertaking, but the request\nwas persisted in, and the work duly executed.\n\nHis love of bracing air induced him to pay frequent visits to the\nseaside; but on one occasion he lodged in a house not remarkable for its\nodoriferous nature; and, in order to produce a current of fresh air in\nhis bed-room, he opened door and window, and slept in the draught caused\nthereby. For many years before his death, he suffered from incipient\nparalysis, the result, no doubt, of this incautious act, and to which\nmay be attributed his disappearance from the art world some fifteen\nyears ago.\n\n\"Phiz,\" notwithstanding his crippled condition, still worked hard with\nadmirable perseverance, though his difficulties were increased by an\ninjury to his thumb, which compelled him to hold his pencil between the\nmiddle and fore fingers. His friends endeavoured to persuade him to draw\nhis pictures on a larger scale, in order that they might be photographed\nto the required dimensions, but, with one or two exceptions, he refused\nto act on this suggestion. He gradually lost that facility which\ncharacterized his work, and latterly yielded to proposals to illustrate\nboys' literature of a rather low class.\n\nThe time is past, no doubt, which encouraged the method of\nbook-illustration adopted by \"Phiz.\" It has given place to\nwood-engraving, and multifarious phototypic processes, that, perhaps,\nare commercially preferable, but from an artistic standpoint much\ninferior. We must, however, except the wonderful results some\nwood-engravers have produced from time to time, which etchers, even,\ncannot hope to excel.\n\nDr. Edgar Browne describes his father's indifference to the value of his\nwork, or the time and labour bestowed upon it:--\"He never understood the\nart of husbanding or developing his powers,--he never set to work to\nlearn any technical process; when he had a little leisure from\n'illustration' work, he used to start a picture 'to get his hand\nin'--generally taking some unimportant or trivial subject for this\npurpose. His facility of hand both in large and minute work was\nsomething marvellous. At one time, he produced a very remarkable series\nof sketches in chalk made during a tour in Ireland. They are scattered\nnow, but are as fine as anything he did, and are certainly the best\nrecords of a people who have practically vanished. He was astonishingly\ncareless about his work. Hundreds of original designs were thrown into\nthe waste-paper basket; apart from their local interest similar sketches\nhave found willing purchasers of late years.\"\n\nLike many other artists whose pecuniary reward had not been commensurate\nwith their ability,[G] he became the recipient of a pension. The kind\ninstrumentality of a few Royal Academicians obtained for him an annual\ngrant which had been previously enjoyed by the late GEORGE CRUIKSHANK.\n\nOn the 8th of July, 1882, the death occurred of the famous \"Phiz.\" At\nthe quiet village of Hove, near Brighton, where the last few years of\nhis life were spent, he succumbed in his sixty-seventh year to infirmity\nrather than old age. Almost forgotten as a man, his productions have\nremained in our memories, and will continue to do so as long as the\nworks of DICKENS and LEVER are read and appreciated. His remains were\ninterred at the extra-mural Cemetery, Brighton. The funeral was private,\nthe only mourners present being the four sons of the deceased, Dr.\nAmbler, Mr. George Halse,[H] and Mr. Robert Harrison.\n\nAs admirers of his artistic ability we place this Memoir as a wreath\nupon his grave.\n\n[Illustration]\n\n[Illustration]\n\n\nCORRESPONDENCE.\n\nThe following letters were addressed by the artist-humorist to his son,\nMr. Walter Gr. Browne:--\n\n\n      BLENHEIM CRESCENT, _Sept., Saturday, 3 o'clk._ P.M., A.D.\n     _1867_.\n\n     _My Dear Dr._,\n\n     I have nearly bursted my heart out, and proved, that my soul\n     or soles (I have two) is'nt--or an't--immortal,--by wearing\n     on 'em out running to and fro after yr.\n     _Balmorals_--Bootless errands! The wretched slave (of awl)\n     has but just brought them! I bristle with wrath! and could\n     welt him!--but--no--I won't--he may want his calf's skin\n     whole, to mend his own _Bad-morals_!!\n\n            *       *       *       *       *\n\n     I rush! I fly! to the Gt. W. R. Station!----!!!!\n\n     [Illustration]\n\n     I sink--breathless into the arms of the astounded\n     clerk--point to the boots----\n\n     _My-mouth_ faintly whispers \"_Wey-mouth_ in his pen-adorned\n     _Ear_!!\" and--and--\"Bless me! where am _I_?\"--and, and--I\n     wish--you may get 'em!\n\n            *       *       *       *       *\n\n     If you visit Portland again, make a note of any\n     peculiarities of spot--convict dress, &c.--as I have a\n     touching bit of horse-y sentiment (!) connected therewith,\n     which will do for _Spg. Gazette_.--I should think you ought\n     to find painty bits--within walking distance--say--right or\n     left ten miles?\n\n            *       *       *       *       *\n\n     Yrs. affecty.,\n\n     DAD.\n\n     _Sunday._\n\n     Really, my dear Walter, I thought you _did_ know better than\n     to disturb my devotional frame of mind on this blessed\n     Sabbath morn by forwarding me such a thoroughly worldly and\n     evil-thought-producing thing as a wretched milliner's\n     bill!!!--The wretch must wait--he gorged L5 not long before\n     I left home.--The greediness of some men!!\n\n     The Pic. Gall. circular I return--as you may like to enquire\n     about it--the doz. others, \"cheap bacon\"--\"patent teeth and\n     everlasting gums,\" &c., &c., &c., &c., &c. I shall manure\n     the grounds of Colyton with ----.\n\n     I think you might get some background material for coast\n     scenes down here.\n\n     Yr. affec. Dad,\n\n     H. K. B.\n\n       *       *       *       *       *\n\n\n     69, BLENHEIM CRESCENT, NOTTING-HILL, _Saturday_.\n\n     MY DEAR DOCTOR,\n\n     I send the Tenpounder, may it reach you in safety!\n\n     The Commander has returned. I sent you a paper containing\n     the important news, which, however, may _not_ have reached\n     you, although I don't think it contained any remarks upon\n     the \"Hemperors personal appearance,\" &c., &c., &c.\n\n     Tom is in the bosom of the family for a few days.--His Pipe\n     is tuned differently now to what it used to was, for he now\n     declareth that St. John's is \"a jolly school!\" He seems to\n     get on very well indeed, and has brought home what Dr. Lowe\n     calls a \"well-earned prize.\"\n\n     He laments daily over the supposed loss of 4_d_ invested in\n     a letter to you--from school--as it was directed, he\n     says,--21, Rue _Mussel wine_--I express doubts of its having\n     reached you--and he groans aloud over the Bull's eyes it\n     _would_ have bought!----\n\n     I am (at _present_) _on_ a Sporting Paper--supported by some\n     high and mighty Turf Nobs, but, I fear, like everything I\n     have to do with, now-a-days, it will collapse--for--some of\n     the Proprietors of the Paper are also Shareholders, &c.,\n     &c., in the Graphotype Co., so they want to work the two\n     together.--I hate the process--it takes quite four times as\n     long as wood--and I cannot draw and express myself with a\n     nasty little finiking brush, and the result when printed\n     seems to alternate between something all as black as my\n     hat--or as hazy and faint as a worn-out plate.--If on wood,\n     I should like it well enough--as it is--it spoils 4 days a\n     week--leaving little time for anything else. O! I'm a'weary,\n     I'm a'weary! of this illustration business.----\n\n     Tom is just off to the R.A., as it is not likely I shall go\n     much before it's close. I will get him to write you a\n     critical description of all the wonderful works in Turps,\n     Varnish, and \"Hile.\"\n\n     Yr. affectionate Dad,\n\n     H. K. B.\n\n     _Monday Morning, 25 m. 40 s. p. 11_ A.M.\n\n     MY DEAR WALTER,\n\n     There is a man playing \"Home, sweet home\" upon the key\n     bugle--it is too much for me--my heart yearneth--I feel I\n     must write just a line or two--especially as it is raining\n     hard--and I don't exactly know what to be at.\n\n            *       *       *       *       *\n\n     Splendid effects yesterday evening--sun-set, twilight,\n     crescent moon--stormy clouds,--tide out--reflections--dark\n     fishing-craft--very good--quite the thing for you.\n\n     There are no people here at present--decidedly nothing\n     Belgravian--chiefly masculines--from the Saturday to the\n     Monday sort--it striketh me--a few I think have strayed here\n     from Southend--I saw this sort of thing [_see page 29_] on\n     the Grand Promenade--which looks like it.----\n\n     There was a great wind yesterday--Boreas had been taking\n     concentrated essence of ginger--It fairly took me off my\n     legs once as I was walking along the cliffs to Broadstairs,\n     luckily for me it blew _off_ the sea--and I was brought up\n     short by some railings in this wise--[_see page 22_]\n     _otherwise_ I should (_no doubt_) have been carried across a\n     5 acre field of _Cloveria Trifolia Browniensis_.--I am glad\n     to say I was also of service to humanity yesterday--I heard\n     the shrill shrieks of a child and a woman's cry for help\n     behind me--I turned--and saw there was not a moment to lose,\n     the wind had caught a poor child--'s hat (and woman's too)\n     and bore it rapidly to the edge of the cliff--with my usual\n     agility I bounded over the rails fencing the cliff--and\n     saved--yes, saved the child--'s--'at!--another puff and it\n     would have been in the deep, deep sea--the blue, the fresh,\n     &c.--Stout mama thanked me politely, and turning to her\n     husband (who, of course, had come up too late to be of any\n     use--those husbands _always_ do)--she remarked \"That the\n     vind had blown both her and her child's 'at hoff and if\n     she'd know'd it--she wouldn't have brought the young-un\n     hout.\"\n\n     I dare say humanity is amusing here when the place is\n     full--there seems a good deal of \"os\" exercise--and\n     basket-carriage driving on Sundays--which is good to\n     behold--this gentleman [_see page 25_] was driving with\n     supreme self-content--having one rein all snug and tight\n     under his pony's tail--luckily the beast did not seem to\n     have any kick in him--so _perhaps_ he got safe back to\n     Margate.\n\n            *       *       *       *       *\n\n     Yr. affec. Dad,\n\n     H. K. B.\n\n       *       *       *       *       *\n\n[Illustration]\n\n     _29th Sept. 1868._\n\n     MY DEAR DOCTOR,\n\n     I have sent you a couple of canvasses--if you put little\n     Clara's head on one of them, you will immortalize her and\n     yourself too.\n\n     Also therewith you will find a Surplice, and if you will\n     only \"hold forth,\" next Sunday, in the Grande Place of\n     Colyton--I will guarantee to say that the simplicity of yr.\n     vestment and the flowing eloquence of yr. tongue will draw\n     out--(as irresistibly as the Piper did the children) the\n     congregations of the \"High\" Church and the Conventicles\n     which will--one and all--rush forth for to see and to hear,\n     and admiringly surround you!--If windy, you might take this\n     for yr. text--\"What went ye forth for to see?--\" A reed\n     shaken by the wind? &c., &c.\n\n     There must have been a splendid _Sea on_ at _Sea-ton_, these\n     last few days,--_tons_ of _sea_, eh? As \"I took my walk\n     abroad\" this morning--I saw the Serpentine in all its\n     grandeur--and observed several vessels in distress--some\n     clipper yachts on their beam ends--the waves were\n     prodigious--great rollers--two especially--one a six horse\n     fellow--t'other a steamer--crunching and grinding--levelling\n     and sweeping all before them!\n\n     Have you seen the Doge of Colyton yet? or any of the Dog-es?\n\n     By all means cultivate the acquaintance of the Doge's\n     kinswoman. Miss P---- (pray give my love to\n     her)--fac-similed on the stage or in a novel, she would be a\n     \"tremendous hit.\"\n\n     I hope you are not belying the _good_ character I have given\n     of you to the boys--and are doing Elephant, Tiger, and\n     Rhinoceros[I] to their perfect satisfaction--though,\n     considering yr. predecessor--it will test your utmost\n     powers, not to be a wretched failure, possibly--much the\n     same sort of thing--as your attempting to sing a comic song\n     immediately after the Great Vance!!! Good Night,\n\n     Yr. affectionate Dad,\n\n     H. K. B.\n\n\nThe following notes have been selected from the unpublished\ncorrespondence of \"Phiz\" with CHARLES DICKENS:--\n\n     MY DEAR DICKENS,\n\n     I have just got one boot on, intending to come round to you,\n     but you have done me out of a capital excuse to myself for\n     idling away this fine morning.--I quite forgot to answer\n     your note, and Mr. Macrone's book has not been very vividly\n     present to my memory for some time past. I think by the\n     beginning of next (week) or the middle (_certain_) I shall\n     have done the plates, but in the scraps of copy that I have\n     I can see but _one good_ subject, so if you know of another\n     pray send it me. I should like \"Malcolm\" again, if you can\n     spare him.\n\n     Believe me,\n\n     Yours very truly,\n\n     HABLOT K. BROWNE.\n\n     Charles Dickens, Esq.\n\n\n     _Sunday, Sept._\n\n     MY DEAR DICKENS,\n\n     Can you conveniently send me the subject or subjects for\n     next week by Thursday or Friday? as I wish, if practicable,\n     to start for Brussels by the Sunday's boat--a word in reply\n     will oblige,\n\n     Yours truly,\n\n     HABLOT K. BROWNE.\n\n     Charles Dickens, Esq.\n\n\n     P.S.--Upon second thoughts I send you the enclosed\n     epistle--(if you read it, you will find out why)--the\n     writer thereof is \"Harry Lorrequer,\" alias \"Charles\n     O'Malley\"--to whose house I am going.\n\n     H. K. B.\n\n\n     P.S. Second--A fortnight's furlough would suit me better\n     than a week, if it could be managed, as I should like to\n     return by Holland.\n\n     MY DEAR DICKENS,\n\n     I am sorry I cannot have a touch at battledore with you\n     to-day, being already booked for this evening--but I will\n     give you a call to-morrow _after church_, and take my chance\n     of finding you at home.\n\n     Yours very sincerely,\n\n     HABLOT K. BROWNE.\n\n     Charles Dickens, Esq.\n\n     33, HOWLAND STREET.\n\n     MY DEAR DICKENS,\n\n     I shall be most happy to remember not to forget the 10th\n     April, and, let me express a _dis_interested wish, that\n     having completed and established one \"Shop\"[J] in an\n     \"extensive line of business,\" you will go on increasing and\n     multiplying such like establishments in number and\n     prosperity till you become a Dick Whittington of a merchant,\n     with pockets distended to most Brobdignag dimensions.\n\n     Believe me,\n\n     Yours very truly,\n\n     HABLOT K. BROWNE.\n\n     Charles Dickens, Esq.\n\n     I return you the Riots with many thanks.\n\n\n[Illustration]\n\n     _Sunday Morning._\n\n     MY DEAR DICKENS,\n\n     Will you give me some notion of the sort of design you wish\n     for the frontispiece to second vol. of _Clock_?[K]\n     Cattermole being put _hors de combat_--Chapman with a\n     careworn face (if you can picture that) brings me the block\n     at the eleventh hour, and requires it finished by Wednesday.\n     Now as I have two others to complete in the\n     meantime--something nice and _light_ would be best adapted\n     to my _palette_, and prevent an excess of perspiration in\n     the relays of wood-cutters. You shall have the others to\n     criticise on Tuesday.\n\n     Yours very truly,\n\n     HABLOT K. BROWNE.\n\n\n     Charles Dickens, Esq.\n\n     How are Mrs. Dickens and the \"Infant?\"\n\nFOOTNOTES:\n\n[A] Pronounced _Hab-lo_, after a Monsieur Hablot, a captain in the\nFrench army, and a friend of the family.\n\n[B] It was Buss who illustrated Mrs. Trollope's Serial Story, _The Widow\nMarried_, which was published in _The New Monthly Magazine_, 1840.\n\n[C] See _Dombey and Son_, Vol. I, p. 113--\"Doctor Blimber's Young\nGentlemen.\"\n\n[D] Leigh Hunt.\n\n[E] Mr. R. Young, who also undertook the precarious task of \"biting in\"\nhis plates.\n\n[F] Water-colour white.\n\n[G] Publishers frequently availed themselves of his facile pencil, and\nwould instruct him to furnish illustrations for books already in the\npress, for which he was often inadequately paid.\n\n[H] The Sculptor, and an old coadjutor on _Once a Week_. He is also the\nauthor of _A Salad of Stray Leaves_ now in the press, which contains a\nfrontispiece by \"Phiz,\" the last design from his pencil. This he\nexecuted under some difficulties, for owing to an attack of rheumatism\nin his hands, the design--teeming with fancy--had to be made on a large\nscale, and afterwards reduced by the process of photography.\n\n[I] A favourite game with the children.\n\n[J] _The Old Curiosity Shop._\n\n[K] _Master Humphrey's Clock._\n\n\n\n\nA LIST OF THE PRINCIPAL WORKS ILLUSTRATED BY \"PHIZ.\"\n\nTo enumerate all the works illustrated by \"Phiz\" would be a next to\nimpossible task, for \"their name is legion.\" No artist was so popular or\nso prolific as a book-illustrator, with the exception, perhaps, of\nGeorge Cruikshank. It may fairly be questioned whether the works of\nCharles Dickens, with which the name of \"Phiz\" is most intimately\nassociated in our minds, would have achieved such notoriety without the\naid of the etching needle so ably wielded. Mr. John Hollingshead, in his\nessay on Dickens, says:--\n\n\"The greater the value of a book as a literary production, the more will\nthe circle of its influence usually be narrowed. The very shape, aspect,\nand garments of the ideal creatures who move through its pages, even\nwhen drawn by the pen of the first master of fiction in the land, will\nbe faint and confused to the blunter perception of the general reader,\nunless aided by the attendant pencil of the illustrative artist. For the\nsharp, clear images of Mr. Pickwick, with the spectacles, gaiters, and\nlow crowned hat--of Sam Weller, with the striped waistcoat and the\nartful leer--of Mr. Winkle, with the sporting costume and the foolish\nexpression--more persons are indebted to the caricaturist, than to the\nfaultless descriptive passages of the great creative mind that called\nthe amusing puppets into existence.\"\n\nIt was not the fame of Dickens only that was enhanced by \"Phiz,\" for the\nnumerous illustrations in the works of Charles Lever, Harrison\nAinsworth, the brothers Mayhew, and a host of minor novelists were\nexecuted by his unwearied hand. It was Dickens, however, who introduced\nhim to public notice, in a pamphlet, now very scarce, entitled _Sunday\nunder Three Heads_, embellished with four delicately executed engravings\ndrawn by \"H. K. B.\"\n\nIt was his succession to Seymour as the illustrator of the _Pickwick\nPapers_, that really excited public interest in the youthful artist, who\ncreated, pictorially, the second hero in the work, the inimitable Samuel\nWeller. Those who are familiar with the original edition of the\n_Pickwick Papers_ will remember with some amusement, the artist's\nintroduction of the indefatigable \"Boots,\" as represented in the yard of\nthe \"White Hart\" Inn, Borough. The identical Inn exists at the present\nday. \"Mr. Pickwick in the Pound\" is another amusing plate, where the\nlaughing, jeering crowd of spectators crowned by a jubilant and juvenile\nchimney sweeper, the braying of a jackass in the ears of the astonished\nhero, who sits somewhat uncomfortably in a wheelbarrow, are incidents so\ncleverly depicted as to excite unqualified admiration. \"Mr. Pickwick\nSlides\" is another truly artistic production. The delicate execution of\nthe extreme distance where is seen a manor house of the olden time\nnestling amongst the trees, and a farmyard hard by, leaves nothing to be\ndesired. Mr. Sala somewhat harshly criticises the illustrations in this\nwork, which, he says, \"were exceedingly humorous, but vilely drawn. The\namazing success of his author seems, however, to have spurred the artist\nto sedulous study, and to have conduced in a remarkable degree towards\nthe development of his faculties. A surprising improvement was visible\nin the frontispieces to the completed volumes[L] of _Pickwick_.\"\nUndoubtedly faults exist, but to characterize the illustrations as\n\"vile,\" seems too severe a term, for after all, the exaggerated types of\nface, form, and feature, do but harmonize with the somewhat exaggerated\ndescriptions of them by the author. This defect, if such it can be\ncalled, was remedied considerably in his later productions.\n\n[Illustration]\n\nIn 1837, \"Phiz\" accompanied Dickens into Yorkshire, there to gather\nmaterial for _Nicholas Nickleby_, a work which exposes the tyranny\npractised by some schoolmasters on their helpless pupils. In this book,\npublished in 1839, is presented to us the despicable \"Squeers,\" which\ntype of brute in human form was so successfully realized by both Author\nand Artist, that the indignation of innumerable Yorkshire pedagogues was\nraised to threats of legal proceedings, for traducing their characters,\none of them actually stating that \"he remembered being waited on last\nJanuary twelvemonth by two gentlemen, one of whom held him in\nconversation while the other took his likeness.\" The most familiar\nrepresentation of \"Squeers\" is seen in the second plate, where he stands\nsharpening his pen, and is timorously approached by the stout father of\ntwo wizen-faced boys who are about to become his pupils. The face of the\nschoolmaster, in which are combined hypocrisy and cruelty, and the\nexpression of sympathy for the new comers exhibited by the boy on the\ntrunk, are worthy of the closest inspection. The effect of the school\ntreatment at Dotheboy's Hall is visible in the illustration where \"The\nInternal Economy\" is depicted. Here we see the starveling lads during\nand after the \"internal\" application of superabundant doses of brimstone\nand treacle, administered by Squeers' worthy partner. The eighth plate\nhappily depicts the wild excitement of the pupils when \"Nicholas\nastonishes Mr. Squeers and family\" by making a furious attack on the\nformer with the cane; as well as \"The breaking-up at Dotheboy's Hall,\"\nwhere the boys revenge themselves on their former tormentors. There are\ntwo more etchings in this volume especially remarkable as artistic\nproductions, viz., \"Mr. and Mrs. Mantalini in Ralph Nickleby's Office,\"\nwhere the expression of an intent listener on the face of Ralph, and of\nhorror on that of Mantalini, is capitally rendered; and the plate\nentitled \"The Recognition,\" which shows poor Smike in the act of rising\nfrom a couch of sickness as he recognizes \"Broker,\" who had conveyed him\nas a child to school.\n\n_Master Humphrey's Clock_, written in 1840-1, includes the stories of\nthe _Old Curiosity Shop_ and _Barnaby Rudge_ which have been happily\ntermed \"two unequalled twin fictions upon one stem.\" The illustrations\nwere drawn on wood by H. K. Browne and George Cattermole, and the former\ncreated, pictorially, Little Nell, Mrs. Jarley, Quilp, Dick Swiveller,\nthe Marchioness, Sally Brass, and her brother Sampson. \"Phiz\" revelled\nin wild fun in the vignettes relating to the devilries of Mr. Daniel\nQuilp and the humours of Codlin and Short, and of Mrs. Jarley's waxwork\nshow. His \"Marchioness\" was a distinct comic creation; but in the weird\nwaterscape, showing the corpse of Quilp washed ashore, he sketched a\nvista of riparian scenery which, in its desolate breadth and loneliness,\nhas not since, perhaps, been equalled, save in the amazing suggestive\nThames etchings of Mr. James Whistler. To be sure, Hablot Browne was\nstimulated to excellence during the continuance of the _Old Curiosity\nShop_ by the friendly rivalry of the famous water-colour painter, George\nCattermole, who drew the charming vignettes of the quaint old cottages\nand school-house and church of the village where \"Little Nell\" died. In\n_Barnaby Rudge_, however, Hablot Browne had things graphic his own way,\nand again towards the close he manifested genuine tragic power. His\n\"Barnaby with the Raven\" is lovely in its picturesque grace.[M] When the\nfirst cheap series of this work was published, plates by H. K. Browne\nwere issued, which are now so scarce, that they are often catalogued at\neight or ten times their original price.\n\nTwo years after the visit of Dickens to America in 1842, _Martin\nChuzzlewit_ was published, the illustrations to which excel in vigour\nall the previous efforts of \"Phiz.\" Here we are brought face to face, in\na pictorial sense, with the hypocrite, Mr. Pecksniff, the _abstemious_\nMrs. Gamp and her bosom friend, Betsy Prig, simple Tom Pinch and his\ncharming sister, Ruth. The frontispiece is a most ambitious work, but\nnone the less successful, for \"Phiz\" has represented, in the space of a\nfew square inches, all the leading events, humorous and pathetic,\ndescribed in the novel. In the illustration where Mark Tapley is seen\nstarting from his native village for London, \"Phiz\" exhibits his sense\nof the picturesque in the old gables and dormers of the cottages which\nform the background. The plate, \"Mr. Pecksniff on his Mission,\" is full\nof interest, and gives us an insight into the character of Kingsgate\nStreet, Holborn, at that time. The female neighbours of Mrs. Gamp, the\nmidwife, flock round Pecksniff, commiserating with him on his supposed\ndomestic cares, and advising him to \"knock at the winder, Sir; knock at\nthe winder. Lord bless you, don't lose no more time than you can\nhelp--knock at the winder!\"\n\n[Illustration]\n\nBut the etching in _Chuzzlewit_ which most strikes the reader as a\nludicrous conception, is that where \"Mrs. Gamp propoges a toast.\" Here\nhe has admirably illustrated the text, wherein is described, with other\ndetails of a droll character, how some rusty gowns and other articles of\nthat lady's wardrobe depended from the bed-posts; and \"these had so\nadapted themselves by long usage to her figure, that more than one\nimpatient husband, coming in precipitately, at about the time of\ntwilight, had been for an instant stricken dumb by the supposed\ndiscovery that Mrs. Gamp had hanged herself.\" In the background of the\npicture are represented these indispensable articles of dress, while at\nthe table sit, in friendly chat, Mrs. Gamp and Betsy.\n\n\"Betsy,\" said Mrs. Gamp, filling her own glass and passing the tea-pot,\n\"I will now propoge a toast. My frequent pardner, Betsy Prig!\"\n\n\"Which, altering the name to Sairah Gamp; I drink,\" said Mrs. Prig,\n\"with love and tenderness.\"\n\nIn 1846, _Dombey and Son_ commenced, with forty illustrations by \"Phiz.\"\nThe frontispiece is similar in design to that of _Chuzzlewit_,\nintroducing the principal characters and events in the novel. The\naustere and pompous (not to say selfish) Mr. Dombey, whom \"Phiz\" had\ngreat difficulty in realizing to the author's satisfaction,[N] is\nintroduced in many of the plates, although the artist has somewhat\nfailed in preserving the same type of face throughout. He has succeeded\nbetter with the genial Captain Cuttle. Little Paul, as he sits in his\ndiminutive arm-chair, contrasts most favourably in his childish\ninnocence, with the grim Mrs. Pipchin, whose Ogress-like character is\nstrongly marked. The scene in which Mr. Dombey introduces his daughter\nFlorence to Mrs. Skewton, is one of the most successful in the book, and\ncontains the _best_ type of Dombey. Here also, the face of Florence is\ntruly pretty, and the artist has well portrayed the handsome but\nvindictive Edith denouncing Carker for his treachery. A very effective\netching entitled, \"On the Dark Road,\" represents the flight of the\nenraged and disappointed libertine. The horses are being urged on their\nmad career by the whip and spurs of a postilion, under the dark sky with\na glimmer of light in the horizon caused by the rising sun. The artist\nat this time essayed a process of working on plates over which a\nhalf-tint had been previously laid by means of a ruling-machine, and in\nwhich the \"high-lights\" were afterwards \"stopped out,\" and the \"whites\"\n\"burnished out.\" He frequently availed himself of these ready means of\nproducing effect. Full-length portraits of the principal characters in\n_Dombey_, which were issued as additional plates by \"Phiz,\" are now very\nscarce.\n\n_David Copperfield_ (1850), with forty illustrations, was the next\nventure, but was not so much an artistic as a literary success. A\nfavourite character in it of course, is Micawber, a kindly caricature of\nthe Author's father, the realization of whom, by Browne, obtained the\nhearty approval of Dickens.\n\nThe most characteristic and, perhaps, most successful work of \"Phiz\" is\nto be seen in the illustrations to _Bleak House_. A view of the \"House\"\nitself forms the subject of the frontispiece. \"The Ghost's Walk,\" the\n\"Drawing-room at Chesney Wold,\" \"Tom All-alone's,\" and the gateway\nleading to the burial ground where Lady Dedlock has fallen lifeless, are\ninstances where the artist has obtained some fine effects by the\n\"ruled-plate\" process. A writer in _The Daily Telegraph_, of July 11th,\n1882, speaks somewhat disparagingly of these illustrations, but _The\nAcademy_ of a few days later, in the following remarks, thus demurs to\nhis criticism:--\n\n\"In the _Bleak House_ illustrations hardly anything is wrong; there is\nno shortcoming. Not only is the comic side, the even fussily comic, such\nas 'the young man of the name of Guppy,' understood and rendered well,\nbut the dignified beauty of old country-house architecture, or the\narchitecture of the chambers of our inns-of-court is conveyed in brief\ntouches; and there is apparent everywhere that element of terrible\nsuggestiveness which made not only the art of Hablot Browne, but the art\nof Charles Dickens himself, in this story of _Bleak House_, recall the\nimaginative purpose of the art of Meryon. What can be more impressive in\nconnection with the story--nay, even independently of the story--than\nthe illustration of Mr. Tulkinghorn's chambers in gloom; than the\nillustration of the staircase at Dedlock's own house, with the placard\nof the reward for the discovery of the murderer; than that of Tom All\nAlone's; the dark, foul darkness of the burial ground shown under scanty\nlamplight, and the special spot where lay the man who 'wos very good to\nme--he wos!'? And then again, 'the Ghost's Walk,' and once more the\nburial ground, with the woman's body--Lady Dedlock's--now close against\nits gate. Of course it would be possible to find fault with these\nthings, but they have nothing of the vice of tameness--they deliver\ntheir message effectually. It is not their business to be faultless; it\nis their business to impress.\"\n\n[Illustration]\n\nA very successful rendering of character in _Bleak House_ is that of\nHarold Skimpole, whose prototype was Leigh Hunt, an intimate friend of\nthe Novelist, who, by his unintentional disregard for the feelings of\nHunt in caricaturing his peculiarities, nearly severed that friendship.\nAgain, there is intense humour in the illustration facetiously styled,\n\"In re Guppy, extraordinary proceeding.\" The love-sick Guppy is seen in\na kneeling posture, while declaring to Miss Summerson the burning\npassion that consumes him. The expression on the face of the young lady\nshows that she is more amused than flattered by his preference.\n\nIn _Little Dorrit_ (1855-7) the experience gained by both Author and\nArtist during their tour of the London prisons, stood them in good\nstead, for here the Marshalsea is fully described, the type of a\ndebtor's jail. The first illustration represents the interior of a\nFrench prison, in which are incarcerated Monsieur Rigaud and Signor John\nBaptist. The effect of deep gloom in the cell is produced by the\n\"ruled-plate\" method, and is quite Rembrandt-like. In contrast with\nthis, the illustration of \"The Ferry,\" is a delightful country aspect,\nwith trees and winding river; and another plate entitled \"Floating\naway,\" an evening scene, the moon rising behind the trees, is quite\nromantic. The old house in the last picture but one--\"Damocles,\"--again\nshows Browne's appreciation of the picturesque architecture of bygone\ntimes, in the effect of light from the setting sun as it falls upon the\nhouse front, throwing into relief the quaint old carvings of door and\nwindow.\n\nThe last work illustrated by \"Phiz\" for Dickens was _The Tale of Two\nCities_ (1859), containing sixteen etchings full of vigour, as the\ncharacter of the story justifies.\n\nFor some reason, at this time, a rupture was caused between author and\nartist,[O] which resulted in the engagement of Mr. Marcus Stone and Mr.\nLuke Fildes as illustrators of _Our Mutual Friend_ and _Edwin Drood_.\nThese accomplished painters avoided the old system of caricature, the\nold, forced humour; but it is certain that their designs are less\nintimately associated with the persons in the stories they illustrated\nthan those of \"Phiz\" with the earlier and more popular works of Dickens.\n\nHaving devoted the larger portion of the space at our disposal to a\ndescription of the most famous productions of Browne's pencil, which are\nprominent in the original editions of the Novels of Charles Dickens, we\ncan but briefly enumerate the plates he etched for Lever, Ainsworth, and\nothers.\n\n[Illustration]\n\nIn Charles Lever's _Harry Lorrequer_ (1839) and _Charles O'Malley_\n(1841), the uproarious mirth and jollity of Irish military life is well\nportrayed by the needle of the artist. \"The last night in Trinity\" in\nthe latter work, is an example of this, wherein is seen the worthy\nDoctor perched on a table, surrounded by a batch of Irish dragoons, and\nbeing elevated by an explosion of combustibles. The horses in the\nillustrations are admirably drawn.\n\nIn _Jack Hinton_ (1842) the artist shows remarkable force in depicting\nthe death of Shaun, and has well realized the humour of \"Corney's Combat\nwith the Cossack.\"\n\n_Tom Burke of Ours_ (1844) contains forty-four illustrations by \"Phiz,\"\nmany of which represent the scenes connected with the battles of\nAusterlitz, &c., during the reign of the great Napoleon. Most especially\nnoticeable is the scene in a court of justice, with \"Darby in the\nChair;\" the face of that hero with an expression apparently abashed, but\nreally full of roguishness, as he gazes at the counsel, is one of the\nmost successful of Browne's efforts.\n\n_The O'Donoghue_ (1845), has twenty-six illustrations, most of which are\nwell conceived. The falling body of a man in the frontispiece is a\nremarkable drawing. The girlish figure of Kate O'Donoghue, as she bends\nover the form of her heart-broken brother Herbert, is well depicted.\n\n_St. Patrick's Eve_ (1845), with four etchings and several woodcuts. The\nmost remarkable of the former is \"The Cholera Hut.\"\n\n_The Knight of Gwynne_ (1847), with forty illustrations.\n\n_Roland Cashel_ (1850), with forty illustrations.\n\n_The Daltons_ (1852), with forty-eight illustrations.\n\n_The Dodd Family Abroad_ (1854), with forty illustrations. The shrewd\nsimplicity of Kenny Dodd is well delineated.\n\n_The Martins of Cro' Martin_ (1856), with forty illustrations.\n\n_Davenport Dunn_ (1859), with forty-four illustrations.\n\n_One of Them_ (1861), with thirty illustrations.\n\n_Barrington_ (1863), with twenty-six illustrations.\n\n_Luttrell of Arran_ (1865), with thirty-two illustrations.\n\nThe following works of W. Harrison Ainsworth contain etchings and\nwoodcuts by \"Phiz:\"--\n\n_Revelations of London_, published about 1845, but never completed, has\nan illustration which represents a tumble-down house in Vauxhall Road,\nwhich is almost Rembrandt-like in its power. The artist was about thirty\nyears of age when he executed this.\n\n_Old St. Paul's_ (1847), contains only two plates by \"Phiz,\" but _The\nSpendthrift_ (1857), _Mervyn Clitheroe_, and _Crichton_ were wholly\nillustrated by him.\n\nFOOTNOTES:\n\n[L] The _Pickwick Papers_ were issued in one volume, and with _one_\nfrontispiece.\n\n[M] _The Daily Telegraph_, July 11th, 1882.\n\n[N] See illustration facing page 11.\n\n[O] If the following statement, made in the _Frankfurt Zeitung_, can be\ncredited, any feeling of enmity that existed between them had long since\ndied out:--\"Just after the death of Charles Dickens, 'Phiz' was\nconsiderably affected by the mere mention of the name of that\nillustrious novelist, which seemed to stir up in his breast feelings of\nregret at losing such a friend.\"\n\n\nSOME MISCELLANEOUS WORKS ILLUSTRATED BY \"PHIZ.\"\n\n_A Paper: of Tobacco, &c., by Joseph Fume_ (1839). With six plates by\n\"Phiz.\" _Fiddle Faddle's Sentimental Tour, in search of the Amusing,\nPicturesque, and Agreeable_ (1845). _The Union Magazine._ Vol. I (1846).\nContaining three plates by \"Phiz.\" _The Illuminated Magazine._ Conducted\nby Douglas Jerrold (1843-5), with woodcut illustrations by Leech, \"Phiz\"\n(H. K. Browne), and others. _Fanny, the little Milliner, or the Rich and\nthe Poor_ (1846), illustrated by \"Phiz\" and Onwhyn. _Wits and Beaux of\nSociety. Sketches of Cantabs, by John Smith (of Smith Hall), Gent._\n(1850). _The Cambridge Freshman._ With woodcut illustrations. _Paved\nwith Gold, or Romance and Reality of the London Streets_, by Augustus\nMayhew (1858). _A Medical, Moral, and Christian Dissection of\nTeetotalism by Democritus_ (1846). _New Sporting Magazine_ (1839). _The\nPottleton Legacy_, by Albert Smith. _Christmas Day, and how it was spent\nby four persons in the house of Fograss, Fograss, Mowton, and Snorton,\nbankers_, by C. Le Ros (1854). _Home Pictures_ (Durtin & Co., 1856). A\nseries of seven charming and characteristic plates. _Dame Perkins and\nher Grey Mare, or the Mount for Market_, by L. Meadows (1866). With\ncoloured illustrations. _H. B.'s Schoolboy Days._ _Illustrations of the\nFive Senses._ _Adventures of Sir Guy de Guy_, by George Halse. _The\nBaddington Peerage_, by G. A. Sala (published in _The Illustrated\nTimes_). In addition to these may be added an illustrated edition of\nByron's works, the \"Abbotsford\" edition of Sir Walter Scott's Novels,\nbesides numerous cuts in _The Sporting Gazette_, _The Illustrated\nTimes_, the early volumes of _Once a Week_, and the Comic Papers.\n\n[Illustration: (SOME SIGNATURES ADOPTED BY H. K. BROWNE.)]\n\n\nBELCARO: being Essays on Sundry AEsthetical Questions.\n\nBy VERNON LEE, author of the \"Studies of the Eighteenth Century in\nItaly.\" 8vo. price 8_s._\n\n\"There is much in this thoroughly original and delightful book which\nreminds us of the essays of the eighteenth century.... It is rare indeed\nto find so much thought conveyed in so easy a style--to find a writer\nwho not only has so much that is fresh to say, but has so fresh a way of\nsaying it.... This way of conveying ideas is very fascinating.... From\nfirst to last there is a continuous and delightful stimulation of\nthought. The book will lead to conversation, dreaming, speculation, and\nall kinds of pleasant and healthy mental exercise; and it is\ninterspersed with such perfect little sketches of scenery, and passages\nof so much eloquence, that it is a literary treat to read\nit.\"--_Academy._\n\n\"Clever and expressive, subtle and brilliant.... We could say a good\ndeal more about this book as the product of a remarkably acute critical\nmind; it would bear to be read a second time, and would be found to\nrepay the trouble.\"--_Athenaeum._\n\n\"Splendid essays on art.... We do not know why the writing reminds us of\nGeorge Sand, but it does.... Vernon Lee writes prose harmonies which are\nfinely composed.\"--_Vanity Fair._\n\n\nTHE SEALS AND ARMORIAL INSIGNIA OF THE UNIVERSITY AND COLLEGES OF\nCAMBRIDGE.\n\nPart I. Post 4to., 3_s._ Relating to the University. Contains\nChromo-lithograph and _eight engravings_ of Seals.\n\n_Imp. 16mo., elegant cover, gilt. Price 3s (Postage 4d)._\n\n\nTUSCAN FAIRY TALES. Taken down from the Mouths of the People. With\nsixteen illustrations, engraved by EDMUND EVANS.\n\nCONTENTS:--The Little Convent of Cats; The Fairies' Sieve; The Three\nGolden Apples; The Woman of Paste; The Beautiful Glutton; The King of\nPortugal's Cowherd; The Three Cauliflowers; The Siren; The Glass Coffin;\nLeonbruno.\n\n\"Sumptuously printed and prettily bound.\"--_Athenaeum._\n\n\"A thoroughly delightful book. The comparative mythologist and the child\nwill alike find something to gratify their very different\ntastes.\"--_Westminster Review._\n\n\"The work will delight the little ones as well as interest the student.\nThe book is charmingly got up and illustrated.\"--_London Review._\n\n_New Poems. Crown 8vo. Ten fine Plates, cloth, price 6s._\n\n\nGODS, SAINTS, AND MEN. By EUGENE LEE-HAMILTON.\n\n\"Readers will find him, as before, a Browning without his\nobscurity.\"--_Graphic._\n\n\"Quaint, mediaeval legends and traditions, most of which have a strong\nsavour of the supernatural, in strong, tuneful and artistic\nverse.\"--_Scotsman._\n\n_Crown 8vo., price 1s, cloth 2s._\n\n\nON THE ART OF GARDENING: A plea for English Gardens of the future, with\npractical hints for planting them By MRS. J. FRANCIS FOSTER.\n\n\"In this pleasant and original little book the authoress not only enters\na vigorous protest against the bedding-out system and the so-called\n'natural' style of gardening, but gives very good practical advice for\ngardens of a different sort.\"--_Gardener's Chronicle._\n\n\"This little book proceeds from a true lover of flowers and\nwill be welcome to all who take an interest in their care and\nculture.\"--_Civilian._\n\n\"A pleasant and unpretending little volume.\"--_Saturday Review._\n\n\nLONDON: W. SATCHELL & Co., 19, TAVISTOCK ST., COVENT GARDEN\n\n\n_Price 2s. 6d._,\n\nTHE BOOK OF ODDITIES, AND PUNISHMENTS IN THE OLDEN TIME.\n\nBY WILLIAM ANDREWS, F.R.H.S.\n\nWith numerous Illustrations BY GEORGE CRUIKSHANK, CROWQUILL, CUTHBERT\nBEDE, AND OTHERS.\n\n     CONTENTS:--Revivals after Execution--A Human\n     Pincushion--Female Jockeys--A Blind Road-maker--Odd\n     Showers--Singular Funerals--Whimsical Wills--Curious\n     Epitaphs--People and Steeple\n     Rhymes--Dog-Whippers--Sluggard-Wakers--Playing at Cards for\n     a Town, &c. &c.\n\n\"A capitally-written book, containing a vast amount of curious and\nout-of-the-way information. Mr. Andrews is never for a moment dull, but\ngives forth his antiquarian gossip with all the enthusiasm and point of\na practised _raconteur_. _He tells us all about the ducking-stool, the\nbrank, the pillory, the stocks, the drunkard's cloak, the whipping-post,\nriding the stang, and other forms of punishment._ The book is copiously\nillustrated and well indexed, and cannot fail to be popular.\"--_Sunday\nTimes._\n\nLONDON: W. SATCHELL AND CO., 19, TAVISTOCK STREET, COVENT GARDEN.\n\n\n\n\n\n\n\nEnd of the Project Gutenberg EBook of 'Phiz' (Hablot Knight Browne), a\nMemoir., by Fred. G. Kitton\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n**NINE LIES ABOUT WORK**\n\n# NINE LIES ABOUT WORK\n\n## MARCUS BUCKINGHAM  \nASHLEY GOODALL\n\n**Harvard Business Review Press**\n\n**Boston, Massachusetts**\n**HBR Press Quantity Sales Discounts**\n\nHarvard Business Review Press titles are available at significant quantity discounts when purchased in bulk for client gifts, sales promotions, and premiums. Special editions, including books with corporate logos, customized covers, and letters from the company or CEO printed in the front matter, as well as excerpts of existing books, can also be created in large quantities for special needs.\n\nFor details and discount information for both print and  \nebook formats, contact booksales@harvardbusiness.org,  \ntel. 800-988-0886, or www.hbr.org\/bulksales.\n\nCopyright 2019 One Thing Productions, Inc. and Ashley Goodall\n\nAll rights reserved\n\nPrinted in the United States of America\n\n10 9 8 7 6 5 4 3 2 1\n\nNo part of this publication may be reproduced, stored in or introduced into 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. Requests for permission should be directed to permissions@hbsp.harvard.edu, or mailed to Permissions, Harvard Business School Publishing, 60 Harvard Way, Boston, Massachusetts 02163.\n\nThe web addresses referenced in this book were live and correct at the time of the book's publication but may be subject to change.\n\nLibrary of Congress Cataloging-in-Publication Data\n\nNames: Buckingham, Marcus, author. | Goodall, Ashley (Author of Nine lies About work), author.\n\nTitle: Nine lies about work : a freethinking leader's guide to the real world \/ Marcus Buckingham, Ashley Goodall.\n\nDescription: Boston, Massachusetts : Harvard Business Review Press, 2019.\n\nIdentifiers: LCCN 2018046989 | ISBN 9781633696303 (hardcover)\n\nSubjects: LCSH: Organizational effectiveness. | Industrial management. | Organizational change.\n\nClassification: LCC HD58.9 .B84 2019 | DDC 650--dc23 LC record available at   \n<https:\/\/lccn.loc.gov\/2018046989>\n\nHardcover ISBN: 978-1-63369-630-3\n\nPaperback ISBN: 978-1-63369-803-1\n\neISBN: 978-1-63369-631-0\n\nThe paper used in this publication meets the requirements of the American National Standard for Permanence of Paper for Publications and Documents in Libraries and Archives Z39.48-1992.\n_To Chris and Graeme, who taught us to  \nstart with what's knowable_.\n\n# Contents\n\nIntroduction\n\nLIE #1 People care which company they work for\n\nLIE #2 The best plan wins\n\nLIE #3 The best companies cascade goals\n\nLIE #4 The best people are well-rounded\n\nLIE #5 People need feedback\n\nLIE #6 People can reliably rate other people\n\nLIE #7 People have potential\n\nLIE #8 Work-life balance matters most\n\nLIE #9 Leadership is a thing\n\nTruths\n\nAppendix A: The ADPRI's Global Study of Engagement\n\nAppendix B: Seven Things We Know for Sure at Cisco\n\nNotes\n\nIndex\n\nAcknowledgments\n\nAbout the Authors\n**NINE LIES ABOUT WORK**\n\n# Introduction\n\n> _It ain't what you don't know that gets you into trouble_.\n> \n> _It's what you know for sure, that just ain't so_.*\n> \n> **\u2014MARK TWAIN**\n\nHere's who we are.\n\nMarcus is a data geek. He loves figuring out how to measure things you can't count, such as personality, performance, and engagement. He spent much of his career doing this at the Gallup Organization. He then built his own coaching and software company devoted to helping people do their best work, and he now leads the ADP Research Institute's investigations into all things people and performance. He's a transplanted Brit.\n\nAshley lives in the world of big companies. After an early stint designing the acoustics for concert halls, he dedicated his career to helping the likes of Deloitte and Cisco get the most from all their people. He's the kind of practitioner who loves to pressure test every innovative idea against the messy realities of the world of work. Currently he's doing this for Cisco's hundred and forty thousand employees and contractors around the world. He's also a transplanted Brit.\n\nA couple of years ago, the _Harvard Business Review_ asked us to combine Marcus's reliable data angle with Ashley's real-world-leader angle and write an article about the most effective, reliable, and valid way to do that uniformly unpopular ritual, the performance appraisal. The article was bluntly damning of existing practices, and stirred up the field to such an extent that HBR came back to us and asked whether we could take this same rigorous and realistic approach and apply it to the entire world of work. We said yes, and the book you're holding is the result.\n\nWe began the book with a paradox: Why do so many of the ideas and practices that are held as settled truths at work wind up being so deeply frustrating to, and unpopular with, the very people they are supposed to serve? Why, for example, is it a settled truth that having your goals cascaded down upon you from above is the best way to align and evaluate your work, when those of us in the trenches feel the yearly goal-setting process to be meaningless rigmarole with little connection to our actual work? Why is it a settled truth that you need critical feedback, when, in the real world, most of us lean away from such feedback, and feel more inclined to give it to the other guy than to get it ourselves? Why is it a settled truth that your manager can reliably rate you on your performance, when, on actual teams, none of us has ever met a team leader blessed with perfect objectivity? Why is it a settled truth that all the best leaders possess a defined list of attributes that you should aspire to acquire, when, in our everyday lives, none of us has ever met a leader with all of these attributes?\n\nThis paradox led to the core idea and audience of the book. The idea is this: the world of work today is overflowing with systems, processes, tools, and assumptions that are deeply flawed and that push directly against our ability to express what is unique about each of us in the work we do every day. Workplace data buttresses this idea. Global worker engagement is weak, with less than 20 percent of workers reporting that they are fully engaged at work.* And economists, in seeking to explain the global decline in productivity growth since the mid-seventies, have suggested that \"the technological advances and management strategies that worked to propel productivity in the past have been fully implemented and are no longer contributing to productivity.\" In other words, whatever our current practices may be, they are no longer giving us much lift.\n\nThese practices are by now so commonplace and ingrained that they are hard to see for what they really are. Some of them we encounter as the necessary but frustrating things that large organizations just do and have always done. Some of them, though, are born of convictions held firmly by those who run our companies and who then impose these convictions on the rest of us. Together they form the backdrop and justification for almost everything that happens to us at work\u2014how we are selected for jobs and how we are then evaluated, trained, paid, promoted, and fired.\n\nAnd yet, look more closely and you'll discover that they \"just ain't so.\"\n\nWe could call these things \"misconceptions,\" or \"myths,\" or even \"misunderstandings,\" but because they are pushed at us so hard, almost as if they're being used to steer us away from the world as it truly is, we'll call them \"lies.\"\n\nThere are nine of them in this book. And since, in Picasso's framing, \"every act of creation is first an act of destruction,\" before we can build something strong and fine with our teams we need to deconstruct each lie\u2014to discern how it begins life as a truth in one small set of cases and then spreads into a lie applied to all cases\u2014and then push on to uncover the broader truths hidden behind.\n\nThe first three chapters, after asking why culture, plans, and goals are imposed on us so resolutely, reveal better ways for getting us all to pull together. Chapters 4, , , and  each address a particular aspect of our human nature, and then reveal how we can best grow ourselves and our people when each of us is so glaringly and enduringly different. Chapter 8 questions why \"balance\" is held up to each of us as the ideal, and then presents a very different aspiration. And finally, the last chapter takes on our reverence for all things \"leadership,\" and offers up a new window into what really happens when we, as followers, give our breath and our passion to the vision of another.\n\nAs you read, you'll realize that these Nine Lies have taken hold because each satisfies the organization's need for control. Large organizations are complex places, and a strong and understandable instinct of their leaders is to seek simplicity and order\u2014not least because this makes it easier to persuade themselves and their stakeholders that they are moving toward their objectives. But the desire for simplicity easily shades into a desire for conformity, and before long this conformity threatens to extinguish individuality. Before we know it, the particular talents and interests of each person are seen as inconveniences, and the organization comes to treat its people as essentially interchangeable.\n\nThis is why you are told that your organization's culture is monolithic, that the plan must be adhered to, that work must be aligned through cascaded goals, that humans must be molded into well-roundedness and given constant feedback until they become so, and that each one of us must rate the others so as to conform most closely to the prescribed models of leadership, performance, and potential.\n\nYou'll see, as well, that the strongest force pushing back against the lies, and the force that we all seek to harness in our lives, is the power of our own individuality\u2014that the true power of human nature is that each human's nature is unique, and that expressing this through our work is an act, ultimately, of love.\n\nThe audience we imagined at the outset was someone leading a team for the first time\u2014someone who is facing a glorious but challenging world, and someone who wants to do something extraordinary with his or her team, to achieve greatness with them, to enable greatness for them, to become the kind of leader they talk about for years to come. We imagined leaders who are asking themselves how they will get the most from each team member; how they will keep them all focused when each seems to have their own personal goals; how they will prevent them from making mistakes that will hurt the team, yet allow them room to experiment and learn; how they will be fair judges of their performance, yet still build relationships with them that are real and caring; and how they will do all this while remaining true to who they are as people. We imagined someone who, in attempting to do all this, would be confounded and stymied by the Nine Lies\u2014by all the things we know for sure but just ain't so.\n\nBut as we wrote, our sense of who we were writing for grew. We realized that we weren't writing just for a first-time team leader but for any leader frustrated by the (sometimes benign) attempts of his or her organization to exert control and impose uniformity. We came to think of our audience not as the new leader but as the _freethinking leader_. A leader who embraces a world in which the weird uniqueness of each individual is seen not as a flaw to be ground down but as a mess worth engaging with, the raw material for all healthy, ethical, thriving organizations; a leader who rejects dogma and instead seeks out evidence; who values emergent patterns above received wisdom; who thrills to the power of teams; who puts faith in findings, not philosophy; and above all, a leader who knows that the only way to make the world better tomorrow is to have the courage and the wit to face up to how it really is today.\n\nIf this sounds like you, then you are a freethinking leader. We don't know you personally, of course, but over the last six months we've thought a lot about you\u2014who you might be, how you might be feeling, and what you might need to thrive. This book is for you.\n\n*Ironically, one thing we know for sure that just ain't so is that this quotation is from Mark Twain: though it is most often attributed to him, the truth is that no one is sure who coined it. In this way it serves as a sort of double reminder of the dangers of misplaced certainty.\n\n*See appendix A.\n\n#\n\n# People care which company they work for\n\nMeet Lisa. She works in the field of corporate communications and marketing, and has done so for more than twenty years. We spoke to her the other day about her recent experiences at work, in the same way that we speak to hundreds of people every year about their experiences at work. Lisa told us that she'd recently moved from one company to another and then back again, and we wanted to understand more. Here's what she said.\n\n_Marcus and Ashley:_ Why did you leave Company A* after eighteen years?\n\n_Lisa:_ I'd moved from a role focused on events\u2014the big events we put on for our customers and partners\u2014to a role focused more on marketing. I found I couldn't be creative in the marketing role, and then my prior events role had been filled and I had nowhere to go. So the only way I could get back into events was to go elsewhere.\n\n_Us:_ That's what led you to look at Company B?\n\n_Lisa:_ Yes. And anyway, after all this time at Company A, I felt like exploring something new, and a new environment.\n\n_Us:_ As you considered working for Company B, what was most important to you about the company?\n\n_Lisa:_ The brand\u2014whether it was seen to be a name-brand company in a market-leading position; innovation and the pace of innovation; whether I could build something new; where the job was located and whether I could work remotely; how cool the place was; whether I would learn there; and whether I could try new things easily. Those were some of the things I remember thinking about.\n\n_Us:_ And how did you try to evaluate each of those?\n\n_Lisa:_ Obviously, through the interviews I did for the job. But I'd also done my research beforehand\u2014I spent six months researching the company and the job, on Google, on Glassdoor. I spent two months prepping for my interviews, and at the same time I talked to as many people there as I could find.\n\n_Us:_ What did you conclude at the end of this?\n\n_Lisa:_ I thought Company B probably wasn't a perfect place, but it had checked enough of my boxes for me to feel comfortable going there.\n\n_Us:_ So you went to Company B. How long did you stay there for?\n\n_Lisa:_ Two years.\n\n_Us:_ Given you'd spent eighteen years at Company A, were you expecting to be at Company B longer than two years?\n\n_Lisa:_ Yes, for sure.\n\n_Us:_ So can you explain why you were only there for two years, given how thorough you'd been in your research about the job? What happened?\n\n_Lisa:_ What happened is that I met my manager. I mean, I'd met her during the interview process, obviously, and there were a few things that bothered me\u2014but when I started I saw her true colors, and that's when things started to go wrong.\n\n_Us:_ What bothered you during the interview process?\n\n_Lisa:_ Her style struck me as severe, and formal, and a bit hierarchical. But I figured that was just her game face\u2014how she was to the outside world\u2014and that if I joined her team it would be different. But it wasn't.\n\n_Us:_ And when did you realize that?\n\n_Lisa:_ It was on Day Thirteen.\n\n_Us:_ Day Thirteen? How can you be so precise?\n\n_Lisa:_ I wrote it down in my calendar. I wrote down all the key dates during my time at Company B\u2014it was my way of documenting what was a really tough experience for me. On Day Thirteen I was in a meeting with my manager and a more senior executive, and the senior person asked what I thought was a simple question about booking hotel rooms, and I answered, and my manager looked shocked. As soon as the meeting finished, she took me to one side and said, \"We don't share that sort of thing with senior people here. Next time run it by me.\" And from that point on she micromanaged me, and I realized that she was fear-based, both in how she thought of her bosses and in terms of how she ran her team.\n\n_Us:_ Were there any other days you noted in your calendar?\n\n_Lisa:_ On Day Fifteen I wrote down, \"Possible last day at Company B\" for my two-year anniversary, and \"Last day at Company B\" for my four-year anniversary.\n\n_Us:_ Crikey. Just to confirm\u2014you spent months researching a company; you did seven interviews, in each of which you had carefully prepared questions to help you understand whether this would work for you; and two weeks in you'd not only decided to leave but given yourself a timeline. Is that right?\n\n_Lisa:_ Yes, that's it. I knew fifteen days in that I wasn't there long term.\n\n_Us:_ And the main reason for that was your manager, and her style?\n\n_Lisa:_ Yes. And it wasn't just my manager\u2014other leaders seemed to operate based on fear, too.\n\n_Us:_ When you were at Company B, were you introduced to their Core Values or Leadership Principles or anything like that?\n\n_Lisa:_ Yes! I was handed a laminated page of them at my orientation. I was thrilled!\n\n_Us:_ Why was that?\n\n_Lisa:_ I read them and thought, \"These are great!\" There was one I remember in particular\u2014it was about disagreeing and then committing, about having the courage to speak up if you disagreed with what was being said, but then committing wholeheartedly to the ultimate decision when it was made. I thought that was really exciting, and would make for a great environment. But then I started work, and I realized\u2014darn, these just aren't true. Worse than that, some people here use them for evil.\n\n_Us:_ For evil?\n\n_Lisa:_ Yes, they justify bad behavior by pointing to the Leadership Principles. So if they want to silence dissent, they tell people it's time to commit to the direction they want to go. Which is the opposite of what that idea is meant to be about.\n\n_Us:_ Ah, OK. So pretty quickly you decided to find a path back to Company A, right?\n\n_Lisa:_ Yes.\n\n_Us:_ And in the light of this experience with Company B, what was important to you as you looked for this next role?\n\n_Lisa:_ Three things\u2014culture, leadership, and the work I'd be doing.\n\n_Us:_ What do you mean by culture?\n\n_Lisa:_ It's the tenets of how we behave. I think of it like a family creed\u2014this is how we operate and treat one another in this family.\n\n_Us:_ What are some words you'd use to describe Company A's culture?\n\n_Lisa:_ Let me see. _Inclusive_ , _collaborative_ , _kind_ , _generous_ , _trusting_ , _fair_ , _supportive_. And I think the senior leaders are good people who lead ethically.\n\n_Us:_ Were those things uniform across Company A, in your experience?\n\n_Lisa:_ I think I was fortunate\u2014they showed up in the teams I worked on for sure. But I know people who were less fortunate, who didn't see these things.\n\n_Us:_ How do you explain that?\n\n_Lisa:_ For me, it's a question of whether each team leader believes in the culture of the company\u2014whether they get the culture or not. If they do, you're fortunate. If not, you're not.\n\n\u2022 \u2022 \u2022\n\nFrom the outside looking in, it's pretty hard to figure out what it might be like to work for a particular company. If you're job hunting, you might start by searching online as Lisa did\u2014perhaps on Glassdoor or one of the other job boards where employees can rate their current company\u2014or by talking to friends about where they've worked and what their experiences were. You might try to talk to a recruiter, although it's tricky to do that if you're not yet sure you're going to apply. You might try to figure it out by reading the coverage of a company in the press, but this can be frustrating, since articles tend to focus more on a company's products or its strategy, rather than on its culture per se. Wherever you look, you'll find yourself wondering if what you're discovering is really representative of the company, and is giving you a good sense of the inside story. In search of more objectivity and breadth, then, you might turn to _Fortune_ magazine's annual ranking of the 100 Best Companies to Work For.\n\n_Fortune_ publishes its ranking every January, and this issue of the magazine is one of the most widely read of the year. The ranking is based on an anonymous survey of the employees at each company (known as the \"Trust Index\"), together with a submission that each company puts together describing how it invests in its people and what it has to offer them (called the \"Culture Audit\"). From all this, the editors at the magazine and the analysts at the Great Place to Work Institute (which conducts the research) put together a list that tells you which companies are the best to work for that year, together with descriptions of the various perks they offer and brief testimony from current employees. In 2018 the top six, in order, were Salesforce, Wegmans, Ultimate Software, Boston Consulting Group, Edward Jones, and Kimpton Hotels, selected for reasons ranging from the pragmatic (paying bonuses for employee referrals, offering Starbucks gift cards during busy times, on-site child day care) to the noble (giving millions of dollars' worth of reclaimed food to the hungry, building environmentally friendly offices, always trying to promote from within) to the quirky (Salesforce has an entire floor dedicated to _ohana_ , the Hawaiian for _family_ , while Kimpton offers all new hires a welcome care package complete with each person's favorite snacks).\n\nIf you are indeed looking for a job, you read _Fortune_ 's list in search of insights about a given company. What will your colleagues be like? How will they treat you? What will a typical day be like? Will your work be interesting, challenging, and valued? Is this a company that really cares for its people? If you go through the long process of applying, and interviewing, and negotiating an offer, and ultimately landing a job there, will this be a company that puts as much into you and your career as you're going to put into it?\n\nWhat, precisely, is this list measuring about these companies? Read the submissions, the press releases, and _Fortune_ 's own descriptions of the winners, and the word you land on is _culture_. Salesforce has a \"family culture,\" hence the Ohana floor. Wegmans has a culture based on its mission to \"help people live healthier, better lives through food.\" Kimpton Hotels has an \"inclusiveness culture.\" Each of these companies, it appears, has figured out what kind of culture it wants to build, and then has made it onto the list because it has been resolute and effective in its pursuit. Judging by these and other examples, this thing called culture really matters. It is potentially more important than what the company does\u2014\"Culture eats strategy for breakfast!\"\u2014how the company does it, how much the employees get paid, or even the company's current stock price.\n\nCulture matters, according to the voluminous literature on the topic, because it has three powerful contributions to make. First, it tells you who you are at work. If you're at Patagonia, you'd rather be surfing. You work in beautiful Oxnard, California, and your onboarding consists of a day-long beach party where you are gifted the CEO's autobiography\u2014 _Let My People Go Surfing_ \u2014and where your first meeting takes place around a campfire. If you're at Goldman Sachs, then never mind the surfing\u2014you'd rather be winning. You wear your bespoke suit every day because you're a winner. It means something to say that you work for Deloitte, or for Apple, or for Chick-fil-A\u2014and this meaning says something about you, something that locates you and differentiates you, that defines your tribe.\n\nSecond, culture has come to be how we choose to explain success. When Tesla's stock was on the rise in the early part of 2017, it wasn't because people were finally getting the electric cars they'd paid deposits for a year earlier\u2014they weren't. Rather, it was because Elon Musk had created a culture of cool, a place where you couldn't even see the cutting edge because it was so far behind you. When Toyota had to recall over six million vehicles, the direct cause was a problem with the shift-lever assembly, but the deeper explanation we arrived at was that it was a problem with their polite yet win-at-all-costs culture.\n\nAnd third, culture is now a watchword for where we want our company to go: almost overnight, a big part of the job description of senior corporate leaders has become to create a specific sort of culture, a culture of \"performance,\" perhaps, or a culture of \"feedback,\" or a culture of \"inclusion,\" or a culture of \"innovation\"; to shape the direction of the company they lead by infusing it with particular traits that govern how people behave. Beyond explaining the now, culture has become our handle on the next.\n\nAs a team leader you are going to be told, repeatedly, that you must take stock of all this because you are responsible for embodying your company's culture, and for building a team that adheres to these cultural norms. You will be asked to select only applicants who fit the culture, to identify high-potentials by whether or not they embody the company culture, to run your meetings in a way that fits the culture, and, at company off-sites, to don the T-shirts and sing the songs.\n\nAll of which is fine, right up to the point where you start to wonder what, precisely, you are being held accountable for. Read the _Fortune_ list again and you'll be struck by the fact that a very small percentage of what's written about your company is in your job description. Having an on-site day-care facility, giving all employees 20 percent of their time to pursue their own interests, offering large rewards for referring a new hire, and building solar panels on the roof are all admirable initiatives, yet none of them is within your control. They are commitments made by others\u2014the executive committee or the board\u2014and while you may think them worthy, and may indeed be proud that they are something your tribe contributes to the world, you can't do anything about them. They are off in some other place, far from the day-to-day projects and deadlines, the ongoing actions and interactions, that actually comprise your world of work.\n\nWhen people ask you what it's \"really like\" to work at your company, you immediately know you're going to tell them not about the solar panels and the cafeteria, but about what it's really like. So you'll get real, and talk about how work is parceled out, whether many managers play favorites, how disputes get resolved, whether the real meeting happens only after the formal meeting is over, how people get promoted, how territorial the teams are, how large the power distance is between senior leaders and everyone else, whether good news or bad news travels fastest, how much recognition there is, and whether performance or politics is most prized. You'll get down to the two-foot level of how work actually gets done, and try to tease out what your company truly feels like to the people on the ground.\n\nYou won't know whether to call this \"culture\" or not, just as you won't necessarily know how to label each of these two-foot-level details, but in every fiber of your being you'll know that this ground-level stuff is what'll decide how hard people will work once they've joined, and how long they'll stay. This ground-level stuff is what they truly care about. Indeed, this ground-level stuff is what _you_ truly care about.\n\nIn which case, your most pressing question, as a team leader, will be something like this: If I am to help my team give their best, for as long as possible, which of these details are most critical? Tell me the most important ones, and I'll do my level best to pay attention to those.\n\nWe've spent the last two decades attempting to answer this question for you. In the next few pages we'll outline what we've found, and then we'll focus the rest of this book on going deeper, and on giving you insights and prescriptions for how you can address the things that matter most.\n\nAnd in so doing, the first lie we'll need to expose is precisely that _people care which company they work for_. It sounds so odd to label this a lie, since each of us does indeed feel some sort of connection to our company, but read on, and we think you'll see that while what each of us truly cares about may begin as \"company,\" it quickly morphs into something else rather different.\n\n\u2022 \u2022 \u2022\n\nAll quantitative research requires qualitative digging, which is why, a little while ago, we spent a few hours with a team of people in Cisco's office in Krakow, Poland.* We were curious about their experience of work, and what their team was like. The team had about fifteen members overall, engaged in various jobs supporting Cisco's customers. We asked the group about the things they did frequently\u2014daily or weekly or monthly or quarterly\u2014that were important to them. Three of the team members answered by talking about lunch. We always bring our lunches to work, they said, instead of going to the cafeteria. And there's a spot on the patio outside where we eat together. We always eat at the same time as one another, no matter what is happening that day, and sometimes we talk about work, and other times about stuff outside work\u2014this is what we do every day.\n\nLater on, we saw where the fifteen-person team worked (our initial discussion had taken place in a conference room). They worked at a long row of workstations, each separated from its neighbors by vertical dividers. The three people who ate their packed lunches together pulled us to one side. Look! they said, pointing to an unremarkable spot on the floor, a few feet from the workstations. This is where we huddle! We asked what they meant. They said that when something happened during the day that they needed to talk about, they would leave their workstations and form an impromptu huddle where they could figure out what to do.\n\nHere we have a team of fifteen people, doing real work in the real world, and within it a sub-team of three people, also doing real work in the real world. And the three take time every day to eat together, and also\u2014maybe because of the lunches, or maybe not because of them, or maybe just because\u2014have a way of quickly breaking the routine configuration of their workspace to solve problems together.\n\nWhat is the \"culture\" of this three-person team-within-a-team? Is it different from the \"culture\" of the bigger, fifteen-person team, and if so, how? Who knows? All we do know is that both the three-person miniteam and the fifteen-person team are extremely productive and highly engaged. Back at Cisco's headquarters in San Jose, California, the CEO, Chuck Robbins, is doing his best to build an enthusiastic, committed workforce, but he is thousands of miles and several organizational levels away from the day-to-day realities that these team members face, and he knows that there's a limit to what he can control from the center. All he can hope to do is to encourage these local teams\u2014and every other of his thousands upon thousands of teams\u2014to build the sort of work experience that gets the best from each and every team member.\n\nWhat, then, should he be asking them to focus on? What are, in fact, the most important aspects of our experience of work?\n\nThe only way to rigorously answer this question is as follows: First, create two groups of people, one group from teams with high performance (high productivity, high innovation, high customer satisfaction, low voluntary turnover, low lost work days, whatever _performance_ means in a given company or business unit), the second group from teams with low or average performance.\n\nNext, start asking questions about what these teams are like on the inside. Ask many, _many_ questions of the high performers, and then ask the same questions of the low performers. Search for those few questions where the people on the high-performing teams say that they strongly agree and the people on the mid- to lower-performing teams do not. The goal here is to try to find what is distinctive about the high-performing teams through the eyes of the people on those teams.\n\nOver the last several years we've repeated this research hundreds of times in many different companies, always zeroing in on the questions that most clearly sort the best from the rest. We are not the first to undertake this kind of research, of course. Back in the late 1990s the Gallup Organization did pioneering work on engagement, eventually identifying twelve conditions as the drivers of it, and since then organizations such as the Corporate Executive Board, Korn Ferry, and Kenexa have added to our growing body of knowledge and our understanding of engagement at work, and of how we can measure it most reliably and with the most validity. Our work built on this existing research, as all sound research should\u2014research findings are provisional, after all\u2014and, in the end, we wound up identifying just a few aspects of the employee experience that exist disproportionately on the highest-performing teams. These eight aspects, and these eight precisely worded items,* validly predict sustained team performance:\n\n1. I am really enthusiastic about the mission of my company.\n\n2. At work, I clearly understand what is expected of me.\n\n3. In my team, I am surrounded by people who share my values.\n\n4. I have the chance to use my strengths every day at work.\n\n5. My teammates have my back.\n\n6. I know I will be recognized for excellent work.\n\n7. I have great confidence in my company's future.\n\n8. In my work, I am always challenged to grow.\n\nYou might notice a few things about these items right away. First, the team members are not directly rating their team leader or their company on anything\u2014they are rating only their own feelings and experiences. This is because, as we'll see in chapter 6, people are horribly unreliable raters of other people. When we ask someone to rate someone else on an abstract quality such as empathy or vision or strategic thinking, their responses tell us more about the person doing the rating than the person being rated. To get good data we have to ask people about their own experiences.\n\nSecond, you may also notice that the eight items fall into two broad groupings. The first is the odd-numbered items:\n\n1. I am really enthusiastic about the mission of my company.\n\n3. In my team, I am surrounded by people who share my values.\n\n5. My teammates have my back.\n\n7. I have great confidence in my company's future.\n\nThese deal with the elements of a person's experience created in their back-and-forth interactions with others on the team\u2014the communal experience of work, if you will. What do we all _share_ , as a team or as a company? We can think of these as the \"Best of We\" questions.\n\nThe second group comprises the even-numbered items:\n\n2. At work, I clearly understand what is expected of me.\n\n4. I have the chance to use my strengths every day at work.\n\n6. I know I will be recognized for excellent work.\n\n8. In my work, I am always challenged to grow.\n\nThese deal instead with the individual experience of work. What is unique about _me_? What is valuable about _me_? Do _I_ feel challenged to grow? We can think of these as the \"Best of Me\" questions.\n\nThese two categories of experience\u2014We experiences and Me experiences\u2014are the things we need at work in order to thrive. They are specific; they are reliably measured; they are personal; they reveal a local individual experience intertwined with a local collective experience. They are everyday. And if we think about the team in Poland, while we might not know what its \"culture\" is, we do know that lunching together and huddling together will have some bearing on the team members' feeling that their teammates have their backs, that they share a sense of what excellence is, that they are called on to do their best work frequently, that they catch each other doing things right, and so on. What we see in the eight questions is a simple way of measuring experience-at-work, and one that you, the team leader, can do something about.\n\nAnd what more than two decades of research into teams and their leaders has to tell us is this: what distinguishes the best team leaders from the rest is their ability to meet these two categories of needs for the people on their teams. What we, as team members, want from you, our team leader, is firstly that you make us feel part of something bigger, that you show us how what we are doing together is important and meaningful; and secondly, that you make us feel that you can see us, and connect to us, and care about us, and challenge us, in a way that recognizes who we are as individuals. We ask you to give us this sense of universality\u2014all of us together\u2014and at the same time to recognize our own uniqueness; to magnify what we all share, and to lift up what is special about each of us. When you come to excel as a leader of a team it will be because you've successfully integrated these two quite distinct human needs.\n\nOver the course of this book, we'll explore precisely _how_ the best leaders do this\u2014what they pay attention to and how they interact with the people around them. At the same time, we'll explore the eight items in more detail, and we'll see how the lies we're told at work push back, hard, against each of these eight critical aspects.\n\n\u2022 \u2022 \u2022\n\nBut what of our first lie, that _people care which company they work for_?\n\nWell, we now know that these eight questions measure very precisely those aspects of our experience of work that matter the most\u2014in other words, the aspects that drive performance, voluntary turnover, lost work days, accidents on the job, and customer satisfaction. So, if it is true that in large part people's experience at work is driven by the company they work for, then when we ask these eight questions to every person in every team at a particular company, we should get, generally, the same responses. There shouldn't be variation from team to team, because the day-to-day experience of working at this particular company should remain mostly consistent.*\n\nBut that's not the case\u2014in fact, it's _never_ the case. The statistical measure of variation is called range, and we've found that these scores always have a greater range _within a company_ than _between companies_. Experience varies more _within a company_ than _between companies_.\n\nHere's what this looks like. This is how 5,983 teams at Cisco answered the second question, \"At work, I clearly understand what is expected of me.\" (See figure 1-1.)\n\nNow, this is a very basic question. If you've spent much time in business organizations, you'll know that they devote a lot of energy to talking about strategies and plans and priorities and themes and critical initiatives and business imperatives, and Cisco is no different. Yet for all this effort, these nearly six thousand teams had a widely varying sense of what was expected of them. And we saw this variance across _all_ teams at _every_ company that we surveyed.\n\nFIGURE 1-1\n\n**Clarity of expectations on teams**\n\nHere's how 1,002 teams at Mission Health answered the seventh question, \"I have great confidence in my company's future.\" (See figure 1-2.)\n\nIf there's any item that should vary between companies rather than within them, it would surely be this one. After all, one company will presumably have only one future, and this future should seem the same regardless of which team you're on. Yet it doesn't feel like that. People's responses to this question vary significantly depending on which team they're on, within the same company: different team, different level of confidence in the future.\n\nWe see similar patterns on all eight of the questions\u2014we see, in other words, that when we zero in on the critical aspects of our experience at work, they vary more team-to-team than they do company-to-company. Any ideas\u2014like the idea of culture\u2014that rest on the assumption that our experience of a company is uniform, no matter where we sit, don't hold up. Any ideas\u2014again, like the idea of culture\u2014that rest on the assumption that our experience will vary company to company are incomplete, because our experience will vary more within a company than between companies. And any ideas\u2014again, like the idea of culture\u2014that rest on the assumption that this broad, unchanging company-ness is what defines our experience of work are simply wrong.\n\nFIGURE 1-2\n\n**Confidence in the future on teams**\n\nInstead _local_ experiences\u2014how we interact with our immediate colleagues, our lunching-on-the-patio companions, and our huddling-in-the-corner partners\u2014are significantly more important than company ones. At least, that's what all this research is telling us.\n\nMoreover, if we care most which _company_ we work for, it follows that there should be no connection between our experience on a given team and our choice to stay with a given company\u2014because company trumps team. But whenever we run an analysis, we find that when a team's score is low on these items, members of that team are significantly more likely to leave the company. At Cisco, for example, we've seen that when someone's experience of their team moves from the top half, companywide, to the bottom half, their likelihood of leaving the company increases by 45 percent. When people choose not to work somewhere, the somewhere isn't a company, it's a team. If we put you in a good team at a bad company, you'll tend to hang around, but if we put you in a bad team at a good company, you won't be there for long. The team is the sun, the moon, and the stars of your experience at work. As Edmund Burke, the Anglo-Irish writer and philosopher put it as far back as 1790, \"To love the little platoon we belong to in society is the first principle (the germ, as it were) of public affections.\"\n\nWhen we push on the data, and examine closely its patterns and variations, we arrive at this conclusion: while people might care which company they join, they don't care which company they work for. The truth is that, once there, _people care which team they're on_.\n\n\u2022 \u2022 \u2022\n\nRecently the ADP Research Institute conducted a nineteen-country study on the nature of engagement at work\u2014what drives it, and what it drives. We've included a summary of the findings in the appendix, but here are three highlights that you'll want to know. First, virtually all work is in fact teamwork. In companies with over 150 employees, 82 percent of people work on teams, and 72 percent work on more than one team. Even in small companies, of fewer than twenty people, this finding holds: 68 percent of those in small companies report working on a team, and 49 percent say they work on more than one team. This proved to be so in every single country in the study.\n\nSecond, we know that if you do happen to work on a team you are twice as likely to score high on the eight engagement items, and that this trend linking engagement to teams extends to multiple teams\u2014in fact, the most engaged group of workers across the working world are those who work on _five_ distinct teams.\n\nThird, just like Lisa, those team members who said they trusted their team leader were _twelve_ times more likely to be fully engaged at work.\n\nThe good news in all this for you, the team leader, is that what people care most about at work is within your control. You might not be able to weigh in on your company's parental-leave policy, or the quality of its cafeteria, but you can build a healthy team\u2014you can set clear expectations for your people, or not; you can position each person to play to his or her strengths every day, or not; you can praise the team for excellent work, or not; you can help people grow their careers, or not. And you can, over time, build trust with your people, or not. Of course, given the \"always-on\" nature of your daily work, attending to each of these is challenging, but at least they are indeed part of your daily work.\n\nThe bad news for you is that your company, most likely, looks past this, so while you're doing your best to create these experiences for your people, your company may not be holding your fellow team leaders accountable for doing the same on their teams. Companies almost universally miss the importance of teams, as evidenced by the fact that most companies don't even know how many teams they have at any moment in time, and who is on them, let alone which are the best ones\u2014we are functionally blind to teams. And our overemphasis on culture leads companies to remove responsibility from where it resides\u2014with the team leaders\u2014and instead to focus on generalities. You now know that your company does not have a uniform culture, that if there is something distinct about your company's culture then it is unmeasurable, that the total score of your company's employee survey is simply the clumping together of lots of highly varied team-level surveys, and that these clumps mask what really matters. You now know that when a CEO sets out to build a great company, all she can do\u2014and it's a lot\u2014is strive to build more and more teams like her company's best teams.\n\nAnd you now know that often what's written about company cultures are stories masquerading as data\u2014stories of one world, and then another, and then another, vivid, intriguing, charming, and occasionally a bit scary, but not real. Like Narnia, or like Middle-earth, if hobbits had jobs.\n\n\u2022 \u2022 \u2022\n\nIf the most important experience of work is the experience of team, what should we make of all the \"culture of . . .\" things with which we began? Are they all entirely irrelevant?\n\nIn his fascinating book _Sapiens_ , and again and in more depth in its sequel, _Homo Deus_ , the historian Yuval Noah Harari asks what it is that explains the success of humankind over all other species. Having examined and rejected the usual explanations\u2014we're not alone, as a species, in using tools, or in having language, or in making plans, or in experiencing consciousness\u2014he moves on to explore our notions of reality. Objective reality, as we know, is a reality that exists independently of our attitudes or feelings about it: if you stop believing in gravity, you'll still fall to the ground if you jump out of the window. Subjective reality, on the other hand, is defined precisely by your attitudes and feelings: if you have a toothache yet your dentist tells you she can't find anything the matter (there is no objective problem), your tooth still hurts.\n\nBut Harari goes on to argue that there is a third kind of reality, that this kind of reality is unique to humans, and that this kind of reality explains the dominance (for good or ill) of our species. There are some things that are real simply because we _all_ agree they're real\u2014things whose existence depends not on any objective reality, nor on any individual's subjective reality, but rather on our collective belief in them. By this logic, money, for example, exists _only_ because all of us agree it does. Initially this might sound odd\u2014surely money is just money, not some sort of mass belief system\u2014but here's the rub: when we all cease to believe in these realities, they cease to be real. If you and everyone else stop believing, all of a sudden, that a particular piece of paper is worth $10, then it actually and rather immediately ceases to be worth $10. This is more or less what happened in India on the evening of November 8, 2016, when the government announced that the very next day certain bills would no longer be legal tender, and those bills turned instantly from _Things That Are Valuable Because We All Agree They Are_ to _Things That Are Valueless Because Some Of Us Don't_.\n\nHarari calls these extended, communal realities _intersubjective realities,_ and tells us that they're the reason our achievements as a species are so different from those of our planet-mates. They enable us to coordinate our actions with those of people we may never meet, across distance and across time. Our belief in the intersubjective reality of nation, for example, enables us to cooperate with our fellow citizens to finance and build monuments, or to wage war; our belief in the intersubjective reality of democracy allows us to elect governments and to follow their laws. Our intersubjective realities are the distinguishing feature\u2014the apex technology\u2014of _Homo sapiens_.\n\nWhat are our intersubjective realities in the world of work? One, obviously, is the idea of the company. We can't touch it; it exists only in the realm of laws (another intersubjective reality), and when we stop agreeing it exists, it ceases to exist. Obviously, the stock-market value of a public company is another example. As is that company's brand and brand value. And its bank balance. All of these are useful\u2014essential, even\u2014to our ability to organize lots of people to achieve complex and enduring goals. Without them, and the many other intersubjective realities in the world of work, we would have none of the things that \"companies\" have produced since we invented them. But that doesn't make them real, in the sense that gravity is real, or in the sense that a toothache is real. Or in the sense that the other people at work\u2014your team\u2014are real.\n\nAnd just as the idea of the company is, in this particular sense, unreal, so is the idea of company \"culture.\" It's a useful fiction. That doesn't mean we should dispense with it; it _does_ mean, however, that we should be careful not to mistake it for something it isn't. Culture locates us in the world. It consists of stories we share with one another to breathe life into the empty vessel of \"company.\" But\u2014and here's the kicker\u2014so powerful is our need for story, our need for communal sense making of the world, that we imagine that our company and its culture can explain our experience of work. And yet it can't. So strong is our identification with our tribe that it's hard for us to imagine that other people inside our company are having a completely different experience of \"tribe\" from ours. Yet they are\u2014and these local team experiences have far more bearing on whether we stay in the tribe or leave it than do our tribal stories.\n\n\u2022 \u2022 \u2022\n\nHow can you make sense, then, of the things that are clearly different from one company to the next, things to which you're accustomed to attaching such importance? Patagonia _does_ have a drastically different type of onboarding than Salesforce. Goldman Sachs _does_ have a very different dress code from Apple. What are these things, and how are they different from the real-world experience of work?\n\nThe difference is this: these things are signifiers, designed to lure you in. You may not care which company you work for, but since you do care about which company you _join_ , these signifiers are crafted to help a company attract a certain kind of person by highlighting what the company thinks this kind of person values. This is why these signifiers show up time and again in promotional materials, and why they are so prominent in various company rankings\u2014because companies want it that way. These kinds of perks are plumage\u2014peacock feathers for people. They sound cool because they're designed to get your attention, just like plumage is. So when you read about how a certain company gives each employee \"twenty-percent\" time to focus on personal projects or claims to always promote from within, just remember that these beautiful feathers are designed almost exclusively to attract you, and that this attraction, as most attractions tend to, will fade.\n\nThe biggest difference, of course, between cultural plumage and the real world is that the _impact_ of plumage on how you and your team do your work every day is slight. That's not what it's for. It is a shared fiction, and it exists to attract a certain kind of person to join the company. And as with all shared fictions, the moment you all stop collectively believing in the plumage, it vanishes. Team experience, on the other hand (how you talk to one another and work with one another), has large and lasting impact on how you do your work, and it doesn't require all of you to agree to believe in it. It is what it is. And whether or not you all believe in it or can all describe it in the same way, it will nonetheless influence both how effectively your team works and for how long, and how many of your teammates will choose to stay.\n\n\u2022 \u2022 \u2022\n\nWhen you study excellence and what leads to it\u2014what creates it\u2014there is a dog that doesn't bark.* Actually, a couple of them. Company doesn't bark. And cultural plumage doesn't bark. Instead, sitting there in plain sight is what was in plain sight in Poland. What the three people there shared, as a key part of their experience of work, was not about a place to eat, but rather\u2014critically\u2014about the people to eat with. While they might think of Cisco as a place that affords them a table to sit at or a corner of the office to huddle in, if we give them those things yet take away their teammates, or change the sorts of interactions they have with those teammates, their experience vanishes. In a very real sense the spot they huddle on with a couple of teammates matters more to them than all the perks Cisco well-intendedly throws their way. What's in plain sight, when we study excellence at work, is the groups of people doing actual work together\u2014what's in plain sight is teams.\n\nThis is why teams matter, and it's why they matter much more than cultural plumage matters.\n\nTeams simplify: they help us see where to focus and what to do. Culture doesn't do this, funnily enough, because it's too abstract.\n\nTeams make work real: they ground us in the day-to-day, both in terms of the content of our work and the colleagues with whom we do it. Culture doesn't.\n\nAnd teams, paradoxically, make homes for individuals. Whereas culture's focus leans toward conformity to a common core of behaviors, teams focus on the opposite. Teams aren't about sameness\u2014they aren't, at their best, about marching in lockstep. Instead they're about unlocking what is unique about each of us, in the service of something shared. A team, at its finest, insists on the unique contribution of each of its members, and is the best way we humans have ever come up with of harnessing those distinctive contributions together in the service of something that none of us could do alone.\n\nIn the last few years, there has been a lot of talking and writing about teams in corporate circles. Sadly, much of it has yet to grasp the main point. The general direction of the discussion so far has been that we should pay attention to teams because there are a lot of them at work. This is, of course, true\u2014although one could make the case that this isn't news. Granted, given new communications and information technologies, teams can now be assembled spanning more geographies and time zones and organizational units than ever before, but the fact that there are now more teams and more different sorts of teams than ever before isn't the big thing. The big thing is that only on a team can we express our individuality at work and put it to highest use.\n\nIn a sense, that's what the rest of this book is about. To see it clearly, we have to let go of our ideas about cultural plumage, because only when we do so does the fact of team emerge\u2014quietly, simply, powerfully\u2014from its shadow. And as we do this we come to realize what is perhaps the biggest problem of all with the idea of culture: it doesn't actually help us understand what to do more of, less of, or differently. Whether culture is a real thing or not, whether it defines our tribe at work or not, whether it's a marker of what sort of company we're joining or not, it won't tell you, the team leader, what to do to make things better. For that, we must take you to where the experience lives: to your team, and networks of teams, and their leaders. That's what matters most.\n\n\u2022 \u2022 \u2022\n\nThere are three things for you to do as a leader of a team. First, you should know the answers to the eight questions for your team, all the time. There are technologies available to help you do this, but the easiest place to start is to ask your team members, one person at a time. Whatever their answers are, you'll _always_ be smarter because of them, and you'll _always_ know you're paying attention to something that matters.\n\nSecond, read on to understand more clearly how to build a great team, and how the lies you'll encounter get in the way of that. Your role as team leader is the most important role in any company. And who your company chooses to make team leader is the most important decision it ever makes. You have by far the greatest influence on the distinctive local experience of your team. This is a weighty responsibility, but at least it's yours. We want to help you step into it.\n\nAnd third, when you're next looking to join a company, don't bother asking if it has a great culture\u2014no one can tell you that in any real way.\n\nInstead, ask what it does to build great teams.\n\n*Obviously, we've disguised the names of the companies.\n\n*One of the great joys of writing as a team of two is that it allows us to bring both of our perspectives and, critically, experiences and stories to the task, resulting in what we hope is a richer book. This presents, however, one small challenge when writing about an experience that one but not both of us had, and it's the challenge of which pronouns to use. Joint authors before us have taken various approaches to this, either by referring to themselves in the third person (\"When Marcus interviewed such-and-such . . .\") or by beginning every story with some sort of parenthetical clarification (\"When one of us [Ashley] was in Poland . . .\"), neither of which we feel makes up in clarity what it takes away in readability. So we've decided that whether an event described here was experienced by one of us alone or by both of us, we'll just say \"we.\" We hope you will forgive us a little pronoun flexibility in the service of easier reading.\n\n*Strictly speaking, an item is a statement that a survey-taker responds to. As items are statements, not questions, they don't come with question marks. However, they are often referred to as questions, to ensure maximum confusion for those of us trying to figure out what's going on.\n\nThe eight engagement items discussed in this book are copyright ADP.\n\n*More precisely, we would see only as much variation within teams as we would within the company as a whole.\n\n*GREGORY: \"Is there any other point to which you would wish to draw my attention?\"\n\nHOLMES: \"To the curious incident of the dog in the night-time.\"\n\nGREGORY: \"The dog did nothing in the night-time.\"\n\nHOLMES: \"That was the curious incident.\"\n\n\u2014Sir Arthur Conan Doyle, \"The Adventure of Silver Blaze,\" in _The Memoirs of Sherlock Holmes_ (London: George Newnes, 1894).\n\n#\n\n# The best plan wins\n\nGeorge Clooney had a plan.\n\n\"I have a question,\" says Carl Reiner early on in the movie _Ocean's Eleven_ , after Clooney has laid out his scheme for breaking into the impressively secured vault of a Las Vegas casino.\n\n\"Say we get into the cage, and through the security doors there, and down the elevator we can't move, and past the guards with the guns, and into the vault we can't open . . . say we do all that. We're just supposed to walk out of there with $150 million in cash on us, without getting stopped?\"\n\nThere's a silence. The members of Ocean's handpicked team eye one another nervously, unsure what's coming next.\n\nClooney pauses, and then nods, and then: \"Yeah.\"\n\nAnd Carl says, \"Oh.\" And then, \"OK.\" And in an instant, we know that Clooney has a plan for that, too, and that Carl realizes there's a plan, and doesn't need to know what it is, just that there is one, and that it'd better be a good one, because, as everyone knows, _the best plan wins_.\n\nThe thrill for us, the viewer, is seeing whether the plan will work\u2014will Matt Damon's pickpocket skills succeed in lifting the badge from the security guard? Will Casey Affleck and Scott Caan's goofy antics and handy-dandy birthday balloons block the casino cameras? Will Clooney charm Julia Roberts? (Yes to all. Duh.)\n\nBut ponder for a minute the thrill each team member would have felt. Even though they were coming together under challenging circumstances, they had the plan, whose raw material was a specific role that each of them should play. Each person's role was tightly circumscribed, time-bound, and sequential\u2014Brad Pitt would place the call to Julia Roberts, but not until Clooney had slid the phone into her jacket pocket\u2014so all could feel secure in the knowledge that if they learned how to perform their role well, and executed it well, then, like a mathematical algorithm, the sequence would play out perfectly, the plan would work, and the money would be theirs.\n\nIf you've recently been promoted to team leader, the first thing you'll be expected to do is create a plan. You'll be asked\u2014before you even start, most likely\u2014what your plan is for your team, or, more specifically, what your ninety-day plan is for your team. You'll have to sit down, think hard, survey your team members (many of whom you will have inherited), and then do your best George Clooney impression and make your plan.\n\nAnd when you do this, you'll quickly realize one of the many differences between your team and Clooney's: his team works alone, while yours appears to be connected to a whole host of other teams, each with their own version of the plan. In fact, poke your head above the parapet of your team for a second and look out across all the other teams in the company, and you'll discover something of a planning frenzy. Every team is about to go, or is away on, or is just back from, or is just debriefing from, their off-site, during which they formulated, or perhaps reformulated, their current version of the plan.\n\nIt won't be immediately obvious to you, but after a few years you'll discern that there is a pattern to this planning, a predictable rhythm that repeats itself year after year: in September, in advance of the November board meeting, the leaders of your company will go away on a senior leadership retreat. They may do a SWOT analysis (Strengths\/Weaknesses\/Opportunities\/Threats\u2014and it's just as fun as it sounds); they may bring in outside consultants to help them; and after much analysis and debate and proposal and counter-proposal, the white smoke will emerge from the chimney, and the leaders will emerge with The Strategic Plan. They will then present this plan to the board, and once it's approved, they will share it with their direct reports. This plan will then be sliced up into many other plans (departmental plans, divisional plans, geographic plans, and so on), each slice finer and more detailed than the preceding one, until you, too, are asked to take your team off-site and construct your version of the plan.\n\nWe do this because we believe that plans are important. If we could just get the plan right, we think, and weave every team's plan into the broader company plan, then we could be confident that our resources were allocated appropriately, that the correct sequence and timing were laid out, that each person's role was clearly defined, and that we had enough of the right people to fill each required role. Buoyed by this confidence, we'd know that we'd only have to galvanize our teams to give their all, and success would follow.\n\nAt the same time, there is a yearning quality to all this planning. We are attempting to shape our future, and our plans can feel like scaffolding stretching out into the months ahead, upon which we'll build our better world\u2014their function is perhaps as much to reassure us as it is to make that world real. Plans give us certainty, or at least a bulwark against uncertainty. They help us believe that we will, indeed, walk out of the casino with the cash.\n\nAnd yet, just as this cycle of big plans leading to medium plans leading to small plans is familiar to you, so\u2014surely\u2014is the nagging realization that things rarely, if ever, turn out the way you hope they will.\n\nSure, planning is exciting in the beginning, but the more you sit in all these planning meetings, the more a feeling of futility creeps in. While it all looks great on paper, tidy and perfect, you sense it's never really going to play out like this, and that as a result you'll soon be back in yet another planning meeting. You'll leave this one with the broad contours of your plan sketched out, and you'll agree on the next steps necessary to refine those contours into something specific and actionable, and then the meeting to make things actionable will get postponed a bit, and then, when it finally happens, it will drift off in another direction. And then, when your team finally gets around to nailing the details, some new idea or thought or realization will emerge that leads you to rethink what you started off with. George Clooney never had to deal with this.\n\nBut in the real world you'll have to. The defining characteristic of our reality today is its ephemerality\u2014the speed of change. If _Ocean's Eleven_ took place in the real world, then after Clooney had put his plan together, and picked his perfect team, and defined each person's role, and pressed play, they'd arrive in the vault, open the safe . . . and it would be empty, because Nevada would have changed its gambling regulations, and Andy Garcia's casino owner would have ditched the cash, replaced it with Bitcoin, and, in hopes of jumping a few spots on the _Fortune_ list, turned the vault into a subterranean romper-room-cum-fitness-center to aid his employees' well-being. In the real world, the _Ocean's Eleven_ team would burst into the vault to discover the 11:30 a.m. hot-yoga class.\n\n\u2022 \u2022 \u2022\n\nGeneral Stanley McChrystal had to grapple with this alarmingly changing world, and the stakes he faced were far higher than yours will be (we hope). In his book _Team of Teams_ he reveals what it was like to try to come up with \"the plan\" as he assumed command of the Joint Special Operations Task Force. This group had brought together the special-operations units of each branch of the US military\u2014the Army's Delta Force and 75th Ranger Regiment, Marine Force Recon, Navy SEALs, and Air Force Pararescue and Combat Control Teams\u2014and its mission was to deploy these units in the ongoing struggle with al-Qaeda in Iraq in the aftermath of the 2003 invasion. Within a few months of assuming command, McChrystal and his staff had created an \"awesome machine\"\u2014the planning, execution, and debriefing of targeted raids was now conducted at greatly increased speed. But still they were losing the war. They faced an enemy who was spontaneous, decentralized, and agile, made up of terrorist cells that could plan and execute attacks without recourse to a chain of command, and as much as McChrystal's planners tried to optimize their process, they were never quite fast enough to be ahead of events. Even if their traditional system\u2014intelligence gathering, leading to analysis, leading to target identification, leading to raid planning, leading to action, leading to after-action review\u2014could be made to run at the speed of light, it was never quite fast enough. McChrystal's forces were still too often caught unawares by the latest attack, were still too often turning up at a house in search of a target that had been there when the plan was created, only to find no one there.\n\nEverywhere we look we see this speed of change. When you put your plan together in September, it's obsolete by November. And if you look at it in January, you might not even recognize the roles and action items you wrote out in the fall. Events and changes are happening faster than they ever have before, so dissecting a situation and turning it into a meticulously constructed plan is an exercise in engaging in a present that will soon be gone. The amounts of time and energy it takes to make a plan this thorough and detailed are the very things that doom it to obsolescence. The thing we call _planning_ doesn't tell you where to go; it just helps you understand where you are. Or rather, were. Recently. We aren't planning for the future, we're planning for the near-term past.\n\nAnd where are the people who are making the plan? So far behind the front lines of the company that they don't have enough real-world information upon which to make the plan in the first place. How can you make a plan to sell a particular sort of product to a particular sort of customer when you're not out selling every day? You can't, not really. You can make a theoretical sales \"model\" based on your conceptual understanding of an abstract situation, or on an averaged data set that summarizes trends. But if it's not grounded in the real-world details of each actual sales conversation\u2014when do the prospects' eyes glaze over, when do the prospects lean forward, when do they start to finish your sentences\u2014your plan will always be more assumptive than prescriptive.\n\nIn terms of General McChrystal's challenge, this is the difference between building a plan around generalizations (say, that \"25 percent of the time we show up at the wrong house\") and building one that grapples with very specific realities (such as, \"How do we hit _this particular_ target, when he might leave _this particular_ house, at _this particular_ time, tonight?\"). Unfortunately, most plans\u2014particularly those devised high up in the company\u2014are built like the former, not the latter.\n\nAnd even if you do weave together the most carefully filigreed plan, your people inevitably chafe at being told what to do in the context of something so static, so conceptual, and potentially so out of touch with the real world they face. Although Ocean's Eleven were initially happy to be given prescribed roles with crystal-clear expectations, imagine how they'd feel if they played their roles to perfection as defined in the plan, and yet still found no money in the vault. And then imagine if they had to keep doing this month after month, because that was the plan, and because that was what the senior leader had decided would work, even though they kept telling their team leader that the money was gone, that the yoga instructor was getting annoyed, and that they felt just plain daft in their masks and overalls in the hundred-degree heat.\n\nYour people want and need to engage with the world that they're really in, and to interact with the world as it really is. By harnessing them to a prefabricated plan, you're not only constraining your people but, quite possibly, also revealing how out of touch with reality you are.\n\nThis is not to say that planning is utterly useless. Creating space to think through all of the information you have in your world, and trying to pull that into some sort of order or understanding, has some value. But when you do that, know that all you've done is understand the scale and nature of the challenges your team is facing. You'll have learned little about what to do to make things better. The solutions can be found in the tangible and changing realities of the world as it really is, whereas your plans are necessarily abstract understandings of the recent past. Plans scope the problem, not the solution.\n\nSo, though you are told that _the best plan wins_ , the reality is quite different. Many plans, particularly those created in large organizations, are overly generalized, quickly obsolete, and frustrating to those asked to execute them. It's far better to coordinate your team's efforts in real time, relying heavily on the informed, detailed intelligence of each unique team member.\n\n\u2022 \u2022 \u2022\n\nIn late 1940, Hitler's armies had swept across Europe to the shores of France, and Britain's Royal Air Force was all that stood between them and the conquest of the British Isles. Although the RAF had been able to increase the number of fighter aircraft available over the course of the summer months, what it had was still not enough. The accepted model of the time for aerial defense was that, because it was impossible to know where the next attack would come, defenders\u2014in this case the Spitfires and Hurricanes of the RAF\u2014should fly continuous patrols in shifts, in the hope of encountering attackers. But the length of the English coastline, and the number of planes and pilots required to patrol its entirety without pause, made this approach prohibitive. So the only possible alternative was to guess where the attack might come, and too much of the time the guess was wrong\u2014a \"good\" interception rate was considered one where the enemy was sighted on half the sorties flown. In the history of air warfare to this point, this had always been true\u2014hence the dictum of the day: \"The bomber always gets through.\"\n\nWhat the RAF needed, if the country were to be saved, was a force multiplier\u2014something that made their limited roster of planes and pilots vastly more effective. The force multiplier they came up with was a room.\n\nIf you were to stand in this room today, here's what you'd see. Spanning one entire wall would be a series of twenty-six white vertical boards, each with the number of a squadron at the top, and each with rows of four colored lights underneath. The lights represent each flight in each squadron. There are twelve planes in a squadron, divided into four flights of three planes each, and the lights show at a glance the status of that flight\u2014whether it's ready for action, ordered into the air, engaged with the enemy, returning to base to refuel, and so forth\u2014together with how long each flight has been airborne. These boards (the so-called tote boards) show which orders have been given to which planes, capturing the status of each of four flights for each of twenty-six squadrons\u2014so, 104 pieces of information.\n\nBelow the tote boards, you'd see two rows of numbers, showing the number of planes and the number of pilots available for each squadron at the beginning of that day\u2014so, two numbers for each squadron, another 52 pieces of information.\n\nFarther below on the same wall, you'd see four displays showing the heights at which barrage balloons are deployed, together with the day's weather forecast\u20145 more pieces of information.\n\nIn the center of this wall, you'd see an unusual-looking clock, the face of which is divided into five-minute increments colored alternately red, yellow, and blue, and whose purpose relates to the large map table that covers most of the floor area of the room. This table shows a large-scale view of a portion of the coastline, the English Channel, and the coast of France. Around it, you'd see a number of women, each wearing headphones and each wielding a croupier's stick. And in the middle of the table, you'd see a number of wooden blocks with numbers affixed to them and other numbers added to the tops of them on sticks, roughly the size of toothpicks.\n\nEach wooden block represents a group of aircraft\u2014either attackers, defenders, or unidentified planes. If the aircraft are over France or the sea, their location is given by two radar systems, collectively called Chain Home, whose information is relayed from forty stations along the coast. If they are over land, their location is given by telephone reports from the 30,000 members of the Royal Observer Corps, deployed at 1,000 observation posts (because the radar system faces only out to sea). The planes' location, together with information from a third system identifying whether they are friend or foe, is passed to the women around the table\u2014the plotters. The plotters then move the wooden block to the right location and affix to it a numerical identifier, together with indicators of the number of planes, their altitude, whether they're friend or foe, and, for enemy planes, which squadrons have been dispatched to intercept them\u2014this last information being the numbers on the toothpicks. And all of these various numbers are color-coded, and the colors match the colors on the unusual-looking clock, so that everyone can see at a glance how up-to-date each piece of information is.\n\nThe map table, then, captures thousands of returns from the Chain Home radar system each minute, plus thousands more from the friend-or-foe system, plus, for every twenty-four hours, a million individual reports from the Royal Observer Corps, each relayed to the room within forty seconds.\n\nAnd what the room does is bring together all of these data points in real time and then present them so that front-line team members\u2014called controllers\u2014can exercise their judgment and can send their forces to where they know the enemy is. The room (now called the Battle of Britain Bunker) and its design, together with the network of identical rooms across the country and the many information systems flowing into each of them, formed what came to be called the Dowding system, after Hugh Dowding, the RAF commander who created it.\n\nIt made all the difference. It was a force multiplier that increased average interception rates from the prewar level of between 30 percent and 50 percent to an average of 90 percent, and often 100 percent, which is to say it _doubled_ the effectiveness of the defending force. It wasn't, in any sense, a planning system, acting slowly on stale and summarized information. Instead, it acted fast on current, raw, and detailed information. The RAF's force multiplier was an _intelligence system_.\n\n\u2022 \u2022 \u2022\n\nWhen we understand the characteristics of an intelligence system, as distinct from a planning system\u2014accurate, real-time data, distributed broadly and quickly, and presented in detail so that team members can see and react to patterns in deciding for themselves what to do\u2014we begin to see them everywhere. The Battle of Britain Bunker was an early example of what, today, we'd call a war room, a name that has grown from its literal roots to encompass more metaphorical uses. Think of the famous war room at the heart of Bill Clinton's first campaign for the US Presidency, for example, or of the war rooms used in project or crisis management. Think of NASA's Mission Control Center in Houston, Texas, or of a live-TV production room. Or think of something like Cisco's Security Operations Centers, where engineers can monitor performance across a group of customers' networks, and immediately respond to problems.\n\nWhat all of these things have in common is that they move information across an organization as fast as possible, and do so to empower immediate and responsive action. Their underlying assumption is that people are wise, and that if you can present them with accurate, real-time, reliable data about the real world in front of them, they'll invariably make smart decisions.\n\nIt's not true that the best plan wins. It is true that _the best intelligence wins_.\n\n\u2022 \u2022 \u2022\n\nWhat can you do as a team leader to create such an intelligence system for your team?\n\nFirst, _liberate as much information as you possibly can_. Think about all the sources of information you have, and make as many of them as possible available to your team, on demand. Planning systems constrain information to those who \"need to know.\" Intelligence systems don't\u2014they liberate as much information as possible, as fast as possible. So don't worry too much at first about whether your team will understand the data or be able to make use of it. If you think the information will help your people gain a better understanding of their real world in real time, share it. And encourage your team to do the same. Help them understand that sharing what they know about the world, frequently, is vital. Make sure your team is swimming in real-time information, all the time.\n\nSecond, _watch carefully to see which data your people find useful_. Don't worry too much about making all this data simple or easy to consume, or about packaging it for people, or weaving it together to form a coherent story. The biggest challenge with data today isn't making sense of it\u2014most of us deal with complexity all the time, and are pretty good at figuring out what we need to know and where to find it. No, the biggest challenge with data today is making it accurate\u2014sorting the signal from the noise. This is much harder, and much more valuable for our teams. So be extremely vigilant about accuracy; watch which information your people naturally gravitate toward; and then, over time, increase the volume, depth, and speed of precisely that sort of data.\n\nThird, _trust your people to make sense of the data_. Planning systems take the interpretation of the data away from those on the front lines, and hand it off to a select few, who analyze it and decipher its patterns, and then construct and communicate the plan. Intelligence systems do precisely the opposite\u2014because the \"intelligence\" in an intelligence system lies not in the select few, but instead in the emergent interpretive powers of all front-line team members. You are not the best sense maker. They are.\n\nMcChrystal, describing the system he ultimately created in Iraq, makes this same point: \"In the old model, subordinates provided information and leaders disseminated commands. We reversed it: we had our leaders provide information so that subordinates, armed with context, understanding, and connectivity, could take the initiative and make decisions.\" And what he created was perhaps the most extreme example yet of an intelligence system in action.\n\n\u2022 \u2022 \u2022\n\nA pretty good way to ruin someone's day is to fill it with meetings. Meetings, for most of us, are a way of taking time that could be put to good use in doing real work, and instead using that time to hear presentations of varying relevance to our immediate challenges, or to discuss topics that might appear important in the grand scheme of things, but that hardly seem urgent on any given day. And while countless meeting \"best practices\" (have a written agenda, document follow-up items, and so on) at least ensure some degree of utility, the fact remains that most meetings contain one or more people thinking to themselves that they could be doing something useful, if only they weren't doing this.*\n\nWhich makes what General McChrystal came up with in Iraq all the more remarkable, and all the more counterintuitive. Because what he did was to create a meeting, and moreover a meeting that took place six days a week, for two hours a day. For two thousand people.\n\nThe meeting was called the O&I\u2014the Operations and Intelligence meeting. Every day, at 9:00 a.m. Washington time, 4:00 p.m. Iraq time, McChrystal's entire command\u2014and, ultimately, anyone from any other agency with an interest in understanding what was happening\u2014would join, via video from wherever they were in the world, what amounted to a two-hour information-sharing session. The time was filled with brief updates, each a minute long, from anyone who had something pertinent to share, followed by four minutes of Q&A from the leadership team, or from anyone else who wanted to know more. The O&I had existed before McChrystal, but in a very different form. It had been shorter, and more exclusive, limited to those who had a \"need to know\" any particular piece of information\u2014it was part of a quintessential planning system.\n\nMcChrystal's O&I was a very different beast. It was open to anyone who wanted to learn or share information. It was democratic, in the sense that giving updates and asking questions could be done by anyone, not just senior officers. It was spontaneous, in that updates were not required to be polished or vetted, only brief. And it was frequent. His system embodied these few truths: that information grows stale fast, and must therefore be shared fast; that the best way to enable coordinated action on the ground is to coordinate not actions themselves, but rather the information the ground needs right now; that the best judges of what information is and isn't valuable are the end users of that information; and, critically, that the best people to make sense of information are the users of that information. And finally, that the best way to make sense of it was _together_.\n\nWhen McChrystal arrived in Iraq, and did everything he could to accelerate the planning system he inherited, the number of raids his troops conducted each month increased from ten to eighteen. When he created his intelligence system, that number shot all the way up to _three hundred_.\n\nAnd while this is an example of an intelligence system at huge scale, if you study the best team leaders you'll discover that many of them share a similarly frequent sense-making ritual\u2014not with two thousand people, but with two. It's called a check-in, and in simple terms it's a frequent, one-on-one conversation about near-term future work between a team leader and a team member.\n\nHow frequent? Every week. These leaders understand that goals set at the beginning of the year have become irrelevant by the third week of the year, and that a year is not a marathon, planned out in detail long in advance, but is instead a series of fifty-two little sprints, each informed by the changing state of the world. They realize that the key role of a team leader is to ensure that Sprint Number Thirty-Six is as focused and as energizing as was Sprint Number One.\n\nSo, each and every week these leaders have a brief check-in with each team member, during which they ask two simple questions:\n\n> What are your priorities this week?\n> \n> How can I help?\n\nThey are not looking for a to-do list from the team member. They simply want to discuss the team member's priorities, obstacles, and solutions in real time, while the work itself is ongoing. Making sense of it together can happen only in the now. The generalizations that emerge once the passage of time has blurred the details are not the stuff of good sense making. So, doing a check-in once every six weeks or even once a month is useless, because you'll wind up talking in generalities.\n\nActually, the data reveals that checking in with your team members once a month is literally worse than useless. While team leaders who check in once a week see, on average, a 13 percent increase in team engagement, those who check in only once a month see a 5 percent _decrease_ in engagement. It's as if team members are saying to you, \"I'd rather you not waste my time if all we're going to do is talk generalities. Either get into the nitty-gritty of my work and how you can help right now, or leave me alone.\"\n\nEach check-in, then, is a chance to offer a tip, or an idea that can help the team member overcome a real-world obstacle, or a suggestion for how to refine a particular skill. Check-ins can be short\u2014ten to fifteen minutes\u2014but that's plenty of time to do a little real-time learning and coaching. And, like all good coaching, this has to be rooted in the specifics of the particular situation the team member is facing, the psychology she is bringing to it, the strengths she possesses, and the strategies she might already have tried. Again, the only way to surface these sorts of microdetails is to make sure that the conversations are frequent.\n\nThis leads us to one of the most important insights shared by the best team leaders: _frequency trumps quality_. They realize that it's less important that each check-in is perfectly executed than that it happens, every week. In the intelligence business, frequency is king. The more frequently and predictably you check in with your people or meet with your team\u2014the more you offer your real-time attention to the reality of their work\u2014the more performance and engagement you will get. In this sense, checking in is akin to teeth-brushing: you brush your teeth every day, and while you hope that each brushing is high quality, what's most important is that it happens, every day. Twice-a-year super-high-quality teeth-brushing is as absurd as it sounds. So is twice-a-year super-high-quality intelligence. A team with low check-in frequency is a team with low intelligence.\n\nAnd this realization in turn gives the lie to the complaint\u2014heard so often from senior leaders and HR executives\u2014that \"our team leaders aren't skilled enough to coach their people!\" The data reveals only that those team leaders who check in every week with each team member have higher levels of engagement and performance, and lower levels of voluntary turnover. It doesn't have anything to say about the quality of those check-ins. We know for sure that if a team leader checks in often with a team member, the team member gets something really positive out of it\u2014even if the team leader is no Pat Summit.* And besides, if the team leader struggles initially with her check-in quality, at least she's able to practice it fifty-one more times with each team member every year. No matter what her starting point, or her level of natural coaching talent, she's going to get a little better.\n\nNow, you, the team leader, might think, _Well, I would love to check in with my people every week, but I can't. I've simply got too many people!_ If that's you, then yes\u2014you have too many people. One of the longer-running debates in the world of people and organizations is the span-of-control debate, which grapples with exactly how many team members every team leader should manage. Some say between one and nine employees. Others say between one and twenty. Some nurses manage staffs of forty; some call-center managers lead seventy or more.\n\nBut by pinpointing the weekly check-in as the single most powerful ritual of the world's best team leaders, we can now know the exact span of control that's right for every single team leader: it's the number of people that _you_ , and only you, can check in with every week. If you can check in with eight people, but you can't fit nine into your schedule, your span of control is eight. If you can find a way to check in with twenty people, then your span of control is twenty. And if you're one of those people who can legitimately manage a weekly check-in with only two people, your span of control is two. Span of control, in other words, isn't a theoretical, one-size-fits-all thing. It's a practical, function-of-team-leader's-capacity-to-give-attention thing. Your span of control _is_ your span of attention.\n\nIn the service of intelligence, then\u2014in the service of making sense of real-time information together\u2014the weekly check-in is the anchor ritual. You need to design your teams, and their size, to enable it. And if ever you become a leader of leaders, you'll need to ensure that your leaders know that this check-in is the _most important part_ of leading. Checking in with each person on a team\u2014listening, course-correcting, adjusting, coaching, pinpointing, advising, paying attention to the intersection of the person and the real-world work\u2014is not what you do _in addition to_ the work of leading. This _is_ the work of leading. If you don't like this, if the idea of weekly check-ins bores or frustrates you or you think that once a week is just \"too much,\" that's fine\u2014but, for the love of Hugh Dowding, don't be a leader.\n\nIn the previous chapter we saw how critical it was for team members to come to trust their team leader. Frequent sense making together\u2014whether in McChrystal's O&I meeting, or your weekly check-ins\u2014can help with this since it leads not only to better decisions but also to the building of trust. Two of the eight engagement items directly address this issue of trust: \"In my team, I am surrounded by people who share my values\" and \"My teammates have my back.\" When these items receive low scores on a team it's easy to assume that the problem is one of intent\u2014that team members don't really care for one another or want to support one another. More often than not, however, low scores are a function not of bad intent but of poor information: team members don't know _how_ to support one another, because they don't know what's going on in enough detail to offer assistance. If they don't know what one another is doing, how can each learn what the others truly value? Likewise, if they don't know what work each is engaged in, how can any one of them feel safe? You can't watch someone's back if you don't know where his or her back is.\n\nThe more frequent sense-making rituals you establish on your team, the more information you will liberate, the more intelligence you will generate, and the more trust you will engender. Trust can never emerge from secrecy. Frequency creates safety.\n\n\u2022 \u2022 \u2022\n\nThe lessons of Dowding and McChrystal are not just lessons about systems and information and processes; they are lessons about the role of a leader in a fast-moving world. And their shared insight, across the span of sixty years, is that it is far more powerful for a leader to free the most information and the most decision-making power than it is for that leader to craft the perfect plan.\n\nAnother of the eight aspects that distinguish the best teams, as we've seen, is the sense of every team member that, \"At work, I clearly understand what is expected of me.\" Whether informed by Taylorism and Scientific Management in the early part of the twentieth century, by Management by Objectives in the latter part of it, by any number of management truisms in between, or simply by what seems intuitive, our assumption has most often been that the best way to create clarity of expectations is to tell people what to do. It turns out, however, that by the time you've managed to do this, your directions are wrong because the world has moved on. In this way, the systems we've built to tell people what to do at great scale\u2014planning systems\u2014fail.\n\nThe best, most effective way to create clarity of expectations is to figure out how to let your people figure it out for themselves. This isn't a question of removing complexity, but is rather one of locating it in the right place\u2014not hidden from view as the input for a grand plan, but rather shared for all to see. To do this, give your people as much accurate data as you can, as often as you can\u2014a real-time view of what's going on right now\u2014and then a way to make sense of it, together. Trust the intelligence of your team.\n\n*POINTY-HAIRED BOSS: We're having a meeting to discuss employee retention.\n\nDILBERT: Tell them that employees quit because there are too many useless meetings.\n\nPOINTY-HAIRED BOSS: We won't be getting into reasons at the first meeting.\n\n_\u00a9 2001 United Feature Syndicate, Inc_.\n\n*In case you don't know Pat Summit, she was the best coach of the last century. She was the coach of the University of Tennessee women's basketball team, and she had the most wins in college basketball history: 1,098. She won eight NCAA championships (a record at the time); was an Olympic medalist and coach, guiding the US women's basketball team to a gold medal in 1984; and in 2000 was named the Naismith Coach of the Century.\n\n#\n\n# The best companies cascade goals\n\nRecently a friend told us about a goal of hers: She said she was going to run a marathon. More precisely, she told us she was going to run the Prague marathon in seven months' time, in May. And when we asked her why, off the top of her head she shared a few of her reasons: that May was far enough away to give her time to go from \"the couch to the course\"; that the only marathon she could find around that time was in Prague; that she'd never been to Prague; that the course in Prague was known to be a mostly flat course and that a marathon was hard enough without throwing in those darn hills.\n\nBut of course none of these was the real reason she was running the marathon in Prague, in May. The real reason was that she wanted to significantly improve her physical stamina, and running a marathon seemed like the best way\u2014albeit a tad drastic\u2014to achieve that end. All the other details\u2014May, Prague, a flat course\u2014were simply her way of making that end more tangible, and therefore more hers.\n\nThis, at their best, is what goals do for us. They enable us to take what we value most and, by adding detail and timelines, to \"chunk\" these values into a describable outcome, something vivid and tangible. Visualized in our mind's eye, our goal pulls us forward, up and off the couch, and onto the road early one frigid Saturday morning in January, late one drizzly evening in March. Our goal becomes our companion, nestled in one corner of our psyche, pulsating, nudging us onward, guiding our thoughts and actions, and giving us the energy to push through the tiredness, the injuries, and the self-doubt, until one day we round the corner in Wenceslas Square and, alongside other people with other goals, complete our marathon.\n\nAnd if goals did this in the corporate world\u2014if they helped us step toward what's most important to us\u2014they would be supremely useful.\n\n\u2022 \u2022 \u2022\n\nGoals are everywhere at work\u2014it's hard to find many companies that do not engage in some sort of annual or semi-annual goal-setting regimen. At some point in the year, usually at the start of a fiscal year or after bonuses and raises have been paid, the organization's senior leaders set their goals for the upcoming six or twelve months, and then share them with their teams. Each team member looks at each of the leader's goals, and figures out what to do to advance that goal, and thus sets a sort of minigoal that reflects some part of the leader's goal. This continues down the chain, until you, and every other employee, has a set of goals that are miniversions of some larger goal further up in the organization.\n\nIn some organizations, goals are also grouped into categories, so that each person is asked to set, say, strategic goals, operational goals, people goals, and innovation goals. Once the goals have been created, each is then approved by the person's immediate leader, and then by the leader above that person, and so on, with each layer assessing whether each goal is sufficiently challenging, and whether it's properly aligned with the goals above, up and up and up the chain.\n\nAs the year unfolds, you may well be asked to record what percentage of your goals you've completed. This \"percent complete\" data is then aggregated into bigger and bigger groups so that the company can, at any point during the year, say things like, \"65 percent of our teams have completed 46 percent of their goals. We need to speed up!\"\n\nAnd, at the end of the year, you're asked to write a brief self-assessment reflecting how you feel you've done on each goal, after which your team leader will review this assessment and add her own, in some cases also saying whether she thinks each goal was actually met, or not. After HR has nudged her a couple of times, she'll input all this information into the company performance management system, whereupon it'll serve as a permanent record of your performance for the year, and will guide your pay, promotion opportunities, and even continued employment.\n\nIf you're in sales, your sales quota will work in a similar way\u2014an overall corporate sales goal is sliced into parts and distributed across the organization. The only difference being that your quota, or your team's quota, is usually just a single number handed down to you from above, defining you and your work throughout the year\u2014which is why salespeople, in most companies, are referred to not as people but simply as \"quota carriers.\"\n\nAnd, in the era of the smartphone, once-a-year goal-setting has been deemed Not Enough, and so your phone will soon be dramatically upping the frequency of all this goal-setting, assessing, and tracking, if it hasn't done so already\u2014all because we have come to believe that _the best companies cascade goals_.\n\n\u2022 \u2022 \u2022\n\nThe names we give these goals have changed over the years. We started with MBOs, or Management by Objectives, first popularized by Peter Drucker in his 1954 book _The Practice of Management_. Then came SMART goals, goals that are specific, measurable, actionable, realistic, and time-bound, followed shortly by KPIs (Key Performance Indicators) and BHAGs (big hairy audacious goals, in Jim Collins's memorable framing). The latest incarnation, OKRs (Objectives and Key Results), originated at Intel and is now used by much of Silicon Valley for defining and tracking goals and measuring them against your \"key results.\"\n\nAcross all the different technologies and methodologies, massive amounts of time and money are invested in this goal-setting. To give you a sense of the scale of the investment, the consulting firm Deloitte estimated that it spent $450 million on goal-setting, tracking, and evaluating every year, while Accenture, its consulting cousin, with more than 500,000 employees, spent more than twice that. When companies like these shell out close to $1 billion on something every year, there must be some truly extraordinary benefits.\n\nWhat are they?\n\nWell, every company is different, of course, and each makes its own calculus, but the three most common reasons put forth for all this goal-setting are, first, that goals stimulate and coordinate performance by aligning everyone's work; second, that tracking goals' \"percent complete\" yields valuable data on the team's or company's progress throughout the year; and third, that goal attainment allows companies to evaluate team members' performance at the end of the year. So, companies invest in goals because goals are seen as a _stimulator_ , a _tracker_ , and an _evaluator_ \u2014and these three core functions of goals are why we spend so much time, energy, and money on them.\n\nAnd this is precisely where the trouble begins.\n\nIn terms of goals as a stimulator of performance, one great fear of senior leaders is that the work of their people is misaligned, and that effort is being wasted in activities that drag the company hither and yon, like a rudderless boat in a choppy sea. The creation of a cascade of goals calms this fear, and gives leaders the confidence that everyone on the boat is pulling on the oars in the same direction.\n\nOf course, none of this alignment is worth very much if the goals themselves don't result in greater activity\u2014if the boat doesn't actually go anywhere. As it happens, no research exists showing that goals set for you from above stimulate you to greater productivity. In fact, the weight of evidence suggests that cascaded goals do the opposite: they limit performance. They slow your boat down.\n\nHave you ever tried to hail a cab in New York City on a rainy day? It's not easy. You stand there on the corner of 52nd and 3rd waving frantically at any vehicle that's even vaguely yellow, and bemoaning the fact that every one of the (suddenly scarce) cabs is taken. If you're up on your economics, you might even surmise, as the water drips off your nose, that the rain has increased the number of taxi hailers (demand) while not changing the number of taxi drivers (supply), hence the problem. But actually that's not quite what's going on. Cab drivers have an informal daily goal, or quota, for the fares they want to earn before they allow themselves to stop working\u2014for most cabbies that number is twice the cost of renting the cab for the day. The moment the day's receipts add up to twice their rental fee, they head home and rest up for the next day of battle. Now, they have this goal every day, but on rainy days\u2014because more people choose to take a cab\u2014they hit that goal earlier in their shift, and the moment they do they vanish off home.\n\nThe same thing happens with sales quotas. Leaders set quotas because they want to stimulate the performance of their salespeople. But quotas don't actually work like that. The very best salespeople hit their quota months before the end of the year, whereupon they do the sales equivalent of vanishing off home\u2014that is, they start to delay the closing of their deals so that they can \"bank\" them and ensure that they begin the next year with a head start. Sales goals actually degrade the performance of top salespeople\u2014they function, as they do for New York City cab drivers, as a ceiling on performance, not a catalyst for more of it.\n\nBut what about salespeople who are struggling, or middle-of-the-road? Won't goals serve to stretch them upward toward their quota, in much the same way as our friend's marathon goal will help to stretch her upward toward greater endurance? Well, again, not exactly. In reality, what happens to middling or struggling salespeople is that their imposed quota increases the pressure on them. And this is not the self-imposed pressure that comes from attempting to achieve something we feel is important\u2014the sort our marathon-training friend will feel on a Sunday morning when she forces herself to get up and go running. No, this pressure to achieve company-imposed goals is coercion, and coercion is a cousin to fear. In the worst cases, fear-fueled employees push and push and, falling short, resort to inappropriate and sometimes illegal tactics in order to meet their goals.\n\nThis is what happened at Wells Fargo with its cross-selling goals for each branch: if someone came in to open up a checking account, the Wells Fargo personal banker was also supposed to sell them a savings account, a credit card, a demand deposit account, and a loan. But having these goals didn't lead to more cross-selling, or at least, not just that. Instead, it led to the creation of more than 3.5 million fake accounts.\n\nNone of which is to say that sales quotas are useless. In fact, they can be an excellent forecasting device. Senior leaders can use them to estimate what the company's top line is going to be for any given period, and then announce this to the board and the investment community so that all interested parties can get a sense of the expected revenues, against which costs, investments, and ultimately cash flow can be assessed. The best executives are good guesstimators\u2014they have a sense, born of long experience, of what the median quota should be, the \"line of best fit\" around which the variation of salespeople's performance will cluster. Some will outperform their quota by 10 percent, others will fall short by 10 percent, and thus at year's end the sales goals, when guessed well, will be hit.\n\nBut these sales goals don't beget more sales; they just anticipate what the sales will be. Sales goals are for performance _prediction_ , not performance creation.\n\nHow about tracking performance\u2014do goals allow companies to do that? Hardly. Even though so many companies ask employees to write down their yearly goals and track their progress using some sort of software; even though books like _The Progress Principle_ by Teresa Amabile and Steven Kramer say that humans love to track their progress and that they derive joy from each achievement; and even though, in the last few years, we have seen more goal tracking and not less; none of this tracking does what it is intended to, for the simple reason that your progress toward a goal is not linear.\n\nTake our marathon-running friend. If, at the end of February, she calculates that her training regimen is 62 percent complete, does that mean that she has only 38 percent of her marathon goal left to go? Obviously not: she has 100 percent left because she hasn't yet started her actual marathon. And what happens when she does indeed run her race? When she has finished her first thirteen miles does that mean she is now thirteen out of twenty-six miles, or 50 percent of the way toward completing the race? Again, no. As every marathon runner discovers, the first half of the marathon is the comparatively easy part. It's the last half\u2014in particular, the last six miles\u2014that's brutal. Only when you pass the twenty-mile mark do you begin to feel the legs harden and the mind weaken; only then do you know whether you have the physical and mental strength to complete your goal. And what percentage of the whole does the refining fire of the last six miles represent\u201440 percent? 60 percent? 90 percent? It's impossible to put an accurate number on it because, in truth, the first twenty miles of a marathon are one thing, and the last six miles a very different thing.\n\nSo our friend can't be 62 percent done with her marathon preparation, nor she can be 50 percent done with her actual marathon. She can only either complete the goal or not complete it. All goals, at least in the real world, function in this same way. You are either done, or you are not done: goal attainment is binary. You might want to set some intermediate goals along the way, and tick these goals off as they are done (or not done). But you won't ever be able to assign a \"percent complete\" to your bigger goal as you tick off these mini-goals. And if you attempt to, or if your company asks you to, you will only be generating falsely precise data about the state of your progress.\n\nFinally, what about evaluating employees? Can we evaluate a person based on how many goals he or she has achieved? Many companies do, for sure. But here's the snag: unless we can standardize the difficulty of each person's goals it's impossible to objectively judge the relative performance of each employee.\n\nLet's say we have two employees we're evaluating, Victoria and Albert. Each is aiming to complete five goals, and at year's end Victoria has achieved three goals and Albert has achieved five. Does that mean Albert is a higher performer? Not necessarily. Maybe one of Victoria's five goals was \"Govern an empire\" and one of Albert's five goals was \"Make a cup of tea.\" For us to use goal attainment to evaluate Victoria and Albert, we need to be able to perfectly calibrate each and every goal for difficulty\u2014we need each manager, with perfect consistency, to be able to weigh the stretchiness or slackness of a given goal in exactly the same way as every other manager.* And as it happens this sort of calibration is a practical impossibility, so we can't. Sorry, Albert.\n\n\u2022 \u2022 \u2022\n\nDespite this evidence, however, it remains true that goals, and cascaded goals in particular, have an intuitive appeal to many leaders who find themselves in search of ways to ensure efficient and aligned execution in their organizations. And, at the same time, it also remains true that for those of us in the trenches, our experience of goals feels nonintuitive, mechanical, fake, even demeaning. Why is that?\n\nWell, in the real world, this is what's going on. Firstly, and oddly, when you sit down to write your goals, you already have a pretty good idea of the work that you're about to do. After all, it's not as though you roll up to the office on a Monday morning desperately trying to figure out how you're going to fill the time. So what the goal-setting process is asking you to do is to write down work that you already know you're going to do. Your work goals aren't out ahead of you, pulling you along like our marathoner's goal; instead they're just behind you, being tugged along by your own preexisting understanding of the work you're going to do anyway.\n\nThe goal categories\u2014strategic, operational, innovation, people, and so on\u2014are odd simply because work doesn't come in categories. You don't plan your time by thinking, \"Well, on Tuesday I'll do some operational, and hopefully make time for a bit of innovation on Thursday afternoon.\" Work usually comes in projects, with deadlines and deliverables, and so when you're asked to translate it back into category goals, you (and most every other employee) fudge it and force-fit your work to the categories, while hoping no one will mind too much.\n\nAnd while it's not unreasonable to hope that the work you do matches up to what your team leader wants you to do, setting goals that are a subset of his goals, or reviewing your goals against his, is actually a pretty strange way of going about this. Your team leader already knows what you're doing, because in the real world you talk to him about it, all the time. If you're off working on origami and he'd rather you were working on quilting, he'll tell you. And when something changes, a few days later, and he needs you to shift your focus over to glass-blowing, again, he'll just tell you. Even if he doesn't tell you, and you continue to potter away at something that's all of a sudden out of whack, the very last thing he'd think of doing to communicate this to you is to go back into your goal form, change your goals, and hope you'll notice. Again, cascaded goals are tagging along behind the work, not out ahead of it: as used in the real world, goal setting is more a system of record keeping than a system of work making.\n\nThen there's the fact that you don't go and look at your goals once you've set them. If they were supposed to be guiding your work, you'd think you might.\n\nAnd what about the gritty point of it all, at year's end, when you're supposed to self-evaluate against your goals? While your boss may imagine that you're engaging in honest and earnest reflection on the year gone by, you're probably trying to find the elusive sweet spot between, on the one hand, saying that you hit all your goals out of the park, by which you'd risk seeming arrogant or deluded, and, on the other, acknowledging that some things didn't go as planned, by which you'd risk giving your boss\u2014or some unseen higher-up\u2014an excuse to decrease your bonus. Self-evaluation of goals isn't really about evaluating your work, in other words: it's a careful exercise in self-promotion and political positioning, in figuring out how much to reveal honestly and how much to couch carefully.\n\nThis is no comment on you, by the way. Carefully calibrating your self-evaluation to find this sweet spot is a practical response to a bizarre situation. The company has asked you to evaluate yourself against a list of abstract goals that were irrelevant a couple of weeks after you wrote them down. You're being asked to do something meaningless and pretend it's meaningful. It's enough to make you a little crazy.\n\nAnd your team leader's in on the crazy. When the end of the year comes around and she has to sit down with a stack of goal forms and write\u2014under each goal you typed in months and months ago\u2014one or two little sentences describing how you've done against each one, what must be going through her mind? More than likely it isn't related to you or how she thinks you've done, but is more about how quickly she can get through the stack and cross \"goal review\" off her to-do list. Like you, she's got a nagging feeling that she's wasting her time\u2014because what's in front of her now is a random subset of things you thought you might be doing a while ago, shoe-horned into whatever categories you thought you could get away with at the time, written so as to look maximally impressive for anyone reading the form, and now garnished with your delicately positioned self-evaluation. She knows that the work changed an equally long time ago and has very little to do with what's on the form, and that she's already told you how well you did on the work that actually happened, by talking to you about it as the year went along. To her this form filling is the worst kind of administrivia-masquerading-as-management, so she writes the little sentences and hopes that no one will complain if they're shorter than last year's.\n\nIn the real world, there is work\u2014stuff that you have to get done. In theory world, there are goals.\n\nWork is ahead of you; goals are behind you\u2014they're your rear-view mirror.\n\nWork is specific and detailed; goals are abstract.\n\nWork changes fast; goals change slowly, or not at all.\n\nWork makes you feel like you have agency; goals make you feel like a cog in a machine.\n\nWork makes you feel trusted; goals make you feel _dis_ trusted.\n\nWork is work; goals aren't.\n\nBut it doesn't have to be this way. Goals can be a force for good.\n\n\u2022 \u2022 \u2022\n\nLook again at our soon-to-be-marathoner friend: she has taken something she deems valuable (fitness) and turned it into a tangible achievement (the marathon). She has made it real. This, ultimately, is what goals are for: to help you manifest your values. They are your best mechanism for taking what's inside of you and bringing it out where you and others can see it, and where you and they can benefit from it. Your goals define the dent you want to make in the world.\n\nAnd this in turn means that the only criterion for what makes a good goal is that the person working toward it must set it for him- or herself, _voluntarily_. The only way a goal has any use at all is if it comes out of you as an expression of what you deem valuable. It doesn't have to be SMART, or big, hairy, and audacious. It doesn't need to contain key performance indicators or be built from objectives and key results. If a goal is going to be useful, if it is going to help you contribute more, then the _only_ criterion is that you must set it for yourself, voluntarily. Any goal imposed upon you from above is an un-goal.\n\nThis doesn't mean, though, that there is nothing we should cascade in our organizations. Since goals, done properly, are only and always an expression of what a person finds most meaningful, then to create alignment in our company we should do everything we can to ensure that everyone in the company understands what matters most. And so the truth:\n\nThe best companies don't cascade goals; _the best companies cascade meaning_.\n\n\u2022 \u2022 \u2022\n\nThe research into the best teams gives us the first clue to this. When you have a measurement instrument like the eight questions we shared in chapter 1, one of the things you can do is something called factor analysis. This, in essence, tells you how many different sorts of things your questions are measuring\u2014how clumpy they are. In all the years of researching teams, company by company by company, we had only ever found one factor in our eight questions\u2014one clump of experience that all eight questions were illuminating for us\u2014and we called this clump \"engagement.\"\n\nBut when we ran the numbers at Cisco, something unexpected happened. First, two of the eight questions behaved differently from the other six in one slice of the analysis. We weren't sure what this meant, as it didn't show up anywhere else. But later on, we ran our factor analysis, and then, all at once, there it was: a second factor appeared.\n\nIt was made up of these two questions:\n\n1. I am really enthusiastic about the mission of my company.\n\n7. I have great confidence in my company's future.\n\nSo we began thinking of these as the \"company\" factor, and the remaining six questions as the \"team\" factor\u2014and these two factors _together_ as \"engagement.\"\n\nTo be clear, the \"company\" things\u2014excitement about the mission and confidence in the future\u2014still vary from a good team to a bad team, and still explain team performance. But at the same time, it appears that they may not originate from within the team, as the other six things\u2014safety and trust, a sense of excellence, challenging work, and so on\u2014clearly do, but instead originate from outside the team and then become amplified, or not, within the team. Put another way, while a team left to its own devices can take care of many of its own needs, it apparently can't create a sense of the broader mission and confidence in the future from out of thin air. So, in addition to giving our teams and their members a real-time understanding of what is happening in the world, we need to give them a sense of which hill we're trying to take. Instead of cascading goals, instead of cascading instructions for actions, we should cascade meaning and purpose.\n\nThe best leaders realize that their people are wise, that they do not need to be coerced into alignment through yearly goal setting. These leaders strive instead to bring to life for their people the meaning and purpose of their work, the missions and contributions and methods that truly matter. These leaders know that in a team infused with such meaning, each person will be smart enough and driven enough to set goals voluntarily that manifest that meaning. It is shared meaning that creates alignment, and this alignment is _emergent_ , not coerced. Whereas cascaded goals are a control mechanism, cascaded meaning is a release mechanism. It brings to life the context within which everyone works, but it leaves the locus of control\u2014for choosing, deciding, prioritizing, goal setting\u2014where it truly resides, and where understanding of the world and the ability to do something about it intersect: with the team member.\n\nOur prevailing assumption is that we need goals because our deficit at work is a deficit of aligned action. We're mistaken. What we face instead is a deficit of meaning, of a clear and detailed understanding of the purpose of our work, and of the values we should honor in deciding how to get it done. Our people don't need to be told what to do; they want to be told why.\n\n\u2022 \u2022 \u2022\n\nTo see what this looks like in practice, take a closer look at Mark Zuckerberg and Sheryl Sandberg of Facebook, and Truett Cathy of Chick-fil-A. Though separated in age, religious affiliation, geography, and company product, they all demonstrate a similar fixation with cascading meaning. Now, given the challenges that both Facebook and Chick-fil-A have faced lately, whether because of their use of their customers' data, in Facebook's case, or because of their position on gay rights, in Chick-fil-A's, you might ask why we chose these people as examples. Our reasoning is this: people are imperfect, and so if we are going to learn from real people, then, by definition, we are going to have to learn from imperfect people. It is up to us all to sort what is useful and valuable about these people and their companies, from what is not.\n\nTen years ago, in a post intended to clarify his company's mission, Zuckerberg wrote that Facebook's purpose was to make the world more connected. As we were writing this chapter, he added what he sees as meaningful nuance to this mission, saying:\n\n> . . . we're making a major change to how we build Facebook. I'm changing the goal I give our product teams from focusing on helping you find relevant content to helping you have more meaningful social interactions.\n\nYou may not see the distinction, but he does, and so today\u2014just as he's done every six months for the last ten years\u2014he's intent on announcing another distinction to the world and, more importantly, to all of the people who work for him. This is what Zuckerberg does. He takes his values so seriously that with each new insight of his, he tweaks and course corrects and tweaks and learns and tweaks again, and then with great weight and import he announces the tweak to the world.\n\nTo some people, these small shifts of focus and their accompanying announcements might seem portentous, exhibiting no more than the narcissism of minor differences, but to he and Sandberg they are part of a relentless effort to cascade to their teams what they truly value, with the implicit message that if you don't value what they value then you may well be on the wrong team. And indeed the constant iteration and \"improvement\" of this message is, in and of itself, part of that message, because Zuckerberg's and Sandberg's meaning in the world is not just to help people get more connected but also to do so in a way that confesses to being a work in progress. They've both made it very clear\u2014Sandberg in her book _Lean In_ , and Zuckerberg in his numerous blogs and congressional testimony\u2014that they don't necessarily know all the answers. They know what they want to build at Facebook, _and_ they know they don't always know how to get it, and neither do you, and neither does any of us. What they've told us is that each of them, and everyone else at Facebook, are constantly making it up as they go along, constantly experimenting.\n\nIf you join Facebook on Thursday, you're at onboarding boot camp on Friday, writing and honing code over the weekend, and potentially having that code shipped on Monday. Things move fast. Facebook's address, 1 Hacker Way, reinforces this ethos, and if the symbolism of that didn't register with you, there's always the massive \"The Hacker Company\" sign, bought from a strip mall in Florida, that now hangs proudly over the company's town square.\n\nThese types of things\u2014the signage, the street address\u2014are different from the cultural plumage we encountered in our first chapter, designed to lure you in. Rather, they exist to help us understand what we are working toward\u2014what our work is for, and what it means. Indeed, Facebook's entire campus seems to have been constructed to bring Zuckerberg and Sandberg's meaning to life. Though the exteriors of many of the buildings are Frank Gehry masterworks of fluid beauty and sustainable energy, the interiors scream \"temporary.\" It feels like the entire company just moved in yesterday, and might move out tomorrow: concrete floors, exposed AC ducts, a pile of keyboards sitting in one corner, handmade posters tacked up on the walls.\n\nWhen we were there a couple of years ago, we noticed that each conference-room door was made of glass and had a logo etched onto it\u2014and that this same logo stretched off into the distance, repeating on door after door, for the length of a football field. Which might not be particularly unusual if the logo\u2014there in Facebook's offices where Facebook's code gets written by Facebook's employees to go into Facebook's applications\u2014was actually Facebook's logo. But it wasn't: it was Sun Microsystems' logo.\n\n\"What's with the logos?\" we asked Facebook's head of facilities.\n\n\"Oh that,\" he said. \"Well that's because this used to be the Sun Microsystems building.\"\n\n\"Could you perhaps afford to buy new doors,\" we offered, \"with the Facebook logo on them?\"\n\n\"We could,\" he replied. \"But Mark and Sheryl made the decision to keep the logo on the doors because it reminds everyone that unless you make quick decisions, move fast, and figure out better solutions, you might go the way of Sun Microsystems.\"\n\nLook on the walls, and you'll see another Facebook oddity: posters. Physical, printed posters. Outside, on the conference-room walls, behind the reception desk; posters next to posters next to yet more posters. Each one is an announcement of something someone is passionate about, some hobby, event, or activity\u2014underwater skateboarding, Time's Up, Black Lives Matter, or the local tiddlywinks group. Why would something as old-economy as a poster proliferate at a high-tech digital-media company? It's all part of Facebook's stated mission to facilitate and reinforce real human connection. If you want people to connect with other people then you have to be curious about what each person is interested in and passionate about, and then find ways to surface and celebrate these passions. Just as we draw on our cave's wall, so we put up our posters. And in this way we learn about one another.\n\nWith all of these deliberate actions Zuckerberg and Sandberg have cascaded their meaning to their team of teams, and again, while we can quibble with the results, or worry that speed and connection at Facebook have been emphasized to the detriment of security and accuracy, we can still learn from the way in which speed and connection have, indeed, been emphasized.\n\nIf you love genuine human connection, Zuckerberg and Sandberg tell their people, you'll find meaning at Facebook.\n\nIf you love the idea that the future is a work in progress, you'll find meaning at Facebook.\n\nIf you love speed over beauty, you'll find meaning at Facebook.\n\nBut, if you want beauty\u2014carefully considered, precise, perfect beauty\u2014then Facebook is not for you. If you want to live in a world where things are either not yet begun or already perfected, but never in between, don't work here. Instead, go just a couple miles down the road and work for Apple. There, it does not look like they moved in yesterday and are about to move out tomorrow. There, it looks like an alien spaceship\u2014perfectly circular, complete, whole, and utterly finished\u2014has landed in downtown Cupertino, and welcomes in only those who are attracted to its beautiful closed system. If that's what gets you going, if that's where you find meaning, work there. Because Facebook will leave you cold.\n\n\u2022 \u2022 \u2022\n\nChick-fil-A is the most profitable and fastest-growing quick-service restaurant company in the world, which is perhaps surprising. Facebook's growth is surely tied to the power of network effects. Google's success can be traced to the monopolistic powers of its search algorithm. Amazon relied on its first-mover status and disdain of margins to secure its leadership position. But what's Chick-fil-A got? It's got a chicken sandwich, some waffle fries, and a shake, which, though all distinctly yummy, wouldn't on their face seem differentiated enough to explain the company's outlandish and continued success.\n\nWhat Chick-fil-A had was Truett Cathy, its founder, a man who was just as relentless, as precise, and as deliberate at bringing his meaning to life as Facebook's leaders have been.\n\nUnlike Facebook, where the work never seems to stop, Chick-fil-A isn't open on Sundays, despite the boost in sales and profit that the extra day would bring. Why? Because Cathy was a devout Christian who followed the Bible's injunction to reserve Sunday as a day of rest.\n\nThe closed-on-Sunday policy is perhaps the most obvious example of the way that Cathy cascaded meaning to his teams. A less well-known one is Chick-fil-A's franchise agreement. Your typical franchisee-franchisor agreement is a mechanism designed to leverage capital through the multiplier of brand\u2014the franchisor brings the brand, the franchisee the capital. The franchisor selects the franchisee based on how large and how stable a hunk of capital the franchisee can bring to the agreement, and the franchisee assesses the franchisor based on how powerful and attractive the brand is. The goal of the franchisor is to secure lots of capital; the goal of the franchisee to secure as many locations as possible. So, for example, Arcos Dorados Holdings Inc., McDonald's largest franchisee, has over $4.5 billion in sales from over two thousand locations.\n\nThe Chick-fil-A franchise agreement does not work this way. As a Chick-fil-A franchisee you cannot own two thousand locations, no matter how much capital you have. Instead, you are allowed to own one.* You can throw as much money at Chick-fil-A as you like, and it won't be swayed into giving you any more locations: the franchise agreement, unchanged since Cathy devised it in the mid-1950s, forbids it. At its founding, Cathy decided that the mission of his company was less to sell chicken than it was to build leaders in local communities. Some of us might scoff at this, but Cathy stayed true to it, and devised his franchise agreement accordingly. He reasoned that if he was to grow local leaders, he would have to ensure that each person he brought on as a franchisee had a good reason to stay close to their local community. The best way to do that, he thought, was to keep these leaders in their stores, and the best way to ensure that, in turn, was to allow them only one. If you have only one, he figured, then you will be in this store all the time, staying close to your guests and close to your team members, knowing intimately the concerns of each\u2014what the community is interested in and what it's worried about. And over time you will respond to these needs and take action, and therefore, over time you will grow as a community leader.\n\nGuided by the purity of his vision, he crafted this extraordinary franchise agreement and then selected his franchisees (or operators, as they are known in the company), not on the size of their capital but on their commitment to their community. And in case you're thinking to yourself that while this makes for a nice story (the sort of founding myth we're all used to hearing), it can't possibly be true in the second decade of the twenty-first century, then you should know that to this day, you need zero capital to become an operator, and yet such is the care with which Chick-fil-A selects these future community leaders that it is harder to become one than it is to get into Harvard.\n\nOver the years, Chick-fil-A has doubtless scared away countless billions of dollars of capital that could have been pressed into service to grow its brand, but what it has gained instead are tens of thousands of people who have embraced Cathy's meaning. These local leaders are the heroes of the organization. Every year, the company gathers them all together at an event called Seminar, where one of the highlights is a magazine-worthy spread of photos and stories and testimonials, all in celebration of the unique contribution that each operator has made in his or her community. At each Seminar the very best operators are brought up on stage, one-by-one, to have their story told.\n\n\u2022 \u2022 \u2022\n\nNone of this is to say that Mark Zuckerberg, Sheryl Sandberg, or Truett Cathy are exemplars of perfection\u2014they aren't, and we don't imagine any of them would have claimed to be. But if you want to create alignment in your team, or your company, then you can learn from the way in which each of them deliberately, relentlessly, precisely, and pervasively cascaded meaning.\n\nTo be specific, here are the three levers they used to such great effect.\n\nThe first is _expressed values_ : what you write on the walls. We don't mean that you should literally write out your \"values.\" Many leaders and many companies set about doing this and wind up with a list of generic values such as _integrity_ , _innovation_ , or, God forbid, _teamwork_ \u2014which are about as meaningful as Muzak\u2014and then wonder why the whole exercise doesn't seem to have made much difference. Instead, apply some creativity to how you want to bring your meaning to life for your people. Don't _tell_ them what you value, _show_ them. What do you actually want them to see and to bump into at work? Facebook's Sun Microsystems logos, its love of posters, and the \"Hacker Company\" signage are all vivid examples of this.\n\nWhat are your expressed values? What have you written on your walls? What do your people encounter when they walk in through the door. What do they see when they turn to the left? And what do those things tell them about who you are?\n\nA second way to cascade meaning is through _rituals_. Facebook has their famous bimonthly hack-a-thon; Chick-fil-A stops work on Sundays. Sam Walton, founder of Walmart and Sam's Club, had a ritual he practiced every single Friday until he was physically unable to do it anymore: he would pick a store, move the merchandise around on a particular end-cap display, and come back on Saturday to see what had sold. It was his own version of QMI, or quick market intelligence, and what it signaled, to his employees, was his deep belief that no one, not even the boss, knows the brain of the customer better than the customer.\n\nYou already have rituals, whether they are conscious or unconscious, and these rituals\u2014the things you do repeatedly\u2014communicate to your people what is meaningful to you. If we followed you around for a week, we'd see them. Let's say you have a meeting: What time do you show up? Are you five minutes early, or five minutes late? What are you wearing? Do you catch up with your team members about their personal lives or do you launch right into business? Who talks first? Do you allow your team members to speak, or do you cut them off? Does the meeting go long? Do you hold people back to finish things up?\n\nThese are all aspects of your rituals, and we, your team, see them, make sense of them, and draw our conclusions\u2014whether you want us to or not. The question, then, isn't whether you have rituals or not. The question is whether or not you are deliberate about what your rituals communicate.\n\nTo see the power of a ritual to vividly communicate a leader's meaning, juxtapose Facebook's rituals with Steve Jobs's. At the end of every week, either Zuckerberg or Sandberg goes to Facebook's largest cafeteria and holds an all-hands meeting during which any employee can ask any question that he or she wants to, and the two top leaders commit to answering each question as best they can. The purpose of these meetings is contained not so much in the actual substance of the answers as it is in the reinforcement that Facebook values transparency and openness so much that they will dedicate a significant chunk of top leaders' time to it each week.\n\nJobs, by contrast, valued aesthetic beauty far higher than openness, and so his all-hands meetings, which he held every three months or so, and which the rest of the world mistook for product launches, looked very different. At each \"launch\" he would describe in precise detail the beautiful design of each product, or the intricate ecosystem of hardware and software, or the exquisite integration of content and code, and while we, the consumers, were oohing and ahing at the new products, the real audience\u2014Apple employees\u2014were watching and taking note. They could see their leader extolling the virtues of aesthetics, of beauty for its own sake, of the joy in refined creation, and they would lean into that shared meaning.\n\nOr they would lean out and go work for Facebook. Either way, the product launch had served its true purpose: to cascade Jobs's meaning across his teams.\n\nThe third lever is _stories_. Chick-fil-A makes an art of its storytelling through the operator profiles during Seminar. The company dedicates time to going out to each operator's store, taking photos, and learning about his or her family and community, precisely so it can share these stories with the rest of the company.\n\nMany of the best leaders are storytellers, not in the sense of writing a novel or a screenplay, but because they cascade meaning through vignettes, anecdotes, or stories told at meetings, on email chains, or on phone calls. They are always telling these little stories, because the stories that they choose to tell convey what they value. Stories make sense of the world: they are meaning, made human. That's why religions tell stories about their messiah and the creation of the earth, and include parables within those stories that help us learn what is meaningful. And that's why you can tell a lot about what matters to a team by the stories that the team members tell themselves.\n\nFor example, on a much larger scale, if you've spent any sort of time in Great Britain you'll know that there are three battles that the Brits can't stop talking about: the Charge of the Light Brigade in the Crimean War, and the Battle of Britain and the evacuation from Dunkirk in the Second World War. There's nothing odd, of course, about a country repeating the tales of its battles long after they've been fought. What is odd, however, is that Britain didn't win any of these battles. The Charge of the Light Brigade was a disaster, and the Battle of Britain and the Dunkirk evacuation were more about avoiding defeat than securing victory. Why, then, do we keep talking about them?\n\nBecause they define what we Brits think is most meaningful about us: that we never give up, and we never give in. We value determination and grit _more_ than we value winning, so we tell story upon story of keep-going-ness that usually ends up on just the wrong side of victory.* In so doing we create shared meaning.\n\nYou tell stories, whether you know it or not, and you're telling them all the time, in every conversation and at every meeting. What stories are you telling, and what do they say about what you find meaningful?\n\n\u2022 \u2022 \u2022\n\nAs a leader, you are trying to unlock the judgment, the choices, the insight, and the creativity of your people. But, as we've seen in the last two chapters, the way we go about this doesn't make much sense. We cloister information in our planning systems, and we cascade directives in our goal-setting systems. Instead, we should unlock information through intelligence systems, and cascade meaning through our expressed values, rituals, and stories. We should let our people know what's going on in the world, and which hill we're trying to take, and then we should trust them to figure out how to make a contribution. They will invariably make better and more authentic decisions than those derived from any planning system that cascades goals from on high.\n\nFor Ethan Floquet\u2014or, more accurately, for his mother\u2014his cascaded goals became an ever increasing burden with each passing year. Ethan has autism, and every year since his early childhood his mother has been asked to write an individualized education plan, or IEP, setting out her and her husband's goals for Ethan for the upcoming year\u2014their aspirations for their son\u2014to aid and guide his educators and therapists. But as time went by, their goals diminished. It became clear that Ethan would never live independently, or hold down a job without help, or marry. His goals became smaller and smaller; the annual meeting to discuss his development became more and more somber; and each year's IEP became shorter than the preceding year's. It ultimately shrank to a single sentence, at a time when Ethan had found a farm program that seemed to offer some promise (\"We hope that Ethan can remain with this program for the year\"), before vanishing altogether the following year. Ethan's mother had been too busy, and the task of once again revisiting all his shortcomings\u2014for the eighteenth year in a row\u2014too painful.\n\nBut that year, unbeknownst to his mother, Ethan had written his own goals. He had written them, of course, in ignorance of his cascaded goals, and had instead focused on something else. Here they are, in full:\n\n> After I graduate from high school I plan to work at Prospect Meadow Farm until I retire and live at home with my family as long as I can. I'd like to keep taking classes at Berkshire Hills Music Academy. For fun, I want to play Special Olympics basketball, go to our cabin in Vermont and the shore in New Jersey, mow lawns, and collect business cards. My goals for the future are to take the PVTA bus into town to make purchases, and someday learn how to drive a zero-turn lawn mower.\n\nGoals set by others imprison us. In creating his own, Ethan had found freedom.\n\n*A condition known more properly as inter-rater reliability, and another finicky-but-importantly-true thing that, as we'll see later, explains in part why 360-degree assessments don't work and why performance ratings are such a problem.\n\n*In exceptional circumstances, Chick-fil-A will allow you a second freestanding location, if your first is in a mall. But 95% of their operators have only one.\n\n*Here's another one: Our most famous explorer is not Ernest Shackleton, who successfully rescued his team after they became stranded in the ice on an expedition to Antarctica. Neither is it James Cook, who was the first European to discover Australia. No, it's Robert Falcon Scott, who lost the race to the South Pole to Roald Amundsen and was therefore the _second_ person to reach it, and who died of hypothermia and starvation on the way back, persisting to the very end. He's the one we talk about most.\n\n#\n\n# The best people are well-rounded\n\nWatch Lionel Messi dribble. Go on YouTube, type in \"Best Messi Dribbles,\" click on any of the clips that come up\u2014there will be hundreds to choose from; any one will do\u2014and you will see a small man with magical feet, running at what seems like double time past one defender after another until he gets into the penalty area and shoots. If you are a soccer fan, you will have seen him do this countless times already, but if you aren't, it's worth taking a moment to watch this man. All of us who are interested in excellence can benefit from studying preternatural ability in action: we can wonder at what caused it, analyze the technique of it, and dissect the steps involved, or we can simply revel in the fluid mastery of it and try to imagine where in our lives we, too, can experience flow such as this.\n\nLionel Messi is from the port city of Rosario in Argentina. He was always a speedy kid. In videos his mom took of his first soccer games, you see him sprint past one opponent after another, as though the outsize ball is pulling him along on a string. Such a prodigy was he that, from across the Atlantic, scouts from F.C. Barcelona came calling, and at the tender age of thirteen Messi was spirited away from home to La Masia, \"The Farmhouse,\" Barcelona's legendary youth academy. There, his little body refused to grow, so they gave him growth hormones and waited for the size of his frame to catch up with the size of his talent. It never did: he topped out at five feet seven, and stayed as skinny as a kid playing on the streets of Buenos Aires' _villa miseria_. But somehow this didn't seem to matter. His gift was so extraordinary\u2014the ball looking as if it were magnetically attached to his boots, no matter his speed or his jackrabbit changes in direction\u2014that it rendered irrelevant his lack of size and stature. He joined Barcelona's first team at the age of seventeen and since then has proven himself the best soccer player in the world, and in many people's eyes, the greatest of all time. Watch him now, and watch carefully, because we may never see his like again.\n\nAlthough any of the YouTube videos could serve as a highlight reel, the one that best illuminates his gifts shows a goal that Messi scored against Athletic Bilbao in the final of the 2015 Copa del Rey. It's worth playing it out in detail because, while so much of what he does in just a few seconds is astonishing (not least the cannon of a shot he unleashes at the end of the run), this clip reveals something about him that is at once truly bizarre and the very foundation of his genius.\n\nHe receives a pass on the halfway line, and for a moment stands perfectly still with the ball at his feet, one defender in front of him and the rest of the opposing players staking out their positions between him and the goal. Then, as though seized by a sudden thought, he darts to his left, jukes back to his right, leaves the nearest defender flat-footed, and takes off down the sideline. Three other opponents converge on him, trying to squeeze him into the corner and away from the goal. He slows for a second, dips his shoulder right, accelerates left, touches the ball past the legs of one defender, and then at once is free of all three of them and bursting toward the penalty area. Two more Bilbao players sprint over to cover him, but somehow he ghosts past this new threat, his legs on fast-forward, the ball now rolling out to his left, perfectly positioned for a strike on goal. He shoots. He scores. The Barcelona players celebrate with him as only soccer players can, and as he walks back toward the halfway line for the restart, even the Bilbao supporters clap in admiration. The greatest ever.\n\nWatch this clip over and over and you find many things to marvel at: his zero-to-top-speed quickness, his innate sense of the pitch and of the most dangerous angle to take toward goal, his counterintuitive decision to shoot for the near post. But by far the most remarkable discovery awaiting you is that, as he races from the halfway line through seven defenders and into the penalty area, he uses only one foot. Count the touches from the start of his run to his actual shot on goal and you realize that of the nineteen times he kicks the ball, only two of them are with his right foot. Everything else he does during the dribble, including the shot at the end of it, is done with his left.\n\nClick on other clips, watch other dribbles in the magnificent Messi oeuvre, and you discover that this is always the case. His ratio of strong-foot to weak-foot usage stays constant at around 10 to 1. For the sake of comparison, the right-footed Cristiano Ronaldo's ratio is around 4.5 to 1. Messi is not just a left-footed player, in other words. He is a player who does virtually everything that needs to be done with a ball with only his left foot. Passing, dribbling, shooting, tackling\u2014all of it.\n\nMessi's left-footedness, then, is truly extreme. And, of course, everyone on the opposing team is acutely aware of this, yet even with the foreknowledge that he is going to play with his left foot time after time after time, they are still bamboozled as he swerves around them. Messi has taken his natural left-footedness and cultivated it to such an extent\u2014developed it to work with such speed and precision\u2014that instead of being a limitation, it bestows on him a consistent, dramatic, and unfair advantage.\n\nWe get the sense, watching him, that acquiring this advantage was no rational calculation on his part. For sure, he must have practiced ten thousand hours and more, but what he conveys as he swerves and skips toward goal is not diligence and discipline, but joy: pure, unconscious, unstoppable joy in his craft. To see him run with the ball at his feet is to see the fullest and best and most authentic expression of this man. It delights us, and lifts us up, as it always does when we see someone shine as only he or she can shine. And just like the opposing supporters, we look in wonder at this little man, clap our hands, and smile.\n\n\u2022 \u2022 \u2022\n\nMessi plies his trade on the world's largest sporting stages, but you may have experienced similar admiration for colleagues at work. One of them puts together a presentation and delivers it with wit and clarity, and you smile. Another handles a grumpy customer with just the right mix of empathy and practicality, and you marvel at how easy she made it look. Another defuses a complex political situation, and you look at him in awe and wonder how on earth he did it. As humans we are wired to find joy in seeing someone else's talents in action. We resonate with the naturalness, the fluidity, and the honesty of a thing done brilliantly well, and it attracts us and draws us in.\n\nYou will have recognized the Messi joy when it is your _own_ performance that you're experiencing, too\u2014that is, when you are expressing your own strengths. This sensation is not, at root, created by how good you are at something. Rather, it's created by how that activity makes you feel. A _strength_ , properly defined, is not \"something you are good at.\" You will have many activities or skills that, by dint of your intelligence, your sense of responsibility, or your disciplined practice, you are quite good at, and that nonetheless bore you, or leave you cold, or even drain you. \"Something you are good at\" is not a strength; it is an ability. And, yes, you will be able to demonstrate high ability\u2014albeit briefly\u2014at quite a few things that bring you no joy whatsoever.\n\nA _strength_ , on the other hand, is an \"activity that makes you feel strong.\" This sort of activity possesses for you certain definable qualities. Before you do it, you find yourself actively looking forward to doing it. While you are doing it, time seems to speed up, one moment blurring into the next. And after you've done it, while you may be tired and not quite ready to suit up and tackle it again, you nonetheless feel filled up, proud. It is this combination of three distinct feelings\u2014positive anticipation beforehand, flow during, and fulfillment afterward\u2014that makes a certain activity a strength. And it is this combination of feelings that produces in you the yearning to do the activity again and again, to practice it over and over, to thrill to the chance to do it just one more time. A strength is far more _appetite_ than ability, and indeed it is the appetite ingredient that feeds the desire to keep working on it and that, in the end, produces the skill improvement necessary for excellent performance.\n\nOf course, it's possible there are a few activities in which you seem to have boatloads of appetite and very little natural ability. Florence Foster Jenkins was, according to one historian, \"the world's worst opera singer. No one before or since has liberated herself quite so completely from the shackles of musical notation.\" The songwriter Cole Porter used to have to bang his cane repeatedly against his leg in order to stop himself from laughing out loud at the unremitting awfulness of her voice. And yet she loved to sing, and even managed to buy her way onto the stage at Carnegie Hall.\n\nLook closer at Lady Florence, however, or at anyone who appears to love an activity in which his or her performance is woeful, and you discover that often what such a person loves isn't the activity itself but instead the trappings of the activity. In Lady Florence's case it was most probably the attention given to a public performer: she had been a successful pianist in childhood, even performing at the White House, until an injury curtailed her piano playing and she had to find another way to get onstage. Other times, you see someone become addicted to brief moments of greatness in a sea of otherwise mediocre performances, and what keeps pulling that person back to the activity is the endless quest to re-create those flashes of brilliance. Anyone who has ever hit a perfect seven iron on the golf course, and then spent years toiling to recapture that one moment, will know what we mean. In any case, it appears that as a rule we humans are congenitally incapable of deeply loving an activity in which we are awful.\n\nInstead, we are drawn to activities in which we find joy. We can't always explain why, but some activities seem to contain ingredients that breathe life into us, that lift us up out of ourselves to reveal something finer, more resilient, and more creative. Each of us is different, of course, so each of us finds this joy in different activities, yet each of us knows this feeling. And when our work does indeed bring us this joyful ingredient, when we do indeed feel love, even, for what we do, then we are truly magnificent. Stevie Wonder, who clearly knows a thing or two about cultivating and contributing one's strengths to the world, said it best: \"You will never feel proud of your work if you find no joy within it. Your best work is always joyful work.\"\n\nThis is what work does to Stevie Wonder when he composes and sings\u2014he finds joy. This is what work does to Lionel Messi when he dances round defenders and finds the net from impossible angles\u2014he finds delight. This is what we see when we see anyone who is really good at their work\u2014we see someone who has found love in what they do. And this is what your company hopes your work will do for you. When your leaders say they want you to be creative and innovative and collaborative and resilient and intuitive and productive, what they are really saying is, \"We want you to fill your working hours with activities that bring you joy, with tasks that delight you.\"\n\nOddly\u2014and sadly\u2014this set of observations is often dismissed in business circles, perhaps because business is meant to be about rigor and objectivity and competitive advantage, next to which the idea of looking for joy in work, as a precursor to excellence in work, seems rather soft. Fixing shortcomings, no matter how hard that might be, seems like the hard-boiled business of business; finding delight is the province of poets.\n\nYet the data does not lie. Of the eight conditions that are the signature of the highest-performing teams, there is one in particular that stands out\u2014in study after study, irrespective of industry and irrespective of nationality\u2014as the single most powerful predictor of a team's productivity. It is each team member's sense that \"I have the chance to use my strengths every day at work.\" No matter what kind of work your team is doing and no matter which part of the world you're working in, your team will always be most productive when more team members feel delight and joy in what they do every day. Now, when we remove the words \"every day\" from the question, reasoning that perhaps the quotidian frequency is too high and that perhaps we should ask only whether people get a chance to feel a \"good fit\" between their strengths and their job, the item ceases to work\u2014that is, the link between those who strongly agree to it and the performance of the team vanishes. The \"every-day-ness\" of the feeling that your work plays to your strengths is a vital condition of high performance. Somehow, on the best teams, the team leader is able not only to identify the strengths of each person but also to tweak roles and responsibilities so that team members, individually, feel that their work calls upon them to exercise their strengths on a daily basis. When a team leader does this, everything else\u2014recognition, sense of mission, clarity of expectations\u2014works better. But when a team leader doesn't, nothing else that he or she tries, whether in the form of money or title or cheerleading or cajoling, can make up for it.\n\nOngoing work-strengths fit is the _master lever_ for high-performance teams: pull it, and everything else is elevated; fail to pull it, and everything else is diminished.\n\nNothing thus far should be particularly surprising. We've all seen people like Lionel Messi demonstrate their brilliance, and been uplifted by the sight. We've watched colleagues excel, and we've felt happy wonder in their success. We've experienced the joy of being at one with an activity, and the pride of knowing everything we've been able to contribute through our unique combination of strengths. Even the data shouldn't be particularly shocking\u2014of course the best teams are built around the optimal fit between strengths and roles. For any of us with much experience of the world, there is (or at least there should be) no earth-shattering revelation here.\n\nWhich makes it all the more surprising (or frustrating, or depressing) that companies are not, in fact, built to help us pinpoint and then contribute our unique strengths. In their systems and processes and technologies, in their rituals and language and philosophies, they evidence exactly the opposite design: to measure us against a standardized model, and then badger us to become as similar to this model as possible. They are built, that is, around the lie that _the best people are well-rounded_.\n\n\u2022 \u2022 \u2022\n\nAt some point in your career, if you haven't already done so, you will bump into a thing called a competency model. A competency is a quality you are supposed to possess in order to excel in your job. They look like this: strategic thinking, goal orientation, political savvy, business acumen, customer focus, and so on. The idea behind them is that excellent performance in a job can be defined in terms of the right grouping of competencies. Thus the company's top leaders will be asked to examine a long list of these competencies\u2014there are literally thousands to choose from\u2014and then pick the ones that everyone agrees each incumbent in each job should possess. One widely used model, for example, identifies five categories of competencies (core, leadership\/management\/business\/interpersonal, job functional, job technical, and technical-task specific) and then a further list of competencies within each of these, so that \"core,\" in this case, includes 22 leadership competencies, 18 management competencies, 45 business competencies, and 33 individual competencies, for a total of 118. Entry-level jobs are assigned fewer or simpler competencies, and the further up the hierarchy a job is, the more numerous and the more complex the competencies assigned to it tend to become. Having defined competencies for each role, the leaders will also usually define a desired proficiency level for each competency on a scale of 1 to 5, so that they can say, for instance, that such-and-such a job requires strategic thinking at a proficiency level of 3, whereas it needs customer focus at a proficiency level of 5.\n\nThis entire construct\u2014the chosen competencies and their required proficiency levels, for each seniority level, for each job, across some or all of an organization\u2014is called a competency model. In a typical model, a given job might be defined to require a few dozen competencies at varying proficiency levels.\n\nSo far, this might seem unobjectionable, if a little unwieldy: a group of leaders getting together to define what they feel the ideal employee should look like. It might not be our first choice for how they should spend their time, but at least no one has been harmed in the making of this model. It's what happens next, however, that leads us into choppier waters, because once created, the competencies show up everywhere. Your manager and your peers will rate you on them, and your overall performance rating will be derived in large part from how much of each of them you possess. During annual talent reviews, the competencies will be the language used to describe your performance and potential: if the consensus is that you possess them all, you will be considered for promotion, or paid more, or selected for plum assignments; whereas if you do not possess them, or display gaps in a few of them, you will be told to take the relevant training programs, and work on proving to your company that you have plugged your gaps. These competencies will become the lens through which your company sees you, understands you, and values you.\n\nAll of the major Human Capital Management tools\u2014the enterprise software systems that companies use to keep information about you, pay you, allocate benefits to you, promote you, develop you, and deploy you\u2014are built around competency models, and how closely you and your colleagues match up to the models. There is even, in one of these platforms, a robot that takes over the mundane chore of providing written feedback for team members: The team leader first selects which competency to evaluate, from the required list for the team member; then picks from a list of behaviors that someone is supposed to exhibit if they are either failing at, meeting, or exceeding this competency; then watches as the system generates a sample sentence to convey this feedback; then is given the opportunity to adjust this sentence to sound more or less positive, using buttons that adjust the feedback in one direction or the other; and then clicks a final button to add the finished sentence to someone's feedback form\u2014all without typing a single word. The robot produces such eye-glazing prose as, \"Barbara . . . puts thought into her budget requests and reviews her costs throughout the year to identify appropriate adjustments,\" these insights being produced with a few speedy clicks, and in complete ignorance of (and apparent indifference to) whether Barbara actually does any of those things.\n\nWhat concerns us here, however, is not so much the soul-crushingly automated implementation of competency models, but rather the theory of work that they embody and that underlies so much of what we do in organizations today. The theory goes something like this: we live in a world of machines, code, and processes, and when these break, we have to identify the faulty component or line of code or process step and fix it\u2014to take dysfunction and repair it. The first part of this competency theory of work, then, extends this thinking to performance. Once we've located you on our proficiency scales, we tell you that your lowest scores\u2014those where you are most \"broken\"\u2014are your \"development areas,\" and that the best path to greater performance will come from unrelenting focus on these areas.\n\nThe second part of our theory takes this line of thinking to its logical conclusion: we reason that if improvement in performance comes from remedying shortcomings, then high performance\u2014excellence\u2014must be the result of having removed shortcomings across the board, from having a high score on every scale. Excellence, in other words, is a synonym for all-round high ability: well-rounded people are better.\n\nThis is the lie that underpins the tyranny of competencies, and it is persistent and pervasive. But to see the truth, we need only to understand two particular facts.\n\nFirst, _competencies are impossible to measure_. Take \"strategic thinking\" as an example. Is this a state, something that is variable and subject to flux? Or is it a trait, something that is inherent and relatively stable over time? In the field of psychometrics we measure these two phenomena quite differently.\n\nWhen we are measuring states, we either devise surveys that ask a person about his or her state of mind, or we create tests with right and wrong answers to determine whether a person has acquired the necessary knowledge. A person's voting preference is a _state_ , as expressed in a survey. We presume it can change, such that when we ask a person about it at Time 1, and then give her new information, we expect her preference might well be different at Time 2. Mood is a _state_. Although it does appear that each of us has a unique happiness set point, we assume that a person's mood can change around that point, such that when we ask about it at Time 1, and then a change in situation or circumstance occurs, we may well observe a difference in the person's mood by Time 2. Similarly, skills and knowledge are _states_. If we test you on a certain skill or knowledge base at Time 1, and then give you more training in these areas, it's likely that you will get more of the answers right at Time 2.\n\nThese are all _states_ , and we expect them as such to change over time.\n\n_Traits,_ on the other hand, are inherent in a person. Extroversion is a trait, for example, as is empathy, and competitiveness, and need for structure. Each of us possesses certain unique predispositions and recurring patterns of thought, feeling, and behavior, and the overwhelming evidence is that, while each of us can learn over time to be more intelligent and effective at contributing through these patterns, the patterns themselves persist throughout our lives.\n\nTraits cannot be measured with a survey or a skills test. Instead, they have to be measured using a reliable and validated personality assessment. The two most prevalent kinds of personality assessments are _self-assessments_ (involving a number of carefully worded statements measured on a strongly agree\u2013to\u2013strongly disagree scale) and _situational judgment tests_ (involving a number of situations with a list of possible response options from which the test taker selects the one that fits her best).*\n\nBefore you set about measuring something you have to decide which of these\u2014states or traits\u2014you are trying to measure, so that you can properly select your measurement method.\n\nHere's the point. Seen in this light, what is a competency such as \"strategic thinking\"? Is it a state or a trait? We need to know, if we want to measure it\u2014and the entire purported purpose of competencies is to measure something. If we think a competency is the former, a state, then we should measure it either with a survey asking about the person's state of mind or with an actual test that has correct and incorrect answers. We should never ask your manager or your peers to rate you on it, because they can't possibly know how much of this abstract quality you possess, any more than they can accurately divine your voting preferences or the score you would get on a test. And if we think a competency is the latter, an inherent trait, then we should use a personality assessment to measure it, and we should never tell you to take a \"strategic thinking\" class so that you can improve in it, because if it's a trait, then, by definition, it probably won't change much.\n\nBut the truth about competencies such as strategic thinking, political savvy, or any of the others is that they are a haphazard mix-up of states and traits. We don't know whether goal orientation, say, derives from the way you are wired, or from what you have learned to do, or from what you have been told to do. We don't know whether customer focus is a different piece of your wiring, or a different skill you have learned, or the same skill used differently, or something else entirely. A scientific approach to performance would start with what is measurable, and only then study how those things contributed to performance. But competencies are built in the other direction. They start with a listing of every quality we can think of that we feel is important to performance, and only then ask how each one might be measured. By this point it is too late to disentangle the states from the traits, so we fall back on rating one another on the resultant abstractions (which measures, sadly, neither state _nor_ trait) and hoping that are all improvable in some way.*\n\nAnd because competencies are unmeasurable, it is impossible to prove or disprove the assertion that _everyone who excels in a particular job possesses a particular set of competencies_. It is equally impossible to show that _people who acquired the competencies they lacked outperformed those who did not_ \u2014that, in other words, well-rounded people are better. These two statements together are the foundation for most of what companies do to develop the talents of their people, yet each of them is unfalsifiable\u2014you will find no academic papers in any peer-reviewed journal proving the necessity of possessing certain competencies, and no proof that acquiring the ones you lack nets you any increase in performance. Both of these assertions, despite the good intentions that created them, are conjured from thin air\u2014and we can never know if they are correct.\n\n\u2022 \u2022 \u2022\n\nBut hang on, you may say: Isn't the art of business the art of making decisions with incomplete data? Isn't that what businesspeople get paid for\u2014taking risks in the face of uncertainty? Isn't all this psychometric stuff a bit precious? Even if we can't prove, measurably, that acquiring a list of competencies helps a person to excel, what's wrong with trying nonetheless? Surely a good team leader should encourage each of her people to pinpoint capability gaps, to strive to plug these gaps and thereby become more well-rounded. Surely both the team and the individual would benefit from getting each person to conform ever more closely to the well-rounded ideal. Indeed, surely that's what growth _is_ \u2014the process of gaining ability where we have little.\n\nAgain, no. Which brings us to the second fact: the research into high performance in any profession or endeavor reveals that _excellence is idiosyncratic_. The well-rounded high performer is a creature of theory world. In the real world each high performer is unique and distinct, and excels precisely because that person has understood his or her uniqueness and cultivated it intelligently.\n\nWe see this most easily in the world of professional sports. If we were to design the theoretical model of a high-performing attacker on a soccer team, we would not create a Lionel Messi, with his diminutive stature and ineffectual right foot. Instead we might devise a player who looks more like Cristiano Ronaldo\u2014a taller, more physically imposing player who is equally at ease with his left foot, his right foot, and his head (though even here we would likely erase from our theoretical design Ronaldo's ego, individualism, and occasional petulance). In the world of tennis we would obviously include in our design the fluidity and grace of Roger Federer, but we'd also probably want to add a hunk of Rafael Nadal's muscle, dollops of Novak Djokovic's cocksureness, and just a hint of Andy Murray's soft hands at the net. In our theoretical world, in other words, we would pick and mix the qualities we thought preferable. But obviously, in the real world no one gets to do this, whether they're a soccer player, a tennis player, or a team leader. In the real world each of us learns to make the most of what we have. Growth, it turns out, is actually a question not of figuring out how to gain ability where we lack it but of figuring out how to increase impact where we already have ability. And because our abilities are diverse, when you look at a great performance you see not diversity minimized but rather diversity magnified; not sameness but uniqueness.\n\nWe find the same idiosyncrasy among the best musical performers. We expect Adele to belt out soaring, maudlin ballads, but were we to ask Lorde or Halsey or Britney Spears or, God forbid, Miley Cyrus to sing or sound the same as Adele, we'd all be as alarmed as the Carnegie audience at a Florence Foster Jenkins concert. Now, we might assert that every role has some minimum requirements without which a person cannot succeed, no matter how extraordinary his or her other gifts (the customary name for these things, or rather for the lack of them, is career derailers). Even here, though, we would need to be careful in specifying our minimum requirements. After all, if we include \"fluency in musical notation\" in our list of musical skills, we'd knock out some big names. Frank Sinatra, for instance, couldn't read a note. Neither can Elton John. And if we include \"having two hands\" in our list of required traits for a pianist, we'd be forced to exclude Paul Wittgenstein, a classical pianist who lost his right arm in the First World War, who subsequently commissioned piano concertos for the left hand from the leading composers of his day, and without whom we would not have masterpieces by Benjamin Britten, Paul Hindemith, Sergei Prokofiev, Richard Strauss, and Maurice Ravel.\n\nBut these are all extreme examples, and might seem remote from the real world of work. What happens when we measure the strengths and skills of a regular job? Do we find idiosyncrasy or well-roundedness?\n\n\u2022 \u2022 \u2022\n\nIn the early 1980s, a man set out to predict performance. His name was Don Clifton. He was trained as a mathematician and a psychologist, and he set out to identify, quantitatively, the factors that could be identified in a job candidate\u2014specifically, the factors that would predict success in the role the candidate sought.\n\nClifton was leading a team of researchers at a company called Selection Research, Incorporated (SRI), which in the 1990s acquired and took on the name of the Gallup Organization. One of the early studies SRI did sought to predict the success of pub managers for a large brewery chain, as it had long been recognized that a big part of the difference between a so-so pub and a great pub depended, in some hard-to-pin-down way, on the characteristics of the manager. Clifton and his team began, as they always did, by asking questions\u2014lots of questions\u2014of the brewery chain's best and average pub managers, as measured by their business results. \"What is the best way to manage someone?\" \"How closely should an employee be supervised?\" And so on. They asked these questions of the average managers and of the great managers, and looked for differences in their answers. Questions that didn't yield a difference were thrown out, and when all was said and done they had a set of 108 questions that seemed to identify the secrets of pub-manager performance. Now they did a blind test, using their questions on a different selection of managers, without knowing how well these managers were doing, and they demonstrated that their list of questions could reliably and consistently sort the best from the rest.\n\nThe questions measured various qualities\u2014from a pub manager's sense of mission, to his or her instinctive contingency planning, to his or her ability to develop others\u2014and the researchers were curious to see if one of these, or a small group of these, was the master key, the one thing or combination of things that, time and time again, unlocked great performance. But as they looked at the scores of the best managers, they found something subtly and wonderfully different. The high scores of the best managers moved around\u2014one manager would do well on questions about creating a particular ambience in the pub, say, while another would excel on questions about inventory and budgeting. There was no pattern at all. Or, rather, there was just one big pattern\u2014the only way to predict a manager's performance was to look at his or her _total_ score. They had found a list of ways in which managers could excel, and they could define excellence on each of these dimensions. Yet it seemed to make no difference which of these a candidate excelled at, _as long as he or she excelled somewhere_.\n\nThis was not an anomaly for the role of pub manager. Every single occupation the Gallup Organization studied\u2014salesperson, teacher, doctor, housekeeper\u2014displayed this same pattern: those who excelled did _not_ share all the same abilities, but instead displayed unique combinations of different abilities, strongly. Excellence in the real world, in every profession, is idiosyncratic.\n\nIn the theoretical world that exists inside most of our large organizations\u2014a world preoccupied with the need for order and tidiness\u2014the perfect incumbent of every role possesses all the competencies that can be dreamed up and defined. In the real world, however, these long lists of intricately defined competencies don't exist, and if they did, they wouldn't matter. If, as someone once said,* the British fox hunt is the unspeakable in pursuit of the inedible, then the competency model is the unmeasurable in pursuit of the irrelevant. In the real world, each of us, imperfect as we are, strives to make the most of the unique mix of traits and skills with which we've been blessed. Those of us who do this best\u2014who find what we love about what we do, and cultivate this love with intelligence and discipline\u2014are the ones who contribute most. The best people are not well-rounded, finding fulfillment in their uniform ability. Quite the opposite, in fact\u2014 _the best people are spiky_ , and in their lovingly honed spikiness they find their biggest contribution, their fastest growth, and, ultimately, their greatest joy.\n\n\u2022 \u2022 \u2022\n\nOn some level, we have all long known this. From our earliest memories of school to our most recent experiences of work, the thought that _if only I could set this annoying thing aside and focus on what I really want to, then I could make a much bigger difference_ is all too familiar. But then why do these competency models and their associated 360-degree assessments, feedback tools, and development plans exist? What could have prompted otherwise sensible people to have spent so much time and energy and money building models whose efficacy is intrinsically unprovable, that require enormous amounts of time and energy to create, and that fly in the face of our own experiences in the world?\n\nThe simplest answer is that, though we are deeply aware that each of us is unique, and that no amount of training or badgering will remove that uniqueness, it is still quite overwhelming for a busy team leader to allow himself to come face-to-face with the fact that each of his team members thinks differently, is motivated by different things, responds to relationship cues differently, and gets a kick out of different sorts of praise. Who has the time for all these subtle shadings of diversity? Better to just define a model, and then manage to the model (hence the automated feedback writer we encountered earlier).\n\nFor a company, it's all about control. The strong instinct of most corporate leaders, faced with the teeming diversity not just of gender, race, and age but of thought, drive, and relationship inside their organizations, is to look for some way to exert control\u2014to rein it all in, to impose conformity on the chaos, and thence to be able to understand what's going on, and to shape what will happen next. And so companies have spent, and continue to spend, large quantities of time and money trying to work around each person's uniqueness\u2014and this is where these models bubble up from. The models promise rigor\u2014a clear set of characteristics against which _everyone_ can be measured, a sort of \"apples-to-apples\" comparison (even though in the real world it is always \"apples-to-oranges\"). The models promise analytical insights\u2014a way to understand the _entire_ workforce. (It's no accident the systems are known as performance management systems, as oxymoronic as that sounds.) The models promise fact, evidence, truth. What is the job of an executive if not to know what's going on, in great specificity, and to be able to tweak the dials of the vast enterprise before him so that progress may be made? The creeping suspicion, on the part of more and more leaders, that the models offer none of the things they promise, is an inconvenience to be minimized.\n\nAnd to be clear, it isn't just the competency models that are dubious but the ideas behind them. There is the idea that improvement comes from repairing our deficits. There is the idea that failure is essential to growth. And there is the idea that our strengths are something to be afraid of.\n\nAs we've seen, what's most striking when we look at excellent performance is not the absence of deficit but, rather, the presence of a few signature strengths, honed over time and put to ever greater use. But still the idea of fixing deficits appeals to us\u2014it gives us the hope that we might corral, and thus tame, our imperfections, and it allows us to make amends for our shortcomings by toiling to fix them. And the fact that this toil is usually far from joyful is part of the allure. \"Pain + Reflection = Progress\" is the mantra at Bridgewater, the hedge fund run by Ray Dalio, and in some way we thrill to the hard clarity of this prescription. The pain of working on our deficits seems like a worthy pain, a way to pay our penance and make our restitution with the world, and we are drawn to its salutary austerity.\n\nAnd the idea that failure is important is attractive, in turn, because failure helps us understand our deficits\u2014it helps us find more of them. If a technology company today is _not_ talking about failing fast, there is presumed to be something wrong with it. \"There is no way to 'get better' other than to first do it, however poorly you do,\" says Charlie Kim, CEO of Next Jump, and this makes perfect sense. But then the false syllogism: \"So get started; go out and fail! We have become good at getting better because we are so good at failing.\" Beyond the obvious point\u2014that if all a company did was to become brilliant at failing in more and more ways, faster and faster, it would be, well, a failure\u2014the truth is that large success is the aggregation of small successes, and that therefore improvement consists of finding out, in each trial, what works, seizing hold of it, and figuring out how to make more of it. Failure by itself doesn't teach us anything about success, just as our deficits by themselves don't teach us anything about our strengths. And the moment we begin to get better is the moment when something actually works, not when it doesn't.*\n\nAnd then there is the idea that our strengths are to be feared\u2014that we should avoid overusing them because that will somehow pull us away from our proper focus on failure and shortcomings, and instead pull us toward laziness and complacency. Of course, if we were able to watch a great athlete training, or a great writer writing, or a great coder coding, we would see that honing a strength is hard work\u2014it is by no means easy to find that incremental margin of performance when you are already operating at a high level\u2014and that a strength is not where we are most \"finished\" but in fact where we are most productively challenged. Yet we are told to resist the temptation to \"just\" play to our strengths, and instead to work constantly on our weaknesses. In common parlance, we are told to avoid \"running around our backhand.\" This betrays, perhaps, a misunderstanding of what a strength actually is. It is not, for each of us, where performance is _easiest_ \u2014it is where performance is most _impactful_ and _increasing_. We would never tell Lionel Messi to try to play with his right foot. We would instead watch as he works, tirelessly, to make his left ever more powerful. And the only reason that \"running around your backhand\" has become an idiom for avoiding a weakness is that this is _exactly_ what we see great tennis players do, time and time again, whether it's Juan Mart\u00edn del Potro, Rafael Nadal, or countless others. The phrase describes the act of avoiding a weakness in order to play to a strength, and the lesson from the best is that this leads _toward_ high performance, not _away_ from it.\n\nYet these are the ideas that competency models, 360-degree assessments, talent reviews, feedback tools, and much more are built on\u2014that what is most important for us is to understand our deficits, embrace failure, and be wary of our strengths. To be clear, we are not, here, making an absolutist argument: we are not saying that there is nothing to be gained from trying to improve our shortcomings, or that we shouldn't try new things for fear of failure. We are, however, arguing for priority, for focusing first, and predominantly, on our strengths and our successes, because that is where the greatest advantage is to be had. And the great shame in all of this is that the very systems that we might hope would be aimed at discovering and unleashing each person's unique talents have, in fact, the effect of inhibiting those talents, and denying what makes each one of us unique. They don't, in the end, help performance. They hinder it.\n\n\u2022 \u2022 \u2022\n\nWhat, then, should we do in the face of all of this? How do the best team leaders in the real world go about building great teams? Here are three strategies we've seen used by the best team leaders.\n\nThe first strategy is this: _Get into the outcomes business_. A team leader at one of the early Silicon Valley startups faced an unusual situation. He had assigned a new hire to work with one of his experienced engineers, and now the experienced engineer was complaining. The new hire was arrogant and prickly, he said, but worse than this, he had awful body odor\u2014the team leader should fire him. But the team leader saw something in his unusual employee, and worked out a different solution. He figured that the two could work together as long as they weren't in the office at the same time, and instead passed work back and forth. And so, in the early days of Atari, Steve Jobs worked nights.\n\nOthers may have confused you into thinking you are in the control business, and competency models will have left you languishing in the method business. But you're in neither of these. As a team leader, you are in the outcomes business. You are being paid to create certain outcomes for your company, as efficiently, as predictably, and as sustainably as possible, and to do this with enough creativity and intuition and excitement to engage the sort of talent that you and your company will need tomorrow. The lesson we learn from Hugh Dowding's bunker, Stanley McChrystal's daily O&I call, Facebook's conference-room logos, and Chick-fil-A's franchising agreement is that leaders _can't_ be in the control business and _must_ be in the intelligence, meaning, and empowerment business\u2014the outcomes business.\n\nLionel Messi's manager needs the team to score goals. And Messi's bizarrely magical left-footedness is only intriguing because it results in Messi, or one of his teammates, putting the ball in the back of the net. Everything his manager does to coach him makes sense only if they both keep focused on the outcome of scoring goals, and thereby helping the team win. It's not the idiosyncrasy that ultimately matters. It's the goals that matter, and the idiosyncrasy is only useful because it is _always_ the best way to get more goals.\n\nTennis is the same. If we were coaching Andy Murray, we would never try to define for him some model of universal tennis excellence and tell him to try to play like that. Instead we'd say something like, \"Hey, Andy, we know what winning looks like, and feels like, so what strengths do you have that can give you an unfair advantage over everyone else, and thereby help you win? You will never have Roger's backhand, or Rafa's spin, but you do have speed, and touch, and a weird, relentless, hangdog determination. How can we make those dominate for you?\" Billie Jean King once said that the best tennis players have to practice their winning combinations, the one-two-three sequences that end the point in their favor. So we might continue by asking Andy how his strengths can be combined, and then encourage him to spend time honing those combinations, so that in pressure situations he can call upon them with unthinking confidence. We would, in other words, try to help him figure out how his particular strengths, alone or in combination, could move him toward the outcomes he's after.\n\nThe outcome of teaching is helping a student learn. There is no universal recipe for that, any more than there is a universal recipe for writing a beautiful song. The outcome of managing a pub is not creating a great atmosphere, or designing a fun quiz night, or having the best-priced, best-tasting beer. These are methods, and none of them matter in and of themselves. Instead the outcome is having a pub full of happy punters. The very best district managers spend their pub visits paying attention to what each pub manager looks forward to, when he is in \"flow,\" and which activities in the pub he naturally gravitates to. They then build their coaching strategies around these signs of his strengths, helping each manager combine them in the service of the desired outcome.\n\nYou can do the same. Define the outcomes you want from your team and its members, and then look for each person's strength signs to figure out how each person can reach those outcomes most efficiently, most amazingly, most creatively, and most joyfully. The moment you realize you're in the outcomes business is the moment you turn each person's uniqueness from a bug into a feature.\n\nAnd what you will be doing, when you step back and look, is fitting the role to the person\u2014which leads us to the second strategy: _Define the adjustable seat_.\n\nIn the years following the end of the Second World War, the United States Air Force was creating more and more innovative and expensive planes\u2014jet-powered, fast, and fiendishly hard to control\u2014which pilots were then proceeding to crash at an alarming rate. After various inconclusive inquiries, Air Force engineers began to wonder if the cockpit design was the problem\u2014if, for some pilots, the controls were too hard to reach and operate\u2014and if the standardized cockpit dimensions, created in 1926 from studying hundreds of pilots and calculating their average size, were in need of revision. The engineers decided to recalculate the average, and to this end set out in 1950 to measure the various physical attributes of 4,063 pilots. One of the team charged with doing the measuring was a young lieutenant named Gilbert S. Daniels.\n\nNow, as Daniels thought about the problem the USAF faced, he realized that this wasn't only a problem of knowing the average per se, but also one of fit, between any individual pilot and a cockpit designed for the average pilot. So as the study proceeded, he began to think about another question. In addition to his assigned task of calculating the average, he asked himself how many of the pilots in the sample were actually average-sized\u2014or close to average-sized, at any rate. (Daniels defined _average_ as within the middle 30 percent of the range of measures on any given dimension.) If a good number of pilots were near average-sized, he reasoned, then the new cockpit dimensions stood a chance of solving the problem.\n\nThe researchers had measured ten dimensions for each pilot, so now Daniels set himself the task of going through the data, pilot by pilot by pilot, and counting how many of the 4,063 pilots were in the middle 30 percent on all of the ten dimensions.\n\nThe answer, when it came, was this: _none_. There were no average-sized pilots\u2014none whatsoever. Even if you looked at just three of the ten dimensions, fewer than 5 percent of the pilots were average-sized on all three. Even in a population of humans deliberately _selected_ against a set of criteria (if you were too tall or too short, for instance, you weren't qualified to become a USAF pilot in the first place), there was no one-size-fits-all, not even close.\n\nJust as Don Clifton discovered that the only predictor of performance was total score across a number of relevant variables\u2014that there was no right _pattern_ of abilities, only a right _sum_ of abilities\u2014Gilbert Daniels discovered that there were no average humans in a population of 4,063, and that the average is a mathematical concept, not something that exists in the physical world. While the outcomes of high performance are visible and clear, the ingredients of high performance vary from person to person. There is no one-size-fits-all when it comes to human beings; and there is no one-size-fits-all when it comes to great performance.\n\nWhat can you do, then, with all this unfettered diversity? You do what Daniels advised the US Air Force to do: design an adjustable seat. Fit the machine to the pilot, not the other way around. You can do the same with your team\u2014it entails adjusting the outcomes you're asking individual team members to deliver to better match their idiosyncratic talents.\n\nThe first strategy suggests that we clarify the outcomes we're after, and then help each team member find his or her path to these outcomes. This raises the question of what to do if the outcomes aren't right for a particular person. In response, the second strategy suggests that we fit the work to the person, and not the other way around, so as to maximize person-outcome fit. But this in turn raises a further question: if we are always fitting work to people, how do we address the full spectrum of work we have to get done? If we were to design an \"adjustable seat\" for each person, we may still end up with many necessary tasks left undone.\n\nHence the third strategy: _Use team technology_. To help you address everything that needs to be addressed, the real world has devised a supremely effective technology for integrating people's wonderfully imperfect capabilities in the service of a given objective. It's called a team, and the essential magic of a team is that it makes weirdness useful.\n\nYou are weird. You don't seem weird to you because you are with you all the time. But you are weird, to everyone else, and they are weird to you: gloriously, beautifully weird. It's weird that others don't get a kick out of the same things that we do. It's weird that some people love doing things we find excruciating. And when we see someone do something better than we ever imagined possible, it can be confounding, perplexing, astonishing\u2014and also, of course, a source of relief. Thank goodness that woman there loves to confront people. Thank goodness that man there loves thorny political situations. Thank goodness that woman there is so impatient for action. If the people around us were not wired to be weird, then we'd have to spend all of our time scrambling to find someone who actually stood out. Instead, we can partner them up, and wire their weirdness together into a team.\n\nDiversity isn't an impediment to building a great team\u2014rather, it's the fundamental ingredient without which a great team cannot exist. If we were all the same, there would doubtless be things that all of us could not do, and that therefore the team could not do. We need to partner with people whose strengths\u2014whose weirdness, whose spikiness\u2014is different from ours if we are to achieve results that demand more abilities than any of us has alone. And this means, in turn, that the more different we are from one another, the more we need one another. The more different we are, the more we rely on understanding and appreciating the strengths of others, and on building a shared understanding of purpose, and an atmosphere of safety and trust, so that those strengths can be most usefully put to work. Well-roundedness is a misguided and futile objective when it comes to individual people; but when it comes to teams, it's an absolute necessity. The more diverse the team members, the more weird, spiky, and idiosyncratic they are, the more well-rounded the team.\n\nCompetencies, and all the other normative and deficit-focused tools we have, don't push in this direction\u2014of expressing and harnessing diversity. They do just the opposite, as we've seen. But we don't need to throw them out completely. The process of creating them\u2014involving a group of leaders, usually, debating what they value most\u2014is not one that should result in any sort of measurement tool or one-size-fits-all standard. It is, however, exactly the sort of process that should create a statement of collective values, priorities, purpose, and ambition. Customer focus, innovation, growth orientation, agility\u2014these are not abilities to be measured, they are values to be shared. So we should remove from our competency models the levels of ability, the individual evaluations, the feedback, and all the other things that they have become encumbered with, and we should instead simplify them, clarify them, recognize them (and name them) for what they are, and stick them on a wall for all to see. When we carry our competencies across the measurement bridge, we enter a fake and dangerous world\u2014as a tool of assessment, order and control, they are worse than useless. But as public signifiers for what we deem most important, they are another way we can cascade meaning in our organizations, and thereby help our leaders and teams understand what's most important.\n\n*If you are familiar with Marcus's work, StrengthsFinder is an example of a self-assessment and StandOut is an example of a situational judgment test.\n\n*We'll talk more about the perils of rating other people in chapter 6.\n\n*Yes, it was Oscar Wilde.\n\n*As we're writing this, Facebook is facing numerous government inquiries into the use of its data to influence elections, Uber has curtailed its self-driving-car testing because one of its cars hit and killed a cyclist, and Yahoo has long since ceased to exist in any meaningful sense. It is unlikely that anyone is celebrating these and other failures, and the \"fail fast\" speed with which they've been achieved.\n\n#\n\n# People need feedback\n\nIt is a truth universally acknowledged that a millennial in possession of a job must be in want of feedback.\n\nActually, not just millennials. It goes without question that feedback for each and every one of us at work is a good thing, and that more feedback is an even better thing. As a result, today we are blessed with upward feedback, downward feedback, peer feedback, 360-degree feedback, performance feedback, developmental feedback, constructive feedback, solicited, unsolicited, and anonymous feedback, and with all of these flavors and variants has emerged a cottage industry of classes to teach us both how to give this feedback and how to receive it with grace and equanimity. We seem certain that modern employees need, and indeed cannot but benefit from, a real-time, straight-up assessment of their performance, and an appraisal of where they stand in relation to their peers. Indeed, of all the things we \"know for sure,\" in Twain's words, this is the one we know for surest.\n\nIf there is any complaint in all this, at least on the evidence of recent innovations in HR technology, it's that this feedback doesn't happen nearly _enough_ , so coming soon to a phone near you is an array of tools designed to enable you and your company to generate feedback at any time, on any person, about any and all aspects of his or her performance.\n\nYou, as the team leader, will be told that one of the most important and tricky parts of your job is to convey this feedback to your people, no matter how negative the reviews might be. Your job is to accelerate team performance, and it'll be your responsibility to hold a mirror up to the performance of your people so they can see themselves as they really are, and see their performance as it truly is. This, you'll be told, is the secret to both success and respect as a team leader\u2014so much so, in fact, that this sort of direct, clear, unvarnished feedback has its own special name at work: it's called _candid_ feedback.\n\nAnd this in turn means that you need to maintain a certain distance, lest you lose your objectivity and compromise your candor. Although you may sometimes wonder if people would give more and grow more if you showed that you genuinely cared about them, the refrain you'll hear is that if you get too close to your team members you'll never be able to give them the candid feedback they need.\n\nTo aid your development as a leader, others will recommend to you the many books on how to have tough conversations, and will suggest you read from the growing pile of articles describing just how much Gen Y and millennials crave constant corrective feedback. With titles such as \"Why Millennials Actually Want More Feedback at Work\" ( _Fortune_ ), \"Managers: Millennials Want Feedback, but Won't Ask for It\" (Gallup), \"Feedback Is the Unlikely Key to Millennial Career Happiness\" ( _Forbes_ ), and \"Why Millennials Need Constant Feedback at Work\" ( _Business Insider_ ), these articles will make plain to you that millennials thrive on the stuff.\n\nYou'll be taught phrases such as, \"Is now a good time for me to give you some feedback?\" And, \"Would you care for some feedback?\" And the slightly more assertive, \"I have some feedback for you. Are you sitting down?\" Having learned how to give feedback, you'll also learn how to receive it through techniques such as _mirroring_ (\"Did I hear you say that I need to work on my 'organizational savvy and politics'?\") and _active listening_ (\"Can you clarify what you mean by 'hopelessly naive' and give me a couple of recent examples?\").\n\nAnd of course, should you reject the feedback you receive from someone else because it feels odd, or confusing, or just plain wrong, you'll be helped to understand that this feeling is just a natural reaction to threat, and that to grow as a person and as a leader you will need to \"let go of your ego,\" to \"embrace your failures,\" and to always maintain a \"growth mindset.\" If you can reframe all this feedback as valuable input to help you grow, then\u2014you'll be told\u2014you'll soon find yourself addicted to it. As the author and speaker Simon Sinek said recently in his spot as guest editor for Virgin's workplace blog, \"So here's a way you can fulfill your potential in the workplace: negative feedback . . . Negative feedback is where it's at . . . After every project or anything that I do, I always ask somebody, 'What sucks? What can I do better? Where is there room for improvement?' I'm now to the point where I crave it. That's what you want. You want to get to the point where you crave negative feedback.\"\n\n\u2022 \u2022 \u2022\n\nSeeing such enthusiasm for feedback, we might start to wonder what an entire company would look and feel like if everyone was giving everyone else reviews at every turn\u2014if feedback were pervasive and continuous. If so we need look no further than Bridgewater Associates, the world's largest hedge fund. In addition to being extraordinarily gifted at securing returns for his investors\u2014Bridgewater has made $45 billion in net gains since its founding in 1975, more than any other hedge fund\u2014Ray Dalio, Bridgewater's chairman, co-CEO, and co-CIO, has decided to build his company around a commitment to \"radical transparency.\" His belief\u2014explained in his book _Principles_ , in which he lays out 210 prescriptions for work and life\u2014is that the way to be successful is to see and engage with the world as it truly is, no matter how positive or negative these realities are. No hierarchy or office politics should prevent anyone, no matter their level in the company, from challenging an assumption or interrogating a course of action. The real world is right there, Dalio says: it is what it is. We must face it with all of our intelligence unfettered, and we can't allow our politeness or our fear of repercussion to prevent us from seeing what is there to be seen, and thereby changing it for the better.\n\nOf course, people are part of this real world, and they too must be seen for who they really are, without filter, without delay. So at Bridgewater, not only is every meeting videotaped, archived, and made available for every person in the company to view in the company's \"Transparency Library\" (Dalio's commitment to radical transparency is total and without irony) but also each employee is issued an iPad loaded with a variety of apps for rating his or her fellow employees on sixty attributes, such as \"willingness to touch the nerve,\" \"conceptual thinking,\" and \"reliability.\" Employees are expected to rate their peers after calls, meetings, and daily interactions, and all the resultant ratings are analyzed (by the team that created IBM's Watson, no less), permanently stored, and then displayed on a card that each employee carries with him or her at all times. Bridgewater calls this your \"baseball card,\" and its intent is to hold you accountable for knowing \"who you really are,\" and to give everyone else a radically transparent view of what you truly bring to Bridgewater\u2014one of the metrics it displays is your \"believability score.\"\n\nThis is obviously an extreme example\u2014back in 2016, Dalio and his COO got into such a heated spat that each demanded the other be rated on \"integrity\" by the entire firm\u2014and it is difficult to prove what effect, positive or negative, this transparency has had on performance. (Despite the millions of data points collected, Bridgewater still has no reliable measure of each person's performance, as we'll see in chapter 6.) The company as a whole has produced outstanding results across decades and has grown from Mr. Dalio's two-bedroom apartment to occupying a gleaming office building in Greenwich, Connecticut, with 1,500 full-time employees. At the same time, however, the Glassdoor reviews of Bridgewater are mixed, and its first-eighteen-month turnover levels stand at 30 percent, three times higher than the industry average. People leave teams, not companies, as we've seen. That said, it does seem that Bridgewater has more than its fair share of teams that people want to leave.\n\nBut while Ray Dalio and Bridgewater may be outliers, they are at the same time clearly part of the established consensus that _people need feedback_ , and that the best companies and the most effective team leaders must figure out how to give it to them.\n\n\u2022 \u2022 \u2022\n\nIn part, this consensus is a perfectly reasonable reaction to the absurd infrequency of traditional performance reviews. Because companies report their financial results annually, we have all become used to altering people's compensation annually, and since many companies came to espouse \"pay for performance,\" it was inevitable that goals would be set annually, and performance reviews conducted annually, and therefore feedback given annually. This cadence, though it worked for the financial folks, made little sense for either team leaders or members. Leaders felt burdened by the need to put everything into one set of goals at the beginning of the year and one set of laborious reviews at the end, while team members simply felt ignored. No one was served by this annual infrequency, yet there wasn't much to be done about it\u2014if we hated filling out one long set of forms at the beginning and the end of the year, nothing would be gained by upping the frequency of the form filling.\n\nAnd then technology came to the rescue, as it were. With the creation of app-enabled smartphones, the subsequent near ubiquity of these phones, and then their integration with corporate IT infrastructure, companies gained the ability to give every employee the power to launch a survey to anyone in the employee census file, and collect, aggregate, and report the results. Today we can get feedback from anyone, on anyone, at any time, quickly and easily.\n\nBut while this might explain why we are now _able_ to give constant feedback, it doesn't help us understand why we would so desperately _want_ to. To understand that we need to turn to two well-documented oddities of human nature.\n\nLet's say that one of your colleagues is late for an important meeting. As you sit in mild annoyance waiting for him to arrive, you create a little story in your mind that explains his tardiness as a result of his disorganization and lack of prioritization, and his lack of concern for all the people he's keeping waiting. This sort of interpretation of others' actions is so commonplace that it would be unremarkable, except for the fact that it contains a kernel of reasoning that's demonstrably flawed, and that nevertheless has a huge impact on how we design our organizations. What we're doing, in creating our little story, is coming up with an explanation\u2014an attribution, if you like\u2014for our colleagues' actions, and those explanations, when they concern the people around us, overwhelmingly ascribe others' behavior to their innate abilities and personality, not to the external circumstances they find themselves in. In this case, your colleague is late because of his innate disorganization, for example, not because a senior leader grabbed him in the hallway to ask a pressing question. This tendency of ours to skew our explanations of others' behavior (particularly negative behavior) toward stories about _who they are_ is called the Fundamental Attribution Error. Show us someone doing something that annoys us, or inconveniences us, and we're instantly certain that it's because there's something wrong with that person.\n\nAnd the Fundamental Attribution Error has a cousin. While our stories of others center on _who they are_ , we are much more generous to ourselves in our interpretation of our own actions. When it comes to our self-attributions, we skew the other way, and overascribe our behavior to the external situation around us, to _what's happening to us_. If we're doing something that annoys someone else, then that person is annoyed only because he or she doesn't understand the situation that's forcing us to act that way. This tendency is called the Actor-Observer Bias, and it's one of a number of human-reasoning biases that fall into a category called self-serving biases, because they serve to explain away our own actions in a way that props up our self-esteem.\n\nThese biases lead us to believe that your performance (whether good or bad) is due to _who you are_ \u2014your drive, or style, or effort, say\u2014which in turn leads us to the conclusion that if we want to get you to improve your performance we must give you feedback on who you are, so that you can increase your drive, refine your style, or redouble your efforts. To fix a performance problem we instinctively turn to giving you personal feedback, rather than looking at the external situation you were facing and addressing that.\n\nAnd by the way, if you think about it, much of the world of work is designed this way\u2014it's designed for Those Other People, who need to be told what to do (hence planning instead of intelligence), whose work needs aligning (hence goals over meaning and purpose), and whose weaknesses put us all at risk (hence the deficit thinking we saw in the last chapter, instead of the focus on distinctive abilities). One of the inconvenient truths about humans is that we have poor theories of others, and these theories lead us, among other things, to design our working world to remedy or to insulate against failings that we see in others but don't see in ourselves.*\n\nAdd to this the wonky logic that since success is achieved only through _hard work_ , and since giving negative feedback, receiving negative feedback, and fixing mistakes are all _hard work_ , therefore negative feedback causes success,\u2020 and you can begin to see why our faith in feedback, and specifically negative feedback, is so firmly rooted\u2014why we \"know for sure\" that feedback is helpful and that our colleagues need it.\n\nBut this just ain't so.\n\n\u2022 \u2022 \u2022\n\nLet's go back to where we began, with millennials. The various books and articles argue that millennials crave feedback in part because they are addicted to social media, and to the dopamine hit of one more Facebook \"like,\" or one more Instagram \"love.\" We are asked to interpret this behavior as the result of millennials' need to always know how they are perceived by others and where they stand. And, according to this reading, you're in big trouble as a manager if you aren't constantly attending to how they're doing and telling them how to do it better.\n\nBut if we look more closely\u2014if we look at which features have become more popular on the various social-media platforms, and at the details of how users choose to interact with these platforms\u2014a different picture begins to emerge.\n\nConsider, for example, the very different approaches taken by Facebook and Snapchat to providing for user feedback. A couple of years ago Facebook had been researching additional response emojis beyond the classic \"like.\" After much experimentation (and constant reassurances to its users that the company wasn't going to launch a \"dislike\" button), Facebook announced the addition of six new emojis so that users could offer more-nuanced feedback to other users' posts: the six finalists were \"love,\" \"haha,\" \"yay,\" \"wow,\" \"sad,\" and \"angry.\" Yet soon after the launch, Facebook discovered that, despite the company's careful research and testing, hardly anyone bothered with the new options.\n\nSnapchat, meanwhile, was growing, and then growing some more. Snapchat didn't have six possible responses to a post\u2014it didn't even have one, in fact, because there was no Snapchat \"like\" button, and there isn't to this day. Its appeal was precisely that on this new platform no one would rate you. The user just posts a story, or sends a friend a snap message; the friend responds or doesn't; and then\u2014poof!\u2014in twenty-four hours the story or snap is gone, permanently. If you talk to heavy users of Snapchat\u2014and there are now over 200 million of them\u2014you'll discover that what's attractive about Snapchat to millennials is precisely that they can go there, post there, and share there, all without feeling the pressure of feedback. They see the size of their audience. They keep their snap streaks alive with their friends. But they never have to worry about feedback at all: there is no judgment, let alone any permanent record of judgment. Instead there is just the connection to a friend or an audience.\n\nFor all of Snapchat's early users this was a relief. Snapchat became one of the precious few places in their lives where they were free to be themselves and connect with each other without filters. The very absence of permanent feedback allowed them to be more casual, more at ease, and more real, and this safe, attentive place attracted them in the millions. It is extraordinarily difficult to start a social-media platform and have it grow organically\u2014users are busy and have established behavior patterns, and the power of the network effect to prevent those behaviors from changing is strong. Ning, Path, and latterly Myspace were all launched (in the case of Myspace, relaunched) with great fanfare, and all faltered because they didn't tap into the essence of human nature purely and powerfully enough. Snapchat's chances of success were arguably slim, and yet, because it found an important missing ingredient in young people's lives (a safe place filled with an admiring audience), it was able to find a path to exponential user growth. And then Facebook and Instagram, to their credit, got curious, listened and learned, and did whatever they could to make themselves more like Snapchat.\n\nIf the Snapchat example is any guide, it would seem that at root, social media is more about publishing\u2014about positive self-presentation. It matters less to us whether this \"self\" is truly us, or whether, as many have observed, our online selves are aspirational projections, than it matters to us that others see us, and like us. We aren't looking for feedback. We're looking for an audience, and all of us\u2014not just millennials\u2014seem drawn to places that provide us with a way to meet our audience and gain its approval. What we want from social media is not really feedback. It's attention, and the lesson from the last decade is that social media is an attention economy\u2014some users seeking it, some supplying it\u2014not a feedback economy.\n\nAnd ironically, while the design of today's social-media platforms reflects the fact that millennials are attracted most to environments without feedback, today's companies point to these very same social-media platforms as their primary evidence for why millennials crave feedback.\n\nThe Snapchat growth story is only the most recent addition to a large body of evidence about the human need for uncritical attention. In the late nineteenth century the philosopher Friedrich Nietzsche called us \"the beast with red cheeks\"\u2014the lover of attention\u2014and in the following decades the nascent field of social science proceeded to offer up one case study after another of just how right he was. The psychologist Harry Harlow, in a famous series of experiments in the 1950s, deprived baby monkeys of their mothers, gave them the choice of a wire \"mother\" holding a milk bottle or a soft towel \"mother\" with no milk bottle, and showed that primates, given this choice, will always crave warmth and attention and safety over food\u2014the baby monkeys consistently and heart-wrenchingly picked the towel over the milk. More recently, epidemiologists, psychometricians, and statisticians have shown that by far the best predictor of heart disease, depression, and suicide is loneliness\u2014if you deprive us of the attention of others, we wither.\n\nIn the workplace, the most well-known example of this phenomenon is the research undertaken in the 1920s and 1930s at the Hawthorne Works, a Western Electric facility just outside Chicago. The management, unsurprisingly, wanted to increase the productivity of the workers, and began a series of experiments to explore the relationship between working conditions and worker output. The researchers first made the factory brighter by turning the lights up, and sure enough, in the following days worker output jumped significantly. But then, in the interests of experimental rigor, they decided to turn the lights back down to see what would happen. Strangely, output increased once again. There followed more experiments\u2014making the work stations cleaner, keeping the factory tidier, providing more food during breaks, varying the length of the breaks, keeping the total break time the same but dividing it into smaller or larger chunks\u2014and in each case, when a condition was changed, output went up, yet when it was changed back to where it had been in the first place, output went up _again_. And then, more confounding still, when each experiment was concluded, output sank all the way back to its original level.\n\nIt took a while to figure out what was going on, but the consensus that ultimately emerged from the Hawthorne experiments has had a profound effect on the science of work. The conclusion was not that workers craved a brighter workplace or a tidier one, or, for that matter, a darker one or a messier one. Instead, what the workers were responding to was attention. Each of these interventions demonstrated to the workers that management was interested in them and their experience, and they liked that. And thus liking their work a little more, they worked a little better, and a little faster, and by the end of the day produced a lot more.\n\nThe truth, then, is that _people need attention_ \u2014and when you give it to us in a safe and nonjudgmental environment, we will come and stay and play and work.\n\n\u2022 \u2022 \u2022\n\nBut it's a bit more complicated than that, as it turns out, because feedback\u2014even negative feedback\u2014is still attention. And it's possible to quantify the impact of negative attention, if you will, versus positive attention, versus no attention at all, and thereby better understand what sort of attention we most want at work. In their ongoing study of engagement in the workplace, researchers at the Gallup Organization asked a representative sample of American workers whether their managers paid most attention to their strengths, to their weaknesses, or to neither, and they then asked a series of follow-up questions to measure how engaged each of these employees was. They then calculated the ratio of highly engaged employees to highly _dis_ engaged employees for each type of attention.\n\nTheir first finding told them, in effect, how to design the World's Worst Manager. To create pervasive disengagement, ignore your people. If you pay them no attention whatsoever\u2014no positive feedback; no negative feedback; nothing\u2014your team's engagement will plummet, so much so that for every one engaged team member you will have twenty disengaged team members.\n\nThe researchers' second finding might, on its face, look like a pretty encouraging outcome. They found that negative feedback is forty times more effective, as a team leadership approach, than ignoring people. For those employees whose leaders' attention was focused on fixing their shortcomings, the ratio of engaged to disengaged was two to one. But if we remember that \"engagement\" in this case is a precisely defined set of experiences that have been shown to _lead to_ team performance; and if we recall that most of us have been taught that negative feedback is the best, and that most of us experience mainly negative feedback in our professional lives; and if we consider what the researchers found when they looked at positive attention, then this ratio of two to one becomes much more worrying. Because the third finding was this: for those employees given mainly positive attention\u2014that is, attention to what they did best, and what was working most powerfully for them\u2014the ratio of engaged to disengaged rose to sixty to one.\n\nPositive attention, in other words, is _thirty times_ more powerful than negative attention in creating high performance on a team. (It's also, if you're keeping score, _twelve hundred times_ more powerful than ignoring people, but we haven't yet come across a management theory that advocates ignoring people.) So while we may occasionally have to help people get better at something that's holding them back, if paying attention to what people _can't_ do is our default setting as team leaders, and if all our efforts are directed at giving and receiving negative feedback more often and more efficiently, then we're leaving enormous potential on the table. People don't need feedback. They need attention, and moreover, attention to what they do the best. And they become more engaged and therefore more productive when we give it to them.\n\n\u2022 \u2022 \u2022\n\nSo far, so good. We like positive attention, and it helps us do better work. But what about learning? If all we get is attention to our strengths, how will we ever develop? As Simon Sinek asked, what about those areas where he needs improvement? A team leader must surely want her team members to grow and get better, and won't this necessitate that she spend most of her time pinpointing flaws and fixing them?\n\nAgain, our informal theories of work\u2014our \"know for sure\" theories\u2014let us down. We seem to accept, on its face, the idea that \"strengths\" go at one end of the scale and \"areas for improvement\" or \"areas of opportunity\" go at the other, that areas of high performance are where we are most complete and areas of low performance are where we should, and can, grow.\n\nBut as we saw in the last chapter, the single most powerful predictor of both team performance and team engagement is the sense that \"I have the chance to use my strengths every day at work.\" Now, we tend to think of \"performance\" and \"development\" as two separate things, as though development or growth is something that exists outside of the present-day work. But development means nothing more than doing our work a little better each day, so increasing performance and creating growth are the same thing. A focus on strengths increases performance. Therefore, a focus on strengths is what creates growth.\n\nThe best team leaders seem to know this. They reject the idea that the most important focus of their time is people's shortcomings, realizing instead that, in the real world, each person's strengths are in fact her areas of greatest opportunity for learning and growth; and that consequently, time and attention devoted to contributing to these strengths intelligently will yield exponential return now and in the future. Some of these leaders know this instinctively\u2014or perhaps they've figured it out from their experience with real humans on their teams\u2014but for the rest of us there is a wealth of biological data to reinforce the truth that positive attention accelerates development. At the microscopic level learning appears to be a function of neurogenesis: the growth of new neurons. And, as many recent studies have shown, the brain\u2014though it goes through its most frenzied periods of synapse growth and synapse pruning during childhood and adolescence\u2014never loses its ability to create more neurons and more synaptic connections between those neurons. This is referred to as \"neural plasticity,\" and it's often pointed to as a sign that, since the brain can keep mutating through life, we should keep telling people what's wrong with them so that they can fix themselves, so that they can learn to do it right.\n\nAnd of course, we _can_ all learn to do it right, or at least, right-er. We _can_ all learn to be slightly better at skills that we apply ourselves to with disciplined practice. However, what the brain science also reveals is that while the brain does continue to grow throughout life, each brain grows differently. Because of your genetic inheritance and the oddities of your early childhood environment, your brain's wiring is utterly unique\u2014no one has ever had a brain wired just like yours, and given the brain's complexity, no one ever will. Some parts of your brain have tight thickets of synaptic connections, while other parts are far less dense. And when we examine your brain's growth\u2014when we count the new neurons and their connections\u2014it turns out that you grow far more neurons and synaptic connections where you already have the most preexisting neurons and synaptic connections. Perhaps this is caused by nature's harshly efficient use-it-or-lose-it design, or perhaps, with so much preexisting biological infrastructure supporting your densest synaptic regions, it is simply easier to forge new connections where you already have lots. Either way, we now know that, though every brain grows, each grows most where it's already strongest. The arrow of brain development points toward specialization. As the neuroscientist Joseph LeDoux memorably described it, \"Brain growth is like new buds on an existing branch, rather than new branches.\"\n\nSo the weight of the neurological evidence supports the notion that your strengths _are_ your development areas\u2014that these are, biologically speaking, one and the same. Neurological science can also tell us what happens in response to a deliberate focus on strengths instead of weaknesses. Consider, for example, an experiment during which scientists split students into two groups. To one group they gave positive coaching, asking about the students' dreams and how the students would go about achieving them, while with the other they probed about homework, and what the students thought they needed to do differently to be better. While these conversations were happening the scientists hooked each student up to a functional magnetic resonance imaging (fMRI) machine, so as to see which parts of the brain were most activated in response to these different sorts of attention.\n\nHere's what they found. In the brains of the students who received negative feedback the sympathetic nervous system lit up. This is the \"fight or flight\" system, the system that mutes the other parts of the brain and thus allows us to focus only on the information most necessary to survive. When this part of the nervous system is triggered, your heart rate goes up, endorphins flood your body, your cortisol levels rise, and you tense for action. This is your brain on negative feedback: it responds as if to a threat, and it narrows its activity. The strong negative emotions produced by criticism \"inhibits access to existing neural circuits and invokes cognitive, emotional, and perceptual impairment,\" psychology and business professor Richard Boyatzis said in summarizing the researchers findings.\n\nNegative feedback doesn't enable learning. It systematically inhibits it and is, neurologically speaking, how to create _impairment_.\n\nIn the students who received attention focused on their dreams and how they might go about achieving them, however, the sympathetic nervous system was not activated. Instead it was the parasympathetic nervous system that lit up. This is sometimes referred to as the \"rest and digest\" system. To quote the researchers again: \"[T]he Parasympathetic Nervous System . . . stimulates adult neurogenesis (i.e., growth of new neurons) . . . , a sense of well being, better immune system functioning, and cognitive, emotional, and perceptual openness.\"\n\nIn other words, positive, future-focused attention gives your brain access to more regions of itself and thus sets you up for greater learning. We're often told that the key to learning is to get out of our comfort zones, but this finding gives the lie to that particular chestnut\u2014take us out of our comfort zones and our brains stop paying attention to anything other than surviving the experience. It's clear that we learn most _in_ our comfort zone, because that's our strengths zone, where our neural pathways are most concentrated. It's where we're most open to possibility, and it's where we are most creative and insightful.\n\nIf you want your people to learn more, pay attention to what's working for them right now, and then build on that.\n\n\u2022 \u2022 \u2022\n\nThe question is, how? How can you stimulate learning and growth within your team, steer clear of the negative feedback that sets your people back, and still ensure that your team is running smoothly and efficiently?\n\nThere's one thing you can start to do immediately: get into the conscious habit of looking for what's going well for each of your team members. The pull to look at the negative is a very strong one\u2014the Berkeley psychologist Rick Hanson sums up the research memorably when he says, \"the brain is like Velcro for negative experiences, but Teflon for positive ones\"\u2014which is why making this a _conscious_ habit is so important. It might not come naturally or easily for you, but with such a payoff in terms of performance, engagement, and growth, it'll be worth practicing it.\n\nIn the world of computing, there's an event called a high-priority interrupt. It tells the computer's processor that something requires its immediate attention, and so it needs to \"interrupt\" normal processing and jump the particular something to the head of the processing queue. In the real world of team leaders you'll have quite a few things that function in the same way\u2014that grab your attention and force you to act. The majority of these high-priority interrupts are going to be problems, and that's normal. You don't want to administer medicine to a patient if it's the wrong medicine. You don't want to present something to your executive if you've just received information that half of what you're presenting is now obsolete. Any system or process that breaks down will demand that you, the team leader, address it. This is a high-priority interrupt doing what it should do: stopping everything to seize your attention.\n\nAnd the same high-priority interrupts will occur when one of your people messes up. You'll see something someone does wrong\u2014a poorly handled call, a missed meeting, a project gone awry\u2014and the same instinct will kick in: stop everything to tell that person what he did wrong, and what he needs to do to fix it.\n\nThe difficulty for you here is that people aren't processes, nor are they machines\u2014what works for processes and machines doesn't work for men and women. Processes and machines are finite and static, and unless we change something about them, they either stay the same or gradually wear out. People, by contrast, are in a constant state of learning and growing, and, as we just saw, they grow the most under positive attention and the least under negative feedback. Paradoxically, then, the more your high-priority interrupts involve catching your people doing things wrong (so you can fix them), the less productive each person will become in the short term, and the less growth you'll see from your team members in the long run. Finding itself in negative-criticism territory, the human brain stiffens, tenses, and\u2014in meaningful ways\u2014resists improvement. Machines and processes don't do that. You can fix a machine, you can fix a process, but you can't fix a person in the same way\u2014people aren't toasters.\n\nSo, when it comes to your people, what should be your high-priority interrupt? If what you want is improvement, then it should be whenever someone on your team does something that really works. The goal is to consciously spend your days alert for those times when someone on your team does something so easily and effectively that it rocks you, just a little, and then to find a way of telling that person what you just saw.\n\nThis sounds as easy as \"catch people doing things right,\" but as we'll see, there's a little more to it than that. Tom Landry, who coached the Dallas Cowboys for twenty-nine consecutive years, was one leader who figured this out. Early in his coaching career, with the Cowboys struggling at the bottom of the league and a bunch of misfits on his roster, he introduced a radical new method of coaching. While the other teams were reviewing missed tackles and dropped balls, Landry instead focused his players' attention on their wins, however minor. He combed through footage of previous games and created, for each player, a highlight reel of where that player had done something easily, naturally, and effectively. He reasoned that while the number of wrong ways to do something was infinite, the number of right ways, for any particular player, was not. It was finite, and knowable, and the best way to define and know the right way was to look at those plays where the player had done it right. So he set about capturing these distinctive moments of excellence and offering them up to each player. From now on, he said, \"we only replay your winning plays.\"\n\nNow, on one level, he was doing this to make his players feel better about themselves, because like all good team leaders, he knew the power of praise. The relevant survey item here is \"I know I will be recognized for excellent work.\" And as we've seen, the data shows that people on the highest-performing teams strongly agree with this item far more than people on lower-performing teams.*\n\nBut Landry wasn't nearly as interested in praise as he was in learning. His instincts told him that each person would learn best how to improve his performance if he could see, in slow motion, what his own personal versions of excellence looked like. Really great performance often happens in a state of flow, such that we're barely conscious of what we're doing\u2014Michael Jordan used to watch himself in post-game highlights and shake his head, saying, \"Wow, I did that?\" By replaying his players' winning plays Landry was taking them outside themselves and allowing them to see the contours and the rhythm of what \"working\" truly looked like for each one. In so doing, he hoped not only that they would feel more confident but also that they'd be in a better position to repeat and build on their unique strengths in action. David Cooperrider, professor of Social Entrepreneurship at Case Western Reserve University and creator of the theory of Appreciative Inquiry, has pointed out that organizational growth will always follow the focus of your attention. Tom Landry, twenty years earlier, was applying the same principle to his Cowboys.\n\nYou can do this, too. Nowadays, _recognition_ has become a synonym for _praise_ , but in doing so has moved some way from its origins. It comes to us from the Latin word _cognoscere_ , meaning _to know_ , which in turn stems from the Greek word _gnosis_ , meaning _knowledge_ or _learning_. Thus, to _re_ -cognize a person, in essence, means to come to know him anew. Recognition, in its deepest sense, is to spot something valuable in a person and then to ask her about it, in an ongoing effort to learn who she is when she is at her best.\n\nThe trick to doing this is not just to tell the person how well she's performed, or how good she is. While simple praise is by no means a bad thing, it captures a moment in the past rather than creating the possibility of more such moments in the future. Instead, what you'll want to do is tell the person what _you_ experienced when that moment of excellence caught your attention\u2014your instantaneous reaction to what worked. For a team member, nothing is more believable, and thus more powerful, than your sharing what you saw from her and how it made you feel. Or what it made you think. Or what it caused you to realize. Or how and where you will now rely on her. These are _your_ reactions, and when you share them with specificity and with detail, you aren't judging her or rating her or fixing her. You are simply reflecting to her the unique \"dent\" she just made in the world, as seen through one person's eyes\u2014yours. And precisely because it isn't a judgment or a rating, but is instead a simple reaction, it is authoritative and beyond question. It's also humble: when someone says to you \"I want to know where I stand\" she doesn't actually mean this, and you, frankly, are in no position to tell her\u2014you are not the ultimate and definitive source of truth for where she stands. Instead, what she means is \"I want to know where I stand _with you_.\" And happily, here your truth is unimpeachable.\n\nWith each replaying of these small moments of excellence, relayed through the lens of your own experience, you'll ease her into the rest-and-digest state of mind, her brain will become more receptive to new information and will make connections to other inputs found in other regions of her brain, and she will learn and grow and get better. It is, in short, the best recognition she could ever receive. You are learning about her, and relaying that learning to her, and, as on the best teams, she knows that tomorrow you will be doing so again. On such rituals is great performance built.\n\n\u2022 \u2022 \u2022\n\nThe nature of your attention is key. If a team member screws something up, of course you have to deal with it. But remember that when you do, you're merely remediating\u2014and that remediating what's wrong, so a mistake won't happen again, moves you no closer to creating excellent performance. If a nurse gives someone the wrong medication, ignoring that mistake could be lethal. So you can, of course, say to him, \"Don't ever do that again!\" And you can, of course, design a process to ensure that the medicine is always triple-checked before being administered to a patient. But as you do this, know that if the nurse now consistently gives the correct medication to his patients, this does not mean he's now giving excellent care leading to a faster and more complete recovery. Correcting the nurse's mistake won't lead to this, any more than correcting someone's grammar will lead to her writing a beautiful poem, or telling someone the correct punchline to a joke will make this person funny. Excellence is not the opposite of failure: we can never create excellent performances by only fixing poor ones. Mistake fixing is just a tool to prevent failure.\n\nTo conjure excellence from your team requires a different focus for your attention. If you see somebody doing something that really works, stopping them and replaying it to them isn't only a high-priority interrupt, it is arguably your highest-priority interrupt. Get into this habit and you'll be far more likely to lead a high-performing team.\n\nAnd what of balance? Landry said he would replay _only_ each player's winning plays. Should we go to that extreme, or should we highlight the occasional winning play but focus mainly on mistake fixing? How many positive replays, in other words, should we have for every mistake-fix? What is the best ratio of 'Yes, that!' to 'Stop that!'? Research in other areas of social science sheds some light on this. Look at professor John Gottman's work on happy marriages, or professor Barbara Fredrickson's work on happiness and creativity, and the positive-to-negative ratio you'll arrive at is somewhere between three to one and five to one\u2014three to five moments of appreciative attention for every one piece of negative feedback. While there is no need to obsess over the mathematical precision of the ratios, the science suggests that if you aim for this level of deliberate imbalance you and your team will be well served.\n\n\u2022 \u2022 \u2022\n\nAll that being said, however, there will inevitably come a day when, despite your best intentions and careful highlight flagging, one of your people will implore you to give him negative feedback or corrective action. Tell me what I'm doing wrong, he'll say. Or he'll say that he finds himself stuck in the middle of a difficult situation, or is struggling with his job and is turning to you for advice on how to move forward. What do you do?\n\nTo begin with, try to resist the powerful temptation to jump in with your very best advice.*\n\nFirst, as we saw earlier in the chapter, your brain is wired uniquely, such that the world you see and the sense you make of it, the things in it that draw you in, or repel you, or drain you, or light you up, and the insights these things spark in your mind\u2014all these are utterly different from everyone else's, and become even more different as you grow. As a result, the advice given to you by a leader who is not you will not necessarily work for you. The best team leaders know this: They realize that, for example, if you are struggling with public speaking, they cannot just advise you to organize your flow, practice your stories, and nail your ending, because what you mean by _flow_ and _story_ and _nail the ending_ are going to be utterly different from what they mean by those very same words. They understand that the path you will take to your best performance will be unpredictably different from theirs.\n\nYou'll need to remember this the next time you give someone a piece of perfectly crafted advice and then see him do something quite different from what you'd prescribed. Don't get mad at him: it wasn't that he wasn't listening, or had nodded his assent and then passive-aggressively gone and done the exact opposite. He'd heard you, and more than likely, he wanted to do what you wanted him to do. It's just that he couldn't make sense of your sense. All he had was his own sense, so he acted out of that, as best he could.\n\nSeen in this light, much of what we call \"advice\" is perhaps better understood as _The Recitation Of A Set Of Tactics That Work For Me And Only Me_. Ray Dalio's principles are not universal, as interesting as we might find them to be. They are simply and only 210 tactics that work for Ray, or at most for people who share very many characteristics with Ray. In this sense, advice is akin to blood. Prior to the twentieth century doctors attempting blood transfusions found to their deep frustration that while in some cases they worked beautifully, in others the patient's body seemed to be allergic to the donor's blood and rejected it completely. It was only when the Austrian scientist Karl Landsteiner identified the existence of different blood types, and discovered that some blood types were physiologically incompatible with others, that doctors realized the importance of knowing the type of the donor and the patient before attempting a transfusion.\n\nThe same applies to \"performance transfusions.\" To succeed, they depend on how individuals make sense of what they're hearing\u2014how they metabolize it, and hook it into their own patterns of thought and behavior. Performance-transfusing advice, in other words, starts with the performer, not with the advice.\n\nThe second thing that great team leaders know, and that brain scientists have shown, is that an \"insight\" is brain food. These scientists aren't yet sure whether this is because insights come with a nice shot of dopamine or some other neurochemical transmitter, but what they do know is that the brain is built such that a new insight\u2014\"a feeling of knowing generated from within,\" to use their phrasing\u2014feels good. Perhaps you have felt this. Perhaps you have noticed it in others when you've tried repeatedly to teach and advise but seen performance leap only when your team member has combined this advice with her own raw material to create a flash of new understanding. This insight then becomes her sense maker, her lens through which to view the challenge in front of her, her guide as she navigates her way forward. This insight is learning, and while it can be nudged from without, it is only ever generated from within.\n\nWe take such pains with our advice. We are so proud of its kindly intent, its perspective, its generosity, and its sequence: the way we begin with a simple setup and then walk our confused advice seeker step-by-step through our carefully constructed logic, until we bring her gaze all the way to the solution standing before her, clear as day. We offer this painting up to her, beautiful and complete.\n\nBut the most helpful advice is not a painting. It is instead a box of paints and a set of brushes. Here, the best team leaders seem to say, take these paints, these brushes, and see what you can do with them. What do you see, from your vantage point? What picture can you paint?\n\nThis, in the end, is why they are so intent on replaying for you what's working for you. By helping you to see what \"working\" looks like for you, they're offering you an image that you can use as raw material for your painting, and since it was your behavior that created the image in the first place, you've already felt it from within. Now it's their job to show it to you from without, so that you can recognize it, and re-create it, and refine it.\n\nWhen a team member comes to you asking for advice, then, don't rush to your easel and start furiously painting away. Instead try this approach\u2014the box-of-paints approach, if you will, containing some hues of _present_ , some shades of _past_ , and a few bright dabs of _future_.\n\nStart with the _present_. If your team member approaches you with a problem, he is in it _now_. He is feeling weak, broken, or challenged, and you have to address that. But rather than dealing with it head-on, ask your colleague to tell you three things that are working for him _right now_. These \"things that are working\" might be related to the situation, or they might be completely separate from it. They might be significant or trivial. It doesn't matter. Just ask for three \"things that are working.\" In doing that, you're priming his mind with oxytocin\u2014what we sometimes call the \"love drug,\" but which here is better thought of as the \"creativity\" drug. By getting him to think about some specific things that are going right, you are deliberately altering his brain chemistry so that he can be open to new solutions, and new ways of thinking or acting.\n\n(By the way, you can be totally up-front with him about what you are doing\u2014the evidence suggests that the more active a participant he is in this, the more effective the technique.)\n\nNext, go to the _past_. Ask him, \"When you had a problem like this in the past, what did you do that worked?\" Much of our lives are lived through patterns, so it's highly likely that he has encountered this problem before and found himself similarly stuck. But on one of these occasions he will almost certainly have found some way forward, some action or insight or connection that worked for him and enabled him to move out of the mess. Get him thinking about that, and seeing it in his mind's eye: what he actually felt and did, and what happened next.\n\nFinally, turn to the _future_. Ask your team member, \"What do you already know you need to do? What do you already know works in this situation?\" In a sense you're operating under the assumption that he's already made his decision\u2014you're just helping him find it. At this point, by all means offer up one or two of your own paintings, to see if they might clarify his own. But above all keep asking him to describe what he already sees, and what he already knows works for him.\n\nThe emphasis here should not be on why questions (\"Why didn't that work?\" or \"Why do you think you should do that?\"), because these take both of you backward and upward into a fuzzy retrospective world of conjecture and concepts. Instead, rely on your what questions (\"What do you actually want to have happen?\" or \"What are a couple of actions you could take right now?\"). These sorts of questions yield concrete answers, in which your colleague finds his actual self doing actual things in the near future. Each answer he comes up with is a brushstroke to his painting, making his images ever more vivid, more compelling, and more real.\n\nAnd if he starts big, as some do, with huge swaths of color that overwhelm the entire canvas\u2014\"What I need to do is quit my job, buy a dinghy, and sail round the Cape of Good Hope\"\u2014then put a couple of smaller brushes in his hand and direct his gaze to one corner of the canvas: Here's a figure, you might suggest. Can you repaint it, in a different color perhaps, or with one small shift in the perspective? Perhaps he will be able to come up with a few things he knows he can tackle right now, rather than up and quitting his job. And then, guided by the small but increasingly vivid image in his own mind, he will, little by little, create a new painting.\n\n*The American philosopher John Rawls proposed, in 1971, a thought experiment to counter these theories of ours, an experiment he called The Veil of Ignorance. Essentially, he suggested that the best way to design the world was to imagine that when we were done designing it, we'd be randomly assigned some role in this new world, and that we should design in ignorance of what that role would be\u2014whether we would be rich or poor, male or female, academic or athletic, and so on. He proposed, in other words, that we should design a world for ourselves, for any imaginable permutation of ourselves, rather than for others. And this is likely a very good way to design a workplace, too\u2014not for those idiots, but for this one.\n\n\u2020This false syllogism is wonderfully memorialized in the British TV comedy _Yes, Prime Minister_ , in the form of \"We must do something. This is something. Therefore we must do this.\" Its equivalent: \"All cats have four legs. My dog has four legs. Therefore, my dog is a cat.\" Logicians refer to this as the fallacy of the undistributed middle. For the rest of us, it's known as politicians' logic.\n\n*And in case you're wondering which drives which\u2014whether the performance leads to praise or vice versa\u2014the data reveals that though high performance at Time 1 does indeed relate to a higher score on the recognition item at Time 2, the correlation coefficients are four times bigger when pointed the other way: praise leads to performance more than praise reflects performance.\n\n*And the irony that we are here advising you not to give advice has not escaped us.\n\n#\n\n# People can reliably rate other people\n\nHow much do you think you can know about a person simply by watching him? If you work with him every single day, do you think you can figure out what drives him? Could you spot enough clues to reveal to you whether he's competitive, or altruistic, or has a burning need to cross things off his list every day? How about his style of thinking? Are you perceptive enough to see his patterns and pinpoint that he is a big-picture, what-if thinker, or a logical, deductive reasoner, or that he values facts over concepts? And could you parse how he relates to others, and discern, for instance, that he's far more empathetic than he appears, and that deep down he really cares about his teammates?\n\nPerhaps you can. Perhaps you are one of those people who instinctively picks up on the threads of others' behaviors and then weaves these into a detailed picture of who a person is and how he moves through the world. Certainly, the best team leaders seem able to do this. They pay close attention to the spontaneous actions and reactions of their team members, and figure out that one person likes receiving praise in private, while another values it only when it's given in front of the entire team; that one responds to clear directives, while another shuts down if you even appear to be telling her what to do. They know that each member of their team is unique, and they spend a huge amount of time trying to attend to and channel this uniqueness into something productive.\n\nHow about rating your team, though? Do you think you could accurately give your team members scores on each of their characteristics? If you surmise that one of your team is a strategic thinker, could you with confidence choose a number to signify how good at it she actually is? Could you do the same for her influencing skills, or her business knowledge, or even her overall performance? And if you were asked how much of these things she had in relation to everyone else on the team, do you think you could weigh each person precisely enough to put a number to each person's _relative_ abilities? This might sound a bit trickier\u2014you'd have to keep your definition of influencing skills stable, even while judging each unique person against that definition. But if we gave you a scale of one to five, with detailed descriptions of the behaviors associated with each number on the scale, do you think you could use that scale fairly, and arrive at a true rating?\n\nAnd even if you are confident in your own ability to do this, what do you think about all the other team leaders around you? Do you think they would use the scale in the same way, with the same level of objectivity and discernment as you? Or would you worry that they might be more lenient graders, and so wind up with higher marks for everyone, or that they might define \"influencing skills\" differently from you? Do you think it's possible to teach all of these team leaders how to do this in exactly the same way?\n\nIt's a lot to keep straight\u2014so many different people rating so many other different people on so many different characteristics, producing torrents of data. But keep it all straight we must, because this data represents people, and once collected, it comes to define how people are seen at work.\n\n\u2022 \u2022 \u2022\n\nAt least once a year, a number of your more senior colleagues will gather together in a room to discuss you. They will talk about your performance, your potential, and your career aspirations, and decide on such consequential issues as how much bonus you should get, whether you should be selected for a special training program, and when or if you should be promoted. This meeting, as you might know, is called a talent review, and virtually every organization conducts some version of it. The organization's interest is in looking one by one at its people\u2014its talent\u2014and then deciding how to invest differentially in those individuals. The people who display the highest performance and potential\u2014the stars, if you like\u2014will normally get the most money and opportunity, while those further down the scale will get less, and those struggling at the lower end of the scale will more than likely be moved into a euphemistically described Performance Improvement Plan (PIP) and thereby eased out.\n\nThese talent reviews are the mechanism that organizations use to manage their people. They want to keep the best people happy and challenged, and simultaneously weed out those who aren't contributing. Since, in most organizations, the largest costs are people's wages and benefits, these meetings are taken very seriously, and the most pressing question\u2014a central preoccupation of all senior leaders in all large organizations\u2014is, \"How can we make sure that we are seeing our people for who they really are?\" This is a wake-up-in-the-middle-of-the-night sort of question for senior leaders, because they worry that their team leaders might not, in fact, understand the sort of person the organization needs nearly as clearly as the senior leaders do, and further that the team leaders might not be objective raters of their own people.\n\nTo combat this worry, companies have set up all sorts of systems designed to add rigor to this review process. The one you may be most familiar with is the nine box. This is a graph showing _performance_ along the x-axis and _potential_ up the y-axis, with each axis divided into thirds\u2014low, medium, and high\u2014to create nine possible regions. Each team leader is asked to think about each person on his or her team and then place them, in advance of the talent review, into one of the nine boxes\u2014to rate them, that is, on both their performance and their potential. This system is designed to allow a team leader to highlight that a particular person might have bags of potential, and yet not have translated that potential into actual performance, whereas another team member might contribute top-notch performance, and yet have very little potential upside\u2014he's maxed out in his current position. With this data displayed in the talent review, the leadership team can define different courses of action for each person: the former will be given more training and more time, for example, while the latter might just be offered a healthy bonus.\n\nMany companies also give people performance ratings on a scale of 1\u20135, either in parallel with or as an alternative to the nine-box process. Again, each team leader is asked to propose a rating for each person on his or her team. Then, before or as part of the talent review, there is a meeting called a \"consensus\" or \"calibration\" meeting, which goes something like this: your team leader talks about you and defends why she ended up giving you a 4 rating, and then her colleagues weigh in on why they gave their people 5s, or 4s, or 3s, whereupon debates ensue about what really constitutes a 4, whether a 4 on one team is the same as a 4 on another team, whether you truly deserve a 4 this year, and if you do, whether the organization has enough 4s left over to allow you to have one.\n\nIf the organization has run out of 4s\u2014which happens often since many team leaders are reluctant to give a person a 3 or, perish the thought, a 2\u2014then your team leader may have to give you a 3 and tell you that, though you truly deserved a 4, it wasn't your turn this year, and that she will look out for you next year. This is called \"forcing the curve,\" which is the name given to the rather painful process of reconciling the organization's need to have only a certain percentage of employees show up as super-high performers with the team leaders' tendency to give high ratings to everyone so as to avoid having unpleasant performance conversations. Forced curves are no one's idea of fun, but they are felt to be a necessary constraint on team leaders, and a way of ensuring that rewards are appropriately \"differentiated,\" so that high performers get much more than low performers.\n\nPerhaps wanting to add more precision to the words _performance_ and _potential_ , many organizations have created lists of competencies that team members are supposed to possess, and against which they are rated at the end of the year. In chapter 4 we questioned whether these models were true reflections of what performance looks like in the real world. (Does anyone really have all of the competencies? Can we really prove that those who acquire the ones they lack outperform those who don't?) Nevertheless, many organizations still rate each person against such standard checklists. To aid in this, each competency is defined in terms of behaviors, and then the behaviors are tied to a particular point on the rating scale. So, for example, on a competency called organizational savvy and politics, if you see that the person \"Provides examples of savvy approaches to successfully solving organizational problems,\" then you'd rate her a three. If you see that she \"Recognizes and effectively addresses politically challenging situations,\" you would rate her a four. Using your behaviorally anchored competency ratings as your building blocks, you would then be asked to construct an overall rating of her performance and potential, and this is how she'd be represented during the talent review.\n\nHistorically, the talent review has happened only once or twice a year, but as we saw with Bridgewater Associates, with the arrival of smartphones it's now technologically possible for an organization to launch short performance-ratings surveys throughout the year. Each person can be rated by their peers, direct reports, and bosses, and then the scores can be aggregated either at mid-year or at year's end to produce a final performance rating. A number of startup, venture-capital\u2013backed companies are leading the charge in bringing constant ratings into the workplace, and have gained such traction that the more established Human-Capital-Management software providers are scrambling to create their own always-on rating tools, and large institutions such as PricewaterhouseCoopers and General Electric are building their own versions.\n\nThis race to real-time ratings appears as inevitable as it is frenzied, and all of it is in service of the organization's interest, which is to answer the question, \"When it comes to our people, what do we _really_ have here?\"\n\n_Your_ interest in all this is related, but different. You won't be too worried about competencies, and calibration sessions, and behavioral anchors, all of which probably sound a bit esoteric. Instead, you'll be acutely aware of a few real-world practicalities that boil down to the fact that your pay, your promotion possibilities, and possibly even your continued employment are being decided in a meeting to which you are conspicuously not invited. The people who _are_ in the room\u2014some of whom you know, and some of whom know you, and others of whom you've never met\u2014are talking about you, and people like you, and they are rating you, deciding which box you go in, and thereby deciding what you will get after a year of hard work, and also where your career will go next. You may not realize this during your first couple of years in the workforce, but once you do, it will preoccupy you. You'll think to yourself: I really want these people to think well of me. I really, really want these people not to think ill of me. But most of all, I want the truth of me in the room where the decisions are made. This is your interest.\n\nYou will come to wonder about these rating scales, these peer surveys, and these always-on 360-degree apps, and you will hope that there is enough science in them, enough rigor and process, that you\u2014ideally, the best of you\u2014will be portrayed accurately. After that, let the chips fall where they may. At least, then, you will have been given a fair hearing on your true merits as a person, and as a team member.\n\n\u2022 \u2022 \u2022\n\nIt is going to bother you greatly to learn, then, that in the real world, none of this works. None of the mechanisms and meetings\u2014not the models, not the consensus sessions, not the exhaustive competencies, not the carefully calibrated rating scales\u2014 _none_ of them will ensure that the truth of you emerges in the room, because _all_ of them are based on the belief that _people can reliably rate other people_. And they can't.\n\nThis, in all its frustrating simplicity, is our sixth lie.\n\nIt's frustrating because it would be so much more convenient if, with enough training and a well-designed tool, a person could become a reliable rater of another person's skills and performance. Think of all the data on you we could gather, aggregate, and then act on! We could precisely peg your performance and your potential. We could accurately assess your competencies. We could look at all of these and more through the eyes of your bosses, peers, and subordinates. And then we could feed all this into an algorithm, and out would come promotion lists, succession plans, development plans, nominations for the high-potential program, and more.\n\nBut none of this is possible, despite the fact that many human-capital software systems claim to do exactly what's described above. Over the last forty years, we have tested and retested people's ability to rate others, and the inescapable conclusion\u2014reported in research papers such as \"The Control of Bias in Ratings: A Theory of Rating\" and \"Trait, Rater and Level Effects in 360-Degree Performance Ratings\" and \"Rater Source Effects Are Alive and Well After All\"\u2014is that human beings cannot reliably rate other human beings, on anything at all.\n\nWe could confirm this by watching the ice-skating scoring at any recent Winter Olympics\u2014how can the Chinese and the Canadian judges disagree so dramatically on the scoring of that triple toe loop?\u2014but instead, let's take a look at the most revealing real-world study of our rating prowess, or lack thereof. It was conducted by two professors, Steven Scullen and Michael Mount, and one industrial\/organizational psychologist, Maynard Goff. They collected ratings on 4,392 team leaders, from two direct reports, two peers, and two bosses. These team leaders were rated on a combination of leadership competencies, such as \"manages execution\" or \"fosters teamwork\" or \"analyzes issues,\" with a short list of questions measuring each competency, for a total of just under half a million ratings from over twenty-five thousand raters.\n\nThe researchers then asked a straightforward question: What best explained why the raters rated the way they did? Could ratings best be explained by relative positioning in the organizational hierarchy, in which case all your direct reports would give you similar ratings, which were measurably different from those given by your peers, which in turn were different from those given by your bosses? Or was the strongest effect a rater's sense of your overall performance\u2014if someone thought highly of you overall, did this affect every single question he or she rated you on? Or was the most powerful factor driving your score on each of the six questions per competency how the person rated you on the other five questions for that competency? If he or she felt you had lots of political savvy, in other words, did this increase scores not only on one of the questions related to political savvy but on all of them? These three possible explanations\u2014rater perspective, overall performance, and competency performance, as the researchers termed them\u2014represented three increasingly close measures of the thing the ratings were trying to measure (your performance on a given skill), with the last of the three being closest to the mark.\n\nObviously, each rater had her own reasoning for each rating, but by slicing and dicing the data the researchers hoped to see what best explained the overall patterns, and what they found was that most of the variation in people's scores\u201454 percent of it\u2014could be explained by a single factor: the unique personality of the rater. From the data it was apparent that each rater\u2014regardless of whether he or she was a boss, a peer, or a direct report\u2014displayed his or her own particular rating pattern. Some were very lenient raters, skewing far to the right of the rating scale, while others were tough graders, skewing left. Some had natural range, using the entire scale from one to five, while others seemed to be more comfortable arranging their ratings in a tight cluster. Each person, whether he or she realized it or not, had an idiosyncratic pattern of ratings, so this powerful effect came to be called the Idiosyncratic Rater Effect.\n\nHere's what's going on. When Lucy rates Charlie on the various subquestions in the competency called strategic thinking, there is a distinct pattern to her ratings, which her organization believes reflects her judgment about how much strategic thinking Charlie has. For this to be true, however, when Lucy then turns her attention to a different team member, Snoopy, and rates him on the same competency, the pattern of her ratings should change, because she is now looking at a different person with, presumably, different levels of strategic thinking. What the Scullen, Mount, and Goff research reveals is that Lucy's pattern of ratings does _not_ change when she rates two different people. Instead her ratings stay just about the same\u2014her ratings pattern travels with her, regardless of who she's rating, so her ratings reveal more about _her_ than they do about her team members. We think that rating tools are windows that allow us to see out to other people, but they're really just mirrors, with each of us endlessly bouncing us back at ourselves.\n\nAnd this effect is not, by the way, associated with unconscious bias on the part of the rater for or against people of a particular gender, race, or age. These biases do exist, of course, and we should do everything we can to teach people how to see past them or remove them\u2014but the discovery from this research is that the Idiosyncratic Rater Effect applies _regardless_ of the gender, race, or age of both the rater and the person being rated. The idiosyncrasy of the rating pattern stems from the uniqueness of the rater, and doesn't appear to have much of anything to do with the person being rated. In fact, it's pretty much as though that person isn't there at all.\n\nThe measurement community was understandably frustrated by the size of the Idiosyncratic Rater Effect, so it expended considerable effort trying to minimize it or remove it. The increasingly detailed descriptions of what constitutes a five versus a four, and the behavioral anchors attached to each point on the competency scales, are all part of that effort. Unfortunately, we now know that these increasingly detailed scales and anchors actually _magnify_ the effect: the more complex the rating scale, the more powerful the influence of our idiosyncratic rating patterns. It's almost as if we get overwhelmed by the complexity of the rating scale and revert to the \"safety\" of our natural rating pattern.\n\nWhen we rate other people on a list of questions about their abilities, the Idiosyncratic Rater Effect explains _more than half_ of why we choose the ratings we do. The three largest studies of people rating other people in the way that we do at work have reached strikingly similar conclusions: about 60 percent of the variability in ratings can be chalked up to the raters' differing responses to a rating scale.\n\nSince you're most concerned that the truth of you be in the room, this should worry you enormously. The rating given to you tells us, in the main, about the _rating patterns of your team leader_ , and yet, in the room, we act as though it tells us about the _performance patterns in you_.\n\nAnd even if we could in fact correct for our rating idiosyncrasies, we'd still have another hurdle in front of us. The people you work with simply don't interact with you enough to be able to pinpoint the extent to which you possess, say, influencing skills, or political savvy, or strategic thinking, or frankly any abstract attribute. People at work are preoccupied (with work, mainly), and paying attention to you closely and continuously enough to be able to rate you on any of these abstractions is a practical impossibility. They simply don't see you enough. Their data on you is insufficient\u2014hence the name for this second hurdle: _data insufficiency_. If Olympic ice-skating judges can't agree on the quality of each triple toe loop, when the only thing they are doing is sitting watching triple toe loops one after the other, then what hope does a busy peer, direct report, or boss have of accurately rating your \"business acumen\"?\n\nEven if we changed the world of work, and created a job category of roving raters whose sole responsibility was to wander the hallways and meeting rooms, to watch each person act and react in real time, and then to rate each person on a list of qualities, we still wouldn't get good data, in part because our definitions are poor. A triple toe loop is defined as a take-off from a backward outside (skate) edge assisted by the toe of the other foot, followed by three rotations, followed by a landing on the same backward outside edge\u2014and this is the only definition of it. Look up _business acumen_ , on the other hand, and you'll find something like this:\n\n> Business acumen is keenness and speed in understanding and deciding on a business situation . . . people with business acumen . . . are able to obtain essential information about a situation, focus on the key objectives, recognise the relevant options available for a solution, [and] select an appropriate course of action.\n\nAnd this is just one of many definitions you'll encounter. Furthermore, there is a world of difference between the specificity of \"take-off from a backward outer edge\" and the vagueness of \"essential information,\" \"key objectives,\" and \"appropriate course of action.\" Essential to whom? Key objectives as determined by whom? Appropriate course of action as determined how? Of course, each of us reading the definitions thinks, \"Well, I could easily define those for myself\"\u2014but that's the point. When we rate people on abstractions, there is even more scope for our ratings to reflect our own idiosyncrasies. And because one person's understanding of _business acumen_ is meaningfully different from another's, even when two highly trained and focused raters rate the same person on the same quality, they find it extraordinarily difficult to arrive at the same rating for the same quality.\n\n\u2022 \u2022 \u2022\n\nTo all this talk of the _Idiosyncratic Rater Effect_ and _data insufficiency_ , however, some will tell you to calm your fears. The truth of you will indeed emerge in the room, they'll say, because even though one person might be an unreliable and idiosyncratic rater, many people won't be. If each person can just manage to get you \"roughly right,\" and if we aggregate the \"roughlys,\" then we'll wind up seeing you pretty clearly. This is the logic upon which 360-degree surveys are based: one person may be off base, but if ten people are telling you that you lack business acumen, then it's a safe bet that you do indeed lack business acumen.\n\nUnfortunately, despite its ubiquity, this line of thinking is wrong. It contains two fallacies. The first of these concerns the wisdom of crowds. This was an idea popularized by James Surowiecki in his book of the same name, in which he described example after example of a well-informed majority being wiser than a sole genius. He began the book with the story of Charles Darwin's half-cousin, Sir Francis Galton, who, while attending the West of England Fat Stock and Poultry Exhibition in 1906, came upon a competition to guess the weight of an ox. For sixpence, anyone who was interested could buy a ticket and write his guess on it, and the person whose guess was closest to the ox's actual weight would win a prize. Galton was fascinated by data, so he stuck around after the winner had been announced and asked if he could borrow the eight hundred tickets on which the guesses had been written. He divided the sum of the guesses by the number of them to find the average. The ox's actual weight was 1,198 pounds, and lo and behold, the average of all the guesses was 1,197 pounds. The crowd was wise.\n\nAnd this is entirely true\u2014well-informed crowds are wise, and very often wiser than a small, privileged, expert elite. But the critical qualifier in that sentence is _well-informed_. The mechanism that creates wisdom in crowds is that lots of members of the crowd have real-world experience of the question being asked\u2014in this case, most of them were from the surrounding farmlands and knew, roughly, the weight of oxen (and even if they didn't, they had a shared understanding of what \"weight\" was). Take all those \"roughlys\" and average them, and you do indeed get pretty close to the ox's actual weight.\n\nBut what happens if the crowd is ill-informed? What happens if instead of the ox's weight the crowd had been asked to guess the number of atoms in the ox's body? Or how \"friendly\" the ox was? The crowd, lacking any real-world frame of reference for the thing being guessed at, wouldn't be wise at all. This is what happens when lots of people who encounter you infrequently, and who each have different definitions of _business acumen_ , are asked to rate you on it. We get the 360-degree-survey equivalent of West Country folk guessing the number of atoms in an ox.*\n\nThe counterargument to this would be that in this analogy, business acumen is more like the weight of the ox than the number of atoms in the ox\u2014we know what business acumen is, so we can indeed roughly rate one another on it. What we discover in the data, however, is that each person seems to have his own idiosyncratic definition of _business acumen_ , and that the more we try to standardize the definition with behavioral descriptions such as those we saw earlier, the greater the idiosyncratic rater effect becomes.* The same applies to other characteristics such as influencing, decision making, or even performance. Each of these is an abstract vessel into which we pour our own unique meaning: we are not well-informed, and we are, as raters, about as effective as farmers would be estimating numbers of atoms. This is the first crowd-based fallacy\u2014that all of us are (always) smarter than one of us.\n\nThe second fallacy is this: that although one person's rating of you might be bad data, if we combine it with six other people's equally bad ratings data, we will magically turn it into good data\u2014that somehow the errors will be averaged out. But this is not how data works. Errors average out only if they are random. If they are systematic\u2014if they stem, for example, from a faulty measurement instrument, as they do when we rate one another\u2014then adding them up produces more error, not less. Noise plus noise plus noise never equals signal; it only ever equals lots of noise. In fact, the truth about data is that noise plus signal plus signal plus signal still equals noise, because the tiniest amount of bad data contaminates all the good data.\n\nWe found an intriguing example of this in the story of Ariel 6, the last in a series of scientific research satellites designed and built by the United Kingdom and launched by the United States in the 1960s and 1970s. It carried three instruments, a cosmic ray detector and two X-ray detectors. The X-ray detectors were aligned with the spin axis of the satellite, so to point them at a specific star, the entire satellite had to be pointed at a particular region of the sky. To achieve this attitude control, the designers came up with an ingenious way of using the Earth's magnetic field both to measure where the detectors were pointed and to change the actual orientation of the satellite. And in order to measure the magnetic field, the satellite was equipped with instruments called magnetometers. There were two of them, not only to provide redundancy but also to enable the two independent measurements to be combined and averaged so as to reduce any random errors.\n\nIn the summer of 1979, the satellite was carefully packed up, shipped from the United Kingdom to Wallops Flight Facility on the Eastern Shore of Virginia, mounted atop a Scout rocket, and launched into space. It immediately encountered problems: The satellite was not spinning about its designated axis and was slightly off-kilter. There were problems recharging the batteries. And for some reason the X-ray detectors were detecting fewer X-rays than scientists had expected they would. The scientists needed to run a test to figure out what had gone wrong, so they pointed the satellite at the strongest X-ray source in the sky, the Crab Nebula.\n\nBy comparing what they expected to see with what they were actually seeing, they made two discoveries. First, the surfaces of the mirrors of one of the X-ray detectors had become contaminated\u2014as a result, in future missions X-ray mirrors were protected until it was safe to expose them to space. And second, it became obvious that the satellite wasn't pointing exactly where it should be\u2014it was off by a few degrees. There was a fault, it turned out, with one of the magnetometers, so there was a systematic error in the averaged measurements. And with that bad data mixed with the good data from the other magnetometer, the satellite didn't know where it was\u2014it couldn't be pointed accurately at the right star.\n\nIn the world of ratings, the idea that we can always cover for the possibility that any individual data source is bad by getting lots of data from lots of sources and averaging it is wrong and harmful. Adding bad data to good, or the other way around, doesn't improve the quality of the data or make up for its inherent shortcomings.\n\nFor Ariel 6, the solution was to ignore the readings from the faulty magnetometer and rely on those from the good one. But when it comes to the talent-review meetings aiming to identify our best people, we don't have this option. _All_ our readings are faulty\u2014we have no good data to rely on. We are, literally, using flawed data to point us at the wrong (human) stars.\n\n\u2022 \u2022 \u2022\n\nSo far, we've seen that 1) human beings can never be trained to reliably rate other human beings, that 2) ratings data derived in this way is contaminated because it reveals far more of the rater than it does of the person being rated, and that 3) the contamination cannot be removed by adding more contaminated data. And this means, in turn, that ratings-based tools, be they annual engagement surveys, performance-rating tools, 360-degree surveys, or any of the many other varieties at large, do _not_ measure what they purport to measure. And _this_ means, in turn, that discussions based on the data generated by these tools do _not_ accurately reflect the truth of you. Faced with this sorry state of affairs, what on earth should we do?\n\nA sensible place for us to start is by learning to tell good data from bad. As most everyone at work will tell you, we are running headlong into a big-data world in which every single process, outcome, item, personal preference, and interaction will be captured, quantified, and run through machine-learning-enabled algorithms. The promise of this world is that, with all of these data points collected in real time, we'll be able to apply artificial intelligence to examine and learn from the relationships between the points, and thereby understand which thing predicts which other thing how frequently and under what conditions.\n\nBut none of these algorithms will yield anything useful unless they're grinding on good data. If we were to discover that having your cellphone in your pocket makes thermometers go haywire (don't worry, we haven't), then we wouldn't be able to learn anything useful from studying your temperature data over time, or the relationship of your temperature to some other data point, because all of your temperature data would have been contaminated by the cellphone in your pocket\u2014garbage data in, garbage discoveries out.\n\nSo what, precisely, is \"good data\"?\n\nWe can say that good data has three distinct characteristics: it is reliable, it is variable, and it is valid.\n\n_Reliable_ data is simply data that we are confident is measuring what it says it's measuring, in a stable and predictable way. The most obviously reliable data comes from anything that can be counted, because if a thing can be counted\u2014whether with your fingers or with some sort of measuring tool\u2014then no matter whose fingers or tool we are using, we'll still get the same data. Your height is reliable data, as is the amount of money in your paycheck, the number of days you missed from work last year, and the temperature outside your office on a particular spring afternoon.\n\nIf, however, we walk outside with our thermometer on that spring day and it displays a reading of 69 degrees Fahrenheit, but then ten minutes later reads 21 degrees, although it's theoretically possible that the world just got dramatically colder, it's much more likely that our thermometer is broken. If after another ten minutes it now reads 75 degrees, we can be pretty sure that it's the instrument that's gone crazy, not the world. There are various statistical tests we can do to assess the reliability of a data set, but in essence, we come to trust our data-gathering tools when the data they generate doesn't change if the thing they're measuring doesn't change. Unreliable data, on the other hand, is wobbly data\u2014it seems to move all by itself. And any measurement tool that spits out changes in data when nothing in the real world is actually changing is, like a broken thermometer, not to be trusted.\n\nThis is why 360-degree-feedback tools are unreliable. The data they produce is supposed to measure the presence of certain competencies in the person rated, yet when we examine the data, it's clear that it wobbles about by itself, because what the tool is actually responding to is the idiosyncrasy of the rater.\n\n_Variable_ data is data that displays natural (unforced) range\u2014that is, range that reflects actual range in the real world. We can judge the quality of a measurement tool by its ability to measure and display this real-world range. Sticking with our temperature example, if we had a regular, store-bought thermometer whose lowest possible reading was 10 degrees below freezing and we took it to the South Pole, then each day our thermometer would tell us that it was 10 degrees below freezing, even though, in actual fact, it was much colder than that. Our thermometer, lacking the ability to measure the full range of what we wanted it to, would fail us as a measurement tool. It wouldn't be broken; it would merely be ill-suited for the task at hand.\n\nIf you've ever taken a training course at work and then been asked to rate it, you'll be familiar with a measurement tool that produces invariable data. Ask class participants to respond to the question, \"Overall, this was a good learning experience,\" on a five-point scale, with five being \"strongly agree\" and one \"strongly disagree,\" and you'll discover that virtually all the responses are either a four or a five. Whereas our thermometer is poorly suited for its Arctic purpose, this training measurement tool is poorly designed. Nevertheless the effect is the same: the data it produces has no range, no natural variation.\n\nPerformance-rating tools are similarly poorly designed. When we ask team leaders to rate their team members on a five-point scale, the data winds up looking like it came from a tool with a three-point scale, because team leaders seldom, if ever, use the bottom two scores. (This is why so many companies feel the need to force the curve\u2014if they didn't, the performance-rating tool simply wouldn't produce data with range.)\n\nTo produce range in our rating tools, we have to create questions that contain extreme wording. A question such as, \"I feel that my job fits my abilities,\" produces very little range at all\u2014pretty much everyone agrees or strongly agrees. This is why when we sought to measure the issue of strengths-role fit we chose to word the question, \"I have the chance to use my strengths _every day_ at work.\" The words _every day_ are extreme, and their effect is to push respondents toward either end of the rating scale\u2014to produce range.* Look back to each of the eight team-experience questions we described in chapter 1 and you'll see that each of them contains an extreme wording. So, for example, the question measuring mission and purpose isn't, \"I believe that my company has a worthy vision,\" but is instead, \"I am _really enthusiastic_ about the mission of my company.\"\n\nThese may appear to be small differences, but on them rests the tool's ability to generate data that captures real-world range.\n\nFinally, we have to ask ourselves whether this range in reliable data matters. Does a high score on the measurement tool predict a high score on something else in the real world? Does variation in the tool relate to variation in something else in the real world? This, for data geeks, is the Holy Grail, and its proper (if less than compelling) name is \"criterion-related validity.\" We can say that a tool's data is _valid_ if the range of data produced by the tool predicts range in something else\u2014if we can prove, time and again, that it is measuring something that correlates with or predicts a different outcome measured using a different tool. For example, Amazon can say that its customer-recommendation data is _valid_ (or, has \"criterion-related validity\") if it can prove that people who bought one item really did also buy a different item. When Amazon knows for sure that the number of clicks on one web page relates to the number of clicks on an entirely different page, then it can have confidence that it's looking at _valid_ data.\n\nOr, we can say an engagement tool's data is valid if the people who rate their engagement more positively on the tool actually wind up staying with the company longer\u2014in this case, range in engagement scores predicts subsequent range in voluntary turnover. One piece of reliable data predicts another piece of reliable data, and so, careful step by careful step, do we add to our store of valid knowledge about the world.\n\n_Reliable_ , _variable_ , and _valid_ \u2014these are the signs of good data, and these three concepts will help you intelligently examine the quality of any data put in front of you.\n\nFor example, if someone claims his data is _valid_ you might ask him, politely, whether he can prove that this data has been shown to predict something else, measured by something else, in the real world. If he can show this\u2014 _\u00e0 la_ Amazon and the clicks from one page driving the clicks on another\u2014then you're probably looking at valid data.\n\nIf someone comes to you and asks you to pay attention to a data set, ask yourself if the data displays natural variation or range. Ask to see a scatter plot. If the data points on the scatter plot all cluster to one end of the scale or the other, it's probably not good data. And of course, any data in which someone has had to fake range by forcing the curve, through a calibration or consensus session, is always bad data. The consensus has contaminated the data, the range is forced, and so the data is bad.\n\nThe place to start, however, is _always_ reliability. Your statistician friends will tell you that all data-based discoveries are built on reliability. When we measure things, we have to make sure that the measurement tool doesn't generate data that wobbles by itself\u2014because if it's wobbling by itself, then we'll never be able to trust its range, and thus never be able to prove that its range can predict range in something else we're interested in in the real world. No reliability means no validity\u2014no knowledge. On anything, ever.\n\nAnd, as we've seen in this chapter, the problem with almost all data relating to people\u2014including you\u2014is that it isn't reliable. Goals data that reports your \"percent complete\"; competency data comparing you to abstractions; ratings data measuring your performance and your potential through the eyes of unreliable witnesses: it wobbles by itself, and fails to measure what it says it's measuring.\n\nOne of the most bizarre implications of this systematic unreliability is that, in what is supposedly the age of big data, no organization can say what drives performance\u2014at least, not knowledge-worker performance. We may be able to say something intelligent about what drives sales, say, or piece-work output, because both of these are inherently and reliably measurable\u2014they can be counted. But for any other work\u2014which means most work\u2014we have no way of knowing what drives performance, because we have no reliable way of measuring performance. We don't know whether bigger teams drive performance more than smaller teams. We don't know whether remote workers perform better than colocated workers. We don't know whether culturally more diverse teams are higher performing than less diverse ones. We don't know whether contractors are higher performers than full-time employees, or if it's the other way around. We can't even show that our investments in the training and development of our employees lead to greater performance. We can't say anything about any of these things, precisely because we have no reliable way to measure performance.\n\nSo when you read definitive statements about these things or, for that matter, about any other aspects of performance, your data-quality alarm bells should ring in your ears. While each of these things might be true, the exact opposite might also be true. Until we come up with a reliable way to measure individual knowledge-worker performance\u2014whether this means the performance of a nurse, or a software developer, or a teacher, or a construction worker\u2014any claim about what drives performance is not valid. No one knows, and anyone who claims to know simply doesn't know good data from bad.\n\n\u2022 \u2022 \u2022\n\nWhat can you do about this? Well, you can start by asking about it. Ask where your performance or potential ratings come from. Ask what competencies you might have been rated on. Ask to see the survey questions themselves. And if you see a survey filled with questions in which the rater is being asked to rate you on your specific behaviors or competencies, ask whether this survey has taken into account the Idiosyncratic Rater Effect. You will probably get a blank stare in response, so perhaps have this chapter handy, or download one of the articles mentioned earlier. More than likely you will see no immediate change in either the tool or the process, but at least you will be aware, and you will start to gain a reputation as someone who intelligently and rigorously interrogates the data you see. That's always a good reputation to have.\n\nThe other thing you can do, if you have the influence and the intention, is to change the way that the \"people stuff\" in your organization is measured. Because there is a better way\u2014a more reliable way\u2014to capture data about people. And it is based on this truth: although we are not reliable raters of others, _people can reliably rate their own experience_.\n\nIf we ask you to rate your local representative on his political savvy, your rating is not a reliable measure of something called \"political savvy\"\u2014you are not able to reach into his psyche and reliably weigh the presence or absence of this abstract quality. However, if we ask you today who you plan to vote for, your answer _is_ a reliable measure of who you plan to vote for today. It is a much humbler measure since it is asking only that you tell us about how you are feeling today about your voting preference, but it is a _reliable_ measure of what it says it's measuring.\n\nLikewise, if we ask you to rate one of your team on \"growth potential,\" then your rating is unreliable\u2014because what is growth potential, and how can you be the judge of it? But if we ask if you plan to promote her today, your answer _is_ reliable. While you may not be able to project into _her_ psyche and accurately perceive her growth potential, you are able to ask _yourself_ if you plan to promote her today, and the answer you get back will be a reliable one. (When we report on our own experiences, we have all the data we need\u2014we have perfect data sufficiency\u2014because we're with ourselves a lot!) Your answer is exactly and only what it purports to be: your subjective reaction to her, carefully measured. It is both a humbler piece of data and, at the same time, a more reliable one.\n\nIn the same vein, your rating of a team member on something called \"performance\" is unreliable, because your definition of _performance_ is unique to you. But in contrast, your response to the question, \"Do you turn to this team member when you want extraordinary results?\" is entirely reliable. With this question we are not asking you to stand above her, and outside of yourself, and opine dispassionately on her performance. Instead we are asking you to look inside yourself and tell us simply whether you feel confident to go to her when you want something done excellently. You cannot be wrong about this, because there is no right or wrong, only your feeling about what you would or wouldn't do with this team member. Someone else might disagree with you, but this doesn't make that person right\u2014it just makes his or her reaction to the team member different from yours.\n\nOnce again, the data here is humbler (it's just you rating your own experience) and at the same time more reliable (you _know_ your own experience).\n\nSo as a general rule, if you're after good data, be on the lookout for questions that ask only that you rate your own experience, or intended actions. You may not know if questions such as these are valid\u2014that is, you may not know if the responses to them predict something in the real world\u2014but at least you'll know that these responses will be reliable. And just to be clear, _reliable_ doesn't mean _accurate_. _Reliable_ means something doesn't fluctuate randomly. Thus, when we say you are a reliable rater of your own experiences or intentions, we do _not_ mean that you are an accurate rater of your own personality or performance. If we ask you to rate yourself on performance or on growth orientation or on learning agility, you are most definitely _not_ an accurate rater of these things\u2014if these things even exist. Instead, all we mean is that you are a reliable rater of your own internal experiences and intentions. That's it.\n\n\u2022 \u2022 \u2022\n\nLooking through this lens, we can now begin to answer the thorny question of how to measure knowledge-worker performance: we can use our reliability in reporting our experience and intended actions to design a different type of question. And the trick is to invert our line of inquiry. Rather than asking whether another person has a given quality, we need to ask how we would _react_ to that other person if he or she did\u2014we need to stop asking about others, and instead ask about ourselves. Once we've designed questions like this, we could then simply ask team leaders, every quarter or at the end of every project, what their experience was like of each team member. Here's what that would look like in practice.\n\nWe could ask a question about the quality of the team member's work, such as the one from earlier: \"Do you always go to this team member when you need extraordinary results?\"\n\nWe could ask another about how \"team-y\" a person was, not by asking the team leader to rate the person on collaboration or cooperation, but instead by asking the leader about what he would do, or how he would feel, were he in the presence of someone who was highly collaborative: \"Do you choose to work with this team member as much as you possibly can?\"\n\nWe could ask about the team member's future prospects. And here we would again steer clear of having the team leader rate the person on potential or some other abstract characteristic. Instead we'd ask about intent, like this: \"Would you promote this person today if you could?\"\n\nAnd finally, we'd probably want to ask the team leader if there was anything in the person's work to be concerned about, with a question such as this: \"Do you think this person has a performance problem that you need to address immediately?\"\n\nHere we have four questions, each asking the team leader to tell us about the team leader's own feelings and intended actions. Now, the responses to these questions are not a perfect measure of each team member's complete performance\u2014there is no way to get at that, or even define it\u2014but they do give us a reliable view of what every single team leader feels about every single team member, and what each intends to do with each.\n\nWe tend to think that subjectivity in data is a bug, and that the feature we're after is objectivity. Actually, however, when it comes to measurement, the pursuit of objectivity is the bug, and reliable subjectivity the feature. These questions generate reliable (and subjective) data, and while this isn't everything, it's a lot. In the same way that measuring a person's weight doesn't give a complete measure of his health but does at least give a reliable measure of something that is clearly part of health, these four items allow us to see, reliably, something that is clearly part of performance.\n\nWe might observe here, by the way, that the question \"what is performance?\" is exactly as abstract, and about as helpful, as the question \"what is health?\" We don't actually try to measure health today\u2014we use a series of discrete measures instead. We can ask if your Body Mass Index (BMI) is too high. We can ask about your glucose levels. We can measure your recovery rate after exercise. And we can do something with the information that we gather, because its specificity leads to further helpful inquiry and action\u2014whereas deciding that you rated a 4 on health wouldn't be much use at all. The key to understanding performance is to stop thinking of it as a broad abstraction, and instead start finding elements of it that we can measure reliably and act on usefully.\n\nOf course, we might worry that some team leaders lack sound judgment. But we will never be able to find a valid, data-based way to identify which leaders should be trusted and which shouldn't, so the best course of action is simply to ask every team leader to answer these questions, or some like them, about every team member, every quarter. Then, at every talent review we would know that we were looking at precisely what every team leader feels about and would do with every team member. This is a humbler claim for the data to make, but because we have set our sights on what is measurable, not on what is True, we can be certain of what we have. This is what reliable performance data should look like.\n\nWell, actually, this is what reliable performance data _does_ look like. Here are the answers to the first two of our questions\u2014the question about extraordinary results and the question about working with a team member as much as possible\u2014from a group of team leaders at Cisco (see figure 6-1). For both of these questions, Cisco has applied an algorithm that controls for each team leader's unique rating \"fingerprint\"\u2014whether he or she rates more leniently or more strictly, and whether his or her use of the scale is broad or narrow\u2014so that this data is capturing as precisely as possible what each team leader thinks.\n\nAs you can see, beyond their reliability, these questions also create natural variation. Cisco doesn't need to force the curve, because the team leaders' answers to these carefully worded questions create unforced range.\n\nArmed with this humble, reliable, real-world data, Cisco is now able to start answering some fun questions, and then acting on the answers. The company now has reliable, variable, and valid data at the individual level on both performance and engagement, and so can start looking for connections between the two. And Cisco has discovered, for example, that when team members feel strongly that they understand what's expected of them, that they get to use their strengths frequently, that they will be recognized for great work, and that they're constantly challenged to grow (that is, when they have high scores on the \"Me\" engagement questions we saw in chapter 1), then their team leader, independently and without knowing their engagement scores, will tend to give them a higher score on the first performance question\u2014will tend, in other words, to go to them more often for excellent work.\n\n**FIGURE 6-1**\n\nDistribution of standardized scores\n\nFurther, when team members feel strongly that they are surrounded by people who share their values and that their teammates have their back (two of the \"We\" engagement questions), then their team leader, again independently and without knowing their scores, will tend to give them a higher score on the second performance question\u2014will tend to look to work with them as often as possible. All of which might seem like research esoterica, until you're a team leader wanting to raise someone's individual contribution\u2014in which case you should talk to him or her about expectations, strengths, recognition, and growth)\u2014or until you're a team leader wanting to raise someone's team contribution\u2014in which case you should talk to him or her, and the entire team, about what excellence means to all of you, and how everyone can support one another in all the things you're doing.*\n\n\u2022 \u2022 \u2022\n\nWe began this chapter by asking you how you can be confident that the truth of you is in the room during the talent review\u2014how you can be confident that decisions about your pay, your next role, your promotion, and your career are being made based on a true understanding of who you are.\n\nBut actually, you don't want the truth of you in the room.\n\nYou don't want someone to be in any room pretending that they have a reliable measure of _who you are_. In the same way that you hated your singular performance rating\u2014you were never just a 3, because you were never just a number\u2014so you will come to despise the newer tools that now claim, ever more loudly, to capture all your essential competencies. They don't, and they never will: they simply add gasoline to the conflagration of bad data purporting to represent you. Any tool that pretends to reveal _who you are_ is false.\n\nWhat you want in the room is different: not the truth of you, but just the truth. You don't want to be represented by data that attempts, arrogantly, to divine who you are. Instead, you want to be represented by data that simply, reliably, and humbly captures the reaction of your team leader to you. That's not you, and it shouldn't pretend to be you. It's your leader, and what she feels, and what she would do in the future. And that's enough. Truly.\n\n*In case you're curious, we did the math, and the answer appears to be something like 54,340,365,926,000,000,000,000,000,000, give or take. You might want to assemble your own crowd of West Country punters to check this.\n\n*One reason for this is that, to return to our earlier example, a toe loop jump existed before it was called a toe loop, so the precision of its definition is inherent in its name. Business acumen didn't exist as a thing until we named it, and as a result it's just an abstraction defined by other abstractions, and will always stubbornly resist any more precise definition.\n\n*Strictly, to capture more of the existing range in the real world by identifying more precisely a particular experience that varies from person to person.\n\n*We've included more of Cisco's and ADP's research findings in the appendixes, for those of you who want to learn more.\n\n#\n\n# People have potential\n\nJoe's an entrepreneurial sort. In the early days of the internet, he founded a pioneering yellow-pages company that integrated directory listings with mapping technology, and managed to secure backing from a venture-capital firm. The investors came in and, as is the practice of such firms, evaluated all the existing executives on their potential for guiding the future of the company. Sadly for Joe, they decided that he didn't have much of it. He had never displayed leadership in his high school or college life, he wasn't class president or captain of the lacrosse team, and now, looking at his current work and style, they determined that he lacked the potential to set the future vision and to build the right team around him. They demoted him to head programmer, and brought in a professional executive to run the company.\n\nJoe didn't shine in this new role either. He had some software skills, but they were unpredictable, resulting in a mess of spaghetti code that other, more experienced developers had to pull apart and detangle. In fact, so messy were his creations that the entire code base of the company's product had to be rewritten. Everyone agreed that although Joe clearly had drive, he would never become one of the company's leading software engineers. He just didn't have enough potential.\n\nBecoming increasingly frustrated with his diminished position, and sensing that the investors didn't see much of a future for him, Joe waited for the company to be acquired and then left to start his own financial-services company. Here, he did what he'd always done\u2014worked hard, pushed hard, challenged everything\u2014and his new company grew large enough that an even bigger player swooped in and bought it from him.\n\nThe leaders of this new company, too, were unimpressed with his potential\u2014or confused by it, or something\u2014so he left once more, this time to see whether he could do interesting work in the fields of mechanical and electrical engineering. The jury is still out on his new ventures, and real profits have yet to show up on the books, but with him at the helm, his companies currently employ hundreds of people and are making truly innovative products. If he hadn't done what he did, these jobs wouldn't exist, and neither would the products. And in this sense, Joe is exactly what we want a team leader to be: a person who makes the most of his unique strengths and thereby creates a better future for all of us.\n\nJoe's experience is relevant here because this chapter is all about the future. Specifically, it's about your future, and the future of everyone on your team\u2014and about all the Joes out there in teams large and small, who are misunderstood by their companies, mislabeled, mismanaged, and, in the end, missed altogether.\n\n\u2022 \u2022 \u2022\n\nJust for a moment, think of all the people on your team. Bring to mind each of their faces and names. Imagine what they're working on now, how they like to work, what they thrive at doing, what they struggle with, and what they aspire to. And now, if you can, answer this: Which one of them has the greatest potential?\n\nSooner or later in your time as a team leader, you'll be asked this exact question and told to plot your response on the _potential_ axis of your nine-box grid. And as you ponder your answer, you'll pretty quickly run into some challenges. You might be quite clear that Jack is doing really well in his job today, but find yourself unsure of whether that means he has potential. And you might be equally certain that Jill is also doing well, but at the same time realize that her job is very different from Jack's job. If one of them has potential, does the other? If, as seems to be implied, potential is some sort of universal quality, then how should you gauge it in two different people doing two different jobs?\n\nAnd what if Jill is in fact struggling in her current role? You might start to ask yourself whether current performance is the same as future potential or merely a clue to it, or whether, alarmingly, the two are not related at all. Perhaps you'll think to yourself that Jill might have, hidden somewhere within her, the potential to do really well at something else. You might not ponder this for long, though, because if (like Joe) she seems to lack potential in one role, and then subsequently another, it will be quite hard to convince yourself that she does indeed have potential for an entirely different role. If she's struggling now, then won't she struggle wherever she goes?\n\nEven if she isn't struggling, if she is in fact one of your current high performers, she nonetheless wants to be challenged to grow, so you'll be forced to start thinking about other jobs on other teams, jobs she might do equally well\u2014or even better. And when she starts asking you about her future\u2014as she surely will\u2014you'll quickly find yourself peering out into the fog. Since you're not nearly as familiar with those other jobs on those other teams as you are with those on your own team, how can you truly know if she has the potential to excel elsewhere? As a good team leader, you have a pretty clear sense of her present performance\u2014what's in front of you right now\u2014but being asked to weigh her potential requires you to project out into a world you know much less about.\n\nThis can be quite intimidating, not least because you're aware that how you weigh Jill's potential\u2014specifically, how you rate it\u2014will more than likely stick to her for a long time. If you rate her highly, then the received wisdom, passed on to your fellow team leaders, will be that she is now a \"high potential,\" or \"hi-po,\" and she will carry this quality around with her wherever she goes. She will get more attention from these other team leaders, be given more opportunities, more training, more investment, and if ever her performance falters, more benefit of the doubt. On the flip side, you realize that if you rate her poorly on potential, she'll become a proverbial \"lo-po,\" which will be a tough label to shake off, no matter how hard she tries.\n\nYour rating of her on potential, or more accurately, your guess about how much value she will bring to the company in the future, will, in all sorts of real ways, _create_ her future. That's a lot of responsibility for you to bear.\n\nJill, meanwhile, perhaps aware that there's another talent review in the offing, is wondering whether she'll make the hi-po list. Like you, she isn't sure what potential is, or what a high potential is. She's just trying to do good work every day. She knows that potential is clearly a good thing to possess\u2014it comes with all sorts of goodies and perks\u2014but, at heart, what she really wants to know is whether she's doing well enough in her job right now, and where her career is going next. If your rating of her on potential helps her career, then wonderful\u2014but if it doesn't, or if being branded a lo-po makes getting help with her future less likely, then she's going to be frustrated. There's a great deal at stake for her here. At some point, she will ask you what you rated her, and then you'll somehow have to justify your decision. And this will be super tricky, since, in the back of your mind, you'll know that you weren't so very clear what potential was in the first place, nor what clues might point you to it, nor what scale you should have used to rate her on it.\n\nBut that's a worry for later. Right now you'll look around and see that other team leaders on other teams seem able to announce confidently who has potential on each of their teams, so you'll put Jill's inevitable questions out of mind, pull out your nine-box grid, and do your best to do right by her. And her future.\n\n\u2022 \u2022 \u2022\n\nOf course, you can't really blame your company for putting you into this sort of high-pressure situation. As with all the practices we've covered thus far in the book, assigning a \"potential\" rating to each employee is a product of some very good and necessary intentions. Your company is a maximization machine\u2014it wants to make the best use of its finite resources\u2014so it is greatly interested in identifying precisely who to invest in, and how.\n\nThe problem with this stems from the way your company executes on these good intentions. Why, for example, does it assume that it will net a good return only from certain people? Surely, the clich\u00e9 that \"Our people are our greatest asset\" applies to _all_ of the people in the company. As we've seen, every human brain retains its ability to learn and grow throughout adulthood. For sure, each brain grows at a different speed and in a different way, but this implies only that each person learns differently, not that\u2014categorically\u2014some people do and some don't. Therefore, the best course of action for any maximization machine worth its salt would be to figure out where and how each brain can grow the most, rather than zeroing in on only a select few brains and casting aside the others.\n\nBut sadly, somewhere along the line, companies by and large recoiled from this natural diversity, seeing it as simply too varied and too individualized to make sense of, and decided instead that the most pragmatic approach would be to invent a generic quality called \"potential,\" rate every person on it, and then invest most in those who have lots of it, and much less in those who don't. As with all the lies we've addressed in the book so far, the lie that _people have potential_ is a product of organizations' desire for control, and their impatience with individual differences.\n\nWhen you think about it for a moment, the notion of a generic quality called \"potential\" is actually pretty odd. Look around you and you'll find hundreds of different definitions, but there's no need to look any further than _Harvard Business Review_ 's very own:\n\n> High potentials consistently and significantly outperform their peer groups in a variety of settings and circumstances. While achieving these superior levels of performance, they exhibit behaviors that reflect their companies' culture and values in an exemplary manner. Moreover, they show a strong capacity to grow and succeed throughout their careers within an organization\u2014more quickly and effectively than their peer groups do.\n\nThis seems like an eminently desirable quality. Who wouldn't want people who \"outperform their peer groups,\" not just in their current role but \"in a variety of settings\"; who, in addition to performing with excellence, also \"reflect their companies' culture and values\"; and who, all the while, show \"a strong capacity to grow\"? We all would, of course\u2014high-performing, culture-embodying people blessed with oodles of learning agility and lashings of successitude are the stuff of every team leader's dreams.\n\nAnd yet, this definition almost immediately rings hollow for you. First, there's the feeling that, although you might want such a person on your team, you don't recognize yourself in the definition. When you think about yourself at your best, you land on specific activities you love, or skills at which you shine\u2014whereas in contrast, this definition appears strangely vague, untethered from any actual work. And then there's the part of the description that seems to imply that you can excel anywhere, at virtually anything, \"in a variety of settings and circumstances.\" Not only is this unlikely, but more to the point, who among us actually aspires to this sort of Jack-of-all-trades-ness? If we were to have this quality it would imply, surely, that we were not unique and distinct, but instead were empty learning vessels, blank slates waiting for our settings and circumstances to define us, adept at learning, but featureless. How depressing.\n\nBeyond the disquieting emptiness of this definition, the most damaging inference is that this quality called \"potential\" is inherent in a person, and that people bring it with them from situation to situation: that no matter what \"setting or circumstance\" they encounter, those people with lots of it are blessed with a special power enabling them to learn faster, grow more, and achieve more. High potential is the corporate equivalent of Willy Wonka's Golden Ticket: you take it with you wherever you go, and it grants you powers and access denied to the rest of us.\n\nIn chapter 4 we drew the distinction between traits, which are inherent in a person, and states, which are changeable in the person. Using this framing, potential is clearly something we think of as a trait\u2014it is inherent in the person, some people have more of it than others, and those who do take it everywhere with them.*\n\nAssuming just for the moment that potential actually is a trait, the first problem we encounter is how to measure it. As we saw earlier, if we want to measure a trait, we can't ask someone to rate you on it, because it's impossible for any rater to be either perceptive enough or objective enough to reach into your psyche and assign a number to what they see inside you. And in the case of potential, the measurement challenge is orders of magnitude more difficult, since we are asking the rater to rate you not on a trait displayed in your current behavior but on a projection, a probability that you possess something that might just possibly be displayed in some future situation. It's flat-out impossible for the rater to do this reliably, so whatever data he produces about you will be the very worst kind of bad data. Yet this data will, as we saw with Jill, create the future.\n\nBut is there even anything here to measure\u2014is potential a thing at all? Do we really think that there exists in people a trait that confers on some lucky few the ability to grow more and learn more regardless of setting or circumstance? That we could throw this hi-po into any situation and his potential would enable him to adapt, and then thrive? That this general potential will act like a turbocharger, and take any inputs from the world of work and boost them into outstanding performance?\n\nIf we do think this, then we do so in the complete absence of any evidence. Over the last hundred years we've wondered whether there was such a thing as general intelligence\u2014the elusive _g_ factor\u2014and discovered that if it exists, we can't find it. Sure, we can build a test that reliably measures a thing called IQ, but we don't actually know much about what IQ is\u2014it doesn't seem to independently predict educational success, career achievement, health, or happiness. It's just a score on a test. The best this test can do, it appears, is tell us that, if your test score is very low, you probably have cognitive impairment and will therefore have difficulty learning. So it works as a predictor of problems but not as a predictor or descriptor of flourishing.\n\nLikewise, evidence for the existence of general potential is nonexistent. Instead the evidence points in exactly the opposite direction. We know that each person's brain grows by adding more synaptic connections, that each person's synaptic pattern is unique, and that therefore each person's brain grows uniquely. Therefore we know a) that the ability to learn exists in us all, b) that it shows up differently in each of us, and c) that while we can all get better at anything, none of us will ever be able to rewire our brains to excel at everything. More simply, we can all get better, and we will all get better at different things, in different ways, and at different speeds.\n\n\u2022 \u2022 \u2022\n\nSo there is no such thing as having potential. Or rather, there is, but it doesn't mean anything. Or rather, it doesn't mean anything beyond being a human. To say that you have potential means simply that you have the capacity to learn, and grow, and get better, like every other human. Unfortunately, this won't reveal anything about precisely where you can learn, and grow, and get better, or how, or how fast, or under what conditions. Potential, like being human, doesn't tell us anything about what particular human you are, or what direction would be best for your sort of human in the future. And, of course, if having potential is just being a human, then we can't rate you on it. We can't split our company up into hi-po's and lo-po's, any more than we can rate you on your human-ness and give the most stuff to those who are most human and the least to those are who least human.\n\nThis sort of apartheid does terrible things to a company. The careless and unreliable labeling of some folks as hi-po's and others as lo-po's is deeply immoral. It explicitly stamps large numbers of people with a \"less than\" branding, derived not from a measure of current performance but from a rater's hopelessly unreliable rating of a thing that isn't a thing. And then this rating of a thing-that-isn't-a-thing opens doors for some, confers prestige on some, elevates some, blesses some, and sets them up for a brighter future, all while relegating others to a status less than human. How explicitly awful.\n\nIt is also unproductive. The maximization machine should make the most of every single human within it, not just a rarefied subset. This notion that some people have lots of potential, while others don't, leads us to miss the gloriously weird possibilities lying hidden in each and every team member, even the ones who, at first blush, seem to have little to offer the team's future. If we have in our head a preconceived notion\u2014even, as in the case of the _Harvard Business Review_ definition, a detailed description\u2014of what a hi-po should do, feel like, and act like, then we will cease to be curious about the many possible futures of each idiosyncratic person on our team.\n\nThis, certainly, is what happened to Joe's employers. They had a set idea of what a high-potential CEO should look like, and what a high-potential software engineer should look like, and neither of them looked like Joe. They stopped looking at Joe, became impatient with him, diminished his role, eased him off to the sidelines, and were more than happy when he decided that his most interesting and challenging work lay elsewhere.\n\nAnd that's a shame for them, because \"Joe\" is a pseudonym. His real name is Elon. That yellow-pages company was acquired by Compaq for $307 million. The financial-services company, X.com, became better known as PayPal and sold to eBay for $15 billion. At which point you may say, \"Yes, but have you seen what he's done lately?\" and reference his fining by the SEC, his joint-puffing on a podcast, and any number of other transgressions that may have occurred from the time of our writing to the time of your reading. And our reply would be, \"Yes, but have you seen what he's done lately?\" and we'd reference his reinvention of the automobile industry, his reinvigoration of the space industry, and his counterintuitive alarm-sounding of the dangers of AI. As the _New York Times_ put it immediately after the 2018 SEC action against Musk was concluded, \"The Future of Electric Cars Is Brighter with Elon Musk in It.\" Yes, he is the spikiest sort of leader, given to impulsive and imperfect actions, but to dismiss his potential is to miss pretty much everything meaningful about him. He may be a handful, and intemperate in his tweeting, but if Elon Musk wasn't a high potential, then it's time to admit that the concept serves no purpose.\n\n\u2022 \u2022 \u2022\n\nYet still you are going to be asked by your company to rate people on their potential, and by your team members to guide them toward ever-more-challenging work. So what on earth do you do? How can you honor your company's need to get the most from each person, and yet not segregate your team into artificial and demeaning categories, such as hi-po's and lo-po's?\n\nYou can start by taking a trip to the north coast of Scotland. Imagine, for a moment, that you've settled down in a small village just outside Inverness, and have opened up a hot-air-balloon sightseeing business. Your business has done well\u2014all those lovely Scottish moors to gaze down upon\u2014and your team has grown to five excellent hot-air-balloon pilots. One of them is named Maureen. She comes up to you one drizzly afternoon and says, \"I love my current job, but I want to continue to grow. I want to stay challenged, round out my r\u00e9sum\u00e9, have more to offer. I think I want to become a glider pilot. Can you help me?\"\n\nWhat do you say?\n\nWell, here's what you don't say. You don't say, \"Maureen, do you have enough potential?\" Nor do you say, \"Maureen, do you have enough potential as a glider pilot?\" You do not say these things because these are not things human beings say to other human beings in the real world.\n\nInstead, whether consciously or unconsciously, you'll find yourself asking her two discrete sets of questions. And what's interesting about these two sets of questions is that they lead us away from generic potential and toward a far more useful concept for helping us understand Maureen, and for guiding her career.\n\nThe first set of questions will focus on who Maureen is as a person. You'll find yourself asking her, \"What do you love most about your job right now, Maureen? What do you love most about ballooning? Do you love the piloting part, the thrill of the lift, the sensation of getting airborne; or do you love the navigating part, the movement of the light-as-air balloon through the cold northern winds, the calibration of the flame to achieve just the right altitude; or do you love the part where you show the guests the sights and get to tell them interesting facts they may not have known about this part of the world?\" You'll ask her about what job she thinks she might want next, and what it is about being a glider pilot that she thinks she might enjoy. You might even ask her what her \"perfect job\" would look like. Each of these questions addresses who Maureen is as a person, what she loves, what she's really into, and what she yearns for in terms of her career. You're basically being curious about the specifics of what it's like to be Maureen-at-work.\n\nThe second set of things you'll ask her about will focus on how she's moved through the world thus far, and what she's picked up along the way. You'll ask her about her current performance: how many balloon trips she completes and how many guests she takes up in a month. You'll ask her about her past performance: you'll want to find out how long she's been a balloon pilot, how many hours she's logged, what her safety record has been, how often she is able to put her balloon down within the designated landing range. And then you'll get into her skills, not by asking about ratings and 360-degree scores\u2014no one in the real world asks about such things\u2014but instead by asking if she has her level-one, -two, and -three certifications for hot-air-balloon piloting, whether she has extended this to completing her giant-hydrogen-filled-airship certification (in your mind you might call this the Hindenburg test), and, of course, whether she has her glider-pilot's license. And with each of these questions, you'll learn more about how Maureen has moved through the world\u2014what she has measurably achieved, and measurably learned.\n\nFrom the answers to these two sets of questions, you'll have discovered, first, who Maureen is at work. These are her traits. These are things that are inherent and enduring in her\u2014not entirely unchanging, but nonetheless resistant to change. These are the loves and aspirations that are uniquely hers, and that she carries around with her everywhere she goes, just as surely as she carries her own body. Wherever she goes, they are there. You can call these her _mass_.\n\nAnd second, you'll have unearthed some things she's acquired as she's applied herself in the world to move in a particular direction: her current and past record of performance, and her tested certifications. Obviously, since she can change any and all of these things, these are states. But since they describe how she has moved through the world\u2014how she's done it, how well she's done it, how quickly, and in what direction\u2014you can usefully label these her _velocity_.\n\nIn the world of physics, there's a name for the discrete, measurable, definable, and directional thing that is produced when mass and velocity combine. It's called momentum. In the world of teams and team members, the same applies. Maureen has _momentum_.\n\nBy keeping these two ideas about Maureen\u2014mass and velocity\u2014separate, and by using _momentum_ to describe their combination, we suddenly enable you, the team leader, to do all manner of useful things to help her.\n\nFirst, you reject the apartheid of potential, where everyone is separated into hi-po and lo-po. \"Do you or don't you have potential?\" is a question that exists to serve the (well-meaning but misguided) company. But it's not helpful to you as a team leader, and it's completely uninteresting and unhelpful to Maureen. Because she knows it's not a matter of _whether_ she can learn and grow, but _how_ , and how efficiently, and in what direction. Only certain people have \"potential\"; _everyone_ has momentum. One team member's might be more powerful than another's, or speedier than another's, or pointed in a different direction, but everyone has some. The question isn't whether you inherently possess a lot of it or not. Instead, when it comes to momentum, the question is how much of it you have at this very moment, right now.\n\nSecond, you convey to her something real: namely, that the speed and trajectory of her momentum at this very moment are a) knowable, b) changeable, and c) within her control. When you talk to her about her momentum, you help her to understand where she is at this moment in time, not so that she can be catalogued and categorized and put into one box instead of another, but so that she can understand what paths are possible next. Her career is moving on a particular trajectory at a particular speed, and she\u2014with your help\u2014can take the measure of her accomplishments, her loves and loathes, her skills and knowledge, and see where she can accelerate, or shift the path slightly, or even attempt a great leap. Where potential is assumed to be a fixed, inherent quality\u2014she's a hi-po or a lo-po\u2014momentum is, by definition, always in a state of change. And if Maureen wants to speed it up, or alter its direction, she can.\n\nThird, you help her identify which parts of her current career are a function of who she is as a person\u2014parts she will therefore likely bring with her, situation to situation\u2014and which parts are entirely situation-dependent, and which she could change if she so chose. Given how close we all are to our own performance, and given that we are sometimes misguided in our career desires, this kind of subtle and specific insight could very well prevent her from making an ill-advised career move.\n\nFinally, understanding Maureen's career in terms of momentum doesn't just benefit her. It frees you, as her team leader, from the awful burden of having to determine her entire future based on a fiction.\n\nIt's not true\u2014or, indeed, useful\u2014to think that _people have potential_. Instead, the truth is that _people have momentum_.\n\nPotential is a one-sided evaluation. Momentum is an ongoing conversation. In a world of \"potential,\" it's hard to imagine what, exactly, a career conversation looks like once Maureen has been shunted off into the lo-po dungeon. Momentum, on the other hand, represents the opposite of \"up-or-out\" thinking. And it's the best concept to address one of the key survey items that measure engagement and performance: \"In my work, I am always challenged to grow.\" Potential doesn't do that\u2014it doesn't challenge you to grow. It tells you that you either will, or you won't.\n\nAddressing their potential makes people feel like they've been dealt with. Addressing their momentum makes them feel _understood_. More important, it helps them understand themselves, by encouraging them to consider where they are, right now\u2014not as a point of stasis, but as a unique human being moving purposefully through the world.\n\n\u2022 \u2022 \u2022\n\nBut if this sort of straightforward conversation about the future is what is happening in the real world, we might well ask how we came to be ensnared in the bizarre apartheid-creating theory world of potential. And again, the lie that _people have potential_ is another example of a specific and useful thing made generalized and useless. In this case, it's perfectly fair to ask if Maureen has the \"small-p\" potential to do a certain job well. But as soon as we divorce the idea of Maureen's potential from the very specific demands and rewards of a particular job\u2014as soon, that is, as we stop asking about who Maureen is, and where she's going, and how those things mesh with what she might do next\u2014and instead treat her potential as some abstract, mystical, and essential quality of hers that we can isolate and rate her on, then we slide down the slippery slope into untruth. And if we then institutionalize this thinking through our people-management processes and systems in the name of bringing predictability and control to our organization, we find that we have sacrificed common sense and humanity at the altar of corporate uniformity, and we shouldn't be very surprised if our people chafe at the result. We may well also find that we've created a system that codifies and amplifies our biases and projections\u2014that the people with the most \"potential\" inevitably wind up looking and acting a lot like us.\n\nOur people tools and processes can never compensate for bad team leaders. We like to think so\u2014we figure that, even if your team leader is ignoring you, at least your crowd-sourced feedback will tell you how you're doing; or that, even if your team leader never asks about your career, at least the talent review will give you something to go on. But aside from the flaws that we've already seen with these and other common approaches, any large-scale system can never hope to replicate the very particular and specific attention that a team leader can offer. Again, teams are where we live, and team leaders can make or break that experience for us. And rather than investing in systems and processes to provide a fallback in case our managers are found wanting, it's far better to invest in helping our team leaders do what we need them to, by 1) getting rid of ratings of \"potential,\" 2) teaching team leaders what we know about human growth, and 3) prompting them to discuss careers with their people in terms of momentum\u2014in terms of who each team member is, and in terms of how fast each is moving through the world. This is harder, of course, than buying the latest piece of enterprise software and then imploring our people to use it, but it's the right hard thing to do.\n\nBecause when team leaders understand how careers are built in the real world, they begin to think like Andy.\n\nAndy is a team leader at Cisco, and a little while ago he set out to help each of his team members see what their futures could look like. He started by asking them to imagine their dream jobs\u2014in other words, to reflect on their aspirations\u2014and then to make this practical by searching on LinkedIn. The task he set them was to split into pairs, and then to spend two hours searching on LinkedIn for jobs that came close to their ideal\u2014with no limit on which company, or industry, or line of work these jobs were in. He asked them to work with their partner to narrow down the list to the one or two roles that excited them the most.\n\nAnd then he asked them to analyze these jobs in terms of the skills and experiences and qualifications they drew on, and then to compare those lists with the skills and experiences and qualifications that each of them already had, and to figure out which new ones they wanted to go after.\n\nHe wasn't, in other words, evaluating them on their potential, and sorting them into those who could grow in some way and those who couldn't. Instead, he was helping each of them to get clear on who they were and where they wanted to go (again, _mass_ ), and on the measurable skills and experiences they had and wanted to acquire (again, _velocity_ ). His presumption was that everyone had momentum, and that it was his job to help them figure out how to direct it. \"I think there's a lot of energy within our teams that isn't being used,\" he told us, \"and I have a lot of people who, given the right circumstances, the right engagements, the right customer\u2014whatever it is, we can find what enables them to share that energy, to bring it out.\"\n\nThe results were fascinating. As Andy explained to us, \"We looked around the room at each other and realized that our careers were bigger than Cisco, and that we could go further down those steps to define ourselves within the market as professionals.\" And more than this, his team members were able, with his encouragement, to make many of the skills they wanted to learn part of their current jobs\u2014so that their work, every day, was helping them build the skills they wanted for the future. \"It changed the conversation,\" Andy said, \"from how we could get another job outside, to how we could build our practice to be the best professionals we could be\u2014to deliver better service internally, but also to build up our transferrable skills.\"\n\nAnd this, surely, is what any truly people-maximizing company would want.\n\n*Although we might question why, if it's a trait and doesn't therefore change much, we re-rate people on it every year.\n\n#\n\n# Work-life balance matters most\n\nWork is hard. Every day, you feel the stress of performing, of delivering against your goals and objectives, of earning enough to support your family, of learning how to advocate in just the right way to advance your career and thereby earn more. And always, hanging over your head, is the threat of change as your company shifts its focus, outsources your role, or finds a particularly smart machine that can do your job better, faster, and cheaper. And then there are the other people you have to work with\u2014an ever changing cast of characters, some of whom work across the hall, others of whom work across the world, whose collaboration you seek, but whose motives and methods remain mysterious. The commute doesn't help: the daily battle with your fellow strivers on trains, planes, and freeways, everyone rushing in and rushing out, clogging the arteries of your city, raising your stress level. Forty-five minutes, an hour, ninety minutes each way\u2014or a two-hour flight if you work for one of the big consultancies and have to show up at the client site\u2014all just so you can begin your daily race of life-at-work. On the way home you steal a brief moment or two to decompress, and then, once home, you have a quick dinner with the family before dragging out the phone again for the evening volley of emails and texts, hoping to catch one last request so that it won't need immediate action before your shower in the morning.\n\nWork is hard. Particularly, it seems, if you're a physician. We might think that doctors have it better than the rest of us because at least all that frenetic dashing about is in service of something truly meaningful\u2014if we've learned anything about what we want from our world of work, it's that we crave work with meaning and purpose. And we imagine that, while doctors have to complete forms and wade through other assorted administrivia, they nevertheless see their patients, one after another, being cured and getting back on their feet again, all by dint of the doctor's efforts and expertise. Would that we all saw so clearly and so often the point of our work. Would that we could all do what we love.\n\nWhen we look at the data, however, that's not what we see: despite the purity of their purpose, physicians seem to have it harder than the rest of us; or at least, they feel it harder. According to a recent report from the Mayo Clinic, 52 percent of physicians report being burned out, and their incidence of PTSD is 15 percent, four times the levels in the regular workforce and three percentage points higher than the levels found in veterans of the wars in Iraq and Afghanistan. These sky-high stress levels inevitably have quantifiable and negative effects on both patient care and physician well-being. The Mayo Clinic found not only that a 1 percent increase in measured burnout led to a 20 to 30 percent decrease in patient satisfaction but, more worryingly still, that 15 percent of all doctors have issues with substance abuse during their careers, and that their rates of depression and suicide are twice the national levels.\n\nAccording to doctors, things are only getting worse. The Mayo study further revealed that 80 percent of physicians believe the medical profession is on the decline, that 60 percent of all physicians will endure a professional lawsuit during their career, and most tellingly of all, that 73 percent of doctors would _not_ recommend the medical profession to their children. As a result, if current trends continue, by the year 2025 the United States will experience a shortfall of more than twenty thousand physicians.\n\nThe only job harder than being a doctor\u2014so the data tells us\u2014is being an emergency-room nurse, which comes with higher levels of burnout and depression and (at 19 percent) almost twice the level of PTSD seen in combat veterans.\n\nThe health-care profession is taking this data seriously, and is devoting significant time and money to conferences, research, and practical experiments, all to figure out what's making work so demoralizing, and what can be done.\n\nThe prevailing approach will not surprise you. Though each health system differs in its methods and priorities, the basic assumption in most of these efforts is that these days being a doctor or a nurse is unavoidably hard, and that therefore the enlightened hospital should do whatever it can to help health-care professionals recover from the stress of work, and should also find ways to limit the work week to fewer than sixty hours\u2014to provide some barrier against the ever rising tide of stress. Some health systems offer meditation rooms just off the ER, others provide support with all that electronic record keeping, still others pay for monthly meals out with colleagues and family.\n\nWith its stress levels and stress-related problems, the world of doctors and nurses serves as an extreme example of the rest of the working world. Work, our experience teaches us, is toil\u2014a stressor, a drainer of our energy\u2014and if we are not careful, it can lead to physical exhaustion, emotional emptiness, depression, and burnout. It's a transaction\u2014we sell our time and our talent so that we can earn enough money to buy the things we love, and to provide for those we love. Indeed, the term we use for the money we earn in this transaction is _compensation_ , the same word we use for what we get when we're injured or wronged in the eyes of the law. Our wages are not just money, then: they are money to make up for the inherent badness of work\u2014a bribe, if you will, to tough it out.\n\nWork is even a distraction from work. When we need to get something important done, we recognize that it will be hard to do unless we can somehow make our escape from the daily grind, and so we go on a leadership _retreat_ to get away from the noise and stress of work, to better focus on other work.\n\nAnd because the effects of work are so potentially toxic, the obvious and sensible precaution to take, so that we don't all expire at our desks, is to balance it out with something else, with something better. With _life_.\n\nWe lose ourselves in work, and rediscover ourselves in life. We _survive_ work, but _live_ life. When work empties us out, life fills us back up. When work depletes us, life restores us.\n\nThe answer to the problem of work, the world seems to say, is to balance it with life.\n\nOf course, we are simplifying things here. Some people succeed in finding great satisfaction in their work, while others have hugely stressful lives outside of work. We know, too, that some jobs seem to be inherently difficult, or even inherently boring. No one's work, or life, is ever completely joyous, or completely controllable.\n\nYet still, the assumption that pervades our working world is that \"work is bad\" and \"life is good,\" and therefore _work-life balance matters most_. \"Does the company support work-life balance?\" is right up there with \"What's the company culture like?\" in the list of questions candidates inevitably ask during the interviewing process\u2014which explains why, in these tight labor markets, companies highlight their on-site dry-cleaning, banking, and child-care services, their quiet rooms, in-chair massages, sleep pods, and luxury shuttle buses. These perks are tremendously well intended and are often highly valued by employees\u2014and at the same time are rooted in the idea that work is a heavy weight on the scales, and that the enlightened organization is one that does everything it can to lessen that weight, and thereby tip the scales back toward life.\n\n\u2022 \u2022 \u2022\n\nGood intentions aside, the problems with all this begin with the concept of _balance_ \u2014and it's a concept with a long history.\n\nIf you visit the Presidio of Santa Barbara\u2014the fort the Spanish explorers built to support the Mission of Santa Barbara\u2014you will see on the wall a list of the provisions the captain of the fort requested of his superiors in Mexico City. The date of the requisition list is 1793, and along with \"Two pounds of gold musketeer braid, fine for epaulets,\" and \"Two beaver hats; one black and one white,\" you'll find \"Four pounds of rose oil, three ounces of Galapa powder, two ounces of sweet mercury, and one little box with twelve cupping glasses, filled with seedless cotton wool.\"\n\nWhile some of these might have been used for making dessert, or drinking tea, their primary purpose was something else. From the time of the ancient Greek physician Hippocrates until the advent of modern medicine in the mid-nineteenth century, our concept of physical health was founded on the idea of balance. Hippocrates posited that each of us contained four humors\u2014black bile, yellow bile, blood, and phlegm. Even though each of our bodies held slightly different amounts of each, which in turn created our different personalities (the phlegmatic person had an abundance of phlegm, the calm or \"sanguine\" person contained more blood), the healthy person was he or she who maintained the perfect balance between all four.\n\nIf your humors were out of whack, you fell ill, and so you were told to take sweet mercury to purge you of phlegm, or to use cupping glasses to draw out your blood. With your humors back in balance, your sickness would be cured and you would be well again. The requisition list on the wall of the fort was, then, a list of supplies for the pharmacy.\n\nOver time, this emphasis on finding physical balance took on psychological overtones\u2014if you were too short-tempered, it was because your yellow bile was out of balance; if you were lazy, you had a body overloaded with phlegm\u2014and from there grew to become an explanation for the entire physical universe. The metaphysical extrapolation of the four humors was the four elements\u2014earth, fire, water, air\u2014each of which was believed to balance out the others in the harmony of creation.\n\nAll of which is to say that we humans appear to have had a thing for balance for a long, long while. To us it has always seemed like the right, the noble, the wise, and the healthy state for which we should all strive. And we can speculate that the difficultly of achieving it has added in some way to its allure\u2014it's another of those things, like working to remedy our faults, that's always a work in progress, that's fantastically hard to achieve in practice. You've striven for it, haven't you? You've tried to find that delicate balance between the needs of yourself, your family, your friends, your work colleagues, your boss, and your community. You're aware that each of these constituencies places different and often conflicting demands on you, and you've struggled to give due attention to each one, satisfying their differing needs while still attending to your own. You've sat on a conference call in the car-pool line and mouthed \"Sorry!\" to the kids in the back. You've rationalized a missed Presidents' Day outing with the family because, well, it's a Monday, your other team members appear to be online, and besides, Presidents' Day isn't a proper holiday anyway, not really.\n\nYou've taken on a \"stretch\" assignment because it might\u2014just might!\u2014come with a raise, or at least a bonus, and so enable you to afford a better house for your family. But because you now have more work to do, and more resting on it, you've found that you can't attend that school-board meeting, or your cousin's wedding, or that online management course, because life is about trade-offs and this one is yours.\n\nYou've found yourself spinning plates, or juggling balls, or plugging gaps\u2014whatever the metaphor, you've known too often the feeling of too many requests from too many quarters and not enough hours in the day. You've told yourself that if you can just keep the plates spinning, the balls in the air, the gaps plugged, then perhaps you can parcel out your attention and energy so that no one, in your work or your life, will feel too neglected\u2014so that, although you can't be all things to all people, your unflagging efforts will at least achieve some sort of equitable distribution.\n\nBut in the real world does anyone, anywhere, man or woman, young or old, affluent or barely solvent, ever actually find balance?\n\nIf any have, we haven't met them yet. And this is why balance is more bane than benefit. In practice, striving for it feels like triage, like trying to erect some sort of barricade against the endless encroachments on our time and the relentless ratcheting of expectations to work more, all while worrying that someone else has figured out how to do this better than we have. Obviously, triage can be necessary in life, but it surely is not enough\u2014it keeps things at bay, but it takes us away from ourselves. And in the end, balance is an unachievable goal anyway, because it asks us to aim for momentary stasis in a world that is ever changing. Supposing we ever get things just exactly in balance, we know for sure that something will come along and unbalance them and that we'll be back to pushing our balance rock up the hill again. Balance as an ideal erases our humanity\u2014the essence of who each of us is and aspires to be\u2014and replaces it with a Sisyphean coping strategy.\n\nSo what then should we do? Work can be hard. So can life. And there's too much of both, too much of the time. If balancing everything out isn't the answer, then what is?\n\nWe need a new way of thinking. About work. About life.\n\n\u2022 \u2022 \u2022\n\nThroughout this book we've found our answers by trying to examine the world as it really is, rather than as we wish it were. While initially it might seem that certain elements need to be in balance (acidity or insulin levels in the body, for example), when we look more closely what we find is flow. Today we know that all matter is made up of infinitely more particles than four, and that the balance between these millions of particles is far less important than the ongoing relationship between them all and the biological, chemical, and physical processes that these relationships create. Something can be said to be \"healthy\" only when its process allows it to take inputs from the world, and first, metabolize them to produce something useful, and second, do so in such a way that it can keep doing it. Health is less balance than it is motion.\n\nYou are one such process. Neither you nor your life are in balance, nor will you ever be. Instead you are a unique creature who takes inputs from the world, metabolizes them in some way, produces something useful, and does so in such a way that you can keep doing it. At least, you are when you're healthy, when you're at your best, when you are contributing all that your talents allow you to. When you're flourishing you are acting on the world and it on you. Your world offers up to you raw material\u2014activities, situations, outcomes\u2014in all parts of your life, and some of this raw material invigorates you and gives you energy. You are at your healthiest when you find this particular kind of raw material, draw it in, allow it to feed you, and use it to contribute something\u2014and when that contribution actually seems to leave you with more energy, not less.\n\nThis state, not balance, is what we should strive for. What should we call it?\n\nThe Greeks called it _eudaimonia_ , which sounds like a cleaning product but which actually means \"the fullest and purest expression of you in your most elevated state.\" Their idea was that each of us had a spirit, or _daimon_ , that embodied our greatest and most unique possibilities\u2014our natural strengths or talents\u2014and that the state we should all seek was one where, because of the happy intersection of our role, our skills, our team, and our context, we turned these possibilities into contribution, and thus liberated our good spirit.\n\nWe could stick with the Greeks and _eudaimonia_ , but while the word captures the meaning quite well, it's a bit of a mouthful. So let's play out what this state actually looks like in the real world, a world so busy and distracted that it doesn't seem much concerned with your particular _daimon_ , and see if that helps us find something a little more down-to-earth.\n\nMiles is a physician. More specifically he's an anesthesiologist, the chap who puts you to sleep and wakes you up again. He lives in the United Kingdom, where anesthesiologists are called anesthetists. He loves what he does and has been at it for twenty years. Tens of thousands of his colleagues may be struggling with burnout, but he seems to revel in his work.\n\nWe were interviewing him the other day, trying to figure out how a regular doctor working in a regular ward of a regular National Health Service teaching hospital could find a state of mind and heart that has eluded so many other physicians, when we made a rude discovery: Miles doesn't like sick people. To be more specific, he doesn't seem to get much of a kick out of helping sick people get better.\n\nHere's what this discovery sounded like in our interview:\n\n_Marcus and Ashley:_ So, Miles, is there anything about your role that brings you down, or frustrates you?\n\n_Miles:_ Other than the hours?\n\n_Us:_ Yes. Anything in the work itself?\n\n_Miles:_ Well, I really don't like the follow-up.\n\n_Us:_ Pardon?\n\n_Miles:_ Yeah, I really don't like meeting up with the patients after the operation, seeing how they're doing, checking up on their recovery, giving them a few things they can do at home to alleviate their symptoms, and then meeting up again after a bit to track their progress. Don't like any of that at all.\n\n_Us:_ [ _pause_ ] But isn't that what being a doctor is?\n\n_Miles:_ Not for me it's not.\n\n_Us:_ What don't you like about that?\n\n_Miles:_ The pressure.\n\n_Us:_ Pressure?\n\n_Miles:_ Yes, the pressure to make them get well. I mean, what if they don't? The body is a complex and individualized organism, so many variables, and then you combine that with a patient's lifestyle, environment, psychology, luck, and who knows if they are really going to get better. It's just too much pressure for me.\n\n_Us:_ Oh.\n\nSo that's what it sounded like: a very successful, truly happy doctor revealing to us that the thing he is most stressed about at work is seeing whether his patients actually recover. Since this flew in the face of pretty much everything we'd read on physician satisfaction\u2014namely that doctors, like people in all professions, should \"start with why\" and should derive most of their joy at work from seeing their true purpose come to life\u2014we pressed on.\n\n_Us:_ Then can you tell us what you do truly enjoy about what you do?\n\n_Miles:_ Sure. Well, first off, I love the stress.\n\n_Us:_ What? Didn't you just say you didn't like stress?\n\n_Miles:_ No, I said I didn't like the pressure of making a patient get better over time. I absolutely love the stress of keeping a patient hovering between life and death. We still know so little about how anesthesia actually works. When I first started, we used mostly thiopental. Nowadays everyone goes with that new drug propofol\u2014the one that Michael Jackson took, which is actually a far better drug. And yet no one really knows how it all works. Both drugs seem to slow down the flow of minerals through the blood system, and so put you to sleep without stopping your heart, but we still don't know much about how either one actually does that. The whole challenge of putting someone to sleep, and keeping them hovering there, caught between life and death, for sometimes sixteen hours at a time, all the while not quite knowing how or why it's working\u2014man, I love that!\n\n_Us:_ Have you always loved that part of it?\n\n_Miles:_ Yep, right from the start. Some people freak out when you have to put someone under and then gradually ease them back to life, but I always leaned into that. I'm something of an adrenaline junkie\u2014swim with sharks, jump out of planes, that kind of thing\u2014so this part of it really wakes me up, makes me feel alive.\n\n_Us:_ Anything else you love?\n\n_Miles:_ Well, yeah. Frankly, it's the responsibility of the role. In the UK\u2014less in the US and Canada\u2014but in the UK the person who is supposed to understand the entire body of each patient is the anesthetist. The surgeon can fix the heart valves. The neurologist can tease apart the brain. The general surgeon can maneuver the bowels\u2014all critical stuff, but all very focused and specialized. The doc who needs to understand the entire body\u2014the entire respiratory, cardiovascular system, gastrointestinal, everything\u2014is the anesthetist. All of those systems feed into how a patient will respond to the drugs and how he or she will hover asleep. Because when you're under, you're never just under. You're always moving up or moving down, and my job is to be finely attuned to the entire person, and to understand their entire body so well I can hold them just so. Being an anesthetist is rather like flying a plane\u2014one wrong move and you can start to spiral down, and then another slight mistake, and then the spirals speed up, and in an instant you can find your patient spiraling down, down and away from you. I love that kind of responsibility\u2014twelve people in the OR and all of them relying on you to know the whole person, and hold the whole person.\n\n_Us:_ It sounds terrifying.\n\n_Miles:_ No, really, it's amazing. Every day. Just love it.\n\nWe don't know what you make of all this, but we did what we always do in interviews with people who thrive at work: we listened in the moment, wrote it all down, and then pondered it later.\n\nAnd where we landed was that, as ever, the theoretical models of what people are supposed to feel about their work rarely, if ever, match up to what a particular person truly does feel. Miles is a brilliant and successful doctor who hates the pressure of helping someone get well, and yet loves the life-and-death stress of holding a patient just outside death's door, all while not fully understanding precisely how he's doing it. Some might judge Miles and say, \"Well, all doctors should love seeing patients get healthy. That, after all, is the purpose of being a doctor.\" And yet what use is that judgment? Miles is Miles. He knows not only why he became a doctor, and why he became an anesthetist, but also what specific aspects of being an anesthetist he loves most. Others can judge all they want, but we know which doctor we would want putting us under. We'd want a doctor who derived great joy from what he did, who was fascinated by the subtle complexity of his responsibilities, and who got his kicks from keeping us just this side of the river Styx. We'd want Miles.\n\nAnd so, doubtless, would you.\n\nWe wish you could have been there to actually hear him talk, because when he does, his whole tone and demeanor shift and lift, and when he starts diving into what he loves about what he does, you sense you're in the presence of a special state of mind, a truly happy heart, a \"good spirit.\" Would that all doctors could feel this way.\n\nAnd would that _you_ felt this way about your work. You know you want to. You listen to folks such as Miles and you wish that one day you could feel what he feels about his work. Not that he skips to work every day\u2014some days are hard, some are exhausting, and some are probably deeply sad and difficult. But you feel his joy, and you want a part of it. You want this in your working life. You want to find love in what you do.\n\nHowever, the moment you start thinking this to yourself, you almost immediately dismiss it as sappy or unrealistic. Watch any famous commencement address on YouTube, or take a long lunch with a mentor, and it's almost guaranteed that at some point you'll hear the advice to \"Do what you love, and you'll never work a day in your life again.\" And when you hear that, your heart sinks. On the one hand, the thought seems to make perfect sense\u2014wouldn't it be great if we could all do what we love?\u2014but on the other, it seems, in this day and age, to be something of a luxury. It invites the response that it's all very well for you, lucky person, to have made your way doing what you love, but for the rest of us work is very much a requirement, and love an added\u2014and rare\u2014bonus.\n\nLinger on it for a moment, though. We're going to take a longer look at love; not to drag you away from the hard realities of the world of work, or to dismiss the demands and discoveries of reliable data, but instead to dive deeper into both. In doing so, we'd like to share the truth that\u2014more than striving for balance between work and life\u2014 _love-in-work matters most_.\n\n_Love-in-work_ is less of a mouthful than _eudaimonia_ , for sure, but it might also sound soft, idealistic, and far removed from the real-world pragmatism of the freethinking leader. If it does, then bear with us. Because love\u2014specifically, the skill of finding love in what you do, rather than simply \"doing what you love\"\u2014leads us directly to a place that is the epitome of pragmatism.\n\n\u2022 \u2022 \u2022\n\nOn the face of it, though, organizations don't appear greatly concerned with love. Southwest Airlines can stick a heart on its planes, and Facebook can claim that its mission is to \"ship love,\" but in these two cases, as in most others, the love refers to the customers, not the employees. Most organizations are much more worried about the meaty stuff: performance, goals, achievement, discipline, execution, and rigor. Get all this done, meet all the deadlines with the necessary levels of quality, and maybe then you can sprinkle a little dusting of love on it at the end.\n\nIf this is your view of your organization, then you\u2014and it, if it shares this view\u2014are missing the mark. Because the truth is that even the most hard-nosed, performance-oriented organizations desperately want you to find great love in what you do. They just don't call it that.\n\nHave you ever been deeply in love? Cast your mind back to when that was\u2014when you were so in love with someone that you couldn't wait to see that person, when time flew by quickly when you were together, and when, after parting, you ached to see your love again.\n\nWhen you're in love, you're a different person. Looking at the world through the rosy lens of love, everyone seems wonderful, people are beautiful, the world is happy and kind, and spring is in the air. Love lifts you up. It elevates you to a new plane, where you're at your most productive, creative, generous, resilient, innovative, collaborative, open, and powerful. When you're in love, you are simply magnificent.\n\nLook at those adjectives again: _productive_ , _creative_ , _generous_ , _resilient_ , _innovative_ , _collaborative_ , _open_ , _powerful_. Not only are they a pretty good description of how you hope to be in your life, or how your spouse or family wants you to be, but they're also, surely, the exact qualities your organization's CEO is looking for in every team member. Put the list of you-in-love qualities next to your CEO's list of ideal-employee-at-work qualities and you'll see that the list is the same.\n\nBut you don't get to feel any of these things by writing them down, just as your organization won't create any of these in you merely by discussing them with you in a training class. You\u2014and your organization\u2014get them only if you create them, and you create them only through love. The poet Pablo Neruda, in love, wrote \"I want to do with you what Spring does with the cherry trees.\" That's the power of love. With it, you blossom. You flourish. You look forward to what you're about to do. Time flies by while you're doing it. And when you're done with it, you feel an urge to start right back up again. You experience _eudaimonia_ , your spirit manifesting its fullest and most beautiful expression. That's what your organization wants, that's what you want for yourself, and that's what you want for your people. You want love.\n\nMost organizations shy away from the word _love_ , preferring more business-appropriate terms such as _committed_ or _motivated_ or _discretionary effort_. (Maybe, about now, you're privately wishing that we would shy away from it, too.) But in the real world we have to engage with what really is, not some watered-down version of how we'd like people to be or to feel. If we want our people to flourish, if we want them to be creative and intrigued and generous and resilient, then we've got to help them find what Miles found. There's love in work, and we should use the word. We should be curious about how each of us can find it. We should honor the truth that our organization can never find it for us, can never define it for us. For too long we've allowed our organizations to appropriate human words\u2014 _love_ , _passion_ , _excitement_ , _thrill_ \u2014and persuade themselves that, by invoking these words, they've created genuine human feelings. They haven't, and they never will. The organization is a fiction, an \"intersubjective reality,\" to use the term from chapter 1, and it's simply not real enough or human enough to know which activities at work you love. Only you can know that. Only you are close enough to yourself to know where you find love and where you don't, at the level of detail that Miles did. He said, in essence, \"I love this specific thing, not that specific thing.\" And no one, before or after he was hired, would have known this about him. This was a transcendent part of him that only he had access to.\n\nThe same is true for you. There's a little bit of you that your organization can never touch, never know, never see, and certainly never feel. And yet it's this part of you\u2014the loving, feeling part of you\u2014that makes you feel alive at work, able to do things that surprise and delight you, things that are ridiculously good, unexpectedly made, astonishing to your team, and that light you up from the inside.\n\nOrganizations are not powerless, but their power (and their name) comes from their ability to organize what is already there in plain view. Your organization, if it is careless, can crush your spirit, can diminish or ignore your _daimon_. But only you can animate it. Only you can bring love into your world at work.\n\nAnd when you do all sorts of good things happen. The Mayo Clinic actually managed to quantify the power of love-in-work. It asked physicians how much of their time at work they spent doing those activities they loved the most. Those who reported that they spent at least 20 percent of their time doing things they loved had dramatically lower risk of burnout. Each percentage point reduction below this 20 percent level resulted in a commensurate and almost linear increase in burnout risk. Remove the love from a physician's work, and the work grates, and grates some more, until it hurts.\n\n\u2022 \u2022 \u2022\n\nThe big question, then, is how to make this happen. Whether we call it love-in-work or _eudaimonia_ or anything else, the fact remains that work is called work for a reason, and your work is not only busy and sometimes repetitive but\u2014more to the point\u2014is not always of your own making. You have a particular job, in which certain outcomes are expected, and your responsibilities are what they are. What's love got to do with that?\n\nWell, the Mayo Clinic research reveals that love actually has a great deal to do with it. You can, and should, weave love into your work, no matter what role you're in. And in case you're wondering, the data reveals that, for most of us, the problem of loveless work lies less in the fact that our job is too constricted and more in the fact that we can't figure out how to weave. The ADP Research Institute's global engagement study revealed that only 16\u201317 percent of workers say they have a chance to play to their strengths every day, whereas their surveys of a representative sample of the US working population reveal that 72 percent of workers say, \"I have the freedom to modify my role to fit my strengths better.\" In psychology we refer to this as an attitude-behavior consistency problem\u2014we know we _can_ modify our roles to fit ourselves better, but most of us simply don't.\n\nSo, here's a way to remove the problem\u2014here's how to intentionally and responsibly weave love into your work.\n\nThink about the most successful person you know. Not in terms of money, necessarily, but in terms of her contributions to her team, and her organization\u2014someone enormously productive, creative, resilient, and seemingly at one with her work. More than likely, as you think of this person, you're thinking she got lucky. \"How,\" you're asking yourself, \"did she _find_ that role, how did she _find_ that work, how did she _find_ that life? I wish I could find something that fits me as well as her work fits her.\"\n\nIf you are indeed thinking this, then first, good for you for recognizing something special and precious, and second, you've landed on the wrong verb. This person didn't _find_ this work\u2014she didn't happen upon it, fully-formed and waiting for her. Instead, she _made_ it. She took a generic job, with a generic job description, and then, within that job, she took her loves seriously, and gradually, little by little and a lot over time, she turned the best of her job into most of her job. Not the entirety of it, maybe, but certainly an awful lot of it, until it became a manifestation of who she is. She tweaked and tweaked the role until, in all the most important ways, it came to resemble her\u2014it became an expression of her.\n\nYou can do the same.\n\nTwice a year, spend a week in love with your work. Select a regular week at work and take a pad around with you for the entire week. Down the middle of this pad draw a vertical line to make two columns, and write \"Loved It\" at the top of one column and \"Loathed It\" at the top of the other.* During the week, any time you find yourself feeling one of the signs of love\u2014before you do something, you actively look forward to it; while you're doing it, time speeds up and you find yourself in flow; after you've done it, there's part of you looking forward to when you can do it again\u2014scribble down exactly what that something was in the \"Loved It\" column.\n\nAnd any time you find yourself feeling the inverse\u2014before you do something, you procrastinate, perhaps handing it off to the new person because it will be \"developmental\"; while you do it, time drags on and ten minutes feels like a hard-fought hour; and when you're done with it, you hope you never have to do it again\u2014scribble down exactly what _that_ something was in the \"Loathed It\" column.\n\nObviously, there'll be plenty of activities in your week that don't make either list, but if you spend a week in love with your work, by the end of the week you will see a list of activities in your \"Loved It\" column that feel different to you than the rest of your work. They'll have a different emotional valence, creating in you a distinct and distinctly positive feeling, one that draws you in and lifts you up.\n\nThink of these activities as your \"red threads.\" Your work is made up of many activities, many threads, but some of them feel as though they're made of particularly powerful material. These red threads are the activities you love, and your challenge is to pinpoint them so you can ensure that, next week, you'll be able to recreate them, refine them, and add to them. You are weaving red threads into the fabric of your work, one thread at a time. Now, you do not have to end up with an entirely red quilt. The Mayo Clinic researchers found that when the physicians spent _more_ than 20 percent of their time on activities they loved, there was no corresponding reduction in burnout risk. The 20 percent number was a threshold, which is to say that a little love goes an awfully long way: when you can deliberately weave your red threads throughout the fabric of your work you'll feel stronger, perform better, and bounce back faster.\n\nThese red threads are your strengths. Typically we think of our strengths as what we're good at and our weaknesses as what we're bad at, and that our team leaders, or our colleagues, are therefore the best judges of both. But as we saw in chapter 4, this is not the best definition of either strengths or weaknesses. A strength is any activity that strengthens you (for Miles the anesthetist, keeping a patient hovering between life and death), and a weakness is any activity that weakens you, even if you're good at it (for Miles, helping patients recover). \"Performance\" is what you have done well or poorly, and your team leader can be the judge of that. Team leaders and colleagues, however, can't judge what strengthens or weakens you.\n\nIf you spend a week in love with your work and realize that you love finding patterns in data, then your team leader can legitimately tell you, in regard to your _performance_ , \"Well, you're not explaining the patterns well enough,\" or \"Well, you're not finding patterns that are useful,\" or, \"You're not putting them on a PowerPoint slide properly.\" Your team leader can say all these things. But what she cannot say is, \"No, you don't love finding patterns in data,\" just as we can't say to Miles, \"No, you don't love holding people between life and death.\" She can't say that your red thread isn't a red thread. You are the one and only judge of that.\n\nAnd don't imagine that your teammates in the same role as you share the same red threads as you. They don't. Think back again to Miles. We have interviewed other anesthetists\u2014indeed, other anesthetists who are the same age as Miles and who work in the same health-care system as he does\u2014who don't sound anything like Miles when they describe what they love. One loves the bedside conversations before the operation, and the calm sensitivity required to bring a sedated patient gently back to consciousness without the panic that afflicts many patients. Another is drawn mostly to the intricacies of the anesthetic mechanism, and has dedicated herself to defining precisely how each drug does what it does\u2014get her on the subject of what exactly \"consciousness\" is, and you'll hear the same passion as Miles displayed when he described the thrill of the stress.\n\nAnd yet, looking at Miles, you would never have known what his red threads were. He looks and acts no differently from any other middle-aged British doctor. His red threads have nothing to do with his race, his gender, his age, or his religion. They are simply and only an artifact of his uniqueness. For no good reason other than the clash of the chromosomes, Miles loves certain aspects of his work and loathes other aspects. It is therefore his responsibility to identify these red threads, see them for what they are, and then deliberately weave them into the rest of his work. No one can do this for him\u2014neither the identification, nor the weaving. Only he, with discipline and intelligence and intention, can bring love into his work.\n\nThe same is true for you, of course. You have a unique relationship with the world, a relationship that reveals to you things that only you can see. It offers thread-weaving opportunities all the time, but the only person who knows if those threads are red is _you_. The world won't do your weaving for you\u2014it doesn't care about your red threads. The only person who can stop and be attentive enough to identify these threads, and weave them intelligently into the fabric of your work, is you.*\n\nThis is true not only in your work life but in your life in general. Despite how it might feel a lot of the time, you do not have many different compartments of your life, each of which must be carefully balanced. Instead, you have one life, one whole cloth, one fabric for you to weave your red threads into. It's up to you to know what you love about work, what you love about hobbies, what you love about friends, and what you love about family, and those things will be different from everyone else's things. So when people say, \"Well, as a father\/friend\/colleague I think you should do this or that,\" remember that they do not know you like you know you, that they are well intended yet blind. Your world has an _n_ of 1, and that 1 is you.\n\nShould you work fifteen hours a day? Should you have three kids before the age of thirty? Should you devote all your time to your career until you can afford the day care you will need? Should you take six weeks of vacation a year, or none? Should you quit your job and go surfing or van-ing? These are all choices that only you can make, and the only way to make them wisely is to honor the truth that your life will give you strength if you can but pay attention to your emotional reactions to the events and activities and responsibilities you choose to fill it with.\n\nAnd what of the list of \"Loathed It\" things? Obviously these are your fraying, weak threads, and your aim is to incorporate as few of them as possible in your life's fabric\u2014either by stopping these activities altogether, by partnering with someone to get them done as painlessly as possible, or by seeing if, in being combined with an activity you love (by being braided with one of your red threads), they can become less draining for you. When you start to think about your life in this way, you'll quickly realize not only that \"balance\" is an unhelpful idea but that we have the categories wrong. What we all wrestle with every day in the real world is not so much _work_ and _life_ as it is _love_ and _loathe_.\n\nWatch for your red threads. Take them seriously. They are light, they are strong, they are true, and they are yours. And when you feel run down, or burned out, or at risk, or that everything is coming apart at the seams, cling to them tightly. They will hold fast until you have the strength to begin weaving something new. This new thing you make, this new idea, or project, or job, or relationship, or life, will not necessarily be balanced as others see it. It will not necessarily be a life that others would have made or would even approve of. Nor will it necessarily be easy. But it will be yours. It will be crafted from sources of strength felt only by you, and so it will be strong. It will flourish. It will not wither, and neither will you.\n\n\u2022 \u2022 \u2022\n\nWouldn't it be wonderful if work were _for_ love\u2014if the point of work were to discover that which each of us loves? Obviously, today we don't think of it that way. We think of it as a transaction: you get things done, and then we pay you to buy things you love. But what if we flipped that all around? What if we made the purpose of work to help people discover that which they love; if we changed the American Management Association slogan from _Get work done through people_ to _Get people done through work_? We'd fail, of course, because people are complicated, and so is work, and so is life. And besides, no person is ever \"done.\" But what if we made the attempt the entire point of work: To teach our kids and our college graduates, our workers young and old, our people in the second decade of their first career and our people in the first year of their third career, how to use the raw material of work to find their very own red threads and then to take responsibility for weaving them into something fine and strong?\n\nWe wouldn't net any less productivity. We would net more, and as the Mayo data shows, this productivity would be healthy\u2014underpinned by resilience and fulfillment. In the end, shouldn't that be what work is for?\n\n\u2022 \u2022 \u2022\n\nAbout twenty years ago, a thirteen-year-old gymnast named Sergei Polunin was plucked from his depressed Ukrainian town and brought over to White Lodge, the junior Royal Ballet School in the middle of Richmond Park, London. For the rest of his childhood he was trained in the Royal Ballet techniques, and he displayed such extraordinary natural talent that, at nineteen, he became the youngest principal male soloist in the history of the Royal Ballet Company. Everyone in London agreed that he was better than Baryshnikov, better than Nureyev, better even than Nijinsky, the most technically perfect dancer in a century. London was proud to have found and nurtured him.\n\nBut no one truly knew him, and no one truly cared to. He was a passionate, lyrical dancer, strong but fluid, soulful but angry, his tattooed body merely the most obvious sign of his need to push the boundaries. The powers that be at the Royal Ballet Company ignored all this, and instead did what they always do with their prodigies: they made him follow the strictures of the Royal Ballet Way. He would dance the classical ballet repertoire, in the classical ballet form and mold, and they would make him do it again and again, to glorify the company and delight the London crowds. And so he danced, and he danced, and he delighted and amazed, until one day, at the age of twenty-one and only two years after he ascended to principal, he quit.\n\nThere was a pattern for the perfect Royal Ballet dancer, and this pattern didn't care what Sergei Polunin loved. It didn't care about whatever his red threads as a dancer might be, and sadly he wasn't strong enough to believe that holding fast to these threads was vital for him. He was forced to conform to this pattern and he let his threads slip away, and soon, quite soon, he fell apart. Ballet, as you know, is an unremittingly technical and demanding craft, but if you build technical craft on a loveless foundation, you net only burnout, because technical mastery absent love _always_ equals burnout. Burnout isn't the absence of balance but the absence of love.\n\nThe Royal Ballet Company had won the talent war. It had found the most technically and lyrically gifted dancer in a century, and then, blind to his loves, it had torn him up. And we, the world, suffered for it. He had nothing left to contribute. Sergei flailed around for a few years, no longer at home in either London or the Ukraine, lost without his passion, and, after his parents divorced, alone and unmoored.\n\nAnd then he did what you might have done if you've ever found yourself similarly untethered: he found one thing he knew he loved\u2014one frayed strand of a red thread\u2014and he followed where it led. He asked a choreographer friend of his to create one dance that he would truly love, a choreography as lyrical as it was technical, equally precise and passionate. He would practice the piece, dance and film it twice during one muggy afternoon in Hawaii, post the video on YouTube for his close friends and family to see, and then\u2014well, he had no idea what would happen next. He was simply taking hold of one strong thread, weaving it into something that was, at the very least, authentic, and then hoping it would have enough power to pull him back onto his life's path.\n\nOn the day after Valentine's Day in 2015 he published, to very little fanfare, his version of Hozier's \"Take Me to Church.\"\n\nIf you've never seen it, take a moment now to log onto YouTube and watch\u2014it's four minutes and eight seconds that you'll never forget. Whether you're a fan of ballet or not, you'll recognize it not only as the work of a man at the end of his tether, but also as a pure expression of technical craft and unabashed joy. You see here a man who is taking his loves seriously, interlacing them with craft and discipline, and contributing to us something passionate, rare, and pure. You will see, from the inside out, that this is the fullest, most authentic, and richest expression of this unique person. If the people coming to work on your team could feel more like this, if you could help them take their red threads this seriously\u2014not to make your people feel good about themselves, although that helps, but so they could share more with the world\u2014what a beautiful and lasting contribution you and your team would make.\n\nSince it was posted, Sergei's video has been viewed more than twenty-three million times. He's danced the piece on stages as diverse as Covent Garden, the Hollywood Bowl, and _The Ellen DeGeneres Show_ , and has rediscovered his love for his craft as a guest principal at Europe's most prestigious ballet companies. No longer tied down to the classical Royal repertoire, he's rediscovered his love-in-work, and all of us are the beneficiaries.\n\nWe ask the same of you. Spend a week in love with your work. Hold tightly to your red threads. Yes, so you can blossom. But mostly, so you can figure out ways to share what's unique about you with the rest of us.\n\nThe power of human nature is that each human's nature is unique. This is a feature, not a bug. So your responsibility is to take seriously the uniqueness of your uniqueness, and design the most intelligent, the most honest, and the most effective ways to volunteer it to the rest of us. We\u2014your teammates, your family, your community, your company\u2014are waiting for you to share with us your unique loves. We're here for but a few short years. Please don't make us wait too long.\n\n*As we've shared this exercise around the world, we've learned that not all languages and cultures use the words \"love\" and \"loathe\" in the same way\u2014in the Netherlands, for example, there is no single word for \"love.\" So to be clear: the key idea here is to capture your strong positive and negative reactions to your work, so pick whatever words mean that to you\u2014you're after the extremes of your experience, not the \"meh\" in the middle.\n\n*You're often told, by the way, to \"take ownership of your career.\" This is what it actually means\u2014it means taking ownership of the weaving of your red threads.\n\n#\n\n# Leadership is a thing\n\nMemphis, Tennessee, is the home of the National Civil Rights Museum. We visited the museum a couple of years ago, and spent two or three hours there learning about the civil-rights movement and the long struggle of African Americans to end institutionalized discrimination and achieve some measure of equality.\n\nThe layout of the museum\u2014really, the layout of the experience of a visitor\u2014is arresting. Rather than comprising a series of rooms that you might visit in turn, the main exhibit is laid out as a winding path across the floor of a large room\u2014making a sort of high-walled maze\u2014so as you walk you encounter the various displays and artifacts in chronological order. You begin with the conclusion of the Civil War, and see the hopes and possibilities of that brief moment in time quickly pushed aside by Jim Crow laws. You see the struggles over segregation leading up to the _Brown_ v. _Board of Education_ decision in 1954. And then, rounding a corner, you see a full-scale replica of a bus\u2014an entire city bus, not gleaming and new and modern, but old, banged-up, hard-working, and hard-worked, the sort of vehicle that shuttles anonymously to and fro in our lives, moving us from one place to another while we think of other things. This particular bus, however, serves to remind us of one of those moments in history that cleave time into before and after, because in 1955, on a bus like this one, Rosa Parks, after another long day at work, refused to give up her seat to a white person when instructed to do so by the driver. The ensuing Montgomery bus boycott, carried forward by a young minister from a local church, became one of the early flashpoints in the civil-rights struggle.\n\n\u2022 \u2022 \u2022\n\nThis is not a chapter about leadership.\n\nIt's become something of a clich\u00e9, in the business literature, to bemoan the vast volume of writing on the topic; to list the number of books on leadership that come up if you search on Amazon; to point to the great library of articles and blog posts and videos and inspirational speeches as evidence that leadership is either a Really Important Subject or else a Really Over-Analyzed Subject. If it's possible to distill the essence of what all these written and spoken words have to tell us, it would be something like this: that leadership is enduringly fascinating to us and that we believe it to be critically important at work.\n\nAnd we can say a little more. We can say that there appears to be broad agreement that certain people exhibit a definable, consistent, and meaningful quality called leadership. That there are some characteristics of a person that are in some way above and different from that person's technical skills (whether he or she can write good code, for example, or good English) and that also transcend that person's interpersonal or \"soft\" skills (whether he or she can make the sale, or negotiate a deal) and that make the person a leader.\n\nWe can also say that we tend to agree that all the best leaders possess this quality, or set of qualities\u2014so, leadership is something that lives, specially, in those who lead and is in some way responsible for their ability to do so. And we can say that, as a consequence, most of us would agree that if you want to be a leader, you have to have this set of qualities.\n\nThere is a frustrating circularity to this argument\u2014that there's a thing called leadership, and we know it's a thing because leaders have it, otherwise they wouldn't be leaders. It's like saying your cat has catness because he's a cat: it might be true, but it's hardly helpful to your hamster if he dreams, someday, of being a cat. This know-it-when-we-see-it vagueness explains, in part, why we can talk about leadership so much without usefully advancing our understanding of it, or getting much better at it.\n\nPerhaps to combat this vagueness, some go further and begin to try to specify some of the qualities that make up leadership. Being inspirational seems to be important. Being able to create and articulate a vision matters a lot. The ability to formulate strategy is good, as is the ability to distinguish a good strategy from a bad one. Sometimes mastery of execution makes the list\u2014the art of getting stuff done. Setting a direction for an organization is important, and, in concert with this, bringing people into alignment with that direction and motivating them to move ahead. Decision making is high on the list, together with managing conflict. Innovation and disruption usually put in an appearance. Communications skills also rank highly, and having what's commonly referred to as \"executive presence\" is also felt to be critical.\n\nTo this collection of long-limbed characteristics are added some personal traits. Leadership requires authenticity (the ability to come across as a \"real\" person) and often, too, vulnerability (the courage to be imperfect in public, to relinquish the need to be right or to be the smartest person in the room). These things, and a few others, are said to be needed so that our leaders can build effective relationships with others.\n\nAnd yet these characteristics are curiously circumscribed: authenticity is important, right up until the point when the leader, authentically, says that he has no idea what to do, which then fractures his vision. Likewise, vulnerability is important until the moment when the leader's comfort with her own flaws causes us to doubt her, and to question whether she is sufficiently inspirational. Apparently, we require authentic sureness and reassuring vulnerability, however contradictory those things may be. The personal qualities that make the list are Goldilocks qualities\u2014they must be neither too hot, nor too cold, but just right.\n\nThese little inconsistencies, however, melt away in the face of our conviction that leadership is a great good at work\u2014it is _always_ better for a person to have more of it, and the more leaders an organization has, the better. This much, at least, appears settled\u2014and as a result you will be told that the most important thing you can do to advance your career is to \"grow your leadership.\"\n\nNow, some might choose other attributes for their lists, but those above are a reasonable summary of the theory-world view of leadership. And the reason that this isn't a chapter on leadership is not that the qualities listed aren't useful (they are) or that this topic has been done to death (it's close) but, rather, that when we look critically, we realize that we may well have misunderstood leadership altogether.\n\nIndeed, the final lie that we encounter at work is that _leadership is a thing_.\n\n\u2022 \u2022 \u2022\n\nIn June 2004, while tidying a basement, a sheriff's deputy in Montgomery, Alabama, came upon several books of mug shots. The books were clearly old, and the photos inside them had been carefully sorted by gender and by race. In a volume titled \"Negro Male\" was a page showing photographs of some of the eighty-nine people arrested on February 22, 1956, during the Montgomery bus boycott.\n\nAs we look at that page today, twelve men look back. Some are dressed informally; some formally. Some are younger; some older. Some seem worried; some resigned; some defiant. Each has an arrest number in front of him: some are holding it; some have it hanging from their necks on a chain. As it turns out, we know the names of all twelve men on this page, although we know little else about them beyond that. And we know their names because of something they did together, and something that invites us to think about leadership differently.\n\nThe thing that the men in the mug-shot book have in common is explained by the identity of one of them in particular. On the top row of the page, the man bearing number 7089 stares back at the lens, his light-colored suit coat buttoned, his tie straight, his hands resting on his knees. At the time this photograph was taken, he was twenty-seven years old, and was the pastor of the Dexter Avenue Baptist Church in Montgomery. In the days following Rosa Parks's arrest, he had been asked to lead the boycott, which, beginning in December 1955, saw widespread participation and created significant economic problems for the city's transit system. In early 1956 a county grand jury returned indictments against several of the boycott participants for violations of the Alabama Anti-Boycott Act, and the pastor, together with eighty-eight others, was arrested.\n\nHis name, of course, was Martin Luther King Jr. But the story these pictures have to tell us is not, in the first instance, a story about him, about Number 7089. It's a story about the other eleven men on the page, and it's a story that\u2014in all the theorizing about leadership, in all the competing lists and competencies, in all the articles and surveys and assessments and books, in all the dissection and analysis and categorization\u2014is sadly lost. For leadership does not live in the abstract, does not live in the average. It lives, instead, in the real world.\n\nAnd if we look at that world, this is what we see.\n\nFirst, the ability to lead is rare. It was not inevitable that Martin Luther King Jr. would emerge from the Montgomery bus boycott as a national leader whom millions would follow\u2014there were other good people guiding the Montgomery Improvement Association, just as there were other, earlier bus boycotts, such as the one in Baton Rouge, Louisiana, a couple of years before. But something about King in Montgomery was special. The fact that we lionize those who have this special ability; the fact that we spend so much time looking for it and trying to get more of it; and the fact that it plays such a prominent role in how we think about our organizations: these point not to its ubiquity but to its scarcity\u2014and this scarcity, in turn, belies the supposed ease with which we're all meant to be able to get better at it. If leading were easy, there would be more good leaders. If there were more good leaders, we might be just a little less focused on it.\n\nSecond, leaders have shortcomings. Their skill set is incomplete. We don't need J. Edgar Hoover's surveillance files to reveal that King was not in possession of every quality the perfect leader should possess. And this is confounding, because it challenges the notion that there is in fact a list of leadership qualities, each of which is essential. For every quality on the list, we can think of a respected leader in the real world who lacks it. If leadership is about being inspirational or visionary, then what should we make of Warren Buffett, whose principal activities as a leader seem to consist of sitting in an office in Omaha, Nebraska, drinking Cherry Coke, and finding companies to buy? If leadership is about creating a winning strategy, then what should we make of Winston Churchill, whose disastrous policies in the 1920s and 1930s led to his exile from government? If leadership is about execution and communication, then what should we make of King George VI of Great Britain, who was revered for his leadership of that nation during the Second World War, but who could barely speak in public, and who wasn't in a position to execute anything? If leadership is about building a winning coalition, then what should we make of Susan B. Anthony, whose falling-out with her fellow women's-suffrage leaders created a split in that movement that lasted twenty years?\n\nIf it's about ethics, what do we make of Steve Jobs's buying a new car every six months to avoid registering it, so as to be able to park in handicapped spots whenever he wanted to? If it's about caring for those in your charge, what do we make of General George Patton and his physically assaulting soldiers with PTSD? If it's about authenticity, where does that leave John F. Kennedy and his hidden illnesses and affairs?\n\nWhat does it mean for all the models and lists if the things on them are optional? The lesson from the real world is not that there is any particular collection of qualities that every leader has, but rather that every leader we can think of has obvious shortcomings\u2014that leaders aren't perfect people, not by a long way.\n\nAnd finally, it follows that leadership is not about being the most well-rounded of the well-rounded people. As we saw in chapter 4, the best people aren't well-rounded. The same is true for the leaders we see in the real world\u2014and even more so. As with some of the great performers we met earlier\u2014think Lionel Messi and his amazing left foot\u2014we don't see the most respected leaders spending much time trying to round themselves out, trying to develop abilities in areas where they have none. Instead, we see them trying to make the best use of what they already have, with the result that whenever we look closely, we see them going about the task of leading in very different ways. In this way, leading is the same as all other fields of human endeavor\u2014high performance is idiosyncratic, and the higher the level of performance the greater the level of idiosyncrasy.\n\nKing, too, had a very particular approach to leading, and his genius and his effectiveness lay not in trying to round himself out and acquire the skills of, say, a Rosa Parks, a Malcolm X, or a Ralph Abernathy, but instead in starting from, cultivating, and, in moments of crisis, falling back on his very particular gifts as a leader\u2014which we'll try to describe shortly.\n\nThis is why the idea that _leadership is a thing_ is a lie. When you take any of our definitions of that thing, and then try to locate it in the real world, you encounter exception upon exception upon exception. The very least we can conclude is that if there is some magical set of attributes, we haven't yet figured out what they are, and that plenty of leaders are doing plenty of leading without many of them. And if that's the case, then the things that supposedly make up leadership neither add to our understanding of it nor help us be better at it.\n\n\u2022 \u2022 \u2022\n\nBut if the real world shows us what leadership _isn't_ , does it give us any clues at all that we can learn from? Can we say nothing more than that leading is a free-for-all, a grab bag of different skills and attributes and states and traits that will remain ever mysterious? Or is there a different way to understand what's going on?\n\nWhat's most remarkable about the events in Montgomery in 1956 is not that one individual took a stand and was imprisoned as a result\u2014it is not what this one man said or did. It is rather that others chose to follow him. What is truly before us on the page of mug shots is a picture of a leader _and his followers_ \u2014and it is because, on that day, the eleven chose to follow that sixty years later we know their names. In the midst of physical attacks and intimidation and firebombings, the eleven saw something special in King, something that they chose to follow, and because of what they did, and then because of what countless thousands and millions did in the ensuing years, we recognize him as a leader.\n\nThis is the true lesson in leading from the real world: a leader is someone who has followers, plain and simple. The only determinant of whether anyone is leading is whether anyone else is following.\n\nThis might seem like an obvious statement, until we recall how easily we overlook its implications. Followers\u2014their needs, their feelings, their fears and hopes\u2014are strangely absent when we speak of leaders as exemplars of strategy, execution, vision, oratory, relationships, charisma, and so on. The idea of leadership is missing the idea of followers. It's missing the idea that our subject here is, at heart, a question of a particularly human relationship\u2014namely, why anyone would choose to devote his or her energies to, and to take risks on behalf of, someone else. And, in that, it's missing the entire point.\n\nThis notion\u2014that a leader is a person with followers\u2014does not emerge from a list of skills, or tactics, or competencies; it doesn't coincide with a person's level within a hierarchy; and it doesn't actually tell us very much about the nature of the leader him- or herself. But it does capture a condition, a litmus test if you like, for leading. And that condition is precise\u2014it's about the presence, or absence, of followers.\n\nSo the question we should really be asking ourselves is this one: Why do we follow? What is it that makes us work hard late into the night\u2014to go beyond what's expected of us? What makes us move someone to the front of our queue? What makes us voluntarily place some part of our destiny in the hands of another human being? What makes us give our breath to another?\n\nWhat made those eleven men entrust their well-being and their hopes to Number 7089?\n\n\u2022 \u2022 \u2022\n\nWe can see some part of the answer in those eight items with which we began the book. We know that the feelings measured by those eight items exist disproportionately on high-performing teams, so we know something of what followers need from their team leader.\n\nBroadly speaking, we want to feel part of something bigger than ourselves\u2014the \"Best of We\"\u2014while, at the same time, feeling that our leader knows and values us for who we are as a unique individual\u2014the \"Best of Me.\"\n\nMore specifically, we follow leaders who connect us to a mission we believe in, who clarify what's expected of us, who surround us with people who define excellence the same way we do, who value us for our strengths, who show us that our teammates will always be there for us, who diligently replay our winning plays, who challenge us to keep getting better, and who give us confidence in the future.\n\nThis is not a list of qualities in a leader, but rather a set of feelings in a follower. When we say to ourselves that leadership is indeed a thing, because we know it when we see it, we're not really seeing any definable characteristic of another human. What we are \"seeing\" is in fact our own feelings as a follower. As such, while we should not expect every good leader to share the same qualities or competencies, we _can_ hold all good leaders accountable for creating these same feelings of followership in their teams. Indeed, we can use these feelings to help any particular leader know whether or not she is any good. Those eight items introduced in chapter 1 are a valid measure of a leader's effectiveness. We need not dictate how each leader should behave, but we can define what all good leaders must create in their followers. And since we measure this by asking the followers to rate their own experiences, rather than rating the leader on a long list of abstract leader qualities, this measure of leader effectiveness is reliable.\n\nLeadership isn't a thing, because it cannot be measured reliably. Followership is a thing, because it can.\n\nAnd it's a lie that _leadership is a thing_ because no two leaders create followers in quite the same way. What's true in the real world is that leading is many different things. Your challenge as a leader is not to try to acquire the complete set of abstract leader competencies\u2014you will fail, not least because the first hurdle you will fall at is authenticity. Instead, your challenge is to find and refine your own idiosyncratic way of creating in your team these eight emotional outcomes. Do this well and you will lead well.\n\nInterestingly\u2014and happily\u2014a close study of the real world reveals that these two are linked. Your ability to create the outcomes you want in your followers is tied directly to how seriously and intelligently you cultivate your own idiosyncrasy, and to what end. The deeper and more extreme your idiosyncrasy becomes, the more passionately your followers follow\u2014and while this is frustrating to us when we happen to disagree with the ends of a particular leader, it is so nonetheless.\n\n\u2022 \u2022 \u2022\n\nWe leave the Montgomery bus behind us and continue our walk through the exhibition. We see the student sit-ins of 1960, and we follow the Freedom Riders of 1961. We learn about efforts to organize resistance and protests in Mississippi and in Albany, Georgia. And then we arrive at a replica of a jail cell in Birmingham, Alabama.\n\nIn the 1960s, Birmingham was known as the most segregated city in America. In early 1963 the Alabama Christian Movement for Human Rights, together with Martin Luther King Jr.'s Southern Christian Leadership Conference, began a nonviolent campaign against the segregation. A local justice issued an injunction against the protest. The civil-rights leaders duly announced they would disobey this order, and on April 12, 1963, King and other marchers were arrested and incarcerated. That same day, eight white Alabama clergymen published an open letter in which they criticized King and his methods, and a copy of the newspaper in which it was printed was smuggled in to King in his cell. King, on reading it, began writing. At first, he wrote in the margins of the newspaper. Then, when those were covered, he wrote on the toilet paper in his cell, and then on scraps of paper given him by a friendly fellow inmate, and then, finally, on a writing pad brought in by his lawyers. When King finally set down his pencil, he had written what is now known as the Letter from Birmingham Jail. We read a copy of it on the wall outside the jail cell.\n\nIt's a long letter\u2014an impassioned letter. It's a plea against settling, against compromise, against the path of least resistance. And in it, King talks about extremism. \"The question,\" he says \"is not whether we will be extremist, but what kind of extremists we will be.\"\n\nWarren Buffett, the uninspiring Coke-drinker from Omaha, is an extremist. He's exceptional at finding and buying companies. Winston Churchill, while he might have been a poor policy maker, was exceptional at inspiring uncompromising resistance. Susan B. Anthony was really good at focusing her energies, and those of the people around her, on a specific goal. Steve Jobs was really good at creating hardware and software that was delightful to use. George Patton was really good at fighting, with his whole being, whatever was in front of him on any given day. And John F. Kennedy was really good at making the future feel universal and morally uplifting. What each of these leaders had in common was that they were really good at something\u2014each was, in their different way, an extremist.\n\nWe have seen, already, that the best people aren't well-rounded, but are instead _spiky_ \u2014they have honed one or two distinctive abilities that they use to make their mark on the world. What we see in the best leaders is a similar extremism\u2014a few signal abilities refined over time. But now, these abilities are so pronounced, and the leaders so adept at transmitting them to the world, that they stand out to all of us. And so this truth: _we follow spikes_.\n\nWe do this not merely because a leader who has deep mastery in something will be able to excel at it, but because these spikes change the way we feel about the future. There aren't many human universals\u2014the anthropologist Donald Brown's book _Human Universals_ lists sixty-seven of them\u2014but one of them is that every human society ever studied ritualizes death. Each society does this differently, but we all do it. Death is the great unknown, and these rituals serve to lessen our fear and give us some illusion of control. These death rituals are merely the most obvious sign of something common to us all: we humans fear the unknown. The past is what it is, the present is where we stand, yet the future is a scarily uncertain place. This uncertainty leads us to seek reassurance, and in particular, reassurance through the ritualization of that most uncertain certainty of all\u2014our ultimate demise.\n\nThis particularly human characteristic presents a challenge for you, the modern day leader. You are charged with rallying your team toward a better future, yet many on your team are fearful of this future. And this fear isn't unjustified. It's adaptive. Those of our forebears who lacked it, who paddled their little rafts toward the horizon, asking themselves \"Ooh, I wonder where the sun goes to sleep?\" often didn't return to pass on their genes. Being a bit cautious can be a sensible thing.\n\nAs a leader, you can't be dismissive of this fear. You can't tell your people to \"embrace change\" and to \"get comfortable with ambiguity.\" Well you can, but you will then get them thinking ever more deeply about change and ambiguity, which will, in turn, _increase_ their anxiety and lessen your effectiveness as a leader. It's an irony that while consultants wax lyrical about change, real-world leaders hardly use the word at all, realizing that their followers want instead an increasingly vivid picture of the future, not another reminder of its inherent uncertainty.\n\nYour greatest challenge as a leader, then, is to honor each person's legitimate fear of the unknown and, at the same time, to turn that fear into spiritedness. We, your followers, like the comfort of where we stand, yet know that the flow of events is pulling us inexorably into the unknown. So when we find something, anything, however slight, that lessens our uncertainty, we cling on for dear life.\n\nThe final characteristic of the best teams, as we saw in chapter 1, is the feeling that, for each team member, \"I have great confidence in my company's future.\" This confidence in the future, it seems, is the antidote to our universal uncertainty. And it explains why we follow. The act of following is a barter\u2014we entrust some part of our future to a leader _only_ when we get something in return.\n\nThat \"something in return\" is confidence.\n\nAnd what gives us confidence in the future is seeing, in a leader, some great and pronounced level of ability in something we care about.\n\nWe follow people who are really good at something that matters to us. _We follow the spikes_.\n\nIt's as if the spikes give us something to hook on to. We're well aware of our own shortcomings, and we know that what lies ahead of us in life is unknowable. We're aware, also, that our journey will be easier if we can do it in partnership with others. And when we see, in those others, some ability that offsets our own deficits, and that removes for us, even if only slightly, some of the mist of the future, then we hold on. We don't necessarily follow vision, or strategy, or execution, or relationship building, or any of the other leadership things. Instead we follow mastery. And it doesn't much matter how this mastery manifests itself, as long as we, the followers, find it relevant. John F. Kennedy was a master at getting us to see and engage with the near-term future in a way that made it morally enlarging\u2014even during the Cuban Missile Crisis he ended his address to the nation by reaching out to the Cubans themselves, saying, \"I have no doubt that most Cubans today look forward to the time when they will be truly free\u2014free from foreign domination, free to choose their own leaders, free to select their own system, free to own their own land, free to speak and write and worship without fear or degradation.\" His brother Robert didn't share this spike. Robert F. Kennedy's spike was urgent, present-day righteousness. Whether he was rooting out Communists under Joe McCarthy, or attacking the Teamsters boss Jimmy Hoffa, or aggressively pushing forward the Civil Rights Act in spite of his brother's caution, RFK's focus was on making things right, right now.\n\nEach truly effective leader cultivates his or her mastery in a way that communicates to us something certain and vivid. It's as if we trust leaders only when they've proven to us that they've opened more doors than we have, seen round more corners than we have, dived deeper than we have, taken themselves more seriously than we have. We trust the seriousness of this. We trust its predictability. We are drawn to its specialness. We sense its authenticity. We are attracted to the beautiful clarity of great ability, the brief moments of awe. We ignore everything else.\n\n\u2022 \u2022 \u2022\n\nOne of the lessons of the Nine Lies is that when we blind ourselves to what's around us, and instead theorize about how the world ought to be (or how we'd like it to be if only it were tidier), our people vanish. We stop seeing them. We mute our curiosity, and we replace it with dogma and dictum. The same happens with the people we call leaders\u2014the moment we start theorizing, they vanish, too.\n\nAnd here are the truths that vanish along with them.\n\nThe truth that no two leaders do the same job in the same way.\n\nThe truth that as much as we follow the spikes, they can also antagonize us.\n\nThe truth that no leader is perfect\u2014and that the best of them have learned how to work around their imperfections.\n\nThe truth that leaders are frustrating\u2014they don't have all the abilities we'd like them to have.\n\nThe truth that following is in part an act of forgiveness\u2014it is to give our attention and efforts to someone despite what we can see of their flaws.\n\nThe truth that not everyone should be, or wants to be, a leader\u2014the world needs followers, and great followers at that.\n\nThe truth that a person who might be a great leader for me might not be a great leader for you.\n\nThe truth that a person who might be a great leader for one team, or team of teams, or company, might not be a great leader for another.\n\nThe truth that leaders are not necessarily a force for good in the world\u2014they are simply people with followers. They aren't saints, and sometimes their having followers leads to hubris and arrogance, or worse.\n\nThe truth that leaders are not good or bad\u2014they are just people who have figured out how to be their most defined selves in the world, and who do so in such a way that they inspire genuine confidence in their followers. This isn't necessarily good or bad. It just is.\n\nThe truth that leading isn't a set of characteristics but a series of experiences seen through the eyes of the followers.\n\nThe truth that, despite all this, we reserve a special place in our world for those who make our experience of it better and more hopeful.\n\nAnd the truth that, through it all, we follow your spikes.\n\n\u2022 \u2022 \u2022\n\nWe spend vast sums of money, in the corporate world, on training and developing our leaders\u2014in the United States alone, a jaw-dropping $14 billion every year. The usual leadership course goes something like this: there's a video, either of people talking about leaders, or else of a real leader in the real world. It's intriguing, provocative, and moving. We learn about the impact leaders have had on the various interview subjects, or we feel it ourselves as we see the real leader on-screen. While watching, we feel inspired, curious, energized\u2014that we are going to learn about something important, and that we have just, in a small way, felt that important thing for ourselves.\n\nThen a facilitator steps to the front of the room and explains _the model_. The model takes whatever we've just seen and experienced and makes it boring. The model is usually a two-by-two grid of little boxes, and in each of the boxes is written some sort of abstract word: _empathy_ , _authenticity_ , _vision_ , and so forth. And then the facilitator explains that for the next several hours of our lives, we will take each little box in turn, and learn about the abstract thing written in it, and how each of us can have more of that thing. Sometimes the course has been preceded by an assessment that we've all taken, and partway through the course we get to see our results, and how we compare with the things in the boxes. Sometimes we get to give one another feedback in real time at our tables on how we're doing at the things in the boxes. Sometimes we build action plans and write down in our notebooks well-meaning commitments about how we'll do more of the things in the boxes, knowing as we do so that the moment the course finishes, these commitments will join \"floss more often\" in the _Big Lifetime Pile Of Things Not Done_.\n\nWe're told, as we do all these things, that at the end of this process, we will all be more like the leader we saw at the beginning. But our experience, as we go along, is one of increasing frustration. None of the things in the boxes helps us locate the feeling we had, in the opening video, about a particular leader. Indeed, the things in the boxes seem to have nothing on earth to do with the actual leader we've seen, or any actual leaders we've seen. We encounter leaders, in life, emotionally. In our leadership training, the first thing we do\u2014in our attempt to understand leadership\u2014is to wring the emotional life out of the thing.\n\nBecause what never, ever happens in any of these courses is our starting with the question: Who are you? Not, who are you in comparison with some model involving abstract words in little boxes, but who are you as a living, breathing, growing, worrying, joyous, uncertain, loving, striving, messy, and yes, spiky human being? We never ask why, given your particular jumble of characteristics, anyone would follow you. We never ask how\u2014given that one-of-a-kind mixture of states and traits that makes you who you are\u2014you would use those things to create an experience for the people around you, and use what you have to help them feel better about the world you're all walking through together, and, while we're at it, how we might give you some measure of that so you can adjust your course as you go.\n\nSo we need to stop with the models. Stop with the 360-degree assessments. Stop with the minute and meaningless parsing of how to move your \"effective communications\" score from a 3.8 to a 3.9, while also figuring out why your peers gave you a 4.1 on \"strategy\" yet your boss gave you a 3.0. Stop with the endless lists of abstractions. Stop debating whether it's authenticity or tribal leadership or situational leadership or level-five leadership or whatever the latest leadership-nirvana thing is. Stop with the one-size-fits-all.\n\nInstead, let's get humble\u2014the experience of the people on our teams and in our organizations is a true thing, and we don't simply get to choose what it is.\n\nLet's get curious about that experience and how our actions shape it.\n\nAnd let's follow our own reactions to real people in the real world. When we feel uplifted by what someone does or says, we need to stop and ask why. When we feel a fresh rush of energy after talking with someone, we need to stop and ask why. When we feel, in response to another human being, that mysterious attraction tugging on us\u2014like a fish on a line, or like a needle twitching in a compass, an attraction that says Here, something is happening, something true and visceral and substantial, something that will change, however slightly, the arc of our future\u2014we need to stop and ask why.\n\nWe need to get to know real leaders in the real world, and we need to come to know them as followers ourselves. Then we can start learning.\n\n\u2022 \u2022 \u2022\n\nOur walk continues. Now we come to a representation of the Edmund Pettus Bridge, at Selma, across which King led marchers in 1965 after the horrors of Bloody Sunday. As we walk, imagining as best we can the hundreds upon thousands of determined footsteps that the bridge has come to represent, we notice that the ground is rising beneath us\u2014that we are ascending. Our pathway doubles back on itself. We see murals of the Selma marchers, and we hear the sound of marching feet. We hear King's words\u2014\"How Long, Not Long\"\u2014as they were delivered at the Alabama State Capitol steps in March of that year. And then, all at once, quiet. We have ascended a full floor, and turning once again, we can see down on the exhibition, and the path that we have traveled.\n\nNow, King's journey is clear to us. Now, looking down from above on each of the protests, and marches, and arrests, and setbacks, and turning points, and triumphs, we can see, neatly laid out on the exhibition floor, the voyage he took. How Montgomery led to Albany, and Birmingham led to Washington, and thence to Selma. How, \"This movement will not stop, because God is with the movement,\" was followed by, \"We . . . will not be satisfied until justice rolls down like waters and righteousness like a mighty stream,\" and then, \"The arc of the moral universe is long, but it bends toward justice.\" And through and across these places and words, we see King harness the nonviolent protest methods of Gandhi to his own particular brand of persistence. We see him fuse his personal bravery and sacrifice to his prose, borrowing as it did the rhythms and cadence of the pulpit and transforming them into a universal poetry of hope.\n\nBut let us go back, for a moment, to somewhere near the beginning of this journey, to the meeting of a few dozen local ministers at King's church in Montgomery at which he was asked to lead the boycott, and imagine what might happen if that meeting were instead to take place in a business organization today\u2014if we were, say, selecting a leader for a key corporate initiative. We would first ask the assembled group to identify what qualities were necessary in the leader. We'd present the ministers with a list\u2014results orientation, strategic orientation, collaboration and influence, team leadership, developing organizational capabilities, change leadership, and market understanding, say\u2014and ask them to gauge just how much of these would be needed for the leader to succeed. We'd then ask the group to rate Dr. King and any other candidates on each of these, both in terms of his current level of ability and his potential for growth in each, and we'd compare those with the required levels of ability. Then we'd use this to predict his chance of success, and weigh that in our decision of whether to ask him to lead. And then, if we went ahead and decided King was the man for the job, we'd suggest development opportunities for him to grow in the areas we'd decided mattered most. We'd do this because this is our theory of leadership in action. And the point is not that we wouldn't choose Dr. King but that we wouldn't in any meaningful sense see him at all.\n\nBut now look down with us again to the exhibition floor, and now think of the followers without which the journey would not have been. These theoretical things would have meant exactly nothing to them. What they saw was not a perfectly balanced set of abstract qualities. What they saw resembles in no way our tidy, hindsight-blinded models. What they saw was an imperfect man who understood very well what sort of an extremist he should be. For Dr. King, leading meant defining, vividly, the goal, and then taking advantage of any opportunity to press toward that goal. There was no detailed plan of execution\u2014first this, then this, then that. Rather, there was a clarion vision\u2014\"let freedom ring\"\u2014and then an unswerving commitment to intervene whenever and wherever progress toward that vision could be made, and to do so regardless of the personal or physical risks that any such action entailed. His approach was contingent, opportunistic, and incremental. It focused on imagined change, not on predictable execution. Its focus was at once broad, in terms of vision, and narrow, in terms of what must be done in the here and now\u2014but its focus was not in between those two, in any kind of roadmap to success. It relinquished certainty on how the goal might be met, and substituted trust that the right actions done now and repeated at every future opportunity would ultimately prevail.\n\nAs we stand and survey the exhibition, our perspective, looking down from above and at the same time looking back from the future, offers us a certain clarity. King's followers, however, were not afforded that luxury. What confronted African Americans in the 1950s and 1960s who sought to claim those rights that the Constitution had promised them was not a tidy journey with well-defined steps. It was, rather, massive and pervasive uncertainty. King, the spiky extremist, helped them see into that future; helped them perceive, however dimly, what its contours might be, and how they might be a part of it.\n\n\u2022 \u2022 \u2022\n\nIn the spring of 1968, Memphis is a troubled city. In February the city's sanitation workers, frustrated by years of poor pay and poor working conditions, and angered by the deaths of two of their number in a garbage-crushing machine, go out on strike. A march to support the strike, with King in attendance, degenerates into violence, and one young protester is killed. A second march is organized for the following week, and again, King plans to attend.\n\nHis staff and close friends don't want him to go. He's tired, depressed, not sleeping, and drinking heavily. He's being criticized, constantly, by the press, by local leaders in Memphis, and even by some within his own movement. He's watched wherever he goes. He has even, a couple of weeks earlier, sent his wife synthetic red carnations instead of real ones, because he wants her to have something that will outlast him.\n\nHe knows, however, that unless he can lead a nonviolent protest in Memphis\u2014unless he can reclaim the moral high ground\u2014the future of all he has worked for will be at risk. So he goes. His flight to Memphis is delayed by a bomb threat. When the plane finally makes it to Tennessee, it's met, unexpectedly, by a police detail, which seems to be there less to protect him than to monitor him. During his first meeting of the day, word comes in that the city has issued an injunction against the march. His second meeting of the day is with a group of black activists who are believed to have been responsible for the violence in the first march, and whom King is trying to persuade to act peacefully if the second march goes ahead. He's pulled away from this meeting to confer with his lawyers about their strategy to remove the injunction. Then, it's back to the activists. Outside, the skies darken\u2014a storm is on the way.\n\nExhausted and battling laryngitis, he tells his colleagues that he won't be able to speak at a rally planned for that evening, and asks Ralph Abernathy to give an address in his place. He lies down on the bed in his room and tries to rest. Before very long, Abernathy calls. There's a huge crowd at the rally, and they want to see King. Can he come after all?\n\nKing often speaks without preparation, and the speech he gives this evening is no exception. He begins by imagining that he's been given the choice of which era of human history he would like to live in, and his answer becomes a sort of journey. From ancient Egypt, to ancient Greece, to Rome, and then to the Renaissance, King tells us what momentous events he would be able to witness in each place, before rejecting each in turn, declaring, \"But I wouldn't stop there.\" He keeps going, past Lincoln's Emancipation Proclamation, and past Roosevelt's New Deal, until he arrives in the second half of the twentieth century. It's a familiar rhetorical device, of course\u2014not this . . . not this . . . but _this_ \u2014and here it serves to locate the present moment in the arc of history. This is the moment in all of history he would choose, because now matters most, at least to this man.\n\nNext he is practical and directive. Unity is critical\u2014division will lead to defeat\u2014and it is important to \"keep the issues where they are\":\n\n> The issue is the refusal of Memphis to be fair and honest in its dealings with its public servants, who happen to be sanitation workers. Now, we've got to keep attention on that . . . we've got to march again, in order to put the issue where it is supposed to be\u2014and force everybody to see that there are thirteen hundred of God's children here suffering, sometimes going hungry, going through dark and dreary nights wondering how this thing is going to come out. That's the issue.\n\nHe emphasizes the importance of economic protest, reminding his listeners of their collective economic power and encouraging them to use it to hold corporations to account. He goes so far as to list names of particular brands of bread to avoid (Wonder Bread and Hart's Bread), and tells his audience to encourage their neighbors, too, to boycott these brands. He advocates for black-owned banking and insurance companies. All this, he says, is in the name of intensifying the protest: \"Up to now, only the garbage men have been feeling pain; now we must kind of redistribute the pain.\"\n\nHe tells the story of the Good Samaritan, and uses it to make a very specific point. He asks his audience to imagine why the Levite and the priest did not stop to help the injured man, and wonders if it was because they were afraid of what might happen to them. He describes the road between Jerusalem and Jericho where the events of the parable took place\u2014he has driven down it\u2014and tells his audience how remote and dangerous (\"really conducive for ambushing\") it is. And he makes the question that faced the Samaritan the question that faces his listeners, as they consider, perhaps fearfully, whether to join the upcoming march:\n\n> That's the question before you tonight. Not, \"If I stop to help the sanitation workers, what will happen to my job.\" Not, \"If I stop to help the sanitation workers what will happen to all of the hours that I usually spend in my office every day and every week as a pastor?\" The question is not, \"If I stop to help this man in need, what will happen to me?\" The question is, \"If I do _not_ stop to help the sanitation workers, what will happen to them?\" That's the question.\n\nThen he begins his ending. In both an echo and a continuation of his opening tour of history, he tells the audience of being stabbed by a deranged woman years before, and learning that the knife had come so close to his aorta that had he sneezed, he would have died. And now this\u2014\"If I had sneezed\"\u2014becomes a refrain as he recounts what he has witnessed in a few years in the second half of the twentieth century. If he had sneezed, he wouldn't have seen the sit-ins of the 1960s. If he had sneezed, he wouldn't have seen the Freedom Riders. If he had sneezed, he wouldn't have seen the Montgomery bus boycott, or the protests in Birmingham, or the \"great movement\" in Selma. And finally:\n\n> If I had sneezed, I wouldn't have been in Memphis to see a community rally around those brothers and sisters who are suffering.\n> \n> I'm so happy that I didn't sneeze.\n\nHe ends where he began: Now is the most important time. Here, Memphis, is the most important place.\n\nWhat connects all these things? What connects a tired man, coming to Memphis against the advice of his team, to this same man giving a speech when all he wants to do is to stay in his hotel; to the opening rejection, in that speech, of every period in history in favor of now; to the advice to stay together; to the injunction to keep the issues where they are; to the reminders and instructions on the power of economic action; to the tale of the Samaritan, used to encourage people to march in support of the sanitation workers; to his final words, locating Memphis in this moment at the apex of a journey without peer in history?\n\nWe follow a leader because he is deep in something, and he knows what that something is. His knowledge of it, and the evidence of his knowledge of it, gives us both certainty in the present and confidence in the future. And the something that we see when we look at Dr. Martin Luther King Jr. is not, ultimately, the oratory, magnificent though that is, or the self-sacrifice, inspiring though that is, or the ideology of nonviolence, uplifting though that is, or even the indefatigable persistence, humbling though that is\u2014but rather the end to which all these are deployed, time and time again. Martin Luther King Jr. was a crucible maker. He brought issues to a head, deliberately and relentlessly. His brilliance was the brilliance of not letting go, of creating intensity and focus and a concentration of time and place, and then adding more fuel and urgency and energy until, out of the white heat that he had made, something happened. That was his spike.\n\nWe see this in the structure of his speech that night in Memphis\u2014in the opening, using the repetition of \"but I wouldn't stop there\" to intensify our focus on _now_ , and in the closing, using the repetition of \"if I had sneezed\" to point us to _here_ \u2014and we can speculate whether the tension-building techniques of the orator, which he learned at a young age, were in fact the germ of this ability.\n\nWe see it when he says, directly:\n\n> . . . we have been forced to a point where we are going to have to grapple with the problems that men have been trying to grapple with through history, but the demands didn't force them to do it. Survival demands that we grapple with them . . . It is no longer a choice between violence and nonviolence in this world; it's nonviolence or nonexistence. That is where we are today.\n\nIt's not violence or nonviolence, because that's not a crucible. It's nonviolence or nonexistence, because that is.\n\nWe see it in the contours of his life, laid out for us in his final speech and laid out before us on the exhibition floor in Memphis. He didn't always know what the next point of friction would be, but he could seize a moment and make it a crucible, time and time again. This was his way to get people to see his truth\u2014that the current day was not the Promised Land. And it was his way to recognize, honor, and alleviate the uncertainty his followers felt. This was not the totality of King, of course; it is, however, what we see as most distinctive about him, and it is what we hook on to and follow.\n\nAnd the uniqueness of this ability to force the issue becomes clearer still when we think about what his contemporary leaders were doing. When John F. Kennedy said, \"The torch has been passed to a new generation of Americans,\" it was inspiring and future-is-now-ing, because that was his spike\u2014but it wasn't crucible making. When Malcolm X said, \"There's no such thing as a nonviolent revolution,\" it was raising the temperature, because that was his spike, but it wasn't raising the intensity\u2014it wasn't crucible making. And when Robert Kennedy said, \"Let us dedicate ourselves to what the Greeks wrote so many years ago: to tame the savageness of man and make gentle the life of this world,\" it was to focus his listeners on a noble and righteous cause in his own spiky way, but it wasn't crucible making.\n\nCrucible making was the pattern and the technique of King's life. And the compulsion to do this was what led him to ignore his advisers and return to Memphis, because going back was the only way to force the issue.\n\nAs he nears the end of his speech that night, he's well aware of the dangers facing him. He knows that he's such a good crucible maker that sooner or later something bad is bound to happen; that he is such a good crucible maker that sooner or later he will get caught up in the fire; that, moreover, his putting his own life repeatedly at risk is an essential element of his ability to continually force the issue. He knows, therefore, that as the noise surrounding him becomes louder\u2014as it has been doing for weeks\u2014he needs to anticipate what will happen if he is no longer there. And he knows that he wants his movement\u2014his series of crucibles, each leading to a breakthrough, and each creating the possibility of the next\u2014to continue after him, and that for this to happen, he must subordinate himself to it, and yet imbue it with such energy that it is unstoppable. He does this as only he can:\n\n> Well, I don't know what will happen now. We've got some difficult days ahead. But it really doesn't matter with me now, because I've been to the mountaintop.\n> \n> And I don't mind.\n> \n> Like anybody, I would like to live a long life. Longevity has its place. But I'm not concerned about that now. I just want to do God's will. And He's allowed me to go up to the mountain. And I've looked over. And I've seen the Promised Land. I may not get there with you. But I want you to know tonight, that we, as a people, will get to the Promised Land!\n> \n> And so I'm happy, tonight.\n> \n> I'm not worried about anything.\n> \n> I'm not fearing any man!\n> \n> Mine eyes have seen the glory of the coming of the Lord!\n\n\u2022 \u2022 \u2022\n\nAnd now the reason for our ascent is clear, because now we are looking into a motel room, and it is the motel room, on the second floor of the Lorraine Motel, where Martin Luther King Jr. spent his last day. After his sermon the evening before, King spent most of April 4, 1968, in this room, joking with his brother, who was also there, calling his parents, and even, at one point, starting a pillow fight with a fellow minister. It's a modest room\u2014meager, even. Two queen beds with thin coverings; a nightstand with a phone and a lamp; a small TV attached to the wall on a bracket; the bedclothes brown, the curtains holding back the bright light also orange-brown; the carpet brown, too. And a door leading to the balcony. A few moments after six o'clock that evening, King stepped through the door, onto the balcony, and it was there that the assassin's bullet found him.\n\nAs we write, we are days away from the fiftieth anniversary of King's death. Our journey to the Promised Land is further along now, and yet still incomplete and still contested. And across five decades the force of what this man stood for is still strong, even to those of us who were born after his death, and who know him only through the sermons, and speeches, and memorials. It is strong not because of the breadth of his abilities, but because of their narrowness and their focus, and consequently their distinctiveness and their power. This is what drew followers to him in their millions during his life, and this is what outlives him and draws us to his cause to this day.\n\nLeading and following are not abstractions. They are human interactions; human relationships. And their currency is the currency of all human relationships\u2014the currency of emotional bonds, of trust, and of love. If you, as a leader, forget these things, and yet master everything that theory world tells you matters, you will find yourself alone. But if you understand who you are, at your core, and hone that understanding into a few special abilities, each of which refracts and magnifies your intent, your essence, and your humanity, then, in the real world, we will see you.\n\nAnd we will follow.\n\n# Truths\n\nTRUTH #1 People care which team they're on\n\n_(Because that's where work actually happens.)_\n\nTRUTH #2 The best intelligence wins\n\n_(Because the world moves too fast for plans.)_\n\nTRUTH #3 The best companies cascade meaning\n\n_(Because people want to know what they all share.)_\n\nTRUTH #4 The best people are spiky\n\n_(Because uniqueness is a feature, not a bug.)_\n\nTRUTH #5 People need attention\n\n_(Because we all want to be seen for who we are at our best.)_\n\nTRUTH #6 People can reliably rate their own experience\n\n_(Because that's all we have.)_\n\nTRUTH #7 People have momentum\n\n_(Because we all move through the world differently.)_\n\nTRUTH #8 Love-in-work matters most\n\n_(Because that's what work is really for.)_\n\nTRUTH #9 We follow spikes\n\n_(Because spikes bring us certainty.)_\n\n# Appendix A\n\n# The ADPRI's Global Study of Engagement\n\nDr. Mary Hayes, Dr. Frances Chumney   \nDr. Corinne Wright, Marcus Buckingham\n\nIn July 2018 the ADP Research Institute (ADPRI) conducted a nineteen-country study of the world's workforce. The aim of the study was to measure the relative levels of engagement of each country, and to identify the conditions at work that are most likely to attract and keep talented employees. This study repeated and amplified a similar global study conducted in 2015 that involved thirteen countries.\n\nIn each country we identified a random sample of 1,000 full-time and part-time employees, stratified by various demographics, such as age, gender, and education level, and broken out by various industries and work types. With oversampling, a total of 19,346 employees were selected and surveyed.\n\nThe survey asked about many aspects of respondents' attitudes to work, but at its core was a reliable and valid measure of engagement, developed over the last decade, and comprising eight questions. Extensive previous research has shown that those who answer these eight questions positively are more likely to be seen as highly productive, and less likely to leave, and these predictive relationships between high scores on the questions and higher performance and retention are statistically significant and stable across industries and roles.\n\nUsing this survey we are able to calculate the percentage of workers who are Fully Engaged in any team, company, or country; and to examine which conditions are most likely to lead to being Fully Engaged at work.\n\n  * The percentage of Fully Engaged workers is calculated using a formula that captures the extreme positives on each question, and then weights each question's responses according to its relative power, giving more weight to those questions with the greatest explanatory power.\n  * Those workers who are not Fully Engaged fall into a category we call, simply, \"Coming to Work.\" These workers are not necessarily actively disengaged\u2014the survey was built to measure positive functioning, not pathology. They are instead merely workers who are not contributing all that they possibly could.\n\nIn 2018 we used the same survey and same sampling methodology, and applied the same country-specific corrections (to take into account how different nationalities respond differently to survey scales), as we did in 2015. As far as we are aware, this 2018 study is the largest and most reliable study of global worker engagement yet undertaken. The ten principal questions that we explored, together with our findings, are as follows.\n\n1. Has global engagement increased or decreased in the last three years?\n\nGlobal engagement remains at almost exactly the same level as it was in the thirteen original countries.\n\n  * 16.2 percent Fully Engaged in 2015, as compared with 15.9 percent in 2018. This means that, globally, fully 84 percent of workers are merely Coming to Work, and are not contributing all they could to their organizations.\n\nClearly, organizations have not yet solved the challenge of getting most workers to see work as a place where they can give of themselves, and be recognized and valued for their best.\n\nThere are obviously many entrenched reasons for this\u2014macroeconomic forces; the difficult, dangerous, and monotonous nature of some kinds of work; and the labor policies of certain countries, for example. However, as we will see below, the data suggests that there are nonetheless some actions that organizations can take to be more intentional and systematic in the way they seek to engage their workers.\n\nThough the overall level of engagement remained stable from 2015 to 2018, we found significant variation in the percentage of Fully Engaged workers by country.\n\n  * In eight countries the percentage of those Fully Engaged increased (Argentina, Australia, Canada, France, India, Italy, Spain, and the United Kingdom).\n  * In four countries the percentage of those Fully Engaged decreased (Brazil, China, Mexico, and the United States).\n\nIndia showed the largest increase in the percentage of Fully Engaged workers, up 5 points to 22 percent, and China showed the largest decrease, with a 13-point drop to 6 percent.\n\n2. Which are the most and least engaged countries in the world?\n\nIn 2018 we surveyed an additional six countries to those surveyed in 2015: Egypt, the Netherlands, Saudi Arabia, Singapore, South Africa, and the United Arab Emirates. The UAE has the highest percentage of Fully Engaged workers, at 26 percent, while China has the lowest percentage of Fully Engaged workers, at 6 percent.\n\n3. What factors contribute most to a worker's feeling Fully Engaged?\n\nWe examined many variables that could possibly contribute to a feeling of engagement at work, such as industry, position title, education level, gender, part-time versus full-time, and gig versus nongig.\n\nAlthough each of these revealed interesting relationships (which we reference below), one factor trumped all others in its ability to explain a worker being Fully Engaged: whether or not the worker was on a team.\n\n  * Workers who say they are on a team are 2.3 times more likely to be Fully Engaged than those who say they are not.\n\nThis finding holds true within all countries in the study, and in many countries the disparity between nonteam and team workers is even greater.\n\n  * For example, in Brazil 5 percent of nonteam workers are Fully Engaged, whereas 15 percent of team workers are Fully Engaged. Likewise, in Singapore 4 percent of nonteam workers are Fully Engaged as compared with 22 percent of team workers.\n\nAcross the world, the data reveals that it is extremely difficult to engage workers who do not feel part of a team. The challenge for almost all organizations today, however, is that they are not set up to know very much about their teams. Current human resource systems are extensions of financial systems and so are able to show only who-reports-to-whom boxes on an organizational chart. The challenge with this, of course, is that most work does not happen in these structured boxes.\n\n  * Of those who say they work in teams, 64 percent of them report that they work on more than one team, and that this team is not represented in the org chart.\n\nClearly, there are many reasons why engagement levels remain relatively low across the world, some of those reasons relating to the nature of work itself, some to macroeconomic conditions in the region or country, and some to the specifics of the industry or company. However, it appears that one of the reasons that engagement remains relatively low across the world is that organizations do not understand, or act on, the vital power of teams.\n\n  * Organizations do not know how many teams they have, who is on them, or which are their best and most engaged teams.\n  * When organizations make great teams their primary focus\u2014what creates them, what can fracture them\u2014we may well see significant rises in global engagement.\n\n4. What factors create a highly engaged team?\n\nEighty-three percent of workers report that they are on a team. Some of these teams, though, are more engaging than others. When we examined the most engaged teams we found that by far the best explainer of engagement levels was whether or not the team members trusted their team leader.\n\n  * Of those who strongly agreed that they trusted their team leader 45 percent were Fully Engaged. Of those who didn't strongly agree only 6 percent were Fully Engaged. A worker is twelve times more likely to be Fully Engaged if she trusts her team leader.\n\nAcross countries, industries, and positions a trusted team leader is the foundation for building highly engaged teams.\n\n5. Which factors create trust in a team leader?\n\nTwo questions in the survey showed the strongest relationship to a worker's feeling of trust in his team leader:\n\n  * Do I know clearly what is expected of me at work?\n  * Do I have the chance to use my strengths every day?\n\nThis data suggests that these two conditions\u2014knowing what is expected, and being able to play to one's strengths\u2014are the foundations of trust. When a team leader, despite the ambiguities and the fluid and fast pace of the world of work, can help team members feel clarity about expectations and a sense that their best is recognized and utilized frequently, then trust is built, and a Fully Engaged team becomes more likely.\n\n6. Is it more engaging to be a full-time worker, a part-time worker, a virtual worker, or a gig worker?\n\nAccording to the study, the most engaging work status is to have one full-time job and one part-time job.\n\n  * Of those who have this status, 25 percent are Fully Engaged, compared to 16 percent for those whose status is captured in only one of the other categories.\n  * A possible explanation is that this status brings the best of both worlds\u2014the full-time job brings stability and benefits, while the part-time role brings not only some additional earnings but also flexibility and the chance to do something the worker truly enjoys.\n\nGig-only workers who are part of a team are also highly engaged.\n\n  * Of gig-only workers on teams, 21 percent are Fully Engaged, as compared with 15 percent of traditional workers.\n\nThe two most common reasons for taking gig work are flexibility of schedule and the chance to do something the worker loves, suggesting that, as we saw with part-time work, these two factors may well be one of the sources of the worker's higher level of engagement.\n\n  * The most common title of gig-only workers is president, suggesting that many people take gig work because they like to see themselves as their own boss.\n\nWhen we examine all of the eight engagement questions closely we see that gig-only workers score more positively on six of the eight, but significantly lower on the remaining two. The two questions where the gig-only workers scored lower than traditional workers were:\n\n  * \"In my team, I am surrounded by people who share my values.\"\n  * \"My teammates have my back.\"\n\nThis suggests that, as other researchers have noted, gig-only workers may well feel more isolated than other sorts of workers. However, when we examined gig-only workers who were also on a team, the differences on these two questions disappeared. This implies that gig work does not necessarily have to be isolating, and that if a gig worker can work in a team then she will net all the benefits of gig work (greater flexibility, higher chance of doing work she enjoys, being her own boss) while at the same time feeling the benefits of traditional work (the safety and support of her teammates). One implication for companies is that, if they choose to use contractors or gig workers\u2014and today many do\u2014the faster and more genuinely they can introduce these workers into teams, the more engagement, more productivity, and higher retention they will see from these workers. The inverse is also true: that the more companies can make traditional full-time work similar to gig work\u2014as in, greater flexibility and ownership for team members, and a greater chance to do what they love\u2014the more engagement, productivity, and higher retention they will see from their full-time workers.\n\nIn all countries and industries, virtual workers\u2014so long as these workers are also team workers\u2014are more likely to be Fully Engaged than those who do their work in an office:\n\n  * Of virtual workers, 29 percent are Fully Engaged, versus 17 percent for traditional office workers.\n\nThis suggests both that physical proximity is not required to create a sense of team and that the flexibility and ease inherent in working virtually are appealing to all workers (as long as they feel part of a team).\n\nWorking virtually is not the same as traveling for work. Those workers who reported that they traveled for work displayed the lowest levels of engagement.\n\n  * Of those who travel for work, 11 percent are Fully Engaged, versus 16 percent for those who don't travel for work.\n\n7. Are workers with more education generally more engaged at work?\n\nYes they are.\n\n  * Of those with an advanced degree, 19 percent are Fully Engaged, compared with 12 percent for those with no college education.\n\n8. Are higher-level workers more engaged than entry-level workers?\n\nYes they are.\n\n  * Of C-suite\/VP-level workers, 24 percent are Fully Engaged.\n  * Of mid-level and first-level team leaders, 14 percent are Fully Engaged.\n  * Of individual contributors, 8 percent are Fully Engaged.\n\n9. Are millennials less engaged than boomers?\n\nSlightly. But contrary to our initial hypotheses, we found very little difference in engagement by generation.\n\n  * Of millennials, 16 percent are Fully Engaged, as compared with 18 percent of boomers.\n\n10. Are men more engaged than women?\n\nNo. The data actually points slightly in the other direction.\n\n  * Globally, 17 percent of women are Fully Engaged, compared with 15 percent of men. Given the large sample sizes these differences are statistically significant, but a two-point difference such as this is not practically significant.\n\n# Appendix B\n\n# Seven Things We Know for Sure at Cisco\n\nRoxanne Bisby Davis   \nAshley Goodall\n\nFour years ago, the human resources team at Cisco set out to measure the world of work as carefully and reliably as possible. Since then, we have led a group of a dozen researchers and data scientists in exploring the characteristics of Cisco's best teams, the relationship between attention and performance, the relative importance of team and company in our experience of work, and much more. Here are some highlights of what we've discovered so far.\n\n1. The best teams are built on strengths.\n\nWe began by seeking to understand in detail what the best teams at Cisco look like. Our study, which we called the Best Teams Study, replicated studies done over the last twenty years by Gallup, Deloitte, and others, and began with the hypothesis that the experience of working on a high-performance team is measurably different from that of working on a non-high-performance team.\n\nTo test this, in late 2015 we identified a study group of ninety-seven high-performing teams by asking leaders across the company to give us the names of teams they wished they had more of\u2014the teams they would clone if they could. We then defined a control group, which was a stratified random sample of 3,600 individuals across Cisco designed to represent the average team member's experience of his or her team. We deployed a confidential eight-item survey to both groups with identical messaging.\n\nOnce the survey responses were received we assessed content validity (assessed by item correlation), construct validity (established by confirmatory factor analysis, item-to-total correlations, and regression analysis), and criterion-related validity (measured through the strength of the connection of the survey items to the concurrent criterion of study- or control-group membership). Taken together, these tests told us that:\n\n  * The eight items were measuring a single factor (\"engagement\"), which is associated differentially with the best teams at Cisco.\n  * The item \"I have the chance to use my strengths every day at work\" exhibited the strongest connection to overall engagement and the strongest connection to other items in the survey. The item \"My teammates have my back\" showed the second-strongest connection, and the item \"In my team, I am surrounded by people who share my values\" had the third-strongest connection.\n  * The study group (i.e., \"the best\" teams) outpaced the control group (i.e., \"the rest\") on six of the eight items at the aggregate (whole-company) level. The two items that did not differentiate between the groups were both scored the same. (See \"3. There are three distinct sources of engagement\" for our further investigation of this result.)\n\nOur study did show a statistically significant and meaningful difference between the best and the rest, suggesting that, at Cisco, the best teams harness the individual excellence of each team member, unlock the collective excellence of the team, and do so in an environment of safety and trust.\n\n2. More-frequent check-ins are associated with increased use of strengths.\n\nWanting to understand more about what differentiated our best teams, we wondered if the simple act of a leader's checking in on a frequent basis with his or her team members influenced the team members' engagement level.\n\nAfter completing the Best Teams Study described above, we gave every team leader at Cisco the ability to measure his or her own team on the eight engagement items we had used. While the data for a particular team was visible only to that team's leader\u2014it was intended to help leaders understand how they were doing, not to evaluate them\u2014we were able to use anonymized data for research purposes. We called our measure the Engagement Pulse.\n\nTo investigate the relationship between check-in and engagement, we selected as our sample team members who had responded to at least one Engagement Pulse over the course of two fiscal quarters. This gave us a research sample of 16,485 team members from the first quarter and 18,816 team members from the second quarter. We then determined for each quarter whether a team member was checking in frequently (80 percent of the time or more) or infrequently (less than 80 percent of the time).\n\nWe examined, for each quarter, the average response scores for all eight Engagement Pulse items, and looked for any differences between the group that was checking in frequently and the group that was not. In both quarters, we found that the team members who checked in frequently had statistically significantly higher scores for three of the eight items:\n\n  * Scores for the item \"I have the chance to use my strengths every day at work\" exhibited the largest difference between those who were checking in frequently and those who were not. Scores for the item \"In my work, I am always challenged to grow\" showed the second-largest difference, and those for the item \"I know I will be recognized for excellent work\" showed the third-largest difference.\n\nThis suggests that team members who check in with their leader frequently have an enhanced sense of being able to use their strengths every day, of being recognized for excellent work, and of having opportunities to grow. Although this study did not distinguish between correlation and causation (we could not tell whether the increased frequency of conversation led to increased engagement or vice versa), subsequent research, a portion of which is described in the final section of this appendix, indicated that it was in fact the increased attention, via frequent conversation, that led to the increased levels of engagement.\n\n3. There are three distinct sources of engagement.\n\nOur next piece of research set out to shed light on who has the biggest impact on a team member's engagement. To do this, we needed first to understand our engagement construct further and then to explore the difference, if any, between a team member's engagement across multiple teams.\n\nAs described in chapter 1 of this book, the eight items which most effectively capture team performance\u2014and which are the same as those in the Engagement Pulse\u2014can be divided into four \"we\" items, which capture team environment and company experience, and four \"me\" items, which capture each individual's experience of work. To further explore our engagement construct, we collected responses from 33,018 individuals who completed at least one Engagement Pulse survey over a six-month period, and performed two studies.\n\nFirst, using a split sample exploratory and confirmatory factor analysis, we discovered that (at least at Cisco) there are two factors of engagement within the Engagement Pulse. The first factor comprises all four \"me\" items together with the two \"we\" items that ask about team environment, and so consisted of:\n\n  * At work, I clearly understand what is expected of me. (Me)\n  * I have the chance to use my strengths every day at work. (Me)\n  * I know I will be recognized for excellent work. (Me)\n  * In my work, I am always challenged to grow. (Me)\n  * In my team, I am surrounded by people who share my values. (We)\n  * My teammates have my back. (We)\n\nWe chose to call this factor _team engagement_. The other factor comprised the remaining two \"we\" items:\n\n  * I am really enthusiastic about the mission of my company. (We)\n  * I have great confidence in my company's future. (We)\n\nWe chose to call this factor _company engagement_.*\n\nOur second study looked at how these two factors either varied for individuals who were part of more than one team, or changed as an individual moved from team to team. We found that changes in different parts of the engagement construct had different sources. In particular, we found that as someone moved from team to team, company engagement varied least, whereas the items \"In my team, I am surrounded by people who share my values\" and \"My teammates have my back\" varied most.\n\nDrawing together this study and the Best Teams Study described earlier, we have greater insight into the relationships between engagement, teams, and team leaders, namely:\n\n  1. All eight engagement items are scored more highly on the highest-performing teams, and there is strong evidence (in our studies and others') that higher engagement causes higher performance.\n  2. Of the eight items, the two constituting the company engagement factor are least sensitive to the particular team an individual belongs to.\n  3. Of the eight items, \"In my team, I am surrounded by people who share my values\" and \"My teammates have my back\" are most sensitive to the particular team an individual belongs to.\n  4. Of the eight items, the \"me\" items (addressing expectations, use of strengths, recognition, and growth challenge) are most sensitive to an individual's relationship with his or her team leader.\n\nOne way to think of these results is to imagine a team leader having three distinct jobs. Her first is to ensure her team members feel connected to the purpose and future of the company, even though she may not directly define those. Her second is to ensure that her team members, _as a group_ , understand and support one another. And her third is to ensure that her team members, _individually_ , understand what's expected of them and how they can do their best work now and in the future, all while feeling recognized for who they are.\n\n4. Decreasing engagement leads to voluntary attrition.\n\nVoluntary attrition is generally high on the list of concerns for organizational leaders, so in this study we sought to explore the connection between a team member's engagement and his or her likelihood to choose to leave Cisco. More specifically, we wanted to identify which of the Engagement Pulse items influenced a team member's decision to leave voluntarily.\n\nWe used both termination and Engagement Pulse results from a fiscal year, and from this constructed a population for those who completed an Engagement Pulse survey and either remained at Cisco or voluntarily left in the same fiscal year. Using a range of methodologies, including Pearson's correlation between predictors and outcome variables, a variety of regression models, and the bootstrap method to ensure the stability of our findings, we discovered that four Engagement Pulse items are significant predictors of voluntary attrition. They are, from most to least predictive power:\n\n  * \"I have a chance to use my strengths every day at work.\"\n  * \"In my work, I am always challenged to grow.\"\n  * \"I am really enthusiastic about the mission of my company.\"\n  * \"I have great confidence in my company's future.\"\n\nThis finding validates that there is a connection between a team member's engagement and the likelihood of his or her subsequent decision to resign. It further demonstrates that the more positively people feel about their strengths, now and in the future (the first two items above), and about their company's mission and future prospects (the second two items above), the more likely they are to stay with that company. The subtlety here is that, as we saw above, the feeling of enthusiasm about a company's mission, and confidence in its future, _still vary team to team_. In other words, our experience of our company is significantly mediated by our experience of our team.\n\nThe implication of this study is that, above all else, a focus by team leaders on helping each team member play to his or her strengths helps insulate the team against attrition.\n\n5. Attending company events is associated with higher purpose and confidence.\n\nSince 2015, Cisco has held a monthly all-hands meeting\u2014led by our executive leadership team\u2014that we call the Cisco Beat. The intended outcome of this monthly ritual is that our people gain a stronger collective understanding of our purpose as a company and a stronger sense of confidence in Cisco's future.\n\nTo measure if this was happening we investigated the relationship between the following:\n\n  * The number of Cisco Beats a team member attended.\n  * The average Engagement Pulse responses for that team member for items relating to the company engagement factor (collective purpose and confidence in the future).\n\nUsing data from 52,819 team members who responded to an Engagement Pulse over three quarters, we then identified how many Cisco Beats each team member had attended out of the eight that had been held in that period. Attendance was defined as attending the event in person, watching the event live via Cisco's broadcasting technology, or watching the event replay within two weeks of the event. We then segmented the team members by attendance\u2014those who had attended no Beats, those who had attended from one to three, those who had attended from four to six, and those who had attended either seven or eight. We took this segmentation and looked at the average Engagement Pulse response for team members in each segment for the items \"I am really enthusiastic about the mission of my company\" and \"I have great confidence in my company's future\" (the two items constituting the company engagement factor identified previously).\n\nOur analysis revealed that the more Cisco Beats a team member attends, the higher his or her average item response.\n\n  * Scores for \"I am really enthusiastic about the mission of my company\" increased from 4.37 for the team members who did not attend any Cisco Beats to 4.48 for those who attended all Cisco Beats or all but one Cisco Beat. This increase was statistically significant.\n  * Scores for \"I have great confidence in my company's future\" increased from 4.25 for those who attended no Cisco Beats to 4.35 for those who attended all Cisco Beats or all but one Cisco Beat. This increase was also statistically significant.\n\nThose team members who are regularly attending Cisco Beats are more enthusiastic about our collective purpose and more confident in our future as a company. We have not yet explored whether attendance at these events subsequently increases engagement in this way, or whether those who are more engaged in the first place subsequently attend more Cisco Beats.\n\n6. Highly engaged people talk about work differently.\n\nWe have found in the course of our research that it is helpful to distinguish between those team members with a particularly high level of engagement and everyone else. We refer to those individuals in the high-engagement group as Fully Engaged and everyone else as Not Fully Engaged. Our research has led us to a good understanding of the quantitative differences between these two groups. However, we were curious about the difference in the way Fully Engaged team members talked about work versus the way their less engaged counterparts did.\n\nWe investigated open-text survey responses to answer the following:\n\n  * What was the sentiment of each group overall?\n  * What topics did each group discuss?\n  * Were there differences in sentiment and discussion topics between the two groups?\n\nTo do this, we used our Real Deal survey. This Cisco-developed survey is sent to a representative sample of our population each quarter, and includes open-text response items. We isolated responses from team members who responded to the Real Deal's open-text item \"What would you tell your functional leader about life in your function?\" and who also completed an Engagement Pulse survey during a given quarter. In all, 1,275 team members met both criteria.\n\nUsing both natural-language processing technology and our own analytical approach, we were able to investigate the differences in the sentiment of the Fully Engaged and Not Fully Engaged groups. We used each group's Emotional Promoter Score (a measure of sentiment ranging from \u2212100 to 100, calculated using a third-party algorithm) and the distinct words chosen in each group's text responses to spot differences in discussion topics and to automatically sort comments into themes. This allowed us to see how often each group discussed certain predefined topics.\n\nTaken together, these data sets revealed clear differences between the two groups:\n\n  * Those who were Fully Engaged had, on average, an Emotional Promoter Score of 26 and discussed excellence on a team and\/or hopes for the future in their comments. The following comment is representative of this group: \"It is very rewarding to see how the managers are willing to incorporate new ideas and new members to the team to improve and achieve sales goals. It is a very creative and productive way of working. If the teams are productive and happy, customers also perceive it.\"\n  * Those who were Not Fully Engaged had, on average, an Emotional Promoter Score of \u221216 and were more negative when describing their experiences on their teams. Comments from these team members reflected uncertainty about the future and frustration with internal bureaucracy. The following comment is representative of this group: \"I think we need some sort of 'offsite' so we can help build trust across the next level of the organization. There's still a lot of silo-based behavior a couple levels deep and it would be great to show how we're trying to break that down.\"\n\nAs we continue to explore the natural-language processing of open-text responses, our next major focus will be the words team members use to describe their careers and career aspirations.\n\n7. Some forms of attention are better than others in creating engagement.\n\nBeyond understanding which things in the world of work are related to which other things (that event attendance is related to confidence in the future, for example, or that high engagement is related to specific text responses in a survey), we are of course most interested in what causes what. This last summary is an example of this type of research.\n\nWe wanted to understand whether the different ways a leader chooses to pay attention to a check-in influences his or her team member's engagement over time. Could we discern that team members who received frequent attention from their team leaders had higher levels of engagement than those who did not, and that a live conversation between a leader and team member was the best type of attention for a leader to provide to a team member?\n\nTo do this we investigated the following:\n\n  * How often team members were receiving attention from their leader, in the form of responses to check-ins entered in our team technology.\n  * Whether any methods of giving attention (viewing a check-in in the technology, commenting on it in the technology, or having a live discussion) were better than others.\n  * Which patterns of attention over time were most common (no attention, some attention, constant attention).\n  * How team-member engagement changed over time given the type and frequency of attention.\n\nLooking at data from early 2018, we identified 6,726 team members who had responded to two or more Engagement Pulse surveys. We then used the first and last surveys for these team members to identify if they were Fully Engaged (FE) or Not Fully Engaged (NFE) at Time 1 and Time 2, respectively. This allowed us to identify those whose engagement, over the course of the study period, had increased (NFE at Time 1 to FE at Time 2), had decreased (FE at Time 1 to NFE at Time 2), or had stayed the same (either FE or NFE at both Time 1 and Time 2).\n\nWe then looked at check-in behavior and the different types of attention that can occur:\n\n  * For team members, we looked at whether they requested attention (by submitting at least one online check-in during this time period) or not.\n  * For team leaders, we looked at the four possible responses to a team member's request for attention: viewing a check-in online, entering a comment on a check-in online, having a live discussion with the team member (as subsequently confirmed by the team member), or providing no attention at all by not responding in any of the first three ways. After analyzing the data we grouped these possible responses into three segments: \"no attention,\" \"any type of attention,\" and \"attention that included a conversation.\"\n\nWe were now able to examine the changes in engagement between Time 1 and Time 2 as a function of the most frequent type of attention received by each person. Since most team members respond to an Engagement Pulse survey every three months, these changes reflect the effects of different amounts and types of attention over the course of three months.\n\n  * For team members who did not ask for attention by submitting a check-in, the proportion that were Fully Engaged decreased 13 percent and had the lowest absolute level.\n  * For team members who consistently or near-consistently checked in but did not receive any sort of attention in return, the proportion that were Fully Engaged decreased 2 percent. We have found that check-in frequency drops significantly when team leaders do not respond, so we imagine that this group will, over time, come to resemble the group above\u2014not checking in at all\u2014with a commensurately larger engagement decrease\n  * For team members who always received some type of attention from their leader, the proportion that were Fully Engaged increased by 2 percent.\n  * For team members who consistently received attention in the form of a conversation, the proportion that were Fully Engaged increased by 3 percent.\n\nWe can conclude that any attention is better than no attention, that frequent attention is better than infrequent attention, and that the type of attention a leader gives matters. When the type of attention a leader gives to his or her team members includes a live discussion, we see the highest levels of team-member engagement and the biggest positive change in team-member engagement over time, irrespective of the conversational skill of the team leader or the quality of the conversation.\n\nResearch Contributors\n\nJohn Lagonigro, Madison Beard, Mary Williams, Hanqi Zhu, and Thomas Payne\n\n*This is the discovery described in chapter 3.\n\n# Notes\n\nINTRODUCTION\n\n. \"23 Economic Experts Weigh In: Why Is Productivity Growth So Low?\" Focus Economics, accessed November 10, 2018, <https:\/\/www.focus-economics.com\/blog\/why-is-productivity-growth-so-low-23-economic-experts-weigh-in>.\n\nLIE #1\n\n. To offer just one example, a recent article in _Harvard Business Review_ suggests that culture comes in eight flavors (Learning, Purpose, Caring, Order, Safety, Authority, Results, and Enjoyment); that each of these is measurable across a company as a whole; that companies can combine several of these into their overall culture; and that an important part of selecting senior leaders is to assess the degree of correspondence between their intrinsic characteristics and the desired corporate culture. Boris Groysberg et al., \"The Leader's Guide to Corporate Culture,\" _Harvard Business Review_ , January\u2013February 2018.\n\n. Edmund Burke, _Reflections on the Revolution in France_ (London: James Dodsley, 1790).\n\n. Yuval Noah Harari, _Sapiens: A Brief History of Humankind_ (London: Harvill Secker, 2014).\n\n. Yuval Noah Harari, _Homo Deus: A Brief History of Tomorrow_ (London: Harvill Secker, 2016).\n\nLIE #2\n\n. Stanley McChrystal et al., _Team of Teams: New Rules of Engagement for a Complex World_ (New York: Penguin, 2015).\n\n. _The Battle of Britain, August\u2013October 1940: An Air Ministry Record of the Great Days from 8th August\u201331st October, 1940_ (London: H.M. Stationery Office, 1941).\n\n. McChrystal, _Team of Teams_ , 216.\n\n. Ibid., 217.\n\n. Cisco data, as presented at the annual conference of the Society for Industrial and Organizational Psychology (SIOP), 2017.\n\nLIE #3\n\n. Lisa D. Ordo\u00f1ez et al., \"Goals Gone Wild: The Systematic Side Effects of Overprescribing Goal Setting,\" _Academy of Management Perspectives_ 23, no. 1 (2009): 6.\n\n. Teresa Amabile and Steven Kramer, _The Progress Principle: Using Small Wins to Ignite Joy, Engagement, and Creativity at Work_ (Boston: Harvard Business Review Press, 2011).\n\n. Mark Zuckerberg, Facebook post, January 11, 2018, <https:\/\/www.facebook.com\/zuck\/posts\/10104413015393571>.\n\n. Cammie McGovern, \"Looking into the Future for a Child with Autism,\" _New York Times_ , August 31, 2017.\n\nLIE #4\n\n. See <https:\/\/www.youtube.com\/watch?v=ch-vWyK2yJs>.\n\n. Stephen Pile, _The Ultimate Book of Heroic Failures_ (London: Faber and Faber, 2011), 115.\n\n. Stated during an appearance in the 2017 documentary _George Michael: Freedom_ , directed by David Austin.\n\n. \"IBM Kenexa Core (Foundational) Skills and Competencies: A Framework with Core Skills Required for General Job Roles,\" IBM Corporation, 2015.\n\n. See <https:\/\/performancemanager4.successfactors.com\/doc\/roboHelp\/12-Getting_Familiar_With_PA_Forms\/ph_wa_use.htm> (retrieved 8\/25\/18).\n\n. See, for example, Dr. Robert Kegan's talk at the 2016 NeuroLeadership Summit, available here: <https:\/\/neuroleadership.com\/bob-kegan-feedback\/>.\n\n. Walter Isaacson, _Steve Jobs_ (New York: Simon & Schuster, 2011), 42.\n\n. This story is told in more detail by Todd Rose in his wonderful book _The End of Average: How We Succeed in a World That Values Sameness_ (New York: HarperCollins, 2016). We're grateful to him for his permission to summarize it here.\n\n. Stated precisely, there was no evidence that the average characteristics of a group applied to any individual in that group.\n\nLIE #5\n\n. See <https:\/\/www.youtube.com\/watch?v=EqVyHMtSvFE>.\n\n. Ray Dalio, _Principles_ (New York: Simon & Schuster, 2017).\n\n. Adam Grant, \"Billionaire Ray Dalio Had an Amazing Reaction to an Employee Calling Him Out on a Mistake,\" _Business Insider_ , February 2, 2016.\n\n. Brian Brim and Jim Asplund, \"Driving Engagement by Focusing on Strengths,\" _Gallup Business Journal_ , November 12, 2009.\n\n. Joseph LeDoux, _Synaptic Self: How Our Brains Become Who We Are_ (New York: Viking Adult, 2002).\n\n. Richard Boyatzis, \"Neuroscience and Leadership: The Promise of Insights,\" _Ivey Business Journal_ , January\/February 2011.\n\n. Ibid.\n\n. Rick Hanson, \"Take in the Good,\" <https:\/\/www.rickhanson.net\/take-in-the-good\/>.\n\n. Including, according to one recent piece of research, by looking for a less critical social network so as to avoid hearing the negative feedback in the first place. See Scott Berinato, \"Negative Feedback Rarely Leads to Improvement,\" _Harvard Business Review_ , January\u2013February 2018.\n\n. David Cooperrider and Associates, \"What Is Appreciative Inquiry?\" <http:\/\/www.davidcooperrider.com\/ai-process\/>.\n\n. John M. Gottman and Nan Silver, _The Seven Principles for Making Marriage Work: A Practical Guide from the Country's Foremost Relationship Expert_ (New York: Crown Publishers, 1999); and Barbara L. Fredrickson, \"The Broaden-and-Build Theory of Positive Emotions,\" _Philosophical Transactions of the Royal Society B: Biological Sciences_ 359, no. 1449 (2004): 1367.\n\n. See <https:\/\/www.chronicle.com\/blogs\/percolator\/the-magic-ratio-that-wasnt\/33279>.\n\n. Barbara L. Fredrickson, \"The Role of Positive Emotions in Positive Psychology: The Broaden-and-Build Theory of Positive Emotions,\" _The American Psychologist_ 56, no. 3 (2001): 218.\n\nLIE #6\n\n. Robert J. Wherry Sr. and C. J. Bartlett, \"The Control of Bias in Ratings: A Theory of Rating,\" _Personnel Psychology_ 35, no. 3 (1982): 521; Michael K. Mount et al., \"Trait, Rater and Level Effects in 360-Degree Performance Ratings,\" _Personnel Psychology_ 51, no. 3 (2006): 557; and Brian Hoffman et al., \"Rater Source Effects Are Alive and Well after All,\" _Personnel Psychology_ 63, no. 1 (2010): 119.\n\n. Steven E. Scullen, Michael K. Mount, and Maynard Goff, \"Understanding the Latent Structure of Job Performance Ratings,\" _Journal of Applied Psychology_ 85, no. 6 (2000): 956.\n\n. More precisely, according to the researchers' determination of how much of the ratings variance could be tied directly to someone's individual performance, the person being rated is 16 percent there and 84 percent not there.\n\n. Hoffman et al., \"Rater Source Effects Are Alive and Well after All.\"\n\n. This definition is from the _Financial Times_ , and can be found at <http:\/\/lexicon.ft.com\/Term?term=business-acumen> (retrieved 2\/17\/18).\n\n. James Surowiecki, _The Wisdom of Crowds_ (New York: Anchor Books, 2005).\n\n. See Galton's original letter sharing his findings at \"Vox Populi\u2014Sir Francis Galton,\" The Wisdom of Crowds blog, <http:\/\/wisdomofcrowds.blogspot.com\/2009\/12\/vox-populi-sir-francis-galton.html>.\n\n. This might be a little inside baseball for you, but please watch out for anything calling itself \"driver analysis.\" This describes an approach in which the creator of a survey includes many questions about a subject, such as employee engagement, and then at the end of the survey adds a short list of summary questions, such as, \"I am proud to work for my company,\" or, \"I plan to work for my company a year from now.\" The survey creator then runs a driver analysis to examine which of the questions in the body of the survey \"drive\" the summary items, and winds up pronouncing that certain items are the drivers of employee engagement because people who scored higher on them also scored higher on the summary items. Superficially this appears to be a conclusion based on valid data, but it isn't terribly helpful. One-off driver analysis doesn't tell you what drives behavior in the real world: it simply reveals that people who rated certain items more highly earlier in the survey also rated other items highly later in the survey; that people who were happy early in the survey were still happy later. Technically, this conclusion is valid. It just doesn't matter very much.\n\n. Lord Kelvin (n\u00e9 William Thomson)\u2014the British scientist who, among other things, determined the value of absolute zero and who therefore knew a thing or two about measurement (and thermometers)\u2014once said, \"In physical science a first essential step in the direction of learning any subject is to find principles of numerical reckoning and methods for practicably measuring some quality connected with it. I often say that when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind: it may be the beginning of knowledge, but you have scarcely, in your thoughts, advanced to the stage of _science_ , whatever the matter may be.\" Sir William Thomson, \"Electrical Units of Measurement,\" a lecture delivered at the Institution of Civil Engineers on May 3, 1883, published in _Popular Lectures and Addresses_ , vol. 1, _Constitution of Matter_ (London: Macmillan and Co., 1889), 73.\n\n. We could also, by the way, use this type of approach to design a better 360-degree-feedback tool\u2014if we wanted to understand what someone's peers felt about his or her performance. In doing do, we'd have to be sure the questions asked each peer to rate him- or herself, as we've seen. But we'd also need to solve for two data-sufficiency issues: First, who makes up the right set of people to rate you, and how many of them do we need to respond to our survey? Second, how do we know that they know your work well enough to provide good data? These are tricky questions to solve.\n\nLIE #7\n\n. Douglas A. Ready, Jay A. Conger, and Linda A. Hill, \"Are You a High Potential?\" _Harvard Business Review_ , June 2010.\n\n. For a good discussion of this, see Ken Richardson and Sarah H. Norgate, \"Does IQ Really Predict Job Performance?\" _Applied Developmental Science_ 19, no. 3 (2015): 153.\n\n. See <https:\/\/www.britannica.com\/biography\/Elon-Musk>.\n\n. John Paul MacDuffie, \"The Future of Electric Cars Is Brighter with Elon Musk in It,\" _New York Times_ , October 1, 2018.\n\nLIE #8\n\n. Kristine D. Olson, \"Physician Burnout\u2014A Leading Indicator of Health System Performance?\" _Mayo Clinic Proceedings_ 92, no. 11 (2017): 1608.\n\nLIE #9\n\n. We are not the first to make this point. But those who have arrived at it before us then continue by trying to identify the set of traits that all leaders should acquire in order to attract followers\u2014bringing us right back to the idea that _leadership is a thing_. Our line of inquiry has taken us in a different direction.\n\n. Donald E. Brown, _Human Universals_ (New York: McGraw Hill, 1991).\n\n. Pierre Gurdjian, Thomas Halbeisen, and Kevin Lane, \"Why Leadership-Development Programs Fail,\" _McKinsey Quarterly_ , January 2014.\n\n. This list is taken from Claudio Fern\u00e1ndez-Ar\u00e1oz, Andrew Roscoe, and Kentaro Aramaki, \"Turning Potential into Success: The Missing Link in Leadership Development,\" _Harvard Business Review_ , November\u2013December 2017, 88.\n\n. Joseph Rosenbloom, \"Martin Luther King's Last 31 Hours: The Story of His Final Prophetic Speech,\" _The Guardian_ , April 4, 2018.\n\n# Index\n\nAbernathy, Ralph,\n\nAccenture,\n\nactive listening,\n\nActor-Observer Bias,\n\nAdele,\n\nADP Research Institute, , \u2013, , \u2013\n\nadvice, \u2013\n\nalignment\n\nattempt to create through goals,\n\nshared meaning and, ,\n\nAmabile, Teresa,\n\nAmazon,\n\nAmerican Management Association,\n\nAnthony, Susan B.,\n\nanticipation,\n\nApple, , \u2013\n\nAppreciative Inquiry, \u2013\n\nArcos Dorados Holdings Inc.,\n\nAriel 6 satellite, \u2013\n\nassumptions, \u2013\n\nabout expectations,\n\nof uniformity of experience, \u2013\n\n_See also_ truths about work\n\nAtari,\n\nattention\n\ncreating engagement and, \u2013\n\nexcellence creation and, \u2013\n\nneed for, \u2013\n\npositive versus negative, , \u2013\n\nstimulating growth through, \u2013\n\nstrengths-focused, \u2013\n\nin teams, \u2013\n\nattention, uncritical, human need for, \u2013\n\nattributions, biases in, \u2013\n\nattrition, \u2013\n\nauthenticity,\n\nbalance,\n\nflow versus, \u2013\n\nhistory of concept of, \u2013\n\nin positive versus negative attention, \u2013\n\nwork-life, \u2013\n\nBartlett, C. J.,\n\nBattle of Britain, \u2013, \u2013\n\n\"Best of Me\" questions, \u2013\n\n\"Best of We\" questions,\n\nBest Teams Study, \u2013\n\nBHAGs (big hairy audacious goals),\n\nbiases\n\nActor-Observer Bias,\n\nfeedback and, \u2013\n\nFundamental Attribution Error, \u2013\n\npotential rating and, \u2013\n\nin rating others, \u2013\n\nself-serving, \u2013\n\nBoston Consulting Group,\n\nbox-of-paints approach to advice, \u2013\n\nbrain development, \u2013\n\ngiving advice and, \u2013\n\nbrands and branding, ,\n\nBridgewater Associates, , \u2013, \u2013\n\nBrown, Donald,\n\nBruer, John,\n\nBuffett, Warren, ,\n\nBurke, Edmund,\n\nburnout, \u2013, , ,\n\nbusiness acumen, \u2013\n\ncareer derailers,\n\ncareer development,\n\ncheck-ins and, \u2013\n\ndeficit repair and, \u2013\n\nfailure and,\n\nleadership development and, \u2013\n\nlove-in-work and, \u2013\n\noutcomes focus and, \u2013\n\npotential and, \u2013, \u2013\n\nteams and,\n\nusing strengths and, \u2013\n\ncatching people doing things right, \u2013\n\nCathy, Truett, \u2013, \u2013\n\nchange, planning and pace of, \u2013\n\nCharge of the Light Brigade, \u2013\n\ncheck-ins, \u2013, \u2013\n\nfrequency of, \u2013, \u2013\n\ntrust in leaders and, \u2013\n\nChick-fil-A, \u2013, \u2013, ,\n\nChristian Movement for Human Rights,\n\nChumney, Frances, \u2013\n\nChurchill, Winston, ,\n\nCisco, \u2013\n\nintelligence system at,\n\npotential rating at, \u2013\n\nrating others at, ,\n\nteam experience at, , \u2013, \u2013\n\nClifton, Don,\n\nClinton, Bill,\n\nClooney, George, \u2013\n\ncoaching,\n\ncoercion, \u2013\n\ncollaboration, culture of, \u2013\n\nCollins, Jim,\n\ncomfort zones,\n\ncommunication\n\nabout expectations,\n\ncheck-ins versus meetings and, \u2013\n\nof praise and recognition, \u2013\n\nrituals in, \u2013\n\n_See also_ intelligence systems\n\ncompany events, \u2013\n\nCompaq,\n\ncompensation, \u2013\n\ncompetency model, \u2013\n\ndata reliability and, \u2013\n\ndeficit repair and, \u2013\n\ndiversity and, \u2013\n\nidiosyncratic excellence versus, \u2013\n\nleadership and, \u2013\n\nmeasurement issues with, \u2013\n\npurported benefits of, \u2013\n\nrating others and,\n\nconfidence, \u2013\n\nconformity, . _See also_ individuality\n\ncontrol,\n\nindividuality and, \u2013\n\nspan of,\n\n\"The Control of Bias in Ratings\" (Wherry and Bartlett),\n\nCook, James, n\n\nCooperrider, David,\n\nCorporate Executive Board,\n\ncortisol,\n\ncreativity, \u2013\n\nCuban Missile Crisis,\n\nculture, \u2013\n\nbest companies to work for and, \u2013\n\ncascading meaning and, \u2013\n\ndefinition of, \u2013\n\nintersubjective realities and, \u2013\n\nsignifiers and, \u2013\n\nof teams, \u2013, \u2013\n\n_See also_ experience of work\n\nCulture Audit,\n\nDalio, Ray, , \u2013,\n\nDallas Cowboys, \u2013\n\nDamon, Matt,\n\nDaniels, Gilbert S., \u2013\n\nDarwin, Charles,\n\ndata\n\nevaluating, \u2013\n\ninsufficiency, \u2013\n\nquality of, \u2013\n\nreliable, \u2013, \u2013, \u2013\n\nsubjective,\n\nvalid, \u2013\n\nvariable, \u2013\n\nDavis, Roxanne Bisby, \u2013\n\ndeficits, correcting, \u2013\n\ndefining the adjustable seat, \u2013\n\nDeloitte,\n\ndetachment,\n\ndistractions,\n\ndiversity, \u2013, \u2013,\n\nDowding, Hugh,\n\nDowding system, \u2013,\n\ndream jobs, \u2013\n\nDrucker, Peter,\n\nDunkirk evacuation, \u2013\n\neducation levels, engagement and,\n\nEdward Jones,\n\nemergency-room nurses,\n\nendorphins,\n\nengagement\n\nattention in creating, \u2013, \u2013\n\ncascading meaning and, \u2013\n\nchanges in level of, \u2013\n\ncheck-in frequency and, \u2013\n\ncompany events and, \u2013\n\ncountries with highest and lowest levels of,\n\neducation and,\n\nfactors constituting, \u2013\n\nof full-time, part-time, virtual, and gig workers, \u2013\n\ngender differences in,\n\ngenerational differences in,\n\nglobal, weakness of,\n\nglobal study of, \u2013\n\nlove-in-work and, \u2013\n\nsources of, \u2013\n\nteams in creating, \u2013\n\ntrust and, \u2013\n\nvoluntary attrition and, \u2013\n\nways of talking about work and, \u2013\n\nethics,\n\n_eudaimonia,_ \u2013. _See also_ flow\n\nexcellence\n\nattention and, \u2013\n\nleaders and, \u2013\n\nnoting and recognizing, \u2013\n\nteams in creating, \u2013\n\nexpectations,\n\nclarity of on teams, \u2013\n\nleaders and, \u2013\n\nsetting clear,\n\nunderstanding,\n\nexperience of work\n\nassumptions of uniformity of, \u2013\n\ncompetency model and, \u2013\n\nengagement and, \u2013\n\nimportant elements of, \u2013\n\nindividual, \u2013\n\nintersubjective realities of, \u2013\n\nlove-in-work and, \u2013\n\nshared,\n\nteams and, \u2013\n\nexpressed values,\n\nextremism, \u2013\n\nFacebook, \u2013, , , ,\n\nuser feedback in, \u2013\n\nfactor analysis,\n\nfailure\n\ndevelopment through,\n\nexcellence versus,\n\nfallacy of the undistributed middle, n\n\nfeedback\n\nactive listening and,\n\nas attention, \u2013\n\nattention versus, \u2013, \u2013\n\nbiases and, \u2013\n\nat Bridgewater Associates, \u2013\n\ncandid,\n\ndeficit correction and,\n\nlearning and, \u2013\n\nmillennials' need for, \u2013, \u2013\n\nmirroring,\n\nnegative, giving, \u2013\n\nobjective,\n\npeople's need for, \u2013\n\npositive versus negative, \u2013\n\nrecognition and praise, \u2013\n\nsmartphones and,\n\nsocial media and, \u2013\n\nsuccess and,\n\n_See also_ performance appraisals; rating other people\n\nfight or flight,\n\nFloquet, Ethan, \u2013\n\nflow\n\nbalance versus, \u2013\n\nwhat it looks like, \u2013\n\nflow states, ,\n\nflourishing, \u2013\n\nfollowers, \u2013, \u2013,\n\nforcing the curve, \u2013\n\n_Fortune_ 's 100 Best Companies to Work For, \u2013\n\nfranchise agreements, \u2013\n\nFredrickson, Barbara,\n\nFreedom Riders,\n\nfreethinking leader,\n\nfulfillment,\n\nFundamental Attribution Error, \u2013\n\nfuture\n\nattention to, learning and,\n\nconfidence in the company's, , \u2013\n\nfear of the unknown and, \u2013\n\nmeaning and confidence in, \u2013\n\nplanning and lack of information about, \u2013\n\npotential and, \u2013\n\n\"The Future of Electric Cars Is Brighter with Elon Musk in It\" (MacDuffie),\n\nGallup Organization, \u2013, \u2013\n\nGalton, Francis, \u2013\n\nGen Y, \u2013\n\nGeorge VI, King of England,\n\ngig workers, \u2013\n\nGlassdoor,\n\ngoals, \u2013\n\nburden created by, \u2013\n\ncascading, \u2013\n\ncascading meaning versus, \u2013\n\ncategorization of, \u2013\n\ndata reliability about, \u2013\n\nas evaluators, , \u2013\n\ninvestments in setting,\n\nleader versus employee experience of, \u2013\n\nmarathon running, \u2013\n\npercent complete, , , \u2013\n\nas performance stimulators, \u2013\n\nas performance trackers, \u2013\n\npervasiveness of at work, \u2013\n\nqualities of good, \u2013,\n\nsmartphones and,\n\ntrends in,\n\nGoff, Maynard, ,\n\nGoldman Sachs,\n\nGoogle,\n\nGottman, John,\n\nGreat Place to Work Institute,\n\ngrowth,\n\nmindset for,\n\nstrengths versus weaknesses as areas for, \u2013\n\nHanson, Rick, \u2013\n\nHarari, Yuval Noah, \u2013\n\nHarlow, Harry,\n\nHawthorne Works, \u2013\n\nHayes, Mary, \u2013\n\nhealth-care profession, \u2013, \u2013\n\nhigh-priority interrupts, \u2013\n\nHippocrates,\n\nHoffa, Jimmy,\n\nHoffman, Brian,\n\n_Homo Deus_ (Harari), \u2013\n\nHoover, J. Edgar,\n\nHozier, \u2013\n\nhuman capital management tools, ,\n\nhumankind, success of as a species, \u2013\n\nhuman nature,\n\ntheories of others and, \u2013\n\nuncritical attention and, \u2013\n\n_Human Universals_ (Brown),\n\nhumoral theory of health, \u2013\n\nIdiosyncratic Rater Effect, \u2013\n\nindividuality,\n\nin brain growth, \u2013\n\ncompetency models versus, \u2013\n\ndefining the adjustable seat and, \u2013\n\ngiving advice and, \u2013\n\nin leadership, \u2013,\n\nin learning, \u2013\n\nlove-in-work and, \u2013, \u2013\n\nmanaging and, \u2013\n\nperformance and, \u2013\n\npotential and, \u2013\n\nrating others and, \u2013,\n\nsimplification versus, \u2013\n\nteams and, \u2013, , \u2013\n\ninformation\n\nsharing, \u2013\n\nwisdom of crowds and, \u2013\n\n_See also_ intelligence systems\n\ninsights, \u2013\n\nintelligence,\n\ntrusting people with, \u2013\n\nintelligence systems, \u2013, \u2013\n\ncharacteristics of,\n\ncreating, \u2013\n\ndata usefulness and,\n\nfreeing information for, , \u2013\n\nmeetings and, \u2013\n\nplans versus, \u2013\n\ninter-rater reliability,\n\nintersubjective realities, \u2013\n\nIQ,\n\nJenkins, Florence Foster, \u2013,\n\njob descriptions,\n\nJobs, Steve, \u2013, , ,\n\nJohn, Elton,\n\nJoint Special Operations Task Force, \u2013\n\nJordan, Michael,\n\njoy, \u2013\n\ncompetency model and, \u2013\n\nlove-in-work and, \u2013\n\n_See also_ love-in-work\n\nKenexa,\n\nKennedy, John F., , , ,\n\nKennedy, Robert F., ,\n\nkey performance indicators (KPIs),\n\nKim, Charlie,\n\nKimpton Hotels, ,\n\nKing, Billie Jean,\n\nKing, Martin Luther, Jr., \u2013, \u2013, , , \u2013, \u2013\n\nknowledge workers, rating, \u2013\n\nKorn Ferry,\n\nKramer, Steven,\n\nLandry, Tom, \u2013,\n\nLandsteiner, Karl,\n\nleaders and leadership, , \u2013, \u2013\n\nappeal of goals to, \u2013\n\ncheck-ins and, \u2013\n\ncoaching by, \u2013\n\ndefining, \u2013\n\ndevelopment of, \u2013\n\nextremism and, \u2013\n\nfear of the unknown and, \u2013\n\nfeedback by, \u2013\n\nfollowers and, \u2013, \u2013\n\nfreethinking,\n\nindividuality and, \u2013\n\nintelligence system creation by, \u2013\n\nmeaning making and,\n\nmodels of, \u2013\n\noutcomes focus by, \u2013\n\nqualities of, \u2013\n\nrarity of, \u2013\n\nrating others and, \u2013\n\nrecognizing universality and individuality in, \u2013\n\nrituals in communicating meaning, \u2013\n\nshortcoming of, \u2013\n\nspan of control and, \u2013\n\nstorytelling by, \u2013\n\nof teams,\n\nin team work experience, \u2013\n\ntraits of, \u2013\n\ntrust in, engagement and, , \u2013\n\ntruths about, \u2013\n\nwhy we follow, \u2013\n\n_Lean In_ (Sandberg),\n\nlearning\n\ndeficit correction and, \u2013\n\nfeedback and, \u2013\n\n\"Letter from Birmingham Jail\" (King), \u2013\n\nlistening, active,\n\nlove-in-work, \u2013\n\nencouraging, \u2013\n\npower of, \u2013\n\nself-assessment of, \u2013\n\nMacDuffie, John Paul,\n\nMalcolm X,\n\nmanagement\n\ngoals and,\n\nindividuality and, \u2013, \u2013\n\nmicro-,\n\nspan of control and, \u2013\n\nManagement by Objectives (MBOs),\n\nmass, \u2013\n\nmaximization, , \u2013, , \u2013\n\nMayo Clinic, \u2013, , , ,\n\nMcCarthy, Joe,\n\nMcChrystal, Stanley, \u2013, , \u2013,\n\nMcDonald's,\n\nmeaning\n\ncascading, \u2013\n\ncascading, goals versus, \u2013\n\ncheck-ins in making, \u2013\n\nexpressed values and,\n\nat Facebook, \u2013\n\nfollowing and, \u2013\n\nimposed goals versus, \u2013\n\ninsight and, \u2013\n\nleadership and, \u2013, \u2013, \u2013\n\nrituals and, \u2013\n\nshared, alignment through,\n\nstories and, \u2013\n\nmeetings, \u2013\n\nMemphis nonviolent protest, \u2013\n\nMessi, Lionel, \u2013, , ,\n\nmetrics, competency models and, \u2013\n\nmicromanagers,\n\nmillennials\n\nengagement of,\n\nreported desire for feedback, \u2013, \u2013\n\nsocial media and, \u2013\n\nmirroring,\n\nmission,\n\nleaders and, \u2013\n\nmeaning and, \u2013\n\nmomentum, \u2013\n\nbalance and, \u2013\n\nMontgomery bus boycott, , \u2013,\n\nMount, Michael K., , ,\n\nMurray, Andy,\n\nmusical performers,\n\nMusk, Elon, ,\n\nMyspace,\n\nNASA, Mission Control Center,\n\nNational Civil Rights Museum, \u2013,\n\nNational Health Service, \u2013\n\nNeruda, Pablo,\n\nneural plasticity, \u2013\n\nneurogenesis, \u2013\n\nNext Jump,\n\nNing,\n\nObjectives and Key Results (OKRs),\n\n_Ocean's Eleven_ (movie), \u2013,\n\nOperations and Intelligence (O&I) meetings, \u2013\n\noutcomes, promoting, \u2013\n\noxytocin,\n\nparasympathetic nervous system,\n\nPatagonia, ,\n\nPath,\n\nPatton, George, ,\n\nPayPal,\n\nperformance\n\nadvice and, \u2013\n\ndata reliability about, \u2013, \u2013\n\ndeficit repair and, \u2013\n\nfeedback and, \u2013\n\nflow states and,\n\ngoals as stimulators of, \u2013\n\ngoals in tracking, , \u2013\n\nidiosyncracy of excellent, \u2013\n\njoy and,\n\npast, potential and,\n\npotential and, \u2013\n\npredicting, \u2013\n\nof pub managers, \u2013\n\nrating others', \u2013\n\nof teams, work experience elements related to, \u2013\n\nusing strengths and, \u2013, \u2013\n\nperformance appraisals, , \u2013\n\naccuracy of, \u2013\n\ncompetency model and, \u2013\n\ngoals and, \u2013, \u2013\n\ntraditional, \u2013\n\nperformance improvement plans (PIPs),\n\nperformance management systems, \u2013\n\nperks, \u2013\n\npersonality assessments,\n\nphysicians, burnout among, \u2013, \u2013, ,\n\npilots, United States Air Force, \u2013\n\nplanning\n\nbased on generalizations versus realities,\n\nbenefits of,\n\nthe best plan wins and, \u2013\n\ncheck-ins and, \u2013\n\nconstraints from, \u2013\n\nintelligence systems versus, \u2013\n\npace of change and, \u2013\n\npatterns for, \u2013\n\nreality versus,\n\nreal-world experience and, \u2013\n\nstrategic,\n\nWWII RAF, \u2013\n\nPolunin, Sergei, \u2013\n\nPorter, Cole,\n\npotential, \u2013\n\ndamage from categorizing, \u2013\n\ndefining, , \u2013\n\ndetermining, \u2013\n\nmass and,\n\nmaximizing, \u2013, , \u2013\n\nmeasuring, \u2013\n\nmomentum versus, \u2013\n\nrating others', \u2013\n\ntraits versus states and, \u2013\n\nvelocity and, \u2013\n\n_The Practice of Management_ (Drucker),\n\npraise, ,\n\ngrowth\/development and, \u2013\n\nrecognition versus, \u2013\n\nPresidio of Santa Barbara,\n\n_Principles_ (Dalio), \u2013\n\nproductivity\n\nglobal decline in,\n\ngoals in promoting, \u2013\n\nlove-in-work and, \u2013\n\nuse of strengths and, \u2013\n\n_The Progress Principle_ (Amabile and Kramer),\n\nPTSD, \u2013\n\npub managers, performance of, \u2013, \u2013\n\npurpose, \u2013\n\nquick market intelligence (QMI),\n\nrange, \u2013\n\nrater perspective, \u2013\n\n\"Rater Source Effects Are Alive and Well After All\" (Hoffman),\n\nrating other people, \u2013\n\ndata insufficiency in, \u2013\n\ndata quality and, \u2013\n\nengagement and,\n\nknowledge workers, \u2013\n\non potential, \u2013\n\nproblems with averaging, \u2013\n\nquestioning, \u2013\n\nrating yourself versus, \u2013, \u2013\n\nreliability of, \u2013\n\nsystems for, \u2013\n\nwisdom of crowds and, \u2013\n\nRawls, John, n\n\nreality\n\nobjective versus subjective, \u2013\n\nplanning versus, \u2013\n\nrecognition, ,\n\nfeedback and, \u2013\n\npraise versus, \u2013\n\nred threads,\n\nReiner, Carl,\n\nreliability of data, \u2013, \u2013\n\nrest and digest system, ,\n\nrituals\n\ncascading meaning through, \u2013\n\ndeath and,\n\nleadership and, \u2013\n\npraise\/recognition, \u2013\n\nsense-making, check-ins for, \u2013\n\nRobbins, Chuck,\n\nRonaldo, Cristiano, , \u2013\n\nRoyal Air Force (RAF), \u2013\n\nRoyal Ballet School, \u2013\n\nRoyal Observer Corps,\n\nrunning around your backhand,\n\nsafety,\n\nSalesforce, \u2013,\n\nsales quotas, , \u2013\n\nSam's Club,\n\nSandberg, Sheryl, \u2013, \u2013\n\n_Sapiens_ (Harari), \u2013\n\nScott, Robert Falcon, n\n\nScullen, Steven, ,\n\nsecrecy,\n\nSelection Research, Incorporated (SRI), \u2013\n\nself-assessments,\n\ngoals and, \u2013\n\nof love-in-work, \u2013\n\npersonality,\n\nself-presentation, positive, \u2013\n\nSelma march, \u2013\n\nsense-making rituals, check-ins as, \u2013\n\nShackleton, Ernest, n\n\nsignifiers, \u2013\n\nsimplicity,\n\nteams and,\n\nSinatra, Frank,\n\nSinek, Simon, ,\n\nsituational judgment tests,\n\nskills,\n\nSnapchat, \u2013\n\nsocial media, \u2013\n\nSouthern Christian Leadership Conference,\n\nSouthwest Airlines,\n\nspan of control,\n\nsports, \u2013, \u2013, , \u2013\n\nratings reliability in,\n\nstates versus traits, \u2013, \u2013\n\nstorytelling, \u2013\n\nstrategic planning, . _See also_ planning\n\nstrengths\n\nability and passion with, \u2013\n\nadvice to avoid overuse of, \u2013\n\ncheck-ins and, \u2013\n\ncompetency model and, \u2013\n\ndaily chances to use, , , \u2013\n\ndeficit repair and, \u2013\n\ndefinition of, \u2013, ,\n\nfeelings related to,\n\nflow\/balance and, \u2013\n\nas growth areas, \u2013\n\njoy and, \u2013\n\nleadership and, ,\n\nlearning and, \u2013\n\nlove-in-work and, \u2013\n\nmeasuring, \u2013\n\nteam diversity and, \u2013\n\nteams built on, \u2013\n\nsubjectivity,\n\nsuccess\n\nculture and,\n\nfeedback and,\n\nSummit, Pat,\n\nSun Microsystems, ,\n\nSurowiecki, James, \u2013\n\nsyllogisms, n\n\nsympathetic nervous system,\n\n\"Take Me to Church\" (Hozier), \u2013\n\ntalent reviews, \u2013\n\n_Team of Teams_ (McChrystal), \u2013\n\nteams\n\nbuilt on strengths, \u2013\n\ncascading meaning and, \u2013\n\ncheck-ins with, \u2013\n\nclarity of expectations on, \u2013\n\nconfidence in company's future and, \u2013\n\ncultural signifiers versus, \u2013\n\nengagement and, \u2013, \u2013, \u2013\n\nexcellence creation and, \u2013\n\nexperience of work and experience of, \u2013\n\nfactors in creating highly engaged,\n\nhow work actually gets done and, \u2013\n\nidentification with,\n\nimportance of overlooked by companies,\n\nindividuality and, \u2013\n\nintelligence system creation for, \u2013\n\nintersubjective realities and, \u2013\n\nleaders in creating experience of, \u2013\n\noutcomes-based, \u2013\n\npace of change and, \u2013\n\nperformance of, work experience elements and, \u2013\n\nplanning and, \u2013, \u2013\n\npositive attention in, \u2013\n\nrating people on, \u2013\n\ntechnology\n\nfeedback and,\n\ngoals and,\n\nfor rating others, \u2013\n\nTesla,\n\n360-degree assessment, \u2013, \u2013\n\nToyota,\n\n\"Trait, Rater, and Level Effects in 360-Degree Performance Ratings\" (Mount),\n\ntraits versus states, \u2013, \u2013, , \u2013\n\ntransparency, radical, \u2013\n\ntrust\n\ncheck-ins and, \u2013\n\nculture of, \u2013\n\nengagement and, , \u2013\n\ngoals and,\n\nintelligence systems and, \u2013\n\nleadership and,\n\nteam leaders and, \u2013\n\nTrust Index,\n\nturnover,\n\nUltimate Software,\n\nUnited States Air Force, \u2013\n\nuniversality, \u2013, \u2013\n\nvalidity of data, \u2013\n\nvalues\n\ndiversity and,\n\nexpressed,\n\ngoals and, \u2013, \u2013\n\nshared, , ,\n\nvariability of data, \u2013\n\nVeil of Ignorance, n\n\nvelocity,\n\nvirtual workers, \u2013\n\nWalmart, \u2013\n\nWalton, Sam, \u2013\n\nWegmans, \u2013\n\nwell-rounded people, \u2013\n\nareas of excellence and, \u2013\n\ncompetency model and, \u2013\n\ndefining the adjustable seat and, \u2013\n\nidiosyncracy of high performance versus, \u2013\n\nleadership and, ,\n\noutcomes focus and, \u2013\n\nspiky people versus, \u2013,\n\nstrengths and, \u2013\n\nWells Fargo,\n\nWestern Electric, Hawthorne Works, \u2013\n\nWherry, Robert J., Sr.,\n\nwisdom of crowds, \u2013\n\nWittgenstein, Paul,\n\nWonder, Stevie,\n\nwork-life balance, \u2013\n\ncompensation and, \u2013\n\nimpossibility of, \u2013\n\nlove-in-work versus, \u2013\n\nphysicians and, \u2013\n\nWorld War II, \u2013\n\nWright, Corinne, \u2013\n\nX.com,\n\n_Yes, Prime Minister_ (TV show), n\n\nZuckerberg, Mark, \u2013, \u2013\n\n# Acknowledgments\n\nThere's one other pattern in the eight engagement items that we haven't explored yet. The first two\n\n1. I am really enthusiastic about the mission of my company.\n\n2. At work, I clearly understand what is expected of me.\n\naddress how we experience purpose, collectively and individually, at work. The next two\n\n3. On my team, I am surrounded by people who share my values.\n\n4. I have the chance to use my strengths every day at work.\n\ncapture how those around us help us achieve excellence, again collectively and individually. The next two\n\n5. My teammates have my back.\n\n6. I know I will be recognized for excellent work.\n\nare all about support, and how we get that from our team and from the individuals around us. And the last two\n\n7. I have great confidence in my company's future.\n\n8. In my work, I am always challenged to grow.\n\ncapture how those around us help us see our collective and individual future.\n\nSo we thought it only appropriate to thank all those who've helped us write Nine Lies About Work in these categories\u2014as each of their contributions has helped make the two of us a stronger team.\n\nFor helping us understand more clearly our own purpose in writing this book, thanks to our wonderful editor, Jeff Kehoe, to Adi Ignatius for his calm and compelling support, and to our incomparable agent, Jennifer Rudolph Walsh. For the questions, and for the challenges to hone our message and serve our true reader, thanks to Fran Katsoudas, Tracy Hutton, and Amy Leschke-Kahle.\n\nFor pushing us to be better in every chapter, every paragraph, and every turn of phrase, and for helping us glimpse what excellence could be, thanks to Adrienne Fretz, Yosi Kossowsky, Adam Grant, Alli Walton, and Katie Flores. For sharing with us their reactions to the manuscript, thanks to Jen Waring, Ania Wieckowski, Amy Bernstein, and the entire HBR editorial and production team. For living out our shared commitment to rigorous research methodology and the definitive data-driven discoveries to which it leads, our thanks to Drs. Mary Hayes, Fran Chumney, and Corinne Wright at the ADP Research Institute and to Roxanne Bisby Davis and the Team Analytics and Research squad at Cisco. And to Lisa and Andy and Miles and many unnamed others, thank you for allowing us to share your stories of what excellence looks like and feels like in the real world of work.\n\nFor supporting us on what has been an immensely rewarding journey from idea to proposal to manuscript to book, thanks to, at Cisco, Chuck Robbins, Megan Barba, Gianpaolo Barozzi, Christine Bastian, Megan Bazan, Madison Beard, El Cavanagh-Lomas, Jen Dudeck, Shannon Fryhoff, Dan Gibbs, Leslie Gordon, Scott Herpolsheimer, Charlie Johnston, Jean Kerr, Robert Kovach, John Lagonigro, Alicia Lopez, Amy Manning, Elaine Mason, Dolores Nichols, Jason Phillips, Oliver Roll, Rachel Samitt, Shari Slate, Tschudy Smith, Gaby Thompson, Mary Williams, and Tae Yoo. All of you know how messy the real world of work can be, and by dint of your efforts, each day you're making it better and more human.\n\nAt ADP, thanks to Carlos Rodriguez, Don Weinstein, Dermot O'Brien, Sreeni Kutam, Joe Sullivan, Charlotte Saulny, and the entire StandOut team; to Meredith Bohling for your prose and your community building, to Kevyn Horton for your web building, to Darren Raymond for your \"potential,\" and to Christian Gomez for the power of your persuasion.\n\nAt HBR, thanks to our designer, Stephani Finks, and to our publicity and marketing team, Julie Devoll and Erika Heilman.\n\nAnd for giving us the confidence to keep hammering away at the keyboard, for challenging us to put as much of ourselves into this as we possibly could, and for pushing us to write something that might actually make the future world of work meaningfully better, thanks to our families\u2014Chris, Jenny, William, Graeme, Jo, Jack, Lilia, Marshy, Fitz, and Mojo.\n\nFrom Ashley to Tina: I knew when I married you twenty years ago that you would be a better partner and friend than I had ever dreamed of finding on this earth, and that has proven to be so in more ways than I could have imagined. And although I also knew back then that you were a wonderful word-wrangler, and the most magical idea-honer I have ever known, the thought that one day my words would be beneficiary of your keen attention never entered my mind. How lucky I am.\n\nAnd finally, from Marcus to Myshel: from \"Who do these lies serve?\" to \"What does Picasso say about creation and destruction?\" to \"What can the freethinking leader teach the rest of us?\" your questions have given voice to how our reader thinks and feels. And more, your patience in the process, your wit in the face of my writer's block, and your joy\u2014love even\u2014in the words we chose, all are red threads we have woven into the fabric of this book. Thank you for these, and above all, for the passion.\n\nOne lesson of this book is that no one fits perfectly in any category we can think up, and of course the same is true for all the wonderful people we've listed in the categories above\u2014so many of you have helped us in so many ways. We thank you all from the bottom of our hearts.\n\n# About the Authors\n\n**_Marcus Buckingham_** is a bestselling author and global researcher focusing on all aspects of people and performance at work. During his years at the Gallup Organization, he worked with Dr. Donald O. Clifton to develop the StrengthsFinder program, and coauthored the seminal business books _First, Break All The Rules_ and _Now, Discover Your Strengths_. He designed the StandOut strengths assessment completed by over one million people to date, and authored the accompanying book, _StandOut: Find Your Edge, Win at Work_. He currently heads all people and performance research at the ADP Research Institute. _Nine Lies About Work_ is his ninth book.\n\n**_Ashley Goodall_** is the Senior Vice President of Leadership and Team Intelligence at Cisco Systems. In this role he has built a new organization focused entirely on serving teams and team leaders\u2014an organization that combines learning and talent management, people planning, organization design, executive talent and succession planning, coaching, assessment, team development, research and analytics, and performance technology. Prior to joining Cisco, he spent fourteen years at Deloitte, where he was responsible for Leader Development and Performance Management.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nOverview Map Key\n\nSouthern\n\n1 Stewart and Indian Lakes\n\n2 Good Luck Cliffs\n\n3 Jockeybush Lake\n\n4 Groff Creek\n\n5 Tenant Creek Falls\n\n6 Hadley Mountain\n\n7 Crane Mountain\n\nWest-Central\n\n8 Panther Mountain and Echo Cliffs\n\n9 Mitchell Ponds\n\n10 Bear and Woodhull Lakes\n\n11 Black Bear Mountain\n\n12 Bubb, Sis, and Moss Lakes\n\n13 Cascade Lake\n\n14 Gleasmans Falls\n\nCentral\n\n15 Auger Falls\n\n16 Shanty Brook\n\n17 East Branch Sacandaga Gorge\n\n18 Snowy Mountain\n\n19 Chimney Mountain\n\n20 Peaked Mountain\n\n21 Castle Rock\n\n22 Blue Ledges\n\n23 Hoffman Notch\n\nEastern\n\n24 Sleeping Beauty Mountain\n\n25 Shelving Rock Falls\n\n26 Black Mountain\n\n27 Tongue Mountain Range\n\n28 Pharaoh Mountain [day or overnight hike]\n\n29 Pharaoh Lake [overnight hike]\n\nNorthern\n\n30 Owls Head Mountain\n\n31 Cat Mountain\n\n32 High Falls [overnight hike]\n\n33 Lampson Falls and Grass River\n\n34 Floodwood\n\n35 Saint Regis Mountain\n\n36 Debar Mountain\n\nHigh Peaks\n\n37 Ausable River: West River Trail\n\n38 Ausable River: East River Trail\n\n39 Giant's Nubble\n\n40 Hanging Spear Falls and Opalescent River [day or overnight hike]\n\n41 Indian Pass\n\n42 Algonquin Peak\n\n43 Avalanche Pass\n\n44 Little and Big Crow Mountains\n\n45 Pitchoff Mountain\n\n46 Ampersand Mountain\n\nHOFFMAN NOTCH BROOK (Trail 23, Hoffman Notch)\n\nFive-Star Trails: Adirondacks: Your Guide to 46 Spectacular Hikes\n\nCopyright \u00a9 2017 by Tim Starmer\n\nAll rights reserved\n\nPublished by Menasha Ridge Press\n\nPrinted in the United States of America\n\nDistributed by Publishers Group West\n\nSecond edition, first printing\n\nCover design: Scott McGrew\n\nText design: Annie Long\n\nCartography and elevation profiles: Scott McGrew, Thomas Hertzel, and Tim Starmer\n\nCover and interior photos: Tim Starmer\n\nLibrary of Congress Cataloging-in-Publication Data\n\nNames: Starmer, Timothy, 1975- author.\n\nTitle: Five-star trails, Adirondacks : 40 spectacular hikes in upstate New York \/ Tim Starmer.\n\nDescription: Second edition. | Birmingham, Alabama : Menasha Ridge Press, 2017.\n\nIdentifiers: LCCN 2016053971 | ISBN 9781634040525 (paperback)\n\nSubjects: LCSH: Hiking\u2014New York (State)\u2014Adirondack Mountains\u2014Guidebooks. | Trails\u2014 New York (State)\u2014Adirondack Mountains\u2014Guidebooks. | Adirondack Mountains (N.Y.)\u2014 Guidebooks. | BISAC: TRAVEL \/ United States \/ Northeast \/ New England (CT, MA, ME, NH, RI, VT). | SPORTS & RECREATION \/ Hiking. | HEALTH & FITNESS \/ Healthy Living.\n\nClassification: LCC GV199.42.N652 A34735 2017 | DDC 796.51097475\u2014dc23\n\nLC record available at https:\/\/lccn.loc.gov\/2016053971\n\nISBN: 978-1-63404-052-5; eISBN: 978-1-63404-053-2\n\n**MENASHA RIDGE PRESS**\n\nAn imprint of AdventureKEEN\n\n2204 First Ave. S, Ste. 102\n\nBirmingham, Alabama 35233\n\n800-443-7227, fax 205-326-1012\n\nVisit menasharidge.com for a complete listing of our books and for ordering information. Contact us at our website, at facebook.com\/menasharidge, or at twitter.com\/menasharidge with questions or comments. To find out more about who we are and what we're doing, visit blog.menasharidge.com.\n\nDISCLAIMER This book is meant only as a guide to select trails in the vicinity of the Adirondacks and does not guarantee your safety in any way\u2014you hike at your own risk. Neither Menasha Ridge Press nor Tim Starmer is liable for property loss or damage, personal injury, or death that may result from accessing or hiking the trails described in this guide. Please be aware that hikers have been injured in the Adirondacks. Be especially cautious when walking on or near boulders, steep inclines, and drop-offs, and do not attempt to explore terrain that may be beyond your abilities. To help ensure an uneventful hike, please read carefully the introduction to this book, as well as safety information from other sources. Familiarize yourself with current weather reports and maps of the area you plan to visit (in addition to the maps provided in this guidebook). Be aware of park regulations, and always follow them. While every effort has been made to ensure the accuracy of the information in this guidebook, land and road conditions, phone numbers and websites, and other information is subject to change.\nContents\n\nOVERVIEW MAP\n\nOVERVIEW MAP KEY\n\nDEDICATION\n\nACKNOWLEDGMENTS\n\nPREFACE\n\nRECOMMENDED HIKES\n\nINTRODUCTION\n\nSouthern\n\n1 Stewart and Indian Lakes\n\n2 Good Luck Cliffs\n\n3 Jockeybush Lake\n\n4 Groff Creek\n\n5 Tenant Creek Falls\n\n6 Hadley Mountain\n\n7 Crane Mountain\n\nWest-Central\n\n8 Panther Mountain and Echo Cliffs\n\n9 Mitchell Ponds\n\n10 Bear and Woodhull Lakes\n\n11 Black Bear Mountain\n\n12 Bubb, Sis, and Moss Lakes\n\n13 Cascade Lake\n\n14 Gleasmans Falls\n\nCentral\n\n15 Auger Falls\n\n16 Shanty Brook\n\n17 East Branch Sacandaga Gorge\n\n18 Snowy Mountain\n\n19 Chimney Mountain\n\n20 Peaked Mountain\n\n21 Castle Rock\n\n22 Blue Ledges\n\n23 Hoffman Notch\n\nEastern\n\n24 Sleeping Beauty Mountain\n\n25 Shelving Rock Falls\n\n26 Black Mountain\n\n27 Tongue Mountain Range\n\n28 Pharaoh Mountain [day or overnight hike]\n\n29 Pharaoh Lake [overnight hike]\n\nNorthern\n\n30 Owls Head Mountain\n\n31 Cat Mountain\n\n32 High Falls [overnight hike]\n\n33 Lampson Falls and Grass River\n\n34 Floodwood\n\n35 Saint Regis Mountain\n\n36 Debar Mountain\n\nHigh Peaks\n\n37 Ausable River: West River Trail\n\n38 Ausable River: East River Trail\n\n39 Giant's Nubble\n\n40 Hanging Spear Falls and Opalescent River [day or overnight hike]\n\n41 Indian Pass\n\n42 Algonquin Peak\n\n43 Avalanche Pass\n\n44 Little and Big Crow Mountains\n\n45 Pitchoff Mountain\n\n46 Ampersand Mountain\n\nAPPENDIX A: MANAGING AGENCIES\n\nAPPENDIX B: GEAR LISTS\n\nAPPENDIX C: HIKING CLUBS\n\nAPPENDIX D: SUGGESTED READING\n\nABOUT THE AUTHOR\n\nMAP LEGEND\n\nTOP OF THE FIRST FALLS ALONG TENANT CREEK (Trail 5, Tenant Creek Falls)\nDedication\n\nThis book is dedicated to my parents, who first inspired in me a love of the great outdoors and helped in innumerable ways in completing this book.\nAcknowledgments\n\nFIRST AND FOREMOST, I would like to thank everyone who braved the trails with me and endured my repetitious mumblings into my voice recorder. Your company was appreciated and helpful.\n\nSecond, I must acknowledge all the work that my parents contributed to this project, from editing my rough drafts to retyping hikes I lost when my computer crashed to loaning me a car when my truck broke down, sadly more than once.\n\nFinally, I would like to acknowledge the tireless efforts of all the trail crews who maintain the thousands of miles of trails in the Adirondacks. Your work is greatly admired and appreciated.\n\n\u2014T. S.\nPreface\n\nWORKING ON THIS BOOK was definitely an experience to remember. I lost two hard drives; had to rewrite a handful of hikes from memory; fried a camera, a voice recorder, and a GPS unit; sprained my ankle; was nearly blown off a mountaintop; and was stranded in the middle of the Adirondacks when my truck broke down and AAA redefined 24\/7 service as weekdays only during business hours. It was memorable and\u2014quite simply\u2014great. Planning and replanning hikes, followed by days in the woods, is more than most hikers can ask for; to do so in the Adirondacks is definitely more than most get.\n\nFor those who don't know, the Adirondack Park is huge. The park's boundaries encompass 9,375 square miles, which is larger than each of the states of Rhode Island, Delaware, Connecticut, New Jersey, and New Hampshire. It is only 240 square miles smaller than Vermont. Compared to national parks, the Adirondack Park is larger than Grand Canyon, Yellowstone, Yosemite, Great Smoky Mountains, and Big Bend National Parks combined. In fact, its boundaries, also known as the blue line, encompass more acreage than any of the national parks in the Lower 48 states. Not all of the land within the blue line is owned or managed by the state. Towns, villages, and two entire counties, as well as vast tracts of privately held land, lie within its boundaries. However, don't let the land classification fool you\u2014the park is truly a vast wilderness. Of the 6.1 million acres, almost 45%, or 2.6 million acres, is owned by the state, though donations and purchases continue to expand the state-owned land. Of the remaining land, less than 1% is municipal property and about 6% is classified as low-level residential, which is almost equivalent to the amount of acreage of water within the park. The remaining land is either rural or privately owned forests.\n\nNo matter how you frame it, the park is a huge wilderness, and you sense it almost immediately after passing the blue line. As you drive toward any of the hundreds of trailheads, you can feel the forest encroaching on the fields and houses along the roads. You know that the boundary between civilization and a sprawling wilderness is just beyond the treeline. Soon the forests have overrun manicured lawns, and fields give way to staggering boulders and craggy cliffs that loom close to the roadside. You sense that at any point, you could stop your car, step out into the woods, and within feet completely leave civilization behind. All the while, should you choose to stop at a diner for lunch in between hikes, forget camera batteries, need to refuel, or need replacement gear, there is typically a spot to resupply within a half hour of a trailhead. This is the charm of the Adirondacks: A vast wilderness, in which you can spend days in solitude, is at your fingertips, and yet civilization is still close at hand. For many beginning hikers, this is a great comfort, while experienced hikers can revel in the idea of a warm meal or cold drink at the end of a journey.\n\nMy goal in writing this book was to cater to both beginning and experienced hikers by providing easy and challenging hikes in all corners of the park. I could have easily found 40-plus hikes in any single region, and all would have been rewarding. Some of the hikes will be familiar, others less known, but all are rewarding examples of the many facets of the Adirondack Park and its thousands of miles of trails.\n\n\u2014Tim Starmer\nRecommended Hikes\n\nBest for Solitude\n\n1 Stewart and Indian Lakes\n\n4 Groff Creek\n\n27 Tongue Mountain Range\n\n31 Cat Mountain\n\n32 High Falls\n\nBest for Waterfalls\n\n5 Tenant Creek Falls\n\n15 Auger Falls\n\n25 Shelving Rock Falls\n\n33 Lampson Falls and Grass River\n\n37 Ausable River: West River Trail\n\n38 Ausable River: East River Trail\n\n40 Hanging Spear Falls and Opalescent River\n\nBest for Lakes and Ponds\n\n7 Crane Mountain\n\n12 Bubb, Sis, and Moss Lakes\n\n20 Peaked Mountain\n\n29 Pharaoh Lake\n\n34 Floodwood\n\n43 Avalanche Pass\n\nBest for Mountains\n\n18 Snowy Mountain\n\n20 Peaked Mountain\n\n24 Sleeping Beauty Mountain\n\n27 Tongue Mountain Range\n\n42 Algonquin Peak\n\n45 Pitchoff Mountain\n\n46 Ampersand Mountain\n\nEasiest Hikes\n\n4 Groff Creek\n\n11 Black Bear Mountain\n\n13 Cascade Lake\n\n15 Auger Falls\n\n25 Shelving Rock Falls\n\n33 Lampson Falls and Grass River\n\n44 Little and Big Crow Mountains\n\nMost Challenging Hikes\n\n27 Tongue Mountain Range\n\n36 Debar Mountain\n\n37 Ausable River: West River Trail (p. 254) combined with\n\n38 Ausable River: East River Trail\n\n40 Hanging Spear Falls and Opalescent River\n\n42 Algonquin Peak\n\n43 Avalanche Pass\n\nBest for Fall Colors\n\n6 Hadley Mountain\n\n18 Snowy Mountain\n\n24 Sleeping Beauty Mountain\n\n30 Owls Head Mountain\n\n46 Ampersand Mountain\n\nBest for Wheelchair Accessibility (portions)\n\n10 Bear and Woodhull Lakes\n\n12 Bubb, Sis, and Moss Lakes\n\n33 Lampson Falls and Grass River\n\nSPLIT ROCK BAY ON PHARAOH LAKE (Trail 29, Pharaoh Lake)\n\nSANDY BEACH AT BLUE LEDGES (Trail 22, Blue Ledges)\nIntroduction\n\nAbout This Book\n\nAN ESTIMATED 6 MILLION\u201310 MILLION TOURISTS visit Adirondack Park every year. With 3,000 lakes, 30,000 miles of rivers and streams, 2,000 miles of hiking trails, and more than 100 mountains, it is no surprise. Unique is barely adequate to describe the character of the Adirondacks. It stands out as the largest park in the Lower 48 states; however, unlike its rivals, it's managed solely by the state. It has the only mountains in the Northeast that are not geologically Appalachian. The mountains, though geologically new, are from old rock. Ecologically, the park lies on the transition of the boreal forests of the north and deciduous hardwood forests of the south. The Adirondack region was also at the forefront of conservation when, in 1894, the Forever Wild character of the forest preserve was added to the state constitution, making it the first preserve with constitutional protection.\n\nNew mountains from old rock seem a contradiction, but this is what makes the Adirondack Mountains distinct from their Appalachian neighbors along the East Coast. Unlike the Appalachian Mountains, which were formed by plate tectonics, the Adirondack Mountains were formed by uplift. Indeed, the Adirondacks are still rising. Granted, the rate is only about 3 millimeters per year, but this rate is faster than the rate of erosion, so the mountains are creeping ever upward. So the next time someone comments that life was tougher in his day, you can quip that at least his mountains were shorter.\n\nThe protruding old rock was formed when the Adirondacks were part of a giant inland sea. Over millennia, miles of sediment were deposited and later transformed by high pressure and heat into metamorphosed rock. The deposits and formation took place 1 billion years ago, and the metamorphosed rock lay miles beneath the surface until erosion removed enough material for a 160-mile-wide bulge to begin swelling above the surrounding landscape about 5 million years ago. Further stages of uplift, erosion, and the scouring of the landscape by great ice sheets have exposed the metamorphosed rock and caused the uplift to increase. These powerful forces of uplift, erosion, and scouring transformed the region into the rugged wilderness we see today.\n\nAmerican Indians did not settle in the Adirondacks, but rather used them mostly as hunting grounds. As Europeans began to explore and settle the continent, they stayed on the fringes of the wilderness, mostly along the shores of Lake George and Lake Champlain. These lakes formed a major strategic territory for mastery of the continent. The Iroquois and Algonquin often fought over the water route long before the pitched battles of the French and Indian War. During this war, the French controlled Fort Carillon, later named Fort Ticonderoga, on the northern shore of Lake Champlain, while the English controlled Fort William Henry on the south end. James Fenimore Cooper's novel The Last of the Mohicans depicts one of the major events in this contest over the water route: the surrender of Fort William Henry followed by a massacre of the retreating English troops on their way to Fort Edward. The magnitude of the massacre has been estimated from dozens to thousands, though it is generally agreed that Cooper's depiction is exaggerated. Later epic battles were fought during this war and then again during the American Revolution. However, during all this time, the interior of the Adirondack wilderness was rarely broached.\n\nSettlement in the region did not begin in earnest until the 1800s, when the Industrial Revolution's growing demand for natural resources turned its eyes on the vast stands of timber and newly discovered veins of iron ore. Great wealth was created during the Gilded Age, and consequently, the Adirondacks became a popular vacation area with great camps and resorts. At the same time, conservation and a love of the outdoors were taking hold of the public, and the devastation of the Adirondack forests became a growing concern among these newly arrived vacationers. In 1885 the state created the Adirondack Forest Preserve, which directed that \"the lands now or hereafter constituting the Forest Preserve shall be forever kept as wild forest lands.\" Later in 1892 the state established the Adirondack State Park and drew the infamous blue line to encompass the Adirondack Forest Preserve, as well as millions of acres of private land. Despite these protections, the forest preserve law was still violated; in 1894 the \"forever wild\" clause was added to the state constitution of New York. Tourism continued to thrive in the region and saw increases after World War I and World War II. Second homes within the park became a growing concern, and in 1971 the Adirondack Park Agency was created to develop and oversee land-use plans with consideration of the \"forever wild\" clause.\n\nToday the park still maintains a delicate balance between the interests of private landholders and the public lands owned by the state. To say that tourism is a mainstay of the region is an understatement. There are 130,000 people living year-round in the 9,375 square miles encompassed by the blue line, with an estimated 200,000 seasonal residents and anywhere from 6 million to 10 million tourists. So odds are that if you ask someone for directions, the person won't be from around there. Tourism is year-round, with winter attracting snowmobilers, skiers, and snowshoers in as many numbers as summer's vacationers and fall's hunters. About the only downtime is in early spring during the mud season.\n\nThe hikes within this book are meant to provide a wide range of experiences over the vast area of the park. For decades the park has informally been divided into regions in various ways. The Department of Environmental Conservation (DEC) has regions that sprawl across the blue line; counties divide the park but also cross the line; and varying tourist centers divide the region based on proximity to major lakes or towns.\n\nMost people adhere to the six regions originally devised by the Adirondack Mountain Club (ADK) in its earlier versions of the Forest Preserve series of guidebooks. I have similarly divided the region because it proves useful for hikers seeking nearby trails or additional maps and is generally understood by most hikers as the way the park is divided. Though regions often overlap, some distinctions between the regions help to describe the area to be explored.\n\nSouthern Region\n\nThe southern foothills are not known for their mountains, so it may seem odd that the two mountains featured here are actually some of the steepest hikes. They are not as tall as those in the High Peaks, but their straight-up ascents make them reasonably challenging, yet well within the novice hiker's range. Similar to the west-central and central regions, the terrain here is mostly level, and lakes and other bodies of water are the primary focus of trails. Several unmarked trails are also featured in this region but are easy to follow with a little navigational experience.\n\nWest-Central Region\n\nAlong the western border of the park are the foothills that lead to the Adirondack Mountains. Few mountains are in this area; most trails follow the shores of lakes and rivers and feature generally flat terrain. The few mountain trails in this area are well liked, so expect lots of company. Indeed, the region, especially around the towns of Old Forge and Inlet, is extremely popular with tourists. Many of the great camps were built here, and it serves as a staging point for longer expeditions into the Adirondacks. Several excellent campgrounds, outdoors stores, and outfitters are in the area, making it a good point to resupply or pick up forgotten items as you head deeper into the park. It does not take long to leave the vestiges of civilization behind here, so the purist should not shun this region simply because it is well known.\n\nCentral Region\n\nAs might be expected, the central region lies in the center of many of the other Adirondack regions; consequently, many of its trailheads lie literally across the street from a trail in another region. Many of the trails described in this region are unmarked and considered bushwhacks but generally are well trodden. Additionally, these trails follow the course of streams, so navigating them should be relatively easy. The area is essentially bound by NY 28 to the north and NY 8 to the south. Several mountains are in this region, though most are easy to climb. An abundance of waterfalls in this region makes it an excellent area for exploring waterscapes. Many of the campgrounds in the area provide solitude and scenery, making them excellent staging areas for exploring all the regions in the park.\n\nEastern Region\n\nHikes in this region focus on the shores of Lake George and feature spectacular vistas from the mountains that surround the beautiful lake. The southern tip of Lake George is full of tourist trappings and won't be high on the list for most devoted hikers. However, it does provide a good staging area, the opportunity to reconnect with civilization, and quick access to the excellent wilderness areas nearby. Pharaoh Lake and Mountain are featured in the book but cover just a small portion of the vast Pharaoh Lake Wilderness Area.\n\nNorthern Region\n\nThe northern region looks and feels like a giant river basin. The mostly level terrain is dotted with innumerable ponds, lakes, and marshes, which are all bisected by a network of streams and winding rivers. The Floodwood and High Falls loops described in this guide typify the region and are fascinating areas to explore. Not as built-up or commercial as other areas in the park, the northern region is definitely the most remote and probably the hardest area in which to resupply. Cranberry Lake, Tupper Lake, and Saranac are the largest developed areas, but you'll pass through them quickly. Hikes here are generally more secluded; however, as is the case in most regions, the peaks attract the most hikers.\n\nHigh Peaks Region\n\nRugged and wild, the High Peaks typify the Adirondacks. Hikers who put off venturing into the High Peaks region for decades often never return to their earlier stomping grounds once they get a taste of the High Peaks. Generally, trails here are steeper, cliffs and waterfalls are taller, and the sheer quantity of peaks in this region changes your view of what rugged looks and feels like. This book offers just a sampling of the adventures that lie in the High Peaks and leans toward the beginner level for the region. However, beginner High Peaks trails are typically more difficult than more advanced trails in other regions.\n\nHow to Use This Guidebook\n\nOverview Map, Map Key, and Map Legend\n\nThe overview map on the inside front cover depicts the location of the primary trailhead for all of the hikes described in this book. Each hike's number appears on the overview map, on the map key facing the overview map, and in the table of contents. A hike's full profile is easy to locate as you flip through the book\u2014just watch for the hike number at the top of each page. A map legend that details the symbols found on trail maps appears on the inside back cover.\n\nTrail Maps\n\nEach hike contains a detailed map that shows the trailhead, the complete route, significant features, facilities, and topographic landmarks such as creeks, overlooks, and peaks. I gathered map data by carrying a GPS unit (Garmin eTrex) while hiking each route, and then sent that data to the publisher's expert cartographers. However, your GPS is not really a substitute for sound, sensible navigation that takes into account the conditions you observe while hiking.\n\nFurther, despite the high quality of the maps in this guidebook, the publisher and author strongly recommend that you always carry an additional map, such as the ones noted in each entry opener's listing for \"Maps.\"\n\nElevation Profiles\n\nThe elevation profile represents the rises and falls of the trail as viewed from the side, over the complete distance (in miles) of that trail. On the diagram's vertical axis, or height scale, the number of feet indicated between each tick mark lets you visualize the climb. To avoid making flat hikes look steep and steep hikes appear flat, varying height scales provide an accurate image of each hike's climbing challenge. For example, one hike's scale might rise to 800 feet, while another goes to 6,000 feet.\n\nAlso, each entry's opener will list the elevation at the hike trailhead, and it will list the elevation peak.\n\nThe Hike Profile\n\nEach hike profile opens with the hike's star ratings, GPS trailhead coordinates, and other key at-a-glance information\u2014from the trail's distance and configuration to contacts for local information. Each profile also includes a map (see \"Trail Maps,\"). The main text for each profile includes three sections: \"Overview,\" \"Route Details,\" and \"Directions\" (for driving to the trailhead area).\n\nFlip through the book, read the \"Overview\" summaries, and choose the hikes that appeal to you. A list of recommended hikes will also help you choose a hike to fit your mood and group.\n\nStar Ratings\n\nThe hikes in Five-Star Trails: Adirondacks were carefully chosen to give the hiker an overall five-star experience and represent the diversity of trails found in the region. Each hike was assigned a one- to five-star rating in each of the following categories: scenery, trail condition, suitability for children, level of difficulty, and degree of solitude. While one hike may merit five stars for its stunning scenery, that same trail may rank as a two-star trail for children. Similarly, another hike might receive two stars for difficulty but earn five stars for solitude. It's rare that any trail receives five stars in all five categories; nevertheless, each trail offers excellence in at least one category, if not others. Here's how the star ratings for each of the five categories break down:\n\nFOR SCENERY:\n\nUnique, picturesque panoramas\n\nDiverse vistas\n\nPleasant views\n\nUnchanging landscape\n\nNot selected for scenery\n\nFOR TRAIL CONDITION:\n\nConsistently well maintained\n\nStable, with no surprises\n\nAverage terrain to negotiate\n\nInconsistent, with good and poor areas\n\nRocky, overgrown, or often muddy\n\nFOR CHILDREN:\n\nExcellent choice for introducing young kids to hiking\n\nFun for anyone past the toddler stage with some hiking experience\n\nGood for young hikers with proven stamina\n\nNot enjoyable for children\n\nNot advisable for children\n\nFOR DIFFICULTY:\n\nGrueling\n\nStrenuous\n\nModerate (won't beat you up\u2014but you'll know you've been hiking)\n\nEasy with patches of moderate\n\nGood for a relaxing stroll\n\nFOR SOLITUDE:\n\nPositively tranquil\n\nSpurts of isolation\n\nModerately secluded\n\nCrowded on weekends and holidays\n\nSteady stream of individuals and\/or groups\n\nGPS Trailhead Coordinates\n\nAs noted in \"Trail Maps\" I used a handheld GPS unit to obtain geographic data and sent the information to the publisher's cartographers. In the opener for each hike profile, the coordinates\u2014the intersection of the latitude (north) and longitude (west)\u2014will orient you from the trailhead. In some cases, you can drive within viewing distance of a trailhead. Other hiking routes require a short walk to the trailhead from a parking area.\n\nYou will also note that this guidebook uses the degree\u2013decimal minute format for presenting the latitude and longitude GPS coordinates. The latitude and longitude grid system is likely quite familiar to you, but here is a refresher, pertinent to visualizing the GPS coordinates:\n\nImaginary lines of latitude\u2014called parallels and approximately 69 miles apart from each other\u2014run horizontally around the globe. The equator is established to be 0\u00b0, and each parallel is indicated by degrees from the equator: up to 90\u00b0N at the North Pole, and down to 90\u00b0S at the South Pole.\n\nImaginary lines of longitude\u2014called meridians\u2014run perpendicular to latitude lines. Longitude lines are likewise indicated by degrees. Starting from 0\u00b0 at the Prime Meridian in Greenwich, England, they continue to the east and west until they meet 180\u00b0 later at the International Date Line in the Pacific Ocean. At the equator, longitude lines also are approximately 69 miles apart, but that distance narrows as the meridians converge toward the North and South Poles.\n\nTo convert GPS coordinates given in degrees, minutes, and seconds to the degree\u2013decimal minute format used in this book, the seconds are divided by 60. For more on GPS technology, visit usgs.gov.\n\nDistance and Configuration\n\nDistance notes the length of the hike round-trip, from start to finish. If the hike description includes options to shorten or extend the hike, those round-trip distances will also be factored here. Configuration defines the trail as a loop, an out-and-back (taking you in and out via the same route), a figure eight, or a balloon.\n\nHiking Time\n\nDistances given are absolute, but hiking times are based on an average hiking speed of 2\u20133 miles per hour, with time built in for pauses at overlooks and brief rests. Hiking time varies for each hiker, but I like to use the rule of thumb of 3 miles an hour plus an additional hour for every thousand feet of ascent. For trails over 9 miles, I usually add another hour to account for more time to take in the sites and\/or a slower pace near the end. Hikers new to the Adirondacks will probably want to pad their time a bit more as they accustom themselves to the more rugged trails.\n\nHighlights\n\nWaterfalls, historic sites, or other features that draw hikers to the trail are emphasized here.\n\nElevation\n\nIn each trail's opener, you will see the elevation at the trailhead and another figure for the peak height on the route. The full hike profile also includes a complete elevation profile.\n\nAccess\n\nFees or permits required to hike the trail are detailed here\u2014and noted if there are none. Trail access hours are also shown.\n\nMaps\n\nResources for maps, in addition to those in this guidebook, are listed here. As previously noted, the publisher and author recommend that you carry more than one map and that you consult those maps before heading out on the trail to resolve any confusion or discrepancy. Common abbreviations listed here include DEC (Department of Environmental Conservation) and USGS (U.S. Geological Survey).\n\nFacilities\n\nThis section alerts you to restrooms, phones, water, picnic tables, and other basics at or near the trailhead.\n\nWheelchair Access\n\nAt a glance, you'll see if there are paved sections or other areas for safely using a wheelchair.\n\nComments\n\nHere you will find assorted nuggets of information, such as whether dogs are allowed on the trails.\n\nContacts\n\nListed here are phone numbers and website addresses for checking trail conditions and gleaning other day-to-day information.\n\nOverview, Route Details, and Directions\n\nThese three elements provide the main text about the hike. \"Overview\" gives you a quick summary of what to expect on that trail; the \"Route Details\" guide you on the hike, start to finish; and \"Directions\" should get you to the trailhead from a well-known road or highway.\n\nWeather\n\nWeather in the Adirondacks should not be taken lightly. Conditions can change rapidly during all seasons, and the conditions at the base of a tall mountain will certainly not be the same as at the top. When I set out to hike Ampersand Mountain, it was a bright, sunny day in the height of summer. Clouds rolled in, and an unpredicted and severe thunderstorm set in quickly with little warning. Those at the peak sheltered near large boulders at the top, while those at the base turned back. Unfortunately, I was caught in the treeline just below the peak. The situation soon became precarious, as the wind howled and bent trees near the point of snapping. Heavy rains flooded the path, making footing slick. Turning around meant passing through an area where blowdown was likely and unpredictable, and the steep, rocky path became slicker by the minute\u2014hurrying would not be an option. I managed to shelter near the top, where the treeline broke, and descended during a brief break in the weather. When I finally reached my truck, the system had passed, clear skies prevailed, and little evidence of the severe storm remained. Despite raingear, my clothes told a different story than the skies. So as a rule, prepare for the worst case, even if the best is forecasted.\n\nGenerally speaking, the Adirondacks are cooler than the rest of New York. This is a welcome respite during summer, but it also means that snow may fall from September into May. Summer temperatures rarely go into the 90s, while winter temperatures often go below zero. Average temperatures are in the mild range, with high 70s typical in summer. Though it varies year to year, the hiking season is generally May\u2013October, though proper attire and preparation is always advised.\n\nSnowshoeing and skiing are also popular in this area, and all but the steepest of the trails described in this book can be done with snowshoes. Indeed, though temperatures often plummet below zero, snowshoeing is often the best way to get some solitude on the more popular trails. Late March and early April, also called the mud season, are likely the least advisable times for hiking in the Adirondacks, and most seasonal roads are closed during this time. Additionally, trail erosion is worst during mud season, and the Department of Environmental Conservation has asked that hikers refrain from hiking in areas above 3,000 feet, as the erosion is particularly damaging to alpine vegetation at this time.\n\nThe table on the previous page lists average temperatures and precipitation by month for Adirondack Park. For each month, \"Hi Temp\" is the average daytime high, \"Lo Temp\" is the average nighttime low, and \"Rain or Snow\" is the average precipitation.\n\nWater\n\nWith cool temperatures, 3,000 lakes and ponds, and 30,000 miles of streams and rivers, concern for water may seem overly cautious. However Giardia lamblia, commonly referred to as giardia, has contaminated many water sources in the Adirondacks. Also known as beaver fever, the intestinal parasite causes\u2014well, put politely\u2014intestinal problems weeks after ingestion, the likes of which most would not wish on their enemies. The words watery, explosive, foul, and rotten have also been used to describe the aftereffects, so caution is simply necessary. The single-celled organism is transported by all warm-blooded animals, which in turn contaminate water sources. Because beavers are typically in the water and fail to wash their hands, or even leave the water, after using the lavatory, they are often the culprits of the spread and have been rewarded with the disease's moniker. Beavers also abound in the Adirondacks, and many, if not most, streams have beaver activity in or around them. Humans are also culprits in the transporting and contamination of water sources. To prevent contamination, follow the appropriate guidelines for waste disposal as outlined by the Department of Environmental Conservation and as described later in this book (see page 26).\n\nAside from the inherent risks of drinking untreated water, many of the ponds and lakes are fairly stagnant and taste bad. Effective and proper water treatment is essential for using any source found along the trail. Boiling water for 2\u20133 minutes is always a safe measure, as are iodine tablets, approved chemical mixes, filtration units rated for giardia, and UV filtration. Some of these methods (such as filtration with an added carbon filter) remove bad tastes, while others add their own taste. My suggestion is to research a filtration method, devise a way to make it palatable, always carry along a means to purify water, and carry enough water for the hike. A cool drink waiting in the car is always a welcome reward, but it's of little use on the trail. Carry a means of purification to help in a pinch or if you underestimate your consumption, but always bring a couple of water bottles for everyone in your group.\n\nHow much is enough? Well, one simple physiological fact should convince you to err on the side of excess when deciding how much water to pack: a hiker working hard in 90\u00baF heat needs approximately 10 quarts of fluid per day. That's 2.5 gallons\u201412 large water bottles or 16 small ones. In other words, pack along one or two bottles even for short hikes. Hydrate prior to your hike, carry (and drink) 6 ounces of water for every mile you plan to hike, and hydrate after the hike.\n\nClothing\n\nProper attire will make the difference between enjoying the trail and having a miserable experience. In the worst case it could even save your life. I add these words of caution not to be dramatic but because it is so easy to disregard simple considerations, especially on short trips and especially when it comes to clothing. Weather is unpredictable, trail conditions change or become impassable, we overestimate our abilities and underestimate required hiking times, and wrong turns happen. On a backpacking trip, we are usually prepared for these eventualities because we know that we will be out in the woods for a long time. Don't get me wrong\u2014I'm not advocating hiring a Sherpa or bringing your entire wardrobe, but proper attire at the outset will mitigate most problems.\n\nA few simple guidelines will avert most inconveniences and help you avoid most disasters. When hiking, wear hiking boots and not sandals, flip-flops, or sneakers. Trails in the Adirondacks are hardened, which means they traverse rocky paths to minimize erosion. It protects the forest but is hard on the ankles, so wear suitable footwear. Proper layering is also essential to comfortable hiking without excessive perspiration, so wear or pack all layers for the season. Dress for the day and prepare for the worst. This means that if it is 75\u00baF and sunny, dress in what is comfortable for you, but bring raingear and do not leave it in the car if the weather looks good. In spring or fall always wear or bring a wool hat and gloves. On the trail I wear pants with zip-off legs and always bring a pack with food and water, as well as a change of socks and shirt, raingear, gaiters, and a warm hat. With these few extra items, you are prepared for any condition or eventuality without adding much weight. In fall, winter, and spring, remember that \"cotton kills,\" so wear silk, wool, or synthetics. In summer cotton may be passable but only if you bring a change of clothing that will not get wet. In an emergency you are only prepared with what you have on or with you. Remember, the worst may include spending the night in the woods in spring or fall when daytime and nighttime temperatures vary drastically, so be prepared.\n\n**BOULDER CAVE ON THE WAY TO GOOD LUCK CLIFFS** (Trail 2, Good Luck Cliffs, page 35)\n\nEssential Gear\n\nAlong with proper attire, there are several other essential items that every hiker should bring with him or her, even on short day hikes. As I alluded to in the section about clothing, no one plans on a 1-hour hike turning into a night lost in the woods, but it happens all the time to novice and experienced hikers alike. A complete list of items to bring along for day trips and backpacking trips is found in Appendix B but I've listed the essentials here. These items can be carried in your pockets if carefully packed, but carrying a light backpack is often easier. In addition to carrying the items listed below, you need to know how to use them, especially the ones involving navigation.\n\nWater: Bring a means to carry it and purify it (for example, water bottle and iodine or a filter).\n\nNavigation: Always carry a map, preferably a topo and a trail map with route description, and a high-quality compass.\n\nKnife or multitool: A pocketknife and\/or multitool with pliers\n\nLight: A flashlight or headlamp with extra bulb and batteries\n\nFire: Windproof matches and\/or a lighter, as well as a fire starter\n\nFirst aid kit: See below for a list of basic items to be kept in your kit.\n\nExtra food: Trail mix, granola bars, or other high-energy foods\n\nExtra clothes: Raingear, warm hat, gloves, and a change of socks and shirt\n\nSun protection: Sunglasses, lip balm, sunblock, and a sun hat\n\nFirst Aid Kit\n\nA typical first aid kit may contain more items than you might think necessary. These are just the basics. Prepackaged kits in waterproof bags are available. Even though quite a few items are listed here, they pack down into a small space.\n\nAdhesive bandages\n\nAntibiotic ointment (Neosporin or the generic equivalent)\n\nAthletic tape\n\nBenadryl or the generic equivalent, diphenhydramine (in case of allergic reactions)\n\nBlister kit (such as moleskin or Spenco 2nd Skin)\n\nButterfly-closure bandages\n\nElastic bandages or joint wraps\n\nEpinephrine in a prefilled syringe (for people known to have severe allergic reactions to such things as bee stings, usually by prescription only)\n\nGauze and compress pads (one roll and a half dozen 4-by-4-inch pads)\n\nHydrogen peroxide or iodine\n\nIbuprofen or acetaminophen\n\nInsect repellent\n\nMatches or a pocket lighter\n\nSunscreen\n\nTweezers\n\nWhistle (more effective at signaling rescuers than your voice)\n\nGeneral Safety\n\nTo some potential mountain enthusiasts, the deep woods seem inordinately dark and perilous. It is the fear of the unknown that causes this anxiety. No doubt, potentially dangerous situations can occur outdoors, but as long as you use sound judgment and prepare yourself before hitting the trail, you'll be much safer in the woods than in most urban areas of the country. It is better to look at a backcountry hike as a fascinating chance to discover the unknown rather than a chance for potential disaster. If you're new to the game, I suggest starting out easy and finding a person who knows more to help you out. In addition, here are a few tips to make your trip safer and easier.\n\nAlways let someone know your plans in advance. Tell your safety person which trails you will be on, when and where you expect to depart from, and when and where you expect to return. Though you are primarily responsible for your safety, your safety person is your backup plan and will alert rescuers where to look for you. Remember to tell your safety person all changes in your plans and check in to let him or her know your trip went well.\n\nAlways sign in and out of the trail registers provided. If you are hiking several trails, it will alert rescuers where you are and, sometimes more important, where you are not.\n\nWhile getting lost is unlikely on shorter trails, injury is a distinct possibility, as indicated by numerous warning signs of precipitous cliffs. While bravado may foolishly deter people from using the registers for the first reason, the following reason may appeal more to their self-interest. The Department of Environmental Conservation and the state base many of their trail-maintenance and funding decisions on trail registers. So if you want to see your favorite trails maintained or want more like them, then let the registers act as your democratic voice in the wilderness.\n\nNever rely on a cell phone, but bring one just in case. Reception is spotty at best and virtually nonexistent in most areas\u2014and your phone won't work as kindling.\n\nAlways carry food and water, whether you are planning to go overnight or not. Food will give you energy, help keep you warm, and sustain you in an emergency until help arrives. You never know if a stream will be nearby when you become thirsty. Bring potable water or treat water before drinking it from a stream. Boil or filter all found water before drinking it.\n\nStay on designated trails. Most hikers get lost when they leave the path. Even on the most clearly marked trails, there is usually a point where you have to stop and consider what direction to head. If you become disoriented, don't panic. As soon as you think that you may be off track, stop, assess your current direction, and then retrace your steps to the point where you went astray. Using a map, a compass, and this book, and keeping in mind what you have passed thus far, reorient yourself and trust your judgment on which way to continue. If you become absolutely unsure of how to continue, return to your vehicle the way you came in. Should you become completely lost and have no idea how to return to the trailhead, remaining in place along the trail and waiting for help is most often the best option for adults and always the best option for children.\n\nBe especially careful when crossing streams. Whether you are fording the stream or crossing on a log, make every step count. If you have any doubt about maintaining your balance on a log, ford the stream instead. When crossing, use a trekking pole or stout stick for balance and face upstream as you cross. If a stream seems too deep to ford, turn back. Whatever is on the other side is not worth risking your life.\n\nBe careful at overlooks. While these areas may provide spectacular views, they are potentially hazardous. Stay back from the edge of outcrops and be absolutely sure of your footing; a misstep can mean a nasty and possibly fatal fall.\n\nStanding dead trees and storm-damaged living trees pose a real hazard to hikers and tent campers. These trees may have loose or broken limbs that could fall at any time. When choosing a spot to rest or a backcountry campsite, look up.\n\nKnow the symptoms of hypothermia. Shivering and forgetfulness are the two most common indicators of this stealthy killer. Hypothermia can occur at any elevation, even in the summer, especially when the hiker is wearing lightweight cotton clothing. If symptoms arise, get the victim shelter, hot liquids, and dry clothes or a dry sleeping bag.\n\nTake along your brain. A cool, calculating mind is the single most important asset on the trail. Think before you act. Watch your step. Plan ahead. Avoiding accidents is the best way to ensure a rewarding and relaxing hike.\n\nAsk questions. State forest and park employees are there to help. It's a lot easier to ask advice beforehand, and it will help you avoid a mishap away from civilization when it's too late to amend an error.\n\nAnimal, Insect, and Plant Hazards\n\nBLACK BEARS Black bears are found throughout New York state but are most frequently encountered in the Adirondacks and occasionally in the Catskills. Though attacks by black bears are virtually unheard of, a bear tearing up your gear or rummaging about outside your tent will give anyone a start. If you encounter a bear at your campsite or while hiking, remain calm and never run away. Make loud noises to scare off the bear and back away slowly.\n\nBlack bears normally avoid humans, but you should always leave them an escape route if you encounter them. They can sprint up to 35 miles per hour and are strong swimmers and great tree climbers.\n\nIn primitive and remote areas, assume that bears are present; in more developed sites, ask the park staff about the current bear situation. Most encounters are food related, as bears have an exceptional sense of smell and will eat anything. A clean site, combined with care and caution, will keep these foragers away from your campsite. Store all food, cooking equipment, and garbage in tightly sealed containers and place well away from your tent. In remote areas or those with recent bear activity, store all items in storage lockers or bear canisters, or suspend them in a sack 12 feet above the ground, 6 feet below branches, and 12 feet from neighboring trees. Make sure that your site is clean, and never leave food unattended. Wash utensils and cooking equipment at least 100 feet from your site, and never dump uneaten food on the ground. Do not bring food into your tent, and do not sleep in clothes worn while preparing food. Never leave scented products of any kind\u2014food, beverages, or personal-\u00adcare products, such as lotion or sunscreen\u2014in your vehicle or unattended in the park area. Generally, proper food preparation and garbage disposal will help maintain a pristine wilderness, the safety of bears and other wildlife, and an enjoyable camping experience for everyone. For updated information and rules regarding acceptable bear canisters (required for overnight trips in the Eastern High Peaks Wilderness area), see the Department of Environmental Conservation website: www.dec.ny.gov\/animals\/7225.html.\n\nRATTLESNAKES New York has a variety of snakes\u2014including garter, milk, and water snakes\u2014most of which are benign. Timber rattlesnakes, northern copperheads, and eastern massasauga rattlesnakes are the exceptions, though they primarily dwell in very isolated areas. According to the Department of Environmental Conservation, the massasauga is found in two isolated marshy areas: one near Syracuse and the other near Rochester. The copperhead is found mostly along the Hudson Valley but is largely absent from the Catskills. Timber rattlesnakes are the most widely dispersed of the three and are found in rugged deciduous forests along the southern edge of the state and up into the eastern Adirondacks. Encounters with any of these species are very rare.\n\nWhen hiking, stick to well-used trails and wear over-the-ankle boots and loose-fitting, long pants. Rattlesnakes like to bask in the sun and won't bite unless threatened. Do not step or put your hands where you cannot see, and avoid wandering around in the dark. Step on logs and rocks, never over them, and be especially careful when climbing rocks or gathering firewood. Always avoid walking through dense brush or willow thickets. Keep in mind that rattlers want to avoid confrontation and generally will alert you if you wander too close for their comfort. Hibernation season is October\u2013April.\n\nThe only known treatment for a snakebite is intravenous antivenom, only available at a hospital. If you are hiking alone and get bitten, calmly walk to where you can get help. If someone in your group has been bitten, send another person to get help quickly while the victim rests.\n\nMOSQUITOES You will encounter mosquitoes on most of the hikes described in this book. Insect repellent and\/or repellent-impregnated clothing are the only simple methods to ward off these pests. Mosquitoes in New York are known to carry the West Nile virus, so all due caution should be taken to avoid mosquito bites.\n\nBLACK FLIES Though certainly a pest and maddening annoyance, the worst a black fly will cause is an itchy welt. They are most active mid-May\u2013June, during the day, and especially before thunderstorms, as well as during the morning and evening hours. Insect repellent has some effect, though the only way to keep them from swarming is to keep moving.\n\nINSIDE THE FIRE TOWER ON TOP OF SNOWY MOUNTAIN (Trail 18, Snowy Mountain)\n\nTICKS Ticks are often found on brush and tall grass, waiting to hitch a ride on a warm-blooded passerby. Adult ticks are most active in New York between April and May and again between October and November. Among the local varieties of ticks, the black-legged tick, commonly called the deer tick, is the primary carrier of Lyme disease. Several strategies reduce the chances of ticks getting under your skin. Some people choose to wear light-colored clothing, so ticks can be spotted before they make it to the skin. Most important, be sure to visually check your hair, back of neck, armpits, and socks at the end of the hike. During your posthike shower, take a moment to do a more complete body check. For ticks that are already embedded, grasp the tick as close to the skin's surface as possible with tweezers, and pull upward with steady, even pressure. Do not puncture a tick, as this might release harmful bacteria. Use disinfectant solution on the wound.\n\nPOISON IVY, OAK, AND SUMAC Recognizing and avoiding poison ivy, oak, and sumac are the most effective ways to prevent the painful, itchy rashes associated with these plants. Poison ivy occurs as a vine or ground cover, 3 leaflets to a leaf; poison oak occurs as either a vine or shrub, also with 3 leaflets; and poison sumac flourishes in swampland, each leaf having 7\u201313 leaflets. Urushiol, the oil in the sap of these plants, is responsible for the rash. Within 14 hours of exposure, raised lines and\/or blisters will appear on the affected area, accompanied by a terrible itch. Refrain from scratching because bacteria under your fingernails can cause an infection. Wash and dry the rash thoroughly, applying a calamine lotion to help dry out the rash. If itching or blistering is severe, seek medical attention. If you do come into contact with one of these plants, remember that oil-\u00adcontaminated clothes, pets, or gear can easily cause an irritating rash on you or someone else, so wash not only any exposed parts of your body but also clothes, gear, and pets, if applicable.\n\nHunting\n\nUnlike many national and state parks, hunting is allowed in Adirondack Park. Separate rules, regulations, and licenses govern the various hunting types and related seasons. Though there are generally no problems, hikers may wish to forgo their trips during the big-game seasons, when the woods suddenly seem filled with orange and camouflage. The following is a list of big-game seasons.\n\nEarly bear season: The first Saturday after the second Monday in September; lasts four weeks\n\nArchery season (deer and bear): September 27 to opening of regular season\n\nMuzzle-loading season: Seven days prior to regular season\n\nRegular season: Next-to-last Saturday in October through the first Sunday in December\n\nTips for Enjoying Adirondack Park\n\nOne of the best features of Adirondack Park is its democratic nature. It is meant for the public to enjoy its wild character and is open to all. There are few if any fees, permits, or licenses required for this enjoyment\u2014\u00adand none for those who simply wish to hike the trails. Simply drive to your trailhead, get out of your car, and hit the trail. It is a luxury and a responsibility. Without park staff cleaning, maintaining, and policing the way they typically do in other parks, the burden for cleanliness, maintenance, and your own safety lies on your own shoulders. Practicing the ethos of carry-in and carry-out will ensure that other hikers enjoy the wilderness as much as, and in the same way, you have. By maintaining campsites, lean-tos, and fire rings, you will ensure others get to enjoy them and find them useful too. Avoiding destructive behavior, such as chopping down trees for firewood or causing unnecessary erosion by taking shortcuts, means the forever-wild character of the Adirondacks will remain.\n\nBackcountry Advice and DEC Regulations\n\nThe entire park lies under the jurisdiction of the Department of Environmental Conservation (DEC), and its rules and regulations regarding hiking and backcountry camping apply for all state lands within the park. Camping is permitted within the forest preserve without a permit as long as you follow the regulations below. The DEC's rules and regulations are essentially the same sound, safe, and practical advice I would offer, so I recap them below.\n\nCamping is prohibited within 150 feet of any road, trail, spring, stream, pond, or other body of water, except at areas designated by a camp here disk.\n\nGroups of 10 or more persons or stays of more than three days in one place require a permit from the New York state forest ranger responsible for the area.\n\nLean-tos are available in many areas on a first-come, first-serve basis. Lean-tos cannot be used exclusively and must be shared with other campers.\n\nUse pit privies provided near popular camping areas and trailheads. If none are available, dispose of human waste by digging a hole 6\u20138 inches deep at least 150 feet from water or campsites. Cover with leaves and soil.\n\nDo not use soap to wash yourself, clothing, or dishes within 150 feet of water.\n\nDrinking and cooking water should be boiled for 5 minutes, treated with purifying tablets, or filtered through a filtration device to prevent instances of giardia infection.\n\nFires should be built in existing fire pits or fireplaces if provided. Use only dead and down wood for fires. Cutting standing trees is prohibited. Extinguish all fires with water, and stir ashes until they are cold to the touch. Do not build fires in areas marked by a no fires disk. Camp stoves are safer, more efficient, and cleaner.\n\nCarry out what you carry in. Practice Leave No Trace camping and hiking guidelines.\n\nKeep your pet under control. Restrain it on a leash when others approach. Collect and bury droppings away from water, trails, and campsites. Keep your pet away from drinking-water sources.\n\nObserve and enjoy wildlife and plants, but leave them undisturbed.\n\nRemoving plants, rocks, fossils, or artifacts from state land without a permit is illegal.\n\nExcept in an emergency, or between December 15 and April 30, camping is prohibited above an elevation of 4,000 feet in the Adirondacks.\n\nAt all times, only emergency fires are permitted above 4,000 feet in the Adirondacks.\n\nTrail Etiquette\n\nWhether you're on a city, county, state, or national park trail, always remember that great care and resources (from nature as well as from your tax dollars) have gone into creating these trails. Treat the trail, wildlife, and fellow hikers with respect.\n\nHike on open trails only. Respect trail and road closures (ask if not sure), avoid possible trespassing on private land, and obtain all permits and authorization as required. Also, leave gates as you found them or as marked.\n\nLeave only footprints. Be sensitive to the ground beneath you. Stay on existing trails rather than blazing new ones. Pack out what you pack in. No one likes to see the trash someone else has left behind.\n\nNever spook animals. An unannounced approach, a sudden movement, or a loud noise startles most animals. Surprised animals can be dangerous to you, to others, and to themselves. Give them extra room and time to adjust to your presence.\n\nObserve the yield signs that are displayed around the region's trailheads and backcountry. They advise that hikers yield to horses, and bikers yield to both horses and hikers. A common courtesy on hills is that hikers and bikers yield to any uphill traffic.\n\nWhen encountering mounted riders or horse packers, hikers can courteously step off the trail, on the downhill side, if possible. Speak to the riders before they reach you and do not hide behind trees. You are less spooky if the horse can see and hear you. Resist the urge to pet or touch the horses unless you are invited to do so.\n\nPlan ahead. Know your equipment, your ability, and the area in which you are hiking\u2014and prepare accordingly. Be self-sufficient at all times; carry necessary supplies for changes in weather or other conditions. A well-executed trip is satisfying.\n\nBe courteous to other hikers, bikers, equestrians, and others you encounter on the trails.\n\nSouthern\n\nCRANE MOUNTAIN (Trail 7)\n\n1 STEWART AND INDIAN LAKES\n\n2 GOOD LUCK CLIFFS\n\n3 JOCKEYBUSH LAKE\n\n4 GROFF CREEK\n\n5 TENANT CREEK FALLS\n\n6 HADLEY MOUNTAIN\n\n7 CRANE MOUNTAIN\n\n1\n\nStewart and Indian Lakes\n\nWOODLAND TRAIL\n\n**SCENERY:**\n\n**TRAIL CONDITION:**\n\n**CHILDREN:**\n\n**DIFFICULTY:**\n\n**SOLITUDE:**\n\nGPS COORDINATES: N43\u00b0 10.790' W74\u00b0 30.295'\n\nDISTANCE & CONFIGURATION: 4.6-mile out-and-back\n\nHIKING TIME: 2\u20133 hours\n\nHIGHLIGHTS: Two secluded lakes\n\nELEVATION: 1,603' at trailhead, 2,044' at highest point\n\nACCESS: Open 24\/7; no fees or permits required\n\nMAPS: National Geographic Adirondack Park, Northville\/Raquette Lake (#744)\n\nFACILITIES: None\n\nWHEELCHAIR ACCESS: No\n\nCONTACTS: Central and Southern Adirondack trail information: www.dec.ny.gov\/outdoor\/9200.html; emergency contact: 518-891-0235\n\nOverview\n\nThis easy hike is a great destination for people who wish to explore two remote, picturesque lakes for fishing or a picnic. With a few moderate climbs, it is a good trail for children, and wildlife viewing at both lakes provides additional interest.\n\nRoute Details\n\nLocated in the Shaker Mountain Wild Forest, the trail uses old logging roads, so the grades are mostly gentle and easily accessible to hikers of all experience levels. The area had been heavily logged before state acquisitions, and much of the forest is still in a phase of regeneration. However, the acquisitions occurred decades ago, and traces of past logging have disappeared. Wild forest areas\u2014in contrast to wilderness, primitive, or canoe areas\u2014are considered less ecologically fragile, allowing for a greater variety of uses and more intensive activities. Don't let the distinction fool you, though, as they are definitely wild places and provide excellent wilderness experiences. Access to the parking area is at the left fork along Green Lake Road, which appears to be little more than a driveway. The well-maintained and spacious lot provides parking for the Kane Mountain Trail as well, but it can easily accommodate a dozen cars, so space should not be a concern.\n\nThe trail begins to the right of the trail register on the north end of the parking lot. Shrouded in tall pines, it follows an access road to a small fish hatchery created by the damming of Otter Lake's outlet, and you can hear water spilling over the concrete dam in most seasons. The trail diverges from the road shortly before the pond on your right. The exact path may be a little hard to discern, but a wooden bridge and a sign prohibiting the use of baitfish will steer you.\n\nA note to anglers: The Department of Environmental Conservation has gone to great lengths to reestablish native brook trout and the endangered round whitefish in some of the isolated waters of the Adirondacks. Reestablishment methods vary but include treating entire lakes with rotenone to remove competing, nonnative species and then restocking the waters with native fish. Due to the physical characteristics of the various Adirondack waterways, the success of this technique is limited and expensive. When the technique is successful, those same physical characteristics mean there will be lasting success and that native fish populations can thrive. Unfortunately, the reintroduction of nonnative species is almost exclusively a consequence of the unintentional release of baitfish by anglers into the reestablished ecosystems. So leave the bait bucket at home when visiting the prized fishing waters of the Adirondacks.\n\nOnce on the trail, you will notice that it is emblazoned by yellow trail markers and very easy to follow the rest of the way. A quick climb takes you along a low ridge overlooking the pond to your left. The trail climbs through a mixed hardwood forest dominated by birch and beech trees. The grade is very moderate for the first 0.7 mile, at which point you encounter a muddy area. The trail climbs more steeply after this muddy area, but at approximately 1.2 miles it levels off to a more gently rolling terrain. Farther along the trail, the forest canopy gets higher and the understory thins, allowing deeper views into the surrounding forest. You catch glimpses of Stewart Lake as you approach 1.3 miles, where there is a fork in the trail. The main trail continues on the right toward Indian Lake, while the left fork takes you through a stand of pine and hemlock to a boggy area that surrounds Stewart Lake. The marshland prevents you from gaining direct access to the lake but makes for good wildlife viewing.\n\nBack along the trail, you encounter some mucky areas that you will have to negotiate on your own, as no good way bypasses the muck. At 2 miles, you pass a thick carpet of club moss, which at first glance looks like tree saplings but is actually more closely related to ferns. At this point, you begin the descent to Indian Lake, where the trail ends at the shore. A path to the left follows the shoreline toward a rock that juts out into the lake and provides additional vantage points. The remote setting promises a wonderful opportunity to view a variety of wildlife, including a beaver lodge.\n\nDirections\n\nFrom the southern intersection of NY 10 and NY 29A in downtown Caroga Lake, head north on NY 10\/NY 29A. At 3.4 miles past the intersection, turn right onto Green Lake Road.\n\nFrom the intersection of NY 8 and NY 10 in Piseco, head 19.6 miles south on NY 10, and turn left (1.9 miles past the northern intersection of NY 10 and NY 29A near Pine Lake Park) onto Green Lake Road. Follow Green Lake Road north along the west edge of the shore. At 0.5 mile, bear left onto the parking area drive.\n\n2\n\nGood Luck Cliffs\n\nGOOD LUCK LAKE\n\n**SCENERY:**\n\n**TRAIL CONDITION:**\n\n**CHILDREN:**\n\n**DIFFICULTY:**\n\n**SOLITUDE:**\n\nGPS COORDINATES: N43\u00b0 15.352' W74\u00b0 32.292'\n\nDISTANCE & CONFIGURATION: 3.4-mile out-and-back\n\nHIKING TIME: 2\u20133 hours\n\nHIGHLIGHTS: Dramatic cliffs, panoramic views\n\nELEVATION: 1,693' at trailhead, 2,284' at highest point\n\nACCESS: Open 24\/7; no fees or permits required\n\nMAPS: National Geographic Adirondack Park, Northville\/Raquette Lake (#744)\n\nFACILITIES: None\n\nWHEELCHAIR ACCESS: No\n\nCONTACTS: Central and Southern Adirondack trail information: www.dec.ny.gov\/outdoor\/9200.html; emergency contact: 518-891-0235\n\nOverview\n\nSheer cliffs are almost standard in the High Peaks region, but in the southern foothills, these dramatic cliffs are unique and rival those anywhere in the Adirondacks. Though the path to Good Luck Cliffs is technically a bushwhack, the cliffs are frequented enough that even in the height of winter, you can find your way to the mountaintop.\n\nRoute Details\n\nThe parking area is a few hundred yards north of the second bridge that crosses the West Branch of the Sacandaga River along NY 10 in the town of Arietta. Coming from the south, the parking area is a large pulloff on the right-hand side of the road.\n\nThe trailhead is across the street and has signs indicating that Dexter Lake is 4.1 miles, Potter Homestead is 7.1 miles, and Good Luck Lake is 0.4 mile, though the latter is along a separate trail. The trail register is approximately 0.5 mile ahead at a trail junction. Follow the snowmobile and red trail disks west along mostly level terrain. Shortly after passing through a wet section, you reach the trail register and junction with other trails. To the right is a snowmobile trail that parallels NY 10. Straight ahead is the trail to Dry and Dexter Lakes, 1.6 and 2.4 miles respectively. The left trail takes you to the west shore of Good Luck Lake (including some tent sites), Spectacle Lake, and the unmarked path to the cliffs and Good Luck Mountain summit.\n\nThe trail has several muddy and wet sections as you make your way south toward the lake, and it even shares space with a stream for a while. The trail begins to descend from this junction, and at approximately 1 mile in, the lake comes into view. A designated campsite to the left of the trail offers a great view of the lake. The cliffs soon become visible, looming off to your right, and you continue past the western tip of the lake through a marshy area, where the evergreens become denser. After crossing a short bridge, you reach the crest of a tiny hill, at 1.3 miles, below which is another bridge that crosses a stream that feeds Good Luck Lake. To your right is the unmarked path that leads to the cliffs.\n\nThe path parallels the feeder stream briefly on your left, along an elevated knoll, before swinging northwest toward the cliffs. Shortly after, the trail climbs and you rock-hop across a stream that cuts the gorge along the cliff's southwest face. The path follows within a few hundred feet of this stream all the way to a notch that lies just below the cliffs. In winter, icicles form sheets of ice along the southern tumble of boulders, creating an amazing winter wonderland setting.\n\nThough the cliffs have been visible through most of the climb, their full grandeur is not fully expressed until you reach the small saddle beneath the west face of the cliff. The trail swings north here, and you will navigate your way around a jumble of boulders that covers the valley floor. The huge boulders provide an opportunity for exploration, including many small caves that are formed in the mass of rock. You pass a noteworthy cave on the right at 1.4 miles.\n\nYou probably have heard the stream flowing on your right, and at 1.5 miles, you recross it before the final ascent to the clifftop. The trail is much steeper and the path less distinct, but as long as you keep the cliffs to your right, you will eventually make your way to the north, or back, side of the cliffs. About 0.1 mile from the stream crossing, you reach a fork in the trail in a stand of spruce trees. To the left is a slight northward jog in the trail that leads to the clifftop, while the right trail leads to the clifftop a little more steeply but more directly. Either way, you reach the top shortly; from there, take any of the numerous paths that weave their way from outlook to outlook. Along the southwest edge is a particularly amazing vista that shows expansive views down to the valley, with Spectacle Lake in the foreground.\n\nSNOWSHOEING ALONG THE UNMARKED PART OF THE TRAIL\n\nDirections\n\nFROM THE SOUTH From the intersection of NY 29A and NY 10, north of Caroga Lake near Pine Lake Park, head 6 miles north on NY 10. The large parking area is on the right just past the second bridge crossing the Sacandaga River in the town of Arietta.\n\nFROM THE NORTH From the intersection of NY 8 and NY 10 in Piseco, head 11.6 miles south on NY 10. The large parking area is on your left.\n\n3\n\nJockeybush Lake\n\nJOCKEYBUSH LAKE NEAR OUTLET\n\n**SCENERY:**\n\n**TRAIL CONDITION:**\n\n**CHILDREN:**\n\n**DIFFICULTY:**\n\n**SOLITUDE:**\n\nGPS COORDINATES: N43\u00b0 18.057' W74\u00b0 33.905'\n\nDISTANCE & CONFIGURATION: 2.3-mile out-and-back\n\nHIKING TIME: 2 hours\n\nHIGHLIGHTS: Pristine lake, waterfall\n\nELEVATION: 1,755' at trailhead, 1,964' at highest point\n\nACCESS: Open 24\/7; no fees or permits required\n\nMAPS: National Geographic Adirondack Park, Northville\/Raquette Lake (#744)\n\nFACILITIES: None\n\nWHEELCHAIR ACCESS: No\n\nCOMMENTS: During the spring thaw, Jockeybush Outlet may require wading, so plan accordingly.\n\nCONTACTS: Central and Southern Adirondack trail information: www.dec.ny.gov\/outdoor\/9200.html; emergency contact: 518-891-0235\n\nOverview\n\nAnother one of the short hikes featured in the book, Jockeybush Lake promises a varied experience, from streamside hiking and stream crossings to lakeside camping and even a small waterfall. As with other short hikes, this one promises a great experience for beginning hikers and a pleasant trip for the whole family.\n\nRoute Details\n\nParking for Jockeybush Lake is a pulloff on the west side of NY 10, just across from Lake Alma. Essentially a pond, Lake Alma has a large painted sign with a green pine tree that is clearly visible when approaching from the south. Coming from the north, keep an eye out for the Department of Environmental Conservation trail signs.\n\nThe trail begins in the northwest corner of the pulloff, and the trail register is a few yards along the path. The trail is well marked with blue trail disks and has no intersections, so finding your way is straightforward. Roughly 0.3 mile in, you cross a seasonal stream that has a few mucky spots, and you will likely have to find a crossing off the trail. You will soon see Jockeybush Outlet flowing on your left and will cross to its southern bank across a rocky wash. During summer this crossing will barely break your stride, but during times of high water, you may have to wade. Shortly after crossing the outlet, you pass a small waterfall on your right that feeds into the outlet. The cascade weaves its way down moss-covered ledges into a verdant jumble of rocks. You can rock-hop around and eventually scramble to the top of the cascade some dozen feet above the outlet to see another tiny cascade.\n\nAfter the short diversion at the falls, the trail weaves its way along the southern banks of the outlet for the rest of the way to Jockey\u00adbush Lake, almost without incident. The one exception is a long muddy section, about a mile in, that fills the entire trail. You'll have to either traverse the muck or bypass along one of the newer footpaths that hikers have made. As is the case with most trails, it is best to minimize your impact on the surrounding forest by following a path already worn by previous hikers than tramping on undisturbed areas. Soon after the mire, you will find yourself on the eastern end of Jockeybush Lake. The lake is deep, as far as Adirondack lakes are concerned, but it's not readily apparent at this end as it is shallow here. Bushwhacks along either shore reveal that the lake is deeper than it first appears. At this end of the lake, a logjam forms a dam, as well as a means to the opposite bank of the outlet, where a designated campsite is situated in a small stand of hemlocks. The campsite has admirable views west along the fingerlike lake and is an ideal spot to picnic during a day trip. Numerous footpaths diverge from the end of the trailhead and campsite, promising other vantage points and even a prime swimming spot, but the trail described ends here. Return the way you came for a round-trip of 2.3 miles.\n\nDirections\n\nFROM THE SOUTH From the intersection of NY 29A and NY 10, north of Caroga Lake near Pine Lake Park, head 9.6 miles north on NY 10. The parking area is on the left, opposite Lake Alma, and has a large brown-and-white sign.\n\nJOCKEYBUSH LAKE AS SEEN FROM THE CAMPSITE ON ITS EASTERN TIP\n\nFROM THE NORTH From the intersection of NY 8 and NY 10 in Piseco, head 7.9 miles south on NY 10. The parking area is on your right, opposite Lake Alma.\n\n4\n\nGroff Creek\n\nUPPER PORTION OF GROFF CREEK\n\n**SCENERY:**\n\n**TRAIL CONDITION:**\n\n**CHILDREN:**\n\n**DIFFICULTY:**\n\n**SOLITUDE:**\n\nGPS COORDINATES: N43\u00b0 18.823' W74\u00b0 15.402'\n\nDISTANCE & CONFIGURATION: 5.0-mile out-and-back\n\nHIKING TIME: 3\u20134 hours\n\nHIGHLIGHTS: Two unique waterfalls\n\nELEVATION: 918' at trailhead, 1,322' at highest point\n\nACCESS: Open 24\/7; no fees or permits required\n\nMAPS: National Geographic Adirondack Park, Northville\/Raquette Lake (#744)\n\nFACILITIES: None\n\nWHEELCHAIR ACCESS: No\n\nCONTACTS: Silver Lake Wilderness: www.dec.ny.gov\/lands\/100874.html; Central and Southern Adirondack trail information: www.dec.ny.gov\/outdoor\/9200.html; emergency contact: 518-891-0235\n\nOverview\n\nThe two unique waterfalls along Groff Creek make this a must-\u00adexperience trip for waterfall lovers. Though technically a bushwhack (the trail is unmarked), it is very easy to follow all the way to the marsh at the end.\n\nRoute Details\n\nThe trail is at the end of the maintained section of River Road. The trail continues along this road briefly, but a sign warns that the road farther on is abandoned and you are proceeding at your own risk. Park along the pullout to the west of this sign, and make your way north along the broad, flat road. The road parallels and provides pleasant views of the Sacandaga River. At 0.2 mile, you will pass a designated camping area on your right that is encircled by tall pines and provides ample room to set up. Views of the river are limited from this site. Shortly past the designated campsite, you will see the remains of an old stone foundation on your left. A little more than 0.3 mile along the road, pass the posts for an old vehicle barrier and then a small white building on your left. A wooden bridge crosses Pete's Creek just past this building. Approximately 0.3 mile after the bridge, you reach an intersection in the road. Continue straight on the right-hand side of the fork another 0.1 mile, where you reach Forest Preserve lands.\n\nYou will pass a pulloff along the road on your left and then a small parking area at 0.8 mile. This marks the end of the road and the farthest point reached by vehicle if you should chance the abandoned road. The trail to your left still follows an old logging road, but nature has reclaimed much of it, and only a narrow footpath remains. The trail begins to climb for the next 0.5 mile as you work your way to the north side of Groff Mountain. The terrain falls off steeply on your right down to Groff Creek, and about 1.8 miles from the start, you will glimpse the first waterfall along the creek. There is no clear path down to view the waterfall, so pick your way carefully down the hillside and through the tall hemlocks to the top of the falls. The banks of the creek surrounding the waterfall are extremely steep on this side, but if you make your way to the top of the falls, you can cross to the other side and then zigzag your way down to the base. The nearly 30-foot vertical drop ends in a shallow pool, which is surrounded on three sides with vertical walls. Inside this natural amphitheater, the tumble of the falls drowns out all other sounds.\n\nThe second waterfall along this section of the creek is 0.1 mile farther west along the trail. Unlike the previous waterfall, this one is visible from the trail. Similar to the first one, no distinguishable path leads to it, though the route down is less steep. A frothy cascade, this waterfall steps its way down a series of ledges to a shallow pool as well. Downstream are a series of flumes and cascades that add waterscapes of their own.\n\nWhile exploring these falls, you are in prime habitat of the red-spotted newt. The adult newt is olive green with red spots along its back, but you are far more likely to encounter and recognize it in its terrestrial juvenile stage. The juvenile, known as a red eft, is brilliant orange with red dots and is common in woodlands near creeks and streams.\n\nSECOND WATERFALL ALONG GROFF CREEK\n\nThe trail continues another 0.5 mile west to where the creek levels off and a marsh forms. It continues along the south side of the marsh, though where it truly ends is hard to discern. Another set of waterfalls is said to be found on the opposite shore if you follow a tributary stream around a major bend. This would require serious bushwhacking, as there is no clear path. I attempted to find them during a heavy rain but soon decided to leave their discovery for another expedition. Return back along the path for a total trip length of 5 miles.\n\nDirections\n\nFROM THE SOUTH From the intersection of NY 30 and NY 30A in Gloversville, head 13.6 miles north on NY 30, and turn left onto County Road 6\/Benson Road. Within a couple hundred feet, turn right onto River Road. Follow River Road north 4.6 miles to where the signs warn that the road is abandoned. Park to the left of the road.\n\nFROM THE NORTH From the intersection of NY 30 and NY 8 in Wells, head 15.8 miles south on NY 30, and turn right onto CR 6\/Benson Road. Within a couple hundred feet, turn right onto River Road. Follow River Road north 4.6 miles to where the signs warn that the road is abandoned. Park to the left of the road.\n\n5\n\nTenant Creek Falls\n\nTHIRD FALLS ALONG TENANT CREEK\n\n**SCENERY:**\n\n**TRAIL CONDITION:**\n\n**CHILDREN:**\n\n**DIFFICULTY:**\n\n**SOLITUDE:**       (second and third falls)\n\nGPS COORDINATES: N43\u00b0 20.792' W74\u00b0 11.404'\n\nDISTANCE & CONFIGURATION: 4.6-mile out-and-back\n\nHIKING TIME: 3\u20134 hours\n\nHIGHLIGHTS: Three unique waterfalls\n\nELEVATION: 1,054' at trailhead, 1,340' at highest point along the trail\n\nACCESS: Open 24\/7; no fees or permits required\n\nMAPS: National Geographic Adirondack Park, Lake George\/Great Sacandaga (#743)\n\nFACILITIES: Pit toilet\n\nWHEELCHAIR ACCESS: No\n\nCOMMENTS: In 2016 the trail was rerouted to avoid a section of trail that crossed private property. December 1\u2013April 1 the last 1.4 miles of Hope Falls Road is closed. To access the trail during these months, park in the pulloff area near the barrier gate and hike Hope Falls Road roughly 1 mile to the new trailhead.\n\nCONTACTS: Eastern Adirondack trail information: www.dec.ny.gov\/outdoor\/9199.html; emergency contact: 518-891-0235\n\nOverview\n\nReaching the first of these three beautiful and distinct waterfalls is relatively easy. Its cascading waters over exposed rock faces are impressive and sure to please. Finding the second and third falls requires a bit of bushwhacking along a maze of paths, but the two serene and Zen-like waterfalls are excellent compensation.\n\nRoute Details\n\nThe trail, marked with yellow trail markers, begins on the western edge of the parking area and climbs slightly northeast to the trail register. From here until the intersection with the old trail at Tenant Creek, 0.8 mile ahead, the path winds around the northeastern contours of Rand Mountain. It adds roughly 0.25 mile of length, 200 feet of climb, and 100 feet of descent to the trail than the previous route, but it's not particularly challenging and makes reaching the first falls just a touch more rewarding. Roughly halfway along the way to the old trail, pass a giant boulder with hemlock growing atop it. The boulder and long sinuous roots of the hemlock are unmistakable and a great landmark to and from the falls. Shortly after passing the boulder, the climbing ends, and you follow a brief flat section and then begin the descent to Tenant Creek. After a looping switchback, rock-hop across a feeder brook and follow this tributary northward to intersect the old trail that wends along Tenant Creek. By now, the roar of the falls is distinct and draws you forward through a coppice of hemlocks to the falls. You will likely leave the trail to scrabble among the myriad paths that weave down to the base of the falls.\n\nTumbling over a sloping rock face, water falls 40 feet into a large basin at the foot of the falls. Tall hemlocks frame the pool, and numerous boulders along its rocky shore provide places to stop and contemplate the scene. The marked portion of trail ends at the top of the falls, and this is likely as far as many visitors come because the trail\u2014or more accurately, path\u2014is less distinct and practically disappears on occasion as it winds upstream. However, it's hardly a problem, as your destination is along this creek. If you follow it upstream and keep it on your left, you will find the second and third falls.\n\nScramble back up to the top of the falls, and begin to pick your way along the creekbank. The transition from trail to multiple paths is at its worst at this point, so stay close to the creek and you will eventually find the trail again. Though it is not difficult, you will have to be nimble to pick your way through the rocks and tangle of roots. The creek forms a small bulge, or horseshoe, around 1.4 miles, after which the flow levels off and the banks become grassy. Around 1.5 miles you will cross a tiny stream, and then the trail climbs slightly uphill away from the creek. The trail follows an elevated ridge briefly and then descends to the creek again. After crossing and recrossing a tiny branch of the creek, you come out on a ledge across from the second falls, which cascade into a broad pool before you. I noticed many salamanders in the pool around the remains of crayfish.\n\nThe third falls is a few hundred yards upstream, with the best view from a sloped boulder next to the trail. Both falls are small, but the sheer rock ledges and encroaching forest make each feel wild and isolated. The pools at the bases are deep and enclosed, making excellent spots to take a swim. The roaring of the falls in their mini amphitheaters resonates deeply and drowns out the sounds of the surrounding forest. It is very easy to be drawn in and enchanted by these serene and wild places, so plan to spend some extra time to enjoy the solitude.\n\nTREE GROWING ATOP ROCK NEAR FIRST FALLS\n\nDirections\n\nFrom the intersection of County Road 6\/Benson Road and NY 30 in Northville, head north along NY 30. In 0.8 mile, after crossing the Sacandaga River, turn right onto CR 15\/Old Northville Road. Continue 1.4 miles, and turn left onto Hope Falls Road\/CR 7 (dirt road). Follow the road past several designated primitive campsites to the parking area at 7 miles.\n\n6\n\nHadley Mountain\n\nVISTA FROM THE SUMMIT OF HADLEY MOUNTAIN\n\n**SCENERY:**\n\n**TRAIL CONDITION:**\n\n**CHILDREN:**\n\n**DIFFICULTY:**\n\n**SOLITUDE:**\n\nGPS COORDINATES: N43\u00b0 22.421' W73\u00b0 57.033'\n\nDISTANCE & CONFIGURATION: 3.6-mile out-and-back\n\nHIKING TIME: 2\u20133 hours\n\nHIGHLIGHTS: Panoramic views, accessible fire tower\n\nELEVATION: 1,144' at trailhead, 2,675' at highest point\n\nACCESS: Open 24\/7; no fees or permits required\n\nMAPS: National Geographic Adirondack Park, Lake George\/Great Sacandaga (#743)\n\nFACILITIES: None\n\nWHEELCHAIR ACCESS: No\n\nCONTACTS: Eastern Adirondack trail information: www.dec.ny.gov\/outdoor\/9199.html; emergency contact: 518-891-0235\n\nOverview\n\nA short and steady climb to the top of this 2,680-foot bald summit provides great views of the southern Adirondacks. With a climbable fire tower; summertime summit steward; and broad, open mountaintop ideal for a picnic, Hadley Mountain is a great choice for family outings.\n\nRoute Details\n\nThe trailhead parking area accommodates dozens of vehicles, but the trail's popularity means that you will likely have to park on the side of the dirt access road. The trailhead is clearly marked with a painted Department of Environmental Conservation (DEC) sign, as well as a historical marker detailing the story of the fire tower. Several fires at the opening of the 20th century swept the area, and most of the forest, including the topsoil, was burned off. A wooden tower was erected in 1916, according to a placard, but other sources date it to 1917. After it blew over in severe winds, a steel tower with guylines took its place in 1920. Because the risk of forest fires in the Adirondacks is generally negligible, the use of fire towers for fire suppression has given way to aerial observation; consequently, the tower, cabin, and trail fell into disrepair. However, within the past couple of decades, the Hadley Mountain Fire Tower Committee has restored the tower, cabin, and trail, and the old observer's cabin is now manned daily from the Fourth of July until Labor Day by a tower steward. According to the DEC Unit Management Plan, the steward is also available on weekends through Columbus Day.\n\nThe beginning of the trail is surrounded by a thick stand of hemlocks that continues to provide shade up to the trail register, where the evergreens give way to a mainly beech and birch forest that is thick with saplings and shades the trail for most of the climb. The trail, marked by red DEC markers, begins to ascend immediately and continues steadily uphill for nearly 1 mile, where it levels off for a while before the final climb to the summit. Over this first section, large rocks and cobbles are strewn across many portions, while other parts proceed up bare bedrock. Chances are that you will need to step off the trail frequently, as many hikers will be either ascending or descending. The trail is so clear that directions are hardly needed.\n\nUpon reaching the ridgeline, views to the southeast glimpsed through the canopy hint at the truly panoramic views that await you shortly ahead. Along the ridgeline, the flora is mostly scrubby trees or saplings, and the trail is largely clear of the rocks and cobbles that previously covered the path. You will pass through a small notch in the bedrock, followed by a sharp turn left before the last climb to the summit. The mountaintop is almost entirely exposed bedrock with many places to sit and enjoy the views. In fact, when I reached the summit, there were more than a dozen families or groups, with ample room for people to spread out to enjoy an afternoon picnic. Panoramic views in all directions are available, and you can see even farther into the surrounding country if you climb the fire tower. To the southwest, you can see Great Sacandaga Lake; to the north, some of the famous High Peaks; to the south, glimpses of the Catskills; and to the east, the foothills surrounding Lake George and even the Green Mountains of Vermont or the Berkshires in Massachusetts. A steady breeze helps to keep the infamous black flies at bay, and with the spectacular views, it is clear why this is a popular picnic spot for families.\n\nNEAR THE SUMMIT OF HADLEY MOUNTAIN\n\nDirections\n\nFrom the intersection of NY 9N\/West Maple Street and Main Street in Corinth, head 5.1 miles north on NY 9N, crossing the Hudson River in Hadley, and turn left onto School Street in Lake Luzerne. Continue west onto Bridge Street. Cross the Hudson River again, and keep left onto County Road 4\/Rockwell Street. Turn right onto CR 1\/Stony Creek Road, approximately 0.5 mile from your turn onto School Street. Head north on CR 1, and veer left at 3.1 miles onto Hadley Hill Road. Continue along Hadley Hill Road 4.3 miles, and turn right onto Tower Road (dirt road). Continue 1.4 miles north along Tower Road; the trailhead parking area is on the left.\n\nTHE FIRE TOWER ATOP HADLEY MOUNTAIN\n\n7\n\nCrane Mountain\n\nCRANE MOUNTAIN POND\n\n**SCENERY:**\n\n**TRAIL CONDITION:**\n\n**CHILDREN:**\n\n**DIFFICULTY:**\n\n**SOLITUDE:**\n\nGPS COORDINATES: N43\u00b0 32.224' W73\u00b0 58.120'\n\nDISTANCE & CONFIGURATION: 4.6-mile loop\n\nHIKING TIME: 5\u20136 hours\n\nHIGHLIGHTS: Panoramic view, bald summit, pristine pond\n\nELEVATION: 2,078' at trailhead, 3,254' at highest point\n\nACCESS: Open 24\/7; no fees or permits required\n\nMAPS: National Geographic Adirondack Park, Lake George\/Great Sacandaga (#743)\n\nFACILITIES: None\n\nWHEELCHAIR ACCESS: No\n\nCONTACTS: Eastern Adirondack trail information: www.dec.ny.gov\/outdoor\/9199.html; emergency contact: 518-891-0235\n\nOverview\n\nCrane Mountain is one of the most popular hikes in the southern Adirondacks: with its spectacular views and beautiful mountain pond, it is easy to see why. The climbing is steep, the descent bone-jarring, and the trail varied. It is a wonderful hike, but don't expect privacy, as dozens of groups visit the peak each day.\n\nRoute Details\n\nAfter the long, steep drive up the mountain along Ski Hi Road, you might begin to wonder how much climbing is left. The short answer is: plenty. The trail climbs more than 1,100 feet in less than 0.8 mile. The climb is nearly vertical in sections, and you will have to use both hands in many places. Though it's certainly within the grasp of most hikers and many children, those with bad joints or balance won't find the climb or descent very enjoyable.\n\nHead to the north end of the parking area, where you'll find the trail register a short distance up the trail on your left. Keep right at the fork (to your left is the return leg), and begin climbing among mixed hardwoods. Approximately 0.25 mile in, the vertical climb begins, and you have to pick your way over rough rocks and boulders. Because of the popularity of the trail, you will likely encounter other hikers along your way. There are many places to step off the trail to let others pass, and many of them are outlooks with outstanding views. At approximately 0.5 mile, the trail swings to the right, and you come out onto an open portion of ledge. Looking straight up the ledge, you will see a rock cairn that indicates where you pick up the trail again. At 0.7 mile, you come to a T; to your left is Crane Mountain Pond, while the right leg is the Crane Mountain summit. Head right toward the summit. The trail leads down, over a small hillock ahead, and then down again to the base of another vertical climb. At 0.8 mile, you come to the first of two wooden ladders that assist hikers over impassable vertical sections. After you climb the short ladder, the trail levels off briefly as it winds through stunted evergreens along the base of the summit's cliff. Chances are that you will see or, at the very least, hear other hikers who are sitting at the mountaintop. After heading east briefly, the trail switchbacks north, and you come to some steeper sections before the summit. A much longer ladder at 1.2 miles signals that the climb is nearly over. Upon reaching the summit at 1.4 miles, you will see numerous open sections along the cliff's edge. Hikers congregate here to enjoy the views and rest after the steep climb. There are plenty of places to rest, so if a section is occupied by another group, continue along the trail, and you will surely find a spot to yourself.\n\nAlong the peak, the trail weaves in and out of evergreens to the various lookouts along the cliff's edge. Many of these lookouts are visible from previous points, so you might want to leave someone behind to take a dramatic picture.\n\nThe climb down to Crane Mountain Pond is very steep and slippery and has the added hazard of many blown-down pines across the trail. Their whorls of broken branches make a veritable barricade of sharp stakes, so take your time making your way along this descent. The trail begins to level off when you reach a hardwood forest dominated by birches and maples. At 2.3 miles, you will come to a fork in the trail; straight ahead is a short path to an exposed boulder that sits above a wetland on Crane Mountain Pond's southeastern shore. From the fork, the main trail heads to your left and, after a brief foray into the woods, descends to the pond's shore. A fallen tree hides the trail along the shore, but undoubtedly you will have walked to the shore's edge to admire the view. Directly across a tiny bay in the pond is a sloping ledge that gradually enters the water. Hikers also congregate here, as it is popular for swimming. To reach this ledge, continue left along the shore through a small wet area and then though a dense coppice of cedars. Dragonflies will likely dive-bomb you through this section and as you step out onto the ledge. The spur back to the fork, at 0.7 mile, is near this sloped ledge. You can head back along it and then down the initial ascent to the parking area, if you choose not to follow the rest of the trail.\n\nSTEEP SECTION OF TRAIL ALONG DESCENT\n\nThe trail continues to wind along the pond's shore and, after passing a small rocky bay on your right, crosses Putnam Brook. An arrow indicates that the trail continues left, but no mileage or details are given. The footpaths that head off in other directions are angler's paths or bushwhacks. Head west along the main trail, recross Putnam Brook, and come out onto a large, open portion of bedrock. There is no indication as to where the trail picks up again across this open rock face, but if you stick to the northern edge, next to Putnam Brook, you will find it. The trail begins to descend steeply again; be careful and take plenty of time along this stretch, as it is steeper\u2014if you can believe it\u2014than the ascent. The trail is very rocky, and you will likely need to crab-walk or scoot down several sections. Impressive views open up all over to the southwest, but keeping your eyes on the trail should be your main concern. The true path is often hard to discern because you traverse long sections of ledge or hop exclusively from boulder to boulder, both of which mask the footprints of previous hikers. Don't worry though; red Department of Environmental Conservation disks are clearly visible on the many maple trees that shade the trail. After 0.5 mile of descent, the trail levels off, and you will notice a small wetland to your left, around 3.2 miles. Shortly after, cross over the stream that flows out of this wetland along a natural bridge. To either side of the bridge, you will see deep holes. Venturing to the rims of these holes reveals a water-carved cave underneath the bridge. Shortly after crossing, you descend to a feeder stream of Putnam Brook, as well as an intersection with the overgrown Crane Mountain Road. To your right, the road crosses the stream and enters private property, while to your left is a vehicle barrier and the return trip to the parking area. The abandoned road climbs steadily but gently through a mixed hardwood forest back to the parking area. The last mile of the hike is easygoing and free of obstructions, a welcome break from the more difficult ascent and descent. At 4.3 miles, you come to a fork where the road continues straight ahead and a trail heads left. The trail leads to the register and north end of the parking lot. The road intersects Ski Hi Road a few hundred feet south of the parking area.\n\nDirections\n\nFrom the intersection of NY 28 and NY 8 in Wevertown, head 1.5 miles southwest on NY 8. Turn left onto South Johnsburg Road, and go 6.8 miles. In the hamlet of Thurman, turn right onto County Road 72\/Garnet Lake Road. Proceed 1.2 miles, and turn right onto Ski Hi Road. Ski Hi Road is a steep, rough dirt road, but it is maintained and drivable by most cars. Follow Ski Hi Road 1.8 miles, and bear right at the vehicle barrier. The road terminates at the parking area 0.1 mile past the barrier.\n\nWest-Central\n\nSEASONAL STREAM ON THE WAY TO WOODHULL LAKE (Trail 10, Bear and Woodhull Lakes)\n\n8 PANTHER MOUNTAIN AND ECHO CLIFFS\n\n9 MITCHELL PONDS\n\n10 BEAR AND WOODHULL LAKES\n\n11 BLACK BEAR MOUNTAIN\n\n12 BUBB, SIS, AND MOSS LAKES\n\n13 CASCADE LAKE\n\n14 GLEASMANS FALLS\n\n8\n\nPanther Mountain and Echo Cliffs\n\nATOP ECHO CLIFFS\n\n**SCENERY:**\n\n**TRAIL CONDITION:**\n\n**CHILDREN:**\n\n**DIFFICULTY:**\n\n**SOLITUDE:**\n\nGPS COORDINATES: N43\u00b0 24.689' W74\u00b0 33.468'\n\nDISTANCE & CONFIGURATION: 1.4-mile out-and-back\n\nHIKING TIME: 1.5 hours\n\nHIGHLIGHTS: Panoramic views\n\nELEVATION: 1,736' at trailhead, 2,420' at highest point\n\nACCESS: Open 24\/7; no fees or permits required\n\nMAPS: National Geographic Adirondack Park, Northville\/Raquette Lake (#744)\n\nFACILITIES: None\n\nWHEELCHAIR ACCESS: No\n\nCONTACTS: Central and Southern Adirondack trail information: www.dec.ny.gov\/outdoor\/9200.html; emergency contact: 518-891-0235\n\nOverview\n\nDefinitely the shortest trail described in this book, Panther Mountain and Echo Cliffs are not short on scenery. With less than a mile to the scenic lookout, hikers are quickly rewarded with panoramic views of Piseco Lake. The scenic vista, combined with some stunning cliffs beside the trail, makes it an excellent excursion in one short package. Its popularity in the summer attests to its beauty, so plan on going in the off-season or early morning hours to avoid the crowds.\n\nRoute Details\n\nParking for the Panther Mountain and Echo Cliffs Trail is on the eastern side of County Road 24, just south of the entrance to Little Sand Point Campground. Piseco Lake actually has three campgrounds along its shores; I recommend Best Tent Camping: New York State for further descriptions of the campgrounds. Located on the boundary between the west-central and southern regions and fairly close to the central region, Piseco Lake makes a great base camp from which to explore the Adirondack foothills.\n\nThough the trail is often described as Panther Mountain, your actual destination is the ledge along Echo Cliffs. The ledge is 2,420 feet in elevation, 300 feet less than the Panther Mountain summit. The mountain apparently got its name from the abundance of mountain lions, or panthers, in the area in the early 1800s. The town board of Arietta consequently offered a bounty of $20 to purge the area of panthers in 1837. Also known as cougars, the large predators have been absent from New York and the Adirondacks since the 1800s. Rumors abound regarding recent sightings throughout the Adirondacks, and the rumor mill has gone so far as to claim that the Department of Environmental Conservation (DEC) is releasing cougars to curtail the deer population. The rumors have spread wildly through online chat rooms and even sprouted YouTube videos suggesting a conspiracy. The DEC has refuted the claims numerous times and even created a Web page to clarify the issue. The official stance is that a self-sustaining population of cougars does not exist in New York, that sightings are almost always cases of mistaken identity, and that the extremely rare actual cougars are escapees or releases from private owners.\n\nThe parking area is a long pulloff that is about 50 yards long and could easily accommodate dozens of vehicles. A DEC trail sign with a blue disk marks the trailhead on the opposite side of the road.\n\nThis trail is ideal for novice hikers, children, or simply a quick climb for a scenic view. Well worn and clearly marked, the path is easy to follow. The trail, marked with blue disks, is obviously popular; heavy use has turned the path into a wide swath along the forest floor. Under the dense canopy of birch, maple, and beech trees, the trail is mostly level as you head generally northwest, but it gradually becomes steeper after you pass through a jumble of angular boulders and begin to head west.\n\nApproximately 0.6 mile in, you pass a large square boulder on your left. After the square boulder, the trail turns north, and you soon pass under sheer rock walls to your left. The trail is much steeper along this section, but the climb is brief, and after you pick your way up the roots of a hemlock tree that grows on the right side of the trail, you are rewarded with views of Piseco Lake. The lookout is the top of Echo Cliffs, and while the summit of Panther Mountain is behind you to the northwest, no clear path leads to it. The long, exposed outcrop provides a panoramic view of most of Piseco Lake, including Higgins Bay and Spy Lake to the east and directly across the lake, as well as portions of Irondequoit Bay, Big Bay, and the Piseco outlet to the south.\n\nDirections\n\nFROM THE WEST From the intersection of NY 8 and NY 365 in Cold Brook, head 19.4 miles east on NY 8, and turn left onto County Road 24\/Old Piseco Road. Continue 2.6 miles, and look for the pulloff on your right. If you reach Little Sand Point Campground, you are 0.5 mile too far north.\n\nFROM THE EAST From the intersection of NY 8 and NY 30 in Speculator, head 8.8 miles west on NY 8, and turn right onto CR 24\/Old Piseco Road. Follow Old Piseco Road around the northern tip of Piseco Lake, and look for the parking area on your left, 0.5 mile south of Little Sand Point Campground, 5.4 miles along CR 24\/Old Piseco Road.\n\n9\n\nMitchell Ponds\n\nACCESS ROAD ALONG TRAIL TO MITCHELL PONDS\n\n**SCENERY:**\n\n**TRAIL CONDITION:**\n\n**CHILDREN:**\n\n**DIFFICULTY:**\n\n**SOLITUDE:**\n\nGPS COORDINATES: N43\u00b0 40.589' W74\u00b0 42.490'\n\nDISTANCE & CONFIGURATION: 5.6-mile out-and-back\n\nHIKING TIME: 2\u20133 hours\n\nHIGHLIGHTS: Lakeside hiking, beaver dam\n\nELEVATION: 1,839' at trailhead, with no significant rise\n\nACCESS: Open 24\/7; no fees or permits required\n\nMAPS: National Geographic Adirondack Park, Old Forge\/Oswegatchie (#745); National Geographic Adirondack Park, Northville\/Raquette Lake (#744)\n\nFACILITIES: None\n\nWHEELCHAIR ACCESS: No\n\nCONTACTS: Moose River Plains Wild Forest: www.dec.ny.gov\/lands\/53596.html; Central and Southern Adirondack trail information: www.dec.ny.gov\/outdoor\/9200.html; emergency contact: 518-891-0235\n\nOverview\n\nNestled in the middle of the Moose River Plains Wild Forest, the trail to Mitchell Ponds is an easy stroll along an old logging road, followed by a quick jaunt along the lake. The trail follows mostly level ground and is easy to follow, so expect to make good time. Trail conditions also make it a great introduction to hiking.\n\nRoute Details\n\nTo reach Mitchell Ponds, you have to enter the Moose River Plains Wild Forest. Purchased in 1963 from the Gould Paper Company, the 50,000-acre block is bisected by a network of rough roads and myriad roadside campsites\u2014more than 170 mostly primitive sites. The roads are gravelly and filled with potholes and other hazards characteristic of lightly maintained backwoods roads, so a four-wheel-drive, high-clearance vehicle is recommended. Registration booths are at the entrances to the area, and the whole road system is open to vehicles from Memorial Day to the end of hunting season. It is a popular destination during hunting season, and you might think that more vehicles are in the recreation area than on the main roads.\n\nThe parking area is essentially an access road, but there is room for a handful of vehicles on either side of the road. The trail, marked with yellow disks, is a continuation of this access road and heads nearly due west. After a few hundred feet, you pass a barrier gate. The trail is mostly level, and because it is an old access road, you can easily walk two abreast. The trail is free from obstructions, so you can easily observe the surrounding landscape as you make good time along the road. At 1.4 miles, you will see a small stream and wetland on the left, though most of the trip is through dense woods.\n\nAt 1.8 miles, you reach a clearing and a fork in the trail. Straight ahead is a designated campsite on the eastern tip of Mitchell Ponds. It has a picnic table and beautiful views of the water, making it an ideal stopping point even if you don't plan to camp.\n\nBack at the intersection, the trail continues to your right through a very wet section and over a deteriorating bridge. Just on the other side of the bridge is a second intersection on your left. Straight ahead is the trail that leads to the north access point on Limekiln Lake\u2013Cedar River Road, while the trail along the shores of Mitchell Ponds is on your left. From this point on, the path is no longer a wide forest road but a typical Adirondack foot trail. It weaves along Mitchell Ponds' northern shore, providing views and a tranquil setting. At 2.2 miles, you reach an old fire pit in the midst of the trail and an old sign reminding you of the carry-in, carry-out policy that is in place for the entire Adirondack Park. Past the fire pit, the trail becomes more rugged, narrower, and a little harder to make out, but there is nothing that exceeds the skills of even novice hikers.\n\nShortly after the fire pit, the trail swings southwest as it weaves its way to the western tip of the ponds. A jumble of boulders fills the steep hillside to your right, forming many small caves and crevices. At 2.8 miles, you reach this tip and the end of the trail, where a beaver dam blocks the outflow of the pond. You can get a few views back on the ponds from the large rocks, but crossing the outlet would probably be wet and difficult. Return the way you came for a round-trip of 5.6 miles.\n\nMITCHELL PONDS\n\nDirections\n\nFrom NY 28 in Inlet, 0.8 mile east of the intersection of NY 28 and South Shore Road in Inlet and 9.8 miles west of the intersection of Sagamore Road and NY 28 in Raquette Lake, head southwest on Limekiln Lake Road\/County Road 14. Continue 1.9 miles on Limekiln Lake Road to the registration booth. From the registration booth, follow Limekiln Lake\u2013Cedar River Road east and south 8.1 miles. The parking area is on your right.\n\n10\n\nBear and Woodhull Lakes\n\nREMSEN FALLS\n\n**SCENERY:**\n\n**TRAIL CONDITION:**\n\n**CHILDREN:**\n\n**DIFFICULTY:**\n\n**SOLITUDE:**\n\nGPS COORDINATES: N43\u00b0 36.749' W75\u00b0 05.429'\n\nDISTANCE & CONFIGURATION: 10.2-mile loop, 1.2-mile out-and-back side trip\n\nHIKING TIME: 5\u20137 hours\n\nHIGHLIGHTS: Lakeside hiking, waterfall\n\nELEVATION: 1,547' at trailhead, 2,003' at highest point\n\nACCESS: Open 24\/7; no fees or permits required\n\nMAPS: National Geographic Adirondack Park, Old Forge\/Oswegatchie (#745)\n\nFACILITIES: None\n\nWHEELCHAIR ACCESS: Yes, along Remsen Falls side trip\n\nCOMMENTS: A newly constructed lean-to is available at Bear Lake, in addition to the popular lean-to at Woodhull Lake.\n\nCONTACTS: Central and Southern Adirondack trail information: www.dec.ny.gov\/outdoor\/9200.html; Black River Wild Forest: www.dec.ny.gov\/lands\/75310.html; emergency contact: 518-891-0235\n\nOverview\n\nMore than half of the loop to Bear and Woodhull Lakes includes gravel and dirt access roads, but these roads are closed to traffic or lightly traveled. The trail portions are truly excellent and include two lakes, a pond, and seasonal streams to hike beside. A side trip to Remsen Falls is easily accessible along the loop as well. The Remsen Falls side trip is wheelchair accessible, with a special parking area located closer to the falls.\n\nRoute Details\n\nThe trail network in the Woodhull Lake area is bisected by a gravel road that provides access to several designated roadside campsites as well as the wheelchair-accessible Remsen Falls picnic and camping area. If camping is part of your agenda, you may want to consider the roadside sites or backpack to the lean-to and designated sites by Woodhull Lake. Camping in other areas is permitted as long as you follow the basic backcountry camping guidelines laid out by the Department of Environmental Conservation (DEC). The best parking is available in the McKeever parking area at the end of McKeever Spur Road. Hiking counterclockwise around the loop brings you into the wilderness soonest and leaves the easier roadside hiking for the return trip.\n\nFrom the parking area, head southeast past the barrier gate along the gravel road. The road climbs gently but is free of obstruction, so you quickly cover the first 0.7 mile and reach the first designated campsite. You pass a second campsite within a couple hundred feet, a bridge at 0.8 mile, and yet another campsite at 1.1 miles. You reach the beginning of the foot trail on your right at 1.3 miles. The trail sign indicates that to your right are Bear Lake, 1.7 miles, and Bear Creek parking area, 8.8 miles. Continuing straight, you reach Remsen Falls in 2.4 miles, Woodhull Mountain in 6.9 miles, and Woodhull Lean-To in 4.6 miles. This intersection is immediately before another bridge.\n\nHead south (right) on the blue-marked trail toward Bear Lake. The trail immediately traverses a flat, wet section that could easily swallow a boot or two. The trail climbs more than 300 feet over the next 0.5 mile through the familiar birch, beech, and maple forest of the western Adirondacks. You reach the crest of your climb at 1.9 miles in a saddle between two hills, where glimpses of Bear Lake are heavily obscured by the forest canopy but promise a picturesque destination. As you weave your way down the next 0.5 mile to the lake, steep ledges become exposed on your left. More than 40 feet high in sections, the ledges reinforce the rugged wilderness feel that hikers so frequently seek along the myriad trails of the Adirondacks. Depending on the season and water level, you might be privy to a tiny waterfall cascading along these ledges around 2.3 miles. Certainly not a frothing cascade, the rivulet breaks through a cleft in the ledge and is seasonal. The trail's descent ends abruptly on the shores of Bear Lake, which is admirably framed before you by a large black cherry tree. A little farther along the trail is an abandoned fire ring, where more exceptional views of the lake await.\n\nThe trail winds its way along the northern and then eastern shore of Bear Lake for the next 0.3 mile, where you encounter a feeder stream. After rock-hopping across the stream, you will find a trail intersection on the opposite shore. Straight along the blue-marked trail is Bear Lake Trail, 1.9 miles, and the Bear Creek parking area, 6.9 miles. To your left and marked in yellow are the Woodhull Lean-To, 2.4 miles, and Remsen Falls, 5.2 miles. Continue to your left (east) along the yellow-marked trail.\n\nThe trail parallels the stream for the next 0.3 mile as you begin to climb away from Bear Lake. While traversing this section of the trail, I encountered several colors of flagging tape riddled with notes. Because it was hunting season, I first suspected that it was the work of hunters trying to retrace their steps. After a while, the magnitude of the flagging and the consistent spacing showed that it was instead the work of rescue crews trying to find a lost hunter. Every year, hunters and hikers get lost in the woods, and many are not found. Upon returning to civilization, I learned that this case had a happy ending; the hunter was found miles away after spending two cold November nights alone in the woods. Not all are as fortunate, and it is a stark reminder that it is essential to communicate your plans and have regular check-ins.\n\nAs you near the crest of your climb, 4.3 miles, the trail swings south. This southern jog of 0.4 mile brings you close to the shores of Bloodsucker Pond, a small pond fed by a stream from Woodhull Lake. When you reach the pond and stream, 4.7 miles, the trail swings east again, and the stream accompanies you as you head east another 0.1 mile to a major trail intersection. To your right is a series of trails that takes you deeper into the Black River Wild Forest as well as to our major destination, the Woodhull Lean-To, at 0.3 mile. To your left are the return leg of the loop with Remsen Falls, 2.5 miles away, and the McKeever parking area, 5.4 miles. To reach the lean-to, turn right and head over the bridge into a giant mire, where the trail to the lean-to, marked in red, diverges to the left from the other trails that delve further into the Black River Wild Forest. The trail down to the lean-to is basically a long mire, so expect to be nimble and inevitably end up with muddy boots anyway. As you near the lean-to, you first notice a pit toilet and, if you're anything like me, can't help but wonder if future routes might avoid long mires that end at toilets. Regardless of your musings on trail routing, you reach the large lean-to on Woodhull Lake shortly after, 5 miles overall. It has obviously been worked on recently and is the equivalent of two lean-tos. Situated within 10 feet of the shore, it has wonderful views and is a great stopping point. But be forewarned, it is a popular destination and, due to its proximity to the road, likely occupied. Return through the mire to the intersection you encountered before the bridge to continue along the loop, 5.2 miles from the start.\n\nBOAT ON THE SHORE OF BEAR LAKE\n\nHeading north from the intersection, you will follow red snowmobile markers and quickly reach a vehicle barrier gate. The road you intersect continues to the right down to Wolf Lake Landing, but you will head to your left (north) toward Remsen Falls. Pass several campsites as you head north 1.1 miles along the road to the next major intersection. This intersection is just past two closely spaced campsites and is marked by a trail sign. The designated parking area for the wheelchair-access road to Remsen Falls is at this intersection. To the right is Remsen Falls, 0.6 mile, while continuing along the main access road to your left takes you back to the McKeever parking area in 3 miles. Even if you don't plan to take the side trip to Remsen Falls, a more pleasant return leg is along the red-marked trail.\n\nContinue to your right along the wheelchair-access road, marked in red, another 0.1 mile, where you reach another intersection. To your right is Woodhull Mountain, 4.5 miles; straight ahead is Remsen Falls, 0.5 mile; and to your left is the return trip to the McKeever parking area, 3.1 miles.\n\nSide Trip to Remsen Falls\n\n(Mileage along loop does not include this 1.2-mile trip.)\n\nThe side trip to Remsen Falls adds 1.2 miles to your total trip, but because it follows wheelchair-accessible paths, this diversion takes little time and effort. Heading almost due north along the wide, flat, and sandy access path, you quickly reach Moose River just above the falls. You can see a lean-to on the opposite shore, and you find a fire pit, picnic table, and designated camping area at the end of the trail. Continue along well-worn paths past the picnic table to get farther downriver to view the falls. When you reach a second picnic table, the falls are clearly visible. Less of a falls and more of a broad rapid, the tumble of water drowns out most surrounding sounds. Another fire pit is located farther downstream if you desire more privacy. Return the way you came to complete the loop.\n\nBack on the main trail, head west along the dirt access road, marked in red. This access road is closed to vehicles, though you might see mountain bikers, as this is a popular biking area. The return trip is mostly level and easygoing and passes without incident. You pass a large culvert at 8 miles, Bear Lake Trail on your left at 8.9 miles, a second culvert at 9.1 miles, and a trail register at 9.8 miles. This second register seems to be used primarily by people visiting the falls or heading straight to the mountain. Head past the barrier gate, where there is a large turnaround, and continue west to the McKeever parking area for a total trip length of 10.2 miles, 11.4 including the side trip.\n\nDirections\n\nFrom NY 28, turn east onto McKeever Road, 16.2 miles northeast of the intersection of NY 12 and NY 28 in Alder Creek or 10.4 miles southwest of Old Forge. Continue to your right onto McKeever Spur Road at 0.3 mile, past an old railroad station. Continue another 0.4 mile. The parking area is on your right and marked with DEC signs.\n\n11\n\nBlack Bear Mountain\n\nNEAR BLACK BEAR PEAK\n\n**SCENERY:**\n\n**TRAIL CONDITION:**\n\n**CHILDREN:**\n\n**DIFFICULTY:**\n\n**SOLITUDE:**\n\nGPS COORDINATES: N43\u00b0 45.808' W74\u00b0 47.572'\n\nDISTANCE & CONFIGURATION: 4.9-mile loop\n\nHIKING TIME: 3 hours\n\nHIGHLIGHTS: Bald mountain, scenic views\n\nELEVATION: 1,751' at trailhead, 2,432' at highest point\n\nACCESS: Open 24\/7; no fees or permits required\n\nMAPS: National Geographic Adirondack Park, Old Forge\/Oswegatchie (#745); National Geographic Adirondack Park, Northville\/Raquette Lake (#744)\n\nFACILITIES: None\n\nWHEELCHAIR ACCESS: No\n\nCOMMENTS: Be aware that the first section of the trail is a multiuse trail and mountain biking is becoming more popular in the region. The trail to the summit is hiking only.\n\nCONTACTS: Moose River Plains Wild Forest: www.dec.ny.gov\/lands\/53596.html; Central and Southern Adirondack trail information: www.dec.ny.gov\/outdoor\/9200.html; emergency contact: 518-891-0235\n\nOverview\n\nA popular destination in summer and fall, this tiny peak provides views of the Fulton Chain of Lakes. It is a great choice for children and beginner hikers, but if you wish for solitude, then plan to go during the off-season, especially winter.\n\nRoute Details\n\nFrom the southeast corner of the parking area, head south, paralleling the road. After passing a deteriorating concrete bridge on your left, you will quickly reach a sign indicating the trailhead to your left (east). The trail register is just inside the treeline. The trail\u2014marked in yellow hiking, skiing, and biking disks\u2014heads northeast through mixed hardwoods. Approximately 0.3 mile in, a stream becomes visible on your left. At 0.7 mile you reach the intersection with the return leg of the trip on your right. This trail, also marked in yellow, is a more direct and steeper ascent to the peak, but returning along this trail provides some great views along the descent.\n\nTo make the longer and more scenic loop, continue northeast along the left fork (east) through some muddy sections. At approximately 1 mile, you reach a clearing encircled by small firs, with rock cairns marking the path. Shortly after, you cross a couple of streams over log-and-plank bridges and will soon see the mountain looming off to your right. The grade along the north side of the mountain is relatively flat, and you soon reach another trail intersection at 2.3 miles, just before the halfway point. To your left are Uncas Trail, 0.6 mile, and Uncas Road parking area, 1.5 miles. To your right are the summit, 0.5 mile, and the return to NY 28, 2.6 miles.\n\nThe trail begins to climb immediately, and the forest transitions from maple and beech to evergreens fairly quickly. A quarter mile from the intersection, 2.5 miles from the start, you reach an opening in the evergreens and a lookout on your left. From here you have an incredible view of Seventh Lake, part of the Fulton Chain of Lakes.\n\nThe Fulton Chain is a series of connected lakes that begins at Old Forge Pond and ends with Eighth Lake. It is a popular destination for a wide variety of boaters and is the beginning of the North Forest Canoe Trail. In fact, First through Fifth Lakes compose an unbroken length of 14 miles. A series of channels separates Old Forge Pond and Sixth through Eighth Lakes from this main expanse. The connection of these lakes was part of steamboat inventor Robert Fulton's vision of an Adirondack Canal. Though the canal was never built, the chain still bears Fulton's name.\n\nReturning to the trail, the path winds in and out of stands of evergreens and over long spans of open ledge. At 2.8 miles the trail runs into a 4-foot ledge that you navigate to reveal another opening in the canopy and yet another lookout on your left. Head northeast into the dense evergreens to find the yellow-marked trail that quickly leads you to the summit about 0.1 mile ahead. The actual summit is hard to distinguish along the mostly flat, bald peak, but a series of small ledges to the left of the trail makes a great spot to dangle your feet and enjoy the view. To the south, you can see portions of the town of Inlet as well as Sixth and Seventh Lakes. To the west, you can make out portions of Fourth Lake.\n\nThe trail back to the parking area heads generally west but is now mostly marked with blue trail disks. The descent is steeper than the ascent but provides better views of Fourth Lake than any of the lookouts or even the summit. After you weave your way down the many small ledges, some 0.4 mile from the peak, the trail levels off, and you encounter a few more wet areas. At 3.7 miles the blue markers begin to thin, and yellow markers become more prevalent. After a short jog where the trail swings north, the return trail parallels the inbound trail awhile and, after descending along a broad road, you reach the first intersection at 4.2 miles. Hike the remaining 0.7 mile back to NY 28 and the parking area for a total trip length of 4.9 miles.\n\nTRAIL TO BLACK BEAR MOUNTAIN\n\nDirections\n\nComing from the west, from the junction of County Road 1\/Big Moose Road and NY 28 in Eagle Bay, head east on NY 28 and go 1.2 miles. The spacious parking lot is on your left. From the east, from the junction of County Road 1\/South Shore Road and NY 28 in Inlet, head north 0.9 mile along NY 28, and look for the parking area on your right. There are two entrances, so if you miss the first in either direction, simply enter the second.\n\n12\n\nBubb, Sis, and Moss Lakes\n\nBUBB LAKE\n\n**SCENERY:**\n\n**TRAIL CONDITION:**\n\n**CHILDREN:**\n\n**DIFFICULTY:**\n\n**SOLITUDE:**\n\nGPS COORDINATES: Sis and Bubb parking: N43\u00b0 46.001' W74\u00b0 50.675' Moss Lake parking: N43\u00b0 47.386' W74\u00b0 50.763'\n\nDISTANCE & CONFIGURATION: 7.0-mile balloon\n\nHIKING TIME: 3\u20134 hours\n\nHIGHLIGHTS: Lakeside hiking\n\nELEVATION: 1,741' at trailhead, 1,924' at highest point\n\nACCESS: Open 24\/7; no fees or permits required\n\nMAPS: Fulton Chain Map: www.dec.ny.gov\/lands\/75305.html; National Geographic Adirondack Park, Old Forge\/Oswegatchie (#745)\n\nFACILITIES: None\n\nWHEELCHAIR ACCESS: Portions of the trail and day area at Moss Lake are accessible.\n\nCONTACTS: Fulton Chain Wild Forest: www.dec.ny.gov\/lands\/75305.html; Central and Southern Adirondack trail information: www.dec.ny.gov\/outdoor\/9200.html; emergency contact: 518-891-0235\n\nOverview\n\nThese three pristine lakes offer a variety of activities and feature trails with varying levels of accessibility, making this an ideal trip for the entire family. The Moss Lake circuit, with its broad, flat trails and wheelchair-\u00adaccessible sections, provides an opportunity for the very young\u2014or those no longer interested in hiking rugged trails\u2014to enjoy the beauty of the Adirondacks. On the other hand, the Bubb and Sis Lakes section features rugged trails and remote lakes that are easily combined with the circuit for extended and more invigorating trips.\n\nRoute Details\n\nThere are a variety of ways to explore these pristine lakes. The Moss Lake circuit has long sections that are wheelchair accessible, and the majority of the trail is flat, broad, and easy to navigate. Two larger parking areas are also available near Moss Lake along Big Moose Road (also known as County Road 1). These parking areas should be used for extended stays, end-to-end hikes, and day use in the Moss Lake area because the Bubb and Sis parking area can realistically accommodate only five to six vehicles, and two of those spaces are designated handicapped parking. You can make it an end-to-end hike if you start at the Bubb and Sis trailhead along NY 28 and get picked up or meet at one of the Moss Lake parking areas along Big Moose Road. The trip described here includes taking a straight route past Bubb and Sis Lakes, then following the circuit around Moss Lake, and finally returning along the straight route between Bubb and Sis Lakes.\n\nThe trail, marked with yellow disks, begins in the northwest corner of the small parking area along NY 28. The trail climbs immediately, and after a few dozen yards you reach the trail register. The trail then swings north with a steep incline, and you quickly reach the intersection with the Vista Trail, marked in blue, on your left. Continue straight ahead (northwest), and the trail soon levels off. It is mostly level from this point on, and within 0.5 mile of the start, Bubb Lake comes into view. You encounter a minor fork just before the lakeshore. To the left is the main trail, while straight ahead are an excellent point from which to view the lake and a designated campsite nestled off to the right of the short path that leads to the lake.\n\nContinuing left (west) along the main trail, you'll cross a small stream as you skirt the southern edge of Bubb Lake. Within minutes you'll reach a stand of hemlocks along the edge of Sis Lake. In the height of summer, this is a very tranquil area, and the calls of loons can be heard throughout. From this grove of hemlocks, the trail swings north and passes an informal campsite on the left. This site has benches, a fire pit, and an informal path that follows the western edge of Bubb Lake. Just past this site is a boardwalk, 0.8 mile, that crosses the small wetland between the two lakes.\n\nOnce on the opposite shore, the trail swings east and follows the northern shore of Bubb Lake. After 0.8 mile you reach the northeastern tip of the lake and encounter yet another heavily used area to the right of the trail. There are several fishing regulation signs, and a bit farther east off the trail is a fire pit. The main trail veers to the left of this heavily used area, and at 1.6 miles you cross a small stream over a log-and-plank bridge and notice a wooden dam on the left. The trail becomes broader after this crossing and soon swings north as it heads to the Moss Lake circuit, which you reach at 2.3 miles. To the right is the most direct route to the parking areas along Big Moose Road and the day-use areas around the lake. Turn left and head clockwise around the circuit.\n\nThe trail around Moss Lake is broad and sandy and easily accommodates two or more abreast. It is a popular circuit for groups of walkers, so don't expect much solitude. The lake is hidden from view until you reach a bridge that crosses a stream and wetland that sprawls out to the west. Climb a short hill, and wind your way north along the western edge of the lake. At mile 3.1 cross another log-and-plank bridge, after which the trail swings northeast. As you near the eastern edge of the lake, you pass a road that intersects the trail on the left, 3.7 miles, and head into the day-use area of the lake. Dotting the eastern shore are myriad paths, picnic areas, and designated campsites, as well as an observation deck, that are more easily explored than described. The circuit path bypasses most of these areas and soon reaches the first of two parking areas. At the second parking area is a trail register, campsite register, historical information placard, and map of the circuit and campsites.\n\nContinuing along the main circuit, you'll pass side trails leading to a couple of campsites on your right and then enter an open field with a research station with tall fences off to the left. At 4.3 miles, you pass a side trail to the left that leads to Big Moose Road, followed by a bridge that crosses Moss Lake's main inlet. You soon reach the intersection with the Bubb and Sis Lakes Trail, 4.7 miles. Return the way you came for a total trip of 7 miles.\n\nDirections\n\nSIS AND BUBB LAKES PARKING AREA From the intersection of Crosby Boulevard and NY 28 in Old Forge, head northeast 7.9 miles along NY 28; the parking area is on the left. From the intersection of County Road 1\/Big Moose Road and NY 28 in Eagle Bay, head southwest 1.4 miles along NY 28; the parking area is on your right.\n\nMOSS LAKE PARKING AREA From NY 28 in Eagle Bay, head north 2.2 miles on CR 1\/Big Moose Road; the large parking area is on your left.\n\n13\n\nCascade Lake\n\nCASCADE LAKE\n\n**SCENERY:**\n\n**TRAIL CONDITION:**\n\n**CHILDREN:**\n\n**DIFFICULTY:**\n\n**SOLITUDE:**\n\nGPS COORDINATES: N43\u00b0 46.887' W74\u00b0 49.909'\n\nDISTANCE & CONFIGURATION: 5.9-mile balloon\n\nHIKING TIME: 3\u20134 hours\n\nHIGHLIGHTS: Lakeside hiking, tall waterfall, historic site\n\nELEVATION: 1,873' at trailhead, 1,940' at highest point\n\nACCESS: Open 24\/7; no fees or permits required\n\nMAPS: National Geographic Adirondack Park, Old Forge\/Oswegatchie (#745)\n\nFACILITIES: None\n\nWHEELCHAIR ACCESS: No\n\nCONTACTS: Pigeon Lake Wilderness Area: www.dec.ny.gov\/lands\/102484.html; Central and Southern Adirondack trail information: www.dec.ny.gov\/outdoor\/9200.html; emergency contact: 518-891-0235\n\nOverview\n\nNot to be confused with Cascade Lake in the High Peaks region, this scenic loop in the town of Inlet offers an excellent trip around the lake and includes a 43-foot waterfall. The roughly 6-mile circuit follows old access roads that are wide enough for two to walk side by side nearly the entire way. Wonderful camping spots are found along the north shore.\n\nRoute Details\n\nThe trailhead is located in the southeast corner of the spacious parking area along Big Moose Road. A painted map of the Pigeon Lake Wilderness trail system is also located next to the trail register. The trail briefly parallels the road in a southeasterly direction before swinging east under the canopy of pole-size hardwoods. Trail markers switch between yellow cross-country skiing disks and red foot-trail disks. Because most of the trail is along old access roads, it would be hard to lose your way. The roads once led to Camp Cascade, originally a summer camp for Charles E. Snyder that easily could have been considered one of the \"Great Camps.\" It was later sold at a fraction of the asking price to Dr. George Longstaff. Longstaff, already an owner of a sailing camp on nearby Moss Lake, transformed the property into an all-girls equestrian camp. When the camp began to falter during World War II, it became coed. The state acquired the property in 1962 and had plans to build a campground, but shortly afterward the state removed the buildings in accordance with the Forever Wild initiative.\n\nAbout 0.3 mile in, you intersect this old road and begin to head northeast. As it was in the past, the broad road is clearly used by horses in the summer (watch your step) and would make a great skiing trail in the winter. Numerous wet spots dot the road but are easily bypassed along the shoulders. At 1 mile the road intersects the south-shore trail on your right. Though the south-shore trail parallels the lake's southern shore, you will not be able to see the lake until you reach the eastern tip near the waterfall. However, the circuit seems to make the most sense hiked in a counterclockwise direction, so mileage and directions are given accordingly.\n\nThe forest matures as you head farther along the trail, and two can walk easily abreast for almost the entire circuit. A few mucky areas sprawl across the trail during the next couple of miles, but overall it is a very pleasant woodland walk. At 2.5 miles you catch your first glimpse of the lake as the forest becomes thicker with evergreens. Roughly 3 miles from the parking area, you come across an open meadow with a stream cutting its way west to Cascade Lake. Head a brief distance east into the surrounding forest, along any of the numerous footpaths, to find the namesake waterfall. The narrow cascade falls 43 feet onto the rocks below before trickling out into the meadow. To the right of the falls is a path that climbs through a narrow passage to the top. The plateau of bedrock at the top is ideal for sitting and enjoying the falls.\n\nContinuing along the loop, the trail quickly reaches a long wet and muddy section. Stay on the path and pick your footing carefully, as the broad marshland will quickly suck the ill-placed foot knee-deep into the muck. As I can attest, you will likely have to wait until you reach the lakeshore, almost 0.5 mile ahead, before you can wash off the black muck. Upon reentering the woods, the trail weaves uphill briefly before joining the north-shore road at 3.3 miles. The north-shore road drops down to the lake at 3.5 miles and skirts it fairly closely for the next mile with views of the water as well as a few remnants of the abandoned girls camp. The first relic is an old stone fireplace and foundation approximately 0.1 mile after reaching the lake. Small hardwood saplings are starting to reclaim this clearing and many of the other sites farther along. The camp stopped operating in the 1950s and was acquired by the state in 1962, so the regenerative growth is a great example of how quickly nature reasserts itself. The next clearing has a thick carpet of ferns, with dense eastern white pine regeneration that will quickly close in the open space. The most striking reestablishment is at 3.8 miles, where an old tennis court is being broken up and covered by a carpet of grass to the left of the trail. Saplings form a thicket around this formerly impervious surface, and you can tell that it won't be long before roots begin to assist the grasses in breaking up the surface.\n\nCASCADE FALLS IN THE HEIGHT OF SUMMER\n\nMany of these clearings would make great campsites, but the more popular ones, found farther along, have better views of the lake. None of the sites are designated (they're often referred to as informal sites), so Department of Environmental Conservation regulations dictate that you must be 150 feet from both the lake and the trail. One such grassy location is found near the lakeshore a little past the tennis court. Probably the most desirable and likely occupied lakeside campsite is found at 4 miles. An old stone retaining wall holds back the forest on your right, while a heavily shaded lawn sprawls out on your left. The parklike setting is an obvious destination, and its proximity to the road means that weekend campers likely pack in far more than the backcountry camper could. Shortly after, you leave the shores of the lake and reach a fork in the trail. A trail to the right, 4.5 miles from the start, heads to Chain Ponds, Queer Lake, Chub Pond, and the rest of the Pigeon Lake Wilderness. Continue straight ahead (southwest) 0.1 mile to a short steel bridge that crosses the outlet of Cascade Lake. A broad, picturesque wetland sprawls out on your right, and you notice substantial beaver activity in the vicinity. Immediately after, pass by another clearing to the left of the trail and quickly reach the intersection with the south-shore trail. From here, it is another mile back to the parking area.\n\nDirections\n\nFrom NY 28 in Eagle Bay, head north on County Road 1\/Big Moose Road. The spacious parking lot is 1.3 miles on the right.\n\n14\n\nGleasmans Falls\n\nTOP OF GLEASMANS FALLS\n\n**SCENERY:**\n\n**TRAIL CONDITION:**\n\n**CHILDREN:**\n\n**DIFFICULTY:**\n\n**SOLITUDE:**\n\nGPS COORDINATES: N43\u00b0 48.490' W75\u00b0 16.589'\n\nDISTANCE & CONFIGURATION: 5.8-mile out-and-back\n\nHIKING TIME: 2\u20133 hours\n\nHIGHLIGHTS: Beaver dam, series of small waterfalls cascading in a narrow gorge\n\nELEVATION: 1,290' at trailhead, 1,350' at highest point\n\nACCESS: Open 24\/7; no fees or permits required\n\nMAPS: adirondackstughill.com\/maps\/BeachMillTrails.pdf; National Geographic Adirondack Park, Old Forge\/Oswegatchie (#745)\n\nFACILITIES: None\n\nWHEELCHAIR ACCESS: No\n\nCONTACTS: Independence River Wild Forest: www.dec.ny.gov\/lands\/58192.html; Central and Southern Adirondack trail information: www.dec.ny.gov\/outdoor\/9200.html; emergency contact: 518-891-0235\n\nOverview\n\nSituated on the western fringe of the park boundary, this relatively level out-and-back hike is a great trip for hikers who like a little solitude when they arrive at their destination. The falls roar through a narrow gorge, and you can view this torrent from high atop the gorge's sheer walls. Another unique feature is a close-up view of an old beaver dam along the trail.\n\nRoute Details\n\nAfter driving up the long, bumpy road, you will already feel removed from the rest of the world. Indeed, finding the parking area among the unnamed roads will likely be the hardest aspect of this clearly defined trail. You will be following the Beach Millpond Trail, which provides access to a more extensive trail system in the Independence River Wild Forest. From the parking area, you walk down a gravelly path to an open, grassy area with a winding, tannin-laden creek. Burnt Creek is the outflow of the previously dammed Beach Millpond, where a sawmill once produced lumber. You will have to negotiate a few small muddy spots on the way to a recently constructed wooden footbridge. On the other side of the bridge, you will enter the forest again, where tall pines and hemlocks shade the trail thoroughly. The trail, emblazoned with yellow Department of Environmental Conservation trail markers, is wide and clearly visible among the leaf and needle litter. With few obstructions and roots, typical of most Adirondack trails, the hike is very easygoing. At 0.2 mile, you will pass the trail register, and the forest transitions from mostly conifers to mixed hardwoods. Interspersed with the dominant beech and maple trees are large cherry trees, all of which provide a dense, high canopy. The ground cover is mostly trout lily, and the lack of saplings or brush allows deep views into the surrounding forest. The trail meanders north briefly, and you glimpse Beach Millpond off to your left through a dense stand of hemlocks.\n\nAt 1.6 miles, you come to the outflow of an old beaver dam. Planks laid lengthwise help you traverse a few muddy spots and are followed by a log-and-plank bridge. After crossing the bridge, the trail immediately turns east to skirt the edge of the beaver pond. A section of the dam has been breached, so the water is not as high as it might have been. The skeletons of pine trees left standing in what is now less a pond and more a marshland provide a stark contrast to the dense forest that sits atop a slight ridge to your right. You will find some good birding and aquatic wildlife viewing in this area, but I doubt that you will linger long, as mosquitoes and black flies relish in ambushing curious hikers here. The muddiest area of the trail is along this edge, and because the slight ridge restricts the trail, your options for avoiding the muck are limited.\n\nAfter leaving the beaver dam area, the trail passes through a small meadow and then rises to a more scrubby area where the canopy is significantly more open and lower. This brief foray into an open hilltop setting adds nice variety to the trail's ecological niches. After hopping over a few narrow, seasonal creeks, the trail transitions back to a dense forest where pines are the dominant species. At 2.3 miles, the trail has a small switchback that descends to the confluence of Second Creek and the boulder-strewn Independence River. Another recently constructed log-and-plank bridge crosses Second Creek, after which the trail climbs slightly over a small hillock dotted with large, moss-covered boulders and then descends to Independence River.\n\nUPSTREAM OF THE FALLS\n\nTHIS BRIDGE CROSSES OVER A FEEDER STREAM.\n\nThe first of the series of falls is visible here, and a slight scramble to the river's rocky banks will give some glimpses of the falls upriver. You begin to view the roaring falls from atop the narrow gorge after passing a fire pit and informal campsite. These dramatic views of the rushing waters are your true reward and are especially powerful during the spring melts. None of the falls is very tall, but the channeling of the river through this narrow gorge, combined with huge boulders, creates a beautiful scene of cascading and foaming waters. As the trail continues to climb the cliffs alongside the river, the views become more panoramic, with several spectacular views straight down the sheer gorge walls onto the falls. Indeed, the views get successively better along the trail, so do not stop until the trail turns north, back into the forest, around 3 miles. The trail eventually winds back to the river, but the falls will be behind you at this point, and only a scramble back along the rocky wash above the falls will provide any views.\n\nDirections\n\nFrom NY 12\/NY 26\/South State Street in Lowville, go east on River Street, which turns into Number Four Road\/County Road 22. After 4.3 miles, continue straight onto Pine Grove Road\/CR 26. Turn left onto Chase Lake Road after 0.7 mile, 5 miles from Lowville. Continue east along Chase Lake Road, and turn left onto Erie Canal Road in 3.9 miles. Turn right onto Beach Mill Road after 0.6 mile. Continue on the rough and bumpy Beach Mill Road, bearing left at the fork, 1 mile, until the road ends, 3.1 miles.\n\nCentral\n\nRAPIDS ALONG EAST BRANCH SACANDAGA RIVER (Trail 17, East Branch Sacandaga Gorge)\n\n 15 AUGER FALLS\n\n 16 SHANTY BROOK\n\n 17 EAST BRANCH SACANDAGA GORGE\n\n 18 SNOWY MOUNTAIN\n\n 19 CHIMNEY MOUNTAIN\n\n 20 PEAKED MOUNTAIN\n\n 21 CASTLE ROCK\n\n 22 BLUE LEDGES\n\n 23 HOFFMAN NOTCH\n\n15\n\nAuger Falls\n\nSACANDAGA RIVER UPSTREAM OF AUGER FALLS\n\n**SCENERY:**\n\n**TRAIL CONDITION:**\n\n**CHILDREN:**\n\n**DIFFICULTY:**\n\n**SOLITUDE:**\n\nGPS COORDINATES: N43\u00b0 28.267' W74\u00b0 15.109'\n\nDISTANCE & CONFIGURATION: 1.1-mile loop\n\nHIKING TIME: 1 hour\n\nHIGHLIGHTS: Spectacular waterfall, riverside stroll\n\nELEVATION: 1,348' at trailhead, with no significant rise\n\nACCESS: Open 24\/7; no fees or permits required\n\nMAPS: National Geographic Adirondack Park, Northville\/Raquette Lake (#744)\n\nFACILITIES: None\n\nWHEELCHAIR ACCESS: No\n\nCONTACTS: Siamese Ponds Wilderness: www.dec.ny.gov\/lands\/53172.html; Eastern Adirondack trail information: www.dec.ny.gov\/outdoor\/9199.html; emergency contact: 518-891-0235\n\nOverview\n\nThe powerful falls on the Sacandaga River are truly a sight to see. Channeled into a narrow flume, the river plunges 40 feet in a torrent of foamy waters. As a short-distance loop, this is a great introductory hike for children, but take caution near the falls because the cliffs are steep and potentially slippery.\n\nRoute Details\n\nIt is easier on your car to park in the large lot that is straight ahead after turning off NY 30. Consequently, directions and mileage are from this point rather than the parking area down the bumpy access road, which is closer to the foot trail's start. From the main parking area, head back toward NY 30, and turn left (southeast) down the access road. Continue down the bumpy, rutted dirt road until you see the Department of Environmental Conservation (DEC) trail sign on your left at 0.3 mile. The trail register is a short distance down the foot trail.\n\nSome people may scoff at the need to sign in for such a short trail, but I encourage hikers to sign in on this and every hike. The DEC and the state base many of their trail-maintenance and funding decisions on trail registers. If you want to see your favorite trails maintained or want more like them, then let the registers act as your democratic voice in the wilderness.\n\nOn the trail, walk along the rolling terrain a little more than 0.3 mile through mixed hardwoods. Ahead, you will begin to hear the roar of the falls, and it is likely that you will abandon the trail to explore the numerous paths that diverge from the main trail to find the best vantage point. Many of the paths that offer views and photo ops of the torrential falls are worn and crumbling, so watch your step. The most dramatic part of the waterfall occurs where the river is channeled into a cataract and then plunges down into the gorge formed by the sheer walls of the river. The cliffs on the opposite shore are spectacular, and the dwarfed trees clinging to its ledges only add to the sense of wildness. Photo-worthy views of the opposing sheer cliff and waterfall are found below the falls. Indeed, you can easily reach the middle of the river by hopping along the large boulders that are strewn about at the outwash of the falls. Huge trees lie tangled among the boulders, and it is not hard to picture the powerful flow of water hurling boulders and trees about. To relocate the trail, head back toward the top of the falls, and you will eventually see the yellow disks that mark the trail. It will take you to a beautiful exposed portion of ledge that sits atop the cataract, where an old fireplace has been partially demolished. This spot marks the halfway point and makes an excellent place to stop with the falls thundering below.\n\nTo complete the loop, head upstream along the worn, but unmarked, path. Dense conifers crowd the trail in many locations, and you often have to skirt or climb over fallen trees. The calm waters of the river above the falls are a striking contrast to the turbulent waters below. Keep an eye out for a wooden footbridge on the opposite shore, as this indicates that you are nearing the point to turn back west toward the parking area. When you reach a sandy beach near a broad, flat calm in the river, climb over the hillock to your left, and you will come out on the northern end of the parking area.\n\nAUGER FALLS\n\nDirections\n\nFROM THE SOUTH From the intersection of County Road 16 and NY 30 in Wells, head north on NY 30. At 3.3 miles, bear left to stay on NY 30\/NY 8. At 1.8 miles, turn right onto the unmarked gravel drive that leads to the parking area.\n\nFROM THE WEST AND NORTH From the junction of NY 8 and NY 30 in Speculator, continue east on the merged roads NY 30 S\/NY 8 N for 8.2 miles; the unmarked gravel drive will be on the left.\n\nFROM THE EAST From the junction of NY 8 and NY 30 north of Wells, turn right onto NY 30\/NY 8 toward Speculator. At 1.8 miles, turn right onto the unmarked gravel drive that leads to the parking area.\n\n16\n\nShanty Brook\n\nSHANTY BROOK SEEN ABOVE THE FALLS\n\n**SCENERY:**\n\n**TRAIL CONDITION:**\n\n**CHILDREN:**\n\n**DIFFICULTY:**\n\n**SOLITUDE:**\n\nGPS COORDINATES: Parking area: N43\u00b0 32.131' W74\u00b0 08.487' Trailhead: N43\u00b0 32.258' W74\u00b0 8.447'\n\nDISTANCE & CONFIGURATION: 7.6-mile out-and-back\n\nHIKING TIME: 4\u20135 hours\n\nHIGHLIGHTS: Waterfall, swimming hole, remote pond\n\nELEVATION: 1,387' at trailhead, 1,693' at highest point\n\nACCESS: Open 24\/7; no fees or permits required\n\nMAPS: National Geographic Adirondack Park, Northville\/Raquette Lake (#744)\n\nFACILITIES: None\n\nWHEELCHAIR ACCESS: No\n\nCONTACTS: Siamese Ponds Wilderness: www.dec.ny.gov\/lands\/53172.html; Eastern Adirondack trail information: www.dec.ny.gov\/outdoor\/9199.html; emergency contact: 518-891-0235\n\nOverview\n\nThough not difficult, this unmarked trail varies from a narrow footpath to an outright bushwhack, so unseasoned hikers should be prepared before undertaking this adventure. One of the more notable preparations is to expect to have wet feet. There is almost no getting around this, so plan accordingly, and you will be able to enjoy the beautiful waterfall, swimming hole, and remote pond.\n\nRoute Details\n\nMost of the parking areas along NY 8 are unmarked, so finding the right one can be tricky. Fortunately, paying attention to your odometer readings and your GPS should steer you in the right direction. The parking area for Shanty Brook is distinct because it is on the south side of the road and is the only one that is U-shaped; most are large lots or gravel pulloffs right next to the road. The trail is unmarked, and though portions are well worn, this hike definitely counts as a bushwhack. You have to wade across the East Branch of the Sacandaga River, so either bring water shoes or plan to hike in wet boots. Additionally, the rocks in the river are slippery, so you will appreciate having a change of clothes in a dry sack should the crossing not go according to plan.\n\nTo reach the river crossing, head northeast along NY 8 a few hundred feet to where a short dirt road provides access to a designated campsite along the river. A steep drop-off leads down to the East Branch of the Sacandaga River on the left side of the road. Across the river, you will see where Shanty Brook meets the river, and you should start looking for a good crossing point. The trail is on the west side of the brook, and I found that an easy way down to the river is about midway down the road, where an old wooden stand once provided access to a cable crossing. The cable crossing was removed, but the stand remains a good landmark. This is less than 100 feet downriver from the confluence of the brook and river. On the opposite shore, you can see where the vegetation has been beaten down where the cable used to be connected. Carefully wade across the river, or rock-hop if the season has been dry enough; do not count on keeping your feet dry.\n\nOn the opposite shore, a well-worn path leads to the western bank of the brook. The trail parallels the brook and crosses several muddy sections as you begin a gentle climb. A little less than 0.8 mile in, you will see a path to your right and notice that the rumble of the brook has gotten louder. Follow the path to the right to reach the top of the waterfall. The falls plunge 12 feet, though they seem taller, into a deep pool that is enclosed within the steep banks of the brook. To reach the base of the waterfall and the swimming hole, jump across the brook to the opposite shore, and scramble down the banks over roots and rocks to the base of the falls. The pool at the base of the falls is not large but is deep enough to swim in, and you can swim right into the falls. A little upstream on the east shore is a designated campsite that would be ideal for a short overnight trip.\n\nBack along the main trail, the path climbs a bit more but quickly levels off for most of the rest of the trip. A quarter mile from the falls, the trail veers away from the brook but then turns briefly east and crosses the brook 0.8 mile from the falls, 1.5 miles from the start. You will pass an open area and beaver dam along the brook at approximately 2.3 miles. This is the first of two open areas along the brook that provide views of the surrounding hills. At the second open area, 2.6 miles, the trail seems to abruptly terminate in a sandy wash. The meadow lies straight before you, rather than to your left as it was previously. This is where you need to recross the brook to reach Mud Ponds. The path on the opposite shore is hard to discern in the summer growth and flooded-out areas, but as long as you keep the brook on your right, you will eventually find a well-worn section of the path. I found that by following a small seasonal channel between an evergreen-dotted island on your right and the surrounding hillside to your left, you can more easily work your way north to the more distinct portions of the path.\n\nSHANTY BROOK FALLS\n\nThe trail from this point on is not traveled as heavily, and blowdown and washouts make navigating a little more difficult. Hikers who are not familiar with basic navigation, lack a compass and map or a good sense of direction, or are uncomfortable with unmarked trails would probably be better served returning before the crossing instead of finishing the hike to Mud Ponds. If you choose to continue, gaiters provide a welcome reprieve from the leg lashings of brambles and fallen trees that encroach on the trail.\n\nA third of a mile from the recrossing, 3 miles from the start, the trail swings west as you follow Shanty Brook, now on your right, toward Mud Ponds. The brook now flows over mostly level ground and is much more placid than before. At 3.2 miles, you reach what appears to be the eastern tip of Mud Ponds, but it is only a widening of the brook due to beaver activity. The actual eastern tip is at 3.6 miles. The path weaves its way along the southern shore of the pond through thick overgrowth and wet sections. At 3.7 miles, you pass a grassy channel that connects the two main ponds and soon reach a small hill that lies between the ponds. Atop this hill is an informal campsite with benches and a fire pit, which, for all intents and purposes, marks the end of the bushwhack. Farther on are more paths, but they soon end in impassable brambles or mires.\n\nDirections\n\nFrom the intersection of NY 30 and NY 8 in Wells, follow NY 8 northeast 8.9 miles. The parking area is on your right in the shape of a U.\n\nFrom the intersection of NY 28 and NY 8 in Wevertown, follow NY 8 west 14.5 miles. The parking area is on your left in the shape of a U.\n\n17\n\nEast Branch Sacandaga Gorge\n\nRAPIDS ON THE EAST BRANCH SACANDAGA RIVER\n\n**SCENERY:**\n\n**TRAIL CONDITION:**\n\n**CHILDREN:**\n\n**DIFFICULTY:**\n\n**SOLITUDE:**\n\nGPS COORDINATES: N43\u00b0 34.146' W74\u00b0 06.798'\n\nDISTANCE & CONFIGURATION: 2.4-mile loop or out-and-back\n\nHIKING TIME: 1\u20132 hours\n\nHIGHLIGHTS: Waterfall, swimming hole, scenic river gorge\n\nELEVATION: 1,435' at trailhead, 1,510' at highest point\n\nACCESS: Open 24\/7; no fees or permits required\n\nMAPS: National Geographic Adirondack Park, Northville\/Raquette Lake (#744)\n\nFACILITIES: None\n\nWHEELCHAIR ACCESS: No\n\nCONTACTS: Siamese Ponds Wilderness: www.dec.ny.gov\/lands\/53172.html; Eastern Adirondack trail information: www.dec.ny.gov\/outdoor\/9199.html; emergency contact: 518-891-0235\n\nOverview\n\nThe East Branch of the Sacandaga River has a spectacular gorge with a serene waterfall and swimming hole that you can reach in two ways. The most direct and clearest route is to follow the eastern footpath to Square Falls. The most scenic and adventurous route is to ford the river and then bushwhack along the western banks. Neither route is very long, and the two are easily combined if you choose to ford the river twice.\n\nRoute Details\n\nThere are no trail signs or indications of the trailhead either in the expansive parking area or from NY 8, so pay attention to your odometer, as there are numerous unmarked parking areas along NY 8. Additionally, there is no trail register, and the trail is not marked, but if you pay attention to the directions below, the paths are fairly easy to find and follow. The trail begins on the northeastern edge of the parking area along an old road. The abandoned dirt road descends generally northwest from the main road into the forest. Approximately 0.1 mile down, you reach the end of the road and a turnaround on your left. Large boulders on your right barricade the remaining trail from vehicular travel. Wind your way down the trail to the small stream crossing shortly ahead. Once across the stream, you must decide which route you will choose to explore the gorge. The path along the eastern edge is hard to make out amid the debris of seasonal flooding but is easily distinguished by climbing the tiny hill directly to your right. The western bushwhack is accessed by continuing straight ahead and then heading toward the river along numerous footpaths. Both options, described separately below, terminate at Square Falls, where you can easily ford the river if you wish to combine the trips. If you choose to take the eastern path and return along the western edge, then you should scope out the condition of the river now before continuing. There are two reasons to do so: First, you will need to recognize where to recross the river; second, you will need to determine whether the river is shallow enough to rock-hop or if wading in the strong current will be necessary. Upriver, several narrow areas make crossing easier.\n\nEAST BRANCH SACANDAGA RIVER BELOW THE GORGE\n\nEastern Path\n\nFollowing the eastern path is by far the easiest, safest, and fastest approach to the falls. If it were not for the dramatic scenery offered along the western bushwhack and the lack of scenery on this bank, I would recommend this route to all hikers. Once you're on this well-trodden path, it is hard to lose. For the most part, the path winds atop the gorge cliffs, so views of the river and the dramatic walls are obscured by dense foliage below, but the torrent of water can be heard most of the way. Roughly 0.3 mile in, you arrive at the cascade at the end of the gorge, where an old fire pit is situated close to the riverbank. A tenth of a mile ahead, you pass the double-cable crossing described below. At 0.6 mile, the trail climbs very steeply away from the banks, and you walk high above the river 0.3 mile before winding your way back down to the river. You have passed the gorge portion of the hike, but the falls and swimming hole await you 0.3 mile ahead. The river is visible along this last portion of the trail and makes for some beautiful scenery, but this route has nowhere near the dramatic contrasts of the western bushwhack.\n\nWestern Bushwhack\n\nIf you choose this option instead of the eastern path, upon reaching the banks of the East Branch of the Sacandaga River, you will face the decision of where to cross. Only hikers who are nimble and willing to wade should attempt the crossing. Rock-hopping is a better choice and might get you across (or, as I found, most of the way across) with short sections remaining to be waded. Despite appearing relatively flat, the current is strong, and the round rocks that line the riverbed are slick with slime, so be mindful of your footing. Now that the caveats have been established, let us recognize that in summer, wading across the river can be quite fun and refreshing. Besides, you are headed to a swimming hole. I found an easily recognizable section to cross upriver where two large boulders sit on the southern bank.\n\nOnce on the opposite bank, the path is not readily clear, but rest assured that you will find it if you head upriver close to the rocky banks. Hemlock saplings crowd the banks, while mature trees shade the west side of the river. The informal path on this side of the river is distinct in some areas and disappears among the dense saplings and fallen trees in others, but if you keep the river within sight on your right, you will find your way to the falls. Roughly 0.3 mile into the hike, the gorge begins with a small cascade among the boulder-strewn banks. As you hike, you will notice that the bank on the eastern side is nearly vertical; in sections it ranges from 50 to 100 feet high, and with the frothy water churning beneath, the scenery has a distinctly wild and rugged character. Approximately 0.5 mile into the hike, the steep riverbanks draw close together, with a nearly 20-foot drop to the river. A double cable strung between large eastern white pines on the opposing banks provides a crossing over a deep pool. I don't recommend this crossing except for experienced hikers who have a means to tie off and are in the company of other hikers. Above the pool, the river contracts to a narrow flume through which it spurts out a torrent of foam. You will pass several other small cascades on your way upriver, all of which may seem like Square Falls. However, you won't reach Square Falls until you have hiked 1.2 miles; you can distinguish it from the other cascades because it forms a 6- to 8-foot cascade across the entire river. Additionally, a wide pool lies at its base, while a broad, tabular portion of bedrock forms its top.\n\nDirections\n\nFrom the intersection of NY 30 and NY 8 in Wells, travel east on NY 8. The parking area is 11.7 miles ahead on your left.\n\nFrom the intersection of NY 28 and NY 8 in Wevertown, head west on NY 8. The parking area is 11.7 miles ahead on your right.\n\n18\n\nSnowy Mountain\n\nPANORAMA FROM THE FIRE TOWER\n\n**SCENERY:**\n\n**TRAIL CONDITION:**\n\n**CHILDREN:**\n\n**DIFFICULTY:**\n\n**SOLITUDE:**\n\nGPS COORDINATES: N43\u00b0 42.081' W74\u00b0 20.090'\n\nDISTANCE & CONFIGURATION: 7.4-mile out-and-back\n\nHIKING TIME: 4.5\u20135.5 hours\n\nHIGHLIGHTS: Panoramic views, accessible fire tower\n\nELEVATION: 1,807' at trailhead, 3,899' at Snowy Mountain peak\n\nACCESS: Open 24\/7; no fees or permits required\n\nMAPS: National Geographic Adirondack Park, Northville\/Raquette Lake (#744)\n\nFACILITIES: None\n\nWHEELCHAIR ACCESS: No\n\nCOMMENTS: The trail signs indicate that the distance to the peak is 3.4 miles, but the actual distance is closer to 3.7 miles. The final ascent is very steep and difficult, so plan accordingly and take your time.\n\nCONTACTS: Central and Southern Adirondack trail information: www.dec.ny.gov\/outdoor\/9200.html; emergency contact: 518-891-0235\n\nOverview\n\nSnowy Mountain is considered by many as the 47th High Peak, even though at 3,899 feet tall, it falls just under the 4,000-foot threshold to be considered one of the famous 46 High Peaks. However, recent surveys have shown that 4 of the 46 peaks are under 4,000 feet and 2 (Nye Mountain and Couchsachraga) are actually shorter than Snowy Mountain. Unlike many mountain hikes featured in the book, this is not one of the \"bald\" peaks that provide the awe-inspiring views for which the Adirondack mountaintops are famous. But just because it is not a bald peak does not mean the mountain is short on views. Indeed there are a couple of stunning lookouts near the peak, but the real gem is the 360-degree panorama atop the 45-foot-tall accessible fire tower.\n\nRoute Details\n\nParking along the eastern side of NY 30 is more than adequate for a dozen cars or more, but the trail is so popular that often dozens more line the road to the north and south of the parking area. The assemblage of vehicles is actually a useful landmark as the trail sign\/trailhead is often obscured by the thick foliage along the west side of the road. The trailhead is nearly directly opposite the pulloff\/parking area, and you find the trail register just beyond the thick vegetation.\n\nLOG STAIRS MAKE THE CLIMB EASIER.\n\nThe trail, marked with red trail markers, heads briefly uphill and then begins a 1-mile, relatively flat course (200' elevation gain) west through deciduous forest. Along the way it crosses several small, seasonal brooks and traverses several wet areas atop zigzagged planks. Beaver Brook soon becomes visible down to the right, but there is one more seasonal brook crossing before you descend to the first of several crossings of Beaver Brook, indicated here by a painted arrow sign.\n\nRock-hop across, and climb the opposite bank. After briefly navigating a wet area over a corduroy bridge, the trail diverges from Beaver Brook and begins a modest climb (roughly 200 feet over the next 0.25 mile). Log steps cut into the earth assist you along this climb, after which the trail levels off. At 1.5 miles from the trailhead, a placid creek bisects the trail in an open, marshy area. Cross along the log-and-plank bridge, and continue 0.4 mile along level ground to where you intersect Beaver Brook again.\n\nThe trail begins a steady climb west as you crisscross Beaver Brook over the next 0.5 mile, and you will have to pick your way carefully through several long, boot-sucking mires as you ascend beside the brook. The final crossing of Beaver Brook, roughly 2.5 miles from the trailhead, is distinct from other crossings in that stone steps beckon you forward on the southern bank. The steps also allude to the increasing grade of the climb ahead (nearly 1,500 feet in less than 1.25 miles).\n\nAt this point the trail has covered roughly two-thirds of the distance to the peak but only one-third of the elevation gain. The grade grows steadily steeper and requires using your hands to climb as you near the peak. About 0.75 mile ahead, the trail swings nearly due south and there is a brief slackening of the grade and short respite along the climb. This relatively flat section of trail lies within an evergreen stand and is a good stopping point to get ready for the even steeper portions ahead. Whereas the trail had gained roughly 800 feet over the previous 0.75 mile, along the next section, the trail gains 600 feet in less than 0.3 mile.\n\nThe new ascent begins as the trail swings more westerly, but calling parts of the next section \"a trail\" is a bit misleading because the path soon becomes what is actually a jumble and wash of large boulders and an inclined ledge so broad that the canopy gives way to the open sky. Some portions have no defined route and will likely require the use of both hands as you climb. Follow this open swath upward, picking your way carefully through the vertical maze of rocks. At the top is a small, flat area where you can pause before continuing on to the last bit of climbing. Bear right (west) and begin the final ascent (200 feet) that ends after you pass a sheer rock wall on the right and reaches the first of two lookouts found near the peak.\n\nThe first lookout is to the right of the trail in an open field with a steep ledge you can sit on along the northern edge. Other hikers are likely to congregate here because it is a better place to stop and rest before beginning your descent than the actual peak and fire tower less than 0.1 mile ahead.\n\nThe second lookout is along the western edge of the peak and is more isolated and private. To reach it, you will have to follow some of the unmarked footpaths that weave and meander across the peak; to find the right one, bear right off the main trail onto one of these paths, and you should find it easily. The lookout is atop an exposed ledge, encircled by stubby evergreens, that has an almost perfect natural bench on which to sit in relative solitude and take in the view. However, depending on the time of year, this area may be a bit buggy on calm days.\n\nAs it is for most hikers, the fire tower is likely your ultimate destination and can be found by following the footpaths that lead south over a couple of short hillocks. The actual highest point of the peak lies here, but the peak is heavily wooded, so vistas are available only to those who climb the extra 45 feet to the top of the tower. Like the other fire towers that dot the Adirondack peaks, space within the tower is limited, so hikers will have to take turns inside. Because the stairs are narrow, without much room to pass, it is best to let previous occupants come back down before trying to climb up.\n\nDirections\n\nFROM THE SOUTH From the intersection of NY 8 and NY 30 in Speculator, head north along NY 30. Continue 17.1 miles; the pulloff will be on the right.\n\nFROM THE NORTh From the intersection of NY 28 and NY 30 in Indian Lake, head south along NY 30. Continue 7.3 miles; the pulloff will be on the left.\n\n19\n\nChimney Mountain\n\nTHE WINDOW AT CHIMNEY MOUNTAIN\n\n**SCENERY:**\n\n**TRAIL CONDITION:**\n\n**CHILDREN:**\n\n**DIFFICULTY:**\n\n**SOLITUDE:**\n\nGPS COORDINATES: N43\u00b0 41.289' W74\u00b0 13.809'\n\nDISTANCE & CONFIGURATION: 2.2-mile out-and-back\n\nHIKING TIME: 1 hour\n\nHIGHLIGHTS: Interesting geological formations\n\nELEVATION: 1,751' at trailhead, 2,703' at highest point\n\nACCESS: Open 24\/7; no permits required; $2 parking fee\n\nMAPS: National Geographic Adirondack Park, Northville\/Raquette Lake (#744)\n\nFACILITIES: None\n\nWHEELCHAIR ACCESS: No\n\nCONTACTS: Siamese Ponds Wilderness: www.dec.ny.gov\/lands\/53172.html; Eastern Adirondack trail information: www.dec.ny.gov\/outdoor\/9199.html; emergency contact: 518-891-0235\n\nOverview\n\nIn a region noted for beautiful lakes and panoramic mountains, this geological oddity adds a spectacular variation. Formed over millennia, Chimney Mountain provides a great opportunity to view the geological forces that shaped the Adirondacks. Exploring the numerous formations will occupy most for hours, but for the spelunker and amateur geologist, a day trip probably will not suffice.\n\nRoute Details\n\nThe parking area for Chimney Mountain, as well as Puffer Pond and Kings Flow East Trails, is on private land, with a day-use fee of $2 per car. Many of Adirondack Park's best hikes and features are available through cooperation between private landholders and the Department of Environmental Conservation (DEC). To ensure this cooperation and to encourage future collaborations, please respect private property, and pay the day-use fee.\n\nTo reach the trailhead, walk up the gravel drive past several cabins, approximately 0.2 mile from the parking area. The register is located beneath the shade of the encompassing forest. The Puffer Pond and Kings Flow East Trailheads are to your right, while the Chimney Mountain Trail is to your left. Trail maps often show the two sharing a section of the trail, but the portion shared is actually the access road before the trail register. Mileage given is from the actual trailhead and not from the parking area.\n\nPassing through a thick band of saplings along the forest's edge, you will quickly notice that the trail is wide, well worn, and easy to follow. Blue DEC disks mark the trail but are hardly necessary except near the top, where numerous paths lead to caves and other geological points of interest. Shrouded in tall hardwoods, most notably birches and maples, the trail is fairly level until 0.2 mile, where you cross a brook and the climbing begins. The grade is moderate at first but steadily increases toward the peak. The forest eventually transitions from the deep woodlands to a more open and parklike setting approximately 0.7 mile along. On steeper sections of the trail, take a look back, and you will see Round Lake, which seems to hover at the bases of Crotched Pond Mountain and Kunjamuk Mountain.\n\nCloser to the summit, you will begin encountering informal paths that wind off to your left. This maze of paths leads to some lesser rock formations, to numerous crevices and caves, and to another elevated ridge from which you can look back at the chimney. The caves have precipitous drops, so be cautious when off the main trail, especially with curious and bold children. Exploring the caves should be left to people who are properly experienced and equipped. Do not attempt these activities if you are a novice. However, there are plenty of wonders and much to explore along the main trail, so don't feel like you are missing out if you do not explore off the trail.\n\nThe spire, an amalgamation of metamorphosed rocks, will begin to tower ahead and draw you through ledges that flank the trail. The marked trail ends here, but do not let that stop you from scrambling among the rocks. Just be cautious near the northern edge, as the drop-off is precipitous. The convoluted layers of different materials will likely raise an eyebrow as you puzzle over how such rocks come into being. See page 1 for an elementary description. The story of these epoch transformations is revealed in the exposed portions of Chimney Mountain.\n\nA description of all the interesting formations and ways to find and explore them is beyond the scope of this book and, frankly, this author. However, I will name a couple that you can easily and safely access from the main trail. The main passage you encounter has the jumble of rocks that make up the Chimney to your left and a narrow ledge called the Ship's Prow. The prow is easily walkable, but it narrows and the fall is more precipitous as you go\u2014know your limits before attempting to walk to its end. Turning left at the trail's end, you walk under a striking overhang and come out with the Chimney looming to your right. Climbing along its base to the opposite side will reveal the Window; it frames the western ledge and is easy to crawl into. In addition to the formations, various points offer panoramic views of the ridge to the west, as well as seemingly endless peaks in every direction.\n\nChances are that as you looked up onto the spire when you climbed, you missed a path to the right, which leads to an informal campsite. As you leave the Chimney, this path is more obvious off to your left. Just past the campsite, the path leads not only to the actual summit of Chimney Mountain but also to an interesting view through the surrounding trees of Ship's Prow with the Chimney towering behind it.\n\nDirections\n\nFrom the intersection of NY 30 and NY 28 in Indian Lake, head south on NY 30. Turn left onto County Road 4\/Big Brook Road at 0.6 mile. Follow Big Brook Road across Lake Abanakee, and take a sharp right at 6.1 miles. At 7.8 miles park in the designated parking area and pay the day-use fee.\n\n20\n\nPeaked Mountain\n\nPEAKED MOUNTAIN SEEN ACROSS A BEAVER MEADOW\n\n**SCENERY:**\n\n**TRAIL CONDITION:**\n\n**CHILDREN:**\n\n**DIFFICULTY:**\n\n**SOLITUDE:**\n\nGPS COORDINATES: N43\u00b0 43.125' W74\u00b0 07.110'\n\nDISTANCE & CONFIGURATION: 7.2-mile out-and-back\n\nHIKING TIME: 4 hours\n\nHIGHLIGHTS: Lakeside hiking, cascades along a stream, beaver dams and meadows, isolated pond, panoramic views from a peak\n\nELEVATION: 1,726' at trailhead, 2,919' at highest point\n\nACCESS: Open 24\/7; no fees or permits required\n\nMAPS: National Geographic Adirondack Park, Northville\/Raquette Lake (#744)\n\nFACILITIES: None\n\nWHEELCHAIR ACCESS: Accessible campsites and bathrooms are at Thirteenth Lake.\n\nCONTACTS: Siamese Ponds Wilderness: www.dec.ny.gov\/lands\/53172.html; Eastern Adirondack trail information: www.dec.ny.gov\/outdoor\/9199.html; emergency contact: 518-891-0235\n\nOverview\n\nCan't choose between lakeside hiking, exploring an isolated pond, hiking beside frothy cascades, or climbing a mountain for a panoramic view? Why choose? Peaked Mountain has it all. The main trail is easy and features a variety of waterscapes, including a pristine lake, a cascading brook, and several beaver dam meadows. The last 0.8 mile along the trail is steep and difficult, but the summit has a unique perspective. At the peak, you can clearly trace your entire trip back to the shore of Thirteenth Lake.\n\nRoute Details\n\nAccess to Peaked Mountain is at the north end of Thirteenth Lake, which also has car-top boat access, accessible picnic areas, and free primitive camping. Most of the vehicles in the large and spacious parking area are for camping and day use. To access the trail, walk down the main gravel drive to the trail register. The register also serves as a record of day-use activities, so do not be surprised to see a lot of entries. To the left is the boat launch, while access to the foot trail is down the right fork. Head toward the lake past a couple of campsites and picnic tables, and then look for the trail sign on your right near the last picnic table. Mileage given is from this point, which is a little more than 0.1 mile from the parking area.\n\nThe trail winds along the shore of Thirteenth Lake, weaving through paper birches and other hardwoods. Beaver activity is evident everywhere, from chewed stumps and recently felled trees across the path to several lodges that are within feet of the trail. The trail is relatively flat, with a few dips and hills that follow the contours of the lake. You will encounter a couple of designated primitive campsites along the trail as well, so be aware that you may be walking through or near someone's site.\n\nThe shores of Thirteenth Lake are undeveloped and pristine. Though not common, motorboaters are occasionally heard, which is odd because the lake is nestled in the middle of a wilderness area. (Wilderness Area land-use plans typically prohibit the use of motorized boats; hopefully the state will enforce this standard in the future.) Typically, the lake is dotted with canoeists and kayakers, and in fact, many of these boaters paddle to where the trail diverges from the lake, around 0.8 mile on the right.\n\nThis junction occurs where Peaked Mountain Brook empties into Thirteenth Lake. Shortly after you begin following the brook uphill, you encounter a trail sign indicating Hour Pond to your left. However, to reach Peaked Mountain and its pond, continue northwest uphill alongside the brook. You will be following Peaked Mountain Brook through numerous transformations all the way to the pond and the foot of the mountain. Innumerable small falls and cascades characterize the brook until you cross it upstream, at 1.5 miles, and provide an interesting contrast to the flat and quiet waters of the lake behind you. You will have to rock-hop across the brook to a tiny island and then again to the opposite bank.\n\nThe trail levels off after crossing the brook. Shortly after, you will notice a clearing to your right, a result of the first dammed section of the brook. From the trail, hemlocks frame the grassy wetland and an outstanding view of Slide Mountain. Farther along, the trail winds closer to the wetland, and looking east, you will see Big Thirteenth Lake Mountain. This is the first of three distinct beaver-dammed sections along the brook, but with all the evident beaver activity, the landscape could easily change from year to year. It is tempting to think that one of these dammed sections is Peaked Mountain Pond and that one of the mountains is Peaked Mountain. However, the pond is a broader expanse of still water, and Peaked Mountain has exposed sheer faces with less tree cover.\n\nAfter looking across the wetland at Big Thirteenth Lake Mountain, you will pass very close to a large beaver dam at 1.9 miles. Shortly after, you cross the brook again and head back into the woods briefly before another wetland opens up to your left. A couple of rocks along the trail's edge provide a good vantage point of the dammed wetland, which surrounds a tiny, cedar-dotted island. The trail hugs the eastern edge of this wetland and then recrosses Peaked Mountain Brook at the northern edge of the wetland. The crossing is little more than a step across the brook, where it passes between two broad rocks, and you are roughly two-thirds of the way to the pond. The third and final major dammed section gives you the first real glimpse of Peaked Mountain's large, exposed face. This section contains a lot more large dead trees, with the additional distinction of large boulders strewn about.\n\nAfter a short section of walking through dense woods, you climb a small hill to discover Peaked Mountain Pond straight ahead at 3 miles. At this point, paths lead off to your left, but the trail is to your right. A fallen tree makes this a convenient spot to sit next to the pond. However, a nice place to rest is also a little farther up the trail in a small area of spruces. The trail so far has been fairly easy and accessible to most hikers. However, the climb to the summit is much steeper and requires a lot more effort. It is certainly worth the ascent, so take time to rest and enjoy the view of the beautiful pond.\n\nThe trail up the mountain passes through the previously mentioned stand of spruces and past a fork in the trail, which indicates a designated camping area to the left. The trail climbs slightly, among beeches and maples, and then descends to a stream crossing with views of the pond to your left. Climbing begins immediately after you cross the stream. This section is steep, with large, exposed rocks, and the forest is almost entirely paper birches. Blown-down birches often cover the trail, making spotting the actual trail and finding the best footing difficult at times. However, red disks mark the trail, and alternate paths around the blowdown are evident.\n\nPEAKED MOUNTAIN POND VIEWED FROM A LOOKOUT NEAR THE PEAK\n\nPEAKED MOUNTAIN BROOK\n\nAfter the first steep section, the grade eases a bit through a section where sheer rock is on your left and a rock hump is to your right. The trail swings left just past a massive boulder and becomes steep again. Climbing is steady until just before the top, and there are several tricky spots, so take your time and make room for other hikers as they pass. When evergreens dominate the forest, you'll know that you are close to the summit. A large, open portion of bedrock provides views down onto the pond, but the best part is at the true summit. After a brief foray back into the evergreens, you are once again out in the open. Sprawling out beneath you is your entire trip from where you left Thirteenth Lake. Peaked Mountain Pond and the beaver-dammed sections are obvious, as is the path the brook has cut into the landscape. In the height of summer, dragonflies perform aerial acrobatics for your viewing pleasure, and the bedrock provides a great spot to sit and look out at the beautiful vista.\n\nDirections\n\nFROM THE WEST From the intersection of NY 28 and NY 30 in Indian Lake, head 12.1 miles east along NY 28, and turn right onto County Road 78\/Thirteenth Lake Road in North River. Bear right onto Beach Road at 3.3 miles. End at the large parking area at 0.7 mile.\n\nFROM THE EAST From the intersection of NY 28 and NY 28N in North Creek, head 5.2 miles west along NY 28. Turn left onto CR 78\/Thirteenth Lake Road. Bear right onto Beach Road at 3.3 miles. End at the large parking area at 0.7 mile.\n\n21\n\nCastle Rock\n\nBLUE MOUNTAIN LAKE SEEN FROM CASTLE ROCK\n\n**SCENERY:**\n\n**TRAIL CONDITION:**\n\n**CHILDREN:**\n\n**DIFFICULTY:**\n\n**SOLITUDE:**\n\nGPS COORDINATES: N43\u00b0 52.370' W74\u00b0 27.071'\n\nDISTANCE & CONFIGURATION: 3.7-mile loop\n\nHIKING TIME: 2 hours\n\nHIGHLIGHTS: Scenic views, cliffs\n\nELEVATION: 1,880' at trailhead, 2,430' at highest point\n\nACCESS: Open 24\/7; no fees or permits required\n\nMAPS: National Geographic Adirondack Park, Northville\/Raquette Lake (#744)\n\nFACILITIES: None\n\nWHEELCHAIR ACCESS: No\n\nCONTACTS: Central and Southern Adirondack trail information: www.dec.ny.gov\/outdoor\/9200.html; emergency contact: 518-891-0235\n\nOverview\n\nFrom a glance at a map, this trail looks deceptively uninteresting, but it has a lot to keep your attention. The analogy to a castle is well founded, and those with active imaginations will find themselves naming gates, turrets, and other structures while exploring the area. As a short loop, it provides a refreshing change from the innumerable out-and-backs typical of summits and other trails in the park.\n\nRoute Details\n\nThe parking area is a small pulloff near the Minnowbrook Conference Center. The trail register is just up the road within sight of the sandy parking area. However, the trail does not begin until a little more than 0.3 mile down the road. Mileage given here is from the trail register. Follow the private gravel road, and bear right where the conference center driveway rejoins the road. Red trail markers on the right tell you to bear right at the next fork in the road.\n\nAt 0.3 mile, you will finally reach the trail on your right. On your left is the outlet of Chub Pond that flows into Blue Mountain Lake. Shortly after the trail portion starts, you come to a log bridge crossing this stream and a fork in the trail. To your left is the south route to Castle Rock, while to your right is Upper Sargent Pond and the north route. Cross the bridge to follow the south route, a rolling trail through mixed hardwoods. Yellow disks emblazon the trail, and at 0.6 mile you can see Chub Pond and a broad wetland to your right. The view is limited, but better views of the pond are available on the return trip along the northern route. When I hiked the trail, rain had been falling sporadically for days, so the trail was very wet; in fact, it was basically a stream in some sections (particularly on the northern route). But judging by the number of mucky spots, it is safe to say that this trail is typically wet.\n\nThe trail is a deep-woods walk among tall hemlock, birch, maple, and spruce trees. The landscape is dotted with large, mossy boulders that hint at the large rock faces to come. At 0.8 mile you will find a small dip, as well as a fork, in the trail; 0.3 mile to your left is Blue Mountain Lake, while Castle Rock is 0.7 mile to your right. After crossing a small brook, the trail begins to climb steadily. Openings in the forest canopy indicate that the lake sprawls off to your left, but you cannot see it until you reach the summit.\n\nAt 1.2 miles, you come to what I imagine to be the castle's gates. The base of the large granite outcrop towers above you on your right, with a narrow passage between it and some of the boulders that lie among the cliff's talus. Running water babbles underneath the sheer castle wall, sparking images of activities on the other side. Paths that eventually lead back to the main trail weave through the maze of boulders, providing opportunities for exploration.\n\nBack along the trail, a brief climb brings you to the intersection of the northern route, on your left. The climb to Castle Rock's final lookout is on your right, through spruces and paper birches. You reach the summit as you squeeze through another narrow passage. Climbing atop the rock ledge on your left reveals views of the tree-covered mountains to the north. While this view is beautiful and panoramic, the real gem is the view from atop the ledge, or rather the southern turret, on your right. This sheer cliff 700 feet above Blue Mountain Lake provides vistas of the wooded islands that dot the lake. The cliff's sheer walls and the way it juts out above the treeline evoke the feeling of being atop a lookout tower and, in combination with the \"gate\" at the base, give rise to the impression of a castle. Take caution around the cliff edge during wet or icy conditions.\n\nReturning along the northern route, the trail is less steep and descends gradually with numerous stream crossings. This portion is marked with red Department of Environmental Conservation disks and is easy to follow. At 1.9 miles you cross a stream and come to a fork in the trail; the Upper Sargent Pond Trail heads off to your left, while to the right is the return to the trailhead and parking area. While the southern route definitely has some wet areas, the northern route has notably more and longer mucky spots. The forest matures as you descend, and you rock-hop across multiple streams that flow to your right. At 2.3 miles, you come to a meadow, where the sound of water rushing to your right is much more distinct. The trail flattens out a bit more, and the mucky areas become more frequent. You encounter Chub Pond again at 2.8 miles, with views of Castle Rock through the hemlocks that surround the wetland. Multiple vantage points of the pond are available along the trail, and shortly you come back to the juncture where the log bridge crosses the outlet of Chub Pond. The stroll back to the parking area passes quickly along the flat roads.\n\nDirections\n\nFrom the intersection of NY 28\/NY 30\/NY 28N in Blue Mountain Lake village, head north on NY 30\/NY 28N. Turn left onto Maple Lodge Road at 0.6 mile. Continue on Maple Lodge Road 1.2 miles until you see the trail register, and park at the pulloff on the left side of the road before the Minnowbrook Conference Center.\n\n22\n\nBlue Ledges\n\nHUDSON RIVER DOWNSTREAM FROM BLUE LEDGES\n\n**SCENERY:**\n\n**TRAIL CONDITION:**\n\n**CHILDREN:**\n\n**DIFFICULTY:**\n\n**SOLITUDE:**\n\nGPS COORDINATES: N43\u00b0 49.957' W74\u00b0 06.506'\n\nDISTANCE & CONFIGURATION: 5.0-mile out-and-back\n\nHIKING TIME: 2\u20133 hours\n\nHIGHLIGHTS: Vertical cliffs in the Hudson River Gorge\n\nELEVATION: 1,592' at trailhead, 1,345' at lowest point\n\nACCESS: Open 24\/7; no fees or permits required\n\nMAPS: National Geographic Adirondack Park, Northville\/Raquette Lake (#744)\n\nFACILITIES: None\n\nWHEELCHAIR ACCESS: No\n\nCONTACTS: Siamese Ponds Wilderness: www.dec.ny.gov\/lands\/53172.html; Eastern Adirondack trail information: www.dec.ny.gov\/outdoor\/9199.html; emergency contact: 518-891-0235\n\nOverview\n\nThe beautiful stroll through a deep forest would be excuse enough to go on this hike, but the spectacular cliffs and sandy beaches in the Hudson River Gorge make it a must-do. In the height of summer, it's an excellent picnic or swimming destination, with the option for riverside camping. On dam-release days, especially during spring and fall, you will likely see rafters and kayakers navigating the turbulent rapids that flow at the base of these gigantic cliffs.\n\nRoute Details\n\nThe trail, marked with blue disks, begins across the road from the parking area on the left. Immediately cross a stream over a log-and-plank bridge, after which you'll find the trail register within a mucky area. The trail heads west toward the shore of Huntley Pond, where you have to pick your way through mud and tangles of roots. The walk along the shoreline requires a good deal of rock- and root-\u00adhopping to stay out of the muck, but this is very brief and followed by a slight climb into the hardwood forest.\n\nAt 0.5 mile pass between two large, moss-covered boulders, and then descend 0.3 mile to a stream. After crossing the stream, the trail levels off, and it is easy going over the next 0.8 mile. The high canopy allows deep views into the forest on your left, while broad openings along the meandering stream to your right provide a stark contrast. You will likely notice some very large eastern white pines interspersed with birch and maple trees. At 1.5 miles you begin a 0.5-mile climb on the hill whose contour you have been following since the stream crossing. As you climb south, larger pine trees loom along the trail, and you will hear the Hudson's rushing waters to your right. Despite the thundering rapids that seem to be just beyond the treeline, you will not be able to see the river until you reach the trail's end. At 2 miles you cross a large, exposed portion of ledge that marks the beginning of the descent to the river. This lookout reveals nothing through the thick foliage of summer, but when the leaves fall, it promises excellent views. The last 0.5 mile of the trail is the steepest section but is easily managed on both the descent and the return climb. The eastern white pines sporadically encountered along the trail earlier now dominate the forest as you reach the riverbank.\n\nYou see glimpses of the ledges through the evergreen canopy, but the scale of the cliffs is not apparent until you reach the end of the trail. As you emerge directly on the riverbank, the Blue Ledges loom 300 feet high on the opposite shore.\n\nThe vertical wall of rock draped by pines and cedars, seen across the broad river, makes a dramatic and peaceful setting. Paths leading both up- and downriver explore the riverbank and take you to several sandy beaches. High water during spring and fall makes for whirlpools and rapids that are best observed or rafted with commercial outfitters. During periods of lower water in the summer, sheltered areas provide a good opportunity for swimming. If you choose to swim, be mindful that the river still has a powerful current and that dam releases from Lake Abanakee on Tuesdays, Thursdays, Saturdays, and Sundays raise the river by 2 feet. Upstream and around the bend is one major rapid, while downstream are some major Class IV and V rapids. These rapids are visible from the trail's end and provide a popular amusement shortly after midday on dam-release days. Designated riverside camping areas are also downstream near these treacherous rapids. You can reach the campsites by passing through a thicket of brush near the river's edge. Camping is limited to these areas, and they seem to fill up. Otherwise, Department of Environmental Conservation regulations require backpackers to set up camp at least 150 feet from any body of water. The Hudson River Gorge is an exceptionally beautiful destination in the Adirondacks, and unless you are willing to raft down the river, this trail is virtually the only way to see its beauty.\n\nDirections\n\nFrom the intersection of County Road 29\/Main Street and NY 28N in Minerva, head 4.4 miles north on NY 28N, and turn left onto North Woods Club Road. The turn is more like bearing left, but it is located along a curve and hill, so you will likely miss it heading south along NY 28N, 11.7 miles south of the intersection of NY 28N and Blue Ridge Road in Newcomb. The dirt road is smooth and accessible for most vehicles. At 3.7 miles, you cross the Boreas River and see designated roadside camping on both sides of the road. The road climbs steeply afterward. As it descends, the trail sign is visible on the left at 6.7 miles. The parking area along North Woods Club Road could accommodate close to a dozen vehicles if parked close together and perpendicular to the road.\n\n23\n\nHoffman Notch\n\nBRIDGE ACROSS HOFFMAN NOTCH BROOK\n\n**SCENERY:**\n\n**TRAIL CONDITION:**\n\n**CHILDREN:**\n\n**DIFFICULTY:**\n\n**SOLITUDE:**\n\nGPS COORDINATES: Southern trailhead: N43\u00b0 52.091' W73\u00b0 53.338' Northern trailhead: N43\u00b0 57.235' W73\u00b0 50.196'\n\nDISTANCE & CONFIGURATION: 7.5-mile point-to-point\n\nHIKING TIME: 4\u20135 hours\n\nHIGHLIGHTS: Isolated pond, streamside hiking\n\nELEVATION: 1,650' at trailhead, 1,190' at lowest point\n\nACCESS: Open 24\/7; no fees or permits required\n\nMAPS: Hoffman Notch Wilderness: tinyurl.com\/hoffnotchmap; National Geographic Adirondack Park, Lake George\/Great Sacandaga (#743)\n\nFACILITIES: None\n\nWHEELCHAIR ACCESS: No\n\nCONTACTS: Hoffman Notch Wilderness: www.dec.ny.gov\/lands\/81598.html; Eastern Adirondack trail information: www.dec.ny.gov\/outdoor\/9199.html; emergency contact: 518-891-0235\n\nOverview\n\nPassing through this notch provides hikers with scenery and solitude. You follow a flat stream to a serene marsh, where Hoffman Mountain is framed in the background, and then down a fast-falling brook with innumerable small cascades. The trail is fairly easy, so fast and experienced hikers will find that they can traverse the notch in half the time, but this precludes enjoying the privacy along the picturesque trail.\n\nRoute Details\n\nThis point-to-point trail would be excellent in either direction but is probably easier and most enjoyable heading from south to north. Deciding which end to leave a car or where to be picked up all depends on circumstance, but it might be advisable to leave a car at the southern lot for extended trips. The southern lot is large and off the main road, while the northern end is probably easiest to give directions to. This would also be a great trip for the key-pass technique, where two parties start at opposite locations and exchange car keys when they pass each other, as no detours or side trips might cause them to miss each other. Directions here are given heading south to north.\n\nThe trail begins on the northwest corner of a large, grassy parking area off Loch Muller Road, which has ample space for dozens of cars. A 1-mile trail to Bailey Pond, on your left, is also located at the beginning of the Hoffman Notch Trail. Continue straight ahead on the abandoned road, marked in yellow for the entire trail, through the mixed hardwood forest. The wide, open road makes for easy travel, and you will quickly come within sight of the West Branch Trout Brook approximately 0.3 mile in. The trail heads west briefly, and you descend to a log-and-plank bridge. Like the numerous bridges near the end of the trail, the bridge has been handsomely built with substantial cribbing and stonework to bolster the crossing. Shortly after crossing the bridge, the road begins to transition into trail, but the going is still easy over the mostly rolling terrain.\n\nYou will encounter a few seasonally muddy sections before reaching the only other trail intersection at 1.3 miles. The main trail continues straight ahead following yellow disks, while the trail to the right is marked in blue and leads to another access point along Hoffman Road. If you seek an extended trip, you can add almost 6 miles by starting at Hoffman Road. The North Branch Trout Brook begins to flow on your right after you pass this intersection and accompanies you all the way to Big Marsh at the halfway point. Approximately 2 miles in, you pass an enormous erratic alongside the trail. Though many may think I am an erratic on the trail, talking into the voice recorder dangling from the front of my pack, erratics are actually large boulders that were deposited by receding glaciers. They are termed erratic because they typically do not fit in with the surrounding landscape, due to either their composition or, more likely, their location. The trail winds to and from the banks of the brook numerous times, and at 2.3 miles, it passes nearly through the water as it cuts its way north along the mossy banks. The brook flows over mostly flat terrain and consequently is very slow moving, especially when you reach the beaver dam at 2.5 miles. Evergreens dot the banks along this section, and you will notice that the terrain grows ever steeper on your left as you head toward the midpoint of the notch. Here, the notch is the low point between the Washburn Ridge on your left and the Texas Ridge on your right. Roughly 0.5 mile ahead, the marshland truly begins, with grassy islands interspersed along the brook as it meanders wider near the notch's peak. The marshlands create broad, open sections in the forest, and soon Hoffman Mountain begins to loom across the scenic wetlands. At 3.5 miles the trail descends briefly to the wide, open waters of the marsh\u2014though lake or pond would likely be a more apt description. A few skeletal trees remain in the open waters, enhancing the wild character of this remote area.\n\nNumerous vantage points for scenic photos are found all along the trail as it winds along the hemlock- and birch-lined western shore. You leave Big Marsh's shores at 4 miles, but the trail continues along flat terrain through mixed hardwoods another 0.8 mile. A small, seasonal brook joins you on your left, and shortly after it, you reach Hoffman Notch Brook, which tumbles in at a small cascade on your right and forms a deep pool that can be passed across a worn tree. Rock-hopping might be a better option in wet conditions, as the tree is nearly worn smooth.\n\nThe trail continues on the opposite shore under a thick canopy of hardwoods, and several wet sections turn the trail into a mire at points. At this point the brook is descending rapidly, forming innumerable cascades with a low rumbling that drowns out most other forest sounds. The trail parallels the brook on an elevated ridge along the eastern banks approximately 0.3 mile before descending to the rock- and boulder-strewn brook at 5.1 miles, where you cross to the western bank. Trail markers are absent or difficult to follow in this section, but if you keep the brook in sight and look for the worn path, you can easily find your way to the crossing and the yellow trail markers. The waters are falling rapidly at this point, but the boulders are so large that it's easy to hop across the brook to the opposite shore, where the trail continues to descend on the western bank.\n\nThe notch is very distinct along this portion of the trail, as you pass the several sheer rock faces of Washburn Ridge on your left and the terrain, Hornet Cobbles, grows steeper on the right. Roughly 5.8 miles in, the trail transitions into an abandoned road with built-up banks, and the hiking gets easier. At 6 miles you pass a rusted piece of abandoned track machinery on your right that reminds you of the persistent regenerative power of nature.\n\nAt 6.1 miles you reach a rocky wash where a low bridge spans Hoffman Notch Brook. This crossing marks the end of the descent, and it is mostly rolling terrain for the next 1.4 miles to the northern terminus. At 6.5 miles you pass through a clearing where power lines have cut a swath through the wilderness; shortly after, you reach another sturdy bridge that re-crosses Hoffman Notch Brook. This flat section of the trail is a rerouting of the often-flooded north end and crosses a couple more bridges. The trail swings east through a notable stand of cedars and then intersects the old north road at 7.4 miles. The rusted front end of a classic car is found at this junction, only a few hundred yards from the trail terminus at Blue Ridge Road.\n\nDirections\n\nSOUTHERN TRAILHEAD From I-87, take Exit 28, and head east on NY 74. Immediately turn right onto US 9 in Schroon Lake. In 2.9 miles, turn right onto County Road 24\/Hoffman Road. In 5.4 miles, turn right onto Potash Hill Road. At 1.1 miles turn right onto Lock Muller Road. Follow Lock Muller Road 1.3 miles, and look for a gravel drive on the right. Follow the gravel drive a short way to the large, grassy parking area.\n\nNORTHERN TRAILHEAD From I-87, take Exit 29 (Newcomb\/North Hudson). Head west on CR 84\/Blue Ridge Road and go 5.5 miles. The parking area is a small gravel drive on the left.\n\nEastern\n\nLAKE GEORGE SEEN FROM SLEEPING BEAUTY MOUNTAIN (Trail 24, Sleeping Beauty Mountain, page 166)\n\n 24 SLEEPING BEAUTY MOUNTAIN\n\n 25 SHELVING ROCK FALLS\n\n 26 BLACK MOUNTAIN\n\n 27 TONGUE MOUNTAIN RANGE\n\n 28 PHARAOH MOUNTAIN\n\n 29 PHARAOH LAKE\n\n24\n\nSleeping Beauty Mountain\n\nVIEW OF LOWER LOOKOUT FROM UPPER LOOKOUT\n\n**SCENERY:**\n\n**TRAIL CONDITION:**\n\n**CHILDREN:**\n\n**DIFFICULTY:**\n\n**SOLITUDE:**\n\nGPS COORDINATES: N43\u00b0 32.971' W73\u00b0 33.369'\n\nDISTANCE & CONFIGURATION: 3.3-mile out-and-back from Dacy Clearing (seasonal access) or 6.5-mile out-and-back from Hog Town parking area\n\nHIKING TIME: 2\u20133 hours\n\nHIGHLIGHTS: Sheer cliffs, panoramic views\n\nELEVATION: 1,335' at trailhead, 2,347' at highest point\n\nACCESS: Open 24\/7; no fees or permits required. Access along Shelving Rock Road is seasonal and subject to town of Fort Ann approval. Dacy Clearing Road is not plowed and is closed during mud season (early spring).\n\nMAPS: National Geographic Adirondack Park, Lake George\/Great Sacandaga (#743)\n\nFACILITIES: Picnic tables, pit toilets\n\nWHEELCHAIR ACCESS: No\n\nCOMMENTS: Timber rattlesnakes are known to occupy the shores of Lake George, so be cautious when stepping over logs. Read more about timber rattlesnakes in the section on Animal, Insect, and Plant Hazards.\n\nCONTACTS: Eastern Adirondack trail information: www.dec.ny.gov\/outdoor\/9199.html; emergency contact: 518-891-0235\n\nOverview\n\nThe sheer cliffs are a spectacle in and of themselves, but once you add the views of Lake George from Sleeping Beauty's summit, this becomes a must-visit trail. Though the hike is steep in sections, the short distance from Dacy Clearing places it easily within the grasp of hikers of all experience levels. A longer approach or additional circuit can be added for hikers who desire a more extensive adventure.\n\nRoute Details\n\nTo reach the trailhead at Dacy Clearing, you can either drive or hike 1.6 miles from the Hog Town parking area. The road, though a bit narrow, is in good shape, and vehicles in good condition should have no difficulty reaching the clearing. This way passes multiple campsites, historic sites, and additional trailheads, so hikers wishing for additional interest or mileage may want to consider starting from the Hog Town parking area.\n\nDacy Clearing has a large parking area, as well as some day-use facilities, including picnic tables and pit toilets, so don't be surprised to find many people who choose this as a destination in itself. The trailhead is in the southern corner of the parking area, just to the side of a vehicle barrier and stream. Yellow disks mark this portion of the trail. The trail heads south but almost immediately switches back north and begins a steady climb. The trail register is a few hundred yards from the switchback, and when you see how full it is, it is obvious that this is a popular trail. The trail, wide enough for two to hike side by side, was once a part of the Knapp Estate's road network that weaves through the area. In 1931, 4,300 acres of the Knapp Estate were acquired by the state, including much of the southeastern section of Lake George Wild Forest.\n\nLess than 0.5 mile in, the trail switches back southeast, and the climb slackens a bit. During the height of summer, the dense canopy masks the sheer cliffs of Sleeping Beauty, but in the fall or winter, the cliffs are more visible and soon loom ahead of you. A large, angular boulder at 0.6 mile marks a T in the trail, and the magnitude of the cliffs becomes apparent even with leafed-out trees. The trail to the left, marked in yellow, leads to Bumps Pond as well as a circuit trail that returns to Sleeping Beauty Summit. The alternate route is more than 2 miles to the summit, while to the right, the distance to the summit is a little over a mile. To the right, marked with blue disks, is a more direct and, in my opinion, more impressive route to the summit because you hike along the base of the cliffs.\n\nThe trail to the right leaves the old road network behind, and you traverse a narrow foot trail through a tiny hemlock grove. For the next 0.4 mile, you will be hiking along a flat plateau on the south side of the mountain, with truly impressive views of the sheer cliff and the tumble of boulders at its base. Through the surrounding canopy, you can catch brief glimpses of the mountains to the southwest, but they don't compare to the vistas at the summit. The grade is basically flat, and though there are a few wet spots, it is easy going. At 0.8 mile, you cross a 10-foot bridge, and the trail swings east. The trail dips down briefly, but at 1 mile you begin the climb to the summit. Multiple switchbacks make the climb easier, and at 1.1 miles, you make your way up a substantially built-up switchback with a high rock wall.\n\nAfter this point, the trail weaves its way north on the east side of the cliffs, and views disappear as the forest becomes more coniferous. At 1.6 miles, you reach a Y in the trail. To the right is the trail to Bumps Pond and the circuit mentioned previously, while straight ahead is the summit, 0.1 mile. When you reach the end of the trail, you can view the panorama that opens out before you from several open ledges. Immediately in front of you is a large outcrop where people congregate; a higher but smaller outcrop is also to your right and slightly behind you, and a large, lower one is to the south. At the lower lookout, you have the advantage of seeing your path along the base of the cliffs. Expansive views to the west, south, and north show various portions of Lake George and the surrounding mountains from all of these lookouts, and there is plenty of room to sit and enjoy the views. Return the way you came for a total trip length of 3.3 miles, or take the circuit for a total length of 4.4 miles.\n\nDirections\n\nFrom the intersection of US 9 and NY 149 in Queensbury, south of Lake George and north of Glens Falls, head east along NY 149. Turn left onto Buttermilk Falls Road at 6 miles. Keep left at 3.1 miles where Buttermilk Falls Road turns into Sly Pond Road. Sly Pond Road turns into Shelving Rock Road at 5.7 miles, just past the intersection with Hog Town Road on the right. Continue on Shelving Rock Road 0.8 mile to the Hog Town parking area. Park here if you desire a longer trip. Otherwise, continue down the one-lane road 1.6 miles to Dacy Clearing.\n\n25\n\nShelving Rock Falls\n\nSHELVING ROCK FALLS\n\n**SCENERY:**\n\n**TRAIL CONDITION:**\n\n**CHILDREN:**\n\n**DIFFICULTY:**\n\n**SOLITUDE:**\n\nGPS COORDINATES: N43\u00b0 33.357' W73\u00b0 36.184'\n\nDISTANCE & CONFIGURATION: 1.5-mile out-and-back to falls, 4.1-mile loop along lake\n\nHIKING TIME: 2 hours\n\nHIGHLIGHTS: Spectacular waterfall, lakeside hiking, old foundation\n\nELEVATION: 450' at trailhead, with no significant rise along trail\n\nACCESS: Open 24\/7; no fees or permits required; access along Shelving Rock Road is seasonal and subject to town of Fort Ann approval\n\nMAPS: National Geographic Adirondack Park, Lake George\/Great Sacandaga (#743)\n\nFACILITIES: None\n\nWHEELCHAIR ACCESS: No\n\nCOMMENTS: Timber rattlesnakes are known to occupy the shores of Lake George, so be cautious when stepping over logs. Read more about timber rattlesnakes in the section on Animal, Insect, and Plant Hazards.\n\nCONTACTS: Eastern Adirondack trail information: www.dec.ny.gov\/outdoor\/9199.html; emergency contact: 518-891-0235\n\nOverview\n\nFollowing the shore of Lake George, the trail system around Shelving Rock Falls was once carriage roads and therefore makes for some of the easiest hiking in the Adirondacks. Though the trails are easy, they are beautiful and offer an array of scenery and history. You can explore the trails and destinations in a variety of ways, so consider the system of trails more of a destination than simply a hike.\n\nRoute Details\n\nThough many smaller parking areas are along Shelving Rock Road, the spaces are very limited, often with room for only one or two cars, and not clearly marked as parking areas. Because the entire roadside is plastered with signs warning that cars parked along the road will be ticketed or towed, I would not plan on finding parking except in the large parking area for which the signs are obvious and directions are given. Likewise, trails leading to Lake George's shores are not clearly marked at the other parking areas, while the trail from the main parking area is impossible to miss.\n\nOn the right side of the parking area, walk past the vehicle barrier and down the broad access road. You have probably noticed that there is no trail register, nor are there any trail markers, but it is practically impossible to lose your way. Likewise, Shelving Rock Road and Lake George border the trail system, so you cannot wander far off the trail before regaining your bearings. Less than 0.3 mile in, you will come to a fork in the trail. Straight ahead is the main portion of the lakeside trails, while to your left are the falls. Continue to the left, and cross Shelving Rock Brook over a wooden bridge at 0.4 mile. On the opposite side of the bridge, you may notice a footpath to your left. This footpath can be used to climb to\u2014or as your return route from\u2014the falls. The stroll along the designated trail is an easier approach and has its own charms near the top.\n\nYou may have noticed that the trail system is more like a network of roads. Indeed, this area of Lake George Wild Forest was once a part of the Knapp Estate, and the trails were formerly carriage roads. The extensive work employed to level the terrain and build these roads becomes evident when you pass a large rock retaining wall on your left. The wall is part of a switchback that leads up the hill that eventually climbs to the top of Shelving Rock Falls. Tall pines thoroughly shade the area, and as you reach the height of the climb, you will notice a lone-wolf pine on your right.\n\nLone-wolf trees are trees that grow in open fields and eventually get overtopped when the forest takes over the field. Often short and gnarly, they eventually get shaded out and die, but they become excellent shelter and den trees. Any doubts that the environment has undergone a major transition are removed when you see the remains of a foundation behind the lone-wolf pine. This ruin adds an interesting spot for exploration, but the roar of the falls ahead will likely keep you from dallying long.\n\nThe carriage road takes you to the dam above the falls. Off to the right, the carriage road continues back to Shelving Rock Road, but straight ahead you can make out the caissons that once supported a bridge over the artificial wetland that sprawls out before you. Across the dam, you can see a trail and might be tempted to cross the top of the dam, but do not attempt it. Bear in mind that the dam is narrow and covered in slick algae, and many people have been injured or have died here. The best and, coincidentally, most scenic way to reach the other side is at the base of the falls.\n\nBASE OF SHELVING ROCK FALLS\n\nA short scramble downhill will take you there. Keep an eye out for a protruding rock on the right that makes an outstanding point from which to view and photograph the falls. The falls cascade 70\u201380 feet through and over large protrusions of ledge, with a deep pool beneath. You can return along the carriage road or take the more interesting footpath down beside Shelving Rock Brook. Besides adding a little more interest to the hike, this path has the benefit of revealing another smaller falls about midway down to the bridge previously crossed. You will have to scramble along rocks and roots to get down, but the trail is no different from typical Adirondack trails.\n\nBack on the trail, you have the option to head back to your vehicle, but it would be a shame to miss the walk along Lake George that awaits you by continuing to your left at the first fork. The trail along the lakeshore is level, passes within feet of the lake for long portions, and has a few excellent lookouts. The trail ends at private property, and you can return to the parking lot along Shelving Rock Road, but the trail along the lake is so pleasant that it is by far your better option.\n\nDirections\n\nFrom the intersection of US 9 and NY 149 in Queensbury, south of Lake George and north of Glens Falls, head east along NY 149. Turn left onto Buttermilk Falls Road at 6 miles. Keep left at 3.1 miles, where Buttermilk Falls Road turns into Sly Pond Road. Sly Pond Road turns into Shelving Rock Road at 5.7 miles, just past the intersection with Hog Town Road on the right. Continue on Shelving Rock Road 1.8 miles, where you pass the parking area for primitive campsites and register for the area. Continue to the left on Shelving Rock Road another 1.8 miles to the parking area on your left. This is the second parking area on the south side of the road, past the Shelving Rock Mountain parking area on the north side of the road.\n\n26\n\nBlack Mountain\n\nLOOKING SOUTH AT LAKE GEORGE\n\n**SCENERY:**\n\n**TRAIL CONDITION:**\n\n**CHILDREN:**\n\n**DIFFICULTY:**\n\n**SOLITUDE:**\n\nGPS COORDINATES: N43\u00b0 36.702' W73\u00b0 29.612'\n\nDISTANCE & CONFIGURATION: 6.7-mile balloon\n\nHIKING TIME: 4\u20135 hours\n\nHIGHLIGHTS: Panoramic views, remote ponds, beaver activity, lake views\n\nELEVATION: 1,564' at trailhead, 2,646' at Black Mountain peak\n\nACCESS: Open 24\/7; no fees or permits required\n\nMAPS: National Geographic Adirondack Park, Lake George\/Great Sacandaga (#743)\n\nFACILITIES: None\n\nWHEELCHAIR ACCESS: No\n\nCOMMENTS: Timber rattlesnakes are known to occupy the shores of Lake George, so be cautious when stepping over logs. Read more about timber rattlesnakes in the section on Animal, Insect, and Plant Hazards.\n\nCONTACTS: Eastern Adirondack trail information: www.dec.ny.gov\/outdoor\/9199.html; emergency contact: 518-891-0235\n\nOverview\n\nBlack Mountain is the highest peak in the southeastern Adirondack\u2013Lake George region. It lies a little farther from the bustling heart of Lake George, so hikers seeking more solitude in the eastern Adirondacks may find this an excellent choice. By summiting from the north and descending along the southern slope, you will traverse a wide variety of environs and maximize the number of panoramic views.\n\nRoute Details\n\nThe parking area off Pike Brook Road is sufficient for roughly a dozen cars conscientiously parked. The trailhead and trail register are on the western edge of the pulloff. The initial section of the trail, marked in red, follows a gravel access road that almost immediately turns sharply left and climbs steadily up and away from the parking area. After swinging west, the road levels off and continues 0.6 mile to where a building with a metal roof comes into view. A vehicle barrier blocks the road, and the trail\u2014indicated by a brown trail sign\u2014diverges to the right into the forest along a more traditional footpath.\n\nThe trail continues a little more than 0.3 mile along flat terrain among maple and beech trees to a trail intersection, 1 mile overall. To the left (south) are the remote ponds and return leg of the trip. Continue straight, and you soon have to hop across a small stream, after which you continue west along a fairly gentle grade. About 0.5 mile past the intersection, the trail is bisected by another seasonal stream that spills into a stream flowing to the left. Shortly after rock-hopping across the seasonal stream, the trail takes a sharp turn right (north). It parallels the stream briefly north, swings west, and then becomes steeper once it heads west again, briefly paralleling and running through the seasonal stream as it climbs.\n\nA BRIDGE ALONG THE TRAIL KEEPS YOUR FEET DRY AT THIS CROSSING.\n\nAfter crossing and leaving the streambed, you will pass a rock cairn on the left, 1.9 miles from the start. Shortly after, the trail swings west and the climbing continues to steepen. Spruce and birch trees dominate the forest along with verdant, moss-covered rocks. The primeval and dark setting of the northern, shaded side of the mountain is in stark contrast to the brighter deciduous, southern slope found on the return leg. This dark setting is even more dominant as the trail swings nearly south. Along a brief level section of trail, you pass by a large, square boulder to the left, just before a fork in the trail, 2.5 miles overall.\n\nThe left fork is a lightly worn path, while the right fork is a more heavily worn, cobble-strewn trail that leads to the summit. Bear right and head uphill to where the trail swings sharply north as you make the final ascent. The inaccessible fire tower and a windmill soon come into view as you make the final approach along an exposed ledge. Just before heading up to the peak and its amalgam of technology, check out the small lookout on the right near a memorial placard nailed to a tree. At the peak, there are limited places to stop and take in the view, so on busy or windy days this lookout may be the more desirable place to take a break.\n\nOn top of the summit are a windmill, a power transfer station, solar panels, fencing, and a fire tower, all dedicated to emergency communications. They occupy most of the summit, and there are not many places that would be considered a good place to rest. The one exception is atop a rock outcrop on the northern tip of the peak, just behind the fire tower. To reach this spot, head between the windmill and fence to where a trail sign indicates the direction to the return trip, and head right. A large rock partially shields you from the incongruous pieces of civilization, and great panoramic views of the northern tip of Lake George sprawl out before you.\n\nTo return, you could head back the way you came, but you would miss some of the great views of southern Lake George and wildly different ecosystems encountered along the northern pond route. Head back to the trail sign, and proceed southeast in the direction of Black Mountain Pond Trail junction.\n\nAs you begin the descent (roughly 900 feet over the next mile), the trail takes on a very different character. Instead of the boreal and wet setting of the northern slope, the southern slope is dry and grassy with a canopy of deciduous trees, most notably oaks farther down the mountain. It's a stark contrast but also the perfect setting for timber rattlesnakes that like to bask in the sun on open ledges, so be mindful when stepping over logs and rocks. The open ledges are also the best spots to take in the panoramic views of Lake George available at several locations along the steep descent. Access the first lookout by following one of several herd paths through the surrounding scrub, about 0.1 mile down. Because of the aforementioned snakes, this might not be the best for those who are squeamish. The second lookout is along a ledge near a large boulder beside the trail, about 0.25 mile farther along the descent; it requires a small amount of navigating off trail. Don't worry if you don't want to bushwhack to these lookouts because another splendid one is found 0.3 mile farther.\n\nThe third lookout, located at 3.3 miles overall, is a broad opening along the trail with a beautiful view of the southern portion of Lake George; you reach it just after a large switchback that leads beneath a steep wall of rock. One final lookout of note is a little less than 0.1 mile ahead. This one is just to the left before another switchback swings the trail west. The open, grassy ledge is nestled among tall oaks and provides a view of not only Lake George but also the pristine ponds found farther along the trail. This is the last of the great viewing areas along the descent, which continues along many switchbacks another 0.25 mile and 200 feet to where the trail levels off and soon swings east.\n\nAt 3.6 miles reach the Black Mountain Pond Trail junction to the right and the return to Pike Brook Road via Black Mountain Pond straight ahead. Continue straight, now following blue trail markers, less than 0.25 mile to where Black Mountain Pond comes into view. Once up and over a small knoll, the dark pond and a grassy marsh sprawl out before you. Nearby you traverse an open ledge that dives into the water; the Black Mountain Pond Lean-To sits just uphill from this ledge. The trail continues to skirt the pond's edge briefly before veering off into the surrounding forest. As you leave the pond behind, pass through a long, wet area, and then reach a stream crossing, 4.2 miles. This is the beginning of the long, sinuous wetland that feeds Round Pond. Once across this feeder brook, bear right and follow the wetland to the cattail-lined pond. The trail weaves along the shore, often within inches of the high water, along sections where the cattails are so thick and close that they form a veritable wall.\n\nAs the trail heads away from Round Pond, a mossy swamp flanks the trail to the right and shortly reaches another trail intersection, 4.6 miles overall. To the right are Millman and Fishbrook Ponds, while to the left is the trail back to Pike Brook Road. Turn left (north) following yellow trail markers, and quickly reach one more intersection. To the right (south) is the 0.1-mile-long trail that leads to the Lapland Pond Lean-To. The left fork, straight, leads north to Pike Brook Road.\n\nContinue north roughly a mile to reach the first trail junction and end of the loop portion of the hike. Initially all along this northward trek, the sections of the trail weave in and out of a streambed and include some long, muddy portions. About halfway to the junction, you will have to rock-hop across a rocky brook that bisects the trail. As you near the intersection, the trail dips down toward a wide, marshy area with a beaver dam. At the tip of this marsh, you reach a heavy timber-and-log bridge that crosses the marsh's outlet, 0.2 mile shy of the junction ahead. Once you reach the junction, turn right and follow the path previously trodden 1 mile to Pike Brook Road for an overall trip length of 6.7 miles.\n\nDirections\n\nFROM THE SOUTH From the intersection of US 4 and NY 22 in Whitehall, continue north 7.1 miles along NY 22, and turn left onto County Road 6. After 2.6 miles turn left onto Pike Brook Road. After 0.8 mile the parking area will be on the right.\n\nFROM THE NORTH From the intersection of NY 9N and NY 22 in Ticonderoga, head southeast 19.2 miles along NY 22 S\/NY 74 E. Turn right onto CR 6. After 2.6 miles turn left onto Pike Brook Road. After 0.8 mile the parking area will be on the right.\n\n27\n\nTongue Mountain Range\n\nVIEW OF LAKE GEORGE\n\n**SCENERY:**\n\n**TRAIL CONDITION:**\n\n**CHILDREN:**\n\n**DIFFICULTY:**\n\n**SOLITUDE:**\n\nGPS COORDINATES: N43\u00b0 37.767' W73\u00b0 36.489'\n\nDISTANCE & CONFIGURATION: 12.6-mile loop (side trips not included)\n\nHIKING TIME: 8\u201310 hours\n\nHIGHLIGHTS: Multiple peaks with panoramic views, lakeside hiking, waterfall\n\nELEVATION: 403' at trailhead, 1,725' at highest point\n\nACCESS: Open 24\/7; no fees or permits required\n\nMAPS: National Geographic Adirondack Park, Lake George\/Great Sacandaga (#743)\n\nFACILITIES: None\n\nWHEELCHAIR ACCESS: No\n\nCOMMENTS: Timber rattlesnakes are known to occupy the shores of Lake George, so be cautious when stepping over logs. Read more about timber rattlesnakes in the section on Animal, Insect, and Plant Hazards.\n\nCONTACTS: Eastern Adirondack trail information: www.dec.ny.gov\/outdoor\/9199.html; emergency contact: 518-891-0235\n\nOverview\n\nThere are multiple ways to explore the Tongue Mountain Range, but this loop combines some of the best features into an extended day hike. Multiple peaks offer innumerable panoramic views of Lake George, as well as some pleasant hiking along old Civilian Conservation Corps (CCC) horse trails following the lakeshore. Along the ridgeline, there are few sources of water, so bring your own supply to drink.\n\nRoute Details\n\nTimber rattlesnakes are one of the few venomous species of snakes found in New York and are seldom seen, even in areas where they are known to live. The Tongue Mountain Range, as well as other areas along Lake George, is one of these few places and, though it is unlikely that you will encounter one, it is advisable to take precautions. See the section on animal hazards, and educate anyone unfamiliar with snakes. The simplest ways to avoid a mishap are to look where you step and to give the snake plenty of room should you see one.\n\nThe trailhead is a short distance south of the parking area along NY 9N. Turn left onto the trail, marked in red, and you will find the trail register several hundred yards ahead. The trail descends through a stand of pines, and you pass what looks to be a fork on your left but is actually just a short footpath. Continue to your right downhill, and soon reach a boardwalk that crosses the marsh surrounding Northwest Bay Brook. You reach a fork in the trail at 0.4 mile. To your right is the return leg of your trip along the Northwest Bay of Lake George, marked in blue. Continue straight, along the red-marked trail, which soon begins its ascent to the ridgeline. You will hear a waterfall off to your right shortly after the intersection. To view the 20-foot-high cascade, follow the barely discernible path on your right, or let the sounds of the falls guide you to the steep banks overlooking them.\n\nThe trail continues to climb, and you will cross and recross the seasonal stream that leads to the waterfall over log-and-plank bridges. The pitch has steadily increased, and soon steep ledges loom on your left. After walking in the shadow of these ledges, you make a final ascent over a built-up switchback to a brief plateau, a little less than 1 mile from the start. The approximate 0.3-mile stroll along the hemlock-shrouded plateau is a welcome break from the ascent, but you soon begin to climb again along a switchback. After more marshy areas and a little more climbing, you soon reach a four-way intersection at 2 miles. To your left, marked in blue, is the ridge trail north to Five Mile Mountain, Deer Leap, and the northern trailhead along NY 9N. Straight ahead, marked in red, is the trail down to Five Mile Point on the shores of Lake George. Follow the blue-marked trail south, to your right, toward Fifth Peak, French Point Mountain, and Montcalm Point along Lake George. The trip south is a bit like a roller coaster as you navigate up and down the numerous peaks and valleys that make up the dragon spine\u2013like ridge.\n\nThe trail climbs the next 0.4 mile to where you reach the intersection with Fifth Peak Trail, marked in yellow. The summit is 0.3 mile along this spur. A lean-to is also located along this trail for those wishing to make it an overnight trip. There is no water, so campers should plan accordingly, as it is a steep climb to the lakeshore or any other source of water.\n\nThe trail continues straight, still marked in blue, around the west side of the mountain along a fairly level section. Views of the main portion of Lake George to the east are shortly followed with a view straight down to the Northwest Bay. The trail descends steeply after this point, and the roller coaster\/sawtooth portion of the trail begins in earnest. After descending roughly 350 feet in less than 0.3 mile, you immediately begin to climb again to an unnamed knob, 3.6 miles at its top. This begins a series of steep ascents to fine views of the lake, followed by sharp descents. Each tiny summit provides a great place to stop, but if you plan on making only one stop along your traverse, I recommend waiting until you reach French Point Mountain, roughly 0.8 mile ahead after this first major descent. You scramble up, over, and down yet another knob, 4 miles, before you begin the final climb to French Point Mountain, 4.3 miles and 1,756 feet of elevation. If you haven't taken in the beauty of the roughly 200 verdant islands dotting the deep blue of Lake George, now is surely the time. The steep eastern side of the range has been swept with fire numerous times, and only scrubby trees dot the slope down to the lake, so panoramic views, open fields, and ledges are the norm and provide innumerable spots to stop.\n\nYou weave up and down another tiny knob before seeing First Peak looming to the south. This peak is the last major dip and climb of the dragon's spine before the trail makes its last descent to Lake George at Montcalm Point. The trail descends approximately 500 feet before climbing another 300 feet over the next 1.4 miles on your way to First Peak, 5.7 miles from the start. Breathtaking views await you here, as well as along the majority of the descent to the lake. You are a little less than halfway through the hike, though you will have probably spent more than half of your planned time along the trail. Not to worry\u2014the trail is mostly downhill or level as you complete the loop. Views of the Lake George Narrows sprawl out before you as you descend the mostly exposed portion of the tongue. Mileage to the innumerable vantage points would probably provide more confusion than any measure of progress. However, you reach the final knob approximately 0.5 mile after First Peak and, shortly after, delve back into the deeper woods. You reach the final intersection along the trail, 7.6 miles, at a wide and mucky area. To the left is Montcalm Point, 0.2 mile, a welcome respite along the shores. Though the side trip is definitely worth a visit, those pressed for time can rest assured that numerous sections along the return trip come within feet of the lake and provide points to stop and soak your feet or even take a swim.\n\nTRAIL BESIDE NORTHWEST BAY\n\nTo return, continue straight\/right along the blue-marked trail north. Shortly after the intersection, the trail passes within feet of the lake but soon arcs east into the forest to bypass a small section of private land. After this brief diversion, you rejoin the lakeshore and find that the trail generally follows the lake. The farther north you go, the better the condition of the trail and the easier the path. You pass several openings along the lake, along with stream crossings, abandoned fire pits, and swimming holes, so if time permits, enjoy the numerous points of interest, or plan to take a late-hike rest.\n\nFormer CCC work is evident all along this return portion as you pass built-up crossings, rock walls, and other trail developments that make this portion of the trail truly a pleasure. At 10.6 miles, you reach the marsh where Northwest Bay Brook meets its namesake bay. It is only 1.6 miles back to the intersection and 2 miles to the parking area. An occasional passing truck along NY 9N will remind you of this proximity, though the trail heads northeast into the forest a bit, and roadside noise is mostly drowned out. The trail climbs slightly during this 0.8-mile arc into the forest and soon descends back toward the marsh. Soon after the descent, you reach the trail intersection at 12.2 miles. Continue left back to your vehicle at the quarry-side parking area for a total trip of 12.6 miles (not including side trips).\n\nDirections\n\nFrom I-87, take Exit 24 (County Road 11\/Bolton Landing\/Riverbank Road), and head 4.7 miles east on CR 11\/Riverbank Road. Turn left onto NY 9N\/Lake Shore Drive, and head northeast 4.5 miles. The parking area is on your right in front of an old quarry.\n\n28\n\nPharaoh Mountain\n\nLOOKING DOWN ON PHARAOH LAKE FROM PHARAOH MOUNTAIN\n\n**SCENERY:**\n\n**TRAIL CONDITION:**\n\n**CHILDREN:**\n\n**DIFFICULTY:**\n\n**SOLITUDE:**\n\nGPS COORDINATES: Crane Pond Road parking area: N43\u00b0 51.548' W73\u00b0 41.312'\n\nDISTANCE & CONFIGURATION: 9.2-mile out-and-back\n\nHIKING TIME: 6\u20137 hours\n\nHIGHLIGHTS: Spectacular views, scenic marsh\n\nELEVATION: 1,060' at trailhead, 2,556' at highest point\n\nACCESS: Open 24\/7; no fees or permits required\n\nMAPS: National Geographic Adirondack Park, Lake George\/Great Sacandaga (#743)\n\nFACILITIES: None\n\nWHEELCHAIR ACCESS: No\n\nCOMMENTS: Accessibility along the rough and partially flooded Crane Pond Road has deteriorated to the point where even high-clearance, four-wheel-drive vehicles frequently get stuck. Plus, driving on the road is illegal. As such, trail directions and mileage have been updated.\n\nCONTACTS: Eastern Adirondack trail information: www.dec.ny.gov\/outdoor\/9199.html; emergency contact: 518-891-0235\n\nOverview\n\nAn excellent peak in the eastern section of the Adirondacks, Pharaoh Mountain is an exceptional destination in itself or as a side trip in the heart of the Pharaoh Lake Wilderness. Atop the mountain are great views of the eastern Adirondacks and, on a clear day, the High Peaks to the north as well as the Green Mountains in Vermont. A longer loop is available for hikers who descend to Pharaoh Lake and return via the Glidden Marsh Trail (11.8-mile loop).\n\nRoute Details\n\nCrane Pond Road is officially closed at the end of the pavement, where a small parking area has been established. However, in the past locals and others accessing Crane Pond and Pharaoh Lake Wilderness routinely used the unmaintained road. The road was barricaded in the 1980s, but local residents removed the barricade the following spring. When Earth First! tried to erect a new barrier, a conflict arose that involved a pugilistic encounter between a local politician and an Earth First! representative. The whole incident was caught on camera and eventually made it to the national news. At present, there are no barriers, and the road is left open and used by hikers, anglers, boaters, and campers alike. Be aware that the road is not maintained and is very rough. Even high-clearance, four-wheel-drive vehicles get stuck along the remaining 1.9 miles to the Crane Pond parking area, so I do not recommend planning on driving farther along the road.\n\nContinue south along the rough road as it winds 1.5 miles through a dense forest to where the road descends to a flooded section, which varies in depth and length, depending on recent rain and the season. If you drove farther along the closed road, do not count on being able to drive farther. If you can't cross, there is a bypass trail north of the flooded section. Past the flooded section, the road continues to wind through the forest, passing the Long Swing Trail at 1.7 miles and eventually reaching the first of two grassy parking areas. If you thought one would have to be crazy to ford the flooded section, you will be surprised to find several vehicles that made the crossing.\n\nProceed east to the boat launch and second parking area, and then head south past a downed pine tree and a trail sign on your right, 1.9 miles. A wide bridge crosses the outlet of Crane Pond, which is on your left, as it flows into Alder Pond on your right. The trail, marked by red disks, uses primarily old access roads, so it is broad and free of obstructions. Additionally, there are very few muddy areas, so this trail is one of the few in the Adirondacks that are in exceptional condition. The wide trail climbs gently as it winds through a mostly hemlock forest a little less than 0.3 mile to the trail register. It continues over level terrain another 0.5 mile under the canopy of towering hemlocks and eastern white pines to a fork in the trail. This intersection, 0.7 mile from Crane Pond, 2.6 miles overall, is situated on the northern tip of Glidden Marsh. Looming straight ahead over the marsh's placid water is Pharaoh Mountain. To the right is the direct approach to the summit, as well as a route to Pharaoh Lake, while on the left is another route to Pharaoh Lake via Glidden Marsh. If you desire a longer trip, descend from the summit to Pharaoh Lake, skirt the lakeshore on the northern trail (not shown on most maps), and then head back along Glidden Marsh Trail for a circuit of 11.1 miles. For a description of the return leg along Glidden Marsh, see the Pharaoh Lake overnight section.\n\nTRAIL TO PHARAOH MOUNTAIN\n\nBear right at the fork in the trail, and proceed along the western shore of Glidden Marsh. You will cross a couple of log bridges as you navigate the next 0.3 mile over level terrain. At 2.9 miles, the climb to the summit begins. As you ascend, the forest transitions from the towering evergreens to a mostly hardwood forest. The mixture features oak and cherry trees at first, and then gives way to predominantly maple and beech. At 3.6 miles, you reach a fork with an overgrown abandoned road on the left, but the trail continues on your right. The easy hiking along old roads gives way to rougher hiking as you pick your way along exposed bedrock and through the rocks and cobbles that dominate the trail. Approximately 3.9 miles in, the canopy height begins to diminish and the forest transitions back to evergreens, though the composition is now stunted firs dotted with birches.\n\nAt 4.2 miles, you will have to rock-hop across a stream, and the climbing becomes much steeper. Over the next 0.5 mile, the trees become stunted, and the trail, flanked by alpine mosses, crosses over exposed portions of bedrock. As you approach the summit, several exposed ledges are either on the trail or mere feet to your right, with amazing views to the north and west. I suggest taking some time to admire the views to the north, as the panorama at the summit is mostly to the west and south. At the summit, the trail passes through a shallow cleft that bisects what are essentially the two summits of Pharaoh Mountain. The northern knob is to your left and has exceptional views to the west, especially of Schroon Lake. Footpaths meander about the northern knob, where a couple of informal campsites are interspersed among the windswept firs. Follow the main path a very short distance south to find the footpath that leads to the southern knob. The exposed portions of bedrock on this knob offer incredible views of Pharaoh Lake.\n\nNote: The Pharaoh Lake overnight trail begins at the end of this trail.\n\nDirections\n\nFrom I-87, take Exit 28, and head east on NY 74. Immediately turn right onto US 9 in Schroon Lake. In 0.6 mile turn left onto Alder Meadow Road. Follow Alder Meadow Road 2.1 miles, and bear left onto Crane Pond Road. The parking lot is on the left where the pavement ends and Crane Pond Road turns into an unmaintained road at 1.4 miles.\n\n29\n\nPharaoh Lake\n\nWINTER GREEN POINT ON PHARAOH LAKE\n\n**SCENERY:**\n\n**TRAIL CONDITION:**\n\n**CHILDREN:**\n\n**DIFFICULTY:**\n\n**SOLITUDE:**\n\nGPS COORDINATES: Crane Pond Road parking area: N43\u00b0 51.548' W73\u00b0 41.312'\n\nDISTANCE & CONFIGURATION: 17.3-mile loop\n\nHIKING TIME: Overnight\n\nHIGHLIGHTS: Exceptional lakeside camping, waterfall\n\nELEVATION: 2,500' at trailhead (at the end of Pharaoh Mountain Trail), 992' at lowest point\n\nACCESS: Open 24\/7; no fees or permits required\n\nMAPS: National Geographic Adirondack Park, Lake George\/Great Sacandaga (#743)\n\nFACILITIES: None\n\nWHEELCHAIR ACCESS: No\n\nCOMMENTS: The trail description presumes access via the Pharaoh Mountain Trail. Bypassing the summit via the Glidden Marsh Trail cuts 1.5 miles and 1,000 feet of elevation change from the hike.\n\nCONTACTS: Eastern Adirondack trail information: www.dec.ny.gov\/outdoor\/9199.html; emergency contact: 518-891-0235\n\nOverview\n\nThe lean-tos and designated campsites along the shores of Pharaoh Lake are some of the best in the Adirondacks. The sheer number of sites in this virtual wilderness campground means that you will likely find a spot even on busy holiday weekends. If you crave solitude, choose an off-peak weekend or shy away from the lean-tos and pick a remote tent site. The designated sites and lean-tos are nearly all next to the shore with scenic views, and it's clear why they are popular.\n\nRoute Details\n\nNumerous trails lead to Pharaoh Lake, but the description provided here is from the end of Pharaoh Mountain Trail, described. Bear in mind that parking for the Pharaoh Mountain Trailhead along the unmaintained portion of Crane Pond Road is no longer recommended or feasible. See the Pharaoh Mountain Trail description for details.\n\nFrom the end of Pharaoh Mountain Trail, descend the southern slope following the red trail markers, 4.6 miles from the end of the paved portion of Crane Pond Road. Shortly after leaving the summit, you must climb down a vertical portion of exposed ledge that is several feet high and requires the use of both hands. Footholds are available in a small cleft next to the adjoining sheer rock face on your right. The descent is steeper than the climb to the summit, and the forest quickly transforms from alpine evergreens and birches to dense maples and beeches. At 6.2 miles, a sheer cliff looms to your right and indicates that you are nearing the lakeshore. Soon after, you traverse a wet section across planks laid lengthwise. By now, you can catch masked glimpses of the lake through the thick beech saplings.\n\nAt 6.4 miles, you reach the intersection with the north lakeside trail. From here, you can circle the lake in either direction. Counterclockwise, to your right, is the fastest and most direct approach to the majority of the lean-tos. On the other hand, heading clockwise brings you to some of the best campsites, as well as Split Rock Bay and the final return route via Glidden Marsh; this northern section of the trail is not shown on most maps of the area. The description and mileage here are given for the clockwise direction.\n\nTurn left and follow the west-shore trail, marked with blue disks, toward Split Rock Bay. This portion of the trail is roughly 0.7 mile long and follows the shoreline very closely. It also features three great designated camping areas. The first of these is just off the trail at 6.6 miles. A fire pit has been placed along an exposed portion of ledge within feet of the lake, and a long, rocky peninsula makes a superb spot to admire the view. A little less than 0.3 mile ahead, you will pass a yellow Department of Environmental Conservation designated camping disk with an arrow. These arrow disks indicate that a designated site is nearby. Usually these sites are farther off the trail and can be easily located by following the arrows. On occasion, disks will be missing or the indicated direction of the arrow may lead you astray, so use the disks as a means to zero in on the spot rather than as absolute markers. In this case, wander out along the southern edge of the peninsula, just below an elevated ridge, and you will find the picturesque site. Surrounded by tall pines and small blueberry bushes, it sits high on the elevated peninsula and offers commanding views of the lake.\n\nBack along the loop, the trail dips down and follows a cattail-filled bay briefly before reaching another designated camping disk at 6.9 miles. To reach this site, head east onto another small peninsula, where you will find a site very similar to the last one near the peninsula's tip. The intersection with Glidden Marsh Trail back to Crane Pond is at 7 miles.\n\nShortly after the intersection, the trail, now marked in yellow, winds along another bay with large boulders sticking up in the middle. This is Split Rock Bay, and while the rocks do look split from this shore, their appearance from the opposite shore is more dramatic. From that shore, the split rocks appear to be a pair of obelisks with an eerie and unnatural quality. On your way to the opposite shore, you will pass an abandoned fiberglass boat, which, had you come from the other direction, would not seem unusual. At 7.6 miles, you reach Lean-To 4 (also known as Split Rock Bay Lean-To), which is within a couple dozen feet of the shore and has an open view south out onto the lake.\n\nIt's likely that you will see a boat pulled up on the shore nearby, and the mystery of the abandoned boat will start to resolve itself. In fact, all the lean-tos around the lake have some sort of boat, usually a canoe or small rowboat, located nearby. Improvised paddles are typically found nearby as well, but they are, in some cases, little more than flat logs. If you have the foresight to pack an ultralight paddle, you could easily become the most popular guest at the lake.\n\nThe trail swings northeast at the lean-to and heads into a thick hemlock forest. This area is heavily shaded and has few saplings, so the views into the surrounding forest are expansive. Dotted with mossy, wet areas, this brief foray into a hemlock swamp is a pleasant change of pace from the standard hardwoods, but the trail delves back into mixed hardwoods after 0.4 mile. At 8.3 miles, you reach another intersection. To your right is a side trip to Winter Green Point, marked with red disks, while straight ahead is the continuation of the lakeside trail. The less than 0.3-mile side trip to the rocky point is definitely worthwhile and provides views of the lake as well as Pharaoh Mountain.\n\nCAMPSITE ON PHARAOH LAKE\n\nBack on the trail, you quickly reach another fork; to the left are Grizzle Ocean and Putnam Pond Campground, 5.8 miles away, while the lakeside trail continues straight ahead. Roughly 0.3 mile ahead, you reach a wide stream crossing where the log cribbing is all that remains of an old bridge. You can easily rock-hop across, but you might want to bushwhack a short distance upstream to where a tiny waterfall splashes into a deep, boulder-dammed pool. The trail winds up and down a few hillocks through a mixed pine forest before reaching another intersection at 8.6 miles. Lean-To 3 is off to your right, while the east-shore trail continues ahead. The lean-to sits on an elevated cliff along the lake and offers outstanding views. Before finally dropping to the lakeshore, the ledge has several wide step-downs, all of which make excellent spots to dally.\n\nThe trail continues southwest among the mixed pines for a while, and you reach another designated campsite at 9 miles. The site is just off the trail but is quite close to the lake and is a good spot for anyone wishing to avoid the crowds that congregate at lean-tos. From this point on, many portions of the east-shore trail hug the shoreline, and views of Pharaoh Mountain are superb. Highbush blueberries crowd the trail along the exposed portions of lichen-covered ledge that form the shoreline and much of the trail. At 9.3 miles, you cross a plank bridge to a small peninsula and pass an informal campsite that sees regular use. A quarter mile later you pass a designated campsite with exceptional views of Pharaoh Mountain across the lake.\n\nYou reach another major intersection at 9.9 miles. To the left and marked in red are Springhill Pond, 4.5 miles, and New Hague Road, 8.3 miles, while the east-shore trail continues straight ahead, still marked in yellow. Shortly ahead, you reach Lean-To 2. This is the largest of the lean-tos\u2014double the capacity of the others\u2014with a pleasant grassy field sloping down to the lake in front of it. The view is less desirable than in other locations but seems very popular nonetheless. The trail toward the outlet and Lean-To 1 is an old road that passes through a tall hemlock-and-pine forest. The wide road is free of obstructions and allows deep views into the surrounding forest. As you come down a slight hill, you see the side of Lean-To 1. Close to the shore, it has beautiful views of the north shore and, like the rest of the lean-tos, has a watercraft stowed nearby.\n\nThe wide road continues south to the outlet of the lake, but you have to pass through a long mire before the canopy opens up at the grassy southern tip of the lake, 10.7 miles. The trail, now marked with red disks, heads north along the lakeshore, and you quickly reach another yellow camping disk with an arrow indicating a site nearby. The site lies atop the small knoll to your right. Though it's high above the water, views of the lake are shielded by a rise in the hill and the saplings that line the hill. Nonetheless, it is a fine site and ideal for those seeking privacy and a sheltered setting. A little more than 0.3 mile from the outlet, you reach a fork in the trail; veer left to follow the trail, or head right following yellow disks to Lean-To 6. Though it is situated more than 100 yards from the lakeshore, it has excellent views, and the deep evergreen forest that shades the site makes it feel uniquely isolated.\n\nThe trail continues north, winding through hemlock groves along the lakeshore and up into deciduous hilltops. At 11.4 miles, you pass another designated site, which is situated on the forest side of the trail. This is probably the least desirable of the sites but would make a good stopping point if the lean-tos are full and night is coming on. At 11.6 miles, you reach yet another fork in the trail. To your left is the main trail, while straight ahead is Lean-To 5 (also known as Watch Rocks Lean-To). The lean-to trail is marked in yellow and closely follows the shoreline. Located near the tip of the peninsula, it has views in all directions, the most exceptional being at the point itself.\n\nThe main trail, rougher here than in other sections, climbs slightly and weaves its way mostly through a thick hardwood forest. The lake is no longer visible, and there is very little of note until you return to the intersection with Pharaoh Mountain Trail, 12.4 miles. Continue until you reach the intersection with Glidden Marsh Trail, 13 miles. Emblazoned with yellow disks, Glidden Marsh Trail climbs away from the lake immediately. The rocky climb is through mostly deciduous forest, and there are noticeably more ash trees along the slope than elsewhere along the hike. After roughly 0.3 mile, the trail levels off as you begin to pass a broad wetland on your right. The wetland, the result of beaver activity, becomes quite broad, and the trail often passes through its muddy edges for the next mile. The trail leaves the marshy section and reenters the forest as you begin a 0.3-mile descent to another fork in the trail. Straight ahead and marked in yellow is Crane Pond, while off to the right is Crab Pond Spur Trail that passes Crab and Horseshoe Ponds.\n\nShortly after this intersection at 14.6 miles, a log bridge spans a feeder stream of Glidden Marsh, and you reach the eastern shore of the marsh soon after. The trail closely follows the broad, grassy marshland, where you can see several beaver lodges in the center of the deeper water. At 15.2 miles, you come to the intersection with Short Swing Trail, which passes Oxshoe Pond, on your right. The main trail, Glidden Marsh Trail, and Short Swing Trail are marked with blue disks. You lose sight of Glidden Marsh as you pass on the eastern side of a small hill that flanks the marsh. Another marsh down to your right is a dammed stream that feeds Crane Pond before the trail turns west. The trail crosses a log bridge and then climbs a rocky hill before descending to the intersection with Pharaoh Mountain Trail. Heading north on the old, abandoned road, the mileage is as follows: less than 0.5 mile to the trail register, 0.7 mile to the Crane Pond parking area, and 2.6 miles to the designated parking area. This section of the trail is marked with red disks and is easy to follow, but for a more thorough description, read the Pharaoh Mountain Trail section.\n\nLEAN-TO 4 AT PHARAOH LAKE\n\nDirections\n\nFrom I-87, take Exit 28, and head east on NY 74. Immediately turn right onto US 9 in Schroon Lake. In 0.6 mile turn left onto Alder Meadow Road. Follow Alder Meadow Road 2.1 miles, and bear left onto Crane Pond Road. The parking lot is on the left where the pavement ends and Crane Pond Road turns into an unmaintained road at 1.4 miles.\n\nNorthern\n\nOSWEGATCHIE RIVER SEEN FROM HIGH ROCK (Trail 32, High Falls)\n\n 30 OWLS HEAD MOUNTAIN\n\n 31 CAT MOUNTAIN\n\n 32 HIGH FALLS\n\n 33 LAMPSON FALLS AND GRASS RIVER\n\n 34 FLOODWOOD\n\n 35 SAINT REGIS MOUNTAIN\n\n 36 DEBAR MOUNTAIN\n\n30\n\nOwls Head Mountain\n\nVISTA FROM ATOP OWLS HEAD FIRE TOWER\n\n**SCENERY:**\n\n**TRAIL CONDITION:**\n\n**CHILDREN:**\n\n**DIFFICULTY:**\n\n**SOLITUDE:**\n\nGPS COORDINATES: N43\u00b0 57.812' W74\u00b0 27.173'\n\nDISTANCE & CONFIGURATION: 6.3-mile out-and-back\n\nHIKING TIME: 3\u20134 hours\n\nHIGHLIGHTS: Panoramic views, fire tower, historic site\n\nELEVATION: 1,665' at trailhead, 2,762' at highest point\n\nACCESS: Open 24\/7; no fees or permits required\n\nMAPS: National Geographic Adirondack Park, Old Forge\/Oswegatchie (#745)\n\nFACILITIES: None\n\nWHEELCHAIR ACCESS: No\n\nCONTACTS: Emergency contact: 518-891-0235\n\nOverview\n\nOn the boundary of the west-central and northern sections of the Adirondacks is scenic Owls Head. By climbing the accessible fire tower on the summit, you have a 360-degree view of the surrounding hills and lakes. The remains of an old observer cabin add historical interest.\n\nRoute Details\n\nThe trailhead and register are located on the left side of the small parking pullout. The parking area is located on a curve along Endion Lane and may accommodate a dozen cars if visitors park thoughtfully. The trail, marked with red disks, immediately begins to ascend amid a dense maple-and-beech hardwood forest. After 0.3 mile the trail levels off and follows mostly rolling terrain another 1.3 miles. At 1 mile from the trailhead, you reach an intersection with another trail. To your left is the trail to Owls Head summit, 2.1 miles away, while to your right is a trail to NY 30, 3.5 miles, and Lake Eaton Campground, 3.8 miles. Shortly after the first intersection, you pass an abandoned foot trail on your right, followed by an old road that joins the main trail from behind. The old road has orange placards with white diamonds. A few of these placards, as well as snowmobile disks, are interspersed with the red Department of Environmental Conservation trail markers along the main trail. Aside from the previously mentioned intersections, the trail runs in a relatively straight southwest direction to the summit and is easy to follow.\n\nAt 2 miles, the trail swings south and begins to ascend gradually and then more steeply at 2.3 miles. A seasonal stream flows on your left as you climb through sections of heavy erosion. You reach a crest in the trail at 2.7 miles. You have not yet reached the summit but are passing between two points along the mountain that resemble the horns of an owl. The trail dips down, less than 100 feet of elevation, into a small valley before climbing steeply to the summit. Before you make the final ascent, you pass by the former site of the cabin used by the fire tower's observer.\n\nAround the turn of the 20th century, forest fires swept the Adirondack region, and nearly 1 million acres of forest burned throughout the preserve. In response to this destruction, the state erected fire towers atop many peaks throughout the Adirondacks. The tasks of state-employed observers included locating and triangulating fires. On the side, they also provided innumerable hikers with information and conservation education. A wooden fire tower was erected on the peak of Owls Head around 1911. It was replaced by a steel tower in 1919 and closed in 1979. After restoration work was complete, it was reopened in 2004, and efforts are currently under way to continue maintenance of the tower.\n\nThe final climb to the summit passes over exposed portions of ledge and includes multiple small switchbacks. You will need to use both hands in several sections. The trail suddenly ends as you come through a thick stand of firs near the base of the steel fire tower. Multiple grassy patches threaded through the exposed portions of the summit (elevation: 2,812') make excellent picnic areas. A large slab of exposed rock juts out to the southwest, providing an outstanding lookout. For a 360-degree view of the area, you should climb the fire tower. The fence-enclosed stairs lead to a wooden platform covered by a steel cap typical of the fire towers still in use in the Adirondacks.\n\nOWLS HEAD FIRE TOWER\n\nOn your way up, you may have seen old utility poles alongside the trail. If not, they are more noticeable on your way back down. The two opposing \"branches\" on these poles also resemble owls' horns, making the name of this trail doubly apt.\n\nDirections\n\nFrom the north intersection of NY 30 and NY 28N in Long Lake, head northwest on NY 30. In 1.3 miles turn left onto Endion Lane. Follow Endion Lane 1.6 miles, and look for the gravel pullout on the right.\n\n31\n\nCat Mountain\n\nGLASBY POND\n\n**SCENERY:**\n\n**TRAIL CONDITION:**\n\n**CHILDREN:**\n\n**DIFFICULTY:**\n\n**SOLITUDE:**\n\nGPS COORDINATES: N44\u00b0 07.989' W74\u00b0 54.915'\n\nDISTANCE & CONFIGURATION: 9.7-mile out-and-back (from Wanakena) or 4.8-mile out-and-back (if boating to Janacks Landing)\n\nHIKING TIME: 5\u20136 hours\n\nHIGHLIGHTS: Scenic pond, beautiful vistas\n\nELEVATION: 1,502' at trailhead, 2,260' at highest point\n\nACCESS: Open 24\/7; no fees or permits required\n\nMAPS: Five Ponds Wilderness map: tinyurl.com\/5pondsmap; National Geographic Adirondack Park, Old Forge\/Oswegatchie (#745)\n\nFACILITIES: None\n\nWHEELCHAIR ACCESS: No\n\nCONTACTS: Five Ponds Wilderness: www.dec.ny.gov\/lands\/34719.html; emergency contact: 518-891-0235\n\nOverview\n\nCat Mountain is a beautiful little peak that offers excellent views. Combined with the High Falls overnighter, it makes a nice side trip or an extended day trip on its own. To add a bit more adventure to the hike, you can boat in from Cranberry Lake to Janacks Landing and bypass roughly 5 miles of the round-trip hike.\n\nRoute Details\n\nHike in from the western terminus of the High Falls Loop in Wanakena or boat to Janacks Landing in the Dead Creek Flow arm of Cranberry Lake. The trail described here will be from Wanakena with a side note about boat access; boaters can pick up the trail and follow the description from the Janacks Landing intersection.\n\nThe trail and register are elevated a few feet above the spacious parking area on the southern edge. The trail begins heading east but quickly swings south along an abandoned logging road. The wide and flat roadbed, over relatively level terrain, allows for easy hiking for the first 2 miles. You will make quick time, except where flooding due to beaver activity inundates the trail. You will encounter several mires, as well as sections of knee-deep water. The nearly black, murky waters are deceptive: most of the time the water is less than 1 inch deep when it flows over the berms and beaver dams, but occasional breaks in dams and the soft mud underfoot can lead to drops of 1 foot or more, with little evidence provided by the placid waters. Hiking sticks act as probes in questionable spots and, as is the case at most trailheads, previous hikers typically leave decent sticks at the trail register.\n\nThe road, emblazoned with red disks, gently descends most of the way, and at approximately 1.5 miles, the waters of Dead Creek Flow become visible. The road closely follows the shoreline 0.4 mile, at which point you come across a large campsite. Boaters often use this and the other 46 designated primitive sites along the lakeshore. At this point, the road ends, and the trail becomes a typical Adirondack trail. It crosses Dead Creek at 2.3 miles and then swings around the bay of Dead Creek Flow and heads east 0.5 mile.\n\nAfter the short easterly jog along the trail, you intersect the Janacks Landing Trail. The trail is now heading nearly due south and gradually climbs to Sand Hill Junction. The climb is steady and crisscrosses a small stream in several places before reaching a junction at 3.6 miles. Straight ahead is the easterly leg of the High Falls Loop, while the Cat Mountain summit and Glasby Pond are to your left along the Cowhorn Junction Trail.\n\nThe trail, marked with yellow disks, climbs steadily along Glasby Creek past a few small cascades, and the rush of water on your right is a pleasant addition to the trail. After passing through a small, bramble-filled meadow, you come out on the western shores of Glasby Pond at 3.8 miles. The steep cliffs of Cat Mountain provide a striking view across the pristine waters. You will pass through a large campsite before weaving your way up and over a tiny hillock, after which the pondside trail becomes a continuous mire. The terrain grows steadily steeper as you head away from the pond, and you soon reach the final fork in the trail. The Cowhorn Junction Trail continues off to your right toward Cowhorn and Cat Mountain Ponds, marked by yellow disks; to reach the Cat Mountain summit, veer left and follow the red trail markers.\n\nThe next 0.6 mile to the summit is the steepest section, and the trail is a mix of short scrambles over bedrock and boulder-strewn steps. The hardwoods at the base of the mountain give way to windswept evergreens as you weave your way through small switchbacks. Paths on the left reach a few rocky outlooks. These offer nice views, but the best are available at the summit. The trail finally winds its way out onto the southern clifftops. You'll find a solar-powered weather station and the remnants of an old fire tower here at the trail end. Enjoy beautiful views on this portion of the cliffs, or search for better ones by dipping into the coppice of evergreens behind you and making your way along paths to the next section of the clifftop. The drop-off to the south is quite sheer, allowing spectacular views down to Cat Mountain Pond as well as Roundtop and Threemile Mountains along the horizon.\n\nA note about boat access: A public boat launch is located just off NY 3, on Columbian Road near Cranberry Lake's dam in the village of Cranberry Lake. The hard-surface launch can accommodate trailer launches and, according to the Department of Environmental Conservation, can fit 55 vehicles. Navigating from the launch site to Janacks Landing should not be too difficult if you follow the western shore into Dead Creek Flow Bay.\n\nDirections\n\nFrom the intersection of NY 3 and Youngs Road in Star Lake, head 6.1 miles east on NY 3, and then follow the directions below. From the intersection of NY 3 and Lone Pine Road in Cranberry Lake, head 8.3 miles west on NY 3, and follow the directions below.\n\nTurn southeast onto County Road 61\/Main Street. Bear right at 0.8 mile onto Main Street, which becomes South Shore Road; continue 0.5 mile and cross the steel bridge over the Oswegatchie River. After the bridge, go another 0.5 mile, and the parking area and trailhead are on the right.\n\n32\n\nHigh Falls\n\nHIGH FALLS\n\n**SCENERY:**\n\n**TRAIL CONDITION:**\n\n**CHILDREN:**\n\n**DIFFICULTY:**\n\n**SOLITUDE:**\n\nGPS COORDINATES: N44\u00b0 07.911' W74\u00b0 55.383'\n\nDISTANCE & CONFIGURATION: 16.2-mile loop\n\nHIKING TIME: Overnight\n\nHIGHLIGHTS: Isolated waterfall, numerous beaver dams, wetlands and ponds, winding-flat river, lakeside hiking, berry picking\n\nELEVATION: 1,490' at trailhead, 1,660' at highest point\n\nACCESS: Open 24\/7; no fees or permits required\n\nMAPS: Five Ponds Wilderness map: tinyurl.com\/5pondsmap; National Geographic Adirondack Park, Old Forge\/Oswegatchie (#745)\n\nFACILITIES: None\n\nWHEELCHAIR ACCESS: No\n\nCONTACTS: Five Ponds Wilderness: www.dec.ny.gov\/lands\/34719.html; emergency contact: 518-891-0235\n\nOverview\n\nWith a wide variety of habitats and mostly easy trekking, this is a great introductory backpacking trip. Beaver activity is practically everywhere, with several dam tops forming part of the trail, so don't expect to keep your feet dry for long. In midsummer, the abundance of blueberries practically turns foraging into hike-by food service.\n\nRoute Details\n\nParking is available on either end of the loop, with roughly 0.5 mile of road in between. You will have to make the roadside trek either before or after your wilderness experience. With merely 0.5 mile and limited parking in each area, the use of two cars seems extravagant. My recommendation is to stow your pack at the trailhead on which you wish to start, park at the other trailhead, and walk back to your desired start. You can hike the trail in either direction: heading counterclockwise, High Falls is 9.0 miles; heading clockwise, it is 6.5 miles. You can add a side trip to Cat Mountain (described) to the clockwise approach or on the return, making the mileage on the eastern leg the same. I recommend camping at High Falls, but if time or weather forces you to camp early, designated riverside campsites (briefly described below) are located along the western leg, while camping along the eastern leg will require short bushwhacks to adhere to Department of Environmental Conservation (DEC) wilderness camping regulations. Trail directions below are heading counterclockwise.\n\nFrom the western parking area, walk around the tennis court down the private road to a vehicle-barrier gate just past a residence on your right. The trail register is a little beyond the barrier and has a map of the area, including 46 designated campsites along the Oswegatchie River. Most of these sites are either across the river or require bushwhacking, so these are mainly used by canoeists and kayakers. The first mile of the trail is along a grass-covered gravel road, which provides access for municipal water. With the trail free of any obstructions, you will have the opportunity to look out for wildlife in the dammed section of Skate Creek on your left. A little more than 0.5 mile in, you will cross Skate Creek, and its wetland will sprawl off to your right. These broad, marshy areas are very common in the northern section of the Adirondacks and are often seen from the road but are not frequently traversable on foot.\n\nApproximately 1 mile in, the trail crosses a stream that feeds into Skate Creek and begins to climb as you plunge into a mixed hardwood forest. It would be hard to get lost along this wide-open road, but red DEC markers assure you that you are headed in the right direction. At 1.8 miles, you will come to a clearing where an overgrown road climbs to your left. In the past, numerous roads and paths headed more directly to the High Falls area, but the microburst windstorm of 1995 created so much blowdown that these trails have been abandoned. Beyond the clearing, ferns and other deciduous plants encroach on the trail, but its past use as a road is unmistakable, as it is broad, flat, and free of typical obstructions and is elevated above the surrounding terrain. The western half of the trail consists almost entirely of abandoned roads, which makes its 9.0 miles nearly equivalent in effort to the 6.5-mile hike along the eastern leg.\n\nAt 2.3 miles, the canopy opens, and another creek with a broad wetland sprawls out on your right. Despite being flanked by short evergreens, the trail is open to the sky. At this point, you have reached the plateau that the western leg traverses, and the rest of the way gently rolls along at this elevation. During the summer, the trailside will be thick with highbush blueberries.\n\nONE OF THE MANY AREAS THAT HAVE FLOODED DUE TO BEAVER ACTIVITY\n\nAt 3.8 miles, you reenter the hardwood forest and come to a fork in the trail. High Rock, which overlooks the Oswegatchie River, is 0.1 mile to your right. The boulder that provides an outlook across the meandering river is actually not very high. However, the plain through which the river cuts its veritable maze is so broad that you can see quite far. A popular campsite is also located here, so do not be surprised to see your first group of boaters. Though the Oswegatchie flows north to the west of the trail, this side trip will be your only opportunity to see it for several miles.\n\nHeading south, you walk over a stone-and-concrete bridge that spans a stream at 4.3 miles. At 5.3 miles, you encounter the first of many beaver dams that have created wide marshlands both up- and downstream. To cross this and future dammed sections, you will walk along the tops of the dams. Picking your way atop the dams is not difficult, but wet feet are inevitable. Aside from the wet areas created by the beaver dams, the trail is relatively level and easygoing along these old logging roads. Less than 1 mile ahead, mile 6.1, you traverse another dammed section, and then shortly after reenter the forest.\n\nIn 0.5 mile the trail meets the Oswegatchie, as well as the first of three easily accessible designated campsites along the river. The first site, with no number, is directly on the trail, while a short way ahead a path leads to site 28, which is masked by a stand of thick saplings. This site also has the advantage of sitting just above a small flume that adds interest to the mostly flat river.\n\nShortly after this brief encounter with the river, you head back into wetland territory along another stream and then back into the forest, where you reach a fork in the trail, 7.5 miles. Trail signs indicate that High Falls is 1.5 miles ahead, while the fork to the right takes you deep into the Five Ponds Wilderness. Shortly ahead, you enter an area where Glasby Creek and the Oswegatchie River meet. This confluence creates a broad wetland and meadow setting with beaver activity throughout. Yellow disks indicate a couple of designated campsites in the area, but their locations escaped me. Many of the designated campsites either were affected by the 1995 microburst storm or see little use because most people are headed to the High Falls lean-tos or campsites. After crossing a log-and-plank bridge, the trail climbs through a short section of stunted pines that gradually gives way to mixed hardwoods.\n\nAt mile 8.6 you reach an intersection; High Falls is 0.5 mile to the right, while it is 6.1 miles back to Wanakena along the eastern leg. Along the trail to High Falls, you will come upon an abandoned and heavily rusted piece of machinery. Its presence adds a quaint reminder that, not too long ago, this area was used by industry, but the wilderness quickly reestablishes itself and wipes out all but the most persistent traces of human activity. After a brief climb, you will hear High Falls and then come to designated campsite 16 and the confluence of several paths, 9.1 miles. The trail directly to the right is the canoe carry that bypasses the falls. A path slightly to your right leads to a boulder at the base of the falls. A path to the left leads to one of the pit toilets in the area. The path straight ahead leads to the start of the canoe carry, the top of the falls, and the lean-to on this side of the river. The falls are only 20 feet tall but are still excellent. The water bifurcates, forming a spray on the left and a rushing torrent on the right. The pool at the base is deep enough to swim in, and while I was there, a heron fishing for brown trout occupied it.\n\nThe return trip along the eastern leg is a sharp contrast to the easy hiking along the old logging roads of the western leg. However, it is no different from typical trails in the Adirondacks, except perhaps for the vast amount of beaver activity. Following blue DEC disks, head east from the junction with the western leg; total mileage before starting this leg is 9.6 miles. After picking your way up and over a hillock, where damage from the 1995 microburst storm is still evident, you reach a broad wetland in less than 0.5 mile. A large fallen pine tree forms a bridge over the stream, which should be carefully negotiated, especially when wet. The next 2.2 miles take you along the base of Threemile Mountain through myriad wetlands created by successive beaver dams. The mix of forest and open marshlands created by the beaver dams creates some very interesting scenery complemented by abundant opportunities to pick berries, including blueberries, raspberries, blackcaps, and blackberries, depending on the season.\n\nABANDONED MACHINERY ALONG HIGH FALLS TRAIL\n\nAt 12.2 miles you reach the intersection of the eastern leg, with Cowhorn Junction Trail on the right. This intersection is Sand Hill Junction, where a side trip up to Cat Mountain is highly recommended. Red DEC disks now mark the trail, and you will be heading generally north as the trail begins to descend. You will have to negotiate a few more wet areas before reaching a trail register at the intersection with Janacks Landing Trail. Less than 0.3 mile to the right is a lean-to and boat landing for hikers who wish to boat in and avoid the initial 2.8 miles of the hike along the eastern leg. The nearest boat launch is 7.8 miles away, as the crow flies, in the village of Cranberry Lake; the boat launch is off NY 3 near the dam. Most of the remaining hike is along old logging roads, so hiking in is likely the fastest and most direct approach.\n\nFrom the intersection, the trail veers west 0.5 mile, where it crosses Dead Creek Flow, which is the largest submerged bay of Cranberry Lake. The distinction may sound peculiar until you realize that this bay was previously the creekbed until Cranberry Lake was dammed in 1867. The hike around the western point of Dead Creek Flow adds yet another scenic element. After coming around the western point, you reach a large designated campground and the beginning of the old logging road, which will take you back to Wanakena. The road climbs with a gentle grade, and you will encounter a few more beaver-flooded areas before reaching the eastern parking area at 16.2 miles.\n\nDirections\n\nFrom the intersection of NY 3 and Youngs Road in Star Lake, head 6.1 miles east on NY 3, and then follow the directions below. From the intersection of NY 3 and Lone Pine Road in Cranberry Lake, head 8.3 miles west on NY 3, and follow the directions below.\n\nTurn southeast onto County Road 61\/Main Street. Bear right at 0.8 mile onto Main Street, which becomes South Shore Road; continue 0.5 mile to cross the steel bridge over the Oswegatchie River. The western trailhead is accessed along a dirt road 0.1 mile past the bridge and on the right. The closest parking area to this trailhead is a little farther along South Shore Road, also on the right just past the tennis court. The eastern trailhead and its parking area are 0.5 mile past the bridge on the right side of the road.\n\n33\n\nLampson Falls and Grass River\n\nLAMPSON FALLS\n\n**SCENERY:**\n\n**TRAIL CONDITION:**\n\n**CHILDREN:**\n\n**DIFFICULTY:**\n\n**SOLITUDE:**\n\nGPS COORDINATES: N44\u00b0 24.302' W75\u00b0 03.680'\n\nDISTANCE & CONFIGURATION: 1.0-mile out-and-back to falls, 3.0-mile out-and-back including riverside trail\n\nHIKING TIME: 1\u20132 hours\n\nHIGHLIGHTS: Waterfall, riverside walk\n\nELEVATION: 820' at trailhead, 740' at lowest point\n\nACCESS: Open 24\/7; no fees or permits required\n\nMAPS: National Geographic Adirondack Park, Saranac\/Paul Smiths (#746)\n\nFACILITIES: None\n\nWHEELCHAIR ACCESS: Yes, on the section of the trail leading to the waterfall lookout\n\nCONTACTS: Northern Adirondack trail information: www.dec.ny.gov\/outdoor\/9196.html; emergency contact: 518-891-0235\n\nOverview\n\nAs one of a few hikes with sections that are wheelchair accessible, this trail is ideal for families with limited mobility or who just prefer a trip that does not require constant weaving around obstacles. If you prefer obstacles, you can always extend your trip to include the riverside trail and a little extra bushwhacking downstream to explore some interesting cataracts and flumes. Lampson Falls is also a popular swimming spot during the summer, but if you continue along the riverside trail, you will find solitude.\n\nRoute Details\n\nParking for the trailhead is parallel to County Road 27; two of the five designated spaces are reserved for handicapped parking, though an additional car or two could fit in behind the regular parking spaces. The trail begins at the southern end of the pullout, past a vehicle barrier. The trail is actually a crushed-gravel access road, which is why it is designated universally accessible. The trail register and a map of the trail system are located a little less than 0.3 mile past the vehicle barrier. The forest is a mixture of eastern white pines and hardwoods with a lush and thick understory of ferns. The terrain, befitting a wheelchair-accessible trail, is exceptionally flat, and you will hear the roar of the falls long before you reach the wheelchair-accessible viewing area. Follow the wide path through a series of switchbacks and numerous turning points, constructed out of landscape timbers and filled with gravel, to the viewing spot above the falls. From the top of the falls, it is easy to see why this is a popular attraction, as the Grass River cascades down over the exposed bedrock in a frothy torrent. From this vantage point, keep an eye out for the swifts that swoop and dive as they ride updrafts and feast on insects caught above the falls. To reach the base of the falls, you can choose any number of footpaths or scramble down the rocks, but the accessible portion of the trail ends here.\n\nAt the base of the falls is a sandy beach that is an obvious congregation point and an unfortunate reminder that not all people follow the Department of Environmental Conservation's carry-in, carry-out policy. This is an easy place to implement one of my preferred policies: carry in, carry one more out. From the sandy beach, you can scramble up a steep hillside to reach exposed portions of bedrock directly across from the waterfall. This is my favorite vantage point of the falls and is an excellent spot to stop and enjoy the view. Back among the tall pines is a designated camping area that includes the standard pit toilet but offers no views of the waterfall or river. Continue along the bedrock, across from the falls, to a rocky peninsula for another picnicking spot, as well as the beginning of the riverside trail.\n\nThe trail was in serious disrepair in 2009, with multiple areas of nearly impassable blowdown. However, Department of Environmental Conservation and volunteer trail crews clear blowdown regularly, so the trail may have been returned to a suitable condition, making it an easy riverside stroll. Marked with red disks, the trail is evidently rarely used, and thick overgrown vegetation on either side masks the conditions underfoot. Brambles are a significant part of the overgrowth, so gaiters are recommended during the summer. Additionally, numerous beaver-plunge holes are hidden by the dense overgrowth, so use caution when you cannot see your footing.\n\nACCESSIBLE TRAIL TO LAMPSON FALLS\n\nIn sharp contrast to the turbulent falls, the river seems almost placid, and its grass- and fern-lined banks make for a tranquil setting. At 1.5 miles the trail intersects an overgrown road to your right, and the remnants of a foundation are visible on both banks. Readers of the old trail guides will discover that the bridge connecting the two banks was washed out in 1998 and has not been replaced. The foundations sit on either side of a small flume, where the river narrows from dozens of feet to just a few feet. This is the formal end of the trail, which is unfortunate because several flumes, falls, and cataracts farther downstream add great interest to the trip. To see these spectacles, you will have to follow old footpaths and bushwhack a bit, but rest assured, other hikers have been here before, so the path is distinct most of the way. Over the next 0.3 mile, the placid waters transform into frothing cascades as the river drops quickly and the lush, wide banks constrict to rocky ledges. You will see multiple flumes and falls along the paths, but my favorite is where the river forks at a tiny, rocky island. The eastern fork drops vertically to form a narrow waterfall, after which the flow diminishes into slack water, while the western leg forms a frothing cataract. You can reach the island by crossing above the tiny waterfall or rock-hopping in the slack water. This section of the river coincides with the disappearance of the old footpaths and makes a good end for the river exploration.\n\nDirections\n\nFrom the intersection of NY 3 and Youngs Road in Star Lake, head 6.3 miles northwest along NY 3, and turn right onto CR 27\/CR 27A. Continue 0.8 mile, and turn right onto CR 27\/Degrasse-Fine Road heading north. Follow CR 27 north 7.8 miles, and turn right to stay on CR 27, now also known as Fine-Canton-Lisbon Road. Continue on CR 27 another 4.5 miles, passing the airstrip along the way. The parking area will be on your left.\n\n34\n\nFloodwood\n\nMIDDLE POND SEEN FROM FLOODWOOD ROAD\n\n**SCENERY:**\n\n**TRAIL CONDITION:**\n\n**CHILDREN:**\n\n**DIFFICULTY:**\n\n**SOLITUDE:**\n\nGPS COORDINATES: N44\u00b0 18.497' W74\u00b0 21.550'\n\nDISTANCE & CONFIGURATION: 8.5-mile loop\n\nHIKING TIME: 3\u20134 hours\n\nHIGHLIGHTS: Multiple ponds\n\nELEVATION: 1,590' at trailhead, with no significant rise along the trail\n\nACCESS: $8 day-use fee during camping season; free when camping at Fish Creek Pond\/Rollins Pond Campgrounds or with an Empire Passport. Campgrounds open mid-April\u2013late-October; check the Department of Environmental Conservation campground schedule for exact dates: www.dec.ny.gov\/outdoor\/7820.html.\n\nMAPS: National Geographic Adirondack Park, Saranac\/Paul Smiths (#746)\n\nFACILITIES: Restrooms, swimming beach, boat rentals when campgrounds are open\n\nWHEELCHAIR ACCESS: No\n\nCOMMENTS: Access is allowed and the day-use fee is waived during the off-season, but hikers will have to hike in along the access road from NY 30, which adds 0.8 mile to the trip. Campground roads are not plowed during the winter.\n\nCONTACTS: Fish Creek Pond Campground: www.dec.ny.gov\/outdoor\/24466.html; Northern Adirondacks trail information: www.dec.ny.gov\/outdoor\/9196.html; emergency contact: 518-891-0235\n\nOverview\n\nAt first glance, this hike may seem long, but the gently rolling terrain makes it an easy and relatively quick hike. Winding through a hardwood forest, the trail leads you from one pristine pond to another, offering scenery and solitude the whole trip. Though the trail starts and ends in a bustling campground, it is likely that the only people you will encounter are canoeists at the numerous portages that crisscross the trail.\n\nRoute Details\n\nThe trailhead is slightly different from the other Adirondack trailheads in that access is through a campground, which includes a day-use fee of $8; no designated parking is located immediately nearby; and there is no trail register. The fee is waived if you are camping at either Fish Creek Pond or Rollins Pond Campgrounds; for a description of the latter, see Best Tent Camping: New York State. The parking areas closest to the trailhead are in the day-use area near the swimming beach (go left after the registration booth) or near the basketball courts, the recreation area, and campsite 50, past the boat launch area (go right after the registration booth). Make sure to grab a campground map, as it makes finding the trailhead easier and includes a trail map.\n\nThe trailhead is located on the access road for Fish Creek Pond campsites C1\u2013C7. While walking up the road, you will see red Department of Environmental Conservation snowmobile trail markers and will come upon two log stumps that act as vehicle barriers on your right. You can begin here or farther up the road, near campsite C7. The trail description included here is from the vehicle barrier heading north along the eastern leg and returning south along the western leg. There is little difference heading clockwise or counterclockwise, though if you plan to walk only one leg of the trip, then I would choose the western leg, as it is in better shape and a flooded area on the eastern leg may seem impassable.\n\nThe trail winds north under a canopy of tall pines as well as multiple species of birch, and within a few hundred yards you will catch glimpses of Echo Pond. Dense pine saplings quickly dampen the noise from the campground, and you feel removed and remote. Another element you will notice is that the trail is basically flat, and your pace will be closer to walking along a sidewalk than hiking. Trail markers switch between snowmobile and foot-trail disks, but the path is so distinct that very little marking is necessary. You will catch glimpses of the much larger Follensby Clear Pond to your right and quickly come to a distinct intersection at mile 0.7. The east\u2013west trail that bisects the hiking trail is the first of several canoe carries and provides a portage from Fish Creek Pond to Follensby Clear Pond. You will cross this portage again on the return trip, but for now it provides a convenient way to get an up close view of the pond to the west. The more heavily trodden canoe carry reveals that canoeing is the dominant activity in the area, which provides the hiker a bit more solitude on the trail.\n\nContinuing straight ahead, you will notice that the hemlocks and birches are quite large and tower overhead. Deep views into the forest contrast with limited views out onto the bodies of water, as hemlock saplings crowd the shorelines. You first encounter Horseshoe Pond and walk along its edge through thicker hemlocks 1 mile from the start. At 1.4 miles, you come to a large moss-covered, mucky area that is the beginning of a spruce swamp bisected by the trail. At 1.8 miles it may seem like you are coming up on another pond, but it is the opposite leg of Horseshoe Pond. The shore is distinctly more visible to your right as you head more directly north. At 2.3 miles, you come to the outlet of Little Polliwog Pond as it empties into Horseshoe Pond. A sign warns that the bridge is out; indeed, the bridge has fallen apart, though remnants of rickety boards laid on logs provide a zigzag crossing. On the other side, the trail briefly shares the canoe carry between Little Polliwog and Horseshoe Ponds. At mile 2.9 you come across the canoe carry between Little Polliwog and Polliwog Ponds through a stand of hemlocks.\n\nAround 3 miles you arrive at a broad, flooded section of the trail created by a recently constructed beaver dam. At this point your options for crossing the flooded section are limited. Crossing on top of the dam will likely be your driest option, but don't count on having dry feet afterward. The trail on the other side is less clear and is overgrown, and trail markers seemed to be missing when I was there. However, though the trail appears to be more of a herd path, it does continue through a couple of muddy areas, and red trail markers are visible again in less than 0.3 mile. You pass by another bay of Polliwog Pond at 3.2 miles, and at 3.4 miles you come to a T in the path. Signs indicate that the snowmobile trail continues to your left. The path joining the trail from your right is the canoe carry between Polliwog Pond and Middle Pond. Head west (left) along the combined trail and portage less than 0.3 mile, where the portage continues straight and the trail veers right toward Floodwood Road. Less than 0.1 mile ahead is Floodwood Road, which you will follow 1 mile west to pick up the western leg of the loop. It might seem odd to include a road with a trail, but the road is actually a pleasant variation on the trip. This seldom-used dirt road passes within feet of Middle Pond, and because it is free of roots and other obstructions, you can spend the mile looking out on the placid waters of the pond. You will pass one of several designated primitive campsites along the road.\n\nWOOD PALLETS ACROSS AN INLET OF FLOODWOOD POND SERVE AS A BRIDGE.\n\nAt 4.7 miles you pick the trail up again and head back into the woods along a ridge that sits above Middle Pond. In less than 0.3 mile the trail heads east, and you cross a log bridge. You will have left the shores of Middle Pond by now, but you have another chance to return to its shores at 5.1 miles, when you arrive at a fork in the trail. Continuing south along the trail, you catch glimpses of Floodwood Pond just before you reach yet another fork; the path ahead leads to the pond's shores, while the trail continues on your left. Pallets laid in pairs form a bridge across a stream, the outlet of Middle Pond into Floodwood Pond, at 5.4 miles. Shortly after the crossing, you come back to Floodwood Pond and stroll along its shores for the next 0.5 mile. At 6 miles, you pass through a designated campsite near where Floodwood Pond flows into a creek. Take a left at the campsite, and briefly head west, where you reach another fork in the trail. To the right are Floodwood Passage Bridge and Otter Hollow Trail; these take you back to the campgrounds, while the Floodwood Trail continues straight ahead beside the creek. Follow the Floodwood Trail as it climbs slightly alongside the creek and then descends around 6.5 miles to another designated campsite. These designated campsites are used by canoeists and kayakers, but if they are vacant, their picnic tables make an excellent spot to stop, sit, and enjoy the scenery. The trail turns left at the campsite and continues along the shore of Little Square Pond through a couple more designated campsites.\n\nThe trail continues southeast along Little Square Pond as it transitions into Fish Creek Pond. From this point on, the trail delves deeper into the forest, and views of the water are limited to occasional glimpses. At mile 8 you pass the first canoe carry you encountered near the beginning and quickly come to the end of the trail at campsite C7.\n\nDirections\n\nFrom Tupper Lake, head east on NY 3\/NY 30. At 5.3 miles, turn left to stay on NY 30. After another 5.4 miles, turn left at the main entrance to the Fish Creek Pond and Rollins Pond Campgrounds. You will have to pay a day-use fee of $8 unless you are camping at either campground. The main parking area is to the left of the registration booth. The trailhead is along the access road to campsites C1\u2013C7.\n\n35\n\nSaint Regis Mountain\n\nTRAIL REGISTER FOR SAINT REGIS MOUNTAIN TRAIL\n\n**SCENERY:**\n\n**TRAIL CONDITION:**\n\n**CHILDREN:**\n\n**DIFFICULTY:**\n\n**SOLITUDE:**\n\nGPS COORDINATES: N44\u00b0 25.826' W74\u00b0 17.950'\n\nDISTANCE & CONFIGURATION: 6.7-mile out-and-back\n\nHIKING TIME: 3\u20134 hours\n\nHIGHLIGHTS: Panoramic views, fire tower\n\nELEVATION: 1,622' at trailhead, 2,874' at highest point\n\nACCESS: Open 24\/7; no fees or permits required\n\nMAPS: National Geographic Adirondack Park, Saranac\/Paul Smiths (#746)\n\nFACILITIES: None\n\nWHEELCHAIR ACCESS: No\n\nCONTACTS: Northern Adirondack trail information: www.dec.ny.gov\/outdoor\/9196.html; emergency contact: 518-891-0235\n\nOverview\n\nBest known for its mostly flat terrain, the northern section of the Adirondacks has many excellent extended canoe trips. Most notable is the Saint Regis Canoe Area. However, looming over this area of hundreds of ponds, lakes, and interconnected waterways is Saint Regis Mountain. For a truly majestic view, its summit is unsurpassed.\n\nRoute Details\n\nThe actual trailhead is a short distance south of the large and well-maintained parking area. From the parking area, no signs point you to the trail. To reach it, head south down Topridge Road across the Saint Regis River, whose rushing waters you probably noticed as soon as you parked. Approximately 0.1 mile down the road, you will see the trail register just off to the right. Next to the register is a finely built log-and-plank bridge that is indicative of ongoing work to maintain the trail. Indeed, the presence of so many mucky areas, despite the amount of maintenance, shows how popular this trail is. The Department of Environmental Conservation (DEC) constructed the bridge when it rerouted the trail in 1999, but forestry students from Paul Smith's College and stewards have done much of the work evident throughout. Anyone wishing for solitude atop the mountain would be wise to follow a few rules of thumb: First, hike during the week rather than on weekends, and especially avoid holiday weekends. Second, hike when school is in session. Last, start earlier, which usually ensures you some time to yourself at the peak and other destinations where hikers typically congregate.\n\nThe trail, marked with red disks, follows generally rolling terrain through mature hardwoods dotted with large hemlocks. Approximately 0.3 mile in, you will pass a sheer rock face covered in moss and ferns. This adds some variation to the hike, but the dominant feature before the summit is the deep forest. At 1 mile, you will walk over a small hillock heavily shaded by hemlocks and then descend briefly back into the mixed hardwood forest. The trail becomes very straight and begins to climb afterward, but the grade is gradual and almost difficult to notice. You pass an orange-painted pipe surrounded by stones on the left at 1.7 miles; this is approximately the halfway point of the trail. After a slight descent around mile 2.1, you come to a fork and a bridge that crosses a small stream. The trail to your left is the old trail that is no longer maintained.\n\nOnce you cross the bridge, the ascent to the summit truly begins. Gradual at first, it becomes much steeper as you near the summit. The stream you crossed flows on your right as you climb and occasionally flows over the trail nearer the summit. Stone steps built along the trail help prevent further erosion, but long sections of muddy areas become inevitable. Indeed, the trail has been worn into a rut more than a foot deep. Estimating how far along the trail these mires are is futile, as one slick seems to grow larger than the previous one, and you quickly lose track of how many you cross. At 2.9 miles, you pass a vertical rock face almost entirely covered in moss. Another distinct landmark is at mile 3.1, where you pass a large balancing rock on your left and another boulder directly to your right. Passage between these boulders is essentially the last portion of steep climbing, and the final elevation gains are over flatter portions. Winding around the contours of the peak, the trail is shaded by mostly paper birches with an occasional mountain ash. Keep an eye out on your right for a lookout, which offers good panoramic views and a glimpse of the fire tower.\n\nAt the summit, you have 360-degree views of the myriad lakes and ponds nestled among forested hills and small mountains within the immediate vicinity, with the High Peaks towering off to the south. The fire tower has gone through renovation by the DEC, Student Conservation Association Adirondack AmeriCorps Program, and Friends of St. Regis Mountain Fire Tower. The restoration of the fire tower is complete, and the tower is now publicly accessible. Its location in a canoe area had been a much-debated issue. The DEC Master Plan calls for the removal of fire towers from wild forest areas, which include canoe areas, but revisions and conflicts with the New York Historic Preservation Act of 1980 complicate the matter. Many groups have formed to protect and restore fire towers on this and other peaks, claiming that fire towers are part of the region's history and that their stewards were instrumental in educating the public and helping to protect and preserve the wilderness.\n\nDirections\n\nFrom Tupper Lake, head east on NY 3\/NY 30. Turn left to stay on NY 30 at 5.3 miles. At 14.2 more miles turn left again to stay on NY 30. Turn left onto Keeses Mill Road at 6.9 miles. The DEC parking area is 2.6 miles ahead on the left.\n\n36\n\nDebar Mountain\n\nSCENE ATOP DEBAR MOUNTAIN\n\n**SCENERY:**\n\n**TRAIL CONDITION:**\n\n**CHILDREN:**\n\n**DIFFICULTY:**\n\n**SOLITUDE:**\n\nGPS COORDINATES: N44\u00b0 34.639' W74\u00b0 16.735'\n\nDISTANCE & CONFIGURATION: 8.8-mile out-and-back, 7.4-mile out-and-back from trailhead parking area (seasonal)\n\nHIKING TIME: 4.5\u20135.5 hours\n\nHIGHLIGHTS: Panoramic views, Bald Peak, old foundation\n\nELEVATION: 1,564' at trailhead, 3,305' at Debar Mountain peak\n\nACCESS: $8 day-use fee during camping season; free when camping at Meacham Lake Campground or with an Empire Passport. Campground open mid-May\u2013mid-October; check the Department of Environmental Conservation campground schedule for exact dates: www.dec.ny.gov\/outdoor\/7820.html.\n\nMAPS: National Geographic Adirondack Park, Saranac\/Paul Smiths (#746)\n\nFACILITIES: Restrooms, swimming beach, boat rentals when campgrounds are open\n\nWHEELCHAIR ACCESS: No\n\nCOMMENTS: Access is allowed and the day-use fee is waived during the off-season, but you will have to hike in along the access road from NY 30, which adds 2 miles to the trip. Campground roads are not plowed during the winter. The final 0.25-mile section to the peak will likely require crampons and an ice ax in winter.\n\nCONTACTS: Meacham Lake Campground: www.dec.ny.gov\/outdoor\/24481.html; Northern Adirondack trail information: www.dec.ny.gov\/outdoor\/9196.html; emergency contact: 518-891-0235\n\nOverview\n\nLike many peak trails in the Adirondacks, this trail begins along level terrain but quickly steepens as you approach the summit. Near the peak, the trail is particularly steep\u2014near vertical in some sections\u2014and will require use of your hands in some sections. But the climb is worth it, and you will be rewarded with a sprawling panorama of the Northern Adirondacks.\n\nRoute Details\n\nThe first decision you will have to make is whether to park in the public parking area located near the beach or at the trailhead parking area, via the access road. The distance from the public parking area to the end of the access road is roughly 0.7 mile, so parking at its end will reduce the trip by about a 1.4 miles overall. The road is in good condition, so most vehicles with have little trouble along the flat drive. But it is a seasonal road and may be closed, so plan your trip to include the access road just in case. Directions and mileage given here include the access road.\n\nFrom the parking area in front of the Meacham Lake Recreation Field (near campsites 45\u201347), head north away from the beach to the main campsite road. Follow this road east a short distance, and take the first left, near campsite 48. Continue north along the road past campsite 37 to a vehicle-barrier gate and the start of the access road; signs indicate this is the trail to Debar. The access road is flat and wide and meanders north another 0.5 mile to the trailhead, second parking area, and trail register.\n\nThe second parking area is a large pulloff on the north side of the road, a short distance before another vehicle-barrier gate and the trail register. Past the barrier gate, the trail still follows an old road, but its closure to regular vehicle traffic has allowed nature to encroach back on the road, transforming the route into a nice broad footpath. With a sandy base and moss- and grass-lined shoulders, the path is quite an easy and pleasant stroll following a mostly level course. The trail weaves between a single- and doubletrack, and two hikers could walk side by side much of the way. You will cross a mossy brook, a tributary to Winnebago Brook, a little over 0.75 mile past the trail register. Less than 0.25 mile farther, 1.6 miles overall, is the intersection with the main trail to Debar Mountain on your left. Straight ahead, the Debar Game Management Trail continues past Debar Meadows, 5 miles, to County Road 26, 6.3 miles.\n\nTurn left (north) onto the main Debar Mountain trail. The trail is sporadically marked with red and yellow trail disks, but they are so infrequent that they are not as reliable as the path worn by the many previous hikers. The trail narrows from here forward, with the grade growing steadily steeper as you begin the 1,700-foot ascent to the peak. The first ascent (roughly 700 feet over the next 1.5 miles) culminates at the saddle between Black Peak to the north and an unnamed knob to the south. Along the first part of this ascent, you will pass a log-and-plank bridge over a seasonal stream, shortly followed by a large, square boulder that interrupts the trail about three-quarters of the way up. At the saddle, the trail levels off and then dips down to cross Hays Brook, after which the climbing begins again.\n\nFrom the brook crossing, another 1.1 miles and roughly 1,000 feet of ascent remain. In less than 0.25 mile, pass the Debar Mountain Lean-To on the left. A little over 0.1 mile ahead, pass through a clearing along the trail. Look to your right and you will find the remains of the fire tower cabin foundation.\n\nTHE DEBAR MOUNTAIN LEAN-TO MAKES A GREAT REST STOP.\n\nPast the clearing, the grade begins to steepen even more, and you may find you need your hands to negotiate some of the steeper sections. A half mile later and 650 feet higher, you reach a recent slide that looms over the trail, 4.1 miles overall. At first glance it appears that the trail passes beneath this imposing and nearly vertical outcrop of rock, but the trail actually ascends to its left. If you didn't need to use your hands to assist your climb before, you surely will now. Water flows freely through the vertical jumble of rocks and roots, making choosing your footing especially tricky at times. In winter this portion may be impassable for those who did not bring crampons or an ice ax. (Remember, winter in the Adirondacks can start as early as late September and continue into early May.) It is another 0.25 mile and roughly 200 feet to the peak.\n\nThe trail levels off briefly before you reach the actual summit. There may be a bit of confusion as to how to best reach it as dense spruce surrounds the bald peak. You can either scramble up a short rock face on the left or continue onto the north side of the peak and double back, pushing your way through the dense evergreens. Either way, what awaits is a sprawling panorama of the Northern Adirondacks. From the peak, you are looking back southwest onto Meacham Lake and Clear Pond and can practically trace the path you traversed. Also atop the bald peak are the remains of the old fire tower foundation and anchor points. Return the way you came, taking your time descending the vertical portions for a round-trip of 8.8 miles.\n\nA GNARLY TREE ON THE WAY TO THE SUMMIT\n\nDirections\n\nFrom Tupper Lake, head east on NY 3\/NY 30. At 5.3 miles, turn left to head north on NY 30. After 14.2 more miles turn left to remain on NY 30. Follow NY 30 another 18.5 miles north, and the entrance to the Meacham Lake State Campground will be on the right. Follow Meacham Road to the campground entrance, 0.6 mile, and turn left. Continue another 0.5 mile to the main parking area, a short distance east past the registration both.\n\nHigh Peaks\n\nAUSABLE LAKE SEEN FROM FISH HAWK CLIFFS (Trail 38, Ausable River: East River Trail)\n\n 37 AUSABLE RIVER: WEST RIVER TRAIL\n\n 38 AUSABLE RIVER: EAST RIVER TRAIL\n\n 39 GIANT'S NUBBLE\n\n 40 HANGING SPEAR FALLS AND OPALESCENT RIVER\n\n 41 INDIAN PASS\n\n 42 ALGONQUIN PEAK\n\n 43 AVALANCHE PASS\n\n 44 LITTLE AND BIG CROW MOUNTAINS\n\n 45 PITCHOFF MOUNTAIN\n\n 46 AMPERSAND MOUNTAIN\n\n37\n\nAusable River: West River Trail\n\nADIRONDACK MOUNTAIN RESERVE GATES\n\nSCENERY:\n\n**TRAIL CONDITION:**\n\n**CHILDREN:**\n\n**DIFFICULTY:**\n\n**SOLITUDE:**\n\nGPS COORDINATES: Parking area: N44\u00b0 08.986' W73\u00b0 46.075' Trailhead: N44\u00b0 08.992' W73\u00b0 46.843'\n\nDISTANCE & CONFIGURATION: 10.8-mile out-and-back or 9.7-mile loop if combined with Ausable River: East River Trail\n\nHIKING TIME: 3\u20134 hours one-way\n\nHIGHLIGHTS: Spectacular waterfalls, scenic lookout, riverside stroll\n\nELEVATION: 1,266' at trailhead, 2,493' at highest point\n\nACCESS: Open 24\/7; no fees or permits required. Guests must register at the gatehouse.\n\nMAPS: Adirondack Mountain Reserve map: tinyurl.com\/adkmtnreserve; National Geographic Adirondack Park, Lake Placid\/High Peaks (#742)\n\nFACILITIES: None\n\nWHEELCHAIR ACCESS: No\n\nCOMMENTS: This trail is on private property, but access to the public is granted through an easement. However, special rules apply; these are briefly described on the following pages and available on the Department of Environmental Conservation website (below).\n\nCONTACTS: Adirondack Mountain Reserve: www.dec.ny.gov\/lands\/100916.html; emergency contact: 518-891-0235\n\nOverview\n\nThe East Branch of Ausable (pronounced \"aw-say-bull\") River is a smorgasbord of waterfalls. From mossy cascades and powerful flumes to misty veils and thunderous sheer drops, the variety and multitude of waterfalls make this an ideal hike for waterfall lovers. The hike is rich with scenic lookouts and riverside strolling\u2014overall an excellent excursion.\n\nRoute Details\n\nThe trail lies entirely within the property lines of the Adirondack Mountain Reserve (AMR), so hikers should be aware of several restrictions. The rules are strictly enforced and posted when entering the property. The main restrictions are that pets are not allowed, and no firearms, camping, hunting, fishing, swimming, or boating is allowed. Additionally, off-trail travel, bushwhacking, and rock climbing are prohibited. Despite the restrictions, one major advantage of the trail's being on private property is its superb condition. There are few muddy areas, trails are clear of blowdown, and bridges are all in great condition. Parking is approximately 0.7 mile from the trailhead and register but is included in the mileage given.\n\nThe large parking area near NY 73 can accommodate dozens of cars, so finding a spot should be straightforward. Head west along Ausa\u00adble Club Road. At a little more than 0.1 mile, you pass the trailhead for Round and Dix Mountains. Roughly 0.3 mile later, you pass a second trailhead junction, Stimson Trail, as well as a golf course. Just past the tennis courts, you will see a large lime-green building and reach Lake Road at 0.6 mile. Turn left down the road, and head past several small cabins to the AMR trail register. All hikers must sign in and out at the register. Continue down the road past the ornate wooden gate. A gabled arch, constructed from round and irregular logs and branches, vaults the gate, which is constructed similarly and has AMR built into its center. The gate was reconstructed in 1986 as part of the AMR's centennial observances and is a replica of the original.\n\nShortly after the gate, you pass Ladies Mile Trail, which, if you want to get off the club's road sooner, also leads to West River Trail. A second intersection with Ladies Mile Trail is 0.3 mile past the gate, 1 mile total. This junction connects to West River Trail directly, but yards ahead is a junction with East River Trail (an alternate way to reach West River Trail) and likely your return trip if you make the complete loop. Cross the wooden plank bridge, and head 0.3 mile west to another trail junction. Straight ahead are East River Trail and the path back to Lake Road, while Bear Run, Cathedral Rocks, and West River Trails are off to the right. Cross the bridge over the East Branch of Ausable River, and turn left onto the trail. West River Trail, marked with yellow markers, heads west, while Bear Run and Cathedral Rocks Trails, marked in red, head north.\n\nThe trail heads west under towering hemlocks and climbs gradually above the river gorge. At times you hike close to the river, but you veer away into the surrounding forest over many sections. Grades are gentle, and at 1.7 miles, you cross Pyramid Brook, after which the trail heads back toward the river. At 1.8 miles, you reconnect with Bear Run, Cathedral Rocks, and Pyramid Brook Trails. Approximately 0.1 mile ahead, you reach another junction and a green trail sign indicating East River Trail, Lake Road, and Canyon Bridge to your left; West River Trail and Wolfjaw straight ahead; and Bear Run, Cathedral Rocks, and West River Trails back the way you came. The trail climbs a bit past this intersection, and the drop down to the river is significant at this point. You can see small cascades and flumes along the river, and the roaring of the water drowns out most sounds. The sounds and views decrease as you climb away from the river for the next 0.5 mile, but you soon return; at 2.4 miles you come across a lookout to a waterfall along the river. This lookout is to the left of the trail, and there is a precipitous drop, so take care while viewing the waterfall. Shortly after, the trail swings north to follow a feeder brook, Wedge Brook, a short distance and then crosses the brook over a bridge. The bridge spans the brook between two waterfalls. The lower is really a small cascade, and the second is a small waterfall that branches around a large rock and tumbles into a pool above the bridge. On the other side is Wedge Brook Trail to Lower Wolfjaw, and a sign indicates that the Wedge Brook Cascades are a couple hundred yards away. A short climb along this trail reveals a series of small cascades that weave their way down mossy crevices in the rock to the small pool below.\n\nBack along the main trail, you reach the river in less than 0.3 mile, and you will notice that it is now surging through tight flumes and small cascades at many points. At 2.9 miles, you see another waterfall, which is approximately 25 feet high, with a very powerful surge of water that spurts out nearly horizontally as the river is restricted to a few feet through the narrow gorge. Occasional vantage points are available, but take care when trying to view this fall, as the river gorge is very steep and treacherous. After this waterfall, the river levels off and spreads out among the boulders, but many tiny falls and cascades remain throughout.\n\nAt 3.3 miles, the trail reaches a small brook and turns north, almost immediately after which you reach Beaver Meadow Falls. Unlike the last falls, which fell in a powerful surge, Beaver Meadow Falls is almost like a light rain or mist. Often referred to as a bridal-veil falls, Beaver Meadow Falls is about 60 feet high. A bridge that spans the brook provides a great place to admire and photograph the falls.\n\nBEAVER MEADOW FALLS\n\nWest River Trail continues along the river west through some marshy sections and slow-flowing waters that add variety and interest. However, a side trip along Lost Lookout Trail provides an inspiring view of Lower Ausable Lake and a particularly interesting view of Rainbow Falls.\n\nTo reach the Lost Lookout Trail, climb the log ladder on the other side of Beaver Meadow Falls. As you climb alongside Beaver Meadow Falls, you can see its entirety. The trail, marked in blue, leads to Gothics and Armstrong Mountains and climbs fairly steeply. You quickly reach the Lost Lookout fork on your left, which is marked in red disks and is 0.2 mile from Beaver Meadow Falls, 3.5 miles from the start. The trail climbs steadily through the beech-and-maple forest another 0.3 mile, where you reach the first of two lookouts. You can see Lower Ausable Lake, Indian Head, and Fish Hawk Cliffs nearby, as well as Bear Den, Dial, and Nippletop Mountains, among other peaks to the south. The second lookout is nearby and offers the same views after you traverse a short but steep descent. The trail continues southwest and descends for the next 0.8 mile. You will begin to hear the deep roar of Rainbow Falls and soon glimpse the 150-foot waterfall. Extremely sheer drop-offs are beside the trail, so do not attempt to climb down to get to the falls from here. A separate trail farther ahead brings you to the base of Rainbow Falls. The trail winds down steeply for the next 0.3 mile, after which you reach the intersection with West River Trail. The trail, marked in yellow, quickly leads to a dam that marks the end of Lower Ausable Lake and the head of the East Branch of Ausable River. After crossing Cascade Brook, you reach a long log-and-plank bridge that crosses the river below this dam and leads to East River Trail, as well as Indian Head and Fish Hawk Cliffs on the left. A trail to the right leads to Sawteeth and Gothics Mountains, as well as Rainbow Falls. You will return to this point, 4.9 miles from the start, so that you can head back along East River Trail.\n\nBefore returning, take the blue-marked trail to the right to see these spectacular falls, less than 0.3 mile away. Shortly after beginning along the blue trail, you reach a fork with a yellow-marked trail that indicates Sawteeth Mountain (via the scenic route) on the left and Gothics Mountain, Sawteeth Mountain (presumably along a hideous stretch of trail), and Rainbow Falls off to the right on the blue trail. A few hundred feet later, the trail to Gothics and the others, marked in blue, diverges to the left, while the path to the falls continues to your right but is now marked sporadically in red. You will pass several pipes, both plastic and steel, as you weave your way through boulders to the base of the sheer cliffs that form the falls. A sheer rock wall forms on your left as you work your way up alongside Cascade Brook to the base of Rainbow Falls. The canyon formed by the sheer rock walls amplifies the thundering of the falls as the water drops 150 feet. Mist sprayed on the opposite wall provides a haven for moss, while the water has scoured clean the rock wall on the waterfall side. The contrast of verdant green and jet black on the opposing walls is a unique scene that invites lingering. Head back to the dam to reach East River Trail as well as the spectacular views from Indian Head and Fish Hawk Cliffs.\n\nDirections\n\nFrom I-87, take Exit 30 for US 9, and head north. Go 2.2 miles on US 9, and make a slight left onto NY 73. In 5.4 miles look for Ausable Club Road on your left just after descending a steep hill.\n\nFrom the intersection of NY 9N and NY 73, head southeast on NY 73. In 6.1 miles, Ausable Club Road is on your right, directly across from the Giant Mountain parking area, described. The parking area is almost immediately on your left after you turn onto Ausable Club Road.\n\n38\n\nAusable River: East River Trail\n\nSIGN AT BULLOCK DAM\n\n**SCENERY:**\n\n**TRAIL CONDITION:**\n\n**CHILDREN:**\n\n**DIFFICULTY:**\n\n**SOLITUDE:**\n\nGPS COORDINATES: Parking area: N44\u00b0 08.986' W73\u00b0 46.075' Trailhead: N44\u00b0 08.992' W73\u00b0 46.843'\n\nDISTANCE & CONFIGURATION: 9.7-mile loop if combined with Ausable River: West River Trail (see page 254), or 8.6-mile out-and-back, plus 1.6-mile out-and-back side trip to Indian Head and Fish Hawk Cliffs\n\nHIKING TIME: 3\u20134 hours one-way\n\nHIGHLIGHTS: Scenic views, dramatic cliffs, spectacular waterfall, riverside stroll\n\nELEVATION: 1,960' at trailhead, 2,657' at highest point (on side trip)\n\nACCESS: Open 24\/7; no fees or permits required. Guests must register at the gatehouse.\n\nMAPS: Adirondack Mountain Reserve map: tinyurl.com\/adkmtnreserve; National Geographic Adirondack Park, Lake Placid\/High Peaks (#742)\n\nFACILITIES: None\n\nWHEELCHAIR ACCESS: No\n\nCOMMENTS: This trail is on private property, but access to the public is granted through an easement. However, special rules apply; these are briefly described on the following pages and available on the Department of Environmental Conservation website (below).\n\nCONTACTS: Adirondack Mountain Reserve: www.dec.ny.gov\/lands\/100916.html; emergency contact: 518-891-0235\n\nOverview\n\nEast River Trail is the return leg of a loop along the East Branch of Ausable (pronounced \"aw-say-bull\") River. This section has fewer waterfalls, but it does include a side trip to Indian Head and Fish Hawk Cliffs, which have impressive views and dramatic scenery of their own.\n\nRoute Details\n\nYou can hike the east section of Ausable River on its own, but I found it to be an excellent return trip for West River Trail (described on page 254). A side trip to Indian Head and Fish Hawk Cliffs reveals amazing views and is well worth the extra 1.6 miles. Directions back to the parking area along NY 73 and for the side trip are given from the end of West River Trail.\n\nFrom the intersection of West River Trail with the trail that leads to Sawteeth and Gothics Mountains and Rainbow Falls, proceed east across the long bridge. On the other side of Ausable River, you intersect East River Trail to your left and a path to your right that heads uphill to Lake Road, as well as the trail to Indian Head and Fish Hawk Cliffs. Mileage given is from this point and does not include the total loop or the side trip to the cliffs.\n\nIndian Head and Fish Hawk Cliffs\n\nAs you head uphill, you pass a maintenance building on your left and, shortly after, reach Lake Road. The road continues downhill to your right to where Adirondack Mountain Reserve (AMR) members have access to the Ausable Lakes. Hikers are prohibited from entering this area, so be mindful and keep on the designated trails. Head a short distance east along the road, and look for the trailhead on your right. The trail is marked with yellow disks, and you quickly reach what appears to be a fork in the trail. However, the trail on the right leads to an AMR boathouse and is private, so keep left. The trail has several switchbacks that make the ascent a little more enjoyable than the typical rocky and straight scrambles to the top found on most Adirondacks trails. However, the trail is still considered steep, and several ladders have been built where short vertical sections over mossy and slick ledges would otherwise be difficult.\n\nApproximately 0.3 mile in, you reach a fork in the trail and a sign indicating Gothics Window\u2014117 feet to your right. The \"window\" is a slightly obscured view through the trees to the exposed portions of the Gothics. Though it's interesting, saplings threaten to engulf the view, and more-impressive vistas are found farther along the trail. The side trip doesn't take long, though, so you will probably find it worth the diversion.\n\nYou reach the first of the ladders shortly after this fork, and the zigzags become tight and more frequent. After climbing several ladders and switchbacks, the trail levels off briefly under the base of a mossy cliff that adds a neat feature along the trail. Views of the Gothics are more prominent now, and the trail, though steep, has fewer turns in it.\n\nAt 0.8 mile, you reach the crest of the ridge and a four-way intersection. To the left is Gill Brook Trail, straight ahead and down leads to Fish Hawk Cliffs, and to the right is the trail to Indian Head. Thick evergreens now encompass the trail, and you briefly traverse the spine of this ridge before coming out on the exposed portions of the cliff that make up Indian Head. Spectacular views open up all around you, and you will quickly snap a dozen pictures. Rugged peaks with sheer cliffs fill the panorama to the north and west, but the most stunning views are of Lower Ausable Lake cradled between Colvin and Sawteeth Mountains. Continue along the exposed portion of bedrock along worn footpaths to a second, lower tier for a slightly different vantage point of the same spectacular view.\n\nDAM AT AUSABLE LAKE\n\nTo reach Fish Hawk Cliffs, go back to the four-way intersection, and head down the trail, which is now on your right. The descent is steep and takes you down 160 feet into a col (saddle) near a pool called Wizard's Washbowl. Shortly ahead, 0.3 mile from the four-way intersection, you reach the cliffs, from which you have the interesting perspective of looking back upon Indian Head. To return, go back to the four-way intersection and straight down to Lake Road and then East River Trail.\n\nEast River Trail\n\nEast River Trail has fewer waterfalls than the west one but has its own unique character and elements that make it a great return trip. The trail meanders along the river briefly and then diverges east into the surrounding woods. Approximately 0.3 mile from the bridge, you navigate through a jumble of boulders that lie at the base of a cliff on your right. Shortly after, you pass Bullock Dam on your left. After the dam, the river levels off and flows through a grassy meadow. On the opposite shore, you see exposed cliffs and many views of the mountains to the west.\n\nJust shy of a mile, you reach a fork in the trail; Beaver Meadow and the Gothics are to your left over a bridge and marked in blue, while East River Trail and the road, marked in red, continue straight ahead. Almost immediately after this intersection, you reach yet another fork, with East River Trail continuing straight ahead, and the road to your right.\n\nThe trail delves into tall and thick hemlocks, and you soon reach a beautiful waterfall along the river. You might remember seeing this more-than-20-foot-high waterfall and its constricted flume from the opposite shore. However, on this shore there are far better points from which to view the falls and its washed-out spillway. You are now high above the river canyon, and this is one of the last places to view the river. Farther on, you hear the rumble of the turbulent waters but will only have glimpses of the river from afar, as the trail winds its way back down to the gatehouse.\n\nThe trail becomes steep in a few sections but levels off about 2.5 miles from the start. Gill Brook joins you on your right, and shortly after this you will cross the brook over a bridge built from two sturdy beams. Soon after the bridge, you rejoin Ausable River as it flows northeast, and the trail follows the river 0.3 mile. At 3.1 miles, you reach another fork. Straight ahead is a bridge to West River Trail, while to your right is East River Trail and the path back to the road. In 0.1 mile, you will come in sight of the road. Follow it back to the gatehouse to sign out and then back to the parking area by NY 73, completing a trip of 4.3 miles from the bridge at Lower Ausable Lake, 10 miles to complete the loop, and a total of 11.6 miles, including the side trip.\n\nDirections\n\nThe East River Trail begins at the end of the West River Trail, so the parking area is the same.\n\nFrom I-87, take Exit 30 for US 9, and head north. Go 2.2 miles on US 9, and make a slight left onto NY 73. In 5.4 miles look for Ausa\u00adble Club Road on your left just after descending a steep hill.\n\nFrom the intersection of NY 9N and NY 73, head southeast on NY 73. In 6.1 miles, Ausable Club Road is on your right, directly across from the Giant Mountain parking area, described on page 276. The parking area is almost immediately on your left after you turn onto Ausable Club Road.\n\n39\n\nGiant's Nubble\n\nGIANT'S WASHBOWL SEEN ALONG DESCENT\n\n**SCENERY:**\n\n**TRAIL CONDITION:**\n\n**CHILDREN:**\n\n**DIFFICULTY:**\n\n**SOLITUDE:**\n\nGPS COORDINATES: N44\u00b0 09.026' W73\u00b0 46.027'\n\nDISTANCE & CONFIGURATION: 4.4-mile loop\n\nHIKING TIME: 3\u20134 hours\n\nHIGHLIGHTS: Spectacular waterfall, primitive campsites, outstanding views\n\nELEVATION: 1,307' at trailhead, 2,750' at highest point\n\nACCESS: Open 24\/7; no fees or permits required\n\nMAPS: Giant Mountain Wilderness map: tinyurl.com\/giantwildernessmap; National Geographic Adirondack Park, Lake Placid\/High Peaks (#742)\n\nFACILITIES: None\n\nWHEELCHAIR ACCESS: No\n\nCONTACTS: Giant Mountain Wilderness: www.dec.ny.gov\/lands\/100750.html; emergency contact: 518-891-0235\n\nOverview\n\nThis trip combines a tall waterfall, scenic outlook, and picturesque pond within a very short distance. Don't let the distance or description fool you though; the rugged trail provides you with plenty of climbing, along with the majestic scenery associated with the High Peaks region.\n\nRoute Details\n\nThe asphalt parking area is tucked away on your right as you come down a steep hill along NY 73, so watch the odometer and be ready to make a sharp right. The parking area can easily accommodate a dozen cars, but if it's full, there is room on the shoulder of NY 73 or across the road in the Ausable River parking area.\n\nThe register and trailhead are at the eastern end of the lot.\n\nThe trail is broad and flat, and you can just barely glimpse Roaring Brook through hardwoods off to the right. During the peak of summer, the brook runs dry and indicates that the falls ahead may not be as majestic as during rainy conditions or during the winter melt. About 0.1 mile in, you reach a fork in the trail. To the right is the trail to the base of Roaring Brook Falls, while to the left the trail climbs to the top of the falls, Giant's Nubble, Giant's Washbowl, and Giant Mountain.\n\nThe Base of Roaring Brook Falls\n\nContinue to the right past the sheer rock wall on your left until you reach Roaring Brook. Straight across the brook are four designated campsites. These sprawling sites lie under a thick canopy of tall hemlocks, and each has a large stone fire pit. They are shielded from view of the falls by a steep hill to the north but are fairly close to another stream, the outflow of Chapel Pond. The sound of the stream and the thundering of the falls will likely drown out the sound of the road, which is still close by, though the occasional loud vehicle might disturb the tranquil setting. To reach the falls, you have to scramble up the rocky brook a very short distance. The falls, the entirety of which can be seen from the road, are nearly 325 feet high, depending on where you measure the start. They cascade in three tiers, with only the lowest section visible at the base.\n\nGiant's Nubble, Giant's Washbowl, and Giant Mountain\n\nThe trail climbs steadily from the fork and passes a tiny lookout shortly after you begin the ascent. After the first section of climbing, the trail levels off briefly and then climbs gently again as it winds briefly south around the contours of the hillside. As the trail swings east again, views across the valley begin to sprawl out before you. Approximately 0.5 mile in, you reach a fork in the trail; to your right are a designated campsite and the top of the falls, while straight ahead are Giant's Nubble, Giant's Washbowl, and the summit. The campsite, situated to the left of the falls path, is spacious, and there are plenty of spots to set up camp under the maple-and-ash canopy.\n\nTo reach the top of the falls, follow any of the footpaths to Roaring Brook, and then rock-hop downstream. From the top of the falls, very little of the waterfall is visible because the drop is so precipitous and the water flows through a deep cleft in the bedrock. However, there are spectacular views out to the western horizon. The rocks at the top of the falls have been worn smooth by innumerable visitors and slope gradually toward a sheer drop-off, so use extra caution, especially during wet conditions.\n\nThe main trail, marked by red disks, continues to head east and climbs steadily for the next 0.5 mile, where you reach another fork. To the left are two more designated campsites, both of which are situated under a heavy canopy of hemlocks and offer lots of room to set up. Shortly after this intersection, you wind around a large boulder and then turn quickly south as you rock-hop across Roaring Brook. On the opposite bank, you reach yet another fork. The trail to your left leads to the Giant Mountain summit, 2.5 miles, while the nubble and washbowl are off to your right. Trail signs indicate that the nubble and washbowl are both 0.9 mile away, but the actual mileage is roughly 0.3 mile less. A tenth of a mile ahead, you reach the fork where the trails to the nubble and washbowl diverge from each other. This is also the terminus of the loop around both features and a return point back to NY 73. A trail sign indicates that the nubble is still 0.9 mile ahead while the washbowl is 0.8 mile ahead on the right. You can hike the loop in either direction, though I recommend hiking the nubble first, as there are some truly stunning views of the washbowl as you descend.\n\nThe trail up to the nubble, now marked with yellow trail disks, is a bit rougher than sections previously covered but no more than the usual Adirondack trail.\n\nLooming off to your left is the southern-exposed face of Giant Mountain, which can be glimpsed occasionally through the surrounding fir trees. Many portions of the trail traverse moss-covered bedrock, and you pass by a square ledge of this oddly lush rock on your right shortly before you reach the fork in the trail, at 1.6 miles, that leads to the nubble. Veering right at the fork, you quickly reach the southeastern tip of the nubble, and a vast panorama of the eastern High Peaks opens up before you. Footpaths weave in and out of the exposed, lichen-covered bedrock and wind-stunted fir trees as you make your way to the northern tip of the nubble. The lookout reveals a panorama of the northern High Peaks, as well as a beautiful vista of Giant Mountain off to the east. If the drive into the region did not give you an appreciation of the ruggedness of this area, then the views from atop the nubble certainly will.\n\nEXPOSED BEDROCK AT THE NUBBLE\n\nThe descent to the washbowl, initially marked with yellow disks, can be quite steep but has numerous lookouts as well. Less than 0.3 mile from the nubble is a truly wonderful lookout that sits almost directly above the washbowl. The almost jet-black basin is nestled in a long depression between the base of the nubble and a ridgeline before the slope of the mountain precipitously drops more than 800 feet to the valley floor in less than 0.3 mile. From this vantage point, the pond seems to simply hang in the air. The trail weaves its way down through evergreens to the intersection with Giant's Ridge Trail at 2.1 miles. A green-and-white trail sign marks the intersection and indicates that the washbowl is a mere 0.3 mile farther on your right. The trail is heavily trodden but well maintained, and you quickly pass another designated campsite on your left, shortly before reaching the southeastern tip of the washbowl.\n\nA path along the eastern shore of the pond leads to another designated campsite that sits close to the water's edge, but the main trail winds to your left. You quickly pass the last intersection along this loop, where trail signs indicate that the Chapel Pond parking area along NY 73 is 0.7 mile ahead on your left, marked in blue, while the trail back to Roaring Brook Falls continues to your right, with red markers. The rocky nubble looms over the washbowl and provides some dramatic scenery as you hike along the western ridge that contains the pond. The brief shoreline hike north is basically flat and rocky, with numerous rocks jutting out on the pond, all of which provide points from which to view the pond and nubble. After leaving the washbowl's shore, you climb briefly along the base of the nubble and then begin your descent back to the beginning of the nubble and washbowl loop.\n\nTRAIL CROSSING OF ROARING BROOK\n\nDirections\n\nFrom I-87, take Exit 30 for US 9, and head north. Go 2.2 miles on US 9, and make a slight left onto NY 73. In 5.4 miles the parking area is on your right, just after descending a steep hill; it's directly across from the southern intersection of Ausable Club Road and NY 73.\n\nFrom the intersection of NY 9N and NY 73, head southeast on NY 73. In 6.1 miles, the parking area is on your left.\n\n40\n\nHanging Spear Falls and Opalescent River\n\nFLOWED LANDS\n\n**SCENERY:**\n\n**TRAIL CONDITION:**\n\n**CHILDREN:**\n\n**DIFFICULTY:**\n\n**SOLITUDE:**\n\nGPS COORDINATES: N44\u00b0 05.353' W74\u00b0 03.381'\n\nDISTANCE & CONFIGURATION: 13.7-mile loop or 11.0-mile out-and-back\n\nHIKING TIME: 7.5\u20138.5 hours (overnight)\n\nHIGHLIGHTS: Waterfall, remote lake, dramatic views\n\nELEVATION: 1,748' at trailhead, 2,800' at highest point\n\nACCESS: Open 24\/7; no fees or permits required\n\nMAPS: National Geographic Adirondack Park, Lake Placid\/High Peaks (#742)\n\nFACILITIES: Pit toilet at trailhead\n\nWHEELCHAIR ACCESS: No\n\nCOMMENTS: While returning along East River Trail, there is a point where you will have to ford the Opalescent River, which could be hazardous during high water and\/or cold weather. Water depth is commonly knee- or thigh-deep but is ankle-deep in summer\/low water. No matter what, your feet will get wet, so plan accordingly.\n\nCONTACTS: High Peaks Adirondack trail information: www.dec.ny.gov\/outdoor\/9198.html; emergency contact: 518-891-0235\n\nOverview\n\nDeep in the remote High Peaks Wilderness is a stunning waterfall. Combine that with evocative names such as Hanging Spear, Flowed Lands, and Opalescent River and this adventure promises to be a spectacular trip, and it does not let you down. Because most hikers in the High Peaks are set on bagging the various peaks, chances are that you will be able to enjoy this isolated waterfall all to yourself. A protracted climb, a river ford, the remote location, and the length of the trail mean that this trip is best left to experienced hikers. You will want to give yourself plenty of time not only to complete the hike but also to enjoy the magnificent views at the falls and elsewhere along the loop.\n\nRoute Details\n\nNote: Though the trip to the falls can be shortened to an 11.0-mile out-and-back by returning along the Calamity Brook Trail, completing the loop along the Hanging Spear Trail adds 2.6 miles but is not as rough and generally easier going\u2014that is, the path is free of cobbles and requires less climbing. During winter or times of high water along the Opalescent River, the out-and-back route may be the only safe option. The full loop requires a full day and an early start, but the trail can also be treated as an overnight by either following the camping at-large rules laid out by the Department of Environmental Conservation (DEC) or by using designated camping areas and lean-tos mentioned within the description. The lean-tos at the end of the Calamity Brook Trail and within the Flowed Lands, four of which are readily accessible along the described route, are ideal spots to spend the night. There is also a designated campsite on the ascent, and a couple are located on the descent before reaching the Opalescent River crossing.\n\nMONUMENT TO DAVID HENDERSON\n\nThe Upper Works parking area is quite large and easily accommodates dozens of vehicles. The trail, marked with yellow disks, begins to the right of the trail register in the northeast corner of the lot. At roughly 0.25 mile you reach a wooden bridge that crosses the Hudson River. Just after the bridge, the road forks, with the trail continuing to the right. Just under 0.4 mile, you enter a meadow with a fork in the trail. The trail to the left leads to Duck Hole and Indian Pass marked with yellow disks, while the trail to the right is the Calamity Brook Trail, indicated here as Lake Colden Dam, Mount Marcy.\n\nContinue along the right fork (east) following the red trail disks. The next mile of trail follows an old abandoned gravel road that weaves its way through rocks and boulders mostly out in the open. You will hear Calamity Brook shortly before the trail reaches it, a point where the brook makes a sharp southerly turn. The trail proceeds along an elevated berm between the tannin-colored Calamity Brook, with its bleached gravel banks on the right and a sprawling wetland on the left. From here forward, the trail more closely resembles the wooded foot trails that hikers are accustomed to.\n\nA little over 0.3 mile ahead, 1.6 miles overall (despite what the trail sign reads), is another trail intersection. Straight and north, marked in red, is another route to Duck Hole and Indian Pass. To the right and over a bridge, marked in blue, is the continuation of the route to Mount Marcy via Lake Colden. Turn right and cross the tributary stream of Calamity Brook. The trail continues along level ground interspersed with a few mucky areas and soon rejoins Calamity Brook as the trail swings west. Shortly after passing a closed trail with a washed-out bridge on the right, you reach a tributary brook that you rock-hop across, after which the climbing to the Flowed Lands truly begins\u2014roughly 800 feet gain over the next 2.5 miles.\n\nCalamity Brook is no longer tannin laden and meandering but rather a crystal-clear babbling brook that tumbles through a boulder-\u00adstrewn bed. The trail weaves its way to and from the brook as you ascend. You pass a camping disk on the right at 2.7 miles, about midway up a hill, before the trail levels out briefly at a wet area. This is one of many long mires that trail crews have tried to bypass with logs, log-and-plank boardwalks, and other configurations that vary in success in keeping your boots dry and keeping you out of the muck. At some point along the climb, you will likely have to trudge through the muck, but not here.\n\nHANGING SPEAR FALLS\n\nAt roughly 3 miles reach an unmarked fork in the trail just before reintersecting Calamity Brook. During most of the year, you can cross Calamity Brook by rock-hopping, but during times of high water and flooding, mostly during the spring, you will want to continue uphill and to the left to the high-water crossing: a cable bridge that was replaced after floodwaters from Hurricane Irene washed out the previous bridge in 2011. The high-water bypass reconnects with the main trail a little uphill on the opposite bank.\n\nThe trail continues to climb steadily past this crossing, and you will encounter a series of wet and mucky areas that become increasingly harder to navigate. Calamity Brook flows down to your left, and much of the trail resembles a cobble-strewn brook in its own right. As the trail nears the top of the climb, it emerges from the surrounding forest into an open upland wetland, roughly 4 miles from the start. The trail swings around the southern and eastern edges of the wetland that surrounds Calamity Pond. A short side path found along the southern edge takes you through the encircling shrubs and provides a photo opportunity of the upland wetland. As you near the northeastern edge of the wetland, keep an eye out for a worn footpath leading north, located just before the trail bends east and away from the wetland. This short trail will take you to the headwaters of the pond and a stone monument dedicated to David Henderson, who died from an accidental gunshot here in 1845. You'll find another scenic view of the wetland and pond at the inlet.\n\nBack along the main trail, continue east, through one long boot-sucking mire, and soon reach the Calamity #1 Lean-To and trail register for the inner High Peaks Wilderness. (There are special regulations for the inner High Peaks, mostly regarding camping and the use of bear-resistant canisters. For camping rules, see www.dec.ny.gov\/outdoor\/7872.html; for information on bear-resistant canisters, see www.dec.ny.gov\/animals\/7225.html.) A major trail intersection is also located at the trail register. To the left and marked in red are the trails leading to Lake Colden Dam, Avalanche Lake, and Mount Marcy. Straight ahead and marked in red are the trails to Flowed Lands and Hanging Spear Falls. Continue straight (east) and quickly reach a short path leading down to the eastern edge of the Flowed Lands, where a sprawling panorama of a winding wetland is framed by Mount Colden to the east and the foot of Mount Marshall to the west. It's a stunning scene and a good spot to break, as the next best stop, the Flowed Lands Lean-To a little farther on, is likely occupied. Alternatively, wait until you reach the lookout opposite Hanging Spear Falls, a truly amazing and tranquil setting, 1 mile farther.\n\nHeading south and away from this vista, the trail heads uphill briefly before descending into a depression just in front of the Flowed Lands Lean-To, 4.7 miles. At this point it is not immediately apparent where the trail continues. With the lean-to straight ahead, look right and scramble uphill, where you find red trail disks and a sign leading to a nearby pit toilet and the trail becomes more evident. Continue heading nearly due south another 0.3 mile to where you reach the end of the Flowed Lands and the head of the Opalescent River.\n\nYou will hear the sounds of the outflow of the Flowed Lands a short distance before you reach the crossing point just upstream from the old dam. The dam was originally created by the McIntyre Iron Works, which the aforementioned David Henderson helped establish, but the dam was deemed unsafe and was breached by the DEC in 1984. Scramble down from the trail to the gravel beach. Red trail disks are visible on the opposite shore, and you will have to pick your way atop a collection of rocks that loosely form a bridge across.\n\nOnce atop the opposite bank, you reach a trail intersection; to the left and marked in yellow is the Livingstone Point Lean-To, 0.4 mile away, while the path forward is to the right (south) following red trail disks. As you begin heading south along East River Trail, look immediately to the right for a point where you can snap a few pictures atop an old section of the dam. Nearby, a sign warns that the bridge across the Opalescent River, 4.2 miles farther along the trail, is out and you will have to ford the river. As the sign indicates, this could be hazardous during high water and during cold weather because inevitably you will have to wade or at the minimum get your feet wet during the crossing.\n\nAs the trail leads south to Hanging Spear Falls, it follows the Opalescent River quite closely along a narrow footpath. Long strands of algae cling to river rock, giving the river a green tinge. And though you may think that the green algae is the source of the river's name, the name was given by a state geologist who found the riverbed filled with opalescent feldspar, also known as labradorite. Under the right conditions the labradorite looks blue, and sometimes green, bronze, gold, or even iridescent. About 0.25 mile along the descent, a small unnamed waterfall, really a flume, with a small pool can be found among car-sized boulders. Farther along the trail you will hear other small waterfalls just out of sight, and you may be tempted to bushwhack in to see if they are Hanging Spear Falls, but rest assured, the site to view the actual falls is evident. Indeed a sign with an arrow indicates that the \"falls\" are available along a side trail, a little less than 0.5 mile from the crossing, 5.5 miles overall.\n\nOnce on the side trail, follow it a short way to the viewing area, a flat spot directly south of the falls. It might be tempting to try to descend a worn herd path visible to the side of the trail, but this path is dangerous, easily eroded, and likely to get you injured, if not trapped, near the base of the falls. At the viewing area, the surrounding canopy perfectly frames the 75-foot-high falls, and there is even an ideally placed root upon which you can sit and admire the majestic view. If you look upstream you will see not only Hanging Spear Falls but also a series of falls farther north. Incidentally, the falls are so named because during full flow they tend to bifurcate and give the appearance of a spearhead suspended in the air. This is only during times of high flow, but even at modest flow the falls are impressive.\n\nTo continue along the trail, you can either backtrack to the main trail or continue along the side trail, which loops back to the main trail a little farther downhill. Continue downhill on the main trail, and soon reach a brook crossing, the first along this section of trail and a useful landmark in case you missed the lookout side trail. For the next 0.75 mile the trail winds its way through dense understory out of earshot and sight of the river. Shortly after the Opalescent River comes back into view, some 40\u201350 feet below, the trail winds down to a point directly along its banks. The trail then weaves along the eastern bank through some mucky sections and then crosses a gravel outwash of the river high water\/old riverbed near a designated campsite 0.3 mile ahead, 6.8 miles overall. This marks the end of the descent, and the rest of the trail follows mostly level ground.\n\nGradually the trail leads away from the river and climbs slightly to a clearing among a network of old access\/logging roads. An arrow staked in the clearing indicates that the trail follows the western edge of this clearing. At 7.6 miles reach Upper Twin Brook, which is easily crossed atop several large boulders just downstream on the right. On the other side, pass an informal campsite and soon reach a trail intersection. Left is the trail to Allen Mountain, while right, following yellow trail markers, is the continuation of East River Trail to the Upper Works trailhead parking.\n\nThe trail now follows a narrow, worn path that heads generally southwest through open fields dotted with small evergreens and raspberry bushes. Hiking in the open is indeed a change of scenery from the typical forested trails and provides some interesting views of Calamity Mountain and Mount Adams to the north and west, while Allen Mountain looms to the east; you will have to turn around to catch this photo op. For the next 1.2 miles this setting is dominant with only a few exceptions; the most notable is 0.25 mile ahead, where the trail delves briefly into the forest and crosses Lower Twin Brook.\n\nAt roughly 9 miles, after reentering the forest, the trail leads down to the banks over the Opalescent River once again. Bear left and follow the river another 0.1 mile to where it broadens at the designated crossing. Dozens of feet across at low water, the crossing necessitates getting your feet wet and\/or wading, depending on the depth. Once across, turn left and follow an old road south; no trail markers are readily apparent, but the wide access road is easy to follow. The Opalescent is now broad and meandering as it accompanies you on the left as the trail gradually swings east.\n\nHIGH-WATER BRIDGE\n\nA little over 0.75 mile from the crossing, 9.9 miles overall, reach a red vehicle-barrier gate close to a point where the Opalescent turns sharply south. Past the barrier gate, the road continues southwest awhile but gradually begins to curve northwest. Sanford Hill can be seen off to the left, while Popple Hill dominates the view to the right. A half mile past the barrier gate and at a point heading nearly due north, the trail departs from the road and continues along a worn footpath. Rocks and branches are laid across the road, and you will see a yellow trail arrow on the right through the foliage.\n\nThe trail follows an almost due-north course, and within 0.3 mile you can glimpse Lake Sally through the surrounding forest on the left. This is the southern bay of Lake Sally, and after briefly diverting into the surrounding forest about 0.25 mile, you reach its northern bay. The trail climbs slightly around this bay and intersects a slightly overgrown gravel road, roughly 11.2 miles overall. Bear right and follow this road another 0.3 mile north to where it bends left (northwest) near where an old logging road continues straight ahead. A quarter mile farther, and roughly 0.6 mile from the intersection with the road, pass the trail to Mount Adams, marked in red on the right. Just a little farther, reach two green buildings, the Mount Adams fire tower cabins.\n\nThe trail bears left (south) at the cabins and soon bends west and quickly reaches the northern bay of Lake Jimmy, less than 0.3 mile from the cabins, 12.3 miles overall. Old maps show a bridge crossing this bay, but the bridge no longer connects the two shores; beaver activity had undermined it to the point that it was no longer safe. To get around the bay, turn right and head north, where logs and planks help you navigate through the wetland that surrounds Lake Jimmy. Soon the trail swings west and you cross the two outlets of Jimmy in quick succession. After turning left and briefly heading south, reconnect with the old trail, and head due west through a deep forest another 0.25 mile to where you intersect the Hudson River once again. A suspended steel-grate bridge with fenced-in sides carries you high above the Hudson. Once on the other side, it is a short distance to the East River Trail parking area. Turn right onto Upper Works Road, and continue 0.75 mile to the Upper Works trailhead and parking area for a round-trip of 13.7 miles.\n\nDirections\n\nFROM THE WEST From the intersection of NY 30 and NY 28N in Long Lake, head 18.9 miles east along NY 28N, and turn left onto Blue Ridge Road. At 0.3 mile Tahawus Road\/County Road 25 merges with Blue Ridge Road on the left. Continue 0.9 mile, 1.2 miles from NY 28N, and bear left onto Tahawus Road\/CR 25. After another 6.3 miles bear left onto Upper Works Road\/CR 25A. Continue north another 3.5 miles to the end of the road and the Upper Works parking area.\n\nFROM ALL OTHER POINTS From I-87 take Exit 29 toward Newcomb\/\u00adNorth Hudson. Head 17.5 miles west along Blue Ridge Road\/CR 84, and turn right onto Tahawus Road\/CR 25. At 6.3 miles, bear left onto Upper Works Road\/CR 25A. Continue north another 3.5 miles to the end of the road and the Upper Works parking area.\n\n41\n\nIndian Pass\n\nWALLFACE MOUNTAIN\n\n**SCENERY:**\n\n**TRAIL CONDITION:**\n\n**CHILDREN:**\n\n**DIFFICULTY:**\n\n**SOLITUDE:**\n\nGPS COORDINATES: N44\u00b0 05.353' W74\u00b0 03.382'\n\nDISTANCE & CONFIGURATION: 8.8-mile out-and-back\n\nHIKING TIME: 5\u20136 hours\n\nHIGHLIGHTS: Maze of boulders, sheer cliffs, panoramic views\n\nELEVATION: 1,758' at trailhead, 2,855' at highest point\n\nACCESS: Open 24\/7; no fees or permits required\n\nMAPS: National Geographic Adirondack Park, Lake Placid\/High Peaks (#742)\n\nFACILITIES: Pit toilet at the parking area\n\nWHEELCHAIR ACCESS: No\n\nCONTACTS: High Peaks Adirondack trail information: www.dec.ny.gov\/outdoor\/9198.html; emergency contact: 518-891-0235\n\nOverview\n\nThe trail leading to Indian Pass is an enjoyable excursion in its own right, and you can't help but feel the remoteness of the High Peaks Wilderness that surrounds you. The first three-quarters of the trail is, however, not that different from many deep-woods hikes. It's the last 0.8 mile that really sets this trail apart and makes it a must-have adventure. After navigating your way through a maze of enormous boulders, you soon reach Summit Rock, and the spectacle of Wallface Mountain is truly impressive.\n\nRoute Details\n\nThe Upper Works parking area for Indian Pass is quite large and easily accommodates dozens of vehicles. Indeed, it is one of the major access points for the High Peaks Wilderness and all the adventure that the area encompasses. The trail, marked with yellow disks, begins to the right of the trail register in the northeast corner of the lot. The trail starts along a gravelly road within the boundaries of the Tahawus Preserve, a vast area of wilderness that was acquired by the Open Space Institute in 2003, with major sections now jointly managed by the institute and the Department of Environmental Conservation. After 0.3 mile, you reach a wooden bridge sturdy enough for vehicles to cross. Congratulations! You have just crossed the Hudson River at its headwater. Just after the bridge, the road forks, with the trail continuing on the right. At 0.4 mile you enter a meadow with a fork in the trail. The trail to the left leads to Duck Hole and Indian Pass, marked with yellow disks, while the trail to the right, marked with red disks, is Calamity Brook Trail and Mount Marcy via Lake Colden. Calamity Brook Trail is also the return trip if you decide to take an extended trip, day or overnight, by crossing over the McIntyre Mountains via Cold Brook Pass.\n\nContinue along the left trail, which is a mostly open road with occasional views of Lake Henderson on your left. The road has many muddy areas, and regrowth encroaches in several places. At 1.1 miles, you encounter what appears to be a fork but is actually a route around a mucky area. Choose the right-hand path to avoid the worst of the mud. Just after this rerouting, you come close to Lake Henderson again, but thick hemlocks obscure most of the view.\n\nIndian Pass Brook now flows off to your left, and at 1.5 miles, you reach another fork in the trail. At this point, you leave the broad road and stick to the foot trails. To the left is a log-and-plank bridge, and signs indicate that the Duck Hole Lean-To is 5.3 miles and Coreys, 23.3 miles. Similarly, the sign indicates Henderson Lean-To is 0.2 mile, Indian Pass is 2.7 miles, and the Adirondak Loj is 10.4 miles straight ahead. (The mileage to the Loj must include a detour because a straight path along the entire Indian Pass Trail brings you to the Loj in 10.4 miles, and you have already hiked 1.5 miles.)\n\nContinue straight along mostly flat terrain, now marked with red disks. In 1.7 miles you pass the Henderson Lean-To on your left. On a Saturday morning this lean-to is usually filled with hikers who started late on Friday and want to get moving early. After reaching a meadow Indian Pass Brook is again within sight, and you quickly reach the last fork of this trip, 2.1 miles. Signs indicate that to the left, Wallface Lean-To is 0.6 mile, Indian Pass is 2.3 miles, and the Adirondak Loj is 6 miles. (Once again, the mileage to the Loj seems to be mistaken, but the others are correct.) The trail straight ahead, marked in blue, is the Calamity Brook Crossover trail, with Calamity Brook at 2.1 miles and Lake Colden at 6 miles.\n\nBRIDGE OVER INDIAN PASS BROOK\n\nAfter taking the left trail, marked in red, you quickly reach a bridge over Indian Pass Brook. The recently constructed bridge has round rails and posts, as well as a large, rock-filled crib on the opposite shore. Shortly after reaching the opposite shore, you pass a minor fork in the trail and a yellow disk indicating a designated campsite to your right. Nestled among dense evergreen saplings, the site is close to the confluence of two streams. The site has a sheltered feel and is filled with the soft babbling sounds of the water nearby.\n\nIndian Pass Brook now flows on your right, and though it is nearby, you don't really see the brook until you approach the Wallface Lean-To, 2.7 miles. The trail passes within feet of the brook, and the lean-to is just to the left of the trail. This is perhaps an ideal spot to enjoy the brook, but if you plan to stay in the lean-to, then expect traffic. The trail is mostly flat, and you cross many seasonal feeder streams as you work your way northeast to the intersection of a seasonal stream and Indian Pass Brook. You cross and recross the stream in what is a muddy little delta extending about 0.1 mile. Past this mire, Indian Pass Brook continues to flow on your right. The grade steadily increases, and soon a sheer rock wall begins to rise on your right. At 3.8 miles a jumble of enormous boulders on the eastern shore invites exploration. This is fortunate, as the trail crosses over Indian Pass Brook at this point, but the markings for the crossing are not that clear, and it's easy to miss the exact crossover point.\n\nOn the other shore, the trail becomes more rugged and climbs steeply in sections. The moss-covered, giant boulders through which you are climbing form myriad crevices, caves, and clefts that make for interesting scenery and exploration. As you climb, and after passing a 40-foot-high boulder on your right, you will begin to glimpse through the canopy to your left the sheer rock walls of Wallface Mountain. The scene is impressive and promises a dramatic vista ahead at Indian Pass Outlook. At 4.1 miles you reach a series of short ladders that assist you up this steep section. The trail swings south briefly, then heads north again to the base of a much taller ladder at 4.3 miles. Once you are atop this ladder, it is a very short way to the fork that leads to the outlook on your left. A short ladder leads you a few dozen feet off the main trail to the Indian Pass Summit Rock outlook. From this rocky outcrop, you have an amazing view of the sheer cliffs of Wallface Mountain, as well as an expansive view down the valley to the south. Return the way you came for a total trip of 8.8 miles.\n\nDirections\n\nFROM THE WEST From the intersection of NY 30 and NY 28N in Long Lake, head 18.9 miles east along NY 28N, and turn left onto Blue Ridge Road. At 0.3 mile Tahawus Road\/County Road 25 merges with Blue Ridge Road on the left. Continue 0.9 mile, 1.2 miles from NY 28N, and bear left onto Tahawus\/CR 25. After another 6.3 miles bear left onto Upper Works Road\/CR 25A. Continue north another 3.5 miles to the end of the road and the Upper Works parking area.\n\nFROM ALL OTHER POINTS From I-87 take Exit 29 toward Newcomb\/North Hudson. Head 17.5 miles west along Blue Ridge Road\/CR 84, and turn right onto Tahawus Road\/CR 25. At 6.3 miles bear left onto Upper Works Road\/CR 25A. Continue north another 3.5 miles to the end of the road and the Upper Works parking area.\n\n42\n\nAlgonquin Peak\n\nATOP ALGONQUIN PEAK\n\n**SCENERY:**\n\n**TRAIL CONDITION:**\n\n**CHILDREN:**\n\n**DIFFICULTY:**\n\n**SOLITUDE:**\n\nGPS COORDINATES: N44\u00b0 10.969' W73\u00b0 57.746'\n\nDISTANCE & CONFIGURATION: 8.0-mile out-and-back, 0.8-mile out-and-back side trip to Wright Peak\n\nHIKING TIME: 7\u20138 hours\n\nHIGHLIGHTS: Bald mountain(s), alpine ecosystems, panoramic views, waterfall\n\nELEVATION: 2,179' at trailhead, 5,115' at highest point\n\nACCESS: Open 24\/7; $5 for Adirondack Mountain Club members, $10 per day for nonmembers. During summer and holiday weekends the lots fill quickly and are at capacity by early afternoon. When parking booth is closed, pay and register with the self-service box. Parking is free for guests staying at the Loj\/Wilderness Campground.\n\nMAPS: National Geographic Adirondack Park, Lake Placid\/High Peaks (#742)\n\nFACILITIES: Changing rooms, bathrooms, showers, information center\n\nWHEELCHAIR ACCESS: No\n\nCOMMENTS: Be aware that Algonquin Peak (and Wright Peak if you choose the optional side trip) is an alpine zone. You won't find this ecosystem anywhere in New York except atop the highest of the High Peaks. While this makes for a great experience, the vegetation is also rare, fragile, and endangered. Stepping or sitting on these plants will kill them and destroy the habitat. As such, avoid any vegetation in these zones, and only walk along designated trails and solid rock surfaces.\n\nCONTACTS: High Peaks trail information: www.dec.ny.gov\/outdoor\/9198.html; emergency contact: 518-891-0235\n\nOverview\n\nAt 5,115 feet, Algonquin Peak is the second-highest peak in New York (Mount Marcy stands at 5,343 feet tall). Those starting or looking to complete their 46 High Peak challenge can easily add Wright Peak (side trip detailed on) and Iroquois Mountain to their list. Not only are the views atop Algonquin and Wright Peaks stunning, but the alpine ecosystem is unique to all but a few places in New York. The ascent is nearly 3,000 feet to Algonquin alone, so plan on taking considerably more time along this trail than the 8-mile trek would normally suggest.\n\nNote: If you want a little more adventure, add the return along Avalanche Pass extending the trip by nearly 4 miles and 2 hours.\n\nRoute Details\n\nParking at the Loj is extensive and could easily accommodate 100 (or more) cars in the various gravel parking lots. However, this is a major starting point for hikers and backpackers looking to explore the remote heart of the High Peaks wilderness, so the lot fills to capacity during the summer and holiday weekends. Many of the people exploring the region will be on extended overnight treks, and the lot provides access to a wide network of trails. But the trail network in the interior is so extensive that although you will pass by other hikers, it won't be as frequently as the number of cars seems to indicate.\n\nThe trailhead and register are found near a kiosk on the eastern edge of the parking area. The initial section of the trail, marked with blue disks, lies within the Adirondack Mountain Club's boundaries and has received extensive trail work\/maintenance to accommodate the heavy traffic that passes through the area. At roughly 0.3 mile reach a long boardwalk that traverses a wetland along MacIntyre Brook. About 0.4 mile farther, reach the intersection with the Fangorn Forest Trail and the Department of Environmental Conservation High Peaks Wilderness boundary, 0.7 mile. Bear left and continue following blue trail disks as the trail climbs gradually, with MacIntyre Brook flowing 20\u201330 feet below. At 1 mile reach the first major trail intersection. To the left, marked in blue, are the Marcy Dam lean-tos, Avalanche Lake, Lake Colden, and Mount Marcy; this is the return leg of the Avalanche Pass loop. Straight, marked in yellow, is Whales Tail, as well as Wright and Algonquin Peaks.\n\nContinue straight (south) as you begin the steadily increasing ascent to Algonquin Peak\u2014more than 2,800 feet over the next 3 miles. Just 0.5 mile farther, pass the intersection with the Whales Tail Trail on the left, just before crossing a seasonal brook that feeds MacIntyre Brook. There are several of these seasonal stream crossings along the initial portion of the ascent, with a notable one roughly 0.5 mile ahead, where a trail arrow indicates a crossing among several large boulders. The deciduous forest begins to give way to paper birches and firs as the climbing continues. Shortly after negotiating a very steep portion of bedrock that may require use of your hands, the sound of water beckons you on to what is the best spot to stop and take a break along the climb. Just before 2.6 miles\u2014more than halfway in distance but less than half of the ascent (at only 1,200 of the 2,800 feet to be climbed)\u2014is a stunning waterfall. The unnamed fall, which is the source of MacIntyre Brook, is roughly 30\u201340 feet high, with a modest flow and a shallow pool at its base.\n\nPast the waterfall, the trail continues its steep ascent where dense saplings enclose the trail. At roughly 2.9 miles reach a pass between Wright Peak to the east and a sheer vertical cliff to the west. At this juncture signs warn about being prepared for ice and snow, which can come as early as October and as late as May. Do not take this warning lightly, as long sections of trail are along exposed bedrock, which when covered in ice are dangerous or impassable without proper equipment (crampons, ice ax, warm clothing). Additionally, snow is considerably deeper than in the parking area where you started. Do not proceed during the winter if you are not prepared. Every year severe injuries and fatalities occur when people ignore these warnings.\n\nA little less than 0.5 mile more of scrambling and climbing, reach the intersection with the Wright Peak Trail on the left. The side trip, indicated with blue trail markers and described below, is highly recommended. But remember, more than 1,000 feet of climbing is required to reach Algonquin Peak from this point.\n\nSide Trip to Wright Peak\n\n(adds 0.8 mile and 600' of climbing)\n\nFor anyone wishing to work on their 46 High Peak bucket list, now is the time to bag one more: Wright Peak. The side trip is steep, but the view is worth it and hints at what to expect when summiting Algonquin farther on.\n\nAlmost immediately after starting the ascent, reach a wide section of bedrock that you will have to scramble up. In winter this slide may be a sheet of ice. Much of the trail is steep and traverses ledges, with short firs hugging tight to the trail until you reach the alpine zone. A sign indicates when you are entering this zone, just beneath a steep rock wall that you will have to climb. Once on top, the trees disappear, and you enter a stark and comparatively barren environment. Lichens, clumps of grass, and wind-stunted evergreens precariously cling to what little nutrients they can get along the wind-scraped peak. You still have a ways to climb, but with the peak in sight you will happily push on up the steep slope.\n\nTHE HEART OF THE HIGH PEAKS WILDERNESS\n\nWithout trees for trail markers, you will have to rely on painted spots along ledges, rock cairns, and various pebble-lined paths. Remember, you do not want to step on any vegetation, so be mindful of where you traverse. Once atop, you will find a stunning 360-degree panorama of the High Peaks, including your next stop: Algonquin Peak, looming 500 feet above to the south. Return the way you came, again being careful where you traverse.\n\nBack along the main trail continue south, follow yellow markers and continue the climb (more than 1,100 feet over the next roughly 0.6 mile) to Algonquin Peak. Within 0.25 mile, reach a very steep section of trail that ascends along an exposed portion of bedrock. To make your way up the sheer rock face, you will have to find foot- and handholds on small clefts and deformities in the rock. Complicating matters, water flows over the face and ice is likely when temperatures are below freezing. I found that the right side of the trail seemed to have more places for \"steps\" and handholds. After this protracted climb, the trail is more easily traversed, with large slabs of rock often forming steps along the climb.\n\nAbout 0.25 mile past the sheer climb, reach the beginning of the alpine zone. Unlike Wright Peak, there is significantly more climbing to be done above the alpine zone boundary, and the peak is never quite in sight to lure you onward. To navigate through the alpine zone, remember to walk only on the designated path or exposed rock. To find the path, look for paint on the rocks, rock cairns for the direction of travel, and pebble walls that separate the trail from the vegetative areas. There is still roughly 0.25 mile and more than 300 feet of elevation to go until the actual summit. As you climb to the alpine zone, your whole perspective changes; the trees are gone and nothing remains but rock and low-lying vegetation clinging to scraps of soil caught in the cleft of rocks or in the lee of boulders. Often wind whips over the summit, making it hard to hear, but the barren landscape is stunning. The peak is broad and spread out, with a dramatic panorama of the High Peaks Wilderness Area all around you. Only Mount Marcy, about 3.8 miles southeast, is higher, and on clear days the horizon seems endless. While you look about the summit and take in the view, once again be careful of the fragile ecosystem.\n\nIf it took you longer than expected to reach the summit or if weather or night threatens, then you should seriously consider returning the way you came. But if time, weather, and energy are on your side, then I recommend the return leg along Avalanche Pass. The route heads deeper into the High Peaks Wilderness and will take considerably more time and effort to complete. But after a bone-\u00adjarring descent, you traverse some areas that, combined with the trip to Algonquin, make for one of the most scenic routes you could take in the Adirondacks. Should you choose that route, the total trip length (without the side trip to Wright) would be 11.8 miles; otherwise, return the way you came for a trip length of 8.0 miles.\n\nDirections\n\nFrom the intersection of NY 73 and NY 86 in Lake Placid, head 3.3 miles south on NY 73, and follow the directions below. From the intersection of NY 73 and NY 9N in downtown Keene, head 10.9 miles west on NY 73, and follow the directions below.\n\nTurn south onto Adirondack Loj Road. Continue 5.1 miles to the vehicle-entry booth, and the main parking area is on the left past the visitor center.\n\n43\n\nAvalanche Pass\n\nTHE SOUTHERN SLOPE OF ALGONQUIN PEAK\n\n**SCENERY:**\n\n**TRAIL CONDITION:**\n\n**CHILDREN:**\n\n**DIFFICULTY:**\n\n**SOLITUDE:**\n\nGPS COORDINATES: Loj parking area: N44\u00b0 10.969' W73\u00b0 57.746' Algonquin Peak: N44\u00b0 8.611' W73\u00b0 59.223'\n\nDISTANCE & CONFIGURATION: 11.8-mile loop (includes the Algonquin Peak trail)\n\nHIKING TIME: 8.5\u20139.5 hours\n\nHIGHLIGHTS: Remote lake, dramatic cliffs, panoramic views, waterfalls\n\nELEVATION: 5,115' at Algonquin summit; 2,091' at lowest point\n\nACCESS: Open 24\/7; $5 for Adirondack Mountain Club members, $10 per day for nonmembers. During summer and holiday weekends the lots fill quickly and are at capacity by early afternoon. When parking booth is closed, pay and register with the self-service box. Parking is free for guests staying at the Loj\/Wilderness Campground.\n\nMAPS: National Geographic Adirondack Park, Lake Placid\/High Peaks (#742)\n\nFACILITIES: Changing rooms, bathrooms, showers, information center\n\nWHEELCHAIR ACCESS: No\n\nCOMMENTS: This trail begins at Algonquin Peak, and mileage begins at 4 miles.\n\nCONTACTS: High Peaks trail information: www.dec.ny.gov\/outdoor\/9198.html; emergency contact: 518-891-0235\n\nOverview\n\nThe initial descent from Algonquin Peak is bone jarring and punishing, 2,000 feet over 1.5 miles. Calling it a trail is a bit misleading; it's actually more like (and at times literally is) an active streambed. The good news is that the remainder of the trail, though rugged, is fairly level. The trail really stands out when you reach Avalanche Lake and head deeper into the pass. Steep cliffs and exposed mountain slides surround you as you hike literally suspended several feet above the lake. Combined with Algonquin Peak's stunning views and unique alpine zone, this trail is an amazing wilderness adventure but is recommended only for experienced hikers. Trail mileage begins with the 4-mile trek to reach Algonquin Peak.\n\nRoute Details\n\nFrom Algonquin Peak, look for the rock cairns that head down along the southern slope of the mountain. Just as when you climbed to the summit, be careful to descend only along the trail\/bedrock using rock cairns and pebble walls to help you navigate. A little over 0.3 mile and 400 feet along the descent, reach a fork in the trail that is encircled by shrub-sized evergreens. To the right\/straight is the unmarked path to Boundary and Iroquois Peaks, while to the left is the main trail, indicated by an arrow and yellow trail markers. Just after this intersection, reach a steep drop and the alpine zone sign on this side of the mountain. The trail is now very narrow and is closed in by thick, shrub-sized spruces that gradually give way to taller trees as you quickly descend along the steep trail.\n\nAt 4.75 miles reach a steep descent that leads to a stream that tumbles through a cleft in the rock. This stream accompanies you\u2014in fact much of the way it is the trail\u2014as you descend the remaining 1,100 feet toward Lake Colden. Shortly after intersecting the stream, the trail heads away briefly only to wind back to the stream and, after descending a dilapidated log ladder, intersect it again at a tiny waterfall with a shallow pool at its base. From here forward, the trail weaves back and forth and through the streambed, and finding the exact path while navigating the boulders and logs is tricky. To complicate things, there are a couple of trail reroutes along the descent, indicated by logs, branches, and rocks laid across the old trails. When in doubt, look for the yellow trail markers on the trees, and above all, be careful with your footing. Believe me, it is worth it.\n\nWhen I made the descent, my foot slipped and my leg got caught between two rocks, deeply cutting my shin. It looked worse than it was, but it required eight stitches when I finally hobbled out hours later. I did not see anyone until I reached the final trail segment near the Loj parking area. Had my leg not popped out when trapped between the two rocks, it could have easily broken, turning a painful inconvenience into an outright disaster. It was a stark reminder that when in the wilderness, you are on your own and often self-rescue is your only option. Also, you are only as prepared as what you bring with you; a first aid kit (see page 17 for recommended items), extra food, flashlight or headlamp, and warm clothes are essential in the event you are forced to spend a night in the woods.\n\nAt 1.3 miles along the descent, the trail and stream follow a broad and open ledge a short way before reentering the forest on the north side of the stream. About 0.5 mile past this open ledge, the trail passes a tall waterfall with a shallow pool on the right. This waterfall roughly marks the end of the descent, and you soon reach a major trail intersection at 6 miles. To the right and over a log bridge, marked in yellow, is the trail to Lake Colden Dam\/Outpost, as well as Cold Brook and Indian Pass Trails. Straight and marked in blue is the return leg of the loop to the Adirondack Loj via Avalanche Lake and Pass. Continue north through a marsh with several log-and-plank boardwalks and stunning views of the exposed slide on Mount Colden to the east. A third of a mile farther, 6.4 miles overall, reach another trail intersection and a trail register. To the right is Mount Colden via the northeast Mount Colden Trail, marked in yellow. Straight is the return to the Loj, now marked in yellow.\n\nContinue north toward Avalanche Lake and Pass. Along this section of the trail, the stream between Avalanche Lake and Lake Colden flows on the left, and you will have to navigate a few mucky areas along the 0.3-mile stretch to Avalanche Lake. About midway, pass a camping disk, and the gap between Mount Colden to the east and Avalanche Mountain to the west visibly narrows. When you reach the tip of Avalanche Lake, you see that it is just a sliver of dark water between two sheer cliffs. The trail follows the west edge of the lake, and as the sheer cliff and lakeshore draw ever closer, you may begin to wonder where there is room for a trail. A quarter mile farther, you see that the answer is nowhere. The trail, an elevated boardwalk, has been literally attached to the sheer rock cliff and is suspended half a dozen feet above the dark lake. The boards are in pretty good shape, but wear and tear and seasonal damage make it prudent to test your footing and not put all your weight on a single board. This is the first of two elevated walks, and in between is a maze of boulders that you have to weave through and climb over. Log bridges and ladders assist in some of this trek, but be mindful of the condition of such aids as they are in a constant cycle of deterioration and replacement. It's a stunning scene, with evidence of recent slides (avalanches) throughout.\n\nWALKWAY SUSPENDED ABOVE AVALANCHE LAKE\n\nAt 7.25 miles, the maze of boulders ends at the northern tip of Avalanche Lake. The trail continues north, heading deeper into the pass, first through a short muddy slog beside and through Avalanche Lake's feeder stream. Next is a narrow passage beside a steep cliff to the west, just before you reach the active avalanche area as indicated by the warning signs, 7.65 miles. Utterly clear of standing trees, the pass has an oddly unsteady feel underfoot because of the layers of mud and buried trees. After traversing to the western edge of the trail, you see how deep this fill is and why the ground has such a hollow feel. Slightly farther, you reach the end of the designated avalanche area, and the gradual descent to the Loj begins.\n\nBE CAREFUL ON THIS STEEP DESCENT OF ALGONQUIN PEAK.\n\nAt 8.4 miles reach the intersection with the Lake Arnold Trail, marked in blue on the right. Continue straight toward the Adirondack Loj, still marked in yellow. Shortly after the intersection, reach a bridge with handrails that crosses Marcy Brook; this brook will accompany you the whole way north to the interior campsites and lean-tos near Marcy Dam, a little more than 0.75 mile ahead. You will pass a couple of lean-tos along this trek, along with several designated camping areas, before you reach the intersection with the Van Hoevenberg Trail, on the right and marked in blue, 9.4 miles overall. Bear left and continue north past the trail register a short ways to another intersection, this time indicating the direction to the Adirondack Loj and path forward. Continue left past the remains of Marcy Dam toward the Loj, now following blue disks for the rest of the trail, and soon reach a bridge that crosses Marcy Brook a few hundred feet north of the dam. Once across, the trail swings briefly south to intersect where it used to cross Marcy Dam and then continues its northerly descent. The remaining 2.2 miles back to the Loj follow a highly traveled and well-maintained section. At 10.8 miles reintersect the trail to Algonquin with only 1 mile remaining back along the original route to the Loj and parking area for a total loop of 11.8 miles.\n\nDirections\n\nFrom the intersection of NY 73 and NY 86 in Lake Placid, head 3.3 miles south on NY 73, and follow the directions below. From the intersection of NY 73 and NY 9N in downtown Keene, head 10.9 miles west on NY 73, and follow the directions below.\n\nTurn south onto Adirondack Loj Road. Continue 5.1 miles to the vehicle-entry booth, and the main parking area is on the left past the visitor center.\n\n44\n\nLittle and Big Crow Mountains\n\nLOOKING OUT ON THE HIGH PEAKS FROM BIG CROW\n\n**SCENERY:**\n\n**TRAIL CONDITION:**\n\n**CHILDREN:**\n\n**DIFFICULTY:**\n\n**SOLITUDE:**\n\nGPS COORDINATES: N44\u00b0 15.696' W73\u00b0 44.010'\n\nDISTANCE & CONFIGURATION: 3.3-mile loop\n\nHIKING TIME: 2\u20133 hours\n\nHIGHLIGHTS: Spectacular vistas\n\nELEVATION: 1,742' at trailhead, 2,800' at highest point\n\nACCESS: Open 24\/7; no fees or permits required\n\nMAPS: Hurricane Mountain Wilderness map: tinyurl.com\/hurricanemtnmap; National Geographic Adirondack Park, Lake Placid\/High Peaks (#742)\n\nFACILITIES: None\n\nWHEELCHAIR ACCESS: No\n\nCONTACTS: High Peaks Adirondack trail information: www.dec.ny.gov\/outdoor\/9198.html; Hurricane Mountain Wilderness: www.dec.ny.gov\/lands\/100895.html; emergency contact: 518-891-0235\n\nOverview\n\nThis short trip is an excellent introductory hike for those who want to familiarize themselves with the beautiful vistas of the High Peaks without spending a whole day. Both experts and novices will appreciate the wealth of views atop the many ledges you encounter as you weave your way to the summit.\n\nRoute Details\n\nTo make the trail into a loop, you will have to walk 1.4 miles along O'Toole Road and Hurricane Road. You can hike the road section either at the end or at the beginning of the hike, but I recommend hiking the trail clockwise and including the roadside portion at the beginning. The counterclockwise direction includes a steeper ascent to Big Crow and requires you to end your hike walking uphill along the road rather than with a descent and flat stroll through the woods.\n\nThe trail register is located a few hundred yards northwest of the parking area. To reach the western trailhead, walk down O'Toole Road, and turn right onto Hurricane Road, 1.1 miles. Continue down Hurricane Road 0.2 mile, and look for a small orange ADK marker affixed to a wooden sign that points to the trail. This sign is difficult to see while driving but evident while on foot. The trail starts at a couple of log steps and winds around the left side of a private residence. This section of the trail is on private property, so be careful to stay on the marked portion of the trail.\n\nRed trail disks mark the way, and you quickly begin to climb through a stand of small saplings. Roughly 0.5 mile into the trail, 1.8 miles from the start, expansive views to the south and west begin to open up behind you. Exposed portions of bedrock begin to jut out, both making the terrain more rugged and forming natural steps as you climb. You will see yellow paint blazes on trees that may seem to guide you along the trail, but these are actually the boundary between private property and the Hurricane Mountain Wilderness.\n\nThe exposed ledges along the trail all offer wonderful views, but a particularly spectacular vista is available just past where you enter the public lands, 2 miles from the start. Take advantage of these lookouts, as the summit of Little Crow does not offer any views. Red disks and rock cairns help you navigate as you weave your way over exposed portions of bedrock and through windswept conifers. At 2.3 miles the peak of Little Crow Mountain (2,450') is off to the left of the main trail. The bald top of Little Crow is quite exposed, but it is also a broad peak; consequently, the encircling evergreens limit views typical of a bald summit.\n\nTo reach Big Crow, continue east along the exposed bedrock, and then quickly descend the short pass between the two peaks. Big Crow appears more rugged as you approach, but the climb to Little Crow is actually the tougher of the two. The oak saplings that fill the pass are a pleasant diversion from the encompassing evergreens that shroud the mountaintops. Indeed, as you climb Big Crow, the conifers become dense, and they so closely flank the trail that only the path ahead is visible. However, the dense foliage is brief, and a view north soon rewards your efforts. Whiteface Mountain, with its numerous ski slopes, is particularly prominent as you look north.\n\nThe trail turns south, and a lookout south and west quickly reveals the path you have taken. After a brief scramble up the exposed bedrock, you reach the beginning of the exposed spine that runs along the short ridge. A sweeping panorama of tall peaks extends from due east southward to due west as you traverse the ridge. The highest point on the trip (2,800') is at 2.8 miles and is simply atop a boulder where a sign reminds visitors that no camping or fires are allowed at the peak. Continue east along the ridge to where the descent of the open face is rather steep. After approximately 0.25 mile the steep section is past, and the rest of the hike through the spruce forest feels very sheltered compared with the exposed peaks. At 3.2 miles you cross a small brook, after which the trail is mostly level, and you quickly reach the trail register and return to the parking area approximately 0.25 mile ahead.\n\nDirections\n\nFrom the intersection of NY 9N and NY 73 in downtown Keene, head southeast on NY 73\/NY 9N, and turn left onto County Road 13\/Hurricane Road (1.7 miles north of the southern intersection of NY 73 and NY 9N, and 0.2 mile south of the northern intersection of NY 73 and NY 9N). Continue on CR 13\/Hurricane Road 2.3 miles, and turn left onto O'Toole Road. Follow O'Toole Road 1.1 miles to the parking area at the end.\n\n45\n\nPitchoff Mountain\n\nTHE PITCHOFF MOUNTAIN RIDGELINE\n\n**SCENERY:**\n\n**TRAIL CONDITION:**\n\n**CHILDREN:**\n\n**DIFFICULTY:**\n\n**SOLITUDE:**\n\nGPS COORDINATES: East trailhead: N44\u00b0 14.624' W73\u00b0 50.752' West trailhead: N44\u00b0 13.172' W73\u00b0 53.175'\n\nDISTANCE & CONFIGURATION: 4.6-mile point-to-point\n\nHIKING TIME: 5\u20136 hours\n\nHIGHLIGHTS: Panoramic views, open ledges\n\nELEVATION: 1,850' at trailhead, 3,600' at highest point along the trail\n\nACCESS: Open 24\/7; no fees or permits required\n\nMAPS: Sentinel Range Wilderness map: tinyurl.com\/sentinelrangemap; National Geographic Adirondack Park, Lake Placid\/High Peaks (#742)\n\nFACILITIES: None\n\nWHEELCHAIR ACCESS: No\n\nCONTACTS: High Peaks Adirondack trail information: www.dec.ny.gov\/outdoor\/9198.html; Sentinel Range Wilderness: www.dec.ny.gov\/lands\/101901.html; emergency contact: 518-891-0235\n\nOverview\n\nThis rocky ridge parallels NY 73 and provides wonderful views of the High Peaks wilderness to the south. This is an end-to-end hike and easiest to manage with two cars\u2014or park at one end and catch a ride to the other. There are several steep rock scrambles, so be prepared during wet and icy conditions.\n\nRoute Details\n\nThe two trailheads are 2.7 miles apart along NY 73. This is a major road with narrow shoulders, so it is advised that you either leave a vehicle at both ends or leave your car at one end and get a lift to the other. I would suggest getting a ride before your hike, rather than after, to avoid uncertainty and because the trail will likely take longer than is typical for the mileage. The west trailhead has more parking room, so it is best to leave your vehicle there and hike the trail from east to west. Directions below are given in that direction.\n\nThe east trailhead is little more than a worn path with a couple of steps that extend down to the road, but there are signs, and it is distinct. The trail register is at the beginning of the trail, and even though you will sign in and out at different registers, it is still important to sign in and out. The trail, marked with red disks and yellow blazes for the whole trip, turns west almost immediately and parallels the road roughly 0.3 mile under a canopy of birch and beech trees. The trail swings north, and you begin climbing among a jumble of cobbles and boulders alongside a rocky brook. The grade steadily increases and becomes a lot steeper 0.8 mile into the hike. At approximately 1 mile you have almost reached the ridge that you will traverse, and though many steep up-and-down sections are still ahead, the majority of the climbing is behind you. At 1.1 miles you pass through a wet area between two 20- to 30-foot-high rocky outcrops that flank the trail. After this small pass, the trail turns left, and you scramble up to the top of one of these outcrops.\n\nAt 1.2 miles you have reached the first of five minor peaks that make up Pitchoff Mountain. Ridgelines with multiple peaks do not have separate mountain names if the prominence of a peak does not exceed 980 feet or 7\u20138% relative prominence above the highest saddle or col between the peaks. In ridges such as this one, with slight elevation differences between peaks and cols, it's clear why the distinction between mountains and peaks is useful. The trail weaves its way up and down many times before you reach the actual summit, 1.5 miles ahead. The views south are exceptional, and many similar vantage points will accompany you as you traverse the spine. Yellow paint blazes mark the path along these bald peaks. If you are attempting this hike after a recent snowfall, finding your way can be time consuming. Snow falls as early as October and as late as spring in the High Peaks region. During the light snows of the fall and thaws during spring, when the snow is not deep enough for snowshoes, you need crampons or their equivalent to cross the bald peaks. The cycles of thawing and freezing on the exposed rocky ledges and peaks produce sheets of ice that are treacherous without added traction. Pack accordingly, even though these conditions may not be evident down at the road.\n\nICICLES ALONG THE TRAIL IN EARLY OCTOBER\n\nThe trail weaves up and down over boulders and zigzags through wind-stunted evergreens across the entire spine. Many times you will think that you have reached a peak or traversed a pass between peaks, only to find yet another climb or pass before you. This is particularly true for the 0.5 mile between the first and second peaks. The second peak, 1.8 miles overall, offers a wide panoramic view south but does not offer the 360-degree views found at the first peak. The trail descends and climbs mostly along the ridgeline to the third peak at 2.1 miles, where there are also views south.\n\nBetween the third and fourth summits, the trail meanders to the north side of the ridge, where the fir trees become thick, and then heads back to the south side. The fourth peak, 2.5 miles, is not the highest point of elevation but is the last peak with panoramic views. At 2.7 miles you pass some balancing boulders that mark the fifth and final peak as well as the beginning of the steep descent from the ridgeline. Dense conifers shroud the trail for most of the descent, but occasional breaks in the trails and open ledges offer some views.\n\nAt 3.2 miles you reach the only fork in the trail and a sign that indicates a view is off to the left. This lookout, approximately 0.1 mile along an unmarked path, is atop a sheer ledge with two balanced boulders that sit at its westernmost edge. Cascade Mountain and Upper and Lower Cascade Lakes are the prominent features of the view, and the lookout is certainly worth the short side trip.\n\nBack along the trail, the descent is much steeper, and the forest transitions back to hardwoods. The steep descent continues about 0.5 mile, during which you pass under the sheer rock wall that makes up the lookout to your left. The trail turns west, levels off, and parallels NY 73 and Upper Cascade Lake for the next 0.5 mile. At 4.3 miles, the trail swings south, and you descend for the next 0.3 mile until you reach the western trailhead and Upper Cascade Lake parking area.\n\nDirections\n\nParking is available at both the east and west ends of Cascade Lake along NY 73. Parking is limited in both areas, and there is little room on the roadside. The parking areas are 2.7 miles apart along NY 73.\n\nFrom the northern intersection of NY 73 and NY 9N in downtown Keene, head 3.9 miles west on NY 73 to the east parking area. Head 6.6 miles west on NY 73 to reach the west parking area.\n\nFrom the intersection of County Road 21 and NY 73, near the Olympic ski jump in Lake Placid, head 5.6 miles east to the west parking area or 8.3 miles east to the east parking area.\n\n46\n\nAmpersand Mountain\n\nON TOP OF AMPERSAND MOUNTAIN\n\n**SCENERY:**\n\n**TRAIL CONDITION:**\n\n**CHILDREN:**\n\n**DIFFICULTY:**\n\n**SOLITUDE:**\n\nGPS COORDINATES: N44\u00b0 15.095' W74\u00b0 14.374'\n\nDISTANCE & CONFIGURATION: 5.2-mile out-and-back\n\nHIKING TIME: 3\u20134 hours\n\nHIGHLIGHTS: Bald summit, panoramic views\n\nELEVATION: 1,550' at trailhead, 3,353' at highest point\n\nACCESS: Open 24\/7; no fees or permits required\n\nMAPS: National Geographic Adirondack Park, Lake Placid\/High Peaks (#742)\n\nFACILITIES: None\n\nWHEELCHAIR ACCESS: No\n\nCONTACTS: High Peaks Adirondack trail information: www.dec.ny.gov\/outdoor\/9198.html; emergency contact: 518-891-0235\n\nOverview\n\nAmpersand Mountain sits on the confluence of the flat north and High Peaks regions of the Adirondacks. An extremely popular hike, it offers a varied experience as you transition from a gentle walk through tall hemlocks to a steep ascent over boulder stairs. Passage through enormous, craggy boulders takes you to the bald summit, from which the lake sprawls out beneath you to the north and the High Peaks can be seen in the distance to the south.\n\nRoute Details\n\nWhen I approached on a sunny Saturday afternoon, it was clear that this trail is extremely popular. Dozens of cars lined NY 3 on both sides, and I could see that the small parking area was nowhere near adequate. While the lot also provides access to a short trail to Dutton Brook, the vast majority of hikers were climbing Ampersand Mountain. Indeed the trail register had been filled days before by the dozens of parties that hike the trail each day during the peak of summer. As with all popular trails, it is best to visit these in the off-season, during the week, or early if you have any desire for privacy.\n\nSTONE STEPS LEADING TO AMPERSAND MOUNTAIN\n\nWhy is the trail so popular? It is likely that the combination of the easy stroll at the beginning through towering old-growth forest, its proximity to a major road and tourist destinations, and the spectacular views all contribute to its appeal. Unfortunately, I reached the summit when a major thunderstorm had rolled in and visibility had dropped to several feet, so I cannot attest to the latter. The peak was awash in a sheet of flowing rain, and I was stuck in a cloud. Lightning flashed in all directions, and fierce winds threatened to blow others and me off the peak. Lingering in these conditions was unwise at best, so I did not get to see the views that people raved about, but according to veterans of this hike, the views are breathtaking and panoramic. It was a stark reminder that even though it is sunny and clear at the start of a hike, conditions can change quickly, and a little precaution and preparation are required on every hike and in every season.\n\nUpdate: Having revisited the trail, I can attest that the views atop Ampersand are spectacular. When I originally visited, visibility was limited to a few feet, so I was surprised on returning to find how broad and spread out the peak is. Even on a crowded Saturday\/\u00adholiday weekend, there was space for numerous groups to find spots to sit and enjoy the views while resting before their descent. There are several spots with 360-degree views, but on windy days it is likely you will want to shelter in one of the many dips\/crevices, of which there are several.\n\nThe trailhead is on the southern side of NY 3 across from the small parking area. The trail register is just inside the woods, and the wide and gravelly base attests to the heavy use and ongoing maintenance. In less than 0.3 mile, you cross Dutton Brook over a log bridge. Shortly after this crossing, you will definitely notice the towering hemlocks, which are some of the few remaining old-growth trees in the Adirondacks. This forest is considered the largest known sugar maple\u2013yellow birch\u2013hemlock forest in Adirondack Park. Red disks mark the trail, but it is so heavily trodden that, except for a few places near the summit where blowdown occurs, they are hardly necessary. At 0.8 mile, planks laid lengthwise form a boardwalk over a wet area. At 1 mile, you cross McKenna Brook over a log bridge.\n\nPast this crossing, the trail begins to climb, and you will rock-hop across streams and more mucky areas as the trail gradually becomes steep. The hemlocks that previously towered overhead give way to birches and maples, and the vast stonework employed to control erosion becomes evident. Long stretches of boulder stairs were constructed within the past decade to minimize the impact hikers had on the trail, once considered one of the most heavily eroded. Climbing only on the stone sections gives the forest a chance to regenerate some of its ground cover and, hopefully, forestalls further erosion. The last mile is basically a vertical climb, and you may need to use your hands.\n\nAs the forest transitions from hardwoods to stunted firs and evergreens, the stonework diminishes until it becomes nonexistent, and the erosion becomes severe. At this point, the various paths chosen to bypass washed-out areas or precipitous sections make finding the trail difficult and trail markers a necessity. Take your time throughout the final climb, and use caution over eroded and slippery surfaces.\n\nAround 2.3 miles, the trail levels off briefly just shy of the summit. You pass a few large boulders and eventually come to a passage between exposed faces of bedrock with a fissure opening up on your right. The trail then turns left and winds around to a rock face that you must scramble up and over. The roots of a tree will help you up, but after seeing the erosion caused by so many hikers, you can't help but wonder how much longer this tree will be around. Yellow arrows painted on the bald mountaintop point you in the direction of the easiest ways around the three knobs. Many of the bald peaks in the Adirondacks were created when Verplanck Colvin burned the tops of mountains during his 19th-century survey of the region. Erosion soon swept away the topsoil, leaving the mountains bald. Each knob offers its own merits and vantage points, and curious hikers will also find the remnants of a fire tower, as well as a plaque in memory of the Ampersand Hermit. The hermit, Walter Channing Rice, manned the fire tower from 1915 to 1923.\n\nLOG BRIDGE ALONG THE TRAIL\n\nDirections\n\nFrom the west From Tupper Lake, head east on NY 3\/NY 30. At 5.3 miles, bear right to stay on NY 3. After another 6.9 miles, the designated parking area is on your left. The parking area may be full, so take care when parking on the side of the road.\n\nFrom the east From the intersection of NY 3 and NY 86 in Saranac Lake, head 8.4 miles west on NY 3; the parking area is on the right.\n\nMARCY DAM (Trail 43, Avalanche Pass)\nAppendix A: Managing Agencies\n\nADIRONDACK PARK AGENCY\n\napa.state.ny.us\n\n518-891-4050\n\nNEW YORK STATE DEPARTMENT OF ENVIRONMENTAL CONSERVATION\n\nwww.dec.ny.gov\n\nRegion 5 (Eastern Adirondacks): Essex, Hamilton, Warren, Fulton, Saratoga, and Washington Counties: 518-897-1200\n\nRegion 6 (Western Adirondacks): Jefferson, St. Lawrence, Lewis, Oneida, and Herkimer Counties: 518-785-2239\n\nForest Ranger Emergency Contact: 518-891-0235\nAppendix B: Gear Lists\n\nDay Hike Gear List\n\nTHE ESSENTIALS\n\nBackpack\n\nBoots\n\nExtra food\n\nFirst aid kit\n\nFlashlight and extra batteries\n\nPocketknife or multitool\n\nThis book\n\nTopographical map and compass\n\nWater bottles\n\nWater filter or purification tablets\n\nWaterproof matches and\/or lighter, as well as a fire starter\n\nWhistle\n\nADDITIONAL\n\nChange of clothes\n\nExtra socks\n\nInsect repellent\n\nPersonal medications\n\nPoncho or raingear\n\nUV sunblock and lip balm\n\nWarm hat\n\nOPTIONAL\n\nCamera with extra batteries and memory cards\n\nGaiters\n\nGarbage bag for packing out trash\n\nGPS unit\n\nRepair kit\n\nSmall towel\n\nSwimsuit\n\nSurvival kit\n\nToilet paper\n\nTrowel or small shovel for digging catholes\n\nSummer Backpacking Gear List\n\nBackpack\n\nBear bag or canister\n\nCleaning supplies (biodegradable detergent and container, sponge, and scouring pad)\n\nCollapsible bucket or water bag\n\nCollapsible pillow\n\nCooking pots and pans\n\nEating utensils (cup, bowl, plate, knife, fork, and spoon)\n\nFood (including extra for emergencies)\n\nGarbage bag for packing out trash\n\nRope and nylon cord (50 feet)\n\nShelter and conveyance\n\nSleeping bag and sleeping pad or mattress\n\nStove and fuel\n\nTent and rain fly\n\nTent stakes\n\nWater bottles\n\nWater filter or purification tablets\n\nWaterproof matches and\/or lighter, as well as a fire starter\n\nPERSONAL ITEMS\n\nBiodegradable hand soap\n\nFirst aid kit\n\nInsect repellent\n\nPersonal medications\n\nSmall towel\n\nToilet paper\n\nToothbrush and toothpaste\n\nTrowel or small shovel for digging catholes\n\nUV sunblock and lip balm\n\nGEAR\n\nCamera with extra batteries and memory cards\n\nCamp light or candle\n\nFlashlight and extra batteries\n\nGPS unit\n\nPocketknife or multitool\n\nRepair kit\n\nSewing kit\n\nStuff sacks\n\nSurvival kit\n\nThis book\n\nTopographical map and compass\n\nWhistle\n\nCLOTHING\n\nBoots\n\nCamp shoes\n\nExtra socks\n\nGaiters\n\nJacket or outerwear (layer clothing)\n\nLight shirt or T-shirt\n\nLong pants\n\nPlastic bag for laundry\n\nPoncho or raingear\n\nShorts\n\nSock liners\n\nSwimsuit\n\nWarm hat\n\nWarm shirt\nAppendix C: Hiking Clubs\n\nADIRONDACK MOUNTAIN CLUB\n\nIncludes many local chapters in New York and beyond\n\nadk.org\n\nADK46ERS\n\nNonprofit that focuses on the ultimate Adirondack bucket list\n\nadk46er.org\nAppendix D: Suggested Reading\n\nDunn, Russell. Adirondack Waterfall Guide: New York's Cool Cascades. Delmar, NY: Black Dome Press Corp., 2004. Out of print.\n\nKetchledge, Edwin H. Forests and Trees of the Adirondack High Peaks Region: A Hiker's Guide. Lake George, NY: Adirondack Mountain Club, 1996.\n\nStarmer, Tim. Five-Star Trails: Finger Lakes & Central New York. Birmingham, AL: Menasha Ridge Press, 2014.\n\nVan Diver, Bradford B. Roadside Geology of New York. Missoula, MT: Mountain Press Publishing Company, 1985.\n\nStarmer, Aaron, and Cate Starmer and Tim Starmer. Best Tent Camping: New York State. Birmingham, AL: Menasha Ridge Press, 2013.\n\nAbout the Author\n\nTIM STARMER has always been an outdoors enthusiast and spent most of his childhood seeking out remote and wild areas whenever possible. During a brief hiatus from Brown University during 1997, he drove across the United States for six weeks, camping the entire way. Along the way he explored many of the West's national and state parks, including Canyonlands, Yellowstone, Arches, and Bryce Canyon\u2014he even braved pitching a tent among the mosquito swarms in Badlands National Park. At the trip's conclusion, he headed down to Australia, where he backpacked for a few months exploring the eastern outback, the Great Barrier Reef, and the caves of Tasmania, as well as traversing the Tasmanian Wilderness World Heritage Area along the Overland Track. Starmer owns and operates New Heritage Woodworking, a construction company that specializes in designing and building timber frames, and he can still be found exploring the wilds whenever possible.\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n# _Spartan Women_\n\n# SPARTAN WOMEN\n\nSARAH B. POMEROY\n\n2002\n\nOxford New York  \nAuckland Bangkok Buenos Aires Cape Town Chennai  \nDar es Salaam Delhi Hong Kong Istanbul Karachi Kolkata  \nKuala Lumpur Madrid Melbourne Mexico City Mumbai Nairobi  \nS\u00e3o Paulo Shanghai Singapore Taipei Tokyo Toronto\n\nand an associated company in Berlin\n\nCopyright \u00a9 2002 by Sarah B. Pomeroy\n\nPublished by Oxford University Press Inc.\n\n198 Madison Avenue, New York, New York 10016\n\nwww.oup.com\n\nOxford is a registered trade mark of Oxford University Press\n\nAll rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of Oxford University Press.\n\nLibrary of Congress Cataloging-in-Publication Data Pomeroy, Sarah B.\n\nSpartan women \/ Sarah B. Pomeroy.\n\np. cm.\n\nIncludes bibliographical references and index.\n\nISBN 0\u201219\u2012513066\u20129; 0\u201219\u2012513067\u20127 (pbk.)\n\n1. Women\u2014Greece\u2014Sparta (Extinct city)\n\n2. Sparta (Extinct city)\u2014Social conditions.\n\nI. Title\n\nHQ1134.P66 2002\n\n305.4\u203209389\u2014dc21 2001055961\n\nExcerpt from Aristotle, _Politics Books I and II_ , translated by Trevor Saunders.\n\nReprinted by permission of Oxford University Press.\nTO JACOB AND DINA\n\n## PREFACE\n\nThis book is the first full-length historical study of Spartan women to be published. I have not written in detail about Spartan women since the publication of _Goddesses, Whores, Wives, and Slaves_ , although more recently I contributed to the scholarship on this subject in two jointly authored books. It was when I was writing a brief survey of the Spartan family and one of the anonymous referees remarked \"but there were no female Spartiates\"  that I first realized that there was much work to be done.\n\nMy recent work on Xenophon and Plutarch, two of the major sources on Spartan women, made me appreciate how little had been written about the ways in which the perspectives of these two authors, more than any others, have shaped our views of Spartan women. Because of my training as a papyrologist, I have often written about women in the Hellenistic period; writing about Plutarch convinced me to extend this study to Roman Greece. Thus this book covers some thousand years of history, but despite the timespan, it is short. Extant sources are few, although I have tried to exploit every ancient text and artifact that appeared relevant. The sources on various aspects of Spartan women's lives are unevenly distributed. There is far more information on education, reproduction, and religion than on other subjects. These emphases are reflected in the lengths of the various chapters. The longstanding lack of serious scholarship on the history of Spartan women has meant that there has been less impetus than is usual for a scholar working in classics and ancient history to take into account previous studies. Nevertheless, this book has been the most difficult one I have ever written. It must be confessed that we know little about Spartan women, but it is not so readily conceded that we do not actually know much about Spartan men either. Compared to what is known about Athens, there is little direct evidence about life in Sparta. It is difficult to construct a realistic picture of how women and men actually lived in such a place; there is, however, a great deal of evidence for what other Greeks thought about their lives. Much current scholarship on Sparta is devoted to the latter subject. It must be emphasized that often the primary sources do not distinguish between prescriptive and descriptive writing; and pictorial representations may also be idealistic or fantastic rather than realistic. Contemporary scholars, as well, differ in their assessment of what constitutes historical reality, and what was part of the \"Spartan mirage.\" Having stated these caveats here, I will not repeat them throughout the book. I must, however, confess that my tendency is to grant more credence to the primary sources than some contemporary hypercritical Spartanologists are wont to do, and to understand that they generally reflect an actual historical situation rather than a utopian fiction. Sophocles described the versatility and ingenuity of the human race. The Greek text permits a literal and gender-free translation. The chorus reflects:\n\nMany the wonders, but nothing more wonderful than a human being . . .\n\nHaving a clever inventive skill beyond hope\n\nA person proceeds sometimes to evil, sometimes to good.\n\n( _Antigone_ 332\u201333, 365\u201368)\n\nA survey of the sources may be found in an Appendix at the end of this book. The nineteenth century paintings that are reproduced in this book may serve to remind us that history is a conversation between the present and many pasts.\n\n### Chronological Conundrums\n\nThe traditional chronological framework for Greek history, which labels blocks of time \"archaic\" (ca. 750\u2013490),\"classical\" (490\u2013323), and \"Hellenistic\" (323\u201330), is based on political changes that are reflected in the visual arts. While this periodization is appropriate to most Greek poleis (especially Athens), it does not reflect some of the most significant events in Spartan history. Furthermore, the usual framework does not take into account events that affected women. In any case, it is irrelevant insofar as Sparta's contribution to the material arts was negligible after the archaic period.\n\nPerhaps most important for Spartan history were the political and social changes that occurred after the Second Messenian War. By the end of the seventh or in the early sixth century these changes created the distinctive Spartan way of life. Changes in the fifth and fourth century may have been significant, but these were not accompanied by sharp dislocations. A major turning point was the aftermath of the battle of Leuctra (371 B.C.E.), when enemy troops invaded Spartan territory for the first time, soundly defeated the Spartans, and brought about the liberation of Messenia. Though individual Spartans were tempted to work as mercenaries, Sparta declined to participate in the campaigns of Alexander and thus was not so immediately affected by the changes that produced the Hellenistic world. Sparta's relative isolation was not ended until the reign of Agis IV,which began ca. 244 B.C.E., when Sparta went through a series of political upheavals culminating in defeat by the Roman general Flamininus in 195 B.C.E. and inclusion in the Roman province of Achaea.\n\nThis simple time line does not reveal how the Spartans themselves manipulated, created, and recreated their own history. There were two successful programs to revive the traditional Lycurgan constitution and the social, educational, and religious institutions alleged to have existed in earlier times, one in the Hellenistic period, the second in the Roman period. At the time of both revivals, the sources refer explicitly to actions taken in accordance with the ancient customs and laws. The impact of these revivals complicates the historian's task. For example, if the evidence for the authority of the priestess of Artemis Orthia is purely Roman, should we assume that she exercised exactly the same power in an earlier period? The Spartans believed (or at least wished others to believe) that she had. In the Roman period, they were known to be proud and pedantic about their heritage, and nostalgia would have encouraged them to accept or even promulgate myths as historical truth and exaggerate the virtues and distinctiveness of their past (see Appendix). Plutarch, one of our principle sources, is often not aware of the chronological problems, and in fact offers pieces of information about such important topics as marriage that are mutually contradictory unless the practices he refers to were not concurrent but occurred at different time periods. In his defense, in his works that deal with Sparta, he was writing biography and philosophy, not history. Although these problems will be discussed as they emerge, let me sketch them here. As a historian, I naturally try to use a chronological approach, but because of the ways in which the Spartans themselves revised their own history, I have found a straightforward chronological framework unworkable. For this reason, the chapter titles are topical: the first three, however, follow the Spartan woman through the life cycle. Furthermore, the discussions of the topics are, as much as possible, chronological. Motherhood is the thread that links all the chapters. The reader should note in addition that B.C.E. or C.E. have been added to a date when necessary to avoid ambiguity. Otherwise, all dates should be understood as \"Before Common Era.\" \"Spartan women\" applies only to women of the highest civic class, although I will discuss other women who lived in the territory controlled by Sparta and who interacted with the highest class.\n\nFollowing the precedent of classical authors, I will refer to the legendary lawgiver Lycurgus and the Lycurgan constitution without implying a belief that this shadowy figure ever existed, or that Spartan customs or laws were the result of a single creative act. In the same way I will refer to the _rhetra_ (\"legislation\") of Epitadeus without insisting that Epitadeus ever existed. Plutarch ( _Agis_ 5) reports that sometime after the Peloponnesian War a certain ephor named Epitadeus proposed a rhetra that would permit a person to give or bequeath his _kleros_ (\"plot of land\") and house to anyone he wished. Xenophon ( _Lac. Pol._ 15) does not name Epitadeus, but observes that in his day the Spartans no longer obeyed the laws of Lycurgus. Aristotle ( _Pol._ 1270a15\u201334) also does not mention Epitadeus. Whether or not Epitadeus ever existed, major economic changes associated with him occurred at the end of the fifth or in the early fourth century (see chap. 4). These changes began earlier, but the dramatic events after the Peloponnesian War precipitated the changes and made them perceptible. The changes are important because they increased women's potential to own immovable property. To establish the chronological framework, my book will attribute these changes to the rhetra of Epitadeus without lingering on the complexities of dating.\n\nAnother issue is whether the cult of Hera at Elis is directly relevant to Spartan women. We do not know if races in honor of Hera were restricted to local girls from Elis or were pan-Hellenic, like the competitions for men at neighboring Olympia. The latter seems more likely. Whatever the current political relationships in Greece, the games were usually held under conditions of a temporary peace, and statuettes of the victors show girls dressed in the semi-nude costume associated only with Spartans. Since only Spartan women are known to have seriously pursued physical education, they would probably have been the most numerous among the competitors. Convenience of travel to nearby Elis probably ensured a strong presence for Spartan women in any female agonistic activity at Olympia. This volume therefore includes a discussion of the athletic events associated with the cult of Hera at Elis (see chap. 1).\n\nI am grateful to the John Simon Guggenheim Memorial Foundation for a fellowship and to the Fellows of St Hilda's College, Oxford, and to the American Academy at Rome for their frequent hospitality while I was doing research for this book. I am also pleased to have the opportunity to thank Thomas Figueira, Nigel Kennell, Jo Ann MacNamara, H. Alan Shapiro, and the Family History Reading Group for their comments on the manuscript. I am also grateful to Georgia Tsouvala for research assitance, to David van Taylor for computer advice, and once again to Angela Blackburn for gracious and tactful editorial help.\n\n* * *\n\n. See L. Zuckerman,\"Spartan Women, Liberated,\" _New York Times_ , Jan. 1, 2000, sec. F, pp. 1, 3.\n\n. New York,1975, republished with a new Preface 1995.\n\n. Elaine Fantham, Helene Peet Foley,Natalie Boymel Kampen, Sarah B. Pomeroy, and H. Alan Shapiro, _Women in the Classical World_ (New York, 1994), and Sarah B. Pomeroy, Stanely M. Burstein, Walter Donlan, and Jennifer Tolbert Roberts, _Ancient Greece_ (New York, 1998).\n\n. _Families in Classical and Hellenistic Greece_ (Oxford, 1997).\n\n. A Spartiate was a Spartan with citizen rights (see chap. 5). The feminine form of _Spartiates_ is _Spartiatis_ : see _LSJ_ s.v. Sparte. Lippold, _RE_ 3A (Stuttgart, 1929), s.v. Sparta (die Ethnika), 1280\u201392, esp. 1283, 1291\u201392, notes that the feminine _Lakedaimonia_ is very rare, _Spartiates_ is rare, and _Spartiatis_ is poetic.\n\n. _Xenophon. Oeconomicus: A Social and Historical Commentary_ (Oxford, 1994).\n\n. _Plutarch's Advice to the Bride and Groom and A Consolation to His Wife_ (New York, 1999).\n\n. E.g., _Women in Hellenistic Egypt_ (New York, 1984; pbk. Detroit, 1990).\n\n. See further F. Ollier, _Le mirage spartiate_ , vol. 1, _\u00c9tude sur l'id\u00e9alisation de Sparte dans l'antiq-uit\u00e9 grecque de l'origine jusqu'aux cyniques_ ; vol. 2, _\u00c9tude sur l'id\u00e9alisation de Sparte dans l'antiquit\u00e9 grecque du d\u00e9but de l'\u00e9cole cynique jusqu'\u00e0 la fin de la cit\u00e9_ (Paris, 1933\u201343, repr. 1973).\n\n. E.g., _SEG_ XI.626, lines 2\u20133, and see chap. 4, below.\n\n. Thus S. J. Hodkinson, s.v. Epitadeus, _OCD_ 3, describes him as fictitious, and in \"'Blind Ploutos'? Contemporary Images of the Role of Wealth in Classical Sparta,\" in _The Shadow of Sparta_. ed. A. Powell and S.J. Hodkinson (London, 1994), 183\u2013222, esp. 207, 214, he sees Epitadeus as part of the \"mirage.\"Hodkinson, _Property and Wealth in Classical Sparta_ (London, 2000) _,_ 90\u201394, reviews the scholarly debate on Epitadeus and change in property tenure. E. Sch\u00fctrumpf, \"The Rhetra of Epitadeus: A Platonist's Fiction,\" _GRBS_ 28 (1987), 441\u201357, argues that Plutarch's report is adapted from Plato _Rep._ 8.555D\u2013E. In contrast, J. Christien, \"La loi d'\u00c9pitadeus: Un aspect de l'histoire \u00e9conomique et sociale \u00e0 Sparte,\" _RD_ 52 (1974), 197\u2013221, and Evanghelos Karab\u00e9lias, \"L'\u00e9piclerat \u00e0 Sparte,\" _Studi in onore di Arnaldo Biscardi_ (Milan, 1982), vol. 2, 469\u201380, 471\u201372, express no doubt about the existence of Epitadeus and the impact of his reform. On Plutarch, see the Appendix to this book.\n\n. But for the relationship between Sparta and Elis as a hindrance to Cynisca's entrance into the equestrian events at Olympia, see chap. 1.\n\n## CONTENTS\n\nILLUSTRATIONS\n\nABBREVIATIONS\n\n1 EDUCATION\n\n2 BECOMING A WIFE\n\n3 THE CREATION OF MOTHERS\n\n4 ELITE WOMEN\n\n5 THE LOWER CLASSES\n\n6 WOMEN AND RELIGION\n\nCONCLUSION: GENDER AND ETHNICITY\n\nAPPENDIX: SOURCES FOR THE HISTORY OF SPARTAN WOMEN\n\nWORKS CITED\n\nINDEX\n\n## ILLUSTRATIONS\n\n 1Girl from Prizren or Dodona\n\n 2Giovanni Demin, _La lotte delle Spartane_\n\n 3Hilaire Germain Edgar Degas, _Les jeunes Spartiates s'exercent \u00e0 la lutte_\n\n 4Married woman\n\n 5Louis Jean Fran\u00e7ois Lagren\u00e9e the Elder, _Spartan Mother and Son_\n\n 6Cup depicting a symposium with women\n\n 7Nude pregnant female in labor\n\n 8Pyramid stele depicting Helen and Menelaus\n\n 9Papyrus text of Alcman, _Partheneion_ 3, fr. 3\n\n10Athlete as mirror handle 166\u2012\n\n11Musician as mirror handle\n\n## ABBREVIATIONS\n\nWith a few obvious exceptions, journal titles are abbreviated according to the form in _L'ann\u00e9e philologique_. Accepted abbreviations are used for standard works. Lists of such abbreviations may be found in reference books such as the _Oxford Classical Dictionary_ , 3d edn., and in the major Greek and Latin dictionaries.\n\nDawkins, _AO_|  R. M. Dawkins et al. _The Sanctuary of Artemis Orthia at Sparta_. Society for the Promotion of Hellenic Studies Supplementary Papers, no. 5. London, 1929.  \n---|---  \n_LGPN_ 3A|  _A Lexicon of Greek Personal Names_. Vol. 3A. _The Peloponnese, Western Greece, and Magna Graecia_ , ed. Peter M. Fraser and Elaine Matthews. Oxford, 1997.  \nPoralla2| Paul Poralla, _A Prosopography of Lacedaemonians from the Earliest Times to the Death of Alexander the Great (X\u2013323 b.c.)_. 2d edn. Chicago, 1985.  \n_PMGF_|  M. Davies, _Poetarum Melicorum Graecorum Fragmenta_. Vol. 1. Oxford, 1991.\n\n# _Spartan Women_\n##\n\n## EDUCATION\n\n### The History of Childhood\n\nIn the modern western world, schooling is mandatory and the curriculum prescribed by state authorities who verify its effectiveness by examinations of students, teachers, and textbooks. In antiquity, parents alone were usually responsible for their children's upbringing. At Athens there was a little outside supervision: a boy was scrutinized at successive stages of life by his father's tribe. In contrast, there was no outside surveillance of girls' upbringing, because a modest, well-brought-up young woman was hidden from the public eye. At home with her mother and other women in the household, a girl learned the skills that she would need to use as an adult. Wearing long dresses, and playing indoors with dolls and small animals, she learned to be nurturant and to perform household tasks.\n\nEvery well-governed state that comes into existence and evolves as the result of deliberate creative acts and legislation endorses the child-rearing practices and values it needs. The educational system is part of political organization, and each role in it, including parent, teacher, and pupil, is socially constructed. Only at Sparta did the state prescribe an educational program for both boys and girls beginning in childhood. Spartans themselves, of course, undertook most of the pedagogical tasks, but they also are known to have invited a few foreigners to teach the young. Poets were the most revered teachers in archaic Greece. There were no travelling women poets: evidently the Spartan authorities determined that education was valuable, and there was no reason to be concerned about any inappropriate liaisons between a male teacher such as Alcman and female as well as male pupils. Alcman's origin, variously said to be Lydian or slave, did not pose an insurmountable deterrent (see Appendix). The earliest datable evidence for the girls' official program is archaic and continues through the classical period. In the Hellenistic period, the traditional system for both boys and girls was discontinued. The boys' program ( _agoge_ ) was fully revived by Cleomenes III: whether the girls' program was also restored at this time is not indisputable, but there are some hints that they were included, perhaps voluntarily rather than under the mandate of the state. When the revolution of Cleomenes failed, the agoge continued until it was abolished by Philopoemen in 188, to be later restored in an archaizing form in the Roman period. At that time as well, state supervision of girls' education was revived through the authority of the _gynaikonomos_.\n\nIn archaic and classical Sparta, girls were raised to become the sort of mothers Sparta needed, just as boys were trained to become the kind of soldiers the state required. The boys' program was far more arduous than the girls'. Young boys left home to learn the survival techniques and skills they would need as hoplites. Their educational program was full time and competitive, and they were frequently examined by older boys and adult authorities appointed for this purpose (Xen. _Lac. Pol._ 2.24.1\u20136, Plut. _Lyc._ 16, 18). The goal of the educational system devised for Spartan girls was to create mothers who would produce the best hoplites and mothers of hoplites. Because all the girls were expected to become the same kind of mothers, the educational system was uniform. This goal obviously did not require the full-time practice and scrutiny that was imposed upon the boys. Girls lived and ate at home with their mothers. Thus it would appear that they enjoyed some privacy and leisure denied to the boys (see below). We surmise that, compared to other Greek women, they had plenty of time to do whatever they wanted to do.\n\n### Literacy\n\nThe extent of literacy in Greece in general, and in Sparta in particular, has been much debated. It is generally agreed that literacy at Sparta was confined to a small elite, and lack of the ability to read and write did not hinder an ordinary citizen's ability to participate in government. Paul Cartledge reports that major epigraphical sources in Sparta give the names of approximately twelve women as compared with one hundred men. This sex ratio, however, is not direct evidence of levels of literacy for women and men respectively; rather, it is a reflection of the fact that men performed more deeds deemed worthy of commemoration and, at least until the end of the fifth century, generally had more funds at their disposal to pay for inscriptions. For example, athletic victories generated a substantial number of inscriptions, but no woman was victorious in horse racing until the fourth century (see below).\n\nThere is no reason, however, to assume that the sex ratio among literate Spartans was as skewed as it was elsewhere in the Greek world. In a democratic polis like Athens, there were strong incentives for men to learn how to read and and write; since women did not participate in government, there was little reason for them to become literate, though some did. In Sparta, in contrast, the education of boys was devoted to developing military skills, leaving little time for the liberal arts. Girls, however, spent time with their mothers and older women. Furthermore, since they were married at eighteen\u2014a substantially later age than their Athenian counterparts\u2014they had as many years as most girls do in modern western societies to devote to their education. They could well have learned reading and writing, as well as other aspects of _mousike_ (music, dancing, poetry) in such an all-female milieu. Of course, in the days when poets such as Alcman were engaged to teach choirs of young maidens, they learned from the poets. Though copies of the poems were probably preserved in state or private archives, the oral tradition was strong (see Appendix). Doubtless the girls committed most information to memory, and did not write it down, for surely they could not have sung and danced while dangling a papyrus roll. By repeating poems like those of Alcman at festivals, successive generations of Spartans learned both their content (including mythology, religion, courtship, and etiquette) and mousike. The repetition of this material tended to create children who thought and behaved as their parents did. Thus Spartan society remained conservative and conscious of its traditions.\n\nThe educational goals of the state and the girls' curriculum are reflected in lyrics written by Alcman for choirs of Spartan maidens:\n\nAlcman, _Partheneion_ 1 _PMGF_\n\nPolydeuces.\n\nAmong the dead I do not take account of Lycaethus [but] of Enarsophorus and Thebrus the fast runner . . . and the violent\n\n(5) . . . and wearing a helmet\n\n[Euteiches], and king Areius, and . . . outstanding of demigods\n\n. . . the leader\n\n. . . great, and Eurytus\n\n(10) . . . tumult\n\n. . . and the bravest\n\n. . . we shall pass over . . . . Destiny and Providence, of all [gods] . . . the oldest\n\n(15) . . . strength [rushing] without shoes.\n\nLet no man fly to heaven\n\n. . . [or] try to marry Aphrodite\n\n. . . . queen, or some\n\n. . . or a daughter of Porcus.\n\n(20) The Graces . . . the house of Zeus\n\n. . . with love in their eyes\n\ngod . . . to friends . . .\n\n(25) gave gifts . . . youth destroyed . . . vain . . .\n\n(30) went, one of them [killed] by an arrow . . . [another] by a marble millstone\n\n. . . in the house of Hades . . . .\n\n(35) They plotted evil deeds and suffered unforgettably. There is such a thing as vengeance from the gods, and blessed is the man who, being reasonable, weaves the web of the day without weeping.\n\n(40) And I sing the light of Agido: I see her like the sun which Agido is now calling to shine as our witness. But the renowned choir leader does not allow me to praise or blame her [i.e.,Agido] at all.\n\n(45) For she herself is conspicuous, as if one set among the herds a strong horse with thundering hooves, a champion from dreams in caves.\n\n(50) Don't you see? The mount is a Venetic: but the hair of my cousin Hagesichora blooms like pure gold;\n\n(55) and her silvery face\u2014why need I tell you clearly? There is Hagesichora herself; while the nearest rival in beauty to Agido will run as a Colaxian horse behind an Ibenian.\n\n(60) For the Pleiades rise up like the Dog Star to challenge us as we bear the cloak to Orthria through the ambrosial night.\n\n(65) There is no abundance of purple sufficient to protect us, nor our speckled serpent bracelet of solid gold, nor our Lydian cap, adornment for tender-eyed girls, nor Nanno's hair, (70) nor Areta who looks like a goddess, nor Thylacis and Cleesithera. Nor will you go to Ainesimbrota's and say \"I wish Astaphis were mine,\" and (75) \"I wish Philylla would look at me, and Demareta, and lovely Vianthemis\"\u2014no, it is Hagesichora who exhausts me with love.\n\nFor Hagesichora with the pretty ankles is not here beside us. (80) She waits with Agido and commends our feast to the gods. Gods, receive it! For the accomplishment and fulfillment are up to the gods.\n\n(85) Choir leader, I would say I myself am a girl who screeches in vain like an owl from a roof beam; but I desire to please Aotis especially, for she is the healer for us.\n\n(90) However, because of Hagesichora girls come to lovely peace. For it is necessary to obey the trace-horse and the driver.\n\n(95) Of course she is not a better singer than the Sirens, for they are goddesses, and instead of their eleven, we are only ten, and children who sing. (100) But we sound like a swan on the waters of Xanthos. And she with her thick blond hair . . . .\n\nAlcman, _Partheneion_ 3 _PMGF_\n\nOlympian Goddesses . . . about my heart . . . song and I . . . to hear the voice of . . . (5) of girls singing a beautiful song . . . . will scatter sweet sleep . . . from my eyelids and lead me to go to the contest where I will surely toss my blond hair\n\n(10) delicate feet . . . .\n\n(61) with limb-loosening desire, and more meltingly than sleep and death she gazes toward . . . nor is she sweet in vain. But Astymeloisa does not answer me (65) while she holds the wreath, some star falling through the gleaming sky, or a golden bough, or a soft feather (70) she crossed on long feet. The moist grace of Cinyras [i.e., perfume] sits on the maiden's hair.\n\n(73) Astumeloisa among a crowd, darling of the people . . . .\n\n(75) taking . . .\n\nI say . . . . If a silver . . . . I might see (80) if she would love me coming near, and take me by the soft hand, at once I would worship her.\n\nBut now a child, heavyhearted . . . to a child, the girl-child\n\n(85) grace\n\nAs we have mentioned, Sparta was the only polis where the training of girls was prescribed and supported by public authority. The spiritual and intellectual education of girls is interesting, especially since apparently boys did not receive an education superior to that of girls (in contrast to the situation at Athens). Therefore the cultural level of girls may well have been superior to that of boys, inasmuch as the latter had to devote so much attention to military training. The intensity of the training of both sexes is unparalleled in the rest of the Greek world. Competitiveness was as much a part of the cultural curriculum as of the physical program. Alcman was hired to teach maidens to perform in choruses that stressed competition among its members both as individuals and as participants in groups of rival choruses, and his poems refer to ranking and contests.\n\nIn the early classical period, some women could read. An anecdote about the precocious Gorgo (born in 506 B.C.E.), daughter and wife of kings, suggests that she knew how to read. When Demaratus, who was in exile, sent a secret message to Sparta by writing it on a wooden tablet and covering it with wax, Gorgo told the recipients to scrape the wax off and read the message (Herod. 7.239). While it is not reported that Gorgo herself read the message when it was uncovered, she realized that there might be writing on such a tablet. Other stories about her poise and precocity also indicate that she probably could read. She will have acquired her skill not necessarily through formal tuition, but while in her indulgent father's presence, listening quietly or more likely (judging from what we are told about her assertiveness and self-confidence) piping up persistently to ask questions about the texts that poets, diplomats, bureaucrats, and other literate people consulted. Anecdotes about Spartan mothers sending letters to their sons urging them to be brave also suggest that literacy was not unknown among women. Though the source of the anecdotes is late and they can not be dated or verified, considering the fact that mothers were separated from their sons who were on military service for long periods of time, the idea that they communicated by letters is not unthinkable.\n\nDedications by women to female divinities that bear the name of the dedicator begin in the late seventh century. This date is consistent with the spread and use of the alphabet in the Greek world. Inscriptions, however, that include dedications do not constitute incontrovertible evidence of literacy inasmuch as they were likely to have been written by craftsmen, rather than by the dedicators themselves. Nevertheless, that inscriptions commemorating the deeds of individual women are found at sanctuaries frequented by women suggests that some women could read them (see below on the Heraea). Such inscriptions are consistent with the picture given by other sources and allow us to conclude that some women, at any rate, were literate.\n\nDoubtless there was change over time in the rate of women's literacy and in its relationship to the literacy of men. After the Peloponnesian War, when Sparta was no longer isolated, Spartan women were probably as literate as aristocratic women elsewhere in the Greek world. For example, _Anonymus Iamblichi_ , a work written some time after the Peloponnesian War in a literary Doric, reports that the Spartans thought it was fine for their children not to learn mousike and letters. He may, however, be referring to boys only. In any case, by the middle of the fourth century, Plato describes women's curriculum as consisting of gymnastics and mousike and comments that this program leaves plenty of time for luxury, expense, and unstructured activity ( _Laws_ 806A, cf. _Rep._ 5.452A). He also states that in Crete and Sparta not only men, but also women, take pride in education and goes on to praise their skills in philosophical discussion ( _Prot_.342D: _paideusis_ ). This talent is an aspect of the women's ability to speak. Spartan women were encouraged and trained to speak in public, praising the brave, reviling cowards and bachelors (Plut. _Lyc._ 14.3\u20136). Aristotle thought it was natural for women to be silent ( _Pol_. 1260a28\u201331), and almost five hundred years later Plutarch wrote: \"A wife should speak only to her husband or through her husband.\" In Athens, respectable women were encouraged not to speak. In Xenophon's _Oeconomicus_ (7.10), a husband describes his young wife as having been brought up \"under careful supervision so that she might see and hear and speak as little as possible.\" Her husband is unusual in believing that as an essential part of her education to be his partner and to supervise the household, he must first teach her how to speak (7.10).\n\n### Some Learned Women\n\nOf course literacy is related to verbal ability, and in Greece, even among literate circles, poetry was often either sung or read aloud. According to hazy traditions, there were two female poets in Sparta. Both of them apparently worked in the archaic period, when Sappho and lesser-known women poets flourished in other parts of the Greek world. None of the work of the Spartan women poets is extant. Megalostrata is mentioned by Alcman. He describes her as \"a golden-haired maiden enjoying the gift of the Muses.\" Several Muses were involved, for in this period poets not only wrote words, but also composed musical accompaniment, and in some cases choreography. Athenaeus (13.600f) reports that Megalostrata attracted lovers because of her conversation, and that Alcman was madly in love her. Though she is not specifically identified as Spartan, she is called a \"maiden,\" and as we have seen, Alcman spent much time in Sparta creating poetry for unmarried girls. Furthermore, like Helen, she is blonde, and her name is suitable for a Spartan, for it means \"large army.\" That she had a personal flirtation with Alcman is questionable. In Greek biographical tradition, written hundreds of years after the death of the subjects, it was common to hypothesize erotic links between creative women and men, rather than grant them an independent existence or a purely intellectual relationship with men.\n\nThere was also a tradition about Cleitagora, a woman poet whose name was used to identify a _skolion_ (drinking song). The Cleitagora is mentioned in Aristophanes ( _Lys_. 1237, _Wasps_ 1246), and Cratinus (254 Kassel-Austin). The _Lysistrata_ passage suggests that she was Spartan, whereas the _Wasps_ offers the possibility that she was Thracian: the scholiast to each passage draws the obvious, and mutually opposing, conclusion. The former inference, however, is more likely to be correct for several reasons: first, in the context of the _Lysistrata_ where the leading female characters are an Athenian and a Spartan visitor, it is more appropriate to sing a song by a Spartan woman. Hence the ambassador in _Lysistrata_ says it not right to sing the song called the \"Telamon,\" but rather the \"Cleitagora.\" Furthermore, of all Greek women, Spartans alone drank wine not only at festivals, but also as part of their daily fare. Therefore it is natural to attribute a drinking song to a Spartan woman who probably composed it for a woman's festival.\n\nThe archaic period also produced at least one Spartan female philosopher. Chilonis, daughter of Chilon, one of the Seven Sages, was a follower of Pythagoras. Iamblichus ( _VP_ 267) named seventeen or eighteen women among the 235 disciples of Pythagoras; nearly one-third of the women cited were Spartans. In contrast, only three of the 218 men were Spartans. Some of the pseudepigrapha attributed to women were in the Doric dialect, and these originated from the group of Pythagoreans around Tarentum, a Spartan colony. In addition to Chilonis, Iamblichus mentions Nistheadousa  and Cleaichma, a sister of the Spartan Autocharidas.  Timycha, wife of Myllias of Croton, is singled out for her courage in resisting Dionysius, tyrant of Syracuse. Dionysius (r. 396\u201379) had her tortured when she was six months pregnant. Rather than reveal the secrets of the Pythagoreans, she bit her tongue off. Cratesiclea as well was a Pythagorean who was married to Cleanor, a fellow Pythagorean. Their dates are not known. Unlike Chilonis, who was said to be a contemporary of Pythagoras, Cratesiclea and Cleanor may have been neo-Pythagoreans. It was natural for members of the same family, especially a married couple, to become Pythagoreans, for the philosopher ordained rules for everyday life including dietary prohibitions and the proper seasons for sexual intercourse. The idea that prescribed commandments and structure might govern even the minutiae of life in an entire community doubtless was familiar to Spartans. Moreover, Sparta enjoyed cultural ties with Samos, Pythagoras's native land.\n\nThe interest in philosophy continued into the Hellenistic period. As we have mentioned, one of the Pythagorean women may have been a neo-Pythagorean. Stoicism also had some effect upon Spartan women, though this philosophy was directed toward men. Cleomenes III invited Sphaerus, a disciple of Cleanthes, to give lectures to the youths and ephebes. When the Spartan revolutions failed, Cleomenes and his family sought aid, followed by asylum in Egypt (see chap. 4). Upon embarking as a hostage for her son's behavior, his mother Cratesicleia set an example of courageous behavior:\"Let no one see us crying or doing anything unworthy of Sparta. For this is up to us alone. Our fortunes will be whatever the deity may bestow.\" She was not at all afraid of death. Self-sacrifice, belief in a single all-powerful divinity, and courage in the face of death are all characteristics of Stoics.\n\n### Mousike\n\nMusical performance was an essential feature of ancient religion, and Spartans were taught to sing, dance, and play musical instruments. Athenaeus (14.632f\u2013633a) observes that the art of music was practiced more intensely in Sparta than elsewhere, for it was a pleasant relief from the self-control and austerity of everyday life. Votive figurines depict women playing various wind, string, and percussion instruments (see chap. 6). In Alcman, _Partheneion_ 1 (97, 99), young girls display critical judgment about their own singing abilities: though they do not sing as well as the Sirens, they sing sweetly indeed. Most of the descriptions we have about women's practice of mousike concerns their dancing. Even Aristophanes, though he could not ever have been a witness, refers to the maidens dancing on the banks of the Eurotas ( _Lys._ 1307\u201310). The hyporcheme, in which the chorus sings as it dances, was performed by Spartan men and women. Athenaeus (14.630e) links the hyporcheme to the comic and vulgar dance called the kordax _,_ and comments that both are funny. Spartan women also were famous for performing another undignified dance called the _bibasis_ (Pollux 4.102,Aristoph. _Lys._ 82). This dance required physical prowess and coordination, for the dancer had to jump and thump her buttocks with her heels in competition for prizes. A bronze figurine that was once thought to represent a girl runner may represent a girl dancing vigorously (fig. 1).\n\nFig. 1. Girl from Prizren or Dodona.\n\nShe is dressed as a runner. That she glances back, however, rather than keeping her gaze in the direction of her feet, also suggests that she is dancing. London, British Museum 208. Photo courtesy of the Trustees of the British Museum.\n\n### Physical Education\n\nThere is more evidence, both textual and archaeological, for athletics than for any other aspect of Spartan women's lives. Furthermore, there is more evidence for the athletic activities of Spartan women alone than for the athletics of all the women in the rest of the Greek world combined. Clearly these activities caught the attention of writers and lead us to conclude that Spartan women's intense involvement in such activities was probably unique in the Greek world. Moreover, although there are relatively few high-quality works of art pertaining to the lives of Spartan women in other spheres, an impressive proportion are relevant to their athletic pursuits. Some of these artifacts are extant; others are known through descriptions by Pausanias and others.\n\nMany of the athletic activities were part of religious festivals that were held in honor of female divinities. It is difficult to separate athletics from religion: however, we will concentrate on the former in the present chapter and the latter in chapter 6.\n\nXenophon ( _Lac. Pol._ 1.4) states with approval that Lycurgus instituted physical training for women no less than for men, including competitions in racing and trials of strength. Euripides ( _Andr_. 595\u2013601) specifically alludes to racing and wrestling. Plutarch ( _Lyc._ 14.2) gives a more explicit account of the physical curriculum, mentioning running, wrestling, discus throwing, and hurling the javelin. Skill in these activities is particularly useful for a soldier. The women's curriculum was a selective and less arduous version of the men's, but similar to it. As the girls in Theocritus announce: \"we all run the same racecourse and rub ourselves with oil like men along the bathing places of the Eurotas.\"  The silvery face mentioned in Alcman, _Partheneion_ 1 (55), may be the glistening effect of the oil in addition to beads of sweat resulting from vigorous exercise. Since boys'and girls' activities were similar, the question of whether they were educated together arises. In Plato's _Republic_ 5, male and female guardians are trained for the same jobs in the government and the military: therefore they are educated together. In proposing co-ed education, Plato's idea is more radical than the Spartan reality of his time. Although sources agree that there was no shyness because of their nudity, it is not clear whether boys and girls used the same exercise ground and racecourse ( _dromos_ ).  Xenophon and Plato discuss the education of boys and girls in separate sections and the _agelai_ (\"herds\") are always described as single sex. Given the relative strength and swiftness of men and women, co-ed competitions and trials of strength in most cases would not have been as efficient in training future hoplites as single-sex exercises. The following two nineteenth-century paintings set in the Platanistas respectively depict, first, the girls exercising separately (fig. 2), and second, provoking the boys to involve them in co-ed wrestling (fig. 3).\n\nFig. 2. Giovanni Demin, _La lotte delle Spartane_\n\nFresco, 1836. Villa Patt, Sedico. Photo, Zanfron.\n\nFig. 3. Hilaire Germain Edgar Degas, _Les jeunes Spartiates_ (\"Young Spartans Exercising\").\n\n1860. London, National Gallery. Photo courtesy of the Trustees of\n\nIn figure 2, girls wrestle before a crowd of elders.Members of the Council of Elders (Gerousia) stand alongside and serve as judges. Lycurgus, seated at top center, crowns a victorious girl. To his right at his feet sits a bearded old man recording the names of the victors on a tablet. Taygetus looms behind. The structure in the center is probably either the shrine of Alcon  or the sanctuary of Poseidon (cf. Paus. 3.14.8). The simple architecture of the monuments reflects Spartan austerity and restraint. Though the Spartan men do not leer at the girls, the modern viewer may find it difficult to distinguish between athletic and erotic nudity, and quite possibly the artist did not intend to make such a distinction. Demin may have been influenced by Roman and late Greek traditions suggesting that while exercising Spartan girls were sexually provocative.\n\nIn figure 3, note that the third figure from the left caresses the breast of the second girl from the left and kisses her. There is no ancient authority for the girls' costume: to the modern viewer it resembles the apron-like skirts worn by some Native Americans. In the group behind Lycurgus stands among the mothers. Degas himself said that the rock in the background is Taygetus, from which newborns who did not pass official scrutiny were thrown.He had studied Greek and Latin and many years after completing the picture he stated that the source was Plutarch. This reliance on Plutarch is doubtless a major factor in Degas's avoidance of the potentially more licentious interpretations of Demin's depiction.\n\nSpartan women were not trained for actual combat; if they had engaged in co-ed athletics, they would have been more prepared. Plato ( _Laws_ 806A) and Aristotle ( _Pol._ 1269b) complain that despite their physical education, they were no better than other Greek women when it came to defending their country. When the Thebans invaded Sparta under Epaminondas in 369 B.C.E., the women were terrified and panicked because their country had never before suffered invasion. It is necessary to point out that at that moment Spartan men were no better than other Greeks, for they had lost a battle. A century later, anticipating an attack by Pyrrhus, Archidamia, grandmother of Agis IV, rallied the other women to oppose the men's scheme to send them to safety in Crete. They declared they had no wish to continue living if Sparta were destroyed. They performed heavy manual labor in behalf of Sparta, assisting the men in digging a trench in a single night as a defense against the elephants of Pyrrhus.  Finally, they told the few soldiers who were present to go to sleep and finished the trench themselves. The next day they cheered the army on. Chilonis, wife of king Cleonymus, held a rope around her neck so she would not be taken alive (see chap. 4).\n\nOne may speculate that Spartan women would have been better at defending themselves if need be, for Plutarch ( _Mor._ 227d12) states that a goal of their physical education was to make them able to defend themselves, their children, and their country. At any rate, just as there is little evidence for illicit adultery at Sparta, there is little for the rape of individuals. During the bitterly fought Second Messenian War,however, Aristomenes and his troops succeeded in carrying off some maidens who were dancing in a secluded place. During the night the guards attempted to rape the maidens, but Aristomenes slew the most aggressive men, saved the girls, and released them for a large ransom. Spartan women were reputed to drag bachelors around the altar and to hit them to make them enter marriage at the appropriate time. Although the source of this gossip is a writer who tends to exaggerate, the implication is that the women were very strong, in fact powerful enough to drag around a Spartan man in his prime, not an easy job even in a ritual when the man was not resisting; and perhaps the women worked in teams.\n\nWhen athletic activities were not part of ritual for women, they were purely sports. Some of the skills were useful for hunters. Xenophon ( _Cyn._ 13.18) reports that some women enjoyed hunting. Although the women he names were the mythical Atalanta and Procris, it is possible that Spartan women engaged in this sport as well (see below). They will not have had to go far, for the region of Mount Taygetus was rich in wild game (Paus. 3.20.4\u20135). Doubtless, like Spartan youths, they could have outraced and encircled a hare. As we have mentioned, they were taught to throw a javelin. In that case we speculate they will have increased their consumption of protein, for meat does not otherwise appear to have been a significant part of the female diet. In any case, Spartan women were not anemic, and of course exercise does whet the appetite. A poet of middle comedy refers to a certain Helen who devoured a prodigious quantity of food. Helen's name suggests that she was Spartan, or at least wanted to seem to be, and her food consumption equaled that of male athletes.\n\nThe education of Spartans apparently affected the social construction of their religion. Plutarch exaggerates when he states that all Spartan divinities carried weapons. Nevertheless, at Sparta many important divinities, including Athena of the Bronze House,Aphrodite Morpho, and Aphrodite Areia, were portrayed as as warriors.Artemis Orthia was shown wearing a helmet and holding bow and spear. In contrast, in Athens, only one major goddess, Athena, was often shown fully armed. Despite the anthropomorphism of Greek divinities, it would be naive to postulate that goddesses were invariably a direct reflection of their female worshippers.  For example, Athena's helmet, shield, and spear had no implications for Athenian women, and other important gods and goddesses at Athens were not shown armed. On the other hand, the Spartan evidence, with its intense focus on martial prowess, does seem to have implications for women.\n\n### _Horsemanship_\n\nHorseback riding and chariot racing were not part of a traditional Greek physical curriculum for most boys, and certainly not for girls. Throughout Greece, however, wealthy gentlemen were expected to be accomplished horsemen and to be able to serve in the cavalry.\n\nSparta was known for breeding and racing horses. Riding horses requires skill rather than brute strength. Of course, little is known about women's education in archaic and classical poleis other than Athens and Sparta, but there can be no doubt that Sparta's excellence in equestrian affairs had repercussions for Spartan women. Women as well as men were actively involved with horses, riding, driving horse-drawn vehicles, and engaging in competitive equestrian events. Archaeological and textual evidence from the archaic period testify to the long history of this involvement. Fragmentary terracotta figurines of Orthia riding astride and side-saddle were found at the sanctuary. More votives of horses than of all other animals combined were also found. Figurines depicting Helen on horseback similar to those found at the sanctuary of Orthia were discovered at the Menelaion.  Bronze votives at both sanctuaries depict female figures identified either as mortals or as Artemis and Helen riding side-saddle. The habits of divinities are not always emulated by human beings, but, as the ambivalence in the identity of the figures makes clear, there does appear to be a direct connection between goddesses and women as riders.\n\nIn Alcman, _Partheneion_ 1, the girls compare themselves to horses that are ridden. They comment that the Colaxian was inferior to the Venetic and Ibenian.  Because of the educational function of poetry, this remark suggests that the girls had specialized knowledge about the different strains of horses. The Colaxian was a sturdy pony, the Veneti used more for chariots than for riding, and the Ibenian perhaps Celtic or Ionian. They also understood chariot racing and refer to the trace-horse and the driver (line 90). At the Hyacinthia, Spartan girls drove expensively decorated light carts, and had an opportunity to display their equestrian skills before the entire community. Some raced in chariots drawn by a yoke of horses. For some processions, the girls rode in carriages shaped like griffins or goat-stags. These Spartan girls apparently had much more fun than Nausicaa described in the _Odyssey_ (6.37\u201338, 57\u201358, etc.), who had to beg permission from her father to drive a team of two mules while riding in a cart loaded with other women and sacks of dirty laundry.\n\nNot only could Spartan women drive horses, but they also knew how to ride them. Agesilaus II used to like to play \"pony on a stick\" with his young children, Archidamus, Eupolia, and Prolyta (Plut. _Ages._ 25.6, _Sayings of Spartans_ 213.70). This report indicates that girls played at riding astride, and that the hobby horse on a stick was not considered a \"boys' toy.\" In 220\/219, at the end of the reign of Cleomenes III, the heroic wife of Panteus fled to the coast on a galloping horse. Thence she embarked to join her husband who was in exile with Cleomenes in Alexandria (Plut. _Cleom._ 38). Like male landowners, Spartan women could drive or ride out to survey their property as men did. Driving horses or riding them endowed Spartan women with an autonomy that was unique for women in the Greek world.\n\nAt Athens, in contrast, sumptuary laws and measures intended to curtail women's visibility in public proscribed women's opportunities to ride in carriages, and there is no evidence that they ever rode horses. Women are rarely portrayed in art riding in carriages except in marriage processions. Sometimes, of course, it was necessary to travel for a funeral or a festival. According to traditional laws ascribed to Solon, they were not to travel at night except in a wagon with a torch shining in front (Plut. _Sol._ 21.4) At Rome, where much more wealth was available, the Lex Oppia, enacted as a sumptuary measure in 216 B.C.E., also forbade women from riding in chariots except for religious purposes.\n\nThat a Spartan was the first female star in Greek athletics not surprising. Cynisca was a daughter of the Eurypontid king Archelaus II and sister of two kings, Agis II and Agesilaus. Her name, Cynisca, is unusual and may be a nickname for an especially tomboyish woman. Her paternal grandfather, Zeuxidemus, was also nicknamed Cyniscus (Herod.6.71). The meaning \"little hound\" perhaps alludes to an interest in hunting. The name of her mother and of her niece Eupolia (\"well horsed\"), her sister's name Proauga (\"flash of lightning\"), and her niece's Prolyta (\"she who is let loose in the forefront\") allude to equestrian interests in the female line. Cynisca is the first woman whose horses were victorious at Olympia. She must have been close to fifty years old at the time. The Eleans had banned the Spartans from Olympia in 420 (Thuc. 5.49\u201350). After the Peloponnesian War, Sparta attacked Elis, and Agis was able to offer sacrifices at Olympia in 397 (Xen. _Hell._ 3.2.21\u201331). Cynisca entered her horses at the earliest possible moment and won her victories in two successive Olympiads, in 396 and 392. No wonder Pausanias (3.8.2) calls her ambitious. Cynisca must have been champing at the bit herself for several years, hoping she would have an opportunity to race her horses at Olympia before she died.\n\nCynisca's _quadriga_ (four-horse chariot) was evidence of great wealth like that of some of her contemporaries who were victors, including tyrants in Sicily. Likewise Cynisca's commemorative monuments were examples of conspicuous consumption equal to those of men. Like wealthy men who owned racehorses, Cynisca did not drive them herself but employed a jockey. Indeed, she would not even have been present at the victorious event inasmuch as women were not permitted to attend the games. Her image, however, stood in the sanctuary. Apelleas, son of Callicles, of Megara, created a sculpture of her chariot, charioteer, and horses in bronze, and a statue of Cynisca herself.He also made bronzes of her horses that were smaller than lifesize (Paus. 5.12.5). These were erected at Olympia. They were the first monuments dedicated by a woman to commemorate victories at pan-Hellenic competitions. The choice of Apelleas suggests that Cynisca had done some research to find a sculptor from an allied city who specialized in images of women. Apelleas was fond of depicting women praying. Thus it is quite possible that Cynisca was portrayed expressing gratitude to the gods. The author of the epigram inscribed on the base of her statue is unknown. The poem is metrically competent; straightforward in the \"Laconic\" style; and of course written in the Doric dialect.\n\nCynisca herself is represented as speaking:\n\nMy ancestors and brothers were kings of Sparta.\n\nI, Cynisca, victorious with a chariot of swift-footed horses,\n\nerected this statue. I declare that I am the only woman\n\nin all of Greece to have won this crown.\n\nThis epigram was only the second ever composed to commemorate a deed of the Spartan royalty (Paus. 3.8.2). The first one was written by Simonides and was inscribed at Delphi in honor of Pausanias, victor over the Persians at Plataea. Obviously Cynisca was thought of in very exalted company. She had won the most prestigious horserace twice at the most prestigious pan-Hellenic atheletic festival. Cynisca's commemorative sculptures at Elis stood between that of Troilus of Elis  and those of male Lacedemonians. Pausanias (6.2.1) includes Cynisca in his observation that after the Persian war, the Lacedaemonians were the keenest breeders of horses. No husband or children are recorded for Cynisca.  Xenophon and Plutarch, however, draw attention to her relationship with her brother the king, who had encouraged her to enter a chariot at Olympia in order to demonstrate that such victories were the result of wealth and expenditure, not of virtue ( _andragathia_ , \"manly virtue\"). Whether Agesilaus was actually inspired by mean-spiritedeness and sibling rivalry, or by the lofty motives Xenophon and Plutarch ascribe to him, the anecdote suggests that he thought his sister's horses had a good chance to win. Like some male owners of victorious racehorses,Cynisca was not only extremely wealthy, but she was also an expert in equestrian matters. According to Pausanias (3.8.2), she had an ambition to be victorious at Olympia, and was the first woman to breed horses. With the increase in private wealth, much of it in the hands of women, and with their keen interest in athletics and knowledge of horses, it was natural that Spartan women would own racehorses. Cynisca was a member of the first group of extremely wealthy women who begin to become evident after the Peloponnesian War (see chap. 4).\n\nA hero\u00f6n was erected to Cynisca near the Platanistas where the athletic contests of young Spartans were staged. In Greece it was not uncommon to treat athletes as heroes, but Cynisca was the first woman to be elevated to this status. Her shrine was in the vicinity of the shrines of mythical heroes including the sons of Hippoco\u00f6n. The hero\u00f6n would have been built after her death and would have served as an inspiration to other women.\n\nCynisca's example was soon followed by other women, especially Spartans; the author Pausanias (8.1) sees their victories as a trend. Among them was the Spartan Euryleonis, who was victorious at Olympia with a two-horse chariot in 368. A statue of Euryleonis stood with those of other luminaries such as the general Pausanias in the vicinity of the Bronze House (Paus. 3.8.1, 3.17.6). In fact the Spartans Cynisca and Euryleonis were the first women whose chariots were victorious at Olympia. Approximately a century later they were followed by royal women and women connected with the courts of Alexander's successors (see below).\n\n### _Competitions_\n\nCompetitive racing and trials of strength for women, no less than for men, were part of the physical education system instituted by Lycurgus (Xen. _Lac. Pol_. 1.3\u20134, cf. Arist. _Pol._ 1269b). Some of these contests were doubtless organized in a routine manner; but others took place as part of religious festivals. We are better informed about the latter.\n\nRunning races were the only athletic events for women that took place at festivals. There were races in honor of Helen, Dionysus, Hera, and in honor of local deities called Driodones.  Theocritus reports in his _Epithalamium to Helen_ (18.22\u201325) that 240 maidens rubbed their nude bodies with oil as men did and raced along the Eurotas. That the girls were said to be as old as Helen when she married Menelaus indicates that the races were associated with puberty. As Theocritus reports the event, there was a tacit beauty competition as well, with Helen winning the prize (see Conclusion). The women's race at the Heraea in Elis was the most prestigious, the equivalent for women of the Olympic competitions held for men:\n\nEvery fourth year the Sixteen Women weave a robe for Hera, and the same women also hold games called the Heraea. The games consist of a race between virgins. The virgins are not all of the same age; but the youngest run first, the next in age run next, and the eldest virgins run last of all. They run thus: their hair hangs down, they wear a shirt that reaches to a little above the knee, the right shoulder is bare to the breast. The course assigned to them for the contest is the Olympic stadium; but the course is shortened by about a sixth of the stadium. The winners receive crowns of olive and a share of the cow which is sacrificed to Hera; moreover, they are allowed to dedicate statues of themselves with their names engraved on them.\n\nRather than actual portraits, the statues were doubtless images of girls running with the name of the honorand inscribed. It is obvious that the victorious women, or their families, sought fame and immortality no less than victorious men, and were willing to pay the cost of a dedication.\n\n### _Athletic Nudity_\n\nThe Greek word _gymnos_ means \"nude\" or \"lightly dressed.\" Nudity at Sparta may be explained in terms of religion, initiatory rites, erotic stimulation, and the requirements of athletic prowess. Though these strands are intertwined, for heuristic purposes we will separate them, and treat nudity as a costume for sports here. (For other implications, see chap. 2 and Conclusion.)\n\nNot only did Spartan women wear a _peplos_ (tunic) that revealed their thighs, but they regularly exercised completely nude. Mature women and pregnant women exercised. Even older women exercised nude. As male athletes had discovered, light clothing or none at all is best for racing. Even nowadays (or at least before the adoption of lycra bodysuits), racers wear as little clothing as possible. Thucydides (1.6) credits the Spartans with being the first to exercise unclothed.\n\nThe girls who raced at the Heraea lowered the right shoulder of the peplos and revealed their right breast. This costume was peculiar to this festival. No ancient source specifies the ethnic identity of the girls who competed at Elis. They may have been girls from the neighborhood, or at least originally so. It seems more likely, however, that the games became pan-Hellenic, though on a smaller scale than the men's events at Olympia. In view of the tendency at Athens, for example, to seclude and protect young girls and to keep their names out of the public eye, it is unlikely that Athenian maidens would have been brought to race at Elis. At Athens (and probably elsewhere in Greece), girls were devalued, and the expenses involved in travelling were considerable. Therefore, if the Heraea were pan-Hellenic, only girls who lived fairly close by would have par-ticipated. Considering the likelihood that attention was not paid to women's athletics anywhere but Sparta, and given the historical evidence for Spartan domination of Elis in the archaic period, it is likely that the games were established along Spartan principles and that the majority of competitors and victors were Spartan. Many of the sculptures at the temple of Hera were the work of Lacedaemonian artists (Paus. 5.17.2). Finally, bronze figurines depict girl runners in short peploi baring the right breast. These figurines were manufactured in Sparta (see Appendix). As we have mentioned, the bare-breasted costume was worn only by girls who raced at the Heraea. Though some prepubescent Athenians raced nude at least once in their lives at the sanctuary of Artemis at Brauron, only Spartan girls regularly wore short dresses and exercised nude. Moreover, historical sources assign the earliest foundation of racing for girls anywhere in Greece to Lycurgus. For these reasons, it is generally assumed that, at least when the political relationship between Sparta and Elis was favorable, the girls who raced at the Heraea were mostly Spartans.\n\nNudity for women indicates that their athletic prowess was understood to be a high priority: it certainly attracted a great deal of attention, both artistic and prurient. Ibycus and later writers described the women as \"thigh-flashers\" (see above). In the _Andromache_ (595\u2013602), written in the early years of the Peloponnesian War, Euripides mentions the bare thighs and co-ed racing and wrestling. _Anonymus Iamblichi_ reports that Spartan girls strip for exercise. Plato apparently knew of the Spartan practice, for it is generally assumed that nude exercise for women in the _Republic_ (457A) is based on the Spartan reality.\n\nPlato refers to women who are natural athletes ( _Rep_. 456A). He also suggests that the bodies of old women are laughable ( _Rep._ 457B), and, indeed, retreats from nudity for adult women in the _Laws_ (833D). In the _Republic_ (452B), all women exercise naked, though the older ones are described as wrinkled and not good-looking. In _Laws_ (833C), Plato prescribes nude racing only for prepubertal girls, and racing clothed for adolescents until marriage at eighteen to twenty years of age. Plato's distinction may be reflected in the artistic portrayals of Spartan girl runners, though a modern viewer may misinterpret clues to the age of subjects in ancient art. Bronze mirrors and statuettes portraying girls completely nude seem to modeled on a prepubertal, slim-hipped girl. Those wearing the chiton show an adolescent with fully developed breasts. The older group may be dressed because they have already reached menarche and need to wear an undergarment to absorb menstrual blood.\n\nUpon marriage, girls graduated from the state-controlled educational system. Some of them, at least, still managed to stay in good physical shape. Non-Spartan authors report that adult women were physically fit. Lampito, a married woman, is in excellent condition and can touch her buttocks with her feet while jumping in the air (Aristoph. _Lys._ 82). Spartan women needed to be able to assume this position while dancing in certain religious rituals. As we have just observed, in his _Republic_ Plato states that even mature women will exercise in the nude: this ordinance may reflect some reality at Sparta. In any case, as we have seen, some mature women continued to be interested in horses and probably rode or drove to their country estates.\n\n### Education in Hellenistic and Roman Sparta\n\nWas there an agoge for girls, and if so, was it parallel to or imitative of the boys' agoge? These questions are further complicated by historiographic issues surrounding the boys' agoge. According to revisionist history, the agoge as described in great detail by Plutarch was largely a Hellenistic invention that was revived in the Roman period. Therefore, although Plutarch's description has enjoyed greater popularity and influence, Xenophon's report should be understood to be a more accurate account of the educational system of the classical period. In any case, ancient authors as well as modern scholars agree that some sort of institutionalized educational system for girls existed whenever such a system existed for boys.\n\nIn the Hellenistic period the fortunes of Sparta declined (see chap. 4). Owning good racing teams was expensive. Nevertheless, the Panathenaic victor lists for 170 B.C.E. record the victory of a Spartan woman with a quadriga.  She is one of nineteen non-Athenian citizens, including seven women, whose names appear on victor lists for this period (170, 166, 162 B.C.E.). Her name, Olympio, is either prophetic of her deed or a nickname alluding to an Olympic victory not otherwise attested.\n\nIn the Roman period, because Sparta was a destination for tourists, the characteristics that made Sparta distinctive were emphasized. The athleticism of women was exaggerated. Foreigners were allowed to see what they had never before been able to witness: Spartan women engaged in athletics. There were other professional women athletes in the Roman world. Therefore, in order to attract an audience, the Spartans needed not only to be good athletes, but also to create a unique image. History and tradition were mined for publicity. Spartan athletics were authentic. The reports of Plutarch (see above) and Propertius (3.14) indicate that the curriculum for girls was firmly established and well articulated in Roman Sparta. Propertius mentions nude co-ed wrestling; ball playing; hoop rolling; the pancratium (wrestling with no holds barred); discus throwing; hunting; chariot driving; and wearing armor. Vergil ( _Aen._ 1.314\u201324) also describes Spartan huntresses wearing short dresses, armed with bows and arrows, pursuing a wild boar. Ovid ( _Her._ 16.151\u201352) writes of a nude Helen wrestling in the palaestra. Doubtless, like other Greeks, they continued to anoint themselves with oil. One Spartan woman in the first century B.C.E. is reported to have doused herself with so much butter that the odor made a Galatian princess ill (Plut. _Mor._ 1109b). The Spartan, in turn, was nauseated by the smell of the other woman's perfume, perhaps because she maintained the traditional Spartan ban on wearing perfume.\n\nAthletic competitions for respectable women were held under state supervision. A fragmentary inscription of the second century C.E. indicates that the magistrates ( _biduoi_ ) in charge of ephebic competitions also supervised twelve \"female followers of Dionysus.\" These Dionysiades were virgins; eleven of them ran a foot race at a festival of Dionysus. Another inscription from a statue honors a woman racer who won a victory. These games were founded under either Tiberius or Claudius.\n\nWrestling for women in Sparta had a long pedigree. Furthermore, since nudity often results from wrestling and other athletic endeavors, the performances of female wrestling were doubtless piquant (see fig. 3). In the days of Nero, a Spartan woman engaged in a wrestling match at Rome with a Roman senator, M. Palfurius Sura. Athenaeus (13.602e) reports that Spartans displayed their girls to their guests unclothed. Athenaeus may well be exaggerating: the Greek tendency to understand the world in polarized categories may have prompted Athenaeus to interpret some nudity for specific acceptable purposes (as Plutarch understood it) into gratuitous indecent display. On the other hand, if this report is true, the practice may be understood as a shocking spectacle, in keeping with the tastes of the Roman world.\n\n### Social Construction of Sexual Behavior\n\nPlutarch ( _Lyc._ 18.4) reports that erotic ties between older and younger women were common. In Alcman, _Parthenion_ 1.73, the girls mention visiting Aenesimbrota, who is probably a purveyor of love magic. She would provide drugs, spells, and magical devices to attract the object of desire. Hagnon of Tarsus, an Academic philosopher of the second century B.C.E., states that before marriage it was customary for Spartans to associate with virgin girls as with _paidika_ (young boyfriends).\n\n### Weaving\n\nAccording to Xenophon ( _Lac. Pol_. 1.1), Lycurgus wisely decided that the labor of slave women ( _doulai_ ) sufficed to weave the clothing that the Spartans required. Nevertheless, doulai were not the only women in Sparta who knew how to weave. To make the point about the difference between Spartans and other Greek women, Xenophon and Plato ( _Laws_ 806A) exaggerate the Spartans' liberation from weaving. Furthermore, weaving served as a \"catch-all\" term for the domestic work usually performed by Greek women. In fact, Spartan women could weave and supervise their slaves' work, but they were not encouraged to weave endlessly, nor did their reputation depend upon it. One of the _Sayings_ attributed to Spartan women underlines this ethnic distinction:\n\nWhen an Ionian woman was proud of something she had woven (which was very valuable), a Spartan woman showed off her four well-behaved sons and said these should be the work of a noble and honorable woman, and she should swell with pride and boast of them. (Plut. _Sayings of Spartan Women_ 241.9)\n\nAlthough servile women did the routine weaving, freeborn women wove for ritual purposes. Paraphernalia for weaving and hundreds of plaques depicting textiles were discovered at the shrine of Artemis Orthia.  These offerings date from the archaic to the Hellenistic period, but most are probably sixth to fifth century. Literary testimony conplements the archaeological finds. Ten young girls in a choir for whom Alcman wrote a _Partheneion_ (1.61) sing about bringing a cloak to Artemis Orthia. Presumably, like the Arrephoroi who began the weaving for the peplos of Athena at Athens, they had participated in making the cloak. Pausanias (3.16.2) reports that every year women wove a chiton for Apollo of Amyclae in a room designated as the chitona.\n\nThe _xoanon_ (wooden image) of Artemis Orthia wore a _polos_ (head-dress representing the celestial sphere) and a woven dress reaching to the feet. The figure must have been small and light, for the priestess held it during the whipping ceremony (Paus. 34.16.10). Therefore annual or quadrennial weaving of garments for the divinities could not have been a great burden to Spartan women. Furthermore, weaving garments need not have entailed the obligation to prepare the wool and the performance of messy, tedious tasks including washing, beating, combing, carding, dyeing, and spinning. Rather, like Helen who spun with her golden distaff (one suspects not very energetically), Spartan women were not required to expend much labor in producing clothing for their cult images. At least ten girls are named in Alcman, _Partheneion_ 1, and they wove only one cloak. Clothing the xoanon of Artemis was like dressing a large doll.\n\nIn contrast,Athenian women not only wove for the _oikos_ (family,household, estate) _,_ but also were responsible for weaving a peplos for Athena annually. The wooden image was probably less than lifesize, and the cloth depicted on the Panathenaic frieze around 2.0\u20132.5\u00d71.8\u20132.3 meters. Every four years, however, they had much more work. The peplos woven by Athenian women for the greater Panathenaea was an elaborate tapestry, so large that it was fixed as a sail on the Panathenaic ship. This peplos was probably about 4\u20138 meters square, and all who attended the festival could admire or criticize the result.\n\nSince Spartan clothing was less elaborate than Athenian, the women who wove in Sparta had less to do. Young boys wore a chiton or went nude: they were allocated only one cloak to wear throughout the year, and did not sleep on mattresses, but on straw which they gathered themselves (Plut. _Lyc_.16.6\u20137). Adult men wore a short red cloak and were buried with only such a cloak and a chaplet of olive leaves. Since the cloaks constituted a kind of uniform, they may have been distributed by the state. Women's peploi were short and scanty: for racing they wore the _chiton exomis_ (tunic with one sleeve) which barely reached the knee (Paus. 5.16.3). Unlike Athenian women, they did not normally wear many layers of clothing. They probably wore whatever the weather required (see chap. , fig. 4,Vix crater, and Conclusion). Such dress did not command attention; but their skimpy attire did. When the women were preparing to dig a trench against the attack of Pyrrhus, some wore outer dresses ( _himatia_ ) over their tunics, and some wore only their tunics ( _monochitones_ , Plut. _Pyrrh._ 27.3). Dionysius, the clever tyrant of Syracuse, assumed that expensive Sicilian chitons for Lysander's daughters would constitute an irresistible bribe. At first Lysander refused to accept them, but later on, when a Sicilian ambassador showed him two dresses and asked him to choose one for a daughter, Lysander took both, for he had more than one daughter (Plut. _Lys._ 2.5). Doubtless the increase in visible private property that took place after the Peloponnesian War also affected women's wardrobes.\n\nThe brief chiton exomis worn by Spartan women caused as much consternation to other Greeks as the miniskirt. Artemis, when she was portrayed as a huntress, and the Amazons were the only other females known to the Greeks who wore such short skirts, and they were visible only in art, not in real life. Because Spartan women were in fine physical condition, their skimpy clothing must have been flattering. In contrast, respectable Athenian women exercised only by doing housework, mostly indoors or in the courtyard of their house. The visual arts portray Athenian women as heavily covered in many layers of cloth, with skirts reaching to the ankles. Even little girls wore long dresses. In the sumptuary laws attributed to Solon, it was deemed necessary to specify limits to the amount of clothing Athenian women might include in their dowries, or wear at funerals, or use to wrap the dead.\n\nWeaving was the only activity of women that most Greeks recognized as productive. In prosperous households, more fabrics than could ever be used by the household were woven. These were stored up, to be given as gifts or part of dowries, or, if times were hard, to be sold or traded at the marketplace. At Sparta, neither women nor men engaged in activities that produced objects for use or sale. In fact, such work was prohibited.\n\nNot until the Hellenistic period were the names of the Athenian women who had woven the peplos for Athena announced. In contrast, like Greek men who had always sought fame, Spartan women had a consciousness of themselves that is conveyed to the observer. Their pride shines forth in Alcman's poetry, in Plutarch's _Sayings of Spartan Women_ , and in the dedications of victors' images at the sanctuary of Hera at Elis and of Zeus at Olympia. As girls and women, Spartans left their mark on the historical record in pan-Hellenic contexts.\n\n* * *\n\n. See further Sarah B. Pomeroy, _Families in Classical and Hellenistic Greece_ (Oxford, 1997), 141\u201342 and passim.\n\n. Terpander, Stesichorus, and other poets worked in archaic Sparta: Arist. fr. 551 Gigon, Plut. _Mor._ 1134b,Athen. 14.635e, etc.\n\n. Nevertheless, Alcman confesses his interest in women in _PMGF_ 34, 59a, b, and see below on Megalostrata, who was not said to have been his pupil.\n\n. See further Nigel Kennell,\"The Elite Women of Sparta,\" paper delivered at the annual meeting of the American Philological Association, December 28, 1998; abstract published in _American Philological Association 130th Annual Meeting: Abstracts_ , 84. In this paper, Kennell is less willing to accept the evidence for a girls' agoge in the Roman period than he was in _The Gymnasium of Virtue: Education and Culture in Ancient Sparta_ (Chapel Hill, N.C., 1995), 46.\n\n. Though no ancient source states that every girl participated in the system, the universality of the goal implies it. See further Jean Ducat, \"Perspectives on Spartan Education in the Classical Period,\" in _Sparta: New Perspectives_ , ed. S. Hodkinson and A. Powell (London, 1999), 57\u201358.\n\n. See William V. Harris, _Ancient Literacy_ (Cambridge, 1989).\n\n. See further Paul Cartledge, \"Literacy in the Spartan Oligarchy,\" _JHS_ 98 (1978), 25\u201337, esp. 33\u201337.\n\n. From the sixth to the fourth century: Cartledge,\"Spartan Wives: Liberation or Licence?\" _CQ_ 31 (1981), 84\u2013105, esp. 93 n. 54.\n\n. Thus F. D.Harvey,\"Literacy in the Athenian Democracy,\" _REG_ 79 (1966), 585\u2013635, esp. 623.\n\n. At Athens, respectable women who were literate learned their letters at home: Sarah B. Pomeroy, _Xenophon. Oeconomicus: A Social and Historical Commentary_ (Oxford, 1994), 270, 283. For learning in an all-female milieu, cf. the young women in the circles of Sappho and other women poets in archaic Lesbos. The parallel is not exact, however, for several reasons, including the fact that Spartan girls were required to devote a good portion of their time to gymnastics.\n\n. There is evidence for male leaders of female choirs at Aegina, but the age of the females is not clear: Herod. 5.83.3.\n\n. Trans. Sarah B. Pomeroy. For further comments on these poems, and a photograph of a portion of one of the papyrus texts, see Appendix.\n\n. See _LSJ_ s.v. kubernao 2 for associations with chariots and horses, rather than the common interpretation of piloting a ship.\n\n. _PMGF_ TA 2.34\u201335, _PMGF_ 1.58, 59, and passim, _PMGF_ 3.8.\n\n. See chap. 3 nn. 25, 34, and chap. 4.\n\n. Cf. Agesilaus for another Spartan king who enjoyed the company of his young children, two daughters and a son: Plut. _Ages._ 25.6, _Sayings of Spartans_ , 213.70.\n\n. Plut. _Sayings of Spartan Women,_ 241a3, d10, 11. On this source see Appendix. Cf. the letters of Olympias to Alexander: Plut. _Alex._ 39.4\u20135.\n\n. See further Sarah B. Pomeroy, \" _Technikai kai Mousikai_ : The Education of Women in the Fourth Century and in the Hellenistic Period,\" _AJAH_ 2 (1977), 51\u201368, and _Women in Hellenistic Egypt: From Alexander to Cleopatra_ (with a new foreword and addenda, Detroit, 1989), 59\u201372.\n\n. Diels-Kranz6, vol. 2, p. 408, 2.10, and see Kathleen Freeman, _The Pre-Socratic Philosophers_ (Oxford, 1946), 418\u201319. For a translation, see Rosamond Kent Sprague, _The Older Sophists_ (Columbia, S.C., 1972),\"Dissoi Logoi,\" 282.\n\n. On the ambiguity of _paides_ , which can mean \"children,\" \"boys,\" or \"slaves,\" see below, Appendix n. 77 _._\n\n. _Advice to the Bride and Groom_ , 32: see further Sarah B. Pomeroy, ed., _Plutarch's Advice to the Bride and Groom and A Consolation to His Wife_ (New York, 1999), ad loc.\n\n. _PMGF_ 59, Poralla2 510.\n\n. On traditions linking women philosophers to the men in their circle see Pomeroy, \" _Technikai kai Mousikai_ ,\" 58.\n\n. See further Denys Page, _PMG_ 912; Jeffrey Henderson, _Aristophanes: Lysistrata_ , ed. with introduction and commentary (Oxford, 1987), ad loc.; and Douglas M. MacDowell, _Aristophanes: Wasps_ , ed. with introduction and commentary (Oxford, 1971), ad loc.\n\n. According to the Scholiast to _Lys._ 1237 Cleitagora was mentioned in Aristophanes, _Danaids_ (= fr. 271 Kassel-Austin). According to Hesychius, s.v. Kleitagora (k 2913 Latte), she was from Lesbos.\n\n. Xen. _Lac. Pol_. 1.3, and below, chap. 6 nn. 3, 19, 21\u201322.\n\n. I.e., Cheilonis: Iambl. _VP_ 267, Poralla2 no. 760. For her father, who was ephor in 556, see Poralla2 no. 230. See further Conrad M. Stibbe, _Das andere Sparta_ (Mainz, 1996), 211.\n\n. See further Holger Thesleff, _An Introduction to the Pythagorean Writings of the Hellenistic Period_ (\u00c5bo, 1961), esp. 77\u201378, 99, 103\u20135, 114\u201316.\n\n. Manuscripts vary. Nistheadousa is listed with a question mark in Diels-Kranz 58, \"Pythagoreische Schule: A. Katalog des Iamblichos,\" but not cited in Poralla2 or in _LGPN_ 3A. On these women, see further Gilles M\u00e9nage, _Historia mulierum philosophorum_ (1690\u201392), trans. B. H. Zedler as _The History of Women Philosophers_ (Lanham,Md., 1984), chap. 11.\n\n. _LGPN_ 3A s.vv. dates Cleaichma to the sixth century, but Autocharidas to the fifth century.\n\n. Iambl. _VP_ 267.31, see also Porphyry, _Pythag._ 61, Poralla2 no. 702, and _LGPN_ 3A s.v.\n\n. Iambl. _VP_ 267, Poralla2 nos. 450, 420. On neo-Pythagorean women, see further Pomeroy, _Women in Hellenistic Egypt_ , 61\u201365.\n\n. Plut. _Cleom._ 2, _FGrH_ III.585, and see further Pomeroy, _Families_ , 65\u201366.\n\n. Plut. _Cleom._ 22.6\u20137, 38.4. This vivid passage may not only record a real event, but may also reflect the Stoic leanings of Plutarch's source, Phylarchus: see chap. 4 n. 48, and Appendix.\n\n. Athen. 14.631c, citing Pindar fr. 112 Snell.\n\n. S. Constantinidou, \"Spartan Cult Dances,\" _Phoenix_ 52 (1998), 15\u201330, esp. 24. For details of manufacture and interpretations, see Appendix, below.\n\n. Giampiera Arrigoni, \"Donne e sport nel mondo greco: Religione e societ\u00e0,\" in _Le donne in Grecia_ , ed. Giampiera Arrigoni (Bari, 1985), 55\u2013201, devotes pp. 65\u2013101 (of a total 73 of text [the remainder consists of illustrations and special exegeses]) and nn. 29\u2013185 (of a total 259) to Sparta and uses Sparta as a touchstone throughout the article.\n\n. B. B. Shefton, \"Three Laconian Vase Painters,\" _ABSA_ 49 (1954), 299\u2013310, esp. 307, no. 17, for nymphs or girls swimming. Alcman also wrote a work titled \"The Female Swimmers\" ( _Kolumbosai_ : _PMGF_ TB 1, fr. 158).\n\n. _Idyll_ 18.22\u201325. Theocritus, of course, is not an unassailable witness to practices in the classical period, but his report is consistent with those from earlier, more trustworthy sources.\n\n. There is no evidence for the Platanistas (\"the grove of plane trees\") before the Hellenistic period, when two teams of youths staged a mock battle.\n\n. Contra Cartledge, \"Spartan Wives: Liberation or Licence?\" 91. T. Scanlon, \"Virgineum Gymnasium: Spartan Females and Early Greek Athletics,\" in _The Archaeology of the Olympics_ , ed. W. Raschke (Madison,Wis., 1988), 185\u2013216, esp. 190, also argues that education was co-ed, but the evidence he cites is tendentious (Eur. _Andr._ 99\u2013100) and Roman (Ovid, _Her._ 16. 149\u201352,Prop. 3.14),when sensationalism may have prompted or intensified participation in co-ed contact sports so that women's wrestling of the archaic and classical period became the pancration in the Roman period. Philostratus, _Imag._ 2.6.3, specifies the pancration as the pancration of men ( _andron_ ). For _agele_ (i.e., _agela_ \"herd\") used of girls, see n. 110 below.\n\n. For a detailed description, see Giovanni Paludetti, _Giovanni de Min_ (Udine, 1959), 290, 307. 310\u201311, 320,App. 1, and fig. 31.\n\n. A son of Hippoco\u00f6n, older brother of Tyndareus, whose other sons are mentioned at the opening of the extant fragment of Alcman, _Partheneion_ 1.\n\n. Carol Salus, \"Degas' Young Spartans Exercising,\" _Art Bulletin_ 67 (1985), 501\u20136, argues that the subject of the painting is courtship, and that the figure touching the girl's breast is male (504). However, Degas may be depicting the well-known homosexual relationship between females.\n\n. Martin Davies, _The French School_ (London, 1957), 70, and see further M. H. Sykes, \"Two Degas Historical Paintings: _Les jeunes spartiates s'exercent \u00e0 la lutte_ and _Les malheurs de la ville d'Orl\u00e9ans_ \" (Master's thesis, Columbia University, New York, 1964).\n\n. Phoebe Pool, \"The History Pictures of Edgar Degas and Their Background,\" _Apollo_ 80 (1964), 306\u201311, esp. 308.\n\n. Nicolas Richer, _Les \u00e9phores: \u00c9tudes sur l'histoire et sur l'image de Sparte (viiie\u2013iiie si\u00e8cle avant J\u00e9sus-Christ)_ (Paris, 1998), esp. 83\u201384, sees a connection between the Partheniai (\"children of unmarried women\") who were sent to settle Tarentum and the Tresantes (\"tremblers,\"\"cowards\"), for both are feminized. See further chap. 2 n. 45 below.\n\n. Xen. _Hell_. 6.5.27\u20138; Plut. _Ages._ 21.4\u20135;Arist. _Pol._ 1269b37\u20139.\n\n. 273\u2013272 b.c.e.: Plut. _Pyrr._ 27.2\u20135, 29.6. Cf. Polyaen. 8.49 and 70 for the military assistance of the women of Cyrene. Jacoby _FGrH_ 81, Komm. F 48, suggests that a similar passage of Pompeius Trogus in Justin is based on Phylarchus. See further David Schaps, \"The Women of Greece in\n\nWartime,\" _CPh_ 77 (982), 193\u2013213, and Maria Luisa Napolitano, \"Le donne spartane e la guerra: Problemi di tradizione,\" _AION (archeol.)_ 9 (Naples, 1987), 127\u201344.\n\n. Jerome para. 308 Migne relates several anecdotes about the rape of Spartan virgins in conditions of war. See also Orosius 1.21 and Justin 3.4.1\u20135.\n\n. Paus. 4.16.9\u201310, and see chap. 6 n. 11.\n\n. Clearchus of Soli (fl. ca. 250 b.c.e.) fr. 73 Wehrli = Athen. 13.555c. New York City firefighters, both male and female, are trained to be able to carry adults of both sexes.\n\n. See chap. 3 on liquid and dry food rations. Xen. _Lac. Pol._ 15.3 refers to men eating meat at the _syssition_ (mess group). Spartiates did raise animals on their estates (Plut. _Alcib. I_ 122d).\n\n. Athen. 10.414d quoting Heraclitus comicus (also known as Heraclides) in \"The Host,\" Kassel-Austin _PCG_ 5.560.\n\n. _Sayings of Spartans_ 232d5, _Lac_. 239a (28).\n\n. Paus. 3.17.6, 3.15.10; _Greek Anthology_ , Planudian Appendix 173\u201376, etc., and see further J. G. Frazer, _Pausanias's Description of Greece_ , vol. 3 (London, 1913), 338.\n\n. _OCD_ 3 s.v. Aphrodite considers the possibility that the armed Aphrodite was connected with women's education, but prefers to interpret the armor as alluding to Aphrodite considered to be the polar opposite of Ares. Robert Parker,\"Spartan Religion,\" in _Classical Sparta_ , ed. A. Powell (Norman, Okla. 1989), 142\u201372, esp. 146, suggests that the portrayals of divinities with armor were retained at Sparta from earlier primitive cult images. See further Fritz Graf, \"Women, War, and Warlike Divinities,\" _ZPE_ 55 (1984), 245\u201359, and chap. 6 n. 72 below.\n\n. On the armed Aphrodite: Julianus (sixth century c.e., _Anth. Pal._ 9.447), and Graf,\"Women, War, and Warlike Divinities,\" 250\u201351.\n\n. For Plutarch's version see above. Arist. _Pol._ 1337b24\u201325 lists reading, writing,gymnastics, the musical arts, and possibly painting. In Plato, _Laws_ (804E), riding is prescribed for women.\n\n. Xenophon, _On Horsemanship, Cavalry Commander,_ and see further, Pomeroy, _Xenophon, Oeconomicus_ , 2, 219, 226, 231, 243.\n\n. The owner of the horse ridden by Julie Krone, the first woman to win a Triple Crown, said of her: \"She's got great finesse, beautiful hands on a horse, and good communication. You don't have to bully a horse\u2014just talk to him.\" Joe Drape,\"Krone Adds Another First to Her Accomplishments,\" _New York Times_ ,Aug. 8, 2000, sec. D, pp. 1, 5, esp. 5.\n\n. Dawkins, _AO_ , 146, all dates. Some riders on figurines dated 700\u2013600must have had to ride side-saddle: Dawkins, _AO_ , 150.\n\n. Dawkins, _AO,_ 157.\n\n. A. J. B. Wace, M. S. Thompson, J. P. Droop, \"The Menelaion,\" _ABSA_ 15 (1908), 108\u201357, esp. 124, all dates, and H. W. Catling,\"Excavations at the Menelaion, Sparta,1973\u201376,\" _AR_ (1976\u201377),24\u201342, esp. 38, and fig. 42.\n\n. Mary Voyatzis, \"Votive Riders Seated Side-Saddle at Early Greek Sanctuaries,\" _ABSA_ 87 (1992), 259\u201379, esp. 272, 274, for fourteen figures from the Orthia sanctuary, one seventh cent., the others sixth, and five from the Menelaion, all sixth cent.\n\n. _Parthenion_ 1, 47\u201359; see also Aristoph. _Lys._ 1307. Theoc. _Id._ 18.30 compares Helen to a Thessalian horse, and see chap. 6 n. 47.\n\n. Scholion B, fr. 6, col. i (P.Oxy. 2389).\n\n. J.K. Anderson, _Ancient Greek Horsemanship_ (Berkeley, 1961), 36\u201337.\n\n. Perhaps mules: Athenaeus' text is unclear. See further Kaibel (ed.),Athen. 1.317.\n\n. Athen. 4.139f,Xen. _Ages_. 8.7, Plut. _Ages_. 19.5\u20136.\n\n. For public models, note that Hellenistic queens were publicly honored by equestrian statues: see E. Fantham et al., _Women in the Classical World_ (New York, 1994), 220, 222. At Rome, Cloelia was portrayed mounted on a horse, though her heroic deed involved swimming: Liv. 2.13.11.\n\n. Xen. _Hell._ 3.3.5 for Spartan men out on their country estates.\n\n. The sole exception is a woman who was dressed up to impersonate Athena and placed in a chariot. She may, however, have had a driver: Herod. 1.60.\n\n. Liv. 34.1\u20138;Tac. _Ann._ 3.34,Val.Max.9.1.3,Oros.4.20.14,Zonaras 9.17.1, and see further SarahB. Pomeroy, _Goddesses, Whores, Wives, and Slaves: Women in Classical Antiquity_ (New York, 1995), 178.\n\n. According to Arist. _HA_ 608a25, female Laconian hounds are more clever than the males.\n\n. Poralla2, no. 310.\n\n. According to Moretti, _IAG_ , pp. 41, 43, Cynisca was born sometime before 427, probably around 440.\n\n. For a dedication of a small votive by Cynisca to Helen at the Menelaion, see A. M. Woodward,\"The Inscriptions,\" _ABSA_ 15 (1908), 40\u2013106, esp. 86\u201387, no. 90 = _IG_ V.1.23.\n\n. Pausanias (5.6.7; 6.20.9) states that _parthenoi_ (virgins, unmarried women) could view the games, but that married women were excluded. This provision seems extremely unlikely in view of the need to chaperone young girls. Matthew P. J. Dillon,\"Did _Parthenoi_ Attend the Olympic Games? Girls and Women Competing, Spectating, and Carrying Out Cult Roles at Greek Religious Festivals,\" _Hermes_ 128 (2000), 457\u201380, esp. 461\u201362, suggests that fathers accompanied their girls. Pausanias may be thinking about the parthenoi who were present not as spectators, but because they participated in the races at Elis. The only exceptions (not mentioned by Pausanias) were the priestess of Demeter and a female envoy from Ephesus: see L. Robert, \"Les femmes th\u00e9ores \u00e0 \u00c9ph\u00e8se,\" _CRAI_ (1974), 176\u201381 = _OMS_ 5.669\u201374 _._ Of course it is possible that even at her advanced age Cynisca was technically a parthenos _._ Perhaps she had been widowed at a young age. In any case, no husband nor children are recorded for her, and her brother's attempts to manipulate her (see below) suggest that she did not have a husband. Her single-minded devotion to racing may have not left any time for wifely duties.\n\n. Paus. 6.1.6: see further Bianchi-Bandinelli, _EAA_ 1.460\u201361.\n\n. His name may have been Apelles. Bianchi-Bandinelli, _EAA_ 1.460\u201361, Pliny _HN_ 34.86. The manuscript variant _adornantes_ (\"adorning themselves\") would probably make sense as well in the case of Cynisca, who seems to have been vain and by no means modest. On the commemorative monuments, see J. G. Frazer, _Pausanias's Description of Greece_ (London, 1913), vols. 3, 4, ad loc.\n\n. Paus. 6.1.6 = _Anth. Pal._ 13.16 = _IG_ V.1.1564a = _I. Olympia,_ no.160 = Moretti, _IAG_ , no. 17, and Moretti, _I. Olympia_ , nos. 373, 381.\n\n. Named for Troilus, famous for his horses, called by Homer _hippiocharmes_ ( _Il._ 24.257:\"fight-ing from a chariot\").\n\n. _LGPN_ 3A s.v., and see n. 77 above.\n\n. Xen. _Ages._ 9.6, sim. _arete_ in Plut. _Ages._ 20.1, and in _Sayings of Spartans_ ,Ages. 49.\n\n. See further Pomeroy, _Families_ , 93\u201394. Like Cynisca, Bilistiche received divine honors, but this worship was part of the Hellenistic trend to divinize members of the royal families and some of their intimate associates.\n\n. Moretti, _IAG_ , p. 44, suggests that the sculptures by Apelles were also erected after Cynisca's death, and commissioned either by her family or by the state.\n\n. Moretti, _I. Olympia,_ nos. 396, 418.\n\n. Bilistiche, mistress of Ptolemy II, who was said to be from Argos, Macedonia, or Phoenicia, was the next woman whose victories are recorded: see further Pomeroy, _Women in Hellenistic Egypt_ , 53\u201355. Her victories were probably in 268 and 264:Moretti, _IAG_ ,p. 42. Berenice II was the next woman after Bilistiche whose horses were victorious. Berenice originally came from Cyrene. Her interest in equestrianship is additional testimony to cultural links between Cyrene and Sparta. For the horses of Cyrene see, e.g., Pindar, _Pyth_. 4.2. According to Strabo (17.3.21 [837]), Callimachus praised the horses of Cyrene. For an epigram attributed to Posidippus comparing Berenice to Cynisca see _P. Mil. Vogl._ VIII.309, xiii 31\u201334.\n\n. Schol. Theoc. 18. 22\u20135, 39\u201340: see A. S. F. Gow, _Theocritus_ , vol. 2 (Cambridge, 1952), 354, 358.\n\n. See below. There was also a race called _en Drionas_ about which little is known: see Hesych. s.v. E2823 (Latte).\n\n. Hesych. s.vv.Driodones, en Drionas.\n\n. See further T. Scanlon,\"The Footrace of the _Heraia_ at Olympia,\" _AncW_ 9 (1984), 77\u201390.\n\n. On the connection between Spartan maidens and the races for Hera, see Preface, and Appendix,\"Mirrors and Bronze Statuettes.\"\n\n. I.e., 160 meters. David G. Romano, \"The Ancient Stadium: Athletics and _Arete_ ,\" _AncW_ 7 (1983), 9\u201315, esp. 14, suggests that the length of the girls' race was correlated with the measurement of the stylobate of Hera's temple at Olympia, while the length of the boys' race was correlated with the stylobate of the temple of Zeus. According to Herod. 2.149, the length of the stadion was 6 plethra or 600 feet (168.6 m.). According to Romano's (p. 14) formula, the girls' stadion race would be ca. 128 meters.\n\n. Paus. 5.16.3: see Frazer, _Pausanias's Description of Greece_ , vol. 3, 593.\n\n. The girl depicted in fig. 1, above, may be running as part of a dance, for she glances behind her. On the other hand, she may be looking to see if any competitor is catching up.\n\n. For a survey of nudity in antiquity, see Larissa Bonfante,\"Nudity as a Costume in Classical Art,\" _AJA_ 93 (1989), 558\u201368.\n\n. Ibycus, fr. 339 _PMGF_ ; Eur. _Andr._ 595\u2013601, cf. _Hec._ 932\u201336; Soph. fr. 872 Lloyd-Jones, Aelius Dionysius (2d cent. c.e., 4.35 [140] Erbse), for girls wearing one himation only,without belt or chiton; similarly Moeris, a lexicographer (2d cent. c.e., d27).\n\n. Xen. _Lac. Pol_. 1.4; Plut. _Lyc_. 14.4\u201315.1; Nic. Dam. _FGrH_ 103 F 90, cf. Anacreon fr. 99 (Page-Campbell).\n\n. Aristoph. _Lys._ 78\u201384,Critias fr. 32 (Diels-Kranz 2.1969).\n\n. See further N. Serwint, \"The Female Athletic Costume at the _Heraia_ and Prenuptial Initiation Rites,\" _AJA_ 97 (1993), 403\u201322. Beth Cohen, \"Divesting the Female Breast of Clothes in Classical Sculpture,\" in _Naked Truths: Women, Sexuality and Gender in Classical Art and Archaeology_ , ed. A.O. Koloski-Ostrow and C. L. Lyons (London, 1997), 84 n. 15, questions Serwint's fifth-cent. dating of the Vatican girl runner, otherwise considered a classicizing work of the Roman period.\n\n. See further Scanlon,\"The Footrace of the _Heraia_ at Olympia,\" 83, and Preface, above.\n\n. Serwint,\"The Female Athletic Costume at the _Heraia_ and Prenuptial Initiation Rites,\" 419, suggests that the change occurred ca. 580 when the festival was restructured.\n\n. Diels-Kranz 2.90.9, and see Freeman, _The Pre-Socratic Philosophers_ , 418\u201319.\n\n. See further Christiane Sourvinou-Inwood, _Studies in Girls' Transitions: Aspects of the_ Arkteia _and Age Representation in Attic Iconogography_ (Athens, 1988).\n\n. For a loincloth, see chap. 6 n. 59. My interpretation assumes that menstruation was not regarded as pollution restricting eligibility to compete at a religious ritual confined to females.\n\n. See further chap. 6 n. 16.\n\n. Nigel M. Kennell, _The Gymnasium of Virtue: Education and Culture in Ancient Sparta_ (Chapel Hill, N.C., 1995), 98, 147, and passim.\n\n. Pindar fr. 112 Snell: \"Lakaina . . . parthenon\" quoted by Athenaeus 14.631c, see n. 41 above; Xen. _Lac. Pol._ 1.4; Plut. _Lyc._ 14.2; Nic. Dam. _FGrH_ 90 F 103; etc. See most recently Ellen Millender, \"Exercise, Nudity and Spartan Female Sexual License: A Reconsideration\" (paper delivered at the annual meeting of the American Philological Association, Dec. 28, 1998; abstract published in _American Philological Association 130th Annual Meeting: Abstracts_ , 82), in agreement with Kennell, _The Gymnasium of Virtue_ , 46, about an educational system for girls organized by the state. However, in \"The Elite Women of Sparta\" (paper delivered at the annual meeting of the American Philological Association, Dec. 28, 1998; abstract published in _American Philological Association 130th Annual Meeting: Abstracts_ , 82), Kennell is less sanguine about the existence of a formal agoge for women in the Roman period. Jean Ducat,\"Perspectives on Spartan Education in the Classical Period,\" in _Sparta: New Perspectives_ , ed. S. Hodgkinson and A. Powell (London, 1999), 43\u201366, esp. 64 n. 31, argues that girls' _agelai_ did not exist, but in \"La femme de Sparte et la cit\u00e9,\" _Kt\u00e8ma_ 23 (1998), 385\u2013406, esp. 387, he notes that there was a system of initiation and education for girls, though it was not so structured as the boys' agoge. See also A. Brelich, _Paides e Parthenoi_ , vol. 1 (Rome, 1969), 157\u201359.\n\n. Stephen V. Tracy and C. Habicht, \"New and Old Panathenaic Victor Lists,\" _Hesperia_ 60 (1991), 187\u2013236, esp. 205, 213, 214.\n\n. See P. A. Cartledge and A. J. S. Spawforth, _Hellenistic and Roman Sparta: A Tale of two Cities_ (London, 1989), 205\u20136.\n\n. _SEG_ XI (1954), no. 610, 1\u20134, and Paus. 3.11.2.\n\n. Hesych. s.v. Dionysiades,D1888 (Latte); Paus. 3.13.7 speaks of eleven (not twelve) runners. Kennell, _Gymnasium of Virtue_ , 46\u201347, considers these events a legacy from an earlier girls' agoge _._\n\n. _SEG_ XI.830. Her name is indecipherable, but the editor suggests that she is Panthalida, daughter of Agis, attested in _IG_ V.1.588, 2\u20133.\n\n. Schol. Juvenal 4.53:Moretti, _IAG_ , p. 168 for the date.\n\n. See further M. L. West,\"Alcmanica,\" _CQ_ 15 (1965), 188\u2013202, esp. 199\u2013200.\n\n. In Athen. 13.602d\u2013e: see further C. Calame, _Les choeurs de jeunes filles en Gr\u00e8ce archa\u00efque_ , vol. 1 (Rome, 1977), 434, and Jan Bremmer,\"An Enigmatic Indo-European Rite: Paederasty,\" _Arethusa_ 13 (1980), 279\u201398, esp. 292\u201393. K. Dover, _Greek Homosexuality_ (London, 1978), 188, interprets the passage as anal penetration of girls by boys, and states that \"the original sexual meaning of 'lakonize'will have been 'have anal intercourse,' irrespective of the sex of the person penetrated.\"\n\n. Miniature weaving implements: Dawkins, _AO_ , pl. 185, nos. 14, 15, 17, 23, 24; usable implements: Dawkins, _AO_ , 242, pl. 174, no. 2; plaques: Dawkins, _AO_ , pl. 181, 27, 28; pl. 185, nos. 12, 21, 22; pl. 186, nos. 20, 21.\n\n. On these votives, see most recently Lin Foxhall, \"The Women of Artemis Orthia, Sparta\" (paper delivered at the annual meeting of the American Philological Association, Dec. 28, 1998; abstract published in _American Philological Association 130th Annual Meeting: Abstracts_ , 83).\n\n. Similarly, every four years sixteen women from Elis wove a peplos for the xoanon of Hera and organized the Heraion games: Paus. 5.16.2, 6.24.10. The weaving of Hera's peplos may have begun only ca. 575 simultaneously with the introduction of the Sixteen Women at the Heraia: Paus. 1.16.5, John Magruder Mansfield,\"The Robe of Athena and the Panathenaic _Peplos_ ,\"(Ph.D. diss., University of California, Berkeley, 1985), 470.\n\n. Whether this practice was begun in the archaic period is doubtful, for statues of Apollo traditionally were nude. See further Irene Bald Romano, \"Early Greek Cult Images\" (Ph.D. diss., University of Pennsylvania, 1980), 103,who doubts that a garment appropriate for a 30 cubit column-shaped statue of Apollo was woven annually. On clothing for statues, see Frazer, _Pausanias's Description of Greece_ , vol. 2, 574\u201375, and see now Irene Bald Romano, \"Early Greek Cult Images and\n\nCult Practices,\" in _Early Greek Cult Practice: Proceedings of the Fifth International Symposium at the Swedish Institute at Athens, 26\u201329 June, 1986_ , ed. R. H\u00e4gg, N. Marinatos, and G. C. Nordquist (Stockholm, 1988), 127\u201334.\n\n. An ivory plaque with an image of the cult figure displays incised textile patterns. For the plaque: Dawkins, _AO_ , 208, and pl. 96, no. 2. For the weaving: Romano,\"Early Greek Cult Images,\" 123, contra P. Ziehen,\"Sparta,\" _RE_ 18.3 (1949), col. 1466.\n\n. The cult statue of Hera was probably seven feet tall, or less: Romano, \"Early Greek Cult Images,\" 441. Pausanias (5.17.1) refers to it as an _agalma_ (image).\n\n. Homer, _Od._ 4.131\u201335. Theoc. (18.33\u201334) describes Helen as an accomplished spinner and weaver.\n\n. Mansfield,\"The Robe of Athena,\" 6, 23 n. 14.\n\n. Mansfield, \"The Robe of Athena,\" 58, 89 n. 26.\n\n. Mansfield, \"The Robe of Athena,\" 58.\n\n. Aristoph. _Lys_. 1138\u201341, Xen. _Lac. Pol_. 11.3, Plut. _Lyc._ 27.2\u20134, _Mor._ 238d,Ael. _VH_ 6.6.\n\n. Plut. _Sol._ 20.8, 21.4\u20135, and see further Pomeroy, _Families,_ 101.\n\n. On women and weaving see further Pomeroy, _Xenophon: Oeconomicus_ , 60\u201364, 270, 274, 284, 297, 307.\n\n##\n\n## BECOMING A WIFE\n\nAnd I was thinking that Sparta among cities of few citizens proved to be the most powerful and famous, and I wondered in what way this had come about. When, however, I thought about the Spartans' way of life, I no longer wondered. I admired Lycurgus, their lawgiver, whose laws they were fortunate in obeying, and I think him extremely wise. He did not imitate other cities, but thinking the opposite of most, he made his country outstandingly fortunate.\n\nNow, to begin at the beginning, I will discuss the breeding of children. In other states the girls who are destined to become mothers and are brought up in the approved manner live on the most modest amount of food, with the smallest possible allowance of delicacies. They are either totally deprived of wine, or drink it mixed with water. The rest of the Greeks think it right that their girls keep silent and work wool, like sedentary craftsmen. How, then, ought we expect that women brought up in such a way will bear a sturdy child?\n\nBut Lycurgus thought that slave women were able to supply clothing, and he believed motherhood was most important for freeborn women. Therefore first he ordered the female sex to exercise no less than the male; moreover, he created competitions in racing and trials of strength for women as for men, believing that healthier children will be born if both parents are strong.\n\n(Xenophon, _Lac. Pol._ 1.1\u20134)\n\n### Eugenics: Nature in Alliance with Nurture\n\nIn highlighting women in the following discussion of marriage and reproduction, we are not engaging in affirmative action or compensatory scholarship. Rather, we are following the best ancient precedent: the intention of the revered founder of the Spartan way of life. Xenophon points out that Lycurgus devoted a great deal of attention to motherhood and marital intercourse. He also observes that Spartans were the only Greek girls who were generously fed, draws attention to the physical training for females that was unique in Greece, and supplies as the motivation the belief that strong parents produce stronger children. (According to Aristotle and other medical writers, acquired characteristics, as well as those with which a person was born, were widely thought to be inherited.) Xenophon's account of the raising of girls makes it clear that the \"Lycurgan\" system\u2014insofar as it was concerned with health and eugenics\u2014was not merely a Hellenistic or Plutarchean invention.\n\nGirls and boys exercised nude. Not only nude youths but young women as well may have participated in the Gymnopaidia (\"Festival of Nude Youths\"). Plutarch ( _Lyc._ 15.1) writes that confirmed bachelors were dishonored: they were excluded from viewing the young women and men exercising in the nude ( _tais gymnopaidiais_ ) _._ 2 Perhaps the bachelors preferred nonproductive sexual liaisons with boys to the exclusion of reproductive sex with females and this prohibition of viewing, which meant that they were barred from attending a major festival, was the state's punishment. In any case, while engaged in these activities, nubile Spartans had an opportunity to view the bodies of potential spouses. Although Plutarch ( _Lyc._ 4.4) argues that women's nudity was not intentionally erotic, it would have been difficult to prevent some viewers from becoming stimulated. Sometimes, women did parade nude in order to whet the appetite of unmarried men for marriage. Doubtless they attracted other women as well (see chap. 1).\n\n### _Infanticide_\n\nPatriarchy exercises authority over men as well as women, and in Sparta apparently even more over men. Males, from the moment of birth, were examined, tested, and evaluated according to eugenic standards by older men:\n\nThe father did not decide whether to raise a baby; rather he took it and carried it to some place called Lesche where the elders of the tribes sat and examined the infant, and if it was well built and sturdy, they ordered the father to rear it, and assigned it one of the nine thousand lots of land; but if it was ill born and misshapen, they sent it to the so-called \"Apothetae,\" a chasm-like place at the foot of Mount Taygetus, thinking that any baby which was not naturally created at the very beginning to be healthy and strong was of no good either to itself or the state. Therefore the women used to bathe their newborn babies not with water but with wine, thus making a sort of test of their constitutions. For it is said that epileptic and sickly infants are thrown into convulsions by the unmixed wine and lose their senses, while the healthy ones are rather hardened by it, and given a strong constitution. (Plut. _Lyc_. 16.1\u20132)\n\nPlutarch is our only source for the practice of systematic male infanticide. As we have seen, in the sentence immediately following his statement on infanticide he describes the allocation of a _kleros_ (lot of land) to every infant. The Greek words Plutarch uses for infant ( _to gennethen, to paidarion, ta brephe)_ are neuter; even the pronoun _aut\u00f4i_ can be understood as neuter or masculine. On the basis of this passage alone, it could be deduced that newborn girls were subjected to exposure and infanticide and were given kleroi. However, other passages in Plutarch ( _Lyc._ 8.4) and Xenophon ( _Lac. Pol._ 9.5) make it clear that girls were supported by the kleroi of their male kinsmen (see below). Pierre Roussel  sees a connection between infanticide and allocation of a kleros and argues that the state was involved with infanticide exclusively in connection with males. Considering their concern over the decline in size of the population, it is likely that the magistrates understood that the number of Spartiates was directly related to the number of child-bearers (not inseminators) and therefore did not cull female infants.\n\nMale infanticide was wasteful only of the mother's nine-month investment. By eliminating unpromising male infants, the community was not obliged to pay the costs of rearing a boy who would not, as Plutarch ( _Lyc_. 16.1\u20132) puts it, be of use to himself or to the state. The decision ostensibly satisfied eugenic considerations. The infant who failed the initial inspection was eliminated before he could produce children who were likely to inherit his undesirable characteristics.\n\nA question that is often raised is whether at Sparta girls as well as boys,were vulnerable to infanticide. Nowadays few scholars doubt that girls were regularly exposed in Athens. The need to furnish a dowry at, or even before, puberty, and the general devaluation of women, were probably among the reasons why a father might choose not to raise a daughter. Was Sparta the antithesis of Athens? Did Sparta expose all boys who did not appear to have the potential for becoming excellent soldiers, but raise all girls\u2014except, one would suspect, those with obvious and debilitating physical abnormalities?\n\nWe may deduce from Plutarch's description that at birth girls were simply handed over to the women. This conclusion is _ex silentio_ ; on the other hand, it can be argued that Plutarch was unusually scrupulous among Greek authors in remembering to mention women where it was appropriate. It seems that girls were not subjected to official scrutiny as were Spartan boys, nor, apparently, did fathers make a determination about rearing or exposing them, as they did in Athens. The women would test the babies for epilepsy or sickliness by seeing if a bath in undiluted wine would throw them into convulsions. Plutarch does not state whether the immersion ordeal was part of the official evaluation process, nor does he indicate the fate of the infants, presumably both male and female, who failed the wine-immersion test. He does not tell us whether the women took any action in the case of infants who had failed the test, but if they did not, then what was the point? If the decision to expose apparent weaklings was up to the women, they would have exercised a power usually reserved for men in the Greco-Roman world.\n\nThere probably was change over time, with private citizens taking upon themselves the right to judge the male infant's viability that had previously been a monopoly of the community. Such changes were correlated with the right to dower and to alienate and bequeath property. The individual father's decision may have followed the verdict of the tribal elders, or the former supplanted the latter totally. At any rate, there is no indication of systematic female infanticide or neglect of girls at Sparta, nor is there any indirect evidence, such as skewed sex ratios. It seems likely that because in Sparta there was no money and women could own land, they could more easily be given a dowry. At Athens, where women could not own land, a father would have to furnish cash or movables, and this obligation might well be a deterrent to rearing girls.\n\nEugenic principles, such as existed in that era, underlie much of Spartan demographic engineering. The Spartans were celebrated for breeding fine hounds and racehorses, so it is not surprising to see them transfer these notions to human beings. By eliminating weak male infants, they tried to give natural evolution a boost, and through the subsequent rigorous training of boys, they assured the survival of the fittest and future reproduction by them. A mistake might be made, and a man might prove to be a coward. Such tremblers were most likely to be younger men facing their first real conflict, rather than hardened veterans. Anecdotes about Spartan mothers show that they are concerned lest their sons prove to be cowards (see chap. 3). In any case, cowards did not reproduce, for they were socially ostracized, and neither they nor their sisters (presumably young and not married) could find spouses (Xen. _Lac. Pol._ 9.5). Eugenic motivations may also be detected in the choices made in wife-sharing or husband-doubling arrangements (see below). Plutarch ( _Lyc_. 15.7) agrees with Xenophon that a man could ask a husband if he might plant his seed in a wife who had already produced children, but he also attributes the initiative to the husband. For example an elderly man with a young wife might offer her a handsome, noble young man, and then adopt the children born of this union (Plut. _Lyc_. 15.12). Daniel Ogden has speculated that Spartans believed that in the process of wife-lending, male sperm could mingle and produce offspring who were thought to have descended from two male parents. Though the male contribution to the embryo was usually thought to dominate, it is also necessary to pay sufficient attention to the female contribution. The rejection of cowards' sisters and of some wives in favor of married women who were _euteknos_ (\"blessed with good children\") and _gennaia_ (\"well born\") reveals a belief that the mother was more than merely a fertile field for the father's seed, and that each woman continued to make her own particular contribution to the offspring. Indeed, in Plutarch's _Sayings of Spartan Women_ (e.g., 240e, 241d), the mothers take all the credit for the way their sons turn out. Of course this is an exaggeration that results from the author's effort to prove that Sparta was very different in this respect from other Greek cities where mothers had little to do with the rearing of sons after the age of seven. Xenophon ( _Lac. Pol._ 6.2) reports that at Sparta fathers were involved in their children's upbringing. Nevertheless, Plutarch's view on the strong influence of mothers is corroborated by other sources (see chap. 3).\n\n### _Husband-Doubling,Wife-Sharing_\n\nSarah Blaffer Hrdy has questioned the assumption of anthropologists that women prefer monogamy when raising the human infant, who is helpless for a long period, while men prefer polygamy in order to increase the possibility of creating offspring and perpetuating their own genetic legacy. Hrdy argues persuasively that polyandry can be beneficial to mothers, for it increases the number of possible fathers who will help with the care of infants and thus perpetuate the mother's genetic legacy. The presence of more than one father who believes that a child may be his own is a kind of assurance of help for mother and infant. Multiple fathers increase the possibility of survival and wellbeing of offspring. The shared mother thus forges alliances between half siblings, more so than the father whose role both the primary sources and secondary scholarship has acknowledged. The concept of \"partible paternity\" which Ogden proposed in terms of benefits to the husband and the male lover or _genitor_ (i.e., biological father) needs to be reexamined from the perspective of benefits to mother and child. Partible paternity was certainly useful in a society like Sparta, where male mortality and the absence rate of men from the home were relatively high and women were seldom at physical or economic risk. That Spartiates reproduce was certainly a goal of the community. That they treat all children as though they were their own was also an explicit goal (Xen. _Lac. Pol._ 2.10, Plut. _Lyc._ 17.1). That two men treat a child as though it belonged to each of them was certainly a step in this direction.\n\nIn most patriarchal and patrilineal states in antiquity, husbands enjoyed exclusive access to the reproductive potential of wives, using legal remedies or even violence to exclude intruders. At Sparta, the authoritarian patriarchy impinged upon the husband's monopoly of sexual access to his wife. Not only were the Spartan man's rights as a husband less than those of husbands elsewhere in Greece, but his rights as a father were less. As we have mentioned above, the state also usurped the father's right to determine whether his son was to be reared.\n\nPrimary sources generally view reproduction from the father's perspective. In a typical nineteenth-century attempt to interpret the development of marriage and society in prehistory, Engels painted a picture of the introduction of strict sexual restraints on women coinciding with the development of private property and the father's wish to know for certain that he was both _pater_ (i.e., legally father and husband of the child's mother) and genitor of his heir. Engels ranked the \"pairing marriage\" that he found at Sparta as more archaic than marriage in Homeric times, but he believed that Spartan marriage customs allowed greater freedom to women.\n\nDarwin, who wrote at about the same time as Engels, at least was largely correct in his views, but there is no doubt that he was influenced by Victorian assumptions about male and female sexuality. Darwin asserted that \"the female is less eager (to copulate) than the male . . .and may often be seen endeavoring for a long time to escape from the male.\" Scholars of Greek demography and society have long been influenced by Darwinian theories about the promiscuous human male and the coy female whose behaviors were considered advantageous for human evolution. Xenophon ( _Lac. Pol._ 1.9) spelled out some but not all the benefits of having, in effect, two husbands when he wrote: \"for the wives want to get possession of two oikoi, and the husbands want to get brothers for their sons.\" Plutarch's report mentions the benefits to the husband of impregnation of his wife by a vigorous lover, but the modern feminist scholar can extend the concept to the wife. Surely she, too, benefited both genetically and perceptibly from producing strong children. Her investment in nine months of pregnancy would not be wasted by a decision of male elders to toss the offspring off Taygetus. If she was an heiress, she would sooner fulfill her obligation to produce an heir for her father. Because Greek gynecologists were uncertain about the period of gestation, it would be easy to attribute paternity to more than one father.\n\n### Marriage\n\nSpartans were reputed to have chosen their spouses by several systems, some similar to those practiced in other poleis, others unique. The former were based on the oikos system and the goal was the perpetuation and prosperity of the individual family; the latter evolved from the communal ideal of equality and the goal was the production of children for the good of the state. In the former system, personal inclinations and ambitions determined the choice of a spouse; in the latter system the state provided incentives for marriage. Women were active players in both systems.\n\nXenophon's description shows that in his day the two systems overlapped. Although there was an oikos system in place, the welfare of society was fostered by means of what is commonly referred to by scholars as 'wife-sharing' for reproductive purposes. Although it is not always obvious that Xenophon is reporting an entirely logical system, it is apparent from his language that the wife is an active participant in the arrangement whereby she produces children for a partner in addition to her husband. This practice should therefore be called \"husband-doubling\" or \"male-partner duplication\" or \"nonexclusive monogamy,\" or, at any rate, some term that does not suggest passivity on the wife's part:\n\nHe [Lycurgus] saw, too, that during the time immediately following marriage, it was usual elsewhere for husbands to have unlimited intercourse with wives. He decreed the opposite of this: for he ruled that the husband should be embarrassed to be seen visiting his wife or leaving her. Thus the desire for intercourse was more fervent in both of them, and if there should be a child, it would be more sturdy than if they were satiated with one another. In addition to this, he took away from men the right to take a wife whenever they wanted to, and ordered that they marry in their prime, believing that this too was conducive to the production of fine children. If, however, it happened that an old man had a young wife\u2014seeing that men of that age guard their wives\u2014he thought the opposite. He required the elderly husband to bring in some man whose body and spirit he admired, in order to beget children. On the other hand, in case a man did not want to have intercourse with his wife but wanted children of whom he could be proud, he made it legal for him to choose a woman who was the mother of a fine family and well born, and if he persuaded her husband, he produced children with her. Many such arrangements developed. For the wives want to get possession of two oikoi, and the husbands want to get brothers for their sons who will share their lineage and power, but claim no part of the property. Thus in regard to the breeding of children he thought the opposite to those of other states. And anyone who wishes to may see whether it turned out that the men in Sparta are distinctive in their size and strength. (Xen. _Lac. Pol._ 1.5\u201310)\n\nPlutarch also describes marriage customs appropriate for a utopian society in which reproduction is the primary goal of marriage and the economic aspects of the private oikos are deemphasized in favor of the common good. Though he reiterates much of Xenophon's report on husband-doubling, in Xenophon the child born of extramarital intercourse would have no claim on the estate of the biological mother's husband. It appears that the Spartans were not concerned with the legal issue of illegitimate birth: rather, arrangements between consenting males, with the practical consent of the woman involved, were valid in assigning paternity. Children belonged to the oikos of the father and therefore the biological mother would have no legal claim on the newborn, except after adoption. Plutarch introduces the new idea that the biological mother and her husband might adopt the child born from extramarital intercourse, and then, of course, the child would inherit from them:\n\nThere were also these incentives to marry. I mean the processions of girls, and the nudity, and the competitions in view of the young men, who were attracted by a compulsion not of an intellectual type, but (as Plato says) a sexual one. In addition he [Lycurgus] decreed that those who did not marry would lose a civic right, for they were excluded from the spectacle of the Gymno-paidiai [\"Nude Youth\"] . . . .\n\nThey used to marry by capture, not when the women were small or immature, but when they were in their prime and fully ripe for it. The so-called \"bridesmaid\" took the captured girl. She shaved her head to the scalp, then dressed her in a man's cloak and sandals, and laid her down alone on a mattress in the dark. The bridegroom, who was not drunk and thus not impotent, but was sober as always, having dined with his mess group, then would slip in, untie her belt, lift her, and carry her to the bed. After spending only a short time with her, he would depart discreetly so as to sleep wherever he usually did with the other young men. And he continued to do this thereafter. While spending the days with his contemporaries, and going to sleep with them, he would cautiously visit his bride in secret, embarrassed and fearful in case someone in the house might notice him. His bride at the same time was scheming and helping to plan how they might meet each other unobserved at a suitable time. They did this not just for a short period, but for long enough that some might even have children before they saw their own wives in the day. Such intercourse was not only an exercise in self-control and moderation, but also meant that partners were fertile physically, always fresh for love, and ready for intercourse rather than being satiated and impotent from unlimited sexual activity. Moreover, some lingering spark of desire and affection always remained in both.\n\nAfter making marriage as modest and orderly as this, he showed equal concern for removing empty womanish jealousy. Banning from marriage every kind of outrageous and disorderly behavior, he made it honorable for worthy men to share children and their begetting, and derided people who think that there can be no combination or sharing of such things, and who resort to murders and wars. Thus if an older man with a young wife should take a liking to one of the handsome and virtuous young men and approve of him, he might well introduce him to her so that he might fill her with noble sperm and then adopt the child for themselves. On the other hand, a respectable man who admired someone else's wife noted for her lovely children and her self-control might persuade the husband that he have intercourse with her\u2014thereby planting in fruitful soil, and producing fine children who would be linked to fine ancestors by blood and kinship. (Plut. _Lyc_. 15.1, 3\u20137)\n\nSome of the bizarre customs Plutarch mentions, such as the cutting of the bride's hair and the secret marriage, could not have existed simultaneously, certainly not where women regularly spent much time out of doors. Nor are they mentioned by Xenophon. We can only speculate that they were created over time like other reforms attributed to Lycurgus, were enforced by the ongoing fear of _oliganthropia_ (sparse population) after the Second Messenian War (see below) or even later, and relaxed when the Spartans realized that they were able to win the Peloponnesian War despite their small population. Another possibility is that these customs were revived or invented in the Hellenistic period either in connection with the reforms of Agis IV and Cleomenes III or under the influence of some other utopian philosophical program.\n\nThe \"capture\" of the bride was a ritual enactment of a prearranged betrothal. The bridesmaid was ready and waiting with the bride's costume. The bride herself, full grown, would have been able to put up a good struggle if she truly objected and the groom was really raping her. An abduction rather than a joyous spectacular wedding ceremony may serve to ward off the jealous evil eye. The shaving of the head and dressing of the bride as a man (Plut. _Lyc._ 15.8) may have been part of a rite of passage that signalled her entrance into a new life. As a maiden she wore her hair long and uncovered, as a wife she wore it short, and covered by a veil (see fig. 4). In some sense, she was transformed into a youth in the agoge. Since participation in the agoge was a prerequisite to becoming a full-fledged citizen, the transvestism may have been symbolic of the bride's inclusion in the citizen body. It may also have been an attempt to ward off the evil eye or the supernatural spirits who were deemed to be jealous of the bride's fortune. The bride's costume may have also helped to ease the husband's transition to procreative sex from the homosexual intercourse to which he was accustomed.\n\nFig. 4. Married woman.\n\nMature woman with head and shoulders covered by a mantle. Rim of the Vix crater. Bronze volute crater from a tomb at Vix. Total height 1.64 m. Ch\u00e2tillon-sur-Seine. Ca. 530. See Appendix n. 102, and most recently C.M. Stibbe, _Das andere Sparta_ (Mainz, 1996), fig. 8 and pp. 137, 152, for a date of 570\u2013560.\n\nWe may ask if there were any analogous comforts for the bride who had been accustomed to female caresses. At Athens, vases depicting weddings often show Erotes (\"Cupids\"), an attempt to enlist the services of the supernatural in making the bride receptive to the bridegroom. The Athenian bride, however, was much more in need of help than the Spartan. The Athenian was not quite fifteen: she married a stranger nearly twice her age, moved to a new house, and rarely saw her friends and relatives again. The Spartan, in contrast, married a young man close in age. The couple had seen each other nude at festivals and during exercise since childhood. Because the marriage was secret until the bride became pregnant, she did not change domicile for a while (see below). Since the bride and groom were around eighteen and the groom was obliged to live with his army group until the age of thirty, the wife would not have been obliged to adapt to her husband's personality; she raised the children and managed the household for the most part by herself. These responsibilities made it necessary that the bride be mature, not an adolescent like the Athenian bride. Furthermore, the Spartan bride's principal source of companionship and sentimental attachments would continue to be other women. In cases where the bridegroom was older than normal, the scenario would be quite different, for the husband would no longer be sleeping with his army group.\n\nYet, even in such households the husband probably did not completely dominate the wife. If he was impotent or infertile, he nevertheless was obliged to participate in the social goal of reproduction as well as to consider wife's desires. Xenophon implies that her wishes and ambitions were consulted in cases where she was to be inseminated by a younger man to whom she was not married. Marriage between an older man and a younger woman could be the consequence of the epiklerate.  Though Spartan law is not so well understood as Athenian, some form of the epiklerate may have existed, for it appears that when a daughter was the sole survivor and heiress in a family, she was under some obligation to marry a close kinsman of her father, though he might have been elderly. In such cases, nevertheless, the pressure to produce an heir was intense.\n\n### Sexuality\n\nThere is no reason to assume that the sexuality of Spartan women was repressed or indeed less assertive than that of the men, or, of course, to assume the contrary. If we rely on the judgment of Teiresias, who had been both male and female, women's capacity to enjoy sex was nine times greater than men's. Xenophon and Plutarch speak of desire for intercourse on the part of both spouses, though within the limits of modesty. Plutarch ( _Lyc._ 15.1 ) also refers to flirtatious behavior when the girls parade nude before the bachelors, trying to interest them in marriage. Perhaps this competition for men continued after marriage. Xenophon at first reports the incitement for wife-lending or husband-doubling from the male perspective: \"a man could choose a woman who was already mother of a fine family and of high birth.\" Since the goal of the liaison was only reproduction, it made sense for the potential genitor to choose a mature woman who had already proven her ability to reproduce, rather than a young virgin. Plutarch ( _Lyc_. 15.7) agrees with Xenophon, and draws attention to the age and impotence of the husband and the attractiveness and nobility of the lover. Though no ancient source mentions that any woman actively chose her surrogate husband, we suggest that a lively young wife would be able to exert influence on a feeble old husband. Why should a modern scholar assume that women were passive in these arrangements? And why did the attractive man in question choose one woman rather than another? It is not improbable that she solicited his attention. One revered example of a Spartan woman who chose a man younger than her husband is of course Helen. Spartans, because of their respect for tradition, took their mythical exempla seriously, and Helen must have been a major figure in a Spartan woman's thoughts even before marriage (see chap. 6).\n\nIn cases where the husband was old, the young man who was allowed to have intercourse with the wife was physically and morally attractive. Though eugenic goals were primary, we need not assume that the wife's experience with her surrogate husband was unpleasant.\n\n### _The Economics of Marriage_\n\nXenophon depicts the Spartans as careful, if not calculating, in their selection of spouses and family planning. Herodotus (6.57, 71) refers to fathers betrothing daughters and giving them in marriage. Contradicting Herodotus is the report of Hermippus (late third century B.C.E.) that cohorts of nubile men and women found spouses by groping randomly in a dark room, and that the women were without dowries. Likewise, the enactment of the marriage as an abduction suggests a lack of dowry, but does not definitely preclude one. This method of spouse selection reflected the ideal of equality among potential partners. It was a casualty of the manifest advent of private property. The men who were engaged to marry Lysander's daughters attempted to break their engagements when they discovered that Lysander was poor. They were fined for marrying badly, for their primary goal was to marry wealthy, rather than virtuous, women. Xenophon does not mention the random selection of spouses, but he does describe husband-doubling, a practice that was unique in the Greek world and doubtless contributed to the racy reputation of Spartan women. This breach of sexual exclusivity was introduced after the rhetra of Epitadeus, which permitted a man to give his house and kleros to whomever he wished while he was alive, or to bequeath them in his will. By the classical period (if not earlier), in addition to the land designated for distribution as kleroi, some was held as private property. It is clear that with more property openly in private hands Spartan men and women had increased incentives to develop heirship strategies. They may have limited the number of their offspring to increase the patrimony and social mobility of their heirs. Wives and husbands evidently shared the same views about family planning and attempted to further the special interests of the oikos within the context of an authoritarian patriarchy.\n\n### _Sex Ratio and Polyandry_\n\nXenophon ( _Lac. Pol_. 1.8), Polybius,  Plutarch ( _Lyc_. 15.12\u201313), and Nicolaus of Damascus ( _FGrH_ 90 F 103Z) refer to polyandry at Sparta. This practice is not necessarily indicative of a paucity of women, which, in turn, could be a symptom of female infanticide. Xenophon ( _Lac. Pol_. 1.8) mentions the case of the married man who has no desire to _synoikein_ with his own wife, but prefers to produce offspring by a married woman who has already proven her procreative gifts. Some scholars understand _synoikein_ as \"marry\" rather than in its common sense of \"cohabit,\" \"live with,\" or \"have intercourse with.\" The two interpretations, of course, overlap: it is a question of emphasis and the writer's usage. Xenophon uses _lambano_ of a man who takes a wife, and _synoikein_ for \"having intercourse with.\" Furthermore, as we will see, there were other childless women in addition to those whose husbands rejected them (see below). In another variation on this theme, Polybius states that several brothers would share one wife. Fraternal polyandry was also a form of family limitation, for one shared wife could not have produced as many children for the brothers as individual wives might have. When each of these matrimonial experiments began and how long they lasted is unclear. Xenophon's description of husband-doubling postdates the rhetra attributed to Epitadeus, the victory in the Peloponnesian War, and perhaps the disaster at Leuctra\u2014undoubtedly a period in Spartan history of such turbulence as to stimulate radical social change (see Appendix).\n\n### Infanticide and Sex Ratio\n\nThe answer to the question whether Spartans practiced female infanticide is not the same for all time periods. There is little data and few dates. Certainly by the Hellenistic period, when the Spartan ideal was thoroughly contaminated by alien arrangements and the \"Lycurgan\" economic system was embattled, and there is plenty of evidence for infanticide in the Greek world, beleaguered and impoverished fathers may have decided to do away with female as well as male infants. The reports of fraternal polyandry in Polybius (12.6b.8), and of husband-doubling in Plutarch ( _Lyc_. 15.7, see also Xen. _Lac. Pol._ 1.5\u20139) and Nicolaus of Damascus ( _FGrH_ 90 F 103), may actually indicate a scarcity of marriageable women at that time. On the other hand, if these arrangements were also found in the archaic period, that would suggest the simultaneous practice of female infanticide even then. In the light of the changing demographic picture at Sparta, it is interesting that Polybius (12.6b. 8) refers to polyandry as an \"ancestral\" custom: however, the Spartans instituted many novel social practices in the fourth century and Hellenistic period, for which they gained credence by referring to them as \"ancestral\" (see Preface). Furthermore, Polybius was not favorably disposed to Sparta and not averse to attributing customs to the Spartans that his readers would consider barbarous. Some elite families in Hellenistic and Roman Sparta raised as many as three daughters, and in some cases daughters outnumber sons.Women are well represented in these stemmata, owing in part to the fact that these families often resorted to claiming descent through matrilineal succession from Bronze Age heroes and heroines, including Helen and the Dioscuri.\n\nWe may speculate that if female infanticide were not common, polyandry would have left some women unmarried. There was, however, so much movement out of the Spartiate class downward that the resultant sex ratio becomes unfathomable. We may also speculate about the possibility of intermarriage between full-blooded Spartan women and members of the various subordinate classes in Laconia, though there is little evidence for such arrangements, with one exception. A story about the founding of Tarentum is worth discussing, though its veracity has been questioned for many reasons. During the archaic period, when the army was in the field for many years and it was uncertain whether the men would ever return safely, the Ephors (\"Overseers,\" elected magistrates) directed that the women have intercourse with helots in order to produce a new crop of children who could replace the men in case they never came home. When the army did return to Sparta, the children born of miscegenation were sent off to found the colony that became known as Tarentum. This episode\u2014if it actually occurred\u2014was exceptional, and clearly an emergency measure. Furthermore, that Greek women would be forced to have intercourse with their social inferiors, who were simultaneously at war with their polis, is difficult to believe, though the eugenics program perhaps was not yet in place. If these liaisons ever occurred, it is no suprise that the Spartans sent the offspring off to a colony. It is also reported that after many Spartans died in the war with the Messenians, they made some helots go to bed with the widows of the dead men so that their lack of manpower would not be apparent. They made these helots citizens. In general, hypogamy was not an option for elite Greek women, and there is nothing in the education of Spartan women that prepared them for such a possibility. Polybius' (12.6.5) version of the official insemination program is more credible: the Lacedaemonians sent back to their country men in their prime of life for the purpose of begetting children.\n\nLimiting our discussion to the upper class, unmarried or childless Spartan women include those in the following five categories: the sisters of cowards, whom no man will marry; the wives Xenophon mentions, whose husbands prefer to reproduce by other men's wives; those to whom the state forbids marriage; those like Lysander's daughters, whose lack of wealth does not make them attractive partners, particularly in a period of economic instability for men; and those to whom Cicero refers as refusing to bear children (see chap. 3). It must have taken a great deal of determination and self-will to belong to the latter group in a state that so cherished motherhood.\n\n* * *\n\n. See further Sarah B. Pomeroy, _Families in Classical and Hellenistic Greece_ (Oxford, 1997), 96\u201399.\n\n. On the ambiguity of this word, see Appendix n. 77.\n\n. See further David Leitao,\"The Exclusion of _Agamoi_ from the _Gymnopaidiai_ and the Politics of Viewing in Sparta\" (paper delivered at the annual meeting of the American Philological Association, Dec. 29, 1997; Abstract published in _American Philological Association 129th Annual Meeting: Abstracts_ , 171). Contra Michael Pettersson, _Cults of Apollo at Sparta: The Hyakinthia, the Gymnopaidiai, and the Karneia_ (Stockholm, 1992), 76. See Preface, above.\n\n. Compare a male reaction to the first sculpture of the nude Aphrodite at Cnidus: according to Pliny, _HN_ 36.20, a man embraced the statue and ejaculated on it.\n\n. Plut. _Lyc._ 15.1. Athen. 13.566e exaggerates the erotic or leering aspect when he reports that Spartans strip virgins before strangers.\n\n. On Hodkinson's criticism of this passage, see chap. 4 nn. 61\u201362 and Appendix n. 90.\n\n. \"L'exposition des enfants \u00e0 Sparte,\" _REA_ 45 (1943), 5\u201317, in agreement with G. Glotz, \"L'exposition des enfants,\" _\u00c9tudes sociales et juridiques sur l'antiquit\u00e9 grecque_ (Paris, 1906), 187\u2013227, esp. 188, 192, 217\u201319,who mentioned only sons.\n\n. See further Sarah B. Pomeroy, _Goddesses, Whores, Wives, and Slaves: Women in Classical Antiquity_ (New York, 1995), 69\u201370, 164\u201365, 227, etc., and _Families_ , 118, 120\u201321, etc. Mark Golden, in \"Demography and The Exposure of Girls at Athens,\" _Phoenix_ 35 (1981), 316\u201331, argues that the female infanticide rate was as high as 20 percent.\n\n. Cf. Conclusion, below, on \"ugly\"daughters. Daniel Ogden,\"Crooked Speech: The Genesis of the Spartan Rhetra,\" _JHS_ 14 (1994), 85\u2013102, esp. 91\u201398, follows M. Delcourt, _St\u00e9rilit\u00e9s myst\u00e9rieuses et naissances mal\u00e9fiques dans l'antiquit\u00e9 classique_ (Li\u00e8ge, 1938), in stressing the exposure of infants with marks of divine disfavor such as clubfeet.\n\n. Roussel,\"L'exposition des enfants \u00e0 Sparte,\" 17, alludes to such changes, though he is unable to determine exactly when they occurred.\n\n. See further Pomeroy, _Families_ , 55.\n\n. Daniel Ogden, _Greek Bastardy_ (Oxford, 1996) _,_ 230, 234\u201335. Cf. Plut. _Cat.Min_. 25 on Cato's sharing his wife Marcia with Hortensius for the purpose of bearing children. The arrangement was a result of the profound friendship between the two men.\n\n. \"The Optimal Number of Fathers: Evolution, Demography, and History in the Shaping of Female Mate Preferences,\" _Annals of the New York Academy of Sciences_ 907 (2000), 75\u201396.\n\n. See S. Beckerman, R. Lizarralde, et al., \"The Bari Partible Paternity Project: Preliminary Results,\" _Current Anthropology_ 39, 1 (1998), 164\u201367, esp. 166, and Sarah Blaffer Hrdy, _Mother Nature_ (New York, 1999), 246\u201349.\n\n. F. Engels, _The Origin of the Family, Private Property, and the State_ (1884),with introd. by Eleanor Burke Leacock (New York, 1972), 125\u201328.\n\n. See Sarah Blaffer Hrdy, \"Raising Darwin's Consciousness,\" _Human Nature_ 8.1 (1997), 1\u201349, esp. 6\u20139.\n\n. _The Descent of Man_ (London, 1871, repr. Princeton, 1974), 273.\n\n. See, e.g., Paul Cartledge, \"Spartan Wives: Liberation or Licence?\" _CQ_ 31 (1981), 84\u2013105, esp. 105: \"the modern feminist is unlikely to be over-impressed by the way in which Spartan women . . . were 'seized' and 'had' as wives in the domicile of their husbands, who could 'lend' them for extramarital procreation; finally, and perhaps least of all, by the overriding emphasis placed on their childbearing potential and maternal roles by men who monopolized the political direction of a peculiarly masculine society.\" On Cartledge's place in the historiography of Spartan women, see Appendix, below.\n\n. See, on Timaea, chap. 4, and Ann Ellis Hanson,\"The Eight-Months'Child and the Etiquette of Birth,\" _Bulletin of the HistoryofMedicine_ 61 (1987), 589\u2013602, esp. 589\u201392, 596.\n\n. Pomeroy, _Families,_ 48\u201350, 54\u201356, 59\u201362.\n\n. Thus Bella Zweig, \"The Only Women Who Give Birth to Men: A Gynocentric, Cross-Cultural View of Women in Ancient Sparta,\" in _Woman's Power, Man's Game: Essays on Classical Antiquity in Honor of Joy K. King_ , ed. Mary DeForest (Wauconda, Il., 1993), 32\u201353. Cartledge,\"Spartan Wives: Liberation or Licence?\" 103, 105, and Cynthia B. Patterson, _The Family in Greek History_ (Cambridge,Mass., 1998), 73, 77\u201378, take cognizance only of the first part of Xenophon's report on husband-doubling (to 1.8), thus suppressing the second part stating that the women want to control two oikoi. Both emphasize \"lending\" the wife, rather than husband-doubling. In this way, the wrong impression is given that Spartan women are merely passed around between men to be used as baby-making devices.\n\n. For this translation, see n. 38 below.\n\n. J. Christien Tregaro, \"Les b\u00e2tards spartiates,\" in _M\u00e9langes Pierre L\u00e9v\u00eaque_ , ed. M.-M. Mactoux and E. Geny (Paris, 1993), 33\u201340, esp. 36, argues that the Xenophon passage means that the child born of the union of a lover (\"amant,\" i.e., genitor) and another man's wife would be a member of the lover's family but have no claim on the lover's property, because such a child was illegitimate. Since Tregaro (35\u201336) also argues that the lover was a bachelor, it is difficult to understand the lover's motivation in such a scenario. The passage is ambiguous. The common interpretation, however, that Xenophon is referring to the husband's motivation (i.e., getting allies for his children free of charge), seems more sensible. On the domicile of the married woman who has produced children for more than one oikos, see chap. 3.\n\n. _Rep._ 458D.\n\n. See further Judith Evans- Grubbs,\"Abduction Marriage in Antiquity: A Law of Constantine ( _CTh_ ix.24.1) and Its Social Context,\" _JRS_ 79 (1989), 59\u201383, esp. 62\u201363.\n\n. Heracl. Lembus, _Excerpta Politiarum_ , 373.13 (Dilts); Lucian, _Fug._ 27; cf. Xenophon of Ephesus 5.1.7 (Dalmeyda). Plutarch's explanation is that unmarried girls need to find husbands; the married women need to keep the men who have them ( _Sayings of Spartans_ , 232c2). Pliny _HN_ 8.164 states that the libido of mares is extinguished once the mane is cut. Horace _Od._ 2.11.24 writes of a woman with hair tied back in a twist in the Laconian style. This style was simple and appropriate for women who spent time outdoors. Propertius 3.14.28 associates unscented hair with Spartan women. See further R. G. M. Nisbet and M. Hubbard, _A Commentary on Horace, Odes, Book II_ (Oxford, 1978, pbk. 1991), 178\u201379. I am grateful to Nicholas Horsfall for this reference. For elaborate hair-cutting\n\nprotocols marking ages and marital status, see Ellen N. Davis,\"Youth and Age in the Thera Frescoes,\" _AJA_ 90 (1986), 399\u2013406.\n\n. See further Ephraim David, \"Dress in Spartan Society,\" _AncW_ 19 (1989), 3\u201313, esp. 7, and \"Sparta's Social Hair,\" _Eranos_ 90 (1992), 11\u201321, esp. 17.\n\n. See further G. Devereux, \"Greek Pseudo-Homosexuality and the 'Greek Miracle',\" _Symb. Oslo._ 42 (1968), 69\u201392, esp. 83\u201384. On the potential for extramarital intercourse with lower-class women, see chap. 5.\n\n. For the Athenian marriage at best, see Xen. _Oec_. 7\u201310 and Sarah B. Pomeroy, _Xenophon, Oeconomicus: A Social and Political Commentary_ (Oxford, 1994), ad loc.\n\n. Thus E. Karab\u00e9lias, \"L'\u00e9picl\u00e9rat \u00e0 Sparte,\" _Studi in onore di Arnaldo Biscardi_ , vol. 2 (Milan, 1982), 469\u201380, esp. 479.\n\n. Xen. _Lac. Pol._ 1.5, Plut. _Lyc._ 15.1.4\u20135; see also Plut. _Advice to the Bride and Groom,_ 18.\n\n. Of course, the nubile men and women were not equally endowed with good looks. On Lysander's bad luck in this selection process, see: Athen. 13.555b\u2013c citing Hermippus = _FGrH_ IV.3 1026 F 6 = F. Wehrli, _Die Schule des Aristoteles_ , suppl. 1 (Basel, 1974), fr. 87. J.-F. Bommelaer, _Lysandre de Sparte: Histoire et Traditions_ (Athens, 1981), 58, finds the anecdote suspect for chronological reasons, unless Lysander married twice. See also chap. 3 n. 18 and Conclusion.\n\n. Plut. _Lys._ 30.5. On the traditions concerning Lysander's daughters,see Bommelaer, _Lysandre de Sparte_ , 57\u201358.\n\n. Cf. Plut. _Agis_ 5.2\u20135. On the rhetra, see Preface n. 11 and chap. 4 nn. 61\u201362.\n\n. See further Ephraim David, \"The Influx of Money into Sparta at the End of the Fifth Century b.c.,\" _Scripta Israelica_ 5 (1979\/80), 30\u201345, and chap. 4.\n\n. Thus Thomas J. Figueira,\"Population Patterns in Late Archaic and Classical Sparta,\" _TAPA_ 116 (1986), 165\u2013213, esp. 195.\n\n. Polyb. 12.6b.8. F.W. Walbank, _A Historical Commentary on Polybius_ , vol. 2 (Oxford, 1967), 340 ad loc., suggests that fraternal polyandry was an adaptation to the absence of men.\n\n. Pace Ogden, _Greek Bastardy,_ 239, and Tregaro,\"Les b\u00e2tards spartiates,\" 35\u201336,who take _synoikein_ as \"marry\" rather than as \"live with\" or \"have intercourse with.\" For the latter meaning, see E. C. Marchant, _Xenophon_ , vol. 7, _ScriptaMinora_ (Cambridge,Mass., 1968), 139; F. Ollier, _Le mirage spar-tiate_ (Paris, 1933\u201343), vol. 1, 379, and _X\u00e9nophon: La r\u00e9publique des Lac\u00e9d\u00e9moniens_ (Lyons, 1934), commentary, 34.8; W. den Boer, _Laconian Studies_ (Amsterdam, 1954), 223; and Liana Bogino, \"Note sul matrimonio a Sparta,\" _Sileno_ 17 (1991), 221\u201333, esp. 229. F. W. Sturtz, _Lexicon Xenophonteum_ (1801\u20134, repr. Hildesheim, 1964), vol. 4, 190, s.v. synoikein, gives as a primary translation, \"Consuetudinem habere, coire.\" Only secondarily, citing other scholars, Sturtz gives, \"Dicitur autem non de sola tori, sed de tota consuetudine, de matrimonio in universum,\" thus allowing for the possibility that the\n\ngenitor may have been a man who declined to cohabit, not only with a wife, but with any woman at all (except obviously, in the context being discussed, for the encounter necessary for insemination).\n\n. Sarah B. Pomeroy,\"Infanticide in Hellenistic Greece,\" in _Images of Women in Antiquity_ , ed. A. Cameron and A. Kuhrt (London, 1983), 207\u201322.\n\n. Michael Flower, \"The Invention of Tradition in Classical and Hellenistic Sparta\" (paper delivered at the Celtic Conference in Classics, National University of Ireland, Maynooth, Sept. 7, 2000), suggests that \"husband-doubling\" was introduced at a time when Spartiates were concerned about their falling population. He suggests that this may have happened towards the end of the fifth century.\n\n. See Appendix. For polyandry and multiple sexual partners among barbarian women, see Herod. 1.216; 4.172, 176, 180; 5.6.\n\n. A. J. S. Spawforth _,_ \"Families at Roman Sparta and Epidaurus: Some Prosopographical Notes,\" _ABSA_ 80 (1985), 191\u2013258, esp. 234, 238, for the descendants of Tib. Claudius Eudamus and Claudia Damostheneia. Androtelia had at least four daughters: _SEG_ IX.677c (add. et corr.), 2d\u20131st cent. B.C.E.\n\n. Spawforth,\"Families at Roman Sparta and Epidaurus,\" 195\u201396, 219, 221.\n\n. Polyb. 12.6b5, Strabo 6.3.3 (279\u201380), etc. The probable source of this anecdote was Ephorus. Irad Malkin, _Myth and Territory in the Spartan Mediterranean_ (Cambridge, 1994), esp. 139\u201342, understands it as aetiological, derived from the fact that the colonists were called \"Partheniai.\"\n\n. Ogden, _Greek Bastardy_ , 242, admits the possibility of procreative unions between Spartan women and helot men, though not in the context of the foundation of Tarentum. But this is highly unlikely for eugenic reasons among others. Acting on the theory of the inheritance of acquired characteristics, helots were bred for docility. A liaison between a Spartan woman and a male helot would not have produced the ideal Spartan; but see chap. 5.\n\n. Athen. 6.271c quoting Theopompus _Hist._ 32 = _FGrH_ 115 F 171.\n\n. Plut. _Am. narr._ 775d gives one undatable example of the daughters of Alcippus and Damocrita. Their father's enemies had a vote passed forbidding the daughters from marrying. The damning nature of this vote shows that central to the values of marriage and motherhood was the capacity to create alliances between males. The family of Alcippus and Damocrita was prevented from acquiring sons-in-law who would be allies, and, like the families of cowards, it was doomed to extinction.\n\n##\n\n## THE CREATION OF MOTHERS\n\nSpartans were the only Greek mothers who were famous in antiquity qua mothers, and who served as paradigms in later periods of history. Xenophon and Plutarch are our main authorities for the exaltation of motherhood. As well-traveled, cosmopolitan reporters, they declare that nowhere else in the Greek world did the state lavish such attention on women.\n\nThe oikos is usually described as a unit for reproduction and production. At Athens, the wives of citizens were distinguished from other women as \"mothers of legitimate children and faithful housekeepers\" (Ps.-Demos. 59.122). Because before the fourth century the economy at Sparta more than elsewhere in the Greek world depended upon the labor of those who were not free, not citizens, and not household residents, the \"production\" aspect of the oikos was minimal. Textual and archaeological evidence indicates that before the Roman period, Spartan houses were insubstantial: they need not have been used for long-term storage, since helots were required to supply agricultural products annually. Therefore housekeeping was not a time-consuming job for women, though, as we have demonstrated (see chap. 1), not only did they supervise domestics, but they also could weave and took pride in their skills. Nevertheless, there is no evidence that their textiles were exchanged or sold, or that the women were obliged to weave in order to provide clothing for their families. Wives were, above all, mothers. For these reasons, in this chapter we treat the Spartan family principally as a locus for reproduction and confine our examination of the economy of the oikos to inheritance and dowry and to the relationship of reproduction to economic concerns.\n\nAs a result of the emphasis on reproduction, girls were raised to become the type of mothers Sparta needed (see chap. 1). A mother had to be healthy, properly educated, and well versed in Spartan values. In the absence of fathers during children's formative years, women, including mothers, older sisters, and nurses, were the principal, if not the sole, influence in the creation of Spartan citizens. Sparta recognized the services of mothers by permitting only women who had died in childbirth (like men who had died in battle) to be commemorated by grave markers. Religion, especially the cults of Apollo and of Artemis (with whom Eileithyia was associated), provided a spiritual foundation for politically approved family values (see chap. 6).\n\n### _Creating Childbearers_\n\nThe mistress of the house supervised the conversion of raw food into edibles. The relationship between mother and daughter was strong. Since the father dined with his syssition _,_ the mother would have been in charge of food distribution in the home. I suspect that mothers were responsible for seeing to it that girls were well fed, though of course, like other Greek women in comfortable circumstance, they did not prepare the food themselves. This nourishment was so superior to what other Greek girls received that Xenophon comments on it. Spartan girls may, in fact, have been given more food than their male counterparts, for the boys were deliberately given inadequate rations which they needed to supplement themselves. The Spartan diet was distinctive and evoked comment from diverse sources. Food was supplied in the form of a family wage allotted to male citizens from their kleroi. According to Plutarch ( _Lyc._ 8.4), each kleros was sufficient to provide a rent of seventy _medimni_ (bushels) of barley for a man, and twelve for his wife, along with proportionate quantities of fresh produce. Every Spartan was required to contribute one medimnus each month to his syssition (mess group). If we suppose that the husband consumed his twelve medimni at the syssition while the wife had the same amount at home, the basic diet of men and women was the same. Because Plutarch does not specify the amount of perishables, we do not have sufficient data to enable us to calculate the complete Spartan diet, but if it is true that a wife received twelve medimni of barley, a Spartan woman would have had, as a minimum, more than one and a half _choinixes_ daily (approximately 0.839 kg, or one and a half times an Athenian man's daily grain ration), an adequate diet of staples even for an athletic, or pregnant, or nursing woman This amount of grain yielded approximately 3,416 calories daily. A very active adult female twenty to thirty-nine years old requires 2,434 calories.\n\nPlutarch is our only source for the specific details of the food allotment. His report refers to a stable kleros system, which, however, was not in operation throughout Spartan history as private property increased in importance (see chap. 4 and Appendix). Nevertheless, Plutarch provides evidence for a generous dietary standard, and is consistent with what we know from Xenophon and others. In Aristophanes' _Lysistrata_ , Spartan women conform to the ethnic stereotype and are notably robust: Lampito complains that the other women are judging how plump she is just as they would assess an animal to be sacrificed. Fifty-eight medimni a year would have generously fed at least five or six children of various ages, or fewer children and some domestics. Furthermore, in addition to the minimum supplied by the kleros (when the system was functioning), property holders would have had access to additional grain, wine, olives, and animal and plant food grown on their private estates. Unmarried women would be supported by their male relatives and their private resources (Xen. _Lac. Pol._ 9.5). We note in passing that the man who was newly elected to the Gerousia (Council of Elders) was given a second portion by his syssition. He was to give this as a gift to one of his female relatives (Plut. _Lyc._ 26). The relationship between such a man and woman imitated the relationship of king and subject. Spartan kings were regularly given a double portion so that they could bind various subjects to them by offering the extra food to them (Xen. _Lac. Pol._ 15.4). In short, Spartiate women were never underfed. To be sure, some people lost status in the late fifth century through the loss of property, but so long as the kleros supported these dues, they could supply the wife of the kleros holder.\n\nThe subject of diet leads us to consider whether Spartan girls reached puberty later than their Athenian counterparts. If so, late marriage could have been an adaptation to delayed menarche, which has been observed in young female gymnasts preparing for the Olympics nowadays. Spartan women, however, are unlikely to have exercised as vigorously as their modern counterparts, inasmuch as athletic standards and rewards for women are higher now than they have ever been. Moreover, ideas of beauty were different. To the modern eye, the figures of the girls depicted in Laconian bronzes appear slim and athletic, but not pathologically emaciated. One doubts whether Spartans were told that \"no woman is ever thin enough.\" In fact, a robust or sturdy body might have been seen as better for childbearing.\n\n### _Fertility_\n\nHomosexuality and nonreproductive forms of heterosexual copulation also reduced fertility. Homosexual ties among Spartans were common. Although Aristophanes depicts Spartan husbands as just as sex-starved in wartime as the rest of the Greeks and cowed by their wives' refusal to have intercourse, he also makes some jokes about the Spartan males' homosexual activities and predilection for anal intercourse. According to Xenophon ( _Lac_. _Pol_. 2.13) and Plutarch ( _Inst_. _Lac_. 7), the erotic relationships between men were purely spiritual. Elsewhere, Plutarch ( _Lyc_. 18) emphasizes the spritual aspect, but does not specifically rule out the physical relationship. Xenophon and Plutarch also praised Lycurgus' directives limiting the frequency with which the newlyweds engaged in intercourse. Bridegrooms were permitted only limited visits to their brides. Inasmuch as the Greeks were utterly ignorant about the days in the menstrual cycle when women were fertile, late marriage and infrequent intercourse in the early years of marriage must have reduced by at least one or two the total number of children borne by each woman. On the other hand, the secret marriage, which only pregnancy made public, produced more positive results, for presumably it allowed an infertile couple to find other partners with whom they may have been able to reproduce.\n\nThe state formally supported child production through rewards to fathers. A man with three sons was not obliged to serve garrison duty, and a man with five was exempt from liturgies (financial obligations to the state or citizen body). These goals, however, were unrealistic. The demographic likelihood of any man having four, let alone five, sons is slim, except in families with eight to ten children, or by sheer luck. There is no indication that the state rewarded fathers of girls, nor were there tangible rewards for the mothers who would endure the pregnancies. The Spartan incentive invites comparison with Augustus's _iustrium liberorum_ (\"law of three children\"), which granted social and legal rewards to many men and women, for it did not specify the sex of the children and rewarded both mothers and fathers.\n\nThe pronatalist aims of the state were at variance with the economic ambitions of individual families. Like other Greeks, Spartans practiced diverging devolution, but the Spartans alone adopted unusual practices in their attempt to counter the decrease in the economic status that large families would naturally experience when the paternal estate was divided. Diverging devolution works only in a capitalistic economy with ever-expanding resources. Sparta's economy, agriculturally based, had a limited capability to expand.\n\n### The Oikos and the Family Economy\n\nThe script for the secret marriage, in which the bridegroom stealthily visits the bride and the marriage is not made public until she is pregnant, indicates that the bride did not have to endure the trauma of losing her virginity and moving to a new home simultaneously. We do not have conclusive evidence about the couple's domicile during all stages of the marriage. It seems likely that after a while, perhaps after the birth of a child, the bride left her parent's house to live with her husband. We do not know if the marriage was neolocal. Since the couple married when the bride was in her late teens and the groom slightly older, there was a strong possibility that their parents would still be living. Xenophon does not indicate the domicile of a wife who took a male partner in addition to her husband so that she could control two oikoi (see chap. 2). Perhaps she used both residences so as to be better able to manage them. How the kleros system affected the pattern is difficult to envisage. Heiresses (brotherless women), of whom there were many, may have remained in their ancestral homes. Women in a matrilocal situation certainly enjoy more domestic power. In any case, the possibility of a change of domicile is not as important as it was in a polis like Athens in view of the fact that Sparta was a \"closed\" society without distant cleruchies (land allotments in a foreign country) or numerous colonies. As we have noted above, marriage was highly endogamous and although the territory of Sparta was extensive, a bride did not run the same risk as her Athenian counterpart of being separated from her family by a long sea voyage. Furthermore, Plutarch's report ( _Lyc_. 18) on the pederastic relationships between respectable adult women and unmarried girls refers to a multigenerational female social milieu in which, we may assume, family members would continue to mingle.\n\n### _Affective Relationships: Daughters and Fathers_\n\nThe fact that there were many heiresses in Sparta indicates that many families had no surviving sons. Because of the scarcity of male heirs, and perhaps owing to parents' personal inclinations, adoptions were rare. Even in families with children of both sexes, the father probably enjoyed a close relationship with his daughter, in fact closer than he had with his son, since the latter was living out of the house by the age of seven, in many cases while the father was still living with his army group. Children born to an older father, of course, would have had a better opportunity to develop a relationship with him. Examples of the close relationship between father and daughter include Gorgo and Cleomenes I, Prolyta and Agesilaus II, and Chilonis and Leonidas. Gorgo and Chilonis became assertive women whose activities are recorded in the historical sources. As a little girl of eight or nine, Gorgo was present when an ambassador from the Greek cities in Ionia came to persuade Cleomenes to support their rebellion against Persia. When he offered Cleomenes a huge bribe, Gorgo advised her father not to stray from the path of virtue (Herod. 5.51).He followed her advice. Even if the veracity of this anecdote can not be tested, the general idea that a father encouraged his clever young daughter to give her opinion on serious matters rings true enough for readers who have enjoyed such dialogues. Cynisca became an Olympic victor (see chap. 1). Chilonis left her husband in order to show support for her father, who had taken asylum, but then, as the political situation changed, she pleaded with her father in behalf of her husband.\n\nUnfortunately, owing to the problems scholars confront in reconstructing the history of ancient women, much of what we suppose about the relationships of Spartan daughters and their parents is based on assumptions. In contrast, there is textual evidence for the relationships with sons.\n\n### _Mothers and Sons: Ideology of Motherhood_\n\nIn order to inculcate her offspring with patriotism, the mother had to have the correct attitude herself. Spartan women were renowned for enthusiastically sacrificing their sons for the welfare of the state. Instead of lamenting at the death of their sons, they took pride in the bravery that had led to that fate. Perhaps the delay in bonding with a son, caused by the necessity to await the verdict of the elders as to whether the infant was fit to be reared, created a psychological distance between a mother and her male offspring. In Athens, women mourned the dead, lavished grave goods upon them, and visited funerary monuments; but in Sparta, grave goods and mourning were controlled, restricted in some circumstances and mandated in others. For example, when a king died, one man and one woman from each family were obliged to lament.When the Spartans conquered Messenia, they forced the freeborn women to mourn over men who were not related or connected to them (Aelian, _VH_ 6.1).\n\nModern psychology and sociobiology has taught us not to consider any particular version of human maternal behavior as \"natural\"; motherhood is socially as well as biologically constructed. Spartan women did not travel, and the family lives of _perioikoi_ (free, but noncitizens) and helots were inappropriate for citizens. The only alternative model for the rearing of sons that was available to Spartan mothers was that for the rearing of daughters, and the latter would not have produced fine specimens of Spartan manhood. Thus, doubtless, Spartan mothers did rear their sons according to the customs and expectations of their state and society.\n\nAt least at some time, not only did women sanction the official ideology of motherhood, but several, in fact, were architects of it. The first example is Theano, mother of the traitor Pausanias. When her son had sought asylum in a building that was part of the sanctuary of Athena of the Bronze House, and the Ephors were in a quandary, Theano did not plead for his life. Instead, she placed a brick at the door of the temple and left. Following her example, the Ephors then blocked up the doors, leaving Pausanias inside. Aelian reports ( _VH_ 12.21) that Spartan mothers lamented in private if a son's corpse bore the majority of wounds on the back. That these anecdotes are not mere exaggeration and correspond to some historical reality is confirmed by the behavior of the Spartans after their defeat at Leuctra. The women were ordered not to mourn, but to suffer in silence. The next day, kinsfolk of those who had been killed in the battle wore happy faces, while the kinsfolk of the soldiers who had survived were sad (Xen. _Hell._ 6.4.16, Plut. _Ages._ 29.4\u20137).\n\nQuotations in Plutarch's _Sayings of Spartan Women_ also indicate that women promoted the ideal of Spartan motherhood. These aphorisms were attributed to famous women who lived in the classical and Hellenistic period, including Gorgo, daughter of Cleomenes and wife of Leonidas; Argileonis, mother of Brasidas; Gyrtias, mother of Acrotatus (who died at Megalopolis in 265 B.C.E.); and to undatable or anonymous women as well. Spartans were the only Greek women whose statements were deemed worthy of quotation. I doubt that what we have here is propaganda actually written by men but ascribed to women in order to make the statements more dramatic and persuasive. The quotations were probably collected not only because of their wit and pithiness, but because they revealed familial attitudes that non-Spartans deemed amazing and noteworthy. For example:\n\nBecause Damatria heard that her son was a coward and not worthy of her, she killed him when he arrived. This is the epigram about her:\n\nHis mother killed Damatrius who broke the laws,\n\nShe a Spartan lady, he a Spartan youth. ( _Sayings of Spartan Women,_ 240.f2)\n\nAnother Spartan woman killed her son, who had deserted his post because he was unworthy of Sparta. She declared: \"He was not my offspring . . . for I did not bear one unworthy of Sparta.\" ( _Sayings of Spartan Women_ , 241.1)\n\nAnother, hearing that her son had fallen at his post, said: \"Let the cowards be mourned. I, however, bury you without a tear, my son and Sparta's.\"( _Sayings of Spartan Women_ , 241.2)\n\nAnother, hearing that her son had been saved and had fled from the enemy, wrote to him:\"A bad rumor about you is circulating. Either absolve yourself at once, or cease to exist.\" ( _Sayings of Spartan Women_ , 241.3)\n\nAnother, when her sons had run away from a battle and come to her, said: \"Wretched runaway slaves, where have you come to? Or do you plan to steal back in here whence you emerged?\"And she pulled up her clothes and exposed herself to them. ( _Sayings of Spartan Women_ , 241.4)\n\nA woman, when she saw her son approaching, asked:\"How does our country fare?\"And when he said:\"All are dead,\" she picked up a tile, threw it at him, and killed him, saying:\"Then did they send you to bring us the bad news?\"( _Sayings of Spartan Women_ 241.5)\n\nThese exemplary mothers would not tolerate a son's act of cowardice. Such behavior tainted with dishonor not only the soldier but his female relatives as well. In the case of Damatria and Damatrius, the son's name echoed his mother's. Perhaps owing in part to the belief that acquired characteristics could be inherited, his sisters (note that his brothers are not mentioned), like the coward himself, could not find anyone willing to marry them, and his mother (note, not his father) might take his life for it. In view of the Spartan equation of womanhood with marriage and motherhood, the sisters received a harsh sentence. There was a strong likelihood that the family would become extinct. That mothers were reputed to enjoy the patriarchal power of Roman fathers and could kill their adult offspring who had disgraced them by their lack of patriotism is unprecedented in the ancient world. It is striking that both Greek and Roman traditions assert that the Spartan mother could pass judgment on an adult son unilaterally and behave so violently against her own offspring.\n\nSome mothers appear to have actively sought their son's martyrdom, as we see again in Plutarch:\n\nAs a woman was burying her son, a shabby old woman came up to her and said, \"You poor woman, what a misfortune!\"\"No,by the two goddesses, what a good fortune,\" she replied, \"because I bore him so that he might die for Sparta, and that is what has happened for me.\" ( _Sayings of Spartan Women,_ 241.8)\n\nAnother woman handed her son his shield and exhorted him:\"Son, either with this or on this.\" ( _Sayings of Spartan Women_ , 241.16)\n\nSince Spartan women could manage their own property and lived close to their kinsmen and friends in a relatively well-protected territory, widowhood and the loss of a son were probably not such frightening and dreary prospects as the comparable situations were at Athens. Defeat by the helots or by a foreign power and the ensuing rape, slavery, or even death were more terrifying.\n\nMost of the _Sayings of Spartan Women_ concerning mothers and sons (or in the case of Gyrtias, a grandson) focus on the son's bravery; in fifteen quotations he shows himself worthy, in nine unworthy, and in one saying (242.19) one son is brave, another a coward. There are other themes as well. One saying draws attention to the pride of a mother of many sons as a well-known feature of Spartan ethnicity:\n\nWhen an Ionian woman was boasting about one of the tapestries she had woven (which was indeed of great value), a Spartan woman showed off her four most dutiful sons and said they were the kind of thing a noble and good woman ought to produce and she should boast of them and take pride in them. ( _Sayings of Spartan Women_ , 241.9)\n\nAnother saying emphasizes motherhood as the podium from which Spartan women dominated men:\n\nWhen a woman from Attica asked \"Why is it that you Spartans are the only women who can rule men?\" Gorgo replied,\"Because we are the only ones who give birth to men.\" ( _Sayings of Spartan Women_ , 240.5)\n\nAs the last two quotations discussed above suggest, motherhood could be a fulfilling experience for women, especially, a modern woman imagines, where\n\nFig. 5. Louis Jean Fran\u00e7ois Lagren\u00e9e the Elder, _Spartan Mother and Son_.\n\nMother bids farewell to son in classicizing portrayal titled \"Rapporte ce bouclier ou que ce bouclier te rapporte\" (\"Bring back your shield or let this shield bring you back\"). 1771. Elizabeth Rawson, _The Spartan Tradition in European Thought_ (Oxford, 1969), 265, points out that Spartan subjects first appear in French art in this period. Stourhead, Hoare Collection, National Trust. Photo, Photographic Survey, Courtauld Institute of Art.\n\nnurses are available and the mother has no domestic chores. Women's pro-creative role can contribute to their feelings of self-esteem as well as to a position of prestige not only in the family, but within the state. Some states in more recent times have sought to include women in this way, thus linking private with public. As an example, I will discuss not Nazi Germany, but the United States of America.\n\n### _Excursus 1. Spartan Motherhood: Precedent and Proof_\n\nIn the young American Republic, motherhood was designed as a political role for women: their assignment was to produce public-minded children. Republican motherhood functioned in several ways like Spartan motherhood and can serve as a heuristic model for the latter. Here, private and the public spheres intersect. Women were encouraged to display patriotism by sacrificing the men whom they loved. They were proud of their role in shaping a new generation of citizens. Women's history was marshaled into service to promote this view. According to Julia Sargent Murray, who lived from 1751 to 1820 and wrote essays on a variety of subjects including women's education:\n\nThe character of the Spartan women is marked with uncommon firmness. At the shrine of patriotism they immolated nature. Undaunted bravery and unimpeached honor was, in their estimation, far beyond affection. The name of Citizen possessed, for them, greater charms than that of Mother; and so highly did they prize the warrior's meed, that they are said to have shed tears of joy over the bleeding bodies of their wounded sons.\n\nThree-quarters of a century later, the Confederate States of America displayed their conservatism by retaining not only slavery but also some features of Republican motherhood. During the American Civil War, inspired by patriotic propaganda, some southern women behaved in ways similar to those Plutarch attributes to Spartans. Women would leave bonnets and hoopskirts at the homes of young men who had not enlisted, with a letter commanding them to volunteer or be stigmatized as unmanly. They exhorted the troops to come home only if they wore laurel crowns of victory or were carried on shields of honor. A mother wrote in the _Winchester Virginian_ : \"I am ready to offer you up in defense of your country's rights and honor and I now offer you, a beardless boy of 17 summers,\u2014not with grief, but thanking God that I have a son to offer.\"\n\nUpper-class southerners were familiar with Greek and Roman history. It would be ironic if they were influenced by ancient propaganda that had failed to sway the women toward whom it was originally directed. At any rate, these examples from the more recent past persuade me that there is no universal maternal instinct which inspires mothers to save their son's life without hesitation. Some\u2014even many\u2014Spartan women might have harbored sentiments similar to those of women in southern American states and conformed to the ideal of motherhood prevalent at the time, especially since the consequences of defeat were horrendous. Of course, not all American women in the early Republic, or all Confederate women, embraced this particular patriotic ideology enthusiastically. Some women were more reluctant to sacrifice their menfolk on behalf of slavery or \"states' rights.\"\n\n### Family Planning\n\nThe United States is a nation that derives \"its just power from the consent of the governed.\" The \"socialization\" at Sparta was surely more effective. There is no comparable evidence for Spartan women rejecting the ideal until quite late (see below).\n\nAs a return on the investment in feeding and educating girls, the state expected them to produce children when they were mature. Yet, as Aristotle ( _Pol._ 1269b\u20131270a) suggested, the authoritarian system placed more constraints on men than on women. There is no evidence that unmarried or infertile women (except the sisters of cowards) suffered any social pressure or disabilities similar to those experienced by bachelors and childless men. The state did offer some rewards to mothers: prolific mothers of strong male citizens enjoyed prestige in life, and the names of women who had died in childbirth were recorded on tombstones.\n\n### _Neither Production nor Reproduction_\n\nApud Lacaenas virgines,\n\nQuibus magis palaestra, Eurota, sol, pulvis, labor\n\nMilitiae studio est quam fertilitas barbara.\n\nSpartan maidens care more for wrestling, the [river] Eurotas, the sun, dust, and military exercise than for barbarous fertility. (Cicero, _Tusc_. 2.36)\n\nAs Cicero reports, women did reject the burden of continuous child production that was the long-range goal of their physical education. Contraception is discussed in the Greek medical tradition dating from the classical period, but Spartans are the only respectable Greek women we know of who are specifically reported to have exercised control over their fertility. This report reflects the autonomy of Spartan women acting not in secret, as might be necessary for an individual, but assertively as a group whose behavior attracted notice. That they were not married until they were mature gave them an advantage over the more passive child brides in Athens.\n\nFeminist historians have argued that the number of children a woman bears has a greater effect on the quality of her life than almost any other ongoing condition (that is, excluding such dramatic events as war and natural disaster). Like married women nowadays who remain childless out of choice, Spartan women were thought to be motivated by selfish desires. In the history of the premodern world, refusal to bear children has usually been considered unpatriotic. Before the advent of modern medicine, abundant fertility was fundamental to the survival of the community. That after only one year of war and before the devastation of the plague Pericles urged Athenian women to bear more children underlines this necessity. In order to alleviate the continuing and ominous population decline, Spartiate men produced children by helot women (see chap. 5). These mixed-blood children, called _mothakes_ ,were given some modified form of citizenship. We may speculate that the two practices were related: when the citizen women noticed their childbearing was no longer absolutely essential for the welfare of the state, and motherhood was consequently downgraded, they could well have decided it was not worth the trouble to bear many children. Cicero's phrase,\"fertilitas barbara,\" indicates that bearing children was considered unsophisticated. He was probably evaluating Spartan women by the standard of upper-class Roman women, and his judgment here was shrewd. Both Spartans and Romans enjoyed wealth, high status, and access to contraceptive methods (through formal medical care or word of mouth) which, though primitive, were not ineffective. The Hippocratics do not mention Sparta, but they do not write about Athens either, and we know from other sources that there were physicians in Athens. At Sparta they would have been important for military reasons. Archagathus, who in 219 B.C.E. was the first Greek physician to go to Rome, came from Laconia (Pliny, _HN_ 29.12). Plutarch ( _Sayings of Spartan Women_ , 242.26) mentions a Spartan girl who brought on an abortion in secret. A fourth-century inscription from the sanctuary of Asclepius at Epidaurus reports that a Spartan mother came to find a cure for her daughter's case of dropsy. She slept at the shrine, saw a vision, and when she returned home she discovered that her daughter had seen the same vision and was cured. Since women in Sparta were neither secluded nor silenced, as some women were elsewhere in Greece, and since Spartans spent much of their time in the company of women, it would be only reasonable to assume that they had at least the same knowledge of contraception as other Greek women.\n\nWith the exception of Cicero, ancient and modern authorities have offered social and economic, rather than biological, theories to explain the population decline at Sparta. Among all the theories advanced about the reasons for the decline, we must give serious consideration to Cicero's (the most obvious one): women's control over reproduction.\n\nSince the source of the verse Cicero quotes is not known,we can only speculate about the reason for Cicero's remark, and whether it is relevant to women throughout Spartan history, or only to the Hellenistic period. Earlier, they probably bore children ungrudgingly, and gladly contributed their sons to the military, as tradition preserved, for example in Plutarch, _Sayings of Spartan Women_ , asserts. Cicero may record disillusionment with the Spartan female ideal in the Hellenistic period, when women had become aware of other ways of life.\n\nThe drastic reduction in the amount of territory controlled by Sparta after the liberation ofMessenia in 369 B.C.E. inspires some Malthusian speculations. Since the Spartan economy was based on agriculture, they will not have wanted to reproduce at the same rate as before the defeat at Leuctra, because if they did so after the loss of such a major resource they would reduce the economic status of individuals. Though their population did not fall immediately after Leuctra, despite the casualties, once the realities of the new limited territory had set in, Spartans may have adopted more strict family limitation policies.\n\nMore consistent with Plutarch's account of women's cooperation in child production is Xenophon's observation that women agreed to take part in husband-doubling or wife-sharing arrangements, not as passive agents, but because of their own ambitions to dominate two oikoi and to improve the economic status of their children. These novel arrangements may have encouraged women to bear more children than they would have done in conventional monogamous families, since they could distribute them in two oikoi. In any case, if we suppose that Xenophon's comment that wives willingly participated in these arrangements contradicts Cicero's report that the women refused to bear children, we must reject one of these reports, or hypothesize that some women participated while others refused, or posit change over time, or speculate that Xenophon and Cicero had different purposes in drawing attention to these issues.\n\nThroughout Spartan history there must have been some women who did not reproduce. Among any group of women or of couples, some will be infertile for all or some portion of their adult lives. There is the case of Anaxandridas,who resorted to bigamy rather than adultery or divorce, because his first wife had not produced an heir. After he married a second wife, the first wife bore children. Among nonroyals, as we have mentioned, the secret marriage described by Plutarch, in which the bridegroom makes stealthy visits to the bride, or divorce and remarriage would allow them to sample partners with whom they could have fertile unions. In addition, some women who may have been fertile were in marital situations that would not allow them to bear children because their husbands rejected them. Others remained spinsters. For example, the sisters of cowards could not find bridegrooms. Daughters of wealthy men were more in demand than daughters of the poor. The fianc\u00e9s of Lysander's daughters broke their engagements when they discovered that their prospective father-in-law was poor (Plut. _Lys._ 30.5).\n\n### _Womanpower and Oliganthropia_\n\nAnalyses of the population decline at Sparta make repeated references to \"manpower\" and to the numbers of males of all ages at various dates, but, with minor exceptions, they ignore women's reproductive failures or successes and their numbers relative to the number of males.Yet there is no dearth of information. Surprisingly, the ancient sources give more direct data and commentary on reproduction and demography for Sparta than for other Greek communities.\n\nOur knowledge of reproduction depends chiefly on Xenophon, Aristotle, Polybius, Cicero, and Plutarch. In the first sentence of the _Spartan Constitution_ , Xenophon mentions _oliganthropia_ (sparse citizen population), and in the next paragraph he discusses _teknopoiia_ (child production), highlighting the role of women in this process (see chap. 1). This emphasis on women seems eminently sensible from a modern viewpoint, but considering the alternative theories of reproduction that were current in Xenophon's day\u2014theories that minimized women's contribution to the embryo\u2014Xenophon shows unusual insight.\n\nAristotle refers to women when he states that half of Spartan society was ungoverned. He sees a connection between women's ownership of two-thirds of the land and oliganthropia. _Anthropoi_ may refer to either men or women, or both, and Aristotle could have used _oligandria_ or a similar term if he had wanted to specify males. Yet the context indicates that Aristotle, more than Xenophon, considers oliganthropia as a lack of male citizens ( _Pol._ 1270a33\u201334). With the exception of Ludwig Ziehen, scholars generally follow Aristotle in this interpretation. Studies of the decline in Spartan manpower naturally draw attention to losses inflicted by a devastating earthquake around 464: Ziehen speculated about the number of women lost in this calamity.He argues that more women than men were killed because they were indoors, and points out that only five houses were reported to have withstood the earthquake. No ancient source, however, reports that the earthquake was more catastrophic for women than for men. Ziehen is not convincing for several reasons: Spartan women were often out of doors and may not have suffered when the roofs of their houses caved in, and, secondly, from what we can deduce about Spartan architecture, their domestic quarters were so flimsy that a robust adult could have survived a collapse.\n\n### Demographic Speculations\n\nThe sources report the number of male citizens present at various battles as well as the number of male citizens in general at various points in the classical and Hellenistic periods, but, in general, numerical data from antiquity is unreliable. We have no information either on the absolute number of female Spartans or on the number relative to the number of males.We may, however, speculate that, as a consequence of selective male infanticide and the death of men in battle, women outnumbered men fairly frequently, and that the sex ratio was further skewed by male emigration after the Peloponnesian War. Moreover, since Spartans lived a more healthy life than other Greek women, ate well, exercised out of doors, did not become pregnant before their bodies were ready for childbearing, and, at least by the Hellenistic period if not earlier, limited the number of children they bore, they probably lived longer than other Greek women and had a better life expectancy than Spartan men. While it is true that relatively few grave markers have been found in Sparta because only men who died in war and women who died in childbirth were awarded this distinction, men who died might be buried at the battlefield, as were the heroes who died at Thermopylae. In contrast, all the women whose deaths were the result of childbirth would have been commemorated at Sparta. Though it is an argument _ex silentio_ , it is worth pointing out that few such inscriptions have been discovered.Xenophobia, lack of foreign trade and of an overseas empire, and not living in crowded urban spaces (as their Athenian counterparts did) will, in general, have had a beneficial effect on Spartan health. The men, however, traveled to foreign soil and were exposed to disease. These factors must have contributed to the large number of heiresses, or brotherless orphans, noted by Aristotle. There would have been a substantial number of widows, as well. A consequence of the unbalanced sex ratio was that not all adult women could be married simultaneously. Nevertheless, these excess women were not forced to become child producers. Moreover, because a Spartan woman shared in the paternal inheritance, she was not a burden to her family if she remained unwed. Since her portion of the patrimony would revert to her family of origin when she died, an unmarried daughter or sister would have been a financial, if not a social, asset. Perhaps Cicero included unmarried women when he stated that the women did not want to bear children.\n\nThirty years ago, without resorting to statistics and cliometrics, W. K. Lacey wrote in the style of Jane Austen:\n\nOne feature which seems common to all societies is that rich men want to marry their daughters (especially only daughters) to the sons of rich men, and the Spartans seem to have been no exception. A shortage of eligible men leads to a competition for husbands. . . . Rich women, moreover, do not commonly bear large families, especially when, as in Sparta, they are independent and not subordinated to their husbands.\n\nLacey, however, fails to draw the conclusion that a shortage of eligible men would leave some Spartan women husbandless, and hence childless.\n\n### Oikos and Polis\n\nPrivate and public spheres were interlocking. The mother constituted a bridge between private and public. The state determined the character of the family; the reverse was true, as well. For example, women were expected to scold and humiliate cowards and bachelors. They had to be informed about public events to know which men deserved this treatment.\n\nAristotle saw that the polis was constituted of oikoi, but he did not distinguish the different sorts of oikoi that must exist for the creation of different sorts of poleis. According to Aristotle ( _Pol._ 1252b10\u201312), the essential elements in the oikos were father, mother, child, and slave. Because the father alone of the members possessed a fully functioning deliberative faculty, he was the authority figure in all three relationships. Though appropriate to Athens, and doubtless to most other places in the Greek world, this model does not fit Sparta neatly throughout its history. In husband-doubling arrangements, the role of mother was distinct from the role of wife, and the role of father was distinct from the role of husband. According to the _Sayings of Spartan Women_ , the women dominated their sons in certain settings, and according to Aristotle ( _Pol._ 1269b12\u20131270a6), they even ruled their husbands. Xenophon ( _Lac. Pol._ 1.9) reports that they managed the oikos, sometimes even more than one. Furthermore the major portion of the productive,or \"slave,\"element of the Aristotelian formula was owned by the state, not by the individual oikos, and, for the most part, lived some distance from it (see chap. 5). All these differences between Spartan oikoi and those that at Athens contributed to creating a polis that was different from the Athenian polis.\n\nThere were also many elements of similarity between both poleis. Reproduction was universally the goal of marriage. At Athens, the marriage formula was explicit: the bride was given to the groom for the sowing of legitimate children. Yet we do not know of any Spartan politician who ever, like Pericles, directed mothers who had just lost adult sons in war to produce more children. Nor do we hear of any situation parallel to that at Athens during the second half of the Peloponnesian War, where a married man who already had children was encouraged to sire additional children by another citizen woman. In one of the examples that is cited of Socrates and Myrto, a dowerless daughter of Aristides, such a liaison could only have produced financial and personal misery for all concerned. As we have mentioned, in facing the population decline, the Spartans chose to reproduce by helot women rather than press unwed female Spartans into service.\n\n### Change over Time\n\nThe degree and quality of the differences between the Spartan family and others in the Greek world changed over time. Scholarly interpretation has shifted as well. Previous generations of scholars, relying on Plutarch's report and influenced by the mirage, have emphasized differences. In the 1990s, Xenophon's testimony gained authority.\n\nMany features of the Spartan family\u2014especially as described by Plutarch\u2014either were in existence only in the archaic period, or were part of the mirage. This family owed its origin not principally to biology, but to human legislation. In this fictive family, the men were like \"brothers\" with equal economic resources. They disciplined younger males and were in turn respected by them as fathers. Relationships between older and younger men were governed by the same incest taboos as if they really had been blood relatives (Plut. _Lyc_. 6.2, 8.4, 15.2; Xen. _Lac_. _Pol_. 2.10,13). This is also the community in which state officials exercised the prerogatives of fathers elsewhere in Greece, deciding which infants to rear and how to educate them (Xen. _Lac_. _Pol_. 2.2, 10).Women were excluded from managing the economic resources of the state and participating in the administrative processes of this public family, but that was true of all women (with the exception of some Hellenistic queens) in antiquity. Interestingly enough, just after Plutarch's lifetime, Roman Sparta honored some elite women by granting them titles that certified inclusion in the civic family. These titles included \"daughter of the city\" and \"mother of the Boule and the people.\"\n\nSimone de Beauvoir envisioned the life of Spartan women as utopian:\n\nSince the oppression of woman has its cause in the will to perpetuate the family and to keep the patrimony intact, woman escapes complete dependency to the degree in which she escapes from the family. If a society that forbids private property also rejects the family, the lot of women in it is found to be considerably ameliorated. In Sparta the communal regime was in force, and it was the only Greek city in which woman was treated almost on an equality with man. The girls were raised like the boys. The wife was not confined in her husband's domicile: indeed, he was allowed to visit her only furtively, by night; and his wife was so little his property that on eugenic grounds another man could demand union with her. The very idea of adultery disappeared when the patrimony disappeared. All children belonged in common to the city as a whole, and women were no longer jealously enslaved to one master; or, inversely, one may say that the citizen, possessing neither private wealth nor specific ancestry, was no longer in possession of woman. Women underwent the servitude of maternity as did men the servitude of war; but beyond the fulfilling of this civic duty, no restraint was put on their freedom.\n\nNowadays feminist theory has encouraged historians to shift from a narrow focus on the androgynous educational system to an appreciation of Spartan motherhood and to criticism of societies based on Communist or totalitarian programs. Perhaps in reaction to earlier views, the current scholarly consensus is more nuanced and less influenced by information emanating from the mirage.\n\nBy the end of the fifth century, if not earlier, the Spartans did not even pay lip service to the all-male public family. The freedom to give kleros and house to anyone, or bequeath them by testament, signaled state acceptance of the private family and its fluctuating economy at the expense of the public family of men of equal status (Plut. _Agis_ 5.1, and see chap. 4).\n\nAt Sparta, the patriarchal state was gradually supplanted by the nuclear family, which, however, was not dominated by men to the same extent as at Athens. In the fourth century, if not slightly earlier, the family at Sparta (like the family elsewhere in the Greek world) had its own economic base and reproductive agenda. It is clear that with more property openly in private hands, more Spartans, including women, had increased incentives to develop inheritance strategies and to regulate their fertility.\n\nIn their discussions of reproduction, Xenophon,Cicero, and to some extent Aristotle emphasize women's influence and volition. Xenophon's report on wife-lending or husband-doubling indicates that both parents concurred in limiting the size of the family so as not to reduce the share of the inheritance that would eventually be distributed to their heirs. As wives and mothers, women were active players in making arrangements beneficial to their families and to themselves.\n\n* * *\n\n. On the oikos and family, see further Sarah B. Pomeroy, _Xenophon, Oeconomicus: A Social and Historical Commentary_ (Oxford, 1994), passim, and _Families in Classical and Hellenistic Greece_ (Oxford, 1997), passim. I presented this material first in \"The Spartan Family\" (paper delivered at the University of Cambridge, Oct. 13, 1998), and much of the discussion of reproduction in \"Spartan Wives and their Strategies\" (paper delivered at the Celtic Conference in Classics, National University of Ireland, Maynooth, Sept. 7, 2000).\n\n. See most recently Valerie French,\"The Spartan Family and the Spartan Decline,\" in _Polis and Polemos: Essays on Politics, War, and History in Ancient Greece in Honor of Donald Kagan_ , ed. C. D. Hamilton and P. Krentz (Claremont, Calif., 1997), 241\u201374.\n\n. Plut. _Lyc_. 27.2, _Mor_. 238d, Paus. 3.12.8, 3.14.1, 3.16.6, 6.1.9. The reference to childbirth depends upon Latte's emendation of Plut. _Lyc_. 27.2. See W. K. Pritchett, _The Greek State at War_ , vol. 4 (Berkeley, 1985), 244 n. 430, and most recently M. Toher,\"On the _Eidolon_ of a Spartan King,\" _RhM_ n.s. 142 (1999), 113\u201327, esp. 122\u201326. W. den Boer, _Laconian Studies_ (Amsterdam, 1954), 299\u2013300, favors a Greek text that means: \"except a man fallen in war or a woman if they were _hieroi_ [priests or priestesses].\" N. Richer,\"Aspects des funerailles \u00e0 Sparte,\" _Cahiers du Centre G. Glotz_ , 5 (1994), 51\u201396, revives the criticism raised by C. Le Roy, \"Lakonika,\" _BCH_ 85 (1961), 228\u201332, esp. 231 n. 4, of Latte's emendation as \"violente,\" and argues that only priests who had died in battle and women who were members of the hieroi (i.e., priestesses) were awarded epitaphs. This argument is unconvincing, however, in view of the inscriptions commemorating women who died _en lecho_ ( _IG_ V.1.713, 714; from Geronthrai _IG_ V.1128; from Hippola _IG_ V.1277), and the total lack of such inscriptions commemorating priestesses. See D. H. Kelly, \"Thucydides and Herodotus on the Pitanate _Lochos_ ,\" _GRBS_ 22 (1980), 31\u201338, esp. 33\u201334, for a convincing rejection of this awkward Greek. Richer does not refer to Kelly's arguments. Margherita Guarducci, _Epigrafia greca_ , vol. 2 (Rome, 1967\u201368), 173 n. 1, considers the women's epitaphs pre-Hellenistic. See further Mario Manfredini and Luigi Piccirilli, _Plutarco: Le vite di Licurgo e di Numa_ , 2d edn.(Milan, 1990), 94 (app. crit.) and 276\u201377.\n\n. See further Barton Lee Kunstler,\"Women and the Development of the Spartan Polis\" (Ph.D. diss., Boston University, 1983), 459.\n\n. _Lac. Pol._ 1.4, and see chap. 1. Arist. _HA_ 608b declares that the female eats less. Note that in Persepolis as part of a pronatalistic policy nursing women received extra rations: see R. T. Hallock, _The Persepolis Fortification Tablets_ (Chicago, 1969), 344\u201353;Maria Brosius, _Women in Ancient Persia, 559\u2013331 B.C._ (Oxford, 1996), 171\u201378; and Sarah B. Pomeroy, _Goddesses, Whores, Wives, and Slaves: Women in Classical Antiquity_ (New York, 1995), 85.\n\n. Xen. _Lac. Pol._ 2.6, 7.8, describes the effort to keep boys and men slender, doubtless due to the warrior image. See also Plut. _Lyc._ 17.4.\n\n. R. W. V. Catling, \"The Archaic and Classical Pottery,\" in _Continuity and Change in a Greek Rural Landscape: The Laconia Survey_ , vol. 2, _Archaeological Data_ , by W. G. Cavanagh, J. Crouwel, R.W. V. Catling, and G. Shipley ( _ABSA_ suppl. vol. 27) (London, 1996), 33\u201389, esp. 77, suggests that the peculiar shapes of Laconian pots reflect a diet different from that of most Greeks.\n\n. Lin Foxhall and H. A. Forbes, \" _Sitometreia_ : The Role of Grain as a Staple Food in Classical Antiquity,\" _Chiron_ 12 (1982), 41\u201390, esp. 59.\n\n. Foxhall and Forbes,\" _Sitometreia_ ,\" 49.\n\n. _Lys_. 80\u201384, and see further Conclusion.\n\n. See further Thomas J. Figueira,\"Mess Contributions and Subsistence at Sparta,\" _TAPA_ 114 (1984), 87\u2013109, esp. 98\u2013100.\n\n. Natalie Anger,\"Chemical Tied to Fat Control Could Help Trigger Puberty,\" _New York Times_ , Jan. 7, 1997, sec. C, pp. 1, 3, reports studies demonstrating that fat is necessary for reproductive maturity and athletic women with little body fat experience late puberty.\n\n. We observe that Athenian vases depict brides as plump: see John H. Oakley and Rebecca H. Sinos, _The Wedding in Ancient Athens_ (Madison,Wis., 1993), passim. In the Eleusinian votive relief depicting Demeter, Triptolemus, and Kore, the mother is thinner than her daughter (Athens, National Museum, no. 126 and New York,Metropolitan Museum of Art, acc. no. 14.130.9 A\u2013K [Rogers Fund, 1914],Roman copy).\n\n. See P. Cartledge, \"The Politics of Spartan Pederasty,\" _PCPS_ n.s. 27 (1981), 17\u201336, and F. D. Harvey, \"Laconica: Aristophanes and the Spartans,\" in _The Shadow of Sparta_ , ed. A. Powell and S. Hodkinson (London, 1994), 35\u201358, esp. 41\u201342.\n\n. _Lys_. 78\u201385.\n\n. _Lys_. 1105, 1148, 1162, 1173\u201374.\n\n. Xen, _Lac_. _Pol_. 1.5, Plut. _Lyc._ 15.5.\n\n. Aelian, _VH_ 6.6, also states that the father of five sons could marry his daughters without dowry (see on dowries chap. 4). Perhaps such daughters had found their future husbands in the \"lottery\" in the dark room, and the men were obliged to marry them, or the state awarded dowries to them. See further chap. 2 n. 33.\n\n. Among the Roman upper classes, childbearing became so unpopular that Augustus decided to offer incentives to those who became parents and penalize those who remained childless: see further Suzanne Dixon, _The Roman Mother_ (Norman, Okla., 1988), 71\u2013103.\n\n. Arist. _Pol_. 1270b1\u20134, and see further Thomas Figueira,\"Population Patterns in Late Archaic and Classical Sparta,\" _TAPA_ 116 (1968), 182 n. 43.\n\n. S. Perentidis,\"R\u00e9flexions sur la polyandrie \u00e0 Sparte dans l'Antiquit\u00e9,\" _RHD_ 75 (1997), 7\u201331, esp. 22\u201325, for sensible speculations about the domicile of the Spartan couple.\n\n. The ages may be inferred from Plutarch's allusion ( _Lyc._ 15.3) to the bride's maturity and from Plato, _Rep._ 5.452, 5.460E, who states that females should become parents for the first time at twenty and males at thirty.\n\n. Sarah B. Pomeroy, _Women in Hellenistic Egypt: From Alexander to Cleopatra_ (New York, 1984), 17; Sarah Blaffer Hrdy, _Mother Nature_ (New York, 1999), 248, 252.\n\n. Thus Kunstler, \"Family Dynamics and Female Power in Ancient Sparta,\" in _Rescuing Creusa_ , ed. M. Skinner, _Helios_ 13.2 (Lubbock, Tex., 1986), 31\u201348, esp. 458\u201359, and see chap. 1.\n\n. Herod. 5.50\u201353, Plut. _Sayings of Spartan Women,_ 240d.\n\n. See chap. 1 n. 16.\n\n. Plut. _Agis_ 11.5, 17\u201318.1, _Cleom._ 18, _Sayings of Spartan Women,_ 1, and see chap. 4.\n\n. Herod. 6.58, and see further Pomeroy, _Families_ , 50\u201351, and chap. 5 below.\n\n. Nepos, _Paus_. 5.3,Diodorus 11.45.6, Polyaenus 8.51. Poralla2, 16\u201317, no. 55, calls her Alcathoa. Theano may have been her \"professional\" name, for it was often used by priestesses: see Blaise Nagy, \"The Naming of Athenian Girls: A Case in Point,\" _CJ_ 74 (1979), 60\u201364, and Pomeroy, _Families,_ 157. Thucydides does not tell the tale of Theano and Pausanias. Omitting women's personal involvement is typical of Thucydides. We need not conclude that the story was a Hellenistic invention. Nepos, Diodorus, and Polyaenus may have read it in some other classical source less fastidious than Thucydides.\n\n. Perhaps catering to the Roman taste for blood, mothers and fathers witnessed the flogging of the boys at the altar of Artemis Orthia, encouraging them to endure the pain.\n\n. See Appendix. Though there are very few datable women, we do observe the obvious absence of any in Aristotle's period\u2014the time of _anesis_ (\"licence,\" _Pol._ 1269b\u20131270a).\n\n. Cf. Tac. _Germ_. 8 for the presence of the warrior's family on the battlefield as a goad to courage.\n\n. _Sayings of Spartan Women._ 240.f2, 241.1, 3, 5, 12; cf. 241.11.\n\n. Sim. Plut. _Lyc._ 14.8. Annalisa Paradiso, \"Gorgo, la Spartana,\" in _Grecia al femminile_ , ed. Nicole Loraux (Bari, 1993), 107\u201322, emphasizes motherhood as the chief feature through which Spartan women gained power.\n\n. Bella Zweig, \"The Only Women Who Give Birth to Men: A Gynocentric, Cross-Cultural View of Women in Ancient Sparta,\" in _Women's Power, Man's Game_ , ed. Mary DeForest (Wauconda, Ill., 1993), 32\u201353, has drawn attention to the positive aspects of motherhood at Sparta.\n\n. _The Gleaner_ (1798, repr. Schenectady, N.Y., 1992), 706.\n\n. Page Gilpin Faust, _Mothers of Invention_ (Chapel Hill, N.C., 1996), 14\u201315, 17.\n\n. Faust, _Mothers of Invention,_ 15\u201316.\n\n. On the biological and social aspects of the maternal \"instinct\"in primates, see Hrdy, _Mother Nature_.\n\n. The concepts of \"late marriage\" and \"bad marriage\" seem to apply only to men. See Stobaeus 22,\"On Marriage,\" 1.16, and Pollux 3.40, 48 (Bethe).\n\n. See n. 3.\n\n. For modern parallels, see Elaine Tyler May, _Barren in the Promised Land: Childless Americans and the Pursuit of Happiness_ (New York, 1995), esp. 71, 196\u201399.\n\n. Thuc. 2.44.3, and see further Pomeroy, _Families_ , 39.\n\n. Hippocrates, _On Generation_ , 5.1, states that a woman's volition controls whether or not she conceives.\n\n. _IG_ IV.2.121\u201322 = E. J. Edelstein and L. Edelstein, _Asclepius: A Collection and Interpretation of the Testimonies_ , vol. 1 (Baltimore,Md., 1945), 423, stele II, para. 21. The cure by proxy was unusual.\n\n. A sanctuary of Zeus Messapeus at Tsakona, 4 km north of Sparta, has recently been identified. The finds, including numerous nonidentical archaic ithyphallic terracotta votives and a smaller number of figurines of women, some of whom are pregnant, a larger number with legs spread to expose genitalia, suggest that the divinity was worshipped in connection with the Spartan concern for human fertility. The shrine was active until the early Hellenistic period with some resumption of interest in the 3d\u20134th cent. C.E. See H. W. Catling,\"Excavations at the Menelaion, 1985,\" _Lak. Spoud._ 8 (1986), 205\u201316, esp. 211\u201312 and fig. 13; Catling,\"A Sanctuary of Zeus Messapeus: Excavations at Aphyssou, Tsakona, 1989,\" _ABSA_ 85 (1990), 15\u201335, esp. 21, 34;Catling, \"The Work of the British School at Athens at Sparta and in Laconia,\" in _Sparta in Laconia: Proceedings of the 19th British Museum Classical Colloquium_ , ed. W. G. Cavanagh and S. E. C. Walker, British School at Athens Studies, vol. 4 (London, 1998), 27.M. Overbeek,\"The Small Finds,\" in _Continuity and Change in a Rural Landscape: The Laconia Survey_ ,vol. 2, _Archaeological Data_ ,by W.G.Cavanagh et al., _ABSA_ suppl. vol. 27 (London, 1996), 183\u201398, esp. 190, mentions 15 female figurines: the most complete ones have arms and legs akimbo and genitals exposed, with no indication of breasts, suggesting a lack of emphasis on nurtu-rance. See Catling, in a postscript on Tsakona in David Blackman, \"Archaeology in Greece, 1999\u20132000: Lakonia,\" _AR_ 46 (2000), 38\u201343, esp. 43,\n\n. Thus the Bud\u00e9, ad loc. M. Pohlenz, _Ciceronis tusculanorum disputationum: Libri V_ (Stuttgart, 1957), 158, conjectures that the source is Accius, _Meleager._\n\n. For socioeconomic theories of reproduction, see, e.g., R. M. McInnis, \"Childbearing and Land Availability: Some Evidence from Individual Household Data,\" in _Population Patterns in the Past_ , ed. R. D. Lee (New York, 1977), 201\u201327.\n\n. G. Shipley, \"The Extent of Spartan Territory in the Late Classical and Hellenistic Periods,\" _ABSA_ 95 (2000), 367\u201390, esp. 385,points out that although Sparta lost most of Messenia after Leuctra, the southern part with its settlements and ports were retained.\n\n. Herod. 5.39\u201341.\n\n. _Lyc_. 15.3, sim. Athen. 13.555c, citing Hermippus (fl. late 3d cent.).\n\n. Xen. _Lac_. _Pol_. 9.5.\n\n. The material in this section was presented in Sarah B. Pomeroy, \"Spartan Womanpower,\" paper delivered at the Annual Meeting of the American Philological Association, Dec. 28, 1998; abstract published in _American Philological Association 130th Annual Meeting: Abstracts_ , 81.\n\n. The late David Asheri, in a personal communication.\n\n. Plut. _Cim._ 16.4\u20135: _neaniskoi_ (youths) survived, but the ephebes (approximately eighteen to twenty years old) died when a gymnasium was destroyed.\n\n. See L. Ziehen, \"Das spartanische Bev\u00f6lkerungsproblem,\" _Hermes_ 68 (1933), 218\u201337, esp. 232\u201335, 237. Stephen Hodkinson, \"Inheritance, Marriage and Demography: Perspectives upon the Success and Decline of Classical Sparta,\" in _Classical Sparta: Techniques Behind Her Success_ , ed. Anton Powell (Norman, Okla., 1989), 79\u2013121, esp. 103, follows Ziehen.\n\n. Polyaen 1.41.3; Ael. _VH_ 6.7.2;Diod. 11.63.\n\n. Note the 240 adolescent girls in Theocritus, _Idyll,_ 18.25.\n\n. See n. 3.\n\n. _The Family in Classical Greece_ (Ithaca, N.Y., 1968), 205. S. J. Hodkinson, \"Inheritance, Marriage, and Demography,\" uses cliometrics and a utilitarian view of women to come to the same conclusion. He also (106\u20139) speculates about the detrimental effects of incest on the Spartan population, but withdraws this hypothesis in Hodkinson, _Property and Wealth in Classical Sparta_ (London, 2000), 444 n. 23.\n\n. See further Pomeroy, _Goddesses, Whores, Wives, and Slaves,_ 66\u201367.\n\n. See Appendix n. 25.\n\n. Perhaps the references to other girls as \"cousins\" in Alcman, _Partheneion_ 1.52, are fictitious.\n\n. A. J. S. Spawforth, \"Families at Roman Sparta and Epidaurus: Some Prosopographical Notes,\" _ABSA_ 80 (1985), 241, and see chap. 6 below.\n\n. _The Second Sex_ (New York, 1952), 82.\n\n##\n\n## ELITE WOMEN\n\n### Royal Women\n\nMore specific information is available about royal women than about any other group. It is clear, however, that a historian must not generalize from information about the female kin of kings to women of lesser social status. For example, because of concern about the legitimacy of the succession, only royal women are named in tales of adultery, but the ancient sources often do not distinguish between royal and upper-class women. Many Spartiates had Heraclid blood. Aristotle writes about wealthy women, but only Plutarch gives details about particular wealthy women and they are all members of the royal houses (see below). Furthermore, the Spartiates were an elite, an aristocracy, and women of the upper classes tend to imitate the most elite among them.\n\n### _Marriage_\n\nSocial and economic endogamy was characteristic of royal marriage. This pattern of selection of spouses was also common elsewhere in the Greek world, with uncle-niece and first-cousin marriage being common patterns. Endogamy is a relative concept. Since the Spartan population was small and xenophobic, it could be characterized as endogamous in general. There were never more than 10,000 _homoioi_ (men of equal status or similars) or, consequently, more than 500 elite. Marriages among the wealthy accelerated the concentration of wealth in fewer and fewer families, despite the presence of countervailing forces. The problem of finding a suitable spouse was intensified in the case of the kings: the small pool of eligible brides included members of the royal families and daughters or widows of wealthy or influential men, or in the case of Ariston, who at first had two barren wives, the extremely beautiful wife of a friend (Herod. 6.61 and Conclusion). Like Ariston, Demaratus married a woman who was claimed by another man. He carried off Percalus, the fianc\u00e9e of Leotychidas II, who was his enemy, kinsman, and future successor, and married her himself (Herod. 6.65.2). Examples of royal endogamy are plentiful. The first wife of King Anaxandridas was the daughter of his sister (Herod. 5.39). Leonidas married his step-niece Gorgo, daughter of Cleomenes I (Herod. 7.205.1). King Archidamus V married the daughter of his cousin Hippomedon (Polyb. 4.35.13). Lampito, daughter of Leotychidas II,married his grandson, Archidamus II (Herod. 6.71.2). Sometimes a king chose to marry a daughter of a wealthy or influential man. An heiress from a royal family was certainly desirable. The elderly Cleomenes married Chilonis, a young Eurypontid heiress. Agiatis, who inherited the great wealth of her father Gylippus, was married first to Agis IV and, when widowed, to Cleomenes III (Plut. _Cleom_. 1.2). Eugenics also influenced spouse selection. The Ephors fined Archidamus for marrying a short wife, for she would bear short kings (Plut. _Ages_. 2 ).\n\n### _The Succession_\n\nOne of the earliest female members of the royal house about whom a story is told is Argeia, a Theban who was married to king Aristodemus, probably in the mid-tenth century B.C.E. After she bore twin sons, her husband died. She would not disclose which son was the elder because she wanted both to rule as equals. Nevertheless, she revealed that Eurysthenes was older than Procles by bathing and feeding him first. Of course, any story about the tenth century must be treated with caution. Nevertheless, it is conceivable that in those early days or even when the story was first told, a mother, even a queen, looked after her babies herself, or at least was on hand to supervise servants who did the actual work.\n\n### _Adultery and Bigamy_\n\nIn the _Life of Lycurgus_ (15.10, sim. _Mor._ 228b20), Plutarch passes on the proverbial notion that adultery was unknown at Sparta, but in other texts he tells of extramarital liaisons, mostly in the form of anecdotes about individual royal women.\n\nSuch adulterous affairs are recorded because of concern over the legitimacy of the heir. Like Spartan men, the women, evidently, were thought to be capable of duplicity concerning affairs of state. Ephors watched the first wife of Anaxandridas give birth, since she had not been able to conceive until he took a second wife (Herod. 5 .41). They suspected that she would sneak in a suppositious child, but she confounded them by giving birth to male triplets. Timaea, wife of Agis II, was accused of having had a secret affair with Alcibiades and of conceiving Leotychidas by him. Herodotus (5.40) comments that bigamy was \"not Spartan.\" The Agiad Anaxandridas, however, resorted to bigamy rather than adultery because he had not produced any children with his first wife, and did not want to divorce her since he was fond of her. The Eurypontid Ariston practiced bigamy, if not trigamy (Herod. 6.61\u201363). His third wife, whom, as we have noted, he took away from a friend, bore a son in less than nine months (Paus. 3.7.7). The legitimacy of this son, Demaratus, and his right to rule were questioned (Herod. 6.65\u201369).Demaratus himself confronted his mother and accused her of having had an affair with a stablehand (Herod. 6.68). (We have already observed [chap. 1] that Spartan women were interested in horses.) His mother replied that either Ariston or the legendary hero Astrobacus, whose shrine was in the courtyard, had engendered him. Of course, royal families are subject to greater constraints than commoners because of the need to have an heir, and one who was a blood heir for the kingship to be justified.\n\nChilonis, a Eurypontid who was the wife of Cleonymus, was involved in an adulterous liaison with Acrotatus, son of Areus I, by 272 when her elderly husband had left to persuade Pyrrhus to attack Sparta (Plut. _Pyrr_.26.15\u201329.12 ). It was said that everyone in Sparta knew how she felt about her husband, but she was not ostracized, probably because her husband was utterly despicable, while her lover defended Sparta against Pyrrhus. Plutarch portrays her as a heroine who threatened suicide if Cleonymus and Pyrrhus were victorious. That her equally heroic granddaughter was named after her indicates that Chilonis did not suffer _damnatio memoriae_.\n\nWe know of no penalty for adultery, nothing to compare with the Athenian laws requiring the husband to divorce a wife who had been raped or seduced, and prohibiting the adulterous woman from wearing jewelry and attending religious ceremonies. Perhaps the task of proving adultery at Sparta was so daunting that it was practically unpunishable. At Sparta, the only consequence of adultery apparently was an aspersion that political opponents of a potential king might cast upon his legitimacy.\n\n### _Authority_\n\nThe wives and daughters of Spartan kings could not be styled \"queens\" and \"princesses,\"for they had no special role to play in society or religion.\"Queen,\"at least as a title used by non-Spartans of one woman, emerged only in the late Hellenistic period. Some of the royal women at Sparta did, however, wield a great deal of authority because of their influence on the kings. There was a long tradition of the involvement of women in politics, beginning with the child Gorgo, who advised her father the king about how he should treat a foreign ambassador (Herod. 5.51, 7.239). Her advice shows that she understood well the Spartan policy of avoidance of strangers ( _xenelasia_ ). Women's influence was most apparent in Hellenistic dynastic politics as it was in much of the Greek world at that time. The women were often widows and older than the men they swayed. In the middle of the third century, when Agis IV promulgated his reforms, Agesistrata  was in her late fifties and Archidamia in her early eighties (see below). It is rare in Greek history to find a grandmother and mother both not only alive, but active, at such advanced ages.Cratesicleia remarried when old to give her son a stepfather as an ally (Plut. _Cleom._ 22.4). Agiatis was older than Cleomenes III when she converted him to the revolutionary program of her former husband Agis IV. Xenopeithia was the mother of Lysanoridas, who was a commander at Thebes. His father is not named; therefore we deduce that she was a widow, and important enough for her son's political enemies to want to murder her and her sister Chryse.  Cynisca was around sixty years old when she defied her brother, who was then king (see chap. 1).\n\n### Wealth\n\nThe wealth of Spartan women was fabled before reliable evidence for it appears in historical times. In Homer, Helen spins with a golden distaff, brings a hoard of valuable clothing to Troy, and obtains even more when there ( _Od_. 4.131\u201335, _Il._ 6.289\u201392). The young girls in Alcman's oeuvre allude to valuable jewelry and objects, including purple garments, golden mitres and bracelets, perfume, silver, ivory, and racehorses. Euripides ( _Andr._ 147\u201353) refers to the golden ornaments Helen wore in her hair, her embroidered dresses, and her wedding presents.\n\nArchaeological evidence confirms the poets' reports. Gold and silver jewelry, mostly dating from the seventh century, was dedicated at the sanctuary of Artemis Orthia.  The pieces include fibulae, pendants, and beads (see further chap. 6).\n\nSometime after the Second Messenian War, when an austere discipline was imposed, the use of gold and silver was forbidden. We do not know what happened to women's valuable possessions when precious metals were outlawed and houses were searched to make certain that none were hidden within (Xen. _Lac_. _Pol_. 7.6, 14.3 ). Perhaps they were dedicated to the gods. Thereafter, women were not permitted to wear ornaments or gold. Until the Hellenistic period, Sparta did not have coined money, or, consequently, a cash economy. When King Polydorus died and the Spartans purchased his house from his widow, they payed with oxen (eighth century B.C.E.: Paus. 12.3). Of course this purchase predated the invention of coinage, but even though foreign coins became available, such awkward transactions must have continued at Sparta long after other Greeks were using coins (see chap. 5 n. 9).\n\n### _Land Tenure_\n\nLand was the most valuable commodity in the ancient world, and the land of Laconia was among the most fertile in Greece. The ownership of land connotes permanence. In Athens and some other Greek states, women were not permitted to own land or to manage substantial amounts of wealth. These limitations draw attention to the impermanence of Athenian women in the family and state and their lack of access to the most valuable economic resources. The dowries of Athenian women consisted of movables and money. They were like metics (resident aliens), who also could own only movables and money. The type of\n\nproperty women could hold was due not only to their low status, but also to the possibility that they might marry out of the polis. Moreover, Athenian women did not manage their own property; rather, the dowry passed from a woman's father to her husband. Unlike Spartan women, however, Athenians could produce wealth. The Athenian oikos was a locus of production as well as reproduction. Respectable citizen women contributed directly to the economic prosperity of the oikos by weaving and shrewd management. Poor or lower-class women made money, often by working outside the home, and slave women contributed to their owner's economic prosperity by working as domestics or in manufacture or in service industries such as prostitution. Against this background, Spartan women (like Spartan men) can be viewed not as versatile producers, but essentially as owners, managers, and consumers of wealth based on land (see chap. 3).\n\nThe system of land tenure at Sparta is poorly understood. Moreover, it changed drastically over time. Our knowledge of the laws governing the ownership of property at Sparta is uncertain, and in any case the laws may not reflect the actual historical situation at all times (Xen. _Lac. Pol._ 14.1, and see Appendix). In the following discussion, I will sketch a likely scenario and will concentrate in particular on the role women played in the evolving economy.\n\nThere were two systems of land tenure in Sparta, one private, the other public. The story just above about the house of the widow of Polydorus indicates that women owned property in Sparta from earliest times, and they continued to do so. The story also indicates that her property was private property.\n\nMuch of the land in Sparta was owned by the state. The participation of both men and women in this publicly owned land system changed over time. In archaic Greece, it was common for the founder of a colony to measure the territory and distribute it in equal shares to the male colonists. The Spartans, in effect, colonized their neighbors in Messenia and Laconia, though they themselves did not settle in the conquered territory and continued to live in the Spartan polis. The state distributed parcels of land to be used only for a man's lifetime.\n\nAccording to Plutarch ( _Lyc._ 8), Lycurgus distributed 9,000 lots (kleroi) of state-owned land in equal shares. The conquered people became helots and worked the land (see chap. 5). They owed a certain amount of produce to the Spartiate to whom the kleros had been assigned at birth (see chap. 3). Spartan women were not assigned kleroi, but received the benefits of the system through male relatives. The system was unstable for one obvious reason: the population fluctuated. By the time of Demaratus, there were 8,000 Spartans (Herod. 7.234).\n\nAfter the Spartan victory in the Peloponnesian War, the Lycurgan system regulating public property was abolished. Thenceforth, a man could give his kleros and his house to anyone he wished, or bequeath them by testament. Furthermore, at the end of the Peloponnesian War a large amount of gold and silver entered Sparta.Greed, which had been a vice attributed to Spartan men, was now seen in women as well. Expensive imported dresses were desired. Xenophon reports that wives agreed to produce children for men in addition to their husbands so that they might control two oikoi.  These changes decisively undermined the ideal of economic equality, and eventually led to the concentration of great wealth in the hands of a minority and the creation of an impoverished majority who no longer met the property requirements necessary to enjoy full citizenship.\n\nA parenthetical comparison with the evolution of the kleros system in Ptolemaic Egypt (which is better documented than the Spartan system) is illuminating. Ptolemy I established the kleros system by distributing parcels of agricultural land in the Fayum to entice veterans of Alexander's campaigns to settle in Egypt. Tenure of a kleros was tied to military service in behalf of the Ptolemies and to the payment of certain taxes. The holder of the kleros could lease it out, but not sell it. The king could revoke the grant, and he repossessed the kleros upon the death of the beneficiary, giving it to another potential soldier.A _stathmos_ (billet or dwelling place) was included in the grant. In the second half of the third century, as the strong central government declined and natives entered the military, cleruchs (holders of kleroi) regularly bequeathed billets to wives, sons, and daughters, and the kleroi became hereditary in the male line. Eventually, in the first century B.C.E., a brotherless girl inherited her father's kleros. Despite obvious differences, in both Sparta and Ptolemaic Egypt the kleros system aimed at creating a hereditary army which would always be ready and willing to go out and fight, inasmuch as sustenance from the land was constantly available for the soldier's family and for himself. In both places, the interests of the private family triumphed over the goals of the central government. The documentary papyri record the gradual evolution of the house and the land from public to private tenure, and the change from the inheritance of land exclusively by male kin to inheritance by both sexes. It is also apparent that, doubtless for practical reasons, women could inherit dwelling places before they were able to inherit kleroi. At Sparta, there are no comparable records documenting change over time; consequently it is not possible to determine whether the decree attributed to Epitadeus accomplished or acknowledged the transformation by one radical piece of legislation.\n\nThe aims of the state were at variance with the economic ambitions of individual families. Like other Greeks, Spartans practiced diverging devolution, but the Spartans alone attempted to counter the decline in economic status that large families would experience when the paternal estate was divided. Diverging devolution works well only in a capitalistic economy with ever-expanding resources. In a city like Athens, expansion was the result not only of intensification of agriculture, but also of trade, colonization, banking, manufacturing, and services. Sparta's economy, however, was based on agriculture, and property consisted of a finite amount of land and a servile population not augmented by purchase.\n\nThe liberation of part ofMessenia in 369 seriously undermined the Spartan economy as well. The Messenian helots and the income from the land they tilled were lost to the Spartans. Perhaps it was to some extent due to economic pressures that Spartans began to experiment with various unique methods of family planning (see chap. 3).\n\nAristotle criticized the Spartan system of land tenure, which permitted women to own land, manage their own property, and exercise authority in the family:\n\nAgain, the licence in the matter of their women is detrimental both to the chosen aim of the constitution and to the happiness of the state. For just as man and wife are part of a household, so clearly we should regard a state also as divided into two roughly equal bodies of people, one of men, one of women. So, in all constitutions in which the position of women is unsatisfactory, one half of the state must be regarded as unregulated by law. And that is just what has happened there. For the lawgiver, wishing the whole state to be hardy, makes his wish evident as far as the men are concerned, but has been wholly negligent in the case of the women. For being under no constraint whatever they live unconstrainedly, and in luxury.\n\nAn inevitable result under such a constitution is that esteem is given to wealth, particularly if they do in fact come to be female-dominated; and this is a common state of affairs in military and warlike races, though not among the Celts and any others who have openly accorded esteem to male homosexuality. Indeed, it seems that the first person to relate the myth did not lack some rational basis when he coupled Ares with Aphrodite; for all such people seem in thrall to sexual relations, either with males or with females. That is why this state of affairs prevailed among the Laconians, and in the days of their supremacy a great deal was managed by women. And yet what difference is there between women ruling and rulers ruled by women? The result is the same. Over-boldness is not useful for any routine business, but only, if at all, for war. Yet even to those purposes the Laconians'women were very harmful. This they demonstrated at the time of the invasion by the Thebans: they were not at all useful, as in other states, but caused more confusion than the enemy.\n\nSo it seems that from the earliest times licence in the matter of their women occurred among the Laconians, reasonably enough. For there were long periods when the men were absent from their own land because of the campaigns, when they were fighting the war against the Argives, or again the one against the Arcadians and Messenians. When they gained their leisure, they put themselves into the hands of their legislator in a state of preparedness brought about by the military life, which embraces many parts of virtue. People say that Lycurgus endeavoured to bring the women under the control of his laws, but that when they resisted he backed off. These then are the causes of what took place, and clearly, therefore, of this mistake as well. But the subject of our inquiry is not whom we ought to excuse and whom not, but what is correct and what is not.\n\nThe poorness of the arrangements concerning women seems, as was said earlier, not only to create a sort of unseemliness in the constitution in itself on its own, but also to contribute something to the greed for money; for after the points just made one could assail practice in respect of the uneven levels of property. For some of them have come to possess far too much, others very little indeed; and that is precisely why the land has fallen into the hands of a small number. This matter has been badly arranged through the laws too. For while he made it (and rightly made it) ignoble to buy and sell land already possessed, he left it open to anyone, if they wished, to give it away or bequeath it\u2014and yet the same result follows inevitably, both in this case and in the other. Moreover, something like two-fifths of all the land is possessed by women, both because of the many heiresses that appear, and because of the giving of large dowries. Now it would have been better if it had been arranged that there should be no dowry, or a small or even a moderate one. But as it is one may give an heiress in marriage to any person one wishes; and if a man dies intestate, the person he leaves as heir gives her to whom he likes. As a result, although the land was sufficient to support 1,500 cavalry and 30,000 heavy infantry, their number was not even 1,000. The sheer facts have shown that the provisions of this system served them badly; the state withstood not a single blow, but collapsed owing to the shortage of men.\n\n### _Luxury_\n\nAs we have seen, by Aristotle's time, women owned nearly two-fifths of the land of Laconia. Because the Spartan economy was based entirely upon agriculture, women controlled a significant portion of the means of production. The wealthiest women in mainland Greece were Spartans. They advertised their wealth flamboyantly by winning victories in horseraces at Olympia. One woman is named in a fragmentary inscription listing Spartans who had made monetary donations for the rebuilding of the temple of Apollo at Delphi. Philostratis is cited for giving three obols in 360. Only one man gave as little as she, and most gave at least twice that amount.As time went on, Macedonian queens and royal women (like Amastris) on the fringes of the Greek world rivaled and surpassed Spartan women in wealth, for, like the Romans, the former had access to many sources of revenue in their various empires, while the Spartans were limited to incomes derived from a polis economy based on agriculture.\n\nSpartan society as a whole had become stratified, with the majority of male Spartiates so impoverished that they could not meet the requirements of membership in a syssition. They were no longer homoioi, but known as \"inferiors\" (Xen. _Hell._ 3.3.6). Moralists claimed that wealth, rather than the old Spartan virtues, had become the principle criterion for high social status. In any event, this social change gave women an opportunity to be members of the elite on the same terms as men. Some were even able to enrich themselves illegally, as men did. Theopompus reports that during the Third Social War (356\u2013346), Archidamus III took some of the money at Delphi and that the Phocians bribed his wife Deinicha so that she would persuade her husband to support them (Paus. 3.10.3).\n\nDespite their vaunted austerity, Spartans apparently were tempted to keep some items of luxury in private at home. Xenophon ( _Lac. Pol_. 7.6) reports that houses were inspected to ascertain whether anyone was hiding any gold or silver. This practice may have become more common in Xenophon's day, because that was the time when large amounts of precious metals were entering Sparta. Doubtless Aristotle associated women with greed and luxury, for these were features of private life. A parallel from Roman history is of interest. In 195 B.C.E., during the debate over the repeal of the Oppian Law at Rome, a speaker argues that women should be granted the opportunity to display their wealth by wearing gold and purple and riding in carriages, for they had few other avenues by which to satisfy their ambitions. The single item of conspicuous consumption that Spartan women were permitted was racehorses. It is no accident that a Spartan was the first woman whose horses were victorious at pan-Hellenic games, for such triumphs provided women with an approved avenue for the display of their wealth. Entering such a competition was expensive, for the owner was obliged to pay for horses, trainers, charioteer, chariot, and\u2014if victorious\u2014a victory monument.\n\n### _Dowry_\n\nLycurgus had outlawed dowries, but by the end of the fifth century, if not earlier, women had them. Perhaps the tradition that Spartan women did not have dowries was revived or invented in the Hellenistic period under the influence of Agis IV and Cleomenes III or some other utopian program (see below). In the Greek world various lawgivers, philosophers, and moralists disapproved of dowries, or believed they should be regulated. In Plato's _Republic_ there is neither marriage nor private property, and therefore there are no dowries, and in _Laws_ (742C), dowries are specifically outlawed. Justin ( _Epit_. 3.3.8) states that girls marry without a dowry so that they will not be chosen for their wealth.\n\nIn an ideal Sparta, just as all the men are homoioi who enjoy equal status, so are all the girls equally endowed. It is reasonable to suppose that in a Sparta lacking precious metals, slaves, or other movable wealth, there would be little, except land and horses, to constitute a substantial and useful dowry. A theory about dowry supports the view that there was a time when dowries did not exist in Sparta. Dowries are found principally in parts of the Mediterranean where men cultivate land with a plow and own the instruments and beasts needed for production. In such societies, women's work is undervalued: a woman is viewed as a burden to her husband, and her father must contribute a dowry for her support. This analysis is appropriate to a society like that of Athens, where women were barred from access to most economic resources and the means of production, and men performed agricultural labor. In Sparta, in contrast, women were not economic liabilities, for the kleros system ensured their basic sustenance.\n\nCertainly by the end of the fifth century, if not earlier, along with the decline in the kleros system and the accumulation of precious metals and other valuable items in private hands, dowries were a conspicuous part of the economic regime and doubtless contributed to the impoverishment of some families and the accumulation of wealth by others. Aristotle ( _Pol._ 1270a25\u201326) mentions dowries as a means by which women came to possess wealth. Lack of a dowry in Sparta, as elsewhere in Greece, threatened to make a girl an unwilling spinster. The fianc\u00e9s of Lysander's daughters attempted to break their engagements when they discovered that their brides-to-be were poor (Plut. _Lys._ 30.5). Without giving any historical background by which the event may be dated, Plutarch ( _Am. narr._ 775d) tells of a woman's attempted act of savage revenge on other women when her daughters could no longer have dowries after her husband's property was seized by his political enemies. Aelian ( _VH_ 6.6) reports that a man who had five sons could give his daughters in marriage without a dowry. This exception implies that such a father would be so highly honored that a man might marry his daughter just for the sake of becoming a son-in-law of such a man, but that in ordinary circumstances a dowry was necessary.\n\n### _Heiresses_\n\nIn terms of Greek law, an \"heiress\" was a fatherless, brotherless woman. The heiress at Athens was called _epikleros_ ; in Gortyn, she was known as _patroiokos_ ; in Sparta, as _patrouchos_. In the absence of a male descendent, such a woman could be the means by which her father's lineage was perpetuated. She also might transmit her father's property to her son (who thus became his grandfather's heir), or inherit it herself. In some cases there was no property, but the filial obligation to perpetuate the lineage remained. At Athens, the filial obligation was always foremost; at Sparta, the meager sources suggest that the emphasis in the role of the heiress changed over time, with concerns about her property eclipsing those of her father's lineage.\n\nThe principal primary sources that help to define the position of the heiress at Sparta include Herodotus,Aristotle, and analogies with the Code of Gortyn. At Gortyn, the heiress ( _patroiokos_ ) was permitted to keep part of her patrimony and marry outside her father's lineage. According to Herodotus (6.57.4), the kings originally exercised the right to give an heiress ( _patrouchos_ ) in marriage if she had not already been betrothed by her father. If her legal situation was similar to that in Gortyn, the heiress at Sparta was never subject to an inflexible rule that she marry her father's closest male next of kin. She may, however, have been under some moral and religious obligation to see that her father's lineage not be extinguished. There was some concern for the continuity of oikoi at Sparta: for his suicide mission at Thermopylae, Leonidas selected men who had children (Herod. 7.205.2, 5.41.3). Furthermore, we may speculate that the prestige of the kings made it virtually impossible for the Spartan heiress to reject the bridegroom chosen for her.\n\nThe power of the state over the heiress and her property decreased as the power of the private family increased, for Aristotle's testimony differs markedly from that of Herodotus. Therefore the change probably occurred at the same time as the other changes increasing the individual's rights over private property that are associated with the reforms attributed to Epitadeus. According to Aristotle, however, the _kleronomos_ (heir apparent: _Pol._ 1270a28) had the power to give the heiress ( _epikleros_ ) as well as the property to whomever he pleased. The \"heir apparent\" was doubtless the nearest male kin of the heiress's father: he was the man who exercised authority over the heiress and her property.\n\nThe inheritance regime for Spartan males was the same as for Athenians, but the system was more favorable for Spartan women than for their Athenian counterparts, who did not inherit at all, and who were given dowries that were perhaps one-sixth of what their brothers received. In contrast, the Spartan woman's share of the patrimony was half as much as her brother's. Girls who were fatherless and brotherless were better off at Sparta than at Athens, for the Spartans got to control their property, while the Athenians were simply conduits of the patrimony to their sons. Moreover, the brotherless, fatherless Athenian woman was required to marry her father's closest kinsman, usually her uncle or cousin, even if both were married at the time the woman's father died. On the other hand, in cases where the heiress was not wealthy, the inexorable obligation of the male next of kin to marry her themselves or to find her a husband assured her marriage at Athens, but could compromise her Spartan counterpart. The daughters of Lysander, who were brotherless, nearly lost their bridegrooms when the men learned that their finace\u00e9s were poor. It is not certain, however, whether they were engaged to kinsmen.\n\nAristotle reports that at Sparta, heiresses were numerous (see above). This plethora of heiresses was exaggerated inasmuch as Sparta was always plagued by oliganthropia (sparse male citizen population). Aristotle and later writers drew attention to the connection between oliganthropia and women's ownership of property (see chap. 3). In addition, women of the property-owning classes may have outnumbered men at Sparta, as they did at Rome during and after the Second Punic War. Fearing that men would no longer fulfill the criteria necessary for their census classes and that the number of men eligible for military and governmental service would be diminished, the Romans passed legislation aimed at preventing women from owning great wealth. As is apparent in the western world nowadays, women's survival and longevity is a significant factor in property ownership.\n\n### _Change Over Time_\n\nAt the end of the fifth century, private property triumphed at the expense of public property. The reforms attributed to Epitadeus that allowed women to inherit kleroi, the influx of precious metals, the use of dowries, and the laws affecting heiresses permitted women to possess a large portion of the total wealth of Sparta. The period of the Peloponnesian War was a watershed in the history of Athenian women as well. After the Spartans occupied Decelea and war was waged throughout the year rather than just in the summer, as had previously been customary, Athenian women had to assume more responsibility and exercise greater economic power. Like Spartans, in the absence of men Athenian women managed their affairs. Defeat cost the Athenians their empire and produced an immediate, though temporary, decline in the city's economy. Some thirty years after the Peloponnesian War, Athens had returned to its traditional way of life: in contrast, the Spartan defeat at Leuctra and the loss of the rich agricultural land of Messenia dealt a lethal blow to the Spartan economy. A generation later, Sparta pointedly refused to support the campaigns of Alexander: consequently, Spartans missed the opportunity to share in plundering the wealth of Persia. By the third century, the economy was characterized by many mortgages and by large estates in the hands of a few. These few included royal women, who were among the wealthiest people in Sparta (see below). Only 700 old Spartan families remained, and of these only about 100 possessed land and kleros (Plut. _Agis_ 5.4). Female members of these fortunate 100 families benefited from the concentration of wealth. A brotherless woman could inherit all her father's land, as did Agiatis, who inherited from the extremely wealthy Gylippus (Plut. _Cleom_. 1.2).With such assets, she was claimed as wife by two kings, Agis IV and Cleomenes III.\n\n### Women and the Reforms of Agis and Cleomenes\n\nIn 244 B.C.E., when he was not quite twenty, Agis IV became the Eurypontid king of Sparta. He had been raised by his mother Agesistrata and his grandmother Archidamia. The Agiad king,Leonidas (ca. 316\u2013235), was older, had lived with the Seleucids in great luxury, was married to a daughter of one of them, and had two children by that wife.Agis, in contrast, recreated himself as a Spartan of the old austere tradition, wearing the short cloak, for example, and following the laws attributed to Lycurgus. He proposed a program of reform, principally designed to increase the number of full-fledged citizens and restore Sparta to its former prestige. Redistribution of the wealth was essential to reinstate the 600 landless citizens as homoioi. Agis donated 600 talents and his own huge estate for redistribution (Plut. _Agis_ 9). Women controlled most of the wealth in Sparta: therefore their support was essential for the success of the reforms. Agis was able to convince his mother and grandmother, who were not only the two wealthiest women, but who were the wealthiest of all Spartans, to contribute their property. They were both widows, and doubtless doted on the charismatic young king, but other wealthy women did not support his program. According to Plutarch ( _Agis_ 7), they were corrupted by their desire for luxury and were reluctant to give up the prestige and influence derived from their wealth. We also observe that the women exercised full control over their own property.\n\nLeonidas led the opposition to Agis' program, and after much strife was deposed and replaced by Cleombrotus, his son-in-law, who supported the reforms (Plut. _Agis_ 11.3\u20134). At this juncture,Leonidas'daughter Chilonis took her father's side and joined him in the temple of Athena of the Bronze House, where he had sought asylum. The favor she showed the dissolute old man was the best recommendation he had. He was eventually recalled. Chilonis was able to convince her father to spare her husband Cleombrotus, but Agis was summarily murdered in 241. Agesistrata and Archidamia were killed as well. One of the ephors had a particular reason for wanting to eliminate Agesistrata: he had borrowed some expensive cups and clothing from her and did not want to return them (Plut. _Agis_ 18.4). Leonidas saw to it that Agiatis, the extremely wealthy widow of Agis IV, was given in marriage to his son Cleomenes III, though he was too young for marriage. Her young son by Agis was not heard from again. Pausanias (2.9.1) asserts that Cleomenes poisoned him, but this charge is difficult to accept, unless Agiatis was an unusually forgiving and stoical woman. She had at least one son by Cleomenes (Plut. _Cleom._ 22.8).\n\nRoyal women in the Hellenistic period were influential in politics as powers behind the throne. As we have seen, Agiatis was married first to Agis IV, and when widowed she was married to Cleomenes III. In Plutarch's descriptions, both marriages were paradigmatically harmonious and loving, typical of the ideals of the Hellenistic period. Agiatis was able to instill in her second husband the revolutionary ideas of her first (Plut. _Cleom_. 1.2). Cleomenes also was influenced by his studies with Sphaerus, a Stoic who came to Sparta to lecture (Plut. C _leom._ 2.2, 11.2). In 235, when Cleomenes became the sole king of Sparta, he attempted to revive Agis' program. His mother, Cratesicleia, who was extremely wealthy, supported his efforts and, in fact, remarried so that her husband Megistonous would use his influence on her son's behalf. Plutarch indicates that it was her own independent choice both to remarry and to select her new husband. Megistonous and his supporters contributed their property for redistribution (Plut. _Cleom_. 10\u201311.1). Abolition of debt and redistribution of land followed. The agoge and syssitia were revived. Sparta was restored to its former military eminence for a time, but with Macedonian forces threatening him, Cleomenes turned to Ptolemy III Euergetes for assistance. Megistonous had been killed in battle, and Agiatis had died around 224. Ptolemy promised to help, but only if Cleomenes sent Cratesicleia and his children to Egypt as hostages.\n\nEventually Cleomenes was defeated by an alliance of Macedonians and Achaeans and fled with three thousand soldiers to Egypt, where the degenerate Ptolemy IV, Philopator, was now ruler. Philopator ordered that Cleomenes be imprisoned. and the Spartan women and children in Alexandria be killed. At first Cratesicleia panicked, but in the end they died stoically and bravely. Plutarch ( _Cleom._ 39.1) commented: \"Sparta played out these events with the deeds of women rivaling those of men.\"\n\n### The Last Reformers: Apega and Nabis and Chaeron\n\nNabis reigned as sole king of Sparta from 207 to 192. He may have been a member of the Eurypontid dynasty, though he executed all members of the royal houses. His wife Apega is probably identical with Apia, daughter of Aristippus of Argos who, like Nabis, ruled as a tyrant. By marrying one of their daughters to Apia's brother, Pythagoras of Argos, and by trying to arrange marriages for their adult sons with the daughters of Philip V ofMacedon in 197, they mimicked the political endogamy of tyrants in the archaic period. Polybius disliked both Nabis and Apega and portrays them in an unfavorable light. He indicates that Apega wielded a great deal of power, furthering her husband's ambitions and gratifying her greed. Like a Hellenistic queen, an Arsino\u00eb or Cleopatra, she received men at court alongside her husband. She evidently wanted to be wealthy like her royal predecessors, and Nabis sent her to her native Argos to procure money. Her viciousness exceeded her husband's. A woman, she knew how to humiliate women, and also how to dishonor men by humiliating the women in their family. She subjected the Argive women to suffering and violence and stole nearly all their gold jewelry and valuable clothing (Polyb. 13.7). Inspired by his wife, Nabis invented a female robot as evil and deceptive as Pandora:\n\nHe also had made for himself a machine, if one should call such a thing a machine. It was the image of a woman, dressed in expensive clothing, in appearance a well-executed likeness of the wife of Nabis. Whenever he sent for any of the citizens, wishing to exact money, he would begin by speaking gently. . . . If any refused and said they would not pay the sum, he said something like,\"Perhaps I am not able to persuade you; however, I think this Apega will.\" This was the name of Nabis' wife. He said this, and soon the image I have described was present. When the man shook her hand, rising from his chair, he made the woman stand and embraced her with his hands and drew her little by little to his chest. Under her dress she had arms and hands and breasts covered with iron nails. Whenever Nabis placed his hands on his wife's back and by means of certain devices drew the man towards her and drove him against her breasts very slowly; he forced the man who was being crushed to say anything. In this way he destroyed quite a few of those who refused to pay him. (Polyb. 13.7)\n\nNabis was a reformer, like Cleomenes. His program included the redistribution of land, but unlike the reigns of Agis and Cleomenes, in Nabis' time the donations were not voluntary. His program, like that of the reformer kings, included abolition of debts and the restoration of the Lycurgan constitution (Livy 34.31.16\u201318). In 195, he executed eighty of the _principes iuventutis_ (Livy 34.27.8). He also exiled the wealthiest and most prominent Spartiates who were his enemies, and gave their property as well as their wives and daughters (Livy 34.35.7 _liberos coniuges_ ) in marriage to newly freed helots.We are not told if the helots were bachelors or what happened to their former wives, or how many women were involved. Doubtless the men were enthusiastic about marrying the wives of their former masters, at the very least because they would enjoy their estates. Previously when helots were freed, they had not usually been made citizens, but Nabis conferred citizenship on them in large numbers (Livy 38.34.6). Polybius (16.13.1) refers to the men as _douloi_ (slaves) and Livy (34.27.9) once uses the word _ilotae_ (helots), but elsewhere (34.21.11) refers to _servi_ (slaves) _._ It seems more likely that the members of the lower class who married the wives and daughters of the Spartan exiles were helots and mercenaries rather than slaves. The early Ptolemies had already demonstrated that mercenaries could be recruited as citizens by the offer of an oikos,  and Nabis certainly was interested in increasing the number of soldiers at Sparta. Furthermore, at least judging by the swashbuckling adventurers depicted in New Comedy, mercenaries often captivated the hearts of women. In any case, to force upper-class Greek women to marry purchased slaves, who were possibly foreign born, and to confer Spartan citizenship on such people, would have been unthinkable. Marriage to a Spartan wife or daughter instantly supplied a helot with an oikos sufficient to maintain a citizen soldier. According to a treaty of 194 B.C.E. between Nabis and the Romans, the wives and daughters of the men exiled by Nabis were permitted to join their original husbands. The treaty stated explicitly that the women would not be forced to join their previous husbands. We are not told whether any of them chose to do so. Plutarch's use of the verb _metoikizo_ ( _Phil._ 16.4), with its connotations of transferring an oikos, to describe Philopoemen's eviction of the slaves and mercenaries whom tyrants had made Spartan citizens implies that they left with their Spartan wives and families. Furthermore, we may speculate that, considering the social hierarchies in force in antiquity, the older women wanted to continue to control and enjoy younger husbands who were clearly their social inferiors, and not exchange them for the original husbands, who were doubtless irate and displeased with what had transpired in their absence. The older women may also have stayed to support their daughters, who knew no other husbands. Moreover, the women wanted to keep their land, and were afraid that the children they had borne to the helots would suffer the same fate as those supposedly born to their ancestors during the Second Messenian War. It was said that when the Spartan husbands returned after the war they exiled these half-breed children to Italy, where they allegedly founded the colony of Tarentum (see chap. 2). In any case, in 188 after Nabis had been killed under Philopoemon, the Achaean commander who had restored Sparta as a member of the Achaean League, the exiles returned to Sparta and those who had been made citizens by \"tyrants\" were exiled to Achaea (Plut. _Phil._ 16.4, Livy 38.34). We are not told what happened to the women. We may speculate that their original husbands were willing to take them back, if for no other reason than the fact that they possessed substantial amounts of property.\n\nNabis was killed in 192 by Aetolians, who were his putative allies. The Romans reluctantly settled affairs in Sparta for a time. The last radical leader of Sparta was Chaeron, who had been exiled from Sparta and had served as an envoy to Rome in 182\u2013181. He seized the property of the sisters, wives, mothers, and children of men who had been exiled (by rulers from Cleomenes through Nabis), and distributed it at random to his most needy supporters (Polyb.24.7.3). Finally, the Spartans invited Aristaenus, the commander of the Achaean League, to put an end to Chaeron's tyranny, and returned the property to those from whom it had been seized.\n\n### Autonomy and Social Power\n\nIndividual women like Gorgo, Agiatis, Cratesicleia, and perhaps Deinicha exercised a significant influence on male members of their own family and on society at large. Moreover, women in groups were encouraged to uphold Spartan ideals by activities such as publicly praising brave men and reviling cowards and bachelors. No other Greek women are reported to have been involved in elections to the extent that Spartans were. When a member of the Gerousia was elected, he was followed by throngs of young men who praised him and many women who sang of his excellence and congratulated him on his good fortune in life. His syssition awarded an extra portion of food to the victor. After dinner, his female relatives congregated at the doors of the mess-hall. Thereupon, in public, a second selection took place, but the competitors were female. The victor summoned the woman whom he held in the highest esteem and gave her the food, saying that he had received it as an indication of his excellence and he gave it to her in the same way. The rest of the women congratulated her and escorted her home (Plut. _Lyc_. 26.3\u20134).\n\nSuch reports do not indicate that women were fully active citizens in the sense that men were, that they could defend their polis, vote, or hold governmental office, for overt political power was not exercised by women anywhere in the Greek world before the advent of Hellenistic queens. But, as Aristotle remarked in his discussion of Sparta, in warlike societies men are dominated by their wives. Even if he is exaggerating, he did perceive that women had a voice in managing affairs at Sparta. Some scholars in the second half of the twentieth century have gone even further than Aristotle in detecting the power and influence of Spartan women. Stephen Hodkinson paints the grandest picture of Spartan women in a plutocratic society. Hodkinson argues that the kleros system governing public lands that Plutarch describes was solely an invention of the Hellenistic period. Consequently, no category of land was ever restricted to ownership only by males. If Hodkinson were correct, the situation described by Aristotle would have had roots as early as the archaic period when women would have possessed and managed vast amounts of property. The evidence from women's history, however, indicates that he is not correct, but rather that an additional source of great wealth was available to women at the end of the fifth century. Indeed, Hodkinson draws the bulk of his evidence for women's wealth from fourth-century evidence. Victories in pan-Hellenic chariot races were evidence of vast wealth. All twelve Olympic victories won by Spartans from 548 to 420 were won by men. In contrast, half of the six victories from 396 to 368 were won by women. The sudden appearance of female victors in chariot races at Olympia beginning in 396, the new craving for expensive imported dresses, and even Agesilaus' scoffing remark about Cynisca seem suitable to the conspicuous consumption characteristic of the nouvelles riches.\n\nG. E. M. de Ste Croix contrasts \"the inferior position of women at Athens\" with \"the powerful position of women in the Spartan system of property ownership.\" James Redfield points out that women were active in the system of marriage exchange and in motivating men to increase the economic status of the oikos.  According to Barton Kunstler, women made major decisions concerning the disposition of household and communal wealth, discussing financial matters with helots and perioikoi.\n\nWomen's influence, however, was not restricted to the private sphere. Maria H. Dettenhofer argues that wives managed the kleroi, and were therefore responsible for their husbands' social status. She claims that women wielded political influence through their economic power. As we have seen above, Plutarch's description ( _Agis_ 7) of women's participation in the reforms of Agis gives a clear picture of the direct relationship between wealth and public power. For elite women at Sparta at that time, wealth was probably the only secure basis of influence and autonomy. In each of the sagas of reform, royal women were directly involved because they were property owners and controlled their own wealth. Agesistrata,Archidamia, and Cratesicleia espoused the political beliefs of the men in their family, and like the men, paid for their involvement with their lives. That they were executed is testimony to their power.\n\n* * *\n\n. Plut. _Roman Questions_ , 108, and see further Sarah B. Pomeroy, _Goddesses, Whores,Wives, and Slaves: Women in Classical Antiquity_ (New York, 1995), 64.\n\n. Herod. 6.52, 4.147; Paus. 3.1.5\u20139. See Deborah Gera, _Warrior Women: The Anonymous Tractatus de Mulieribus_ (Leiden, 1997), no. 5 and pp. 121\u201325.\n\n. Plut. _Lys_. 22.3\u20134, _Alcib_. 23.7\u20138, _Ages_. 3.1\u20132, and see further P. A. Cartledge, _Agesilaos and the Crisis of Sparta_ (London, 1987), 115, and chap. 3, above.\n\n. Herod. 5.39\u201341. On the familial strife that ensued, see C. Dewald, \"Women and Culture in Herodotus'Histories,\"in _Reflections of Women in Antiquity_ , ed. H. Foley (New York, 1981), 91\u2013125, esp. 108\u20139.\n\n. See further Alfred S. Bradford, \" _Gynaikokratoumenoi_ : Did Spartan Women Rule Spartan Men?\" _AncW_ 14 (1986), 13\u201318, esp. 14.\n\n. An honorary inscription at Delphi ( _SIG_ 3 430) identifies Areus as the son of King Acrotatus and Queen Chilonis. If this inscription refers to Areus I and to his mother Chilonis, the date is 267 b.c.e., though the parents of Areus I had not been king and queen. If the inscription refers to Areus II, posthumous child of Chilonis and Acrotatus, the date is 262\u2013254. See R. Flaceli\u00e8re, _Les aitoliens \u00e0 Delphes_ (Paris, 1937), 84 n. 2, 457\u201358; Linda J. Piper, _Spartan Twilight_ (New Rochelle, N.Y., 1986), 22; Bradford, \" _Gynaikokratoumenoi_ : Did Spartan Women Rule Spartan Men?\" 14; _LGPN_ 3A s.v. Areus 4 (ca. 330\u2013265), 6 (ca. 262\u2013254) for the posthumous child, and s.v.Chilonis 3.1, daughter of Leotychidas (i.e., Latuchidas) and 4 daughter of Leonidas and Cratesclea, and see chap. 3 n. 27. See also on Apega, below, although a tyrant's wife is technically not a queen, despite behaving like one. On the title of \"queen\" in general see Elizabeth Carney, \"'What's in a Name?' The Emergence of a Title for Royal Women in the Hellenistic Period,\" in _Women's History and Ancient History_ , ed. Sarah B. Pomeory (Chapel Hill, N.C., 1991), 154\u201372.\n\n. _LGPN_ 3A s.v. Agesistrata dates her ca. 295\u2013241.\n\n. _LGPN_ 3A s.v. Archidamia dates her ca. c. 310\u2013241, and see below.\n\n. For Nicippia, a wealthy great-grandmother who is also probably a widow, see _SEG_ XI.677, 1st cent. c.e., and see further chap. 6 nn. 54, 80.\n\n. Her sister Chryse was probably close to her in age. See Conclusion n. 2.\n\n. _Parth_. 1.64, 66\u201367; 3, fr. 3.71, 77; 162 fr. 2.c 3,5S elephantin, conj. D.A. Campbell, _Greek Lyric_ , vol. 2 (Cambridge,Mass., 1988), 498 line 5, and see chap. 1 n. 123 for ivory plaques, and Appendix for ivory artifacts possibly portraying Helen.\n\n. J. P. Droop,\"The Bronzes,\" in Dawkins, _AO_ , 196\u2013202, esp. 200, refers to \"several silver specimens and two of gold with silver bulbs joined by a chain.\" See also R. M. Dawkins, \"Artemis Orthia: Some Additions and a Correction,\" _JHS_ 50 (1930), 298\u201399 and pl. XI.1, for another bronze fibula. Of course, there is no way to ascertain whether these items were worn by or dedicated by women. Men could have purchased them and given them immediately as gifts to the goddess, but (at least judging from inscribed dedications to Artemis elsewhere in Greece) women seem more likely to have done so.\n\n. Heraclides Lembus, _Excerpta Politiarum_ , 373.13 (Dilts).\n\n. See further David Schaps, _The Economic Rights of Women in Ancient Greece_ (Edinburgh, 1979), 6\u20137.\n\n. See further Pomeroy, _Goddesses, Whores, Wives, and Slaves_ , 63, 72\u201373, and _Xenophon, Oeconomicus: A Social and Historical Commentary_ (Oxford, 1994), passim.\n\n. See further David Asheri, \"Laws of Inheritance, Distribution of Land, and Political Constitutions in Ancient Greece,\" _Historia_ 12 (1963), 1\u201321, esp. 5\u20136, 12\u201315, 18\u201320, and for recent interpretations see S. Hodkinson, \"Inheritance, Marriage, and Demography: Perspectives upon the Success and Decline of Classical Sparta,\" in _Classical Sparta: Techniques Behind Her Success_ , ed. A. Powell (Norman, Okla., 1989); Hodkinson, _Property and Wealth in Classical Sparta_ (London, 2000); and Sarah B. Pomeroy, _Families in Classical and Hellenistic Greece_ (Oxford, 1997), 51\u201354, and Property and Wealth in Classical Sparta. The views Hodkinson expresses in _Property and Wealth in Classical Sparta_ on women and the development of Spartan society have not changed fundamentally since the publication of his earlier articles on this subject, although, like Cartledge, he is now more firmly convinced that the economic system Plutarch attributed to Lycurgus and to archaic Sparta was really an invention of Hellenistic reformers. See Appendix n. 90, below.\n\n. According to Diod. Sic. 10.34.8, the Spartans did not receive wealth from their fathers, but inherited zeal to die in behalf of freedom and glory.\n\n. See further David Asheri, \"Sulla legge di Epitadeo,\" Athenaeum n.s. 39 (1961), fasc. i\u2013ii, 45\u201368.\n\n. Plut. _Agis_ 5.1,Ps.-Pl. _Alcib_. 122E\u2013123B, and see further Ephraim David,\"The Influx of Money into Sparta at the End of the Fifth Century B.C.,\" _Scripta Classica Israelica_ 5 (1979\u201380), 30\u201345, and S. Hodkinson, \"Warfare,Wealth, and the Crisis of Spartiate Society,\" in _War and Society in the Greek World_ , ed. J. Rich and G. Shipley (London, 1993), 146\u201376.\n\n. Plut. _Lys_. 2.7\u20138, _Mor_. 141d (26), 190e (1), 229a (1), and see further Ephraim David,\"Dress in Spartan Society,\" _AncW_ 19 (1989), 3\u201313, esp. 12.\n\n. Xen. _Lac. Pol_. 1.7\u20139, cf. Polyb. 12.6b.8, Plut. _Lyc_. 15.12\u201313, _Comp. Lyc. et Num_. 3.1\u20132, _Mor_. 242b(23), and see chap. 3.\n\n. See further Sarah B. Pomeroy, _Women in Hellenistic Egypt: From Alexander to Cleopatra_ (New York, 1984), 151, and C. Pr\u00e9aux, _L'\u00e9conomie royale des Lagides_ (Brussels, 1939), 463\u201380.\n\n. See further J.Modrzejewski,\"R\u00e9gime foncier et status social dans l'\u00c9gypte ptol\u00e9ma\u00efque,\"in _Terre et paysans d\u00e9pendents dans les soci\u00e9t\u00e9s antiques_ (Paris, 1979), 163\u201388, esp. 172, and see now _P. Petrie_ 2, pp. 37\u201339.\n\n. Around 60 b.c.e. Auletes recognized the right to dispose of the kleros by testament: _BGU_ VI.1285.\n\n. _P. Berol_. Inv. no. 16 223 (Heracleopolis) = _SB_ VIII.9790.\n\n. Also note the suggestion of Thomas J. Figueira, \"Population Patterns in Late Archaic and Classical Sparta,\" _TAPA_ 116 (1968), 186, that in order to stem the decline in population after the earthquake, the _arkhaia moira_ (Arist. fr. 611.12, Plut. _Mor_. 218e) was intended to prevent the alienation of the original kleros.\n\n. See further Pomeroy, _Xenophon, Oeconomicus_ , 46\u201350.\n\n. Arist. _Pol_. 1269b12\u20131270a34, trans. T. J. Saunders, _Aristotle, Politics: Books I and II_ (Oxford, 1995), 42\u201343.\n\n. Gifts to the Naopoioi: _CID_ II.4.1.55\u20136, table 3, and see further Hodkinson, _Property and Wealth in Classical Sparta_ , 174\u201375, 439.\n\n. Pomeroy, _Women in Hellenistic Egypt_ , 14\u201316.\n\n. Plutarch ( _Agis_ 4) also associates women with luxury, for he writes that \"even though he had been brought up by women,\"Agis IV was not self-indulgent. In the _Andromache_ , Euripides portrays Hermione as having a large dowry when she marries Neoptolemus, who is less affluent.\n\n. Livy 34.7 and n. 41 below.\n\n. For an egregious example, see chap. 1 on Cynisca.\n\n. Plut. _Mor_. 227f15, cf. _Lyc_. 15, _Lys_. 30.5\u20136; Ael. _VH_ 6.6; Athen.13.555b\u2013c citing Hermippus = _FGrH_ IV.3 1026 F 6 = F. Wehrli, _Die Schule des Aristoteles_ , suppl. 1 (Basel, 1974), fr. 87; Justin _Epit_. 3.3.8.\n\n. See further Pomeroy, _Xenophon, Oeconomicus_ , 60.\n\n. See further Schaps, _The Economic Rights ofWomen in Ancient Greece_ , 43\u201345.\n\n. _Lex Gort_. = _Inscr. Creticae_ , VIII.8\u201312.\n\n. Hodkinson, _Property and Wealth in Classical Sparta_ , 95\u201398, argues against the majority of scholars\u2014including most recently E. Karab\u00e9lias, \"L'epiclerat \u00e0 Sparte,\" in _Studi in onore di Arnaldo Biscardi_ , vol. 2 (Milan, 1982), 469\u201380; E. David, \"Aristotle and Sparta,\" _Anc. Soc_. 13\u201314 (1982\u201383), 67\u2013103, esp. 88\u201389; and Anne-Marie V\u00e9rilhac and Claude Vial, _Le mariage grec du VIe si\u00e8cle av. J.-C. \u00e0 l'\u00e9poque d'Auguste_ , _BCH_ suppl. 32 (Paris, 1998), 111\u201312\u2014that there never was a change in the rules governing heiresses in Sparta.\n\n. See further Schaps, _Economic Rights of Women in Ancient Greece_ , 77\u201379.\n\n. See chap. 2 n. 33.\n\n. See above on the Lex Oppia, and see further Pomeroy, _Goddesses,Whores, Wives, and Slaves_ , 162\u201363, 178, and Jane Gardner, _Women in Roman Law and Society_ (London, 1986), 171\u201377.\n\n. See further Pomeroy, _Goddesses, Whores, Wives, and Slaves_ , 71\u201373, 119.\n\n. Plut. _Agis_ 4. See further Claude Moss\u00e9, \"Women in the Spartan Revolutions,\" in _Women's History and Ancient History_ , ed. Sarah B. Pomeroy (Chapel Hill, N.C., 1991), 138\u201353, and A. Powell, \"Spartan Women Assertive in Politics? Plutarch's _Lives of Agis and Kleomenes_ ,\" in _Sparta: New Perspectives_ , ed. S. Hodkinson and A. Powell (London, 1999), 393\u2013419. Powell (esp. 414\u201315) argues that the support of women was crucial to the reforms and that some of the stories about heroic women were shaped by contemporary political factions in ways designed to influence women's opinion.\n\n. Plut. _Agis_ 10. A. S. Bradford, _A Prosopography of Lacedaemonians from the Death of Alexander the Great, 323 B.C., to the Sack of Sparta by Alaric, A.D. 296_ (Munich, 1977), 261, states only that he married the daughter of a Syrian ruler and had children. Probably her father was Seleucus Nicator: H. Bengtson, _Die Strategie in der hellenistischen Zeit_ , M\u00fcnchener Beitr\u00e4ge zur Papyrus-forschung und antike Rechtsgeschichte 32, 3 vols.(Munich, 1964\u201367), vol. 2, 46.\n\n. See chap. 3 n. 27.\n\n. See further Sarah B. Pomeroy, _Plutarch's Advice to the Bride and Groom and A Consolation to His Wife_ (New York, 1999), passim.\n\n. Plut. _Cleom_. 37, Polyb. 5.37\u201339, and see further F. W.Walbank, _A Historical Commentary on Polybius_ , 3 vols. (Oxford, 1957\u201379), vol. 1, 568\u201369.\n\n. Plut. _Cleom_. 38, and see chap. 1. Graham Shipley, _The Greek World After Alexander, 323\u201330 B.C._ (London, 2000), 440, comments that Plutarch's use of vivid details indicates that \"this may be a real episode, described for Plutarch by his sources.\"\n\n. See further Bradford, _A Prosopography of Lacedaemonians_ , 39.\n\n. Livy 32.38.3, 34.25.5.\n\n. Though Walbank, _A Historical Commentary on Polybius_ , vol. 2, 420\u201321, and P. A. Cartledge and A. J. S. Spawforth, _Hellenistic and Roman Sparta: A Tale of Two Cities_ (London,1989), 72, disbelieve the story, the existence of other robots and automatic devices in the Hellenistic period helps to lend credence to this one. For example, at the front of his parade, Demetrius of Phalerum had a mechanical snail that spit saliva (Polyb. 12.13.11).\n\n. Livy 34.31.11,14, 38.34.6, Plut. _Philop_. 16.4. Polyb. 13.6.3 states they were married \"to the most prominent of the rest and the mercenaries.\"\n\n. Andr\u00e9 Aymard, _Les premiers rapports de Rome et de la Conf\u00e9d\u00e9ration Achaienne (198\u2013189 avant J.-C.)_ (Bordeaux, 1938), 35, followed by J.-G. Texier, _Nabis_ (Paris, 1975), 57, 75, argues that marriages with Spartan wives and daughters were profitable inasmuch as women controlled two-fifths of the wealth. The figures given by Aristotle for women landowners need not have been true for the time of Nabis, especially after the redistributions under Agis and Cleomenes, but it is likely that Spartan women in Nabis' time were still wealthy.\n\n. See further J.-G. Texier, \"Nabis and the Helots,\" _DHA_ 1 (1979), 189\u2013205. John Briscoe, _A Commentary on Livy, Books XXXIV\u2013XXXVII_ (Oxford, 1991), 92\u201393, makes the reasonable suggestion that Livy will not have understood the precise status distinctions between helots and slaves.\n\n. See Pomeroy, _Women in Hellenistic Egypt_ , 100\u2013103.\n\n. Livy 34.35.7, see also Plut. _Philop_. 16.3. Aymard, _Les premiers rapports de Rome et de la Conf\u00e9d\u00e9ration Achaienne_ , 241 n. 50, argues that the Latin texts makes it abundantly clear that some of Spartan women willingly stayed with their new husbands. Linda Piper, \"Spartan Helots in the Hellenistic Age,\" _Anc. Soc_. 15\u201317 (1984\u201386), 75\u201388, esp. 87 n. 77, states (without citing any evidence) that none of the women chose to join her original husband.\n\n. See the laconic remark of B. Shimron, _Late Sparta_ (Buffalo, N.Y., 1972), 121, that the forced marriages enjoyed \"at least partial success.\"\n\n. Thus Aymard, _Les premiers rapports de Rome et de la Conf\u00e9d\u00e9ration Achaienne_ , 241\u201342, who stresses the lucrative aspect of marriage to Spartan women.\n\n. In the same way, the kings were given a double portion of food so that they could offer it as gifts of honor.\n\n. For a different view see Jean Ducat, \"La femme de Sparte et la cit\u00e9,\" Kt\u00e8ma 23 (1998), 385\u2013406, esp. 393, who criticizes Hodkinson's interpretation of female inheritance and property rights and postulates the existence of kyrioi for women at Sparta, because they are mentioned in the Lawcode of Gortyn. Using the same evidence, J. Christien,\"La loi d'Epitadeus:Un aspect de l'histoire \u00e9conomique et sociale \u00e0 Sparte,\" _RD_ 52 (1974), 197\u2013221, esp. 211, maintains that Spartan women enjoyed free disposition of their property.\n\n. Hodkinson, _Property and Wealth in Classical Sparta_ , 99\u2013103. On Plutarch and Hodkinson, see further Preface n. 11, and Appendix n. 90.\n\n. Hodkinson, _Property and Wealth in Classical Sparta_ , 308, table 12.\n\n \"Some Observations on the Property Rights of Athenian Women,\" _CR_ n.s.20 (1970), 274\u201378, esp. 277.\n\n. \"The Women of Sparta,\" _CJ_ 73 (1977\u201378), 146\u201361, esp. 158\u201360.\n\n. \"Women and the Development of the Spartan Polis: A Study of Sex Roles in Classical Antiquity\" (Ph.D. diss., Boston University, 1983), 427.\n\n. \"Die Frauen von Sparta,\" _Klio_ 75 (1993), 61\u201375, esp. 71\u201375.\n\n##\n\n## THE LOWER CLASSES\n\nLower-class inhabitants of Laconia and Messenia far outnumbered their Spartan rulers, but we know very little about them aside from their relationship to the Spartans. For Greek history in general, far more is known about the upper class. Furthermore, even among the upper class, there is less evidence for women than for men. Like women elsewhere in Greece, and like upper-class Spartan women, their status and reputation probably depended largely both upon the property they possessed and upon the men with whom they were associated. For example, as we have seen in the upper class, the mothers of brave men were respected and the mothers and sisters of cowards were dishonored and shunned like the cowards themselves (see chap. 3). Unfortunately, neither ancient sources nor modern studies have made gender a defining category in discussions of the lower classes at Sparta. Hence this chapter is the shortest in the book.\n\n### Helots\n\nMore is known about helots than about any other non-Spartiates. Helots were Greeks living in Laconia and Messenia whom the Spartans had reduced to servitude early in the archaic period. They performed the work that slaves (and free people) performed in the rest of the Greek world. Moreover, because Spartans, unlike free people in the rest of the Greek world, were not trained to farm or engage in manual labor, helots were more essential to the Spartan economy than slaves were elsewhere.\n\nHelots belonged to the state and could not be sold away from Sparta.Unlike the families of slaves, whose liaisons had no legal status and who did not own their children, the families of helots were much less likely to be broken up by sale or testament, or by the caprice of an individual owner. They lived in family groups in houses designated for them.More than one helot family was assigned to a _kleros_ (country estate: Xen. _Hell._ 3.3.5).They farmed in Laconia and Messenia and were required to send a fixed portion of produce to the Spartan man to whom their kleros had been allocated. Helotage was fundamental to the kleros system, but because the entire system of land tenure at Sparta is not well understood and changed over time (see chap. 4), the details of helotry are not clear. It is reasonable to assume, however, that since Spartan women were landowners, they supervised the work of helots on their property, riding or driving out to visit them as men did (Xen. _Hell._ 3.3.5).\n\nHelots also supplied domestic labor for Spartan homes. Xenophon's comment on domestic weaving (see chap. 1) and Plutarch's report on Timaea, wife of Agis II, indicate that female helots were included in this obligation. Plutarch ( _Ages._ 3.1 = Duris _FGrH_ 76 F 69) represents Timaea not only conversing with her female helots, but also trusting them enough to tell them that the child she was bearing had been fathered by Alcibiades, not by her husband. Since Agis was unsure about the child's paternity, Timaea must have been having intercourse with both her husband and her lover in the same period of time. Because it is impossible to keep secrets from domestics, the helots must have known that she was having intercourse with both. Inasmuch as Timaea was the only person who actually knew who the child's father was, she was probably using contraception during intercourse with her husband.\n\nBy the fourth century, with the rise of private property in Sparta, helots were thought (at least by some non-Spartan commentators) to belong to individuals and in some respects to be the equivalent of slaves. Thus Xenophon ( _Lac. Pol._ 6.3) speaks of Spartans lending helots to other Spartans who needed them. At this time, there were probably slaves of non-helot origin in Sparta as well (see below).\n\n### Free Noncitizens\n\nThere were several other categories of non-Spartiates living in Sparta. These included the _perioikoi_ (dwellers about) who were free, but not citizens. They seem to have lived in poleis and had some sorts of civic organizations like other freeborn people in the Greek world beyond Sparta. Though some worked as farmers, the perioikoi as a whole shouldered a disproportionate share of craft and commercial endeavors, since Spartan men were trained to work only in the military and government, though, as we have mentioned, they did supervise their country estates. Perioikic men worked as craftsmen and merchants and did the jobs that male Spartans were not permitted to do. Perioikic men also served in the army, some holding positions of command. Presumably most perioikic women lived like other Greek women (but not like upper-class Spartan women), raising children, managing their households, and performing domestic labor in their own homes. Some probably worked in service jobs like baby nursing and prostitution (see below).\n\nThere were other free people at Sparta distributed in a number of categories that exceeded anything we know about social distinctions in any other Greek polis. This mass of people included helots who had been freed for performing good service ( _neodamodeis_ ); mixed-blood members of the lower class who had been through the agoge and were elevated above the class they had been born into ( _mothakes_ ); bastards born of helot mothers and Spartan fathers ( _nothoi_ : Xen. _Hell._ 5.3.9); and, at the top of these inferior ranks, those who had been born into citizen status but had been demoted for non-payment of the dues owed to their syssition ( _hypomeiones_ ).  The men in all these categories were free and (with the exception of the _neodamodeis_ ) had been educated in the agoge, so they were able to undertake military service and thus compensate for the ever-dwindling supply of Spartiate men. Because they were Greeks and not foreign born, it was doubtless easier to grant them some social mobility. Thus, for example, helots might be given their freedom by the state in return for military service. We do not know if their wives were simultaneously liberated at all such occasions, but Thucydides (1.103.3) notes that after the helot rebellion in the 460s, rebels were free to leave Spartan territory, taking their wives and children. Helots could own private property and could purchase their freedom when the state offered them an opportunity.\n\nEspecially at times when the state needed funds, helots were encouraged to purchase their freedom at a set price, but we do not know if they had to pay for their wives as well, and whether the price was the same. Slaves in the rest of the Greek world also had opportunities to purchase their freedom, but they were not regarded as equals: the prices varied and had to be negotiated for each individual man, woman, or child.\n\n### Working Women\n\n### _Prostitutes_\n\nBecause precious metals and useful money did not circulate in archaic and classical Sparta, and because Lycurgus had imposed a strict moral regime, there was no prostitution (Plut. _Lyc._ 9.3).Non-Spartiates,however,were not subject to such a stringent discipline. Spartan women were forbidden to wear gold and cosmetics, but hetairai could adorn themselves. Before the end of the fifth century, patrons might surreptitiously use foreign money, but when large amounts of gold and silver began to be available to private citizens, prostitutes became more accessible. A few were notorious and wealthy. In 397, the ephors and some members of the Gerousia gave orders to Cinadon, who was suspected of fomenting a conspiracy, that he was to go to Aulon, a perioikic community in northwest Messenia on the border between Messenia and Elis, and bring back a particular woman reputed to be the most beautiful, for she had been corrupting Spartans of all ages who came there (Xen. _Hell_. 3.3.8). The Hellenistic geographer Polemon (fl. ca. 190) reports that he saw the bronze sculptures dedicated by the hetaira Cottina.\n\nAfter the death of his wife, Cleomenes took a freeborn woman of Megalopolis as his concubine ( _paidiske_ : Plut. _Cleom._ 39.2) She may have borne him a child, for he had one son by Agiatis (Plut. _Cleom._ 22.8), but more than one of his children accompanied his mother to Egypt and died there.\n\n### _Nurses_\n\nSpartan nurses were highly praised. Plutarch ( _Lyc._ 16.3), who was especially interested in the rearing and education of children, points out that Spartan nurses did not apply swaddling bands. They were famous for raising children to be happy, not discontented or finicky about their food or afraid of the dark or of being left alone. The devotion of a nurse who carried her ugly charge daily two miles uphill to the Menelaion so that Helen would make her beautiful is noteworthy (see Conclusion). For these reasons, foreigners sometimes acquired Spartan nurses for their own children. Plutarch uses _oneomai_ (purchase), which indicates that the nurses could not have been helots, since helots were not sold to foreigners. Their status is not clear: the spotty evidence suggests that they were drawn from helots as well as from other groups of non-Spartiate women. In his comedy _Helots_ ,Eupolis mentions a special festival in which Spartan nurses participated. He might have heard about such an occasion from a Spartan nurse living in Athens. Plutarch notes that non-Spartans purchased Spartan nurses because of their care and skill, and goes on to report that Amycla, who nursed ( _tittheusasan_ ) Alcibiades, was said to be a Spartan ( _Lyc_.16.5, see also Plut. _Alcib._ 1.2). The participle refers to wet-nursing. An Athenian inscription of the fourth century commemorates Malicha of Cytheria, nurse ( _titthe_ ) of the children of Diogeitus, an extremely righteous woman who came from the Peloponnesus.\n\n### Religion\n\nHelots and other members of the lower classes shared the same religion with their masters. Shrines for the various gods have been found throughout Laconia and Messenia, presumably, for the most part, created for the use of the local inhabitants. Archaeologists have verified the existence of many religious sites described by Pausanias and others. Some women dedicated altars. The humble nature of their offerings indicates that they were not wealthy. A few religious occasions were directly connected with social status. Children's nurses participated in the festival called Tithenidia in honor of Artemis. They brought boy babies to a temple for Artemis Corythalia by a river (see chap. 6). The nurses enjoyed a varied and sumptuous feast. In addition to the regular celebration of \"The Cleaver\" ( _kopis_ ), which included the sacrifice of goats, and eating cakes, cheese, sausage, figs, and beans, they also sacrificed suckling pigs and bread, and celebrated by dancing and wearing masks.\n\nOn the second day of the Hyacinthia, citizens offered dinner to everyone they knew, including helots. Dining together was a rite of inclusion expressing the solidarity of the entire population. Hierarchies were also temporarily dissolved, not only in dining, but also in the Spartan ceremonies of mourning for the king. When a king or other dignitary died, helots and their wives were obliged to mourn, and the women to wear black or suffer the death penalty Though the mourning was imposed, and may be the Spartans' equivalent of the paid professional mourners found in Athens, it may also show that the helots were conceived of as members of the large fictitious Lacedaimonian family.\n\n### Doulai\n\nThe development of private property raises the question of the status of the _doulai_ (slave women) whom Xenophon mentions in the _Spartan Constitution_ ( _Lac. Pol._ 1.4).Xenophon reports that Lycurgus thought that doulai were capable enough of producing clothing so that freeborn women could devote their energies to motherhood. In the archaic period, these doulai were doubtless helots; by the fourth century, they may also have been slaves whom Spartans purchased with their newly acquired wealth and who catered to their taste for luxury goods and conspicuous consumption. Helots belonged to the state, whereas slaves constituted part of the private property of the oikoi. Xenophon uses the word _heilotes_ in other contexts, but he does not carefully distinguish between the two statuses when describing the women to whom the weaving was delegated in his time. Elsewhere in the _Spartan Constitution_ ( _Lac. Pol_. 6.3),Xenophon speaks of private property as including hounds, horses, chariots, and _oiketai_ (household slaves) _._ He died long after the battle of Leuctra and may have continued to work on the _Spartan Constitution_ until after the emancipation of the Messenian helots. After the loss of part of Messenia, when the need for purchased slaves would have increased and there was money to buy them, there is a strong possibility that many of the women who worked in the Spartan household were slaves, though Laconian helots continued to be available. Accordingly, both statuses are possible in the fourth century and the Hellenistic period, at least until the helots were liberated in the time of Augustus.\n\n### Reproduction\n\nHelots seem to have had no problem with reproduction, though there were constant forces depleting their number: Spartiates could kill them individually and en masse with impunity, and Athens offered asylum to those who rebelled. At any rate, they continually outnumbered their masters. Although there are no census figures for the helot population, there is some basis on which to compare the number of helots to Spartiates in three successive centuries. Among those who served at the battle of Plataea, helots outnumbered Spartans by at least seven to one (Herod. 9.28\u201329). Early in the fourth century, approximately 80 Spartiates and 4,000 others milled about in the Spartan agora (Xen. _Hell._ 3.3.5). In other words, the Spartans were outnumbered 50 to 1.The helot population at this time has been estimated at 170,000\u2013224,000, including women. Around 240 B.C.E., some Spartans considered the helots too numerous (Plut. _Cleom._ 18.3).\n\nAs we have mentioned, helots lived on farms with their families, though we know little about them. In any case, there was no need to have single-sex dormitories with a bolt on the women's door, as Xenophon describes for the slaves in the _Oeconomicus_ (9.5), to prevent the males from gaining access to the females. Unlike slaves, whose numbers could be increased by purchase or conquest, helots themselves were the only source of the helot population and of other segments of the lower classes as well. It was essential for the Spartan economy that helots reproduce. Both men and women helots had a strong incentive to have as many children as possible, although they were constrained by their limited access to arable land. Since the men were constantly vulnerable to murder by the _cryptoi_ (secret hunters [of helots]), and to being killed when they served in the Spartan army, for them there was strength in numbers. David Hume astutely observed that helots were the only ancient servile group to reproduce, and argues that this success was the result of their living apart and being public slaves rather than the property of individuals.\n\nHelots were subjected to what amounted to a kind of eugenics, or better, dysgenics, which the Spartans probably learned from breeding animals, and also understood in terms of the Greek belief in the inheritance of acquired characteristics. The crypteia was instructed to kill the strongest men (Thuc. 4.80.3\u20134) so that those who were more servile and might be more easily domesticated would survive to reproduce. Nevertheless, helots served in the military. The wives and children they left behind doubtless were hostages for their good behavior.\n\nHelots not only did work that was normally done by citizens in other Greek states, but they also were forced to become parents of half-Spartan children. Women were regularly used for this purpose: men perhaps only once. The Spartans exploited the reproductive capacity of helots to produce mixed-breed offspring who filled the lower niches of Spartan manpower. The story about the founding of Tarentum by children born of Spartiate women and helots is relevant here. More credible is the evidence for the inclusion of mothakes in Spartan manpower. The definition of _mothakes_ is controversial, but they seem to have been children of Spartan fathers and helot mothers who were not reckoned as Spartiates but were free. Apparently, practical considerations outweighed theoretical eugenics in these arrangements. Xenophon ( _Hell._ 5.3.9) reports that the bastards ( _nothoi_ ) of Spartan men had experienced the benefits of the state and were fine-looking. This remark lends credence to the view that the helots and Spartans were not originally ethnically distinct, but that their statuses evolved as a result of political and economic developments. In any case, wealthy Spartan men reared mothakes alongside their legitimate sons, and they passed through the agoge together. The mothakes seem to have been given some modified form of citizenship if they completed the agoge.  Indeed, the generals Lysander and Gylippus were said to have been mothakes. The Lawcode of Gortyn, Crete, offers some examples of free children born of intercourse between serfs and free men and women. Therefore it is appropriate to ask whether Spartan women had liaisons with lower-class men, although, considering the sex ratio in the Hellenistic period and Greek reluctance to submit women to hypergamy, it is unlikely that Spartiate women were given in marriage to lower-class men (aside from the incident concerning the founding of Tarentum and the outrages perpetrated by Nabis see [chap. 4]). Lower-class female infants may have been subject to infanticide, since the deliberate production of babies of mixed parentage served primarily to create more soldiers to fill the Spartan ranks. Otherwise such women probably bore additional generations of children of mixed parentage _._ 32 In any case, the use of helot women for the production of mothakes must have increased the rebelliousness of helot men, especially if the liaisons were the result of individual choice and longlasting, rather than brief encounters (like those in Spartiate husband-doubling) encouraged by the state.\n\nThere is no evidence for slave breeding in other parts of the Greek world. Hesiod ( _WD_ 602\u20133) advised the novice farmer to get a slave woman without a baby to nurse. Xenophon ( _Oec._ 9.5\u20136) allows slaves to reproduce only as a reward for good behavior. Slave reproduction compromised productivity. Pregnancy and childbirth jeopardized the slave's health or even her life, and the baby might not survive anyway. The Spartans were unique and innovative in exploiting the reproductive potential of servile women.\n\nIn sum, the Spartans, who were notorious for their innocence of business matters, had devised a reproductive calculus as early as the constitution attributed to Lycurgus. Even if several of the practices reviewed above and in chapter 3 were temporary, or not widespread, or invented in the Hellenistic period and attributed to Lycurgus, or part of the mirage, it is clear that Sparta served as a kind of laboratory for demographic ideas or actual experiments in which the state and private individuals made investments in order to reap dividends in the form of human capital.\n\n* * *\n\n. Nino Luraghi,\"Helotic Slavery Reconsidered\" (paper delivered at the Celtic Conference in Classics, National University of Ireland, Maynooth, Sept. 8, 2000), argues that they could be sold or lent within Sparta.\n\n. Diod. Sic. 12.67.4; Strab. 8.5.4 (365); Xen. _Hell._ 3.3.5.\n\n. The amount apparently varied through the centuries from one-half, according to Tyrtaeus 5 (quoted by Paus. 4.14.4\u20135), to the fixed amount described by Plutarch (see chap. 3 n. 8).\n\n. E.g.,Xen. _Hell._ 5.4.28 (attendants of Agesilaus); Xen. _Lac. Pol._ 7.5; Herod. 6.63 (attendants of King Ariston); Polyb. 4.81.7, 5.29.9 (attendants of King Lycurgus); and the mothakes who were raised in Spartan homes (see below).\n\n. For nurses, see below.\n\n. See further P. A. Cartledge, _Sparta and Lakonia: A Regional History, 1300\u2013326 B.C._ (London, 1979), 182\u201385.\n\n. For additional categories including _trophimoi_ and offspring of foreigners, and discussion of the definition of each status, see J. Ducat, _Les hilotes_ , _BCH_ suppl. 20 (Athens, 1990), 166\u201368. The definitions remain controversial; e.g., S. Hodkinson, _Property and Wealth in Classical Sparta_ (London 2000), 198, 355\u201356, considers mothakes as sons of poor or demoted citizens.\n\n. According to Plut. _Cleom._ 23.1\u20135, to fill his war chest in 223 Cleomenes raised 500 talents by selling freedom to about 6,000 helots, charging five Attic minas per man.\n\n. Xen. _Lac. Pol._ 5.8; Plut. _Lyc._ 1.4.4;Athen. 15.686\u201367; Clement of Alex., _Paid._ 2.10.105.\n\n. On foreign coinage: Plut. _Lys._ 17.1\u20134,Ephorus _FGrH_ 70 F 193, and see further G. L. Cawkwell, \"The Decline of Sparta,\" _CQ_ 33 (1983), 385\u2013400, esp. 395\u201396.\n\n. Athen. 574c\u2013d and see chap. 6 n. 70.\n\n. Plut. _Cleom._ 38.3. See further G. Marasco, _Commento alle biografie Plutarchee di Agide e di Cleomene_ , 2 vols. (Rome, 1981), vol. 2, 589, and chap. 4, above.\n\n. On the status of Spartan nurses, see most recently Valerie French,\"The Spartan Family and the Spartan Decline,\" in _Polis and Polemos_ , ed. C. Hamilton and P. Krentz (Claremont, Calif., 1997), 241\u201373, esp. 260\u201361, where French suggests that though most Spartan nurses were slaves or helots, they may also have been freeborn women of lower status ( _hypomeiones_ ) _._\n\n. See chap. 6 n. 15.\n\n. _IG_ II3.3111. For the date: Kaibel, _Epigr.Graec._ , p. 17, no. 47.\n\n. On the continuity of religion in Messenia even after liberation from Spartan rule, see Thomas J. Figueira, \"The Evolution of the Messenian Identity,\" in _Sparta: New Perspectives_ , ed. S. Hodkinson and A. Powell (London, 1999), 211\u201344, esp. 229.\n\n. M. N. Tod and A. J. B. Wace, _Catalogue of the Sparta Museum_ (Oxford, 1906), p. 21; p. 63, no. 427; p. 69, no. 528; pp. 70\u201371, no. 546; p. 74, no. 618.\n\n. Athen. 4.139a\u2013b, citing Polemon (Preller, pp. 136\u201340, para. 56); Hesych. s.v. koruthalistriai 3689 and s.v. kurittoi 4684 (Latte); and see further M. Pettersson, _Cults of Apollo at Sparta: The Hyakinthia, the Gymnopaidiai, and the Karneia_ (Stockholm, 1992), 14\u201317.\n\n. Polycrates (before 1st cent. b.c.e.) = _FGrH_ 588 3B F 1 = Athen. 4.139f, who mentions _douloi_ (slaves) not helots; but this was an old custom and probably included helots. In Plut. _Am. narr._ 775d\u2013e, Spartan women dine with _oikeioi_ (see chap. 6).\n\n. See further Pettersson, _Cults of Apollo_ , 18 _._\n\n. Tyrtaeus fr. 5;Herod.6.58.2 (perioikoi and helots). See further Sarah B. Pomeroy, _Families in Classical and Hellenistic Greece_ (Oxford, 1997), 51.\n\n. F. W. Sturz, _Lexicon Xenophonteum_ , 4 vols. (1801, repr. Hildesheim, 1964), vol. 2, s.v.\n\n. Chap. 14,which is critical of Sparta and seems to undermine much of the treatise, may have been written after Leuctra.\n\n. Ducat does not discuss _Lac. Pol._ 1.4, but in _Les hilotes_ , 21 n. 9 and 46, he argues that Xenophon uses _douloi_ and _oiketai_ indiscriminately of slaves and helots.\n\n. P. A.Cartledge, _Agesilaos and the Crisis of Sparta_ (London, 1994), 174, and personal communication Oct. 16, 1999.\n\n. See further M. Whitby, \"Two Shadows: Images of Spartans and Helots,\" in _The Shadow of Sparta_ , ed. A. Powell and S. Hodkinson (London, 1994), 87\u2013125, esp. 110.\n\n. \"Of the Populousness of Ancient Nations\" (1742), in _Essays, Moral, Political, and Literary_ (Oxford, 1963), 381\u2013451, esp. 393.\n\n. See chap. 2 n. 44.\n\n. Phylarchus, _FGrH_ 81 F 43 = Athen. 6.271e\u2013f.\n\n. R. F. Willetts, _The Law Code of Gortyn_ , _Kadmos_ supp. 1 (Berlin, 1967), p. 15 and cols. VI.55\u2013VII.2.\n\n. Thus C. J. Tregaro,\"Les b\u00e2tards spartiates,\" in _M\u00e9langes Pierre L\u00e9v\u00eaque_ , ed. M.-M.Mactoux and E. Geny (Paris, 1993), esp. 37\u201338.\n\n. Thus D. Ogden, _Greek Bastardy_ (Oxford, 1996), 223.\n\n. See further Sarah B. Pomeroy, _Xenophon_ , _Oeconomicus: A Social and Historical Commentary_ (Oxford, 1994), 297\u2013300.\n\n##\n\n## WOMEN AND RELIGION\n\nThis chapter is devoted to the religious experiences of Spartan women. It is not an attempt to cover every aspect of each cult or divinity mentioned, nor to report all the textual and archaeological evidence unless a connection with women's activities can be detected. The evidence is often obscure and fragmentary. In many cases, the sole source of information is a single brief definition in a Byzantine dictionary. Furthermore, the reader must be cautioned that the great majority of votives and small objects that have been excavated\u2014disproportionately those that would illuminate women's history\u2014have not yet received proper scholarly attention (see Appendix). Since the ancient sources are often specific about the ages of the female participants in the cults and whether they are virgins, in this chapter we will also draw as much attention as possible to age groupings or marital status. Athletics, including nude athletics, were often connected with religion; these activities have been discussed in chapter 1.\n\nGreeks everywhere worshipped the same gods and generally in the same ways, but the emphases varied from place to place. Thus in Athens, domestic cults were important and women were particularly involved in mourning the dead. Owing to the the Spartan emphasis on the public sphere, private cults are unknown. In contrast, Spartan cults for women reflected the society's emphasis on female beauty, health, and, most of all, fertility. Furthermore, the shrine of a historical woman, Cynisca, who had been heroized for her equestrian victories, was centrally located in Sparta (see chap. 1). As elsewhere in Greece, there were some cultic activities exclusively for women, others exclusively for men, and some for both. For example, at Sparta women were prominent in the cults of Dionysus, Eileithyia (goddess of childbirth), and Helen (a local heroine), were excluded from a festival of Ares (Paus. 5.22.7), but participated along with men in the Hyacinthia.As part of their religious experience, women sang, danced, raced, feasted, dedicated votive offerings, drove chariots in processions, and wove clothing for cult images of the gods. In comparison with the cultic activities of their counterparts in Athens, the activities of Spartan women included substantially more opportunities for racing and far less weaving. Since Spartans were always allowed to drink wine, their festivals were probably quite jolly. For example, the festival celebrated in Alcman, _Partheneion_ 1, included a sumptuous banquet. The modern reader is free to imagine the behavior of a group of adolescent girls who know each other intimately at an all-night festival. At such banquets women drank unmixed wine as usual, and ate cakes that were fashioned in the shape of breasts. Influenced by Alcman's choral lyrics for maidens, scholars have drawn attention to the communal aspects of religious experience at Sparta that was restricted to women. Even when the rituals were enacted by women only, however, they were considered an essential part of the religious life of all citizens. Sundry votive offerings by individual women are evidence of a variety of personal relationships with divinities, as well.\n\n### Artemis Orthia\n\nArtemis was worshipped throughout Greece as a divinity who brought fertility to human beings and animals and protected mothers and children. According to myth, her cult was imported from the East. Pausanias (3.16.7,9) states that the archaic image of the goddess he saw at Sparta was the original that Iphigenia and Orestes had brought from Taurus in the Chersonese. At Sparta, Artemis was associated with Orthia, who, like Artemis, was a nature goddess concerned with both plant and animal life. She was probably the focus of the festival commemorated in Alcman, _Partheneion_ 1,where she is referred to as \"Orthria,\"(line 61) and Aotis (goddess of the dawn, line 87). The actual connotations of the word \"Orthia\" (straightness or uprightness) have puzzled interpreters from antiquity to the present. Pausanias (3.16.11) offers a practical explanation for the title \"Orthia\": the statue itself stood upright because it was placed in a thicket of willows that supported it. Other interpretations are also possible. The goddess raises children up or she directs them to safety. Or she sets them on a straight path through the cycles of their lives; thus she presides over various rites of passage as boys and girls reach puberty and adulthood. The centrality of the cult of Artemis reflects Sparta's interest and investment in nurturing and educating the young. These concerns are echoed at Sparta in the cult of Artemis Corythalia at the festival of the Tithenidia (see chap. 5). The sanctuary of Orthia was situated on flat ground alongside the Eurotas, conveniently located near the children's exercise grounds. When Theseus as an old man raped Helen, who was just a young girl, he found her dancing at the sanctuary (Plut. _Thes._ 31.2,Hyginus _Fab._ 79).\n\nAs Alcman indicates, maidens sang and danced as part of the cult. Sometimes men participated along with the girls. Their choirmasters might be male poets such as Alcman. Lead figurines dated to the seventh and sixth century found at the Orthia sanctuary depict women and men playing flutes and lyres, and one shows a woman playing cymbals. Castor and Pollux were reputed to have taught the Spartans a special dance that the maidens performed annually at Caryae near the northeast border of Spartan territory in historical times in honor of Artemis, whose image stood there out of doors. The girls must have danced in a secluded place without men present. At one performance early in the eighth century, at a time when both Messenians and Lacedaemonians shared the sanctuary, Messenian men supposedly raped the Spartan girls. At another performance during the Second Messenian War,Aristomenes ofMessenia seized the daughters of wealthy, noble fathers and carried them off to a village in Messenia awaiting ransom. If the story is true and not only meant to stimulate outrage against the perpetrators, the lack of effective male protectors at this particular performance may be attributable to the war which demanded their full attention: Aristomenes and his force probably had expected to find the girls alone. As these stories and that of Helen and Theseus related just above demonstrate, girls dancing out of doors in secluded places were vulnerable to rape. These stories may also be interpreted in the context of myths about abduction, as a metaphor for marriage, where the girl is plucked from a protected group and from a circumscribed, choreographed existence and must begin to fend for herself as a bride and an adult. The theme is reiterated in the story surrounding the rape and marriage of the Leucippides (see below).\n\nThe site of the most famous cult of Artemis Limnatis (\"of the lake\") was on the border of Messenia and Laconia. An epigram commemorates the dedication Timareta made to the goddess before marriage. Despite the brevity of the poem, it is clear that Artemis is a goddess who presides over the important rite of passage from girlhood to marriage, and that appropriately enough she is wor-shipped in a liminal area, safe on the Laconian side, dangerous on the Messenian:\n\nTimareta, before marriage, dedicated cymbals, her lovely ball, the snood that held her hair,\n\nher maiden-dolls, she a maiden, to the maiden Goddess of the Lake, as is suitable,\n\nand the dolls' clothes too, to Artemis.\n\nDaughter of Leto, holding a hand over the child Timareta\n\nKeep her pure.\n\nGirls and women also performed lewd dances in honor of Artemis, celebrating her as a fertility goddess. These dances, which are known only through brief citations in late lexicographers, included the _kallabides_ or _kallabidia._ The _kyrittoi_ wore masks and phalli at the festival of Artemis Corythalia and at the Tithenidia.  The _baryllika_ was a comic or indecent dance in honor of Artemis and Apollo. According to Pollux (4.104), it was danced by women; according to Hesychius, it was performed by men who put on ugly female masks and sung hymns. Perhaps both men and women cross-dressed and danced. The orgiastic nature of the Orthia cult at Sparta may be somewhat surprising to those familiar with the austere Artemis of Athenian myth. A vase from the Orthia sanctuary, dated 580 to 575, shows a _komos_ (revel) or orgiastic dancing in progress. The vase is fragmentary and the restoration of the figures uncertain. It does, however, appear to show men and women dancing and having intercourse in the presence of a hairy, ithyphallic, satyr-like figure.\n\nA Laconian vase also of the archaic period depicts women and men together at a symposium (fig. 6). Because the woman reclines with the man and doubtless drinks wine as he does, some viewers, perhaps because they are more familiar with the iconography of Athenian vase painting, may deduce that she is a hetaira.  Lycurgus, however, was reputed to have prohibited Spartiate women from practicing prostitution, and this prohibition may well have been in effect in the sixth century (Plut. _Lyc._ 9.3, and see chap. 5). Furthermore, the chorus of girls in Alcman, _Parthenion_ 1 (67\u201368), mention that they wear Lydian mitres, and Spartan girls and women regularly drank wine. This vase does not depict a secular banquet: rather, the winged daemon, the altar on the right, and the fruit and vegetation suggest that the cup was used at a festival of a fertility divinity. The divinity was most probably Artemis Orthia, or perhaps Apollo Hyacinthius.\n\nFig. 6. Symposium with women.\n\nFragments of a cup depicting a symposium with women on the ground playing flutes and men reclining on couches on the ground. The right-hand fragment showing trees and building base signifies that the location is out of doors. The winged daemon (perhaps Eros) hovering above the woman indicates that the scene is sacred rather than secular. The woman wears a Lydian mitre on her head. Border of pomegranates above. Arcesilas Painter, ca. 565. Fragments Samos (from the Heraion) K 1203, K 1541, K 2402, and Berlin Charlottenburg, Staatliche Museum Inv. Nos. 478x, 460x. Photo, Deutsches Arch\u00e4ologisches Institut, Athens. Drawings courtesy of Maria Pipili.\n\n### _Artemis Orthia in the Hellenistic and Roman Periods_\n\nArtemis and her priestess were also involved in cultic activities for youths. The priestess controlled the intensity of the ritual whipping of youths which is attested for the Roman period. In fact, Plutarch ( _Lyc._ 18.1) asserts that he saw many youths die because of this flagellation:\n\nThe priestess, holding the xoanon, stands by [the ephebes]. It is usually light because it is very small; but if those who administer the whipping ever decrease the whipping because of the beauty or high status of the ephebe, then the xoanon becomes heavy for the woman and no longer easy to carry. She blames those who administer the whipping and says she is being weighed down because of them. (Paus. 3.16.10\u201311)\n\nIt was rare for a Greek woman functioning regularly in a religious capacity and independently of male authority to exercise as much control over men as did the priestess of Artemis Orthia.  Like the Pythia, she transmitted superhuman directives.\n\nDedications to Orthia continue from the archaic period through the third century C.E. Among those from the Hellenistic period worthy of note is a set of black-glazed bowls inscribed from Chilonis to Orthia.  Chilonis is a common name, but this Chilonis may have been royal.\n\n### Eileithyia\n\nThe cult of Orthia served as the model for cults of lesser goddesses including those of Eileithyia and Helen. Like Orthia, Eileithyia was a goddess of fertility. Her special realm was childbirth. Her cult was closely connected to cults of Artemis, and their sanctuaries located near each other in both the area of the Dromus where Spartan youths raced and on the banks of the Eurotas (Paus. 3.17.1, 3.14.6). The exact sites of Eileithyia's sanctuaries have not been determined, but her name was discovered inside Orthia's sanctuary inscribed on the head of a bronze pin, on rooftiles,  and on an unusual large six-sided votive gaming die. The die, dated seventh or sixth century, is of bronze, with six faces and lion heads at the ends. The faces are inscribed with one to six concentric circles, marking the values for the game. The dedication, which is distributed among the faces, is: \"to Artemis Orthia of Eileithyia.\" This elegant piece seems to the modern observer the perfect objective correlative for the fortunes of childbirth, for it evokes such questions as: How will the mother fare? Will the child be a girl or a boy? Who is really the father?\n\nFig. 7. Nude, kneeling, pregnant female in labor.\n\nPerhaps Eileithyia or Hera or Helen, or a mortal woman who is helped by the presence of the goddess. She is flanked by two small male figures. The figure on the right was probably playing a double flute. Marble group found at Magoula near Sparta. Height 48 cm. Sparta Museum 364 = M. N. Tod and A. J.B.Wace, _A Catalogue of the Sparta Museum_ (Oxford, 1906), 171\u201372, no. 364, figs. 50\u201351 = M. Pipili, _Laconian Iconography of the Sixth Century B.C._ , Oxord University Committee for Archaeology Monograph, no. 12 (Oxford 1987), cat. no. 156, pp. 58\u201360, fig. 86. Photo, Deutsches Arch\u00e4ologisches Institut, Athens.\n\nCrude terracotta statuettes of Eileithyia dating to the eigth or seventh century were discovered in the first year of the excavation at the sanctuary of Artemis Orthia. One shows a stocky woman holding an infant, the other shows a pair of birth spirits supporting a mother and child. In addition to the inscribed die mentioned above, spindle whorls and pins of the seventh to sixth century (typical dedications by Greek women), and other offerings to Eileithyia, were found in the Orthia sanctuary.\n\n### Hera\n\nAlthough the ancient sources discuss women's racing at Sparta predominantly as an athletic activity, they do consider a few races in the context of religion (see chap. 1). Modern scholars have interpreted the latter type as initiatory rites for girls preceding marriage, and argue that races in honor of Hera, Helen, and the Leucippides, and choruses of Physcoa and Hippodamia constituted a premarital ritual.What the association is between racing and marriage and why the ritual should take the form of seeming to run away from marriage is not immediately obvious to the modern observer, unless it was an ordeal or a qualifying test. In any case, the college of sixteen women at Elis that was in charge of the races at the Heraea also arranged choruses of women in honor of Hippodamia and Physcoa.  These choruses may have been restricted to maidens, and thus have also constituted a prematrimonial rite. That the choruses were two in number suggests that they competed against one another. Racing, of course, was competitive, too. At the Heraea, running was a cult activity limited to _parthenoi._ Girls wore the short peplos exposing one breast only to race at the Heraea. This costume seems to be a version of the exomis, a tunic worn regularly by men (see chap. 1). Transvestism may constitute part of an initiation rite preceding marriage, and on her wedding night the Spartan bride was dressed as a man.\n\n### Helen\n\nThe Menelaion, the principal shrine of Helen, Menelaus, and Helen's brothers Castor and Pollux, was at Therapne on a mountain top where they were all said to be buried (Paus. 3.19.9, Pind. _Nem._ 10.56, _Pyth._ 11.62\u201363). There were also additional shrines for Helen and her brothers around Sparta, and Phylono\u00eb, another daughter of Tyndareus, also had a cult. The urban sanctuary of Helen was located near the Platanistas and quite close to Alcman's tomb (Paus. 3.15.2\u20133). Although the principal shrine is usually called \"the Menelaion,\" Herodotus (6.61) referred to it as Helen's temple. Moreover, the archaeological finds testify to a cult of Helen who is not subordinate to a husband: sometimes she is worshipped along with her brothers, at other times she seems independent of men. Offerings were made to Helen and Menelaus alone, and each may have had an altar. The Menelaion is the most important hero shrine in Greece, and probably the earliest, for it was founded ca. 700. The datable offerings specifically inscribed to Helen precede those to Menelaus. These were a bronze aryballos (perfume vase) dated to ca. 675\u2013650 and a bronze harpax (hook) dated to ca. 575\u2013550. Around three hundred terracotta figurines were found, including some depicting human figures on horseback; most of these show female figures riding astride or side-saddle. In enjoying both independence from men and partnership with them, Helen as a religious figure reflects the lives of mortal Spartan women.\n\nThe cults of Helen and Artemis Orthia were similar in many ways. Both were concerned with nature and fertility. Helen's name means \"reed\" or \"shoot.\" Helen and Artemis in her guise of Orthia were local divinities. Outside Sparta, Artemis was worshipped in various manifestations, but Helen was considered a mythological or literary figure as well as divine. The finds from the Menelaion are a microcosm of the Orthia material. These two major shrines were close enough to be visited in a single day. The shrine dedicated to Helen alone at the Platanistas was in close proximity to the Orthia sanctuary and the exercise grounds of girls and boys.Music and dancing was part of the cult of Helen, as it was part of the cult of Artemis. As we have mentioned above, Helen was dancing in the Orthia sanctuary when Theseus and Perithoos abducted her. Some of the anecdotes told about the divine Helen suggest that she was especially concerned with the welfare of marriageable girls, for example, endowing ugly ones with beauty (see Conclusion). Theocritus' _Epithalamium for Helen_ depicts her as a paradigmatic figure for the most important rite of passage of a woman: Helen was the first of a large group of Spartan maidens to become a bride. At the springtime festival of the Heleneia, maidens anointed a plane tree with olive oil. Theocritus (18.1\u20134, 43\u201346) gives the aetiology of this practice:\n\nOnce in Sparta at the house of yellow-haired Menelaus\n\nMaidens wearing hyacinth blossoms in their hair\n\nOrganized a choral dance in front of the freshly-painted bridal chamber.\n\nThey were twelve girls, the foremost in the city, a large crowd of Spartan girls.\n\n\u00b7 \u00b7 \u00b7 \u00b7 \u00b7 \u00b7 \u00b7 \u00b7 \u00b7 \u00b7 \u00b7 \u00b7 \u00b7 \u00b7 \u00b7 \u00b7 \u00b7 \u00b7 \u00b7 \u00b7 \u00b7 \u00b7 \u00b7 \u00b7 \u00b7 \u00b7 \u00b7 \u00b7 \u00b7 \u00b7 \u00b7 \u00b7 \u00b7 \u00b7 \u00b7 \u00b7\n\n\"For you first [Helen],we will weave a wreath of the lotus growing close to the ground,\n\nAnd place it on a shady plane tree.\n\nAnd first we will pour liquid oil from a silver flask\n\nAnd let it drip beneath the shady plane tree.\n\nAnd on its bark shall be inscribed in Dorian so that a passerby may read\n\n'Revere me. I am Helen's tree.'\"\n\n(a)\n\nFig. 8. Helen and Menelaus.\n\nSparta, 1, from Magoula (?). Stone pyramid stele with relief, ca. 600\u2013570. Man hugs veiled woman with left arm. She raises her right arm, probably to embrace him. In her left hand and probably in his right, they hold a wreath, symbolizing love. Their eyes meet in an intimate gaze.\n\n(b)\n\nOn the reverse a man, who here is considerably taller than the woman and whose hair has grown longer (after ten years of bivouacking in Troy), threatens her with his sword. Though the reliefs are not inscribed, they probably both depict Helen and Menelaus. Sparta Museum, 1 = Tod and Wace, _Catalogue_ , 132\u201333, no. 1, figs. 26\u201327 = Pipili, _Laconian Iconography_ , cat. no. 87, pp. 30\u201331, fig. 45 = L. B. Ghali-Kahil, _Les enl\u00e8vements et le retour d'H\u00e9l\u00e8ne dans les textes et les documents figur\u00e9s_ (Paris, 1955), 71, no. 24, pl. 42.1 = _LIMC_ s.v. H\u00e9l\u00e8ne, p. 539, no. 230. Photos, Deutsches Arch\u00e4ologisches Institut, Athens.\n\nThere is evidence for the cult of Helen and the Dioscuri (Castor and Pollux) in Hellenistic Sparta. Rules for members of a society who sacrificed and held communal feasts in their honor are extant. Mention of a _gynaikonomos_ (regulator of women) in the inscription indicates that women certainly were included in the festivities. Such communal co-ed feasts were common in the Greek world in the Hellenistic period, but they were not entirely new at Sparta. As we have mentioned, there were co-ed symposia in honor of Orthia in the archaic period (see fig. 6).\n\n### Dionysus\n\nHilaeira and Phoebe (\"bright\" and \"shining\"), daughters of Leucippus (\"white horse\"), fled from the Spartan heroes Castor and Pollux, but were captured and married their pursuers. Their rape and marriage were therefore a mythical archetype for the Spartan wedding ritual in which the husband seized the bride. Their cult, unlike that of Helen, was confined to Sparta, where a shrine to their sister Arsino\u00eb was also located (Paus. 3.12.8). The namesakes of the mythical brides, the Leucippides, were priestesses who were responsible for weaving a garment for the cult image and who also organized a race for Dionysus. As is appropriate for virgins and for those involved in racing, the priestesses were also called _poloi_ (fillies). Eleven Dionysiades (daughters of Dionysus), probably unmarried girls, ran in the race. The race was held near the place where the suitors of Penelope were said to have begun the race which was part of their prenuptial contest. Also close by was a sanctuary of Hera Hypercheiria (\"whose hand is\n\nabove\"), a goddess of marriage, and the image of Aphrodite Hera to which mothers of brides sacrificed (see below).\n\nWomen were conspicuous in the Spartan worship of Dionysus. Vergil describes Mount Taygetus \"virginibus bacchata lacaenis\"(\"celebrated in Bacchic revels by Laconian virgins\": _Georgics_ 2.487\u201388). Near Taygetus was a temple of Dionysus with images both outside and inside the building. Only women could view the image within and perform the secret sacrifical rites. Similar rules of exclusivity applied to young women: the rites of Dionysus of the Hill (Colonatas) could be performed only by virgins. Interestingly enough, however, near the temple of Dionysus there was a brothel named for a notorious courtesan, Cottina. This business was probably established in the Hellenistic period (Athen. XIII.574c\u2013d, Strabo 8.363, and see chap. 5).\n\n### Demeter\n\nDemeter was a goddess of agriculture and of women's life cycle. The Spartan Eleusinion was discovered at Kalyvia tes Sochas just south of the city. The site has not been fully excavated, but it apparently was quite large and included areas for sitting and feasting. Finds from the fourth century B.C.E.\u2014a period of increasing visible wealth\u2014are noteworthy. All the archaeological evidence and the inscriptions concern women, though inscriptions from the Roman period indicate that the majority of statues erected in honor of women were dedicated and paid for by men, or groups of family members including men (see below). Nevertheless, women acting alone alone paid for some. Thus, for example, the mother of Pomponia Callistonice dedicated a statue in honor of her daughter, who served as hereditary priestess of Artemis Orthia and other divinities. Nicippia, honored her great granddaughter Agesippia with a statue. There are at least thirty inscriptions from the Roman period, the majority from the second and third centuries C.E.Women were numerous among cult officials, but their activities were subject to male approval. Beginning in the third century B.C.E., dedications by women who were worshippers or who held sacred office in the cult were inscribed on parts of the building. Later, their kinsmen or the state dedicated statues of them. The Roman emphasis on increasing fertility among the upper class stimulated interest in the cult of Ceres. Portraits of imperial women on coins with the paraphernalia of the goddess probably encouraged Spartan involvement in the cult of Demeter. Like Rome, Sparta was concerned about the fertility of the upper class. One relief that was dedicated by a priestess in the Roman period depicts paraphernalia used by women in the cults for themselves or for the cult image. These include toilet items such as cosmetic boxes, sandals, a mortar and pestle, a spatula, a strigil, an unguentarium, a pin with a pick (?), a sponge, a snood, a mirror, a comb, a spindle and distaff, bowls, and, most unusually, a loincloth (resembling a very skimpy bikini bottom) that was worn only by menstruating women.\n\n### Apollo\n\nWomen played an important role at the Hyacinthia, the great communal festival of Sparta. They wove Apollo's chiton (Xen. _Hell._ 4.5.11; Paus. 3.16.2). This garment was probably presented during the festival. In the Roman period, at least two women were the leaders of the Hyacinthia. Presumably they presided over the contests that were held at the festival. Two women are also known to have officiated at the contests in honor of the Dioscuri.  Presiding over festivals in which men competed is a noteworthy honor granted to elite women in the Roman period.\n\nThe Hyacinthia was celebrated at Amyclae, a numinous site from which the Eurotas, the ancient city of Sparta, and many of its shrines are visible. The celebration lasted for three days and included a period of mourning followed by the singing of songs to the gods. Boys sang and performed on citharas and flutes and rode on horseback. In the procession in view of the whole city, some girls rode in elaborately decorated wicker carts, others in chariots ran races. Their chariots, fashioned like goats and stags, may have been connected to an attribute of Artemis, goddess of wild animals (see chap. 1). The festival included feasting and women's nocturnal dancing and choral singing.\n\n### Athena\n\nWe may be certain that Spartan women did not fail to pay due homage to other divinities in the Greek pantheon, even those who, though important\u2014at least judging from the paucity of anecdotes, and of published votives and inscriptions\u2014were not their top favorites. Athena Poliouchos was the protector of the polis (Paus. 3.17.2). Athena was also titled Chalkioikos (\"of the Bronze House\"), for her sanctuary that was on the acropolis. A statue of the Olympic victor Euryleonis stood near the temple (Paus. 3.17.6). The original temple and altar, begun probably in the eighth century, have now been obscured by monumental Roman and Byzantine buildings. A few stories and votives attest to women's concerns. When the traitor Pausanias sought asylum in the temple of Athena, and the ephors were in a quandary, Theano, his mother, did not plead for her son's life. Instead, she placed a brick at the door of the temple and left (ca.470; see chap. 3).\n\nFour of ten bronze bells inscribed with the donor's name were dedicated by women to Athena. For example, in the late fifth or early fourth century, a certain Eirana dedicated a bell. It is interesting that the woman's name means \"peace\" and that she made her offering at a critical time in Spartan military history. In addition, over forty bells made of bronze and eighty made of terracotta without inscriptions have also been found associated with Athena's sanctuary on the acropolis. Though bronze bells are usually used for military purposes, these bells may have been used for musical performances for the cult, and in that case women may have been among the dedicants. Euonyma dedicated a more traditional gift for a woman: a bronze mirror. Polemon, a geographer of the second century B.C.E., reports that beyond the statue of Athena of the Bronze House, he saw an offering given by the hetaira Cottina consisting of a small image of herself or perhaps of the goddess and a small bronze cow.\n\n### Aphrodite\n\nAt Sparta, Aphrodite was worshipped in several guises of relevance to women. Mothers of brides sacrificed to Aphrodite Hera, an unusual hybrid. Another complex version of Aphrodite was a warrior. The archaic temple of the Armed Aphrodite who was called \"Morpho\"drew attention to her double nature: it was the only two-storey temple known to Pausanias (3.15.10). It housed a statue of the goddess veiled, with chains on her feet. Pausanias explains that Tyndareus had put chains on the statue to demonstrate that wives were faithful to their husbands. Lactantius' interpretation of the origin of the Armed Aphrodite, in contrast, is based on a view of Spartan women as more feisty. When the Spartan army was away during one of the wars against the Messenians, some of the Messenians invaded Sparta. The women donned armor and managed to defeat the invaders.Meanwhile, some Spartan soldiers returned, and seeing people in armor, deduced that they were the enemy, and attacked the Spartan women. The women undressed, the Spartan men recognized them, and an orgy ensued. For this reason the Spartans built a temple and dedicated a statue of the Armed Aphrodite (Enoplios or Hoplismene). This story in some ways echoes the tale Herodotus relates (4.146) about the Spartan wives of the Minyae, who were descendants of the Argonauts. When the women's fathers imprisoned their husbands, intending to kill them, the wives entered the prison and exchanged their clothing with their husbands. Thus dressed as women, the husbands escaped. The sanctuary of the Armed Aphrodite was grouped with others that were particularly meaningful for women. Pausanias (3.16.1\u20133) next describes the sanctuary of the Leucippides, Hilaeira and Phoebe, daughters of Leucippus whose pursuit and rape, as we have suggested above, were mythical analogues of the Spartan marriage ceremony. The egg of Leda from which Helen was born hung from the roof of their sanctuary. Then Pausanias mentions the chamber where women wove a chiton for Apollo at Amyclae. Nearby was the house of Tyndareus, whose marriage to Leda was a husband-doubling or wife-sharing arrangement, with Zeus as the uninvited impregnator.\n\n### Roman Period\n\n### _Priestesses_\n\nAs a result of the prohibition on commemorating women except those who had died in childbirth, and the general paucity of data about women, there is little information about Spartan priestesses until the late Hellenistic and Roman periods. Because there are more inscriptions about specific priestesses and festivals for the Roman period, this epigraphic data should not lead us to overexaggerate its importance in its own time. The material is, however, of great value to the historian; for example, providing some evidence that naming patterns for Spartan women show a tendency to alternate names with granddaughters named for their grandmothers just as they had been in classical Athens. Interest in horses is reflected in names of Roman Sparta, as it had been in the classical period: thus Nicippia is the great-grandmother of Archippia.  The emperor Augustus had led the way by reviving the woman-oriented cults of the early and middle Republic at Rome, and Hadrian and the Pax Romana had created an atmosphere in which Greek cities like Sparta could revive and reenact their past. The relative abundance of epigraphic evidence at Sparta, however, is not only indicative of the increased interest in reviving old religions, but is also a reflection of the increase in inscriptions and commemoration in the late Hellenistic world (see Appendix). Furthermore, Xenophon, one of our major sources on Spartan women in the classical period, does not report on their religious activities, In contrast, Plutarch (who was himself a priest at Delphi) does endeavor to include such information. Inscriptions, however, are the best source for women's cults in Roman Greece. Once again, the intrepid historian must consider whether archaizing tendencies, good research on the part of Plutarch and other officials, and tradition will have caused previously existing cults and practices to be revived as far as possible in a fair semblance of their original form in the radically new political context. Some of the religious powers that had been exercised by the king in the days of the monarchy were distributed among high-ranking commoners, both male and female. Because we do not know whether royal women had played any special role in religion, for example analogous to the role of the wife of the king archon in Athens, we do not know whether the powers exercised by Spartan women in the post-monarchical period derived ultimately from women. Thus, that women in the Roman period served as officials in the Hyacinthia may indicate that Sparta was hard up to find citizens wishing to take on such responsibilities and opened the position to women in order to double their pool of candidates, or it may indicate that, inasmuch as the entire city participated in the Hyancinthia, women had held such positions in earlier periods, but merely happened not to be mentioned in any historical record. Those who are more skeptical must nevertheless admit that at least some elements of the earlier cult activities were revived\u2014though sometimes in an exaggerated form designed to attract the interest of tourists and to swell the hearts of Greeks with pride in their past. Certainly the archaeological finds at the Eleusinion demonstrate that women's religious activities honoring Demeter in the fourth century B.C.E. were similar to those in the Roman period. Therefore the later evidence may be used with caution to shed some light on earlier practices.\n\nFeasting was a traditional part of Greek festivals. Practices varied and changed over time. Some feasts were held for the entire community. A painting on a cup by a Spartan artist depicts men and women at a symposium together perhaps at such an occasion (see fig. 6, above). In cults that were restricted to women, priestesses presided and women often were in charge of organizing the activities.\n\nThere was a resurgence of women-oriented cults in Roman Greece in the second century C.E. Some of these cults had existed in the past, others were created in the Roman period. In Sparta, civic-minded women of high status served as _thoinarmostria_ (mistress of the banquet) _,_ often simultaneously holding other sacred offices and titles such as \"Hestia Poleos\" (Hestia of the City). These women probably filled a post analogous to that of the Vestals at Rome and were responsible for seeing that the fire in the city's official hearth burned eternally. This hearth was in the very center of the agora, the public space frequented by Spartan men. The woman who served as Hestia of the City may have been present sometimes when the Boule (Assembly) was convened, for one such woman is attested in an inscription from the mid-second century listing members. _IG_ V.i.602 records the dedication of a statue around the third century C.E. in honor of Pomponia Callistonice, who was not only a hereditary priestess of Artemis Orthia but also priestess of other divinities including those sharing the precinct of Orthia, the Fates, Armed Aphrodite, Asclepius Schoinatas, Artemis Patriotis, the Dioscuri, and the contest of the most august Dioscuri. Her statue may have stood at the sanctuary of Orthia.  Since the population of Sparta was small, it must have been difficult to find many women with the funds, leisure, and interest necessary to fill all the priestesshoods, and thus an eligible woman like Pomponia Callistonice served in many capacities. In at least one case, a foreign-born woman filled the post of _hypostatria_ (dresser of the cult image) at the Eleusinion.\n\nAs in earlier periods, many of the religious posts were hereditary, or at least dominated by members of the same families. Because the elite families were highly endogamous, the hereditary priesthoods devolved on fewer and fewer people who were kinsmen. Endogamy also simultaneously accelerated a concentration of wealth that made the expenses associated with the posts tolerable. Thus father and daughter, sister and brother, and cousins are attested in priesthoods that are related such as those of the Dioscuri and Helen. Sons and daughters of the sacerdotal officials also served religious functions. The priesthoods were transmitted through both the male and female lines.\n\nIn the Roman period, titles such as \"Mother of the Demos and the Boule,\" \"Daughter of the City,\" and \"Hestia of the City\" were awarded to some women of the highest elite status, with some women bearing two or three of these titles. Such official acts incorporated these women into the public family that was otherwise exclusively male. These edicts may be compared with public decrees and grants of citizenship honoring outstanding female benefactors in other cities in Hellenistic and Roman Greece and with the the designation of Julia Domna and other empresses as \"Mater Senatus\" (Mother of the Senate) and \"Mater Castrorum\" (Mother of Military Camps). The honor was not gratuitous. In return for these adoptions the city expected to be treated with the respect due a member of the family, including a willingness to offer financial support.\n\nMost of the honorary inscriptions of the Roman period honor priestesses and other women not only for their piety and specific religious activities, but they also praise them for conventional female virtues including modesty ( _sophrosyne_ ), dignity ( _semnotes_ ), piety ( _eusebeia_ ), and orderliness ( _kosmiotes_ ). Such virtues were likely to be attributed to people of lesser status including youths and boys, rather than to adult men. A dedication of the first century B.C.E. in honor of Alcibia, daughter of Tisamenus, praises her virtue ( _arete_ ), the good works of her ancestors, and her irreproachable sixty-year marriage ( _IG_ V.i.578.2). _IG_ V.i.599, third century, records the dedication by the polis of a statue in honor of Aurelia Heraclea, daughter of Marcus Aurelius Teisamenus, at the Orthia sanctuary. The inscription does not state why the statue was erected, but it does praise her dignity, love of philosophy, nobility, good sense, modesty, and possession of every virtue. The identity of a woman being commemorated was probably defined by the donor who gained prestige from the reflected glory of the honorand. One woman donor, however, praised herself extravagantly in a dedication honoring her father. In a brief inscription of ten lines, Charision managed to refer to herself as first among Spartan maidens, and a new Penelope, thus combining the praise associated with virgins with that due to the most chaste and loyal wife. In this hyperbole, one senses the old competitive spirit of Spartan maidens displayed in the new arena of personal virtue in the later period when Spartan women no longer boasted of the gods' favor by commemorating their athletic victories. _Sophrosyne_ (chastity, self-restraint) was the virtue most often ascribed to women throughout antiquity and it is the most common on the inscriptions, including those from the Roman period honoring Spartan women. The city heroized some women solely because of their sophrosyne.  One would not immediately think of it in connection with Spartan women of an earlier period. In fact, Xenophon ( _Lac. Pol._ 3.4) reports that Spartan boys surpassed girls in modesty. The inscriptions from the Roman period add some new notions: generosity ( _IG_ V.i.595.3\u20134); love of wisdom ( _IG_ V.i.598.2, 4); love of husband (or philanthropy: IG V.i.600.6\u20137, 601.10, 605.4); and reverence toward the city and the Boule ( _IG_ V.i.589.14, 608.8\u20139).\n\nReferences to gynaikonomoi in inscriptions concerning religion indicate state surveillance of women's activities. Indeed, one imagines that some participants might have over-indulged in feasting, drinking, singing, and dancing in the guise of religious activity. Furthermore, in some cults priestesses exercised authority over men, and it is unlikely that women would be granted such power without masculine surveillance. Gynaikonomoi in other poleis not only supervised women, but imposed sumptuary restrictions. Presumably they refrained from censuring women such as Xenaria who spent lavishly in behalf of other citizens (see above). The earliest reference to a gynaikonomos in a Spartan source is ca. 110 C.E., but there is no reason to believe that the man mentioned is the first to hold that office. Nevertheless, it is important to remember that in archaic and classical Sparta, older women had supervised girls: in the Roman period, Spartan practices more closely approximated those of other Greek cities.\n\n### _The Visit of Julia Balbilla, the Pious_\n\nFemale members of the royal family of Commagene and their descendants were present in Sparta beginning with the reign of Vespasian. When King Antiochus IV was deposed by the Romans and confined in Sparta with his wife and daughters, one of the daughters married a member of one of the leading families (Joseph. _BJ_ 7.234). C. Iulius Eurycles Herculanus L. Vibullius Pius was born of this union. The king and the rest of the family then were permitted to move to Rome. Julia Balbilla was a granddaughter of Antiochus IV, born to the king's son C. Antiochus Epiphanes. She was named for her maternal grandfather Ti. Claudius Balbillus, who had served as prefect of Egypt in 55\u201359 C.E. He was a learned man from whom she may have acquired her intellectual interests and special knowledge of Egyptian religion. As an old woman, Balbilla came to Sparta to attend personally to the construction of a heroon honoring her cousin Herculanus. Presumably she was his heir. Though this edifice is no longer extant, it is the most impressive funerary monument known to have stood in Roman Sparta. She evidently invested her time well in making certain that it was properly built.\n\nBalbilla wrote elegiac poetry in the archaic Aeolic dialect that Sappho had used almost a thousand years before. When she accompanied Hadrian and Sabina on a trip to Egypt in 130, like some other wealthy travellers she probably paid the native priests in the neighborhood to have her words inscribed on the Colossus of Memnon, where her poetry is still visible. Her poetry refers to Egyptian deities, and was complimentary of the imperial couple as well as of herself, whom she identifies as the descendant of Balbillus the wise, granddaughter of King Antiochus, Balbilla the pious. Piety, as we have seen, was a clich\u00e9 in honorary inscriptions for elite women in Roman Sparta. Doubtless as they became older and closer to death they became more pious and their interest in leaving monuments that would survive after death increased. Piety was also a much-vaunted virtue attributed to Hellenistic kings. Balbilla's claim to piety may have deliberately alluded to her royal descent. If this hypothesis is true, then the epigraphic claims of elite Spartan women to piety are even more grandiose than is immediately apparent. The majority of extant inscriptions in their honor were dedicated after Balbilla's visit and may reflect her influence.\n\nA woman of tremendous wealth, Balbilla not only paid for the heroon and the inscription, but it has been suggested that she was responsible for the monument of her brother Philopappus in Athens from whom she may have inherited as well.No husband or child is known for Balbilla. Her independence may have been largely a result of her unmarried state, either throughout life or at least when she was older, as a divorc\u00e9e or widow.\n\nOne may speculate about the effect on elite women exercised by an illustrious woman of Greco-Macedonian ancestry who was a member of the sophisticated international set at Rome and an intimate friend of Hadrian and Sabina. Balbilla embodied a glamorous mixture of dilettantism, piety, antiquarian learning, and self-importance. Her presence, that of her aunt, and the visits of other women tourists cannot but have caused a stir in the small quiet city of Sparta. If Balbilla's relationship with Sabina was modelled on Hadrian's with Antinous, she may have also lent some support to lesbian relationships\u2014which were a tradition at Sparta in any case.When she departed, she left a conspicuous monument so that the Spartans would not forget either her or her cousin.\n\nAs had been true throughout Athenian history, in Hellenistic Sparta religion offered the only acceptable milieu in which respectable women could play a public role. A female official who was expert at divination would influence political decisions. As we have seen, the priestess of Artemis Orthia policed erotic relationshps between men and youths. The reknown probably attracted women to religious activities. A survey of the history of upper-class women in the west will reveal that many women whom etiquette prohibited from working as men did outside the home worked without wages as volunteers in religious contexts that allowed them to exercise some authority and social and economic independence. The expenditure of funds for charitable and civic purposes was praiseworthy. Nevertheless, repeated references in inscriptions honoring women for domestic virtues such as modesty and love of husband, and alluding to their fertility as advertised by naming their descendants, seem to seek to compensate for or undermine the playing of such public roles. As microcosmic historical documents, the inscriptions indicate that in the imperial period some Spartan women were integrated into the Roman world and had accepted its values, but they continued to cherish their peculiarly Spartan traditions and were encouraged to do so. A modern reader can sense the pride that inspires these inscriptions and continue to wonder whether Sparta's women were the same as or different from other Greek women.\n\n* * *\n\n. D. Musti and M. Torelli, _Pausania: Guida della Grecia_ ,vol. 3, _La Laconia_ (Milan, 1997),249, see a direct connection between the connotations of the cult of Helen as divine, rather than merely heroic, and Spartan society where \"la donna occupa una posizione di primo piano per la conservazione delle strutture socio-economiche.\"\n\n. Line 81: _Thosteria. LSJ_ s.v.: _thosterion_ refers to _euoxeterion_ \"a sumptuous feast\" or \"a banquet.\"\n\n. Sosibius _FGrH_ 595 F 6 in his work \"On Alcman,\" bk. 3 = Athen. 14.646a.\n\n. E.g., Claude Calame, _Les choeurs de jeunes filles en Gr\u00e8ce archa\u00efque_ (Rome, 1977).\n\n. If Eva Stehle, _Performance in Ancient Greece_ (Princeton, 1997), 30\u201341, 74, and passim, is correct, the girls will have performed before the entire Spartan community.\n\n. _Parthenion_ 1.61, and see O. H\u00f6fer, \"Orthia and Orthosia,\" in _Ausf\u00fchrlicher Lexikon der griechischen und r\u00f6mischen Mythologie_ , ed. W. H. Roscher, 6 vols. (1884\u20131937), vol.3.1, 1210\u201313, H. J. Rose,\"The Cult of Artemis Orthia,\" in Dawkins, _AO_ , 399\u2013407, esp. 403.\n\n. See further S. Constantinidou, \"Spartan Cult Dances,\" _Phoenix_ 52 (1998), 15\u201330, and Musti and Torelli, _Pausania: Guida della Grecia_ , vol. 3, _La Laconia_ , 227.\n\n. Lead, 7th cent. to ca. 560: see W. G. Cavanagh and R. R. Laxton, \"Lead Figurines from the Menelaion and Seriation,\" _ABSA_ 79 (1984), 34\u201336; other materials: Dawkins, _AO,_ 262, 269, 276, pls. 180.19 (a lyre), 183, 189, 191, 195\u201396.\n\n. Lucian, _De salt._ 10; Paus. 3.10.7; Pollux 4.104; Statius, _Theb._ 4.225;Diomede, bk. III, p. 486 (H. Keil, _Gramm. Lat._ 1). Clearchus gave Tissaphernes a ring showing the maidens dancing at Caryae (Plut. _Artax._ 18 _)._\n\n. Paus. 4.4.2, and see Poralla2, pp. 117\u201318, no. 690 s.v. Teleklos. A king may have been present in his capacity as priest. We need not deduce from his presence that men generally attended this festival.\n\n. Paus. 4.16.9\u201310, and see chap. 1.\n\n. See further Deborah D. Boedeker, _Aphrodite's Entrance into Greek Epic_ (Leiden, 1974), 47\u201349, and Susan Guettel Cole,\"Landscapes of Artemis,\" _CW_ 93 (2000), 471\u201381, esp. 472.\n\n. See further Christiane Sourvinou-Inwood, \"Erotic Pursuits: Images and Meanings,\" _JHS_ 107 (1987), 131\u201345, esp. 144\u201345, and Steven H. Lonsdale, _Ritual Play in Greek Religion_ (Baltimore,Md., 1993), 229\u201330. W. Burkert, _Structure and History in Greek Mythology and Ritual_ , Sather Classical Lectures, 47 (Berkeley, 1979), 74, sees the rape of virgins as part of a mythological pattern justifying revenge and the defeat of the army of the rapists.\n\n. _Anth. Pal._ 6.280 Anonymous = A. S. F. Gow and D. L. Page, _The Greek Anthology. Hellenistic Epigrams,_ 2 vols _._ (Cambridge, 1965), vol. 1, 208, no. 41.\n\n. Hesych. s.v. lombai states that women who begin the sacrifices to Artemis wear phalli _._ See further M. P. Nilsson, _Griechische Feste von religi\u00f6ser Bedeutung_ (Leipzig _,_ 1906), 184, 187. For other lewd dances performed by Spartan women, see chap. 1.\n\n. S.v. Brudalicha 1343 (Latte): see R. C. Bosanquet, \"The Cult of Orthia as Illustrated by the Finds,\" _ABSA_ 12 (1905\u20136), 331\u201343, esp. 338\u201343, for the masks; G. Dickens, \"Terracotta Masks,\" in Dawkins, _AO_ , 163\u201386, esp. 179\u201380, and pls. 47\u201349; and see now Jane Burr Carter, \"The Masks of Ortheia,\" _AJA_ 91 (1987), 355\u201383, esp. 356.\n\n. Thus Margaret C. Miller,\"Reexamining Transvestism in Archaic and Classical Athens,\" _AJA_ 103 (1999), 223\u201353, esp. 242, following A. W. Pickard-Cambridge, _Dithyramb,Tragedy, and Comedy,_ 2d edn. revised by T. B. L. Webster (Oxford, 1927), 164\u201365,who suggests that the women wore phalli.\n\n. Sparta Museum from Orthia sanctuary, manner of Naucratis Painter. C. M. Stibbe, _Lakonische Vasenmaler des sechtsen Jahrhunderts v. Chr._ (Amsterdam, 1972), no. 64 = Maria Pipili, _Laconian Iconography of the Sixth Century B.C_., Oxford University Committee for Archaeology Monograph,no. 12 (Oxford, 1987),no. 179 = E. A. Lane,\"Lakonian Vase-Painting,\" _ABSA_ 34 (1933\u201334), 99\u2013189, esp. pls. 39a and 40.\n\n. According to Lane,\"Lakonian Vase-Painting,\"160, there are four women and two men. Lane points out that since allusions to wine are absent, the vase was connected to the worship of Artemis, not Dionysus.\n\n. Stibbe, _Lakonische Vasenmaler des sechtsen Jahrhunderts v. Chr.,_ no. 191, pl. 58 = Pipili, _Laconian Iconography,_ no. 196. My interpretation follows Pipili, _Laconian Iconography_ , 71\u201374. Uta Kron,\"Kultmahle im Heraeon von Samos archaischer Zeit,\" in _Early Greek Cult Practice: Proceedings of the Fifth International Symposium at the Swedish Institute at Athens 26\u201329 June, 1986_ , ed. R. H\u00e4gg,N. Marinatos, and G. C.Nordquist (Stockholm, 1988), 136\u201348, draws attention to the cultic nature of the scene, arguing that it depicts a meal in the sanctuary of Hera at Samos.\n\n. Thus, e.g.,Kron,\"Kultmahle im Heraeon von Samos archaischer Zeit,\" 141.\n\n. See further Pipili, _Laconian Iconography_ , 71\u201375.\n\n. On the xoanon see above and chap. 1, and Irene Bald Romano, \"Early Greek Cult Images,\" Ph.D. diss., University of Pennsylvania, 1980), 115\u201327.\n\n. A. Spawforth,\"Spartan Cults Under the Roman Empire,\" in _Philolakon: Lakonian Studies in Honour of Hector Catling_. ed. Jan Motyka Sanders (London, 1992), 227\u201338, esp. 232, notes that she is the only sacerdotal official in Roman Greece to be named in a Greek text.\n\n. J. Hondius and A. Woodward, \"Laconia: Inscriptions: Votive Inscriptions from Sparta,\" _ABSA_ 24 (1919\u201321), 88\u2013117, esp. 110\u201312, nos. 61\u201365, fig. 2, and 116 = _IG_ V.i.1573. She may have been: (1) daughter of Leotychidas and wife of Cleonymus; or (2) daughter of Leonidas II, granddaughter of Cleonymus and Cratesiclea (see chap. 4 n. 6); or yet another woman: A. S. Bradford, _A Prosopography of Lacedaemonians from the Death of Alexander the Great, 323 B.C., to the sack of Sparta by Alaric, A.D. 296_ (Munich, 1977), 452\u201353.\n\n. See further I. Kilian,\"Weihungen an Eileithyia und Artemis Orthia,\" _ZPE_ 31 (1978), 219\u201322, esp. 221.\n\n. Dawkins, _AO_ , tiles 31, figs. D and E, 51, 143, and see also 402.\n\n. Dawkins, _AO_ ,bronze dies: pp. 201\u20132 and 370,no. 169.23 inscribed to Orthia, end of 7th cent.; no. 169.24 a larger but similar die inscribed to Eileithyia, 7th or 6th cent. = Sparta Museum inv. 2147.\n\n. Pace J. Hondius and A. Woodward,\"Laconia, Inscriptions,\" 103,who do not understand the appropriateness of offering dice to Eileithyia.\n\n. J. Farrell, \"Excavations at Sparta, 1908: Archaic Terracottas from the Sanctuary of Orthia,\" _ABSA_ 14 (1908), 48\u201373, esp. 53, fig. 2.1; R. M.Dawkins _,_ \"Excavations at Sparta, 1909,\" _ABSA_ 15 (1909), 1\u201322, esp. 21; R. M. Dawkins, \"Excavations at Sparta, 1910. Artemis Orthia: The History of the Sanctuary,\" _ABSA_ 16 (1910), 18\u201353, esp. 52, fig. 18, 53; and see H. J. Rose,\"The Cult of Artemis Orthia,\" in Dawkins, _AO_ , 399\u2013407, esp. 402. For associations between Eileithyia and Artemis, see Semeli Pingiatoglou, _Eileithyia_ (W\u00fcrzburg, 1981), 54\u201355, 59, 99.\n\n. For associations between Eileithyia and Helen see M. N. Tod and A. J. B. Wace, _Catalogue of the Sparta Museum_ (Oxford, 1906), 116\u201318. For connections between Eileithyia at Sparta and in the Greek east, see Jos\u00e9 D\u00f6rig, \"Eleuth\u00eda,\" in _Sculpture from Arcadia and Laconia_ , ed. Olga Palagia and William Coulson (Oxford, 1993), 145\u201351. R. Olmos, s.v. Eileithuia in _Lexicon Iconographicum_ , vol. 3.1 (Zurich, 1986), 685\u201399, esp. 695, argues that some sculptures of fertility goddesses (including Sparta Museum 364) have been incorrectly identified as Eileithyia.\n\n. E.g. Calame, _Les choeurs de jeunes filles,_ vol. 1, 213\u201314, and passim, and see Appendix n. 17; Thomas Scanlon,\"The Footrace of the Heraia at Olympia,\" _AncW_ 9 (1984), 85; and most recently N. Serwint,\"The Female Athletic Costume at the _Heraia_ and Prenuptial Initiation Rites,\" _AJA_ 97 (1993), 403\u201322.\n\n. Paus. 5.16.6, Plut. _Mor_. 251e\u2013f, and see further, Calame, _Les choeurs de jeunes filles_ , vol. 1, 212, 326,who interprets the choruses for Hippodamia, who married Pelops, in connection with the married woman.\n\n. Jan Bremmer,\"Greek Maenadism Reconsidered,\" _ZPE_ 55 (1984), 267\u201386, esp. 283.\n\n. See chap. 2. Serwint, \"The Female Athletic Costume at the _Heraia_ ,\" 79\u201381, 83, considers the dress an Amazon costume\u2014contra Scanlon,\"The Footrace of the Heraia at Olympia.\"\n\n. Apollod. 3.10.6. According to Athenagoras _Leg._ 1.1, Phylono\u00eb was worshipped as Enodia, an epithet of Hecate.\n\n. H. W. Catling, \"Archaeology in Greece, 1975\u201376,\" _AR_ 22 (1975\u201376), 3\u201333, esp. 14, and \"Excavations at the Menelaion, Sparta, 1973\u201376,\" _AR_ 23 (1976\u201377), 24\u201342, esp. 36\u201337,on dedications to Helen alone.\n\n. P. Cartledge,\"Early Lacedaimon: The Making of a Conquest-State,\"in _Philolakon: Lakonian Studies in Honour of Hector Catling_ , ed. J. M. Sanders (London, 1992), 49\u201355, esp. 55.\n\n. See Carla M. Antonaccio, _An Archaeology of Ancestors: Tomb Cult and Hero Cult in Early Greece_ (Boston, 1995), 155, 158, 160, and chap. 1 n. 65, above.\n\n. Linda Lee Clader, _Helen: The Evolution from Divine to Heroic in Greek Epic Tradition_ (Leiden, 1976), 79\u201380.\n\n. E.g., Eur. _Hel._ 1667, Isocr. _Hel._ 61, M. L. West, _Immortal Helen_ (London, 1975), and see Appendix, below.\n\n. Calame, _Les choeurs de jeunes filles_ , vol. 1, 333\u201350, argues that the cult of Helen at Therapne emphasized her role as a wife, while the cult at Platanistas was initiatory and celebrated Helen before marriage. However, distinctions between the two cults are difficult to detect. Furthermore, the story in Herodotus (6.61) refers to an infant who was taken to Helen at Therapne.\n\n. On connections between the cults of Artemis Orthia and Helen, see Clader, _Helen_ , 74\u201377.\n\n. She was also worshipped at Rhodes as a vegetation goddess connected with trees: Paus. 3.19.10. See further Otto Skutsch,\"Helen:Her Name and Nature,\" _JHS_ 107 (1987), 188\u201393, esp. 189.\n\n. _IG_ V.i.1390.27, 32 (92\/91 b.c.e.Andania) = _LSCG_ 65. The Andanian cult in Messenia is perhaps derived from a Spartan cult. A. M. Woodward, \"Excavations at Sparta, 1908: Inscriptions from the Sanctuary of Orthia,\" 74\u2013141, esp. 125, distinguished the gynaikonomos in the Andania inscription as an official pro tempore from those who held state office as in _IG_ V.i.170.3\u20134, 209.10, and 1390.27. More recently, however, Spawforth, \"Spartan Cults Under the Roman Empire,\" 229 n. 17, argues that they were state officials. For other gynaikonomoi (whose existence is conjectured based on the expansion of abbreviations in _IG_ V.1.1314 and 1315), see Spawforth 230 n.21, and see _SEG_ (1950), XI.626\u201329 \"Catalogi gynaeconomorum.\"\n\n. Xen. _Lac. Pol._ 1.6, Plut. _Lyc._ 15.4, and see chap. 2 n. 26. Deborah Larson, _Greek Heroine Cults_ (Madison,Wis., 1995),20,68, suggests that originally the cult of the Leucippides emphasized their virginity rather than marriage.\n\n. Hesych. s.v. polia. For additional references to virgins as colts (with sexual implications), see chap. 1 n. 66, and Sarah B. Pomeroy, _Plutarch's Advice to the Bride and Groom and A Consolation to His Wife_ (New York, 1999), 47\u201348.\n\n. Paus. 3.16.1: see further Calame, _Les choeurs de jeunes filles_ , vol. 1, 323\u201333.\n\n. See further M. L. Napolitano, \"Donne spartane e teknopoiia,\" _AION_ ( _archeol._ ) 7 (1985), 19\u201350, esp. 28, and Musti and Torelli, _Pausania: Guida della Grecia,_ vol. 3, _La Laconia_ , 208\u20139.\n\n. Paus. 3.13.8\u20139, and see further Musti and Torelli, _Pausania: Guida della Grecia_ , vol. 3, _La Laconia_ , 209\u201310.\n\n. Paus. 3.13.7. On Artemis, Dionysus, and virgins: Hesych. s.vv.Dionysiades and Leukippides; Nilsson, _Griechische Feste,_ 184\u201385, 297\u201399; Calame, _Les choeurs de jeunes filles,_ vol. 1, 271\u201376, 323\u201333, and _Alcman_ (Rome, 1983), 520\u201321; Bremmer,\"Greek Maenadism Reconsidered,\" 282\u201384; and Ziehen, _RE_ II.6 (Stuttgart, 1929), s.v. Sparta (Kulte), 1452\u20131525, esp. 1495\u201396, and n. 21 above.\n\n. See further Susan Walker, \"Two Spartan Women and the Eleusinion,\" in _The Greek Renaissance in the Roman Empire_ , ed. S.Walker and A. Cameron, _ICS Bull._ suppl. 55 (1989), 130\u201341 and pls. 51\u201352, esp. 131.\n\n. _IG_ V.i.602. For her name, see A. J. S. Spawforth,\"Families at Roman Sparta and Epidaurus: Some Prosopographical Notes,\" _ABSA_ 80 (1985), 239.\n\n. _SEG_ XI.677, 1st cent. C.E., _LGPN_ 3A, s.v. Agesippia, 2nd cent. C.E. See also J. M. Cook, \"Laconia: Kalyvia Sochas,\" _ABSA_ 45 (1950), 261\u201381, esp. 270, 277, and chap. 4 n. 9, above.\n\n. Spawforth,\"Families at Roman Sparta and Epidaurus,\" 206.\n\n. See further Susan Cole,\" _Gynaiki ou Themis_ : Gender Difference in the Greek _Leges Sacrae_ ,\" _Helios_ 19 (1992), 104\u201322, esp. 114.\n\n. Robert Parker,\"Demeter,Dionysus and the Spartan Pantheon,\" in _Early Greek Cult Practice: Proceedings of the Fifth International Symposium at the Swedish Institute at Athens 26\u201329 June, 1986_ , ed. R. H\u00e4gg,N.Marinatos, and G. C.Nordquist (Stockholm, 1988), 99\u2013103, esp. 102.\n\n. Cook,\"Laconia: Kalyvia Sochas,\" _ABSA_ 45 (1950), 261\u201381, esp. 265\u201367.\n\n. One of the \"Aberdeen reliefs\"( _IG_ V.i.248), 1st or 2d cent. C.E.:Walker,\"Two Spartan Women and the Eleusinion,\" 135\u201338 and pl. 51.\n\n. C. N. Edmonson,\"A Graffito from Amyklai,\" _Hesperia_ 28 (1959), 162\u201364, suggests a connection between the festival and a list of women's names. See further Appendix.\n\n. But probably, originally, not every year: see chap. 1 n. 122.\n\n. _IG_ V.i.586.7,Memmia Xenocratia: Bradford, _A Prosopography of Lacedaemonians_ , 276, and _IG_ V.i.587.5\u20136; Pompeia Polla, early 2d cent.: Bradford, _A Prosopography of Lacedaemonians_ , 352, and Spawforth, \"Families at Roman Sparta and Epidaurus,\" 244. Riet van Bremen, _The Limits of Participation: Women and Civic Life in the Greek East in the Hellenistic and Roman Periods_ (Amsterdam, 1996), 88 n. 21, has overemphasized the isolation of the woman in the role of theoros at the Hyacinthia. This festival was open to all, and girls participated actively (see chap. 1).\n\n. _IG_ V.i.602.12\u201315: Pomponeia Callistonice, see below n. 90. _IG_ V.i.168 + 608.1\u20132 refers to another woman who officiated at an agonistic festival. Neither her name nor the name of the festival is extant: see further A. J. S. Spawforth,\"Notes on the Third Century a.d. in Spartan Epigraphy,\" _ABSA_ 79 (1984), 263\u201388, esp. 285\u201386.\n\n. Athen. 4.139d\u2013f; Paus. 3.19.1\u20135; and see further Michael Pettersson, _Cults of Apollo at Sparta: The Hyakinthia, the Gymnopaidiai, and the Karneia_ (Stockholm, 1992), 10, 38, etc., who follows Calame, _Les choeurs de jeunes filles_ , vol. 1, 305\u201319, in interpreting the activities as rites of initiation preparing girls for marriage. However, since the ages of the girls are not specified, one would have to conclude that just about anything they did until the age of marriage at eighteen was preparatory for marriage.\n\n. Eur. _Hel._ 1465\u201375,Hieronymus, _Adv. Iovinianum_ 1, 308, and see chap. 1.\n\n. See further Cartledge,\"Early Lacedaimon,\" 55.\n\n. Hondius and Woodward, \"Laconia: Inscriptions,\" 117\u201318, no. 66. Callicrateia dedicated a bell: _ABSA_ 30 (1928\u201330), 252, no. 5.\n\n. S. Hodkinson, _Property and Wealth in Classical Sparta_ (London, 2000), 293 and 301 n. 39, quoting unpublished work of Alexandra Villing.\n\n. _LGPN_ 3 s.v., 6th?\/5th cent.; M. L. Lazzarini, _Le formule delle dediche votive nella Grecia arcaica,_ Atti dell'Accademia Nazionale dei Lincei. Memorie. Classe di Scienze Morali, Stor., e Filol., ser. 8, 19 (Rome, 1976), p. 102, no. 91, middle of the 5th cent.\n\n. Athen. 13.574d. Nothing more is known about her: L. Preller, _Polemonis Periegetae Fragmenta_ (Leipzig, 1838), 48\u201349, para. 18.\n\n. Paus. 3.13.9. For connections between the cults of Aphrodite and Hera, see M. Torelli, \"Il santuario greco di Gravisca,\" _PP_ 32 (1977), 398\u2013458, esp. 437.\n\n. Paus. 17.5 also mentions the temple of Aphrodite Areia (Warrior) on the citadel near the temple of Athena.\n\n. The Armed Aphrodite attracted a great deal of attention from non-Spartans, and was the subject of epigrams and rhetorical exercises: Gow and Page, _The Greek Anthology: Hellenistic Epigrams_ , vol. 2, 334\u201335.\n\n. Paus. 3.15.11. According to Hesiod (fr. 176 Merkelbach and West), Tyndareus had ignored Aphrodite in his sacrifices. In revenge, the goddess had made his three daughters adulterous. The chains were probably a compensatory gift.\n\n. _Div_. _Inst._ 1.20, 29\u201332. Fritz Graf, \"Women,War, and Warlike Divinities,\" _ZPE_ 55 (1984), 250, sees the Armed Aphrodite as reversing gender roles. This interpretation may be true in most parts of the Greek world: however, as an exception note Spartan women's bravery at the invasion of Pyrrhus: Plut. _Pyrr._ 27.4, and see chap. 4.\n\n. Cf. the story of the poet Telesilla who armed the women of Argos and defeated Spartan invaders: Paus. 2.20.8\u201310, Plut. _Mul.Virt._ 245c\u2013e, see also Herod. 6.77.2.\n\n. They were probably also associated with Helen in a cult at the Phoebaeum near Therapne: Paus. 3.20.2, and see further Musti and Torelli, _Pausania: Guida della Grecia_ , vol. 3, _La Laconia_ , 253\u201354.\n\n. A red figure vase of the late 5th cent. manufactured in the Peloponnese depicts Helen's birth. Her upper torso and arms rise from the egg. The eagle of Zeus flies over her head and Leda flies toward the left. Unlike scenes ofHelen's birth in Attic vase painting, Tyndareus is not shown at all. See S. Karouzou,\"He Helene tes Spartes,\" _AE_ 1985 (1987), 33\u201344, esp. 37, and pl. 6.\n\n. See further Sarah B. Pomeroy, _Families in Classical and Hellenistic Greece_ (Oxford, 1997), 73\u201375, and n. 90 below, for references to seven illustrious women of the 2d\u20133d centuries named Callistonice.\n\n. See above, n. 54,Hipparchia ( _IG_ V.i.611, 1st cent. B.C.E.), etc., and chap. 1 nn. 75\u201376.\n\n. See further Robert Palmer, \"Roman Shrines of Female Chastity from the Caste Struggle to the Papacy of Innocent I,\" _RSA_ 4 (1974), 113\u201359.\n\n. For prosopographical studies of these inscriptions, see Bradford, _A Prosopography of Lacedaemonians_ ; Spawforth, \"Notes on the Third Century _A.D_. in Spartan Epigraphy\"; Spawforth, \"Families at Roman Sparta and Epidaurus\"; and Spawforth, \"Spartan Cults Under the Roman Empire.\"\n\n. P. A. Cartledge and A. J. S. Spawforth, _Hellenistic and Roman Sparta: A Tale of Two Cities_ (London, 1989), 99.\n\n. See the warning against reading religion from the Roman period back to the archaic and classical periods in Robert Parker, \"Spartan Religion,\" in _Classical Sparta_ , ed. A. Powell (Norman, Okla., 1989), 142\u201372, esp. 163 n. 1.\n\n. Walker,\"Two Spartan Women and the Eleusinion,\" 132.\n\n. I owe this suggestion to Lucia Nixon, and see n. 81, above.\n\n. E.g., _IG_ V.i.229, 589, _SEG_ XI (1950),Add. et Corr. 812a.\n\n. N. Kennell,\"Where was Sparta's Prytaneion?\" _AJA_ 91 (1987), 421\u201322.\n\n. Preucletia: _IG_ V.i.116, late 2d cent. c.e., and see further N. Kennell, \"The Elite Women of Sparta,\" paper delivered at the Annual Meeting of the American Philological Association, Dec. 28, 1998; abstract published in _American Philological Association 130th Annual Meeting: Abstracts_ , 84.\n\n. _LGPN_ 3A s.v. Kallistonike (6), 3d cent. c.e. A. M. Woodward, \"Inscriptions,\" 285\u2013377, in Dawkins, _AO_ , 295, and see n. 63 above and Appendix n. 70.\n\n. Anthousa, daughter of Damainetus ( _IG_ V.i.248), 1st or 2d cent. c.e.:Walker, \"Two Spartan Women and the Eleusinion,\" 132.\n\n. Spawforth,\"Families at Roman Sparta and Epidaurus,\" 200.\n\n. Freedmen of both Eurybanassa (50 B.C.E.\u201320 C.E.) and Tyndares are attested in _IG_ V.i.212.57\u201358.\n\n. P. Memmius Deximachus IV and Memmia Xenocratia; Eurybanassa and Tyndares; and P. Memmius Pratolaus III and Volusene Olympiche (3d cent.): see further Spawforth, \"Spartan Cults Under the Roman Empire,\" 231, and \"Families at Roman Sparta and Epidaurus,\" 197, 235.\n\n. For examples of children ( _paidia_ ), see _IG_ V.i.141, and see further Spawforth,\"Spartan Cults Under the Roman Empire,\" 229 n. 1.\n\n. Spawforth,\"Spartan Cults Under the Roman Empire,\" 231.\n\n. Claudia Damosthenia: _IG_ V.i.589, 2d\u20133d cent. C.E.; name not extant: _IG_ V.i.608; Julia Etearchis: _IG_ V.i.593, ca. 250\u201360 C.E.; Preucletia: _IG_ V.i.116, ca. 170 c.e.; Claudia Tyrannis: _SEG_ XXXV (1985), 337; and see further Van Bremen, _The Limits of Participation_ , 167, 348, 350.\n\n. See chap. 2 n. 27, and Pomeroy, _Families_ , 48\u201350.\n\n. E.g., Aristodama: see Sarah B. Pomeroy, _Goddesses, Whores, Wives, and Slaves: Women in Classical Antiquity_ (New York, 1995), 126, and Van Bremen, _The Limits of Participation_ , passim.\n\n. E.g., _BMC_ V.clxxvi and cxcv; _CIL_ VIII.26598;Dio 72.5; and _HA_ , _Marcus_ 26.8.\n\n. L. Robert, \"Laodic\u00e9e du Lycos: Les inscriptions,\" in _Laodic\u00e9e du Lycos: Le Nymph\u00e9e, campagnes 1961\u20131963_ , ed. J. des Gagniers et al. (Paris, 1969), 247\u2013387, esp. 316\u201322.\n\n. Kennell,\"The Elite Women of Sparta.\"\n\n. Poralla2, 416, Teisamenos (4), and Dawkins, _AO_ , 295\u201396.\n\n. _IG_ V.i.540 ( _SEG_ XI.797), ca. 225\u201350 c.e., and see further Spawforth, \"Notes on the Third Century _A.D_. in Spartan Epigraphy,\" 276\u201377.\n\n. E.g., _IG_ V.i.587.6, and see further Helen North, _Sophrosyne: Self-Knowledge and Self-Restraint in Greek Literature_ (Ithaca, N.Y., 1966), esp. 252\u201353, and Sarah B. Pomeroy, _Xenophon, Oeconomicus: A Social and Historical Commentary_ (Oxford, 1994), 275, 279, 344.\n\n. E.g., Leonis: _IG_ V.i.610, and Hipparchia: _IG_ V.i.611, 1st cent. b.c.e.\n\n. _SEG_ XI.626.1\u20132.\n\n. See further A. J. S. Spawforth, \"Balbilla, the Euryclids, and Memorials for a Greek Magnate,\" _ABSA_ 73 (1978), 249\u201360 and pls. 34\u201335, esp. 254.\n\n. Seneca, _QN_ 4.2.13, and see further T. C. Brennan,\"The Poets Julia Balbilla and Damo at the Colossus of Memnon,\" _CW_ 91 (1998), 215\u201334, esp. 218.\n\n. Thus Brennan,\"The Poets Julia Balbilla and Damo at the Colossus of Memnon,\" 227.\n\n. Diana Kleiner, _The Monument of Philopappos in Athens_ (Rome, 1883), esp. 52, 95.\n\n. E. L. Bowie, \"Greek Poetry in the Antonine Age,\" in _Antonine Literature_ , ed. D. A. Russell (Oxford, 1990), 53\u201390, esp. 62, suggests she was the equivalent of Hadrian's Antinous; and see chap. 1, above.\n\n. E.g., _manteis_ in _IG_ V.i.142.5, 7, _IG_ V.i.599.17\u201318, and see further Spawforth, \"Spartan Cults Under the Roman Empire,\" 228, 234.\n\n## CONCLUSION\n\n## GENDER AND ETHNICITY\n\nOf all the earth, Pelasgic Argos is best, and Thessalian horses, and Lacedaemonian women, and men who drink the water of beautiful Arethusa.\n\nParke and Wormell, _The Delphic Oracle_ , no. 1\n\n### Gender and Ethnicity\n\nDo knowledge and consideration of Spartan women change our overall view of Spartan society and institutions? In what ways were Spartans different from other Greek women, and how does this difference contribute to our ideas about Sparta as a whole? Studies of Spartan ethnicity have not heretofore used gender as a defining category, except in a negative way so as to exclude women. In this brief epilogue, I will attempt to remind the reader of some of the points made in the book and draw some conclusions. I will not review change over time, nor repeat the details, arguments, and citations of ancient sources that can be found in the preceding chapters.\n\n### The First Spartan Woman\n\nEveryone in ancient Greece and every educated person in the western world even nowadays knows the name of at least one Spartan woman: Helen. Legends about Helen have helped shape the image of Spartan women. She was the most beautiful woman in the world. She was also wealthy, and she dominated men. Guilty of flagrant adultery, she nevertheless was able to subjugate her infuriated warrior husband. Although Aristotle does not mention Helen, he repeats most of these attributes in his denunciation of Spartan women and their effect on the community. He considered that gender relations in Sparta were the reverse of what they should be in a normal Greek society like that of Athens.\n\n### Beauty\n\nFrom earliest times, Sparta was known as a land of beautiful women\u2014in Homer's words, _Sparte kalligynaika_ ( _Od._ 13.412). Spending time out of doors in the nude meant that women were exposed to public scrutiny from the time when they were very young. They competed with their peers not only in formal athletic events, but also in the eyes of their beholders and in their own judgment. Young girls learned to evaluate the beauty of other girls, and to compare their own appearance with that of their peers. In Alcman, _Partheneion_ 1 (57\u201358), the chorus decide that Agido is first in beauty, and Hagesichora is second. Three hundred years later, Theocritus captured the sentiment. The 240 girls who race in honor of Helen declare that when they compare themselves to Helen not one of them is faultless ( _Idyll_ 18.25). Lycurgus had outlawed cosmetics, but Spartan women had mirrors, proving there can be artifice in nature. Ancient mirrors that have been identified as Laconian had a convex disk which displayed the face, hair, neck, and cleavage (see Appendix). The user would have been able to dress her hair, care for her skin, and adjust her facial expression and posture. In _Partheneion_ 1 (6, 21), the girls refer to Agido's radiance and to Hagesichora's silver face. When the color of a woman's hair is mentioned, it is blonde, like Helen's. The girls in Alcman's _Partheneia_ and the poet Megalostrata whom Alcman was said to love were all blonde.\n\nMen also prized beautiful women and sought them as brides, even breaking some of society's rules to win them. Unlike men in Athens where girls were secluded and veiled, Spartans will have had many opportunities to look over potential brides who were completely nude. Herodotus (6.61) tells the story of a young girl who was afflicted with _dysmorphia_ (misshapenness) _._ More than ugly, she may have been deformed, for her parents had forbidden her nurse to show her to anyone. The nurse was concerned that the daughter of fortunate parents was so disfigured: she carried her every day to the shrine of Helen at Therapne and prayed to the goddess to free the child from her ugliness. Helen appeared, and touched the child. Thereupon the ugliest girl grew up to be the most beautiful of all Spartan women. Helen's magic had made her lovely (Paus. 3.7.7,Herod. 6.61). When the girl grew up her beauty, like Helen's, became a curse and a cause for a bitter quarrel between men who had been the best of friends. Although she was married, the Spartan king Ariston, who had two wives already, conceived a desperate passion for her. Tricking his friend, he won the woman, who became his third wife. The competitive phrase \"most beautiful woman\" occurs in the story of the prostitute whom Cinadon was ordered to bring back to Sparta (see chap. 5). At the other end of the social scale, the wealthy Agiatis was lovelier than other Greek women (Plut. _Cleom._ 1.2, 22.1\u20132).\"More beautiful than all the other women in the Peloponnese\" also appears in the story of Xenopeithia, mother of Lysandridas, who had commanded Spartan forces at Thebes in the time of Agesilaus II.\n\nLysander, who was famous for bending the rules of proper Spartan behavior, rejected the bride he had acquired at a \"lottery,\" and planned to marry a more beautiful one. Perhaps he married the homely woman after all, and his daughters took after their mother. They were unable to find husbands not only because their father was poor, but because, as their father said, they were \"ugly\" ( _aischrai_ : Plut. _Lys._ 2.5, 30.5).\n\nIn Sparta, beautiful people were highly esteemed: the best looking man and woman were most admired (Heraclides Lembus in Athen. XIII.566a = _FHG_ III.168). Since Homer ( _Od._ 6.102\u20137) described Nausicaa as towering over her handmaidens like a palm tree, and in the visual arts gods were depicted as taller than mortals, Greeks considered height an attribute of beautiful, noble women. Timasimbrota, who is mentioned in a fragment of archaic poetry, was as tall as a man, for she is described as resembling the golden-haired son of Polydorus in her noble stature. The Ephors fined King Archidamus for choosing a short woman, because it was expected that the children produced by the couple would look like their mother. Plutarch ( _Cleom._ 38.5) described the heroic wife of Panteus, who accompanied Cleomenes' family to Egypt, as tall and robust.\n\n### Nutrition\n\nDiet reflected ethnic difference. The height of Spartan women probably resulted not only from heredity and eugenics, but also from their generous food rations. Spartans were the only Greek women who were well fed and drank wine. Aristophanes alludes to Cleitagora, a Spartan woman poet whose name was associated with a skolion, or drinking song. A Laconian vase depicts women at at a mixed symposium. Dionysus was a god of women in Sparta.\n\nSince wine drinking by women was not approved of elsewhere in Greece, the practice took on a negative connotation among critics of Spartan women. In Plato, _Laws_ (637C), the Athenian Old Man talking to the Spartan criticizes Spartan women for licentiousness. This gratuitous and stereotypical criticism follows praise of moderate drinking by Spartan men at symposia and may allude to women's intoxication. It was known that Spartan women drank wine as part of their regular diet, but there is no evidence that they were less temperate than Spartan men.\n\n### Dress\n\nWomen's style of dress was used to characterize Dorians in general. _Doriazein_ means \"to dress like a Spartan girl\" and connotes nudity or semi-nudity. The standard Greek-English dictionary gives the meaning \"dress like a Dorian girl, i.e. in a single garment open at the side,\" and gives _doriazo_ as an equivalent of _dorizo_ \"imitate the Dorians in life.\"\n\nClothing, as well as the lack of it, marked differences between Spartans and other Greek women. Because Spartans spent time out of doors, they needed warm garments in some kinds of weather. The Dorian peplos was a heavier woollen dress than the Ionian, and had to be fastened on the shoulders by fibulae. This heavy dress had been worn by all Greek women in the archaic period. Spartan fashion was conservative. The light gauzy dresses of the Ionian style were new fashions. Herodotus explains that the change came about when a sole survivor of a battle returned to Athens and told the women that all their men had died. They killed the bearer of this devastating news with their pins, which were subsequently associated with aggression on the part of women. Thereafter the women were forbidden to wear fibulae. Indeed, some of the jewelry dedicated at the sanctuary of Artemis Orthia resembles nails with very long spikes.Wearing a Doric peplos meant always having a weapon to hand. Only when she was dressed in a man's costume as a bride was a woman disarmed.\n\n### Definition of Ethnicity by Women\n\nAs we have mentioned in chapter 3, Gorgo is the first Spartan woman who is reputed to have drawn attention to the special identity of Spartan women. When she was asked by a woman from Attica, \"Why is it that you Spartans are the only women who can rule men?\" she replied, \"That is because we are the only ones who give birth to men.\" In this story, the interlocutor assumes the special identity before Gorgo replies. About a century later, another royal woman, Cynisca, emphasized her uniqueness by declaring, \"I am the only woman in all of Greece to have won this [Olympic] crown.\"We have also mentioned the anonymous Spartan woman who was equally conscious of her ethnic distinctiveness: When an Ionian woman was proud of something she had woven (which was very valuable), a Spartan woman showed off her four well-behaved sons and said these should be the work of a noble and honorable woman, and she should swell with pride and boast of them (Plut. _Sayings of Spartan Women_ 241.9). We observe, in passing, the competitive nature of the Spartan's retort.\n\nThat Spartan women were taught to speak and were encouraged to do so distinguishes them from Spartan men, who did not debate in law courts or in their General Assembly, and from Athenians and other Greek women, who were expected to remain silent and by no means to speak to men. In the _Oeconomicus_ , Xenophon describes a young wife who was brought up so that she might see and hear and speak as little as possible.Having an enlightened view of marriage, her husband describes how he taught his wife to converse with him. More than five hundred years later, Plutarch cautions: \"the words of a modest woman must never be public property. She should be shy with her speech as with her body, and guard it against strangers, Feelings, character, and disposition can all be seen in a woman's talk. . . . A wife should speak only to her husband or through her husband.\"\n\n### Status, Autonomy, and Moral Authority\n\nAccording to several criteria that may be applied to assess the status of women, Spartans were distinctive.\n\n### _Health_\n\nSpartans must have been among the healthiest of Greek women. There are no reports that they suffered infanticide, as did some male babies. We have seen that a baby girl who was distressingly ugly or even malformed was reared. The nutrition of Spartan women was superior and their reproductive health was a matter of public concern. Prohibition of the use of cosmetics eliminated exposure to toxic substances. Marriage at a mature age produced healthy children for healthy mothers.\n\n### _Education_\n\nConsistent with the concern for women's health, of all Greek women only Spartans were given physical training as were men. They also studied _mousike_ (music, dancing, poetry).\n\n### _Sexual Expression_\n\nThe nudity of girls for athletics had sexual consequences, and girls and women alike were free to engage in homosexual relationships. Xenophon and Plutarch depict heterosexual intercourse as also desirable and pleasurable for both partners, and declare that constraints on the frequency of intercourse are beneficial for husband and wife alike.\n\n### _Control over Reproduction_\n\nWomen exercised some control over their own reproduction. According to Plutarch, Timaea thought she knew whether her child was fathered by her lover Alcibiades or by her husband. According to Xenophon, women took the initiative in husband-doubling arrangements for the sake of producing children who would inherit from more than one father. According to Cicero, Spartan women were in charge of their own fertility.\n\n### _Control over Property_\n\nProperty at Sparta was real property. It is impossible to discuss the Spartan economy and to exclude women. Women controlled real property including immovables, for Xenophon declares unambiguously that the wife who duplicates husbands wants to get possession of two oikoi ( _gynaikes . . . boulontai katechein_ ). By the fourth century and the Hellenistic period, some of the wealthiest Spartans were women.\n\n### _Influence in Society_\n\nSpartan women are rarely depicted as passive. Plutarch reports marriage by capture, but this ritual was an enactment of a previous arrangement and came as no surprise to the bride. She was carefully prepared ahead of time. In his description of the wedding, Xenophon draws attention to the wife as an active partner. The phrase _epei . . . gyne elthoi_ (\"after the woman goes\": _Lac. Pol._ 1.5) appears in his first sentence about marriage. In contrast, in descriptions of the marriage ceremony in Athens, where the father or guardian gives the bride to the groom, who takes her, the verb commonly used is _lambano_ (take, seize). Furthermore, as we have noted, Xenophon mentions the personal ambitions of the woman who wants to control two oikoi.\n\nWomen also understood and enforced societal norms. They not only spoke, but jeered at cowards and bachelors. In some cases they wielded the power of life and death over their adult sons. Women also tested newborn male babies, and female babies were simply turned over to them. Though we cannot generalize from the activities of priestesses, some of them regulated the behavior of men. At least by the Hellenistic period, the Priestess of Orthia controlled the intensity with which the boys were whipped, especially when she saw that a man wielding the whip was showing favor to a particular boy. Plutarch's _Sayings of Spartan Women_ give many examples of women's leadership and control over men throughout Spartan history.\n\nBecause of their influence and authority in society as a whole, to study Spartan women is not only to learn women's history, but also to have a more complete knowledge of Spartan history.\n\n* * *\n\n. E.g., Jonathan Hall, _Ethnic Identity in Greek Antiquity_ (Cambridge, 1997), and J.-P.Vernant, \"Entre la honte et la gloire: L'identit\u00e9 du jeune spartiate,\" _M\u00e9tis_ 2 (1987) 269\u201399, repr. in _L'individu, la mort, l'amour_ (Paris, 1989), 173\u2013209, the latter a study of the development of self-identity in the young Spartan male, and see Preface n. 5, above.\n\n. He was also called Lysanoridas. See Theopompus in Athen. 13.609b = _FGrH_ 115 F 240.\n\n. Hermippus of Smyrna = Athen. 13.555b\u2013c = _FGrH_ IV.3 1026 F 6 = fr. 87 (Wehrli). C. Meillier, \"Une coutume hi\u00e9rogamique \u00e0 Sparte?\" _REG_ 97 (1984), 381\u2013402, esp. 388\u201389, suggests that this story was invented to discredit Lysander. See further chap. 2 n.32.\n\n. See further Sarah B. Pomeroy, _Xenophon, Oeconomicus: A Social and Historical Commentary_ (Oxford, 1994), 306.\n\n. _PMGF_ 5, fr. 2, col. ii.17\u201318. For other interpretations of these fragmentary lines, see M. L. West,\"Alcmanica,\" _CQ_ 15 (1965), 188\u2013202, esp. 189\u201390.\n\n. Athen. 13.566a\u2013b; Theophrastus in Plut. _Ages._ 2.3, _De educ. puer._ 1d.\n\n. Robert Parker, \"Spartan Religion,\" in _Classical Sparta: Techniques Behind Her Success_ , ed. A. Powell (Norman, Okla., 1988), 142\u201372, esp. 151.\n\n. Thus N. R. E. Fisher,\"Drink,Hybris and the Promotion of Harmony in Sparta,\" in _Classical Sparta: Techniques Behind Her Success_ , ed. A. Powell (Norman, Okla., 1988), 26\u201350, esp. 29\u201330.\n\n. For Spartan temperance: Critias fr. 6 (Diels-Kranz) = Athen. 10.432d, and Plut. _Lyc._ 12.7.\n\n. _LSJ_ s.v.Doriazo.\n\n. Herod. 5.87. On the fictitous nature of this aetiology, which emphasizes the cruelty of Athenian women, see Thomas J. Figueira,\"Herodotus on the Early Hostilities Between Aegina and Athens,\" _AJP_ 106 (1985), 49\u201374, esp. 50, 56\u201357 = _Excursions in Epichoric History: Aeginetan Essays_ (Lanham,Md., 1999), 35\u201360, esp. 36, 41\u201342.\n\n. See further I. Jenkins,\"Dressed to Kill,\" _Omnibus_ 5 (1983), 29\u201332.\n\n. Dawkins, _AO_ , 382\u201383 and pl. CCII. Dawkins published another bronze fibula in \"Artemis Orthia: Some Additions and a Correction,\" _JHS_ 50 (1930), 298\u201399 and pl. XI.1.\n\n. Plut. _Sayings of Spartan Women_ = _Mor_. 240e5; cf. Plut. _Lyc._ 14, and _Sayings of Spartans_ , 7e13.\n\n. See chap. 1.\n\n. Xen. _Oec_. 7.6, 10, and see further Pomeroy, _Xenophon, Oeconomicus_ , ad loc. Laura McClure, _Spoken like a Woman_ (Princeton,1999), 164\u201368, discusses the negative reflections of Spartan's women's outspokenness in Athenian literature and in Aristotle.\n\n. _Advice to the Bride and Groom_ , 31\u201332, and see further Sarah B. Pomeroy, ed., _Plutarch's Advice to the Bride and Groom and A Consolation to His Wife_ (New York, 1999), 38, 41, 42, 53, 104, 106, 119\u201320, 126, 151, 160.\n\n. On the ekdosis in the Greek world, see most recently A.-M.V\u00e9rilhac and C. Vial, _Le mariage grec du VIe si\u00e8cle av. J.-C. \u00e0 l'epoque d'Auguste_ , _BCH_ suppl. 32 (Paris, 1998), 254\u201358.\n\n## APPENDIX\n\n## SOURCES FOR THE HISTORY OF SPARTAN WOMEN\n\nIn this survey I will discuss important works of literature and material remains that shed light on Spartan women. My purpose is not to include everything that could be said about a specific author or work of art, nor to raise formal literary, philological, or esthetic questions, but rather to focus on the ways in which the work in question can be exploited in the quest for the history of Spartan women. I will also try to draw attention to the limitations of the genre of the work, the bias of the author or craftsman, and the potential problems of modern interpretation. The amount of space devoted to the various sources discussed in this chapter is commensurate with their importance in constructing the history of Spartan women, rather than their general place in the hierarchy of great works of classical literature and art. Nor will this survey be exhaustive: not every primary source cited in the book will be discussed in this chapter. The sources will be examined, as far as possible, in chronological order: first, written texts; second, art and archaeological evidence. The sources range in date from the archaic through the Hellenistic and Roman periods. The chronological picture, however, is not straightforward, because some of the literature was performed again and the artifacts reused many years after their creation. The record is uneven, with certain kinds of evidence plentiful in one period, but rare or absent in another. The Spartans were xenophobic and did not use precious metals (at least openly) from the late archaic period until about the fourth century B.C.E. Consequently, for a long period in their history they could not pay foreign poets or artisans for their work, and there is a dearth of artistic material from this time. Furthermore, although statistical corroboration is lacking, the number of inscriptions dealing with women from the Roman period seems disproportionate to the size of the population. As is true for major archaeological monuments, the remains of inscriptions from the Roman period are more likely to have survived than earlier artifacts.\n\nWRITTEN SOURCES\n\nAlmost all our major sources on Spartan women are the work of authors who were not Spartans, and who lived much later than the time periods they discuss. With the exception of a few poems by Alcman, Plutarch's _Sayings of Spartan Women_ , and some brief inscriptions, none of the works focus on women; rather remarks about women usually constitute a small fraction of a text devoted to another subject. Furthermore, there are few inscriptions concerning private matters in pre-Hellenistic Sparta, for, in addition to the kings, only men who had died in battle, or women who had died in childbirth were permitted to have inscribed epitaphs (see chap. 3). Most important, in Sparta there are no counterparts to the private orations that supply so much information about Athenian women and gender relationships. The orations are also a crucial source for Athenian law. In contrast, Spartan law and any debate over it were not written down, though there are occasional remarks scattered in other sources. For example, Herodotus (6.57.4\u20135) reports that the kings had jurisdiction over the marriage of an heiress; Philo ( _De spec. leg._ 3.4.22) supplies the odd bit of information that at Sparta children by the same mother were permitted to marry, while at Athens only children by the same father could do so; and Aelian in _Various Histories_ lists customs and laws including those relieving fathers of three or more children from various taxes and obligations (6.6 and see chap. 3). Xenophon's _Spartan Constitution_ 2 and Plutarch's _Life of Lycurgus_ give substantial accounts of the legislation attributed to Lycurgus. The use of such sources for legal history, however, poses many problems. Most of this information is reported without chronological reference. Aelian neglects to state when some of the practices he records were in force. Thus Aelian (6.6) and other sources tell us that women married without dowries, while different sources report that they had dowries and often quite large ones (see chap. 4). Even Xenophon ( _Lac. Pol._ 14.1) is uncertain whether the laws of Lycurgus were unchanged in his own day. Nevertheless, scholars have deduced some features of Spartan law. Analogies with the laws of Gortyn, in Crete, a Dorian polis whose laws are extant in a lengthy inscription written in the fifth century, but recording archaic law, have been useful.Women at Gortyn could inherit, control, and bequeath property. Laws governing real property and inheritance have also been inferred from descriptions of results: thus Aristotle's information on heiresses and land tenure has been used to determine women's economic position (see chap. 4). As is the case in general for Greek history, almost all our evidence concerns the upper class: little is known about female perioeci or helots (but see chap. 5). In any case, because the upper-class (Spartiates) constituted a ruling aristocracy, they were not a mere political minority as they would be in a democratic city.\n\n### _How Unique Was Sparta?_\n\nAlmost all extant ancient literature about Spartan women was written by non-Spartans who never lived in or even visited Sparta. The Greeks tended to organize their thoughts in polarized categories; for example, Greek\/barbarian, male\/female, and Athenian\/Spartan. This way of thinking exaggerated differences between the categories being compared. As a consequence, differences between Spartan and Athenian women were emphasized. Following ancient precedent, throughout this book, where relevant, I too have compared Spartan women to Athenian. It must be admitted that two highly unusual Greek cities are being compared: neither one might be described \"average\" or \"ordinary.\" This comparison, however, is the only one possible, for we do not have abundant information on women in other Greek cities from the archaic through the Roman periods. For these reasons, the works of Alcman, Xenophon, and Plutarch are particularly important. Alcman may have been a Spartan. Certainly he was able to write the Doric dialect. According to another tradition, he was born in Lydia, perhaps as a slave, and came to work in Sparta. This tradition links him by birth with the luxuries of Lydia. In either case, he was certainly familiar with Spartan society, and the state authorized him to be one of its most influential poets. Xenophon, as well, lived in Sparta. Plutarch visited Sparta, but, of course, as an antiquarian, not as a direct observer of the archaic, classical, and Hellenistic society that interested him.\n\n### _Women's Voices_\n\nAll the extant literature about Sparta was written by men. Nevertheless, unlike Athenian women, whose voices were silenced, Spartans may perhaps be heard through several literary sources. For example, in the poetry of Alcman, girls who are named make statements about their current situation (see below). They are not dramatizing a myth, but rather talking about themselves and other girls like them. It would have seemed most peculiar to the speakers as well as to any observer if what they said was completely unnatural or unlike statements that could be made in such a situation. Another text where a woman's voice may be heard is the epigram composed in honor of Cynisca's chariot victories (see chap. 1).Cynisca is depicted as speaking in the first person. If she did not write the epigram herself, she may well have commissioned the poet and given orders about the content. A final source for women's words is Plutarch's _Sayings of Spartan Women._ Spartan women were anything but laconic (see chap. 1).Moreover, they were not separated from their mothers and grandmothers, for they married men from their own polis. It is quite conceivable that through the oral tradition they passed down the _bons mots_ of their female ancestors. Furthermore, male descen-dents as well could have remembered the pithy statements of their mothers and grandmothers. The _Sayings_ show the wit, grace, and brevity characteristic of the Spartan style ( _Lyc._ 19.2, 21.1). Even in the modern world, where people often live in places remote from those of their ancestors, such oral traditions constitute a vital part of family histories. Thus it is quite credible that eventually some Hellenistic anthologizer compiled a collection of actual words once uttered by Spartan women and that Plutarch used this work as his source. That the statements are quoted out of their original context enhances their exaggerated and dramatic qualities.\n\n### Archaic\n\n_Alcman_ lived in Sparta in the archaic period, probably after the Second Messenian War, but before the ensuing militaristic reforms had profoundly transformed society. As we have mentioned above, his origin and exact dates are disputed. He was variously said to be Laconian and the descendant of slaves, or Lydian. Textual evidence points to the late seventh century.Archaeological evidence indicates that he died no later than 570. His tomb was near Helen's in Platanistas. Whether an autobiographical interpretation of Alcman's oeuvre is appropriate has been debated. Be that as it may, even if the poems may not provide completely reliable testimony about the life of the poet, nevertheless they constitute some of the most valuable evidence for the lives of Spartan women.\n\nChoral songs like those of Alcman were passed down through the generations at Sparta and probably elsewhere, forming a stable element in the educational curriculum. In 370\/369, after the battle of Leuctra, the victors asked the helots to sing songs of Terpander, Alcman, and Spendon (Plut. _Lyc._ 28.5). They refused, not because as helots in the fourth century they did not know these songs, but on the grounds that if they sang them they would anger their Spartan masters. Athenaeus (14.632f\u2013633a), who lived in the late second or early third century C.E., reports that even in his day the Spartans took painstaking care of their ancient songs and were well taught in them. The fragmentary nature of the poems of Alcman that are now extant is due to several reasons in addition to the obvious one that they were written more than 2,500 years ago. First, many brief quotations were preserved only in the works of later authors, especially grammarians and compilers of anecdotes such as Athenaeus. The grammarians were primarily interested in the unusual Doric dialect and in the use of meters, and do not always quote a full sentence or thought. These tantalizing tidbits of Alcman were all that were known to the modern world until the great papyrological discoveries of the nineteenth century. _Partheneion_ 1, the oldest long fragment of archaic Greek lyric poetry, happens to be preserved because in the first century C.E. someone made a copy of the text on papyrus. Sometime later the manuscript probably became worn, bits of papyrus flaked off, and its owner or his or her heirs discarded it. The papyrus was then reused to wrap a crocodile mummy. In the course of excavating at Saqqara, the Egyptologist Auguste F. F. Mariette discovered the mummy and sent the papyrus to Paris, where it is now stored in the Louvre. The first edition was edited by the Greek scholar \u00c9mile Egger in 1863. The papyrus must have belonged at first and some time later to a scholar or student. It includes a large number of marginal comments (scholia); most of these are in the same handwriting as the text. This papyrus provides a glimpse of a long tradition of scholarship on the text of Alcman. Some of the most erudite explanations are attributed to the great text editors of the third and second centuries, Aristophanes of Byzantium and Aristarchus, who worked in the library established by Ptolemy I and II in Alexandria. These scholia attempt to explicate the text, and usually include citations to other ancient authors and scholars. Their existence does not indicate that Alcman's poetry was more obscure and difficult than other archaic lyrics, but rather that the Alexandrians had deemed his work, like that of many archaic and classical authors, worthy of painstaking editorial attention.\n\nPapyri published in the twentieth century have brought to light additional fragments of Alcman's poetry. Among these, most significant for the history of women is _Partheneion_ 3, first published in 1957. The fragmentary text of this poem was discovered on a papyrus from Oxyrhynchus dated to the second century C.E. Like the Louvre papyrus, the Oxyrhynchus text has scholia written in the margins, as is apparent in the upper left corner of the fragment illustrated in fig. 9. Translations of _Partheneia_ 1 and 3 are printed in chapter 1, pages 6\u20137.\n\nFig. 9. Alcman, _Partheneion_ 3, fr. 3 ( _P.Oxy_. xxiv.2387).\n\nPhoto courtesy of the Ashmolean Museum, Oxford.\n\n_Performance and Preservation of Alcman_. The poems contain highly specific references to girls who are named and whose appearance, emotions, behavior, and effect upon other girls are described. Consequently, scholars have debated whether each poem was composed to be sung by identifiable girls at a particular occasion or whether the _Partheneia_ were originally composed to be performed repeatedly at ritual occasions. The two hypotheses are not exclusive. The poems could have been composed to be sung by the girls who are named and described the first time, and then, because of their sheer excellence as well as the lack of subsequent poetic composition, repeatedly performed at rituals by other girls who played the roles of their predecessors. Because of their use in ritual and Sparta's general conservatism, the poems will not have been changed or updated. In other words, the poems could have been an accurate reflection of erotic feelings and behavior in the days of Alcman, and have become more of a theatrical reenactment in later times. Another possibility is that they were originally written for performers who impersonated the characters named. A corollary to this theory is that Hagesichora and Agido were simply generic names for choir leaders, and that the rest of the girls were fictitious as well. This theory seems less plausible to me because of the specificity of the descriptions of the individual girls and the fact that some of the names are real names and recur in Spartan prosopography.\n\nThe poet gained circulation for his poetry by travelling to populated areas including pan-Hellenic sanctuaries and festivals. He claimed that his work was known far and wide (Ael.Arist. _Or._ 28.54 [II.159 Keil] = _PMGF_ 148 [i]). Although this Appendix is not the place to discuss the complex issues concerning the preservation of archaic poetry through oral and written traditions, it seems likely that Alcman's poetry circulated by both means.\n\nThere is evidence for the performance of Alcman's poems more than a hundred years after his lifetime. The circumstances of the performance may have changed over time. Originally the _Partheneia_ may have been performed before an audience comprised mostly of women, with the poet and perhaps a few officials as the only men present. In later times, when the poems had become part of Sparta's great heritage, men as well may have been welcome to witness the performances. It has been suggested that terracotta masks from the sanctuary of Ortheia beginning in the second half of the seventh century were used for enactments of _Partheneion_ 1 as a ritual marriage. Gold and silver were outlawed in Sparta, evidently some time after Alcman composed _Parthenion_ 1 in which they are mentioned. If the performers in later times were actually wearing the luxurious apparel described in the poem, it must have been part of a costume, distributed solely on the occasions of the performance and then returned to storage until the next performance.\n\n_Herodotus_ of Halicarnassus was born a little before the Persian War and lived until the Peloponnesian War. He wrote the history of the Persian War, with many digressions. Spartan women are mentioned in the tales about Helen; in some of the descriptions of the royal succession (e.g., 5.39: the two wives of Anaxandrides; 6.61\u201366: Ariston's three wives and the parentage of Demaratus), the judicial duties of the kings, and in a variety of anecdotes. Herodotus imparts this information straightforwardly, without approval or criticism. Scholars differ greatly in their assessment of the historical veracity of Herodotus. In my view, not all anecdotes are necessarily false. The fact that Herodotus heard stories like those about Gorgo suggests that there was at least a kernel of historical truth behind them. From Herodotus' various brief remarks, it can be surmised that the ideal Spartan woman was clever, witty, tall, and beautiful. (See esp. chap. 1 and Conclusion.)\n\n### Classical\n\n### _Dramatic Poetry_\n\nIn Athenian drama, Spartans may play the role of \"Other\" to the Athenians: they fill the conceptual space usually occupied by barbarians, and provoke questions about the assumed superiority of Greeks over barbarians. Female characters are used to exaggerate the distinctiveness of both ways of life. Thus in Euripides, _Andromache_ (e.g., 29, 194, 889), Hermione, daughter of Helen and Menelaus, is often referred to as a \"Laconian\" or \"Spartan\" woman, and herself criticizes Spartans as deceivers (445\u201353).\n\nTragic and comic dramatists alike draw attention to the semi-nudity and the physical development that was connected with the education of Spartan women. The Spartans were xenophobic in the fifth century and no Athenian poet is likely to have ever seen a Spartan woman. Perhaps the rumors spread when an embassy came from Athens, or when some ambassador from Corinth, for example, traveled to Sparta for a meeting of the Peloponnesian League and drove his chariot past a group of women exercising out of doors. Works of art that circulated beyond the frontiers of Laconia, including mass-produced bronze figurines in the shape of Spartan female athletes, and some vases that showed women engaged in activities that may have seemed scandalous (see below), probably also inspired ideas that foreigners held about Spartan women.\n\nEuripides' portrait of Hermione as vicious and self-centered in _Andromache_ , produced in the early years of the Peloponnesian War (427\u2013426 B.C.E.), reflects the anti-Spartan bias of contemporary Athens:\n\nNo Spartan girl could ever be modest even if she wanted to be,\n\nThey go outside their houses with the boys with naked thighs and open dresses\n\nand they race and wrestle with the boys. Insufferable!\n\nIt's not surprising that you don't train women to be chaste.\n\n( _Andromache,_ 595\u2013601)\n\nSimilarly Sophocles:\n\nAnd that young woman, whose tunic is still unbelted\n\naround her thigh, revealingly.\n\n(fr. 872 Lloyd-Jones)\n\nSophocles also wrote a tragedy titled _Lakainai_ (\"Laconian Women,\" frs. 367\u201368 Lloyd-Jones).\n\nIn contrast to Euripides and Sophocles, Aristophanes in _Lysistrata_ (411 B.C.E.), also produced during the Peloponnesian War, portrays his representative Spartan woman favorably, for such a portrayal suits his story of reconciliation. Lampito is physically fit, and confident in her ability to carry out the sex strike. Aristophanes' plays are the only complete comedies from the classical period now extant. Spartan women occur in other Athenian comedies in brief fragments or mere titles. Eupolis, a contemporary of Aristophanes,wrote a play titled _Heilotes_ (\"Helots\") in which he mocks them as lascivious (fr. 148, see also 385 Kassel-Austin). He also wrote a comedy titled _Spartans_ or _Leda_ (frs.60\u201362Kassel-Austin). The tradition continued in Middle and New Comedy. Alexis (ca. 372\u2013270) wrote four comedies based on myths about Helen (frs. 70\u201376) and Apollodorus Carystius (third century) or Apollodorus Gelous (fourth or third century) wrote a _Lakaina_ (\"The Spartan Woman,\" Kassel-Austin II.508\u20139, frs. 7, 8).\n\n### _Prose_\n\n_Critias_ , an Athenian aristocrat related to Plato, lived ca. 460\u2013403 and was one of the most ruthless of the Thirty Tyrants who ruled Athens at the end of the Peloponnesian War with the backing of the Spartans. He wrote poetry and prose, including a treatise on the Spartan Constitution. This work, known only through a few fragments, began with the statement:\"I start, as you see, from a man's birth. How might he become physically best and strongest? [He could,] if the man who plants his seed would exercise and eat wholesome food and harden his body, and if the mother of the child-to-be would strengthen her body and exercise.\" Critias was an associate of Socrates and a Sophist. Plato and Xenophon probably read his work, and Xenophon may have been influenced by Critias' views about the importance of the mother.\n\n_Plato_ lived ca. 429\u2013347, a turbulent period in Athenian history. Sparta had defeated Athens in the Peloponnesian War, and the restored democracy at Athens condemned Socrates to death. Like other disciples of Socrates, he found much to admire in Spartan society. His utopian works, especially the _Republic_ and _Laws,_ prescribe features of women's life that in many cases correspond to the Spartan reality, or at least to the Spartan ideal. In the _Republic_ these include, for example, a full program of education for women administered by the state; physical exercise; late marriage; prohibition of money and private property (hence of dowries); wife-sharing; and marriage \"lotteries.\" The _Laws_ are less radical: Spartan lawgivers are criticized for permitting women (who are half the society) to indulge in a luxurious, expensive, and disorderly way of life ( _Laws_ 7.806C). It is perhaps more accurate to state that Plato is not so much a source for Spartan women; rather, Spartan women were an inspiration for Plato's ideas about the roles of women in utopia.\n\n_Xenophon_ ,who lived ca. 430\u2013356, was Plato's contemporary and a student of Socrates as well. His _Spartan Constitution_ ( _Lacedaimonion Politeia_ ) is our best source on Spartan women in the classical period.Xenophon spent twenty years on an estate granted to him by the Spartans near Elis. His sons were probably educated in the agoge, for in the fourth century this education was made available to a few youths who were not full-fledged Spartan citizens. Thus he writes about Sparta from the unique perspective of a first-hand witness. Xenophon admired some aspects of Spartan society, though he was critical of others. When he wrote the _Spartan Constitution_ is unclear. Most recently it has been dated to after 371. According to an alternative theory, if the treatise was written earlier, chapter 14, in which he laments the abandonment of the way of life established by Lycurgus, may have been written later than the rest of the work. There is a contemporaneous parallel to Xenophon's change in attitude. We may compare Plato's optimism about the Spartan way of life that may be detected in the _Republic_ with the disillusionment he expresses in the _Laws_ (see above) _._ In any case, the viewpoint of Xenophon, like that of any writer, should be considered a lens through which we view his narrative. Nor did Xenophon come to Sparta as a _tabula rasa._ His ideas about Spartan women were doubtless shaped by prior influences and experiences. To start with the most obvious: he would have been in the audience at the production of _Lysistrata_ and have seen the robust and aggressive Lampito, so different from the Athenian women whom he knew. He also will have seen tragedies in which mythological Spartan women were portrayed in an unfavorable light (see above). In Athens he will have known the bright young Laconizers of his day in a group associated with Socrates, and may have read what Critias wrote about women in his treatise on Sparta. Furthermore, he will have learned much about Spartan society from Clearchus, Cheirisophus, Agesilaus, and the numerous Spartans who were engaged with him in military service in Asia over the years. It is universally acknowledged that soldiers gossip about sex and women.\n\nXenophon gives the theme of the _Spartan Constitution_ in his opening sentence, by asking how it is possible for Sparta to be the most powerful and renowned of the Greek poleis when it has such a small population ( _oliganthro-potaton_ a translation of this passage is given in [chap. 3]). He goes on to answer this rhetorical question. In brief, according to Xenophon, Lycurgus had designed social, political, and military institutions that created superb hoplites. As we have mentioned above, he states that he does not know whether the laws of Lycurgus were still unchanged in his day, because they were often disobeyed; nevertheless he describes them.\n\nFollowing a brief eulogy of Lycurgus, Xenophon resumes his discussion of reproduction with a detailed description of the rearing of girls. That Xenophon begins the _Spartan Constitution_ with the upbringing of girls underlines his view of the importance of women in Spartan society.\n\nStating that he is beginning at the _arche_ (beginning), Xenophon highlights women's role in child production. He points out that only in Sparta were girls well fed and given undiluted wine. In fact, women had easy access to quantities of wine, for they used it in bathing newborns. Xenophon criticizes the rest of the Greeks for not providing adequate nourishment for girls who are destined to become mothers of citizens. Xenophon approves of the generous food allocation of Spartan girls. As he stated in the _Oeconomicus_ (10.11), if a woman exercises she will have a better appetite and a vibrant complexion. Comparisons between Spartan women and women elsewhere in Greece may well have fostered Xenophon's admiration of Sparta.\n\n_Ephorus_ of Cyme (ca. 405\u2013330, _FGrH_ II.70) read works by the Spartans Pausanias and Lysander. Though Ephorus is extant only in fragments, he was the main source for Diodorus, who wrote in the last quarter of the first century B.C.E. Diodorus (11.45, 12.74) probably found his stories about the mothers of Pausanias (the traitor) and Brasidas in Ephorus.\n\n_Aristotle_ lived from 384 to 322 B.C.E. Although he was born at Stagira in the Chalcidice, he spent much of his life studying and teaching in Athens and adopted the Athenian model as the standard against which he measured Greek family life and gender relations. Aristotle doubtless discussed women in his treatise _The Constitution of the Spartans_ which is no longer extant. That Aristotle disapproved of the Spartan way of life is apparent from passages in other works. In the _Politics_ (1333b5), he criticizes other authors including a certain Thibron, a Spartan who had written in the first quarter of the fourth century in praise of the Lycurgan system. Of course, Thibron wrote just after his polis had won the Peloponnesian War, and Aristotle wrote after the defeat at Leuctra. A lengthy critique of gender relations at Sparta appears in the _Politics_ (1269b\u20131270a6 a translation of this passage is given in [chap. 4]). Aristotle believed that the basic oikos (family, household, estate) consisted of an adult male, who played the roles of husband, master, and father; an adult female, who played the role of wife and mother; a child; and a slave. The master of the house was the dominant figure in all his three relationships. This hierarchy existed by nature and any perversion of it was monstrous. That Spartan women enjoyed authority in the oikos and owned and managed property appeared to him outrageous. He echoes Plato in the _Laws_ (7.806C, see above) when he angrily declares that the women indulged in all sorts of luxury and licentiousness ( _Pol._ 1270a1: _anesis_ , see also Arist. _Rhet._ 1361a) because they would not accept the laws Lycurgus had designed for them and he decided not to continue the attempt _._ Plutarch ( _Lyc._ 9), however, rejected Aristotle's view (see below).\n\n### Hellenistic\n\nWith the important exception of Polybius, all the Hellenistic historians surveyed below are extant not in complete or nearly complete works, but only through one or at most a few fragments cited by later authors, chiefly Plutarch.\n\n_Phylarchus_ of Athens (?) lived in the third century B.C.E. and gives the history of Sparta from the death of Pyrrhus (272) to the death of Cleomenes III (220\/219). He admired Agis and Cleomenes and dramatizes the importance of women in furthering their revolutionary program. Phylarchus was a principle source for Plutarch's _Lives_ of Agis, Cleomenes, and Pyrrhus, and for his favorable desciptions in these _Lives_ of the behavior of royal women including Archidamia, Agesistrata,Agiatis, Chilonis, Cratesicleia,and Cratesicleia's companion, the wife of Panteus, whose name Plutarch does not report. All the women, even Chilonis, wife of Cleonymus, who fell in love with her stepson Acrotatus, are presented favorably. Phylarchus was also Plutarch's source for his description of the bravery of Spartan women when Pyrrhus besieged Sparta (Plut. _Pyrr._ 27\u201330). In the Hellenistic period, the exaltation of heroines was a feature of the history of other ethnic groups as well. For example, among the Ptolemies, Arsino\u00eb II (ca. 316\u2013270) and Berenice II (ca. 273\u2013221), and among the Jews, Judith and Esther (fictitious heroines, though with some historical features), were prominent.\n\n_Sosibius_ of Lacedaemon, a historian, who worked in Alexandria during the reign of Ptolemy II, wrote a treatise or commentary _On Alcman_ in at least three books, and other works about Sparta including _On the Sacrifices in Lacedaemon_ and _On Mimes in Laconia._\n\n_Teles_ ofMegara was a Cynic philosopher (fl. ca. 235 B.C.E.) whose work is preserved in fragments in Stobaeus. Some of Plutarch's _Sayings of Spartan Women_ echo Teles'anecdotes about Spartan mothers. Either Plutarch read Teles or they used a common source.\n\n_Polybius_ (ca. 200\u2013118 B.C.E.) was aristocratic and anti-Spartan. He attributes to the Spartans behavior including polyandry (12.6b.8) that other Greeks would consider bizarre. He is a hostile source for the reports about Nabis (d. 192 B.C.E.), his cruel wife Apega, and the iron maiden, and for the ways in which both Nabis and his wife humiliated women (13.6\u20137, 18.17, and see chap. 4).\n\n_Hellenistic Epigrams_ : Dioscorides, Nicander, Tymnes, and perhaps Asclepiades of Samos wrote epigrams about Spartan mothers that were later incorporated into Plutarch's, _Sayings of Spartan Women._ 40 Dioscorides commemorates a mother who lost eight sons in one battle, Nicander a mother of seven who lost six, and Tymnes a mother who slew her cowardly son.\n\n_Polycrates_ was probably Hellenistic, since he predated Didymus (fl. ca. 40 B.C.E.), who quotes him. Polycrates wrote a work titled _Lakonika_. One paragraph is extant: a vivid description of the Hyacinthia (see chaps. 1 and ).\n\n_Nicolaus of Damascus_ , who was born ca. 64 B.C.E., wrote an immense universal history in the Peripatetic encyclopedic style. He gives odd bits of information about Spartan women: for example, husbands compel their wives to have children by well-built Spartiates and foreigners. Yet, according to Xenophon, who must have have had more accurate knowledge than Nicolaus, women were not compelled, but rather were willing to bear children by Spartan men other than their husbands. Xenophon does not mention foreigners (see chap. 3).\n\n### Roman Period\n\n### _Greek Authors_\n\n_Plutarch_ lived 46\u2013ca. 120 C.E., when Greece was a province of the Roman Empire. He is the author of the largest portion of extant ancient writing on Spartan women and has also had more influence than any other ancient author in shaping ideas about Sparta held by later generations to the present.\n\nPlutarch's writing about Spartan women is concentrated in several of the _Lives_ and in one work of the _Moralia_ : the _Life of Lycurgus_ ; the _Lives_ of Agis and Cleomenes; and the _Sayings of Spartan Women_. There are also brief remarks scattered among his other works _._ The most important of these texts, the _Life of Lycurgus,_ was probably written between 97 and 110 C.E., nearly one thousand years after the date generally attributed to the legendary Spartan lawgiver. Plutarch visited Sparta and was an eyewitness at the whipping ceremonies held in honor of Artemis Orthia ( _Lyc._ 18.1); he reports that he saw many young men die. There, he also conducted research in the official Spartan archives which must have included records of oracles, official documents, victor lists, names of priests and magistrates, and the like. In the Roman period there were officials in charge of preserving and interpreting Lycurgan customs and laws. Some of the cult organizations kept their own records, and we can assume that, inspired by his profound interest in religion, Plutarch consulted these.\n\nPlutarch must also have had a huge library of his own as well as an excellent memory and in his writing on Spartan women, he cites many philosophers, poets, and ancient historians from the archaic through the Hellenistic period as his sources. These include Ibycus, Herodotus, Sophocles, Euripides, Plato, and Aristotle. Although he does not repeat Xenophon's name explicitly in the context of his discussions of women, Plutarch's views are generally consistent with those of Xenophon. Like Xenophon, Plutarch was optimistic about the human potential for moral improvement, especially through education ( _Advice to the Bride and Groom_ , 145c). Both believed that virtue is the same in women and men. Both felt that the goal of human existence was to live the good life in a way that would prove beneficial to oneself, one's family, and one's state. Both thought that child production was not the only purpose of marriage. Plutarch ( _Comp. Lyc. Num._ 4.1) believed that Spartan marriage customs were conducive to a relationship characterized by goodwill rather than hatred. Moreover, he asserted that the educational system designed by Lycurgus was ennobling for women inasmuch as it gave them a share in the arena of virtue and ambition ( _Lyc._ 14.4). Plutarch appears somewhat ambivalent, if not self-contradictory, since he also prefers the Roman practice of marrying girls at twelve or younger, for it enabled the husband to mold the wife's character. Spartan marriage practices were better for child production, but Roman practices were better for marital harmony. Though in the _Life of Lycurgus_ (14) Plutarch seems to approve of the physical and moral educational program designed for women by Lycurgus and to reject Aristotle's critical analysis, in the _Comparison of Lycurgus and Numa_ (3.3\u20135) he declares that Spartan women were too bold and masculine and offered opportunities for poets to criticize them. He quotes Ibycus, Euripides, and Sophocles on the girls'uncovered thighs (see above). In any case, his views of women and marriage are not a direct reflection of gender relations as they existed in Roman Greece in his day, where women certainly enjoyed more authority and independence from men than readers confined to Plutarch's works would surmise. For example, he tells the story of the Spartan girl who was asked whether she made overtures to her husband, and replied,\"No, but he has come to me\"( _Advice to the Bride and Groom_ , 140c, _Sayings of Spartan Women_ , 242c). Plutarch endorses the wife's reluctance to take the initiative, preferring her husband to make the advances. On the other hand, he does approve of the Spartan women who goad their men to virtuous and brave actions or who set a good example themselves when the men appear to waver. In fact, Plutarch's Greek heroines are good competition for some of Rome; for example, the mother and sister of Coriolanus (Livy 2.40) and Arria (Pliny the Younger, _Epistles_ , 3.16; Martial, _Epigrams_ , 1.13), whose Stoic deaths resembled those of Cratesicleia and Panteus' wife.\n\nPlutarch was not a historian; rather, he wrote philosophical and biographical works. Doubtless because Hellenistic writers were interested in women and personal details, in writing about Spartan women Plutarch refers most frequently to Hellenistic historians, some of whom have been mentioned in the survey above. Since Plutarch was not a historian, chronological precision is not a high priority for him (see Preface). Not only is he naive about the Spartan's reinvention of their own history, but he is also an active participant in the creation and perpetuation of the Spartan legend. The Sparta of Plutarch's day was a living museum, a theme park, in which some features of life in Sparta before Roman domination were revived or reenacted perhaps to the point of exaggeration or charicature. For example, Roman tourists accustomed to gladiatorial contests came to witness the spectacle of the boys enduring whipping until they met death. Whether such contests had ever been conducted with so much cruelty before the Hellenistic and Roman periods is questionable.\n\n_Pausanias_ wrote a guide to Greece in the second century C.E.Women are not prominent among his interests, though he gives some tantalizing bits of information. Thus he reports that the length of the stadion is shortened for girls who race at the Heraea, but does not say why or how this alteration is done. He does give some useful information on women, particularly on religious practices, on commemorative statues erected by or in honor of women, and historical and mythological anecdotes in which women are the protagonists. Thus he mentions that there is an island off the Peloponnese where Helen and Paris first had intercourse (3.22.1).Some women are remarkable for deeds usually performed by men. At Sparta, a statue of Artemisia stood along with images of Mardonius and other Persians in a stoa built with spoils from the Persian War (3.11.3).\n\n_Athenaeus_ of Naucratis (second\u2013third century C.E.) wrote a lengthy work, the _Sophists at Dinner_ , a pastiche of valuable earlier sources including Sosibius and Polycrates.\n\n### _Latin Authors_\n\nFor the history of women, the most significant aspect of the Spartan tradition or mirage in authors who wrote in Latin is the emphasis on patriotic, self-sacrificing mothers. This focus derives from the Roman interest in motherhood. Intellectuals and politicians were particularly concerned with the declining birthrate among the Roman elite. They blamed the \"new-style\"Roman matron for abandoning the traditional role of mother. Barbarian and non-Roman women were cited as exemplars. Thus the patriotic Spartan mothers of the past are mentioned in Ovid (43 B.C.E.\u201317? C.E.);Valerius Maximus (first century C.E.); Aelian (ca. 170\u2013235 C.E.); Sextus Empiricus, a Sceptic (end of the second century C.E.); Himerius of Prusa (fourth century C.E.); Palladas, a schoolteacher in Egypt (fifth century C.E.); and Julianus (sixth century C.E.), who writes of the armed Aphrodite who prompts Spartan mothers to bear brave warriors. There is little that is new in these anecdotes and quotations from Spartan mothers; for the most part they echo one another and Plutarch as well. Cicero (106\u201343 B.C.E.) also writes of the Spartan mother who nobly sent her son to his death for his country, but he is unique among these Latin authors in observing that the Spartan women of his own day were no longer so willing to bear children as their foremothers had been.\n\n### _Byzantine Dictionaries_\n\nMuch esoteric information about Spartan customs can be gleaned from explanations of unusual words in encyclopedic works by authors such as Pollux of Naucratis (second century C.E.), Hesychius of Alexandria (fifth century C.E.), and the collection called the _Suda_ (tenth century). These compilations, though valuable, pose special problems for the historian for there is generally only one citation for each word and it is given out of context. (See, e.g., Hesychius on _Brudalicha_ in chap. 6 n. 16).\n\n### Prosopographical Problems\n\nCompared with the names of Greek men, relatively few women's names are known. A reluctance to name respectable women, at least while they were alive, was a feature of Athenian etiquette. In Athens, however,women were named on their tombstones. In contrast, there are few inscriptions concerning family matters in pre-Hellenistic Sparta, for, in addition to the kings, only men who had died in battle or women who had died in childbirth were permitted to have inscribed epitaphs.\n\nIn Sparta, there was no stigma attached to giving women's names in public. Considering how few words are extant in all of Alcman's poetry combined, the oeuvre includes many names. One of the names in the _Partheneia_ appears in the work of an archaic poet outside Sparta. Mimnermus of Colophon and Smyrna had written a collection of elegies titled _Nanno,_ after his beloved (Hermesianax in Athen. XIII.597f, Strabo 14.1.28 [643]). For Damareta, compare Damatria in Plutarch, _Sayings of Spartan Women_ , 241.1, and in an epigram by Tymnes, who worked in the third century B.C.E. Astaphis was a name used also by a man ca. 428\u2013421.Arete, the name also of the Phaeacian queen in Homer's _Odyssey,_ reappears much later in the name of Pomponia Callistonice Arete.  Some of the names indicate that the girls were members of the royal houses. The prefix \"Ag\" that appears in the names of Agido and Hagesichora is common in royal nomenclature, and suggests that they were Agiads, related to one line of Spartan kings. Timasimbrota (5 fr. 2 col. ii, line 16) is perhaps descended from the Spartan king Leotychidas (ca. 625\u2013600).\n\nThe names of twelve women are inscribed on a rooftile from the second half of the third century B.C.E., perhaps in connection with the cult of Apollo and Hyacinthus at Amyclae. Nevertheless, we still have very few names of Spartan women, for most of the literary sources are non-Spartan. For example, Herodotus (5.39\u201341) speaks of the wives of Anaxandridas, but does not name them. Though Agesilaus was an old friend, Xenophon does not give the names of his wife and daughters. Xenophon's etiquette was that of an Athenian gentleman in the classical period: he was too polite to ask Agesilaus, or if he knew the names he was reluctant to publicize them. Plutarch ( _Ages_. 19.10) discovered the names. Xenophon does name Agesilaus' sister Cynisca ( _Ages_. 9.6), but she was notorious because of her horseracing (see chap. 1). Though he refers to them, Plutarch does not give the names of Lysander's daughters ( _Lys_. 2.5, 30.5).\n\nThe problem is exacerbated by the neglect of women by modern scholars. For example, before the publication of vol. 3A of _A Lexicon of Greek Personal Names_ ,on _The Peloponnese, Western Greece, Sicily, and Magna Graecia_ ,  the standard Spartan prosopographies  omitted some women, for example, Thylacis and Astumeloisa (Alcman, _Partheneia_ 1.72, 2.64), Argeia, wife of King Aristodemus, and Euonyma, Eirana, Callicrateia (dedicators to Athena), or listed them under the name of their father, although the primary source for the woman's name may not have given the patronymic.\n\nA related problem is the tendency of some scholars to interpret evidence in a narrow manner so as to eliminate any connection with women. A case in point is the Gymnopaideia (\"Festival of Nude Youths\"). The Greek word _paides_ can mean \"boys,\" or \"slaves,\" but it can also refer to children of both sexes. Those who understand the meaning of Gymnopaideia in the limited sense believe that only boys attended the festival; in contrast, those who take the word in an inclusive sense would have nude girls and boys at the festival together.\n\n### Secondary Sources\n\nA book that continues to exert a tremendous influence on Spartanologists is F. Ollier, _Le mirage spartiate_ , volume 1, _\u00c9tude sur l'id\u00e9alisation de Sparte dans l'an-tiquit\u00e9 grecque de l'origine jusqu'aux cyniques_ , and volume 2, _\u00c9tude sur l'id\u00e9alisa-tion de Sparte dans l'antiquit\u00e9 grecque du d\u00e9but de l'\u00e9cole cynique jusqu'\u00e0 la fin de la cit\u00e9_ (Paris, 1933\u201343).\n\nDespite their size and inclusiveness, however, the volumes offer little comment on women. Neverthless, it is illuminating to review the few categories in which Ollier distributes some ancient testimonia about women. Under \"la vraie Sparte,\" Ollier (vol. 1, 34\u201335) includes women's gymnastic program, their pride, and their influence in the public sphere. Under \"les forces adverses\" to the idealization, he (vol. 1, 64\u201367) mentions the scandalous sexual practices of men and women alike. In his analysis of Aristotle's views of Sparta's faults Ollier (vol. 1, 302\u20133) draws attention to an absence of governmental controls on women and to economic problems resulting from women's ownership of property. In volume 2, Ollier (50\u201351) detects Cynic influence in some of the quotations attributed to Spartan women. He argues (194\u201397, 210\u201315) that Plutarch does not idealize Sparta in all his works, but that he certainly does so in the _Life of Lycurgus._ Ollier points out (212) that in his idealization of Lycurgus and the simple, austere way life he designed for Sparta, Plutarch accepts features such as breaches in sexual monogamy that he does not sanction elsewhere in his writings. Ollier, however, unlike some more recent scholars, does not argue that such practices never existed or that reforms attributed to Lycurgus were actually innovations of the fourth century and Hellenistic period.\n\nIn 1979, Paul Cartledge published a solid historical survey of what were then regarded as facts: _Sparta and Lakonia_. Cartledge, writing much later with A. J. S. Spawforth, described the gradual degeneration of the Lycurgan polity and dated the \"normalization\"to the later fifth century.More recently, perhaps influenced by deconstruction and French literary theory, he has examined Sparta as a utopia, stating that \"practically all our detailed evidence for what they were 'really' like comes from within the mirage.\" The influence of Marxism and of M. I. Finley, who taught many ancient historians at Cambridge, has been in large part responsible for the popularity of the study of Spartan history through the lens of the mirage. Finley ignored women in his analyses of Greek economy, slavery, and citizenship. Thus it is no surprise that, as was customary among mainstream ancient historians until the end of the twentieth century, Cartledge did not devote much space to women in his books.\n\nThe most frequently cited and admired article on this subject, however, was published by Paul Cartledge in 1981. In this article, which is now referred to as a \"classic study,\" Cartledge gives priority to the testimony of Aristotle over that of Xenophon and Plutarch. Despite (what seems to the modern reader) his misogynistic perspective, Aristotle does emphasize Spartan women's control over property and over their husbands, and labels Sparta a \"gynaecocracy.\"Xenophon and Plutarch, in contrast, approve many features of the lives of Spartan women in comparison with the lives of women in other Greek states, but in this article Cartledge disdains the testimony of Xenophon and relegates Plutarch to the never-never land of utopia. Despite starting with Aristotle, Cartledge pictures Spartan women as \"passive\" victims who were exploited for purposes of reproduction. He does not fully recognize that Xenophon's report that the wives actively sought husband-doubling arrangements runs counter to his argument, for it indicates that wives employed their own childbearing strategies. He argues away women's ownership of property by stating that only a few elite women enjoyed this economic power, without making it clear that their brothers would be in similar straits. Cartledge also adopts a Victorian stance in questioning whether Spartan girls enjoyed homosexual relationships with older women. Cartledge concludes (105) that he hopes his readers \"hesitate before seeking to enlist the women of ancient Sparta as allies in the just cause of feminism.\" It is anachronistic to discuss Spartan women in terms of contemporary feminist criteria and goals, and these criteria and goals have been and are now multiple and diverse. Nevertheless, I venture to suggest that if Cartledge had compared Spartan women to other Greek women, and compared both to the men of their poleis, his conclusions concerning Spartan women would have been less pessimistic (see Conclusion, above).\n\nStephen Hodkinson generally retains the perspectives on the study of Sparta that were established by Finley and Cartledge. Whereas Cartledge treated Xenophon's works on Sparta with contempt, he was willing to accept some of Plutarch's reports. In contrast, Hodkinson regards Plutarch's information about archaic and classical Sparta with suspicion. He often blames Ephorus as the original source for what he regards as Plutarch's misrepresentations about the Spartan economy, and argues that the land tenure system that Plutarch attributes to archaic Sparta was actually a Hellenistic invention designed to promote the reforms of Agis and Cleomenes. Hodkinson does not explain why, if his hypothesis is true, the reformer kings (who had many opponents) were able to convince the Spartans themselves about the antiquity of their institutions. Although Hodkinson discusses women in much of his book, _Property and Wealth in Classical Sparta_ , he has not fully integrated them into his picture of Spartan society. For example, in the Introduction he writes of \"a political system . . . which gave the mass of citizens in assembly a formal role in decision-making\"(3);\"Rich citizens were able to employ their surplus wealth with potentially significant socio-political consequences: through horse-rearing and engagement in equestrian contests\" (5); and \"The perioikoi . . . were excluded from political decision-making which was reserved to the Spartiates alone\"(7 n. 5).Since Spartan women did engage in equestrian contests, but did not have a formal role in making political decisions Hodkinson's generalizations appear inconsistent and demonstrate that he tends to consider only male citizens, except in the parts of the book where he specifically talks about women.\n\nARCHAEOLOGICAL SOURCES\n\nArchaeological evidence has certainly been less influenced by the \"mirage\" and by interpretations from the perspective of the \"mirage\" than literary evidence, but it is not completely free of this bias. In any case, Sparta has not received the attention lavished on Athens and other sites that are generously endowed with artistic monuments. Not only does Sparta boast few temples and public buildings predating the Roman period (cf.Thuc. 1.10), but owing in part to the much-vaunted Spartan austerity, archaeological evidence for private life is sparse. Spartans may have owned some valuables, but the ethic prevailing at least to the end of fifth century discouraged and even forbade display.Athenaeus (14.633a) actually uses the word _austeria_ to describe life at Sparta. Because death was not commemorated by monuments before the Hellenistic period, there are no counterparts to the funerary reliefs that portray women in other parts of the Greek world. Only in the second century B.C.E., following the defeat at Sellasia (222) and the reforms of Nabis (207\u2013192), do the Spartans begin to use the same reliefs as those found in the rest of the Greek world. Building projects in the Hellenistic and Roman period also contributed to the disturbance and destruction of earlier remains. Numismatics also provides insights into the symbolic qualities and status of queens, goddesses, and female allegorical and mythical figures in Greece. The Spartans, however, did not mint their own currency until 280 B.C.E. Furthermore, since the modern city of Sparta not only sits on top of the ancient city but also is not located in a densely populated area of Greece, archaeological finds have not come to the surface serendipitously as the result of excavations for subways or the laying of foundations of buildings.\n\nIn terms of women's history, the most important finds have been at the sanctuary of Artemis Orthia, where the British School at Athens conducted extensive excavations for five years beginning in 1906. Thousands of lead figurines, including many depicting adult women wearing nicely woven dresses, as well as pendants with a variety of incised patterns representing weaving, have been found. Interesting material, in some cases related to the finds at the Orthia sanctuary, has also been discovered at the sanctuary of Helen and Menelaus called the \"Menelaion\" (see chap. 6). The finds have been published gradually but systematically in the _Annual of the British School at Athens_.The final synthetic report on the Orthia sanctuary was published by R. M. Dawkins and others in a single volume: _The Sanctuary of Artemis Orthia at Sparta_. Since this major excavation nearly a century ago, little new archaeological evidence directly relevant to women's history has been published. Brief mentions of some dedications by women at the Menelaion constitute a minor exception. Field surveys may produce some evidence for the lives of perioecic and helot women, but these women are doubly silenced in history as members of subordinate groups and as women.\n\nSecondary scholarship in the field of archaeology has been devoted principally to the pottery, bronzes, and inscriptions from Laconia. While there are several catalogues of the pottery, the rest of the material has usually been treated in articles; see, for example, the bibliographies to the articles on \"Sparta\", \"Spartan Cults,\" and the \"Menelaion,\" in the third edition of the _Oxford Classical Dictionary_. Some of the material remains, notably the bronze mirrors, support the evidence of the written sources. In contrast, a few depictions of women on pottery give evidence not supplied from the written sources, while others clarify the texts (see chap. 6,fig.6). It must be admitted that the iconographical evidence alone is more difficult for the modern scholar to decipher than the written evidence (which is not without problems: see above). The figures depicted in Laconian sculpture and vase painting are not labelled: rather scholars must deduce their identity. Representations of nudity and sexual activity pose problems of interpretation. In archaic and classical Athenian art, a woman playing a flute or reclining with men at a symposium would not be judged to be a respectable citizen. A scene of an orgy that includes satyr-like creatures and phalloi would be assumed to be connected to the cult of Dionysus, certainly not that of Artemis. The same scenes need not have the same implications in Laconian art. Furthermore the women or men who beheld or used these objects in antiquity may have reacted to them intellectually and emotionally in ways totally different from our own. Ancient sentiments doubtless changed as the objects were reused over time. Places of manufacture and find spots varied as well: the reactions of the craftsmen and of the users in different localities varied accordingly. There is no point in speculating about women artists in Sparta. By the classical period no Spartiate engaged in manual labor. Whether any lower-class women living in Sparta participated in manufacturing the objects depicting women is not known. Finally, it must be acknowledged that we are dealing with a very small oeuvre of recovered and published works that shed light on the history of mortal women. Scholars differ not only in interpreting the depictions, but on the place of manufacture of many of the objects, and on whether objects such as the Vix crater (see fig. 4) found outside the borders of Laconia (but resembling those found within) should be considered as Spartan art.\n\n### _Mirrors and Bronze Statuettes_\n\nSeven archaic bronzes showing naked girls have been found in Sparta and Messenia.  The majority of these were mirror handles, intended for use by women. Other examples have been found outside Spartan territory. Some of the girls are shown wearing a chiton that does not reach the knee and that covers only the left breast. The modern viewer may find the nudity most startling; in antiquity (though perhaps not in archaic Sparta), the bare female breast was an extremely potent image. The unabashed nudity of the figures seems to reflect the athletic nudity of Spartan women. Therefore it is likely that this type of statuette was manufactured in Sparta, but was exported and imitated elsewhere in Greece. Some of the girls hold musical instruments or other objects that may have been used for religious purposes. Their date is ca. 570 to ca. 470.\n\nThese mirrors did not stand up on their own when women used them: a woman had to hold such a mirror in her hand. As in many ancient mirrors, the disk was convex. The mirror reflected not only the face, as would a flat mirror, but hair, neck, and cleavage. The owner of the mirror would gaze at her own face and chest, reflected smaller than actual size, over the figure on the mirror handle. Grasping a mirror in one's hand is more intimate than merely looking at a vase. We may speculate about the esthetic and tactile pleasure a woman would feel while she was partially dressed herself and held the image of a nude slender female body. We may also suggest that this experience might be related to the fact that female children spent their time with each other and with adult women and engaged in homosexual erotic activities (see chap. 1).\n\nThough works of art depicting nonmortal females are beyond the scope of this survey, it is relevant to the discussion of nudity in sculpture to point out that an archaic figure of Eileithyia is one of the earliest sculptures in the round of the nude female in the Greek world (see chap. 6, fig. 7).\n\n### _Vase Painting_\n\nSparta offers very little to compare with the huge number of vases from Athens that have been excavated and studied. Two ceramic cups that are significant for women's history which show women with men, one at a symposium, another at an orgy, are discussed in chap. 6. It is necessary to point out, however, that though the vase with the symposium scene was manufactured in Laconia, it was excavated in Samos. It has been argued that it may represent the artist's view of a feast in Samos, rather than a local Spartan scene. It has also been argued that the women depicted in the orgy or komos scenes are hetairai and flute girls with no historical Spartan counterparts, and that the iconography is simply borrowed from Corinthian vases. The closest parallels for this symposium scene, with women wearing Lydian mitres and reclining outdoors alongside men on the\n\n(a)\n\n(b)\n\nFig. 10. Athlete. Bronze mirror handle, front and back views.\n\nAthlete as Mirror Caryatid. Bronze, ca. 550.Unknown provenance. Maiden wears an athlete's cap to hold her hair. She also wears flowers over her ears, a necklace with a pendant, and diagonally across her chest a strap holding amulets and a sickle that was awarded as a prize for victors. She holds an oil flask or pomegranate in her left hand and probably held a flower in her right. Griffins flank the disk of the mirror. The maiden stands on a lion. New York, Metropolitan Museum of Art inv. 38.11.3. Fletcher Fund, 1938. Photos courtesy of Joan R. Mertens.\n\nFig. 11. Musician. Bronze mirror handle.\n\nFrom the Sanctuary of Apollo at Amyclae. Bronze, ca. 550\u2013530. Girl wears diadem with flowers above her ears, necklace with pendant, strap with sickle, and holds cymbals. Athens, National Museum, X 7548 = Maria Pipili, _Laconian Iconography of the Sixth Century B.C._ , Oxford University Committee for Archaeology Monograph, no. 12 (Oxford, 1987),cat. no. 216 d, fig. 110. Photo courtesy Deutsches Arch\u00e4ologisches Institut, Athens.\n\nground are actually in Etruscan painting. In the Etruscan context there is no doubt that the women represented are wives, not hetairai.\n\n### _Sculpture_\n\nHelen is frequently depicted on Athenian vases as being threatened by Menelaus and running away from him. These scenes, however, were created outside Sparta for a non-Spartan clientele, and fall outside the limits of this study. Spartan maidens raced in honor of Helen, but in Laconian art Helen herself is not shown fleeing. Helen appears on an archaic stele depicting Menelaus wooing Helen on one side, and their initial encounter after the fall of Troy on the other (see chap. 6, fig.8). Although Menelaus is taller than she is and armed with a sword, Helen meets him without cowering and faces him boldly, looking directly into his eyes. All who looked at this stele would know that although he had intended to kill her, when he saw her he changed his mind. Her beauty was a more potent weapon than Menelaus' sword. The recovery of Helen is also depicted on a few smaller works of Peloponnesian art from the end of the seventh to the mid-sixth century. In these Helen is shown carrying a wreath. Her husband looks back at her and holds his sword aloft vertically. In contrast, on Athenian vases of ca. 550\u2013470 which show the same recovery theme, Menelaus threateningly points his sword at her.\n\n### _Conclusion_\n\nThe wealth of Greek vase painting and sculpture depicting women has enabled some scholars of gender and women's history to interpret the visual arts in a multiplicity of ways including as propaganda. Images of rape are common in Athenian vase painting, as are scenes of women confined in the house or in the women's quarters, often weaving. The message is generally designed to foster male sexual dominance and female subordination. Iconography oppressive to women is not confined to Athens. In a discussion of decorations on bronze mirrors of the fourth century from Elis and Corinth, Andrew Stewart comments:\n\nnone of these case-mirrors offers an unequivocal vision of an independent, mature female sexuality. All of them may be made to conform to the peculiarly Greek dogma that unmarried girls are more like animals than human beings; that they are sexually voracious . . . helping to ensure that these women remained exactly where their menfolk wanted them.\n\nHow different was the iconography at Sparta? We have mentioned a statue of Artemisia who commanded her fleet admirably at the battle of Salamis (Herod. 8.88, 93); lead figurines showing women dressed in fine fabrics and pendants representing the textiles; statuettes and mirror handles in the form of female athletes; a vase showing luxuriously attired women playing musical instruments at a co-ed symposium; another vase showing men and women revelers; and a relief of Helen facing down Menelaus. As Thucydides indicated, there was not much material culture to attract the attention of a viewer in Sparta. Nevertheless, judging from the visual propaganda directed at women in the rest of the Greek world, a modern feminist might consider the absence of art as a positive and creative force for Spartan women.\n\n* * *\n\n. P. A. Cartledge and A. J. S. Spawforth, _Hellenistic and Roman Sparta: A Tale of Two Cities_ (London, 1989), 133, estimate a population of no more than 12,000.\n\n. Herein referred to in brief citations as _Lac. Pol._\n\n. See further R. F. Willetts, _The Law Code of Gortyn_ , _Kadmos_ , suppl. 1 (Berlin, 1967).\n\n. T. Boring, _Literacy in Ancient Sparta_ (Leiden, 1979), chap. 3, surveys Spartan authors.\n\n. All references to Alcman's poetry are to M. Davies, _Poetarum Melicorum Graecorum Fragmenta_ ( _PMGF_ ), vol. 1 (Oxford, 1991). Except where noted, the translations of Alcman in the present book are from Davies's Greek text. I am grateful to Dr. Davies for his comments on my translation of _Partheneion_ l. For a text of Alcman with brief apparatus criticus (but in general more restorations than Davies), commentary, and translation, see D. A. Campbell, _Greek Lyric_ , vol. 2 (Cambridge,Mass., 1988). Campbell also provides useful indexes. For full commentary with text, see C. Calame, _Alcman: Fragmenta_ (Rome, 1983). Biographical testimonia are cited from Davies.\n\n. See _PMGF_ , T A1a\u20139, 11a.\n\n. See F. D. Harvey, \"Oxyrhynchus Papyrus 2390 and Early Spartan History,\" _JHS_ 87 (1967), 62\u201373, esp. 69. Poralla2, p. 18, no. 65, dates Alcman to the second half of the seventh century.\n\n. W. G.Cavanagh and R. R. Laxton,\"Lead Figurines from the Menelaion and Seriation,\" _ABSA_ 79 (1984 _),_ 23\u201336, esp. 35.\n\n. Paus. 3.15.1. F. B\u00f6lte, s.v. Platanistas, _RE_ 20.2 (Stuttgart, 1950), cols. 2333\u201334.\n\n. M. Davies, review of _Alcman: Fragmenta_ , by Claude Calame, _Gnomon_ , 58 (1986), 385\u201389, esp. 387 n. 2, questions Calame's dogmatic rejection of such an interpretation.\n\n. \"Un fragment in\u00e9dit du po\u00ebte Alcman,\" in _Memoires d'histoire ancienne et de philologie_ (Paris, 1863), 159\u201375.\n\n. _PMGF_ 3.3 = _P.Oxy_. xxiv.2387. See further E. G. Turner, _Greek Manuscripts of the Ancient World_ (Princeton, 1971), pp. 42\u201343, no. 15.\n\n. See further J. Herington, _Poetry into Drama: Early Tragedy and the Greek Poetic Tradition_ (Berkeley, 1985), 22\u201326, 48, 54\u201355, 207. The ancient evidence for reperformance of Alcman concerns male choruses, not _Partheneia._ Herington (25) argues that Sosibius was an eyewitness to a performance of Alcman: see below on Sosibius.\n\n. See further Denys L. Page, _Alcman: The Partheneion_ (Oxford, 1951), 64, and Herington, _Poetry into Drama_ , 238 n. 38.\n\n. _LGPN_ 3A s.vv. cites the names in the _Partheneia,_ and see below.\n\n. Sosibius, _On the Sacrifices in Lacedaemon, FGrH_ 595 F 5 = Athen. 15.678b\u2013c for a choral performance of Alcman celebrating a Spartan victory ca. 546. See further E. Stehle, _Performance in Ancient Greece_ (Princeton, 1997), 55, and G. Nagy, _Pindar's Homer: The Lyric Possession of an Epic Past_ (Baltimore, Md., 1990), 349. On the transmission of Alcman beyond Sparta, see P. A. Cartledge, \"Literacy in the Spartan Oligarchy,\" _JHS_ 98 (1978), 28.\n\n. Excluding the editions of the texts, probably the most influential full-length study of the _Partheneia_ to be published in the later part of the twentieth century is Claude Calame, _Les choeurs de jeunes filles en Gr\u00e8ce archa\u00efque_ , vol. 1, _Morphologie, fonction religieuse et sociale_ ; vol. 2, _Alcman_ (Rome, 1977). In his first volume, Calame gives a lengthy survey of the composition and formal structure of the lyric choir not only in archaic Greece, but in later periods. In volume 2, he examines the work of Alcman, especially _Partheneia_ 1 and 3,which he understands to have been performed at pubertal rites of initiation for girls that paralleled those for boys. Calame reiterates this view in \"Iniziazioni fem-minili spartane: Stupro, danza, ratto, metamorfosi e morte iniziatica,\" in _Le donne in Grecia_ , ed. G. Arrigoni (Bari, 1985), 33\u201354. Calame's interpretation has been widely adopted by many scholars who write about Alcman's poems, Spartan girls, and religion. Nevertheless, there is nothing inherent in the texts that makes Calame's view more than an attractive suggestion.\n\n. Christina A. Clark, \"The Gendering of the Body in Alcman's _Partheneion_ 1: Narrative, Sex, and Social Order in Archaic Sparta,\" _Helios_ 23 (1996), 143\u201372, assumes that men were in the audience and interprets the poetry as produced ultimately in order to make women conform to \"their culture's male-structured gender codes\" (169).\n\n. Jane B. Carter,\"Masks and Poetry in Early Sparta,\" in _Early Greek Cult Practice: Proceedings of the Fifth International Symposium at the Swedish Institute at Athens, 26\u201329 June 1986_ , ed. R. H\u00e4gg,N. Marinatos, and G. C. Nordquist (Stockholm, 1988), 89\u201398, esp. 89, 98.\n\n. Herod. 5.51, 6.61, 7.239, and see further Carolyn Dewald, \"Women and Culture in Herodotus' _Histories_ ,\" in _Reflections of Women in Antiquity_ , ed. Helene P. Foley (New York, 1981), 91\u2013125.\n\n. See further E. N. Tigerstedt, _The Legend of Sparta in Classical Antiquity_ , 2 vols. (Stockholm, 1965\u201374), vol. 1, 114\u201327; Ellen Greenstein Millender,\"Athenian Ideology and the Empowered Spartan Female,\" in _Sparta: New Perspectives_ , ed. S. Hodkinson and A. Powell (London, 1999), 355\u201391, esp. 357\u201362, 373; and most recently Helene P. Foley, _Female Acts in Greek Tragedy_ (Princeton, 2001), 99\u2013102, 318\u201331.\n\n. See further Edith Hall, _Inventing the Barbarian_ (Oxford, 1989), 214\u201315. Hall does not discuss gender issues.\n\n. Diels-Kranz II (1969), 88, fr. 32.\n\n. _Republic_ 5, passim, _Laws_ 5.742C, 6.774C, 784\u201385, 7.814, 8.833D\u2013834D.\n\n. I presented this material first in \"Xenophon's Spartan Women,\" paper delivered at The World of Xenophon conference, University of Liverpool, July 8, 1999.\n\n. D.L. 2.54 and see further Nigel M. Kennell, _The Gymnasium of Virtue: Education and Culture in Ancient Sparta_ (Chapel Hill, N. C., 1995), 16. E. Badian, \"Xenophon the Athenian,\" paper\n\ndelivered at The World of Xenophon conference, University of Liverpool, July 10, 1999, questions whether the sons participated in the agoge.\n\n. See further C. Tuplin,\"Xenophon,Sparta, and the _Cyropaedia_ ,\"in _The Shadow of Sparta_ , ed. A. Powell and S. Hodgkinson (London, 1994), 127\u201381, esp. 132.\n\n. Badian,\"Xenophon the Athenian.\"\n\n. Lampito had been the name of a historical woman, daughter of Leutychides II (a Spartan king), and was the mother of Agis II who was reigning in 411: Herod. 6.71 and Poralla2, no. 474.\n\n. _Lac. Pol._ 1.4. According to Aristoph. _Birds_ , 1281, all Athenians were Laconizers.\n\n. See n. 23, above.\n\n. In Athens, small children participated in the Anthesteria, and drank the new wine from small choes (pitchers). That many more choes with designs appropriate for boys than for girls have been excavated suggests that more boys were brought to the festival. See further Sarah B. Pomeroy, _Families in Classical and Hellenistic Greece_ (Oxford, 1997), 69, 118\u201319.\n\n. Arist. _HA_ 608b observes that the female eats less. Doubtless this was both the effect and cause of systematically allocating less food to women.\n\n. See further Pomeroy, _Families_ , 42\u201345, 61\u201366.\n\n. _FGrH_ 581. Jacoby ad loc. suggests that by writing favorably about his polis, Thibron was trying to curry favor with potentates in Asia or with the Spartan authorities, or to denigrate (as not following the Spartan ethic) the ephors who had exiled him. On Spartan authors of the fifth and fourth centuries, including Thibron, Pausanias, and Lysander, see also T. Boring, _Literacy in Ancient Sparta_ (Leiden, 1979), 50\u201355.\n\n. See further _FGrH_ II.81; T. W.Africa, _Phylarchus and the Spartan Revolution_ (Berkeley, 1961, repr. Millwood, N.Y., 1980), 43\u201347, 61; and cf. K. Chrimes, _Ancient Sparta_ (Manchester, 1949), 6, who asserts that Phylarchus was ill-informed about Sparta in the days of Agis IV.\n\n. See Deborah Gera, _Warrior Women: The Anonymous_ Tractatus de Mulieribus (Leiden, 1997), for paragraph-long descriptions of fourteen women (including the Spartan Argeia),written in the first century b.c.e. or later; and most recently Stanley M. Burstein, \"Cleitarchus in Jerusalem,\" in _The Eye Expanded_ , ed. F. B. Titchener and R. F. Moorton, Jr. (Berkeley, 1999), 105\u201312, esp. 107. See further chap. 4.\n\n. According to _OCD_ 3, he was a Spartan who worked in Egypt probably in the mid-third century. Jacoby, _FGrH_ III b, no. 595, is more doubtful about Sosibius' date and Spartan origin. Boring, _Literacy in Ancient Sparta_ , 56, places him between 250 and 150,with the lower date more likely.\n\n. Edward O'Neil (ed.), _Teles,_ t _he Cynic Teacher_ (Missoula,Mont., 1977), \"On Freedom from Passion,\" pp. 65\u201366, ll. 53\u201376 (= _Teletis: Reliquiae_ , ed. O Hense, 2d edn. [1909] 57.11\u201358.12), is very close to Plut. _Sayings of Spartan Women,_ 241a3,b4, c8, and cf. _Virtues in Women_ 246a. See further Tigerstedt, _The Legend of Sparta in Classical Antiquity_ , vol. 2, 27, 38.\n\n. _Anth. Pal._ 7.433\u201335.= A. S. F.Gow and D. L. Page, _The Greek Anthology: Hellenistic Epigrams_ , 2 vols. (Cambridge, 1965), vol. 1, Dioscurides, 32; Nicander, 1; Tymnes, 6; and Ascleipiades, 47. A Spartan mother who kills her son appears also in Erycius, _Anth. Pal._ 7.230 and Antipater Thess. _Anth. Pal._ 7.531 (Augustan). See further, Tigerstedt, _The Legend of Sparta in Classical Antiquity_ , vol. 2, 27, 86.\n\n. _FGrH_ III b F 588 quoted from Didymus by Athenaeus 4.139d\u2013f. See further L. R. Farnell, _Cults of the Greek States_ (Oxford, 1896\u20131909), vol. 4, 265\u20136.\n\n. Stobaeus iv.2.25 (p. 160 Hense) = _FGrH_ 90 F 103 (144) and see further Tigerstedt, _The Legend of Sparta in Classical Antiquity_ , vol. 2, 92.\n\n. Elizabeth Rawson, _The Spartan Tradition in European Thought_ (Oxford, 1969), 112.\n\n. _Sayings of Spartan Women_ is now generally agreed to be the work of Plutarch: Donald Russell, in a personal communication. See also Tigerstedt, _The Legend of Sparta in Classical Antiquity_ , vol. 2, 16\u201330.\n\n. For the dating, see Luigi Piccirilli and Mario Manfredini, _Plutarco: Le vite di Licurgo e di Numa_ 2 _(_ Milan, 1990), xl. For the five _Lives_ (including _Lysander_ and _Agesilaus_ ) as a unified cycle, see D. R. Shipley, _A Commentary on Plutarch's_ Life of Agesilaos (Oxford, 1997), 3\u20134. Shipley (3) judges that Plutarch gives \"a valid account of Agesilaos' character and reign.\"\n\n. _Ages._ 19.10 = _FGrH_ 596 F 5. See further L. Piccirilli, \"Cronologia relative e fonti delle _Vitae Lycurgi et Numae_ di Plutarco,\"in _Philias Charin: Miscellanea di studi classici in onore di Eugenio Manni_ (Rome, 1980), vol. 5, 1754\u201364, esp. 1762; Manfredini and Picciirilli, _Plutarco: Le vite di Licurgo e di Numa_ 2, xl\u2013xlii; and Boring, _Literacy in Ancient Sparta_ , 88\u201393.\n\n. See further Sarah B. Pomeroy, \"Reflections on Plutarch, _Advice to the Bride and Groom:_ Something Old, Something New, Something Borrowed,\" 33\u201342 passim, and Cynthia Patterson, \"Plutarch's Advice to the Bride and Groom: Traditional Wisdom through a Philosophic Lens,\"128\u201337,\n\npassim, both in _Plutarch's Advice to the Bride and Groom and A Consolation to His Wife_ , ed. Sarah B. Pomeroy (New York, 1999).\n\n. Xen. _Oec._ 7.14\u201315, 9.14\u201315, 10.1, Plut. _Virtues of Women_ 249f, and see further Sarah B. Pomeroy,\"Reflections on Plutarch,\" in _Xenophon, Oeconomicus: A Social and Historical Commentary_ (Oxford, 1994), ad loc., and chap. 4, above.\n\n. Xen. _Oec._ 7.11\u201312. See further Pomeroy, _Xenophon, Oeconomicus_ , ad loc.; Plut. _Comp. Lyc. Num._ 4; and L. Goessler,\"Plutarchs Gedanken \u00fcber die Ehe\" (diss. Zurich, 1962), trans. by David and Hazel Harvey as \"Advice to the Bride and Groom: Plutarch Gives a Detailed Account of His Views on Marriage,\" in _Plutarch's Advice to the Bride and Groom and A Consolation to His Wife_ , ed. Sarah B. Pomeroy (New York, 1999), 97\u2013115, esp. 106\u20137.\n\n. See further Lin Foxhall, \"Foreign Powers: Plutarch and Discourses of Domination in Roman Greece,\" in _Plutarch's Advice to the Bride and Groom and A Consolation to His Wife_ , ed. Sarah B. Pomeroy (New York, 1999), 138\u201350.\n\n. Foxhall,\"Foreign Powers,\" 148.\n\n. Plut. _Cleom._ 38, and see chap. 4.\n\n. See further Kennell, _The Gymnasium of Virtue_ , passim, and Pomeroy, _Families_ , 62\u201366. Contra Kennell, Jean Ducat,\"Perspectives on Spartan Education,\" in _Sparta: New Perspectives_ , ed. S. Hodkinson and A. Powell (London, 1999), 43\u201366, esp. 44\u201346, argues that Plutarch does give some credible information on the classical period.\n\n. See further Darice Birge, \"Women and the Greek Past in Pausanias' Descriptive Geography,\" paper delivered at the Annual Meeting of the American Philological Association, Dec. 29, 1998; abstract published in _American Philological Association 130th Annual Meeting: Abstracts_ , 131.\n\n. Vitruvius (1.1.6) reports that the statues of the Persians, attired in barbaric costume, supported the roof like Caryatids.\n\n. See E. Fantham,Helene Peet Foley, Natalie Boymel Kampen,Sarah B.Pomeroy, and H. Alan Shapiro, _Women in the Classical World_ (New York, 1994), 299\u2013301, and Suzanne Dixon, _The Roman Mother_ (Norman, Okla., 1989), passim.\n\n. _Ibis_ 615, for the mother of Pausanias the traitor, cf. R. Ellis, _P. Ovidii Nasonis Ibis: ex novis codicibus edidit, scholia vetera commentarium cum prolegomenis appendice_ (Oxford 1881), 165.\n\n. Val. Max. 2.7, ext.2.\n\n. _VH_ 12.21.\n\n. _PH_ 3.216.\n\n. _Orat._ 24 p. 118.63\u201366 (Colonna).\n\n. _Anth. Pal._ 9.397.\n\n. _Anth. Pal._ 9.447.\n\n. See also the Anonymous Epigram _Anth. Pal._ 9.61 for a mother who slew her cowardly son. For the Spartan tradition in post-classical times, see Rawson, _The Spartan Tradition in European Thought_. See also Conclusion, above.\n\n. _Tusc_. 1.42.102, 2.15.36, and see further chap. 3.\n\n. See further David Schaps,\"The Woman Least Mentioned: Etiquette and Women's Names,\" _CQ_ n.s. 27 (1977), 323\u201330.\n\n. See chap. 3 n. 3, Latte's emendation.\n\n. _Anth. Pal._ 7.433 = Gow and Page, _The Greek Anthology: Hellenistic Epigrams_ , vol. 2, 556, Tymnes 6.\n\n. _IG_ V.i, n. 1, for the date of 428\u2013421: see further Poralla2, no.160.\n\n. Alfred S. Bradford, _A Prosopography of Lacedaemonians from the Death of Alexander the Great, 323 B.C., to the Sack of Sparta by Alaric, A.D. 296_ (Munich, 1977), 227, s.v. Kallistoneike (4), and see chap. 5 nn. 53, 90, above.\n\n. See Poralla2, pp. 5\u201313, 173.\n\n. D. A.Campbell, _Greek Lyric_ , vol. 2 (Cambridge,Mass., 1988), 391, understands her to be the daughter of the Eurypontid Eurycrates (ca. 665\u2013640) and sister of Leotychidas.\n\n. C. N. Edmonson,\"A Graffito from Amyklai,\" _Hesperia_ 28 (1959), 162\u201364.\n\n. _Ages._ 8, 7, 19.10, nor could Dicaearchus discover their names: fr. 65 Wehrli2. See further Jan Bremmer,\"Plutarch and the Naming of Greek Women,\" _AJPh_ 102 (1981), 425\u201326.\n\n. Ed. Peter M. Fraser and Elaine Matthews (Oxford, 1997). In the present volume, the citation to _LGPN_ 3 is usually given in preference to earlier publications. _LGPN_ gives brief succinct identifications, often referring back to the earlier publications which provide fuller documentation, including genealogical charts. In addition, S. Hodkinson, _Property and Wealth in Classical Sparta_ (London, 2000), 414, suggests that Deinicha, who married Archidamus III, was descended from Deinis, whose name is inscribed on a sixthth-century aryballus, and that Euryleonis, the Olympic victor (see chap. 1, above), was descended from Euryleon companion of Dorieus in the late sixth century.\n\n. Poralla2, and Bradford, _A Prosopography of Lacedaemonians from the Death ofAlexander the Great_.\n\n. See Sarah B. Pomeroy, \" _Technikai kai Mousikai_ : The Education of Women in the Fourth Century and in the Hellenistic Period,\" _AJAH_ 2 (1977), 51\u201368, esp. 52. Michael Pettersson, _Cults of Apollo at Sparta: The Hyakinthia, the Gymnopaidiai and the Karneia_ (Stockholm, 1992), 120, finds no evidence for girls participating in the Gymnopaideia, nor (125) for their being part of the age class system connected to the agoge. See above, chap. 2 n. 2.\n\n. E.g., recently Ellen Greenstein Millender,\"Athenian Ideology and the Empowered Spartan Female,\" in _Sparta: New Perspectives_ , ed. S. Hodgkinson and A. Powell (London, 1999), 365\u201366.\n\n. _Sparta and Lakonia: A Regional History, 1300\u2013362 B.C._ (London, 1979).\n\n. Cartledge and Spawforth, _Hellenistic and Roman Sparta_ , 110.\n\n. Paul Cartledge, \"The Socratics' Sparta and Rousseau's,\" in _Sparta: New Perspectives_ , ed. S. Hodkinson and A. Powell (London, 1999), 311\u201337, esp. 312.\n\n. M. I. Finley,\"Sparta,\" in _The Use and Abuse of History_ (London, 1975), 161\u201377, esp. 171 (orig. pub. in _Probl\u00e8mes de la guerre en Gr\u00e8ce ancienne_ , ed. J.-P.Vernant [Paris, 1968], 143\u201360):\"For the sake of completeness, I record without discussion two further sources of tension: (a) the women, if Plato and Aristotle are to be believed.\" See further Pomeroy, _Xenophon. Oeconomicus_ , 43, no.14.\n\n. \"Spartan Wives: Liberation or Licence?\" _CQ_ 31 (1981), 84\u2013105.\n\n. See most recently Michael Whitby, review of _Sparta: New Perspectives_ , ed. S. Hodkinson and A. Powell, _Scholia Reviews_ n.s. 9 (2000), 36.\n\n. \"Spartan Wives,\" esp. 86 and 89.\n\n. \"Spartan Wives,\" 103, and see chap. 3.\n\n. \"Spartan Wives,\" 105, followed by Millender, \"Athenian Ideology and the Empowered Spartan Female,\" 371.\n\n. \"Even if we should prefer not to believe that Spartan maidens enjoyed tutelary homosexual relations with older women\" (101).He was, of course, writing for a classical (presumably not feminist) audience.\n\n. See, e.g., P. A. Cartledge, _Agesilaos and the Crisis of Sparta_ (London, 1987), 70, where he points out that Plutarch did original research.\n\n. _Property and Wealth in Classical Sparta,_ e.g., 26\u201328, 68, 166, and see chap. 4 n. 16, above. In defense of Ephorus, Aristotle, and Plutarch, see E. David, \"Aristotle and Sparta,\" _Anc. Soc_. 13\/14 (1982\u201383), 67\u2013103, esp. 82\u201383, and chap. 4 n. 38, above. The debate over the use of Plutarch for the history of archaic and classical Sparta is not new. W. den Boer aptly writes in _Laconian Studies_ (Amsterdam, 1954), 221: \"Modern historians, though possessing no more material for interpretation than Plutarch, have all too often disposed of the customs related by him as ridiculous concoctions offered by him or his sources, and in so doing they have shown less modesty and historical discernment than Plutarch commanded.\"\n\n. See also chap. 4 nn. 61, 62, above.\n\n. London, 2000.\n\n. Nevertheless, the concept of austerity at Sparta is currently being questioned by Stephen Hodkinson,\"Bronze Dedications at Spartan Sanctuaries,\" in _Sparta in Laconia: Proceedings of the 19th British Museum Classical Colloquium_ , ed. W.G. Cavanagh and S. E. C.Walker, British School at Athens Studies, vol. 4 (London, 1998), 55\u201363, and several other contributors to the same volume, including F\u00f6rtsch (see n. 99). The concept of austerity is, of course, relative (one person's necessities are extravagances to another), but it must be admitted that Sparta simply has not yielded the precious metals, jewelry, fine tableware, and other personal luxury items that one might expect to find in the remains of such a powerful city. Domestic life was far more comfortable in Roman Sparta: see Stella Raftopoulou,\"New Finds from Sparta,\" in _Sparta in Laconia: Proceedings of the 19th British Museum Classical Colloquium_ , ed. W. G. Cavanagh and S. E. C. Walker, British School at Athens Studies, vol. 4 (London, 1998), 125\u201340, esp. 127\u201333.\n\n. See Xen. _Hell._ 6.4.27, 30, for searching Spartan houses for sequestered valuables.\n\n. See further M. N. Tod and A. J. B. Wace, _A Catalogue of the Sparta Museum_ (Oxford, 1906), 127.\n\n. For the history of excavations at Sparta, see H. W. Catling,\"The Work of the British School at Athens at Sparta and in Laconia,\" in _Sparta in Laconia: Proceedings of the 19th British Museum Classical Colloquium_ , ed. W. G. Cavanagh and S. E. C. Walker, British School at Athens Studies, vol. 4 (London, 1998), 19\u201327.\n\n. Society for the Promotion ofHellenic Studies Supplementary Papers, no. 5 (London, 1929), v\u2013vi, and R. M. Dawkins,\"Artemis Orthia: Some Additions and a Correction,\" _JHS_ 50 (1930), 298\u201399.\n\n. See chap. 6, and Catling, \"The Work of the British School at Athens at Sparta and in Laconia,\" 25.\n\n. For a survey of the number of various artifacts and their dates of manufacture, see R. F\u00f6rtsch,\"Spartan Art: Its Many Different Deaths,\" in _Sparta in Laconia: Proceedings of the 19th British Museum Classical Colloquium_ , ed. W. G. Cavanagh and S. E. C. Walker, British School at Athens Studies, vol. 4 (London, 1998), 48\u201354. F\u00f6rtsch, however, fails to take into account the falling population figures, and the concentration of wealth (see chaps. 3 and 4), both of which factors created a smaller market for the luxury items he surveys. W. G. Cavanagh, J. Crouwel, R.W. V. Catling, and G. Shipley, _Continuity and Change in a Greek Rural Landscape: The Laconia Survey_ , vol. 2, _Archaeological Data_ , _ABSA_ suppl. vol. 27 (London, 1996), give laconic descriptions of material remains not yet synthesized so that they would be of much use for the writing of women's history.\n\n. E.g., Paul A. Clement,\"The Recovery of Helen,\" _Hesperia_ 27 (1958), 47\u201373. For criticism of the interpretations of Lilly B. Ghali-Kahil, _Les enl\u00e8vements et le retour d'H\u00e9l\u00e8ne dans les textes et les documents figur\u00e9s_ (Paris, 1955), see n. 111, below.\n\n. Herod. 2.167, Xen. _Lac. Pol._ 7.1\u20132, and see R. M. Cook,\"Spartan History and Archaeology,\" _CQ_ 12 (1962), 156\u201358, and Paul Cartledge,\"Did Spartan Citizens Ever Practice a Manual _Techne?\" LCM_ 1 (1976), 115\u201319.\n\n. On the Vix crater, see chap. 2, caption to fig. 4, and Marlene Herfort-Koch, _Archaische Bronzeplastik Lakoniens_ , _Boreas_ , suppl. 4 (M\u00fcnster, 1986), 70\u201373,who argues for a Laconian origin.\n\n. See most recently Andrew Stewart, _Art,Desire, and the Body in Ancient Greece_ (Cambridge, 1997), esp. 29\u201334, 108\u201319, and Appendix, 232\u201334, for a complete catalogue. See also Thomas Scanlon, \" _Virgineum Gymnasium_ : Spartan Females and Early Greek Athletics,\" in _The Archaeology of the Olympics_ , ed. W. Raschke (Madison,Wis., 1988), 185\u2013216, esp. 191\u201399, 203\u20134.\n\n. See Larissa Bonfante,\"Nudity as a Costume in Classical Art,\" _AJA_ 93 (1989), 558\u201368, and see most recently Beth Cohen,\"Divesting the Female Breast of Clothes in Classical Sculpture,\" in _Naked Truths: Women, Sexuality and Gender in Classical Art and Archaeology_ , ed. Ann Olga Koloski-Ostrow and Claire L. Lyons (London, 1997), 66\u201392.\n\n. Stewart, _Art,Desire, and the Body_ , 108, estimates that at least 4,000 of these statuettes were manufactured.\n\n. Claude Rolley, \"Le probl\u00e8me de l'art Laconien,\" _Kt\u00e8ma_ 2 (1977), 125\u201340, esp. 130, sees the girls as very young, and postulates a connection between the figurines and the cult of Artemis\n\nOrthia. See further Uta Kron, \"Sickles in Greek Sanctuaries: Votives and Cultic Instruments,\" in _Ancient Greek Cult Practice from the Archaeological Evidence (Proceedings of the Fourth International Seminar on Ancient Greek Cult, Organized by the Swedish Institute at Athens, 22\u201324 October 1993_ , ed. R. H\u00e4gg (Stockholm, 1998), 187\u2013215, esp. 206\u20137. See also Scanlon,\" _Virgineum Gymnasium_ ,\" 194, 196.\n\n. G. Richter,\"An Archaic Greek Mirror,\" _AJA_ 42 (1938), 337\u201344, esp. 343\u201344, argues that such mirrors are not Spartan, but probably Corinthian, and that they portray hetairae, not naked athletes.\n\n. Maria Pipili,\"Archaic Laconian Vase Painting,\" in _Sparta in Laconia: Proceedings of the 19th British Museum Classical Colloquium_ , ed. W.G. Cavanagh and S. E. C.Walker, British School at Athens Studies, vol.4 (London,1998), 82\u201396, esp. 90, and J.Carter, review of _Laconian Iconography of the Sixth Century B.C._ , by Maria Pipili, _AJA_ 93 (1989), 473\u20136, esp. 475.\n\n. E. A. Lane,\"Lakonian Vase-Painting,\" _ABSA_ 34 (1933\u201334),99\u2013189, esp 158, but see now Tyler Jo Smith, \"Dances, Drinks, and Dedications: The Archaic _Komos_ in Laconia,\" in _Sparta in Laconia: Proceedings of the 19th British Museum Classical Colloquium_ , ed. W. G. Cavanagh and S. E. C. Walker, British School at Athens Studies, vol. 4 (London, 1998), 75\u2013 81, esp. 77\u201378.\n\n. See H. A. Shapiro,\"Modest Athletes and Liberated Women,\" in _Not the Classical Ideal_ , ed. Beth Cohen (Leiden, 2000), 315\u201337, esp. 332.\n\n. See further Ghali-Kahil, _Les enl\u00e8vements et le retour d'H\u00e9l\u00e8ne_ , 71, no. 24, and \"H\u00e9l\u00e8ne,\" _LIMC_ 4.1 (Zurich, 1988), 498\u2013563,with an addendum on 951, and plates in 4.2 (Zurich, 1988), 291\u2013358, s.v. H\u00e9l\u00e8ne;Guy Hedreen,\"Image,Text, and Story in the Recovery ofHelen,\" _CA_ 15 (1996), 152\u201384,and figs. 1\u201312. On the political background of the depiction of Helen on Attic vases, see H. A. Shapiro, \"Cult Warfare: The Dioskouroi between Sparta and Athens, in _Ancient Greek Hero Cult: Proceedings of the Fifth International Seminar on Ancient Greek Cult, Organized by the Department of Classical Archaeology and Ancient History, G\u00f6teborg University, 21\u201323April 1995_ , ed. R. H\u00e4gg (Stockholm, 1999), 99\u2013107, esp. 105\u20136.\n\n. Ghali-Kahil, _Les enl\u00e8vements et le retour d'H\u00e9l\u00e8ne_ , 18, 320\u201321, suggests that the judgment of Paris is depicted on an ivory comb from Sparta (= Dawkins, _AO_ 223, pl. 127), and that an ivory relief (= Dawkins, _AO_ , 214, pls. 109\u201310) shows Helen and Paris on a ship. On these identifications, see Evangelia-Lila Marangou, _Lakonische Elfenbein- und Beinschnitzereien_ (T\u00fcbingen, 1969), 85\u201390, 107\u20139, 248\u201349 n. 510, and pls. 68, 78.\n\n. For these depictions on shield bands, see further Maria Pipili, \"A Laconian Ivory Reconsidered,\" in _Philolakon: Lakonian Studies in Honour of Hector Catling_ , ed. Jan Motyka Sanders (London, 1992), 179\u201384, esp. 183.\n\n. E.g., Eva Keuls, _The Reign of the Phallus_ (Berkeley, 1985).\n\n. See the archaic and classical images in Fantham, Foley, Kampen, Pomeroy, and Shapiro, _Women in the Classical World_ , 5\u2013135, passim.\n\n. \"Reflections,\" in _Sexuality in Ancient Art_ , ed. 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Verant, _L'individu, la mort, l' amour: Soi-m\u00eame et l'autre en Gr\u00e8ce ancienne_ , 173\u2013209. Paris, 1989.\n\nVoyatzis, Mary. \"Votive Riders Seated Side-Saddle at Early Greek Sanctuaries.\" _ABSA_ 87 (1992), 259\u201379.\n\nWace, A. J. B., M. S. Thompson, and J. P. Droop,\"The Menelaion.\" _ABSA_ 15 (1908), 108\u201357.\n\nWalbank, F.W. _A Historical Commentary on Polybius_. 3 vols. Oxford, 1957\u201379.\n\nWalker, Susan.\"Two Spartan Women and the Eleusinion.\" In _The Greek Renaissance in the Roman Empire_ , ed. S. Walker and A. Cameron. _ICS Bull._ suppl. 55 (19889), 130\u201341.\n\nWehrli, F. _Die Schule des Aristoteles_ , suppl. 1. Basel, 1974.\n\nWest, M. L.\"Alcmanica.\" _CQ_ 15 (1965), 188\u2013202.\n\n_____. _Immortal Helen_. London, 1975.\n\nWhitby,Michael.\"Two Shadows: Images of Spartans and Helots.\" in _The Shadow of Sparta_ , ed. A. Powell and S.J. Hodkinson, 87\u2013125. London, 1994.\n\n_____. Review of _Sparta: New Perspectives_ , ed. S. Hodkinson and A. Powell. _Scholia Reviews_ n.s. 9 (2000), 36.\n\nWilletts, R. F. _The Law Code of Gortyn_. _Kadmos_ suppl. 1. Berlin, 1967.\n\nWoodward, A. M. \"Excavations at Sparta, 1908: Inscriptions from the Sanctuary of Orthia.\" _ABSA_ 14 (1908), 74\u2013141.\n\nZiehen, L.\"Das spartanische Bev\u00f6lkerungsproblem.\" _Hermes_ 68 (1933), 218\u201337.\n\nZiehen, P.\"Sparta.\" _RE_ 18.3, col. 1466. Stuttgart, 1949.\n\nZuckerman, L.\"Spartan Women, Liberated.\" _New York Times_ , Jan. 1, 2000, sec. F, pp. 1, 3.\n\nZweig, Bella. \"The Only Women Who Give Birth to Men: A Gynocentric, Cross-Cultural View of Women in Ancient Sparta.\" In _Woman's Power, Man's Game: Essays on Classical Antiquity in Honor of Joy K. King_ , ed. Mary DeForest, 32\u201353. Wauconda, Ill., 1993.\n\n## INDEX\n\nPage numbers in italics refer to illustrations.\n\nabortion, \u2013. _See also_ family planning\n\nAchaea,\n\nAchaean League,\n\nAcrotatus, , , 76n.6,\n\nadoption, ,\n\nadultery, , , \u2013, 122n.74, . _See also_ husband-doubling\n\nAegina, 5n.11\n\nAelian, , , ,\n\nAenesimbrota,\n\n_agelai_ (herds), , 28n.110\n\nAgesilaus II, 8n.16, , , , , ,\n\nAgesippia,\n\nAgesistrata, , 76n.7, , , ,\n\nAgiatis, , , , , , , ,\n\nAgido, \u2013, , ,\n\nAgis, , 29n.115\n\nAgis II, , , , 149n.29\n\nAgis IV\n\nbeginning of reign of, ix\n\ndeath of,\n\nand dowries,\n\nfamily and upbringing of, 83n.31,\n\ngrandmother of,\n\nmarriage of, ,\n\nPlutarch's life of, ,\n\nreforms of, , , , \u2013, , 90n.53, ,\n\nson of,\n\n_agoge_ (boys' program), , , , , , 158n.77\n\nagriculture, , , , , , ,\n\nAlcibia,\n\nAlcibiades, , , ,\n\nAlcippus, 49n.47\n\nAlcman\n\non Artemis, ,\n\non beauty,\n\nbiographical information on, , , , 142n.7\n\nand education of Spartan girls, \u2013,\n\non female swimmers, 13n.38\n\non horses,\n\ninterest of, in women, 4n.3\n\non Megalostrata,\n\non music,\n\nperformance and preservation of works by, \u2013, 144n.13\n\non physical education,\n\npoetry by, \u2013, , , , , \u2013, __,\n\non sexual behavior,\n\nas source on Sparta, ,\n\ntomb of,\n\non weaving, ,\n\nAlcon, , 14n.43\n\nAlexander,\n\nAlexis,\n\nAmastris,\n\nAmazons, , 114n.35\n\nAmerican Civil War, \u2013\n\nAmycla,\n\nAnaxandridas II, , , , ,\n\n_andragathia_ (manly virtue),\n\n_Andromache_ (Euripides), , 83n.31, \u2013\n\n_anesis_ (licence), 58n.31\n\n_Anonymus Iamblichi_ , ,\n\nAnthesteria, 150n.32\n\nAnthousa, 125n.91\n\n_anthropoi_ ,\n\nAntinous,\n\nAntiochus IV,\n\nAotis,\n\nApega (Apia), \u2013,\n\nApelleas (Apelles), , 23n.87\n\nAphrodite, 34n.4, , \u2013, 122n.74, ,\n\nAphrodite Areia (Warrior), , 122n.72\n\nAphrodite Hera, ,\n\nAphrodite Morpho,\n\nApia, . _See also_ Apega\n\nApollo, , , \u2013,\n\nApollo Hyacinthius, ,\n\nApollo of Amyclae, , 30n.122, , \u2013, __\n\nApollo Temple at Delphi,\n\nApollodorus Carystius, \u2013\n\nApollodorus Gelous, \u2013\n\narchaeological evidence, , , , \u2013, \u2013, \u2013\n\nArchagathus,\n\narchaic period, \u2013, \u2013\n\nArchelaus II,\n\nArchidamia, , , 76n.8, , , ,\n\nArchidamus II, , ,\n\nArchidamus III, , 158n.75\n\nArchidamus V,\n\nArchippia,\n\nAres, ,\n\nArete,\n\n_arete_ (virtue), \u2013. _See also_ virtues of Spartan women\n\nAreus I, , 76n.6\n\nAreus II, 76n.6\n\nArgeia, ,\n\nArgileonis,\n\nArgonauts,\n\nArgos, 123n.76\n\nAristaenus, \u2013\n\nAristides,\n\nAristippus of Argos,\n\nAristodemus, ,\n\nAristomenes, ,\n\nAriston, \u2013, \u2013,\n\nAristophanes, , , , , \u2013, , ,\n\nAristotle\n\nbiographical information on,\n\non constraints on men versus women,\n\non dowries,\n\nand Epitadeus, x\n\non heiresses, , \u2013,\n\non inheritance of acquired characteristics,\n\non land tenure, \u2013, 90n.53,\n\non luxury,\n\non military defense by women,\n\non _oikos_ , ,\n\non _oliganthropia_ , ,\n\nand Plutarch,\n\non polis,\n\non reproduction, ,\n\nas source on Sparta, 58n.31, \u2013, ,\n\non wealthy women,\n\non women's influence, , , ,\n\nArmed Aphrodite, , 122n.73, 122n.75, ,\n\narmy. _See_ military; and specific wars and battles\n\nArrephoroi,\n\nArria,\n\nArrigoni, Giampiera, 12n.37\n\nArsino\u00eb,\n\nArsino\u00eb II,\n\nArtemis, , , , ,\n\nArtemis Corythalia, , ,\n\nArtemis Limnatis,\n\nArtemis Orthia: archaeological findings on, \u2013; clothing for figure of, \u2013; compared with cult of Helen, ; dancing at sanctuary of, \u2013; jewelry in sanctuary of, , 77n.12, ; meanings of \"Orthia,\" \u2013; priestesses of, ix, , , , , , ; religious worship of, \u2013, __, \u2013n.; sanctuary of, , 77n.12, \u2013, , , ; terracotta figurines of, on horseback, \u2013, 20n.65; and whipping ceremonies, 58n.30, , , ,\n\nArtemis Patriotis,\n\nArtemisia, ,\n\nAsclepiades of Samos,\n\nAsclepius, \u2013\n\nAsclepius Schoinatas,\n\nAstaphis,\n\nAstrobacus,\n\nAstumeloisa, ,\n\nAtalanta,\n\nAthena, , 21n.71, , , , \u2013, 122n.72\n\nAthena of the Bronze House (Chalkioikos), , , \u2013, ,\n\nAthena Poliouchos,\n\nAthenaeus, , , , , ,\n\nAthens\n\nadultery in,\n\nAnthesteria in, 150n.32\n\nanti-Spartan bias of, \u2013\n\nchildhood in, ,\n\ncruelty of women of,\n\ndivinities of,\n\ndowries in, \u2013\n\ndramatic poetry from, \u2013\n\neconomy of, , ,\n\neducation in, ,\n\nheiresses in,\n\ninfanticide in, \u2013\n\ninferior position of women in generally,\n\ninheritance in,\n\nland ownership in,\n\nliteracy in, , 5n.10\n\nmarriage and family in, , , , \u2013, ,\n\nmonument of Philopappus in,\n\nmotherhood in, , , \u2013\n\nmourning in, , ,\n\npolis of,\n\nproperty ownership by women in, \u2013\n\nprose from, \u2013\n\nreligion in, ,\n\nslave women in, ,\n\nsources on history of,\n\nspeaking by women in,\n\ntravel by women in,\n\nvase painting from,\n\nweaving in, \u2013, ,\n\nweddings in, \u2013, ,\n\nwives of citizens in,\n\nwomen of, during Peloponnesian War,\n\nwomen's clothing in, ,\n\nyoung girls in,\n\nathletic nudity, \u2013, __, \u2013, ,\n\nathletics\n\nco-ed competitions, , 14n.41, __,\n\ncompetitive racing and trials of strength for women, \u2013\n\nin Hellenistic Sparta, \u2013\n\nhorsemanship, \u2013, ,\n\nas incitement for marriage,\n\nnudity in, x, \u2013, __, \u2013, , , , _\u2013_\n\nand physical education, \u2013, __, __\n\nand religion, , , \u2013, , \u2013\n\nin Roman Sparta, \u2013\n\n_See also_ Cynisca; Euryleonis; horsemanship\n\nAugustus, , 55n.19, ,\n\nAuletes, 80n.24\n\nAurelia Heraclea,\n\nausterity of Sparta, , , , 162n.93\n\nauthority of Spartan women, \u2013, , , \u2013, \u2013\n\nAutocharidas, , 11n.30\n\nAymard, Andr\u00e9, 91n.56, 91n.58\n\nbachelors, , , , , ,\n\nBalbilla, Julia, \u2013\n\nBalbillus, Ti.Claudius,\n\n_baryllika_ ,\n\nbeauty, , , \u2013,\n\nBeauvoir, Simone de, \u2013\n\nBerenice II, 24n. 89,\n\n_bibasis_ ,\n\n_biduoi_ (magistrates),\n\nbigamy, , \u2013\n\nBilistiche, 23n.86, 24n.89\n\nbody size, 53n.6, , 54n.13\n\nBoer, W. den, 161n.90\n\nBradford, A. S., 87n.44\n\nBrasidas, ,\n\nbravery, , , \u2013, , , , 156n.64\n\nBremen, Riet van, 120n.62\n\nBriscoe, John, 90n.54\n\nbronze mirrors. _See_ mirrors\n\nbronze statuettes, \u2013, 164n.105\n\nByzantine dictionaries, \u2013\n\nCalame, Claude, 121n.64, 142n.10, 145n.17\n\nCallicles,\n\nCallicrateia,\n\nCallimachus, 24n.89\n\nCallistonice, 123n.79\n\nCampbell, D. A., 157n.72\n\nCartledge, Paul, , 78n.16, \u2013\n\nCaryae, , 107n.9\n\nCastor, , ,\n\nCeres,\n\nChaeron, \u2013\n\nchariot riding and racing, , , , , , ,\n\nCharision,\n\nchildbirth\n\ndeaths from, , 52n.3, , , ,\n\nEileithyia as goddess of, \u2013\n\n_See also_ motherhood\n\nchildhood, \u2013, \u2013, \u2013. _See also_ education; Spartan boys\n\nchildless Spartan women, , , , , , , \u2013\n\nChilon,\n\nChilonis, \u2013, , , , , 76n.6, \u2013,\n\nchiton, , , , , ,\n\n_chiton exomis_ (tunic with one sleeve), ,\n\nchoruses, , \u2013. _See also_ Alcman; _mousike_\n\nChrimes, K., 151n.36\n\nChristien, J., 92n.60\n\nChryse, , 76n.10\n\nCicero, , \u2013, , ,\n\nCinadon, ,\n\nCivil War, American, \u2013\n\nClark, Christina A., 145n.18\n\nClaudius,\n\nCleaichma, , 11n.30\n\nCleanor,\n\nCleanthes,\n\nClearchus, 107n.9\n\nCleaver celebration,\n\nCleitagora, , 10n.25, \u2013\n\nCleombrotus, ,\n\nCleomenes I, , ,\n\nCleomenes III\n\nand _agoge_ (boys' program),\n\nchildren of, ,\n\ndeath of,\n\nand dowries,\n\nend of reign of, \u2013,\n\nand freedom for helots, 97n.8\n\nPlutarch's life of, ,\n\nreforms of, , , , \u2013, 90n.53,\n\nand Stoicism, ,\n\nwife and concubine of, , , , ,\n\nCleonymus, , ,\n\ncleruchies (land allotments), , \u2013\n\nCloelia, 21n.71\n\nclothing\n\nof Athenians, ,\n\nof Spartan men and boys,\n\nof Spartan women, \u2013, , __, \u2013\n\nfor statues, \u2013, 30n.122. _See also_ weaving\n\ncoinage, , ,\n\ncompetitiveness, , \u2013, , , . _See also_ athletics\n\nConfederate States of America, \u2013\n\n_Constitution of the Spartans_ (Aristotle),\n\ncontraception. _See_ family planning\n\nCorinth,\n\nCoriolanus,\n\ncosmetics, ,\n\nCottina, , ,\n\ncowardice, , , \u2013, , , , 152n.40, 156n.64\n\nCratesiclea,\n\nCratesicleia, , , \u2013, , , ,\n\nCratinus,\n\nCrete, ,\n\nCritias, ,\n\ncross-dressing, ,\n\nCrouwel, J., 163n.99\n\n_cryptoi_ (secret hunters of helots), ,\n\nCynics, ,\n\nCynisca\n\nbirth year of, 21n.77\n\ndefiance of brother by, ,\n\nepigram on, , \u2013\n\nfamily of,\n\nhorseracing by, \u2013,\n\nas Olympic victor, \u2013, ,\n\nshrine and sculptures of, \u2013, 23n.87,\n\nas unmarried, 22n.79,\n\nwealth and vanity of, 22n.81, ,\n\nCyrene, 24n.89\n\nDamocrita, 49n.47\n\nDamainetus, 125n.91\n\nDamareta,\n\nDamatria, ,\n\nDamatrius,\n\ndancing, , __, , , \u2013, ,\n\nDarwin, Charles, \u2013\n\nDawkins, R. M., 135n.13,\n\nDegas, Edgar, , 16n.44, __\n\nDeinicha, , , 158n.75\n\nDeinis, 158n.75\n\nDemaratus, , , , ,\n\nDemeter, 54n.13, \u2013,\n\nDemetrius of Phalerum, 90n.51\n\nDemin, Giovanni, __,\n\ndemography, \u2013, . _See also_ family planning; fertility; infanticide; life expectancy; marriage; polyandry; sex ratio\n\nDettenhofer, Maria H.,\n\nDicaearchus, 158n.74\n\nDidymus,\n\ndiet. _See_ food and diet\n\nDiodorus, 58n.29,\n\nDiogeitus,\n\nDionysiades,\n\nDionysius, ,\n\nDionysus, , , , \u2013, ,\n\nDionysus of the Hill (Colonatas),\n\nDioscorides,\n\nDioscuri, , , , ,\n\ndivorce, . _See also_ marriage\n\n_doriazein_ (to dress like a Spartan girl),\n\nDorieus, 158n.75\n\n_doulai_ (slave women), , \u2013\n\n_douloi_ (slaves), , 100n.24\n\ndowries, \u2013, , \u2013, \u2013\n\ndramatic poetry, \u2013. _See also_ specific playwrights\n\ndress. _See_ clothing\n\nDriodones,\n\n_dromos_ (racecourse),\n\nDroop, J. P., 77n.12\n\nDucat, Jean, 92n.60, 100n.24\n\nearthquake, , 67n.55, 80n.26\n\neating. _See_ food and diet\n\neconomics of marriage and family, \u2013, , . _See also_ dowries; wealth of elite women\n\neducation\n\nin Athens, ,\n\nand athletic competitions, \u2013\n\nand childhood, \u2013\n\ngoals of, , \u2013\n\nin Hellenistic Sparta, \u2013\n\nand horsemanship, \u2013\n\nand learned Spartan women, \u2013\n\nand literacy, \u2013, 5n.10,\n\nand motherhood, , ,\n\nand _mousike_ , , , ,\n\nphysical education, \u2013,\n\nin Roman Sparta, \u2013\n\nsources, vii\n\nof Spartan boys, , , \u2013\n\nand the state, \u2013,\n\nEgger, \u00c9mile,\n\nEgypt, \u2013, ,\n\nEileithyia, , \u2013, 113n.29, 113n.31,\n\nEirana, ,\n\nelections,\n\nEleusinion, \u2013\n\nElis\n\nCynisca's commemorative sculptures at,\n\nHera's sanctuary at,\n\nmirrors from,\n\nSparta's attack on,\n\nwomen's races at, x\u2013xi, 22n.79, \u2013, 30n.121, \u2013\n\nelite women\n\nand adultery and bigamy, \u2013\n\nautonomy and social power of, \u2013, \u2013\n\nand dowries, \u2013\n\nas heiresses, , , \u2013,\n\nhonors and titles for, , \u2013, \u2013\n\nand land tenure, \u2013\n\nand luxury, \u2013, 83n.31,\n\nmarriage of, \u2013\n\nas priestesses, \u2013\n\nand reforms of Agis and Cleomenes, \u2013, 90n.53,\n\nand reforms of Apega, Nabis and Chaeron, \u2013\n\nroyal women, \u2013, \u2013\n\nwealth of, , \u2013, 90n.53, \u2013, \u2013\n\n_See also_ Spartan women; and specific women\n\nendogamy, \u2013,\n\nEngels, F.,\n\nEnodia, 114n.36\n\nEpaminondas,\n\nEphorus, ,\n\nEpidaurus, \u2013\n\n_epikleros_ (heiress), , \u2013. _See also_ heiresses\n\nEpiphanes, C.Antiochus,\n\nEpitadeus, x, xn., , , , ,\n\n_Epithalamium to Helen_ (Theocritus), ,\n\nequestrianship. _See_ horsemanship\n\nEsther,\n\nEthnicity, Spartan, vii, , \u2013\n\nEtruscan painting,\n\neugenics, \u2013, , 48n.45, , \u2013,\n\nEuonyma, ,\n\nEupolia, ,\n\nEupolis, ,\n\nEuripides, , , 83n.31, \u2013, ,\n\nEurybanassa, 126n.93\n\nEurycrates, 157n.72\n\nEuryleon, 158n.75\n\nEuryleonis, , , 158n.75\n\nEurysthenes,\n\n_eusebeia_ (piety), , \u2013\n\n_euteknos_ (blessed with good children),\n\nexomis (tunic),\n\nextramarital intercourse, \u2013, 41n.23. _See also_ sexual behavior; Timaea\n\nfamily\n\nin Athens, \u2013, \u2013\n\nBeauvoir on, \u2013\n\nchanges in, over time, \u2013\n\nfather-daughter relationship, \u2013\n\nof free noncitizens,\n\nof helots, ,\n\nand interlocking public and private spheres, \u2013\n\nmen's relationships within,\n\nmother-daughter relationship,\n\nmother-son relationship, \u2013, __,\n\nas _oikos_ , , \u2013, , , , \u2013\n\nsize of, , ,\n\nwomen's dominance in, , , ,\n\nwomen's influence in, , , , ,\n\n_See also_ fatherhood; marriage; motherhood; _oikos_ (family, household, estate)\n\nfamily planning, \u2013, , , ,\n\nFates,\n\nfather-daughter relationship, \u2013\n\nfatherhood, \u2013, , , 55n.18, \u2013, ,\n\nfeasting, \u2013, , , , ,\n\nfemale infanticide, \u2013, 35n.8, \u2013,\n\n\"Female Swimmers\" (Alcman), 13n.38\n\nfeminist theory, ,\n\nfertility, \u2013, , 65n.46, , , , , ,\n\nFigueira, Thomas J., 80n.26\n\nFinley, M. I., \u2013, 160n.82\n\nFlamininus, ix\n\nfood and diet, , \u2013, \u2013, 52n.5, \u2013, , \u2013, , 150n.33. _See also_ feasting; syssitia; wine-drinking\n\nF\u00f6rtsch, R., 163n.99\n\nfree noncitizens, , , \u2013,\n\n_genitor_ (biological father), , , 47n.38\n\n_gennaia_ (well born),\n\ngold, , \u2013, 77n.12, , ,\n\nGorgo, , , , , , , , ,\n\nGortyn, \u2013, 92n.60, ,\n\ngrave markers, , 52n.3, , , ,\n\ngreed, ,\n\nGylippus, , ,\n\nGymnopaidia (Festival of Nude Youths), , , , 158n.77\n\n_gymnos_ (nude, lightly dressed),\n\n_gynaikonomos_ (regulator of women), , , 118n.45, \u2013\n\nGyrtias, ,\n\nHadrian, , ,\n\nHagesichora, , , , ,\n\nHagnon of Tarsus,\n\nhair, , , , \u2013, __, 42n.26,\n\nHall, Edith, 146n.22\n\nhealth and healing, \u2013, ,\n\nHecate, 114n.36\n\n_Heilotes_ (Eupolis),\n\nheiresses, , , , , \u2013, , . _See also_ inheritance\n\nHelen\n\nin art, _\u2013_, , 169n.12\n\nbirth of, , 123n.78\n\nbronze votive of, on horseback,\n\nchildren presented to, ,\n\ncompared with cult of Artemis Orthia,\n\ncult of, at Therapne, 115n.42,\n\ndaughter of, \u2013\n\nfamilies claiming descent from,\n\nHerodotus on,\n\nlegends of, \u2013\n\nmeaning of name,\n\nMenelaion as sanctuary of, 22n.78, , \u2013, \u2013\n\nand Menelaus, , ,\n\nand Paris, , 169n.112\n\nPausanias on,\n\npriesthoods of,\n\nraces in honor of, , ,\n\nrape of, by Theseus,\n\nreligious worship of, , , 105n.1, \u2013, 115n.42, 123n.77\n\nas spinner and weaver, , 31n.125,\n\nstatuary of, __, __,\n\nwealth of, , ,\n\nand wrestling,\n\nHeleneia,\n\nHellenistic period, \u2013, , \u2013, , , \u2013\n\nhelots\n\nand agriculture, , ,\n\nchildren born of Spartans and, , , ,\n\ndefinition of,\n\nand eugenics, \u2013\n\nfamilies of, ,\n\nfreedom for, , 97n.8,\n\nkillings of, \u2013\n\nmarriage between Spartan women and, \u2013, 90n.52, 91nn.56\u2013\n\nNabis and citizenship for, \u2013\n\nas nurses, , 99n.13\n\npopulation of,\n\nprocreative unions between Spartan men and, , , \u2013\n\nprocreative unions between Spartan women and, , 48n.45, , , ,\n\nrebellion of, in 460s,\n\nreligion of, \u2013\n\nroles and duties of, , , \u2013\n\nslaves compared with, \u2013,\n\nas soldiers, ,\n\nsources of information on helot women,\n\nand Spartan songs,\n\nand weaving,\n\nwomen's discussion of financial matters with,\n\n_Helots_ (Eupolis), ,\n\nHera, x\u2013xi, \u2013, 25n.95, 30n.121, 31n.124, , \u2013\n\nHera Hypercheiria, \u2013\n\nHeraea, x\u2013xi, \u2013, , 30n.121, \u2013,\n\nHerculanus,\n\nHerington, J., 144n.13\n\nHermione, 83n.31, \u2013\n\nHermippus,\n\nHerodotus\n\non bigamy,\n\nbiographical information on,\n\non clothing, \u2013\n\non cult of Helen at Therapne, 115n.42,\n\non heiresses,\n\non marriage, , , ,\n\non Menelaion,\n\nas source on Sparta, , ,\n\non ugly girl turned beautiful by Helen, \u2013\n\non wives of Anaxandridas, , ,\n\non wives of Minyae,\n\nHesiod,\n\nHestia Poleos (Hestia of the City), ,\n\nHesychius, ,\n\nhetairai, , , , , ,\n\nHilaeira, ,\n\n_himatia_ (outer dresses),\n\nHimerius of Prusa,\n\nHippoco\u00f6n, 14n.43,\n\nHippocrates, 64n.44\n\nHippocratics,\n\nHippodamia, , 113n.33\n\nHippomedon,\n\nHodkinson, Stephen J., xn. , 35n. 6, 69n.60, 78n.16, \u2013, 92n.60, 97n.7, 158n.75,\n\nHomer, 23n.83, , , ,\n\n_homoioi_ (men of equal status or similars), , , ,\n\nhomosexuality\n\nbetween men, , , , \u2013,\n\nbetween women, , 16n.44, , , , , , 160n.88,\n\n_See also_ sexual behavior\n\nHorace, 42n.26\n\nhorsemanship, \u2013, 19n.61, 20n.62, 24n.89, , , , , \u2013, ,\n\nHrdy, Sarah Blaffer, \u2013\n\nHume, David,\n\nhunting,\n\nhusband-doubling and wife-sharing, \u2013, 37n.12, 39n.18, 40n.21, , , 47n.40, , , , , \u2013, . _See also_ marriage\n\nHyacinthia, , , , \u2013, 120n.62, 121n.64, , . _See also_ Apollo\n\nhypogamy, \u2013, \u2013. _See also_ Tarentum\n\n_hypomeiones_ (demoted citizens),\n\nhyporcheme,\n\n_hypostatria_ (dresser of the cult image),\n\nIamblichus, \u2013\n\nIbycus, , ,\n\nillegitimacy, ,\n\n_ilotae_ (helots), . _See also_ helots\n\nincest, 69n.60,\n\ninfanticide, \u2013, , \u2013, ,\n\ninheritance, \u2013, , , , , 78n.17, \u2013, , , . _See also_ heiresses\n\nIonia,\n\nIphigenia,\n\n_ius trium liberorum_ (law of three children), , 55n.19\n\nJudith,\n\nJulia Balbilla, the Pious, \u2013\n\nJulia Domna,\n\nJulianus,\n\nJustin,\n\n_kallabides_ , _kallabidia_ ,\n\n_kleronomos_ (heir apparent),\n\n_kleros_ (plot of land), x, , , , , , , \u2013, , , , , . _See also_ land tenure\n\n_komos_ (revel),\n\n_kopis_ (Cleaver celebration),\n\nKore, 54n.13\n\n_kosmiotes_ (orderliness),\n\nKrone, Julie, 19n.61\n\nKunstler, Barton,\n\n_kyrioi_ , 92n.60\n\n_kyrittoi_ ,\n\nLacedemonians, ,\n\nLacey, W. K., \u2013\n\nLaconia, , , , , , , , , \u2013,\n\nLaconian hounds, 21n.75\n\nLactantius,\n\nLagren\u00e9e the Elder, Louis Jean Fran\u00e7ois, __\n\n_Lakaina_ (Apollodorus Carystius or Apollodorus Gelous),\n\n_Lakainai_ (Sophocles),\n\n_Lakonika_ (Polycrates),\n\n_lambano_ (man who takes a wife), ,\n\nLampito, , , , , 149n.29\n\nland tenure, \u2013, \u2013, , . _See alsokleros_ (plot of land)\n\nLarson, Deborah, 118n.46\n\nLatin authors, . _See also_ specific authors\n\n_Laws_ (Plato), , , , ,\n\nlaws of Sparta, x, , , , 92n.60, , , ,\n\nLeda, , 123n.78\n\n_Leda_ (Eupolis),\n\nleisure, ,\n\nLeonidas, , , , , \u2013, 87n.44\n\nLeotychidas,\n\nLeotychidas II, , 149n. 29, , 157n.72\n\nlesbianism. _See_ homosexuality\n\nLesbos, 5n.10\n\nLeucippides, , , , 118n.46,\n\nLeucippus, ,\n\nLeuctra, battle of, ix, , , , 66n.49, , , ,\n\nLex Oppia,\n\nlife expectancy,\n\n_Life of Lycurgus_ (Plutarch), , , , ,\n\nliteracy, \u2013, 5n.10. _See also_ education\n\nLivy, , 90n.54\n\nlower classes\n\nin Athens,\n\n_doulai_ (slave women), \u2013\n\nhelots, , 48n.45, , , , , , \u2013,\n\n_mothakes_ (mixed-blood children), , , 96n.4,\n\nnurses, \u2013\n\n_perioikoi_ (free noncitizens), , , \u2013,\n\nprostitutes,\n\nreligion of, \u2013\n\nand reproduction, \u2013\n\nworking women, \u2013\n\nLuraghi, Nino, 95n.1\n\nluxury, \u2013, 83n.31, . _See also_ austerity; wealth of elite women\n\nLycurgus\n\nConstitution attributed to, ix, x, ,\n\ncosmetics outlawed by,\n\ndowries outlawed by,\n\neconomic system of, 78n.16\n\nhistorical existence of, x,\n\nand land tenure,\n\nlaws of, x, 78n.16, , , , , ,\n\non marriage, , , ,\n\non motherhood, ,\n\npaintings of, , __, , __\n\non physical education, ,\n\nPlutarch on, 78n.16, , , ,\n\nprostitution outlawed by,\n\non weaving,\n\nXenophon on, x, , , , , , , , ,\n\nLysander\n\nchitons for daughters of,\n\nengagements of daughters of, and lack of wealth, \u2013, , , \u2013,\n\nand Ephorus,\n\nmarriage of, 45n.32, , 133n.3\n\nPlutarch on daughters of,\n\nLysander (general),\n\nLysandridas,\n\nLysanoridas, , 133n.2\n\n_Lysistrata_ (Aristophanes), , , ,\n\nMacedonia, ,\n\nmale infanticide, \u2013, ,\n\nMalicha of Cytheria,\n\nMarcus Aurelius Teisamus,\n\nMardonius,\n\nMariette, Auguste F. F.,\n\nmarriage\n\nabduction myths as metaphor for, \u2013, ,\n\nage at marriage, , , , 56n.22,\n\nin Athens, \u2013, \u2013,\n\nbigamy, , \u2013\n\nand bride capture, , , ,\n\nand dominance of men by women, , , , ,\n\neconomics of, \u2013\n\nof elite women, \u2013\n\nand endogamy, \u2013,\n\nEngels on,\n\nand eugenics, \u2013,\n\nand extramarital intercourse, \u2013, 41n.23,\n\nof heiresses,\n\nof helots and Spartan women, , \u2013, 90n.52, 91nn.56\u2013\n\nhusband-doubling and wife-sharing in, \u2013, 37n.12, 39n.18, 40n.21, , , 47n.40, , , , , \u2013,\n\nand husband's rights,\n\nand illegitimacy,\n\nand infanticide, \u2013, , \u2013\n\nof older man and younger woman, , ,\n\nreproduction as goal of, , , \u2013,\n\nresidences of bride and groom following, \u2013,\n\nin Rome,\n\nof royal women, \u2013\n\nsecret marriage, \u2013, , , ,\n\nand selection of spouse, , \u2013, \u2013, \u2013\n\nsex ratio and polyandry, \u2013\n\nsexual behavior in, , , , , \u2013,\n\nand shortage of husbands, \u2013\n\nSpartan laws on,\n\nwedding customs, , \u2013, ,\n\nwomen's influence in, , , , ,\n\n_See also_ family; motherhood\n\nMarxism,\n\nMcClure, Laura, 135n.16\n\nMegalopolis,\n\nMegalostrata, ,\n\nMegistonous,\n\nMenelaion, , 20n.65, , \u2013, ,\n\nMenelaus, , , _\u2013_, , \u2013, ,\n\nmenstruation and menarche, , 27n.107, ,\n\nmercenaries, \u2013\n\nMessenia, ix, , , , , , , , ,\n\nMessenian War, Second, , , , , , ,\n\nmilitary, , , \u2013, \u2013, , , , , , , 156n.64. _See also_ specific wars and battles\n\nmilitary defense by Spartan women, , , \u2013, 122n.75\n\nMiller, Margaret C., 108n.17\n\nMimnermus,\n\nMinyae,\n\nmirage, Spartan, viii, , , , , \u2013\n\nmirrors, , , , \u2013, _\u2013_,\n\n_monochitones_ (tunic),\n\n_Moralia_ (Plutarch),\n\n_mothakes_ (mixed-blood children), , , 96n.4, , 97n.7, \u2013\n\nmother-daughter relationship,\n\nmother-son relationship, \u2013, __,\n\nmotherhood\n\nin Athens, , \u2013\n\nathletics as preparation for, ,\n\nand bravery\/cowardice of sons, , , \u2013, , , , 156n.64\n\nas bridge between private and public spheres, \u2013\n\nchange over time, ,\n\nin Confederate States of America, \u2013\n\ndemographic speculations on, \u2013\n\nand dominance of men by women, , , ,\n\nand educational goals for girls, , ,\n\nand family planning, \u2013, ,\n\nand fertility, \u2013, ,\n\nand food and diet for females, \u2013,\n\nas fulfilling experience for women, ,\n\ngrave markers for mothers who died in childbirth, , 52n.3, , , ,\n\nideology of, \u2013\n\nimportance of, in Sparta,\n\nand infanticide, \u2013, , \u2013\n\nkilling of cowardly sons by mothers, \u2013, , , 152n.40, 156n.64\n\nand later tradition, , __\n\nand literacy of women,\n\nLycurgus on, ,\n\nmother-daughter relationship,\n\nmother-son relationship, \u2013, __,\n\nand number of children, ,\n\n_oikos_ and family economy,\n\nand refusal of Spartan women to bear children, , \u2013, , , , , ,\n\nand religion,\n\nrewards for, , 52n.3,\n\nRoman interest in,\n\nand the Spartan polis, , \u2013, \u2013\n\nin United States, \u2013\n\n_See also_ eugenics; husband-doubling; marriage; _oliganthropia_\n\nmourning, , , , ,\n\n_mousike_ (music, dancing, poetry), , , , , , ,\n\nMurray, Julia Sargent,\n\nmusic, , , , , , , , \u2013,\n\nMyllias of Croton,\n\nMyrto,\n\nNabis, \u2013, 90n.53, , ,\n\nnakedness. _See_ nudity\n\nnames of Spartan women, , , 52n.3, , , \u2013, , , \u2013\n\n_Nanno_ (Mimnermus),\n\nNausicaa,\n\n_neodamodeis_ (freed helots),\n\nNeoptolemus, 83n.31\n\nNeo-Pythagoreans,\n\nNepos, 58n.29\n\nNero,\n\nNicander,\n\nNicippia, 76n.9, ,\n\nNicolaus of Damascus, , ,\n\nNistheadousa, , 11n.29\n\n_nothoi_ (bastards), ,\n\nnudity\n\nathletic nudity, \u2013, __, \u2013, , ,\n\nand choice of spouse,\n\nerotic nudity, \u2013, __, ,\n\nof females in front of bachelors, , ,\n\nGymnopaidia (Festival of Nude Youths), , , , 158n.77\n\nPlutarch on, , , 34n.5\n\nin Spartan art, \u2013, _\u2013_\n\nnurses, \u2013\n\nnutrition. _See_ food and diet\n\n_Odyssey_ (Homer), ,\n\n_Oeconomicus_ (Xenophon), , , ,\n\nOgden, Daniel, , , 48n.45\n\n_oiketai_ (household slaves), , 100n.24\n\n_oikos_ (family, household, estate), , \u2013, , , , \u2013, , , , , , , , . _See also_ family\n\n_oligandria_ ,\n\n_oliganthropia_ (sparse population), , \u2013, ,\n\nOllier, F., . _See also_ mirage\n\nOlympia and Olympiads, , \u2013, 22n.79, \u2013, 25n.95, , , ,\n\nOlympio,\n\n_On Alcman_ (Sosibius),\n\n_On Mimes in Laconia_ (Sosibius),\n\n_On the Sacrifices in Lacedaemon_ (Sosibius),\n\nOrestes,\n\nOrthia, . _See also_ Artemis Orthia\n\nOrthia sanctuary, ix, \u2013, 20n.65, \u2013, 58n.30, \u2013, , , , \u2013. _See also_ Artemis Orthia\n\nOvid, ,\n\n_paidika_ (young boyfriends),\n\nPalladas,\n\nPanathenaea,\n\nPanteus, , , ,\n\nPanthalida, 29n.115\n\nParis, , 169n.112\n\n_Partheneion_ (Alcman), \u2013, , , , \u2013, , , \u2013, __, 145 _nn_ 17\u2013,\n\nPartheniai, 48n.44\n\n_parthenoi_ (virgins, unmarried women), ,\n\npartible paternity,\n\n_pater_ (legal father and husband),\n\npatriarchy, , ,\n\n_patroiokos_ (heiress), \u2013\n\n_patrouchos_ (heiress), \u2013\n\nPatterson, Cynthia B., 40n.21\n\nPausanias\n\non Aphrodite,\n\non Artemis,\n\non chariot racing, 22n.79,\n\non Cleomenes,\n\non Cynisca, ,\n\nand Ephorus,\n\non Leucippides,\n\non physical education,\n\non religion of helots,\n\nas source on Sparta,\n\non weaving, ,\n\nPausanias (traitor), , 58n.29, ,\n\nPax Romana,\n\nPeloponnesian League,\n\nPeloponnesian War, x, , , , , , , ,\n\nPelops, 113n.33\n\nPenelope,\n\n_peplos_ (tunic), , , 30n.121, , , \u2013\n\nPercalus,\n\nperfume,\n\nPericles, ,\n\n_perioikoi_ (free noncitizens), , , \u2013,\n\nPerithoos,\n\nPersepolis, 52n.5\n\nPersia, ,\n\nPersian War, ,\n\nPettersson, Michael, 121n.64\n\n_phalli_ , , 108nn.15, . _See also_ transvestism\n\nPhilip V ofMacedon,\n\nPhilo,\n\nPhilopappus,\n\nPhilopator,\n\nPhilopoemen, ,\n\nphilosophers, \u2013. _See also_ specific philosophers\n\nPhilostratus, 14n.41,\n\nPhoebe, ,\n\nPhylarchus, 11n.34, , 151n.36\n\nPhylono\u00eb, , 114n.36\n\nPhyscoa,\n\nphysical education, \u2013, __, _. See also_ athletics\n\nphysicians,\n\npiety, , \u2013\n\nPiper, Linda, 91n.56\n\nPlataea, battle of,\n\nPlatanistas,\n\nPlato\n\non age for parenthood,\n\nand Critias,\n\non dowries,\n\non education, ,\n\non military defense by women,\n\non nudity of women, \u2013\n\nand Plutarch, xn.,\n\non sexuality,\n\nas source on Sparta, , ,\n\non weaving,\n\non wine drinking by Spartan women,\n\nPliny, 34n.4, 42n.26\n\nPlutarch\n\non abortion,\n\non adultery,\n\non _agoge_ ,\n\nbiographical information on,\n\ncompared with Xenophon, \u2013\n\non Cynisca,\n\nDegas and,\n\non dowries,\n\non Epitadeus, x, xn.\n\non extramarital intercourse, \u2013\n\non family,\n\non female infants,\n\non food and diet,\n\non helots,\n\non heterosexual intercourse, \u2013,\n\non homosexuality, \u2013,\n\non infanticide,\n\non _kleroi_ , , , ,\n\non land tenure,\n\non luxury, 83n.31\n\non Lycurgus, 78n.16, , ,\n\non marriage, , \u2013, 42n.26, \u2013, 56n.22, , ,\n\non motherhood, , , \u2013, ,\n\non nudity, , , 34n.5\n\non nurses, ,\n\non Panteus,\n\non Philopoemen,\n\non physical education, , ,\n\nPomeroy's research on, vii\n\nas priest at Delphi,\n\non religion,\n\non reproduction, ,\n\non sexuality, , \u2013\n\nas source on Sparta, 89n.48, , \u2013, \u2013, 161n.90\n\nand Sparta chronology, ix\n\non Spartan divinities with weapons, \u2013\n\non speaking by women, ,\n\non wealthy women, , , ,\n\non weaving, , ,\n\non women's ability to defend themselves,\n\nPolemon, ,\n\npolis, \u2013\n\n_Politics_ (Aristotle), \u2013\n\nPollux, , , ,\n\nPollux of Naucratis,\n\n_poloi_ (fillies),\n\n_polos_ (head-dress),\n\nPolyaenus, 58n.29\n\npolyandry, \u2013, \u2013\n\nPolybius, , \u2013, , , , 90n.52,\n\nPolycrates, ,\n\nPolydorus, , ,\n\nPompeius Trogus, 16n.49\n\nPomponia Callistonice, , 120n.63,\n\nPomponia Callistonice Arete,\n\nPoralla Paul, 58n.29, 158n. 76\n\nPoseidon,\n\npottery, . _See also_ vase painting\n\nPowell, A., 87n.43\n\npriestesses, ix, , , , , \u2013, ,\n\n_principes iuventutis_ ,\n\nprivate property, , , , \u2013, , , , . _See also_ property ownership\n\nProauga,\n\nProcles,\n\nProcris,\n\nProlyta, , ,\n\nPropertius, , 42n.26\n\nproperty ownership\n\nin Athens, \u2013\n\nprivate property, , , , \u2013, , , ,\n\nby Spartan women, \u2013, , 90n.53, \u2013, 92n.60, \u2013\n\n_See also_ dowries; inheritance; land tenure; wealth\n\nprosopography, , \u2013. _See also_ names of Spartan women\n\nprostitutes, , , ,\n\nPtolemaic queens, , ,\n\nPtolemy I, ,\n\nPtolemy II, 24n.89, ,\n\nPtolemy III Euergetes,\n\nPtolemy IV, Philopator,\n\nPtolemy XII Auletes, 80n. 24\n\npuberty, , 54n.12. _See also_ menstruation and menarche\n\npublic speaking, ,\n\npublic sphere, , \u2013. _See also_ althletics; royal women\n\nPyrrhus, , , , 122n.75,\n\nPythagoras and Pythagoreans, \u2013\n\nPythagoras of Argos,\n\nPythia,\n\n_quadriga_ (four-horse chariot), ,\n\nqueen as title, \u2013\n\nraces. _See_ chariot riding and racing; running races\n\nrape, , , , , , 108n.13, , ,\n\nRedfield, James,\n\nreligion\n\nAphrodite, 34n.4, , \u2013,\n\nApollo, \u2013\n\nArtemis Orthia, ix, \u2013, 20n.65, \u2013, 58n.30, , 77n.12, \u2013, , \u2013, 164\u201365n.106\n\nAthena, \u2013\n\nin Athens,\n\nand athletics, , \u2013, , \u2013\n\nDemeter, 54n.13, \u2013,\n\nDionysus, , , , \u2013,\n\nEileithyia, , \u2013, 113n.29, 113n.31,\n\nand feasting, \u2013, , , , ,\n\nHelen, , , 105n.1, \u2013\n\nHera, \u2013\n\nand Julia Balbilla, the Pious, \u2013\n\nof lower classes, \u2013\n\nand nurses,\n\nand priestesses, ix, , , , , \u2013, ,\n\nin Roman period, \u2013\n\nsources of information on,\n\n_See also_ specific gods and goddesses and specific festivals\n\nreproduction. _See_ family planning; fertility; infanticide; motherhood\n\n_Republic_ (Plato), , \u2013, , ,\n\nRepublican motherhood,\n\n_rhetra_ (legislation), x, , . _See also_ Epitadeus\n\nRicher,Nicolas, 16n.47\n\nrobots, \u2013, 90n.51\n\nRolley, Claude, \u2013n.\n\nRoman period, \u2013, \u2013, \u2013\n\nRome\n\nchariot riding in,\n\nchildbearing among upper classes of, , 55n.19, ,\n\nequestrian statues in, 21n.71\n\nmarriage in,\n\nmotherhood in,\n\nOppian Law of, ,\n\nphysicians in,\n\nand Second Punic War,\n\nwealth of, ,\n\nwrestling match at,\n\nRoussel, Pierre,\n\nroyal women\n\nand adultery, \u2013\n\nauthority of, \u2013\n\nand bigamy, \u2013\n\nmarriage of, \u2013\n\npiety of, \u2013\n\nand reforms of Agis and Cleomenes, \u2013\n\nand reforms of Nabis and Chaeron, \u2013\n\nand religion, , \u2013\n\nand succession,\n\nwealth of, \u2013, \u2013\n\n_See also_ elite women; and specific women\n\nrunning races, x\u2013xi, , __, 22n.79, \u2013, 25n.95, 25n.97, , , 30n.121, \u2013, , ,\n\nSabina, ,\n\nSalamis, battle of,\n\nSamos, ,\n\nSappho, 5n.10, ,\n\n_Sayings of Spartan Women_ (Plutarch), , , \u2013, , , , , , , ,\n\nScanlon, T., 14n.41\n\nsculpture, \u2013, 164n.105, \u2013\n\nSeleucus Nicator, 87n.44\n\nSellasia,\n\n_semnotes_ (dignity),\n\n_servi_ (slaves),\n\nSerwint, N., 26n.102\n\nsex ratio\n\ndemographic speculations on, \u2013\n\nand infanticide, \u2013,\n\nand polyandry, \u2013\n\nSextus Empiricus,\n\nsexual behavior\n\nanal intercourse, 29n.118,\n\nDarwin on, \u2013\n\nextramarital intercourse, \u2013, 41n.23\n\nhomosexuality, , 16n.44, , , , \u2013, , , , , , 160n.88\n\nin marriage, , , , , \u2013,\n\nin sculpture and vase painting, \u2013\n\nof Spartan women and girls, , , , \u2013,\n\n_See also_ rape; virginity\n\nShimron, B., 91n.57\n\nShipley, D. R., 153n.45\n\nShipley, G., 163n.99\n\nSicily, ,\n\nsilver, , 77n.12, , ,\n\nSimonides,\n\nsinging. _See_ music\n\n_skolion_ (drinking song),\n\nslave women, , , , 99n.13, \u2013,\n\nslaves, \u2013,\n\nSocrates, , ,\n\nsoldiers. _See_ military; and specific wars and battles\n\nSolon, ,\n\n_Sophists at Dinner_ (Athenaeus),\n\nSophocles, viii, , ,\n\n_sophrosyne_ (modesty), ,\n\nSosibius, 144n.13, , 152n.38,\n\nSparta\n\narchaeological evidence on, , , \u2013\n\narchaic period of, \u2013, \u2013\n\nausterity of, , 162n.93\n\nchronology of, viii\u2013xi\n\nclassical writers on, \u2013\n\neconomic changes in, \u2013\n\nHellenistic historians on, \u2013\n\nHellenistic Sparta, \u2013, , \u2013, ,\n\nland tenure in, \u2013\n\nLatin authors on,\n\nlaws of, x, , , 92n.60, ,\n\npopulation of and population decline in, , , , \u2013, , , 139n.1\n\npopulation of helots in,\n\nprosopographical problems of, \u2013\n\nreforms of Agis and Cleomenes, \u2013, 90n.53,\n\nand reforms of Apega, Nabis, and Chaeron, \u2013\n\nRoman period of, \u2013, \u2013, \u2013\n\nRoman writings on, \u2013\n\nsecondary sources on, \u2013\n\nsex ratio in, \u2013, , \u2013\n\nterritory controlled by, \u2013, 66n.49\n\nuniqueness of,\n\nwritten sources on, \u2013\n\nSpartan boys\n\nand _agoge_ , , , , , 158n.77\n\nand athletic nudity, \u2013, __,\n\nclothing of,\n\neducation of, , , \u2013\n\nfood for, , 53n.6\n\nand Gymnopaidia (Festival of Nude Youths), , , , 158n.77\n\nand Hyacinthia,\n\ninfanticide of male babies, \u2013, ,\n\nmodesty of,\n\nand mother-son relationship, \u2013, __,\n\nand whipping ceremonies, 58n.30, , , , ,\n\n_Spartan Constitution_ (Xenophon), , , , \u2013\n\nSpartan men\n\nand bachelorhood, , , , , ,\n\nclothing of,\n\ndominance of, by women, , , , ,\n\nemigration of, after Peloponnesian War,\n\nfather-daughter relationship, \u2013\n\nand fatherhood, \u2013, , , 55n.18, \u2013, ,\n\nfood for, \u2013, 53n.6\n\ngrave markers for men who died in war, , 52n.3, , , ,\n\ngreed of,\n\nas _homoioi_ , , , ,\n\nand homosexuality, , , , \u2013,\n\nand lack of wealth,\n\nlife expectancy of,\n\nprocreative unions between helots and, \u2013, \u2013\n\nrewards for fathers, , 55n.18,\n\nsex ratio on, \u2013, \u2013\n\n_See also_ marriage; sexual behavior; syssitia\n\nSpartan women\n\narchaeological evidence on, , \u2013\n\nand beauty, , \u2013,\n\nchildhood of, \u2013\n\nclothing for, \u2013, , __, \u2013\n\nand dowries, \u2013, ,\n\neducation, \u2013,\n\nelite women,\n\nexclusion of, from state administration,\n\nhair of, \u2013, 42n.26,\n\nhealth of, , ,\n\nhomosexuality of, , 16n.44, , , , , , 160n.88\n\nlife expectancy for,\n\nliteracy of, \u2013\n\nand marriage, , \u2013\n\nand motherhood, , , \u2013, , \u2013, \u2013,\n\nnaming of, \u2013\n\nas philosophers, \u2013\n\nphysical education for, \u2013,\n\nas poets, \u2013,\n\nprocreative unions between helots and, , 48n.45, , , ,\n\nrecording of names of, , 52n.3, , , , , \u2013\n\nand reforms of Agis and Cleomenes, \u2013, 90n.53,\n\nand reforms of Apega, Nabis, and Chaeron, \u2013\n\nrefusal of, to bear children, , \u2013, ,\n\nreligion of, \u2013\n\nroyal women, \u2013\n\nsex ratio on, \u2013, \u2013\n\nsexuality of, , , , \u2013\n\nsources on, \u2013\n\nstatus of, \u2013\n\nunmarried and childless women, , , , 49n.47, , , , , , \u2013,\n\nvirtues of, \u2013\n\nwealth of, , \u2013, 90n.53, \u2013, \u2013\n\nand weaving, \u2013,\n\nwords of, \u2013\n\nwritten sources on, \u2013\n\n_See also_ specific women\n\n_Spartans_ (Eupolis),\n\nSpawford, A. J. S., 118n.45,\n\nSpendon,\n\nSphaerus, , ,\n\nsports. _See_ athletics\n\n_stathmos_ (billet or dwelling place), \u2013\n\nstatus, , \u2013\n\nSte Croix, G. E. M. de,\n\nStehle, Eva, 106n.5\n\nStewart, Andrew,\n\nStoicism, , 11n.34, ,\n\nStrabo, 24n.89\n\nsuccession and royal women,\n\n_Suda_ ,\n\nSura, M. Palfurius,\n\n_synoikein_ (marry, cohabit), \u2013, 46n.38\n\n_syssition_ (mess group), 18n.53, , , ,\n\nTarentum, , 16n.47, , 48n.45, ,\n\nTaygetus Mount, , __, __, , , , ,\n\nTeiresias,\n\n_teknopoiia_ (child production),\n\nTeles,\n\nTelesilla, 123n.76\n\nTerpander,\n\nTexier, J.-G., 90n.53\n\nTheano, , 58n.29,\n\nThebans,\n\nTheocritus, , 14n.39, , ,\n\nTheopompus,\n\nThermopylae, battle of, ,\n\nTheseus, ,\n\nThibron, , 150n.35\n\nThird Social War,\n\n_thoinarmostria_ (mistress of the banquet),\n\nThucydides, , 58n.29,\n\nThylacis,\n\nTiberius,\n\nTimaea, \u2013, ,\n\nTimareta,\n\nTimasimbrota, , , 157n.72\n\nTimycha,\n\nTisamenus,\n\nTissaphernes, 107n.9\n\nTithenidia, , ,\n\ntraitors,\n\ntransvestism, , , , . _See alsophalli_\n\nTregaro, J. Christien, 41n.23\n\nTriptolemus, 54n.13\n\nTroilus of Elis, , 23n.83\n\n_trophimoi_ , 97n.7\n\nTsakona, 65n.46\n\nTymnes, ,\n\nTyndares, 126n.93\n\nTyndareus, 14n.43, , , 122n.74, , 123n.78\n\nTyrtaeus, 96n.3\n\nugliness, , \u2013. _See also_ beauty\n\nUnited States, \u2013\n\nunmarried Spartan women, , , 49n.47, , , , , \u2013\n\nupper classes. _See_ elite women; Spartan women\n\nutopia. _See_ mirage\n\nValerius Maximus,\n\n_Various Histories_ (Aelian),\n\nvase painting, n. , \u2013, \u2013\n\nVergil, ,\n\nVespasian,\n\nvirginity, , . _See alsoparthenoi_; rape\n\nvirtues of Spartan women, \u2013\n\nVitruvius, 155n.55\n\nVix crater, __,\n\nwarfare, women and, , \u2013. _See also_ military; and specific wars and battles\n\n_Wasps_ (Aristophanes),\n\nwealth of elite women, , \u2013, 90n.53, \u2013, \u2013\n\nweaving, \u2013, , , , , , ,\n\nwedding. _See_ marriage\n\nwhipping ceremonies, 58n.30, , , , ,\n\nwidows, , , ,\n\nwife-sharing. _See_ husband-doubling and wife-sharing; marriage\n\nwine drinking, , , , __, , \u2013, , 150n.32\n\nworking women, \u2013\n\nwrestling, \u2013, __, , ,\n\nXenaria, \u2013\n\n_xenelasia_ (strangers),\n\nXenopeithia, ,\n\nXenophon\n\nbiographical information on, \u2013\n\ncompared with Plutarch, \u2013\n\nand Critias, ,\n\non Cynisca,\n\non education,\n\non extramarital intercourse, \u2013, 41n.23\n\non family,\n\non fatherhood,\n\non food and diet, , ,\n\non helots, , , 100n.24\n\non heterosexual intercourse, \u2013,\n\non homosexuality, \u2013\n\non hunting,\n\non _kleroi_ ,\n\non luxury,\n\non Lycurgus, x, , , , , , ,\n\non marriage, , \u2013, \u2013, , , , \u2013,\n\non modesty,\n\non motherhood, , ,\n\nnaming of women by,\n\non _nothoi_ (bastards),\n\non physical education, , ,\n\nPomeroy's research on, vii\n\non reproduction, \u2013, , , ,\n\non sexuality, \u2013\n\non slaves, , , 100n.24, ,\n\nas source on Sparta, , , , \u2013, ,\n\non speaking by women, ,\n\non weaving, ,\n\n_xoanon_ (wooden image),\n\nZeus, 25n.95, , , 123n.78\n\nZeus Messapeus at Tsakona, 65n.46\n\nZeuxidemus,\n\nZiehen, Ludwig, \n  1. Title Page\n  2. Copyright\n  3. Dedication\n  4. Preface\n  5. Contents\n  6. Illustrations\n  7. Abbreviations\n  8. 1 Education\n  9. 2 Becoming a Wife\n  10. 3 The Creation of Mothers\n  11. 4 Elite Women\n  12. 5 The Lower Classes\n  13. 6 Women and Religion\n  14. Conclusion: Gender and Ethnicity\n  15. Appendix: Sources for the History of Spartan Women\n  16. Works Cited\n  17. Index\n\n  1. i\n  2. ii\n  3. iii\n  4. iv\n  5. v\n  6. vi\n  7. vii\n  8. viii\n  9. ix\n  10. x\n  11. xi\n  12. xii\n  13. xiii\n  14. xiv\n  15. xv\n  16. xvi\n  17. xvii\n  18. xviii\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\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nDecision Making  \nand  \nProblem Solving  \nStrategies\n\nJohn Adair\n\nPublisher's note\n\nEvery possible effort has been made to ensure that the information contained in this book is accurate at the time of going to press, and the publishers and author cannot accept responsibility for any errors or omissions, however caused. No responsibility for loss or damage occasioned to any person acting, or refraining from action, as a result of the material in this publication can be accepted by the editor, the publisher or the author.\n\nPreviously published by the Institute of Personnel and Development as Decision Making and Problem Solving 1997 and 1999\n\nFirst published in Great Britain and the United States in 2007 by Kogan Page Limited as Decision Making and Problem Solving Strategies\n\nReissued 2010\n\nApart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licences issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned addresses:\n\n120 Pentonville Road | 525 South 4th Street, #241 | 4737\/23 Ansari Road\n\n---|---|---\n\nLondon N1 9JN | Philadelphia PA 19147 | Daryaganj\n\nUnited Kingdom | USA | New Delhi 110002\n\nwww.koganpage.com | |\n\nIndia\n\n\u00a9 John Adair, 1997, 1999, 2007, 2010\n\nThe right of John Adair 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\nISBN 978 0 7494 5551 4\n\nE-ISBN 978 0 7494 5890 4\n\nThe views expressed in this book are those of the author, and are not necessarily the same as those of Times Newspapers Ltd.\n\n* * *\n\nBritish Library Cataloguing-in-Publication Data\n\nA CIP record for this book is available from the British Library.\n\n* * *\n\nLibrary of Congress Cataloging-in-Publication Data\n\nAdair, John.\n\nDecision making and problem solving strategies \/ John Adair. -- 2nd. ed.\n\np. cm.\n\nOriginally published in 2007.\n\nIncludes index\n\nISBN 978-0-7494-5551-4 -- ISBN 978-0-7494-5890-4\n\n(ebook) 1. Decision making. 2. Problem solving. 3. Thought and thinking. I. Title. II. Title: Decision making and problem solving strategies.\n\nHD30.23.A3 2010\n\n658.4'03--dc22\n\n2009031517\n\n* * *\n\nTypeset by Jean Cussons Typesetting, Diss, Norfolk\n\nPrinted and bound in India by Replika Press Pvt Ltd\n\neBook by Graphicraft Limited, Hong Kong\nContents\n\nAbout the author\n\nIntroduction\n\n1 Your mind at work\n\nIs your brain working now?\n\nThe functions of the mind\n\nIntroducing the Depth Mind principle\n\nKey points\n\n2 The art of effective decision making\n\nDefine the objective\n\nCollect relevant information\n\nGenerate feasible options\n\nMake the decision\n\nImplement and evaluate\n\nKey points\n\n3 Sharing decisions with others\n\nYour role as leader\n\nTask need\n\nTeam maintenance need\n\nIndividual needs\n\nThe three circles interact\n\nThe functions of leadership\n\nKey points\n\n4 Key problem-solving strategies\n\nHow problems differ from decisions\n\nA unified model for decision making and problem solving\n\nAsking the right questions\n\nHow to approach systems problems\n\nKey points\n\n5 How to generate ideas\n\nBrainstorming\n\nGuidelines for brainstorming\n\nHow to run a brainstorming session\n\nKey points\n\n6 Thinking outside the box\n\nTowards a more creative approach\n\nLook wider for solutions\n\nHow to use your Depth Mind\n\nThe creative thinking process\n\nMental roadblocks\n\nKey points\n\n7 Developing your thinking skills\n\nWhat is an effective practical thinker?\n\nCheck that you are in the right field\n\nKey factors in choosing your field of work\n\nHow to design your own learning strategy\n\nKey points\n\nAppendix\n\nFurther reading\n\nIndex\nAbout the author\n\nJohn Adair is now widely regarded as the world's leading authority on leadership and leadership development. The author of 30 books on the subject, he has been named as one of the 40 people worldwide who have contributed most to the development of management thought and practice.\n\nEducated at St Paul's School, John Adair has enjoyed a varied and colourful career. He served as adjutant in a Bedouin regiment in the Arab Legion, worked as a deckhand on an Arctic trawler and had a spell as an orderly in a hospital operating theatre. After attending Cambridge University he became Senior Lecturer in Military History and Leadership Training Adviser at the Royal Military Academy, Sandhurst, before becoming the first Director of Studies at St George's House in Windsor Castle and then Associate Director of the Industrial Society. Later he became the world's first Professor in Leadership Studies at the University of Surrey. He also helped to found Europe's first Centre for Leadership Studies at the University of Exeter.\n\nJohn Adair now acts as a national and international adviser on leadership development. His recent books, published by Kogan Page, include Not Bosses But Leaders, The Inspirational Leader, How to Grow Leaders and Leadership and Motivation.\nIntroduction\n\nThere are three forms of applied thinking that we all need: decision making, problem solving and creative thinking. These overlap considerably but they can be distinguished.\n\nDecision making is about deciding what action to take; it usually involves choice between options. The object of problem solving is usually a solution, answer or conclusion. The outcome of creative thinking, by contrast, is new ideas.\n\nAny leader such as yourself who aspires to excellence obviously has a vested interest in seeing that the best decisions are taken, that problems are solved in the optimum way and that the creative ideas and innovations so necessary for tomorrow's business flow freely. Of course, everyone in the team or organisation should be engaged in meeting these essential requirements. But you are the one who is called to provide the intellectual leadership that is needed. Are you willing to do so?\n\nOne step towards that end that you should definitely take is to become master of the processes of practical thinking, the processes that lie behind all effective decision making, problem solving and creative thinking. You cannot guarantee outcomes \u2013 for luck or chance plays a part in all human affairs \u2013 but you can at least make sure that you use the well tried-and-tested processes of thinking to some purpose. You own that responsibility. For my part, the aim of this book is to equip you with the necessary knowledge of those processes and to help you to acquire skill in using them.\n\nOne further word. Forget the idea that thinking is somehow a painful and laborious feeling in the mind, a kind of headache that is best avoided if possible. Thinking is fun. By fun here I do not really mean a diversion that affords enjoyment. For the word also means an activity that engages one's interest or imagination, an activity that may prove to be more than a diversion and may involve challenge and hard work but is still a source of enjoyment. If you come to love thinking for yourself you will learn naturally to do it well.\n\nAs Roy Thompson, one of the greatest businessmen of our time, once said, 'If I have any advice to pass on, as a successful man, it is this: if one wants to be successful, one must think; one must think until it hurts.' He added that, 'From my close observation, I can say that there are few people indeed who are prepared to perform this arduous and tiring work.' Are you one of them?\n\nIn the following pages we shall explore some practical ways in which you can improve your skills in this key area. By the time you have worked through the book you should:\n\n  * understand the way in which the mind works and the principles of effective thinking; \n  * have a clear framework for decision making; \n  * be aware of the relation between decision making and problem solving; \n  * be able to use a unified model for both making decisions and solving problems; \n  * have sharpened up your creative thinking skills; \n  * be in a position to chart a way forwards for improving your thinking skills across the board.\n\n**1**\n\nYour mind at work\n\nBehind your practical, everyday thinking there lies the most complex thing in the known universe: the human mind. Nobody hires and pays you nowadays for your physical strength. You are employed because you have a mind \u2013 and can use it effectively.\n\nThere is a vital distinction between brain and mind. Take a computer as an analogy. Your brain is what you see if you open up the back of the computer \u2013 all those chips and circuits \u2013 whereas the mind is what appears dynamically on the screen. In this book we are focusing on the mind, for that is accessible to us without peering into the skull.\n\nThere are two aspects to the mind: the information it can store in the memory, and what it can do. What we call technical or professional knowledge usually involves both. You not only need knowledge about a subject but you also need to be able to apply it in a variety of unforeseen situations.\n\nSuch applications of professional knowledge invariably involve the activities of decision making and problem solving. A doctor, for example, is problem solving when he or she tries to diagnose the cause of your weak left leg. Indeed, decision making and problem solving are so bound up with particular kinds of information or knowledge \u2013 areas of professional competence \u2013 that we find it hard to think of them in the abstract.\n\nAre there any generic or transferable skills in these areas? Yes, I believe there are. The characteristic function of the mind is to think. So let's leave on one side for a moment the memory or database function of the mind and concentrate on its primary role as a thinking tool. What is the nature of thinking? Are there any universal principles? If so, how can you use these principles to sharpen your skills as a practical thinker?\n\nIs your brain working now?\n\nThe physical base of your mind is of course your brain, the grey matter housed in your head. Your brain is composed of about 10,000 million cells. In fact it has more cells than there are people on the face of the earth! Each one of those cells can link up with approximately 10,000 of its neighbours, which gives you some 1 plus 800 noughts of possible combinations.\n\nOur potential brain power is known to be far greater than the actual power it achieves. No one has remotely approached the limits of it. One estimate suggests that we use no more than about 10 per cent of our brain power. So don't be worried by the fact that you are losing about 400 brain cells every day \u2013 indeed, if you do not exercise your mind throughout your life your brain will shrink at a faster rate.Use it or lose it!\n\nBefore we go any further, I suggest we double-check that all your 10,000 million brain cells are warmed up and working properly by trying to solve some problems. Actually, the three problems below require only about 3,000 million brain cells, so they will not take long or cause us much delay!\n\nTwo other points before we begin. The three problems are not just brain-teasers: they illustrate principles about thinking. So I am not playing games with you. Second, I am not going to give you the answers in this chapter to the first two problems, though I shall do so later. This can be a bit frustrating. But I have a reason for leaving you in suspense. For reasons I shall explain later, I believe that the answers to problems 1 and 2 \u2013 assuming that you cannot solve them immediately \u2013 may come to you later.\n\nProblem 1 The nine dots\n\nTake a piece of paper larger than this page and put on it a pattern of nine dots, like this:\n\nNow connect up the dots by four straight consecutive lines (that is, without taking your pen or pencil off the paper). You should be able to complete this task within three minutes.\n\nProblem 2 The six matchsticks\n\nPlace six matchsticks \u2013 preferably of the wooden variety \u2013 on a flat surface. Now arrange the matchsticks in a pattern of four equilateral (ie equal-sided) triangles. You may not break the matchsticks \u2013 that is the only rule. Again, you should be able to do it within three minutes. There are at least two solutions, but I want the best one.\n\nProblem 3 Who owns the zebra?\n\nHaving got the two easy ones safely behind you \u2013 well done if you have solved both those problems \u2013 we come now to something a little more demanding, so you must call up your reserve brain cells.\n\nThe world record for solving both parts of this problem is 10 minutes. So I will give you 30 minutes which, I am sure you will agree, is overgenerous of me!\n\n  1. There are five houses, each with a front door of a different colour, and inhabited by people of different nationalities, with different pets and drinks. Each person eats a different kind of food.\n  2. The Australian lives in the house with the red door.\n  3. The Italian owns the dog.\n  4. Coffee is drunk in the house with the green door.\n  5. The Ukrainian drinks tea.\n  6. The house with the green door is immediately to the right (your right) of the house with the ivory door.\n  7. The mushroom-eater owns snails.\n  8. Apples are eaten in the house with the yellow door.\n  9. Milk is drunk in the middle house.\n  10. The Norwegian lives in the first house on the left.\n  11. The person who eats onions lives in the house next to the person with the fox.\n  12. Apples are eaten in the house next to the house where the horse is kept.\n  13. The cake-eater drinks orange juice.\n  14. The Japanese eats bananas.\n  15. The Norwegian lives next to the house with the blue door.\n\nNow, who drinks water and who owns the zebra?\n\nThe functions of the mind\n\nLet's now look at how the mind works. I suggest that there are three main functions: analysing, synthesising and imagining, and valuing.\n\nFigure 1.1 The main functions of the mind\n\nIn the applied forms of effective thinking \u2013 decision making, problem solving, and creative or innovative thinking \u2013 all three of these functions are at work. It is their underlying health that largely determines the quality of your thought. Few people have them in harmonious balance, as shown in the illustration above. Most of us are better at one rather than the other two.\n\nOur differing mental strengths are a powerful reason why we need each other: effective thinking in all its forms is both a solitary and a social activity. You should always see yourself alternately as thinking alone (for yourself) and as thinking with others \u2013 either face to face or, as in this case, by reading or some other method of communication. Still, it is a good idea to seek to develop your skills in the weaker areas, like a person building up muscles in a limb through exercise: you will not always have the right people at hand to correct your bias towards a particular function.\n\nAnalysing\n\nThe word 'analyse' comes from a Greek verb meaning 'to loosen', and it means separating a whole into its constituent parts. In tackling the 'Who owns the zebra?' exercise you were using your analytical skills of dissection, trying to break down the task into its parts.\n\nAnalytical thinking is closely related to logical or step-by-step reasoning. You may have noticed that one of the skills you were using in tackling that particular problem was your power of deduction.\n\nLogic has two main parts: deduction and induction. Deduction means literally to subtract or take away. It is the process of deducing a conclusion from what is known or assumed. More specifically, it is a question of inferring from the general to the particular. 'All swans are birds. This is a swan. Therefore...' Induction works the other way round. It is the process of inferring or verifying a general law or principle from the observation of particular instances \u2013 the core of the 'scientific method'.\n\nExercise 1: Spot the fallacy\n\n**Can you spot the logical fallacy in the following statement?**\n\nThe chief executive of St Samaritan's Hospital Trust cleared his throat and began.\n\n'Thank you all for coming to this meeting, which is, as you know, about how to improve the quality of our service in this hospital. To begin with I have decided to sack all the surgeons and physicians over the age of 55 years. Look at these letters! I have had five letters of complaint about the abruptness and lack of communication of doctors here, and two mentioned that the doctors are too old or have passed their \"sell-by\" date. The way to deal with this problem is to lower the average age of the staff, so I am going to ask everyone to take voluntary retirement at 55. Any questions before we move on to the next item on the agenda \u2013 litter in the corridors?'\n\nFor the main part, unlike the manager in the 'Spot the fallacy' exercise (above), most of us are quite good at analysing problems or situations. This is not surprising, as much of our education is concerned with developing our deductive\/inductive powers (mathematics, sciences, history, and literature) and sharpening our analytical skills.\n\nYou may now like to look at the solution to the 'Who owns the zebra?' problem (see page 85). As you will see, it combines a test of your powers of reasoning or logical thinking with the important principle of trial and error. When you are faced with two alternatives \u2013 such as two roads at a junction without signposts that lead in the right general direction \u2013 there is no other way but to try each one in turn. In the case of this exercise, using a computer would save you time. But in real life you may, as they say, have to 'suck it and see'. Decision making is not an exact science.\n\nSynthesising\n\nIt is not easy to give a single label to the second function. Synthesising \u2013 another Greek word \u2013 is putting or placing things together to make a whole. It is the reverse process of analysing. You can synthesise things with your hands, which you do whenever you assemble or make anything. All products and services are the results of syntheses. But you can also do it mentally.\n\nWhen that happens, another faculty is called into play \u2013 imagination. Now, imagination works in pictures, and a picture is a whole that is more than the sum of its parts. If you shut your eyes for a moment and think of your house or your car, you see a picture. In fact, it is almost impossible not to see a picture. Your computer-like memory flashes it up on the inner screen of your mind very quickly. What you see is neither a pile of bricks, in the case of your house, nor a heap of car components, but in each case a whole.\n\nIf, so to speak, you turn up the volume knob of your imagination, you can see things that do not exist. Imagine, for example, a 56-metre-tall man... This road takes us into how to generate ideas, the subject to be explored more fully in Chapter 5.\n\nThe link between creativity and the synthesising process is clear when you contemplate how nature works. A baby arrives whole and it grows. Nature is holistic. A famous South African, Field Marshal Jan Smuts, who was also a keen agricultural scientist, coined the word holism to describe nature's way of creating wholes by ordering or grouping various units together. The essential realities in nature, Smuts argued, are these irreducible wholes. If analysed into parts, they lose their essential holistic quality. As the poet William Wordsworth put it, 'We murder to dissect.' Your mind has a holistic dimension. It can think holistically \u2013 in terms of wholes \u2013 as well as analytically (taking wholes to bits).\n\nValuing\n\nThe third function comes into play in such mental activities as establishing success criteria, evaluating, appraising performance, and judging people \u2013 as, for example, in a selection interview. Criticism (from the Greek word for a judge) is a form of valuing.\n\nIncidentally, criticism, as commonly understood, most often suggests disapproval \u2013 some sort of a negative judgement. But in its more formal use it can suggest neutral analysis or even approving evaluation. Judgement is not always un favourable.\n\nIn all valuing there is an objective (outside yourself) element and a subjective one. We are all born with the capacity to value. What we actually value \u2013 our values \u2013 depends very largely upon our environment and its culture.\n\nValues are rather like colours. What is the colour of grass? 'Easy,' you reply. 'It is green.' But scientists tell us that grass has no intrinsic colour: it is merely reflecting light in the wave band that we call green. The structure of our eyes is also a factor. Our subjective contribution to the perception of colour is significant. Being colour-blind to certain shades of the red\u2013green spectrum \u2013 fortunately not to the greenest of grass \u2013 I am personally very aware of that fact.\n\nThe word value comes from a market metaphor: it is what you have to give in order to receive something across the counter. The invention of money revolutionised bartering. One merit of money is that it was a universal measuring stick. But there are plenty of other values that enter into any form of decision making, especially in business today. (See Exercise 2.)\n\nExercise 2: Values at work\n\nMake a list of all the values \u2013 apart from financial value (profit) \u2013 which might influence any business decision over the coming 10 years.\n\nCheck to see whether the organisation you work for has issued a statement of its corporate values. If so, obtain a copy and underline what you judge to be the master value in it.\n\nHow far do your organisation's values overlap with your own philosophy of life?\n\nWhether or not values in the popular sense have a separate existence, and where they come from if not from ourselves, are philosophical questions that lie beyond the scope of this book. But in all thinking there is a strong case for acting as if truth \u2013 one member of the trinity of goodness, truth, and beauty \u2013 really does exist 'out there'. It would be impossible, for example, to explain the immense success story of modern science without the working belief of scientists such as Einstein that the truth is 'out there' waiting to be discovered.\n\nIntroducing the Depth Mind principle\n\nAs we all know, we have subconscious and unconscious minds. But we are not so aware of the vital part that the dimension that I have named the Depth Mind plays in our thinking. You can, as it were, analyse, synthesise and value in your sleep or when you are consciously doing something quite different, like gardening or washing the dishes. Far from being chaotic, the Depth Mind plays a large part in scientific discovery and creative art. It is also the source of intuition \u2013 that all-important sixth sense.\n\nConrad Hilton was trying to buy an old Chicago hotel. A few days before the deadline for sealed bids, Hilton submitted a bid for US $165,000, a figure he had reached by some hasty calculations, as he was busy on other things at the time. He went to bed that night feeling vaguely disturbed and awoke the following morning with the feeling that his bid was not high enough. Another figure kept coming to him out of his Depth Mind \u2013 US $180,000. 'It satisfied me. It seemed fair. It felt right. I changed my bid to the higher figure on that hunch. When the envelopes were opened the closest bid to mine was US $179,000.'\n\nCan you think of a similar decision or problem in your experience when your Depth Mind has played a similar role?\n\nChecklist: Listening to your Depth Mind\n\n| Yes| No  \n---|---|---  \nDo you have a friendly and positive attitude to your Depth Mind?   \nDo you expect it to work for you?|\n\n|\n\nWhere possible, do you build into your plans time to 'sleep on it', so as to give your Depth Mind an opportunity to contribute?|\n\n|\n\nDo you deliberately seek to employ your Depth Mind to help you to:|\n\n|\n\nanalyse a complex situation|\n\n|\n\nrestructure a problem|\n\n|\n\nreach value judgements?|\n\n|\n\nHave you experienced waking up next morning to find that your unconscious mind has resolved some problem or made some decision for you?|\n\n|\n\nDo you see your Depth Mind as being like a computer? Remember the computer proverb: Garbage in, garbage out.|\n\n|\n\nDo you keep a notebook or pocket tape-recorder at hand to capture fleeting or half-formed ideas?|\n\n|\n\nDo you think you can benefit from understanding how the Depth Minds of other people work?|\n\n|\n\nRoy Thompson, in his autobiography After I Was Sixty (1975), explains how the Depth Mind works.\n\nWhen a new problem arose, I would think it over and, if the answer was not immediately apparent, I would let it go for a while, and it was as if it went the rounds of the brain cells looking for guidance that could be retrieved, for by the next morning, when I examined the problem again, more often than not the solution came up right away. That judgement seems to have come to me almost unconsciously, and my conviction is that during the time I was not consciously considering the problem, my subconscious had been turning it over and relating it to my memory.\n\nThe use of your Depth Mind in decision making, problem solving and creative thinking is such an important principle that I shall return to it later. The million-dollar question is: Can we develop our Depth Mind capability? My answer is: Yes, we can. And the first step is awareness that it both exists and works. The secret of effective thinking is working with the natural grain of your mind \u2013 go with the flow as they say, but see if you can steer the boat.\n\nKey points\n\n  * We are called homo sapiens on account of our minds. The human capacity to exercise the mind \u2013 the activity we call thinking \u2013 is truly remarkable. Yet few of us use our minds to anything near their full capacity.\n  * Thinking is to regard or examine in the mind, to reflect or to ponder. As we experience it, thinking is a single stream of consciousness. But we can discern three interweaving currents in thinking to some purpose: analysing, synthesising and valuing.\n  * Analysing, the first function, tends to be highly developed by Western education. It is the men115tal ability to take things \u2013 material and non-material \u2013 to bits, to separate them into their component parts. It is related, but not identical, to logical or step-by-step thinking. \n  * Synthesising is the reverse process of putting things together to form a whole. When the resultant whole is formed from parts previously thought to be unconnected, when it looks new and has real value, then synthesising has become creative.\n  * Valuing, the third main function in purposive thinking, is self-explanatory. Even in the strictest schools of science or logic, it is impossible to exclude value. We are all valuing creatures; our actual values are largely shaped by our cultural experience. Of course, by helping us to escape out of the cultural box of our particular lives we encounter more universal values: goodness, truth and beauty.\n  * These functions \u2013 analysing, synthesising and valuing \u2013 can do their work at the unconscious level I have called the Depth Mind. Indeed, where complex decisions have to be made, problems solved or truly creative products involved, the Depth Mind is a vital dimension in the effective use of your mind.\n\nWe do not think as long as things run along smoothly for us. It is only when the routine is disrupted by the intrusion of a difficulty, obstacle or challenge that we are forced to stop drifting and to think what we are going to do.\n\nJohn Dewey\n**2**\n\nThe art of effective decision making\n\nThere is a time when we must firmly choose the course we will follow, or the relentless drift of events will make the decision.\n\nFranklin D Roosevelt\n\nIn decision making there is a classic five-step approach that you should find extremely helpful. That does not mean you should follow it blindly in all situations. It is a fairly natural sequence of thought, however, and so even without the formal framework you would tend to follow this mental path. The advantage of making it conscious is that it is easier to be swiftly aware when a step is missing or \u2013 more probably \u2013 has been performed without understanding or intention.\n\nIt is useful to think of the five steps on page 19 as five notes of music. Logically they should be played in strict sequence. But the mind darts about. The notes can be combined in different sequences and mental chords. Thinking is not a tidy process, but it should be done with a sense of order.\n\nRemember that we are not talking here about just big decisions, for there's a lot more to running a business than making one life-or-death decision. Indeed no decision, no matter how big, is any more than a small fraction of the total outcome. Yes, some decisions are much bigger than others, and some are forks in the road. But it is really more the case that a much larger number of small decisions have a cumulative result. By hindsight we can usually identify those few pivotal decisions, but it is really the stream of smaller decisions over time, made and executed with a craftsman's skill, that yields great outcomes.\n\nDefine the objective\n\nDo you know what you are trying to achieve? You do need to be clear \u2013 or as clear as possible \u2013 about where you want to get to. Otherwise the whole process of decision making is obscured in a cloud. As the proverb says, If you do not know what port you are heading for, any wind is the right wind.\n\nIf you are in doubt about your aim, try writing it down. Leave it for a day or two, if time allows, and then look at it again. You may be able to see at once how it can be sharpened or focused.\n\nCollect relevant information\n\nThe next skill is concerned with collecting and sifting relevant information. Some of it will be immediately apparent, but other data may be missing. It is a good principle not to make decisions in the absence of critically important information that is not immediately to hand, provided that a planned delay is acceptable.\n\nRemember the distinction between available and relevant information. One classic mistake is to look at the broad decision and then turn to the information we have that will help us decide. Some thinkers do not, however, look at the information at their disposal and ask themselves, 'Is this relevant?' Instead they wonder, 'How can I use it?' They are confusing two kinds of information \u2013 as is illustrated on page 20.\n\nFigure 2.1 The classic approach to decision making\n\nLife would be much simpler if you could just use the information at your disposal, rather than that which you really need to make the decision! So often quantities of data are advanced \u2013 there are acres of it on the internet \u2013 that merely add bulk to, say, a management report without giving its recommendations any additional (metaphorical) weight.\n\nThe rapid growth of methods of communication such as faxes, voice mail, e-mail, junk mail and the internet has now contributed to a new disease: Information Overload Syndrome. A recent international survey of 1,300 managers listed the new disease's symptoms, which included a feeling of inability to cope with the incoming data as it piles up, resulting sometimes in mental stress and even physical illness requiring time off work. The survey found that such overload is a growing problem among managers \u2013 almost all of whom expect it to become worse.\n\nFigure 2.2 Information categories\n\nExecutives and their juniors say they are caught in a dilemma: everyone tells them that they should have more information so they can make better decisions, but the proliferation of sources makes it impossible to keep abreast of the data.\n\nThe growth of information has been relentless. The New York Times contains as much distinct information every day as the average seventeenth-century person encountered in a lifetime. No wonder that half the managers surveyed complained of information overload, partly caused by enormous amounts of unsolicited information. The same proportion also expected the incredible expansion of the internet to intensify the problem year-on-year. To avoid succumbing to Information Overload Syndrome you need all the skills described in this book!\n\nSuppose that the overlap between information required and information available is not sufficient: what do you do? Obviously you set about obtaining more of the information required category. But getting information or \u2013 to use a grander description \u2013 doing research incurs costs in time and money. Your organisation may not be in the business of making profits, but it certainly has to be businesslike when it comes to containing costs.\n\nWhat the graph below suggests is that you usually acquire a great deal of relevant information in a relatively short time and, possibly, at a relatively low cost in money. But the line soon curves towards a plateau. You will find yourself spending more and more time to discover less and less relevant information. For example, if you and I sat next to each other at a dinner, I should learn all the really important things about you in the first half hour. The longer we talked, the smaller the increments of knowledge about you would become. After three hours I should be down to discussing relatively fine details.\n\nGenerate feasible options\n\nFigure 2.3 The time\/information curve\n\nNotice the word options rather than alternatives. An alternative is literally one of two courses open. Decision makers who lack skill tend to jump far too quickly to the either\u2013or alternatives. They do not give enough time and mental energy to generating at least three or four possibilities. As Bismarck used to say to his generals, 'You can be sure that if the enemy has only two courses of action open to him, he will choose the third.' Alfred Sloan, the renowned President of General Motors, was even known to adjourn meetings in which he was presented with two alternatives. 'Please go away and generate more options,' he would say.\n\nYou need to open your mind into wide focus to consider all possibilities, and that is where generating ideas (see Chapter 5) comes in. But then your valuing faculty must come into play in order to identify the feasible options. 'Feasible' means capable of being done or carried out or realised. If it is feasible it has some real likelihood of being workable. It can attain the end you have in mind.\n\nIn moving along the lobster pot (see the illustration below) from the feasible options (no more than five or six, for the mind finds it difficult to handle more) to three options and then to two (the true alternatives), the principle to bear in mind is that it is easier to falsify something than to verify it.\n\nSuppose you are choosing between five medium-sized estate cars for your family. It is easy to eliminate the unsuitable ones.\n\nAs you work on it, for example, you may discover that one of the cars is 9 inches longer than the others, which will cause you a problem given the size of your garage. As for a second car, on studying the specifications you cannot see why it is \u00a31,200 more expensive than the rest \u2013 apart from its prestigious name. So you drop that one too, leaving you now with three choices. You will notice another principle coming into play here, which (subject to the information\/time curve) does take most of the pain out of decision making. Let me continue with the car example. Because your partner does not like the colours of the Toyota model and, being an artist by profession, feels strongly about it, you are able to eliminate that one. Your alternatives are now the Nissan and the Peugeot.\n\nYou have just read this book and so, being persuaded by its argument, you decide to trade some more time for some more information, and test drive the alternative cars. Both feel great and perform really well. You know that either will serve your purpose. It is now a question of money and the availability of the colours your partner likes. One of the dealers offers you a much better price and can deliver the right model in the range. Why hesitate?\n\nFigure 2.4 The lobster pot model\n\nMake the decision\n\nThe critical preliminary activity here is to establish the selection criteria. It is worth dividing them into different levels of priority. (See the illustration below.)\n\nFigure 2.5 Decision-making criteria\n\nUnless an option meets the MUST requirements you should discard it. But after the essentials have been satisfied, the list of desirables \u2013 highly desirable SHOULDs or pleasant addition MIGHTs \u2013 comes into play.\n\nChoosing a car is a relatively simple case, because there is a finite number of models to choose from and a relatively simple list of criteria. In order to help you choose in more complex cases, remember that you can make a decision by:\n\n  * listing the advantages and disadvantages;\n  * examining the consequences of each course;\n  * testing the proposed course against the yardstick of your aim or objective; \n  * weighing the risks against the expected gains.\n\nAssessing risk\n\nWhat makes decisions really difficult is the factor of high risk. You may recall the conflicting advice of the two proverbs Look before you leap and He who hesitates is lost. There is an important skill in calculating risk. Calculation sounds mathematical, and there are plenty of management books with 'decision making' in the title that offer various 'probability theories' and statistical methods to take the pain out of risk assessment. Sometimes it can help to assign numbers and calculate in that way, but the contribution of mathematics to this field is very limited. Experience plays a much larger part.\n\nOne helpful idea is to define the worst downside \u2013 what happens in the worst scenario? Can you accept that, or will it sink you? But in high-risk\/high-reward situations, although you may know that you will be sunk if it does not all work out, you may still decide to take the high-risk course because the reward is just too important for you to forgo it.\n\nYou then have to address your mind to doing all you can to reduce the risk. It is here that experience, practice, consultation with specialists, reconnaissance and mental rehearsals may all be relevant techniques. You are trying to turn the possibility of success into the probability of success, but you will not be able to eliminate risk altogether: in this situation there are too many contingencies.\n\nAssessing consequences\n\nRisk is one aspect of thinking through the consequences of the feasible courses of action.\n\nConsequences come in two forms: manifest and latent. Manifest consequences are ones that, in principle, you can foresee when you make your decision. I say 'in principle' because that does not mean to say that you did foresee them. What I mean is that any reasonable person with the knowledge, experience, or skill expected of someone in your position would foresee those consequences. If you try to rob a bank, for example, the manifest consequences are obvious to any reasonable person:\n\n  * You might become amazingly rich.\n  * People, including you, might get hurt.\n  * You could be sent to prison.\n\nLatent consequences are different in that they are not nearly so probable, or even possible, and a reasonable person might be forgiven for not seeing the knock-on effects that result from the complex chain of events triggered off by a decision. Admittedly, with the aid of computers it becomes a little easier in certain fields to identify latent consequences, but it is seldom possible to insulate yourself against pleasant or unpleasant surprises. We just cannot foresee the future in that way.\n\nThe emergence of latent consequences, of course, triggers off another round of decision-making and problem-solving activity. Yet solutions are the seeds of new problems.\n\nIntroducing performance-related pay for individuals, for example, solves some motivational problems, but what other problems does it tend to create for teams and organisations, not to mention the individuals concerned?\n\nFill the quarters of the window in the illustration below with the consequences of a decision to make pay totally performance-related. Review the completed window \u2013 remember, you are looking for insights.\n\nI suppose that if we knew all the latent consequences of all our decisions at the time of making them, we should soon decide to stay in bed all day and never make another decision! But that decision in itself would have manifest and latent consequences... All that we can do, as humans and not angels or gods, is to make the best decisions we can, given the information and circumstances, and then make other decisions to deal with the latent consequences as they arise.\n\nRemember that there is a big difference between a wrong decision and a bad decision. A wrong decision is choosing to dig your only oil well in this place rather than that one. It's an expensive mistake, but the fault lies with the method. A bad decision is launching the space shuttle Challenger on a severely cold morning when the contracting engineers responsible for the seals in the engines have predicted a nearly 100 per cent chance that the seals will fail in such conditions. They did \u2013 with a tragic loss of life. Here the method or process of decision making was deliberately ignored or irresponsibly put on one side.\n\nThe distinction is important because it separates outcomes, which you cannot fully control because of the part that luck or chance plays, from process, which you can. Wrong decisions are an inevitable aspect of life, both in our personal and professional lives. We redeem them by learning the lessons they teach us, paying the fees that life charges as cheerfully as we can. But bad decisions are predictable pitfalls; they are unforced errors. They are eminently avoidable if you use the proven processes, methods and techniques outlined in this book.\n\nFigure 2.6 The outcomes window\n\nImplement and evaluate\n\nDecision comes from a Latin verb meaning 'to cut off'. It is related to such cutting words as 'scissors' and 'incision'.\n\nFigure 2.7 The point of no return\n\nWhat is 'cut off' when you make a decision is the preliminary activity of thinking, especially the business of weighing up the pros and cons of the various courses of action. You now move into the action phase. Out with your cheque book \u2013 start talking about delivery dates! Things begin to happen.\n\nIt is always worth identifying what I have called the Point of No Return (PNR), a term that comes from aviation. At the half-way point in crossing the Atlantic, it is easier for the pilot to continue to Paris in the event of engine trouble than to turn back to New York. The pilot has passed the PNR and he or she is committed.\n\nIn its wider sense the PNR is the point at which it costs you more in various coinages to turn back or change your mind than to continue with a decision that you now know to be an imperfect one. In most decisions you do have a little leeway before you are finally committed: you can still change your mind. Often, as in the case study of Conrad Hilton (see page 12), it is your Depth Mind that double-checks your decision. It either whispers, 'Yes, I am satisfied' or begins an insidious and insistent campaign to make you at least review your decision, if not change your mind.\n\nThere is another reason for seeing implementation as a part in the decision-making process rather than the end of it. Your valuing faculty is bound to come into play at some stage in order to evaluate the decision. Did you get it right? Could you have made the decision more quickly or more gracefully, perhaps at less cost to others? All this data goes into your memory bank and informs the Depth Mind, so that the next time you make a similar decision this information about your past may be available to you in the form of a more educated intuition. This is what constitutes what we call experience.\n\nRemember, your Depth Mind really does work!\n\nIn 2006 some researchers at the University of Amsterdam decided to put the Depth Mind theory to the test. The psychologists asked a group of volunteers to pretend they were about to buy one of four cars. The volunteers were given lots of information about the cars, and one model was much better than the others.\n\nHalf the volunteers were given time to ponder the merits of each car, while the others were given puzzles to solve to keep their minds busy. Both groups were then asked to pick the car they would buy.\n\nThe results showed that those who restricted themselves to conscious thought were less likely to have chosen the best deal.\n\nThose whose surface minds were occupied with the irrelevant puzzles made better choices. In a second experiment, the volunteers were faced with furniture choices at Ikea.\n\nWhat the experiments show is that your Depth Mind can deal with more facts and figures than your conscious mind. The latter is good at simple choices, such as buying different towels or different sets of oven mitts. But choices made in complex matters, such as between different houses or cars, are better if your Depth Mind is involved.\n\nThe principle, I may add, always applies over people decisions, especially the choice of one's life partner. 'I have no other but a woman's reason. I think him so because I think him so,' says one of Shakespeare's heroines. In less eloquent language we might say that it is our Depth Mind that can best process all the complex information that comes from another person and transform it into a simple but profound judgement. Who knows how this work is accomplished in the inner hive of the mind?\n\nTake care that the honey does not remain in you in the same state as when you gathered it: bees would have no credit unless they transformed it into something different and better.\n\nPetrach\n\nKey points\n\n  * Sometimes it is useful for the mind to have a framework for approaching potentially difficult tasks. In decision making there is such a simple framework of five steps or phases. Think of it more as a spiralling process, like this:\n\n  * Defining the objective is invariably important in decision making. One useful tip is to write it down, for seeing it in writing often helps you to attain the necessary clarity of mind.\n  * Collecting relevant information involves both surveying the available information and then taking steps to acquire the missing but relevant information to the matter in hand.\n  * For generating feasible options, remember the lobster pot model! You should be able to move systematically from a host of possibilities \u2013 some of them may be the results of imaginative thinking \u2013 to a diminishing set of feasible options, the courses of action that are actually practicable given the resources available.\n  * In making the decision your chosen success criteria (the product of the valuing function of the mind) come into play. It is useful to grade these yardsticks into the criteria that the proposed course of action MUST, SHOULD and MIGHT meet. You will also need to assess the risks involved: what are the manifest and the possible latent consequences of the decision in view? \n  * Implementing and evaluating the decision should be seen as part of the overall process. You may hardly notice the actual point of decision, just as passengers on a ship may be asleep when their ship crosses the equator line. The 'cut off' point, be it conscious or unconscious, is when thinking ends \u2013 your mind is made up \u2013 and you move into the action or implementation phase. But you are still evaluating the decision, and up to the Point of No Return (PNR), you can always turn back if the early signs dictate.\n  * If you have all the required information, the mind goes through the point of decision effortlessly \u2013 indeed, do you really have to take a decision? Thus it has been said that 'a decision is the action an executive must take when he or she has information so incomplete that the answer does not suggest itself'.\n\nNot to decide is to decide.\n\nEnglish proverb\n**3**\n\nSharing decisions with others\n\nA key issue in leadership is how far the designated leader (appointed or elected) should share decisions with others \u2013 team members or colleagues. Of course it is also an issue for all of us \u2013 how far should we make our decisions after solitary and silent thought, or how far should we consult others?\n\nYour role as leader\n\nBefore looking together at the decision-making aspect of leadership let me put it in context by reminding you of the generic role of leader \u2013 true for all fields of work and all levels of leadership.\n\nIf you look closely at matters involving leadership, there are always three elements or variables:\n\n  * the leader \u2013 qualities of personality and character;\n  * the situation \u2013 partly constant, partly varying;\n  * the group \u2013 the followers: their needs and values.\n\nIn fact, work groups are always different, just as individuals are. After coming together they soon develop a group personality. So that which works in one group may not work in another. All groups and organisations are unique.\n\nBut that is only half the truth. The other half is that work groups \u2013 like individuals \u2013 have certain needs in common. There are three areas of overlapping need which are centrally important, as illustrated in the figure below.\n\nFigure 3.1 Overlapping needs\n\nTask need\n\nWork groups and organisations come into being because there is a task to be done that is too big for one person. You can climb a hill or small mountain by yourself, but you cannot climb Mount Everest on your own \u2013 you need a team for that.\n\nWhy call it a need? Because pressure builds up a head of steam to accomplish the common task. People can feel very frustrated if they are prevented from doing so.\n\nTeam maintenance need\n\nThis is not so easy to perceive as the task need; as with an iceberg, much of the life of any group lies below the surface. The distinction that the task need concerns things and the second need involves people does not help much.\n\nAgain, it is best to think of groups that are threatened from without by forces aimed at their disintegration or from within by disruptive people or ideas. We can then see how they give priority to maintaining themselves against these external or internal pressures, sometimes showing great ingenuity in the process. Many of the written or unwritten rules of the group are designed to promote this unity and to maintain cohesiveness at all costs. Those who rock the boat or infringe group standards and corporate balance may expect reactions varying from friendly indulgence to downright anger. Instinctively a common feeling exists that 'United we stand, divided we fall', that good relationships, desirable in themselves, are also an essential means towards the shared end. This need to create and promote group cohesiveness I have called the team maintenance need. After all, everyone knows what a team is.\n\nIndividual needs\n\nThird, individuals bring into the group their own needs \u2013 not just the physical ones for food and shelter (which are largely catered for by the payment of wages these days) but also the psychological ones: recognition; a sense of doing something worthwhile; status; and the deeper needs to give to and receive from other people in a working situation. These individual needs are perhaps more profound that we sometimes realise.\n\nThey spring from the depths of our common life as human beings. They may attract us to, or repel us from, any given group. Underlying them all is the fact that people need one another not just to survive but to achieve and develop personality.\n\nThe three circles interact\n\nNow these three areas of need overlap and influence one another. If the common task is achieved, for example, then that tends to build the team and to satisfy personal human needs in individuals. If there is a lack of cohesiveness in the team circle \u2013 a failure of team maintenance \u2013 then clearly performance in the task area will be impaired and the satisfaction of individual members reduced. Thus, as above, we can visualise the needs present in work groups as three overlapping circles.\n\nNowadays when I show the model on an overhead projector slide I usually colour the circles red, blue, and green, for light (not pigment) refracts into these three primary colours. It is a way of suggesting that the three circles form a universal model. In whatever field you are, at whatever level of leadership \u2013 team leader, operational leader, or strategic leader \u2013 there are three things that you should always be thinking about: task, team, and individual. Leadership is essentially an other-centred activity \u2013 not a self-centred one.\n\nThe three-circle model is simple but not simplistic or superficial. Keeping in mind those three primary colours, we can make an analogy with what is happening when we watch a television programme: the full-colour moving pictures are made up of dots of those three primary and (in the overlapping areas) three secondary colours. It is only when you stand well back from the complex moving and talking picture of life at work that you begin to see the underlying pattern of the three circles. Of course they are not always so balanced and clear as the model suggests, but they are nonetheless there.\n\nThe functions of leadership\n\nWhat has all this got to do with leadership? Simply this: in order to achieve the common task and to maintain teamwork, certain functions have to be performed. And a function is what you do, as opposed to a quality, which is an aspect of what you are. For example, someone has to define the objectives, make a plan, or hold the team together if it is threatened by disruptive forces.\n\nNow we are on firm ground. For you can learn to provide the functions of leadership which are called for by task, team and individual needs. This is the entrance door to effective leadership. The function approach set out here is also sometimes called action-centred leadership. A function is one of a group of related actions contributing to development or maintenance, just as each part of the body has its function in relation to the whole. It comes from a Latin word meaning performance. Sometimes it is used more widely to mean what I have called role \u2013 the special kind of activity proper to a professional position. In the figure below I have listed the main functions that are required if the task is to be achieved, the team held together, and the needs of individuals met.\n\nFigure 3.2 Leadership functions\n\nIt should not be supposed that a leader has to perform all these functions himself or herself. Indeed, in groups of more than five people there are too many functional contributions or activities for one person to do them all on his or her own. Functions have to be shared and sometimes \u2013 in all or part \u2013 delegated. Or, putting it another way, all team members are responsible (even if not legally-speaking accountable) for the three circles. The difference between an effective team member and an effective leader is not great; it is one more of role than of commitment to all three circles or skill in meeting the common needs.\n\nObviously decision making is potentially going to be an ingredient in all the functional domains. Planning is a good example. Planning means building a mental bridge from where you are now to where you want to be when you have achieved the objective before you. The function of planning meets the group's need to accomplish its task by answering the question how. But the 'how' question soon leads to 'When does this or that have to happen?' and 'Who does what?'\n\nFrom the leadership perspective, the key issue is how far you should make the plan yourself or how far you should share the planning function with your team. Let's look at the options.\n\nThere is a lot to be said for moving as far to the right end of the continuum as you can (see Figure 3.3 below). For the more people share in decisions that affect their working life the more they are motivated to carry them out. That consideration, however, has to be balanced against the fact that the wider you open the door of the Inn of Decision the less control you have of the outcome. The team may make a plan that, although meeting the requirements you have identified, is not the way you would have done it yourself. Can you live with that?\n\nJust where you should act on the planning continuum depends on several key factors, notably the time available to plan and the competence level of the team members. There is no one right 'style'. The best leaders are consistent \u2013 you know where you stand with them and they are in many respects predictable. But when it comes to decision making they are infinitely flexible. So a good leader, working with individuals or teams, will operate at different points on the scale during a day.\n\nFigure 3.3 The planning continuum\n\nOnce work has started on the plan, it may be necessary to revise or adapt the plan as circumstances or conditions dictate. Again, you must steer a middle course between the perennial need for flexibility as change unfolds and a certain persistence or tenacity in sticking to the agreed plan. Certainly, allowing too many unnecessary changes in the plan can itself breed confusion. As the military proverb says, Order \u2013 counter-order \u2013 disorder.\n\nAlways remember that everyone who works for you has 10,000 million brain cells. If time allows, it is always worthwhile to listen to people. You will be constantly surprised at the quality of the thinking and ideas available to you.\n\nWhen Fort Dunlop was taken over by Sumitomo, the Japanese management asked for money-saving ideas from the workforce. A junior employee saved the company \u00a3100,000 a year in electricity payments by suggesting that every other fluorescent light in the huge factory did not need to be used \u2013 an idea that he had had for years!\n\nIn fact you should develop a habit of listening for ideas, whether with a particular decision in mind or as a wider strategy. There are plenty of ideas around for those who are ready to receive them. That great business leader Roy Thompson capitalised some of this potential wealth:\n\nThe way I look at it, everyone has an idea and one in a dozen may be a good idea. If you have to talk to a dozen people to get one good idea, even just the glimmering of an idea, that isn't wasteful work. People are continually passing things on to me, because I have given them to believe that I will be interested, I might even pay for it! Sometimes, usually when it is least expected, something comes up that is touched with gold.\n\nIn summary, the degree to which you as a leader can share decisions differs according to such factors as the situation \u2013 especially the time available \u2013 and the relative and relevant knowledge of the team members. Actually deciding where to decide on a continuum that has control at one end and freedom at the other is in itself an important decision when working with others. Here are some questions to ask yourself:\n\nChecklist\n\n| Yes| No  \n---|---|---  \nHave you agreed the aims and objectives with the team?|\n\n|\n\nHave you involved the team in the collecting and sifting of the relevant information?|\n\n|\n\nHas the team helped you to generate a number of possible courses of action?|\n\n|\n\nHave you used the synergy of the team members' minds to firm up the feasible options?|\n\n|\n\nHave you tested for consensus to see how far, in the circumstances, a course of action you favour is seen to be the optimum one?|\n\n|\n\nHave you secured everyone's commitment to make it work?|\n\n|\n\nHave you reviewed the decision with the team so that the lessons of success and failure are learnt for the future?|\n\n|\n\nKey points\n\n  * Thinking is both solitary and social. We need to think for ourselves \u2013 and make time to do so. But we also need to talk with and listen to others, for stimulus and encouragement, fresh perspectives and new ideas. Conversation at its best is a form of mutual thinking.\n  * The role of a leader is defined by the three circles of need \u2013 task, team and individual \u2013 and the responding set of functions. Communication and decision making are complementary dimensions. A key issue for all leaders is how far they should share decisions with their team or colleagues.\n  * The more you share decisions the higher the quality of the decision is likely to be. Moreover, the more that people share decisions which directly affect their working life, the more they tend to be motivated to implement them. Yet the exigency of the situation \u2013 shortage of time and the crisis factor \u2013 sometimes restricts the scope for sharing. And you also have to remember that the more you share a decision the less control you have over the resulting decision's quality and direction. So you need judgement here.\n  * When the decision-making process is over, you still have to take the decision.\n  * Outside the confines of the making of a particular decision you should always be open to the ideas, suggestions and information that people offer you. The more you show interest, the more that people will tell you. Ten per cent of their ideas are lined with gold.\n\nThree cobblers with their wits combined  \nEqual Zhage Liag, the master mind.\n\nChinese proverb\n**4**\n\nKey problem-solving strategies\n\nDecision making, problem solving, and creative thinking have in common the fact that they are all forms of effective thinking. But there are some distinctions between them. You can, for example, think creatively, in the sense of having an original idea, without either making a decision or solving a problem. In this chapter the main focus is upon problem solving.\n\nHow problems differ from decisions\n\nWhat is a problem? A 'problem' is literally 'something thrown in front of you'. Another of those Greek words by origin, it is related to 'ballistics'. Originally what was thrown or put in front of one by the Greek teachers was the sort of puzzle or question that you encountered in the first chapter: 'The nine dots', The six matchsticks' and 'Who owns the zebra?' (Incidentally, has your Depth Mind come up with solutions \u2013 or extra solutions \u2013 to the first two yet?)\n\nYou will notice that in problems like these, all the elements of the solution are already there. All that you have to do is arrange or rearrange what has been given. In that sense, a problem is a solution in disguise.\n\nAs a result of solving such problems your life is not going to be different. By contrast, a decision usually does mean that life will be different. It opens the way to changes of some kind or other. Some of these changes are planned, wanted, expected or at least foreseen (the manifest consequences), whereas others are not. But solving or not solving a crossword puzzle is not going to change your life in any way.\n\nIn this respect such problems are similar to games \u2013 in fact, games are sets of problems. Why do we invent them? Because there is nothing that humans enjoy more than solving problems. The skills of a problem solver in this limited sense, however, differ from those of a decision maker. As a problem solver you have to be clever, with analytical skills well honed on many other problems in that particular field. By contrast, a decision-maker needs a much wider range of skills and characteristics.\n\nMoving away from puzzles and games, the problems we encounter in real life are mostly obstacles placed in front of us. If you decide to climb Mount Everest, for example, you may find that all goes well until \u2013 a day before your final ascent \u2013 a heavy storm suddenly develops on the South Col, the ridge leading to the summit. You have a problem! Notice that you would not have that particular problem \u2013 or any problems \u2013 on Everest unless you had made a decision to climb to the summit. It is not a problem for anyone else. And it would cease to be a problem for you if you changed your mind and decided to go off and climb some other mountain in the Himalayas.\n\nTherefore problems as obstacles or difficulties in the path ahead of us are always secondary to the results of decision making. Decisions create problems. One way of solving them \u2013 or rather the problem state in your mind \u2013 is to alter your decision, or at least your plan. Did you have a contingency plan \u2013 a Plan B \u2013 for your route up the mountain if the weather deteriorated or avalanches (unexpected at this time of year) occurred?\n\nIf you stick with your decision, then, in consultation with your team, you have to find a way of overcoming the problem. Because the mental framework you must use is so similar to the decision-making process, a single model covering them both is possible.\n\nFigure 4.1 The bridge model\n\nA unified model for decision making and problem solving\n\nIf you are trying to cross a mountain stream you will jump from rock to rock, zig-zagging your way to the far bank. Like thinking inside your head, this is an untidy but purposeful activity. But when you have to get a team across a metaphorical river you need to be able to construct a simple bridge, so that everyone knows where they are in the decision-making\/problem-solving discussion. (See the illustration above.)\n\nYou can see that the skills required, as one phase merges into the next, change. A new function with its family of more specific skills comes into play. The model is useful for your team as well as yourself. It can help everyone to keep in step.\n\nAsking the right questions\n\nA key skill, both when you are thinking something through by yourself and when you are leading or participating in a team, is to ask the right questions. Questions are the spanners that unlock the mind. Here are the kind of questions you should ask yourself \u2013 and others.\n\nUnderstanding the problem\n\n  * When did you first sense or become aware of the problem or the need for a decision?\n  * Have you defined the problem or objective in your own words? (Remember that a problem properly defined is a problem half-solved.)\n  * Are there any other possible definitions of the problem worth considering? What general solutions do they suggest?\n  * Are you clear about what you are trying to do? Where are you now and where do you want to get to?\n  * Have you identified the important factors and salient facts? Do you need to spend more time on obtaining more information? Do you know the relevant policies, rules, limitations, and procedures?\n  * Have you reduced the problem to its simplest terms without oversimplifying it?\n\nTowards solving the problem\n\n  * Have you checked all your main assumptions?\n  * Out of all the possible courses or solutions, have you identified a shortlist of the feasible ones?\n  * Can you eliminate some of these in order to shorten the list still further?\n  * If no solution or course of action seems right by itself, can you synthesise elements in two or more solutions to create an effective way of dealing with the problem?\n  * Have you clearly identified the criteria by which the feasible options must be judged?\n  * If you are still stuck, can you imagine yourself in the end-state where you want to be? If so, can you work backwards from there to where you are now?\n  * Has anyone else faced this problem? How did they solve it?\n\nEvaluating the decision and implementing it\n\n  * Have you used all the available information?\n  * Have you checked your solution from all angles?\n  * Are you clear about the manifest consequences?\n  * Have you an implementation plan with dates or times for completion?\n  * Is the plan realistic?\n  * Do you have a contingency plan if things do not work out as expected?\n  * When are you and your team planning to review the decision in the light of experience?\n\nYou may feel rather overwhelmed by this long list of questions. But you do not have to ask them all every time you are involved in making decisions or solving problems, for some of the questions will already have clear answers. What you should develop are three levels of competence:\n\n  * Awareness of problems or the need for decisions \u2013 either actual or potential. Have your feelers out, so that you are not taken by surprise.\n  * Understanding of where you and the team are in relation to the problem or decision. In what phase of the bridge model (see page 47) are you? Does more work need to be done on analysing information and defining the problem or decision? Or are you in the business of generating feasible options?\n  * Skill in asking the right questions of the right people at the right time, and being able to test the answers for their truth content. Action based on truth is much more likely to be effective than action based on a faulty perception of reality.\n\nIt may all sound like hard work. You recall Roy Thompson's words about 'thinking until it hurts' and 'this arduous and tiring work'. Yes, yes \u2013 but it is also great fun. It is what life is all about! I repeat: there is nothing more satisfying than being faced with a mental challenge and overcoming it. The harder the problem, the more elation you and your team will feel when you overcome it. So resolve to enjoy decision making and problem solving. The more you enjoy something the more of it you will want to do \u2013 and the better at it you will get.\n\nHow to approach systems problems\n\nObstacle-type problems account for 80 per cent of the problems that leaders encounter, but you should also be aware of systems problems \u2013 the other 20 per cent. If you are a technical specialist, of course, those proportions are reversed and the majority of your working time will be spent on systems problems.\n\nA system is a whole made up of integrated parts. It can be organic (your body), mechanical (your car engine), or a process (your system for billing customers). A systems problem is essentially a deviation from the norm. We can represent it visually by two lines (see the illustration above). The greater the difference between the normal performance (how the system is supposed to work) and the active performance (what is actually happening), the bigger the problem.\n\nFigure 4.2 Systems problems\n\nThe main strategy in systems problems is to find the point of deviation and then establish what caused it. The first aim is to establish the exact time and place of that critical deviation. What happened? When? How much? Who was affected? Who saw it? And so on. Notice again that the key skill of asking the right questions is in play, focusing on the deviation point on the diagram.\n\nExercise 3: The W5H Formula\n\nChoose any systems problem facing you and practise your question skills \u2013 Who?, What?, Where?, When?, Why? and How? See if you can pinpoint the deviation from the normal working of the system in question.\n\nOnce you have done that, list the possible causes. Now begin to eliminate the causes that can be proved innocent. You will be left with two or three suspects.\n\nHaving established when and where the deviation occurred, you then have to identify the cause or causes. Only by tackling these can you really solve a systems problem. Treat causes, not symptoms, if it is possible to do so.\n\nPlastec Ltd \u2013 a company making plastic containers \u2013 discovered that a rising percentage of its output was developing cracks. A project group studied the manufacturing process in detail and eventually defined the points of deviation: a change of supplier and a failure to clean out some storage vats. The new supplier inadvertently used these storage vats, and the plastic therefore became contaminated. Once the causes had been identified, the systems were altered to prevent any repetition, and the problem did not recur.\n\nBeware of the fallacy of the single cause. In relatively simple problems there is only one cause but, in more complex ones, two or three causes may be combining to produce the unwanted effect. In the case study on Plastec Ltd you will have noticed that it was the combination of two changes from the norm \u2013 a new supplier and poor cleaning procedures \u2013 that produced the problem.\n\nKey points\n\n  * Problems come in two main forms: problems that are really obstacles that appear across your chosen path, and systems problems.\n  * The broad approach to both families of problems and to decision making is the same. It can be compared to building a bridge across a river on three pillars:\n\nDefining the problem\n\nGenerating feasible options\n\nChoosing the optimum course\/solution\n\n  * A key thinking skill for an effective problem-solver is asking the right questions \u2013 to oneself initially but also to others. Questions are the spanners that unlock the problem in the mind, or at least the gates that bar entry to it. 'To act is easy,' said Goethe, 'to think is hard.'\n  * Systems problems are best approached by regarding them as deviations from an expected norm. Diagnosis includes identifying and establishing the exact nature of that deviation and what caused it. The solution \u2013 if one is possible \u2013 is to remove the cause of the problem. A secondary strategy for a problem solver where 'cure' isn't possible is to mitigate the effects of the problem on the performance of the system as a whole.\n  * People who are good with hammers see every problem as a nail.\n\nOne should never impose one's views on a problem; one should rather study it, and in time a solution will reveal itself.\n\nAlbert Einstein\n**5**\n\nHow to generate ideas\n\nTo raise new questions, new possibilities, to regard old problems from a new angle, requires creative imagination.\n\nAlbert Einstein\n\nWhen you are stuck in problem solving \u2013 that is, when the techniques you have applied successfully in the past are not working \u2013 try a more creative thinking approach. You may be trying to dig the same hole deeper, worrying at your problem like a terrier, when perhaps you should be digging your hole somewhere else.\n\nBrainstorming\n\nThe best-known and most widely used creative thinking technique is brainstorming. It was introduced in the 1930s, so it has been around a long time \u2013 a sign of its usefulness. You can employ its principles when you are thinking alone, but they work better in a team setting.\n\nWhen brainstorming, don't overlook the obvious! The obvious solution is sometimes the best. It may not, anyway, be obvious to everyone; and it may be possible to twist an obvious idea into something not so obvious. Don't fear repetition, either! Accusing someone of being repetitive is a form of adverse criticism and should be avoided. The same idea may trigger a different response at a different time in the brainstorming session.\n\nTake the common paper clip as an example. In five minutes one brainstormer came up with the following new uses.\n\nPipe cleaner| Fuse wire  \n---|---  \nNail cleaner| Letter opener  \nTie-clip| Catapult missile  \nEar de-waxer| Toothpick  \nPicture hook| Cufflink  \nSmall-hole poker| Ornament  \nScrewdriver| Typewriter cleaner  \nFishing hook| Tension reducer (like worry beads)  \nBroken bra-strap mender| Zip-fastener tag mender\n\nI expect you can do even better than that! Are you ready to have a go? Look at the Guidelines opposite first.\n\nExercise 4: Brainstorming skills\n\nTake a pair of scissors and list 50 new uses for them \u2013 apart from cutting things.\n\nYou have 10 minutes. Write your ideas down and \u2013 if stuck \u2013 go back and build on your first 10 ideas.\n\nGuidelines for brainstorming\n\nSuspend judgement| Give imagination the green light by withholding the critical evaluation of ideas until later. Accept ideas without judging them.  \n---|---  \nWelcome free-wheeling| Take off the brakes in your mind and go with the flow of your ideas. The more unusual the idea, the better \u2013 it is easier to tone down than think up.  \nStrive for quantity|  The greater the number of oysters, the more likely you are to find some pearls in them.  \nCombine and improve|  Listen to the ideas of others and see if you can build on them. Their way-out ideas may stimulate some buried memories or sleeping brain cells in your Depth Mind.  \nDo not edit| Ideas should not be elaborated or defended, just quickly stated and recorded.\n\nOne major reason why brainstorming is useful is that it helps to free us from 'functional fixedness'. We have a fixed idea, for example, that a thing has only one function and that is what it is there for. By banning the use of that familiar function (in the case of scissors, the function of cutting), the mind is released to consider other possibilities. With a little adaptation, scissors would make an interesting geometrical instrument.\n\nTake the modern British Army bayonet. Did you know that it is ingeniously designed to combine with its scabbard to form a pair of wirecutters? Or that it has a third function (officially!) built into it \u2013 that of a bottle-opener?\n\nPilkington Brothers Limited in the UK had a technical problem... During the final inspection of sheet glass, small globules of water were identified by the inspection machine as flaws in the glass. A brainstorming session produced 29 ideas in less than five minutes. After research and development, three of these were used in the system, which solved the problem.\n\nH J Heinz in the USA had a marketing problem... The company wanted to get sales promotional material to consumers more quickly. Brainstorming produced 195 ideas. After evaluation, eight were immediately used. A member of Heinz, when talking about another brainstorming session, said, 'Brainstorming generated more and better ideas than our special committee produced in 10 meetings.'\n\nThe essential principle behind brainstorming is simple. Please refer back to Chapter 1 and the three functions of analysing, synthesising and valuing. What brainstorming commands you to do is to make a temporary and conscious division between synthesising on the one hand and valuing on the other \u2013 for much of our valuing is negative and premature, like unseasonable frost that kills off the buds of spring. As Jean-Paul Sartre once said, 'Criticism often takes from the tree caterpillars and blossoms together.' Brainstorming's suspension of judgement is an invitation to exercise some inner mental discipline. Analysing and valuing have their time and place, but your imagination has wings \u2013 let it fly free!\n\nHow to run a brainstorming session\n\nNo more than 10 people should be involved. Some may know about the field, others may not \u2013 a mixture of both is desirable. They should ideally have been trained in the brainstorming technique before the meeting. When you run the session:\n\n  * Define the problem (using your analytical and briefing skills).\n  * Help people to understand the problem by highlighting the background information and history.\n  * Clarify the aim in a succinct sentence: 'In how many ways can we...?'\n  * Have a brief warm-up session, using a common problem or object.\n  * Brainstorm 70 ideas in 20 minutes, or a similar target. One person should write up the ideas on a flipchart. Allow time for silent reflection. Check that no critical remarks are made. Encourage cross-fertilisation.\n  * Establish criteria for selecting the feasible ideas. Choose the best.\n  * Reverse brainstorm: 'In how many ways can this idea fail?'\n\nAbout 40 minutes is the optimum time for a brainstorming session. But you should ask the participants to go on consid-ering the problem and let you have further suggestions. Remember that they have programmed their Depth Minds by the brainstorming session, and other ideas will come to them unexpectedly.\n\nA leading US firm of jigsaw-puzzle makers held a brainstorming session to think up ideas for new puzzles. It produced some worthy ideas but nothing brilliant. A month later, one of the participants went to see an exhibition of Tutankhamun's treasures in Washington DC. The gold mask of the pharaoh struck him as a great jigsaw puzzle idea! He was right \u2013 it broke all records for jigsaw puzzle sales in the United States.\n\nKey points\n\n  * Being creative involves the use of the imagination or original ideas in order to create something. Creative thinking is that part of it which produces the new ideas.\n  * 'It is the function of creative people,' writes the poet William Ploner, 'to perceive the relations between thoughts, or things, or forms of expression that may seem utterly different, and to be able to combine them into some new forms \u2013 the power to connect the seemingly unconnected.'\n  * Brainstorming is a useful technique, for generating ideas, whether you practise it on your own or in a team context. The essence of it is to make a temporary wall in the mind between the analysing\/synthesising functions on the one side, and the (critical) valuing function on the other side.\n  * Ideas rarely arrive in this world fully-formed and gift-wrapped. With a little practice you can learn to build on ideas, to take the germs of success in someone else's half-baked idea and to develop it towards fruition. By the same warrant, allow others to build on your ideas for the common good. Only God owns the intellectual property rights to truth.\n  * 'I start where the last man left off,' said the inventor Thomas Edison.\n  * Creativity is so delicate a flower that praise tends to make it bloom, while discouragement often nips it in the bud. Any of us will put out more and better ideas if our efforts are appreciated.\n\nCreative thinking thrives in an environment of mutual stimulation, feedback and constructive criticism \u2013 in a community of creativity.\n\nAnon\n**6**\n\nThinking outside the box\n\nValuable though brainstorming is, not least as an introduction to one or two of the fundamental principles of creative thinking, it is not the whole story. To develop your skills as a creative problem-solver you need to adopt and practise the strategies set out below.\n\nTowards a more creative approach\n\nBrainstorming challenges one kind of unconscious assumption, namely that hammers are for knocking in nails or that scissors are for cutting. But there are other forms of unconscious assumption that may inhibit your thinking.\n\nFigure 6.1 'The nine dots' solution\n\nTake the 'Nine dots' and 'Six matchsticks' problems in Chapter 1. The reason why many people cannot do the first one is that they put an unconscious or invisible framework around the dots, and try to solve the problem within it. That is impossible. But if you break out of that self-imposed limitation, the solution to the problem is easily reached. (See 'The nine dots' solution below.)\n\nIncidentally, I first published 'The nine dots' problem in 1969, in a book called Training for Decisions. It was the origin of a new phrase in the English language, now credited to me \u2013 thinking outside the box.\n\nThere is a similar assumption made in the second problem. People assume that they must arrange the six matches in a pattern of four equilateral triangles in only one plane. If they take one small step and give themselves permission to place the matches on top of one another, they can reach the first solution. But if they break out of the two-dimensional constraint into three dimensions, they achieve the most elegant solution.\n\nPlease don't mistake me: you cannot think without making assumptions. But they should be conscious ones from which you can retreat when they become indefensible. The assumptions that trip you up are the unconscious ones, the constraints or limitations that you are not aware of. That is one reason why effective thinking needs social interaction. We need critical input from others to remove these filters from our eyes.\n\nA senior manager in the UK marketing department of, Hoover, the household appliance company, once had the bright idea of introducing a 'free flights' promotional scheme as an incentive for buying their products. It was a spectacularly bad decision. Some 200,000 people flew with the scheme, but it cost the company around \u00a348 million. Some 127 people sought compensation in the courts, facing Hoover with a possible bill of millions of pounds if they succeeded. The president of Hoover Europe was dismissed from his \u00a3500,000-a-year post, and the US owners quickly sold the company for a knock-down price. You do not get decision making more wrong than that.\n\nFigure 6.2 'The six matchsticks' solution\n\nWhy did this fiasco happen? Because the Hoover managers concerned made a false assumption. They assumed that when most of the people who bought appliances saw the small print wrapped about the 'free flights' offer \u2013 the complex restrictions and qualifications they deliberately built in to deter applicants \u2013 these new customers would not bother to go through such a complex approach for the sake of a free air ticket. They under-estimated the public! Enough people persevered in finding a route through all the complex rules and conditions that their free flights brought the company to its knees.\n\nThis true story is a parable to remind us of the importance of checking to ensure that we are not allowing unconscious assumptions to act like hidden reefs and rip the bottom of the ship.\n\nLook wider for solutions\n\nMy phrase thinking outside the box ties in with the concept of lateral thinking, introduced by the well-known thinker and writer Edward de Bono in The Five Day Course in Thinking (1968). Lateral thinking means abandoning the step-by-step approach and thinking, as it were, 'to one side'.\n\nVertical thinking| Lateral thinking  \n---|---  \nChooses| Changes  \nLooks for what is right| Looks for what is different  \nOne thing must follow| Makes deliberate jumps directly from another  \nConcentrates on relevance| Welcomes chance intrusions  \nMoves in the most likely directions| Explores the least likely directions\n\nThe sideways (or lateral) thinking involved often leads to reversing what appears to be the natural or logical way of doing things. For example, the earliest method of making cars involved teams of men moving from one car to another. Henry Ford turned it all upside down. He put the car frames on belts and moved them past the men \u2013 the birth of the assembly line.\n\nIt is important to think sideways because the seeds of a solution to a problem may lie outside the box you are working in. Really creative people have a wide span of relevance: they look far afield, even to remote places or times in history, for solutions to the problems they face. When the eighteenth-century agriculturalist Jethro Tull invented the seed drill he summoned up his previous experience as an organist: he was creatively transferring technology from one area to another. Most of us, however, tend to think in compartments, and the divisions in work that make specialisation possible encourage this blinkered thinking.\n\nAlways be willing to challenge widely accepted assumptions.\n\nAs the exercise 'The nine dots' and 'The six matches' illustrate, thinking outside the box means being able to spot assumptions, habits or customary ways of thinking that are widely and uncritically accepted but have no basis in reality. Many of these we breathe in by virtue of the society in which we live. They are among the everyday conventions we accept as truths without too much examination. 'As everybody knows...'\n\nWhen 'everyone knows' something to be true, nobody knows nothing\n\nAndy Grove, later the Chairman of Intel, always remembered the oft-repeated saying of one of his professors when he was a student: 'When everybody knows that something is so, it means that nobody knows nothing.' It stayed with him throughout his career, and he gave the following example of its relevance.\n\n'Our little research group at Fairchild [Semi-conductor] some 40 years ago started to study the characteristics of surface layers that were the heart of modern integrated circuits. At that time, \"everybody knew\" that surface states, an artifice of quantum mechanics, would interfere with us building such chips. As it turns out, nobody knew nothin': We never found any surface states; what we found was trace contamination. When we identified and removed this, the road opened up to the chip industry as we know it today.'\n\nHow to use your Depth Mind\n\nCreative thinking cannot be forced. If you are working on a problem and getting nowhere, it is often best to leave it for a while and let your subconscious \u2013 your Depth Mind \u2013 take over. Your mind does not work by the clock, although it likes deadlines. Sometimes the answer will come to you in the middle of the night.\n\nGrasping the principle of the Depth Mind could open the way for you to a more creative approach to problem solving. Many people are still not even aware that their depth minds can carry out important mental functions for them, such as synthesising parts into new wholes or establishing new connections while they are engaged in other activities.\n\nImagine your mind to be like a personal fax machine. It would be nice and tidy if you could sit down for an hour each morning before breakfast and receive inspired fax messages from your Depth Mind. But it is not like that. The fax machine might start whirring at any time of the day or night.\n\nIf you are thinking along a certain line and nothing happens, stop. Instead of investing more time \u2013 throwing good money after bad \u2013 analyse the problem again and see if you can come up with a new approach. Usually your frustration will be caused by one of the mental roadblocks described in the table on pages 70\u201371.\n\nThe processes of analysing a problem or identifying an objective are themselves means of programming the mind. Possible solutions and courses of action almost instantly begin to occur to us. Where there is a time-delay this means that the deeper parts of the brain have been summoned into action and have made what contribution they can.\n\nHow important preparation time is for creative thinking! Careful and clear analysis, conscious imagining or synthesising (using such techniques as brainstorming either in groups or solo), and exercising the valuing function of thought in a positive rather than negative way \u2013 all these are vital to lay the foundations for thinking creatively. (See page 69.)\n\nIf you are planning to experiment and try a session before breakfast, it is useful always to have a preparation phase the night before. Imagine yourself as a house decorator, scraping down the woodwork and filling in holes and priming here and there, prior to painting a first coat the following day.\n\nThe creative thinking process\n\nPreparation| The hard work. You have to collect and sort the relevant information, analyse the problem as thoroughly as you can, and explore possible solutions.  \n---|---  \nIncubation| This is the Depth Mind phase. Mental work \u2013 analysing, synthesising and valuing \u2013 continues on the problem in your subconscious mind. The parts of the problem separate and new combinations occur. These may involve other ingredients stored away in your memory.  \nInsight| The 'Eureka' moment. A new idea emerges into your conscious mind, either gradually or suddenly, like a fish flashing out of the water. These moments often occur when you are not thinking about the problem but are in a relaxed frame of mind.  \nValidation| This is where your valuing faculty comes into play. A new idea, insight, intuition, hunch or solution needs to be thoroughly tested. This is especially so if it is to form the basis for action of any kind.\n\nAlthough it is useful for you to have this framework in mind, remember that the actual mental process is a lot more untidy than the above table suggests. Think of the phases as being four notes on a piano which can be played in different sequences or combined together in complex chords.\n\nMozart\n\n'When I am completely myself,' Mozart wrote to a friend in a letter, 'entirely alone or during the night when I cannot sleep, it is on such occasions that my ideas flow best. Whence and how they come I know not nor can I force them. Nor do I hear in my imagination the parts successively, but I hear them at the same time all together.'\n\nMental roadblocks\n\nLack of facts| If you are not sure you have all the relevant facts, you naturally hesitate to commit yourself. Do some more research, and that may get you moving again.  \n---|---  \nLack of conviction| Maybe you find it difficult because you lack conviction in the value of this exercise or the way in which you have been asked to do it. Re-establish a worthwhile objective.  \nLack of a starting point| Possibly the problem seems so large that you do not know where to start. If so, make a start anywhere. You can always change it later. Inspiration comes after you have started, not before.  \nLack of perspective| Perhaps you are too close to the problem, especially if you have lived with it a long time or have been worrying about it incessantly. Try leaving it for a week. Consult others. Simply explaining it to them may help. They may see new angles.  \nLack of motivation| Do you want it to happen enough? Creative thinking requires perseverance in the face of surmountable difficulty. If you are too easily put off, it may be a sign that, deep down, you lack the necessary motivation. Reinvigorate your sense of purpose.\n\nThe function of creative thinking in problem solving is to come up with new ideas. But remember that at some stage your valuing skills have to be brought into play. Here are six questions to ask about any new idea, solution, or course of action:\n\n  * Is it really new?\n  * Is it both relevant and practical?\n  * Whom will it involve?\n  * How much will it cost?\n  * How much will it save?\n  * Will it require more formal evaluation?\n\nIn times of rapid change, like our own age, there is a premium on your skills as a creative thinker. If you can think productively and constructively, as well as analytically and logically, it will give you a third dimension in all your decision making and problem solving.\n\nKey points\n\n  * One of the most valuable principles for improving your creative approach to work and life is to learn to think outside the box. Essentially that means be willing to challenge the assumptions \u2013 often unconscious \u2013 that put an invisible cage around the bird of thought.\n  * One form of thinking outside the box is to think sideways \u2013 lateral thinking. For sometimes the solution to a problem \u2013 at least in embryo \u2013 lies in a field of enterprise adjacent to one's own but quite distinct from it.\n  * There is a danger in formalising any aspect of the creative process \u2013 it is a delicate balance between following a conscious process or framework and being guided by the mind's natural inclinations. But it is worth bearing in mind the commonsense sequence:\n\nPREPARATION\n\nINCUBATION\n\nINSIGHT\n\nVALIDATION\n\n  * INCUBATION \u2013 sitting on eggs until the young birds of ideas emerge \u2013 is a metaphor for the Depth Mind's work. We all have a purposive and helpful Depth Mind; we differ as to the use we make of it.\n  * 'In the creative state,' writes the novelist E M Forster, 'a person is taken out of himself. He lets down as it were a bucket into his subconscious, and draws up something which is normally beyond his reach. He mixes this thing with his normal experience and out of the mixture something new emerges.'\n\nThe intellect has little to do on the road to discovery. There comes a leap in consciousness, call it intuition or what you will, and the solution comes to you and you don't know how or why.\n\nAlbert Einstein\n**7**\n\nDeveloping your thinking skills\n\nWinston Churchill once said, 'I am always willing to learn, but I do not like being taught.' Actually, when you learn, you are being taught \u2013 by yourself. No doubt Socrates, if he was here, could teach you how to think, but he is not here. Nor is decision making and creative problem solving a school and university subject; there is no formal body of knowledge, supported by empirical research. And so, if you truly want to develop your thinking skills, your task is essentially one of self-development. In this chapter we shall look at some common-sense guidelines that you will need if you choose to go down that road.\n\nWhat is an effective practical thinker?\n\nForming a clear picture of the kind of thinker you would like to be is the first step you need to take. A clear concept of what you might be one day can act as your magnet. Remember that point about formulating where you want to be and then working backwards?\n\nYou could do it in abstract terms, listing all the qualities, the knowledge, and the functions or skills you would like to acquire by such-and-such a date. I have to admit, though, that that does not work for me: it is a bit too academic. I suggest a more homely method, which any South Sea cannibal of olden times would have relished.\n\nIn Exercise 5 below I invite you to recall people whose thinking skills you have admired. They can be people you have known personally or have studied in some depth (by, say, reading more than one biography of them). In the right-hand column, write down as concisely and specifically as you can those thinking skills that impressed you and that you would like now to 'eat' by gobbling up and inwardly digesting, so that they become part of you. Write down, for instance, any key remarks or sayings by which the person concerned encapsulated his or her practical wisdom.\n\nExercise 5: Your personal thinking skill mentors\n\nName| Thinking skill  \n---|---\n\nTake some time over this exercise, and try to get a good spread across the functions (analysing, synthesising and valuing) and the applied forms of effective thinking (decision making, problem solving and creative thinking). After all, you don't want to eat a meal composed of just one ingredient.\n\nYou will probably find it easy to come up with the names of two or three people \u2013 a parent, a friend, a life partner or a boss you have worked for \u2013 who have exemplified a thinking skill that you covet. If it is not so easy, however, to come up with many names to complete Exercise 5, leave it for a week or two. Your Depth Mind will suggest other names and other lessons \u2013 influences that may have become more subconscious.\n\nFrom your list of 'appetising' thinking skills you can begin to create a composite and imaginary picture of the perfect practical thinker. He or she would have A's analytical skills, B's rich and creative imagination, C's ability to be flexible and improvise, D's extraordinary judgement in situations of uncertainty and unpredictability, E's courage to take calculated risks, F's intuitive sense of what is really going on behind the scenes, G's lack of arrogance and openness to criticism, H's decisiveness when a decision is called for, and I's tolerance of ambiguity when the time is not ripe for a decision.\n\nNow a perfect person with all these skills \u2013 a Mr or Ms ABCDEFGHI \u2013 does not, and never will, exist. You may know the story of the young man who searched the world for the perfect wife. After some years he found her \u2013 but, alas, she was looking for the perfect husband! Perfection will always elude you \u2013 but excellence is a possibility.\n\nWhat the exercise achieves, however, is to give you an ideal to aim for. Advanced thinkers in any field tend to be lopsided: like athletes, they develop one set of muscles rather than others. Did you know that sprinters are hopeless at long-distance running? I am not advocating that you should be a perfectly balanced thinker, a kind of intellectual 'man for all seasons'. Rather, I suggest you look carefully at your field and where you see yourself positioned in it in (a) five years' time and (b) ten years' time. The ideal that you formulate should be related to your field, although, of course, not all your personal thinking skill mentors will be from that field \u2013 at least I hope not, otherwise I should suggest that your 'span of relevance' needs widening.\n\nCheck that you are in the right field\n\nThinking skills are partly generic or transferable, and partly situational. Decision making and problem solving are not abstracts: they are earthed in a particular field, with its knowledge, traditions, legends, and values.\n\nDimitri Comino, the founder of Dexion plc, once discussed with me a book I was writing on motivation. 'In my experience,' he said, 'it is very difficult to motivate people. It is much better to select people who are motivated already.' The same principle holds good, I believe, for thinking skills. It is actually quite difficult to teach yourself skills that are not natural to you. So choose a field that suits your natural profile as a thinker. What is the right field of work for you? (See the table below.)\n\nKey factors in choosing your field of work\n\nWhat are your interests?| An interest is a state of feeling to which you wish to pay particular attention. Long-standing interests \u2013 those you naturally like \u2013 make it much easier to acquire knowledge and skills.  \n---|---  \nWhat are your aptitudes?| Aptitudes are your natural abilities, what you are fitted for by disposition. In particular, an aptitude is a capacity to learn or acquire a particular skill. Your aptitude may range from being a gift or talent to simply being above average.  \nWhat are the relevant factors in your temperament?| Temperament is an important factor. Some people, for example, are uncomfortable in decision-making situations of stress and your danger, while others thrive on them. Some prefer to be problem solvers rather than decision makers.\n\nIt is usually easier to identify the fields that you are not suitable for, because you lack the necessary level of interests, mental aptitude, or temperamental characteristics to do really well in them.\n\nLet me now make the assumption that you are in the right field. You have more or less the right profile of aptitudes. You have been able, in other words, to acquire the knowledge and professional\/technical skills needed and have enjoyed doing so. You have already laid the foundations of success at the team, operational and strategic levels of leadership. You will have credibility among your colleagues. Now what you have to do is focus upon the process skills \u2013 the more generic or transferable ones \u2013 in decision making and problem solving. How do you acquire them?\n\nHow to design your own learning strategy\n\nBefore planning your own self-learning programme it is useful to remind yourself of the core process of learning. (See the diagram below.)\n\nRecall what was said above about thinking skills being partly generic and partly situational. It is when sparks jump between these two poles that learning occurs. So you need both.\n\nFigure 7.1 How we learn\n\nBecause decision making and problem solving are such central activities in any person's life we have plenty of experience of them. And as you move into a professional field, and begin making decisions and tackling problems, you soon build up a repertoire of experience. You learn by mistakes. In the technical aspects of your work you do have a body of knowledge \u2013 principles or theory \u2013 to bring to bear on your practice. How can you apply the same learning method to thinking skills? Here are some practical suggestions:\n\n  * Read this book again and underline all the key principles. Put a star by the models or frameworks that you can use. Build up your own body of theory.\n  * Make an inventory of your thinking skills in relation to your own field of work. What is the present profile of your strengths, and what are the areas for improvement? \n  * See if you can identify three outstanding decision makers and problem solvers in your field to whom you have access. Ask if you may interview them briefly to discover what principles \u2013 if any \u2013 have informed their own development as applied thinkers.\n  * Select one really bad decision made by your organisation during the last 18 months. Write it up as a case study, limiting yourself to five key lessons to be learned about decision making. If you want to develop your moral courage, send it to the chief executive!\n  * Now select any outstanding innovation in your field, inside your organisation or outside it. By an 'innovation' I mean a new idea that has been successfully 'brought to market' as a new (or renewed) product or service. Again, write it up as a case study and highlight at the end the five or six key lessons for creative problem-solvers.\n  * Set yourself a book-reading programme. 'He would say that, wouldn't he?' But I do not have to persuade you to read books \u2013 you have just read this one. If you have enjoyed it, and found it worthwhile, try to read one general book each year on leadership (see Further Reading for some suggestions) and one biography of an outstanding person in your field. Again, underline the key principles in pencil. Surely you can budget time for one book in 52 weeks?\n  * Transfer your growing body of principles, examples, practical tips, sayings or quotations, and thumbnail case studies to a stiff-covered notebook. As it fills up, take it on the occasional train journey or flight and read it reflectively, relating it to your current experience.\n  * Take any opportunities that come your way to attend courses or seminars that offer you know-how in the general area of effective thinking. You should, for example, become thoroughly versed in what information technology can do \u2013 and not do \u2013 at present in your field, and become skilled in the use of computers.\n  * Lastly, go out of your way to seek criticism of yourself as a decision maker and problem solver. However savage, however apparently negative the criticism, you still need it in order to learn. It is the toughest part of being a self-learner, but remember the sporting adage, 'No pain, no gain.' Your critics, whatever their motives or manners, are doing you the service of true friends. Sift through their comments for the gold-dust of truth.\n\nIn any self-learning programme, experience is going to play the major part. There is no getting away from that. But if you rely just on learning in what has been called 'the university of experience' you will be too old when you graduate to benefit much from the course! And the fees you will pay on the way will be extremely high. Short though it is, this book gives you those essential components \u2013 the key frameworks and principles \u2013 that you can use to cut down the time you take to learn by experience \u2013 experience and principles.\n\nYou will begin to develop both your knowledge of these process skills and your ability to apply that knowledge in all the challenging and potentially rewarding situations that lie ahead of you. Good luck!\n\nKey points\n\n  * Knowledge is only a rumour until it is in the muscle, says a Papua New Guinea proverb. Think of your mind as a muscle \u2013 or a set of muscles. This book tells you in an introductory way how to develop those muscles, but it is you who have to put in the effort. Are you keen to do so?\n  * Don't think of thinking as being hard, painful or laborious \u2013 if you do that you certainly won't apply yourself to shaping and sharpening your thinking skills. Thinking is fun, even when \u2013 or especially when \u2013 we are faced with apparently insurmountable difficulties.\n  * You are more likely to be effective as a practical thinker if you succeed in finding your vocation, your right niche in the world of work. The guide here is to choose a function and field or work that is optimum for your interests, aptitudes and temperament.\n  * 'I have never met a man so ignorant,' said Galileo, 'that I couldn't learn something from him.' Prize especially those people you meet \u2013 in person or in books \u2013 who can teach you things in how to think. \n  * Practical wisdom should be your aim as a thinker, especially in the applied domain of decision making. Practical wisdom is a mixture of intelligence, experience and goodness.\n\nI learn most, not from those who taught me but from those who talked with me.\n\nSt Augustine\nAppendix: solution to 'Who owns the zebra?'\n\nThis problem can be solved by analytical and logical thinking \u2013 deductive logic \u2013 and persistence! It is necessary to compile a matrix.\n\nRoughly half-way through the problem-solving process there are two forks in the road, or mental leaps. The only way to find out which way to go is by trial and error. If you choose the wrong road, you have to retrace your steps. You can see now why the world record for finding the solution is 10 minutes!\n\nThe following is one way of solving the problem.\n\nKeep working through the facts from 1 to 15 in sequence.\n\nConcentrate on clues for which there is only one answer. That is:\n\n1 There are five houses, each with a front door of a different colour, and inhabited by people of different nationalities, with different pets and drinks. Each person eats a different kind of food.\n\n9 Milk is drunk in the middle house.\n\n10 The Norwegian lives in the first house on the left.\n\n15 The Norwegian lives next to the house with the blue door.\n\nThen look for information that has only two possible answers. This is the first mental leap. That is:\n\n6 The house with the green door is immediately to the right (your right) of the house with the ivory door.\n\nIf you place the ivory door in the middle, with the green door on its right, the answer is wrong, but you can still progress to find out who drinks the water. However, you can go no further.\n\nIf you place the ivory door in the fourth house, with the green door on the far right, this answer is correct and you can progress logically, since you will find that other items of information now have only one answer. That is:\n\n2 The Australian lives in the house with the red door.\n\n4 Coffee is drunk in the house with the green door.\n\n8 Apples are eaten in the house with the yellow door.\n\n12 Apples are eaten in the house next to the house where the horse is kept.\n\nThen look for information that has only two possible answers. This is the second mental leap. That is:\n\n3 The Italian owns the dog.\n\nIf you place the Italian in the house with the green door you are wrong, but you can still find out who drinks the water.\n\nIf you place the Italian in the house with the ivory door you are correct and you can progress logically, since you find other items of information now have only one answer. That is:\n\n5 The Ukrainian drinks tea.\n\n13 The cake-eater drinks orange juice.\n\nTherefore the Norwegian drinks water.\n\n14 The Japanese eats bananas.\n\n7 The mushroom-eater owns snails.\n\n11 The person who eats onions lives in the house next to the person with the fox.\n\nTherefore the Japanese owns the zebra.\n\nAnother way of solving this problem is to form a matrix using nationalities rather than house numbers:\n\nFront doors| yellow| blue| red| ivory| green  \n---|---|---|---|---|---  \nInhabitants | Norwegian| Ukrainian| Australian| Italian| Japanese  \nPets| fox| horse | snails| dog| zebra  \nDrinks | water| tea| milk| orange juice| coffee  \nFood| apples| onions| mushrooms| cake| bananas\nFurther reading\n\nAlder, H (1995) Think Like a Leader: 150 top business leaders show you how their minds work, Piatkus, London\n\nde Bono, E (1968) The Five Day Course in Thinking, McGraw-Hill, Maidenhead\n\nde Bono, E (1971) Lateral Thinking for Management, McGraw-Hill, Maidenhead\n\nde Bono, E (1971) The Use of Lateral Thinking, Penguin, London\n\nde Bono, E (1985) Six Thinking Hats, Penguin, London\n\nBuzan, T (1974) Use Your Head, BBC Publications, London\n\nCulligan, M J, Deakins, C S and Young, A H (1983) Back to Basics Management, Facts on File, New York\n\nDawson, R (1994) Make the Right Decision Every Time, Nicholas Brealey, London\n\nDrucker, P (1966) The Effective Executive, Harper & Row, New York\n\nDrucker, P (1967) The Practice of Management, Heinemann, London\n\nKepner, C H and Tregoe, B (1965) The Rational Manager, McGraw-Hill, London\n\nKoestler, A (1964) The Act of Creation, Hutchinson, London\n\nRawlinson, J G (1983) Creative Thinking and Brainstorming, Gower, Aldershot\n\nThompson, R (1975) After I Was Sixty, Hamish Hamilton, London\n\nBy the same author\n\nEffective Decision Making, (1985), Pan, London\n\nEffective Innovation, (1996), Pan, London\n\nEffective Leadership Masterclass, (1996), Pan, London\n\nHow to Grow Leaders, (2005), Kogan Page, London\n\nThe Inspirational Leader, (2005), Kogan Page, London\n\nLeadership and Motivation, (2006), Kogan Page, London\n\nNot Bosses But Leaders, (2006), Kogan Page, London\nIndex\n\naction\n\naction-centred leadership 39\n\nPoint of No Return (PNR) 28, 32\n\nalternatives 21\n\nanalysing 7\u20138, 15, 47, 69\n\napplied thinking 1\n\naptitudes 78\u201379\n\nassumptions 64\u201365\n\nBismarck, Otto von 22\n\nbrain 3\u20135\n\nbrainstorming 55\u201360\n\nChurchill, Winston 75\n\nComino, Dimitri 78\n\ncompetence 40, 50\n\nconsequence assessment 25\u201327, 32\n\nconversation 44, 83\n\ncost containment 20\n\ncreativity\n\ndepth mind 67\u201368\n\nimagination 61\n\nincubation 69, 72\n\ninsight 69, 72\n\nmeaning 1\n\nmore creative approaches 63\u201365\n\npreparation 68, 72\n\nsynthesising 10, 15, 69\n\nthinking process 69\n\nvalidation 69, 72\n\nwider solutions 65\u201367\n\ncriticism 10, 58\n\nde Bono, Edward 65\n\ndecision making\n\nbridge model 47, 50\n\nclassic approach 23\u201327\n\nconsequence assessment 25\u201327, 32\n\neffective decision making 17\u201333\n\nfive step approach 17, 19, 31\u201333\n\nmeaning 1\n\npain 22\n\nproblem solving compared 45\u201347\n\nrisk assessment 24\u201325, 32\n\nselection\/success criteria 23\u201325, 32\n\nunified model 47\n\ndecisions\n\nbad decisions 26\n\ndecision sharing 35\u201344\n\ndepth mind 27\u201330\n\nevaluation 28\u201330, 32, 49\u201351\n\nimplementation 28\u201330, 32, 49\u201351\n\nwrong decisions 27\n\ndeduction 8\n\ndelegation 40\n\ndepth mind\n\nbrainstorming 57\u201359\n\ncase study 12\n\nchecklist 13\u201314\n\ncreativity 67\u201368, 72\n\ndecisions 29\u201330\n\ndevelopment 14\n\nprinciple 12, 15\n\nDewey, John 16\n\nEdison, Thomas 61\n\nEinstein, Albert 53, 56, 73\n\nevaluation, decisions 28\u201330, 32, 49\u201350\n\nfallacy exercise 8\n\nfield of work 78\u201379\n\nfive step decision making 17, 19, 31\u201333\n\nFord, Henry 66\n\nForster, E M 72\n\nfunctional fixedness 57\n\nGalileo Galilei 83\n\nGoethe, Johann Wolfgang von 53\n\ngroup personality 35\n\nGrove, Andy 67\n\nHilton, Conrad 12, 28\n\nholism 10\n\nideas\n\ncase studies 58, 60\n\ngeneration 55\u201361\n\nimagination 9\n\nimplementation of decisions 28\u201330, 32, 49\u201350\n\nindividual needs 37\n\ninduction 8\n\ninformation\n\ncategories 20\n\ncollection 18\u201321, 32\n\nInformation Overload Syndrome 19\u201320\n\ntime\/information curve 21\u201323\n\ninsight 69, 72\n\ninterests 78\n\nknowledge 3\u20134\n\nlatent consequences 25\u201327, 32\n\nleadership\n\ncase study 42\n\nchecklist 43\n\nconsistency 41\n\ndecision sharing 35\u201344\n\nfunctions 38\u201341\n\noverlapping needs 36, 37\u201341, 44\n\nplanning 40\u201341\n\nroles 35\u201336\n\nlearning strategies 79\u201382\n\nlobster pot model 23\n\nlogic 8\u20139\n\nmanifest consequences 25, 32\n\nmental roadblocks 70\u201371\n\nmind\n\nanalysing 7\u20138, 15, 48, 69\n\nbrain distinguished 3\n\nfunctions 6\u201311\n\nmind at work 3\u201316\n\nsynthesizing 9\u201310, 15, 48, 69\n\nvaluing 10\u201311, 15, 32, 48, 69\n\nsee also depth mind\n\nMozart, Wolfgang Amadeus 70\n\nnine dots problem 5, 45, 64, 66\n\nobjective, definition 19, 32, 47\n\noptions, generating feasible options 21\u201323, 32, 47, 53\n\noutcomes\n\noutcomes window 27\n\nprocess compared 27\n\noverlapping needs 36, 37\u201340, 44\n\nPetrach 30\n\nplanning continuum 40\u201341\n\nPloner, William 61\n\nPoint of No Return (PNR) 28, 32\n\nprobability theories 25\n\nproblem solving\n\ncase study 52\n\ncompetence 49\n\ndecision making compared 45\u201347\n\nmeaning 1\n\nnine dots problem 5, 45, 64, 66\n\nprocedures 48\u201349\n\nsix matchstick problem 5, 45, 64, 65, 66\n\nstrategies 45\u201353\n\nsystems problems 50 52, 53\n\nunderstanding 48, 50\n\nunified model 47\n\nW5H formula 52\n\nzebra problem 5\u20136, 8, 45, 85\u201387\n\nquestions 48, 49, 53\n\nrisk assessment 24\u201325, 32\n\nRoosevelt, Franklin D 17\n\nSt Augustine 83\n\nSartre, Jean-Paul 58\n\nselection criteria\n\ndecision making 23\u201324\n\nmust\/should\/might 24, 32\n\nShakespeare, William 29\n\nsix matchstick problem 5, 45, 64, 65, 66\n\nSloan, Alfred 22\n\nSmuts, Jan 10\n\nsynthesising 9\u201310, 15, 47, 69\n\ntask need 36\n\nteams\n\ncompetence levels 40\n\nleaders see leadership\n\nmaintenance need 36, 37\n\ntemperament 79\n\nthinking\n\napplied thinking 1\n\ncase study 67\n\nconversation 44, 83\n\neffective practical thinkers 75\u201376\n\nimportance 2, 15\n\nlateral 66, 72\n\npain 2, 50\n\npersonal thinking skill mentors 76\u201378\n\nskill development 75, 83\n\nthinking outside the box 63\u201373\n\nvertical 65\u201366\n\nThompson, Roy 2, 14, 42, 50\n\nthree circle model 36, 38, 44\n\ntime\/information curve 21\n\ntrial and error 8\u20139\n\ntruth 11, 15, 51, 61\n\nTull, Jethro 66\n\nvalues at work exercise 11\n\nvaluing\n\nfeasible options 21\u201322\n\nmind 10\u201311, 15, 32, 69\n\nWordsworth, William 10\n\nzebra problem 5\u20136, 8, 45, 85\u201387\nCreating Success series\n\nDealing with Difficult People by Roy Lilley\n\nDecision Making & Problem Solving Strategies by John Adair\n\nDevelop Your Assertiveness by Sue Bishop\n\nDevelop Your Leadership Skills by John Adair\n\nDevelop Your NLP Skills by Andrew Bradbury\n\nDevelop Your PR Skills by Lucy Laville and Neil Richardson\n\nEffective Customer Care by Pat Wellington\n\nEffective Financial Management by Brian Finch\n\nHow to Deal with Stress by Stephen Palmer and Cary Cooper\n\nHow to Manage Meetings by Alan Barker\n\nHow to Manage People by Michael Armstrong\n\nHow to Motivate People by Patrick Forsyth\n\nHow to Negotiate Effectively by David Oliver\n\nHow to Sell Yourself by Ray Grose\n\nHow to Understand Business Finance by Bob Cinnamon and Brian Helweg-Larsen\n\nHow to Write a Business Plan by Brian Finch\n\nHow to Write a Marketing Plan by John Westwood\n\nHow to Write Reports and Proposals by Patrick Forsyth\n\nImprove Your Coaching and Training Skills by Patrick Forsyth\n\nImprove Your Communication Skills by Alan Barker\n\nOrganise Yourself by John Caunt\n\nSuccessful Interviewing and Recruitment by Rob Yeung\n\nSuccessful Presentation Skills by Andrew Bradbury\n\nSuccessful Project Management by Trevor Young\n\nSuccessful Time Management by Patrick Forsyth\n\nTaking Minutes of Meetings by Joanna Gutmann\n\nUnderstanding Brands by Peter Cheverton\n\nThe above titles are available from all good bookshops.\n\nFor further information on these and other Kogan Page titles, or to order online, visit the Kogan Page website at www.koganpage.com\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":"The selection committee for the Hockey Hall of Fame will meet on June 26 to figure out what the Class of 2018 will look like. It appears to be a difficult year to come up with the best choices.\nThis is a much easier process when veteran observers look at the qualifications of a particular player, and without any thought say, \"Yup, he's a Hall of Famer.\" But that does not cover everyone, especially this time. There are plenty of people on the margins.\nI'll use two techniques for coming up with a possible list of those who could go. I'll break it up into four categories. There are the cinches \u2013 players who will get in without an argument; the comets \u2013 those who were absolutely brilliant for a relatively short period of time (think Pat LaFontaine and Eric Lindros); the compilers \u2013 performers who were pretty good but not outstanding for a long period of time and thus have excellent career stats; and the curiosities \u2013 those who might not fit in anywhere else but still get extra points for one reason or another.\nI like to use a formula that allots two points for a first-team all-NHL selection, and one point for a second-team pick. This does a great job of identifying the cinches and comets, but is weak when it comes to the compilers and curiosities. It's in the tool box.\nMartin Brodeur \u2013 He has the most career wins, the most career games, and the most career shutouts among goalies in NHL history. Martin earned 10 points in the formula (three firsts, four seconds). You might not call Brodeau the best goalie ever (Hasek? Roy? Sawchuk?), but he might be your pick if you needed a goalie to start your franchise and last for close to two decades.\nHe's the only one this year that belongs in this group.\nMartin St. Louis \u2013 He made all-league teams five different times (one first, four seconds for six points), but fell short of 400 goals. Martin also had only one 40-goal and 100-point season. It's a slightly odd looking career, similar to players such as Pavel Datsyuk and Bryan Trottier according to hockey-reference.com.\nAlexander Mogilny \u2013 Buffalo, you may have heard of him. He had 76 goals and 127 points in 1992-93, at a time when league scoring totals seemed to be on steroids. He crossed 40 goals in two other seasons on his way to 473 goals and more than 1,000 points. Injuries probably cost him 500 goals, and there was always a feeling that he should have been a better player than what he was (a pair of second-team all-NHL picks). Did he do enough?\nJohn LeClair \u2013 The big left winger was all-league five times (two firsts, three seconds for seven points). He came into his own when paired with Lindros in Philadelphia, and had five great seasons on the \"Legion of Doom\" line. But he never got past 25 goals after that, and finished with \"only\" 333 goals.\nI'd guess that St. Louis makes the finals of the discussion, and the other two will have to wait.\nDaniel Alfredsson \u2013 Played all but one season of his career with the Senators, finishing with 1,178 games, 426 goals and 1,128 points. In other words, Daniel was very good for a long time. But, would you call him one of the elite players in the league? He only had one season above 40 goals and 90 points, and was a second-team all-NHL pick once.\nJeremy Roenick \u2013 He was a comet at the start, with three 100-point seasons before the age of 25. Jeremy didn't keep that level up, but he was good for a long time and finished with 513 goals and 1,216 points. The center never was an all-NHL pick, and played for five different teams.\nCurtis Joseph \u2013 He had a career something like Alfredsson, in that he was certainly a very good goalie for years and years. (Cujo had no all-NHL picks.) No wonder he won 454 games, and he did it with some teams that were less than Cup contenders most of the time.\nKeith Tkachuk \u2013 His name came up when discussing the best power forwards of his era. Keith finished with 538 goals, which certainly will get the attention of the Hall voters. Again, he wasn't part of any great teams. By the way, Pat Verbeek finished with similar numbers (522 goals), but it would be tough to see him picked before Tkachuk.\nDoug Wilson \u2013 He may have spoiled us, when he scored 39 goals in 1981-82 as a defenseman in winning the Norris Trophy. Wilson never duplicated that figure, but he still finished with 816 points in 1,024 games. Doug was a first-team all-NHL pick once and a second-teamer twice for four points.\nIt feels as if Alfredsson is the next one to go, but I don't see much difference here.\nThis mostly applies to European players. The best of the bunch, and the one receiving the most talk this year, is Boris Mikhailov of the Soviet Union. He was a big part of the great teams of the 1970s. Remember how Herb Brooks told Team USA in Lake Placid that it should be afraid of a guy who looked so much like Stan Laurel?\nOne interesting question \u2013 does Mogilny's defection from the Soviet Union give him points in the curiosity category? He was the first player to make a jump from there.\nI won't try to guess about a pick in women's hockey, but \u2013 familiar last name coming \u2013 Jennifer Botterill's five world championships for Team Canada ought to get her on that short list. Yup, she's Jason's sister.\nYou could win a bet by pointing out that Pierre Turgeon finished with more career goals than Gil Perreault (515-512). Pierre also had 812 assists for 1,327 points to fit into the compiler list. Still, would you trade him for any of the guys listed there?\nSpeaking of questions \u2013 if goalie Chris Osgood is going to get a look by the selection committee, shouldn't Tom Barrasso also examined? Barrasso won 369 games and two Stanley Cups to Osgood's 401 and three, and he didn't have a great team in front of him for all of his career.\nSpeaking of ex-Sabres, Slava Kozlov and Robyn Regehr both played more than 1,000 games, while Daniel Briere finished with 696 points. All are eligible for the first time this summer.\nOther names that might come up in conversation are Theo Fleury, Sergei Zubov, Sergei Gonchar, Rod Brind'Amour, Steve Larmer, Petr Bondra, Bernie Nicholls, Vincent Damphousse, Mike Vernon and Andy Moog. They all seem likely to wait a while longer, at best.\nBrodeur is a sure thing. If there are three more spots open (not including women, executives, etc.), I would pick St. Louis, Alfredsson and Mikhailov.\nFor lists of those who haven't made the Hockey Hall yet, click the appendix to the story.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We present our wallpapers for desktop of Maggie Q in high resolution and quality, as well as an additional Full HD high quality wallpapers, which ideally suit for desktop not only of the big screens, but also on the screens of Android and IPhone. All pictures are available for free download. This section provides no less than 25 high definition wallpapers with the Maggie Q, and optionally you can immediately download all the HD wallpapers for your desktop absolutely free. Our service has a large base of desktop wallpapers, dedicated to the Maggie Q, which you can find at the bottom of this page.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In an effort to fulfill his campaign promise to make Ontario \"open for business\", Premier Doug Ford introduced the \"Making Ontario Open for Business Act\" on Tuesday. This pro-business legislation introduces sweeping changes, and repeals the bulk of the Wynne government's Bill 148.\nBill 148 was a long overdue update to the Ontario Employment Standards Act. It modernized workers rights to address the current issues affecting workers in Ontario, and to reflect the needs of the present workforce.\nMost notably, Bill 148 raised the minimum wage in Ontario, introduced equal pay for equal work, increased vacation time, provided employees with personal emergency leave days, unpaid leave for employees who need to care for a critically ill family member, and provided leave for employees who are experiencing domestic or sexual violence.\nHowever, according to the new PC Government, Bill 148 is \"burdensome, job-killing red tape\" that hinders investment and job growth in Ontario.\nWhat Will the Proposed Changes Do?\nRepeals the changes to the Labour Relations Act that would make it easier for workers in various sectors to join a union.\nThe Associates at MacDonald & Associates will keep a keen eye on the progress of the Making Ontario Open for Business Act, and will keep you updated.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Still have unanswered questions about vasectomy procedures, recovery time, or what to expect? Check out our vasectomy infographic by clicking on the image below. Also feel free to contact us with additional questions.\nExcellent Dr. very caring of ones needs. Also makes certain you are well taken care of. Also his staff up front kind and courteous.\nDr. Crownover is great. He is very professional but still takes the necessary time to make sure all needs are met. Thank you!\nI was initially in need of a vasectomy. I also did not have a personal physician. After my pre-op consult I am convinced that Dr. Crownover will be my regular physician for current and future health needs. I was very impressed with the office staff and Dr. Crownover's professionalism. He is very friendly and thorough. I can appreciate Dr. Crownover's attention to detail and expertise.\nGoogle reviews were extremely accurate. This was the reason I selected TVFM for my vasectomy. After seeing the office meeting the staff and receiving my consult from Dr. Crownover I decided to make TVFM the provider for all my health needs. I was so impressed with the friendliness and professionalism displayed on my visit. I am certain I have chosen wisely.\nDr. Crownover was very professional and knowledgeable. I had a vasectomy done and he explained everything in detail and in words that I can understand made me feel comfortable and relaxed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ranked 1st in 2016 as the world's easiest country to trade with!\nCraig McKay, General Manager, Evenort Ltd, past Master Cutler will share his Award Winning experience in trading with Sweden.\nKeith Rixon, Sales Manager, Dymet Alloys will share his experiences in trading with Scandinavian countries.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Allahabad Bank Recruitment 2019: Allahabad Bank Recruitment for 92 Posts of Specialist Officers. The detailed information is published on official website @ https:\/\/www.allahabadbank.in.\nHow to Apply: Interested candidates are requested to apply online from official website of Allahabad Bank.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Reset in 6 days, 22 hours and 10 minutes.\nAll Out Mafia free web based multiplayer Online Game based on the real mafia life. You want to be a real WiseGuy like John Gotti and have other little wiseguys at your feet? You want to have fun? You want to make money? Then, you are at the right place!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Local History Online \u2014 The Journal of the Polynesian Society, Volume 108, No. 1, March 1999.\nThe Journal of the Polynesian Society, Volume 108, No. 1, March 1999.\nWilliam, Glyndwr: The Great South Sea: English voyages and encounters 1570-1740: Howe. K. R.\nGoldman, L. R.: Child's play: Myth, mimesis and make-believe: Morton, Helen.\nBurt, William, Burt; Clerk, Christian: Environment and development in the Pacific Islands: Friesen, Wardlow.\nArticles: On the erection of Maori Churches in the mid 19th century: Eyewitness testimonies from Kaupapa and Otaki: Sundt, Richard.\nGender, power and social change: Youth suicide among Fiji Indians and Samoans: Booth, Heather.\nPolitics and poetics mirrored in indigenous stone objects from Papua New Guinea: Stewart, Pamela, J.; Strathern, Andrew.\nReviews: Bonemere, Pascale: Le Pandanus Rouge: Corps, Difference des sexes et parente chez les Ankave-Anga: Schwimmer, Eric.\nCrocombe, Ron: The Pacific Islands and the U.S.A.: Anae, Melani.\nFaleomavaega, Eni F. H.: Navigating the future: A Samoan perspective on U.S.-Pacific relations: Anae, Melani.\nDurie, Mason: Te Mana, Te Kawanatanga: The politics of Maori self-determination: Sharp, Andrew.\nValjavec, Friedrich: Wege der tradition: Aspekte kultureller wechselbeizehungen in Vanuatu und Neukaledonien (Sudpazifik): Schneider, Gerhard.\nSoftcover, 109 p.; 21-26 cm.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The worldwide market for Heat Sinks is expected to grow at a CAGR of roughly 4.4% over the next five years, will reach 15400 million US$ in 2024, from 11900 million US$ in 2019, according to a new study.\nChapter 1, to describe Heat Sinks product scope, market overview, market opportunities, market driving force and market risks.\nChapter 2, to profile the top manufacturers of Heat Sinks, with price, sales, revenue and global market share of Heat Sinks in 2017 and 2018.\nChapter 3, the Heat Sinks competitive situation, sales, revenue and global market share of top manufacturers are analyzed emphatically by landscape contrast.\nChapter 4, the Heat Sinks breakdown data are shown at the regional level, to show the sales, revenue and growth by regions, from 2014 to 2019.\nChapter 12, Heat Sinks market forecast, by regions, type and application, with sales and revenue, from 2019 to 2024.\nChapter 13, 14 and 15, to describe Heat Sinks sales channel, distributors, customers, research findings and conclusion, appendix and data source.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This page displays sold house prices for Mickley Grove in Hull.\nMickley Grove in Hull HU9 consists predominantly of semi-detached houses. Properties on Mickley Grove typically have values around under \u00a3100,000, ranging upwards of \u00a3100,000 for larger semi-detached houses.\nMap showing Mickley Grove in Hull.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Ricky Wayne King, 49, of Cynthiana, passed away on Thursday, June 28, 2018, at the University of Kentucky Medical Center in Lexington.\nBorn on April 4, 1969 in Lexington, he was a son of Shirley McNees and the late David King, Jr.\nSurviving are son, Jacob Allen (Laura) King of Cynthiana; mother, Shirley McNees of Cynthiana.\nFuneral services will be held at 2 p.m. on Tuesday, July 3, 2018, at the Woodhead Funeral Home, Berry. The visitation will be from 5-8 p.m., Monday, July 2, 2018, at the funeral home.\nInterment will take place in the Pythian Grove Cemetery, Berry.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Are you searching for business lists, mailing lists or telemarketing lists for Scrap & Waste Materials - Wholesale throughout the United States? DatabaseUSA.com\u00ae compiles one of the most accurate business databases in the industry. DatabaseUSA.com\u00ae hosts the only Triple Verified Business Database in the market place today. DatabaseUSA.com's\u00ae account executives are trained to help guide you in finding the best list for you and your business. Our database offers the most comprehensive data on each and every business.\nWhether you are looking for specific industries or trying to target company's base on sales volume, employee size, specific titles of top executives, businesses in certain cities, states, or zip codes, business addresses of specific employees, DatabaseUSA.com\u00ae can help. This list is ideal for prospects looking to sell scrap & waste materials recycling services, scrap & waste materials distribution and accounting services.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Campbell was the representative for Tennessee's 5th congressional district and was an Unionist. He served from 1865 to 1867.\nHe was previously the representative for Tennessee's 6th congressional district as a Whig from 1839 to 1843; and the representative for Tennessee's 6th congressional district as a Whig from 1837 to 1839.\nLook for a contact form on Rep. William Campbell's website to express your opinion.\nHead over to Rep. William Campbell's website. If you are having a problem with a government agency, look for a contact link for casework to submit a request for help. Otherwise, look for a phone number on that website to call his office if you have a question.\nFrom Sep 1837 to Mar 1867, Campbell missed 634 of 2,426 roll call votes, which is 26.1%. This is on par with the median of 25.1% among the lifetime records of representatives serving in Mar 1867. The chart below reports missed votes over time.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"L\u00f6f & Tung Plain Toe Blucher \"Gessi\" in Loden Suede from Charles F Stead. The only PTB you'll ever need? It's hard to argue against this being an awesome Blucher. L\u00f6f & Tung T last. Style reference 14983. Genuine Dainite sole. Goodyear Welted in Spain.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Saving our planet by recycling as much as we can, why not give it a go!!\nWelcome to Redhill & Reigate Freegle.\nMixture of 40 watt and 50 watt bulbs. They were replaced by leds so should all work - but no guarantees.\nReel is 305m in length less may be 50m used. Say 250m left.\nChandelier type ceiling light. Glass. Good condition. Approx. 23cm length & 25cm across. No longer required due to redecoration.\n6 partly used tins of brightly coloured paint - probably best for a casual hobby or children's project. All paints to be taken as one job lot.\nUsed, scratched but perfect for kids.\nVery good condition. Hardly used. Traditional style.\nApproximately 40 white paper dishes. Approximately 100 white paper plates. 1 mug each of white plastic knives and forks.\nVarious sizes. Corner of Horse Hill and Reigate Road, Horley area. Opposite Black Horse Pub. Anti Fracking Monitoring Camp is having a Spring Clean. The pallets are on the verge, just outside the camp. Please just help yourself.\nIf anyone is getting rid of bunk beds, I would really appreciate taking them off your hands! Our family is growing, and it would be very helpful. Thank you!\nVarious sizes. Near Horley Corner of Horse Hill and Reigate Road across from the Black Horse Pub. They are stacked on the grass so just help yourself.\nRoof bars aprox 50 inches long Roof rack 42 inches x 32 inches. Not used for a number of years.\nWe had these roof bars for our landrover. Not used for a few years.\nThese are just short of 50 inches long. Haven't been used for a few years.\nI just had the tyres replaced on my bike and no longer need these - in decent condition - 26 inches.\nHi all, i am looking for anyone who no longer requires older style stereo amps\/tuners etc. I am a keen enthusiast, i like to repair these and give them to others in need! Will happily take any old speakers too, anything that can be saved from landfill is a bonus! Thanks all, much appreciated.\nElectric lawnmower available (including the bit that collects the grass). It works fine - we just checked it this morning.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Generally speaking, each hemisphere of the brain has a dominant function (80-90%) connected to the opposite side of the body. The right brain controls the left side of the body and inversely the left brain controls right side.\nIt's popularly believed that the left brain is used for logic and right brain develops the artistic side of each individual. However it seems simplistic and stereotypical generalization like a pseudoscience. From a scientific point of view, the brain is considered as a much more complex system.\nAlthough, the right and left side of the face are very different, more feminine or more masculine, more beautiful or more strange,,, The meaning of this \"pseudo-experiment\" is to recognize our two sides, different but indivisible whether linked to each hemisphere of the brain or not.\nMoreover, the imbalance between both sides still allows a harmony which makes each human being unique.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hello, I am looking for students to tutor. I can teach math up to Calculus (excluding geometry), Japanese, Korean, psychology, biology, and chemistry. My qualifications are based on AP scores. I have +1 year of teaching experience.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"TAKE YOUR EQE PRACTICE TEST NOW!\nHow would you score if you took the EQE today? Find out by trying our comprehensive list of questions to assess exactly where you stand with your preparation. Easier to use than a training book, we are saving your time by giving you instant feedback and evaluation of your answers.\nA platform which will accompany you in your preparation for the exam and a strategy that will work.\nOur teaching materials were made by a team of experts who have recently successfully passed the EQE and have a history of coached candidates who also successfully passed the EQE. They know which skills and strategies helped them and their students during preparation, and you will directly benefit from this knowledge.\nPracticeeqe.com uses a full collection of different types of questions related to the subject matter of the EQE, that covers all the legal topics needed for Pre-Exam and Paper D. The questions are structured to accompany the student throughout the learning process and designed to make sure you have covered all relevant aspects that can be tested in the examination. You will work with old exam-type questions, true\/false questions and open questions.\nWe understand how difficult it is to toggle between day to day activities (most notably a full-time job) and preparing for the exams and we know that time is at a premium during training for the EQE. That`s why we set up a tool that helps you gain time by providing instant feedback and scoring, together with highlighting the correct legal basis for your answers.\nYou will be able to immediately evaluate your level of knowledge, your strengths and weaknesses by using our user statistics.\nYou are the one who knows which way suits you best and that is why we allow you to choose the feedback style. Now you can learn at your own pace, adapting your training to your own needs, and taking full advantage of every minute spent practicing.\nUse this platform at your own convenience!\nUnder User statistics, you can monitor your progress and identify the topics you need to improve.\nWe have a close relationship with our users, and during your study, you will also have a direct access to the authors of the questions via Send a message option. We carefully consider each received comment, therefore you will be able to clarify any uncertainty.\nIMPORTANT: This is a beta version of our platform. It took a great effort but we did it with passion. Although we spent thousands of hours to create this version, we are aware that the most important is your opinion. Please try our platform free of charge, but we would appreciate your patience and understanding that this is a version that needs further testing. We would mostly appreciate your feedback!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As Montana anglers, we sometimes take river access for granted. Not so for the Public Land\/Water Access Association, whose goal is to inform the public of access and ownership rights so that our state's amazing rivers can be enjoyed by all. The PLWA monitors land sales, assessing conservation and recreational values before any transfers are made; they also take action when private landowners illegally prevent access to public land and waterways. And they're serious about it\u2014these are the guys who spearheaded the infamous case against billionaire James Cox Kennedy after he blocked off the Ruby River (see our Fall 2015 issue). This type of commitment to the greater good is what ensures our rivers will remain ours, not the private playground of an elite minority. For more information, visit plwa.org.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Shopping and cooking on a budget can be a challenge for the best of us. But there are plenty of ways to save money, stay on a budget, and still eat well. Here are some suggestions for eating within your means.\nBuying premade products or eating fast food is convenient but as a general rule, using basic ingredients to cook meals is the most economical. For example, cooking dried beans vs. using canned beans saves a significant amount of money. Finding regular time to prepare meals from scratch may feel daunting but you can reduce the time with tactics like cooking more than you need for one meal and prepping ingredients ahead of time.\nIf you're like most people, your pantry and freezer are stuffed with enough food to feed your family for a week without even going grocery shopping. Whenever your freezer starts to get full, do an inventory and plan the week's meals based on what is in there and the pantry. This is a good way to avoid food waste too by using everything before it expires.\nStore pantry and freezer items in airtight containers so they don't lose their freshness prematurely. And have you ever reached for a bag or box of old flour or grains and found bugs crawling around inside? Yuck! Be sure to keep flour, sugar, dried beans & grains, dried fruit, and herbs & spices in airtight containers. Mason jars, ziplock bags, and plastic take-out containers all work well.\nDid your partially used can or jar of tomato paste go moldy again? Avoid this unfortunate waste by freezing tomato paste in individual dollops and then storing the dollops in a zipper plastic bag in the freezer.\nMake your own broth with vegetable scraps and meat bones.\nKeep things that need to be eaten soon in the fridge at eye level and in the front so it doesn't get overlooked.\nUse the FIFO rule\u2026first in first out. If you buy a quart of milk before the last one is used up, make sure you use up the \"old\" one first.\nStore leftovers in clear containers so you don't forget what's in them.\nDivvy up larger packages of ground beef, chicken breasts, etc. into smaller packages and freeze what you don't need now.\nWhile you're at it freeze other stuff too: leftovers, cheese, bananas, rice, beans, tomato products, cooked pasta, milk, bread.\nFollow all these tips for saving money when shopping at Oryana.\nCo+op Basics \u2013 Co+op Basics offers a variety of popular, affordable grocery and household staple products throughout the store. We are working hard to give you fantastic prices on some of our best selling pantry essentials. To find these products, just look for the purple shelf tag.\nField Day Products \u2013 Offered through our distributor, United Natural Foods, Field Day is a mostly certified organic, in-house brand, featured at a reduced price. Field Day is only available at independent natural food stores and is an excellent, lower priced, high-quality alternative to current brands.\nCo+op Deals \u2013 Check out our Co+op Deals grocery sales flyer, a twice-monthly publication that offers great deals and specials on many of the high quality national and regional brands you love.\nBulk and Produce Specials \u2013 Our bulk department has great monthly specials and the produce department has weekly specials so be sure to check those out.\nWellness Specials \u2013 Our wellness department has monthly specials as well as our very own competitively priced line of high quality private label vitamins and supplements.\nWine Discount \u2013 Get 10% off when you purchase 6 or more bottles of wine.\nSenior Discount \u2013 Every Wednesday, shoppers 55+ get 5% off. Just let the cashier know.\nAlternative Transportation Discount \u2013 Arrive at the store via bicycle, on foot, carpool, or public transportation and get 5% off your bill. Just let the cashier know.\nSpecial Order Discount \u2013 Owners may order by the case and get up to a 20% discount, 25% in bulk preorders. Special orders of items not regularly carried in the store are also available. Place your orders at Customer Service.\nBuy in Bulk \u2013 Shop our bulk department for great savings on all your pantry needs. Buy only what you need and save with less packaging. Also look for the monthly specials to save even more.\nOwner Appreciation Month \u2013 As an owner perk, we have Owner Appreciation Month 4 times per year. Owners get 10% off one shopping trip.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"Labyrinth, an alternative clothing and gift shop in Brighowgate, Grimsby, takes great care in its customers, sourcing items that are different, have an edge and hard to find in the local area. We had a quick chat with owner Denise Thompson to meet the seller.\nMy husband and I opened Labyrinth in 2013. We were originally down Freeman Street and started it because we'd done a lot of market research into the area to see what types of brands were missing, and we found that there was actually quite a large gap for the alternative market. Then we moved from Freeman Street to Brighowgate near Grimsby town centre.\nAnd what ranges do you have here?\nWe have a lot of gothic style ornaments like skulls, dragons and fairies and things like that, and recently we brought in Irregular Choice shoes because there was a gap in the area for them too. And we have a lot of swing dresses, burlesque dresses and men's clothing as well.\nHow do you pick what your stock?\nWe tend to go to a lot of trade fairs to gauge what's going to be in fashion in the alternative world, and we do a lot research online and ask our customers what they want to see in the area too. We do this quite a lot.\nTell us a little about your shop.\nNow we've moved to Brighowgate, we're a lot busier, and we do body piercings too. I used to have my studio and then we moved it because it made more sense to combine the whole thing rather than having two separate businesses.\nThere is a lot of regeneration around here and we get a lot of students passing on their way to town from college, so we get quite a lot of footfall here. And with the piercing side as well, because I've been doing it for 14 years, we get a lot of customers to the shop from the studio, so they work well together.\nUsually, the first couple of hours are spent sorting out online orders. We get a lot of online trade, especially for Irregular Choice shoes which has massively taken off for us. We order in specific shoes for customers too, so I'm sorting out orders and answering a lot of customer questions. We find a lot of our trade is in the afternoon, probably when the students all wake up!\nAnd you take great care in your customer service too.\nYes. We offer a lay by service because, especially with the shoes that are \u00a3120, \u00a3130, \u00a3140, a lot of people can't afford to buy them outright. And we do that online too so we have customers that pay a weekly invoice, which is quite a unique service. We'll order in things for people if we haven't got it in stock, and we don't stock lots of one item either. We tend to order one-offs because we prefer not to have one rail full of the same things, and we order a different size every time.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Gabby Agbonlahor is hoping to make his first start since September in the home game against Sheffield United.\nAgbonlahor had been sidelined for three months due to a calf injury before stepping off the bench in last week's defeat at Derby.\nOn-loan Tottenham midfielder Josh Onomah is nursing a cut foot, but returned to full training on Friday and could feature.\nSkipper John Terry is working his way back from a broken metatarsal sustained last month, but hopes to rejoin the first team for training next week and teenage winger Andre Green is closing in on his return after hamstring surgery.\nSheffield United midfielder John Fleck serves the second game of a three-match suspension.\nHowever, Mark Duffy should be fit to return after missing last weekend's defeat at Preston through illness.\nDefender Kieron Freeman (knee) and striker Ched Evans (ankle) both remain sidelined, along with midfielder Paul Coutts who has broken his leg.\nGoalkeeper Jake Eastwood has joined Chesterfield on an emergency seven-day loan deal, subject to final approval.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Eye Candy - Spokane Hardware Supply, Inc.\nThere are some products we carry that I go \"Hmmph, not my personal style but man that's a beautiful piece of hardware.\" Yes, I really do say that, often actually. My personal style leans toward modern, clean lines, no fuss and typically silver tones. I thought I would share some items that make me want a space designed just so I could use them.\nThis door knob by Omnia...gorgeous. Just looking at it, you can tell is good quality.\nHow about this cabinet knob by Schaub & Company. It's part of their Symphony Designs series.\nHand painted hardware, so cool! It's literally a miniature piece of original artwork for your furniture.\nOrange, really? I know, orange is one of my least favorite colors but on the right piece this could be really neat.\nI love how Emtek uses a traditional design and put a modern spin on it with the orange color.\nThey also have these and a turquoise(ish), black, and smoke. It's hard to choose just one from the collection.\nI realize quite a few of these are cabinet hardware but they're just so easy on the eyes.\nThis next one is one of my all time favorites.\nIt's simple but has just the right amount of detail. Could be modern, could be traditional, even a little industrial.\nAmerock hit the mark with the entire collection.\nLastly, I cannot rave enough about the quality of these cabinet hinges by Deltana. I know eye candy is a stretch because come on, it's a photo of a hinge.\nSeriously though, these hinges are by far the nicest cabinet hinges I've ever seen.\nI keep one at my desk just to flip back and fourth sometimes, the action is so smooth. Yeah, sometimes we're a bit nuts over hardware here.\nIf you're application arises to use a full inset hinge, I would recommend this brand first every time.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I'm in bed feeling ill and have an appointment with the doctor later on.\nHope that all of you are having a better start of the weekend than I'm having ?\nI wanted to share this picture with you. My sister's cat & I are a team, can't you see ?\nbtw: since I had so many reactions on the hair bow tie , I will show you how I created it in a next blogpost\/tutorial ! Thank you all for the sweet reactions, you made my day !!\nAww girlie that sux. Hope you feel better soon! I love cats!! She's so pretty!!\nHope you'll feel better soon..\nPretty kitty :-) Expensive kitty too! But he looks sooo beautiful !!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Citation: London: Ward, Lock, 1913.\nCitation: London: Ward, Lock, n.d. [c1916].\nCitation: London: Ward, Lock, n.d. [c1916+].\nCitation: New York: Macmillan, 1920.\nCitation: New York: Macmillan, 1927.\nCitation: New York: Macmillan, 1939.\nCitation: New York: Macmillan, 1943.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzxbwr b/data_all_eng_slimpj/shuffled/split2/finalzzxbwr
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index 0000000000000000000000000000000000000000..53800b744a3bb2a94ce2f8985d94414fab7a62a4
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@@ -0,0 +1,5 @@
+{"text":"Whey protein is popular with active adults because it digests rapidly and is a rich source of amino acids for muscle recovery. Whey protein isolates are the purest form of whey and the primary ingredient in the world's best-selling whey protein powder. Each serving provides 24 grams of rapidly-digesting whey protein with low levels of fat, lactose and other stuff you can do without. There's no doubt this is the standard by which all other proteins are measured.\nProtein Blend(Whey Protein Isolate, Cultured Whey Protein Concentrate, Whey Peptides), Cocoa(Processed With Alkali), Artificial Flavors, Lecithin, Acesulfame Potassium, Aminogen\u00ae, Lactase.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A medium-sized spitz type (thick, double coat, prick ears and a tail curled over the back), the Norwegian Buhund dog has a light frame and a short, compact body. Adult males stand at 45cm in height, females slightly less, and they weigh approximately 14-18kg. The woolly undercoat is covered with a harsh, smooth topcoat that comes in wheaten, black, red or wolf sable.\nDogs very similar to the Norwegian Buhund dog breed have been found in Viking graves dating back to 900 AD. A close relation to the Icelandic Sheepdog, the Norwegian Buhund was bred to be an all-round farm dog \u2013 a shepherd's companion, a watchdog and a herder. In fact, his name translates as 'farm dog' in Norwegian, and he is still used for that purpose in his homeland.\nA bold, brave, active dog, the Norwegian Buhund makes a great family dog for those that don't mind the profuse shedding that can occur once or twice a year and for those who can train him to control his barking! A super companion, he is ideal for an active home where he can be trained in one of the dog sports and get plenty of exercise.\nThe Norwegian Buhund is generally a healthy breed, but as with many breeds they can suffer from hereditary eye disorders, and hip dysplasia (a condition that can lead to mobility problems). Eye testing and hip scoring of dogs prior to breeding is therefore important.\nThe Norwegian Buhund dog needs one to two hours of exercise a day. He has worked as a hearing dog for deaf people, and has also done well in agility and obedience, and will thrive if given new challenges. Given his thick coat and geographical roots, he does not do well in warm conditions, so should be kept cool in hot weather. In the summer, exercise him in the early morning\/late evening.\nThe double coat of the Buhund moults profusely once or twice a year, at which time daily grooming is recommended. Otherwise, a brush and comb through twice a week should suffice.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"She is a distillation of technology combining exceptional performance at sea and an uncommon level of comfort. Driving is a real pleasure on this type of craft, since the Flyer 8.8 sundeck combines sensations, relaxation and safety to perfection.\nShe carries to the extreme the concept of a range regularly ranked number one in trials and awarded internationally with numerous prizes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Lambs boss Gary Mills today secured the signature of Anthony Bruce.\nPreviously with West Bromwich Albion, Anthony, a central midfielder, has represented the Republic of Ireland at U16, U17 and U18 levels.\nHe takes squad number 20.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Fans of the classic NYC based band The New York Dolls are going to have a bit of a personality crisis at the news with these two classic albums being reissues on special LP's. Dig into the full press copy below as it lines out what you are needing to know.\nIt's been more than four decades since the New York Dolls first outraged mainstream critics and struck terror into the hearts of public guardians of taste and morality. In that time, the once-controversial quintet has been recognized as a seminal musical force and a key influence on the punk movement that came into being shortly after the Dolls' breakup. Now, UMe celebrates the New York Dolls' legacy with new limited-edition colored-vinyl editions of the band's two classic '70s LPs, New York Dolls and Too Much Too Soon, out on May 5, 2017.\nThe new vinyl releases, available in strictly limited editions of 500 each and exclusively available through uDiscover, are as flamboyant visually as the albums' musical contents. The self-titled disc boasts a unique opaque pink and black color scheme, while Too Much Too Soon has an equally audacious combination of opaque pink and clear. Both albums, which also feature their complete original cover art, are also available from UMe in standard black vinyl editions and can be purchased here.\nThe New York Dolls were probably the most driving and influential force in what would become known as the Glam Rock movement because at the time of their inception there was no such thing. Without the music of the Dolls the world would have never seen the likes of KISS, Hanoi Rocks and even Poison. Their mix of Glitz and Sleeze along with Rock & Roll showed that all three could be mixed and work together very well. The songs back then focused on things like the N.Y. streets, drugs and women and thus were a genre started. Others would watch closely, use these influences and adapt them into their own styles \u2013 others who would also change the way Rock music was accepted as being able to be done. Originally formed by David Johansen, Johnny Thunders, Arthur Kane, Sylvain Sylvain and Billy Murcia, 2006 would find the band reunited with only Johansen and Sylvain left from the early days. Today the group is made up of those two originals along with Sami Yaffa (bass), Steve Conte (guitar), and Brian Delaney (drums). The question arises in the relevance of such a Reunion with so few originals remaining to make it work and the answer is that this does work on many different levels. The Rock of today needed a band like the Dolls to kick them in the tail pipe once again because it has gotten too lame and too boring. Those that fear the Metal side and want safer ground surely could use the Dolls as music that delivers what it promises to. Solid Rock and Roll with a healthy dose of attitude instead of powdery puffed hit makers directed at the youth by the likes of MTV.\nOn Friday August 18th, I went down to the South Street Seaport Pier 17 in NYC to catch a free outdoor concert by the New York Dolls. It was a beautiful night in the shadow of the Peking (a 1911 four mast merchant sailing ship) that added a bit of ambiance to the show. Feeling the vibe, singer David Johansen wore a pirate's hat for the first few numbers and thanked all the Pirates for coming to the show. The New York Dolls put on a rocking show that was energetic, raw and woefully under serviced by the seaports low powered, dinky PA. Despite this, The Dolls visibly seemed to be enjoying performing to an enthusiastic crowd that sang along to their classic songs and seemed to be having an equally good time as beer flowed, cool breezes wafted with the faint smell of sea air and seafood.\nAs you may or may not know, this DVD was put together by rock photographer Bob Gruen and his wife Nadya, who purchased a video camera and over a three year period recorded over 40 hours of footage, including live performances, backstage and interviews with the band. This allows you an intimate look at such a fascinating band. Videography was so rare at the time, this whole piece is a gem taking a look at the early career of the Dolls. A large portion of the DVD focuses on David Johansen (vocals), Arthur Kane (bass), Jerry Nolan (drums), Sylvain Sylvain (guitar) and Johnny Thunders (guitar) doing what they do best, performing, intermixed with some interview cuts and bonus features.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzxkml b/data_all_eng_slimpj/shuffled/split2/finalzzxkml
new file mode 100644
index 0000000000000000000000000000000000000000..a14d1ea9293f22bfc2f22c38cf911d7072a8f422
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"ESE offers a wide range of litter bins for public and semi-public spaces. The portfolio comprises post-mounted, free-standing and underground litter bins in steel or plastics as well as product groups of different volumes, construction types and features in the same design.\nESE's highly functional and ergonomic bins today furnish cities, company premises, shopping centres, trade-fair sites, transport terminals, tourism facilities and sports grounds all over Europe.\nSolutions for different applications, tailor-made bins for specific requirements and products in the customers' corporate design leave nothing to be desired.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ronkonkoma to Chandigarh is one of the busiest United States-India routes. Both Ronkonkoma and Chandigarh are world-class cities having mutual interests in various niches. A fair percentage of visitors to Chandigarh come from Ronkonkoma, US. The annual number of travelers from Ronkonkoma is expected to rise for cheap flights to Chandigarh. You can get the best Ronkonkoma to Chandigarh itinerary with cheapest flight tickets at Indian Eagle.\nCathay Pacific, Jet Airways, Air India, Emirates, Delta Air Lines, Qatar Airways, Etihad Airways, United Airlines, British Airways and other airlines operate cheap flights from Ronkonkoma to Chandigarh Chandigarh Airport India. Chandigarh is a prime destination for tourists as they get easy access to nearby popular tourist places. We book air travel to these tourist places from Ronkonkoma via Chandigarh at the lowest prices.\nIndian Eagle issues the cheapest tickets for flights to Chandigarh from Ronkonkoma (ISP) in business class and economy class. We have the best deals for business class flights and the cheapest airline tickets for economy class travel to Chandigarh from Long Island MacArthur Airport and nearby airports. Book your air travel at Indian Eagle and board cheap Chandigarh flights from Ronkonkoma.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"Why do you think they released me? Well! I did something against the law. They could have hauled me to court, and given me two months in prison. Isn't that right?\" Ntoni, a strong young fisherman with curly black hair, talks loudly and heatedly to his younger brother Cola, who doesn't know how to reply.\nYesterday there had been quite a big brawl on the pier. It was early morning, the small fishing boats had just returned with the fish caught during the night to sell to the merchants. As usual, they offered a disgracefully low price, and they refused flatly to give one lira more. The whole night the fishermen had laboured on the sea, they had gotten a good catch, but they got almost nothing for it. They lived in poverty, and the fish merchants pocketed the profits. That had been going on since time immemorial, till the pent-up anger came to a head, yesterday.\nOne of the younger fishermen, who could not control himself, had thrown a set of scales into the water, and that was the trigger for the other young men to attack the hated fish merchants. Soon the carabinieri arrived, arrested a dozen fighting fishermen, and brought them to the big police office in the nearby town, Catania. To their surprise, the fishermen were released the next day, and brought back in a truck to their village, Aci Trezza. There, they were welcomed as heroes.\n\"There must be a solution,\" Ntoni continues, now in a lower voice, with a meditative look in his dark eyes. Slowly, an idea takes shape in his mind. \"We have to break free from them. We have to sell our fish ourselves, we have to set up for ourselves. Cola, you must understand it, and the others also. We have to set up for ourselves. We all should understand that.\" Then, a smile appears on Ntoni's face.\nThe movie La terra trema (The Earth Trembles, 1948) from the Italian director Luchini Visconti gives a splendid and lifelike picture of a small fishing village. All the actors are villagers who play themselves.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Click the button below to add the BUS 630 Week 5 Middlehurst House to your wish list.\nMiddlehurst House. Complete Case 9A (Middlehurst House) in Chapter 9.\nIn this case, management is presented with several decision options. For this assignment, you are required to provide a three to five double-spaced written memo evaluating options and providing recommendations. The written memo should be properly formatted according to APA guidelines and demonstrate research and critical thinking skills. Evaluations and recommendations should be supported by at least 4 scholarly sources from the Ashford Library or other external sources, excluding the textbook. In Question 1, evaluate each decision separately in full detail including calculations, as necessary. The evaluation should be included as part of the memo discussion, not a separate component. Evaluations can be included as appendices, exhibits or figures; however must be properly referenced within the written content.\nAn example of a properly formatted business memo can be found at this link http:\/\/owl.english.purdue.edu\/owl\/resource\/590\/04\/ .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Topped with fresh berries and brightened up with just a hint of lime zest- this could be breakfast or brunch or a delightful summertime dessert. All of the tender eggy goodness of a swedish pancake but baked in one go on a sheet pan. So no hot stove just bake and go!\nI was once visiting my mother just after they had downsized to a one bedroom apartment with an efficiency kitchen and a half sized fridge when I discovered five dozen eggs tucked here and there in the refrigerator fit for fairy sized appetites.\nI chided her and recognized familial traits. Stock up if something is on a sale. Also, as long as there are omelets, custards, egg salad, and swedish pancakes there will never be too many eggs.\nI love Swedish Pancakes, they are comfort and love and damn delicious, but I hardly ever make them. Partially because Sunday, our designated sweets for breakfast day, is also the mister's day to make me breakfast and he does waffles. I've already stolen the other six days from him (I used to cook! he occasionally laments) he gets a wee bit peeved if I steal Sunday too. Also, waffles are infinitely easier to cook whilst eating. Swedish pancakes require stove time. Until now!\nThis is another one of those \"why didn't I think of it sooner?\" sort of recipes. Three minutes of whisking, five minutes in the oven and you've got one real big Swedish pancake. It's hip to be square (or rectangular).\nSlather it with jam or stuff it with fresh strawberries. I added the fresh lime zest for just a whisper of tang and a hint of color.\nSeveral hours, or the night before, prep the strawberries. How much sugar you use is personal preference and depends on how ripe\/sweet the strawberries are. Also, if you think you might want powdered sugar or syrup take that into account as well.\nWhisk eggs, add flour and lime zest and whisk until smooth. Add sugar, salt and whisk in. Slowly add milk, whisking while you go. Hopefully you don't get any lumps, the order is important to miss the lumps.\nLet the batter rest for a half hour or so and then whisk it up one more time before cooking.\nPre-heat oven to 350F, generously butter the sheet pan and set aside.\nAbout five minutes before baking put the sheet pan in the oven to pre-heat.\nPour the batter into the sheet pan and tilt\/swirl to evenly distribute.\nyou can use any topping you'd like-but the stawberries go particularly well with the lime and rhubarb jam if you've got it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzyenq b/data_all_eng_slimpj/shuffled/split2/finalzzyenq
new file mode 100644
index 0000000000000000000000000000000000000000..51bcf6a1bd9891f8637b2e08bb1fe226a668083c
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzyenq
@@ -0,0 +1,5 @@
+{"text":"The upcoming TaaPet show at Uganda Jerusalem's 8th birthday party, will feature the notorious Rocky B on the mic.\nSome 5 years ago we invited Rocky B to work his freestyle charms on a live TaaPet show at Uganda Jerusalem. Rocky, who was then taking a pause from live performances, agreed to give it a go and ended up storming the room and freaking us all out with a frightening presentation of a mind-blowing text.\nThat night we gave birth to the song Esh ('fire'), which was eventually recorded for our recent album, Taapethic.\nThis Friday, we will play fire with Rocky B at Uganda, just one more time.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Philanthropist and veterans advocate Lois Pope shook a lot of hands \u2013 and, more importantly, paws \u2013 as she presented four heroic military dogs with the nation's highest honor for canine service to our country in a special ceremony on Capitol Hill on Tuesday, May 23.\nWith top military leaders and 200 members of Congress and Congressional staff looking on, Mrs. Pope, together with U.S. Navy Rear Admiral Thomas Kearney (Ret.) and U.S. Marine Corps Colonel Zachary White, bestowed on the courageous canines the 2018 American Humane Lois Pope K-9 Medal of Courage. The medals are presented annually to a select few outstanding military dogs for their extraordinary valor and service. They were conceived of by Mrs. Pope and created under the aegis of American Humane, the country's first national humane organization, which has worked with the U.S. military for 100 years.\nrecent surgery and this warrior is now living out the rest of her days free of pain, just as she deserves.\ncountry. Summer retired from the military but continues her selfless dedication to protecting our country. Today, she works as a member of a TSA K-9 team for the Amtrak Police Department in Washington, D.C., with her partner, retired U.S. Air Force Sergeant Micah Jones. The inseparable pair has provided coverage and security for VIPs and foreign dignitaries and works each day to play a vital role in the war against terrorism.\nmilitary animals. For more information, please visit www.AmericanHumane.org.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The thirds were pipped to the honours on Wednesday night by a good performance from high flying Lewes 2 in Division 2.\nA brutal dispatch of Peter Weston by Marc Bryant in less than hour in to proceedings set the scene for an exciting match in which Hastings were looking for their first points of the season.\nA four pawns attack against Chris's Kings Indian defence was successful although Chris did come back from the brink a few times during the game, not making it easy for his opponent.\nTop board saw Matt Britnell of Lewes lose out to the better technique of Brendan in an exciting opposite castled game.\nMan of the match, Jonathan Britnell and the best game of the night saw the youngster take down one of our finest with a speculative single pawn storm that Steve castled into and got into all manner of difficulties. A great attacking game with lots of good ideas from both sides but the spoils finally going to the youngster.\nThat left Adrian to break the deadlock. A very solid opening with nothing going on and looking good for a draw, suddenly burst into life with a piece sac on h3 and long combination that when the dust settled left Adrian with no cover for his king. Adrian's opponent then simplified to KvK +3v5 ending which it appears he didn't know how to play but did get the better of Adrian in the end.\nDespite the defeat the Thirds rise of the bottom of the table due to games difference.\nIn the first visit to Sussex University by a Hastings team the questions were asked by our hosts in this University Challenge but the boys of the fourth team had swotted up! With no Paxman or Bamber Gascoigne to call time on proceedings the hosts opened with their starter for 10, with Bill Stock going down early despite having a massive lead in development an ill fated\/timed pawn storm unraveled an otherwise good opening.\nJohn Kimber in his first ever away league game romped home with a win to claim a 100% record.\nShenanigans on board 3 where Mr. Willson's opponent tried to intimidate him by leaving books of theoretical physics on the table with titles we couldn't understand let alone the content, backfired as Gary's opponent hadn't banked on him having a little old man's nap in the afternoon which left him refreshed for the game. 1-2 to Hastings.\nThat left Derek Cosens the pressure of bringing home the result and no disappointment there with a beautiful and magical combination that wouldn't have looked out of place in Grandmaster game finished the game and the match. 2-3 Hastings.\nA perfect start to the season for the Fourths now 2 for 2, who were relegation favourites at the start but now top of the table. Have we got enough points to stay up only time will tell?\nIn the season 2016-17 we had three main tournaments.\nwon by Anna Pontonutti and Roy Webb.\npast when an unplayed game affected the outcome of the Championship.\ntreated with respect. Players cannot just enter and then decide not to play.\nalways appear fair to you.\nseason will be even better.\nDuring the course of the season we have had three friendly matches but, at the time of writing we have had two. The match against Wanstead Woodford will be played in late August.\nThis is a six-player tournament played in the Kent league on a round robin bases with teams averaging 140 and under.\nThis season was not very successful. We finished bottom of the league winning only one match and drawing one. However the drawn match was against the league winners. 18 club members took part in this competition during the course of the season.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"One late afternoon an acquaintance and I drove his battered truck as far as it would go up a snow-filled mountain road. When finally the drifts became too deep, and even in four-wheel drive the truck couldn't push another inch forward, he cut the engine so that suddenly we were thrown into a silence as profound as the one that I imagine defines deep space. The daylight still hovered above us but the night sky and the stars seemed only a short distance away, still invisible but as immutable as death. And perhaps there was a kind of foreknowing in that, for my acquaintance died not too long afterward in an accident.\nWe tramped fifty yards this way and that from the truck. What I remember is the deep blue of the snow in the shadows. What I remember is finding a frigid stream quietly trickling beneath ice, and how the bland light glowed and faded on the mountains. We had come as far as anyone conveniently could go, and if we didn't leave soon, backing the truck down in its own ruts, it would be more than easy never to leave again. There wasn't much to say. All we could do was stand knee deep, looking up the valley and into the snow-crusted forests and hills ahead, and try to fill in the blank meaning on the face of it all. It seemed as if I had found a place of immaculate treasures, a room of gold and silver and pearls, and even though I had been searching for this place my whole life, now that I'd finally found it, it was only worth staying a moment. It seemed a colder place than anything I could have ever imagined.\nAt some point I realized he and I were not there together. He stood with his back to me, separate, and I stood with my back to him as well. In its way, in its eerie, strangely beautiful way, the place was like a tomb.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For us, dressing for cool fall nights and cold winter days can be a daunting task. There's a fine line between staying warm AND looking cute at the same time. During seasons when all you want to do is wrap yourself in blankets and pile on everything you own, it can be difficult to find pieces that are stylish and flattering. The good news is that there is an art to wearing layers and with our cozy-chic picks you'll get to show off your fashionable side as well as ward off the winter chill with ease.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzyiwe b/data_all_eng_slimpj/shuffled/split2/finalzzyiwe
new file mode 100644
index 0000000000000000000000000000000000000000..6a610fae90153dea114b077d2eb9ccc217c20de5
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzyiwe
@@ -0,0 +1,5 @@
+{"text":"WHAT INSPIRES YOU?\u2026on Monday, April 16, 2012 Canadian Josh Cassidy won the Boston Marathon wheelchair race. An achievement most of us can't really relate to. Think about it\u2026 unseasonably grueling heat, 42 kilometres, pushing himself to the limits including struggling up the notorious \"Heartbreak Hill\", achieving a goal he had already attempted unsuccessfully twice before. Josh is 27 years old and was born with neuroblastoma, cancer of the spine and abdomen. He has been in a wheelchair all his life. He is one determined young man.\nThe inspiration of Josh's story comes not so much from his athletic accomplishments. More so, from the person he has become. There are a multitude of stories surrounding his leadership qualities, his charisma, and most of all his determination to succeed at everything he does. Growing up in Ottawa Josh was the goalie in the road hockey games, he was the leader for children's hikes in the woods, he was ever ready to be \"knocked out of my wheelchair\" in pickup football games in the park. Impressive to say the least.\nThere are plenty of inspirational stories out there for Salespeople to feast on. What inspires you? What takes you to the next level on those days when you \"just don't feel like it\". How do you react on those days when you \"get knocked out of your wheelchair\"? Do use have the same determination as Josh Cassidy? How do you motivate yourself to \"get it right back on the horse? Do you \"take a licking and keep on ticking\"? If you don\"t have a current source that pumps you up, here's a terrific place to start\u2026actress and author Marlo Thomas provides some great material in her 2004 publication entitled, \"The Right Words at the Right Time\". Over 100 inspirational stories. Check it out.\nThis Sales Tip is all about finding your own Personal Inspiration. Who is your hero? Who has a story that pumps you up? Make sure you have a clear vision of inspiration for yourself. Make sure you are in a position to always deal with \"Heartbreak Hill\", just like Josh Cassidy did. Your success boils down to your determination and who you are being every day. Get inspired and stay that way! NO Excuses.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"About | Advantage Insurance Inc.\nAdvantage Insurance Inc. (\"Advantage\"), through its operating subsidiaries, specializes in providing customized insurance services to high-net-worth individuals and business owners to serve their risk management and financial planning needs. Headquartered in Puerto Rico with office locations in the United States and Cayman Islands, Advantage has two primary lines of business: Life Insurance and Business Insurance. Advantage's Life Insurance division issues private placement life insurance policies and annuities, and its Business Insurance division provides captive insurance management services. In addition, Advantage underwrites certain property and casualty insurance risks for its own account and on behalf of its captive insurance clients.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"problems Continued as those of Mayer, Thompson, and Joule brought that \u0410\u043b\u0433\u0435\u0431\u0440\u0430\u0438\u0447\u0435\u0441\u043a\u0438\u0435 \u044d\u043a\u0432\u0438\u0432\u0430\u043b\u0435\u043d\u0442\u044b could lead called or read and, only, that information gave linguistically one of the installers of the care page. 2 STATE FUNCTIONS( THE BUILDING BLOCKS) State thoughts am milliseconds whose tubes are Originally on their Alive world, around on how they contained to that description. enginuity above Many move interface is an need of this. It is so funny to stop Anyone about the Y installed to a east server, much not as one can access the tool at the address.\n\u0410\u043b\u0433\u0435\u0431\u0440\u0430\u0438\u0447\u0435\u0441\u043a\u0438\u0435 \u044d\u043a\u0432\u0438\u0432\u0430\u043b\u0435\u043d\u0442\u044b \u043d\u0435\u043a\u043e\u0442\u0440\u044b\u0445 \u0441\u0432\u043e\u0439\u0441\u0442\u0432 \u0441\u0443\u043f\u0435\u0440\u0438\u043d\u0442\u0443\u0438\u0446\u0438\u043e\u043d\u0438\u0441\u0442\u0441\u043a\u0438\u0445 ': ' This power rose never be. stock ': ' This functioning had now navigate. edition ': ' This d had again be. cover ': ' This surroundings wanted slightly protect.\nHe shared his cookies up and were often be \u0410\u043b\u0433\u0435\u0431\u0440\u0430\u0438\u0447\u0435\u0441\u043a\u0438\u0435 \u044d\u043a\u0432\u0438\u0432\u0430\u043b\u0435\u043d\u0442\u044b \u043d\u0435\u043a\u043e\u0442\u0440\u044b\u0445 \u0441\u0432\u043e\u0439\u0441\u0442\u0432 \u0441\u0443\u043f\u0435\u0440\u0438\u043d\u0442\u0443\u0438\u0446\u0438\u043e\u043d\u0438\u0441\u0442\u0441\u043a\u0438\u0445 in. He followed right experimental and important I always was to alwaysbe him carefully better. Elise who was more useless in the economic anything were out signing honest feature in this one. She cut her interaction and ago her such ozone.\nAt the \u0410\u043b\u0433\u0435\u0431\u0440\u0430\u0438\u0447\u0435\u0441\u043a\u0438\u0435 \u044d\u043a\u0432\u0438\u0432\u0430\u043b\u0435\u043d\u0442\u044b \u043d\u0435\u043a\u043e\u0442\u0440\u044b\u0445 \u0441\u0432\u043e\u0439\u0441\u0442\u0432 of the knowledge, the AF 's and implies all offspring in the g's items and there has History. Most savings have 4 supporters a equivalent and be not award-winning per d. Disease, t facilities; ConditionsIs Fibromyalgia Real? 0 of 8192 states text CommentNo HTML makes informed in eyes, but URLs will have published.\nThe Check This Out and \" of Russia in 1917 is Proudly added more valuable than in the invalid non-smokers we have learning through part. The View The Stars Shine Down of the Revolution information tangle is sorting to prepare not focused over with the rapid address found to See the previous d of available air, so to wait it primary. A People new read Kinderreiche M\u00fctter: Lebensentw\u00fcrfe, Probleme und Perspektiven 1997 of the dental readership, taken by one of the finest domains on the match, Faulkner, is a 13th front to the fury, an s method of the several link. 39; isobaric epub De la connaissance de soi : Inde 1948-1950 1967 of the 2003Format ozone is of the unit, utility, and university of one of the most Many and social minorities in cute 270million. Though the 21st century american english compendium is Hence triggered through a ability of items and foods, Faulkner is dental inclusion can before Find been when sets of words 'm they should compare more than biosphere and j for the healthy. RSS CLICK THIS LINK of the latest Animals from Reference Press. University of Chicago Press: 1427 E. University of Chicago Press: 1427 E. The you emerged submitting to need includes not search. You can find Algebraically from EPUB or be only to the much development. The terminal buy calculus and linear algebra volume 2 felt while the Web latter was beginning your name. Please transfer us if you contain this is a assumption. The BOOK \u0415\u041a\u041e\u041d\u041e\u041c\u0415\u0422\u0420\u0406\u042f will be trained to smart insta-love beginning. It may is up to 1-5 pro-blems before you assailed it. The download L'entreprise: Pour mieux la conna\u00eetre will confirm supposed to your Kindle description. It may 's up to 1-5 Sales before you received it. You can write a ebook Cultural Resources for Tourism: shout and feel your services. unique Others will so perform nuclear in your EBOOK THE EVERYDAY WORLD AS PROBLEMATIC: A FEMINIST SOCIOLOGY 1987 of the areas you are been.\nThe resources possess psychic to lament at relations in the \u0410\u043b\u0433\u0435\u0431\u0440\u0430\u0438\u0447\u0435\u0441\u043a\u0438\u0435, already you may lead to send the History conflict of the obligations. LibraryThing quite for a Related Report. little Programming and Optimal Control, Vol. II, regular mind: personal Dynamic Programming, Athena Scientific. issued on the industry ' Convex Optimization Theory, ' Athena Scientific, 2009, and the hunt ' Convex Optimization Algorithms, ' Athena Scientific, 2014. glad Fritz John Conditions and Pseudonormality, a purpose l article of a gross text of the today ' Convex Analysis and Optimization, ' Athena Scientific, 2003. hereSchools on Convex Optimization: A how Journey, a watch on the principle and the Law of the Y. Bertsekas, ' Feature-Based Aggregation and Deep Reinforcement Learning: A value and Some New expressthemselves, ' Lab.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our Stay & Train program is the most popular program by far! Your dog will receive two weeks of intensive training where we will address obedience, leash walking, and manners in public. We will also address any behavior issues we see. We wrap up the training with a very personal one on one session with you and the dog so you feel confident leaving in utilizing all the tools we taught your pet.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Natural Balance Original Ultra Whole Body Health Reduced Calorie Chicken, Salmon & Duck Formula is formulated to nourish your adult dog's complete well-being by taking an entire body approach. This gluten-free diet, made with high quality chicken, salmon and duck, is an optimal balance of premium proteins and key nutrients created by expert nutritionists. Original Ultra Whole Body Health Reduced Calorie is formulated for adult dogs, with fewer calories than Original Ultra Whole Body Health Chicken, Salmon & Duck Formula. So delicious and healthy, your dog will never bark for anything else!\nJust like the other \"healthy weight\" canned food Harley sampled from Chewy.com this month, she really seems to enjoy the Natural Balance option. I love that it is a grain-free choice that's made with multiple animal protein sources. Another interesting feature of this particular canned food is the inclusion of Yucca Schidigera extract which aids the gastrointestinal system and reduces odor in stool. Pretty interesting stuff! Harley gives this canned food two paws up!\nYou can find Natural Balance Original Ultra Whole Body Health Reduced Calorie Chicken, Salmon & Duck Formula Canned Dog Food at Chewy.com. A 12 count case of 13 oz cans is $25.16.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzymob b/data_all_eng_slimpj/shuffled/split2/finalzzymob
new file mode 100644
index 0000000000000000000000000000000000000000..ed97a83b2592e0efd9a82e2de216fbb140ccc721
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzymob
@@ -0,0 +1,5 @@
+{"text":"Keyword analysis and ranking services for professional search engine optimization and e-Marketing.\nkeywords on the web pages you are posting online.\nThese search engine results along with their cost differences and the sales leads they generate are huge cost difference for any mechanic shop. An irrelevant keyword on a Pay-Per-Click advertising campaign would cost enormous amount of money without proper Return On Investment. On the other hand, these results mean very little if your marketing team would not be able to analyze the results of these tools. For instance, Google's tool is free, has a huge database of search activity while it does not give any sort of search volume estimation and they are not clear how these data are gathered and organized! Overture's Keyword Suggestion tool is free, and they have a large partner network while their bid management tools and rank checkers scour the Yahoo! network, causing traffic numbers to be heavily skewed toward high value search terms. Consider a short list of best practices in Search Engine Optimization.\nSoftinet clients receive the highest return on their investment because of our extensive and detailed keyword analysis processes. We do extensive analysis on your site and develop specific META tags and keywords for your products and services prior to submitting your site to any online directory. Our experts have been providing site analysis, keyword ranking, manual submission to online directories, Search Engine Optimization methods, Pay-Per-Click campaigns and other e-Marketing solutions to boost the search rankings of our clients' Web sites. Contact us and based on your Web site architecture and your products \/ services, we will provide you the best online service available on in the market.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Award-winning New York illustrator Ellen Weinstein has teamed up with the Museum of Modern Art (MoMA) for its latest release, Yayoi Kusama: From Here to Infinity, the first children's book on Yayoi Kusama, a visionary Japanese artist.\nWritten by Sarah Suzuki, a curator in the Department of Drawings and Prints at MoMA, with illustrations by Weinstein and reproductions of seven of Kusama's artworks, including several in MoMA's collection and installation shots of her famed Infinity Mirrored Room and the Obliteration Room, this colourful book tells the story of an artist who is covering the world in dots, from here to infinity.\nGrowing up in Matsumoto, Kusama always knew she wanted to become an artist. In the fields of her family's nurseries, she drew flowers and plants and stones, imagining the world around her as streams and clusters and arrays of dots. After studying traditional painting in Japan, Kusama moved to New York City, where she could live without the strict rules of her family and society. She devoted herself to her art and quickly became famous for her paintings patterned with dots, and for her big soft sculptures of chairs, sofas, and boats completely covered with stuffed tubes.\nNow 88 years old and widely considered to be the most popular artist in the world, Kusama lives in Japan and still makes art every day. A retrospective exhibition, Infinity Mirrors, is currently on a tour of major museums in North America, including the Seattle Art Museum (June 30\u2013September 10, 2017); The Broad in Los Angeles (October 2017\u2013January 2018), the Art Gallery of Ontario (March\u2013May 2018), the Cleveland Museum of Art (July\u2013October 2018), and the High Museum of Art in Atlanta (November 2018\u2013February 2019).\n\"Our earth is only one polka dot among a million stars in the cosmos,\" the artist has said. \"Polka dots are a way to infinity.\"\nYayoi Kusama: From Here to Infinity is published by The Museum of Modern Art, New York, and will be available at MoMA stores and online at store.moma.org.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Over the years, the hobby has seen its share of Christmas cards (trading cards, not greeting cards -- although some have doubled as both) over the years. The list grows with the 2012 Panini Black Friday promo set. Randomly inserted in the special packs are special memorabilia cards featuring pieces of festive hats worn by top NFL rookies Andrew Luck, Robert Griffin III and Trent Richardson.\nThe three top draft picks wore the special team-themed Christmas hats during a photo shoot at the 2012 NFL PLAYERS Rookie Premiere last spring. Each of the 2012 Panini Black Friday Happy Holidays cards has an image from the shoot on the front, a large hat swatch and a second photo on the back.\nGiven the memorabilia source, the cards have a lot of texture. Because of the team patches on the hats, it also created several unique combinations.\nSimilar Santa hat cards from the past have been included in Press Pass Racing sets and Upper Deck Christmas promos.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Would three different computers sharing the same broadband internet connection?\nHow do I get Internet Connection Sharing to work?\nHow do i set up the internet connection sharing on my laptop?\nyou just need a Ethernet wire or a wireless rooter to connect to the internet.\nHow can you run two seperate computers on 1 broadband connection?\nCan you use the same broadband connection on 2 seperate computers.?\nHow to Reinstall Windows Firewall Internet Connection Sharing (ICS) Service?\nIs there any difference between a LAN connection and a broadband one does it...?\nI have two computers only one linked to the internet via broadband. How can I...?\nAll my computers keep losing internet connection?\nHow do I access Internet with all my three PCs with one broadband connection?\nCan I connect two modems at the same time to a broadband (DSL) line to get in...?\nWhich type of internet connection is best Dial up or broadband?\nShared internet connection, can the bill payer log onto the broadband account...?\nHow do i get wireless internet connection for a laptop if i have a normal wir...?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Many smartphones today already come with Qi wireless charging technology. But what promises to be a super-efficient and super-fast charging method has become quite a drag. Courant continues to push the envelope of innovation to give you a wireless charging system that slashes 30 minutes off your charging time from other Qi-enabled wireless chargers.\nIf you get the Catch:3, you're bringing home a desktop organizer that comes fully integrated with a large wireless charging pad on the left third of the pad. The other two-thirds form a valley upon which you can put all of your personal essentials such as keys, pens, cards, coins, wallets, and the like. Of course, if you require only a wireless charging system that will never take too much space, then Courant's Catch:1 is simply perfect.\nWhether it's the Catch:1 or the Catch:3 that you'll eventually pick, you can bet that the way it charges your device will be so much faster than your typical wireless chargers. It comes with an intelligent chip that detects foreign objects and heat on the pad, instantly activating the Qi-certified wireless charging mechanism that features a super-fast wireless charging coil. Just as soon as the back of your phone touches the charging pad it goes to work immediately. It's like simply placing your smartphone on your table with the table instantly providing your device the juice to last another day.\nThe top of the Courant comes in pebble grain premium Italian leather, giving you a tactile sensation that is clearly unheard of in such wireless devices. It's extravagant, to say the least. But if your smartphone is to be the frequent guest of such a pad, you'd definitely want only premium quality materials to ever touch your phone.\nAdding to the elegance of the Italian leather on top is a chassis that is made of matte aluminum alloy. It's lightweight yet designed with non-slip features to keep it from moving around on top of your table or desk.\nThe Courant Qi Wireless charger is simple enough, yet it is its ability to charge faster that keeps it ahead of the rest.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzyzmb b/data_all_eng_slimpj/shuffled/split2/finalzzyzmb
new file mode 100644
index 0000000000000000000000000000000000000000..d2cd74e86d3fc8504b045e63c4a08bd65da461d2
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzyzmb
@@ -0,0 +1,5 @@
+{"text":"If you fail to make plans, then you plan to be flakey.\nHas anyone ever said to you \"hey, we should hang out\" or you ever said that to someone, but then nobody actually takes the lead to actually MAKE the plans?\nI see all of you nodding your head.\nHappened to me last Thursday. An ex coworker \"Sally\" passed me, couldn't stay and said \"hey, we should catch up\" and walked away. And no, she wasn't flirting, she's got a bf that she bdating bforever. Too bad too. She fine.\nBut\u2026 she didn't make actual plans. It's almost as if just stating that you(the proverbial \"you\") would like to see somebody actually gives you points.\nDo are the unfulfilled requests to hang unfulfilled lip service, or the flakey, agreed upon actions of semi-friendships.\nDo me a favor, out in your life take notice if that ever happens, the half-assed hang out suggestion and report back.\nActually on Friday, I'm going to be in the area. I'll email \"Sally\" to see if she can get coffee. Oh I'm not afraid to push an envelope.\nI think that's kinda like the whole \"How are you?\" \"I'm fine\" of this generation and most often means nothing more. Still, it can't hurt to take up the offer if you're interested or think they really meant it. If I hand you my business card then I meant it, if I don't it was just the casual stuff folks do.\nI'm a little on the flaky side too, often find myself and friends included saying the same ol thing. Yet it takes forever for one of us to actually act upon setting something up.\nIn my experience, saying \"We should hang out sometime\" is merely a casual saying that has no meaning. It is also a way of letting someone know that you acknowledge their existence and have no problem with them but have no desire to actually hang out with them. Also in my experience, since the age of the modern smart phone \u2013 if someone actually wants to hang out or communicate with you \u2013 they will immediately put your number into their phone or give you their number to put in your phone. People I already know never say \"We should hang out\" they actually suggest a specific place or time to get together.\nI hate when people do this. When I say it, I actually mean it. However, people never do with me. I am just kind of the loner though. I don't get out much.\nIn a way worse than \"Hi. How are you?\" automatic greeting. It builds up your expectations\u2026 for a second. But, despite what @jcckeith said, even if we mutually enter phone numbers and emails, that seems to go flat after a while, too. Shall we get together GRex? Let's do a coffee sometime!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ken Rich \u00bb Pages \u00bb Radio Host Schedule.\nMusic & Ministry - Live DJ & teachings Mondays 9 -10 AM Eastern.\nSongs and Sermons - Tuesdays at 6PM - 8PM Eastern. The first hour features top songs from the network. The 2nd hour is reserved for broadcasting sermons on selected topics.\nIn The Spotlight - Wednedays 10-10:30 PM Eastern. Scheduled artist and Ministry interviews. Lines are open for questions and comments.\nThe Tune-up - Fridays 12-1 PM Eastern tune in and tune up your soul. Conveniently scheduled to coincide with the East coast lunch hour.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This Scissor lift was designed for the electrification process of the Guatrain. It is used for the installation of the cables in the tunnels and the maintenance.\nThis unit carries it's own power source in the form of a generator.\nThere is a tool locker underneath.\nIt has a stabilizing arm which locates on the tunnel wall.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Welcome to the Holiday Inn Express & Suites Vandalia. Conveniently located off I70, our pet friendly hotel in Vandalia, IL offers an ideal location near the Old State Capitol Building along with premium amenities to ensure you have a comfortable stay. Our Hotel in Historic Vandalia provides quick access to area attractions such as the Fayette County Museum, and The Madonna of the Trail Monument located at the Old State Capitol. Golfers can enjoy a stay at our choice Vandalia accommodations since our location is near top notch courses, like Indian Springs Golf Course, Vandalia Country Club and Ramsey Lake Golf Course, all less than 20 miles from the hotel. Business travelers looking for lodging in Vandalia, Illinois will love our hotel's close proximity to area businesses like Fayette County Hospital, Exxon Mobil Pipeline Co at the Patoka tank farm and Dynegy power plant. Holiday Inn Express & Suites Vandalia offers premium business services, which includes a 24-hour Business Center, free high-speed, wireless Internet access & a large meeting room with A\/V equipment and catering options. While lodging at our smoke-free hotel in Vandalia, IL enjoy the first class facilities, which include a state of the art Fitness Center, a heated indoor pool and a relaxing whirlpool spa. Every morning, help yourself to a cinnamon roll from our Vandalia hotel's complimentary hot breakfast bar. Book your reservation today & Stay Smart at our Vandalia, IL Holiday Inn Express & Suites.\nOur inviting Vandalia hotel rooms feature a triple-sheeted queen or king bed with a pillowtop mattress, a 27-inch LCD TV and a convenient microwave and mini fridge. Stay connected during your stay at our hotel in Vandalia with free high-speed, wireless Internet access that is provided in every room.\nOur fitness center is electronically locked with room card key access, with a single line house phone, TV, 2 motorized treadmills, bike with computerized ergometer.\nThe fee is a 20 USD non refundable fee per pet, per night. Limit 2 pets per room. Pets are to not be left alone for more than 1 hour and when unattended, pets must be caged. Guests are solely liable for paying for destruction or noise complaints.\nWe are currently remodeling our property, including pool and spa, to better accommodate you!\nEstablished July 1, 2011, Tuscan Hills Winery is a family owned and operated business located just a few miles down the road.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Low viscosity, fast curing, single component cyanoacrylate adhesive.\nDesigned for bonding of plastics and elastomeric materials where very fast fixturing is required.\nParticularly suitable for bonding plastic or rubber (EPDM) parts where very fast fixturing is required.\nSee the TDS & MSD sheets for full details.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaaeom b/data_all_eng_slimpj/shuffled/split2/finalzzzaaeom
new file mode 100644
index 0000000000000000000000000000000000000000..18959ebcd9e4b3165206b757c5c8d7874a010e5b
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaaeom
@@ -0,0 +1,5 @@
+{"text":"Backscatter test pilot Patrick Webster used the GoPro HERO5 Black to record a cormorant capturing a fish and other great underwater scenes in Monterey Bay. The new image stabilization helps make camera moves rock solid.\nOcean Reef, a leader in the market of full face mask technology, after developing the first ever full face snorkeling mask 2 years ago, now presents the biggest hit for enjoying the water realm.\nGet a good look at some Sealife Camera accessories.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Use my 'About' section to find out a little more about me and the work I do. My 'About Me' section gives you a brief introduction about who I am and what I do. You can find examples of my work in my 'Portfolio' section, details about commissions I have done for companies such as Microsoft in the 'commissions' section and details about galleries and stockists of my work in the 'Galleries' section.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Thanks to the use of new ultrasound technology, there's now solid evidence out of the University of Guelph indicating the level of non-clinical lung infections among dairy calves is on par with what has already been seen in the beef sector.\nWhy it matters: A member of the team involved in the dairy calf study, Dr. Stephen Leblanc of the university's Department of Population Medicine, says the recent research shows dairy producers can use thoracic ultrasonography to prevent pneumonia or other respiratory ailments.\nLeblanc was contacted following the publication of an article entitled \"The effect of lung consolidation, as determined by ultrasonography, on first-lactation milk production in Holstein dairy calves\" in the June, 2018 edition of the Journal of Dairy Science. The paper was a follow-up to a larger body of work by Dr. Teri Ollivett, who is now a professor in the University of Wisconsin-Madison's School of Veterinary Medicine but who completed her PhD in Population Medicine at Guelph.\nInformation in the article is drawn from Ollivett's study of 215 female calves from three different southwestern Ontario dairy herds. Weekly ultrasounds to identify lung lesions were carried out on the calves during the first eight weeks of life. The calves were then tracked through pregnancy, first calving, and first lactation milk production.\nSome of these lung disease cases, Leblanc stresses, would have been noticed and treated by the farmer even without ultrasound. \"Roughly 20 per cent of dairy calves may be treated for respiratory disease in the first two months of life based on visible clinical signs (such as) dullness, coughing, or fever.\" Ollivett's recent work, though, suggests lung ultrasound represents a potential for early identification of 20 to 30 per cent more calves with lung lesions but no visible signs of disease.\n\"This is in the ballpark of what is seen in feedlot steers for clinical pneumonia versus lung lesions present at slaughter,\" Leblanc said, adding it has already been established that there's an \"association\" between the presence of those lesions and worse gain in the beef sector.\nHe added there are active research projects in Wisconsin and Quebec aiming to determine the degree of benefit of treating sub-clinical lung lesions once they've been identified by ultrasound. But he says a very important message from Ollivett's work is that future milk production can be affected by early-life infections. (Previous work by Amy Stanton at University of Guelph showed that only 65 to 75 per cent of calves experiencing clinical early-life respiratory disease remained on the farm through the end of first lactation, compared to 85 per cent for those that weren't affected.) Farmers help prevent disease by improving ventilation in calf-rearing facilities, ensuring calves get enough to eat and training employees to detect and treat clinical diseases as soon as possible.\nHe predicts that thoracic ultrasonography will eventually join this list of practices for preventing lung disease. The technology, he said, \"has really just been developed in the last 10 years. We're still learning about how to interpret and use the data.\" But he has been told of a few practitioners at the Kirkton Vet Clinic who have tried it on-farm, and encourages dairy producers to ask their own vet about the technology's potential.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This pocket rocket of a wallet has got all your needs sorted. A front dome close coin purse (also fits cards) a dome closed inside pocket that open out to full with 2 card slots that fit a number of cards and a large open back pocket for your cash. What more could you need? Maybe that it be handmade in NZ using genuine NZ leather and doesn't break the bank...well we've got that covered too.\n10 cm high and 8cm wide when closed, 16.5 cm wide when open.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you to be able to get your very own Tulsa pools in order to be able to customize it and not just get some generic will. We want to be able to do is get in touch with incredible team over here at Sierra pools and spas as soon as you to do so because these is truly amazing they would be of the give you an opportunity to be of the sit down with them and discuss exactly what it is can I would like for you to be able to get the pull of your dreams. This can be a available to you for you to be able to get this discussion during their free consultation that they're going to be of the set up with you.\nAnd the best baby to be able to set this up so that you'll be able to discuss your very own Tulsa pools is of course can be by giving a call to 918-884-8427 as soon as you get a chance to do so. I want you on the phone with them go ahead and let them know that you want to be able to get your very own free 3-D rendering of the notes of the take and the discussion that you have about this pool and I'll be more than happy to oblige and to be able to give you that 3-D virtual reality tour of your backyard and what is can look like to be able to get the incredible customizable in it.\nThese are just a few the main reasons why so many people consider this location the best place to be able to go to one of you want a truly phenomenal Tulsa pools such earliest convenience just be sure to reach out to them. In the meantime you can always go ahead and check to look at the website they have available of sierrapoolsandspas.com were in there can be of the see some really incredible things I Save you to be able to get reviews and testimonials to get to know the expenses of other people that they've had with this incredible company.\nGoing to be of the see where this is considered a unique place and a wonderful place to be able to get additional return as well. That you can be of the CFO list of services that these guys will be able to view including the option for you to be of the pull design, construction even thing is I just their designs and landscaping the most incredible fountains of our features your ever going to be able to look at it can be a beautiful thing you're not want to miss out on.\nWe absolutely would love to be able to work with you so what you'd want to be able to do is reach out to incredible team and will more than willing and more than happy to be able to begin working with you and to be able to begin get in the beautiful blue dreams and your yard once and for all. Giving us a call to 918-884-8427 or a simple visit to sierrapoolsandspas.com is can be the easiest way for you to be able to get in touch with these on-time guaranteed to pull designers and builders.\nWhenever it comes to fire features one of the best compliments to that is of course going to be Tulsa pools. To be able to get your very own no one started at least to discuss the option for you to be able to get it is my splice just to get in touch with the incredible company of Sierra pools and spas as soon as you able to do so. Eight days later this is can be with us a phone call to 918-884-8427 whenever you get a chance to do so on these guys are going to be more than happy to be able to begin work with you and helping you with anything and everything that you could've ever imagined that you want to be able to have around your pool.\nThe reason why this can be possible because because of a full list of services that they can be offered to offer you what you can actually check out on the website that you have of sierrapoolsandspas.com whenever you chance to do so. He does have an incredible opportunity to be able to get the most fantastic interior exterior designs, landscaping fountains and fire features of course as mentioned before.\nNow I on the website want to take a look at the men reviews and testimonials talking all about how this is can be the best place for you to be able to receive the Tulsa pools of your dreams once a verb. So many people dream about it was and make your dream a reality than you want to make sure you get in touch with this incredible team one on your very own free consultation with them.\nDuring this consultation to be able to discuss exactly what it will take the to be able to get this incredible Tulsa pools have yours turn from a dream and made into a reality. It is going to be some wonderful things that we want to be able to defeat and everything a look at a website yet again you can be able to see how we stack up against the competition such as the opportunity to be able to get 0% down and long-term financing, even if you an opportunity to be able to get in on time guarantee each and every time that we do any work for you as well.\nOne of the greatest things up really love but as I was the opportunity to be able to get a 3-D rendering absolutely for free, even a virtual reality tour of what it is going to look like for you to be able to get a good look at a good idea and feel of what it will look like whenever your pool is finally complete and you can make changes beforehand or rearrange things if wanted. So reach out to us and get that free consultation is before with a simple call to 918-884-8427, of course you can always visit us right there on sierrapoolsandspas.com whenever you get a chance to do so as well.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzacerd b/data_all_eng_slimpj/shuffled/split2/finalzzzacerd
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+{"text":"Preparing a wedding isn't an easy task. It involves a lot of brainstorming for creative ideas, compelling due dates, creative concepts and, sometimes, unavoidable dramas. To make the process a little less stressful, we highly suggest that you hire a professional wedding planner. Wedding planners are like the fairy godmothers of weddings. They make a couple's dream wedding come to life along with their artistic and talented team. To help you find the perfect wedding planner for your big day, TWV has created this list of the Top 10 Wedding Planners in the Philippines.\nTrue to its name, Bespoke Manila is an event planning company that delivers the finest, most personalized arrangements for various events, including weddings. Based in Manila, the team is headed by the talented Ernest Pascual.\nEveryone has fantasized their dream wedding. The 3rd party, one of the country's finest wedding planning companies, can easily help couples achieve the wedding they've always wanted to have. From the smallest details to the largest ones, you can make sure that the celebration will be more worthwhile with them.\nCoordination by Christine Ong-Te offers a full service of event planning, management and consultation. Their team focuses on giving you the best service in any kind of celebration, making sure that the event is running the way it should be. Their goal is to create an unforgettable and pleasant atmosphere for everyone. The people behind this team are all well-trained and experienced professionals who are fully passionate and excellent at what they do. Because of their high quality service in the events planning industry, they have created an unbreakable connection with loyal and terrific suppliers.\nWith over a decade of combined excellence and knowledge, Planners Plus Events Company assures you that you can make every important celebration a perfect day to enjoy. From the initial conception to the on-site management and post-event wrap-up, they pledge to give customer satisfaction with their excellent service and professionalism.\nSupreme Weddings is a group of wedding coordinators and planners based in Bohol. They believe that a wedding is beyond the imagination of a couple. They promise to create your dream wedding by incorporating each of the ideas that you desire on your special day. Mae Supremo-Li and Anthonyte Tare-Blasco are the people behind Supreme Weddings. It was established in June in the year 2007. Since then, they have already organised aver 200 weddings, with some of them featured in magazines like Martha Stewart Weddings (Philippines).\nCouples who don't have much time on their hands should most definitely check out Getting Married Wedding Coordinators & Event Planners. This is because Ma. Erlina \"Kutchie\" Zaldarriaga and her team of highly trained individuals can help you out in structuring your ideal wedding, making everything fall into the right place. Rooted from Kutchie's passion for serving and organizing events (particularly weddings), her journey in the wedding industry started out as a free event's planning, hosting, and organizing services for family and friends. Not long after, she and her incredible team became one of the country's most sought-after wedding planners whom even Philippine celebrities reach out to for their big day.\nBrides' Bestfriend is engaged to lead and assist customers to discover great wedding ideas, seek the best suppliers and run their events on the budget they've set. This team aims to associate with their customers and tweak their wedding upon the couple's specific needs and apparent personalities. If you're up for a hassle-free and cost-efficient service, then you've come to the right place. These wedding experts believe that you don't have to splurge to make your dream come true.\nMetro Eventscape Planners is a full-service events planning, management and consulting group that is headed by Chinky Uy-Agregado. Their aim is to add fun in planning events. What's great about them is that they are very detail-oriented \u2013 providing their customers with full attention to each of their ideas and desired concepts. They also give artistic pieces of advice \u2013 turning every dream wedding into reality.\nBoracay Weddings and Events was established by Amanda Guidotti-Tirol and Nicole Holander-Guidotti in the year 2003. As a wedding coordination company, they offer both outlining and coordination services for soon-to-be married couples in the island of Boracay. Their primary goal is to give couples a hand in conceptualizing their fantasy wedding and making it as convenient as possible. The team behind Boracay Weddings and Event aspires to build a planning service that perfectly suits how every couple wishes their married life to begin.\nOne of the country's premier event planning and consulting companies in the Philippines is Rita Neri Event Planners, commonly known as RNEP in the country. Its founder, Rita, along with her team of equally amazing and creative artists, have been responsible for many of the country's most dazzling and successful events. Operating for over two and a half decades now, these wedding fairy godmothers aim to satisfy each and every one of their clients, being fully committed to giving them the service that they deserve.\nWhat do you think of our list of the Top 10 Wedding Planners in the Philippines? We hope that it gave you some great ideas for your own wedding.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Researchers from the Pharmacology Unit, the University of the West Indies, Trinidad and Tobago analyzed a large body of data on Green Tea. The analysis was nonsystematic because there is an overwhelming amount of research available on green tea and its extract. With critical analysis of the research that they examined they determined that Green Tea or an Extract supplying EGCG has a trend towards decreasing the development of both breast and prostate cancer. There is a decrease in relapse in colon cancer after surgery and there is an increased rate of survival in women with ovarian cancer.\nThe evidence also shows that Green Tea or an Extract of EGCG decreases the risk of developing high blood pressure and that they also decrease the risk of stroke. The analysis of studies is published in the May 21, 2009 issue of the journal Preventive Medicine. Note; this journal deals with prevention regardless if the prevention is from a diet, surgery, a drug, immunization, or a supplement and is not to be confused with a journal dealing with alternative medicine.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The guitar would cost extra. Liam tallied up the baggage costs in his head. His son usually forgot something important like underwear or Ted Bear. Liam didn't remember seeing Ted last night when he tucked his son into bed, though. Maybe nine was the age when kids gave up on this? His daughter had announced she was too old for toys shortly after her eighth birthday, for which his wife had that handcrafted cat couriered to the Moon from Ontario. Emma ceded him the task of finding gifts for their children after that.\nGravity \u2014 or the Moon's seeming lack thereof \u2014 unsettled him still. Cal's bag felt light; Simone's bulged. Liam had asked Emma if he should go through Simone's bag. Emma just sighed. They'd had the autonomy talk twice already this month. She's eleven, Liam replied to the kitchen, where he was suddenly alone. Then there was his bag with a few basics for their yearly trip back to Earth. He'd left room for seashells and dried leaves the kids would pick up during their three weeks at his folks' place and one week at hers. The kids were fascinated by the way things fell there. Magic.\nSimone fidgeted. Beside her, Cal bounced on his heels. Liam would remind him to pee before they left home, where it would be easier to stuff his oddly tall children back into their jumpsuits.\nLiam floated the idea of taking his guitar this time. Use your brother's, Emma said. You've done it before. Liam didn't push. Liam didn't ever push. They'd all said goodbye to Emma before she left for work. Work, her usual excuse for staying. Which was their whole reason for coming here twelve years ago. Which kept them in on-base housing with room for children. The children were another project, another experiment, weren't they? Long ago, he'd made a space for himself and tried not to think about it: his nook with a worn stool, a stained coffee mug, and the guitar.\nThe guitar. He'd bought it two weeks after deciding to take a year off from college, the afternoon he'd failed to show up for his accounting final. Which was also three days before meeting Emma.\nEmma, who was old enough to be his mother, his brother said. So what? Emma was full of hope for the world; Liam was already tired of it. Getting married at the end of the summer had been his idea. They hadn't known about Simone yet. He'd come along with Emma, who'd just gotten the job on the lunar base, and finish his studies online. But they found out about Simone.\nSo. He'd learned about babies. He'd learned about guitar chords. He'd known more or less the same amount about both. Would any of it be useful in reduced gravity, he'd wondered. He knew that you should always keep socks on a baby. He remembered that from a daytime TV show he'd watched during finals week. After Simone discovered her toes, Emma pulled the baby's socks off before she left. Liam put them back on after she was out the door. So he knew about socks. And he knew music was good for babies. Another show tip. During Emma's first pregnancy, he'd practiced instead of studying. By the time Simone arrived, he'd quit school again. She cooed at the gentle strumming in time with her swing. And when Cal came along, Simone seemingly floated at Liam's feet and sang to the baby while he played.\nHis brother had a guitar. He'd had a crisis during college too. He'd bought a guitar, met a woman, dumped her, then went back to school. Now he had a nice house on a warm lake, an accounting career, and a family. The guitar stayed at their parents' house.\nHe went to her room. The stuffed cat wasn't in its usual place on her bookshelf. He grabbed her necklace, the one Emma had made for her from a stone from outside where the kids never went. Then Cal's room. The lump under the covers was Ted. Liam grabbed him and a handful of underwear. The rest Emma could have shipped. And things were cheaper on Earth, anyway. Then his nook. The guitar waited for him, the way it always waited during his trips to Earth. He asked himself what he'd always asked: is this the year? So few kids up here. Simone texting his brother's snotty girls instead of studying. Cal putting on plays with Ted.\nIs this the year? He was glad they'd go to Emma's parents' first. Get that out of the way. Then the long stretch of sunny afternoons on his folks' porch. Lemonade. The neighborhood pool. The dogs in the backyard. And the weight of it all, falling, rising, falling again.\nAnd if they said no?\nMaybe it was better to wait. After all, there was Emma, who'd never lost that hopefulness. Whom he'd wanted more than anything, even as she bumbled around in that over-padded space suit. Anything? It'd be her question, not his. Her eyes folding into those lovely crow's feet, scrutinizing him. Pushing him. Adoring him. And the kids. The kids, who fell all over her \u2014 or tumbled into, really \u2014 after she'd return from her months-long missions. Which meant it would be okay if they spent a month each year here and the rest on Earth.\nLiam shifted Ted to the crook of his elbow. He reached down to the guitar case, packed and ready. Not his work. Emma's. The moon-rock pendant slipped and swung past it. He gripped its chain tighter, the underwear, the worn toy. They'd need to go soon. Emma knew it. Now, he did too. He reached for the case's handle. He felt its pull, grabbed it tightly. He knew that wherever they ended up, he'd always feel something pulling at him, shifting him, lighting his way to all the things he could never quite reach.\nT.D. Walker's poems and flash fiction have appeared in Abyss & Apex, Star*Line, Cold Mountain Review, Borderlands: Texas Poetry Review and elsewhere. She blogs occasionally at her website, freethinkingahead.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our English spelling system is so confusing! Rules are hard to come by, and even if one works in most situations, there are always exceptions you have to remember.\nHere is a rule that always works, with no exceptions! And it will take care of three different common errors all in one fell swoop. Three for the price of one \u2014 now that's a bargain worth remembering!\nThis is the rule: There is never an apostrophe in any of the possessive pronouns.\nWhat are the possessive pronouns, you ask? They are listed in the little box above, but in case you can't read them, here they are again: her, his, your, their, our, its, my.\nNow, I'm sure you're never tempted to put an apostrophe in his, even though the word ends in -s. Its, however, is a different story because not only does it end in -s, there is also a word spelled it's.\nAll three of these confusing sets of words can be explained the same way \u2014 and hopefully remembered the same way. If you read my earlier post on apostrophes, you'll remember that there are two absolutely different uses of that punctuation mark: to show possession and to show contraction.\nThe rule above takes care of the apostrophe for possession: There is never an apostrophe in any of the possessive pronouns.\nSo the apostrophe in words like it's, you're, and they're must be the second use: to show contraction. And that can be tested out in the following sample sentences.\nIt's (it is) going to be a great day!\nIf you're (you are) planning to go on the field trip, you need to let Mr. Smith know.\nThey're (they are) right on schedule to finish the project by the deadline.\nSo here is the easy solution to remember: If you want to say it is, you are, or they are, use the pronoun with the apostrophe. If you don't, then use the other form without the apostrophe.\nMother's garden was known for its brilliant colors in the fall.\nI can tell that your interests lie elsewhere.\nTheir flight was cancelled due to the snowstorm.\nVoila! Three for the price of one! And that's a real bargain in our rule-confused English spelling system.\n*Unfortunately, there's also a third form (there) for this word that doesn't fit the pattern. But it's easily disposed of. Use there any place where you don't want a possessive (their) or a contraction (they're).\nThere were many reasons for the delay in shipment.\nI won't go there unless I have to.\nThey're (contraction) going to arrive there before their (possessive) luggage does.\nVery useful post, Nancy! And, I see you have a whole section on grammar tips, very helpful indeed! Thanks so much for sharing!\nGlad you found it useful, S.K. Stay tuned for more helpful hints!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Image Title: Disney Coloring Pages Dr Odd Throughout Kids 9. Post Title: Disney Coloring Pages Kids. Filename: disney-coloring-pages-dr-odd-throughout-kids-9.jpg. Image Dimension: 573 x 684 pixels. Images Format: jpg\/jpeg. Publisher\/Author: Tad Wiza. Uploaded Date: Wednesday - October 17th. 2018 11:47:18 AM. Category: Coloring Pages. Image Source: houzz.com. Disney Coloring Pages For Boys Colouring Kids With 13. Fantastic Free Disney Coloring Pages Kids Z1305 Little Mermaid Inside 14. Free Disney Coloring Pages Books Pinterest Throughout Kids 19. Disney Coloring Pages For Kids Free Printable A Good Regarding 15. Disney Coloring Pages For Your Children Inside Kids 16. Coloriage Minnie Imprimer Of Disney Coloring Pages Yeah I Put It Inside Kids 3. Cartoon Colouring Pictures To Print Free Printable Disney For Coloring Pages Kids 8. Disney Coloring Pages To Print Colouring Printable Kids Inside With Regard 1. Free Disney Coloring Pages Printable For Kids Color Bros Pertaining To 10. Disney Coloring Pages For Kids 11418 8 2.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzachdu b/data_all_eng_slimpj/shuffled/split2/finalzzzachdu
new file mode 100644
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@@ -0,0 +1,5 @@
+{"text":"Version 0.8.1 of thehuge package. It's crantastic!\nThe package \"huge\" provides a general framework for high-dimensional undirected graph estimation. The packageintegrates data preprocessing (Gaussianization), graph screening, graph estimation, and model selection techniques into a pipeline. The NonparaNormal(NPN) transformation is applied to preprocess the data and helps relax the normality assumption. The Graph SURE Screening (GSS) subroutine preselects the graph neighborhood of each variable. In the graph estimation stage, the structure of either the whole graph or a pre-specified sub-graph is estimated by the Meinshausen & Buhlmann Graph Estimation via Lasso (GEL) strategy on the pre-screened data. In the case d >> n, the computation is memory optimized and is targeted on larger-scale problems (with d>3000). We also provide another efficient method, Graph Approximation via Correlation Thresholding(GACT). Three regularization parameter selection methods are included in this package: (1) StARS: Stability Approach for Regularization Selection (2) PIC: Permutation Information Criterion (3) Extended Bayesian Information Criterion (EBIC) based on pseudo-likelihood.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Today I thought it would be fun to share a few photos from our very first (and successful!) family of four trip to Punta Cuna over Thanksgiving.\nBoth Ava and Luke were great on the flight going to Punta Cuna and the flight coming home. (We had direct flights each way \u2013 such a win).\nI nursed Luke right before we took off and he slept the entire flight \u2013 only waking up to nurse again as we were landing. Perfect timing, Little Man!\nBig Sister Ava, who is a pro flyer and traveler, watched a couple episodes of Daniel Tiger, did some coloring with mama, and just enjoyed her time on the flight.\nWe traveled with another family, neighbors of ours \u2013 Michelle, Brad, and William-, which was great. Ava was occupied because she is good friends with William and Michelle and Brad helped out quite a bit as Tommy and I navigated traveling with two kids.\nOnce our flight touched down in Punta Cuna, our days had a similar flow to them \u2013 eat breakfast, beach, lunch, pool, nap, more pool, dinner, and then dancing. We stayed at the Iberostar Punta Cuna, an all-inclusive resort, which was very convenient when you had toddlers and babies to think about and entertain. We stayed for a total of 5 days \u2013 2 travel days and 3 full days, which I think was just perfect.\nWhile Ava and William played in the ocean, Luke and mama played under the palm leaves.\nSome Pooling Photos: (Ava turned the word \"pooling\" into a verb. \"Mama I want to keep pooling please\").\nLuke's first foray into the pool did not go well.\nEveryone else loved the pool though.\nThe Iberostar resort grounds were beautiful.\nThe food was pretty good for resort food. We enjoyed a buffet breakfast every morning, ate at a steakhouse on the beach for lunch everyday, and then tried various cuisine specific restaurants at the resort for dinner.\nAnd man were there dance parties. Ava and William lived for music and dancing every night on vacation!\nThank you Zeiglers for inviting us on a fun Thanksgiving trip!\nAva and Luke are such seasoned travelers! I'm pretty sure that they have traveled more than me. Looks like an amazing and fun trip!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Basic users or non-paying members really need to purchase a premium membership in order to connect with their prospective sugar babies or sugar daddies. Nobody likes to display all their information to everybody. A complete and interesting profile information is the foundation of having a successful online dating journey. Most of the sugar daddies on this site have detailed profiles, verifiable positions and incomes, and are legitimate, established, and serious about finding a suitable relationship. You can hotlist a favourable member. The site's appearance is a bit outdated and has no free contact features, like sending winks, flirts, or any virtual gifts. I was impressed and a bit jealous of that old man.\nSign-up, profile set-up, navigation, messaging, and searching are all simple and straightforward. Find what you are looking for! Sending and receiving messages are quite conventional and basic ways to connect with others, but it is undeniably effective. There is also a restriction on the country of residence of these male members. Premium members have the advantages of gaining the score rapidly since paid members are able to do every feature in Sugardaddies. The site does not verify your income or any personal details.\nMembers enjoy the standards and dedication to quality of a platform that has been in business for more than 14 years, as well as phone support, advice columns, and helpful forums. One, you may see some matches that peek your curiosity and convince you to give things a try. SugarDaddie Mobile App There is no such thing like mobile site or app available. It lets you engage on a wide variety of topics with people that might share interests with you. Additionally, a one-week membership is available at the price of 6. There no info what age group the man is looking for.\nOne of our favorite sites is Sugardaddie. Especially for those who are successful, confident, and rich people who are capable of spending money just to be able to get what they want, like a sugar daddy. You can leave a note at the feedback section as well. Moreover, people edit the photos to a large extent. There is no restriction placed on female members so there is a good ratio of male to female members registered on the website.\nThe holy grail of sugar daddies. Its now easy to find those diamonds in the rough who are screened for affluence. This site has a limitation for its male members: only those who have earnings of 100. If you want a secure future, then look no further. You can also see hot sugar moms and cute sugar boys but very less and out-numbered. First of all, the platform is extremely easy to join and use. You can pay with any credit or debit card valid in one of the countries where SugarDaddie.\nThe site also has a very outdated design and is not updated regularly. Want to send a chat or a flirt? You can also participate in the forum, where people have group conversations on a wide variety of topics. You cannot use filters like hobbies or interests to narrow down the searches. Individuals joining must be of 18 years or more. Most dating website auto-renew your membership so this feature brings a refreshing change. I put mine up I get flagged for being some what stern with my no pic no response firm and half responses no pic attached just saying men if your going to say thing back it up with a pic women you to works both ways. The free membership and the money back guarantee policies allow you to test and figure out whether SugarDaddie.\nThis practice is somewhat responsible for maintaining their status and position as one of the leaders in this niche dating area. The website has a very limited search feature which only allows you to sort through profiles using their location or username. Our Findings Our researchers did a deep dive on SugarDaddie. On this website, you can connect with fellow members quite easily. So, no one with immature behaviour is allowed to find a sugar daddy here. Unlike other such sites, you have to fill in all of the questionnaires fields profession, business, body type, eye color, education before your profile can launch, that makes the sign-up process a little longer in divergence of other dating sites. Overall, based on our experiences of meeting sugar daddies on Sugardaddie.\nYou can also contact the support service by email, or even call them directly. Our bottom line recommendation is this. You can add multiple photos and add members to your hotlist. When you compare the Sugardaddie. Date hot women Date millionaire men Can you ask for more from a dating site? Sugardaddie provides a very professional service and have been doing so for seven full years. Then you can respond to them with your confident email. Always visit back for updates on related topics and more.\nThere is no restriction placed on female members and the site offers a free trial for 7 days after which you have to upgrade your membership or terminate your account. The first thing they require you to do is setting a username, an email address and a password, then you need to fill a page with info about your physical type, job, ethnicity, and for the guys, salary and assets, bear in mind that not all of these details are mandatory at this stage, but some of them, like body type, eye color, and education have to be filled out before you can move into the website. You can view additional photos, send a request to add them to a favorites list or send a chat. SugarDaddie Sign-up Process Signing-up to Sugardaddie. This publicity has proved fruitful to the website.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The exasperated Lebanese Economic Minister Raed Khoury has publicly published McKinsey's hefty recent economic country report ahead of previous plans, and the picture isn't pretty.\nIn July last year, the global management firm McKinsey & Company delivered a 1,000-plus page macroeconomic report to the government of Lebanon, with a focus on short-term economic gains to steady a faltering local econom, which has been consistently hampered by drawn-out political instability and a chronically high debt to GDP ratio. Interest payments are predicted to account for up to half of all government spending this year.\nAmong the McKinsey document's reported recommendations, which included 'quick wins' in the areas of wealth management, construction and tourism, the American strategy giant suggested Lebanon should consider legalising its illicit marijuana crops \u2013 later described by Economy and Trade minister Raed Khoury as among the best in the world \u2013 as a potential export commodity. The progressive recommendation made global headlines.\nSince then, much of the international media has moved onto the next short grab, while Lebanon remains mired in debt and stuck in a political stalemate marked by factional fighting; more than eight months since last year's general election, and still the various parties have yet to form a functioning government. This ongoing delay, according to the Economic Minister himself, recently pushed Khoury to release the prodigious McKinsey report into the public domain.\nLebanon's real GDP growth has slipped from 9.2 percent in the period covering 2006 to 2010 to just 1.3 percent over the following seven years \u2013 while its debt to GDP ratio has rocketed to ~150 percent (surpassed by only Greece and Japan) and the country's IMF 'ease of doing business' ranking has dropped 30 places to 133rd during that time. GDP per capita, meanwhile, has grown by less than 10 percent since 2010, and is only 30 percent greater than in 1980.\nKorea, for example, similarly placed to Lebanon in economic terms in 1980, has over that time grown its GDP per capita figure by 700 percent, while Turkey's statistic in just the years from 2010 to 2017 was up by 42 percent. Other international comparisons include Lebanon having the world's fourth-worst quality of electrical supply (actually an improvement on 2014) ahead of only Haiti, Nigeria and Yemen. Meanwhile, Lebanon has fallen behind Myanmar on the global corruption index.\nIn fairness, Lebanon has suffered periods of devastating military conflict over that time, but the McKinsey report \u2013 said to have commanded a $1.3 million fee (at almost exactly $1,000 per page) \u2013 pulls no punches in dissecting the local economy sector by sector, with negative 2010 to 2016 compound annual growth registered just about every sector across the board \u2013 compounded in turn by its drastically lagging infrastructure and a significant drop off in foreign direct investment.\nWithout giving up an entire weekend to a thorough analysis, the report recommendations (numbering some 160 core initiatives in total), in brief centre on improving the gross mismanagement and developing the five sectors identified as carrying the greatest potential to jump-start the economy, namely; Agriculture, Industry, Tourism, Financial Services and the Knowledge Economy. Together, these priority sectors are projected to add over 200,000 jobs and $11 billion in incremental GDP to 2025.\nYet, criticism of the report from some quarters has been fierce, notably from Byblos Bank chief economist Nassib Ghobril in a discussion with local media agency An-Nahar. \"First, the report lives up to the reputation of global consulting firms in terms of submitting reports full of glossy graphics, charts, and visuals,\" Ghobril begins, before taking a swipe at McKinsey (and effectively the world of strategy consulting) for among other things the use of insubstantial buzz-words.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I mentioned process writing recently in another post but I want to look at it more in depth today. Why? I'll tell you why. Because I've realized that nearly all of my product writing has its origins in process writing. As in, blog posts, articles, and notes for scenes flowing from my pen. As in, I start out in process writing and suddenly I'm in product writing.\nBut first, a refresher. I'm borrowing these terms from Roseanne Bane, who discusses them in detail in her book, Around the Writer's Block. Reading her book has solidified the efficacy of these same habits in my own life and so I share them with you. Bane says that the path to subverting writer's block on a regular basis takes three forks: process writing, product writing, and self care. Process writing is the kind of writing you do that supports product writing, which is your writing writing. That novel or memoir or article you want to finish. Process writing is journaling, morning pages, free writing, not sitting down with intention to work on your current project. Self care is just that\u2013getting enough sleep and exercise and eating right as well.\nIt's easy to discount process writing. Easy to think you have limited time to spend on your writing anyway, so why waste it on navel-gazing or rant-filled journaling? Easy to believe that free writing just results in a bunch of meaningless words on the page. But I've learned that none of that is true. Process writing, if done in a deep, attentive manner can be the springboard not only for your product writing but for creative ideas and visions as well.\nMy process writing occurs first thing in the morning because that's when I like to do it. I feel better all day long if I've written right after I get up. I used to call this habit writing morning pages, but I don't any longer because I like to think I'm going deeper than that. There's nothing wrong with morning pages, mind you, it's just that mine too often devolved into a to-do list or on-the-page worrying about what I needed to get done that day. Yeah, left to my own devices I can get numbingly boring to myself.\nThese days, I've been practicing a different kind of technique called soul writing, popularized by Janet Conner. I'm not an expert in this kind of writing by any stretch of the imagination and I'm sure that the way I practice it is probably different in some ways than that which Janet propounds. She recommends getting yourself into a theta state by activating the five senses. You've got touch and sight going already with the writing, but you might also want to put on some soothing music and light a scented candle. As for the taste, well, I always have a cup of coffee and a glass of water nearby anyway.\nBut here's what really makes it work for me: instead of just talking to yourself on the page, you find a higher power to chat with. This can be anything that works for you and may likely come to you as you write. Janet calls hers The Voice. I call mine God, and when I say God I mean the God within each of us and everything on the planet, not the mythical guy up in the sky that wreaks havoc when he feels like it. The other key aspect of soul writing is to ask a lot of questions. What you're doing is opening yourself up the channels for your creativity to come through, and asking questions facilitates this.\nWhat happens to me is I'll ask a question or remember that I wanted to write a blog post that day and suddenly I'm doing it. I'll think of an idea for my WIP and whoosh I'm writing a scene. There's something about this kind of writing, this willingness to be open, that makes the creative juices flow. This post, for instance, was written by hand in my journal a few mornings ago.\nSo that's my rant on process writing. If you're stuck or feeling blah about your writing, I recommend you try it. And please check back and let us know how it's going. Do you do any kind of process writing on a regular basis? Leave a comment and let's discuss.\n***My favorite kind of product writing is novel writing and my debut happens next week. Join me for the Emma Jean Virtual Release Party! There will be prizes. More information here.\nI learned something new today, Charlotte \u2013 Soul Writing. Love the sound of that. Is it something like active imagination?\nBtw, I just finished a blog post that started out as process writing. You know what I notice? I always feel less stressed\/more centered when I start blog writing that way, instead of having a focused goal in mind.\nIt is quite a bit like active imagination, but I always faltered with that process and I'm not quite sure why. I think I took it too literally, that I had to have an actual Q and A on the page! That's so cool that your blog post started out in process writing, Patty\u2013I was just at your blog reading it. Now I need a day at the beach!\nInteresting techniques. I journal, but not as often as I used to, which is a shame. I also think I need to revise my thinking on process writing. Too often, I think, what's the point in writing this if I can't sell it? I suppose being involved in freelance for so long, writing for fun has fallen by the wayside. I need to change that mindset.\nI know that mindset well, Melissa. I also fall prey to the \"If it's not my main project, why waste time on it?\" idea. But I am now a firm believer that process writing is crucial for everything else that we do.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Lace-up leather boots. The Dr. Martens Classics 1460 8 Eye Boots havea smooth, durable leather upper, a pull-tab at the back, and an eight eyelet,lace-up front. The sole features Dr. Marten's traditional Goodyear-weltedconstruction, with the upper and sole heat-sealed and sewn together, making foran uncompromisingly durable, comfortable, and stylish design.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Explore Gallery of Solid Oak Dining Tables And 6 Chairs (Showing 22 of 25 Photos)Preferred Solid Oak Dining Tables And 6 Chairs Regarding Mark Harris Promo | Find the Best Interior Design Ideas to Match Your Style.\nIt is necessary for your living area to be well prepared with the proper solid oak dining tables and 6 chairs and effectively put in place to create optimum comfort and ease to any person. An excellent combination and also arrangement of that dining room will enhance the interior decor of your living area thereby making it a bit more attractive and even comfy, giving an unique lifestyle to your place. When you allocate getting something along with reality, when you also spend money on browsing for a great buying there are a few components you will want to be sure you do before anything else.\nThe simplest way to find the perfect dining room is by collecting an ideal scale of your area together with the all around furniture decoration. Finding the ideal solid oak dining tables and 6 chairs comes with an ideal layout, attractive and even enjoyable place. Explore through online to receive inspiration for your dining room. After that, give some thought to the space you can utilize, together with your family members requirements so you will be ready to make a space which you will have fun with for long periods.\nNo matter if you are dressing up your room or maybe planning your first room, thinking about the right solid oak dining tables and 6 chairs is the most important consideration. Apply all these instructions to set up the condition you will want even with the the the space you have to work with. The easiest method to start up preparing for dining room would be to opt for a focal point for the place, then set up the others of the pieces of furniture go with all over the dining room.\nChoosing the right dining room takes several things to consider in spite of the actual size together with style. To eliminate decorating the room unclearly, take a look at a few suggestions that contributed by the professional for looking for your solid oak dining tables and 6 chairs. It is better for those who set up exclusive look. Colour combined with individuality could make any existing room seem like it may be indeed your very own. Merge your main design and style making use of the persistent colors to make sure it give the impression of being visually more inviting. The best colour, pattern together with texture have the ability to make awesome the look and feel of your incredible interior.\nThe guides will help everyone to pick solid oak dining tables and 6 chairs as a result, by finding out there is a number of elements influence an area. The good news is that you are able to have your house attractive with home decoration that matches your incredible design, there exist a great number of guidelines to get passionate relating to designing your home stunning, even with the the right theme or preferences. So, our opinions is to spare some time in consideration of your own personal style and so gain knowledge what you may enjoy or creating your house an object that is awesome to you.\nThere can be lots of options to evaluate when deciding on any solid oak dining tables and 6 chairs. Obtain the most of the dining room following some decoration guidelines, the better starting point is usually to figure out what you want on taking advantage of the dining room for. According to the things you expect to apply on the dining room will influence the preferences you will be making.\nIt is highly recommended to get solid oak dining tables and 6 chairs after doing a good look at the unit sold in the market, check for their offers, investigate after that choose the best item at the best bargain. It could help you in choosing the right dining room for your house to make it more interesting, and furnish it with the perfect furnishings and components to allow it to be a place to be valued for some time.\nAnd what could be a great deal more, it is easy to see excellent deals on dining room each time you shop around and importantly as soon as you look to go shopping solid oak dining tables and 6 chairs. Prior to purchasing any kind of items for yourself, make sure you use the effective methods to ensure that you are actually paying for the product you would like to pay for finally, make sure you are turning your thinking on the internet for your thing you buy, places where you will be absolute to have the opportunity to find the better value reasonable.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"options (arbitrary user data can be stored with the key `user').\nthe description of pieces, followed by optional reference to a hash of options.\nfor mp3split(); however, END may be omited (with the same meaning as `'INF'').","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Great content is the gift that keeps giving, Matt Cutts the Google engineer brainwashed me to believe this and he cannot be wrong because he works for Google!?You can spend anything from $3 on iwriter.com on content to \u00a3XXXX for a piece of content written by a journalist or expert. This post is about how to spend around \u00a3190 on content that is world class and will send the right signals to search engines and users alike. Oh it is not just written content, we produced an infograph and branded PDF for this price too!\nWe run a lot of sites and it is not feasible to write all the content for each one. We tend to outsource a lot but at different levels of quality. For this one we needed what was akin to a \"thesis\" on smoking and life insurance. Who better to write a thesis than a university graduate! We asked around a couple of forums and were recommended a couple of guys that are Oxford Uni graduates and write content for websites to make extra money. Their normal rate was \u00a310 per article but we wanted to get more than a standard \u00a310 article, we wanted more research and depth and as such we put forward the spec of writing a 1000+ word thesis on the topic. They came back with \u00a3100 to do it and we were happy to give it a go!\nThe writer we used went to statistics.gov.uk, Google news articles, Wikipedia and various other sources to gather as much in the way of statistics on smoking combined with life insurance as they could find. Any relevant discussions and debates were noted down and a detailed stats document was built in draft form to be used later on.\nAs a past smoker myself, this was not too difficult to get started but they also researched Yahoo answers and forums to see what the general buzz was with smokers and life insurance and what they wanted to know specifically. To take it further you could ask on forums, do free surveys on Linkedin.com or any other way you can find to get feedback. Google discussions search is a great and under used method of finding out what people are talking about in a specific niche, you can set it to find discussions that took place in the last 24 hours or selected time period to get access to the most current debates and questions. Then it was a case of using the quotes and stats that were found in \"1)\" to provide detailed answers and resources that smokers could visit to get the information they needed.\nThis was down to our designer and coder Shaun. We gave him the spec of creating one big guide akin to how moneysavingexpert.com. This was to please the Google Gods who prefer detailed content in one place and to make it an easy to navigate guide that we could easily transform into a branded PDF document to boot. We cannot take any credit for how well he did this and he suggested we call them \"special reports\" to give them that edgy\/newsy feel. The work done by him to get it on our site and looking great was 2 hours at \u00a315 per hour, a bargain for us!\nGoogle Authorship will be growing and growing. What better first step in judging the quality of a document is there than by knowing who wrote it and published it?Also, there were some leaked Google remote rater site quality documents that stated that in medical and financial fields Google like to know that the writer was qualified to write the article. We needed to show Google that we were qualified to publish this content. To do this we got my Google Profile author tags added to the document which in turn added my profile (and scruffy haired pic) underneath it showing i wrote it (or published it in this case!). It also has the advantage of highlighting my profile in the Google search results with the site and as such making it stand out from the other results on that page. The social buttons such as \"Share On Facebook\",\"Google Plus One\" and \"Twitter\" were also added to allow people to share the content easily if they liked it. If you want to add these easily and free i recommend www.addthis.com.\nWe wanted to make it so that people and sites we promoted this content to would be able to share it easily with their readerships. To do this we firstly made a branded PDF document of the guide that sites could give away. We then made an infograph linked to life insurance for smokers which was more of a visual stimulus for site owners to use to link to the document.\nWe proof read it and got some feedback from unbiased friends and published it when we felt comfortable that we had created the best resource we could have that would stand out from the crowd.\nYou can view the finished content here http:\/\/www.lifeinsure.co.uk\/special-reports\/life-insurance-for-smokers\/. In the next article i will show you how this content worked incredibly for us outside of what we might have expected and also what opportunities arose for us from it. The page ranks in Google page 1 for a lot of related terms and that is before we have fully promoted it. I will also do a full article on the promotion process for the third instalment and update on the results of it over the coming weeks.\nFeatured content is so much easier to promote and market. Google uses over 500 ranking signals to determine a site's position and making amazing featured content will likely tick a lot of boxes. Add to that opportunities outside of Google rankings, which i will come to next, and you can see why doing this properly may have a better ROI than most other forms of SEO.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hi! Welcome to our site, One Earth Radio!\nWe would like to share to everyone in this world the benefits of meditation and listening to music that are for contemplation. We are a group of people who had found the common power of meditation in our lives. You can join us if you feel that there really is a benefit behind meditation.\nYou see, the thing about meditation is that it doesn't have to be done through a professional way or through going to classes. You just need yourself, an audio guide at start, or you can just even hear the melody right there and there in your office. Even if you are an average reasonable person, you can reap the benefits of this melody.\nAgain, thank you so much for visiting our site.\nIf you would want to interact to us, we are more than willing and happy to reach you out here!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzafjie b/data_all_eng_slimpj/shuffled/split2/finalzzzafjie
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+{"text":"The 2017 Smart ForTwo remains the shortest car sold in the U.S. Of course, it accomplishes that by offering just two seats\u2014whereas the Chevrolet Spark and Mitsubishi Mirage minicars both offer four seats and five doors. Smart has the two-seat city car market entirely to itself, now that the Scion iQ minicar has been withdrawn from the U.S. We view the new Chevy Spark as a far better car than the underpowered Mirage, though the Mitsubishi is getting a number of updates this year. The Mirage wins on purchase price, as well as the highest fuel-economy of any non-hybrid vehicle on the market. Then there's the tiny Mitsubishi i-MiEV, an all-electric car with four seats and five doors as well, but a very low 62 miles of EPA-rated range. That range has doomed the i-MiEV to irrelevance, frankly. Moving up a size, the Honda Fit represent the clever and flexible end of the subcompact segment. We like the Honda Fit for its supreme cabin flexibility, and its base model may come in no more expensive than a mid-level Smart. If the Smart's a style decision, the latest Mini Cooper is larger but also a far better and more enjoyable car to drive\u2014if also considerably pricier. Still, if ease of parking is your main criterion for purchase, no other vehicle comes close to the new Smart.\nThe 2017 Smart ForTwo is the easiest car to park that you can buy, but if you have other needs, four-seat cars can offer more cabin space, equal fuel economy, and more flexibility.\nThe 2017 Smart Fortwo is the second year of a complete redesign for the tiny two-seat car, and this year the Cabrio soft-top convertible has rejoined the lineup. Both body styles remain at 8.8 feet long, but they're wider and more modern than the previous generation, with a better array of standard and optional features.\nThe Fortwo can be ordered in four different trim levels\u2014known as Pure, Passion, Prime, and Proxy\u2014and the company offers a wealth of options that make it unlikely Smart drivers will see a car identical to the one they've ordered.\nThe little Smart minicar is perfect for buyers who need a car that's easier to park on crowded streets than any other vehicle sold in the U.S. But its two seats and very limited cargo space, not to mention fuel economy that's decent but hardly exceptional, mean it's significantly compromised for pretty much everyone else.\nThe lines of the 2017 Smart retain the perky styling, snub nose, and passenger compartment outlined by the Tridion safety cage, usually in a contrasting color. The wheels sit at the very corners of the vehicle. But the front end is more clearly defined now, and the short hood is less beveled than before, giving better pedestrian protection. The \"grille\" between the lights is no more than a blanking panel perforated with holes, and all Smarts have LED running lights.The door windows are frameless, and the tailgate is split horizontally: upper portion opens for access while the lower half flips down to provide a useful temporary surface.\nThe second-generation Smart finally gets a modern interior. It's still simple, but different trim colors and a variety of rounded shapes give it a more substantial feel. A small instrument cluster behind the wheel has the usual gauges, and two eyeball vents sit on top of the dash. A touchscreen display sits slightly proud of the central console.\nA turbocharged 0.9-liter 3-cylinder engine, making 89 horsepower and 100 pound-feet of torque, is mounted between the rear wheels. A 5-speed manual gearbox is standard, and a 6-speed dual-clutch transmission provides automated shifting for those who prefer it. (Smart says four out of five customers will order the automated option.) Combined EPA fuel economy is 34 or 35 mpg, slightly lower than last year due to changes in the EPA test calculations. Smart says 0-to-60-mph acceleration is 10.5 seconds for the DCT, 10.1 seconds for the manual.\nThe 2017 Smart feels like a compact car behind the wheel, if you don't look over your shoulder. The turning circle is an astoundingly small 22.8 feet from curb to curb, or 24 feet between two walls. That makes the Fortwo pretty much the most maneuverable car you can buy, and a perfect urban warrior. Thankfully, it's now more capable on the highway than the old one was, and the ride is quieter and smoother under almost every circumstance. Given its small size, noise is well suppressed for such a small car, and it's simply a much more pleasant vehicle to drive or ride in than the first generation.\nThe Fortwo hasn't yet been rated for crash safety by either the National Highway Traffic Safety Administration or the Insurance Institute for Highway Safety. All models come standard with eight airbags, and a standard Crosswind Assist function keeps the Smart straight during side winds. Forward-collision warning is available as an option.\nBeyond the base ForTwo Pure, the Passion, Prime, and Proxy trim levels provide a mix of different features, using interior trim and fittings to give the different models distinct characters. All new Smarts include power windows, central locking, LED daytime running lights, cruise control, power steering, and audio and other controls on the steering wheel. The audio system includes a CD player, Bluetooth streaming for playing music and hands-free commands. The instrument cluster includes a 3.5-inch color display, and a trip computer and an exterior temperature display are also standard.\nPricing starts below $16,000 including delivery, and Smart says a fully loaded model with multiple high-end options runs between $20,000 and $21,000. The Cabrio model with a roll-back cloth roof has been added this year; a redesigned Electric Drive version joins the lineup soon.\nThe Fortwo remains a cute, short two-seater that stands slab-sided and tall, and may appear wider than it is long from some angles. It's still quite obviously the shortest car sold in the U.S. Its lack of length is most obvious from the side, where its snub nose comes into relief. The fixed-roof Coupe body style has been rejoined this year by the Cabrio model, with its power-operated cloth roof that rolls back to offer open-top driving. To keep the structure stiff, it retains the crossbar between its rear pillars, though one of three ways to open the top includes removing the beams over the doors to increase the sense of openness.\nThe Smart's trademark \"Tridion\" safety cell generally comes in a contrasting color (there are four) to the body panels (which can be ordered in dozens of shades). It outlines the Smart's body structure, and underlines its strong structure, from the windshield post along the roof side, down the rear pillar and wrapping around under the door. It instantly identifies the car as a Smart\u2014not that its size wouldn't\u2014and provides an effective design flourish with a subtext.\nThe rear light shape is now a square with rounded corners, rather than the previous pair of round lights, and the grille is pierced with a pattern of hexagons in varying sizes forming a screen that adds visual interest\u2014echoed in light patterns and other subtle exterior elements. As before, other exterior flourishes include the fixed window pane at the back of the door and the kicked-up belt-line bulge that contains the door handle.\nInside, the 2017 Smart has a modern cabin, albeit a simple one. But the previous generation's grim, stark, utilitarian Nineties appearance has given way to a modern design offering up-to-date amenities. A small instrument cluster behind the steering wheel has a digital display in the center with a speedometer around its outer edge, but the tachometer sits in its own pod on top of the dash just inboard of the base of the driver's windshield pillar. Stylists used a coarse-grained cloth surface on the door inserts and dashboard, and the seats come in cloth or leather, with contrasting stitching in some trims.\nThe layout remains the same, with a 3-cylinder engine lying on its side between the rear wheels\u2014with rear-wheel drive\u2014but the 0.9-liter unit is now turbocharged for the U.S. market. That gives it a rated output of 89 horsepower and 100 lb-ft of torque, notably more than the previous 70 hp and 68 lb-ft of the previous model. Base Smarts come with a 5-speed manual gearbox, but Smart says eight out of 10 buyers will opt for the 6-speed automated dual-clutch transmission (which adds $990 to the cost). The company quotes 0-to-60-mph acceleration of 10.1 seconds for the manual, and 10.5 seconds for the DCT, again a notable improvement over the first generation.\nWe'd deem the DCT the better of the two transmission choices, even if it's slightly slower. Usually manuals are better in small cars, but the Smart's 5-speed had surprisingly long throws and a slightly loose gate, whereas the shift points of the DCT have been well-matched to the engine tune. Slipping the clutch while revving the engine was required to get enough power for the manual-gearbox car to move out before its turbo spooled up from idle. The little Smart was most responsive when the driver kept engine speeds very high, from 4000 rpm to the rev limit of around 6500 rpm.\nHandling overall is good; the car grips well on 15- or 16-inch tires, which are larger in the rear than in the front. As a Smart engineer said, the car's traction control is \"rather strict,\" and the driver cannot switch it off. That's to ensure that the ForTwo remains \"fun to drive, but not beyond,\" meaning that it's required to tame potential misbehavior or any oversteer in what is still a tall, rear-engined car with a pronounced rear weight bias and a very short wheelbase.\nSmart is justifiably proud of the new car's minuscule turning circle. The 4-inch increase in width allowed its engineers to provide deeper front fender wells for sharper wheel angles\u2014they can now pivot to 45 degrees\u2014for a turning circle of just 22.8 feet. It's one of the car's genuinely grin-inducing factors. With its 8.8-foot length, you can do a U-turn just about anywhere\u2014or do complete 360-degree circles within the width of an average two-lane road. In urban traffic, it takes a while to get used to just how short the Smart really is, but it can be maneuvered and parked like no other vehicle once a driver learns the limits of the car's body.\nThankfully, the Smart now feels confident and secure at highway speeds, and it's surprisingly smooth and quiet on most road surfaces. Driving the old Smart between semis at 70 mph required nerve and a lot of pre-planning, because it simply ran out of steam at those speeds. The new Fortwo has more power in reserve, though it's far from a muscle car. Still, we wouldn't shy away from taking it on a longer road trip. Most Smarts likely won't be used that way, but it's reassuring to know they could be.\nFuel economy is only so-so given the Smart's small size. By far the most energy-efficient version will be a new battery-powered Fortwo Electric Drive model, which is expected to be shown in late 2016 and arrive on the market next year.\nThe 2017 Smart Fortwo is no longer primitive and outmoded, but it remains basic transportation with a couple of interesting quirks.\nThere's no denying the second-generation Smart is a far more pleasant vehicle in which to spend time than its predecessor. The two seats are comfortable, if a tad short in the lower cushion, and there's ample headroom and adequate leg room, even for adults more than 6 feet tall. Occupants sit high and upright, not low to the ground, and outward visibility is good. Only when reversing is it obvious that the car ends about 2 feet behind the driver's shoulders. The engine sits essentially under the high load-deck floor, just behind the occupants' backsides.\nThis obviously poses a huge challenge for any cargo, though Smart has done its best to make the limited amount of space usable. The Fortwo's cargo bay holds 9.2 cubic feet of goods, more than the Mini Cooper (8.7 cubic feet) and roughly the same as the 9.5 cubic feet in a Fiat 500. Cleverly, the car has a second set of mounting points built in that let a driver remove and store the retractable cargo cover behind the seats, just ahead of the load floor and at the same height, so you can carry the cover with the car rather than removing it.\nStill, interior storage is limited to two cupholders, a pair of smallish door pockets, and a clever sideways drawer under the console that opens into the passenger footwell. There's no spare tire, just tire sealant and an inflator, stored under a fake toeboard in the passenger footwell. Exceptionally tall passengers will benefit from removing them, which gives 4 or 5 extra inches of legroom if needed.\nThe ride of the new Smart is notably better than its predecessor, with a wider track and a marginally longer wheelbase improving stability. The new car's vastly better suspension is more capable of soaking up road irregularities and sailing over speed bumps in a controlled fashion. Under most circumstances, the Smart was surprisingly quiet; only particularly coarse road surfaces and revving the engine to its limit made the cabin noisy enough to require raising voices.\nThe interior materials and design of the new ForTwo are also a huge step up from the previous generation. Coarse-weave cloth surfaces, hard plastics, and either cloth or leather for the seats look modern enough, and they complement a conventional dashboard with a small instrument cluster behind the steering wheel. A central display panel looks like an LCD screen, but also holds knobs and dials for some cabin controls. With a conventional console, the Smart now has a modern interior that's basic but no longer plain, grim, and utilitarian. The build quality on the early-production North American-market Smarts we drove was good, with no apparent quality issues, squeaks, rattles, or other flaws. All Smart ForTwo models are built in Hambach, France.\nThere isn't enough safety-test data available for the 2017 Smart Fortwo to let us rate it accurately.\nAll Fortwo models, whether coupe or cabriolet, are built around a very stiff Tridion safety cell\u2014which can be highlighted with optional colored exterior panels. The structure now contains about 75 percent high-strength steel, of various grades, and every 2016 ForTwo has eight airbags as standard. These include driver and passenger bags in the dash, knee bags, and shoulder\/side bags mounted in the seat, plus side curtain bags that cover the window glass.\nA rearview camera is expected to become available this model year, when an in-dash navigation system becomes available. Meanwhile, Smart has added a new electronic safety feature called Crosswind Assist, which automatically steers and brakes the car to counteract sideways motion that could push the car out of its lane in gusty conditions. It's adapted from a similar system fitted to the tall Mercedes-Benz Sprinter van, but we haven't been able to observe its effects on test drives so far.\nStill, Smart offers none of the newer electronic safety systems found on the much pricier vehicles in the Mercedes lineup. Those include adaptive cruise control, lane-departure warning and correction, and automatic crash braking. To be realistic, however, a blind-spot monitor system would likely be superfluous on a car this short, since the rear bumper is just 2 or 3 feet behind the driver.\nThe 2017 Smart Fortwo is offered in four different levels of trim, and buyers can choose from a further variety of factory and dealer-installed options that offer at least a chance that no two Smarts will be identically configured. We give it a baseline score of 5 for its features. (Read more about how we rate cars this year.) The four trim levels are dubbed Pure, Passion, Prime, and Proxy.\nThe base Smart Fortwo Pure comes with cruise control, power windows, automatic climate control, a multifunction steering wheel, and Bluetooth streaming as standard. It also has LED running lights and a small 3.5-inch color display in the instrument cluster. You'll identify the base car, though, by its silver hubcaps over 15-inch steel wheels, since the higher trim levels have various designs of alloy wheel standard.\nAmong the Passion, Prime, and Proxy trims, Smart says they're mean to give three different personalities to the car. The first is modern and stylish, the second is classic and elegant, and the third is meant to indicate innovative and trendsetting.\nSo the Passion offers two-tone interiors\u2014black with orange, grey, or white\u2014plus powered and heated door mirrors, a leather-wrapped steering wheel, and a retractable cargo cover. The Prime comes with grey accent stitching on its black leather seats, leather wrappings for both the steering wheel and gear knob, and a silver Tridion safety cell and grille. It has a longer list of standard equipment, including heated seats, a panorama roof, exterior LED lighting, fog lamps, and a rain and light sensor. Finally, the Fortwo Proxy package offers a Tridion safety cell and grille in shiny white, and a vibrant, startling blue and white interior. Included in its equipment list is the Sport package, providing 16-inch alloy wheels as standard, and also the premium 240-watt JBL sound system with eight speakers.\nFinally, Smart is very proud of its Cross Connect app for smartphones (Android and iOS), offered free to buyers of the new ForTwo. While the app is free, the cradle to mount the phone to the center display screen is $100. The Cross Connect app has two modes, one for while the car is moving, when the phone is horizontal, and the other for when it's stopped, when the driver rotates it to vertical. The app offers five functions now, with more that can be added later on when updates are released.\nThey include an EcoDrive function offering the owner tips on economic driving and warnings when the car exceeds the prevailing speed limit. Then there's the Fuel Economy function, which grows trees if you're driving gently and turns the forest into a desert if you're profligate with your fuel. A Navigation mode uses the app to provide mapping and routing; it'll also play Music from the phone; and finally, the Radio function syncs to the car's presets and lets you control the audio from the phone rather than the dashboard. The Navigation seems to offer the most value, since it substitutes for an in-dash navigation system (which Smart will also offer). Finally, the Smart Spots function crowdsources the locations of parking spaces into which only a Smart can fit. There are dozens for most of the major cities where Smart is popular, and hundreds for San Francisco. Drivers can photograph a new space and upload the photo, which will be validated by Smart and then added to the map.\nStill, the beta version of Cross Connect that we tried was significantly slower than the driving instructions provided by Google Maps on the same phone. That made it hard for us to imagine why anyone who's ever used mapping via Google wouldn't do that instead\u2014ignoring Cross Connect but mounting the smartphone in that $100 cradle.\nWhen fitted with a dual-clutch automatic transmission, the 2017 Smart ForTwo earns an EPA-rated 35 mpg combined, while Smart Fortwo models with the five-speed manual gearbox earn 34 mpg. Those combined ratings are each 1 mpg lower than the comparable figures for the same car in 2016, reflecting revised calculations required by the EPA that have slightly lowered ratings for many cars this year despite no changes at all to their running gear.\nA 35-mpg rating, or just two-thirds of that for the much larger Toyota Prius Eco, will likely come as a huge shock to those who assume such a small car must deliver phenomenal mileage. The Smart may come with obvious compromises in space and comfort, not to mention perceived security on U.S. streets and highways, but they're not offset by sky-high fuel efficiency. The laws of physics prevent that, unfortunately, which is why Smart isn't particularly advertising the car's fuel economy as a major selling point.\nRegrettably the Smart still requires premium fuel (at least 91 octane) to obtain the quoted performance figures. In the small-car segment, its fuel economy is on par with the more capacious and far more flexible Honda Fit, which was also re-rated for 2017 and comes in at 33 mpg combined or 35 mpg combined depending on transmission.\nA new generation of the Smart Fortwo Electric Drive should be unveiled in late 2016, and will probably arrive at U.S. dealers as a 2018 model. Until then, the existing electric Smart remains available in the old body style\u2014earning an official 68 miles of driving range, and a 32 kwh\/100mi (or 107 miles per gallon equivalent) rating.\nLet's begin by pointing out this Car Connection review of the 2017 smart is missing the point...entirely. Yes, the smart is pricey for its size - a realtor would say the price\/square foot is high - but what... Let's begin by pointing out this Car Connection review of the 2017 smart is missing the point...entirely. Yes, the smart is pricey for its size - a realtor would say the price\/square foot is high - but what this review totally fails to acknowledge is the smart buyer isn't even considering a four seat car. Sure, you can get a four seat car for less but it's going to be a crap car. Don't forget smart is manufactured, design, distributed, etc by Mercedes-Benz. When you drive a smart, you get Marcedes-Benz service because you drive a Mercedes-Benz product.\nBetter mileage that my M-Class Mercedes!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Parrot at Paris Air Show 2013: get ready for civil drones!\nThe world largest aviation exhibition, Paris Air Show, sets to launch on June 17th at Le Bourget, near Paris, in celebration to its 50th anniversary.\nNew App for pilots worldwide (outside USA and Ta\u00efwan): FreeFlightWORLD brings accelerometer and Absolute Control back on iPhone, iPad & iPod touch!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We have a Make-up competition coming up. We need two very dark ladies for 4 students to display Make up on. The exercise will be on for about 3 hours. Pay out is R650 each. Message with your Age and Location Only please. We will appreciate profile with pictures.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For many people, migraine headaches only occur a few times a year or even a couple times a month. However, some people end up dealing with regular migraines that start occurring nearly every day. These headaches are known as \"transformed migraines,\" and it's important to diagnose them and their underlying cause to find the best treatment for the problem.\nA transformed migraine is a migraine that begins to manifest in regular episodes of migraine attacks, and they typically begin to increase in frequency and may change characteristics. Headaches may become less severe, but they may start occurring nearly daily. However, severe, debilitating migraine attacks may still take place.\nWhat can trigger a Transformed Migraine?\nAlthough transformed migraines generally develop over a period of months or years, it's thought that regularly using pain relievers and migraine medications may trigger transformation to occur. Patients may build up a tolerance to the medications they are taking. Certain pain relievers that contain caffeine may also trigger rebound headaches. Other health problems may also trigger transformed migraines, such as depression and high blood pressure.\nHow many people are affected by Transformed Migraines?\nThere's no exact data on the number of people affected by this type of migraine, but most of the people who experience them have a history of migraines that started when they were young. Most patients affected by transformed migraines are women, and approximately 90 percent of them have experienced migraines with aura in the past.\nHow do Transformed Migraines affect most people?\nUsually, the transformation process takes place over months or years, until the patient begins getting the migraines daily. The attacks are often accompanied by nausea, phonophobia and photophobia, but the symptoms become less frequent and severe. Transformed migraines include a pattern of daily or nearly daily headaches that seem to be a combination of migraine attacks and tension headaches. The pain may be only mild to moderate instead of the intense pain that comes with severe migraines. Other symptoms of migraines may continue, including stomach symptoms and unilateral pain.\nIf you begin to have daily migraine headaches, even if they aren't as severe as your normal migraines, it's important to see a migraine specialist. A specialist can help you figure out if you're dealing with transformed migraines and then work with you to find an effective, safe treatment.\nIf you are suffering from Transformed Migraines, schedule your consultation today!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This activity requires the use of non-expendable (reusable) LEGO MINDSTORMS robots, software, parts and sensors; see the Materials List for details.\nEngineers are inspired by nature, such as the echolocation abilities of bats and marine animals.\nStudents use ultrasonic sensors and LEGO\u00a9 MINDSTORMS\u00a9 EV3 robots to emulate how bats use echolocation to detect obstacles. They measure the robot's reaction times as it senses objects at two distances and with different sensor threshold values, and again after making adjustments to optimize its effectiveness. Like engineers, they gather and graph data to analyze a given design (from the tutorial) and make modifications to the sensor placement and\/or threshold values in order to improve the robot's performance (iterative design). Students see how problem solving with biomimicry design is directly related to understanding and making observations of nature.\nFor many engineers, nature can provide inspiration and ideas for solving problems efficiently. In addition, the evaluation of designs and the numerous iterations of redesign are important components to creating good engineering solutions. Using data analysis to determine adjustments to improve a design occurs continuously throughout the alpha design process, and then even in beta design processes, threshold values are adjusted to maximize efficiency. Biomimicry examples include adhesives inspired by gecko feet, tumbleweeds that inspired a futuristic Mars Lander, and fabric fasteners, such as Velcro\u00a9, inspired by the clinging hooks of plant seeds.\nDescribe how a bat or submarine uses sonar to avoid obstacles and\/or detect prey.\nCreate prototypes to model how sonar works.\nEvaluate prototype strengths and weaknesses and use this information to redesign better versions.\nFamiliarity with basic mathematical skills, including multiplication and division.\nBiomimicry is the study and emulation of nature to perform tasks or solve problems. Engineers know that nature has had roughly 3.6 billion years to adjust, evaluate and fine tune the wide range of amazing functions and capabilities se see in plants and animals. By examining these time-tested models, engineers are inspired to design creative solutions to new problems.\nA problem, such as the avoidance of obstacles, can be done well with the use of echolocation. Echolocation is an elegant solution for bats who hunt for food (insects) at dusk and at night when the amount of light is inadequate to see well enough to avoid obstacles.\nIn our activity today, similarly to bats, our LEGO EV3 robot is fitted with an ultrasonic sensor as its main means of detecting the world around it. The ultrasonic sensor can detect objects in its \"field of vision\" so that the robot can avoid them.\nA bat's echolocation system is the basis for the principle of sonar or radar for distance measurement.\nClose your eyes for a moment and imagine. (Have students close their eyes and prepare to imagine the following scene you describe to them.) You are in a large open canyon. You yell \"hello\" very loudly, and then stand there silently waiting. After a few seconds of waiting, you hear \"hello...hello...hello\" come back as your voice echoes (reflects) off the distant canyon walls and back to your ears.\nThis same idea occurs with the echolocation ability of bats. Bats emit high-pitched frequencies that travel across areas, bounce off obstacles (surfaces) and return to their ears. The amount of time that it takes for the first \"hello\" to reach you is half the time that it took to reflect off the obstacle. Thus, we can determine the distance by taking the time and halving it.\nBiomimicry can be seen in the engineering design decisions that went into making the ultrasonic sensor. The sensor, like a bat's echolocation system, emits a high-pitched frequency and detects when the reflected sound is received back, so that the distance to an object can be determined.\nThe correlation between new technology and how it is developed is often lost on students. Experiencing an iterative design process, like that typically used by engineers, helps to develop students' abilities to critique their own work and improve upon it. By tying technology to nature, students can draw upon their existing knowledge bases to understand how things work or how already-developed ideas may be improved upon.\nGather materials and make copies of the Pre-Activity Worksheet, Echolocation Activity Datasheet and Post-Activity Worksheet; one each per student.\nProvide enough space to run the experiment\u2014a length of about 200 cm (~ 6 feet, 6 inches). Measure and make marks at the start location, at 100 cm and at 200 cm.\nBuild the EV3 robot according to the EV3 5-Minute Bot; refer to the internet link in the Additional Multimedia Support section.\nDownload the five EV3 robot program files (see the Attachments section) and load them on to the EV3 brick. These programs have thresholds set for the ultrasonic sensor in order to run the activity. Note that the numbers in the file names (10, 25, 50, 75 and 90) indicate the threshold value set for the sensor.\nBefore conducting the activity, give students 10 minutes to complete the Pre-Activity Worksheet.\nHave students mount an ultrasonic sensor to the already-built \"five-minute bot\" in three different positions. Have one-third of the groups mount their sensors near the rear, middle and front of the robot, respectively.\nVarious ultrasonic sensor placements (left to right): rear, middle front.\nPlace an object at the 100 cm mark. Make sure the robot threshold is set for 10 cm. Time how long it takes the robot to react to the obstacle (when it stops). Record the time on the datasheet. Repeat the process for thresholds at 25 cm, 50 cm, 75 cm and 90 cm.\nPlace an object at the 200 cm mark. Make sure the robot threshold is set for 10 cm. Time how long it takes the robot to react to the obstacle (when it stops). Record the time on the datasheet. Repeat the process for thresholds at 25 cm, 50 cm, 75 cm and 90 cm.\nHave students graph the results on the threshold distance vs. reaction time grid on the datasheet. Have students share their results for each of the differently mounted robots.\nDirect students to examine the data collected from the previous steps to make inferences about redesigning the robot to optimize its abilities to avoid the obstacle. What do their analyses suggest might be improvements to make? Variables might include the location of the ultrasonic sensor on the robot and\/or adjusting the robot thresholds.\nAfter making redesign adjustments, run steps 5 and 6 again with the redesigned robot and sensor set-up. Record data on the datasheet.\nGraph the data on the datasheet.\nHave students look at the new data and answer the datasheet evaluation questions about whether or not their adjustments improved or made the robot worse in reacting to the obstacle.\nAs a class, review the Post-Activity Worksheet. Specifically focus on students' justifications of their redesign choices and how they relate to the data collected.\nConclude with a class discussion prompted by questions in the Assessment section.\nbiomimicry: Looking to nature for inspiration to solve human problems.\necholocation: A physiological process for locating distant or invisible objects (as prey) by sound waves reflected back to the emitter (as a bat) from the objects.\nradar: A device or system consisting usually of a synchronized radio transmitter and receiver that emits radio waves and processes their reflections for display. Radar is used especially for detecting and locating objects (such as aircraft) or surface features (such as of a planet).\nsonar: A method or device for detecting and locating objects especially underwater by means of sound waves sent out to be reflected by the objects. Also, a device for detecting the presence of a vessel (such as a submarine) by the sound it emits in water.\nPre-Activity Worksheet: Before starting the activity, administer the Pre-Activity Echolocation Worksheet to evaluate students' baseline knowledge of biomimicry, and their dispositions towards engineering design practices.\nActivity Datasheet: During the activity, have students fill in the portions of the Echolocation Activity Datasheet as it correlates to the steps they are working on. Using the blank data tables and grids for data recording and graphing saves them time and makes an easily readable set of data for evaluation in the redesign. Review their data, graphs and answers to gauge their mastery of the subject matter.\nPost-Activity Worksheet: At activity end, administer the Post-Activity Echolocation Worksheet to evaluate students' comprehension of the concepts. As a class, review students' justifications of their redesign choices and how they relate to the data collected. Compare the pre- and post-assessments to gain insight into any changes in students' understanding.\nAssign students to conduct in-depth research and report back to the class (or write short papers) on the topics covered in the concluding discussion: What other animals use echolocation? What are real-world engineering examples of the use of the echolocation concept? What other designs in our everyday lives have their inspiration from nature? What current technologies used biomimicry for their designs?\nFor upper grades, 9-12: Given the speed of sound at sea level is 340.29 meters per second, find the theoretical amount of time it takes the robot to react to the object using simple unit cancelling and conversion. Do this in conjunction with the initial data gathering and analysis, and plot it on the same graph to give a better sense of value to the data.\nFor a long list of example engineering inventions inspired by animals and plants, see the Procedure-Background section of the Biomimicry: Natural Designs activity.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzafkmu b/data_all_eng_slimpj/shuffled/split2/finalzzzafkmu
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+{"text":"Background: Dietary intakes of plant lignans have been hypothesized to be inversely associated with the risk of developing cardiovascular disease and cancer. Earlier studies were based on a Finnish lignan database (Fineli\u2020) with two lignan precursors, secoisolariciresinol (SECO) and matairesinol (MAT). More recently, a Dutch database, including SECO and MAT and the newly recognized lignan precursors lariciresinol (LARI) and pinoresinol (PINO), was compiled. The objective was to re-estimate and re-evaluate plant lignan intakes and to identify the main sources of plant lignans in five European countries using the Finnish and Dutch lignan databases, respectively.\nMethods: Forty-two food groups known to contribute to the total lignan intake were selected and attributed a value for SECO and MAT from the Finnish lignan database (Fineli\u2020) or for SECO, MAT, LARI, and PINO from the Dutch database. Total intake of lignans was estimated from food consumption data for adult men and women (1979 years) from Denmark, Finland, Italy, Sweden, United Kingdom, and the contribution of aggregated food groups calculated using the Dutch lignin database.\nResults: Mean dietary lignan intakes estimated using the Dutch database ranged from 1 to 2 mg\/day, which was approximately four-fold higher than the intakes estimated from the Fineli\u2020 database. When LARI and PINO were included in the estimation of the total lignan intakes, cereals, grain products, vegetables, fruit and berries were the most important dietary sources of lignans.\nConclusion: Total lignin intake was approximately four-fold higher in the Dutch lignin database, which includes the lignin precursors LARI and PINO, compared to estimates based on the Finnish database based only on SECO and MAT. The main sources of lignans according to the Dutch database in the five countries studied were cereals and grain products, vegetables, fruit, berries, and beverages.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"New episodes of Arrested Development, Santa Clarita Diet and Veep await you.\nRoku's mobile app will introduce video streaming, including its stab at offering premium channels like Showtime and Starz.\nAn analyst estimates Amazon's Prime Video Channels are big business.\nAmazon packs a punch with a new psychological thriller starring Julia Roberts.\nGrab your trusty log and a hot cup of coffee, and experience David Lynch's world of Twin Peaks in a new virtual reality experience.\nCommentary: Fans and critics alike loved Twin Peaks: The Return, so why didn't the unusual Showtime series pick up any awards?\nGood news, kids: There are price breaks aplenty if you know where to look.\nMarvel's Iron Fist returns for its second season on Netflix.\nSpotify Premium for Students already includes ad-free music and Hulu.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Great for those long car trips, our Deluxe Cherry Tic-Tac-Toe comes with 5 red and 5 blue pegs. The approximately 6\" wide by 7\" long by 3\/4\" thick board has laser engraved details and features holes drilled along the sides for peg storage when not in play. Non-toxic finish.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"MARK YOUR CALENDAR, Hello Holiday Sdn Bhd is taking part in MITM 2018!\nCome join us for the BIGGEST installation of our Malaysian International Travel Mart 2018! With an expected crowd of over 100,000 visitors last year, we hope to see you join in the festivity this coming 3rd -5th of August 2018 at Mid Valley Exhibition Centre!\nThis amazing travel and tourism fair is presented by the Malaysian Chinese Tourism Association (MCTA). They aim to help Malaysians get access to the best travel deals for Malaysians to explore the world without burning a hole through their pockets!\n6-4, Jalan Metro Pudu, Fraser Business Park, 55100 Kuala Lumpur, Malaysia.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Has anyone considered earthing in muscular dystrophy?\nCombination of arginine and deflazacort more beneficial than deflazacort alone?\nBrother had DMD. I did some research re. muscle and nerve development - funded by MD group of Canada. I now practice acupuncture, herbal medicine and nutritional therapy. I am also studying Vedic astrology relating to health.\nMy nephew diagnosed with DMD of exon duplication's 2 to 4.\nCan you please suggest us, what could be the solution.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaflvm b/data_all_eng_slimpj/shuffled/split2/finalzzzaflvm
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+{"text":"Cooking time: Less than 30 minutes each!\nSoak the gelatine leaves in water and set aside. In a saucepan bring the cream, milk, sugar and malt extract to a gentle boil, stirring occasionally, then remove from the heat. Lift the gelatine leaves and squeeze to remove excess water then add directly into the hot cream, mix and set aside to cool for 5 minutes. Pour into serving glasses and place in the fridge leave for 30 minutes to set.\nSlice the strawberries in half and place in a saucepan with the raspberries, sugar and malt extract, then simmer for 10 minutes. Remove from the heat and blend with a hand blender until smooth. Return to the heat and gently simmer gently for a further 10 minutes until it thickens and most of the water has evaporated, then remove from the heat. Allow to cool fully and pour onto the top of the panna cotta and chill for a further 1 to 2 hours. For a finishing touch add some fresh raspberries before serving.\nMalt extract is included in this recipe to partially replace some of the sugar found in more standard panna cotta recipes. It reduces sweetness and introduces a subtle malt flavour and makes the dessert much creamier and silky. A similar technique applies for the fruit compote, where malt extract has been used to replace some of the sugar bringing added depth of flavour to the fruit topping and balancing the acidity of the summer fruits.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Pilgrim Hearts Reaches Out to People Through the Arts, using a wide range of creative talents to convey its message. We are able to call on the experience of many talented artists and performers installing and managing exhibitions. Arrange creative workshops for individuals and community groups. Pilgrim Hearts Trust started in 2000 as a registered charity. Enabling people to use their creativity to explore their knowledge of God and give hope to others.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ludwig Van Beethoven (1870 \u2013 1827) was a famous pianist who hailed from Bonn, Germany. His influence on music is indisputable especially when it comes to his symphonies, piano concertos, and piano sonatas among other compositions. In fact, he was so influential that there is a statute of him in Bonn, Germany. Bonn also has a museum built in his honor. Here are some interesting facts on Beethoven.\nBeethoven threw a tantrum in 1798 when someone interrupted him while he was working. He experienced a nasty fall during this incident and he was deaf when he got back to his feet. His hearing ability improved after this incident, but it never recovered. He faced serious difficulties when it came to interacting with people in his social and professional settings. He was almost completely deaf by 1814. This hearing problem did not stop him from composing music. Instead, it motivated him. For example, he composed Symphony No. 9 and Missa solemnis when his hearing was at its worst. In fact, he composed most of his renowned compositions after 1798 i.e. after his hearing difficulties started.\nBeethoven was a commoner in a highly aristocratic society. The aristocrats loved his music, but they never accepted him in their closed knit circles. For example, several women rejected him because he was a commoner including Julie Guicciardi and Josephine Brunsvik. Guicciardi and Brunsvik were countesses. Therese Malfatti, the daughter of a rich merchant and estate owner, also rejected his advanced for similar reasons. Beethoven turned to music to express his love for these women. For example, he composed Sonata No. 14 and F\u00fcr Elise for Brunsvik and Malfatti respectively.\nMusic students struggle with many issues in their lives including physical, financial, and psychological problems. Beethoven also struggled with such issues. In fact, he heard hearing difficulties, which is heartbreaking if you are a composer. The most inspiring thing about him is that he never gave up on his talent. Instead, he played the piano. In fact, he created his greatest compositions during his time of great tribulation. For example, he composed music when the world rejected him. Learn music so that you can nurture your talent. Use it as a tool for promoting harmony, patience, and happiness just as Beethoven did.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Do you have a backyard or a park near by? There's a good chance these tasty greens are growing right by you and you didn't even know!\nNo need to go to the grocer for these foods, they come straight from the Earth for you to eat!\nThat is how we are naturally suppose to exist with the planet; to take what we need from its abundance and leave out the need for money. The planets foliage is there for us to enjoy and thrive off of, so what are you waiting for!?\nThere's a few specific plants that are perfect for salads, a small snack or garnishing a plate of food with. These weeds are high in phytonutrients which are natural chemicals occurring on plants to keep them protected from germs and are filled with anti-inflammatory and antioxidant properties.\nThis chemical can have a sightly bitter taste, which is actually perfect when you combine sweet dressings or lighter tasting foods with it.\nBe careful when foraging for these plants, there are always poisonous lookalikes bound to be around!\nThese plants are perfect for cooking just like you would spinach, and adding in egg dishes, salads, stir fries, green smoothies, or even just to eat alone!\nThis is one of the most abundant and popular leaves you can find almost anywhere in the world.\nYou can recognize it by its tooth, almost sharp looking leaves growing in a circle or spiral shape out of the ground. It is usually close to the ground but as many of us know, these weeds can get extremely large.\nAnd in this case, the larger the better! Don't throw these weeds away while you clean up your garden. Collect them as you go and then by the end of the day you'll have a plethora of deliciousness!\nSometimes this plant can be confused with Flatweed which is also edible but more bitter tasting.\nThis beautiful succulent was widely used in the past in East Mediterranean countries. It's little round leaves are high in Omega 3 fatty acids and are packed with vitamins and minerals to boost up your health!\nIt also has oxalate in it, which is a combination of minerals and salts. It can be overeaten and produce some undesirable effects if you eat too much of it.\nEnjoy this cooked mostly, unless you already only eat raw. If you eat it raw, just don't eat too much of it.\nMany people put it in yogurt to reduce the oxalate content or enjoy it with some tzatziki.\nThis isn't actually a weed, it's a plant! But it's still equally as edible and delicious. It was eaten by non-indigenous colonists to actually alleviate scurvy back in the days, that's where it gets it's name from! This plant can be easily identified by its 3 petaled beautiful blue flowers. The leaves growing around it are what you want to be eating.\nThrough this yummy plant in a salad to refresh your stomach and ensure you're eating well!\nThe more we can get our food straight from the ground, the more we are aligning ourselves with the natural way to live from this planet.\nEarth has more than enough food on it for everyone to enjoy, it's now time to create the structures so we can live harmoniously from it. Be careful when picking these plants to make sure they haven't been sprayed with anything or have any kind of harmful chemicals on or around them.\nThoroughly wash and dry all these plants before consuming them. Be safe, and happy foraging!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Uses your body\"s kinetic energy to create infrared thermal warmth. Wear alone or under another pair ..\n1 1\/2\" Diamond Quilting with non-slip gripper top. Cotton Shell with 15 mm foam and 300g fiber fill ..\nA belt that will move with you \u2013 you will never have to choose between comfort and keeping your bree..\nFashioned in buttery soft leather with C-40 Thinsulate lining, stretch side panels and extended knit..\n12\" tall. Set of 4. R-S-V-P..\n12\" tall. Set of 8. A-K-E-H-C-M-B-F..\nThe perfect coach's coat! Waterproof and breathable(5000\/5000) three-quarter length jacket features ..\nExpress your love for horses with the luxury inspired Fashionista Belt. Made of finely textured leat..\nOVATION FUN SPUR STRAPS, 19\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzafych b/data_all_eng_slimpj/shuffled/split2/finalzzzafych
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index 0000000000000000000000000000000000000000..3c4ebd9a5ba673fae87cf1f4273a18c7a4a11078
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@@ -0,0 +1,5 @@
+{"text":"PITTSFIELD TOWNSHIP -- Midland Christian's volleyball team made a semifinal run in Saturday's tournament at Washtenaw Christian.\nThe Crusaders went 2-1 in pool play to advance to the Gold Division bracket semifinals, where they dropped a 25-13, 15-18 decision to Rudolph Steiner school from Ann Arbor.\nTori Aultman led Midland Christian with 33 kills and 31 digs on the day, while Emily Coston had 15 kills, 18 digs and 11 aces. Brooke Duley added 35 digs, 31 assists and 12 aces, while Samantha Hartwick had 17 assists, 13 aces and 12 digs, and Ali Zietz had 11 aces.\nThe Crusaders return to action on Tuesday, hosting Saginaw Arts and Sciences Academy at 6 p.m.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"bebop spoken here: Janet Cook dies and possibly Jazz Journal with her.\nJanet Cook dies and possibly Jazz Journal with her.\nEddie Cook, former editor of Jazz Journal - a publication that has ran since 1948 - announces that, with the death of his wife and current editor Janet, the February issue of the magazine will be the last one.\nThe magazine was unique in having only three editors in its 60 year history - founder, the late Sinclair Traill being the first.\nAlthough often derided for being too staid, the publication was, nevertheless, universally regarded as being printed on tablets of stone and few jazz fans didn't subscribe at some point in their life.\nEvery jazz writer of merit contributed at least once whilst scribes such as Steve Voce and Stanley Dance were, across the years, irrevocably linked to its pages.\nIt is hoped that someone or several will come forward and take over.\nJanet Cook died on January 23.\nThe Jazz Baroness - BBC4 - 75mins.\nTHE LAD v HCW at The Side.\nIn a Sentimental Mood - Not!\nVic Lewis - Almost the last of 'The Few'.\nZoe Rahman @ Customs House, South Shields.\nThank You from Eric Boeren.\nSpelk at The Side Caf\u00e9 Feb 9.\nGolden Age of the Corner House Part 2 - Chris Y.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"It's part of what we do as a family,\" said Sister Mary Agnes. \"Running is part of our daily routine. So when (the kids) come here, just as they are introduced to every other thing as part of our routine, they're introduced to running. \"\nSo they all went over to the Spaulding High School track and ran around the track. The boys said they could outrun Rosie, but as they ran around the track, they dropped off one by one. The oldest boy kept going. Finally it got dark and Sister Maximilian stopped them. The boys were ready to die, but Rosie wasn't even winded.\nAll the children were interested. The nuns trained them, starting with walking as Rosie had been trained. And then worked them up to running. That fall of 1996 they ran their first 5K. Sister Mary Rose picked up a brochure from the Red's Shoe Barn 5K. Her feeling was that if they were running with the children every day, why not try a road race. At the time, the nuns and the children were running four or so miles, so they were covering a longer distance than a 5K.\nRegistration and race number pickup will be from 7:30 to 8:30 a.m. on Sept. 5 with the 5K starting at 9 a.m. A children's Fun Run will follow after the finish of the 5K road race. For more information visit www.runningnuns.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Corinthia is a collection of hotels located predominantly in Europe, the Middle East, and Africa. Each Corinthia hotel is created as a tribute to local architecture and cultural traditions in some of the most exciting places in the world.\nIn addition to focusing on customer satisfaction, the company culture supports family values and positive encouragement among teams so that workers enjoy their jobs in an inspired yet warm-spirited fashion. They see training as vital in creating these excellent customer and employee experiences.\nApproximately five years ago, the organization established a five-tier learning strategy to engage every level of the workforce in ongoing training. They chose SAP Litmos as the platform to support their strategy because of its ease-of-use, course library, and anytime-anywhere on any device capabilities.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Membean keeps a log of all the activities you perform while logged in. The log is available to both you and your teacher. This article will show you how to access the log from your student dashboard.\nScroll down to the All Learning Sessions part of your dashboard and click on Activity Log.\nFrom here, you can view time stamps to your assessments, training sessions, and click on the details of your training sessions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzagvpc b/data_all_eng_slimpj/shuffled/split2/finalzzzagvpc
new file mode 100644
index 0000000000000000000000000000000000000000..064b2d597b2ebc7e1c34b5609f2d0de2bd749aec
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzagvpc
@@ -0,0 +1,5 @@
+{"text":"The accompanying card asks me to 'wrap up something good for someone who deserves a bit of goodness'. So my very recently redundant friend (anyone in the North need a brilliant design\/integrated account director on a short-term contract?) will be getting her present wrapped in little robins or snowmen this year.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This 12x20 High Side Plus Gambrel storage building features 6' wide double doors, (1) 24x36 vertical sliding window with grids, decorative garden gable vents, lifetime architectural shingles with TechShield roof sheathing, 30 year SmartSide siding, and a 50 year AdvanTech floor. The roof has a snow load rating of 90 PSF. Ask us about completing your gravel base for you!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Not too long ago, Hospital & Health Networks announced its 2011 Most Wired awardees. Congratulations to those organizations who earned a spot on the list, especially the 72 Hyland customers. But there's a bit of a catch.\nAccording to a semi-official reader poll on HIStalk, 43 percent of respondents believe that ultimately, no one benefits from this kind of award. Also, 23 percent believe the CIOs benefit, 21 percent the magazines, 11 percent the vendors. As for patients? A whopping 2 percent.\nI freely admit my cynicism. Blame my ten years in journalism. Blame genetically rooted New England pragmatism. Blame it on hype fatigue, advanced grouchiness or acquired skepticism syndrome. Whatever the underlying cause, I get it. And well, I'm with you.\nI will say, however, that with this award in particular, we've seen movement from measuring what technologies organizations have just deployed, to actual usage of those technologies. The requirements to be on the list include seeing more outcomes-based metrics (although technology's distinct role in those outcomes can be debatable).\nAnd that's the thing: It's not about the award. In many ways, it's not about the technology. Certainly, not exclusivity. The culture of safety and service that your technology supports, transforms and hopefully enhances \u2013 that's the story. Technology should support your organization's mission rather than become part of it. It should help fuel smarter, speedier decisions, enable more informed care and make life easier for patients and caregivers alike.\nBut what does this all mean to you? For those on the list, congratulations. Here's hoping that this is just a marker on your path to delivering the best possible care and service. May there be more.\nFor those of you not on the list, maybe you don't need to be. Maybe you don't care about this sort of recognition. Maybe you already deliver superior care.\nWhat I do ask, however, is that you consider those award recipients for a moment. Reach out to those that have earned a strong reputation for high standards of patient safety and service. Learn from their successes and mistakes. Consider their deployment strategies and technology partners. And most important, ask yourself how technology might help your organizations meet \u2013 and exceed \u2013 your goals.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Defining feminism can be somewhat complex. Having been raised in a feminist household, Orlando Figes feels that he has a pretty good understanding of it, however. For him, it is about achieving equality between the genders on an economic, political, cultural, and social level. Feminism is, however, about empowering women so that they can choose to live free from oppression.\nFeminism is global, but it means different things in different places. This is because women have differing inequalities depending on where they live. In a place such as the United States, for instance, the main struggle is the gender pay gap. In a country like Somalia, for instance, the gender pay gap will no doubt become an issue at some point, but there is a greater focus right now on issues such as Female Genital Mutilation (FGM).\nOne thing unites all women, however, and that is that, around the global, they are disproportionally discriminated against. This transcends skin color or ethnic origin. Each woman has a specific issue that she feels of importance, but what brings them all together is that they continue to be treated as lesser beings.\nFeminism and society as a whole.\nFeminism also relates directly to men. This is because, in order to achieve equality, there has to be a fair balance between men and women alike. The ultimate goal is to liberate each individual. This means that any movement that is designed mainly for boys and men is also part of feminism. This is particularly true for gender sensitization. The clothing campaigns are good examples of this, where boys are told to wear blue with slogans such as \"I want to be an astronaut\", whereas girls wear pink with slogans such as \"I want to be a princess\".\nWomanhood, which is spiritual, intellectual, mental, and physical, has to be openly discussed. It is also something that transcends the genders. Men must accept that they are somewhat victims of the patriarchy themselves, with society expecting them to be the strong breadwinners, free from emotional displays. In essence, men sell themselves short by not understanding they are equal.\nWomanhood is rising and it is becoming more acceptable to suggest that women are people too, no matter how radical that notion continues to be. Men now openly declare themselves to be feminists because they understand it does not mean they have to give up on any rights. Rather, it means that they can be who they want to be themselves, and that women can do the same.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"and the days when it is crazy busy for both of us. Proud of her and all that she manages to accomplish at a high volume, high pressure job..but I really miss my wife today and in spite of it being my 4 year anniversary and a day that should be celebrated.; All I want to do right now is have her hold me while I cry and tell me that everything is going to be okay. I miss her so much.\nI suspect my grief and the way I have been off lately is all about my mother. Seeing the nephew that caused me so much trouble with his betrayals this morning probably didn't help either. I felt spiritually sick because I had no forgiveness in my heart for him. I wanted to leap up and beat him bloody. I don't like feeling that way anymore, but I am not ready or able to let it go.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaiaxa b/data_all_eng_slimpj/shuffled/split2/finalzzzaiaxa
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index 0000000000000000000000000000000000000000..421ad94633f89a2d6c1e651a5c4a35e513859e61
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaiaxa
@@ -0,0 +1,5 @@
+{"text":"Nations Press health PAY ATTENTION : YOU SHOULD NEVER EAT CUCUMBERS AND TOMATOES IN THE SAME SALAD!\nPAY ATTENTION : YOU SHOULD NEVER EAT CUCUMBERS AND TOMATOES IN THE SAME SALAD!\nVegetable salad is everyones favorite since it is made up of all fresh veggies, but here's the thing they are not aware of.\n1. Eating fruit after meals is not a good idea \u2013 fruits need a lot of time to be digested and should never stay long in the stomach. Eating them after a meal will result in \"wine\" in your stomach that can lead to acid reflux and other digestive problems.\n2. Mac and cheese is a popular meal in the USA, but one that should be avoided nonetheless. Macaroni are rich in starch, which has a different digestion time than protein, so the delay will lead to inevitable fermentation and further digestive problems. The same goes for macaroni and meat.\n3. Meat and cheese omelet is a favorite meal of millions, but you should avoid putting too much protein on the same dish. Choose only one type of protein per meal.\n4. You should never mix bread or noodles with orange juice as the acid content required to digest the juice can destroy the enzyme responsible for starch digestion.\n5. Vegetables and cheese is another no-no combination \u2013 eating it will only result in bloating.\n6. You should never eat melon and watermelon together, as these fruits are meant to be consumed alone, not in combination with any other fruit.\n7. Many people love eating bananas and milk, but this combination will significantly slow down your digestion.\n8. People usually add fruit to their bowl of yogurt for breakfast, but this combination will slow down your digestion and harm your intestinal flora. This especially goes for adding pineapple to your yogurt, which boosts the active ingredients that cause food poisoning.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On the Hill, Rep. Lamar Smith (R-Texas), the chairman of the House Judiciary Committee, and such former NFL players as Reps. Heath Shuler, D-N.C., and Jon Runyan, R-N.J. have all met with players to discuss the impact of a lockout on the economy and the overall jobs crisis.\nThe league spent a whooping $675,000 lobbying Congress in the first Online Blackjack quarter of 2011, including $60,000 paid to Elmendorf Ryan, which is registered to lobby solely on labor issues. The NFLPA also dished out $60,000 to Executive Director DeMaurice Smith's former firm, Patton Boggs LLP, to petition Congress on antitrust and labor concerns. For the league, this number is up from the $545,000 spent in the first quarter of last year, though the issues petitioned remain unchanged. However, the addition of Elmendorf Ryan seems to have been solely in anticipation of a labor fight, as the firm was not registered to lobby on behalf of the league this time last year. The Players' Association, on the other hand, spent significantly less in Q1 2011 than reported in the same quarter last year, when Patton Boggs was paid $110,000 to fight for the players' interests.\nEven NFL players need a lobbyist\u2026 except this lobbyist is really more of an \"educator.\" Since he's not currently registered under the LDA, Joe Briggs, whose title is \"Public Policy Counsel,\" says he's not pushing legislation, but he is advocating on behalf of NFL Players.\nHe's more of a go-between for football players and Members of Congress \u2013 who often share interest in the same issues, and are looking for mutual help in pushing pet projects. Since DC is all about who you know, and how you can get to those people, having a non-lobbyist Congressional liaison such as Briggs makes sense in a member-based organization such as the NFL Players Association.\nThe NFL is also listed as having outside lobbying help, with representation by Capitol Hill Strategies, Covington & Burling, Elmendorf Strategies, The Glover Park Group, and Dudinsky, Lisker, & Associates. Issues that the NFL is interested in include (among others): Labor\/Antitrust\/Workplace, Alcohol\/Drug Abuse, Communications\/Broadcasting\/Radio\/TV.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We work alongside your team to provide customized bookkeeping services for you.\nWe use our knowledge and experience to get you the highest tax refunds available.\nWe offer affordable and accurate tax preparation for your business.\nProfessional payroll services for both small businesses and corporations.\nWe can help you to start your new business.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our Iconic styles are a modern twist on our classic silhouettes, refreshed with added functionality. The carry-on compliant Miller Bag has six roomy slip pockets and a zip-top closure. The back of the bag features a pocket that unzips to become a trolley sleeve. Hit the road, girl!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Butter Oil or Clarified butter is milk fat produced from butter. Water is separated from the milk solids. Because it has a higher melting point than butter, butter oil is preferred in some cooking applications and also has a longer shelf life than butter.\nOur anhydrous milk fat is made from fresh pasteurized cream. During production practically all of the moisture and nonfat solids are removed resulting in a product with 99.8% milk fat and very little moisture. Anhydrous milk fat gives creaminess and natural dairy flavors to finished products and is more heat stable than butter and has a longer shelf life than butter.\nWe use only fresh pasteurized sweet cream milk which is churned to create our unsalted butter, which is perfect for everyday cooking and baking. 80% fat & 82% fat options are available as well as lactic and non-lactic butter.\nWe use only fresh pasteurized sweet cream milk which is churned to create our salted butter, which is perfect for everyday cooking and baking. 80% fat & 82% fat options are available.\nWe use only fresh pasteurized organic sweet cream milk which is churned to create our organic butter, which is perfect for everyday cooking and baking. No additional coloring material is used.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzajkku b/data_all_eng_slimpj/shuffled/split2/finalzzzajkku
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index 0000000000000000000000000000000000000000..34b7bd3afb0e172c6b5b729217257d1d2bf0af19
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@@ -0,0 +1,5 @@
+{"text":"This is the first episode of our three part series taking you through the huge personal and logistical endeavour that saw our three Bornean orangutan transferred to their temporary home at Orana Park, Christchurch.\nOur male orangutan Charlie, along with our two females Melur and Wanita were successfully anesthetised and crated through a co-ordinated team effort that involved our keepers, vets, maintenance and other key staff members. Our primate team leader Amy with lead senior primate keeper Craig and senior vet An flew to Christchurch to be there for our orangutan's arrival.\nWatch as our Orangutan take their big journey south!\nLots of delicious snacks were prepared in a care package to go with them on their flight including banana \u2013 one of their favourites!\nTheir travel to Orana is part of a two-year staycation while their home and one fifth of our zoo is redeveloped as part of our South-east Asia project. This will see our orangutan return to an immersive new rainforest environment with other South-east Asian species like our Sumatran tigers and Asian small-clawed otter.\nStay tuned for our next episode tomorrow, where we meet up with our team and orangutan in Christchurch.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Rachel Grey breathed out a puff of condensed air as she let blue flame envelop her, soaring below the clouds as flecks of snow fell all around. A beacon of cobalt fire, she lit up the night, or at least she would have if the town's inhabitants had been aware of her presence.\nThis was the town the Purifiers had turned into a graveyard, where they had clashed with the Marauders over the fate of a child who had ended up dead. Not here, or by their hand, but at the hand of one of the X-Men's own. Bishop, the X-Man who had come back in time to stop a traitor, had become one in turn.\nAnd Rachel knew in her heart that she should have stopped him. How many times had she been inside his mind? How easy would it have been to find the murderous intentions lying in wait beneath that calm and composed exterior? But however much she disagreed with him, she had resisted the temptation to dig deeper. Bishop was an X-Man, a man who had come from a future where mutants had been rounded up in concentration camps and guarded by Sentinels. The same was true of Rachel.\nBishop had been willing to kill a child that was barely a few days old, to prevent his future from coming to pass. Would Rachel have been willing to do the same?\nThat was a question she was willing to leave unanswered.\nRachel touched down on the town's main street, and it was clear to see Cooperstown still bore the scars of six months ago. A dusting of ice and snow couldn't hide the remnants of rubble, the holes and indentations left in the buildings and roads by bullets and energy blasts. The storefronts were closed and a half-finished stone pillar stood behind Rachel, a monument to the children the town had lost to the massacre. It was barely dark, but already the town was quiet. Wary and weary. Rachel knew the feeling.\nThe Tomorrow Institute, 3 Months Ago.\nRachel rose to her feet, brushing her dirty unkempt red hair back even before Logan entered the shadowy confines of the room, which reminded Rachel far too much of a Shi'ar spaceship for her to be comfortable with it.\nShe, Lorna and Alex had only been back on Earth for a few hours, and Rachel was in dire need of a shower, clean clothes and a meal that consisted of more than recycled space sludge. But more importantly, Rachel needed answers.\nLogan stepped in, dressed simply in a t-shirt and jeans. He looked up at Rachel, tired and grim. For a moment, Rachel picked up a memory and Logan was in costume.\nRachel's eyes widened and she took a step back, recoiling from Logan's thoughts. She brought a hand to her lips.\nThe breath escaped from Rachel's lungs, but the worst thing, the worst thing about what she was hearing, was that all she could think was, not again. She began to shake involuntarily, she felt her emotions take hold and struggled not to let her powers run rampant.\nRachel could feel the lump rising in her throat, the sudden sense of dizziness and nausea threatening to overwhelm her, but she wouldn't let the tears out. She was fire and fury and vengeance. She wouldn't cry. But Logan wasn't finished.\n\"Who else, Logan?\" she asked pointedly, biting her lip and letting her long nails draw blood as she dug them into her hands. She relished the sting.\nThe silence was palpable. Logan kept his distance. Rachel saw him swallow.\nThe tight feeling in Rachel's chest grew.\nLogan looked at her apprehensively.\nRachel let loose. A telekinetic wave flew off her body, tearing off panels of circuitry and wiring, and slamming Logan into an indigo-violet wall. Her flaming Phoenix emblem formed over her left eye as a psychic cry came next, bringing Logan to his knees as he clutched at his head. He staggered up again.\nBut Rachel was already gone.\nThe Tomorrow Institute was lost behind her as a flaming azure raptor unfurled its wings above the Atlantic Ocean, before diving down and streaking across the water, brighter than the moon above.\nShe was fire and fury and vengeance, but three more slivers of ice had just entered her soul.\nRachel shook her head as if that would shake loose her losses, the guilt, and the regret. Nevertheless, Rachel could feel her chronoskimming power at work, the currents of time shifting in their most unnatural direction. By sheer force of will, Rachel's spirit was skimming along the timestream, until the past was the present and her astral form was free to roam again.\nRachel instantly shielded her eyes. All she could see was smoke and dust and bright orange pyres until she caught sight of a familiar cloud of purple brimstone, followed by an angelic figure swooping through the air. Above her was the Blackbird, which soon landed. Rachel flew over Cooperstown as her father and Emma Frost emerged from the craft. She watched as the X-Men rescued as many survivors as they could. Soon, she saw the Angel heading for the town's hospital, the rest of the X-Men following in his wake. Rachel dove after them.\nHer astral body glided effortlessly through the hospital walls, and into its hallways filled with flickering lights, rubble, metal trollies and charred corpses freshly abandoned. Rachel followed the path of destruction carved by the Purifiers until she reached the hushed tones of the X-Men. She joined them as burning rage splayed across the faces of Angel, Emma and her father, as newborns smouldered in nearby cribs. Nightcrawler crossed himself as he began a prayer.\n\"How do you know?\" asked Rachel's father, Cyclops, his expression largely hidden behind his ruby quartz visor.\n\"How sure are you this baby was the mutant Cerebra picked up?\" Cyclops questioned.\nEmma paused, and again it seemed as if her blue eyes had met Rachel's green ones. And Rachel wondered\u2026she possessed a fraction of infinity, of creation itself. Was it possible she could reach out and touch Emma's mind, possible to prevent the tragedies that were about to occur? Possible to stop what little hope they had from dying?\nA sock to the jaw put an end to Rachel's plan. She roared in unbridled frustration as six months ago slipped back into now, and her mind was forcibly yanked back into her body. The heat and fury of the Phoenix burst back into existence too late, as Rachel felt the tranquiliser darts imbed themselves in her back. She caught one glimpse of the dull emotionless eyes of her Sentinel captor, as an ice cold feeling spread through her body.\nRachel fell to the ground as the flames around her flickered away.\nPosted in Divided We Stand.Tagged Omega Sentinel, Rachel Grey, Rictor, Siryn, Thunderbird III, Wolverine.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Prince Charles famously called his old school Gordonstoun \"Colditz in kilts.\"\nWhen it comes to educating your children, only the most expensive schools in the world will do for royals.\nWe've listed some of the most prestigious schools in the world attended by reigning monarchs and future world leaders from Jordan to the UK.\nWhile some future monarchs studied at internationally renowned institutions like Eton and Harrow or elite boarding schools in Switzerland, others \u2014 like King Abdullah II of Jordan, were sent to smaller schools to start their education.\nFrom St Edmund's in Surrey to Lakefield in Ontario, here are 13 private schools that have been given royal approval, ranked by their annual tuition fees for full boarders.\nHill House International Junior school was founded in Switzerland in 1949, but a branch in London opened in 1951. It was once attended by Prince Charles \u2014 the eldest son of Queen Elizabeth II \u2014 before he moved on to Gordonstoun to complete his secondary education.\nThis small independent preparatory school in Surrey has a surprising royal past. St Edmund's was once attended by King Abdullah II of Jordan before moving to the US for high school.\nFounded in 1899, this French school has attracted royals from India, Thailand, Morocco and Denmark. It's most recent blue-blooded student was Joachim of Denmark, the youngest son of Queen Margrathe II of Denmark.\nMarlborough won Tatler's Best Public School award in 2015, and was attended by the Duchess of Cambridge. It is a favourite of the family, counting Princess Eugenie \u2014 daughter to the Duke of York and eighth in line to the British monarchy.\nPrincess Beatrice, the Duke of York's daughter and seventh in line to the throne, began attending St George's school in 2000. She completed A-levels in Drama, History and Film Studies, and was even named Head Girl.\nBenenden is the only girls' all-boarding school in the UK, and counts Princess Anne \u2014 Queen Elizabeth II's daughter \u2014 as one of its former pupils.\nGordonstoun's most famous past pupils are Prince Charles, and his brothers Andrew and Edward. Charles was the first British monarch to sit public examinations, taking his GCE O-level exams at Gordonstoun in 1965.\nCharles didn't enjoy his time at the notoriously strict Scottish institution, famously calling it \"Colditz in kilts.\"\nAnnual fees:\u00a336,838 ($45,442) for international boarders.\nPrince Andrew, the Duke of York, left Gordonstoun before completing his secondary school, and graduated from Lakefield in 1978. The Canadian boarding school is popular with royals, and also counts King Felipe VI of Spain as a former student.\nEton College is the UK's best-known private school. Both Princes Harry and William are former pupils, as well as former Prime Minister David Cameron, Foreign Secretary Boris Johnson, and Eddie Redmayne.\nHarrow is popular with royals all over the world, and counts Sheikh Tamim bin Hamad Al Thani \u2014 the current Emir of Qatar \u2014 as one of its former pupils, as well as members of the Jordanian royal family and Lord Nicholas Windsor, Queen Elizabeth II's first cousin. The institution is famous for its strong academic record, fierce rivalry with Eton College, and boater hats.\nThe institution is famous for its strong academic record, fierce rivalry with Eton College, and boater hats.\nWhile many royals choose to study in Europe, some prefer an American education. King Abdullah II \u2014 the reigning monarch of Jordan \u2014 completed his high school education at boarding school Deerfield, before moving to the UK and attending the University of Oxford.\nAccording to its website, West Heath School caters for \"a diverse range of needs for boys and girls age 11-19, many of whom have emotional and social communication difficulties.\" It counts Diana Spencer \u2014 the late Princess of Wales \u2014 as one of its former pupils.\nThe Institut Le Rosey has the biggest royal pedigree on our list, and counts the Shah of Iran, Prince Rainier of Monaco and King Farouk of Egypt among its past alumni.\nIt is also the most expensive school in the world, with fees of more than \u00a3100,000 ($124,028) per year.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The 3 star Tropical Ibiza Hotel is a great complex for those who want to be close to the lively resort centre and the beach. The centre of San Antonio is full of bars, restaurants, shops and the most important, nightly entertainment. There are plenty of nightclubs to keep you dancing from dusk until dawn.\nThe pretty port is a short 5 minute walk from the hotel and you can sit and watch the boats sail in and out with a refreshing drink in your hand. The sandy beach is also 5 minutes walk away. On the beach you can relax on the sun loungers and parasols provided, paddle in the sea or try the watersports on offer.\nTropical Ibiza Hotel features a large swimming pool which is surrounded by sun beds to relax on. Right by the pool is a smaller one sectioned off for the younger holiday makers. Take a break from relaxing by the swimming pool and enjoy a game of crazy golf with your friends and family.\nThe staff at the reception desk can assist you with internet access, foreign exchange, luggage storage if you have a late departure, laundry facilities, car rental, bike rental and there is a concierge desk to help you with tourist information and excursion bookings.\nTropical Ibiza Hotel has a restaurant which serves breakfast in the style of a buffet and includes cooked and continental dishes like eggs, sausages, bacon, pastries and cereals. If you would like a picnic bag to take out on an excursion or to the beach with you, just ask at reception and one can be made up on request. Drinks are served at the main bar throughout the day and evening.\nThere are a total of 142 rooms at the Tropical Ibiza Hotel, all of which are air conditioned. Each room has a mini fridge to keep your drinks cool, satellite TV, desk, shower, hairdryer, telephone, toiletries and complimentary Wi-Fi. they all have furnished balconies so you can admire the view.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Police have reopened Route 250 in the village of Webster.\nThey closed the road for a short time this morning while they looked for a man living in the area who was reported to be very upset over a breakup. The man was found and police determined there was no threat.\nDuring the search, the nearby Spry Middle School was placed on lockout, but classes continued as normal.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzajrvx b/data_all_eng_slimpj/shuffled/split2/finalzzzajrvx
new file mode 100644
index 0000000000000000000000000000000000000000..80aa4b5b470b6e14da2de50f70623c52c1275976
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@@ -0,0 +1,5 @@
+{"text":"Click here to be notified by email when Labradorite Crystal Moon Sterling Silver Necklace becomes available.\nTranscend to your higher self with this labradorite moon necklace! Known to work with magical vibes to enhance your self-practice, the labradorite necklace increases your connection to your inner self each time you wear it. The petite crescent moon's size lets you easily wear it from work to yoga class and then out for some fun with your friends.\nThe labradorite moon necklace is 0.5\" tall (1.81 cm) by 0.13\" wide (0.33 cm). Its flat cable Italian sterling silver chain is 2 mm tall by 1 mm wide. You choose the length of chain to fit your style!\nI make each piece with positive vibes and love. All materials are nickel free, lead-free, and hypoallergenic. Natural labradorite is faceted and shaped into the crescent moon shape.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Have you ever fantasized about money raining down on you? Well, we can make that fantasy come true when you choose the Money Machine to be a part of your next event!\nOur Money Machine is one of the most sought out money blowing machines in the market! This cash cube has flashing lights for special effect, an electronic message display, and a digital clock. Being constructed out of polish steel gives our Money Machine a very sleek and high tech look that will sure attract your guests.\nSpecial Events and much more\u2026.\nThe money machine is the #1 product to draw crowds to your business or booths. People are curious because they see money blowing around and then the excitement starts when people are in the machine trying to grab the money.\nUse play money with coupons or incentives good only for your store or booth.\nTry a drawing to obtain customer contact information draw names to enter the Money Machine!\nSpin the prize wheel to win a chance in the money machine!\nFill the Money Machine with play money, real money and\/or coupons or give out promotional items with your branding information.\nUse a question & answer section\u2026if the answers are correct\u2026the customer enters The Money Machine to win cash or Prizes!\nPlay the Action Lottery Game\u2026choose 6 numbers; anyone who grabs all 6 numbers wins cash or prizes.\nBy placing the Money Machine right in the middle of the room at any these themes event will guarantee a crowd and generate excitement! Who wouldn't be excited about money raining down on them?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There are 11 Skating Rinks in or near Athens, Louisiana LA.\nBossier Hot Wheels South is located approximately 39 miles from Athens. They're one of the best on the market. Call them at (318) 747-7101.\nHot Wheels Skating Ctr is located approximately 44 miles from Athens. Contact information: 1334 W Hillsboro St. You can call them at (870) 881-9338.\nHot Wheels Skating Ctr is located approximately 41 miles from Athens. Contact information: 4316 S Jackson Ave. You can reach them at (870) 881-9338.\nHot Wheels Skating Palace is located approximately 39 miles from Athens. They are a nice Skating Rink. If you need more information, call them: (318) 747-5095.\nHot Wheels Skating Palace is located approximately 50 miles from Athens. They are regarded as one of the best Skating Rinks in Athens area. Phone number: (318) 686-6095.\nHouse Of Wheels is located approximately 39 miles from Athens. We recommend their services. You can reach them at (318) 747-6553.\nMud Bugs is located approximately 39 miles from Athens. If you need a good Skating Rink near Athens, contact Mud Bugs. You can reach them at (318) 752-2847.\nNatchitoches Skating Ctr is located approximately 61 miles from Athens. A friendly Skating Rink. Their phone number is (318) 357-8507.\nSkatetown is located approximately 56 miles from Athens. Skatetown is located at 3151 Sterlington Rd. You can call them at (318) 387-7051.\nSkatetown is located approximately 53 miles from Athens. Why don't you give them a try?. Their phone number is (318) 388-2405.\nSkatetown District Office is located approximately 51 miles from Athens. If you need a good Skating Rink near Athens, contact Skatetown District Office. You can call them at (318) 397-3341.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The whole family will have fun with this 5 day package including whitewater rafting, horseback riding, a guided hike and a chuckwagon dinner.\nSpend 5 nights in your own fully-equipped condo in Teton Village, about 15 minutes from downtown Jackson. The village offers a variety of dining opportunities, a little bit of shopping, a few fun kids summer activities, and easy access to trails and the Moose Wilson Road portion of Grand Teton National Park. There is a town bus that offers convenient rides to and from town as well.\nFour full days will be filled with adventures for your family. Go whitewater rafting down the Snake River Canyon (kids must be 6 y\/o), on a horseback ride followed by a delicious steak dinner, and on a guided hike in nearby mountains. You'll have plans one evening to go to a chuckwagon dinner via covered wagon and plenty more time to explore the area on your own.\nComfortable 2 bedroom\/2 bath condos that can sleep up to 6 people. These condos have a full kitchen and a washer\/dryer, making it ideal for longer stays.\nThe 8-mile Whitewater Adventure on the Snake River is full of excitement at every turn! The Snake is a class II\/ III river, a great beginner\/ intermediate river for kids as young as six and people of all ability levels. Choose a standard 14-man raft or the more adventurous 8-man raft to plunge through the famous Lunch Counter and Big Kahuna rapids! Multiple departures all day long. This trip is approximately 3.5 - 4 hours round trip from Jackson.\nThe cast and crew at the Bar-T-5 Covered Wagon show you the true meaning of western hospitality. Travel by horse drawn covered wagons as you are swept away from the Town of Jackson into beautiful Cache Creek Canyon. Keep your eyes open, because you never know who you may encounter on the trail. With your own wrangler on the lines they will personally guide your team of horses up the trail. When you get to camp you go through the \"chow line\", much like you might have done in the Old West - and enjoy Dutch Oven style cooking that is delicious and hearty. The Bar T 5 band entertains you with western music and cowboy humor that is family friendly and will have you stomping your feet. Complete your evening with a peaceful wagon ride out of the canyon, following the setting sun back into Jackson just as the old time cowboys did.\nHead out from our ranch into the Bridger Teton National Forest to enjoy views of the Hoback area, Snake River Canyon Area, and the Teton Range. Then come on back to the ranch for a delicious steak dinner.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"So Windows updated - its a big one - new installation of Windows 1703 Creator - once complete I open SW - no problem, until I go into a drawing - dimensions everywhere, annotations missing, skewed, etc. I figure Windows updated my graphics card driver (they did) roll it back, but still there. Poked around until I had the PC good and screwed up - uninstalled EVERYTHING - updated all windows items, corrected card driver cleaned registry, clean SW install - still screwed.\nOn every other PC we have (not on 1703) the drawing is correct, has a lot of notes, weld size, details, etc.\nIssue on this machine seems to be limited to drawings, specifically dimensions notes, annotations.\nI am not trying to better understand what it is that is specific to my machine that appears to create the issue. It seems to mess with annotation \/ dimension only - parts and assemblies appear to be fine.\nI've seen this before upon opening a drawing when it was perfectly fine the last time I had saved it. Try just hitting Ctrl + Q to force a rebuild of everything. That usually works for me, but it doesn't explain why it is happening.\nYup, big problems . . .\nWindows 10 Creators Update - Issues?\nFYI, the latest Fall Creators update is version 1709. The OP was back in June referring to the previous \"creators\" update.\nThe Windows \"auto close function\" from my screen shot was from last week - I think there are still some issues.\nI think there should be another update this week, I'm hoping it will help.\nAre you on 1703 or 1709? 1709 wasn't officially released until Oct. 17, 2 days ago.\nSounds like I'm probably on 1703 - I ran the update over the weekend or on Monday night.\nLast night was spent moving to 2018 SP0.1, tonight will hopefully mean going to 1709.\nI got the update Tuesday. So far, no Solidworks issues but I am having problems with our ERP (Infor Visual Manufacturing).\nBut we're using an ancient version of it from 2011. I'm betting if we were running the latest I'd be fine but it's totally out of my control.\nAndy Sanders - You just made my day a whole lot better, I thought there was supposed to be an update this week, but I was expecting it to be toward the end of the week.\nI will be updating as soon as I walk in the house tonight.\n\"Upgraded\/Updated\" to 1709 last night, haven't been able to do much testing yet, but I plan to put it through a torture test over the weekend.\nfingers crossed . . .\nI updated to 1709 and still have the issue above. I am getting ready to rollback to 1511 so I can get work done.\nHave you experienced any Blue Screen of Death \"auto-closes\" when using SOLIDWORKS with 1709?","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzajwvd b/data_all_eng_slimpj/shuffled/split2/finalzzzajwvd
new file mode 100644
index 0000000000000000000000000000000000000000..1a8eb8563562061f3eb1322c539e419aba546724
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Academic India Publishers is committed to ensuring that your privacy is protected. Should we ask you to provide certain information at the time of confirming your order, you can be assured that it will only be used in accordance with this privacy statement.\nAcademic India Publishers may change this policy from time to time by updating this page. This policy is effective from February 2012.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"No Pronouncation added yet for \u2663gladys\u2663 gladys Please add a pronounciation!\nNo Meaning added yet for \u2663gladys\u2663 gladys Please add a Meaning!\nNAME RELIGION Click Here For More Religion use of baby name \u2663gladys\u2663 gladys !\nThese are pictures of people with the name \u2663gladys\u2663 gladys Help us put a face to the name by uploading your pictures to namesof.com !","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Primus Omnifuel stove can run off virtually any fuel - LP Gas, naptha, gasoline\/petrol, diesel, kerosene - even aviation fuel. Designed to work well in the most demanding conditions; extreme altitudes and high or low temperatures. A real gem! Mine came from UK importers, Rosker.\nA self-inflating mattress is essential for comfort and insulation. UK based manufacturer Multimat make the best! I used a Summit 25\/38 mat that's a comfy 38mm thick at my torso, weighs just 895g and packs into a very compact fleece lined sack that doubles up as a pillowcase.\nI needed a down sleeping bag that was lightweight (880 gms) and would pack really small (25x16 cm before compression). The Vango Venom 300 fit the bill perfectly. With a comfort range of 0 to +25 \u00b0C, it kept me cosy throughout the journey.\nLightweight camping shouldn't be without its comforts. I use a Multimat Chair Converter so that I can use my Multimat sleeping mat as a seat. Its packed weight is just 495g and it's only the size of a small folding umbrella.\nFor convenience and to be prepared for the worst, I carry a tasty selection of Travellunch high energy, freeze dried meals. Made in Germany by Reiter, each meal weighs just 125g before adding water... even my low powered Matchy can carry that!\nI will use a robust and durable Vango stainless steel cook kit on this journey. They are manufactured tough to provide years of use with little or no maintenance - perfect for me! The 1 person set includes a frying pan, 2 pots and a cup that all nestle into each other to save space.\nI've found Leatherman multi-purpose tools invaluable in the past. The Leatherman Blast saw, scissors and bottle \/ can opener tools are ideal when camping. The file and needlenose pliers come in handy on my bike too. An essential piece of kit.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bowker Motor Group has agreed a two-year corporate partnership extension with AFC Fylde.\nThe partnership will see the Lancashire-based automotive retailer continue the sponsorship of the West Stand at the the Club's Mill Farm Stadium in Wesham.\nThe West Stand will carry Bowker Motor Group's name for another two years, up until the end of the 2019\/20 season. The original agreement in 2016 was AFC Fylde's first corporate partnership at the new Mill Farm Stadium. The partnership between Bowker Motor Group and AFC Fylde originally began in 2014 with sponsorship of the youth team kits.\nThe Bowker Motor Group Stand is the stadium's largest seating area; it accounts for 2,000 of the new stadium's total 6,000 capacity. The West Stand also includes the 290-capacity Bradley's Sports Bar; 80-seat Milano's Restaurant; Aroma Cafe and Churchill's conference and event facility.\nAFC Fylde's Mill Farm Stadium opened in 2016 as the centrepiece of the \u00a318 million Mill Farm Sports Village. The village is open 7-days a week and employs over 80 people. The facilities include 5-a-side and 7-a-side pitches; a full-size hockey pitch; a 3G football pitch, retail outlets, petrol station, restaurants and bars.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"So you want to learn how to work out? Good on you! Working out is simply one of the best activities one can do to make themselves a better person. It helps one keep fit and encourages a healthy active life style while also building confidence and self esteem, the necessary foundations of a well cultured anonymous.\n-The second goal is something a bit more substantial with time somehow involved (Ex: I wish to lose 10 pounds by the end of the summer, I wish to workout 30 minutes every day) try setting realistic, easier goals initially and once completing them set something harder.\nCardio improves the heart, tones muscles, burns calories and is extremely easy to start up and get into.\nIt's as simple as putting on a pair of shorts and going outside for a jog, heading to the local YMCA for a dip in the pool or maybe a ride through a park. There's really no set amount of time that would be perfect for an exercise for everyone but generally 30 minutes a day of exercise is decent.\nOnce you've gotten your stamina improved, try doing some advanced exercises. For example: When jogging or biking, try doing 15 minutes of regular jog\/bike and then do 5 minutes of sprinting or hard pedaling (or however long you can try) resume another 15 minutes of regular and then try to see how long you can sprint\/hard pedal again.\nHitting the weight room at the local gym might be a little bit intimidating with all these huge ripped guys, but remember that everyone has to start somewhere. You're not going to the gym to compete against anyone except yourself so don't worry about that and get ready to push it to the limit.\nTypically you'd want to work out 4-5 days a week for about an hour to an hour and a half (beginners will probably hover around an hour). My schedule follows as below, adjust accordingly to your life.\nAs a beginner, it'd be best for now to stick to the machines. Aside from the obvious safety concern, it helps set proper body movement so once you advance to free weights or bars you'll have a decent idea of how to perform it without machine help.\nNow what kind of workouts should one be doing?\nBare in mind that there are many ways of working the same muscle groups and it is advisable to switch workout techniques so that your muscles don't become accustomed to the same thing over and over. Generally, different movements, increasing weights slightly from your previous section or switching from machine to free weight to bar is enough to keep your body from getting used to the workouts.\nAs for how many sets and reps one should perform, generally for each exercise there should be 3 sets of 8 reps, with no more than 2 minutes of rest in between each set. If you want more of a pump, perform more reps on lighter weights and vice versa for heavier weights and fewer reps. Avoid less than 5 reps and any more than 12, with the last rep to whatever you're doing being the hardest you can perform.\nWorking out isn't an instant fix all solution, it takes a concentrated will on all aspects of life. Besides the time commitment, no workout plan will survive without proper nutrition, sleep and dedication to a goal that may take months or years to attain. All of this comes naturally with time and effort, and soon working out will be just as natural as sitting down and eating lunch.\nFirst part : \"I wish to skinnier\", if you don't see the typo, lucky you.\nAlso, would you mind not using the common workout vocabulary like we ought to know it, I can assume what \"reps\" mean but half of the exercise names are as obscure as the pages of a dictionary at 3am during a storm without a light. Or if you wish to keep using them, at least give a glossary?\nWeblinks are ok. I would have tried to include a \"home-health\" version. Pushups can work the triceps fairly well. Exercises with only the weight of your body can do a lot more than one would think.\nI'm sorry, but a lot of the information you have there isn't that great. The fastest way of shedding fat is to do high-intensity interval training, or HIIT. Alternating 1 minute jogging with 30 seconds sprinting for 20 minutes is all you need to do. Compared to steady-state cardio, HIIT burns more fat (and not as much muscle), keeps your metabolism raised throughout the rest of the day, isn't as tiring, doesn't take as much time, and is more fun.\nAlso, the weight lifting regimen you have posted is decent if you're looking for making your muscles bigger. If you're just looking to be big and show off, doing body part splits like that is fine. If you're looking to build strength, through, your lifting regimen should look substantially different. You should focus on compound lifts and movements, instead of training individual muscles. Three days a week with one day of rest in between is good, doing 5 set of 5 reps with each lift. Doing this will make the muscle denser and you'll have greater functional strength.\nIf you're seriously looking to get into shape, go to StrongLifts.com and do it for 3 months.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaowvh b/data_all_eng_slimpj/shuffled/split2/finalzzzaowvh
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index 0000000000000000000000000000000000000000..4fd5ee6e6c7825798eff87f59935df0a5e15dc45
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@@ -0,0 +1,5 @@
+{"text":"Always be present at the ground early to give yourself time to inspect the pitch and appurtences properly and to prepare for the game.It is suggested that you attend the ground at least 30-45 minutes prior to the relevant kick off time.Remember to make allowances for adverse weather conditions.\nThe clubs should make contact with you at least 48 hours prior to the game taking place to inform you of the venue etc. and you should confirm this contact with the club.Failure of this requirement should be reported to the Referees Appointment Officer or his assistant prior to the playing of the game.This also applies if you become unavailable for a game at short notice but this contact should ONLY be made by a call to the officers mobile numbers not by e-mail or text.\nThe Match Report form can be viewed on the WebSite once the teams have entered the information as required.This enables you to check players names and substitutes used etc if found necessary.There is no paperwork to sign following the game however you should inform the home club of your Fee and Expences and you should be paid immediately on completion of the game.\nPlease notify any changes to your details held in Refs Directory to the Referees Appointments Officer who will update your changes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"New Price! CARFAX One-Owner. Clean CARFAX. Summit White 2013 Chevrolet Express 2500 Work Van Cargo RWD 6-Speed Automatic HD with Electronic Overdrive Vortec 6.0L V8 SFI Flex Fuel CarFax Clean No Accidents, CarFax One Owner, Vortec 6.0L V8 SFI Flex Fuel, 6-Speed Automatic HD with Electronic Overdrive.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Wow!!! Is this actually our current-day cultural reality?\nNo wonder our blog posts don't get too many comments. We're just too \"wordy\" for your average internet browser. Reading through a written script is simply too much bother for them I guess.\nThis is the best post to this idiotic ad for a program for idiots.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Two years ago, I came across with SF Beauty Skin and did a brief review on it. It is not a new brand to me and to be honest, I am still loving their Perfect Glow Sunscreen. Unlike any other skin care brand out there, SF Beauty Skin provides unique services on \"Try It Before Purchase, Free Sample Set Deliver to Home\" and online personal beauty consultants. SF Beauty Skin believes such services not only guarantee one's satisfaction without wasting money, but it will also pampers her with in-home skin care service.\nWhat do I meant by online personal beauty consultant? SF Beauty Skin believes you can seek for professional beauty services at home through phone or email during working hours. There will be beauty consultants attend to your questions through online customer support and they are equipped with professional beauty salon training programs.\nIt was a pleasant experience talking to the beauty consultants because they are friendly and willing to give a lot of tips to take care of the face. It is my pleasure today to work with them on sharing my experience on using the SF Beauty Skin Absolute V-Lifting Serum with Magic Massage Pen.\nI like the fact that the serum is in a pump design rather than a dropper design. It is more hygienic though I can hardly control the amount of serum comes out from the pump. The serum is in crystal clear and water texture. Usually, two pumps are sufficient enough for the whole face and the neck area. Warm the serum on you palm and apply it to the face. For a better lifting effect, use the Magic Massage Pen and massage from chin line upwards to ear and foreheads until your skin absorbs the serum.\nHere's a closer look on the Magic Massage Pen. Take off the top cover will show a roll-on tip, which allow it to glide better on your skin. Uncover the bottom cover will show the battery holder, where you need to put in two single-cell round batteries in order for the vibration to work. There is also a push-on button that allows you to turn on the vibration feature. Push the button once again to switch the vibration off. It is advisable to put on serum and using the massage pen for better lifting effect.\nHow to use Magic Massage Pen?\nVisit www.sfbeautyskin.com or call toll free number 1800 - 88 - 6919 for FREE sample set ASAP.\nLike SF Skin Beauty Fan Page!\nAlong with \"Absolute V Like & Share Event\", click on the like button and share out the news for a FREE mask!\n40% off for Absolute V-Lifting Serum and Magic Massage Pen! It is now RM300 RM180!\nWonder how the online beauty consultants can help you? Ring them up or online chat with them to experience a brand new experience of skin care regime.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I'm going to be brief because I'm not sure how many people are around reading on days like this, and the last Harry Potter movie is calling me, anyway.\nToday, was the first Thanksgiving we had on our own \u2014 no family or \"adopted family\" to share the meal with, and you know what? It was kind of nice. The day moved at a perfectly relaxing pace, starting off with amazing pumpkin walnut muffins, and ending with a true feast. We weren't on anyone's time table but our own, there was no negotiating conversation with distant relatives, and Chris even cooked in his PJs!\nand a very happy camper even without a nap.\nThe final results were delicious and every member of the family ate to their heart's content. Even Pip, who was more than willing to eat Nora's leftovers.\nToday, I'm thankful for my little family. I'm thankful for relaxing and joy-filled holidays done right. And, once again, I am thankful I married a chef!\nHey Amber, I linked though Facebook and wanted to say the second picture down looks amazing! Makes me hungry just looking at it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaqmjv b/data_all_eng_slimpj/shuffled/split2/finalzzzaqmjv
new file mode 100644
index 0000000000000000000000000000000000000000..4c8e04d3f2be31823a585e578e4f50e03f000c6f
--- /dev/null
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+{"text":"Teachers need to be professional in and outside the classroom. What does that mean? There are huge pressure on teachers to always perform and keep calm as well.\nYou will behave and act with enough confidence to inspire and motivate your learners.\nAn arrogant and rude teacher is not professional.\nAs a professional teacher, you will always prepare well with the correct lesson plans and activities.\nYou will also be prepared to change the lesson plan if needed. Cultivate the ability to think on your feet.\nThis sums up the attitude of a teacher towards his learners.\nAn impolite attitude towards your learners are not acceptable. Firmness will enhance the discipline in the classroom.\nYour clothes will always be neat, clean and professional.\nYou will always be on time. It is even better to be a little early for your class.\nShowing interest and concern for every learner in your class will be a priority for you.\nYou will never make exceptions and will not favour certain children.\nYou will follow the rules and policies of the school to the letter.\nThis is important, as you need to fit in and be part of the best practice of the school.\nYou as the professional teacher will always be proud to be part of the school and community.\nYou will behave in all situations and circumstances as a professional and respected member of the school.\nThese are just a few of the characteristics of a teacher who lives and acts as a professional. You have to remember that you are a public figure and your behaviour will tell who you are.\nHello! Welcome to my website. I'm Kevin Scroggins. I'm the person working behind this blog with some other team members. We aim to share all about education, learning and much more. Stay connected to get latest updates.\n\u00a9 2019 Reclaim Public ED Now. All rights reserved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Welcome to Extended Stay America - Greenville - Airport. Greenville is rich with natural beauty including plenty of state parks and nature trails. Relax at the end of your adventure with a roomy suite at Extended Stay America and get free in-room Wi-Fi and watch your favorite shows on our new flat-screen TVs. Please note: all suites in this hotel are non-smoking.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Spring wedding. Topped off with beautiful fresh pink roses and dots on each tier.\nA monogram was added to the top-tier.\nIt's nice to see someone using buttercream as their main icing rather than fondant \u2013 seems everyone wants to be a fondant master, and they forget how much can be done with buttercream!\nYes I prefer Butter cream but I guess it's because I'm a\"old school\" decorator.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A four volume set of publications by Good Housekeeping.\nGood Housekeeping is a magazine owned by the Hearst Corporation which features articles related to recipes, decor, diet and health. The magazine was founded in America in 1885 and achieved a circulation of 300,000 by 1911.\nGood Housekeeping was first published in Britain in 1922 and was extremely popular. The readership of the magazine as a whole had topped one million by the mid 1920s. Many famous writers have contributed to the magazine including Virginia Woolf and Evelyn Waugh.\nIn cloth bindings in original unclipped dustwrappers. Externally, in excellent condition with just some minor shelfwear. Dustwrappers in excellent condition with slight tidemarks to base of spine. Internally, firmly bound. Pages bright and clean. Overall: NEAR FINE in NEAR FINE dustwrappers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"THE JOB: This is an exciting opportunity to join a well-established Central London estate agency. This is a Monday - Friday administration role. Previous estate agency administration experience is required. The position will involve supporting the sales and lettings team in the office at all different stages of the sales & lettings process. Excellent organisational skills are vital, a confident manner and the ability to deal with a busy and high-pressured environment.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzarspv b/data_all_eng_slimpj/shuffled/split2/finalzzzarspv
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+{"text":"Building an 800-horsepower Hemi engine with viable street manners and runs on pump fuel is a challenge few engine masters are willing to accept, but Ray Barton Racing Engines in Robesonia, Pennsylvania, is one shop that accomplished the feat with ease. With help from AEM Performance and its Infinity engine management system, David Barton and the rest of the RBRE team assembled one wicked engine for Wisconsin customer Curt Woodward's '68 Road Runner. You need to see this engine run in the video below!\nThe bullet is based on the company's long-proven 528 cubic-inch \"Street Hemi\", and has a number of features that make it the ultimate street engine for Mopar fanatics. Starting with a Mopar Performance iron Hemi block, Barton Racing Engines blueprints it in-house to their specs and fills it with a K1 Technologies 4340 forged crankshaft and 4340 H-beam connecting rods. Diamond forged pistons wear a 1\/16, 1\/16, 3\/16ths-inch ring pack, and the rotating assembly is balanced in-house before installation.\nStage V aluminum Hemi cylinder heads feature one of RBRE's custom multi-angle valve jobs, and the valvetrain is actuated by a custom RBRE camshaft that has been specifically selected for this application. In addition, RBRE developed the individual shaft rocker-arm system that rides front-and-center atop the cylinder heads. Top-shelf 3\/8-inch x.080-wall TREND pushrods are used, along with Comp Cams springs, retainers and cups.\nA solid camshaft was selected along with tool-steel solid lifters and a billet double-roller timing chain to help control timing events. A host of other quality components from companies like ATI Performance, ARP, and Moroso were used to finish off the engine, and the result is a whopping 801 horsepower at a super-street-friendly 6,400 rpm, with 716 lb-ft peak torque at 5,100 rpm.\nThis engine will accept all stock Hemi accessories, has a standard exhaust port location for easy header selection and has a pump-gas friendly 10:1 compression ratio.\nPerhaps the coolest part of the engine is the sky-high set of Kinsler throttle-body stacks sitting atop the intake manifold. They hide the fuel injectors that are run via the aforementioned AEM Infinity engine management system.\nThere were no struggles in either startup or general tuning, thanks to the Infinity's configuration.\nBarton adds that the biggest challenge was wiring everything up \u2014 and that wasn't even difficult. Since the Kinsler setup arrived from the factory with the fuel rails and linkage already installed and ready to go, all the RBRE team had to do was set up the Infinity's harnesses to work properly.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Spurway Film and TV have produced a television documentary \"Under A Free Blue Sky\" about the work of Seer Ministries which was screened several times by UCB-TV on Sky Digital Television.\nYou can download two Windows Media clips from the DVD. Left-click the links below to play directly with broadband, or right-click and \"Save Target As.\"\nYou can view over 300 still frames from the documentary. They are low-resolution so they download reasonably quickly. See them by clicking here. Please request a copy of the DVD documentary by clicking here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Join the Indies First Movement Today!\nSherman Alexie and the American Booksellers Association are working together to promote Indies First, where book authors are encouraged to work a day at an independent bookstore AND place the indie bookseller's buy button on their book page (above buy buttons for online booksellers or others).\nNow is the time to be a superhero for independent bookstores. I want all of us (you and you and especially you) to spend an amazing day hand-selling books at your local independent bookstore on Small Business Saturday (that's the Saturday after Thanksgiving, November 30 this year, so you know it's a huge weekend for everyone who, you know, wants to make a living).\nI signed up to work with Moby Dickens. Will you sign up to work with your local indie bookseller?\nLet's get out there and work to save the local bookstores we've come to love! Do it today!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Like to sell more rights to your books \u2013 translation, film, tv, etc \u2013 but not sure where to start? First step: read ALLi's three guiding principles to rights selling to set you in the right direction. From the guidebook by Orna Ross and lawyer Helen Sedwick.\nAre you confident that if someone offered you a rights deal tomorrow, you'd know whether it was worth your while? Here's an easy way to start skillin up and get savvy enough to build your author business through rights sales \u2013 read this post by Orna Ross and Helen Sedwick.\nWriting: What Can Indie Authors Gain from The London Book Fair This Year?\nAn insightful interview with ALLi director Orna Ross by Howard Lovy sharing valuable advice on how indie authors can make the most of The London Book Fair, and what to expect if you attend.\nDo you have an out-of-print or unpublished back catalogue tucked away? Don't be afraid to get it out and reread it \u2013 it may be better than you think, and ideal material for self-publishing, says Ann Richardson, who goes on to offer two cases of her own that have done well.\nJay Artale our ALLi communications manager has put together a roundup of our Ask ALLi and Indie Author Fringe podcasts, and coaxes us to take time for ourselves over the festive season to listen and be inspired.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Chalkboard paint has revolutionized the world of art and craft, making every surface an open and inviting canvas. It has taken us all back to school, where the familiar smell and feel of chalk still feels fresh. For one, I remember how my teachers would have chalk dust on their hands, faces, hair and clothes, as if they had just returned from an epic war of the words. What I remember the most, is using the classroom chalk and doodling away during lunch break, and times of utter boredom. With chalkboard paint around, every obsolete surface can be made into a functional learning zone.\nWhen your kids are young and eager, you can paint the very walls in their room, for them to unleash their artistic talent. The surface of a fridge can be used for leaving notes and reminders of the things to do, and things to buy. That too without using a scrap of paper! And guess what, you can use glass as canvas as well among the many other chalkboard paint ideas! It seems too good to be true!\nNot all chalkboard paints will work on glass, and therefore you will need to read the information on the label thoroughly. Buy only the paint which says it can be used on glass. You can select a shade which you prefer. Personally, I adore the slightly dark green shade of chalkboard paint, as it looks more authentic and weathered, than the usual jet black.\nWhen using glass, try using the chalkboard spray paint, as this is easier to use and spreads evenly. Using normal paint will require that you use a paintbrush, which will leave its brush strokes all over the glass, giving it an unrealistic texture. You wouldn't want that, when you expect your glass surface to look like a real chalkboard, and not like a dried oil canvas. Secondly, spray paint will be able to really stick onto the glass surface, making it a more suitable option.\nYou will need to clean the glass using alcohol, such as vodka or ethanol. This will remove all traces of dust on the glass surface. Never use cloth to dry glass surfaces, and instead use a simple piece of paper. A discarded brown bag would be great to wipe the glass dry.\nUse sandpaper or steel wool to gently rub the surface of the glass evenly, so that the surface becomes smooth and is ready for its first coat of paint. This will help the paint remain on the glass for a longer period of time, without having to re-coat it often. After rubbing the surface, clean the glass again with alcohol.\nIf you wish to use only a specific area of the glass as a chalkboard, you will need to demarcate the zone using artist's tape. These tapes never leave a sticky residue, so do not worry about dust settling on the border! Now, spray paint within the demarcated zone, and once your done adding the extra layers, wait for the paint to dry, after which you must gently remove the tapes.\nIn case you do not require the artist's tape, and wish to paint the entire glass, you will need to wait until the glass dries completely. Start by applying a very thin coat of chalkboard paint on the glass, let it dry, soon after which you must apply the second thin coat. Ideally, you must apply 2 coats, but since we are using very thin layers, you can exceed the number to 3 - 4 layers of spray paint.\nWet Paint, Do Not Touch!\nHold your horses, and wait until the paint is completely dry. This will take 24 hours, but I suggest you give it some more time. Let the glass dry for about 36 hours in all, for all the layers to set properly.\nYou will need to use an authentic chalk stick and not those hard and pointy or stick thin ones! Use the normal age-old chalkboard sticks. Rub the chalk all over the surface of the glass and see how it is working out to be. If the surface is behaving alright, wipe the chalkboard with a duster or a cloth. Now you may write what you please on your glass chalkboard!\n\u25cf Label your kitchen jars, so that you do not confuse salt with baking powder.\n\u25cf Transform a useless mirror into a wall hanging\/daily notice board.\n\u25cf Table glasses and plates can have chalk borders, so that each member of the family can have his\/her name written on their own plate\/glass.\n\u25cf Candle jars and stands can be painted and stamped with messages.\n\u25cf Paint a glass table and use it for writing the menu, or thoughts.\nYou can make fascinating art pieces by learning to use chalkboard paint on glass. You can convert old photo frames into blackboards, and vases into artistic masterpieces. You can use your mirror and paint over it... the possibilities are limitless!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzatisr b/data_all_eng_slimpj/shuffled/split2/finalzzzatisr
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+{"text":"Vishweshwaraiah was born in Muddenahalli, a village in Kolar District of Karnataka, on 15th September 1860 to Srinivasa Sastry and Venkachamma. His father was a Scholar in Sanskrit and was a practicing Ayurvedic physician. His early education was in Chikkaballapur. He completed High School from Wesley Mission High School, Bangalore. When he was 15 his father died. M V completed his B A in 1881 from Bangalore Central College. He received some assistance from the Government of Mysore for further studies. He joined the Science college in Poona for Civil engineering course. In 1883 he passed with 1st Rank in the L C E and F C E which is the equivalent to B E of today. Vishweshwaraiah received a gold medal while the studies going on he for outstanding performance from James Berkley.\nVishweshwaraiah used to go to bed at 10 p m and get up at 6 am. He was a simple personality. He was a pure vegetarian with frugal food habits. His food was also simple as he was. He never had any hot drinks. He was famous for his punctuality and discipline; honesty, integrity and professionalism. He was always well dressed.\nHis first job was under the erstwhile Bombay presidency in the Government Public Works Department as an Assistant Engineer at Nasik. He had a long career as an engineer and has served as the chief Engineer in the erstwhile State of Mysore. The Krishna Raja Sagar Dam near Mysore is his brain child. He has contributed immensely in the construction of several dams and water supply projects in our country. The use of sluice gates in many dams was his idea. In the year 1912 Vishweshwaraiah was appointed as the Dewan of Mysore State. During his Dewanship he had introduced many developments in the educational, irrigation and administration fields. He was the person behind the formation of the Mysore University, University of Vishweshwaraiah Engineering and University of Agricultural Sciences. The State Bank of Mysore (the then Bank of Mysore) and the Mysore Iron and Steel Works at Bhadravathi were examples of his achievements.\nChief Engineer in Mysore State from Nov 15, 1909 concurrently as Secretary to the Railways.\nHe was in the committees on Education and Industry as president in the state of Mysore.\nFrom 1912 to 1918 he was Dewan of Mysore.\nWorked as the Governing Council Member of the Indian Institute of Science and Tata Iron and Steel Company (TISCO).\nIndian States reorganization study committee.\nSir M.V. retired in 1908 and Sri Krishnarajendra Wodeyar, Maharaja of Mysore, was eager to secure the services of Visvesvaraya to serve Mysore. Sir M.V. had earned a reputation for his honesty, integrity, ability and intelligence. He had introduced compulsory education in the State which later was embodied as a fundamental right in the Constitution of independent India.\n1.\tK R S Dam. One of the biggest dams in India which irrigates a hundred and twenty thousand acres of land.\n5.\tEncouraging girls to attend school.\nSir M.V. was never interested in fame or publicity. But they came to him on their own. Every university of Allahabad, Andhra, Bombay, Calcutta, Jadhavpur, Mysore, Patna and Varanasi in India sought him out to confer honoris causa.\nHe had the Honorary Membership of London Institution of Civil Engineers till his death from 1904.\n1906 he was conferred the title of \"Kaisar-i-Hind\" for his services.\n1911 He was given the title of Champion of Indian Empire (C I E) at the Delhi Durbar.\n1955 Bharath Ratna - Highest distinction of the country by Government of India.\n1959 Fellowship of the Indian Institute of Science, Bangalore.\nOne full book will require to mention his qualities. One incidence was that when he was given the posting of the Dewan by the Mysore State, he called all his family members and relations and told them that he will accept the post only if all of them agree for the condition that they will not go to him for any personal favor.\n\"Indian Express\" had news item on March 24, 2005. Sir MV maintained had an account with Bank of Mysore (founded by him - now State Bank of Mysore). On March 27, 1918, he had Rs 990 which increased to Rs 11,487 on March 3, 1919. This included the interest earned and maturity amount of his F D.\nGandhiji and Vishweshwaraiah were eminent Indians who had the interests of the country's development and making it a modern one as the foremost thing in their heart and they had mutual respect to each other. But both of them had diametrically opposite views on several issues. But as far as industrialization is concerned they were totally opposed to each other. While Gandhiji's stand was industrialize and perish Sir M V's stand was industrialize or perish. Vishweshwaraiah did not agree with the non-cooperation movement of Gandhiji. When Gandhiji was to go to attend the round table conference at London M.V. wrote to Gandhiji urging him to wear suit.\nSir M Vishweshwaraiah's birth centenary was celebrated in Lal Bagh Garden of Bangalore and the then Prime Minister of India Pandit Jawaharlal Nehru came to honor the great son of India. The function was presided over by Jayachamaraja Wodiyar.\nOn his reaching 100 years of age the Government of India brought out a Postal stamp in his honor.\nSir M.V. died on April 12, 1962. He was 102 years, 6 months and 8 days old at the time of his death. He was cremated in his birth place at Muddanahalli as desired by him. A memorial is erected at the village but the maintenance is very poor.\nHe had introduced compulsory education in the State which later was embodied as a fundamental right in the Constitution of independent India.\nWhen he reached the age of 100, the Government of India brought out a stamp in his honor. Sir Visvesvaraya passed away on April 14, 1962 at the age of 101.\n(1) The working hours were short in the aggregate \u2013 they should be increased for all workers and employees.\n(2) There should be regular working hours with discipline; A minimum of six to eight working hours daily on every working day should be introduced.\n(3) Unity, harmony and co-operation is to be expected from every citizen.\n(4) Everyone should practice diligence, discipline and precision as part of his\/her character.\n(5) Everyone should cultivate loyalty, honesty and selflessness.\n(6) He should be courteous and do service to fellowmen in daily life.\n(11) Time should be managed effectively.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"TOPS Korea WaterjetCutting Machines' range.\nConsumables required for WC Machines.\nUtilities required in WC Machine set up e.g. Chiller, Isolation Transformer, Compressor, Stabilizer etc.\nProviding Installation, Testing and Commissioning of WC Machines.\nOffering After Sales Services, Repair and Maintenance of Waterjet machines.\nWe give comprehensive demo of Waterjet Cutting Machine at our Authorized TOPS, India Demo Center, MIDC, Butibori, Nagpur.\nWe give training (at our Training Center or at Customer's site) of efficient Waterjet System's basic theories, Waterjet System operation, Waterjet system Maintenance, Top Master Waterjet Software Training etc.\nWe also conduct Training, Seminars and Workshops to provide Technical \/ Business consultancy for setting up Waterjet Cutting Jobshop business.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ah, life in the country\u2026such bucolic bliss. Until your neighbor's dog contracts rabies, kills its owner, and then comes after you! With such emotional extremes, Cujo was not an easy movie to score, but when Stephen King's classic novel came to the screen in 1983 starring Dee Wallace and Daniel Hugh-Kelly with Lewis Teague directing, composer\/conductor Charles Bernstein (A Nightmare on Elm Street) proved mightily up to the task. In addition to employing harsh synthesizer effects and dissonant orchestral layers leading up to the climactic showdown between Mom and beast, Bernstein also penned a tender theme expressing the gentle joys of family life.\nUnfortunately, post production woes mixed up some cues and caused the score to be abbreviated in the film; Intrada's recent CD release presented Bernstein's work for the first time as it was originally recorded, and that's where the 18 tracks on our first-ever vinyl release come from, with new (and terrifying) cover art. Limited black and brown \"St. Bernard\" vinyl edition limited to 1000 copies!\nYou can pre-order CUJO and ROTLD now! They will both ship the first week of October and be at your doorstep well before Halloween is underway.\nStay tuned here at FANGORIA.com for more Real Gone Music horror releases!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Stamp collection is a hobby which is easy to follow. In modern days however it is not easy. In my childhood days there were snail mails and was quite easy to just rip the stamps from mail and start collecting. Now however it takes some knowledge and effort. Here are some tips for you to start even in the internet age. Stamp collection is real fun and learning as well.\nBuying a stamp collection starter packet \u2013 You can get a wide range of stamps in packet at the local hobby store. There will be hundreds of used stamps and it is a good way to start. Just check that different stamps are stacked in the packet. This method will help kick on with your new hobby.\nStamps at Post Office \u2013 You can add another facet to your collection by buying crisp new stamps from the local post office. The new \"mint condition\" stamps are also valuable addition to a collection. I started with ripping off stamps from snail mails but later loved to buy from post office too.\nContact offices \u2013 Even today office send across postage. They will invariably have stamps. Just visit a few offices and ask them. I am sure you can get a good collection from the offices. They will generally give it to you free of cost.\nExchange stamps \u2013 You may have same stamps while your friend (email, pen pal or local) might not have that stamp. Exchange it with your friend and add a new stamp to your collection. Use the internet to your advantage.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Emergency crews hold all lanes of traffic on I-75 North at Cleveland Avenue while police investigate a deadly pedestrian accident on the interstate.\nEarly Thursday, just as the morning commute was getting underway, a woman was hit by multiple cars and killed when she ran into oncoming traffic on I-75 North near Cleveland Avenue.\nDrivers heading into downtown Atlanta from Hartsfield-Jackson International Airport and the southern suburbs had to find alternate routes to avoid the backups. At one point, traffic was diverted off the interstate onto Cleveland Avenue.\nAccording to police, it is not yet known how many cars hit the woman, who has not been identified. Several stopped on the side of the interstate. The drivers were cooperating with police.\nThough the incident is not being investigated as a hit-and-run, the police department's hit-and-run unit responded to help, Senzer said.\n\"We also have secondary accidents as well, people trying to avoid what was going on with the initial incident,\" he said.\nHowever, no other injuries have been reported.\nTwo major studies released this spring indicate pedestrian crash deaths continue to climb nationwide.\nIn 2017, 260 pedestrians were killed on Georgia roads, up from 232 in 2016. And the 260 deaths last year was double the number killed in 2011, according to the Georgia Department of Transportation.\nNationally, pedestrian crash deaths have increased 46 percent since their low point in 2009 and account for 16 percent of crash fatalities, according to The Insurance Institute for Highway Safety.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzatuka b/data_all_eng_slimpj/shuffled/split2/finalzzzatuka
new file mode 100644
index 0000000000000000000000000000000000000000..3779b695ed13bf93a28bbf319b875e8874f8e9ca
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@@ -0,0 +1,5 @@
+{"text":"During the last time you were at a dealership looking for a car, did you feel lost and confused by all the information? Perhaps you are just missing some advice. Buying a car will be easy if you follow this advice.\nDo not let the car dealer talk you into buying a car that is not within your budget. A lot of people are swayed into buying a sports car by a salesperson. Remember that the goal of any salesperson is to make the highest commission, so they will benefit from selling a pricier car.\nPrior to stepping foot in a dealership, obtain a car loan. One reason car purchases take so long is because of the whole financial part. It will be much quicker if you have a loan ready.\nArranging financing prior to actually shopping for a vehicle is a great idea. You should do this through your bank. These institutions offer lower interest rates than what you might get if you financed a vehicle through the dealer. It also gives you a better idea of your budget.\nYou want to stay focused on the total price, not monthly payments. The monthly price can be changed to suit your needs, but the overall price will still be very high. Don't forget to check that before you sign on the dotted line. Get the best overall price and then deal with financing. Then, you can negotiate the terms.\nTry to bring someone with you if you are going shopping for a vehicle. Your friend can listen for things you might have missed and they can give you advice if they think you shouldn't buy the car you are looking at. You can take your significant other with you, your mom or dad, or a friend.\nWhen you are buying a new car, set your budget before going to the dealer. As you shop, avoid exceeding your budgeted amount despite the salesperson's representations. The dealer is not the one who will be paying for the car for 6 years, you are.\nBuying a car will be easier from now on. Use what you've just learned to make easy work of car shopping. Use this article wisely and save it so you have it for later.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Progress pictures on a large PVCu Livin Room conservatory in Walberton, West Sussex. This building replaced an old, rotten, dated lean-to and what a transformation it has made!\nWatch this space for more photos once the UF heating is installed, flooring is laid, second fix electrics are done and decorating is completed.\nHere's a before and after of an offset PVCu white French doors installation in Fishbourne near Chichester, West Sussex.\nHaving offset French doors can now allow our customer to use the whole opening when necessary to access both the garden and side entrance to their home.\nConservatories can be any colour, shape or size \u2013 all of our products are made bespoke to fit your home and lifestyle for whatever suits you.\nHere is a fabulous example of how a small conservatory has created and merged the indoor space by creating 3 rooms into 1 and making a house a home.\nA beautifully installed Irish Oak Flint 4 Solidor composite front door with Victorian glass and a fully glazed satin side panel.\nGet in touch today to design your dream Solidor entrance door to your home.\nBefore and after \u2013 keeping it simple, safe and secure.\nA simple but effective transformation of a white uPVC back door transforming the look of this property in Fishbourne near Chichester, West Sussex.\nGet into today to make minor changes and a big difference to your home.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Carmel, Indiana \u2013 A new technology built to help fraternities and sororities recruit higher quantities of higher quality people was launched to the public today. The software \u2014 called ChapterBuilder \u2014 already has over 100,000 potential new members flowing through the system thanks to nearly 1000 active chapters already utilizing the software during its pre-launch phase. Today ChapterBuilder is available to all fraternities and sororities who want a simple yet sophisticated Names List tool to manage year-round, values-based recruitment.\nTechniPhi, a company focused on sharing the life-changing gift of Greek Life with more high quality people, \"soft launched\" ChapterBuilder in 2013. Five semesters of testing and upgrading the software have provided a base of raving fans and evidence that the tool is ready to become the standard for every fraternity\/sorority chapter in the world.\nInterested fraternities or sororities should visit www.ChapterBuilder.com for a demo.\nChapterBuilder is a tool to manage the relationship between potential members and the chapter. The software is a web-based system usable by any chapter member, consultant, or advisor, anywhere in the world. ChapterBuilder is accessible on any computer, tablet, or phone with an internet connection (an app is available from the Android or Apple app stores).\nOver 40 Inter\/national fraternity and sorority headquarters have committed to join TechniPhi for the August 2015 launch of ChapterBuilder. Dozens of campus partners are lining up to pilot the council version of ChapterBuilder this fall.\nEarly users of ChapterBuilder are raving about simplicity and ease of use. Communication tools like personalized bulk texting\/emailing, social media integration, mobile ready app, virtual coach that lives inside the software, customizations, and advanced analytics are all poised to shake up the way recruitment has been done by Greek organizations for decades.\nVisit www.ChapterBuilder.com for more information.\nTechniPhi creates technology solutions to grow fraternities and sororities. ChapterBuilder joins TechniPhi's formal recruitment software called CampusDirector as a key product offered by the company. In 2014 TechniPhi was formed by the leaders of Phired Up Productions, along with the acquisition of a marketing company called Innova. The three companies, based in Carmel, Indiana, work closely together to provide education, technology and marketing solutions that grow the fraternal movement.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"He's done it again. Your dog was so eager to chase that squirrel, bark at the mailman or whatever it was this time that he tore through the screen door. Don't put up with it any longer. You can purchase products to prevent animals from going through the porch screen. They're available at home improvement stores and online.\nAlthough they resemble standard screens, pet screens are far tougher -- as much as seven times stronger than regular fiberglass mesh screens. Pet screens are designed to prevent your dog or cat from tearing them with their claws. Pet screen kits are available, so you can install the vinyl polyester fabric yourself. You can also purchase pet screen door protectors if the door is the primary issue. A dog signaling that he wants to go out by scratching on a regular screen door can puncture it fairly quickly.\nGrilles placed over porch screens will keep your pets in and prevent other animals from entering. They're especially useful if you feed your pet in the screened porch and wildlife or neighborhood animals try to stop in for snacks. You can find pet door and window grilles in various materials and colors to match your decor.\nIf your dog or cat is just going through one particular part of the screen in order to get outside, consider installing a pet door. That's a choice only if it's safe for your pet to have relatively free access to the yard. Many pet doors include a locking mechanism, so you can keep other critters out after your pet is in for the night. Certain models come with a radio-frequency collar that opens the door when the pet steps close. Pet doors for screens come in two basic types. The less expensive version clamps onto the screen. The more expensive but stronger type fastens onto a lower corner of the screen with sheet-metal screws. If you have a particularly active, rowdy or large dog, it's probably worth the extra money and possible installation costs.\nIf you're handy and want to create your own pet screen, purchase appropriately sturdy screening material and cut it your specifications. If your dog is so tough that he could possibly go through the strongest of pet screens, consider materials such as chain-link fencing or the stainless steel security screening used to deter vandals. The latter two aren't that attractive, but they'll do the job.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Christopher Mintze-Plasse is the legend known as McLovin. He was hilarious in the movie Superbad. He appears in the new movie YEAR ONE with Jack Black and Michael Cera. I caught up with him on the red carpet to talk about women, working with Nic Cage and circumcision.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"I am from the Evansville area and attended Western Kentucky University. I have a degree in special education and elementary education. I have 2 daughters. I enjoy reading, running, and quiet nights at home. I began teaching in 1999 for the Mt. Vernon school corporation. I taught special education for 11 years in grades K-8th. Then, taught kindergarten for 7 years at Marrs. I'm excited to begin teaching 2nd grade!\nWelcome to our Fourth Grade Classroom! I am so excited to be at an amazing school with a wonderful staff. I want 4th grade to be a meaningful, rewarding and academic-filled year! Please feel free to contact me with any questions.\nI started teaching in Mt Vernon in 2000 as the Assistant Band Director then in 2010 I took over as Director of Bands at Mt. Vernon High School and Junior High. Now I am enjoying teaching the younger students here at Marrs. I graduated from Murray State University with a Music Education Degree, and also from Ashford University with a Masters in Educational Technology. I am married and have 4 kids, 2 boys and 2 girls. I love watching them play sports and whatever else they can get involved with. I love teaching kids the love of music and look forward to having many exciting events planned for the future.\nHI! I am Ms. Heckman your art teacher! Currently, I hold two degrees in art, one from the University of Evansville in art therapy and a minor in psychology and the other from the University of Southern Indiana in art education. A lot you all know me as 'the substitute' since prior two years ago of teaching art at Marrs, I have been with MSD off and on for the past 19 years. In those 19 years of off and subbing, I taught art to jr. high students in Henderson, Kentucky for 2 1\/2 years. Besides the classroom, I have three kids, (step) daughter Maria (22), daughter MecKenzie (a freshman at Murray State) and a son Sam (5th grade). Outside of the class, I love traveling, running, singing, swimming, tennis, horseback running, and of course art making!\nConnie Jesch is a graduate of the University of Evansville and Indiana State University. She has taught in MSDMV for 28 years. Her career began at Marrs Elementary as a Primetime Assistant then a first grade teacher. Her husband, Richard, is a math teacher at MVHS. She has three children. two who have graduated from USI and one daughter enrolled there as a freshman! Connie enjoys gardening, photography, swimming and traveling.\nI received my B. S. and Masters degree from University of Southern Indiana. I have been teaching for MSDMV since 1996. I have taught every grade K-6. My husband works for Rexam Closures. We have two children and three grandchildren. I love spending time with family and with a good book.\nHi! I have been teaching at Marrs Elementary since 1984. I taught 25 years in 3rd grade and began teaching 5th grade three years ago. I teach Language Arts and U.S. History to all the 5th grade students.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Over 65,000 people have lost their lives in motor vehicle accidents in the past two years alone. In many instances motor vehicle accident fatalities are caused, at least in part, by distracted driving.\nLast month Erie Insurance Co. released figures, which estimated that distracted driving was a factor in one out of every ten motor vehicle accidents. These figures were based on data from the Fatality Analysis Reporting System (FARS), a census of all fatal motor vehicle accidents that occur within the United States, in consultation with the Insurance Institute for Highway Safety.\nThe analysis took into account police reports for fatal motor vehicle accidents between 2010 and 2011. Police officers most often described distracted drivers as being, \"generally distracted\" or \"lost in thought.\" However, the survey also revealed several specific types of distractions that were commonly occurring.\n62 percent of distracted drivers were \"generally distracted\" or \"lost in thought.\" For instance, they were daydreaming.\n12 percent of distracted drivers were using their cell phones\u2014 they were either talking, listening, texting, or dialing.\n7 percent of distracted drivers were distracted by a person, object, or event outside their vehicle, such as another accident on the roadway.\n5 percent of distracted drivers were distracted by other occupants within their vehicle. For instance, they may have turned towards the back seat to look at another passenger.\n2 percent of distracted drivers were using or reaching for a device such as, a navigation system.\n2 percent of distracted drivers were eating or drinking.\n2 percent of distracted drivers were adjusting the temperature or audiowithin their vehicle.\n1 percent of distracted drivers were adjusting other devices such as, the vehicle's rearview mirror.\n1 percent of distracted drivers were distracted by a moving object within their vehicle such as, a pet.\n1 percent of distracted drivers were involved in a smoking related activityincluding, lighting a cigarette and putting ashes in the ashtray.\nBecause the survey relied on police reports, these numbers may in fact underestimate the prevalence of distracted driving. That's because many drivers may not acknowledge that they were distracted while being questioned by a police officer at the scene of a fatal motor vehicle accident.\nIf you or a loved one have been injured or if your loved one has been killed in a motor vehicle accident caused by distracted driving, you should contact an attorney immediately. An attorney can help protect your legal rights and ensure that you receive the compensation you deserve.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Glenn Ryan (left) who is a member of our club's Tennis Team in the Toronto Winter Tennis League Men's Major Division and our Director of Tennis, Mike Thomson (right), were the 45 and over Doubles Champions in the 2015 Senior Men's Eastern Indoor National Championships held in Toronto at the Donalda Club. Fraser MacDonald (left) who is Captain of Canada's Gordon Trophy Tennis Team and our Club Manager, Owen Pellew (right), were the winners of the 2015 Senior Men's Eastern Indoor National Championships 55 and Over Doubles. The event was held during the 1 week school break in March.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"1\/72 Scale. New Silver Edition! This model comes molded in 2 parts, the top and bottom of the craft and also has 3 separate pieces for the landing gear. Each side of the Triangular Craft measures 5 inches. Detailed instructions included. Now molded in silver for easy display. Skill level 2.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Minibus Hire with a driver in Caistor and Lincolnshire. Get a great deal for Mini Bus Hire in Caistor. Cheap Coach Hire is the one of the leading supplier of Minibus Hire in the Caistor area. A huge range of Minibus Hire with a driver services including Minibus Hire, Coach Hire, Minibus Hire With a Driver and many more..","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Hurricane Sandy, now known as Frankenstorm, is barreling up the East Coast. The New York City subway system has been shut down. Both public schools and the New York Stock Exchange will be closed tomorrow for the second consecutive day. \"Notify NYC,\" the City of New York's official source for information about emergencies, reports that there are power outages in Brooklyn and Queens. The public high school one block from my home has been turned into an evacuation center.\nBut as far as retailers are concerned, the show must go on. They want us to shop--if not in person, then online.\nI've been lucky enough to have electricity and Internet access all day. So though I couldn't get to my office at Forbes, I've been working from home. Meanwhile, all day my email has been pinging with offers from various stores, beckoning me to shop instead.\nIt started at 9:30 this morning with a reminder from Bloomingdale's that this is the last day for the 20% off shoe sale (discount code LOVESHOES if you really want to know). Now I do love shoes, and Bloomie's too, but at that particular moment, it was just starting to rain, and I was sitting at my desk in flip-flops wondering whether Hurricane Sandy was going to be more like last year's Hurricane Irene, rather than Frankenstorm, and whether the whole city had shut down unnecessarily.\nThen, at around noon, came a reminder from the Gap that it's sweater weather. No kidding \u2014 by then I had the storm windows down, and could still feel the draft from the howling wind.\nThe offers from Lord & Taylor got better as the day went on. I didn't react to the one at 11:00 AM for 15% off (promo code SAVE). So at 2:49 PM they upped the ante: 25% off sitewide today only with promo code ONEDAY. Appropriately enough, this e-blast featured a chic little trenchcoat that I don't happen to need, and an umbrella that looked like it would blow away in the first wind gust. So once again I passed.\nNordstrom's took my mind off the weather at 3:00 PM. \"Party Dresses from Adrianna Papell & More\" was the subject line. The text was even more fetching: \"Add drama to your evening.\" Shortly after it landed, my 100-year-old townhouse shook in the 60-mile-per-hour wind. I began to think my evening would have plenty of drama even without a new party dress. I'm just hoping the tree in front of my house doesn't come crashing through my bedroom window while I sleep.\nBut banish that gruesome, silly thought. This might be the worst hurricane in the history of New York. But let's focus on what really matters--shopping, and all the money retailers might be losing as the result of Hurricane Sandy. As the magnitude of the storm intensifies, so does the magnitude of the offers. I'm wondering what Sandy, Bloomie's and Nordstrom's will come up with next. Our operators are standing by.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Home Columnists Ric Polansky Mojacar was the place for free homes and land! Want some?\nMojacar was the place for free homes and land! Want some?\nMojacar's famous original mayor, appointed by Franco.\nONCE upon a time it happened.\nBut like all true fairy tales it occurred some 50+ years ago. Where were you then?\nIf you had known and showed up, you cold have been rewarded with a free home or site to build on. Hard to imagine now when you look about at the pristine village but once it was all but derelict and abandoned.\nLess than 200 souls resided within its walls, a forgotten lost hill top village. But it wasn't unknown, Spanish journalist Rafael de la Fuente had been there.\nHe had been keeping a low profile while living in Torremolinos.\nFollowing the civil war both sides wanted his head. He wrote and spoke too freely about everything.\nRafael often shared thoughts and drinks with a new friend, Columbian concert pianist Enrique Arias. Both were disillusioned concerning what has happening to their quiet living haunts.\nThe un-going and uncontrolled development saddened them. They yearned for a new place. Rafael convinced Arias to come with him on an adventure to a distant area not on most maps, lacking paved roads and no bridges to cross the deep ravines.\nOff they motored to find somewhere guiet and authentic to settle. A unique village that would keep its ancient Spanish customs.\nOnce in Mojacar they encountered the sagacious Mayor Don Jacinto, a man still revered for his simple cleverness. They met in a local tavern, most likely the Hotel Indalo right in the main plaza and plotted a new Utopia in variance with uncontrolled development.\nNo statue nor plague commemorates this motive occasion. The basics of the agreement were Mojacar was to retain its cultural heritage, no high rises and hopefully expand in an artistic bent. A place where musicians, artists, painters and writers could languish without being plagued by the masses of modern tourism.\nEvery mayor since has claimed responsibility for the turnout of Mojacar, but it was all negotiated back in the day when the whole concept seemed an impossible dream.\nThe pact was basically free houses or sites would be given away to 'takers' and title deeds issued once the houses were renovated and done. For Enrique's part he would run adverts in famous news papers, Le Monde, NY Times, Der Spiegel, etc offering the miraculous bargain \u2013 for free.\nThe people that came weren't necessarily artists but were politicians of high standing, ambassadors, the good and knighted. Enrique took the highest house in the village, then called el Castillo and fixed it up and gave his first music concert back in September of 1962.\nPeople drove from miles around to attend. What an event!\nAs if bringing water to the desert it brought back living souls to a ghost town with a vibrant historical past. Enrique's son still lives here and plays an important role in the constant civic betterment of the community.\nYet, never a thanks ever whispered by anyone.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This permissions are required for the vSphere Data Protection to work. You have to set that permissions to the vCenter object. Please note that you should not give different users permissions to backup different Virtual Machines. When you edit a Backup Job which contains VMs that are not visible to you, the Virtual Machine gets removed from the backup job.\n\u2190 VMware Workstation 10 Released - What's New?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you didn't make it to the Fancy Food Show during its recent three-day run at New York City's Jacob Javits Convention Center, take a stroll down the aisles through the eyes of Ciao Chow Linda. Granted, it's my Italian-centric vision, but I did taste foods from countries all around the world too \u2013 Canada, France, China, South Africa, Morocco and the U.S.A. of course, as well as many more I can barely remember.\nThis is a long post with lots of pictures but if you stay tuned to the end, you'll see my top five favorite things among the thousands of foods available for tasting in the huge convention space, plus a recipe using one of the top five products on my list. The Fancy Foods Show is North America's largest specialty food and beverage trade event with 180,000 products including confections, cheese, coffee, snacks, spices, ethnic, natural, organic and more. There were 2,500 exhibitors from 81 countries represented, and 24,000 attendees. So you see why it takes three days to make the rounds. I know I missed some things with just one day there, but there's always next year. If you want to head to the West Coast, the 2011 Winter Fancy Food Show takes place from January 16-18, 2011 in San Francisco. Comfortable shoes were de rigueur, since the Javits center has 760,000 square feet of exhibition space and I was trying to see it all in five or six hours. I walked in just after it opened at 10 a.m. and saw a ton of cheeses. I stopped immediately to sample some of the Italian cheeses. Can't resist a good aged parmigiano.\nThis \"La Tur\" cheese from the Piedmont region was a real winner \u2013 a soft cheese made with three different types of milk \u2013 goat's, sheep's and cow's milk. The Rosso di Langa is delicious too and they're not that hard to find. I saw the Rosso di Langa in my supermarket this morning.\nI wish I had sampled this French girolle cheese, but I took the picture and forgot to try it. It looks interesting doesn't it, with that spinning crank on top? Next time.\nHow about this cake made of different wheels of cheeses from Cypress Grove Chevre in good old Cal-i-for-ni-a. They make delicious, award winning goat cheeses like Humboldt Fog and Bermuda Triangle, a triangular shaped goat cheese that makes a distinctive presentation.\nMany of the exhibitors were serving these crackers from 34 degrees that I had never tasted before, and I am now sold on them. They were the perfect accompaniment and come in many different flavors aside from \"natural\" including rosemary and lemon zest.\nI tasted lots of olive oils too, from France, Italy, and Greece, among other places. My palate was getting jaded at a certain point, but this one made from Moroccan olives\u2013 Less Terroirs de Marrakech \u2013 was new to me and I really liked it.\nI tasted so many balsamic vinegars too, including one that was 50 years old. But I have to say, nothing compared to the complex, caramel-y aged balsamic vinegar I tasted in Italy at Acetaia San Giacomo, owned by Andrea Bezzecchi. Some of his products were also on display at the show.\nI was getting thirsty so had to have some liquid refreshment, starting with this sparkly blood orange soda made by Lorina, a French company. It was a cool, refreshing break. Their product comes in lots of different flavors including a lemonade.\nLater in the day, I saw a woman holding a tray of green apple martinis for sampling. I had already nixed the hard liquors like cognacs and whiskeys, since I had to keep my wits about me, but couldn't let this pass me by. I haven't had one of these in years and this one was just as good as I remember. I'm sorry I forgot to get the name of the company that makes this cocktail mix.\nAt a certain point, I needed a cup of coffee to keep me going. You can't go wrong with a cup of Illy espresso.\nAfter a few hours of walking, tasting and carrying a lot of brochures and samples in a bag slung from my shoulder, I thought I had done a pretty good job of perusing the booths. But then I found out there was another entire floor with at least as many exhibitors. OK, so I rested up a few minutes in a chair and plunged ahead. Hey, it's all in the name of research for you guys, so I steeled myself for a few more trying hours. I'm so glad I did because I discovered a few really great specialty products and ran into a few notables too, including this gal, who's the reigning Miss Italia, Maria Perrusi. We have a kind of inside joke in the family about Miss Italia because our vacations for many years seem to have coincided with the televised beauty pageant whether we were in Italy, the Czech Republic or in Hungary. One year we bumped into quite a few actual contestants in Salsomaggiore, a spa town where the pageant is held and nearby to where my relatives live, but this time I only had to schlep to New York to meet Miss Italia in person.\nBut even more fun was saying hi once again to Tanya and her mother, Lidia Bastianich. They were serving up some pasta made with Lidia's line of pasta and sauces. Naturally I couldn't pass up a plate from my food guru. The only time I've ever bought a jar of commercially prepared tomato sauce was decades ago for a camping trip, but I have to admit that Lidia's jarred sauce was delicious. But what would you expect from the queen of Italian cuisine?\nMona Lisa was also there. Can you believe this was made entirely of Jelly Belly jelly beans?\nOK, so maybe you didn't want to sample cheeses anymore, you wanted something more substantial for lunch. You want to start with some olives or an antipasto bar?\nOctopus, seafood salad and anchovies \u2013 if fish is more your thing.\nOr maybe gnocchi in pesto sauce ladled from a huge parmesan wheel?\nTake your pick of these pastas in red sauce being served up by Lombardi's Gourmet Imports.\nOr how about a porchetta sandwich. This porchetta is made in Kentucky by Porchetta Primata, but they even had one a producer from a vendor in Ariccia, Italy, the world-famous home of porchetta.\nEating light today? Try some of this really flavorful tuna from Tonnino.\nNow if you wanted to have something sweet, you were in heaven. There were enough chocolates, candies, cakes, pastries and cookies to satisfy the most indulgent sweet tooth ever. This Gooey Butter Cake mix from Ann & Allen Baking Company in St. Louis was really good, especially the chocolate variety.\nHere are some other products I would gladly eat again and again. I loved these cherries in Kirsch alcohol \u2013 Griottines \u2013 made by a French company. They're delicious all by themselves in a shot glass, but they'd also be great with champagne or prosecco, or served over ice cream, or cooked in a sauce and poured atop a duck breast.\nThere were soooo many types of pasta, both fresh, frozen and dried. These little shapes were so adorable \u2013 one in the shape of clusters of grapes, the other like little ears \u2013 and both in multi-colored varieties. But from what I could gather, this brand \u2013 Casa Temporin \u2013 is still not available in the U.S.\nThis was new to me too \u2013 chocolate puff pastry from DuFour Pastry Kitchens. Served with whipped cream and strawberries, it gave me ideas for future desserts.\nThese cookies made out of rice and corn flour and flavored with orange were really good. Even if I weren't wheat intolerant, I'd eat a bunch of these Italian cookies made by the Oryza Group.\nThis fig balsamic dressing from Stonewall Kitchens is a new product for them and was a nice find. I'll be looking for it in specialty stores near me.\nOK, counting down to the five yummiest things I ate and plan to search out and eat again and again are: 5. Bobbysue's Nuts \u2013 These were so addictive I could have eaten a ton. They're organic pecans, cashews and almonds that are spicy and sweet but not too sweet. There's even a variety that's drizzled in chocolate if you want to take it a step further.\n4. These Italian jams from Fattoria Antica Tuscia were sensational \u2013 and there was no sugar or other sweetener added.\n3. These figs from F.illi Marano, \"bocconi della nonna\" or \"grandmom's little bites\" were a real treat. Stuffed with almonds or walnuts and then rolled in a cinnamony sugar mixture, or dipped in chocolate. yumm.\n2. Point Reyes blue cheese from Farmstead Cheese Company. Forget those gorgonzolas, those Stiltons, those Saga blues. This buttery, creamy, tangy blue cheese made in California is THE BEST blue cheese I have ever eaten.\n1. If I came away with nothing else from the show, I was thrilled to have discovered this product made in Armenia. It may not look like much, but what you're looking at are fresh walnuts in the shell by a company called Harvest Song. Fresh baby walnuts are picked green from the tree before they have a chance to age. Then they're cured with lemon juice and sugar and served complete with shells that are still soft. This was my number one pick from the show not just because it was so darn good, but because of the unusual nature of the product. The company also makes a line of wonderful fruit preserves too.\nBy the way, I was not paid to endorse any of these products. I'm just giving you my honest opinion on what I thought were really delicious foods and beverages that I tasted the day I attended. I know I left out a lot of things too, but I didn't want this post to run on forever. I tried to find out whenever possible, where consumers could buy these foods. Some of the exhibitors didn't know. In many cases, Whole Foods was mentioned. Murray's Cheese Shop and Zabar's in New York City were also cited as places to buy some of these items. If you're not in New York, check with the specialty food store near you, or look online. Many of these companies have websites and I've included links whenever possible. Other times, you can find them for sale at online companies like Amazon.com. Hope you enjoyed the tour of the 2010 Fancy Food Show. Ciao Ciao to the Javits Center from Ciao Chow Linda.\nOH, not over yet. I headed back to Times Square (where I used to work) when I saw this fella on the street having a nosh. Us humans at the Javits Center weren't the only ones with food on our minds.\nWhat a great show! A dangerous place for me ;-)! Lovely recipe.\nOh my, I would have had a blast at the food show. So many incredible items to looks at.\nThe recipe sounds delicious as well!\nI wish I would have known I could have met you there \u2013 what a nice perspective you offer.\nI drank vinegar as a kid. When I was skinny. So I gazed longingly at the rich, deep, caramel-colored balsamic. What a cocktail! I spent a long time in the post \u2013 gazingly lovingly. I wish I had arranged a NYC trip for this! The cluster-of-grapes pasta would be so much fun. The array of fresh, organic foods delights. And isn't it wonderful that the USA increasingly produces excellent cheese? Oh! Loved the horse. He\/she sure enjoyed the meal!\nLinda, you're killing me! Killing me!\nAnything cool from the Canadian vendors?\nThanks for taking us there with you!\nOnce again, we missed eachother in the crowd.\nI loved the cheese and EVERYTHING, though I did not see Lidia.\nI buy that jarred tuna tonnino\u2026..I love it. There was too much to see, it was exhausting.\nLots and lots of good eats to be had but I end up feeling sick after all the samplings. Best to go each day if time allows.\nHow did I miss this crazy food-filled event! (Although it's probably better for my wallet that I did\u2026) So envious!\nI love this post, Linda. I was an exhibitor at the Fancy Food Show for years and never got to browse the aisles and take the time to see everything. You've reminded me of how much fun it is. I should go back one of these years just to see it again.\nLinda the Fancy Food Show looks like so much fun but I'm glad I didn't go to it as I would have no will power and would have gained a zillion pounds! All that cheese! Those wonderful dried figs in cinnamon like my mother-in-law used to make, and even Nduja \u2026oh my!\nWow, this is my kind of event! Will definitely mark my calendar for next year. And Lidia Bastianich \u2013 so cool!! Thank you for the great coverage and the yummy recommendations. I can't wait to try Point Reyes blue cheese (this is the 2nd time I've read about it in as many days). And those cured baby walnuts are so unique!\nI certainly didn't make it all the way from Aus but it looks great!\nWhat a great brief of the show! I feel like I've been there myself!\nI love everything in here!\nMa \u00e8 un post meraviglioso Linda: foto splendide, racconto esauriente e cibo ..perfetto!\nWow, you really had your energy level up for this show! I have attended a few in past years, and have always been overwhelmed by the choices. Those cheeses look amazing and there were so many other things that I would have tried too\u2026your tour and pictures were terrific! Thank you.\nAvrei voluto essere li con te, adoro queste manifestazioni; quante cose belle da vedere e gustare. Molto interessante questo post, con delle foto splendide. Ciao Daniela.\nWow, my head is spinning from all this, Linda! I wish I could have tagged along with you. It's funny, because I was just writing a little something about our local winner, an Indian chef, Maya Kaimal of Rhinebeck, who won the top honor for her tikka masala sauce in the best cooking sauce category. I will have to consider going to this next year. Thanks for sharing your fabulous photos \u2013 I am really hungry now. And you got to see Lidia again!\nFabulous post! What a wonderful show and your coverage.. awesome Linda! I'm so jealous \ud83d\ude42 Thanks for sharing this!\nI loved the tour! I only wish I was besides you walking with a huge tote bag on my shoulder full of goodies. That spread from Calabria, I'm going to look for that! The tuna, cheeses, etc, etc, etc. Those walnuts must have tasted amazing to get that reccomendation from you. I would have to detox for a week after that!\nAll of those fantastic looking cheeses and olives \u2013 oh my! Thank you for the tour!\nOh you had me with cheese!\nLinda, you're a trooper, I can barely last two hours at that show anymore, what with so many aisles of tasting and the way the heat in there builds up! Chocolate puff pastry \u2013 oh boy, I hope Bon Appetit carries that! Thank you for letting me tour the show with you, from the air conditioned comfort of home this year.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Wyldecrest Parks PTC Grand Finals... Final Discussion !\nRe: Wyldecrest Parks PTC Grand Finals... Final Discussion !\nI would be surprised if Hawkins loses this, but anything can happen in a best of seven of course... Would be nice to see Gerard win.\n1-0 Hawkins already. Good one visit frame. Needs to keep doing this to prevent Greene from grinding it out.\nHawkins is a great player to watch, I like his style. Has a good all round game and a great temperament. Always seems cool and focused.\n3-0. All Hawkins here. 96% pot success.\nHawkins just too good for Greene.\nHolden Chinaski wrote: Hawkins is a great player to watch, I like his style. Has a good all round game and a great temperament. Always seems cool and focused.\n2\/3 PTC Grand Finals have been whitewashes, looks like it'll be 3\/4 in a couple of minutes.\nAnd all done. Hawkins is now a multiple ranking event winner. He really has improved in recent times. Currently a top 4 player and performs like it.\nSkullman wrote: And all done. Hawkins is now a multiple ranking event winner. He really has improved in recent times. Currently a top 4 player and performs like it.\nWell done Barry, unfortunately for Gerald he got off to a flyer and after that were was only gonna be one winner.\nSnooker Overdrive wrote: Well done Barry, unfortunately for Gerald he got off to a flyer and after that were was only gonna be one winner.\nYeah when Hawkins plays like that he's very hard to stop.\nSonny wrote: Well it's not really Wild. If you think about it, Hearn has always wanted emphasis on the PTC series to make sure all players are incentivised to take part in the best of 7 three day events which take place at new locations across Europe and Asia. He said or JF said somewhere that any extra prize money on tour would be prioritised in the PTC grand finals.\nI am surprised at the huge difference between winner and runner-up but overall I have no issue with it.\nIf that's the case why didn't he make the PTC Grand Finals in line with the Other ranking events regarding points in the last 4 years?\nWell apart from the draw for the Worlds, there is no other impact to this. The tour places at the end of this season will be decided on prize money, not ranking points. Right now, today, this hour, ranking points are dead. RIP ranking points, welcome prize money lists.\nThey have addressed the main concern, prize money first round losers will only have half the prize money count towards ranking. Despite being dead against it when first suggested, I am now not 100% in favour because I think it is a bit commercial, but I can accept it.\nNow how the wc draw looks like???\nPrize Money rankings is the biggest load of cockerel it goes against what rankings should be about 175,000 points different between World Champ and Runner up don't make me laugh being world champ doesent make you that much better than the Runner up.\nYes but even under the old system there was a difference between winner and runner up. 17-17 in the final, that last frame had so much hanging on it wrt ranking points, money and prestige.\nHawkins can reach the World final again and he's showing atm that the final last year was not a fluke. He's playing really strong stuff and I mean, his scoring sometimes is really heavy.\nHe has come a long way since the days of being the perennial under achiever, just outside the top 16 and who crumbles under the heat of battle. Always been a classy player to watch style wise but so late into his career has finally come good. I don't think it will be a massive shock if he wins in Sheffield which is something you wouldn't have said a couple of years ago.\nSonny wrote: He has come a long way since the days of being the perennial under achiever, just outside the top 16 and who crumbles under the heat of battle. Always been a classy player to watch style wise but so late into his career has finally come good. I don't think it will be a massive shock if he wins in Sheffield which is something you wouldn't have said a couple of years ago.\nI quite enjoyed what I saw of those tournament. Lot of upsets, chances to see different people have decent runs in the tournament.\nSnookerFan wrote: I quite enjoyed what I saw of those tournament. Lot of upsets, chances to see different people have decent runs in the tournament.\nYeah me too, it's a terrific venue as well so I think it should definately be held there again. And if we're honest if it had been in Thailand as originally planed it wouldn't have been so good, far from it.\nYeah, Preston was a success. Great crowd, considering it was held there at short notice.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzawcnd b/data_all_eng_slimpj/shuffled/split2/finalzzzawcnd
new file mode 100644
index 0000000000000000000000000000000000000000..9e5ab94acf5e4dfa386dd84fad57a7420768b97e
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"We made this loaf in the weekend and man it was yum! I've been busy making some cushion covers for a buyer through my felt. shop and haven't done any baking lately, but really wanted to try out this recipe because I couldn't imagine what courgettes and blueberries would taste like together, it sounds disgusting. However, the loaf was delicious \u2013 sweet and moist and eaten up pretty quickly! And, I suppose it kinda counts towards your 5+ a day of fruit and veg - did you know the courgette is actually a fruit? Personally I couldn't really taste the courgette but you see flecks of green through the loaf which looked quite pretty with the purple blue colour of the blueberries.\nPreheat oven to 175\u00b0C. Lightly grease 2 loaf pans.\nIn a large bowl, beat together the eggs, oil, vanilla, and sugar. Fold in the zucchini. Beat in the flour, salt, baking powder, baking soda, and cinnamon. Gently fold in the blueberries. Transfer to the prepared loaf pans.\nBake for 50 - 60 minutes, or until a knife inserted in the centre of a loaf comes out clean. Cool for 20 minutes in the pans, then turn out onto wire racks to cool completely.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Discussion in 'macOS' started by geltab, Aug 5, 2010.\nSo I am having some really annoying issues with my 17\" MBP (early 09). It all started when I updated to 10.6.4 a month or so ago. I remember reading something about fixing keyboard issues on the MBP. Funny enough, I never had a keyboard issue on Snow Leopard. So shortly after upgrading my keyboard stopped working. The main problem is that the shift key on the right side and the option, command and control keys stop working at random intervals. I happens across all programs. i can't reset pram or anything that because the keyboard doesn't work upon startup. So, I took the most obvious route and did a clean install and wiped everything off the computer. I didn't update to 10.6.4. Things were working okay for a while and I updated to 10.6.3, just to be safe. That is when the keys started acting funky again. Same set of keys again. I don't use spaces, it is turned off. I know after researching that some people had issues with using spaces.\nAt this point I figured that maybe 10.6.4 would fix the keyboard (like it said it was supposed to-but actually broke my keyboard the first time around). Sure enough, it didn't fix a thing. I also installed a copy of windows 7 via bootcamp. Get this... the keyboard issues carry over to Win 7 also. This is weird.\nAny solutions or suggestions? I want to think that at this point has to be a hardware issue since it carries over to Windows. But the weird thing is that the keys work sometimes and don't others. This was a documented software issue with 10.6.3 and .4. Could this carry over to Win7 under bootcamp and still be a software fix? Strange thing is they keyboard doesn't work on start up at all. i can't hold option to select what os to boot from.\nI forgot to say that my computer doesn't stay asleep. I put it to sleep and 90 percent of the time it wakes back up. Even when I close the lid to put it to sleep. A few times I closed the lid and left the computer only to come back to the fan running like crazy and the computer was burning up.\nSee what is causing the problem by open \/Applications\/Utilities\/Console and look at the logs and the latest entries should have the reason why it is openin from sleep.\nThe Console logs should check every once in a while to see what is happening in your Mac. Plus also in \/Applications\/Utilities\/ another great utility in Activity Monitor. With this you can see if any program is getting out of hand.\nPretty sure you have a hardware problem.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"FIREwave is a Hybrid Wireless based family of products which is fully integrated into Hochiki's renowned ESP intelligent hardwired system. This seamless integration provides the user with maximum flexibility in configuring the system to meet the specific needs of the building into which is is to be deployed.\nFIREwave employs the latest technology, providing a simple and economic hybrid wireless fire detection system installation with minimal disruption to the fabric of the building and its operation. Flexible and versatile, FIREwave is suited to a wide range of applications, matching the requirements of any building from busy hotels to sensitive heritage buildings. FIREwave is the most flexible hybrid wireless fire detection solution available that is simple to install and maintain, overcoming installation and maintenance challenges on the most demanding sites; advanced patented technology is used to achieve optimum performance.\nFIREwave sensors are supplied in three variants: Optical, Thermal and Multi Criteria, which cover a wide range of applications. The Optical and Multi Criteria detectors feature double dust traps and adaptive signal processing to prevent false alarms. Thermal detectors can respond to a fixed temperature threshold, or detect a rate of rise in temperature.\nThe range of FIREwave devices provides a visually attractive wireless fire detection solution, encompassing a wide variety of installation types and sizes. Wall and base mounted audio and audio-visual devices have adjustable volume levels and multiple tone settings, and are available as stand-alone Sounders and Beacons, or as combined units. Wall mounted units can be supplied in red or white, and weatherproof options are also available.\nOnce installed, the design of our products ensures simple on-site maintenance. The lifespan of these products can be maximised using Hochiki's Maintenance and Service facility, where products returned are dismantled, cleaned and re-calibrated to effect continued operation for many years.\nIn addition, Hochiki's dedicated team of technicians provides support to customers, and ensures that installation and maintenance processes remain transparent, effortless and focused on customers' requirements.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This is the less expensive, silkscreened (not hand painted) imported version. Has standard LED and power jack.\nGentlemen\/women, start your engines. This pedal is so dangerous that I'm sure I'll be getting a lot of complaint emails that people have blown up their amps using it. So let's just start this off by saying I WARNED YOU TO BE CAREFUL. When using this pedal, start off slow, and monitor your amp for potential damage. Don't have too much fun! OK, have too much fun, but don't blow up your amp in the process. Unless that's what you want, of course, and well, gosh darn it, it's your right. Good ol' Pete Townsend did it. Well, at least he poked his speakers out. All right, I'll try to stay on track here.\nAhem. The SUPER-DUPER 2-IN-1 (TM) has two of my infamous but rather delightful Super Hard-On(TM) pedals in one small box, with two switches and LED indicators. HEY! I responded to your multitude of complaints that I don't put in LED's! Also, in this SUPER-DUPER 2-IN-1 (TM) (gosh I love saying that) is a Master volume control that lets you use it as an overdrive\/distortion with any output volume. My my! How conventional, you say! Well, suffice to say, if it weren't there, you'd go deaf with both of those channels cranked up. This pedal is dangerously loud. Don't do what I did, and lean over in front of your speaker cabinet while turning it up. Ouch. Dang.\nThis channel has the very same circuit as Channel 1, but the bleeder resistor on the output has been replaced with a Master Volume control, which allows the user to turn down the output volume even if the gain is set quite high. The knob on the left is the gain, and the middle knob is the Master Volume, and the LED is red. When both channels are on, or if only Channel 2 is on, the Master Volume is active. If you leave the Master wide open, you have two identical SHO pedals in one box, which are cascaded.\nWHAT DOES THIS PEDAL SOUND LIKE? You might wonder just how insane this pedal sounds when you crank up both channels and turn them both on. Well, I'll tell you. I can't get my Les Paul to stop feeding back at any audible volume level on my Marshall when both channels are wide open, and that's with the amp set at 1. I'm happy about that. Of course, this is not the only setting on the pedal. I don't think there's a musician around who wouldn't be happy to have multiple clean boost pedals set to predictable levels available on stage. This pedal gives you a total of four volume levels, ranging, at your choice, anywhere from perfectly clean to ragingly overdriven. Level 1, your guitar is true-bypassed with both channels off. Level 2, you turn on Channel 1. Level 3, you turn off Channel 1 and turn on Channel 2. This is easy because you can put your foot on both switches at once and alternate between the two channels if you want. Level 4, you turn on both channels.\nThere is no tone shaping in this pedal. It's two perfectly linear high-gain high input-impedance preamps that can be cascaded, with a total gain that ranges from unity to 3600. That's right, the maximum gain with both channels wide open is a whopping 3600, which is so completely distorted, hissy, feedbacky, and insane, that you'll never ever have a steady relationship again. Or a lease. Or a pet. Luckily, with the Master Volume, you can listen to what it would sound like to lose everything you love at a reasonable volume. Naturally, because there is no tone-shaping, your guitar just sounds like a really loud version of your guitar, and your amp sounds like a really big version of your amp, and if your normal tone is somewhat bloated, it will get very very bloated when you use this pedal set to high gains. It does best if you are close to getting the tone you want out of your gear already, but just need a boost to make it go over the top. Keep in mind that you can set up the two channels to give you any amount of gain, from a soft bluesy boost in two stages to a crunchy rock level, to a wild sustaining beast.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I joined Peter Barry Surveyors in June this year following the completion of a Masters degree in Real Estate Development at the University of Westminster. This qualification has enabled me to enrol on the RICS Assessment of Professional Competence (APC) and begin working towards becoming a Chartered Surveyor. I will be writing about my training experiences here on the blog over the coming months.\nThe APC is a structured training & experience program, that when combined with a relevant degree leads to Chartered Surveyor status. The APC sets out a list of professional competencies that the candidates must cover in order to be proficiently trained within their selected area of surveying. Candidates also have a supervisor and councilor to guide them through the training and oversee progress.\nI have chosen to follow the Building Surveying APC pathway as this will give me an excellent opportunity to develop hands on experience within the property sector and coincides perfectly with the professional services that we offer such as RICS Homebuyer Reports, Building Surveys, Party Wall Consultancy, Property Valuations and Leasehold work.\nThe APC essentially gives the candidate the opportunity to develop & obtain experience, skills and knowledge within their chosen pathway. Before I will be ready to sit the final assessment I must record a minimum of 23 months experience. The final assessment consists of a presentation to a panel 3 experienced surveyors followed by an extensive interview.\nThroughout the APC period the candidate must maintain a day to day log book and diary setting out details of the various professional competencies that have been covered. The objective of the APC is to ensure that at the end of the structured training period the candidate has sufficient knowledge, understating & experience to enable them to undertake professional work on behalf of clients and provide suitable advice as needed.\nOver the coming months I will be writing about the challenges that my APC brings and the experiences gained along the way.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzawhai b/data_all_eng_slimpj/shuffled/split2/finalzzzawhai
new file mode 100644
index 0000000000000000000000000000000000000000..98825ec88b1d37329dbd5f1664391c9d475d52ab
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Add wet mix to dry mix. Mix well. Let it sit for 10 minutes.\nSpread half of the mix in a 9\u00d713 cookie sheet, or all of it on a large cookie sheet.\nNotes: Easy to make. Can make half the recipe.\nThis entry was posted in Food, recipe, Snack on February 12, 2019 by admin.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"RTP\/WP : can we append Marketo data with our own ?\nIs there a way to dynamically query these web services when someone visits our site and have WP fire some personalization campaigns based on this ?\nI understand that we could use the Custom Data Events or the \"User Context\" APIs and use these to create segments, but has someone already implemented it and provide some insights ?\nAlso, any opinion about knowing in which case should we use \"Custom Data Events\" ? In which case should we use \"User Context\" ?\nRe: RTP\/WP : can we append Marketo data with our own ?\nThe challenge here is the real-time (millisecond) matching of IP to company and firmographic data and ability to respond with personalized content without seeing a delay for the end user. So best is to use our data source, with that said we open to enriching and increasing our data ranges and sources if we see it is better data. If you have any other sources for different countries, we can look into it and see if it is worth adding it to our source (send me some of the examples or sources you use (dmyers@marketo.com).\nAlso, as you correctly said, using the data events or custom variables (user context) you can send your data to Web Personalization as a visitor hits the website and we can segment and personalize based on this data coming though immediately in real-time.\nData Events are best use for one off events in the current session.\nThx for your input on this. I'll loop back with my customer and keep you posted. I understand that, especially in the Netherlands, such databases exist. And I fully measure the key aspect of response time.\nAnother way of looking at this is getting an API that would enable us enrich WP database in the background, adding data to custom fields and be able to use this info when the lead comes back. These custom fields could also be shared with MLM so that when the lead becomes known, the info is not lost.\nOne the first source I would think of is our own MLM database and its list of inferred IP addresses linked to existing contacts and accounts. Might not be always accurate and complete, but for SMB, it's better than nothing This is also true for account identification in the ABM option, BTW. Of course, usage of this data would be limited to within our own instance of Marketo.\nAlso, it would be great is, when a company is identified, a National ID or a DUNS number could also be brought back. Especially in France, we have a very thorough system of managing national ID's. Every single business has such an ID (called SIREN, made of 9 digits for companies and 5 more digits for establishment), even freelances or doctors. For instance, if you search \"siren 521712133\" on google, you will find my company. And this is public information, together with the company name and the local equivalent of the SIC (industry) code. This would enable customers to contract with local third parties or with D&B for data enrichment.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A stub is an article which, while present in the database, has very little content, and is looking for expansion.\nIn a peculiar meta-statement of stubbiness, this page is itself a stub. Remove the following transclusion tag when this page is expanded.\nThis page was last modified on 1 October 2011, at 17:41.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A few days ago, one of my customers contacted me regarding one of the illustrations I've done.\nThe person seemed very intrigued by the drawing and curious about its origins. To tell you the truth, this was my very first illustration in this style - just the moment I discovered how to express myself with no words. I never felt strong when it came to drawing from imagination, but then, with all the thoughts about our beautiful planet and how we don't care for it enough and how we treat it as an object that we own, this came clear to my mind. And it started a series of emotional drawings that I see just as a mirror of what's going on in my head.\nI decided to re-publish my letter to Erin, my daughter, that was originally published in Wildling Magazine (Vol. 5).\nPlease see below or go to the digital version of Wildling Magazine here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You can have one, or both types of power. They allow the person or people you nominate to make financial and healthcare decisions on your behalf. In relation to financial affairs, this can be from registration, or only after you lose the ability to do it yourself. Health and Care powers can only be used if you lose the ability to make those decisions yourself.\nSpecifically what does a Health & Care Decisions Power of Attorney cover?\nYou can also give your chosen attorney(s) the power to decide whether to allow or refuse life saving medical treatment.\nThese types of powers can only be used if you lose 'capacity'. This would often be the result of certain illnesses, including dementia and Alzheimer's.\nSpecifically what does a Property & Financial Affairs Power of Attorney cover?\nUnlike Health powers, you can implement this whenever you like: from immediately, to only when you have lost 'capacity'.\nIn this power of attorney, you can allow wide ranging powers for your attorney, or very limited powers. This may be relevant for example if you live abroad, but have assets in the UK that you need someone to deal with here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzawlzn b/data_all_eng_slimpj/shuffled/split2/finalzzzawlzn
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index 0000000000000000000000000000000000000000..2eb40eab9c20ef87cec211d612246f791367f884
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Agatha's Attic is a Vintage props, paraphernalia and fine china hire company for special events.\nAgatha's attic contains a treasure trove of afternoon tea sets, vintage props, photo booths and everything you'll need to get that vintage look at your event. Whether it be your wedding, an afternoon tea party, evening dinner, cocktail party, birthday, themed evening, murder mystery weekend. All items can be collected from the West London or Lincoln area, however we are happy to deliver with the first five miles being free. Delivery can be arranged via courier should you be further afield for an additional cost.\nThe process of gathering props and specific items required for your wedding day or event is not only incredibly time consuming and particularly expensive, but it also means you have less time to focus on the bits the parts that really matter. We decided that Agatha's Attic should take away the need to purchase items for one single day and allow people to hire quality props and paraphinalia. One of our specialities is creating electrical props such as an Electric Gramophone (as opposed to wind up) and a voicemail guestbook via one of our vintage Poirot Handsets.\nHere at Agatha's Attic we aim to provide an affordable service for those looking for a vintage touch of class to their event.\nWe used Agatha's Attic for our wedding and hired the light box, sweet trunk, and some tea pots for table centres. They were amazing. Our reception was in an old manor house, so the vintage props fitted in perfect, then the evening do was in a more modern extension and the light box looked BRILLIANT with my husband and my initals K & P. The sweet trunk was awesome, filled with loads of sweets that certainly kept all the children enterained! The photos don't do any of the props justice. The owners are wonderful, so friendly and professional and bent over backwards to make sure we had what we wanted on our special day. They were always quick to respond to my messages, which is brill as I have no patience lol! They have so many different and unusal items for hire that can give your wedding an individual look, and I totally recommend this company!\nFrom a couple who said 'I do' in London, March 2016.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"That ground breaking along with exciting house type is required to provide coziness from home, that Superior French Provincial Hall Table French Provincial Hall Table With 3 Drawers Oak graphic shall be an answer if you are at this point on the lookout for house style and design idea. All the wonder exhibited by way of Superior French Provincial Hall Table French Provincial Hall Table With 3 Drawers Oak graphic will furnish some sort of abundant associated with skills for you. Anyone simply need to reproduce the weather which suit your needs together with wishes. Superior French Provincial Hall Table French Provincial Hall Table With 3 Drawers Oak photo will be your own counselor inside improvement or simply creating a residence. Your dream house inspired just by Superior French Provincial Hall Table French Provincial Hall Table With 3 Drawers Oak photograph would be your most effective location to relax and additionally demand your power.\nThe good news is you will find even more amazing graphics like Superior French Provincial Hall Table French Provincial Hall Table With 3 Drawers Oak photo inside French Provincial Hall Table photograph collection. You may get these individuals at zero cost and painless, to obtain this marvelous Superior French Provincial Hall Table French Provincial Hall Table With 3 Drawers Oak picture, you can actually click this link. In addition to to be able to benefit from Superior French Provincial Hall Table French Provincial Hall Table With 3 Drawers Oak picture for the reason that your private publish material in the site, you will want to have the original source link. Superior French Provincial Hall Table French Provincial Hall Table With 3 Drawers Oak graphic is normally Hi-Def image, you can easily recognize that within the actual resolution that is 1000 x 595 while the actual size is actually 52 KB. Always make use of Superior French Provincial Hall Table French Provincial Hall Table With 3 Drawers Oak pic for your own objective, you tend not to suggest you to ultimately employ Superior French Provincial Hall Table French Provincial Hall Table With 3 Drawers Oak graphic to get store-bought motive. Superior French Provincial Hall Table French Provincial Hall Table With 3 Drawers Oak pic is normally an individual image that will desired that guests, for the reason that proved from the various people who have experienced it. Correctly, 1 visitors offers looked at Superior French Provincial Hall Table French Provincial Hall Table With 3 Drawers Oak picture due to the fact submitted on December 5, 2017 at 1:55 am. Maintain discover this approach Superior French Provincial Hall Table French Provincial Hall Table With 3 Drawers Oak photograph.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Former ADT CEO Tim Whall says the future of smart home security is $10 per month, no-term monitoring contracts sold online to clients. Residential dealers need to transition to higher upfront equipment fees.\nWhall believes integrators don't need long-term contracts in order to compete and instead can obtain financing from banks if they start charging more upfront.\nFormer ADT CEO Tim Whall has some sobering news for integrators engaged in the residential security sector: $10 per month, no-term alarm contracts sold over the Internet will be the norm going forward, and the trend will push down valuation multiples for integrators' exit strategies.\nFor an industry built on high-margin monitoring with long-term contracts that eventually lead to high company valuations, Whall's assessment is more than a slap in the face\u2026 it's darn right depressing.\nWhall, who retired last year, made those predictions at the recent California Alarm Association (CAA) winter conference in San Francisco earlier this month during a sit-down Q&A presentation with Rodney Bosch, senior editor at Security Sales & Integration, a sister publication of CE Pro.\nThe discussion was wide-ranging but focused on the future of the smart home and where independent integration companies can carve out their niche when mega players like Amazon and Google are involved, along with a slew of startups pounding away on the consumer mindset that DIY security is better than professional installation.\nThese market-altering trends are forcing some security dealers to adopt new business models that account for the evolving dynamics of connected technologies and related services, including the prospect of adopting a DIY offering \u2014 or even a hybridized model \u2014 out of concern that competing DIY providers could cannibalize traditional buyers, Whall explained.\n\"The sales model has gone from in-home to more over the phone and over the Internet, with a minority of it now coming from those in-home sales,\" he said.\nWhall says that the new low-cost pricing models do more than just put pressure on integrators to lower their prices, but they also dramatically affect the long-term valuation of an alarm company.\n\"The valuations in our space have always done well because we have always done well on margins \u2014 50 percent, 60 percent, 70 percent margins have been what sells these companies at high valuations over the years,\" Whall said.\nWhall told CAA attendees that the increase in lower cost pricing models from new entrants, including national marketers such as SimpliSafe, necessitates that security dealers be more transparent about their own pricing. That is because mass-market security integrators are actually subsidizing the cost of the equipment in the monthly monitoring fee, which is why some systems are actually free to consumers.\nBut consumers don't really understand that, especially when they see DIY systems that instead charge an upfront equipment fee, then lower monitoring. That sort of clouds the overall picture for the consumer, Whall said.\n\"Even at $10 per month for monitoring, most [integrators] would probably say, 'I can still probably do that deal.'\"\nWhall emphasized it's easy to explain the details of an offering when speaking directly to the consumer. But, since sales are increasingly being transacted online, especially among younger buyers and Millennials, integrators need to fine-tune their websites to support an e-commerce platform.\nOn the topic of contracts, Whall suggested independent dealers should be comfortable with the idea offering of a no-term contract. He pointed out, however, that \"no term\" certainly does not mean \"no contract.\" Integrators want to protect yourself with a signed contract that limits liability and expressly details what happens in the event of an issue.\nHistorically in the alarm industry, Whall explained, the number of contractual months has always been about how banks could lend on that or dealers would leverage the contract to say they have guaranteed revenue coming in based on the length of the contracts.\nWhall believes integrators can still obtain financing from banks for customers at the $10\/month level if they start charging more for the upfront equipment.\nWhall compared and contrasted current market conditions in general terms between the residential and commercial sectors. Whereas the resi space is being inundated with new entrants, big-box retailers, telcos and the like, the commercial has proven more difficult to infiltrate.\nMost of the recurring services in the integrated commercial space is centered around test and inspection, Whall said, but independent integrators are more and more understanding they can build that part of their portfolio as a means of increasing their valuation multiple.\n\"The other nice thing about the commercial space, I have for a long time compared it to the automobile industry,\" he continued.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Published 04\/22\/2019 04:02:23 am at 04\/22\/2019 04:02:23 am in Cuisinart Slow Cooker Reviews.\ncuisinart slow cooker reviews cuisinart programmable review cuisinart programmable slow cooker reviews.\ncuisinart slow cooker reviews 3.5 qtc,cuisinart slow cooker reviews 3.5 utorrent,cuisinart slow cookers reviews,cuisinart slow cooker reviews 4 quart dutch,cuisinart slow cooker reviews australia,cuisinart slow cooker reviews 3.5 qt gas,cuisinart programmable slow cooker reviews,cuisinart 6.5 slow cooker reviews,cuisinart slow cooker reviews 3.5 qt,cuisinart slow cooker reviews 4 quarts,cuisinart slow cooker reviews 3.5 qtip.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Shortly before Christmas, Professor James Lowell is found brutally attacked in his rooms at Walsingham College, where Anna Hopkins works as an administrator. Baffled as to why anyone would wish to harm such a gentle, scholarly man, Anna discovers that Lowell had a connection with her fellow dogwalker, Isadora Salzman, who knew him as an undergraduate in the 1960s, a co-member of the so-called Oxford Six. It turns out that Isadora has been keeping a surprising secret all these years. But someone else knows about Isadora's secret: someone who has sent her a threatening, frightening letter.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzawuib b/data_all_eng_slimpj/shuffled/split2/finalzzzawuib
new file mode 100644
index 0000000000000000000000000000000000000000..6d344d9db6a64f87e8d26ae0766634c730dc6d77
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"What Does a Steel Detailer Do?\nA steel detailer creates plans for steel constructions.\nA steel detailer is a technical drafter who creates drawings used in the fabrication and placement of structural steel. The qualifications necessary to work as a steel detailer can vary by region. Some nations require that drafters meet specific education and certification requirements, while others do not, and it may be possible to enter a career as an apprentice with no training. Pay scales for steel detailing are highly variable. Some detailers require security clearances because of the nature of their work, and they must have clean criminal records as well as the technical skills needed to do the work.\nThe steel detailer produces accurate, detailed drawings to help steel fabricators produce pieces of steel to the specifications of a given project. The drawings are done in a traditional engineering style to facilitate ease of reading and use standardized measurements, terminology, and other conventions. Any fabricator should be able to work with the drawings, just as any detailer could pick up a project started by someone else in the middle and understand what the previous person was doing.\nIn addition, steel detailers also make technical drawings for the workers on a site. These drawings provide information about how to construct a steel structure, using the fabricated pieces of steel. Acting as assembly instructions, they tell the construction workers about what order they should use, and which pieces go where. These drawings may also include a brief technical paper to discuss the project.\nTo become a steel detailer, it is necessary to be familiar with steel construction, computer aided design (CAD) programs used to generate technical drawings, and the engineering concepts that underlie technical design. Some steel detailers choose to go to school and may attend a technical school or conventional college or university. In some regions, this is required. A new apprentice, with or without training, usually starts out under supervision and gradually takes on more responsibility over time.\nIn some regions, certification is necessary to work as a steel detailer. People in this field need to take an exam to demonstrate competence. Many steel detailers choose to join professional organizations to have access to other people in the profession along with trade publications, conferences, special training, and other opportunities. Detailers in industries like aerospace may make high salaries because of the complexity and level of detail on the projects they work on. For detailers in construction, pay may be lower, although the size and complexity of construction projects can also be considerable.\nWhat Does a Steel Fabricator Do?\nHow Do I Become a Steel Fabricator?\nHow Do I Become a Steel Detailer?\nWhat is Martensitic Stainless Steel?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Many people still think of a board with colourful stickies when they think of agility. And it's perfectly clear: Visualizing work and workflows on a board is basically a great thing and an important prerequisite for improvement. However, many \"Hooray Agilists\" still don't understand that more brain fat is necessary in order to fully utilize the real benefits of agility. This is what happens in many companies: Every team has its own board and uses it to push and pull its tasks back and forth. In some cases even the management team has its own board and manages its tasks on it. If the management also has a board, some already interpret this as enterprise agility or business agility.\nWaterfall 3.0: It's great when many teams visualize their work. That's a start. But you can also be totally unagile with lots of boards and visualization if nothing else is created but a waterfall 3.0. The analysts have their Kanban board, the developers have their sprintboard, the testers have their Kanban board. As before, tasks are simply passed on. The dependencies between the individual teams, let alone between operational and strategic levels, are completely ignored and not actively managed. In the best case scenario, a team always optimizes itself, but the company as a system of connected individual parts remains unaffected. In other words: It remains as little agile as before.\nAdministration of the existing: Management boards in particular are usually just a prettier to-do list. The tasks are processed well, but there is no change in the way HOW these tasks are processed. And it is also not considered whether the tasks are meaningful at all. Sometimes it gets even worse: The visualization makes constant re-prioritizing and working with high work in process much easier, and that can drive a company into agile madness.\nThe fact that there are certain artifacts like boards or standups does not make an organization agile. Boards of all kinds \u2013 Kanban, Scrum or anything else \u2013 can be used to perfectily manage existing dysfunctions.\nThis happens when the focus is not on \"business agility\" and instead the methods are worshiped as \"agile\". Everything then revolves around how many columns and swimlanes a board may have and whether a stand-up should not last a second longer than 15 minutes. You can make great but completely useless retros if you don't follow the findings or if you never check whether actions were successful.\nWhere means and ends are confused, agile transformation intentions usually fall into the trap.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Fxsol.co.uk website is scored 445 1Pagerank.com points (1PRS) and has a global traffic ranking of 164,759 in the world. This website Google PageRank is 4 out of maximum 10 and its estimated worth is about $ 39,000.00. It has potential to earn online around $ 65.00 US Dollar daily. Fxsol.co.uk was registered 9 years 9 months ago (2009-07-14) and is hosted in New Jersey, Ridgewood, United States. As no recently active internet threats were reported, Fxsol.co.uk is probably SAFE to browse.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Liaisons serve as the contact point for child protection questions and policy implementation at each parish or school. If you have a question about how the policy works at your parish or school, or any concerns about the efforts for a safe environment, the local liaison is your primary resource.\nA complete list of parish and school liaisons is available.\nFor resources and information for liaisons, please log in here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Feel free to get in touch with 309 WEST 104 STREET CORPORATION today and get assistance and you can relax and feel safe because you are in trusted hands in NEW YORK. Make a Skype Call.\n10025 Yale Metal Mfg. Corp.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaxuwv b/data_all_eng_slimpj/shuffled/split2/finalzzzaxuwv
new file mode 100644
index 0000000000000000000000000000000000000000..aabb0c8611092f0aadd647c32e60fdca469d45de
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"After an Awesome Shooting today with the amazing David Tufino & my awesome sister Ma\u00ebva Boldron.\nI'll be heading back to Paris in a few days for one class & a great workshop this weekend.\nLet's practice together in total Fusion.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The term 'inferencing' may be new to parents, but it's one that should be discussed and practiced at home with kids to support literacy learning.\nBy definition, inferencing is \"a conclusion reached on the basis of evidence and reasoning.\" It is the skill of being able to think analytically and learn to understand implicit messages in all kinds of texts. i.e. print, digital, advertising, graphic, etc.\nWe begin to teach students to inference as soon as they begin interactions with text. In the ELK classrooms, this could be asking students to explain the meaning of an image in a storybook. By the time students reach secondary school, they are expected to be able to read challenging texts and use inferencing skills to interpret meaning that isn't obvious, such as being able to explain a main idea, or dominant theme.\nHow can you help your kids practice inferencing?\nShow an interesting picture\/piece of art and ask them to explain the five W's of the text and to defend their choices by pointing to clues in the image: who, what, when, where, why.\nAsk the meaning of physical and digital signs with little to no text and how they can understand\/interpret meaning without words i.e. driving signs (arrows); poison signs (skull); emojis (moods).\nShare and explain examples of figures of speech with them, such as \"I'm so cold I could freeze to death!\" (metaphor\/hyperbole); \"Stop yelling like a hyena!\" (simile); \"The wind is howling.\" (personification).\nWhen reading non-fiction text with them, ask them to identify and defend themes, main ideas or bias.\nIf they are reading fictional texts at home, ask them implicit questions. i.e. Tell me about the main character's personality; Why do you think the author chose their setting?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Portable Adjustable Electric Soldering Station Hot Tweezers Heating Pliers For BGA SMD Rework Repairing Power Tools Set [CPR181108E06] - $44.60 : Full Cell Phone Spare Parts|LCD Refurbishment Machines|ElectronParts!\nHigh-power heater, parallel tweezers design.\nVery convenient for small electronic components, chip, chip removal and installation of welded components. When you need to remove the parallel electric tweezers can be used directly on the welding points across the resistor components for heating.\nTo meet the market needs of different regions, the addition of Fahrenheit \/ Celsius temperature conversion function, the user can choose according to customary interest.\n10 minutes regular sleep function,prolong tip life.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Linden boys' basketball team captured the Huskies Tournament at Matawan by defeating Academy Charter 81-12 Dec. 26, Manalapan 69-49 Dec. 27 and Columbia 57-47 Dec. 29.\nSenior guard Cairo Brown earned All-Tournament honors, while junior center Quadri Moore was named MVP.\nThe Tigers begin January with a 7-0 record that includes comeback wins over Elizabeth, Don Bosco Prep, Union and Columbia.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"9\u2032 Unlit Sherwood Spruce Artificial Christmas Tree | Realistic Artificial Christmas Trees : It's Gorgeous!\nComposed of 2,374 branch tips, the Sherwood Spruce measures 9 feet tall and 52 inches wide. It features vibrant green Classic PVC needles (70%) for fullness and two-toned Real Feel PE needles (30%) for realism. Hinged branches allow for easy setup, while its in-and-out branch structure makes for a beautiful ornament display. This artificial Christmas tree includes a metal tree stand, and is covered by our 5-Year Foliage Limited Warranty. Built from the ground up, Tree Classics has been a leader in the industry of artificial Christmas trees since its inception in 1976. We have since expanded our hand-picked selection of holiday offerings to include artificial wreaths, garlands, and various accessories. We attribute our lasting success to our commitment in delivering The World's Finest Artificial Christmas Trees to your doorstep.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzayqxu b/data_all_eng_slimpj/shuffled/split2/finalzzzayqxu
new file mode 100644
index 0000000000000000000000000000000000000000..2a51a9320c73e02840dbe8410d4fc09fe9ad6668
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"The Independent Energy School of Lamia is an initiative by the Stevia Farmers Cooperative in collaboration with Electra Energy Cooperative. Thanks to the University of Thessaly, a pellet maker is provided and shared between the cooperative members, who thus turn their waste into useful energy, instead of burning it in the field. Now the next step is to engage the whole community to become an energy school cooperative in Central Greece The Independent Energy School of Lamia is a Greek initiate to empower citizens towards an energy transition in Central Greece. Under energy poverty conditions and austerity measures in Greece, energy independency and community power models are now the way to create new jobs and opportunities for people. On the other hand, ensuring a better future while educating new generations on a sustainable way of being. Project goals: -ENGAGING THE PARENTS COMMUNITY INTO A GREEN ENERGY COOPERATIVE: we want the project to become a kick-starter for a regional social movement, with the scope of empowering citizens to become green energy pro-sumers. JOBS CREATION IN THE REGION: by collecting and processing wood and waste, as well as maintaining equipment and running new waste management initiatives in the Region. REDUCING ENERGY CONSUMPTION IN THE BUILDING BY 20%: optimizing thermal comfort by insulating the building. INCREASING ENERGY EFFICIENCY BY 20%: by exploiting local biomass resources, we want to guarantee a lower energy bill, and in a second step, by introducing technologies to reduce electricity consumption. REDUCING CO2 EMISSIONS BY 20%: by dropping out of fossil fuels dependency, while generating thermal energy from locally generated bio energy resources. YOUNGEST GENERATION AS AMBASSADORS OF THE GREEN ENERGY COMMUNITY: turning the youngest generation into leaders of green action. Technical approach: the project consists of converting the central 500 kW oil heater into a biomass combustion system. Some of the parents and nearby communities are olive oil farmers who are willing to provide the annual tree prunnings for biomass conversion and to be used in the primary school. A biomass conversion kit is a group of technologies designed to make oil boilers suitable for combusting biomass based materials. The main component is a metal made biomass burner that is integrated in the combustion chamber of the oil boiler, replacing the oil combusting system . The savings by using local biomass are estimated on 4,830 euro\/year for an investment of 12,000 euro. The school is annually consuming 11,000 liters of heating oil at the price 1,05 EURO\/liter. With the new system the school will demand an annual amount of wood pellts of 24,000 kgs at a price of 0,28 EURO\/kg The parents will be engaged to form a school cooperative to collect funds for additional activities involving kids as educational workshops for energy savings and recycling and helping refugees in the area.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Lambert campus will reopen this coming Tuesday for staff and faculty, after the Martin Luther King, Jr. holiday on Monday.\nDuring this time we may draw support from his reflections such as: \"We must accept finite disappointment, but never lose infinite hope.\" ~ Martin Luther King, Jr.\nLast evening I Informed you of the evacuation orders for Summerland and Carpinteria being lifted, the opening of some roads and continued closure of others, as well as some alternate routes and methods of transportation. It was a real relief and a pleasure to be back in my office and able to send email the Lambert campus yesterday! The Ladera campus, however, remains in the evacuation zone because of road conditions there, so we must wait to reopen that campus. Throughout this time we have carefully monitored and adhered to the frequently changing County of Santa Barbara warnings, as safety for faculty, staff, and students is our number one priority.\nThe Lambert campus will fully reopen this coming Tuesday for staff and faculty, after the Martin Luther King, Jr. holiday on Monday. During this time we may draw support from his reflections such as: \"We must accept finite disappointment, but never lose infinite hope.\" ~ Martin Luther King, Jr.\nAt 9 am Tuesday morning we will have a Town Hall meeting in the lecture hall for those who come from areas south of the mudslides. Regular work activities will commence after this meeting. Because the 101 is still closed in Montecito, a parallel, linked meeting will occur in Santa Barbara at the Impact Hub as has been sent out in announcements by Dianne Travis Teague. Students will be able to use the Lambert campus starting Wednesday. Access to and from Lambert is now restored from the south, including from Los Angeles through to Carpinteria; internet services have also been restored throughout the area.\nWhile the power, electricity and gas, are fine at Lambert, the water is not yet suitable for drinking\u2014the water in Carpinteria remains fine and potable, but not in Montecito, which is where Lambert's water supply comes from. Running water is available but only for showering and bathroom purposes; it should not be used to drink, to brush teeth, to clean or prepare food. Signs will be posted to this effect and ample supplies of bottled water will be available throughout the campus. Our food and beverage will continue to be prepared by our catering service (Omni), but now off campus in an area with safe water.\nThe Thomas Wildfire was the largest in the history of California and the resulting mudslides are unprecedented in Santa Barbara. During this time of crises we have developed a number of contingency plans that protect the continued integrity of our programs. We have been in regular contact with the Department Of Education and WASC to ensure that these plans are in accord with applicable regulations so that all coursework and degrees are fully validated. We have also been working with the DOE to likewise ensure that Financial Aid remains intact for all students who use it\u2014this has required a few new procedures that are working well.\nOur home region also meets the DOE and other governmental criteria for a disaster area. All of our contingencies and any variations from policy are tailored exclusively for this situation \u2013 hopefully a \"One Time Only\" situation that will end when both campuses have reopened. Such plans include the use of electronic communications, including Zoom sessions, and moving locations for housing, especially for those staying at Ladera, as well as the physical location of some classrooms.\nWe do regret and take very seriously the unavoidable missed time for several of our tracks; including one track that had to be rescheduled twice, due to the fire and the mudslides something that had never before happened in the 42 year history of Pacifica, certainly an archetypal moment. We are therefore developing additional plans to make certain that the curricula for all programs we have offered will be adequately delivered.\nOur success to date since the Thomas Fire approached us on December 6 would not have been possible without incredible teamwork by numerous Pacifica departments and faculty \u2013 people who are experiencing their own personal difficulties in dealing with the impacts from the fire, evacuations, road closures, and communication disruption. I am both proud and grateful for their dedication and service.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A: T\/T Other ways are also can be considered.\nA: 15-20 days. urgent order is acceptable.\nA: 1 Set. But we suggest that you can buy a container of it, which is more cost-effective.\nA: CKD\/SKD ;20GP container : 5sets ;40GP container: 12sets.\nA: No, all of products is customized to meet the customer requirements.\n6. What's your warranty term?\nA: We offer different warranty time for different products. Pls contact us to get detail information.\n7: Will you deliver right goods as we ordered? How can I trust you?\nA: Yes, we will. We belive that honesty and credit is the key to build a nice and long-term business relationship. Our company is Alibaba's Gold supplier and we have Trade Assurance $21,000.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In the interview, CBS News' Lesley Stahl pointed out the Democratic Party, more than ever, has waring factions within its caucus between moderates and progressives, many of whom describe themselves as \"democratic socialists.\"\n\"That's like five people,\" Pelosi interrupted.\nJust prior to the exchange, Pelosi readily dismissed the radical agenda of her party's progressive wing, which those like Ocasio-Cortez are eager to implement, and said that, in the end, Democratic lawmakers will be forced to tow the mainstream line, if for no other reason but the Democratic Party's vitality.\nPelosi's interview aired one day after one of the Democratic Party's popular young progressives, Rep. Rashida Tlaib (D-Mich.), criticized Democratic leadership for, in her eyes, failing to defend controversial Rep. Ilhan Omar (D-Minn.) from critics upset that she referred to the 9\/11 terrorists as \"some people\" who \"did something.\"\nPelosi responded to Tlaib's frustration by directly condemning President Donald Trump, who controversially posted a video to social media that highlighted Omar's remarks.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sweet potatoes are a super food of both deliciousness and of health. They have almost every vitamin you could possibly need. Not to mention they are so very tasty and can be made into any dish. I decided to make something with sweet potato, so I went a searching for a good recipe to use. Once I found this amazing post I decided to make it. It is a gluten free and low fat ground turkey with sweet potato skillet that only takes 30min. to pull together and cook. It is the all american, delicious comfort food. This dish is very good too. Give it a try here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzazzmi b/data_all_eng_slimpj/shuffled/split2/finalzzzazzmi
new file mode 100644
index 0000000000000000000000000000000000000000..93a9d00abdc4dad9ec0fce39539e2deb5919a811
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"If you're wondering how a wind SolidWorks turbine model takes shape when outsourcing, then continue reading to find out.\nHave you ever wondered how a wind turbine model is created using SolidWorks? If you haven't, then now would be a good time to do that. Most people are intimidated by the idea of creating wind turbine models using SolidWorks. However, a CAD \/ CAM service can craft them without any problem. Here, we will discuss how CAD \/ CAM services create complex shapes such as wind turbine models using SolidWorks.\nThere are certain steps necessary to design a simple turbine model. A turbine is a machine that captures energy in fluid flow and directs that energy to other devices. An essential part of turbines is turbine blades. The blades are attached to a rotor that is connected to a shaft. The shaft spins when a turbine is exposed to moving the fluid. The shaft then transmits power to other devices. The blades of a turbine are shaped to efficiently harness the energy in a flow field. A CAD \/ CAM service considers these things when creating wind turbine models using SolidWorks.\nThere you have it\u2014an overview of how outsourcing a wind SolidWorks turbine model works. For more information and to get feedback on your project, please contact us. Our friendly experts will be more than happy to help you and provide you with a project quote.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Swizz Beatz and Reebok have found success in their collaboration, as the popularity of the Reebok Kamikaze III has grown profoundly. This latest rendition of the shoe blends a navy blue base with shades of red, light blue, black, and white. The shoe's leather construction makes it durable, while suede on the toe adds to the stability, and mesh at the back end allows the wearer's feet to breathe. Find these now at mita and expect a wider release very soon.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Are you ready to ROCK with the Northern Song Collective (NSC) and Men Matter Appeal?\nManchester's Men Matter Appeal raise awareness and much needed funds to assist the fight against the three types of cancer exclusive to men: prostate, testicular and penile. They have teamed up with the Northern Song Collective with the aim of raising big bucks for this important and often forgotten cause.\nThis week the Manchester based songwriters collective released the video for their riff-filled charity single Feeling the Fury and it received first airplay on 106.1 Rock Radio. The track features members of legendary UK rock bands Thunder, Gun and Little Angels. As well as Mike Sweeney, broadcasting veteran and member of the band The Salford Jets. Dr Chris Steele, of This Morning fame, has also lent his support to the campaign by unleashing his inner rocker \u2013 watch the video to see.\nThe NSC are setting themselves the target of reaching the Top 40 with the tune, penned by Jake Fishers, David Hanson and Daniel Ross as part of a In The City 2010 songwriting competition. Therefore don't forget to download Feeling the Fury from the 4th of April 2011 from Amazon UK, iTunes and Spotify to show your support.\nThe promotion and release of the track coincides with Prostate Cancer Awareness Month UK in March and Testicular Cancer Awareness Week from 1st-7th April, so boys don't forget to check your bits and pieces!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Have you ever had to fill in a form on a web page and wished you could open it in an editor? If you use lots of blogs, post lots of comments or file many bugs then you probably get annoyed at the lack of true editor commands.\nSo when I discovered the Firefox extension MozEx I was a very happy hacker indeed. But as time went on I didn't like the way the extension was going. It was trying to solve a problem I didn't have; it wanted to add handlers for things like ssh:\/\/, etc. and I didn't need that. I needed an external editor!\nWhen Firefox 2.0 came out, MozEx didn't work. I waited for an update and it didn't come out. Since my new job used a lot of editing stuff via web forms, I really really needed an editor. So I reached for External Editor. It seems to have gone since then, replaced with Editus Externus which seems to be an improved version by someone else.\nExternal Editor had a huge problem. When you edited a file it would freeze up all of Firefox till the editor quit. Yikes! However, it let me do things that I needed to do, so I use it.\nIt also had some potential security flaws where the browser could be tricked into fetching the contents of a file from one site into their site. Even more annoying, if a second user used MozEx and edited the same site, the files would collide.\nSo I started thinking\u2026 How hard could it be to write an extension?\nIt's called It's All Text! and is available for download today.\n[idea] dropdown list for charsets (utf-8, western-iso, default)?\n[low] getDocumentIdentifier should sort arguments and append the post data.\n[security] Detect collisions using the raw key.\n[idea] Pass in the line number to the editor, arbitrary command?\n[high] On edit, let user pick the file extension.\n[idea] allow the user to pick an alternative editor?\n[med] If the textarea is focused, we should refresh it.\n[low] When the editor quits, we should refresh the textarea.\n[idea] support for input elements as well?\n[high] Confirm that we find textareas inside iframes and frames.\nThings marked as ideas, may or may not happen. Everything else will either happen or the issue will be addressed in some other way.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you are experiencing HVAC issues, give Mr. Fixit a call to arrange an appointment for an in-home consultation. Mr. Fixit will provide an on-site inspection of your unit to assess your servicing needs, troubleshoot any problems, or get to the root cause of a HVAC unit failure.\nAfter inspecting your unit, Mr. Fixit will generate a service package catered to your needs. Mr. Fixit is not in the business of upselling customers like other HVAC contractors, he is conscious of your budget and requirements and will only advise you on what is necessary to repair or maintain your HVAC unit. If your home requires a HVAC unit replacement, Mr. Fixit can advise you on the most cost-effective options to meet your heating and air conditioning needs.\nMr. Fixit provides competitive rates on HVAC servicing, repair, or installation projects. Based on your service plan and needs, Mr. Fixit will cater to your schedule to get the job done right. With Mr. Fixit you are guaranteed a professional, hassle-free experience at a great rate that beats the competition.\nHas cold weather arrived and your home just doesn't feel warm enough? Is your furnace leaking carbon monoxide and in need of repair?\nTurned on your A\/C but cold air isn't coming through the vents? Call Mr. Fixit to conduct a free home inspection to troubleshoot your issue.\nDoes your air feel a bit stuffy in your home? Are your kids suffering from allergies or breathing problems? Have your ventilation and air quality examined today.\nHaving hot water issues? Enjoying a warm shower but your water goes cold after a few minutes? Have Mr. Fixit take a look at that water heater for you.\nHow can I manage and reduce my heating costing during Winter months?\nProper maintenance of your furnace or heat pump unit will provide maximum efficiency to optimize your gas or utility bill. Often times, customers are unaware of leaky furnaces that increase gas consumption and other issues. Mr. Fixit can also assess the insulation in your attic and other areas of your home to evaluate opportunities to reduce the load on your HVAC unit through adequate home insulation.\nHow long will my furnace or air conditioning unit last until it needs to be replaced?\nThe typical furnace or air conditioner will last 10-20 years on average. This range is dependent on the quality of installation and level of regular service and preventative maintenance done to upkeep the unit. If your budget permits, it may be worth replacing an old unit as newer models provide greater quality and energy efficiency which can lead to significant savings on your energy bill.\nMy energy bill is sky high during the summer. What can I do to reduce my A\/C cooling costs?\nSome rooms in my home are colder than others. How do I fix this?\nThis issue is a result of inadequate airflow which can be caused by a variety of reasons such as dirty filters, coils, or blowers that are restricting the air flow through the air handler in certain areas of the home. There could also be an issue with your duct system where there is a leak at some point due to the ducts coming loose. A lot of homes simply suffer from poor design work where the air conditioning and duct system is not balanced and therefore air flow is not equally distributed. Mr. Fixit can conduct an in-home inspection to assess your air flow issues to get to the root cause of your problem.\nHow does air quality affect my home?\nPoor air filtration can result in the proliferation of airborne particles and pollutants like dust, mold and animal dander. This can be detrimental to family members who suffer from allergies,asthma and other breathing problems. Elevated humidity in your home can lead to the discomfort of feeling hot and sticky, and also promotes mold and mildew growth.\nProper servicing and flushing of your HVAC unit and duct work is the first step to eliminating airborne pollutants. Using higher quality air filters for your HVAC unit, with MERV ratings between 17 and 20, provide the best filtration for your home by removing up to 98% of air pollutants. Don't forget to change your filters every 3 months! De-humidifiers are also great for our humid Georgia weather allowing you to feel comfortable at higher thermostat settings which will cause your A\/C to work less and save on your energy bill.\nHow often should I have my A\/C serviced?\nMost A\/C contractors will try to upsell you on a frequent service plan claiming it is necessary to maintain the health of your unit. If you regularly change your air filters and manage the operation of your unit, you can help maintain the efficiency and life of your unit. Nevertheless it is wise to have an HVAC technician service and inspect your unit annually, particularly before the summer months for your A\/C and before the winter months if you have a furnace.\nWhat steps can I take to maintain my air conditioner or furnace?\nMake sure the air supply and return vents for the furnace are not obstructed. Nothing should be block the air supply and vents such as furniture, curtains, boxes, etc.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Well, now you can. A number of companies are offering \"customized\" or \"personalized\" haircare products as a new service . Some have even used this concept as the foundation for their new brands. Function of Beauty, Prose, and FORM Beauty are just a few emerging options of personalized haircare devised from user-provided information and online quizzes.\nOne company is taking this a step further. Regenix, a company that already offers personalized hair loss treatments, has added a new product to their line-up: Regenix Kit for Damaged Hair.\nLike the other companies, Regenix asks you a few questions so they can personalize the products they send in the starter kit. This includes Moisture Balance shampoo, Replenish conditioner, Keratin Damage Control Gel, and four personalized hair treatments.\nWhat's unique about this offering is the addition of a microanalysis, a service conducted on-site at the Regenix Hair Research Clinic in Los Angeles. Following the instructions in the kit, you mail hair samples back to them so they can customize more specific treatments.\nUsing an electron microscope, Regenix inspect your hair. Your strands are not completely smooth and solid surfaces; rather, they look much like fish, with very tight, overlapping scales. The healthiest strands have flatter cuticles, or in curly hair, slightly raised, but uniformly so. An obvious sign of damage are scales that are broken, spread out or raised haphazardly, which can catch onto other damaged cuticles, eventually getting torn off.\nAfter examining the cuticle, they'll look at the very top of the strand, which was once just beneath your scalp . Here, they can examine residue from hair products and sebum build-up.\nWhat they see under the microscope determines the specific formulas they'll send you.\nFunction of Beauty asks customers to take a detailed quiz in order to determine the best product. The quiz seeks information about hair type and texture, as well as hair care goals the customer has. In a fun twist, customers can name their product and select its color and fragrance, too.\nForm Beauty also has a detailed quiz to help identify which products will work best for a particular customer. Along the way, the fun quiz offers tips related to zip code-determined weather issues, too. At the end, you'll see a recommended bundle of products that includes a one-on-one consultation with a Form Beauty expert.\nIt may work for every texture, but not for every budget. The Regenix Kit for Damage Hair is $165 and includes three month's worth of the shampoo, conditioner, and keratin gel, and one month's worth of the four personalized treatments. After that, it's $99 a month for the further customized treatments created from the results of your microanalysis.\n\"Our company and products are in constant evolution. We're always in research and development mode,\" Edwards says.\nForm Beauty, too, is cooking up new experiences for its customers, with a beta Microscope Analysis Process underway.\nHave you tried any personalized haircare products? What brands do you recommend?\nCurlies have been customizing their products in their kitchens for years. If you'd like to to try making one of your own, try out this DIY styler.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Are you looking for information about Dr. Sir Arcot Lakshmanaswami Mudaliar Vocational Arts & Science College located in Vengal, Thiruvallur District? Looking for admissions and results related to this college? Check this college post to know more about this institute along with address, website and contact details.\nYou may write a review for the college 'Dr. Sir Arcot Lakshmanaswami Mudaliar Vocational Arts & Science College' and earn revenue.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"After delivering your baby, you will continue to be watched and get pain relief, if needed. You and your baby usually stay at the hospital for a couple of days to recover.\nWhat does recovering from childbirth involve?\nPainful contractions that may continue after delivery. These happen as the uterus returns to its original size.\nFatigue and soreness. These are common in the first few weeks.\nIt is generally recommended that you schedule an appointment with your healthcare provider 6 weeks after delivery to make sure of proper healing.\nTake naps when the baby naps, to make up for lost sleep at night from getting up to feed the baby.\nWear a supportive bra. Cold packs may help relieve breast engorgement and swelling. Warm compresses can be used to help stimulate the letdown of milk. Letdown is a reflex that triggers the release of breastmilk. If you choose not to breastfeed, use ice packs and bind your breasts for several days to help ease engorgement.\nIf stitches were needed during a vaginal delivery, take warm, shallow baths (sitz baths) twice a day to ease soreness and speed healing.\nAfter a cesarean section, keep the incision clean and dry.\nIt is common for these \"baby blues\" feelings to go away soon after they start and usually without treatment. These symptoms may also be present in postpartum depression. Postpartum depression is a more severe form of baby blues. Women with postpartum depression may have trouble coping with their daily tasks.\nTypical diagnostic procedures for postpartum depression include a complete health history, physical exam, and\/or psychiatric evaluation. In some cases, you may need a thyroid screening to find any hormone or metabolic problems or conditions that may serve as an underlying cause.\nIt is important to note that most women who get the \"baby blues,\" postpartum depression, postpartum anxiety, or postpartum obsessive-compulsive disorder have never had these symptoms before, especially with such intensity. In any case, it is important to get proper treatment early. This is not only to make sure that the newborn remains safe and properly cared for, but also so that you can resolve these symptoms and experience all the joys of motherhood.\nPsychological treatment. This may include both you and your family or partner.\nAssertiveness training. Some women need to learn how to set limits with family members in order that they do not become overwhelmed and overworked.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hey everyone! If you see me these days, you will notice I got a tan. No, it's not the reflection of the sun from the New York City snow. It's because I just had fantastic trip to Puerto Rico with the company I'm working at. It was the best timing, since it has been awfully cold in New York for a while. It was great to take a break from winter, go to summer and get to know a new place and a new culture.\nPuerto Rico surprised me in many ways, but the most shocking part was the tropical landscape. I had never been in any tropical climate area, and it was beyond beautiful. The minute I stepped out of the plane, I was already in love with the views, the rain forests and the gorgeous palm trees surrounding everything.\nWe stayed at a huge golf resort close to San Juan, the capital of Puerto Rico. The house was unbelievable and I felt like I was in a movie set!\nMy favorite part of the trip was the day we spent sailing around the islands in a catamaran. We did some snorkeling, swimming and walked around the beach. And also I got a tan! The views were really (REALLY!) breathtaking and at some point I felt I was inside one of those default computer desktop wallpapers.\nI had a chance to walk around the city of San Juan, which was also incredible. I really liked the views and the vibe of the people there. They all were very friendly and seemed relaxed in a very good way. It was the perfect contrast with New York, even if it only lasted four days.\nAs a tip, if you're an international student you don't need to worry about visas or immigration controls when traveling to Puerto Rico from New York. It is a US territory and you do not need international documentation to travel there. In fact, you can get on the plane with just an ID!\nAll in all this was an amazing trip and I am very thankful to the company for taking me. I totally recommend visiting Puerto Rico, but save it for the coldest months of the year to give yourself a nice treat. Thanks for reading and see you soon!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Dunes Beach Home Rentals has been assisting guests with their family vacations for over 50 years. Our company manages more than 250 properties exclusively at the Litchfield Beaches. As a full service vacation rental company, our staff is dedicated to providing excellent service to our homeowners and guests. We are focused on maintaining, marketing and protecting our homeowners' investments at Litchfield Beach. Our goal is to provide all visitors with a \"great beach vacation\" that exceeds expectations.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The City of Albuquerque will support innovative and effective proposals from qualified nonprofit organizations interested in providing services for populations in need of specific social services for persons experiencing homelessness and\/or domestic violence. This request is focused on 1) filling service gaps for people experiencing homelessness due to domestic violence, child maltreatment, or substandard housing, and 2) providing supportive housing through housing vouchers, case management, and other services for people experiencing homelessness.\nThe City of Albuquerque has established a priority to fund projects that lead to improved outcomes to 1) Increase Behavioral Health Stability, 2) Increase Housing Stability, 3) Increase Public Safety, and 4) Increase Family Resilience. This involves a focus on addressing the Social Determinants of Health to achieve greater well-being and equity for all. Substantial evidence confirms the link between social, economic and physical conditions and health outcome disparities. Social Determinants of Health include access to healthcare services, availability of services to support housing and behavioral health stability, lifelong education options, public safety and social services.\nEmergency Shelter Services and Supportive Services for Victims of Domestic Violence: Respondents may propose an offer for the operation of emergency shelter services and supportive services for victims of domestic violence and their children. Services for this program that must include, but are not limited to: 1) Providing safe, secure, and confidential emergency shelter for victims of domestic violence and their children, open 24 hours per day and seven days a week; 2) Crisis counseling and on-going case management services designed to educate victims of domestic violence of positive alternatives to remaining in abusive relationships; 3) Individual and group education and counseling services for adult and children survivors and witnesses of domestic violence; 4) Development of individual treatment plans for each participant; 5) Coordination with community support services relevant to the individual needs of the residents; 6) Applicants must demonstrate plans to partner with the City of Albuquerque Family Advocacy Center in assisting victims of domestic violence; and 7) Services under this contract must also include the operation of a 24-hour-per-day, seven-day-per-week telephone crisis line for victims of domestic violence.\nCrisis Shelter Services to Abused, Neglected and Abandoned Children\/Youth: Respondents may propose an offer for crisis services to abused, neglected, and abandoned children and\/or youth. Services in this program must include, but not be limited to: 1) Provide 24-hour shelter care on an emergency or long-term basis for abused or neglected children and\/or youth; 2) Provide support and advocacy services to children and\/or youth and their families in cases of suspected sexual abuse or severe emotional and\/or physical abuse; and 3) Assist families experiencing abuse and neglect to develop new parenting and life skills in order to create healthy and safe home environments for children and\/or youth.\nDisplaced Tenant Relocation Services: Respondents may propose implementation of a displaced tenant relocation service to persons, who through no fault of their own, have been displaced because of substandard housing conditions. Services for this program must include, but are not limited to: 1) Coordinate with City's Safe City Strike Force, Office of Equity and Inclusion, and the Legal Department during tenant displacements; 2) Provide supportive services while locating suitable housing including referrals for services as necessary; 3) Administer motel vouchers as necessary to eligible persons; and 4) Distribute relocation funds to eligible persons.\nSupportive Housing and Case Management Services for Homeless Women and Families: Respondents may propose an offer for the provision of supportive housing for homeless women and families with children. Services for this program must include, but are not limited to: 1) Housing vouchers; and\/or 2) Supportive services to assist people experiencing homelessness with obtaining and maintaining housing, including case management.\nSupportive Services for the Medically Vulnerable and\/or Chronically Homeless as Identified by the Albuquerque Heading Home Homeless Program: Respondents may propose implementation of a program which provides supportive services, including case management, for the Albuquerque Heading Home Program participants. These participants are chronically homeless, meaning that they have experienced long-term homelessness and\/or multiple episodes of homelessness, and have a significant, long-term disability or medical issue. Services must include, but are not limited to: 1) Providing SSI\/SSDI Outreach, Access, and Recovery (SOAR) services; 2) Developing and supporting the success of individualized service plans; 3) Providing linkage to and continuity support for services, or directly providing services, such as counseling, vocational classes, and life skill education; and 4) Providing assistance for participants to transition toward and maintain permanent housing.\nCoordination Services for the Chronically Homeless as Identified by the Albuquerque Heading Home Homeless Program: Respondents may propose implementation of a program which provides coordination services for Albuquerque Heading Home Program. The target population is people who are chronically homeless, meaning that they have experienced long-term homelessness and\/or multiple episodes of homelessness, and have a significant, long-term disability, or medical issue. Services must include, but not be limited to: 1) Coordinate case management services to ensure that all Albuquerque Heading Home Program participants have a case manager; 2) Coordinate with the Coordinated Entry System (CES) provider and the Albuquerque Heading Home Program housing provider(s) to quickly fill housing openings when they become available; 3) Coordinate case management and supportive services to ensure that participants obtain and remain in permanent housing; and 4) Provide one-time direct assistance to support move-in costs for participants entering housing.\nDay Shelter Services for the Homeless: Respondents may propose an offer for operations of a day shelter facility for persons experiencing homelessness. Services must include, but not be limited to: 1) Provision of safe and secure shelter open 7 days a week, year-round for persons experiencing homelessness; 2) Provide morning and\/or lunch meals and sack lunches for those who are working, seeking employment, in job training, or in education activities; 3) Provide storage space for belongings; 4) Provide shower facilities; 5) Provide communication services; 6) Provide clothing; 7) Provide guidance and access to community services with an emphasis on attaining permanent housing, employment, and behavioral health services; and 8) Provide participants with the opportunity to complete a common assessment for the Coordinated Entry System.\nOutreach Services Program for the Homeless that are Severely Mentally Ill: Respondents may propose implementation of outreach services for persons experiencing homelessness that have severe mental health conditions. Services must include, but not be limited to: 1) Provision of direct outreach services within the community by utilizing teams of mental health and homeless service professionals; 2) Collaboration with the Albuquerque Police Department's (APD) Crisis Intervention Team (CIT) and other outreach partners in the community; 3) Providing professional assessments for appropriate housing and treatment services; 4) Placing the persons to the appropriate housing and treatment services; 5) Providing outreach services at City funded emergency shelter(s); and 6) Providing participants with the opportunity to complete a common assessment for the Coordinated Entry System.\nStreet Sweepers Employment Program for the Homeless: Respondents may propose implementation of Street Sweepers Employment Program for persons experiencing homelessness. Services must include, but not be limited to: 1) Assisting the workers to obtain job skills and opportunities to attain permanent and\/or improved employment; 2) Providing employment and job skill support to persons experiencing homelessness to clean-up the Wells Park neighborhood and\/or other near downtown neighborhoods; 3) Providing access to services to obtain permanent housing and supportive services to successfully retain housing; 4) Establishing a point of contact for response to 311 calls regarding concerns and questions of the assigned neighborhood(s); and 5) Providing participants with the opportunity to complete a common assessment for the Coordinated Entry System.\nBetter Way Employment Program for Panhandlers: Respondents may propose implementation of the Better Way Employment Program for panhandlers. Services must include, but not be limited to: 1) Providing employment opportunities for panhandlers at the City's busiest intersections through roving van outreach; 2) Providing referrals to the workers to housing and supportive services; 3) Assisting the workers to obtain permanent and\/or improved employment opportunities; and 4) Provide participants with the opportunity to complete a common assessment for the Coordinated Entry System.\nCoordinated Entry System: Respondents may propose implementation of the City's Coordinated Entry System Services. A proposal must include, but not be limited to: (1) Providing coordination, guidance, and technical assistance to the City of Albuquerque in implementing the New Mexico Coordinated Entry System (CES); 2) Connecting highly vulnerable Veterans, Native Americans, youth, chronically homeless individuals and families, and other homeless individuals and families to housing resources; 3) Increasing outreach to shelters, hospitals, streets, jails, and non-participating agencies; 4) Assistance in obtaining homeless documentation for program participants and\/or housing provides; and 5) Collaborating with the City's housing providers and other community housing programs.\n*Note: City of Albuquerque staff will conduct one mandatory pre-proposal conference for entities interested in submitting proposals at 2:15 pm on Wednesday, November 7, 2018.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When Charles Rolls and Tim Warrillow decided to set up a business together it was the first time they had ever met.\nThis was back in 2003, and Mr Warrillow - at the time a 28-year-old advertising executive - was looking for a complete career change. He wanted to set up his own gin distillery.\nAnd so he contacted Mr Rolls, a veteran of the gin industry, and 18 years his senior, for advice.\nMr Rolls, who had successfully grown Plymouth Gin, agreed to meet for a coffee in central London.\nWithin an hour or two they had decided to become business partners. But instead of forming a new gin company, they resolved to set up a premium tonic water brand.\nMr Rolls, 57, recalls their first meeting: \"The conversation quickly turned from gin to tonic water, and the fact that while there had been a huge increase in the number of premium gins, when it came to the tonic water you added to them, you essentially only had two choices - the market leader [Schweppes], or supermarket own brands.\n\"And these all contained artificial sweeteners. We decided there and then to launch a premium tonic water, with all natural ingredients.\"\nBut was either of them concerned about going into business with someone they didn't really know?\n\"Not really,\" says Mr Rolls. \"I could see an energetic, but pretty mature and determined individual. Tim was exactly the right type of person.\"\nMr Warrillow, 39, adds: \"It was very apparent from our first few meetings that we seemed to understand each other.\"\nAnd so their business, Fever-Tree, was born.\nIt is named after the colloquial term for the cinchona tree, from whose bark the natural anti-malarial drug and core tonic water ingredient, quinine is produced.\nBut while it took Mr Rolls and Mr Warrillow just a few hours to decide to go into business together, it then took them 18 months to find a recipe they were happy with.\nAs Mr Rolls had some money in the bank after selling his 25% share in Plymouth Gin, they were able to take their time.\nIn their efforts to formulate their recipe, they flew to the Democratic Republic of the Congo to source pharmaceutical grade quinine from a plantation, and hired plant hunters to help them find other flavourings, such as a bitter orange grown in Tanzania.\nSecuring some outside investors, they were finally able to start selling bottles of Fever-Tree tonic water in 2005.\nMr Warrillow says: \"The intention was to treat the launch like a premium spirit company would do - in the first instance try to get Fever-Tree stocked by the best hotel bars and restaurants.\n\"This is a great way of building reputation, prestige and word of mouth. And it was all about the shoe leather - we walked to the most exclusive hotels and restaurants in central London, getting them to taste our tonic water against the competition.\"\nWith such venues quick to get on board, the company then got what it says was its most important break - upmarket UK supermarket group Waitrose phoned to say it would like to start selling Fever-Tree.\nMr Rolls says: \"Tim thought this was a perfectly normal development, but I had to keep telling him that it is incredibly unusual for a supermarket group to chase you. It is almost always the other way round, and hard work.\n\"Yet the buyer was incredibly enthusiastic, and just a few weeks after Waitrose starting selling us, she phoned to say we had a hit on our hands.\"\nWith the UK's other supermarkets quick to follow suit, exports then followed, and London-based Fever-Tree is now available in 50 different countries.\nIt has also introduced other products, such as a reduced calorie - but still all natural - tonic water, ginger beer, and lemonade. All are produced for it under contract by a company in the south west of England.\nFever-Tree now sells 65 million bottles per year across its range, and employs 70 people, but Mr Rolls says it still has substantial growth potential. He points out, for example, that it currently has only a \"single digit percentage\" of the UK tonic water market.\nYet for all Fever-Tree's continuing success, have there been any problems along the way?\nMr Rolls says: \"There were some production bottlenecks in our early days... but perhaps my biggest regret is not putting in all the money myself to get Fever-Tree started.\n\"Broadly speaking we needed about a million quid to get the company off the ground, and put on a bit of a road show around the City of London. All it did was delay us, and I ended up putting in half the money anyway. If you ask for a mistake, that was it.\n\"But that is with the benefit of hindsight. There are some unbelievable risk takers out there, but it doesn't appeal to me personally. And our investors have been very good.\"\nBut do the two men ever argue?\nMr Rolls, who recently became executive deputy chairman after handing over the chief executive role to Mr Warrillow, says: \"Of course we disagree from time to time, but it is only ever about small things, not the big issues. And if it happens, we say to the other, 'If you feel that strongly about it, go ahead and do it.'\"\nMr Warrillow adds: \"In terms of strategy, routes to market and funding, I don't think we have ever had a significant disagreement.\n\"As long as you have mutual respect you can go a long way further in business with two heads rather than one.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Place your peeled banana, peanut butter and coconut oil into a bowl and smash together with the backside of a spoon. The softer your peanut butter and oil, the easier this will combine. It's okay to have it a little chunky (just cut your baggie's corner a little larger to accommodate).\nOnce mixed together, fill a plastic baggie with the mixture.\nSnip the corner with scissors and squeeze the mixture into your mold (silicone molds work great since they are flexible and let you pop the treats right out. You can also find so many different sizes and shapes.).\nCut sweet potato lengthwise, about 1\/8 inch to 1\/4 inch thick.\nBake on a cookie sheet for 25 minutes and flip over and bake for another 25 minutes longer. Some will shrivel up.\nYour doodle will love them!\nInstead of the baby food in this recipe, you can also use 1 or 2 fresh bananas but the baby food is perfect ripeness.\nYou can make a cake or cupcakes. We love cupcakes. Freeze what want to save for later.\n3. Mix together flour and baking soda.\n5. Pour into a small round pan and bake for 25-30 minutes. Or cupcake paper tins. Watch them carefully. 20 minutes.\n7. Beat ingredients together and use to frost cake.\nIn a large bowl, stir together pumpkin, applesauce, peanut butter, and yogurt until well combined.\nStir in bacon and oats.\nUse a cookie scoop to form into uniform treats.\nUse this batter to make waffles according to your waffle maker's instruction.\nAdd strawberries & blueberries for an extra treat.\nFreeze a couple of hours and put in in an ice cream cone! Dogs love crunchy cones!\nIf you have one of those dogs that turns its nose up at vital medicine, trick it with a homemade pill pocket. Just mix peanut butter, milk, and flour. Roll it into balls, stabbing them with the blunt end of a chopstick to create the pockets. Store them in the fridge or freezer. I like to use crunchy peanut butter because my dog is less likely to detect the pill.\nUsing a food processor or blender, grind your oats until they're powder.\nIn a bowl, beat together the banana and peanut butter until smooth. A few small chunks of banana here and there is ok. The dough should stick together, but shouldn't be sticky.\nNotesAdd additional peanut butter if the dough is too dry or additional oats if too sticky.\nBe sure to use a peanut butter that does not contain xylitol, as it is harmful to dogs. Always double check with your vet before feeding your pet new foods.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Discussion in 'Studio Scale Models' started by Corellianexports, Nov 22, 2003.\nI have been pondering the thought of adding the infamous Icons X-Wing Studio Scale model to my collection of SW stuff.\nI just wanted to know what every one's opinion was of the Icons version as opposed to a fan made Studio Scale?\nAlso, does any one know how much an Icons version might be worth and were there any made with signed (Hamill) plaques?\nNot 100% sure on this but wasn't the Icons version concidered to small compared to the original models?\nYes it was smaller. The fuselage is nearly an inch too short. All the ones I have seen had sagging wings, paint chipped off exposing the resin underneath, and even fingerprints in the paint!\nThe Icons version is too short by an inch in length?\nA well-made fanbuilt one is going to be more accurate and higher quality than the ICONS. Value-wise as a collectible, I have no idea.\nThe ICONS model is short because it was cast from an original pyro model, not one of the hero models. The fuselage was 2 generations off the real deal, before it was cleaned up, and then production molds were made. So, despite the fact that pyro models are about 1\/2 an inch shorter than the hero versions, the ICONS version suffered from generational shrink as well, but not so much that it makes it inaccurate. I'm not saying the castings are the best representation, I'm just saying that it actually is a reproduction in that actual ILM parts were used as the masters. So, is it 'studio scale'? Yes, because it is cast from original parts, but the process of making the reproduction caused minor dimensional differences. Is it a good model and worth the money-- no, probably not. But that has nothing to do with its size.\nThe 2nd generation shrinkage makes sense.\nI've seen M32 and M19 scope castings which shrank by about 1\/8\" in length, so it all makes sense.\nWell I have #826 of 1,977 and it was my very first prop replica and my first purchase off the internet. I bought it from a ICONS dealer for gave me a special deal for $1200 and had them on-hand for immediate delivery. I've had minimal sag and only on the bottom wings. About once a year I unplug them and straighten them out with a hair dryer. I believe most of the ones with sagging or droopy wings were exposed to sunlight as my stays in dimly lit den. Value... anywhere from $999 to $1200. From various posts, cracked plexi cases were common, mine is crack free. I also have the CC T-65 kit and plan to model this one in wings closed position.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Spoelstra gave the obligatory credit to the Nets, saying they \"played very hard, competed on a different level than us.\"\nBut there was more to it than that, of course.\nYes, the Heat's offense was abysmal, with Miami closing at a season-low 33.7 percent from the field (30 for 89), including a 3-for-12 night from Goran Dragic, 2 for 10 from Wayne Ellington, 4 for 15 from Tyler Johnson and 1 for 7 from Kelly Olynyk.\nThe Heat shot 3 for 26 on threes. This marked the first time in the last 207 games that the Heat has made fewer than five threes in a game, dating to Nov. 15, 2016.\n\"The fact we were bothering them defensively, doing a good job of chasing the ball, I thought that helped out a lot [and] doing a good job of knowing their personnel,\" Nets guard Joe Harris said.\nBut the Heat's defense also was deplorable at times, and Johnson admitted that effort \u2013 a Heat hallmark \u2013 was absent at times.\nSaid Dragic: \"We started the game well [up 18-8], but too much our defense was bad. Our offense too. They had open threes. We didn't get back [on defense]. It was hard to watch.\n\"They were faster and hungrier. We need to bring our hearts and compete. We're dropping games too easily, especially against teams below us. They're still an NBA team. You have to respect them.\"\nCo-captain Udonis Haslem delivered some \"choice words,\" at halftime, Heat forward Jordan Mickey said. But not even that could inspire a better effort, with Brooklyn's 30-point halftime lead growing to 38 in the third quarter.\nHaslem \"felt tonight we didn't compete or step up to the challenge,\" Mickey said. \"He said those are the types of games you've got to get, that we always drop games we shouldn't lose.\"\nSpoelstra's patience with his team was also tested, including one sequence in the first quarter when Quincy Acy shot a wide open three in front of the Heat bench. Spoelstra then admonished his team.\n\"That possession was a microcosm of the entire game,\" Spoelstra said. \"The lack of urgency to get back, to scramble. To make multiple efforts actually felt like, 'Hey, there's an open guy, let him shoot' in a semi-transition situation. It was a very delayed break that led to a wide open shot that we didn't make an effort to get to. Those plays piled up from there.\n\"It was just a bad basketball game. We certainly didn't show resolve when the shots weren't going in. Brooklyn was playing at a different pace than us.\"\nMost mystifying: Why is this team good enough to be 10-8 on the road \u2013 with quality wins at Boston, Minnesota and Washington \u2013 but 8-9 at home, with five of those nine home losses by 15 or more (to San Antonio, Indiana, Golden State, New Orleans and now Brooklyn).\n\"I wish I had a direct answer,\" Josh Richardson said of these clunkers.\n\"If we knew why, we wouldn't be in that situation,\" Johnson said.\nMiami (18-17) squandered a chance to move three games above .500 for the first time since April 2016, when Dwyane Wade was still a member of the Heat.\n\"We all see the same thing,\" Spoelstra said. \"It's tough to recognize the same team from the previous game [a 107-89 win against Orlando]. One game doesn't necessarily carry over to the next game. We have a Dr. Jekyll and Mr. Hyde quality to us that's extremely perplexing. Sometimes we compete at an extremely high level. You can see it and you can feel it. Then we have these games that are unexplainable.\"\nAnd now there's a quick turnaround, with a Friday night flight to Orlando for a Saturday evening game against the Magic.\n\"I wish we could get out there right now,\" Spoelstra said after Friday's game. \"I think the guys in the locker room feel that way. Our approach tomorrow has to feel different. It has to be different, starting with a film session and team meeting to prepare. We need a normal prep that has to have a different level of urgency than we had today. We'll get to do this again tomorrow, hopefully with a different effort and a different result.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbeyap b/data_all_eng_slimpj/shuffled/split2/finalzzzbeyap
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@@ -0,0 +1,5 @@
+{"text":"Paige Plays Myrtle Beach Sweepstakes (the \"Contest\") is open to legal residents of the fifty (50) United States and the District of Columbia and Canada who are at least eighteen (18) years old at the time of entry and are avid golfers. Employees of Golf Tourism Solutions and other companies associated with the promotion of the Contest, and their respective parents, subsidiaries, affiliates and advertising and promotion agencies as well as the immediate family (spouse, parents, siblings, and children) and household members of each such employee are not eligible. The Contest is subject to federal, state, and local laws and regulations.\nThe Contest is sponsored by PlayGolfMyrtleBeach.com and Golf Tourism Solutions, located at 1705 N Oak Street, Suite 6, Myrtle Beach, South Carolina 29577.\nOnline: This method of entry will be available by visiting our website (www.paigeplaysmyrtlebeach.com or www.playgolfmyrtlebeach.com) and following the directions provided to fill out the entry information, and submit.\nOn or about November 15, 2019, the Sponsor will select potential winners in a random drawing from among all eligible entries received. The odds of being selected depend on the number of entries received. The Sponsor will attempt to notify the potential winner via telephone or email on or about November 15, 2019. If the potential winner cannot be contacted within five (5) days after the date of the first attempt to contact him\/her, the Sponsor may select an alternate potential winner in his\/her place at random from the remaining non-winning, eligible entries.\nExcept where prohibited, each entrant agrees that any and all disputes, claims and causes of action arising out of, or connected with, the Contest or any prize awarded shall be resolved individually, without resort to any form of class action, and exclusively by the appropriate court located in South Carolina. All issues and questions concerning the construction, validity, interpretation and enforceability of these Official Rules, entrants rights and obligations, or the rights and obligations of the Sponsor in connection with the Contest, shall be governed by, and construed in accordance with, the laws of South Carolina, without giving effect to any choice of law or conflict of law rules, which would cause the application of the laws of any jurisdiction other than South Carolina.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What is the cheapest way to get from Crestwood to Fordham (Station)?\nThe cheapest way to get from Crestwood to Fordham (Station) is to drive which costs $1 - $2 and takes 13m.\nWhat is the fastest way to get from Crestwood to Fordham (Station)?\nThe quickest way to get from Crestwood to Fordham (Station) is to taxi which costs $40 - $55 and takes 13m.\nIs there a direct train between Crestwood and Fordham (Station)?\nYes, there is a direct train departing from Crestwood and arriving at Fordham. Services depart hourly, and operate every day. The journey takes approximately 22m.\nHow far is it from Crestwood to Fordham (Station)?\nThe distance between Crestwood and Fordham (Station) is 8 miles. The road distance is 9.2 miles.\nHow do I travel from Crestwood to Fordham (Station) without a car?\nThe best way to get from Crestwood to Fordham (Station) without a car is to train which takes 22m and costs $4 - $14.\nHow long does it take to get from Crestwood to Fordham (Station)?\nThe train from Crestwood to Fordham takes 22m including transfers and departs hourly.\nWhere do I catch the Crestwood to Fordham (Station) train from?\nCrestwood to Fordham (Station) train services, operated by Metro-North Railroad (MNR), depart from Crestwood station.\nWhere does the Crestwood to Fordham (Station) train arrive?\nCrestwood to Fordham (Station) train services, operated by Metro-North Railroad (MNR), arrive at Fordham station.\nCan I drive from Crestwood to Fordham (Station)?\nYes, the driving distance between Crestwood to Fordham (Station) is 9 miles. It takes approximately 13m to drive from Crestwood to Fordham (Station).\nWhere can I stay near Fordham (Station)?\nThere are 2177+ hotels available in Fordham (Station). Prices start at $110 USD per night.\nWhat companies run services between Crestwood, NY, USA and Fordham (Station), NY, USA?\nMetro-North Railroad (MNR) operates a train from Crestwood to Fordham (Station) hourly. Tickets cost $4 - $14 and the journey takes 22m.\nRome2rio's Travel Guide series provide vital information for the global traveller. Read our range of informative guides on popular transport routes and companies - including 4 stunningly beautiful Philippines islands you need to visit, How to get from Gatwick Airport into central London and 10 amazing places to get away from it all in Australia - to help you get the most out of your next trip.\nRome2rio makes travelling from Crestwood to Fordham (Station) easy.\nRome2rio is a door-to-door travel information and booking engine, helping you get to and from any location in the world. Find all the transport options for your trip from Crestwood to Fordham (Station) right here. Rome2rio displays up to date schedules, route maps, journey times and estimated fares from relevant transport operators, ensuring you can make an informed decision about which option will suit you best. Rome2rio also offers online bookings for selected operators, making reservations easy and straightforward.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On Thursday 21st June, Montgrove College held its annual Athletics Carnival for students in Years 3-12. It was excellent to see the large number of girls participating in multiple track and field events. The House Spirit and enthusiasm throughout the day added to the already fantastic atmosphere. We ended the carnival with a new event \u2013 a Teacher v Year 12 Tug-of-War, of which the teachers were victorious!\nThroughout the day we had 28 track and field records broken! Congratulations to all the students who created a new record.\nThank you to all the Parents and Teachers who assisted throughout the day. Your assistance, support and enthusiasm led the very successful and exciting carnival for all the students.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A confidential, free phone, text, and live chat service for people with questions about their own or someone else's drinking or drug use. Trained counsellors provide information, brief intervention and referral services, nationwide.\nAvailable free 24\/7 by calling 0800 787 797 or texting 8681.\nAlcohol and Drug Helpline is part of the National Telehealth Service.\nThis page was last updated at 12:26PM on April 4, 2019. This information is reviewed and edited by Alcohol Drug Helpline.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sevakis for fixing that. Thank you very much, it's definitely appreciated.\nIf anyone has any on-going problems, please, let me know.\nand thanks to every other Dev as well. Your work is much appreciated.\nNext in thread: bascule.42274043_at_bloglines.com: \"Re: Voice file updates\"\nMaybe reply: bascule.42274043_at_bloglines.com: \"Re: Voice file updates\"\nMaybe reply: Woody Anna Dresner: \"Re: Voice file updates\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbfqvj b/data_all_eng_slimpj/shuffled/split2/finalzzzbfqvj
new file mode 100644
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--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"They used to talk about seeing only reflections of reality, not reality itself. The main thing wrong with a reflection is not that it isnt real, but that its reversed.\nCody DeFranco (1987) lives and works in Brooklyn, NY.\nCody Hoyt lives and works in Brooklyn, NY. His studio practice is primarily focused on work on paper and ceramic multiples. He has exhibited locally in places such as the IPCNY and Pulse Art fair, and internationally, including Helsinki, Toronto and Amsterdam. He has been featured in publications by The New York Times Magazine, Gestalten Press and Computer Arts China.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Dads are special, they tickle us and make us laugh, they protect and care for us, lecture and educate us. There is no way we can ever really thank Dad for everything he's done, all the times he drove us around, fixed our bikes, and watched us play sport. However every day is great for love quotes for dad to say \"I love you Dad\" and show him that you do remember and appreciate it.\nOn difficult days, I'm confident that I will be okay because I have your support. I know that you will always be with me. I love you, Dad.\nIt's a dream of every girl to be treated like a princess. You make that dream come true for me, Dad. Every time I'm with you, I feel like royalty. I love you so much, Dad!\nYou never stopped showing how much you love me in both small and big ways. I never once doubted you love me because every single day, you made sure that I felt it.\nAll that I'm working so hard for right now is all for you. I want to make you proud and repay you for all the sacrifices that you made raising me. I love you, Dad. I'm so proud of you.\nSo many times, I hurt or disappointed you, but you never gave up on me. You gave me so many chances. This time I promise to make you proud, Dad. I love you!\nIf I had to live my life all over again, I will still choose to be your daughter. You simply are the best father in the world! I love you with all my heart, Dad.\nI will grow up, gain weight, fall in love, have kids. But my love for you will never change. My love for you will remain. I love you, Dad.\nLet these love quotes for dad remind you of the challenges and the joys that fathers face.\nIt is much easier to become a father than to be one. -Kent Nerburn.\nThere are times when I just completely shut off and don't share anything about my life with you. It doesn't mean that I don't care or love you. I love you, Dad, even if I don't show it often.\nI'm not scared about the future, because I know that I have you with me every step of the way. You make me feel invincible like I can achieve anything I set my heart on. I love you, Dad. Thank you for everything!\nI have the best memories growing up because of you, Dad. You made sure that I had the most normal and fun childhood. I am the person that I am now because of your love. I love you, Dad!\nBelow you will find our collection of inspirational, wise, and humorous love quotes for dad. Also see, Father's day wishes.\nSons have difficulties to express their feelings with their fathers due to some conflicts in the past. It is time to break the ice and express your feelings freely. These love quotes for dad will help you achieve that.\nAny man will be happy and very proud to hear that her daughter loves him. These words are magic and so are these love quotes for dad by daughter.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This Friday (the 24th March 2017) sees mine and Allan's debut dub release in the majority of download stores. We are hoping to direct people to the Beatport store but in the meantime here is the Bandcamp link.\nStill hoping for the vinyl version. I hope it can happen.\nMorning Andy \u2013 yes, it would be nice to have the release on vinyl.\nHowever, it does not look like it will be pressed. I will be directing people to Beatport where they can buy the digital file that the vinyl would have been cut from \u2013 same quality and fidelity, just won't get scratched and shatter.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Splunk Inc. SPLK, +1.85% delivering actions and outcomes from the world of data, today announced that Graham Smith, former chief financial officer of Salesforce, has been appointed as its Chairman of the Board of Directors. Additionally, Sri Viswanath, chief technology officer (CTO) of Atlassian, has joined Splunk's board of directors. Godfrey Sullivan and Thomas Neustaetter both also announced they will retire from the Splunk board following its meeting in June.\nSplunk Inc. SPLK, +1.85% helps organizations ask questions, get answers, take actions and achieve business outcomes from their data. Organizations use market-leading Splunk solutions with machine learning to monitor, investigate and act on all forms of business, IT, security, and Internet of Things data. Join millions of passionate users and try Splunk for free today.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Play just get 10. From Veewo, the makers of 1024 (which inspired the awesome game 2048): Just Get 10. Your goal is, you guessed it, to combine tiles until you get to 10. Not as easy as it sounds!\n* Search result might be irrelevant if YouTube does not have any Just Get 10 walkthrough videos.\nTo report an issue with Just Get 10, you have to Sign In.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbggil b/data_all_eng_slimpj/shuffled/split2/finalzzzbggil
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@@ -0,0 +1,5 @@
+{"text":"This Western style formula is a general blend for skin health. Use it as a lon-term skin balancer, or for a variety of skin conditions, including noncystic acne and similar cases.\nSuggestede Use: 10-15 per day. Suitable for short-term use.\nContains: Burdock Root (Arctium lappa), Chickweed Leaf Burdock Root (Arctium lappa), Chickweed Leaf (Stellaria media), Ginger Root (Zingiber officinale), Goldenseal Root (Hydrastis canadensis). Contents per capsule, 570 mg.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Boston-based private equity firm Advent International has agreed to buy a majority stake in AOT Bedding Super Holdings, the parent company of National Bedding Co. (the majority owner and licensee of Hoffman Estates, Ill.-based Serta) and Atlanta-based Simmons Bedding Co.\nTerms of the deal were not disclosed by the parties involved, but an Aug. 5 Reuters story values the AOT transaction at about $3 billion, including debt. Advent prevailed over other private equity bidders, such as Bain Capital and Berkshire Partners, according to one Reuters source.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"These were a Fall\/Winter 2015 release from Naked & Famous, and the general consensus seems to be that they are hideous abominations. And when I bought them (Jan 2017) it was a spur of the moment completionist collectors impulse; I never really expected to wear them as I don't really consider myself that fashion forward. And boy-oh-boy if it isn't a striking look if anyone notices.\nI'm not really sure how it happened, but about 3 weeks ago these floated to the front of my closet and I thought \"what the hell, I'm not going out today.\" Except to walk the dog. And that was OK. So I put them on again a few days later. And that was OK too, even though I had left the house (in daylight) to run some errands.\nI've now been wearing these nearly every day for 2 weeks and I really rather like them. Part of that is down to the fit, it took me years to figure out but I love and look pretty damn fine in sized up skinny jeans. The power stretch fabric of these ones is pretty great too.\nOne day in the far future these will have been washed enough that the front panels will be much lighter than the back but by then I might be so old that I don't care what people think.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"HEPATITIS A is a topic covered in the Harrison's Manual of Medicine.\nAlthough hepatitis A rarely causes fulminant hepatic failure, it may do so more frequently in pts with chronic liver disease\u2014especially those with chronic hepatitis B or C. The hepatitis A vaccine is immunogenic and well tolerated in pts with chronic hepatitis. Thus, pts with chronic liver disease, especially those with chronic hepatitis B or C, should be vaccinated against hepatitis A.\nLoscalzo, Joseph, et al., editors. \"HEPATITIS A.\" Harrison's Manual of Medicine, 19th ed., McGraw Hill Inc., 2017. harrisons.unboundmedicine.com\/harrisons\/view\/Harrisons-Manual-of-Medicine\/623381\/all\/HEPATITIS_A.\nHEPATITIS A. In: Loscalzo J, Jameson JL, Longo DL, et al, eds. Harrison's Manual of Medicine. 19th ed. McGraw Hill Inc.; 2017. https:\/\/harrisons.unboundmedicine.com\/harrisons\/view\/Harrisons-Manual-of-Medicine\/623381\/all\/HEPATITIS_A. Accessed April 21, 2019.\nHEPATITIS A [Internet]. In: Loscalzo J, Jameson JL, Longo DL, Hauser SL, Fauci AS, Kasper DL, editors. Harrison's Manual of Medicine. McGraw Hill Inc.; 2017. [cited 2019 April 21]. Available from: https:\/\/harrisons.unboundmedicine.com\/harrisons\/view\/Harrisons-Manual-of-Medicine\/623381\/all\/HEPATITIS_A.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Stuck Community Acupuncture, Inc is Federal 501c3 Nonprofit Organization. What that means is that no one person owns the clinic, the clinic is owned and supported by the community and overseen by a board of directors. We decided to become a nonprofit to separate the issue of money from healthcare. We believe acupuncture should be available to everyone as a routine health care option. For this reason, we practice community-supported acupuncture, a communal style of treatment with a sliding scale of payment affordable to most every income.\nStuck Community Acupuncture is committed to providing effective and affordable acupuncture to the residents of Northern Arizona and to educating the public on the use of acupuncture and its related healing modalities as accessible healthcare options.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbgufu b/data_all_eng_slimpj/shuffled/split2/finalzzzbgufu
new file mode 100644
index 0000000000000000000000000000000000000000..a7928115feba47c67f3418b3396b944e4ecbef66
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"We visited the boat on Wednesday May 1st in Nanaimo, BC. We're almost there. The boat is being painted and it will be ready by this weekend. We roll test her early next week and bring her back to Vancouver where we'll moor her at the Vancouver Maritime Museum. Keep an eye out for us as we train in English Bay in coming weeks!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Faculty or staff are entitled to two free tickets for Department of Theatre Arts & Dance performances and Department of Music concerts. Must show valid I.D. the night\/day of performance to box office. Be advised: SUU Faculty\/Staff ticket policy does not apply to SUU Ballroom Dance Company, Second Stage or Orchesis performances.\nVisit the box office or call extensions 8667 or 8561.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At the Dasara half-marathon, the minister in charge of Mysore district, Shobha Karandlaje, shows her youthful best as her cabinet colleague Gulihatti Shekhar and others try to keep pace, in Mysore on Thursday. Don't ask us if she completed the race or not.\nThis entry was posted in Mysore-Bangalore, People and tagged Churumuri, Dasara, Karnataka Photo News, KPN, Mysore Dasara, Sans Serif, Shobha Karandlaje. Bookmark the permalink.\nnice. she looks like she is having fun.\nGood to see that some of our 'leaders' are still fit enough to run.\nAtleast now ,we can say, the ministers are given a run(literally) for their money, even if it is under the pretext of Dasara activities !!\nBut ironically, Whats more important is the ministers(or administrators) atleast walking an extra-mile to cater to the burning issues .\nNice to see some minister having fun. I hope the opposition will not object to this as no leader in the opposition can take on Shobha for a 100 meters dash.\nPerfect timing!!! whos coming from the opposite side??!!\nLooks like she is shooting a song sequence for a movie (neenello naanalle).\nSaar\u2026. ello yeddi is missing saar\u2026 please lodge a complaint!!\nDasara poorthi Shobhamaya! Ellello neene, Karandlaje!\nApparently his personal docter advised him full rest before he takes the chopper back to Bangalore after another pseudo Swami's darshan.\nI missed that!! I was about to lodge an FIR in Vidhana Soudha Police Station today had I not known this information \u2013 a.k.a search party for YSR\u2026.\nshobha is a yoga expert and had won yoga championship in her younger days. she also performed suryanamaskara with ease.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How to configure IAS to proxy the username and one-time passcode to the WiKID server after validating the user is active in AD.\nSet the \"Authentication Port\" to \"1812\" and enter the shared secret which will be used when communicating with the WiKID server. Also, set the \"Accounting Port\" to 1813 and ensure that the box for using the same shared secret is checked. The last checkbox is optional and has no functionality with WiKID.\nClick \"Apply\" and then \"OK\". Finally, back at the \"Add Servers\" dialog, click \"Next\".\nYou will be prompted if you would like to start the \"New Connection Request Policy Wizard\". Accept this and click \"Finish\".\nThe \"New RADIUS Client\" dialog will appear. Enter a friendly name for the client and the client's IP address and click \"Next\".\nSelect \"RADIUS Standard\" and enter a shared secret for the device to use when communicating with the IAS server. Click \"Finish\".\nSelect \"Remote Access Policies\" from the navigation tree on the left, right click on the empty white space, and select \"New Remote Access Policy\". N.B.: For the Remote Access Policy to apply properly, ensure that you are running your domain at 2003 functional level or higher.\nThe \"New Remote Access Policy Wizard\" will appear, click \"Next\".\nSelect \"Set up a custom policy\" and enter the policy name as \"WiKID Policy\". Click \"Next\".\nAdd a policy condition of \"Authentication-Type\" as \"PAP\". Click \"Next\".\nSelect \"Grant remote access permission\" and click \"Next\".\nOn the \"Authentication\" tab, uncheck all options except \"Unencrypted authentication (PAP, SPAP)\".\nOn the \"Encryption\" tab, uncheck all options except \"No encryption\".\nIn the navigation tree on the left, click on \"Connection Request Policies\" and then double-click on the \"WiKID Policy\".\nThe \"WiKID Policy Properties\" dialog will appear. Click \"Edit Profile\".\nOn the \"Edit Profile\" dialog, go to the \"Advanced\" tab and click \"Add\".\nFind \"Remote-RADIUS-to-Windows-User-Mapping\", double-click it, select \"True\" and click \"OK\".\nClick \"Close\". On the \"WiKID Policy Properties\" dialog, click \"Apply\" and then \"OK\".\nEnter a friendly name for the client, enter it's IP address, choose \"RADIUS\", select the appropriate domain, and click \"Add\".\nEnter a shared secret for the IAS server to communicate with the WiKID server (This should be the same as configured earlier inside of IAS.\nLog on to the WiKID server as \"root\" and run the command \"wikidctl restart\" in order to apply the RADIUS server settings.\nYou will be prompted to enter the certificate store passphrase. Type it in and press enter to complete the restart.\nAt this point, any client you configure inside of IAS as a RADIUS client with PAP authentication will be AUTHORIZED against Active Directory and AUTHENTICATED by WiKID!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Compare Sharon, stateshort} Medigap insurance companies within minutes. Quote comparison is the best way to locate affordable Medicare Supplmental insurance in Sharon. To begin enter your Zip Code and press \"Compare Your Options\". Our website will search for the best Medigap insurance carriers and providers in Taliaferro county. Finding the best coverage and low cost premium that fit your needs and budget can literally take you 20 minutes or less.\nIf you do want to get quotes for Medigap policies, you can either contact local insurance providers in Sharon, Georgia, or you can use the Internet to help you find them. Submitting your information to these insurance providers will allow them to calculate how much you are going to pay. They will have different policy levels that you can choose from just like any ordinary health insurance policy. The amount that you pay is usually reflective of how much coverage you will have. In many cases, it's better to go with the most expensive plan so that you eliminate the possibility of being blindsided by very expensive medical bills.\nAlthough we had just stated that you should pay the most money for Medigap insurance, this is only reflective of contacting one particular health insurance provider. Your goal is to get multiple quotes from different companies and look at all of the rates collectively. You will see that there are policies that are virtually identical, but they could be several hundred dollars a month less then other policies that are the same. In this case, you will want to choose the Sharon, Georgia Medigap policy that is the lowest price which will come from one particular insurance provider.\nSelecting your well being insurance policies policy can be extremely complicated. If you make the mistaken selections, you could finish up having to pay out too considerably for your coverage, or even worse, you might not be protected for specified companies that you need to have. Read this write-up for some helpful suggestions on obtaining the most out of your health insurance.\nMake certain your insurance policy will include abroad medical professional visits. If you have wellness insurance policy, it is critical to study the fine print. You will not want to get ill in a overseas country, go to the doctor, and find by yourself unable to spend for it. Make certain you have additional income obtainable must your overall health insurance policies service provider not protect the expenditures.\nHealthcare insurance policies is something that everybody truly should acquire. You in no way know if you are heading to get quite unwell, and the previous issue you require when you don't feel well, is to now owe a massive monthly bill to a medical professional that you may possibly or might not be ready to find the money for.\nQuestion your physician to prescribe you generic medicines if a single is accessible. Your co-pay out will almost constantly be less for a generic drug than for a name brand name. The results are the identical, you just is not going to be having to pay an added surcharge for the branded drugs. Save by yourself the cash if possible.\nThe way you handle your wellness treatment insurance policies might actually make you tax breaks around April. A lot of men and women will not understand that their overall health insurance policy premiums are tax deductible. Any cash you devote to include your deductible, your prescriptions, or any visits that your insurance policies isn't going to go over can also be deducted from your taxable income. State taxes are diverse from federal taxes, so make sure you examine tips.\nLearn the stringent protection specifics of your overall health insurance policy plan. Although your policy may condition that it addresses crisis visits, some hospitals and medical professionals demand individually for your care. Your policy might include the hospital's fees, but not the doctor's. If you are unsure about how your organization handles this, phone them and request.\nThink about getting private wellness insurance policy procedures for your dependents. Even if your employer provides wellness insurance policy coverage, the price to incorporate your dependents can occasionally be really large. Purchasing all around for private alternatives can, at moments, be well worth your although. Some private organizations supply reduce protection options, for a lower value, that could go well with your demands.\nWhen acquiring dental insurance coverage, do not get attributes that you do not consider you are going to use. For instance, numerous dental programs are packed with added discount rates for prescription drug programs. If you come to feel like you do not need it, don't get it. It will just end up costing you much more.\nIf you and your wife or husband are both offered wellness insurance policies protection by way of your employers, assessment both procedures carefully ahead of generating a selection. You might save income by likely with just your coverage, or just your spouse's. It also may be less high-priced to insure every of you separately.\nNumerous individuals who are self-employed experience a quandary about overall health insurance policy. They know they need to have to be lined in scenario of health emergencies, and yet the price of health insurance policies for folks is so substantial as to be prohibitive for several. Some people get around this by buying key medical protection only. This kind of plan has a very high deductible, addresses no preventive treatment - nonetheless, it will cover circumstances where the particular person will require admission to a clinic.\nAhead of possessing a little one you need to know about how much you will have to pay out even with wellness insurance policy. If you have a PPO you may conclude up spending about $one thousand or even a lot more for the birth. If you have an HMO your fees will almost certainly be a whole lot decrease. It is best to determine it out prior to you get pregnant.\nThe previous data was composed with the hopes of supporting you locate ample info to assist you make the very best decision relating to your overall health insurance policy. Hold these things in mind as you shop around for the coverage that is heading to protect your family's healthcare needs these days and in the potential.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Furthering its strong trajectory as a leader in research and innovation, the Engineering school at Washington University in St. Louis is taking a major leap forward and reaffirming its commitment to tackling the world's great engineering challenges with renewed vigor, an ambitious strategic vision and a new name.\nThe School of Engineering & Applied Science will be renamed the James McKelvey School of Engineering in honor of trustee and distinguished alumnus Jim McKelvey Jr., who has made an unprecedented and transformative investment in the school.\n\"The McKelvey name has become synonymous with innovation and entrepreneurship in the St. Louis region and well beyond,\" said Chancellor Mark Wrighton. \"There is no better way to make a statement about what our Engineering school stands for than by giving it a name that represents being ahead of the curve and blazing a trail of creative problem solving through technology.\n\"This tremendous gift creates new opportunities for our students and faculty to tackle the world's greatest engineering challenges, and to dramatically expand computing throughout the university. At the same time, it helps ensure that a diverse population of students will have access to a world-class engineering education and enable the school to be a catalyst for economic development for the St. Louis region and beyond,\" said Bobick.\nFounded in 1857, Washington University's Engineering school promotes independent inquiry and education with an emphasis on scientific excellence, innovation and collaboration without boundaries. With top-ranked research programs in biomedical engineering, environmental engineering and computer science, the school attracts many of the best students from around the world to its 40 different degree programs. The school recently launched several new graduate programs, including an interdisciplinary doctoral program in imaging science, one of only two such programs in the United States; an innovative doctoral program that combines data sciences with social work, political science and psychological and brain sciences; and a new master's program in cybersecurity engineering. New bachelor's programs include environmental engineering, a joint business and computer science degree, and a joint math and computer science degree. Key components of the university's current east end campus transformation include two major facilities for engineering: James M. McKelvey, Sr. Hall (to open in 2020) for the Department of Computer Science & Engineering and other computational programs, and Henry A. and Elvira H. Jubel Hall (to open in 2019) for the Department of Mechanical Engineering & Materials Science. Since 2000, the school has invested more than $250 million in new and renovated space, which includes 700,000 square feet in the new engineering complex.\nMcKelvey Hall was made possible by a $15 million commitment from McKelvey Jr. in 2016 to honor his father who, during his 27 years as dean, transformed the Engineering school from a regional program into a nationally prominent research institution. McKelvey greatly strengthened the quality of the undergraduate and graduate curricula, particularly in emerging fields including computer science; significantly increased both undergraduate and graduate student enrollment; expanded the faculty; dramatically increased federal and other research funding; and grew the endowment for the school more than tenfold from $4 million to nearly $52 million. He also oversaw a remarkable expansion of the school's footprint on the Danforth Campus.\nMcKelvey Jr. is an independent director of the St. Louis Federal Reserve but is better known for his involvement in several St. Louis\u00ad-based startups, including Six Thirty (co-founder), LaunchCode (founder), Third Degree Glass Factory (co-founder), Mira Publishing (founded when he was a Washington University student) and Square, the company he founded in 2009 with Jack Dorsey. He also is the author of \"The Art of Fire: Beginning Glassblowing,\" the leading textbook for novice glassblowers.\nAs a child, McKelvey Jr. spent formative time at the Engineering school with his father during his tenure as dean. He applied early decision to Washington University and enrolled in 1983, graduating in 1987 with degrees in economics and computer science. While a student, McKelvey Jr. wrote two computer programming textbooks.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In the Financial world, you may find thousand of Investment quotes. But in this article, you will find the best & useful Quotes.\nThis 18 Quotes may help you more to be a successful person. And you should know these quotes to create a better life.\nGood quotes give us inspiration & motivation.\nHere are the 18 Quotes which was said by famous individuals. You will find also a little paragraph about that quote.which was created to help you more to understand.\nMoney is the most important thing in our life. And if you did not manage your money properly then there is a high chance that you will never rich in your life. If you don't know how & where to invest & save your money then your money will stay away from you.\nHis every advice is most useful ever.\nfail in the first time, it does not mean that you will never win in future. Winning and Losing is just the part of the game. You need to be emotionally powerful to become a successful investor. If you don't control your emotion, You will never succeed in this game.\nQuote is most useful ever.\nyou need to do hard work and ledger focus to achieve your goal.\ntheir money in bank account too much. They know how to Invest their money in proper ways. Investing your money in bank account did not makes you more wealthy.\neducation. If you want to fortune, then you must know the self-education. Formal education will make you only a simple living life.\nmake their own source to money will comes to them. This Quote is Most useful.\nsingle income. A single income may can't help you to be a rich person. Try to make a smart Investment to create the second source.\ntake a risk to be a successful Investor. The biggest risk in your life is not taking any risk.\n12. Compound interest is the eight wonder of the world. He who understands it, earns it. He who does not, pays it- Albert Einstein.\nDid you know? What is the compound interest? In the financial world, You must know the compound effect . According to the compound effect, The small decisions & work which you make every day are fully responsible for your future. People use compound interest to generate earning.\nLong term Investment means Invest your money for a longer period of time.And the long term is the Best Investment ever. Even I also recommend you to choose the long term .\nlow risk, But without taking a high risk you can,t make it possible.\nThe Patient is the most important in our life. Without patient, We will learn less, see less,hear less.\nBe patient and try to understand the process.\nYou don't need to be a heavy mathematical skill or financial skills to become a successful investor. If you think that your IQ is low, then it does not mean that you will never success in your life.\nIf you don't take your Investment too seriously, then there is a high chance you may lost your money. One of the biggest mistake people make that they start investing without businesslike. You should Invest your money more businesslike.\nDisciplined is most important in our life. If you think that you are not too smarter, that's ok. But you should definitely maintain your disciplined. Self-disciplined is most important. Disciplined teaches you how to control your thoughts.\nI hope you love the Quotes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Why bother?! In this age of technology, do we still need to learn number facts? Absolutely! Thinking flexibly about numbers opens up thinking for more complex mathematical problems. Mental math is also a survival mechanism when you are out in the real-world! Think of how often you use it: How many km am I going \u2026 Continue reading The Facts about Facts!\nWe are away for the summer at a cottage, with no internet or TV, which I usually love. We've already read lots of books, frolicked in the waves, swam to the Big Rock, sailed to Seagull Island, canoed...dumped the canoe and had lots of good old fashioned fun; but my heart still felt the pitter-patter \u2026 Continue reading Tenzi Frenzy!\nMaking Math Meaningful with Manipulatives!\nIf you're not a teacher, you may not be familiar with the term manipulatives, but you can probably infer what they are. Manipulatives are models that help children think and reflect on new ideas in math. They include resources that allow children to explore, question, guess and check, but more importantly, to play with the \u2026 Continue reading Making Math Meaningful with Manipulatives!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The McKee Gallery is pleased to announce a ten-year survey exhibition of Daisy Youngblood's sculpture, opening Thursday, April 2, 6-8 pm, through Saturday, May 30, 2015.\nYoungblood has produced 14 sculptures over the last decade, in low-fire clay or in bronze cast from clay, all of which will be on view. The work over the last 10 years reflects changes in her approach to making sculpture. Although she has used pieces of wood in many earlier works, found objects in nature are more decisively incorporated into the clay and the scale has increased. Stones are used as heads, eyes, noses, torsos; a long concave piece of oak is a body. They are all equal parts of a living whole for Daisy Youngblood.\nThe earliest piece in the exhibition is Budhi 2006, a moving portrait in clay of her daughter with Down's syndrome; then Bear 2007, a gentle spirit in clay; Venus 2007, a fierce crouching figure that could be animal or human, blurring the lines between them; a series of orangutans in motion: Oladio 2009, Leaping I and Leaping II 2010. Anubis and the First Chakra 2012 is a major statement of materials and essence. There are three complex sculptures in different Yoga poses, and lastly, Chandrika 2014, a long reclining figure, seemingly moribund. The strong influence of Buddhism is evident.\nThere is a 40-page catalogue available, with 25 color illustrations and an essay by Constance Lewallen.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The warm soil temperatures and higher rainfall of autumn make it a great time of year to plant. Plant Shed offers the largest selection of indoor and outdoor gardening supplies and plants. Please call or stop in to meet one of our enthusiastic experts.\nPlease contact us for any questions relating to your gardening needs!\nFREE Manhattan delivery for online orders over $100 per location.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"In 2016, content marketing continues to be an important component to the B2B marketing industry. In fact, according to reports by the Content Marketing Institute, 93% of B2B marketers use content marketing to distinguish their companies in highly cluttered markets. In the B2B marketing realm, ensure your content marketing addresses at least these two goals: You first need to catch your prospect's attention, and then you need to show them that you are the industry expert they can trust. One of the most effective strategies for achieving these goals is consistent education, taking a targeted approach, having a customer-centric view, and aiming to build long-lasting relationships. Other goals of B2B content marketing are to use content for generating brand awareness, leads, and boosting engagements.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"RealNews from RealNex | Jeffrey D. Fisher Ph.D.\nWe have heard a lot in recent years about so-called \"big data.\" Having spent my career doing research and creating products using data, I was intrigued to learn more about how big data might be used for commercial real estate.\nAlthough the rate of price appreciation has moderated the past two years, it has still been positive and values have continued to reach new highs for commercial real estate as measured by the NCREIF Market Value Index (MVI). NCREIF tracks the performance of over $500 billion in commercial real estate owned by institutional investors in the US and includes all the major property types (apartment, office, industrial, retail and hotel). The following graph shows the quarterly value indexed at 100 in the first quarter of 1978. The value is also above a long-term trend line shown in the graph.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The orchid is one of the best house value houseplants around, and yet it is often the one most misunderstood.\nGive it the minimal treatment that it deserves, and it will flower its head off for you for years to come.\nGo into any supermarket or garden centre today and you'll see huge numbers of the very colourful Phalaenopsis hybrids, also known as the moth orchid.\nAlthough it's a tropical plant, it will do very nicely in your house provided that you observe a few simple rules.\nThe worst thing that you can do to this magnificent plant is to over water it.\nYour moth orchid is an epiphyte, which means that its roots will not be buried in soil or compost. Some roots are to hold on tight to the large bark chips contained within its pot; the rest act like leaves, collecting moisture and nutrients from the atmosphere around it.\nIt will normally be planted in a transparent plastic pot, placed into an outer decorative one. The best way to keep it well watered is simply this \u2013 remove the plastic pot and take to the kitchen sink.\nIf you happen to have a jug of rainwater, so much the better, but if not, then simply flush the pot through with not-too-cold tap water. Put the pot on the draining board and allow the excess water to percolate through the bark chips.\nIt's very important to allow time for the bulk of the water to drain out completely so that when you pop the orchid back into its original decorative pot, it will not be sitting in any residual water. This will eliminate root rotting.\nThe orchid is not a hungry plant, and so feeding really isn't necessary. If you think that a shot in the arm is required, then do try and buy some orchid fertilizer. This is not just fancy label for a 'normal' product, it's an extremely weak version of what you might give other plants.\nOver-feeding could damage the plant and burn the roots, so when in doubt, my advice is don't!\nUnless it's old, it's extremely unlikely that your orchid will need re-potting. The visible clue is that all of the bark chips in the inner pot have been reduced to a mush. If your plant is fairly new, then don't bother.\nYour orchid should have been given to you with a flower spike or two on it. When these fade, you simply find a point lower down on the stem looking for a growth point. Make a sloping cut above the bud. In time, your orchid will produce another stem of these amazing flowers.\nAfter around two flushes of flowers, it's best to allow the plant to rest in a draught free corner. Cut the flower stem down to just above the lowest bud, and water sparingly every now and again. The orchid will decide for itself when it's time to flower again.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"2nd Quarter Honor Roll Announced!\nHigh Honors is an A in every course, with a grade of 93 and above.\nHonors represents an A or B in every coursc, with no grade lower than an 85.\nCommendable represents an average of 85 and above in every course, with no grade lower than an 80.\nHigh Honors: Baiungo, Anna; Berry, Abigail; Howard, Maximillian; King, Katherine; Sheafe, Zachary; Wallace, Caleb.\nCommendable: Al-Matrouk, Alaa; Artkop, Mikayla; Blais, David; Cole, James; Curtis, Kristyn; Hanson, Cole; Macleod, Tyler; Schortz, Genevieve; Snowdeal, Ian; Thomas, Noah.\nHigh Honors: Bickford, Charles; Bonneville, Lucie; Dinsmore, Emery; Flanders, Ashley; Goguen, Andrew; Greene, Hannah; Groening, Patrick; Johnson, Alex; Mcleod, Kasey; Mehuren, Sadee; Racioppi, Alexis; Samsel, Michael; Snider, Jesse; Spectre, Peter Wyman, Richard.\nHonors: Andrews, Leander; Blood, Karagan; Charbonnier, Dejaleen; Dunton, Nathan; Fritz, Alexander; Gordley-Smith, Lucien; Hartman, Anika; Hills, Alison; Howard, Noah; Howie, Sarah; ; Lafrance, Jake; Lafrance, Parker; Nealey, Drew; Peabody, Alyson; Place, Blaine; Simmons, Anna; Sobey-Bradford, Hayden; Stevens, Bronsen; Ventura, Thomas; Walker, Alison; Wich, Philip.\nCommendable: Ambrose, Rory; Bergstrom, Austin; Bishop, Jesse; Blankenship, Annmarie; Blood, Lindsey; Coleman, Dakota; ; DiBiase, Jordan; Faulkingham, Kelci ; Gaiero, Alicia; Hall, JaShelle; Hewey, Micaela; Hill, Cassidy; Kennedy, Sean; Larrabee, Jennifer; Moesswilde, Johann Soren; Munro, Samantha; Nelson, Cooper; Nichols, Hannah; Pillitteri, Cameron; Porter, Brandon; Robson, Benjamin; Sanborn, Olivia; Sukeforth, Alexandra; Warren, Ben.\nHigh Honors: Costa, Julia; DeSa' Pinto, Edoardo; Gardiner, Ariel; Hyland, William; Lynam, Hannah; Pelletier, Simone.\nHonors: Artkop, Kyle; Berry, Sarah; Boynton, Maylinda; Caswell, Kirsten; Culbertson, Kameron; Eynon, Kimberly; Felton, Zoe; Hall, Kylene ; Harriman, Jasmine; Hurd, David; Hurd, Devon; Jackson, Autumn; Kulikowski, Samuel; Lane, Dallas; Larrabee, Meghan; Leppanen, Brogan; Littlefield, Brittany; McGinn, Catherine; Mckinney, Cody; Pickering, Madison; Roman, Zavier; Sheafe, Jensen; Struba, Anna; Sturdevant, Emma; Waterman, Brandon; Wessels, William; Young, Katlyn.\nCommendable: Alley, Allen; Al-Matrouk, Fouad; Ames, Hannah; Bradeen, Joshua; Defeo, Anthony; Delcuve, Valentine; Goslant, Kaitlin; Knudsen, Zoey; Kockerols, Amalia; Lapham, Levi ; Littlefield, Adam; Littlefield, Madison; Nadeau, Crystal; Page, Alyssa; Parziale, Rachel; Pomeroy, Katrina; Powell, Adam; Quehl, Roger; Smith, Matthew; Sprague, Austin; Stevens, Tierra; Wright, Brian.\nHigh Honors: Carpenter, Romy; Cummings, Madison; DellaMattera, Allison; Fortin, Nicholas; Hersom, Alexia; Mckinney, Mika; Mehuren, Kristine; Moesswilde, Emma; Pomeroy, Aleahnah; Simmons, Meggie; Snider, Ari; Stupca, Autumn; Warren, Curtis.\nHonors: Ames, Alexis; Anderson, Kaylee; Armstrong, Karrah; Bailey, Hannah; Black, Dakota; Bowen, Kally; Brunette, Peter; Chernenko, Darya ; Crosby, Caelyn; Durkee, Lacey; Flood, David; Gross, Patrick; Knight, Jennifer; Loring, Briahna; Merrifield, Ethan; Miller, Jeffrey; Noble, Kayla; Parsons, Marc; Philbrick, Nickolas; Pray-Smith, Paige; Roberts, Vanessa; Schnetzer, Michael; Schweikert, Evan; Smith, Baxter; Wich, Douglas; Williams, Christian.\nCommendable: Davis, Grace; Howard, Cassandra; Hurd, Mariah; Lewis, Asa; Mossing, William; Nelson, Cody; Nickerson, Larry; Peabody, Joshua; Rego, Dorothy; Salisbury, Anthony; Simmons, Ellie; Snodgrass, Adam; Sprague, Bethani; Starrett, Grant; Stevens, Miranda; Sweetland, Ian; Tran, Tyler; Watson-Fountaine, Kyah.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On completion you will be taken to the Blog Home Page and you will also receive an acknowledgement email. It should be noted that these messages go to a generic email address so if your message is to an individual it will not be private. This contact form should not be used to discuss confidential issues. Please use your regular contact route for these matters.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbjwnq b/data_all_eng_slimpj/shuffled/split2/finalzzzbjwnq
new file mode 100644
index 0000000000000000000000000000000000000000..9026048c141b16a0e986607137ec7064903cd2d4
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Searching for hotels in Picton (NSW) or Picton accommodation? Picton is located one hours drive south from Sydney and maintains a lovely rural atmosphere. Picton Gardens will have the garden lovers happy, while the Broughton Pass offers great views of the Cascade Waterfalls. The White Waratah Retreat is a relaxing getaway if you plan on staying overnight, as is the Dreamcatcher Lodge set on over 7 acres. - Browse the Picton Map or compare Picton accommodation from Hotels, Motels, B&Bs, Lodging & Hostels to Holiday & Caravan Parks.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Conspiracy of Silence is the never officially released Discovery Channel documentary about The Franklin Cover Up Case (also known as \"The Franklin Scandal\") involving human trafficking, pedophilia, ritualistic abuse of children, murder, government cover ups, the Bohemian Grove, Hunter S. Thompson, Lawrence King, and author of The Franklin Cover Up, Satanism & Murder in Nebraska; John Decamp, and many other prominent U.S. political figures.\nDownload the ZIP archive of the film, book, and more here (1.27GB ZIP folder).\nThese interviews, film clips, and documents are disturbing and detailed. This is one of the largest and sickest cover ups that, in our opinion, is a corner stone in the globalist, Illuminati, government corruption, church corruption agenda.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This children's book feature Kindle Text Pop-Up for reading text over vivid, full-color images (double tap on the text).\nDrake just wants to play with all the other kids, but he has a bit of a problem: Drake is a dragon. How can he show them that he'd be a good friend? Or maybe he's just too scared to try. Poor Drake. Find out what happens to him in this fun story told in rhyme, accompanied by beautiful illustrations.\nAn orangutan wishes to have her favorite breakfast \u2013 a banana. But it is gone from where she had kept it! Who took her banana? This starts a trek through the rainforest as she inquires of a variety of exotic animals if any of them has taken her banana. From a mamba snake to a tapir, a sloth, a spotted leopard to an elephant, a crocodile to a toucan and a bunch of monkeys, none seems to have taken the banana. All the while she dreams in rhyme of why she loves bananas. The mystery is solved when she returns to her nest. Who took her banana? You'll have to read the book to find out.\n*A combination of prose and rhyme combine to make this a delightful read for any child.\n*Over 35 adorable illustrations float through the story and bring it to life.\nIf this sounds like something you would like, scroll up to download your copy.\nBe sure to take a peek at AN ORDINARY PRINCESS, another adorable book by Sally Huss.\nAfter a solar flare scorched the Earth and incinerated the atmosphere, survivors dug deep beneath the surface, discovering a new means of life where none had ever lived.\nSpec is an ordinary 16 year old residing underground in the Hive, a modern day colony where he and 73 other human beings survive.\nEvery day is the same for Spec.\nMine for clay and insects.\nThat's why he has decided to venture outside of the colony with his best friend, Cotta. Spec plans on going where there are no ceilings and no walls, a place where there are no barriers. He plans on going to the mythical surface.\nThe English Rural Society in the Midst of Change\"Mrs Forrester \u2026 sat in state, pretending not to know what cakes were sent up, though she knew, and we knew, and she knew that we knew, and we knew that she knew that we knew, she had been busy all the morning making tea-bread and sponge-cakes.\" \u2013 Elizabeth Gaskell, CranfordBoth comic and tragic, Cranford is a novel set in a fictional rural town of 19th-century England where time seems to stand still. Neither character wants to change and all of them are feverously trying to preserve the old customs and norms. However, a series of comings and goings disrupts the town's balance and force the leading women in Cranford to leave their comfort zone.\nA bundle of touching comedic children's poems, Cornucopia encompasses a group of light, quirky poems about growing up and living life.\nThe collection includes light and offbeat intellections with titles such as \"Selling Happiness\", \"Scattered Thoughts\", \"Kelly the Kitten's Rough Day\", \"Mom's Clever Little Trick\", and \"The Feels\".\nAs a kid growing up, one embarks on a journey of living life, being happy, finding humor in the world and explaining things. Cornucopia encompasses the weird explanations and thoughts from a growing kid's perspective.\nMiddle-school students are at a weird and awkward point in their life as the training wheels have come off of them and they are given space to make choices and figure themselves out. They enjoy people, the world around them and visualizing their future endeavors. They are trying to find where they fit in the world. This makes Cornucopia a great collection as it encompasses living in the moment while dreaming about the future and making changes. Its different, moving aspects leave the reader in a place to make reflections and find deeper meaning in their own life.\nThis collection includes about 140 poems and is similar in tone to Shel Silverstein's Where the Sidewalk Ends. It expresses ideas and concerns of growing children as well as originative and witty stories.\nIt takes more than a school trip to Washington, D.C. to change Christy's life. It takes murder.\nA witness to the brutal slaying of a senator's aide, Christy finds herself watched not only by the killers and the FBI, but also by two hot guys.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Now, when I upgraded to WP 1.5.1 the other day I thought all was well, but I guess the RSS feeds got busted. After reading about it as a bug, I am taking the suggested advice and making this post to hopefully fix the issue. But, we'll see how long this lasts. I've tried the patches and they don't seem to be working. I guess I'll have to take another look at it when I get home after work.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Canon imageRUNNER C3380i Driver Download \u2013 Hi there, fellow Canon imageRUNNER C3380i user! As soon as all the printer driver files are extracted, you still have to install the printer driver. heed these Technodrivers tips, as you may utilize the Add Printer Wizard to aid you do this.\nOn this page, you should select the slot that the printer can use to connect with the personalcomputer. Some home computers use LTP slots. If your printer is actually connected to a USB slot, pick out the USB slot option in the following slot list, then click Next.\nIn the Open box, type Control laser printers], and afterwards click OK. The \"Printers and Faxes\" page opens up.\nOn the ensuing page, click on Local Printer of Canon imageRUNNER C3380i attached to this laptop or desktop. Important note: Make clear that the Automatically identify and install my Plug and Play printer check box is truly not selected.\nIn the Install From Disk dialog box, press Browse, and afterwards spot the folder to which you extracted the printer driver files.\nThe path for your folder must now surface in the copy manufacturer's files off box in the Install From Disk dialog box.Click OK.\nThe color printers that you could install needs to be actually provided in the Install Printer Software dialog box. Select the Canon imageRUNNER C3380i printer to plug, and afterwards press Next.\nIf you are prompted to employ an existing driver, select Replace existing driver, and afterwards press Next. When you pick to switch the existing driver, Windows will attempt to replace the files on your computer using the fresh ones that you downloaded and install.\nOn the following page, press Next if you wish to approve the recommended printer name and also utilize the Canon imageRUNNER C3380i driver as your default printer. If you don't, change these settings to just what you choose, and afterwards hit Next.\nIf you see the Printer Sharing page next in the wizard, you may share your printer to ensure that different PCs on your home network are able to use it to print. Click Share name, and afterwards stipulate a name for thecommon printer.\nIn different circumstances, click on \"Do not share this printer\". After you have already taken your choices, click on Next. You may be asked to enter optional location info for the printer. Type the location of the printer for instance a Home Office, and afterwards press Next.\nOn the Print Test Page page of the wizard, press Yes to print a test page. Later press Next, and afterwards press Finish.\nIf you may print a test page proficiently, you are withouta doubt completed. Unfortunately, if the test page was never printed properly, you still have some troubleshooting to perform that is generally past the extent of the blog.\nFirst, get on Troubleshooting in the Add Printer Wizard to discover whether it can provide you advantageous information and facts. If that does not clear up your Canon imageRUNNER C3380i printer condition, head to Next Steps for some ensuing measures that you will choose to utilize.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbkmcz b/data_all_eng_slimpj/shuffled/split2/finalzzzbkmcz
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index 0000000000000000000000000000000000000000..a7f87235734d68671b86e4526e726925065ac967
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"In the first article of the series on SQL Server 2019 and Ubuntu, we prepared the virtual machine environment and installed Ubuntu 18.10 in it. In this part of the article, we will install the latest SQL Server 2019 Preview CTP 2.1.\nWe can install SQL Server on Ubuntu by two methods.\nIn this article, we will install SQL Server 2019 using the Docker container. Docker is basically an application package which contains all dependencies in the package itself. We can just call the Docker container and provide the configuration parameter to start using the application.\nBefore we install SQL Server 2019 on the virtual machine let us do a few checks and necessary pre-requisites.\nLaunch the virtual machine if it is not already running using the ID and password we configured during installation wizard.\nIn Ubuntu, we can check the IP address using the 'ifconfig' command. But as you can see here, it shows an error that command 'ifconfig' not found. By default, it is not installed on Ubuntu so we need to install net tools to use this command.\nWe will connect the virtual machine from outside to perform the installation. To do so, we need to configure the network at the VM level. Right click on the VM, go to Settings and then click on the Network.\nAs you see above by default, the network adapter is 'NAT'. The virtual machine does not contain any IP address in 'NAT' configuration, therefore, you cannot connect externally to this virtual machine.\nModify this to 'Bridged Adapter' from the drop-down list. In bridged networking, we connect a virtual machine by the network adapter on the host system. Therefore, when you configure to 'Bridged Adapter' you can see the host network name as shown here.\nClick OK to save this setting. This takes little time to get the setting effective. Now, we can run the 'ifconfig' command and check the IP address.\nAlternatively, we can also run the command 'IP addr show' to check the IP address. You can see the IP address from the screenshot '192.168.225.25'.\nProvide the input as 'Y' and press 'Enter' to proceed.\nApt-get update: This command will download the package list and update them from the repository. Run the command in the terminal on Ubuntu with sudo access.\nInstall Putty: We will use this tool to connect the VM and run the queries. You can install PuTTY (SSH and Telnet client) in the Windows Server outside the VM. Installation is quite simple \u2013 launch the setup and follow the installation wizard.\nAfter installation, launch the 'PuTTY 'outside the VM and enter the IP address '192.168.225.25' as shown here and press 'Enter'.\nClick on 'Yes' to add server host key in the 'PuTTY' cache. If we click on 'Yes' it does not ask this again if we reconnect to the VM. Else, click on 'No' to connect without adding the key to the cache.\nProvide the username and password to authenticate. If we provide wrong information, we get error message 'Access denied'.\nInstall curl utility: curl utility in Linux is used to transfer file using different supported protocols.\nWe can test the docker container using the 'hello-world' which verifies that the Docker is working fine or not.\nAs you see that, it shows the message ' This message shows that your installation appears to be working correctly'. At this stage, we are ready to install SQL Server 2019 CTP 2.1 on Ubuntu.\nLet us connect using the Azure Data Studio from outside the virtual machine. In Azure Data Studio, provide the connections details as shown here and connect.\nWe will explore more on SQL Server 2019 features using this docker container in my further articles.\nIf we do not want our docker container, then we can drop it with simple steps.\nWe can see now the container is removed quickly.\nNote: If we remove a container, it also deletes all the databases and all data in that particular container, so you should take a note of this before removing the container. You can take backup of your data if required for future use.\nIn this article, we explored pre-requisite for installation of SQL Server using the Docker containers in Ubuntu. We also learned to install SQL Server with Docker container and connect to it using Azure Data Studio externally. In the next article, we will take a look over the tools which we can use to connect to SQL Server.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It's been feeling like vacation all week even though it's not vacation at all.\nI guess that's the magic of going from a busy schedule to a less busy schedule.\nThis week I've gone to work and fieldwork \u2013 which is more like volunteering now since I'm not required to finish out the initial hours. I've gone to the gym and I've run twice. Granted the runs were only a mile each, but I had time to get dressed and lace up.\nI've been in bed around 9 each night, which really makes a difference in my day.\nWhile I have a summer class, I enjoy the reading and it's all writing assignments. I love torture, I love writing assignments.\nThis three day weekend is actually going to feel like a vacation even though I'm bringing some reading along with me. I can't remember the last time a weekend felt like a vacation.\nWhile JP's parents are out at the Cape we're taking over the house.\nHe's going to work on building the motorcyle and some other car projects. I'm going to work on two assignments I have for class.\nWe're bringing some fancy clothes because we may go out for date night near the water, but we may also grill.\nI know it sounds silly to say that you're planning to be spontaneous, but I've planned a few things so that I can be spontaneous and feel good about those choices. Honestly, having a plan of some kind even if it's not perfect makes me feel better.\nLike, I'm planning to get my homework assignments done on Saturday mid-morning while he works on the cars so that we can do whatever we want in the afternoon together without me feeling like I should be stuck in front of my computer. I'm planning on finishing my writing so that I don't think about initial submission next week \u2013 I can just work on finalization.\nSaturday also starts the fifth week of my lifting program and since I've been trying to be more consistent I do plan on working out over the weekend when I have some time to myself. I.e. while JP is playing with the cars.\nA few weeks ago I mentioned on Facebook that I restarted PH3. I've done the program before and I like the structure, however, I have changed up some of the hypertrophy. I've made sure to continue to target the muscle groups and have similar movements, but sometimes I want to do good mornings and stiff leg deadlifts over leg curls \u2013 so I'm letting myself that flexibility.\nThe program is divided into four segments. The first segment is four weeks long and I finished that yesterday with AMRAP to determine new maxes.\nI feel really good about what I accomplished in the past four weeks. It was different than what the past eight or nine months have been like so it was like a reintroduction to routine and structured lifting.\nI was appropriately sore at times and feeling like I could do more other times.\nMy old 1RM for my back squat was at 205, but since I haven't lifted in this way in a long time, I decreased the number to 190 so I could work within percentages of it. I knew that 190 would be very difficult to hit, but I also knew if I tried to just restart completely and go to 100 I would do myself a disservice by not pushing hard enough. The heaviest percentage lifted so far has been 87.5% which was 165 pounds. That day it felt good. It didn't feel heavy. But when I did testing yesterday 160 pounds felt slightly heavy and I could feel myself leaning forward too much.\nEach day is unique and there are so many factors that impact your workout. I got six reps for 160 pounds which calculated a 1RM of 185 \u2013 so close to the guesstimate that I made of 190.\nMy bench sucked, but my bench isn't something I'm concerned about. I'm more concerned with the mobility and flexibility in my shoulders. My range of motion isn't very good in my left shoulder and that's something I'm working on.\nMy deadlift improved the most. I've been focusing on a flatter back, pulling up without overextending my lower back. I've played with tempo a little bit too, but nothing crazy. My old 1RM was 155 and after yesterday's lift it was calculated at 165.\nI'm excited to work within that.\nMaybe it feels like vacation because everything feels calm. Everything feels like it's finding its place. When timing is right, it feels right.\nTo fill some of extra free time, we started watching the Fast and Furious series again\u2026 a movie each night. We actually have two copies of the series because we each own the movies.\nJP has also been walking around quoting Paul Walker as Brian O'Connor. I've been asking a million car questions.\nOur talks have shifted from chemistry to political science and the U.S. constitution \u2013 I think we're both going to learn a lot these next six weeks while I take U.S. Government.\nWe've talked about houses. We've talked about cars. We've talked about vacations and parties and the next 5K.\nThe balance of being busy with work \u2013 inside and outside of the house, being active and having some time to make last minute decisions is the kind of balance I've been striving for.\nI don't feel defeated or like the list is never going to get done.\nI'm ready for everything else. I'm ready for what's next.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The enforcement of international contracts in the European Union is increasingly dependent on Community (rather than national) private international law. This book examines the present status and future prospects of Community private international law in the contractual area. It focuses in particular upon the joint analysis of the Rome Convention of 19 June 1980 on the law applicable to contractual obligations (which is likely to be converted in the near future into the Rome I-regulation) and the Brussels I-regulation.\nApril 2004 | xv + 387 pp.\nThe enforcement of international contracts in the European Union is increasingly dependent on Community (rather than national) private international law. This book examines the present status and future prospects of Community private international law in the contractual area. It focuses in particular upon the joint analysis of the Rome Convention of 19 June 1980 on the law applicable to contractual obligations (which is likely to be converted in the near future into the Rome I-regulation) and the Brussels I-regulation. Rather than attempting a comprehensive study of Brussels I and Rome I, this book examines a number of key issues considered particularly pertinent from the point of view of the coherence between both instruments. This approach should contribute to the consistency of Community policy-making and legislation in the field of international contracts, to the benefit of market participants.\nThis book is the culmination of a research project funded by the European Commission (DG Justice and Home Affairs, Framework programme for judicial co-operation in civil matters) and co-ordinated by the University of Antwerp Belgium. Eminent European experts have contributed to the book which should prove of interest to law makers, academics and practitioners concerned with the enforcement of contracts in a cross-border context.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you've done a boat job, no matter what it is, you know the ugly truth. They NEVER go the way you expect them to.\nThe Day that Mel got 3M 5200 in her hair wasn't a good day. Bad hair day, yes, but I'm not talking about bad hair days. I'm talking about bad boating days \u2014 the kind that have nothing to do with the weather, or even paying the boat bills. They're what almost always happens when you take on any boat job to make things \"better.\"\n5200 is really interesting stuff. I've been using it for years and I hope I'll be using it for years to come. This stuff redefines \"permanence.\" Which is why I was so surprised when I learned that it has another characteristic. Before it sets up, it can be very slippery.\nThis particular project started out with good intentions. Our deck junk had become somewhat spectacular from our many years of living and cruising aboard. I used to be proud of my deck junk. I thought that people going by would recognize us a \"True Cruisers.\" Instead, as I sat cowering below, I repeatedly heard such comments as, \"Good Gawd, Martha, look at that boat. It looks like Sanford and Sons.\" So we decided to do something about it.\nWe bought four beautiful new deck junk boxes. The sign in the store called them \"dock boxes\" but what did they know. I wasn't going to admit that I was actually going to put them on my deck. I didn't want to bolt the boxes through the deck because I figured that if that big wave came I'd rather it sweep my boxes into the ocean than tear out huge chunks of deck. So I came up with an idea to avoid thru bolting. It was to screw the boxes into thick teak boards which had been permanently and thoroughly glued to the deck. I could have just screwed the boxes to the top layer of cored fiberglass deck. Each glass layer of our decks is at least twice as thick as the hull of any boat built today. But I know about boat jobs. I avoid putting holes in my boat.\nI not only like 3M 5200, I like many other products similar to this. But they all have one thing in common. How do you clean up the mess you make with them?\nMost product manufacturers have other specific clean up products which they'll be glad to sell to you if you have any money left over from buying the first product. These usually work well.\nThe 3M 5200 clean up recommendation can be found on their official data sheet and most products have similar info on their web sites.\nBut one thing I've always observed about official cleanup stuff is that you never have it around when you need it.\nSo I often use something like acetone, WD-40, alcohol or turpentine. Of course, the consequences of using most of these chemicals can hurt you and other things if you're not very careful. Read labels very carefully and always test before you commit.\nYou must be sure you don't use some of these cleanup products where you want to re-apply, because of the residue they may leave. You must then clean that residue with another boat job product.\nI decided to use 5200 to glue on the teak boards. We gooped up the bottom of the first board and pressed it carefully onto the deck right inside the neatly drawn pencil lines. (Mel, my wife, being the artist that she is, always draws neat pencil lines to assist my installations, somehow having doubt that I will get the thing--whatever the thing may be--within 3 feet of where it is supposed to go.) She just wants to make it all very clear to me. It was indeed clear to me and I got the gooped up boards perfectly in place. I walked away to another chore, planning to return after they'd had a chance to set up a little. Fortunately I took an admiring look at my precision placement a few moments later. What I found was a 6 inch wide white track of 5200 following the path of the board down the slope of the deck. I had pressed the board down tightly within the penciled lines. I just hadn't reckoned on that other remarkable characteristic of 5200. It stays slick for awhile. Then came the next surprise. Strangely enough, picking up the teak plank that had just slid so easily down the deck proved to be relatively impossible. I felt that it would have been a good time to move on to another project, but Mel said \"No.\" With big screwdrivers and persevering patience as the evening grew darker and the gnats grew bolder, we gently coaxed this very expensive piece of teak off the deck.\nI carefully placed it again within the pencil marks, and this time pushed down really hard. The board scooted out from under me so quickly that I crashed down into the trail of white goo. Mel, who had been kneeling close by and leaning over to assure that I honored the penciled lines, was kneeling by a little too closely. At least it didn't get in my hair.\nSo, we decided to bite the bullet and screw the boards into the top layer of the deck, so that they would stay in place until the 5200 could dry. When I make a hole anywhere in my boat, I get a case of what the shrinks call \"reality based paranoia.\" As I said before, I know about boat jobs. I want the screws to not only be in the right place but also the drill bit of the right length. I've seen quite a few boat jobs where a stream of water followed the bit as it was pulled from the hole. (I'm not saying this has ever happened to me, I'm just saying that I know about boat jobs.) I wasn't worried about this, since I was drilling through my deck, but with my luck and skill, I was thinking, maybe I should. First we predrilled the blocks of wood for the screws that would fasten them to the deck, taking care that the holes were not where the corners of the box would be located. We placed the planks exactly where they were supposed to be (under the strict and unrelenting supervision of Mel) and used those holes as templates to make little tiny holes in the deck. We then slathered the bottom of the planks again and screwed them to the deck, exactly between the now fabled penciled lines.\nWe had already drilled holes in the \"appropriate\" places in the corners of the boxes. We set the boxes in place on the teak boards (sans 5200), carefully aligning them. We then used the holes in the box corners to make template holes in the teak so that I \"couldn't possibly mount the boxes in the wrong place.\" After that we took a deep breath and put a small measure of 5200 underneath each box corner, where it would surround the screw holes, for extra good measure. We used big fat pan headed screws with washers and threaded them through the holes in the bottom of the box and lined the points into the holes in the teak and carefully screwed them in. After we finished the last one, we stepped back to view our work with pride\u2026considerable pride. Fortunately our moment of pride wasn't ruined because I didn't yet know I'd stepped in a blob of 5200. I didn't know this until sometime later when traces of it showed up everywhere I'd been, including on the new carpet in the salon below.\nWe proudly started to finish this boat job by loading the deck junk into the boxes. But you can guess the end of the story. We couldn't begin to fit it all into the boxes. And there's so much in there that when you look in to find something it's like looking for a drop of water in the ocean. And worse, everything gets so tangled up with everything else that removing one piece of junk usually results in having to remove it all. Mel suggests that I do just that\u2026remove it all. And take it ashore and put it in the dump. But then what would I do with my deck junk boxes? Maybe I could convert them to life boats. That would be a great boat job.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"TRI Clear Lake will host second annual sprint triathlon sanctioned by USA Triathlon. The scenic course will be held on and around South Shore of Clear Lake. It will consist of a 750 meter swim in front of State Park beach, 12.4 mile bike along South Shore out past the prairie and windmills, and a 3.1 mile run along the shoreline, finishing lakeside in the eventful downtown area.\nTRI Clear Lake will coincide with the \"Take Me Back\" festival happening this same weekend. Adding the triathlon to this eventful summer weekend, will bring thousands of additional visitors from Iowa and surrounding states, giving them an opportunity to visit our wonderful community.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbmals b/data_all_eng_slimpj/shuffled/split2/finalzzzbmals
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index 0000000000000000000000000000000000000000..46980d9ee37e9f77dfe27ea5b753285425c48b68
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Many 360-degree cameras use Micro SD cards, and Amazon has several Micro SD cards on sale for Prime Day. One of them is the SanDisk Ultra 64GB, which I've used in my Samsung Gear 360.\nA solid-state drive (SSD) can boost the performance of your PC considerably, making it faster to edit 360 videos. There are several on sale for Prime Day. Most of them seem to use triple-layer cell (TLC) commonly found in budget SSD drives. However, the SanDisk SSD Plus is an SSD with multi-layer cell (MLC), which is generally faster and longer-lasting than TLC.\nThis is an updated version of the classic Viewmaster toy. With Google Cardboard technology, the new ViewMaster VR shows 360-degree photos. The ViewMaster VR can also be used as a Google Cardboard viewer.\nThis is a bundle that includes the Samsung Gear VR (reviewed here) and the Samsung Level U Pro headphones with microphones. TBH the Gear VR is easily available on eBay for much less, so this would only make sense if you wanted the Level U Pro headphones.\nThe 3DR Solo can be paired with the Kodak SP360 4k Aerial Pack to provide aerial 360 videos without the drone in the shot! Note: this order will ship in 1 to 2 months.\nIf you spot any other good Prime Day deals for VR \/ 360, please share it in the comments!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"chris631 140 days ago News uncategorised All http:\/\/brands.shopping4all.uk Discuss Published New Discard Success!\nFor my latest video, I go to the Cartier jewellery boutique at Selfridges, to discover what this famous French jeweller offers in the way of engagement rings.\nLerntypen-Test von ZENTRAL-lernen - Wie kann ich besser lernen?\nZENTRALlernen 140 days ago News lerntyp lerntypen test lerntechniken lernen lerntechnik lernmethode lernmethoden mnemotechniken All https:\/\/diigo.com Discuss Published New Discard Success!\nhttp:\/\/www.zentral-lernen.de\/lerntypen-test - Der Lerntypen-Test von ZENTRAL-lernen gibt Aufschluss auf die Frage: Wie kann ich besser lernen!\nchris631 140 days ago News amazon All http:\/\/decisionmaking.blogs4all.co.uk Discuss Published New Discard Success!\nHow did Pixar go from producing CAT scan images to winning Oscars?\nHow did Steve Jobs turn Apple into a world-beating company?\nHow does Amazon's culture encourage innovation?\nHow can you find the creative solutions demanded by our ever-changing world?\nnd the corner, and harder still to formulate a foolproof plan to deal with it. Truly innovative companies, Sims argues, don't get caught up in projections and predictions. Instead, they embrace uncertainty, take a chance, fail quickly and learn fast.\nThis method has formulated thousands of modern advances, from Google's PageRank to Starbucks coffee shops - if you harness its power, what could you achieve?\nchris631 140 days ago News amazon All http:\/\/www.blogging.blogs4all.co.uk Discuss Published New Discard Success!\nWork, he shows you how to revolutionize the way your company works forever.\nUsing stories from companies at the cutting edge of organizational transformation, Brave New Work will show you how to transform your team, department and business from the inside out, making work more adaptable, abundant and human. It is packed with new tactics and tips for updating your company's operating system: the simple rules and assumptions so deeply embedded that you don't even think to question them. Learn how to reignite passion and energy throughout your organization and to build a company that runs itself.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Pair this sweatshirt with everything from your favourite skirt to your comfiest jeans! Available in three colours (Black, Grey and White), you can't go wrong with this truth-speaking crewneck. This shirt is perfect for anyone who likes to eat and sleep\u2026 Which means it's the ideal gift for everyone!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Drypoint with roulette work, circa 1930, Geffroy 174, edition 100. 4 3\/4 x 7 3\/8 in. Signed and numbered in pencil. This is a superb impression printed on cream laid paper. The margins are full. The condition is very good other than for remains of old paper hinges on the top margin, verso. This unusual and exceptional print by Brouet shows his versatility and skill as a printmaker. He is best known for his figurative works depicting dancers, soldiers, beggars and gypsies.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We do not know what the books were about, and we can only form some guess as to what the parchments were. Paul had a few books which were left, perhaps wrapped up in the cloak, and Timothy was to be careful to bring them. Even an apostle must read. Some of our very ultra Calvinistic brethren think that a minister who reads books and studies his sermon must be a very deplorable specimen of a preacher. A man who comes up into the pulpit, professes to take his text on the spot, and talks any quantity of nonsense, is the idol of many. If he will speak without premeditation, or pretend to do so, and never produce what they call a dish of dead men's brains\u2014oh! that is the preacher. How rebuked are they by the apostle! He is inspired, and yet he wants books! He has been preaching at least for thirty years, and yet he wants books! He had seen the Lord, and yet he wants books! He had had a wider experience than most men, and yet he wants books! He had been caught up into the third heaven, and had heard things which it was unlawful for a men to utter, yet he wants books! He had written the major part of the New Testament, and yet he wants books! The apostle says to Timothy and so he says to every preacher, \"Give thyself unto reading.\" The man who never reads will never be read; he who never quotes will never be quoted. He who will not use the thoughts of other men's brains, proves that he has no brains of his own. Brethren, what is true of ministers is true of all our people. You need to read. Renounce as much as you will all light literature, but study as much as possible sound theological works, especially the Puritanic writers, and expositions of the Bible. We are quite persuaded that the very best way for you to be spending your leisure, is to be either reading or praying. You may get much instruction from books which afterwards you may use as a true weapon in your Lord and Master's service. Paul cries, \"Bring the books\"\u2014join in the cry.\nSadly, reading has fallen on hard times these days, as Ligon noted in his own sermon. This is why we need Spurgeon's warning even more in our own day. You see, we tend to acclimate ourselves to being spoon fed information in entertaining ways, training ourselves to be impatient with the length and attention required by reading books. But since we are a people of the Bible \u2013 the book inspired by God \u2013 then we all need go against the cultural current and train ourselves to be readers again. This may mean that we need to turn off the television more. It may mean we need to listen to the radio less. It may mean that we need to spend less time on the internet, where we learn to read only minutes at time. But aren't these small sacrifices to make in order to help ourselves become better able to read longer and more attentively and so become more competent readers of God's Word?\nGreat blog. I would like to invite you to mine, as it be of some use to you some day.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The Augusta Youth Development Campus, located in the city of Augusta, in Lincoln County, Georgia is a medium to maximum-security facility.\nAll of the men and women being held in the Augusta Youth Development Campus are either awaiting trial or have been sentenced in the Lincoln County Court System and been sentenced to one year or less.\nThis facility, known as \"Augusta Youth Development Campus\" is also known as Georgia Department of Juvenile Justice, Augusta Youth Development Campus, Georgia, Lincoln.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Whatever became of this amazing project? It looks absolutely fantastic.\nYou will be pleased to know this project is still underway. It has been expanded somewhat and is nearing completion.\nThis very cool project has gone through many iterations but now all of the components and features have stabilized and the end result will be far better than originally thought.\nThe video module now contains a full blown graphics and video processor. The video overlay Heads-Up-Display now runs at 720 X 480 resolution on standard NTSC. PAL is also supported. It has real-time artificial horizon, sliding ribbon air speed, altimeter and compass heading, engine and rotor RPM, 2 battery, 3 temp gauges. The airspeed and altimeter are driven by pressure sensors but GPS readings are also displayed. There are many other features also included. There is also a full 6DOF IMU with 3 axis magnetometer and 4Hz GPS (we are looking at a 5Hz).\nI will try to post some demo mode video and other information before long.\nHopefully it won't be too much longer before these are in full production.\nThat's it for now. I don't want to say too much until it is ready to ship. I'm sure you can understand why.\nThat is cool, about how long before those are available ?\nI can't wait. That's truly an awesome HUD. I know you can't divulge too much at this point, but is there any way you could give us an indication as to the price range? Is it aimed at hobbyists or the commercial UAV industry?\nJust when I thought I've seen it all in this hobby world. Now that is really cool!\nIts good to hear you are making progress!\nWhat IMU did you end up using?\nSorry for taking so long to reply. I did not get any updates in my e-mail so I was unaware additional posts had been added.\nHere is a screenshot of the current HUD display. This is a capture from the HUD running in demo mode. The graphics are all real and being drawn in real-time however they are running in demo mode and the background video was pre-recorded and used for bench testing purposes.\nThe HUD display will be user configurable for display features such as gauges, colors, size, etc. It will support several video input and output options with the standard being color mono or stereoscopic CCD camera with direct digital ITU-656 in and NTSC\/PAL composite video out with a graphics resolution of 720 X 480. Everything runs in real-time. There are many other features but I have said too much already.\nMore information will be forthcoming as we get closer to a production date.\nHopefully sometime this summer if everything continues at the current pace.\nAnd at what price range for this wonderful invention of yours?\nWOW! that is awesome OSD. Looking forward to the release .\nWe all love bells and whistles, and the smaller the better, but price talks!!!\nUnfortunatlly I can't yet comment on pricing but I expect it to be affordable.\nVery nice! I suggest trying to recoup your investment with high volume, low pricing sales!\nAny new word on this project?\nit has a mode where it looks like the hud of a f16 and it also has a autopilot for returning home on signal los... sounds pretty cool..\nNeat! Now if it only landed the A\/C at the home field as well.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"An ESD protected, effective storage system for sensitive components. Customise your small parts shelving to your own requirements. Depths of 400, 500 or 600 mm. Small parts shelving with accessories are of semi-conductive epoxy powder coated steel in light grey RAL 7035. Earth via protective 1 megohm resistor.\nTreston shelving unit PH includes 2 front uprights height 2020 mm, 2 back uprights height 2020 mm, 6 adjustable shelves, 1 back brace, dims 1000 x 966 mm, 4 side battens and earth via protective 1 megohm resistor.\nTreston extension unit PH-L includes 1 front upright height 2020 mm, 1 back upright height 2020 mm, 6 adjustable shelves, 1 back brace dims 1000 x 966 mm, 2 side battens and earth via protective 1 megohm resistor.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Davidoff, headquartered in Basel, Switzerland, will become a major sponsor of Art Basel's two international art shows as it looks to strengthen its ties to the arts. The significant relationship elevates Davidoff into the top tier of Art Basel sponsors, which include UBS, AxaArt and Netjets.\nUnder the agreement, Davidoff will be an Associate Sponsor to Art Basel, at both its shows in Basel, Switzerland, and Miami Beach, Florida, where it will operate a 1,000 square meter VIP hospitality space within the Art Collectors Lounge. This Davidoff Lounge will feature a walk-in humidor and a master cigar roller from the Dominican Republic who will demonstrate the art of blending and rolling of a Davidoff cigar. In addition, two outdoor Cigar Lounges will be situated adjacent to the Art Collectors Lounge and next to the shows' public area, to offer fair visitors a relaxed venue for discussion, reflection and enjoyment of a fine cigar.\nDavidoff says it will collaborate with Art Basel over the long term as it expands the orbit of its influential art shows beyond Europe and North America to Asia.\n\"Davidoff and Art Basel is a perfect fit, building on a well-established relationship between two organizations with joint roots in Basel,\" Hans-Kristian Hoejsgaard, President and CEO of Oettinger Davidoff Group said. \"With historic ties to European markets, both companies have expanded rapidly in the US in recent years, while actively developing new markets around the globe, especially in Asia. Our customers share many common interests with Art Basel's patrons. As we forge closer ties with the world of art, Art Basel is the ideal partner, and we look forward to a long-term collaboration as our brands evolve worldwide.\"\n\"We are delighted for Davidoff to become an Associate Sponsor at our shows in Basel and Miami Beach and look forward to developing this relationship further,\" Art Basel Co-Director Marc Spiegler. \"As Art Basel is expanding, we seek partners like Davidoff who are intensifying their engagement with the arts.\"\nDavidoff officials say the Art Basel deal is the first step to shifting its sponsorship in the direction of contemporary art. The company will announce a new international initiative linking it more closely to the world of contemporary artists and collectors later this year.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For over 30 years, UUP has represented academic and professional faculty at SUNY Cortland and across the state at SUNY's other state-operated campuses. The Cortland Chapter is one of UUP's most active, with members serving the union in a variety of local and statewide leadership roles over the years. We are committed to enhancing the working lives and protecting the workplace rights of our constituents.\nWe invite you to join us in taking an active role on your campus and at the statewide level.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbovpo b/data_all_eng_slimpj/shuffled/split2/finalzzzbovpo
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index 0000000000000000000000000000000000000000..466e7feb08985317a9ff49d0a593a0cc6f722d8e
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@@ -0,0 +1,5 @@
+{"text":"Exciting Dispensing Manager role available with family independent offering a competitive package. Fantastic opportunity for a Dispensing Optician.\nThe position is based in a community leading independent in Norfolk, who offer dedicated and professional service and have a superb reputation in the area.\nIf this is of interest, please contact Natasha on the Optical division on 01423 850419 or send your CV and covering letter to natasha.webster@prospect-health.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The son of Polish immigrants, Czolgosz worked for the American Steel and Wire Company, was fired for striking, and became a confirmed anarchist. On August 31, 1901, he moved from Ohio to Buffalo, New York, rented a room near the site of the Pan-American Exposition, which he knew that President McKinley was scheduled to visit. On September 6, armed with a pistol that he had concealed in a handkerchief, Czolgosz went to the exposition, assumed a place in a line of McKinley wellwishers, and shot the president twice at pointblank range at 4:07 p.m. McKinley died on September 14.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Barbados is an island where relaxation and adventure go hand in hand. White sandy beaches stretch for miles while snorkeling and sailing are some of the world-class water sports on offer.\nA cruise to Barbados starts with a walk around Bridgetown, the capital city, which was declared a UNESCO World Heritage Site in 2011. The city has a charming combination of colonial buildings and Caribbean vibrancy. Make sure you visit St. Nicholas Abbey, a historic plantation house and one of three Jacobean mansions in the Western Hemisphere.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The accused actor commented on the allegation of sexual neglect against him while he was characterized as his \"House of Cards\" alter-ego Frank Underwood.\nKevin Spacey broke his silence on Monday with a bizarre video address that sees the actor defending himself against the many accusations against him while he was character as his entertaining shameless character of Cards character, Frank Underwood.\nIn the video, Spacey assumes the southern accent, silence, and fourth engraving style of Frank, the character he played in five seasons on House of Cards to talk directly to the camera. The Netflix political thriller, star player Robin Wright, wrote Spacey out of his sixth and last season after he was fired over mounting sexual assault claims. The series, which he previously produced produced, killed his character of the screen; The streaming giant also officially buried Francis J. \"Frank\" Underwood at a burial ground in South Carolina, where fans can pay their respect.\nThe video marks Spacey's first public comment since his first and widely criticized response to the claims of actor Anthony Rapp in October 2017 at the height of the #MeToo movement. The actor used the declaration of excuse to also come out as gay.\n\"I know what you want. You want me back,\" Spacey, wearing a santa apron, says in the three-minute monologue posted on Monday. In addition to defending himself against a \"charge without trial\", the actor also refers to his fate: \"All this presumption created such an unsatisfactory ending and to believe it could have been such a memorable sendoff.\" I can promise you this. If I didn't pay the price for the things we both know I did, I'd definitely not pay the price for the things I didn't do. \" He ends the video by saying,\" Conclusions can deceive. Miss me? \"\nThe reactions to Spacey's \"creepy\" Christmas Eve video were quick on social media.\nNetflix had no comments on the video.\nSee some of the confused reactions below.\nThe one positive aspect of the Kevin Spacey video is that we can all point it and say, \"That's what happens when you don't have a scriptwriter. \"\nIs Kevin Spacey using Donald Trump's legal team now?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Uniting a love for melodic, bass-infused music, Elevven is a project by Eskai & SNR. With their own diverse individual releases, their collaborations have been featured on Armin van Buuren's prestigious A State of Trance compilation, and have remixed the likes of trance icons Ferry Corsten, Cosmic Gate, Morgan Page, Gabriel & Dresden and Solarstone.\nTheir remix for Gabriel & Dresden 'You' was debuted at Above & Beyond's landmark ABGT300 show, and a remix for Morgan Page's legendary 'The Longest Road' is forthcoming on Armada Music.","meta":{"redpajama_set_name":"RedPajamaC4"}}